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ivalued function can and should be considered a specification for functions: A function f: S → T fulfills the specification $F\colon S {\rightrightarrows}T$ if for any s ∈ dom(F) it holds that f(s) ∈ F(s). In this case we say that f is a choice for F. The operations on multivalued functions are chosen such that they behave well with the interpretation as specifications. For instance, the composition F ∘ G of two multivalued functions $G\colon R{\rightrightarrows}S$ and $F\colon S {\rightrightarrows}T$ is of type $R {\rightrightarrows}T$ and its value sets are given by (F ∘ G)(r) : = {t ∈ T ∣ G(r) ⊆ dom(F) ∧ ∃s: t ∈ F(s) ∧ s ∈ G(r)}. This should be compared to (F ∘ RG)(r) : = {t ∣ ∃s: t ∈ F(s) ∧ s ∈ G(r)}, which is what is commonly used as composition for relations. Both the relational and the multifunction composition are associative operations. The domain condition added in the multifunction composition is a modifier that addresses the difference in interpretations. For the multifunction composition it is true that if f is a choice for F and g is a choice for G then f ∘ g is a choice for F ∘ G, which may fail for the relational composition as illustrated in Figure [fig: comp]. From the same figure it can be seen that the multifunction composition is not symmetric under changing the direction of the multifunctions while for relations this is the case. The relational and the multifunction composition produce identical results if the multifunction that is applied last is total (`comp_rcmp`) resp. the first one is singlevalued (`sing_comp`). [.48] [thick,scale=1.5] at (0,.8) R; at (1,.8) S; at (2,.8) T; (0,0) – node[above]g (1,-.5); (1,.5) – (2,.5); (1,-.5) – node[below]f (2,-.5); (0,0) – (2,-.5); at (1.35,-.15) f ∘ g; (0,0) circle (.05cm); (1,.5) circle (.05cm); (1,-.5) circle (.05cm); (2,.5) circle (.05cm); (2,-.5) circle (.05cm); (1.1,0) – (1.1,-.675) – (-.2,0) – node[above]G (1.1,.675) – (1.1,0); (1.5,.4) – (0.9,.4) – (0.9,.6) – node[above]F (2.1,.6) – (2.1,.4) – (1.5,.4); (1,.15) – (-.075, -.1) – (-.1,.075) – (2.075,.6) – (2.1,.4) – (1,.15); at (1.4,.1) F ∘ RG; [.48] [thick,scale=1.5] (0,0) – (0,2); (0,0) – (3,0); at (2.8,-.2) S; at (-.2,2.2) T; at (1.25,1.5) G; at (0.15,.85) f; (1,1.25) to [curve through =(1,1.25).. (1.9,1.2) ] (2,1.25); (2,1.25) – (2,.5); (1,.5) to [curve through =(1.2,.45).. (1.9,.4) ] (2,.5); (1,1.25) – (1,.5); (.5, 1) – node[above]F (.5,.75); (.5,1) to [curve through =(1,1).. (1.9,1.1) ] (2.5,1); (.5,.75) to [curve through =(1,.65).. (2,1) ] (2.5,1); (0,.5) to [curve through =(.5,.85).. (1,.8).. (2,1.05).. (2.5,1) ] (3,.8) ; Functions and partial functions are special cases of multifunctions. A function f: S → T may be identified with the specification that on input s it makes f(s) the only eligible return value, i.e. with s ↦ {t ∈ T ∣ t = f(s)}. The multifunction associated to a function is always total and singlevalued (`F2MF_tot` and `F2MF_sing`) and assuming that T is not empty and an appropriate choice principle, each total singlevalued multifunction arises in this way (`fun_spec`). We call this connection between functions and total singlevalued multifunctions the `F2MF` correspondence. This construction can be extended to partial functions by assigning to g: ⊆ S → T the multifunction function s ↦ {t ∈ T ∣ g(s)is defined and equals t}. The resulting multifunction is singlevalued (`PF2MF_sing`) but need not be total. Note that we are being unspecific about how to encode partial functions here. One may use functions to an option type as outlined previously but should keep in mind that this can be understood to imply the domain of the partial function to be decidable which is rarely the case in our applications. It is not difficult to check that both relational and multifunction composition restrict to regular composition of functions or partial functions when considered only on singlevalued resp. total and singlevalued multifunctions (`F2MF_comp` and `PF2MF_comp`). Any multifunction can be assigned a reverse multifunction by switching input and output. Each property of a multifunction has a co-version that requires the same property for the reverse multifunction. Many of the co-properties have nice characterizations for the special cases of functions. For instance, a function is injective if and only if the associated multifunction is co-singlevalued (`mfinv_inj_sing`). For later reference we list: [`PF2MF_cotot`][resu: PF2MFcotot] A partial function is surjective if and only if its associated multifunction is co-total. For multifunctions $F, G\colon S {\rightrightarrows}T$ we say that F tightens G if it is more restrictive as a specification. This can be spelled out elementary as dom(G) ⊆ dom(F) and ∀s ∈ dom(G): F(s) ⊆ G(s). Under weak additional assumptions this is equivalent to the statement that each choice function for F is also a choice function G (`icf_tight` and `tight_icf`, see Figure [fig: tight]). A function is a choice for a multifunction F if and only if its associated multifunction tightens F (`icf_spec`). For a partial function we say that it is a partial choice for F if its associated multifunction tightens F. This can be spelled out to still mean that f is a partial choice for F if whenever s ∈ dom(F) then f(s) is defined and an element of F(s). Many further properties of multifunctions are proven in the mf library and we just pick an example: [`tight_comp`][lem: tight comp] If F tightens Fʹ and G tightens Gʹ then F ∘ G tightens Fʹ ∘ Gʹ. Capturing the computable partial functions and relativization ------------------------------------------------------------- Next we discuss another way to generate multifunctions from functions that is related to the standard way of modeling diverging computation in intuitionistic type theories. Recall that for a type T the option type optT adds a single element to T. That is, each t ∈ T corresponds to an element Somet ∈ opt(T) and there is a single additional element None ∈ opt(T). Given a function $N \colon{\mathbb N}\times S \to {\operatorname{opt}}T$ define a multifunction $\Phi\_N\colon S {\rightrightarrows}T$ via ΦN(s) : = {t ∈ T ∣ ∃n: N(n, s) = Somet}. We call the additional input $n\in{\mathbb N}$ the effort parameter and the reason for this should become apparent from the next paragraph where it takes the role of the number of steps taken by a Turing machine. Consider the special case where $S = {\mathbb N}= T$. Given a partial computable function f: ⊆ S → T we want to argue that there exists a primitive recursive N that recovers it through the Φ-assignment above. Indeed fix some Turing machine that computes f and consider the function N that on input (n, s) returns Somet if the machine on input s terminates within the first n time-steps and returns t, and None otherwise. For this function N it holds that ΦN(s) is empty if the machine diverges and otherwise consists of the single value f(s), thus ΦN(s) = f if we identify partial functions with their induced multifunction. Although there are technical differences, it should be clear that the core idea behind this is a version of the Kleene normal-form theorem. The above makes the Φ assignment particularly interesting to us as any primitive recursive function is definable in Coq. Even for primitive recursive N, the multifunction ΦN need neither be total nor singlevalued. In the case where S and T are the natural numbers, N(n, s) : = None leads ΦN to be the empty function and N(n, s) : = Some(n) results in the total specification for that renders any return value eligible. However, for any N, it is possible to obtain a singlevalued tightening of the form ΦNʹ: Choose Nʹ to be the function that on input n and s evaluates N on inputs (0, s), …, (n, s) and returns the first value returned by N if such exists and None otherwise. This Nʹ is monotone in the sense that if it returns a value on some effort, then it returns the same value on all bigger efforts. Whenever a function is monotone in this sense, the corresponding multifunction is singlevalued. As the procedure of searching is computable, it is also true that for any computable N the function ΦN has a computable choice function. Thus, assuming that the functions definable in Coq are computable (in the sense in which this was discussed in the introduction), we may use the Φ assignment to capture computability in Coq. Since we are interested in specification of partial operators, we relativize the construction. That is, we start from the Φ assignment and add an additional argument that reaches over Baire space. For a fixed φ from Baire space we thus capture computability by an oracle Turing machine relative to φ just as the original assignment captured computability by a regular Turing machine. As the use of the word oracle indicates, φ need not be computable. In case the oracle is incomputable, the functions computable relative to it may be incomputable as well. However, if one does not consider φ to be fixed but instead as an input to the computation, one recovers a realistic model of computation and this is the model that is at the basis of computable analysis. We further adapt to the conventions of computable analysis by interpreting the discrete input to the oracle Turing machine as a curried function input. That is, for a given functional input φ, an oracle Turing machine is considered to return an element of Baire space if the relativized computation with oracle φ terminates for each possible input. If the computation on oracle φ diverges for any of the possible inputs, the return value is considered undefined. To make this precise, first recall that we use the notations ${\mathcal B}= {A}^{Q}$ and ${\mathcal B}' = {A}'^{{Q}'}$. Given a function $M: {\mathbb N}\times {\mathcal B}\times {Q}' \to {\operatorname{opt}}{A}'$ we consider the multifunction $F\_M \colon {\mathcal B}{\rightrightarrows}{\mathcal B}'$ defined by $$F\_M(\varphi) := \{ \psi \in {\mathcal B}' \mid \forall q', \exists n\colon M(n, \varphi, q') = {\operatorname{Some}}(\psi(q'))\}.$$ One may repeat the discussion for the Φ assignment above to verify that for each operator computable by an oracle Turing machine, there exists some M definable in Gödels system T such that FM is the multivalued function assigned to the operator computed by the oracle Turing machine. In the same way as was done for the unrelativized case, one may force singlevaluedness. It is also still true that for any M that is computable by an oracle Turing machine this singlevalued selection of FM is computable by an oracle Turing machine. Thus, under the assumption that all functions definable in Coq are computable, and whenever all the question and answer types come with canonical bijections with the natural numbers, we may use a definition of M as certificate for the computability of an operator. In the rest of the paper, whenever we mention computability, we mean computability in this sense. As an informal example consider sections of elements of Baire space, i.e. let all question and answer sets be the natural numbers so that ${\mathcal B}$ and ${\mathcal B}'$ both are ${\mathbb N}^{\mathbb N}$. Consider the function M that on input of an effort $n\in {\mathbb N}$, some $\varphi \in {\mathcal B}= {\mathbb N}^{\mathbb N}$ and a $q'\in {Q}' = {\mathbb N}$ returns the value $M(n, \varphi, q') \in {\operatorname{opt}}({A}') = {\operatorname{opt}}({\mathbb N})$ given by $$M(n, \varphi, q') := \begin{cases} {\operatorname{Some}}(n) & \text{if } \varphi(n) = q' \\ {\operatorname{None}}& \text{otherwise.} \end{cases}$$ Then the values of the corresponding operator FM are given by $$F\_M(\varphi) = \{\psi \in {\mathbb N}^{\mathbb N}\mid \forall q'\!,\exists a'\!\colon \varphi (a') = q' \!\land {\operatorname{Some}}(a') = {\operatorname{Some}}(\psi(q'))\} = \{\psi \mid \forall q'\!\colon \varphi(\psi(q')) = q'\}.$$ That is FM(φ) is the set of all sections of φ. Consequentially, the domain of FM is the collection of all surjective elements of Baire space. As a given surjective function can have several different sections, FM is not singlevalued. However, one can make the values unique by deciding on the minimal section, where a section ψ of a surjection φ is called minimal if it is pointwise less or equal to any other section. An Mʹ such that FMʹ is the specification of being a minimal section can be found by following the construction of a singlevalued selection for the Φ assignment given above. A version of this example that is of particular interest for us, we have made formal: [`examples/continuous_search.v`][ex: continuous search 1] Consider the case where A : = {0, 1} are the Booleans, Qʹ : = { ⋆ } is the canonical one point set and the other question and answer sets are the natural numbers. This means that ${\mathcal B}= \{0,1\}^{\mathbb N}$ is Cantor space and ${\mathcal B}' = {\mathbb N}^{\{\star\}}$ can be understood as the natural numbers by identifying $\psi \in {\mathcal B}'$ with $\psi(\star) \in {\mathbb N}$. Consider the operator $F\colon {\subseteq {\mathcal B}} \to {\mathcal B}'$ that on input of some χ from Cantor space that is not the constant function returning 1, returns the first $q' \in {\mathbb N}$ such that χ(qʹ) = 0. Then F = FM if we choose for $M\colon {\mathbb N}\times{\mathcal B}\times \{\star\} \to {\operatorname{opt}}({\mathbb N})$ the function such that M(n, χ, ⋆ ) equals Some(k) if k is the smallest number no bigger than n such that χ(k) = 0 and None if no such k exists. Just like in the informal example above, this M may be constructed by first considering the multifunction returning any zero and then moving to a singlevalued selection function. Note that the Φ and the F assignments differ considerably in their interpretation of what is considered input and output. As sketched above this is not relevant for finding a singlevalued selection. However, it renders different operations natural for the Φ and the F assignment and the natural operations for the latter tend to be more problematic than for the former. As illustration of this, and as an additional motivation for the next section, let us briefly look into composition. First consider the Φ assignment. Given functions $N \colon {\mathbb N}\times R \to {\operatorname{opt}}S$ and $N'\colon {\mathbb N}\times S \to {\operatorname{opt}}T$ one may define a new function $N'\circ\_{\Phi} N\colon {\mathbb N}\times R \to {\operatorname{opt}}T$ via $$N' \circ\_{\Phi} N (\langle n, m\rangle, r) := \begin{cases} N'(n,t) & \text{if } N(m,r) = {\operatorname{Some}}t \\ {\operatorname{None}}& \text{otherwise,} \end{cases}$$ where ⟨ ⋅ , ⋅ ⟩ is the Cantor (or any standard) pairing function. This captures the relational composition in the sense that ΦNʹ ∘ ΦN = ΦNʹ ∘ RΦN which in turn tightens ΦNʹ ∘ ΦN. Under the assumption that N is monotone one may further simplify the construction by replacing the pair by the maximum. Let us try to adapt the above to the setting of the F assignment. Given Mʹ and M one would most likely be interested in FMʹ ∘ FM. For simplicity let us assume that FMʹ and FM are singlevalued so that the values of the composition are given by (FMʹ ∘ FM)(φ) : = FMʹ(FM(φ)). A straightforward adaption of the proof for the Φ assignment does not give information about this operator but instead about an operation that would be more natural to consider for functionals. Namely given functionals $\mathcal F\colon {\mathcal B}\times {Q}' \to {A}'$ and $\mathcal G\colon {\mathcal B}\times {Q}'' \to {Q}'$ it makes a statement about the functional $(\varphi, q'') \mapsto \mathcal F(\varphi, \mathcal G(\varphi, q''))$. The construction for functionals that corresponds to what we were originally interested in is “functional substitution”, spells out $(\varphi, q'') \mapsto \mathcal F(\lambda q'. \mathcal G(\varphi, q'), q'')$ in the language for functionals and requires different typing of $\mathcal F$ and $\mathcal G$. Let us thus assume we are given $M\colon {\mathbb N}\times {\mathcal B}\times {Q}' \to {\operatorname{opt}}{A}'$ and $M'\colon {\mathbb N}\times {\mathcal B}'\times {Q}''\to {\operatorname{opt}}{A}''$ and let us try to find some $M' \circ\_F M \colon {\mathbb N}\times {\mathcal B}\times {Q}'' \to {\operatorname{opt}}{A}''$ such that FMʹ ∘ FM is a tightening of FMʹ ∘ FM. Due to the definition of the F assignment, we can recover from M for each fixed effort n and input $\varphi \in {\mathcal B}$ an approximation M(n, φ, ⋅ ): Qʹ → opt(Aʹ) to possible functional inputs to Mʹ. However, Mʹ expects an input of type Qʹ → Aʹ and not of type Qʹ → opt(Aʹ) and to move from the latter to the former one has to pick a value whenever None occurs as return-value. Without further information about Mʹ it is not clear why the choices of these values should be irrelevant. And in particular it is not clear why the return values of Mʹ on an arbitrary extension should have anything to do with the return-value of the composition. However, Coq is consistent with functional extensionality and any functions defined in Coq without use of non-computational axioms can be evaluated in a finite amount of time. One might thus tend to believe that if Mʹ is defined in this way, then its value on fixed discrete input only relies on a finite number of the return values of its functional input. If the original φ is from the domain of FM it is thus possible to choose the effort big enough for the way in which we extend to be irrelevant. The additional information that is needed about Mʹ for being able to carry out this kind of composition concretely is effective information about its continuity as presented in the upcoming section. Continuity of partial operators between naming spaces ----------------------------------------------------- r.145 (0,0) rectangle node F (1.5,1); (.5,-.25) – (.5,0); (1,0) – (1,-.25); (.25,1.25) rectangle node φ (1.25,1.75); (.5, 1) – (.5,1.25); (1,1.25) – (1,1); [fig: continuous operator] This section presents an information theoretic development of a notion of continuity of operators between naming spaces. The Incone library provides proofs that the definitions presented here are equivalent to more traditional notions of continuity, but the discussion of these equivalences is postponed to Section [sec: metric spaces] since it requires some background about metric spaces and topology that are not necessary for the presentation in the current section. For the following fix some types Q, A, Qʹ and Aʹ and set ${\mathcal B}:= {A}^{Q}$ and ${\mathcal B}':={A}'^{{Q}'}$. Intuitively continuity says that the values of an operator $F\colon {\mathcal B}\to {\mathcal B}'$ interpreted as functional of type $F\colon {\mathcal B}\times {Q}' \to {A}'$ only depend on finite information about the values of the functional input from ${\mathcal B}$ and thus can be thought of as being represented by a diagram as depicted in Figure [fig: continuous operator]. Mathematically, continuity can be described as follows: A function $F\colon {\mathcal B}\to {\mathcal B}'$ is continuous if for any element φ of ${\mathcal B}$ and any qʹ ∈ Qʹ there exists a certificate, i.e. a finite list $\mathbf q\in {\operatorname{seq}}{Q}$ such that for any ψ that coincides with φ on $\mathbf q$ it holds that F(ψ)(qʹ) = F(φ)(qʹ). Here, two functions are said to coincide on a finite list $\mathbf q$ if φ(q) = ψ(q) for any q that appears in $\mathbf q$. A partial operator $F\colon {\subseteq {\mathcal B}} \to {\mathcal B}'$ is continuous if for all φ ∈ dom(F) and qʹ ∈ Qʹ there exists a certificate, i.e. a finite list $\mathbf q \in {\operatorname{seq}}{Q}$ such that the above statement holds for any ψ ∈ dom(F) that coincides with φ on $\mathbf q$. [`examples/continuous_search.v`] Consider the function $F\_0\colon {\mathbb N}^{\mathbb N}\to {\mathbb N}^{\mathbb N}$ defined by F0(φ)(n) : = φ(n) + φ(0). Then for any inputs φ and n the finite list ${\mathbf}q := (n, 0)$ is a certificate and thus F0 is continuous. For the operator F1(φ)(n) : = φ(φ(n)) the list ${\mathbf}q:= (\varphi(n), n)$ is appropriate. Most functions that can be explicitly defined are continuous. For instance any function definable in Gödels system T is continuous and as long as no strictly non-computational axioms are involved, it is reasonable to assume that the functions definable in Coq are computable and therefore continuous (compare discussion in Section [sec: comp-cont]). r.145 (0,0) rectangle node M (1.5,1); (.5,-.25) – (.5,0); (1,0) – (1,-.25); (.25,1.25) rectangle node φ (1.25,1.75); (.5, 1) – (.5,1.25); (1,1.25) – (1,1); The same remains true for operators whose specification can be given as FM for some function $M\colon {\mathbb N}\times {\mathbb N}^{\mathbb N}\times {\mathbb N}\to {\operatorname{opt}}{\mathbb N}$ such that M has a Coq-definition that does not rely on non-computational axioms. For instance for the search operator F from Example [ex: continuous search 1] the function M can be defined in Coq only using very elementary means and the list (0, …, F(φ)( ⋆ )) is a certificate for functional input φ and discrete input n. The search operator does not have a continuous total extension. Thus, one should not expect such an extension to be definable in Coq without reliance on non-computational axioms. The definition of continuity in the Incone library follows the mathematical definition given earlier mostly literally. It superficially looks more complicated due to the use of multifunction to substitute partial functions, but the definition is chosen such that it implies a continuous multifunction to be singlevalued and does thus only really apply to partial functions. Another difference is that instead of a list for each question the definition of continuity in Incone uses a Skolem-function L: Qʹ → seqQ. This switches the order of the corresponding existential and universal quantification. Whenever an appropriate choice principle is available, these definitions are equivalent (`choice_cont`). The definition used in Incone has the advantage that it allows for a fully constructive proof of the fact that the composition of continuous operators is continuous. [`cont_comp`][resu: contcomp] Let $F\colon {\subseteq {\mathcal B}} \to {\mathcal B}'$ and $G\colon{\subseteq {\mathcal B}'} \to {\mathcal B}''$ be continuous partial operators. The operator $F \circ G \colon {\subseteq {\mathcal B}} \to {\mathcal B}''$ is continuous. [summary] The idea behind the proof is that the certificate functions L and Lʹ whose existence is guaranteed by the continuity of F and G can be interpreted as multivalued functions and composed relationally to obtain a certificate function for the composition of the operators. Furthermore, the needed relational composition can be realized constructively on the level of combining lists. As we compare different notions of continuity in the later chapters, let us briefly discuss sequential continuity on naming spaces. Let ${\mathcal B}= {A}^{Q}$, note that ${\mathcal B}$ is a naming space if Q and A are countable and non-empty but for the following definition we do not need these assumptions. An element $\varphi\in {\mathcal B}$ is said to be the limit of a sequence $(\varphi\_n)\_{n \in {\mathbb N}} \subseteq {\mathcal B}$ if for each fixed argument q ∈ Q the sequence (φn(q)) is eventually constantly φ(q). Formally $$\lim\nolimits\_{\mathcal B}(\varphi\_n) = \varphi \quad \iff\quad \forall q, \exists n\_0, \forall n\geq n\_0 \colon \varphi\_n(q) = \varphi(q).$$ If a sequence in Baire space has a limit, this limit is uniquely determined (`lim_sing`) and thus the above defines a partial function $\lim\_{\mathcal B}\colon{\subseteq {\mathcal B}^{\mathbb N}}\to {\mathcal B}$. A partial operator $F\colon{\subseteq {\mathcal B}} \to {\mathcal B}'$ is called sequentially continuous if for any φ ∈ dom(F) and any sequence $(\varphi\_n)\_{n \in {\mathbb N}} \subseteq {\operatorname{dom}}(F)$ such that $\lim\_{\mathcal B}(\varphi\_n) = \varphi$ it also holds that $\lim\_{{\mathcal B}'}(F(\varphi\_n)) = F(\varphi)$. It is well known that the topological structure of Baire space ${\mathbb N}^{\mathbb N}$ is such that sequential continuity of partial operators on Baire space is equivalent to their continuity and the Incone library includes a formal proof of this and that it remains true for naming spaces. However, this is a classical fact and constructively sequential continuity need not imply continuity, thus the library separates the equivalence into two implications. [`cont_scnt` and `scnt_cont`][resu: contscnt] A partial operator between naming spaces is continuous if and only if it is sequentially continuous. Section [sec: bairemetric] discusses how to prove that any naming space is metrizable, or more specifically it defines a metric on ${\mathcal B}$ from each enumeration of Q. It also presents proofs that the notions of convergence and continuity induced by this metric coincide with those given above. Thus, the formal proofs that continuity and sequential continuity are equivalent to what is described in Section [sec: continuities] imply the above theorem. However, metric spaces use real numbers, which leads to the axioms of the real numbers appearing in the assumptions printed when inspecting the proofs. This is even though the proofs do not use these axioms in an essential way. Thus, the above statement is given a separate proof in the Incone library. Construction of a universal and some of its properties ------------------------------------------------------ Recall that a naming space is a space of functions whose target and argument spaces are countable and non-empty. A continuous universal, or just universal, is an assignment that for each pair ${\mathcal B}$ and ${\mathcal B}'$ of naming spaces provides another naming space ${\mathcal B}''$ and an operation $F\_{M(\cdot)}\colon {\mathcal B}'' \to {\mathcal B}{\rightrightarrows}{\mathcal B}'$ such that each of its values is continuous and for each continuous $F\colon {\subseteq{\mathcal B}}\to {\mathcal B}'$ there exists an element $\psi \in {\mathcal B}''$ such that FM(ψ) tightens F. That is: A universal provides a way to code any continuous operator between naming spaces by an element of another naming space. We call such a code, i.e. some ψ such that FM(ψ) tightens F, an associate of F with respect to the universal or just an associate if the universal is clear from the context. Note that the type of ψ can be inferred from the universal together with the type of F. Let us give some motivation for the terminology chosen here. To justify the term “universal” note that replacing all the naming spaces by the set of finite binary strings and the word “continuous” by “computable” one recovers a specification that is fulfilled by the universal Turing machine. While the construction of a universal Turing machine takes some effort, continuous universals can be chosen very simple: Classically any naming space can be replaced by Baire space and modulo this one may use Kleene-Kreisel associateship to obtain a universal. A more popular variant in computable analysis is to move from Baire space to the space of infinite binary strings, i.e. Cantor space and use Weihrauch’s η operator. The former of these is conceptually more well adapted to our setting and there are several excellent sources to read up about its background. A previous version of Incone’s universal closely followed generalization of Kleene-Kreisel associateship first presented by van Oosten. The current universal modifies that construction slightly by moving away from the idea that questions should be asked sequentially and allowing for a finite number of questions to be asked in parallel. A mathematical description of Incone’s universal can be given as follows: for fixed naming spaces ${\mathcal B}= {A}^{Q}$ and ${\mathcal B}' = {A}'^{{Q}'}$ set Qʺ : = seqA × Qʹ and Aʺ : = seqQ + Aʹ. That is let ${\mathcal B}'':= ({\operatorname{seq}}{Q}+ {A}')^{{\operatorname{seq}}{A}\times {Q}'}$. That ${\mathcal B}''$ is a naming space, i.e. that Qʺ and Aʺ are countable and non-empty, follows directly from ${\mathcal B}$ and ${\mathcal B}'$ being naming spaces. Assign to $\psi \in {\mathcal B}'$ the multifunction $F\_{U(\psi)}\colon {\mathcal B}{\rightrightarrows}{\mathcal B}'$ defined as follows: φʹ ∈ FU(ψ)(φ) if and only if for any qʹ ∈ Qʹ there exists an N and a finite sequence of lists $({\mathbf}a\_i)\_{i\in\{1,\ldots,N\}} \subseteq {\operatorname{seq}}{A}$ such that for i < N it holds that $\psi({\mathbf}a\_i, q') = {\operatorname{?}}{\mathbf}q$ for some ${\mathbf}q \in {\operatorname{seq}}{Q}$ (where? denotes the left inclusion in the sum) and ${\mathbf}a\_{i+1} = {\mathbf}a\_i {\ensuremath{{}+ \!\! +{}}}(\varphi(q\_1),\dots, \varphi(q\_{{\|{\mathbf}q\|}}))$ and $\psi({\mathbf}a\_N,q') = {\operatorname{!}}\varphi'(q')$ (where! denotes the right inclusion of the sum). The above is best understood as running a small while program: For given functional input φ and input qʹ, the universal attempts to extract a value from ψ by first calling it on input (ε, qʹ), and then branching according to the return value: if the return value is a list of questions it updates the list in the first argument with the respective answers according to φ. If the return value is an answer, it interprets this answer as the return value of the operator (cf. Figure [fig:U]). The notation we used strongly suggests that the universal can be specified by means of a function $U\colon {\mathcal B}'' \to {\mathbb N}\times {\mathcal B}\times {Q}\to {\operatorname{opt}}{A}'$ where the universal as described above can be recovered using the operator assignment M ↦ FM described in Section [sec: computability] via ψ ↦ FU(ψ). Indeed, Incone defines such a function `U` using only very elementary means. We refrain from writing out the exact definition of U here and point the interested reader to the Incone library, where the definition can be printed and a formal proof that it fulfills the above specification can be found (`FU_spec`). Furthermore, $\verb$U(ψ) is always monotone in the sense of the previous section (`U_mon`) and in particular FU(ψ) is always singlevalued (`FU_sing`). (0,0) rectangle (4,3); (-2.6,-.1) rectangle (5,3.1); at (4.5,1.5) FU(ψ); at (3.5,1) U; at (1,-.5) qʹ; (1,-.3) – (1,0); (1,0) – (1,1) – (.35,1); at (1.85,.5) b : = (ε, qʹ); (-2.5,0) rectangle (-.5,3); at (-1.5,1.5) ψ; (-.5,2) – (0,2); at (.2,2) d; (.35,2) – (.5,2)–(.5,2.8) – (.8,2.8); at (1.15,2.35) $d = {\operatorname{?}}{\mathbf}q$; at (1.15,1.65) d = !aʹ; (.5,2) – (.5,1.25) – (3,1.25) – (3,0); at (1,2.8) ${\mathbf}q$; (0,1) – (-.5,1); at (.2,1) b; (0,3.5) rectangle (3,5.5); at (1.5,4.5) φ; (2,3.5) – (2,3); at (2.6,2.75) $\varphi(q\_1)\ldots\varphi(q\_{{\|{\mathbf}q\|}})$; (2,2.5) – (2,1) – (1,1); at (2.9,2) update list; (1,3) – (1,3.5); (3,0) – (3,-.3); at (3,-.5) aʹ; [fig:U] That this actually defines a universal can be separated into two statements. Let us first argue that all operators of the form FU(ψ) are continuous. We actually prove the stronger statement, that from an associate ψ one can obtain a self modulating modulus of continuity for FU(ψ). Let us start by introducing the notion of a modulus of continuity for a multifunction $F\colon {\mathcal B}{\rightrightarrows}{\mathcal B}'$. Recall from Section [sec: computability] that we decided to call a finite list a certificate for $\varphi\in{\mathcal B}$ and qʹ ∈ Qʹ if for any ψ that coincides with φ on this list the return-values of F are identical. Note that previously we assumed F to be a partial function and here we talk about a multifunction, so we have to elaborate. Call a list certificate for φ and qʹ if for each ψ that coincides with φ on this list and any φʹ ∈ F(φ) and ψʹ ∈ F(ψ) it holds that φʹ(qʹ) = ψʹ(qʹ). Note that if F is singlevalued, being a certificate in this sense is equivalent to being a certificate for the corresponding multivalued function. Furthermore note that the existence of a certificate implies that the elements of F(φ) can only take one possible value in qʹ. In particular, if continuity of multifunctions is defined as before, i.e. by requiring a certificate to exist for all inputs, the corresponding notion implies singlevaluedness and should be understood as a notion of continuity for partial functions specified by relations and not as a notion for multifunctions (cf. ). Define the multivalued modulus of continuity $C\_F\colon {\mathcal B}{\rightrightarrows}{\operatorname{seq}}{Q}^{{Q}'}$ of a multifunction $F\colon {\mathcal B}{\rightrightarrows}{\mathcal B}'$ by CF(φ) : = {L: Qʹ → seqQ ∣ ∀qʹ: L(qʹ)is a certificate for $\varphi$ and $q’$}. The definition of CF makes sense for any multifunction $F\colon {\mathcal B}{\rightrightarrows}{\mathcal B}'$ but CF(φ) can only be non-empty if F(φ) has at most one element. The continuous partial operators $F\colon {\subseteq {\mathcal B}} \to {\mathcal B}'$ can be specified as exactly those multifunctions such that the domain of CF is a super-set of the domain of F (`cont_spec`). If F is continuous, then we call any partial choice function $\mu\colon{\subseteq {\mathcal B}} \to {\operatorname{seq}}{Q}^{{Q}'}$ of CF a modulus of continuity for F. Note that if μ is a modulus of continuity of F, then Cμ has the same type as CF. Thus it makes sense to call a modulus of continuity μ for F self-modulating if it is continuous and a modulus of continuity for itself, that is if it is also a choice function for Cμ. Let ψ be an associate of a partial operator F with respect to the universal U from above, i.e. let ψ be such that FU(ψ) tightens F. A self-modulating modulus for F can readily be obtained from ψ by tracking the queries in the evaluation of the universal. The same can be done for the values that the universal calls the function ψ on and one defines functions UQ and US such that FUQ(ψ) and FUS(ψ) are the corresponding operators. [`FqM_mod_FU`, `FqM_mod_FqM` and `FqM_mod_FsM`][resu: FUcont] For any fixed ψ of appropriate type the operator FUQ(ψ) is a modulus of continuity for FU(ψ), for itself and for FUS(ψ). The universal is used in Incone to construct exponentials in the category of represented spaces or, more simply put, to construct spaces of functions. The above Theorem [resu: FUcont] in particular implies that for any ψ the operator FU(ψ) is continuous. Its more general statement is enough to provide what is needed to prove the evaluation procedure on the constructed space of functions to be a continuous operation. The functions UQ and US are of more theoretical than practical importance. For the purpose of inspecting the evaluation of an associate Incone provides a more useful function `gather_queries` that on input of an associate ψ, a functional input φ, a discrete input qʹ and an effort n returns the list of all queries posed up to the n-th loop of the evaluation of the universal on these values (see `examples/KleeneKreisel.v` for examples). To finish the proof that U is an universal it is left to show that any partial continuous operator has an associate with respect to U. [`U_universal`][resu: Uuniversal] Any partial continuous operator $F\colon{\subseteq {\mathcal B}} \to {\mathcal B}'$ between naming spaces has an associate with respect to the universal U described above. I.e. there exists some ψ such that FU(ψ) tightens F. We do not give a lot of details for this proof here, but let us sketch the most important parts and point out some interesting details. Let μ be any function that chooses through the multivalued modulus of continuity CF of F and let f be any function that chooses through F. For any fixed enumeration $(q\_i)\_{i \in {\mathbb N}}$ of Q, one can attempt to define an associate ψ for F as follows: On input $(\mathbf a, q')$ interpret the list $\mathbf a$ as a partial function by assuming that its elements are the return values on the first ${\|\mathbf a\|}$ elements mentioned in the enumeration of Q, i.e. interpret it as the finite function such that qi ↦ ai for all $0 < i \leq {\|{\mathbf}a\|}$. Extend this function to a total function $\varphi\_{\mathbf a}$ that is from the domain of F if this is possible. If $\mu(\varphi\_{{\mathbf}a}, q') \subseteq (q\_1,\ldots, q\_{{\|{\mathbf}a\|}})$ then return ${\operatorname{!}}f(\varphi\_{{\mathbf}a})(q')$, otherwise ask for the $({\|{\mathbf}a\|} +1)$-st element mentioned in the enumeration. As μ is a modulus for F, evaluating this associate using the universal U only results in correct return values. That μ(φ, qʹ) must be contained in some initial segment of the enumeration and the sequence of functions $\varphi\_{{\mathbf}a}$ converges to φ gives hope that the iteration may often be finite. Without further assumptions about μ, however, this can not be proven and the function ψ defined above might fail to be an associate of F. Since it is always possible to extract a self-modulating modulus from an associate it may not be surprising that the proof can be completed if the modulus is additionally known to be self-modulating. In the library the existence of a self-modulating modulus is proven by picking μ to be minimal with respect to subset inclusion under the additional condition that it can only return initial segments with respect to the enumeration of Q. We call such a modulus a minimal modulus with respect to the enumeration. [`mod_minmod`] A minimal modulus is always self-modulating. The reason for the indefinite article in this lemma is that the existence of a minimal modulus is not constructively provable, and indeed the Coq proof of its existence is classical and relies on a fairly strong choice principle, namely one that covers naming spaces and is thus strictly stronger than countable choice. [`exists_minmod`][lem: exists minmod] For any continuous partial operator there exists a minimal modulus of continuity. A minimal modulus is unique up to choice of the sequence and as it is known to be impossible to constructively prove the existence of an extensional way to obtain a modulus of continuity, the proof is inherently classical. In particular there are computable operators on Baire space whose minimal modulus with respect to the identity as enumeration is not computable. By contrast, from a computable associate a computable self-modulating modulus can be read of. Incone thus makes some efforts to avoid the use of the minimal modulus and instead allow to construct a self-modulating modulus from additional information about the operator. The details of this leave the scope of this paper we do not elaborate further on this aspect. Finally, the library defines a function D that exchanges the arguments of the universal. [`D_spec`][resu: Dspec] For all φ and ψ it holds that FU(ψ)(φ) = FU(Dφ)(ψ). Here, the types have been purposefully omitted, details can be found in the library. Note that, while U(ψ) has the more complicated type and is interpreted as FU(ψ) using the operator assignment, D can be directly interpreted as a function or using the associated multifunction. The above theorem is interesting because it is related to the Cartesian closure of the category of represented spaces (see Section [sec: represented spaces] for details on the category). However, it falls short in strength as it only considers a special case in which it is not necessary to talk about tupling of elements of naming spaces. [`FsM_mod_FU`, `FsM_mod_FsM` and `FsM_mod_FqM`] Theorem [resu: FUcont] remains true if UQ and US are exchanged and ψ and φ are exchanged. Represented spaces and continuous realizability =============================================== Computable analysis is the theory of computation on sets of continuum cardinality. To make such spaces available to computation, computable analysis considers encodings of such sets over Baire space ${\mathcal B}= {\mathbb N}^{\mathbb N}$. Typically, such an encoding is understood to be a surjective partial function $\delta\colon {\subseteq {\mathcal B}} {\rightrightarrows}X$ from Baire space to the set that is called a representation. Instead of a partial function, the representation can also be considered a singlevalued multifunction and we will go back and forth between these two views seamlessly. According to Lemma [resu: PF2MFcotot] the requirement of being surjective translates to the corresponding multifunction being co-total. In the formal development in Incone everything is formulated using singlevalued, co-total multifunctions. Moreover, instead of using Baire space, ${\mathcal B}$ is allowed to be any naming space in the sense of the previous chapter, i.e. any space of the form ${\mathcal B}= {A}^{Q}$ where A and Q are countable and non-empty. A representation of a set X is defined as a naming space ${\mathcal B}$ together with a singlevalued, co-total multifunction $\delta\colon {\mathcal B}{\rightrightarrows}X$. Note that ${\mathcal B}$ can be inferred from δ. This justifies the fact to just call δ the representation. Just like in computable analysis we call those $\varphi \in {\mathcal B}$ such that x ∈ δ(φ) the names of x. The surjectivity of δ can now be reformulated as each element of the space having at least one name and the singlevaluedness as each name uniquely identifying the element. An alternate maybe more descriptive phrase for “name” would be “description”. As ${\mathcal B}$ is a naming space it is of the form ${\mathcal B}= {A}^{Q}$ for some countable and non-empty sets Q of questions and A of answers. A useful interpretation of φ being a name is that it provides on-demand information about x. If φ is a name of x then for each question q ∈ Q about the abstract object x the value φ(q) ∈ A can be considered a valid answer to that question. A represented space is a pair ${\mathbf X}=(X, \delta\_{\mathbf X})$ of a set X and a representation $\delta\_{\mathbf X}$ of X. When appropriate we decorate the naming space of the representation and its question and answer sets with indices as well, i.e. we assume $\delta\colon{\mathcal B}\_{\mathbf X}{\rightrightarrows}X$, where ${\mathcal B}\_{\mathbf X}= {A}\_{\mathbf X}^{{Q}\_{\mathbf X}}$ and ${Q}\_{\mathbf X}$ and ${A}\_{\mathbf X}$ are countable and non-empty. As an example let us equip the real numbers with a representation. [`examples/Q_reals.v`][ex: qreals] Let $X := {\mathbb R}$ be the set of real numbers and pick the question and answer sets to be the rational numbers, i.e. ${Q}\_{\mathbb R}= {A}\_{\mathbb R}:= {\mathbb Q}$ and thus ${\mathcal B}\_{\mathbb R}= {\mathbb Q}^ {\mathbb Q}$. Clearly ${\mathbb Q}$ is countable and non-empty so that ${\mathcal B}\_{\mathbb R}$ is a naming space. Let $\delta\_{\mathbb R}\colon {\subseteq {\mathcal B}\_{\mathbb R}} \to {\mathbb R}$ be the partial function specified by $$\delta\_{\mathbb R}(\varphi) = x \quad \iff \quad \forall \varepsilon \in {\mathbb Q}, 0 < \varepsilon \implies \|x - \varphi(\varepsilon)\| \leq \varepsilon.$$ Then $\delta\_{\mathbb R}$ is a representation of ${\mathbb R}$. Indeed, using the axiomatization of the real numbers provided by Coq’s standard library $\delta\_{\mathbb R}$ can be proven singlevalued and surjective (`rep_RQ_sing` and `rep_RQ_sur`) and we refer to the represented space $({\mathbb R}, \delta\_{\mathbb R})$ simply as ${\mathbb R}$. r0.275 [fig:realizer] Computability and continuity of partial operators on naming space can be used to define computability and continuity of functions between represented spaces by means of realizers. For represented spaces ${\mathbf X}$ and ${\mathbf X}'$, a partial operator $F\colon{\subseteq {\mathcal B}\_{\mathbf X}} \to {\mathcal B}\_{{\mathbf X}'}$ is a realizer of a function $f\colon {\mathbf X}\to {\mathbf X}'$ if for each name φ of x the value F(φ) is defined and a name of f(x) (cf. Figure [fig:realizer]). A function between represented spaces is continuous if it has a continuous realizer and computable if it has a computable realizer. The represented spaces form a Cartesian closed category both if the continuous functions are used as morphisms, and if the computable functions are used. The use of the word “continuous” here is often contested and many say it should be reserved for the topological concept. Others argue that the above is the correct notion and that the topological one is only a syntactic approximation of it. It is one of the objectives of this paper to give a formal proof that the above and topological continuity have a considerable overlap in that they coincide for metric spaces. For instance for the real numbers represented as indicated in Example [ex: qreals] it is true that a function from ${\mathbb R}\to {\mathbb R}$ has a continuous realizer if and only if it is continuous in any of the more traditional ways to make sense of being continuous. This is more generally true for functions between metric spaces and we go into detail about this in Section [sec: metric spaces]. It is possible to extend the definition of being a realizer to allow for both the realizer and the realized function to be multivalued. We refrain from stating the elementary definition as for the purpose of this paper it is sufficient to know that the extended definition fulfills the following specification: [`rlzr_spec`][resu: rlzrspec] A multifunction $F\colon {\mathcal B}\_{\mathbf X}{\rightrightarrows}{\mathcal B}\_{{\mathbf X}'}$ realizes another multifunction $f\colon {\mathbf X}{\rightrightarrows}{\mathbf X}'$ if and only if $\delta\_{{\mathbf X}'}\circ F$ tightens $f\circ \delta\_{\mathbf X}$. The above lemma can also be used backwards to express the notion of a tightening as a special case of being a realizer by using the identity function as representation (`id_rlzr_tight`). One may even further extend the definition of a realizer by dropping the requirement that the representations are singlevalued. In the terminology of computable analysis this would mean dealing with multi-representations and while in the manipulation of discrete data the use of non-singlevalued encodings is fairly common, in computable analysis the use of multirepresentations is rare. The above characterization does not generalize to multirepresentations. For the full definitions we point the interested reader to the Rlzrs library. Here, we are only interested in representations and mostly in continuous, and therefore singlevalued, realizers. However, in Section [sec: closed choice] we discuss closed choice on the natural numbers as an elementary example of a multifunction between represented spaces. We call a multifunction between represented spaces continuously realizable if there exists a continuous realizer in the sense of the previous lemma. Note that under the usual identification of a function with the multifunction that uniquely specifies it, continuity is a special case of continuous realizability and we sometimes use the latter to distinguish it from other notions of continuity if confusion is possible. One reason that the use of continuously realizable multifunctions is common in computable analysis is that continuity of functions often fails for extensionality reasons. For instance, one may formulate a multifunction corresponding to the parallel or on the space introduced in Section [sec: closed choice] as Sierpinski space to see that there exist continuously realizable functions that do not have any continuous choice function. A special case where exactly the opposite behavior appears is that of a naming space equipped with the identity function as a representation: a multifunction between naming spaces is continuously realizable if and only if it has a singlevalued tightening that is continuous in the sense of Section [sec: continuity] and therefore allows for a continuous choice function. That continuity and continuous realizability is preserved under composition follows from content of the Rlzrs library together with the fact that continuity of operators on Baire space is preserved under composition (Theorem [resu: contcomp]). [`comp_cont` and `comp_hcr`] The composition of continuous functions is continuous and the composition of continuously realizable multifunctions is continuously realizable. Examples and basic constructions such as products and sums ---------------------------------------------------------- Now that we can talk about continuity and computability on the real numbers, a reasonable next step is to attempt to prove addition and multiplication computable. Both of these functions are of type ${\mathbb R}\times {\mathbb R}\to {\mathbb R}$ and to make sense of continuity of functions of these types we need to specify how ${\mathbb R}\times{\mathbb R}$ should be made a represented space. The Incone library automatically generates such a represented space ${\mathbf X}\times {\mathbf Y}$ from arbitrary represented spaces ${\mathbf X}$ and ${\mathbf Y}$ by using the query type ${Q}\_{{\mathbf X}\times{\mathbf Y}}:= {Q}\_{\mathbf X}+ {Q}\_{\mathbf Y}$, the answer type ${A}\_{{\mathbf X}\times{\mathbf Y}} := {A}\_{\mathbf X}\times {A}\_{\mathbf Y}$ and the representation $\delta\_{{\mathbf X}\times{\mathbf Y}}$ defined by $$\delta\_{{\mathbf X}\times{\mathbf Y}}(\psi) = (x, y) \quad \iff \quad \delta\_{{\mathbf X}}({\operatorname{fst}}{} \circ \psi \circ {\operatorname{inl}}) = x \wedge \delta\_{{\mathbf Y}}({\operatorname{snd}}{} \circ \psi \circ {\operatorname{inr}}) = y.$$ This can be decoded as follows: A name of the pair (x, y) should be a pair (φ, φʹ) of a name for x and a name for y. Since the set of pairs ${\mathcal B}\_{\mathbf X}\times {\mathcal B}\_{\mathbf Y}$ does not have the type that we required a naming space to have, we embed it into the naming space ${\mathcal B}\_{{\mathbf X}\times{\mathbf Y}}:=({A}\_{\mathbf X}\times {A}\_{\mathbf Y})^{{Q}\_{\mathbf X}+ {Q}\_{\mathbf Y}}$. There are several possible choices for ${\mathcal B}\_{{\mathbf X}\times{\mathbf Y}}$, but for the one picked by Incone the projection function $\pi\colon {\mathcal B}\_{{\mathbf X}\times {\mathbf Y}} \to {\mathcal B}\_{\mathbf X}$ can particularly conveniently be expressed by the natural operations on the question and answer spaces, namely π(ψ) : = fst ∘ ψ ∘ inl and the formula for the second projection is similar. [`prod_rep_sing`, `prod_rep_sur` and `prod_uprp_cont`] For any represented spaces ${\mathbf X}$ and ${\mathbf Y}$ the space $(X\times Y, \delta\_{{\mathbf X}\times{\mathbf Y}})$ is a represented space and it is the product of ${\mathbf X}$ and ${\mathbf Y}$ in the category of represented spaces. Part of verifying the universal property of a product in the category of represented spaces with continuous resp. computable functions as morphisms is to prove the projections computable resp. continuous (`fst_cont` and `snd_cont`). [`examples/Q_reals.v`][resu: R operations] Addition and multiplication of real numbers is computable (`Rplus_cont` and `Rmult_cont`). As described in more detail in Section [sec: computability] this should be taken to mean that the operations are continuous and the realizers can be explicitly specified as Coq-functions whose definitions contain no axioms. Indeed, the realizers are defined not through the more complicated operator assignment but more directly using the `F2MF` correspondence. Furthermore, their definition only uses very simple tools and the operations should therefore even be considered primitive recursive. Incone additionally proves some other basic functions on product spaces computable. Most notably it provides the possibility to glue continuous functions $f\colon {\mathbf X}\to {\mathbf X}'$ and $g\colon {\mathbf Y}\to {\mathbf Y}'$ together to obtain another continuous function $f \times g\colon {\mathbf X}\times {\mathbf Y}\to {\mathbf X}' \times {\mathbf Y}'$ (`fprd_cont`). More generally, such a construction is provided for continuously realizable multifunctions. As another basic example of a represented space that is needed below let I be any countable and non-empty set. Set ${Q}\_{\mathbf I}:= \{\star\}$ and ${A}\_{\mathbf I} := I$. Then the function $\delta\_{\mathbf I}(\varphi) := \varphi(\star)$ makes the pair $\mathbf I:= (I, \delta\_{\mathbf I})$ a represented space that is discrete in the following sense: [`cs_id_dscrt`][resu: csiddscrt] For any countable, non-empty set I the represented space $\mathbf I$ described above is discrete in the sense that any function that has $\mathbf I$ as its domain is continuous. Moreover, any multivalued function with $\mathbf I$ as input space is continuously realizable. In particular, the natural numbers can be assigned a discrete represented space. We denote both the set and the represented space of natural numbers by ${\mathbb N}$. Let us briefly mention a couple of additional constructions. The Incone library proves that the represented space $\mathbf 1$ constructed from the unit type as above is a terminal object in the category of represented spaces. It defines for each pair of represented spaces ${\mathbf X}$ and ${\mathbf Y}$ a space ${\mathbf X}+{\mathbf Y}$ that is proven to be the category-theoretical sum. It gives a separate option type construction and proves the resulting space to be isomorphic to ${\mathbf X}+ \mathbf 1$. There is also an elementary construction of the space of finite lists of elements from a represented space ${\mathbf X}$, where asking a question about an element of x results in a list of answers to question for each of the elements of the list. The space of infinite sequences, limits and pointwise operations ---------------------------------------------------------------- Let I be a countable, non-empty set and let ${\mathbf X}$ be a represented space. Define a represented space $\prod\_{I} {\mathbf X}$ whose underlying set are the functions of type I → X by setting ${Q}\_{\prod\_{I}{\mathbf X}}:= I \times {Q}\_{\mathbf X}$, ${A}\_{\prod\_I}:= {A}\_{\mathbf X}$ and $$(x\_i) \in \delta\_{\prod\_I{\mathbf X}}(\varphi) \quad \iff\quad \forall i \colon I, x\_i \in\delta\_{\mathbf X}(q \mapsto \varphi(i,q)),$$ where (xi) is short for the function i ↦ xi. [`rep_Iprod_sing`, `rep_Iprod_sur` and `cprd_uprp_cont`][resu: cprd] Let I be countable and non-empty. Then $\prod\_{I}{\mathbf X}:= (X^I, \delta\_{\prod\_I {\mathbf X}})$ is a represented space and ${\mathbf X}^\omega:= \prod\_{\mathbb N}{\mathbf X}$ is a countably infinite product in the category of represented spaces and continuous functions. The use of the symbol ω instead of ${\mathbb N}$ is to differentiate the space ${\mathbf X}^\omega$ of infinite sequences in ${\mathbf X}$ from the space of functions from the natural numbers to ${\mathbf X}$ that is discussed in the next section. The proof that $\delta\_{\prod\_I{\mathbf X}}$ is singlevalued assumes functional extensionality and the proof of surjectivity needs a choice principle over I. Since $I={\mathbb N}$ is by far the most common use-case and I is assumed to be countable anyway, this usually boils down to the axiom of countable choice. The proof of the universal property relies on stronger choice principles and the law of excluded middle. Since the category of represented spaces with computable functions fails to have countably infinite products, the corresponding result is inherently inefficient. Given a sequence of computable functions $f\_n\colon {\mathbf Y}\to {\mathbf X}$ the function $F\colon {\mathbf Y}\to {\mathbf X}^{\mathbb N}$ that glues them together need only be continuous. Thus we only prove the existence of its realizer. To obtain a computable realizer for F one needs one algorithm that uniformly computes the fn and not just the existence of an algorithm for each fn. How far our use of axioms can be optimized in this case is difficult to tell at this point in time since the current proof uses a part of the library that has not yet been optimized in terms of axiom use. Since it is more a sanity result than something that may actually be of use, optimizations here are not our highest priority. An example of a partial function whose natural domain is a subset of the space of sequences is the limit operator. Consider the multivalued function $\lim\_{\mathbf X}\colon {\mathbf X}^\omega {\rightrightarrows}{\mathbf X}$ where $x \in \lim\_{\mathbf X}(x\_n)$ if and only if there is a convergent sequence of names $(\varphi\_n) \subseteq {\mathcal B}\_{\mathbf X}$ and some φ such that φ is a name of x, each φn is a name for xn and the sequence (φn) converges to φ in ${\mathcal B}\_{\mathbf X}$, i.e. $\lim\_{{\mathcal B}\_{\mathbf X}}(\varphi\_n) = \varphi$ where the limit in ${\mathcal B}\_{\mathbf X}$ is taken point-wise as explained in Section [sec: continuity]. While the limit operator on Baire space is singlevalued, this need not be true for the limit operator on a general represented space, as can be seen at the example of Sierpinski space that is discussed in Section [sec: closed choice]. In most spaces that are relevant for numerical analysis, the limit operator is singlevalued but discontinuous. It is often the case that computability of the limit operator can be recovered by restricting it to an appropriate set of efficiently convergent sequences. [`examples/Q_reals.v`][resu: R limits] The limit operator $\lim\_{\mathbb R}$ where ${\mathbb R}$ is represented as in Example [resu: R operations] is discontinuous (`lim_not_cont`). Its restriction to those sequences (xn) that are efficiently Cauchy in the sense that ∣xn − xm∣ ≤ 2− n + 2− m is computable (`lim_eff_hcr`). To be strict, the example file proves these properties with respect to the metric notion of convergence that is introduced later in Section [sec: continuities]. To really obtain what is claimed here one has to additionally use that metric convergence and convergence in the represented space of real numbers are equivalent. The major part of this equivalence is proven in Theorem [lem:lim-mlim] and the rest, namely that the representation of real numbers introduced in Example [ex: qreals] is equivalent (in the sense introduced shortly) to the representation of real numbers as metric space, is also provided by the Incone library. A function $f\colon{\mathbf X}\to{\mathbf Y}$ between represented spaces is called sequentially continuous if it preserves limits, i.e. if $\lim\_{\mathbf X}x\_n = x$ implies that $\lim\_{\mathbf Y}f(x\_n) = f(x)$. In general, sequential continuity need not imply continuity and this is not a matter of whether one works constructively or not. From the point of view of a classical mathematician there exist sequentially continuous functions between represented spaces that are simply not continuously realizable. The difference can be eliminated by making additional assumptions about the involved represented spaces. One such assumption is referred to as admissibility and while we do not go into the details here, this condition is of major importance throughout different parts of computable analysis. Most spaces encountered in practice are admissible, in particular the space of real numbers from Example [ex: qreals], all the representations of metric spaces considered in Section [sec: metric spaces], and also the representations of hyper-spaces that are topic of Section [sec: closed choice] are admissible. The relation between sequential continuity and continuous realizability is discussed in more detail in Section [sec: continuities]. The Incone library proves some further lemmas about spaces of sequences that might be useful in applications and should thus not go unmentioned. Two represented spaces ${\mathbf X}$ and ${\mathbf Y}$ are isomorphic, in symbols ${\mathbf X}\simeq {\mathbf Y}$, if there exists a continuous bijection with continuous inverse. The spaces are computably isomorphic if there exists a computable bijection with computable inverse. In the special case where the underlying set of two represented spaces are identical we call their representations equivalent if the identity function is an isomorphism. While mathematicians are usually fairly liberal in identifying sets and equivalence of representations is a widely used concept, in our formal development isomorphy is by far the more common notion. [`cprd_prd`] For any represented spaces ${\mathbf X}$, ${\mathbf Y}$ and countable, non-empty set I it holds that $\prod\_{I}({\mathbf X}\times {\mathbf Y}) \simeq \big(\prod\_{I}{\mathbf X}\big) \times \big(\prod\_{I}{\mathbf Y}\big)$, the spaces are even computably isomorphic. The realizers are defined using very limited means and interpreted using the `F2MF` assignment, thus they should be considered primitive recursive. Any function $f\colon {\mathbf X}\to {\mathbf Y}$ can be extended to a function $f^I \colon \prod\_I {\mathbf X}\to \prod\_I {\mathbf Y}$ that applies f pointwise, i.e. is defined by fI((xi)i ∈ I) : = (f(xi))i ∈ I. [`ptw_cont`] Whenever f is a continuous function, then also fI is continuous. This lemma has a multivalued variant (`ptw_hcr`). If the realizer of f can be expressed as a Coq-function through the `F2MF` interpretation, then so can fI. The same should hold true if a realizer of f can be expressed via a Coq function through the operator assignment M ↦ FM from Section [sec: computability], although we have not carried out the details yet. Another common situation is that an operation $\\colon {\mathbf X}\times {\mathbf Y}\to {\mathbf Z}$ is used to construct an operation $\^I\colon \prod\_I {\mathbf X}\times \prod\_I {\mathbf Y}\to \prod\_I {\mathbf Z}$ via (xi) \* I(yi) : = (xi \* yi). For instance the natural operations on diverse spaces of sequences of real numbers are introduced by pointwise applying the arithmetic operations of the real numbers. More concretely, the most common vector space structure on the space ${\mathbb R}^\omega$ of sequences in the real numbers is to set ${\mathbf X}={\mathbf Y}={\mathbf Z}:={\mathbb R}$ in the above and use $+^{\mathbb N}$ where + is addition of real numbers. A proof that this extension also preserves continuity can be directly obtained from the previous two lemmas. [`cptw_op_cont`] If \* is a continuous operation, then the corresponding pointwise operation \* I is also continuous. As mentioned before, examples where these results are useful is when reasoning about pointwise addition and multiplication as operations on ${\mathbb R}^\omega$. The operation of multiplying a sequence with a scalar is also covered by first embedding the scalars into the sequences as the constant sequences and then using pointwise multiplication. Function spaces and infinite sequences as functions --------------------------------------------------- Let ${\mathbf X}$ and ${\mathbf Y}$ be represented spaces and denote by ${\mathbf Y}^{\mathbf X}$ the collection of all continuous functions from ${\mathbf X}$ to ${\mathbf Y}$. Recall that Section [sec: universal] gave an explicit description of a continuous universal U. This universal can be used to equip the space ${\mathbf Y}^{\mathbf X}$ with a representation as follows: Let ${\mathcal B}\_{{\mathbf Y}^{\mathbf X}}$ be the space that the universal U points to for input naming spaces ${\mathcal B}\_{\mathbf X}= {A}\_{\mathbf X}^{{Q}\_{\mathbf X}}$ and ${\mathcal B}\_{\mathbf Y}= {A}\_{\mathbf Y}^{{Q}\_{\mathbf Y}}$. This expands to ${Q}\_{{\mathbf Y}^{\mathbf X}} = {\operatorname{seq}}{A}\_{\mathbf X}\times {Q}\_{\mathbf Y}$ and ${A}\_{{\mathbf Y}^{\mathbf X}} = {\operatorname{seq}}{Q}\_{\mathbf X}+ {A}\_{\mathbf Y}$, where ${\mathcal B}\_{{\mathbf Y}^{\mathbf X}} = {A}\_{{\mathbf Y}^{\mathbf X}} ^ {{Q}\_{{\mathbf Y}^{\mathbf X}}}$. Note that a function (as opposed to a partial or multifunction) is uniquely determined by each of its realizers. Thus there exists a unique partial function $\delta\_{{\mathbf Y}^{\mathbf X}}\colon {\mathcal B}\_{{\mathbf Y}^{\mathbf X}}{\rightrightarrows}{\mathbf Y}^{\mathbf X}$ specified by $$\delta\_{{\mathbf Y}^{\mathbf X}}(\psi) = f \quad \iff \quad F\_{U(\psi)} \text{ realizes } f.$$ Put differently, although $\delta\_{{\mathbf Y}^{\mathbf X}}$ is specified as a multifunction, it is singlevalued. Any $f \in {\mathbf Y}^{\mathbf X}$ is continuous which by definition means that there exists some continuous $F\colon{\subseteq {\mathcal B}\_{\mathbf X}} \to {\mathcal B}\_{\mathbf Y}$ that realizes f. Because U is a universal there exists some ψ such that FU(ψ) tightens F (cf. Theorem [resu: Uuniversal]). As being a realizer is preserved under tightening also FU(ψ) realizes f. Thus $\delta\_{{\mathbf Y}^{\mathbf X}}$ is surjective and co-total when considered a multifunction. [`fun_rep_sing` and `fun_rep_sur`][prop:fun-rep] For any represented spaces ${\mathbf X}$ and ${\mathbf Y}$ the space $({\mathbf Y}^{\mathbf X},\delta\_{{\mathbf Y}^{\mathbf X}})$ of continuous functions as defined above is a represented space. The proof of singlevaluedness assumes proof irrelevance and functional extensionality. Here proof irrelevance is another commonly assumed axiom that asserts a Coq property (`Prop`) and that over the internal logic of Coq is implied by the law of excluded middle. The following theorem uses the finite products as they were constructed in Section [sec: simple types] and can be proven from the continuity properties of the universal from Theorem [resu: FUcont]. [`eval_cont`][resu: evalcont] Evaluation as operation ${\mathbf Y}^{\mathbf X}\times {\mathbf X}\to {\mathbf Y}$ is computable. The function space construction overlaps in its scope with the space of infinite sequences: For a countable, non-empty index set I and a represented space ${\mathbf X}$, the set underlying the space $\prod\_I {\mathbf X}$ is the set of functions from I to ${\mathbf X}$. Recall what it meant for the discrete space $\mathbf I$ generated from I as in Section [sec: simple types] to be discrete: Proposition [resu: csiddscrt] says that any function starting from $\mathbf I$ is continuous. As a consequence, the sets underlying $\prod\_I {\mathbf X}$ and ${\mathbf X}^{\mathbf I}$ are identical. Indeed these spaces are computably isomorphic. [`sig_iso_fun`][resu: sigisofun] For any represented space ${\mathbf X}$ and any countable, non-empty set I the space $\prod\_{I} {\mathbf X}$ from the last section is computably isomorphic to the function space ${\mathbf X}^{\mathbf I}$, where $\mathbf I$ the discrete space constructed from I as described in Section [sec: simple types]. [sketch] Recall that ${A}\_{{\mathbf X}^{\mathbf I}} = {\operatorname{seq}}{Q}\_{\mathbf I} + {A}\_{\mathbf X}$ and that we decided to refer to the left inclusion into this sum by? and to the right inclusion by!. A realizer $T\colon {\mathcal B}\_{\prod\_I{\mathbf X}} \to {\mathcal B}\_{{\mathbf X}^{\mathbf I}}$ that translates a name of a sequence to a name of the corresponding function can be directly specified via $$T(\varphi)({\mathbf}a,q') := \begin{cases}?(\star) & \text{if } {\mathbf}a = \epsilon \\!\varphi(a\_1,q') & \text{, where } {\mathbf}a = (a\_1, \ldots, a\_{\|{\mathbf}a\|}). \end{cases}$$ To see that this realizer is correct we need to prove that whenever φ is a $\prod\_I{\mathbf X}$-name of some (xi), then FU(T(φ)) is a realizer of the function $i \mapsto x\_i\colon \mathbf I \to {\mathbf X}$. The former means that for any question q about xi, φ(i, q) is a valid answer. To check that FU(T(φ)) is a realizer note that ⋆ ↦ i is the only valid name of i in $\mathbf I$ and that for the correctness of T it is thus enough to argue that FU(T(φ))( ⋆ ↦ i) on any input q returns φ(i, q). Now the values of FU(T(φ)) are obtained by evaluating the universal. If the evaluation of U(T(φ)) is started on functional input ⋆ ↦ i and input q it will first call T(φ) on input (ε, q) and be returned?( ⋆ ). This will lead U to evaluate ⋆ ↦ i in ⋆ and call T(φ) on the updated input ((i), q). As T(φ) returns!φ(i, q) on this input U will finish its run by returning φ(i, q) and we have proven correctness of the translation. The translation in the other direction, i.e. constructing a sequence from a continuous function proceeds by using a variation of the realizer of evaluation. Note that the first translation can be defined by very elementary means but is specific to the details of the universal U. The translation in the other direction is independent of the implementation of the universal but only relies on availability of an algorithm for evaluation. The algorithm for evaluation, however, executes the universal which requires an unbounded search and is considerably less elementary. More specifically, the first translation can be defined as a function that is interpreted directly or as its associated multifunction, while the second translation requires the use of the more complicated operator assignment discussed in detail in Section [sec: computability]. Indeed, the second translation need in general not have a total continuous realizer and thus one should not expect it to be possible to give a definition of a translation as elementary as the one for the first direction. The above theorem is a good example of a result that is usually stated as an equivalence but that was formulated as an isomorphism in our formal development. The sets underlying ${\mathbf X}^\omega$ and ${\mathbf X}^{\mathbb N}$ are considered equal from a mathematical point of view, but formally the elements of the latter are pairs of a function and a proof that this function is continuous. The set of such pairs is still in bijection with its first components because we work in a setting where different proofs of the same property are considered equal. More specifically the equality of these proofs is what the axiom of proof irrelevance states and over the logic of Coq proof irrelevance is implied by the law of excluded middle that we regularly assume anyway. The bijection can thus be given as the function that just drops the proof in the second component of the pair and for the inverse one lifts an arbitrary function by adding the proof obtained from the fact that ${\mathbb N}$ is discrete. Theorem [resu: sigisofun] above is an important building block for obtaining the results about hyper-spaces that are presented Section [sec: closed choice]. As a more basic example of an application of this theorem let us give some additional justification for our previous choice to use the identity function as canonical representation of a given naming space. More specifically let us prove that the structure thus obtained is the same as considering a naming space as space of functions from its question set as discrete space to its answer set as such. For any naming space ${\mathcal B}= {A}^{Q}$ it holds that $({\mathcal B},\mathrm{id}\_{\mathcal B}) \simeq \mathbf A^{\mathbf Q}$. By the previous theorem $\mathbf A^{\mathbf {Q}} \simeq \prod\_{Q}\mathbf A$. The equivalence of the representation of $\prod\_{Q} \mathbf A$ with the identity representation is straightforward to define directly. To be exact, T(φ)(q, ⋆ ) : = φ(q) continuously translates from the identity representation to that of $\prod\_{Q}\mathbf A$ and Tʹ(φʹ)(q) : = φʹ(q, ⋆ ) continuously translates back. Application to continuity in metric and hyper spaces ==================================================== A map $d\colon M \times M \to {\mathbb R}$ is called a pseudo-metric on a set M if it returns non-negative values, is symmetric under exchange of its arguments and fulfills d(x, x) = 0 and the triangle inequality d(x, z) ≤ d(x, y) + d(y, z). It is called a metric if it is additionally true that d(x, y) = 0 implies x = y. A pair (M, d) is called a pseudo-metric space if d is a pseudo-metric on M and a metric space if d is a metric. Every pseudo-metric space comes with a topology that is generated by the open balls with respect to the pseudo-metric and therefore with notions of continuity of functions on and convergence of sequences in this space. The notion of convergence is of particular importance since any pseudo-metric space is first-countable and thus knowing the limits of sequences is sufficient for characterizing continuity. The notion of a metric captures the properties that one expects a notion of distance to have in a very general sense. Metric spaces are a widely applicable tool for talking about continuity on many spaces of practical interest and a common sight in many branches of mathematics. As such, metric spaces have received considerable attention in their formal treatment. A definition of continuity that does not require any knowledge of topology can be given using the ɛ-δ-criterion. A function f: M → Mʹ between pseudo-metric spaces (M, dM) and (Mʹ, dMʹ) is called continuous in x ∈ M if $$\forall \varepsilon > 0, \exists \delta > 0, \forall y\colon d\_{M} (x, y) \leq \delta \implies d\_{M'}(f(x),f(y)) \leq \varepsilon.$$ Here, ɛ and δ are real numbers but replacing them by rational numbers results in an equivalent definition. The function is called continuous if it is continuous in every point of M. An element x of a pseudo-metric or metric space (M, d) is said to be the limit of a sequence (xn) in M, in symbols lim(M, d)(xn) = x, if ∀ɛ > 0, ∃N, ∀n ≥ N: d(x, xn) ≤ ɛ. Again, ɛ is taken to be a real number but may equivalently be restricted to be rational. A function f between pseudo-metric spaces is said to be sequentially continuous in x if for each sequence xn in M with lim(M, dM)(xn) = x it holds that lim(Mʹ, dMʹ)(f(xn)) = f(x) and sequentially continuous if it is sequentially continuous in every point of M. Note that above for metric spaces we reuse the terms that we already used for represented spaces: We give an alternative definition of what it means to be continuous, a limit and sequentially continuous. Since we make a clear distinction of whether the spaces we operate on are metric spaces or represented spaces, this overloading rarely leads to confusion. Note that while the names of the concepts are identical, there are significant differences for their behavior in a metric and a represented space context respectively: A function between metric spaces is continuous if and only if it is sequentially continuous while for represented spaces the backward implication can fail. Recovering continuity on Baire space from a metric structure ------------------------------------------------------------ Let ${\mathcal B}$ be a naming space, i.e. ${\mathcal B}= {A}^{Q}$ for some countable, non-empty sets Q and A. Since Q is countable and non-empty there exists a surjective function from ${\mathbb N}$ to Q. Such a function may be understood as an enumeration $\mathbf q = (q\_n)\_{n \in {\mathbb N}}$ of Q. For each such enumeration $\mathbf q$ define a mapping $d\_{\mathbf q}\colon {\mathcal B}\times {\mathcal B}\to {\mathbb R}$ by $$d\_{\mathbf q}(\varphi, \psi) := \begin{cases} 0 & \text{if } \varphi = \psi \\ 2^{-\min\{n\in {\mathbb N}\mid \varphi(q\_n) \neq \psi(q\_n)\}} & \text{otherwise.} \end{cases}$$ To argue that this definition is well-formed note that the second case is only assumed if it is not true that φ = ψ but in this case there exists some q such that φ(q) ≠ ψ(q) and since $\mathbf q$ is an enumeration of Q there exists some n such that q = qn and thus φ(qn) ≠ ψ(qn). Thus, the set that the minimum is taken over is not empty and the minimum takes a natural number value. [`dst_pos`, `dst_sym`, `dstxx`, `dst_trngl`, `dst_eq`] $({\mathcal B}, d\_{\mathbf q})$ is a metric space. Let us briefly discuss some details specific to our Coq proof of this proposition. The discussion before the proposition implies that the definition of $d\_{\mathbf q}$ branches over the undecidable proposition whether two elements of a naming space are equal and thus a priori only specifies a relation and not a function. Now, the axiomatization of the real numbers implies a choice principle that is strong enough to move from this relation to an actual function. To be able to prove that the thus defined function is a metric it is additionally necessary to assume functional extensionality. Another tool that is used in the proof to deal with the occurring minima somewhat efficiently is a function named `search`. There are several adaptations of such a function in other developments some of which could have been reused by assumption of rather mild additional axioms. The $\verb$search function made an earlier appearance in this paper: It is what is behind the implementation of the search operator from Example [ex: continuous search 1]. Recall that in Section [sec: continuity] we introduced for any naming space ${\mathcal B}$ the limit operator $\lim\_{\mathcal B}$ corresponding to pointwise convergence. We can now compare this limit operator to the one that corresponds to the metric notion of convergence and was introduced as lim(M, d) in the introduction of this section. [`lim_lim`][thm: limlim] Let ${\mathcal B}$ be a naming space and $\mathbf q$ an enumeration of its question set. Convergence with respect to $d\_{\mathbf q}$ is exactly pointwise convergence. I.e. $\lim\_{({\mathcal B},d\_{\mathbf q})} = \lim\_{\mathcal B}$. As sequential continuity is defined directly from the notions of convergence, the above theorem implies that the notion of sequential continuity for naming spaces as introduced in Section [sec: continuity] is also reproduced. A function between naming spaces is sequentially continuous if and only if it is sequentially continuous as function between the corresponding metric spaces. It is possible to generalize the above corollary to apply not only to total but also to partial operators. We omit the generalized statement here and instead formulate a theorem about the non-sequential version of continuity. [`cont_cont`][resu: contcont] Let ${\mathcal B}$ and ${\mathcal B}'$ be naming spaces and $\mathbf q$ and $\mathbf q'$ enumerations of their respective question sets. A partial operator $F\colon {\subseteq {\mathcal B}} \to {\mathcal B}'$ is continuous in the sense of Section [sec: continuity] if and only if it is continuous as a function from $({\operatorname{dom}}(F),d\_{\mathbf q})$ to $({\mathcal B}', d\_{\mathbf q'})$. We omit details of the straightforward proofs of Theorem [thm: limlim] and Theorem [resu: contcont]. [`examples/continuous_search.v`][ex: continuous search 3] The traditional notion of continuity of partial operators on Baire space ${\mathbb N}^{\mathbb N}$ is captured by the continuity introduced in Section [sec: continuity] if all of the questions and answer sets are taken to be the natural numbers. Comparing continuity in represented and in metric spaces -------------------------------------------------------- @!C@C+15ex@R1ex & @<1ex>@=>[ddddd]@<1ex>@=>[r] & @<1ex>@=>[l] @<1ex>@=>[r] @<1ex>@=>[uuuuu] & @=>[uuuuu] @<1ex>@=>[l] [fig: implications] A metric space is called separable if there exists a dense sequence $(r\_n)\_{n\in {\mathbb N}}$ in it. Here, density of a sequence can equivalently be taken to mean that any open ball in the metric space contains an element of the sequence or that for any point of the metric space there exists a subsequence of (rn) that converges to that point. Separable metric spaces are well investigated in computable analysis. To each dense sequence $\mathbf r = (r\_n)\_{n \in {\mathbb N}}$ in a metric space (M, d) assign a multifunction $\delta\_{{\mathbf M}\_{\mathbf r}}\colon {\mathbb N}^{\mathbb N}{\rightrightarrows}M$ defined by $$x \in \delta\_{{\mathbf M}\_{\mathbf r}}(\varphi) \quad \iff \quad \forall n, d(x, r\_{\varphi(n)}) \leq 2^{-n}.$$ Clearly ${\mathbb N}^{\mathbb N}$ is a naming space and $\delta\_{{\mathbf M}\_{\mathbf r}}$ can be proven singlevalued and surjective (`mrep_sing` and `mrep_sur`). Thus $\delta\_{{\mathbf M}\_{\mathbf r}}$ defines a representation and we denote the corresponding represented space by ${\mathbf M}\_{\mathbf r} := (M, \delta\_{{\mathbf M}\_{\mathbf r}})$. Note that the construction of ${\mathbf M}\_{\mathbf r}$ is very similar to how we chose to represent the real numbers through rational approximations in Example [ex: qreals]. A name of an element of a metric space produces an index of an approximation from an accuracy requirement. The only difference is that for real numbers we decided to minimize the number of types involved by using rational numbers as both accuracy requirements and approximations while for metric spaces we use accuracy requirements of the form 2− n to minimize the number of distinct types by matching the type of indices and setting both question and answer types to be ${\mathbb N}$. A sequence (xn) in a separable metric space (M, d) converges to a limit x from M if and only if the same is true in the represented space ${\mathbf M}\_{\mathbf r}$ or in symbols if and only if it holds that $\lim\_{{\mathbf M}\_{\mathbf r}}(x\_n) = x$. This remains true irrespectively of the choice of the dense sequence that is used for constructing the representation and can more concisely be formulated as $\lim\_{(M,d)} = \lim\_{{\mathbf M}\_{\mathbf r}}$. Furthermore, if (Mʹ, dʹ) is another separable metric space, then a function f: M → Mʹ is continuous as a function between metric spaces if and only if it is continuously realizable as a function $f\colon {\mathbf M}\_{\mathbf r} \to {\mathbf M}'\_{\mathbf r'}$ and this remains true irrespectively of the choices of the dense sequences $\mathbf r$ and $\mathbf r'$. This section describes our formal proofs of statements comparing the two continuity notions and their sequential versions (cf. Figure [fig: implications]). The proofs have been kept as constructive as possible. Since the definition of a metric space relies on the axiomatic reals, only one of the implications is fully constructive, the others are conservative over the background theory of real numbers and do not rely on the axioms of the real numbers in an essential way. Let us reiterate that the construction of a represented space from a separable metric space explicitly relies on the choice of a dense sequence. I.e. for each dense sequence $\mathbf r$ in the metric space one obtains a represented space ${\mathbf M}\_{\mathbf r}$. In the following we usually drop the index $\mathbf r$ for brevity. This is justified in a continuity setting as different choices of dense sequences lead to isomorphic represented spaces. As always, the situation is more complicated if computability is considered and in this case one should assume in the following that two metric spaces with dense sequences are fixed. In the formal development, the dense sequences are always present as parameters. Let us first describe the proof of the equivalence of the notions of sequential continuity. The main part of the proof is that the notions of limit in the metric space and the corresponding represented space coincide. [`lim_mlim`][lem:lim-mlim] Whenever (M, d) is a separable metric space and ${\mathbf M}$ as above then $$\lim\nolimits\_{(M,d)} = \lim\nolimits\_{\mathbf M}.$$ [idea] The proof that convergence in the represented space implies convergence in the metric space is straightforward. The idea behind the other direction can be sketched as follows: If (xn) converges to x in the metric space then there exists a modulus of convergence, i.e. some $\mu\colon {\mathbb N}\to {\mathbb N}$ such that ∀n, m ≥ μ(n): d(xm, x) ≤ 2− n. From an arbitrary sequence (φʹm) of names of xm and a name φʹ of x an appropriate convergent sequence of names can be defined by $$\varphi\_m(n) := \begin{cases} \varphi'(n+1), & \text{if } \mu(n+1) \leq m \\ \varphi'\_{m}(n), & \text{otherwise} \end{cases}$$ and its limit is given by φ(n) : = φʹ(n + 1) which is clearly a name of x again. The availability of a modulus of convergence as a function relies on a use of the axiom of countable choice. This could probably be eliminated by appropriate assumptions about the values of the metric being approximable on the elements of the dense sequence (i.e. by working with computable metric spaces). That the sequential notions of continuity on metric and represented spaces coincide follows immediately from the above Theorem [lem:lim-mlim]. As the proof of each direction requires to translate limits in both directions, either of the directions is as constructive or non-constructive as the worse direction of the previous theorem. [`scnt_mscnt`][resu: scntmscnt] Let (M, d) and (Mʹ, dʹ) be separable metric spaces and ${\mathbf M}$ and ${\mathbf M}'$ the derived represented spaces. Then f : M → Mʹ, is sequentially continuous as a function between metric spaces if and only if it is sequentially continuous as function $f\colon {\mathbf M}\to {\mathbf M}'$. Next let us state the equivalence of continuous realizability and ɛ-δ-continuity. One implication, namely proving continuous realizability from ɛ-δ-continuity needs stronger assumptions and for the Incone library we thus separated the proofs. [`cont_mcont` and `mcont_cont`][resu: contmcont] Let (M, d) and (Mʹ, dʹ) be two separable metric spaces. A function f: M → Mʹ is ɛ-δ-continuous if and only if $f\colon {\mathbf M}\to {\mathbf M}'$ is continuous. While the proof that continuous realizability implies ɛ-δ-continuity is straightforward, the proof of the other implication required some work and we sketch some of the details. Interestingly, the tools needed for this proof are in spirit fairly close to those that were used to prove the existence of associates in Section [sec: universal], more specifically we also use minimal moduli. Call a function $m\colon {\mathbb N}\to {\mathbb N}$ a metric modulus of continuity of f in x ∈ M if $$\forall y\in M\colon d(x, y) \leq 2^{-m(n)} \implies d'(f(x),f(y)) \leq 2^{-n}$$ and call such a modulus minimal if it is minimal in the obvious way. This notion generalizes the one used to prove the existence of associates in Section [sec: universal] if the naming spaces are equipped with the metric space structures that were introduced in Section [sec: bairemetric] and reasonable assumptions about the enumeration used for this are made. [`exists_minmod_met`] For any continuous function f between metric spaces and any argument x for f there exists a minimal modulus of f in x. This lemma implies its version for naming spaces that we have earlier proven in Section [sec: universal]. As we noted in the discussion following Lemma [lem: exists minmod] this weaker version cannot be proven constructively. It follows that the same remains true for the lemma above. The key idea behind the rest of the proof is to also use some notion of being self-modulating. This is slightly complicated by the fact the notion that made sense for moduli of continuity on naming spaces does not translate to the metric setting. Moreover, in general the minimal modulus of continuity fails to have some of the nice properties that we relied on for naming spaces. The function assigning to each x the minimal modulus function of f in x is usually not even continuous. This is, for instance, not possible if (M, d) is connected as the function takes values in a totally disconnected space and functions from connected to totally disconnected spaces can only be continuous if they are constant. One might suspect that the awkward typing is the cause for this, and that quantifying the continuity of f in x by use of a function of type ${\mathbb R}\to {\mathbb R}$ instead would help, but it does not. It is known that also in this case returning the minimal modulus of continuity need not be a continuous operation. A construction of a continuous such assignment, while possible in general, takes considerably more effort. Thus, for our proof that ɛ-δ-continuity implies continuous realizability we rely on a notion of being almost self-modulating instead: The value of the minimal modulus on slightly disturbed input from the metric space can be bounded in terms of a shift of the minimal modulus in the original value. Sierpiński space and closed choice on the naturals -------------------------------------------------- As a somewhat orthogonal class of examples of represented spaces that are far from being metrizable let us consider certain hyper spaces. A hyper space is a space whose underlying set consists of subsets of a given represented space and depending on the application very different ways to represent such sets of subsets can be appropriate. An important tool for introducing representations of hyper spaces is Sierpinski space ${\mathbb S}$. The base set of ${\mathbb S}$ is the two point set {⊤, ⊥} equipped with the total representation $\delta\_{\mathbb S}$ with names of type ${\mathbb N}\to {\mathbb{B}}$ specified by $$\delta\_{\mathbb S}(\varphi) = \top\quad \iff \quad \exists n \in {\mathbb N}\ \varphi(n) = {\mathrm{true}}.$$ The elements of Sierpinski space should be understood as symbols for termination and divergence. More specifically: From any kind of computational process for which the notion of an elementary computational step is meaningful one may obtain a sequence of Booleans by taking the n-th element to be the truth-value of the statement “the process finishes within the first n basic steps”. This way one obtains a name of ⊤ if the process terminates and ⊥ otherwise. The final topology of this representation is $\{\emptyset,\{\top\},{\mathbb S}\}$ and as a non-Hausdorff space this space is not metrizable. Without going into details let us mention that also for Sierpinski space there is a tight connection between continuous realizability and topological continuity. To connect Sierpinski space to hyperspaces consider for a subset $U \subseteq {\mathbf X}$ its characteristic function $\chi\_U\colon {\mathbf X}\to {\mathbb S}$ defined by $$\chi\_U(x) := \begin{cases} \top\ \text{ if } x \in U, \\ \bot \ \text{otherwise.} \end{cases}$$ For illustration let us assume that we can also safely equip ${\mathbf X}$ with the final topology of the representation without changing the notion of continuity. Recall that a function is topologically continuous if and only if preimages of open sets are open. From the topology of Sierpinski space specified above it is evident that in this setting a subset of ${\mathbf X}$ is open if and only if its characteristic function is continuous. Inspired by this we may generalize and follow sources such as in identifying the space ${\mathcal O}({\mathbf X})$ of open subsets of ${\mathbf X}$ with the space of continuous functions ${\mathbb S}^{\mathbf X}$ as defined in Section [sec: function spaces]. Similarly, the space ${\mathcal A}({\mathbf X})$ of closed subsets of ${\mathbf X}$ can be introduced as the space of complements of opens. If ${\mathbf X}$ is taken to be a concrete space it is often possible to give simpler descriptions of ${\mathcal O}({\mathbf X})$ and ${\mathcal A}({\mathbf X})$. For instance if ${\mathbf X}:= {\mathbb N}$ is the discrete represented space of natural numbers one may make use of a space of infinite sequences as introduced in Section [sec: sequences] and in particular of the special case $I = {\mathbb N}$ and ${\mathbf X}= {\mathbb S}$ of Lemma [resu: sigisofun] which guarantees that ${\mathcal O}({\mathbb N}) = {\mathbb S}^{\mathbb N}\simeq \prod\_{\mathbb N}{\mathbb S}= {\mathbb S}^\omega$. There exists a fully concrete description of ${\mathcal O}({\mathbb N})$ that is often used for reasoning about this space in computable analysis: The enumeration representation, where a name of an open set enumerates its elements. We call the corresponding space $\mathcal O\_{\mathbb N}$. The representation of the corresponding concrete space $\mathcal{A}\_{\mathbb N}$ of closed subsets encodes sets as functions $\varphi\colon {\mathbb N}\to {\mathbb N}$ via $$\delta\_{\mathcal{A}\_{\mathbb N}}(\varphi) := {\mathbb N}\setminus\{ n \in {\mathbb N}\mid \exists m\colon \varphi(m) = n+1\}.$$ That is: the information that a name φ specifies about a closed set is an enumeration of its complement. We provide a formal proof that the enumeration representations of the open and closed subsets of the natural numbers capture the abstract structure of these spaces through the exponential in the category of represented spaces. [`AN_iso_Anat`, `ON_iso_Onat` and `clsd_iso_open`][resu: ONisoOnat] ${\mathcal A}({\mathbb N}) \simeq {\mathcal O}({\mathbb N})$ and the concrete spaces of open and closed sets above are correct, i.e. ${\mathcal A}({\mathbb N})\simeq\mathcal A\_{\mathbb N}$ and ${\mathcal O}({\mathbb N}) \simeq \mathcal O\_{\mathbb N}$. The first of these isomorphies may look surprising but is evident on closer inspection: the isomorphism is taking the complement and it is realized by the identity function. The isomorphism of ${\mathcal O}({\mathbb N})$ and $\mathcal O\_{\mathbb N}$ is proven by first replacing ${\mathcal O}({\mathbb N})$ by ${\mathbb S}^\omega$ as described above. The realizers for the isomorphisms between ${\mathbb S}^\omega$ and $\mathcal O\_{\mathbb N}$ can be defined as functions directly by relying on the Cantor paring function or any standard pairing function. We chose to use the standard pairing function provided by the mathematical components library. As an application let us consider choice operators. To implement a choice operator one has to select an element of a closed set. As closed sets are represented as complements of open sets and thus by providing positive information about their complement, a realistic implementation of a choice operator is rarely possible. Such an implementation still solves a problem that can be described by fairly basic means and thus asserting the existence of a solution is popular for classifying unsolvability of other tasks according to their logical strength. More details about this can be found in survey articles about Weihrauch reducibility such as, more concrete examples of classifications where the problems discussed here make an appearance can be found for instance in. Let us give a precise formulation of what it means to select an element from a closed subset. For a fixed represented space ${\mathbf X}$ let closed choice on ${\mathbf X}$ be the task of finding a realizer of the multifunction $C\_{\mathbf X}$ defined by $$C\_{\mathbf X}\colon {\mathcal A}({\mathbf X}) {\rightrightarrows}{\mathbf X}, \quad a \in C\_{\mathbf X}(A) \iff a \in A.$$ Or in words: a is an acceptable return value of $C\_{\mathbf X}$ on input A if and only if a is an element of A. This in particular means that the domain of $C\_{\mathbf X}$ are the non-empty subsets of ${\mathbf X}$. This is relevant as it means that a potential realizer of $C\_{\mathbf X}$ can behave arbitrarily when handed a name of the empty set. The realizer need not produce any name of an element of ${\mathbf X}$ in this case and may even be undefined. Consider the special case where ${\mathbf X}={\mathbb N}$. While the argument space of the multivalued function $C\_{\mathbb N}$ is ${\mathcal A}({\mathbb N})$ we may use the same definition to obtain a multifunction $C\_{\mathbb N}'\colon \mathcal A\_{\mathbb N}{\rightrightarrows}{\mathbb N}$. A mathematician may even consider it pointless to give this function a new name as ${\mathcal A}({\mathbb N}) \simeq {\mathcal A}\_{\mathbb N}$ and isomorphic spaces are regularly identified. Indeed, for the question of whether $C\_{\mathbb N}$ has a continuous realizer the space ${\mathcal A}({\mathbb N})$ may be substituted with $\mathcal A\_{\mathbb N}$ for this exact reason. [`CN_CN’_hcr`][lem:cn-cn’-hcr] $C\_{\mathbb N}$ has a continuous realizer if and only if $C\_{\mathbb N}'$ does. On the concrete space ${A}\_{\mathbb N}$ a standard argument can be used to see that a continuous realizer cannot exist. [`CN’_not_cont`][lem:cn’-not-cont] $C\_{\mathbb N}'$ does not have a continuous realizer. [outline] The proof proceeds by contradiction. Assume that to the contrary $C\_{\mathbb N}'$ is continuous and that F is a continuous realizer. Pick any name φ of the one point set {0}. As F is a realizer, it has to return a name of 0 on input φ, i.e. F(φ)( ⋆ ) = 0. Since F is continuous there is a list ${\mathbf}q \subseteq {\mathbb N}$ such that F(φ)( ⋆ ) = F(ψ)( ⋆ ) for all $\psi\colon {\mathbb N}\to {\mathbb N}$ that coincide with φ on ${\mathbf}q$. Consider the name φʹ of the non-empty set $A := {\mathbb N}\setminus \left ( \{n \ \| \ \exists m \in {\mathbf}q, \varphi(m) = n+1 \} \cup \{0\} \right )$ defined by $$\varphi'(n) := \begin{cases} \varphi(n), & \text{if } n \in {\mathbf}q \\ 1, & \text{otherwise.} \end{cases}$$ On one hand, F(φʹ)( ⋆ ) ∈ A since F is a realizer. On the other hand F(φʹ)( ⋆ ) = F(φ)( ⋆ ) = 0 as φ and φʹ coincide on ${\mathbf}q$. Now since $0 \notin A$ we arrive at a contradiction and this completes the proof. Combining the theorem with the lemma we conclude that closed choice on the natural numbers does not have a continuous realizer: [`CN_not_cont`][resu: CNnotcont] Closed choice on the natural numbers does not have a continuous solution. Clearly the existence of a continuous solution also rules out the possibility that a computable solution exists. Conclusion ========== The Incone library provides general tools for enriching abstract mathematical structures of interest with computational structures and for comparing different such encodings. It can be used to investigate reasonability of encodings in a fully formal setting, to provide computational content for mathematical statements or to prove it impossible to do so. We feel that the examples from this paper showcase these capabilities well. Moreover, they involve many of the most prominent features of Incone: The abstract definition of the space of open subsets is based on the library’s function space construction and our proof of isomorphy relies on the fact that in this specific case the corresponding space of functions is isomorphic to a space of sequences. We believe the Incone library to be reasonably accessible for the computable analysis community and hope that its combination with methods from that community could help to make parts of it more accessible to the numerical analysis community. Let us make an effort to outline some relations to other approaches. For a given representation, i.e. singlevalued and co-total (surjective) multifunction $\delta\colon {\mathcal B}{\rightrightarrows}X$, one may define a partial equivalence relation on Baire space via φ ∼ ψ ⇔ ∃x ∈ X: x ∈ δ(φ) ∧ x ∈ δ(ψ). If δ is considered a partial function the right-hand side can be replaced by δ(φ) = δ(ψ) if the equality is interpreted in the appropriate way. The resulting structure $({\mathcal B}, \sim)$ is what is called a setoid in constructive mathematics. Conversely, for each such setoid one may take X to be the set of equivalence classes with respect to ∼ . This makes the quotient mapping a representation and one may check that the constructions are “inverse” in an appropriate sense. Other notions for representations can be translated accordingly. For instance, a function that preserves equivalence induces a function on the equivalence classes and realizes this function with respect to the quotient mappings as representations. If the function does not preserve the equivalence relation, it induces a multifunction on the equivalence classes that it still realizes. While mathematicians take quotients without thinking twice, in a type theoretic setting quotients are a problematic operation. Our proof assistant of choice, Coq, does not support quotient types and there are reasons why it refrains from doing so. As setoids allow to still reason about the quotient without really taking it, they are a very popular tool in the Coq community and constructive mathematics in general. The specification of an appropriate type X and a representation that acts as quotient mapping is additional information that has to be provided by the user. This extra information is exclusively used for specification purposes and we aim to give the greatest possible amount of freedom in how the user wants to present this information. For instance our choice to treat representations as relations and not as functions can be justified by this. This comes with the drawback that definability of a function on the description of the quotient need no longer be related to it being realistically implementable. For instance consider Sierpinski space: In this case the description of the quotient is a discrete two element set which means that all self-maps are definable. However, the one that switches ⊤ and ⊥ is not implementable on the level of names. As the information from the description of the quotient is computationally irrelevant, another way of introducing it is via an axiomatization similar to that of the real numbers from Coq’s standard library that was used in this paper. Ideally such an axiomatization should avoid stating unrealistic assumptions as computationally relevant facts but the example of the reals shows that even if it does, this can be worked around. The idea behind the constructions provided by Incone is to support a user in conveniently obtaining descriptions of quotient types by piecing them together from basic building blocks. As basic spaces Incone defines discrete spaces, separable metric spaces with a distinguished dense sequence and Sierpinski space. One may combine these basic spaces via building products, sums, infinite sequences and spaces of functions. The space ${\mathcal O}({\mathbb N})$ is an example for such a combination as it is constructed as a space of functions between elementary defined spaces. This particular space, sometimes also called P(ω), is a common sight in work on constructive or computational analysis. This is even though as a space it is very different from the objects that are actually of interest in these fields. However, many spaces of practical importance can be identified with a subspace of this space: Any quasi-Polish space in the sense of de Brecht and therefore in particular any Polish space embeds isomorphically into ${\mathcal O}({\mathbb N})$ as a subspace of low descriptive complexity. This covers all of the spaces considered in Section [sec: metric spaces]. Let us give a concrete example by outlining of how to isomorphically embed the real numbers into ${\mathcal O}({\mathbb N}) \simeq {\mathbb S}^{\mathbb N}$. First enumerate the open rational intervals and then send a real number x to the function that on input of a natural number n returns ⊤ if x is contained in the n-th interval and ⊥ if this is not the case. As checking inclusion of a real number into an open interval is semi-decidable relative to availability of approximations to the real number, we obtain an element of ${\mathcal O}({\mathbb N})$ that continuously depends on the real number. This can be checked to lead to a continuous, and even computable, embedding of the reals into ${\mathcal O}({\mathbb N})$. Of course, this is nothing but the identification of a real number with a Dedekind cut. For future directions in the development of Incone we plan to work on additional applications. One particularly fitting extension of the contents of this paper would be a proof that $C([0,1]) \simeq {\mathbb R}^{[0,1]}$. This statement is called the Computable Weierstraß Theorem : Here, C([0, 1]) are the continuous real-valued functions on the unit interval represented as separable metric space with supremum norm and the rational polynomials as dense sequence. The function-space ${\mathbb R}^{[0,1]}$ has the same underlying set but uses a different representation that the theorem states to be computably equivalent. Other possibilities include: * Results about ODE solving. This may be done by providing an interface with C-CoRn, parts of it could also be done separately by relying on libraries such as Coq-Interval. * Duality theory for spaces of summable sequences (ℓp-spaces) which provide a pool of examples where sub-spaces of exponentials can be treated complexity theoretically. Additionally it constitutes a step towards capturing popular methods for solving partial differential equations. * A characterization of continuity via preimages of open sets, general considerations about admissibility, discreteness, compactness and many other similar results. There are also still some gaps in the Incone library that we plan to fill. Currently there is no complete proof that the category of represented spaces is Cartesian closed as the corresponding universal property is missing. The library proves a restricted case by providing a duality operator, but a full proof would be desirable. Further steps would be to use a formalization of a model of computation to make it possible to talk about computability without moving to the meta level and using the Prop/Type distinction. This would have the additional advantage of opening the possibility to do complexity theory. Even without reliance on a model of computation it should be possible to add the capability to do qualitative complexity theory in terms of tracking the decrease in accuracy of approximations. As the Incone library keeps close to recent work about complexity theory for computable analysis, this should be possible with relatively low effort. A full treatment of step-counting complexity might become available in the not too distant future due to recent progress on the formalization of models of computation and methods from implicit complexity theory. Another way to gain insight into such efficiency considerations would be to capture the trace of the basic feasible functionals on the operators on Baire space. An overview of the Incone library ================================= The goal of the Incone library is to provide a Coq formalization of some of the most important concepts and basic results from computable analysis. The library uses notations that allow to state theorems similar to how they would be formulated in natural language and therefore should be readable for a mathematician familiar with computable analysis without having a deep knowledge of Coq. For example, Lemma [lem: tight comp] corresponds to the following statement in the library. ``` Lemma tight_comp R S T (f f': T ->> R) (g g': S ->> T): f \tightens f' -> g \tightens g' -> (f \o g) \tightens (f' \o g'). ``` Here, `tight_comp` is the name of the lemma and the symbols following this name until the colon at the end of the line are parameters of its statement. These parameters can alternatively be thought of as universally quantified. The phrase `(f f': T ->> R)` thus translates to the mathematical language used in the paper as “for all $f, f'\colon T {\rightrightarrows}R$”, or “let f and fʹ be multifunctions from T to R”. That R and T are sets can be omitted as it can be inferred from their use. The notation for multifunctions may seem unfortunate at first but the closer match of a notation `=>` seemed unwise to use as it is commonly overloaded with a number of different meanings. The colon is followed by the body of the lemma, where `->` is the logical implication and `_ \o _` and `_ \tightens _` are the notations for composition and the tightening order on multivalued functions. Many of the infix notations in the library start in a `\` to avoid blocking too many keywords. Most statements in this paper and many additional results can be found in the files of the Incone library. In the paper, we labeled each theorem and lemma with the names of the relevant statements in the library. This should make it easy to locate and inspect them in Incone. The structure of this paper and the formal development of the concepts are quite similar. For most parts making the connection is about as complicated as indicated by the example above. The most important exception where one needs additional information for the translation is that the library uses the expression “continuity space” for what is usually referred to as “represented space”. We decided to use this notion since represented spaces are tied to computability theory, which the Incone library avoids except on the meta-level. This has another important consequence for the translation between the results of the paper and the formal development: The formal versions will only prove continuity where the paper claims computability. The user can verify the computability claim by inspecting the definitions of the realizers. Whenever a result proves a function with name `fun_name` continuous, there will be a function whose name contains the string `fun_name_rlzr`. It can be checked that this definition does not rely on any assumptions, so that it certifies computability. How to check this manually is described in more detail below. The smaller differences between the formal and the informal development of concepts include the definition of the space of functions ${\mathbf Y}^{\mathbf X}$: While in this paper we chose the underlying set to be the continuous functions, in the formal development we use the co-domain of the function-space representation for the definition. In Incone the space of functions is denoted by `cs_fun` with the notation `_ c-> _` and that its underlying set contains exactly the continuous functions is proven separately (`ass_cont`). The reason for this choice is to minimize the strength of the axioms that are automatically assumed whenever spaces of functions are used. The proof that every continuous function has an associate requires a considerably stronger background theory than most of the other basic results one would be interested in. Another design choice for the sake of minimizing the use of axioms is that Incone does not use the Coq-internal notion of equality for equality of multivalued functions. Instead, it defines this equality as an equivalence relation `_ =~= _`. This avoids the need to assume the axiom of propositional extensionality that, while being theoretically justified and fairly commonly used, is known for having caused inconsistencies in the past. As such equivalence relations are a common sight in constructive mathematics, Coq provides the necessary infrastructure to make most operations equally effortless for equalities and equivalence relations. Thus this detail can mostly be ignored by the user. The Incone library has some sublibraries that are of separate interest and have fewer dependencies. These parts can be obtained individually and we give an overview over their purpose in Section [sec: sublibs]. Before we do so, let us sketch what steps need to be taken to get an installation of Incone up and running and verify the correctness of the results described in this paper. We will assume that the user has successfully installed Coq on his machine, instructions of how to do so are available online. How to get started ------------------ The paper describes release version 1.0 of Incone. As Incone is an ongoing project, newer versions will become available and for a description of the newest version, the GitHub repository can be consulted. For readers only interested in verifying the results of the paper, we give instructions how to install and use version 1.0 of the library here. This version of Incone has been tested with version 8.9.0 of the Coq proof assistant. Additionally, the Coquelicot library (version 3.0) and the ssreflect package from math-comp (version 1.7.0) needs to be installed. To install Incone, the libraries mf, Rlzrs, Metric and Incone have to be downloaded and installed in this order. They can be found in their respective GitHub repositories. Alternatively, the paper’s project page <https://holgerthies.github.io/continuity> offers a package containing all libraries. All of the claims the paper makes about the incone library can be checked by installing the library, opening a new Coq file with the following includes in the preamble. ``` From mf Require Import choice_mf. From rlzrs Require Import all_rlzrs. From metric Require Import all_metric Qmetric. From incone Require Import all_cs classical_func classical_cont classical_mach Duop Q_reals baire_metric. ``` Each result in the paper states the name of the corresponding lemma in Incone in brackets after the number of the result. The formal lemmas and theorems can be inspected via `Check lem_name`, definitions can be detailed using `Print def_name` and notations by `Locate "not_name"`. In the case where multiple results share a name, `Locate "lem_name"` lists all lemmas with that name and unique identifiers and `Print Assumptions lem_name` shows the axioms that the result assumes. To list all results involving a concept Coq’s search function can be used via `Search _ (concept)` and `Search "phrase"` might be useful where `phrase` is an expected substring of a lemma’s name. Coq’s search function can also be used with multiple arguments and the underscore above is due to the first argument being treated differently. Omitting it will lead to only functions returning `concept` being listed as a result and a warning that this may not be the intended behavior. For example, one of the theorems from the paper has the name `cont_comp`. Typing ``` Check cont_comp. ``` prints the following statement: ``` cont_comp : forall (X Y Z : cs) (f : Y -> Z) (g : X -> Y), f \is_continuous -> g \is_continuous -> (f \o_f g) \is_continuous ``` Here, `cs` is a short notation for `continuity_space`, which is the Incone equivalent of a represented space. The notation `_ \o_f _` is for composition of functions, the shorter `_ \o _` is preserved for composition of multivalued functions. The rest of the notations should be self-explanatory but it may be worth noting that the use of a ∖ for non-prefix notations is to avoid blocking too many keywords. The user may first use `Locate "_ \is_continuous"` to find that this is a notation for the definition `continuous`. Then the details of the definition can be printed using the command `Print continuous`. The above may or may not be the desired theorem from the paper. Typing ``` Locate cont_comp. ``` shows that there are several lemmas with this name and the location of the files that contain them. For users that are not only interested in verifying the results from the paper, the examples folder is a good starting point. Most of the the examples are documented via comments in the files and contain some rudimentary proofs that can be used to get familiar with using the library. A reader familiar with Coq will notice right away that these proofs heavily use the ssreflect proof language. This is true for the whole of the development and a comprehensible overview over this language and its advantages can be found in. The structure of Incone and its sub-libraries --------------------------------------------- Some concepts described in this paper are expected to have applications that are unrelated to computable analysis and the most important ones have been exported to small libraries that can be installed separately and have less dependencies. For instance, due to its potential applications outside of Incone the development of a convenient environment for manipulation of multifunctions was exported and can be obtained separately as the mf-library. Already the Incone library uses multifunctions for several different purposes: Through the Rlzrs library for the formulation of realizability, but also for dealing with partiality issues in Coq and in their traditional role in computable analysis as formalization of computational tasks. Let us list all the packages that we decided to export from Incone with a short description of their contents. * mf: The mf library contains statements from the theory of multivalued functions. The formalization closely follows the description in the first part of Section [sec: operators]. Several useful notations are introduced, for example `_ ->> _` for multivalued functions and `_ \tightens _` for the tightening relation. An exhaustive list of the important concepts can be found in the preamble of the file `mf.v` in the library. * Rlzrs: A library for basic realizability using the question and answer structure described in the beginning of Section [sec: represented spaces]. It defines basic notations such as `_ \is_name_of _` or `_ \realizes _` and heavily relies on the mf library. * Metric: A classical formalization of metric spaces. This library contains results that are independent from the theory of represented spaces. It also defines the metric notion of a limit, an efficient limit and proves some statements about continuity. The layout of Metric roughly follows that of the Coquelicot library. However, the definitions are kept close to the classical mathematical treatment and are thus most similar to the metric spaces that can be found in Coq’s standard library. ![Internal and external dependencies of the incone library.](incone "fig:")[fig: dependencies] The libraries above can, and have to be, installed separately but should be considered part of the Incone system. However, Incone also has some external dependencies. As mentioned before it uses the ssreflect proof language. This language originates from the mathematical components development and Incone also uses other parts of that library. Another external dependency that Incone shares with Metric is Coquelicot, a popular formalization of some facts from analysis that is conservative over the axiomatization of the real numbers in the standard library. See Figure [fig: dependencies] for an overview of Incone, its sublibraries and dependencies. The mf and Rlzrs libraries are fairly manageable and their few files can be navigated by hand and are well documented. The Metric library is less accessible and heavily interleaved with both the Coquelicot library and the development of metric spaces in Coq’s standard library. Thus, we take some space to sketch some of its more important features and give additional justification of its existence. The library for metric spaces ----------------------------- Metric spaces are available in Coq’s standard library and there are several other developments that include a treatment of metric spaces or related concepts. The original intention of the Metric library was to provide an interface between the the metric spaces from the standard library and a similar concept provided by Coquelicot. Coquelicot is a popular library for doing analysis in Coq and is conservative over the axiomatization of the real numbers in the standard library. The concept from Coquelicot that we are referring to is named a “uniform space” in Coquelicot. However, these spaces are more restrictive than what a mathematician would expect to be called a uniform space and instead closely resemble pseudo-metric spaces. For Coquelicot the choice of pseudo-metric spaces over metric spaces is due to neither Coq, nor the axioms of the real numbers, implying functional extensionality. This makes it challenging to define a metric on any kind of space of functions. A pseudo-metric can often be defined in a straightforward manner. The definitions of limits and continuity used in Coquelicot rely on filters instead of sequences. As the “uniform spaces” are Coquelicot’s most general structure and are first-countable, the definitions via filters are equivalent to those using sequences. However, it is not clear whether the equivalence of the derived concepts can be proven in the setting that Coquelicot works in. The Metric library provides interfaces with both the standard library of Coq (`MS2M_S`, `M_S2MS`, `Uncv_lim`, `cont_limin`, etc.) and the Coquelicot library (`US2MS`, `MS2US`, `cntp_cntp`, etc.) so that it is possible to reuse results proven there (for instance the lemmas `limD`, `limM`, `R_cmplt` that assert the limit to commute with additional multiplication and the real numbers to be complete are proven this way). The definitions of continuity that the metric library uses are as stated in this paper in the chapter about metric spaces. While these definitions quantify over real numbers ɛ and δ, restricting the quantifiers to only reach over a discrete subset of the real numbers often leads to equivalent definitions. For an easy back and forth between the different convergence statements Incone provides a line of lemmas that contain the phrase “tpmn” (for “two to the power minus n”) in their name (`tpmnP`, `lim_tpmn`, `dns_tpmn`, etc.). For instance `tpmnP` proves that 2− n ≤ 2− m on real numbers reflects m ≤ n on natural numbers and `lim_tpmn` says that in the definition of the limit one may replace ɛ by 2− n and thereby quantification over ${\mathbb R}$ by quantification over ${\mathbb N}$. A similar set of lemmas is provided for replacement of ɛ and δ with rational numbers. For rational numbers Incone also provides a constructive instantiation of the restriction of the up function (`upQ`, `limQ`, `archimedQ`, etc.). The function `upQ`, for instance, is useful for recovering computational content from proofs in the standard library, as it may in some cases be used as substitute for the non-computable function `up` defined on all reals. Finally, the Metric library proves some results that are not proven in Coquelicot, most of these rely on the axioms that our development assumes over the assumptions made by Coquelicot. It also defines an interface for handling sub-spaces as continuity of partial functions is important for our purposes. In the case of metric spaces both continuity and its sequential variant can be recovered from analogous point-wise notions while for represented spaces this is only the case for sequential continuity. The availability of point-wise notions is not always an advantage and can introduce subtle problems in the treatment of sub-spaces. Even in the most well-behaved cases there is a difference between a function being continuous in each point of the subset and the restriction of the function being continuous. For instance, the characteristic function of the closed unit interval has a continuous restriction but is not continuous in either end-point. This can easily lead to confusion and as a result some of the statements of important theorems about continuous functions on the real numbers from the standard library differ slightly from what a mathematician would expect them to say. Examples for this include the mean value and the intermediate value theorem. --- 1. <https://github.com/coq/coq/wiki/The-Logic-of-Coq#axioms>[↩](#fnref1)	arxiv_0000377
mesoscopic* structure, i.e. they appear as a combination of components fairly independent of each other. These components are called communities, modules or clusters and the problem of how to reveal them plays a significant role in understanding the organization and function of complex networks. Over the last decade, a great number of algorithms for community detection (CD) have been proposed to address the problem in a variety of settings, such as undirected/directed, unweighted/weighted networks and non-overlapping/overlapping communities (for a comprehensive survey, see ). These approaches, however, are adopted in a non-private manner, i.e. a data collector (such as Facebook) knows all the contributing users and their relationships before running CD algorithms. The output of such a CD, in the simplest form, is a clustering of nodes. Even in this case, i.e. only a node clustering (not the whole graph) is revealed, contributing users privacy may still be put at risk. In this paper, we address the problem of CD from the perspective of differential privacy. This privacy model offers a formal definition of privacy with a lot of interesting properties: no computational/informational assumptions about attackers, data type-agnosticity, composability and so on. By differential privacy, we want to ensure the existence of connections between users to be hidden in the output clustering while keeping the low distortion of clusters compared to the ones generated by the corresponding non-private algorithms. As far as we know, the problem is quite new and only mentioned in the recent work where Mülle et al. use a sampling technique to perturb the input graph so that it satisfies differential privacy before running conventional CD algorithms. This technique (we call it EdgeFlip afterwards) is classified as input perturbation in differential privacy literature (the other two categories are algorithm perturbation and output perturbation). Similarly, TmF approach can apply to the true graph to get noisy output graphs as in the work of Mülle et al. Earlier, 1k-Series, Density Explore Reconstruct (DER) and HRG-MCMC are the best known methods for graph structure release under differential privacy. These methods can be followed by any exact CD algorithm to get a noisy clustering satisfying differential privacy. We choose Louvain method as such a CD algorithm. However, as we will see in the experiments, the output clusterings by the aforementioned methods have very low modularity. This fact necessitates new methods for CD problem under differential privacy. Our main contributions are the new schemes LouvainDP (input perturbation) and ModDivisive (algorithm perturbation) which perform much better than the state-of-the-art. LouvainDP is a high-pass filtering method that randomly groups nodes into supernodes of equal size to build a weighted supergraph. LouvainDP is guaranteed to run in linear time. ModDivisive is a top-down approach which privately divides the node set into the k-ary tree guided by the modularity score at each level. The main technique used in ModDivisive is the Markov Chain Monte Carlo (MCMC) to realize the exponential mechanism. We show that ModDivisive’s runtime is linear in the number of nodes, the height of the binary tree and the burn-in factor of MCMC. The linear complexity enables us to examine million-scale graphs in a few minutes. The experiments show the high modularity and low distortion of the output clusters by LouvainDP and ModDivisive. Our contributions are summarized as follows: * We analyze the major challenges of community detection under differential privacy. We explain why techniques borrowed from k-Means fail and how the difficulty of ε-DP recommender systems justifies a relaxation of ε. * We design an input perturbation scheme LouvainDP that runs in linear time using the high-pass filtering technique from and Louvain method. * We propose an algorithm perturbation scheme ModDivisive as a divisive approach by using the modularity-based score function in the exponential mechanism. We prove that modularity has small global sensitivity and ModDivisive also runs in linear time. * We conduct a thorough evaluation on real graphs of different sizes and show the outperformance of LouvainDP and ModDivisive over the state-of-the-art. The paper is organized as follows. We review the related work for community detection algorithms and graph release via differential privacy in Section [sec:related]. Section [sec:pre] briefly introduces the key concepts of differential privacy, the popular Louvain method and the major challenges of ε-DP community detection. Section [sec:input] focuses on the category of input perturbation in which we propose LouvainDP and review several recent input perturbation schemes. We describe ModDivisive in Section [sec:algo]. We compare all the presented schemes on real graphs in Section [sec:eval]. Finally, we present our conclusions and suggest future work in Section [sec:conclusion]. Table [tab:notation] summarizes the key notations used in this paper. [htb] [tab:notation] \| Symbol \| Definition \| \| --- \| --- \| \| G = (V, EG) \| true graph with n = ∣V∣ and m = ∣EG∣ \| \| Gʹ = (V, EGʹ) \| neighboring graph of G \| \| G̃ = (V, EG̃) \| sample noisy output graph \| \| G1 = (V1, E1) \| supergraph generated by LouvainDP \| \| k \| fan-out of the tree in ModDivisive \| \| K \| burn-in factor in MCMC-based algorithms \| \| λ \| common ratio to distribute the privacy budget \| \| C \| a clustering of nodes in G \| \| Q(G, C) \| modularity of the clustering C on graph G \| Related Work ============ Community Detection in Graphs ----------------------------- There is a vast literature on community detection in graphs. For a recent comprehensive survey, we refer to. In this section, we discuss several classes of techniques. Newman and Girvan propose modularity as a quality of network clustering. It is based on the idea that a random graph is not expected to have a modular structure, so the possible existence of clusters is revealed by the comparison between the actual density of edges in a subgraph and the density one would expect to have in the subgraph if the nodes of the graph were connected randomly (the null model). The modularity Q is defined as $$\label{eqn:modularity} Q = \sum\_{c=1}^{n\_c}\left[\frac{l\_c}{m} - \left(\frac{d\_c}{2m} \right) ^2 \right]$$ where nc is the number of clusters, lc is the total number of edges joining nodes in community c and dc is the sum of the degrees of the nodes of c. Many methods for optimizing the modularity have been proposed over the last ten years, such as agglomerative greedy, simulated annealing, random walks, statistical mechanics, label propagation or InfoMap, just to name a few. The recent multilevel approach, also called Louvain method, by Blondel et al. is a top performance scheme. It scales very well to graphs with hundreds of millions of nodes. This is the chosen method for the input perturbation schemes considered in this paper (see Sections [subsec:louvain] and [sec:input] for more detail). By maximizing the modularity, Louvain method is based on edge counting metrics, so it fits well with the concept of edge differential privacy (Section [sec:diffpriv]). One of the most recent methods SCD is not chosen because it is about maximizing Weighted Community Clustering (WCC) instead of the modularity. WCC is based on triangle counting which has high global sensitivity (up to O(n)). Moreover, SCD pre-processes the graph by removing all edges that do not close any triangle. This means all 1-degree nodes are excluded and form singleton clusters. The number of output clusters is empirically up to O(n). Graph Release via Differential Privacy -------------------------------------- In principle, after releasing a graph satisfying ε-DP, we can do any mining operations on it, including community detection. The research community, therefore, expresses a strong interest in the problem of graph release via differential privacy. Differentially private algorithms relate the amount of noise to the computation sensitivity. Lower sensitivity implies smaller added noise. Because the edges in simple undirected graphs are usually assumed to be independent, the standard Laplace mechanism is applicable (e.g. adding Laplace noise to each cell of the adjacency matrix). However, this approach severely deteriorates the graph structure. The state-of-the-art try to reduce the sensitivity of the graph in different ways. Density Explore Reconstruct (DER) employs a data-dependent quadtree to summarize the adjacency matrix into a counting tree and then reconstructs noisy sample graphs. DER is an instance of input perturbation. Xiao et al. propose to use Hierarchical Random Graph (HRG) to encode graph structural information in terms of edge probabilities. Their scheme HRG-MCMC is argued to be able to sample good HRG models which reflect the community structure in the original graph. HRG-MCMC is classified as an algorithm perturbation scheme. A common disadvantage of the state-of-the-art DER and HRG-MCMC is the scalability issue. Both of them incur quadratic complexity O(n2), limiting themselves to medium-sized graphs. Recently, Nguyen et al. proposed TmF that utilizes a filtering technique to keep the runtime linear in the number of edges. TmF also proves the upper bound O(lnn) for the privacy budget ε. At the same time, Mülle et al. devised the scheme EdgeFlip using an edge flipping technique. However, EdgeFlip costs O(n2) and is runnable only on graphs of tens of thousands of nodes. A recent paper by Campan et al. studies whether anonymized social networks preserve existing communities from the original social networks. The considered anonymization methods are SaNGreeA and k-degree method. Both of them use k-anonymity rather than differential privacy which is the focus of this paper. Preliminaries ============= In this section, we review key concepts and mechanisms of differential privacy. Differential Privacy -------------------- Essentially, ε-differential privacy (ε-DP) is proposed to quantify the notion of indistinguishability of neighboring databases. In the context of graph release, two graphs G1 = (V1, E1) and G2 = (V2, E2) are neighbors if V1 = V2, E1 ⊂ E2 and ∣E2∣ = ∣E1∣ + 1. Note that this notion of neighborhood is called edge differential privacy in contrast with the notion of node differential privacy which allows the addition of one node and its adjacent edges. Node differential privacy has much higher sensitivity and is much harder to analyze. Our work follows the common use of edge differential privacy. The formal definition of ε-DP for graph data is as follows. [def:e-dp] A mechanism A is ε-differentially private if for any two neighboring graphs G1 and G2, and for any output O ∈ Range(A), $$Pr[\mathcal{A}(G\_1) \in O] \leq e^{\epsilon} Pr[\mathcal{A}(G\_2) \in O] \nonumber$$ Laplace mechanism and Exponential mechanism are two standard techniques in differential privacy. The latter is a generalization of the former. Laplace mechanism is based on the concept of global sensitivity of a function f which is defined as Δf = maxG1, G2∣∣f(G1) − f(G2)∣∣1 where the maximum is taken over all pairs of neighboring G1, G2. Given a function f and a privacy budget ε, the noise is drawn from a Laplace distribution $Lap(\lambda): p(x\|\lambda) = \frac{1}{2\lambda} e^{-\|x\|/\lambda}$ where λ = Δf/ε. (Laplace mechanism ) [def:lap-mech] For any function f : G → Rd, the mechanism A $$\mathcal{A}(G) = f(G) + \langle Lap\_1(\frac{\Delta f}{\epsilon}),...,Lap\_d(\frac{\Delta f}{\epsilon})\rangle$$ satisfies ε-differential privacy, where $Lap\_i(\frac{\Delta f}{\epsilon})$ are i.i.d Laplace variables with scale parameter $\frac{\Delta f}{\epsilon}$. □ Geometric mechanism is a discrete variant of Laplace mechanism with integral output range Z and random noise Δ generated from a two-sided geometric distribution $Geom(\alpha): Pr[\Delta = \delta\| \alpha] = \frac{1-\alpha}{1+\alpha} \alpha^{\|\delta\|}$. To satisfy ε-DP, we set α = exp( − ε). We use geometric mechanism in our LouvainDP scheme. For non-numeric data, the exponential mechanism is a better choice. Its main idea is based on sampling an output O from the output space O using a score function u. This function assigns exponentially higher probabilities to outputs of higher scores. Let the global sensitivity of u be Δu = maxO, G1, G2∣u(G1, O) − u(G2, O)∣. (Exponential mechanism ) [def:exp-mech] Given a score function u : (G × O) → R for a graph G, the mechanism A that samples an output O with probability proportional to $\exp(\frac{\epsilon.u(G,O)}{2\Delta u})$ satisfies ε-differential privacy. □ Composability is a nice property of differential privacy which is not satisfied by other privacy models such as k-anonymity. (Sequential and parallel compositions ) [theorem:composition] Let each Ai provide εi-differential privacy. A sequence of Ai(D) over the dataset D provides Σi = 1nεi-differential privacy. Let each Ai provide εi-differential privacy. Let Di be arbitrary disjoint subsets of the dataset D. The sequence of Ai(Di) provides maxi = 1nεi-differential privacy. □ Louvain Method -------------- Since its introduction in 2008, Louvain method becomes one of the most cited methods for the community detection task. It optimizes the modularity by a bottom-up folding process. The algorithm is divided in passes each of which is composed of two phases that are repeated iteratively. Initially, each node is assigned to a different community. So, there will be as many communities as there are nodes in the first phase. Then, for each node i, the method considers the gain of modularity if we move i from its community to the community of a neighbor j (a local change). The node i is then placed in the community for which this gain is maximum and positive (if any), otherwise it stays in its original community. This process is applied repeatedly and sequentially for all nodes until no further improvement can be achieved and the first pass is then complete. [fig:louvain] We demonstrate Louvain method in Fig.[fig:louvain] by a graph of 13 nodes and 20 edges. If each node forms its own singleton community, the modularity Q will be -0.0825. In the first pass of Louvain method, each node moves to the best community selected from its neighbors’ communities. We get the partition [{0, 1, 2}, {3, 4}, {5, 6, 11, 12}, {7, 8, 9, 10}] with modularity 0.46375. The second phase of first pass builds a weighted graph corresponding to the partition by aggregating communities. The second pass repeats the folding process on this weighted graphs to reach the final partition [{0, 1, 2}, {3, 4, 5, 6, 11, 12}, {7, 8, 9, 10}] with modularity 0.47. This simple agglomerative algorithm has several advantages as stated in. First, its steps are intuitive and easy to implement, and the outcome is unsupervised. Second, the algorithm is extremely fast, i.e. computer simulations on large modular networks suggest that its complexity is linear on typical and sparse data. This is due to the fact that the possible gains in modularity are easy to compute and the number of communities decreases drastically after just a few passes so that most of the running time is concentrated on the first iterations. Third, the multi-level nature of the method produces a hierarchical structure of communities which allows multi-resolution analysis, i.e the user can zoom in the graph to observe its structure with the desired resolution. In addition, Louvain method is runnable on weighted graphs. This fact supports naturally our scheme LouvainDP as described in Section [subsec:louvaindp]. Challenges of Community Detection under Differential Privacy ------------------------------------------------------------ In this section, we explain why community detection under differential privacy is challenging. We show how techniques borrowed from related problems fail. We also advocate the choice of ε as a function of graph size n. The problem of differentially private community detection is closely related to ε-DP k-Means clustering and recommender systems. The ε-DP k-Means is thoroughly discussed in. However, techniques from ε-DP k-Means are not suitable to ε-DP community detection. First, items in k-Means are in low-dimensional spaces and the number of clusters k is usually small. This contrast to the case of community detection where nodes lie in a n-dimensional space and the number of communities varies from tens to tens of thousands, not to say the communities may overlap or be nested (multi-scale). Second, items in ε-DP k-Means are normalized to [ − 1, 1]d while the same preprocessing seems invalid in ε-DP community detection. Moreover, the output of k-Means usually consists of equal-sized balls while this is not true for communities in graphs. Considering the graph as a high-dimensional dataset, we tried the private projection technique in which is followed by spectral clustering, but the modularity scores of the output are not better than random clustering. Recent papers on ε-DP recommender systems show that privately learning the clustering of items from user ratings is hard unless we relax the value of ε up to logn. Banerjee et al. model differentially private mechanisms as noisy channels and bound the mutual information between the generative sources and the privatized sketches. They show that in the information-rich regime (each user rates O(n) items), their Pairwise-Preference succeeds if the number of users is Ω(nlogn/ε). Compared to ε-DP community detection where the number of users is n, we should have ε = Ω(logn). Similarly, in D2P scheme, Guerraoui et al. draw a formula for ε as $$\epsilon\_{D2P}^{(p,0,\lambda)} = \ln(1+\frac{(1-p).\mathcal{N}\_E}{p.\|\mathcal{G}\_\lambda\|})$$ where λ is the distance used to conceal the user profiles (λ = 0 reduces to the classic notion of differential privacy). NE is the number of items which is exactly n in community detection. ∣Gλ∣ is the minimum size of user profiles at distance λ over all users (∣Gλ∣ = o(NE) except at unreasonably large λ). Clearly, at p = 0.5 (as used in ), we have εD2P(0.5, 0, λ) ≈ lnn. The sampling technique in D2P is very similar to EdgeFlip which is shown ineffective in ε-DP community detection for ε ∈ (0, 0.5lnn) (Section [sec:eval]). Note that ε-DP community detection is unique in the sense that the set of items and the set of users are the same. Graphs for community detection are more general than bipartite graphs in recommender systems. In addition, modularity Q (c.f. Formula [eqn:modularity]) is non-monotone, i.e. for two disjoint sets of nodes A and B, Q(A ∪ B) may be larger, smaller than or equal to Q(A) + Q(B). To further emphasize the difficulty of ε-DP community detection, we found that IDC scheme using Sparse Vector Technique is hardly feasible. As shown in Algorithm 1 of, to publish a noisy graph that can approximately answer all cut queries with bounded error m0.25n/ε0.5, IDC must have B(α) “yes” queries among all k queries. B(α) may be as low as $\sqrt{m}$ but k = 22n. In the average case, IDC incurs exponential time to complete. To conclude, ε-DP community detection is challenging and requires new techniques. In this paper, we evaluate the schemes for ε up to 0.5lnn. At ε = 0.5lnn, the multiplicative ratio (c.f. Definition [def:e-dp]) is $e^\epsilon = e^{0.5\ln n} = \sqrt{n}$, a reasonable threshold for privacy protection compared to ε = lnn (i.e. eε = n) in ε-DP recommender systems discussed above. The typical epsilon in the literature is 1.0 or less. However, this value is only applicable to graph metrics of low sensitivity O(1) such as the number of edges, the degree sequence. The global sensitivity of other metrics like the diameter, the number of triangles, 2K-series etc. is O(n), calling for local sensitivity analysis (e.g. ). For counting queries, Laplace/Geometric mechanisms are straightforward on real/integral (metric) spaces. However, direct noise adding mechanisms on the space P of all ways to partition the nodeset V are non-trivial because ∣P∣ ≈ nn and P is non-metric. That is another justification for the relaxation of ε up to 0.5lnn. Input Perturbation ================== In this section, we propose the linear scheme LouvainDP that uses a filtering technique to build a noisy weighted supergraph and calls the exact Louvain method subsequently. Then we discuss several recent ε-DP schemes that can be classified as input perturbation. Fig.[fig:input-perturb] sketches the basic steps of the input perturbation paradigm. [b]0.42![image](input-perturb) [fig:input-perturb] [b]0.42![image](algo-perturb) [fig:algo-perturb] LouvainDP: Louvain Method on Noisy Supergraphs ---------------------------------------------- The basic idea of LouvainDP is to create a noisy weighted supergraph G1 from G by grouping nodes into supernodes of equal size k. We then apply the filtering technique of Cormode et al. to ensure only O(m) noisy weighted edges in G1. Finally, we run the exact community detection on G1. In, Cormode et al. propose several summarization techniques for sparse data under differential privacy. Let M be a contingency table having the domain size m0 and m1 non-zero entries (m1 ≪ m0 for sparse data), the conventional publication of a noisy table Mʹ from M that satisfies ε-DP requires the addition of Laplace/geometric noise to m0 entries. The entries in Mʹ could be filtered (e.g. removing negative ones) and/or sampled to get a noisy summary Mʺ. This direct approach would be infeasible for huge domain sizes m0. Techniques in avoid materializing the vast noisy data by computing the summary Mʺ directly from M using filtering and sampling techniques. In our LouvainDP, the supergraph G1 is an instance of sparse data with the domain size $m\_0 = \frac{n\_1(n\_1+1)}{2}$ where n1 is the number of supernodes and m1 non-zero entries corresponding to non-zero superedges. We use the one-sided filtering to efficiently compute G1 with O(m) edges in linear time. ### Algorithm LouvainDP can run with either geometric or Laplace noise. We describe the version with geometric noise in Algorithm [algo-louvaindp]. [algo-louvaindp] [1] undirected graph G, group size k, privacy budget ε noisy partition C̃ $G\_1 \leftarrow \varnothing$, $n\_1 = \lfloor \frac{\|V\|}{k}\rfloor -1$, V1 ← {0, 1, .., n1} ε2 = 0.1, ε1 = ε − ε2, α = exp( − ε1) get a random permutation Vp of V compute the mapping M : Vp → V1 compute superedges of G1: E1 = {e1(i, j)} where i, j ∈ V1 m1 = ∣E1∣ + Lap(1/ε2), $m\_0 = \frac{n\_1(n\_1+1)}{2}$ $\theta = \lceil \log\_{\alpha} \frac{(1+\alpha)m\_1}{m\_0-m\_1} \rceil $ $s = (m\_0 - m\_1) \frac{\alpha^{\theta}}{1 + \alpha}$ e1(i, j) = e1(i, j) + Geom(α) add e1(i, j) to G1 draw w from the distribution Pr[X ≤ x] = 1 − αx − θ + 1 add edge e1(i, j) with weight w to G1 run Louvain method on G1 to get C̃1 compute C̃ from C̃1 using the mapping M C̃ Given the group size k, LouvainDP starts with a supergraph G1 having $\lfloor \frac{\|V\|}{k}\rfloor$ nodes by randomly permuting the nodeset V and grouping every k consecutive nodes into a supernode (lines 1-4). The permutation prevents the possible bias of node ordering in G. The set of superedges E1 is easily computed from G. Note that m1 = ∣E1∣ ≤ m due to the fact that each edge of G appears in one and only one superedge. The domain size is $m\_0 = \frac{n\_1(n\_1+1)}{2}$ (i.e. we consider all selfloops in G1). The noisy number of non-zero superedges is m1 = ∣E1∣ + Lap(1/ε2). Then by one-sided filtering, we estimate the threshold θ (line 7) and the number of passing zero superedges s (line 8). For each non-zero superedge, we add a geometric noise and add the superedge to G1 if the noisy value is not smaller than θ. For s zero superedges $e\_1(i,j) \notin E\_1$, we draw an integral weight w from the distribution Pr[X ≤ x] = 1 − αx − θ + 1 and add e1(i, j) with weight w to G1 if w > 0. ### Complexity LouvainDP runs in O(m) because the loops to compute superedges (Line 5) and to add geometric noises (lines 9-12) cost O(m). We have $s = (m\_0 - m\_1) \frac{\alpha^{\theta}}{1 + \alpha} \leq \frac{m\_0 - m\_1}{1+\alpha} \frac{(1+\alpha)m\_1}{m\_0-m\_1} = m\_1$ (see Line 7). So the processing of s zero-superedges costs O(m). Moreover, Louvain method (line 17) is empirically linear in m1. We come up with the following theorem. [lem:louvaindp-edges] The number of edges in G1 is not larger than 2m. LouvainDP’s runtime is O(m) ### Privacy Analysis In LouvainDP, we use a small privacy budget ε2 = 0.01 to compute the noisy number of non-zero superedges m1. The remaining privacy budget ε1 is used for the geometric mechanism Geom(α). Note that getting a random permutation Vp (line 3) costs no privacy budget. The number of nodes n is public and given the group size k, the number of supernodes n1 is also public. The high-pass filtering technique (Lines 6-16) inherits the privacy guarantee by. By setting ε1 = ε − ε2, LouvainDP satisfies ε-differential privacy (see the sequential composition (Theorem [theorem:composition])). Alternative Input Perturbation Schemes -------------------------------------- 1K-series, DER, TmF and EdgeFlip are the most recent differentially private schemes for graph release that can be classified as input perturbation. While 1K-series and TmF run in linear time, DER and EdgeFlip incur a quadratic complexity. DER and EdgeFlip are therefore tested only on two medium-sized graphs in Section [sec:eval]. The expected number of edges by EdgeFlip is $\|E\_{\tilde{G}}\| = (1-s)m + \frac{n(n-1)}{4} s$ (see ) where $s = \frac{2}{e^{\epsilon} + 1}$ is the flipping probability. Substitute s into ∣EG̃∣, we get $ \|E\_{\tilde{G}}\| = m + (\frac{n(n-1)}{4} - m)\frac{2}{e^{\epsilon} + 1}$. The number of edges in the noisy graph G̃ generated by EdgeFlip increases exponentially as ε decreases. To ensure the linear complexity for million-scale graphs, we propose a simple extension of EdgeFlip, called EdgeFlipShrink (Algorithm [algo-edgeflipshrink]). Instead of outputting G̃, EdgeFlipShrink computes Ĝ that has the expected number of edges m by shrinking EG̃. First, the algorithm compute the private number of edges m̃ using a small budget ε2 (Lines 2-3 ). The new flipping probability s̃ is updated (Line 5). The noisy expected number of edges in the original EdgeFlip is shown in Line 6. We obtain the shrinking factor $p = \frac{\tilde{m}}{m\_0}$ (Line 7). Using p, every 1-edge is sampled with probability $\frac{1-\tilde{s}}{2} p$ instead of $\frac{1-s}{2}$ as in. The remaining 0-edges are randomly picked from EG as long as they do not exist in Ĝ (Lines 14-19). The expected edges of Ĝ is E[Ĝ] = E[m̃] = m and the running time of EdgeFlipShrink is O(m). [algo-edgeflipshrink] [1] undirected graph G, flipping probability s anonymized graph Ĝ $\hat{G} \leftarrow \varnothing$ ε2 = 0.1 m̃ = m + Lap(1/ε2) $\epsilon = \ln(\frac{2}{s}-1) - \epsilon\_2$ $\tilde{s} = \frac{2}{e^\epsilon + 1}$ $m\_0 = (1-\tilde{s})\tilde{m} + \frac{n(n-1)}{4}\tilde{s}$ $p = \frac{\tilde{m}}{m\_0}$ // process 1-edges n1 = 0 add edge (i, j) to Ĝ with prob. $\frac{1-\tilde{s}}{2} p$ n1 + + // process 0-edges n0 = m̃ − n1 random pick an edge $(i,j) \notin E\_G$ add edge (i, j) to Ĝ n0- - Ĝʹ Algorithm Perturbation ====================== The schemes in the algorithm perturbation category privately sample a node clustering from the input graph without generating noisy sample graphs as in the input perturbation. This can be done via the exponential mechanism. We introduce our main scheme ModDivisive in Section [subsec:moddivisive] followed by a variant of HRG-MCMC runnable on large graphs (Section [subsec:hrg-fixed]). Fig.[fig:algo-perturb] sketches the basic steps of the algorithm perturbation paradigm. ModDivisive: Top-down Exploration of Cohesive Groups ---------------------------------------------------- ### Overview In contrast with the agglomerative approaches (e.g. Louvain method) in which small communities are iteratively merged if doing so increases the modularity, our ModDivisive is a divisive algorithm in which communities at each level are iteratively split into smaller ones. Our goal is to heuristically detect cohesive groups of nodes in a private manner. There are several technical challenges in this process. The first one is how to efficiently find a good split of nodes that induces a high modularity and satisfies ε-DP at the same time. The second one is how to merge the small groups to larger ones. We cope with the first challenge by realizing an exponential mechanism via MCMC (Markov Chain Monte-Carlo) sampling with the modularity as the score function (see Theorem [def:exp-mech]). The second challenge is solved by dynamic programming. We design ModDivisive as a k-ary tree (Fig.[fig:mod-divisive]), i.e. each internal node has no more than k child nodes. The root node (level 0) contains all nodes in V and assigns arbitrarily each node into one of the k groups. Then we run the MCMC over the space of all partitions of V into no more than k nonempty subsets. The resultant subsets are initialized as the child nodes (level 1) of the root. The process is repeated iteratively for each child node at level 1 and stops at level maxL. Fig.[fig:mod-divisive] illustrates the idea with k = 3 for the graph in Fig.[fig:louvain]. ![image](mod-divisive_bestcut) [fig:mod-divisive] ### Algorithm Algorithm [algo-moddivisive] sketches the main steps in our scheme ModDivisive. It comprises two phases: differentially private sampling a k-ary tree of depth maxL which uses the privacy budget ε1 and finding the best cut across the tree to get a good clustering of nodes which consumes a budget maxL.εm. The first phase (lines 1-14) begins with the creation of eA, the array of privacy budgets allocated to levels of the tree. We use the parameter λ ≥ 1 as the common ratio to form a geometric sequence. The rationale behind the common ratio is to give higher priority to the levels near the root which have larger node sets. By sequential composition (Theorem [theorem:composition]), we must have ∑ieA[i] = ε1. All internal nodes at level i do the MCMC sampling on disjoint subsets of nodes, so the parallel composition holds. Subsection [subsubsec:priv] analyzes the privacy of ModDivisive in more detail. We use a queue to do a level-by-level exploration. Each dequeued node r’s level will be checked. If its level is not larger than maxL, we will run ModMCMC (Algorithm [algo-modmcmc]) on it (line 9) to get a partition r.part of its nodeset r.S (Fig.[fig:mod-divisive]). Each subset in r.part forms a child node and is pushed to the queue. The second phase (line 15) calls Algorithm [algo-bestcut] to find a highly modular partition across the tree. [algo-moddivisive] [1] graph G, group size k, privacy budget ε, max level maxL, ratio λ, BestCut privacy at each level εm noisy partition C̃ compute the array eA[0..maxL − 1] s.t. ∑ieA[i] = ε1, eA[i] = eA[i + 1] \* λ where ε1 = ε − maxL.εm initialize the root node with nodeset V root = NodeSet(G, V, k) root.level = 0 queue Q ← root r ← Q.dequeue() r.part = ModMCMC(G, r.S, k, eA[r.level]) Pi = NodeSet(G, Si, k) Pi.level = r.level + 1 r.childi ← Pi Q.enqueue(Pi) C̃ ← BestCut(root, εm) C̃ *Differentially Private Nodeset Partitioning* Let P be the space of all ways P to partition a nodeset A to no more than k disjoint subsets, the direct application of exponential mechanism needs the enumeration of P. The probability of a partition P being sampled is $$\frac{\exp(\frac{\epsilon\_p}{2\Delta Q} Q(P,G))} {\sum\_{P'\in \mathcal{P}} \exp(\frac{\epsilon\_p}{2\Delta Q} Q(P',G))}$$ However, ∣P∣ = ∑i = 1kS(∣A∣, i) where S(∣A∣, i) is the Stirling number of the second kind, $S(n,k) \approx \frac{k^n}{k!}$. This sum is exponential in ∣A∣, so enumerating P is computationally infeasible. Fortunately, MCMC can help us simulate the exponential mechanism by a sequence of local transitions in P. The space P is connected. It is straightforward to verify that the transitions performed in line 3 of ModMCMC are reversible and ergodic (i.e. any pair of nodeset partitions can be connected by a sequence of such transitions). Hence, ModMCMC has a unique stationary distribution in equilibrium. By empirical evaluation, we observe that ModMCMC converges after K∣r.S∣ steps for K = 50 (see Section [subsec:eval-div]). Each node r in the tree is of type NodeSet. This struct consists of an array r.part where r.part[u] ∈ {0..k − 1} is the group id of u. To make sure that ModMCMC runs in linear time, we must have a constant time computation of modularity Q(P) (line 4 of Algorithm [algo-modmcmc]). This can be done with two helper arrays: the number of intra-edges r.lc[0..k − 1] and the total degree of nodes r.dc[0..k − 1] in each group. The modularity Q is computed in O(k) (Formula [eqn:modularity]) using r.lc, r.dc. When moving node u from group i to group j, r.lc and r.dc are updated accordingly by checking the neighbors of u in G. The average degree is a constant, so the complexity of ModMCMC is linear in the number of MCMC steps. [algo-modmcmc] [1] graph G, nodeset r.S, group size k, privacy budget εp sampled partition r.part initialize r.part with a random partition P0 of k groups pick a neighboring partition Pʹ of Pi − 1 by randomly selecting node u ∈ r.S and moving u to another group. accept the transition and set Pi = Pʹ with probability $min(1,\frac{\exp(\frac{\epsilon\_p}{2\Delta Q} Q(P',G))} {\exp(\frac{\epsilon\_p}{2\Delta Q} Q(P\_{i-1},G))})$ // until equilibrium is reached a sampled partition r.part = Pi *Finding the Best Cut* Given the output k-ary tree R with the root node root, our next step is to find the best cut across the tree. A cut C is a set of nodes in R that cover all nodes in V. As an example, a cut C in Fig.[fig:mod-divisive] returns the clustering [{0,1,2,3}, {4}, {5,6,11}, {8,9,10,7,12}]. Any cut has a modularity score. Our goal is to find the best cut, i.e. the cut with highest modularity, in a private manner. We solve this problem by a dynamic programming technique. Remember that modularity is an additive quantity (c.f. Formula [eqn:modularity]). By denoting opt(r) as the optimal modularity for the subtree rooted at node r, the optimal value is opt(root). The recurrence relation is straightforward $$opt(r) = \max \{Q(r), \sum\_{t \in r.children} opt(t)\} \nonumber$$ Algorithm [algo-bestcut] realizes this idea in three steps. The first step (lines 1-6) uses a queue to fill a stack S. The stack ensures any internal node to be considered after its child nodes. The second step (lines 7-17) solves the recurrence relation. Because all modularity values are sensitive, we add Laplace noise Laplace(ΔQ/εm). The global sensitivity ΔQ = 3/m (see Theorem [theorem:delta-Q]), so we need only a small privacy budget for each level (εm = 0.01 is enough in our experiments). The noisy modularity modn is used to decide whether the optimal modularity at node r is by itself or by the sum over its children. The final step (lines 18-25) backtracks the best cut from the root node. [algo-bestcut] [1] undirected graph G, root node root, privacy budget at each level εm best cut C stack $S \leftarrow \varnothing$, queue Q ← root r ← Q.dequeue() S.push(r) Q.enqueue(ri) dictionary $sol \leftarrow \varnothing$ r ← S.pop(), r.modn = r.mod + Laplace(ΔQ/εm) sol.put(r.id, (val = r.modn, self = True)) sm = ∑ri ∈ r.childrensol[ri.id].modn sol.put(r.id, (val = sm, self = False)) sol.put(r.id, (val = r.modn, self = True)) list $C \leftarrow \varnothing$, queue Q ← root r ← Q.dequeue() C = C ∪ {r} Q.enqueue(ri) C ### Complexity ModDivisive creates a k-ary tree of height maxL. At each node r of the tree other than the leaf nodes, ModMCMC is run once. The run time of ModMCMC is O(K \* ∣r.S∣) thanks to the constant time for updating the modularity (line 4 of ModMCMC). Because the union of nodesets at one level is V, the total runtime is O(K \* ∣V∣ \* maxL). BestCut only incurs a sublinear runtime because the size of tree is always much smaller than ∣V∣. The following theorem states this result The time complexity of ModDivisive is linear in the number of nodes n, the maximum level maxL and the burn-in factor K. ### Privacy Analysis We show that ModMCMC satisfies differential privacy. The goal of MCMC is to draw a random sample from the desired distribution. Similarly, exponential mechanism is also a method to sample an output x ∈ X from the target distribution with probability proportional to $\exp(\frac{\epsilon u(x)}{2\Delta u})$ where u(x) is the score function (x with higher score has bigger chance to be sampled) and Δu is its sensitivity. The idea of using MCMC to realize exponential mechanism is first proposed in and applied to ε-DP graph release in. In our ModMCMC, the modularity Q(P, G) is used directly as the score function. We need to quantify the global sensitivity of Q. From Section [sec:diffpriv], we have the following definition (Global Sensitivity ΔQ) [def:sensitivity] ΔQ = maxP, G, Gʹ∣Q(P, G) − Q(P, Gʹ)∣ We prove that ΔQ = O(1/m) in the following theorem [theorem:delta-Q] The global sensitivity of modularity, ΔQ, is smaller than $\frac{3}{m}$ [proof:delta-Q] Given the graph G and a partition P of a nodeset Vp ⊆ V (for any node of the k-ary tree other than the root node, its nodeset Vp is a strict subset of V), the neighboring graph Gʹ has EGʹ = EG ∪ e. We have two cases Case 1. The new edge e is an intra-edge within the community s. The modularity Q(P, G) is $\sum\_{c}^{k} (\frac{l\_c}{m} - \frac{d\_c^2}{4m^2})$. The modularity Q(P, Gʹ) is $\sum\_{c\neq s}^{k} (\frac{l\_c}{m+1} - \frac{d\_c^2}{4(m+1)^2}) + (\frac{l\_s+1}{m+1} - \frac{(d\_s+2)^2}{4(m+1)^2})$. The difference $d\_1 = Q(P,G') - Q(P,G) = \frac{1}{m+1} -\frac{1}{m(m+1)} \sum\_{c}^{k}l\_c + \frac{2m+1}{4m^2(m+1)^2} \sum\_{c}^{k}d\_c^2 - \frac{d\_s+1}{(m+1)^2}$ Because ΔQ is the absolute value of d1, we consider the most positive and the most negative values of d1. Remember that ∑ckdc ≤ 2m, so the positive bound $d\_1 < \frac{1}{m+1} + \frac{(2m+1)4m^2}{4m^2(m+1)^2} < \frac{3}{m+1}$. For the negative bound, we use the constraints ∑cklc ≤ m and ds ≤ 2m, so $d\_1 > \frac{1}{m+1} - \frac{m}{m(m+1)} - \frac{2m+1}{(m+1)^2} > \frac{-2}{m+1}$. As a result, $\Delta Q = \|d\_1\| < \frac{3}{m+1} < \frac{3}{m}$. Case 2. The new edge e is an inter-edge between the communities s and t. Similarly, we have $Q(P,G) = \sum\_{c}^{k} (\frac{l\_c}{m} - \frac{d\_c^2}{4m^2})$ while $Q(P,G') = \sum\_{c\neq s,t}^{k} (\frac{l\_c}{m+1} - \frac{d\_c^2}{4(m+1)^2}) + (\frac{l\_s}{m+1} - \frac{(d\_s+1)^2}{4(m+1)^2}) + (\frac{l\_t}{m+1} - \frac{(d\_t+1)^2}{4(m+1)^2})$. The difference $d\_2 = Q(P,G') - Q(P,G) = -\frac{1}{m(m+1)} \sum\_{c}^{k}l\_c + \frac{2m+1}{4m^2(m+1)^2} \sum\_{c}^{k}d\_c^2 - \frac{2d\_s+2d\_t+2}{4(m+1)^2}$. Again, we consider the most positive and the most negative values of d2, using the constraint ∑ckdc ≤ 2m, the positive bound $d\_2 < \frac{(2m+1)4m^2}{4m^2(m+1)^2} < \frac{2}{m+1}$. For the negative bound, we use the constraints ∑cklc ≤ m and ds + dt ≤ 2m, so $d\_2 > - \frac{m}{m(m+1)} - \frac{4m+2}{4(m+1)^2} > \frac{-2}{m+1}$. As a result, $\Delta Q = \|d\_2\| < \frac{2}{m+1} < \frac{2}{m}$. To recap, in both cases $\Delta Q < \frac{3}{m}$. A Variant of HRG-MCMC --------------------- Similar to DER and EdgeFlip, HRG-MCMC runs in quadratic time due to its costly MCMC steps. Each MCMC step in HRG-MCMC takes O(n) to update the tree. To make it runnable on million-scale graphs, we describe briefly a variant of HRG-MCMC called HRG-Fixed (see Fig.[fig:hrg-fixed]). Instead of exploring the whole space of HRG trees as in HRG-MCMC, HRG-Fixed selects a fixed binary tree beforehand. We choose a balanced tree in our HRG-Fixed. Then HRG-Fixed realizes the exponential mechanism by sampling a permutation of n leaf nodes (note that each leaf node represents a graph node). The next permutation is constructed from the current one by randomly choosing a pair of nodes and swap them. The bottom of Fig.[fig:hrg-fixed] illustrates the swap of two nodes a and d. The log-likelihood L still has the sensitivity ΔL = 2lnn as in HRG-MCMC. Each sampling (MCMC) operation is designed to run in O(d̄.logn). By running K.n MCMC steps, the total runtime of HRG-Fixed is O(K.m.logn), feasible on large graphs. The burn-in factor K of HRG-Fixed is set to 1,000 in our evaluation. [fig:hrg-fixed] Experiments and Results ======================= In this section, our evaluation aims to compare the performance of the competitors by clustering quality and efficiency. The clustering quality is measured by the modularity Q and the average F1-score in which the modularity is the most important metric as we aim at highly modular clusterings. The efficiency is measured by the running time. All algorithms are implemented in Java and run on a desktop PC with $Intel^{\circledR}$ Core i7-4770@ 3.4Ghz, 16GB memory. Two medium-sized and three large real graphs are used in our experiments [1](#fn1). `as20graph` is the graph of routers comprising the internet. `ca-AstroPh` and `dblp` are co-authorship networks where two authors are connected if they publish at least one paper together. `amazon` is a product co-purchasing network where the graph contains an undirected edge from i to j if a product i is frequently co-purchased with product j. `youtube` is a video-sharing web site that includes a social network. Table [tab:dataset] shows the characteristics of the graphs. The columns Com(munities) and Mod(ularity) are the output of Louvain method. The number of samples in each test case is 20. [htb] [tab:dataset] \| \| Nodes \| Edges \| Com \| Mod \| \| --- \| --- \| --- \| --- \| --- \| \| as20graph \| 6,474 \| 12,572 \| 30 \| 0.623 \| \| ca-AstroPh \| 17,903 \| 196,972 \| 37 \| 0.624 \| \| amazon \| 334,863 \| 925,872 \| 257 \| 0.926 \| \| dblp \| 317,080 \| 1,049,866 \| 375 \| 0.818 \| \| youtube \| 1,134,890 \| 2,987,624 \| 13,485 \| 0.710 \| The schemes are abbreviated as 1K-series (1K), EdgeFlip (EF), Top-m-Filter (TmF), DER, LouvainDP (LDP), ModDivisive (MD), HRG-MCMC and HRG-Fixed. Quality Metrics --------------- Apart from modularity Q, we use $\bar{F\_1}$, the average F1-score, following the benchmarks in. The F1 score of a set A with respect to a set B is defined as the harmonic mean H of the precision and the recall of A against B. We define $prec(A,B) = \frac{\|A \cap B\|}{\|A\|}$, $recall(A,B) = \frac{\|A \cap B\|}{\|B\|}$ $$F\_1(A,B) = \frac{2.prec(A,B).recall(A,B)}{prec(A,B) + recall(A,B)} \nonumber$$ Then the average F1 score of two sets of communities C and Cʹ is defined as $$\begin{gathered} F\_1(A,C) = \max\_{i} F\_1(A,c\_i), \;\; c\_i \in C = \{c\_1,..,c\_n\} \\ \nonumber \bar{F\_1}(C,C') = \frac{1}{2\|C\|} \sum\_{c\_i\in C} F\_1(c\_i,C') + \frac{1}{2\|C'\|} \sum\_{c\_i\in C'} F\_1(c\_i,C) \nonumber\end{gathered}$$ We choose the output clustering of Louvain method as the ground truth for two reasons. First, the evaluation on the real ground truth is already done in and Louvain method is proven to provide high quality communities. Second, the real ground truth is a set of overlap communities whereas the schemes in this paper output only non-overlap communities. The chosen values of ε are {0.1lnn,0.2lnn,0.3lnn,0.4lnn,0.5lnn}. Performance of LouvainDP ------------------------ We test LouvainDP for the group size k ∈ {4, 8, 16, 32, 64}. The results on `youtube` are displayed in Fig. [fig:louvaindp]. We observe a clear separation of two groups k = 4, 8 and k = 16, 32, 64. As k increases, the modularity increases faster but also saturates sooner. Similar separations appear in avg.F1Score and the number of communities. Non-trivial modularity scores in several settings of (k, ε) indicate that randomly grouping of nodes and high-pass filtering superedges do not destroy all community structure of the original graph. At ε = 0.5lnn and k = 4, 8, the total edge weight in G1 is very low ( < 0.05m), so many supernodes of G1 are disconnected and Louvain method outputs a large number of communities (Fig. [fig:louvaindp-youtube-com]). The reason is that the threshold θ in LouvainDP is an integral value, so causing abnormal leaps in the total edge weight of G1. We pick k = 8, 64 for the comparative evaluation (Section [subsec:eval-comp]). [b]0.32 [fig:louvaindp-youtube-mod] [b]0.32![](louvaindp-youtube-f1 "fig:") [fig:louvaindp-youtube-f1] [b]0.32![](louvaindp-youtube-com "fig:") [fig:louvaindp-youtube-com] [fig:louvaindp] Performance of ModDivisive -------------------------- The effectiveness of ModDivisive is illustrated in Fig. [fig:moddivisive] for graph `youtube` and λ = 2.0, K = 50. We select six pairs of (k, maxL) by the set {(2,10),(3,7),(4,5), (5,4),(6,4),(10,3)}. Modularity increases steadily with ε while it is not always the case for avg.F1Score. The number of communities in the best cut is shown in Fig. [fig:moddivisive-youtube-com]. Clearly, the small number of communities indicates that ModDivisive’s best cut is not far from the root. The reason is the use of λ = 2.0, i.e. half of privacy budget is reserved to the first level, the half of the rest for the second level and so on. Lower levels receive geometrically smaller privacy budgets, so their partitions get poorer results. [h]0.32 [fig:moddivisive-youtube-mod] [h]0.32![](moddivisive-youtube-f1 "fig:") [fig:moddivisive-youtube-f1] [h]0.32![](moddivisive-youtube-com "fig:") [fig:moddivisive-youtube-com] [fig:moddivisive] We choose λ = 2.0 to obtain a good allocation of ε among the levels. Fig. [fig:ratio-amazon] shows the modularity for different values of λ. Note that λ = 1.0 means ε is equally allocated to the maxL levels. By building a k-ary tree, we reduce considerably the size of the state space P for MCMC. As a result, we need only a small burn-in factor K. Looking at Fig. [fig:burnfactor-amazon], we see that larger K = 100 results in only tiny increase of modularity in comparison with that of K = 50. [b]0.22![image](ratio-amazon) [fig:ratio-amazon] [b]0.22![image](burnfactor-amazon) [fig:burnfactor-amazon] [fig:moddiv-ratio-burnfactor] Comparative Evaluation ---------------------- In this section, we report a comparative evaluation of LouvainDP and ModDivisive against the competitors in Figures [fig:all-as20graph], [fig:all-caAstroPh], [fig:all-amazon], [fig:all-dblp] and [fig:all-youtube]. The dashed lines in subfigures [fig:all-as20graph-com], [fig:all-caAstroPh-com], [fig:all-amazon-com], [fig:all-dblp-com] and [fig:all-youtube-com] represent the ground-truth number of communities by Louvain method. ModDivisive performs best in most of the cases. On `as20graph` and `ca-AstroPh`, HRG-MCMC outputs the whole nodeset V with the zero modularity while 1K-series, TmF, DER also give useless clusterings. EdgeFlip produces good quality metrics exclusively on `ca-AstroPh` while LouvainDP returns the highest modularity scores on `as20graph`. However, the inherent quadratic complexity of EdgeFlip makes Louvain method fail at ε = 0.1lnn and 0.2lnn for `ca-AstroPh` graph. On the three large graphs, ModDivisive dominates the other schemes by a large margin in modularity and avg.F1Score. LouvainDP is the second best in modularity at k = 64. Our proposed HRG-Fixed is consistent with ε and has good performance on `dblp` and `youtube`. Note that HRG-MCMC is infeasible on the three large graphs due to its quadratic complexity. Again, 1K-series, TmF and EdgeFlipShrink provide the worst quality scores with the exception of 1K-series’s avg.F1Score on `youtube`. The runtime of the linear schemes is reported in Fig. [fig:runtime]. EdgeFlipShrink, 1K-series, TmF and LouvainDP benefit greatly by running Louvain method on the noisy output graph G̃. ModDivisive and HRG-Fixed also finish their work quickly in O(K.n.maxL) and O(K.m.logn) respectively. [b]0.22![image](runtime-all) [fig:runtime-all] [b]0.22![image](runtime-moddiv-amazon) [fig:runtime-moddiv-amazon] [fig:runtime] Conclusion ========== We have given a big picture of the problem ε-DP community detection within the two categories: input and algorithm perturbation. We analyzed the major challenges of community detection under differential privacy. We explained why techniques borrowed from k-Means fail and how the difficulty of ε-DP recommender systems enables a relaxation of privacy budget. We proposed LouvainDP and ModDivisive as the representatives of input and algorithm perturbations respectively. By conducting a comprehensive evaluation, we revealed the advantages of our methods. ModDivisive steadily gives the best modularity and avg.F1Score on large graphs while LouvainDP outperforms the remaining input perturbation competitors in certain settings. HRG-MCMC/HRG-Fixed give low modularity clusterings, indicating the limitation of the HRG model in divisive CD. The input perturbation schemes DER, EF, 1K-series and TmF hardly deliver any good node clustering except EF on the two medium-sized graphs. For future work, we plan to develop an ε-DP agglomerative scheme based on Louvain method and extend our work for directed graphs and overlapping community detection under differential privacy. [t]0.32![](all-as20graph-mod-3 "fig:") [fig:all-as20graph-mod] [t]0.32![](all-as20graph-f1-3 "fig:") [fig:all-as20graph-f1] [t]0.32![](all-as20graph-com-3 "fig:") [fig:all-as20graph-com] [fig:all-as20graph] [t]0.32![](all-caAstroPh-mod-3 "fig:") [fig:all-caAstroPh-mod] [t]0.32![](all-caAstroPh-f1-3 "fig:") [fig:all-caAstroPh-f1] [t]0.32![](all-caAstroPh-com-3 "fig:") [fig:all-caAstroPh-com] [fig:all-caAstroPh] [t]0.32![](all-amazon-mod-3 "fig:") [fig:all-amazon-mod] [t]0.32![](all-amazon-f1-3 "fig:") [fig:all-amazon-f1] [t]0.32![](all-amazon-com-3 "fig:") [fig:all-amazon-com] [fig:all-amazon] [t]0.32![](all-dblp-mod-3 "fig:") [fig:all-dblp-mod] [t]0.32![](all-dblp-f1-3 "fig:") [fig:all-dblp-f1] [t]0.32![](all-dblp-com-3 "fig:") [fig:all-dblp-com] [fig:all-dblp] [t]0.32![](all-youtube-mod-3 "fig:") [fig:all-youtube-mod] [t]0.32![](all-youtube-f1-3 "fig:") [fig:all-youtube-f1] [t]0.32![](all-youtube-com-3 "fig:") [fig:all-youtube-com] [fig:all-youtube] --- 1. http://snap.stanford.edu/data/index.html[↩](#fnref1) Detecting Communities under Differential Privacy ================================================ Complex networks usually expose community structure with groups of nodes sharing many links with the other nodes in the same group and relatively few with the nodes of the rest. This feature captures valuable information about the organization and even the evolution of the network. Over the last decade, a great number of algorithms for community detection have been proposed to deal with the increasingly complex networks. However, the problem of doing this in a private manner is rarely considered. In this paper, we solve this problem under differential privacy, a prominent privacy concept for releasing private data. We analyze the major challenges behind the problem and propose several schemes to tackle them from two perspectives: input perturbation and algorithm perturbation. We choose Louvain method as the back-end community detection for input perturbation schemes and propose the method LouvainDP which runs Louvain algorithm on a noisy super-graph. For algorithm perturbation, we design ModDivisive using exponential mechanism with the modularity as the score. We have thoroughly evaluated our techniques on real graphs of different sizes and verified their outperformance over the state-of-the-art. Introduction ============ Graphs represent a rich class of data observed in daily life where entities are described by nodes and their connections are characterized by edges. Apart from microscopic (node level) and macroscopic (graph level) configurations, many complex networks display a mesoscopic structure, i.e. they appear as a combination of components fairly independent of each other. These components are called communities, modules or clusters and the problem of how to reveal them plays a significant role in understanding the organization and function of complex networks. Over the last decade, a great number of algorithms for community detection (CD) have been proposed to address the problem in a variety of settings, such as undirected/directed, unweighted/weighted networks and non-overlapping/overlapping communities (for a comprehensive survey, see ). These approaches, however, are adopted in a non-private manner, i.e. a data collector (such as Facebook) knows all the contributing users and their relationships before running CD algorithms. The output of such a CD, in the simplest form, is a clustering of nodes. Even in this case, i.e. only a node clustering (not the whole graph) is revealed, contributing users privacy may still be put at risk. In this paper, we address the problem of CD from the perspective of differential privacy. This privacy model offers a formal definition of privacy with a lot of interesting properties: no computational/informational assumptions about attackers, data type-agnosticity, composability and so on. By differential privacy, we want to ensure the existence of connections between users to be hidden in the output clustering while keeping the low distortion of clusters compared to the ones generated by the corresponding non-private algorithms. As far as we know, the problem is quite new and only mentioned in the recent work where Mülle et al. use a sampling technique to perturb the input graph so that it satisfies differential privacy before running conventional CD algorithms. This technique (we call it EdgeFlip afterwards) is classified as input perturbation in differential privacy literature (the other two categories are algorithm perturbation and output perturbation). Similarly, TmF approach can apply to the true graph to get noisy output graphs as in the work of Mülle et al. Earlier, 1k-Series, Density Explore Reconstruct (DER) and HRG-MCMC are the best known methods for graph structure release under differential privacy. These methods can be followed by any exact CD algorithm to get a noisy clustering satisfying differential privacy. We choose Louvain method as such a CD algorithm. However, as we will see in the experiments, the output clusterings by the aforementioned methods have very low modularity. This fact necessitates new methods for CD problem under differential privacy. Our main contributions are the new schemes LouvainDP (input perturbation) and ModDivisive (algorithm perturbation) which perform much better than the state-of-the-art. LouvainDP is a high-pass filtering method that randomly groups nodes into supernodes of equal size to build a weighted supergraph. LouvainDP is guaranteed to run in linear time. ModDivisive is a top-down approach which privately divides the node set into the k-ary tree guided by the modularity score at each level. The main technique used in ModDivisive is the Markov Chain Monte Carlo (MCMC) to realize the exponential mechanism. We show that ModDivisive’s runtime is linear in the number of nodes, the height of the binary tree and the burn-in factor of MCMC. The linear complexity enables us to examine million-scale graphs in a few minutes. The experiments show the high modularity and low distortion of the output clusters by LouvainDP and ModDivisive. Our contributions are summarized as follows: * We analyze the major challenges of community detection under differential privacy. We explain why techniques borrowed from k-Means fail and how the difficulty of ε-DP recommender systems justifies a relaxation of ε. * We design an input perturbation scheme LouvainDP that runs in linear time using the high-pass filtering technique from and Louvain method. * We propose an algorithm perturbation scheme ModDivisive as a divisive approach by using the modularity-based score function in the exponential mechanism. We prove that modularity has small global sensitivity and ModDivisive also runs in linear time. * We conduct a thorough evaluation on real graphs of different sizes and show the outperformance of LouvainDP and ModDivisive over the state-of-the-art. The paper is organized as follows. We review the related work for community detection algorithms and graph release via differential privacy in Section [sec:related]. Section [sec:pre] briefly introduces the key concepts of differential privacy, the popular Louvain method and the major challenges of ε-DP community detection. Section [sec:input] focuses on the category of input perturbation in which we propose LouvainDP and review several recent input perturbation schemes. We describe ModDivisive in Section [sec:algo]. We compare all the presented schemes on real graphs in Section [sec:eval]. Finally, we present our conclusions and suggest future work in Section [sec:conclusion]. Table [tab:notation] summarizes the key notations used in this paper. [htb] [tab:notation] \| Symbol \| Definition \| \| --- \| --- \| \| G = (V, EG) \| true graph with n = ∣V∣ and m = ∣EG∣ \| \| Gʹ = (V, EGʹ) \| neighboring graph of G \| \| G̃ = (V, EG̃) \| sample noisy output graph \| \| G1 = (V1, E1) \| supergraph generated by LouvainDP \| \| k \| fan-out of the tree in ModDivisive \| \| K \| burn-in factor in MCMC-based algorithms \| \| λ \| common ratio to distribute the privacy budget \| \| C \| a clustering of nodes in G \| \| Q(G, C) \| modularity of the clustering C on graph G \| Related Work ============ Community Detection in Graphs ----------------------------- There is a vast literature on community detection in graphs. For a recent comprehensive survey, we refer to. In this section, we discuss several classes of techniques. Newman and Girvan propose modularity as a quality of network clustering. It is based on the idea that a random graph is not expected to have a modular structure, so the possible existence of clusters is revealed by the comparison between the actual density of edges in a subgraph and the density one would expect to have in the subgraph if the nodes of the graph were connected randomly (the null model). The modularity Q is defined as $$\label{eqn:modularity} Q = \sum\_{c=1}^{n\_c}\left[\frac{l\_c}{m} - \left(\frac{d\_c}{2m} \right) ^2 \right]$$ where nc is the number of clusters, lc is the total number of edges joining nodes in community c and dc is the sum of the degrees of the nodes of c. Many methods for optimizing the modularity have been proposed over the last ten years, such as agglomerative greedy, simulated annealing, random walks, statistical mechanics, label propagation or InfoMap, just to name a few. The recent multilevel approach, also called Louvain method, by Blondel et al. is a top performance scheme. It scales very well to graphs with hundreds of millions of nodes. This is the chosen method for the input perturbation schemes considered in this paper (see Sections [subsec:louvain] and [sec:input] for more detail). By maximizing the modularity, Louvain method is based on edge counting metrics, so it fits well with the concept of edge differential privacy (Section [sec:diffpriv]). One of the most recent methods SCD is not chosen because it is about maximizing Weighted Community Clustering (WCC) instead of the modularity. WCC is based on triangle counting which has high global sensitivity (up to O(n)). Moreover, SCD pre-processes the graph by removing all edges that do not close any triangle. This means all 1-degree nodes are excluded and form singleton clusters. The number of output clusters is empirically up to O(n). Graph Release via Differential Privacy -------------------------------------- In principle, after releasing a graph satisfying ε-DP, we can do any mining operations on it, including community detection. The research community, therefore, expresses a strong interest in the problem of graph release via differential privacy. Differentially private algorithms relate the amount of noise to the computation sensitivity. Lower sensitivity implies smaller added noise. Because the edges in simple undirected graphs are usually assumed to be independent, the standard Laplace mechanism is applicable (e.g. adding Laplace noise to each cell of the adjacency matrix). However, this approach severely deteriorates the graph structure. The state-of-the-art try to reduce the sensitivity of the graph in different ways. Density Explore Reconstruct (DER) employs a data-dependent quadtree to summarize the adjacency matrix into a counting tree and then reconstructs noisy sample graphs. DER is an instance of input perturbation. Xiao et al. propose to use Hierarchical Random Graph (HRG) to encode graph structural information in terms of edge probabilities. Their scheme HRG-MCMC is argued to be able to sample good HRG models which reflect the community structure in the original graph. HRG-MCMC is classified as an algorithm perturbation scheme. A common disadvantage of the state-of-the-art DER and HRG-MCMC is the scalability issue. Both of them incur quadratic complexity O(n2), limiting themselves to medium-sized graphs. Recently, Nguyen et al. proposed TmF that utilizes a filtering technique to keep the runtime linear in the number of edges. TmF also proves the upper bound O(lnn) for the privacy budget ε. At the same time, Mülle et al. devised the scheme EdgeFlip using an edge flipping technique. However, EdgeFlip costs O(n2) and is runnable only on graphs of tens of thousands of nodes. A recent paper by Campan et al. studies whether anonymized social networks preserve existing communities from the original social networks. The considered anonymization methods are SaNGreeA and k-degree method. Both of them use k-anonymity rather than differential privacy which is the focus of this paper. Preliminaries ============= In this section, we review key concepts and mechanisms of differential privacy. Differential Privacy -------------------- Essentially, ε-differential privacy (ε-DP) is proposed to quantify the notion of indistinguishability of neighboring databases. In the context of graph release, two graphs G1 = (V1, E1) and G2 = (V2, E2) are neighbors if V1 = V2, E1 ⊂ E2 and ∣E2∣ = ∣E1∣ + 1. Note that this notion of neighborhood is called edge differential privacy in contrast with the notion of node differential privacy which allows the addition of one node and its adjacent edges. Node differential privacy has much higher sensitivity and is much harder to analyze. Our work follows the common use of edge differential privacy. The formal definition of ε-DP for graph data is as follows. [def:e-dp] A mechanism A is ε-differentially private if for any two neighboring graphs G1 and G2, and for any output O ∈ Range(A), $$Pr[\mathcal{A}(G\_1) \in O] \leq e^{\epsilon} Pr[\mathcal{A}(G\_2) \in O] \nonumber$$ Laplace mechanism and Exponential mechanism are two standard techniques in differential privacy. The latter is a generalization of the former. Laplace mechanism is based on the concept of global sensitivity of a function f which is defined as Δf = maxG1, G2∣∣f(G1) − f(G2)∣∣1 where the maximum is taken over all pairs of neighboring G1, G2. Given a function f and a privacy budget ε, the noise is drawn from a Laplace distribution $Lap(\lambda): p(x\|\lambda) = \frac{1}{2\lambda} e^{-\|x\|/\lambda}$ where λ = Δf/ε. (Laplace mechanism ) [def:lap-mech] For any function f : G → Rd, the mechanism A $$\mathcal{A}(G) = f(G) + \langle Lap\_1(\frac{\Delta f}{\epsilon}),...,Lap\_d(\frac{\Delta f}{\epsilon})\rangle$$ satisfies ε-differential privacy, where $Lap\_i(\frac{\Delta f}{\epsilon})$ are i.i.d Laplace variables with scale parameter $\frac{\Delta f}{\epsilon}$. □ Geometric mechanism is a discrete variant of Laplace mechanism with integral output range Z and random noise Δ generated from a two-sided geometric distribution $Geom(\alpha): Pr[\Delta = \delta\| \alpha] = \frac{1-\alpha}{1+\alpha} \alpha^{\|\delta\|}$. To satisfy ε-DP, we set α = exp( − ε). We use geometric mechanism in our LouvainDP scheme. For non-numeric data, the exponential mechanism is a better choice. Its main idea is based on sampling an output O from the output space O using a score function u. This function assigns exponentially higher probabilities to outputs of higher scores. Let the global sensitivity of u be Δu = maxO, G1, G2∣u(G1, O) − u(G2, O)∣. (Exponential mechanism ) [def:exp-mech] Given a score function u : (G × O) → R for a graph G, the mechanism A that samples an output O with probability proportional to $\exp(\frac{\epsilon.u(G,O)}{2\Delta u})$ satisfies ε-differential privacy. □ Composability is a nice property of differential privacy which is not satisfied by other privacy models such as k-anonymity. (Sequential and parallel compositions ) [theorem:composition] Let each Ai provide εi-differential privacy. A sequence of Ai(D) over the dataset D provides Σi = 1nεi-differential privacy. Let each Ai provide εi-differential privacy. Let Di be arbitrary disjoint subsets of the dataset D. The sequence of Ai(Di) provides maxi = 1nεi-differential privacy. □ Louvain Method -------------- Since its introduction in 2008, Louvain method becomes one of the most cited methods for the community detection task. It optimizes the modularity by a bottom-up folding process. The algorithm is divided in passes each of which is composed of two phases that are repeated iteratively. Initially, each node is assigned to a different community. So, there will be as many communities as there are nodes in the first phase. Then, for each node i, the method considers the gain of modularity if we move i from its community to the community of a neighbor j (a local change). The node i is then placed in the community for which this gain is maximum and positive (if any), otherwise it stays in its original community. This process is applied repeatedly and sequentially for all nodes until no further improvement can be achieved and the first pass is then complete. [fig:louvain] We demonstrate Louvain method in Fig.[fig:louvain] by a graph of 13 nodes and 20 edges. If each node forms its own singleton community, the modularity Q will be -0.0825. In the first pass of Louvain method, each node moves to the best community selected from its neighbors’ communities. We get the partition [{0, 1, 2}, {3, 4}, {5, 6, 11, 12}, {7, 8, 9, 10}] with modularity 0.46375. The second phase of first pass builds a weighted graph corresponding to the partition by aggregating communities. The second pass repeats the folding process on this weighted graphs to reach the final partition [{0, 1, 2}, {3, 4, 5, 6, 11, 12}, {7, 8, 9, 10}] with modularity 0.47. This simple agglomerative algorithm has several advantages as stated in. First, its steps are intuitive and easy to implement, and the outcome is unsupervised. Second, the algorithm is extremely fast, i.e. computer simulations on large modular networks suggest that its complexity is linear on typical and sparse data. This is due to the fact that the possible gains in modularity are easy to compute and the number of communities decreases drastically after just a few passes so that most of the running time is concentrated on the first iterations. Third, the multi-level nature of the method produces a hierarchical structure of communities which allows multi-resolution analysis, i.e the user can zoom in the graph to observe its structure with the desired resolution. In addition, Louvain method is runnable on weighted graphs. This fact supports naturally our scheme LouvainDP as described in Section [subsec:louvaindp]. Challenges of Community Detection under Differential Privacy ------------------------------------------------------------ In this section, we explain why community detection under differential privacy is challenging. We show how techniques borrowed from related problems fail. We also advocate the choice of ε as a function of graph size n. The problem of differentially private community detection is closely related to ε-DP k-Means clustering and recommender systems. The ε-DP k-Means is thoroughly discussed in. However, techniques from ε-DP k-Means are not suitable to ε-DP community detection. First, items in k-Means are in low-dimensional spaces and the number of clusters k is usually small. This contrast to the case of community detection where nodes lie in a n-dimensional space and the number of communities varies from tens to tens of thousands, not to say the communities may overlap or be nested (multi-scale). Second, items in ε-DP k-Means are normalized to [ − 1, 1]d while the same preprocessing seems invalid in ε-DP community detection. Moreover, the output of k-Means usually consists of equal-sized balls while this is not true for communities in graphs. Considering the graph as a high-dimensional dataset, we tried the private projection technique in which is followed by spectral clustering, but the modularity scores of the output are not better than random clustering. Recent papers on ε-DP recommender systems show that privately learning the clustering of items from user ratings is hard unless we relax the value of ε up to logn. Banerjee et al. model differentially private mechanisms as noisy channels and bound the mutual information between the generative sources and the privatized sketches. They show that in the information-rich regime (each user rates O(n) items), their Pairwise-Preference succeeds if the number of users is Ω(nlogn/ε). Compared to ε-DP community detection where the number of users is n, we should have ε = Ω(logn). Similarly, in D2P scheme, Guerraoui et al. draw a formula for ε as $$\epsilon\_{D2P}^{(p,0,\lambda)} = \ln(1+\frac{(1-p).\mathcal{N}\_E}{p.\|\mathcal{G}\_\lambda\|})$$ where λ is the distance used to conceal the user profiles (λ = 0 reduces to the classic notion of differential privacy). NE is the number of items which is exactly n in community detection. ∣Gλ∣ is the minimum size of user profiles at distance λ over all users (∣Gλ∣ = o(NE) except at unreasonably large λ). Clearly, at p = 0.5 (as used in ), we have εD2P(0.5, 0, λ) ≈ lnn. The sampling technique in D2P is very similar to EdgeFlip which is shown ineffective in ε-DP community detection for ε ∈ (0, 0.5lnn) (Section [sec:eval]). Note that ε-DP community detection is unique in the sense that the set of items and the set of users are the same. Graphs for community detection are more general than bipartite graphs in recommender systems. In addition, modularity Q (c.f. Formula [eqn:modularity]) is non-monotone, i.e. for two disjoint sets of nodes A and B, Q(A ∪ B) may be larger, smaller than or equal to Q(A) + Q(B). To further emphasize the difficulty of ε-DP community detection, we found that IDC scheme using Sparse Vector Technique is hardly feasible. As shown in Algorithm 1 of, to publish a noisy graph that can approximately answer all cut queries with bounded error m0.25n/ε0.5, IDC must have B(α) “yes” queries among all k queries. B(α) may be as low as $\sqrt{m}$ but k = 22n. In the average case, IDC incurs exponential time to complete. To conclude, ε-DP community detection is challenging and requires new techniques. In this paper, we evaluate the schemes for ε up to 0.5lnn. At ε = 0.5lnn, the multiplicative ratio (c.f. Definition [def:e-dp]) is $e^\epsilon = e^{0.5\ln n} = \sqrt{n}$, a reasonable threshold for privacy protection compared to ε = lnn (i.e. eε = n) in ε-DP recommender systems discussed above. The typical epsilon in the literature is 1.0 or less. However, this value is only applicable to graph metrics of low sensitivity O(1) such as the number of edges, the degree sequence. The global sensitivity of other metrics like the diameter, the number of triangles, 2K-series etc. is O(n), calling for local sensitivity analysis (e.g. ). For counting queries, Laplace/Geometric mechanisms are straightforward on real/integral (metric) spaces. However, direct noise adding mechanisms on the space P of all ways to partition the nodeset V are non-trivial because ∣P∣ ≈ nn and P is non-metric. That is another justification for the relaxation of ε up to 0.5lnn. Input Perturbation ================== In this section, we propose the linear scheme LouvainDP that uses a filtering technique to build a noisy weighted supergraph and calls the exact Louvain method subsequently. Then we discuss several recent ε-DP schemes that can be classified as input perturbation. Fig.[fig:input-perturb] sketches the basic steps of the input perturbation paradigm. [b]0.42![image](input-perturb) [fig:input-perturb] [b]0.42![image](algo-perturb) [fig:algo-perturb] LouvainDP: Louvain Method on Noisy Supergraphs ---------------------------------------------- The basic idea of LouvainDP is to create a noisy weighted supergraph G1 from G by grouping nodes into supernodes of equal size k. We then apply the filtering technique of Cormode et al. to ensure only O(m) noisy weighted edges in G1. Finally, we run the exact community detection on G1. In, Cormode et al. propose several summarization techniques for sparse data under differential privacy. Let M be a contingency table having the domain size m0 and m1 non-zero entries (m1 ≪ m0 for sparse data), the conventional publication of a noisy table Mʹ from M that satisfies ε-DP requires the addition of Laplace/geometric noise to m0 entries. The entries in Mʹ could be filtered (e.g. removing negative ones) and/or sampled to get a noisy summary Mʺ. This direct approach would be infeasible for huge domain sizes m0. Techniques in avoid materializing the vast noisy data by computing the summary Mʺ directly from M using filtering and sampling techniques. In our LouvainDP, the supergraph G1 is an instance of sparse data with the domain size $m\_0 = \frac{n\_1(n\_1+1)}{2}$ where n1 is the number of supernodes and m1 non-zero entries corresponding to non-zero superedges. We use the one-sided filtering to efficiently compute G1 with O(m) edges in linear time. ### Algorithm LouvainDP can run with either geometric or Laplace noise. We describe the version with geometric noise in Algorithm [algo-louvaindp]. [algo-louvaindp] [1] undirected graph G, group size k, privacy budget ε noisy partition C̃ $G\_1 \leftarrow \varnothing$, $n\_1 = \lfloor \frac{\|V\|}{k}\rfloor -1$, V1 ← {0, 1, .., n1} ε2 = 0.1, ε1 = ε − ε2, α = exp( − ε1) get a random permutation Vp of V compute the mapping M : Vp → V1 compute superedges of G1: E1 = {e1(i, j)} where i, j ∈ V1 m1 = ∣E1∣ + Lap(1/ε2), $m\_0 = \frac{n\_1(n\_1+1)}{2}$ $\theta = \lceil \log\_{\alpha} \frac{(1+\alpha)m\_1}{m\_0-m\_1} \rceil $ $s = (m\_0 - m\_1) \frac{\alpha^{\theta}}{1 + \alpha}$ e1(i, j) = e1(i, j) + Geom(α) add e1(i, j) to G1 draw w from the distribution Pr[X ≤ x] = 1 − αx − θ + 1 add edge e1(i, j) with weight w to G1 run Louvain method on G1 to get C̃1 compute C̃ from C̃1 using the mapping M C̃ Given the group size k, LouvainDP starts with a supergraph G1 having $\lfloor \frac{\|V\|}{k}\rfloor$ nodes by randomly permuting the nodeset V and grouping every k consecutive nodes into a supernode (lines 1-4). The permutation prevents the possible bias of node ordering in G. The set of superedges E1 is easily computed from G. Note that m1 = ∣E1∣ ≤ m due to the fact that each edge of G appears in one and only one superedge. The domain size is $m\_0 = \frac{n\_1(n\_1+1)}{2}$ (i.e. we consider all selfloops in G1). The noisy number of non-zero superedges is m1 = ∣E1∣ + Lap(1/ε2). Then by one-sided filtering, we estimate the threshold θ (line 7) and the number of passing zero superedges s (line 8). For each non-zero superedge, we add a geometric noise and add the superedge to G1 if the noisy value is not smaller than θ. For s zero superedges $e\_1(i,j) \notin E\_1$, we draw an integral weight w from the distribution Pr[X ≤ x] = 1 − αx − θ + 1 and add e1(i, j) with weight w to G1 if w > 0. ### Complexity LouvainDP runs in O(m) because the loops to compute superedges (Line 5) and to add geometric noises (lines 9-12) cost O(m). We have $s = (m\_0 - m\_1) \frac{\alpha^{\theta}}{1 + \alpha} \leq \frac{m\_0 - m\_1}{1+\alpha} \frac{(1+\alpha)m\_1}{m\_0-m\_1} = m\_1$ (see Line 7). So the processing of s zero-superedges costs O(m). Moreover, Louvain method (line 17) is empirically linear in m1. We come up with the following theorem. [lem:louvaindp-edges] The number of edges in G1 is not larger than 2m. LouvainDP’s runtime is O(m) ### Privacy Analysis In LouvainDP, we use a small privacy budget ε2 = 0.01 to compute the noisy number of non-zero superedges m1. The remaining privacy budget ε1 is used for the geometric mechanism Geom(α). Note that getting a random permutation Vp (line 3) costs no privacy budget. The number of nodes n is public and given the group size k, the number of supernodes n1 is also public. The high-pass filtering technique (Lines 6-16) inherits the privacy guarantee by. By setting ε1 = ε − ε2, LouvainDP satisfies ε-differential privacy (see the sequential composition (Theorem [theorem:composition])). Alternative Input Perturbation Schemes -------------------------------------- 1K-series, DER, TmF and EdgeFlip are the most recent differentially private schemes for graph release that can be classified as input perturbation. While 1K-series and TmF run in linear time, DER and EdgeFlip incur a quadratic complexity. DER and EdgeFlip are therefore tested only on two medium-sized graphs in Section [sec:eval]. The expected number of edges by EdgeFlip is $\|E\_{\tilde{G}}\| = (1-s)m + \frac{n(n-1)}{4} s$ (see ) where $s = \frac{2}{e^{\epsilon} + 1}$ is the flipping probability. Substitute s into ∣EG̃∣, we get $ \|E\_{\tilde{G}}\| = m + (\frac{n(n-1)}{4} - m)\frac{2}{e^{\epsilon} + 1}$. The number of edges in the noisy graph G̃ generated by EdgeFlip increases exponentially as ε decreases. To ensure the linear complexity for million-scale graphs, we propose a simple extension of EdgeFlip, called EdgeFlipShrink (Algorithm [algo-edgeflipshrink]). Instead of outputting G̃, EdgeFlipShrink computes Ĝ that has the expected number of edges m by shrinking EG̃. First, the algorithm compute the private number of edges m̃ using a small budget ε2 (Lines 2-3 ). The new flipping probability s̃ is updated (Line 5). The noisy expected number of edges in the original EdgeFlip is shown in Line 6. We obtain the shrinking factor $p = \frac{\tilde{m}}{m\_0}$ (Line 7). Using p, every 1-edge is sampled with probability $\frac{1-\tilde{s}}{2} p$ instead of $\frac{1-s}{2}$ as in. The remaining 0-edges are randomly picked from EG as long as they do not exist in Ĝ (Lines 14-19). The expected edges of Ĝ is E[Ĝ] = E[m̃] = m and the running time of EdgeFlipShrink is O(m). [algo-edgeflipshrink] [1] undirected graph G, flipping probability s anonymized graph Ĝ $\hat{G} \leftarrow \varnothing$ ε2 = 0.1 m̃ = m + Lap(1/ε2) $\epsilon = \ln(\frac{2}{s}-1) - \epsilon\_2$ $\tilde{s} = \frac{2}{e^\epsilon + 1}$ $m\_0 = (1-\tilde{s})\tilde{m} + \frac{n(n-1)}{4}\tilde{s}$ $p = \frac{\tilde{m}}{m\_0}$ // process 1-edges n1 = 0 add edge (i, j) to Ĝ with prob. $\frac{1-\tilde{s}}{2} p$ n1 + + // process 0-edges n0 = m̃ − n1 random pick an edge $(i,j) \notin E\_G$ add edge (i, j) to Ĝ n0- - Ĝʹ Algorithm Perturbation ====================== The schemes in the algorithm perturbation category privately sample a node clustering from the input graph without generating noisy sample graphs as in the input perturbation. This can be done via the exponential mechanism. We introduce our main scheme ModDivisive in Section [subsec:moddivisive] followed by a variant of HRG-MCMC runnable on large graphs (Section [subsec:hrg-fixed]). Fig.[fig:algo-perturb] sketches the basic steps of the algorithm perturbation paradigm. ModDivisive: Top-down Exploration of Cohesive Groups ---------------------------------------------------- ### Overview In contrast with the agglomerative approaches (e.g. Louvain method) in which small communities are iteratively merged if doing so increases the modularity, our ModDivisive is a divisive algorithm in which communities at each level are iteratively split into smaller ones. Our goal is to heuristically detect cohesive groups of nodes in a private manner. There are several technical challenges in this process. The first one is how to efficiently find a good split of nodes that induces a high modularity and satisfies ε-DP at the same time. The second one is how to merge the small groups to larger ones. We cope with the first challenge by realizing an exponential mechanism via MCMC (Markov Chain Monte-Carlo) sampling with the modularity as the score function (see Theorem [def:exp-mech]). The second challenge is solved by dynamic programming. We design ModDivisive as a k-ary tree (Fig.[fig:mod-divisive]), i.e. each internal node has no more than k child nodes. The root node (level 0) contains all nodes in V and assigns arbitrarily each node into one of the k groups. Then we run the MCMC over the space of all partitions of V into no more than k nonempty subsets. The resultant subsets are initialized as the child nodes (level 1) of the root. The process is repeated iteratively for each child node at level 1 and stops at level maxL. Fig.[fig:mod-divisive] illustrates the idea with k = 3 for the graph in Fig.[fig:louvain]. ![image](mod-divisive_bestcut) [fig:mod-divisive] ### Algorithm Algorithm [algo-moddivisive] sketches the main steps in our scheme ModDivisive. It comprises two phases: differentially private sampling a k-ary tree of depth maxL which uses the privacy budget ε1 and finding the best cut across the tree to get a good clustering of nodes which consumes a budget maxL.εm. The first phase (lines 1-14) begins with the creation of eA, the array of privacy budgets allocated to levels of the tree. We use the parameter λ ≥ 1 as the common ratio to form a geometric sequence. The rationale behind the common ratio is to give higher priority to the levels near the root which have larger node sets. By sequential composition (Theorem [theorem:composition]), we must have ∑ieA[i] = ε1. All internal nodes at level i do the MCMC sampling on disjoint subsets of nodes, so the parallel composition holds. Subsection [subsubsec:priv] analyzes the privacy of ModDivisive in more detail. We use a queue to do a level-by-level exploration. Each dequeued node r’s level will be checked. If its level is not larger than maxL, we will run ModMCMC (Algorithm [algo-modmcmc]) on it (line 9) to get a partition r.part of its nodeset r.S (Fig.[fig:mod-divisive]). Each subset in r.part forms a child node and is pushed to the queue. The second phase (line 15) calls Algorithm [algo-bestcut] to find a highly modular partition across the tree. [algo-moddivisive] [1] graph G, group size k, privacy budget ε, max level maxL, ratio λ, BestCut privacy at each level εm noisy partition C̃ compute the array eA[0..maxL − 1] s.t. ∑ieA[i] = ε1, eA[i] = eA[i + 1] \* λ where ε1 = ε − maxL.εm initialize the root node with nodeset V root = NodeSet(G, V, k) root.level = 0 queue Q ← root r ← Q.dequeue() r.part = ModMCMC(G, r.S, k, eA[r.level]) Pi = NodeSet(G, Si, k) Pi.level = r.level + 1 r.childi ← Pi Q.enqueue(Pi) C̃ ← BestCut(root, εm) C̃ *Differentially Private Nodeset Partitioning* Let P be the space of all ways P to partition a nodeset A to no more than k disjoint subsets, the direct application of exponential mechanism needs the enumeration of P. The probability of a partition P being sampled is $$\frac{\exp(\frac{\epsilon\_p}{2\Delta Q} Q(P,G))} {\sum\_{P'\in \mathcal{P}} \exp(\frac{\epsilon\_p}{2\Delta Q} Q(P',G))}$$ However, ∣P∣ = ∑i = 1kS(∣A∣, i) where S(∣A∣, i) is the Stirling number of the second kind, $S(n,k) \approx \frac{k^n}{k!}$. This sum is exponential in ∣A∣, so enumerating P is computationally infeasible. Fortunately, MCMC can help us simulate the exponential mechanism by a sequence of local transitions in P. The space P is connected. It is straightforward to verify that the transitions performed in line 3 of ModMCMC are reversible and ergodic (i.e. any pair of nodeset partitions can be connected by a sequence of such transitions). Hence, ModMCMC has a unique stationary distribution in equilibrium. By empirical evaluation, we observe that ModMCMC converges after K∣r.S∣ steps for K = 50 (see Section [subsec:eval-div]). Each node r in the tree is of type NodeSet. This struct consists of an array r.part where r.part[u] ∈ {0..k − 1} is the group id of u. To make sure that ModMCMC runs in linear time, we must have a constant time computation of modularity Q(P) (line 4 of Algorithm [algo-modmcmc]). This can be done with two helper arrays: the number of intra-edges r.lc[0..k − 1] and the total degree of nodes r.dc[0..k − 1] in each group. The modularity Q is computed in O(k) (Formula [eqn:modularity]) using r.lc, r.dc. When moving node u from group i to group j, r.lc and r.dc are updated accordingly by checking the neighbors of u in G. The average degree is a constant, so the complexity of ModMCMC is linear in the number of MCMC steps. [algo-modmcmc] [1] graph G, nodeset r.S, group size k, privacy budget εp sampled partition r.part initialize r.part with a random partition P0 of k groups pick a neighboring partition Pʹ of Pi − 1 by randomly selecting node u ∈ r.S and moving u to another group. accept the transition and set Pi = Pʹ with probability $min(1,\frac{\exp(\frac{\epsilon\_p}{2\Delta Q} Q(P',G))} {\exp(\frac{\epsilon\_p}{2\Delta Q} Q(P\_{i-1},G))})$ // until equilibrium is reached a sampled partition r.part = Pi *Finding the Best Cut* Given the output k-ary tree R with the root node root, our next step is to find the best cut across the tree. A cut C is a set of nodes in R that cover all nodes in V. As an example, a cut C in Fig.[fig:mod-divisive] returns the clustering [{0,1,2,3}, {4}, {5,6,11}, {8,9,10,7,12}]. Any cut has a modularity score. Our goal is to find the best cut, i.e. the cut with highest modularity, in a private manner. We solve this problem by a dynamic programming technique. Remember that modularity is an additive quantity (c.f. Formula [eqn:modularity]). By denoting opt(r) as the optimal modularity for the subtree rooted at node r, the optimal value is opt(root). The recurrence relation is straightforward $$opt(r) = \max \{Q(r), \sum\_{t \in r.children} opt(t)\} \nonumber$$ Algorithm [algo-bestcut] realizes this idea in three steps. The first step (lines 1-6) uses a queue to fill a stack S. The stack ensures any internal node to be considered after its child nodes. The second step (lines 7-17) solves the recurrence relation. Because all modularity values are sensitive, we add Laplace noise Laplace(ΔQ/εm). The global sensitivity ΔQ = 3/m (see Theorem [theorem:delta-Q]), so we need only a small privacy budget for each level (εm = 0.01 is enough in our experiments). The noisy modularity modn is used to decide whether the optimal modularity at node r is by itself or by the sum over its children. The final step (lines 18-25) backtracks the best cut from the root node. [algo-bestcut] [1] undirected graph G, root node root, privacy budget at each level εm best cut C stack $S \leftarrow \varnothing$, queue Q ← root r ← Q.dequeue() S.push(r) Q.enqueue(ri) dictionary $sol \leftarrow \varnothing$ r ← S.pop(), r.modn = r.mod + Laplace(ΔQ/εm) sol.put(r.id, (val = r.modn, self = True)) sm = ∑ri ∈ r.childrensol[ri.id].modn sol.put(r.id, (val = sm, self = False)) sol.put(r.id, (val = r.modn, self = True)) list $C \leftarrow \varnothing$, queue Q ← root r ← Q.dequeue() C = C ∪ {r} Q.enqueue(ri) C ### Complexity ModDivisive creates a k-ary tree of height maxL. At each node r of the tree other than the leaf nodes, ModMCMC is run once. The run time of ModMCMC is O(K \* ∣r.S∣) thanks to the constant time for updating the modularity (line 4 of ModMCMC). Because the union of nodesets at one level is V, the total runtime is O(K \* ∣V∣ \* maxL). BestCut only incurs a sublinear runtime because the size of tree is always much smaller than ∣V∣. The following theorem states this result The time complexity of ModDivisive is linear in the number of nodes n, the maximum level maxL and the burn-in factor K. ### Privacy Analysis We show that ModMCMC satisfies differential privacy. The goal of MCMC is to draw a random sample from the desired distribution. Similarly, exponential mechanism is also a method to sample an output x ∈ X from the target distribution with probability proportional to $\exp(\frac{\epsilon u(x)}{2\Delta u})$ where u(x) is the score function (x with higher score has bigger chance to be sampled) and Δu is its sensitivity. The idea of using MCMC to realize exponential mechanism is first proposed in and applied to ε-DP graph release in. In our ModMCMC, the modularity Q(P, G) is used directly as the score function. We need to quantify the global sensitivity of Q. From Section [sec:diffpriv], we have the following definition (Global Sensitivity ΔQ) [def:sensitivity] ΔQ = maxP, G, Gʹ∣Q(P, G) − Q(P, Gʹ)∣ We prove that ΔQ = O(1/m) in the following theorem [theorem:delta-Q] The global sensitivity of modularity, ΔQ, is smaller than $\frac{3}{m}$ [proof:delta-Q] Given the graph G and a partition P of a nodeset Vp ⊆ V (for any node of the k-ary tree other than the root node, its nodeset Vp is a strict subset of V), the neighboring graph Gʹ has EGʹ = EG ∪ e. We have two cases Case 1. The new edge e is an intra-edge within the community s. The modularity Q(P, G) is $\sum\_{c}^{k} (\frac{l\_c}{m} - \frac{d\_c^2}{4m^2})$. The modularity Q(P, Gʹ) is $\sum\_{c\neq s}^{k} (\frac{l\_c}{m+1} - \frac{d\_c^2}{4(m+1)^2}) + (\frac{l\_s+1}{m+1} - \frac{(d\_s+2)^2}{4(m+1)^2})$. The difference $d\_1 = Q(P,G') - Q(P,G) = \frac{1}{m+1} -\frac{1}{m(m+1)} \sum\_{c}^{k}l\_c + \frac{2m+1}{4m^2(m+1)^2} \sum\_{c}^{k}d\_c^2 - \frac{d\_s+1}{(m+1)^2}$ Because ΔQ is the absolute value of d1, we consider the most positive and the most negative values of d1. Remember that ∑ckdc ≤ 2m, so the positive bound $d\_1 < \frac{1}{m+1} + \frac{(2m+1)4m^2}{4m^2(m+1)^2} < \frac{3}{m+1}$. For the negative bound, we use the constraints ∑cklc ≤ m and ds ≤ 2m, so $d\_1 > \frac{1}{m+1} - \frac{m}{m(m+1)} - \frac{2m+1}{(m+1)^2} > \frac{-2}{m+1}$. As a result, $\Delta Q = \|d\_1\| < \frac{3}{m+1} < \frac{3}{m}$. Case 2. The new edge e is an inter-edge between the communities s and t. Similarly, we have $Q(P,G) = \sum\_{c}^{k} (\frac{l\_c}{m} - \frac{d\_c^2}{4m^2})$ while $Q(P,G') = \sum\_{c\neq s,t}^{k} (\frac{l\_c}{m+1} - \frac{d\_c^2}{4(m+1)^2}) + (\frac{l\_s}{m+1} - \frac{(d\_s+1)^2}{4(m+1)^2}) + (\frac{l\_t}{m+1} - \frac{(d\_t+1)^2}{4(m+1)^2})$. The difference $d\_2 = Q(P,G') - Q(P,G) = -\frac{1}{m(m+1)} \sum\_{c}^{k}l\_c + \frac{2m+1}{4m^2(m+1)^2} \sum\_{c}^{k}d\_c^2 - \frac{2d\_s+2d\_t+2}{4(m+1)^2}$. Again, we consider the most positive and the most negative values of d2, using the constraint ∑ckdc ≤ 2m, the positive bound $d\_2 < \frac{(2m+1)4m^2}{4m^2(m+1)^2} < \frac{2}{m+1}$. For the negative bound, we use the constraints ∑cklc ≤ m and ds + dt ≤ 2m, so $d\_2 > - \frac{m}{m(m+1)} - \frac{4m+2}{4(m+1)^2} > \frac{-2}{m+1}$. As a result, $\Delta Q = \|d\_2\| < \frac{2}{m+1} < \frac{2}{m}$. To recap, in both cases $\Delta Q < \frac{3}{m}$. A Variant of HRG-MCMC --------------------- Similar to DER and EdgeFlip, HRG-MCMC runs in quadratic time due to its costly MCMC steps. Each MCMC step in HRG-MCMC takes O(n) to update the tree. To make it runnable on million-scale graphs, we describe briefly a variant of HRG-MCMC called HRG-Fixed (see Fig.[fig:hrg-fixed]). Instead of exploring the whole space of HRG trees as in HRG-MCMC, HRG-Fixed selects a fixed binary tree beforehand. We choose a balanced tree in our HRG-Fixed. Then HRG-Fixed realizes the exponential mechanism by sampling a permutation of n leaf nodes (note that each leaf node represents a graph node). The next permutation is constructed from the current one by randomly choosing a pair of nodes and swap them. The bottom of Fig.[fig:hrg-fixed] illustrates the swap of two nodes a and d. The log-likelihood L still has the sensitivity ΔL = 2lnn as in HRG-MCMC. Each sampling (MCMC) operation is designed to run in O(d̄.logn). By running K.n MCMC steps, the total runtime of HRG-Fixed is O(K.m.logn), feasible on large graphs. The burn-in factor K of HRG-Fixed is set to 1,000 in our evaluation. [fig:hrg-fixed] Experiments and Results ======================= In this section, our evaluation aims to compare the performance of the competitors by clustering quality and efficiency. The clustering quality is measured by the modularity Q and the average F1-score in which the modularity is the most important metric as we aim at highly modular clusterings. The efficiency is measured by the running time. All algorithms are implemented in Java and run on a desktop PC with $Intel^{\circledR}$ Core i7-4770@ 3.4Ghz, 16GB memory. Two medium-sized and three large real graphs are used in our experiments [1](#fn1). `as20graph` is the graph of routers comprising the internet. `ca-AstroPh` and `dblp` are co-authorship networks where two authors are connected if they publish at least one paper together. `amazon` is a product co-purchasing network where the graph contains an undirected edge from i to j if a product i is frequently co-purchased with product j. `youtube` is a video-sharing web site that includes a social network. Table [tab:dataset] shows the characteristics of the graphs. The columns Com(munities) and Mod(ularity) are the output of Louvain method. The number of samples in each test case is 20. [htb] [tab:dataset] \| \| Nodes \| Edges \| Com \| Mod \| \| --- \| --- \| --- \| --- \| --- \| \| as20graph \| 6,474 \| 12,572 \| 30 \| 0.623 \| \| ca-AstroPh \| 17,903 \| 196,972 \| 37 \| 0.624 \| \| amazon \| 334,863 \| 925,872 \| 257 \| 0.926 \| \| dblp \| 317,080 \| 1,049,866 \| 375 \| 0.818 \| \| youtube \| 1,134,890 \| 2,987,624 \| 13,485 \| 0.710 \| The schemes are abbreviated as 1K-series (1K), EdgeFlip (EF), Top-m-Filter (TmF), DER, LouvainDP (LDP), ModDivisive (MD), HRG-MCMC and HRG-Fixed. Quality Metrics --------------- Apart from modularity Q, we use $\bar{F\_1}$, the average F1-score, following the benchmarks in. The F1 score of a set A with respect to a set B is defined as the harmonic mean H of the precision and the recall of A against B. We define $prec(A,B) = \frac{\|A \cap B\|}{\|A\|}$, $recall(A,B) = \frac{\|A \cap B\|}{\|B\|}$ $$F\_1(A,B) = \frac{2.prec(A,B).recall(A,B)}{prec(A,B) + recall(A,B)} \nonumber$$ Then the average F1 score of two sets of communities C and Cʹ is defined as $$\begin{gathered} F\_1(A,C) = \max\_{i} F\_1(A,c\_i), \;\; c\_i \in C = \{c\_1,..,c\_n\} \\ \nonumber \bar{F\_1}(C,C') = \frac{1}{2\|C\|} \sum\_{c\_i\in C} F\_1(c\_i,C') + \frac{1}{2\|C'\|} \sum\_{c\_i\in C'} F\_1(c\_i,C) \nonumber\end{gathered}$$ We choose the output clustering of Louvain method as the ground truth for two reasons. First, the evaluation on the real ground truth is already done in and Louvain method is proven to provide high quality communities. Second, the real ground truth is a set of overlap communities whereas the schemes in this paper output only non-overlap communities. The chosen values of ε are {0.1lnn,0.2lnn,0.3lnn,0.4lnn,0.5lnn}. Performance of LouvainDP ------------------------ We test LouvainDP for the group size k ∈ {4, 8, 16, 32, 64}. The results on `youtube` are displayed in Fig. [fig:louvaindp]. We observe a clear separation of two groups k = 4, 8 and k = 16, 32, 64. As k increases, the modularity increases faster but also saturates sooner. Similar separations appear in avg.F1Score and the number of communities. Non-trivial modularity scores in several settings of (k, ε) indicate that randomly grouping of nodes and high-pass filtering superedges do not destroy all community structure of the original graph. At ε = 0.5lnn and k = 4, 8, the total edge weight in G1 is very low ( < 0.05m), so many supernodes of G1 are disconnected and Louvain method outputs a large number of communities (Fig. [fig:louvaindp-youtube-com]). The reason is that the threshold θ in LouvainDP is an integral value, so causing abnormal leaps in the total edge weight of G1. We pick k = 8, 64 for the comparative evaluation (Section [subsec:eval-comp]). [b]0.32 [fig:louvaindp-youtube-mod] [b]0.32![](louvaindp-youtube-f1 "fig:") [fig:louvaindp-youtube-f1] [b]0.32![](louvaindp-youtube-com "fig:") [fig:louvaindp-youtube-com] [fig:louvaindp] Performance of ModDivisive -------------------------- The effectiveness of ModDivisive is illustrated in Fig. [fig:moddivisive] for graph `youtube` and λ = 2.0, K = 50. We select six pairs of (k, maxL) by the set {(2,10),(3,7),(4,5), (5,4),(6,4),(10,3)}. Modularity increases steadily with ε while it is not always the case for avg.F1Score. The number of communities in the best cut is shown in Fig. [fig:moddivisive-youtube-com]. Clearly, the small number of communities indicates that ModDivisive’s best cut is not far from the root. The reason is the use of λ = 2.0, i.e. half of privacy budget is reserved to the first level, the half of the rest for the second level and so on. Lower levels receive geometrically smaller privacy budgets, so their partitions get poorer results. [h]0.32 [fig:moddivisive-youtube-mod] [h]0.32![](moddivisive-youtube-f1 "fig:") [fig:moddivisive-youtube-f1] [h]0.32![](moddivisive-youtube-com "fig:") [fig:moddivisive-youtube-com] [fig:moddivisive] We choose λ = 2.0 to obtain a good allocation of ε among the levels. Fig. [fig:ratio-amazon] shows the modularity for different values of λ. Note that λ = 1.0 means ε is equally allocated to the maxL levels. By building a k-ary tree, we reduce considerably the size of the state space P for MCMC. As a result, we need only a small burn-in factor K. Looking at Fig. [fig:burnfactor-amazon], we see that larger K = 100 results in only tiny increase of modularity in comparison with that of K = 50. [b]0.22![image](ratio-amazon) [fig:ratio-amazon] [b]0.22![image](burnfactor-amazon) [fig:burnfactor-amazon] [fig:moddiv-ratio-burnfactor] Comparative Evaluation ---------------------- In this section, we report a comparative evaluation of LouvainDP and ModDivisive against the competitors in Figures [fig:all-as20graph], [fig:all-caAstroPh], [fig:all-amazon], [fig:all-dblp] and [fig:all-youtube]. The dashed lines in subfigures [fig:all-as20graph-com], [fig:all-caAstroPh-com], [fig:all-amazon-com], [fig:all-dblp-com] and [fig:all-youtube-com] represent the ground-truth number of communities by Louvain method. ModDivisive performs best in most of the cases. On `as20graph` and `ca-AstroPh`, HRG-MCMC outputs the whole nodeset V with the zero modularity while 1K-series, TmF, DER also give useless clusterings. EdgeFlip produces good quality metrics exclusively on `ca-AstroPh` while LouvainDP returns the highest modularity scores on `as20graph`. However, the inherent quadratic complexity of EdgeFlip makes Louvain method fail at ε = 0.1lnn and 0.2lnn for `ca-AstroPh` graph. On the three large graphs, ModDivisive dominates the other schemes by a large margin in modularity and avg.F1Score. LouvainDP is the second best in modularity at k = 64. Our proposed HRG-Fixed is consistent with ε and has good performance on `dblp` and `youtube`. Note that HRG-MCMC is infeasible on the three large graphs due to its quadratic complexity. Again, 1K-series, TmF and EdgeFlipShrink provide the worst quality scores with the exception of 1K-series’s avg.F1Score on `youtube`. The runtime of the linear schemes is reported in Fig. [fig:runtime]. EdgeFlipShrink, 1K-series, TmF and LouvainDP benefit greatly by running Louvain method on the noisy output graph G̃. ModDivisive and HRG-Fixed also finish their work quickly in O(K.n.maxL) and O(K.m.logn) respectively. [b]0.22![image](runtime-all) [fig:runtime-all] [b]0.22![image](runtime-moddiv-amazon) [fig:runtime-moddiv-amazon] [fig:runtime] Conclusion ========== We have given a big picture of the problem ε-DP community detection within the two categories: input and algorithm perturbation. We analyzed the major challenges of community detection under differential privacy. We explained why techniques borrowed from k-Means fail and how the difficulty of ε-DP recommender systems enables a relaxation of privacy budget. We proposed LouvainDP and ModDivisive as the representatives of input and algorithm perturbations respectively. By conducting a comprehensive evaluation, we revealed the advantages of our methods. ModDivisive steadily gives the best modularity and avg.F1Score on large graphs while LouvainDP outperforms the remaining input perturbation competitors in certain settings. HRG-MCMC/HRG-Fixed give low modularity clusterings, indicating the limitation of the HRG model in divisive CD. The input perturbation schemes DER, EF, 1K-series and TmF hardly deliver any good node clustering except EF on the two medium-sized graphs. For future work, we plan to develop an ε-DP agglomerative scheme based on Louvain method and extend our work for directed graphs and overlapping community detection under differential privacy. [t]0.32![](all-as20graph-mod-3 "fig:") [fig:all-as20graph-mod] [t]0.32![](all-as20graph-f1-3 "fig:") [fig:all-as20graph-f1] [t]0.32![](all-as20graph-com-3 "fig:") [fig:all-as20graph-com] [fig:all-as20graph] [t]0.32![](all-caAstroPh-mod-3 "fig:") [fig:all-caAstroPh-mod] [t]0.32![](all-caAstroPh-f1-3 "fig:") [fig:all-caAstroPh-f1] [t]0.32![](all-caAstroPh-com-3 "fig:") [fig:all-caAstroPh-com] [fig:all-caAstroPh] [t]0.32![](all-amazon-mod-3 "fig:") [fig:all-amazon-mod] [t]0.32![](all-amazon-f1-3 "fig:") [fig:all-amazon-f1] [t]0.32![](all-amazon-com-3 "fig:") [fig:all-amazon-com] [fig:all-amazon] [t]0.32![](all-dblp-mod-3 "fig:") [fig:all-dblp-mod] [t]0.32![](all-dblp-f1-3 "fig:") [fig:all-dblp-f1] [t]0.32![](all-dblp-com-3 "fig:") [fig:all-dblp-com] [fig:all-dblp] [t]0.32![](all-youtube-mod-3 "fig:") [fig:all-youtube-mod] [t]0.32![](all-youtube-f1-3 "fig:") [fig:all-youtube-f1] [t]0.32![](all-youtube-com-3 "fig:") [fig:all-youtube-com] [fig:all-youtube] --- 1. http://snap.stanford.edu/data/index.html[↩](#fnref1)	arxiv_0000383
All the Way There and Back: Inertial-Based, Phone-in-Pocket Indoor Wayfinding and Backtracking Apps for Blind Travelers ======================================================================================================================= We introduce two iOS apps that have been designed to support wayfinding and backtracking for blind travelers navigating in indoor building environments. Wayfinding involves determining and following a route through the building’s corridors to reach a destination, and assumes that the app has access to the floor plan of the building. Backtracking one’s route, on the other hand, requires no map knowledge. Our apps only use the inertial and magnetic sensors of the smartphone, and thus require no infrastructure modification (e.g., installation and support of BLE beacons). Unlike systems that use the phone’s camera, users of our apps can conveniently keep their phone tucked inside a pocket, while interacting with the apps using a smartwatch. Routing directions are given via speech. Both apps were tested in a user study with seven blind participants, who used them while navigating a campus building. <ccs2012> <concept> <conceptid>10010520.10010553.10010562</conceptid> <conceptdesc>Human-centered computing Accessibility technologies </conceptdesc> <conceptsignificance>600</conceptsignificance> </concept> <conceptdesc>Human-centered computing Human-centered computing Interaction devices </conceptdesc> <conceptsignificance>400</conceptsignificance> </concept> </ccs2012> Introduction ============ Navigating a building can be confusing and disorienting for anyone who is visiting the building for the first time. For those who are blind, this experience can be even more challenging, as these travelers cannot rely on visual feedback. While multiple research efforts have focused on accessible technology to support independent travel in indoor space, at the time of this writing there are no commercially available systems ready for widespread use. Indoor spaces are challenging because GPS cannot be relied upon. This doesn’t mean that indoor localization is not feasible: a variety of techniques have been demonstrated, some of which have been adapted for navigation by blind individuals. One example is NavCog in its various versions , which uses Bluetooth Low Energy (BLE) for accurate localization. BLE localization is a mature technology. Unfortunately, BLE-based navigation is only possible if a specific infrastructure (a possibly large set of BLE beacons) has been installed, and a laborious calibration procedure (fingerprinting) has been conducted. This raises questions of scalability, as the success of this technology hinges on the good will of the agency that manages the building. It would thus be desirable that the technology powering the navigational aid would not depend on external dedicated infrastructure. In recent years, and with the advent of powerful AI, there has been intense interest in systems that use visual sensors (e.g., the camera embedded in a smartphone) to extract positional data, and to provide users with information about the world around them. For example, navigation apps built on Apple’s ARKit have been developed specifically for use by blind travelers . Visual-inertial odometry technology, coupled with Particle Filtering, can produce extremely accurate localization. Its core strength, the use of visual data, is unfortunately also its main drawback. Users of these apps must hold the smartphone (in their hand, or perhaps attached to a lanyard ) in such a way that the camera has a good view of the environment. This is not always possible, convenient, or desirable. Blind travelers normally use a long cane or a dog guide, and thus have one hand already occupied maneuvering the cane or holding the dog. Indeed, it has been often observed that navigational aid for blind walkers should be hands-free . In this article, we describe an experiment we conducted with 7 blind travelers, who tested two apps created in our lab that were designed to assist with navigation in buildings characterized by a network of corridors. Both apps utilize data from the inertial sensors of a smartphone, and thus require no external infrastructure. Since the apps don’t use data from the smartphone camera, users can conveniently keep the phone tucked in a pocket (indeed, this is how the apps have been tested in our experiment). The first one is a Wayfinding app: it has knowledge of the floor plan of the building to be navigated, computes the shortest length route to destination, and provides navigational support in an accessible way. The second one is a Backtracking app. Its sole purpose is to help an individual, who has previously walked on a certain route (e.g., from the front entrance of a building to a certain office room), to trace back their steps. While arguably less useful than a full blown wayfinding application, backtracking support can help a blind traveler feel more confident in some situations. Importantly, the Backtracking app requires no prior knowledge of the building layout: it only uses data recorded during the first route traversal (way-in) to generate support for the user when walking back (return). Both apps have provision for error recovery (i.e., can provide corrective directions if a participant missed a turn, or took the wrong turn). These are the main contributions of this work: * We demonstrate an accessible iPhone app (Wayfinding) that uses two different localization technologies, complemented by Particle Filtering, for inertial-based localization in a building with known floor plan. * We propose a new hybrid approach for backtracking that uses magnetic and inertial data, with no knowledge of the building layout (and thus usable in any venue even without access to a floor plan). This algorithm was implemented and tested in the Backtracking app. * We demonstrate the effectiveness of a simple speech-based user interface mechanism, consistent for both apps, designed to mitigate the unavoidable inaccuracy associated with the dead-reckoning nature of inertial-based localization. * We describe a user study with 7 blind travelers. The participants first traversed three routes (292 meters and 13 turns in total) using our Wayfinding app; then, they traversed the same routes, in the opposite direction, using the Backtracking app. This article is structured as follows. After reviewing the related work in Sec. [sec:RW], we describe our Wayfinding and Backtracking apps (including their shared user interface mechanisms) in Sec. [sec:Apparatus]. Our experiment with seven blind participant is described in Sec. [sec:Experiment], and results, along with limitations of our approach, are discussed in Sec. [sec:Discussion]. Sec. [sec:Conclusions] has the conclusions. Related Work ============ Multiple research considered smartphone app implementations to enhance the mobility of individuals with visual impairments, especially in GPS-denied indoor environments. These apps leverage a variety of technology, including BLE beaconing, Wi-Fi, IMUs, video cameras, or sensor fusion, to offer navigational support . As an example, NavCog incorporates a precise localization algorithm based on BLE beacons, and offers customizable voice and non-vocal sound instructions for user interaction. Additionally, LuzDeploy , built upon NavCog, has adopted crowdsourcing technology to maintain the localization infrastructure. Subsequent versions integrated BLE and inertial data positioning, and Particle Filtering (NavCog3 ). A virtual navigation app (VirtualNav ) was built upon the foundation of NavCog3, to explore the transfer of virtually acquired route knowledge to practical navigation. Other projects using BLE-based localization include ASSIST) and GuideBeacon . Alternative technologies considered for localization include RFID tags (e.g., PERCEPT ), color tags (e.g., NaviLens ), Wi-Fi beaconing , and magnetic navigation . In recent work, visual-inertial odometry (VIO) has been shown to provide accurate localization without relying on dedicated infrastructure . Backtracking apps using similar localization technologies have also been developed . As mentioned earlier, these system require use of a camera with unoccluded visual field. Research on navigation using inertial sensing (the technology considered in this article) includes . Riehle et al. described a classic pedestrian dead-reckoning system (PDR) with explicit turn detection, which was tested with 8 blind or low vision participants in a simple indoor route. Participants could request assistance from the experimenter if they felt they needed it. Compared to a condition in which the route was described at the beginning, but no guidance was provided during the trial, it was shown that use of the PDR system resulted in fewer requests for assistance made and in more instances of successful localization of the route’s destination. Fallah et al. and Apostolopoulos et al proposed an inertial-based navigation system designed to overcome inherent localization inaccuracy through the “user as a sensor modality”. Routes were represented via a sequence of perceivable landmarks. Participants were asked to follow directions to the next landmark; once arrived at the landmark, they were asked to confirm it on the app. This ingenious strategy allowed the localization system to reset itself at the correct location at each detected landmark. The inertial system was based on step counting (through an accelerometer) and orientation sensing (through a compass). A Particle Filter was used to track the state (location and orientation) of the user. In addition, different methods were considered to estimate the user’s step length using the Particle Filter. An experiment with 6 blind and low vision participants was conducted on two floors of a building with 11 routes, most of which contained one or two turns. One route included climbing a staircase, while another route included taking an elevator. These routes contained multiple landmarks that needed to be discovered by the participants, including doors that needed to be counted, hallways, stairs, ramps. Peng et al. demonstrated an indoor/outdoor wayfinding system, designed for navigation in a transit hub, that tracked the user’s location through fusion of inertial and GPS data. Along with localization, research has focused on the design of accessible user interface modalities for navigational systems , using modalities such as speech , sound , and vibration . Apparatus ========= We developed two iOS apps for this study: Wayfinding and Backtracking. The Wayfinding app has access to a floor plan of the building to be navigated. It is designed to find a route from the current user’s location to the desired destination, update the route as necessary during traversal, and provide directions to the user as they are following the route. The Backtracking app has no prior information about the building layout. Similarly to, Backtracking operates in two phases. In the first phase (way-in), it simply tracks the route taken by the user to reach a destination. Then, in the return phase, it produces directions to help the user re-trace the same way-in route. To do so, the app progressively matches the partial return route with the way-in route (in reverse). As will be clear in the following, the Backtracking app needs to use different localization strategies than the Wayfinding app, as a consequence of its lack of building layout information. The user interface, however, was designed to be almost identical for the two apps, shielding the user from the different underlying mechanisms. Wayfinding ---------- ### Localization We implemented two different pedestrian dead-reckoning (PDR) algorithms for user localization and tracking using data from the phone’s inertial sensors. These two techniques are termed Azimuth/Steps and RoNIN, respectively . A Particle Filter is applied to the output of either system. During the tests, we ran both systems in parallel, though only one of them was used to provide guidance to the user. The reason for running both algorithms in parallel was twofold. First, we wanted to have a “fail safe” mechanism: if an algorithm failed to correctly track the participant, we could resort to switching to the other algorithm. Note that this only happened once in the whole experiment. Second, it allowed us to comparatively assess localization data from both algorithms in a variety of situations. Both algorithms compute the user’s locations in terms of an arbitrary (but fixed) world reference frame, with the Z axis pointing downwards (in the direction of gravity). We used a simple initial calibration procedure (described below) to find the angle of the rotation around the Z axis that brings the world reference frame to the frame used to define the floor plan (the floor plan frame). *Azimuth/Steps (A/S). This algorithm produces, at each detected step (heel strike), a 2-D vector Δp* with a length equal to the estimated step length, and direction given by the phone’s azimuth (or heading) angle. This vector is then fed as input to the Particle Filter (discussed later in this section). Heel strikes (steps) are computed by an LSTM-based algorithm that processes data from the phone’s inertial sensors . Each participant’s step length was estimated by the following calibration procedure. The participant was asked to walk along a straight corridor path (38.25 meters in length). By dividing the length of this path by the number of steps (from the step counter), we obtained the user’s approximate step length. A similar step length calibration was used in . Note that after calibration, the step length is further updated by the Particle Filter (in a similar fashion to ). In spite of its simplicity, A/S was shown to produce good path reconstruction results when coupled with a Particle Filter, which can mitigate the drift effect inherent in the dead-reckoning nature of the algorithm . One drawback of this approach is that the orientation of the vector Δp (azimuth) is that of the phone, rather than that of the walker. This means that moving the phone to a different location on one’s body (e.g., pulling it out of the pocket to take a call) may incorrectly be interpreted as a change in walking direction. In our experiment, participants kept the phone tucked inside their back pocket throughout the trials, hence the orientation of the phone with respect to the user’s body could be considered to be constant. *RoNIN. RoNIN is a machine learning algorithm for dead-reckoning from inertial data. Its feasibility for the reconstruction of paths taken by blind walkers, using a long cane or a dog guide, was demonstrated in . RoNIN produces velocity vectors at a rate of 25 Hz. We used the authors’ open-source ResNet18 implementation[1](#fn1), and integrated the velocity vectors over individual step periods to obtain displacement vectors Δp. Unlike the A/S algorithm, RoNIN produces displacement vectors that are defined with respect to the world frame (rather than the phone’s frame). This means that the heading direction estimated by RoNIN is independent of the phone’s orientation with respect to one’s body . Hence, the user is not constrained to hold the phone in a fixed location. However, RoNIN is as liable as A/S to orientation drift due to the integration of noisy sensor data. Experiments reported in showed that while RoNIN worked remarkably well for some individuals, it produced less satisfactory results for others. In particular, the magnitude of the velocity vector returned in output for certain users was found to be either smaller or larger than their actual speed. To account for this user-dependent error, we employed the same calibration procedure described above to find an adjustment coefficient (RoNIN multiplier) for each participant. In practice, we divided the length of the path traversed during calibration by the estimation of the same path length produced by RoNIN. Then, during the trials with the same participants, we multiplied the velocity vectors produced by the RoNIN multiplier thus computed. Particle Filtering (PF). Particle Filtering is a form of Bayesian filtering that is commonly used for spatial tracking in the presence of prior information or constraints . In our case, knowledge of the floor plan allows us to define ``impenetrable walls" that are unlikely to be traversed by a traveler . Using a Particle Filter, the posterior distribution of the walker’s location is expressed by means of a set of samples (particles). In our experiments, we used 500 particles, which was found to be a reasonable trade-off between computational speed and accuracy of tracking. Each i-th particle is characterized by a X-Y location p*i, a drift angle value Δθ, i, and, for A/S localization, a step length value si. At the beginning of a trial, all particles are located at the (known) initial user location. The drift angles associated with the particles are sampled from a zero-mean Gaussian distribution with σ = 30∘, while the step lengths are sampled from a Gaussian distribution with σ = 6 cm centered at the step length found during the initial calibration. Each particle has a (positive) weight wi. Weights are all initialized to 1/500. At each time period (in our case, at each detected step), each particle is spatially propagated by a vector that is a function of the displacement vector Δp produced by A/S or by RoNIN. Specifically, Δp is first rotated by the particle’s drift angle Δθ, i. For A/S localization, its length is changed to si. The particle’s drift angle Δθ, i is updated by addition of Gaussian noise (σ = 1∘). Then, Gaussian noise is added to its modulus (σ = 0.5 ⋅ si) and phase (σ = 0.5∘). The particle’s location is then updated by adding the resulting vector to pi. Additionally, the particle weights can be modified during a positional update as described later in this section (after re-weighting, the weights are normalized to sum to 1.) At each time step, a new set of 500 particles is resampled with replacement from the current set of particles. This means that, for 500 times, a particle j in the current set is picked at random, with the probability of a particle being sampled equal to its weight. A new particle is generated with the same characteristics (location pj, angular drift Δθ, j, and, for A/S, step length sj). Gaussian noise (σ=10 cm) is added to the X and Y components of its location. The old particles are then discarded, and the new ones are retained until the next update. The user’s location is taken to be the weighted sum of the particles’ locations: p = ∑iwipi. Particles are re-weighted at each time period according to the following rules: 1. If a particle is found to be crossing a wall, its weight is set to 0. 2. If a particle is at a distance from the weighted average p larger than a threshold value D, its weight is set to 0. 3. If at any time a particle is found to be inside a room, its weight is multiplied by 0.9. \| \| \| \| --- \| --- \| \| image \| \| \| (a) \| (b) \| [fig:PF] Provision (1) reduces the risk of trajectories going through impenetrable walls. Compactness of the particle set is enforced by (2). This is particularly useful for large open spaces, including long corridors, where the particle set could otherwise expand boundlessly. The threshold values D for provision (2) were set (based on trial-and-error experiments) to 5.5 m for A/S localization and to 3.5 m for RoNIN. Provision (3) reduces the likelihood that a user walking in a corridor be mistakenly localized inside a room. It is important to note that all rooms in the building are considered “open” (see Fig. [fig:PF] (a)). Even though the selected routes are defined on corridors, walkers could indeed enter a room through an open door (as it happened with participant P7; see Fig. [fig:R3W] (d)), and should be tracked in those spaces as well. However, we noted that without provision (3), sizeable amounts of particles would often enter through the open doors (and get temporarily stuck in) one or more rooms when a participant was walking on a corridor nearby (Fig. [fig:PF] (a)). By reducing their weights, we decrease the likelihood that particles in these isolated clusters be selected for resampling. Resampling also ensures that only angular drift values (and, for A/S, step lengths) that lead to ``legitimate" trajectories are preserved. Explicitly modeling the drift angle was shown in to effectively mitigate the effect of accumulating angular drift, a well-known issue associated with dead-reckoning. For A/S localization, we also found that, through resampling, the particles’ step length values tended to coalesce towards a value that was often different from that measured during initial calibration, as discussed later in Sec. [sec:population]. In our iPhone implementation, Particle Filtering updates are performed each time a footstep is detected. An update normally requires between 0.01 to 0.02 seconds to complete. *Initial Orientation Calibration. For each trial in our experiment, we assumed that participants would start from a known location, and that they initially walked along a known direction. After 6 steps, we found the angle between the reconstructed trajectory (defined, as mentioned earlier, with respect to an arbitrary world reference frame) and the known walking direction, as defined in the floor plan frame. ### Routes and Waypoints The walkable area in a building characterized by a network of corridors can be represented by a set of waypoints* (located at the corridors’ junctions) and route segments, which join any two waypoints if there is a traversable straight path between the two. A route joining any two waypoints is a sequence of interconnected route segments. At each time t, the localization algorithm (whether A/S or RoNIN, followed by Particle Filtering) produces an estimated location p(t) of the user. Based on this location, the app issues notifications to the user as appropriate. However, rather than directly using the 2-D locations p, we consider the projected locations $\bar p(t)$ onto its associated route segments. This approach is justified by the nature of typical buildings with networks of corridors, and simplifies the logic used to produce notifications, which is described in Sec. [sec:notifications]. When users walk on a long corridor, far from junctions with other corridors, projecting their location onto the route segment associated with that corridor is a safe operation. More care must be taken when in the proximity of a junction, due to potential localization errors leading to incorrect assignments. For example, as shown in Fig. [fig:PF] (b) (see locations marked by blue arrows), associating the user’s location with the closest route segment may lead to incorrect results when the distance of the user to the junction is comparable with the radius of localization uncertainty. To deal with such situations, we adopted the following mechanism. A walker whose position is currently associated with a certain route segment, maintains this association until the projected location $\bar p$ is closer than a threshold distance T to a junction with other route segments (T was set to 1.5 m in our experiment). At that point, route segment association becomes ambiguous, and p is no longer associated with any segment. Segment association is resumed when the projection of p onto any of the route segments ending at that junction is at a distance of T (or larger) from the junction. From that point on, the walker is associated with this new segment (see Fig. [fig:PF] (b)). Our wayfinding app determines the shortest route to a destination using the iOS GameplayKit toolkit[2](#fn2). Routes are constantly updated as the user moves, and re-routing is computed as necessary. In addition to waypoints, we defined a number of “landmarks”, whose presence was communicated to the walker when nearby. These landmarks, whose location is shown in Figs. [fig:R1W]– [fig:R3W] (a), are listed in Tab. [tab:landmarks]. Backtracking ------------ As mentioned earlier, our Backtracking app is structured in two distinct phases. In the way-in phase, the app is in charge of recording the route taken by the user (who may be accompanied by a sighted guide, or perhaps may be exploring the environment on their own). During the return phase, the app is in charge of tracking the path of the walker and matching it with the way-in path. In particular, the system needs to determine when the walker is getting close to a location where a turn was taken during the way-in, in which case it can produce an appropriate notification. ### Path Reconstruction: Way-in Unlike the Wayfinding app, impenetrable wall constraints cannot be used to correct for orientation drift, because the Backtracking app has no knowledge of the floor map of the building. In buildings characterized by networks of corridors, however, it is conceivable that walkers would proceed along relatively straight paths until they turn at a corridor junction. The angle made by two intersecting corridors, for typical buildings, is often equal to ± 90∘ or to a multiple of 45∘. This geometric structural constraint can be leveraged to sidestep orientation drift. Following , we represent both way-in and return paths as a sequence of straight segments interleaved with discrete angle turns. We use the robust turn detector described in , which processes azimuth information using a Mixture Kalman Filter . Although this algorithm was shown to work well even for multiples of 45∘ turns, we constrained detection to multiples of 90∘ for the purpose of this experiment (this reflects the type of junctions found in the building considered for our tests, which were all of ± 90∘). The way-in path can be depicted as a 2-D polygonal chain (polyline), where the length of each segment is equal to the number of steps taken in that segment, and consecutive segments have an angle as measured by the turn detector (see e.g. Fig. [fig:projection]). Steps are detected using the same LSTM network used for A/S localization. ### Path Matching: Return Our strategy for matching the return path with the way-in path is based on the coordination of two different algorithms: projected return sequence and sequence alignment. Projected Return Sequence. This algorithm builds a 2-D trajectory as a polyline, as described above for the way-in path. This simple algorithm works reasonably well in “ideal” conditions but fails if (1) the walker’s step length is different between way-in and return (leading to a different number of steps for the same segment; see Fig. [fig:projection] (a)); (2) the turn detector fails to recognize a turn (e.g., because taken too slowly) or produces a false positive (due to an irregular trajectory); or (3) the walker does not follow exactly the same route (in reverse) as during way-in, e.g. because they missed a turn then turned around. All of these situations may be expected, especially when someone walks without visual feedback. \| \| \| \| --- \| --- \| \| \| \| \| (a) \| (b) \| [fig:projection] Sequence Alignment. As described in , one may cast the path matching problem as one of (sub)sequence alignment. Assume that the sequence of way-in measurements has been reversed, which is convenient since the route is being backtracked. At each time during return, we determine the initial way-in subsequence of measurements that best matches the current sequence of return measurements. In symbols: given the (reversed) way-in sequence of measurements (observations) W = (ow(1), …, ow(N)), and the current sequence of return measurements R = (or(1), …, or(J)), the goal is to find a sequence of indices i1, …, iJ such that (ow(i1), …, ow(iJ)) best matches R under an appropriate criterion. For real-time guidance, we are interested in the last matching point iJ: we will assume that the walker at the current return time index J is in the same location they were at time index iJ during way-in. Standard dynamic programming approaches (e.g. Dynamic Time Warping ) can then be used to find an optimal match. The measurements we consider are magnetic field vectors and turns detected. Step detection is also considered implicitly: for both way-in and return, the sequences of time indices are defined such that there are three regularly spaced time intervals between two consecutive detected steps. We found that this choice gives enough spatial granularity for magnetic field matching, while ensuring parsimonious sampling (e.g., no samples are recorded when the user is stationary). It is well known that the magnetic field recorded in different positions within a building is not uniform, due to reasons such as the presence of large metallic objects and magnetic field generating appliances. “Magnetic signatures” can thus be attached to specific locations, enabling sophisticated localization mechanisms (e.g., IndoorAtlas’ magnetic positioning[3](#fn3).) The magnetic field at a certain location can be measured by a 3-axis magnetometer, of the type embedded in any modern smartphone. (Prior calibration of the magnetometers is necessary for good results.) From the measured 3-D magnetic field vector we derive a 2-D vector that is invariant to the orientation of the phone as described in . Given the measurements, one can create a directed graph with nodes indexed as (i, j), where i is a way-in time index, and j is a return time index. (Note that the graph is constantly expanded as the walker progresses along the return path.) A node (i, j) in the graph has only three edges, to (i + 1, j), (i + 1, j + 1), and (i, j + 1), respectively. This is consistent with the assumption that the walker is normally moving in the same direction as in the (reversed) way-in path, but possibly with a different step length (resulting in steps detected in either phase that cannot be matched in the other phase, which are accounted for by the edges to (i + 1, j) and (i, j + 1) ). The edges to (i + 1, j) and (i, j + 1) carry a non-null cost, while node costs are defined as a function of the magnetic field vector discrepancy (measured as the Euclidean difference between the recorded vectors) between the measurements ow(i) and or(j). Sequence alignment is obtained by finding the minimum cost path starting from (0, 0) and arriving at a node (i, J). Unfortunately, this simple approach, originally proposed in , did not produce satisfactory results in our preliminary tests, especially if the return path is not perfectly identical to the way-in path. Indeed, we found that, in some cases, the magnetic field can vary rather dramatically when moving across the width of a building corridor. An improvement to pure magnetic-based alignment can be obtained by considering the turns taken by the walker, which, in an ideal case, should be matched between way-in and return. We use the mechanism described in , which considers the difference in walking direction between way-in and return by structuring the graph as a sequence of layered planar graphs, where each layer represents a possible orientation discrepancy between way-in and return. The minimum cost path in the graph is recomputed at each new return sample. We use the incremental Dynamic Time Warping (iDTW) proposed by Riehle et al. , which uses a sliding window defined around the endpoint of the previously found optimal path (we set the window size equal to 200 samples). Although this algorithm produces a suboptimal solution, it represents a good compromise between precision and computational cost. Hybrid Matching Strategy. The sequence alignment method described above was shown to work well even for moderately complex situations. However, since the optimal path is recomputed at each iteration, the estimated user location at the current time may become “jittery”. This may cause instability when producing guidance notifications, complicating the task of deciding when exactly to notify the walker of an upcoming turn. We propose a simple solution to this problem, based on the notion of last reliable match. We use local properties of the current minimum cost graph path to decide, at the current time J, whether the match (iJ, J) can be considered “reliable”, meaning that it is likely to be preserved even after later observations are recorded. In practice, we look at the terminal part (the last 21 samples) of the minimum cost path in the graph ending at (iJ, J), and judge this last match to be reliable if this path segment is well approximated by a line with unitary slope. This simple algorithm was shown to work well in most cases. When a reliable match is detected, a projected return sequence is initialized and continuously updated, as discussed earlier, until a new reliable match is found. This projected sequence starts from the last reliable match, with an initial direction equal to the current walker orientation (as defined by the sequence of turns taken during return). Guidance notifications are produced based on the walker’s location identified using this projected sequence. As soon as a new reliable match is detected, the projected sequence is re-initialized at that match (see Fig. [fig:projection] (b)). Examples of reliable matches and projected paths are shown later in Sec. [sec:TGPB]. ### Way-in Path Simplification In some cases, the way-in path contains redundant turns or loops. This may happen when, for example, if the walker took a short detour (perhaps because they were unclear about the route to take), or if they followed a zig-zag course instead of walking straight. It would be desirable to remove these redundant features before backtracking. In our experiment, we implemented a very simple “path simplification” algorithm for the way-in path. In short, we define a conservative radius of uncertainty D equal to 7 steps. We then merge together nearby parallel edges that are at a distance of D or less from each other. We do the same with turn locations (vertices). Then, we re-create a simplified path as a shortest length polyline that goes through the remaining edges and vertices. An example of successful way-in path simplification is shown in Fig. [fig:optimization] (a). Note that the original path (shown to the left) contains a short detour with a 180∘ turn, which was due to the walker originally missing a turn. This spurious piece was correctly removed by the simplification algorithm. Fig. [fig:optimization] (b) shows a case with a complex way-in path, containing multiple loops. Although our algorithm was able to remove these loops, and the resulting simplified path maintained the correct path geometry, its first segment turned out to be substantially shorter than in the original path, which led to an unsuccessful backtracking trial. \| \| \| --- \| \| \| \| (a) \| \| \| \| (b) \| [fig:optimization] User Interface -------------- We list in the following the general principles that guided our interface design for the two apps. *1. Consistency.* Even though the two apps utilize different localization technologies, the user interface is almost identical between the two. Both apps ultimately provide directions towards a destination in similar environments (networks of corridors), thus it is only natural that they should afford similar user experiences. The small differences (highlighted in Sec. [sec:notifications]) are a consequence of the different prior knowledge the apps can rely on. *2. Robustness to localization inaccuracy.* The localization system of both apps relies on dead reckoning from a smartphone’s inertial sensors. In spite of mitigation techniques (Particle Filtering for Wayfinding, sequence alignment for Backtracking), localization errors of up to 2-3 meters should be expected. Neglecting these potential errors when issuing notifications may lead to catastrophic results. For example, suppose that the app issues a notification shortly before it locates the walker at a junction. As shown in Fig. [fig:inaccuracies] (a), the walker’s real position may happen to be at some distance before or after the junction. In either case, if the walker were to turn, they would face a wall. What’s worse, they would have no way to know whether to search for the junction to their right or to their left. To account for the expected localization inaccuracy, we made the following design choice: the presence of a junction where a turn should be taken is announced with substantial advance notice, enough to ensure that, with high likelihood, the walker has not yet reached the junction. Although conceptually very simple, it was not obvious a priori that this approach would be viable and acceptable. The notification is given at a variable distance from the junction (depending on the current localization error). Users are thus in charge of searching for the next available opening upon hearing the notification, which involves active exploration with the long cane or with the dog guide. An important goal of our experiments was to verify whether our participants would be successful in this task, and, importantly, whether they would find this strategy acceptable. Based on preliminary tests, we decided that a notification should be issued when the walker is located by the app at 7 meters from the next waypoint (for the Wayfinding app), or 14 steps from the next turn point in the reversed way-in path (for the Backtracking app). While this distance may seem large, it is important to observe that our participants often kept walking while listening to the notification; by the time the notification is completed and they have processed it, they may have already advanced by 2-3 meters. \| \| \| \| --- \| --- \| \| \| \| \| (a) \| (b) \| [fig:inaccuracies] *3. Minimal disruption.* Walking without visual feedback in an unknown space requires a good deal of concentration. It is important that the notifications generated during walking be designed to minimize the required cognitive load for processing, and that they are not distracting. Our apps issue notifications in the form of short synthetic speech sentences (Sec. [sec:notifications]). For the most time-critical notifications (e.g., announcing an upcoming turn), the notification is preceded by a short chime, and accompanied by a short vibration on the Watch. The user can, at any time, have the last notification repeated, or hear a description of the remaining route. In addition, walkers heard a short sound at each footstep. Originally implemented for debugging purposes, we decided to leave this feature in the apps, giving each participant the option to turn it off. *4. Watch-based control.* In our experiment, we asked our participants to keep the phone in a pocket throughout the trials. We also asked our participants to wear an Apple Watch, that they used to control the app. These are the gestures considered for the Watch: * Before the beginning of a trial: Participants were asked to select a specific route (e.g., “Path number 2”) from a list. The list could be traversed in both directions through left and right swipes on the Watch’s face, with the name of the current item in the list read by VoiceOver (see Fig. [fig:pictures] (b)). * To start the app: Participants were asked to start the either app by rotating the Watch’s crown, and keep rotating (in either direction) until they heard a “ding” sound from the Watch, signifying that the app had started. At that time, participants heard the notification “Please start walking.” * To hear the last notification again: At any time during a trial, participants could do a right swipe on the Watch’s face to hear the last notification issued by the app. * To hear a route description: At any time during a trial, participants could do a left swipe on the Watch’s face to hear a description (in terms of route segments and turns) of the remaining route, from their current location till destination. * To stop the app: Upon arrival at destination, or if the trial was aborted, participants were asked to again rotate the Watch’s crown to stop the app. ### Notifications Notifications are produced based on the current route and the location of the user. More specifically, we consider the distance of the user’s location (projected on the associated route segment for the Wayfinding app, or to the associated way-in path segment for the Backtraking app) to the next waypoint in the route or to nearby landmarks (Wayfinding), or to the next turn in the reversed way-in (Backtracking). Our apps support up to four distinct types of notifications. Note that, in case of conflict (a notification triggered while a prior notification was being delivered), the ongoing notification is never interrupted, except in case the new notification is of type (1) as listed below (alerting the user of an upcoming turn). This type of notification is considered time-critical, and therefore takes priority. Also note that the same notification is never repeated, except for notifications of type (3). 1. When the next waypoint (Wayfinding) or turn (Backtracking) is at less than 7 meters. If a turn should be taken at that waypoint, the notification is: “At the upcoming [X,left/right L,T] junction, turn [left, right].” Note that the junction morphology (X, L, T) requires access to a floor plan, and for this reason, this detail is not announced by the Backtracking app. If the walker is not supposed to turn at the next waypoint, the message is simply: “Keep walking straight.” This last notification is meant to reduce the risk that the walker, having found a junction, may mistakenly turn there. Note that this information is not available for the Backtracking app, which is only aware of the turns taken during the way-in. If the next waypoint is the destination (final) waypoint, or the end of the way-in route, the notification is: “Approaching your destination, Your destination is just ahead of you”. If the route ends at a wall or at a closed door, this notification is followed by: “Please stop when you find a wall”. 2. Upon entering a new route segment (Wayfinding) or when the last reliable match is updated to a new segment(Backtracking). Notification: “Walk straight for about XX [meters/feet/steps]. Then, you will turn [left, right].” The last sentence was removed for the Backtracking app after participant P4, for reasons that will be made clear later. Some route segments may contain protruding obstacles (e.g., a steel cabinet) on either side. In this case, the notification is preceded by “Please keep to the [left/right]”. 3. When the user has been walking in the wrong direction on a route segment for at least 4.5 meters (Wayfinding), or is at a distance of more than 16 steps from the way-in path (Backtracking). “You are walking in the wrong direction. Please turn around and start walking again.” This length (4.5 m) was chosen through trial-and-error in initial experiments, and is consistent with the expected radius of uncertainty of localization. This notification is repeated if they continue walking in the wrong direction for another 4.5 meters. 4. When the next landmark is at less than 2 meters (Wayfinding only). “Landmark nearby. [Landmark name] to the [left/right].” Note that a shorter threshold distance than for approaching junctions is used here. This is justified by the fact that, while a junction needs to be announced before the walker has reached it, advance notice is less critical for announcing nearby “landmarks”. Experiment ========== [ht] [tab:participants] \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| \| \| \| Blindness \| \| Step length (cm) \| Step length (cm) \| RoNIN \| Preferred \| \| \| \| \| onset \| \| (calibration) \| (final) \| multiplier \| units \| \| P1 \| F \| 73 \| L \| Dog \| 48 \| – \| 0.96 \| Steps \| \| P2 \| M \| 69 \| B \| Cane \| 51 \| – \| 1.08 \| Feet \| \| P3 \| M \| 53 \| B \| Cane \| 54 \| – \| 1.14 \| Feet \| \| P4 \| F \| 69 \| B \| Cane \| 51 \| 44 \| 1.0 \| Feet \| \| P5 \| M \| 75 \| L \| Cane \| 44 \| 41 \| 1.21 \| Meters \| \| P6 \| F \| 76 \| L \| Cane \| 40 \| 40 \| 1.11 \| Steps \| \| P7 \| F \| 72 \| L \| Dog \| 63 \| 58 \| 1.08 \| Feet \| Population ---------- We recruited 7 participants for this experiment. The participants’ characteristics are summarized in Tab. [tab:participants]. All participants were blind, with at most some residual light perception. All participants were experienced independent walkers, although P6 had recently switched to using a long cane after many years of walking with a dog guide, and was still re-learning to use the cane. P5 had a hearing impairment and used hearing aids. All participants were iPhone users, except for P7, who used a cell phone with a physical keypad. Only P1 regularly wore a smartwatch (Apple Watch). Tab. [tab:participants] also shows the average step length values measured during the initial calibration, as well as the average step length from the particles at the end of the trials (remember that step length was added as a state in the Particle Filter only beginning with P4). Note that the step length measured in the initial calibration is always larger than or equal to that found by the Particle Filter. This suggests that participants may have walked with a different step length during calibration (which was conducted in a “safe”, straight corridor stretch) and during the actual test, where they might have felt less confident and therefore walked with shorter steps. \| \| \| \| --- \| --- \| \| image \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| \| image \| image \| \| (e) \| (f) \| [fig:pictures] Procedure --------- ### Environment The trials were conducted on the second floor of a campus building. Three routes were defined, as shown in Figs. [fig:R1W]–[fig:R3W] (a). When testing the Wayfinding app, participants first traversed route R1W (Fig. [fig:R1W] (a); length: 123 m), then route R2W (Fig. [fig:R2W] (a); length: 97 m), and finally route R3W (Fig. [fig:R3W] (a); length: 72 m). Note that the endpoints of these routes were at the location of a closed exit door, and that the endpoint of each route coincided with the starting point of the next route. R1W traversed one L-junction, two T-junctions, and two X-junctions, and included 4 turns. R2W contained two L-junctions, two T-junctions, and two X-junctions, with 5 turns. R3W contained two L-junctions and two X-junctions, with 4 turns. The width of the corridors ranged from 1.9 m to 6.0 m. The building was generally quiet, with a small number of students and researchers occasionally encountered in the corridors. As mentioned earlier, all junctions in this building were at 90∘. A few potentially challenging situations are shown in Fig. [fig:pictures], and include: a narrow door (always kept open) that had to be traversed (a); multiple alcoves that had to be negotiated (c); a side staircase in a narrow corridor that had to be avoided (e); a wide (10 m × 6 m) open space, partially visible in (d). When testing the Backtracking app, the same routes were traversed in the opposite direction and in the reverse order (these routes are called R1B, R2B, and R3B). So in practice, each participant first walked R1W, R2W, and R3W, using the Wayfinding app, while the Backtracking app (in a different smartphone) collected way-in data for each route. Then, they tested the Backtracking app by traversing R3B, R2B, and R1B. We would like to emphasize that the Backtracking app did not have knowledge of the building layout. A separate building was used for the practice trial when participants were shown how to use the two apps. The practice trial route was simpler, with only 2 turns, for a total length of 63 meters. The two buildings were at a short distance from each other. ### Modalities After being consented, each participant was explained the purpose of both apps and their functions and encouraged to ask questions if something was not clear. Particular care was taken to impress upon the participants that notifications about upcoming turns would be produced with advance notice. We explained that, upon hearing such a notification, they would need to search for the first place where they could take a turn. This place could be in their close proximity, or a few meters down the way. After this initial phase, participants underwent the simple calibration procedure described in Sec. [sec:localization]. Then, we walked with them to the location where the practice trial would start. Each participant carried two iPhones in their pants pocket (note that we asked the participants prior to the experiment to please wear pants with pockets on the experiment day). Both phones needed to be carried for the initial (Wayfinding) trials: while one of them (an iPhone 12) ran the Wayfinding app, the second one (an iPhone XR) recorded way-in data. Participants also wore a wireless bone conduction headset (Shokz OpenRun), through which they could receive notifications from apps. They also wore an Apple Watch Series 8, used to interact with the apps. Before beginning the practice trial, we made sure that the VoiceOver speed and sound volume were set to the desired level. We asked each participant whether they preferred directions to be given in units of meters, feet, or steps, and set the apps’ parameters accordingly (see Tab. [tab:participants]). Note that for the first three participants (P1–P3), due to an implementation mistake, the Backtracking app produced distances only in units of steps. We asked the participants to practice making left and right swipes on the Watch. All participants eventually learned how to do this, though P2 had some difficulties at the beginning, as he interpreted “right” or “left” as referring to the axis of his left arm (where he wore the Watch), i.e. in a direction orthogonal to the forearm, rather than parallel to it. At this point, the practice trial started: the practice route was traversed using the Wayfinding app; once at destination, it was traversed in the reverse direction using the Backtracking app. After the practice trial, participants were asked whether they wanted to turn off the sound of individual detected footsteps. All participants chose to leave the footstep sound on, with some commenting that hearing it made them feel reassured that the system was functioning. We also asked them whether they wanted to turn off notification of nearby landmarks (a few landmarks were announced during the practice trial). They all chose to leave landmark notifications on. Upon completion of the practice trials, the participant and the experimenters moved to the building where the actual experiment was to take place, and in particular to the starting point of the initial route (R1W). From there, the sequence of trials described in Sec. [sec:env] was started. At the beginning of each trial, the participant was accompanied to the starting location of the route, and their body was oriented along the initial walking direction. Then, they were asked to select the next route from the list, using the Watch. Once the route was selected, they were asked to rotate the Watch’s crown to start the app. We also asked them to make a left swipe on the Watch, to hear a description of the whole route. Then, the participant started walking the route. Upon arriving at destination, participants were asked to stop the app by rotating the Watch’s crown. They were asked if they wanted to rest a bit before starting the next trial. Then, they were re-positioning to the starting point of the next route (same as the endpoint of the previous route), and oriented correctly, before starting the next route. During the trials, the experimenters followed the participants at a safe distance, taking care not to influence their routing decision. At the transition between trials with the Wayfinding app and trials with the Backtracking app, participants were handed a new headset and Watch (same models), and asked to substitute the old ones with these new ones. Both devices can be paired with one iPhone at a time; since the Backtracking app ran on a different iPhone than the Wayfinding app, this switch was necessary. At the end of the last trial (route R1B with the Backtracking app), participants and experimenters walked back to the first building, where participants were asked to participate in a questionnaire, involving the ten System Usability Scale (SUS) questions, as well as a number of open ended questions. [h] [tab:performance] c\|c\|c\|c\|c\|c\|c\|c\|\|c\| &P1&P2&P3&P4&P5&P6&P7&Length *R1W&261 (R, E)&355 (R)&297 (R)& 216&271&223&206 &123 m R2W& 304 (E)&209&134&163&211&262&171&97 m R3W& 125&170&98&127&144&139&330&72 m R3B& 180&$\varprod$&115&134 (E)&136&$\varprod$&$\varprod$ &72 m R2B& 182 (E)&232&$\varprod$& 238&173&154&149 &97 m R1B& 187&206&149&167&$\varprod$, 163 (E)&184&$\varprod,\varprod$&123 m*************************** \| \| \| \| --- \| --- \| \| \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| [fig:R1W] \| \| \| \| --- \| --- \| \| \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| [fig:R2W] \| \| \| \| --- \| --- \| \| \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| [fig:R3W] [h!] [tab:landmarks] \| Landmark ID \| Landmark type \| \| --- \| --- \| \| 1 \| alcove \| \| 2 \| benches \| \| 3 \| staircase \| \| 4 \| photocopiers \| \| 5 \| alcove \| \| 6 \| alcove \| \| 7 \| pillar \| \| 8 \| couches \| \| 9 \| pillar \| \| 10 \| pillar \| \| 11 \| door \| \| 12 \| cabinets \| Observations ------------ ### Tracking and Guidance Performance: Wayfinding Tab. [tab:performance] reports the duration of successful route traversals. The average speed (route length divided by traversal time) was 0.50 m/s for all three routes. The first route (R1W) required an intervention from the experimenters for participants P1, P2, and P3. For P1, the localization algorithm used to generate notifications (A/S by default) had to be switched to RoNIN after the first turn, because A/S was unable to correctly track the participant (see Fig. [fig:R1W] (b), blue line). For P2 and P3, we had to manually reset the user’s location in the app. In all three cases, the remainder of the Wayfinding trials were completed without any issues (except for P1 in R2W, as discussed later). It is important to note that for these three participants, the step length update si had yet to be incorporated in our Particle Filter implementation (Sec. [sec:localization]), hence A/S relied solely on the step length measured during calibration. The first route segment of R1W is a long (more than 40 meters) and narrow corridor, with several obstacles and a staircase (see Fig. [fig:pictures] (e)) on its right side (participants were advised not to walk on the staircase). This caused the participants to walk with a substantially smaller stride length than during calibration. By the time they reached the end of the corridor, the accumulated length error was such that, even with Particle Filtering, tracking became exceedingly challenging. For P1, switching to RoNIN (which produced excellent results in this case; Fig. [fig:R1W] (b), red line) did the trick. However, RoNIN also occasionally gave poor results in the same area (e.g., it was unable to track P2). After we upgraded the Particle Filtering to include step length values, this problem no longer occurred, and we were able to use A/S successfully for all other participants. Tab. [tab:performance] also shows (cells filled in gray) situations in which participants successfully completed the trial, but occasionally had to trace back their path (as prompted by the app) because they had missed a turn. These situations are clearly visible in Fig. [fig:R1W] (d), Fig. [fig:R2W] (c) and (d), and Fig. [fig:R3W] (c) and (d). Later analysis, using the recorded video and logged data from the app, brought to light the reasons for these missed turns. In some cases, participants were distracted by passer-by (P5 in R1W), or they were speaking at the time notification was issued (P2 in R2W; Fig. [fig:R2W] (c)). P6 missed a turn in R3W while walking in a large open space (Fig. [fig:R3W] (c)). Probably due to the lack of a nearby wall, she might have been unsure about where she should turn exactly, so she continued walking until she perceived a wall to the right. By that time, she was notified to turn around. P7, who missed one turn in R1W and one in R2W, as well as multiple turns in R3W, walked very fast with a dog guide. In these situations, she had already walked past the junction by the time the notification was completed and she could process it. Of note, on route R3W, soon after the beginning of the trial, P7 walked straight into a small room whose door was open (Fig. [fig:R3W] (d)). She then missed other turns in the same route, as shown in the figure, one of which multiple times, until eventually she found the correct final route segment. Only one time (P6 in R2W) did the app issue a notification shortly after the participant had passed the junction, causing her to miss the turn. This was due to localization algorithm momentarily undershooting her position. However, after being notified that she should turn around, she successfully negotiated the turn. Participants using a long cane appeared to generally react proactively to early advance notice of upcoming turns. They typically would begin walking closer to the wall on the side where they were expected to turn, and tap the wall with their cane until they found an opening (Fig. [fig:pictures] (f)). In some cases, an alcove (wall recess) was located right before the corridor junction, and some participants got “stuck” in this alcove, before finding their way out and proceeding to the junction (Fig. [fig:pictures] (c)). A remarkable exception was represented by the mobility technique of P2. Rather than scanning the ground surface in front of himself, P2, a seasoned independent traveler, used the cane to periodically tap the floor surface, and made judgments about nearby surfaces based on the sound produced by this tapping. Upon hearing a notification of an upcoming junction, he would move somewhat closer to the corresponding side of the corridor (without tapping the wall) until he perceived the presence of an opening to his left or to his right, at which point he would make a 90∘ turn into it. However, in one situation in the R3W route, he missed a turn and had to then turn back as guided by the system (he then commented “I knew where that was exactly”, indicating that he may have indeed perceived the opening the first time around.) The two participants with a dog guide displayed very different behaviors. Since dog guides are normally trained to understand “left” or “right” directions, the participants simply needed (in principle) to convey the direction from the app verbally to their dogs. For the case of P7, this mechanism worked well, except for the fact that, as mentioned earlier, she walked really fast with her dog, thereby often passing by a junction to then receive a “turn around” notification. P1, on the other hand, worked with a slower and more cautious dog. For example, in route R1W, right after the second turn, they got stuck in an alcove to the right, and needed help from the experimenter to get out of it (marked as E in Tab. [tab:performance]). It is conceivable that if P1 had been using a cane, she could have searched left and right to find a way out of the alcove. In two other situations, her dog initially refused to turn into a specific corridor. In particular, at the beginning of R2W (Fig. [fig:R2W] (d)), she missed the first turn because, by the time she gave her dog a “right” command after the notification, they had already walked past the intersection. Afterwards, they correctly turned around as advised by the app. However, as she issued a “left” command to return to the route, the dog refused to turn and instead walked straight towards the exit door. At that time, we intervened, lest they would leave the building. It took some time before P1 managed to convince her dog to walk again on that corridor. After the study was completed, P1 explained to us that her dog might have felt nervous due to the circumstances of the trials. Both inertial sensing algorithms (A/S and RoNIN) were able to successfully track the participants through the routes, except for the initial part of R1W for the first three participants before the implementation of the adaptive step length mechanism, as discussed earlier. Occasionally, the reconstructed path “cut” some corners (see e.g. Fig. [fig:R1W] (c), RoNIN) or momentarily overshot the location (Fig. [fig:R3W] (b), A/S), but the drift tracking mechanism of our Particle Filtering implementation seemed to have worked well, at least for these trials. In particular, it is remarkable that both algorithms were able to track the rather chaotic path taken by P7 in R3W (Fig. [fig:R3W] (d)). ### Tracking and Guidance Performance: Backtracking As shown in Tab. [tab:performance], six trials with the Backtracking app had to be aborted, because the app failed to track the walker during the return route. In one of these trials (P3 in R2B), the participant had taken a wrong turn (for a reason described later in this section). The app correctly issued a “turn around” notification. However (as he later told us), P3 decided to instead keep walking, as he mistakenly thought that he remembered the path he had taken. As he kept walking away from the route, the app soon became unable to track him. Two trials had to be aborted for route R3B. One of them (with P7) was due to incorrect way-in path simplification, consequent to a “meandering” way-in path (Fig. [fig:optimization] (b)). In the case of P2 and P6, the culprit was a large magnetic field discrepancy (visible for P6 in Fig. [fig:Backtracking-bad]; note the white horizontal line). The first two segments of R1B are located in very wide corridors (Fig. [fig:pictures] (d)), and we noticed that the magnetic field changes dramatically from one side to the other of these corridors (as already remarked in Sec. [sec:localization]). The different locations of the walker in the corridors between way-in and return likely caused this large discrepancy. Another common situation occurred in route R2B. As can be seen in Fig. [fig:R2W] (a), the return route (starting from the star in that figure) goes through two L-junction, then crosses an X-junction. In our original implementation of the Backtracking app, the notification issued upon entering a new segment was similar to that of the Wayfinding app: “Walk straight for about XX [meters/feet/steps]. Then, you will turn [left, right].” This is the type of notification that was issued after the second L-turn in the return route R2B, indicating a final right turn after walking for approximately 15 meters. What happened with P2, P3, and P4 is that, rather than walking through the X-junction, they turned right on it, presumably because they knew they had to turn right at some point. This situation did not happen with the Wayfinding app (in the reverse route) because, as discussed in Sec. [sec:notifications], in the proximity of the X-junction the Wayfinding app would notify the walker to “Keep walking straight”. This option is not available to the Backtracking app. The reason is that the Backtracking app is only aware of the way-in route: it cannot possibly know that the user traversed an X junction at some point, and thus cannot remind the user, on the return route, to “walk straight” when in proximity of the junction. We thus decided to slightly change the notification format, by removing the last sentence (“Then, you will turn [left, right]”) when entering a new segment, and used this new implementation from P5 on. This ostensibly minimal interface modification was sufficient to avoid incorrect turns at this junction for the remaining participants. Participant P5 was not able to complete R1B, a route that initially proceeds straight through an X-junction (Fig. [fig:R1W] (a)). P5 soon started veering to the left, hit a wall, then started walking around trying to find his way, taking multiple turns that the app was unable to match against the way-in route. We decided to abort that trial and to try again from the starting point. This time, P5 turned right instead of going straight. He told us that he knew that he had to turn right at some point, based on the initial route description provided (through a left swipe on the Watch), which stated that he should walk for 27 meters, then turn right. Note that this is a similar situation as experienced in the X-junction for R2B, as described above. One experimenter made him notice that he had not walked for 27 meters yet. At that point, P5 changed course, and was able to successfully complete the trial. While traversing route R1B, participant P2, after being notified by the app of an upcoming left turn, encountered difficulties when trying to locate the turn, and ended up getting trapped in an alcove before reaching the junction. After leaving the alcove, P2 faced another alcove at the other side of the corridor. Eventually, and following directions from the app, the participant was able to return to the correct route. The trail for P7 on route R1B had to be aborted; a second trial was likely unsuccessful. P7, as mentioned earlier, walked with a fast and very confident dog, and zipped through the X-junction where she was supposed to turn right. By the time she processed the notification asking her to turn around, she was already a long way down the corridor. Unfortunately, the Backtracking app was not able to recover tracking afterwards. Examples of successful trials with the Backracking app are shown in Figs. [fig:Backtracking-good]–[fig:Backtracking-medium]. In these plots, the way-in route (shown with a thick purple line) has segments whose length is obtained by multiplying the number of steps taken in the segments by the step length, computed through initial calibration. For this reason, these segments are not necessarily co-located with the corridors shown in the underlying floor plan. Note that this is not a problem in itself: the goal of the Backtracking app is simply to match the walker’s location during return with the same location during the way-in, in order to produce correct guidance notifications. Whether these reconstructed routes are metrically consistent with the floor plan is irrelevant for our purposes. \| \| \| \| --- \| --- \| \| \| \| \| \| \| [fig:Backtracking-good] \| \| \| \| --- \| --- \| \| \| \| \| \| \| [fig:Backtracking-medium] \| \| \| \| --- \| --- \| \| \| \| [fig:Backtracking-bad] ### Final Questionnaire The responses of the participants to the System Usability Scale questionnaire, administered at the end of the trials, are reported in Tab. [tab:SUS]. The overall score was 80.36. Though the interpretation of SUS scores is the object of debate , this score converts to a percentile rank of 90% based on the distribution of scores reported in . The open-ended questions, along with a summary of the responses, are listed below. 1. Do you think that the system always knew your location? Most participants replied with variations of “yes, most of the times” (99% of the times according to P7, 80% according to P1). However, P2 and P3 replied with “No”. According to P2, there were moments in which localization was correct. P3 felt that localization was sensitive to how one walks. 2. Do you think that the system gave you correct directions? P4–P7 replied “yes”. P2 thought that it gave correct directions most of the times (80% of the times according to P1). P3 said: “Yes, when it knew what it was doing.” [h] [tab:SUS] r\|c\|c\|c\|c\|c\|c\|c\|\|c\| &P1&P2&P3&P4&P5&P6&P7&Mean &3&4&1&5&5&5&4&3.86 & 2&1&1&2&2&4&1&1.86 & 4&4&5&5&5&5&5&4.71 &1&1&1&1&4&3&1&1.71 of a technical person to be able to use this system.&&&&&&&& &4&2&1&5&4&5&5&3.71 &3&3&1&1&2&4&1&2.14 &3&3&5&4&5&5&5&4.29 to use this system very quickly.&&&&&&&& &2&1&1&1&2&1&1&1.29 &4&4&3&5&4&5&5&4.29 &2&1&1&2&1&4&1&1.71 3. The system often gives turning directions (such as “at the next junction, turn right”) with some advance notice, which means that you need to find the turn using your cane/dog. Was this a problem for you? All participants stated that this was not a problem in general, with some adding specific comments. P1 said: “Once I got used to it, I sort of got it.” P2 commented that it was not a problem, unless there are obstacles he could bump against when moving closer to the wall in preparation for the turn. P4 said that only one time this created a problem, and precisely when, in R2B, she took a right turn too soon (a situation discussed in Sec. [sec:TGPB]). She then added that she realized that she should have waited for a notification prompting her to take the turn. P6 said that, while not a problem in itself, it would have been preferable if the notification times were more “consistent”, in the sense of producing a prompt always at the same distance from the junction (whereas, due to localization errors, this distance was often variable). P7 commented that this was the only thing in the system that was not as accurate as she thought. 4. Were the notifications understandable? Too many notifications? Too few? All participants found that the notifications were “fine” or “just right”. P2 elaborated on his reply, commenting that the system produced exactly the type of information he needed: it gave him a rough distance to the next turn, then it gave him a heads-up before the turn. P1 added that she would have liked to know if there is a door to open. (Note the routes considered in the study did not require walkers to open any doors.) P4 and P6 commented very positively about landmark notifications. For example, P6 mentioned two announcements (“stairwell to the left”, “bench to the right”) as examples of useful information. Likewise, P4 was enthusiastic about the landmark notifications, and thought they could be useful in practice. For example, she mentioned that sometimes she may feel thirsty, and it would be nice to know if she was passing by a drinking fountain. She also mentioned that landmark notification would be particularly useful when someone visits a place for the first time; this type of notification could be turned off once one is more familiar with the environment. Similarly, P2 thought that the amount of landmarks announced in the trials was just right. However, if he was really concentrated on the route, he probably would want fewer notifications. 5. Was it easy for you to use the Watch? All participants except for P2 said that they found it easy. P2 said that “it was an adventure” but also commented that with some more practice, he would have found it easier to use the Watch gestures. 6. What would you like to have in this app that is not already there? Both P2 and P5 said that they would like to have more contextual information about the space they are visiting. P1 asked about how one would find the starting point, and especially the correct direction when starting a route. P3 mentioned that he would like to be able to scroll through a route description one step at the time. (In the current implementation, a left swipe produces a description of the remaining route, but does not let the user stop and repeat each step). P4 wanted the ability to manually add landmarks on the fly (e.g., when passing by a certain location of interest). P5 also mentioned that he would like to know which direction he was facing at each time, as this information would help him when he was confused. 7. Did you notice any difference between the Wayfinding system and the Backtracking system? In general, the participants found that the two apps were very consistent with each other (with P5 mentioning that if we hadn’t told him that they were different apps, he would not have noticed.) P2 and P3 lamented the fact that the two apps used different units. As mentioned earlier, this was due to an implementation mistake, which was later corrected. 8. Do you think that using this app would make you feel safer or more confident when traveling alone in a new place? [Asked for each app separately.] All participants except for P3 replied with an enthusiastic “yes”. P3 thought that he could use the Backtracking app for situations such as when at a conference with multiple tables in a large hall, if for example one wanted to go to the restroom then come back to the same table. P2 envisioned using it when visiting a new place (e.g., a large medical building). He would like to be able to have a menu from which he could narrow down the place he would need to go. He also mentioned that, even if the app were not able to take him precisely there, there would be a lot of value in at least helping him to get closer. P4 thought that the Wayfinding app would help even for buildings that she had visited before. She mentioned an example of a large office building she had started visiting recently. While walking there, she stopped to think about where to go next, and immediately several bystanders came to her, offering unrequested (and unwelcome) help. This app would help her in these situations of uncertainty. P4 also said that using this app would relieve her from having to constantly focus on keeping track of her position and deciding where to go - her brain would have more “bandwidth” for other things. This is particularly the case in noisy and distracting environments. She also mentioned a practical example of application of the Backtracking app: walking with a sighted guide in a medical building to reach a doctor’s office, then having to walk back when there is no one around to help. P5 thought that the apps helped him create a mental map of the whole route, and enjoyed the fact that he didn’t have to memorize the whole path because the apps would give him prompts when traversing it. Discussion ========== Inertial-based localization is attractive because it does not rely on external infrastructure, and because it doesn’t require use of the smartphone’s camera, which would force the user to hold the phone in one hand while walking, or attach it to their clothing or to a lanyard. Prior wayfinding systems for blind travelers using inertial sensors relied on the “user as a sensor” modality to overcome localization errors due to accumulating drift. While the “user as a sensor” modality can be a powerful tool for mitigation of ambiguous situations (see e.g. , where it was tested successfully in an indoor/outdoor wayfinding system), it can be demanding for the user, who is tasked with detecting and identifying the landmarks. With our work, we decided to “push the envelop” and see whether a Wayfinding or Backtracking system that only relies on inertial sensor data, without input from the user, could be feasible. Our experimental results suggest that, at least for buildings characterized by networks of corridors, this is indeed the case. Of course, confirmation from users when they reached some well perceivable landmarks could be integrated for improved robustness. The environment in which we tested our systems contained narrow and wide corridors, as well as fairly large open spaces. It thus can be considered representative of many public spaces (office buildings, schools, health centers). However, larger buildings (e.g., airports, transit hubs, large shopping malls) may be challenging for inertial-only wayfinding systems: in very large opens spaces, particles disperse isotropically, and thus the Particle Filter may not be able to successfully correct for accumulating drift. The building considered in our study did not have junctions with corridors intersecting at other than 90 degrees. While arbitrary intersection angles may not represent a problem for Particle Filtering (which can adapt to any wall layout), a higher turning angle granularity may increase the error rate of the turn detector used for the Backtracking app (though 45 degrees turns were not shown to create problems in simulations with the WeAllWalk data set ). In our study, interaction with the apps was only enabled through the Watch, a mechanism that was very well received by the participants. In practical situations, though, users may want to take the phone out of their pocket, e.g. to pick up a call or to send a text. These “natural” types of interactions will be allowed (and encouraged) in future studies. As mentioned in Sec. [sec:localization] the A/S algorithm assumes that the phone does not change orientation with respect to one’s body, and therefore may produce incorrect values for the walking direction if one re-orients the phone with respect to their body. In these cases, RoNIN would be a preferable localization algorithm, enabling “natural” phone interactions. Qualitative analysis of the trajectories recorded from A/S and RoNIN (see e.g. Figs. [fig:R1W]-[fig:R3W]) showed no clear winner, and it appears that both techniques can be used successfully, if followed by a carefully designed Particle Filter, whose purpose is to ensure that the reconstructed routes are consistent with the building layout. For the Backtracking app, Particle Filtering was not an option, because the layout of the walls in the building is unknown. Our approach is to use structural geometric constraints (representing paths as sequences of straight segments and turns at discrete angles), graph-based sequence alignment, and magnetic field consistency, to match a return path with its way-in counterpart. The relatively large number of failed trials (6 out of 21) highlights the difficulty of the task, and the need for further research. In general, we found that our Backtracking app works well when users do not deviate too much from the way-in path. Large detours from the original route are difficult for the algorithm to handle, as it becomes liable to mismatches. Another problem is related to the spatial variability of the magnetic field within the width of a large corridor or hallway, which may cause mismatches when, during return, the user walks on a different trajectory within the same space. In future work, we will consider more adaptive statistical models for the magnetic anomalies in indoor environments (also relying on existing magnetic field data sets ). One of the goals of our investigation was to evaluate whether the simple interface mechanism introduced in Sec. [sec:UI], which notifies walkers of an incoming turn with large advance notice, would be feasible and acceptable. Prior work (e.g. ) studied optimal sonification techniques to ensure that walkers begin turning at the right time. These mechanisms are appropriate for localization systems (e.g. using computer vision or BLE beacons) that are substantially more accurate than inertial sensing . In our case, the user interface must accommodate for localization errors, hence the early notification. We were not sure whether participants would have difficulty discovering the exact location of the junction, or whether they would find this modality burdensome or annoying. We were please to find that, by and large, our participants were able to negotiate these situation successfully, even in the face of structural impediments (e.g., when they would get stuck in an alcove when looking for an opening). What’s more, participants, when asked whether they thought that advance notification was a problem (Sec. [sec:FQ], question 3.), they overwhelmingly asserted that this was not the case. The positive usability scores recorded from the SUS responses are another indication that our participants were generally satisfied with the interface design. We are aware, however, that the advance notification approach could fail, for example, if there are two openings close to each other on the same side of the corridor (say, within two meters from each other). In this case, more contextual information could be produced by the system (e.g., notifying the user that they will encounter two junctions, and that they should take the second one). Perhaps the most critical limitation of our approach is that users of both systems must start a route from a certain location, and begin walking in a certain direction. This initialization phase is necessary to geometrically calibrate the user’s path with the floor plan frame. Note that the same requirement is found in prior work with inertial-based wayfinding , and is a consequence of the dead-reckoning nature of this method. Note that in principle it is possible for the system to function even without knowledge of starting position and orientation. As shown in , one may distribute a large number of particles uniformly across the environment, then after a certain period (several minutes) of walking time, the surviving particles would converge with good likelihood around the user’s position. However, this would require the user to walk aimlessly and without guidance from the system for a while, something that may not be desirable in practice. We envision two possible solutions to the problem of starting location/orientation determination. One possibility is to use the “user as a sensor” paradigm. Users may start from a location that is well perceivable, such as the main entrance of building or other identifiable landmarks such as the end of a staircase, and be advised to start walking in a certain known direction (in our example, straight across building entrance or coasting a wall to the left or to the right of the staircase entrance.) A different solution, one that we are currently exploring, is to create a hybrid system that can use visual data (e.g. via automatic landmark recognition ) for sporadic “fixes” using computer vision techniques when the location and orientation of the user needs to be ascertained. After that, the user may move the smartphone back to their pocket, and be tracked by the inertial system. Finally, we were heartened to hear from our participants that they found that the apps they tested could make them feel safer and more confident in their independent travel. While we are aware of the limitations of our systems, this type of feedback (along with the good scores from the SUS questionnaire) confirms that our proposed technology has serious potential for practical use as a navigational aid. Conclusions =========== We have presented the results of an experiment with seven blind participants who used two custom-designed iPhone apps for wayfinding and backtracking. These apps use different inertial-based mechanisms to track the user in a known floor plan (Wayfinding) or to match the user’s path with a previously taken path (Backtracking). Our experiments showed that inertial-based tracking, coupled with a carefully designed user interface, is a suitable technology for wayfinding in a building characterized by a network of corridors when the initial position/orientation of the user is known. Out study also showed that our approach to backtracking (which assumes no knowledge of the building’s layout) has good potential, but improvements are needed to increase its robustness in practical scenarios. --- 1. https://github.com/Sachini/ronin[↩](#fnref1) 2. https://developer.apple.com/documentation/gameplaykit[↩](#fnref2) 3. https://www.indooratlas.com[↩](#fnref3) All the Way There and Back: Inertial-Based, Phone-in-Pocket Indoor Wayfinding and Backtracking Apps for Blind Travelers ======================================================================================================================= We introduce two iOS apps that have been designed to support wayfinding and backtracking for blind travelers navigating in indoor building environments. Wayfinding involves determining and following a route through the building’s corridors to reach a destination, and assumes that the app has access to the floor plan of the building. Backtracking one’s route, on the other hand, requires no map knowledge. Our apps only use the inertial and magnetic sensors of the smartphone, and thus require no infrastructure modification (e.g., installation and support of BLE beacons). Unlike systems that use the phone’s camera, users of our apps can conveniently keep their phone tucked inside a pocket, while interacting with the apps using a smartwatch. Routing directions are given via speech. Both apps were tested in a user study with seven blind participants, who used them while navigating a campus building. <ccs2012> <concept> <conceptid>10010520.10010553.10010562</conceptid> <conceptdesc>Human-centered computing Accessibility technologies </conceptdesc> <conceptsignificance>600</conceptsignificance> </concept> <conceptdesc>Human-centered computing Human-centered computing Interaction devices </conceptdesc> <conceptsignificance>400</conceptsignificance> </concept> </ccs2012> Introduction ============ Navigating a building can be confusing and disorienting for anyone who is visiting the building for the first time. For those who are blind, this experience can be even more challenging, as these travelers cannot rely on visual feedback. While multiple research efforts have focused on accessible technology to support independent travel in indoor space, at the time of this writing there are no commercially available systems ready for widespread use. Indoor spaces are challenging because GPS cannot be relied upon. This doesn’t mean that indoor localization is not feasible: a variety of techniques have been demonstrated, some of which have been adapted for navigation by blind individuals. One example is NavCog in its various versions , which uses Bluetooth Low Energy (BLE) for accurate localization. BLE localization is a mature technology. Unfortunately, BLE-based navigation is only possible if a specific infrastructure (a possibly large set of BLE beacons) has been installed, and a laborious calibration procedure (fingerprinting) has been conducted. This raises questions of scalability, as the success of this technology hinges on the good will of the agency that manages the building. It would thus be desirable that the technology powering the navigational aid would not depend on external dedicated infrastructure. In recent years, and with the advent of powerful AI, there has been intense interest in systems that use visual sensors (e.g., the camera embedded in a smartphone) to extract positional data, and to provide users with information about the world around them. For example, navigation apps built on Apple’s ARKit have been developed specifically for use by blind travelers . Visual-inertial odometry technology, coupled with Particle Filtering, can produce extremely accurate localization. Its core strength, the use of visual data, is unfortunately also its main drawback. Users of these apps must hold the smartphone (in their hand, or perhaps attached to a lanyard ) in such a way that the camera has a good view of the environment. This is not always possible, convenient, or desirable. Blind travelers normally use a long cane or a dog guide, and thus have one hand already occupied maneuvering the cane or holding the dog. Indeed, it has been often observed that navigational aid for blind walkers should be hands-free . In this article, we describe an experiment we conducted with 7 blind travelers, who tested two apps created in our lab that were designed to assist with navigation in buildings characterized by a network of corridors. Both apps utilize data from the inertial sensors of a smartphone, and thus require no external infrastructure. Since the apps don’t use data from the smartphone camera, users can conveniently keep the phone tucked in a pocket (indeed, this is how the apps have been tested in our experiment). The first one is a Wayfinding app: it has knowledge of the floor plan of the building to be navigated, computes the shortest length route to destination, and provides navigational support in an accessible way. The second one is a Backtracking app. Its sole purpose is to help an individual, who has previously walked on a certain route (e.g., from the front entrance of a building to a certain office room), to trace back their steps. While arguably less useful than a full blown wayfinding application, backtracking support can help a blind traveler feel more confident in some situations. Importantly, the Backtracking app requires no prior knowledge of the building layout: it only uses data recorded during the first route traversal (way-in) to generate support for the user when walking back (return). Both apps have provision for error recovery (i.e., can provide corrective directions if a participant missed a turn, or took the wrong turn). These are the main contributions of this work: * We demonstrate an accessible iPhone app (Wayfinding) that uses two different localization technologies, complemented by Particle Filtering, for inertial-based localization in a building with known floor plan. * We propose a new hybrid approach for backtracking that uses magnetic and inertial data, with no knowledge of the building layout (and thus usable in any venue even without access to a floor plan). This algorithm was implemented and tested in the Backtracking app. * We demonstrate the effectiveness of a simple speech-based user interface mechanism, consistent for both apps, designed to mitigate the unavoidable inaccuracy associated with the dead-reckoning nature of inertial-based localization. * We describe a user study with 7 blind travelers. The participants first traversed three routes (292 meters and 13 turns in total) using our Wayfinding app; then, they traversed the same routes, in the opposite direction, using the Backtracking app. This article is structured as follows. After reviewing the related work in Sec. [sec:RW], we describe our Wayfinding and Backtracking apps (including their shared user interface mechanisms) in Sec. [sec:Apparatus]. Our experiment with seven blind participant is described in Sec. [sec:Experiment], and results, along with limitations of our approach, are discussed in Sec. [sec:Discussion]. Sec. [sec:Conclusions] has the conclusions. Related Work ============ Multiple research considered smartphone app implementations to enhance the mobility of individuals with visual impairments, especially in GPS-denied indoor environments. These apps leverage a variety of technology, including BLE beaconing, Wi-Fi, IMUs, video cameras, or sensor fusion, to offer navigational support . As an example, NavCog incorporates a precise localization algorithm based on BLE beacons, and offers customizable voice and non-vocal sound instructions for user interaction. Additionally, LuzDeploy , built upon NavCog, has adopted crowdsourcing technology to maintain the localization infrastructure. Subsequent versions integrated BLE and inertial data positioning, and Particle Filtering (NavCog3 ). A virtual navigation app (VirtualNav ) was built upon the foundation of NavCog3, to explore the transfer of virtually acquired route knowledge to practical navigation. Other projects using BLE-based localization include ASSIST) and GuideBeacon . Alternative technologies considered for localization include RFID tags (e.g., PERCEPT ), color tags (e.g., NaviLens ), Wi-Fi beaconing , and magnetic navigation . In recent work, visual-inertial odometry (VIO) has been shown to provide accurate localization without relying on dedicated infrastructure . Backtracking apps using similar localization technologies have also been developed . As mentioned earlier, these system require use of a camera with unoccluded visual field. Research on navigation using inertial sensing (the technology considered in this article) includes . Riehle et al. described a classic pedestrian dead-reckoning system (PDR) with explicit turn detection, which was tested with 8 blind or low vision participants in a simple indoor route. Participants could request assistance from the experimenter if they felt they needed it. Compared to a condition in which the route was described at the beginning, but no guidance was provided during the trial, it was shown that use of the PDR system resulted in fewer requests for assistance made and in more instances of successful localization of the route’s destination. Fallah et al. and Apostolopoulos et al proposed an inertial-based navigation system designed to overcome inherent localization inaccuracy through the “user as a sensor modality”. Routes were represented via a sequence of perceivable landmarks. Participants were asked to follow directions to the next landmark; once arrived at the landmark, they were asked to confirm it on the app. This ingenious strategy allowed the localization system to reset itself at the correct location at each detected landmark. The inertial system was based on step counting (through an accelerometer) and orientation sensing (through a compass). A Particle Filter was used to track the state (location and orientation) of the user. In addition, different methods were considered to estimate the user’s step length using the Particle Filter. An experiment with 6 blind and low vision participants was conducted on two floors of a building with 11 routes, most of which contained one or two turns. One route included climbing a staircase, while another route included taking an elevator. These routes contained multiple landmarks that needed to be discovered by the participants, including doors that needed to be counted, hallways, stairs, ramps. Peng et al. demonstrated an indoor/outdoor wayfinding system, designed for navigation in a transit hub, that tracked the user’s location through fusion of inertial and GPS data. Along with localization, research has focused on the design of accessible user interface modalities for navigational systems , using modalities such as speech , sound , and vibration . Apparatus ========= We developed two iOS apps for this study: Wayfinding and Backtracking. The Wayfinding app has access to a floor plan of the building to be navigated. It is designed to find a route from the current user’s location to the desired destination, update the route as necessary during traversal, and provide directions to the user as they are following the route. The Backtracking app has no prior information about the building layout. Similarly to, Backtracking operates in two phases. In the first phase (way-in), it simply tracks the route taken by the user to reach a destination. Then, in the return phase, it produces directions to help the user re-trace the same way-in route. To do so, the app progressively matches the partial return route with the way-in route (in reverse). As will be clear in the following, the Backtracking app needs to use different localization strategies than the Wayfinding app, as a consequence of its lack of building layout information. The user interface, however, was designed to be almost identical for the two apps, shielding the user from the different underlying mechanisms. Wayfinding ---------- ### Localization We implemented two different pedestrian dead-reckoning (PDR) algorithms for user localization and tracking using data from the phone’s inertial sensors. These two techniques are termed Azimuth/Steps and RoNIN, respectively . A Particle Filter is applied to the output of either system. During the tests, we ran both systems in parallel, though only one of them was used to provide guidance to the user. The reason for running both algorithms in parallel was twofold. First, we wanted to have a “fail safe” mechanism: if an algorithm failed to correctly track the participant, we could resort to switching to the other algorithm. Note that this only happened once in the whole experiment. Second, it allowed us to comparatively assess localization data from both algorithms in a variety of situations. Both algorithms compute the user’s locations in terms of an arbitrary (but fixed) world reference frame, with the Z axis pointing downwards (in the direction of gravity). We used a simple initial calibration procedure (described below) to find the angle of the rotation around the Z axis that brings the world reference frame to the frame used to define the floor plan (the floor plan frame). *Azimuth/Steps (A/S). This algorithm produces, at each detected step (heel strike), a 2-D vector Δp* with a length equal to the estimated step length, and direction given by the phone’s azimuth (or heading) angle. This vector is then fed as input to the Particle Filter (discussed later in this section). Heel strikes (steps) are computed by an LSTM-based algorithm that processes data from the phone’s inertial sensors . Each participant’s step length was estimated by the following calibration procedure. The participant was asked to walk along a straight corridor path (38.25 meters in length). By dividing the length of this path by the number of steps (from the step counter), we obtained the user’s approximate step length. A similar step length calibration was used in . Note that after calibration, the step length is further updated by the Particle Filter (in a similar fashion to ). In spite of its simplicity, A/S was shown to produce good path reconstruction results when coupled with a Particle Filter, which can mitigate the drift effect inherent in the dead-reckoning nature of the algorithm . One drawback of this approach is that the orientation of the vector Δp (azimuth) is that of the phone, rather than that of the walker. This means that moving the phone to a different location on one’s body (e.g., pulling it out of the pocket to take a call) may incorrectly be interpreted as a change in walking direction. In our experiment, participants kept the phone tucked inside their back pocket throughout the trials, hence the orientation of the phone with respect to the user’s body could be considered to be constant. *RoNIN. RoNIN is a machine learning algorithm for dead-reckoning from inertial data. Its feasibility for the reconstruction of paths taken by blind walkers, using a long cane or a dog guide, was demonstrated in . RoNIN produces velocity vectors at a rate of 25 Hz. We used the authors’ open-source ResNet18 implementation[1](#fn1), and integrated the velocity vectors over individual step periods to obtain displacement vectors Δp. Unlike the A/S algorithm, RoNIN produces displacement vectors that are defined with respect to the world frame (rather than the phone’s frame). This means that the heading direction estimated by RoNIN is independent of the phone’s orientation with respect to one’s body . Hence, the user is not constrained to hold the phone in a fixed location. However, RoNIN is as liable as A/S to orientation drift due to the integration of noisy sensor data. Experiments reported in showed that while RoNIN worked remarkably well for some individuals, it produced less satisfactory results for others. In particular, the magnitude of the velocity vector returned in output for certain users was found to be either smaller or larger than their actual speed. To account for this user-dependent error, we employed the same calibration procedure described above to find an adjustment coefficient (RoNIN multiplier) for each participant. In practice, we divided the length of the path traversed during calibration by the estimation of the same path length produced by RoNIN. Then, during the trials with the same participants, we multiplied the velocity vectors produced by the RoNIN multiplier thus computed. Particle Filtering (PF). Particle Filtering is a form of Bayesian filtering that is commonly used for spatial tracking in the presence of prior information or constraints . In our case, knowledge of the floor plan allows us to define ``impenetrable walls" that are unlikely to be traversed by a traveler . Using a Particle Filter, the posterior distribution of the walker’s location is expressed by means of a set of samples (particles). In our experiments, we used 500 particles, which was found to be a reasonable trade-off between computational speed and accuracy of tracking. Each i-th particle is characterized by a X-Y location p*i, a drift angle value Δθ, i, and, for A/S localization, a step length value si. At the beginning of a trial, all particles are located at the (known) initial user location. The drift angles associated with the particles are sampled from a zero-mean Gaussian distribution with σ = 30∘, while the step lengths are sampled from a Gaussian distribution with σ = 6 cm centered at the step length found during the initial calibration. Each particle has a (positive) weight wi. Weights are all initialized to 1/500. At each time period (in our case, at each detected step), each particle is spatially propagated by a vector that is a function of the displacement vector Δp produced by A/S or by RoNIN. Specifically, Δp is first rotated by the particle’s drift angle Δθ, i. For A/S localization, its length is changed to si. The particle’s drift angle Δθ, i is updated by addition of Gaussian noise (σ = 1∘). Then, Gaussian noise is added to its modulus (σ = 0.5 ⋅ si) and phase (σ = 0.5∘). The particle’s location is then updated by adding the resulting vector to pi. Additionally, the particle weights can be modified during a positional update as described later in this section (after re-weighting, the weights are normalized to sum to 1.) At each time step, a new set of 500 particles is resampled with replacement from the current set of particles. This means that, for 500 times, a particle j in the current set is picked at random, with the probability of a particle being sampled equal to its weight. A new particle is generated with the same characteristics (location pj, angular drift Δθ, j, and, for A/S, step length sj). Gaussian noise (σ=10 cm) is added to the X and Y components of its location. The old particles are then discarded, and the new ones are retained until the next update. The user’s location is taken to be the weighted sum of the particles’ locations: p = ∑iwipi. Particles are re-weighted at each time period according to the following rules: 1. If a particle is found to be crossing a wall, its weight is set to 0. 2. If a particle is at a distance from the weighted average p larger than a threshold value D, its weight is set to 0. 3. If at any time a particle is found to be inside a room, its weight is multiplied by 0.9. \| \| \| \| --- \| --- \| \| image \| \| \| (a) \| (b) \| [fig:PF] Provision (1) reduces the risk of trajectories going through impenetrable walls. Compactness of the particle set is enforced by (2). This is particularly useful for large open spaces, including long corridors, where the particle set could otherwise expand boundlessly. The threshold values D for provision (2) were set (based on trial-and-error experiments) to 5.5 m for A/S localization and to 3.5 m for RoNIN. Provision (3) reduces the likelihood that a user walking in a corridor be mistakenly localized inside a room. It is important to note that all rooms in the building are considered “open” (see Fig. [fig:PF] (a)). Even though the selected routes are defined on corridors, walkers could indeed enter a room through an open door (as it happened with participant P7; see Fig. [fig:R3W] (d)), and should be tracked in those spaces as well. However, we noted that without provision (3), sizeable amounts of particles would often enter through the open doors (and get temporarily stuck in) one or more rooms when a participant was walking on a corridor nearby (Fig. [fig:PF] (a)). By reducing their weights, we decrease the likelihood that particles in these isolated clusters be selected for resampling. Resampling also ensures that only angular drift values (and, for A/S, step lengths) that lead to ``legitimate" trajectories are preserved. Explicitly modeling the drift angle was shown in to effectively mitigate the effect of accumulating angular drift, a well-known issue associated with dead-reckoning. For A/S localization, we also found that, through resampling, the particles’ step length values tended to coalesce towards a value that was often different from that measured during initial calibration, as discussed later in Sec. [sec:population]. In our iPhone implementation, Particle Filtering updates are performed each time a footstep is detected. An update normally requires between 0.01 to 0.02 seconds to complete. *Initial Orientation Calibration. For each trial in our experiment, we assumed that participants would start from a known location, and that they initially walked along a known direction. After 6 steps, we found the angle between the reconstructed trajectory (defined, as mentioned earlier, with respect to an arbitrary world reference frame) and the known walking direction, as defined in the floor plan frame. ### Routes and Waypoints The walkable area in a building characterized by a network of corridors can be represented by a set of waypoints* (located at the corridors’ junctions) and route segments, which join any two waypoints if there is a traversable straight path between the two. A route joining any two waypoints is a sequence of interconnected route segments. At each time t, the localization algorithm (whether A/S or RoNIN, followed by Particle Filtering) produces an estimated location p(t) of the user. Based on this location, the app issues notifications to the user as appropriate. However, rather than directly using the 2-D locations p, we consider the projected locations $\bar p(t)$ onto its associated route segments. This approach is justified by the nature of typical buildings with networks of corridors, and simplifies the logic used to produce notifications, which is described in Sec. [sec:notifications]. When users walk on a long corridor, far from junctions with other corridors, projecting their location onto the route segment associated with that corridor is a safe operation. More care must be taken when in the proximity of a junction, due to potential localization errors leading to incorrect assignments. For example, as shown in Fig. [fig:PF] (b) (see locations marked by blue arrows), associating the user’s location with the closest route segment may lead to incorrect results when the distance of the user to the junction is comparable with the radius of localization uncertainty. To deal with such situations, we adopted the following mechanism. A walker whose position is currently associated with a certain route segment, maintains this association until the projected location $\bar p$ is closer than a threshold distance T to a junction with other route segments (T was set to 1.5 m in our experiment). At that point, route segment association becomes ambiguous, and p is no longer associated with any segment. Segment association is resumed when the projection of p onto any of the route segments ending at that junction is at a distance of T (or larger) from the junction. From that point on, the walker is associated with this new segment (see Fig. [fig:PF] (b)). Our wayfinding app determines the shortest route to a destination using the iOS GameplayKit toolkit[2](#fn2). Routes are constantly updated as the user moves, and re-routing is computed as necessary. In addition to waypoints, we defined a number of “landmarks”, whose presence was communicated to the walker when nearby. These landmarks, whose location is shown in Figs. [fig:R1W]– [fig:R3W] (a), are listed in Tab. [tab:landmarks]. Backtracking ------------ As mentioned earlier, our Backtracking app is structured in two distinct phases. In the way-in phase, the app is in charge of recording the route taken by the user (who may be accompanied by a sighted guide, or perhaps may be exploring the environment on their own). During the return phase, the app is in charge of tracking the path of the walker and matching it with the way-in path. In particular, the system needs to determine when the walker is getting close to a location where a turn was taken during the way-in, in which case it can produce an appropriate notification. ### Path Reconstruction: Way-in Unlike the Wayfinding app, impenetrable wall constraints cannot be used to correct for orientation drift, because the Backtracking app has no knowledge of the floor map of the building. In buildings characterized by networks of corridors, however, it is conceivable that walkers would proceed along relatively straight paths until they turn at a corridor junction. The angle made by two intersecting corridors, for typical buildings, is often equal to ± 90∘ or to a multiple of 45∘. This geometric structural constraint can be leveraged to sidestep orientation drift. Following , we represent both way-in and return paths as a sequence of straight segments interleaved with discrete angle turns. We use the robust turn detector described in , which processes azimuth information using a Mixture Kalman Filter . Although this algorithm was shown to work well even for multiples of 45∘ turns, we constrained detection to multiples of 90∘ for the purpose of this experiment (this reflects the type of junctions found in the building considered for our tests, which were all of ± 90∘). The way-in path can be depicted as a 2-D polygonal chain (polyline), where the length of each segment is equal to the number of steps taken in that segment, and consecutive segments have an angle as measured by the turn detector (see e.g. Fig. [fig:projection]). Steps are detected using the same LSTM network used for A/S localization. ### Path Matching: Return Our strategy for matching the return path with the way-in path is based on the coordination of two different algorithms: projected return sequence and sequence alignment. Projected Return Sequence. This algorithm builds a 2-D trajectory as a polyline, as described above for the way-in path. This simple algorithm works reasonably well in “ideal” conditions but fails if (1) the walker’s step length is different between way-in and return (leading to a different number of steps for the same segment; see Fig. [fig:projection] (a)); (2) the turn detector fails to recognize a turn (e.g., because taken too slowly) or produces a false positive (due to an irregular trajectory); or (3) the walker does not follow exactly the same route (in reverse) as during way-in, e.g. because they missed a turn then turned around. All of these situations may be expected, especially when someone walks without visual feedback. \| \| \| \| --- \| --- \| \| \| \| \| (a) \| (b) \| [fig:projection] Sequence Alignment. As described in , one may cast the path matching problem as one of (sub)sequence alignment. Assume that the sequence of way-in measurements has been reversed, which is convenient since the route is being backtracked. At each time during return, we determine the initial way-in subsequence of measurements that best matches the current sequence of return measurements. In symbols: given the (reversed) way-in sequence of measurements (observations) W = (ow(1), …, ow(N)), and the current sequence of return measurements R = (or(1), …, or(J)), the goal is to find a sequence of indices i1, …, iJ such that (ow(i1), …, ow(iJ)) best matches R under an appropriate criterion. For real-time guidance, we are interested in the last matching point iJ: we will assume that the walker at the current return time index J is in the same location they were at time index iJ during way-in. Standard dynamic programming approaches (e.g. Dynamic Time Warping ) can then be used to find an optimal match. The measurements we consider are magnetic field vectors and turns detected. Step detection is also considered implicitly: for both way-in and return, the sequences of time indices are defined such that there are three regularly spaced time intervals between two consecutive detected steps. We found that this choice gives enough spatial granularity for magnetic field matching, while ensuring parsimonious sampling (e.g., no samples are recorded when the user is stationary). It is well known that the magnetic field recorded in different positions within a building is not uniform, due to reasons such as the presence of large metallic objects and magnetic field generating appliances. “Magnetic signatures” can thus be attached to specific locations, enabling sophisticated localization mechanisms (e.g., IndoorAtlas’ magnetic positioning[3](#fn3).) The magnetic field at a certain location can be measured by a 3-axis magnetometer, of the type embedded in any modern smartphone. (Prior calibration of the magnetometers is necessary for good results.) From the measured 3-D magnetic field vector we derive a 2-D vector that is invariant to the orientation of the phone as described in . Given the measurements, one can create a directed graph with nodes indexed as (i, j), where i is a way-in time index, and j is a return time index. (Note that the graph is constantly expanded as the walker progresses along the return path.) A node (i, j) in the graph has only three edges, to (i + 1, j), (i + 1, j + 1), and (i, j + 1), respectively. This is consistent with the assumption that the walker is normally moving in the same direction as in the (reversed) way-in path, but possibly with a different step length (resulting in steps detected in either phase that cannot be matched in the other phase, which are accounted for by the edges to (i + 1, j) and (i, j + 1) ). The edges to (i + 1, j) and (i, j + 1) carry a non-null cost, while node costs are defined as a function of the magnetic field vector discrepancy (measured as the Euclidean difference between the recorded vectors) between the measurements ow(i) and or(j). Sequence alignment is obtained by finding the minimum cost path starting from (0, 0) and arriving at a node (i, J). Unfortunately, this simple approach, originally proposed in , did not produce satisfactory results in our preliminary tests, especially if the return path is not perfectly identical to the way-in path. Indeed, we found that, in some cases, the magnetic field can vary rather dramatically when moving across the width of a building corridor. An improvement to pure magnetic-based alignment can be obtained by considering the turns taken by the walker, which, in an ideal case, should be matched between way-in and return. We use the mechanism described in , which considers the difference in walking direction between way-in and return by structuring the graph as a sequence of layered planar graphs, where each layer represents a possible orientation discrepancy between way-in and return. The minimum cost path in the graph is recomputed at each new return sample. We use the incremental Dynamic Time Warping (iDTW) proposed by Riehle et al. , which uses a sliding window defined around the endpoint of the previously found optimal path (we set the window size equal to 200 samples). Although this algorithm produces a suboptimal solution, it represents a good compromise between precision and computational cost. Hybrid Matching Strategy. The sequence alignment method described above was shown to work well even for moderately complex situations. However, since the optimal path is recomputed at each iteration, the estimated user location at the current time may become “jittery”. This may cause instability when producing guidance notifications, complicating the task of deciding when exactly to notify the walker of an upcoming turn. We propose a simple solution to this problem, based on the notion of last reliable match. We use local properties of the current minimum cost graph path to decide, at the current time J, whether the match (iJ, J) can be considered “reliable”, meaning that it is likely to be preserved even after later observations are recorded. In practice, we look at the terminal part (the last 21 samples) of the minimum cost path in the graph ending at (iJ, J), and judge this last match to be reliable if this path segment is well approximated by a line with unitary slope. This simple algorithm was shown to work well in most cases. When a reliable match is detected, a projected return sequence is initialized and continuously updated, as discussed earlier, until a new reliable match is found. This projected sequence starts from the last reliable match, with an initial direction equal to the current walker orientation (as defined by the sequence of turns taken during return). Guidance notifications are produced based on the walker’s location identified using this projected sequence. As soon as a new reliable match is detected, the projected sequence is re-initialized at that match (see Fig. [fig:projection] (b)). Examples of reliable matches and projected paths are shown later in Sec. [sec:TGPB]. ### Way-in Path Simplification In some cases, the way-in path contains redundant turns or loops. This may happen when, for example, if the walker took a short detour (perhaps because they were unclear about the route to take), or if they followed a zig-zag course instead of walking straight. It would be desirable to remove these redundant features before backtracking. In our experiment, we implemented a very simple “path simplification” algorithm for the way-in path. In short, we define a conservative radius of uncertainty D equal to 7 steps. We then merge together nearby parallel edges that are at a distance of D or less from each other. We do the same with turn locations (vertices). Then, we re-create a simplified path as a shortest length polyline that goes through the remaining edges and vertices. An example of successful way-in path simplification is shown in Fig. [fig:optimization] (a). Note that the original path (shown to the left) contains a short detour with a 180∘ turn, which was due to the walker originally missing a turn. This spurious piece was correctly removed by the simplification algorithm. Fig. [fig:optimization] (b) shows a case with a complex way-in path, containing multiple loops. Although our algorithm was able to remove these loops, and the resulting simplified path maintained the correct path geometry, its first segment turned out to be substantially shorter than in the original path, which led to an unsuccessful backtracking trial. \| \| \| --- \| \| \| \| (a) \| \| \| \| (b) \| [fig:optimization] User Interface -------------- We list in the following the general principles that guided our interface design for the two apps. *1. Consistency.* Even though the two apps utilize different localization technologies, the user interface is almost identical between the two. Both apps ultimately provide directions towards a destination in similar environments (networks of corridors), thus it is only natural that they should afford similar user experiences. The small differences (highlighted in Sec. [sec:notifications]) are a consequence of the different prior knowledge the apps can rely on. *2. Robustness to localization inaccuracy.* The localization system of both apps relies on dead reckoning from a smartphone’s inertial sensors. In spite of mitigation techniques (Particle Filtering for Wayfinding, sequence alignment for Backtracking), localization errors of up to 2-3 meters should be expected. Neglecting these potential errors when issuing notifications may lead to catastrophic results. For example, suppose that the app issues a notification shortly before it locates the walker at a junction. As shown in Fig. [fig:inaccuracies] (a), the walker’s real position may happen to be at some distance before or after the junction. In either case, if the walker were to turn, they would face a wall. What’s worse, they would have no way to know whether to search for the junction to their right or to their left. To account for the expected localization inaccuracy, we made the following design choice: the presence of a junction where a turn should be taken is announced with substantial advance notice, enough to ensure that, with high likelihood, the walker has not yet reached the junction. Although conceptually very simple, it was not obvious a priori that this approach would be viable and acceptable. The notification is given at a variable distance from the junction (depending on the current localization error). Users are thus in charge of searching for the next available opening upon hearing the notification, which involves active exploration with the long cane or with the dog guide. An important goal of our experiments was to verify whether our participants would be successful in this task, and, importantly, whether they would find this strategy acceptable. Based on preliminary tests, we decided that a notification should be issued when the walker is located by the app at 7 meters from the next waypoint (for the Wayfinding app), or 14 steps from the next turn point in the reversed way-in path (for the Backtracking app). While this distance may seem large, it is important to observe that our participants often kept walking while listening to the notification; by the time the notification is completed and they have processed it, they may have already advanced by 2-3 meters. \| \| \| \| --- \| --- \| \| \| \| \| (a) \| (b) \| [fig:inaccuracies] *3. Minimal disruption.* Walking without visual feedback in an unknown space requires a good deal of concentration. It is important that the notifications generated during walking be designed to minimize the required cognitive load for processing, and that they are not distracting. Our apps issue notifications in the form of short synthetic speech sentences (Sec. [sec:notifications]). For the most time-critical notifications (e.g., announcing an upcoming turn), the notification is preceded by a short chime, and accompanied by a short vibration on the Watch. The user can, at any time, have the last notification repeated, or hear a description of the remaining route. In addition, walkers heard a short sound at each footstep. Originally implemented for debugging purposes, we decided to leave this feature in the apps, giving each participant the option to turn it off. *4. Watch-based control.* In our experiment, we asked our participants to keep the phone in a pocket throughout the trials. We also asked our participants to wear an Apple Watch, that they used to control the app. These are the gestures considered for the Watch: * Before the beginning of a trial: Participants were asked to select a specific route (e.g., “Path number 2”) from a list. The list could be traversed in both directions through left and right swipes on the Watch’s face, with the name of the current item in the list read by VoiceOver (see Fig. [fig:pictures] (b)). * To start the app: Participants were asked to start the either app by rotating the Watch’s crown, and keep rotating (in either direction) until they heard a “ding” sound from the Watch, signifying that the app had started. At that time, participants heard the notification “Please start walking.” * To hear the last notification again: At any time during a trial, participants could do a right swipe on the Watch’s face to hear the last notification issued by the app. * To hear a route description: At any time during a trial, participants could do a left swipe on the Watch’s face to hear a description (in terms of route segments and turns) of the remaining route, from their current location till destination. * To stop the app: Upon arrival at destination, or if the trial was aborted, participants were asked to again rotate the Watch’s crown to stop the app. ### Notifications Notifications are produced based on the current route and the location of the user. More specifically, we consider the distance of the user’s location (projected on the associated route segment for the Wayfinding app, or to the associated way-in path segment for the Backtraking app) to the next waypoint in the route or to nearby landmarks (Wayfinding), or to the next turn in the reversed way-in (Backtracking). Our apps support up to four distinct types of notifications. Note that, in case of conflict (a notification triggered while a prior notification was being delivered), the ongoing notification is never interrupted, except in case the new notification is of type (1) as listed below (alerting the user of an upcoming turn). This type of notification is considered time-critical, and therefore takes priority. Also note that the same notification is never repeated, except for notifications of type (3). 1. When the next waypoint (Wayfinding) or turn (Backtracking) is at less than 7 meters. If a turn should be taken at that waypoint, the notification is: “At the upcoming [X,left/right L,T] junction, turn [left, right].” Note that the junction morphology (X, L, T) requires access to a floor plan, and for this reason, this detail is not announced by the Backtracking app. If the walker is not supposed to turn at the next waypoint, the message is simply: “Keep walking straight.” This last notification is meant to reduce the risk that the walker, having found a junction, may mistakenly turn there. Note that this information is not available for the Backtracking app, which is only aware of the turns taken during the way-in. If the next waypoint is the destination (final) waypoint, or the end of the way-in route, the notification is: “Approaching your destination, Your destination is just ahead of you”. If the route ends at a wall or at a closed door, this notification is followed by: “Please stop when you find a wall”. 2. Upon entering a new route segment (Wayfinding) or when the last reliable match is updated to a new segment(Backtracking). Notification: “Walk straight for about XX [meters/feet/steps]. Then, you will turn [left, right].” The last sentence was removed for the Backtracking app after participant P4, for reasons that will be made clear later. Some route segments may contain protruding obstacles (e.g., a steel cabinet) on either side. In this case, the notification is preceded by “Please keep to the [left/right]”. 3. When the user has been walking in the wrong direction on a route segment for at least 4.5 meters (Wayfinding), or is at a distance of more than 16 steps from the way-in path (Backtracking). “You are walking in the wrong direction. Please turn around and start walking again.” This length (4.5 m) was chosen through trial-and-error in initial experiments, and is consistent with the expected radius of uncertainty of localization. This notification is repeated if they continue walking in the wrong direction for another 4.5 meters. 4. When the next landmark is at less than 2 meters (Wayfinding only). “Landmark nearby. [Landmark name] to the [left/right].” Note that a shorter threshold distance than for approaching junctions is used here. This is justified by the fact that, while a junction needs to be announced before the walker has reached it, advance notice is less critical for announcing nearby “landmarks”. Experiment ========== [ht] [tab:participants] \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| \| \| \| Blindness \| \| Step length (cm) \| Step length (cm) \| RoNIN \| Preferred \| \| \| \| \| onset \| \| (calibration) \| (final) \| multiplier \| units \| \| P1 \| F \| 73 \| L \| Dog \| 48 \| – \| 0.96 \| Steps \| \| P2 \| M \| 69 \| B \| Cane \| 51 \| – \| 1.08 \| Feet \| \| P3 \| M \| 53 \| B \| Cane \| 54 \| – \| 1.14 \| Feet \| \| P4 \| F \| 69 \| B \| Cane \| 51 \| 44 \| 1.0 \| Feet \| \| P5 \| M \| 75 \| L \| Cane \| 44 \| 41 \| 1.21 \| Meters \| \| P6 \| F \| 76 \| L \| Cane \| 40 \| 40 \| 1.11 \| Steps \| \| P7 \| F \| 72 \| L \| Dog \| 63 \| 58 \| 1.08 \| Feet \| Population ---------- We recruited 7 participants for this experiment. The participants’ characteristics are summarized in Tab. [tab:participants]. All participants were blind, with at most some residual light perception. All participants were experienced independent walkers, although P6 had recently switched to using a long cane after many years of walking with a dog guide, and was still re-learning to use the cane. P5 had a hearing impairment and used hearing aids. All participants were iPhone users, except for P7, who used a cell phone with a physical keypad. Only P1 regularly wore a smartwatch (Apple Watch). Tab. [tab:participants] also shows the average step length values measured during the initial calibration, as well as the average step length from the particles at the end of the trials (remember that step length was added as a state in the Particle Filter only beginning with P4). Note that the step length measured in the initial calibration is always larger than or equal to that found by the Particle Filter. This suggests that participants may have walked with a different step length during calibration (which was conducted in a “safe”, straight corridor stretch) and during the actual test, where they might have felt less confident and therefore walked with shorter steps. \| \| \| \| --- \| --- \| \| image \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| \| image \| image \| \| (e) \| (f) \| [fig:pictures] Procedure --------- ### Environment The trials were conducted on the second floor of a campus building. Three routes were defined, as shown in Figs. [fig:R1W]–[fig:R3W] (a). When testing the Wayfinding app, participants first traversed route R1W (Fig. [fig:R1W] (a); length: 123 m), then route R2W (Fig. [fig:R2W] (a); length: 97 m), and finally route R3W (Fig. [fig:R3W] (a); length: 72 m). Note that the endpoints of these routes were at the location of a closed exit door, and that the endpoint of each route coincided with the starting point of the next route. R1W traversed one L-junction, two T-junctions, and two X-junctions, and included 4 turns. R2W contained two L-junctions, two T-junctions, and two X-junctions, with 5 turns. R3W contained two L-junctions and two X-junctions, with 4 turns. The width of the corridors ranged from 1.9 m to 6.0 m. The building was generally quiet, with a small number of students and researchers occasionally encountered in the corridors. As mentioned earlier, all junctions in this building were at 90∘. A few potentially challenging situations are shown in Fig. [fig:pictures], and include: a narrow door (always kept open) that had to be traversed (a); multiple alcoves that had to be negotiated (c); a side staircase in a narrow corridor that had to be avoided (e); a wide (10 m × 6 m) open space, partially visible in (d). When testing the Backtracking app, the same routes were traversed in the opposite direction and in the reverse order (these routes are called R1B, R2B, and R3B). So in practice, each participant first walked R1W, R2W, and R3W, using the Wayfinding app, while the Backtracking app (in a different smartphone) collected way-in data for each route. Then, they tested the Backtracking app by traversing R3B, R2B, and R1B. We would like to emphasize that the Backtracking app did not have knowledge of the building layout. A separate building was used for the practice trial when participants were shown how to use the two apps. The practice trial route was simpler, with only 2 turns, for a total length of 63 meters. The two buildings were at a short distance from each other. ### Modalities After being consented, each participant was explained the purpose of both apps and their functions and encouraged to ask questions if something was not clear. Particular care was taken to impress upon the participants that notifications about upcoming turns would be produced with advance notice. We explained that, upon hearing such a notification, they would need to search for the first place where they could take a turn. This place could be in their close proximity, or a few meters down the way. After this initial phase, participants underwent the simple calibration procedure described in Sec. [sec:localization]. Then, we walked with them to the location where the practice trial would start. Each participant carried two iPhones in their pants pocket (note that we asked the participants prior to the experiment to please wear pants with pockets on the experiment day). Both phones needed to be carried for the initial (Wayfinding) trials: while one of them (an iPhone 12) ran the Wayfinding app, the second one (an iPhone XR) recorded way-in data. Participants also wore a wireless bone conduction headset (Shokz OpenRun), through which they could receive notifications from apps. They also wore an Apple Watch Series 8, used to interact with the apps. Before beginning the practice trial, we made sure that the VoiceOver speed and sound volume were set to the desired level. We asked each participant whether they preferred directions to be given in units of meters, feet, or steps, and set the apps’ parameters accordingly (see Tab. [tab:participants]). Note that for the first three participants (P1–P3), due to an implementation mistake, the Backtracking app produced distances only in units of steps. We asked the participants to practice making left and right swipes on the Watch. All participants eventually learned how to do this, though P2 had some difficulties at the beginning, as he interpreted “right” or “left” as referring to the axis of his left arm (where he wore the Watch), i.e. in a direction orthogonal to the forearm, rather than parallel to it. At this point, the practice trial started: the practice route was traversed using the Wayfinding app; once at destination, it was traversed in the reverse direction using the Backtracking app. After the practice trial, participants were asked whether they wanted to turn off the sound of individual detected footsteps. All participants chose to leave the footstep sound on, with some commenting that hearing it made them feel reassured that the system was functioning. We also asked them whether they wanted to turn off notification of nearby landmarks (a few landmarks were announced during the practice trial). They all chose to leave landmark notifications on. Upon completion of the practice trials, the participant and the experimenters moved to the building where the actual experiment was to take place, and in particular to the starting point of the initial route (R1W). From there, the sequence of trials described in Sec. [sec:env] was started. At the beginning of each trial, the participant was accompanied to the starting location of the route, and their body was oriented along the initial walking direction. Then, they were asked to select the next route from the list, using the Watch. Once the route was selected, they were asked to rotate the Watch’s crown to start the app. We also asked them to make a left swipe on the Watch, to hear a description of the whole route. Then, the participant started walking the route. Upon arriving at destination, participants were asked to stop the app by rotating the Watch’s crown. They were asked if they wanted to rest a bit before starting the next trial. Then, they were re-positioning to the starting point of the next route (same as the endpoint of the previous route), and oriented correctly, before starting the next route. During the trials, the experimenters followed the participants at a safe distance, taking care not to influence their routing decision. At the transition between trials with the Wayfinding app and trials with the Backtracking app, participants were handed a new headset and Watch (same models), and asked to substitute the old ones with these new ones. Both devices can be paired with one iPhone at a time; since the Backtracking app ran on a different iPhone than the Wayfinding app, this switch was necessary. At the end of the last trial (route R1B with the Backtracking app), participants and experimenters walked back to the first building, where participants were asked to participate in a questionnaire, involving the ten System Usability Scale (SUS) questions, as well as a number of open ended questions. [h] [tab:performance] c\|c\|c\|c\|c\|c\|c\|c\|\|c\| &P1&P2&P3&P4&P5&P6&P7&Length *R1W&261 (R, E)&355 (R)&297 (R)& 216&271&223&206 &123 m R2W& 304 (E)&209&134&163&211&262&171&97 m R3W& 125&170&98&127&144&139&330&72 m R3B& 180&$\varprod$&115&134 (E)&136&$\varprod$&$\varprod$ &72 m R2B& 182 (E)&232&$\varprod$& 238&173&154&149 &97 m R1B& 187&206&149&167&$\varprod$, 163 (E)&184&$\varprod,\varprod$&123 m*************************** \| \| \| \| --- \| --- \| \| \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| [fig:R1W] \| \| \| \| --- \| --- \| \| \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| [fig:R2W] \| \| \| \| --- \| --- \| \| \| image \| \| (a) \| (b) \| \| image \| image \| \| (c) \| (d) \| [fig:R3W] [h!] [tab:landmarks] \| Landmark ID \| Landmark type \| \| --- \| --- \| \| 1 \| alcove \| \| 2 \| benches \| \| 3 \| staircase \| \| 4 \| photocopiers \| \| 5 \| alcove \| \| 6 \| alcove \| \| 7 \| pillar \| \| 8 \| couches \| \| 9 \| pillar \| \| 10 \| pillar \| \| 11 \| door \| \| 12 \| cabinets \| Observations ------------ ### Tracking and Guidance Performance: Wayfinding Tab. [tab:performance] reports the duration of successful route traversals. The average speed (route length divided by traversal time) was 0.50 m/s for all three routes. The first route (R1W) required an intervention from the experimenters for participants P1, P2, and P3. For P1, the localization algorithm used to generate notifications (A/S by default) had to be switched to RoNIN after the first turn, because A/S was unable to correctly track the participant (see Fig. [fig:R1W] (b), blue line). For P2 and P3, we had to manually reset the user’s location in the app. In all three cases, the remainder of the Wayfinding trials were completed without any issues (except for P1 in R2W, as discussed later). It is important to note that for these three participants, the step length update si had yet to be incorporated in our Particle Filter implementation (Sec. [sec:localization]), hence A/S relied solely on the step length measured during calibration. The first route segment of R1W is a long (more than 40 meters) and narrow corridor, with several obstacles and a staircase (see Fig. [fig:pictures] (e)) on its right side (participants were advised not to walk on the staircase). This caused the participants to walk with a substantially smaller stride length than during calibration. By the time they reached the end of the corridor, the accumulated length error was such that, even with Particle Filtering, tracking became exceedingly challenging. For P1, switching to RoNIN (which produced excellent results in this case; Fig. [fig:R1W] (b), red line) did the trick. However, RoNIN also occasionally gave poor results in the same area (e.g., it was unable to track P2). After we upgraded the Particle Filtering to include step length values, this problem no longer occurred, and we were able to use A/S successfully for all other participants. Tab. [tab:performance] also shows (cells filled in gray) situations in which participants successfully completed the trial, but occasionally had to trace back their path (as prompted by the app) because they had missed a turn. These situations are clearly visible in Fig. [fig:R1W] (d), Fig. [fig:R2W] (c) and (d), and Fig. [fig:R3W] (c) and (d). Later analysis, using the recorded video and logged data from the app, brought to light the reasons for these missed turns. In some cases, participants were distracted by passer-by (P5 in R1W), or they were speaking at the time notification was issued (P2 in R2W; Fig. [fig:R2W] (c)). P6 missed a turn in R3W while walking in a large open space (Fig. [fig:R3W] (c)). Probably due to the lack of a nearby wall, she might have been unsure about where she should turn exactly, so she continued walking until she perceived a wall to the right. By that time, she was notified to turn around. P7, who missed one turn in R1W and one in R2W, as well as multiple turns in R3W, walked very fast with a dog guide. In these situations, she had already walked past the junction by the time the notification was completed and she could process it. Of note, on route R3W, soon after the beginning of the trial, P7 walked straight into a small room whose door was open (Fig. [fig:R3W] (d)). She then missed other turns in the same route, as shown in the figure, one of which multiple times, until eventually she found the correct final route segment. Only one time (P6 in R2W) did the app issue a notification shortly after the participant had passed the junction, causing her to miss the turn. This was due to localization algorithm momentarily undershooting her position. However, after being notified that she should turn around, she successfully negotiated the turn. Participants using a long cane appeared to generally react proactively to early advance notice of upcoming turns. They typically would begin walking closer to the wall on the side where they were expected to turn, and tap the wall with their cane until they found an opening (Fig. [fig:pictures] (f)). In some cases, an alcove (wall recess) was located right before the corridor junction, and some participants got “stuck” in this alcove, before finding their way out and proceeding to the junction (Fig. [fig:pictures] (c)). A remarkable exception was represented by the mobility technique of P2. Rather than scanning the ground surface in front of himself, P2, a seasoned independent traveler, used the cane to periodically tap the floor surface, and made judgments about nearby surfaces based on the sound produced by this tapping. Upon hearing a notification of an upcoming junction, he would move somewhat closer to the corresponding side of the corridor (without tapping the wall) until he perceived the presence of an opening to his left or to his right, at which point he would make a 90∘ turn into it. However, in one situation in the R3W route, he missed a turn and had to then turn back as guided by the system (he then commented “I knew where that was exactly”, indicating that he may have indeed perceived the opening the first time around.) The two participants with a dog guide displayed very different behaviors. Since dog guides are normally trained to understand “left” or “right” directions, the participants simply needed (in principle) to convey the direction from the app verbally to their dogs. For the case of P7, this mechanism worked well, except for the fact that, as mentioned earlier, she walked really fast with her dog, thereby often passing by a junction to then receive a “turn around” notification. P1, on the other hand, worked with a slower and more cautious dog. For example, in route R1W, right after the second turn, they got stuck in an alcove to the right, and needed help from the experimenter to get out of it (marked as E in Tab. [tab:performance]). It is conceivable that if P1 had been using a cane, she could have searched left and right to find a way out of the alcove. In two other situations, her dog initially refused to turn into a specific corridor. In particular, at the beginning of R2W (Fig. [fig:R2W] (d)), she missed the first turn because, by the time she gave her dog a “right” command after the notification, they had already walked past the intersection. Afterwards, they correctly turned around as advised by the app. However, as she issued a “left” command to return to the route, the dog refused to turn and instead walked straight towards the exit door. At that time, we intervened, lest they would leave the building. It took some time before P1 managed to convince her dog to walk again on that corridor. After the study was completed, P1 explained to us that her dog might have felt nervous due to the circumstances of the trials. Both inertial sensing algorithms (A/S and RoNIN) were able to successfully track the participants through the routes, except for the initial part of R1W for the first three participants before the implementation of the adaptive step length mechanism, as discussed earlier. Occasionally, the reconstructed path “cut” some corners (see e.g. Fig. [fig:R1W] (c), RoNIN) or momentarily overshot the location (Fig. [fig:R3W] (b), A/S), but the drift tracking mechanism of our Particle Filtering implementation seemed to have worked well, at least for these trials. In particular, it is remarkable that both algorithms were able to track the rather chaotic path taken by P7 in R3W (Fig. [fig:R3W] (d)). ### Tracking and Guidance Performance: Backtracking As shown in Tab. [tab:performance], six trials with the Backtracking app had to be aborted, because the app failed to track the walker during the return route. In one of these trials (P3 in R2B), the participant had taken a wrong turn (for a reason described later in this section). The app correctly issued a “turn around” notification. However (as he later told us), P3 decided to instead keep walking, as he mistakenly thought that he remembered the path he had taken. As he kept walking away from the route, the app soon became unable to track him. Two trials had to be aborted for route R3B. One of them (with P7) was due to incorrect way-in path simplification, consequent to a “meandering” way-in path (Fig. [fig:optimization] (b)). In the case of P2 and P6, the culprit was a large magnetic field discrepancy (visible for P6 in Fig. [fig:Backtracking-bad]; note the white horizontal line). The first two segments of R1B are located in very wide corridors (Fig. [fig:pictures] (d)), and we noticed that the magnetic field changes dramatically from one side to the other of these corridors (as already remarked in Sec. [sec:localization]). The different locations of the walker in the corridors between way-in and return likely caused this large discrepancy. Another common situation occurred in route R2B. As can be seen in Fig. [fig:R2W] (a), the return route (starting from the star in that figure) goes through two L-junction, then crosses an X-junction. In our original implementation of the Backtracking app, the notification issued upon entering a new segment was similar to that of the Wayfinding app: “Walk straight for about XX [meters/feet/steps]. Then, you will turn [left, right].” This is the type of notification that was issued after the second L-turn in the return route R2B, indicating a final right turn after walking for approximately 15 meters. What happened with P2, P3, and P4 is that, rather than walking through the X-junction, they turned right on it, presumably because they knew they had to turn right at some point. This situation did not happen with the Wayfinding app (in the reverse route) because, as discussed in Sec. [sec:notifications], in the proximity of the X-junction the Wayfinding app would notify the walker to “Keep walking straight”. This option is not available to the Backtracking app. The reason is that the Backtracking app is only aware of the way-in route: it cannot possibly know that the user traversed an X junction at some point, and thus cannot remind the user, on the return route, to “walk straight” when in proximity of the junction. We thus decided to slightly change the notification format, by removing the last sentence (“Then, you will turn [left, right]”) when entering a new segment, and used this new implementation from P5 on. This ostensibly minimal interface modification was sufficient to avoid incorrect turns at this junction for the remaining participants. Participant P5 was not able to complete R1B, a route that initially proceeds straight through an X-junction (Fig. [fig:R1W] (a)). P5 soon started veering to the left	arxiv_0000395
Computing with Polynomial Ordinary Differential Equations ========================================================= In 1941, Claude Shannon introduced the General Purpose Analog Computer (GPAC) as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later on electronic) machines of that time. Following Shannon’s arguments, functions generated by the GPAC must satisfy a polynomial differential algebraic equation (DAE). As it is known that some computable functions like Euler’s Γ(x) = ∫0∞tx − 1e− tdt or Riemann’s Zeta function $\zeta(x)=\sum\_{k=0}^\infty \frac1{k^x}$ do not satisfy any polynomial DAE, this argument has often been used to demonstrate in the past that the GPAC is less powerful than digital computation. It was proved in, that if a more modern notion of computation is considered, i.e. in particular if computability is not restricted to real-time generation of functions, the GPAC is actually equivalent to Turing machines. Our purpose is first to discuss the robustness of the notion of computation involved in, by establishing that many natural variants of the notion of computation from this paper lead to the same computability result. Second, to go from these computability results towards considerations about (time) complexity: we explore several natural variants for measuring time/space complexity of a computation. Quite surprisingly, whereas defining a robust time complexity for general continuous time systems is a well known open problem, we prove that all variants are actually equivalent even at the complexity level. As a consequence, it seems that a robust and well defined notion of time complexity exists for the GPAC, or equivalently for computations by polynomial ordinary differential equations. Another side effect of our proof is also that we show in some way that polynomial ordinary differential equations can actually be used as a kind of programming model, and that there is a rather nice and robust notion of ordinary differential equation (ODE) programming. Analog Computation Continuous-Time Computations General Purpose Analog Computer Real Computations Introduction ============ Claude Shannon introduced in the General Purpose Analog Computer (GPAC) as a model for Differential Analysers, which are mechanical (and later on electronic) continuous time analog machines, on which he worked as an operator. The model was later refined in,. It was originally presented by Shannon as a model based on circuits. Basically, a GPAC is any circuit (loops are allowed[1](#fn1)) that can be built from the 4 basic units of Figure [fig:gpaccircuit], which implement constants, addition, multiplication and integration, all of them working over analog real quantities (that were corresponding to angles in the mechanical Differential Analysers, and later on to voltages in the electronic versions). Note that the set of allowed constants will generally be restricted, for example to rational numbers, to avoid pathological issues. Given such a circuit, the function which gives the value of every wire (or a subset of the wires) over time is said to be generated by the circuit. In Definition [def:gpacgenerableext], we consider an extension of this notion. [shift=(0,0),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) k; (.7,.35) – (1,.35); at (1,.35) k; at (.35, -.3) A constant unit; [shift=(4,0),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) + ; (.7,.35) – (1,.35); (-.3,.175) – (0,.175); (-.3,.525) – (0,.525); at (1,.35) u + v; at (.35, -.3) An adder unit; at (-.3,.525) u; at (-.3,.175) v; [shift=(0,-2),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) × ; (.7,.35) – (1,.35); (-.3,.175) – (0,.175); (-.3,.525) – (0,.525); at (1,.35) uv; at (.35, -.3) A multiplier unit; at (-.3,.525) u; at (-.3,.175) v; [shift=(4,-2),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) ∫; (.7,.35) – (1,.35); (-.3,.175) – (0,.175); (-.3,.525) – (0,.525); at (1,.35) $w=\int u\thinspace dv$; at (.35, -.3) An integrator unit; at (-.3,.525) u; at (-.3,.175) v; [fig:gpaccircuit] An important aspect of this model is that despite the apparent simplicity of its basic blocks, sophisticated functions can easily be generated. Figure [fig:gpacexamplesin] illustrates how the sine function can be generated using two integrators, with suitable initial states. Incidentally, the sine function is also the solution of a very simple ordinary differential equation. Shannon itself realized that functions generated by a GPAC are nothing more than solutions of a special class of polynomial differential equations. In particular it can be shown that a function $f: \R \to \R$ is generated by Shannon’s model, if and only if it is a (component of the) solution of a polynomial initial value problem (PIVP) of the form: $$\label{eq:ode} \left\{\begin{array}{@{}r@{}l}y'(t)&=p(y(t))\\y(t\_0)&=y\_0\end{array}\right.,\qquad t\in\R$$ where p is a vector of polynomials and y(t) is vector. In other words, f(t) = y1(t), and yiʹ(t) = pi(y(t)) where pi is a multivariate polynomial. [shift=(-4.2,0),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) − 1; (-3.5,.35) – (-3.15,.35) – (-3.15,.175) – (-2.8,.175); [shift=(-2.8,0),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) × ; (-2.1,.35) – (-1.75,.35) – (-1.75,.525) – (-1.4,.525); [shift=(-1.4,0),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) ∫; (-.7,.35) – (-.35,.35) – (-.35,.525) – (0,.525); [shift=(0,0),rotate=0] (0,0) – (0.7,0) – (0.7,0.7) – (0,0.7) – (0,0); at (.35,.35) ∫; (.7,.35) – (1.4,.35); at (1.4,.35) sin(t); at (-1, -0.5) $\left\lbrace \begin{array}{@{}c@{}l} y'(t)&=z(t)\\ z'(t)&=-y(t)\\ y(0)&=0\\ z(0)&=1 \end{array} \right.\Rightarrow\left\lbrace\begin{array}{@{}c@{}l} y(t)&=\sin(t)\\ z(t)&=\cos(t) \end{array}\right.$; (1,.35) – (1,1) – (-3.15,1) – (-3.15,.525) – (-2.8,.525); (1,.35) circle[radius=.07]; (-4.9,.35) – (-4.55,.35) – (-4.55,-.3) – (-.3,-.3) – (-.3,.175) – (0,.175); (-1.75,-.3) – (-1.75,.175) – (-1.4,.175); (-1.75,-.3) circle[radius=.07]; at (-4.9,.35) t; [fig:gpacexamplesin] Intuitively, the link between a GPAC and a PIVP is the following: the idea is just to introduce a variable for each output of a basic unit, and write the corresponding ordinary differential equation (ODE), and observe that it can be written as an ODE with a polynomial right hand side. While many of the usual real functions are known to be generated by a GPAC, a notable exception is Euler’s Gamma function Γ(x) = ∫0∞tx − 1e− tdt function or Riemann’s Zeta function $\zeta(x)=\sum\_{k=0}^\infty \frac1{k^x}$,, which are known not to satisfy any polynomial DAE, i.e. they are not solutions of a system of the form. If we have in mind that these functions are known to be computable under the computable analysis framework, the previous result has long been interpreted as evidence that the GPAC is a somewhat weaker model than computable analysis. In 2007, it was proved that this is more an artifact of the notion of real-time generation considered by Shannon than a true consideration about the computational power of the model. Indeed, Shannon assumes the GPAC computes in “real time” - a very restrictive form of computation: at time t the output of the machine must be Γ(t). If we change this notion of computability to the kind of “converging computation” used in recursive analysis, or in modern computability theory, then the Γ function becomes computable, and more generally all functions over a bounded domain, computable in the sense of computable analysis, are actually GPAC computable (and conversely). The idea used in, to compute a function $f:\R\rightarrow\R$ is to define a polynomial initial-value problem (PIVP) such that the argument x of f is provided to the PIVP via the initial condition, and the system has a component which converges to f(x). Moreover, the convergence rate of the component to f(x) is known and we know exactly how much time we have to wait to get a desired accuracy when computing f(x). More precisely, the following was proved: [GPAC computable function] [def:joc2007] $f:\R\rightarrow\R$ is called GPAC-computable if there are polynomials p and q with computable coefficients such that for any $x\in\R$, there exists (a unique) $y:I\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = q(x) and yʹ(t) = p(y(t)) $\blacktriangleright$ y satisfies a PIVP * if t ≥ 1 then ∣y1(t) − f(x)∣ ≤ e− t$\blacktriangleright$ y1 converges to f(x) [] [prop:avant] Let a and b be some computable reals. A function $f:[a,b] \to \R$ is computable[2](#fn2) if and only if it is GPAC-computable. In this paper our purpose is twofold: first explore natural variations on the notion of computability presented in Definition [def:joc2007] and, second, go towards complexity theory and not only computability theory, by introducing some natural ways to measure complexity. It is important to understand that talking about time complexity for continuous-time systems is known to be a non-trivial issue. Indeed, defining a robust (time) complexity notion for continuous time systems is a well known open problem with no generic solution provided at this day. In short, the difficulty is that the naive idea of using the time variable of the ODE as a measure of “time complexity” is problematic, since time can be arbitrarily contracted in a continuous system due to the “Zeno phenomena” (e.g. by using functions like arctan which contract the whole real line into a bounded set). It follows that all computable languages can then be computed by a continuous system in time O(1) (see e.g. ,,,,,,,,, ). Two first natural quantities will be considered: first, the time variable of the ordinary differential equation, that we will sometimes call time, and a bound on the norm of the involved variables, that we will sometimes call space. As a reparameterization of the time variable of an ordinary differential equation leads to a new ordinary differential equation with the same solution curve, but which is traveled along time at a different speed, a natural idea is to try to consider quantities that are kept invariant by reparameterization. A natural choice for such quantity is the length of the curve. We recall that the length of a curve $y\in C^1(I,\R^n)$ defined over some interval I = [a, b] is given by $\glen{y}(a,b)=\int\_I\infnorm{y'(t)}dt.$ Definition [def:joc2007] leads then naturally to consider the following natural two variants of computability of functions over $\R^n$ given below. Given $x\in\R^n$, we write xi for the ith component of x and xi..j for the vector (xi, xi +, …, xj). $\Rpoly$ denotes the set of polynomial-time computable reals. $\K[\R^n]$ denotes polynomial functions with n variables and with coefficients in $\K$, where variables live in $\R^n$ and $\Rp=[0,+\infty[$. In this document, f : ⊆ X → Y denotes a partial function, i.e. f : Z → Y where X ⊆ Z. We also take $\pastsup{\delta} f(t)=\sup\_{u\in[t,t-\delta]\cap\R\_+}f(t)$. The intuition is that in Definition [def:joc2007] we can reparameterize the time variable, but this will happen at the cost of space. Hence, if we want to know how many resources are needed to compute f(x) with some accuracy μ, we should measure not only the time but also the space needed to obtain this accuracy. This is done in Definition [def:gc], while Definition [def:glc] is a variant which, instead of using of measuring accuracy against time and space, does this by measuring accuracy against the length of the solution curve needed to achieve that accuracy. Figures [fig:glc] and [fig:gc] illustrate those definitions. [The space $\K$ of the coefficients] In this paper, the coefficients of all considered polynomials will belong to $\K$. Formally, $\K$ needs to a be generable field, as introduced in. However, without a significant loss of generality, the reader can consider that $$\K=\Rpoly$$ which is the set of polynomial time computable real numbers. All the reader needs to know about $\K$ is that it is a field and it is stable by generable functions (introduced in Section [sec:shannon]), meaning that if $\alpha\in\K$ and f is generable then $f(\alpha)\in\K$. It is shown in that there exists a small generable field $\Rgen$ lying somewhere between $\Q$ and $\Rpoly$, with probable strict inequality on both sides. We now get to our first and main notion of computable function: [Analog Length Computability][def:glc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$ and $\Omega:\Rp^2\rightarrow\Rp$. We say that f is Ω-length-computable if and only if there exist $d\in\N$, and $p\in\K^d[\R^d],q\in\K^d[\R^n]$ such that for any $x\in\dom{f}$, there exists (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = q(x) and yʹ(t) = p(y(t))$\blacktriangleright$ y satisfies a PIVP * for any $\mu\in\Rp$, if $\glen{y}(0,t)\geqslant\Omega(\infnorm{x},\mu)$ then $\infnorm{y\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$ $\blacktriangleright$ y1..m converges to f(x) * $\infnorm{y'(t)}\geqslant1$$\blacktriangleright$ technical condition: the length grows at least linearly with time[3](#fn3) We denote by $\glc{\Omega}$ the set of Ω-length-computable functions, and by $\gplc$ the set of $(\poly)$-length-computable functions, and more generally by $\cglc$ the length-computable functions (for some Ω). [Analog Time-Space computability][def:gc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$ and $\Upsilon,\Omega:\Rp^2\rightarrow\Rp$. We say that f is (Υ, Ω)-time-space-computable if and only if there exist $d\in\N$, and $p\in\K^d[\R^d],q\in\K^d[\R^n]$ such that for any $x\in\dom{f}$, there exists (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = q(x) and yʹ(t) = p(y(t))$\blacktriangleright$ y satisfies a PIVP * for all $\mu\in\Rp$, if $t\geqslant\Omega(\infnorm{x},\mu)$ then $\infnorm{y\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$ $\blacktriangleright$ y1..m converges to f(x) * $\infnorm{y(t)}\leqslant\Upsilon(\infnorm{x},t)$, for all t ≥ 0$\blacktriangleright$ y(t) is bounded We denote by $\gc{\Upsilon}{\Omega}$ the set of (Υ, Ω)-time-space-computable functions, by $\gpc$ the set of $(\poly,\poly)$-time-space-computable functions, and by $\cgc$ the set of time-space-computable functions. [domain=1:11,samples=500,scale=1] (0.9,-0.1) grid (11.1,4.1); (1,-0.1) – (1,4.2); (0.9,0) – (11.2,0) node[right] $\glen{y}$; plot[id=fn1] function 1+exp((x-1)\(7-x)/10)+(1+sin(10\x))/(1+exp(x-3))-2\exp(-(x-1)) ; (0.9,1) – (11.1,1); (11,1) node $f(\textcolor{blue}{x})$; (1.0,0.4) – (0.9,0.4); (0.95,0.4) node[left] $q\_1(\textcolor{blue}{x})$; (3.3,2.7) node y1; (7.0,0) – (7.0,-0.2); (7.0,1) – (7,2) node[midway,black,left] $e^{-\textcolor{mygreen}{0}}$; (7,-0.1) node[below] $\Omega(\textcolor{blue}{x},\textcolor{mygreen}{0})$; (8.4,1) – (8.4,1.36) node[midway,black,left] $\scriptstyle e^{-\textcolor{mygreen}{1}}$; (8.4,0) – (8.4,1); (8.4,0) – (8.4,-0.2); (8.4,-0.1) node[below] $\Omega(\textcolor{blue}{x},\textcolor{mygreen}{1})$; [domain=1:11,samples=500,scale=1] (0.9,-0.1) grid (11.1,4.1); (1,-0.1) – (1,4.2); (0.9,0) – (11.2,0) node[right] t; plot[id=fn2] function 1+exp((x-1)\(7-x)/10)/2+(1+sin(10\x))/(1+exp(x-3))-1.5\exp(-(x-1)) ; (0.9,1) – (11.1,1); (11,1) node $f(\textcolor{blue}{x})$; (1.0,0.4) – (0.9,0.4); (0.95,0.4) node[left] $q\_1(\textcolor{blue}{x})$; (1.6,2.5) node y1; (5.75,0) – (5.75,-0.2); (5.75,1) – (5.75,2) node[midway,black,left] $e^{-\textcolor{mygreen}{0}}$; (5.75,-0.1) node[below] ${\scriptstyle t\_0=}\Omega(\textcolor{blue}{x},\textcolor{mygreen}{0})$; (5.75,0) – (5.75,1); (7.45,1) – (7.45,1.36) node[midway,black,left] $\scriptstyle e^{-\textcolor{mygreen}{1}}$; (7.45,0) – (7.45,1); (7.45,0) – (7.45,-0.2); (7.45,-0.1) node[below] ${\scriptstyle t\_1=}\Omega(\textcolor{blue}{x},\textcolor{mygreen}{1})$; plot[id=fn3] function 1.5+x/5+sin(x\2)/2; (1.0,2.15) – (0.9,2.15); (0.95,2.15) node[left] $q\_2(\textcolor{blue}{x})$; (8.5,2.5) node y2; (1,2.92) – (5.75,2.92); (5.75,2) – (5.75,2.92); (0.9,2.92) – (1,2.92); (.9,2.92) node[left] $\Upsilon(\textcolor{blue}{x},t\_0)$; (1,3.42) – (7.45,3.42); (7.45,1.36) – (7.45,3.42); (0.9,3.42) – (1,3.42); (.9,3.42) node[left] $\Upsilon(\textcolor{blue}{x},t\_1)$; Indeed, Proposition [prop:avant] can be reformulated as: Let a* and b be some computable reals. A function $f:[a,b] \to \R$ is computable[4](#fn4) if and only if it is length-computable if and only if it is time-space-computable. More surprisingly, we prove that it turns out that both classes are the same, even at the complexity level. [th:main-weak] $\gplc=\gpc$. This turns out suprisingly to also be equivalent with many variants, both at the computability and complexity level. For example, the error could also be given as input, via an initial condition. The intuition behind the following definition is that the initial condition also depends on the accuracy μ. Hence, instead of what happens in Definition [def:gc], we are not guaranteed that a component converges to f(x), only that it stays in a e− μ-vicinity of f(x) after some time, and that the space used is bounded. [Analog weak computability][def:gwc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$, $\Omega:\Rp^2\rightarrow\Rp$ and $\Upsilon:\Rp^3\rightarrow\Rp$. We say that f is (Υ, Ω)-weakly-computable if and only if there exist $d\in\N$, $p\in\K^d[\R^d],q\in\K^d[\R^{n+1}]$ such that for any $x\in\dom{f}$ and $\mu\in\Rp$, there exists (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = q(x, μ) and yʹ(t) = p(y(t))$\blacktriangleright$ y satisfies a PIVP * if $t\geqslant\Omega(\infnorm{x},\mu)$ then $\infnorm{y\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$ $\blacktriangleright$ y1..m approximates f(x) * $\infnorm{y(t)}\leqslant\Upsilon(\infnorm{x},\mu,t)$ $\blacktriangleright$ y(t) is bounded We denote by $\gwc{\Upsilon}{\Omega}$ the set of (Υ, Ω)-weakly-computable functions, by $\gpwc$ the set of $(\poly,\poly)$-weakly-computable functions, and by $\cgwc$ the set of weakly-computable functions. Or we could consider a notion of online-computation, the intuition behind it being that if some external input x(t) approaches a value x̄ sufficiently close, then by waiting enough time, and assuming that the external input stays near the value x̄ during that time interval, we will get an approximation of f(x̄) with some desired accuracy. This process is illustrated in Figure [fig:goc]. By constantly changing the external input x(t) and “locking it” during some time near some value, we are able to compute approximations of f(x) for several arguments in a single “run” of the GPAC. [domain=1:11,samples=500,scale=1] [shift=(0,5.5)] (0.9,-0.1) grid (11.1,3.1); (1,-0.1) – (1,3.2); (0.9,0) – (11.2,0) node[right] t; plot[domain=1:4,id=fn4] function2.25+sin(x\7\pi)/4\(3+tanh((x-3)\(x-3)))/2; (0.9,2.25) – (4.8,2.25); (0.9,2.25) node[left] x̄; (1,2.75) – (5.2,2.75); (1,1.75) – (5.2,1.75); (5,1.75) – (5,2.75); (4.95,2.25) – (5.05,2.25); (5,2.45) node[right] $e^{-\Lambda(\textcolor{blue}{\bar{x}},\textcolor{mygreen}{1})}$; (4,2.25) – (7,.5); plot[domain=7:11,id=fn5] function0.5+sin(x\3\(3+tanh((x-9)\(x-9)))/4\pi)/4\(3+tanh((x-10)\(x-10)))/4; (6.2,0.75) – (11,0.75); (6.2,0.25) – (11,0.25); (6.4,0.25) – (6.4,0.75); (6.35,.5) – (6.45,.5); (6.4,.5) node[left] $e^{-\Lambda(\textcolor{blue}{\bar{x}'},\textcolor{mygreen}{2})}$; (6.6,.5) – (11.1,.5); (11.1,.5) node[right] x̄ʹ; [shift=(0,4.75)] (1,0) – (3.2,0); (3.2,0) – (4.87,0); (6.54,0) – (9.2,0); (9.2,0) – (11,0); (4.85,0) – (4.85,7.25); (3.18,0) – (3.18,4.75); (2.1,4.7) node[below,myyellow] undefined; (4,4.7) node[below,myyellow] accurate; (3,4.8) node[above,myyellow] stable; (5.7,4.8) node[above,myyellow] unstable; (5.7,4.7) node[below,myyellow] undefined; (6.55,0) – (6.55,6.25); (9.18,0) – (9.18,4.75); (7.9,4.7) node[below,myyellow] undefined; (10.1,4.7) node[below,myyellow] accurate; (9.1,4.8) node[above,myyellow] stable; (0.9,-0.1) grid (11.1,4.1); (1,-0.1) – (1,4.2); (0.9,0) – (11.2,0) node[right] t; plot[id=fn6] function 1+exp((x-1)\(7-x)/10)/2+(1+sin(10\x))/(1+exp(x-2.8))-1.5\exp(-(x-1)) +(1+tanh(100\(x-6)))/2\sin(10\x)\exp((6-x)/1.75) ; (0.9,2.42) – (4.85,2.42); (0.95,2.42) node[left] $f(\textcolor{blue}{\bar{x}})$; (1.0,0.4) – (0.9,0.4); (0.95,0.4) node[left] y0; (2.6,1.5) node y1; (3.2,2.8) – (6,2.8); (3.2,2.08) – (6,2.08); (5.75,2.08) – (5.75,2.8) node[midway,black,left] $e^{-\textcolor{mygreen}{1}}$; (5.7,2.44) – (5.8,2.44); (1,0) – (1,-0.2); (1,-.1) node[below] t1; (3.2,0) – (3.2,-0.2); (3.2,-0.1) node[below] ${\scriptstyle t\_1+}\Omega(\textcolor{blue}{\bar{x}},\textcolor{mygreen}{1})$; (9.2,1.05) – (11.1,1.05); (11,1.05) node[right] $f(\textcolor{blue}{\bar{x}'})$; (6.55,0) – (6.55,-0.2); (6.55,-.1) node[below] t2; (9.2,0) – (9.2,-0.2); (9.2,-0.1) node[below] ${\scriptstyle t\_2+}\Omega(\textcolor{blue}{\bar{x}'},\textcolor{mygreen}{2})$; (9.2,1.19) – (11,1.19); (9.2,0.91) – (11,0.91); [Online computability][def:goc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$ and $\Upsilon,\Omega,\Lambda:\Rp^2\rightarrow\Rp$. We say that f* is (Υ, Ω, Λ)-online-computable if and only if there exist δ ≥ 0, $d\in\N$ and $p\in\K^d[\R^d\times\R^{n}]$ and $y\_0\in\K^d$ such that for any $x\in C^0(\Rp,\R^n)$, there exists (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = y0 and yʹ(t) = p(y(t), x(t)) * $\infnorm{y(t)}\leqslant\Upsilon\big(\pastsup{\delta}{\infnorm{x}}(t),t\big)$ * for any $I=[a,b]\subseteq\Rp$, if there exist $\bar{x}\in\dom{f}$ and μ̄ ≥ 0 such that for all t ∈ I, $\infnorm{x(t)-\bar{x}}\leqslant e^{-\Lambda(\infnorm{\bar{x}},\bar{\mu})}$ then $\infnorm{y\_{1..m}(u)-f(\bar{x})}\leqslant e^{-\bar{\mu}}$ whenever $a+\Omega(\infnorm{\bar{x}},\bar{\mu})\leqslant u\leqslant b$. We denote by $\goc{\Upsilon}{\Omega}{\Lambda}$ the set of (Υ, Ω, Λ)-online-computable, by $\gpoc$ the set of $(\poly,\poly,\poly)$-online-computable functions and by $\cgoc$ the set of online-computable functions. [th:main] All notions of computations are equivalent, both at the computability level: $$\cglc=\cgc=\cgwc=\cgoc$$ and at the complexity level: $$\gplc=\gpc=\gpwc=\gpoc$$ The rest of the current paper is devoted to prove these equivalences between definitions. In Section [sec:shannon] we recall some results established by, and generalize several of them to multivariate functions. The proof of the previous Theorem [th:main] then follows but is however rather involved, and requires the introduction of other equivalent intermediate classes. We show several inclusions between these classes which will guarantee the result of Theorem [th:main]. First we show that $\gpc \subseteq \gpwc$, which follows from the fact that it is possible to rescale the system using the length of the curve as a new variable to make sure it does not grow faster than a polynomial (Section [app:gpcsubsetgpwc]). The other direction ($\gpwc \subseteq \gpc$) is really harder: the first step is to transform a computation into a computation that tolerates small perturbations of the dynamics ($\gpwc \subseteq \gprc$, Section [app:weaktorobust]). The second problem is to avoid that the system explodes for inputs not in the domain of the function ($\gprc \subseteq \gpsc$, Section [app:robusttostrong]). As a third step, we allow the system to have its inputs (input and precision) changed during the computation, but we require that the system has a maximum delay to react to these changes ($\gpsc \subseteq \gpuc$, Section [app:strongtounaware]). Finally, as a fourth step, we add a mechanism that feeds the system with the input and some precision. By continuously increasing the precision with time, we ensure that the system will converge when the input is stable. The result of these 4 steps is a lemma yielding a nice notion of online-computation ($\gpuc \subseteq \gpoc$, Section [app:unawaretoonline]). Equality $\gpc=\gpwc=\gpoc$ follows because time and length are related for polynomially bounded systems. A side effect of the closure properties of these classes, and of our proofs, is that programming with (polynomial length) ODE becomes a pleasant exercise, once the logic is understood. For example, simulating the assignment y : = g∞ corresponds to the dynamics of y(0) = y0, $y'(t)=\reach(y(t),g(t))+E(t)$, for a fixed function $\reach$, tolerating bounded error E(t) on dynamics, and g fluctuating around g∞. Other example: from a $\gpc$ system computing f, just adding the corresponding $\gpoc$-equations for g, yields a computing g ∘ f, by feeding the output of the system computing f to the (online) input of g. The PIVP Class ============== This sections recalls some known results about the class of functions generated by polynomial initial value problems. We omitted the proofs but this section contains all the necessary definitions and theorems needed to make this paper self-contained. Other lemmas related to the PIVP class are introduced in the paper when needed to avoid a long list of lemmas. A much more complete and detailed analysis of this class, with all the proofs, can be found in but we give a short overview below. Terminology is important here: the functions of this class are called generable, and should not be confused with the notion of computable function introduced earlier. Informally, the main results on this class are the following: * this class is stable by arithmetic operations and composition; * this class contains many useful functions such as trigonometric functions; * if yʹ = f(y) where f in this class, then y is also in this class. The general idea is that working directly with polynomial differential equations is a perilous exercise but it becomes easier if we can use more than polynomials. For example, assume that the above results are true, and consider the following differential equation: y(0) = 1, yʹ(t) = sin(y(t)). It can be seen that sin is generable so it follows that y is generable. Another example is the following differential equation: y(0) = 1, yʹ(t) = tanh(y(t)2). It can be seen again that tanh is generable, and polynomials are also generable so x ↦ tanh(x2) is generable, thus y is generable. Hopefully these two examples will convince the reader that this class gives us a lot of flexibility when writing differential systems. Another important aspect of this class is the growth of the functions. Without restrictions, it is very easy to build fast-growing functions, such as towers of exponentials. In this work, we crucially need to bound the growth of functions to limit the power of our systems. A necessary condition for this is that we should only write differential equations of the form yʹ = f(y) where f is generable and bounded by a polynomial. Of course this condition is trivially satisfied by polynomials but is also verified by many other functions such as sin or tanh. The following concept can be attributed to : a function $f: \R \to \R$ is said to be a function if there exists a system of the form with f(t) = y1(t) for all t, where y1 denotes the first component of the vector y defined in $\R^d$. We need in our proof to extend this concept to talk about (i) multivariable functions and (ii) the growth of these functions. This leads to the following: [Generable function ][def:gpacgenerableext] Let $d,e\in\N$, I be an open and connected subset of $\R^d$, $\mtt{sp}:\R\rightarrow\Rp$ and $f:I\rightarrow\R^e$. We say that $f\in\gval[\K]{\mtt{sp}}$ if and only if there exist n ≥ e, $p\in\MAT{n}{d}{\K}[\R^n]$, $x\_0\in\K^d$, $y\_0\in\K^n$ and $y:I\rightarrow\R^n$ satisfying for all x ∈ I: * y(x0) = y0 and $\jacobian{y}(x)=p(y(x))$ (i.e. ∂jyi(x) = pij(y(x))) $\blacktriangleright$ y satisfies a differential equation * f(x) = y1..e(x)$\blacktriangleright$ f is a component of y * $\infnorm{y(x)}\leqslant \mtt{sp}(\infnorm{x})$$\blacktriangleright$ y is bounded by $\mtt{sp}$ [Polynomially bounded generable function] The class of generable functions with polynomially bounded value is called $\gpval$: $$f\in\gpval\Leftrightarrow\text{ there exists a polynomial $\mtt{sp}$ such that }f\in\gval{\mtt{sp}}$$ The following closure properties can be seen as extensions of the results from to multivariate functions: [Arithmetic on generable functions ][lem:gpacextclassstable] Let d, e, n, $m\in\N$, $\mtt{sp},\ovl{\mtt{sp}}:\R\rightarrow\Rp$, $f:\subseteq\R^d\rightarrow\R^n\in\gval{\mtt{sp}}$ and $g:\subseteq\R^e\rightarrow\R^m\in\gval{\ovl{\mtt{sp}}}$. Then: * $f+g, f-g\in\gval{\mtt{sp}+\ovl{\mtt{sp}}}$ over $\dom{f}\cap\dom{g}$ if d = e and n = m * $fg\in\gval{\max(\mtt{sp},\ovl{\mtt{sp}},\mtt{sp}\thinspace\ovl{\mtt{sp}})}$ if d = e and n = m * $f\circ g\in\gval{\max(\ovl{\mtt{sp}},\mtt{sp}\circ\ovl{\mtt{sp}})}$ if m = d and $g(\dom{g})\subseteq \dom{f}$ Our key result is that the solution to an ODE whose right-hand side is generable, and possibly depends on an external and C1 control, may be rewritten as a GPAC. A corollary of this result is that the solution of a generable ODE is generable. [Generable ODE rewriting ][prop:gpacextivpstablepre] Let $d,n\in\N$, $I\subseteq\R^n$, $X\subseteq\R^d$, $\mtt{sp}:\Rp\rightarrow\Rp$ and $(f:I\times X\rightarrow\R^n)\in\gval[\K]{\mtt{sp}}$. Define $\ovl{\mtt{sp}}=\max(\idfun,\mtt{sp})$. Then there exist $m\in\N$, $(g:I\times X\rightarrow\R^m)\in\gval[\K]{\ovl{\mtt{sp}}}$ and $p\in\K^m[\R^m\times\R^d]$ such that for any interval J, $t\_0\in\K\cap J$, $y\_0\in\K^n\cap J$, y ∈ C1(J, I) and x ∈ C1(J, X), if y satisfies: $$\left\{\begin{array}{@{}r@{}l@{}}y(t\_0)&=y\_0\\y'(t)&=f(y(t),x(t))\end{array}\right.$$ then there exists $z\in C^1(J,\R^m)$ such that: $$\left\{\begin{array}{@{}r@{}l@{}}z(t\_0)&=g(y\_0,x(t\_0)) \\z'(t)&=p(z(t),x'(t))\end{array}\right. \qquad \left\{\begin{array}{@{}r@{}l@{}}y(t)&=z\_{1..d}(t)\\\infnorm{z(t)}&\leqslant\ovl{\mtt{sp}}(\infnorm{y(t),x(t)})\end{array}\right.$$ A simplified version of this lemma shows that generable functions are closed under ODE solving. [Closure under ODE of generable functions ][cor:gpacextivpstable] Let $d\in\N$, $J\subseteq\R$ an interval, $\mtt{sp},\ovl{\mtt{sp}}:\Rp\rightarrow\Rp$, $f:\subseteq\R^d\rightarrow\R^d$ in, $t\_0\in\K\cap J$ and $y\_0\in\K^d\cap\dom{f}$. Assume there exists $y:J\rightarrow\dom{f}$ satisfying for all t ∈ J: $$\left\{\begin{array}{@{}r@{}l@{}}y(t\_0)&=y\_0\\y'(t)&=f(y(t))\end{array}\right. \qquad\infnorm{y(t)}\leqslant\ovl{\mtt{sp}}(t)$$ Then $y\in\gval{\max(\ovl{\mtt{sp}},\mtt{sp}\circ \ovl{\mtt{sp}})}$ and is unique. It follows that many polynomially bounded usual analytic[5](#fn5) functions are in the class $\gpval$. We will also need the following results, which tell us how the solution of a GPAC varies if there is a slight change in the parameters defining it. In the next theorem $\sigmap{p}$ denotes the sum of the absolute values of the coefficients of the polynomial p. [Parameter dependency ][th:dependencyrightside] Let I = [a, b], $p\in\R^n[\R^{n+d}]$, $k=\degp{p}$, $e\in C^0(I,\R^d)$, $x,\delta\in C^0(I,\R^n)$ and $y\_0,z\_0\in\R^d$. Assume that $y,z:I\rightarrow\R^d$ satisfy: $$\left\{\begin{array}{@{}r@{}l}y(a)&=y\_0\\y'(t)&=p(y(t),x(t))\end{array}\right.\qquad \left\{\begin{array}{@{}r@{}l}z(a)&=z\_0\\z'(t)&=e(t)+p(z(t),x(t)+\delta(t))\end{array}\right.\qquad t\in I$$ Assume that there exists ɛ > 0 such that for all t ∈ I, $$\begin{gathered} \mu(t):= \\ \left(\infnorm{z\_0-y\_0}+\int\_{a}^t\infnorm{e(u)}+k\sigmap{p}M^{k-1}(u)\infnorm{\delta(u)}du \right) \exp \left(k\sigmap{p}\int\_{a}^tM^{k-1}(u)du\right) \\ <\varepsilon\end{gathered}$$ where $M(t)=\varepsilon+\infnorm{y(t)}+\infnorm{x(t)}+\infnorm{\delta(t)}$. Then for all t ∈ I, $$\infnorm{z(t)-y(t)}\leqslant\mu(t)$$ [Modulus of continuity ][prop:generablemodcont] Let $\mtt{sp}:\Rp\rightarrow\Rp$, $f\in\gval{\mtt{sp}}$. There exists $q\in\K[\R]$ such that for any $x\_1,x\_2\in\dom{f}$, if $[x\_1,x\_2]\subseteq\dom{f}$ then $$\infnorm{f(x\_1)-f(x\_2)}\leqslant\infnorm{x\_1-x\_2}q(\mtt{sp}(\max(\infnorm{x\_1},\infnorm{x\_2}))).$$ In particular, if $f\in\gpval[\K]$ then there exists $q\in\K[\R]$ such that if $[x\_1,x\_2]\subseteq\dom{f}$ then $$\infnorm{f(x\_1)-f(x\_2)}\leqslant\infnorm{x\_1-x\_2}q(\max(\infnorm{x\_1},\infnorm{x\_2})).$$ After these statements, we can go to the proof of Theorem [th:main]. This is done by proving various implications. Proof that ALP is ATSP ====================== The purpose of the current section is to show the following. $\gpc=\gplc$. Some remarks ------------ We start by a remark: [Norm function, ][lem:norm] There is a family of functions $\norm\_{\infty,\delta}\in\gpval$ such that, for any $x\in\R^n$ and δ ∈ ]0, 1], we have: $$\infnorm{x}\leqslant\norm\_{\infty,\delta}(x)\leqslant\infnorm{x}+\delta.$$ The proof --------- In one direction the proof is simple because if the system uses polynomial time and space then there is a relationship between time and length and we only need to add one variable to the system to make sure that the technical condition holds. The other direction is more involved because we need to rescale the system using the length of the curve to make sure it does not grow faster than a polynomial, which is ensured by the technical condition. Let $f\in\gc{\Upsilon}{\Omega}$ where Υ and Ω are polynomials, which we assume to be increasing functions. Apply Definition [def:gc] to get d, p, q, let $k=\degp{p}$ and define: $$\Omega^\(\alpha,\mu)=\Omega(\alpha,\mu)\left(1+\sigmap{p}\max\big(1,\Upsilon(\infnorm{x},\Omega(\alpha,\mu))\big)^k\right)$$ Let $x\in\dom{f}$ and consider the following system: $$\left\{\begin{array}{@{}r@{}l} y(0)&=q(x)\\z(0)&=0\end{array}\right.\qquad \left\{\begin{array}{@{}r@{}l} y'(t)&=p(y(t))\\z'(t)&=1\end{array}\right.$$ Note that z(t) = t* (this variable is there only to ensure that the length of z grows at least linearly). Let $t,\mu\in\Rp$ and assume that $\glen{z}(0,t)\geqslant\Omega^\(\infnorm{x},\mu)$. We will show that $t\geqslant\Omega(\infnorm{x},\mu)$ by contradiction. Assume the contrary and let u* ∈ [0, t]. By definition: $$\begin{aligned} \infnorm{y(u),z(u)} &\leqslant1+\infnorm{y(u)}\leqslant1+\Upsilon(\infnorm{x},t)\\ &<1+\Upsilon(\infnorm{x},\Omega(\infnorm{x},\mu))\end{aligned}$$ and thus $$\begin{aligned} \infnorm{y'(u),z'(u)}&=\infnorm{1,p(y(u))}\\ &<1+\sigmap{p}\big(1+\Upsilon(\infnorm{x},\Omega(\infnorm{x},\mu)))^k.\end{aligned}$$ Consequently: $$\glen{y,z}(0,t)<t\sup\_{u\in[0,t]}\infnorm{y'(u),z'(u)}\leqslant\Omega^\(\infnorm{x},\mu)$$ which is absurd. Since $t\geqslant\Omega(\infnorm{x},\mu)$, by definition we get that $$\infnorm{y\_{1..m}(t)-f(x)}\leqslant e^{-\mu}.$$ Finally, $\infnorm{y'(t),z'(t)}\geqslant\infnorm{z'(u)}\geqslant1$ for all $t\in\Rp$. This shows that that $f\in\glc{\Omega^\}$ where Ω\* is a polynomial. Let $f\in\glc{\Omega}$ where Ω is a polynomial, which we assume to be an increasing function. Apply Definition [def:glc] to get Ω, d, p, q. Also assume that the polynomial Ω is an increasing function. Let $k=\degp{p}$. Apply Lemma [lem:norm] to get that $g(x)=\norm\_{\infty,1}(p(x))$ belongs to $\gpval$. Apply Definition [def:gpacgenerableext] to get the corresponding m, r, x0 and z0. Let $x\in\dom{f}$. For the analysis, it will be useful to consider the following systems: $$\left\{\begin{array}{@{}r@{}l} y(0)&=q(x)\\z(x\_0)&=z\_0 \end{array}\right.\qquad \left\{\begin{array}{@{}r@{}l} y'(t)&=p(y(t))\\ \jacobian{z}(x)&=r(z(x))\end{array}\right.$$ Note that by definition z1(x) = g(x). Define ψ(t) = g(y(t)) and ψ̂(u) = ∫0uψ(t)dt. Now define the following system: $$\left\{\begin{array}{@{}r@{}l} \hat{y}(0)&=q(x)\\\hat{z}(0)&=z(q(x))\\\hat{w}(0)&=\frac{1}{g(q(x))} \end{array}\right.\qquad \left\{\begin{array}{@{}r@{}l} \hat{y}'(u)&=\hat{w}(u)p(\hat{y}(u))\\ \hat{z}'(u)&=\hat{w}(u)r(\hat{z}(u))p(\hat{y}(u))\\ \hat{w}'(u)&=-\hat{w}(u)^3r\_1(\hat{z}(u))p(\hat{y}(u)) \end{array}\right.$$ where by r1 we mean the first row of r. We will check that ŷ(u) = y(ψ̂− 1(u)), ẑ(u) = z(ŷ(u)) and ŵ(u) = (ψ̂− 1)ʹ(u). We will use the fact that for any h ∈ C1, $(h^{-1})'=\frac{1}{h'\circ h^{-1}}$. Also note that ψ̂ʹ = ψ. * ŷ(0) = y(ψ̂− 1(0)) = y(0) = q(x) * ŷʹ(u) = (ψ̂− 1)ʹ(u)yʹ(ψ̂− 1(u)) = ŵ(u)p(y(ψ̂− 1(u))) = ŵ(u)p(ŷ(u)) * ẑ(0) = z(ŷ(0)) = z(q(x)) * $\hat{z}'(u)=\jacobian{z}(\hat{y}(u))\hat{y}'(u)=\hat{w}(u)r(z(\hat{y}(u)))p(\hat{y}(u))=\hat{w}(u)r(\hat{z}(u))p(\hat{y}(u))$ * $\hat{w}(0)=\frac{1}{\hat{\psi}'(\hat{\psi}^{-1}(0))}=\frac{1}{\psi(0)}=\frac{1}{g(q(x))}$ * $\hat{w}'(u)=\frac{-(\hat{\psi}^{-1})'(u)\hat{\psi}''(\hat{\psi}^{-1}(u))}{\cramped{(\hat{\psi}'(\hat{\psi}^{-1}(u)))^2}} =-\hat{w}(u)^3\psi'(\hat{\psi}^{-1}(u))=\scalarprod{\grad{g}(y(\hat{\psi}^{-1}(u)))}{y'(\hat{\psi}^{-1})}$ and since $\grad{g}(x)=\transpose{r\_1(z(x))}$ (transpose of the first row of the Jacobian matrix of z because g = z1) then $$\hat{w}'(u) =-\hat{w}(u)^3\scalarprod{\transpose{r\_1(z(y(\hat{\psi}^{-1}(u))))}}{p(y(\hat{\psi}^{-1}(u)))} =-\hat{w}(u)^3r\_1(\hat{z}(u))p(\hat{y}(u))$$ We now claim that this system computes f quickly and has a polynomial bound. First note that by Lemma [lem:norm]: $$\infnorm{y'(t)}\leqslant g(y(t))\leqslant\infnorm{y'(t)}+1$$ and thus $$\glen{y}(0,t)\leqslant\hat{\psi}(t)\leqslant\glen{y}(0,t)+t.$$ Thus $$\begin{aligned} \glen{\hat{y}}(0,u)&=\int\_0^u\infnorm{\hat{y}'(\xi)}d\xi= \int\_0^{\hat{\psi}^{-1}(u)}\infnorm{\hat{w}(\hat{\psi}(t))p(\hat{y}(\hat{\psi}(t)))}\hat{\psi}'(t)dt \\&=\int\_0^{\hat{\psi}^{-1}(u)}\infnorm{(\hat{\psi}^{-1})'(\hat{\psi}(t))\hat{\psi}'(t)p(y(t))}dt \\ &=\int\_0^{\hat{\psi}^{-1}(u)}\infnorm{p(y(t))}dt=\glen{y}(0,\hat{\psi}^{-1}(u))\leqslant\hat{\psi}(\hat{\psi}^{-1}(u))\leqslant u.\end{aligned}$$ It follows that $$\infnorm{\hat{y}(u)}\leqslant\infnorm{\hat{y}(0)}+u\leqslant\infnorm{q(x)}+u\leqslant\poly(\infnorm{x},u).$$ Similarly: $$\infnorm{\hat{z}(u)}=\infnorm{z(\hat{y}(u))}\leqslant\poly(\infnorm{x},u)$$ because $z\in\gpval$ and thus is polynomially bounded. Finally, $$\infnorm{\hat{w}}=\frac{1}{\psi(\hat{\psi}^{-1}(u)}=\frac{1}{g(\hat{y}(u))}\leqslant\frac{1}{\infnorm{y'(\hat{\psi}^{-1}(u))}}\leqslant1$$ because by hypothesis, $\infnorm{y'(t)}\geqslant1$ for all $t\in\Rp$. This shows that indeed $\infnorm{(\hat{y},\hat{z},\hat{w})(u)}$ is polynomially bounded in $\infnorm{x}$ and u. Now let $\mu\in\Rp$ and $t\geqslant1+\Omega(\infnorm{x},\mu)$ then $$\begin{aligned} \glen{\hat{y}}(0,t)&=\glen{y}(0,\hat{\psi}^{-1}(t))\\ &\geqslant\hat{\psi}(\hat{\psi}^{-1}(t))-\hat{\psi}^{-1}(t)\\ &\geqslant t-\hat{\psi}^{-1}(t)\\ &\geqslant 1+\Omega(\infnorm{x},\mu)-\frac{1}{\psi(\hat{\psi}^{-1}(t))}\\ &\geqslant \Omega(\infnorm{x},\mu)\end{aligned}$$ because, as we already saw, $\infnorm{\psi(\hat{\psi}^{-1}(t))}\geqslant1$. Thus by definition: $$\infnorm{\hat{y}\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$$ because ŷ(t) = y(ψ̂− 1(t)). This shows that $f\in\gpc$. Proof that ALP implies AWP ========================== The purpose of the current section is to state the following. $\gpc=\gpwc$. The inclusion $\gpc\subseteq\gpwc$ is immediate from Definitions [def:gc] and [def:gwc]. The other inclusion will follow from the results of the other sections. Proof that AWP implies ARP ========================== The purpose of the current section is to state the following: [th:weaktorobust] $\gpwc\subseteq\gprc$. i.e. that it possible to transform a computation into a computation that tolerates small perturbations of the dynamics, where: [Analog robust computability][def:grc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$, $\Theta,\Omega:\Rp^2\rightarrow\Rp$ and $\Upsilon:\Rp^3\rightarrow\Rp$. We say that f is (Υ, Ω, Θ)-robustly-computable if and only if there exist $d\in\N$, and $(h:\R^d\rightarrow\R^d),(g:\R^n\times\Rp\rightarrow\R^d)\in\gpval$ such that for any $x\in\dom{f}$, $\mu\in\Rp$, $e\_0\in\R^d$ and $e\in C^0(\Rp,\R^d)$ satisfying $$\infnorm{e\_0}+\int\_0^\infty\infnorm{e(t)}dt\leqslant e^{-\Theta(\infnorm{x},\mu)},$$ there exists (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = g(x, μ) + e0 and yʹ(t) = h(y(t)) + e(t)$\blacktriangleright$ y satisfies a generable IVP * if $t\geqslant\Omega(\infnorm{x},\mu)$ then $\infnorm{y\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$ $\blacktriangleright$ y1..m approximates f(x) * $\infnorm{y(t)}\leqslant\Upsilon(\infnorm{x},\mu,t)$ $\blacktriangleright$ y(t) is bounded We denote by $\grc{\Upsilon}{\Omega}{\Theta}$ the set of (Υ, Ω, Θ)-robustly-computable functions, and by $\gprc$ the set of $(\poly,\poly,\poly)$-robustly-computable functions. Intuitively, this definition says that even if the initial condition and the ODE defining the PIVP are (slightly) perturbed or have (small) errors in Definition [def:gc], the PIVP is still capable of computing an approximation of f(x). Actually, we prove in this section that $\gpwc\subseteq\gprc$. Then the equality will follow from results of other sections. Some remarks ------------ [Domain of definition of g and h] There is a subtle but important detail in this definition: we more or less replaced the polynomials p and q by generable functions g and h. It could have been tempting to take this opportunity to restrict the domain of definition of g to $\dom{f}\times\Rp$ and that of h to a subset of $\R^d$ where the dynamics takes place. We kept the entire euclidean space for good reasons. First it makes the definition simpler. Second, it makes the notion stronger and more useful. This last point is important because we are going to use robust computability (and the next notion of strong computability) in cases where we have less or no control over the errors and thus over the trajectory of the system. On the downside, this requires to check that g and h are indeed defined over the entire space! The examples below show how to build robustly-computable functions. In the first example, we only need to define Θ so that it works, whereas in the second case, careful design of the system is needed for it to be robust. [Polynomials are robustly-computable] In order to make polynomials robustly-computable, we will play with the choice of Θ and see that this is enough to make the system robust. Let $q\in\K[\R^d]$ be a multivariate polynomial: we will show that $q\in\gprc$. Let $x\in\R^d$, $\mu\in\Rp$, $e\_0\in\R$ and $e\in C^0(\Rp,\R)$. Assume that ∣e0∣ + ∫0∞∣e(t)∣dt ≤ e− μ and consider the following system for $t\in\Rp$: y(0) = q(x) + e0 yʹ(t) = e(t) We claim that this system satisfies Definition [def:grc]: * The system is of the form $y(0)=\poly(x)+e\_0$ and $y'(t)=\poly(y(t))+e(t)$ where the polynomials have coefficients in $\K$. * For any t ≥ 0, we have: $$\infnorm{y(t)-q(x)}\leqslant \|e\_0\|+\int\_0^t\|e(u)\|du\leqslant\|e\_0\|+\int\_0^\infty\|e(u)\|du\leqslant e^{-\mu}$$ so we can take Ω(α, μ) = 0. * For any $t\in\Rp$, we have: $$\infnorm{y(t)}\leqslant\infnorm{q(x)}+\|e\_0\|+\int\_0^t\|e(u)\|du\leqslant\infnorm{q(x)}+1\leqslant\poly(\infnorm{x})$$ so we can take Υ to be any polynomial such that $\Upsilon(\infnorm{x},\mu)\geqslant\infnorm{p(x)}+1$. This shows that $q\in\grc{\Upsilon}{\Omega}{\Theta}$ where Θ(α, μ) = μ. In the previous example, we saw that we could modify the associated system of some computable functions to make them robustly-computable. It appears that this is not a coincidence but a general fact. To understand how the proof works, one must first understand the problem. Let us consider a computable function $f:\subseteq\R^d\rightarrow\R$ in $\gwc{\Upsilon}{\Theta}$ and the associated system for $x\in\dom{f}$ and $\mu\in\Rp$: y(0) = q(x, μ) yʹ(t) = p(y(t)) This system converges to f(x) very quickly: $\infnorm{y\_1(t)-f(x)}\leqslant e^{-\mu}$ when $t\geqslant\Omega(\infnorm{x},\mu)$ and y(t) is bounded: $\infnorm{y(t)}\leqslant\Upsilon(\infnorm{x},\mu,t)$. Let us introduce some errors in the system by taking $e\_0\in\R^d$ and $e\in C^0(\Rp,\R^d)$ such that $\infnorm{e\_0}+\int\_0^\infty\infnorm{e(u)}du\leqslant e^{-\Theta(\infnorm{x},\mu)}$ for some unspecified Θ and consider the perturbed system: z(0) = q(x, μ) + e0 zʹ(t) = p(z(t)) + e(t) The relationship between this system and the previous one is given by Theorem [th:dependencyrightside] and can be informally written as: $$\begin{aligned} \label{eq:rel\_weak:informal\_error} \infnorm{z(t)-y(t)}&\leqslant\left(\infnorm{e\_0}+\int\_0^t\infnorm{e(u)}du\right)e^{\int\_0^tk\sigmap{p}\infnorm{y(u)}^{k-1}du}\\ &\leqslant\left(\infnorm{e\_0}+\int\_0^\infty\infnorm{e(u)}du\right)e^{\int\_0^tk\sigmap{p}\Upsilon(\infnorm{x},\mu,u)^{k-1}du}\tag\{using the bound of $y(t)$}\\ &\leqslant e^{k\sigmap{p}t\Upsilon(\infnorm{x},\mu,t)^{k-1}-\Theta(\infnorm{x},\mu)}\tag\{assuming that $\Upsilon$ is increasing}\end{aligned}$$ One observes that this bound grows to infinity whatever we choose for Θ because of the dependency in t. On the other hand, we do not need to simulate y for arbitrary large t: as soon as $t\geqslant\Theta(\infnorm{x},\mu)$ we can stop the system and get a good enough result. Unfortunately, one does not simply stop a differential system, however we can slow it down. To this end, introduce $\psi(t)=(1+\Theta(\infnorm{x},\mu))\tanh(t)$ and w(t) = z(ψ(t)). If we show that w satisfies a differential system, then we are almost done. Indeed $\psi(t)\leqslant1+\Theta(\infnorm{x},\mu)$ for all $t\in\Rp$ and if $t\geqslant1+\Theta(\infnorm{x},\mu)$ then $\psi(t)\geqslant\Theta(\infnorm{x},\mu)$, so the system “kind of stops” between $\Theta(\infnorm{x},\mu)$ and $\Theta(\infnorm{x},\mu)+1$. Furthermore, if $t\geqslant1+\Theta(\infnorm{x},\mu)$ then: $$\begin{aligned} \infnorm{w\_1(t)-f(x)}&\leqslant\infnorm{z(\psi(t))-y(\psi(t))}+\infnorm{y\_1(\psi(t))-f(x)}\tag\{use the triangle inequality}\\ &\leqslant e^{k\sigmap{p}\psi(t)\Upsilon(\infnorm{x},\mu,\psi(t))^{k-1}-\Theta(\infnorm{x},\mu)}+e^{-\mu} \tag\{using \eqref{eq:rel\_weak:informal\_error}}\\ &\leqslant e^{k\sigmap{p}(1+\Theta(\infnorm{x},\mu))\Upsilon(\infnorm{x},\mu,1+\Theta(\infnorm{x},\mu))^{k-1}-\Theta(\infnorm{x},\mu)}+e^{-\mu}\tag\{using the bound on $\psi$}\\ &\leqslant 2e^{-\mu}\tag\{for a suitable choice of $\Theta$}\end{aligned}$$ We are left with showing that w(t) = z(ψ(t)) can be be generated by a generable IVP with perturbations. In the case of no perturbations, this is very easy because wʹ(t) = ψʹ(t)zʹ(t) = x(1 − tanh(t))p(z(t)) which is generable. The following lemma extends this idea to the case of perturbations. [PIVP Slow-Stop][lem:pivpslowstop] Let $d\in\N$, $y\_0\in\R^d$, $T,\theta\in\Rp$, $(e\_{0,y},e\_{0,A})\in\R^{d+1}$, $(e\_y,e\_A)\in C^0(\Rp,\R^{d+1})$ and $p\in\K^d[\R^d]$. Assume that $\infnorm{e\_0}+\int\_0^\infty\infnorm{e(t)}dt\leqslant e^{-\theta}$ and consider the following system: $$\left\{\begin{array}{@{}r@{}l}y(0)&=y\_0+e\_{0,y}\\ A(0)&=T+2+e\_{0,A}\end{array}\right. \quad \left\{\begin{array}{@{}r@{}l}y'(t)&=\frac{1+\tanh(A(t))}{2}p(y(t))+e\_{y}(t)\\ A'(t)&=-1+e\_{A}(t)\end{array}\right.$$ Then there exist an increasing function $\psi\in C^0(\Rp,\Rp)$ and $z:\psi(\Rp)\rightarrow\R^d$ such that: ψ(0) = 0 z(0) = y0 + e0, y zʹ(t) = p(z(t)) + (ψ− 1)ʹ(t)ey(ψ− 1(t)) and y(t) = z(ψ(t)). Furthermore ψ(T + 1) ≥ T and ψ(t) ≤ T + 4 for all $t\in\Rp$. Furthermore, ∣A(t)∣ ≤ T + 3 for all $t\in\Rp$. Let $f(t)=\frac{1+\tanh(A(t))}{2}$ and note that 0 < f(t) < 1 for all $t\in\Rp$. Check that we can integrate A explicitly: A(t) = T + 2 − t + e0, A + ∫0teA(u)du. If we take ψ(t) = ∫0tf(u)du then ψ is an increasing function because f > 0, so it is a diffeomorphism from $\Rp$ onto $\psi(\Rp)$. Note that ψ(t) ≤ t for all $t\in\Rp$. Let t ≥ T + 3, then $$\begin{aligned} A(t)&\leqslant T+2-t+\|e\_{0,A}\|+\int\_0^t\|e\_A(u)\|du\\ &\leqslant T+2+e^{-\theta}-t\leqslant T+3-t\leqslant0\end{aligned}$$ because θ ≥ 0. Apply Lemma [lem:boundstanh] to get that tanh(A(t)) ≤ − 1 + eT + 3 − t and thus $$f(t)\leqslant \frac{1}{2}e^{T+3-t}\text{ for }t\geqslant T+3.$$ Integrating this inequality shows that $$\begin{aligned} \psi(t)&\leqslant\psi(T+3)+\frac{1}{2}\int\_{T+3}^{t}e^{T+3-u}du\\ &\leqslant T+3+\frac{1}{2}(1-e^{T+3-t})\leqslant T+4.\end{aligned}$$ This shows that ψ(t) ≤ T + 4 for all $t\in\Rp$. Let t ≤ T + 1, then by the same reasoning: A(t) ≥ T + 2 − t − e− θ ≥ T + 1 − t ≥ 0 thus tanh(A(t)) ≥ 1 − et − T − 1 and $f(t)\geqslant1-\frac{1}{2}e^{t-T-1}$. Thus: $$\psi(T+1)\geqslant\int\_0^{T+1}1+\frac{1}{2}e^{u-T-1}du=T+1+\frac{1}{2}(1-e^{-1-T})\geqslant T.$$ Finally, apply Lemma [lem:perturbedtimescalegpac] to get that y(t) = z(ψ(t)) where z satisfies for $t\in\psi(\Rp)$: z(0) = y(0) zʹ(t) = p(z(t)) + (ψ− 1)ʹ(t)ey(ψ− 1(t)) The proof --------- The proof of the implication AWP implies ARP of Theorem [th:weaktorobust] is then the following. Let Υ\, Ω\ be polynomials such that $f\in\gwc{\Upsilon^\}{\Omega^\}$. Without loss of generality, we assume they are increasing functions on both arguments. Apply Definition [def:gwc] to get $d\in\N$, $p\in\K^d[\R^d]$, $q\in\K^d[\R^{n+1}]$ and let $k=\degp{p}$. Define: $$\begin{aligned} T(\alpha,\mu)&=\Omega^\(\alpha,\mu+\ln2)\\ \Theta(\alpha,\mu)&=k\sigmap{p}(T(\alpha+1,\mu)+4)(\Upsilon^\(\alpha,\mu,T(\alpha+1,\mu)+4)+1)^{k-1}+\mu+\ln2\\ \Omega(\alpha,\mu)&=T(\alpha+1,\mu)+1\end{aligned}$$ Let $x\in\dom{f}$, $(e\_{0,y},e\_{0,A})\in\R^{d+1}$, $(e\_y,e\_A)\in C^0(\Rp,\R^{d+1})$ and $\mu\in\Rp$ such that $$\infnorm{e\_0}+\int\_0^\infty\infnorm{e(t)}dt\leqslant e^{-\Theta(\infnorm{x},\mu)}.$$ Apply Lemma [lem:pivpslowstop] and consider the following systems (where ψ is given by the lemma): $$\left\{\begin{array}{@{}r@{}l}y(0)&=q(x,\mu)+e\_{0,y}\\ A(0)&=T(\norm\_{\infty,1}(x),\mu)+2+e\_{0,A}\end{array}\right. \quad \left\{\begin{array}{@{}r@{}l}y'(t)&=\frac{1+\tanh(A(t))}{2}p(y(t))+e\_{y}(t)\\ A'(t)&=-1+e\_{A}(t)\end{array}\right.$$ $$\left\{\begin{array}{@{}r@{}l}z(0)&=q(x,\mu)+e\_{0,y}\\ z'(t)&=p(z(t))+(\psi^{-1})'(t)e\_{y}(\psi^{-1}(t))\end{array}\right. \quad \left\{\begin{array}{@{}r@{}l}w(0)&=q(x,\mu)\\ w'(t)&=p(w(t))\end{array}\right.$$ By definition of p and q, if $t\geqslant\Omega^\(\infnorm{x},\mu)$ then $\infnorm{w\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$. Furthermore, $\infnorm{w(t)}\leqslant\Upsilon^\(\infnorm{x},\mu,t)$ for all $t\in\Rp$. Define $T^\=T(\norm\_{\infty,1}(x),\mu)$. Apply Lemma [lem:norm] to get that $$\infnorm{x}\leqslant\norm\_{\infty,1}(x)\leqslant\infnorm{x}+1$$ and thus $$T(\infnorm{x},\mu)\leqslant T^\\leqslant T(\infnorm{x}+1,\mu).$$ By construction, ψ(t) ≤ T\* + 4 for all $t\in\Rp$. Let $t\in\Rp$, apply Theorem [th:dependencyrightside] by checking that: $$\begin{aligned} &\left(\infnorm{e\_{0,y}}+\int\_0^{\psi(t)}\infnorm{(\psi^{-1})'(u)e\_{y}(\psi^{-1}(u))}du\right) e^{k\sigmap{p}\int\_0^{\psi(t)}(\infnorm{w(u)}+1)^{k-1}du}\\ &\leqslant\left(\infnorm{e\_{0,y}}+\int\_0^t\infnorm{e\_{y}(u)}du\right) e^{k\sigmap{p}\int\_0^{\psi(t)}(\Upsilon^\(\infnorm{x},\mu,u)+1)^{k-1}du}\tag\{by a change of variable}\\ &\leqslant e^{k\sigmap{p}\psi(t)(\Upsilon^\(\infnorm{x},\mu,\psi(t))+1)^{k-1}-\Theta(\infnorm{x},\mu)}\tag\{by hypothesis on the error}\\ &\leqslant e^{k\sigmap{p}(T(\infnorm{x}+1,\mu)+4)(\Upsilon^\(\infnorm{x},\mu,T(\infnorm{x}+1,\mu)+4)+1)^{k-1}-\Theta(\infnorm{x},\mu)}\tag\{because $\psi$ is bounded}\\ &\leqslant e^{-\mu-\ln2}\leqslant1\tag\{by definition of $\Theta$}\end{aligned}$$ Thus $$\infnorm{z(\psi(t))-w(\psi(t))}\leqslant e^{-\mu-\ln2}\text{ for all }t\in\Rp.$$ Furthermore, if $t\geqslant\Omega(\infnorm{x},\mu)$ then $$\psi(t)\geqslant \psi(T(\infnorm{x}+1,\mu)+1)\geqslant\psi(T^\+1)\geqslant T^\.$$ By construction ψ(T\) ≥ T\* so $$\psi(t)\geqslant T^\\geqslant T(\infnorm{x},\mu)=\Omega^\(\infnorm{x},\mu+\ln2)$$ thus $$\infnorm{z(\psi(t))-f(x)}\leqslant e^{-\mu-\ln2}.$$ Consequently, we have $$\begin{aligned} \infnorm{y(t)-f(x)}&\leqslant\infnorm{z(\psi(t))-w(\psi(t))}+\infnorm{w(\psi(t))-f(x)}\\ &\leqslant 2e^{-\mu-\ln2}\leqslant e^{-\mu}.\end{aligned}$$ Let $t\in\Rp$, then $$\begin{aligned} \infnorm{y(t)}&=\infnorm{z(\psi(t))}\leqslant\infnorm{w(\psi(t))}+e^{-\mu}\\ &\leqslant\Upsilon^\(\infnorm{x},\mu,\psi(t))+1\leqslant\Upsilon^\(\infnorm{x},\mu,T(\infnorm{x}+1,\mu)+4)+1\\ &\leqslant\Upsilon^\(\infnorm{x},\mu,\Omega^\(\infnorm{x}+1,\mu+\ln2)+4)+1\end{aligned}$$ which is polynomially bounded in $\infnorm{x}$ and μ. Furthermore $$\|A(t)\|\leqslant T^\+4\leqslant\Omega^\(\infnorm{x}+1,\mu+\ln2)+4$$ which are both polynomially bounded in $\infnorm{x}$, μ. Finally, (y, A)(0) = g(x, μ) + e0 and (y, A)ʹ(t) = h(y(t), A(t)) + e(t) where g and h belong to $\gpval[\K]{}$ because $\tanh,\norm\_{\infty,1}\in\gpval$. [Polynomial versus generable][rem:weaktorobustgen] The proof of Theorem [th:weaktorobust] also works if q is generable (i.e. $q\in\gpval$) instead of polynomial in Definition [def:gc] or Definition [def:gwc]. Proof that ARP implies ASP ========================== This section is devoted to prove the following result: it is always possible to avoid that the system in Definition [def:grc]. explodes for inputs not in the domain of the function, or for perturbations of the dynamics which are too big. This motivates the following result and Definition [def:gsc]. [Robust ⊆ strong][th:robusttostrong] $\gprc=\gpsc$. where [Analog strong computability][def:gsc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$, $\Theta,\Omega:\Rp^2\rightarrow\Rp$ and $\Upsilon:\Rp^4\rightarrow\Rp$. We say that f is (Υ, Ω, Θ)-strongly-computable if and only if there exist $d\in\N$, and $(h:\R^d\rightarrow\R^d),(g:\R^n\times\Rp\rightarrow\R^d)\in\gpval$ such that for any $x\in\R^n$, $\mu\in\Rp$, $e\_0\in\R^d$ and $e\in C^0(\Rp,\R^d)$, there is (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$ and $\hat{e}(t)=\infnorm{e\_0}+\int\_0^t\infnorm{e(u)}du$: * y(0) = g(x, μ) + e0 and yʹ(t) = h(y(t)) + e(t)$\blacktriangleright$ y satisfies a generable IVP * if $x\in\dom{f}$, $t\geqslant\Omega(\infnorm{x},\mu)$ and $\hat{e}(t)\leqslant e^{-\Theta(\infnorm{x},\mu)}$ then $\infnorm{y\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$ * $\infnorm{y(t)}\leqslant\Upsilon(\infnorm{x},\mu,\hat{e}(t),t)$$\blacktriangleright$ y(t) is bounded We denote by $\gsc{\Upsilon}{\Omega}{\Theta}$ the set of (Υ, Ω, Θ)-strongly-computable functions, and by $\gpsc$ the set of $(\poly,\poly,\poly)$-strongly-computable functions. Actually, we prove in this section that $\gprc\subseteq\gpsc$. Equality follows from results in other sections. Some remarks ------------ The following Lemma can be proved by providing explicitly such a function: [Max function, ][lem:max] There is a family of functions $\mx\_\delta\in\gpval$ such that: For any $x,y\in\R$ and δ ∈ ]0, 1] we have: $$\max(x,y)\leqslant\mx\_\delta(x,y)\leqslant\max(x,y)+\delta$$ For any $x\in\R^n$ and δ ∈ ]0, 1] we have: $$\max(x\_1,\ldots,x\_n)\leqslant\mx\_\delta(x)\leqslant\max(x\_1,\ldots,x\_n)+\delta$$ The following lemmas can also be established: [Bounds on tanh, ][lem:boundstanh] $1-\sgn{t}\tanh(t)\leqslant e^{-\|t\|}$ for all $t\in\R$. [Perturbed time-scaling][lem:perturbedtimescalegpac] Let $d\in\N$, $x\_0\in\R^d$, $p\in\R^d[\R^d]$, $e\in C^0(\Rp,\R^d)$ and $\phi\in C^0(\Rp,\Rp)$. Let ψ(t) = ∫0tϕ(u)du. Assume that ψ is an increasing function and that $y,z:\Rp\rightarrow\R^d$ satisfy for all $t\in\Rp$: $$\left\{\begin{array}{@{}r@{}l}y(0)&=x\_0\\y'(t)&=p(y(t))+(\psi^{-1})'(t)e(\psi^{-1}(t))\end{array}\right.\qquad \left\{\begin{array}{@{}r@{}l}z(0)&=x\_0\\z'(t)&=\phi(t)p(z(t))+e(t)\end{array}\right.$$ Then z(t) = y(ψ(t)) for all $t\in\Rp$. In particular, $$\int\_0^{\psi(t)}\infnorm{(\psi^{-1})'(u)e(\psi^{-1}(u))}du=\int\_0^t\infnorm{e(u)}du$$ and $$\sup\_{u\in[0,\psi(t)]}\infnorm{(\psi^{-1})'(u)e(\psi^{-1}(u))}=\sup\_{u\in[0,t]}\frac{\infnorm{e(u)}}{\phi(u)}.$$ Use that ϕ = ψʹ, ψʹ ⋅ (ψ− 1)ʹ ∘ ψ = 1 and that ψʹ ≥ 0. On a more technical side, we will need to “apply” Definition [def:grc] over finite intervals and we need the following lemma to do so. [Finite time robustness][lem:finitetimerobust] Let $f\in\grc{\Upsilon}{\Omega}{\Theta}$, I = [0, T], $x\in\dom{f}$, $\mu\in\Rp$, $e\_0\in\R^d$ and $e\in C^0(I,\R^d)$ such that $$\infnorm{e\_0}+\int\_I\infnorm{e(t)}dt<e^{-\Theta(\infnorm{x},\mu)}.$$ Assume that $y:I\rightarrow\R^d$ satisfies for all t ∈ I: y(0) = g(x, μ) + e0 yʹ(t) = h(y(t)) + e(t) where g, h come from Definition [def:grc] applied to f. Then for all t ∈ I: * $\infnorm{y(t)}\leqslant\Upsilon(\infnorm{x},\mu,t)$ * if $t\geqslant\Omega(\infnorm{x},\mu)$ then $\infnorm{y\_{1..m}-f(x)}\leqslant e^{-\mu}$ The trick is simply to extend e so that it is defined over $\Rp$ and such that: $$\infnorm{e\_0}+\int\_0^\infty\infnorm{e(u)}du\leqslant e^{-\Theta(\infnorm{x},\mu)}$$ This is always possible because the truncated integral is strictly smaller than the bound. Formally, define for $t\in\Rp$: $$\bar{e}(t)=\begin{cases}e(t)&\text{if }t\leqslant T\\ e(T)e^{\frac{e(T)}{\varepsilon}(T-t)}&\text{otherwise}\end{cases}$$ $$\qquad\text{where }\varepsilon=e^{-\Theta(\infnorm{x},\mu)}-\infnorm{e\_0}-\int\_I\infnorm{e(t)}dt>0$$ One easily checks that $\bar{e}\in C^0(\Rp,\R^d)$ and that: $$\begin{aligned} \infnorm{e\_0}+\int\_0^\infty\infnorm{\bar{e}(t)}dt&=\infnorm{e\_0}+\int\_0^T\infnorm{e(t)}dt+\int\_T^\infty e(T)e^{\frac{e(T)}{\varepsilon}(T-t)}dt\\ &=e^{-\Theta(\infnorm{x},\mu)}-\varepsilon+\left[-\varepsilon e(T)e^{\frac{e(T)}{\varepsilon}(T-t)}\right]\_T^\infty\\ &=e^{-\Theta(\infnorm{x},\mu)}\end{aligned}$$ Assume that $z:\Rp\rightarrow\R^d$ satisfies for $t\in\Rp$: z(0) = g(x, μ) zʹ(t) = g(z(t)) + ē(t) Then z satisfies Definition [def:grc] so $\infnorm{z}(t)\leqslant\Upsilon(\infnorm{x},\mu)$ and if $t\geqslant\Omega(\infnorm{x},\mu)$ then $\infnorm{z\_{1..m}(t)-f(x)}\leqslant e^{-\mu}$. Conclude by noting that z(t) = y(t) for all t ∈ [0, T] since e(t) = ē(t). The proof --------- The proof of Theorem [th:robusttostrong] is then the following. Let Ω, Θ, Υ be polynomials and $(f:\subseteq\R^n\rightarrow\R^m)\in\grc{\Upsilon}{\Omega}{\Theta}$. Without loss of generality, we assume that Ω, Θ, Υ are increasing functions of their arguments. Apply Definition [def:grc] to get d, h and g. Let $x\in\R^n$, $\mu\in\Rp$, $(e\_{0,y},e\_{0,\ell})\in\R^{d+1}$ and $(e\_y,e\_\ell)\in C^0(\Rp,\R^{d+1})$. Define $\hat{e}(t)=\infnorm{e\_0}+\int\_0^t\infnorm{e(u)}du$, and consider the following system for $t\in\Rp$: $$\left\{\begin{array}{@{}r@{}l} y(0)&=g(x,\mu)+e\_{0,y}\\ y'(t)&=\psi(t)h(y(t))+e\_{y}(t)\\ \ell(0)&=\mx\_1(\norm\_{\infty,1}(x),\mu)+1+e\_{0,\ell}\\ \ell'(t)&=1+e\_{\ell}(t) \end{array}\right.$$ $$\psi(t)=\frac{1+\tanh(\Delta(t))}{2}\qquad \Delta(t)=\Upsilon(\ell(t),\ell(t),\ell(t))+1-\norm\_{\infty,1}(y(t))$$ We will first show that the system remains polynomially bounded. Apply Lemma [lem:max] and Lemma [lem:norm] to get that: $$\begin{aligned} \infnorm{\ell(0)}&\leqslant\max(\infnorm{x}+1,\mu)+1+\infnorm{e\_{0,\ell}}\\ &\leqslant\poly(\infnorm{x},\mu)+\infnorm{e\_{0,\ell}}\end{aligned}$$ Consequently: $$\begin{aligned} \infnorm{\ell(t)}&\leqslant\infnorm{\ell(0)}+\int\_0^t1+\infnorm{e\_{\ell}(u)}du\nonumber\\ &\leqslant \poly(\infnorm{x},\mu)+t+\infnorm{e\_{0,\ell}}+\int\_0^t\infnorm{e\_{\ell}(u)}du\nonumber\\ &\leqslant\poly(\infnorm{x},\mu)+t+\hat{e}(t)\nonumber\\ &\leqslant\poly(\infnorm{x},\mu,t,\hat{e}(t))\label{eq:robust\_to\_strong:ell}\end{aligned}$$ Since $g,h\in\gpval$, there exist two polynomials $\mtt{sp}$ and $\ovl{\mtt{sp}}$ such that $\infnorm{g(x)}\leqslant\mtt{sp}(\infnorm{x})$ and $\infnorm{h(x)}\leqslant\ovl{\mtt{sp}}(\infnorm{x})$ for all $x\in\R^d$ and without loss of generality, we assume that $\mtt{sp}$ and $\ovl{\mtt{sp}}$ are increasing functions. Let $t\in\Rp$, there are two possibilities: * If Δ(t) ≥ 0 then $\norm\_{\infty,1}(y(t))\leqslant1+\Upsilon(\ell(t),\ell(t),\ell(t))$ so apply Lemma [lem:norm] and use to conclude that $\infnorm{y(t)}\leqslant\poly(\infnorm{x},\mu,t,\hat{e}(t))$ and thus: $$\begin{aligned} \infnorm{\psi(t)h(y(t))}&\leqslant\ovl{\mtt{sp}}(\infnorm{y(t)})\tag\{use that $\tanh<1$}\\ &\leqslant\poly(\infnorm{x},\mu,t,\hat{e}(t))\label{eq:robust\_to\_strong:rhs\_one}\end{aligned}$$ If Δ(t) < 0 then apply Lemma [lem:boundstanh] to get that $\psi(t)\leqslant\frac{1}{2}e^{\Delta(t)}\leqslant e^{\Delta(t)}$. Apply Lemma [lem:norm] to get that $\Delta(t)\leqslant\Upsilon(\ell(t),\ell(t),\ell(t))+1-\infnorm{y(t)}$ and thus $\infnorm{y(t)}\leqslant \Upsilon(\ell(t),\ell(t),\ell(t))+1-\Delta(t)$ and thus: $$\begin{aligned} \infnorm{\psi(t)h(y(t))}&\leqslant e^{\Delta(t)}\ovl{\mtt{sp}}(\infnorm{y(t)})\tag\{use the bound on $\psi$}\\ &\leqslant e^{\Delta(t)}\ovl{\mtt{sp}}(\Upsilon(\ell(t),\ell(t),\ell(t))+1-\Delta(t))\tag\{use the bound on $\infnorm{y(t)}$}\\ &\leqslant\poly(\ell(t))e^{\Delta(t)}\poly(-\Delta(t))\tag\{use that $\Upsilon$ is polynomial}\\ &\leqslant\poly(\ell(t))\tag\{use that $e^{-x}\poly(x)=\bigO{1}$ for $x\geqslant0$ and fixed $\poly$}\\ &\leqslant\poly(\infnorm{x},\mu,t,\hat{e}(t))\label{eq:robust\_to\_strong:rhs\_two}\end{aligned}$$ Putting and together, we get that: $$\begin{aligned} \infnorm{y(t)}&\leqslant\infnorm{g(x,\mu)}+\infnorm{e\_{0,y}}+\int\_0^t\infnorm{\psi(u)h(y(u))}+\infnorm{e\_y(u)}du\\ &\leqslant\mtt{sp}(\infnorm{x,\mu})+\int\_0^t\poly(\infnorm{x},\mu,u,\hat{e}(u))du+\hat{e}(t)\\ &\leqslant\poly(\infnorm{x},\mu,t,\hat{e}(t))\end{aligned}$$ We will now analyze the behavior of the system when the error is bounded. Define Θ\(α, μ) = Θ(α, μ) + 1. Define ψ̂(t) = ∫0t*ψ(u)du and note that it is a diffeomorphism since ψ > 0. Apply Lemma [lem:perturbedtimescalegpac] to get that y(t) = z(ψ̂(t)) for all $t\in\Rp$, where z satisfies for $\xi\in\hat{\psi}(\Rp)$: z(0) = g(x, μ) + e0, y zʹ(ξ) = h(z(ξ)) + ẽ(ξ) $$\qquad\text{where} \int\_0^{\hat{\psi}(t)}\infnorm{\tilde{e}(\xi)}d\xi=\int\_0^t\infnorm{e\_y(u)}du$$ Assume that $x\in\dom{f}$ and let $T\in\Rp$ such that $\hat{e}(T)\leqslant e^{-\Theta^\(\infnorm{x},\mu)}$. Then $\hat{e}(T)<e^{-\Theta(\infnorm{x},\mu)}$ and for all t* ∈ [0, T]: $$\begin{aligned} \infnorm{e\_{0,y}}+\int\_0^{\hat{\psi}(t)}\infnorm{\tilde{e}}(u)du&=\infnorm{e\_{0,y}}+\int\_0^t\infnorm{e\_y(u)}du\\ &\leqslant\hat{e}(t)\leqslant e^{-\Theta(\infnorm{x},\mu)}\end{aligned}$$ Apply Lemma [lem:finitetimerobust] to get for all u ∈ [0, ψ̂(T)]: $$\infnorm{z(u)}\leqslant\Upsilon(\infnorm{x},\mu,u)$$ $$\text{if }u\geqslant\Omega(\infnorm{x},\mu)\text{ then }\infnorm{z\_{1..m}(u)-f(x)}\leqslant e^{-\mu}$$ Apply Lemmas [lem:max] and [lem:norm] to get for all t ∈ [0, T]: $$\begin{aligned} \ell(t)&\geqslant \mx\_1(\norm\_{\infty,1}(\infnorm{x},\mu))+1-\infnorm{e\_{0,\ell}}+t-\int\_0^t\infnorm{e\_\ell(u)}du\\ &\geqslant \max(\infnorm{x},\mu)+1+t-\hat{e}(t)\\ &\geqslant\max(\infnorm{x},\mu,t)\tag\{using that $\hat{e}(t)\leqslant1$}\end{aligned}$$ Consequently, using Lemma [lem:norm], for all t* ∈ [0, T]: $$\begin{aligned} \label{eq:robust\_to\_strong:ineq\_Delta} \Delta(t)&\geqslant\Upsilon(\ell(t),\ell(t),\ell(t))-\infnorm{y(t)}\\ &\geqslant\Upsilon(\infnorm{x},\mu,t)-\infnorm{y(t)}\tag\{using that $\ell(t)\geqslant\max(\infnorm{x},\mu,t)$}\\ &=\Upsilon(\infnorm{x},\mu,t)-\infnorm{z(\hat{\psi}(t))}\tag\{using that $y(t)=z(\hat{\psi}(t))$}\\ &\geqslant0\tag\{because $\hat{\psi}(t)\in[0,\hat{\psi}(T)]$}\end{aligned}$$ Consequently for all t* ∈ [0, T]: $$\hat{\psi}(t)=\int\_0^t\psi(u)du=\int\_0^t\frac{1+\tanh(\Delta(u))}{2}du\geqslant\frac{t}{2}$$ Define Ω\(α, μ) = 2Ω(α, μ). Assume that $T\geqslant \Omega^\(\infnorm{x},\mu)$ then $\hat{\psi}(T)\geqslant\Omega(\infnorm{x},\mu)$ and thus $\infnorm{y\_{1..m}(T)-f(x)}=\infnorm{z(\hat{\psi}(T))-f(x)}\leqslant e^{-\mu}$. Finally, (y, ℓ)(0) = g\(x, μ) + e0 where $g^\\in\gpval$. Similarly (y, ℓ)ʹ(t) = h\((y, ℓ)(t)) + e(t) where $h^\\in\gpval$. Note again that both h\* and g\* are defined over the entire space. This concludes the proof that $f\in\gsc{\Omega^\}{\poly}{\Theta^\}$. Proof that ASP implies AXP ========================== This section is devoted to prove the following: in Definition [def:gsc] we defined a class with a high degree of robustness to perturbations and related it to previous classes. However, the value f(x) the system computes still depends on the initial condition (i.e. x is provided via the initial condition). Here we want robustness to errors like in Definition [def:gsc], but we also want to dynamically change the argument x during a computation, as done in Definition [def:goc]. Since these are two exigent requirements, we named this computability form as “extreme”. Here $\indicator{X}$ denotes the function defined by $\indicator{X}(x)=1$ if x ∈ X and $\indicator{X}(x)=0$ otherwise. [Strong ⊆ , $\gpsc\subseteq\gpuc$][th:strongtounaware] $f\in\gpsc$ iff there exist polynomials Υ, Λ, Θ and a constant polynomial[6](#fn6) Ω such that $f\in\guc{\Upsilon}{\Omega}{\Lambda}{\Theta}$. where [ computability][def:guc] Let $n,m\in\N$, $f:\subseteq\R^n\rightarrow\R^m$, $\Upsilon:\Rp^3\rightarrow\Rp$ and $\Omega,\Lambda,\Theta:\Rp^2\rightarrow\Rp$. We say that f is (Υ, Ω, Λ, Θ)--computable if and only if there exist δ ≥ 0, $d\in\N$ and $(g:\R^{d}\times\R^{n+1}\rightarrow\R^d)\in\gpval[\K]{}$ such that for any $x\in C^0(\Rp,\R^n)$, $\mu\in C^0(\Rp,\Rp)$, $y\_0\in\R^d$, $e\in C^0(\Rp,\R^d)$ there exists (a unique) $y:\Rp\rightarrow\R^d$ satisfying for all $t\in\Rp$: * y(0) = y0 and yʹ(t) = g(t, y(t), x(t), μ(t)) + e(t) * $\infnorm{y(t)}\leqslant \Upsilon\left(\pastsup{\delta}{\infnorm{x}}(t),\pastsup{\delta}{\mu}(t), \infnorm{y\_0}\indicator{[1,\delta]}(t)+\int\_{\max(0,t-\delta)}^t\infnorm{e(u)}du\right)$ * For any I = [a, b], if there exist $\bar{x}\in\dom{f}$ and μ̌, μ̂ ≥ 0 such that for all t ∈ I: $$\mu(t)\in[\check{\mu},\hat{\mu}]\text{ and }\infnorm{x(t)-\bar{x}}\leqslant e^{-\Lambda(\infnorm{\bar{x}},\hat{\mu})} \text{ and }\int\_{a}^b\infnorm{e(u)}du\leqslant e^{-\Theta(\infnorm{\bar{x}},\hat{\mu})}$$ then $$\infnorm{y\_{1..m}(u)-f(\bar{x})}\leqslant e^{-\check{\mu}}\text{ whenever } a+\Omega(\infnorm{\bar{x}},\hat{\mu})\leqslant u\leqslant b.$$ We denote by $\guc{\Upsilon}{\Omega}{\Lambda}{\Theta}$ the set of (Υ, Ω, Λ, Θ)--computable functions and by $\gpuc$ the set of $(\poly,\poly,\poly,\poly)$--computable functions. Actually we prove the implication from left to right. The equivalence will follow from other sections. Some remarks ------------ A very common pattern in signal processing is known as “sample and hold”, where we have a variable signal and we would like to apply some process to it. Unfortunately, the processor often assumes (almost) constant input and does not work in real time (analog-to-digital converters are a typical example). In this case, we cannot feed the signal directly to the processor so we need some black box that samples the signal to capture its value, and holds this value long enough for the processor to compute its output. This process is usually used in a τ-periodic fashion: the box samples for time δ and holds for time τ − δ. We will need two intermediate lemmas before introducing sample and hold. [“low-X-high” and “high-X-low”, ][lem:lxhhxl] For every I = [a, b], there exists $\lxh\_I,\hxl\_I\in\gpval$ such that for every $\mu\in\Rp$ and $t,x\in\R$ we have: * $\lxh\_I$ is of the form $\lxh\_I(t,\mu,x)=\phi\_1(t,\mu,x)x$ where $\phi\_1\in\gpval$, * $\hxl\_I$ is of the form $\lxh\_I(t,\mu,x)=\phi\_2(t,\mu,x)x$ where $\phi\_2\in\gpval$, * if $t\leqslant a, \|\lxh\_I(t,\mu,x)\|\leqslant e^{-\mu}$ and $\|x-\hxl\_I(t,\mu,x)\|\leqslant e^{-\mu}$, * if $t\geqslant b, \|x-\lxh\_I(t,\mu,x)\|\leqslant e^{-\mu}$ and $\|\hxl\_I(t,\mu,x)\|\leqslant e^{-\mu}$, * in all cases, $\|\lxh\_I(t,\mu,x)\|\leqslant\|x\|$ and $\|\hxl\_I(t,\mu,x)\|\leqslant\|x\|$. [“periodic low-integral-low”][lem:plil] There is a family of functions $\plil\_{I,\tau}\in\gpval$ where $\mu,\tau\in\Rp$, $I=[a,b]\subsetneq[0,\tau]$ and $x\in\R$ with the following property: there exist a constant K and ϕ such that $\plil\_{I,\tau}(t,\mu,x)=\phi(t,\mu,x)x$ and: * $\plil\_{I,\tau}(\cdot,\mu,x)$ is τ-periodic * for all $t\notin I$, $\|\plil\_{I,\tau}(t,\mu,x)\|<e^{-\mu}$ * for any $\alpha:I\rightarrow\Rp,\beta:I\rightarrow\R$: 1 ≤ ∫abϕ(t, α(t), β(t))dt ≤ K [“periodic low-integral-low”] Let $t\in\R,\tau\in\Rp,\mu,x\in\R,I=[a,b]\subseteq[0,\tau]$ with 0 < b − a < τ and define: $$\plil\_{I,\tau}(t,\mu,x)=\lxh\_J(f(t),\nu,K)x$$ where $$\delta=b-a \qquad \omega=\frac{2\pi}{\tau} \qquad K=\frac{1}{4}+\frac{2}{\delta} \qquad t\_1=\frac{a+b}{2}-\frac{\tau}{4}$$ $$\nu=\mu+2+\ln(1+x^2) \qquad f(t)=\sin(\omega(t-t\_1)) \quad J=\left[f(a),f\left(a+\frac{\delta}{4}\right)\right]$$ [of Lemma [lem:plil]] The τ-periodicity is trivial. Using trigonometric identities, observe that $$f(t)-f(a)=-2\sin\left(\omega\frac{t-b}{2}\right)\sin\left(\omega\frac{t-a}{2}\right)$$ Now it is easy to see that if t ∈ [0, a] then $\omega\frac{t-b}{2},\omega\frac{t-a}{2}\in[-\pi,0]$ thus f(t) ≤ f(a). By the choice of J and Lemma [lem:lxhhxl], we get that $\lxh\_J(f(t),\mu+2,K)\leqslant e^{-\nu}$. Similarly if t ∈ [b, τ] then $\omega\frac{t-b}{2},\omega\frac{t-a}{2}\in[0,\pi]$ and we get the same result. We conclude the first part of the result using that ∣xe− ν∣ ≤ e− μ. Let $\alpha:I\rightarrow\Rp,\beta:I\rightarrow\R$. Let $a'=a+\frac{\delta}{4}$ and $b'=b-\frac{\delta}{4}$. Since $\lxh>0$, we have $\int\_a^b\plil\_{I,\tau}(t,\alpha(t),\beta(t))dt\geqslant\int\_{a'}^{b'}\plil\_{I,\tau}(t,\alpha(t),\beta(t))dt$. Again observe that $$f(t)-f(a')=-2\sin\left(\omega\frac{t-b'}{2}\right)\sin\left(\omega\frac{t-a'}{2}\right)$$ Consequently, if t ∈ [aʹ, bʹ] then f(t) ≥ f(aʹ). By the choice of J and Lemma [lem:lxhhxl], we get that $\lxh\_J(f(t),\nu,K)\geqslant K-e^{-\nu}\geqslant K-\frac{1}{4}$ since ν ≥ 2. Finally $\int\_a^b\plil\_{I,\tau}(t,\alpha(t),\beta(t))dt\geqslant (b'-a')(K-\frac{1}{4})\geqslant1$ and $\int\_a^b\plil\_{I,\tau}(t,\alpha(t),\beta(t))dt\leqslant (b-a)K$ by Lemma [lem:lxhhxl]. Apply Lemma [lem:gpacextclassstable] multiple times to get that $\plil\_{I,\tau}\in\gval{\poly}$. [Sample and hold][lem:sample] There is a family of functions $\sample\_{I,\tau}(t,\mu,x,g) \in \gpval$, where $t\in\R,\mu,\tau\in\Rp,x,g\in\R,I=[a,b]\subsetneq[0,\tau]$, with the following property: let $\tau\in\Rp$, $I=[a,b]\subsetneq[0,\tau]$, $y:\Rp\rightarrow\R$, $y\_0\in\R$, $x,e\in C^0(\Rp,\R)$ and $\mu:\Rp\rightarrow\Rp$ be an increasing function. Suppose that for all $t\in\Rp$: $$y(0)=y\_0\qquad y'(t)=\sample\_{I,\tau}(t,\mu(t),y(t),x(t))+e(t)$$ Then: $$\|y(t)\|\leqslant2+\smashoperator{\int\_{\max(0,t-\tau-\|I\|)}^t}\|e(u)\|du+\max\left(\|y(0)\|\indicator{[0,b]}(t),\pastsup{\tau+\|I\|}\|x\|(t)\right)$$ Furthermore: * if $t\notin I\pmod{\tau}$ then ∣yʹ(t)∣ ≤ e− μ(t) + ∣e(t)∣ * for $n\in\N$, if there exist $\bar{x}\in\R$ and $\nu,\nu'\in\Rp$ such that ∣x̄ − x(t)∣ ≤ e− ν and μ(t) ≥ νʹ for all t ∈ nτ + I then ∣y(nτ + b) − x̄∣ ≤ ∫nτ + I∣e(u)∣du + e− ν + e− νʹ. * for $n\in\N$, if there exist $\check{x},\hat{x}\in\R$ and $\nu\in\Rp$ such that x(t) ∈ [x̌, x̂] and μ(t) ≥ ν for all t ∈ nτ + I then y(nτ + b) ∈ [x̌ − ɛ, x̂ + ɛ] where ɛ = 2e− ν + ∫nτ + I∣e(u)∣du. * for any $J=[c,d]\subseteq\Rp$, if there exist $\nu,\nu'\in\Rp$ and $\bar{x}\in\R$ such that μ(t) ≥ νʹ for all t ∈ J and ∣x(t) − x̄∣ ≤ e− ν for all t ∈ J ∩ (nτ + I) for some $n\in\N$, then ∣y(t) − x̄∣ ≤ e− ν + e− νʹ + ∫t − τ − ∣I∣t∣e(u)∣du for all t ∈ [c + τ + ∣I∣, d]. * if there exists $\Omega:\Rp\rightarrow\Rp$ such that for any J = [a, b] and $\bar{x}\in\R$ such that for all $\nu\in\Rp$, $n\in\N$ and t ∈ (nτ + I) ∩ [a + Ω(ν), b] we have ∣x̄ − x(t)∣ ≤ e− ν, then ∣y(t) − x̄∣ ≤ e− ν for all t ∈ [a + Ω\(ν), b] where Ω\(ν) = max(Ω(ν + ln3), μ− 1(ν + ln3)) + τ + ∣I∣. [Sample and hold] Let $t\in\R,\mu,\tau\in\Rp,x,g\in\R,I=[a,b]\subsetneq[0,\tau]$ and define: $$\sample\_{I,\tau}(t,\mu,x,g)=\plil\_{I,\tau}(t,\hat{\mu},\reach(\check{\mu},x,g))$$ where $$\check{\mu}=\frac{\mu+1}{\min(1,\|I\|)}\qquad\hat{\mu}=\mu+\max(0,\ln(\tau-\|I\|))$$ Let $n\in\N$. Apply Lemma [lem:plil], Lemma [lem:reach] and Remark [rem:reachmult] to get that: * For all t ∈ In = [nτ + a, nτ + b]: $$y'(t)=\phi(t)\reach(\check{\mu}(t),y(t),x(t))+e(t)$$ where ∫Inϕ ≥ 1. Since ∣x(t) − 0∣ ≤ supu ∈ In∣x(u)∣ and $$\int\_{I\_n}\phi\check{\mu}=\int\_{I\_n}\phi\frac{1+\mu}{\|I\|}\geqslant1$$ then $$\begin{aligned} \|y(n\tau+b)-0\| &\leqslant \sup\_{I\_n}\|x(u)\|+\int\_{I\_n}\|e(u)\|du+e^{-1}\\ &\leqslant1+\sup\_{u\in I\_n}\|x(u)\|+\int\_{I\_n}\|e(u)\|du.\end{aligned}$$ * For all t ∈ [nτ + b, (n + 1)τ + a]: ∣yʹ(t)∣ ≤ ∣e(t)∣ + e− μ̂(t) ≤ ∣e(t)∣ + e− ln(τ − ∣I∣) thus $$\begin{aligned} \|y(t)-0\| &\leqslant\int\_{n\tau+b}^{t}\|e(u)\|du+(\tau-\|I\|)e^{-\ln(\tau-\|I\|)}\\ &\qquad+1+\sup\_{u\in I\_n}\|x(u)\|+\int\_{I\_n}\|e(u)\|du\\ &\leqslant2+\sup\_{u\in I\_n}\|x(u)\|+\int\_{n\tau+a}^{t}\|e(u)\|du.\end{aligned}$$ * For all t ∈ In + 1: $$y'(t)=\reach(\phi(t)\check{\mu}(t),y(t),x(t))$$ where ∫Inϕ ≥ 1. Since ∣x(t) − 0∣ ≤ supu ∈ In + 1∣x(u)∣ then $$\begin{aligned} \|y(t)-0\| &\leqslant\max\left(\sup\_{u\in[(n+1)\tau+a,t]}\|x(u)\|,\|y((n+1)\tau+a)-0\|\right)\\ &\qquad+\int\_{(n+1)\tau+a}^t\|e\|\\ &\leqslant2+\sup\_{u\in[n\tau+a,t]}\|x(u)\|+\int\_{n\tau+a}^{t}\|e(u)\|du.\end{aligned}$$ Note that this analysis is a bit subtle: the first point does not give a bound on ∣y(t)∣ over In, it only gives a bound on ∣y(nτ + b)∣. On the contrary the two other points give bounds on ∣y(t)∣ over [nτ + b, (n + 1)τ + b] which cover the whole period so by correctly putting everything together, we get that for all $\|y(t)\|\leqslant2+\sup\_{u\in[t,t-\tau-\|I\|]\cap\Rp}$ ∣x(u)∣ + ∫t − τ − ∣I∣t∣e(u)∣du for all t ≥ b. The case of the initial segment is similar in aspect but uses the other result from Lemma [lem:reach]: * For all t ∈ [0, a]: ∣yʹ(t)∣ ≤ ∣e(t)∣ + e− μ̂(t) ≤ ∣e(t)∣ + e− ln(τ − ∣I∣) thus ∣y(t)∣ ≤ ∫0t∣e(u)∣du + ae− ln(τ − ∣I∣) + ∣y0∣ ≤ ∫0t∣e(u)∣du + 1 + ∣y0∣. * For all t ∈ [a, b]: $$y'(t)=\reach(\phi(t)\check{\mu}(t),y(t),x(t))+e(t)$$ where ∫Inϕ ≥ 1. Since ∣x(t) − 0∣ ≤ supu ∈ [a, t]∣x(u)∣ then $$\begin{aligned} \|y(t)-0\|&\leqslant\max(\sup\_{u\in[a,t]}\|x(u)\|,\|y(a)-0\|)+\int\_a^t\|e(u)\|du\\ &\leqslant1+\int\_0^t\|e(u)\|du+\max(\|y\_0\|,\sup\_{u\in[a,t]}\|x(u)\|).\end{aligned}$$ Finally, we get that for all $t\in\Rp$: $$\|y(t)\|\leqslant2+\smashoperator{\int\_{t-\tau-\|I\|}^t}\|e(u)\|du+\max\left(\|y(0)\|\indicator{[0,b]}(t),\pastsup{\tau+\|I\|}\|x\|(t)\right)$$ The first extra statement is a trivial consequence of Lemma [lem:plil] and the fact that μ̌(t) ≥ μ(t). The second extra statement has mostly been proved already and uses Lemma [lem:plil] and Lemma [lem:reach] again. Let $n\in\N$, assume there exist $\bar{x}\in\R$ and $\nu\in\Rp$ such as described. For all t ∈ In = [nτ + a, nτ + b] we have $$y'(t)=\phi(t)\reach(\check{\mu}(t),y(t),x(t))+e(t)$$ where ∫Inϕ ≥ 1. Since ∣x(t) − x̄∣ ≤ e− ν and $\int\_{I\_n}\phi\check{\mu}=\int\_{I\_n}\phi\frac{1+\mu}{\|I\|}\geqslant\nu'$ then ∣y(nτ + b) − x̄∣ ≤ e− ν + ∫In∣e(u)∣du + e− νʹ. The third statement is a consequence of the previous one: since nτ + I is a compact set and x is a continuous function, it admits a maximum over nτ + I. Apply the previous statement to $\frac{\bar{x}+\sup\_{n\tau+I}x}{2}\geqslant\bar{x}$ to conclude. The last extra statement requires more work. Let ν ≥ 0 and $n\in\N$ such that nτ + a ≥ Ω(ν). Apply Lemma [lem:plil], Remark [rem:reachmult] and Lemma [lem:reach] to get that: * For all t ∈ In: $$y'(t)=\phi(t)\reach(\check{\mu}(t),y(t),x(t))$$ where ∫Inϕ ≥ 1. Since t ≥ nτ + a ≥ Ω(ν) and t ∈ In then ∣x(t) − x̄∣ ≤ e− ν. And since $$\int\_{I\_n}\phi\check{\mu}=\int\_{I\_n}\phi\frac{1+\mu}{\|I\|}\geqslant1+\mu(n\tau+a)$$ then ∣y(nτ + b) − x̄∣ ≤ e− ν + e− μ(nτ + a). * For all t ∈ [nτ + b, (n + 1)τ + a]: ∣yʹ(t)∣ ≤ e− μ̂(t) ≤ e− μ̂(nτ + a) thus $$\begin{aligned} \|y(t)-\bar{x}\| &\leqslant(\tau-\|I\|)e^{-\hat{\mu}(n\tau+a)}+e^{-\nu}+e^{-\mu(n\tau+a)}\\ &\leqslant e^{-\nu}+2e^{-\mu(n\tau+a)}.\end{aligned}$$ * For all t ∈ In	arxiv_0000401
Global propagator for the massless Dirac operatorand spectral asymptotics ========================================================================= ### 1 July 2022 We construct the propagator of the massless Dirac operator W on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals — the positive and the negative propagators — correspond to positive and negative eigenvalues of W, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of W. Keywords: Dirac operator, hyperbolic propagators, global Fourier integral operators, Weyl coefficients. 2020 MSC classes: primary 35L45; secondary 35Q41, 58J40, 58J45, 35P20. Statement of the problem ======================== Let (M, g) be a connected oriented closed Riemannian 3-manifold. Throughout this paper we denote by ∇ the Levi-Civita connection, by Γαβγ the Christoffel symbols and by $$\label{riemannian density} \textstyle \rho(x):=\sqrt{\textcolor{black}{\det} g\_{\alpha\beta}(x)}$$ the Riemannian density. Let us clarify straight away why, when dealing with the massless Dirac operator, we restrict our analysis to the 3-dimensional case. The reason is twofold: on the one hand, dimension three is physically meaningful in that it represents the first step towards a potential future analysis of the relativistic 3+1-dimensional setting, and on the other hand, our method requires the eigenvalues of the principal symbol of our operator to be simple, cf. Section [Preliminary results for general first order systems], which is not the case for the massless Dirac operator in higher dimensions. Let ej, j = 1, 2, 3, be a positively oriented global framing, i.e. a set of three orthonormal smooth vector fields[3](#fn3) whose orientation agrees with the orientation of the manifold. In chosen local coordinates xα, α = 1, 2, 3, we will denote by ejα the α-th component of the j-th vector field. Throughout this paper we use Greek letters for holonomic (tensor) indices and Latin for anholonomic (frame) indices. We adopt Einstein’s convention of summation over repeated indices. Let $$\label{Pauli matrices basic} s^1:= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} =s\_1 \,, \quad s^2:= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} =s\_2 \,, \quad s^3:= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} =s\_3$$ be the standard Pauli matrices and let σα : = sj ejα be their projection along the framing. The quantity σα is a vector-function with values in the space of trace-free Hermitian 2 × 2 matrices. [massless dirac definition] We call massless Dirac operator the operator $$\label{massless dirac definition equation} W:=-i\sigma^\alpha \left( \frac\partial{\partial x^\alpha} +\frac14\sigma\_\beta \left( \frac{\partial\sigma^\beta}{\partial x^\alpha} +\Gamma^\beta{}\_{\alpha\gamma}\,\sigma^\gamma \right) \right) :H^1(M;\mathbb{C}^2)\to L^2(M;\mathbb{C}^2).$$ Here H1 is the usual Sobolev space of functions which are square integrable together with their first partial derivatives. In relativistic particle physics the massless Dirac equation is often referred to as the Weyl equation, which explains our notation. Our operator W appears as the result of separating out the time variable in the relativistic Weyl equation, see for details. Henceforth, we refer to the massless Dirac operator simply as the Dirac operator, which conforms with the terminology adopted in differential geometry. The Dirac operator admits several equivalent definitions. The most common is the geometric definition written in terms spinor bundles. Our analytic Definition [massless dirac definition] is equivalent to the standard geometric one, see. Definition [massless dirac definition] depends on the choice of framing and this issue requires clarification. Let G : M → SU(2) be an arbitrary smooth special unitary matrix-function and let $\widetilde W$ be the Dirac operator corresponding to a given framing. Consider the transformation $$\label{transformation of the Dirac operator under change of frame} \widetilde W\mapsto G^\\widetilde WG:=W,$$ where the star indicates Hermitian conjugation. It turns out that W* is also a Dirac operator, only corresponding to a different framing. Let us now look at the matter the other way round. Suppose that $\widetilde W$ and W are two Dirac operators. Does there exist a smooth matrix-function such that $W=G^\\widetilde WG\,$? If the operators $\widetilde W$ and W* are in a certain sense ‘close’ then the answer is yes, but in general there are topological obstructions and the answer is no. This motivates the introduction of the concept of spin structure, see. The gauge transformation, is the manifestation, at operator level, of the freedom of pointwise rotating the framing in a smooth way, $$\label{gauge transfomation in terms of O} \widetilde{e}\_j \mapsto O\_j{}^k\, \widetilde{e}\_k=:e\_j, \qquad O\in C^\infty(M;\mathrm{SO}(3)),$$ via the double cover $$\mathrm{SU}(2) \overset{2:1}{\rightarrow} \mathrm{SO}(3).$$ The Dirac operator is uniquely determined by the metric and spin structure modulo an SU(2) gauge transformation. The Dirac operator is symmetric with respect to the L2 inner product ⟨u, v⟩ : = ∫Mu\*v ρ dx , u, v ∈ L2(M; C2), where dx = dx1dx2dx3. Furthermore, a simple calculation shows that it is elliptic[4](#fn4). It is well known that the Dirac operator is self-adjoint and its spectrum is discrete, accumulating to + ∞ and to − ∞. Let λk be the eigenvalues of W and vk the corresponding orthonormal eigenfunctions, k ∈ Z. The choice of particular enumeration is irrelevant for our purposes, but what is important is that eigenvalues are enumerated with account of their multiplicity. Note that the Dirac operator has the special property that it commutes with the antilinear operator of charge conjugation $$v= \begin{pmatrix} v\_1\\ v\_2 \end{pmatrix} \mapsto \begin{pmatrix} -\overline{v\_2}\\ \overline{v\_1} \end{pmatrix} =:\mathrm{C}(v),$$ see for details, and this implies that eigenvalues have even multiplicity. [dirac propagator definition] We define the Dirac propagator as U(t) : = e− itW. The Dirac propagator is the (distributional) solution of the hyperbolic Cauchy problem [hyperbolic Cauchy problem] $$\label{main PDE massless Dirac} \left( -i\,\frac{\partial}{\partial t} + W \right) U=0\,,$$ U(0) = Id . It is a time-dependent unitary operator which can written via functional calculus as U(t) = ∑λke− itλk vk ⟨vk , ⋅ ⟩. The Dirac propagator can be written as a sum of three operators U(t) = U+(t) + U0 + U−(t) defined as U+(t) : = ∑λk > 0e− itλk vk ⟨vk , ⋅ ⟩, U0 : = ∑λk = 0vk ⟨vk , ⋅ ⟩, U−(t) : = ∑λk < 0e− itλk vk ⟨vk , ⋅ ⟩. We call the operators, and positive, zero mode and negative propagators, respectively. These are time-dependent partial isometries. Note that the operator U0 is nontrivial only if the Dirac operator has zero modes (i.e. if zero is an eigenvalue). We define the positive ( + ) and negative ( − ) local counting functions as $$\label{local counting functions} N\_\pm(y;\lambda):= \begin{cases} 0 & \text{for }\lambda \le0,\\ \sum\_{0<\pm\lambda\_k<\lambda} [v\_k(y)]^\* \,v\_k(y)& \text{for }\lambda >0. \end{cases}$$ Of course, integration over M gives $$\label{global counting functions} N\_\pm(\lambda) := \int\_M N\_\pm(y;\lambda)\,\rho(y)\,dy = \begin{cases} 0 & \text{for }\lambda \le0,\\ \sum\_{0<\pm\lambda\_k<\lambda} 1& \text{for }\lambda >0. \end{cases}$$ The functions are the ‘global’ counting functions, the only difference with the usual definition being that we count the positive and negative eigenvalues separately. Let μ̂ : R → C be a smooth function such that μ̂ = 1 in some neighbourhood of the origin and suppμ̂ is sufficiently small. Here ‘sufficiently small’ means that suppμ̂ ⊂ ( − T0, T0), where T0 is the infimum of lengths of all the geodesic loops originating from all the points of the manifold. Following the notation of, we write the Fourier transform as Fλ → t[f](t) = f̂(t) = ∫− ∞+ ∞e− itλf(λ) dλ and the inverse Fourier transform as $$\mathcal{F}\_{t\to \lambda}^{-1}[\hat f](\lambda)=f(\lambda)=\dfrac{1}{2\pi}\int\_{-\infty}^{+\infty} e^{it\lambda}\hat{f}(t) \,dt.$$ Accordingly, we put $\mu:=\mathcal{F}^{-1}[\hat \mu]$. It is known that the mollified derivative of the positive (resp. negative) counting function admits a complete asymptotic expansion in integer powers of λ: (N±ʹ \* μ)(y, λ) = c2±(y) λ2 + c1±(y) λ + c0±(y) + … as λ → + ∞. Here \* stands for the convolution in the variable λ. [definition of Weyl coefficients] We call local Weyl coefficients the smooth functions ck±(y) appearing in the asymptotic expansions. 1. Our definition of Weyl coefficients does not depend on the choice of mollifier μ. If $\widetilde{\mu}$ is another mollifier with the same support properties, then $$(N\_\pm'\\mu)(y,\lambda)-(N\_\pm'\\widetilde\mu)(y,\lambda)=O(\lambda^{-\infty})\quad \text{as} \quad \lambda\to+\infty.$$ 2. Our definition of Weyl coefficients is, in a sense, unusual. The standard convention in the literature is to call local Weyl coefficients the functions appearing in the asymptotic expansion of the mollified counting function N \* μ , as opposed to its derivative. The two definitions are, effectively, the same up to integrating factors, $$\begin{gathered} \label{expansion for mollified counting function} (N\_\pm\\mu)(y,\lambda)= \int\_{-\infty}^\lambda (N\_\pm'\\mu)(y,\kappa)\,d\kappa \\ = \frac13\, c^\pm\_{2}(y)\,\lambda^{3} + \frac12\, c^\pm\_{1}(y)\,\lambda^2 + c^\pm\_{0}(y)\,\lambda +\dots \quad \text{as} \quad \lambda\to+\infty,\end{gathered}$$ compare with. As a matter of convenience, we will stick with Definition [definition of Weyl coefficients] throughout this paper. 3. It was shown in that $$\label{first two weyl coefficients} c^\pm\_{2}(y)=\frac1{2\pi^2}\,, \qquad c^\pm\_{1}(y)=0.$$ 4. The unmollified counting functions N±(y, λ) also admit asymptotic expansions as λ → + ∞, but here the situation is more delicate because these functions are discontinuous and one encounters number-theoretic issues. It is known that $$\label{one term asymptotics unmollified} N\_\pm(y,\lambda) = \frac1{6\pi^2}\, \lambda^{3} + O(\lambda^2) \quad \text{as} \quad \lambda\to+\infty$$ uniformly over y ∈ M and, furthermore, under appropriate assumptions on geodesic loops, $$\label{two term asymptotics unmollified} N\_\pm(y,\lambda) = \frac1{6\pi^2}\, \lambda^{3} + o(\lambda^2) \quad \text{as} \quad \lambda\to+\infty.$$ 5. An important topic in the spectral theory of first order elliptic systems is the issue of spectral asymmetry. Let us mention that to observe spectral asymmetry for our Dirac operator one as to go as far as the sixth Weyl coefficients. This follows from the fact that the eta function $$\eta(s):=\sum\_{\lambda\_k\ne 0}\dfrac{\operatorname{sgn} \lambda\_k}{\|\lambda\_k\|^s}=\int\_0^{+\infty} \lambda^{-s} (N'\_+(\lambda)-N'\_-(\lambda))\,d \lambda$$ is holomorphic in the complex half-plane Res > − 2 and has its first pole at s = − 2. The value of the residue of the eta function at s = − 2, which was computed explicitly by Branson and Gilkey, describes the difference ∫M(c− 3+(y) − c− 3−(y)) ρ(y) dy between the sixth (global) Weyl coefficients. Our paper has two main objectives. Objective 1 Construct the propagators U±(t) explicitly, modulo integral operators with infinitely smooth kernels, and do so as a single invariantly defined oscillatory integral global in space and in time. Objective 2 Compute the third Weyl coefficient c0±(y). One cannot, in general, identify the third Weyl coefficient by looking at the asymptotic behaviour of the unmollified counting function. In order to illustrate this point, let us consider the 3-torus equipped with standard flat metric. Already in this simple case the mathematical statement $$\label{two term asymptotics unmollified false} N\_\pm(y,\lambda) = \frac1{6\pi^2}\, \lambda^{3} + c^\pm\_{0}(y)\,\lambda + o(\lambda) \quad \text{as} \quad \lambda\to+\infty$$ is false. This fact can be established by writing down the eigenvalues explicitly as in and using standard results from analytic number theory. Main results ============ The study of Dirac operators in curved space, arguably the most important operators from the point of view of physical applications alongside the Laplacian, has a long and noble history in the mathematical literature. Excellent introductions to the subject can be found in and. Due to the physical significance of the topic, numerous researchers have contributed to our current understanding of the spectrum of Dirac operators on Riemannian manifolds. One can ask, for example, how the eigenvalues behave under perturbations of the metric, how the spectrum depends on the spin structure, whether zero modes exist, et cetera. Later in this paper we will be concerned with the study of the asymptotic behaviour of large (in modulus) eigenvalues of the Dirac operator on a closed 3-manifold. In the case of scalar elliptic operators, such as for example the Laplace–Beltrami operator, a wide range of classical techniques are available in the literature to compute spectral asymptotics. However, if one is interested in a first order system, whose spectrum is, in general, not semi-bounded, the heat kernel method can no longer be applied, at least in its original form, and even resolvent techniques require major modification. A very natural approach in this case is the so-called wave method, going back to Levitan and Avakumovic, which involves recovering information about the eigenvalue counting function from the behaviour of the wave propagator, see. How this can be done will be explained in greater detail later on. This partly motivates our interest in the Dirac propagator, which is also of interest on its own. Of course, the hyperbolic Cauchy problem for W lies at the heart of relevant applications in theoretical physics (e.g., the mathematical description of neutrinos/antineutrinos in curved space). In order to construct the propagator precisely, one needs to know all eigenvalues and eigenfunctions of W, which is unrealistic for a generic Riemannian manifold. However, microlocal techniques allow one to construct the propagator approximately, modulo an integral operator with smooth integral kernel. This fact is well-known and an extensive discussion can be found, for instance, in. There are, however, several issues with this classical construction: (i) it is not invariant under changes of local coordinates, (ii) it is local in space and (iii) it is local in time. The last issue, locality in time, is especially serious: it is to do with obstructions associated with caustics. In practice, constructing a propagator locally in time means that for large times one has to use compositions U(t) = U(t − tj) ∘ U(tj − tj − 1) ∘ ⋯ ∘ U(t2 − t1) ∘ U(t1). The propagator U(t) is a special case of a Fourier integral operator and it is known that handling compositions of such operators is a daunting task. Our goal is to construct the Dirac propagator explicitly, in a global – i.e., as a single oscillatory integral – and invariant (under change of coordinates and gauge transformations) fashion. The key idea, originally proposed by Laptev, Safarov and Vassiliev in and further developed in, is to use Fourier integral operators with complex-valued, as opposed to real valued, phase function. Crucially, this allows one to circumvent the topological obstructions due to caustics. Our work partly builds upon and. In, using the wave method, Chervova, Downes and Vassiliev obtained an explicit formula for the second Weyl coefficient of an elliptic self-adjoint first order pseudodifferential matrix operator, fixing thirty years of incorrect or incomplete publications in the subject, see. In the same authors applied the results from to the Dirac operator. Unlike the current paper, the approach from is not geometric in nature and the complexity of phase functions is not actually put to use. Note that some results from were improved by Strohmaier and Li in, where the authors studied the second term of the mollified spectral counting function of Dirac type operators and characterised operators in this class with vanishing second Weyl coefficient. A fully geometric global construction of the (scalar) wave propagator $e^{-it \sqrt{-\Delta}}$ on closed Riemannian manifolds, as a single oscillatory integral with complex-valued phase function, was recently proposed by the authors and Levitin in, and subsequently extended to the Lorentzian setting in. The publication is the starting point of the current paper. Our main results are as follows. 1. We present a global construction of each of the two propagators, the positive propagator U+(t) and the negative propagator U−(t), as a single invariantly defined oscillatory integral, global in space and in time, with distinguished complex-valued phase function (Theorem [theorem propagator for A]). We provide a closed formula for the principal symbols of the propagators (Theorem [theorem small time expansion principal symbol]) and an algorithm for the calculation of the subprincipal symbols and all asymptotic components of lower degree of homogeneity in momentum (subsection [The algorithm]). 2. We give an explicit small time expansion of principal and subprincipal symbols of positive and negative propagators in terms of geometric invariants (Theorem [theorem invariant expression principal and subprincipal symbols dirac]). 3. We compute the third local Weyl coefficients in the asymptotic expansion of the two eigenvalue counting functions (Theorem [theorem about third Weyl coefficient]). Along the way we prove a number of results about general first order elliptic systems and invariant representations of pseudodifferential operators on manifolds. Note that the third Weyl coefficients can, in principle and with some work, be also obtained by a different method using results available in the literature, see Remark [remark about third weyl coefficient]. Our paper is structured as follows. In Section [Preliminary results for general first order systems] we explain how to construct explicitly positive and negative propagators for a general first order elliptic self-adjoint (pseudo)differential matrix operator, with a rigorous mathematical justification. In Section [Invariant description] we deal with the delicate issue of invariant descriptions of pseudodifferential operators acting on scalar functions. In particular, we examine the relation between our g-subprincipal symbol and the standard notion of subprincipal symbol for operators acting on half-densities. In Section [Global propagator for the massless Dirac operator] we apply the results from Section [Preliminary results for general first order systems] to the Dirac operator. A formula for the principal symbol of positive and negative Dirac propagators is provided in Section [Principal symbol of the global Dirac propagator], whereas small time expansions for principal and subprincipal symbols of positive and negative propagators are obtained in Section [Explicit small time expansion of the symbol]. Our final results are expressed in terms of geometric invariants: curvature of the Levi-Civita connection associated with the metric g and torsion of the Weitzenböck connection generated by the framing defining the Dirac operator. In Section [An application: spectral asymptotics] we use the results from Section [Explicit small time expansion of the symbol] to compute the third local Weyl coefficients for the Dirac operator. Finally, in Section [Examples] we apply our techniques to two explicit examples: M = S3, where formulae are isotropic in momentum, and M = S2 × S1, where they are not. The paper is complemented by two appendices, containing background material and technical proofs. Preliminary results for general first order systems =================================================== In this section we will consider a broader class of first order systems and we will prove fairly general results, which will be later applied to the special case of the Dirac operator. In doing so, we will need some of the technology developed in. The setting of our analysis is somewhat different from that in, in that our operators are differential, as opposed to pseudodifferential (see also Remark [remark about pseudodifferential]), and act on scalar functions on a Riemannian manifold, as opposed to half-densities on a manifold with no metric structure. In particular, the change of the space in which the operator acts raises delicate issues concerning the invariance of the mathematical objects involved. For these reasons we provide here a modified version of some of the results from, adapted to the setting of our paper. Throughout this section, M will be a smooth connected closed Riemannian manifold of dimension d ≥ 2. Let A be an elliptic symmetric (with respect to ) first order m × m matrix differential operator acting on m-columns of smooth complex-valued scalar functions v ∈ C∞(M; Cm) and let Aprin : TʹM → Herm(m, C) be the principal symbol of A, where TʹM : = T\*M \ {0} and Herm(m, C) is the real vector space of m × m Hermitian matrices. We denote by h(j)(x, ξ) the eigenvalues of Aprin(x, ξ) enumerated in increasing order, with positive index j = 1, 2, …, m+ for positive h(j)(x, ξ) and negative index j = − 1, − 2, …, − m− for negative h(j)(x, ξ). We assume that the eigenvalues of the principal symbol Aprin are simple. Clearly, m = m+ + m−, because the ellipticity condition detAprin(x, ξ) ≠ 0 ensures that all eigenvalues are nonzero. In fact, as our operator is differential, one can show that m can only be even and that we have $$\label{m plus and m minus equal m/2} m^+=m^-=\frac{m}2\,.$$ Furthermore, the eigenvalues h(j) of the principal symbol and the corresponding normalised eigenvectors v(j) possess the symmetry $$\label{symmetry eigenvalues and eigenvectors} h^{(-j)}(x,\xi)=-h^{(j)}(x,-\xi), \quad v^{(-j)}(x,\xi)=v^{(j)}(x,-\xi), \qquad j=1,\ldots,\frac{m}2\,.$$ Under the above assumptions the spectrum of A is discrete and accumulates to + ∞ and to − ∞. We denote eigenvalues and orthonormalised eigenfunctions of A by λk and vk, respectively, enumerated with account of their multiplicity. By replacing W with A, one can define the ‘full’ propagator UA(t) for A via, as well as the positive, zero mode and negative propagators via –, which we denote by UA+(t), UA0 and UA−(t), respectively. Each eigenvalue h(j)(x, ξ) of the principal symbol can be interpreted as a Hamiltonian on the cotangent bundle. The corresponding Hamiltonian flow (x(j)(t; y, η), ξ(j)(t; y, η)), i.e. the (global) solution to Hamilton’s equations $$\dot{x}^{(j)}=h\_\xi^{(j)}(x^{(j)},\xi^{(j)}), \qquad \dot{\xi}^{(j)}=-h\_x^{(j)}(x^{(j)},\xi^{(j)})$$ with initial condition (x(j)(0; y, η), ξ(j)(0; y, η)) = (y, η), generates a Lagrangian manifold to which one can, in turn, associate a global Lagrangian distribution. See and references therein for details. In particular, the singularities of the solution to the initial value problem ( − i∂t + A)v = 0, v∣t = 0 = v0 propagate along Hamiltonian trajectories generated by the eigenvalues of Aprin. Positive and negative propagators: an abstract approach ------------------------------------------------------- Our aim is to show that UA+(t) and UA−(t) can be separately approximated by a finite sum of global oscillatory integrals. Before doing so, let us state and prove an abstract preparatory theorem. Let v ∈ C∞(R × Mx × My), (λ, x, y) ↦ v(λ, x, y). We write v = O(∣λ∣− ∞) as λ → ± ∞ if for every α > 0, every k ∈ N and every linear partial differential operator P with infinitely smooth coefficients of order k on Mx × My there exists a positive constant Cα, P such that ∣Pv∣ ≤ Cα, P ∣λ∣− α for\ \, ± λ > 1 , uniformly over Mx × My . [theorem positive and negative propagator abstract approach] Let (T−, T+) ⊆ R be an open interval (possibly, the whole real line) and let u+(t, x, y), u−(t, x, y), $\widetilde{u}^+(t,x,y)$ and $\widetilde{u}^-(t,x,y)$ be elements of C∞(Mx × My; Dʹ(T−, T+)), satisfying 1. $u^+(t,x,y)+u^-(t,x,y)=\widetilde u^+(t,x,y)+\widetilde u^-(t,x,y) \mod C^\infty((T\_-,T\_+)\times M\_x\times M\_y)\,$. Furthermore, assume that for every ζ ∈ C0∞(T−, T+) we have 1. Ft → λ− 1[ζ u±] = O(∣λ∣− ∞) as λ → ∓ ∞, 2. $\mathcal{F}^{-1}\_{t\to \lambda}[\zeta\,\widetilde u^\pm]=O(\|\lambda\|^{-\infty}) \quad \text{as }\lambda\to \mp \infty$. Then $$u^\pm(t,x,y)=\widetilde u^\pm(t,x,y) \mod C^\infty((T\_-,T\_+)\times M\_x\times M\_y).$$ Let ζ ∈ C0∞(T−, T+). Multiplying (a) by ζ(t) we get $$\begin{gathered} \label{proof preparatory theorem formula 1} \zeta(t)\,u^+(t,x,y)+\zeta(t)\,u^-(t,x,y)=\zeta(t)\,\widetilde u^+(t,x,y)+\zeta(t)\,\widetilde u^-(t,x,y) \\ \mod C\_0^\infty(\mathbb{R}\times M\_x\times M\_y).\end{gathered}$$ Applying the inverse Fourier transform Ft → λ− 1 to, letting λ → + ∞ and using assumptions (b) and (c) we obtain $$\label{proof preparatory theorem formula 2} \mathcal{F}^{-1}\_{t\to\lambda}[\zeta\,u^+]=\mathcal{F}^{-1}\_{t\to\lambda}[\zeta\,\widetilde u^+]+O(\|\lambda\|^{-\infty})\quad \text{as }\lambda\to+\infty.$$ Here, when dealing with the remainder from, we used the fact that the Fourier transform of a compactly supported smooth function is rapidly decreasing. The compactness of M ensures a uniform estimate in the spatial variables. Furthermore, (b) and (c) immediately imply $$\label{proof preparatory theorem formula 3} \mathcal{F}^{-1}\_{t\to\lambda}[\zeta\,u^+]=\mathcal{F}^{-1}\_{t\to\lambda}[\zeta\,\widetilde u^+]+O(\|\lambda\|^{-\infty})\quad \text{as }\lambda\to-\infty.$$ Combining and we arrive at $$\label{proof preparatory theorem formula 4} \mathcal{F}^{-1}\_{t\to\lambda}[\zeta\,(u^+-\widetilde u^+)] = O(\|\lambda\|^{-\infty})\quad \text{as }\|\lambda\|\to+\infty,$$ which implies $$\label{proof preparatory theorem formula 5a} \zeta\,(u^+-\widetilde u^+)\in C^\infty(\mathbb{R}\times M\_x\times M\_y).$$ As ζ ∈ C0∞(T−, T+) in the above formula is arbitrary, we conclude that $$\label{proof preparatory theorem formula 5b} u^+-\widetilde u^+\in C^\infty((T\_-,T\_+)\times M\_x\times M\_y).$$ A similar argument gives $$\label{proof preparatory theorem formula 6} u^--\widetilde{u}^- \in C^\infty((T\_-,T\_+)\times M\_x\times M\_y).$$ Construction of positive and negative propagators ------------------------------------------------- [theorem propagator for A] The positive and negative propagators can be written, modulo an infinitely smoothing operator, as a finite sum of oscillatory integrals, global in space and in time. More precisely, we have $$\label{propagator approximated general theorem positive} U^+\_A(t)\overset{\mod \Psi^{-\infty}}{=}\sum\_{j=1}^{m^+} U^{(j)}\_A(t),$$ $$\label{propagator approximated general theorem negative} U^-\_A(t)\overset{\mod \Psi^{-\infty}}{=}\sum\_{j=1}^{m^-} U^{(-j)}\_A(t),$$ where $$\label{oscillatory intergral for U^(j)} U^{(j)}\_A(t):=\frac{1}{(2\pi)^d}\int\_{T'M} e^{i\varphi^{(j)}(t,x;y,\eta)}\,\mathfrak{a}^{(j)}(t;y,\eta)\,\chi^{(j)}(t,x;y,\eta)\,w^{(j)}(t,x;y,\eta)\,\left( \,\cdot\,\right)\rho(y)\,dy\,d\eta$$ and * by $\overset{\mod \Psi^{-\infty}}{=}$ we mean that the operator on the LHS is equal to the operator on the RHS up to an integral operator with infinitely smooth integral kernel; * ( ⋅ ) is meant for insertion of f0(y) when computing (UA(j)f0)(x); * the phase function φ(j) ∈ C∞(R × M × TʹM; C) satisfies 1. φ(j)∣x = x(j) = 0, 2. φxα(j)∣x = x(j) = ξα(j), 3. detφxαηβ(j)∣x = x(j) ≠ 0, 4. Imφ(j) ≥ 0; * the symbol a(j) ∈ Sph0(R × TʹM; Mat(m; C)) is an element in the class of polyhomogeneous symbols of order zero with values in m × m complex matrices, which means that a(j) admits an asymptotic expansion in components positively homogeneous in momentum, $$\label{asympotic expansion symbol for A} \mathfrak{a}^{(j)}(t;y,\eta)\sim \sum\_{k=0}^{+\infty} \mathfrak{a}^{(j)}\_{-k}(t;y,\eta), \qquad \mathfrak{a}^{(j)}\_{-k}(t;y,\alpha\,\eta)=\alpha^{\textcolor{black}{-k}}\, \mathfrak{a}^{(j)}\_{-k}(t;y,\eta), \quad\forall \alpha >0;$$ * the function χ(j) ∈ C∞(R × M × TʹM) is a cut-off satisfying 1. χ(j)(t, x; y, η) = 0 on {(t, x; y, η) ∣ ∣h(j)(y, η)∣ ≤ 1/2}, 2. χ(j)(t, x; y, η) = 1 on the intersection of {(t, x; y, η) ∣ ∣h(j)(y, η)∣ ≥ 1} with some conical neighbourhood of {(t, x(j)(t; y, η); y, η)}, 3. χ(j)(t, x; y, α η) = χ(j)(t, x; y, η) for α ≥ 1 on {(t, x; y, η) ∣ ∣h(j)(y, η)∣ ≥ 1}; * the weight w(j) is defined by $$\label{weight definition general case} w^{(j)}(t,x;y,\eta):= [\rho(x)\,\rho(y)]^{-\frac12} \left[{\det}^2( \varphi^{(j)}\_{x^\alpha\eta\_\beta}) \right]^\frac14,$$ where the smooth branch of the complex root is chosen in such a way that $$w^{\color{black}(j)}(0,y;y,\eta)=[\rho(y)]^{-1}.$$ Note that the weight w(j) is the inverse of a smooth density in the variable y and a smooth scalar function in all other variables. The powers of the Riemannian density ρ in are chosen in such a way that the symbol a(j) and the integral kernel $$\label{integral kernel of U^(j)} u^{(j)}(t,x,y):=\frac{1}{(2\pi)^d}\int\_{T'\_yM} e^{i \varphi^{(j)}(t,x;y,\eta)}\,\mathfrak{a}^{(j)}(t;y,\eta)\,\chi^{(j)}(t,x;y,\eta)\,w^{(j)}(t,x;y,\eta)\,d\eta$$ of the operator are scalar functions in all variables. The fact that the symbol is a genuine scalar function on R × TʹM is a crucial feature of our construction. Taking the square and then extracting the fourth root in serves the purpose of making the weight invariant under inversion of a single coordinate xα or a single coordinate yα. Note, however, that if one works on an orientable and oriented manifold, then one can simplify to read $$\label{weight definition general case oriented manifold} w^{(j)}(t,x;y,\eta)= [\rho(x)\,\rho(y)]^{-\frac12} \left[{\det}\varphi^{(j)}\_{x^\alpha\eta\_\beta}\right]^\frac12.$$ The existence of a phase function satisfying conditions (i)–(iv) is a nontrivial matter. In fact, the space of phase function satisfying these conditions is nonempty and path-connected, see. Let us emphasise that a phase function φ(j) satisfying conditions (i)–(iv) from Theorem [theorem positive and negative propagator abstract approach] automatically satisfies φt(j)(t, x(j); y, η) + h(j)(x(j), ξ(j)) = 0, see, e.g.,. The equation φt(j)(t, x; y, η) + h(j)(x, ∇φ(j)(t, x; y, η)) = 0 is known in the literature as eikonal equation. Note that when x = x(j)(t; y, η) formula turns into. In the classical approach to the construction of hyperbolic propagators, is required to be satisfied in some open neighbourhood of {(t, x; y, η) ∈ R × M × TʹM ∣ x = x(j)(t; y, η)}. This is a fundamental difference with the approach adopted in the current paper, where is only required to be satisfied ‘along the Hamiltonian flow’, i.e., one only needs. Indeed, there is no open neighbourhood of where the special phase functions that will be introduced and used in Section [Global propagator for the massless Dirac operator] — the Levi-Civita phase functions — satisfy. Relaxing the requirements on our phase functions is needed in order to accommodate an imaginary part and, consequently, circumvent obstructions arising from caustics. [Proof of Theorem [theorem propagator for A]] Suppose that we have constructed the symbols a(j) appearing in the oscillatory integrals so that $$\label{sum of ALL the U\_A^(j)} \widetilde{U}\_A(t):= \sum\_jU^{(j)}\_A(t) = \sum\_{j=1}^{m^+} U^{(j)}\_A(t) + \sum\_{j=1}^{m^-} U^{(-j)}\_A(t)$$ satisfies $$\label{proof propagator general case 1} \left( -i\,\frac{\partial}{\partial t} + A \right) \widetilde{U}\_A(t)\overset{\mod \Psi^{-\infty}}{=}0\,,$$ $$\label{proof propagator general case 2} \widetilde{U}\_A(0)\overset{\mod \Psi^{-\infty}}{=} \mathrm{Id}\,.$$ How to achieve this will be explained in subsection [The algorithm]. Put $$\label{sum of positive u^(j)} \widetilde{u}^+(t,x,y):= \sum\_{j=1}^{m^+}u^{(j)}(t,x,y),$$ $$\label{sum of negative u^(j)} \widetilde{u}^-(t,x,y):= \sum\_{j=1}^{m^-}u^{(-j)}(t,x,y),$$ so that the Schwartz kernel of the operator $\widetilde{U}\_A(t)$ reads $$\label{sum of ALL u^(j)} \widetilde{u}(t,x,y) = \widetilde{u}^+(t,x,y) + \widetilde{u}^-(t,x,y).$$ Let u(t, x, y), u+(t, x, y) and u−(t, x, y) be the Schwartz kernels of the operators UA(t), UA+(t), and UA−(t), respectively. Formulae and imply $$\label{u agrees with tilde u} u(t,x,y)=\widetilde u(t,x,y) \mod C^\infty(\mathbb{R}\times M\_x\times M\_y;\mathrm{Mat}(m,\mathbb{C})).$$ This fact can be established as follows. Let $$\label{proof propagator general case 3} u\_\infty(t,x,y):=u(t,x,y)-\widetilde u(t,x,y).$$ From the construction algorithm, we know that $$\label{proof propagator general case 4} \left[ \left( -i\,\frac{\partial}{\partial t} + A^{(x)} \right) u\_\infty \right] (t,x,y) = f(t,x,y),$$ u∞(0, x, y) = ζ(x, y), where f ∈ C∞(R × Mx × My; Mat(m, C)) and ζ ∈ C∞(Mx × My; Mat(m, C)). Here the superscript in A(x) indicates that the differential operator A acts in the variable x. Using functional calculus, we can write the functions u∞, f and ζ in terms of the eigenfunctions of A as u∞(t, x, y) = ∑j, kajk(t) vj(x) [vk(y)]\, f(t, x, y) = ∑j, kbj*k(t) vj(x) [vk(y)]\, ζ(x, y) = ∑j, kcj*k vj(x) [vk(y)]\. Here the smooth functions bjk* and the constants cjk are given, whereas the functions ajk are our unknowns. Substituting – into, we obtain the family of first order ODEs $$\label{proof propagator general case 9} \left[ \left( -i\,\frac{d}{d t} + \lambda\_j \right) a\_{jk} \right] (t) = b\_{jk}(t),$$ ajk(0) = cjk, whose solutions are ajk(t) = e− iλjt(cjk + i∫0teiλjs bjk(s) ds). Let $\widehat\zeta$ be the operator with integral kernel ζ(x, y), $$\widehat\zeta: \ v(x)\mapsto\int\_M\zeta(x,y)\,v(y)\,\rho(y)\,dy\,.$$ Then $$\label{cjk} c\_{jk} = \langle v\_j,\widehat\zeta v\_k\rangle = \frac{1}{\lambda\_j^l\lambda\_k^n} \langle A^l v\_j,\widehat\zeta A^n v\_k\rangle = \frac{1}{\lambda\_j^l\lambda\_k^n} \langle v\_j,(A^l\widehat\zeta A^n)v\_k\rangle.$$ The operator $A^l\widehat\zeta A^n$ is a pseudodifferential operator of order − ∞, so formula and the fact that λk ∼ k1/d when k → ∞ allow one to conclude that the cjk decay faster than any power of j and k as j, k → ∞. A similar argument shows that the bjk(t) and their time derivatives decay faster than any power of j and k as j, k → ∞ uniformly over any bounded open interval in R. Formula now tells us that the same is true for the ajk(t). This, in turn, implies that the series on the RHS of defines a function u∞(t, x, y) which is smooth in all variables. So we arrive at, which gives us assumption (a) in Theorem [theorem positive and negative propagator abstract approach] with (T−, T+) = R. Resorting to standard stationary phase arguments – see, e.g., – and using the properties (i)–(iv) of our phase functions, it is easy to see that u± and $\widetilde{u}^\pm$ satisfy assumptions (b) and (c) of Theorem [theorem positive and negative propagator abstract approach]. Hence, Theorem [theorem positive and negative propagator abstract approach] gives us and. The fact that the construction is global in time is guaranteed by. If one is prepared to give up globality in time, Theorem [theorem propagator for A] and the corresponding proof can be adapted in a straightforward manner to the more customary case of real-valued – as opposed to complex-valued – phase functions. This is achieved by prescribing the phase functions to take values in R, dropping condition (iv) and replacing everywhere in the statement and in the proof the time domain R with the interval (T−, T+), where T+ : = minj inf{t > 0 ∣ detφxαηβ(j)∣x = x(j) = 0, (y, η) ∈ TʹM} , T− : = maxj sup{t < 0 ∣ detφxαηβ(j)∣x = x(j) = 0, (y, η) ∈ TʹM} . The values of T± depend on the choice of particular real-valued phase functions, but we always have T− < 0 < T+ . Observe that Theorem [theorem positive and negative propagator abstract approach] was formulated in such a way that it covers both the case of real-valued and complex-valued phase functions. The reader will have noticed that the zero mode propagator UA0 does not appear in our construction. This is due to the fact that, clearly, $$U^0\_A\overset{\mod \Psi^{-\infty}}{=}0.$$ We end this subsection with the observation that, thanks to the presence of the weight w(j) in formula, the scalar matrix-function a0(j) does not depend on the choice of the phase functions φ(j). This motivates the following definition. [definition principal symbol general case] We call a0(j) the principal symbol of the Fourier integral operator. The above definition agrees with the standard definition of principal symbol of a Fourier integral operator expressed as a section of the Keller–Maslov bundle, see. The algorithm ------------- The integral kernel of UA(j)(t) can be constructed explicitly as follows. Step 1. Choose a phase function φ(j) compatible with Theorem [theorem propagator for A]. We will see later on that for the special case of the Dirac operator we can identify a distinguished phase function, the Levi-Civita phase function. Furthermore, set χ(j) ≡ 1. In fact, the purpose of the cut-off is to localise integration in a neighbourhood of the h(j)-flow and away from the zero section: different choices of χ(j) result in oscillatory integrals differing by an infinitely smooth function. Step 2. Act with the operator − i∂t + A(x) on the oscillatory integral. This produces a new oscillatory integral $$\label{algorithm formula 1} \frac{1}{(2\pi)^d}\int\_{T'\_yM} e^{i \varphi^{(j)}(t,x;y,\eta)}\,a^{(j)}(t,x;y,\eta)\,w^{(j)}(t,x;y,\eta)\,d\eta$$ whose amplitude a(j) ∈ C∞(R × M × TʹM; Mat(m, C)) is given by a(j) : = e− iφ(j)[w(j)]− 1( − i∂t + A(x))(eiφ(j) a(j) w(j)). By making use of the fact that φ(j) and w(j) are positively homogeneous in momentum η of degree 1 and 0, respectively, one can write down an asymptotic expansion for the amplitude a(j) in components positively homogeneous in momentum: a(j)(t, x; y, η) ∼ ∑k = − 1+ ∞a− k(j)(t, x; y, η), a− k(j)(t, x; y, α η) = α− k a− k(j)(t, x; y, η), ∀α > 0. Step 3. As u(j)(t, x, y) is to be the (distributional) solution of the hyperbolic equation $$(-i \partial\_t +A^{(x)})u^{(j)}(t,x,y)\overset{\mod C^\infty}{=}0,$$ one would like to impose the condition a(j)(t, x, y, η) = 0. However, the amplitude a(j), unlike the symbol a(j), depends on x, and doing so would result in an unsolvable system of partial differential equations (PDEs). The current step consists in excluding the dependence of a(j) on x by means of a procedure known as reduction of the amplitude, to the end of reducing the system of PDEs to a system of ordinary differential equations instead. Put[5](#fn5) $$\label{operator L with j} L^{(j)}\_\alpha:=\left[(\varphi^{(j)}\_{x\eta})^{-1}\right]\_\alpha{}^\beta\,\dfrac{\partial}{\partial x^\beta}$$ and define [operators mathfrak S with j] $$\begin{gathered} \mathfrak{S}\_0^{(j)}:=\left.\left( \,\cdot\, \right)\right\|\_{x=x^{(j)}}\,, \label{mathfrak S0 with j}\\ \mathfrak{S}\_{-k}^{(j)}:=\mathfrak{S}\_0^{(j)} \left[ i \, [w^{(j)}]^{-1} \frac{\partial}{\partial \eta\_\beta}\, w^{(j)} \left( 1+ \sum\_{1\leq \|\boldsymbol{\alpha}\|\leq 2k-1} \dfrac{(-\varphi^{(j)}\_\eta)^{\boldsymbol{\alpha}}}{\boldsymbol{\alpha}!\,(\|\boldsymbol{\alpha}\|+1)}\,L^{(j)}\_{\boldsymbol{\alpha}} \right) L^{(j)}\_\beta \right]^k\,,\label{mathfrak Sk with j}\end{gathered}$$ where $\boldsymbol{\alpha}\in \mathbb{N}^d$, $\|\boldsymbol{\alpha}\|=\sum\_{j=1}^d \alpha\_j$ and $(-\varphi^{(j)}\_\eta)^{\boldsymbol{\alpha}}:=(-1)^{\|\boldsymbol{\alpha}\|}\, (\varphi^{(j)}\_{\eta\_1})^{\alpha\_1}\dots ( \varphi^{(j)}\_{\eta\_d})^{\alpha\_d}$. The operator is well defined, because the differential operators Lα(j) commute. Furthermore, the operators S− k(j) are invariant under change of local coordinates x and y. [remark Lalpha] Let f : Rd → Rd, $x\mapsto \widetilde x$, be a (locally) invertible map. Then the operators $$\widetilde L\_\alpha:=[(\nabla\_x f)^{-1}]\_\alpha{}^\beta \dfrac{\partial}{\partial x^\beta}$$ are the pushforward of partial derivatives $\partial/\partial \widetilde{x}^\alpha$ along f− 1. Hence, the operators $\widetilde L\_\alpha$ commute because the partial derivatives $\partial/\partial \widetilde{x}^\alpha$ commute. An adjustment of the above argument to our setting with f = φη(j) (and with account of the fact that φ(j) is complex-valued) provides an alternative explanation for the commutation of the operators Lα(j). The amplitude-to-symbol operator is defined as $$\begin{gathered} \mathfrak{S}^{(j)}:C^\infty(\mathbb{R}\times M \times T'M)\to C^\infty(\mathbb{R} \times T'M)\,, \nonumber \\ \label{definition amplitude to symbol with j} \mathfrak{S}^{(j)}:=\sum\_{j=0}^\infty \mathfrak{S}^{(j)}\_{-k}\,.\end{gathered}$$ When acting on a function positively homogeneous in momentum, the operator S− k(j) excludes the dependence on x and decreases the degree of homogeneity by k. The reduction of the amplitude is achieved by replacing the amplitude a(j) in by S(j)a(j) = : b(j), with b(j)(t; y, η) ∼ ∑k = − 1+ ∞b− k(j)(t; y, η) , b− k(j) = ∑l + s = kS− l(j) a− s(j) . The oscillatory integral $$\label{algorithm formula 6} \frac{1}{(2\pi)^d}\int\_{T'\_yM} e^{i \varphi^{(j)}(t,x;y,\eta)}\,\mathfrak{b}^{(j)}(t;y,\eta)\,w^{(j)}(t,x;y,\eta)\,d\eta$$ differs from only by an infinitely smooth function. We refer the reader to for further particulars and detailed proofs concerning the amplitude-to-symbol operator. Step 4. Set b− k(j) = 0, k = − 1, 0, 1, …. Equations, combined with the initial conditions stemming from the constraint $$\label{algorithm formula 8} \sum\_{j} U^{(j)}(0)\overset{\mod \Psi^{-\infty}}{=}\mathrm{Id},$$ yield a hierarchy of (matrix) transport equations for the homogeneous components a− k(j). For the special case of the massless Dirac operator, the first few equations in the hierarchy are given by –. Let us make a few remarks warranted by formula. The m oscillatory integrals appearing on the RHS of and are not independent of one another, but they ‘mix’ at t = 0 via the initial condition. Now, satisfying involves representing the identity operator on C∞(M; Cm) in a somewhat nonstandard fashion, as $$\label{indentity operator oscillatory integral general case} \mathrm{Id}\overset{\mod{\Psi^{-\infty}}}{=}\sum\_j \frac{1}{(2\pi)^d}\int\_{T'M} e^{i\varphi^{(j)}(0,x;y,\eta)} \,\mathfrak{s}^{(j)}(y,\eta)\,\chi^{(j)}(0,x;y,\eta)\,w^{(j)}(0,x;y,\eta)\,(\,\cdot\,)\,\rho(y)\,dy\,d\eta,$$ with s(j) ∈ Sph0(TʹM; Mat(m; C)). In terms of the symbols a(j), the initial condition reads a(j)(0; y, η) = s(j)(y, η). From the fact that the principal symbol of the identity operator is the identity matrix it follows that ∑ja0(j)(0; y, η) = ∑js0(j)(y, η) = 1m × m. Furthermore, one can show that s0(j)(y, η) = v(j)(y, η) [v(j)(y, η)]\. However, obtaining formulae for subleading components s− 1(j) is already a challenging task, see. In general, lower order components of s(j) depend in a nontrivial manner on the eigenvalues and eigenprojections of the matrix-function Aprin(x, ξ) and on the choice of phase functions φ(j). The invariant representation of the identity operator – and, more generally, of pseudodifferential operators – on manifolds is not a well-studied subject. An initial analysis of the scalar case was carried out in. For the case of the Dirac operator a more detailed examination of will be provided in subsection [identity operator dirac]. [remark about pseudodifferential] All statements and results presented in this section carry over verbatim to the case where A* is an elliptic symmetric first order m × m matrix pseudodifferential – as opposed to differential – operator, with the following exceptions: * formulae and have to be dropped as they are no longer true; * ‘Step 2.’ in subsection [The algorithm] has to be modified to take into account the action of a pseudodifferential operator on an oscillatory integral in an invariant manner, along the lines of. Let us point out that in this section we did not use anywhere the fact that M carries a Riemannian structure. If one replaces the Riemannian density with an arbitrary positive density, all statements and results stay the same. Invariant description of pseudodifferential operators acting on scalar functions ================================================================================ In order to prepare ourselves to address the issue of initial conditions for our transport equations in the case of the Dirac operator, we need to discuss first the more general question of invariant representation of a pseudodifferential operator. We devote a separate section to this, as we believe this matter to be of independent interest. Note that we treat the case of a scalar operator merely for the sake of presentational convenience: all the formulae and arguments in this subsection remain unchanged for matrix pseudodifferential operators acting on m-columns of scalar functions. We call time-independent Levi-Civita phase function the function ϕ ∈ C∞(M × TʹM; C) defined by $$\label{formula time-independent Levi-Civita phase function} \phi(x;y,\eta):=\int\_{\gamma} \zeta\, dz +\frac{i\epsilon}{2} h(y,\eta)\,\left[ \mathrm{dist}(x,y)\right]^2$$ when x lies in a geodesic neighbourhood of y and continued smoothly elsewhere in such a way that Im ϕ ≥ 0. Here γ is the (unique) shortest geodesic connecting y to x, ζ is the parallel transport of η along γ, $$\label{eigenvalues massless dirac} h(y,\eta):=\sqrt{g^{\alpha\beta}(y)\,\eta\_\alpha\eta\_\beta}\,,$$ dist is the geodesic distance and ε is a positive parameter. Let P be a pseudodifferential operator of order p acting on scalar functions over a Riemannian d-manifold. The operator P can be written, modulo an integral operator with smooth kernel, in the form P = ∫TʹMeiϕ(x; y, η) p(y, η) χ0(x; y, η) w0(x; y, η) ( ⋅ ) ρ(y) dy dη, where ϕ is the time-independent Levi-Civita phase function, p ∈ Sphm(TʹM), χ0 is a cut-off localising integration to a neighbourhood of the diagonal and away from the zero section (see also (I)–(III) in Theorem [theorem positive and negative propagator abstract approach]) and $$w\_0(x;y,\eta):= \left[\rho(x)\,\rho(y)\right]^{-\frac12}\left[{\det}^2 \phi\_{x^\alpha\eta\_\beta}(x;y,\eta) \right]^\frac14.$$ Here the smooth branch of the complex root is chosen in such a way that w0(y; y, η) = [ρ(y)]− 1. Note that is, effectively, a special case of with t = 0. Formula provides an invariant representation of the pseudodifferential operator P. We call full symbol of the operator P the scalar function p(y, η) ∼ ∑k = − p+ ∞p− k(y, η). Furthermore, we call the homogeneous functions pp and pp − 1 the g-principal and g-subprincipal symbol, respectively[6](#fn6). The notions of principal and subprincipal symbols of a pseudodifferential operator are nowadays standard concepts in microlocal analysis. The former makes sense for operators acting either on scalar functions or on half-densities, whereas the latter is only defined for operators acting on half-densities. We refer the reader to for further details. Note that the concept of subprincipal symbol was introduced by Duistermaat and Hörmander in. It is easy to see that the concept of principal symbol Pprin and that of g-principal symbol pp coincide. As far as the subprincipal symbol is concerned, the situation is more complicated, in that before drawing a comparison we need to turn our operator into an operator acting on half-densities. Put P1/2 : = ρ1/2 P ρ− 1/2 and let Psub be the subprincipal symbol of the operator defined in accordance with. A natural question to ask is: what is the relation between Psub and pp − 1? [theorem subprincipal symbols for P dirac] The invariant quantities Psub and pp − 1 are related as $$\label{relation between invariant subps} \mathfrak{p}\_{p-1}=P\_\mathrm{sub}+\frac{i}{2}(P\_\mathrm{prin})\_{y^\alpha\eta\_\alpha}+\frac{i}{2}\,\Gamma^\alpha{}\_{\beta\gamma}\left[ \eta\_\alpha(P\_\mathrm{prin})\_{\eta\_\beta}\right]\_{\eta\_\gamma}- \frac\epsilon2\,g\_{\beta\gamma} \left[h \,(P\_\mathrm{prin})\_{\eta\_\beta} \right]\_{\eta\_\gamma}.$$ Theorem [theorem subprincipal symbols for P dirac] implies that, in particular, the two notions of subprincipal symbol coincide when the principal symbol does not depend on η, i.e. when P is a pseudodifferential operator of the type “multiplication by a scalar function plus an operator of order − 1”. Note that the identity operator, whose invariant representation was investigated in, falls into this class. A tedious, yet straightforward, calculation shows that the RHS of is a scalar function on the cotangent bundle. In fact, the second and third summands on the RHS of admit an invariant representation in terms of the Laplace–Beltrami operator associated with the neutral metric n on the cotangent bundle T\*M, which, in local coordinates (x1, …, xd, ξ1, …, ξd), reads $$n\_{jk}(x,\xi)= \begin{pmatrix} -2\, \xi\_\gamma \, \Gamma^\gamma{}\_{\alpha\beta}(x) & \delta\_\alpha{}^\mu\\ \delta^\nu{}\_\beta & 0 \end{pmatrix}, \qquad j,k \in \{1,\ldots, 2d\}.$$ The adjective ‘neutral’ refers to the fact that the metric n has signature (d, d). It turns out that the neutral metric is an effective tool in the development of an invariant theory of pseudodifferential operators on Riemannian manifolds. As the analysis of this matter requires a lengthy discussion and would take us away from the core subject of our paper, we plan to address it in detail elsewhere. See also. [Proof of Theorem [theorem subprincipal symbols for P dirac]] Consider the pseudodifferential operator P and turn it into an operator on half-densities P1/2 via. In what follows we work in an arbitrary coordinate system, the same for x and y. Dropping the cut-off, the integral kernel of P1/2 now reads $$\label{proof general subprincipal dirac temp 1} \frac1{(2\pi)^d} \int\_{T'\_yM} e^{i\phi(x;y,\eta)} \,\mathfrak{p}(y,\eta)\, \sqrt{\det\phi\_{x\eta}}\,d\eta\,.$$ Our phase function admits the expansion $$\label{proof general subprincipal dirac temp 2} \phi(x;y,\eta)=(x-y)^\alpha\eta\_\alpha +\frac12\Gamma^\alpha{}\_{\beta\gamma}\,\eta\_\alpha(x-y)^\beta (x-y)^\gamma +\frac{i\epsilon h}2g\_{\alpha\beta}(x-y)^\alpha(x-y)^\beta + O(\\|x-y\\|^3),$$ which implies that $$\label{proof general subprincipal dirac temp 3} \sqrt{\det\phi\_{x\eta}}=1+ \frac12[ \Gamma^\alpha{}\_{\alpha\beta}+i\epsilon h^{-1}\eta\_\beta ](x-y)^\beta+O(\\|x-y\\|^2).$$ Substituting and into, we get $$\begin{gathered} \label{proof general subprincipal dirac temp 4} \frac{1}{(2\pi)^d}\int e^{i(x-y)^\alpha\eta\_\alpha} \Bigl\{ \mathfrak{p}\_p \\ +\left( \frac12 \left[ i\Gamma^\alpha{}\_{\beta\gamma}\,\eta\_\alpha -\epsilon hg\_{\beta\gamma} \right] (x-y)^\beta (x-y)^\gamma +\frac12 \left[ \Gamma^\alpha{}\_{\alpha\beta}+i\epsilon h^{-1}\eta\_\beta \right] (x-y)^\beta \right) \mathfrak{p}\_p \\ +\mathfrak{p}\_{p-1} +O(\\|\eta\\|^{p-2}) \Bigr\}\, d\eta\,.\end{gathered}$$ Excluding the x-dependence from the amplitude in by acting with the operator $$\label{operator Melrose} \mathcal{S}\_\mathrm{right}(\,\cdot\,):=\left.\left[\exp\left(i\frac{\partial^2}{\partial x^\mu\,\partial \eta\_\mu}\right)(\,\cdot\,)\right]\right\|\_{x=y},$$ we arrive at $$\begin{gathered} \label{proof general subprincipal dirac temp 5} \frac{1}{(2\pi)^d}\int e^{i(x-y)^\alpha\eta\_\alpha} \Bigl\{ \mathfrak{p}\_p \\ -\frac{i}{2}\left[ \eta\_\alpha \,\Gamma^\alpha{}\_{\beta\gamma}\,(\mathfrak{p}\_p)\_{\eta\_\beta}\right]\_{\eta\_\gamma}+\frac\epsilon2 \left[h \,g\_{\gamma\beta}\,(\mathfrak{p}\_p)\_{\eta\_\beta} \right]\_{\eta\_\gamma} \\ +\mathfrak{p}\_{p-1} +O(\\|\eta\\|^{p-2}) \Bigr\}\, d\eta\,.\end{gathered}$$ Computing the subprincipal symbol of and using the fact that pp = Pprin = (P1/2)prin , we obtain. Note that the sign in front of the correction term $$\frac{i}{2}(P\_\mathrm{prin})\_{y^\alpha\eta\_\alpha}$$ is opposite to the usual one, see, for example,. This is due to the fact that in this paper we use the right – as opposed to left – quantization. Global propagator for the Dirac operator ======================================== In this section we will start the analysis of the global propagator for the Dirac operator, specialising Theorem [theorem propagator for A] to the case A = W. We denote by Wprin(y, η) : = σα(y) ηα the principal symbol of W and by $$\label{zero order part dirac definition} W\_0(x):=-\frac{i}4\sigma^\alpha(x)\sigma\_\beta(x) \left( \frac{\partial\sigma^\beta}{\partial x^\alpha}(x) +\Gamma^\beta{}\_{\alpha\gamma}(x)\,\sigma^\gamma(x) \right)$$ its zero order part, see Definition [massless dirac definition]. The principal symbol Wprin(y, η) has eigenvalues h± = ± h, where h is given by, compare with. This fact, which can be easily established by writing down in local coordinates, shows that the Dirac operator is indeed elliptic. It is well-known that the Hamiltonian flow (x+(t; y, η), ξ+(t; y, η)) generated by h is (co-)geodesic. The two flows (x±(t; y, η), ξ±(t; y, η)) are related as (x−(t; y, η), ξ−(t; y, η)) = (x+(t; y, − η), − ξ+(t; y, − η)). Our goal is to write down explicitly the positive and negative propagators and in the form for a distinguished choice of phase functions. To this end, we give the following definition (see also ). [definition LC phase functions with epsilon dirac] We call positive ( + ), resp. negative ( − ), Levi-Civita phase function the infinitely smooth function φ± ∈ C∞(R × M × TʹM; C) defined by $$\label{definition positive negative Levi-Civita phase function} \varphi^\pm(t,x;y,\eta)=\int\_{\gamma^\pm} \zeta^\pm \,dz+\frac{i\,\epsilon}{2}h(y,\eta) {\operatorname{dist}}^2(x,x^\pm(t;y,\eta))$$ for x in a geodesic neighbourhood of x±(t; y, η) and continued smoothly elsewhere in such a way that Imφ± ≥ 0. Here dist is the Riemannian geodesic distance, the path of integration γ± is the shortest geodesic connecting x± to x, ζ± is the result of parallel transport of ξ±(t; y, η) along γ± and ε is a positive parameter. The positive and negative Levi-Civita phase functions are related as $$\label{relation between the two Levi-Civita phase functions} \varphi^-(t,x;y,\eta) = - \overline { \varphi^+(t,x;y,-\eta) }.$$ Let us point out that the way one continues φ± outside a neighbourhood of the flow does not affect the singular part of the propagators. The choice of a different smooth continuation results in an error $\overset{\mod \Psi^{-\infty}}{=}0$, as one can show by a straightforward (non)stationary phase argument. The time-independent phase function ϕ introduced in the previous section is the restriction to t = 0 of the phase functions φ±, ϕ(x; y, η) = φ+(0, x; y, η) = φ−(0, x; y, η). It is easy to see that the positive and negative Levi-Civita phase functions satisfy conditions (i), (ii) and (iv) from Theorem [theorem propagator for A]. Furthermore, implies that condition (iii) is also satisfied. Hence, Theorem [theorem propagator for A] ensures that the integral kernel of U± can be written as a single oscillatory integral $$\label{integral kernel of U pm} u^\pm(t,x,y):=\frac{1}{(2\pi)^3}\int\_{T'\_yM} e^{i \varphi^\pm(t,x;y,\eta)}\,\mathfrak{a}^\pm(t;y,\eta)\,\chi^\pm(t,x;y,\eta)\,w^\pm(t,x;y,\eta)\,d\eta,$$ where φ± is the positive/negative Levi-Civita phase function. [definition full symbols dirac propagator] We define the full symbol of the positive (resp. negative) propagator to be the scalar matrix-function a+ (resp. a−), obtained through the algorithm described in Section [The algorithm] with Levi-Civita phase functions. We define the subprincipal symbol of the positive (resp. negative) propagator to be the scalar matrix-function a− 1+ (resp. a− 1−) obtained the same way. As to the principal symbol, this object was defined earlier, see Definition [definition principal symbol general case]. We stress that the mathematical objects contained in the above definition are uniquely and invariantly defined. They only depend on the phase functions which, in turn, originate from the geometry of M in a coordinate-free covariant manner, cf. Definition [definition LC phase functions with epsilon dirac]. To the best of our knowledge, there is no accepted definition of full symbol or subprincipal symbol for a Fourier integral operator available in the literature to date. The geometric nature of our construction allows us to provide invariant definitions of full and subprincipal symbol of the Dirac propagator, analyse them, and give explicit formulae. This paper, alongside, aims to build towards an invariant theory for pseudodifferential and Fourier integral operators on manifolds. Before moving on to computing the principal and subprincipal symbols of the positive (resp. negative) Dirac propagator, an important remark is in order. In addition to what was discussed in Section [Preliminary results for general first order systems] for the general case, the construction of the Dirac propagator has to be consistent with the gauge transformation,. In particular, the action of the gauge transformation needs to be carefully accounted for by the construction process. The transformation leads to the transformation a±(t; y, η) ↦ G\(x) a±(t; y, η) G(y). in the oscillatory integral. Note that this introduces an x-dependence which has to be handled by means of amplitude-to-symbol reduction. Transport equations ------------------- By acting with the Dirac operator W* on in the variable x and dropping the cut-off, we obtain $$\label{transport equations formula 1} W u^\pm(t,x,y)=\frac{1}{(2\pi)^3}\int\_{T'\_yM} e^{i \varphi^\pm(t,x;y,\eta)}\,a^\pm(t;y,\eta)\,w^\pm(t,x;y,\eta)\,d\eta,$$ where $$\label{transport equations formula 2} \begin{split} a&=-i e^{-i\varphi^\pm} (w^\pm)^{-1} \partial\_t \left(e^{i\varphi^\pm}\,\mathfrak{a}^\pm \,w^\pm\right)+ \left[-i e^{-i\varphi^\pm} (w^\pm)^{-1}\sigma^\alpha \partial\_{x^\alpha} \left(e^{i\varphi^\pm} \,w^\pm \right) + W\_0\right]\mathfrak{a}^\pm\\ &= \left(\varphi^\pm\_t+\sigma^\alpha \varphi^\pm\_{x^\alpha}\right)\mathfrak{a}^\pm -i \mathfrak{a}^\pm\_t+ \left[-i (w^\pm)^{-1}\left( w^\pm\_t+\sigma^\alpha w^\pm\_{x^\alpha} \right)+W\_0 \right]\mathfrak{a}^\pm. \end{split}$$ Put a ∼ ∑k = − 1+ ∞a− k, where a1± : = (φt± + Wprin(x, φx±))a0± and a− k± : = (φt± + Wprin(x, φx±))a− k − 1± − i(a− k±)t + [ − i(w±)− 1(wt± + σαwxα±) + W0]a− k± for k ≥ 0. Note that the a− k±, k ≥ − 1, are positively homogeneous in momentum of degree − k. Our transport equations read $$\begin{gathered} \mathfrak{S}^\pm\_0 a^\pm\_1=0, \label{first transport equation dirac} \\ \mathfrak{S}^\pm\_{-1} a^\pm\_1+ \mathfrak{S}^\pm\_{0}a^\pm\_0=0, \label{second transport equation dirac} \\ \mathfrak{S}^\pm\_{-2} a^\pm\_1+ \mathfrak{S}^\pm\_{-1}a^\pm\_0+\mathfrak{S}^\pm\_0 a^\pm\_{-1}=0, \label{third transport equation dirac} \\\nonumber \dots\end{gathered}$$ Recalling that v± are the normalised eigenvectors of Wprin corresponding to the eigenvalues ± h, denote by P±(y, η) : = v±(y, η) [v±(y, η)]\* the spectral projections along the eigenspaces spanned by v±. Of course, Wprin = h (P+ − P−), Id = P+ + P−, and $$\label{projections in terms of principal symbol dirac} P^\pm=\frac12\left( \mathrm{Id}\pm \frac{W\_\mathrm{prin}}{h} \right).$$ Let us label the transport equations with nonnegative integer numbers in increasing order, so that is the zeroth transport equation, is the first transport equation and so on. Direct inspection of and reveals that * multiplication of the n-th transport equation by P∓(x±, ξ±) on the left allows one to determine P∓(x±, ξ±)a− n±(t; y, η), n ≥ 0, algebraically; * multiplication of the (n + 1)-th transport equation by P±(x±, ξ±) on the left and the use of allows one to determine P±(x±, ξ±)a− n±(t; y, η), n ≥ 0, upon solving a matrix ordinary differential equation in the variable t. Summing up and one obtains a− k±(t; y, η), in view of. Pseudodifferential operators U±(0) ------------------------------------ This subsection is devoted to the examination of operators U±(0). We need to examine these operators because, as explained in subsection [The algorithm], their full symbols determine the initial conditions a− k±(0; y, η) for our transport equations. We have U±(0) = θ( ± W), where $$\label{4 December 2019 equation 2} \theta(\lambda) := \begin{cases} 1\quad\text{for}\quad\lambda>0, \\ 0\quad\text{for}\quad\lambda\le0. \end{cases}$$ We see that the operators U±(0) are self-adjoint pseudodifferential operators of order zero, orthogonal projections onto the positive/negative eigenspaces of the operator W. The operator Id − U+(0) − U−(0) is the orthogonal projection onto the nullspace of the operator W, hence $$\label{sum of propagators at time zero} U^+(0)+U^-(0)\overset{\mod \Psi^{-\infty}}{=} \mathrm{Id}.$$ The principal symbols of the operators U±(0) read [U±(0)]prin = P±(y, η), where P± are the orthogonal projections onto the positive/negative eigenspaces of the principal symbol of the operator W, see. The analysis of the full symbol of U±(0) is a delicate task which was investigated, to a certain extent and in a somewhat different setting, in. In order to develop the ideas from we have to address a number of issues. * We are now dealing with scalar fields as opposed to half-densities. * We are now making full use of Riemannian structure. * We are now working in the special setting of a system of two equations in dimension three with trace-free principal symbol. * Unlike, we are aiming to evaluate the actual matrix-functions [U±(0)]sub and not only their traces. In order to calculate the subprincipal symbols of the pseudodifferential operators U±(0) we will need the following auxiliary result. [theorem covariant derivatives gauge transformation] Fix a point y ∈ M and let $\{\widetilde e\}\_{j=1}^3$ be a framing on M. Let G ∈ C∞(M; SU(2)) be a gauge transformation such that G(y) = Id and let $$\label{new framing generated by G} {e}\_j{}^\alpha:=\frac12 \operatorname{tr}(s\_j\,G^\\,s^k\,G)\,\widetilde{e}\_k{}^\alpha.$$ Then $$\label{covariant derivative of gauge transformation G} \nabla\_\alpha G(y)=-\frac{i}{2} \left[\overset{\}{K}{}\_{\alpha\beta}(y)-\overset{\}{\widetilde{K}}{}\_{\alpha\beta}(y)\right]\,\sigma^\beta(y),$$ where K* (resp. $\widetilde K$) is the contorsion tensor of the Weitzenböck connection (see Appendix [Geometric properties of the Weitzenböck connection]) associated with the framing {ej}j = 13 (resp. $\{\widetilde e\_j\}\_{j=1}^3$), the star stands for the Hodge dual applied in the first and third indices, see formula, and σα(y) is defined by. The proof is provided in Appendix [appendix proof covariant derivative gauge transformation]. [remark about w and tilde w] Let $\{\widetilde e\}\_{j=1}^3$ and {e}j = 13 be a pair of framings related in accordance with, and let $\widetilde W$ and W be the corresponding Dirac operators, see Definition [massless dirac definition]. Then $$\label{relation between w and tilde w} W=G^\\widetilde WG.$$ The following theorem is the main result of this subsection. [theorem about subprincipal symbols of propagators at zero] We have $$\label{4 December 2019 equation 6} [U^\pm(0)]\_\mathrm{sub}(y,\eta) = \pm \frac1{4(h(y,\eta))^3}\, \overset{\}{T}{}^{\alpha\beta}(y)\, \eta\_\alpha\eta\_\beta \,\mathrm{Id}\,,$$ where T is the torsion tensor of the Weitzenböck connection (see Appendix [Geometric properties of the Weitzenböck connection]) associated with the framing {ej}j = 13 encoded within the Dirac operator W (see Definition [massless dirac definition]) and the star stands for the Hodge dual applied in the second and third indices, see formula. Let us fix a point y ∈ M and choose normal geodesic coordinates x centred at y such that ejα(y) = δjα . Consider the (local) operator with constant coefficients $$\label{dirac with constant coefficients} \widetilde W:=-is^\alpha\,\frac{\partial}{\partial x^\alpha}\,,$$ where the sα are the standard Pauli matrices. Let us choose a smooth special unitary 2 × 2 matrix-function G such that G(0) = Id, $$\label{relation between two principal symbols} [W]\_\mathrm{prin}=[G^\\widetilde WG]\_\mathrm{prin}+O(\,\\|\eta\\|\,\\|x\\|^2\,)\,,$$ compare with. It is easy to see that such a matrix-function G(x) exists and is defined uniquely modulo O(∥x∥2). Let us now compare the subprincipal symbols of the pseudodifferential operators θ( ± W) and $\theta(\pm G^\\widetilde WG)$, with $G^\\widetilde WG$ understood as an operator acting in Euclidean space (constant metric tensor gαβ(x) = δαβ). It can be shown that at the origin we have $$[W]\_\mathrm{sub}(0,\eta)=[G^\\widetilde WG]\_\mathrm{sub}(0,\eta).$$ Thus, the proof of the Theorem [theorem about subprincipal symbols of propagators at zero] has been reduced to the case when we are in Euclidean space and the operator W is given by formulae and. We have $$\label{dima1} \theta(\pm\widetilde W) = \frac1{(2\pi)^3} \int\_{T'\mathbb{R}^3} e^{i(x-z)^\alpha\eta\_\alpha}\,P^\pm(\eta)\,(\,\cdot\,)\,dz\,d\eta\,,$$ where $$\label{dima2} P^\pm(\eta) = \frac12 \left( \mathrm{Id} \pm \frac1{\\|\eta\\|}s^\beta\eta\_\beta \right).$$ Formulae and imply that $$\label{dima3} \theta(\pm G^\\widetilde WG) = \frac1{(2\pi)^3} \int\_{T'\mathbb{R}^3} e^{i(x-z)^\alpha\eta\_\alpha}\,Q^\pm(x,z,\eta)\,(\,\cdot\,)\,dz\,d\eta\,,$$ where $$\label{dima4} Q^\pm(x,z,\eta) = G^\(x)\,P^\pm(\eta)\,G(z) = \frac12\, G^\(x) \left( \mathrm{Id} \pm \frac1{\\|\eta\\|}s^\beta\eta\_\beta \right) G(z)\,.$$ Excluding the z-dependence from the amplitude Q± by acting with the operator $$\label{operator Melrose other way round} \mathcal{S}\_\mathrm{left}(\,\cdot\,):=\left.\left[\exp\left(-i\frac{\partial^2}{\partial z^\mu\,\partial \eta\_\mu}\right)(\,\cdot\,)\right]\right\|\_{z=x},$$ compare with, we arrive at $$\label{dima5} \theta(\pm G^\\widetilde WG) = \frac1{(2\pi)^3} \int\_{T'\mathbb{R}^3} e^{i(x-z)^\alpha\eta\_\alpha}\,\mathcal{Q}^\pm(x,\eta)\,(\,\cdot\,)\,dz\,d\eta\,,$$ where Q±(x, η) = Q0±(x, η) + Q− 1±(x, η) + O(∥η∥− 2) , $$\label{dima7} \mathcal{Q}\_0^\pm(x,\eta) = \frac12\, G^\(x) \left( \mathrm{Id} \pm \frac1{\\|\eta\\|}s^\beta\eta\_\beta \right) G(x)\,,$$ $$\label{dima8} \mathcal{Q}\_{-1}^\pm(x,\eta) = -\, \frac i2\, G^\(x) \left( \mathrm{Id} \pm \frac1{\\|\eta\\|}s^\beta\eta\_\beta \right)\_{\eta\_\mu} G\_{x^\mu}(x)\,.$$ In the Euclidean setting the standard formula for the subprincipal symbol reads $$\label{dima9} [\theta(\pm G^\\widetilde WG)]\_\mathrm{sub} = \mathcal{Q}\_{-1}^\pm + \frac i2(\mathcal{Q}\_0^\pm)\_{x^\mu\eta\_\mu}\,.$$ Substituting and into and setting x* = 0, we get $$\begin{gathered} \label{dima10} [\theta(\pm G^\\widetilde WG)]\_\mathrm{sub} = \pm \frac i4 \left[ G^\\_{x^\mu} \left( \frac1{\\|\eta\\|}s^\beta\eta\_\beta \right)\_{\eta\_\mu} - \left( \frac1{\\|\eta\\|}s^\beta\eta\_\beta \right)\_{\eta\_\mu} G\_{x^\mu} \right] \\ = \pm \frac{i( \delta\_\beta{}^\mu\\|\eta\\|^2-\eta\_\beta\,\eta^\mu) } {4\\|\eta\\|^3} \left[ G^\\_{x^\mu} s^\beta - s^\beta G\_{x^\mu} \right].\end{gathered}$$ Theorem [theorem covariant derivatives gauge transformation] tells us that $G\_{x^\mu}=\frac{i}{2}\overset{\}{K}{}\_{\mu\nu}\,s^\nu$. Substituting this into, and using standard properties of Pauli matrices and, we get $$\begin{split} \label{dima11} [\theta(\pm G^\\widetilde WG)]\_\mathrm{sub} & = \pm \frac{ \delta\_\beta{}^\mu\\|\eta\\|^2-\eta\_\beta\,\eta^\mu } {8\\|\eta\\|^3} \left[ s^\nu s^\beta + s^\beta s^\nu \right] \overset{\}{K}{}\_{\mu\nu} \\ & = \pm \frac{ 1 } {4\\|\eta\\|^3} \left( \overset{\}{K}{}^\gamma{}\_\gamma \delta\_{\mu\nu} -\overset{\}{K}{}\_{\mu\nu} \right)\eta^\mu\eta^\nu \operatorname{Id} \\ & = \pm \frac{1}{4\\|\eta\\|^3} \overset{\}{T}{}\_{\mu\nu} \,\eta^\mu\,\eta^\nu \operatorname{Id}. \end{split}$$ The above argument combined with yields. Observe that formula implies $$\label{4 December 2019 equation 6 trace} \operatorname{tr}\, [U^\pm(0)]\_\mathrm{sub}(y,\eta) = \pm \frac1{2(h(y,\eta))^3}\, \overset{\}{T}{}^{\alpha\beta}(y)\, \eta\_\alpha\eta\_\beta\,,$$ which agrees with and. Principal symbol of the global Dirac propagator =============================================== In this section we provide an explicit geometric characterisation of the principal symbols of the positive and negative Dirac propagators. [theorem small time expansion principal symbol] The principal symbols of the positive and negative Dirac propagators are a0±(t; y, η) = ζ±(t; y, η) [v±(y, η)]\, where ζ±(t; y, η) is the parallel transport of v±(y, η) along x± with respect to the spin connection, i.e. $$\label{definition of parallel transport of spinor} \left( \frac{d}{dt} +[\dot{x}^\pm]^\alpha\,\frac14 \sigma\_\beta \left(\frac{\partial\sigma^\beta}{\partial x^\alpha}+ \Gamma^\beta{}\_{\alpha\gamma}\sigma^\gamma \right) \right) \zeta^\pm=0, \qquad \zeta^\pm\|\_{t=0}=v^\pm.$$ It is known that the principal symbols a0± are independent of the choice of the phase function and read a0±(t; y, η) = v±(x±, ξ±) [v±(y, η)]\ e− i ∫0tq±(x±(τ; y, η), ξ±(τ; y, η)) dτ, where $$\label{q appearing in principal symbols} q^\pm=[v^\pm]^\* \,W\_{\mathrm{sub}} \,v^\pm-\frac{i}{2}\{[v^\pm]^\,W\_\mathrm{prin}-h^\pm,v^\pm \}-i\, [v^\pm]^\\{v^\pm,h^\pm\}\,,$$ and $$\label{subprincipal symbol dirac} W\_{\mathrm{sub}}(y):=W\_0(y)+\frac{i}2 \sigma^\alpha(y) \,\Gamma^\beta{}\_{\alpha\beta}(y)+ \frac{i}{2}[W\_\mathrm{prin}(y,\eta)]\_{y^\alpha\eta\_\alpha}.$$ In formula curly brackets denote the Poisson bracket {B, C} : = ByαCηα − BηαCyα and the generalised Poisson bracket {B, C, D} : = ByαCDηα − BηαCDyα on matrix-functions on the cotangent bundle. In formula the second term on the RHS is the result of switching to half-densities, see. Introducing the shorthand q±(t) : = q±(x±(t; y, η), ξ±(t; y, η)), the task at hand is to show that ζ±(t; y, η) = e− i ∫0tq±(τ) dτ v±(x±, ξ±). More explicitly, we need to show that $$\label{proof principal symbols task pt} e^{i\,\int\_0^t q^\pm(\tau)\,d \tau}\left( \frac{d}{dt} +[\dot{x}^\pm]^\alpha\,\frac14 \sigma\_\beta \left(\frac{\partial\sigma^\beta}{\partial x^\alpha}+ \Gamma^\beta{}\_{\alpha\gamma}\sigma^\gamma \right) \right)\left[ e^{-i\,\int\_0^t q^\pm(\tau)\,d \tau}\,v^\pm(x^\pm,\xi^\pm)\right]=0,$$ where we premultiplied our expression by ei ∫0tq±(τ) dτ for the sake of convenience. We shall prove for a0+, which corresponds to the upper choice of signs in. The proof for a0− is analogous. Let us begin by computing $$\label{proof principal symbol dirac 1} \begin{split} e^{i\,\int\_0^t q^+(\tau)} \dfrac{d}{dt}\left( e^{-i\,\int\_0^t q^+(\tau)\,d \tau}\,v^+(x^+,\xi^+)\right) & = -i q^+(t) \,v^++ v^+\_{x^\alpha}[\dot{x}^+]^\alpha+v^+\_{\xi\_\alpha} [\dot{\xi}^+]\_\alpha\\ & = -i q^+(t) \,v^+ + \{v^+,h\}. \end{split}$$ To this end, let us choose geodesic normal coordinates centred at x+(t; y, η) = 0 and such that [ξ+(t; y, η)]α = δ3α. Furthermore, up to a global rigid rotation of the framing, we can assume that ejα(0) = δjα. In our special coordinate system we have $$\label{proof principal symbol dirac 2bis} v^+(0,\xi^+)= \begin{pmatrix} 1\\0 \end{pmatrix}, \qquad v^-(0,\xi^+)= \begin{pmatrix} 0\\1 \end{pmatrix},$$ and we can expand our framing about x+ = 0 as $$\label{proof principal symbol dirac 3} \begin{pmatrix} e\_1{}^1(x)&e\_1{}^2(x)&e\_1{}^3(x)\\ e\_2{}^1(x)&e\_2{}^2(x)&e\_2{}^3(x)\\ e\_3{}^1(x)&e\_3{}^2(x)&e\_3{}^3(x) \end{pmatrix} = \begin{pmatrix} 1&l^3(x)&-l^2(x)\\ -l^3(x)&1&l^1(x)\\ l^2(x)&-l^1(x)&1 \end{pmatrix}+O(\\|x\\|^2) \quad \text{as }x\to 0,$$ where lk(x) = O(∥x∥), k = 1, 2, 3. The fact that ([v+]\*v+)(x, ξ) = 1 implies {[v+]\, P+, v+}(0, ξ+) = [vxα+]\ v+ [v+]\* vξα+ − [vξα+]\* v+ [v+]\* vxα+ = 0, which, in turn, yields {[v+]\, Wprin, v+} = h* {[v+]\, 2P+ − Id, v+} = − h* {[v+]\, v+}. A standard perturbation argument gives us $$\label{proof principal symbol dirac 4} h\,\{[v^+]^\,v^+\}(0,\xi^+)=\left.-\frac i2 \left( \dfrac{\partial l^1}{\partial x^1}+\dfrac{\partial l^2}{\partial x^2}\right)\right\|\_{x=0}$$ and $$\label{proof principal symbol dirac 5} \{v^+,h\}(0,\xi^+)=\left.\frac{i}2 \begin{pmatrix} 0\\ \dfrac{\partial l^1}{\partial x^3}+i\dfrac{\partial l^2}{\partial x^3} \end{pmatrix}\right\|\_{x=0}.$$ Furthermore, combining with and, we get $$\label{proof principal symbol dirac 6} W\_\mathrm{sub}(0)=-\left. \frac12\left(\dfrac{\partial l^1}{\partial x^1}+\dfrac{\partial l^2}{\partial x^2}+\dfrac{\partial l^3}{\partial x^3} \right)\right\|\_{x=0}\,\mathrm{Id}.$$ Substituting, and – into, and then and into, we conclude that $$\label{proof principal symbol dirac 9} e^{i\,\int\_0^t q^+(\tau)} \dfrac{d}{dt}\left( e^{-i\,\int\_0^t q^+(\tau)\,d \tau}\,v^+(x^+,\xi^+)\right)=\left.\dfrac{i}{2}\begin{pmatrix} \dfrac{\partial l^3}{\partial x^3} \\ 0 \end{pmatrix}\right\|\_{x=0} +\left.\dfrac{i}{2}\begin{pmatrix} 0\\ \dfrac{\partial l^1}{\partial x^3}+i\dfrac{\partial l^2}{\partial x^3} \end{pmatrix}\right\|\_{x=0}.$$ Similarly, in our special coordinate system we have $$\label{proof principal symbol dirac 10} \begin{split} \left.[\dot{x}^+]^\alpha\,\frac14 \sigma\_\beta \left(\frac{\partial\sigma^\beta}{\partial x^\alpha}+ \Gamma^\beta{}\_{\alpha\gamma}\sigma^\gamma \right) v^+\right\|\_{x=0,\ \xi=\xi^+} & = \left.\frac14 \sigma\_\beta \left(\frac{\partial\sigma^\beta}{\partial x^3}\right) \begin{pmatrix} 1\\0 \end{pmatrix} \right\|\_{x=0} \\ & = \left.-\frac{i}{2} \begin{pmatrix} \dfrac{\partial l^3}{\partial x^3} \\ \dfrac{\partial l^1}{\partial x^3}+i\dfrac{\partial l^2}{\partial x^3} \end{pmatrix} \right\|\_{x=0}. \end{split}$$ Summing up and we arrive at. Explicit small time expansion of the symbol =========================================== Even though the presence of gauge degrees of freedom represents an additional challenge in the analysis of the propagator, one can put this freedom to use and exploit it to obtain a small time expansion for the propagator. Our strategy goes as follows. 1. Compute the principal and subprincipal symbols of the positive (resp. negative) propagator for a conveniently chosen framing; 2. Using the gauge transformation,, switch to an arbitrary framing with the same orientation[7](#fn7); 3. Express the final result in terms of geometric invariants. Special framing --------------- Let us fix an arbitrary point y ∈ M and let Vj ∈ TyM, j = 1, 2, 3 be defined by Vj : = ej(y). [Levi-Civita framing] [definition Levi-Civita framing] Let U be a geodesic neighbourhood of y. For x ∈ U, let $\widetilde e\_j^{\mathrm{loc}}(x)$, j = 1, 2, 3, be the parallel transport of Vj along the shortest geodesic connecting y to x. We define the Levi-Civita framing generated by {e*j}j = 13 at y* to be the equivalence class of framings coinciding with $\{\widetilde e\_j^{\mathrm{loc}}\}\_{j=1}^3$ in a neighbourhood of y. With slight abuse of notation, in the following we will identify the Levi-Civita framing with one of its representatives, denoted by $\{\widetilde e\_j\}\_{j=1}^3$. The choice of a particular representative does not affect our results. Using the Levi-Civita framing is especially convenient due to the following property. [lemma Levi-Civita framing] In normal coordinates centred at y, the Levi-Civita framing admits the following expansion: $$\label{expansion Levi-Civita framing Dirac} \widetilde e\_j{}^\alpha(x)=e\_j{}^\alpha(y)+\frac16 e\_j{}^\beta(y)\,R^\alpha{}\_{\mu\beta\nu}(y)\,x^\mu x^\nu +O(\\|x\\|^3), \qquad j=1,2,3,$$ where R is the Riemann curvature tensor[8](#fn8). In normal geodesic coordinates centred at y, the unique geodesic connecting y to x can be written as $$\label{8 March 2019 formula 2} \gamma^\alpha(t)=\frac{x^\alpha}{\\|x\\|\_E}\,t,$$ where ∥ ⋅ ∥E is the Euclidean norm, so that γ(∥x∥E) = x. Assuming t and ∥x∥E to be small and of the same order, let us perform an expansion in powers of t of $\widetilde e\_j$. The parallel transport equation defining the framing $\{\widetilde e\_j\}\_{j=1}^3$ reads $$\label{8 March 2019 formula 3} \dot{\widetilde{e}}\_j{}^\alpha(\gamma(t))=-\dot{\gamma}^\beta(t)\, \Gamma^\alpha{}\_{\beta\mu}(\gamma(t))\,\, \widetilde{e}\_j{}^\mu(\gamma(t)), \qquad j=1,2,3.$$ Since $\widetilde{e}\_j(0)=V\_j$ and Γ(0) = 0, at linear order in t we have $\dot{\widetilde e}\_j(\gamma(t))=O(t),$ which implies $$\label{8 March 2019 formula 4} \widetilde e\_j(\gamma(t))=V\_j+O(t^2).$$ Substituting into, we get $$\label{8 March 2019 formula 5} \dot{\widetilde e}\_j{}^\alpha(\gamma(t))=-\frac{x^\beta\,x^\nu}{\\|x\\|\_E^2} \partial\_\nu\Gamma^\alpha{}\_{\beta\mu}(0)\,V\_j{}^\mu\,t+O(t^2),$$ so that $$\label{8 March 2019 formula 6} \widetilde e\_j{}^\alpha(\gamma(t))=V\_j-\frac{1}{2}\frac{x^\beta x^\nu}{\\|x\\|\_E^2} \partial\_\nu\Gamma^\alpha{}\_{\beta\mu}(0) \, V\_j{}^\mu\,t^2+O(t^3)$$ and $$\label{8 March 2019 formula 7} \widetilde e\_j{}^\alpha(x)=\widetilde e\_j{}^\alpha(\gamma(\\|x\\|\_E))=V\_j{}^\alpha-\frac12\partial\_\nu\Gamma^\alpha{}\_{\beta\mu}(0)\,V\_j{}^\mu\, x^\beta x^\nu +O(\\|x\\|^3), \qquad j=1,2,3.$$ Formula follows at once from and the elementary identity $$\label{relation beteween christoffel and riemann dirac} \partial\_\nu\Gamma^\alpha{}\_{\beta\mu}(0)=-\frac13 (R^\alpha{}\_{\beta\mu\nu}+R^\alpha{}\_{\mu\beta\nu})(0).$$ [corollary projection Pauli matrices] In normal coordinates x centred at y, the Pauli matrices $\widetilde\sigma^\alpha(x)$ projected along the Levi-Civita framing (see ) satisfy $$\label{Pauli matrices projected along Levi Civita framing dirac} \widetilde\sigma^\alpha(y)=\sigma^\alpha(y), \qquad [\widetilde\sigma^\alpha]\_{x^\beta}(y)=0, \qquad [\widetilde\sigma^\alpha]\_{x^\mu x^\nu}(y)=\frac16 \left[ R^\alpha{}\_{\nu\beta\mu}(y)+R^\alpha{}\_{\mu\beta\nu}(y) \right] \sigma^\beta(y).$$ [Proof of Corollary [corollary projection Pauli matrices]] Formula follows immediately from. [corollary taylor expansion W0 dirac] Let $\widetilde{W}$ be the Dirac operator corresponding to the choice of the Levi-Civita framing. Then, in normal coordinates centred at y, its zero order part $\widetilde{W}\_0$ (see formula ) admits the following expansion: $$\label{derivative of W\_0 at zero dirac} \widetilde{W}\_0(x)=\frac{i}{4} {\operatorname{Ric}}\_{\alpha\beta}(y)\,\widetilde{\sigma}^\beta(y)\, x^\alpha +O(\\|x\\|^2).$$ Formula is obtained by expanding the RHS of in powers of x in normal coordinates centred at y, substituting and in and performing a lengthy but straightforward calculation. It is a somewhat nontrivial fact that the coefficient of the linear term in turns out to be trace-free. Small time expansion of the principal symbols --------------------------------------------- The first step towards computing small time expansions for principal and subprincipal symbols of W is to obtain an expression for these objects in a neighbourhood of a given point y ∈ M for the choice of the Levi-Civita framing generated by our framing {ej}j = 13 at y. Observe that, as we are after a small time expansion of the symbols, it is enough to restrict our attention to a small open neighbourhood of y. In the following, we will denote with a tilde objects associated with the Dirac operator $\widetilde{W}$ corresponding to the choice of the Levi-Civita framing. [theorem small time expansion principal symbols dirac] For the choice of the Levi-Civita framing, the positive and negative principal symbols are independent of t and read $$\label{small time expansion principal LC dirac} \widetilde {\mathfrak{a}}^{\pm}\_0(t;y,\eta)=\widetilde{P}^{\pm}(y,\eta).$$ In accordance with Theorem [theorem small time expansion principal symbol], the principal symbols are determined by the eigenvectors of $\widetilde{W}\_\mathrm{prin}$ and their parallel transport with respect with the spin connection along the Hamiltonian trajectories. Hence, it suffices to show that $$\label{proof small time expansion principal LC dirac 1} \widetilde{\zeta}^\pm(t;y,\eta)=\widetilde{v}^\pm(y,\eta).$$ Once this is achieved, follows from the fact that $\widetilde{W}\_\mathrm{prin}(y,\eta)=W\_\mathrm{prin}(y,\eta)$. In normal coordinates centred at y the parallel transport equation reads $$\label{new proof principal symbols 1} \left[ \frac{d}{dt} +[\dot{x}^\pm]^\alpha\,\frac14 \widetilde{\sigma}\_\beta(x^\pm)\left( \frac{\partial\widetilde{\sigma}^\beta}{\partial x^\alpha}(x^\pm)+\Gamma^\beta{}\_{\alpha\gamma}(x^\pm)\widetilde{\sigma}^\gamma(x^\pm)\right) \right] \widetilde{\zeta}^\pm=0, \qquad \widetilde{\zeta}^\pm\|\_{t=0}=\widetilde{v}^\pm.$$ We claim that $$\label{new proof principal symbols 2} [\dot{x}^\pm]^\alpha\left( \frac{\partial\widetilde{\sigma}^\beta}{\partial x^\alpha}(x^\pm)+\Gamma^\beta{}\_{\alpha\gamma}(x^\pm)\widetilde{\sigma}^\gamma(x^\pm)\right)=0.$$ In fact, we have $$[\dot{x}^\pm]^\alpha\left( \frac{\partial\widetilde{\sigma}^\beta}{\partial x^\alpha}(x^\pm)+\Gamma^\beta{}\_{\alpha\gamma}(x^\pm)\widetilde{\sigma}^\gamma(x^\pm)\right)= [\dot{x}^\pm]^\alpha (\partial\_{x^\alpha} \widetilde{e}\_j{}^\beta+\Gamma^\beta{}\_{\alpha\gamma}\, \widetilde{e}\_j{}^\gamma)(x^\pm) \,s^j$$ and $$[\dot{x}^\pm]^\alpha (\partial\_{x^\alpha} \widetilde{e}\_j{}^\beta+\Gamma^\beta{}\_{\alpha\gamma}\, \widetilde{e}\_j{}^\gamma)(x^\pm)=0 \quad \text{for}\quad j=1,2,3$$ in view of Definition [definition Levi-Civita framing] and the properties of the Hamiltonian flows x±, i.e. that x+( ⋅ ; y, η) is geodesic and relation. By substituting into we arrive at. Small time expansion of the subprincipal symbols ------------------------------------------------ Let us now turn our attention to the subprincipal symbols $\widetilde{\mathfrak{a}}\_{-1}^\pm$. Unlike the principal symbols, the subprincipal symbols depend on the choice of phase functions. As here we are only interested in small time expansions and the injectivity radius Inj(M, g) is strictly positive, we can work, without loss of generality, in a neighbourhood of y with no conjugate points to y. The absence of conjugate points allows us to construct positive and negative propagators for small times by means of the algorithm described in subsection [The algorithm] for the choice of real-valued Levi-Civita phase functions φ±(t, x; y, η) = ∫γ±ζ± dz, cf. Definition [definition LC phase functions with epsilon dirac] for ε = 0. In the remainder of this subsection we adopt the same coordinates for x and y and we choose normal geodesic coordinates centred at y. We remind the reader that, in such coordinates, $$\label{x plus minus normal coordinates dirac} [x^\pm]^\alpha(t;0,\eta)=\pm\frac{\eta^\alpha}{h}\,t.$$ According to and, we have $$\label{real valued LC phase functions dirac expansion} \varphi^\pm(t,x;0,\eta)=x^\alpha\eta\_\alpha \mp h\,t\pm \frac{\textcolor{black}{t}}{3h} R^\alpha{}\_\mu{}^\beta{}\_\nu(0) \eta\_\alpha\eta\_\beta\,x^\mu x^\nu +O(\\|x\\|^4+t^4)$$ and $$\label{weight real valued LC phase functions dirac expansion} w^\pm(t,x;0,\eta)=1+\frac1{12}\operatorname{Ric}\_{\mu\nu}(0)\,x^\mu\,x^\nu\mp\frac{t}{3h}\operatorname{Ric}^\mu{}\_\nu(0)\,\eta\_\mu\,x^\nu +O(\\|x\\|^3+\|t\|^3).$$ Recall that the weight w is defined by. As explained in subsection [Transport equations dirac], the subprincipal symbols are determined by the first and the second transport equations, and. More precisely, if we are interested in expansions with remainder O(t2), we need to determine up to zeroth order in t and up to first order in t. To this end, we begin by observing that formulae and, see also, imply that the differential evaluation operators S− 2± and S− 1± admit the following expansions in normal coordinates centred at y. [lemma expansion fractur S minus 1 and S minus 2] We have 1. $$\label{expansion S\_-2 order zero dirac} \mathfrak{S}^\pm\_{-2}=\left.\frac12 \left[i \dfrac{\partial^2}{\partial x^\alpha\partial \eta\_\alpha} \right]^2 \left( \,\cdot\,\right)\right\|\_{t=0, \,x=0}+O(t),$$ 2. $$\label{expansion S\_-1 order one dirac} \mathfrak{S}^\pm\_{-1}=i\,\mathfrak{S}\_0^\pm\left(\dfrac{\partial^2}{\partial x^\alpha\partial \eta\_\alpha}\pm\frac{t}{2}\, h\_{\eta\_\alpha\eta\_\beta} \dfrac{\partial^2}{\partial x^\alpha\partial x^\beta} \right) +O(t^2).$$ (a) It is an immediate consequence of, and $$\label{proof taylor expansion fraktur sigmas dirac 1} L^\pm\_\alpha=\dfrac{\partial}{\partial x^\alpha}+O(\\|x\\|+\|t\|).$$ (b) Substituting into with k = 1 and recalling that φη±∣x = x± = 0, we get $$\label{2 March 2019 forumla 3} \mathfrak{S}^{\pm}\_{-1}=\mathfrak{S}\_0^{\pm}\left[ i \dfrac{\partial^2}{\partial x^\alpha\partial \eta\_\alpha}-\frac{i}{2} \varphi^{\pm}\_{\eta\_\alpha\eta\_\beta}L^{\pm}\_\alpha L^{\pm}\_\beta \right]+O(t^2).$$ Formula and the fact that φηαηβ±∣x = x± = ∓ t hηαηβ + O(t3) yield. In order to be able to compute the subprincipal symbols, we need to determine the initial condition $\widetilde{\mathfrak{a}}^\pm\_{-1}\|\_{t=0}$ first. [lemma initial condition subprincipal symbol dirac] For the choice of real-valued Levi-Civita phase functions, the positive and negative subprincipal symbols $\widetilde{\mathfrak{a}}^\pm\_{-1}$ vanish at t = 0: $$\label{initial condition subprincipal symbol dirac} \widetilde{\mathfrak{a}}^\pm\_{-1}(0;y,\eta)=0.$$ The subprincipal symbols are scalar functions, so it enough to establish in one specific coordinate system. Let us choose normal coordinates centred at y = 0 such that $\widetilde{e}\_j{}^\alpha(0)=\delta\_j{}^\alpha$. We observe that the torsion of the Weitzenböck connection generated by the Levi-Civita framing at y vanishes at y, as a consequence of the fact that the first derivatives of the framing are zero, cf. and –. Therefore, Theorem [theorem about subprincipal symbols of propagators at zero] tells us that [U±(0)]sub(0, η) = 0. A straightforward perturbation argument shows that (v±)xα(0, η) = 0. Substituting and into with P = U±(0) and ε = 0 and using the fact that Christoffel symbols vanish at y, we arrive at. We are now in a position to examine the first transport equation. [lemma projection along other eigenspace Dirac] The projection onto the negative (resp. positive) eigenspace of $\widetilde{W}\_\mathrm{prim}$ of the subprincipal symbol of the positive (resp. negative) propagator is given by $$\begin{gathered} \label{projection on other eigenspace subps dirac} \widetilde{P}^\mp(x^\pm,\xi^\pm)\widetilde{\mathfrak{a}}\_{-1}^\pm(t;y,\eta)=\pm it\,\widetilde{P}^\mp(y,\eta)\left[\frac1{8h^3}\,{\operatorname{Ric}}\_{\alpha\beta}(y)\,\eta^\alpha\eta^\beta-\frac1{4h}{\operatorname{Ric}}\_{\alpha\beta}(y)\,\eta^\alpha \widetilde{P}^\pm\_{\eta\_\beta}(y,\eta) \right] \\+O(t^2).\end{gathered}$$ We will establish formula by expanding the first transport equation up to first order in t and then acting with $\widetilde{P}^\mp$ on the left. Recall that a− k± is defined by. Working in normal coordinates centred at y and using –, we obtain $$\label{17 June 2019 formula 2} \begin{aligned} \mathfrak{S}^\pm\_0 \widetilde{a}^\pm\_{0}(t;0,\eta)&= \Bigl\{ \left(\varphi^\pm\_t+\widetilde{W}\_\mathrm{prin}(x,\varphi\_x^\pm)\right)\widetilde{\mathfrak{a}}^\pm\_{-1} -i (\widetilde{\mathfrak{a}}^\pm\_{-0})\_t + \left[-i (w^\pm)^{-1}\left( w^\pm\_t+\sigma^\alpha w^\pm\_{x^\alpha} \right)+\widetilde{W}\_0 \right]\widetilde{\mathfrak{a}}^\pm\_{-0} \Bigr\} \Bigr\|\_{x=x^\pm} \\ &=(\widetilde{W}\_\mathrm{prin}(x^\pm,\xi^\pm)\mp h)\,\widetilde{\mathfrak{a}}^\pm\_{-1}(t;0,\eta)+\frac{i\,t}{3h^2}{\operatorname{Ric}}\_{\alpha\nu}(0) \left(\eta^\alpha\eta^\nu \pm\frac12 h\,\eta^\alpha \,\widetilde{\sigma}^\nu(0)\right)\widetilde{P}^\pm(y,\eta)\\ &\pm \frac{t\,\eta^\alpha}{h}(\widetilde{W}\_0)\_{x^\alpha}(0)\,\widetilde{P}^\pm(y,\eta)+O(t^2). \end{aligned}$$ Furthermore, in view of Theorem [theorem small time expansion principal symbol] and Lemma [lemma expansion fractur S minus 1 and S minus 2](b), we have $$\label{17 June 2019 formula 3} \begin{aligned} \mathfrak{S}^\pm\_{-1}\widetilde{a}^\pm\_{1}(t;0,\eta) & = \left[\dfrac{\partial^2}{\partial x^\alpha\partial \eta\_\alpha}\pm\frac{t}{2}\, h\_{\eta\_\alpha\eta\_\beta} \dfrac{\partial^2}{\partial x^\alpha\partial x^\beta} \right] \left. \left(\varphi^\pm\_t+\widetilde{W}\_{\mathrm{prin}}(x,\varphi^\pm\_x)\right)\widetilde{\mathfrak{a}}\_0^\pm\right\|\_{x=x^\*}+O(t^2)\\ &= -\frac{2{\mathrm{i}}t}{3h^2}{\operatorname{Ric}}\_{\alpha\nu}(0) \,\eta^\alpha\eta^\nu \widetilde{P}^\pm \pm{\mathrm{i}}t \left[\frac{2}{3h}R^{\mu}{}\_\alpha{}^\nu{}\_\beta(0)\,\eta\_\mu\eta\_\nu\,\widetilde{\sigma}^\beta(0)\widetilde{P}^\pm\right]\_{\eta\_\alpha}\\ &\pm it \frac{\eta^\beta}{h}\left[(\widetilde{W}\_\mathrm{prin})\_{x^\alpha x^\beta}(0,\eta)\widetilde{P}^\pm\right]\_{\eta\_\alpha} \\ &+{\mathrm{i}}\,t \left[ \frac{h\_{\eta\_\alpha\eta\_\beta}}{3h}\,R^{\mu}{}\_\alpha{}^\nu{}\_\beta(0)\,\eta\_\mu\eta\_\nu\pm\frac{1}{2}h\_{\eta\_\alpha\eta\_\beta}(\widetilde{W}\_\mathrm{prin})\_{x^\alpha x^\beta}(0,\eta) \right]\widetilde{P}^\pm+O(t^2). \end{aligned}$$ Adding up and and projecting along $\widetilde{P}^{\mp}$, we arrive at $$\label{17 June 2019 formula 4} \begin{split} \widetilde{P}^{\mp}\	arxiv_0000402
Chandra High Energy Transmission Grating Spectrum of AE Aquarii =============================================================== The nova-like cataclysmic binary AE Aqr, which is currently understood to be a former supersoft X-ray binary and current magnetic propeller, was observed for over two binary orbits (78 ks) in 2005 August with the High-Energy Transmission Grating (HETG) onboard the Chandra X-ray Observatory. The long, uninterrupted Chandra observation provides a wealth of details concerning the X-ray emission of AE Aqr, many of which are new and unique to the HETG. First, the X-ray spectrum is that of an optically thin multi-temperature thermal plasma; the X-ray emission lines are broad, with widths that increase with the line energy, from σ ≈ 1 eV ($510~\rm km~s^{-1}$) for to σ ≈ 5.5 eV ($820~\rm km~s^{-1}$) for ; the X-ray spectrum is reasonably well fit by a plasma model with a Gaussian emission measure distribution that peaks at $\log T ({\rm K})=7.16$, has a width σ = 0.48, an Fe abundance equal to 0.44 times solar, and other metal (primarily Ne, Mg, and Si) abundances equal to 0.76 times solar; and for a distance d = 100 pc, the total emission measure $EM=8.0\times 10^{53}~\rm cm^{-3}$ and the 0.5–10 keV luminosity $L\_{\rm X}= 1.1\times 10^{31}~\rm erg~s^{-1}$. Second, based on the f/(i + r) flux ratios of the forbidden (f), intercombination (i), and recombination (r) lines of the He α triplets of,, and measured by Itoh et al. in the XMM-Newton Reflection Grating Spectrometer spectrum and those of,,, and in the Chandra HETG spectrum, either the electron density of the plasma increases with temperature by over three orders of magnitude, from $n\_{\rm e}\approx 6\times 10^{10}~\rm cm^{-3}$ for [$\log T({\rm K})\approx 6$] to $n\_{\rm e}\approx 1\times 10^{14}~\rm cm^{-3}$ for [$\log T ({\rm K})\approx 7$], and/or the plasma is significantly affected by photoexcitation. Third, the radial velocity of the X-ray emission lines varies on the white dwarf spin phase, with two oscillations per spin cycle and an amplitude $K\approx 160~\rm km~s^{-1}$. These results appear to be inconsistent with the recent models of Itoh et al., Ikhsanov, and Venter & Meintjes of an extended, low-density source of X-rays in AE Aqr, but instead support earlier models in which the dominant source of X-rays is of high density and/or in close proximity to the white dwarf. Introduction ============ AE Aqr is a bright (V ≈ 11) nova-like cataclysmic binary consisting of a magnetic white dwarf primary and a K4–5 V secondary with a long 9.88 hr orbital period and the shortest known white dwarf spin period P = 33.08 s. Although originally classified and interpreted as a disk-accreting DQ Her star, AE Aqr displays a number of unusual characteristics that are not naturally explained by this model. First, violent flaring activity is observed in the radio, optical, ultraviolet (UV), X-ray, and TeV γ-rays. Second, the Balmer emission lines are single-peaked and produce Doppler tomograms that are not consistent with those of an accretion disk. Third, the white dwarf is spinning down at a rate ${\dot{P}}= 5.64 \times 10^{-14}~{\rm s~s^{-1}}$. Although this corresponds to the small rate of change of $1.78~\rm ns~yr^{-1}$, AE Aqr’s spin-down is typically characterized as “rapid” because the characteristic time $P/\Pdot\approx 2\times 10^7$ yr is short compared to the lifetime of the binary and because the spin-down luminosity $L\_{\rm sd}=-I\Om{\dot{\Omega}}\approx 1\times 10^{34}~\rm erg~s^{-1}$ (where $I\approx 0.2{M\_{\rm wd}}{R\_{\rm wd}}^2 \approx 2\times 10^{50}~\rm g~cm^2$ is the moment of inertia for a white dwarf of mass ${M\_{\rm wd}}=0.8~{\rm M\_{\odot}}$ and radius ${R\_{\rm wd}}= 7.0\times 10^8$ cm, $\Om = 2\pi/P$, and ${\dot{\Omega}}= -2\pi\Pdot/P^2$) exceeds the secondary’s thermonuclear luminosity by an order of magnitude and the accretion luminosity by two orders of magnitude. Because of its unique properties and variable emission across the electromagnetic spectrum, AE Aqr has been the subject of numerous studies, including an intensive multiwavelength observing campaign in 1993 October [, and the series of papers in ASP Conf. Ser. 85 ]. Based on these studies, AE Aqr is now widely believed to be a former supersoft X-ray binary and current magnetic propeller, with most of the mass lost by the secondary being flung out of the binary by the magnetic field of the rapidly rotating white dwarf. These models explain many of AE Aqr’s unique characteristics, including the fast spin rate and rapid secular spin-down rate of the white dwarf, the anomalous spectral type of the secondary, the anomalous abundances, the absence of signatures of an accretion disk, the violent flaring activity, and the origin of the radio and TeV γ-ray emission. To build on this observational and theoretical work, while taking advantage of a number of improvements in observing capabilities, during 2005 August 28–September 2, a group of professional and amateur astronomers conducted a campaign of multiwavelength (radio, optical, UV, X-ray, and TeV γ-ray) observations of AE Aqr. Attention is restricted here to the results of the X-ray observations, obtained with the High-Energy Transmission Grating (HETG) and the Advanced CCD Imaging Spectrometer (ACIS) detector onboard the Chandra X-ray Observatory. has previously provided an analysis and discussion of the timing properties of these data, showing that: (1) as in the optical and UV, the X-ray spin pulse follows the motion of the white dwarf around the binary center of mass and (2) during the decade 1995–2005, the white dwarf spun down at a rate that is slightly faster than predicted by the spin ephemeris. Here, we present a more complete analysis and discussion of the Chandra data, providing results that in many ways reproduce the results of the previous Einstein, ROSAT, ASCA, and XMM-Newton and the subsequent Suzaku observations of AE Aqr, but also that are new and unique to the HETG; namely, the detailed nature of the time-average X-ray spectrum, the plasma densities implied by the He α triplet flux ratios, and the widths and radial velocities of the X-ray emission lines. As we will see, these results appear to be inconsistent with the recent models of,, and of an extended, low-density source of X-rays in AE Aqr, but instead support earlier models in which the dominant source of X-rays is of high density and/or in close proximity to the white dwarf. The plan of this paper is as follows. In §2 we discuss the observations and the analysis of the X-ray light curve (§2.1), spin-phase light curve (§2.2), spectrum (§2.3), and radial velocities (§2.4). In §3 we provide a summary of our results. In §4 we discuss the results and explore white dwarf (§4.1), accretion column (§4.2), and magnetosphere (§4.3) models of the source of the X-ray emission in AE Aqr. In §5 we draw conclusions, discuss the bombardment model of the accretion flow of AE Aqr, and close with a few comments regarding future observations. The casual reader may wish to skip §2.1–2.2, which are included for completeness, and concentrate on §2.3–2.4, which contain the important observational and analysis aspects of this work. Observations and Analysis ========================= AE Aqr was observed by Chandra beginning on 2005 August 30 at 06:37 UT for 78 ks (). The level 2 data files used for this analysis were produced by the standard pipeline processing software ASCDS version 7.6.7.1 and CALDB version 3.2.1 and were processed with the Chandra Interactive Analysis of Observations (CIAO[1](#fn1)) version 3.4 software tools to convert the event times in the evt2 file from Terrestrial Time (TT) to Barycentric Dynamical Time (TDB) and to make the grating response matrix files (RMFs) and the auxiliary response files (ARFs) needed for quantitative spectroscopic analysis. The subsequent analysis was performed in the following manner using custom Interactive Data Language (IDL) software. First, using the region masks in the pha2 file, the source and background events for the ± first-order Medium Energy Grating (MEG) and High Energy Grating (HEG) spectra were collected from the evt2 file. Second, after careful investigation, ± 0.0030 Å ( ± 0.0015 Å) was added to the ± first-order MEG (HEG) wavelengths, respectively, to account for an apparent shift (by 0.27 ACIS pixels) in the position of the zero-order image. Third, to account for the spin pulse delay measured in the optical, UV, and X-ray wavebands produced by the motion of the white dwarf around the binary center of mass, $-2\, \cos 2\pi \phi\_{\rm orb}$ s was added to the event times t, where the white dwarf orbit phase $2\pi\phi\_{\rm orb}=\Omega\_{\rm orb}(t-T\_0)$, where $\Omega\_{\rm orb}=2\pi/P\_{\rm orb}$ and $P\_{\rm orb}=0.411655610$ d and $T\_0({\rm BJD})= 2445172.2784$ are the orbit ephemeris constants from Table 4 of. Fourth, to account for the Doppler shifts produced by Chandra’s (mostly, Earth’s) motion relative to the solar system barycenter, the event wavelengths λ were multiplied by a factor $[1+v\_{\rm los}/c]$, where $v\_{\rm los}$ is the (time-dependent) line-of-sight velocity between the spacecraft and the source, determined using an IDL code kindly supplied by R. Hoogerwerf (and checked against the line-of-sight velocities derived from the barycentric time corrections supplied by the CIAO tool axbary); during the observation, $v\_{\rm los}$ varied from $-11.9~\rm km~s^{-1}$ at the beginning of the observation, rose to $-11.3~\rm km~s^{-1}$, and then fell to $-11.5~\rm km~s^{-1}$ at the end of the observation. Fifth, a filter was applied to restrict attention to events from two orbital cycles $\phi\_{\rm orb}=20503.9$–20505.9, resulting in an effective exposure of 71 ks. Sixth, white dwarf spin pulse phases were calculated using the updated cubic spin ephemeris of derived from the recent ASCA (1995 October) and XMM-Newton (2001 November) and the current Chandra observation of AE Aqr: $2\pi\phi\_{\rm spin} = \Om\_0(t-T\_{\rm max}) + {1\over2}\Odot(t-T\_{\rm max})^2 + {1\over6}{\ddot{\Omega}}(t-T\_{\rm max})^3$, where $\Om\_0=2\pi/P\_{33}$ and ${\dot{\Omega}}= -2\pi\Pdot\_{33}/P\_{33}^2$ and P33 = 0.00038283263840 d, $\Pdot\_{33}=5.642\times 10^{-14}~\rm d~d^{-1}$, and $T\_{\rm max}({\rm BJD}) = 2445172.000042$ are the spin ephemeris constants from Table 4 of, and ${\ddot{\Omega}}=-1.48\times 10^{-11}~\rm d^{-3}$. The HETG event data are then fully characterized by the event time t, white dwarf orbit phase $\phi\_{\rm orb}$, white dwarf spin phase $\phi\_{\rm spin}$, and wavelength λ. Light curve ----------- Figure 1 shows the background-subtracted count rate light curve derived from the HETG event data. As has been well established by previous X-ray observations, the X-ray light curve of AE Aqr is dominated by flares, although this is by far the longest uninterrupted observation of AE Aqr and hence the clearest view of its X-ray light curve. During the Chandra observation, the flares last between a few hundred and a few thousand seconds, producing increases of up to 3–5 times the baseline count rate of $\sim 0.1~\rm counts~s^{-1}$. To constrain the cause and nature of the flares of AE Aqr, it is of interest to determine if the count rate variations shown in Figure 1 are accompanied by, or perhaps are even due to, dramatic variations in the X-ray spectrum. Previous investigations have indicated that this is not the case. For a flare observed by ASCA, found no significant difference between the quiescent and flare X-ray spectra, although a “hint” of spectral hardening was recognized during the flare. For a flare observed by XMM-Newton, found that the X-ray spectrum at the beginning of the flare was similar to that in quiescence, but that the spectrum became harder as the flare advanced. Our ability to investigate spectral variations during the individual flares observed by Chandra is limited by the relatively low HETG count rate and the relatively fast timescale of the flares, although it is possible to investigate spectral variations for the ensemble of the flares. To accomplish this, the light curve shown in Figure 1 was divided into five count rate ranges: $ I\_1 < 0.14~\rm counts~s^{-1}$, $0.14\le I\_2 < 0.18~\rm counts~s^{-1}$, $0.18\le I\_3 < 0.23~\rm counts~s^{-1}$, $0.23\le I\_4 < 0.29~\rm counts~s^{-1}$, and $ I\_5\ge 0.29~\rm counts~s^{-1}$, where the count rate cuts were set to produce a roughly equal number of counts per count rate range, and the source and background counts were collected in three wavebands: hard (1–8 Å), medium (8–11 Å), and soft (11–26 Å), where the wavelength cuts were set to produce a roughly equal number of counts per wavelength interval. The background-subtracted soft S over hard H versus the total (1–26 Å) count rates are plotted in Figure 2. The dotted line in that figure is the best-fitting constant function S/H = 0.953 for the first three data points (count rate ranges I1–I3). As shown by the figure, the softness ratio of the next higher count rate range I4 is consistent with the lower ranges, while that of the highest count rate range I5 is significantly (3.6σ) less. Consistent with the result of, we find that the X-ray spectrum of the flares of AE Aqr is harder than that in quiescence only at the peak of the flares. Spin-phase light curve ---------------------- Figure 3a shows the background-subtracted spin-phase count rate light curve derived from the HETG event data. It is well fit (χ2 per degree of freedom ≡ χν2 = 7.26/7 = 1.04) by the cosine function $A + B\,\cos 2\pi (\phi\_{\rm spin}- \phi\_0)$, with mean count rate $A=0.191\pm 0.002~\rm counts~s^{-1}$, semi-amplitude $B=0.031 \pm 0.002~\rm counts~s^{-1}$ (hence, relative X-ray pulse semi-amplitude B/A = 16% ± 1%) and, consistent with the updated spin ephemeris, phase offset ϕ0 = 0.00 ± 0.01 (throughout the paper, errors are 1σ or 68% confidence for one free parameter). The above result applies to the observation-average spin-phase light curve, although it is of interest to determine if the X-ray pulse semi-amplitude B and/or the relative semi-amplitude B/A varies with the mean count rate A. As for the investigation of the softness ratio variations, this is best determined as a function of time, using a time resolution sufficient to resolve the flares, but this is not possible with the relatively low count rate HETG data. Instead, background-subtracted spin-phase count rate light curves were derived from the HETG event data for each of the five count rate ranges defined above, and were then fit with the cosine function $A + B\,\cos 2\pi\phi\_{\rm spin}$. The resulting values of the X-ray pulse semi-amplitude B are plotted versus the mean count rate A in Figure 4. The figure demonstrates that B increases linearly with A (B/A is constant with A) in the middle of the count rate range, but that it saturates at both the low and high ends of the range; over the full count rate range, B/A is not well fit (χν2 = 5.82/4 = 1.46) by a constant B/A = 0.162 ± 0.012 (dotted line in Fig. 4). Instead, B/A versus A and B versus A are well fit by a linear relation a + bA with, respectively, a = 0.22 ± 0.03 and b = − 0.25 ± 0.15, with χν2 = 2.77/3 = 0.92 (dashed curve in Fig. 4), and a = 0.011 ± 0.006 and b = 0.10 ± 0.03, with χν2 = 2.38/3 = 0.79 (solid line in Fig. 4). To determine if the observed spin-phase flux modulation is due to photoelectric absorption (or some other type of broad-band spectral variability), background-subtracted spin-phase count rate light curves were derived from the HETG event data for each of three wavebands defined above: hard (1–8 Å), medium (8–11 Å), and soft (11–26 Å). Figure 3b shows the ratio of the resulting soft S over hard H spin-phase count rate light curves. The data is well fit (χν2 = 7.20/9 = 0.80) by a constant S/H = 0.916 ± 0.019, which strongly constrains the cause of the observed spin-phase flux modulation. Specifically, if the observed flux modulation is caused by photoelectric absorption, a variation in the neutral hydrogen column density $\Delta N\_{\rm H}\approx 3\times 10^{21}~\rm cm^{-2}$ is required, whereas the essential constancy of the softness ratio light curves requires $\Delta N\_{\rm H}\lesssim 1\times 10^{20}~\rm cm^{-2}$; a factor of 30 times lower. lccccccccc & velocity& σ& & & & $\log n\_{\rm e}$ Element& ($\rm km~s^{-1}$)& (eV)& Lyman α& f& i& r& G = (f + i)/r& Rʹ = f/(i + r)& ($\rm cm^{-3}$) to 0.4inOto 0.4em.& $-75\pm\phn 51$& 1.12 ± 0.09& $2.91\phn\pm 0.27$& $0.35\phn\pm 0.17\phn$& $0.84\phn\pm 0.24\phn$& $1.12\phn\pm 0.26\phn$& 1.05 ± 0.35& 0.178 ± 0.090& 11.36− 0.32+ 0.42 to 0.4inNeto 0.4em.& $+45\pm\phn 53$& 2.02 ± 0.17& 0.999 ± 0.073& 0.206 ± 0.067& 0.200 ± 0.064& 0.622 ± 0.089& 0.65 ± 0.18& 0.251 ± 0.088& 12.30− 0.36+ 0.35 to 0.4inMgto 0.4em.& + 49 ± 104& 3.85 ± 0.60& 0.188 ± 0.019& 0.070 ± 0.021& 0.040 ± 0.020& 0.193 ± 0.024& 0.57 ± 0.17& 0.300 ± 0.099& 13.04− 0.73+ 0.45 to 0.4inSito 0.4em.& + 95 ± 118& 5.50 ± 0.90& 0.185 ± 0.019& 0.055 ± 0.014& 0.028 ± 0.017& 0.205 ± 0.020& 0.41 ± 0.11& 0.235 ± 0.065& 14.14− 0.36+ 0.34 Spectrum -------- Figure 5 shows the background-subtracted count spectrum derived from the HETG event data, using Δλ = 0.05 Å wavelength bins as a compromise between spectral resolution and sign-to-noise ratio. As is typical of unabsorbed cataclysmic variables (CVs), the X-ray spectrum of AE Aqr is that of a multi-temperature thermal plasma, with emission lines of H- and He-like O, Ne, Mg, Si, and S and L-shell –. However, unlike other CVs, and, in particular, unlike other magnetic CVs, the H-like and He-like lines and the “neutral” fluorescent Fe K line are not apparent in the HETG spectrum. The apparent absence of these features in the HETG spectrum places limits on the maximum temperature of the plasma in AE Aqr and the amount of reflection from the surface of the white dwarf, although higher-energy instruments, such as those onboard Suzaku, are better suited to study this portion of the X-ray spectrum. We conducted a quantitative analysis of the X-ray spectrum of AE Aqr in three steps. First, Gaussians were fitted to the strongest emission lines of H- and He-like O, Ne, Mg, and Si to determine their radial velocities, widths, and fluxes. Second, a global model was fitted to the X-ray spectrum to constrain its absorbing column density, emission measure distribution, and elemental abundances. Third, using the emission measure distribution and the flux ratios of the He α forbidden (f), intercombination (i), and recombination (r) lines, constraints are placed on the density of the plasma. ### Line radial velocities, widths, and fluxes The radial velocities, widths, and fluxes above the continuum of the Lyman α emission lines of,,, and and the He α triplets of,,, and were determined by fitting the flux in the MEG spectrum in the immediate vicinity of each emission feature with a constant plus one (Lyman α) or three (He α) Gaussians, employing the ARF and RMF files to account for the effective area of the spectrometer and its Δλ = 0.023 Å [$690~\rm km~s^{-1}$ at 10 Å] FWHM spectral resolution. For each emission feature, the radial velocity was determined relative to the laboratory wavelengths from the Interactive Guide for ATOMDB version 1.3.[2](#fn2) More specifically, the assumed wavelengths for the Lyman α lines are the mean of the wavelengths of the doublets weighted by their relative emissivities (2:1), whereas the wavelengths for the He α intercombination lines are the unweighted means of their component x and y lines. In the fits to the He α triplets, the radial velocities and widths determined from the fits to the corresponding Lyman α lines were assumed, so that, in all cases, the fits had four free parameters. For these fits, unbinned data were employed and the C statistic was used to determine the value of and error on the fit parameters, which are listed in Table 1. As demonstrated in Figure 6, we find that the widths of the Lyman α emission lines increase with the line energy, from σ = 1.1 ± 0.1 eV for to σ = 5.5 ± 0.9 eV for. For comparison, during two flares of AE Aqr observed with the XMM-Newton Reflection Grating Spectrometer (RGS), found σ ≈ 1.2 and 2 eV for the Lyman α emission lines of and, respectively. The trend shown in Figure 6 is well fit (χν2 = 1.43/2 = 0.71) with a linear function a + bE with a = − 0.80 ± 0.29 and b = 2.9 ± 0.4. Consistent with the non-zero intercept a, the line widths are not constant in velocity units, but increase with the line energy: σ = 512 ± 39, 593 ± 51, 784 ± 121, and $822\pm 135~\rm km~s^{-1}$ for,,, and, respectively. In addition to the line widths, there is some evidence that the radial velocities of the Lyman α emission lines increase with the line energy, from $v=-75\pm 51~\rm km~s^{-1}$ for to $v=+95\pm 118~\rm km~s^{-1}$ for. Note, however, that the velocity difference, $\Delta v= 170\pm 128~\rm km~s^{-1}$, differs from zero by just 1.3σ, is a small fraction of the widths of the lines, and is probably affected by systematic effects. If a common radial velocity offset is assumed in the fits of the Lyman α lines, the derived velocity $v=-1\pm 33~\rm km~s^{-1}$. ### Global model To produce a global model of the X-ray spectrum of AE Aqr, ATOMDB IDL version 2.0.0 software[3](#fn3) was used to construct ATOMDB version 1.3.1[4](#fn4) optically-thin thermal plasma X-ray spectral models for the continuum and for the 12 cosmically abundant elements C, N, O, Ne, Mg, Al, Si, S, Ar, Ca, Fe, and Ni at 40 temperatures spaced uniformly in logT [specifically, $\log T ({\rm K})= 5.0, 5.1, 5.2, \ldots, 8.9$]. Using custom IDL software, these spectral eigenvectors were convolved with a Gaussian, to account for the observed widths of the emission lines as a function of temperature, and multiplied by the grating ARFs and RMFs, to account for the spectrometer’s effective area and spectral resolution. The resulting ± first-order MEG and HEG spectral models were then binned to 0.05 Å and coadded. Finally, the observed MEG plus HEG count spectrum (Fig. 5) and whence the spectral models were “grouped” to a minimum of 30 counts per bin so that Gaussian statistics could be used in the fits. Numerous model emission measure (EM) distributions were tested against the data: 1, 2, 3, and 4 single-temperature models, a cut-off power law (dEM/dlogT ∝ Tα for $T\le T\_{\rm c}$), a power law with an exponential cutoff (dEM/dlogT ∝ Tα for $T\le T\_{\rm c}$ and $dEM/d\log T \propto T^\alpha \exp[(T\_{\rm c}-T)/T\_{\rm f}]$ for $T > T\_{\rm c}$), and a Gaussian (dEM/dlogT ∝ exp[ − (logT − logT0)2/2σ2]), all with photoelectric absorption by a neutral column and variable elemental abundances relative to those of. Among these models, the best (if not a particularly good) fit (χν2 = 382.7/233 = 1.64) was achieved with a model with a Gaussian emission measure distribution with a peak temperature $\log T\_0 ({\rm K})=7.16\pm 0.01$, a width σ = 0.48 ± 0.01, an absorbing column density $N\_{\rm H}=(1.0\pm 0.6)\times 10^{20}~\rm cm^{-2}$, an Fe abundance equal to 0.44 ± 0.02 times solar, and the other metal abundances equal to 0.76 ± 0.03 times solar.[5](#fn5) The X-ray spectrum of this model is shown superposed on the data in Figure 7. For an assumed distance d = 100 pc, the total emission measure $EM=8.8\times 10^{53}~\rm cm^{-3}$ and the unabsorbed 0.5–10 keV luminosity $L\_{\rm X}=1.1\times 10^{31}~\rm erg~s^{-1}$; if this luminosity is due to accretion onto the white dwarf, the mass accretion rate $\dot M = L\_{\rm X}{R\_{\rm wd}}/G{M\_{\rm wd}}\approx 7.3\times 10^{13}~\rm g~s^{-1}$. Before leaving this section, it is useful to note that the Gaussian emission measure distribution derived above extends over nearly two orders of magnitude in temperature — from T ≈ 1.6 × 106 K to 1.3 × 108 K at ± 2σ — but peaks at a temperature T ≈ 1.4 × 107 K or 1.2 keV. Such a low temperature is uncharacteristic of CVs, and in particular magnetic CVs: note for instance that AE Aq had the lowest continuum temperature of any magnetic CV observed by ASCA. The characteristic temperatures of magnetic CVs are the shock temperature, $T\_{\rm s}=3\mu m\_{\rm H} G{M\_{\rm wd}}/8k{R\_{\rm wd}}\approx 4.1\times 10^8$ K or 35 keV, applicable for radial free-fall onto the white dwarf, and the blackbody temperature, $T\_{\rm bb}=(L\_{\rm X}/\sigma A)^{1/4}\approx 19\, (f/0.25)^{-1/4}$ kK (where the radiating area $A=4\pi{R\_{\rm wd}}^2 f$ and our choice for the fiducial value of the fractional emitting area f will be justified in §4.1), applicable for deposition of the accretion luminosity in the surface layers of the white dwarf (e.g., bombardment or blobby accretion). The only evidence for plasma at the shock temperature is supplied by the Suzaku X-ray spectrum of AE Aqr, but argued strongly for a nonthermal origin for this emission. The blackbody temperature, on the other hand, is close to the temperature of the hot spots that derived from the maximum entropy maps of the UV pulse profiles of AE Aqr. More on this below. ### Plasma densities We now consider in more detail the forbidden (f), intercombination (i), and recombination (r) line fluxes of the He α triplets of,,, and derived in §2.3.1 and listed in Table 1. As elucidated by ; ; and, these line fluxes can be used to constrain the electron temperature $T\_{\rm e}$ via the G = (f + i)/r flux ratio and the electron density $n\_{\rm e}$ via the R = f/i flux ratio of each He-like ion. Because the errors on the observed R flux ratios are large, we follow and employ the Rʹ = f/(i + r) flux ratio as an electron density diagnostic in the subsequent discussion. The observed G and Rʹ flux ratios and errors are listed in Table 1. To interpret these results, we derived the Rʹ = RG/(1 + R + G) flux ratios from the G and R flux ratios tabulated by for a collisional plasma for,,,, and for $\log T\_{\rm e} ({\rm K})=6.2$, 6.4, 6.6, 6.8, and 7.0, respectively, where the temperatures are those of the peaks of the He-like triplet emissivities weighted by the Gaussian emission measure distribution, determined in the previous section from the global fit to the X-ray spectrum. With the exception of,[6](#fn6) in each case the assumed electron temperature is consistent with the observed G flux ratio. With these assumptions, the theoretical values of Rʹ are shown in the left panels of Figure 8 as a function of $\log n\_{\rm e}$. In addition to the Rʹ flux ratios of,,, and measured from the Chandra HETG spectrum (Table 1), we added to Figure 8 the Rʹ flux ratios of,, and measured by Itoh et al. from the XMM-Newton RGS spectrum of AE Aqr. Figure 8(b) of this paper corrects an error in Figure 5(h) of Itoh et al., which showed the Rʹ flux ratio extending from 0.09 to 0.39, whereas the data in their Table 4 shows that it should extend only to 0.29. With this correction, the RGS- and HETG-derived values of and errors on the Rʹ flux ratio of are nearly identical. This correction, the significantly smaller error range on the Rʹ flux ratio of, and the HETG results for and indicates that, contrary to the conclusion of Itoh et al., the electron density of the plasma in AE Aqr increases with temperature by over three orders of magnitude, from $n\_{\rm e}\approx 6\times 10^{10}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K})\approx 6.2$] to $n\_{\rm e}\approx 1\times 10^{14}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K}) \approx 7.0$]. In addition to electron density, the R and hence the Rʹ flux ratio is affected by photoexcitation, so we must investigate the sensitivity of the Rʹ flux ratios to an external radiation field. To investigate the conditions under which a low-density plasma can masquerade as a high-density plasma, we derived the Rʹ flux ratios from the G and R flux ratios tabulated by for a low-density ($n\_{\rm e}=1~\rm cm^{-3}$) collisional plasma irradiated by a blackbody with temperature $T\_{\rm bb}$ and dilution factor W. Under the assumption that this flux originates from the surface of the white dwarf, $W={1\over 2}\ \{1-[1-({R\_{\rm wd}}/r)]^{1/2}\}$, where r is the distance from the center of the white dwarf, hence $W={1\over 2}$ on the white dwarf surface. In the right panels of Figure 8 we show contours of the observed Rʹ flux ratios (white curves) and 1σ error envelops (colored polygons) of the various He-like ions as a function of $T\_{\rm bb}$ and W. The figure demonstrates that the observed Rʹ flux ratios of,,,, and can be produced in a low-density plasma sitting on the white dwarf surface if the blackbody temperature $T\_{\rm bb}\approx 7$, 10, 14, 18, and 30 kK, respectively, or at higher temperatures at greater distances from the white dwarf. Conversely, the figure gives the allowed range of distances (dilution factors) for each ion for a given blackbody temperature. For example, for $T\_{\rm bb} = 25$ kK, the volume of plasma in which dominates [$\log T ({\rm K})\approx 7.0$] could be on the white dwarf surface, while that of [$\log T ({\rm K})\approx 6.8$] would have to be at $r\approx 3\, {R\_{\rm wd}}$, that of [$\log T ({\rm K})\approx 6.6$] would have to be at $r\approx 20\, {R\_{\rm wd}}$, and so on for the lower Z ions. Radial velocities ----------------- In the next component of our analysis, we used two techniques to search for orbit- and spin-phase radial velocity variations in the X-ray emission lines of AE Aqr. ### Composite line profile technique In the first technique, similar to that employed by, phase-average as well as orbit- and spin-phase resolved composite line profiles were formed by coadding the HETG event data in velocity space v = c(λ − λ0)/λ relative to laboratory wavelengths λ0 from the Interactive Guide for ATOMDB version 1.3. The lines used in this analysis are those labeled in Figures 5 and 7 shortward of 20 Å. For the H-like lines, the wavelengths are the mean of the wavelengths of the doublets weighted by their relative emissivities (2:1), while for the He-like lines, the wavelengths are for the stronger resonance lines. The resulting phase-average composite line profile is shown in Figure 9. Fit with a Gaussian, its offset $v=25\pm 26~\rm km~s^{-1}$ and width $\sigma=712\pm 27~\rm km~s^{-1}$. Applying the composite line profile technique to the orbit-phase resolved composite line profiles results in the radial velocities shown in Figure 10a. Assuming that, like EX Hya, the orbit-phase radial velocities follow the motion of the white dwarf, these data are well fit (χν2 = 1.29/6 = 0.22) by the sine function $$v(\phi\_{\rm orb}) = \gamma + K {1\over\Delta} \int^{\phi\_{\rm orb}+\Delta/2}\_{\phi\_{\rm orb}-\Delta/2} \sin 2\pi \phi\, d\phi,$$ where Δ = 1/8, with $\gamma =30\pm 25~\rm km~s^{-1}$ and $K=12\pm 36~\rm km~s^{-1}$ (solid curve in Fig. 10a), although they are slightly better fit (χν2 = 1.41/7 = 0.20) with a constant $\gamma =29\pm 25~\rm km~s^{-1}$. For future reference, we note that the 1σ, 2σ, and 3σ (Δχ2 = 1.0, 2.71, and 6.63) upper limits to the orbit-phase radial velocity semi-amplitude K = 48, 72, and $105~\rm km~s^{-1}$, respectively. Applying the composite line profile technique to the spin-phase resolved composite line profiles results in the radial velocities shown in Figure 10b. Unlike the orbit-phase radial velocities, these data are not well fit (χν2 = 18.3/7 = 2.6) by a constant, but they are well fit (χν2 = 2.85/5 = 0.57) by the sine function $$v(\phi\_{\rm spin}) = \gamma + K {1\over\Delta} \int^{\phi\_{\rm spin}+\Delta/2}\_{\phi\_{\rm spin}-\Delta/2} \sin 4\pi (\phi-\phi\_0)\, d\phi,$$ where Δ = 1/8, with $\gamma =29\pm 25~\rm km~s^{-1}$, $K=149\pm 38~\rm km~s^{-1}$, and ϕ0 = 0.060 ± 0.021 (solid curve in Fig. 10b). ### Cross correlation technique The composite line profile technique employed above utilizes the strongest isolated emission lines in the HETG spectrum of AE Aqr, ignoring the many weaker and often blended spectral features shown in Figures 5 and 7. In an attempt to reduce the size of the error bars on the derived spin-phase radial velocities, a cross correlation technique was tested. To accomplish this, spin-phase resolved spectra were formed by adding the HETG event data in wavelength space using bins of constant velocity width $\Delta v =100~\rm km~s^{-1}$ (specifically, λ = 3.000, 3.001, 3.002, …25.002 Å). In the absence of an obvious template against which to cross correlate the resulting spin-phase resolved spectra, the spectrum from the first spin phase bin, $\phi\_{\rm spin}=0\pm \Delta/2$, was used as the template. The resulting spin-phase radial velocities, shown in Figure 10c, are very similar to those derived using the composite line technique (Fig. 10b), but the error bars are smaller by a factor of approximately 40%. These data are reasonably well fit (χν2 = 4.75/4 = 1.19) by equation 2 with $\gamma =22\pm 15~\rm km~s^{-1}$, $K=167\pm 23~\rm km~s^{-1}$, and ϕ0 = 0.045 ± 0.012 (solid curve in Fig. 10c). Note that, given the manner in which this result was derived, the γ velocity is now relative that of the template, the spectrum from the first spin phase, for which the composite line profile technique gave a radial velocity $v=-1\pm 57~\rm km~s^{-1}$ (i.e., consistent with zero). Given this fit to the radial velocities of the spin-phase resolved spectra, it is possible to produce a spin-phase average spectrum that accounts for (removes the effect of) the spin-phase radial velocities. This was accomplished by multiplying the wavelengths of the HETG event data by a factor $1-v(\phi\_{\rm spin})/c$, where $v(\phi\_{\rm spin})=\gamma + K\, \sin 4\pi (\phi\_{\rm spin} - \phi\_0)~\rm km~s^{-1}$ with parameters that are equal to the previous best-fit values: $\gamma =22~\rm km~s^{-1}$, $K=167~\rm km~s^{-1}$, and ϕ0 = 0.045. The spin-phase radial velocities derived using the resulting boot-strapped spin-phase average spectrum as the cross correlation template are shown in Figure 10d. They are very similar to the radial velocities derived using the “vanilla” cross correlation technique (Fig. 10c), but the error bars are smaller by a factor of approximately 30%. These data are now not particularly well fit (χν2 = 10.3/5 = 2.1) by equation 2 with $\gamma =15\pm 10~\rm km~s^{-1}$, $K=163\pm 15~\rm km~s^{-1}$, and ϕ0 = 0.022 ± 0.008 (solid curve in Fig. 10d). The deviations from the fit appear to be consistent with a radial velocity amplitude that is larger on the 0–0.5 spin-phase interval and smaller on the 0.5–1 spin-phase interval. Accordingly, equation 2 was modified to allow this additional parameter, and the data are then well fit (χν2 = 0.82/4 = 0.21) with $\gamma =14\pm 10~\rm km~s^{-1}$, $K\_1=206\pm 20~\rm km~s^{-1}$ (valid on $\phi\_{\rm spin}=0$–0.5), $K\_2=120\pm 20~\rm km~s^{-1}$ (valid on $\phi\_{\rm spin}=0.5$–1), and ϕ0 = 0.023 ± 0.008 (dotted curve in Fig. 10d). Summary ======= As summarized below, our long, uninterrupted Chandra HETG observation provides a wealth of details concerning the X-ray emission of AE Aqr: 1. The X-ray light curve is dominated by flares that last between a few hundred and a few thousand seconds, produce increases of up to 3–5 times the baseline count rate (Fig. 1), and are achromatic except near their peaks (Fig. 2); the white dwarf spin-phase X-ray light curve is achromatic and sinusoidal in shape, with a relative semi-amplitude of approximately 16% (Fig. 3); and the X-ray pulse amplitude increases linearly with the mean count rate in the middle of the range, but saturates at both the low and high ends of the range (Fig. 4). 2. The X-ray spectrum is that of an optically thin multi-temperature thermal plasma (Fig. 5); the X-ray emission lines are broad (Fig. 9), with widths that increase with the line energy, from σ ≈ 1 eV ($510~\rm km~s^{-1}$) for to σ ≈ 5.5 eV ($820~\rm km~s^{-1}$) for (Fig. 6); the X-ray spectrum is reasonably well fit by a plasma model with a Gaussian emission measure distribution that peaks at $\log T ({\rm K})=7.16$, has a width σ = 0.48, an Fe abundance equal to 0.44 times solar, and other metal (primarily Ne, Mg, and Si) abundances equal to 0.76 times solar (Fig. 7); and for a distance d = 100 pc, the total emission measure $EM=8.0\times 10^{53}~\rm cm^{-3}$ and the 0.5–10 keV luminosity $L\_{\rm X}= 1.1\times 10^{31}~\rm erg~s^{-1}$. 3. Based on the f/(i + r) flux ratios of the He α triplets of,, measured by Itoh et al. in the XMM-Newton Reflection Grating Spectrometer spectrum, and those of,,, and in the Chandra HETG spectrum, the electron density of the plasma increases with temperature by over three orders of magnitude, from $n\_{\rm e}\approx 6\times 10^{10}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K})\approx 6.2$] to $n\_{\rm e}\approx 1\times 10^{14}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K})\approx 7.0$] (Table 1 and Fig. 8a–e), and/or the plasma is significantly affected by photoexcitation (Fig. 8f–j). 4. The radial velocity of the X-ray emission lines varies on the white dwarf spin phase, with two oscillations per spin cycle and an amplitude $K\approx 160~\rm km~s^{-1}$ (Fig. 10). Discussion ========== Over the years, two very different models have been proposed for the source of the X-ray emission of AE Aqr. On one hand, based on ROSAT and ASCA data, argued that the X-ray emission, including the flares, must occur close to the white dwarf, so that the gravitational potential energy can heat the X-ray emitting plasma to the observed temperatures. Similarly, based on Ginga and ASCA data, argued that the X-ray emission, both persistent and flare, originates within the white dwarf magnetosphere; came to similar conclusions based on XMM-Newton Optical Monitor (OM) and European Photon Imaging Camera (EPIC) data. On the other hand, based on XMM-Newton RGS data, argued that the f/(i + r) flux ratios of the He α triplets of,, and are consistent with a plasma with an electron density $n\_{\rm e}\sim 10^{11}~\rm cm^{-3}$ and, given the observed emission measure, a linear scale l ≈ (2–3) × 1010 cm. Because this density is orders of magnetic less than the conventional estimate for the post-shock accretion column of a magnetic CV, and because this linear scale is much larger than the radius of the white dwarf, these authors argued that the optically thin X-ray-emitting plasma in AE Aqr is due not to accretion onto the white dwarf, but to blobs in the accretion stream, heated to X-ray emitting temperatures by the propeller action of the white dwarf magnetic field. has taken issue with some of the details of this model, arguing that the detected X-rays are due to either (1) a tenuous component of the accretion stream or (2) plasma “outside the system,” heated by accelerated particles and/or MHD waves due to a pulsar-like mechanism powered by the spin-down of the magnetic white dwarf. The presence of non-thermal particles in AE Aqr is supported by the observed TeV γ-rays and the recent discovery by of a hard, possibly power-law, component in the Suzaku X-ray spectrum of AE Aqr. Finally, have proposed that the observed unpulsed X-ray emission in AE Aqr is the result of a very tenuous hot corona associated with the secondary star, which is pumped magnetohydrodynamically by the propeller action of the white dwarf magnetic field. As we argue below, the results of our Chandra HETG observation of AE Aqr — particularly the orbit-phase pulse time delays, the high electron densities and/or high levels of photoexcitation implied by the He α triplet flux ratios, and the large widths and spin-phase radial velocities of the X-ray emission lines — are not consistent with an extended, low-density source of X-rays in AE Aqr, but instead support earlier models in which the dominant source of X-rays is of high density and/or in close proximity to the white dwarf. Consider first the systemic velocity of the X-ray emission lines. We variously measured this quantity to be $v=-1\pm 33~\rm km~s^{-1}$ from the Lyman α emission lines (§2.3.1), $v =25\pm 26~\rm km~s^{-1}$ from the composite line profile (Fig. 9), $\gamma =29\pm 25~\rm km~s^{-1}$ from the composite line profile fit to the radial velocities (Fig. 10b), and $\gamma =22\pm 15~\rm km~s^{-1}$ from the cross correlation fit to the radial velocities (Fig. 10c). Optimistically assuming that these measurements are neither correlated nor strongly affected by systematic effects, the weighted mean and standard deviation of the systemic velocity of the X-ray emission lines $\gamma\_{\rm X}=21\pm 11~\rm km~s^{-1}$. Relative to the systemic velocity of the optical emission (absorption) lines, $\gamma\_{\rm O}\approx -38\pm 9\> (-64\pm 11)~\rm km~s^{-1}$, the X-ray emission lines are redshifted by $\Delta v=\gamma\_{\rm X} - \gamma\_{\rm O} \approx 59\pm 14~(85\pm 16)~\rm km~s^{-1}$. This result is to be compared to the free-fall velocity $v\_{\rm ff}=(2G{M\_{\rm wd}}/{R\_{\rm wd}})^{1/2}\approx 5500~\rm km~s^{-1}$ onto the surface of the white dwarf, the infall velocity $v\_{\rm in}\le v\_{\rm ff}/4\approx 1375~\rm km~s^{-1}$ below the putative standoff shock, and the gravitational redshift $\Delta v=G{M\_{\rm wd}}/{R\_{\rm wd}}c = 51^{+13}\_{-11}~\rm km~s^{-1}$ from the surface of the white dwarf with a mass ${M\_{\rm wd}}=0.8\pm 0.1~{\rm M\_{\odot}}$ and radius ${R\_{\rm wd}}= (7.0\mp 0.8)\times 10^8$ cm. Although the determination and interpretation of systemic velocities is fraught with uncertainties, the systemic velocity of the X-ray emission lines of AE Aqr is consistent with the gravitational redshift from the surface of the white dwarf, and hence with a source on or near the surface of the white dwarf, rather than a more extended region within the binary. Supporting the proposal that the X-ray emission of AE Aqr is closely associated with the white dwarf is the fact, established previously by, that the X-ray spin pulse follows the motion of the white dwarf around the binary center of mass, producing a time delay Δt = 2.17 ± 0.48 s in the arrival times of the X-ray pulses. Given this result, it is somewhat surprising that the radial velocity of the X-ray emission lines does not appear to vary on the white dwarf orbit phase: the measured orbit-phase radial velocity semi-amplitude $K=12\pm 36~\rm km~s^{-1}$, whereas the expected value $K=2\pi\Delta t c/P\_{\rm orb} = 115\pm 25~\rm km~s^{-1}$ (Fig. 10a). The difference between the measured and expected radial velocity semi-amplitudes is $103\pm 44~\rm km~s^{-1}$, which differs from zero by 2.3σ. While this discrepancy is of some concern, we showed above that the radial velocity of the X-ray emission lines varies on the white dwarf spin phase (Fig. 10b–d), which argues strongly for a source of X-rays trapped within, and rotating with, the magnetosphere of the white dwarf. In contrast, it seems clear that our result for the radial velocity of the X-ray emission lines is not consistent with the proposal, put forward by, that the accretion stream is the dominant source of X-rays in AE Aqr. For the system parameters of AE Aqr, the accretion stream makes its closest approach to the white dwarf at a radius $r\_{\rm min}\approx 1\times 10^{10}$ cm and a velocity $v\_{\rm max}\approx 1500~\rm km~s^{-1}$. Accounting for the binary inclination angle i = 60∘, the predicted accretion stream radial velocity amplitude $v\_{\rm max}\sin i\approx 1300~\rm km~s^{-1}$, whereas the 3σ upper limit to the orbit-phase radial velocity semi-amplitude $K= 105~\rm km~s^{-1}$. Clearly, the accretion stream, if it follows a trajectory anything like the inhomogeneous diamagnetic accretion flow calculated by, cannot be the dominant source of X-rays in AE Aqr. An additional argument against the accretion stream, plasma “outside the system”, a hot corona associated with the secondary star, or any other extended source of X-rays in AE Aqr is the high electron densities $\log n\_{\rm e}\approx 10.8$, 11.4, 12.3, 13.0, and 14.1 inferred from the f/(i + r) flux ratios of the He α triplets of,,,, and, respectively. Given the differential emission measure distribution dEM/dlogT ≈ 6.6 × 1053 exp[ − (logT − logT0)2/2σ2] with $\log T\_0 ({\rm K})\approx 7.16$ and σ ≈ 0.48, the,,,, and He α triplet emissivity-weighted emission measure $EM=\int (dEM/d\log T)\, \varepsilon\, d\log T/\int\varepsilon\, d\log T\approx [0.9,1.6,3.3,4.8,5.8]\times 10^{52}~\rm cm^{-3}$ and the linear scale $l=(EM/n\_{\rm e}^2)^{1/3}\approx 1.4\times 10^{10}$, 6.7 × 109, 2.0 × 109, 7.4 × 108, and 1.4 × 108 cm or 20, 10, 3, 1, and $0.2~{R\_{\rm wd}}$, respectively. Although even the observed f/(i + r) flux ratio can be produced in a low-density plasma suffering photoexcitation by an external radiation field, this requires both high blackbody temperatures ($T\_{\rm bb}\ga 25$ kK) and large dilution factors ($W\la {1\over 2}$), hence small-to-zero distances above the white dwarf surface. We conclude that, if the bulk of the plasma near the peak of the emission measure distribution is not of high density, it must be in close proximity to the white dwarf. White dwarf ----------- To investigate the possibility that the X-rays observed in AE Aqr are associated with the white dwarf, we considered a simple geometric model, similar to that derived by from the maximum entropy maps of the UV pulse profiles measured with the Faint Object Spectrograph onboard the Hubble Space Telescope. The model consists of a rotating white dwarf viewed at an inclination angle i = 60∘ from the rotation axis $\hat z$ with two bright spots, centered 20∘ above and below the equator and separated in longitude by 180∘ (i.e., the upper and lower spots are centered at spherical coordinates [θ, ϕ] = [70∘, 0∘] and [110∘, 180∘], respectively). Such a model naturally produces two unequal flux peaks per spin cycle, with the brighter (dimmer) peak occurring when the upper (lower) spot is pointed toward the observer. Instead of the optically thick assumption applied in the optical and UV, we assume that the X-ray emitting spots are optically thin. In this case, the X-ray flux modulation is produced solely by occultation by the body of the white dwarf, so the brightness of the lower spot must be reduced by approximately 30% to avoid producing a second peak in the X-ray light curve. Assuming that the brightness distribution of the spots is given by a Gaussian function exp( − δθ2/2σ2), where δθ is the polar angle from the center of the spot and the spot width σ = 30∘ (hence $\rm FWHM=70^\circ $ and the fractional emitting area f = ∫exp( − θ2/2σ2)sinθdθ ≈ 0.25, justifying our choice for the fiducial value of this quantity in §2.3.2), the light curves of the upper spot, the lower spot, and the total flux are as shown in Figure 11a. Smaller spots produce squarer light curves, while larger spots produce lower relative oscillation amplitudes; the relative pulse semi-amplitude of the model shown, $(I\_{\rm max}-I\_{\rm min}) /(I\_{\rm max}+I\_{\rm min})=16\%$, is consistent with observations (Fig. 3a). The flux-weighted mean radial velocities of the emission from the upper spot, the lower spot, and the total flux are as shown in Figure 11b for an assumed white dwarf rotation velocity $v\_{\rm rot}=2\pi{R\_{\rm wd}}/P\_{33}=1330~\rm km~s^{-1}$. As observed, the predicted radial velocity of the total flux goes through two oscillations per spin cycle, although the two peaks are not equal in strength, the red-to-blue zero velocity crossings do not occur near $\phi\_{\rm spin}=0.25$ and 0.75, and the maximum velocity amplitude is greater than observed (Fig. 10b–d). Although it is possible to remedy these deficiencies by using two bright spots that are more nearly equal in brightness, this significantly reduces the pulse amplitude of the total flux. Furthermore, the model predicts that the width of the X-ray emission lines should be narrower near $\phi\_{\rm spin}=0$ and 0.5 and broader near $\phi\_{\rm spin}=0.25$ and 0.75, which is not corroborated by the data. Finally, the model predicts that the radial velocity amplitude of the total flux is higher (lower) on the 0.75–1.25 (0.25–0.75) spin phase interval, whereas the boot-strapped cross correlation technique provides evidence that this is the case approximately one quarter of a cycle later: on the 0–0.5 (0.5–1) spin phase interval (ignoring the small phase offset). Accretion Columns ----------------- To place their results in a theoretical framework, interpreted the hot spots derived from their maximum entropy maps as the result of reprocessing of X-rays emitted by the post-shock gas in the accretion columns of a magnetic white dwarf. For an assumed point source of illumination, they found good fits to the UV pulse profiles if the angle between the spin axis and the magnetic axis — the magnetic colatitude — β = 76∘ ± 1∘, the peak spot temperature $T\_{\rm max}=26\pm 2$ kK, the height of the illuminating source above the white dwarf surface $H/{R\_{\rm wd}}= 3\pm 1$, and the X-ray luminosity $L\_{\rm X} = 3\times 10^{33}\, (H/3)^2\, (\eta/0.5)^{-1}~\rm erg~s^{-1}$, where η is the efficiency of conversion of accretion luminosity into 0.1–4 keV X-rays (note that our fiducial value for this quantity is larger by an order of magnitude than that assumed by Eracleous et al. because the mean temperature of the X-ray spectrum of AE Aqr is uncharacteristically low). These results are troubling for two reasons. First, as noted by Eracleous et al., the shock height required to produce the size of the UV spots is quite large (comparable to the corotation radius). Second, the X-ray luminosity required to heat the 26 kK spots by reprocessing is more than two orders of magnitude greater than observed in the 0.5–10 keV bandpass. Adding to these problems, we found that such a model reproduces neither the X-ray pulse profile nor the radial velocity variation observed in AE Aqr. First, the predicted pulse profile has a square waveform, with the X-ray flux rapidly doubling (halving) when one of the illuminating spots emerges from (is hidden behind) the body of the white dwarf. Second (if the illuminating spots are not equal in brightness), the predicted radial velocity variation goes through only one oscillation per spin cycle. Given these problems, we modeled the accretion columns as two uniform emission volumes contained within a polar angle Δθ of the magnetic axis and radii $r/{R\_{\rm wd}}-1=0$–h. For an inclination angle i = 60∘, the pulse profiles and radial velocity variations were investigated for magnetic colatitudes β = 60∘–90∘, opening angles Δθ = 15∘–90∘, shock heights h = 0–3, and infall velocities $v\_{\rm infall}= 0$–$v\_{\rm ff}=0$–$5500~\rm km~s^{-1}$, with and without intensity weighting by a Gaussian function exp( − δθ2/2σ2). Although the parameter space is huge, it did not seem possible to reproduce the observed X-ray pulse profile, let alone the radial velocity variation, with such a model, unless it looked very much like the two-spot model discussed in the previous section (i.e., β ≈ 70∘, σ ≈ 30∘, h ≈ 0, and $v\_{\rm infall}\la 0.01\, v\_{\rm ff}$): even modest infall velocities produced model radial velocity variations with systematic velocities that are greater (in absolute magnitude) than observed, while large shock heights and/or opening angles and/or Gaussian widths produced small relative pulse amplitudes, and, for much of the parameter space, flux maximum occurs at spin phases $\phi\_{\rm spin} = 0.25$ and 0.75, when both of the accretion columns are visible on the limb of the white dwarf. Magnetosphere ------------- We next considered the possibility, argued by, that the X-ray emission of AE Aqr, both persistent and flare, is due to plasma associated with the magnetosphere of the white dwarf. We assume that the magnetosphere is filled with plasma stripped off of the accretion stream, which, as noted above, makes its closest approach to the white dwarf at a radius $r\_{\rm min}\approx 1\times 10^{10}$ cm and a velocity $v\_{\rm max}\approx 1500~\rm km~s^{-1}$. The magnetic field of the white dwarf can control the motion of an ionized component of the stream if the magnetic pressure B2/8π [where $B=B\_\star\, ({R\_{\rm wd}}/r\_{\rm min})^3$ for a dipole field with a surface magnetic field strength B⋆] is greater than the ram pressure $\rho v\_{\rm max}^2$, hence if $B\_\star > (8\pi\mu m\_{\rm H}n\_{\rm e})^{1/2}\, v\_{\rm max}\, (r\_{\rm min}/{R\_{\rm wd}})^3\approx 7\, (n\_{\rm e}/10^{13}~{\rm cm^{-3}})^{1/2}$ MG. If the kinetic energy of the accretion stream is thermalized in a strong shock at its closest approach to the white dwarf, it will be heated to a temperature $T\_{\rm max}=3\mu m\_{\rm H} v\_{\rm max}^2/16k\approx 3.3\times 10^7$ K or approximately twice the peak temperature of the emission measure distribution. Hence, the magnetosphere plausibly could be the source of X-rays in AE Aqr, independent of any other source of energy, if it is fed at a rate $\dot M = 2\, L\_{\rm X}/v\_{\rm max}^2\approx 9.8\times 10^{14}~\rm g~s^{-1}$. The magnetosphere is an attractive source of X-rays in AE Aqr in as much as it would naturally supply a range of densities, linear scales, and velocities, as required by the data. On the other hand, we found that the widths of the X-ray emission lines increase from σ ≈ 1 eV ($510~\rm km~s^{-1}$) for to σ ≈ 5.5 eV ($820~\rm km~s^{-1}$) for (Fig. 6), whereas the line widths predicted by this model are nominally much larger: for plasma trapped on, and forced to rotate with, the white dwarf magnetic field, the projected rotation velocity $v\_{\rm rot}\sin i = 2\pi r\sin i/P\_{33}$, which varies from $1150~\rm km~s^{-1}$ for $r={R\_{\rm wd}}=7\times 10^8$ cm (as in the two-spot model) to $16{,}500~\rm km~s^{-1}$ for $r=r\_{\rm min}=1\times 10^{10}$ cm. However, the observed value for the lines widths should be near the lower end of this range, since, for a magnetosphere uniformly filled with plasma with an inward (radial) velocity $v\_{\rm r}\propto v\_{\rm ff}=(2G{M\_{\rm wd}}/r)^{1/2}$, the plasma density ρ ∝ r− 3/2 and the X-ray emissivity ɛ ∝ ρ2 ∝ r− 3. Conclusion ========== Of the simple models considered above for the source of the X-ray emission of AE Aqr, the white dwarf and magnetosphere models are the most promising, while the accretion column model appears to be untenable: it fails to reproduce the Chandra HETG X-ray light curves and radial velocities, and it requires an X-ray luminosity that is more than two orders of magnitude greater than observed in the 0.5–10 keV bandpass to heat the UV hot spots by reprocessing. A more intimate association between the X-ray and UV emission regions could resolve this problem, and it is interesting and perhaps important to note that such a situation is naturally produced by a bombardment and/or blobby accretion solution to the accretion flow. In particular, if the cyclotron-balanced bombardment solution applies to AE Aqr, it would naturally produce comparable X-ray and UV spot sizes and luminosities, comparable relative pulse amplitudes, comparable orbit-phase pulse time delays, the observed tight correlation between the X-ray and UV light curves, and an accretion region that is heated to the observed T ∼ 107 K. Such a solution to the accretion flow applies if (1) the specific accretion rate ($\dot m =\dot M/A$) onto the white dwarf is sufficiently low that the accreting plasma does not pass through a hydrodynamic shock, but instead is stopped by Coulomb interactions in the white dwarf atmosphere, and (2) the magnetic field of the white dwarf is sufficiently strong that the accretion luminosity can be radiated away by a combination of optically thin bremsstrahlung and line radiation in the X-ray waveband and optically thick cyclotron radiation in the infrared and optical wavebands. According to, the limit for the validity of cyclotron-balanced bombardment is $\dot m\la 1.19^{+0.69}\_{-0.47}\times 10^{-4}\, B\_7^{2.6}~\rm g~cm^{-2}~s^{-1}$ for a white dwarf mass ${M\_{\rm wd}}=0.8\pm 0.2\, {\rm M\_{\odot}}$ and magnetic field strength B7 = B⋆/107 G. Are these conditions satisfied in AE Aqr? Perhaps. First, assuming that the observed X-ray luminosity is driven by accretion onto the white dwarf, the accretion rate $\dot M=L\_{\rm X}{R\_{\rm wd}}/G{M\_{\rm wd}}\approx 7.3\times 10^{13}~\rm g~s^{-1}$, hence the specific accretion rate $\dot m\approx 4.7\times 10^{-5}\, (f/0.25)^{-1}~\rm g~cm^{-2}~s^{-1}$. Second, the strength of the magnetic field of the white dwarf in AE Aqr is uncertain, but based on the typical magnetic moments of intermediate polars ($10^{32}~\rm G~cm^3$) and polars ($10^{34}~\rm G~cm^3$), it should lie in the range $0.3~{\rm MG}\la B\_\star\la 30~{\rm MG}$. More specific estimates of this quantity include $B\_\star \la 2$ MG, based on evolutionary considerations ; B⋆ ∼ 1–5 MG, based on the low levels of, and upper limits on, the circular polarization of AE Aqr ; and B⋆ ∼ 50 MG, based on the (probably incorrect) assumption that the spin-down of the white dwarf is due to the pulsar mechanism. The above limit for the validity of cyclotron-balanced bombardment requires $B\_\star\ga 7^{+1.5}\_{-1.1}\, (f/0.25)^{-0.35}$ MG, which is probably not inconsistent with the circular polarization measurements and upper limits. Conversely, the validity of the bombardment solution in AE Aqr could be tested with a more secure measurement of, or lower upper limit on, the magnetic field strength of its white dwarf. Where do we go from here? Additional constraints on the global model of AE Aqr, and in particular on the connection between the various emission regions, can be supplied by simultaneous multiwavelength observations, including those obtained during our 2005 multiwavelength campaign, the analysis of which is at an early stage. TeV observations with the Whipple Observatory 10-m telescope obtained over the epoch 1991–1995 and the Major Atmospheric Gamma-Ray Imaging Cherenkov (MAGIC) 17-m telescope obtained during our campaign provide only (low) upper limits on the TeV γ-ray flux from AE Aqr, so it would be useful to unambiguously validate or invalidate earlier reports of detections of TeV flux from this source; in particular, a dedicated High Energy Stereoscopic System (HESS) campaign of observations, preferably in conjunction with simultaneous X-ray observations, is badly needed. Similarly, it would be useful to independently validate or invalidate the existence of the high-energy emission detected by in the Suzaku X-ray spectrum of AE Aqr, and to establish unambiguously if it is thermal or nonthermal in nature. Going out on a limb, it is our bet that AE Aqr is not a TeV γ-ray source and that the high-energy X-ray excess is thermal in nature, specifically that it is due to the stand-off shock present when the specific mass accretion rate onto the white dwarf is large (e.g., during flares) and the bombardment solution is no longer valid. Additional high resolution X-ray spectroscopic observations are also warranted, particularly to confirm or refute the high electron densities inferred from the flux ratios of the,, and He α triplets; it is, after all, a priori unlikely that each of these ratios lies on the knee of the theoretical f/(i + r) flux ratio curves, and the trend of increasing density with increasing temperature is opposite to what is expected from a cooling flow (although it is consistent with the opposite, e.g., adiabatic expansion). It would be extremely helpful to apply the various Fe L-shell density diagnostics to AE Aqr, since they are typically less sensitive to photoexcitation, but the 17.10/17.05 line ratio for one is compromised by blending due to the the large line widths, and the other line ratios all require higher signal-to-noise ratio spectra than currently exists. It would be extremely useful to (1) determine the radial velocities of individual X-ray emission lines, rather than all the lines together; (2) resolve the individual lines on the white dwarf spin and orbit phases, (3) investigate the flare and quiescent spectra separately, (4) resolve individual flares, and (5) include the Fe K lines into this type of analysis. Such detailed investigations are beyond the capabilities of Chandra or any other existing X-ray facility, but would be possible with future facilities such as IXO that supply an order-of-magnitude or more increase in effective area and spectral resolution. I thank R. Hoogerwerf for the IDL code used to account for the Doppler shifts produced by Chandra’s motion relative to the solar system barycenter, M. Ishida for verifying the error in Figure 5(h) of, K. Beuermann for an exchange of e-mails regarding the bombardment model, and the anonymous referee, whose report resulted in a significant expansion in the scope of this work and this manuscript. Support for this work was provided by NASA through Chandra Award Number GO5-6020X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Facility: Anders, E., & Grevesse, N. 1989,, 53, 197 Blumenthal, G. R., Drake, G. W. F., & Tucker, W. H. 1972,, 172, 205 Beskrovnaya, N. G., Ikhsanov, N. R., Bruch, A., & Shakhovskoy, N. M. 1995, in ASP Conf. Ser. 85, Cape Workshop on Magnetic Cataclysmic Variable, ed. D. A. H. Buckley & B. Warner (San Francisco: ASP), 364 Buckley, D. A. H., & Warner, B. 1995, ASP Conf. 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R., & Horne, K. 1997,, 286, 436 Venter, L. A., & Meintjes, P. J. 2007,, 378, 681 --- 1. Available at http://cxc.harvard.edu/ciao/.[↩](#fnref1) 2. Available at http://cxc.harvard.edu/atomdb/WebGUIDE/.[↩](#fnref2) 3. Available at http://asc.harvard.edu/atomdb/features\_idl.html[↩](#fnref3) 4. http://asc.harvard.edu/atomdb/[↩](#fnref4) 5. This result is driven primarily by, and should be understood to apply primarily to, Ne, Mg, and Si. Consistent with the quality of the data, no attempt was made to further subdivide the element abundances.[↩](#fnref5) 6. In the tabulation, for for $n\_{\rm e}=10^{14}~\rm cm^{-3}$, G = 0.90, 0.76, 0.67, and 0.56 for T = 5, 7.5, 10, and 15 MK, respectively, but then rises to G = 0.70 for T = 30 MK; the observed G = 0.41 ± 0.11.[↩](#fnref6) Chandra High Energy Transmission Grating Spectrum of AE Aquarii =============================================================== The nova-like cataclysmic binary AE Aqr, which is currently understood to be a former supersoft X-ray binary and current magnetic propeller, was observed for over two binary orbits (78 ks) in 2005 August with the High-Energy Transmission Grating (HETG) onboard the Chandra X-ray Observatory. The long, uninterrupted Chandra observation provides a wealth of details concerning the X-ray emission of AE Aqr, many of which are new and unique to the HETG. First, the X-ray spectrum is that of an optically thin multi-temperature thermal plasma; the X-ray emission lines are broad, with widths that increase with the line energy, from σ ≈ 1 eV ($510~\rm km~s^{-1}$) for to σ ≈ 5.5 eV ($820~\rm km~s^{-1}$) for ; the X-ray spectrum is reasonably well fit by a plasma model with a Gaussian emission measure distribution that peaks at $\log T ({\rm K})=7.16$, has a width σ = 0.48, an Fe abundance equal to 0.44 times solar, and other metal (primarily Ne, Mg, and Si) abundances equal to 0.76 times solar; and for a distance d = 100 pc, the total emission measure $EM=8.0\times 10^{53}~\rm cm^{-3}$ and the 0.5–10 keV luminosity $L\_{\rm X}= 1.1\times 10^{31}~\rm erg~s^{-1}$. Second, based on the f/(i + r) flux ratios of the forbidden (f), intercombination (i), and recombination (r) lines of the He α triplets of,, and measured by Itoh et al. in the XMM-Newton Reflection Grating Spectrometer spectrum and those of,,, and in the Chandra HETG spectrum, either the electron density of the plasma increases with temperature by over three orders of magnitude, from $n\_{\rm e}\approx 6\times 10^{10}~\rm cm^{-3}$ for [$\log T({\rm K})\approx 6$] to $n\_{\rm e}\approx 1\times 10^{14}~\rm cm^{-3}$ for [$\log T ({\rm K})\approx 7$], and/or the plasma is significantly affected by photoexcitation. Third, the radial velocity of the X-ray emission lines varies on the white dwarf spin phase, with two oscillations per spin cycle and an amplitude $K\approx 160~\rm km~s^{-1}$. These results appear to be inconsistent with the recent models of Itoh et al., Ikhsanov, and Venter & Meintjes of an extended, low-density source of X-rays in AE Aqr, but instead support earlier models in which the dominant source of X-rays is of high density and/or in close proximity to the white dwarf. Introduction ============ AE Aqr is a bright (V ≈ 11) nova-like cataclysmic binary consisting of a magnetic white dwarf primary and a K4–5 V secondary with a long 9.88 hr orbital period and the shortest known white dwarf spin period P = 33.08 s. Although originally classified and interpreted as a disk-accreting DQ Her star, AE Aqr displays a number of unusual characteristics that are not naturally explained by this model. First, violent flaring activity is observed in the radio, optical, ultraviolet (UV), X-ray, and TeV γ-rays. Second, the Balmer emission lines are single-peaked and produce Doppler tomograms that are not consistent with those of an accretion disk. Third, the white dwarf is spinning down at a rate ${\dot{P}}= 5.64 \times 10^{-14}~{\rm s~s^{-1}}$. Although this corresponds to the small rate of change of $1.78~\rm ns~yr^{-1}$, AE Aqr’s spin-down is typically characterized as “rapid” because the characteristic time $P/\Pdot\approx 2\times 10^7$ yr is short compared to the lifetime of the binary and because the spin-down luminosity $L\_{\rm sd}=-I\Om{\dot{\Omega}}\approx 1\times 10^{34}~\rm erg~s^{-1}$ (where $I\approx 0.2{M\_{\rm wd}}{R\_{\rm wd}}^2 \approx 2\times 10^{50}~\rm g~cm^2$ is the moment of inertia for a white dwarf of mass ${M\_{\rm wd}}=0.8~{\rm M\_{\odot}}$ and radius ${R\_{\rm wd}}= 7.0\times 10^8$ cm, $\Om = 2\pi/P$, and ${\dot{\Omega}}= -2\pi\Pdot/P^2$) exceeds the secondary’s thermonuclear luminosity by an order of magnitude and the accretion luminosity by two orders of magnitude. Because of its unique properties and variable emission across the electromagnetic spectrum, AE Aqr has been the subject of numerous studies, including an intensive multiwavelength observing campaign in 1993 October [, and the series of papers in ASP Conf. Ser. 85 ]. Based on these studies, AE Aqr is now widely believed to be a former supersoft X-ray binary and current magnetic propeller, with most of the mass lost by the secondary being flung out of the binary by the magnetic field of the rapidly rotating white dwarf. These models explain many of AE Aqr’s unique characteristics, including the fast spin rate and rapid secular spin-down rate of the white dwarf, the anomalous spectral type of the secondary, the anomalous abundances, the absence of signatures of an accretion disk, the violent flaring activity, and the origin of the radio and TeV γ-ray emission. To build on this observational and theoretical work, while taking advantage of a number of improvements in observing capabilities, during 2005 August 28–September 2, a group of professional and amateur astronomers conducted a campaign of multiwavelength (radio, optical, UV, X-ray, and TeV γ-ray) observations of AE Aqr. Attention is restricted here to the results of the X-ray observations, obtained with the High-Energy Transmission Grating (HETG) and the Advanced CCD Imaging Spectrometer (ACIS) detector onboard the Chandra X-ray Observatory. has previously provided an analysis and discussion of the timing properties of these data, showing that: (1) as in the optical and UV, the X-ray spin pulse follows the motion of the white dwarf around the binary center of mass and (2) during the decade 1995–2005, the white dwarf spun down at a rate that is slightly faster than predicted by the spin ephemeris. Here, we present a more complete analysis and discussion of the Chandra data, providing results that in many ways reproduce the results of the previous Einstein, ROSAT, ASCA, and XMM-Newton and the subsequent Suzaku observations of AE Aqr, but also that are new and unique to the HETG; namely, the detailed nature of the time-average X-ray spectrum, the plasma densities implied by the He α triplet flux ratios, and the widths and radial velocities of the X-ray emission lines. As we will see, these results appear to be inconsistent with the recent models of,, and of an extended, low-density source of X-rays in AE Aqr, but instead support earlier models in which the dominant source of X-rays is of high density and/or in close proximity to the white dwarf. The plan of this paper is as follows. In §2 we discuss the observations and the analysis of the X-ray light curve (§2.1), spin-phase light curve (§2.2), spectrum (§2.3), and radial velocities (§2.4). In §3 we provide a summary of our results. In §4 we discuss the results and explore white dwarf (§4.1), accretion column (§4.2), and magnetosphere (§4.3) models of the source of the X-ray emission in AE Aqr. In §5 we draw conclusions, discuss the bombardment model of the accretion flow of AE Aqr, and close with a few comments regarding future observations. The casual reader may wish to skip §2.1–2.2, which are included for completeness, and concentrate on §2.3–2.4, which contain the important observational and analysis aspects of this work. Observations and Analysis ========================= AE Aqr was observed by Chandra beginning on 2005 August 30 at 06:37 UT for 78 ks (). The level 2 data files used for this analysis were produced by the standard pipeline processing software ASCDS version 7.6.7.1 and CALDB version 3.2.1 and were processed with the Chandra Interactive Analysis of Observations (CIAO[1](#fn1)) version 3.4 software tools to convert the event times in the evt2 file from Terrestrial Time (TT) to Barycentric Dynamical Time (TDB) and to make the grating response matrix files (RMFs) and the auxiliary response files (ARFs) needed for quantitative spectroscopic analysis. The subsequent analysis was performed in the following manner using custom Interactive Data Language (IDL) software. First, using the region masks in the pha2 file, the source and background events for the ± first-order Medium Energy Grating (MEG) and High Energy Grating (HEG) spectra were collected from the evt2 file. Second, after careful investigation, ± 0.0030 Å ( ± 0.0015 Å) was added to the ± first-order MEG (HEG) wavelengths, respectively, to account for an apparent shift (by 0.27 ACIS pixels) in the position of the zero-order image. Third, to account for the spin pulse delay measured in the optical, UV, and X-ray wavebands produced by the motion of the white dwarf around the binary center of mass, $-2\, \cos 2\pi \phi\_{\rm orb}$ s was added to the event times t, where the white dwarf orbit phase $2\pi\phi\_{\rm orb}=\Omega\_{\rm orb}(t-T\_0)$, where $\Omega\_{\rm orb}=2\pi/P\_{\rm orb}$ and $P\_{\rm orb}=0.411655610$ d and $T\_0({\rm BJD})= 2445172.2784$ are the orbit ephemeris constants from Table 4 of. Fourth, to account for the Doppler shifts produced by Chandra’s (mostly, Earth’s) motion relative to the solar system barycenter, the event wavelengths λ were multiplied by a factor $[1+v\_{\rm los}/c]$, where $v\_{\rm los}$ is the (time-dependent) line-of-sight velocity between the spacecraft and the source, determined using an IDL code kindly supplied by R. Hoogerwerf (and checked against the line-of-sight velocities derived from the barycentric time corrections supplied by the CIAO tool axbary); during the observation, $v\_{\rm los}$ varied from $-11.9~\rm km~s^{-1}$ at the beginning of the observation, rose to $-11.3~\rm km~s^{-1}$, and then fell to $-11.5~\rm km~s^{-1}$ at the end of the observation. Fifth, a filter was applied to restrict attention to events from two orbital cycles $\phi\_{\rm orb}=20503.9$–20505.9, resulting in an effective exposure of 71 ks. Sixth, white dwarf spin pulse phases were calculated using the updated cubic spin ephemeris of derived from the recent ASCA (1995 October) and XMM-Newton (2001 November) and the current Chandra observation of AE Aqr: $2\pi\phi\_{\rm spin} = \Om\_0(t-T\_{\rm max}) + {1\over2}\Odot(t-T\_{\rm max})^2 + {1\over6}{\ddot{\Omega}}(t-T\_{\rm max})^3$, where $\Om\_0=2\pi/P\_{33}$ and ${\dot{\Omega}}= -2\pi\Pdot\_{33}/P\_{33}^2$ and P33 = 0.00038283263840 d, $\Pdot\_{33}=5.642\times 10^{-14}~\rm d~d^{-1}$, and $T\_{\rm max}({\rm BJD}) = 2445172.000042$ are the spin ephemeris constants from Table 4 of, and ${\ddot{\Omega}}=-1.48\times 10^{-11}~\rm d^{-3}$. The HETG event data are then fully characterized by the event time t, white dwarf orbit phase $\phi\_{\rm orb}$, white dwarf spin phase $\phi\_{\rm spin}$, and wavelength λ. Light curve ----------- Figure 1 shows the background-subtracted count rate light curve derived from the HETG event data. As has been well established by previous X-ray observations, the X-ray light curve of AE Aqr is dominated by flares, although this is by far the longest uninterrupted observation of AE Aqr and hence the clearest view of its X-ray light curve. During the Chandra observation, the flares last between a few hundred and a few thousand seconds, producing increases of up to 3–5 times the baseline count rate of $\sim 0.1~\rm counts~s^{-1}$. To constrain the cause and nature of the flares of AE Aqr, it is of interest to determine if the count rate variations shown in Figure 1 are accompanied by, or perhaps are even due to, dramatic variations in the X-ray spectrum. Previous investigations have indicated that this is not the case. For a flare observed by ASCA, found no significant difference between the quiescent and flare X-ray spectra, although a “hint” of spectral hardening was recognized during the flare. For a flare observed by XMM-Newton, found that the X-ray spectrum at the beginning of the flare was similar to that in quiescence, but that the spectrum became harder as the flare advanced. Our ability to investigate spectral variations during the individual flares observed by Chandra is limited by the relatively low HETG count rate and the relatively fast timescale of the flares, although it is possible to investigate spectral variations for the ensemble of the flares. To accomplish this, the light curve shown in Figure 1 was divided into five count rate ranges: $ I\_1 < 0.14~\rm counts~s^{-1}$, $0.14\le I\_2 < 0.18~\rm counts~s^{-1}$, $0.18\le I\_3 < 0.23~\rm counts~s^{-1}$, $0.23\le I\_4 < 0.29~\rm counts~s^{-1}$, and $ I\_5\ge 0.29~\rm counts~s^{-1}$, where the count rate cuts were set to produce a roughly equal number of counts per count rate range, and the source and background counts were collected in three wavebands: hard (1–8 Å), medium (8–11 Å), and soft (11–26 Å), where the wavelength cuts were set to produce a roughly equal number of counts per wavelength interval. The background-subtracted soft S over hard H versus the total (1–26 Å) count rates are plotted in Figure 2. The dotted line in that figure is the best-fitting constant function S/H = 0.953 for the first three data points (count rate ranges I1–I3). As shown by the figure, the softness ratio of the next higher count rate range I4 is consistent with the lower ranges, while that of the highest count rate range I5 is significantly (3.6σ) less. Consistent with the result of, we find that the X-ray spectrum of the flares of AE Aqr is harder than that in quiescence only at the peak of the flares. Spin-phase light curve ---------------------- Figure 3a shows the background-subtracted spin-phase count rate light curve derived from the HETG event data. It is well fit (χ2 per degree of freedom ≡ χν2 = 7.26/7 = 1.04) by the cosine function $A + B\,\cos 2\pi (\phi\_{\rm spin}- \phi\_0)$, with mean count rate $A=0.191\pm 0.002~\rm counts~s^{-1}$, semi-amplitude $B=0.031 \pm 0.002~\rm counts~s^{-1}$ (hence, relative X-ray pulse semi-amplitude B/A = 16% ± 1%) and, consistent with the updated spin ephemeris, phase offset ϕ0 = 0.00 ± 0.01 (throughout the paper, errors are 1σ or 68% confidence for one free parameter). The above result applies to the observation-average spin-phase light curve, although it is of interest to determine if the X-ray pulse semi-amplitude B and/or the relative semi-amplitude B/A varies with the mean count rate A. As for the investigation of the softness ratio variations, this is best determined as a function of time, using a time resolution sufficient to resolve the flares, but this is not possible with the relatively low count rate HETG data. Instead, background-subtracted spin-phase count rate light curves were derived from the HETG event data for each of the five count rate ranges defined above, and were then fit with the cosine function $A + B\,\cos 2\pi\phi\_{\rm spin}$. The resulting values of the X-ray pulse semi-amplitude B are plotted versus the mean count rate A in Figure 4. The figure demonstrates that B increases linearly with A (B/A is constant with A) in the middle of the count rate range, but that it saturates at both the low and high ends of the range; over the full count rate range, B/A is not well fit (χν2 = 5.82/4 = 1.46) by a constant B/A = 0.162 ± 0.012 (dotted line in Fig. 4). Instead, B/A versus A and B versus A are well fit by a linear relation a + bA with, respectively, a = 0.22 ± 0.03 and b = − 0.25 ± 0.15, with χν2 = 2.77/3 = 0.92 (dashed curve in Fig. 4), and a = 0.011 ± 0.006 and b = 0.10 ± 0.03, with χν2 = 2.38/3 = 0.79 (solid line in Fig. 4). To determine if the observed spin-phase flux modulation is due to photoelectric absorption (or some other type of broad-band spectral variability), background-subtracted spin-phase count rate light curves were derived from the HETG event data for each of three wavebands defined above: hard (1–8 Å), medium (8–11 Å), and soft (11–26 Å). Figure 3b shows the ratio of the resulting soft S over hard H spin-phase count rate light curves. The data is well fit (χν2 = 7.20/9 = 0.80) by a constant S/H = 0.916 ± 0.019, which strongly constrains the cause of the observed spin-phase flux modulation. Specifically, if the observed flux modulation is caused by photoelectric absorption, a variation in the neutral hydrogen column density $\Delta N\_{\rm H}\approx 3\times 10^{21}~\rm cm^{-2}$ is required, whereas the essential constancy of the softness ratio light curves requires $\Delta N\_{\rm H}\lesssim 1\times 10^{20}~\rm cm^{-2}$; a factor of 30 times lower. lccccccccc & velocity& σ& & & & $\log n\_{\rm e}$ Element& ($\rm km~s^{-1}$)& (eV)& Lyman α& f& i& r& G = (f + i)/r& Rʹ = f/(i + r)& ($\rm cm^{-3}$) to 0.4inOto 0.4em.& $-75\pm\phn 51$& 1.12 ± 0.09& $2.91\phn\pm 0.27$& $0.35\phn\pm 0.17\phn$& $0.84\phn\pm 0.24\phn$& $1.12\phn\pm 0.26\phn$& 1.05 ± 0.35& 0.178 ± 0.090& 11.36− 0.32+ 0.42 to 0.4inNeto 0.4em.& $+45\pm\phn 53$& 2.02 ± 0.17& 0.999 ± 0.073& 0.206 ± 0.067& 0.200 ± 0.064& 0.622 ± 0.089& 0.65 ± 0.18& 0.251 ± 0.088& 12.30− 0.36+ 0.35 to 0.4inMgto 0.4em.& + 49 ± 104& 3.85 ± 0.60& 0.188 ± 0.019& 0.070 ± 0.021& 0.040 ± 0.020& 0.193 ± 0.024& 0.57 ± 0.17& 0.300 ± 0.099& 13.04− 0.73+ 0.45 to 0.4inSito 0.4em.& + 95 ± 118& 5.50 ± 0.90& 0.185 ± 0.019& 0.055 ± 0.014& 0.028 ± 0.017& 0.205 ± 0.020& 0.41 ± 0.11& 0.235 ± 0.065& 14.14− 0.36+ 0.34 Spectrum -------- Figure 5 shows the background-subtracted count spectrum derived from the HETG event data, using Δλ = 0.05 Å wavelength bins as a compromise between spectral resolution and sign-to-noise ratio. As is typical of unabsorbed cataclysmic variables (CVs), the X-ray spectrum of AE Aqr is that of a multi-temperature thermal plasma, with emission lines of H- and He-like O, Ne, Mg, Si, and S and L-shell –. However, unlike other CVs, and, in particular, unlike other magnetic CVs, the H-like and He-like lines and the “neutral” fluorescent Fe K line are not apparent in the HETG spectrum. The apparent absence of these features in the HETG spectrum places limits on the maximum temperature of the plasma in AE Aqr and the amount of reflection from the surface of the white dwarf, although higher-energy instruments, such as those onboard Suzaku, are better suited to study this portion of the X-ray spectrum. We conducted a quantitative analysis of the X-ray spectrum of AE Aqr in three steps. First, Gaussians were fitted to the strongest emission lines of H- and He-like O, Ne, Mg, and Si to determine their radial velocities, widths, and fluxes. Second, a global model was fitted to the X-ray spectrum to constrain its absorbing column density, emission measure distribution, and elemental abundances. Third, using the emission measure distribution and the flux ratios of the He α forbidden (f), intercombination (i), and recombination (r) lines, constraints are placed on the density of the plasma. ### Line radial velocities, widths, and fluxes The radial velocities, widths, and fluxes above the continuum of the Lyman α emission lines of,,, and and the He α triplets of,,, and were determined by fitting the flux in the MEG spectrum in the immediate vicinity of each emission feature with a constant plus one (Lyman α) or three (He α) Gaussians, employing the ARF and RMF files to account for the effective area of the spectrometer and its Δλ = 0.023 Å [$690~\rm km~s^{-1}$ at 10 Å] FWHM spectral resolution. For each emission feature, the radial velocity was determined relative to the laboratory wavelengths from the Interactive Guide for ATOMDB version 1.3.[2](#fn2) More specifically, the assumed wavelengths for the Lyman α lines are the mean of the wavelengths of the doublets weighted by their relative emissivities (2:1), whereas the wavelengths for the He α intercombination lines are the unweighted means of their component x and y lines. In the fits to the He α triplets, the radial velocities and widths determined from the fits to the corresponding Lyman α lines were assumed, so that, in all cases, the fits had four free parameters. For these fits, unbinned data were employed and the C statistic was used to determine the value of and error on the fit parameters, which are listed in Table 1. As demonstrated in Figure 6, we find that the widths of the Lyman α emission lines increase with the line energy, from σ = 1.1 ± 0.1 eV for to σ = 5.5 ± 0.9 eV for. For comparison, during two flares of AE Aqr observed with the XMM-Newton Reflection Grating Spectrometer (RGS), found σ ≈ 1.2 and 2 eV for the Lyman α emission lines of and, respectively. The trend shown in Figure 6 is well fit (χν2 = 1.43/2 = 0.71) with a linear function a + bE with a = − 0.80 ± 0.29 and b = 2.9 ± 0.4. Consistent with the non-zero intercept a, the line widths are not constant in velocity units, but increase with the line energy: σ = 512 ± 39, 593 ± 51, 784 ± 121, and $822\pm 135~\rm km~s^{-1}$ for,,, and, respectively. In addition to the line widths, there is some evidence that the radial velocities of the Lyman α emission lines increase with the line energy, from $v=-75\pm 51~\rm km~s^{-1}$ for to $v=+95\pm 118~\rm km~s^{-1}$ for. Note, however, that the velocity difference, $\Delta v= 170\pm 128~\rm km~s^{-1}$, differs from zero by just 1.3σ, is a small fraction of the widths of the lines, and is probably affected by systematic effects. If a common radial velocity offset is assumed in the fits of the Lyman α lines, the derived velocity $v=-1\pm 33~\rm km~s^{-1}$. ### Global model To produce a global model of the X-ray spectrum of AE Aqr, ATOMDB IDL version 2.0.0 software[3](#fn3) was used to construct ATOMDB version 1.3.1[4](#fn4) optically-thin thermal plasma X-ray spectral models for the continuum and for the 12 cosmically abundant elements C, N, O, Ne, Mg, Al, Si, S, Ar, Ca, Fe, and Ni at 40 temperatures spaced uniformly in logT [specifically, $\log T ({\rm K})= 5.0, 5.1, 5.2, \ldots, 8.9$]. Using custom IDL software, these spectral eigenvectors were convolved with a Gaussian, to account for the observed widths of the emission lines as a function of temperature, and multiplied by the grating ARFs and RMFs, to account for the spectrometer’s effective area and spectral resolution. The resulting ± first-order MEG and HEG spectral models were then binned to 0.05 Å and coadded. Finally, the observed MEG plus HEG count spectrum (Fig. 5) and whence the spectral models were “grouped” to a minimum of 30 counts per bin so that Gaussian statistics could be used in the fits. Numerous model emission measure (EM) distributions were tested against the data: 1, 2, 3, and 4 single-temperature models, a cut-off power law (dEM/dlogT ∝ Tα for $T\le T\_{\rm c}$), a power law with an exponential cutoff (dEM/dlogT ∝ Tα for $T\le T\_{\rm c}$ and $dEM/d\log T \propto T^\alpha \exp[(T\_{\rm c}-T)/T\_{\rm f}]$ for $T > T\_{\rm c}$), and a Gaussian (dEM/dlogT ∝ exp[ − (logT − logT0)2/2σ2]), all with photoelectric absorption by a neutral column and variable elemental abundances relative to those of. Among these models, the best (if not a particularly good) fit (χν2 = 382.7/233 = 1.64) was achieved with a model with a Gaussian emission measure distribution with a peak temperature $\log T\_0 ({\rm K})=7.16\pm 0.01$, a width σ = 0.48 ± 0.01, an absorbing column density $N\_{\rm H}=(1.0\pm 0.6)\times 10^{20}~\rm cm^{-2}$, an Fe abundance equal to 0.44 ± 0.02 times solar, and the other metal abundances equal to 0.76 ± 0.03 times solar.[5](#fn5) The X-ray spectrum of this model is shown superposed on the data in Figure 7. For an assumed distance d = 100 pc, the total emission measure $EM=8.8\times 10^{53}~\rm cm^{-3}$ and the unabsorbed 0.5–10 keV luminosity $L\_{\rm X}=1.1\times 10^{31}~\rm erg~s^{-1}$; if this luminosity is due to accretion onto the white dwarf, the mass accretion rate $\dot M = L\_{\rm X}{R\_{\rm wd}}/G{M\_{\rm wd}}\approx 7.3\times 10^{13}~\rm g~s^{-1}$. Before leaving this section, it is useful to note that the Gaussian emission measure distribution derived above extends over nearly two orders of magnitude in temperature — from T ≈ 1.6 × 106 K to 1.3 × 108 K at ± 2σ — but peaks at a temperature T ≈ 1.4 × 107 K or 1.2 keV. Such a low temperature is uncharacteristic of CVs, and in particular magnetic CVs: note for instance that AE Aq had the lowest continuum temperature of any magnetic CV observed by ASCA. The characteristic temperatures of magnetic CVs are the shock temperature, $T\_{\rm s}=3\mu m\_{\rm H} G{M\_{\rm wd}}/8k{R\_{\rm wd}}\approx 4.1\times 10^8$ K or 35 keV, applicable for radial free-fall onto the white dwarf, and the blackbody temperature, $T\_{\rm bb}=(L\_{\rm X}/\sigma A)^{1/4}\approx 19\, (f/0.25)^{-1/4}$ kK (where the radiating area $A=4\pi{R\_{\rm wd}}^2 f$ and our choice for the fiducial value of the fractional emitting area f will be justified in §4.1), applicable for deposition of the accretion luminosity in the surface layers of the white dwarf (e.g., bombardment or blobby accretion). The only evidence for plasma at the shock temperature is supplied by the Suzaku X-ray spectrum of AE Aqr, but argued strongly for a nonthermal origin for this emission. The blackbody temperature, on the other hand, is close to the temperature of the hot spots that derived from the maximum entropy maps of the UV pulse profiles of AE Aqr. More on this below. ### Plasma densities We now consider in more detail the forbidden (f), intercombination (i), and recombination (r) line fluxes of the He α triplets of,,, and derived in §2.3.1 and listed in Table 1. As elucidated by ; ; and, these line fluxes can be used to constrain the electron temperature $T\_{\rm e}$ via the G = (f + i)/r flux ratio and the electron density $n\_{\rm e}$ via the R = f/i flux ratio of each He-like ion. Because the errors on the observed R flux ratios are large, we follow and employ the Rʹ = f/(i + r) flux ratio as an electron density diagnostic in the subsequent discussion. The observed G and Rʹ flux ratios and errors are listed in Table 1. To interpret these results, we derived the Rʹ = RG/(1 + R + G) flux ratios from the G and R flux ratios tabulated by for a collisional plasma for,,,, and for $\log T\_{\rm e} ({\rm K})=6.2$, 6.4, 6.6, 6.8, and 7.0, respectively, where the temperatures are those of the peaks of the He-like triplet emissivities weighted by the Gaussian emission measure distribution, determined in the previous section from the global fit to the X-ray spectrum. With the exception of,[6](#fn6) in each case the assumed electron temperature is consistent with the observed G flux ratio. With these assumptions, the theoretical values of Rʹ are shown in the left panels of Figure 8 as a function of $\log n\_{\rm e}$. In addition to the Rʹ flux ratios of,,, and measured from the Chandra HETG spectrum (Table 1), we added to Figure 8 the Rʹ flux ratios of,, and measured by Itoh et al. from the XMM-Newton RGS spectrum of AE Aqr. Figure 8(b) of this paper corrects an error in Figure 5(h) of Itoh et al., which showed the Rʹ flux ratio extending from 0.09 to 0.39, whereas the data in their Table 4 shows that it should extend only to 0.29. With this correction, the RGS- and HETG-derived values of and errors on the Rʹ flux ratio of are nearly identical. This correction, the significantly smaller error range on the Rʹ flux ratio of, and the HETG results for and indicates that, contrary to the conclusion of Itoh et al., the electron density of the plasma in AE Aqr increases with temperature by over three orders of magnitude, from $n\_{\rm e}\approx 6\times 10^{10}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K})\approx 6.2$] to $n\_{\rm e}\approx 1\times 10^{14}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K}) \approx 7.0$]. In addition to electron density, the R and hence the Rʹ flux ratio is affected by photoexcitation, so we must investigate the sensitivity of the Rʹ flux ratios to an external radiation field. To investigate the conditions under which a low-density plasma can masquerade as a high-density plasma, we derived the Rʹ flux ratios from the G and R flux ratios tabulated by for a low-density ($n\_{\rm e}=1~\rm cm^{-3}$) collisional plasma irradiated by a blackbody with temperature $T\_{\rm bb}$ and dilution factor W. Under the assumption that this flux originates from the surface of the white dwarf, $W={1\over 2}\ \{1-[1-({R\_{\rm wd}}/r)]^{1/2}\}$, where r is the distance from the center of the white dwarf, hence $W={1\over 2}$ on the white dwarf surface. In the right panels of Figure 8 we show contours of the observed Rʹ flux ratios (white curves) and 1σ error envelops (colored polygons) of the various He-like ions as a function of $T\_{\rm bb}$ and W. The figure demonstrates that the observed Rʹ flux ratios of,,,, and can be produced in a low-density plasma sitting on the white dwarf surface if the blackbody temperature $T\_{\rm bb}\approx 7$, 10, 14, 18, and 30 kK, respectively, or at higher temperatures at greater distances from the white dwarf. Conversely, the figure gives the allowed range of distances (dilution factors) for each ion for a given blackbody temperature. For example, for $T\_{\rm bb} = 25$ kK, the volume of plasma in which dominates [$\log T ({\rm K})\approx 7.0$] could be on the white dwarf surface, while that of [$\log T ({\rm K})\approx 6.8$] would have to be at $r\approx 3\, {R\_{\rm wd}}$, that of [$\log T ({\rm K})\approx 6.6$] would have to be at $r\approx 20\, {R\_{\rm wd}}$, and so on for the lower Z ions. Radial velocities ----------------- In the next component of our analysis, we used two techniques to search for orbit- and spin-phase radial velocity variations in the X-ray emission lines of AE Aqr. ### Composite line profile technique In the first technique, similar to that employed by, phase-average as well as orbit- and spin-phase resolved composite line profiles were formed by coadding the HETG event data in velocity space v = c(λ − λ0)/λ relative to laboratory wavelengths λ0 from the Interactive Guide for ATOMDB version 1.3. The lines used in this analysis are those labeled in Figures 5 and 7 shortward of 20 Å. For the H-like lines, the wavelengths are the mean of the wavelengths of the doublets weighted by their relative emissivities (2:1), while for the He-like lines, the wavelengths are for the stronger resonance lines. The resulting phase-average composite line profile is shown in Figure 9. Fit with a Gaussian, its offset $v=25\pm 26~\rm km~s^{-1}$ and width $\sigma=712\pm 27~\rm km~s^{-1}$. Applying the composite line profile technique to the orbit-phase resolved composite line profiles results in the radial velocities shown in Figure 10a. Assuming that, like EX Hya, the orbit-phase radial velocities follow the motion of the white dwarf, these data are well fit (χν2 = 1.29/6 = 0.22) by the sine function $$v(\phi\_{\rm orb}) = \gamma + K {1\over\Delta} \int^{\phi\_{\rm orb}+\Delta/2}\_{\phi\_{\rm orb}-\Delta/2} \sin 2\pi \phi\, d\phi,$$ where Δ = 1/8, with $\gamma =30\pm 25~\rm km~s^{-1}$ and $K=12\pm 36~\rm km~s^{-1}$ (solid curve in Fig. 10a), although they are slightly better fit (χν2 = 1.41/7 = 0.20) with a constant $\gamma =29\pm 25~\rm km~s^{-1}$. For future reference, we note that the 1σ, 2σ, and 3σ (Δχ2 = 1.0, 2.71, and 6.63) upper limits to the orbit-phase radial velocity semi-amplitude K = 48, 72, and $105~\rm km~s^{-1}$, respectively. Applying the composite line profile technique to the spin-phase resolved composite line profiles results in the radial velocities shown in Figure 10b. Unlike the orbit-phase radial velocities, these data are not well fit (χν2 = 18.3/7 = 2.6) by a constant, but they are well fit (χν2 = 2.85/5 = 0.57) by the sine function $$v(\phi\_{\rm spin}) = \gamma + K {1\over\Delta} \int^{\phi\_{\rm spin}+\Delta/2}\_{\phi\_{\rm spin}-\Delta/2} \sin 4\pi (\phi-\phi\_0)\, d\phi,$$ where Δ = 1/8, with $\gamma =29\pm 25~\rm km~s^{-1}$, $K=149\pm 38~\rm km~s^{-1}$, and ϕ0 = 0.060 ± 0.021 (solid curve in Fig. 10b). ### Cross correlation technique The composite line profile technique employed above utilizes the strongest isolated emission lines in the HETG spectrum of AE Aqr, ignoring the many weaker and often blended spectral features shown in Figures 5 and 7. In an attempt to reduce the size of the error bars on the derived spin-phase radial velocities, a cross correlation technique was tested. To accomplish this, spin-phase resolved spectra were formed by adding the HETG event data in wavelength space using bins of constant velocity width $\Delta v =100~\rm km~s^{-1}$ (specifically, λ = 3.000, 3.001, 3.002, …25.002 Å). In the absence of an obvious template against which to cross correlate the resulting spin-phase resolved spectra, the spectrum from the first spin phase bin, $\phi\_{\rm spin}=0\pm \Delta/2$, was used as the template. The resulting spin-phase radial velocities, shown in Figure 10c, are very similar to those derived using the composite line technique (Fig. 10b), but the error bars are smaller by a factor of approximately 40%. These data are reasonably well fit (χν2 = 4.75/4 = 1.19) by equation 2 with $\gamma =22\pm 15~\rm km~s^{-1}$, $K=167\pm 23~\rm km~s^{-1}$, and ϕ0 = 0.045 ± 0.012 (solid curve in Fig. 10c). Note that, given the manner in which this result was derived, the γ velocity is now relative that of the template, the spectrum from the first spin phase, for which the composite line profile technique gave a radial velocity $v=-1\pm 57~\rm km~s^{-1}$ (i.e., consistent with zero). Given this fit to the radial velocities of the spin-phase resolved spectra, it is possible to produce a spin-phase average spectrum that accounts for (removes the effect of) the spin-phase radial velocities. This was accomplished by multiplying the wavelengths of the HETG event data by a factor $1-v(\phi\_{\rm spin})/c$, where $v(\phi\_{\rm spin})=\gamma + K\, \sin 4\pi (\phi\_{\rm spin} - \phi\_0)~\rm km~s^{-1}$ with parameters that are equal to the previous best-fit values: $\gamma =22~\rm km~s^{-1}$, $K=167~\rm km~s^{-1}$, and ϕ0 = 0.045. The spin-phase radial velocities derived using the resulting boot-strapped spin-phase average spectrum as the cross correlation template are shown in Figure 10d. They are very similar to the radial velocities derived using the “vanilla” cross correlation technique (Fig. 10c), but the error bars are smaller by a factor of approximately 30%. These data are now not particularly well fit (χν2 = 10.3/5 = 2.1) by equation 2 with $\gamma =15\pm 10~\rm km~s^{-1}$, $K=163\pm 15~\rm km~s^{-1}$, and ϕ0 = 0.022 ± 0.008 (solid curve in Fig. 10d). The deviations from the fit appear to be consistent with a radial velocity amplitude that is larger on the 0–0.5 spin-phase interval and smaller on the 0.5–1 spin-phase interval. Accordingly, equation 2 was modified to allow this additional parameter, and the data are then well fit (χν2 = 0.82/4 = 0.21) with $\gamma =14\pm 10~\rm km~s^{-1}$, $K\_1=206\pm 20~\rm km~s^{-1}$ (valid on $\phi\_{\rm spin}=0$–0.5), $K\_2=120\pm 20~\rm km~s^{-1}$ (valid on $\phi\_{\rm spin}=0.5$–1), and ϕ0 = 0.023 ± 0.008 (dotted curve in Fig. 10d). Summary ======= As summarized below, our long, uninterrupted Chandra HETG observation provides a wealth of details concerning the X-ray emission of AE Aqr: 1. The X-ray light curve is dominated by flares that last between a few hundred and a few thousand seconds, produce increases of up to 3–5 times the baseline count rate (Fig. 1), and are achromatic except near their peaks (Fig. 2); the white dwarf spin-phase X-ray light curve is achromatic and sinusoidal in shape, with a relative semi-amplitude of approximately 16% (Fig. 3); and the X-ray pulse amplitude increases linearly with the mean count rate in the middle of the range, but saturates at both the low and high ends of the range (Fig. 4). 2. The X-ray spectrum is that of an optically thin multi-temperature thermal plasma (Fig. 5); the X-ray emission lines are broad (Fig. 9), with widths that increase with the line energy, from σ ≈ 1 eV ($510~\rm km~s^{-1}$) for to σ ≈ 5.5 eV ($820~\rm km~s^{-1}$) for (Fig. 6); the X-ray spectrum is reasonably well fit by a plasma model with a Gaussian emission measure distribution that peaks at $\log T ({\rm K})=7.16$, has a width σ = 0.48, an Fe abundance equal to 0.44 times solar, and other metal (primarily Ne, Mg, and Si) abundances equal to 0.76 times solar (Fig. 7); and for a distance d = 100 pc, the total emission measure $EM=8.0\times 10^{53}~\rm cm^{-3}$ and the 0.5–10 keV luminosity $L\_{\rm X}= 1.1\times 10^{31}~\rm erg~s^{-1}$. 3. Based on the f/(i + r) flux ratios of the He α triplets of,, measured by Itoh et al. in the XMM-Newton Reflection Grating Spectrometer spectrum, and those of,,, and in the Chandra HETG spectrum, the electron density of the plasma increases with temperature by over three orders of magnitude, from $n\_{\rm e}\approx 6\times 10^{10}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K})\approx 6.2$] to $n\_{\rm e}\approx 1\times 10^{14}~\rm cm^{-3}$ for [$\log T\_{\rm e} ({\rm K})\approx 7.0$] (Table 1 and Fig. 8a–e), and/or the plasma is significantly affected by photoexcitation (Fig. 8f–j). 4. The radial velocity of the X-ray emission lines varies on the white dwarf spin phase, with two oscillations per spin cycle and an amplitude $K\approx 160~\rm km~s^{-1}$ (Fig. 10). Discussion ========== Over the years, two very different models have been proposed for the source of the X-ray emission of AE Aqr. On one hand, based on ROSAT and ASCA data, argued that the X-ray emission, including the flares, must occur close to the white dwarf, so that the gravitational potential energy can heat the X-ray emitting plasma to the observed temperatures. Similarly, based on Ginga and ASCA data, argued that the X-ray emission, both persistent and flare, originates within the white dwarf magnetosphere; came to similar conclusions based on XMM-Newton Optical Monitor (OM) and European Photon Imaging Camera (EPIC) data. On the other hand, based on XMM-Newton RGS data, argued that the f/(i + r) flux ratios of the He α triplets of,, and are consistent with a plasma with an electron density $n\_{\rm e}\sim 10^{11}~\rm cm^{-3}$ and, given the observed emission measure, a linear scale l ≈ (2–3) × 1010 cm. Because this density is orders of magnetic less than the conventional estimate for the post-shock accretion column of a magnetic CV, and because this linear scale is much larger than the radius of the white dwarf, these authors argued that the optically thin X-ray-emitting plasma in AE Aqr is due not to accretion onto the white dwarf, but to blobs in the accretion stream, heated to X-ray emitting temperatures by the propeller action of the white dwarf magnetic field. has taken issue with some of the details of this model, arguing that the detected X-rays are due to either (1) a tenuous component of the accretion stream or (2) plasma “outside the system,” heated by accelerated particles and/or MHD waves due to a pulsar-like mechanism powered by the spin-down of the magnetic white dwarf. The presence of non-thermal particles in AE Aqr is supported by the observed TeV γ-rays and the recent discovery by of a hard, possibly power-law, component in the Suzaku X-ray spectrum of AE Aqr. Finally, have proposed that the observed unpulsed X-ray emission in AE Aqr is the result	arxiv_0000413
Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors and Fibrations ===================================================================================================== This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and manipulate abstract quivers, categories, groupoids, diagrams, functors and natural transformations, and to perform a vast array of automated abstract algebraic computations using (arbitrary combinations of) the above structures; to manipulate and abstractly reason about arbitrary universal properties, including products, coproducts, pullbacks, pushouts, limits and colimits; and to manipulate, visualize and compute with strict (symmetric) monoidal categories, including full support for automated string diagram rewriting and diagrammatic theorem-proving. In so doing, Categorica combines the capabilities of an abstract computer algebra framework (thus allowing one to compute directly with epimorphisms, monomorphisms, retractions, sections, spans, cospans, fibrations, etc.) with those of a powerful automated theorem-proving system (thus allowing one to convert universal properties and other abstract constructions into (higher-order) equational logic statements that can be reasoned about and proved using standard automated theorem-proving methods, as well as to prove category-theoretic statements directly using purely diagrammatic methods). Internally, Categorica relies upon state-of-the-art graph and hypergraph rewriting algorithms for its automated reasoning capabilities, and is able to convert seamlessly between diagrammatic/combinatorial reasoning on labeled graph representations and symbolic/abstract reasoning on underlying algebraic representations of all category-theoretic concepts. In this first of two articles introducing the design of the framework, we shall focus principally upon its handling of quivers, categories, diagrams, groupoids, functors and natural transformations, including demonstrations of both its algebraic manipulation and theorem-proving capabilities in each case. Introduction ============ As a field of pure mathematics, category theory emerged in the mid-20th century with the work of Eilenberg and Mac Lane on group theory and algebraic topology, as well as the work of Serre and Grothendieck in the closely related area of homological algebra. When conceived as an alternative foundation for mathematics (as exemplified by work on elementary topos theory within mathematical logic), the shift from set theory to category theory necessitates a certain shift in philosophical perspective, wherein one transitions away from identifying mathematical objects based upon their internal structure (such as in Zermelo-Fraenkel set theory, where the axiom of extensionality defines set equality based purely upon the set membership relation), and towards identifying mathematical objects based upon their relationships to other objects of the same “type” (as exemplified by the Yoneda lemma, arguably the most fundamental result in category theory). This more relational perspective on the nature of structure turns out to be a useful lens through which to view many fields, including many outside of mathematics altogether. For instance, in the foundations of quantum mechanics, moving from set-theoretic to category-theoretic foundations inevitably changes one’s view from quantum systems and their states as being the fundamental objects of study (as in the traditional Dirac-von Neumann axioms of quantum mechanics, represented in terms of operator theory on Hilbert space) to quantum processes and their compositions as being the fundamental objects of study (as in the categorical quantum mechanics formalism of Abramsky and Coecke, later developed into a fully diagrammatic theory of quantum information by Coecke and Duncan). Likewise, in computer science, one is able to move from a set-theoretic view in which program state is treated as fundamental (as in the Turing machine picture of computation, and as exemplified by the imperative programming paradigm) to a category-theoretic view in which functions and their compositions are treated as fundamental (as in the λ-calculus picture of computation, and as exemplified by the functional programming paradigm). Such process-theoretic models, in which processes and their algebra of composition are treated on a fundamentally “higher” footing than states and their internal structure, have also proved useful and/or instructive for the foundations of natural language processing, cybernetics, machine learning, complex networks and control theory, computational complexity theory (based on prior work on computational irreducibility), database systems, and many other domains of knowledge. This general process of applying foundational methods and concepts from category theory to fields outside of pure mathematics has come to be known as applied category theory. With the advent and maturation of so many fruitful domains of applied category theory, many software tools have now been developed to facilitate working with category-theoretic data structures in an automated or semi-automated fashion, such as [Catlab.jl](https://algebraicjulia.github.io/Catlab.jl/dev/), built on top of the Julia programming language, which uses the formalism of generalized algebraic theories and dependent types to facilitate automated algebraic manipulation of certain fundamental data structures such as operads and symmetric monoidal categories. There are also many examples of diagrammatic proof assistants, such as the highly general Quantomatic and Cartographer frameworks, or the more specialized PyZX framework, which facilitate the automated rewriting of string diagrams in symmetric monoidal category theory (with, in the latter case, a particular emphasis on diagrammatic quantum information theory). There are even projects, such as the ANR [CoREACT](https://coreact.wiki) project, which aim to formalize certain key aspects of applied category theory, such as compositional/categorical rewriting theory, within existing proof assistant frameworks such as Coq. The aim of the present article is to introduce another such software framework, known as Categorica, which is written principally in the Wolfram Language and is designed to be fully integrated into the Mathematica software system, and which seeks to combine many of the computational algebraic capabilities of existing frameworks such as [Catlab](https://algebraicjulia.github.io/Catlab.jl/dev/) with many of the diagrammatic theorem-proving capabilities of frameworks such as Quantomatic. At its core, the Categorica framework consists of a unified collection of advanced symbolic and diagrammatic algorithms for efficiently converting between purely algebraic representations of categories, diagrams, functors and other key category-theoretic constructions (by employing an entirely presentation-theoretic view of categories, with arrows within an underlying quiver acting as generators, composition acting as the fundamental binary operation, and algebraic equivalences between objects and morphisms acting as relations) and purely graph-theoretic/combinatorial representations of the very same structures (by employing a description of quivers, categories and functors in terms of graphs, hypergraphs and combinatorial rewriting systems). In this manner, Categorica is able to combine the features of a diagrammatic theorem-prover, a higher-order (equational) logic theorem prover, a (hyper)graph rewriting framework, and an abstract computer algebra system, all in an entirely seamless fashion, automatically adopting the most appropriate algorithmic approach and most efficient concrete data structures for any particular problem, and then converting the result back into the desired abstract representation at the end. The first in this series of two articles introducing the design of Categorica will focus primarily on the core algebraic structures of the framework (with a particular emphasis on Categorica’s representation and handling of quivers, categories, functors and natural transformations), some simple diagrammatic theorem-proving capabilities (such as Categorica’s ability to derive and prove necessary and sufficient algebraic conditions for diagrams to commute, for categories to be groupoids, for functors to be faithful, etc.), as well as some more advanced abstract algebraic capabilities (such as the computation of retractions and sections, epimorphisms and monomorphisms, initial and terminal objects, constant and coconstant morphisms, injectivity/surjectivity of functors on both objects and morphisms, and even some simple cases of Grothendieck fibrations). The forthcoming second article in the series will focus upon the extension of these capabilities to the handling of universal properties (especially products and coproducts, pullbacks and pushouts, and limits and colimits), including illustrations both of how to prove theorems using them, and of how to perform algebraic computations involving them, as well as the applications of these methods to the handling of (strict, symmetric) monoidal categories, including capabilities for string diagram representation, manipulation and rewriting. Categorica’s core algorithmic framework is based on the hypergraph rewriting/Wolfram model formalism outlined within, which may be augmented with basic techniques from the theory of graph grammars, and algebraic/compositional graph rewriting theory (and, in particular, the theory of double-pushout rewriting) in order to develop a highly efficient (hyper)graphical/diagrammatic theorem-proving system. We begin in Section [sec:Section1] by introducing the fundamental data structure underlying all abstract category-theoretic objects in Categorica, namely the abstract quiver (a directed multigraph whose vertices are objects and whose edges are arrows), and we show how an `AbstractQuiver` object in Categorica may be used in order to “freely generate” a corresponding `AbstractCategory` object by allowing for certain arrows to be composed (at which point they become morphisms), as well as introducing identity morphisms for each object, in a manner that is consistent with both associativity and identity axioms. We demonstrate how Categorica is able to keep track of all necessary algebraic equivalences between morphisms (as necessitated by the underlying axioms of category theory) automatically, as well as how new algebraic relations between objects and arrows/morphisms may be introduced, in order to construct more general examples non-free categories. We also highlight a very simple application of Categorica’s automated theorem-proving capabilities, by proving that certain categorical diagrams commute (i.e. that, within certain categories, all directed paths yield the same morphism up to algebraic equivalence), and by automatically computing both necessary and sufficient algebraic conditions to force non-commuting categorical diagrams to commute, using the `AbstractCategory` function. In Section [sec:Section2], we proceed to show some examples of simple algebraic computations that can be performed using the `AbstractCategory` function, such as the computation of monomorphisms, epimorphisms and bimorphisms (i.e. the category-theoretic generalizations of injective, surjective and bijective functions in mathematical analysis), and the computation of sections, retractions and isomorphisms (i.e. the category-theoretic generalizations of left-invertible, right-invertible and invertible elements in abstract algebra). We introduce a special kind of category known as a groupoid, in which all morphisms are isomorphisms, and we show how Categorica’s automated theorem-proving capabilities can again be used in order to prove that certain `AbstractCategory` objects are or are not groupoids, and indeed to compute necessary and sufficient algebraic conditions (whenever they exist) to force non-groupoidal `AbstractCategory` objects to become groupoidal. The relationships between monomorphisms vs. epimorphisms and retractions vs. sections are both examples of a far more general category-theoretic notion of duality: the dual of a particular construction can be obtained by simply reversing the direction of morphisms, and therefore reversing the order in which morphisms are composed. Categorica has in-built functionality (via the “DualCategory” property in particular) for making the systematic exploration of category-theoretic dualities completely straightforward, and we illustrate this by investigating some other common examples of dual constructions using Categorica, including initial vs. terminal objects (i.e. the category-theoretic generalizations of bottom and top elements in partially ordered sets), strict initial vs. strict terminal objects (i.e. initial and terminal objects whose incoming/outgoing morphisms are all isomorphisms), and constant vs. coconstant morphisms (i.e. the category-theoretic generalizations of constant functions and zero-maps in mathematical analysis). In Section [sec:Section3], we move on to explore functors (i.e. homomorphisms/structure-preserving maps between categories) using the `AbstractFunctor` function, demonstrating Categorica’s capabilities for handling both covariant and contravariant functors (i.e. functors in which morphism directions are preserved and reversed, respectively). We illustrate how Categorica is able to distinguish between different notions of injectivity, surjectivity and bijectivity for `AbstractFunctor` objects, including injectivity, surjectivity and bijectivity on objects; essential injectivity, essential surjectivity and essential bijectivity (i.e. injectivity, surjectivity and bijectivity on objects, but only up to isomorphism); and faithfulness, fullness and full faithfulness (i.e. injectivity, surjectivity and bijectivity on morphisms). We also show Categorica may be used to investigate an important class of functors known as Grothendieck fibrations (i.e. the category-theoretic generalizations of fiber bundles in topology) from a total category to a base category, including special cases such as discrete fibrations (i.e. Grothendieck fibrations in which the fiber categories are all discrete categories containing only objects and identity morphisms). Finally, in Section [sec:Section4], we introduce the `AbstractNaturalTransformation` function, allowing one to represent arbitrary natural transformations between `AbstractFunctor` objects in Categorica, and showcase its ability to compute (and prove) both necessary and sufficient algebraic conditions to force transformations between functors to be natural. We demonstrate furthermore how the `AbstractNaturalTransformation` framework may be used to detect natural isomorphisms between both objects and functors in a highly general way. We also include a short note regarding some of the internal algorithmic details of the Categorica system, and, in particular, its use of double-pushout rewriting formalism and compositional graph rewriting theory to reduce abstract algebraic and diagrammatic reasoning problems to concrete (hyper)graph rewriting problems. We conclude in Section [sec:Section5] with a summary of directions for future research and development. The majority of the core Categorica functionality presented within this article, and its forthcoming companion article, has been fully documented and exposed via the Wolfram Function Repository, including functions such as [`AbstractQuiver`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractQuiver/), [`AbstractCategory`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractCategory/), [`AbstractFunctor`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractFunctor/), [`AbstractProduct`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractProduct/), [`AbstractCoproduct`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractCoproduct/), [`AbstractPullback`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractPullback/), [`AbstractPushout`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractPushout/) and [`AbstractStrictMonoidalCategory`](https://resources.wolframcloud.com/FunctionRepository/resources/AbstractStrictMonoidalCategory/). However, there is also a significant amount of functionality that has not yet been documented and/or exposed, but which is nevertheless available through the Categorica [GitHub Repository](http://github.com/JonathanGorard/Categorica) (with the overall framework currently consisting of over 20,000 lines of symbolic, high-level Wolfram Language code). All explicit examples of categories C presented within this article are small, in the sense that the object sets ob(C) and morphism sets hom(C) are both actually sets, rather than proper classes. Indeed, the only categories currently supported explicitly by Categorica are strictly finite, which enables us to bypass any considerations of set-theoretic foundations. Quivers, Categories and Diagrams ================================ Every category represented within the Categorica framework has, as its underlying “skeleton”, a directed multigraph called a quiver (i.e. a collection of vertices called objects, connected by directed edges called arrows, which can have arbitrary multiplicity). One important terminological convention that is enforced consistently throughout the Categorica framework is that the directed edges in a quiver are always known as arrows, whereas the directed edges in the corresponding category that is generated by that quiver (potentially freely, potentially with additional algebraic structure) are always known as morphisms. By and large, most conventions regarding the notation and nomenclature used within the Categorica framework have been chosen to reflect the conventions set within Mac Lane’s classic treatise on the subject. If Q is an arbitrary quiver, then we shall denote the set of objects/nodes in Q by ob(Q), and the set of directed edges/arrows in Q by arr(Q). The rules for how to generate a category C (whose set of objects is denoted ob(C) and whose set of morphisms is denoted hom(C), by analogy to sets of homomorphisms in abstract algebra) from the underlying quiver Q are then very straightforward: firstly, for every object X or arrow f : X → Y in the quiver Q, there is a corresponding object X or morphism f : X → Y in the category C: ∀X ∈ ob(Q), ∃X ∈ ob(C), and ∀(f : X → Y) ∈ arr(Q), ∃(f : X → Y) ∈ hom(C); secondly, for every object X in the quiver Q, there exists a corresponding morphism idX : X → X in the category C mapping that object to itself: ∀X ∈ ob(Q), ∃(idX : X → X) ∈ hom(C); and, thirdly, for every pair of morphisms f : X → Y and g : Y → Z in the category C where the codomain of the first morphism matches the domain of the second, i.e. codom(f : X → Y) = dom(g : Y → Z) (in this case, for instance, the domain of the arrow/morphism f : X → Y is X, i.e. dom(f : X → Y) = X, and its codomain is Y, i.e. codom(f : X → Y) = Y), there exists a third morphism g ∘ f : X → Z mapping from the domain of the first morphism to the codomain of the second: ∀(f : X → Y), (g : Y → Z) ∈ hom(C), ∃(g ∘ f : X → Z) ∈ hom(C), or, slightly more abstractly (omitting the arbitrary object labels X, Y and Z): $$\begin{gathered} \forall f, g \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ such that } \qquad \mathrm{codom} \left( f \right) = \mathrm{dom} \left( g \right),\\ \exists \left( g \circ f : \mathrm{dom} \left( f \right) \to \mathrm{codom} \left( g \right) \right) \in \mathrm{hom} \left( \mathcal{C} \right).\end{gathered}$$ The resulting category C is known as the free category generated by the quiver Q (with the arrows of Q acting as the generators, in the algebraic sense, of the morphisms of C). This construction can be illustrated diagrammatically by means of the following minimal example: $$\begin{tikzcd} & Y \arrow[dr, "g"] &\\ X \arrow[ur, "f"] & & Z \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop left, "id\_X"] & & Z \arrow[loop right, "id\_Z"] \end{tikzcd}.$$ As shown in Figure [fig:Figure1], the `AbstractQuiver` and `AbstractCategory` functions in Categorica allow one to represent arbitrary (finite) quivers and categories within the language, with both quivers and free categories being represented internally as lists of objects and associations of arrows/morphisms between those objects (which are then trivially interconvertible with the corresponding combinatorial representations of these structures as labeled directed graphs); every `AbstractCategory` object comes equipped with an associated `AbstractQuiver` object representing its underlying “skeletal” structure. ![image](CategoricaFigure1)![image](CategoricaFigure2) [fig:Figure1] Within the construction of the free category C described above, the morphisms idX : X → X that were introduced for each object X are known as identity morphisms on X, and the abstract binary operation: ∘ : hom(C) × hom(C) → hom(C), that appears in the definition of the morphism g ∘ f : X → Z is known as the composition operation (such that the resulting morphism g ∘ f : X → Z may be referred to as the composition of morphisms f : X → Y and g : Y → Z). The identity morphisms idX : X → X must satisfy the requisite axioms to act as both left and right identities under the operation of morphism composition, such that if f : X → Y is a morphism in the category C, then f ∘ idX : X → Y should be the same as f : X → Y (i.e. idX acts as a right identity on f): ∀(f : X → Y) ∈ hom(C), (f ∘ idX : X → Y) = (f : X → Y), or, illustrated diagrammatically: $$\begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[rr, bend right, swap, "f \circ id\_X"] \arrow[loop left, "id\_X"] & & Y \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X \arrow[rr, "f = f \circ id\_X"] \arrow[loop left, "id\_X"] & & Y \end{tikzcd},$$ and, similarly, idY ∘ f : X → Y should be the same as f : X → Y (i.e. idY acts as a left identity on f): ∀(f : X → Y) ∈ hom(C), (idY ∘ f : X → Y) = (f : X → Y), or, illustrated diagrammatically: $$\begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[rr, bend right, swap, "id\_Y \circ f"] & & Y \arrow[loop right, "id\_Y"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X \arrow[rr, "f = id\_Y \circ f"] & & Y \arrow[loop right, "id\_Y"] \end{tikzcd}.$$ Additionally, the operation of morphism composition itself must satisfy the axiom of associativity, such that if f : X → Y, g : Y → Z and h : Z → W form a set of three composable morphisms in the category C, then the composition (h ∘ g) ∘ f : X → W should be the same as the composition h ∘ (g ∘ f) : X → W: $$\begin{gathered} \forall \left( f : X \to Y \right), \left( g : Y \to Z \right), \left( h : Z \to W \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \left( \left( h \circ g \right) \circ f : X \to W \right) = \left( h \circ \left( g \circ f \right) : X \to W \right),\end{gathered}$$ or, illustrated diagrammatically: $$\begin{tikzcd} & Y \arrow[r, "g"] & Z \arrow[dr, "h"] &\\ X \arrow[ur, "f"] \arrow[rrr, "\left( h \circ g \right) \circ f"] \arrow[rrr, bend right, swap, "h \circ \left( g \circ f \right)"] & & & W \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[r, "g"] & Z \arrow[dr, "h"] &\\ X \arrow[ur, "f"] \arrow[rrr, swap, "\left( h \circ g \right) \circ f = h \circ \left( g \circ f \right)"] & & & W \end{tikzcd}.$$ Categorica automatically keeps track of all algebraic equivalences between morphisms that must be imposed in order to maintain consistency with these identity and associativity axioms, as shown in Figure [fig:Figure2] for the case of two relatively simple `AbstractCategory` objects. Note that the `CircleDot[ {\dots} ]` and `OverTilde[ {\dots} ]` representations of the composition operation ∘ and the identity morphisms idX, respectively, are simply the defaults chosen by Categorica, and can be overridden simply by passing additional arguments to `AbstractCategory`. ![image](CategoricaFigure3)![image](CategoricaFigure4) [fig:Figure2] Although we have considered only free categories so far, it is also possible to construct non-free categories which possess additional algebraic structure in the form of additional equivalences between their objects and morphisms (beyond simply the minimal algebraic equivalences necessitated by the axioms of category theory). For instance, we could consider imposing the equivalence X = Y between objects X and Y in the following simple category: $$\begin{tikzcd} & Y \arrow[dr, "g"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] & & Y \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X = Y \arrow[rr, bend left, "g"] \arrow[rr, bend right, swap, "g \circ f"] \arrow[loop left, "f"] & & Z \end{tikzcd},$$ or the equivalence (f1 : X → Y) = (f2 : X → Y) between morphisms f1 : X → Y and f2 : X → Y in the following, slightly different, simple category: $$\begin{tikzcd} & Y \arrow[dr, "g"] &\\ X \arrow[ur, swap, "f\_1"] \arrow[ur, bend left, "f\_2"] \arrow[rr, swap, "g \circ f\_1"] \arrow[rr, swap, bend right, "g \circ f\_2"] & & Z \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[dr, "g"] &\\ X \arrow[ur, "f\_1 = f\_2"] \arrow[rr, swap, "g \circ f\_1 = g \circ f\_2"] & & Z \end{tikzcd}.$$ Note that we are only permitted to impose algebraic equivalences between morphisms whose domains and codomains both match, i.e., abstractly, using the functions dom : hom(C) → ob(C) and codom : hom(C) → ob(C) introduced above, we have: $$\forall f, g \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \implies \qquad \mathrm{dom} \left( f \right) = \mathrm{dom} \left( g \right), \qquad \text{ and } \qquad \mathrm{codom} \left( f \right) = \mathrm{codom} \left( g \right).$$ Note also that an algebraic equivalence between morphisms, such as (f1 : X → Y) = (f2 : X → Y) in the above, will, in general, automatically imply other algebraic equivalences between morphisms, such as: (g ∘ f1 : X → Z) = (g ∘ f2 : X → Z), as shown in the corresponding diagram, although in this particular case the converse does not necessarily hold, as we shall see in more detail later. In Figure [fig:Figure3], we see the `AbstractCategory` objects corresponding to the two examples presented above (with object equivalence X = Y and morphism equivalence (f1 : X → Y) = (f2 : X → Y) specified, respectively), showing the directed graph representations of the corresponding categories both with and without algebraic equivalences imposed. One of the general design principles of Categorica is that the keyword “Reduced” within property names such as “ReducedLabeledGraph” is used to indicate that all known algebraic equivalences should be applied (as opposed to the keyword “Full” within property names such as “FullLabeledGraph”, which is used to indicate that no algebraic equivalences should be applied, and that the full, “maximally-unreduced” algebraic structure should be presented instead). There is also a related keyword “Simple” (as in the property name “SimpleLabeledGraph”), which can be used to remove all self-loops and multi-edges from the corresponding directed graph representation of any quiver, category, functor, etc. (for instance in order to yield a more presentable form of a particular diagrammatic representation, of the kind that might be shown within an academic paper). Although this automatic simplification feature is frequently very useful when performing practical diagrammatic manipulations, since the purpose of the present article is to be completely explicit and to illustrate (in as much detail as necessary) the underlying design and algorithmic underpinnings of the Categorica framework, we shall not make use of it here. ![image](CategoricaFigure5)![image](CategoricaFigure6) [fig:Figure3] One of the most common applications of non-free categories and algebraic equivalences between morphisms, occurring throughout many fields of mathematics (and especially in abstract and homological algebra), is the formal treatment of commutative diagrams. Indeed, it is often said that commutative diagrams play the same role in category theory and abstract/homological algebra that equations play in more elementary algebra. For instance, when one says that the following diagram commutes: $$\begin{tikzcd} X \arrow[rr, "f"] \arrow[dd, swap, "g"] & & Y \arrow[dd, "i"]\\ \\ Z \arrow[rr, swap, "h"] & & W \end{tikzcd},$$ one is effectively imposing the following equivalence between morphisms: (h ∘ g : X → W) = (i ∘ f : X → W), or, in diagrammatic form, one has: $$\begin{tikzcd} X \arrow[rr, "f"] \arrow[dd, swap, "g"] \arrow[ddrr, bend left = 10, "h \circ g"] \arrow[ddrr, bend right = 10, swap, "i \circ f"] & & Y \arrow[dd, "i"] \\ \\ Z \arrow[rr, swap, "h"] & & W \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X \arrow[rr, "f"] \arrow[dd, swap, "g"] \arrow[ddrr, "h \circ g = i \circ f"] & & Y \arrow[dd, "i"]\\ \\ Z \arrow[rr, swap, "h"] & & W \end{tikzcd}.$$ This condition of commutativity of diagrams may thus be characterized purely combinatorially, namely as the condition that all directed paths through the labeled graph representation of the associated category yield the same morphism (up to algebraic equivalence). Categorica is hence able to fall back to using purely graph-theoretic search algorithms in order to compute the minimum set of morphism equivalences necessary to force this diagram to commute, as well as to prove that these equivalences are indeed sufficient, as shown in Figure [fig:Figure4]. However, although the case of a commutative square is very straightforward, it does not take long before this method of diagram chasing becomes essentially unmanageable for a human algebraist to enact in full detail; for instance, even upon considering the next obvious case, namely a commutative oblong of the form: $$\begin{tikzcd} X\_1 \arrow[rr, "f\_1"] \arrow[dd, swap, "h"] & & Y\_1 \arrow[rr, "g\_1"] \arrow[dd, "i"] & & Z\_1 \arrow[dd, "j"]\\ \\ X\_2 \arrow[rr, swap, "f\_2"] & & Y\_2 \arrow[rr, swap, "g\_2"] & & Z\_2 \end{tikzcd},$$ things have already become quite a bit trickier to analyze than they were for the commutative square, since one not only needs to force the two interior squares to commute: $$\begin{tikzcd} X\_1 \arrow[rr, "f\_1"] \arrow[dd, swap, "h"] \arrow[ddrr, bend left = 10, "i \circ f\_1"] \arrow[ddrr, bend right = 10, swap, "f\_2 \circ h"] & & Y\_1 \arrow[rr, "g\_1"] \arrow[dd, "i"] \arrow[ddrr, bend left = 10, "j \circ g\_1"] \arrow[ddrr, bend right = 10, swap, "g\_2 \circ i"] & & Z\_1 \arrow[dd, "j"]\\ \\ X\_2 \arrow[rr, swap, "f\_2"] & & Y\_2 \arrow[rr, swap, "g\_2"] & & Z\_2 \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X\_1 \arrow[rr, "f\_1"] \arrow[dd, swap, "h"] \arrow[ddrr, "\substack{i \circ f\_1\\ = f\_2 \circ h}"] & & Y\_1 \arrow[rr, "g\_1"] \arrow[dd, "i"] \arrow[ddrr, "\substack{j \circ g\_1\\ = g\_2 \circ i}"] & & Z\_1 \arrow[dd, "j"]\\ \\ X\_2 \arrow[rr, swap, "f\_2"] & & Y\_2 \arrow[rr, swap, "g\_2"] & & Z\_2 \end{tikzcd},$$ i.e: (i ∘ f1 : X1 → Y2) = (f2 ∘ h : X1 → Y2), and (j ∘ g1 : Y1 → Z2) = (g2 ∘ i : Y1 → Z2), but one must also somehow take care of the commutativity of the outer rectangle: $$\begin{tikzcd} X\_1 \arrow[rr, "f\_1"] \arrow[dd, swap, "h"] \arrow[ddrrrr, bend left = 10, "\left( j \circ g\_1 \right) \circ f\_1"] \arrow[ddrrrr, bend right = 10, swap, "\left( g\_2 \circ f\_2 \right) \circ h"] & & Y\_1 \arrow[rr, "g\_1"] & & Z\_1 \arrow[dd, "j"]\\ \\ X\_2 \arrow[rr, swap, "f\_2"] & & Y\_2 \arrow[rr, swap, "g\_2"] & & Z\_2 \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X\_1 \arrow[rr, "f\_1"] \arrow[dd, swap, "h"] \arrow[ddrrrr, "\substack{\left( j \circ g\_1 \right) \circ f\_1\\ = \left( g\_2 \circ f\_2 \right) \circ h}"] & & Y\_1 \arrow[rr, "g\_1"] & & Z\_1 \arrow[dd, "j"]\\ \\ X\_2 \arrow[rr, swap, "f\_2"] & & Y\_2 \arrow[rr, swap, "g\_2"] & & Z\_2 \end{tikzcd},$$ i.e: ((j ∘ g1) ∘ f1 : X1 → Z2) = ((g2 ∘ f2) ∘ h : X1 → Z2). Nevertheless, despite the additional complexity, Categorica is able to handle this case (and, indeed, the case of much larger and more complicated diagrams) in much the same way, as demonstrated in Figure [fig:Figure5]. A fully rigorous definition and treatment of categorical diagrams necessitates the introduction of a certain functor defined over index categories (or schemes), which we shall revisit following our introduction to Categorica’s handling of abstract functor objects in Section [sec:Section3]. Note that `AbstractQuiver` objects in Categorica may also carry algebraic equivalence information on both objects and arrows: these equivalences are then translated into corresponding algebraic equivalences on objects and morphisms upon promotion of the quiver to a full `AbstractCategory` object via the pipeline outlined above. ![image](CategoricaFigure7)![image](CategoricaFigure8) [fig:Figure4] ![image](CategoricaFigure9)![image](CategoricaFigure10) [fig:Figure5] Monos, Epis, Retractions and Sections: The Case of Groupoids ============================================================ The category-theoretic analog of an injective function in mathematical analysis is a monomorphism: a left-cancellative morphism. Specifically, if the morphism f : X → Y in the category C is such that, for all objects Z and all pairs of morphisms g1 : Z → X and g2 : Z → X such that (f ∘ g1 : Z → Y) = (f ∘ g2 : Z → Y), one is able to “cancel on the left” by f : X → Y to obtain that (g1 : Z → X) = (g2 : Z → X), then f : X → Y is a monomorphism: $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is a monomorphism }\\ \iff \qquad \forall Z \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \forall \left( g\_1 : Z \to X \right), \left( g\_2 : Z \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \left( f \circ g\_1 : Z \to Y \right) = \left( f \circ g\_2 : Z \to Y \right) \qquad \implies \qquad \left( g\_1 : Z \to X \right) = \left( g\_2 : Z \to X \right),\end{gathered}$$ or, illustrated diagrammatically, one has that if: $$\begin{tikzcd} & X \arrow[dr, "f"] &\\ Z \arrow[ur, swap, "g\_1"] \arrow[ur, bend left, "g\_2"] \arrow[rr, swap, "f \circ g\_1"] \arrow[rr, bend right, swap, "f \circ g\_2"] & & Y \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & X \arrow[dr, "f"] &\\ Z \arrow[ur, swap, "g\_1"] \arrow[ur, bend left, "g\_2"] \arrow[rr, swap, "f \circ g\_1 = f \circ g\_2"] & & Y \end{tikzcd},$$ then one also necessarily has: $$\begin{tikzcd} & X \arrow[dr, "f"] &\\ Z \arrow[ur, "g\_1 = g\_2"] \arrow[rr, swap, "f \circ g\_1 = f \circ g\_2"] & & Y \end{tikzcd}.$$ The corresponding dual notion to that of a monomorphism (i.e. the construction obtained by reversing the direction of the morphisms, and hence reversing the order of morphism composition, in the diagrams shown above) is that of an epimorphism: a right-cancellative morphism, and hence the category-theoretic analog of a surjective function in analysis. Specifically, if the morphism f : X → Y in the category C is such that, for all objects Z and all pairs of morphisms g1 : Y → Z and g2 : Y → Z such that (g1 ∘ f : X → Z) = (g2 ∘ f : X → Z), one is able to “cancel on the right” by f : X → Y to obtain that (g1 : Y → Z) = (g2 : Y → Z), then f : X → Y is an epimorphism: $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is an epimorphism }\\ \iff \qquad \forall Z \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \forall \left( g\_1 : Y \to Z \right), \left( g\_2 : Y \to Z \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \left( g\_1 \circ f : X \to Z \right) = \left( g\_2 \circ f : X \to Z \right) \qquad \implies \qquad \left( g\_1 : Y \to Z \right) = \left( g\_2 : Y \to Z \right),\end{gathered}$$ or, illustrated diagrammatically, one has that if: $$\begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f"] \arrow[rr, bend right, swap, "g\_2 \circ f"] & & Z \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f = g\_2 \circ f"] & & Z \end{tikzcd},$$ then one also necessarily has: $$\begin{tikzcd} & Y \arrow[dr, "g\_1 = g\_2"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f = g\_2 \circ f"] & & Z \end{tikzcd}.$$ Figure [fig:Figure6] illustrates the basic diagrammatic setup for both monomorphisms and epimorphisms (commonly abbreviated simply to monos and epis in the category-theoretic literature) in Categorica, and demonstrates in these two cases that all morphisms initially, i.e. in the absence of any further algebraic equivalences, correspond to monomorphisms and epimorphisms, respectively. Figure [fig:Figure7] shows that, by imposing the algebraic equivalence (f ∘ g1 : Z → Y) = (f ∘ g2 : Z → Y) in the former case and (g1 ∘ f : X → Z) = (g2 ∘ f : X → Z) in the latter case, one is able to force the morphism f : X → Y to cease to be a monomorphism in the former example, and to cease to be an epimorphism in the latter example. Finally, Figure [fig:Figure8] demonstrates that one is able to restore the status of morphism f : X → Y as a monomorphism (in the former case) or an epimorphism (in the latter case) by imposing the additional algebraic equivalence (g1 : Z → X) = (g2 : Z → X) (for the monomorphism case) or (g1 : Y → Z) = (g2 : Y → Z) (for the epimorphism case). Note that any morphism that is both a monomorphism and an epimorphism is known as a bimorphism (the category-theoretic analog of a bijective function), and Categorica contains in-built functionality for detecting and manipulating bimorphisms in much the same way. ![image](CategoricaFigure11)![image](CategoricaFigure12) [fig:Figure6] ![image](CategoricaFigure13)![image](CategoricaFigure14) [fig:Figure7] ![image](CategoricaFigure15)![image](CategoricaFigure16) [fig:Figure8] An important special case of monomorphisms are sections: morphisms that possess left inverses (and hence which are, themselves, right inverses of some other morphism). Likewise, an important special case of epimorphisms are retractions: dual to sections, these are morphisms that possess right inverses (and hence which are, themselves, left inverses of some other morphism). Specifically, if the morphisms f : X → Y and g : Y → X in the category C are such that f ∘ g : Y → Y is the identity morphism on Y (i.e. (f ∘ g : Y → Y) = (idY : Y → Y)), then f : X → Y is a retraction of g : Y → X, and g : Y → X is a section of f : X → Y: $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is a retraction }\\ \iff \qquad \exists \left( g : Y \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ such that } \qquad \left( f \circ g : Y \to Y \right) = \left( id\_Y : Y \to Y \right),\end{gathered}$$ or, illustrated diagrammatically, a retraction is a morphism f : X → Y such that one has: $$\begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[loop above, "id\_X"] \arrow[loop below, "g \circ f"] & & Y \arrow[ll, bend left, "g"] \arrow[loop above, "id\_Y"] \arrow[loop below, "f \circ g"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[loop above, "id\_X"] \arrow[loop below, "g \circ f"] & & Y \arrow[ll, bend left, "g"] \arrow[loop right, "f \circ g = id\_Y"] \end{tikzcd}.$$ Similarly, if g ∘ f : X → X is the identity morphism on X (i.e. (g ∘ f : X → X) = (idX : X → X)) then f : X → Y is a section of g : Y → X, and g : Y → X is a retraction of f : X → Y: $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is a section }\\ \iff \qquad \exists \left( g : Y \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ such that } \qquad \left( g \circ f : X \to X \right) = \left( id\_X : X \to X \right),\end{gathered}$$ or, illustrated diagrammatically, a section is a morphism f : X → Y such that one has: $$\begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[loop above, "id\_X"] \arrow[loop below, "g \circ f"] & & Y \arrow[ll, bend left, "g"] \arrow[loop above, "id\_Y"] \arrow[loop below, "f \circ g"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[loop left, "g \circ f = id\_X"] & & Y \arrow[ll, bend left, "g"] \arrow[loop above, "id\_Y"] \arrow[loop below, "f \circ g"] \end{tikzcd}.$$ These category-theoretic notions of retractions and sections are so-named because they naturally generalize the corresponding notions in topology, wherein one considers retractions of topological spaces into subspaces to be continuous maps that preserve all points in the subspace (and for which the corresponding inclusion maps of the subspaces into the original spaces, such that the compositions of the two maps always reduce to the identity maps on the subspaces, would be sections). Figure [fig:Figure9] illustrates the basic diagrammatic setup for both retractions and sections in Categorica, and demonstrates that initially (in the absence of any further algebraic equivalences) no morphisms, with the exception of identity morphisms, are either retractions or sections, and therefore that no morphisms, with the exception of identity morphisms, possess either left or right inverses. However, by imposing the algebraic equivalence (f ∘ g : Y → Y) = (idY : Y → Y), one is able to force the morphism f : X → Y to be a retraction (and hence to possess the morphism g : Y → X as its right inverse), as well as to force the morphism g : Y → X to be a section (and hence to possess the morphism f : X → Y as its left inverse). It is therefore straightforward to prove, using Categorica’s diagrammatic theorem-proving capabilities, that every section is necessarily a monomorphism (i.e. that the existence of a left inverse necessarily implies left-cancellativity) and that every retraction is necessarily an epimorphism (i.e. that the existence of a right inverse necessarily implies right-cancellativity). ![image](CategoricaFigure17)![image](CategoricaFigure18) [fig:Figure9] A morphism that acts as both a retraction and a section (and therefore which both is, and also possesses, a left and right inverse) is known as an isomorphism. Specifically, if the morphism f : X → Y in the category C is such that there exists another morphism g : Y → X such that both g ∘ f : X → X is the identity morphism on X (i.e. (g ∘ f : X → X) = (idX : X → X)) and f ∘ g : Y → Y is the identity morphism on Y (i.e. (f ∘ g : Y → Y) = (idY : Y → Y), then f : X → Y is an isomorphism (as, correspondingly, is g : Y → X): $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is an isomorphism }\\ \iff \qquad \exists \left( g : Y \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ such that } \qquad \left( g \circ f : X \to X \right) = \left( id\_X : X \to X \right)\\ \text{ and } \qquad \left( f \circ g : Y \to Y \right) = \left( id\_Y : Y \to Y \right),\end{gathered}$$ or, illustrated diagrammatically, an isomorphism is a morphism f : X → Y such that one has: $$\begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[loop above, "id\_X"] \arrow[loop below, "g \circ f"] & & Y \arrow[ll, bend left, "g"] \arrow[loop above, "id\_Y"] \arrow[loop below, "f \circ g"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} X \arrow[rr, bend left, "f"] \arrow[loop left, "g \circ f = id\_X"] & & Y \arrow[ll, bend left, "g"] \arrow[loop right, "f \circ g = id\_Y"] \end{tikzcd}.$$ A category in which all morphisms are isomorphisms is known as a groupoid, since the morphisms of a groupoid consisting of a single object trivially form a group under the operation of morphism composition (with the associativity and identity axioms deriving from the underlying axioms of category theory, and the inverse axiom deriving from the existence of both left and right inverses for each morphism), and therefore the morphisms of a groupoid consisting of multiple objects may be thought of as forming an abstract generalization of a group in which the group operation is partial, i.e. only well-defined for particular pairs of elements. Groupoids are particularly widely-studied in algebraic topology and homotopy theory, wherein the fundamental groupoid of a topological space generalizes the more traditional fundamental group to the case where one does not necessarily fix a single distinguished base point. Much like in the case of commutative diagrams discussed previously, Categorica is able to use purely graph-theoretic algorithms to compute the minimum set of morphism equivalences necessary to force the category shown above to be a groupoid, as well as to prove that these equivalences are indeed sufficient to do so, as shown in Figure [fig:Figure10]. However, once the initial category is even marginally more complex than this, the task of determining the minimum set of algebraic conditions necessary for the category to be groupoidal quickly evolves to be highly non-trivial; for instance, for the case of a three-object category of the form: $$\begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] &\\ X \arrow[ur, swap, "f"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \end{tikzcd},$$ one must consider not only the conditions necessary to force the morphisms f : X → Y and g : Y → Z (and hence also the morphisms f− 1 : Y → X and g− 1 : Z → Y) to be isomorphisms, namely: $$\begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] \arrow[loop above, "f \circ f^{-1}"] \arrow[loop below, "id\_Y"] &\\ X \arrow[ur, swap, "f"] \arrow[loop left, "id\_X"] \arrow[loop below, "f^{-1} \circ f"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] \arrow[loop above, "f \circ f^{-1} = id\_Y"] &\\ X \arrow[ur, swap, "f"] \arrow[loop left, "f^{-1} \circ f = id\_X"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \end{tikzcd},$$ i.e: (f− 1 ∘ f : X → X) = (idX : X → X), and (f ∘ f− 1 : Y → Y) = (idY : Y → Y), and: $$\begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] \arrow[loop above, "g^{-1} \circ g"] \arrow[loop below, "id\_Y"] &\\ X \arrow[ur, swap, "f"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \arrow[loop right, "id\_Z"] \arrow[loop below, "g \circ g^{-1}"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] \arrow[loop above, "g^{-1} \circ g = id\_Y"] &\\ X \arrow[ur, swap, "f"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \arrow[loop right, "g \circ g^{-1} = id\_Z"] \end{tikzcd},$$ i.e: (g− 1 ∘ g : Y → Y) = (idY : Y → Y), and (g ∘ g− 1 : Z → Z) = (idZ : Z → Z), respectively, but also the conditions necessary to force the resulting pair of composite morphisms g ∘ f : X → Z and f− 1 ∘ g− 1 : Z → X to be isomorphisms as well, namely to collapse: $$\begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] &\\ X \arrow[ur, swap, "f"] \arrow[rr, "g \circ f"] \arrow[loop left, "\left( f^{-1} \circ g^{-1} \right) \circ \left( g \circ f \right)"] \arrow[loop below, "id\_X"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \arrow[ll, bend left, "f^{-1} \circ g^{-1}"] \arrow[loop right, "\left( g \circ f \right) \circ \left( f^{-1} \circ g^{-1} \right)"] \arrow[loop below, "id\_Z"] \end{tikzcd},$$ down to: $$\mapsto \qquad \begin{tikzcd} & Y \arrow[dr, swap, "g"] \arrow[dl, bend right, swap, "f^{-1}"] &\\ X \arrow[ur, swap, "f"] \arrow[rr, "g \circ f"] \arrow[loop left, "\left( f^{-1} \circ g^{-1} \right) \circ \left( g \circ f \right) = id\_X"] & & Z \arrow[ul, bend right, swap, "g^{-1}"] \arrow[ll, bend left, "f^{-1} \circ g^{-1}"] \arrow[loop right, "\left( g \circ f \right) \circ \left( f^{-1} \circ g^{-1} \right) = id\_Z"] \end{tikzcd},$$ in addition to any relevant permutations thereof. Nevertheless, the generality of Categorica’s algebraic reasoning algorithms ensures that it is able to handle such cases in exactly the same way, as demonstrated in Figure [fig:Figure11]. ![image](CategoricaFigure19)![image](CategoricaFigure20) [fig:Figure10] ![image](CategoricaFigure21)![image](CategoricaFigure22) [fig:Figure11] Dual constructions of this general kind, such as monomorphisms vs. epimorphisms, retractions vs. sections, etc., are ubiquitous throughout category theory, and can be investigated systematically using the “DualCategory” property of `AbstractCategory` objects in the Categorica framework (e.g. a morphism that registers as a monomorphism within a particular `AbstractCategory` object will register as an epimorphism in the corresponding `AbstractCategory` object returned by the “DualCategory” property, etc.). A very straightforward example of a dual construction is that of initial vs. terminal objects; an initial object X in the category C is an object such that, for every object P (including X itself) in C, there exists a unique outgoing morphism f : X → P: ∀X ∈ ob(C), Xis an initial object ⇔ ∀P ∈ ob(C), ∃!(f : X → P) ∈ hom(C), or, illustrated diagrammatically: $$\begin{tikzcd} X \arrow[rr, dashed, "\exists! f"] & & \forall P \end{tikzcd}.$$ Dually, a terminal object X in the category C is an object such that, for every object P (including X itself) in C, there exists a unique incoming morphism f : P → X: $$\begin{gathered} \forall X \in \mathrm{ob} \left( \mathcal{C} \right), \qquad X \text{ is a terminal object }\\ \iff \qquad \forall P \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \exists! \left( f : P \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right),\end{gathered}$$ or, illustrated diagrammatically: $$\begin{tikzcd} \forall P \arrow[rr, dashed, "\exists! f"] & & X \end{tikzcd}.$$ Initial and terminal objects generalize many key construction in pure mathematics, such as the bottom/minimal and top/maximal elements ⊥ and ⊤ of partially-ordered sets (since, if one considers a category C constructed from a given poset P whose objects are elements of P and where the morphism f : X → Y exists in C if and only if X ≤ Y in P, then the bottom/minimal element ⊥ is an initial object and the top/maximal element ⊤ is a terminal object in the category C), and the empty and point spaces ${\varnothing}$ and \* in topology (since, if one considers the category Top whose objects are topological spaces and whose morphisms are continuous functions between them, then the empty space ${\varnothing}$ is an initial object in Top because there exists a unique continuous function mapping the empty space to any other topological space, and the point space \* is a terminal object in Top because there exists a unique continuous function mapping any topological space to the point space). If we take the simple triangular diagram case that we have investigated previously: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop left, "id\_X"] & & Z \arrow[loop right, "id\_Z"] \end{tikzcd},$$ then it is easy to see that X here is an initial object, and, dually, that Z is a final object, as illustrated by the elementary Categorica implementation shown in Figure [fig:Figure20]. However, a slightly more subtle refinement of the concept of an ordinary initial object is that of a strict initial object, namely an initial object X for which all incoming morphisms f : Q → X (where Q is an arbitrary object in the category C) must be isomorphisms: $$\begin{gathered} \forall X \in \mathrm{ob} \left( \mathcal{C} \right), \qquad X \text{ is a strict initial object }\\ \iff \qquad \forall P \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \exists! \left( f : X \to P \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ and }\\ \forall Q \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \forall \left( g : Q \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \exists \left( g^{-1} : X \to Q \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ such that},\\ \left( g \circ g^{-1} : X \to X \right) = \left( id\_X : X \to X \right), \qquad \text{ and } \qquad \left( g^{-1} \circ g : Q \to Q \right) = \left( id\_Q : Q \to Q \right),\end{gathered}$$ or, illustrated diagrammatically: $$\begin{tikzcd} \forall Q \arrow[rr, bend left, "\forall g"] \arrow[loop left, "g^{-1} \circ g = id\_Q"] & & X \arrow[ll, bend left, dashed, "\exists g^{-1}"] \arrow[rr, dashed, "\exists! f"] \arrow[loop above, "g \circ g^{-1} = id\_X"] & & \forall P \end{tikzcd}.$$ Dually, a strict terminal object is a terminal object X for which all outgoing morphisms f : X → Q (where Q is again an arbitrary object in the category C) must be isomorphisms: $$\begin{gathered} \forall X \in \mathrm{ob} \left( \mathcal{C} \right), \qquad X \text{ is a strict terminal object },\\ \iff \qquad \forall P \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \exists! \left( f : P \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ and }\\ \forall Q \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \forall \left( g : X \to Q \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \exists \left( g^{-1} : Q \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \text{ such that }\\ \left( g \circ g^{-1} : Q \to Q \right) = \left( id\_Q : Q \to Q \right), \qquad \text{ and } \qquad \left( g^{-1} \circ g : X \to X \right) = \left( id\_X : X \to X \right),\end{gathered}$$ or, illustrated diagrammatically: $$\begin{tikzcd} \forall P \arrow[rr, dashed, "\exists! f"] & & X \arrow[rr, bend left, "\forall g"] \arrow[loop above, "g^{-1} \circ g = id\_X"] & & \forall Q \arrow[ll, bend left, dashed, "\exists g^{-1}"] \arrow[loop right, "g \circ g^{-1} = id\_Q"] \end{tikzcd}.$$ ![image](CategoricaFigure39)![image](CategoricaFigure40) [fig:Figure20] Returning again to the simple triangular diagram above, and introducing either a new incoming morphism h : Q → X to the initial object X, or a new outgoing morphism h : Z → Q from the terminal object Z (along with corresponding morphisms h− 1 : X → Q or h− 1 : Q → Z in the reverse directions, respectively), i.e: $$\begin{tikzcd} & & & & & & Y \arrow[ddrr, "g"] \arrow[loop above, "id\_Y"] & &\\ \\ Q \arrow[rrrr, swap, "h"] \arrow[uurrrrrr, bend left, "f \circ h"] \arrow[rrrrrrrr, bend right, swap, "\left( g \circ f \right) \circ h"] \arrow[loop above, "id\_Q"] \arrow[loop left, "h^{-1} \circ h"] & & & & X \arrow[uurr, swap, "f"] \arrow[uurr, bend left, "\left( f \circ h \right) \circ h^{-1}"] \arrow[rrrr, "g \circ f"] \arrow[rrrr, bend right, "\left( g \circ f \right) \circ \left( h \circ h^{-1} \right)"] \arrow[llll, bend right, swap, "h^{-1}"] \arrow[loop above, "id\_X"] \arrow[loop below, "h \circ h^{-1}"] & & & & Z \arrow[loop right, "id\_Z"] \end{tikzcd},$$ or: $$\begin{tikzcd} & & Y \arrow[ddrr, swap, "g"] \arrow[ddrr, bend left, "\left( h^{-1} \circ h \right) \circ g"] \arrow[ddrrrrrr, bend left, "h \circ g"] \arrow[loop above, "id\_Y"] & & & & & &\\ \\ X \arrow[uurr, "f"] \arrow[rrrr, "g \circ f"] \arrow[rrrr, bend right, "\left( h^{-1} \circ h \right) \circ \left( g \circ f \right)"] \arrow[rrrrrrrr, bend right, swap, "\left( h \circ g \right) \circ f"] \arrow[loop left, "id\_X"] & & & & Z \arrow[rrrr, swap, "h"] \arrow[loop above, "id\_Z"] \arrow[loop below, "h^{-1} \circ h"] & & & & Q \arrow[llll, bend right, swap, "h^{-1}"] \arrow[loop above, "id\_Q"] \arrow[loop right, "h \circ h^{-1}"] \end{tikzcd},$$ respectively, we see that, in the first case, X ceases to be a strict initial object (since the incoming morphism h : Q → X is not an isomorphism), and in the second case, Z ceases to be a strict terminal object (since the outgoing morphism h : Z → Q is also not an isomorphism), as demonstrated in Figure [fig:Figure21]. Indeed, X and Z are no longer initial/terminal objects even in the non-strict sense, due to the existence of multiple (currently algebraically inequivalent) outgoing morphisms: $$\begin{gathered} \left( f : X \to Y \right) \neq \left( \left( f \circ h \right) \circ h^{-1} : X \to Y \right),\\ \text{ and } \qquad \left( g \circ f : X \to Z \right) \neq \left( \left( g \circ f \right) \circ \left( h \circ h^{-1} \right) : X \to Z \right),\end{gathered}$$ from X to Y and from X to Z in the former case, and multiple (currently algebraically inequivalent) incoming morphisms: $$\begin{gathered} \left( g : Y \to Z \right) \neq \left( \left( h^{-1} \circ h \right) \circ g : Y \to Z \right)\\ \text{ and } \qquad \left( g \circ f : X \to Z \right) \neq \left( \left( h^{-1} \circ h \right) \circ \left( g \circ f \right) : X \to Z \right),\end{gathered}$$ from Y to Z and from X to Z in the latter case. However, if we now impose the pair of algebraic equivalences: (h ∘ h− 1 : X → X) = (idX : X → X), and (h− 1 ∘ h : Q → Q) = (idQ : Q → Q), thereby ensuring that: $$\begin{gathered} \left( \left( f \circ h \right) \circ h^{-1} : X \to Y \right) = \left( f : X \to Y \right),\\ \text{ and } \qquad \left( \left( g \circ f \right) \circ \left( h \circ h^{-1} \right) : X \to Z \right) = \left( g \circ f : X \to Z \right),\end{gathered}$$ in the former case, and the pair of algebraic equivalences: (h− 1 ∘ h : Z → Z) = (idZ : Z → Z), and (h ∘ h− 1 : Q → Q) = (idQ : Q → Q), thereby ensuring that: $$\begin{gathered} \left( \left( h^{-1} \circ h \right) \circ g : Y \to Z \right) = \left( g : Y \to Z \right),\\ \text{ and } \qquad \left( \left( h^{-1} \circ h \right) \circ \left( g \circ f \right) : X \to Z \right) = \left( g \circ f : X \to Z \right),\end{gathered}$$ in the latter case, then morphisms h : Q → X and h : Z → Q (as well as their corresponding reverse morphisms h− 1 : X → Q and h− 1 : Q → Z, respectively) now become isomorphisms, collapsing the above diagrams to: $$\begin{tikzcd} & & & & & & Y \arrow[ddrr, "g"] \arrow[loop above, "id\_Y"] & &\\ \\ Q \arrow[rrrr, swap, "h"] \arrow[uurrrrrr, bend left, "f \circ h"] \arrow[rrrrrrrr, bend right, swap, "\left( g \circ f \right) \circ h"] \arrow[loop left, "h^{-1} \circ h = id\_Q"] & & & & X \arrow[uurr, "\left( f \circ h \right) \circ h^{-1} = f"] \arrow[rrrr, swap, "\left( g \circ f \right) \circ \left( h \circ h^{-1} \right) = g \circ f"] \arrow[llll, bend right, swap, "h^{-1}"] \arrow[loop below, "h \circ h^{-1} = id\_X"] & & & & Z \arrow[loop right, "id\_Z"] \end{tikzcd},$$ and: $$\begin{tikzcd} & & Y \arrow[ddrr, "\left( h^{-1} \circ h \right) \circ g = g"] \arrow[ddrrrrrr, bend left, "h \circ g"] \arrow[loop above, "id\_Y"] & & & & & &\\ \\ X \arrow[uurr, "f"] \arrow[rrrr, swap, "\left( h^{-1} \circ h \right) \circ \left( g \circ f \right) = g \circ f"] \arrow[rrrrrrrr, bend right, swap, "\left( h \circ g \right) \circ f"] \arrow[loop left, "id\_X"] & & & & Z \arrow[rrrr, swap, "h"] \arrow[loop below, "h^{-1} \circ h = id\_Z"] & & & & Q \arrow[llll, bend right, "h^{-1}"] \arrow[loop right, "h \circ h^{-1} = id\_Q"] \end{tikzcd},$$ respectively, and therefore causing X and Z to be strict initial and terminal objects, respectively, as illustrated in Figure [fig:Figure22]. Any object that is both an initial object and a terminal object simultaneously is known as a zero object (and, by extension, any object that is both a strict initial object and a strict terminal object simultaneously is known as a strict zero object), and shares certain structural features in common with the point space object \* in the category ${\mathbf{Top\_{\}}}$ of pointed* topological spaces (i.e. topological spaces with a distinguished base point). Categories that contain zero objects are consequently referred to as pointed categories. Categorica also has specialized functionality for handling zero objects, strict zero objects and pointed categories built into the `AbstractCategory` function. ![image](CategoricaFigure41)![image](CategoricaFigure42) [fig:Figure21] ![image](CategoricaFigure43)![image](CategoricaFigure44) [fig:Figure22] The final example of a dual construction that we shall discuss within the context of the present section, due to its close relationship with the monomorphism vs. epimorphism duality discussed previously, is that of constant vs. coconstant morphisms. If the morphism f : X → Y in the category C is such that, for all objects Z and all pairs of morphisms g1 : Z → X and g2 : Z → X, one necessarily has that (f ∘ g1 : Z → Y) = (f ∘ g2 : Z → Y), then f : X → Y is a constant morphism: $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is a constant morphism},\\ \iff \qquad \forall Z \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \forall \left( g\_1 : Z \to X \right), \left( g\_2 : Z \to X \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \text{ one has } \qquad \left( f \circ g\_1 : Z \to Y \right) = \left( f \circ g\_2 : Z \to Y \right),\end{gathered}$$ or, illustrated diagrammatically, one has: $$\begin{tikzcd} & X \arrow[dr, "f"] &\\ Z \arrow[ur, swap, "g\_1"] \arrow[ur, bend left, "g\_2"] \arrow[rr, swap, "f \circ g\_1"] \arrow[rr, bend right, swap, "f \circ g\_2"] & & Y \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & X \arrow[dr, "f"] &\\ Z \arrow[ur, swap, "g\_1"] \arrow[ur, bend left, "g\_2"] \arrow[rr, swap, "f \circ g\_1 = f \circ g\_2"] & & Y \end{tikzcd}.$$ Note that this is identical to the diagram that appears in the hypothesis of the definition of a monomorphism (and therefore we may say, slightly loosely, that a monomorphism is a constant morphism for which one necessarily has that (g1 : Z → X) = (g2 : Z → X)). Dually, if the morphism f : X → Y in the category C is such that, for all objects Z and all pairs of morphisms g1 : Y → Z and g2 : Y → Z, one necessarily has that (g1 ∘ f : X → Z) = (g2 ∘ f : X → Z), then f : X → Y is a coconstant morphism: $$\begin{gathered} \forall \left( f : X \to Y \right) \in \mathrm{hom} \left( \mathcal{C} \right), \qquad \left( f : X \to Y \right) \text{ is a coconstant morphism},\\ \iff \qquad \forall Z \in \mathrm{ob} \left( \mathcal{C} \right), \qquad \forall \left( g\_1 : Y \to Z \right), \left( g\_2 : Y \to Z \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \text{ one has } \qquad \left( g\_1 \circ f : X \to Z \right) = \left( g\_2 \circ f : X \to Z \right),\end{gathered}$$ or, illustrated diagrammatically, one has: $$\begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f"] \arrow[rr, bend right, swap, "g\_2 \circ f"] & & Z \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f = g\_2 \circ f"] & & Z \end{tikzcd}.$$ Note that this is, likewise, identical to the diagram that appears in the hypothesis of the definition of an epimorphism (and therefore we may also say, again slightly loosely, that an epimorphism is a coconstant morphism for which one necessarily has that (g1 : Y → Z) = (g2 : Y → Z)). Constant morphisms are so-named because they generalize the notion of constant functions in mathematical analysis (and, accordingly, coconstant morphisms may be thought of as generalizing the notion of zero-maps in analysis); any morphism that is both a constant morphism and a coconstant morphism is known as a zero morphism, since such morphisms share certain key algebraic properties with morphisms mapping into and out of zero objects (by the same token, constant morphisms share certain algebraic properties with morphisms mapping into terminal objects, and coconstant morphisms share certain algebraic properties with morphisms mapping out of initial objects). Figure [fig:Figure23] illustrates the basic diagrammatic setup for both constant and coconstant morphisms in Categorica, and demonstrates that initially (in the absence of any further algebraic equivalences) the morphism f : X → Y is not a constant morphism (in the former case), or not a coconstant morphism (in the latter case), and hence also not a zero morphism (in either case). Figure [fig:Figure24] shows that, by imposing the algebraic equivalence (f ∘ g1 : Z → Y) = (f ∘ g2 : Z → Y) in the former case and (g1 ∘ f : X → Z) = (g2 ∘ f : X → Z) in the latter case, one is able to force the morphism f : X → Y to be a constant morphism in the former example, and to be a coconstant morphism in the latter example, and therefore a zero morphism in both examples. ![image](CategoricaFigure45)![image](CategoricaFigure46) [fig:Figure23] ![image](CategoricaFigure47)![image](CategoricaFigure48) [fig:Figure24] The `AbstractCategory` function in Categorica also contains functionality for detecting and manipulating many other special types of morphism, including endomorphisms (i.e. morphisms mapping an object X to itself) through the “Endomorphisms” property and its variants, automorphisms (i.e. isomorphisms between an object X and itself) through the “Automorphisms” property and its variants, etc. There are also many other special types of category (beyond simply the groupoids and commutative diagrams discussed thus far) that can be automatically detected and represented using Categorica, with `AbstractCategory` in many cases including specialized functionality for manipulating them, including discrete categories (i.e. categories in which the only morphisms are identity morphisms) through the “DiscreteCategoryQ” property, indiscrete categories (i.e. categories in which there exists a unique morphism between every pair of objects) through the “IndiscreteCategoryQ” property, balanced categories (i.e. categories in which every bimorphism is also necessarily an isomorphism) through the “BalancedCategoryQ” property, etc. Needless to say, many additional properties and capabilities are also planned for future development. In many cases these properties have rather elegant mathematical interpretations; for instance, discrete categories may be thought as being a natural category-theoretic generalization of discrete topological spaces (i.e. topological spaces equipped with a discrete topology, in which every subset is open), since in a discrete topological space the only allowable paths are the identity paths from a point to itself, etc. However, in the interests of brevity, we will not cover these additional capabilities in any detail here. Functors and Fibrations ======================= A homomorphism (i.e. a structure-preserving map) between two categories is known as a functor. More precisely, the map F : C → D from a category C to a category D is a functor if and only if it associates every object X in category C to a corresponding object F(X) in category D, i.e: ∀X ∈ ob(C), ∃F(X) ∈ ob(D), and every morphism f : X → Y in category C to a corresponding morphism F(f) : F(X) → F(Y) in category D, i.e: ∀(f : X → Y) ∈ hom(C), ∃(F(f) : F(X) → F(Y)) ∈ hom(D), in such a way that the identity morphisms are all preserved, such that the identity morphism idX : X → X on object X in category C is always mapped to the identity morphism idF(X) : F(X) → F(X) on object F(X) in category D, i.e: ∀X ∈ ob(C), (F(idX) : F(X) → F(X)) = (idF(X) : F(X) → F(X)), and the composition of morphisms is also preserved, such that the composite morphism g ∘ f : X → Z in category C (where f : X → Y and g : Y → Z are also morphisms in C) is always mapped to the composite morphism F(g) ∘ F(f) : F(X) → F(Z) in category D, i.e: $$\begin{gathered} \forall \left( f : X \to Y \right), \left( g : Y \to Z \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \left( F \left( g \circ f \right) : F \left( X \right) \to F \left( Z \right) \right) = \left( F \left( g \right) \circ F \left( f \right) : F \left( X \right) \to F \left( Z \right) \right).\end{gathered}$$ The functor construction may be illustrated diagrammatically by means of the following minimal example: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( g \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Functoriality turns out to be an extremely powerful algebraic condition, and functors may consequently be thought of as generalizing many central constructions in pure mathematics, including group actions in the context of group theory (wherein one considers functors from single-object groupoids, i.e. groups, to the category Set whose objects are sets and whose morphisms are set-valued functions), group representations in the context of representation theory (wherein one again considers functors from groups/single-object groupoids, but now to the category Vect whose objects are vector spaces and whose morphisms are linear maps), and tangent bundles in differential geometry (wherein one considers functors from the category Manp whose objects are differentiable manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps, to the category Vect(Manp) whose objects are vector bundles over the manifolds in Manp and whose morphisms are vector bundle homomorphisms). Indeed, as mentioned previously, even categorical diagrams themselves may be formalized as functors D : J → C from some index category/scheme J to an arbitrary category C. Power sets in set theory (i.e. functors from the category Set of sets and set-valued functions to itself) and fundamental groups in algebraic topology (i.e. functors from the category ${\mathbf{Top\_{\}}}$ of pointed topological spaces and continuous functions between them, to the category Grp* of groups and group homomorphisms) constitute further examples of key functorial constructions in mathematics that have previously been alluded to within this article. Figure [fig:Figure12] shows the implementation of the minimal example presented above, along with a slightly larger example, in the Categorica framework using the `AbstractFunctor` function; much like with the `AbstractCategory` objects themselves, Categorica automatically keeps track of, and enforces, all necessary algebraic equivalences between objects and morphisms within the codomain category of an `AbstractFunctor` object that must be imposed in order to maintain consistency with the functor axioms described above. ![image](CategoricaFigure23)![image](CategoricaFigure24) [fig:Figure12] Although the formal definition presented above corresponds to the case of covariant functors (namely functors which preserve the directions of morphisms, and hence which also preserve the order of composition of morphisms), there also exists a dual notion of contravariant functors, which obey analogous axioms but which reverse the directions of morphisms, and hence also reverse the order of morphism composition. The definition of the mapping on objects remains unchanged from the covariant case, but now every morphism f : X → Y in category C is mapped to a corresponding morphism F(f) : F(Y) → F(X) in category D, i.e: ∀(f : X → Y) ∈ hom(C), ∃(F(f) : F(Y) → F(X)) ∈ hom(D). The condition on identity morphisms is also unchanged from the covariant case, but now the composite morphism g ∘ f : X → Z in category C (where f : X → Y and g : Y → Z are also morphisms in C) is always mapped to the composite morphism F(f) ∘ F(g) : F(Z) → F(X) in category D, i.e: $$\begin{gathered} \forall \left( f : X \to Y \right), \left( g : Y \to Z \right) \in \mathrm{hom} \left( \mathcal{C} \right),\\ \left( F \left( g \circ f \right) : F \left( Z \right) \to F \left( X \right) \right) = \left( F \left( f \right) \circ F \left( g \right) : F \left( Z \right) \to F \left( X \right) \right).\end{gathered}$$ The contravariant functor construction may, just as before, be illustrated diagrammatically by means of the following minimal example: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( f \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( Z \right) \arrow[ur, "F \left( g \right)"] \arrow[rr, swap, "F \left( f \right) \circ F \left( g \right)"] \arrow[loop below, "id\_{F \left( Z \right)}"] & & F \left( X \right) \arrow[loop below, "id\_{F \left( X \right)}"] \end{tikzcd}.$$ Just as covariant functors from the category Manp of differentiable manifolds and differentiable maps to the category Vect(Manp) of vector bundles and vector bundle homomorphisms may be used to generalize the construction of tangent bundles in differential geometry, the construction of cotangent bundles may correspondingly be generalized using the corresponding contravariant functor from Manp (or, more precisely, from its dual/opposite category (Manp)op) to Vect(Manp). Likewise, the operation of taking dual spaces in linear algebra corresponds to the application of a certain contravariant functor from the category Vect(K) of vector spaces and linear maps over some fixed field K (or, more precisely, from its dual/opposite category (Vect(K))op), to itself. Indeed, the power set construction in set theory may in fact be formalized as either a covariant functor P : Set → Set (as above), or as a contravariant functor P : Setop → Set. The notion of presheaves on a space X in algebraic topology may be formalized as a functor from the dual/opposite category Cop of the category C whose objects are open sets U ⊆ X and whose morphisms are inclusion maps U ⊆ V, to Set; when appropriately extended to the case where the category C (and hence the domain category Cop of the contravariant functor) is arbitrary, this yields the corresponding category-theoretic notion of a presheaf, which is far more general. Figure [fig:Figure13] demonstrates a Categorica implementation of the corresponding contravariant versions of the two functors previously shown in Figure [fig:Figure12], obtained in each case by setting the first argument of `AbstractFunctor` (i.e. the covariance argument) to `False`. At any time, the dual version of any `AbstractFunctor` object may be computed directly using the “SwapVariance” property (analogous to the “DualCategory” property of `AbstractCategory` objects), with Categorica automatically making any required modifications to the order of morphism composition within all morphism equivalence lists, etc. ![image](CategoricaFigure25)![image](CategoricaFigure26) [fig:Figure13] In its full form, and in the absence of any additional variance information, an `AbstractFunctor` object is specified by the `AbstractCategory` object for its domain category, an association of object mappings, an association of morphism (or, more precisely, arrow) mappings, a list of new objects in the codomain category, a list of new morphisms (or, more precisely, arrows) in the codomain category, a list of new object equivalences in the codomain category, and a list of new morphism equivalences in the codomain category (as well as optional additional information regarding choices of composition and identity symbols, as with `AbstractCategory`). Note, in particular, that all mappings take place at the level of the underlying `AbstractQuiver` object, rather than on the `AbstractCategory` object itself, so as to guarantee consistency with the composition/functoriality axioms. All algebraic equivalences between objects and arrows/morphisms in the domain `AbstractCategory` object (or its underlying `AbstractQuiver` object) are automatically translated into corresponding algebraic equivalences in the codomain `AbstractCategory` object. We may therefore regard a functor F : C → D as corresponding to a pair of (set-valued, since we are dealing here with small categories) functions, Fob and Fhom, with the former mapping the set of objects in category C to the set of objects in category D and the latter mapping the set of morphisms in category C to the set of morphisms in category D: Fob : ob(C) → ob(D), and Fhom : hom(C) → hom(D). This, in turn, presents (at least) two distinct ways in which functors may be classified as either injective, surjective or bijective; namely, functors may be injective/surjective/bijective on objects, or on morphisms, or on both, or on neither (depending upon the injectivity/surjectivity/bijectivity properties of the functions Fob and Fhom). A simple way in which a functor may fail to be injective on objects is by having two objects in the codomain category be equivalent which were not previously equivalent in the domain category; for instance, consider the functor: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( g \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd},$$ plus the additional algebraic condition that F(X) = F(Y), imposed on the objects in the codomain category D, such that one instead has: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} F \left( X \right) = F \left( Y \right) \arrow[rr, bend left, "F \left( g \right)"] \arrow[rr, bend right, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop above, "id\_{F \left( X \right)} = id\_{F \left( Y \right)}"] \arrow[loop below, "F \left( f \right)"] & & F \left( Z \right) \arrow[loop right, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Such a functor would no longer be injective, and hence also no longer bijective, on objects, although its injectivity (and therefore bijectivity) on objects could nevertheless be restored by introducing a corresponding algebraic condition X = Y on the objects in the domain category C, such that one instead has: $$\begin{tikzcd} X = Y \arrow[rr, bend left, "g"] \arrow[rr, bend right, swap, "g \circ f"] \arrow[loop above, "id\_X = id\_Y"] \arrow[loop below, "f"] & & Z \arrow[loop right, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} F \left( X \right) = F \left( Y \right) \arrow[rr, bend left, "F \left( g \right)"] \arrow[rr, bend right, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop above, "id\_{F \left( X \right)} = id\_{F \left( Y \right)}"] \arrow[loop below, "F \left( f \right)"] & & F \left( Z \right) \arrow[loop right, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Figure [fig:Figure14] implements this basic example in Categorica, demonstrating how functor injectivity on objects may be removed (by imposing the algebraic equivalence F(X) = F(Y) on objects in the codomain category) and then subsequently reinstated (by imposing the additional algebraic equivalence X = Y on objects in the domain category). Dual to this construction, a simple way in which a functor may fail to be surjective on objects is by having a new object be introduced in the codomain category that did not previously exist in the domain category; for instance, consider now the functor: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( g \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] & & P \arrow[loop above, "id\_P"]\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] & \end{tikzcd}.$$ Such a functor would no longer be surjective, and hence also no longer bijective, on objects, although its surjectivity (and therefore bijectivity) on objects could nevertheless be restored by introducing a new algebraic condition P = F(X) on the objects in the codomain category D, such that one instead has: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( g \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) = P \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)} = id\_P"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Figure [fig:Figure15] implements this dual example in Categorica, demonstrating now how functor surjectivity on objects may be removed (by introducing the new object P in the codomain category) and then subsequently reinstated (by imposing the algebraic equivalence P = F(X) on objects in the codomain category), in much the same way. ![image](CategoricaFigure27)![image](CategoricaFigure28) [fig:Figure14] ![image](CategoricaFigure29)![image](CategoricaFigure30) [fig:Figure15] However, there may nevertheless be cases in which a functor is only injective, surjective or bijective on objects up to isomorphism, which in turn motivates the concepts of essential injectivity, essential surjectivity and essential bijectivity. For instance, consider again the example presented above, in which a functor fails to be injective on objects because of an additional algebraic condition F(X) = F(Y) that has been imposed on the objects in the codomain category D: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} F \left( X \right) = F \left( Y \right) \arrow[rr, bend left, "F \left( g \right)"] \arrow[rr, bend right, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop above, "id\_{F \left( X \right)} = id\_{F \left( Y \right)}"] \arrow[loop below, "F \left( f \right)"] & & F \left( Z \right) \arrow[loop right, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Now, rather than imposing the strict algebraic equality X = Y on the objects in the domain category C, suppose instead that one introduces a new inverse morphism f− 1 : Y → X in C, such that: (f ∘ f− 1 : Y → Y) = (idY : Y → Y), and (f− 1 ∘ f : X → X) = (idX : X → X), and therefore the objects X and Y become isomorphic, thus yielding: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[dl, bend right, swap, "f^{-1}"] \arrow[loop above, "f \circ f^{-1} = id\_Y"] &\\ X \arrow[ur, swap, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "f^{-1} \circ f = id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd},$$ in the domain category, which maps to: $$\mapsto \qquad \begin{tikzcd} F \left( X \right) = F \left( Y \right) \arrow[rr, bend left, "F \left( g \right)"] \arrow[rr, bend right, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop above, "id\_{F \left( X \right)} = id\_{F \left( Y \right)}"] \arrow[loop below, "F \left( f \right)"] \arrow[loop left, "F \left( f^{-1} \right)"] & & F \left( Z \right) \arrow[loop right, "id\_{F \left( Z \right)}"] \end{tikzcd},$$ in the codomain category. Then, the resulting functor would still not be strictly injective, nor strictly bijective, on objects, but it would be essentially injective, and hence essentially bijective (i.e. injective/bijective up to isomorphism). Figure [fig:Figure16] implements this example in Categorica, showing how essential injectivity may be reinstated by imposing the isomorphism X ≅ Y on objects in the domain category. Dually, consider again the previous example of a functor that fails to be surjective on objects because of a new object P that has been introduced in the codomain category: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( g \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] & & P \arrow[loop above, "id\_P"]\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Similarly, rather than imposing the strict algebraic equality P = F(X) on the objects in the codomain category D, we can instead introduce a new pair of morphisms i : P → F(X) and j : F(X) → P, such that: $$\begin{gathered} \left( i \circ j : F \left( X \right) \to F \left( X \right) \right) = \left( id\_{F \left( X \right)} : F \left( X \right) \to F \left( X \right) \right),\\ \text{ and } \qquad \left( j \circ i : P \to P \right) = \left( id\_P : P \to P \right),\end{gathered}$$ and therefore force the objects P and F(X) to be isomorphic, hence yielding: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Y \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} P \arrow[dd, "i"] \arrow[dr, bend left = 40, "F \left( f \right) \circ i"] \arrow[ddrr, bend left = 60, "\left( F \left( g \right) \circ F \left( f \right) \right) \circ i"] \arrow[loop above, "j \circ i = id\_P"] & & &\\ & F \left( Y \right) \arrow[dr, "F \left( g \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[uu, bend left, "j"] \arrow[loop below, "i \circ j = id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Now, the resulting functor is still not strictly surjective, nor strictly bijective, on objects, but it is essentially surjective, and hence essentially bijective (i.e. surjective/bijective up to isomorphism). Figure [fig:Figure17] implements this dual example in Categorica, showing how essential surjectivity may be reinstated by imposing the isomorphism P ≅ F(X) on objects in the codomain category. ![image](CategoricaFigure31)![image](CategoricaFigure32) [fig:Figure16] ![image](CategoricaFigure33)![image](CategoricaFigure34) [fig:Figure17] On the other hand, shifting now from considering the properties of the Fob function to those of the Fhom function, functors that are injective, surjective or bijective on morphisms are known as faithful functors, full functors or fully faithful functors, respectively. A simple way in which a functor may fail to be faithful (i.e. fail to be injective on morphisms) is by having two morphisms in the codomain category be equivalent which were not previously equivalent in the domain category; for instance, consider the functor: $$\begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f"] \arrow[rr, bend right, swap, "g\_2 \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, swap, "F \left( g\_1 \right)"] \arrow[dr, bend left, "F \left( g\_2 \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g\_1 \right) \circ F \left( f \right)"] \arrow[rr, bend right, swap, "F \left( g\_2 \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd},$$ plus the additional algebraic condition that: (F(g1) ∘ F(f) : F(X) → F(Z)) = (F(g2) ∘ F(f) : F(X) → F(Z)), on the morphisms in the codomain category D, such that one instead has: $$\begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f"] \arrow[rr, bend right, swap, "g\_2 \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, swap, "F \left( g\_1 \right)"] \arrow[dr, bend left, "F \left( g\_2 \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g\_1 \right) \circ F \left( f \right) = F \left( g\_2 \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Such a functor would no longer be faithful (i.e. no longer injective on morphisms), and hence also no longer fully faithful (i.e. no longer bijective on morphisms), although its faithfulness, and therefore full faithfulness, could nevertheless be restored by introducing a corresponding algebraic condition: (g1 ∘ f : X → Z) = (g2 ∘ f : X → Z), on the morphisms in the domain category C, such that one now has: $$\begin{tikzcd} & Y \arrow[dr, swap, "g\_1"] \arrow[dr, bend left, "g\_2"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g\_1 \circ f = g\_2 \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, swap, "F \left( g\_1 \right)"] \arrow[dr, bend left, "F \left( g\_2 \right)"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g\_1 \right) \circ F \left( f \right) = F \left( g\_ 2 \right) \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( Z \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Figure [fig:Figure18] implements this elementary example in Categorica, illustrating how the faithfulness of a functor may be removed (by imposing the algebraic equivalence F(g1) ∘ F(f) = F(g2) ∘ F(f) on morphisms in the codomain category) and then subsequently reinstated (by imposing the additional algebraic equivalence g1 ∘ f = g2 ∘ f on morphisms in the domain category). Dual to this construction, a simple way in which a functor may fail to be full (i.e. fail to be surjective on morphisms) is by having a new morphism be introduced in the codomain category that did not previously exist in the domain category; for instance, consider now the functor: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, swap, "F \left( g \right)"] \arrow[dr, bend left, "h"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right)"] \arrow[rr, bend right, swap, "h \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Such a functor would no longer be full (i.e. no longer surjective on morphisms), and hence also no longer fully faithful (i.e. no longer bijective on morphisms), although its fullness, and hence full faithfulness, could nevertheless be restored by introducing a new algebraic condition: (F(g) : F(Y) → F(Z)) = (h : F(Y) → F(Z)), on morphisms in the codomain category D, such that one now has: $$\begin{tikzcd} & Y \arrow[dr, "g"] \arrow[loop above, "id\_Y"] &\\ X \arrow[ur, "f"] \arrow[rr, swap, "g \circ f"] \arrow[loop below, "id\_X"] & & Z \arrow[loop below, "id\_Z"] \end{tikzcd} \qquad \mapsto \qquad \begin{tikzcd} & F \left( Y \right) \arrow[dr, "F \left( g \right) = h"] \arrow[loop above, "id\_{F \left( Y \right)}"] &\\ F \left( X \right) \arrow[ur, "F \left( f \right)"] \arrow[rr, swap, "F \left( g \right) \circ F \left( f \right) = h \circ F \left( f \right)"] \arrow[loop below, "id\_{F \left( X \right)}"] & & F \left( Z \right) \arrow[loop below, "id\_{F \left( Z \right)}"] \end{tikzcd}.$$ Figure [fig:Figure19] implements this dual example in Categorica, illustrating how the fullness of a functor may be removed (by introducing the new morphism h : F(Y) → F(Z) in the codomain category) and then subsequently reinstated (by imposing the algebraic equivalence F(g) = h on morphisms in the codomain category). ![image](CategoricaFigure35)![image](CategoricaFigure36) [fig:Figure18] ![image](CategoricaFigure37)![image](CategoricaFigure38) [fig:Figure19] The final example of a functorial construction that we shall cover within this section, due to its rather foundational significance in the fields of algebraic geometry, algebraic topology and (higher) homotopy theory, is that of a (Grothendieck) fibration. This purely category-theoretic notion of a fibration is effectively a grand generalization of the notion of a fiber bundle (or topological fibration) in topology, wherein one considers a functor of the general form F : E → B, with the domain category E being the total category of the fibration (thus playing an analogous role to that of the total space of a fiber bundle) and the codomain category B being the base category (thus playing an analogous role to that of the base space of the same fiber bundle). For each object X in the base category B, one can consider the fiber category FX at X, which is a category whose object set ob(FX) is the set of objects in the total category E that map to X under the functor F (i.e. the preimage of X under the function Fob) and whose morphism set hom(FX) is the set of morphisms in the total category E that map to the identity morphism idX : X → X on X under F (i.e. the preimage of idX : X → X under the function Fhom), i.e: $$\begin{gathered} \forall X \in \mathrm{ob} \left( \mathcal{B} \right), \qquad F\_X \text{ is the fiber category of functor } F : \	arxiv_0000415
Line tension of a dislocation moving through an anisotropic crystal =================================================================== ### April 16, 2018 Los Alamos National Laboratory, Los Alamos, NM, 87545, USA Army Research Laboratory, Aberdeen Proving Ground, MD, 21005, USA Plastic deformation, at all strain rates, is accommodated by the collective motion of crystalline defects known as dislocations. Here, we extend an analysis for the energetic stability of a straight dislocation, the so-called line tension (Γ), to steady-state moving dislocations within elastically anisotropic media. Upon simplification to isotropy, our model reduces to an explicit analytical form yielding insight into the behavior of Γ with increasing velocity. We find that at the first shear wave speed within an isotropic solid, the screw dislocation line tension diverges positively indicating infinite stability. The edge dislocation line tension, on the other hand, changes sign at approximately 80% of the first shear wave speed, and subsequently diverges negatively indicating that the straight configuration is energetically unstable. In anisotropic crystals, the dependence of Γ on the dislocation velocity is significantly more complex; At velocities approaching the first shear wave speed within the plane of the crystal defined by the dislocation line, Γ tends to diverge, with the sign of the divergence strongly dependent on both the elastic properties of the crystal, and the orientation of the dislocation line. We interpret our analyses within the context of recent molecular dynamics simulations (MD) of the motion of dislocations near the first shear wave speed. Both the simulations and our analyses are indicative of instabilities of nominally edge dislocations within fcc crystals approaching the first shear wave speed. We apply our analyses towards predicting the behavior of dislocations within bcc crystals in the vicinity of the first shear wave speed. Keywords: moving dislocations, line tension, dynamic loading, anisotropy Motivation ========== Metallic plasticity is accommodated by the movement of linear crystalline defects known as dislocations. In regimes of high-rate loading , such as impact , the velocity of dislocations may approach the intrinsic elastic transverse wave speed of the continuum to accommodate high rates of material deformation. A suite of recent molecular dynamics (MD) simulations as well as some experimental data suggest that dislocation velocities can reach or even exceed the transverse wave speed (ct) of the material. Within these dynamic regimes, both the energetics and the stability of dislocations markedly differs from that as predicted within regimes of quasi-static deformation. As an example, upon considering the interaction energy between a solid and a dynamic dislocation moving at velocities (v) approaching the transverse wave speed, discrete dislocation theory predicts that the energy of nominally screw and edge dislocations diverges as ∼ O(1 − (v/ct)2)− 1/2 and ∼ O(1 − (v/ct)2)− 3/2, respectively . Both experiment and numerical simulations suggest that dislocations may traverse at, or even exceed, the transverse wave speed of the continuum. On the other hand, consideration of dislocation theory suggests divergent energies in these regimes. Therefore, the notion of straight, steady state dislocation motion at velocities approaching the transverse wave speed requires reconciliation. Central to stability analyses of dislocations is the dislocation line tension (Γ). Analogous to a taut string, dislocations exhibit a tendency to straighten, minimizing their length, and also their interaction energy with the crystal. The line tension is a scalar quantity related to the configurational force acting to straighten and minimize the length of a dislocation, if the line tension is positive (indicating stability of the configuration) . Conversely, a negative line tension indicates instability, and the related force will drive an unstable dislocation away from its current configuration. The line tension depends upon the orientation, i.e. the angle (ϑ) between the dislocation Burgers vector (b⃗) and the dislocation line direction (t⃗) . Including an orientational dependence in the small bowout limit, as with anisotropic surfaces , or dislocations as considered here, the line tension (Γ) is related to the (dislocation) energy E(ϑ, v) as $$\begin{aligned} {\Gamma}&=\left(1+\frac{{\partial}^2 }{{\partial}{\vartheta}^2}\right)E({\vartheta},v) \,, \label{eq:def-linetension}\end{aligned}$$ where v denotes the dislocation velocity. Eq. is valid for curved as well as for straight dislocations (which are merely a special case of the former) . Within this work, we assume a bow-out type mechanism of instability which occurs along a dislocation glide plane. Independent of the details of the mechanism, for a characteristic dislocation length, L, the dislocation line tension reduces to $$\begin{aligned} {\Gamma}&={\bar{\Gamma}}\frac{\bar{\mu}b^2}{4\pi}\ln \left(\frac{\mathcal{L}}{r\_0}\right)+\Gamma\_0 \,, \label{eq:def-bowout}\end{aligned}$$ where ${\bar{\Gamma}}{\vcentcolon=}{\bar{\Gamma}}({\vartheta},v)$ denotes the dislocation line tension prelogarithmic factor, $\bar\mu$ is the average shear modulus, and r0 represents the cutoff parameter introduced to circumvent divergent dislocation energies. In addition, the term Γ0 depends upon the geometry of the bow-out under consideration (e.g., small bow-out ). The two constants r0 and Γ0 depend upon details of the dislocation core and details of the specific bow-out mechanism under consideration, respectively, both of which are beyond the scope of the present work. In the limit of an infinitely long dislocation (L ≫ r0), $\Gamma\sim(\bar{\mu} b^2/4\pi){\bar{\Gamma}}\ln(R/r\_0)$, with R (of the order of interdislocation spacing) denoting the radius of an outer cylinder surrounding the dislocation core, however the magnitude of ${\bar{\Gamma}}$ does not depend upon L, R, or detailed geometrical features of the bow-out. Within this work we focus our attention on ${\bar{\Gamma}}$ only within eq., since it represents the contribution of the cumulative strain and kinetic energy of the crystal to the dislocation line tension, and at L ≫ r0 is expected to dominate within eq.. The objective of this work is to study the dependence of the dislocation line tension prelogarithmic factor ${\bar{\Gamma}}$ on both the velocity and orientation of the dislocation line (i.e. ${\bar{\Gamma}}({\vartheta},v)$). The dependence of the dislocation line tension on the velocity provides a mechanistic interpretation of the critical velocity at which a moving (straight) dislocation becomes energetically unstable with respect to bow-out. As noted, MD simulations for fcc metals such as show [1](#fn1) that at some critical driving stress the dislocation velocity “jumps” to the transonic regime, exhibiting a discontinuity in the velocity response to an applied stress, (and a similar jump to supersonic happens around the longitudinal sound speed). In fcc metals these rapid shifts in velocity are limited to edge dislocations only, where the Burgers vector is perpendicular to the dislocation line, indicating that the character of the dislocation is important towards understanding the mechanics of dislocation motion within high velocity regimes. Furthermore, there remains sustained interest in delineating the limits of applicability of the dislocation drag coefficient to infinite, straight, dislocations in regimes of high dislocation velocity. Current models assume perfectly straight dislocations and thereby introduce artificial (unphysical) divergences at the transverse sound speed within an infinite isotropic crystal. Some authors have demonstrated, that if terms accounting for acceleration are included into the models of screw dislocation fields in the isotropic limit, the divergence that is otherwise present at the transverse sound speed in the steady state approximation, disappears. While this may explain the smooth transition of screw dislocations into the transonic regime, it does not explain the sudden jump-like acceleration from sub- to transonic observed for edge dislocations at some critical stress . Our plausible interpretation of these findings is as follows. At some critical velocity which is a fraction of the transverse sound speed, the (edge) dislocation line tension (eq. ) becomes negative, favoring bow-out of the dislocation line. With the configurational change of the dislocation line, the drag coefficient which relates driving stress to dislocation velocity also decreases; The dislocation, which is no longer straight, then experiences an abrupt acceleration into the transonic regime. This description is qualitatively in agreement with recent MD simulations, e.g. , where a bowing nucleus is observed to form on a transient dislocation at the instant of acceleration. Hence, the objective presently is to generalize the work on anisotropic line tension of ref. to include a velocity dependence along the same lines as was previously done only within the isotropic case (where edge and screw dislocations decouple making the orientation dependence of ${\bar{\Gamma}}$ trivial). This will provide an initial estimate for the evolution of the line tension with increasing velocity, leading to a critical velocity, beyond which a straight dislocation is energetically unstable [2](#fn2). Our present study will be limited to the line tension calculated from the dislocation self-energy, i.e. we consider a single infinitely long dislocation and infinitesimal bowout, leaving the contribution from dislocation-dislocation (and dislocation-crystal lattice) interactions to future work. The remainder of this work is organized as follows. Within the next section, we begin by reviewing the equations governing the dynamic balance of linear momentum for a linear elastic solid. We then provide a brief review of the elastic field equations for an infinite dislocation situated within a linearly elastic anisotropic crystal of either fcc or bcc structure. With the elastic field equations, and the governing equations of dynamic equilibrium we are able to formulate expressions for both the energy and line tension of a dislocation within an anisotropic crystal with respect to both the orientation (ϑ) and the velocity (v⃗) of the straight dislocation. Within section [sec:isotropic], we consider the limit of elastic isotropy, demonstrating that our expressions for both the energy and line tension reduce to already existent expressions available within the literature . Within section [sec:cubic], we apply our expression for the line tension to several anisotropic fcc and bcc structured crystals examining the dependence of ${\bar{\Gamma}}$ on both the velocity and orientation of the dislocation line. In the final section [sec:conclusion], we provide a brief synopsis of the results of this work. Method ====== We proceed with a brief summary of the general method for deriving displacement gradient fields of dislocations. Upon integrating over the displacement gradient fields, both the dislocation energy and line tension may be obtained. More details can be found within the extensive reviews . [h!t!b] lccccc & ρ[g/ccm] & c11[GPa] & c12[GPa] & c44[GPa] & A fcc & & & & & Al & 2.70 & 106.75 & 60.41 & 28.34 & 1.22 Cu & 8.96 & 168.3 & 122.1 & 75.7 & 3.28 Ni & 8.90 & 248.1 & 154.9 & 124.2 & 2.67 bcc & & & & & Fe & 7.87 & 226.0 & 140.0 & 116.0 & 2.70 Nb & 8.57 & 246.5 & 134.5 & 28.73 & 0.51 Ta & 16.4 & 260.2 & 154.4 & 82.55 & 1.56 W & 19.3 & 522.39 & 204.37 & 160.58 & 1.01 Within the scope of this work, we are concerned with cubic (fcc and bcc) crystals, whose stiffness tensor contains the set of three cubic elastic constants {c11, c12, c44}. Tensors of elastic constants are typically defined in Cartesian coordinates which are aligned with the crystal axes. For orientations of the coordinate system which are not aligned with the crystal bases, it is necessary to rotate the stiffness tensor into the prescribed basis. The tensor of second order elastic constants (i.e., the fourth order elastic stiffness tensor) within the crystal reference frame reads $$\begin{aligned} C\_{ijkl}&=c\_{12}{\delta}\_{ij}{\delta}\_{kl}+c\_{44}\left({\delta}\_{ik}{\delta}\_{jl}+{\delta}\_{il}{\delta}\_{jk}\right)-\left(2c\_{44}+c\_{12}-c\_{11}\right)\sum\_{\alpha=1}^3{\delta}\_{i\alpha}{\delta}\_{j\alpha}{\delta}\_{k\alpha}{\delta}\_{l\alpha} \,,\label{eq:STIFF}\end{aligned}$$ where the sum over repeated indices is implicit, and the last term which includes the explicit summation represents the fourth order identity tensor. Within table [tab:values-metals] we list the material specific elastic constants utilized within this study, as well as the material densities and Zener anisotropy ratio. In the isotropic limit c11 → c12 + 2c44, and the three cubic constants reduce to the two Lamé constants: c12 → λ, c44 → μ. The governing equations to which the displacement gradient field provides a solution are the equations of motion (e.o.m.) and the (leading order) stress-strain relations known as Hooke’s law: $$\begin{aligned} {\partial}\_i{\sigma}\_{ij}&=\rho\ddot u\_j\,, & {\sigma}\_{ij}&=C\_{ijkl}{\epsilon}\_{kl}=C\_{ijkl}u\_{k,l}\,, {\nonumber}\\ {\epsilon}\_{kl}&{\vcentcolon=}\frac12\left(u\_{k,l}+u\_{l,k}\right) \,, \label{eq:Hooke}\end{aligned}$$ where εij denotes the infinitesimal strain tensor, ρ the material density, and the last equality in the first line follows from the elastic constants’ Voigt symmetry. Furthermore, we have introduced the notation [3](#fn3) $u\_{k,l}{\vcentcolon=}{\partial}\_l u\_k$ for the gradient of the displacement field uk, and $\ddot u\_j{\vcentcolon=}\frac{{\partial}^2 u\_j}{{\partial}t^2}$ denoting temporal derivatives. In the static case $\ddot{\vec{u}}=\vec{0}$, however even at constant, finite dislocation velocity a similar simplification may be devised. In the dynamic case with constant velocity, the displacement field may be expressed in terms of a time-dependent shifted position (x⃗ − v⃗t), i.e. uk(xi, t) = uk(xi − vit), and hence its temporal derivative can be expressed in terms of its gradient: $\dot u\_i=-v\_j u\_{i,j}$. Then the e.o.m. (eq. [eq:Hooke]) may be simplified as $$\begin{aligned} 0&={\partial}\_i{\sigma}\_{ij}-\rho\ddot u\_j=C\_{ijkl}u\_{k,il}-\rho v\_iv\_l u\_{j,il} \,,\end{aligned}$$ or upon defining “effective” elastic constants (or the “dynamic stiffness tensor”) : $$\begin{aligned} \hat C\_{ijkl}u\_{k,il}&= 0\,, & \hat C\_{ijkl}{\vcentcolon=}C\_{ijkl}-\rho v\_iv\_l {\delta}\_{jk} \,. \label{eq:eom1}\end{aligned}$$ We hence need only to find uk which satisfies the differential equation. Finding solutions for dislocation displacement gradients -------------------------------------------------------- Throughout this work, we limit our analyses to infinite, straight dislocations. This simplification removes any dependence of the field quantities on the position along the dislocation line. A. N. Stroh describes a method to compute solutions for u⃗ based upon the ansatz [4](#fn4) $$\begin{aligned} u\_k&=\frac{D A\_k}{2\pi i}\ln\left(m\_j x\_j+p n\_j x\_j\right) \,, \label{eq:disp}\end{aligned}$$ where the perpendicular unit vectors m⃗ and n⃗ are conveniently chosen normal to the sense vector t⃗ of the dislocation whose displacement field we are seeking: i.e. t⃗ = m⃗ × n⃗. Inserting this ansatz into the e.o.m. transforms the differential equation into an eigenvalue problem in terms of the unknown coefficients Ak and p. The overall factor D is finally determined by the boundary conditions that a Burgers circuit around the dislocation line has a discontinuity described by the Burgers vector b⃗, and that there be no external forces present at the dislocation core. Due to Voigt symmetry, the eigenvalue problem can be formulated in terms of a 6-dimensional vector $\vec\zeta$ and associated 6 × 6 matrix $\mat N$ comprised of four 3 × 3 blocks, i.e. $\mat N\cdot\vec\zeta=p\vec\zeta$, see for details on this “sextic formalism”. On the other hand, one can reformulate the theory so that this sextic eigenvalue problem is replaced by one of evaluating a set of definite integrals . We will now summarize this (simpler) latter approach in the notation of ref. . The crucial observation is that the unit vectors m⃗, n⃗ are defined only up to an arbitrary angle ϕ. Hence, as one can show, averaging over this angle yields a solution for uj, k in terms of the averaged matrix $$\begin{aligned} \langle \mat N\rangle&=\frac1{2\pi}\int\limits\_0^{2\pi}\mat N d\phi =\begin{pmatrix} \mat S & \mat Q \\ \mat B & \mat S^T \end{pmatrix} \,,\end{aligned}$$ with $$\begin{aligned} \mat S&=-\frac1{2\pi}\int\limits\_0^{2\pi}(nn)^{-1}(nm)\,{\mathrm{d}}\phi \,, & \mat S^T&=-\frac1{2\pi}\int\limits\_0^{2\pi}(mn)(nn)^{-1}{\mathrm{d}}\phi \,,{\nonumber}\\ \mat Q&=-\frac1{2\pi}\int\limits\_0^{2\pi}(nn)^{-1}{\mathrm{d}}\phi \,, & \mat B&=-\frac1{2\pi}\int\limits\_0^{2\pi}\left[(mn)(nn)^{-1}(nm)-(mm)\right]{\mathrm{d}}\phi \,, \label{eq:thematrix}\end{aligned}$$ where we have employed the shorthand notation $(ab)\_{jk}{\vcentcolon=}a\_i \hat C\_{ijkl} b\_l$. In particular, the displacement field reads $$\begin{aligned} u\_j(\abs{\vec{x}},\phi)&=-\frac{b\_l}{2\pi}\left\{S\_{jl}\ln\abs{\vec{x}}-S\_{il}\int\limits\_0^\phi\left[(nn)^{-1}(nm)\right]\_{ji}{\mathrm{d}}\phi'-B\_{il}\int\limits\_0^\phi(nn)^{-1}\_{ji}{\mathrm{d}}\phi'\right\} \,. \label{eq:uj-sol}\end{aligned}$$ We emphasize that the displacement field is comprised of the superposition of a radially dependent term ($\ln\abs{\vec{x}}$) and an angular term ϕ. Upon selecting a coordinate system which rotates with ϕ about the dislocation line (mixi = r, nixi = 0), the displacement gradient computes to : $$\begin{aligned} u\_{j,k}(r,\phi)&=\frac{\tilde{u}\_{j,k}(\phi)}{r}\,, {\nonumber}\\ \tilde{u}\_{j,k}(\phi)&=-\frac{b\_l}{2\pi}\left\{m\_k S\_{jl}-n\_k\left[(nn)^{-1}(nm)\right]\_{ji}S\_{il}-n\_k(nn)^{-1}\_{ji}B\_{il}\right\} \,.\label{eq:ukl-sol}\end{aligned}$$ Notice that the ũj, k(ϕ) dependence on ϕ resides within the unit vectors m⃗, n⃗. The tensors $\mat S$, and $\mat B$ are constant projections of the dynamic stiffness tensor onto the plane with unit normal vector t⃗ as computed from eq. ; Their values depend on the elastic constants as well as (constant) dislocation velocity, and material density, cf.. These integrals are directly amenable to numerical integration for any given set of material constants [5](#fn5), and this numerical integration need be performed only once for a given dislocation velocity, material, and orientation. The dislocation displacement gradients may subsequently be algebraically assembled according to eq.. Note, that this method is limited to the case of steady-state motion leading to the simpler e.o.m.. Its strength is its simplicity; if one were to study acceleration as well (which is beyond the scope of our present work), a more general method using dynamic Green functions would be necessary . In the steady state limit, of course, the same dislocation solution is reproduced as with the current method, see e.g. for the isotropic case (which we discuss in section [sec:isotropic] below). Energy and line tension ----------------------- The concept of line tension, as applied to a dislocation, follows that of the accumulation of strain energy incurred through undulation of a material surface; The surface tension is defined as the change in energy per unit length due to an infinitesimal undulation of a surface, with respect to a nominally flat surface. The sign of this difference in energy indicates if a wavy configuration of the material surface is energetically favorable. When considering a dislocation within this context, the line tension is related to both the energy, and the orientation of the dislocation line through the relation eq., cf. As noted in the introduction, this relation omits a constant which depends upon the three dimensional geometry of the instability under examination . The infinitesimal linear elastic strain energy density surrounding the dislocation core follows as dWs = σijdεij, or upon integration $$\begin{aligned} W^s&=\int{\sigma}\_{ij}{\mathrm{d}}{\epsilon}\_{ij}=\frac12C\_{ijkl}{\epsilon}\_{ij}{\epsilon}\_{kl} \,.\end{aligned}$$ Due to Voigt symmetry of the elastic constants one may substitute the infinitesimal strains ${\epsilon}\_{ij}{\vcentcolon=}(u\_{i,j}+u\_{j,i})/2$ with the displacement gradients. As before, we emphasize that ui, j(ϑ, ϕ) depends on the dynamic stiffness tensor, as defined earlier, while the elastic strain energy density, as written, depends linearly on the nominal (static) elastic stiffness tensor. Upon considering finite velocity, in addition to the linear elastic strain energy density Ws we also account for the kinetic energy density due to the temporal evolution of the displacement field, $$\begin{aligned} W^k&=\frac{\rho}2\left({\frac{{\partial}u\_i}{{\partial}t}}\right)^2 =\frac{\rho}2 v\_i v\_j u\_{k,i} u\_{k,j} \,,\end{aligned}$$ where we have utilized the relation $\dot u\_k=-v\_j u\_{i,j}$. Hence, the total energy density is $$\begin{aligned} W({\vartheta},v)&=W^s+W^k=\left(C\_{ijkl}+\rho v\_i {\delta}\_{jk} v\_l\right) u\_{j,i} u\_{k,l}/2 \,,\label{eq:eng\_dens}\end{aligned}$$ which depends on both the character of the dislocation and the velocity through the displacement gradients. Upon integrating the energy density per unit length through a hollow cylinder defined by inner and outer radii r0 and R, respectively, the energy per unit length E of a moving dislocation is [6](#fn6) $$\begin{aligned} E&=\int\_{r\_0}^{R}r \,{\mathrm{d}}r\int\_{0}^{2\pi}{\mathrm{d}}{\phi}\, W {\nonumber}\\& =\frac12\left(C\_{ijkl}+\rho {\delta}\_{jk} v\_i({\vartheta}) v\_l({\vartheta})\right) \left[\int\_{0}^{2\pi}{\mathrm{d}}{\phi}\, \tilde{u}\_{j,i}({\vartheta}) \tilde{u}\_{k,l}({\vartheta})\right]\ln\left(\frac{R}{r\_0}\right) \,, \label{eq:energy1}\end{aligned}$$ where $\tilde{u}\_{i,j}{\vcentcolon=}r u\_{i,j}$ is independent of r (see eq. ). Since the only radial dependence resides within ui, j ∝ 1/r, the integral ui, juk, lrdr ∝ ln(R/r0) which depends upon an arbitrary constant. Upon insertion of eq. into eq. our final expression for the line tension of a steady state moving dislocation within an anisotropic infinite medium is $$\begin{aligned} {\Gamma}&=\frac12\left(1+\frac{{\partial}^2 }{{\partial}{\vartheta}^2}\right) \left(C\_{ijkl}+\rho {\delta}\_{jk} v\_i({\vartheta}) v\_l({\vartheta})\right) \left[\int\_{0}^{2\pi}{\mathrm{d}}{\phi}\,\tilde{u}\_{j,i}({\vartheta}) \tilde{u}\_{k,l}({\vartheta})\right]\ln\left(\frac{R}{r\_0}\right) \,. \label{eq:linetension}\end{aligned}$$ We note that our expression for Γ does not include an arbitrary constant which is O(1) in the zero velocity limit. This is due to our ansatz in eq. where we have assumed an infinite, straight dislocation. We will apply eq. to dislocations of arbitrary character angle contained within the slip plane with subsonic velocities within both isotropic solids as well as in anisotropic fcc and bcc crystals. Crystal lattice dependence and numerical implementation ------------------------------------------------------- Within the previous subsections, we have detailed the computation of the displacement gradient, the dislocation line energy, and dislocation line tension. Within this section, we relate the two perpendicular vectors (m and n), which the displacement gradient depends on, to the geometry of the crystalline lattice. We employ Cartesian coordinates aligned with the crystal axes, and consider a fixed Burgers vector b⃗ and a unit vector normal to the slip plane under consideration, n⃗0. The material direction of both the Burgers vector and slip plane depend upon the crystal structure. The sense vector of the dislocation t⃗ and its dependence on ϑ depends upon the Burgers vector and slip plane normal as $$\begin{aligned} \vec{t}({\vartheta})&=\frac1b\left[\vec{b}\cos{\vartheta}+\vec{b}\times\vec{n}\_0\sin{\vartheta}\right] \,, \label{eq:def-sensevec}\end{aligned}$$ where ϑ is the angle between t⃗ and b⃗ and $b{\vcentcolon=}\abs{\vec{b}}$. Therefore, ϑ = 0 and ϑ = π/2 correspond to pure screw and pure edge dislocations, respectively. Since we assume that dislocation bow-out will occur within the slip plane spanned by t⃗ and b⃗, we may define a third mutually perpendicular vector m⃗0(ϑ) in the direction of motion as $$\begin{aligned} \vec{m}\_0({\vartheta})=\vec{n}\_0\times \vec{t}({\vartheta}) \,.\end{aligned}$$ We will assume, without loss of generality, that the dislocation moves in that direction, i.e. v⃗(ϑ) = vm⃗0(ϑ). From m⃗0, n⃗0 we construct $$\begin{aligned} \vec{m}({\vartheta},{\phi})&=\vec{m}\_0({\vartheta})\cos{\phi}+ \vec{n}\_0\sin{\phi}\,, {\nonumber}\\ \vec{n}({\vartheta},{\phi})&=\vec{n}\_0\cos{\phi}- \vec{m}\_0({\vartheta})\sin{\phi}\,,\end{aligned}$$ which span the plane normal to t⃗, as illustrated within figure [fig:geometry]. [fig:geometry] In order to evaluate eq. we must resort to numerical methods, since it is analytically insoluble. We have implemented eq. within a numerical code [7](#fn7) which relies upon the stiffness tensor (Cijkl), the Burgers vector direction (b⃗/∣b⃗∣), and the slip plane normal (n⃗0) as input. The integrals over ϕ (i.e. ũi, j, Bij, etc.) are performed using a trapezoidal numerical integrator with a step size of δϕ ≈ π/500; The derivatives with respect to ϑ are evaluated by successively differencing evaluations of E(ϑ, v) (eq. ) over small increments of δϑ (e.g. $\frac{{\partial}E({\vartheta},v)}{{\partial}{\vartheta}} {\vcentcolon=}(E({\vartheta}+\delta {\vartheta}, v) - E({\vartheta}-\delta {\vartheta}, v)) /2\delta {\vartheta}$), where δϑ ≈ π/600. For each of the materials investigated within this study, the resulting Γ has been evaluated at roughly 600 evenly spaced intervals of ϑ where − π/2 < ϑ < π/2, and 625 evenly spaced intervals of ${\beta\_{\bar\mu}}{\vcentcolon=}\sqrt{{\rho v^2}/{\bar{\mu}}}$ within the range $0 \leq {\beta\_{\bar\mu}}\leq 1$, with $\bar\mu{\vcentcolon=}\frac14\left(c\_{11}-c\_{12}+2c\_{44}\right)$ denoting the mean shear modulus. This results in ∼ 375, 000 data points for each material. Throughout the remainder of this work, it will be convenient to show and discuss results in the rescaled, dimensionless line tension prelogarithmic factor ${\bar{\Gamma}}$ via $$\begin{aligned} {\Gamma}&=\left(\frac{\bar\mu b^2}{4\pi}\ln\frac{R}{r\_0}\right){\bar{\Gamma}}\,. \label{eq:linetensionprefactor}\end{aligned}$$ Our results will be presented in terms of ${\bar{\Gamma}}({\vartheta},{\beta\_{\bar\mu}}(v))$ (for which we will use the terms “line tension” and “prelogarithmic factor” interchangeably) as a function of dislocation character ϑ and dimensionless, scaled velocity ${\beta\_{\bar\mu}}$. The isotropic limit =================== We begin by examining the energy, and line tension of a moving dislocation in the isotropic limit. In this limit, the expressions for the energy and line tension are considerably simplified and are analytically tractable, yielding insight towards our anisotropic results. Referring to eq., in the limit that c11 → c12 + 2c44 there is no orientational dependence dictated by the crystal symmetry; the dislocation displacement gradients, energy, and line tension do not depend upon the orientation of the dislocation within the crystal. Upon adopting a coordinate system fixed to the dislocation line with t⃗ oriented along $\hat z$ and n⃗0 oriented along $\hat y$, the ϑ dependence resides entirely within the Burgers vector, and the vectors defined within section [sec:method] reduce to $$\begin{aligned} \vec{t}&=(0,0,1)\,,& \vec{m}&=\left(\cos{\phi},\sin{\phi},0\right)\,,& \vec{n}&=\left(-\sin{\phi},\cos{\phi},0\right)\,,{\nonumber}\\ \vec{v}&=(v,0,0) \,, & \vec{b}&=(b\sin{\vartheta},0,b\cos{\vartheta})\,, & r&=\vec{m}\cdot\vec{x}=\sqrt{x^2+y^2} \,.\end{aligned}$$ One may verify that, in these coordinates, the well known solutions for stationary moving edge and screw dislocations within an isotropic crystal are reproduced: $$\begin{aligned} u\_{x,x}&=\frac{-b\_e y}{\pi {\beta\_{\textrm{t}}}^2 }\left(\frac{1/{\gamma\_{\textrm{l}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right) /{\gamma\_{\textrm{t}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{x,y}&=\frac{b\_e (x-t v)}{\pi {\beta\_{\textrm{t}}}^2 }\left(\frac{1/ {\gamma\_{\textrm{l}}}}{ \left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right)/ {\gamma\_{\textrm{t}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{y,x}&=\frac{b\_e (x-t v)}{\pi {\beta\_{\textrm{t}}}^2 }\left( \frac{1/{\gamma\_{\textrm{l}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{{\gamma\_{\textrm{t}}}\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right) }{ \left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{y,y}&=\frac{b\_e y}{\pi {\beta\_{\textrm{t}}}^2 }\left(\frac{1/{\gamma\_{\textrm{l}}}^3}{ \left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right)/ {\gamma\_{\textrm{t}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{z,x}&=-\frac{b\_s\, y/{\gamma\_{\textrm{t}}}}{2 \pi \left((x-t v)^2+y^2/{\gamma\_{\textrm{t}}}^2\right)}\,, \qquad\qquad u\_{z,y}=\frac{b\_s\, (x-t v)/ {\gamma\_{\textrm{t}}}}{2 \pi \left((x-t v)^2+y^2/{\gamma\_{\textrm{t}}}^2 \right)} \,, \label{eq:u-isotropic}\end{aligned}$$ where we have employed a dislocation fixed coordinate system (x → (x − vt)) and where [8](#fn8) $$\begin{aligned} {\gamma\_{\textrm{t}}}&=\frac{1}{\sqrt{1-{\beta\_{\textrm{t}}}^2}}\,, & {\beta\_{\textrm{t}}}&=\frac{v}{{c\_{\textrm{t}}}}\,, & {\gamma\_{\textrm{l}}}&=\frac{1}{\sqrt{1-{\beta\_{\textrm{l}}}^2}}\,, & {\beta\_{\textrm{l}}}&=\frac{v}{{c\_{\textrm{l}}}} \,,\end{aligned}$$ and ${c\_{\textrm{t}}}=\sqrt{{\mu}/\rho}$, ${c\_{\textrm{l}}}=\sqrt{\left({\lambda}+2{\mu}\right)/\rho}$ are the transverse and longitudinal sound speed, respectively. The edge and screw components of the Burgers vector, $b\_e{\vcentcolon=}b\sin{\vartheta}$ and $b\_s{\vcentcolon=}b\cos{\vartheta}$, are the only variables depending on the angle ϑ. Energy of a moving dislocation in an isotropic crystal ------------------------------------------------------ It is readily verified that even for finite velocities, the edge and screw components of dislocations within isotropic media do not interact energetically, and therefore both the displacements, and resulting energies of an arbitrarily oriented dislocation may be superposed. Specifically, since Cijklui, jedgeuk, lscrew = 0, there is no cross term in the total energy of a mixed dislocation and we are justified superposing the energies of edge and screw components. We note also that εxyedge = (ux, yedge + uy, xedge)/2 vanishes at y = 0 (and hence changes sign across the glide plane) at a dislocation velocity v = cR which is known as the Rayleigh wave velocity (i.e. the velocity of surface waves) . It is defined by the relation γlγt = (1 − βt2/2)− 2 and is always smaller than (but fairly close to) the transverse sound speed, cR < ct, a typical value being cR ≈ 0.93ct if Poisson’s ratio is 1/3 (corresponding to cl = 2ct). An additional simplification occurs in the strain energy of the screw dislocation: $W^s=\frac{{\lambda}}{2}({\epsilon}\_{ii})^2+{\mu}{\epsilon}\_{ij}{\epsilon}\_{ij}$, and since the strain of the screw dislocation is traceless, the dependence on λ will only appear for the edge dislocation, i.e. $$\begin{aligned} {\epsilon}\_{ii}^{\textrm{screw}} & = 0\,, & {\epsilon}\_{ii}^{\textrm{edge}} &=\frac{-b\,{c\_{\textrm{t}}}^2 y}{\pi {c\_{\textrm{l}}}^2 {\gamma\_{\textrm{l}}}\left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)} \,,\end{aligned}$$ see eqs.,. The total energy for an arbitrary dislocation orientation follows as $$\begin{aligned} E^{\textrm{mixed}}({\vartheta},v)&= E\_0\left\{{\gamma\_{\textrm{t}}}\cos^2{\vartheta}+\frac{\sin^2{\vartheta}}{{\beta\_{\textrm{t}}}^2}\left[\frac{8}{{\gamma\_{\textrm{l}}}}+4{\gamma\_{\textrm{l}}}+{\gamma\_{\textrm{t}}}^3\left(1-\frac{6}{{\gamma\_{\textrm{t}}}^2}-\frac{7}{{\gamma\_{\textrm{t}}}^4}\right)\right]\right\} \,, {\nonumber}\\ E\_0&=\frac{{\mu}b^2}{4\pi}\ln\left(\frac{R}{r\_0}\right) \,, \label{eq:energy-mixed}\end{aligned}$$ with the contributions from screw and edge dislocations scaled by cos2ϑ and sin2ϑ, respectively (see also ). In the limit v → 0 the energies for both edge and screw dislocations reduce to the well-known expressions $$\begin{aligned} E^{\textrm{screw}}(0,0)&=E\_0 \,, & E^{\textrm{edge}}(\pi/2,0)&=E\_0\frac{2({\lambda}+{\mu})}{{\lambda}+2{\mu}} =\frac{E\_0}{1-{\nu}} \,,\end{aligned}$$ where Poisson’s ratio is related to the Lamé constants via ν = λ/2(μ + λ). On the other hand, as the velocity approaches the transverse sound speed, both energies diverge as $$\begin{aligned} \lim\limits\_{v\to{c\_{\textrm{t}}}}E^{\textrm{screw}}(0,v)&\to\frac{E\_0}{\sqrt{1-{\beta\_{\textrm{t}}}^2}} \,,{\nonumber}\\ \lim\limits\_{v\to{c\_{\textrm{t}}}}E^{\textrm{edge}}(\pi/2,v)&\to \frac{E\_0}{2\sqrt{2}}\left(\frac{1}{(1-{\beta\_{\textrm{t}}}^2)^{\frac32}}+\frac{37}{4\sqrt{1-{\beta\_{\textrm{t}}}^2}}+\frac{16 (2-\nu)}{\sqrt{1-\nu }}+{\mathcal{O}}\left(\sqrt{1-{\beta\_{\textrm{t}}}^2}\right)\right) \,. \label{eq:isotropic-energy-divergences}\end{aligned}$$ All divergences appearing in the energy as well as the displacement gradients as v → ct can be traced back to the expression (nn)− 1 appearing in and, and are due to $$\begin{aligned} \det\left(nn\right)&=\det\left(\vec{n}\cdot\hat{\mat{C}}\cdot\vec{n}\right) =\det\left(\vec{n}\cdot\mat{C}\cdot\vec{n}-\rho \left(\vec{n}\cdot\vec{v}\right)^2{\mathbb{1}}\right) {\nonumber}\\ &=0 \,,\end{aligned}$$ as v → ct (or v → cl) and ϕ → ± π/2 in which case n⃗ ∥ v⃗. In fact, the leading divergence as v → ct and ϕ → ± π/2 is (nn)− 1 ∼ 1/(1 − βt2). Due to averaging over ϕ, the matrices $\mat{S}$, $\mat{B}$ exhibit a weaker divergence of $1/\sqrt{1-{\beta\_{\textrm{t}}}^2}$. Upon inspecting expression, it is clear that the edge dislocation has a divergence of order 1/(1 − βt2)3/2 in the displacement gradient (ui, j) originating from the collection of terms − nk[(nn)− 1(nm)]jiSil − nk(nn)ji− 1Bil. In the solution for pure screw dislocations, two types of simplification lead to the weaker divergence of $1/\sqrt{1-{\beta\_{\textrm{t}}}^2}$ in the displacement gradient: Firstly, $\mat{S}\cdot\vec{b}=0$ for geometrical reasons, and furthermore $\lim\limits\_{v\to{c\_{\textrm{t}}}}\mat{B}\cdot\vec{b}\propto\sqrt{1-{\beta\_{\textrm{t}}}^2}\to0$. Thus, all but the last term in vanish in the pure screw dislocation case, and that surviving term, nk(nn)ji− 1Bil, furthermore exhibits a weaker divergence than in the edge case. We emphasize that the leading divergences in ui, j outlined above occur only at particular angles, ϕ = ± π/2, i.e. perpendicularly to the direction of dislocation motion. This is also the reason why the degree of divergence in the dislocation energy E (which involves an integration over ϕ) is not twice that of ui, j, despite ui, j entering quadratically. Within this work, we limit our analysis to the subsonic regime. Solutions for the strain fields of straight dislocations have also been derived for the transonic and supersonic regimes . All of these solutions, however, diverge at ct and cl, i.e. they cannot describe the vicinity of the two sound speeds. For more recent literature on supersonic dislocations, see and references therein. Line tension and stability -------------------------- The line tension prelogarithmic factor (${\bar{\Gamma}}$) follows from application of eq. to E(ϑ, v), using our isotropic solution (eq. ) for E (see ref. for the static case). We immediately find $$\begin{aligned} {\Gamma}({\vartheta},v)&= \frac{E\_0}{{\beta\_{\textrm{t}}}^2}\left\{2\cos^2{\vartheta}\left[\frac{8}{{\gamma\_{\textrm{l}}}}+4{\gamma\_{\textrm{l}}}+{\gamma\_{\textrm{t}}}^3\left(1-\frac{6}{{\gamma\_{\textrm{t}}}^2}-\frac{7}{{\gamma\_{\textrm{t}}}^4}\right)-\frac{{\beta\_{\textrm{t}}}^2{\gamma\_{\textrm{t}}}}{2}\right]\right.{\nonumber}\\ &\quad\qquad \left. -\sin^2{\vartheta}\left[\frac{8}{{\gamma\_{\textrm{l}}}}+4{\gamma\_{\textrm{l}}}+{\gamma\_{\textrm{t}}}^3\left(1-\frac{6}{{\gamma\_{\textrm{t}}}^2}-\frac{7}{{\gamma\_{\textrm{t}}}^4}\right)-2{\beta\_{\textrm{t}}}^2{\gamma\_{\textrm{t}}}\right]\right\} .\label{eq:linetension-isotropic}\end{aligned}$$ In the zero velocity limit this expression tends to the classical result given within ref. : $$\begin{aligned} \lim\limits\_{v\to0}{\Gamma}({\vartheta},v)&= \frac{E\_0}{(1-{\nu})}\Big\{(1+{\nu})\cos^2{\vartheta}+(1-2{\nu})\sin^2{\vartheta}\Big\} ,\end{aligned}$$ where ν denotes Poisson’s ratio. For metals of interest, ν ∼ 1/3, which indicates that the line tension of a static screw dislocation is roughly four times larger than for a static edge dislocation, i.e.: ${\Gamma}({\vartheta},v=0,{\nu}=1/3)=E\_0\left(2\cos^2{\vartheta}+\frac12\sin^2{\vartheta}\right)$. Figure [fig:linetension-isotropic] shows the velocity dependence of the line tension ${\bar{\Gamma}}(0,{\beta\_{\textrm{t}}}{c\_{\textrm{t}}})$ in red and ${\bar{\Gamma}}(\pi/2,{\beta\_{\textrm{t}}}{c\_{\textrm{t}}})$ in blue, where, referring to eq. $$\begin{aligned} {\bar{\Gamma}}({\vartheta},v)&{\vcentcolon=}\frac{{\Gamma}({\vartheta},v)}{E\_0} \,,\end{aligned}$$ is the dimensionless, rescaled line tension prelogarithmic factor in the isotropic limit. [fig:linetension-isotropic] What is particularly interesting about the full velocity dependent expression, is that the divergent term γt3 is negative in the pure edge case. This means that the line tension of an edge dislocation (where ϑ = π/2) can become zero (in contrast to the screw case) indicating that it will cease to be stable in a straight configuration. In particular, for ν ≈ 1/3 we find that the line tension of an edge dislocation becomes zero at around 80% transverse sound speed, whereas it remains positive everywhere in the pure screw case — see figure [fig:linetension-isotropic]. The range of velocities where this happens, depending on the value of Poisson’s ratio (which lies in the range − 1 < ν < 1/2), is roughly $0.72\lesssim{\beta\_{\textrm{t}}}^0\lesssim0.83$; this is always beneath the Rayleigh speed. Finally, upon considering specific geometries of bow-out, a ϑ-dependent “constant” (Γ0) may need to be superposed to the expression for the line tension (i.e. corresponding to different constants for edge and screw dislocations), see e.g. ref. . This addition would only change the value of the velocity βt0, but not the main result: that βt0 < ct exists only for edge dislocations and is smaller than (but fairly close to) the transverse sound speed. This observation is in agreement with previous MD simulation results which indicate a sudden jump in velocity versus stress to the transonic regime for edge dislocations, whereas the transition is smooth for screw dislocations. It is also instructive to analyze at which angle ϑ a mixed dislocation in the isotropic limit becomes unstable at high velocities. The change in asymptotic behavior from positive to negative occurs when the coefficient of γt3 becomes zero in eq., i.e. when ${\vartheta}=\arctan\sqrt{2}$ (independent of Poisson’s ratio). Figure [fig:linetension-fcccontour] (a) below shows the regions of stability of such a mixed dislocation in the form ${\bar{\Gamma}}({\vartheta},v)$. For the purpose of this figure we have chosen ν = 1/3. Other values of ν will change the contours within the low to intermediate velocity regime, but not in the asymptotic region βt → 1, since the divergent terms do not depend on ν (see eq. ). Finally, we need to comment on earlier, similar works that have analyzed the stability of dislocations at high velocities: Our result disagrees with the earlier assessment of ref. (see also ), whose author found that screw dislocations become unstable at high velocities whereas edge dislocations remain stable. On the other hand, our results are in perfect agreement with ref. . A somewhat different stability analysis was performed in the anisotropic case for cubic metals in ref. . The authors’ stability analysis was based upon finding the velocity where the force of interaction between two parallel dislocations on the same slip plane changes sign, and the result is that this happens at some velocity lower than the shear wave speed for both edge and screw dislocations. Within the next section, we extend the general stability analysis (${\bar{\Gamma}}({\vartheta},v)$) of dislocations to anisotropic crystals; We concentrate on fcc and bcc crystals. Cubic crystals ============== The isotropic analyses and results presented within the previous section provide insight due to their simple, analytical form (i.e., eq. ). Since realistic material crystals are anisotropic however, the approximated isotropic computations have the capacity to obscure subtle details from our line tension calculations. Within this section, we compute and present results for ${\bar{\Gamma}}$ (eq. ) employing full elastic anisotropy with the elastic constants found within table [tab:values-metals]. Slip systems of dislocations and divergences -------------------------------------------- The atomic structure of fcc and bcc crystals dictates the slip plane normal and Burgers vector with respect to reference cubic crystal axes, $$\begin{aligned} \vec{b}^{\textrm{fcc}}&=\frac{b}{\sqrt{2}}\left(1,1,0\right)\,, & \vec{n}\_0^{\textrm{fcc}}&=\frac{1}{\sqrt{3}}\left(1,-1,1\right) \,, {\nonumber}\\ \vec{b}^{\textrm{bcc}}&=\frac{b}{\sqrt{3}}\left(1,-1,1\right)\,, & \vec{n}\_0^{\textrm{bcc}}&=\frac{1}{\sqrt{2}}\left(1,1,0\right) \,. \label{eq:slipplanes-bcc}\end{aligned}$$ Following section [sec:implementation] we may construct both the dislocation line direction (t⃗) and velocity vector (vm⃗0) within the plane of slip. The velocity vector with respect to the moving dislocation depends on ϑ as $$\begin{aligned} \vec{v}^{\textrm{fcc}}({\vartheta})&=v\vec{m}\_0^{\textrm{fcc}} =\frac{v}{\sqrt{6}}\left(\sqrt{3} \sin ({\vartheta})-\cos ({\vartheta}),\sqrt{3} \sin ({\vartheta})+\cos ({\vartheta}),2 \cos ({\vartheta})\right) \,, {\nonumber}\\ \vec{v}^{\textrm{bcc}}({\vartheta})&=v\vec{m}\_0^{\textrm{bcc}} =\frac{v}{\sqrt{6}}\left(\sqrt{2} \sin ({\vartheta})+\cos ({\vartheta}),-\sqrt{2} \sin ({\vartheta})-\cos ({\vartheta}),\sqrt{2} \sin ({\vartheta})-2 \cos ({\vartheta})\right) \,.\end{aligned}$$ Considering the two limits, we have $\vec{v}^{\textrm{fcc}}(0)=\frac{v}{\sqrt{6}}\left(-1,1,2\right)$, $\vec{v}^{\textrm{bcc}}(0)=\frac{v}{\sqrt{6}}\left(1,-1,-2\right)$ for screw dislocations, and $\vec{v}^{\textrm{fcc}}(\pi/2)=\frac{v}{\sqrt{2}}\left(1,1,0\right)$, $\vec{v}^{\textrm{bcc}}(\pi/2)=\frac{v}{\sqrt{3}}\left(1,-1,1\right)$ for edge dislocations. Before proceeding immediately to the dependence of the line tension on the velocity, crystal structure, and character angle, an important subtlety which requires consideration within anisotropic media is divergences in dislocation energy at high velocities. As in the isotropic limit, these divergences occur at the lowest critical velocity leading to det(nn) = 0 at certain angle(s) ϕ. However, this critical velocity vcrit is not necessarily the lowest transverse sound speed of the crystal, nor is it the lowest transverse sound speed with respect to the direction of dislocation movement, however in between the two. To demonstrate this point, the lowest transverse sound speed in the direction of v⃗ is computed from $$\begin{aligned} \det\left(\vec{v}\cdot\mat{C}\cdot\vec{v}-\rho v^2{\mathbb{1}}\right)\Big\|\_{v=v\_{{\textrm{t}}+}}=0 \,, \label{eq:sound1}\end{aligned}$$ and we denote this velocity vt + because due to (v⃗ ⋅ n⃗)2 ≤ v2 (as we will see shortly) it constitutes an upper bound on the critical velocity responsible for divergences in both the dislocation energy and line tension. Within the plane spanned by n⃗ as ϕ spans the range 0 → 2π, there is an angle ϕ− which yields the lowest transverse sound speed vt − within the plane of the dislocation. At ϕ− the relation $$\begin{aligned} \det\left(\vec{n}\cdot\mat{C}\cdot\vec{n}-\rho v^2{\mathbb{1}}\right)\Big\|\_{{\phi}={\phi}\_-,\,v=v\_{{\textrm{t}}-}}=0 \label{eq:sound2}\end{aligned}$$ is satisfied. Eqs. and provide an upper and lower bound, respectively, for the critical velocity of the divergence. At a crystal dependent critical angle, ϕcrit, within the plane of the dislocation at critical velocity vcrit the following relation is satisfied $$\begin{aligned} \det\left(\vec{n}\cdot\mat{C}\cdot\vec{n}-\rho \left(\vec{n}\cdot\vec{v}\right)^2{\mathbb{1}}\right)\Big\|\_{{\phi}={\phi}\_{\textrm{crit}},\,v=v\_{\textrm{crit}}}=0 \,. \label{eq:def-vcrit}\end{aligned}$$ Since (v⃗ ⋅ n⃗)2 ≤ v2 (where the equal sign corresponds to ϕ = ± π/2), we always have $$\begin{aligned} v\_{{\textrm{t}}-}\le v\_{\textrm{crit}} \le v\_{{\textrm{t}}+} \,,\end{aligned}$$ and in the isotropic limit all three velocities coincide because there is only one shear modulus. We emphasize that ϕcrit, the angle corresponding to the critical velocity is material dependent. Only ϕ− depends only on the crystal structure and may be determined to be ${\phi}\_-^{\textrm{fcc}}=\arcsin(1/\sqrt{3})$ and ϕ−bcc = π/3 for fcc and bcc crystals, respectively. [h!t!b] lccccc fcc & vμ̄[m/s] & $v\_{\textrm{t}}^{\bar{1}12}$[m/s] & $v\_{\textrm{crit}}^{\bar{1}12}$[m/s] & vt110[m/s] & vcrit110[m/s] Al & 3089 & 3032 & 3024 & 2929 & 2929 Cu & 2348 & 2040 & 1994 & 1606 & 1606 Ni & 3098 & 2750 & 2704 & 2288 & 2288 bcc & vμ̄[m/s] & $v\_{\textrm{t}}^{1\bar{1}\bar{2}}$[m/s] & $v\_{\textrm{crit}}^{1\bar{1}\bar{2}}$[m/s] & $v\_{\textrm{t}}^{1\bar{1}1}$[m/s] & $v\_{\textrm{crit}}^{1\bar{1}1}$[m/s] Fe & 3178 & 2814 & 2585 & 2925 & 2745 Nb & 2223 & 2101 & 1997 & 2340 & 2147 Ta & 2032 & 1943 & 1918 & 1957 & 1926 W & 2877 & 2875 & 2875 & 2875 & 2875 Within table [tab:velocities] we list explicit values for vt + and vcrit for the various fcc and bcc metals investigated within this study computed using the densities and elastic constants listed in table [tab:values-metals]. Since the velocity vectors are aligned along different directions for screw and edge dislocations, values for both cases are listed separately. Notice that only for edge dislocations in fcc crystals, the critical velocity happens to equal the lowest shear wave speed corresponding to the direction of dislocation motion, i.e. vcrit110 = vt110 (see table [tab:velocities]). Finally, in order to have a single scaling parameter within all of our plots, we introduce a “velocity” calculated with respect to the mean shear modulus ($\bar{\mu}{\vcentcolon=}(c\_{11}-c\_{12}+2c\_{44})/4$), vμ̄ which is listed for all cases as well for comparison. Line tension of pure screw and edge dislocations ------------------------------------------------ [fig:linetension-fcc] Using the elastic constants listed within table [tab:values-metals] as well as the Burgers vector and slip plane normal specific to fcc and bcc crystal structures, we have computed the line tension for a selection of fcc and bcc metals from eq. with, as described in section [sec:linetension]. Figure [fig:linetension-fcc] illustrates the line tension of three fcc metals (Al, Cu, Ni) versus dimensionless velocity (${\beta\_{\bar\mu}}$), as compared to the isotropic limit with Poisson’s ratio ν = 1/3. We begin with directing our attention to the overall behavior of the curves. As compared to elastic isotropy, one sees relatively similar behavior in that screw dislocations remain stable up to high velocities whereas edge dislocations reach instability at some critical velocity (which might be shifted by a finite but unknown constant due to Γ0 within ). The main differences for increasingly anisotropic fcc metals are the ${\beta\_{\bar\mu}}$ of the divergences, which decrease with increasing anisotropy. Additionally, screw dislocations exhibit a moderate drop in line tension close to ${\beta\_{\bar\mu}}\approx0.8$ before becoming infinitely stable at their critical velocity, vcrit. Focusing our attention specifically on figure [fig:linetension-fcc] (a), Al (in blue) behaves very similar to the isotropic result which is as to be expected, and similar to figure [fig:linetension-isotropic]. With increasing anisotropy, an additional competing divergence (which is negative) becomes evident, which is both weaker and of opposite sign than the positive divergence present at the critical velocity. Since there is an additional arbitrary constant which we have not included within this work, it is possible that both Ni and Cu exhibit negative line tensions over a range of $0.7\leq{\beta\_{\bar\mu}}\leq0.8$. However, comparisons to MD simulation results , suggest that this is not the case, and the constant is likely either small or positive. Within figure [fig:linetension-fcc] (b) both our isotropic analyses, and our three select anisotropic fcc metals exhibit a negative divergence at a fraction of the scaled velocity i.e. $0.7\leq {\beta\_{\bar\mu}}\leq 0.95$, all of which is less than our isotropic example which diverges at ${\beta\_{\bar\mu}}=1$. Therefore, with the inclusion of and increasing degree of anisotropy the predicted velocity of the instability for nominally fcc edge dislocations decreases. The line tension of dislocations in bcc metals, on the other hand, exhibits a significantly more pronounced dependence on the elastic anisotropy, see figure [fig:linetension-bcc]. Starting with figure [fig:linetension-bcc] (a), nominally screw bcc dislocations, niobium and tungsten follow a similar trend to the isotropic curve, specifically, they exhibit only a single diverging term. Both tantalum and more so iron contain at least two diverging terms of opposite sign. Iron exhibits another peculiar feature in screw dislocations between $0.65\lessapprox{\beta\_{\bar\mu}}\lessapprox0.78$: The small dip we saw in the dislocation line tension for fcc metals is much more pronounced in bcc iron and extends far into the negative region making straight screw dislocations unstable. Next, turning our attention to figure [fig:linetension-bcc] (b), the dashed isotropic curve suggests that with increasing velocity nominally edge dislocations become unstable. Surprisingly however, this is only the case with tungsten (A = 1.01). The other three (Fe, Nb, Ta) more anisotropic metals diverge positively, hence becoming infinitely stable at a scaled velocity which scales inversely with the degree of anisotropy; For these three more anisotropic metals the critical velocity is ${\beta\_{\bar\mu}}\leq 0.9$. With the exception of tungsten, figure [fig:linetension-bcc] suggests that all four bcc metals investigated exhibit diverging positive line tensions with increasing subsonic velocities for both nominally edge and screw dislocations. [fig:linetension-bcc] [fig:linetension-fcccontour] [fig:linetension-bcccontour] Line tension dependence on velocity and character angle ------------------------------------------------------- The figures for pure screw and edge dislocations (figures [fig:linetension-fcc] and [fig:linetension-bcc]) are limited to the two extremes of the dislocation character in ϑ and do not capture potentially significant deviations from isotropy that may arise for mixed characters of dislocations. To cover the entire gamut of parameter space for the line tension with respect to ϑ and ${\beta\_{\bar\mu}}$ for cubic anisotropic materials, we show the line tension for all seven metals investigated within this study (i.e. figures [fig:linetension-fcccontour] and [fig:linetension-bcccontour]) as a function of both the scaled velocity (${\beta\_{\bar\mu}}$) as well as dislocation character (ϑ). Compared to elastic isotropy (i.e. figure [fig:linetension-fcccontour] (a)), these figures exhibit a rich structure. In organizing contour figures of ${\bar{\Gamma}}$, in addition to ${\beta\_{\bar\mu}}$, we must consider the range of the character angle, ϑ. A physical interpretation of ϑ dictates that ${\bar{\Gamma}}$ must be π-periodic in ϑ, since ${\bar{\Gamma}}$ should be insensitive to the direction of the line orientation ( ± t). Furthermore, within fcc crystals, the orientation of the Burgers vector ($\vec{b}^{\textrm{fcc}} = (1,1,0)b/\sqrt{2}$) dictates that the dislocation line energy must be symmetric with respect to rotations of the dislocation line of ± δϑ with respect to b⃗fcc contained within the slip plane, n⃗0fcc. For bcc crystals, however, the dependence on ϑ is more subtle. Physically, considering a small increment of ± δϑ of the dislocation line with respect to the bcc Burgers vector direction ($\vec{b}^{\textrm{bcc}} = (1,-1,1)b/\sqrt{3}$), the orientation of the dislocation line is not symmetric with respect to its orientation within the bcc lattice, despite being symmetric with respect to the Burgers vector direction within the same bcc lattice. This is because the slip plane (n⃗0bcc) within a bcc unit lattice has an aspect ratio of $\sqrt{2}$. We may succinctly express the lack of mirror symmetry in the line tension, which is only present for the bcc materials investigated within this study, by expanding the zero velocity dislocation line energy about ϑ = 0, as $$\begin{aligned} \vec{b}^{\textrm{fcc}}\cdot\mat{B}[\vec{t}^{\textrm{fcc}}(\delta {\vartheta})]\cdot\vec{b}^{\textrm{fcc}} \approx \vec{b}^{\textrm{fcc}}\cdot\mat{B}[\vec{t}^{\textrm{fcc}}(0)]\cdot\vec{b}^{\textrm{fcc}}+\sum\_{n=0}^{\infty}\mathcal{O}(\delta {\vartheta})^{2n}\,, {\nonumber}\\ \vec{b}^{\textrm{bcc}}\cdot\mat{B}[\vec{t}^{\textrm{bcc}}(\delta {\vartheta})]\cdot\vec{b}^{\textrm{bcc}} \approx \vec{b}^{\textrm{bcc}}\cdot\mat{B}[\vec{t}^{\textrm{bcc}}(0)]\cdot\vec{b}^{\textrm{bcc}}+\sum\_{n=0}^{\infty}\mathcal{O}(\delta {\vartheta})^{n}\,\end{aligned}$$ Since only the fcc line energy prelogarithmic factor, $\vec{b}\cdot\mat{B}\cdot\vec{b}$ contains only even terms in δϑ of O(2n), we expect ${\bar{\Gamma}}$ to be symmetric about ϑ = 0 for fcc materials only, and we consider 0 ≤ ϑ ≤ π/2. With this symmetry lacking for the bcc materials which we examine, we extend the range from − π/2 ≤ ϑ ≤ π/2. We have numerically verified that these are the appropriate ranges. In addition, we note that the zero velocity limit was previously computed for copper, nickel, niobium, and iron in ref. , and our generalized results are in qualitative agreement with that earlier work at ${\beta\_{\bar\mu}}=0$. Finally, in describing the contour plots for bcc metals, figures [fig:linetension-bcccontour], we note that our choice of slip system influences the dependence on ϑ; With an alternate slip plane, e.g., $\vec{b}^{\textrm{bcc}}=\frac{b}{\sqrt{3}}\left(1,+1,1\right)$, $\vec{n}\_0^{\textrm{bcc}}=\frac{1}{\sqrt{2}}\left(1,-1,0\right)$, as opposed to eq., the ϑ dependence within these plots would be reversed, i.e. ϑ → − ϑ. We proceed by investigating the line tension of several fcc metals, aluminum which is considered to be a “fairly isotropic” fcc metal exhibits a second, additional, unstable region at high velocities for ϑ close to π/5, see figure [fig:linetension-fcccontour] (b). With increasing anisotropy, the additional unstable region becomes even more pronounced, e.g., copper (figure [fig:linetension-fcccontour] (c)) and nickel (figure [fig:linetension-fcccontour] (d)). All three figures [fig:linetension-fcccontour] (b)–(d) are shown for ${\beta\_{\bar\mu}}\leq v\_{\textrm{t}}^{110}/v\_{\bar{\mu}}$, i.e. velocities below and including the lowest divergence which for fcc metals occurs for pure edge dislocations only. Due to the differences in geometry, the line tension of mixed dislocations in bcc metals first diverges at the lowest shear wave speed at a character angle of ${\vartheta}=\arctan\left(\sqrt{2}\right)$. All four figures [fig:linetension-bcccontour] are therefore shown for ${\beta\_{\bar\mu}}$ below that speed, i.e. ${\beta\_{\bar\mu}}\leq \sqrt{(c\_{11}-c\_{12})/2\bar{\mu}}$ for Fe, Ta, and W, and ${\beta\_{\bar\mu}}\leq \sqrt{c\_{44}/\bar{\mu}}$ for Nb. We now turn our attention to those figures: Even the almost isotropic tungsten (figure [fig:linetension-bcccontour] (a)) exhibits some deviations in the stability of mixed “almost edge” dislocations in the high velocity regime compared to the truly isotropic case (figure [fig:linetension-fcccontour] (a)). Within tantalum, pure screw and edge dislocations appear stable, see figure [fig:linetension-bcc]. However, upon inspection of figure [fig:linetension-bcccontour] (b), the existent unstable region splits into two parts which move away from angles characterizing pure edge, as seen within the figure. Additionally, a third unstable region forms close to the screw configuration at angles ϑ > 0, and this explains the small decrease we see in the nominally screw case shown in figure [fig:linetension-bcc] (a). In iron, which is even more anisotropic, the three unstable regions extend further into the lower velocity regime, and the near screw instability overlaps more with the pure screw configuration. Again, these features were not visible in the figure [fig:linetension-bcc] for pure screw and edge. What was visible already there, however, is the development of the third unstable region at small angles ϑ and moderately high velocities leading to the very pronounced decrease in the line tension for nominally screw dislocations. As ${\beta\_{\bar\mu}}\to1$, those (almost) screw dislocations with small positive ϑ become stable once more. Despite an unknown constant in the line tension which may shift the whole graph, it seems unlikely that this decrease (cf. figure [fig:linetension-bcc] (a)) will move into the positive region entirely, considering that for even small ϑ that dip extends orders of magnitude further into negative values. Niobium, finally, behaves slightly different than the other bcc metals we have discussed thus far: The two unstable regions in the range − π/2 < ϑ < 0 have almost connected to one near the lowest transverse shear wave velocity ${\beta}\_{\bar \mu}\to\sqrt{c\_{44}/\bar\mu}\approx0.82$ and are located at intermediate (negative) angles ϑ, see figure [fig:linetension-bcccontour] (c). The reason this figure looks markedly different is that niobium has an anisotropic Zener ratio A which is smaller than one (in contrast to all other metals we have discussed). Once more, this feature was not visible in the plots for pure screw and edge dislocations (figure [fig:linetension-bcc]) which, without further inspection, may give the illusion of all Nb dislocations being stable, independent of subsonic velocity. Lastly, we emphasize that because all elastic wave speeds in anisotropic crystals are direction dependent, defining a border between “subsonic” and “transonic” is nebulous: At sufficiently high velocities, the dislocation will be moving faster than the lowest shear wave speed with respect to one direction in the crystal, but not with respect to the lowest shear wave speed with respect to another direction. We have also seen, that the divergence in the dislocation field is not necessarily equal to the lowest shear wave speed associated to the direction of dislocation motion. Therefore, it makes more sense to distinguish between motions below and above the “critical” velocity where displacement gradients of dislocations are divergent, and this velocity depends upon the character angle ϑ of the dislocation. Within this work, we hence use the term “transonic” as referring to a velocity above this critical velocity, a distinction which is unambiguously defined. Since according to our table [tab:velocities] this velocity is very close to the lowest shear wave speed associated to the direction of dislocation motion, our slight abuse of the term “transonic” seems justified. Discussion and comparison to molecular dynamics simulations ----------------------------------------------------------- Comparing our results (${\bar{\Gamma}}({\vartheta},v)$) to earlier work, our figures [fig:linetension-fcccontour] are in agreement with the notion of instabilities within MD simulations for pure edge dislocations only, where “jumps” into the transonic regime were observed; Screw dislocations transitioned smoothly into the transonic regime) . Within those works it was assumed that in analogy to the isotropic case divergences would occur at the lowest shear wave speed corresponding to the direction of motion of the dislocation. As we have demonstrated, this is not necessarily the case for pure screw and edge dislocations and the actual critical velocity follows from eq., and therefore depends on the geometry of the crystal, the slip plane, and on the character of dislocation. Typically, the critical velocity and the lowest shear wave speed corresponding to the direction of dislocation motion are very close together (see table [tab:velocities]), and are difficult to discern within the resolution of the MD simulations without detailed knowledge of the displacement gradient field as employed within this work. What we presently have found for the line tension prelogarithmic factor (${\bar{\Gamma}}({\vartheta},v)$) of mixed dislocations, is that they diverge at the lowest shear wave speed only for a particular character angle, and for geometrical reasons, this angle is ϑ → ± π/2 for fcc metals and ${\vartheta}\to\arctan\left(\sqrt{2}\right)$ for bcc metals. In a further effort towards quantitative comparison with available data, we initially focus on recent MD simulations performed by Oren and coworkers . The simulations were conducted on moving steady state nominally screw and edge dislocations within a copper sample at velocities both approaching, and exceeding the estimated first shear wave barrier. As with the other simulations examined, the screw dislocations gradually transitioned from subsonic to transonic motion with increasing applied stress. On the contrary, the nominally edge dislocation simulations exhibited the characteristic discontinuity in velocity versus stress at approximately the first shear wave barrier, which was recorded at a velocity of vcrit ≈ 1.63 km/s and is close to our computed value of vcrit ≈ 1.61 km/s (table [tab:velocities]). The authors of graphically report the formation and expansion of nuclei along the dislocation line at this velocity as the mechanism for transonic transition (for dislocations longer than ∼ 10∣b⃗∣). We identify this mechanism as an instability, and proceed to apply our line tension analysis. However, observing figure [fig:linetension-fcc] (b), we notice that the line tension approaches zero at a velocity closer to 1 km/s. We attribute this difference both to the existence of a constant (Γ0) which does not scale logarithmically with the dislocation line length, and also the energetic interaction of the two separated partial dislocations which is beyond the scope of our analyses. For example, a positive constant would translate the whole curve upwards and hence the line tension would become negative at a higher velocity which is closer to the reported critical velocity. The same conclusion may be drawn for nickel when comparing our results to the MD simulations of . Also within these MD simulations a discontinuity in edge dislocation velocity versus stress is observed closer to the critical velocity than within our figure [fig:linetension-fcc] (b). Similarly, due to the omission of an unknown constant in our line tension calculation, a comparison of results can only be made qualitatively. Even a moderate shift of our curves by a constant can move the crossing of the zero axis significantly closer to the critical velocity, especially since the asymptotes become fairly steep. In ref. , MD simulations for Al and Ni lead to the conclusion that the discontinuity in velocity happens slightly below the lowest shear wave speed (which deviates from ours because the interatomic potential employed by the authors yields differing values for their elastic constants) corresponding to the direction of dislocation motion, and therefore agrees somewhat better with our present study. Conclusion and Outlook ====================== We have presented a generalized analysis of the line tension prelogarithmic factor (${\bar{\Gamma}}({\vartheta},v)$) of dislocations within anisotropic media for both an arbitrary character angle, within the slip plane, as well as a constant, finite velocity. We have applied our analyses towards computing the line tension of a suite of fcc and bcc structured crystals over a range of character angles and subsonic velocities. Upon including full elastic anisotropy, we find that certain character angles exhibit a diverging, negative line tension with increasing velocity, suggesting that the concept of straight dislocations within these regimes of ϑ-v space is energetically unfavorable, and likely should not be applied. In addition, we find that the lowest material shear wave velocity which is responsible for the diverging line tension is not necessarily parallel to the direction of dislocation motion, as often assumed , but rather contained within the plane defined by the line direction of the dislocation under consideration. Finally, we have demonstrated that the line tension of bcc structured materials lacks mirror symmetry (independent of velocity) with respect to the dislocation character angle about the nominally screw, or edge orientation. We attribute this asymmetry in the line tension to the geometry of the bcc crystalline lattice (both Burgers vector and slip plane orientation) which lacks symmetry with respect to the reference cubic crystal axes. Based upon our bcc results, figures [fig:linetension-bcccontour] (b)–(d), we may make a prediction about bcc structured Fe, Nb, and Ta. We expect that MD simulations will show that pure edge dislocations in Fe, Nb, and Ta do not “jump” into the transonic regime with increasing velocity; However, mixed dislocations within a range of angles ϑ do; This is particular to the bcc structure, and dependent upon the inclusion of elastic anisotropy (cf. figure [fig:linetension-bcc] (b)). We interpret “transonic” as velocities above those at which the displacement gradients of steady-state moving dislocations diverge. We emphasize that this speed, with the inclusion of elastic anisotropy, is typically close to, but less than the lowest shear wave speed corresponding to the direction of dislocation movement. The two speeds coincide only for select character angles (ϑ) in fcc and bcc metals. Both the analyses and the results of this work delineate regimes of forbidden velocities within a two dimensional dislocation framework, where dislocations are interpreted to be straight, infinite entities. We anticipate that both the existence and identification of these regimes will aid in enhancing our understanding of dynamic plasticity within fully anisotropic crystals under high rates of loading. Acknowledgements ---------------- We thank D. J. Luscher for enlightening discussions. B. A. S. thanks Joshua Crone for fruitful discussions. We also thank the anonymous referee for valuable comments. This work was performed in part under the auspices of the U.S. Department of Energy under contract DE-AC52-06NA25396. In particular, the authors are grateful for the support of the Advanced Simulation and Computing, Physics and Engineering Models Program. In addition, research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-17-2-0224. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. --- 1. One exception is ref. , as the authors discussed only screw dislocations and naturally no jump was found there.[↩](#fnref1) 2. What that new shape is exactly will be more difficult to determine and may be pursued in future studies.[↩](#fnref2) 3. Since we are considering Cartesian coordinates we do not worry about co-/contra-variant indices.[↩](#fnref3) 4. We follow ref. below.[↩](#fnref4) 5. Analytic results for $\mat S$, $\mat B$ are much more difficult to obtain and are known only in the simplest of cases, namely within the limit of isotropy , and for dislocations oriented along $\left<110\right>$ within an fcc lattice .[↩](#fnref5) 6. By considering differences in energy between two similar dislocation configurations, within a sufficiently large crystal the difference in linear elastic strain and kinetic energy will always dominate because it scales logarithmically with the crystal radius, R. We hence neglect effects associated with the dislocation core, not only for simplicity but also assuming that the cores remain similar across two configurations and their effects on the line tension is therefore subleading. For additional details on the energetics of the dislocation core, we refer to, e.g., .[↩](#fnref6) 7. A second suite of computations of was performed within Mathematica for validation purposes. Within this implementation, differentiation w.r.t. ϑ was performed symbolically and only the integrations over ϕ were completed numerically.[↩](#fnref7) 8. Note that our definition of γ matches the one known from special relativity and is related to Weertman’s notation via ${\beta}\_{\textrm{W}}\rightsquigarrow 1/{\gamma}$, $a\_{\textrm{W}}\rightsquigarrow c$.[↩](#fnref8) Line tension of a dislocation moving through an anisotropic crystal =================================================================== ### April 16, 2018 Los Alamos National Laboratory, Los Alamos, NM, 87545, USA Army Research Laboratory, Aberdeen Proving Ground, MD, 21005, USA Plastic deformation, at all strain rates, is accommodated by the collective motion of crystalline defects known as dislocations. Here, we extend an analysis for the energetic stability of a straight dislocation, the so-called line tension (Γ), to steady-state moving dislocations within elastically anisotropic media. Upon simplification to isotropy, our model reduces to an explicit analytical form yielding insight into the behavior of Γ with increasing velocity. We find that at the first shear wave speed within an isotropic solid, the screw dislocation line tension diverges positively indicating infinite stability. The edge dislocation line tension, on the other hand, changes sign at approximately 80% of the first shear wave speed, and subsequently diverges negatively indicating that the straight configuration is energetically unstable. In anisotropic crystals, the dependence of Γ on the dislocation velocity is significantly more complex; At velocities approaching the first shear wave speed within the plane of the crystal defined by the dislocation line, Γ tends to diverge, with the sign of the divergence strongly dependent on both the elastic properties of the crystal, and the orientation of the dislocation line. We interpret our analyses within the context of recent molecular dynamics simulations (MD) of the motion of dislocations near the first shear wave speed. Both the simulations and our analyses are indicative of instabilities of nominally edge dislocations within fcc crystals approaching the first shear wave speed. We apply our analyses towards predicting the behavior of dislocations within bcc crystals in the vicinity of the first shear wave speed. Keywords: moving dislocations, line tension, dynamic loading, anisotropy Motivation ========== Metallic plasticity is accommodated by the movement of linear crystalline defects known as dislocations. In regimes of high-rate loading , such as impact , the velocity of dislocations may approach the intrinsic elastic transverse wave speed of the continuum to accommodate high rates of material deformation. A suite of recent molecular dynamics (MD) simulations as well as some experimental data suggest that dislocation velocities can reach or even exceed the transverse wave speed (ct) of the material. Within these dynamic regimes, both the energetics and the stability of dislocations markedly differs from that as predicted within regimes of quasi-static deformation. As an example, upon considering the interaction energy between a solid and a dynamic dislocation moving at velocities (v) approaching the transverse wave speed, discrete dislocation theory predicts that the energy of nominally screw and edge dislocations diverges as ∼ O(1 − (v/ct)2)− 1/2 and ∼ O(1 − (v/ct)2)− 3/2, respectively . Both experiment and numerical simulations suggest that dislocations may traverse at, or even exceed, the transverse wave speed of the continuum. On the other hand, consideration of dislocation theory suggests divergent energies in these regimes. Therefore, the notion of straight, steady state dislocation motion at velocities approaching the transverse wave speed requires reconciliation. Central to stability analyses of dislocations is the dislocation line tension (Γ). Analogous to a taut string, dislocations exhibit a tendency to straighten, minimizing their length, and also their interaction energy with the crystal. The line tension is a scalar quantity related to the configurational force acting to straighten and minimize the length of a dislocation, if the line tension is positive (indicating stability of the configuration) . Conversely, a negative line tension indicates instability, and the related force will drive an unstable dislocation away from its current configuration. The line tension depends upon the orientation, i.e. the angle (ϑ) between the dislocation Burgers vector (b⃗) and the dislocation line direction (t⃗) . Including an orientational dependence in the small bowout limit, as with anisotropic surfaces , or dislocations as considered here, the line tension (Γ) is related to the (dislocation) energy E(ϑ, v) as $$\begin{aligned} {\Gamma}&=\left(1+\frac{{\partial}^2 }{{\partial}{\vartheta}^2}\right)E({\vartheta},v) \,, \label{eq:def-linetension}\end{aligned}$$ where v denotes the dislocation velocity. Eq. is valid for curved as well as for straight dislocations (which are merely a special case of the former) . Within this work, we assume a bow-out type mechanism of instability which occurs along a dislocation glide plane. Independent of the details of the mechanism, for a characteristic dislocation length, L, the dislocation line tension reduces to $$\begin{aligned} {\Gamma}&={\bar{\Gamma}}\frac{\bar{\mu}b^2}{4\pi}\ln \left(\frac{\mathcal{L}}{r\_0}\right)+\Gamma\_0 \,, \label{eq:def-bowout}\end{aligned}$$ where ${\bar{\Gamma}}{\vcentcolon=}{\bar{\Gamma}}({\vartheta},v)$ denotes the dislocation line tension prelogarithmic factor, $\bar\mu$ is the average shear modulus, and r0 represents the cutoff parameter introduced to circumvent divergent dislocation energies. In addition, the term Γ0 depends upon the geometry of the bow-out under consideration (e.g., small bow-out ). The two constants r0 and Γ0 depend upon details of the dislocation core and details of the specific bow-out mechanism under consideration, respectively, both of which are beyond the scope of the present work. In the limit of an infinitely long dislocation (L ≫ r0), $\Gamma\sim(\bar{\mu} b^2/4\pi){\bar{\Gamma}}\ln(R/r\_0)$, with R (of the order of interdislocation spacing) denoting the radius of an outer cylinder surrounding the dislocation core, however the magnitude of ${\bar{\Gamma}}$ does not depend upon L, R, or detailed geometrical features of the bow-out. Within this work we focus our attention on ${\bar{\Gamma}}$ only within eq., since it represents the contribution of the cumulative strain and kinetic energy of the crystal to the dislocation line tension, and at L ≫ r0 is expected to dominate within eq.. The objective of this work is to study the dependence of the dislocation line tension prelogarithmic factor ${\bar{\Gamma}}$ on both the velocity and orientation of the dislocation line (i.e. ${\bar{\Gamma}}({\vartheta},v)$). The dependence of the dislocation line tension on the velocity provides a mechanistic interpretation of the critical velocity at which a moving (straight) dislocation becomes energetically unstable with respect to bow-out. As noted, MD simulations for fcc metals such as show [1](#fn1) that at some critical driving stress the dislocation velocity “jumps” to the transonic regime, exhibiting a discontinuity in the velocity response to an applied stress, (and a similar jump to supersonic happens around the longitudinal sound speed). In fcc metals these rapid shifts in velocity are limited to edge dislocations only, where the Burgers vector is perpendicular to the dislocation line, indicating that the character of the dislocation is important towards understanding the mechanics of dislocation motion within high velocity regimes. Furthermore, there remains sustained interest in delineating the limits of applicability of the dislocation drag coefficient to infinite, straight, dislocations in regimes of high dislocation velocity. Current models assume perfectly straight dislocations and thereby introduce artificial (unphysical) divergences at the transverse sound speed within an infinite isotropic crystal. Some authors have demonstrated, that if terms accounting for acceleration are included into the models of screw dislocation fields in the isotropic limit, the divergence that is otherwise present at the transverse sound speed in the steady state approximation, disappears. While this may explain the smooth transition of screw dislocations into the transonic regime, it does not explain the sudden jump-like acceleration from sub- to transonic observed for edge dislocations at some critical stress . Our plausible interpretation of these findings is as follows. At some critical velocity which is a fraction of the transverse sound speed, the (edge) dislocation line tension (eq. ) becomes negative, favoring bow-out of the dislocation line. With the configurational change of the dislocation line, the drag coefficient which relates driving stress to dislocation velocity also decreases; The dislocation, which is no longer straight, then experiences an abrupt acceleration into the transonic regime. This description is qualitatively in agreement with recent MD simulations, e.g. , where a bowing nucleus is observed to form on a transient dislocation at the instant of acceleration. Hence, the objective presently is to generalize the work on anisotropic line tension of ref. to include a velocity dependence along the same lines as was previously done only within the isotropic case (where edge and screw dislocations decouple making the orientation dependence of ${\bar{\Gamma}}$ trivial). This will provide an initial estimate for the evolution of the line tension with increasing velocity, leading to a critical velocity, beyond which a straight dislocation is energetically unstable [2](#fn2). Our present study will be limited to the line tension calculated from the dislocation self-energy, i.e. we consider a single infinitely long dislocation and infinitesimal bowout, leaving the contribution from dislocation-dislocation (and dislocation-crystal lattice) interactions to future work. The remainder of this work is organized as follows. Within the next section, we begin by reviewing the equations governing the dynamic balance of linear momentum for a linear elastic solid. We then provide a brief review of the elastic field equations for an infinite dislocation situated within a linearly elastic anisotropic crystal of either fcc or bcc structure. With the elastic field equations, and the governing equations of dynamic equilibrium we are able to formulate expressions for both the energy and line tension of a dislocation within an anisotropic crystal with respect to both the orientation (ϑ) and the velocity (v⃗) of the straight dislocation. Within section [sec:isotropic], we consider the limit of elastic isotropy, demonstrating that our expressions for both the energy and line tension reduce to already existent expressions available within the literature . Within section [sec:cubic], we apply our expression for the line tension to several anisotropic fcc and bcc structured crystals examining the dependence of ${\bar{\Gamma}}$ on both the velocity and orientation of the dislocation line. In the final section [sec:conclusion], we provide a brief synopsis of the results of this work. Method ====== We proceed with a brief summary of the general method for deriving displacement gradient fields of dislocations. Upon integrating over the displacement gradient fields, both the dislocation energy and line tension may be obtained. More details can be found within the extensive reviews . [h!t!b] lccccc & ρ[g/ccm] & c11[GPa] & c12[GPa] & c44[GPa] & A fcc & & & & & Al & 2.70 & 106.75 & 60.41 & 28.34 & 1.22 Cu & 8.96 & 168.3 & 122.1 & 75.7 & 3.28 Ni & 8.90 & 248.1 & 154.9 & 124.2 & 2.67 bcc & & & & & Fe & 7.87 & 226.0 & 140.0 & 116.0 & 2.70 Nb & 8.57 & 246.5 & 134.5 & 28.73 & 0.51 Ta & 16.4 & 260.2 & 154.4 & 82.55 & 1.56 W & 19.3 & 522.39 & 204.37 & 160.58 & 1.01 Within the scope of this work, we are concerned with cubic (fcc and bcc) crystals, whose stiffness tensor contains the set of three cubic elastic constants {c11, c12, c44}. Tensors of elastic constants are typically defined in Cartesian coordinates which are aligned with the crystal axes. For orientations of the coordinate system which are not aligned with the crystal bases, it is necessary to rotate the stiffness tensor into the prescribed basis. The tensor of second order elastic constants (i.e., the fourth order elastic stiffness tensor) within the crystal reference frame reads $$\begin{aligned} C\_{ijkl}&=c\_{12}{\delta}\_{ij}{\delta}\_{kl}+c\_{44}\left({\delta}\_{ik}{\delta}\_{jl}+{\delta}\_{il}{\delta}\_{jk}\right)-\left(2c\_{44}+c\_{12}-c\_{11}\right)\sum\_{\alpha=1}^3{\delta}\_{i\alpha}{\delta}\_{j\alpha}{\delta}\_{k\alpha}{\delta}\_{l\alpha} \,,\label{eq:STIFF}\end{aligned}$$ where the sum over repeated indices is implicit, and the last term which includes the explicit summation represents the fourth order identity tensor. Within table [tab:values-metals] we list the material specific elastic constants utilized within this study, as well as the material densities and Zener anisotropy ratio. In the isotropic limit c11 → c12 + 2c44, and the three cubic constants reduce to the two Lamé constants: c12 → λ, c44 → μ. The governing equations to which the displacement gradient field provides a solution are the equations of motion (e.o.m.) and the (leading order) stress-strain relations known as Hooke’s law: $$\begin{aligned} {\partial}\_i{\sigma}\_{ij}&=\rho\ddot u\_j\,, & {\sigma}\_{ij}&=C\_{ijkl}{\epsilon}\_{kl}=C\_{ijkl}u\_{k,l}\,, {\nonumber}\\ {\epsilon}\_{kl}&{\vcentcolon=}\frac12\left(u\_{k,l}+u\_{l,k}\right) \,, \label{eq:Hooke}\end{aligned}$$ where εij denotes the infinitesimal strain tensor, ρ the material density, and the last equality in the first line follows from the elastic constants’ Voigt symmetry. Furthermore, we have introduced the notation [3](#fn3) $u\_{k,l}{\vcentcolon=}{\partial}\_l u\_k$ for the gradient of the displacement field uk, and $\ddot u\_j{\vcentcolon=}\frac{{\partial}^2 u\_j}{{\partial}t^2}$ denoting temporal derivatives. In the static case $\ddot{\vec{u}}=\vec{0}$, however even at constant, finite dislocation velocity a similar simplification may be devised. In the dynamic case with constant velocity, the displacement field may be expressed in terms of a time-dependent shifted position (x⃗ − v⃗t), i.e. uk(xi, t) = uk(xi − vit), and hence its temporal derivative can be expressed in terms of its gradient: $\dot u\_i=-v\_j u\_{i,j}$. Then the e.o.m. (eq. [eq:Hooke]) may be simplified as $$\begin{aligned} 0&={\partial}\_i{\sigma}\_{ij}-\rho\ddot u\_j=C\_{ijkl}u\_{k,il}-\rho v\_iv\_l u\_{j,il} \,,\end{aligned}$$ or upon defining “effective” elastic constants (or the “dynamic stiffness tensor”) : $$\begin{aligned} \hat C\_{ijkl}u\_{k,il}&= 0\,, & \hat C\_{ijkl}{\vcentcolon=}C\_{ijkl}-\rho v\_iv\_l {\delta}\_{jk} \,. \label{eq:eom1}\end{aligned}$$ We hence need only to find uk which satisfies the differential equation. Finding solutions for dislocation displacement gradients -------------------------------------------------------- Throughout this work, we limit our analyses to infinite, straight dislocations. This simplification removes any dependence of the field quantities on the position along the dislocation line. A. N. Stroh describes a method to compute solutions for u⃗ based upon the ansatz [4](#fn4) $$\begin{aligned} u\_k&=\frac{D A\_k}{2\pi i}\ln\left(m\_j x\_j+p n\_j x\_j\right) \,, \label{eq:disp}\end{aligned}$$ where the perpendicular unit vectors m⃗ and n⃗ are conveniently chosen normal to the sense vector t⃗ of the dislocation whose displacement field we are seeking: i.e. t⃗ = m⃗ × n⃗. Inserting this ansatz into the e.o.m. transforms the differential equation into an eigenvalue problem in terms of the unknown coefficients Ak and p. The overall factor D is finally determined by the boundary conditions that a Burgers circuit around the dislocation line has a discontinuity described by the Burgers vector b⃗, and that there be no external forces present at the dislocation core. Due to Voigt symmetry, the eigenvalue problem can be formulated in terms of a 6-dimensional vector $\vec\zeta$ and associated 6 × 6 matrix $\mat N$ comprised of four 3 × 3 blocks, i.e. $\mat N\cdot\vec\zeta=p\vec\zeta$, see for details on this “sextic formalism”. On the other hand, one can reformulate the theory so that this sextic eigenvalue problem is replaced by one of evaluating a set of definite integrals . We will now summarize this (simpler) latter approach in the notation of ref. . The crucial observation is that the unit vectors m⃗, n⃗ are defined only up to an arbitrary angle ϕ. Hence, as one can show, averaging over this angle yields a solution for uj, k in terms of the averaged matrix $$\begin{aligned} \langle \mat N\rangle&=\frac1{2\pi}\int\limits\_0^{2\pi}\mat N d\phi =\begin{pmatrix} \mat S & \mat Q \\ \mat B & \mat S^T \end{pmatrix} \,,\end{aligned}$$ with $$\begin{aligned} \mat S&=-\frac1{2\pi}\int\limits\_0^{2\pi}(nn)^{-1}(nm)\,{\mathrm{d}}\phi \,, & \mat S^T&=-\frac1{2\pi}\int\limits\_0^{2\pi}(mn)(nn)^{-1}{\mathrm{d}}\phi \,,{\nonumber}\\ \mat Q&=-\frac1{2\pi}\int\limits\_0^{2\pi}(nn)^{-1}{\mathrm{d}}\phi \,, & \mat B&=-\frac1{2\pi}\int\limits\_0^{2\pi}\left[(mn)(nn)^{-1}(nm)-(mm)\right]{\mathrm{d}}\phi \,, \label{eq:thematrix}\end{aligned}$$ where we have employed the shorthand notation $(ab)\_{jk}{\vcentcolon=}a\_i \hat C\_{ijkl} b\_l$. In particular, the displacement field reads $$\begin{aligned} u\_j(\abs{\vec{x}},\phi)&=-\frac{b\_l}{2\pi}\left\{S\_{jl}\ln\abs{\vec{x}}-S\_{il}\int\limits\_0^\phi\left[(nn)^{-1}(nm)\right]\_{ji}{\mathrm{d}}\phi'-B\_{il}\int\limits\_0^\phi(nn)^{-1}\_{ji}{\mathrm{d}}\phi'\right\} \,. \label{eq:uj-sol}\end{aligned}$$ We emphasize that the displacement field is comprised of the superposition of a radially dependent term ($\ln\abs{\vec{x}}$) and an angular term ϕ. Upon selecting a coordinate system which rotates with ϕ about the dislocation line (mixi = r, nixi = 0), the displacement gradient computes to : $$\begin{aligned} u\_{j,k}(r,\phi)&=\frac{\tilde{u}\_{j,k}(\phi)}{r}\,, {\nonumber}\\ \tilde{u}\_{j,k}(\phi)&=-\frac{b\_l}{2\pi}\left\{m\_k S\_{jl}-n\_k\left[(nn)^{-1}(nm)\right]\_{ji}S\_{il}-n\_k(nn)^{-1}\_{ji}B\_{il}\right\} \,.\label{eq:ukl-sol}\end{aligned}$$ Notice that the ũj, k(ϕ) dependence on ϕ resides within the unit vectors m⃗, n⃗. The tensors $\mat S$, and $\mat B$ are constant projections of the dynamic stiffness tensor onto the plane with unit normal vector t⃗ as computed from eq. ; Their values depend on the elastic constants as well as (constant) dislocation velocity, and material density, cf.. These integrals are directly amenable to numerical integration for any given set of material constants [5](#fn5), and this numerical integration need be performed only once for a given dislocation velocity, material, and orientation. The dislocation displacement gradients may subsequently be algebraically assembled according to eq.. Note, that this method is limited to the case of steady-state motion leading to the simpler e.o.m.. Its strength is its simplicity; if one were to study acceleration as well (which is beyond the scope of our present work), a more general method using dynamic Green functions would be necessary . In the steady state limit, of course, the same dislocation solution is reproduced as with the current method, see e.g. for the isotropic case (which we discuss in section [sec:isotropic] below). Energy and line tension ----------------------- The concept of line tension, as applied to a dislocation, follows that of the accumulation of strain energy incurred through undulation of a material surface; The surface tension is defined as the change in energy per unit length due to an infinitesimal undulation of a surface, with respect to a nominally flat surface. The sign of this difference in energy indicates if a wavy configuration of the material surface is energetically favorable. When considering a dislocation within this context, the line tension is related to both the energy, and the orientation of the dislocation line through the relation eq., cf. As noted in the introduction, this relation omits a constant which depends upon the three dimensional geometry of the instability under examination . The infinitesimal linear elastic strain energy density surrounding the dislocation core follows as dWs = σijdεij, or upon integration $$\begin{aligned} W^s&=\int{\sigma}\_{ij}{\mathrm{d}}{\epsilon}\_{ij}=\frac12C\_{ijkl}{\epsilon}\_{ij}{\epsilon}\_{kl} \,.\end{aligned}$$ Due to Voigt symmetry of the elastic constants one may substitute the infinitesimal strains ${\epsilon}\_{ij}{\vcentcolon=}(u\_{i,j}+u\_{j,i})/2$ with the displacement gradients. As before, we emphasize that ui, j(ϑ, ϕ) depends on the dynamic stiffness tensor, as defined earlier, while the elastic strain energy density, as written, depends linearly on the nominal (static) elastic stiffness tensor. Upon considering finite velocity, in addition to the linear elastic strain energy density Ws we also account for the kinetic energy density due to the temporal evolution of the displacement field, $$\begin{aligned} W^k&=\frac{\rho}2\left({\frac{{\partial}u\_i}{{\partial}t}}\right)^2 =\frac{\rho}2 v\_i v\_j u\_{k,i} u\_{k,j} \,,\end{aligned}$$ where we have utilized the relation $\dot u\_k=-v\_j u\_{i,j}$. Hence, the total energy density is $$\begin{aligned} W({\vartheta},v)&=W^s+W^k=\left(C\_{ijkl}+\rho v\_i {\delta}\_{jk} v\_l\right) u\_{j,i} u\_{k,l}/2 \,,\label{eq:eng\_dens}\end{aligned}$$ which depends on both the character of the dislocation and the velocity through the displacement gradients. Upon integrating the energy density per unit length through a hollow cylinder defined by inner and outer radii r0 and R, respectively, the energy per unit length E of a moving dislocation is [6](#fn6) $$\begin{aligned} E&=\int\_{r\_0}^{R}r \,{\mathrm{d}}r\int\_{0}^{2\pi}{\mathrm{d}}{\phi}\, W {\nonumber}\\& =\frac12\left(C\_{ijkl}+\rho {\delta}\_{jk} v\_i({\vartheta}) v\_l({\vartheta})\right) \left[\int\_{0}^{2\pi}{\mathrm{d}}{\phi}\, \tilde{u}\_{j,i}({\vartheta}) \tilde{u}\_{k,l}({\vartheta})\right]\ln\left(\frac{R}{r\_0}\right) \,, \label{eq:energy1}\end{aligned}$$ where $\tilde{u}\_{i,j}{\vcentcolon=}r u\_{i,j}$ is independent of r (see eq. ). Since the only radial dependence resides within ui, j ∝ 1/r, the integral ui, juk, lrdr ∝ ln(R/r0) which depends upon an arbitrary constant. Upon insertion of eq. into eq. our final expression for the line tension of a steady state moving dislocation within an anisotropic infinite medium is $$\begin{aligned} {\Gamma}&=\frac12\left(1+\frac{{\partial}^2 }{{\partial}{\vartheta}^2}\right) \left(C\_{ijkl}+\rho {\delta}\_{jk} v\_i({\vartheta}) v\_l({\vartheta})\right) \left[\int\_{0}^{2\pi}{\mathrm{d}}{\phi}\,\tilde{u}\_{j,i}({\vartheta}) \tilde{u}\_{k,l}({\vartheta})\right]\ln\left(\frac{R}{r\_0}\right) \,. \label{eq:linetension}\end{aligned}$$ We note that our expression for Γ does not include an arbitrary constant which is O(1) in the zero velocity limit. This is due to our ansatz in eq. where we have assumed an infinite, straight dislocation. We will apply eq. to dislocations of arbitrary character angle contained within the slip plane with subsonic velocities within both isotropic solids as well as in anisotropic fcc and bcc crystals. Crystal lattice dependence and numerical implementation ------------------------------------------------------- Within the previous subsections, we have detailed the computation of the displacement gradient, the dislocation line energy, and dislocation line tension. Within this section, we relate the two perpendicular vectors (m and n), which the displacement gradient depends on, to the geometry of the crystalline lattice. We employ Cartesian coordinates aligned with the crystal axes, and consider a fixed Burgers vector b⃗ and a unit vector normal to the slip plane under consideration, n⃗0. The material direction of both the Burgers vector and slip plane depend upon the crystal structure. The sense vector of the dislocation t⃗ and its dependence on ϑ depends upon the Burgers vector and slip plane normal as $$\begin{aligned} \vec{t}({\vartheta})&=\frac1b\left[\vec{b}\cos{\vartheta}+\vec{b}\times\vec{n}\_0\sin{\vartheta}\right] \,, \label{eq:def-sensevec}\end{aligned}$$ where ϑ is the angle between t⃗ and b⃗ and $b{\vcentcolon=}\abs{\vec{b}}$. Therefore, ϑ = 0 and ϑ = π/2 correspond to pure screw and pure edge dislocations, respectively. Since we assume that dislocation bow-out will occur within the slip plane spanned by t⃗ and b⃗, we may define a third mutually perpendicular vector m⃗0(ϑ) in the direction of motion as $$\begin{aligned} \vec{m}\_0({\vartheta})=\vec{n}\_0\times \vec{t}({\vartheta}) \,.\end{aligned}$$ We will assume, without loss of generality, that the dislocation moves in that direction, i.e. v⃗(ϑ) = vm⃗0(ϑ). From m⃗0, n⃗0 we construct $$\begin{aligned} \vec{m}({\vartheta},{\phi})&=\vec{m}\_0({\vartheta})\cos{\phi}+ \vec{n}\_0\sin{\phi}\,, {\nonumber}\\ \vec{n}({\vartheta},{\phi})&=\vec{n}\_0\cos{\phi}- \vec{m}\_0({\vartheta})\sin{\phi}\,,\end{aligned}$$ which span the plane normal to t⃗, as illustrated within figure [fig:geometry]. [fig:geometry] In order to evaluate eq. we must resort to numerical methods, since it is analytically insoluble. We have implemented eq. within a numerical code [7](#fn7) which relies upon the stiffness tensor (Cijkl), the Burgers vector direction (b⃗/∣b⃗∣), and the slip plane normal (n⃗0) as input. The integrals over ϕ (i.e. ũi, j, Bij, etc.) are performed using a trapezoidal numerical integrator with a step size of δϕ ≈ π/500; The derivatives with respect to ϑ are evaluated by successively differencing evaluations of E(ϑ, v) (eq. ) over small increments of δϑ (e.g. $\frac{{\partial}E({\vartheta},v)}{{\partial}{\vartheta}} {\vcentcolon=}(E({\vartheta}+\delta {\vartheta}, v) - E({\vartheta}-\delta {\vartheta}, v)) /2\delta {\vartheta}$), where δϑ ≈ π/600. For each of the materials investigated within this study, the resulting Γ has been evaluated at roughly 600 evenly spaced intervals of ϑ where − π/2 < ϑ < π/2, and 625 evenly spaced intervals of ${\beta\_{\bar\mu}}{\vcentcolon=}\sqrt{{\rho v^2}/{\bar{\mu}}}$ within the range $0 \leq {\beta\_{\bar\mu}}\leq 1$, with $\bar\mu{\vcentcolon=}\frac14\left(c\_{11}-c\_{12}+2c\_{44}\right)$ denoting the mean shear modulus. This results in ∼ 375, 000 data points for each material. Throughout the remainder of this work, it will be convenient to show and discuss results in the rescaled, dimensionless line tension prelogarithmic factor ${\bar{\Gamma}}$ via $$\begin{aligned} {\Gamma}&=\left(\frac{\bar\mu b^2}{4\pi}\ln\frac{R}{r\_0}\right){\bar{\Gamma}}\,. \label{eq:linetensionprefactor}\end{aligned}$$ Our results will be presented in terms of ${\bar{\Gamma}}({\vartheta},{\beta\_{\bar\mu}}(v))$ (for which we will use the terms “line tension” and “prelogarithmic factor” interchangeably) as a function of dislocation character ϑ and dimensionless, scaled velocity ${\beta\_{\bar\mu}}$. The isotropic limit =================== We begin by examining the energy, and line tension of a moving dislocation in the isotropic limit. In this limit, the expressions for the energy and line tension are considerably simplified and are analytically tractable, yielding insight towards our anisotropic results. Referring to eq., in the limit that c11 → c12 + 2c44 there is no orientational dependence dictated by the crystal symmetry; the dislocation displacement gradients, energy, and line tension do not depend upon the orientation of the dislocation within the crystal. Upon adopting a coordinate system fixed to the dislocation line with t⃗ oriented along $\hat z$ and n⃗0 oriented along $\hat y$, the ϑ dependence resides entirely within the Burgers vector, and the vectors defined within section [sec:method] reduce to $$\begin{aligned} \vec{t}&=(0,0,1)\,,& \vec{m}&=\left(\cos{\phi},\sin{\phi},0\right)\,,& \vec{n}&=\left(-\sin{\phi},\cos{\phi},0\right)\,,{\nonumber}\\ \vec{v}&=(v,0,0) \,, & \vec{b}&=(b\sin{\vartheta},0,b\cos{\vartheta})\,, & r&=\vec{m}\cdot\vec{x}=\sqrt{x^2+y^2} \,.\end{aligned}$$ One may verify that, in these coordinates, the well known solutions for stationary moving edge and screw dislocations within an isotropic crystal are reproduced: $$\begin{aligned} u\_{x,x}&=\frac{-b\_e y}{\pi {\beta\_{\textrm{t}}}^2 }\left(\frac{1/{\gamma\_{\textrm{l}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right) /{\gamma\_{\textrm{t}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{x,y}&=\frac{b\_e (x-t v)}{\pi {\beta\_{\textrm{t}}}^2 }\left(\frac{1/ {\gamma\_{\textrm{l}}}}{ \left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right)/ {\gamma\_{\textrm{t}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{y,x}&=\frac{b\_e (x-t v)}{\pi {\beta\_{\textrm{t}}}^2 }\left( \frac{1/{\gamma\_{\textrm{l}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{{\gamma\_{\textrm{t}}}\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right) }{ \left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{y,y}&=\frac{b\_e y}{\pi {\beta\_{\textrm{t}}}^2 }\left(\frac{1/{\gamma\_{\textrm{l}}}^3}{ \left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)}-\frac{\left(1-\frac{{\beta\_{\textrm{t}}}^2}{2}\right)/ {\gamma\_{\textrm{t}}}}{\left((x-t v)^2 +y^2/{\gamma\_{\textrm{t}}}^2\right)}\right) \,,{\nonumber}\\ u\_{z,x}&=-\frac{b\_s\, y/{\gamma\_{\textrm{t}}}}{2 \pi \left((x-t v)^2+y^2/{\gamma\_{\textrm{t}}}^2\right)}\,, \qquad\qquad u\_{z,y}=\frac{b\_s\, (x-t v)/ {\gamma\_{\textrm{t}}}}{2 \pi \left((x-t v)^2+y^2/{\gamma\_{\textrm{t}}}^2 \right)} \,, \label{eq:u-isotropic}\end{aligned}$$ where we have employed a dislocation fixed coordinate system (x → (x − vt)) and where [8](#fn8) $$\begin{aligned} {\gamma\_{\textrm{t}}}&=\frac{1}{\sqrt{1-{\beta\_{\textrm{t}}}^2}}\,, & {\beta\_{\textrm{t}}}&=\frac{v}{{c\_{\textrm{t}}}}\,, & {\gamma\_{\textrm{l}}}&=\frac{1}{\sqrt{1-{\beta\_{\textrm{l}}}^2}}\,, & {\beta\_{\textrm{l}}}&=\frac{v}{{c\_{\textrm{l}}}} \,,\end{aligned}$$ and ${c\_{\textrm{t}}}=\sqrt{{\mu}/\rho}$, ${c\_{\textrm{l}}}=\sqrt{\left({\lambda}+2{\mu}\right)/\rho}$ are the transverse and longitudinal sound speed, respectively. The edge and screw components of the Burgers vector, $b\_e{\vcentcolon=}b\sin{\vartheta}$ and $b\_s{\vcentcolon=}b\cos{\vartheta}$, are the only variables depending on the angle ϑ. Energy of a moving dislocation in an isotropic crystal ------------------------------------------------------ It is readily verified that even for finite velocities, the edge and screw components of dislocations within isotropic media do not interact energetically, and therefore both the displacements, and resulting energies of an arbitrarily oriented dislocation may be superposed. Specifically, since Cijklui, jedgeuk, lscrew = 0, there is no cross term in the total energy of a mixed dislocation and we are justified superposing the energies of edge and screw components. We note also that εxyedge = (ux, yedge + uy, xedge)/2 vanishes at y = 0 (and hence changes sign across the glide plane) at a dislocation velocity v = cR which is known as the Rayleigh wave velocity (i.e. the velocity of surface waves) . It is defined by the relation γlγt = (1 − βt2/2)− 2 and is always smaller than (but fairly close to) the transverse sound speed, cR < ct, a typical value being cR ≈ 0.93ct if Poisson’s ratio is 1/3 (corresponding to cl = 2ct). An additional simplification occurs in the strain energy of the screw dislocation: $W^s=\frac{{\lambda}}{2}({\epsilon}\_{ii})^2+{\mu}{\epsilon}\_{ij}{\epsilon}\_{ij}$, and since the strain of the screw dislocation is traceless, the dependence on λ will only appear for the edge dislocation, i.e. $$\begin{aligned} {\epsilon}\_{ii}^{\textrm{screw}} & = 0\,, & {\epsilon}\_{ii}^{\textrm{edge}} &=\frac{-b\,{c\_{\textrm{t}}}^2 y}{\pi {c\_{\textrm{l}}}^2 {\gamma\_{\textrm{l}}}\left((x-t v)^2 +y^2/{\gamma\_{\textrm{l}}}^2\right)} \,,\end{aligned}$$ see eqs.,. The total energy for an arbitrary dislocation orientation follows as $$\begin{aligned} E^{\textrm{mixed}}({\vartheta},v)&= E\_0\left\{{\gamma\_{\textrm{t}}}\cos^2{\vartheta}+\frac{\sin^2{\vartheta}}{{\beta\_{\textrm{t}}}^2}\left[\frac{8}{{\gamma\_{\textrm{l}}}}+4{\gamma\_{\textrm{l}}}+{\gamma\_{\textrm{t}}}^3\left(1-\frac{6}{{\gamma\_{\textrm{t}}}^2}-\frac{7}{{\gamma\_{\textrm{t}}}^4}\right)\right]\right\} \,, {\nonumber}\\ E\_0&=\frac{{\mu}b^2}{4\pi}\ln\left(\frac{R}{r\_0}\right) \,, \label{eq:energy-mixed}\end{aligned}$$ with the contributions from screw and edge dislocations scaled by cos2ϑ and sin2ϑ, respectively (see also ). In the limit v → 0 the energies for both edge and screw dislocations reduce to the well-known expressions $$\begin{aligned} E^{\textrm{screw}}(0,0)&=E\_0 \,, & E^{\textrm{edge}}(\pi/2,0)&=E\_0\frac{2({\lambda}+{\mu})}{{\lambda}+2{\mu}} =\frac{E\_0}{1-{\nu}} \,,\end{aligned}$$ where Poisson’s ratio is related to the Lamé constants via ν = λ/2(μ + λ). On the other hand, as the velocity approaches the transverse sound speed, both energies diverge as $$\begin{aligned} \lim\limits\_{v\to{c\_{\textrm{t}}}}E^{\textrm{screw}}(0,v)&\to\frac{E\_0}{\sqrt{1-{\beta\_{\textrm{t}}}^2}} \,,{\nonumber}\\ \lim\limits\_{v\to{c\_{\textrm{t}}}}E^{\textrm{edge}}(\pi/2,v)&\to \frac{E\_0}{2\sqrt{2}}\left(\frac{1}{(1-{\beta\_{\textrm{t}}}^2)^{\frac32}}+\frac{37}{4\sqrt{1-{\beta\_{\textrm{t}}}^2}}+\frac{16 (2-\nu)}{\sqrt{1-\nu }}+{\mathcal{O}}\left(\sqrt{1-{\beta\_{\textrm{t}}}^2}\right)\right) \,. \label{eq:isotropic-energy-divergences}\end{aligned}$$ All divergences appearing in the energy as well as the displacement gradients as v → ct can be traced back to the expression (nn)− 1 appearing in and, and are due to $$\begin{aligned} \det\left(nn\right)&=\det\left(\vec{n}\cdot\hat{\mat{C}}\cdot\vec{n}\right) =\det\left(\vec{n}\cdot\mat{C}\cdot\vec{n}-\rho \left(\vec{n}\cdot\vec{v}\right)^2{\mathbb{1}}\right) {\nonumber}\\ &=0 \,,\end{aligned}$$ as v → ct (or v → cl) and ϕ → ± π/2 in which case n⃗ ∥ v⃗. In fact, the leading divergence as v → ct and ϕ → ± π/2 is (nn)− 1 ∼ 1/(1 − βt2). Due to averaging over ϕ, the matrices $\mat{S}$, $\mat{B}$ exhibit a weaker divergence of $1/\sqrt{1-{\beta\_{\textrm{t}}}^2}$. Upon inspecting expression, it is clear that the edge dislocation has a divergence of order 1/(1 − βt2)3/2 in the displacement gradient (ui, j) originating from the collection of terms − nk[(nn)− 1(nm)]jiSil − nk(nn)ji− 1Bil. In the solution for pure screw dislocations, two types of simplification lead to the weaker divergence of $1/\sqrt{1-{\beta\_{\textrm{t}}}^2}$ in the displacement gradient: Firstly, $\mat{S}\cdot\vec{b}=0$ for geometrical reasons, and furthermore $\lim\limits\_{v\to{c\_{\textrm{t}}}}\mat{B}\cdot\vec{b}\propto\sqrt{1-{\beta\_{\textrm{t}}}^2}\to0$. Thus, all but the last term in vanish in the pure screw dislocation case, and that surviving term, nk(nn)ji− 1Bil, furthermore exhibits a weaker divergence than in the edge case. We emphasize that the leading divergences in ui, j outlined above occur only at particular angles, ϕ = ± π/2, i.e. perpendicularly to the direction of dislocation motion. This is also the reason why the degree of divergence in the dislocation energy E (which involves an integration over ϕ) is not twice that of ui, j, despite ui, j entering quadratically. Within this work, we limit our analysis to the subsonic regime. Solutions for the strain fields of straight dislocations have also been derived for the transonic and supersonic regimes . All of these solutions, however, diverge at ct and cl, i.e. they cannot describe the vicinity of the two sound speeds. For more recent literature on supersonic dislocations, see and references therein. Line tension and stability -------------------------- The line tension prelogarithmic factor (${\bar{\Gamma}}$) follows from application of eq. to E(ϑ, v), using our isotropic solution (eq. ) for E (see ref. for the static case). We immediately find $$\begin{aligned} {\Gamma}({\vartheta},v)&= \frac{E\_0}{{\beta\_{\textrm{t}}}^2}\left\{2\cos^2{\vartheta}\left[\frac{8}{{\gamma\_{\textrm{l}}}}+4{\gamma\_{\textrm{l}}}+{\gamma\_{\textrm{t}}}^3\left(1-\frac{6}{{\gamma\_{\textrm{t}}}^2}-\frac{7}{{\gamma\_{\textrm{t}}}^4}\right)-\frac{{\beta\_{\textrm{t}}}^2{\gamma\_{\textrm{t}}}}{2}\right]\right.{\nonumber}\\ &\quad\qquad \left. -\sin^2{\vartheta}\left[\frac{8}{{\gamma\_{\textrm{l}}}}+4{\gamma\_{\textrm{l}}}+{\gamma\_{\textrm{t}}}^3\left(1-\frac{6}{{\gamma\_{\textrm{t}}}^2}-\frac{7}{{\gamma\_{\textrm{t}}}^4}\right)-2{\beta\_{\textrm{t}}}^2{\gamma\_{\textrm{t}}}\right]\right\} .\label{eq:linetension-isotropic}\end{aligned}$$ In the zero velocity limit this expression tends to the classical result given within ref. : $$\begin{aligned} \lim\limits\_{v\to0}{\Gamma}({\vartheta},v)&= \frac{E\_0}{(1-{\nu})}\Big\{(1+{\nu})\cos^2{\vartheta}+(1-2{\nu})\sin^2{\vartheta}\Big\} ,\end{aligned}$$ where ν denotes Poisson’s ratio. For metals of interest, ν ∼ 1/3, which indicates that the line tension of a static screw dislocation is roughly four times larger than for a static edge dislocation, i.e.: ${\Gamma}({\vartheta},v=0,{\nu}=1/3)=E\_0\left(2\cos^2{\vartheta}+\frac12\sin^2{\vartheta}\right)$. Figure [fig:linetension-isotropic] shows the velocity dependence of the line tension ${\bar{\Gamma}}(0,{\beta\_{\textrm{t}}}{c\_{\textrm{t}}})$ in red and ${\bar{\Gamma}}(\pi/2,{\beta\_{\textrm{t}}}{c\_{\textrm{t}}})$ in blue, where, referring to eq. $$\begin{aligned} {\bar{\Gamma}}({\vartheta},v)&{\vcentcolon=}\frac{{\Gamma}({\vartheta},v)}{E\_0} \,,\end{aligned}$$ is the dimensionless, rescaled line tension prelogarithmic factor in the isotropic limit. [fig:linetension-isotropic] What is particularly interesting about the full velocity dependent expression, is that the divergent term γt3 is negative in the pure edge case. This means that the line tension of an edge dislocation (where ϑ = π/2) can become zero (in contrast to the screw case) indicating that it will cease to be stable in a straight configuration. In particular, for ν ≈ 1/3 we find that the line tension of an edge dislocation becomes zero at around 80% transverse sound speed, whereas it remains positive everywhere in the pure screw case — see figure [fig:linetension-isotropic]. The range of velocities where this happens, depending on the value of Poisson’s ratio (which lies in the range − 1 < ν < 1/2), is roughly $0.72\lesssim{\beta\_{\textrm{t}}}^0\lesssim0.83$; this is always beneath the Rayleigh speed. Finally, upon considering specific geometries of bow-out, a ϑ-dependent “constant” (Γ0) may need to be superposed to the expression for the line tension (i.e. corresponding to different constants for edge and screw dislocations), see e.g. ref. . This addition would only change the value of the velocity βt0, but not the main result: that βt0 < ct exists only for edge dislocations and is smaller than (but fairly close to) the transverse sound speed. This observation is in agreement with previous MD simulation results which indicate a sudden jump in velocity versus stress to the transonic regime for edge dislocations, whereas the transition is smooth for screw dislocations. It is also instructive to analyze at which angle ϑ a mixed dislocation in the isotropic limit becomes unstable at high velocities. The change in asymptotic behavior from positive to negative occurs when the coefficient of γt3 becomes zero in eq., i.e. when ${\vartheta}	arxiv_0000419
Graph-Embedded Multi-Agent Learning for Smart Reconfigurable THz MIMO-NOMA Networks =================================================================================== With the accelerated development of immersive applications and the explosive increment of internet-of-things (IoT) terminals, 6G would introduce terahertz (THz) massive multiple-input multiple-output non-orthogonal multiple access (MIMO-NOMA) technologies to meet the ultra-high-speed data rate and massive connectivity requirements. Nevertheless, the unreliability of THz transmissions and the extreme heterogeneity of device requirements pose critical challenges for practical applications. To address these challenges, we propose a novel smart reconfigurable THz MIMO-NOMA framework, which can realize customizable and intelligent communications by flexibly and coordinately reconfiguring hybrid beams through the cooperation between access points (APs) and reconfigurable intelligent surfaces (RISs). The optimization problem is formulated as a decentralized partially-observable Markov decision process (Dec-POMDP) to maximize the network energy efficiency, while guaranteeing the diversified users’ performance, via a joint RIS element selection, coordinated discrete phase-shift control, and power allocation strategy. To solve the above non-convex, strongly coupled, and highly complex mixed integer nonlinear programming (MINLP) problem, we propose a novel multi-agent deep reinforcement learning (MADRL) algorithm, namely graph-embedded value-decomposition actor-critic (GE-VDAC), that embeds the interaction information of agents, and learns a locally optimal solution through a distributed policy. Numerical results demonstrate that the proposed algorithm achieves highly customized communications and outperforms traditional MADRL algorithms. Reconfigurable intelligent surface, THz, MIMO-NOMA, MADRL, distributed optimization. Introduction ============ upcoming 6G era confronts a variety of emerging applications, involving immersive applications such as ultra-high definition (UHD) video and virtual reality/augmented reality (VR/AR), as well as Internet of Things (IoT) applications like wearable devices and smart homes. To meet the unprecedented challenges raised by ultra wide-band communications and massive IoT connectivity, terahertz (THz) massive multiple-input-multiple-output non-orthogonal multiple access (MIMO-NOMA) has become an essential technology for 6G, which can provide 10 Gbps-order ultra-fast transmission speed and support millions of connections. Generally, THz MIMO-NOMA systems utilize the large-scale antenna array with hybrid beamforming structure, which can compensate the severe fading over high-frequency THz bands, and reduce the hardware complexity and power consumption. In addition, assisted with the MIMO-NOMA technology, highly spatial-correlated users can be grouped into one cluster and supported by a single radio frequency (RF) chain, which can significantly improve spectral efficiency and connective density. To employ the massive MIMO-NOMA technology for THz communications, the authors in proposed an energy-efficient user clustering, hybrid precoding, and power optimization scheme, where the blockage probability and unreliability of line-of-sight (LoS) links have been ignored. Nevertheless, due to the high obscuration susceptibility, the application of THz MIMO-NOMA network may suffer from serious transmission unreliability and intermittency resulting from either wall blockage or human-body blockage effect, which may significantly degrade the experiences of unreliability-sensitive 6G immersive applications. Fortunately, the newly-emerged reconfigurable intelligent surface (RIS) technology is regarded as a promising way to overcome the shortcomings in THz MIMO-NOMA communications. Specifically, RISs can dynamically transform and reshape spatial beams, and thus construct virtual LoS links between transmitters and receivers to avoid blockage. Meanwhile, a smart radio environment can also be created to achieve significant spectrum/energy efficiency improvement and flexible scheduling. Given the aforementioned benefits, increasingly research efforts have been devoted to the RIS-aided MIMO-NOMA networks operating in low frequencies. In, the authors proposed a joint passive and active beamforming method for RIS-aided multiple input single output (MISO)-NOMA network, which obtained a locally optimal solution based on a second-order cone programming (SOCP)-alternating direction method of multipliers (ADMM) algorithm. The authors in further studied the joint passive and active beamforming in RIS-aided MISO-NOMA networks under both ideal and non-ideal RIS assumptions. Furthermore, the authors in proposed joint deployment and beamforming frameworks for RIS-aided MIMO-NOMA networks based on deep reinforcement learning. However, the existing methods are inapplicable to 6G THz MIMO-NOMA communications: 1) The existing MIMO-NOMA mechanisms are incapable to deal with the extremely heterogenous quality-of-service (QoS) requirements of 6G users. 2) Compared with low-frequency MIMO communications, THz MIMO communications face a more prominent transmission unreliability problem. 3) Since THz MIMO-NOMA networks usually have high-dimensional spatial channel information, existing centralized and iterative optimization algorithms usually lead to unacceptable high complexity and information exchange overhead to schedule the complicated THz MIMO-NOMA scenarios. Therefore, we aim to propose a novel smart reconfigurable MIMO-NOMA THz framework that can realize customizable and intelligent indoor communications with high energy efficiency and low complexity based on a machine learning mechanism in this work. Here, we consider two types of heterogeneous users, namely IoT users and super-fast-experience (SE) users. Specifically, SE users, like VR/AR and UHD video, require ultra-high-speed and reliable immersive communication experiences, while the densely connected IoT users, like smart cleaners and smart watches, tolerate sporadic and unreliability-tolerant traffic transmission. Different from the conventional systems, we introduce a low-complexity and decentralized learning-based framework that can jointly design the user clustering, NOMA decoding, and hybrid beam reconfiguration schemes in a cooperative setting with multiple APs and RISs: 1) By adaptively aligning users’ equivalent spatial channels as well as customizing NOMA decoding orders based on the QoS requirements, the intra-cluster interference suffered by SE users can be completely eliminated. 2) We adjust the highly-directional hybrid beams through the cooperation among APs and RISs, which can ensure tailored spatial data channels and mitigate both inter-cluster and inter-AP interference via active hybrid beamforming and coordinated passive beamforming. 3) To overcome the non-ideal discrete phase-shift control, we exploit a dynamic RIS element selection structure for the hybrid beam reconfiguration, which can flexibly refrain unfavorable and negative reflections via an element-wise ON/OFF control to enhance energy efficiency. Overall, the proposed framework can realize customizable hybrid spatial and power-domain multiplexing, as well as improving the multi-domain resource utilization. Based on the above framework, we propose a long-term joint RIS element selection, coordinated discrete phase-shift control, and power allocation learning strategy. The objective function is formulated to maximize the system energy efficiency as well as satisfying users’ data rate and reliability, which is an NP-hard mixed-integer nonlinear programming (MINLP) problem. To efficiently solve the non-convex, strongly coupled, and highly complex MINLP problem online, we transfer it into a decentralized partially observable Markov decision process (Dec-POMDP). Thereafter, we introduce a novel cooperative multi-agent reinforcement learning (MADRL) method, namely graph-embedded value-decomposition actor-critic (GE-VDAC), to efficiently coordinate multi-AP and multi-RIS in a distributed manner. The proposed GE-VDAC algorithm can not only improve generalization ability of multi-agent learning, but also reduce information exchange overhead. The main contributions of this work can be summarized as follows. * We propose a novel smart reconfigurable THz MIMO-NOMA framework, which can realize highly customizable and intelligent communications to support ultra-wide bands and ultra-dense connections. The hybrid spatial beams are smartly and cooperatively reconfigured through multi-AP and multi-RIS coordinations, where dynamic RIS element selection, on-demand data enhancement, flexible interference suppression, and efficient hybrid spatial and power-domain multiplexing are allowed. * The long-term joint RIS element selection, coordinated discrete phase-shift control, and power allocation optimization problem is formulated by a Dec-POMDP model. Under customized user clustering, NOMA decoding, and sub-connected hybrid beamforming schemes, the Dec-POMDP model can maximize the expected system energy efficiency while satisfy extremely heterogeneous data rate and reliability requirements for different users. * To efficiently solve the non-convex, strongly coupled, and highly complex MINLP problem online, we propose a novel distributed MADRL algorithm, namely GE-VDAC, which learns the decomposed local policies by embedding the agents’ interaction information into dimension-reduced and permutation-invariant features. We show that the proposed GE-VDAC can not only converge to a locally optimal solution, but also achieve a better coordination and generalization with low information exchange overhead. * We present numerical results to verify the effectiveness of the proposed strategy. The proposed GE-VDAC achieves higher system energy efficiency and faster learning speed than traditional MADRL algorithms. Moreover, both reliable and ultra-high-speed communications can be achieved by SE users despite the increment of connected users. The rest of this paper is organized as follows. Section II describes the smart reconfigurable THz MIMO-NOMA network. Section III presents the formulated Dec-POMDP problem, and Section VI proposes the GE-VDAC based distributed MADRL algorithm. Numerical results are presented in Section V before the conclusion in Section VI. Notation: We denote the imaginary unit by j, and represent vectors and matrices by lower and upper boldface symbols, respectively. $\frac{\partial F}{ \partial x}$ denotes the first partial derivative of function F with respect to x. E[ ⋅ ] represents the statistical expectation. ∣ ⋅ ∣ and ∥ ⋅ ∥ denote the absolute value and the Euclidean norm, respectively. Moreover, ⋆ means the permutation operation. The main notations used throughout this paper is summarized in Table [table:notation]. [t] 1 [table:notation] Smart Reconfigurable THz MIMO-NOMA Framework ============================================ [figsysmodel] As shown in Fig. [figsysmodel], we consider an indoor downlink massive MIMO-NOMA THz network that serves densely distributed SE users and IoT users under multiple pre-installed THz APs and RISs. Denote the set of M APs and J RISs as M = {1, 2, ..., M} and J = {1, 2, ..., J}, respectively. By coordinating and cooperating with neighboring APs and RISs, each AP m serves a set KSm of KSm SE users and a set KUm of KUm IoT users, which have utterly diversified QoS requirements in terms of transmission data rate and reliability. For simplicity, we assume KS1 = ... = KSM = KS and KU1 = ... = KUM = KU. Here, we denote Km = KUm ∪ KSm, Km = KU + KS, and K = K1 ∪ K2 ∪ ... ∪ KM. To reduce hardware complexity and energy dissipation, we assume each user equips with a single antenna and the APs apply the sub-connected hybrid beamforming structure. Moreover, each AP is equipped with NA antennas and NR RF chains, where each RF chain connects to a subset of Nsub antennas via Nsub phase shifters with Nsub = NA/NR. Define the analog and digital beamforming matrixes at each AP m as Vm ∈ CNA × NR and Wm ∈ CNR × NR. Denote the phase-shift matrix of each RIS j by $\bm{\Theta}^j$. Based on the MIMO-NOMA technology, the highly spatial-correlated SE and IoT users can be grouped into one cluster, which is served by a reconfigured beam transmitted from an AP antenna subarray that is connected with a RF chain. Define Unkm as user k in cluster n under AP m, and Knm be the set of Knm users in cluster n under AP m. Furthermore, the whole system is divided into T time slots, indexed by T = {1, 2, ..., T}. In general, we aim to propose a learning-based mechanism for the smart reconfigurable THz MIMO-NOMA network to jointly coordinate both multiple APs and RISs, which has the flow chart as shown in Fig. [figprocedure]. Decentralized scheduling begins after channel estimation, which contains the process of learning-based decentralized policies and low-complexity schemes. In the learning-based decentralized policies, we can determine the dynamic RIS element selection, discrete phase-shift control, and power allocation with incomplete system information. Based on the smartly reconfigured channels, each AP can further obtain QoS-based user clustering, customized NOMA decoding order, and sub-connected hybrid beamformer using low-complexity schemes. Thereafter, data transmissions can be performed. [figprocedure] Without loss of generality, we assume each RIS equips with a low-cost RF chain that can estimate the CSI based on the semi-passive RIS channel estimation methods. Specifically, at the beginning of each time slot, APs would send pilot signals to neighboring users to estimate the direct channels. Meanwhile, RISs would turn off their reflecting elements to sense the channels, where the indirect channels can be estimated by processing the received signals. Both direct and indirect channels can be estimated under compressed sensing or deep learning methods, which are out of the scope of this work. For the sake of expression, we ignore the time slot index t from Section [SecRIS] to [SecPower]. RIS Element Selection and Discrete Phase-Shift Control ------------------------------------------------------ We suppose that each RIS j consists of L reflecting elements, which is controlled by a software-defined RIS controller. Since the continuous phase control is hard to realize in practice, a finite-resolution passive phase shifter is utilized for each reflecting element. To save energy consumption as well as overcoming the non-ideal reflecting effect due to the discrete phase-shift control, we propose to leverage a dynamic RIS element selection structure during transmissions[4](#fn4). By flexibly controlling the ON/OFF states and the phase shifts of passive phase shifters, the dynamic selection structure can prevent unfavorable reflections and achieve higher energy efficiency. Based on the dynamic selection structure, the phase-shift matrix $\bm{\Theta}^j$ on RIS j can be given by $$\bm{\Theta}^{j} = \mathrm{diag}\left(\omega\_{1}^{j}e^{j \theta\_{1}^{j}},...,\omega\_{L}^{j}e^{j \theta\_{L}^{j}}\right),$$ where ωlj denotes the ON/OFF state of the l-th RIS element, given by $$\omega\_{l}^{j}\!=\! \begin{cases} 1, & \text{if reflecting element $l$ at RIS $j$ is turned ON,}\\ 0, & \text{otherwise.} \end{cases}$$ The discrete phase shift of a selected element l at RIS j is determined by an integer βlj ∈ {0, 1, ..., 2BR − 1} and the RIS resolution bit BR, which can be written as $$\theta\_{l}^{j} \in \mathcal{F} = \left\{ 2^{1-b}\pi \beta\_{l}^{j} \big\| \beta\_{l}^{j} \in \{0,1,...,2^{B^{\mathrm{R}}-1}\} \right\}.$$ Channel Model ------------- The equivalent channel vector from AP i ∈ M to user Unkm via multiple RISs is defined as hnkim = [hnkim1, ..., hnkimNR] ∈ C1 × NA, where hnkims ∈ C1 × Nsub is the equivalent channel from subarray s at AP i to user Unkm. Here, hnkim is given by $$\mathbf{h}\_{nk}^{im} = \underset{\text{AP-user}}{\underbrace{\widetilde{\mathbf{h}}\_{nk}^{im}}} +\sum\_{j=1}^{J}\underset{\text{AP-RIS-user}}{\underbrace{\mathbf{f}\_{nk}^{jm}\bm{\Theta}^j\mathbf{G}^{ij}}},$$ where $\widetilde{\mathbf{h}}\_{nk}^{im} \in \mathbb{C}^{1\times N\_{\mathrm{A}}}$, fnkjm ∈ C1 × L, and Gij ∈ CL × NA denote the direct spatial channels of AP i-user Unkm, RIS j-user Unkm, and AP i-RIS j links, respectively. We assume LoS paths always exist between the APs and its neighboring RISs. However, the LoS paths for the AP-user and RIS-user links may be blocked. Therefore, channel $\mathbf{\widetilde{h}}\_{nk}^{im}$ can be modeled by $$\begin{split} & \mathbf{\widetilde{h}}\_{nk}^{im} \!=\! \mathbbm{1}\_{nk}^{im,\mathrm{LoS}}\sqrt{N\_{\mathrm{A}}}\sqrt{L\_{\mathrm{path}}(f,d\_{nk}^{im,\mathrm{AU}})}\varUpsilon\bm{\alpha}^{H}\left(\varphi\_{nk}^{im0}\right) \\ & + \sum\_{l=1}^{n\_{\mathrm{NL}}}\sqrt{N\_{\mathrm{A}}}\sqrt{L\_{\mathrm{path}}(f,d\_{nk}^{im,\mathrm{AU}})}\lambda\_{nk}^{iml}(f)\varUpsilon\bm{\alpha}^{H}\left(\varphi\_{nk}^{iml}\right), \end{split}$$ where $\varUpsilon$ is the antenna gain, Lpath(f, dnkim, AU) is the path loss determined by the frequency f and the distance dnkim, AU between AP i and user Unkm. $\mathbbm{1}\_{nk}^{im,\mathrm{LoS}}\in\{0,1\}$ indicates the existence of the LoS path, which is determined by the LoS probability based on the indoor THz blockage model. Moreover, nNL signifies the number of NLoS paths, and λnkiml is the reflection coefficient for NLoS path l. φnkiml denotes the angle of departure (AoD) for the downlink channel between AP i and user Unkm. Given AoD φ, the array response vector can be denoted by $\bm{\alpha}(\varphi) = \frac{1}{\sqrt{N\_{\mathrm{A}}}}\left[1,...,e^{j\pi[m\sin\varphi]},..., e^{j\pi[(N\_{\mathrm{A}}-1)\sin\varphi]}\right]^T$. According to, the path loss Lpath(f, d) includes both spreading loss and absorption loss, i.e., $$\begin{split} L\_{\mathrm{path}}(f,d)[dB] &\!=\!L\_{\mathrm{spread}}(f,d)[dB]\!+\!L\_{\mathrm{absorption}}(f,d)[dB] \\ &\!=\!20\log\_{10}\!\left(\frac{c}{4\pi fd}\right)\!\!-10k(f)d\log\_{10}e, \end{split}$$ where k(f) is the frequency-dependent medium absorption coefficient, d is the distance, and c is the light speed. User Clustering --------------- In traditional CSI-based massive MIMO-NOMA networks, the user clustering usually contains a cluster-head selection (CHS) procedure that chooses cluster head based on channel conditions, followed by which the remaining users are grouped with the highest channel-correlated cluster heads. However, the CSI-based user clustering is inflexible to guarantee the ultra-high data rate and mitigate interference for SE users. To achieve customizable and intelligent communication experiences, we extend the traditional CSI-based user clustering to a QoS-based user clustering. Relying on the smartly reconfigured beams, users’ spatial channels can be flexibly aligned to achieve adaptive user clustering. Specifically, we group multiple IoT users with a single SE user. In this way, we can ensure multiplexing gain for SE users while enhancing IoT user connections. In light of this, we assume the number of RF chains activated at each AP is equal to its serving SE users, i.e., NR = KS. Let Un1m represent the SE user, and Unkm, ∀k > 1, be the IoT user. The low-complexity user clustering scheme based on the reconfigured channels can be stated as follows. The SE users are firstly selected as the cluster heads of different clusters. Then, we define the channel spatial correlation C(hkmm, hn1mm) between an ungrouped IoT user k and the cluster head Un1m as $$\label{channel\_corr} \mathcal{C}\left(\mathbf{h}\_{k}^{m},\mathbf{h}\_{n1}^{m}\right) = \frac{\left\|\mathbf{h}\_{n1}^{m}\left(\mathbf{h}\_{k}^{m}\right)^{H}\right\|}{\\|\mathbf{h}\_{k}^{m}\\|\\|\mathbf{h}\_{n1}^{m}\\|}.$$ Thereafter, multiple IoT users can be non-orthogonally grouped into the same cluster, where the cluster head achieves the strongest reconfigured channel correlations with them. The computational complexity of the QoS-based clustering scheme is O(KUNR). In comparison, the conventional CSI-based clustering scheme has the computational complexity of O(NR(KU + KS)2). Customized NOMA Decoding ------------------------ Relying on the adaptively aligned channels through RISs, the NOMA user decoding order in each cluster can be flexibly customized based on QoS requirements without degrading the system performance. Exploiting the NOMA successive interference cancellation (SIC), SE users can subtract signals of other users within the same cluster, and completely eliminate intra-cluster interference based on power-domain multiplexing to realize ultra-high-speed and reliability-guaranteed communications. To realize this goal, each SE user with the highest QoS requirements in its cluster would be decoded at last, while the data signals of Knm − 1 IoT users in each cluster n would be decoded first. To ensure the SIC at each IoT users in the same cluster can be successfully carried out, the NOMA decoding order of IoT users are rearranged according to equivalent channel gains as ∣hn2mVmwnm∣2 ≥ ∣hn3mVmwnm∣2 ≥ ... ≥ ∣hnKnmmVmwnm∣2. Thereafter, the SE user would sequentially decode the signals of the former Knm − 1 IoT users to completely cancel intra-cluster interference. To successfully cancel interference from IoT user Unkm, ∀k > 1, imposed to SE user Un1m, we have the following SIC constraint γn1 → nkm ≥ γnk → nkm, ∀k > 1, where γn, k → n, km is the signal-to-interference-and-noise ratio (SINR) of IoT user Unkm for decoding its signal, and γn, 1 → n, km is the SINR to decode the signal of IoT user Unkm at SE user Un1m. We denote $\mathbbm{1}\_{nk}^{m,\mathrm{fail}} = 0$ if the SIC constraint can be satisfied, i.e., the SE user Un1m can cancel the intra-cluster interference from IoT user Unkm. Otherwise $\mathbbm{1}\_{nk}^{m,\mathrm{fail}} = 1$, and the intra-cluster interference from IoT user Unkm would be taken as noise at the SE user. In this way, γn, 1 → n, km can be expressed as $$\label{SINR\_decode}\small \begin{split} &\gamma\_{n,1 \to n,k}^{m} \!=\! {\left\|\mathbf{h}\_{n1}^{mm}\mathbf{V}^{m}\mathbf{w}\_{n}^{m} \right\|^{2} \! p\_{nk}^{m}} \! \times \! \! \bigg[ \! \bigg(\sum\_{k'=1}^{k-1}p\_{nk'}^{m} \!\!+\!\! \sum\_{k'>k} \! \mathbbm{1}\_{nk'}^{m,\mathrm{fail}}p\_{nk'}^{m}\bigg) \! \\& \left\|\mathbf{h}\_{n1}^{mm}\mathbf{V}^{m}\mathbf{w}\_{n}^{m}\right\|^{2} + \sum\_{(m',n')\atop\ne(m,n)}\left\|\mathbf{h}\_{n1}^{m'm}\mathbf{V}^{m'}\mathbf{w}\_{n'}^{m'}\right\|^{2}P\_{n'}^{m'} +\sigma^{2}\bigg]^{-1}, \end{split}$$ where σ2 is the power spectral density of additional white Gaussian noise (AWGN), pnkm is the power allocation coefficient for user Unkm, and Pnm = ∑k = 0Knmpnkm is the power consumption of RF chain n at AP m. Sub-Connected Hybrid Beamforming -------------------------------- We design the analog beamforming at each AP to increase effective data gain of each cluster, and design the digital beamforming for suppressing residual inter-cluster interference that is not mitigated by RISs based on zero-forcing (ZF). The analog beamforming matrix with the sub-connected structure can be formulated as $$\mathbf{V}^m\!=\!\left[\!\begin{array}{ccc} \mathbf{v}\_{m11}^{\mathrm{sub}} &... & \mathbf{v}\_{m1N}^{\mathrm{sub}}\\ \mathbf{v}\_{m21}^{\mathrm{sub}} &... & \mathbf{v}\_{m2N}^{\mathrm{sub}}\\ ... &... &...\\ \mathbf{v}\_{mN1}^{\mathrm{sub}} &... & \mathbf{v}\_{mNN}^{\mathrm{sub}} \end{array}\!\right]\!=\!\left[\!\begin{array}{cccc} \mathbf{v}\_{m11}^{\mathrm{sub}} &... & \bm{0}\\ \bm{0} &... & \bm{0}\\ ... &... &...\\ \bm{0} &... & \mathbf{v}\_{mNN}^{\mathrm{sub}} \end{array}\!\right]\!,$$ where vmnisub = 0 for n ≠ i, and vmnnsub ∈ CNsub × 1 has an amplitude $\frac{1}{\sqrt{N\_{\mathrm{sub}}}}$. The sub-connected analog beamforming is designed to improve the effective gains toward cluster heads that are highly spatial-correlated to the cluster members. Considering BA-bits quantized phase shifters in each antenna subarray, the i-th element of the active analog beamforming vector vmnnsub is given by $$\mathbf{v}\_{mnn}^{\mathrm{sub}}(i) = \frac{1}{\sqrt{N\_{\mathrm{sub}}}} e^{-j \frac{2\pi \varpi\_{mn}^{\}}{2^{B^{\mathrm{A}}}}}, i \in \{1,2,..,N\_{\mathrm{sub}}\},$$ where $\varpi\_{mn}^{\}$ is the quantized phase calculated by $$\label{quantized\_phase} \varpi\_{mn}^{\} = {\arg \min}\_{\varpi \in \{0,1,...,2^{B^{\mathrm{A}}}-1\}} \left\|e^{j\frac{2\pi \varpi}{2^{B^{\mathrm{A}}}}} - \frac{\mathbf{h}\_{n1}^{mm}(i)}{\left\|\mathbf{h}\_{n1}^{mm}(i)\right\|} \right\|.$$ To further suppress the effective inter-cluster interference that hasn’t been canceled by RISs, we design the ZF digital beamforming as follows. Let wnm* ∈ CNR × 1 denote the digital beamforming vector for cluster n at AP m. The cluster center is given as $\widehat{\mathbf{h}}\_n^{mm}= \frac{1}{K\_n^m}\sum\_{k \in \mathcal{K}\_n^m}\left(\mathbf{h}\_{nk}^{mm}\mathbf{V}^{m}\right)^{T} $. Denote $\widehat{\mathbf{H}}^{mm} = \left[\widehat{\mathbf{h}}\_1^{mm},\widehat{\mathbf{h}}\_2^{mm},...,\widehat{\mathbf{h}}\_{N\_{\mathrm{R}}}^{mm}\right]$. Therefore, the ZF digital beamforming is calculated as $$\begin{split} \widehat{\mathbf{W}}^{m} &= \left[\widehat{\mathbf{w}}\_{1}^{m},..., \widehat{\mathbf{w}}\_{N\_{\mathrm{R}}}^{m}\right] \\ &= \left(\widehat{\mathbf{H}}^{mm}\right)^{H}\left[\widehat{\mathbf{H}}^{mm}\left(\widehat{\mathbf{H}}^{mm}\right)^{H}\right]^{-1}. \end{split}$$ By introducing column power normalizing, the baseband precoding matrix can be expressed as $$\mathbf{w}\_{n}^{m} = \frac{\widehat{\mathbf{w}}\_{n}^{m}}{\\|\mathbf{V}^{m}\widehat{\mathbf{w}}\_{n}^{m}\\|\_2}.$$ Transmission Rate ----------------- Define E{snkm(snkm)H} = 1 as the transmitted symbols with normalized power, and $x\_{n}^{m} = \sum\_{k=0}^{K\_n^m}\sqrt{p\_{nk}^{m}}s\_{nk}^{m}$ as the transmitted signal in cluster n. Define znkm as the AWGN. The baseband signal received by each SE user Un1m after SIC can be formulated as $$\small \begin{split} y\_{n1}^{m} &\! = \! \underset{\text{desired signal}}{\underbrace{\mathbf{h}\_{n1}^{mm}\mathbf{V}^m\mathbf{w}\_{n}^{m} \sqrt{p\_{n1}^{m}}s\_{n1}^{m}}} + \! \underset{\text{residual intra-cluster interference}}{\underbrace{\mathbf{h}\_{n1}^{mm}\mathbf{V}^m\mathbf{w}\_{n'}^{m} \! \! \sum\_{k>1} \! \mathbbm{1}\_{nk}^{m,\mathrm{fail}} \! \sqrt{p\_{nk}^{m}}s\_{nk}^{m}}}\\& +\! \underset{\text{inter-cluster interference}}{\underbrace{\sum\_{(m',n') \ne (m,n)}\mathbf{h}\_{n0}^{m'm}\mathbf{V}^{m'}\mathbf{w}\_{n'}^{m'}x\_{n'}^{m'}}} +\underset{\text{noise}}{\underbrace{z\_{n1}^{m}}}. \end{split}$$ Therefore, the SINR of each SE user Un1m can be written as $$\label{SINR\_UHQ} \begin{split} \gamma\_{n1}^{m}&= {\left\|\mathbf{h}\_{n1}^{mm}\mathbf{V}^{m}\mathbf{w}\_{n}^{m}\right\|^{2}p\_{n1}^{m}} \bigg(\sum\_{k>1}\mathbbm{1}\_{nk}^{m,\mathrm{fail}}\left\|\mathbf{h}\_{n1}^{mm}\mathbf{V}^{m}\mathbf{w}\_{n}^{m}\right\|^{2}p\_{nk}^{m}\\& +\sum\_{(m',n')\ne(m,n)}\left\|\mathbf{h}\_{n1}^{m'm}\mathbf{V}^{m'}\mathbf{w}\_{n'}^{m'}\right\|^{2}P\_{n'}^{m'}+\sigma^{2}\bigg)^{-1}. \end{split}$$ Moreover, the SINR of IoT users Unkm, ∀k > 1, can be given by $$\label{SINR\_IoT} \begin{split} \gamma\_{nk}^{m}&= \left\|\mathbf{h}\_{nk}^{mm}\mathbf{V}^{m}\mathbf{w}\_{n}^{m}\right\|^{2}p\_{nk}^{m} \bigg( \sum\_{k'=1}^{k-1}\left\|\mathbf{h}\_{nk}^{mm}\mathbf{V}^{m}\mathbf{w}\_{n}^{m}\right\|^{2}p\_{nk'}^{m}+\\& \sum\_{(m',n')\ne(m,n)}\left\|\mathbf{h}\_{nk}^{m'm}\mathbf{V}^{m'}\mathbf{w}\_{n'}^{m'}\right\|^{2}p\_{n'}^{m'}+\sigma^{2} \bigg)^{-1}. \end{split}$$ Hence, the data rate of each user Unkm at each time slot can be expressed as Rnkm = log2(1 + γnkm), ∀k ∈ K. Power Consumption ----------------- The total power consumption for the proposed networks at each time slot includes both transmit and circuit power, which can be formulated as P = ∑m = 1M∑n = 1NRξPnm + KPD + MPAP + ∑j = 1J∑l = 1LωljPRIS(BR), where ξ denotes the inefficiency of the phase shifter in the THz network, PD is the circuit power consumption at each user, and PRIS(BR) denotes hardware energy dissipated at each RIS element depending on the RIS resolution bit BR. Moreover, PAP is the circuit power consumption at each AP, which is given by PAP = PBB + NRPRF + NA(PPS + PA), where PBB is the baseband power consumption, and PRF, PPS, and PA denote power consumption of per RF chain, per phase shifters and per power amplifies, respectively. Therefore, the network energy efficiency η at each time slot can be formulated as $$\label{EE}\small \eta \!=\! \frac{\sum\_{m=1}^{M}\sum\_{n=1}^{N\_{\mathrm{R}}}\sum\_{k=1}^{K\_n^m}\log\_2\left(1+\gamma\_{nk}^{m}\right)} {\sum\_{m=1}^{M}\sum\_{n=1}^{N} \xi P\_{n}^{m} \!+\! KP\_{\mathrm{D}} \!+\! MP\_{\mathrm{AP}} \!+\! \sum\_{j=1}^{J}\sum\_{l=1}^{L} \omega\_{l}^{j}P\_{\mathrm{RIS}}\left(B^{\mathrm{R}}\right)}.$$ Reliability Model ----------------- Suppose the THz MIMO-NOMA networks maintain a traffic buffer queue qnkm(t) for each user, ∀t ∈ T, which aims to transmit specific data volume in a predefined time. At the beginning of each time slot t, the queue length qnkm(t) of user Unkm can be updated by qnkm(t + 1) = Ankm(t) + [qnkm(t) − Rnkm(t)]+, where Rnkm(t) is the transmission data rate of, Ankm(t) denotes the traffic arrival column at time slot t following the poisson distribution, and [x]+ = max{x, 0}. To ensure reliable transmission, we should guarantee the queue stability and keep the outages below a predefined threshold. The queue stability is ensured by E[qnkm(t)] = limT → ∞∑t = 1Tqnkm(t) < ∞, ∀k ∈ K. Moreover, the outage probability, i.e., the probability that the traffic queue length qnkm(t) of device Unkm exceeds the threshold qnkm, max, is limited by a certain threshold εnkm. Therefore, the reliability conditions can be formulated as Pr(qnkm(t) ≥ qnkm, max) ≤ εnkm, ∀k ∈ K. Lyapunov Optimization Based Dec-POMDP Problem ============================================= In this Section, we first formulate the constrained Dec-POMDP problem, and then transfer the constrained Dec-POMDP problem into a normal Dec-POMDP problem based on the Lyapunov optimization theory. Constrained Dec-POMDP Model --------------------------- We aim to dynamically find a stationary online policy π, which can jointly optimize the dynamic RIS element selection, coordinated discrete phase-shift control, and transmit power allocation by observing the environment state at each time slot t. Under the given user clustering, NOMA decoding, and hybrid beamforming schemes, the joint policy dynamically maximizes the expected system energy efficiency for the smart reconfigurable THz MIMO-NOMA network, while satisfying various QoS requirements in terms of data rate and transmission reliability of users. Therefore, the online objective function can be formulated as follows [P1] $$\mathcal{P}\_{0}: \max\_{\mathbf{a}(t)} \lim\_{T\to \infty} \sum\_{t=0}^{T} \mathbb{E} \left[\eta(t)\right], \tag{\ref{P1}{a}}$$ $$\begin{aligned} {\mathrm{s.t.}}~ & R\_{nk}^{m}(t) \geq R\_{nk}^{m,\min}, \forall k \in \mathcal{K}, t \in \mathcal{T}, \label{constraint\_rate} \tag{\ref{P1}{b}} \\& \sum\_{n=1}^{N\_{\mathrm{R}}} \sum\_{k=1}^{K\_{n}^{m}(t)} p\_{nk}^{m}(t) \leq P\_{\mathrm{AP}}^{\max}, \forall m \in \mathcal{M}, t \in \mathcal{T}, \label{constraint\_power} \tag{\ref{P1}c} \\& \omega\_{l}^{j}(t) \in\{0, 1\},\forall j \in \mathcal{J}, 1 \leq l \leq L, t \in \mathcal{T}, \label{constraint\_bin} \tag{\ref{P1}d} \\& \beta\_{l}^{j}(t) \in \{0,1,...,2^{B^{\mathrm{R}}-1}\}, \forall j \in \mathcal{J}, 1 \leq l \leq L, t \in \mathcal{T}, \label{constraint\_RIS} \tag{\ref{P1}e} \\& \eqref{queue}, \eqref{reliability}, \tag{\ref{P1}f}\end{aligned}$$ where $\mathbf{a}(t) = \left[\bm{\omega}(t), \bm{\alpha}(t), \bm{\beta}(t)\right]$. Constraint ensures the minimum data rate of each user, denotes the total transmit power should be less than the maximum power PAPmax, signifies the binary ON/OFF state of RIS elements, and constraint represents the discrete phase-shift control. Note that the SIC constraint is ignored since it has been implied in the SINR expression via the indicator $\mathbbm{1}\_{nk}^{m,\mathrm{fail}}$. This online policy can be modeled as a constrained Markov Decision Process (MDP). However, solving P0 in a centralized way is computationally inefficient due to the large-scale joint state-action space and the heavy overhead of high-dimensional information exchange from multiple APs and RISs to the centralized controller. To tackle P0 in an efficient and low-complexity manner, we model the long-term energy efficiency optimization problem as a constrained Dec-POMDP. In detail, the POMDP provides a generalized framework to describe an MDP with incomplete information, while Dec-POMDP extends POMDP to the decentralized settings as P = (I, S, Ω, A, Ps → sʹa, π, r, Γ). Here, I denotes a set of agents. S, Ω, and A denote the state, observation, and action spaces. At each slot t, given the true state s(t) ∈ S, each agent i partially observes a local observation, based on which it can determine a local action using decentralized policy π. The joint observation and action vectors of all agents are denoted by \Omega(t) ∈ Ω and a(t) ∈ A, and the decentralized policy π(a(t)∣s(t)) means the probability of taking joint action a(t) at the joint state s(t). Furthermore, Ps → sʹa = P(sʹ∣s, a) is the probability of transferring into a new state sʹ by taking action a at state s. Given the global reward function r(s(t), a(t)) and the discount factor Γ, the agents can cooperate and coordinate to maximize the discounted return. In this work, we model APs and RISs as two types of agents, whose set is denoted as I = I0 ∪ I1. Let Iu denote the set of the u-th type agents, where u = {0, 1} represent APs and RISs respectively. We specify the formulated Dec-PoMDP as follows. ### Observation The joint observation can be expressed as \Omega(t) = [\Omega10(t), ..., \OmegaM0(t), \Omega11(t), ..., \OmegaJ1(t)], where \Omegaiu(t) is the local observation of agent i ∈ Iu that involves part of the environment information, given by $$\label{state\_P1\_AP}\small \mathbf{\Omega}\_{i}^{0}(t) \!=\! \bigg[\! \left\{\bm{\widetilde{\mathrm{H}}}^{ii'}(t)\right\}\_{i'\in\mathcal{N}\_{i}^{0}\cup\{i\}}, \mathbf{q}^{i}(t), \bm{\tau}\_{i}^{0}(t) \! \bigg], \forall i \in \mathcal{I}^{0},$$ $$\label{local\_obs\_RIS}\small \mathbf{\Omega}\_{i}^{1}(t) = \bigg[\left\{\mathbf{F}^{im}(t)\right\} \_{m\in\mathcal{N}\_{i}^{1}}, \left\{ \mathbf{G}^{mi}(t)\right\} \_{m\in\mathcal{N}\_{i}^{1}}\!, \bm{\tau}\_{i}^{1}(t) \bigg], \forall i \in \mathcal{I}^{1},$$ where $\bm{\tau}\_{i}^{u}(t)$ denotes the local action-observation history of agent i, and Ni0 and Ni1 denotes the neighboring agent sets of RIS i ∈ I0 and AP i ∈ I1, respectively. Furthermore, we define $\widetilde{\mathbf{H}}^{ii'}(t)=\left[\left\{\widetilde{\mathbf{h}}\_{nk}^{ii'}(t)\right\}\_{k\in\mathcal{K}^{i'}}\right]$ and Fim(t) = [{fnkim(t)}k ∈ Km] as the vectorized channels. Note that in our settings, the joint observation also represents the true environment state, i.e., \Omega(t) = s(t). ### Action In this work, we consider a long-term joint RIS element selection, coordinated discrete phase-shift control, and power allocation optimization problem. Therefore, we define the joint action vector as a(t) = [a10(t), ..., aM0(t), a11(t), ..., aJ1(t)], where aiu(t) is the local action of agent i ∈ Iu that can be expressed as $$\mathbf{a}\_{i}^{u}(t)=\begin{cases} \mathbf{p}^{i}(t), & \forall i\in\mathcal{I}^{0},\\ \left[\bm{\omega}^{i}(t),\bm{\beta}^{i}(t)\right], & \forall i\in\mathcal{I}^{1}. \end{cases}$$ Here, pi(t) = [p11m(t), p12m(t), ..., pNRKNRm(t)m(t)] is the downlink power allocation vector of AP agent i ∈ I0, and $\bm{\omega}^{i}(t) = \left[\omega\_{1}^{i}(t),\omega\_{2}^{i}(t),...,\omega\_{L}^{i}(t)\right]$ and $\bm{\beta}^{i}(t)=\left[\beta\_{1}^{i}(t),\beta\_{2}^{i}(t),...,\beta\_{L}^{i}(t)\right]$ denote the dynamic reflecting element selection and the discrete phase-shift control of RIS agent i ∈ I1, respectively. ### Decentralized Policy The joint policy is decomposed into multiple decentralized and parallel local policies π(aiu(t)∣\Omegaiu(t)) for each agent i ∈ Iu. At each time slot t, all the agents take local actions aiu(t) based on the observed local observations \Omegaiu(t) to cooperatively maximize the global reward. ### Global Reward Let δm(t) = ∑n = 1NR∑k = 1Knm(t) [Rnkm, min(t) − Rnkm(t)]+ be the data rate constraint violation. Then, the global reward can be defined as r(s(t), a(t)) = η(t) − ξδ(t), which maximizes the network energy efficiency as well as satisfying data rate requirements. Here, δ(t) = ∑m = 1Mδm(t), and ξ is a non-negative parameter that imposes a penalty for data rate violation. However, due to the intractable long-term average queue stability and reliability constraints, it is infeasible to directly use the general MADRL methods to solve the constrained Dec-POMDP problem. Hence, we recast the constrained Dec-POMDP into a standard Dec-POMDP based on the Lyapunov optimization theory. Equivalently Transferred Dec-POMDP ---------------------------------- Based on the Markov’s inequality, we can obtain the upper bound of the outage probability Pr(qnkm(t) ≥ qnkm, max) ≤ E[qnkm(t)]/qnkm, max. Therefore, the reliability constraint can be strictly guaranteed by ensuring E[qnkm(t)] ≤ qnkm, maxεnkm, ∀k ∈ K. To tackle the long-term average constraints and, we apply the virtual queue technique and Lyapunov optimization theory to recast the intractable constrained Dec-POMDP as a normal Dec-POMDP, which can be solved by general MADRL algorithms. To meet the reliability constraints, we construct a virtual queue Ynkm(t) to model the instantaneous deviation of queue length constraints for each user Unkm. Here, Ynkm evolves as follows Ynkm(t + 1) = [Ynkm(t) + (qnkm(t + 1) − qnkm, maxεnkm)]+, where we initialize Ynkm(t) = 0 at t = 1, ∀k ∈ K. Let $\bm{\Xi}\_{nk}^{m}(t)=\left[q\_{nk}^{m}(t), Y\_{nk}^{m}(t)\right]$, ∀k ∈ K. Define $\bm{\Xi}(t) = \left[\bm{\Xi}\_{11}^{1}(t), \bm{\Xi}\_{12}^{1}(t),...,\bm{\Xi}\_{N\_{\mathrm{R}}K^{M}}^{M}(t)\right]$ as the combined queue vector. Then, we introduce the Lyapunov function $\mathcal{L}(\bm{\Xi}(t)) = \frac{1}{2} \bm{\Xi}(t)\bm{\Xi}^{T}(t)$ to measure the satisfaction status of the reliability constraint, which should be maintained under a low value. Moreover, the one-step Lyapunov drift is defined as $\Delta\mathcal{L} (t) = \mathcal{L}(\bm{\Xi}(t+1)) - \mathcal{L}(\bm{\Xi}(t))$, which can be upper bounded by the following Lemma. [LemmaLyapunovDrift] Define $\Delta{\mathcal{L}}\_{nk}^{m}(t) = \mathcal{L}(\bm{\Xi}\_{nk}^{m}(t+1)) - \mathcal{L}(\bm{\Xi}\_{nk}^{m}(t))$, then ΔLnkm(t) can be upper bounded by ΔLnkm(t) ≤ Cnkm + Bnkm(t) + Λnkm(t)(Ankm(t) − Rnkm(t)), where Cnkm = 2Cnkm, q + Cnkm, Y is a constant, with $C\_{nk}^{m,\mathrm{q}}=\frac{1}{2}\left(A\_{nk}^{m,\max}\right)^{2}+\frac{1}{2}\left(R\_{nk}^{m,\max}\right)^{2}$ and $C\_{nk}^{m,\mathrm{Y}}=\frac{1}{2}\left(q\_{nk}^{m,\max}\epsilon\_{nk}^{m}\right)^{2}$. Moreover, $B\_{nk}^{m}(t)=\frac{1}{2}\left(q\_{nk}^{m}(t)\right)^{2}+Y\_{nk}^{m}(t)\left[A\_{nk}^{m}(t)+q\_{nk}^{m}(t)\right]$ and Λnkm(t) = Ynkm(t) + 2qnkm(t) are fixed values at each time slot. See Appendix [ProofLyapunovDrift]. Based on the Lyapunov optimization, the original constrained Dec-POMDP of maximizing the system energy efficiency while ensuring the long-term reliability constraints can be transformed into minimising the following drift-minus-bonus function $$\begin{split} & \Delta\mathcal{L}(t) - \zeta \eta(t) + \xi \delta(t) \leq \sum\_{m=1}^{M}\sum\_{n=1}^{N\_{\mathrm{R}}}\sum\_{k=1}^{K\_n^m(t)} \bigg[ C\_{nk}^{m} + B\_{nk}^{m}(t) \\ & + \Lambda\_{nk}^{m}(t)\left(A\_{nk}^{m}(t)-R\_{nk}^{m}(t)\right) \bigg] - \zeta \eta(t) + \xi \delta(t), \end{split}$$ where ζ denotes the positive coefficient that controls the tradeoff between energy efficiency and transmission reliability, and the inequality is implied by Lemma [LemmaLyapunovDrift] and ΔL(t) = ∑m = 1M∑n = 1NR∑k = 1Knm(t)ΔLnkm(t). By ignoring the independent terms from control variables, P1 can be rearranged as [P2] $$\max\_{\pi} \! \!\lim\_{T \to \infty} \frac{1}{T} \sum\_{t=1}^{T} \mathbb{E}\_{\pi} \! \left[ \zeta \eta(t) \!-\! \xi \delta(t) \!+\! \sum\_{m=1}^{M}\sum\_{n=1}^{N\_{\mathrm{R}}}\sum\_{k=1}^{K\_n^m(t)}\! \Lambda\_{nk}^{m}(t)R\_{nk}^{m}(t) \right], \tag{\ref{P2}{a}}$$ $$\begin{aligned} {\mathrm{s.t.}}~ & \eqref{constraint\_power}-\eqref{constraint\_RIS}. \tag{\ref{P2}{b}}\end{aligned}$$ Based on, the local observation of AP i ∈ I0 in can be transformed as $$\label{local\_obs\_AP}\small \mathbf{\Omega}\_{i}^{0}(t)\!=\! \bigg[ \left\{\bm{\widetilde{\mathrm{H}}}^{ii'}(t)\right\} \_{i'\in\mathcal{N}\_{i}^{0}\cup\{i\}}\!, \bm{\Lambda}^{i}(t), \bm{\tau}\_i^{0}(t) \bigg]\!, \forall i \in \mathcal{I}^{0},$$ where $\bm{\Lambda}^{i}(t) = \left[\Lambda\_{11}^{i}(t), \Lambda\_{12}^{i}(t),...,\Lambda\_{N\_{\mathrm{R}}K\_{n}^{i}(t)}^{i}(t)\right]$. From, the global reward function r(s(t), a(t)) in the equivalent Dec-POMDP is modified into $$\small r\left(\mathbf{s}(t), \mathbf{a}(t)\right)= \zeta \eta(t) - \xi \delta(t) + \sum\_{m=1}^{M}\sum\_{n=1}^{N\_{\mathrm{R}}}\sum\_{k=1}^{K\_n^m(t)} \Lambda\_{nk}^{m}(t)R\_{nk}^{m}(t).$$ GE-VDAC Algorithm for Distributed RIS Configuration and Power Allocation ========================================================================== Review of MADRL Methods ----------------------- The standard Dec-POMDP problem can be solved based on general cooperative MADRL algorithms,. By exploiting the remarkable representation ability of deep neural network (DNN) for approximating arbitrary non-convex functions, MADRL algorithms can learn the mapping from agents’ observations to actions through exploitation and exploration. However, MADRL algorithms commonly suffer partial observability, environment instability, and credit assignment problems. To cope with environment instability, the multi-actor critic framework based on centralized-training and distributed-execution (CTDE) design is proposed, which learns distributed policies with a centralized critics. However, since it utilizes a unified critic value to compute local policy gradients, agents cannot fairly learn to contribute for global optimization, leading to the so-called multiagent credit assignment problem,, also referred to as indolent agent. Therefore, the idea of different rewards has been introduced, which computes the individual reward for each agent to evaluate their contributions and motivate coordinations. QMIX is the widely utilized mechanism that decomposes the joint state-action value into monotonic individual state-action functions for different agents. To overcome the monotonicity limitation of QMIX, QTRAN with higher generalization of value factorization can be leveraged. Nevertheless, both QMIX and QTRAN as extensions of the deep Q-learning are mainly used to deal with the problems with discrete action spaces. In, value-decomposition actor-critic (VDAC) is introduced to incorporate the value decomposition into the advantage actor-critic (A2C) framework, which has continuous action space and higher sample efficiency. In our proposed framework, it’s critical to facilitate coordination among multiple APs and RISs to mitigate interference, improve system energy efficiency, and guarantee diversified QoS requirements. Under the realistic partially observable environment, although the joint design of RIS configuration and power allocation has been decomposed into distributed local policies, information still need to be exchanged among neighboring agents to achieve fully coordination. However, considering the high-dimensional CSI information resulting from the massive MIMO structure, directly exchanging information among interacting neighboring agents during each execution can cause high communication overhead and latency. Therefore, the existing MADRL algorithms are still inefficient to solve the highly coupled Dec-POMDP problem. In this section, we propose GE-VDAC, a novel MADRL algorithm that can tackle the multiagent credit assignment as well as the above problems. The Proposed GE-VDAC Framework ------------------------------ GE-VDAC extends the commonly-used CTDE design in the existing MADRL algorithms, which realizes more efficient cooperative learning by integrating two techniques, i.e., graph embedding and different rewards. The interplay among interacting agents are modeled as a directed communication graph. Instead of directly exchanging high-dimensional information, the neighboring agents exchange low-dimensional embedded features learned by graph embedding, thus requiring fewer information exchange to achieve efficient coordination under partially observable environment. Moreover, the graph-embedded features possess permutation-invariant property. By learning distributed DRL policies over the graph-embedded observation space that has reduced dimension and enjoys permutation-invariant property, the learning speed and the generalization ability can be improved. [figalg] As shown in Fig. [figalg], the GE-VDAC jointly trains three neural networks, i.e., the distributed graph-embedded actors, the distributed critics, and the centralized mixing network. For simplicity, we assume the agents belonging to the same type u share the neural network parameters of both the distributed actor πu and critic Vu. The distributed graph-embedded actor contains a semi-distributed graph embedding module learning graph-embedded features for efficient information exchange, in conjunction with a fully-distributed action generation module that predicts local action based on the local embedded state attained by the graph embedding module. The graph embedding module is implemented as message passing graph neural network (MPGNN),. Moreover, the action generation module mainly comprises a gated recurrent unit (GRU), which is a simplified variant of long-short term memory (LSTM) that can achieve comparative performance. Two fully-connected (FC) layers are connected with the GRU before and after, respectively. On the other hand, the distributed critic evaluates an individual value of the local embedded state, which achieves differential reward to judge each agent’s contribution for global optimization. Meanwhile, a global mixing network is trained to combine the distributed critics, which guarantees that equivalent policy improvement can be achieved based on the value decomposition. Graph-Embedded Actor -------------------- ### Graph Modeling To characterize the interplay among neighboring agents, we model the THz reconfigurable massive MIMO-NOMA networks as a directed communication graph G = [I, E, TN, TE], where agents are modeled as two types of heterogeneous nodes I, and interplay among agents are modeled as edges E. TN : I → CdNu maps a node i to its dNu-dimensional node feature oiu(t). Denote tuple (i, iʹ) as a directed edge from the source node i to the destination node iʹ. TE : E → CdEiiʹ maps an edge (i, iʹ) to the dEiiʹ-dimensional edge feature diiʹ(t). Both oiu(t) and diiʹ(t) are extracted from the local observation of node i ∈ Iu. The node feature of a AP node i ∈ M contains the spatial channel information from AP i to its associated users, the queue information of its associated users, and the local action-observation history of AP i, defined as $$\label{AP\_nodefeature} \mathbf{o}\_{i}^{0}(t) = \left[\mathrm{vec}\left(\bm{\widetilde{\mathrm{H}}}^{ii}(t)\right),\bm{\Lambda}^{i}(t), \bm{\tau}\_i^{0}(t) \right], \forall i \in \mathcal{I}^{0}.$$ Moreover, the node feature of a RIS node i ∈ I1 includes the local action-observation history, i.e., $\mathbf{o}\_{i}^{1}(t) = \bm{\tau}\_i^{1}(t)$. The edge feature diiʹ depicts the interplay effect of agent i to iʹ, which can be mathematically denoted as $$\small \begin{split} &\mathbf{d}\_{ii'}(t)=\\& \begin{cases} \left[\left\{\mathrm{vec}\left(\widetilde{\mathbf{H}}^{im}(t)\right)\right\}\_{m\in\mathcal{N}\_{1}^{i'}}\right], & i\in\mathcal{I}^{0},i'\in\mathcal{I}^{1}\text{, and } i'\in\mathcal{N}\_{i}^{0},\\ \left[\mathrm{vec}\left(\mathbf{G}^{ii'}(t)\right),\mathrm{vec}\left(\mathbf{F}^{ii'}(t)\right)\right], & i\in\mathcal{I}^{1},i'\in\mathcal{I}^{0}\text{, and }i'\in\mathcal{N}\_{j}^{1},\\ \mathrm{vec}\left(\bm{\widetilde{\mathrm{H}}}^{ii'}(t)\right), & i,i'\in\mathcal{I}^{0}\text{, and }i'\in\mathcal{N}\_{i}^{0},\\ \mathbf{0}, & \text{otherwise}. \end{cases} \end{split}$$ Specifically, for an AP-RIS link, the edge feature is the spatial channels from AP i to the users served by neighboring APs of RIS iʹ. For an AP-AP link, the edge feature denotes the channel from AP i to the users associated with AP iʹ. For a RIS-AP link, the edge feature includes the channels between RIS i and both AP iʹ and its associated users. To learn the structured representation of the THz reconfigurable massive MIMO-NOMA networks, we integrate MPGNN into the distributed actor/critic networks for embedding the graph. Therefore, node/edge features of the directed graph over high-dimensional joint state space can be embedded into permutation-invariant features over low-dimensional space. By exchanging the learned low-dimensional and permutation-invariant embedded features among neighboring agents, it’s capable to improve generalization and enhance coordination among APs and RISs, while only requiring low information exchange overhead. ### Graph-Embedded Actor The graph-embedded actor (policy) comprises a semi-distributed graph embedding module which learns the embedding of the graph, followed by a distributed action generation module that outputs action by taking the local embedded sate as inputs. Here, we define πθG as the graph embedding sub-policy, and define πθAu, ∀u, as the action generation sub-policies, which are parameterized as MPGNNs and GRUs, respectively. Let $\bm{\theta}\_{\mathrm{G}}^{u}$ and $\bm{\theta}\_{\mathrm{A}}^{u}$ denote the MPGNN and GRU weight parameters shared among agents i ∈ Iu, respectively. We maintain a MPGNN at each distributed node i ∈ Iu. Similar to the multi-layer perceptron (MLP), MPGNN exploits a layer-wise structure. In each MPGNN layer, each agent first transmits embedded information to its neighbor agents, and then aggregates embedded information from neighbor agents and updates its local hidden state. Define the edges (i, iʹ) and (iʹ, i) as the outbound and inbound edges of node i. Let Ni +u and Ni −u denote the sets of neighbor agents that are linked with agent i through outbound and inbound edges of agent i, respectively. In layer n, each agent i ∈ Iu locally embeds its node feature and the outbound edge feature diiʹ(t) as eiiʹ(n)(t) = ψu(n)(zui(n − 1)(t), diiʹ(t)), ∀iʹ ∈ Ni +u, i ∈ Iu, where zui(n − 1)(t) denotes the hidden state of MPGNN layer n − 1 at agent i ∈ Iu, and ψu( ⋅ ) is the distributed embedding function. Thereafter, each agent i transmits the outbound embedded feature eiiʹ(n)(t) to its neighbor agents iʹ ∈ Ni +u, and receives the inbound embedded feature eiʹi(n)(t) from iʹ ∈ Ni −u. Thereafter, the received embedded features are aggregated with a aggregation function ϕu( ⋅ ), which is a permutation-invariant function such as sum( ⋅ ), mean( ⋅ ) and max( ⋅ ). By combining the local hidden state and the aggregated features using the combining function Ψu( ⋅ ), agent i obtains the hidden state zui(n)(t) of layer n as zui(n)(t) = Ψu(n) ( zui(n − 1)(t), ϕu(n)( {eiʹi(n)}iʹ ∈ Ni −u(t) ) ), ∀i ∈ Iu. Combining and, the update rule of the n-th MPGNN layer at node i ∈ Iu can be rewritten as $$\label{local\_hidden\_state}\small \begin{split} &\mathbf{z}\_{ui}^{(n)}(t)\! = \\ &\!\Psi\_{u}^{(n)}\!\left(\!\mathbf{z}\_{ui}^{(n-1)}(t),\phi\_{u}^{(n)}\!\left(\!\left\{ \psi\_{u}^{(n)}\!\left(\!\mathbf{z}\_{ui'}^{(n-1)}(t),\mathbf{d}\_{i'i}(t)\!\right)\!\right\} \_{i'\in\mathcal{N}\_{i-}^{u}}\!\right)\!\right)\!, \end{split}$$ where zui(n)(t) is the hidden/output state at n-th layer for agent i ∈ Iu, and zui(0)(t) = oiu(t). After the graph embedding module, agent i ∈ Iu would predict local action utilizing GRU based on the output local embedded state $\widetilde{\mathbf{z}}\_{i}^{u}(t)$, which is given by $$\label{embedding} \widetilde{\mathbf{z}}\_{i}^{u}(t) = \left[\mathbf{o}\_{i}^{u}(t), \mathbf{z}\_{ui}^{(N)}(t)\right].$$ The local action aiu(t) for each agent i ∈ Iu is sampled from the action generation sub-policy $\pi\_{\theta\_{\mathrm{A}}}^u\left(\mathbf{a}\_{i}^{u}(t)\|\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$. [H] [1] Distributed actors πθu, ∀u ∈ {0, 1}. Initialize t = 1, qnkm(t) = 0, Ynkm(t) = 0, ∀k ∈ K; Each agent obtains local observation as and ; Each agent i ∈ Iu calculates local feature embedding eiiʹ(n)(t), ∀iʹ ∈ Ni +u, using ; Each agent i ∈ Iu aggregates feature embedding eiʹi(n)(t) from neighbor agents iʹ ∈ Ni −u, and update the hidden state zui(n)(t) based on ; Each agent i ∈ Iu samples local action aiu(t) according to $\pi\_{\theta\_{\mathrm{A}}}^u\left(\mathbf{a}\_{i}^{u}(t)\|\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$; Each AP m calculates QoS-based user clustering, obtains hybrid beamformer Vm(t) and Wm(t), and decides decoding order of IoT users for each user cluster; Observe the global reward r(s(t), a(t)), and calculate qnkm(t + 1) and Ynkm(t + 1) according to and ; $\left\{\mathbf{s}(t),\widetilde{\mathbf{z}}(t),\mathbf{a}(t),r\left(\mathbf{a}(t),\mathbf{s}(t)\right),\mathbf{s}(t+1)\right\}\_{t\in\{1,2,...,T\}}$ [alg:actor] The coordinated scheduling for APs and RISs based on the distributed graph-embedded actors can be summarized as Algorithm [alg:actor]. At each time slot t, APs and RISs exchange embedded features and obtain their local embedded state $\widetilde{\mathbf{z}}\_{i}^{u}$ through MPGNN. Thereafter, each agent i ∈ Iu samples its local action based on the action generation sub-policy $\pi\_{\theta\_{\mathrm{A}}}^{u}\left(\mathbf{a}\_{i}^{u}(t)\|\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$ of the local actor. By observing the reconfigured channels, each AP further decides the user clustering, hybrid beamforming and IoT user decoding order. In this way, the customized user clustering, NOMA decoding and hybrid beamforming schemes are considered as part of the environment feedback, which would be implicitly learned by the distributed actors through exploration and exploitation. The experiences are then joined into the replay memory buffer B for centralized learning. ### Policy improvement Denote the combined parameters of graph embedding module and action generation module in the distributed actor (policy) as $\bm{\theta}\_{\pi}^{u} = \left[ \bm{\theta}\_{\mathrm{G}}^{u}, \bm{\theta}\_{\mathrm{A}}^{u} \right]$. Here, the distributed actors $\bm{\theta}\_{\pi} = [\{\bm{\theta}\_{\pi}^{u}\}\_{u\in\{0,1\}}]$ are trained to maximize the following performance function: $$\label{policy\_loss} L\_{\pi}(\bm{\theta}\_{\pi}^{u}) = \mathbb{E}\_{\mathbf{s}\sim p^{\pi},\mathbf{a}\sim\pi}\left[r(\mathbf{s}(t),\mathbf{a}(t))\right],$$ where pπ is the joint state transition by following the joint policy π. Therefore, we calculate policy gradient based on the advantage function, which is given by $$\label{policy\_gradient} \Delta \bm{\theta}\_{\pi}^{u} (t) = \sum\_{i} \nabla\_{\bm{\theta}\_{\pi}^{u}}\log \pi^{u}\left(\mathbf{a}\_{i}^{u}(t)\|\hat{\mathbf{s}}\_{i}^{u}(t)\right) A \left(\mathbf{s}(t),\mathbf{a}(t)\right),$$ where $ \hat{\mathbf{s}}\_{i}^{u}(t)$ is the actual input of the graph-embedded actor. Moreover, A(s(t), a(t)) denotes the temporal difference (TD) advantage, which is given by $$\label{Advantage} \begin{split} A\left(\mathbf{s}(t),\mathbf{a}(t)\right) &= Q(\mathbf{s}(t),\mathbf{a}(t)) - V\_{tot}(\mathbf{s}\_{t}) \\ &= r(\mathbf{s}(t),\mathbf{a}(t))+\Gamma V\_{tot}(\mathbf{s}\_{t+1}) - V\_{tot}(\mathbf{s}\_{t}). \end{split}$$ Here, Vtot(st) and Q(s(t), a(t)) denote the global state value and the global state-action value, respectively. Value Decomposition ------------------- To address credit assignment problem during training, we rely on the value-decomposition critics to train the distributed actors, which decomposes the global state value Vtot(s(t)) into local critics that are combined with a mixing function fmix as $$\small \begin{split} V\_{\mathrm{tot}}(\mathbf{s}(t)) =& f^{\mathrm{mix}}\bigg(\mathbf{s}(t), V^{0}\left(\widetilde{\mathbf{z}}\_{1}^{0}(t)\right), V^{0}\left(\widetilde{\mathbf{z}}\_{2}^{0}(t)\right),..., V^{0}\left(\widetilde{\mathbf{z}}\_{M}^{0}(t)\right), \\& V^{1}\left(\widetilde{\mathbf{z}}\_{1}^{1}(t)\right), V^{1}\left(\widetilde{\mathbf{z}}\_{2}^{1}(t)\right),...,V^{1}\left(\widetilde{\mathbf{z}}\_{J}^{1}(t)\right)\bigg), \end{split}$$ where $V^{u}\left(\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$ is the local state value for agent i ∈ Iu. Now, we are ready to introduce the following Lemma. [LemmaVD] Any action aiu that can improve agent i’s local critic value at local embedded state $\widetilde{\mathbf{z}}\_{i}^{u}$ will also improve the global state value Vtot, if the other agents stay at the same local embedded states taking actions a− iu, and the following relationship holds $$\label{V\_partial\_derivation} \frac{\partial V\_{\mathrm{tot}}\left(\mathbf{s}(t)\right)}{\partial V^{u}\left(\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)} \geq 0, \forall i\in\mathcal{I}^{u}, u\in\{0,1\}.$$ Based on, the global reward monotonically increase with $V^{u}\left(\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$ when the other agents stay at the same local embedded states taking actions a− iu(t). Therefore, if a local action aiu(t) is capable to improve $V^{u}\left(\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$, obviously it can also improve Vtot(s(t)). Based on Lemma [LemmaVD], it’s rational to approximately decompose the high-complexity global critic into distributed critics combined by a mixing network with non-negative network weight parameters, thus reducing complexity and achieving difference rewards. During centralized training, each agent attains a differential reward $V^{u}\left(\widetilde{\mathbf{z}}\_{i}^{u}(t)\right)$ based on the local graph-embedded features to evaluate its contribution for global reward improvement, which can further facilitate agents’ coordination. The weights of mixing network are generated through separate hypernetworks. Each hypernetwork takes the joint state and state-value as input to compute the weights of one layer of the mixing network. To guarantee that the output weights are non-negative, the absolute activation function is exploited at the hypernetworks. Define $\bm{\theta}\_{\mathrm{V}}^{u}$ as the weight parameters of distributed critic Vθu that are shared among agents i ∈ Iu, and denote $\bm{\mu}$ as the weights of the mixing network fμmix. The distributed critic and mixing network are optimized by mini-batch gradient descent to minimize the following critic loss: $$\label{Critic\_Loss} \begin{split} & L\_{V}\left(\bm{\theta}\_{\mathrm{V}}^{u}(t), \bm{\mu}(t)\right) = \left[\hat{r}\_t\left(\mathbf{s}(t),\mathbf{a}(t)\right) -V\_{\mathrm{tot}}\left(\mathbf{s}(t)\right)\right]^{2} \\ \! &= \! \bigg[ \hat{r}\_t\left(\mathbf{s}(t),\mathbf{a}(t)\right) - \! \! f\_{\mu}^{\mathrm{mix}} \!\bigg(\! \mathbf{s}(t), \! V\_{\theta}^{0}\left(\widetilde{\mathbf{z}}\_{1}^{0}(t)\right), \!..., \! V\_{\theta}^{1}\left(\widetilde{\mathbf{z}}\_{J}^{1}(t)\right) \!\bigg) \!\bigg]^{2} \!, \end{split}$$ where r̂t(s(t), a(t)) = ∑i = 1nΓi − 1rt + i + ΓnVtot(st + n) is the n-step return bootstrapped from the last state, with n upper-bounded by T. Therefore, the mixing networks can be updated by $$\label{Mixing\_Update} \bm{\mu} \leftarrow \bm{\mu} -\kappa\_{\mu}\sum\_{t}\nabla\_{\bm{\mu}} L\_{\mathrm{V}}\left(\bm{\theta}\_{\mathrm{V}}^{u}(t), \bm{\mu}(t)\right),$$ where κμ is the learning rate for mixing network update. To reduce complexity, we further share the weight parameters of non-output layers between distributed critics and actors for agents of the same type, similar to. Denote the combined weight parameters of the distributed actors and critics networks as $\bm{\theta}^{u} = \left[\bm{\theta}\_{\mathrm{G}}^{u}, \bm{\theta}\_{\mathrm{A}}^{u}, \bm{\theta}\_{\mathrm{V}}^{u}\right]$. Therefore, the critic gradient with respect to $\bm{\theta}^{u}$ can be given by $$\label{Critic\_Gradient} \Delta \bm{\theta}\_{\mathrm{V}}^{u}(t) = \nabla\_{\bm{\theta}^{u}} L\_{\mathrm{V}}\left(\bm{\theta}\_{\mathrm{V}}^{u}(t), \bm{\mu}(t)\right).$$ Hence, the update rule of the distributed actor/critic networks can be derived as $$\label{Theta\_Update} \bm{\theta}^{u} \leftarrow \bm{\theta}^{u} + \sum\_{t} \left( \kappa\_{\pi} \Delta \bm{\theta}\_{\pi}^{u}(t) - \kappa\_{\mathrm{V}} \Delta \bm{\theta}\_{\mathrm{V}}^{u}(t)\right),$$ where κπ and κV are the learning rate for policy improvement and critic learning, respectively. The whole GE-VDAC based MADRL algorithm is summarized in Algorithm [alg:seq]. [H] [1] Initialize the parameters of distributed actors/critics $\bm{\theta}^{u}, \forall u$, and the mixing network $\bm{\mu}$; Initialize the learning rates κμ, κπ, κV; Initialize replay buffer B = ∅; Execute the semi-distributed multi-AP multi-RIS scheduling for T steps based on algorithm [alg:actor]; Add experiences $\big\{\mathbf{s}(t),\widetilde{\mathbf{z}}(t),\mathbf{a}(t),r\left(\mathbf{a}(t),\mathbf{s}(t)\right),\mathbf{s}(t+1)\big\}\_{t\in\{1,2,...,T\}}$ to replay buffer B; Draw a batch of experiences from buffer B; Calculate the policy gradient $\bm{\theta}\_{\pi}^{u},\forall u$, based on ; Calculate the critic loss based on ; Update mixing network weights $\bm{\mu}$ based on ; Calculate the critic gradients $\Delta \bm{\theta}\_{\mathrm{V}}^{u},\forall u$, based on ; Update distributed graph-embedded actors and critics based on. [alg:seq] Theoretical Analysis of GE-VDAC ------------------------------- ### Permutation invariance Firstly, we show that GE-VDAC has permutation-invariant property, which can lead to better generalization. Consider a graph G, whose node and edge features are denoted by O and D. Let ν denote the permutation operator. Let ν ⋆ I and ν ⋆ (O, D) denote the permutation of nodes (agents) and graph features, respectively. Given two graphs G and Gʹ, if their exists a permutation ν satisfying (O, D) = ν ⋆ (Oʹ, Dʹ), then G and Gʹ are isomorphic, denoted as G ≅ Gʹ. Given two isomorphic communication graphs G ≅ Gʹ with permutation (O, D) = ν ⋆ (Oʹ, Dʹ), the distributed actors/critics are permutation invariant if the output joint action vector a, aʹ and joint critic vector v, vʹ satisfy a = ν ⋆ aʹ, v = ν ⋆ vʹ. Traditional centralized/multi-agent actor-critic algorithms usually exchange information directly and utilize neural networks such as MLP, convolutional neural network (CNN) and recurrent neural network (RNN) to learn actor and critic, which cannot achieve the permutation invariance property. Despite a sample and all of its agents’ permutations actually represent the same environments, they would be viewed as utterly different samples. Since traditional algorithms cannot adapt to agents’ permutations intelligently, all permutations of one sample should be fed to train actors and critics, leading to poor generalization and learning speed. However, by extending the traditional DRL algorithms, the graph embedding based distributed actors and critics in GE-VDAC enjoy the permutation invariance property, as shown below. [PropositionPermutationInvariance] If the aggregation functions ϕu( ⋅ ), ∀u, utilized in and are permutation invariant, the distributed actors/critics based on GE-VDAC satisfy permutation invariance. Considering the update rule of hidden states in, if aggregation functions ϕu( ⋅ ), ∀u, are permutation invariant, the graph embedding features obtained by MPGNN πθGu, ∀u, enjoy permutation invariant property, i.e., πθGu(ν ⋆ O, ν ⋆ D) = ν ⋆ (πθGu(O, D)). Given agents’ permutation (Oʹ, Dʹ) = (ν ⋆ O, ν ⋆ D), we have (zu)ʹ = πθGu(ν ⋆ O, ν ⋆ D) = ν ⋆ (πθGu(O, D)) = ν ⋆ zu, ∀u. Therefore, we can obtain $\left(\widetilde{\mathbf{z}}\_{i}^{u}\right)' = \widetilde{\mathbf{z}}\_{\nu(i)}^{u}, \forall i, \forall u$, which implies the permutation equivalence of the local embedded states. Considering the definition of the local critics, we can achieve $$\mathbf{v}'=\left\{ V\_{\theta}^{u}\left(\left(\widetilde{\mathbf{z}}\_{i}^{u}\right)'\right)\right\} \_{i\in\left(\nu\star\mathcal{I}\right)} =\left\{ V\_{\theta}^{u}\left(\widetilde{\mathbf{z}}\_{\nu(i)}^{u}\right)\right\} \_{i\in\left(\nu\star\mathcal{I}\right)} =\nu\star\mathbf{v}.$$ Moreover, since the local actions are sampled from $\mathbf{a}\_{\nu(i)}^{u}\sim\pi\_{\theta\_{\mathrm{A}}}^{u}\big(\cdot\|\widetilde{\mathbf{z}}\_{\nu(i)}^{u}\big)$ and $\left(\mathbf{a}\_{i}^{u}\right)'\sim\pi\_{\theta\_{\mathrm{A}}}^{u}\big(\cdot\|\left(\widetilde{\mathbf{z}}\_{i}^{u}\right)'\big)$, we have $$\!\pi\_{\theta\_{\mathrm{A}}}^{u}\!\left(\left(\mathbf{a}\_{i}^{u}\right)'\|\left(\widetilde{\mathbf{z}}\_{i}^{u}\right)'\right)\! \!=\! \pi\_{\theta\_{\mathrm{A}}}^{u}\big(\mathbf{a}\_{\nu(i)}^{u}\|\widetilde{\mathbf{z}}\_{\nu(i)}^{u}\big), \forall i \!\in\! \mathcal{I}^{u}, u\!\in\!\{0,\!1\},$$ which signifies aʹ = ν ⋆ a. This ends the proof. Based on Proposition [PropositionPermutationInvariance], the learned distributed actors/critics in GE-VDAC are insusceptible to the agents’ permutations. Therefore, GE-VDAC can improve the generalization and the learning speed in multi-agent cooperative learning. ### Convergence Guarantee We also demonstrate that the proposed GE-VDAC algorithm can achieve the locally optimal policy. The convergence of the GE-VDAC algorithm can be theoretically guaranteed with the following theorem. [Convergence] The proposed algorithm can converge to a locally optimal policy for the Dec-POMDP, i.e., $$\label{optimum} \lim \inf\_t \\|\nabla\_{\bm{\theta}\_{\pi}} L\_{\pi}(\bm{\theta}\_{\pi})\\|=0, w.p.1.$$ See Appendix [ProofConvergence]. ### Complexity Analysis The computational complexity of the proposed GE-VDAC algorithm mainly comes from MPGNNs and GRUs. For each MPGNN, the local embedding function ψ( ⋅ ) and the combining function Ψ( ⋅ ) are parameterized as two-layer MLPs with Sψ and SΨ neurons in each MLP layer, respectively. Thus, the computational complexity of local embedding, aggregation, and combination at agent i ∈ Iu in MPGNN layer n can be respectively written as O (degi +u(SZ(n − 1)Sψ + SESψ)), O (degi −uSE), and O (SΨ(SZ(n − 1) + SE) + SΨSZ(n)), where the outdegree/indegree degi ±u = ∣Ni ±u∣ indicates the number of outbound/inbound edges of agent i ∈ Iu, SE is the size of the local embedding eiiʹ(n), and SZ(n) represents the size of the output vector at layer n that is initialized as SZ(0) = (dNu + dEiiʹ). Since all the agents can be calculated in parallel, the computational complexity of the whole N MPGNN layers is O(∑n = 1NCmax(n)), where Cmax(n) implies the maximal complexity of agents in MPGNN layer n, i.e., $$\begin{gathered} \label{complexity\_MPGNN} C\_{\max}^{(n)} = \max\_{i \in \mathcal{I}^{u}, \atop u\in\{0,1\}} \bigg[ \mathrm{deg}\_{i+}^{u} S\_{\psi}\left(N\_{\mathrm{Z}}^{(n-1)}+S\_{\mathrm{E}}\right)\\ + \mathrm{deg}\_{i-}^{u}S\_{\mathrm{E}} + S\_{\Psi}\left(S\_{\mathrm{Z}}^{(n-1)}+S\_{\mathrm{E}}+S\_{\mathrm{Z}}^{(n)}\right) \bigg].\end{gathered}$$ On the other hand, since the complexity of GRU per weight and time step is O(1), its computational complexity depends on the number of GRU weight parameters. For agent i ∈ Iu, the total parameter number of one GRU cell is 3(Sinu + SGRU)SGRU, where Sinu = SZ(N) + dNu denotes the size of the GRU input defined in, and SGRU is the number of neurons for each gate in the GRU cell. Based on the above analyses, the computational complexity of the proposed GE-VDAC algorithm can be obtained by O(∑n = 1NCmax(n) + 3(SZ(N) + maxu ∈ {0, 1}dNu + SGRU)SGRU). In comparison, the computational complexity of the traditional VDAC algorithm (without information interaction) can be obtained by modifying the GRU input size and ignoring the MPGNN part in the GE-VDAC algorithm as O(3[SinVDAC + SGRU]SGRU), where $S\_{\mathrm{in}}^{\mathrm{VDAC}} = \max\_{{i\in\mathcal{I}^{u},\atop u\in\{0,1\}}}\left(d\_{\mathrm{N}}^{u}+\sum\_{i'\in\mathcal{N}\_{i+}^{u}}d\_{\mathrm{E}}^{ii'}\right)$ is the maximal size of the GRU input at all agents. On the other hand, an advanced VDAC algorithm with direct information exchange among agents, denoted as IE-VDAC, has the complexity of O(3[SinIE-VDAC + SGRU]SGRU), where $S\_{\mathrm{in}}^{\mathrm{IE-VDAC}} = \max\_{{i\in\mathcal{I}^{u},\atop u\in\{0,1\}}}\left(d\_{\mathrm{N}}^{u}+\sum\_{i'\in\mathcal{N}\_{i+}^{u}}d\_{\mathrm{E}}^{ii'}+\sum\_{i'\in\mathcal{N}\_{i-}^{u}}d\_{\mathrm{E}}^{i'i}\right)$. Numerical Results ================= In this section, we present numerical results to demonstrate the effectiveness of the proposed smart reconfigurable massive MIMO-NOMA networks. We consider a network with M = 3 APs uniformly deployed on the ceiling of an 8 × 5 m2 indoor room. Furthermore, we respectively consider J = {1, 2, 4} RISs mounted on the walls. When J = 1, RIS is deployed in the middle of one wall. When J ∈ {2, 4}, RISs are symmetrically and uniformly spaced on walls, respectively. Each AP m equips NR = 4 RF chains to serve KS = 4 SE users and KU = Km − KS IoT users. For simplicity, we assume that the numbers of users associated with each AP are the same. The minimum rate requirements for SE users and IoT users are 2 Gbit/s and 0.1 Gbit/s, respectively. We further assume both SE and IoT users arrive the network under the Poisson distribution with the density of 10 Gbit/s and 0.2 Gbit/s, respectively. Moreover, the maximum queue lengths for SE users and IoT users are set as 25 Gbit/s and 10 Gbit/s, and the violation probability is limited by 0.1. On the other hand, each RIS equips with L = 20 cost-effective RIS elements with quantization bit BR = {1, 2}. The LoS probabilities between AP-device and RIS-device links are calculated according to the human blockage model in. Moreover, the reflection coefficient of NLoS path is given by λ = ρFρR, where $\rho^{\mathrm{F}} = \frac{\cos\varphi\_{\mathrm{in}} - \sqrt{n\_{\mathrm{r}}^{2}-\sin^2\varphi\_{\mathrm{in}}}}{\cos\varphi\_{\mathrm{in}} + \sqrt{n\_{\mathrm{r}}^{2}-\sin^2\varphi\_{\mathrm{in}}}}$ denotes the Fresnel reflection coefficient, and $\rho^{\mathrm{R}} = e^{-\frac{1}{2}\left(\frac{4\pi f\sigma\_{\mathrm{s}}\cos\varphi\_{\mathrm{in}}}{c}\right)}$ is the Rayleigh roughness factor. φin represents the angle of incidence and reflection, nr = 1.922 + 0.0057j means the refractive index, and σs = 0.05e-3 denotes the standard deviation of the surface characterizing the material roughness. Furthermore, the AWGN power spectral density is σ2 = − 174 dBm/Hz. The simulation parameters and the neural network (NN) settings are summarized in Table [table:param]. For GE-VDAC, we empirically learn N = 2 MPGNN layers with local embedding size SE = 32 and output size SZ(1) = SZ(2) = 48. [t] 1 [table:param] [figconvergence] [figK] We compare the proposed GE-VDAC algorithm with the following three baselines: * Central critic: the multi-agent centralized-critic distributed-actor algorithm, which extends the traditional multi-agent actor-critic algorithm to the hybrid discrete and continuous action space. * *VDAC: the fully distributed VDAC* algorithm without information exchange among agents. * *IE-VDAC: the VDAC* algorithm with direct Information Exchange among neighboring agents. The convergence performance comparisons among different algorithms are shown in Fig. [figconvergence], where all of the agents are trained for 150000 time steps, and we set K = 24, NA = 64, b = 1. We can see that the value-decomposition based algorithms (VDAC, IE-VDAC, GE-VDAC) outperform central critic algorithm, since the coordination among agents is enhanced through difference rewards. The fully distributed VDAC algorithm requires no information exchange overhead and takes the least training time, but has lower learning speed and expected reward than both IE-VDAC and GE-VDAC. With less information exchange overhead than IE-VDAC, GE-VDAC can achieve the highest expected reward and learning speed, and the performance gap increases with J, demonstrating its ability to improve generalization and enhance agents’ coordination. Moreover, since GE-VDAC utilizes the dimension-reduced embedded features, it has lower complexity than IE-VDAC and takes less training time, and the time gap is larger as J increases. [RPNA] In Fig. [figK], we show the energy efficiency and reliability performance of the smart reconfigurable THz MIMO-NOMA networks under different associated user numbers K. Here, we further consider conventional THz massive MIMO-NOMA without the aid of RISs as benchmarks, which utilizes equal power allocation under channel-based/QoS-based user clustering and NOMA decoding. We can see that the CSI-based MIMO-NOMA scheme has the lowest reliability of SE users, and the performance gap dramatically grows as the associated users increase. Without the aid of RISs, the QoS-based MIMO-NOMA scheme can provide higher reliability than CSI-based MIMO-NOMA scheme, but leads to the lowest energy efficiency. However, the proposed algorithms can achieve the highest global energy efficiency, while guaranteeing the reliability of SE users. Even when the number of IoT users increases, the reliability of SE users can be guaranteed to be larger than 0.94, which verifies that the proposed reconfigurable THz MIMO-NOMA can provide on-demand QoS for heterogeneous users with efficient spectral utilization. Fig. [RPNA] and Fig. [figNA] present the performance of the proposed networks under different number of RISs NA and RIS quantization bit b. As shown in Fig. [RPNA], when NA and J increase, the power consumption increases due to the higher hardware power dissipation. Meanwhile, the sum rate of users increases with NA but decreases with J, since the spatial stream interferences become more severe as J growing. In Fig. [figNA](a), the energy efficiency declines with NA and J based on the proposed algorithms due to the higher circuit power dissipation. Compared with 1-bit RISs, 2-bit RISs result in higher power dissipation to achieve competitive data rate. Nevertheless, 2-bit RISs can achieve higher energy efficiency when J = 4 and the number of antennas is large, since they have higher flexibility to suppress the severe spatial interference. Furthermore, Fig. [figNA](b) shows the reliability of SE users increases with NA, J, and b, which demonstrates the effectiveness of massive MIMO structure and the cooperative RISs to avoid blockage. [figNA] Conclusion ========== In this work, we propose a novel smart reconfigurable THz MIMO-NOMA framework, which empowers customizable and intelligent indoor communications. We first propose specific schemes of customized user clustering, NOMA decoding, and hybrid beamforming. Thereafter, we formulate a long-term Dec-POMDP problem under a joint dynamic RIS element selection, coordinated discrete phase-shift control, and power allocation strategy, which maximizes the network energy efficiency while ensuring heterogenous data rate and reliability for on-demanded users. To efficiently solve the intractable highly complex MINLP problem, we further develop a decentralized GE-VDAC based MADRL algorithm. We theoretically demonstrate that the GE-VDAC algorithm achieves locally optimal convergence and better generalization. Numerical results show that the proposed algorithm can achieve higher energy efficiency and faster learning speed compared to traditional MADRL methods. Furthermore, we can also obtain highly customized communications through efficient resource utilization. Proof of Lemma [LemmaLyapunovDrift] =================================== Considering the definition of qnkm and ([x]+)2 ≤ x2, we have $$\label{q\_2\_bound} \begin{split} &\frac{1}{2}\left(q\_{nk}^{m}(t+1)\right)^{2} \leq \frac{1}{2}\left[q\_{nk}^{m}(t)-R\_{nk}^{m}(t)+A\_{nk}^{m}(t)\right]^{2} \\& \leq \frac{1}{2}\left(q\_{nk}^{m}(t)\right)^{2}+C\_{nk}^{m,\mathrm{q}}+q\_{nk}^{m}(t)\left(A\_{nk}^{m}(t)-R\_{nk}^{m}(t)\right). \end{split}$$ Similarly, considering the definition of Znkm, we have $$\label{z\_2\_bound} \begin{split} & \frac{1}{2}\left(Y\_{nk}^{m}(t+1)^{2} - Y\_{nk}^{m}(t)^{2} \right) \\& \overset{(a)}{\leq} \frac{1}{2}\left(q\_{nk}^{m}(t+1)\right)^{2}+C\_{nk}^{m,\mathrm{Y}}+Y\_{nk}^{m}(t)q\_{nk}^{m}(t+1), \end{split}$$ where (a) is due to qnkm, maxεnkm and qnkm(t) are non-negative. Substituting into and considering the fact that qnkm(t + 1) = qnkm(t) − Rnkm(t) + Ankm(t) due to the nonempty traffic queue, we can obtain $$\label{z\_2\_bound2} \begin{split} & \frac{1}{2}\left(Y\_{nk}^{m}(t+1)^{2} - Y\_{nk}^{m}(t)^{2} \right) \\& \leq C\_{nk}^{m,\mathrm{q}}+C\_{nk}^{m,\mathrm{Y}} -\left(q\_{nk}^{m}(t)+Y\_{nk}^{m}(t)\right)R\_{nk}^{m}(t)+ B\_{nk}^{m}(t) . \end{split}$$ Substituting and into $\Delta \mathcal{L}\_{nk}^{m}(t)=\frac{1}{2}\big[q\_{nk}^{m}(t+1)^{2}-q\_{nk}^{m}(t)^{2}\big]+\frac{1}{2}\big[Y\_{nk}^{m}(t+1)^{2}-Y\_{nk}^{m}(t)^{2}\big]$ concludes the proof. Proof of the Theorem [Convergence] ================================== For the sake of expression, we ignore the time slot index t in the following proof. The joint policy can be written as the product of the graph embedding sub-policy and the independent action generation sub-policies: $$\label{joint\_policy} \pi\left(\mathbf{a}\|\mathbf{s}\right) =\prod\_{u}\prod\_{i}\pi\_{\theta\_{\mathrm{G}}}\left(\widetilde{\mathbf{\mathbf{z}}}\_{i}^{u}\|\mathbf{O},\mathbf{D}\right) \pi\_{\theta\_{\mathrm{A}}}^{u}\left(\mathbf{a}\_{i}^{u}\|\widetilde{\mathbf{\mathbf{z}}}\_{i}^{u}\right).$$ Therefore, the policy gradient in can be rewritten as $$\label{pg\_2}\small \begin{split} \mathbf{g}\_{\pi} & =\mathbb{E}\_{\pi}\left[\sum\_{u}\sum\_{i}\nabla\_{\bm{\theta}\_{\pi}}\log\left(\pi\_{\theta\_{\mathrm{G}}}\left(\widetilde{\mathbf{\mathbf{z}}}\_{i}^{u}\|\mathbf{O},\mathbf{D}\right) \pi\_{\theta\_{\mathrm{A}}}^{u}\left(\mathbf{a}\_{i}^{u}\|\widetilde{\mathbf{\mathbf{z}}}\_{i}^{u}\right)\right)A\left(\mathbf{a},\mathbf{s}\right)\right] \\& =\mathbb{E}\_{\pi}\left[\nabla\_{\bm{\theta}\_{\pi}}\log \left(\prod\_{u}\prod\_{i}\pi\_{\theta\_{\mathrm{G}}}\left(\widetilde{\mathbf{\mathbf{z}}}\_{i}^{u}\|\mathbf{O},\mathbf{D}\right)\pi\_{\theta\_{\mathrm{A}}}^{u}\left(\mathbf{a}\_{i}^{u}\|\widetilde{\mathbf{\mathbf{z}}}\_{i}^{u}\right)\right) A\left(\mathbf{a},\mathbf{s}\right)\right] \\& \normalsize \overset{\eqref{joint\_policy}}{=} \mathbb{E}\_{\pi}\left[\nabla\_{\bm{\theta}\_{\pi}}\log\pi\left(\mathbf{a}\|\mathbf{s}\right)\left(Q\left(\mathbf{a},\mathbf{s}\right)-V\_{\mathrm{tot}}\left(\mathbf{s}\right)\right)\right], \end{split}$$ where $\bm{\theta}\_{\pi}=\left[\left\{ \mathbf{\bm{\theta}}\_{\mathrm{G}}^{u}\right\},\left\{ \mathbf{\bm{\theta}}\_{\mathrm{A}}^{u}\right\} \right]$ is the joint policy parameter vector. Denoting dπ(s) as the discounted ergodic state distribution, we can obtain, $$\label{baseline} \begin{split} &\nabla\_{\bm{\theta}\_{\pi}}\log\pi\left(\mathbf{a}\|\mathbf{s}\right)V\_{\mathrm{tot}}\left(\mathbf{s}\right) \\ =&\sum\_{\mathbf{s}}d^{\pi(s)}V\_{\mathrm{tot}}\left(\mathbf{s}\right)\nabla\_{\bm{\theta}\_{\pi}}\sum\_{\mathbf{a}}\log\pi\left(\mathbf{a}\|\mathbf{s}\right) \\ =&\sum\_{\mathbf{s}}d^{\pi(s)}V\_{\mathrm{tot}}\left(\mathbf{s}\right)\nabla\_{\bm{\theta}\_{\pi}}1 =\mathbf{0}. \end{split}$$ Substituting into, we have $$\mathbf{g}\_{\pi} =\mathbb{E}\_{\pi}\left[\nabla\_{\bm{\theta}\_{\pi}}\log\pi\left(\mathbf{a}\|\mathbf{s}\right)Q\left(\mathbf{s},\mathbf{a}\right)\right],$$ which leads to a standard policy gradient for single-agent actor-critic algorithm. In, it is proven that an actor-critic based on such gradient can converge to a local maximal expected return, which satisfies. This ends the proof. 1 M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. 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Jiang are with the School of Electronic Information, Wuhan University, Wuhan 430072, China (e-mail: {xiaoxiaxu, chenqimei, jh}@whu.edu.cn).[↩](#fnref1) 2. X. Mu is with School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing, China. (email: [email protected]).[↩](#fnref2) 3. Y. Liu is with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, UK. (email:[email protected]).[↩](#fnref3) 4. Note that dynamic RIS element selection requires a complex design for the RIS array, which might be realized in the near future. In this work, we assume the reflecting elements in RISs can be dynamically turned ON/OFF by controlling the PIN diodes.[↩](#fnref4) Graph-Embedded Multi-Agent Learning for Smart Reconfigurable THz MIMO-NOMA Networks =================================================================================== With the accelerated development of immersive applications and the explosive increment of internet-of-things (IoT) terminals, 6G would introduce terahertz (THz) massive multiple-input multiple-output non-orthogonal multiple access (MIMO-NOMA) technologies to meet the ultra-high-speed data rate and massive connectivity requirements. Nevertheless, the unreliability of THz transmissions and the extreme heterogeneity of device requirements pose critical challenges for practical applications. To address these challenges, we propose a novel smart reconfigurable THz MIMO-NOMA framework, which can realize customizable and intelligent communications by flexibly and coordinately reconfiguring hybrid beams through the cooperation between access points (APs) and reconfigurable intelligent surfaces (RISs). The optimization problem is formulated as a decentralized partially-observable Markov decision process (Dec-POMDP) to maximize the network energy efficiency, while guaranteeing the diversified users’ performance, via a joint RIS element selection, coordinated discrete phase-shift control, and power allocation strategy. To solve the above non-convex, strongly coupled, and highly complex mixed integer nonlinear programming (MINLP) problem, we propose a novel multi-agent deep reinforcement learning (MADRL) algorithm, namely graph-embedded value-decomposition actor-critic (GE-VDAC), that embeds the interaction information of agents, and learns a locally optimal solution through a distributed policy. Numerical results demonstrate that the proposed algorithm achieves highly customized communications and outperforms traditional MADRL algorithms. Reconfigurable intelligent surface, THz, MIMO-NOMA, MADRL, distributed optimization. Introduction ============ upcoming 6G era confronts a variety of emerging applications, involving immersive applications such as ultra-high definition (UHD) video and virtual reality/augmented reality (VR/AR), as well as Internet of Things (IoT) applications like wearable devices and smart homes. To meet the unprecedented challenges raised by ultra wide-band communications and massive IoT connectivity, terahertz (THz) massive multiple-input-multiple-output non-orthogonal multiple access (MIMO-NOMA) has become an essential technology for 6G, which can provide 10 Gbps-order ultra-fast transmission speed and support millions of connections. Generally, THz MIMO-NOMA systems utilize the large-scale antenna array with hybrid beamforming structure, which can compensate the severe fading over high-frequency THz bands, and reduce the hardware complexity and power consumption. In addition, assisted with the MIMO-NOMA technology, highly spatial-correlated users can be grouped into one cluster and supported by a single radio frequency (RF) chain, which can significantly improve spectral efficiency and connective density. To employ the massive MIMO-NOMA technology for THz communications, the authors in proposed an energy-efficient user clustering, hybrid precoding, and power optimization scheme, where the blockage probability and unreliability of line-of-sight (LoS) links have been ignored. Nevertheless, due to the high obscuration susceptibility, the application of THz MIMO-NOMA network may suffer from serious transmission unreliability and intermittency resulting from either wall blockage or human-body blockage effect, which may significantly degrade the experiences of unreliability-sensitive 6G immersive applications. Fortunately, the newly-emerged reconfigurable intelligent surface (RIS) technology is regarded as a promising way to overcome the shortcomings in THz MIMO-NOMA communications. Specifically, RISs can dynamically transform and reshape spatial beams, and thus construct virtual LoS links between transmitters and receivers to avoid blockage. Meanwhile, a smart radio environment can also be created to achieve significant spectrum/energy efficiency improvement and flexible scheduling. Given the aforementioned benefits, increasingly research efforts have been devoted to the RIS-aided MIMO-NOMA networks operating in low frequencies. In, the authors proposed a joint passive and active beamforming method for RIS-aided multiple input single output (MISO)-NOMA network, which obtained a locally optimal solution based on a second-order cone programming (SOCP)-alternating direction method of multipliers (ADMM) algorithm. The authors in further studied the joint passive and active beamforming in RIS-aided MISO-NOMA networks under both ideal and non-ideal RIS assumptions. Furthermore, the authors in proposed joint deployment and beamforming frameworks for RIS-aided MIMO-NOMA networks based on deep reinforcement learning. However, the existing methods are inapplicable to 6G THz MIMO-NOMA communications: 1) The existing MIMO-NOMA mechanisms are incapable to deal with the extremely heterogenous quality-of-service (QoS) requirements of 6G users. 2) Compared with low-frequency MIMO communications, THz MIMO communications face a more prominent transmission unreliability problem. 3) Since THz MIMO-NOMA networks usually have high-dimensional spatial channel information, existing centralized and iterative optimization algorithms usually lead to unacceptable high complexity and information exchange overhead to schedule the complicated THz MIMO-NOMA scenarios. Therefore, we aim to propose a novel smart reconfigurable MIMO-NOMA THz framework that can realize customizable and intelligent indoor communications with high energy efficiency and low complexity based on a machine learning mechanism in this work. Here, we consider two types of heterogeneous users, namely IoT users and super-fast-experience (SE) users. Specifically, SE users, like VR/AR and UHD video, require ultra-high-speed and reliable immersive communication experiences, while the densely connected IoT users, like smart cleaners and smart watches, tolerate sporadic and unreliability-tolerant traffic transmission. Different from the conventional systems, we introduce a low-complexity and decentralized learning-based framework that can jointly design the user clustering, NOMA decoding, and hybrid beam reconfiguration schemes in a cooperative setting with multiple APs and RISs: 1) By adaptively aligning users’ equivalent spatial channels as well as customizing NOMA decoding orders based on the QoS requirements, the intra-cluster interference suffered by SE users can be completely eliminated. 2) We adjust the highly-directional hybrid beams through the cooperation among APs and RISs, which can ensure tailored spatial data channels and mitigate both inter-cluster and inter-AP interference via active hybrid beamforming and coordinated passive beamforming. 3) To overcome the non-ideal discrete phase-shift control, we exploit a dynamic RIS element selection structure for the hybrid beam reconfiguration, which can flexibly refrain unfavorable and negative reflections via an element-wise ON/OFF control to enhance energy efficiency. Overall, the proposed framework can realize customizable hybrid spatial and power-domain multiplexing, as well as improving the multi-domain resource utilization. Based on the above framework, we propose a long-term joint RIS element selection, coordinated discrete phase-shift control, and power allocation learning strategy. The objective function is formulated to maximize the system energy efficiency as well as satisfying users’ data rate and reliability, which is an NP-hard mixed-integer nonlinear programming (MINLP) problem. To efficiently solve the non-convex, strongly coupled, and highly complex MINLP problem online, we transfer it into a decentralized partially observable Markov decision process (Dec-POMDP). Thereafter, we introduce a novel cooperative multi-agent reinforcement learning (MADRL) method, namely graph-embedded value-decomposition actor-critic (GE-VDAC), to efficiently coordinate multi-AP and multi-RIS in a distributed manner. The proposed GE-VDAC algorithm can not only improve generalization ability of multi-agent learning, but also reduce information exchange overhead. The main contributions of this work can be summarized as follows. * We propose a novel smart reconfigurable THz MIMO-NOMA framework, which can realize highly customizable and intelligent communications to support ultra-wide bands and ultra-dense connections. The hybrid spatial beams are smartly and cooperatively reconfigured through multi-AP and multi-RIS coordinations, where dynamic RIS element selection, on-demand data enhancement, flexible interference suppression, and efficient hybrid spatial and power-domain multiplexing are allowed. * The long-term joint RIS element selection, coordinated discrete phase-shift control, and power allocation optimization problem is formulated by a Dec-POMDP model. Under customized user clustering, NOMA decoding, and sub-connected hybrid beamforming schemes, the Dec-POMDP model can maximize the expected system energy efficiency while satisfy extremely heterogeneous data rate and reliability requirements for different users. * To efficiently solve the non-convex, strongly coupled, and highly complex MINLP problem online, we propose a novel distributed MADRL algorithm, namely GE-VDAC, which learns the decomposed local policies by embedding the agents’ interaction information into dimension-reduced and permutation-invariant features. We show that the proposed GE-VDAC can not only converge to a locally optimal solution, but also achieve a better coordination and generalization with low information exchange overhead. * We present numerical results to verify the effectiveness of the proposed strategy. The proposed GE-VDAC achieves higher system energy efficiency and faster learning speed than traditional MADRL algorithms. Moreover, both reliable and ultra-high-speed communications can be achieved by SE users despite the increment of connected users. The rest of this paper is organized as follows. Section II describes the smart reconfigurable THz MIMO-NOMA network. Section III presents the formulated Dec-POMDP problem, and Section VI proposes the GE-VDAC based distributed MADRL algorithm. Numerical results are presented in Section V before the conclusion in Section VI. Notation: We denote the imaginary unit by j, and represent vectors and matrices by lower and upper boldface symbols, respectively. $\frac{\partial F}{ \partial x}$ denotes the first partial derivative of function F with respect to x. E[ ⋅ ] represents the statistical expectation. ∣ ⋅ ∣ and ∥ ⋅ ∥ denote the absolute value and the Euclidean norm, respectively. Moreover, ⋆ means the permutation operation. The main notations used throughout this paper is summarized in Table [table:notation]. [t] 1 [table:notation] Smart Reconfigurable THz MIMO-NOMA Framework ============================================ [figsysmodel] As shown in Fig. [figsysmodel], we consider an indoor downlink massive MIMO-NOMA THz network that serves densely distributed SE users and IoT users under multiple pre-installed THz APs and RISs. Denote the set of M APs and J RISs as M = {1, 2, ..., M} and J = {1, 2, ..., J}, respectively. By coordinating and cooperating with neighboring APs and RISs, each AP m serves a set KSm of KSm SE users and a set KUm of KUm IoT users, which have utterly diversified QoS requirements in terms of transmission data rate and reliability. For simplicity, we assume KS1 = ... = KSM = KS and KU1 = ... = KUM = KU. Here, we denote Km = KUm ∪ KSm, Km = KU + KS, and K = K1	arxiv_0000424
On the Reconstruction of Static and Dynamic Discrete Structures =============================================================== We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the ‘classical’ task of reconstructing finite point sets in Rd). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems. Introduction ============ We begin with an informal definition of the general field of discrete tomography. As a comprehensive treatise of the general field would, however, go far beyond the scope of the present paper, and as we want to limit the overlap with surveys in the literature as much as possible, we will focus on the most fundamental case of reconstructing finite point sets in Rd from some of their discrete X-rays. Our special emphasis will be on (a subjective selection of) recent developments and applications. We conclude the introduction with a few comments on the structure of the paper and its bibliography. What is discrete tomography? ---------------------------- Discrete tomography deals with the problem of retrieving knowledge about an otherwise unknown discrete object from information about its interactions with certain query sets. Of course, this is not a formal definition and, as a matter of fact, the occuring terms leave room for different interpretations. For instance, the discrete object can be any set in some Rd that allows a finite encoding, e.g., a finite point set, a polytope or even a semialgebraic set. Also functions with finite support are included. Knowledge may mean full reconstruction, the detection of certain properties and measures of the object or just a ‘yes/no’ decision whether the object equals (or is close to) a given blueprint. The query sets may be windows, affine spaces or certain families of more general manifolds, and interaction may simply mean intersection but could also refer to a very different probing procedure. While results which, in retrospective, belong to this area go back a long time, the name discrete tomography and the establishment of the so-named field is of more recent origin. In the past decades the focus has been on the issues of uniqueness, computational complexity, and algorithms, first under the theoretical assumption that exact X-ray data were available. Later, stability and instability questions were pursued, and the effect of noise was studied. Discrete tomography has important applications in physics, materials science, and many other fields. It has, however, also been applied in various other contexts, including scheduling , data security , image processing , data compression , combinatorics , and graph theory . Closely connected are also several recreational games such as nonograms , path puzzles , sudokus , and color pic-a-pix . Scope of the present paper -------------------------- In the following we will concentrate mainly on the ‘classical’ task of reconstructing a finite point set in Rd from the cardinalities of its intersections with the lines parallel to a (small) finite number of directions. Already in this restricted form, discrete tomography displays important features also known from the continuous world. In particular, discrete tomography is ill-posed in the sense of Hadamard : the data may be inconsistent, the solution need not be unique (Thm. [thm:nonunique]), and small changes in the data may result in dramatic changes of the solutions (Thm. [thm:instabDT3]). As will become clear, discrete tomography is not simply the discretization of continuous tomography. It derives its special characteristics from the facts that, on the one hand, there are only data in very few directions available, but on the other hand, the classes of objects that have to be reconstructed are rather restricted. Therefore discrete tomography is based on methods from combinatorics, discrete optimization, algebra, number theory, and other more discrete subfields of mathematics and computer science. There exist already books and articles, which give detailed accounts of various aspects of discrete tomography and its applications. We single out. The present article differs from these surveys in various ways: the mathematical focus will be on recent developments (which have not been covered in previous surveys). Further, new applications will play a significant role, i.e., applications to other scientific areas like physics and materials science, but also to inner mathematical fields such as number theory, optimization, and imaging. We will, moreover, include pointers to certain related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems. As general sources for the continuous case and inverse problems, see and, respectively. As a service to the reader and with a view towards a more complete picture we will restate some aspects which are basic for the present article but have been covered before. In order to limit the overlap to other surveys we will, however, neither elude on the tomographic reconstruction of quasicrystals (see ) or polyominoes (see ), nor on the polyatomic case (see ) or point X-rays (see ). Also, we will not study general k-dimensional X-rays (see ) but concentrate on the case k = 1. This means that our exposition will be based on the X-ray transform rather than on the Radon transform (which is the case k = d − 1). Organization of the present paper and its bibliography ------------------------------------------------------ After introducing the basic notation in Sect. [h-DT-basics] we will briefly survey well-known structural results related to the ill-posedness of the problems (Sect. [sec-structural]) and their computational complexity (Sect. [sec:compaspects]). Turning to recent results and applications, Sect. [sect:superresolution] will illustrate some quite unexpected complexity jumps in (a related basic model of) superresolution. Particular emphasis will then be placed on new developments in dynamic discrete tomography involving the movement of points over time which are only accessible by very few of their X-ray images. As Sect. [sect:dynamics] will show aspects of discrete tomography and particle tracking interact deeply. Another more recent issue, which comes up in materials science, is that of multi-scale tomographic imaging. Sect. [sec:tom-grain-mapping] will indicate how different aspects of the reconstruction of polycrystalline materials based on tomographic data lead to very different techniques involving methods from the geometry of numbers, combinatorial optimization and computational geometry. Sect. [sect:numbertheory] deals with some inner mathematical connections between discrete tomography and the Prouhet-Tarry-Escott problem from number theory, and Sect. [sec:concluding-remarks] concludes with some final remarks. Let us point out that (with the exception of some new interpretations and simple observations) the results stated here have all been published in original research papers (which are, of course, cited appropriately). Even more, since we want to use the standard notation and, in particular, a standard framework for expressing the results, some overlap with the above mentioned surveys is unavoidable. Finally, let us close the introcuction with a comment on the bibliography. Due to the character of the present paper we included references of different kinds. Of course, we listed all original work quoted in the main body of the paper. However, we felt that for the generally interested reader it would be worthwhile to add sources for general reading. On the other hand, in terms of the included applications we focussed mainly on outlining those aspects to which discrete tomography can potentially contribute. While this is in line with the scope of the present paper, readers interested in these fields of applications may appreciate pointers to sources for additional reading. Hence we organized the bibliography in six different parts, namely general reading, papers in tomography, and further reading on particle tracking, tomographic grain mapping, macroscopic grain mapping, and the Prouhet-Tarry-Escott problem, respectively. Basic notation ============== As pointed out before we will focus on the ‘classical’ inverse problem of reconstructing a finite point set F in Rd or Zd from the cardinalities of its intersections with the lines parallel to a finite number of directions. There are, however, certain aspects which involve weights on the points of F. Hence we will introduce the basic notions for appropriate generalizations of characteristic functions of finite point sets, partly following. As usual, let N0, N, Z, Q, and R denote the sets of non-negative integers, natural numbers, integers, rationals, and reals, respectively. Further, for n ∈ N, we will often use the notation [n] = {1, …, n} and [n]0 = [n] ∪ {0}. In the following, let d, m ∈ N; d denotes the dimension of the space Rd, and m is the number of directions in which images are taken. To exclude trivial cases, we will usually assume that d, m ≥ 2. In oder to describe the objects of interest, we fix nonempty sets D ⊂ Rd and C ⊂ R and consider functions ψ : D → C with finite support $\textnormal{supp}(\psi)=\{x\in D : \psi(x)\neq 0\}$. In our context, the most relevant pairs (D, C) of a domain and a codomain are those where D = Rd or D = Zd and C = {0, 1}. Other standard codomains are C = N0, C = Z, and also their relaxations [0, 1], [0, ∞[, and R. For any pair (D, C), let F(D, C) denote the class of all functions ψ : D → C with finite support. Of course, for C = {0, 1}, such a function ψ can be viewed as the indicator or characteristic function of a finite set F and can therefore be identified with $\textnormal{supp}(\psi)$. We will write F(D) for F(D, {0, 1}) and identify it with the set of all finite subsets of D. In particular, the case F(Zd) encodes the classical finite lattice sets. Since this case is particularly important we will often abbreviate F(Zd) by Fd. Further, let Sd denote the set of all 1-dimensional subspaces of Rd, while Ld is the set of 1-dimensional lattice lines, i.e., lines through the origin spanned by an integer vector. For S ∈ Sd, we use the notation A(S) for the set of all affine lines in Rd that are parallel to S. The situation of (Fd, Ld) will be referred to as the lattice case. Now, let ψ ∈ F(D, C) and S ∈ Sd. The discrete X-ray of ψ parallel to S (or, in a slight abuse of language, in the direction S) is the function XSψ : A(S) → R defined by $$T \, \longmapsto \, \bigl(X^{}\_{S}\psi\bigr) (T) \, = \sum\_{x\in T}\psi(x).$$ Since ψ has finite support all sums are finite. In the case of C = {0, 1} where ψ can be identified with $F=\textnormal{supp}(\psi)$, we will often write XSF. See Fig. [fig:Xrays] for an illustration. [fig:Xrays] The mapping Xψ on Sd defined by S ↦ XSψ is called the discrete X-ray transform of ψ. (In typical applications only very few values of Xψ are available.) Note that it is straightforward to extend this notation to k-dimensional X-rays. Accordingly, for k = d − 1, we obtain the discrete Radon transform of ψ whose measurements come from hyperplane X-rays. We will, however, focus on the X-rays defined above, which provide 1-dimensional measurements. The basic task of discrete tomography is then to reconstruct an otherwise unknown function ψ ∈ F(D, C) from its X-rays with respect to a finite number m of given lines S ∈ Sd. The X-ray information is encoded by means of data functions. In fact, the lines T ∈ A(S) can be parametrized by vectors t ∈ S⊥ such that T = t + S. Hence, one may regard XSψ as a function on S⊥. For algorithmic purposes, it is often preferable to use other representations and encode XSψ as a finite set of pairs (x, β) with x ∈ D, β ∈ C and XSψ(x + S) = β. Ill-posedness ============= We begin with some results that deal with the basic issues of uniqueness and stability. Uniqueness and non-uniqueness ----------------------------- Given a subset F of F(D), and S ⊂ Sd. We say that two different sets F1, F2 ∈ F are tomographically equivalent with respect to S if XSF1 = XSF2 for all S ∈ S. The pair (F1, F2) will then also be referred to as a switching component. Further, a set F ∈ F is uniquely determined within F by its X-rays parallel to the lines in S if there does not exist any other set Fʹ in F that is tomographically equivalent to F with respect to S. If the context is clear we will simply say that F ∈ F is uniquely determined. The following classical non-uniqueness result, usually attributed to, has been rediscovered several times. [thm:nonunique] For any finite subset L of Ld there exist sets in Fd that cannot be determined by X-rays parallel to the lines in L. Figure [fig:zonotopeconstruction] gives an illustration of the typical construction process to obtain different lattice sets with equal X-rays. [fig:zonotopeconstruction] Note that Thm. [thm:nonunique] is in accordance with similar results in continuous or geometric tomography. In fact, let S ⊂ Sd be finite, and let C, K ∈ Rd be compact and C ⊂ int(K). Further let f : Rd → R be infinitely often differentiable with support K. Then there is a function g with support in K, infinitely often differentiable, but otherwise arbitrary on C such that the continuous X-rays of f and g with respect to all lines in S coincide; for a proof, see . Also, characteristic functions of compact sets, i.e., functions f : Rd → {0, 1} with compact support are not determined by their (continuous) X-rays in any finite number of directions; see . While non-uniqueness is an undesirable feature for many applications it will play a positive role for applications in number theory later; see Sect. [sect:numbertheory]. In contrast to Thm. [thm:nonunique], uniqueness of the reconstruction can sometimes be guaranteed when certain prior knowledge is available. [thm:irrationalunique] There exists a line S ∈ Sd such that every set F ∈ Fd is uniquely determined by its one X-ray XSF. At first glance, this result may seem surprising. However, the used a priori information is that the set is contained in Zd. Then, indeed, the X-ray XSF for any one line in Sd \ Ld determines F uniquely, simply because no translate of S can contain more than one lattice point. As this argument shows, Thm. [thm:irrationalunique] can easily be extended to F(Zd, C). While Thm. [thm:irrationalunique] does not seem to be of great practical use, it shows nonetheless that no matter how fine the lattice discretization might be, one X-ray suffices. In the limit, however, finitely many X-rays do in general not suffice. In this sense, discrete tomography does not behave like a discretization of continuous tomography. Note that Thm. [thm:irrationalunique] is quite different in nature from a result of (see also ) that for almost any finite dimensional space of objects its elements can be distinguished by a single X-ray in almost any direction. In fact, Fd is not finite-dimensional, and moreover, any line S ∈ Sd \ Ld works for any set F ∈ Fd, i.e., S does not depend on the set but is given beforehand. The next result is due to Rényi for d = 2, who attributes an algorithmic proof to Hajós. The generalization to d ≥ 2 was given by Heppes . [ Rényi] [thm:renyi] Let S be a finite subset of Sd. Then every set F ∈ F(Rd) with ∣F∣ ≤ ∣S∣ − 1 is uniquely determined by its X-rays parallel to the lines in S. Since the two color classes of the two-coloring of the vertices of the regular 2m-gon in the plane are tomographically equivalent with repect to the lines parallel to its edges, Thm. [thm:renyi] is best possible. A strengthening for (mildly) restricted sets of directions is given in. But even more: generic directions are much better. [] [thm:matousek] There exist constants c > 0 and m0 ∈ N such that for all m ≥ m0 the following holds: For almost all sets S ⊂ S2 of m directions any F ∈ F(R2) with ∣F∣ ≤ 2cm/log(m) is uniquely determined by its X-rays parallel to the lines of S. Let us point out that in continuous tomography it is well-known that a compactly supported, infinitely differentiable function f : Rd → R, which does not contain ‘details of size 2π/b’ or smaller can be recovered reliably from its X-rays on special sets of m directions, provided that m > b. For a precise statement and proof, see . See also . Such a result can be viewed, to some extent, as an analogue to Rényi’s theorem. (Note, however, that the difference to f is measured in an integral norm and hence the difference may get arbitrarily small without ever reaching 0. For a characterization of the null-space, see .) For the special case m = ∣S∣ = 2, uniqueness within F2 is characterized by the work of Ryser . See Fig. [fig:renyiexample] for an example of a set of 12 points in Z2 that is uniquely determined by its two X-rays in the coordinate directions. For uniqueness results from two directions in geometric tomography, see . [fig:renyiexample] In the lattice case, the specialization of Rényi’s theorem that any set F ∈ Fd is determined by any set of ∣F∣ + 1 lattice lines is only best possible for the cardinalities ∣F∣ = m ∈ {1, 2, 3, 4, 6}. For other m the result can be improved at least by 1. [] [thm:renyiimprovement] Let m ∈ N \ {1, 2, 3, 4, 6} and let Fd(m) be the class of sets in Fd of cardinality less than or equal to m. Let L ⊂ Ld with ∣L∣ ≥ m. Then the sets in Fd(m) are determined by their X-rays parallel to the lines in L. The question of smallest switching components is widely open in the lattice case. What is the smallest number n = n(d, m) such that there exist L ⊂ Ld with ∣L∣ = m and two different lattice sets F1, F2 ∈ Fd of cardinality n that are tomographically equivalent with respect to the lines in L? Probabilistic arguments of show that, in the lattice case, switching components of a size that is polynomial in m exist for each d. All deterministic constructions so far lead to exponential size switching components. Several small switching components are depicted in Fig. [fig:smallswitch]. We remark that switching components seem to have appeared first in the work of Ryser . Later work on switching components includes . Computational investigations related to the explicit construction of switching components can be found in. Switching components for other projection models are considered in . Special types of switching components in the context of superresolution imaging, hv-convex polynomioes, and, in a more algebraic setting, are studied in ,, and , respectively. Quite strong uniqueness results exist for a geometrically motivated more restricted class of lattice sets. A lattice set F ∈ Fd is called convex, if F = conv(F) ∩ Zd. [][thm:uniquenessconv] There are S1, S2, S3, S4 ∈ Ld such that every finite convex lattice set F is uniquely determined by XS1F, …, XS4F. Further, every set of at least seven coplanar lattice lines always suffices. Let us point out that the ‘good’ sets of directions with respect to Thm. [thm:uniquenessconv] need not have coordinates of large absolute value. Examples for d = 2 are $$\{(1,0)^T, (1,1)^T, (1,2)^T, (1,5)^T\} \quad \textnormal{and}\quad \{(1,0)^T, (2,1)^T, (0,1)^T, (-1,2)^T\}.$$ In fact, the sets of four good lattice lines in Thm. [thm:uniquenessconv] are those whose cross-ratio of their slopes does not lie in {4/3, 3/2, 2, 3, 4}. A converse result for the more general class of hv-convex lattice sets is given in . Generalizations of Thm. [thm:uniquenessconv] to so-called Q-convex lattice sets and convex algebraic Delone sets (in the context of quasicrystals) can be found in and, respectively. As a matter of fact, the directions in Thm. [thm:uniquenessconv] are all coplanar. It is not known how exactly the situation changes if we insist that the lines are in general position, i.e., each d of them span Rd. [quest-general] Let d ≥ 3. Is there a finite subset L of lines in Ld in general position such that each convex set in Fd is uniquely determined by its X-rays parallel to the lines in L? If so, what is the smallest cardinality? Is there a smallest number m such that any set L ⊂ Ld of m lines has the property, that each convex set in Fd is uniquely determined by its X-rays parallel to the lines in L? The color classes of a 2-coloring of the vertices of the permutahedron in Rd provide a lower bound on such a universal number, which grows at least quadratically in m . Fig. [fig:truncoct] depicts the 3-dimensional permutahedron, which is a truncated octahedron. [line join=bevel,z=-5.5, scale=0.6] (A0) at (1,0,2); (A1) at (1,0,-2); (A2) at (-1,0,2); (A3) at (-1,0,-2); (A4) at (2,1,0); (A5) at (2,-1,0); (A6) at (-2,1,0); (A7) at (-2,-1,0); (A8) at (0,2,1); (A9) at (0,2,-1); (A10) at (0,-2,1); (A11) at (0,-2,-1); (A12) at (0,1,2); (A13) at (0,1,-2); (A14) at (0,-1,2); (A15) at (0,-1,-2); (A16) at (2,0,1); (A17) at (2,0,-1); (A18) at (-2,0,1); (A19) at (-2,0,-1); (A20) at (1,2,0); (A21) at (1,-2,0); (A22) at (-1,2,0); (A23) at (-1,-2,0); (A0) – (A14) – (A10) – (A21) – (A5) – (A16) – cycle; (A2) – (A12) – (A8) – (A22) – (A6) – (A18) – cycle; (A4) – (A20) – (A8) – (A12) – (A0) – (A16) – cycle; (A7) – (A23) – (A10) – (A14) – (A2) – (A18) – cycle; (A0) – (A12) – (A2) – (A14) – cycle; (A4) – (A16) – (A5) – (A17) – cycle; (A8) – (A20) – (A9) – (A22) – cycle; (d1) at (A0); (d1) at (A2); (d1) at (A4); (d1) at (A5); (d1) at (A6); (d1) at (A7); (d1) at (A8); (d1) at (A9); (d1) at (A10); (d1) at (A12); (d1) at (A14); (d1) at (A16); (d1) at (A17); (d1) at (A18); (d1) at (A20); (d1) at (A21); (d1) at (A22); (d1) at (A23); [fig:truncoct] An analogue to Thm. [thm:uniquenessconv] also holds in the realm of geometric tomography: convex subsets of Rd are determined by their continuous X-rays from sets of four ‘good’ directions, see . Let us point out that it is the codomain {0, 1} which makes the problem difficult. In fact, the case of functions in F(Zd, Z) or, equivalently, lattice sets with integer weights is much simpler, as linear diophantine equations can be solved via the Hermite normal form; see e.g. ). But this also implies that the study of uniqueness for functions ψ ∈ F(Zd, N0) is much easier. In fact, suppose we are given a finite set L ⊂ Ld and a bounded subset B of Zd which will act as a superset of all supports we are allowing. Then the corresponding X-ray problem with data functions all identical to 0 can be formulated as a homogenous system of linear diophantine equations and solved efficiently. Let ψ be a non trivial solution, define ψ+ : Zd → R by +(x)= (x) & 0 & for x ∈ Zd, and set ψ− = ψ+ − ψ. Then, of course, ψ+, ψ− ∈ F(Zd, N0), ψ = ψ+ − ψ−, and XSψ ≡ 0 for all S ∈ L. Hence, ψ+, ψ− are tomographically equivalent with respect to L. The uniqueness problem for functions ψ ∈ F(Zd, N0) also permits an algebraic characterization. The subsequently stated result of Hajdu and Tijdeman uses the following notation. A vector v = (ν1, …, νd)T ∈ Zd is reduced if $\textnormal{gcd}(\nu\_1,\dots,\nu\_d)=1.$ Let v+ and v− denote the vectors whose jth component is νj+ = max{0, νj} and νj− = max{0, − νj}, respectively. With $\boldsymbol{X}$ we abbreviate the vector (X1, …, Xd) of indeterminants. Accordingly, for a = (α1, …, αd)T ∈ N0d, the monomial $X\_1^{\alpha\_1}\cdot\ldots\cdot X\_d^{\alpha\_d}\in \mathbb{Z}[\boldsymbol{X}]$ is denoted by $\boldsymbol{X}^a$. [] [thm:hajdutijdeman] Let ψ, φ ∈ F(N0d, N0), let v ∈ Zd be reduced, and set S = lin{v}. Then XSψ = XSφ if, and only if, the polynomial $$\sum\_{a\in \textnormal{supp}(\psi)}\boldsymbol{X}^a-\sum\_{b\in \textnormal{supp}(\varphi)}\boldsymbol{X}^b$$ is divisible by $\boldsymbol{X}^{v+}-\boldsymbol{X}^{v^-}.$ Note that the assumption that the functions are defined on N0d rather than on Zd is no restriction of generality. Let v(1), …, v(m) ∈ Zd \ {0} be reduced, S = {lin{v(1)}, …, lin{v(m)}} and $$f\_\mathcal{S}=\prod\_{v\in\{v^{(1)},\dots,v^{(m)}\}}(\boldsymbol{X}^{v^+}-\boldsymbol{X}^{v^-}).$$ A consequence of Thm. [thm:hajdutijdeman] is that ψ, φ ∈ F(N0d, N0) are tomographically equivalent with respect to S if, and only if, there is a polynomial p in $\mathbb{Z}[\boldsymbol{X}]$ such that $$\sum\_{a\in \textnormal{supp}(\psi)}\boldsymbol{X}^a-\sum\_{b\in \textnormal{supp}(\varphi)}\boldsymbol{X}^b =p\cdot f\_\mathcal{S}.$$ The algebraic representation by polynomials can be utilized in various ways; examples will be given in Sect. [subsec-stability] (stability) and [sect:numbertheory] (number theory). Additional aspects of uniqueness, in particular, concepts of additivity, are discussed in . For uniqueness results for functions in F(N0d, {0, 1}) with several different types of bounded support, see and the references cited therein. Stability and instability ------------------------- The results of the previous section were based on the assumption that the data functions are given exactly. We will now consider the case that the X-rays may contain errors. In the following we will measure the size of a function ψ ∈ F(D, C) in terms of its ℓ1-norm, i.e., ∥ψ∥1 = ∑x ∈ D∣ψ(x)∣. In particular, given a finite set S ⊂ Sd of lines and two sets F1, F2 ∈ F(Rd) their X-ray difference will be ΔS(F1, F2) = ∑S ∈ S∣∣XSF1 − XSF2∣∣1. The first result in this section shows that at least some (however marginal) stability is present. In fact, the X-ray difference, if not 0, must jump to at least 2(m − 1). This means that either two sets are tomographically equivalent or their X-ray difference grows at least linearly in m. [][thm:stability1] Let S ⊂ Sd, ∣S∣ = m, and F1, F2 ∈ F(Rd) with ∣F1∣ = ∣F2∣. If ΔS(F1, F2) < 2(m − 1), then F1 and F2 are tomographically equivalent with respect to S. The same statement holds in the lattice case. As we will see in Thm. [thm:instabDT3], this result is, in fact, best possible. First, we use it to give ‘noisy’ variants of some of the uniqueness results of the previous section. We begin with a stable version of Thm. [thm:renyi]. [] [thm:stability:renyi] Let S ⊂ Sd, F1, F2 ∈ F(Rd), and ΔS(F1, F2) < 2∣F1∣. Further, let ∣F1∣ = ∣F2∣ and ∣F1∣ + 1 ≤ ∣S∣, or let ∣F1∣ ≤ ∣F2∣ and 2∣F1∣ ≤ ∣S∣. Then F1 = F2. The statement persists in the lattice case. The following result is a stable version of Thm. [thm:uniquenessconv]. [][thm:stabilityconv] There are sets S ⊂ Ld of cardinality 4 for which the following is true: If F1, F2 ∈ Fd are convex, and ∣F1∣ = ∣F2∣, but F1 ≠ F2, then ΔS(F1, F2) ≥ 6. Further, for any set S ⊂ Ld of at least 7 coplanar lattice lines, and sets F1, F2 as before, ΔS(F1, F2) ≥ 2(∣S∣ − 1). The following theorem uses the known characterization of the (rather rare) cases of uniqueness in the special case d = m = 2 to quantify the deviation of solutions for noisy data; it generalizes a previous result from . [] [thm:stabDTDalen] Let S ⊂ L2 with ∣S∣ = 2, let F1, F2 ∈ F2 with ∣F1∣ = ∣F2∣. Further, suppose that F1 is uniquely determined by XSF1 for S ∈ S, and set β = ΔS(F1, F2). Then $$4\|F\_1\cap F\_2\|+(\beta+2)\Bigl(\beta-1 +\sqrt{8\|F\_1\cap F\_2\|+(\beta-1)^2}\Bigr)\geq 4\|F\_1\|.$$ Stability results in the continuous case with finitely many X-rays typically rely on bounds of the variation of the functions, measured in some weighted Sobolev norms; see and (and references therein). In the realm of geometric tomography, Volčič showed that the problem of reconstructing a convex body from its X-rays in four ‘good’ directions (which guarantee uniqueness) is well-posed. Some further stability estimates are given in . In contrast to Thm. [thm:stabDTDalen], the task of reconstructing finite lattice sets from X-rays taken along m ≥ 3 directions is highly instable. In particular the following result shows that Thm. [thm:stability1] is sharp. [] [thm:instabDT3] Let S ⊂ Sd with ∣S∣ ≥ 3, and let α ∈ N. Then there exist F1, F2 ∈ F(Rd) with the following properties: 1. F1 is uniquely determined by XSF1 for S ∈ S; 2. F2 is uniquely determined by XSF2 for S ∈ S; 3. ΔS(F1, F2) = 2(m − 1); 4. ∣F1∣ = ∣F2∣ ≥ α; 5. F1 ∩ F2 = ∅. The statement also holds in the lattice case. The proof for d = 2 is due to, while extends the construction to general d. It is actually possible to show that not even affine transformations help much to increase the overlap of the two sets. Computational aspects ===================== Next we deal with algorithmic aspects of actually reconstructing the, one or all sets that are consistent with the given X-ray data. We will restrict the exposition to functions in F(Zd, C) with C ⊂ Q and lines in Ld since all computational issues can then be studied in the well-known binary Turing machine model; see for background information. Again, emphasis will be placed on the lattice case. Algorithmic problems -------------------- Let S ⊂ Ld be finite. From an algorithmic point of view the following questions are basic: Are the data consistent? If so, reconstruct a solution! Is this solution unique? We will now introduce the correponding problems more precisely. ConsistencyF(Zd, C)(S) Data functions fS for S ∈ S. Does there exist ψ ∈ F(Zd, C) such that XSψ = fS for all S ∈ S? ReconstructionF(Zd, C)(S) Data functions fS for S ∈ S. Determine a function ψ ∈ F(Zd, C) such that XSψ = fS for all S ∈ S, or decide that no such function exists. UniquenessF(Zd, C)(S) A function ψ ∈ F(Zd, C). Does there exist φ ∈ F(Zd, C) \ {ψ} such that XSψ = XSφ for all S ∈ S? Of course, ReconstructionF(Zd, C)(S) cannot be easier than ConsistencyF(Zd, C)(S). Further, note that UniquenessF(Zd, C)(S) actually asks for nonuniqueness in order to place the problem into the class NP; see Thm. [thm:complexity]. For certain codomains such as C = {0, 1} it is reasonable to actually ask for the number of solutions even in the case of non-uniqueness. We will introduce the following problem for geneneral C with the understanding that the (not really interesting) answer ‘∞’ is permitted. #ConsistencyF(Zd, C)(S) Data functions fS for S ∈ S. Determine the cardinality of the set of functions ψ ∈ F(Zd, C) such that XSψ = fS for all S ∈ S. Observe that a given instance I = (fS : S ∈ S) can be consistent only if ∥fS∥1 does not depend on S. Since this condition can be checked efficiently we will in the following often tacitly assume that this is the case and set n = n(I) = ∥fS∥1. Further, for any given instance I = (fS : S ∈ S), the support of all solutions is contained in the grid $$G\, =\, G({\mathcal{I}})\, =\, \bigcap\_{S\in {\mathcal{S}}}\bigl(\textnormal{supp}(f\_S)+S\bigr)$$ associated with I. Of course, G(I) can be computed from I by solving polynomially many systems of linear equations. Hence we can associate a variable xg with every grid point and formulate ConsistencyF(Zd, C)(S) as a linear (feasibility) problem with the additional constraints that xg ∈ C for all g ∈ G. This simple observation shows already that ConsistencyF(Zd, C)(S) is algorithmically easy for C ∈ {[0, 1] ∩ Q, Q} simply because linear programming can be solved in polynomial time, and also for C = Z since systems of linear diophantine equations can be solved in polynomial time; see e.g.. Next we are turning to the other relevant codomains, with a special emphasis on the lattice case. [] [thm:complexity] ConsistencyF(Zd, C)(S) and UniquenessF(Zd, C)(S), C ∈ {{0, 1}, N0}, are both in P if ∣S∣ ≤ 2 whereas they are NP-complete if ∣S∣ ≥ 3. Also, the problem #ConsistencyFd(S) is #NP-complete for ∣S∣ ≥ 3. The complexity status of the counting problem for ∣S∣ = 2 is still open. Is #ConsistencyFd(S), ∣S∣ = 2, a #P-complete problem? Let us now return to the Rényi setting. [] ReconstructionFd(S) is in P if the input is restricted to those instances I = (fS : S ∈ S) with n(I) < ∣S∣. A similar result holds for convex lattice sets when the lattice lines are chose according to Thm. [thm:uniquenessconv]. So, let Cd denote the subset of Fd of convex lattice set, and let ReconstructionCd(S) signify the correponding reconstruction task. [] For any set S ⊂ Ld of at least seven coplanar directions and for suitable such sets of cardinality four ReconstructionCd(S) can be solved in polynomial-time. Let us now turn to the following ‘noisy’ versions of ConsistencyFd(S) and UniquenessFd(S). X-Ray-CorrectionFd(S) Data functions fS for S ∈ S. Does there exist F ∈ Fd such that ${\displaystyle}\sum\_{S \in \mathcal{S}}\|\|X\_SF-f\_S\|\|\_1\le m-1?$ Similar-SolutionFd(S) A set F1 ∈ Fd. Does there exist F2 ∈ Fd with ∣F1∣ = ∣F2∣ and F1 ≠ F2 such that ΔS(F1, F2) ≤ 2m − 3? Nearest-SolutionFd(S) Data functions fS for S ∈ S. Determine a set F\* ∈ Fd such that ${\displaystyle}\sum\_{S\in\mathcal{S}}\|\|X\_SF^\-f\_S\|\|\_1=\min\_{F \in \mathcal{F}^d}\sum\_{S\in\mathcal{S}}\|\|X\_SF-f\_S\|\|\_1.$ Note that X-Ray-CorrectionFd(S) can also be viewed as the task of deciding, for given data functions f*S, S ∈ S, whether there exist ‘corrected’ data functions fʹS, S ∈ S, that are consistent and do not differ from the given functions by more than a total of m − 1. Nearest-SolutionFd(S) asks for a set F\* ∈ Fd that fits the potentially noisy measurements best. The computational complexity of these tasks is as follows. [] [thm:algstability] The problems X-Ray-CorrectionFd(S), Similar-SolutionFd(S), and Nearest-SolutionFd(S) are in P for ∣S∣ ≤ 2 but are NP-complete for ∣S∣ ≥ 3. Algorithms ---------- Several polynomial-time algorithms for ReconstructionF(Zd, C)(S), C ∈ {{0, 1}, N0}, ∣S∣ = 2, can be found in the literature. In addition to Ryser’s algorithm for C = {0, 1}, there are approaches based on network-flows or matroid intersections . Moreover, the problem can be modeled as an integer linear program, which involves a totally unimodular coefficient matrix, and which can therefore be solved as a linear program (see, for instance, ). For further comments, see . In the presence of NP-hardness, one cannot expect to find generally efficient algorithms. There are, however, various techniques from combinatorial optimization that can and have been applied to solve instances to optimality up to certain sizes; see . Similarly as for ∣S∣ = 2, the reconstruction problem ReconstructionF(Zd, C)(S), C ∈ {{0, 1}, N0}, can be formulated als integer linear program for arbitrary ∣S∣. However, for ∣S∣ ≥ 3 the coefficient matrix is in general no longer totally unimodular. Of course, we can still solve the corresponding linear programming relaxation (where {0, 1} is replaced by [0, 1] or N0 by [0, ∞[) efficiently. Unless P = NP, Thm. [thm:complexity] implies that it will, however, in general not be efficiently possible to convert the obtained fractional solution into a required integer one. Since, in general, measured data are noisy anyway, research focused on approximate solutions. It is quite natural to try to solve ReconstructionFd(S) even if ∣S∣ ≥ 3 by using the available polynomial-time algorithms for ∣S∣ = 2 in an alternating approach. First, two of the given ∣S∣ data functions are selected and a solution F0 is computed which is consistent with these. In the jth step, at least one of the two directions is replaced by a different one from S, and a solution is constructed which satisfies the corresponding two constraints and is closest to Fj. While each step of such an alternating direction approach can be performed in polynomial time, there are severe limitations on the guaranteed quality of the produced solution. For an analysis of this and other approaches, see . Despite their theoretical limitations there are several approaches that are reported to work very well in practice. Among these are BART and DART . The former, which is a variant of ART as described in , is implemented in the open-source software SNARK14 (example code can be found in ), the latter is implemented in the open-source ASTRA toolbox . Further algorithms are discussed in and . For applets illustrating several algorithmic tasks in discrete and geometric tomography, see and , respectively. [fig:circuit] Let us, finally, point out that for certain applications full reconstructions are not needed. For instance, in quality control for circuit board productions (see Fig. [fig:circuit]) one may want to certify that the production process actually produced a desired blueprint structure (‘verification’). Then one can, of course, compute data functions from the blueprint and compare them with the measured data from the produced board. If the difference is large one would report an error. If, however, the difference is small, the produced board can still be quite different from the blueprint (particularly if the data do not determine the image uniquely). This ambiguity can be reduced by applying a (polynomial-time deterministic) reconstruction heuristic on both sets of data functions and subsequently comparing the reconstructions. In practice such checks have shown to be able to detect production flaws even on very limited data and quite poor (and very fast) reconstructions algorithms. Superresolution and discrete tomography ======================================= Electron tomography, pioneered originally in the life sciences (see ), is becoming an increasingly important tool in materials science for studying the three-dimensional morphologies and chemical compositions of nanostructures . For various technical reasons, however, atomic resolution tomographic imaging as envision in has not become a full reality yet (favorable instances are reported in ; see also the surveys ). One of the challenges faced by current technology is that tomographic tilt series need to be properly aligned (see, e.g., ). Therefore, and also to prevent radiation damage, one might wonder whether is is possible to proceed in a multimodal scheme. Suppose some reconstruction has been obtained from a (possibly technologically less-demanding) lower-resolution data set. Can one then use limited additional high-resolution data (for instance, acquired from only two directions) to enhance the resolution in a subsequent step? As we will now see the tractability of this approach depends strongly on the reliability of the initial lower-resolution reconstruction. Details of the presented results can be found in . Computational aspects --------------------- We have already remarked that a function ψ ∈ Fd can be viewed as a characteristic function that encodes a finite lattice set. In a different, yet equivalent, model the function ψ can be viewed as representing a binary image. In this interpretation the points x ∈ Zd represent the pixel/voxel coordinates while ψ(x) denotes their colors (typically, values 0 and 1 are considered to represent white and black pixels, respectively); see Fig. [fig:gridimage] for an illustration. Similarly, for l ∈ N, a function ρ ∈ F(Zd, [l]0) can be viewed as representing a gray-scale image with l + 1 different gray levels (values 0 and l typically representing the ‘gray level’ white and black, respectively). [fig:gridimage] For simplicity of the exposition we restrict our discussion to the case d = 2. Now suppose we want to reconstruct a binary image ψ ∈ F2 contained in an n1 × n2 box from low-resolution gray scale information and high-resolution X-ray data. The lower resolution is quantified by some k ∈ N \ {1}, and we assume that n1 and n2 are divisible by k. More precisely, we assume that an n1/k × n2/k low-resolution (gray-scale) image ρ ∈ F(Z2, [k2]0) of ψ is available, and the pixels x in ψ result from a k × k subdivision of the pixels y of ρ. Hence in any such subdivision B we have ∑x ∈ Bψ(x) = ρ(y). For given ρ(y) and unknown ψ(x), x ∈ B, we call the above equation a *k × k block constraint. We say that, for some ɛ* ∈ N0, a block constraint is satisfied within an error of ɛ*, if ρ(y) − ɛ* ≤ ∑x ∈ Bψ(x) ≤ ρ(y) + ɛ. We may think of ρ as being the result of some lower-resolution reconstruction of ψ. In order to increase the resolution we want to utilize additional high-resolution X-ray data XSψ that are available from the two coordinate directions S1 and S2, and we set S = {S1, S2}. Relatively to ρ the data XSψ, S ∈ S, can be considered as k-times finer resolution X-ray data. For given k ≥ 2 and ɛ ∈ N0 the task of (noisy) superresolution is as follows. nSR(k, ɛ) A gray-level image ρ ∈ F(Z2, [k2]0), a subset R of ‘reliable pixels’ of ρ, and data functions fS1, fS2 at a k-times finer resolution. Determine a function ψ ∈ F2 such that [ht]1 XSψ = fS for S ∈ {S1, S2}, all k × k block constraints for the pixels in R are satisfied, and all other k × k block constraints are satisfied within an error of ɛ, or decide that no such function exists. Since our focus is in the following on double-resolution imaging, i.e., on the case k = 2, let us set $\textsc{nDR}(\varepsilon)=\textsc{nSR}(2,\varepsilon),$ for ɛ > 0. In the reliable situation, i.e., for ɛ = 0 we simply speak of double-resolution and set $\textsc{DR}=\textsc{nSR}(2,0).$ (Then, of course, the set R can be omitted from the input.) An illustration is given in Fig. [super:fig:process]. [super:fig:process] As it turns out DR is tractable. [] [super:thm:main1] DR and also the corresponding uniqueness problem can be solved in polynomial time. The algorithm presented in is based on a decomposition into subproblems, which allows to treat the different gray levels separately. If we view DR as the reconstruction problem for m = ∣S∣ = 2 with additionally block constraints we can compare Thm. [super:thm:main1] with Thm. [thm:complexity] and see that bock constraints impose fewer algorithmic difficulties than X-ray data from a third direction. The next result, which deals with the case that some of the gray levels come with small uncertainties depicts (potentially unexpected) complexity jumps. [][super:thm:main2] Let k ≥ 2 and ɛ > 0. 1. nSR(k, ɛ) is NP-hard. 2. The problem of deciding whether a given solution of an instance of nSR(k, ɛ) has a non-unique solution is NP-complete. To put it succinctly: noise in tomographic superresolution imaging does not only affect the quality of a reconstructed image but also the algorithmic tractability of the inverse problem itself. $\textsc{DR}$ without any block constraints boils down to the reconstruction problem for m = 2 and is hence solvable in polynomial-time. $\textsc{DR}$ is, however, NP-hard if several (but not all) block constraints (which are required to be satisfied with equality) are present (Thm. [super:thm:main2]). Possibly less expectedly, if all block constraints are included, then the problem becomes polynomial-time solvable again (Thm. [super:thm:main1]). If, on the other hand, from all block constraints some of the data come with noise at most 1, then the problem becomes again NP-hard (Thm. [super:thm:main2]). And yet again, if from all block constraints all of the data are sufficiently noisy, then the problem is in P (as this is again the problem of reconstructing binary images from X-ray data taken from two directions). Figure [super:fig:complexityjumps] gives an overview of these complexity jumps. ![(From ) Overview of complexity jumps for the problem of reconstructing a binary image from row and column sums and additional 2\times2 block constraints.](complexityjumps "fig:") [super:fig:complexityjumps] It does not seem likely that, but is still open, whether the tractability result of Thm. [super:thm:main1] persists for k ≥ 3. [] Is the problem nSR(k, 0) NP-hard for k ≥ 3? In the realm of dynamic discrete tomography (see Sect. [sect:dynamics]) block constraints play the role of special window constraints which can be used to encoding velocity information for moving points. For additional information on discrete tomography problems involving other kinds of constraints, see . Stability and instability ------------------------- Let us now turn to a discussion of the stability of the solutions to DR. [thm:instabDR] Let ${\mathcal{S}}=\{\textnormal{lin}(1,0)^T,\textnormal{lin}(0,1)^T\},$ and α ∈ N. Then there exist instances I1 and I2 of DR with the following properties: 1. F1 is the unique solution to I1; 2. F2 is the unique solution to I2; 3. ΔS(F1, F2) = 4; 4. ∣F1∣ = ∣F2∣ ≥ α; 5. $\|F\_1 \cap F\_2\|=\frac{1}{2}\|F\_1\|.$ The proof is based on a construction in of an instance I of DR that admits precisely two solutions Fʹ1 ≠ Fʹ2 with ∣Fʹ1∣ = ∣Fʹ2∣ ≥ α + 2. From these two solutions points in one block are deleted to obtain F1 and F2; see Fig. [fig:largeswitchingcomp] for an illustration. A small X-ray error of 4 can thus lead to quite different reconstructions (again, see Fig. [fig:largeswitchingcomp]). It should be noted, however, that the set F2 has a much larger total variation (TV) than F1 (for some background information see, e.g., ). Regularization by total variation minimization, as proposed in , would therefore always favor the reconstruction F1. [fig:largeswitchingcomp] It is instructive to compare DR (Thm. [thm:instabDR]) with its discrete tomography counterparts for m = 2 (Thm. [thm:stabDTDalen]) and m ≥ 3 (Thm. [thm:instabDT3]), which do not involve any block constraints. On the one hand, the reconstruction problem for m = 2 is much more stable than its double-resolution counterpart. In fact, an easy calculation for β = 4 shows that Thm. [thm:stabDTDalen] implies the bound $$\|F\_1\cap F\_2\|\geq \|F\_1\|-5\sqrt{\|F\_1\|}-9.$$ Thus, if the original set F1 is uniquely determined by its X-rays, then any reconstruction F2 from X-rays with error 4 needs to coincide with F1 by an asymptotically much larger fraction than the ∣F1∣/2 provided in Thm. [thm:instabDR] for DR. On the other hand, the instability result for m = 3 (see Thm. [thm:instabDT3]) is stronger than that of Thm. [thm:instabDR] as for the former an X-ray error of 4 can lead to disjoint reconstructions. Hence in terms of the block constraints seem to play a somewhat weaker role than constraints modeling data from a third direction. Dynamics ======== Let us now turn to dynamic discrete tomography, which, in fact, represents rather recent developments in the field (see ). (For dynamic aspects of computerized tomography, see, e.g., and the references cited therein.) We focus here on the task of tomographic particle (or point) tracking, which amounts to determining the paths P1, …, Pn of n points in space over a period of t ∈ N moments in time from X-ray images taken from a fixed number m of directions. This problem comprises, in fact, two different but coupled basic underlying tasks, the reconstruction of a finite set of points from few of their X-ray images (discrete tomography) and the identification of the points over time (tracking). The latter is closely related to topics in combinatorial optimization including matching and k-assigment problems; see for a comprehensive survey on assignment problems. Let us remark that particle tracking methods have been proven useful in many different fields such as fluid mechanics, geoscience, elementary particle physics, plasma physics, combustion, and biomedical imaging (see also the monograph and the references cited therein). Most previous tomographic particle tracking methods (such as ) can be considered as particle imaging velocimetry (PIV) as they aim at capturing several statistical parameters of groups of particles instead of dealing with them individually. The individual tracking considered here is in the literature also sometimes referred to as particle tracking velocimetry (PTV) or low particle number density PIV . For more general background information on particle tracking methods, see the monographs . The exposition in this section will partly follow. Algorithmic problems -------------------- We want to focus here on the interplay between discrete tomography and tracking. Therefore, we will distinguish the cases that for none, some or all of the τ ∈ [t] moments in time, a solution F(τ) ∈ Fd of the discrete tomography task at time τ is explicitly available (and is then considered the correct solution regardless whether it is uniquely determined by its X-rays). The former case will be referred to as the (partially) or (totally) tomographic case while we speak of the latter as positionally determined. It should be noted that the positionally determined case can be viewed as being the generic case in Rd, d ≥ 3, because there any two (affine) lines in general position are disjoint, hence X-ray lines meet only in the points of F(τ). For simplicity we assume in the following that there are no particles disappearing or reappearing within the tracked time interval. When P = {p1, …, pn} denotes the (abstract) set of n particles, we are in the tracking step thus interested in a one-to-one mapping π(τ) : P → F(τ), τ ∈ [t], that identifies the points of F(τ) with the particles. The particle tracks are then given by Pi = (π(1)(pi), …, π(t)(pi)), i ∈ [n]. This identification is referred to as coupling. In typical applications we would like to incorporate prior knowledge about ‘physically likely’ paths. It seems most natural to input such information in terms of the cost c(P1, …, Pn) of the feasible particle tracks. Note, however, that the number of different particle tracks (P1, …, Pn) is (n!)t, hence exponential in n and t. This means that already for moderate problem sizes the costs of all potential tracks cannot be encoded explicitly. There are various ways to deal with this problem. The most general approach is based on the assumption that ‘an expert knows a good solution if she or he sees it.’ More technically speaking, it is enough for an algorithm to have access to the cost c(P1, …, Pn) only when the particle track c(P1, …, Pn) is considered. Accordingly, suggest an oracular model, where such knowledge is available through an algorithm O, called an objective function oracle, which computes for any solution (P1, …, Pn) its cost c(P1, …, Pn) in time that is polynomial in all the other input data. Then the general problem of tomographic particle tracking for S ⊂ Sd, can be formulated as follows. TomTrac(O; S) t ∈ N and data functions fS(τ) with ∥fS(τ)∥1 = n, for S ∈ S, τ ∈ [t]. Decide whether, for each τ ∈ [t], there exists a set F(τ) ∈ Fd such that XSF(τ) = fS(τ) for all S ∈ S. If so, find particle tracks P1, …, Pn of minimal cost for O among all couplings of all tomographic solutions F(1), …, F(t). In the positionally determined case the problem TomTrac(O; S) reduces to the following tracking problem, which can be viewed as a *t-dimensional assignment* problem. Trac(O; d) t ∈ N and sets F(1), …, F(t) ∈ Fd with ∣F(1)∣ = … = ∣F(t)∣ = n. Find particle tracks P1, …, Pn of minimal cost for O among all couplings of the sets F(1), …, F(t). A priori knowledge may be available in various ways and may then lead to different objective function oracles; see. Here we focus on information that is actually explicitly available. For instance, we speak of a path value oracle if the cost c(P1, …, Pn) is just the sum ∑i = 1nw(Pi) of the weights of the individual paths Pi, i ∈ [n]. Note that the number of different weights is bounded by nt, and can hence be encoded explicitly for fixed (and small) t; see Thm. [thm:matching]. If, further, the weights are just the sums of all costs of assigning points between consecutive moments in time the objective function can be described by just (t − 1)n2, i.e., polynomially many numbers. In this case, the objective function is of Markov-type as it reflects only memoryless dependencies. Combinatorial models, which can be viewed as special choices of such parameters, are based on the knowledge that the positions of the particles in the next time step lie in certain windows. A particular such situation has be analyzed in Sect. [sect:superresolution]. For more results on combinatorial models see. Algorithms and complexity ------------------------- We begin with a simple tractability result for the positionally determined case. [][thm:ILP-posdet] For Markov-type objective function oracles O the problem Trac(O; d) decomposes into uncoupled minimum weight perfect bipartite matching problems and can hence be solved in polynomial time. Although the reconstruction problem in discrete tomography for ∣S∣ = 2 directions can be solved in polynomial time (see Thm. [thm:complexity]) it turns out that there are severe limitations of extending the previous result already for the following quite restricted partially tomographic case. In fact, the problem becomes hard even if there is only one time step, i.e., t = 2, and F(1) is explicitly known while the set F(2) of particle positions for τ = 2 is only accessible through its two X-rays XS1F(2) and XS2F(2). [][thm:tomo-matching] Even if all instances are restricted to the case t = 2, where the solution F(1) is given explicitly, TomTrac(O; S), ∣S∣ = 2, for Markov-type objective function oracles O is NP-hard. Also the corresponding uniqueness problem is NP-complete and the counting problem is #P-complete. Unless P = NP, there is thus, in general, no efficient algorithm that provides exact solutions to every instance of TomTrac(O; S), ∣S∣ = 2. A possible remedy is to resort to heuristics, which aim at providing approximate solutions. Before we discuss such a heuristic let us state two additional intractability results, which concern the positionally determined case for non Markov-type function oracles. [][thm:matching] The problem Trac(O; d) is NP-hard, even if all instances are restricted to a fixed t ≥ 3, and O is a path value oracle. The NP-hardness persists if the objective function values provided by O are all encoded explicitly. It turns out that even if the particles are expected to move along straight lines, this a priori knowledge cannot be exploited efficiently (unless P = NP). [] [cor:straightline] For every fixed d ≥ 2 and t ≥ 3 it is an NP-complete problem to decide whether a solution of Trac(O; d) exists where all particles move along straight lines. The proof of Thm. [cor:straightline] given in relies on the hardness of the particular variant A3ap of 3D-Matching established in . For further results and a discussion of their practical implications see. The previous complexity results show that even for t = 3 and even if there is no tomography involved the coupling becomes hard unless it is of the Markov-type, i.e., it only incorporates information that relate not more than two consecutive moments in time (Thm. [thm:matching]). But even for t = 2, which, of course, is of Markov-type, the problem is hard if tomography is involved at one point in time (Thm. [thm:tomo-matching]). This means that there is not much room for efficient algorithms or ‘self-suggesting’ polynomial-time heuristics. There are, however, quite involved heuristics for TomTrac(O; S1, S2), which allow to incorporate varous different forms of a priori knowledge and different levels of ‘particle history’; see. Here we focus only on one basic method, called Rolling Horizon Tomography, which was introduced in and applied to the study of the slip velocity of a gliding arc discharge in . The general idea in Rolling Horizon Tomography is to model the time step from τ to τ + 1 as a linear program, based on the assumption that F(1) is known (hence we are dealing with the partially tomographic case). The constraints encode the X-rays provided by the data functions f1(τ + 1), f2(τ + 1). The variables correspond to the points in the grid G(τ + 1) and are collected in a vector x(τ + 1). The X-ray information is encoded by means of a totally unimodular matrix A(τ + 1) and a right-hand side vector b(τ + 1). Further, each point gi(τ + 1) ∈ G(τ + 1) carries a weight αi(τ + 1), which reflects the ‘distance’ to a best point gi(τ) ∈ F(τ) (which is a likely ‘predecessor’). These weights are collect in a vector a(τ + 1). Various choices of weights are discussed in , which, for instance, model knowledge on the velocity of the particles. The algorithm can then be described as follows. Beginning with F(1), Rolling Horizon Tomography solves successively for τ ∈ [t − 1] the linear program $$\begin{array}{lrcl} &\multicolumn{3}{c}{\min \,\,\bigl(a^{(\tau+1)}\bigr)^T x^{(\tau+1)}}\\[.1cm] \textnormal{s.\,t. }&A^{(\tau+1)}x^{(\tau+1)} & = & b^{(\tau+1)},\\ & x^{(\tau+1)} & \le &{1 \kern -.41em 1 },\\ & x^{(\tau+1)} & \ge & 0,\\ \end{array}$$ in order to determine F(τ + 1) (via its encoding as a 0-1 incidence vector of a basic feasible solution of the linear program). Finally the paths P1, …, Pn are obtained by a routine that computes a perfect bipartite matching in the graph with vertices F(τ) ∪ F(τ + 1) and edges corresponding to the pairs of vertices that realize the distances αi(τ). Rolling Horizon Tomography runs in polynomial time, is exact in the sense that it is guaranteed to return a solution which matches the data. It also allows to incorporate physical knowledge and it is reported to work quite well in practice (see ). However, (and with a view to Thm. [thm:tomo-matching] not surprisingly), it is only a heuristic, which may fail to reconstruct the correct paths. The reason is that the weights used to measure the quality of the assignment do not incorporate the requirement that no two particles can have originated from the same location at the previous moment in time. Explicit example are given in which also gives generalizations that combine the general rolling horizon approach with interpolation and backtracking techniques to provide algorithms that incorporate physical knowledge even better while still running in polynomial time. Tomographic grain mapping ========================= Tomographic grain mapping deals with the problem of characterizing polycrystalline materials from tomographic data. Polycrystalline materials consist of multiple crystals, called grains. These grains, often 10 − 100 micrometer in diameter, are of central interest in many areas of materials science as most metals, ceramics and alloys are such polycrystalline materials. In fact, the grains determine many of the material’s physical, chemical, and mechanical properties (see, e.g., or the monographs ). Diffraction and indexing ------------------------ There are several non-trivial technological and algorithmical challenges involved in tomographic grain mapping on different scales. Typically, only high-energy X-rays will penetrate the material. In fact, the required X-ray energies are often so large that current experiments need to be conducted at modern synchrotron facilities. For many applications the data are acquired by diffraction (as, e.g., in the 3-Dimensional X-ray Diffraction microscopy technique, 3DXRD and in Diffraction Constrast Tomography, DCT ). Diffraction occurs, however, only if the grain is in a ‘favorable’ position. This is governed by Bragg’s law which relates the unit vectors t, s that signify the incoming and the diffraction directions and the wavelength λ of the X-ray with the crystalline structure of the grain encoded by its dual (or reciprocal) lattice L∘. More precicely, Bragg’s law is as follows: $$\frac{t-s}{\lambda}=\ell\in L^\circ\setminus\{0\};$$ (see Fig. [fig:indexingbragg](a) for an illustration). Consequently, tomographic data are typically only available from a small number of directions (often, 8 − 10). [fig:indexingbragg] The limited number of data, and the fact that multiple grains are simultaneously imaged, poses major algorithmic challenges at quite different scales, from the atomic to the macroscopic level. The problem, commonly referred to as indexing , is to group the tomographic data according to their grain of origin. This allows often the determination of grain parameters like the lattice (including its orientation), or the center of mass; see Fig. [fig:indexingbragg](b). Based on the tomographic data acquired for each single grain, the macroscopic geometric structure of the full collection of different grains is then to be determined. Of course, such tasks can be highly interrelated, and there are also possible cases where it is favorable to reconstruct several of the grains simultaneously. More details can be found in . Macroscopic reconstruction -------------------------- As in Sect. [sect:superresolution], a single grain g can be considered as a binary image ψg ∈ F3. The points of $\textnormal{supp}(\psi\_g)$ correspond to the pixels that belong to g. The paper describes one of the first attempts of reconstructing multiple grains, the so-called grain map. In this paper the ART algorithm is used, but it is found that often the reconstructions contain unrealistic void spaces between adjacent grains. To overcome this problem, a Monte-Carlo approach based on Gibbs priors was introduced in . This approach was generalized in (see also ) to deal with the task of reconstructing grain maps of moderately deformed grains. More stochastic approaches to grain map reconstruction can be found in . For alternative approaches, see . In the following we describe a linear-programming based method, introduced in , which returns approximations of grain maps. It is based on only a few input parameters for each grain: (approximations of) its center-of-mass, its volume and, if available, its second-order moments. The centers-of-mass can be determined by the indexing procedure, the grain volume by integration of the respective X-ray data, and the second-order moments by backprojecting the projections acquired from the same grain. The aim is to reconstruct what we call generalized balanced power diagrams (GBPDs). These diagrams generalize power diagrams (which are also known as Laguerre or Dirichlet tessellations), which in turn generalize Voronoi diagrams; see also and. Any GBPD is specified by a set of distinct sites S = {s1, …, sl} ⊂ Rd, additive weights (σ1, …, σl)T ∈ Rl, and positive definite matrices A1, …, Al ∈ Rd × d. The *jth generalized balanced power cell* Pj is then defined by Pj = {x ∈ Rd : ∣∣x − sj∣∣Aj2 − σj ≤ ∣∣x − sk∣∣Ak2 − σk, ∀k ≠ j}, where ∣∣ ⋅ ∣∣Aj, j ∈ [l], denotes the ellipsoidal norm $$\|\| x\|\|\_{A\_j}=\sqrt{ x^T A\_j x}.$$ The generalized balanced power diagram P is the l-tuple P = (P1, …, Pl). The proposed method is able to find optimal σ1, …, σl that guarantee that the volumes of each cell are within prescribed ranges. The concept of GBPDs can be viewed as structure-driven weight balanced clusterings; see . For j ∈ [l] let sj denote the center of the jth grain, and let κj−, κj+ be lower and upper bounds for its volume, respectively. Further, let x1, …, xq be the points of the image that has to be partitioned into the grains, and set γi, j = ∣∣xi − sj∣∣Ai2 for all i, j. Then we can model the assignment problem by the following linear program: $$\begin{array}{lll} \displaystyle \textnormal{(LP)} &\displaystyle\min \sum\_{i=1}^q\sum\_{j=1}^l\gamma\_{i,j}\xi\_{i,j} &\\ \displaystyle \textnormal{subject to} &\displaystyle\sum\_{j=1}^l\xi\_{i,j}=1 &\displaystyle(i\in[q]),\\ \displaystyle &\displaystyle\kappa\_j^-\leq\sum\_{i=1}^q \xi\_{i,j} \leq \kappa\_j^+ &\displaystyle(j\in[l]),\\[4.0ex] \displaystyle &\displaystyle\xi\_{i,j}\geq 0 &\displaystyle(i\in[q];\: j\in[l]). \end{array}$$ In general, the variables ξi, j specify the fraction of the point xi that is assigned to the center sj. Since, however, the coefficient matrix is totally unimodular all basic feasible solutions are binary, and we obtain an optimal assignment of pixels to grains in polynomial time. An example for the quality of reconstruction for planar grain maps (which are easier to visualize) is shown in Fig. [fig:stoyan]. Reports on the favorable performance of the presented approach on various (real-world) data sets can be found in the recent papers . [fig:stoyan] We remark that the clustering approach described above was previously applied (in an ‘isotropic’ fashion) in the context of farmland consolidation . For an application in designing electoral districts where municipalities of a state have to be grouped into districts of nearly equal population while obeying certain politically motivated requirements, see . Switching components and a problem in number theory =================================================== Switching components, i.e., pairs of tomographically equivalent sets as introduced in Sect. [subsec-uniqueness], are strongly related to an old problem in Diophantine number theory, called the Prouhet-Tarry-Escott or PTE-problem, named after Eugène Prouhet , Gaston Tarry , and Edward B. Escott . [Prouhet, 1851; Tarry, 1912; Escott, 1910] Given k, n ∈ N, find two different multisets X = {ξ1, …, ξn} ⊂ Z and Y = {η1, …, ηn} ⊂ Z, such that $$\xi\_1^j+\xi\_2^j+\ldots+\xi\_n^j = \eta\_1^j+\eta\_2^j+\ldots+\eta\_n^j,\qquad \textnormal{for }j\in[k].$$ Pairs (X, Y) satisfying the above equation are called PTE solutions. More precisely, we speak of (k, n)-solutions, and the numbers k and n, respectively, are referred to as the degree and size of the PTE solution. Often the notation $X\stackrel{k}{=}Y$ is used to indicate that (X, Y) is a degree k solution. For instance, as an elementary calculation shows, $$\{0,14,28,56,70,84\}\stackrel{5}{=}\{4,6,40,44,78,80\}.$$ The PTE problem has connections to several other problems in number theory, including the ‘easier’ Waring problem,, the Hilbert-Kamke problem, and a conjecture due to Erds and Szekeres,. There are also connections to Ramanujan identities , other types of multigrade equations , problems in algebra , geometry , combinatorics , graph theory , and computer science . For background information see . The PTE problem can be traced back to a correspondence between Goldbach and Euler. In his 1950 letter Goldbach states the identity $$\begin{gathered} (\alpha+\beta+\delta)^2+(\alpha+\gamma+\delta)^2+(\beta+\gamma+\delta)^2+\delta^2\\ =(\alpha+\delta)^2+(\beta+\delta)^2+(\gamma+\delta)^2+(\alpha+\beta+\gamma+\delta)^2,\label{eq:classic}\end{gathered}$$ where α, β, γ, δ ∈ Z. In other words, $$\{\alpha+\beta+\delta, \alpha+\gamma+\delta, \beta+\gamma+\delta,\delta\}\stackrel{2}{=}\{\alpha+\delta,\beta+\delta,\gamma+\delta,\alpha+\beta+\gamma+\delta\}.$$ It was already known to Prouhet, Tarry, and Escott that there exist (k, 2k)-solutions for every k (see and ). Such solutions can be generated as follows. Express each p ∈ [2k + 1 − 1]0 as a binary number. If this binary expression of p contains an even number of 1’s, then assign p to the set X, otherwise to Y. Then, (X, Y) with X = {ξ1, …, ξ2k} and Y = {η1, …, η2k} is a (k, 2k)-solution. Proofs of this result can be found in . For generalizations, see . On the other hand, there are no (k, n)-solutions whenever n < k + 1. This result, commonly attributed to Bastien , can be derived from the Newton’s identities. A (k, n)-solution is called ideal if n = k + 1. The following is a long-standing open question (see and ). Do there exist ideal PTE solutions for every k? Presently, ideal solutions are only known for k ∈ [11] \ {10}. Concerning upper bounds on n, the currently best bound (of ) guarantees that for any k there exists a (k, n)-solution with $n\leq \frac{1}{2}(k^2-3)$ if k is odd and $n \leq \frac{1}{2}(k^2-4)$ if k is even. The proofs are non-constructive. In fact, all currently known constructive proofs yield bounds that are exponential in k. PTE solutions from switching components --------------------------------------- The following explicit connection between the PTE problem and switching components first appeared in. Following , we will focus on the case d = 2 (for general d see ). For given M ⊂ Zd and c ∈ Zd, let Πc(M) denote the multiset Πc(M) = {cTx : x ∈ X}. Clearly, Πc(M) ⊂ Z. Perhaps more surprisingly, the following result holds if we insert the points of a switching component. [] [thm:PT1] If (X, Y) is an (m + 1)-switching component in Z2 and c ∈ Z2 such that Πc(X) ≠ Πc(Y), then (Πc(X), Πc(Y)) is a degree m solution of the PTE problem. This construction of PTE-solutions from switching components can be exemplified, say, for the switching components depicted in Fig. [fig:smallswitch]. [fig:smallswitch] For instance, if in Fig. [fig:smallswitch](a) the origin is located in the lower left lattice point (which, of course, is an arbitrary choice) the sets X and Y of black and white points are $$\begin{array}{lll} X&=&\left\{\left(\begin{array}{c}0\\2\end{array}\right), \left(\begin{array}{c}1\\0\end{array}\right), \left(\begin{array}{c}2\\5\end{array}\right), \left(\begin{array}{c}4\\1\end{array}\right), \left(\begin{array}{c}5\\6\end{array}\right),\left(\begin{array}{c}6\\4\end{array}\right)\right\},\\[2ex] Y&=&\left\{\left(\begin{array}{c}0\\1\end{array}\right), \left(\begin{array}{c}1\\4\end{array}\right), \left(\begin{array}{c}2\\0\end{array}\right), \left(\begin{array}{c}4\\6\end{array}\right), \left(\begin{array}{c}5\\2\end{array}\right),\left(\begin{array}{c}6\\5\end{array}\right)\right\}. \end{array}$$ For cT = (1, 2) we obtain the PTE solution $$\Pi\_c(X)=\{1,4,6,12,14,17\} \stackrel{5}{=} \{2,2,9,9,16,16\}=\Pi\_c(Y)$$ of degree 5. As a basic ingredient the standard proof of Thm. [thm:PT1] uses the encoding of points by polynomials mentioned in connection with Thm. [thm:hajdutijdeman]. Thm. [thm:PT1] allows to derive explicit constructions of families of PTE solutions; ). As an example let us consider the result of Prouhet, Tarry, and Escott that (k, 2k)-solutions exist for every k. The proof given in extends over two half-pages. The geometric shortcut via Thm. [thm:PT1] just uses the construction of switching components (X, Y) with ∣X∣ = ∣Y∣ ≤ 2k from Fig. [fig:zonotopeconstruction]. Generalizations --------------- The geometric point of view also helps in studying other variants of the PTE problem. Naturally, PTE can be considered over arbitrary rings R. For R = Zd we obtain PTEd** which can be viewed as a d-dimensional or multinomial version of the original PTE problem. [PTEd] Given d, k, n ∈ N, find two different multisets X = {ξ1, …, ξn}, Y = {η1, …, ηn} ⊂ Zd with ξl = (ξl1, …, ξld)T, ηl = (ηl1, …, ηld)T for l ∈ [n] such that ∑l = 1nξl1j1ξl2j2 ⋅ … ⋅ ξldjd = ∑l = 1nηl1j1ηl2j2 ⋅ … ⋅ ηldjd for all non-negative integers j1, …, jd with j1 + j2 + … + jd ≤ k. There are trivial ways of generating PTEd-solution from PTE1-solutions. For instance, if $\{\alpha\_1,\dots,\alpha\_n\}\stackrel{k}{=}\{\beta\_1,\dots,\beta\_n\}$ is a PTE1-solution, then $$\{(\alpha\_1,\dots,\alpha\_1)^T,\dots,(\alpha\_n,\dots,\alpha\_n)^T\}\stackrel{k}{=}\{(\beta\_1,\dots,\beta\_1)^T,\dots,(\beta\_n,\dots,\beta\_n)^T\}$$ is a solution to PTEd. A general method of generating non-trivial solutions for PTE2 is provided by. [] [prpproC] Every (m + 1)-switching component (X, Y) in Z2, is a degree m solution of the PTE2 problem. For instance, for the sets X and Y corresponding to Fig. [fig:smallswitch](a) it is elementary to verify that $$\begin{aligned} 0^i2^j+1^i0^j+2^i5^j+&4^i1^j+5^i6^j+6^i4^j\\ &= 0^i1^j+1^i4^j+2^i0^j+4^i6^j+5^i2^j+6^i5^j\end{aligned}$$ for all non-negative i, j with i + j ≤ 5. Applying Thm. [prpproC] to the known smallest size switching components (an example for k + 1 = 6 is depicted in Fig. [fig:smallswitch](a)), one sees that for every degree k ∈ {1, 2, 3, 5} there exist ideal PTE2 solutions;. The following problem is, however, open already for k = 4. Do there exist ideal PTE2 for every degree k? Concluding remarks ================== The present paper tried to support the following conviction of the authors: Discrete tomography is a broad and interesting field, both, in terms of its methods and its applications. Discrete tomography has strong links to various areas within mathematics which have the potential to provide new insight in older problems. 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Friedel-pair based indexing method for characterization of single grains with hard X-rays., 524(1-2):64–68, 2009.. S. E. Offerman, N. H. van Dijk, J. Sietsma, S. Grigull, E. M. Lauridsen, L. Margulies, H. F. Poulsen, M. T. Rekveldt, and S. van der Zwaag. Grain nucleation and growth during phase transformations., 298(5595):1003–1005, 2002.. H. F. Poulsen and X. Fu. Generation of grain maps by an algebraic reconstruction technique., 36(4):1062–1068, 2003.. P. Reischig, A. King, L. Nervo, N. Viganò, Y. Guilhem, W. J. Palenstijn, K. J. Batenburg, M. Preuss, and W. Ludwig. Advances in X-ray diffraction contrast tomography: Flexibility in the setup geometry and application to multiphase materials., 46(2):297–311, 2013.. L. Rodek, H. F. Poulsen, E. Knudsen, and G. T. Herman. A stochastic algorithm for reconstruction of grain maps of moderately deformed specimens based on X-ray diffraction., 40(2):313–321, 2007.. S. Schmidt, S. F. Nielsen, C. Gundlach, L. Margulies, X. Huang, and D. Juul Jensen. Watching the growth of bulk grains during recrystallization of deformed metals., 305(5681):229–232, 2004.. R. M. Suter, D. Hennessy, C. Xiao, and U. Lienert. Forward modeling method for microstructure reconstruction using X-ray diffraction microscopy: Single-crystal verification., 77(12):123905, 2006.. N. Viganò, W. Ludwig, and K. J. Batenburg. Reconstruction of local orientation in grains using a discrete representation of orientation space., 47(6):1826–1840, 2014.. Further reading: Macroscopic grain mapping: A. Alpers, A. Brieden, P. Gritzmann, A. Lyckegaard, and H. F. Poulsen. Generalized balanced power diagrams for 3D representations of polycrystals., 95(9):1016–1028, 2015.. A. Brieden and P. Gritzmann. A quadratic optimization model for the consolidation of farmland by means of lend-lease agreements. In D. Ahr, R. Fahrion, M. Oswald, and G. 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`cdf`3$\tilde p$ and `cdf`4$\tilde p$ appear to converge approximately at the same rate. Increasing the oversampling factor makes the difference in convergence rates more pronounced. The sparse AZ algorithm ----------------------- [t] [alg:sparseAZ] Input: A, Z ∈ CM × N, b ∈ CM Output: x ∈ CN such that Ax ≈ b [1] Create sparse matrix (I − AZ\)A* Solve (I − AZ\)A*x1 = (I − AZ\)b* using sparse QR x2 ← Z\(b* − Ax1) x ← x1 + x2 Finally, we consider an algorithm that only exploits the sparsity of the matrices A, Z and A − AZ\*A. As for B-splines in, here we use the sparse direct rank-revealing QR decomposition of in the first step of the AZ algorithm instead of a low-rank solver. The sparse AZ algorithm is formulated in Algorithm [alg:sparseAZ]. The first step is the creation of the matrix A − AZ\*A in sparse form. It was shown in that a sparse version $\hat A-\hat A\hat Z^\\hat A$ (holding $\mathcal O(N^{(d-1)/d})$ non-zero values and $\mathcal O(1)$ non-zero elements in each column and row) can be created in $\mathcal O(N)$ operations. Given this sparse matrix $\hat A-\hat A\hat Z^\\hat A$, we can easily create the matrix containing its non-zero columns $\hat A(I-\hat A\hat Z^\)\hat AE$ with $$E(\mathbf k, \mathbf l) = \delta\_{\mathbf k\mathbf l}, \qquad \mathbf k\in I\_{\mathbf N}, \mathbf l\in \K.$$ It remains to construct the sparse matrix E\W− 1 since $$A-AZ^\A = \hat A(I-\hat A\hat Z^\)\hat AEE^\W^{-1}.$$ Provided Assumption [ass:boundary] is satisfied, we deduce from Lemma [lem:dwtstructure] that E\W− 1 has $\mathcal O(N^{(d-1)/d}\log N)$ non-zero elements, i.e., $\mathcal O(N^{(d-1)/d})$ rows with $\mathcal O(\log N)$ elements each. Because of the structure in both matrices the standard sparse matrix-matrix multiply in Julia 1.3 performs the multiplication of $A(I-\hat A\hat Z^\)\hat AE$ with E*W− 1 in $\mathcal O(N^{(d-1)/d}\log(N))$. Hence, for the best possible time complexity in the first step of the sparse AZ algorithm, we need to construct E\*W− 1 in $\mathcal O(JN^{(d-1)/d})$ operations. Here, we present an algorithm that in the worst case is $\mathcal O(N\log N)$. An inspection of §[ss:dwt] using compactly supported sequences and convolutions instead of matrices yields the insight that each column of WJi− 1 (1-D iDWT transform) contains a shifted version of either one of $$\begin{aligned} \label{eq:idwtfilters} &&g^{J},\\\nonumber &&[g^{J-1}]\_{\uparrow 2^1}\star h^{J},\\\nonumber &&[g^{J-2}]\_{\uparrow 2^2}\star [h^{J-1}]\_{\uparrow 2^1}\star h^{J}, \\\nonumber &&\qquad\qquad\quad\vdots\\\nonumber &&[g^{1}]\_{\uparrow 2^{J-1}}\star [h^{2}]\_{\uparrow 2^{J-1}}\star\cdots\star [h^{J-1}]\_{\uparrow 2^1} h^{J}, \\\nonumber &&[h^{1}]\_{\uparrow 2^{J-1}}\star [h^{2}]\_{\uparrow 2^{J-1}}\star\cdots\star [h^{J-1}]\_{\uparrow 2^1} h^{J}\end{aligned}$$ where we left out the dimension subindex to Ji and where [a]↑ q denotes upsampling and aJ periodisation: $$([a]\_{\uparrow q})\_k = a\_{kq}, \qquad (a^J)\_k = \sum\_{l\in\Z} a\_{k+l2^J}.$$ A convolution a ⋆ b can be computed in $\mathcal O(n\log n)$ operations with n the sum of the supports of a and b using the Fast Fourier Transform. The top filter in has constant support, but the support lengths grow steadily up to Ji2Ji. Therefore, the Ji + 1 filters can be computed in $\mathcal O(N\_i\log N\_i)$ operations. If we assume that the degrees of freedom are evenly distributed over all dimensions, then we have $N\_i=\mathcal O(N^{1/d})$ and the total number of nonzero entries in E\*W− 1 is $\mathcal O(JN^{(d-1)/d})$. Hence, knowing their locations, the sparse matrix can also be constructed in $\mathcal O(JN^{(d-1)/d})$ operations. In the worst case, if one dimension has all degrees of freedom, then the cost of the full construction algorithm of the sparse E\*W− 1 may be as large as $\mathcal O (N\log N)$. This is unlikely to be the case in practice. Figure [fig:AZStimings1d-3d] shows that the sparse AZ algorithm appears to be more efficient than the algorithms above, especially in the lower dimensions. In 3-D, the level of sparsity is not yet high enough to show a possible advantageous effect of using a sparse solver. We also compare coefficient norm and residual of the different algorithms. For the coefficient norm, it is known for the vanilla and reduced AZ algorithms that a small-norm coefficient will be returned if it exists, since the solver in step 1 is closely related to a truncated SVD solver. For the direct sparse QR solver of, no analogous error analysis is known, hence the coefficient norm might be large. While this was not the case for spline extension in, coefficients are indeed larger using wavelet extensions as shown in Figure [fig:AZerrors1d-3d]. Both the sparse AZ and a direct sparse QR solver are affected. This has a negative impact on the residual, as is also shown in Figure [fig:AZerrors1d-3d]. Adaptively smoothed wavelet AZ algorithm ======================================== In this final section we compare the use of wavelet-based extensions to the simpler setting of spline-based extensions. The approximation space spanned by a basis of B-splines is exactly the same as that spanned by spline-based wavelets. Indeed, the wavelet transform is merely a change of basis. Therefore, the best approximations are the same: $$\arg \min\_{u \in \SPAN \Phi\_N} \Vert f - u \Vert = \arg \min\_{u \in \SPAN \Psi\_N} \Vert f - u \Vert.$$ In typical applications of wavelets their compression properties play a major role. That is not really the case here: the wavelet least squares matrix A is even somewhat less sparse than the corresponding matrix $\hat A$ using B-splines. Another beneficial property of wavelets is the multiresolution nature of the approximation. In particular, unlike with B-splines, it is possible to associate different weighting factors with different scales. This enables the construction of bases for a range of function spaces with varying smoothness properties. For example, methods for the solution of partial differential equations employ wavelet bases for Sobolev spaces. In our setting, weighing different wavelet scales allows one to obtain smoother approximations. The increased smoothness is only visible in the extension Ξ \ Ω of the wavelet frame, since the approximation always resembles the function itself in the interior Ω. A smooth extension is not guaranteed by the methods described in §[wavs:az]. A least squares solver aims to minimize the residual of the system with a minimal norm solution. Therefore, the resulting wavelets coefficients do not necessarily decrease with increasing scale, even when approximating smooth functions. All coefficients have roughly similar size. In contrast, when approximating with a regular basis, the decrease of wavelet coefficient size is guaranteed for smooth functions, depending on the order of the multiresolution analysis at hand. [h] [alg:smoothedAZ] Input: A, Z ∈ CM × N, b ∈ CM, $W\in \R\_+^{N\times N}$ Output: x ∈ CN such that Ax ≈ b [1] Solve (I − AZ\)AWx1 = (I* − AZ\)b* using a randomized low-rank solver x2 ← Z\(b* − AWx1) x ← Wx1 + x2 Smoothing can be introduced by switching to a weighted least squares formulation. We weigh the wavelet coefficients using the smoothed AZ algorithm (Algorithm [alg:smoothedAZ]). We simply add a diagonal weight matrix in the first step of the AZ algorithm, replacing x1 by Wx1, and leave the other steps unchanged. The diagonal matrix has weights that depend on the scale of the corresponding entries of x1. Much in the same way, the weighted reduced and sparse AZ algorithms can be formulated and implemented. Note that a weighted least squares problem is only being solved in step 1, not in step 2. The dual does not require modifications in this formulation. We aim for a coefficient vector in which the coefficients decrease in size with increasing scale. The logic is as follows. Say a function is approximated on a coarse scale $\mathbf J-1$ with approximation error e1. It can be expected that this approximation can be refined by adding wavelet coefficients on finer scales with size on the order of $\mathcal O(e\_1)$. We ensure that the coefficients on the finer scale have that size simply by choosing the corresponding diagonal entries of W equal to e1. The function is now approximated on the finer scale, say with an approximation error e2. This error can be used to weight the next scale of coefficients, and so on. Thus, we obtain a diagonal weighting operator parametrized with the weights e1, …, eL and size $\mathbf N$, L ≤ log2(Ni), i = 1, …, d: $$W(\mathbf k,\mathbf l; [e\_1, e\_2, \dots, e\_L], \mathbf N) = \delta\_{\mathbf k,\mathbf l}e\_{\min\{1,L-\min\_{i=1,\dots,d}\{\log\_2(N\_i)-L\}\}}\qquad \mathbf k,\mathbf l \in I\_{\mathbf N}.$$ We use the size of the right hand side as an initial weight for the first approximation. This way we arrive at the adaptive Algorithm [alg:adaptivelysmoothedAZ]. [h] [alg:adaptivelysmoothedAZ] Input: $\mathbf N$, $\mathbf\osi$, f, Ω ⊂ Ξ, wavelet types Output: x (the wavelet extension coefficients) [1] $\mathbf n\gets\mathbf N-\min(N)+1$ $e \gets \\|b^{\mathbf\osi}\_{\mathbf n}\\|$ $W \gets W(\cdot,\cdot; e, \mathbf N)$ x ← Apply the smoothed AZ algorithm with $A^{\mathbf\osi}\_{\mathbf n}$, $Z^{\mathbf\osi}\_{\mathbf n}$, $b^{\mathbf\osi}\_{\mathbf n}$, W $e \gets [e; \\|A^{\mathbf\osi}\_{\mathbf n}x-b^{\mathbf\osi}\_{\mathbf n}\\|]$ $\mathbf n \gets 2\mathbf n$ [fig:1Dcoefs] ![image](img/fspline_AZR) ![image](img/fspline_D) ![image](img/f_AZR)![image](img/f_AZWR)![image](img/f_D) ![image](img/2Dcoefs_AZR)![image](img/2Dcoefs_AZWR)![image](img/2Dcoefs_D) We illustrate the adaptive Algorithm with a simple example in 1D first. In Figure [fig:1Dcoefs] we compare to a pivoted QR and the reduced AZ algorithm (both without smoothing). The function f(x) = ex is approximated on the interval [0, 0.6] using a `cdf33` wavelet extension. The weighted algorithms very clearly leads to the smoothest extension (shown in the left panel). In the right panel, the sizes of the wavelet coefficients are compared. The approximation domain [0, 0.6] is visible in all levels of the wavelet coefficients. There, all coefficients have rougly similar size (the three lines overlap) because they approximate the same function. The coefficients differ in the extension [0.6, 1]. There, it is seen that the smoothed approximation (red line) yields significantly smaller coefficients than the non-smoothed approximations. In Figure [fig:2Dcoefs] and Table [tab:comparison] we compare the adaptive Algorithm [alg:adaptivelysmoothedAZ] using both sparse (only in table) and reduced AZ with the original non-smoothed reduced and sparse AZ algorithm. Also we compare with a simple pivoted QR. Finally, we compare wavelet extension with spline extension approximation as in where no adaptive weighting is possible. The smoothed approximation (the middle column of Figure [fig:2Dcoefs] and second row of the table) leads to a visually smoother extension. We also see that the non-smoothed methods tend to result in larger wavelet coefficients at the boundary. For the smoothed result, one can also see that the wavelet coefficients at a coarse scale are larger then those at a finer scale. The approximant itself also takes another shape. The spline extension approximant will drop down to zero outside of Ω. Choosing a wavelet extension instead of a spline extension has no effect on the residual, nor does weighting. As expected, the pivoted QR leads to the smallest coefficient norm. Concluding remarks ================== We have shown that wavelet approximation on general domains is possible and efficient using regular wavelets defined on a bounding box. Compared to existing wavelet literature, we have had to compute a discrete dual scaling sequence. Apart from this construction, with values listed exhaustively in the appendix, well-known scaling functions and wavelets could be used. The ill-conditioning of the extension problem necessitates a least squares formulation with regularization. The proposed algorithms have a complexity of $\mathcal O(N)$ operations in 1-D, $\mathcal O(N^{3/2})$ in 2-D and $\mathcal O(N^{3(d-1)/d})$ in d-D, d > 1. This should be compared to the cubic cost of standard direct solvers. The algorithms required a detailed study of the sparsity structure of all the matrices involved. The use of a general sparse QR function does not require such detailed study of the structure. As such, it is simpler to implement. The results experimentally appear to be more efficient. However, it also seems less stable. Appendix ======== The values of the discrete duals that were used for the experiments in this paper are tabulated in Table [tab:discretevalues]. [email protected] & & & [tab:discretevalues] --- 1. Email: `[email protected]`. Website: `https://people.cs.kuleuven.be/~vincent.coppe`.[↩](#fnref1) 2. Email: `[email protected]`. Website: `https://people.cs.kuleuven.be/~daan.huybrechs`.[↩](#fnref2) 3. Since we intend to employ wavelets on a bounding box Ξ to approximate functions on a subset Ω ⊂ Ξ, the periodicity of the basis on Ξ is not actually a restriction on Ω, as long as the boundaries of Ω and Ξ do not touch. One can use other boundary conditions on Ξ, but periodicity is the simplest one to implement and manipulate.[↩](#fnref3) 4. The values of p and p̃ do not uniquely determine a biorthogonal multiresolution analysis. We use filters corresponding to B-splines as described in. Thus, the meaning of `cdf44` in this paper differs from the widely used CDF filters of primal and dual order 4 in signal processing, e.g., in the JPEG2000 standard.[↩](#fnref4) 5. For completeness, in our implementation we have performed a DWT on a vector of length N with a `NaN` instead of a floating point number at the indices $\K$ (non-zero indices of $\hat A\_{\mathbf N}-\hat A\_{\mathbf N}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N}$). Since a mathematical operation between a `NaN` and a floating point number results in a `NaN`, the resulting vector thus contains `NaN`s at the non-zero indices of $W\_{\mathbf J}(\hat A\_{\mathbf N}-\hat A\_{\mathbf N}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N})$.[↩](#fnref5) Efficient function approximation on general bounded domains using wavelets on a cartesian grid ============================================================================================== ### KU LeuvenDepartment of Computer ScienceCelestijnenlaan 200A3001 Leuven, Belgium Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $\mathcal O(N)$ operations in 1-D and $\mathcal O(N^{3(d-1)/d})$ in d-D, d > 1. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients. Keywords Fourier extension, wavelets, efficient algorithms, frames, ill-conditioning, function approximation, oversampling Mathematics Subject Classification (2010) 65D15, 65T60, 65Y20 Introduction ============ Wavelets have many applications in signal processing. Their most common uses are in compression, edge detection, denoising and other signal enchancements. The wide applicability of wavelets is mostly due to the localization properties of wavelets in time and frequency, such that many signals can be sparsely represented, as well as to the existence of the (bi)orthogonal Discrete Wavelet Transform that can be applied very efficiently. Wavelets are also applied in solution methods for partial differential equations or integral equations using wavelet-based discretizations. Here, too, the localization properties of wavelets are of interest. In this setting, wavelets with suitable level-dependent scalings can generate stable bases for a range of function spaces, and the existence of dual bases with varying smoothness is frequently useful. However, it is in general difficult to create a wavelet basis on a complex geometry, i.e., to create a basis for a domain that is not a hypercube. Several methods have been proposed to enable the use of wavelet on general domains. One method is based on solving a Dirichlet problem with a fictitious domain method. Other methods employ adaptive finite element techniques. However, in it is stated that the required smoothness conditions for efficient adaptive wavelet methods are difficult to impose on domains that do not have product structure. A wavelet frame method is constructed in that can cope with domains that are overlapping unions of subdomains, each of them being the image under a smooth parametrization of a hypercube. Frames generalize a basis in the sense that they allow for redundancy. As in, in this paper we resort to using a wavelet frame rather than a wavelet basis. However, we use a different type of frame and we restrict ourselves to the problem of function approximation rather than the solution of operator equations. The approximation problem we consider is the same as that considered in based on Fourier series and splines respectively. We aim for a fast algorithm for the approximation of a function f on a compact domain Ω that can have an arbitrary shape. Without loss of generality, this bounded domain can be scaled such that Ω ⊂ Ξ with Ξ = [0, 1]d. With Fourier extensions, (tensor products of) Fourier series are used on Ξ, while in an analogous spline extension is introduced based on a periodic spline basis on Ξ. Here, we extend the idea further to wavelets. While it is difficult to create a wavelet basis on Ω, it is easy to create one on Ξ. Consider for example tensor products of Daubechies or CDF wavelets, periodized to the interval [0, 1]. If we restrict the basis to Ω, we naturally arrive at a frame that we will call a wavelet extension frame. In this paper we focus on algorithms, rather than on the properties of a frame. Nevertheless, we recall its definition and the motivation for its use in function approximation. A family of functions Φ = {ϕk}k = 1∞ is a frame for a Hilbert space $\mathcal H$ if $$\label{eq:frame} A\\|f\\|^2\leq\sum\_{k=1}^\infty \|\langle f,\phi\_k\rangle\|^2 \leq B\\|f\\|^2,\qquad \forall f\in\mathcal H$$ for constants A, B > 0. It is more general than a basis, as demonstrated by the construction above. In particular frames may be redundant. In the setting of this paper, redundancy arises since our approximant can take any value in Ξ \ Ω. This leads to apparent ill-conditioning of the approximation problem. However, recent theory indicates that the ill-conditioning of the linear systems to be solved does not prevent stable and highly accurate function approximation if one uses regularization techniques in combination with oversampling. For that reason we consider least squares approximations and develop an efficient regularizing solver. Wavelets are by their nature adaptive. It is possible to extend or refine a wavelet basis by adding basis functions on a finer scale. This is not possible in a spline basis. A translation-invariant spline basis ΦN = {ϕ( ⋅ − hk)}k = − ∞∞ with h > 0 can be refined by dilating the basis functions, but all basis functions change as a result. It is the possibility of adaptivity of wavelets that motivates their study in this paper. However, we will not (yet) fully take advantage of the possibilities. We do use one form of adaptivity at the end of the paper to arrive at a wavelet extension approximation with a smooth extension by choosing level-dependent weights, which is not possible in the context of spline extensions. Though the methods of the paper are general, we consider in our examples the Daubechies and Cohen–Daubechies–Feauveau (CDF) family of wavelets, since they are widely used and have compact support. The duals of these bases are a key ingredient in the construction of efficient solvers. For Daubechies and CDF wavelets, dual bases in $L^2(\R)$ are well studied. They can be used for function approximation using a Galerkin-type approach, i.e., based on (bi)orthogonal projections using inner products. However, inner products with wavelets on general domains are not easily computed, especially not in the multivariate case, since they require the numerical evaluation of integrals on domains of general (and possibly irregular) shape. Instead, we focus in our experiments on a collocation approach based on discrete function samples. Collocation and oversampling necessitate the construction of bases that are dual with respect to a discrete oversampled equispaced grid. We provide such a construction on the bounding box using cartesian grids, taking advantage of their regular structure, and demonstrate how this construction can be used for the efficient solution on the subdomain of general shape. The structure of the paper is as follows. In §[wavs:wavelets], wavelets are introduced along with the discrete wavelet transform. The structure of the latter can be used to create efficient matrix-vector products. We recall these basics in order to modify them later on. In §[wavs:periodicwavelet], bases biorthogonal to periodic wavelet bases on the interval are discussed. We describe the construction of discrete dual bases. In §[wavs:approximationproblem] we discretize the function approximation and arrive at a matrix system. Next, we compare several algorithms to solve this system in §[wavs:az]. Finally, we use the adaptive nature of wavelets to construct a smooth extension in §[wavs:smooth] and end with some concluding remarks in §[wavs:conclusion]. Wavelets ======== Wavelets may be created by dilating and translating a given function. For particular choices of $\psi(t)\in L^2(\R)$, the family $$\label{eq:motherfunction} \psi\_{jk}(t) = {2^{j/2}}\psi(2^{j}t-k), \qquad j,k\in\Z$$ forms a basis for $L^2(\R)$. This family is a wavelet basis and ψ(t) is called the mother function. There exists a great variety of other types of wavelets, some giving rise to a frame rather than a basis. However, we limit ourselves here to orthogonal and biorthogonal wavelet bases that are translation invariant as above, with compact support, and that can be constructed using a multiresolution analysis. We can take advantage of their regular structure to implement efficient operations. Multiresolution analysis ------------------------ A multiresolution analysis in the context of wavelets was introduced in and can be defined as follows. [def:multiresolution] A multiresolution analysis of $L^2(\R)$ is a nested sequence ⋯ ⊂ V− 2 ⊂ V− 1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ ⋯ of closed subspaces of $L^2(\R)$ such that 1. $\lim\_{j\rightarrow\infty}V\_{j}=\overline{\bigcup\_{j\in\Z}V\_j}=L^2(\R)$ and $\lim\_{j\rightarrow-\infty}V\_{j}=\bigcap\_{j\in\Z}V\_j=\emptyset$. 2. f(t) ∈ Vj ⇔ f(2t) ∈ Vj + 1, $\forall j\in\Z$. 3. f(t) ∈ V0 ⇔ f(t − k) ∈ V0, $\forall k\in\Z$. 4. there exists a ϕ(t) ∈ V0 such that $\{\phi(\cdot-k)\}\_{k\in\Z}$ forms a Riesz basis for V0. The first condition states that the sequence of subspaces is a non-redundant approximation of $L^2(\R)$. The second and third condition introduce scale and translation invariance. The last one demands the existence of a translation invariant basis for V0. Similar to the wavelet mother function we call the function ϕ(t) introduced in Definition [def:multiresolution] the father function. Analogously to the mother function it generates a family of functions: $$\label{eq:scalingbases} \phi\_{jk}(t)={2^{j/2}}\phi(2^{j}t-k), \qquad \forall k\in\Z, \quad j\in\Z.$$ For every j, $\{\phi\_{jk}\}\_{k\in\Z}$ forms a Riesz basis of Vj. We call $\{\phi\_{jk}\}\_{k\in\Z}$ a scaling basis of Vj. More specifically, if $\{\phi(\cdot-k)\}\_{k\in\Z}$ forms an orthonormal basis for V0, $\{\phi\_{jk}\}\_{k\in\Z}$ forms an orthonormal basis of Vj for every $j\in\Z$. In that case, we can also define the sequence of orthogonal projections $$\label{eq:projection} {\mathcal P}\_j f = \sum\_{k \in \Z} \langle f, \phi\_{jk} \rangle \, \phi\_{jk}.$$ The scale invariance of the multiresolution analysis in Definition [def:multiresolution] implies the existence of a two-scale relation $$\label{eq:twoscalerelation} \phi(t) = \sqrt 2\sum\_{k\in\Z}h\_k\phi(2t-k)$$ in which hk is a sequence. If we require that $\int\_{\R}\phi(t)\d t\neq 0$, we have that $$\sum\_{k\in\Z}h\_k=1.$$ If we further require that $\{\phi(\cdot-k)\}\_{k\in\Z}$ forms an orthonormal basis for V0, then h satisfies so-called double shift orthogonality conditions: $$\label{eq:doubleshiftorthonormality} \sum\_{k\in\Z}h\_k\overline h\_{k+2n}=\delta\_{0n},\qquad \forall n\in\Z.$$ A wavelet basis that follows from an orthonormal multiresolution analysis is $$\label{eq:orthonormalwaveletbasis} \psi\_{jk}(t)=\sqrt 2\sum\_{l\in\Z} g\_k \phi\_{jl}(2t-k),\qquad \forall k,j\in\Z$$ with $g\_k=(-1)^k\overline h\_{-k+1}$ and h as in . It forms an orthonormal basis for $L^2(\R)$. Biorthogonal multiresolution analysis ------------------------------------- Orthogonality is a rather restrictive requirement. A compactly supported and symmetric sequence h that satisfies double shift orthogonality can only have two non-zero coefficients. That restriction is lifted using biorthogonal wavelets. To construct biorthogonal wavelets we create, as in , a biorthogonal multiresolution analysis. To that end, next to the first (primal) multiresolution , we define a second (dual) one $$\label{eq:dualnestedsubspaces} \cdots\subset \tilde V\_{-2}\subset \tilde V\_{-1}\subset \tilde V\_{0} \subset \tilde V\_{1}\subset \tilde V\_{2}\subset \cdots$$ for which a dual scaling function $\tilde\phi(t)\in\tilde V\_0$ exists such that $\{\tilde\phi(\cdot-k)\}\_{k\in\Z}$ forms a Riesz basis of $\tilde V\_0$. The dual scaling function satisfies the two-scale relation $$\label{eq:dualtwoscalerelation} \tilde\phi(t) = \sqrt 2\sum\_{k\in\Z}\tilde h\_k\tilde\phi(2t-k).$$ We call this basis a dual scaling basis, while the scaling bases in are primal scaling bases. If $$\label{eq:biorthogonalscalingbases} \left\langle\phi(\cdot-k),\tilde\phi(\cdot-l)\right\rangle\_{L^2(\R)}=\delta\_{kl},$$ i.e., the primal scaling and dual scaling bases are biorthogonal to each other, both multiresolution analyses together form a biorthogonal multiresolution analysis. The orthogonal projection of becomes a more general oblique projection, $$\label{eq:dualprojection} {\mathcal P}\_j f = \sum\_{k \in \Z} \langle f, \tilde \phi\_{jk} \rangle \, \phi\_{jk}.$$ Alternatively, with the roles of primal and dual scaling functions interchanged, we also have $$\label{eq:dualdualprojection} \tilde{\mathcal P}\_j f = \sum\_{k \in \Z} \langle f, \phi\_{jk} \rangle \, \tilde \phi\_{jk}.$$ All further analysis in this paper is based on the biorthogonal setting. The orthogonal setting corresponds to $V\_j=\tilde V\_j$ and $\phi=\tilde\phi$. Next to the primal wavelet basis we also define a dual wavelet basis $$\label{eq:dualwavelets} \tilde\psi\_{jk}(t)=\sqrt 2\sum\_{k\in\Z} \tilde g\_k \tilde \phi(2t-k),\qquad \forall k,j\in\Z.$$ To obtain biorthogonal wavelet bases we require the mixed conditions $$\begin{gathered} \left\langle \phi\_{jk},\tilde\phi\_{jl}\right\rangle=\delta\_{kl},\quad \left\langle \phi\_{jk},\tilde\psi\_{jl}\right\rangle=0,\quad \left\langle \psi\_{jk},\tilde\phi\_{jl}\right\rangle=0,\qquad \forall {j,k,l}\in\Z\\ \left\langle \psi\_{ik},\tilde\psi\_{jl}\right\rangle=\delta\_{kl}\delta\_{ij},\qquad\forall {i,j,k,l}\in\Z.\end{gathered}$$ One can verify that these conditions follow from a dual double-shift orthogonality and two mixed alternating flip-relations, $$\begin{gathered} \sum\_{k\in\Z}\overline h\_k\tilde h\_{k+2n}=\delta\_n,\qquad \forall n\in\Z\\ \overline{g}\_k=(-1)^k\tilde h\_{1-k},\quad \tilde g\_k=(-1)^k\overline{h}\_{1-k},\qquad \forall k\in\Z.\end{gathered}$$ In the remainder of the text we will use compactly supported sequences, i.e., sequences a for which there exist $K\_1,K\_2\in\Z$ such that ak = 0 if k < K1 or k > K2. As a result, all associated scaling functions and wavelets have compact support as well. Discrete wavelet transform -------------------------- The sequences $h,\tilde h, g, \tilde g$ that describe the wavelet and scaling bases in the previous section can be used to define the discrete wavelet transform (DWT). We revisit its definition in order to motivate the statements in the complexity analysis of the numerical methods later on. The DWT transforms scaling coefficients of a given function $f(t)\in L^2(\R)$: $v\_{jk}=\left\langle f,\tilde\phi\_{jk}\right\rangle$, $j,k\in\Z$ to its wavelet coefficients: $w\_{jk}=\left\langle f,\tilde\psi\_{jk}\right\rangle$, $j,k\in\Z$. The inverse discrete wavelet transform (iDWT) transforms wavelet coefficients back into scaling coefficients. Both the DWT and iDWT are recursive algorithms. In every step, the DWT transforms scaling coefficients at a given level j + 1 to wavelet and scaling coefficients at a coarser level j, while the iDWT recovers in each step the scaling coefficients of the fine level j + 1 using wavelet and scaling coefficients at level j: $$\begin{gathered} v\_{jk} = \sum\_{l\in\Z}\overline{\tilde h}\_{l-2k}v\_{j+1,l},\quad w\_{jk} = \sum\_{l\in\Z}\overline{\tilde g}\_{l-2k}v\_{j+1,l},\qquad \forall j,k\in\Z\label{eq:dwtstep}\\ v\_{j+1,k} = \sum\_{l\in\Z} h\_{k-2l}v\_{jl}+g\_{k-2l}w\_{jl},\qquad \forall j,k\in\Z\label{eq:idwtstep}.\nonumber\end{gathered}$$ Usually, the DWT is implemented to transform a finite vector of length N = 2J, $\mathbf v\_J=\{v\_{Jk}\}\_{k=0}^{N-1}\in{\mathbb{C}}^{N}$, to a vector $\mathbf w\_J\in{\mathbb{C}}^{N}$: $$\begin{aligned} \mathbf w\_J &= [v\_{00}, w\_{00}, \underbrace{w\_{10}, w\_{11}}\_{\text{2 elements}},\dots,\underbrace{w\_{l,0},\dots,w\_{l,2^l-1}}\_{2^{l}\text{ elements}},\dots, \underbrace{w\_{J-1,0},\dots,w\_{J-1,2^{J-1}-1}}\_{2^{J-1}\text{ elements}} ]\label{eq:waveletvector}\\ &= [\mathbf{v}\_{0}^T,\hat{\mathbf{w}}\_{0}^T,\hat{\mathbf{w}}\_{1}^T,\dots,\hat{\mathbf{w}}\_{J-1}^T]^T\nonumber\end{aligned}$$ with $\hat{\mathbf{w}}\_{j} = \{w\_{j,k}\}\_{k=0}^{2^j-1}$. Boundary conditions deal with the finite nature of the vectors. We will assume a periodic boundary condition[3](#fn3), i.e., $$\label{eq:periodicboundarycondition} v\_{jk} = v\_{j,k+2^j},\quad w\_{jk} = w\_{j,k+2^j}\qquad \forall j\in\Z^+,\forall k\in\Z.$$ To transform $\mathbf v\_J$ into $\mathbf w\_J$ the DWT performs J steps like . If we use [A]↓ q to denote the down-sampling of a matrix, i.e., the selection of every qth row ([A]↓ q)(k, l) = A(qk, l) and use A\* to denote the adjoint of A, one step of the DWT can be represented in matrix notation as $$\begin{bmatrix} \mathbf v\_{j-1}\\\hat{\mathbf{w}}\_{j-1} \end{bmatrix} = \begin{bmatrix} {\tilde{H}\_j^\}\\{\tilde{G}\_j^\} \end{bmatrix}\_{\downarrow 2} \mathbf v\_{j}$$ with matrix H̃j ∈ C2j × 2j $$\begin{aligned} \tilde H\_j(k,l) = \sum\_{m\in\Z}\tilde h\_{k-l + m2^j}, \qquad\forall k,l=0,\dots,2^j-1,\end{aligned}$$ such that [H̃j\]↓ 2 ∈ C2j* − 1 × 2j and $[\tilde{H}\_j^\]\_{\downarrow 2}(k,l)= \sum\_{m\in\Z}\tilde h\_{l-2k + m2^j}$. Note that the summation over m* here is used to incorporate the periodic boundary conditions. The matrices Hj, Gj and $\tilde G\_j$ are defined analogously. The full DWT in matrix notation is $\mathbf w\_J = W\_J\mathbf v\_J$, with WJ ∈ CN × N and $$\label{eq:fulldwt} W\_J=\underbrace{ \begin{bmatrix} \begin{bmatrix} {\tilde{H}\_1^\}\\{\tilde{G}\_1^\} \end{bmatrix}\_{\downarrow 2}&0\\ 0&I\_{2^{J}-2} \end{bmatrix} \begin{bmatrix} \begin{bmatrix} {\tilde{H}\_2^\}\\{\tilde{G}\_2^\} \end{bmatrix}\_{\downarrow 2}&0\\ 0&I\_{2^{J}-2^{2}} \end{bmatrix} \cdots \begin{bmatrix} \begin{bmatrix} {\tilde{H}\_{J-1}^\}\\{\tilde{G}\_{J-1}^\} \end{bmatrix}\_{\downarrow 2}&0\\ 0&I\_{2^{J-1}} \end{bmatrix} \begin{bmatrix} {\tilde{H}\_J^\}\\{\tilde{G}\_J^\} \end{bmatrix}\_{\downarrow 2} }\_{J\text{ terms}}.$$ Similarly, the full iDWT can be written as $\mathbf v\_J = W^{-1}\_J\mathbf w\_J$. The iDWT matrix is the inverse of WJ, WJ− 1. This inverse can be decomposed as $$\label{eq:fullidwt} W^{-1}\_J=\underbrace{ \begin{bmatrix} {{H}\_J^\}\\{{G}\_J^\} \end{bmatrix}\_{\downarrow 2}^\* \begin{bmatrix} \begin{bmatrix} {{H}\_{J-1}^\}\\{{G}\_{J-1}^\} \end{bmatrix}\_{\downarrow 2}^\&0\\ 0&I\_{2^{J-1}} \end{bmatrix} \cdots \begin{bmatrix} \begin{bmatrix} {{H}\_2^\}\\{{G}\_2^\} \end{bmatrix}\_{\downarrow 2}^\&0\\ 0&I\_{2^{J}-2^{2}} \end{bmatrix} \begin{bmatrix} \begin{bmatrix} {{H}\_1^\}\\{{G}\_1^\} \end{bmatrix}\_{\downarrow 2}^\&0\\ 0&I\_{2^{J}-2} \end{bmatrix} }\_{J\text{ terms}}$$ where subsampling takes precedence over taking the adjoint in order to avoid a multitude of brackets. ![image](img/iDWT.png) With the use of cascading filter banks the DWT and iDWT can be implemented in $\mathcal O(N)$ operations. This algorithm is called the fast wavelet transform (FWT) and was introduced in. The same complexity can not be achieved using an ordinary matrix-vector multiply since W* and W− 1 contain $\mathcal O(N\log (N))$ non-zero elements. This is clear by looking at Figure [fig:dwtstructure] and by the following lemma. [lem:dwtstructure] Each column of WJ has $\mathcal O(J)$ non-zero elements while each row of its inverse has $\mathcal O(J)$ non-zero elements. Furthermore, both have $\mathcal O(J2^J)$ non-zero elements. Owing to our periodic setting, in the following we say that a matrix $A\in\R^{N\times N}$ is banded if $$\|m-n\|\mod N>b\Rightarrow A(m,n)=0.$$ Let $A\in\R^{N\times N}$ have bandwidth a and $B\in\R^{2N\times 2N}$ have bandwidth b, then A[B]↓ 2 has bandwidth a + b/2. This is verified by writing (A[B]↓ 2)(m, n) = ∑i = 0N − 1A(m, i)B(2i, n), which is only non-zero if $\|m-i\|\mod N\leq a$ and $\|2i-n\|\mod 2N\leq b$, i.e, if ∣m − n∣ ≤ a + b/2. We can rewrite W in J vertical blocks $$W = \begin{bmatrix} B\_1\\B\_2\\\\\vdots\\\\B\_{J-2}\\B\_{J-1}\\B\_J \end{bmatrix} = \begin{bmatrix} \begin{bmatrix}\tilde H\_{1}^\\\\tilde G\_{1}^\\end{bmatrix}\_{\downarrow 2}[\tilde H\_{2}^\]\_{\downarrow 2}\cdots[\tilde H\_{J-1}^\]\_{\downarrow 2} [\tilde H\_J^\]\_{\downarrow 2}\\ [\tilde G\_{2}^\]\_{\downarrow 2}[\tilde H\_{3}^\]\_{\downarrow 2}\cdots[\tilde H\_{J-1}^\]\_{\downarrow 2} [\tilde H\_J^\]\_{\downarrow 2}\\\\ \vdots\\\\ [\tilde G\_{J-2}^\]\_{\downarrow 2}[\tilde H\_{J-1}^\]\_{\downarrow 2}[\tilde H\_J^\]\_{\downarrow 2}\\ [\tilde G\_{J-1}^\]\_{\downarrow 2}[\tilde H\_J^\]\_{\downarrow 2} \\ [\tilde G\_J^\]\_{\downarrow 2} \end{bmatrix}.$$ The blocks B1 and B2 have size 2 × 2J* and block Bj has size 2j − 1 × 2J for j ≥ 1. First we show that each block has a bounded number of non-zero elements per column. To that end we denote K and L as the number of non-zero elements in the sequences $\tilde h$ and $\tilde g$ respectively. It is clear that $[\tilde H\_j^\]\_{\downarrow 2}$ and $[\tilde G\_j^\]\_{\downarrow 2}$ have $\mathcal O(K/2)$ and $\mathcal O(L/2)$ non-zero elements per column respectively since HJ and GJ are banded with bandwidth K and L respectively. The products also have a bounded number of non-zero elements per column by the first part of the proof. Matrix $[\tilde G\_{J-1}^\]\_{\downarrow 2}[\tilde H\_J^\]\_{\downarrow 2}$ has e.g. L/2 + K/4 non-zero elements per column. Since each block contains $\mathcal O((K+L)2^J)$ non-zero elements with a limited number of non-zero per column, W contains $\mathcal O(2^J J)$ non-zero elements and $\mathcal O(J)$ non-zero elements per column. The proof for W− 1 is entirely analogous; but write W− 1 in J horizontal blocks. If we introduce the dual DWT $\tilde W\_J$ $$\label{eq:dualdwt} \tilde W\_J=\underbrace{ \begin{bmatrix} \begin{bmatrix} {{H}\_1^\}\\{{G}\_1^\} \end{bmatrix}\_{\downarrow 2}&0\\ 0&I\_{2^{J}-2} \end{bmatrix} \begin{bmatrix} \begin{bmatrix} {{H}\_2^\}\\{{G}\_2^\} \end{bmatrix}\_{\downarrow 2}&0\\ 0&I\_{2^{J}-2^{2}} \end{bmatrix} \cdots \begin{bmatrix} \begin{bmatrix} {{H}\_{J-1}^\}\\{{G}\_{J-1}^\} \end{bmatrix}\_{\downarrow 2}&0\\ 0&I\_{2^{J-1}} \end{bmatrix} \begin{bmatrix} {{H}\_J^\}\\{{G}\_J^\} \end{bmatrix}\_{\downarrow 2} }\_{J\text{ terms}}$$ and analogously define the iDWT $\tilde W^{-1}\_J$, we can verify that $$\label{eq:dwtrelations} W\_J^\=\tilde W\_J^{-1}\qquad(W\_J^{-1})^\=\tilde W\_J.$$ Periodic wavelets on the interval and discrete duals ==================================================== Periodization ------------- For simplicity of the exposition we again assume that N = 2J, $J\in\N$. We introduce the periodic and scaled father function with period 1 as $$\label{eq:periodizedfatherfunction} \phi\_{N}(t) = \sum\_{k\in\Z} 2^{J/2}\phi\left(2^{J}(t-k)\right).$$ The periodic scaling basis that consists of N translated father functions $\phi\_{kN}(t)=\phi\_{N}(t-\tfrac kN)$ is ΦN = {ϕkN}k = 0N − 1. Similarly, the dual scaling function in L2(0, 1) is $$\tilde\Phi\_{N} = \{ \tilde\phi\_{kN} \}\_{k=0}^{N-1}.$$ with translates $\tilde\phi\_{kN}(t)=\tilde\phi\_{N}(t-\tfrac kN)$ of the dual father function $$\tilde\phi\_{N}(t) = \sum\_{k\in\Z} 2^{J/2}\tilde\phi\left(2^{J}(t-k)\right).$$ Wavelet bases with period 1 can be devised in the same way as above for the scaling bases, by summing over their translations. Alternatively, we can define them using periodic scaling bases and the iDWT matrices WJ− 1 and $\tilde W^{-1}\_J$: $$\begin{aligned} \label{eq:primalwaveletbasis} \Psi\_{N} = \{\psi\_{kN}\}\_{k=0}^{N-1},\qquad \psi\_{kN}(t)=\sum\_{l=0}^{N-1}\phi\_{lN}(t)(W^{-1}\_J)(l,k),\\ \tilde\Psi\_{N} = \{\tilde\psi\_{kN}\}\_{k=0}^{N-1},\qquad \tilde\psi\_{kN}(t)=\sum\_{l=0}^{N-1}\tilde\phi\_{lN}(t)(\tilde W^{-1}\_J)(l,k)\label{eq:dualwaveletbasis}.\end{aligned}$$ Note that the index k in ψkN incorporates both the scale and the translation of the corresponding wavelet in this notation. The N wavelets are defined on all scales 0, 1, …, J − 1, as in. Compactly supported wavelets and discrete evaluation ---------------------------------------------------- We focus on two well-known families of compactly supported wavelet families. The first family are the Daubechies orthogonal wavelet bases. It was shown in that the scaling function for orthogonal wavelets with p vanishing moments (a regularity condition on the wavelets) has a support of length at least 2p − 1. Daubechies wavelets are optimal in the sense that they have a minimum support length for a given number of vanishing moments. In Figure [fig:daubechieswavelets], ϕ and ψ are shown for `db2`, `db3` and `db4`, i.e., the father and mother functions of the Daubechies wavelet with 2, 3 and 4 vanishing moments. These standard wavelets are widely used in applications. The Daubechies wavelet and scaling functions are defined by their compactly supported sequences h and g. No closed form formula is known for the functions themselves. Note that the functions associated with `db2` are continuous, but nowhere differentiable. However, one can evaluate compactly supported scaling functions at dyadic points k/2j, $k\in\Z$ from the sequence h using the following procedure. We create a matrix system by evaluating the two-scale relation in integer points. For a case where ϕ(k) = 0 for k < 0 or k > 5 we obtain for example $$\begin{aligned} \begin{bmatrix} \phi(0)\\ \phi(1)\\ \phi(2)\\ \phi(3)\\ \phi(4)\\ \phi(5) \end{bmatrix} = \sqrt 2 \begin{bmatrix} h\_0&&&&&\\ h\_2&h\_1&h\_0&&&\\ h\_4&h\_3&h\_2&h\_1&h\_0&\\ &h\_5&h\_4&h\_3&h\_2&h\_1\\ &&&h\_5&h\_4&h\_3\\ &&&&&h\_5\\ \end{bmatrix} \begin{bmatrix} \phi(0)\\ \phi(1)\\ \phi(2)\\ \phi(3)\\ \phi(4)\\ \phi(5) \end{bmatrix}.\end{aligned}$$ In this notation, the two-scale relation implies that the matrix shown has an eigenvalue 1. The corresponding eigenvector represents ϕ evaluated at integer points. To evaluate at finer dyadic levels, it suffices to use the two-scale relation repeatedly, since it also states that $$\phi\left(\tfrac k {2^{j+1}}\right) = \sum\_{l\in\Z}h\_l\phi\left(\tfrac k {2^j}-l\right).$$ The wavelet function may be evaluated at the dyadic points by first evaluating the corresponding scaling function, after which equation can be applied. [fig:cdfwavelet] The second wavelet family we consider are the biorthogonal CDF wavelets. These are compactly supported and symmetric. More specifically, among the family of CDF wavelets we use those that have the centered B-spline as primal father function. All sequences h, g, $\tilde h$ and $\tilde g$ are compactly supported and symmetric as well. Figure [fig:cdfwavelet] shows the primal and dual father and mother functions for `cdf24`, `cdf33` and `cdf35`. The first digit in the name indicates the number of vanishing moments of the dual wavelet, the second digit refers to those of the primal wavelet. Contrary to the Daubechies wavelets, closed-form formulas exist for some primal scaling functions. The primal scaling functions of `cdf`$p\tilde p$ shown in Figure [fig:cdfwavelet] are the centered B-splines of order p. However, the dual scaling function can typically only be evaluated in dyadic points using the above procedure.[4](#fn4) Continuous dual bases --------------------- Both the Daubechies and CDF wavelets have at least one compact dual in the continuous sense, i.e., the dual is biorthogonal in $L^2(\R)$: $$\langle\phi\_{jk},\tilde\phi\_{jl}\rangle\_{L^2(\R)}=\int\_{\R}\phi\_{jk}(t)\tilde\phi\_{jl}(t)\d t=\delta\_{kl},\qquad \forall j,k,l\in\Z.$$ The dual scaling function is compactly supported as well. So, a compactly supported basis biorthogonal to ΦN (ΨN) in the continuous sense is $\tilde\Phi\_N$ ($\tilde\Psi\_N$). Note that there might exist multiple dual bases for the same primal basis. Discrete dual scaling bases --------------------------- The various dual bases of CDF are well understood. In the context of this paper, for the efficient computation of wavelet approximations on irregular domains we will require a notion of discrete duality. We therefore introduce the following bilinear form: $$\label{eq:discreteinnerproduct} \langle f,g\rangle\_{\osi} = \sum\_{m\in\Z} f\left(\tfrac mq\right)g\left(\tfrac mq\right).$$ Here, $\osi \in \mathbb{N}$, $\osi \geq 1$, plays the role of an oversampling factor. We aim for a periodic dual scaling basis $\tilde\Phi^\osi\_N=\{\tilde\phi^\osi\_{kN}\}\_{k=0}^{N-1}$ defined on [0, 1]. In order to construct such duals, we focus first on the samples of the father function on the whole real line. Define the sequence b by sampling ϕ in the oversampled grid, $$b^\osi\_m=\phi\left(\tfrac m\osi\right), \qquad m\in\Z.$$ The sequence b is compactly supported because ϕ is compactly supported. Biorthogonality with respect to between the integer shifts of ϕ and those of a discrete dual father function $\tilde \phi^\osi(t)$, with samples $\tilde b$, translates into the conditions $$\label{eq:discretebiorthogonality} \langle \phi, \tilde \phi^\osi(\cdot-k) \rangle\_\osi = \sum\_{m}b\_m\tilde b\_{m-k\osi} = \delta\_{0k}, \qquad \forall k\in\Z.$$ Note that shifts by an integer k of a continuous function correspond to shifts of $k\osi$ samples of its sampled sequence in the discrete grid, because $\osi$ is the oversampling factor. Once a dual sequence b̃ satisfying is found, it does not immediately give rise to a continous representation of the dual functions $\tilde\phi^\osi(t)$. However, we do know its evaluations in the points $\tfrac m\osi$. We can define suitable discrete periodized dual functions as $$\label{eq:discreteperiodicdual} \tilde \phi^\osi\_{kN}\left(\tfrac mq \right) = N^{-1/2} \sum\_{l \in \Z} \tilde{b}\_{m-\osi k-N\osi l}, \qquad m=0,\ldots,Nq-1.$$ Note that there are Nq samples in [0, 1), and that the summation over l introduces periodization. By construction, these functions satisfy the discrete biorthogonality conditions $$\label{eq:discrete\_dual\_biorthogonality} \langle \tilde\phi^\osi\_{kN},\phi\_{lN}\rangle\_{N, \osi}=\delta\_{kl},$$ where ⟨ ⋅ , ⋅ ⟩N, q is a scaled analogue of restricted to [0, 1]: $$\label{eq:discreteinnerproduct\_N} \langle f,g\rangle\_{N,\osi} = \sum\_{m = 0}^{Nq-1} f\left(\tfrac m{Nq}\right)g\left(\tfrac m{Nq}\right).$$ The discrete dual leads to the discrete projections $$\label{eq:discrete\_projection} {\mathcal P}^\osi\_N f(t) = \sum\_{k=0}^{N-1} \langle f, \tilde \phi^\osi\_{kN} \rangle\_{N,\osi} \, \phi\_{kN}(t) = \sum\_{k=0}^{N-1} v\_{kN}^\osi\, \phi\_{kN}(t).$$ The discrete duals play the role of the continuous dual $\tilde \phi\_{kN}$ in. However, the discrete inner product does not actually require the evaluation of integrals. By construction, the projection is exact on the span of the scaling functions: $${\mathcal P}^\osi\_N f = f, \quad \forall f \in \SPAN \Phi\_N.$$ In the terminology of splines literature, these reproducing projections are examples of a quasi-interpolation method, in which the global approximation is constructed using local approximations. [fig:compactdualsdb] [fig:compactdualscdf] Unlike the Daubechies and CDF scaling functions and their continuous duals, the discrete duals we employ are non-standard. However, it was recently shown by the authors in that discrete compact dual sequences $\tilde b$ always exist for a basis consisting of translates of B-splines, which is exactly the setting of our choice of CDF primal scaling bases. Experiments show that compact duals of Daubechies scaling functions can also be found. Figures [fig:compactdualsdb] and [fig:compactdualscdf] show some of these compact duals, namely those with the smallest possible support. In Table [tab:discretevalues] of the appendix we give the values of the primal and dual sequences used in the experiments below. We observe that the discrete duals for the Daubechies case are somewhat larger than those of the CDF duals. These discrete duals are readily found by solving the linear algebraic conditions , noting that the system is finite because b and b̃ are compactly supported. The size of the discrete duals can be reduced by allowing for a larger support and solving the corresponding conditions in a least squares sense, thereby minimizing the norm of the solution to an underdetermined system. The relevance of the size of the discrete duals is formalized in the following lemma. [lem:discreteerror] Let f ∈ L∞([0, 1]) and let the synthesis operator TNa = ∑k = 0N − 1ak ϕkN of ΦN be a bounded operator from CN to L2([0, 1]) with bound B > 0, i.e., $$\Vert T\_N \mathbf{a} \Vert\_{L^2([0,1])} = \left\Vert \sum\_{k=0}^{N-1} a\_k \phi\_{kN} \right\Vert\_{L^2([0,1])} \leq B \, \Vert \mathbf{a} \Vert\_{\ell\_2({\mathbb{C}}^N)}.$$ Define ‖ ⋅ ‖N, q2 = ⟨ ⋅ , ⋅ ⟩N, q. If the support of the sequence b̃ is less than Nq, then $$\label{eq:discrete\_error} \Vert {\mathcal P}\_N f - \tilde{{\mathcal P}}^\osi\_N f \Vert \leq B \, \Vert f - {\mathcal P}\_N f \Vert\_{N,q} \, \Vert \tilde{b} \Vert\_{\ell\_2}.$$ Using , Cauchy-Schwartz and , we find for u ∈ L∞([0, 1]) the following inequality: $$\begin{aligned} \left\| \langle u, \tilde \phi^\osi\_{kN} \rangle\_{N,\osi} \right\| &\leq \\| u\\|\_{N,\osi}\\|\tilde \phi^\osi\_{kN} \\|\_{N,\osi}\\ &= N^{-1/2} \, \Vert u \Vert\_{N,q} \, \Vert \tilde{b} \Vert\_{\ell\_2}.\end{aligned}$$ In the last line, we have used the assumption on the discrete support of b̃ to note that the periodic copies of b̃ in do not actually overlap, hence the equality of norms. Next, let $g\_N = f - {\mathcal P}\_N f = f - f\_N$, such that f = fN + gN. Since both ${\mathcal P}\_N$ and $\tilde{{\mathcal P}}\_N$ are exact on the span of ΦN, we have that ${\mathcal P}\_N f\_N = \tilde{{\mathcal P}}\_N f\_N = f\_N$. In addition, ${\mathcal P}\_N g\_N = 0$. With norms in L2([0, 1]) unless noted otherwise, this means that $$\begin{aligned} \Vert {\mathcal P}\_N f - \tilde{{\mathcal P}}^\osi\_N f \Vert &= \Vert {\mathcal P}\_N g\_N - \tilde{{\mathcal P}}^\osi\_N g\_N \Vert = \left\Vert \tilde{{\mathcal P}}^\osi\_N g\_N \right\Vert = \left\Vert \sum\_{k=0}^{N-1} \langle g\_N, \tilde \phi^\osi\_{kN} \rangle\_{N,\osi} \, \phi\_{kN} \right\Vert \\ &\leq B \, \left\Vert \left\{\langle g\_N, \tilde \phi^\osi\_{kN} \rangle\_{N,\osi}\right\}\_{k=0}^{N-1}\, \, \right\Vert\_{\ell\_2({\mathbb{C}}^N)} \\ &\leq B \, \sqrt{N} \left\Vert \left\{\langle g\_N, \tilde \phi^\osi\_{kN} \rangle\_{N,\osi}\right\}\_{k=0}^{N-1}\, \, \right\Vert\_{\ell\_\infty({\mathbb{C}}^N)} \\ &\leq B \, \Vert g\_N \Vert\_{N,q} \, \Vert \tilde{b} \Vert\_{\ell\_2}.\end{aligned}$$ The lemma shows that the discrete projection yields a result that is close to the actual continuous dual projection, as long as the dual sequence b̃ does not grow too large in norm. In particular, the convergence rate with N remains the same, only the constant factor is affected. In practice, the assumption on the support of b̃ is not restrictive unless N is very small. Discrete dual wavelet bases? ---------------------------- We have replaced the continuous dual basis with a discrete one. In view of the close correspondence between the continuous inner product coefficients vkN = ⟨f, ϕ̃Nk⟩ and their discrete analogues $v\_{kN}^\osi = \langle f, \tilde{\phi}^\osi\_{Nk} \rangle\_{N,\osi}$, as quantified by Lemma [lem:discreteerror], we forego the construction of a discrete dual wavelet basis. We simply retain the primal and dual scaling functions and wavelets, and we continue to use the wavelet transform given by Wj and W̃j respectively. Thus, the construction involving the discrete sequence b̃ of the previous section can be thought of merely as a quadrature scheme to approximate vkN using equispaced samples of f. Compared to other quadrature schemes for wavelet and scaling coefficients, such as Sweldens quadrature, our quadrature discretization has an additional discrete orthogonality structure that will be used later on. The construction of a fully discrete wavelet basis based on $\tilde{\phi}^\osi\_{kN}$ would have several disadvantages. First of all, the sequence b̃ has no multiscale structure. Therefore, $\tilde{\phi}^\osi\_{kN}$ does not satisfy a two-scale relation and further structure would have to be imposed on b̃. Secondly, and more importantly, changing the dual scaling function from ϕ̃ to $\tilde{\phi}^\osi\_{kN}$ would affect the primal wavelet. Finally, there is no guarantee that a discrete dual wavelet basis exists for a given primal scaling function. We choose to retain the Daubechies and CDF family of scaling functions and wavelets. Note that the number of dual vanishing moments apparent as $\tilde p$ in the naming of the CDF wavelets `cdf`$p\tilde p$ has no influence on the regularity of the discrete duals $\tilde{\phi}^\osi$ defined above. It does change the shape of the primal wavelet basis elements through the alternating-flip relation $g\_k=(-1)^k\overline{\tilde{h}}\_{1-k}$. The higher $\tilde p$, the larger the support of the wavelets. Another effect of the number of dual vanishing moments is seen in the wavelet transforms. If $\tilde p < p$ then the norm of $\tilde W\_J$ may increase significantly. This is illustrated in Table [tab:dwtnorms]. For the purposes of this paper, it seems best to consider $\tilde p \geq p$. [h] The approximation problem ========================= We formally define the approximation problem. To that end, we introduce notation for the multivariate approximation. We largely adopt the same notation as was used for B-splines, but we replace the spline bases with a tensor product of the wavelet bases . Bold letters such as $\mathbf N$ denote a vector of length d, $\mathbf N=(N\_1,\dots N\_d)$, and $I\_{\mathbf N}$ denotes the index set {(i1, …, id) ∣ j = 1…, d, ij = 0, …, Nj − 1}. Furthermore, the tensor product of wavelet bases is written as $$\begin{aligned} \Psi\_{\mathbf N}=\Psi\_{N\_1}\otimes\cdots\otimes\Psi\_{N\_d}=\{\psi\_{\mathbf k\mathbf N}\}\_{\mathbf k\in I\_{\mathbf N}}\qquad \psi\_{\mathbf k\mathbf N}(\mathbf t) = \Psi\_{k\_1N\_1}(t\_1)\times\cdots\times\Psi\_{k\_dN\_d}(t\_d).\end{aligned}$$ The tensor products of scaling bases and dual bases are denoted analogously. Continuous projection --------------------- The approximation problem can be discretized in two ways, namely, using inner products and point evaluation. The former is denoted the continuous projection and leads to the system $A\_{\mathbf N}x=b\_{\mathbf N}$, with $$\begin{aligned} A\_{\mathbf N}(\mathbf k,\mathbf l) &= \langle \psi\_{\mathbf l\mathbf N}, \tilde \psi\_{\mathbf k\mathbf N}\rangle\_{\Omega}, \qquad \mathbf k, \mathbf l\in I\_\mathbf N,\label{eq:contsystem}\\ b\_{\mathbf N}(\mathbf k) &= \langle f,\tilde\psi\_{\mathbf k\mathbf N}\rangle\_{\Omega}\nonumber.\end{aligned}$$ Note that the inner products are defined over Ω, but the wavelet basis has been defined on Ξ. If Ω = Ξ then $A\_{\mathbf N}$ is the identity matrix due to the continuous duality of $\Psi\_{\mathbf N}$ and $\tilde\Psi\_{\mathbf N}$. In our setting Ω ⊂ Ξ, hence the matrix entries of $A\_{\mathbf N}$ may differ from 0/1 if one of the functions involved overlaps with the boundary. It may be difficult to evaluate the inner products on Ω numerically, especially in the multivariate setting. We explicitly define the wavelet and scaling basis elements that overlap with the boundary of a given domain Ω. They are contained in the sets $$\label{eq:splineboundaryset} \K\_{\mathbf N}(\Omega) = \{ {\mathbf k} \in I\_{\mathbf N} \, \| \, \, \mysupp \phi\_{\mathbf k\mathbf N} \cap \Omega \neq \varnothing\text{ and }\mysupp \phi\_{\mathbf k\mathbf N} \cap {\Omega^c} \neq \varnothing \}$$ and $$\label{eq:waveletboundaryset} \L\_{\mathbf N}(\Omega) = \{ {\mathbf k} \in I\_{\mathbf N} \, \| \, \, \mysupp \psi\_{\mathbf k\mathbf N} \cap \Omega \neq \varnothing\text{ and }\mysupp \psi\_{\mathbf k\mathbf N} \cap {\Omega^c} \neq \varnothing \}$$ respectively, where Ωc = Ξ \ Ω is the complement of Ω in Ξ. Discrete projection ------------------- For the fully discrete method, we need to define sampling points. Given a basis $\Psi\_{\mathbf N}$, we sample in a regular (cartesian) grid oversampled by an integer qi > 1 in each dimension: $$\mathcal T^{\mathbf\osi}\_{\mathbf N} = \left\{\left.\left(\tfrac{k\_1}{\osi\_1 N\_1},\dots,\tfrac{k\_d}{\osi\_d N\_d}\right)\, \right\| \, k\_i=0,\dots,N\_i\osi\_i-1,\quad i=1,\dots, d \right\}.$$ Since Daubechies wavelets can only be evaluated in points k/2j, $k\in\Z, j\in\N$, we choose each component of $\mathbf \osi$ to be dyadic, i.e., qi = 2j, $j\in\N\_0$, when approximating using a Daubechies wavelet basis. Only the points in the intersection $\mathcal T^{\mathbf\osi,\Omega}\_{\mathbf N}=\mathcal T^{\mathbf\osi}\_{\mathbf N}\cap \Omega$ are of interest. The number of points in that set, $M=\#\mathcal T^{\mathbf\osi,\Omega}\_{\mathbf N}$, should be larger than N = Πj = 1dNj in order to obtain the oversampled system $A^{\mathbf\osi}\_{\mathbf N}x=b^{\mathbf\osi}\_{\mathbf N}$. That system is given by $$\begin{aligned} A^{\mathbf\osi}\_{\mathbf N}(m,\mathbf l) &= \psi\_{\mathbf l\mathbf N}(\mathbf t\_m), \qquad \mathbf l\in I\_\mathbf N, m\in I\_M\label{eq:discsystem}\\ b^{\mathbf\osi}\_{\mathbf N}(m) &= f(\mathbf t\_m)\nonumber,\end{aligned}$$ where $\mathbf t\_m\in \mathcal T^{\mathbf\osi,\Omega}\_{\mathbf N}$. The discrete nature of the support in the discrete setting is mirrored in the definition $$\mysupp\_{\mathbf\osi} \phi\_{\mathbf l\mathbf N} = \mysupp\phi\_{\mathbf l\mathbf N} \cap \mathcal T^{\mathbf\osi}\_{\mathbf N}.$$ This notion of discrete support is used to determine the number of basis elements that overlap with the boundary of Ω as $$\label{eq:splineboundaryset\_discrete} \K\_{\mathbf N}^{\mathbf\osi}(\Omega) = \{ \mathbf k \in I\_{\mathbf N}, \, \| \, \mysupp\_{\mathbf\osi} \phi\_{\mathbf k\mathbf N} \cap \Omega \neq \varnothing\text{ and }\mysupp\_{\mathbf\osi} \phi\_{\mathbf k\mathbf N} \cap {\Omega^c} \neq \varnothing \}$$ and $$\label{eq:waveletboundaryset\_discrete} \L^{\mathbf\osi}\_{\mathbf N}(\Omega) =\{ \mathbf k \in I\_{\mathbf N}, \, \| \, \mysupp\_{\mathbf\osi} \psi\_{\mathbf k\mathbf N} \cap \Omega \neq \varnothing\text{ and }\mysupp\_{\mathbf\osi} \psi\_{\mathbf k\mathbf N} \cap {\Omega^c} \neq \varnothing \}.$$ [ass:boundary] We assume that the dimension of the boundary of Ω ⊂ Ξ is exactly one less than the dimension of Ω itself. This means that we will not consider fractal domains. In other words, the sets $\K\_{\mathbf N}(\Omega)$ and $\K^{\mathbf\osi}\_{\mathbf N}(\Omega)$ grow in size as $\mathcal O\left(N^{(d-1)/d}\right)$ since we further assume that the oversampling is linear, i.e., M = γN, with γ > 1. Firstly, we have $$\#\L\_{\mathbf N}(\Omega)=\mathcal O(J\#\K\_{\mathbf N}(\Omega))\quad\mbox{and}\quad \#\L^{\mathbf\osi}\_{\mathbf N}(\Omega)=\mathcal O(J\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$$ where J = Πj = 1dJi = log2(N). Secondly, provided Assumption [ass:boundary] is satisfied $$\#\L\_{\mathbf N}(\Omega) = \#\L^{\mathbf\osi}\_{\mathbf N}(\Omega)=\mathcal O\left(N^{(d-1)/d}\log(N)\right).$$ The former is a direct consequence of the compact nature of the scaling basis, while the latter is a combination of the former and Lemma [lem:dwtstructure]. The AZ algorithm ================ In general, the systems introduced in and are severely ill-conditioned. This is the result of the inherent redundancy of extension frame approximations. One illuminating interpretation of the redundancy is that an approximation can take any form outside of Ω while not influencing the behavior on Ω. Extension frames, their ill-conditioning and further implications are studied in detail in . There, it is advised to solve the ill-conditioned systems using regularization and oversampling to obtain an numerically stable and accurate approximation. One can, e.g., use a truncated singular value decomposition (SVD) as a solver. Unfortunately, this solver and other regularized solvers generally have cubic complexity in N. [h] [alg:AZ] Input: A, Z ∈ CM × N, b ∈ CM Output: x ∈ CN such that Ax ≈ b [1] Solve (I − AZ\)A*x1 = (I − AZ\)b* using a randomized low-rank solver x2 ← Z\(b* − Ax1) x ← x1 + x2 The AZ algorithm (Algorithm [alg:AZ]) consists of three simple steps and was introduced in to reduce the computational complexity. It is a generalization of the algorithms proposed in for the more specific Fourier extension problem. There, similar ill-conditioned systems have to be solved since Fourier extension frames are similarly redundant as wavelet extension frames. The cost of Fourier extension was reduced from cubic to $\mathcal O(N\log^2(N))$ in 1-D and to $\mathcal O(N^2\log^2(N))$ in 2-D. The AZ algorithm was also successfully applied in to reduce the cost of spline extension approximations to $\mathcal O(N)$ in 1-D, $\mathcal O\left(N^{3/2}\right)$ in 2-D and $\mathcal O\left(N^{3(d-1)/d}\right)$ in d-D with d > 1. It is this latter application of AZ, and the corresponding analysis in, that we set out to extend to the wavelet case. The AZ algorithm is shown in pseudocode in Algorithm [alg:AZ]. It solves Ax = b with a time complexity $$\label{eq:aztimings} \mathcal O(r\texttt T\_\text{mult}+r^2M),$$ where r is the rank of the system in the first step and Tmult is the time complexity of applying A and Z\* to a vector. The residual of the solution corresponds the approximation error in our setting, and it is equal to the residual of the solution in step one of the algorithm. The matrix Z can in principle be chosen arbitrarily. However, the goal is to choose Z such that the rank r of the system in step 1 is small. Loosely speaking, this corresponds to choosing Z\* as a pseudo-inverse to a large subspace of the range of A. This is where the discrete biorthogonality properties of the discrete duals constructed in §[wavs:periodicwavelet] play a decisive role. The choice of Z ----------------- First, we will simply state our choice of Z for both the continuous and discrete setting, based on analogy to previously studied cases. Later on, we will prove why these choices indeed give rise to a low-rank matrix A − AZ\*A in step 1 of the AZ algorithm. Briefly, we intend to make sure that Z\*A approximates the identity matrix up to a small perturbation. In the continuous setting, it suffices to choose $Z\_{\mathbf N}$ equal to the identity. Indeed, recall from that $A\_{\mathbf N}$ itself is close to the identity matrix in this setting. The perturbation is related to basis functions that overlap with the boundary. In order to reuse the results of, we introduce the scaling system matrices $\hat A$ and $\hat Z$, before we consider the wavelet system matrices A and Z of - above. In, the B-spline system matrices do not contain inner products with the dual basis as in, but with the primal scaling basis itself. Thus, $\hat A\_{\mathbf N}$ is defined as the Gram matrix of ΦN, $$\hat A\_{\mathbf N}(\mathbf k,\mathbf l)= \langle \phi\_{\mathbf k\mathbf N},\phi\_{\mathbf l\mathbf N}\rangle\_{\Omega}, \quad \mathbf k,\mathbf l\in I\_{\mathbf N}.$$ This change is also reflected in the structure of $\hat Z\_{\mathbf N}$. It is not the identity matrix, but rather the Gram matrix of $\tilde\Phi\_{\mathbf N}$: $$\hat Z\_{\mathbf N}(\mathbf k,\mathbf l) = \left\langle \tilde\phi\_{\mathbf k\mathbf N}, \tilde\phi\_{\mathbf l\mathbf N}\right\rangle\_{L^2(0,1)^d}, \qquad \mathbf k,\mathbf l\in I\_{\mathbf N}.$$ If we denote by $W\_{\mathbf J}$ the Kronecker product of the d DWT matrices {WJi}i = 1d, one can verify that the relation between the wavelet system matrix $A\_{\mathbf N}$ and the scaling system matrix $\hat A\_{\mathbf N}$ is given by $$\label{eq:AZtoscaling} A\_{\mathbf N}= W\_{\mathbf J}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N}W\_{\mathbf J}^{-1},$$ using and . Therefore, $Z\_{\mathbf N}^\A\_{\mathbf N}= W\_{\mathbf J}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N}W\_{\mathbf J}^{-1}$ as well. In the discrete setting, we can define both $A^{\mathbf\osi}\_{\mathbf N}$ and our chosen matrix $Z^{\mathbf\osi}\_{\mathbf N}$ in terms of the pointwise evaluations of the discrete dual scaling functions, followed by the continuous dual wavelet transform. Let $$\hat Z^{\mathbf\osi}\_{\mathbf N}(m,\mathbf l)=\tilde\phi^{\mathbf\osi}\_{\mathbf l\mathbf N}(\mathbf t\_m) \quad \mbox{and} \quad \hat A^{\mathbf\osi}\_{\mathbf N}(m,\mathbf l)=\phi^{\mathbf\osi}\_{\mathbf l\mathbf N}(\mathbf t\_m), \qquad m\in I\_M, \mathbf l\in I\_{\mathbf N}, \mathbf t\_m\in \mathcal T^{\mathbf\osi,\Omega}\_{\mathbf N}.$$ Then $$\label{eq:AandZtoscaling} Z^{\mathbf\osi}\_{\mathbf N} = \hat Z^{\mathbf\osi}\_{\mathbf N} W^{-1}\_{\mathbf J}\quad\mbox{and}\quad A^{\mathbf\osi}\_{\mathbf N} = \hat A^{\mathbf\osi}\_{\mathbf N} W^{-1}\_{\mathbf J}$$ such that $\left(Z^{\mathbf\osi}\_{\mathbf N}\right)^\A^{\mathbf\osi}\_{\mathbf N}=W\_{\mathbf J} \left(\hat Z^{\mathbf\osi}\_{\mathbf N}\right)^\\hat A^{\mathbf\osi}\_{\mathbf N}W\_{\mathbf J}^{-1}$ again using . In the wavelet case Tmult of is $\mathcal O(N)$, since all matrices are combinations of matrices that can be applied in $\mathcal O(N)$ operations. This is because they either contain $\mathcal O(N)$ non-zero elements or they can be represented by a (i)DWT. The rank and sparsity structure of A* − AZ\*A ---------------------------------------------------- In order to study the properties of the matrix A − AZ\*A in the wavelet case, we again make use of the results of for the scaling bases. In the current notation, the approximation in the scaling basis leads to the matrix $\hat A-\hat A\hat Z^\\hat A$. We can write A* − AZ\*A as a product of this matrix with DWTs: $$\begin{aligned} A\_{\mathbf N}-A\_{\mathbf N}Z\_{\mathbf N}^\A\_{\mathbf N} &= W\_{\mathbf J}\hat A\_{\mathbf N}W^{-1}\_{\mathbf J}-W\_{\mathbf J}\hat A\_{\mathbf N}W^{-1}\_{\mathbf J} W\_{\mathbf J}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N}W\_{\mathbf J}^{-1} \nonumber\\ &= W\_{\mathbf J}\left(\hat A\_{\mathbf N}-\hat A\_{\mathbf N}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N}\right)W^{-1}\_{\mathbf J}\label{eq:AAZAcontinuousderivation}.\end{aligned}$$ Here, we used and . Similarly, for the discrete projection, $$\begin{aligned} A^{\mathbf\osi}\_{\mathbf N}-A^{\mathbf\osi}\_{\mathbf N}(Z^{\mathbf\osi}\_{\mathbf N})^\A^{\mathbf\osi}\_{\mathbf N} &=\hat A^{\mathbf\osi}\_{\mathbf N}W^{-1}\_{\mathbf J}-\hat A^{\mathbf\osi}\_{\mathbf N}W^{-1}\_{\mathbf J}(\hat Z^{\mathbf\osi}\_{\mathbf N}\tilde W^{-1}\_{\mathbf J})^\\hat A^{\mathbf\osi}\_{\mathbf N}W^{-1}\_{\mathbf J}\nonumber\\ &= \left(\hat A^{\mathbf\osi}\_{\mathbf N}-\hat A^{\mathbf\osi}\_{\mathbf N}W^{-1}\_{\mathbf J}(\tilde W^{-1}\_{\mathbf J})^\(\hat Z^{\mathbf\osi}\_{\mathbf N})^\\hat A^{\mathbf\osi}\_{\mathbf N}\right)W^{-1}\_{\mathbf J}\nonumber\\ &=\left(\hat A^{\mathbf\osi}\_{\mathbf N}-\hat A^{\mathbf\osi}\_{\mathbf N}(\hat Z^{\mathbf\osi}\_{\mathbf N})^\\hat A^{\mathbf\osi}\_{\mathbf N}\right)W^{-1}\_{\mathbf J}\label{eq:AAZAdiscretederivation}\end{aligned}$$ using and . We restate the results of Theorems 6.1 and 6.2 and Corollary 6.6 in using the notation above. The proofs are algebraically tedious, yet conceptually straightforward. They rely on two central observations: (i) the matrices A and Z are highly sparse due to the compact support of the basis functions and (ii) the effect of Ω ⊂ Ξ compared to the case Ω = Ξ is confined to those basis functions that overlap with the boundary. The main technical difficulty is to accurately describe these basis functions and their corresponding index sets. [][lem:scalinglowrank] For the AZ pairs $(\hat A\_{\mathbf N},\hat Z\_{\mathbf N})$ and $(\hat A^{\mathbf\osi}\_{\mathbf N},\hat Z^{\mathbf\osi}\_{\mathbf N})$, the matrix A − AZ\*A has 1. at most $\#\K\_{\mathbf N}(\Omega)$ and $\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega)$ non-zero columns, 2. at most rank $\#\K\_{\mathbf N}(\Omega)$ and $\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega)$, 3. $\mathcal O(\#\K\_{\mathbf N}(\Omega))$ and $\mathcal O(\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$ non-zero rows, 4. $\mathcal O(\#\K\_{\mathbf N}(\Omega))$ and $\mathcal O(\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$ non-zero elements, respectively. The constants in the big $\mathcal O$ notation are independent of N. We will also add a more precise statement on the number of non-zero rows. For this we need to introduce two more index sets: $$\begin{aligned} \nonumber\M\_{\mathbf N}(\Omega) &=& \{ \mathbf k \in I\_{\mathbf N}\, \|\, \forall\mathbf l\in\K\_{\mathbf N}(\Omega), \forall\mathbf i\in I\_{\mathbf N}:\\ &&\qquad\mysupp \tilde \phi\_{\mathbf i\mathbf N}\cap \mysupp \phi\_{\mathbf l\mathbf N}\neq\emptyset\text{ and }\mysupp \phi\_{\mathbf i\mathbf N}\cap\mysupp\phi\_{\mathbf k\mathbf N}\neq\emptyset \}\label{eq:Mcont}\end{aligned}$$ and $$\begin{aligned} \M^{\mathbf\osi}\_{\mathbf N}(\Omega) = \{ m \in I\_{M}\, \|\, \forall\mathbf l\in \K^{\mathbf\osi}\_{\mathbf N}, \forall\mathbf i\in I\_{\mathbf N}: \mysupp\_{\mathbf\osi} \tilde \phi\_{\mathbf i\mathbf N}\cap \mysupp\_{\mathbf\osi} \phi\_{\mathbf l\mathbf N}\neq\emptyset\text{ and }\phi\_{\mathbf i\mathbf N}(\mathbf t\_m)\neq 0 \}.\label{eq:Mdiscr}\end{aligned}$$ The set $\M\_{\mathbf N}(\Omega)$ corresponds to the indices of all dual basis functions that overlap with any primal basis function that overlaps with the boundary. Similarly, the set $\M^{\mathbf\osi}\_{\mathbf N}(\Omega)$ has indices of all points in the support of any dual discrete basis function which overlaps with any primal basis function that overlaps with the boundary. [thm:lowrow] For the AZ pairs $(\hat A\_{\mathbf N},\hat Z\_{\mathbf N})$ and $(\hat A^{\mathbf\osi}\_{\mathbf N},\hat Z^{\mathbf\osi}\_{\mathbf N})$, the matrix A − AZ\*A has non-zero row indices $\M\_{\mathbf N}(\Omega)$ and $\M^{\mathbf\osi}\_{\mathbf N}(\Omega)$ respectively. Furthermore, $\#\M\_{\mathbf N}(\Omega)=\mathcal O(\#\K\_{\mathbf N}(\Omega))$ and $\#\M^{\mathbf\osi}\_{\mathbf N}(\Omega)=\mathcal O(\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$. For the AZ pair $(\hat A\_{\mathbf N},\hat Z\_{\mathbf N})$ we proceed similarly as in the proof of. First we note that: $$\begin{aligned} \label{eq:proofnonzerorows1} \left(I - Z^\{A}\right)(\mathbf k,\mathbf l) = \left\{ \begin{array}{ll} \delta\_{\mathbf k\mathbf l} & \text{if }\mysupp\phi\_{\mathbf N,\mathbf l}\subset\Omega^c,\\ 0 & \text{if }\mysupp\phi\_{\mathbf l\mathbf N}\subset\Omega,\\ \delta\_{\mathbf k\mathbf l}-(\phi\_{\mathbf l\mathbf N},\tilde\phi\_{\mathbf k\mathbf N})\_{L^2(\Omega)} &\text{otherwise}. \end{array}\right.\end{aligned}$$ This is due to the compact support of the basis functions and to the continuous biorthogonality. In case Ω = Ξ all matrix entries would be zero, here they differ if $\phi\_{\mathbf{N},\mathbf l}$ is supported outside of Ω or if it overlaps with the boundary. Secondly, multiplication by A* on the left yields: $$\begin{aligned} \label{eq:plungestructure2} \left( A (I - Z^\A) \right)(\mathbf k,\mathbf l) = \left\{ \begin{array}{ll} 0 & \text{if }\mysupp\phi\_{\mathbf l\mathbf N}\subset\Omega^c,\\ 0 & \text{if }\mysupp\phi\_{\mathbf l\mathbf N}\subset\Omega,\\ a\_{\mathbf k\mathbf l} &\text{otherwise}, \end{array}\right.\end{aligned}$$ where the values $a\_{\mathbf k\mathbf l}$ may or may not be zero. We note that A* − AZ\*A has several columns that are identically zero. The proof of also shows that the non-zero column indices of A − AZ\*A are given by $\K\_{\mathbf N}(\Omega)$, hereafter abbreviated by $\K$. If we let $E\in\{0,1\}^{N\times \#\K}$ be the extension matrix that extends $\K$ to $I\_{\mathbf N}$, $$E(\mathbf k,\mathbf l) = \delta\_{\mathbf k\mathbf l}, \qquad \mathbf k\in I\_{\mathbf N}, \mathbf l\in \K,$$ then the matrix A(I − Z\*A)E contains all non-zero columns of A(I − Z\*A) but has size $N\times\#\K$ instead of N × N. The matrix (I − Z\*A)E contains a small number of non-zero rows since $$\begin{aligned} [(I-Z^\A)E](\mathbf k,\mathbf l) = \left\{ \begin{array}{ll} \delta\_{\mathbf k\mathbf l}-(\phi\_{\mathbf l \mathbf N},\tilde\phi\_{\mathbf k \mathbf N})\_{L^2(\Omega)} & \text{if }\mathbf l\in\K\\ 0 & \text{otherwise} \end{array}\right.\end{aligned}$$ using and . The non-zero row indices of this matrix are in the index set $$\label{eq:I1} I\_1 = \K \cup \{ \mathbf k \in I\_{\mathbf N}\, \|\, \forall \mathbf l\in\K: \mysupp \tilde \phi\_{\mathbf k \mathbf N}\cap \mysupp \phi\_{\mathbf l \mathbf N}\neq \emptyset \}$$ so we can rewrite A(I* − Z\*A)E as AE1E1\(I* − Z\*A)E with $E\_1\in \{0,1\}^{N\times \# I\_{\mathbf N}}$ an extension matrix derived from the index set I1 $$E\_1(\mathbf k,\mathbf l) = \delta\_{\mathbf k\mathbf l}, \qquad \mathbf k\in I\_{\mathbf N}, \mathbf l\in I\_1.$$ The non-zero row index set of A(I − Z\*A) is thus $$\{\mathbf k \in I\_{\mathbf N}\, \|\, \forall\mathbf i\in I\_1: \mysupp \phi\_{\mathbf i \mathbf N}\cap\mysupp\phi\_{\mathbf k \mathbf N}\neq\emptyset \}$$ which is equivalent to $\M\_{\mathbf N}(\Omega)$ in after substituting the earlier found expression for I1. For the AZ pair $(\hat A^{\mathbf\osi}\_{\mathbf N},\hat Z^{\mathbf\osi}\_{\mathbf N})$, I1 retains the form of but the meaning of $\K$ is now the one in . That is why the non-zero row index set of A(I − Z\*A) is . Because of Lemma [lem:scalinglowrank], $\#\M\_{\mathbf N}(\Omega)=\mathcal O(\#\K\_{\mathbf N}(\Omega))$ and $\#\M^{\mathbf\osi}\_{\mathbf N}(\Omega)=\mathcal O(\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$. Lemmas [lem:dwtstructure] and [lem:scalinglowrank], Theorem [thm:lowrow] and the relations between the matrices A and Z for the scaling and wavelet bases - combined lead to the following theorem. [thm:waveletlowrank] For the AZ pairs $(A\_{\mathbf N},Z\_{\mathbf N})$ and $(A^{\mathbf\osi}\_{\mathbf N}, Z^{\mathbf\osi}\_{\mathbf N})$, the matrix A − AZ\*A has 1. at most $\#\L\_{\mathbf N}(\Omega)$ and $\#\L^{\mathbf\osi}\_{\mathbf N}(\Omega)$ non-zero columns, 2. at most rank $\#\K\_{\mathbf N}(\Omega)$ and $\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega)$, 3. $\mathcal O(\#\L^{\mathbf\osi}\_{\mathbf N}(\Omega))$ and at most $\#\M\_{\mathbf N}(\Omega)$ non-zero rows, 4. $\mathcal O(J\#\L\_{\mathbf N}(\Omega))$ and $\mathcal O(J\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$ non-zero elements, respectively. The constants in the big $\mathcal O$ notation are independent of N. The index sets are defined by -. 1. One can follow a similar reasoning as in the proofs of Theorem 6.1, Theorem 6.2 in to obtain this statement. There it is shown that the non-zero column indices of $\hat A-\hat A\hat Z^\\hat A$ are those of the scaling* basis elements that overlap with the boundary, i.e., $\K\_{\mathbf N}(\Omega)$ and $K^{\mathbf\osi}\_{\mathbf N}(\Omega)$. This reasoning applied to A − AZ\*A says that the non-zero column indices are those of the wavelet basis elements that overlap with the boundary, i.e., $\L\_{\mathbf N}(\Omega)$ and $\L^{\mathbf\osi}\_{\mathbf N}(\Omega)$. 2. The rank of A − AZ\*A is less than or equal to the rank of $\hat A-\hat A\hat Z^\\hat A$ since $A-AZ^\A=W(\hat A-\hat A\hat Z^\\hat A)W^{-1}$ or $A-AZ^\A=(\hat A-\hat A\hat Z^\\hat A)W^{-1}$ and W* is of full rank. 3. In the discrete case $\hat A-\hat A\hat Z^\\hat A$ has at most $\#\M^{\mathbf\osi}\_{\mathbf N}(\Omega)$ non-zero rows, so A* − AZ\*A which is the former right-multiplied with a discrete wavelet transform has at most $\#\M^{\mathbf\osi}\_{\mathbf N}(\Omega)$ non-zero rows. In the continuous case, $A-AZ^\A=W(\hat A-\hat A\hat Z^\\hat A)W^{-1}$ where $\hat A-\hat A\hat Z^\\hat A$ has $\mathcal O(\#\K\_{\mathbf N}(\Omega))$ non-zero rows and W* has $\mathcal O(J)$ non-zero elements per column, thus A − AZ\*A has $\mathcal O(J \#\K\_{\mathbf N}(\Omega))=\mathcal O(\#\L\_{\mathbf N}(\Omega))$ non-zero rows. 4. The compact support of the scaling bases ensures that the number of non-zero elements per column and per row of $\hat A-\hat A\hat Z^\\hat A$ is bounded by a constant independent of N. Hence $\hat A-\hat A\hat Z^\\hat A$ has $\mathcal O(\#\K\_{\mathbf N}(\Omega))$ ($\mathcal O(\#\K^{\mathbf\osi}\_{\mathbf N}(\Omega))$) non-zero elements (as shown in Lemma [lem:scalinglowrank]). The bound on the number of non-zero elements per row and right-multiplication of $\hat A-\hat A\hat Z^\\hat A$ with W− 1 results in a matrix with $\mathcal O(J)$ elements per row. Therefore, the matrix has $\mathcal O(J\#\K^{\mathbf\osi}\_{\mathbf N})$ non-zero elements. The same holds for the continuous case. Furthermore, since the matrix still has a bounded number of non-zero elements per column after right-multiplication with W− 1, left-multiplication with W* results in a matrix where the number of non-zero elements per row and per column grow like $\mathcal O(J)$ and A − AZ\*A contains in the continuous case $\mathcal O(J\#\L^{\mathbf\osi}\_{\mathbf N})$ non-zero elements. The vanilla AZ algorithm ------------------------ We will refer to Algorithm [alg:AZ] as the vanilla AZ algorithm. Here, the matrices A and Z, as well as the matrix A − AZ\*A have dimension M × N, where M is the total number of sample points and N is the total number of degrees of freedom. We have already established that this algorithm is not optimal, because the matrix A − AZ\*A has a large number of zero-rows and zero-columns. Still, because of its apparent simplicity, we state the expected computational complexity. Also, a surprising feature of the low-rank solver we have used in our implementation in step 1 is that its computational complexity is actually much better. [thm:aztimings] Provided Assumption [ass:boundary] is satisfied, the AZ algorithm (Algorithm [alg:AZ]) using the AZ pairs $(A\_{\mathbf N},Z\_{\mathbf N})$ and $(A^{\mathbf\osi}\_{\mathbf N},Z^{\mathbf\osi}\_{\mathbf N})$ can be implemented with $\mathcal O(N)$ operations in 1-D, $\mathcal O(N^{2})$ operations in 2-D and $\mathcal O(N^{(3d-2)/d}) $ operations in d-D, d > 1. Recall that the computational of the AZ algorithm, Algorithm [alg:AZ], is $$\mathcal O(r\texttt T\_\text{mult}+r^2M).$$ Assumption [ass:boundary] combined with Theorem [thm:waveletlowrank] shows that $r=\mathcal O(N^{(d-1)/d})$. In §[s:choosingz] it was shown that $\texttt T\_\text{mult}$ is $\mathcal O(N)$. The full AZ-algorithm therefore requires $\mathcal O(N)$ operations in 1-D, and $\mathcal O(N^{(3d-2)/d}) $ operations in d-D, d > 1. Theorem [thm:aztimings] is illustrated in Figure [fig:AZtimings1d-3d]. In these numerical results we only consider the discrete setting, because the inner product integrals in are not trivial to compute. The complete code for this experiment and the following experiments is available online. As a low-rank solver in step 1 of the algorithm we have used the low-rank QR solver implemented in. It is an algorithm that builds the QR factorization of a matrix A by random sampling. This means that A is sampled by multiplying A with r random vectors. It is an adaptive algorithm that determines r adaptively in order to obtain an accurate factorization. The low-rank QR is in general a bit more efficient than the low-rank SVD implemented in the same package. We note in Figure [fig:AZtimings1d-3d], to our surprise, that the AZ algorithm outperforms the expected complexity of Theorem [thm:aztimings]. The experimental time complexity follows the dotted line more closely than the dashed one in 2-D and 3-D. We did not explicitly indicate our knowledge of the low number of non-zero rows (see Theorem [thm:waveletlowrank]) to the low-rank solver. However, the solver apparently took notice automatically and used it to its advantage, lowering the time complexity to that of Theorem [thm:reducedaztimings] in the following section. There, we do assume an algorithm that explicitly uses knowledge of the non-zero rows and columns. The reduced AZ algorithm ------------------------ Following Theorem [thm:waveletlowrank] the non-zero rows and columns are described by the index sets $\L^{\mathbf\osi}\_{\mathbf N}(\Omega)$ (or $\L^{\mathbf\osi}\_{\mathbf N}(\Omega)$ in the discrete setting) and $\M^{\mathbf\osi}\_{\mathbf N}(\Omega)$ ($\M^{\mathbf\osi}\_{\mathbf N}(\Omega)$). To ease notation, we will use $\L$ and $\M$ instead if the meaning is independent of the context or can be deduced out of the context. By iterating once over all wavelet basis functions and checking their support one can determine the indices in $\L$ in $\mathcal O(N)$ operations. Indeed, owing to the completely regular structure of the bases involved, the supports of the basis functions are easily computed. From the definitions - we see that we can compute $\M$ by first computing $\K$ — this in $\mathcal O(N)$ operations by iterating over all scaling basis elements — then checking their overlap with the duals, again in $\mathcal O(N)$ operations. If these index sets are known we can create the (sparsely representable) extension and restriction matrices $E=\{0,1\}^{N\times\#\L}$ and $R=\{0,1\}^{\#\M\times M}$: $$\begin{aligned} E(\mathbf k, \mathbf l) &=& \delta\_{\mathbf k\mathbf l}, \qquad \mathbf k\in I\_{\mathbf N}, \mathbf l\in \L \\ R(k, l) &=& \delta\_{kl}, \qquad k\in\M, l\in I\_m.\end{aligned}$$ [t] [alg:reducedAZ] Input: A, Z ∈ CM × N, b ∈ CM, ε > 0 Output: x ∈ CN such that Ax ≈ b [1] Determine E which extends the index set $\L$ to $I\_{\mathbf N}$ Determine R which restricts IM to the indices of non-zero rows of (I − AZ\)A*E. Solve R(I − AZ\)AEx1 = R(I* − AZ\)b* x2 ← Z\(b* − AEx1) x ← Ex1 + x2 With these matrices we construct the matrix R(I − AZ\)A*E that holds the same information as A − AZ\*A but without all known zero rows and columns. This matrix is used in the reduced AZ algorithm (Algorithm [alg:reducedAZ]). The number of non-zero columns of R(I − AZ\)A*E still grows at a faster rate than its rank, see Theorem [thm:waveletlowrank]. That is why in the following theorem we distinguish between a full direct solver, such as a pivoted QR, and a low-rank direct solver, such as a randomized low-rank QR. Note also that reducing the size does not change the complexity of applying the matrices. Both A − AZ\*A and R(I − AZ\)A*E result in a matrix-vector multiply that takes $\mathcal O(N)$ operations. [thm:reducedaztimings] Provided Assumption [ass:boundary] is satisfied, the reduced AZ algorithm (Algorithm [alg:reducedAZ]) using the AZ pair $(A\_{\mathbf N},Z\_{\mathbf N})$ can be implemented with 1. $\mathcal O(N\log(N))$ operations in 1-D, $\mathcal O(\log(N)^3N^{3/2})$ operations in 2-D and $\mathcal O(\log(N)^3N^{3(d-1)/d})$ operations in d-D with d > 1 if a full direct solver is used in the 3rd step. 2. $\mathcal O(N)$ operations in 1-D, $\mathcal O(\log(N)N^{3/2})$ operations in 2-D and $\mathcal O(\log(N)N^{3(d-1)/d})$ operations in d-D with d > 1 if a low-rank direct solver is used in the 3rd step. Using the AZ pair $(A^{\mathbf\osi}\_{\mathbf N},Z^{\mathbf\osi}\_{\mathbf N})$ the reduced AZ algorithm can be implemented with 1. $\mathcal O(N\log(N))$ operations in 1-D, $\mathcal O(\log(N)^2N^{3/2})$ operations in 2-D and $\mathcal O(\log(N)^2N^{3(d-1)/d})$ operations in d-D with d > 1 if a full direct solver is used in the 3rd step. 2. $\mathcal O(N)$ operations in 1-D, $\mathcal O(N^{3/2})$ operations in 2-D and $\mathcal O(N^{3(d-1)/d})$ operations in d-D with d > 1 if a low-rank direct solver is used in the 3rd step. The proof largely follows, with the exception of a factor $\mathcal O(J)$ in some places and using Theorem [thm:waveletlowrank]. The extension matrix E can be constructed in $\mathcal O(N)$ operations by iterating once over all wavelet basis functions. The matrix R can be constructed in $\mathcal O(N)$ by creating the set $\M$. For the discrete case, this immediately gives the non-zero row indices. For the continuous case, an additional DWT is needed to find these non-zero row indices.[5](#fn5) The matrix R(A − AZ\*A)E has size m × n, where 1. $n=\#\L = \mathcal O(JN^{(d-1)/d})$ 2. $m=\mathcal O(\# \L^{\mathbf\osi})=\mathcal O(N^{(d-1)/d})$ in the continuous setting and $m=\#\M =\mathcal O(N^{(d-1)/d})$ in the discrete setting. However, by Theorem [thm:waveletlowrank] and Assumption [ass:boundary] its rank r is only $\mathcal O(\K) = \mathcal O(N^{(d-1)/d})$. Next, creating and solving the m × n linear system with a direct solver requires $\mathcal O(n\texttt T\_\text{mult}+mn^2)$ operations, while it takes $\mathcal O(r\texttt T\_\text{mult}+mr^2)$ with a low-rank solver. Therefore, the full time complexity of the algorithm is $$\mathcal O(N + n\texttt T\_\text{mult} + nM + mn^2) = \mathcal O(nN + mn^2)$$ with a full direct solver and $$\mathcal O(N + r\texttt T\_\text{mult} + nM + mr^2) = \mathcal O(rN + mr^2)$$ with a low-rank direct solver, where we take linear oversampling $M=\mathcal O(N)$ into account and all matrices can be applied in $\mathcal O(N)$ operations, so $T\_\text{mult}=\mathcal O(N)$. Filling in the results obtained earlier in the proof results in the statement of the theorem. The statements in Theorem [thm:reducedaztimings] are corroborated in Figure [fig:AZR1timings1d-3d]. The difference in time complexity between a full and a low-rank direct solver is only a matter of logarithmic factors. However, the experiments indicate that the low-rank solver is the better choice, at least in our implementation, since apart from slightly improved time complexity its cost seems also substantially lower in absolute terms. As noted at the end of §[ss:vanillaAZ], the reduced AZ algorithm does not perform much better than vanilla AZ (in the dotted lines) since the randomized solver seems to take advantage of the non-zero rows. There are however some advantages to explicitly removing the zero columns and rows. First, the sampling matrix in the low-rank solver reduces in size from $\mathcal O(N\times N^{(d-1)/d})$ to O(N(d − 1)/dlog(N) × N(d − 1)/d). Secondly, all matrices and decompositions stored are smaller. Finally, we do not need to select a solver that implicitly takes care of the zero rows. From Figure [fig:AZR2timings2d] it is clear that Daubechies wavelets are less efficient than CDF wavelets (in our implementation). Furthermore, we see that wavelets with higher $\tilde p$ and hence larger support are less efficient than those with a lower number of dual vanishing moments. Of course, their wider support results in a larger number of basis functions that overlap with the boundary. The left bottom panel of Figure [fig:AZerrors1d-3d] shows the residual of the experiments in Figure [fig:AZR2timings2d], but with $\mathbf\osi=(4,4)$ instead of $\mathbf\osi=(2,2)$. The residual directly corresponds to the approximation error in the point samples. We see the expected algebraic convergence and that wavelets with a higher order converge faster. Note that the oversampling factors $\mathbf\osi=(4,4)$ were needed here to make a clear distinction between the experimental CDF convergence rates. With $\mathbf\osi=(2,2)$, the choice in the previous experiments including Figure [fig:AZR2timings2d], both `cdf`3$\tilde p$ and `cdf`4$\tilde p$ appear to converge approximately at the same rate. Increasing the oversampling factor makes the difference in convergence rates more pronounced. The sparse AZ algorithm ----------------------- [t] [alg:sparseAZ] Input: A, Z ∈ CM × N, b ∈ CM Output: x ∈ CN such that Ax ≈ b [1] Create sparse matrix (I − AZ\)A* Solve (I − AZ\)A*x1 = (I − AZ\)b* using sparse QR x2 ← Z\(b* − Ax1) x ← x1 + x2 Finally, we consider an algorithm that only exploits the sparsity of the matrices A, Z and A − AZ\*A. As for B-splines in, here we use the sparse direct rank-revealing QR decomposition of in the first step of the AZ algorithm instead of a low-rank solver. The sparse AZ algorithm is formulated in Algorithm [alg:sparseAZ]. The first step is the creation of the matrix A − AZ\*A in sparse form. It was shown in that a sparse version $\hat A-\hat A\hat Z^\\hat A$ (holding $\mathcal O(N^{(d-1)/d})$ non-zero values and $\mathcal O(1)$ non-zero elements in each column and row) can be created in $\mathcal O(N)$ operations. Given this sparse matrix $\hat A-\hat A\hat Z^\\hat A$, we can easily create the matrix containing its non-zero columns $\hat A(I-\hat A\hat Z^\)\hat AE$ with $$E(\mathbf k, \mathbf l) = \delta\_{\mathbf k\mathbf l}, \qquad \mathbf k\in I\_{\mathbf N}, \mathbf l\in \K.$$ It remains to construct the sparse matrix E\W− 1 since $$A-AZ^\A = \hat A(I-\hat A\hat Z^\)\hat AEE^\W^{-1}.$$ Provided Assumption [ass:boundary] is satisfied, we deduce from Lemma [lem:dwtstructure] that E\W− 1 has $\mathcal O(N^{(d-1)/d}\log N)$ non-zero elements, i.e., $\mathcal O(N^{(d-1)/d})$ rows with $\mathcal O(\log N)$ elements each. Because of the structure in both matrices the standard sparse matrix-matrix multiply in Julia 1.3 performs the multiplication of $A(I-\hat A\hat Z^\)\hat AE$ with E*W− 1 in $\mathcal O(N^{(d-1)/d}\log(N))$. Hence, for the best possible time complexity in the first step of the sparse AZ algorithm, we need to construct E\*W− 1 in $\mathcal O(JN^{(d-1)/d})$ operations. Here, we present an algorithm that in the worst case is $\mathcal O(N\log N)$. An inspection of §[ss:dwt] using compactly supported sequences and convolutions instead of matrices yields the insight that each column of WJi− 1 (1-D iDWT transform) contains a shifted version of either one of $$\begin{aligned} \label{eq:idwtfilters} &&g^{J},\\\nonumber &&[g^{J-1}]\_{\uparrow 2^1}\star h^{J},\\\nonumber &&[g^{J-2}]\_{\uparrow 2^2}\star [h^{J-1}]\_{\uparrow 2^1}\star h^{J}, \\\nonumber &&\qquad\qquad\quad\vdots\\\nonumber &&[g^{1}]\_{\uparrow 2^{J-1}}\star [h^{2}]\_{\uparrow 2^{J-1}}\star\cdots\star [h^{J-1}]\_{\uparrow 2^1} h^{J}, \\\nonumber &&[h^{1}]\_{\uparrow 2^{J-1}}\star [h^{2}]\_{\uparrow 2^{J-1}}\star\cdots\star [h^{J-1}]\_{\uparrow 2^1} h^{J}\end{aligned}$$ where we left out the dimension subindex to Ji and where [a]↑ q denotes upsampling and aJ periodisation: $$([a]\_{\uparrow q})\_k = a\_{kq}, \qquad (a^J)\_k = \sum\_{l\in\Z} a\_{k+l2^J}.$$ A convolution a ⋆ b can be computed in $\mathcal O(n\log n)$ operations with n the sum of the supports of a and b using the Fast Fourier Transform. The top filter in has constant support, but the support lengths grow steadily up to Ji2Ji. Therefore, the Ji + 1 filters can be computed in $\mathcal O(N\_i\log N\_i)$ operations. If we assume that the degrees of freedom are evenly distributed over all dimensions, then we have $N\_i=\mathcal O(N^{1/d})$ and the total number of nonzero entries in E\*W− 1 is $\mathcal O(JN^{(d-1)/d})$. Hence, knowing their locations, the sparse matrix can also be constructed in $\mathcal O(JN^{(d-1)/d})$ operations. In the worst case, if one dimension has all degrees of freedom, then the cost of the full construction algorithm of the sparse E\*W− 1 may be as large as $\mathcal O (N\log N)$. This is unlikely to be the case in practice. Figure [fig:AZStimings1d-3d] shows that the sparse AZ algorithm appears to be more efficient than the algorithms above, especially in the lower dimensions. In 3-D, the level of sparsity is not yet high enough to show a possible advantageous effect of using a sparse solver. We also compare coefficient norm and residual of the different algorithms. For the coefficient norm, it is known for the vanilla and reduced AZ algorithms that a small-norm coefficient will be returned if it exists, since the solver in step 1 is closely related to a truncated SVD solver. For the direct sparse QR solver of, no analogous error analysis is known, hence the coefficient norm might be large. While this was not the case for spline extension in, coefficients are indeed larger using wavelet extensions as shown in Figure [fig:AZerrors1d-3d]. Both the sparse AZ and a direct sparse QR solver are affected. This has a negative impact on the residual, as is also shown in Figure [fig:AZerrors1d-3d]. Adaptively smoothed wavelet AZ algorithm ======================================== In this final section we compare the use of wavelet-based extensions to the simpler setting of spline-based extensions. The approximation space spanned by a basis of B-splines is exactly the same as that spanned by spline-based wavelets. Indeed, the wavelet transform is merely a change of basis. Therefore, the best approximations are the same: $$\arg \min\_{u \in \SPAN \Phi\_N} \Vert f - u \Vert = \arg \min\_{u \in \SPAN \Psi\_N} \Vert f - u \Vert.$$ In typical applications of wavelets their compression properties play a major role. That is not really the case here: the wavelet least squares matrix A is even somewhat less sparse than the corresponding matrix $\hat A$ using B-splines. Another beneficial property of wavelets is the multiresolution nature of the approximation. In particular, unlike with B-splines, it is possible to associate different weighting factors with different scales. This enables the construction of bases for a range of function spaces with varying smoothness properties. For example, methods for the solution of partial differential equations employ wavelet bases for Sobolev spaces. In our setting, weighing different wavelet scales allows one to obtain smoother approximations. The increased smoothness is only visible in the extension Ξ \ Ω of the wavelet frame, since the approximation always resembles the function itself in the interior Ω. A smooth extension is not guaranteed by the methods described in §[wavs:az]. A least squares solver aims to minimize the residual of the system with a minimal norm solution. Therefore, the resulting wavelets coefficients do not necessarily decrease with increasing scale, even when approximating smooth functions. All coefficients have roughly similar size. In contrast, when approximating with a regular basis, the decrease of wavelet coefficient size is guaranteed for smooth functions, depending on the order of the multiresolution analysis at hand. [h] [alg:smoothedAZ] Input: A, Z ∈ CM × N, b ∈ CM, $W\in \R\_+^{N\times N}$ Output: x ∈ CN such that Ax ≈ b [1] Solve (I − AZ\)AWx1 = (I* − AZ\)b* using a randomized low-rank solver x2 ← Z\(b* − AWx1) x ← Wx1 + x2 Smoothing can be introduced by switching to a weighted least squares formulation. We weigh the wavelet coefficients using the smoothed AZ algorithm (Algorithm [alg:smoothedAZ]). We simply add a diagonal weight matrix in the first step of the AZ algorithm, replacing x1 by Wx1, and leave the other steps unchanged. The diagonal matrix has weights that depend on the scale of the corresponding entries of x1. Much in the same way, the weighted reduced and sparse AZ algorithms can be formulated and implemented. Note that a weighted least squares problem is only being solved in step 1, not in step 2. The dual does not require modifications in this formulation. We aim for a coefficient vector in which the coefficients decrease in size with increasing scale. The logic is as follows. Say a function is approximated on a coarse scale $\mathbf J-1$ with approximation error e1. It can be expected that this approximation can be refined by adding wavelet coefficients on finer scales with size on the order of $\mathcal O(e\_1)$. We ensure that the coefficients on the finer scale have that size simply by choosing the corresponding diagonal entries of W equal to e1. The function is now approximated on the finer scale, say with an approximation error e2. This error can be used to weight the next scale of coefficients, and so on. Thus, we obtain a diagonal weighting operator parametrized with the weights e1, …, eL and size $\mathbf N$, L ≤ log2(Ni), i = 1, …, d: $$W(\mathbf k,\mathbf l; [e\_1, e\_2, \dots, e\_L], \mathbf N) = \delta\_{\mathbf k,\mathbf l}e\_{\min\{1,L-\min\_{i=1,\dots,d}\{\log\_2(N\_i)-L\}\}}\qquad \mathbf k,\mathbf l \in I\_{\mathbf N}.$$ We use the size of the right hand side as an initial weight for the first approximation. This way we arrive at the adaptive Algorithm [alg:adaptivelysmoothedAZ]. [h] [alg:adaptivelysmoothedAZ] Input: $\mathbf N$, $\mathbf\osi$, f, Ω ⊂ Ξ, wavelet types Output: x (the wavelet extension coefficients) [1] $\mathbf n\gets\mathbf N-\min(N)+1$ $e \gets \\|b^{\mathbf\osi}\_{\mathbf n}\\|$ $W \gets W(\cdot,\cdot; e, \mathbf N)$ x ← Apply the smoothed AZ algorithm with $A^{\mathbf\osi}\_{\mathbf n}$, $Z^{\mathbf\osi}\_{\mathbf n}$, $b^{\mathbf\osi}\_{\mathbf n}$, W $e \gets [e; \\|A^{\mathbf\osi}\_{\mathbf n}x-b^{\mathbf\osi}\_{\mathbf n}\\|]$ $\mathbf n \gets 2\mathbf n$ [fig:1Dcoefs] ![image](img/fspline_AZR) ![image](img/fspline_D) ![image](img/f_AZR)![image](img/f_AZWR)![image](img/f_D) ![image](img/2Dcoefs_AZR)![image](img/2Dcoefs_AZWR)![image](img/2Dcoefs_D) We illustrate the adaptive Algorithm with a simple example in 1D first. In Figure [fig:1Dcoefs] we compare to a pivoted QR and the reduced AZ algorithm (both without smoothing). The function f(x) = ex is approximated on the interval [0, 0.6] using a `cdf33` wavelet extension. The weighted algorithms very clearly leads to the smoothest extension (shown in the left panel). In the right panel, the sizes of the wavelet coefficients are compared. The approximation domain [0, 0.6] is visible in all levels of the wavelet coefficients. There, all coefficients have rougly similar size (the three lines overlap) because they approximate the same function. The coefficients differ in the extension [0.6, 1]. There, it is seen that the smoothed approximation (red line) yields significantly smaller coefficients than the non-smoothed approximations. In Figure [fig:2Dcoefs] and Table [tab:comparison] we compare the adaptive Algorithm [alg:adaptivelysmoothedAZ] using both sparse (only in table) and reduced AZ with the original non-smoothed reduced and sparse AZ algorithm. Also we compare with a simple pivoted QR. Finally, we compare wavelet extension with spline extension approximation as in where no adaptive weighting is possible. The smoothed approximation (the middle column of Figure [fig:2Dcoefs] and second row of the table) leads to a visually smoother extension. We also see that the non-smoothed methods tend to result in larger wavelet coefficients at the boundary. For the smoothed result, one can also see that the wavelet coefficients at a coarse scale are larger then those at a finer scale. The approximant itself also takes another shape. The spline extension approximant will drop down to zero outside of Ω. Choosing a wavelet extension instead of a spline extension has no effect on the residual, nor does weighting. As expected, the pivoted QR leads to the smallest coefficient norm. Concluding remarks ================== We have shown that wavelet approximation on general domains is possible and efficient using regular wavelets defined on a bounding box. Compared to existing wavelet literature, we have had to compute a discrete dual scaling sequence. Apart from this construction, with values listed exhaustively in the appendix, well-known scaling functions and wavelets could be used. The ill-conditioning of the extension problem necessitates a least squares formulation with regularization. The proposed algorithms have a complexity of $\mathcal O(N)$ operations in 1-D, $\mathcal O(N^{3/2})$ in 2-D and $\mathcal O(N^{3(d-1)/d})$ in d-D, d > 1. This should be compared to the cubic cost of standard direct solvers. The algorithms required a detailed study of the sparsity structure of all the matrices involved. The use of a general sparse QR function does not require such detailed study of the structure. As such, it is simpler to implement. The results experimentally appear to be more efficient. However, it also seems less stable. Appendix ======== The values of the discrete duals that were used for the experiments in this paper are tabulated in Table [tab:discretevalues]. [email protected] & & & [tab:discretevalues] --- 1. Email: `[email protected]`. Website: `https://people.cs.kuleuven.be/~vincent.coppe`.[↩](#fnref1) 2. Email: `[email protected]`. Website: `https://people.cs.kuleuven.be/~daan.huybrechs`.[↩](#fnref2) 3. Since we intend to employ wavelets on a bounding box Ξ to approximate functions on a subset Ω ⊂ Ξ, the periodicity of the basis on Ξ is not actually a restriction on Ω, as long as the boundaries of Ω and Ξ do not touch. One can use other boundary conditions on Ξ, but periodicity is the simplest one to implement and manipulate.[↩](#fnref3) 4. The values of p and p̃ do not uniquely determine a biorthogonal multiresolution analysis. We use filters corresponding to B-splines as described in. Thus, the meaning of `cdf44` in this paper differs from the widely used CDF filters of primal and dual order 4 in signal processing, e.g., in the JPEG2000 standard.[↩](#fnref4) 5. For completeness, in our implementation we have performed a DWT on a vector of length N with a `NaN` instead of a floating point number at the indices $\K$ (non-zero indices of $\hat A\_{\mathbf N}-\hat A\_{\mathbf N}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N}$). Since a mathematical operation between a `NaN` and a floating point number results in a `NaN`, the resulting vector thus contains `NaN`s at the non-zero indices of $W\_{\mathbf J}(\hat A\_{\mathbf N}-\hat A\_{\mathbf N}\hat Z\_{\mathbf N}^\\hat A\_{\mathbf N})$.[↩](#fnref5)	arxiv_0000443
→ X is called a complete join if it satisfies the following properties: 1. π is injective on individual simplices. 2. For each p-simplex σ = ⟨v0, ⋯, vp⟩ of X, π− 1(σ) is the join π− 1(v0) \* π− 1(v1) \* ⋯ \* π− 1(vp). A simplicial complex X is called weakly Cohen-Macaulay of dimension n if X is (n − 1)-connected and the link of each p-simplex of X is (n − p − 2)-connected. We sometimes shorten weakly Cohen-Macaulay to wCM. The main result regarding complete join that we will use is the following. [prop-join-conn] If Y is a complete join complex over a wCM complex X of dimension n, then Y is also wCM of dimension n. [rem-cjoin] If π : Y → X is a complete join, then X is a retract of Y. In fact, we can define a simplicial map s : X → Y such that $\pi\circ s = {{{\rm id}}}\_X$ by sending a vertex v ∈ X to any vertex in π− 1(v) and then extending it to simplices. The fact that s can be extended to simplices is granted by the condition that π is a complete join. In particular we can also conclude that if Y is n-connected, so is X. Bad simplices argument ---------------------- Let (X, Y) be a pair of simplicial complexes. We want to relate the n-connectedness of Y to the n-connectedness of X via a so called bad simplices argument, see for more information. One identifies a set of simplices in X \ Y as bad simplices, satisfying the following two conditions: 1. [bad-sim-1] Any simplex with no bad faces is in Y, where by a ``face" of a simplex we mean a subcomplex spanned by any nonempty subset of its vertices, proper or not. 2. [bad-sim-2] If two faces of a simplex are both bad, then their join is also bad. We call simplices with no bad faces good simplices. Bad simplices may have good faces or faces which are neither good nor bad. If σ is a bad simplex, we say a simplex τ in Lk(σ) is good for σ if any bad face of τ \* σ is contained in σ. The simplices which are good for σ form a subcomplex of Lk(σ) which we denote by GLσ and call the good link of σ. [prop-bad-sim] Let X, Y and GLσ be as above. Suppose that for some integer n ≥ 0 the subcomplex GLσ of X is (n − dim(σ) − 1)-connected for all bad simplices σ. Then the pair (X, Y) is n-connected, i.e. πi(X, Y) = 0 for all i ≤ n. We can apply the proposition in the following way. [thm–bad-sim] Let Y be a subcomplex of a simplicial complex X and suppose the space X \ Y has a set of bad simplices satisfying [bad-sim-1] and [bad-sim-2] above, then: 1. If X is n-connected and GLσ is (n − dim(σ))-connected for all bad simplices σ, then Y is n-connected. 2. If Y is n-connected and GLσ is (n − dim(σ) − 1)-connected for all bad simplices σ, then X is n-connected. The mutual link trick --------------------- In the proof of, there is a beautiful argument for resolving intersections of arcs inspired by Hatcher’s flow argument. They attributed the idea to Andrew Putman. Recall Hatcher’s flow argument allows one to ``flow“ a complex to its subcomplex. But in the process, one can only ``flow” a vertex to a new one in its link. The mutual link trick will allow one to ``flow“ a vertex to a new one not in its link provided ``the mutual link” is sufficiently connected. To apply the mutual link trick, we first need a lemma that allows us to homotope a simplicial map to a simplexwise injective one. Recall a simplicial map is called simplexwise injective if its restriction to any simplex is injective. See also for more information. [lem:injectifying] Let Y be a compact m-dimensional combinatorial manifold. Let X be a simplicial complex and assume that the link of every p-simplex in X is (m − p − 2)-connected. Let ψ: Y → X be a simplicial map whose restriction to ∂Y is simplexwise injective. Then after possibly subdividing the simplicial structure of Y, ψ is homotopic relative ∂Y to a simplexwise injective map. Note that as discussed in, there is a mistake in the connectivity bound given in that has been corrected later in. [The mutual link trick][lemma-replace-trick] Let Y be a closed m-dimensional combinatorial manifold and f : Y → X be a simplexwise injective simplicial map. Let y ∈ Y be a vertex and f(y) = x for some x ∈ X. Suppose xʹ is another vertex of X satisfying the following condition. 1. f(LkY(y)) ≤ LkX(xʹ), 2. the mutual link LkX(x) ∩ LkX(xʹ) is (m − 1)-connected, Then we can define a new simplexwise injective map g : Y → X by sending y to xʹ and all the other vertices yʹ to f(yʹ) such that g is homotopic to f. The conditions that f is simplexwise injective and f(LkY(y)) ≤ LkX(xʹ) guarantee that the definition of g can be extended over Y and g is again simplexwise injective. We need to prove g is homotopic to f. The homotopy will be the identity outside StY(y). Note that since f is simplexwise injective, we have f(LkY(y)) ≤ LkX(x). Together with Condition (1), we have f(LkY(y)) ≤ LkX(x) ∩ LkX(xʹ). Since LkY(y) is an (m − 1)-sphere and LkX(x) ∩ LkX(xʹ) is (m − 1)-connected, there exists an m-disk B with ∂B = LkY(y) and a simplicial map φ: B → LkX(x) ∩ LkX(xʹ) so that φ restricted to ∂B coincides with ψ restricted to LkY(y). Since the image of B under φ is contained in StX(x) which is contractible, we can homotope g, replacing g∣StY(y) with φ. Since the image of B under f is also contained in LkX(xʹ), we can similarly homotope f, replacing f∣StY(y) with φ. These both yield the same map, so g is homotopic to f. Higman–Thompson groups and their braided versions ================================================= In this section, we first give an introduction to the Higman–Thompson groups and then define their ribbon version. Higman–Thompson groups ---------------------- The Higman–Thompson groups were first introduced by Higman as a generalization of the groups given earlier in handwritten, unpublished notes of Richard Thompson. First let us recall the definition of the Higman–Thompson groups. Although there are a number of equivalent definitions of these groups, we will use the notion of paired forest diagrams. First we define a finite rooted d-ary tree to be a finite tree such that every vertex has degree d + 1 except the leaves which have degree 1, and the root, which has degree d (or degree 1 if the root is also a leaf). Usually we draw such trees with the root at the top and the nodes descending from it down to the leaves. A vertex v of the tree along with its d adjacent descendants will be called a caret. If the leaves of a caret in the tree are leaves of the tree, we will call the caret elementary. A collection of r many d-ary trees will be called a $\emph{$(d,r)$-forest}$. When d is clear from the context, we may just call it an r-forest. Define a paired (d, r)-forest diagram to be a triple (F−, ρ, F+) consisting of two (d, r)-forests F− and F+ both with l leaves for some l, and a permutation ρ ∈ Sl, the symmetric group on l elements. We label the leaves of F− with 1, …, l from left to right, and for each i, the ρ(i)th leaf of F+ is labeled i. Define a reduction of a paired (d, r)-forest diagram to be the following: Suppose there is an elementary caret in F− with leaves labeled by i, ⋯, i + d − 1 from left to right, and an elementary caret in F+ with leaves labeled by i, ⋯, i + d − 1 from left to right. Then we can “reduce” the diagram by removing those carets, renumbering the leaves and replacing ρ with the permutation ρʹ ∈ Sl − d + 1 that sends the new leaf of F− to the new leaf of F+, and otherwise behaves like ρ. The resulting paired forest diagram (Fʹ−, ρʹ, Fʹ+) is then said to be obtained by reducing (F−, ρ, F+). See Figure [fig:reductionV] below for an idea of reduction of paired (3, 2)-forest diagrams. The reverse operation to reduction is called expansion, so (F−, ρ, F+) is an expansion of (Fʹ−, ρʹ, Fʹ+). A paired forest diagram is called reduced if there is no reduction possible. Define an equivalence relation on the set of paired (d, r)-forest diagrams by declaring two paired forest diagrams to be equivalent if one can be reached by the other through a finite series of reductions and expansions. Thus an equivalence class of paired forest diagrams consists of all diagrams having a common reduced representative. Such reduced representatives are unique. [fig:ele-V] [line width=1pt, scale=0.5] [xshift= -4cm] (-12,-2) – (-9,0) – (-6,-2) (-9,0) – (-9,-2) (-11,-1.33) – (-11,-2) (-11,-1.33) – (-10,-2) (-4,-2) – (-1,0) – (2,-2) (-1,0) – (-1,-2); (3.5,0) – (3.5,-3); (-12,-2) circle (1.5pt) (-9,0) circle (1.5pt) (-6,-2) circle (1.5pt) (-9,-2) circle (1.5pt) (-11,-1.33) circle (1.5pt) (-11,-2) circle (1.5pt) (-10,-2) circle (1.5pt) (-4,-2) circle (1.5pt) (-1,0) circle (1.5pt) (-1,-2) circle (1.5pt) (2,-2) circle (1.5pt); at (-12,-2.5) 1; at (-11,-2.5) 2; at (-10,-2.5) 3; at (-9,-2.5) 4; at (-6,-2.5) 5; at (-4,-2.5) 6; at (-1,-2.5) 7; at (2,-2.5) 8; [ xshift=13cm] (-12,-2) – (-9,0) – (-6,-2) (-9,0) – (-9,-2) (-8,-2) – (-9,-1.33) – (-10,-2) (-4,-2) – (-1,0) – (2,-2) (-1,0) – (-1,-2); (-12,-2) circle (1.5pt) (-9,0) circle (1.5pt) (-6,-2) circle (1.5pt) (-9,-2) circle (1.5pt) (-9,-1.33) circle (1.5pt) (-10,-2) circle (1.5pt) (-8,-2) circle (1.5pt) (-4,-2) circle (1.5pt) (-1,0) circle (1.5pt) (2,-2) circle (1.5pt) (-1,-2) circle (1.5pt); at (-12,-2.5) 4; at (-6,-2.5) 6; at (-10,-2.5) 1; at (-8,-2.5) 3; at (-9,-2.5) 2; at (-4,-2.5) 7; at (2,-2.5) 5; at (-1,-2.5) 8; [ yshift=-4.5cm,xshift= -4cm] (-12,-2) – (-9,0) – (-6,-2) (-9,0) – (-9,-2) (-4,-2) – (-1,0) – (2,-2) (-1,0) – (-1,-2); (3.5,0) – (3.5,-3); (-12,-2) circle (1.5pt) (-9,0) circle (1.5pt) (-6,-2) circle (1.5pt) (-9,-2) circle (1.5pt) (-4,-2) circle (1.5pt) (-1,0) circle (1.5pt) (-1,-2) circle (1.5pt) (2,-2) circle (1.5pt); at (-12,-2.5) 1; at (-9,-2.5) 2; at (-6,-2.5) 3; at (-4,-2.5) 4; at (-1,-2.5) 5; at (2,-2.5) 6; [xshift=13cm, yshift=-4.5cm] (-12,-2) – (-9,0) – (-6,-2) (-9,0) – (-9,-2) (-4,-2) – (-1,0) – (2,-2) (-1,0) – (-1,-2); (-12,-2) circle (1.5pt) (-9,0) circle (1.5pt) (-6,-2) circle (1.5pt) (-9,-2) circle (1.5pt) (-4,-2) circle (1.5pt) (-1,0) circle (1.5pt) (2,-2) circle (1.5pt) (-1,-2) circle (1.5pt); at (-12,-2.5) 2; at (-6,-2.5) 4; at (-9,-2.5) 1; at (-4,-2.5) 5; at (2,-2.5) 3; at (-1,-2.5) 6; [fig:reductionV] There is a binary operation \* on the set of equivalence classes of paired (d, r)-forest diagrams. Let α = (F−, ρ, F+) and β = (E−, ξ, E+) be reduced paired forest diagrams. By applying repeated expansions to α and β we can find representatives (Fʹ−, ρʹ, Fʹ+) and (Eʹ−, ξʹ, Eʹ+) of the equivalence classes of α and β, respectively, such that Fʹ+ = Eʹ−. Then we declare α \* β to be (Fʹ−, ρʹξʹ, Eʹ+). This operation is well defined on the equivalence classes and is a group operation. [def:Higman-V-F-T] The Higman–Thompson group Vd, r is the group of equivalence classes of paired (d, r)-forest diagrams with the multiplication \* . The usual Thompson group V is a special case of Higman–Thompson groups. In fact, V = V2, 1. Ribbon Higman–Thompson groups ----------------------------- For convenience, we will think of the forest F+ drawn beneath F− and upside down, i.e., with the root at the bottom and the leaves at the top. The permutation ρ is then indicated by arrows pointing from the leaves of F− to the corresponding paired leaves of F+. See Figure [fig:elementofV] for this visualization of (the unreduced representation of) the element of V3, 2 from Figure [fig:reductionV]. [line width=1pt, scale = 0.5] [xshift =10cm] (-12,-2) – (-9,0) – (-6,-2) (-9,0) – (-9,-2) (-11,-1.33) – (-11,-2) (-11,-1.33) – (-10,-2) (-4,-2) – (-1,0) – (2,-2) (-1,0) – (-1,-2); (-11.9,-2.2) -> (-10.2,-4.8); (-10.9,-2.2) -> (-9.2,-4.8); (-9.9,-2.2) -> (-8.2,-4.8); (-9.2,-2.2) -> (-11.8,-4.8); (-5.8,-2.2) -> (1.8,-4.8); (-4.1,-2.2) -> (-5.9,-4.8); (-1.1,-2.2) -> (-3.8,-4.8); (1.8,-2.2) -> (-0.8,-4.8); (-12,-2) circle (1.5pt) (-9,0) circle (1.5pt) (-6,-2) circle (1.5pt) (-9,-2) circle (1.5pt) (-11,-1.33) circle (1.5pt) (-11,-2) circle (1.5pt) (-10,-2) circle (1.5pt) (-4,-2) circle (1.5pt) (-1,0) circle (1.5pt) (-1,-2) circle (1.5pt) (2,-2) circle (1.5pt); [ xshift = 10cm, yscale=-1, yshift=7cm] (-12,-2) – (-9,0) – (-6,-2) (-9,0) – (-9,-2) (-8,-2) – (-9,-1.33) – (-10,-2) (-4,-2) – (-1,0) – (2,-2) (-1,0) – (-1,-2); (-12,-2) circle (1.5pt) (-9,0) circle (1.5pt) (-6,-2) circle (1.5pt) (-9,-2) circle (1.5pt) (-9,-1.33) circle (1.5pt) (-10,-2) circle (1.5pt) (-8,-2) circle (1.5pt) (-4,-2) circle (1.5pt) (-1,0) circle (1.5pt) (-1,-2) circle (1.5pt) (2,-2) circle (1.5pt); [fig:elementofV] Now in the ribbon version of the Higman–Thompson groups, the permutations of leaves are simply replaced by ribbon braids which can twist between the leaves. [defn-rb] Let I = ∐ i = 1dIi : [0, 1] × {1, ⋯, l} → R2 be an embedding which we refer to as the marked bands. A ribbon braid is a map R : ([0, 1] × {0, 1, ⋯, l}) × [0, 1] → R2 such that for any 0 ≤ t ≤ 1, Rt : [0, 1] × {1, ⋯, l} → R2 is an embedding, R0 = I, and there exists σ ∈ Sl such that R1(t) ∣ Ii = Iσ(i)(t) or R1(t)∣Ii = Iσ(i)(1 − t). The usual product of paths defines a group structure on the set of ribbon braids up to homotopy among ribbon braids. This group, denoted by RBl, does not depend on the choice of the marked bands and it is called the ribbon braid group with l bands. A ribbon braid is pure if σ is trivial and we define PRBl to be the pure ribbon braid group with l bands. If we further assume R1(t) ∣ Ii = Iσ(i)(t), this subgroup is called the oriented ribbon braid group RBl+. Similarly, we have the oriented pure ribbon braid group PRBl+. Note that $RB\_l \cong {{\mathbb {Z}}}^l\rtimes B\_l$ where the action of the braid group Bl with l strings is induced by the symmetric group action on the coordinates of Zl. In particular, for the pure ribbon braid group PRBl, we have PRBl ≅ Zl × PBl, where PBl is the pure braid group with l strings. Under this isomorphism, $RB\_l^+ \cong (2{{\mathbb {Z}}})^l\rtimes B\_l$ and PRBl+ ≅ (2Z)l × PBl. A ribbon braided paired (d, r)-forest diagram is a triple (F−, r, F+) consisting of two (d, r)-forests F− and F+ both with l leaves for some l and a ribbon braid r ∈ RBl connecting the leaves of F− to the leaves of F+. The expansion and reduction rules for the ribbon braids just come from the natural way of splitting a ribbon band into d components and the inverse operation to this. See Figure [fig:Splittingribb] for how to split a half twisted band when d = 2. Note that not only are the two bands themselves twisted but the bands are also braided. Everything else will be the same as in the braided case, so we omit the details here. As usual, we define two ribbon braided paired forest diagrams to be equivalent if one is obtained from the other by a sequence of reductions or expansions. The multiplication operation \* on the equivalence classes is defined the same way as for bVd, r. We direct the reader to [line width=1pt,scale=1.2] (0,0) – (1,0) (0,-2) – (1,-2) (0,0) to [out=-90, in=90] (1,-2) (1,0) to [out=-90, in=90] (0,-2); (0,-0.1) to [out=-90, in=90] (1,-1.9); (0,0) to [out=-90, in=90] (1,-2); (3,-1) -> (4,-1); [xshift=6.5cm] (0,0) – (0.3,0) (0.7,0) – (1,0) (0,-2) – (0.3,-2) (0.7,-2) – (1,-2); (0.3,0) to [out=-90, in=90] (0.1,-0.7) (1,0) to [out=-90, in=60] (0.6,-.7); (0,-.1) to [out=-90, in=110] (0.4,-0.7) (.7,-.1) to [out=-90, in=80] (0.9,-.7) ; (0,0) to [out=-90, in=120] (0.4,-0.7) (0.7,0) to [out=-90, in=90] (0.9,-.7) ; (0.9,-0.7) to [out=-120, in=90] (0.3,-2) (0.6,-0.7) to [out=-120, in=90] (0,-2); (0.1,-0.7) to [out=-90, in=90] (0.7,-2) (0.4,-0.7) to [out=-60, in=90] (1,-2) ; (0.1,-0.7) to [out=-90, in=90] (0.7,-2) (0.4,-0.7) to [out=-60, in=90] (1,-2); [fig:Splittingribb] The ribbon Higman–Thompson group RVd, r (resp. the oriented ribbon Higman–Thompson group RVd, r+) is the group of equivalence classes of (resp. oriented) ribbon braided paired (d, r)-forests diagrams with the multiplication \* . Asymptotic mapping class groups related to the ribbon Higman–Thompson groups ============================================================================ The purpose of this section is to generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman–Thompson groups. In particular, we will build a geometric model for the ribbon Higman–Thompson groups which will be crucial for proving homological stability in Section [sec:homstab]. Our construction is largely based on the ideas in and. d-rigid structure ------------------- In this subsection, we generalize the notion of a rigid structure to that of a d-rigid structure. A *d-leg pants* is a surface which is homeomorphic to a (d + 1)-holed sphere. Recall that the usual pair of pants is a 2-leg pants. We will draw a d-leg pants with one boundary component at the top. In this way, we can conveniently put a counter-clockwise total order on the boundary components, making the top component the minimal one. See Figure [fig-inf-surf-seam] for an example of a 3-leg pants. We proceed to build some infinite type surfaces using some basic building blocks. [defn-inf-surf] Let Σ be an compact oriented surface. Call the boundary components of Σ the based boundary components. Then Σd, r∞ is the infinite surface, built up as an inductive limit of infinite surfaces Σd, r, m with m ≥ 0: 1. Σd, r, 0 is obtained from Σ by deleting the interior of a disk in Σ. When Σ is a disk D, we declare Dd, r, 0 = ∂D. 2. Σd, r, 1 is obtained from Σd, r, 0 attaching a copy of r-leg pants along the newly created boundary of Σd, r, 0. 3. For m ≥ 1, Σd, r, m + 1 is obtained from Σd, r, m by gluing a pair of d-leg pants to every nonbased boundary circle of Σd, r, m along the top boundary of the pants. The surface Σd, r, 1 is called the base of Σd, r∞ and the boundary components of Σd, r∞ coming from the base are the based boundary components. For each m ≥ 1, the nonbased boundary components of Σd, r, m naturally embed in Σd, r∞ and we call these the admissible loops. We call the admissible loops coming from Σd, r, 1 the rooted loops. The surface Σd, r∞ has a natural induced orientation. The two indices d, r in the definition of Σd, r∞ will be used later to define the Higman–Thompson version of the asymptotic mapping class group (see Definition [groupB]), where d is related to the valance of the rooted trees and r is the number of roots in the definition of the Higman–Thompson groups. To define our d-rigid structure, we do not really need Σd, r, 0. But it will be convenient to have Σd, r, 0 later in Definition [defn-forest] for defining the map from Σd, r∞ to the tree Td, r. [defn-add-subsur] A compact subsurface A ⊂ Σd, r∞ is admissible if Σd, r, 1 ⊆ A and all of its nonbased boundaries are admissible. The subsurfaces Σd, r, m are called the standard admissible subsurfaces of Σd, r∞. [rem-emb-model-disk] In the special case where the starting surface is a disk, we will use the notation Σ = D, Σd, r, m = Dd, r, m, and Σd, r∞ = Dd, r∞. See Figure [fig-inf-surf-seam] for a picture of the surface D3, 2∞. In this case, we can think of Dd, r∞ as a subsurface of a disk D. More specifically, let D = {(x, y) ∣ x2 + y2 ≤ 1} and $x\_i= \frac{2i-r-1}{r+1}$, 1 ≤ i ≤ r. We place r disks with center at each (xi, 0) of radius $r\_0=\frac{1}{4(r+1)}$. Denote these disks by D1, ⋯Dr. The complement of the interior of these r disks in D is homeomorphic to the r-leg pants Dd, r, 1. Now for each disk Di, 1 ≤ i ≤ r, we can equally distribute d points in the x-axis inside Di and place a disk with radius $\frac{r\_0}{d^2}$ centered at each. We have the complement of the interior of these d disks in Di are all d-leg pants. We can continue the process inductively. At the end, the disks converge to a Cantor set which we denote by C. In particular Dd, r∞ is homeomorphic to D \ C. We will refer to this as the puncture model for Dd, r∞. See Figure [disk-model-inf-surface] for a picture of D3, 2∞ with this model. The advantage of this model is we can view Dd, r∞ and all its admissible subsurfaces directly as a subsurfaces of D. [rem-emb-model-gen] Now Σd, r∞ can be obtained from Σ by attaching a copy of Dd, r∞ to the nonbased boundary component of Σd, r, 0. In particular, Σd, r∞ is obtained from Σ by deleting a copy of the Cantor set, and any admissible subsurface of Σd, r∞ can be viewed directly as a subsurface of Σ using the puncture model. Recall that any two Cantor sets are homeomorphic, hence, by the classification of infinite surfaces, we have Σd, r∞ is homeomorphic to Σ \ C where C is the standard ternary Cantor set sitting inside some disk in Σ regardless of the choice of d and r. [fig-inf-surf-seam] [disk-model-inf-surface] 1. A suited d-pants decomposition of the infinite surface Σd, r∞ is a maximal collection of distinct nontrivial simple closed curves in the interior of Σd, r∞ \ Σd, r, 1 which are not isotopic to the boundary, pairwise disjoint and pairwise non-isotopic, with the additional property that the complementary regions in Σd, r∞ \ Σd, r, 1 are all d-leg pants. 2. A *d-rigid structure* on Σd, r∞ consists of two pieces of data: * a suited d-pants decomposition, and * a *d-prerigid structure, i.e. a countable collection of disjoint line segments embedded into Σd, r∞ \ Σd, r, 1, such that the complement of their union in each component of Σd, r∞ \ Σd, r, 1 has 2 connected components.These pieces must be compatible* in the following sense: first, the traces of the d-prerigid structure on each d-leg pants (i.e. the intersections with pants) are made up of d + 1 connected components, called seams; secondly, each boundary component of the pants intersects with exactly two components of the seams at two distinct points; thirdly, the seams cut each pants into two components. Note that these conditions imply that each component is homeomorphic to a disk. One says then that the suited d-pants decomposition and the d-prerigid structure are subordinate to the d-rigid structure. 3. By construction, Σd, r∞ is naturally equipped with a suited d-pants decomposition, which will be referred to below as the canonical suited d-pants decomposition. We also fix a d-prerigid structure on Σd, r∞ (called the canonical d-prerigid structure) compatible with the canonical suited d-pants decomposition. See Figure [fig-inf-surf-seam]. Using the puncture model, the seams of the canonical d-prerigid structure are just the intersections of [ − 1, 1] × {0} with each d-pants. The resulting d-rigid structure is called the canonical d-rigid structure on Σd, r∞. Very importantly, for each admissible subsurface, the canonical d-rigid structure induces an order on the admissible boundaries. In Figure [fig-inf-surf-seam], the induced order on the admissible loops are counterclockwise. Using the puncture model, the admissible loops are ordered from left to right. 4. The seams cut each component of Σd, r∞ \ Σd, r, 1 into two pieces, we choose the front piece in each component and these r pieces together form the visible side of Σd, r∞. 5. A suited d-pants decomposition (resp. d-(pre)rigid structure) is asymptotically trivial if outside a compact subsurface of Σd, r∞, it coincides with the canonical suited d-pants decomposition (resp. canonical d-(pre)rigid structure). [rem-comp-disk] It is important that the seams cut each d-pants into two components and each component is homeomorphic to a disk as the mapping class group of a disk is trivial. [defn-asym-rgd] Let Σd, r∞ and $ \bar{\Sigma}^{\infty}\_{d,r'}$ be two surfaces with d-rigid structure and let $\varphi: \Sigma^{\infty}\_{d,r} \to \bar{\Sigma}^{\infty}\_{d,r'}$ be a homeomorphism. One says that φ is asymptotically rigid if there exists an admissible subsurface A ⊂ Σd, r∞ such that: 1. φ(A) is also admissible in $\bar{\Sigma}^{\infty}\_{d,r'}$, 2. φ ∣ A maps the based boundaries to based boundaries, admissible loops to admissible loops and 3. the restriction of $\varphi: \Sigma^{\infty}\_{d,r} \setminus A\to \bar{\Sigma}^{\infty}\_{d,r'}\setminus\varphi(A)$ is rigid, meaning that it respects the traces of the canonical d-rigid structure, mapping the suited d-pants decomposition into the suited d-pants decomposition, the seams into the seams, and the visible side into the visible side. If we drop the condition that φ should map the visible side into the visible side, φ is called asymptotically quasi-rigid. The surface A is called a support for φ. We are not using the word ``support" in the usual sense, as the map outside the support defined above might well not being the identity, but the map is uniquely determined up to isotopy by Remark [rem-comp-disk]. In, they do not actually require that the support must contain the base. This will not make a difference, as one can always enlarge the support so that it contains the base. [rem-iso-rgs] The surface Σd, r + d − 1∞ can be identified with the surface Σd, r∞ such that Σd, r + d − 1, m = Σd, r, m + 1 for any m ≥ 1, and the d-rigid structure of Σd, r∞ coincides with d-rigid on Σd, r + d − 1∞ outside Σd, r, 2. In this way, Σd, r∞ is asymptotically rigid homeomorphic to Σd, r + d − 1∞ through the identity map. [induced-asy-rig-str] Let Σʹ be a subsurface of Σd, r∞ such that there exist an admissible subsurface A of Σd, r∞ satisfying: 1. A ∩ Σʹ is a compact surface, 2. The boundaries of Σʹ are disjoint from the admissible boundary components of A. 3. If an admissible boundary component L of A is contained in Σʹ, then the punctured disk component of Σd, r∞ cutting along L is also contained in Σʹ. Then Σʹ has a naturally induced d-rigid structure. In fact, we can take A ∩ Σʹ to be the base surface and d-rigid structure can simply be inherited from Σd, r∞. One, of course, can choose different A here which may give different induced d-rigid structure, but it is unique up to asymptotically rigid homeomorphism. Asymptotic mapping class groups surjecting to Higman–Thompson groups -------------------------------------------------------------------- Given a (possibly noncompact) surface Σ, recall the mapping class group of Σ is defined to be the group of isotopy classes of orientation preserving homeomorphisms of Σ that fixes ∂Σ pointwise, i.e. Map(Σ) = Map(Σ, ∂Σ) : = Homeo+(Σ, ∂Σ)/Homeo0(Σ, ∂Σ). With this, we can now define the asymptotic mapping class group and the half-twist asymptotic mapping class group. [groupB] The asymptotic mapping class group BVd, r(Σ) (resp. the half-twist asymptotic mapping class group HVd, r(Σ)) is the subgroup of Map(Σd, r∞) consisting of isotopy classes of asymptotically rigid (resp. quasi-rigid) self-homeomorphisms of Σd, r∞. When Σ is the disk, we sometimes simply denote the group by BVd, r (resp. HVd, r). Let A be an admissible subsurface of Σd, r∞, and Map(A) be its mapping class group which fixes the each boundary component pointwise. Each inclusion A ⊆ Aʹ of admissible surfaces induces an injective embedding jA, Aʹ : Map(A) → Map(Aʹ). The collection forms a direct system whose direct limit we call the compactly supported pure mapping class group, denoted by PMapc(Σd, r∞). The group PMapc(Σd, r∞) is naturally a subgroup of BVd, r(Σ) and we denote the inclusion map by j. [defn-forest] Let Fd, r be the forest with r copies of a rooted d-ary tree and Td, r be the rooted tree obtained from Fd, r by adding an extra vertex to Fd, r and r extra edges each connecting this vertex to a root of a tree in Fd, r. There is a natural projection q : Σd, r∞ → Td, r, such that the pullback of the root is Σd, r, 0 and the pull back of the midpoints of any edges are admissible loops. Now any element in BVd, r(Σ) can be represented by an asymptotically rigid homeomorphism φ : Σd, r∞ → Σd, r∞. In particular we have an admissible subsurface A of Σd, r∞ such that φ∣A : (A, ∂bA) → (φ(A), φ(∂bA)) is a homeomorphism. Let F− be the smallest subforest of Fd, r which contains q(A) ∩ Fd, r, and F+ be the smallest subforest of Fd, r which contains q(φ(A)) ∩ Fd, r. Note that F− and F+ have the same number of leaves and their leaves are in one-to-one correspondence with the admissible loops of A and φ(A). Now let ρ be the map from leaves of F− to F+ induced by φ. Together this defines an element [(F−, ρ, F+)] ∈ Vd, r. We call this map π. One can show π is well defined. Similarly to and, we now have the following proposition. [thm-ses-asm] We have the short exact sequences: $$1 \to \operatorname{PMap}\_c(\Sigma\_{d,r}^\infty) \xrightarrow{j} {{\mathcal {B}}}V\_{d,r}(\Sigma) \xrightarrow{\pi} V\_{d,r} \to 1;$$ $$1 \to \operatorname{PMap}\_c(\Sigma\_{d,r}^\infty) \xrightarrow{j} {{\mathcal {H}}}V\_{d,r}(\Sigma) \xrightarrow{\pi} V\_{d,r}({{\mathbb {Z}}}/2{{\mathbb {Z}}}) \to 1.$$ Here, as in, Vd, r(Z/2Z) is the twisted version of the Higman–Thompson group where one allows flipping the subtree below every leaf. See for example for more information. We will prove the proposition for BVd, r(Σ). The other case is essentially the same. First we show the map π is surjective. Given any element [(F−, ρ, F+)] ∈ Vd, r, let T− (resp. T+) be the tree obtained from F− (resp. F+) by adding a single root on the top and r edges connecting to each root of the trees in F− (resp. F+). Furthermore, let Tʹ− (resp. Tʹ+) be the tree obtained from F− (resp. F+) by throwing away the leaves and the open half edge connecting to the leaves. Then let A− = q− 1(Tʹ+) and A+ = q− 1(Tʹ+). We have A− and A+ are both admissible subsurfaces of Σd, r∞. Now one can produce a homeomorphism φ0 : A− → A+ which is identity on the based boundary and maps the admissible loops of A− to the admissible loops of A+ following the information from ρ, mapping the visible part to the visible part for each admissible loop. From here, we extend φ0 to a map φ : Σd, r∞ → Σd, r∞ such that φ is a asymptotically rigid homeomorphism. If an element g ∈ BVd, r(Σ) is mapped to a trivial element π(g) = [(F−, ρ, F+)] ∈ Vd, r, then the two forests F− and F+ are the same and the induced map ρ is trivial. This means we can assume the support A for the asymptotically rigid homeomorphism φg corresponding to g is the same as φ(A) and φ induces identity map on the admissible boundary components. Thus g ∈ PMapc(Σd, r∞). Finally, given any element g ∈ PMapc(Σd, r∞), it is clear that π ∘ j(g) = 1. The mapping class group Map(Σd, r∞) has a natural quotient topology coming from the compact-open topology on Homeo+(Σd, r∞, ∂Σd, r∞). See for more information. In, Aramayona and Funar showed that when Σ is a closed surface, HV2, 1(Σ) is dense in Map(Σ2, 1∞). We improve their result to the following. [thm-dense-asy] The groups BVd, r(Σ) and HVd, r(Σ) are dense in the mapping class group Map(Σd, r∞). The proof in adapts directly to show that HVd, r(Σ) is dense in Map(Σd, r∞) and so we will not repeat it here. To show BVd, r(Σ) is also dense in Map(Σd, r∞), it suffices to show any element in HVd, r(Σ) can be approximated by a sequence of elements in BVd, r(Σ). Note first that any half Dehn twists around admissible loops in Σd, r∞ lies in HVd, r(Σ). In fact, given an admissible loop α, we can choose an admissible subsurface A such that α is an admissible loop of A. Then the half Dehn twist around α is asymptotic quasi-rigid with support A, in fact, it is identity on all the components of Σd, r∞ \ A except at the component containing α where it rotates 180 degree. Now given an asymptotic quasi-rigid homeomorphism f of Σd, r∞ with support Aʹ, we can first compose f with half Dehn twists around those admissible loops of A where f restricted to the component below them switches the front and back. The composition now is an asymptotic rigid homeomorphism. Thus HVd, r(Σ) can be generated by BVd, r(Σ) and half Dehn twists around the admissible loops in Σd, r∞, it suffices now to show that any half Dehn twists around admissible loops in Σd, r∞ can be approximated by a sequence of elements in BVd, r(Σ). Given an admissible loop L, let hL be a half Dehn twist at L. We will construct a sequence of elements xi ∈ BVd, r(Σ) such that for any compact subset K of Σd, r∞, there exists N such that for any j ≥ N, xj and hL coincide on K. Recall we have the map q : Σd, r∞ → Td, r (cf. Definition [defn-forest]) such that the admissible loops are mapped to edge middle points in Td, r. Now consider those admissible loops such that their image under q lying below q(L) have distance i to q(L). Note that there are di such admissible loops. We list them as Li, 1, ⋯, Li, di. Let hLi, k be the half Dehn twists around Li, k and let xi = hLhLi, 1⋯hLi, di, then xi ∈ BVd, r(Σ) and the sequence {xi} has the desired property. Now recall by Remark [rem-emb-model-gen] that Σd, r∞ is homeomorphic to Σ \ C for any d and r, hence we have our first theorem stated in the introduction. The asymptotic mapping class group of the disk punctured by the Cantor set -------------------------------------------------------------------------- In the last subsection, we want to identify the asymptotic mapping class group BVd, r(D) with the oriented ribbon Higman–Thompson groups RVd, r+ and the half-twist asymptotic mapping class group HVd, r(D) with the ribbon Higman–Thompson group RVd, r. The following lemma appears in without a proof, so we provide the details here. Note that what they call the pure ribbon braid group is the oriented pure ribbon braid group in our definition (see Definition [defn-rb]). [lem-pbraid-ribb] Let Dk be the (k + 1)-holed sphere. Then Map(Dk) can be naturally identified with the pure oriented ribbon braid group PRBk+. Note that Dk can be identified with a disk with k holes. Let ∂b denote the boundary of the disk. Let D̄k be a disk with k punctures obtained from Dk by attaching one punctured disk to each hole. The induced map Cap : Map(Dk) → PMap(D̄k) is the capping homomorphism. Note that PMap(D̄k) ≅ PBk. Now applying and the fact that the Dehn twists around the holes of Dk commute, one sees that the kernel K is a free abelian group of rank k generated by these k Dehn twists. Here the capping homomorphism splits. To prove this, we first embed PBk into PRBk+ by viewing the pure braid group of k-strings as the set of ribbon braids on k bands such that the bands have no twists. We can think of Dk as being embedded into R2 with ∂b as the unit circle and the k holes in Dk equally distributed inside ∂b along the x-axis. The intersections of these holes with the x-axis gives k sub-intervals of the x-axis denoted I1, ⋯, Ik. We now put the bands representing a pure braid x ∈ PBk ≤ PRBk+ in D × [0, 1] which starts and ends at I1, ⋯, Ik. Note that the bands here will not twist at all. Now we comb the bands straight from bottom to top. This induces a homeomorphism of Dk × {0} and hence an element in the mapping class group Map(Dk). One checks that this map is a group homomorphism and injective. Since PBk acts on K trivially, we have Map(Dk) ≅ K × PBk ≅ Zk × PBk ≅ PRBk+ where the number of Dehn twists around each boundary component is naturally identified with the number of full twists on each bands. To promote Lemma [lem-pbraid-ribb] such that it works for the ribbon braid group, we need some extra terminology. As in the proof of Lemma [lem-pbraid-ribb], we identify Dk with the unit disk in R2 with k small disks whose centers are equally distributed on the x-axis removed. The x-axis cuts the boundary loops of each deleted disk into two components, providing a cell structure on the loops. We will call the part that lies above the x-axis the visible part. We define the rigid mapping class group RMap+(Dk) of Dk to be the isotopy classes of homeomorphisms of Dk which fix ∂bDk pointwise and map the visible part of the holes to the visible part of the holes. Note elements in ∂bDk are allowed to map one boundary hole to another just as in the definition of the asymptotic mapping class group. If we only assume the cell structure on the loops has to be preserved, the resulting group is called quasi-rigid mapping class group Dk and denoted by RMap(Dk). With these preparations, the following lemma is now clear. [lem-braid-ribb] There is a natural isomorphism between the oriented ribbon braid group RBk+ and RMap+(Dk) (resp. between the ribbon braid group RBk and RMap(Dk)). As in the proof of Lemma [lem-pbraid-ribb], we put the element in the (oriented) ribbon braid group between D × [0, 1], then we comb the bands straight from bottom to top which gives the corresponding element in RMap+(Dk) (resp. RMap(Dk) ). Given two admissible subsurfaces A and Aʹ of Dd, r∞ (possibly with different r) with k admissible boundary components, we want to fix a canonical way to identify a homeomorphism f : A → Aʹ as an element in the ribbon braid group. Note that each boundary loop except the base one inherits a visible side from Dd, r∞. We will use the puncture model for Dd, r∞ going forward. As above, let Dk be the subsurface of D which is the compliment of k disjoint open disks with centers at $a\_i = \frac{2i-k-1}{k+1} $ of radius 2− k for 1 ≤ i ≤ k. Now given any admissible subsurface Ak of Dd, r∞ with k many admissible boundaries, denote the centers from left to right by ci ∈ [0, 1] × {0}, 1 ≤ i ≤ k with radius r1, r2⋯, rk. Now we define an isotopy NAk : D × [0, 1] → D such that ${{\mathcal {N}}}\_{A\_k,0} ={{{\rm id}}}\_D$ and NAk, 1 maps Ak to Dk via a homeomorphism. We first shrink the admissible boundary loops of Ak such that they have radius r, where r = min{r1, ⋯, rk, 2− k}. Then we isotope Ak by moving the centers ci to ai along [0, 1] × {0} in D. And in the last step we enlarge the radius one by one to 2− k. The following lemma is now immediate. [lem-home-brd] Let ϕ : Dd, r∞ → Dd, r∞ be an asymptotically rigid (resp. quasi-rigid) homeomorphism which is supported on the admissible subsurface Ak. Denote Akʹ = ϕ(Ak), then 1. rϕ = NAkʹ, 1 ∘ ϕ∣Ak ∘ NAk, 1− 1 : Dk → Dk gives an element in the oriented ribbon braid group RBk+ (resp. the ribbon braid group RBk). Conversely, given an element r ∈ RBk+ (resp. RBk), we have an asymptotic rigid (resp. quasi-rigid) homeomorphism which is unique up to isotopy, supported on Ak, and map Ak to Akʹ. 2. let Ak + d be the admissible subsurface of Dd, r∞ obtained from Ak by adding a d-leg pants and ϕ(Ak + d) = Ak + dʹ. Then the associated oriented ribbon braid (resp. the ribbon braid) of ϕ can be obtained from rϕ by splitting the corresponding band into d bands. Conversely, if we split one band of the ribbon braids into d bands, the isotopy class of the corresponding asymptotic rigid (resp. quasi-rigid) homeomorphism does not change. Note that for any d ≥ 2, r ≥ 1, we have an natural embedding ιd, r : Dd, r∞ → Dd, r + 1∞ by mapping the rooted boundaries of Dd, r∞ to the first r rooted boundaries according to the order. This induces an embedding of groups iH, d, r : HVd, r → HVd, r + 1, iB, d, r : BVd, r → BVd, r + 1. On the other hand, we also have natural embeddings iR, d, r : RVd, r → RVd, r + 1 and iR+, d, r : RVd, r+ → RVd, r + 1+ induced by inclusion of roots. We have the following. [thm-iden-asym-ribb] There exist isomorphisms fd, r : HVd, r → RVd, r such that fd, r + 1 ∘ iH, d, r = iH, d, r + 1 ∘ fd, r. Restricting to the subgroups BVd, r, one gets isomorphisms fd, r : BVd, r → RVd, r+ with the same property. Since the two cases are parallel, we will only prove the theorem for BVd, r. We will define two maps fd, r : BVd, r → RVd, r+ and gd, r : RVd, r+ → BVd, r such that $f\_{d,r} \circ g\_{d,r} ={{{\rm id}}}$ and $g\_{d,r}\circ f\_{d,r} = {{{\rm id}}}$. Given an element x ∈ BVd, r, we can define fd, r as follows. Let φx be an asymptotically rigid homeomorphism of Dd, r∞ representing x with support Ak, where k is the number of admissible loops. By Proposition [thm-ses-asm], π(x) provides an element [F−, σ, F+] in the Higman–Thompson group Vd, r, where F− and F+ are (d, r)-forests with k leaves. But what we want is a ribbon braid connecting the k leaves. For this we simply apply Lemma [lem-home-brd] (1) to the map φx with support Ak, denote the corresponding element in RBk+ by rφx. We define fd, r(x) = [F−, rφx, F+]. Now given y ∈ RVd, r+, one can define an element in BVd, r as follows. Suppose (F−, ry, F+) is a representative for y, where F− and F+ are (d, r)-forests and r is a ribbon braid between the leaves of F− and F+. Add a root to F− (resp. F+) with an edge connecting to the root of each tree in F− (resp. F+) and then throw away the open half edge connecting to the leaves. Denote the resulting tree by T− (resp. T+). Now q− 1(T−), q− 1(T+) gives us two admissible subsurfaces Ak, Akʹ in Dd, r∞ where k is the number of leaves for F−. And by Lemma [lem-home-brd] (1), the ribbon element ry in RBk+ give us an asymptotic rigid homeomorphism ψy with support Ak and maps Ak to Akʹ. Now one can check that $f\_{d,r} \circ g\_{d,r} ={{{\rm id}}}$ and $g\_{d,r}\circ f\_{d,r} = {{{\rm id}}}$. Therefore, the two groups are isomorphic. The fact that the diagram commutes is immediate from the definition. Homological stablity of ribbon Higman–Thompson groups ===================================================== In this section, we show the homological stability for oriented ribbon Higman–Thompson groups and explain at the end how the same proof applies to the ribbon Higman–Thompson groups. Homogeneous categories and homological stability ------------------------------------------------ In this subsection, we review the basics of homogeneous categories and refer the reader to for more details. Note that we adopt their convention of identifying objects of a category with their identity morphisms. [] A monoidal category (C, ⊕ , 0) is called homogeneous if 0 is initial in C and if the following two properties hold. H1 Hom(A, B) is a transitive Aut(B)-set under postcomposition. H2 The map Aut(A) → Aut(A ⊕ B) taking f to $f \oplus {{{\rm id}}}\_B$ is injective with image $$\operatorname{Fix}(B) := \{ \phi\in {{\mathrm{Aut}}}(A\oplus B) \mid \phi\circ (\imath\_A\oplus {{{\rm id}}}\_B) = \imath\_A \oplus {{{\rm id}}}\_B\}$$ where $\imath\_A\colon 0 \to A$ is the unique map. [] Let (C, ⊕ , 0) be a monoidal category with 0 initial. We say that C is prebraided if its underlying groupoid is braided and for each pair of objects A and B in C, the groupoid braiding bA, B : A ⊕ B → B ⊕ A satisfies $$b\_{A,B} \circ (A\oplus \imath\_B) = \imath\_B\oplus A : A\to B\oplus A.$$ Let (C, ⊕ , 0) be a monoidal category with 0 initial and (A, X) a pair of objects in C. Define Wn(A, X)• to be the semi-simplicial set with set of p-simplices Wn(A, X)p : = HomC(X⊕ p + 1, A ⊕ X⊕ n) and with face map di : HomC(X⊕ p + 1, A ⊕ X⊕ n) → HomC(X⊕ p, A ⊕ X⊕ n) defined by precomposing with $X^{\oplus i} \oplus \imath\_X \oplus X^{\oplus p-i}$ Also call the category C satifies LH3 at a pair of objects (A, X) with slope k ≥ 2 if: LH3[cond:H3] For all n* ≥ 1, Wn(A, X)• is $(\frac{n-2}{k})$-connected.* Quite often, we can reduce the semi-simplicial complex to a simplicial complex. [] [defn:Sn(X,A)] Let A, X be objects of a homogeneous category (C, ⊕ , 0). For n ≥ 1, let Sn(A, X) denote the simplicial complex whose vertices are the maps f: X → A ⊕ X⊕ n and whose p-simplices are (p + 1)-sets {f0, …, fp} such that there exists a morphism f: X⊕ p + 1 → A ⊕ X⊕ n with f ∘ ij = fj for some order on the set, where $$i\_j = \imath\_{X^{\oplus j}} \oplus {{{\rm id}}}\_X \oplus \imath\_{X^{\oplus p-j}} \colon X = 0 \oplus X \oplus 0 \longrightarrow X^{\oplus p+1}.$$ Let Aut(A ⊕ X⊕ ∞) be the colimit of $$\cdots \xrightarrow[]{-\oplus X} Aut(A\oplus X^{\oplus n}) \xrightarrow[]{-\oplus X} Aut(A\oplus X^{\oplus n+1}) \xrightarrow[]{-\oplus X} Aut(A\oplus X^{\oplus n+2})\xrightarrow[]{-\oplus X}\cdots$$ Then any Aut(A ⊕ X⊕ ∞)-module M may be considered as an Aut(A ⊕ X⊕ n)-module for any n, by restriction, which we continue to call M. We say that the module M is abelian if the action of Aut(A ⊕ X⊕ ∞) on M factors through the abelianizations of Aut(A ⊕ X⊕ ∞), or in other words if the derived subgroup of Aut(A ⊕ X⊕ ∞) acts trivially on M. We are now ready to quote the theorem that we will use. [] [thm:hom stab] Let (C, ⊕ , 0) be a pre-braided homogeneous category satisfying [cond:H3] for a pair (A, X) with slope k ≥ 3. Then for any abelian Aut(A ⊕ X⊕ ∞)-module M the map Hi(Aut(A ⊕ X⊕ n); M) → Hi(Aut(A ⊕ X⊕ n + 1); M) induced by the natural inclusion map is surjective if $i \leq \frac{n-k+2}{k}$, and injective if $i \leq \frac{n-k}{k}$. Homogeneous category for the groups RVd, r+ --------------------------------------------------- The purpose of this section is to produce a homogeneous category for proving homological stability of the ribbon Higman–Thompson groups RVd, r+. Note that by Theorem [thm-iden-asym-ribb], it is the same as proving the asymptotic mapping class groups BVd, r have homological stability. This allows us to define our homogeneous category geometrically. The category is similar to the ones produced in. Essentially, we replace the annulus or Möbius band by the infinite surface Dd, 1∞. Recall Dd, r∞ is an infinite surface equipped with a canonical asymptotic rigid structure with boundary component denoted ∂bDd, r∞. Let I = [ − 1, 1] ⊂ ∂bDd, r∞ be an embedded interval as in Figure [fig-braid-monoid](a). Let I− = [ − 1, 0] and I+ = [0, 1] be subintervals of I. Let Dd, 1∞ ⊕ Dd, 1∞ be the boundary sum of two copies of Dd, 1∞ obtained by identifying I+ of the first copy with I− of the second copy. Inductively, we could define similarly ⊕ rDd, 1∞ for any r ≥ 0. Here ⊕ 0Dd, 1∞ is just the standard disk D. Abusing notation, when referring to I− and I+ on ⊕ rDd, 1∞, we will mean the two copies of I− and I+ which remain on the boundary. Thus we have an operation ⊕ on the set ⊕ rDd, 1∞ for any r ≥ 0. See Figure [fig-braid-monoid](b) for a picture of ( ⊕ 2Dd, 1∞) ⊕ ( ⊕ 3Dd, 1∞). In fact, we have (( ⊕ rDd, 1∞), ⊕ ) is the free monoid generated by Dd, 1∞. Note that ⊕ rDd, 1∞ has a naturally induced d-rigid structure and we can identify it with Dd, r∞, which will be of use to us later. We can now define the category Gd to be the monoidal category with objects ⊕ rDd, 1∞, r ≥ 0, ⊕ as the operation, and D as the 0 object. So far it is the same as defining the objects as the natural numbers and addition as the operation. When r = s, we define the morphisms Hom( ⊕ rDd, 1∞, ⊕ sDd, 1∞) = BVd, r which is the group of isotopy classes of asymptotically rigid homeomorphisms of Dd, r∞; when r ≠ s, let Hom( ⊕ rDd, 1∞, ⊕ sDd, 1∞) = ∅. Note that we did not universally define the morphisms to be the sets of isotopy classes of asymptotically rigid homeomorphisms as we want our category to satisfy cancellation, i.e., if A ⊕ C = A then C = 0, see for more information. The category Gd has a natural braiding as in the usual braid group case, see Figure [fig-braid-monoid](c). [fig-braid-monoid] Now applying, we have a pre-braided homogeneous category UGd. The category UGd has the same objects as Gd and morphisms defined as following: For any s ≤ r, a morphism in Hom( ⊕ sDd, 1∞, ⊕ rDd, 1∞) is an equivalence class of pairs ( ⊕ r − sDd, 1∞, f) where f : ( ⊕ r − sDd, 1∞) ⊕ ( ⊕ sDd, 1∞) → ⊕ rDd, 1∞ is a morphism in Gd and ( ⊕ r − sDd, 1∞, f) ∼ ( ⊕ r − sDd, 1∞, fʹ) if there exists an isomorphism g : ⊕ r − sDd, 1∞ → ⊕ r − sDd, 1∞ ∈ Gd making the diagram commute up to isotopy. $$\begin{tikzcd} (\oplus\_{r-s} D^{\infty}\_{d,1}) \oplus (\oplus\_s D^{\infty}\_{d,1}) \arrow[d,"g~\oplus~ {{{\rm id}}}\_{\oplus\_s D^{\infty}\_{d,1}}"'] \arrow[r, "f"] & \oplus\_r D^{\infty}\_{d,1} \\ (\oplus\_{r-s} D^{\infty}\_{d,1}) \oplus (\oplus\_s D^{\infty}\_{d,1}) \arrow[ur, "f'"'] & \end{tikzcd}$$ We write [ ⊕ r − sDd, 1∞, f] for such an equivalence class. Now by Theorem [thm:hom stab], to prove homological stability for the oriented ribbon Higman–Thompson groups, we only need to verify that the category Gd satisfies Condition [cond:H3] at the pair (D, Dd, 1∞). In fact, by Theorem [thm-iden-asym-ribb], proving the oriented ribbon Higman–Thompson groups satisfy homological stability is the same as proving that the the asymptotic mapping class groups BVd, r satisfy homological stability. Now consider the family of groups $${{\mathrm{Aut}}}(A\oplus X) \hookrightarrow {{\mathrm{Aut}}}(A\oplus X^{\oplus 2}) \hookrightarrow {{\mathrm{Aut}}}(A\oplus X^{\oplus 2})\hookrightarrow \cdots \hookrightarrow {{\mathrm{Aut}}}(A\oplus X^{\oplus n}) \hookrightarrow \cdots$$ Where A = D and X = Dd, 1∞. By definition, this gives rise to the family of groups ${{\mathcal {B}}}V\_{d,1} \hookrightarrow {{\mathcal {B}}}V\_{d,2} \hookrightarrow \cdots \hookrightarrow {{\mathcal {B}}}V\_{d,n} \hookrightarrow \cdots$. Now we have shown that the category (Gd, ⊕ , D) is a prebraided homogenuous category, so by Theorem [thm:hom stab], it suffices to verify Condition [cond:H3] at the pair (D, Dd, 1∞) to prove our homological stablity result. As a matter of fact, we will show that Wr(D, Dd, 1∞)• is (r − 3)-connected in the next subsection. First, let us further characterize the morphisms in UGd. Call 0 = I− ∩ I+ the basepoint of ⊕ rDd, 1∞. [defn-qr-emb] Given s < r, an injective map φ : ( ⊕ sDd, 1∞, I+) → ( ⊕ rDd, 1∞, I+) is called an asymptotically rigid embedding if it satisfies the following properties: 1. φ(∂Dd, s∞) ∩ ∂Dd, r∞ = I+. 2. φ maps ⊕ sDd, 1∞ homeomorphically to φ( ⊕ sDd, 1∞) and there exists an admissible surface A ⊂ ⊕ sDd, 1∞ such that φ : ⊕ sDd, 1∞ \ A → φ( ⊕ sDd, 1∞) \ φ(A) is rigid. 3. [defn-qr-emb-cplt] The closure of the complement of φ( ⊕ sDd, 1∞) in ⊕ rDd, 1∞ with its induced d-rigid structure is asymptotically rigidly homeomorphic to ⊕ r − sDd, 1∞. For s < r, the equivalence classes of pairs [ ⊕ r − sDd, 1∞, f] are in one-to-one correspondence with the isotopy classes of asymptotically rigid embeddings of ( ⊕ sDd, 1∞, I+) into ( ⊕ rDd, 1∞, I+). Here isotopies are carried out among asymptotically rigid embeddings. Given an equivalence class of a pair [ ⊕ t − sDd, 1∞, f], we have the restriction map f ∣ ⊕ sDd, 1∞ is an asymptotically rigid embedding. Any two equivalence classes of pairs will induce the same map f ∣ ⊕ sDd, 1∞, hence we have a well-defined map from the set of equivalence pairs to the set of isotopy classes of asymptotically rigid embeddings. We produce the inverse of the restriction map as follows. If we have an asymptotically rigid embedding φ : ( ⊕ sDd, 1∞, I+) → ( ⊕ rDd, 1∞, I+), by part [defn-qr-emb-cplt] of Definition [defn-qr-emb], we also have an asymptotically rigid homeomorphism ϕ : C → ⊕ r − sDd, 1∞ where C is the closure of the complement of φ( ⊕ sDd, 1∞) in ⊕ rDd, 1∞. Up to isotopy, we can assume ϕ− 1 ∣ I+ coincides with φ ∣ I−. Now define a map f̄ : ( ⊕ r − sDd, 1∞) ⊕ ( ⊕ sDd, 1∞) → ⊕ rDd, 1∞ by f̄ ∣ ⊕ r − sDd, 1∞ = ϕ− 1 and f̄ ∣ ⊕ sDd, 1∞ = φ. One can check that f̄ is an asymptotically rigid homeomorphism. Then $(\oplus\_{r-s} D^{\infty}\_{d,1},\bar f)$ gives a representative of an equivalence class of pairs. Higher connectivity of the complex Wr(D, Dd, 1∞)• ----------------------------------------------------------- We want to prove that the complex Wr(D, Dd, 1∞)• is highly connected, see the diagram on Figure [fig:sumrelations] and the paragraph preceeding it for an outline of our general strategy. [rem:firstchange] As explained in the proof of, a simplex of Sr(D, Dd, 1∞) has a canonical ordering on its vertices induced by the local orientation of the surfaces near the parameterized interval in their based boundary. Thus the geometric realization ∣Wr(D, Dd, 1∞)•∣ is homeomorphic to Sr(D, Dd, 1∞). Our first step now is to simplify the complex Sr(D, Dd, 1∞) further. Given r ≥ 2, we call a loop α : (I, ∂I) = ([0, 1], {0, 1}) → ( ⊕ rDd, 1∞, 0) an asymptotically rigidly embedded loop if there exists an asymptotically rigid embedding φ : (Dd, 1∞, I+) → ( ⊕ rDd, 1∞, I+) with φ ∣ (∂Dd, 1∞, 0) = α up to based isotopy. See Figure [ad-loop] for a picture. When r = 1, we just call a loop asymptotically rigidly embedded if it is isotopic to the boundary. [ad-loop] [lem-cha-asre] When r ≥ 2, a loop α : (I, ∂I) → ( ⊕ rDd, 1∞, 0) is isotopic to an asymptotically rigidly embedded loop if and only if there exists an admissible surface A ⊆ ⊕ rDd, 1∞ such that the admissible loops of A are disjoint from α, the number of admissible loops of A that lie in the disk bounded by α is 1 + a(d − 1) for some a ≥ 0 and there exist some admissible loops which do not lie inside the disk bounded by α up to isotopy. It is clear that a loop which is isotopic to an asymptotically rigidly embedded loop has the properties given in the lemma. For the other direction, we can assume up to isotopy that α(I) ∩ ∂( ⊕ rDd, 1∞) = I+. We know that Dd, r∞ is asymptotically rigidly homeomorphic to Dd, r + d − 1∞, thus the surface bounded by the loop α is asymptotically rigidly homeomorphic to Dd, 1∞. Therefore, the number of the boundary components bounded by the complement disk is $r-1\mod d-1$ and thus it is asymptotically rigidly homeomorphic to Dd, r − 1∞. These two facts together imply α is an asymptotically rigidly embedded loop. Now we define the complex Ur(D, Dd, 1∞) which is the surface version of the complex Ur given in. For r ≥ 1, let Ur(D, Dd, 1∞) denote the simplicial complex whose vertices are isotopy classes of asymptotically rigidly embedded loops and a set of vertices α0, ⋯, αp forms a p-simplex if and only if any corresponding asymptotically rigid embeddings ϕ0, ⋯, ϕp form a p-simplex in Sr(D, Dd, 1∞). We denote the canonical map from Sr(D, Dd, 1∞) to Ur(D, Dd, 1∞) by π. The following lemma follows directly from the definition. In fact, given a set of vertices α0, ⋯, αp, if they form a p-simplex, then any corresponding asymptotically rigid embeddings ϕ0, ⋯, ϕp form a p-simplex in Sr(D, Dd, 1∞). But this means for any ψi ∈ π− 1(αi), 0 ≤ i ≤ p, the collection ψ0, ϕ1, ⋯, ψp forms a p-simplex. [lem-cpt-jn] The map π is a complete join. Now by Proposition [prop-join-conn], we only need to show that Ur(D, Dd, 1∞) is highly connected. Similar to, we will produce several other complexes closely related to Ur(D, Dd, 1∞). We first have the following complex which is analogous to the complex Ur∞ in. Let Ur∞(D, Dd, 1∞) be the simplicial complex with vertices given by asymptotically rigidly embedded loops in ⊕ rDd, 1∞ where α0, α1, ⋯, αp form a p-simplex if the punctured disks bounded by them are pairwise disjoint (outside of the based point) and there exists at least one admissible loop that does not lie in those disks. [rem-Un] 1. The (r − 2)-skeleton of Ur∞(D, Dd, 1∞) is the same as that of Ur(D, Dd, 1∞). Notice though that Ur∞(D, Dd, 1∞) is in fact infinite dimensional. 2. Since ⊕ rDd, 1∞ is asymptotically rigidly homeomorphic to ⊕ r + d − 1Dd, 1∞, we have Ur∞(D, Dd, 1∞) is isomorphic to Ur + d − 1∞(D, Dd, 1∞) as a simplicial complex. We also need the another complex which is the surface version of the complex Tr∞ given in. For convenience, we will orient the admissible loops in ⊕ rDd, 1∞ such that they bound the punctured disk according to the orientation. An almost admissible loop is a loop α : (I, ∂I) → ( ⊕ rDd, 1∞, 0) which is freely isotopic to one of the nonbased admissible loops. Note that by Lemma [lem-cha-asre], an almost admissible loop is an asymptotically rigidly embedded loop. [defn-subcpxT] Define the simplicial complex Tr∞(D, Dd, 1∞) to be the full subcomplex of Ur∞(D, Dd, 1∞) such that all its vertices are almost admissible loops. Just as discussed in Remark [rem-Un], we have Tr∞(D, Dd, 1∞) is in fact isomorphic to Tr + d − 1∞(D, Dd, 1∞) as a simplicial complex. We now want to further characterise the almost admissible loops by building a connection to the usual arc complex. We let A be the quotient [0, 2]/1 ∼ 2. This corresponds to identifying the endpoint 1 of the interval [0, 1] with the base point 1 of the circle given by [1, 2]/1 ∼ 2. An injective continuous map L : (A, 0) → (Dd, r∞, 0) is called a lollipop on the surface Dd, r∞ if α ∣ [1, 2] is isotopic to an admissible loop in Dd, r∞ and L ∣ [0, 1] is an arc connecting the base point 0 to the loop L([1, 2]). The map L ∣ [0, 1] is called the arc part of the lollipop L and L ∣ [1, 2] is called the loop part. See Figure [jumping] where the blue curve is a lollipop. Lollipops are examples of what Hatcher-Vogtmann refer to as tethered curves. [lem:loop-arc] The set of isotopy classes of almost admissible loops is in one-to-one correspondence with the set of isotopy classes of lollipops. We define a map g from the isotopy classes of lollipops to the isotopy classes of almost admissible loops and show that the map is bijiective. Given a lollipop L : (A, 0) → (Dd, r∞, 0), we can map it to an almost admissible loop α : [0, 1] → (Dd, r∞, 0) as follows. We define α(0) = 0 and let α(t) run parallel to L outside the region bounded by L. The orientation of α is simply the one coincides with the loop part of L. Since α can be freely homotoped to the admissible loop L ∣ [1, 2], we have α is almost admissible. Any isotopy of L induces an isotopy of α, hence the map is well-defined. Now we show g is surjective. Given any almost admissible loop α : [0, 1] → (Dd, r∞, 0), let A be the admissible loop which is freely isotopic to α. Up to isotopy, we can assume that A lies in the interior of the surface bounded by α. Then the surface bounded by α and A must be an annulus. From here one can produce an arc connecting the base point 0 to a point in A. Together with A, this provides the lollipop. Finally, we argue that g is injective. Suppose L1 and L2 are two lollipops such that g(L1) and g(L2) are isotopic, denote the isotopy by f. By the isotopy extension theorem (see for example ) there exists an isotopy F : Dd, r∞ × [0, 1] → Dd, r∞ such that F ∣ Dd, r∞ × 0 = idDd, r∞ and F ∣ g(L1) × [0, 1] = f. In particular F ∣ Dd, r∞ × 1 maps the almost admissible loop g(L1) to the almost admissible loop g(L2). Hence L1 is isotoped through F to a lollipop which lies in a small neighborhood of L2 and is bounded by the loop g(L2). Therefore, one can then isotope L1 to L2. We now have the following definition of lollipop complex. The lollipop complex Lr∞(D, Dd, 1∞) has vertices as lollipops, and p + 1 lollipops, L0, L1, ⋯, Lp, form a p-simplex if they are pairwise disjoint outside the base point 0 and there exists at least one admissible loop which does not lie inside the disks bounded by the Lis. The following lemma is immediate from Lemma [lem:loop-arc]. [lem-id-T-L] The complex Lr∞(D, Dd, 1∞) is isomorphic to Tr∞(D, Dd, 1∞) as a simplicial complex. [lem-link-loli] Given a p-simplex σ in Lr∞(D, Dd, 1∞), its link Lk(σ) is isomorphic to Lrσ∞(D, Dd, 1∞) for some rσ > 0. By Lemma [lem-id-T-L], we can just prove the lemma for Tr∞(D, Dd, 1∞). Let α0, α1, ⋯, αp be the vertices of σ which are almost admissible loops. Up to isotopy, we can assume they are pairwise disjoint except at the basepoint 0. Now let C be the complement surface of σ, whose based boundary is the concatenation of αp, αp − 1, ⋯, α0 and ∂D. The surface C has a naturally induced d-rigid structure. In particular, C is asymptotically rigidly homeomorphic to Dd, rσ∞ for some rσ > 0. Thus link Lk(σ) is isomorphic to Trσ∞(D, Dd, 1∞). $$\begin{tikzpicture}[scale=0.85,every node/.style={thick,scale=0.85}] \matrix (m) [matrix of math nodes, column sep=1em, row sep=2em, text depth=.5em, text height=1em, ampersand replacement=\&] { \& \& \& U\_r^\infty(D,D^{\infty}\_{d,1}) \& T\_r^\infty(D,D^{\infty}\_{d,1}) \& L\_r^\infty(D,D^{\infty}\_{d,1}) \\ W\_r(D,D^{\infty}\_{d,1})\_\bullet \& S\_r(D,D^{\infty}\_{d,1}) \& U\_r(D,D^{\infty}\_{d,1})\\ \& \& U\_r^{(r-2)}(D,D^{\infty}\_{d,1})\& (U\_r^{\infty}(D,D^{\infty}\_{d,1})^{(r-2)} \\}; \path[->] (m-1-5) edge node[above]{$\supseteq$} (m-1-4) (m-1-5) edge node[above]{$\cong$} (m-1-6) (m-2-2) edge node[above]{$\pi$} (m-2-3) (m-3-3) edge node[left]{$\subseteq$} (m-2-3) (m-3-3) edge node[above]{$\cong$} (m-3-4) (m-3-4) edge node[left]{$\subseteq$} (m-1-4) (m-2-1) edge node[above]{$\simeq$} (m-2-2); \end{tikzpicture}$$ [fig:sumrelations] Let us summarize the relationships we have so far between our various complexes which is illustrated in Figure [fig:sumrelations]. The leftmost homeomorphism between Wr(D, Dd, 1∞)• and Sr(D, Dd, 1∞) comes from Remark [rem:firstchange]. By Lemma [lem-cpt-jn], there is a complete join map π from the complex Sr(D, Dd, 1∞) to the complex of isotopy classes of asymptotically rigidly embedded loops, Ur(D, Dd, 1∞), which implies that both complexes have exactly the same connectivity properties. Thus we can choose to work with Ur(D, Dd, 1∞). Next, Remark [rem-Un](1) demonstrates that the (r − 2) skeleton of Ur(D, Dd, 1∞) is the same as the (r − 2) skeleton of the complex of asymptotically rigidly embedded loops in ⊕ rDd, 1∞, denoted Ur∞(D, Dd, 1∞). Since our goal is to show Wr(D, Dd, 1∞)• is weakly Cohen-Macauley of dimension r − 2 (see Corollary [cor-Sr-conn]), this implies we can again shift our focus to Ur∞(D, Dd, 1∞). Next by Definition [defn-subcpxT], we have that Tr∞(D, Dd, 1∞) is a subcomplex Ur∞(D, Dd, 1∞). In the next pages, we will show that this complex is isomorphic to the lollipop complex Lr∞(D, Dd, 1∞) as a simplicial complex, Lemma [lem-id-T-L], and that the lollipop complex (and hence Tr∞(D, Dd, 1∞)) is contractible with a bad simplices argument, Proposition [prop-Tn-inf]. We then use the contractibility of Tr∞(D, Dd, 1∞) and a bad simplices argument to prove the complex Ur∞(D, Dd, 1∞) is contractible and weakly Cohen-Macaulay of dimension r − 2, Proposition [prop-Un-inf] and Corollary [cor-conn-Ur], implying ultimately that our initial complex is weakly Cohen-Macaulay of dimension (r − 2) as needed. In Proposition [prop-Un-inf], we will deduce the connectivity of Ur∞(D, Dd, 1∞) using the connectivity of the lollipop complex Lr∞(D, Dd, 1∞) by applying a bad simplices argument. Our goal now is to show that Lr∞(D, Dd, 1∞) is highly connected. Let us make some definitions first. Given any lollipop L : (A, 0) → (Dd, r∞, 0), we define the free height hL to be the minimal number m such that L([1, 2]) is contained in Dd, r, m up to free isotopy. We also define the height of an admissible loop to be the minimal number m such that it is contained in Dd, r, m (see Definition [defn-inf-surf]). To analyze the connectivity of Lr∞(D, Dd, 1∞), we need the following lemma which is a direct translation of. [lem-hight-esm] For any r, p, N ≥ 1, there exists a number hr, p, N ≥ 0, such that for any p-simplex σ in Lr∞(D, Dd, 1∞), and any h ≥ hr, p, N, there are at least N lollipops of free height h in Lr∞(D, Dd, 1∞) that are in Lk(σ). Note that for any vertex L in Lr∞(D, Dd, 1∞), L ∣ [1, 2] is an admissible loop in ⊕ rDd, 1∞. Recall the function q defined in Definition [defn-forest] which maps an admissible loop to an edge midpoint in the tree Td, r. Since each edge has a unique descendent vertex, we can instead map the loop to this vertex which lies in the forest Fd, r. Using this connection, we can now choose hr, p, N to be the same as in. Then we have at least N admissible loops of height h ≥ hr, p, N which lie in the complement of the surface corresponding to σ in ⊕ rDd, 1∞. Connecting each of these admissible loops to the base point in the complement surface, we get a set of lollipops in Lk(σ). We now show that the complex Lr∞(D, Dd, 1∞) is in fact contractible. The idea of proof is similar to that of but with significantly more technical difficulty. Intuitively, to define their complex, one only needs information from the loop parts of the lollipops which are much easier to ``make" them disjoint in general, but for us, we also have to deal with the arc parts which could potentially cause more problems. [prop-Tn-inf] The complex Lr∞(D, Dd, 1∞) is contractible for any r ≥ 1. The complex Lr∞(D, Dd, 1∞) is obviously non-empty. We will show by induction that for all k ≥ 0, any map Sk → Lr∞(D, Dd, 1∞) is null-homotopic. Assume Lr∞(D, Dd, 1∞) is (k − 1)-connected. Let f : Sk → Lr∞(D, Dd, 1∞) be a map. As usual, we can assume that the sphere Sk comes with a triangulation such that the map f is simplicial. We first use Lemma [lem:injectifying] to homotope f to a map that is simplexwise injective. For that we need that for every p-simplex σ in Lr∞(D, Dd, 1∞), its link Lk(σ) is (k − p − 2)-connected. But by Lemma [lem-link-loli], Lk(σ) can be identified with Lrσ∞(D, Dd, 1∞) for some rσ ≥ 1, so we have it is (k − 1)-connected and the conditions of Lemma [lem:injectifying] are satisfied. Now since Sk is a finite simplicial complex, the free height of the vertices of Sk has a maximum value. We first want to homotope f to a new map such that all the vertices have free height at least h = hr, k, N where N = v0 + v1 + … + vk + 2, where vi is the number of i-simplices of Sk and hr, k, N is determined by Lemma [lem-hight-esm]. For that we use a bad simplices argument. We call a simplex of the sphere Sk bad if all of its vertices are mapped to vertices in Lr∞(D, Dd, 1∞) that have free height less than h. We will modify f by removing the bad simplices inductively starting by those of the highest dimension. Let σ be a bad simplex of maximal dimension p among all bad simplices. We will modify f and the triangulation of Sk in the star of σ in a way that does not add any new bad simplices. In the process, we will increase the number of vertices by at most 1 in each step, and not at all if σ is a vertex. This implies that, after doing this for all bad simplices, we will have increased the number of vertices of the triangulation of Sk by at most v1 + … + vk. As Sk originally had v0 vertices, at the end of the process its new triangulation will have at most v = v0 + v1 + … + vk vertices. There are two cases. *Case 1: p* = k.** If the bad simplex σ is of the dimension k of the sphere Sk, then its image f(σ) has a complement loop which bounds a surface C asymptotically rigidly homeomorphic to Dd, rσ∞ for some rσ ≥ 1 by Lemma [lem-link-loli]. Now we can choose a lollipop y in C with free height at least h + 1. In particular f(σ) ∪ y form a (k + 1)-simplex. We can then add a vertex a in the center of σ, replacing σ by ∂σ \* a and replacing f by the map (f∣∂σ) \* (a ↦ y) on ∂σ \* a. This map is homotopic to f through the simplex f(σ) ∪ {y}. We have added a single vertex to the triangulation. Because L has free height h + 1, we have not added any new bad simplices, and we have removed one bad simplex, namely σ. Moreover, f remains simplexwise injective. *Case 2: p* < k.** If the bad simplex σ is a p-simplex for some p < k, by maximality of its dimension, the link of σ is mapped to vertices of free height at least h in the complement of the subsurface f(σ). The simplex σ has p + 1 vertices whose images are pairwise disjoint outside the based point up to based isotopy. By Lemma [lem-hight-esm] and our choice of h, there are at least N = v + 2 lollipops y1, …, yN of free height h such that each f(σ) ∪ {yi} form a (p + 1)-simplex. As there are fewer vertices in the link than in the whole sphere Sk, and Sk has at most v vertices, by the pigeonhole principle, the loop part of the vertices in f(Lk(σ)) are contained in at most v punctured disks bounded by the corresponding admissible loops with free height h. As N = v + 2, there are at least two of the above vertices yi and yj of free height h such that any loop parts of vertices in f(Lk(σ)) are disjoint from the loop parts of yi and yj. We can further assume that the arc parts of yi and yj never intersect with any loop part of the vertices in f(Lk(σ)). And up to replacing the loop part of yi and yj with an admissible loop lying inside the disk bounded by the loop parts of yi and yj (note that this may increase the free height of yi and yj), we can further assume that the arc parts of vertices in f(Lk(σ)) are disjoint from the loop part of yi and yj. But unlike the situation in the proof of, a new problem we are facing here is that the arc parts of yi or yj might intersect the arc parts of the vertices in f(Lk(σ)) even up to isotopy. In particular, given a simplex τ lying in the link of σ, f(σ) ∪ f(τ) ∪ yi does not necessarily form a simplex now. For that we want to apply the mutual link trick (cf. Lemma [lemma-replace-trick]) to remove the intersections of f(Lk(σ)) with yi via a sequence of homotopies. In the process, we will only modify f on Lk(σ) and the new map still maps Lk(σ) to LkLr∞(D, Dd, 1∞)(f(σ)). Recall that f is simplexwise injective. Up to isotopy, we can further choose representatives for vertices in f(Lk(σ)) such that the intersection points of vertices in f(Lk(σ)) and yi are isolated. Moreover, we assume the number of intersection points is minimal for each vertex in f(Lk(σ)). Now we choose an intersection point x0 in the arc yi([0, 1]) that is closest to yi(1), denote the corresponding lollipop by β which is the image of some vertex b ∈ Lk(σ). We can choose βʹ to be a variation of β: βʹ coincides with β for the most part, except around the intersection point with yi, we replace it by an arc going around the loop part of yi. See Figure [jumping] for a picture of this. Now we apply Lemma [lemma-replace-trick], for which we need to check the following two conditions: [jumping] 1. f(LkSk(b)) ≤ LkLr∞(D, Dd, 1∞)(βʹ). This follows from our definition of βʹ. If a vertex v in f(LkSk(b)) is disjoint from β, using the fact that the intersection point x0 is the closest one to yi(1) and f(v) is disjoint from the loop part of yi, we have βʹ is also disjoint from v. 2. Lk(β) ∩ Lk(βʹ) is (k − 1)-connected. The lollipops β and βʹ together will bound a disk which contains the loop part of β and yi. In any event, the complement of these is a surface asymptotically rigidly homeomorphic to some surface Dd, rʹ∞ for some rʹ ≥ 1. By our induction, it is (k − 1)-connected. Now Lemma [lemma-replace-trick] says we can homotope f to a new map such that f(b) = βʹ and f(Lk(σ)) has fewer intersection points with yi. Step by step, at the end we have a simplexwise injective map f such that for any vertex in f(Lk(σ)), it only intersects with yi at the base point. In particular for any τ ∈ Lk(σ), we have f(σ) ∪ f(τ) ∪ {yi} forms a simplex in Lr∞(D, Dd, 1∞). We can then replace f inside the star St(σ) = Lk(σ) \* σ ≃ Sk − p − 1 \* Dp by the map (f∣Lk(σ)) \* (a ↦ yi) \* (f∣∂σ) on Lk(σ) \* a \* ∂σ ≃ Sk − p − 1 \* D0 \* Sp − 1. which agrees with f on the boundary Lk(σ) \* ∂σ of the star, and is homotopic to f through the map (f∣Lk(σ)) \* (a ↦ yi) \* (f∣σ) defined on Lk(σ) \* a \* σ ≃ Sk − p − 1 \* D0 \* Dp. Now Lk(σ) \* a \* ∂(σ) has exactly one extra vertex a compared to the star of σ, unless σ is just a vertex, in which case its boundary is empty and it has the same number of vertices. As yi has height at least h, we have not added any new bad simplices. Hence we have reduced the number of bad simplices by one by removing σ. By induction, we can now assume that there are no bad simplices for f with respect to a triangulation with at most v vertices. With this assumption, we want to cone off f just as we coned off the links in the above argument. We have more than N = v + 2 vertices of free height h in Lr∞(D, Dd, 1∞), and at most v vertices in the sphere. The loop parts of these vertices are admissible loops of height at least h. By the pigeonhole principle, we know that there are at least two lollipops zi and zj of free height h such that the punctured disks bounded by their loop parts are disjoint from the punctured disk bounded by any loop part of the lollipops in the vertices of f(Sk). Just as before, we can further assume that the arc parts of zi and zj never intersect with any loop part of the vertices in f(Sk), and the arc parts of vertices in f(Sk) are disjoint from the loop part of zi and zj. But the same problem appears again, as we want vertices of f(Sk) to be disjoint from the whole lollipop zi. For that we apply Lemma [lemma-replace-trick] again and the same proof as before implies that we can homotope f such that its image is disjoint from zi. In particular f(Sk) lies in the link of zi. Hence we can homotope f to a constant map since St(zi) is contractible. [prop-Un-inf] The complex Ur∞(D, Dd, 1∞) is contractible. As Tr∞(D, Dd, 1∞) is a subcomplex of Ur∞(D, Dd, 1∞), we can use a bad simplices argument. We call a vertex of Ur∞(D, Dd, 1∞) bad if it does not lie in Tr∞(D, Dd, 1∞) and a simplex bad if all of its vertices are bad. Given a bad p-simplex σ, we need to determine the connectivity of the good link GLσ (see Subsection [sub-bad-sim] for the definition of GLσ). As in the proof of Lemma [lem-link-loli], we have a complement surface Cσ of σ in Dd, 1∞. Note that Cσ inherits a d-rigid structure and it is asymptotically rigidly homeomorphic to ⊕ rσDd, 1∞ for some rσ > 0. In particular, we can now identify GLσ with Trσ∞(D, Dd, 1∞) which is contractible. Thus by Proposition [prop-bad-sim], we have the pair (Ur∞(D, Dd, 1∞), Tr∞(D, Dd, 1∞)) is i-connected for any i ≥ 0. By Proposition [prop-Tn-inf], Tr∞(D, Dd, 1∞) ≅ Lr∞(D, Dd, 1∞) is contractible, so we also have Ur∞(D, Dd, 1∞)) is contractible. [cor-conn-Ur] The complex Ur(D, Dd, 1∞) is weakly Cohen-Macaulay of dimension r − 2. Note first that a simplicial complex is (r − 3)-connected if and only if its (r − 2)-skeleton is. Since Ur(D, Dd, 1∞) has the same (r − 2)-skeleton as Ur∞(D, Dd, 1∞) and Ur∞(D, Dd, 1∞) is contractible, in particular (r − 3)-connected, we indeed have Ur(D, Dd, 1∞) is (r − 3)-connected. Now let σ be a p-simplex of Ur(D, Dd, 1∞), with vertices ϕ0, ϕ1, ⋯, ϕp. We need to check that the link LkUr(D, Dd, 1∞)(σ) is (r − p − 4)-connected. We can assume p ≤ r − 3 as any space is ( − 2)-connected. Moreover, it suffices to show the (r − p − 3)-skeleton of LkUr(D, Dd, 1∞)(σ) is (r − p − 4)-connected. Since ϕ0, ϕ1, ⋯, ϕp forms a p-simplex, similar to the proof of Lemma [lem-link-loli], we have the complement surface of σ is asymptotically rigidly homeomorphic to some d-rigid surface Dd, kσ∞ for some kσ > 0. Then we can identify the (r − p − 3)-skeleton of LkUr(D, Dd, 1∞)(σ) with the (r − p − 3)-skeleton of Ukσ∞(D, Dd, 1∞). Since Ukσ∞(D, Dd, 1∞) is even contractible, we have the connectivity bound we need. Now by Lemma [lem-cpt-jn] and Proposition [prop-join-conn], we have the following. [cor-Sr-conn] The complexes Sr(D, Dd, 1∞) and Wr(D, Dd, 1∞)• are weakly Cohen-Macaulay of dimension r − 2. Homological stability --------------------- We are finally ready to prove the homological stability result. [thm-hmg-stab-RB+] Suppose d ≥ 2. Then the inclusion maps induce isomorphisms ιR+, d, r : Hi(RVd, r+, M) → Hi(RVd, r + 1+, M) in homology in all dimensions i ≥ 0, for all r ≥ 1 and for all H1(RVd, ∞+)-modules M. From Corollary [cor-Sr-conn], we have that Wr(D, Dd, 1∞) is (r − 2)-connected, hence in particular, the category Gd satisfies property LH3 at the pair of objects (D, Dd, 1∞) with slope k = 3. By Theorem [thm:hom stab], we have for any abelian RV∞+-module M the map Hi(RVd, r+; M) → Hi(RVd, r + 1+; M) induced by the natural inclusion map is isomorphism if r ≥ 3i + 3. But we can improve the stability range as in the proof of by noticing that we have the same canonical isomorphism between RVd, r+ and RVd, r + d − 1+. In fact, denote this isomorphism by Id, r, we have the following commutative diagram: $$\begin{tikzcd} RV^+\_{d,1+ r-1 }\arrow[d,"I\_{d,r}"] \arrow[r, "\iota\_{R^+,d,r}"] & ~ RV^+\_{d,1+(r-1) +1}\arrow[d,"I\_{d,{1+(r-1) +1}}"] \\ RV^+\_{d,{d+{r-1}}} \arrow[r, "\iota\_{R^+,d,r +1}"] & ~RV^+\_{d,{d+{r-1}+1}} \end{tikzcd}$$ Given that the vertical maps are isomorphism, and the bottom horizontal maps induce isomorphism on the i-th homology when d + r − 1 ≥ 3i + 3, we also the top map must also induce isomorphism on the homology as long as r ≥ 3i + 3. This has improved the stable range by d − 1. Step by step, we must have the map ιR+, d, r induce isomorphism on homology in all dimensions i ≥ 0 and for all r ≥ 1. This finishes the proof of Theorem [thm-hmg-stab-RB+]. [thm-hmg-stab-RB] Suppose d ≥ 2. Then the inclusion maps induce isomorphisms ιR, d, r : Hi(RVd, r, M) → Hi(RVd, r + 1, M) in homology in all dimensions i ≥ 0, for all r ≥ 1 and for all H1(RVd, ∞)-modules M. The proof will be exactly the same as that of Theorem [thm-hmg-stab-RB+]. Note first that by Theorem [thm-iden-asym-ribb], it is the same as proving the half-twist asymptotic mapping class groups HVd, r have homological stability. We define the braided monoidal category Gʹd to be the category with objects ⊕ rDd, 1∞, r ≥ 0, ⊕ as the operation, and D as the 0 object. When r = s, we define the morphisms Hom( ⊕ rDd, 1∞, ⊕ sDd, 1∞) = HVd, r which can also be understood as the group of isotopy classes of asymptotically quasi-rigid homeomorphisms of ⊕ rDd, 1∞; when r ≠ s, let Hom( ⊕ rDd, 1∞, ⊕ sDd, 1∞) = ∅. We then have a homogeneous category UGʹd and to prove the homological stability for the sequence of groups HVd, 1 ≤ HVd, 2 ≤ ⋯, we only need to prove the associated space Wr(D, Dd, 1∞)•, in fact the associated simplicial complex Sr(D, Dd, 1∞), is highly connected. At this point, the complex is slightly different from the oriented case, but still the new complex Sr(D, Dd, 1∞) is a complete join over the old complex Ur(D, Dd, 1∞). Hence, the connectivity of Sr(D, Dd, 1∞) again follows from Corollary [cor-conn-Ur] and Proposition [prop-join-conn].	arxiv_0000446
posed into two separate subtasks of sensing and data transmission. Similarly, the transmission of a saved parameter (in memory) can be decomposed into the separate subtasks of reading the attribute from memory and data transmission. In order to minimize energy consumption, the two transmission subtasks can be combined together by grouping the data from these subtasks and transmitting it in one data packet, as shown in Fig. [figdecomp] . Zhu et al. , decompose larger tasks into multiple subtasks depending upon their ability to recombine with other smaller tasks to save energy. In summary, the decomposing and combining technique in consists of the following four phases: ### Decomposition This phase decomposes the energy-intensive tasks into multiple subtasks depending upon their ability to group with other subtasks to conserve energy. As the harvested energy is not sufficient to run the high power tasks continuously, subtasks can be executed with the limited available energy budget. ### Combining This phase combines multiple subtasks that can be executed on the same processor to minimize the energy consumption. In addition, some tasks can be executed concurrently to reduce the idle time of the processor. For example, the tasks of sensing and reading a value from memory can be executed simultaneously as they utilize different resources of the node, depending on the harvested energy. The advantage of the concurrent execution is the reduced delay and smaller latency in data transmission. However, it also demands higher energy which is available only once in a while in. ### Admission control In this phase, the tasks are filtered according to their priority and energy consumption during their execution. Although tasks are combined to save energy, generally, the harvested energy is not enough to run all ready-to-execute tasks. Therefore, an admission controller further filters the tasks based on the following task-specific parameters: * Priority of tasks * Available harvested energy * Energy consumption of tasks Task priority is important in all applications and in particular for time-critical real-world scenarios. Depending upon the application, the task’s deadline is further categorized into two types, i.e. soft deadline and hard deadline. In general, soft deadlines are less critical as compared to hard deadlines and their violation does not harm the functioning of the system. On the contrary, hard deadlines are essential to be respected in all circumstances, which create major loss if ignored or violated . Therefore, the admission controller arranges the tasks according to their priorities and deadlines. In addition to the task priority, the available energy and task energy consumption are also important factors which help in deciding the execution of tasks. If the available energy is lower than the required for the task execution, the task can be delayed so that sufficient energy is accumulated in the (from energy harvesters) to serve the task successfully. ### Optimization This phase optimizes the execution of tasks on the basis of the following attributes: * The additional available energy (available after executing the current task) * The required number of executions of each task * Energy consumption of each task The optimization phase further filters the tasks in order to use the harvested energy efficiently. The additional available energy is important to decide about the execution of future time-critical tasks. In order to avoid deadline violations in the future, a certain minimum amount of energy must be available during all time slots, to serve the future tasks that possess hard deadlines. Authors in evaluate their scheme using as a source of energy. Results show that their technique executes more tasks with fewer missed deadlines as compared to the previous schemes which do not employ the decomposing/combining strategy for energy-intensive tasks. In addition to decomposing and combining of tasks, a model is proposed in to schedule the tasks on the node. It proposes a dynamic optimization model based on to schedule the tasks taking into account their deadlines, energy consumption, and available harvested energy. The decomposed subtasks can be combined together for concurrent execution. It also proposes a less complex greedy scheduling policy which can be implemented on the resource constrained sensor nodes and it consumes less energy as compared to the original model. The simulation results indicate that the proposed algorithm executes higher number of tasks within the same energy budget compared to the previous schemes. As the harvested energy in energy harvesting-based sensors is limited, they can essentially perform only one atomic task at a time. In the particular case of batteryless sensors, the available energy burst (in the ) can not be employed to execute multiple tasks simultaneously. Therefore, combining the tasks to reduce energy consumption is not appropriate solution for energy harvesting based sensors. As a result, there is a potential of alternate task scheduling algorithms that can provide on the sensor nodes with limited energy budget. [t] [figtaskscheduling] Duty cycling ------------ One of the most familiar and common methods for minimizing the energy consumption in is to use a sleep/awake mechanism. When a node is not executing any useful operation, it is switched to the sleep mode, which reduces its energy consumption . In traditional, most of the protocols devise a duty cycling mechanism based on the number of tasks, their energy consumption and remaining energy of the node. However, these methods are not suitable for, due to the variable nature of incoming harvested energy. In, the harvested energy is varying with the environmental conditions due to the intermittent nature of ambient energy harvesting sources . Therefore, the duty cycle of the node depends on the incoming harvested energy to efficiently utilize the energy in executing tasks on the node. In addition to the current harvested energy, the sleep duration of the node also depends on the energy to be harvested in future to proactively plan the consumption of incoming energy for required operations. Therefore, we classify the duty cycling mechanisms into the aforementioned two categories and extensively explore their operations in the following subsections. ### Duty cycling depending on the harvested energy In a network of multiple nodes, duty cycling mechanism depends on the harvested energy, consumed energy, distance between nodes and data aggregator/receiver, and the future harvested energy. In traditional, nodes near the sink exhaust quickly due to the additional burden of relaying the data of far-away nodes in multi-hop communication. On the other hand, in, nodes remain alive as long as they are receiving replenishable energy from the environment using energy harvesters. Kansal et al. present duty cycling mechanisms (for single node as well as multiple nodes) for sustainable performance in. They describe a model for calculating the minimum size of the /battery for sustainable operation of the embedded device. The duty cycle within a sensor node depends upon the average harvested as well as consumed energy in active and sleep modes. The duty cycle can be adjusted such that the overall energy consumption does not exceed the overall harvested energy. Similarly, if the harvested energy is increased, the duty cycle can also be raised to improve the performance within the given energy budget. Authors in show the record of the battery voltage for more than two days which depicts that the node has adjusted its duty cycle in accordance with the harvested energy for sustainable operation. The authors also propose a duty cycling mechanism for a wireless network having multiple nodes. In a network configuration, determining sleep/awake time interval is challenging as the harvested energy for each node can be different due to its geographical location and orientation with respect to the energy source (e.g., solar energy). Therefore, the node attempting to transmit data will send a BEACON packet and wait for a certain time period for an ACK. It repeats the process until an ACK is received from the destination node. A mathematical model for duty cycling the sensor nodes according to the harvested energy is described in. It achieves the and maximizes the system performance by adapting the dynamics of the replenishable energy source at run-time. It employs the algorithm to predict the harvested energy which is used to compute the duty cycle of the node. Bouachir et al. propose a protocol for efficient energy utilization in cooperative. Their scheme uses the nodes’ residual energy and data requirements to decide the active as well as sleep time periods of sensor nodes. As a result, it decreases the active time of nodes near the data aggregator and increases the active time of nodes away from data aggregator with the passage of time to balance the energy consumption of nodes. Therefore, it minimizes the problem of early depletion of nodes near the data aggregator, which reduces the coverage hole dilemma . Yang et al. propose a sensing scheduling scheme which dynamically adapts the sensing rate according to the available energy in the. They also propose a mathematical model for optimal sensing scheduling in energy harvesting sensors . In contrast to the previous works that focus on the energy allocation to the sensors, optimizes the sensing epoch depending on the energy budget. It presents the infinite and finite battery case and suggest an online scheduling policy that approaches the theoretical offline optimal scheduling mechanism. It also proposes a virtual energy harvesting sensing system to analyze the battery level which is helpful in deciding the sensing epoch. An event-driven duty cycling mechanism is proposed in for power management of a road side monitoring unit. It harvests energy from the and transmits data packets according to the events of the traffic flow on the road using an algorithm. This implementation achieves lower energy consumption with longer lifetime of the sensing device. ### Duty cycling depending on the available and predicted harvested energy In certain circumstances, the current harvested energy may not be sufficient to run an energy-intensive task on the node. Therefore, knowledge of the future incoming energy is important to delay the tasks until sufficient energy is available, without missing any deadline as illustrated in Fig. [fig:predictedenergymotivation]. As shown in the figure, task 1 is executed as soon as it arrives at the node due to the higher energy availability than required for the execution of task. However, task 2 arrives when the available energy is lower than required for the execution of task. As there is a prediction of future harvested energy, the task is delayed until sufficient energy is available for its execution. In this way, the predicted harvested energy improves the energy utilization and minimizes the number of missed deadlines of tasks. Moser et al. present an algorithm for task scheduling in environmentally powered. To the best of our knowledge, this is the first detailed work that addresses the task scheduling problem in. In conventional, the sensor nodes have a fixed energy supply (i.e. from a battery) and the only issue is to meet the deadlines of tasks for their execution on a single processor. In this case of abundant energy availability, various scheduling schemes such as or can be incorporated . However, these schemes have certain drawbacks and can violate the deadline of tasks, when implemented on the energy harvesting node, which has intermittent energy availability. In the algorithm, future incoming tasks can experience energy black-out periods (which results in deadline violations), whereas the scheme can miss the deadlines, if the energy is limited near tasks deadlines. The algorithm can solve this problem and schedules the tasks such that the task having an earlier deadline is executed first. However, does not perform well in energy harvesting systems and violates deadlines due to energy consumption for less important tasks. This energy starvation results in the failure of execution of time-critical tasks with shorter deadlines. On the contrary, a scheme that hesitates in executing the tasks (until their deadlines) can perform well by conserving energy for time-critical tasks that have shorter deadlines . The optimal start time of the task can be decided according the available energy and upcoming energy in the future. Therefore, there is an opportunity to devise sophisticated algorithms for the prediction of future harvested energy. Then, this harvested energy profile determines the time slot for the execution of various tasks. Authors in evaluate their technique using the energy from and results show that it offers fewer deadline violations as compared to previous algorithms. Authors in extend the work of with a detailed mathematical model and consider the practical considerations in implementing the algorithm on a real embedded device. They also propose an optimal start time for the execution of tasks depending upon their deadlines, energy consumption, stored energy and future harvested energy. However, an important factor in the calculation of optimal start time is the prediction of energy to be harvested in future. In order to estimate the future harvested energy, authors in propose an Energy Variability Characteristic Curve (EVCC) which gives an upper bound on the harvested energy in a certain time interval Δ. As a result, the harvested energy in any time interval Δ can easily be approximated, according to the given EVCC. However, another problem is to determine how much capacity the should have, for perpetual operation of the sensor node in. The answer lies in the maximum harvested energy, in addition to the consumed power in any particular time interval. This energy consumption can be calculated depending on the number of tasks arriving at the processor with common deadlines. Sommer et al. propose a scheduling framework for various sensors (such as Global Positioning System (GPS), accelerometer, magnetometer, etc.) for perpetual tracking of flying foxes that travel long distances from their foraging camps in the search of food. The sensors are sampled based on the available and the future harvested energy. They also take into account the mobility and activity of flying foxes to trigger the next sensor sample. This technique ensures that maximum tracking accuracy is achieved within the given dynamic energy budget. The scheme in is quite different from traditional tracking schemes that focus on minimizing the energy consumption with a fixed tracking accuracy. On the contrary, emphasizes on achieving maximum tracking accuracy within the available energy budget which is replenished using on-board solar cells. The consideration of current and future energy levels while sampling the sensors ensures optimal time duration between successive samples, such that energy is neither depleted nor overflows (due to limited capacity of ) thanks to the future incoming energy from replenishable energy source (i.e., solar energy). Additionally, sensor sampling based on mobility ensures that enough energy is available to track the rest of the trip of flying foxes using the limited available energy. Györke et al. exploit the knowledge about the environment to schedule non-equidistant samples, both is time as well as in space. The predicted harvested energy is taken into account to devise a conservative sampling approach when the future incoming energy is low. The proposed technique also uses the neighbour’s information in deciding the duty cycle. It increases the duty cycle of nodes in vicinity of a place where an event has occurred. The other nodes operate at their usual duty cycle to conserve energy. [figdecomp] Akgün et al. address the problem of duty cycling in cluster based using slots. Traditional protocols assign fixed slots to the cluster members for their data transmission. In contrast, the technique in uses the predicted harvested energy and available energy to assign slots to the nodes. In based, the node having maximum predicted harvested energy is assigned the next time slot in the day. On the other hand, at night, the node having maximum residual energy is given the next time slot. Kooti et al. present a task scheduling mechanism for based sensor nodes. Their scheme consists of two parts: offline scheduling and online scheduling. Firstly, the tasks are scheduled offline depending on the energy requirements and their criticality. Then, in the online scheduling phase, the tasks are adjusted according to the real-time available energy and future harvested energy, and are executed depending on their deadlines. Renner et al. propose a scheduling algorithm on the basis of predicted harvested energy in sensors. They present a harvested energy prediction model based on the changes in the previous harvested energy pattern in real-time, which reduces the prediction error and is applicable in real-world environments. Instead of periodic sampling, they exploit the periodic nature of the sun to adaptively trace the energy pattern and reduce the number of prediction updates. Sommer et al. propose an accurate method for the estimation of of the battery which can lead towards energy neutral scheduling of tasks. They use the current flow from the energy harvester and the battery voltage to estimate the current. They employ a mathematical model called conflation to combine information from energy harvester as well as the battery to calculate the battery. Cui proposes the execution of tasks on the node using the predicted energy in based. He employs a recurrent long short term memory neural network to forecast the harvested energy. Later, this harvested energy is used to schedule tasks, which offers better performance than conservative and greedy approaches. El Osta et al. propose to schedule periodic and aperiodic tasks separately, based on their priority. The periodic tasks are executed as soon they arrive at the scheduler if sufficient energy is available for their execution. On the other hand, aperiodic tasks are executed only if there is no periodic task in the queue and the is not depleted. The aperiodic task consumes surplus energy that would have been wasted, if there is no other task ready for the execution. It is to ensure that the execution of aperiodic tasks does not affect the execution of future periodic tasks by consuming a significant amount of energy, which depletes the. Authors in use an reader to harvest energy for capturing and transmitting an image. The harvested energy is stored in an to form fixed energy bursts to power the load (i.e., camera). It is more efficient to store the small amount of harvested energy instead of capturing large amount of energy as it may introduce delay in the execution of tasks. The is also non-ideal, causing some leakage, which may reduce the usable energy budget. Therefore, it is more energy efficient to store the limited harvested energy to execute at least one atomic task as it takes a short build-up time and causes less leakage. Instead of fixed energy bursts , Gomez et al. introduce dynamic energy bursts that are matched with the requirements of the load. They introduce an that arranges short energy bursts according to the requirements of the load and transforms small amounts of harvested energy into high-powered short energy bursts. Their tracks the optimal power point of load and schedules the energy bursts accordingly. This results in efficient utilization of limited, variable and transient harvested energy to power the sensor nodes in. [fig:predictedenergymotivation] Zhang et al. present a scheduling scheme for storage-less and converter-less energy harvesting systems in. They formulate a system model for powering the sensor nodes using the to minimize the deadline miss ratio. They develop a task scheduling algorithm using an artificial neutral network based on the harvested energy profile, the tasks priority and the energy budget. They also propose an integer nonlinear programming solution for optimal task scheduling on the node. Severini et al. implement an energy-aware lazy scheduling algorithm on a low-cost and commercially available platform from Texas Instruments. This lazy scheduling algorithm delays the execution of tasks depending on their deadline and according to the available energy. Experiments are performed under varying incoming energy conditions. The results show the effectiveness of the scheduling algorithm and its suitability for real-time implementation in. Liu et al. propose converter-less and storage-less operation of energy harvesting sensors. Due to the lower efficiency of the converter (40-90%) and energy leakage of the, the efficiency of the system is decreased. Authors in design a hardware prototype using, without converters and batteries, which performs better than conventional systems. They also propose an intra-task scheduling algorithm which pre-empts the tasks according to their deadlines. The tasks are performed according to the available energy and their deadlines to meet the execution time requirement and use the energy efficiently. Zhang et al. propose a scheduling mechanism for time-critical energy harvesting sensors. They present harvesting-aware speed selection of processor and radio transmission to manage the harvested energy and to meet the deadlines of tasks. They suggest two versions of their algorithm: central and decentralized, to be implemented according to the application environment. Simulations show that their scheme outperforms in terms of energy reserves than traditional schemes, which do not employ processor speed selection and transmission power control. The available energy budget plays a pivotal role in scheduling the tasks on the nodes in. Table [tablesurveyedpapers] portrays that all of the previous task scheduling algorithms consider the available energy budget while deciding the execution of tasks to enhance the performance and minimizing deadline misses. Fang et al. propose a mathematical model for optimal scheduling in energy harvesting based mobile sensor nodes. As the harvested energy in mobile sensors varies with the change in energy source as well as with the mobility of embedded devices, it is challenging to devise an optimal scheduling algorithm. The sensor is switched to the sleep mode, when there is no task to be executed. After the arrival of new tasks, the corresponding sensor component is switched to active mode depending on the energy budget. Authors in use a Lyapunov optimization framework to make control decisions that greedily minimize a bound on the drift-plus-penalty expression over fixed-length time slots. It has fair computational complexity and is easy to implement in real-time scenarios. Pan et al. suggest a technique for the execution of tasks under unstable energy harvesting conditions. Their algorithm maximizes the progress of tasks using the information about the harvested energy. They also employ a routine which triggers sleep/wake-up events and start the execution of tasks automatically after resuming. Anagnostou et al. propose a power aware hardware scheduler for energy harvesting sensors. Instead of a traditional mechanism, where the harvested energy is stored in an, they switch between the energy harvester directly and the, depending on the harvested energy and task requirements. This mechanism leads to maximum utilization of the harvested energy even when the is full to its capacity. Authors in implement an FPGA based prototype which shows that their algorithm achieves two times higher task execution rate under variable energy harvesting conditions, compared to the conventional schemes. Sanchez et al. propose a hybrid approach for energy management in. This hybrid mechanism consists of battery states and the operating conditions of the nodes such as active, sleep and low duty cycle modes. In addition, they use decentralized control, which reduces the burden of controlling the whole network from a single node, extending the lifetime of the system. The previous task scheduling algorithms employ conventional sensors, which consume energy during their operation. In addition, these sensors perform their operation using a reliable energy source (i.e., a battery) that provides continuous power for a longer duration. However, this scenario is entirely different for energy harvesting based sensing due to the dependence between energy and information, and intermittent execution of sensor node. Firstly, the energy harvester provides both energy as well as context information, and the signal needs to be sampled at suitable time instances according to the amount of context information and available energy. Secondly, the sensor node can run at most one atomic task at one time, due to the limited available energy stored in the capacitor. Therefore, it opens doors for dedicated task scheduling algorithms for energy harvesting based sensors. The objective of these task scheduling algorithms is to maximize the context detection accuracy as well ensure the of sensor nodes in. Most of the previous task scheduling algorithms consider the predicted harvested energy, while scheduling the tasks on the node. This results in maximum utilization of the current and future harvested energy without missing the deadlines of tasks. In order to fully understand the functioning of the task scheduling algorithms, it is important to comprehend the energy prediction algorithms as well. Most of the harvested energy prediction algorithms use the previous harvested energy samples to estimate the future harvested energy profile. We discuss some of the previous energy prediction algorithms in the following subsection in detail. [c]\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\| [tablesurveyedpapers] & & & & & & Difficulty in using Year and & Energy & DVFS & Decomp./ & Duty & Predicted & Deadline & Performance & Performance & it in energy harvesting Reference & harvester & & Comb. & cycling & energy & & metric & assessment & based sensing & & & & & & Difficulty in using Year and & Energy & DVFS & Decomp./ & Duty & Predicted & Deadline & Performance & Performance & it in energy harvesting Reference & harvester & & Comb. & cycling & energy & & metric & assessment & based sensing 2001 & General & & & & & & Remaining energy & Modeling & High 2004 & Solar & & & & & & Latency & Simulation & High 2006 & Solar & & & & (C) & & Energy utilization & Hardware & Medium 2006 & General & & & & (R) & & Deadline violation & Simulation & High 2007 & Solar & & & & (C) & & Deadline violation & Simulation & High 2008 & Solar & & & & & & Deadline miss rate & Simulation & High 2009 & Solar & & & & & & Deadline miss rate & Simulation & High 2010 & Solar & & & & & & Deadline miss rate & Simulation & High 2010 & Solar & & & & & & No. of executed tasks, Average & Hardware & High & & & & & & & accuracy, Energy consumption & & 2010 & Water flow & & & & & & Depleted nodes & Simulation & High 2011 & Solar & & & & & & Deadline miss rate & Simulation & High & & & & & & & Capacity of energy storage & & 2011 & General & & & & & & Average idle time & Simulation & High & & & & & & & Feasible task set & & 2011 & Solar & & & & & & Battery charge level & Simulation & High & & & & & & & Energy violation rate & & 2012 & Solar & & & & (R) & & Number of tasks executed, & Hardware & Medium & & & & & & & missed deadlines & & 2012 & Solar & & & & (C) & & Deadline miss rate & Simulation & High & & & & & & & Minimum storage capacity & & 2012 & General & & & & (R) & & End-to-end delay, BF, & Simulation & High & & & & & & & Packet delivery ratio & & 2012 & Solar & & & & & & Energy consumption & Simulation & High 2012 & General & & & & & & Number of deadline violations & Simulation & High & & & & & & & First deadline violation & & 2012 & Solar & & & & (R) & & QoS violation count & Simulation & High 2012 & Solar & & & & (C) & & Root mean square error & Simulation & High 2013 & Solar & & & & & & Time & energy consumption & Simulation & High 2013 & General & & & & & & Length of idle time & Simulation & High 2013 & Solar & & & & (C) & & Network lifetime & Simulation & High 2013 & Solar & & & & (R) & & Task drop rate & Simulation & High 2013 & Solar & & & & (C) & & Battery SoC & Simulation & High 2014 & Solar & & & & (C) & & No. of executed tasks & Simulation & High 2014 & Solar & & & & (R) & & Time & energy & Simulation & High & & & & & & & constraint & & 2015 & Solar & & & & & & Deadline miss rate & Simulation & High & & & & & & & Number of ready tasks & & 2015 & RFID reader & & & & & & Power consumption & Hardware & Medium & & & & & & & with time & & 2015 & Solar & & & & (R) & & Number of tasks executed & Simulation & High 2015 & General & & & & & & Mean Square Error & Simulation & High 2015 & General & & & & (R) & & Completed tasks & Hardware & Medium 2015 & Solar & & & & & & Deadline miss rate, & Hardware & Medium & & & & & & & Energy utilization & & 2016 & General & & & & & & Average overhead & Simulation & High & & & & & & & Average busy period & & 2016 & Solar & & & & (C) & & Deadline success ratio & Simulation & High 2016 & Solar & & & & & & System efficiency & Hardware & Medium 2016 & General & & & & & & Deadline miss rate, & Simulation & High & & & & & & & energy violation rate & Hardware & 2016 & Solar & & & & & & No. of packets sent to BS, & Simulation & High & & & & & & & Delay & & 2016 & Solar & & & & (C) & & Deadline miss ratio, & Hardware & Medium & & & & & & & Energy utilization efficiency & & 2016 & General & & & & & & Data queue size, & Simulation & High & & & & & & & Energy queue size & & 2016 & General & & & & & & Sensing performance, Infeasible & Simulation & High & & & & & & & sensing epoch, Battery overflow & & 2016 & Solar & & & & & & Power consumption, & Hardware & Medium & & & & & & & Battery lifetime & & 2017 & General & & & & & & Deadline miss rate, & Simulation & High & & & & & & & response time & & 2017 & General & & & & & & Harvested to consumed & Simulation & High & & & & & & & energy rate & & 2017 & General & & & & & & Energy and time overhead & Simulation & High & & & & & & & Energy consumption & & 2017 & General & & & & & & Remaining energy of node & Hardware & Medium 2018 & Solar & & & & & & Expected reward & Simulation & High 2018 & Solar & & & & & & Energy efficiency, & Hardware & Medium & & & & & & & Execution rate & & 2018 & Solar & & & & (C) & & Task completion rate & Simulation & High 2019 & Solar & & & & (C) & & Average tracking error & Simulation & High Algorithms for energy prediction in ----------------------------------- In order to ensure, the sensor nodes need to consume the harvested energy in such a way that the current node operation is not affected, and the future tasks do not run out of energy. Therefore, information about the future harvested energy is important to schedule the energy consumption proactively for sustainable operation of the system. In the literature, there are various harvested energy prediction models that utilise previous energy samples, weather conditions and seasonal trends to compute the future harvested energy in energy harvesting based. Table [tablepredictedenergy] comprehensively presents some of the harvested energy prediction algorithms that employ statistical, probabilistic and machine learning models to predict the future harvested energy. On a normal sunny day, the harvested energy from a solar powered node is highest at noon, and decreases at dawn and dusk, finally reaching zero at night , due to the non-availability of sunlight. Knowing this overall pattern of, the future solar harvested energy can be predicted using the previous energy pattern. Kansal et al. present a harvested energy prediction model based on for solar powered. Their model relies on the intuition that the harvested energy in a particular day at a given time slot is similar to that of the energy harvested in the previous days at the corresponding time slots. Therefore, the harvested energy in a particular time slot is calculated by accumulating the weighted average of harvested energy in the previous days in the same time. algorithm awards higher weight to the recent energy values and exponentially decreases weight for the previous energy samples to calculate the future harvested energy in solar powered. The weight is calculated dynamically using the real previous energy traces, which provide the lowest value of error. However, gives significant prediction error when there is a sudden change in the weather. It is due to the reason that the scheme does not take the seasonal weather trends and diurnal cycles into account. Hassan et al. propose an energy prediction model for solar powered, which takes into account the sudden changes in the environment. It also takes into account the seasonal and diurnal cycles of the solar energy. However, this scheme is more computationally complex as it takes multiple parameters into account and costs more energy as well as processing time, incurring delay in the system. Another technique which considers weather conditions for predicting the harvested energy in solar powered is presented in . It presents a model to predict solar as well as wind harvested energy. The method takes the data from the weather forecast stations to predict the energy to be harvested in future time slots. However, this scheme is also computationally complex and depends on another source, which increases the cost of the system. Additionally, receiving weather data and processing it on an energy-constrained miniaturized sensor node hinders the execution of other time-critical tasks. Piorno et al. present a prediction algorithm for which depends on and takes into account the sudden and abrupt changes in weather conditions. They propose to use the weighting factor depending on the solar conditions of the current day relative to the previous days. Cui proposes a prediction algorithm for sensor nodes in. It employs a recurrent long short term memory neural network to forecast the harvested energy. However, this is a complex method that has higher cost in terms of energy, time, memory requirement and computational resources. [t!] [tablepredictedenergy] llll Year & Reference & Input parameters & Method 2007 & & Previous samples & 2008 & & Previous minimum energy & Worst-case energy & & & prediction 2009 & & Weather conditions from & Weather-conditioned & & recent past samples & moving average 2010 & & Weather forecast & Quadratic solar & & & power model 2011 & & Multiple energy harvesters & Markov model 2012 & & Previous energy profiles & Profile energy & & & prediction 2016 & & Previous energy profiles & Profile energy & & & prediction 2016 & & Past observations, & Q-learning & & Current weather & 2016 & & Previous samples & Curve fitting 2019 & & Global horizontal & Austronomical & & irradiance & model Most of the previous harvested energy prediction schemes , forecast the future solar harvested energy for the next single time slot. However, occasionally, it is also important to estimate the harvested energy for future N (where N > 1) time slots. Moser et al. present a harvested energy prediction algorithm that calculates the harvested energy for future N time slots. Their algorithm considers the harvested energy in previous time slots of length N. Then, the worst case harvested energy in the previous time slots is considered as the predicted energy for the future slots. However, does not take the fast changing weather into account which results in higher prediction errors in swift weather changing environments. It is also a pessimistic approach, as it considers the worst case harvested energy only. Cammarano et al. propose a prediction model using harvested energy profiles of previous days. They store different types of energy profiles (like sunny, partially sunny, cloudy, etc.) and compare the initial values of currently harvested energy with the stored energy profiles. The stored energy profile having highest correlation with the current harvested energy is considered as the predicted energy profile for the rest of the day. However, this scheme also burdens the miniaturized nodes with complex computation and data storage requirements. Ventura et al. present an energy harvesting and consumption algorithm for body sensor networks using a Markov model. Their algorithm considers multiple types of energy harvesters and predicts the future states of nodes in terms of energy level depending on the probabilistic model based on previous energy samples. Authors in propose a Q-learning based solar harvested energy prediction model using previous energy samples and current weather conditions. This results in lower prediction error than conventional. As the harvested energy from depends on luminous intensity, Zou et al. propose to predict the locations of sunlight for next time interval to predict the harvested energy. They use a piecewise least squares curve fitting estimation using the previous samples to estimate the future harvested energy. Kai et al. predict the future solar harvested energy using global horizontal irradiance and solar cell’s characteristics. In order to compensate for deviations from the actual values, their scheme compares the predicted harvested energy values with the previous harvested energy. In summary, there are various energy prediction algorithms for solar powered, which take into account the previous harvested energy values, weather forecasts and previous energy profiles, to correctly predict the future harvested energy, as shown in Table [tablepredictedenergy]. However, these algorithms, when implemented on the node, consume a significant amount of harvested energy during their execution. Therefore, in addition to prediction accuracy, the cost in terms of energy and computational complexity must also be explored. Such an analysis will provide the real picture about the models and will identify the algorithm which provides best energy prediction results, while executing within the limited energy and computational resources. Summary and insights -------------------- We comprehensively survey and analyse previous task scheduling algorithms in to enhance the lifetime of sensor nodes. However, none of the previous algorithms employ energy harvesting based sensing; instead they exploit conventional sensors for monitoring the desired physical parameter. It results in significant energy consumption compared to energy harvesting based sensors as discussed in Section [sensingandenergyharvesting]. Therefore, keeping in view this new class of sensors (i.e., energy harvesting based sensing), the previous task scheduling algorithms need to be revised or new dedicated task scheduling algorithms should be proposed to allow the sustainable operation of sensor nodes in. We critically analyse the previous task scheduling schemes and explore their applicability for energy harvesting based sensors in the following section. In addition, we rigorously discuss the opportunities for transforming the conventional task scheduling algorithms for the emerging class of energy harvesting based sensors. [fig:taskschedulingfuture] Critical Analysis of Task Scheduling Algorithms for Energy Harvesting based Sensing =================================================================================== We discussed that task scheduling algorithms manage the execution of tasks on the node to extend the lifetime of sensor nodes in conventional. However, considering the new category of sensors (i.e., energy harvesting based sensing), we explore the applicability of task scheduling algorithms in this new paradigm. There is a fundamental difference between conventional and energy harvesting based activity sensors in terms of their operation and energy consumption. Conventional sensors consume energy while in operation (see Fig. [fig:taskschedulingfuture](a)), whereas, energy harvesting based sensors generate energy, in addition to providing a signal that can be used for detecting the underlying activities, as depicted in Fig. [fig:taskschedulingfuture](b). As energy and context information from the energy harvester are strongly correlated, aggressive sampling of the harvesting signal may not be required, which opens new directions for devising task scheduling mechanism compared to conventional activity sensors. In addition, the harvested energy can act as a trigger signal to decide the next sensing epoch depending on the variation in the harvesting signal. Furthermore, the variation in the harvesting signal can be translated as the change in ongoing activity, using few initial samples of the energy harvesting signal. The task scheduling becomes more challenging in multi-source energy harvesting based sensing, as shown in Fig. [fig:taskschedulingfuture](c), due to the different amount of energy and context information from each harvester unit. Depending upon the type of application, one energy harvester can provide higher context information compared to others, in certain physical conditions, which needs to be tracked in real-time. This results in higher activity detection accuracy without consuming significant energy, for example, to acquire the signal at a substantially high sampling rate or to sample all energy harvesting signals. In addition, the amount of harvested energy from each energy harvester is unique depending upon the environmental conditions and properties of the transducer. This energy can be combined using DC-DC converters or individual harvesters can be selected according to the amount of generated energy, to avoid the energy conversion losses . Challenges of current task scheduling techniques ------------------------------------------------ Most of the existing task scheduling algorithms are complex and require significant amount of energy during their operation on the miniaturized and resource-constrained sensor node. Decomposing and combining the tasks is not applicable for batteryless energy harvesting based sensors due to their limited energy budget that can, at most, run one atomic tasks at a time. Secondly, algorithms need complex hardware circuits to provide multiple voltage levels to individual components of the sensor node. It may require multiple s in batteryless sensor nodes which increase the energy losses, cost and form factor, and decrease the usable energy. Most of the previous task scheduling algorithms focus on enhancing the lifetime of the system, ignoring the performance in terms of activity detection accuracy. These scheduling algorithms do not take into account the correlation between energy and context information, which may help in deceasing the sampling rate of the harvesting signal, resulting in less energy consumption, in contrast to conventional activity sensors. In addition, current mechanisms rely on sequential functioning of the program, which is not applicable for intermittent operation in batteryless sensors, which experience frequent energy blackouts. Timing failure is another issue in batteryless devices due to frequent power failures and energy scarcity periods, in contrast to conventional battery operated sensing devices. Furthermore, devising task scheduling algorithms for multi-source energy harvesters is more challenging due to the varying amount of harvested energy and context information from each harvester unit in real-time. Moreover, the predicted harvested energy plays an important role in devising task scheduling algorithms, as discussed in Section [Schedulinginehiot]. However, there is no energy prediction algorithm for kinetic, thermal and RF energy, in particular, for applications which involve mobility. Furthermore, there is no existing mechanism to manage the additional harvested energy in energy positive sensors, which may lead towards of sensors. Finally, most of the current task scheduling algorithms are validated through computer simulations without implementing on real hardware test bed. Based on the comprehensive discussion of existing task scheduling algorithms in Section [Schedulinginehiot] and aforementioned challenges, we describe the possible solutions and future directions for incorporating energy harvesting based sensors in the next subsection. Possible solutions ------------------ Task scheduling algorithms should be devised to focus both on enhancing the context information as well as the system lifetime. Among the available options, duty cycling is the most appropriate choice for scheduling the tasks on the miniaturized and resource-constrained sensor node due to its inherent lower complexity and ease of implementation. It is also compatible with the batteryless sensors due to their transient and intermittent operation. In addition, it is also important to devise harvested energy prediction algorithms to allow scheduling of tasks according to the harvested energy. As energy prediction algorithms consume a significant amount of energy during their operation, another approach is to devise task scheduling algorithms without quantifying the future energy, which can reduce the energy consumption of miniaturized sensor nodes, resulting in sustainable operation of the system. It is also important to analyse the complexity and overhead in terms of energy for running task scheduling algorithms before implementation on real resource-constrained sensor nodes. Depending upon the type of application and nature of the environment in which energy harvesters are employed, task scheduling algorithms can be adapted in real-time. Furthermore, in case of multi-source energy harvesters, it is important to switch between the most suitable energy harvesters depending upon the context information and harvested energy from the individual transducer output signals in real-time. In energy harvesting based sensing, context information and the amount of harvested energy are correlated. Therefore, energy can be saved by reducing the sampling rate of the sensing signal during the steady state, without losing context detection accuracy. Moreover, the additional harvested energy can be employed to run other pending tasks on the node, such as data processing and communication, which leads towards the potential of in. Therefore, new operating systems and programming frameworks are needed for running energy harvesting based sensors, which take into account the intermittent operation of batteryless sensors, to resolve the checkpointing and timing failure issues, which are not present in conventional battery-based devices. Finally, there is a potential of task scheduling algorithms for energy harvesting based sensors, to meet the objectives of both maximized sensing performance as well as ensuring of sensor nodes in. Future Research Directions ========================== We discussed that energy harvesters can be employed as a simultaneous source of energy as well as context information. Since this new class of sensors has inherent differences than conventional sensors, it brings opportunities for dedicated energy management algorithms for its perpetual operation. Therefore, conventional task scheduling algorithms need to be revised due to the correlation between energy and information in energy harvesting based sensing. As the harvested energy is limited, task scheduling algorithms are required to execute tasks on the node according to harvested energy profile, for achieving of sensor nodes in. There are various challenges in devising efficient task scheduling algorithms for energy harvesting based sensing due to the varying and intermittent harvested energy that puts further constraints in case of batteryless sensors. The goal of task scheduling algorithms is to enhance the activity detection performance as well as the operational lifetime of sensor nodes in. Based on the previous discussion, future research directions for ensuring higher activity detection accuracy as well as achieving of sensor nodes are described in the following subsections in detail. Exploring optimal sampling frequency of sensing signals ------------------------------------------------------- In order to extract context information, the sensing signals from the energy harvesting circuit are sampled, whereas processing can be done offline or on the device, depending on the energy budget. The sampling frequency of the energy harvesting signal is an important parameter which plays an important role in the activity detection accuracy. Increasing the sampling frequency results in higher energy consumption with the advantage of higher activity detection accuracy. Therefore, there is a potential to study the optimal sampling frequency of various sensing signals in terms of energy consumption as well as activity detection accuracy. As the harvested energy varies according to the type of application, the sampling frequency also varies accordingly which needs to be studied in detail. Maximum Power Point (MPP) tracking ---------------------------------- As the output voltage and current of the energy harvesting transducer changes due to the environmental conditions, its also varies swiftly under the external stimuli. tracking of transducer is more challenging than solar cells, due to rapid output voltage/current fluctuations in the former compared to the relatively slow variations in the latter. Therefore, sophisticated hardware modules are required which can dynamically track its at a high frequency in run-time . However, it will also consume more energy in sampling the signal at a higher frequency to find its voltage. Therefore, there is a potential to study the optimal tracking frequency, amount of energy consumed in tracking and additional harvested energy, to estimate the overall gain. Employing multi-source energy harvesters as a simultaneous source of energy as well as context information ---------------------------------------------------------------------------------------------------------- In order to ensure the sustainable operation of sensor nodes in, the harvested energy should be sufficient to power the sensor’s hardware without the need of any external energy source (i.e., a battery). Occasionally, the harvested energy from a single energy harvester is not sufficient to continuously power each module of the sensor node. For example, can not provide sufficient energy during night and darkness. On the other hand, the harvested energy from is very small during lower vibrations/movements, such as sitting and standing in human activity recognition applications. Therefore, multi-source energy harvesters (e.g.,,, and ) can be employed to harvest higher energy as well as extract rich activity information. The energy from these harvesters can be accumulated to power a sensor node to achieve. In addition, the signals from multi-source energy harvesters can be fused to extract rich context information. For example, while provides information about the movement/activity, can be used to identify the indoor and outdoor environments, depending upon the amount of harvested energy, which can be employed in the localization applications. However, if a signal does not possess information about the underlying activity, its fusion with other signals may increase the cost, complexity and energy consumption of the system. In summary, there is a potential to study the amount of harvested energy and context detection accuracy from multi-source energy harvesters in various applications, such as human activity recognition, gait recognition and transport mode detection. Scheduling framework for energy harvesting based sensing -------------------------------------------------------- In order to ensure of sensor nodes and achieve higher activity detection accuracy, the harvested energy should be consumed efficiently to run a maximum number of tasks on the sensor node in. In energy harvesting based sensors, the execution of tasks is a function of context information, harvested energy and required energy. However, the previous scheduling schemes employ conventional motion sensors, instead of using the energy harvesters as activity sensor and energy source simultaneously. Furthermore, multi-source energy harvesters can be used to extract higher energy as well as rich context information, which brings new challenges due to the different amount of energy and context information from each harvester unit. This new class of sensors (i.e., sensing using energy harvesters) brings opportunities for revised task scheduling schemes, due to the correlation between harvested energy and context information. Eventually, it will enable the perpetual operation of sensor nodes along with higher detection accuracy of the underlying activity, without the need of any external depletable energy source (i.e., a battery). Considering energy harvesters as activity sensors and source of energy simultaneously, the key research challenges for devising task scheduling algorithms are as follows: * Devising a duty cycling algorithm for the sensor node according to the harvested energy profile, which provides highest activity detection accuracy with minimum power consumption. * Studying the relation between the harvested energy and the sampling frequency of the energy harvesting signal. * In order to acquire actual and real pattern of the energy harvesting signal, the energy harvesting transducer can be disconnected from the capacitor/load during sampling the signal for context detection applications. However, it may result in lower harvested energy with the advantage of higher activity detection accuracy, which needs to be studied in detail. * In case of multi-source energy harvesters, there is a need to devise an optimal sampling frequency for each energy harvesting signal. * Exploring the information gain of each energy harvesting signal in multi-source energy harvesters and fusing the most information-rich signals, that provide highest activity detection performance, with minimum energy consumption. * Processing the energy harvesting signals and extracting the dominant features on the resource-constrained sensor node. This may result in lower energy consumption in data transmission (due to smaller feature set) with higher energy consumption in processing the data on the sensor node. * Implementing the classification algorithm on the sensor node for online (real-time) activity detection, resulting in autonomous operation of sensor nodes in. * Devising a scheduling framework in batteryless sensors that work intermittently under limited and un-reliable harvested energy. * Studying the trade-off between the online and offline processing of various sensing signals, collected from the energy harvesting transducers. * In case of lower harvested energy, the collected data or extracted features may not be transmitted to the server for offline processing. Therefore, array of energy harvesters can be employed to harvest higher energy with the overhead of increased cost and form factor. In summary, considering the aforementioned research challenges while devising task scheduling schemes will ensure the ultimate goals of higher context detection accuracy as well as of sensor nodes in. [t!] [tableCompPredEH] ll Solar energy harvesting & Kinetic energy harvesting Periodic & Aperiodic Relatively stable & Relatively unstable Easily predictable & Difficult to predict Higher energy than & Lower energy than Easy to design the power & Difficult to design the power conditioning circuit & conditioning circuit Power is generated in the & Power is generated in the presence of light & presence of vibrations Relatively less noise in the & Relatively more noise in the signal & signal Generates DC voltage & Generates AC voltage Predicting the harvested energy in ---------------------------------- In contrast to, the harvested energy in is highly fluctuating, which varies quickly according to the underlying stress/vibrations. This unstable output voltage poses more challenges to devise a prediction model for the harvested energy. There is a significant difference between the energy pattern in and, as listed in Table [tableCompPredEH]. The harvested energy in is predictable due to its overall known pattern as described in Section [subsec:predictingharvestedenergySEHIoT]. However, in contrast to, the harvested energy in can not be predicted by merely accumulating the previous weighted energy samples, as in , especially for long term energy predictions. The reason behind it lies in the sudden changes in the harvested energy due to the change in the nature of the underlying vibration source. The harvested energy pattern in based depends upon the type of application which demands dedicated models for harvested energy prediction for each use case. If the target application is human activity recognition, the energy is non-identical in different types of activities , including walking, standing, running, going upstairs/downstairs, etc. Similarly, if the target application is transport mode detection , the harvested energy depends on the mode of transportation. Therefore, demands different energy prediction models due to the distinct harvested energy pattern in each application. Furthermore, in contrast to static applications (except ), is typically deployed in a mobility scenario, and has various states in most of the context detection applications, including human activity recognition and transport mode detection. Therefore, in order to predict the harvested energy in, the mobility of the energy harvester must be taken into account to achieve higher prediction accuracy. Similarly, dedicated energy prediction algorithms are required for other energy harvesters, such as and to estimate the harvested energy for execution of the tasks on sensor nodes efficiently. The predicted harvested energy plays an important role in scheduling the tasks in sensor nodes. As the capacity of is limited (due to small-sized capacitors/batteries), the harvested energy can overflow if the stored energy is not consumed in executing the tasks beforehand, which results in the wastage of resources. For example, when the battery is fully charged, there is no room to store the incoming harvested energy, which results in energy wastage, if it is not properly utilised. The solution is to utilise the maximum energy, when the is charged to its capacity as well as there is a prediction of future harvested energy. It results in the efficient employment of resources and the maximum utilization of energy in executing the tasks within their deadlines. Similarly, if the is depleted, the tasks can be delayed, according to the predicted harvested energy, to achieve a minimum performance level. Batteryless ----------- Conventional employ batteries to store the harvested energy and power the sensor nodes. However, batteries are costly, bulky, toxic and have a limited lifetime . A promising solution is to use capacitors to store the harvested energy. As capacitors generally have higher leakage and lower energy storage capacity compared to batteries, task scheduling with a capacitor-based is more challenging. Intermittently charged capacitors restrict the continuous utilization of energy, which further restrain the frequent execution of tasks. Therefore, there is a potential of task scheduling algorithms that take into account the intermittent operation of load/sensor node to enable the autonomous operation of batteryless sensor nodes in with higher context detection performance. Conclusion ========== Energy harvesters are employed to power sensor nodes in to replace the conventional manually rechargeable batteries that hinder their widespread adaptability and pervasive deployment. In addition to energy generation, recently energy harvesters have been used as sensors for context detection. This saves significant energy that would otherwise be used for powering conventional activity sensors. Using the energy harvester as a simultaneous source of energy and information enables energy positive sensing, which harvest higher energy than required for signal acquisition for context detection. However, the harvested energy is still not sufficient to allow the of sensor nodes in. In order to ensure the sustainable operation of sensor nodes, the precious harvested energy needs to be consumed very efficiently for running the operational tasks on the nodes. In this survey paper, we comprehensively analyse the previous task scheduling based energy management algorithms for sensors. We critically analyse the challenges in incorporating the emerging class of energy harvesting based sensors in the conventional task scheduling algorithms. Based on the extensive study of the literature, we rigorously review the need for revised task scheduling algorithms for energy positive sensors and provide potential solutions. Finally, we present future research directions towards the goal of enabling the sustainable and autonomous operation of batteryless sensor nodes in. --- 1. The numbers are obtained from Dimensions Source: <https://app.dimensions.ai/discover/publication> Accessed on: March 16, 2020[↩](#fnref1) 2. In the literature, PMU is also called as Energy Management Unit (EMU)[↩](#fnref2) Task Scheduling for Simultaneous IoT Sensing and Energy Harvesting: A Survey and Critical Analysis ================================================================================================== The has important applications in our daily lives including health and fitness tracking, environmental monitoring and transportation. However, sensor nodes in suffer from the limited lifetime of batteries resulting from their finite energy availability. A promising solution is to harvest energy from environmental sources, such as solar, kinetic, thermal and radio frequency, for perpetual and continuous operation of sensor nodes. In addition to energy generation, recently energy harvesters have been used for context detection, eliminating the need for additional activity sensors (e.g. accelerometers), saving space, cost, and energy consumption. Using energy harvesters for simultaneous sensing and energy harvesting enables energy positive sensing – an important and emerging class of sensors, which harvest higher energy than required for signal acquisition and the additional energy can be used to power other components of the system. Although simultaneous sensing and energy harvesting is an important step forward towards autonomous self-powered sensor nodes, the energy and information availability can be still intermittent, unpredictable and temporally misaligned with various computational tasks on the sensor node. This paper provides a comprehensive survey on task scheduling algorithms for the emerging class of energy harvesting-based sensors (i.e., energy positive sensors) to achieve the sustainable operation of. We discuss inherent differences between conventional sensing and energy positive sensing and provide an extensive critical analysis for devising new task scheduling algorithms incorporating this new class of sensors. Finally, we outline future research directions towards the implementation of autonomous and self-powered. , Wearable, Energy Harvesting, Ubiquitous Computing, Sensing, Scheduling, Task Scheduling, Prediction Introduction ============ the advancements in, low power miniaturized sensors in are becoming popular for monitoring the physical attributes in various applications including surveillance, smart cities, healthcare, exploration of mines, battle field monitoring and even deep sea exploration. However, conventional sensor nodes employ a rechargeable battery, which has limited energy storage capacity and thus hinders the perpetual operation of sensor nodes in. In order to solve this problem, recently, kinetic, solar, thermal, and RF energy harvesters have been used to convert the environmental energy into electrical energy, to extend the battery lifetime of sensor nodes . This relieves the issue of limited lifetime of batteries, and thus allows the autonomous operation of sensor nodes in without human intervention. ![Energy harvesters can be used as context sensors as well as source of energy in ](into_figure_v2.png "fig:") [fig:introfigure] [t!] [tableprevioussurveys] llcccc Year & Reference & Energy harvesting & Sensing using energy harvesting & Simultaneous sensing and energy harvesting & Task scheduling 2006 & & × & × & × & 2008 & & × & × & × & 2010 & & & × & × & × 2013 & & × & × & × & 2014 & & & × & × & × 2016 & & × & × & × & 2016 & & & × & × & × 2018 & & & × & × & × 2019 & & & & × & × 2020 & Proposed & & & & Recently, energy harvesters are also being used as sensors for activity detection , as depicted in Fig. [fig:introfigure], to save sensor-related energy consumption that would otherwise be used for powering conventional activity sensors, such as accelerometers. Energy harvesters can be employed as energy efficient sensors to detect the underlying activity as well as a source of energy to power sensor nodes . In simultaneous sensing and energy harvesting paradigm, the harvested energy can be used to at least acquire the energy harvesting signal (for context detection) leading towards energy positive sensing compared to conventional energy negative sensing. This allows full utilization of energy harvesting capabilities in practical and real-world environments, leading towards autonomous operation of sensor nodes in. Background ---------- Despite the emerging importance of energy harvesting, the amount of generated energy from the environment is still insufficient to enable the of miniaturized sensor nodes , especially for wearable sensing devices with small form factor. There are various mechanisms that can be employed to enable the of miniaturized sensor nodes such as: (i) maximizing the harvested energy by implementing optimal energy harvesting mechanisms or considering multi-source energy harvesters , and (ii) minimizing the energy consumption by using novel low power sensing mechanisms such as energy harvesting-based sensing . In addition to maximum energy harvesting and low power sensing, energy management algorithms can be employed to manage the precious harvested energy, ensuring of sensor nodes . There are different types of energy management algorithms, including transmission power control , scheduling and task scheduling , to name a few. [figresearchpapers] Task scheduling algorithms schedule the broader set of tasks (such as sampling, processing, transmission, etc.) on the sensor node, according to the available energy budget, to prolong its operational lifetime. It is an effective method to minimize the energy consumption on the sensor node, due to its direct interaction with the (i.e., battery/capacitor) and energy consumption (in executing the tasks) on the processor. Task scheduling algorithms are more effective to minimize the energy consumption, than other communication-focused energy management schemes (i.e., transmission power control) due to their interaction with a broader set of tasks including digitizing, sampling, and processing as well as communication. The objective of task scheduling algorithms is to run sensor nodes using the intermittent harvested energy to ensure and achieve maximum sensing performance in applications. Therefore, in this survey, we comprehensively analyse task scheduling algorithms for energy harvesting based sensing to enable the of sensor nodes in. [t!] [tableNomenclature] ll Term & Detail VAC & AC voltage Voc & Open circuit voltage Vrec & Rectified voltage VR & Resistor voltage Vcap & Capacitor voltage Vpmu & voltage Phar & Harvested power Pacq & Signal acquisition power Motivation ---------- In recent years, there has been a growing trend in energy harvesting mechanisms to power sensors and related task scheduling-based energy management schemes, as depicted in Fig. [figresearchpapers][1](#fn1). Previously, energy harvesters have been used for activity detection , as depicted in Fig. [fig:introfigure], to replace conventional activity sensors, which operate using the supplied power. Some of the previous works also use multi-source energy harvesters to harvest higher energy as well as rich context information. However, only one proposal employs the harvested energy to power a system load. In order to make best use of energy harvesters, they must be employed as a simultaneous source of energy and context information . This enables energy positive sensing which harvest enough energy to acquire the energy harvesting signal, as discussed in Section [energypositivesensing]. Finally, task scheduling schemes can be incorporated to ensure of sensor nodes using the limited and unreliable harvested energy. The ability to extract information from energy harvesting signals, and in some cases to gain energy in the process, has the potential to dramatically change the task scheduling landscape for. Scheduling algorithms have to consider the information and energy gain (for energy positive sensing), rather than energy loss, of different sensors to achieve an objective function. This potential warrants a comprehensive survey of task scheduling algorithms to analyse their support for such decisions. State-of-the-art ---------------- There are several works in the literature that explore task scheduling in. Table [tableprevioussurveys] compares our paper to previous survey papers related to with energy focused research. Some of the previous works present extensive surveys on task scheduling schemes to minimize energy consumption in. However, none of theses surveys considered energy harvesting mechanisms to power sensor nodes with the associated opportunities and challenges compared to battery-powered. Others survey papers covered energy harvesting mechanisms to power sensor nodes in order to enhance their operational lifetime and to reduce maintenance cost. Nevertheless, they considered conventional sensors instead of energy harvesting-based sensing. A recent comprehensive survey is the first to cover energy harvesting-based sensing while optimizing sensing, computing and communication for. However, simultaneous sensing and energy harvesting as well as the resulted energy positive sensing concept has not been discussed. Moreover, none of previous survey papers , which focus on energy harvesting research, considered task scheduling as a crucial mechanism to manage execution of tasks under the limited and time-varying harvested energy. Contributions ------------- To address the aforementioned gaps in the literature, this work critically surveys task scheduling schemes to minimize the energy consumption for perpetual operation of sensor nodes in, and analyses their potential to support energy-harvesting based sensors. Our contributions are as follows: * We discuss the implementation of energy harvesters as sensors and energy sources simultaneously, which is more useful in practical environments leading towards self-powered batteryless. We also explore the concept of energy positive sensing, which uses the harvested energy to acquire the energy harvesting signal for sensing, eliminating the need for additional power consuming sensors. * We present an extensive discussion of task scheduling based energy management algorithms for running the tasks on the resource constrained sensor node under limited and varying harvested energy due to unreliable environmental energy (i.e., kinetic, solar, thermal, RF, etc). * Based on an extensive study of the literature, we comprehensively describe the key challenges and potential solutions when integrating energy positive sensing with conventional task scheduling algorithms. * Finally, we present the future research directions to enable the sustainable and autonomous operation of batteryless sensor nodes in. The remainder of this paper is organized as follows: Section [sensingandenergyharvesting] comprehensively describes the mechanism of simultaneous sensing and energy harvesting with the resulting concept of energy positive sensing. Task scheduling algorithms for are presented in Section [Schedulinginehiot] along with energy prediction algorithms, to ensure the perpetual operation of sensor nodes in. Section [sec:challengesandopputunities] critically analyses the challenges and opportunities for devising task scheduling algorithms for energy positive sensing. Future research directions are described in Section [FutureResearchDirections] and finally, Section [conclusion] concludes the paper. Table [tableNomenclature] provides the nomenclature used in this paper. [figsensing] [ht] [tablesensingpapers] \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Year \| Reference \| Energy harvester \| Sensing based on \| Target application \| Energy stored \| Load \| Algorithm \| Accuracy \| \| \| \| \| \| \| \| \| implementation \| \| \| 2012 \| \| Thermal \| Voc & Vcap \| Water flow detection \| No \| No \| Offline \| – \| \| 2013 \| \| Kinetic \| Vrec \| Airflow speed monitoring \| Yes \| No \| Offline \| – \| \| 2014 \| \| Thermal \| Vcap \| Water and appliance metering \| No \| No \| Offline \| – \| \| 2015 \| \| Thermal \| Signal pulse \| Chemical reaction detection \| No \| No \| Offline \| – \| \| 2015 \| \| Kinetic \| Voc \| Human step count \| No \| No \| Offline \| 96 % \| \| 2015 \| \| Kinetic \| Voc \| Calories burnt \| No \| No \| Offline \| – \| \| 2015 \| \| Kinetic \| Voc \| Food intake detection \| No \| No \| Offline \| 86 % \| \| 2015 \| \| Kinetic \| Voc \| Human activity recognition \| No \| No \| Offline \| 99 % \| \| 2015 \| \| Kinetic \| Voc \| Human activity recognition \| No \| No \| Offline \| 83 % \| \| 2016 \| \| Kinetic \| Voc \| Hotword detection \| No \| No \| Offline \| 85 % \| \| 2016 \| \| Kinetic \| Voc \| Human activity recognition \| No \| No \| Offline \| 91 % \| \| 2016 \| \| Kinetic a \| Accelerometer \| Train route detection \| – \| – \| Offline \| 97 % \| \| 2016 \| \| Kinetic \| Voc \| Transport mode detection \| No \| No \| Offline \| 85 % \| \| 2016 \| \| Kinetic \| Voc \| Ball impact on racket \| No \| No \| Offline \| – \| \| 2016 \| \| Kinetic \| Vcap \| Human step count \| Yes \| No \| Offline \| 89 % \| \| 2017 \| \| Thermal \| Signal pulse \| Chemical reaction detection \| No \| No \| Offline \| – \| \| 2017 \| \| Kinetic \| Vcap \| Human activity recognition \| Yes \| No \| Offline \| 96 % \| \| 2017 \| \| Solar \| Voc \| Hand gesture recognition \| No \| Yes \| Offline \| – \| \| 2017 \| \| Kinetic b \| Voc \| Human gait recognition \| No \| No \| Offline \| 95 % \| \| 2017 \| \| RF \| Signal phase \| Touch detection \| No \| No \| Offline \| – \| \| 2017 \| \| Kinetic c \| VR \| Voice demodulation \| No \| Yes \| Offline \| – \| \| 2018 \| \| Kinetic \| Vcap \| Human activity recognition \| Yes \| No \| Offline \| 95 % \| \| 2018 \| \| Kinetic \| VAC & Vcap \| Human gait recognition \| Yes \| No \| Offline \| 86 % \| \| 2018 \| \| Kinetic \| Voc \| Human activity recognition \| No \| No \| Offline \| 95 % \| \| 2018 \| \| Kinetic \| Voc \| Knee surgery monitoring \| No \| No \| Offline \| – \| \| 2018 \| \| RF \| Signal phase \| Hand gesture sensing \| No \| No \| Offline \| – \| \| 2019 \| \| Solar \| Voc \| Hand gesture recognition \| No \| No \| Offline \| 96 % \| \| 2019 \| \| Kinetic \| Vcap \| Transport mode detection \| No \| No \| Offline \| 92 % \| \| 2019 \| \| Kinetic b \| Voc \| Human gait recognition \| No \| No \| Offline \| 96 % \| \| 2019 \| \| Kinetic \| Voc \| Heart beat monitoring \| No \| No \| Offline \| – \| \| 2019 \| \| Solar & Kinetic \| Voc \| Recognizing places \| No \| No \| Offline \| 88 % \| \| 2020 \| \| Kinetic \| Current \| Human activity recognition \| Yes \| Yes \| Offline \| 97 % \| , c Uses the output power measured across a resistor, which connected between both terminals of the transducer Simultaneous Sensing and Energy Harvesting ========================================== In addition to energy harvesting, a recent trend is to use the output signals from energy harvesters for extracting the context information, as shown in Fig. [figsensing](b) compared to using dedicated sensors (e.g. accelerometer, magnetometer, etc.), as depicted in Fig. [figsensing](a). Thanks to the varying nature of the harvested energy, it contains the context information about the environment in which the energy harvester is deployed. For example, the harvested voltage from transducer contains information about the temperature of the environment. Similarly, using the harvested energy from transducer, the indoor and outdoor environments can be differentiated . Likewise, if the transducer is placed on the wrist of human body, the output signal provides information about the type of the underlying activity . It is based on the phenomenon that experiences distinct vibration patterns during different human activities (e.g., walking, running, sitting, standing, etc.) of the human body. These different types of activities leave their distinct signatures in the output signal of. By analysing the output signal from the energy harvester, it is possible to find the type of activity performed. The principal advantage of using the energy harvester for context detection lies in its sensor related power saving, as compared to the conventional activity sensors (such as accelerometers and magnetometers), which operate on the supplied power from the ESU, as illustrated in Fig. [figsensing]. Table [tablesensingpapers] comprehensively summarizes relevant recent works on using energy harvesters as sensors. This table shows interesting observations form the energy harvesting-based sensing literature. Firstly, it shows that energy harvesters are used as a source of context information in various applications, including human gait and activity recognition, health and fitness tracking, transport mode detection, as well as in localization and shadow detection (as shown in column 5). Secondly, most of the previous works focus on sensing using the signal received from or, as shown in column 3. A limited number of works employ and as a source of information due to their limited applications, weaker output signal and significant noise component. One interesting point to note is that most of the previous research is focused on extracting context information using. A small number of previous works consider and other harvesters as information sources. It can be due to the reason that harvested energy from is relatively stable as compared to and does not contain a fluctuating, variable and fine grained pattern compared to the latter. In addition, potentially can only be used in applications where the context information is contained in changing light conditions and thus it is not applicable in dark environments, such as at night. While most of the previous works use single energy harvester, only one work considers multi-source energy harvesting based sensing. Authors in employ and to sense different types of environment (i.e., indoor and outdoor), and types of human activities (sitting, walking, running, etc.) respectively. Multi-source energy harvesting based sensing is useful in applications where the objective is to identify different types of environments that offer changing conditions, such as, light intensity (solar), movement (kinetic) and temperature (thermal). In these applications, one type of energy harvester may not capture the distinct pattern for each type of activity. However, combining the information from multi-source energy harvesters provides rich context information, which can distinguish various types of activities with reasonable accuracy. There are different types of signals in the energy harvesting circuit, such as open circuit voltage Voc, rectified voltage Vrec and capacitor voltage Vcap, as shown in Fig. [figsensingoptions], that contain the required context information . From Table [tablesensingpapers], we can observe that most previous works use open circuit voltage, while a limited number of works employ the capacitor voltage for context information extraction as presented in column 4. Apart from Voc and Vcap, there is only one work which uses rectified voltage Vrec for sensing. Additionally, only a limited number of recent works store the harvested energy in the, as shown in column 6 of Table [tablesensingpapers]. This stored harvested energy can be used to power the sensor node or its selected module, depending upon the amount of harvested energy. In addition, the stored energy can be used as an information source . Furthermore, in addition to sensing using energy harvester, none of previous works use the harvested energy to run a system load except , as illustrated in column 7 of Table [tablesensingpapers]. Although some previous works use the stored energy for sensing, they still do not employ this energy to power any system load , except , which uses two separate transducers one for sensing and the other for energy harvesting. Therefore, it ignores the potential of using both transducers for concurrent sensing as well as energy harvesting. Moreover, and use the harvested energy from and respectively, to transmit a signal without storing it in an. The received signal indicates that the (coarse-grained) activity is triggered. It does not extract information from the harvesting signal, instead it uses the harvesting energy to transmit a pulse. The activity is detected on the receiver end by analysing the characteristics of the activity pulse, such as its strength and time duration. On the other hand, when a load is used with the energy harvester, it changes (distorts) the harvesting voltage waveform. As a result, it may affect the information content in the harvested signal. As energy scavenging is generally the major application of energy harvesters, there is only one work that focuses on simultaneous sensing and energy harvesting to detect the activity and to power the dynamic load respectively. [figsensingoptions] The second last column in Table [tablesensingpapers] depicts that all of the previous works implement their proposed algorithms offline. This is due to the reason that most of the previous works do not harvest energy, instead they use open circuit voltage for context detection. Although some other works store the harvested energy in a capacitor, they manually discharge the capacitor, instead of using a realistic dynamic load. In addition, the harvested energy from tiny energy harvesters is very small and limited. In addition, due to the non-ideal s, some of the harvested energy is wasted via leakage. Furthermore, there is also some energy loss during the charging/discharging process in the. Therefore, it is impractical to implement computationally expensive algorithms on the node, with intermittent and unreliable harvested energy from the environment. However, sophisticated energy harvesting techniques can be incorporated to maximize the harvested energy that can realize the concept of online processing on the sensor node. As the available energy is precious and limited, it must be used efficiently in applications demanding less energy consumption and operating at a low duty cycle. Furthermore, the harvested energy should match the application’s energy consumption profile for of the system. Finally, most of the previous works attain reasonable context detection accuracy ( > 80%) for energy harvesting based sensing using well-known machine learning classification algorithms. It proposes that energy harvesting based sensing can be used in place of conventional power hungry activity sensors (such as accelerometers) to save the energy while attaining reasonable context detection accuracy. In order to further enhance the context detection accuracy, deep learning and neural network based models can be employed , which have shown promising results in various applications, such as speech recognition , face recognition and natural language processing . In contrast to , this survey focuses on the use of energy harvesters as a source of energy and context information simultaneously. This results in full utilisation of energy harvesters, reducing form factor, cost and weight. In addition, we explain the use of harvested energy to acquire the energy harvesting signal (for context detection) leading towards energy positive sensing . In order to provide adequate detail to the reader, we comprehensively explore both of these emerging mechanisms in the following subsections. [t!] [figEHmechanism] Sensing and energy harvesting concurrently ------------------------------------------ Energy harvesters can be used as sensor as well as source of energy to detect the underlying activity and to power the sensor nodes. The generated energy from the energy harvesting transducer is stored in an, which is later used to run a system load. Fig. [figEHmechanism] highlights the general energy harvesting mechanism using a DC-DC boost converter in the [2](#fn2). The harvested energy is first stored in a capacitor and then, it is used to run hardware device. It is evident from Table [tablesensingpapers] that most of the previous work is focused on using AC or capacitor voltages for extracting the activity information from the energy harvester. Although it is a widely used practice in the literature, it is practically inefficient for real-world scenarios. This is because energy harvesters are not being used as a source of energy in the open circuit configuration. Similarly, the previous schemes that employ the capacitor voltage for sensing manually discharge the capacitor, instead of using a real practical system load (such as a sensor node). Fig. [sensingtrend] depicts the basic building blocks of an energy harvesting circuit, which include a transducer, and an. When a load is connected to the energy harvesting circuit, it modifies the shape of the input AC as well as capacitor voltages. It is due to the reason that AC voltage of the transducer depends on the capacitor voltage as: VAC = Vcap + Vpmu where, Vpmu is typically equal to the voltage drop across two diodes in the bridge rectifier. The shape of the harvesting AC voltage depends upon the capacitor voltage, which may impact its embedded information content, compared to the original open circuit AC voltage. Similarly, when a load is connected to the capacitor, it discharges the capacitor after irregular time intervals, instead of fixed manual discharging . The reason behind it lies in the distinct vibration pattern, for example in, during various activities, which produces different voltage levels at the output of the transducer; thus charging the capacitor at a different rate. Therefore, the harvested energy (stored in the capacitor) distorts the generated AC signal according to Eq. [eq:voltageaccappmu], if energy harvesting and sensing is performed using the same. Therefore, the authors in employ two transducers to procure accurate context information in the presence of a distorted AC signal. They design a hardware prototype which is embedded in a shoe and contains two transducers mounted in the front and rear of a shoe. The harvested AC voltage increases in amplitude due to the rise in the capacitor voltage, as depicted in Fig. [figsehssignaldistortion]. It is due to the fact that the charging current decreases when the capacitor is being charged to a higher voltage level. Furthermore, has a significantly high internal resistance of the order of MΩ and the output voltage is determined by the load resistance as well as the internal resistance of . As the current flow is decreased, the voltage on the internal resistance also decreases, enhancing the corresponding output voltage. This modified (and distorted) AC signal may have less information content as compared to the original AC signal. The authors in propose a filtering algorithm to remove the effect of capacitor voltage on the harvesting AC voltage to enhance the gait recognition accuracy. However, this filtering algorithm is difficult to implement on miniaturized and resource constrained sensor nodes, in real practical environments with a limited and time-varying supply of harvested energy. Therefore, more sophisticated algorithms are needed to extract useful information from the harvesting signals in the presence of an energy harvesting circuit, while powering a realistic intermittently-powered load. This intermittent operation is due to the use of capacitors as, which store a small amount of energy to run at most one atomic task, in contrast to batteries which can power nodes continuously for a longer duration. One drawback of employing capacitor voltage for activity detection is that the harvested energy, in some applications, is not enough to quickly charge the capacitor. In other words, sometimes, it takes longer to charge the capacitor up to a certain required load voltage (especially under lower vibrations), which may introduce delay in activity detection. One solution is to employ a smaller capacitor, that can be charged quickly at the cost of smaller energy burst for the load which is enough to run the node for executing at least one atomic task. In summary, it is important to devise a sensing mechanism, which provides activity information using the altered harvested signal in the presence of a capacitor and system load. This configuration will eventually let the energy harvester to work as both source of energy and information simultaneously, leading towards autonomous sensors in. Authors in do not employ the harvested energy to power a system load, instead, they discharge the capacitors manually, which are recharged again with the harvested energy from the transducers. Sandhu et al. employ energy harvesters as sensors and source of energy simultaneously. They use two as load, which are powered using the stored harvested energy from the transducer. In addition, they explore various sensing points in the energy harvesting circuit, that offer two types of signals: current and voltage. They evaluate the sensing potential of various available signals in the energy harvesting circuit and show that the harvesting current signal offers highest activity classification accuracy, while powering a dynamic load. Energy positive sensing ----------------------- Simultaneous sensing and energy harvesting enables energy positive sensing – an important and emerging class of sensors, which harvest higher energy than required for the signal acquisition. This additional harvested energy can be used to power other hardware modules on the node, moving towards. Eventually, it leads towards the design and implementation of autonomous sensor nodes in, that solely operate using the harvested energy and employ the energy harvesting transducers as sensors for a theoretically indefinite lifetime. Sandhu et al. are the first to explore the energy positive sensing concept in transport mode application. The energy harvesting signal needs to be digitized before processing by the embedded device. In digitizing the harvesting signal, an consumes, what we call, signal acquisition power . This signal acquisition power can be lower or higher than the harvested power, depending upon the characteristics of the transducer and environment in which it is employed. The Acquisition Power Ratio (APR) is quantitatively defined as: $$APR = \frac{P\_{har}}{P\_{acq}} \label{eq:apr\_energy\_positive sensing}$$ Depending on the amount of harvested energy, sensor nodes fall into three major categories. * Energy positive sensing (if APR > 1) * Energy neutral sensing (if APR = 1) * Energy negative sensing (if APR < 1) Energy negative sensors have lower harvested energy than required by the to digitize the sensing signal and thus need an external energy source to power the nodes. Energy neutral sensing lies at the boundary between energy negative sensing and energy positive sensing, and has less practical significance. Energy positive sensor is an important and emerging class of sensors, which harvest higher energy than required for the signal acquisition, which can be used to power other hardware modules on the node. Sandhu et al. extensively explore energy positive sensing mechanism using hardware prototypes. They use transport mode detection as a case study and employ two different designs for the energy harvesting circuit: with and without DC-DC boost converter. They collect data from six types of transport modes, including ferry, train, bus, car, tricycle and pedestrian movement. In addition to the harvesting signals, they use the harvested energy to power a dynamic load. Their results show that, on average, DC-DC converter based energy harvesting circuits harvest more energy than required for signal acquisition. Thus, it can lead towards the aforementioned energy positive sensing paradigm, resulting in self-powered and autonomous operation of batteryless sensor nodes in. [sensingtrend] Summary and insights -------------------- Energy harvesters serve the dual purpose of energy generation as well as context detection, addressing the requirement of conventional activity sensors, such as accelerometers, magnetometers and gyroscopes. This mechanism saves energy that would otherwise be consumed in powering the conventional activity sensors. Most of the previous works employ transducers for context detection due to their wide applicability and strong output signal. In order to further save the energy consumption, the stored energy in the can be sampled at a lower sampling rate for context detection, at the cost of higher latency. However, the harvested energy is still not sufficient to continuously power the sensor node using miniaturized energy harvesters, such as placed on the wrist in human fitness tracking applications. This opens the door for energy management algorithms, that schedule the execution of tasks on the node, according to the harvested energy profile, to prolong the operational lifetime of the system. Considering the energy harvesting based sensing mechanism, in the next section we comprehensively survey various task scheduling algorithms that manage the time-varying and limited harvested energy on the miniaturized sensor nodes in to enhance their operational lifetime. Task Scheduling in ================== The harvested energy from miniaturized sensors in is not sufficient to power the sensing hardware continuously without any interruption. Therefore, task scheduling based energy management schemes are required in that ensure the efficient utilization of harvested energy. This section surveys existing task scheduling schemes for, with a particular focus on schemes that have the potential to support sensing modalities that harvest energy and sense simultaneously. These algorithms ensure optimal utilization of harvested energy to extend system lifetime as well to provide highest activity detection/monitoring performance. [figsehssignaldistortion] Task scheduling schemes are devised to manage the execution of tasks on the node to maximize the performance in terms of activity monitoring within the limited available energy budget. This means that the objective of task scheduling schemes is to minimize the energy consumption and deadline miss ratio, maximize the harvested energy and ensure the operation of sensor node for extended time period with higher accuracy of activity monitoring/detection according to the type of application. The most common types of tasks on the node include sampling of information/signal, digitizing, processing, storing, data transmission and reception, as displayed in Fig. [fig:typesoftasks]. According to the type of application, the tasks are scheduled by the task scheduler for the execution by the processor on the sensor node at different frequency. The task scheduler takes various parameters into account, while deciding the execution of upcoming tasks on the node, as depicted in Fig. [figscheduling]. The figure shows that the task scheduler considers the energy budget, task deadline, predicted energy and type of task(s), while scheduling the execution of tasks on the node. If the available energy is insufficient, energy-intensive tasks can be decomposed into smaller subtasks, which consume less energy during their execution. In the previous literature, the task scheduling techniques are based on the following key principles: * Tasks are queued according to their priority and order of their deadlines. * As long as sufficient energy is available, a high priority task is executed depending on the amount of harvested energy. * If the harvested energy is not enough to run a high priority task, the next task in the queue is executed. * Low priority tasks are executed, if there is energy left after the execution of high priority tasks. * If sufficient energy is not available, the tasks are delayed until their deadlines, to allow the transducer to harvest energy. * If a high priority task arrives during the execution of a low priority task, the low priority task is pre-empted (according to the type of application), to execute the high priority task. * Tasks are scheduled according to the available energy, predicted harvested energy, required energy and deadlines of tasks. * Tasks can be broken down into smaller subtasks, which consume small amount of energy and take less time during their execution. Later, these subtasks can be combined, according to their similarity, to reduce the energy consumption in frequent switching of the hardware from idle mode to active state. Fig. [figtaskscheduling] displays a summary of major task scheduling algorithms in. In a nutshell, there are three major strategies employed for devising a task scheduling algorithm for sensor nodes in : * * Decomposing and combining of tasks * Duty cycling Table [tablesurveyedpapers] provides a brief overview of different task scheduling schemes in the literature which use one or more of these strategies. The goal of the scheduling schemes is to run the sensor node within the limited available energy budget to meet tasks deadlines, enhance the performance in terms of activity detection and to achieve. Table [tablesurveyedpapers] comprehensively describes the state-of-the-art related to task scheduling schemes, their performance metrics and their evaluation methods (i.e., simulation or hardware implementation). Some of the previous works employ to manage the harvested energy and schedule the tasks on the node by dynamically adjusting the voltage and frequency in, as discussed in detail in Section [DVSF]. The nodes are equipped with energy harvesters that generate energy to run the tasks, and the node switching is controlled using. Furthermore, the voltage level can be adjusted to power the active hardware module only, instead of the whole node, to minimize the energy consumption. Secondly, larger tasks can be decomposed into smaller subtasks that consume less energy during their execution, as described in Section [Decomposingandcombiningtasks]. Identical tasks can be grouped together to save energy that would otherwise be consumed in repeated switching of the node’s hardware. For example, instead of transmitting two data packets separately, they can be merged to save the energy required to initialize the radio transceiver. Duty cycling is another task scheduling mechanism that allows controlling the consumed energy by the nodes when they are not performing any useful operation, as discussed in Section [subsection:dutycycling]. The nodes in are turned on to execute the tasks according to the harvested energy, type of the task and energy demand. This results in lower energy consumption as nodes are switched to low energy modes during their idle time slots. [fig:typesoftasks] The predicted harvested energy plays an important role in scheduling the tasks on the node, when the available energy is scarce. Table [tablesurveyedpapers] shows that there are two main approaches for considering the predicted harvested energy in the previous works: devising a model for energy computation (C) or using previous available algorithms (R). Most of the previous works devise an energy prediction model, which computes the future harvested energy using the previous harvested energy samples and environmental parameters, as discussed in Section [subsec:predictingharvestedenergySEHIoT] in detail. Low priority tasks that require higher energy can be delayed if there is a prediction of higher harvested energy in the future. This allows to utilize the available limited harvested energy for running the high priority (and low energy) tasks without violating their deadlines. Another objective of scheduling schemes is to meet tasks deadline. High priority tasks are executed ahead of low priority tasks to meet the requirements. Depending on the application, the low priority tasks can be pre-empted during their execution, when high priority tasks arrive at the task scheduler. Finally, tasks are executed within the available energy budget to achieve on the sensor node. Tasks are scheduled according to the available energy and required energy to run the tasks, as shown in Table [tablesurveyedpapers]. If the available energy is lower compared to the requirement of a high energy task, a low energy task is executed, even though it has lower priority, to utilize the valuable harvested energy for useful operation. As illustrated in Table [tablesurveyedpapers], the task scheduling algorithms can be evaluated using two approaches: performing simulations or implementing in a real-world scenario. Table [tablesurveyedpapers] portrays that most of the previous works validate their algorithms using simulations instead of real hardware implementation. The last column of Table [tablesurveyedpapers] depicts the difficulty level in implementing the given task scheduling algorithms on the energy harvesting based sensing device. based algorithms are highly difficult to implement on the tiny and miniaturized sensor nodes, due to the stringent requirement of complex circuitry that provides various voltage levels for different components of the node. Similarly, the algorithms that are validated in simulations need significant effort to be implemented on the real energy harvesting based sensor, due to different hardware design for this new class of sensors that run intermittently without conventional batteries. Furthermore, the algorithms that decompose and recombine tasks may not be suitable due to limited amount of harvested energy that can be used to run, at most, one atomic task at a time. However, duty cycling based scheduling mechanisms that are implemented on real hardware platforms can be implemented on the energy harvesting sensor. It is due to their ease of implementation and compatibility with intermittently powered sensor nodes. However, it needs significant revision as the energy harvester acts as a sensor and source of energy simultaneously without conventional ; therefore, we place them in medium difficulty level, as displayed in Table [tablesurveyedpapers]. Most interestingly, none of the previous task scheduling algorithms employs energy harvesters as activity/motion sensors. All existing task scheduling schemes employ energy harvesters as a source of power and use conventional sensor modules for the target application, consuming considerable energy during their operation. This puts further pressure on the limited stored harvested energy, due to the requirement to power both the sensor as well as the processing unit, which reduces the overall system lifetime. Therefore, there is a potential to devise dedicated task scheduling algorithms that employ energy harvesters as sensors and a source of energy simultaneously . The objective of these task scheduling schemes is to maximize the lifetime of sensor nodes as well as to achieve the required performance level, using the intermittent and limited harvested energy. Interestingly, this harvested energy has some synchronization with the underlying activity and thus energy is harvested when an activity is performed. Later, this harvested energy is employed to sample the energy harvesting signal to identify the type of activity. This leads towards the possibility of autonomous and self-powered systems, due to the elimination of sensor-related energy consumption by exploiting the embedded information in the energy harvesting signals for context detection applications. [ht] [figscheduling] Dynamic voltage and frequency scaling ------------------------------------- The power consumption of the sensor hardware depends on the supplied voltage (as well as the current flow) and the operating frequency (i.e., the clock frequency). Therefore, both of these parameters can be adjusted in real-time to optimize the power consumption of the sensor node. The objective of is to maximize the performance given the limited energy budget or to minimize the energy consumption under a performance bound. Authors in present a task scheduling technique using for rechargeable real-time systems. The tasks are grouped according to their deadlines and are executed if the available energy is higher than a certain minimum threshold. Liu et al. propose a scheduling mechanism which slows the processing of tasks according to the available as well as the predicted harvested energy in the future in based systems. The processor frequency depends on the deadline of tasks and the available energy budget. The task is executed at increasing speed as it approaches its deadline. Authors in decouple the energy and timing constraints to simplify the scheduling problem. In addition to the use of, they delay the execution of high energy tasks, if the stored energy is not enough, until sufficient energy is harvested for the execution of tasks on the node. A algorithm using load matching and considering battery charge/discharge overhead is presented in . The load matching is important for maximum power output from the energy harvesting transducer. Since, some amount of the harvested energy is wasted in battery leakage, the algorithm in decides between the stored energy or direct use of the harvested energy in running the sensor node. This saves the energy that would otherwise be wasted due to the non-ideal. A based task scheduling algorithm for structural health monitoring is presented in . In this scheme, both periodic as well as sporadic tasks are scheduled using a linear regression based algorithm. The sporadic tasks are executed according to their energy and timing constraints to maximize the. Liu et al. combine the static and adaptive scheduling techniques with to attain high performance with energy and timing constraints. Their algorithm adaptively schedules the tasks when there is a prediction of energy overflow, to achieve the maximum benefit from the harvested energy. It also considers the future harvested energy during the scheduling of tasks. Lin et al. propose a task scheduling framework based on. They track the of the to harvest maximum energy from a solar cell. The tasks are scheduled according to the predicted energy, available energy budget and the deadline of tasks to minimize the task drop ratio. Tan et al. model the energy harvesting system as an energy model, a task model and a resource model and present a task scheduling algorithm based on. The method combines the free dispersed time slots together to harvest maximum energy for running the tasks, which results in the execution of a higher number of tasks within the limited energy budget. algorithms are not suitable for scheduling tasks on energy harvesting based sensors due to the resource-constrained hardware. As the energy harvesting circuit is intentionally kept simple (to avoid energy losses), it can not provide various voltage levels to execute tasks on the sensor node. In addition, the miniaturized sensor nodes can easily operate using a fixed voltage level, while drawing different amount of current during various operations, which reduces the power (and energy) wastage. Therefore, alternate task scheduling algorithms are required that work seamlessly without additional overhead in terms of energy and resources, to minimize the energy consumption on energy-constrained sensor nodes. Decomposing and combining of tasks ---------------------------------- As the harvested energy in is limited, it is difficult to run the power intensive tasks continuously. Nevertheless, if such tasks are executed, the remaining energy may not be sufficient to run the future incoming time-critical tasks. A promising solution of this issue is to decompose the energy-intensive tasks into multiple subtasks which demand lower energy during their operation. For example, the task of transmitting the sensed data can be decomposed into two separate subtasks of sensing and data transmission. Similarly, the transmission of a saved parameter (in memory) can be decomposed into the separate subtasks of reading the attribute from memory and data transmission. In order to minimize energy consumption, the two transmission subtasks can be combined together by grouping the data from these subtasks and transmitting it in one data packet, as shown in Fig. [figdecomp] . Zhu et al. , decompose larger tasks into multiple subtasks depending upon their ability to recombine with other smaller tasks to save energy. In summary, the decomposing and combining technique in consists of the following four phases: ### Decomposition This phase decomposes the energy-intensive tasks into multiple subtasks depending upon their ability to group with other subtasks to conserve energy. As the harvested energy is not sufficient to run the high power tasks continuously, subtasks can be executed with the limited available energy budget. ### Combining This phase combines multiple subtasks that can be executed on the same processor to minimize the energy consumption. In addition, some tasks can be executed concurrently to reduce the idle time of the processor. For example, the tasks of sensing and reading a value from memory can be executed simultaneously as they utilize different resources of the node, depending on the harvested energy. The advantage of the concurrent execution is the reduced delay and smaller latency in data transmission. However, it also demands higher energy which is available only once in a while in. ### Admission control In this phase, the tasks are filtered according to their priority and energy consumption during their execution. Although tasks are combined to save energy, generally, the harvested energy is not enough to run all ready-to-execute tasks. Therefore, an admission controller further filters the tasks based on the following task-specific parameters: * Priority of tasks * Available harvested energy * Energy consumption of tasks Task priority is important in all applications and in particular for time-critical real-world scenarios. Depending upon the application, the task’s deadline is further categorized into two types, i.e. soft deadline and hard deadline. In general, soft deadlines are less critical as compared to hard deadlines and their violation does not harm the functioning of the system. On the contrary, hard deadlines are essential to be respected in all circumstances, which create major loss if ignored or violated . Therefore, the admission controller arranges the tasks according to their priorities and deadlines. In addition to the task priority, the available energy and task energy consumption are also important factors which help in deciding the execution of tasks. If the available energy is lower than the required for the task execution, the task can be delayed so that sufficient energy is accumulated in the (from energy harvesters) to serve the task successfully. ### Optimization This phase optimizes the execution of tasks on the basis of the following attributes: * The additional available energy (available after executing the current task) * The required number of executions of each task * Energy consumption of each task The optimization phase further filters the tasks in order to use the harvested energy efficiently. The additional available energy is important to decide about the execution of future time-critical tasks. In order to avoid deadline violations in the future, a certain minimum amount of energy must be available during all time slots, to serve the future tasks that possess hard deadlines. Authors in evaluate their scheme using as a source of energy. Results show that their technique executes more tasks with fewer missed deadlines as compared to the previous schemes which do not employ the decomposing/combining strategy for energy-intensive tasks. In addition to decomposing and combining of tasks, a model is proposed in to schedule the tasks on the node. It proposes a dynamic optimization model based on to schedule the tasks taking into account their deadlines, energy consumption, and available harvested energy. The decomposed subtasks can be combined together for concurrent execution. It also proposes a less complex greedy scheduling policy which can be implemented on the resource constrained sensor nodes and it consumes less energy as compared to the original model. The simulation results indicate that the proposed algorithm executes higher number of tasks within the same energy budget compared to the previous schemes. As the harvested energy in energy harvesting-based sensors is limited, they can essentially perform only one atomic task at a time. In the particular case of batteryless sensors, the available energy burst (in the ) can not be employed to execute multiple tasks simultaneously. Therefore, combining the tasks to reduce energy consumption is not appropriate solution for energy harvesting based sensors. As a result, there is a potential of alternate task scheduling algorithms that can provide on the sensor nodes with limited energy budget. [t] [figtaskscheduling] Duty cycling ------------ One of the most familiar and common methods for minimizing the energy consumption in is to use a sleep/awake mechanism. When a node is not executing any useful operation, it is switched to the sleep mode, which reduces its energy consumption . In traditional, most of the protocols devise a duty cycling mechanism based on the number of tasks, their energy consumption and remaining energy of the node. However, these methods are not suitable for, due to the variable nature of incoming harvested energy. In, the harvested energy is varying with the environmental conditions due to the intermittent nature of ambient energy harvesting sources . Therefore, the duty cycle of the node depends on the incoming harvested energy to efficiently utilize the energy in executing tasks on the node. In addition to the current harvested energy, the sleep duration of the node also depends on the energy to be harvested in future to proactively plan the consumption of incoming energy for required operations. Therefore, we classify the duty cycling mechanisms into the aforementioned two categories and extensively explore their operations in the following subsections. ### Duty cycling depending on the harvested energy In a network of multiple nodes, duty cycling mechanism depends on the harvested energy, consumed energy, distance between nodes and data aggregator/receiver, and the future harvested energy. In traditional, nodes near the sink exhaust quickly due to the additional burden of relaying the data of far-away nodes in multi-hop communication. On the other hand, in, nodes remain alive as long as they are receiving replenishable energy from the environment using energy harvesters. Kansal et al. present duty cycling mechanisms (for single node as well as multiple nodes) for sustainable performance in. They describe a model for calculating the minimum size of the /battery for sustainable operation of the embedded device. The duty cycle within a sensor node depends upon the average harvested as well as consumed energy in active and sleep modes. The duty cycle can be adjusted such that the overall energy consumption does not exceed the overall harvested energy. Similarly, if the harvested energy is increased, the duty cycle can also be raised to improve the performance within the given energy budget. Authors in show the record of the battery voltage for more than two days which depicts that the node has adjusted its duty cycle in accordance with the harvested energy for sustainable operation. The authors also propose a duty cycling mechanism for a wireless network having multiple nodes. In a network configuration, determining sleep/awake time interval is challenging as the harvested energy for each node can be different due to its geographical location and orientation with respect to the energy source (e.g., solar energy). Therefore, the node attempting to transmit data will send a BEACON packet and wait for a certain time period for an ACK. It repeats the process until an ACK is received from the destination node. A mathematical model for duty cycling the sensor nodes according to the harvested energy is described in. It achieves the and maximizes the system performance by adapting the dynamics of the replenishable energy source at run-time. It employs the algorithm to predict the harvested energy which is used to compute the duty cycle of the node. Bouachir et al. propose a protocol for efficient energy utilization in cooperative. Their scheme uses the nodes’ residual energy and data requirements to decide the active as well as sleep time periods of sensor nodes. As a result, it decreases the active time of nodes near the data aggregator and increases the active time of nodes away from data aggregator with the passage of time to balance the energy consumption of nodes. Therefore, it minimizes the problem of early depletion of nodes near the data aggregator, which reduces the coverage hole dilemma . Yang et al. propose a sensing scheduling scheme which dynamically adapts the sensing rate according to the available energy in the. They also propose a mathematical model for optimal sensing scheduling in energy harvesting sensors . In contrast to the previous works that focus on the energy allocation to the sensors, optimizes the sensing epoch depending on the energy budget. It presents the infinite and finite battery case and suggest an online scheduling policy that approaches the theoretical offline optimal scheduling mechanism. It also proposes a virtual energy harvesting sensing system to analyze the battery level which is helpful in deciding the sensing epoch. An event-driven duty cycling mechanism is proposed in for power management of a road side monitoring unit. It harvests energy from the and transmits data packets according to the events of the traffic flow on the road using an algorithm. This implementation achieves lower energy consumption with longer lifetime of the sensing device. ### Duty cycling depending on the available and predicted harvested energy In certain circumstances, the current harvested energy may not be sufficient to run an energy-intensive task on the node. Therefore, knowledge of the future incoming energy is important to delay the tasks until sufficient energy is available, without missing any deadline as illustrated in Fig. [fig:predictedenergymotivation]. As shown in the figure, task 1 is executed as soon as it arrives at the node due to the higher energy availability than required for the execution of task. However, task 2 arrives when the available energy is lower than required for the execution of task. As there is a prediction of future harvested energy, the task is delayed until sufficient energy is available for its execution. In this way, the predicted harvested energy improves the energy utilization and minimizes the number of missed deadlines of tasks. Moser et al. present an algorithm for task scheduling in environmentally powered. To the best of our knowledge, this is the first detailed work that addresses the task scheduling problem in. In conventional, the sensor nodes have a fixed energy supply (i.e. from a battery) and the only issue is to meet the deadlines of tasks for their execution on a single processor. In this case of abundant energy availability, various scheduling schemes such as or can be incorporated . However, these schemes have certain drawbacks and can violate the deadline of tasks, when implemented on the energy harvesting node, which has intermittent energy availability. In the algorithm, future incoming tasks can experience energy black-out periods (which results in deadline violations), whereas the scheme can miss the deadlines, if the energy is limited near tasks deadlines. The algorithm can solve this problem and schedules the tasks such that the task having an earlier deadline is executed first. However, does not perform well in energy harvesting systems and violates deadlines due to energy consumption for less important tasks. This energy starvation results in the failure of execution of time-critical tasks with shorter deadlines. On the contrary, a scheme that hesitates in executing the tasks (until their deadlines) can perform well by conserving energy for time-critical tasks that have shorter deadlines . The optimal start time of the task can be decided according the available energy and upcoming energy in the future. Therefore, there is an opportunity to devise sophisticated algorithms for the prediction of future harvested energy. Then, this harvested energy profile determines the time slot for the execution of various tasks. Authors in evaluate their technique using the energy from and results show that it offers fewer deadline violations as compared to previous algorithms. Authors in extend the work of with a detailed mathematical model and consider the practical considerations in implementing the algorithm on a real embedded device. They also propose an optimal start time for the execution of tasks depending upon their deadlines, energy consumption, stored energy and future harvested energy. However, an important factor in the calculation of optimal start time is the prediction of energy to be harvested in future. In order to estimate the future harvested energy, authors in propose an Energy Variability Characteristic Curve (EVCC) which gives an upper bound on the harvested energy in a certain time interval Δ. As a result, the harvested energy in any time interval Δ can easily be approximated, according to the given EVCC. However, another problem is to determine how much capacity the should have, for perpetual operation of the sensor node in. The answer lies in the maximum harvested energy, in addition to the consumed power in any particular time interval. This energy consumption can be calculated depending on the number of tasks arriving at the processor with common deadlines. Sommer et al. propose a scheduling framework for various sensors (such as Global Positioning System (GPS), accelerometer, magnetometer, etc.) for perpetual tracking of flying foxes that travel long distances from their foraging camps in the search of food. The sensors are sampled based on the available and the future harvested energy. They also take into account the mobility and activity of flying foxes to trigger the next sensor sample. This technique ensures that maximum tracking accuracy is achieved within the given dynamic energy budget. The scheme in is quite different from traditional tracking schemes that focus on minimizing the energy consumption with a fixed tracking accuracy. On the contrary, emphasizes on achieving maximum tracking accuracy within the available energy budget which is replenished using on-board solar cells. The consideration of current and future energy levels while sampling the sensors ensures optimal time duration between successive samples, such that energy is neither depleted nor overflows (due to limited capacity of ) thanks to the future incoming energy from replenishable energy source (i.e., solar energy). Additionally, sensor sampling based on mobility ensures that enough energy is available to track the rest of the trip of flying foxes using the limited available energy. Györke et al. exploit the knowledge about the environment to schedule non-equidistant samples, both is time as well as in space. The predicted harvested energy is taken into account to devise a conservative sampling approach when the future incoming energy is low. The proposed technique also uses the neighbour’s information in deciding the duty cycle. It increases the duty cycle of nodes in vicinity of a place where an event has occurred. The other nodes operate at their usual duty cycle to conserve energy. [figdecomp] Akgün et al. address the problem of duty cycling in cluster based using slots. Traditional protocols assign fixed slots to the cluster members for their data transmission. In contrast, the technique in uses the predicted harvested energy and available energy to assign slots to the nodes. In based, the node having maximum predicted harvested energy is assigned the next time slot in the day. On the other hand, at night, the node having maximum residual energy is given the next time slot. Kooti et al. present a task scheduling mechanism for based sensor nodes. Their scheme consists of two parts: offline scheduling and online scheduling. Firstly, the tasks are scheduled offline depending on the energy requirements and their criticality. Then, in the online scheduling phase, the tasks are adjusted according to the real-time available energy and future harvested energy, and are executed depending on their deadlines. Renner et al. propose a scheduling algorithm on the basis of predicted harvested energy in sensors. They present a harvested energy prediction model based on the changes in the previous harvested energy pattern in real-time, which reduces the prediction error and is applicable in real-world environments. Instead of periodic sampling, they exploit the periodic nature of the sun to adaptively trace the energy pattern and reduce the number of prediction updates. Sommer et al. propose an accurate method for the estimation of of the battery which can lead towards energy neutral scheduling of tasks. They use the current flow from the energy harvester and the battery voltage to estimate the current. They employ a mathematical model called conflation to combine information from energy harvester as well as the battery to calculate the battery. Cui proposes the execution of tasks on the node using the predicted energy in based. He employs a recurrent long short term memory neural network to forecast the harvested energy. Later, this harvested energy is used to schedule tasks, which offers better performance than conservative and greedy approaches. El Osta et al. propose to schedule periodic and aperiodic tasks separately, based on their priority. The periodic tasks are executed as soon they arrive at the scheduler if sufficient energy is available for their execution. On the other hand, aperiodic tasks are executed only if there is no periodic task in the queue and the is not depleted. The aperiodic task consumes surplus energy that would have been wasted, if there is no other task ready for the execution. It is to ensure that the execution of aperiodic tasks does not affect the execution of future periodic tasks by consuming a significant amount of energy, which depletes the. Authors in use an reader to harvest energy for capturing and transmitting an image. The harvested energy is stored in an to form fixed energy bursts to power the load (i.e., camera). It is more efficient to store the small amount of harvested energy instead of capturing large amount of energy as it may introduce delay in the execution of tasks. The is also non-ideal, causing some leakage, which may reduce the usable energy budget. Therefore, it is more energy efficient to store the limited harvested energy to execute at least one atomic task as it takes a short build-up time and causes less leakage. Instead of fixed energy bursts , Gomez et al. introduce dynamic energy bursts that are matched with the requirements of the load. They introduce an that arranges short energy bursts according to the requirements of the load and transforms small amounts of harvested energy into high-powered short energy bursts. Their tracks the optimal power point of load and schedules the energy bursts accordingly. This results in efficient utilization of limited, variable and transient harvested energy to power the sensor nodes in. [fig:predictedenergymotivation] Zhang et al. present a scheduling scheme for storage-less and converter-less energy harvesting systems in. They formulate a system model for powering the sensor nodes using the to minimize the deadline miss ratio. They develop a task scheduling algorithm using an artificial neutral network based on the harvested energy profile, the tasks priority and the energy budget. They also propose an integer nonlinear programming solution for optimal task scheduling on the node. Severini et al. implement an energy-aware lazy scheduling algorithm on a low-cost and commercially available platform from Texas Instruments. This lazy scheduling algorithm delays the execution of tasks depending on their deadline and according to the available energy. Experiments are performed under varying incoming energy conditions. The results show the effectiveness of the scheduling algorithm and its suitability for real-time implementation in. Liu et al. propose converter-less and storage-less operation of energy harvesting sensors. Due to the lower efficiency of the converter (40-90%) and energy leakage of the, the efficiency of the system is decreased. Authors in design a hardware prototype using, without converters and batteries, which performs better than conventional systems. They also propose an intra-task scheduling algorithm which pre-empts the tasks according to their deadlines. The tasks are performed according to the available energy and their deadlines to meet the execution time requirement and use the energy efficiently. Zhang et al. propose a scheduling mechanism for time-critical energy harvesting sensors. They present harvesting-aware speed selection of processor and radio transmission to manage the harvested energy and to meet the deadlines of tasks. They suggest two versions of their algorithm: central and decentralized, to be implemented according to the application environment. Simulations show that their scheme outperforms in terms of energy reserves than traditional schemes, which do not employ processor speed selection and transmission power control. The available energy budget plays a pivotal role in scheduling the tasks on the nodes in. Table [tablesurveyedpapers] portrays that all of the previous task scheduling algorithms consider the available energy budget while deciding the execution of tasks to enhance the performance and minimizing deadline misses. Fang et al. propose a mathematical model for optimal scheduling in energy harvesting based mobile sensor nodes. As the harvested energy in mobile sensors varies with the change in energy source as well as with the mobility of embedded devices, it is challenging to devise an optimal scheduling algorithm. The sensor is switched to the sleep mode, when there is no task to be executed. After the arrival of new tasks, the corresponding sensor component is switched to active mode depending on the energy budget. Authors in use a Lyapunov optimization framework to make control decisions that greedily minimize a bound on the drift-plus-penalty expression over fixed-length time slots. It has fair computational complexity and is easy to implement in real-time scenarios. Pan et al. suggest a technique for the execution of tasks under unstable energy harvesting conditions. Their algorithm maximizes the progress of tasks using the information about the harvested energy. They also employ a routine which triggers sleep/wake-up events and start the execution of tasks automatically after resuming. Anagnostou et al. propose a power aware hardware scheduler for energy harvesting sensors. Instead of a traditional mechanism, where the harvested energy is stored in an, they switch between the energy harvester directly and the, depending on the harvested energy and task requirements. This mechanism leads to maximum utilization of the harvested energy even when the is full to its capacity. Authors in implement an FPGA based prototype which shows that their algorithm achieves two times higher task execution rate under variable energy harvesting conditions, compared to the conventional schemes. Sanchez et al. propose a hybrid approach for energy management in. This hybrid mechanism consists of battery states and the operating conditions of the nodes such as active, sleep and low duty cycle modes. In addition, they use decentralized control, which reduces the burden of controlling the whole network from a single node, extending the lifetime of the system. The previous task scheduling algorithms employ conventional sensors, which consume energy during their operation. In addition, these sensors perform their operation using a reliable energy source (i.e., a battery) that provides continuous power for a longer duration. However, this scenario is entirely different for energy harvesting based sensing due to the dependence between energy and information, and intermittent execution of sensor node. Firstly, the energy harvester provides both energy as well as context information, and the signal needs to be sampled at suitable time instances according to the amount of context information and available energy. Secondly, the sensor node can run at most one atomic task at one time, due to the limited available energy stored in the capacitor. Therefore, it opens doors for dedicated task scheduling algorithms for energy harvesting based sensors. The objective of these task scheduling algorithms is to maximize the context detection accuracy as well ensure the of sensor nodes in. Most of the previous task scheduling algorithms consider the predicted harvested energy, while scheduling the tasks on the node. This results in maximum utilization of the current and future harvested energy without missing the deadlines of tasks. In order to fully understand the functioning of the task scheduling algorithms, it is important to comprehend the energy prediction algorithms as well. Most of the harvested energy prediction algorithms use the previous harvested energy samples to estimate the future harvested energy profile. We discuss some of the previous energy prediction algorithms in the following subsection in detail. [c]\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\| [tablesurveyedpapers] & & & & & & Difficulty in using Year and & Energy & DVFS & Decomp./ & Duty & Predicted & Deadline & Performance & Performance & it in energy harvesting Reference & harvester & & Comb. & cycling & energy & & metric & assessment & based sensing & & & & & & Difficulty in using Year and & Energy & DVFS & Decomp./ & Duty & Predicted & Deadline & Performance & Performance & it in energy harvesting Reference & harvester & & Comb. & cycling & energy & & metric & assessment & based sensing 2001 & General & & & & & & Remaining energy & Modeling & High 2004 & Solar & & & & & & Latency & Simulation & High 2006 & Solar & & & & (C) & & Energy utilization & Hardware & Medium 2006 & General & & & & (R) & & Deadline violation & Simulation & High 2007 & Solar & & & & (C) & & Deadline violation & Simulation & High 2008 & Solar & & & & & & Deadline miss rate & Simulation & High 2009 & Solar & & & & & & Deadline miss rate & Simulation & High 2010 & Solar & & & & & & Deadline miss rate & Simulation & High 2010 & Solar & & & & & & No. of executed tasks, Average & Hardware & High & & & & & & & accuracy, Energy consumption & & 2010 & Water flow & & & & & & Depleted nodes & Simulation & High 2011 & Solar & & & & & & Deadline miss rate & Simulation & High & & & & & & & Capacity of energy storage & & 2011 & General & & & & & & Average idle time & Simulation & High & & & & & & & Feasible task set & & 2011 & Solar & & & & & & Battery charge level & Simulation & High & & & & & & & Energy violation rate & & 2012 & Solar & & & & (R) & & Number of tasks executed, & Hardware & Medium & & & & & & & missed deadlines & & 2012 & Solar & & & & (C) & & Deadline miss rate & Simulation & High & & & & & & & Minimum storage capacity & & 2012 & General & & & & (R) & & End-to-end delay, BF, & Simulation & High & & & & & & & Packet delivery ratio & & 2012 & Solar & & & & & & Energy consumption & Simulation & High 2012 & General & & & & & & Number of deadline violations & Simulation & High & & & & & & & First deadline violation & & 2012 & Solar & & & & (R) & & QoS violation count & Simulation & High 2012 & Solar & & & & (C) & & Root mean square error & Simulation & High 2013 & Solar & & & & & & Time & energy consumption & Simulation & High 2013 & General & & & & & & Length of idle time & Simulation & High 2013 & Solar & & & & (C) & & Network lifetime & Simulation & High 2013 & Solar & & & & (R) & & Task drop rate & Simulation & High 2013 & Solar & & & & (C) & & Battery SoC & Simulation & High 2014 & Solar & & & & (C) & & No. of executed tasks & Simulation & High 2014 & Solar & & & & (R) & & Time & energy & Simulation & High & & & & & & & constraint & & 2015 & Solar & & & & & & Deadline miss rate & Simulation & High & & & & & & & Number of ready tasks & & 2015 & RFID reader & & & & & & Power consumption & Hardware & Medium & & & & & & & with time & & 2015 & Solar & & & & (R) & & Number of tasks executed & Simulation & High 2015 & General & & & & & & Mean Square Error & Simulation & High 2015 & General & & & & (R) & & Completed tasks & Hardware & Medium 2015 & Solar & & & & & & Deadline miss rate, & Hardware & Medium & & & & & & & Energy utilization & & 2016 & General & & & & & & Average overhead & Simulation & High & & & & & & & Average busy period & & 2016 & Solar & & & & (C) & & Deadline success ratio & Simulation & High 2016 & Solar & & & & & & System efficiency & Hardware & Medium 2016 & General & & & & & & Deadline miss rate, & Simulation & High & & & & & & & energy violation rate & Hardware & 2016 & Solar & & & & & & No. of packets sent to BS, & Simulation & High & & & & & & & Delay & & 2016 & Solar & & & & (C) & & Deadline miss ratio, & Hardware & Medium & & & & & & & Energy utilization efficiency & & 2016 & General & & & & & & Data queue size, & Simulation & High & & & & & & & Energy queue size & & 2016 & General & & & & & & Sensing performance, Infeasible & Simulation & High & & & & & & & sensing epoch, Battery overflow & & 2016 & Solar & & & & & & Power consumption, & Hardware & Medium & & & & & & & Battery lifetime & & 2017 & General & & & & & & Deadline miss rate, & Simulation & High & & & & & & & response time & & 2017 & General & & & & & & Harvested to consumed & Simulation & High & & & & & & & energy rate & & 2017 & General & & & & & & Energy and time overhead & Simulation & High & & & & & & & Energy consumption & & 2017 & General & & & & & & Remaining energy of node & Hardware & Medium 2018 & Solar & & & & & & Expected reward & Simulation & High 2018 & Solar & & & & & & Energy efficiency, & Hardware & Medium & & & & & & & Execution rate & & 2018 & Solar & & & & (C) & & Task completion rate & Simulation & High 2019 & Solar & & & & (C) & & Average tracking error & Simulation & High Algorithms for energy prediction in ----------------------------------- In order to ensure, the sensor nodes need to consume the harvested energy in such a way that the current node operation is not affected, and the future tasks do not run out of energy. Therefore, information about the future harvested energy is important to schedule the energy consumption proactively for sustainable operation of the system. In the literature, there are various harvested energy prediction models that utilise previous energy samples, weather conditions and seasonal trends to compute the future harvested energy in energy harvesting based. Table [tablepredictedenergy] comprehensively presents some of the harvested energy prediction algorithms that employ statistical, probabilistic and machine learning models to predict the future harvested energy. On a normal sunny day, the harvested energy from a solar powered node is highest at noon, and decreases at dawn and dusk, finally reaching zero at night , due to the non-availability of sunlight. Knowing this overall pattern of, the future solar harvested energy can be predicted using the previous energy pattern. Kansal et al. present a harvested energy prediction model based on for solar powered. Their model relies on the intuition that the harvested energy in a particular day at a given time slot is similar to that of the energy harvested in the previous days at the corresponding time slots. Therefore, the harvested energy in a particular time slot is calculated by accumulating the weighted average of harvested energy in the previous days in the same time. algorithm awards higher weight to the recent energy values and exponentially decreases weight for the previous energy samples to calculate the future harvested energy in solar powered. The weight is calculated dynamically using the real previous energy traces, which provide the lowest value of error. However, gives significant prediction error when there is a sudden change in the weather. It is due to the reason that the scheme does not take the seasonal weather trends and diurnal cycles into account. Hassan et al. propose an energy prediction model for solar powered, which takes into account the sudden changes in the environment. It also takes into account the seasonal and diurnal cycles of the solar energy. However, this scheme is more computationally complex as it takes multiple parameters into account and costs more energy as well as processing time, incurring delay in the system. Another technique which considers weather conditions for predicting the harvested energy in solar powered is presented in . It presents a model to predict solar as well as wind harvested energy. The method takes the data from the weather forecast stations to predict the energy to be harvested in future time slots. However, this scheme is also computationally complex and depends on another source, which increases the cost of the system. Additionally, receiving weather data and processing it on an energy-constrained miniaturized sensor node hinders the execution of other time-critical tasks. Piorno et al. present a prediction algorithm for which depends on and takes into account the sudden and abrupt changes in weather conditions. They propose to use the weighting factor depending on the solar conditions of the current day relative to the previous days. Cui proposes a prediction algorithm for sensor nodes in. It employs a recurrent long short term memory neural network to forecast the harvested energy. However, this is a complex method that has higher cost in terms of energy, time, memory requirement and computational resources. [t!] [tablepredictedenergy] llll Year & Reference & Input parameters & Method 2007 & & Previous samples & 2008 & & Previous minimum energy & Worst-case energy & & & prediction 2009 & & Weather conditions from & Weather-conditioned & & recent past samples & moving average 2010 & & Weather forecast & Quadratic solar & & & power model 2011 & & Multiple energy harvesters & Markov model 2012 & & Previous energy profiles & Profile energy & & & prediction 2016 & & Previous energy profiles & Profile energy & & & prediction 2016 & & Past observations, & Q-learning & & Current weather & 2016 & & Previous samples & Curve fitting 2019 & & Global horizontal & Austronomical & & irradiance & model Most of the previous harvested energy prediction schemes , forecast the future solar harvested energy for the next single time slot. However, occasionally, it is also important to estimate the harvested energy for future N (where N > 1) time slots. Moser et al. present a harvested energy prediction algorithm that calculates the harvested energy for future N time slots. Their algorithm considers the harvested energy in previous time slots of length N. Then, the worst case harvested energy in the previous time slots is considered as the predicted energy for the future slots. However, does not take the fast changing weather into account which results in higher prediction errors in swift weather changing environments. It is also a pessimistic approach, as it considers the worst case harvested energy only. Cammarano et al. propose a prediction model using harvested energy profiles of previous days. They store different types of energy profiles (like sunny, partially sunny, cloudy, etc.) and compare the initial values of currently harvested energy with the stored energy profiles. The stored energy profile having highest correlation with the current harvested energy is considered as the predicted energy profile for the rest of the day. However, this scheme also burdens the miniaturized nodes with complex computation and data storage requirements. Ventura et al. present an energy harvesting and consumption algorithm for body sensor networks using a Markov model. Their algorithm considers multiple types of energy harvesters and predicts the future states of nodes in terms of energy level depending on the probabilistic model based on previous energy samples. Authors in propose a Q-learning based solar harvested energy prediction model using previous energy samples and current weather conditions. This results in lower prediction error than conventional. As the harvested energy from depends on luminous intensity, Zou et al. propose to predict the locations of sunlight for next time interval to predict the harvested energy. They use a piecewise least squares curve fitting estimation using the previous samples to estimate the future harvested energy. Kai et al. predict the future solar harvested energy using global horizontal irradiance and solar cell’s characteristics. In order to compensate for deviations from the actual values, their scheme compares the predicted harvested energy values with the previous harvested energy. In summary, there are various energy prediction algorithms for solar powered, which take into account the previous harvested energy values, weather forecasts and previous energy profiles, to correctly predict the future harvested energy, as shown in Table [tablepredictedenergy]. However, these algorithms, when implemented on the node, consume a significant amount of harvested energy during their execution. Therefore, in addition to prediction accuracy, the cost in terms of energy and computational complexity must also be explored. Such an analysis will provide the real picture about the models and will identify the algorithm which provides best energy prediction results, while executing within the limited energy and computational resources. Summary and insights -------------------- We comprehensively survey and analyse previous task scheduling algorithms in to enhance the lifetime of sensor nodes. However, none of the previous algorithms employ energy harvesting based sensing; instead they exploit conventional sensors for monitoring the desired physical parameter. It results in significant energy consumption compared to energy harvesting based sensors as discussed in Section [sensingandenergyharvesting]. Therefore, keeping in view this new class of sensors (i.e., energy harvesting based sensing), the previous task scheduling algorithms need to be revised or new dedicated task scheduling algorithms should be proposed to allow the sustainable operation of sensor nodes in. We critically analyse the previous task scheduling schemes and explore their applicability for energy harvesting based sensors in the following section. In addition, we rigorously discuss the opportunities for transforming the conventional task scheduling algorithms for the emerging class of energy harvesting based sensors. [fig:taskschedulingfuture] Critical Analysis of Task Scheduling Algorithms for Energy Harvesting based Sensing =================================================================================== We discussed that task scheduling algorithms manage the execution of tasks on the node to extend the lifetime of sensor nodes in conventional. However, considering the new category of sensors (i.e., energy harvesting based sensing), we explore the applicability of task scheduling algorithms in this new paradigm. There is a fundamental difference between conventional and energy harvesting based activity sensors in terms of their operation and energy consumption. Conventional sensors consume energy while in operation (see Fig. [fig:taskschedulingfuture](a)), whereas, energy harvesting based sensors generate energy, in addition to providing a signal that can be used for detecting the underlying activities, as depicted in Fig. [fig:taskschedulingfuture](b). As energy and context information from the energy harvester are strongly correlated, aggressive sampling of the harvesting signal may not be required, which opens new directions for devising task scheduling mechanism compared to conventional activity sensors. In addition, the harvested energy can act as a trigger signal to decide the next sensing epoch depending on the variation in the harvesting signal. Furthermore, the variation in the harvesting signal can be translated as the change in ongoing activity, using few initial samples of the energy harvesting signal. The task scheduling becomes more challenging in multi-source energy harvesting based sensing, as shown in Fig. [fig:taskschedulingfuture](c), due to the different amount of energy and context information from each harvester unit. Depending upon the type of application, one energy harvester can provide higher context information compared to others, in certain physical conditions, which needs to be tracked in real-time. This results in higher activity detection accuracy without consuming significant energy, for example, to acquire the signal at a substantially high sampling rate or to sample all energy harvesting signals. In addition, the amount of harvested energy from each energy harvester is unique depending upon the environmental conditions and properties of the transducer. This energy can be combined using DC-DC converters or individual harvesters can be selected according to the amount of generated energy, to avoid the energy conversion losses . Challenges of current task scheduling techniques ------------------------------------------------ Most of the existing task scheduling algorithms are complex and require significant amount of energy during their operation on the miniaturized and resource-constrained sensor node. Decomposing and combining the tasks is not applicable for batteryless energy harvesting based sensors due to their limited energy budget that can, at most, run one atomic tasks at a time. Secondly, algorithms need complex hardware circuits to provide multiple voltage levels to individual components of the sensor node. It may require multiple s in batteryless sensor nodes which increase the energy losses, cost and form factor, and decrease the usable energy. Most of the previous task scheduling algorithms focus on enhancing the lifetime of the system, ignoring the performance in terms of activity detection accuracy. These scheduling algorithms do not take into account the correlation between energy and context information, which may help in deceasing the sampling rate of the harvesting signal, resulting in less energy consumption, in contrast to conventional activity sensors. In addition, current mechanisms rely on sequential functioning of the program, which is not applicable for intermittent operation in batteryless sensors, which experience frequent energy blackouts. Timing failure is another issue in batteryless devices due to frequent power failures and energy scarcity periods, in contrast to conventional battery operated sensing devices. Furthermore, devising task scheduling algorithms for multi-source energy harvesters is more challenging due to the varying amount of harvested energy and context information from each harvester unit in real-time. Moreover, the predicted harvested energy plays an important role in devising task scheduling algorithms, as discussed in Section [Schedulinginehiot]. However, there is no energy prediction algorithm for kinetic, thermal and RF energy, in particular, for applications which involve mobility. Furthermore, there is no existing mechanism to manage the additional harvested energy in energy positive sensors, which may lead towards of sensors. Finally, most of the current task scheduling algorithms are validated through computer simulations without implementing on real hardware test bed. Based on the comprehensive discussion of existing task scheduling algorithms in Section [Schedulinginehiot] and aforementioned challenges, we describe the possible solutions and future directions for incorporating energy harvesting based sensors in the next subsection. Possible solutions ------------------ Task scheduling algorithms should be devised to focus both on enhancing the context information as well as the system lifetime. Among the available options, duty cycling is the most appropriate choice for scheduling the tasks on the miniaturized and resource-constrained sensor node due to its inherent lower complexity and ease of implementation. It is also compatible with the batteryless sensors due to their transient and intermittent operation. In addition, it is also important to devise harvested energy prediction algorithms to allow scheduling of tasks according to the harvested energy. As energy prediction algorithms consume a significant amount of energy during their operation, another approach is to devise task scheduling algorithms without quantifying the future energy, which can reduce the energy consumption of miniaturized sensor nodes, resulting in sustainable operation of the system. It is also important to analyse the complexity and overhead in terms of energy for running task scheduling algorithms before implementation on real resource-constrained sensor nodes. Depending upon the type of application and nature of the environment in which energy harvesters are employed, task scheduling algorithms can be adapted in real-time. Furthermore, in case of multi-source energy harvesters, it is important to switch between the most suitable energy harvesters depending upon the context information and harvested energy from the individual transducer output signals in real-time. In energy harvesting based sensing, context information and the amount of harvested energy are correlated. Therefore, energy can be saved by reducing the sampling rate of the sensing signal during the steady state, without losing context detection accuracy. Moreover, the additional harvested energy can be employed to run other pending tasks on the node, such as data processing and communication, which leads towards the potential of in. Therefore, new operating systems and programming frameworks are needed for running energy harvesting based sensors, which take into account the intermittent operation of batteryless sensors, to resolve the checkpointing and timing failure issues, which are not present in conventional battery-based devices. Finally, there is a potential of task scheduling algorithms for energy harvesting based sensors, to meet the objectives of both maximized sensing performance as well as ensuring of sensor nodes in. Future Research Directions ========================== We discussed that energy harvesters can be employed as a simultaneous source of energy as well as context information. Since this new class of sensors has inherent differences than conventional sensors, it brings opportunities for dedicated energy management algorithms for its perpetual operation. Therefore, conventional task scheduling algorithms need to be revised due to the correlation between energy and information in energy harvesting based sensing. As the harvested energy is limited, task scheduling algorithms are required to execute tasks on the node according to harvested energy profile, for achieving of sensor nodes in. There are various challenges in devising efficient task scheduling algorithms for energy harvesting based sensing due to the varying and intermittent harvested energy that puts further constraints in case of batteryless sensors. The goal of task scheduling algorithms is to enhance the activity detection performance as well as the operational lifetime of sensor nodes in. Based on the previous discussion, future research directions for ensuring higher activity detection accuracy as well as achieving of sensor nodes are described in the following subsections in detail. Exploring optimal sampling frequency of sensing signals ------------------------------------------------------- In order to extract context information, the sensing signals from the energy harvesting circuit are sampled, whereas processing can be done offline or on the device, depending on the energy budget. The sampling frequency of the energy harvesting signal is an important parameter which plays an important role in the activity detection accuracy. Increasing the sampling frequency results in higher energy consumption with the advantage of higher activity detection accuracy. Therefore, there is a potential to study the optimal sampling frequency of various sensing signals in terms of energy consumption as well as activity detection accuracy. As the harvested energy varies according to the type of application, the sampling frequency also varies accordingly which needs to be studied in detail. Maximum Power Point (MPP) tracking ---------------------------------- As the output voltage and current of the energy harvesting transducer changes due to the environmental conditions, its also varies swiftly under the external stimuli. tracking of transducer is more challenging than solar cells, due to rapid output voltage/current fluctuations in the former compared to the relatively slow variations in the latter. Therefore, sophisticated hardware modules are required which can dynamically track its at a high frequency in run-time . However, it will also consume more energy in sampling the signal at a higher frequency to find its voltage. Therefore, there is a potential to study the optimal tracking frequency, amount of energy consumed in tracking and additional harvested energy, to estimate the overall gain. Employing multi-source energy harvesters as a simultaneous source of energy as well as context information ---------------------------------------------------------------------------------------------------------- In order to ensure the sustainable operation of sensor nodes in, the harvested energy should be sufficient to power the sensor’s hardware without the need of any external energy source (i.e., a battery). Occasionally, the harvested energy from a single energy harvester is not sufficient to continuously power each module of the sensor node. For example, can not provide sufficient energy during night and darkness. On the other hand, the harvested energy from is very small during lower vibrations/movements, such as sitting and standing in human activity recognition applications. Therefore, multi-source energy harvesters (e.g.,,, and ) can be employed to harvest higher energy as well as extract rich activity information. The energy from these harvesters can be accumulated to power a sensor node to achieve. In addition, the signals from multi-source energy harvesters can be fused to extract rich context information. For example, while provides information about the movement/activity, can be used to identify the indoor and outdoor environments, depending upon the amount of harvested energy, which can be employed in the localization applications. However, if a signal does not possess information about the underlying activity, its fusion with other signals may increase the cost, complexity and energy consumption of the system. In summary, there is a potential to study the amount of harvested energy and context detection accuracy from multi-source energy harvesters in various applications, such as human activity recognition, gait recognition and transport mode detection. Scheduling framework for energy harvesting based sensing -------------------------------------------------------- In order to ensure of sensor nodes and achieve higher activity detection accuracy, the harvested energy should be consumed efficiently to run a maximum number of tasks on the sensor node in. In energy harvesting based sensors, the execution of tasks is a function of context information, harvested energy and required energy. However, the previous scheduling schemes employ conventional motion sensors, instead of using the energy harvesters as activity sensor and energy source simultaneously. Furthermore, multi-source energy harvesters can be used to extract higher energy as well as rich context information, which brings new challenges due to the different amount of energy and context information from each harvester unit. This new class of sensors (i.e., sensing using energy harvesters) brings opportunities for revised task scheduling schemes, due to the correlation between harvested energy and context information. Eventually, it will enable the perpetual operation of sensor nodes along with higher detection accuracy of the underlying activity, without the need of any external depletable energy source (i.e., a battery). Considering energy harvesters as activity sensors and source of energy simultaneously, the key research challenges for devising task scheduling algorithms are as follows: * Devising a duty cycling algorithm for the sensor node according to the harvested energy profile, which provides highest activity detection accuracy with minimum power consumption. * Studying the relation between the harvested energy and the sampling frequency of the energy harvesting signal. * In order to acquire actual and real pattern of the energy harvesting signal, the energy harvesting transducer can be disconnected from the capacitor/load during sampling the signal for context detection applications. However, it may result in lower harvested energy with the advantage of higher activity detection accuracy, which needs to be studied in detail. * In case of multi-source energy harvesters, there is a need to devise an optimal sampling frequency for each energy harvesting signal. * Exploring the information gain of each energy harvesting signal in multi-source energy harvesters and fusing the most information-rich signals, that provide highest activity detection performance, with minimum energy consumption. * Processing the energy harvesting signals and extracting the dominant features on the resource-constrained sensor node. This may result in lower energy consumption in data transmission (due to smaller feature set) with higher energy consumption in processing the data on the sensor node. * Implementing the classification algorithm on the sensor node for online (real-time) activity detection, resulting in autonomous operation of sensor nodes in. * Devising a scheduling framework in batteryless sensors that work intermittently under limited and un-reliable harvested energy. * Studying the trade-off between the online and offline processing of various sensing signals, collected from the energy harvesting transducers. * In case of lower harvested energy, the collected data or extracted features may not be transmitted to the server for offline processing. Therefore, array of energy harvesters can be employed to harvest higher energy with the overhead of increased cost and form factor. In summary, considering the aforementioned research challenges while devising task scheduling schemes will ensure the ultimate goals of higher context detection accuracy as well as of sensor nodes in. [t!] [tableCompPredEH] ll Solar energy harvesting & Kinetic energy harvesting Periodic & Aperiodic Relatively stable & Relatively unstable Easily predictable & Difficult to predict Higher energy than & Lower energy than Easy to design the power & Difficult to design the power conditioning circuit & conditioning circuit Power is generated in the & Power is generated in the presence of light & presence of vibrations Relatively less noise in the & Relatively more noise in the signal & signal Generates DC voltage & Generates AC voltage Predicting the harvested energy in ---------------------------------- In contrast to, the harvested energy in is highly fluctuating, which varies quickly according to the underlying stress/vibrations. This unstable output voltage poses more challenges to devise a prediction model for the harvested energy. There is a significant difference between the energy pattern in and, as listed in Table [tableCompPredEH]. The harvested energy in is predictable due to its overall known pattern as described in Section [subsec:predictingharvestedenergySEHIoT]. However, in contrast to, the harvested energy in can not be predicted by merely accumulating the previous weighted energy samples, as in , especially for long term energy predictions. The reason behind it lies in the sudden changes in the harvested energy due to the change in the nature of the underlying vibration source. The harvested energy pattern in based depends upon the type of application which demands dedicated models for harvested energy prediction for each use case. If the target application is human activity recognition, the energy is non-identical in different types of activities , including walking, standing, running, going upstairs/downstairs, etc. Similarly, if the target application is transport mode detection , the harvested energy depends on the mode of transportation. Therefore, demands different energy prediction models due to the distinct harvested energy pattern in each application. Furthermore, in contrast to static applications (except ), is typically deployed in a mobility scenario, and has various states in most of the context detection applications, including human activity recognition and transport mode detection. Therefore, in order to predict the harvested energy in, the mobility of the energy harvester must be taken into account to achieve higher prediction accuracy. Similarly, dedicated energy prediction algorithms are required for other energy harvesters, such as and to estimate the harvested energy for execution of the tasks on sensor nodes efficiently. The predicted harvested energy plays an important role in scheduling the tasks in sensor nodes. As the capacity of is limited (due to small-sized capacitors/batteries), the harvested energy can overflow if the stored energy is not consumed in executing the tasks beforehand, which results in the wastage of resources. For example, when the battery is fully charged, there is no room to store the incoming harvested energy, which results in energy wastage, if it is not properly utilised. The solution is to utilise the maximum energy, when the is charged to its capacity as well as there is a prediction of future harvested energy. It results in the efficient employment of resources and the maximum utilization of energy in executing the tasks within their deadlines. Similarly, if the is depleted, the tasks can be delayed, according to the predicted harvested energy, to achieve a minimum performance level. Batteryless ----------- Conventional employ batteries to store the harvested energy and power the sensor nodes. However, batteries are costly, bulky, toxic and have a limited lifetime . A promising solution is to use capacitors to store the harvested energy. As capacitors generally have higher leakage and lower energy storage capacity compared to batteries, task scheduling with a capacitor-based is more challenging. Intermittently charged capacitors restrict the continuous utilization of energy, which further restrain the frequent execution of tasks. Therefore, there is a potential of task scheduling algorithms that take into account the intermittent operation of load/sensor node to enable the autonomous operation of batteryless sensor nodes in with higher context detection performance. Conclusion ========== Energy harvesters are employed to power sensor nodes in to replace the conventional manually rechargeable batteries that hinder their widespread adaptability and pervasive deployment. In addition to energy generation, recently energy harvesters have been used as sensors for context detection. This saves significant energy that would otherwise be used for powering conventional activity sensors. Using the energy harvester as a simultaneous source of energy and information enables energy positive sensing, which harvest higher energy than required for signal acquisition for context detection. However, the harvested energy is still not sufficient to allow the of sensor nodes in. In order to ensure the sustainable operation of sensor nodes, the precious harvested energy needs to be consumed very efficiently for running the operational tasks on the nodes. In this survey paper, we comprehensively analyse the previous task scheduling based energy management algorithms for sensors. We critically analyse the challenges in incorporating the emerging class of energy harvesting based sensors in the conventional task scheduling algorithms. Based on the extensive study of the literature, we rigorously review the need for revised task scheduling algorithms for energy positive sensors and provide potential solutions. Finally, we present future research directions towards the goal of enabling the sustainable and autonomous operation of batteryless sensor nodes in. --- 1. The numbers are obtained from Dimensions Source: <https://app.dimensions.ai/discover/publication> Accessed on: March 16, 2020[↩](#fnref1) 2. In the literature, PMU is also called as Energy Management Unit (EMU)[↩](#fnref2)	arxiv_0000450
example to illustrate the scheme and how it is used to extract universal physical properties. In section [sec:bosons] we turn to the application of SDRG to the superfluid insulator transition of bosons in a random one dimensional potential. We contrast the strong-disorder theory with the weak disorder analysis of Giamarchi and Schulz. In section [sec:extensions] we discuss extensions of the SDRG analysis to bosonic systems with ohmic dissipation and to bosons in two dimensions. Brief review of the technique ============================= In the standard application of SDRG to random systems the aim is to solve for the universal properties of the ground state and low energy excitations. As in any RG scheme, we focus on low energies by successively eliminating high energy modes, thereby generating a series of effective Hamiltonians acting on the thinning Hilbert space. Within field theory, this program is often facilitated by the weakness of the non-linear coupling that mixes high and low frequencies. What is the guiding principle that allows to safely eliminate high energy modes of a random system without changing its low energy physics? The key is the local separation of scales effected by the strong randomness. A grain of the system with atypically high energy near the cutoff scale $\W$, is likely to be surrounded by much weaker couplings. The broader the disorder distribution the more the chosen grain sticks out of its surrounding, thereby allowing a perturbative treatment of the coupling between the grain and its lower energy neighbors. Lets see how it works in a concrete example. Dasgupta-Ma decimation in the spin-1/2 Heisenberg chain ------------------------------------------------------- We now explain the procedure through the simple example of a random spin-1/2 Heisenberg chain. H=iJi Sii+1 The motivation to study this model came from experiments done on quasi-1d organic salts, mostly Qn(TCNQ)2. These salts have chains of stacked double benzene rings, with each pair having one excess spin-1/2. The measured susceptibility behaves as a power-law χ ∼ T− α with α < 1 and varying from sample to sample. Recall that the Currie susceptibility of free spins is χ ∼ T− 1. Let us assume that the exchange coupling Ji between the spins is widely distributed. Somewhere on a finite chain there is a bond l having the largest exchange coupling which we denote by Jl = Ω (From now on we denote the upper energy cutoff of the chain by $\W$). Because the distribution is wide, the largest bond is much stronger than a typical bond and in particular it is likely to be much larger than the neighboring bonds, that is Jl ± 1 ≪ Ω. Therefore in our search for the ground state of the chain we can first diagonalize the strongest bond with $H\_0=\W\,{\bf S}\_l\cdot {\bf S}\_{l+1}$ and treat the couplings to the rest of the chain as a perturbation. At zeroth order the degenerate low energy manifold of the chain consists of the spins l, l + 1 frozen to a singlet state, whereas all other spins of the chain are free. The effective Hamiltonian acting within this subspace is obtained through degenerate perturbation theory in all the other couplings. At first order we retrieve all the original nearest neighbor couplings along the chain, except the coupling of the strong pair to their left and right neighbors. A new coupling between the left and right neighbor spins is generated at second order of perturbation theory through virtual occupation of the excited states of the strong pair at energy $\W$: l-1,l+2=Jl-1 Jl+12Sl-1l+2. We thereby eliminate the two strongest interacting spins and reconnect the chain by generating an effective coupling between the spins to the left and right of the decimated bond: Jl-1,l+2=Jl-1,Jl+1[Jeff] We expect this perturbation theory, controlled by the small parameter $J\_{l\pm1}/\W$, to work almost every time in the limit of strong disorder. Crucially, after the decimation step we have exactly the same form of the Hamiltonian: a nearest-neighbor Heisenberg model. The next stage in our analysis must be repeated application of the decimation step. This leads to gradual renormalization of the distribution of the exchange constants Ji upon decreasing the cutoff $\W$. If the disorder increases with repeated decimation, then we are safe. The strong disorder assumption only becomes better and better. This is indeed the case as we will show below. Qualitative ground state picture: the random singlet phase ---------------------------------------------------------- The random singlet phase of a random Heisenberg model. Pairs of strongly interacting sites form non-overlapping singlets in a random fashion. These singlets mostly form between nearest neighbors, but also over an arbitrarily large distance. The long range singlets induce strong correlations between far away sites. [rsfig4] Before going into the formal derivation of the flow equations, let us sketch a qualitative picture of the ground state that may be inferred from the structure of the RG decimation rules. A decimation of a bond essentially freezes two spins in a singlet state. At early stages of the RG, many singlets form between nearest neighbors. However, as more bonds are decimated, and the cutoff scale $\W$ is reduced, the largest couplings may occur on bonds generated at earlier stages of the RG between further neighbors. Eventually as Ω is reduced far below its initial value singlets may form between very far sites. The qualitative picture of the ground state generated in this process shown in Fig. [rsfig4] looks like a random arrangements of non crossing singlet pairs occuring at all scales. Hence the name random-singlet phase. From this simple picture we can infer important information about the nature of the correlations in the system. Suppose we are given a single realization of a random Heisenberg chain. What is the typical correlation $C\_{ij}=\av{{\bf S}\_i\cdot{\bf S}\_j}$ we will measure between two far removed spins i and j? Most likely these specific spins have not formed a singlet and therefore the correlation between them is very small - exponentially suppressed with distance (as we shall see later, with the square root of the distance): Cijtypical~e-a. These are called the typical correlations. What if rather than a single realization the measurement averages over an ensemble of chains. This would be the case, for example, if we could do neutron scattering on a bulk sample of the material Qn(TCNQ)2, which contains a macroscopic number of chains. The average over the many realizations could be dominated by rare instances in which the correlation Cij is atypically strong. Specifically, in the rare possibility that the two spins i, j do happen to form a singlet, the correlation between them would be − 1. How rare is that really? If the two sites considered survive through the decimation procedure until they become nearest neighbors, then they are very likely to form a singlet. The probability of a site to survive to that stage, i.e., until ∣i − j of its original nearest neighborson the side of the other site are removed, is 1/∣i − j∣ (this is the density of survivng sites at that stage). Hence the probability of both sites i and j to survive to that stage is ps ≈ 1/∣i − j∣2. We can now compute the average correlation to be ij=(-1)ps+e-a(1-ps)-ps~1(i-j)2, So despite the localized nature of the ground state, the average correlations fall off only as a power law. This is a good example of Griffiths effects, where the average correlations of a random system are dominated by rare instances with anomalously strong correlations. Master equation for the flow of distribution functions ------------------------------------------------------ Let us now show how detailed and precise information about the ground state and low-energy correlations is obtained, following Fisher. The key step is to translate the Dasgupta-Ma decimation rules into a master equation describing how the repeated decimations renormalize the probability distribution of exchange couplings when they operate on an ensemble of Hamiltonians. Let us denote by ρ0(J) the distribution of J in the physical system of interest. As we decimate more bonds while reducing the cutoff scale from $\W\_0$ to $\W$ the distribution evolves to $\rho(J,\W)$. To derive the master equation that governs this evolution, it is much better to replace J and $\G$ by the dimensionless scaling variable $\zeta\_i=\ln ({\W/ J\_i})$ and RG scale parameter $\G=\ln(\W\_0/\W)$. While J was defined on the changing interval $[0,\W]$, $\z$ is always defined on the interval [0, ∞]. In particular $\z\_i=0$ corresponds to the bond with largest exchange coupling $J=\W$. In these variables the Dasgupta-Ma decimation rule ([Jeff]) takes the additive form: i-1,i+2=i+i+1+2 [DasMaLog] Due to the strong disorder assumption, $\z\_i =\ln (\W/J\_i)$ is almost always much greater than − ln2, and it will therefore be safe to ignore the latter in Eq. ([DasMaLog]). Now we are ready to derive the master equation for the distribution of ζ’s - which we denote $P\_{\G}(\zeta)$. Integrating out the high energy shell $[\W-d\W,\W]$ consists of two stages: (i) Remove all the strong bonds with $0<\z<d\G$ and redefine the remaining ζ’s according to the new cutoff, (ii) add the couplings generated through second-order perturbation theory across the decimated bonds. Let’s consider the contribution of stage (i) to the change of the distribution function. Having reduced the cutoff we need to redefine the ζm on every remaining bond: mm-d==m-=m-d. So the entire distribution P(ζ) moves to the left. This can be expressed mathematically: dP()=d. [m1] Now consider the contribution from stage (ii), which is adding the renormalized bonds. The RG rule ([DasMaLog]) prescribes how the distribution of the new bonds derives from that of the original constituent (left and right) bonds: P()=0dl0dr P(l)P(r)(-l-r-2) The contribution of these bonds to the full distribution should be scaled by their fraction in the total population. Since the new bonds are produced only where we find a strong bond the probability of generating them is the probability to find a strong bond $\z\in[0,\G]$ that is $P(0)d\G$. From this we get the contribution of the newly generated bonds to the full distribution $dP(\z)=d\G P(0)P\_{\text{new}}(\z)$. Putting the two contributions together we obtain the master equation: =+ P(0)0d0dr P()P(r)(--r), [me] where we dropped the $\G$ subscript of P(ζ) and neglected the ln2 in the $\d$-function. There is a point we glossed over. We removed some probability by getting rid of all the probability density at small ζ, and we added probability by adding all the new bonds. Do we need to adjust the normalization of our distribution function? Integrating both sides of Eq. ([me]) reveals that the normalization is unchanged. For each bond we lost, we added a renormalized bond. Solution of the flow equation ----------------------------- The additive nature of the decimation rule ([DasMaLog]), embodied in the $\d$-function that appears in the master equation ([me]) suggests a potential solution in the form of an exponential distribution $P\_\G(\z)=f(\G)e^{-f(\G)\zeta}$. Indeed, plugging this ansatz into ([me]) leads to an ordinary differential equation for $f(\G)$, $\partial\_\G f=-f^2$, which is solved by $f( \G)=1/\G$. Hence we obtain the self similar solution P()=1e-/. We see that the system flows to infinite randomness as the width of the distribution grows without limit. Converting back to the physical variables we obtain a power law distribution of effective exchange coupling (J)= 1 (J)1-1/ [rs-dist] This approaches a non-normalizable distribution ∝ 1/J at the fixed point. Again a sign of infinite randomness. The distribution ([rs-dist]) is an attractor of the RG flow. Moreover, it turns out to be a global attractor. The nearest neighbor Heisenberg chain flows to the same infinite randomness fixed point regardless of the initial distribution as long as the disorder is not correlated. Hence the results we will extract for the low energy physics are universal. Physical properties ------------------- An important step in calculating physical properties is to establish a relation between energy scale $\W$ and length scale. A typical length scale is the distance between surviving spins at the scale $\W$. Lets compute the number of surviving spins on the original chain. Every time we decimate a strong bond, we remove two. Therefore, upon changing the RG scale by $d\G$, the number changes by $dN=-2P\_\G(0)N d\G$. Recall from the above solution that $P(0)=f(\G)=1/\G$; hence $N(\G)\sim N\_0/\G^2$. The average distance between surviving spins is then $L(\G)=l\_0 N\_0/N\sim \G^2$, or converting to energy units: $L(\W)\sim l\_0\ln^2(\W\_0/\W)$, where l0 is the original lattice spacing. This means that the excitation energy of singlets of length ℓ is: J~-, which is contrary to usual quantum-critical point scaling where E ∼ 1/ℓz. this type of scaling is called infinite-randomness scaling. We are now in position to compute the spin susceptibility at temperature T. Lets run the RG decimation from the upper cutoff $\W\_0>>T$ down to $\W=T$. The decimated spins are essentially frozen into singlets and therefore do not contribute to the susceptibility. On the other hand, surviving spins at the scale $\W$ are typically coupled by bonds J ≪ T, and therefore expected to behave as free spins. They contribute a Currie susceptibility χ(T) = n(T)/T, where n(T) is the density of surviving spins on the chain at energy scale $\W=T$. From our previous calculation we have $n(T)=N(T)/N\_0\sim n\_0\ln^{-2} (\W\_0/T)$. Therefore, (T)=. This is not quite T− α, but on a log scale it is indistinguishable from a power law. Random bosons: from infinite to finite randomness ================================================= Let us now turn to the second aim of this review: the application of the SDRG method to interacting bosons propagating in a random potential. Model ----- Our analysis focuses on the quantum rotor Hamiltonian: =j Uj (j-j)2 -j Jj ł(j+1-j), [model] where on each site the phase $\hat\theta\_i$ and charge $\hat n\_i$ are conjugate variables, which obey $\l[\hat{n}\_j,\,\f\_k\rr]=-i\delta\_{jk}$. This model describes an effective Josephson-junction array (see Fig. [fig:model]a) with random Josephson couplings Ji and charging energies Ui. In addition, there is a random offset charge ${\bar n}\_i$ in each grain, which is tantamount to a random gate voltage, $V\_j=2U\_j\overline{n}\_j$. The integer part of $\on\_i$ can be absorbed into the definition of $\hat n\_i$ such that $\on\_i$ is defined on the interval $(-\half,\half]$. In this review we focus on the case of generic disorder, where $\on$ can attain any value in this interval. The generic case, as well as the more restricted $\on$ disorder classes, were considered in Ref.. The different classes of disorder give rise to distinct insulating phases, but cause nearly unnoticeable differences in the behavior at the critical point. Besides Josephson junction arrays there are several other important physical realizations of the model ([model]). In systems of ultra-cold trapped atoms, disorder can be generated by optical speckle patterns, incommensurate lattice potential, or by corrugation in the wire that generates the trapping magnetic field in atom chips. With increasing disorder, the atoms concentrate in small puddles at minima of the potential connected by random Josephson coupling which depends on the potential barrier between them. The Hamiltonian ([model]) can be rigorously derived and the distributions of coupling constants computed ab-initio. Finally, disordered superconducting nano-wires with no unpaired gapless electrons will also be generically described by Eq. ([model]). More generally, the model ([model]) is a good effective description for superfluid to insulator transition driven by phase fluctuations. It can be thought of as a low energy effective theory for the phase degrees of freedom, after integrating out the gapped amplitude fluctuations. The weak disorder limit[wdRG] ----------------------------- Before describing the strong-disorder theory, let us briefly review the common understanding of the superfluid insulator transition in one dimension, taking the weak disorder viewpoint. The weak disorder theory was formulated in a seminal paper by Giamarchi and Schulz (GS) Ref.. The weak disorder expansion is natural to carry out when the action is written in terms of charge variables that live on the bonds rather than on the sites. Define j=ij ni. The current through the i’th bond is then $\dot{\phi}\_i$, and the charge on the i’th site is ϕi − ϕi − 1. Therefore, the charging energy of the site is $E\_{c}=U\_i \l(\phi\_i -\phi\_{i-1}-\on\_i\rr)^2$. What about the Josephson energy? If we neglect phase slips, a Josephson junction is essentially an inductor with inductance Li = 1/Ji. Therefore the Josephson energy in this approximation is: $ E\_J\approx L\_i I^2/2={\dot\phi\_i}^2{/ (2J\_{i})}. $ We are tempted to write the Lagrangian in this representation as 0=i [GS-latt] But this expression does not capture the periodicity of the Josephson energy with respect to the phase difference accross the junction. To account for the periodicity, we must allow for phase slips that change θ → θ ± 2π. Since the phase and charge are canonical conjugates, the translation of all phases to the left of the junction i by ± 2π is achieved by the operator exp[ ± 2πi∑j < inj] = e± 2πiϕi. Such terms should be included and assigned an action cost: ps=- i i ł(2i) ξi is the fugacity (or rate) of phase slips, and strongly depends on both Ji and Ui, Ui + 1. The total Lagrangian is given by L = L0 + Lps. Within the weak disorder limit, the randomness in 1/J and U that appears in L0 is perturbatively irrelevant. Furthermore, the disorder in $\on\_i$ can be absorbed by shifts to ϕ. These shifts then appear in the ξ term, and produce a combination of cosine and sine terms. Rewriting the Lagrangian in the continuum limit, with this in mind, gives dx ( 2-v ()2)-a-1( (x) e2i+(x)\* e-2i) [GS-cont] with ξ(x) a complex number, a is the original lattice constant, v the sound velocity, and K the Luttinger parameter (our definition of K is the inverse of the K appearing in Ref. ). The weak-disorder analysis proceeds by assuming a Gaussian distribution of the disorder with $\overline{\xi(x)\xi(x')^\}=(\pi v)^2 a^{-1} D\delta(x-x')$. The disorder strength is parameterized by the dimensionless parameter D. Next, momentum shell RG is used to find the flow of K* and D. The result, quoted from is: =D (3-2K),=-D K2. These scaling equations imply that the critical point is characterized by a universal value of the Luttinger parameter K = 3/2 irrespective of the initial disorder strength. It is important to note, however, that obtaining the long-wavelength field theory ([GS-cont]), with a single well defined Luttinger parameter K, relied on having weak disorder in J and U in the first place. We will see later what can go wrong with this mapping at strong disorder. Decimation steps for SDRG [rgsteps] ----------------------------------- Our main task now is to develop a strong-disorder RG scheme for the random-boson problem. As in the original application of real space RG to disordered systems, the strategy for finding the ground state and low lying excitations is to iteratively isolate and solve the strongest elements in the Hamiltonian. There are two types of elements in the Hamiltonian: onsite charging energies, and bond Josephson couplings. Roughly, two sites connected by the strongest bond will be converted to a phase-coherent cluster. Similarly, in sites with strong charging interactions we will eliminate all the on-site excited states. First, we must determine what we mean by strong elements of the Hamiltonian. The charging interaction can be considered while ignoring Josephson couplings. It is natural to pick the charging-energy gap separating the ground state from excited charge states as the energy scale for decimation purposes. This gap is given by: i=Uił(1-2\|i\|) [cgap] where $-1/2<\on<1/2$. At the same time, the energy scale, which characterizes a bond is the Josephson coupling, Ji. A strong bond is expected to bind two sites into a phase-coherent cluster. [fig:model] At each step of the RG we eliminate the term responsible to the largest energy scale =i{Ji,i}. [ogap] If the largest energy scale is a gap, Δi, then the site i freezes into the charge state with the lowest energy. Quantum fluctuations induce an effective hopping between sites i + 1 and i − 1: Ji-1,i+1=. This result, obtained by simple application of second order degenerate perturbation theory, is illustrated in Fig. [fig:model]b. The factor $1+2\|{\bar n}\|$ in the denominator varies between 1 and 2; it does not affect any universal features of the transition and we can safely set it to unity. Alternatively, when the strongest coupling in the chain is the bond Ji, a phase-coherent cluster forms, with a phase ${\tilde\theta\_{i,i+1}}=(\hat\theta\_i+\hat\theta\_{i+1})/2$. Since charging energy is the inverse of capacitance, and the capacitance is additive, the effective Ui, i + 1 of the new cluster will be: =+. [urecurs] The offset charge $\overline{n}$ in the cluster that forms in the bond decimation step, is simply the sum of the two offset charges: i,i+1=i+2. This equality, however, is defined modulo adding or subtracting one, so the the result always belong to the interval $\on\_{i,\,i+1}\in(-1/2,1/2]$. This decimation step is illustrated in Fig. [fig:model]c. We note that the energy scale $J=\W$ eliminated in the bond decimation step is the energy barrier that needs to be overcome in order to generate a phase slip which breaks the coherence. The gap to internal fluctuations of the relative phase $\w\_J=\sqrt{\W U\_i}$ is in general much smaller than $\W$. Fortunately virtual occupation of the Josephson plasmon affects only a small change on the Josephson coupling of the joined cluster to the neighboring sites, Ji-1Ji-1(1-18),which we will find is negligible near the critical point. It is important to note here that the above decimation step does not assume long range order; it states that phase fluctuations within the newly-formed cluster are harmonic, and therefore the cluster can not be broken due to phase-slips. These harmonic fluctuations are crucial for the understanding of the properties of the superfluid phase, as explained in Sec. [SFLutt]. Before proceeding, we note that in this discussion we have consciously omitted another possible bond decimation step. Suppose that the two sites linked by the strong bond with J = Ω have large charging energies Ui > Ω, but offset charges $\on\_i$ close to $\half$ such that $\D\_i<\W$. These sites then have two nearly degenerate charging states, and behave effectively as spin-1/2 degrees of freedom rather than rotors. Once connected with a large J of order Ω they will form a singlet state, i.e., a single boson will be shared equally by them. These sites therefore are not joined to make an effective superfluid cluster, but rather are completely eliminated while generating Jeff between the left and right neighbor to the pair. This possibility is treated in Ref. and is very important in describing the insulating phase. In this review we focus on the superfluid phase and the critical point, where the probability of finding two nearby sites with large U is exceedingly small and can be neglected. Flow equations and phase diagram -------------------------------- ### Scaling variables The iteration scheme outlined above translates to a flow of distribution functions for the Josephson couplings, interaction strengths, and offset charges. Finding a solution of the flow equations is made easier if we parametrize the couplings in the Hamiltonain appropriately. Typically, the best parametrization is in terms of variables that make the decimation steps become approximate sum rules. For the Josephson couplings this is achieved by using a logarithmic parametrization: i=/Ji. The charging energies are best expressed through the capacitance, which is additive in the cluster formation step. We define: i=/Ui. Note that $\zeta=2\W C$ with C the capacitance. Offset charges are added up as well upon decimation, and therefore are good variables according to the above criterion. As the dimensionless flow parameter, we define as before: =, where $\W\_0$ is of the order of the largest bare energy scale in the Hamiltonian. ### Flow equations The distribution functions should reflect which Hamiltonian parameters are correlated. Clearly Ui and $\on\_i$ must be correlated, since the decimation procedure eliminates sites with large $\Delta\_i=U\_i(1-2\|\on\_i\|)$. The elimination of sites with a large gap, therefore, introduces correlations between charging energies and offset charges on each site. The Josephson couplings, however, remain uncorrelated with charging energies or offset charges in nearby sites. Thus we can parametrize the coupling distribution functions in terms of two functions: $g\_{\G}(\beta)$ the bond log-coupling distribution, and $F(\zeta,\on)$ the joint distribution function of the inverse charging energy, and the offset charge. For the latter, it is useful to write: F(,)=f(,)(-1+2\|\|), where $\zeta\_i=\W/U$ and the Heaviside step function Θ enforces the constraint $\W/\Delta>1$ or equivalently $\zeta>1-2\|\on\|$. We can now write the master flow equations for the distributions of coupling constants implied by the decimation rules discussed above: &=&g+f1d1 d2 g(1)g(2)(-1+2)+g()(g0 -f1),[bgflow1] &=&+g0 d1d2d1d2 f(1,1)f(2,2) (-1+2)(-1+2) &&+f(1+f1-g0 ).[bgflow2] Let us recount the origin of all terms contributing to the derivatives $dg(\beta)/d\G$ and $df(\zeta,\on)/d\G$. First, a trivial, yet important, effect on the distributions stems from changing the cutoff $\W$. Since βi = lnΩ/Ji, the change of Ω → Ω − dΩ causes a shift βi → βi + dβ with $d\beta=d\W/\W=-d\G$. By the same token, this also shifts ζ → ζ + dζ with $d\zeta=-\zeta d\W/\W=-\zeta d\G$. These simple shifts are captured through the chain rule by the first term in both equations, ${\partial g\over \partial \beta}$ and $\zeta{\partial f\over \partial \zeta}$. The next terms are due to the formation of new bonds and clusters. As $\W\rightarrow \W-d\W$, a bond is decimated if its strength J is within this range, i.e., if $\beta<d\G=d\W/\W$. The probability of this for each bond is $g(\beta=0)d\G$. Out of two clusters with paramters ζ1, 2 and $\on\_{1,\,2}$, a bond decimation produces a new cluster with parameters $\zeta=\zeta\_1+\zeta\_2,\,\on=\on\_1+\on\_2$ (suppressing the mod in the calculation). This is the essence of the convolution term in Eq. ([bgflow2]). [palestinian-flag] The convolution term in Eq. ([bgflow1]) describes the formation of a new bond upon site decimation with the new bond variable $\b=\b\_1+\b\_2$. The probability for such a site decimation event, defined as $f\_1 d\G$, is somewhat more complicated than the probability for bond decimation discussed above. This is because a site is decimated when $\W=\D\_i=U\_i(1-2\|\on\_i\|)$, which occurs at a value of ζ that depends on $\|\bar n\|$. The region in parameter space $(\on,\zeta)$ that would get decimated is shown in Fig. [palestinian-flag] and the corresponding probability is obtained from integrating over it f1=-0.50.5 d(1-2\|\|)f(1-2\|\|,) Note that as the energy scale $\W$ decreases by $d\W$, the width of the decimated region scales as $\zeta d\G$, which shrinks to zero on the $\on=\pm1/2$ lines, since $\zeta\_i=1-2\|\on\_i\|$ on the decimation curve. In the vicinity of the transition, the seemingly complicated expression for f1 simplifies significantly. As more sites are joined, the charge offsets add up (modulu 1), and therefore rapidly pick values uniformly distributed within $(-\half<\on<\half]$. The distribution $f(\zeta,\on)$ becomes essentially independent of $\on$, and since it flows to strong in ζ it is only weakly dependent on ζ for values ζ < 1. Thus, we can simply replace $f(1-2\on,\on)\approx f\_0$, which implies $f\_1\approx \half f\_0$. The remaining terms in Eqs. ([bgflow1]) and ([bgflow2]) are needed to maintain the normalization of the distributions. Normalization can be verified by integrating over the entire range of ζ and $\on$ in Eq. ([bgflow2]) and β in Eq. ([bgflow1]), and making sure that the derivative of the total probability integral on the LHS’s is zero. ### Scaling solution and reduced flow equations Next, we attempt to solve the functional flow equations by employing the following intuition, which is gained from the spin models analyzed in Sec [sec:method]. When the decimation steps follow a sum rule for two positive couplings, the probability distributions for these couplings should consist of exponentials. This leads to the scaling ansazt: g()&=&g0e-g0, [ans1] f(, )&=&f021-e-f0e-f0 [ans2]. This ansatz describes a family of functions that are parametrized by two variables, f0 and g0. Plugging the scaling ansatz back to the flow equations, one hopes, will generate a flow equations in terms of f0 and g0 alone, and without any explicit functional dependence on ζ or β. Approximately, this is indeed the case. We assume that f0 ≪ 1, and the substitution of the scaling ansatz to the flow equations yields: &=&f0 (1-g0)[SFeqa] &=&-f0 g0. [SFeq] The flow lines in the space f0 versus g0, plotted in Fig. [fig:RGflow], are given by f0=2(g0-1-g0)+. The different flow lines are parameterized by ε, which sets the detuning from the critical manifold at $\e=0$. Flows that lie below the critical manifold, $\e<0$, terminate on the line g0 > 1, f0 = 0 marking the superfluid phase. Values above the critical manifold, $\e>0$, flow to the region g0 < 1 and where f0 is relevant and flows to larger and larger values. This flow characterizes the insulating regime of the model. [fig:RGflow] We note that the scaling equations ([SFeq]) have the form of the Kosterlitz-Thouless flow equations, if we were to rewrite them in terms of $\sqrt{f\_0}$ and g0. These scaling variables, however, have a different physical meaning. In particular, the variable that gains a universal value at the transition is the exponent g0 rather than the Luttinger parameter. Below we elucidate the nature of the superfluid phase. We then explain the essence of the critical point in light of the anomalous properties of the superfluid leading up to it. This will also serve to clarify why a theory that is perturbative in the disorder strength, such as the replica treatment of Giamarchi Schulz can fail in this regime. The superfluid phase -------------------- As mentioned above, the line g0 > 1, f0 = 0 marks the terminus of the flows on the superfluid side of the phase diagram. What this fixed line describes, is the formation of a global superfluid cluster where phase-slips are prohibitively rare, and do not disturb the superfluidity. Indeed, the RG is dominated by bond decimations where clusters repetitively coalesce until they connect the two sides of the chain, and, therefore, the stiffness of this cluster is finite. The fact that the interaction parameter f0 flows to zero may at first sight suggest that the superfluid is described by a classical Josephson array with infinite compressibility. To see why this is not the case let us start with a system of length L and continue the decimation until there is only one site left. This site represents a superfluid cluster extending through the full length L of the chain. The typical value of the charging energy of the whole cluster is Ec ∼ Ωf0, which scales as ∼ 1/L in late stages of the RG flow. Hence f0 is irrelevant because it describes the charging energy of clusters that become macroscopic at the end of the flow. So, while the capacitance, which is an extensive quantity, grows as L, the compressibility, or capacitance per unit length, approaches a finite value at the terminus of the flow. Therefore, a superfluid fixed point, written in terms of the phase variable, is essentially a quantum harmonic theory with random “spring constants” Ji distributed according to the power-law distribution P(J) = (g0/Ω) (J/Ω)g0 − 1. On the other hand, the masses of the coupled oscillators are related to the compressibility κ. The elementary excitations of the superfluid described by the harmonic theory on the fixed line are harmonic phonons, which are localized by the disorder at all frequencies $\w>0$. For weak disorder the localization length associated with a single phonon wave-function at frequency $\w$ diverges toward zero frequency as ℓ ∼ 1/ω2. On the other hand, it was shown in Ref. that the distributions obtained by SDRG give rise to anomalous localization properties with: ~ł{ (2)/ & g0=1 1/g0 & 1<g0<2 1/2 &g02. To see how the wide distributions of J lead to an anomalous superfluid phase, consider the superfluid stiffness of an effective harmonic chain of length L, given by s-1=1Li Ji-1, [stiffness] According to the SDRG, the random variables xi = 1/Ji are distributed as p(x) ∼ x− (1 + g0). For g0 < 2 the variance of 1/Ji diverges, as does the variance of the macroscopic variable 1/ρs. This is the origin of the anomalous behavior in the superfluid having 1 < g0 < 2. Now we also see why the weak disorder theory can fail in the strong disorder weak interaction regime of the superfluid. Consider again the dual lattice action ([GS-latt]) that serves as the starting point for Ref.. Now, that the disorder in 1/J is diverging in magnitude, it cannot be brushed away even if it is perturbatively irrelevant. It is important to keep in mind that the power-law distribution with the exponent g0 is not the bare distribution of J’s in the array, but rather the fully renormalized distribution at low energy scale. It is natural to ask at this point if there is a simple observable that can bear witness to this distribution and allow to measure g0. One such quantity is the critical current, which is simply the Josephson energy of the weakest link in a chain. Consider a superfluid chain much longer than the coherence length ξ, which would diverge at the critical point. Renormalize down to the scale ξ, or as we will later show energy scale $\W\_\xi=\W\_0/\xi$. The dependence of ξ on the detuning from the critical point will be discussed in the next section. At this point we have an effective chain with N = L/ξ links. The Josephson couplings distribute as $P(J)=(g\_0 /\W)(J/\W)^{g\_0-1} $. The probability density of the weakest link, Jm, and of the current Ic (an extreme value statistic), is P(Ic)=N g00( Ic0)g0-1This is a power-law at small values of Ic, which is cut of by the exponential. A corollary of this is that the typical value of critical current, defined by the sharp peak of the distribution, vanishes as ${\bar I}\_c\sim (\W\_0/\xi) (L/\xi)^{-1/g\_0}$. That is, it vanishes as a power-law of the system size with power 1/g0 < 1. The superfluid-insulator transition [SFLutt] -------------------------------------------- The critical point which controls the unstable flow toward the insulator sits at g0 = 1 and f0 = 0, and marks the end of the superfluid fixed line. From our discussion above we can gain an intuitive understanding for why the superfluid should break down at g0 = 0. If we consider again the stiffness ([stiffness]) of a classical array (or a harmonic chain), we see that the average $\av{1/\rho\_s}$ diverges as ∼ (g0 − 1)− 1. Hence, the critical point is where the classical Josephson array loses its stiffness. It is, however, important to realize that the phase transition is not classical, and the superfluid stiffness at the transition point is not zero. We can think of the transition as being tuned by crossing the critical manifold at some value of f0 > 0 and g0 > 1. Quantum fluctuations due to sites with large charging energy drive the downward flow of g0 toward the fixed point, where g0 = 1. The superfluid stiffness and compressibility, on the other hand, are not universal properties on the fixed point, but must be integrated along the flow. For example, the stiffness of the final superfluid cluster spanning the chain depends on all the effective internal bonds J̃i that connect the sites making that cluster and were decimated in the process of its formation: ρs− 1 = L− 1∑iJ̃i− 1. Similarly, the compressibility is the sum of capacitances of all sites making up the superfluid cluster: κ = L− 1∑iUi− 1. In both cases L is the total length of the chain. These quantities were computed in Ref. on the critical manifold showing that they approach a constant value that depends on the initial disorder for the flow. For the sake of comparison with the weak disorder theory of Giamarchi and Schulz it is interesting to look at the Luttinger parameter, given by $K=\pi\sqrt{\rho\_s\kappa}$. There, the fully renormalized Luttinger parameter at the transition takes a universal value K = 3/2. On the other hand, for the strong-disorder transition, the SDRG predicts a value of the Luttinger parameter that depends on where the critical manifold was crossed. That value diverges as the crossing point gets closer to the classical fixed point at f0 = 0. The non universal value of the Luttinger parameter at the transition point is the most controversial prediction of the SDRG in the strong disorder regime. It is also worth noting that since the Luttinger parameter does not appear as a natural scaling variable in the SDRG, computing its value requires a rather elaborate integration over the entire critical RG flow from high to low energies. Hence it is not surprising that its value on the critical manifold comes out non universal within this theory. But because the Luttinger parameter is not a natural object to compute within the SDRG it is important to verify this prediction in numerical simulations. Such calculations will be discussed in section [sec:numerics]. Despite this difference, there is at least a formal similarity between the strong-disorder fixed-point and the Kosterlitz-Thouless flow at weak disorder. As we already pointed out, the reduced flow equations are formally identical to the Kosterlitz-Thouless equations, where the parameters g0 and $\sqrt{f\_0}$ of the distribution functions are playing the roles that the Luttinger parameter and phase-slip fugacity play in the Kosterlitz-Thouless flow. As a consequence, length and time scales have the same exponential divergence ~0 ea/ , ~0-1eb/ as a function of the detuning ε from the critical point, that is a hallmark of the KT transition. The insulating phases --------------------- The superfluid phase and the superfluid-insulator critical point discussed above are not affected in any important way by the nature of the initial disorder distributions. But type of disorder in the offset charges ${\bar n}\_i$ does have a decisive effect on the nature of the insulating phases. Different constraints on the distribution of ${\bar n}\_i$ give rise to three distinct insulating phases. The physical properties of these insulators are largely determined by the charging gap distribution of the sites that survive the RG flow deep into the insulating region. The possibilities are as follows: * Bose-glass phase – A generic distribution of offset charges, $-0.5\le\on<0.5$, results in a Bose-glass phase. This is a compressible and gapless state, which is also characterized by a diverging superfluid susceptibility. The gaplessness and compressibility are due to the presence of a finite density of sites with a charging energy arbitrarily close to zero. The diverging superfluid susceptibility, χ, is a typical property of a compressible phase. The form of the divergence is χSF ∼ ln(Ω/hp), with hp the probing proximity field. χ diverges in the limit of hp → 0. Alternatively, for a finite size sample, χSF ∼ lnL. * Mott-glass phase – In a commensurate lattice with disorder only in J and U, the offset charges vanish identically $\on\_i=0$. The system then flows to the Mott-glass phase; an incompressible yet gapless state with a finite superfluid susceptibility. Both the compressibility and the gaplessness are a consequence of the universal distribution of U, Eq. ([ans2]), which in this case also describes the charging gaps. The distribution $f(U)\sim \frac{1}{U^2}e^{-f\_0 \Omega/ U}$ has no support at U → 0, which implies incompressibility. Specifically: ~e-f0()0 where μ is the probing chemical potential. By looking at the smallest charging energy for a system of size L, we find that the gap drops as $\Delta\sim\frac{1}{\ln L}$. The Mott-glass is a classic example of a Griffiths phase: it is incompressible, yet the average temporal auto-correlations of the system will be dominated by the essentially single large cluster of size lnL that determines the gap. * Random-singlet glass – If $\on$ is randomly either $\on=0.5$ or $\on=0$, the gapless insulating phase has both a diverging compressibility and diverging superfluid susceptibility. Only sites with $\on=0.5$ and no charging gap survive late in the flow as two level systems. Thus, the insulator is described by an effective random spin-1/2 xx chain, known to be in the random singlet phase. Both the compressibility and superfluid susceptibility scale in the same way: SF~~. with μ being the probing chemical potential or proximity field. From the random-singlet energy-length scaling, we obtain for a finite size system: $\chi\_{SF}\sim\kappa\sim \frac{e^{-c\sqrt{L}}}{L^{3/2}}$. Numerical tests of the strong disorder transition ------------------------------------------------- The critical point identified using SDRG is characterized by finite randomness. Although the fixed-point analysis is also controlled and justified by the smallness of interactions in its vicinity, it is not as reliable as the analysis of the random singlet phase or the critical point of the random transverse-field Ising model, where the SDRG flow is to infinite randomness. It is, therefore, desirable to seek numerical verification of the theoretically predicted universal physics. Some of the recent work that addressed this question is summarized by Pollet in another review published in this volume. In what follows we will survey the numerical results and comment on the debate surrounding their interpretation. Since we are dealing with a strongly correlated system, a fully quantum calculation, such as quantum Monte Carlo (QMC) or Density Matrix Renormalization Group (DMRG) are needed to extract the critical properties. It is important to understand the fundamental difficulties, which complicate these numerical calculations in the strong disorder regime. First, according to the SDRG, rare weak links generated in the course of renormalization play a crucial role in driving the transition at strong disorder. This implies that a very large ensemble of different realizations of the random chain needs to be analyzed in order to detect the effect of rare events on the physics. Second, when rare weak links are effective they slow down the convergence of numerical methods such as QMC and DMRG. Early QMC studies of the strong disorder regime have not probed directly for the scaling predictions of the SDRG, but rather checked how BKT scaling works in such systems. For systems with moderate disorder Balabanyan et. al. found a good fit to BKT scaling with the appropriate value of the Luttinger parameter (K = 2 for the commensurate system used in that calculation). For stronger disorder, however, the fit did not work as well and strong finite size effects were cited as the reason. A more recent QMC study provided strong, albeit indirect, support for the strong randomness scenario. These simulations show KT-like scaling of the correlation length at the transition both for weak and strong disorder. Above a certain disorder strength, however, the Luttinger parameter as well as the susecptibility-length scaling exponent at the transition were seen to depend on the disorder strength at which the transition is crossed. In agreement with the strong disorder scenario, the critical Luttinger parameter exceeded the universal value predicted by the standard theory. Analytical support to the results of Ref. [] was given in [], which also showed, however, that at very strong disorder the asymptotic value of the Luttinger parameter is only obtained at very large length scales. The Luttinger parameter, which was the focus of these studies is, however, not a natural quantity to characterize the strong-disorder critical point. Rather, the relevant scaling properties are encoded in the RG flow, Eqs. ([SFeqa]) and ([SFeq]), of the variables g0 and f0 associated with the distributions of Josephson couplings and charging energies. An apparent difficulty in extracting this scaling behavior from numerical calculations is that the renormalized coupling distributions are not directly observable quantities. This problem was addressed in Ref., which developed a finite size scaling theory relying on measurable quantities. A direct connection was established there between the renormalized distribution of weak links, parameterized by g0, and the measured distribution of superfluid stiffness on an ensemble of finite clusters. Specifically, the tail of the distribution of the measured value of 1/ρs in a system of size L was shown to follow a power-law with the same exponent as the distribution of the variable xi ≡ 1/Ji at the energy scale Ω(L). Thus, the finite-size scaling behavior of the distribution of 1/ρs, could be compared with finite size scaling formulas obtained from the RG flow. The QMC results obtained for a model of bosons at integer filling with particle-hole symmetric (off diagonal) disorder were found to scale in the manner predicted by the SDRG to within the numerical error bars. Furthermore, finite size scaling assuming a standard KT transition did not fit the data at all (See Suppllementary material of Ref. []). Another recent study by Pollet et. al. [] focused on the anomalous finite-size scaling of the inverse stiffness in the superfluid phase leading up to the critical point. The Monte Carlo simulations carried out in this study also found distributions characterized by broad power-law tails as predicted by the SDRG. Pollet et al., however, made the interesting observation that the scaling of the median value of 1/ρ with L can be fit over a wide regime to the scaling of a classical Josephson array (or a harmonic chain) with a power-law distribution of Josephson couplings $P(J)\sim J^\a$ for small J. Such a classical JJA, or harmonic chain, have inverse stiffness ρs− 1 = L− 1∑iJi− 1, which leads to the scaling law (up to logarithmic corrections) $\rho\_s^{-1}(L)=\rho\_s^{-1}(\infty) + a/L^{\a}$ with $\a$ being a length independent exponent. While Ref. [] did not find noticeable deviations from classical scaling of the median of 1/ρs, the broad power-law distribution of the tail of 1/ρs is generated in the first place by sites with large charging energy, that is, by a quantum effect. Moreover, the power $\tilde\alpha$ extracted from the tail of the distribution rather than from the scaling of the median of 1/ρs is seen to flow downward as a function of L due to such site decimations. Pollet et. al. correctly point out that the tail of the distribution of 1/ρs is not important for the thermodynamic stiffness in the superfluid phase as long as the tail exponent ${\tilde\a}(L\to\infty)>0$. Nevertheless, it is important to note in this regard that the RG flow of $\tilde\a$ to negative values is one mechanism that would necessarily destabilize the superfluid. If the bare disorder is not sufficiently strong compared to the interaction, then this strong disorder mechanism can be preceded by proliferation of phase slips through the standard KT mechanism at a universal value of the Luttinger parameter. Interestingly, for the model, and parameter regime investigated in Ref. [] the value of the thermodynamic Luttiger parameter extrapolated in this way reached the universal value Kc = 3/2 very close (but on the superfluid side) of the classical transition point defined by $\a=0$. It is quite striking that, although the critical value of the Luttinger parameter found in this way is very close to the weak-disorder universal value, the only scaling theory that fits the data is that predicted by the SDRG. A possible explanation is that very strong deviations of the Luttinger parameter from the weak disorder universal value require exceedingly large systems to be observed according to the scaling analysis in Ref. [] (although values of K < 4 should still be reasonably accessible). Regardless, these puzzling results highlight the need for a more complete understanding of the superfluid-insulator transition in the intermediate disorder regime, where both rare weak links and vortex proliferation may play an important role. Further applications of SDRG to bosonic systems =============================================== Dissipative superconductors --------------------------- Systems of Josephson junctions in the presence of dissipation can also be analyzed using SDRG. The analysis considers a large N generalization of Josephson junctions (which would correspond to an O(2) model). In this limit, one can write a quadratic (imaginary-time) action which includes the nearest-neighbor Josphson coupling Ji, as well as dissipation γ, encoded in the frequency-dependent part of the action: S=i ł[-Jiii+1+ł(i+i\|\|2/z)i2]. The εi’s are determined self-consistently for each site by requiring that ⟨ϕ⃗2⟩ = 1. z is a parameter that determines the nature of the dissipation. The case z = 1 coincides with the case we have solved in the previous section, with γ being the capacitance. 2 > z > 1 represents the superohmic regime, while z > 2 is the superohmic regime. The case of z = 2 is precisely the Ohmic regime; furthermore, it coincides with the so-called Hertz-Millis theory for fluctuating order-parameters in a dissipative electronic environments. Speicifically, this action describes the onset of supercondcutivity in disordered thin wires. At z = 2, the ohmic case, one can carry out the same RG steps as outlined in Sec. [rgsteps] for the rotor model. A strong bonds lead to formation of clusters with dissipation γeff = γ1 + γ2, and decimating a site with a large ε2 ∼ e− γ2 produces a weak bond between its neighbors, $J\_{eff}=2\frac{J\_1 J\_3}{\epsilon\_2}$, where we used sites 1,2 and 3 as an example. This analysis, in the O(N) language, results in a phase transition tuned by $\delta=\overline{\ln J-\gamma}$ between an ordered phase with J’s relevant, and a paramagnetic phase with J’s scaling to zero. In an electronic wire, this provides a description of the superconducting-metal transition. The critical point is shown to be a random-singlet fixed point, identical in its universal properties to the transverse field Ising model. For the full details of this interesting and surprising result, we refer the reader to Ref.. Two dimensional rotor model --------------------------- Applying the SDRG to higher dimensions is always challenging, since the decimations alter the geometry of the system. Nevertheless, several works have succeeded in employing SDRG in higher dimensions for the transverse-field Ising model, the random-hopping model. Recently, a SDRG analysis of the 2d, square lattice, rotor model with no offset charge (as in Eq. ([model]) with $\on\_i=0$) was carried out in Ref.. Technically, the SDRG method had to be modified in several ways for the 2d rotor model analysis. First, cluster formation led to the addition of Josephson energy in the case of the sites making up the cluster are both connected to another site. Second, because the system may not flow to a strong-disorder fixed point, or one with nearly no interactions (as happens in 1d), the phase fluctuations, i.e., phonons, within clusters were taken into account as suppressing the Josephson couplings emanating from renormalized clusters. Third, in order to make the analysis numerically tractable, only couplings with strength above an absolute cut-off were recorded at each decimation stage. This is particularly important since the connectivity of the system increases dramatically with site decimations. The 2d rotor-model SDRG analysis revealed a percolation like superfluid-insulator phase transition. The transition was most apparent by looking at the parameter flow in the space of $\Delta J/\overline{J}$ vs. $\overline{U}/\overline{J}$, with $\overline{J}$ the average Josephson couplings, ΔJ the standard deviation of J, and $\overline{U}$ the average Josephson couplings (see Fig. [2DRGflow]). All quantities were calculated from the largest 2Ñ Josephson energies, and charging energies, with Ñ the renormalized size of the system. This is needed to put the analysis on the same footing as the initial model, which only has 2N Josephson energies, and N charging energies. A critical point appears at $\overline{U}/\overline{J}\approx 0.3$ and $\Delta J/\overline{J}\approx 1$. Reassuringly, these values were, by and large, independent of the initial disorder distributions. The transition was established to be of the percolation type. At the transition, a fractal cluster forms, with a Hausdorff fractal diension of df = 1.3 ± 0.2, and a correlation length exponent ν = 1.09 ± 0.04. The calculated exponent ν conforms with the Harris criterion νd > 2. In contrast the standard XY transition relevant for the clean limit is characterized by ν ≈ 0.663 and thus violates the Harris criterion. While the superfluid phase should be conventional, the insulating phase was identified as an incompressible Mott glass, with a gap that falls off as 1/lnL with L the linear size of the system. For complete details we refer the interested readers to Ref.. [2DRGflow] Conclusions =========== In this review we have explored two prominent applications of the SDRG method. First, we considered the Heisenberg model, where this method had its first success. Then, we presented the recent application of the method to the random-boson superfluid-insulator transition. The latter problem has been an unresolved problem for a long time, and research complementary to the SDRG analysis is continuing. The question of the ultimate stability of the strong disorder fixed-point to proliferation of phase-slips in the usual Kosterlitz-Thouless mechanism at very long scales remains open. However, numerical simulations indicate that the scaling predictions of the SDRG are, at strong disorder, more relevant to accessible system sizes than those of the standard weak disorder theory. Even beyond the problems presented in this review, the application of SDRG to low-dimensional quantum systems continues to be a fruitful research direction. A subjective opinion of the authors, is that the next frontier appears to be the detailed analysis of quantum dynamics. The basic principles for all applications of SDRG, however, remain the same as those presented here. Much of the work reviewed here was carried out together with many collaborators, without whom non of it would have come to pass. We would like to especially acknowledge Daniel S. Fisher, Anatoli Polkovnikov, Yariv Kafri, Victor Gurarie, Joel Moore, John Chalker, David Pekker, Shankar Iyer, Susanne Pielawa, and Ronen Vosk. In addition, we would like to express our gratitude for the many discussions we had with Thierry Giamarchi, Pierre le Doussal, David Huse, Thomas Vojta, and Steve Girvin, Nikolay Prokof’ev, Boris Svistunov and Lode Pollet. We are grateful to the Packard Foundation, as well as to the NSF, ISF, the Minerva foundation, the Miller Institute of Science at UC Berkeley, and the Moore Foundation for support through the Caltech IQIM. 41ifxundefined [1] ifx#1 ifnum [1] #1firstoftwo secondoftwo ifx [1] #1firstoftwo secondoftwo ““#1””@noop [0]secondoftwosanitize@url [0]` 12`$12`&12`#12`12`\_12`%12@startlink[1]@endlink[0]@bib@innerbibempty , , and , [Phys. Rev. Lett. 43, 1434 ( 1979)](\doibase 10.1103/PhysRevLett.43.1434) and , @noop Phys. Rev. B 22, 1305 ( 1980) and , [Phys. Rev. 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The focus is on recent applications of the strong disorder renormalization group to the physics of disordered-boson systems and the superfluid-insulator transition in one dimension. The fact that there is also a well understood weak disorder theory for this problem allows to illustrate what aspects of the physics change at strong disorder. In particular the strong disorder RG analysis suggests that the transitions at weak disorder and strong disorder belong to distinct universality classes, but this question remains under debate and is not fully resolved to date. Further applications of the strong disorder renormalization group to higher-dimensional Bose systems and to bosons coupled to dissipation are also briefly reviewed. Introduction ============ Random systems can be broadly classified by the effect disorder has at large length scales. In many cases, the quenched randomness tends to be averaged out on long distances. Disorder then plays only a minor role in determining the universal behavior, and could be understood at a perturbative level. The more interesting alternative is that the disorder remains finite as the system is coarse grained. Finally, at the opposite extreme are systems in which the disorder grows without bound upon coarse graining. Such systems are said to be governed by infinite-randomness fixed points. The strong disorder renormalization group (SDRG) method, which is the focus of this review, provides a way to exploit the strong randomness in order to systematically compute universal aspects of the physics. The technique was originally developed by Dasgupta and Ma to investigate the ground state and low energy behavior of the random Heisenberg spin chain. Later, the SDRG scheme was extended by Bhatt and Lee and formulated rigorously by Daniel Fisher. Fisher showed in particular that the scheme gives asymptotically exact results for the low energy universal behavior of systems controlled by infinite randomness fixed points. For example the random spin-1/2 Heisenberg chain, flows toward a ground state decribed by an infinite randomness fixed point, the random singlet phase, for any amount of bare disorder. Somewhat richer physics is at play in the random transverse field Ising chain. This model exhibits a quantum phase transition between a magnetically ordered and a paramegnetic phase that is controlled by an infinite randomness fixed point. The analysis of the Ising model was extended to two dimensions by Motrunich et. al., who found a similar transition controlled by an infinite randomness fixed fixed point. The SDRG approach has since been applied and extended to address a host of problems spanning different fields. These include classical stochastic dynamics, Dynamic response of spin chains at low temperatures, entanglement in random spin chains and most recently non-equilibrium quantum dynamics. For a comprehensive account of the technique and its various applications, we refer the reader to Ref.. The goal of this brief review is twofold. First, we aim to provide a clear and compact introduction to the SDRG method and the physical considerations involved in its application. The second objective is to review recent progress in understanding the superfluid-to-insulator transition of interacting bosons in a random potential from the standpoint of SDRG. In contrast to other problems mentioned above, it is not immediately clear why the SDRG should be a suitable approach to characterize the superfluid-insulator transition. In fact an accepted theory of boson localization in one dimension is perturbative in the disorder. It predicts a critical point at which disorder is (dangerously) irrelevant. However, it has recently been argued that the superfluid may undergo a very different localization transition if the bare disorder is sufficiently strong. Moreover, as we shall discuss, the superfluid phase itself may become anomalous due to the strong disorder. The rest of the review is structured as follows. In section [sec:method] we review the technique using the spin-1/2 Heisenberg chain as an example to illustrate the scheme and how it is used to extract universal physical properties. In section [sec:bosons] we turn to the application of SDRG to the superfluid insulator transition of bosons in a random one dimensional potential. We contrast the strong-disorder theory with the weak disorder analysis of Giamarchi and Schulz. In section [sec:extensions] we discuss extensions of the SDRG analysis to bosonic systems with ohmic dissipation and to bosons in two dimensions. Brief review of the technique ============================= In the standard application of SDRG to random systems the aim is to solve for the universal properties of the ground state and low energy excitations. As in any RG scheme, we focus on low energies by successively eliminating high energy modes, thereby generating a series of effective Hamiltonians acting on the thinning Hilbert space. Within field theory, this program is often facilitated by the weakness of the non-linear coupling that mixes high and low frequencies. What is the guiding principle that allows to safely eliminate high energy modes of a random system without changing its low energy physics? The key is the local separation of scales effected by the strong randomness. A grain of the system with atypically high energy near the cutoff scale $\W$, is likely to be surrounded by much weaker couplings. The broader the disorder distribution the more the chosen grain sticks out of its surrounding, thereby allowing a perturbative treatment of the coupling between the grain and its lower energy neighbors. Lets see how it works in a concrete example. Dasgupta-Ma decimation in the spin-1/2 Heisenberg chain ------------------------------------------------------- We now explain the procedure through the simple example of a random spin-1/2 Heisenberg chain. H=iJi Sii+1 The motivation to study this model came from experiments done on quasi-1d organic salts, mostly Qn(TCNQ)2. These salts have chains of stacked double benzene rings, with each pair having one excess spin-1/2. The measured susceptibility behaves as a power-law χ ∼ T− α with α < 1 and varying from sample to sample. Recall that the Currie susceptibility of free spins is χ ∼ T− 1. Let us assume that the exchange coupling Ji between the spins is widely distributed. Somewhere on a finite chain there is a bond l having the largest exchange coupling which we denote by Jl = Ω (From now on we denote the upper energy cutoff of the chain by $\W$). Because the distribution is wide, the largest bond is much stronger than a typical bond and in particular it is likely to be much larger than the neighboring bonds, that is Jl ± 1 ≪ Ω. Therefore in our search for the ground state of the chain we can first diagonalize the strongest bond with $H\_0=\W\,{\bf S}\_l\cdot {\bf S}\_{l+1}$ and treat the couplings to the rest of the chain as a perturbation. At zeroth order the degenerate low energy manifold of the chain consists of the spins l, l + 1 frozen to a singlet state, whereas all other spins of the chain are free. The effective Hamiltonian acting within this subspace is obtained through degenerate perturbation theory in all the other couplings. At first order we retrieve all the original nearest neighbor couplings along the chain, except the coupling of the strong pair to their left and right neighbors. A new coupling between the left and right neighbor spins is generated at second order of perturbation theory through virtual occupation of the excited states of the strong pair at energy $\W$: l-1,l+2=Jl-1 Jl+12Sl-1l+2. We thereby eliminate the two strongest interacting spins and reconnect the chain by generating an effective coupling between the spins to the left and right of the decimated bond: Jl-1,l+2=Jl-1,Jl+1[Jeff] We expect this perturbation theory, controlled by the small parameter $J\_{l\pm1}/\W$, to work almost every time in the limit of strong disorder. Crucially, after the decimation step we have exactly the same form of the Hamiltonian: a nearest-neighbor Heisenberg model. The next stage in our analysis must be repeated application of the decimation step. This leads to gradual renormalization of the distribution of the exchange constants Ji upon decreasing the cutoff $\W$. If the disorder increases with repeated decimation, then we are safe. The strong disorder assumption only becomes better and better. This is indeed the case as we will show below. Qualitative ground state picture: the random singlet phase ---------------------------------------------------------- The random singlet phase of a random Heisenberg model. Pairs of strongly interacting sites form non-overlapping singlets in a random fashion. These singlets mostly form between nearest neighbors, but also over an arbitrarily large distance. The long range singlets induce strong correlations between far away sites. [rsfig4] Before going into the formal derivation of the flow equations, let us sketch a qualitative picture of the ground state that may be inferred from the structure of the RG decimation rules. A decimation of a bond essentially freezes two spins in a singlet state. At early stages of the RG, many singlets form between nearest neighbors. However, as more bonds are decimated, and the cutoff scale $\W$ is reduced, the largest couplings may occur on bonds generated at earlier stages of the RG between further neighbors. Eventually as Ω is reduced far below its initial value singlets may form between very far sites. The qualitative picture of the ground state generated in this process shown in Fig. [rsfig4] looks like a random arrangements of non crossing singlet pairs occuring at all scales. Hence the name random-singlet phase. From this simple picture we can infer important information about the nature of the correlations in the system. Suppose we are given a single realization of a random Heisenberg chain. What is the typical correlation $C\_{ij}=\av{{\bf S}\_i\cdot{\bf S}\_j}$ we will measure between two far removed spins i and j? Most likely these specific spins have not formed a singlet and therefore the correlation between them is very small - exponentially suppressed with distance (as we shall see later, with the square root of the distance): Cijtypical~e-a. These are called the typical correlations. What if rather than a single realization the measurement averages over an ensemble of chains. This would be the case, for example, if we could do neutron scattering on a bulk sample of the material Qn(TCNQ)2, which contains a macroscopic number of chains. The average over the many realizations could be dominated by rare instances in which the correlation Cij is atypically strong. Specifically, in the rare possibility that the two spins i, j do happen to form a singlet, the correlation between them would be − 1. How rare is that really? If the two sites considered survive through the decimation procedure until they become nearest neighbors, then they are very likely to form a singlet. The probability of a site to survive to that stage, i.e., until ∣i − j of its original nearest neighborson the side of the other site are removed, is 1/∣i − j∣ (this is the density of survivng sites at that stage). Hence the probability of both sites i and j to survive to that stage is ps ≈ 1/∣i − j∣2. We can now compute the average correlation to be ij=(-1)ps+e-a(1-ps)-ps~1(i-j)2, So despite the localized nature of the ground state, the average correlations fall off only as a power law. This is a good example of Griffiths effects, where the average correlations of a random system are dominated by rare instances with anomalously strong correlations. Master equation for the flow of distribution functions ------------------------------------------------------ Let us now show how detailed and precise information about the ground state and low-energy correlations is obtained, following Fisher. The key step is to translate the Dasgupta-Ma decimation rules into a master equation describing how the repeated decimations renormalize the probability distribution of exchange couplings when they operate on an ensemble of Hamiltonians. Let us denote by ρ0(J) the distribution of J in the physical system of interest. As we decimate more bonds while reducing the cutoff scale from $\W\_0$ to $\W$ the distribution evolves to $\rho(J,\W)$. To derive the master equation that governs this evolution, it is much better to replace J and $\G$ by the dimensionless scaling variable $\zeta\_i=\ln ({\W/ J\_i})$ and RG scale parameter $\G=\ln(\W\_0/\W)$. While J was defined on the changing interval $[0,\W]$, $\z$ is always defined on the interval [0, ∞]. In particular $\z\_i=0$ corresponds to the bond with largest exchange coupling $J=\W$. In these variables the Dasgupta-Ma decimation rule ([Jeff]) takes the additive form: i-1,i+2=i+i+1+2 [DasMaLog] Due to the strong disorder assumption, $\z\_i =\ln (\W/J\_i)$ is almost always much greater than − ln2, and it will therefore be safe to ignore the latter in Eq. ([DasMaLog]). Now we are ready to derive the master equation for the distribution of ζ’s - which we denote $P\_{\G}(\zeta)$. Integrating out the high energy shell $[\W-d\W,\W]$ consists of two stages: (i) Remove all the strong bonds with $0<\z<d\G$ and redefine the remaining ζ’s according to the new cutoff, (ii) add the couplings generated through second-order perturbation theory across the decimated bonds. Let’s consider the contribution of stage (i) to the change of the distribution function. Having reduced the cutoff we need to redefine the ζm on every remaining bond: mm-d==m-=m-d. So the entire distribution P(ζ) moves to the left. This can be expressed mathematically: dP()=d. [m1] Now consider the contribution from stage (ii), which is adding the renormalized bonds. The RG rule ([DasMaLog]) prescribes how the distribution of the new bonds derives from that of the original constituent (left and right) bonds: P()=0dl0dr P(l)P(r)(-l-r-2) The contribution of these bonds to the full distribution should be scaled by their fraction in the total population. Since the new bonds are produced only where we find a strong bond the probability of generating them is the probability to find a strong bond $\z\in[0,\G]$ that is $P(0)d\G$. From this we get the contribution of the newly generated bonds to the full distribution $dP(\z)=d\G P(0)P\_{\text{new}}(\z)$. Putting the two contributions together we obtain the master equation: =+ P(0)0d0dr P()P(r)(--r), [me] where we dropped the $\G$ subscript of P(ζ) and neglected the ln2 in the $\d$-function. There is a point we glossed over. We removed some probability by getting rid of all the probability density at small ζ, and we added probability by adding all the new bonds. Do we need to adjust the normalization of our distribution function? Integrating both sides of Eq. ([me]) reveals that the normalization is unchanged. For each bond we lost, we added a renormalized bond. Solution of the flow equation ----------------------------- The additive nature of the decimation rule ([DasMaLog]), embodied in the $\d$-function that appears in the master equation ([me]) suggests a potential solution in the form of an exponential distribution $P\_\G(\z)=f(\G)e^{-f(\G)\zeta}$. Indeed, plugging this ansatz into ([me]) leads to an ordinary differential equation for $f(\G)$, $\partial\_\G f=-f^2$, which is solved by $f( \G)=1/\G$. Hence we obtain the self similar solution P()=1e-/. We see that the system flows to infinite randomness as the width of the distribution grows without limit. Converting back to the physical variables we obtain a power law distribution of effective exchange coupling (J)= 1 (J)1-1/ [rs-dist] This approaches a non-normalizable distribution ∝ 1/J at the fixed point. Again a sign of infinite randomness. The distribution ([rs-dist]) is an attractor of the RG flow. Moreover, it turns out to be a global attractor. The nearest neighbor Heisenberg chain flows to the same infinite randomness fixed point regardless of the initial distribution as long as the disorder is not correlated. Hence the results we will extract for the low energy physics are universal. Physical properties ------------------- An important step in calculating physical properties is to establish a relation between energy scale $\W$ and length scale. A typical length scale is the distance between surviving spins at the scale $\W$. Lets compute the number of surviving spins on the original chain. Every time we decimate a strong bond, we remove two. Therefore, upon changing the RG scale by $d\G$, the number changes by $dN=-2P\_\G(0)N d\G$. Recall from the above solution that $P(0)=f(\G)=1/\G$; hence $N(\G)\sim N\_0/\G^2$. The average distance between surviving spins is then $L(\G)=l\_0 N\_0/N\sim \G^2$, or converting to energy units: $L(\W)\sim l\_0\ln^2(\W\_0/\W)$, where l0 is the original lattice spacing. This means that the excitation energy of singlets of length ℓ is: J~-, which is contrary to usual quantum-critical point scaling where E ∼ 1/ℓz. this type of scaling is called infinite-randomness scaling. We are now in position to compute the spin susceptibility at temperature T. Lets run the RG decimation from the upper cutoff $\W\_0>>T$ down to $\W=T$. The decimated spins are essentially frozen into singlets and therefore do not contribute to the susceptibility. On the other hand, surviving spins at the scale $\W$ are typically coupled by bonds J ≪ T, and therefore expected to behave as free spins. They contribute a Currie susceptibility χ(T) = n(T)/T, where n(T) is the density of surviving spins on the chain at energy scale $\W=T$. From our previous calculation we have $n(T)=N(T)/N\_0\sim n\_0\ln^{-2} (\W\_0/T)$. Therefore, (T)=. This is not quite T− α, but on a log scale it is indistinguishable from a power law. Random bosons: from infinite to finite randomness ================================================= Let us now turn to the second aim of this review: the application of the SDRG method to interacting bosons propagating in a random potential. Model ----- Our analysis focuses on the quantum rotor Hamiltonian: =j Uj (j-j)2 -j Jj ł(j+1-j), [model] where on each site the phase $\hat\theta\_i$ and charge $\hat n\_i$ are conjugate variables, which obey $\l[\hat{n}\_j,\,\f\_k\rr]=-i\delta\_{jk}$. This model describes an effective Josephson-junction array (see Fig. [fig:model]a) with random Josephson couplings Ji and charging energies Ui. In addition, there is a random offset charge ${\bar n}\_i$ in each grain, which is tantamount to a random gate voltage, $V\_j=2U\_j\overline{n}\_j$. The integer part of $\on\_i$ can be absorbed into the definition of $\hat n\_i$ such that $\on\_i$ is defined on the interval $(-\half,\half]$. In this review we focus on the case of generic disorder, where $\on$ can attain any value in this interval. The generic case, as well as the more restricted $\on$ disorder classes, were considered in Ref.. The different classes of disorder give rise to distinct insulating phases, but cause nearly unnoticeable differences in the behavior at the critical point. Besides Josephson junction arrays there are several other important physical realizations of the model ([model]). In systems of ultra-cold trapped atoms, disorder can be generated by optical speckle patterns, incommensurate lattice potential, or by corrugation in the wire that generates the trapping magnetic field in atom chips. With increasing disorder, the atoms concentrate in small puddles at minima of the potential connected by random Josephson coupling which depends on the potential barrier between them. The Hamiltonian ([model]) can be rigorously derived and the distributions of coupling constants computed ab-initio. Finally, disordered superconducting nano-wires with no unpaired gapless electrons will also be generically described by Eq. ([model]). More generally, the model ([model]) is a good effective description for superfluid to insulator transition driven by phase fluctuations. It can be thought of as a low energy effective theory for the phase degrees of freedom, after integrating out the gapped amplitude fluctuations. The weak disorder limit[wdRG] ----------------------------- Before describing the strong-disorder theory, let us briefly review the common understanding of the superfluid insulator transition in one dimension, taking the weak disorder viewpoint. The weak disorder theory was formulated in a seminal paper by Giamarchi and Schulz (GS) Ref.. The weak disorder expansion is natural to carry out when the action is written in terms of charge variables that live on the bonds rather than on the sites. Define j=ij ni. The current through the i’th bond is then $\dot{\phi}\_i$, and the charge on the i’th site is ϕi − ϕi − 1. Therefore, the charging energy of the site is $E\_{c}=U\_i \l(\phi\_i -\phi\_{i-1}-\on\_i\rr)^2$. What about the Josephson energy? If we neglect phase slips, a Josephson junction is essentially an inductor with inductance Li = 1/Ji. Therefore the Josephson energy in this approximation is: $ E\_J\approx L\_i I^2/2={\dot\phi\_i}^2{/ (2J\_{i})}. $ We are tempted to write the Lagrangian in this representation as 0=i [GS-latt] But this expression does not capture the periodicity of the Josephson energy with respect to the phase difference accross the junction. To account for the periodicity, we must allow for phase slips that change θ → θ ± 2π. Since the phase and charge are canonical conjugates, the translation of all phases to the left of the junction i by ± 2π is achieved by the operator exp[ ± 2πi∑j < inj] = e± 2πiϕi. Such terms should be included and assigned an action cost: ps=- i i ł(2i) ξi is the fugacity (or rate) of phase slips, and strongly depends on both Ji and Ui, Ui + 1. The total Lagrangian is given by L = L0 + Lps. Within the weak disorder limit, the randomness in 1/J and U that appears in L0 is perturbatively irrelevant. Furthermore, the disorder in $\on\_i$ can be absorbed by shifts to ϕ. These shifts then appear in the ξ term, and produce a combination of cosine and sine terms. Rewriting the Lagrangian in the continuum limit, with this in mind, gives dx ( 2-v ()2)-a-1( (x) e2i+(x)\* e-2i) [GS-cont] with ξ(x) a complex number, a is the original lattice constant, v the sound velocity, and K the Luttinger parameter (our definition of K is the inverse of the K appearing in Ref. ). The weak-disorder analysis proceeds by assuming a Gaussian distribution of the disorder with $\overline{\xi(x)\xi(x')^\}=(\pi v)^2 a^{-1} D\delta(x-x')$. The disorder strength is parameterized by the dimensionless parameter D. Next, momentum shell RG is used to find the flow of K* and D. The result, quoted from is: =D (3-2K),=-D K2. These scaling equations imply that the critical point is characterized by a universal value of the Luttinger parameter K = 3/2 irrespective of the initial disorder strength. It is important to note, however, that obtaining the long-wavelength field theory ([GS-cont]), with a single well defined Luttinger parameter K, relied on having weak disorder in J and U in the first place. We will see later what can go wrong with this mapping at strong disorder. Decimation steps for SDRG [rgsteps] ----------------------------------- Our main task now is to develop a strong-disorder RG scheme for the random-boson problem. As in the original application of real space RG to disordered systems, the strategy for finding the ground state and low lying excitations is to iteratively isolate and solve the strongest elements in the Hamiltonian. There are two types of elements in the Hamiltonian: onsite charging energies, and bond Josephson couplings. Roughly, two sites connected by the strongest bond will be converted to a phase-coherent cluster. Similarly, in sites with strong charging interactions we will eliminate all the on-site excited states. First, we must determine what we mean by strong elements of the Hamiltonian. The charging interaction can be considered while ignoring Josephson couplings. It is natural to pick the charging-energy gap separating the ground state from excited charge states as the energy scale for decimation purposes. This gap is given by: i=Uił(1-2\|i\|) [cgap] where $-1/2<\on<1/2$. At the same time, the energy scale, which characterizes a bond is the Josephson coupling, Ji. A strong bond is expected to bind two sites into a phase-coherent cluster. [fig:model] At each step of the RG we eliminate the term responsible to the largest energy scale =i{Ji,i}. [ogap] If the largest energy scale is a gap, Δi, then the site i freezes into the charge state with the lowest energy. Quantum fluctuations induce an effective hopping between sites i + 1 and i − 1: Ji-1,i+1=. This result, obtained by simple application of second order degenerate perturbation theory, is illustrated in Fig. [fig:model]b. The factor $1+2\|{\bar n}\|$ in the denominator varies between 1 and 2; it does not affect any universal features of the transition and we can safely set it to unity. Alternatively, when the strongest coupling in the chain is the bond Ji, a phase-coherent cluster forms, with a phase ${\tilde\theta\_{i,i+1}}=(\hat\theta\_i+\hat\theta\_{i+1})/2$. Since charging energy is the inverse of capacitance, and the capacitance is additive, the effective Ui, i + 1 of the new cluster will be: =+. [urecurs] The offset charge $\overline{n}$ in the cluster that forms in the bond decimation step, is simply the sum of the two offset charges: i,i+1=i+2. This equality, however, is defined modulo adding or subtracting one, so the the result always belong to the interval $\on\_{i,\,i+1}\in(-1/2,1/2]$. This decimation step is illustrated in Fig. [fig:model]c. We note that the energy scale $J=\W$ eliminated in the bond decimation step is the energy barrier that needs to be overcome in order to generate a phase slip which breaks the coherence. The gap to internal fluctuations of the relative phase $\w\_J=\sqrt{\W U\_i}$ is in general much smaller than $\W$. Fortunately virtual occupation of the Josephson plasmon affects only a small change on the Josephson coupling of the joined cluster to the neighboring sites, Ji-1Ji-1(1-18),which we will find is negligible near the critical point. It is important to note here that the above decimation step does not assume long range order; it states that phase fluctuations within the newly-formed cluster are harmonic, and therefore the cluster can not be broken due to phase-slips. These harmonic fluctuations are crucial for the understanding of the properties of the superfluid phase, as explained in Sec. [SFLutt]. Before proceeding, we note that in this discussion we have consciously omitted another possible bond decimation step. Suppose that the two sites linked by the strong bond with J = Ω have large charging energies Ui > Ω, but offset charges $\on\_i$ close to $\half$ such that $\D\_i<\W$. These sites then have two nearly degenerate charging states, and behave effectively as spin-1/2 degrees of freedom rather than rotors. Once connected with a large J of order Ω they will form a singlet state, i.e., a single boson will be shared equally by them. These sites therefore are not joined to make an effective superfluid cluster, but rather are completely eliminated while generating Jeff between the left and right neighbor to the pair. This possibility is treated in Ref. and is very important in describing the insulating phase. In this review we focus on the superfluid phase and the critical point, where the probability of finding two nearby sites with large U is exceedingly small and can be neglected. Flow equations and phase diagram -------------------------------- ### Scaling variables The iteration scheme outlined above translates to a flow of distribution functions for the Josephson couplings, interaction strengths, and offset charges. Finding a solution of the flow equations is made easier if we parametrize the couplings in the Hamiltonain appropriately. Typically, the best parametrization is in terms of variables that make the decimation steps become approximate sum rules. For the Josephson couplings this is achieved by using a logarithmic parametrization: i=/Ji. The charging energies are best expressed through the capacitance, which is additive in the cluster formation step. We define: i=/Ui. Note that $\zeta=2\W C$ with C the capacitance. Offset charges are added up as well upon decimation, and therefore are good variables according to the above criterion. As the dimensionless flow parameter, we define as before: =, where $\W\_0$ is of the order of the largest bare energy scale in the Hamiltonian. ### Flow equations The distribution functions should reflect which Hamiltonian parameters are correlated. Clearly Ui and $\on\_i$ must be correlated, since the decimation procedure eliminates sites with large $\Delta\_i=U\_i(1-2\|\on\_i\|)$. The elimination of sites with a large gap, therefore, introduces correlations between charging energies and offset charges on each site. The Josephson couplings, however, remain uncorrelated with charging energies or offset charges in nearby sites. Thus we can parametrize the coupling distribution functions in terms of two functions: $g\_{\G}(\beta)$ the bond log-coupling distribution, and $F(\zeta,\on)$ the joint distribution function of the inverse charging energy, and the offset charge. For the latter, it is useful to write: F(,)=f(,)(-1+2\|\|), where $\zeta\_i=\W/U$ and the Heaviside step function Θ enforces the constraint $\W/\Delta>1$ or equivalently $\zeta>1-2\|\on\|$. We can now write the master flow equations for the distributions of coupling constants implied by the decimation rules discussed above: &=&g+f1d1 d2 g(1)g(2)(-1+2)+g()(g0 -f1),[bgflow1] &=&+g0 d1d2d1d2 f(1,1)f(2,2) (-1+2)(-1+2) &&+f(1+f1-g0 ).[bgflow2] Let us recount the origin of all terms contributing to the derivatives $dg(\beta)/d\G$ and $df(\zeta,\on)/d\G$. First, a trivial, yet important, effect on the distributions stems from changing the cutoff $\W$. Since βi = lnΩ/Ji, the change of Ω → Ω − dΩ causes a shift βi → βi + dβ with $d\beta=d\W/\W=-d\G$. By the same token, this also shifts ζ → ζ + dζ with $d\zeta=-\zeta d\W/\W=-\zeta d\G$. These simple shifts are captured through the chain rule by the first term in both equations, ${\partial g\over \partial \beta}$ and $\zeta{\partial f\over \partial \zeta}$. The next terms are due to the formation of new bonds and clusters. As $\W\rightarrow \W-d\W$, a bond is decimated if its strength J is within this range, i.e., if $\beta<d\G=d\W/\W$. The probability of this for each bond is $g(\beta=0)d\G$. Out of two clusters with paramters ζ1, 2 and $\on\_{1,\,2}$, a bond decimation produces a new cluster with parameters $\zeta=\zeta\_1+\zeta\_2,\,\on=\on\_1+\on\_2$ (suppressing the mod in the calculation). This is the essence of the convolution term in Eq. ([bgflow2]). [palestinian-flag] The convolution term in Eq. ([bgflow1]) describes the formation of a new bond upon site decimation with the new bond variable $\b=\b\_1+\b\_2$. The probability for such a site decimation event, defined as $f\_1 d\G$, is somewhat more complicated than the probability for bond decimation discussed above. This is because a site is decimated when $\W=\D\_i=U\_i(1-2\|\on\_i\|)$, which occurs at a value of ζ that depends on $\|\bar n\|$. The region in parameter space $(\on,\zeta)$ that would get decimated is shown in Fig. [palestinian-flag] and the corresponding probability is obtained from integrating over it f1=-0.50.5 d(1-2\|\|)f(1-2\|\|,) Note that as the energy scale $\W$ decreases by $d\W$, the width of the decimated region scales as $\zeta d\G$, which shrinks to zero on the $\on=\pm1/2$ lines, since $\zeta\_i=1-2\|\on\_i\|$ on the decimation curve. In the vicinity of the transition, the seemingly complicated expression for f1 simplifies significantly. As more sites are joined, the charge offsets add up (modulu 1), and therefore rapidly pick values uniformly distributed within $(-\half<\on<\half]$. The distribution $f(\zeta,\on)$ becomes essentially independent of $\on$, and since it flows to strong in ζ it is only weakly dependent on ζ for values ζ < 1. Thus, we can simply replace $f(1-2\on,\on)\approx f\_0$, which implies $f\_1\approx \half f\_0$. The remaining terms in Eqs. ([bgflow1]) and ([bgflow2]) are needed to maintain the normalization of the distributions. Normalization can be verified by integrating over the entire range of ζ and $\on$ in Eq. ([bgflow2]) and β in Eq. ([bgflow1]), and making sure that the derivative of the total probability integral on the LHS’s is zero. ### Scaling solution and reduced flow equations Next, we attempt to solve the functional flow equations by employing the following intuition, which is gained from the spin models analyzed in Sec [sec:method]. When the decimation steps follow a sum rule for two positive couplings, the probability distributions for these couplings should consist of exponentials. This leads to the scaling ansazt: g()&=&g0e-g0, [ans1] f(, )&=&f021-e-f0e-f0 [ans2]. This ansatz describes a family of functions that are parametrized by two variables, f0 and g0. Plugging the scaling ansatz back to the flow equations, one hopes, will generate a flow equations in terms of f0 and g0 alone, and without any explicit functional dependence on ζ or β. Approximately, this is indeed the case. We assume that f0 ≪ 1, and the substitution of the scaling ansatz to the flow equations yields: &=&f0 (1-g0)[SFeqa] &=&-f0 g0. [SFeq] The flow lines in the space f0 versus g0, plotted in Fig. [fig:RGflow], are given by f0=2(g0-1-g0)+. The different flow lines are parameterized by ε, which sets the detuning from the critical manifold at $\e=0$. Flows that lie below the critical manifold, $\e<0$, terminate on the line g0 > 1, f0 = 0 marking the superfluid phase. Values above the critical manifold, $\e>0$, flow to the region g0 < 1 and where f0 is relevant and flows to larger and larger values. This flow characterizes the insulating regime of the model. [fig:RGflow] We note that the scaling equations ([SFeq]) have the form of the Kosterlitz-Thouless flow equations, if we were to rewrite them in terms of $\sqrt{f\_0}$ and g0. These scaling variables, however, have a different physical meaning. In particular, the variable that gains a universal value at the transition is the exponent g0 rather than the Luttinger parameter. Below we elucidate the nature of the superfluid phase. We then explain the essence of the critical point in light of the anomalous properties of the superfluid leading up to it. This will also serve to clarify why a theory that is perturbative in the disorder strength, such as the replica treatment of Giamarchi Schulz can fail in this regime. The superfluid phase -------------------- As mentioned above, the line g0 > 1, f0 = 0 marks the terminus of the flows on the superfluid side of the phase diagram. What this fixed line describes, is the formation of a global superfluid cluster where phase-slips are prohibitively rare, and do not disturb the superfluidity. Indeed, the RG is dominated by bond decimations where clusters repetitively coalesce until they connect the two sides of the chain, and, therefore, the stiffness of this cluster is finite. The fact that the interaction parameter f0 flows to zero may at first sight suggest that the superfluid is described by a classical Josephson array with infinite compressibility. To see why this is not the case let us start with a system of length L and continue the decimation until there is only one site left. This site represents a superfluid cluster extending through the full length L of the chain. The typical value of the charging energy of the whole cluster is Ec ∼ Ωf0, which scales as ∼ 1/L in late stages of the RG flow. Hence f0 is irrelevant because it describes the charging energy of clusters that become macroscopic at the end of the flow. So, while the capacitance, which is an extensive quantity, grows as L, the compressibility, or capacitance per unit length, approaches a finite value at the terminus of the flow. Therefore, a superfluid fixed point, written in terms of the phase variable, is essentially a quantum harmonic theory with random “spring constants” Ji distributed according to the power-law distribution P(J) = (g0/Ω) (J/Ω)g0 − 1. On the other hand, the masses of the coupled oscillators are related to the compressibility κ. The elementary excitations of the superfluid described by the harmonic theory on the fixed line are harmonic phonons, which are localized by the disorder at all frequencies $\w>0$. For weak disorder the localization length associated with a single phonon wave-function at frequency $\w$ diverges toward zero frequency as ℓ ∼ 1/ω2. On the other hand, it was shown in Ref. that the distributions obtained by SDRG give rise to anomalous localization properties with: ~ł{ (2)/ & g0=1 1/g0 & 1<g0<2 1/2 &g02. To see how the wide distributions of J lead to an anomalous superfluid phase, consider the superfluid stiffness of an effective harmonic chain of length L, given by s-1=1Li Ji-1, [stiffness] According to the SDRG, the random variables xi = 1/Ji are distributed as p(x) ∼ x− (1 + g0). For g0 < 2 the variance of 1/Ji diverges, as does the variance of the macroscopic variable 1/ρs. This is the origin of the anomalous behavior in the superfluid having 1 < g0 < 2. Now we also see why the weak disorder theory can fail in the strong disorder weak interaction regime of the superfluid. Consider again the dual lattice action ([GS-latt]) that serves as the starting point for Ref.. Now, that the disorder in 1/J is diverging in magnitude, it cannot be brushed away even if it is perturbatively irrelevant. It is important to keep in mind that the power-law distribution with the exponent g0 is not the bare distribution of J’s in the array, but rather the fully renormalized distribution at low energy scale. It is natural to ask at this point if there is a simple observable that can bear witness to this distribution and allow to measure g0. One such quantity is the critical current, which is simply the Josephson energy of the weakest link in a chain. Consider a superfluid chain much longer than the coherence length ξ, which would diverge at the critical point. Renormalize down to the scale ξ, or as we will later show energy scale $\W\_\xi=\W\_0/\xi$. The dependence of ξ on the detuning from the critical point will be discussed in the next section. At this point we have an effective chain with N = L/ξ links. The Josephson couplings distribute as $P(J)=(g\_0 /\W)(J/\W)^{g\_0-1} $. The probability density of the weakest link, Jm, and of the current Ic (an extreme value statistic), is P(Ic)=N g00( Ic0)g0-1This is a power-law at small values of Ic, which is cut of by the exponential. A corollary of this is that the typical value of critical current, defined by the sharp peak of the distribution, vanishes as ${\bar I}\_c\sim (\W\_0/\xi) (L/\xi)^{-1/g\_0}$. That is, it vanishes as a power-law of the system size with power 1/g0 < 1. The superfluid-insulator transition [SFLutt] -------------------------------------------- The critical point which controls the unstable flow toward the insulator sits at g0 = 1 and f0 = 0, and marks the end of the superfluid fixed line. From our discussion above we can gain an intuitive understanding for why the superfluid should break down at g0 = 0. If we consider again the stiffness ([stiffness]) of a classical array (or a harmonic chain), we see that the average $\av{1/\rho\_s}$ diverges as ∼ (g0 − 1)− 1. Hence, the critical point is where the classical Josephson array loses its stiffness. It is, however, important to realize that the phase transition is not classical, and the superfluid stiffness at the transition point is not zero. We can think of the transition as being tuned by crossing the critical manifold at some value of f0 > 0 and g0 > 1. Quantum fluctuations due to sites with large charging energy drive the downward flow of g0 toward the fixed point, where g0 = 1. The superfluid stiffness and compressibility, on the other hand, are not universal properties on the fixed point, but must be integrated along the flow. For example, the stiffness of the final superfluid cluster spanning the chain depends on all the effective internal bonds J̃i that connect the sites making that cluster and were decimated in the process of its formation: ρs− 1 = L− 1∑iJ̃i− 1. Similarly, the compressibility is the sum of capacitances of all sites making up the superfluid cluster: κ = L− 1∑iUi− 1. In both cases L is the total length of the chain. These quantities were computed in Ref. on the critical manifold showing that they approach a constant value that depends on the initial disorder for the flow. For the sake of comparison with the weak disorder theory of Giamarchi and Schulz it is interesting to look at the Luttinger parameter, given by $K=\pi\sqrt{\rho\_s\kappa}$. There, the fully renormalized Luttinger parameter at the transition takes a universal value K = 3/2. On the other hand, for the strong-disorder transition, the SDRG predicts a value of the Luttinger parameter that depends on where the critical manifold was crossed. That value diverges as the crossing point gets closer to the classical fixed point at f0 = 0. The non universal value of the Luttinger parameter at the transition point is the most controversial prediction of the SDRG in the strong disorder regime. It is also worth noting that since the Luttinger parameter does not appear as a natural scaling variable in the SDRG, computing its value requires a rather elaborate integration over the entire critical RG flow from high to low energies. Hence it is not surprising that its value on the critical manifold comes out non universal within this theory. But because the Luttinger parameter is not a natural object to compute within the SDRG it is important to verify this prediction in numerical simulations. Such calculations will be discussed in section [sec:numerics]. Despite this difference, there is at least a formal similarity between the strong-disorder fixed-point and the Kosterlitz-Thouless flow at weak disorder. As we already pointed out, the reduced flow equations are formally identical to the Kosterlitz-Thouless equations, where the parameters g0 and $\sqrt{f\_0}$ of the distribution functions are playing the roles that the Luttinger parameter and phase-slip fugacity play in the Kosterlitz-Thouless flow. As a consequence, length and time scales have the same exponential divergence ~0 ea/ , ~0-1eb/ as a function of the detuning ε from the critical point, that is a hallmark of the KT transition. The insulating phases --------------------- The superfluid phase and the superfluid-insulator critical point discussed above are not affected in any important way by the nature of the initial disorder distributions. But type of disorder in the offset charges ${\bar n}\_i$ does have a decisive effect on the nature of the insulating phases. Different constraints on the distribution of ${\bar n}\_i$ give rise to three distinct insulating phases. The physical properties of these insulators are largely determined by the charging gap distribution of the sites that survive the RG flow deep into the insulating region. The possibilities are as follows: * Bose-glass phase – A generic distribution of offset charges, $-0.5\le\on<0.5$, results in a Bose-glass phase. This is a compressible and gapless state, which is also characterized by a diverging superfluid susceptibility. The gaplessness and compressibility are due to the presence of a finite density of sites with a charging energy arbitrarily close to zero. The diverging superfluid susceptibility, χ, is a typical property of a compressible phase. The form of the divergence is χSF ∼ ln(Ω/hp), with hp the probing proximity field. χ diverges in the limit of hp → 0. Alternatively, for a finite size sample, χSF ∼ lnL. * Mott-glass phase – In a commensurate lattice with disorder only in J and U, the offset charges vanish identically $\on\_i=0$. The system then flows to the Mott-glass phase; an incompressible yet gapless state with a finite superfluid susceptibility. Both the compressibility and the gaplessness are a consequence of the universal distribution of U, Eq. ([ans2]), which in this case also describes the charging gaps. The distribution $f(U)\sim \frac{1}{U^2}e^{-f\_0 \Omega/ U}$ has no support at U → 0, which implies incompressibility. Specifically: ~e-f0()0 where μ is the probing chemical potential. By looking at the smallest charging energy for a system of size L, we find that the gap drops as $\Delta\sim\frac{1}{\ln L}$. The Mott-glass is a classic example of a Griffiths phase: it is incompressible, yet the average temporal auto-correlations of the system will be dominated by the essentially single large cluster of size lnL that determines the gap. * Random-singlet glass – If $\on$ is randomly either $\on=0.5$ or $\on=0$, the gapless insulating phase has both a diverging compressibility and diverging superfluid susceptibility. Only sites with $\on=0.5$ and no charging gap survive late in the flow as two level systems. Thus, the insulator is described by an effective random spin-1/2 xx chain, known to be in the random singlet phase. Both the compressibility and superfluid susceptibility scale in the same way: SF~~. with μ being the probing chemical potential or proximity field. From the random-singlet energy-length scaling, we obtain for a finite size system: $\chi\_{SF}\sim\kappa\sim \frac{e^{-c\sqrt{L}}}{L^{3/2}}$. Numerical tests of the strong disorder transition ------------------------------------------------- The critical point identified using SDRG is characterized by finite randomness. Although the fixed-point analysis is also controlled and justified by the smallness of interactions in its vicinity, it is not as reliable as the analysis of the random singlet phase or the critical point of the random transverse-field Ising model, where the SDRG flow is to infinite randomness. It is, therefore, desirable to seek numerical verification of the theoretically predicted universal physics. Some of the recent work that addressed this question is summarized by Pollet in another review published in this volume. In what follows we will survey the numerical results and comment on the debate surrounding their interpretation. Since we are dealing with a strongly correlated system, a fully quantum calculation, such as quantum Monte Carlo (QMC) or Density Matrix Renormalization Group (DMRG) are needed to extract the critical properties. It is important to understand the fundamental difficulties, which complicate these numerical calculations in the strong disorder regime. First, according to the SDRG, rare weak links generated in the course of renormalization play a crucial role in driving the transition at strong disorder. This implies that a very large ensemble of different realizations of the random chain needs to be analyzed in order to detect the effect of rare events on the physics. Second, when rare weak links are effective they slow down the convergence of numerical methods such as QMC and DMRG. Early QMC studies of the strong disorder regime have not probed directly for the scaling predictions of the SDRG, but rather checked how BKT scaling works in such systems. For systems with moderate disorder Balabanyan et. al. found a good fit to BKT scaling with the appropriate value of the Luttinger parameter (K = 2 for the commensurate system used in that calculation). For stronger disorder, however, the fit did not work as well and strong finite size effects were cited as the reason. A more recent QMC study provided strong, albeit indirect, support for the strong randomness scenario. These simulations show KT-like scaling of the correlation length at the transition both for weak and strong disorder. Above a certain disorder strength, however, the Luttinger parameter as well as the susecptibility-length scaling exponent at the transition were seen to depend on the disorder strength at which the transition is crossed. In agreement with the strong disorder scenario, the critical Luttinger parameter exceeded the universal value predicted by the standard theory. Analytical support to the results of Ref. [] was given in [], which also showed, however, that at very strong disorder the asymptotic value of the Luttinger parameter is only obtained at very large length scales. The Luttinger parameter, which was the focus of these studies is, however, not a natural quantity to characterize the strong-disorder critical point. Rather, the relevant scaling properties are encoded in the RG flow, Eqs. ([SFeqa]) and ([SFeq]), of the variables g0 and f0 associated with the distributions of Josephson couplings and charging energies. An apparent difficulty in extracting this scaling behavior from numerical calculations is that the renormalized coupling distributions are not directly observable quantities. This problem was addressed in Ref., which developed a finite size scaling theory relying on measurable quantities. A direct connection was established there between the renormalized distribution of weak links, parameterized by g0, and the measured distribution of superfluid stiffness on an ensemble of finite clusters. Specifically, the tail of the distribution of the measured value of 1/ρs in a system of size L was shown to follow a power-law with the same exponent as the distribution of the variable xi ≡ 1/Ji at the energy scale Ω(L). Thus, the finite-size scaling behavior of the distribution of 1/ρs, could be compared with finite size scaling formulas obtained from the RG flow. The QMC results obtained for a model of bosons at integer filling with particle-hole symmetric (off diagonal) disorder were found to scale in the manner predicted by the SDRG to within the numerical error bars. Furthermore, finite size scaling assuming a standard KT transition did not fit the data at all (See Suppllementary material of Ref. []). Another recent study by Pollet et. al. [] focused on the anomalous finite-size scaling of the inverse stiffness in the superfluid phase leading up to the critical point. The Monte Carlo simulations carried out in this study also found distributions characterized by broad power-law tails as predicted by the SDRG. Pollet et al., however, made the interesting observation that the scaling of the median value of 1/ρ with L can be fit over a wide regime to the scaling of a classical Josephson array (or a harmonic chain) with a power-law distribution of Josephson couplings $P(J)\sim J^\a$ for small J. Such a classical JJA, or harmonic chain, have inverse stiffness ρs− 1 = L− 1∑iJi− 1, which leads to the scaling law (up to logarithmic corrections) $\rho\_s^{-1}(L)=\rho\_s^{-1}(\infty) + a/L^{\a}$ with $\a$ being a length independent exponent. While Ref. [] did not find noticeable deviations from classical scaling of the median of 1/ρs, the broad power-law distribution of the tail of 1/ρs is generated in the first place by sites with large charging energy, that is, by a quantum effect. Moreover, the power $\tilde\alpha$ extracted from the tail of the distribution rather than from the scaling of the median of 1/ρs is seen to flow downward as a function of L due to such site decimations. Pollet et. al. correctly point out that the tail of the distribution of 1/ρs is not important for the thermodynamic stiffness in the superfluid phase as long as the tail exponent ${\tilde\a}(L\to\infty)>0$. Nevertheless, it is important to note in this regard that the RG flow of $\tilde\a$ to negative values is one mechanism that would necessarily destabilize the superfluid. If the bare disorder is not sufficiently strong compared to the interaction, then this strong disorder mechanism can be preceded by proliferation of phase slips through the standard KT mechanism at a universal value of the Luttinger parameter. Interestingly, for the model, and parameter regime investigated in Ref. [] the value of the thermodynamic Luttiger parameter extrapolated in this way reached the universal value Kc = 3/2 very close (but on the superfluid side) of the classical transition point defined by $\a=0$. It is quite striking that, although the critical value of the Luttinger parameter found in this way is very close to the weak-disorder universal value, the only scaling theory that fits the data is that predicted by the SDRG. A possible explanation is that very strong deviations of the Luttinger parameter from the weak disorder universal value require exceedingly large systems to be observed according to the scaling analysis in Ref. [] (although values of K < 4 should still be reasonably accessible). Regardless, these puzzling results highlight the need for a more complete understanding of the superfluid-insulator transition in the intermediate disorder regime, where both rare weak links and vortex proliferation may play an important role. Further applications of SDRG to bosonic systems =============================================== Dissipative superconductors --------------------------- Systems of Josephson junctions in the presence of dissipation can also be analyzed using SDRG. The analysis considers a large N generalization of Josephson junctions (which would correspond to an O(2) model). In this limit, one can write a quadratic (imaginary-time) action which includes the nearest-neighbor Josphson coupling Ji, as well as dissipation γ, encoded in the frequency-dependent part of the action: S=i ł[-Jiii+1+ł(i+i\|\|2/z)i2]. The εi’s are determined self-consistently for each site by requiring that ⟨ϕ⃗2⟩ = 1. z is a parameter that determines the nature of the dissipation. The case z = 1 coincides with the case we have solved in the previous section, with γ being the capacitance. 2 > z > 1 represents the superohmic regime, while z > 2 is the superohmic regime. The case of z = 2 is precisely the Ohmic regime; furthermore, it coincides with the so-called Hertz-Millis theory for fluctuating order-parameters in a dissipative electronic environments. Speicifically, this action describes the onset of supercondcutivity in disordered thin wires. At z = 2, the ohmic case, one can carry out the same RG steps as outlined in Sec. [rgsteps] for the rotor model. A strong bonds lead to formation of clusters with dissipation γeff = γ1 + γ2, and decimating a site with a large ε2 ∼ e− γ2 produces a weak bond between its neighbors, $J\_{eff}=2\frac{J\_1 J\_3}{\epsilon\_2}$, where we used sites 1,2 and 3 as an example. This analysis, in the O(N) language, results in a phase transition tuned by $\delta=\overline{\ln J-\gamma}$ between an ordered phase with J’s relevant, and a paramagnetic phase with J’s scaling to zero. In an electronic wire, this provides a description of the superconducting-metal transition. The critical point is shown to be a random-singlet fixed point, identical in its universal properties to the transverse field Ising model. For the full details of this interesting and surprising result, we refer the reader to Ref.. Two dimensional rotor model --------------------------- Applying the SDRG to higher dimensions is always challenging, since the decimations alter the geometry of the system. Nevertheless, several works have succeeded in employing SDRG in higher dimensions for the transverse-field Ising model, the random-hopping model. Recently, a SDRG analysis of the 2d, square lattice, rotor model with no offset charge (as in Eq. ([model]) with $\on\_i=0$) was carried out in Ref.. Technically, the SDRG method had to be modified in several ways for the 2d rotor model analysis. First, cluster formation led to the addition of Josephson energy in the case of the sites making up the cluster are both connected to another site. Second, because the system may not flow to a strong-disorder fixed point, or one with nearly no interactions (as happens in 1d), the phase fluctuations, i.e., phonons, within clusters were taken into account as suppressing the Josephson couplings emanating from renormalized clusters. Third, in order to make the analysis numerically tractable, only couplings with strength above an absolute cut-off were recorded at each decimation stage. This is particularly important since the connectivity of the system increases dramatically with site decimations. The 2d rotor-model SDRG analysis revealed a percolation like superfluid-insulator phase transition. The transition was most apparent by looking at the parameter flow in the space of $\Delta J/\overline{J}$ vs. $\overline{U}/\overline{J}$, with $\overline{J}$ the average Josephson couplings, ΔJ the standard deviation of J, and $\overline{U}$ the average Josephson couplings (see Fig. [2DRGflow]). All quantities were calculated from the largest 2Ñ Josephson energies, and charging energies, with Ñ the renormalized size of the system. This is needed to put the analysis on the same footing as the initial model, which only has 2N Josephson energies, and N charging energies. A critical point appears at $\overline{U}/\overline{J}\approx 0.3$ and $\Delta J/\overline{J}\approx 1$. Reassuringly, these values were, by and large, independent of the initial disorder distributions. The transition was established to be of the percolation type. At the transition, a fractal cluster forms, with a Hausdorff fractal diension of df = 1.3 ± 0.2, and a correlation length exponent ν = 1.09 ± 0.04. The calculated exponent ν conforms with the Harris criterion νd > 2. In contrast the standard XY transition relevant for the clean limit is characterized by ν ≈ 0.663 and thus violates the Harris criterion. While the superfluid phase should be conventional, the insulating phase was identified as an incompressible Mott glass, with a gap that falls off as 1/lnL with L the linear size of the system. For complete details we refer the interested readers to Ref.. [2DRGflow] Conclusions =========== In this review we have explored two prominent applications of the SDRG method. First, we considered the Heisenberg model, where this method had its first success. Then, we presented the recent application of the method to the random-boson superfluid-insulator transition. The latter problem has been an unresolved problem for a long time, and research complementary to the SDRG analysis is continuing. The question of the ultimate stability of the strong disorder fixed-point to proliferation of phase-slips in the usual Kosterlitz-Thouless mechanism at very long scales remains open. However, numerical simulations indicate that the scaling predictions of the SDRG are, at strong disorder, more relevant to accessible system sizes than those of the standard weak disorder theory. Even beyond the problems presented in this review, the application of SDRG to low-dimensional quantum systems continues to be a fruitful research direction. A subjective opinion of the authors, is that the next frontier appears to be the detailed analysis of quantum dynamics. The basic principles for all applications of SDRG, however, remain the same as those presented here. Much of the work reviewed here was carried out together with many collaborators, without whom non of it would have come to pass. We would like to especially acknowledge Daniel S. 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Algebraic independence for values of integral curves ==================================================== We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbf{Q}}$). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields. This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series E2, E4, E6. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of Value Distribution Theory. Introduction ============ A theorem of Nesterenko ----------------------- This work was motivated by questions related to the following algebraic independence result. Let E2, E4, and E6 be the classical Eisenstein series, seen as holomorphic functions on the complex unit disk $D \defeq \{q \in \CC \mid \|q\|<1\}$, explicitly defined by $$\begin{aligned} E\_{2}(q) = 1 - 24 \sum\_{j=1}^{\infty}\sigma\_1(j)q^j\text{, }\ \ E\_{4}(q) = 1+240 \sum\_{j=1}^{\infty}\sigma\_3(j)q^j\text{, } \ \ E\_6(q) = 1-504 \sum\_{j=1}^{\infty}\sigma\_5(j)q^j\end{aligned}$$ for every q ∈ D, where $\sigma\_k(j) \defeq \sum\_{d\mid j}d^k \in \ZZ$. Let us also consider the q-expansion of the j-invariant $$\begin{aligned} J(q) = 1728\frac{E\_4(q)^3}{E\_4(q)^3 -E\_6(q)^2} = \frac{1}{q} + 744 + \sum\_{j=1}^{\infty}c(j)q^j\text{.}\end{aligned}$$ [Nesterenko ][thmnes] For every $z \in D\minus\{0\}$, we have $$\begin{aligned} \trdeg\_{\QQ}\QQ(z,E\_2(z),E\_4(z),E\_6(z))\ge 3\text{.} \end{aligned}$$ This result is an improvement of Barré-Sirieix’s, Diaz’s, Gramain’s, and Philibert’s breakthrough concerning the solution of a conjecture of Mahler: for every algebraic $z \in D\minus\{0\}$, J(z) is transcendental. In order to fully motivate our contributions, we next sketch the main steps of Nesterenko’s original proof. In view of an algebraic independence criterion due to Philippon ( Théorème 2.11; see also Lemma 2.5), it suffices to construct a sequence of polynomials with integral coefficients $Q\_n \in \ZZ[X\_0,X\_1,X\_2,X\_3]$, for n ≫ 0, such that degQn = O(nlogn), log∥Qn∥∞ = O(nlog2n) — here, ∥Qn∥∞ denotes the maximum of the absolute values of all the coefficients of Qn —, and $$\begin{aligned} -an^4 \le \log \|Q\_n(z, E\_2(z),E\_4(z),E\_6(z))\| \le - b n^4\end{aligned}$$ for some real constants a > b > 0. For this, Nesterenko implemented a method benefiting from the fact that E2, E4, and E6 have integral Taylor coefficients in their q-expansion and satisfy the so-called Ramanujan equations: $$\begin{aligned} q\frac{d E\_2}{dq} = \frac{E\_2^2 - E\_4}{12}\text{, }\ \ q\frac{d E\_4}{dq} = \frac{E\_2E\_4 - E\_6}{3}\text{, } \ \ q\frac{d E\_6}{dq} = \frac{E\_2E\_6 - E\_4^2}{2} \text{.}\end{aligned}$$ It is also essential in his construction that 1. for each k ∈ {1, 2, 3}, the sequence of Taylor coefficients (E2k(j)(0)/j!)j ≥ 0 grows polynomialy in j, and 2. [1](#fn1) there exists a constant C > 0 such that $$\begin{aligned} \ord\_{q=0}P(q,E\_2(q),E\_4(q),E\_6(q))\le C(\deg P)^4 \end{aligned}$$ for every non-zero polynomial $P \in \CC[X\_0,X\_1,X\_2,X\_3] \minus\{0\}$. The first condition can be easily deduced from the explicit description of the Taylor coefficients of E2k given above. The second, which may be regarded as a strong form of algebraic independence between the functions q, E2(q), E4(q), and E6(q), is a non-trivial consequence of Nesterenko’s *D-property* (Definition [dprop] below; cf. Paragraph 6), an algebraic property concerning the global behavior of the foliation in $\CC^4$ induced by the vector field $$\begin{aligned} \label{vecfield} v \defeq x\_0\frac{\partial}{\partial x\_0} + \frac{(x\_1^2 - x\_2)}{12}\frac{\partial}{\partial x\_1} + \frac{(x\_1x\_2-x\_3)}{3}\frac{\partial}{\partial x\_2} + \frac{(x\_1x\_3-x\_2^2)}{2}\frac{\partial}{\partial x\_3}\text{.}\end{aligned}$$ A considerable part of is devoted to a proof of a stronger form of the estimate in (ii). Nesterenko’s method goes as follows. 1. Using that the Taylor coefficients of E2, E4, and E6 are integers of polynomial growth (property (i) above), we may apply Siegel’s Lemma ( I.1 Lemma 1) to obtain auxiliary polynomials with integral coefficients $P\_n \in \ZZ[X\_0,X\_1,X\_2,X\_3]\minus \{0\}$ such that degPn = n, log∥Pn∥∞ = O(nlogn), and $$\begin{aligned} \ord\_{q=0} P\_n(q,E\_2(q),E\_4(q),E\_6(q)) \ge c n^4\end{aligned}$$ for some constant c > 0. 2. For a fixed $z \in D \minus \{0\}$, the next step consists in proving the existence of a sequence jn = O(nlogn) and of constants α > β > 0 such that the composed function $f\_n(q) \defeq P\_n(q,E\_2(q),E\_4(q),E\_6(q))$ satisfies $$\begin{aligned} -\alpha n^4 \le \log \|f\_n^{(j\_n)}(z) \| \le -\beta n^4 \end{aligned}$$ for n ≫ 0. The main point for obtaining the above lower bound is that, if all the Taylor coefficients of fn at q = z up to a sufficiently large order are too small, then its first non-zero Taylor coefficient at q = 0 will have absolute value < 1, thereby contradicting its integrality. Here, we also make essential use of property (ii) above. This is the most delicate part of the argument. 3. Finally, for n ≫ 0, if we consider the differential operator $$\begin{aligned} v^{[j\_n]} \defeq 12^{j\_n}v \circ (v-1)\circ \cdots \circ (v-(j\_n-1))\text{,} \end{aligned}$$ then the Ramanujan equations imply that $Q\_n \defeq v^{[j\_d]}(P\_n) \in \ZZ[X\_0,X\_1,X\_2,X\_3]$ satisfies $$\begin{aligned} (12q)^{j\_n}f\_n^{(j\_n)}(q) = Q\_n(q,E\_2(q),E\_4(q),E\_6(q)) \end{aligned}$$ for every q ∈ D. The required properties for Qn are now easily deducible from (1) and (2). A puzzling remark ----------------- One of the most striking features of the above method is its generality. Indeed, a close inspection of the previous arguments suggests that, if f1, …, fm are holomorphic functions on the unit disk D with integral Taylor coefficients at q = 0, satisfying some algebraic differential equations with rational coefficients, and verifying conditions akin to (i) and (ii) above, then, mutatis mutandis, the above method applied to the system (f1, …, fm) in place of (E2, E4, E6) would produce another transcendence result. This was certainly known to specialists; see, for instance, Section 3, where the pertinent properties satisfied by E2, E4, and E6 were axiomatized as above — more generally, see Philippon’s notion of *K-functions, introduced in. Clearly, one may produce examples of such f*i ad libitum by algebraically manipulating Eisenstein series, but this procedure does not lead to new transcendence results. The problem on the existence of functions f1, …, fm satisfying the above properties, but not “related” to classical modular forms (in some imprecise sense), was explicitly stated by Zudilin in. Since the publication of and, it became apparent that the phenomenon of Mirror Symmetry provides a large class of functions with integral Taylor coefficients with respect to some canonical coordinate and which satisfy natural algebraic differential equations — see,,,, for integrality issues. In, Zudilin studies some candidates within Mirror Symmetry for playing the role of (f1, …, fm), but the few cases where he is able to prove all the required properties of Nesterenko’s method — those linked to elliptic curves and K3 surfaces — are all of modular nature. It becomes clear in Zudilin’s work that one of the main obstructions in applying this method for such functions is condition (i) (and, in particular, that the radius of convergence is equal to 1), which is not verified in general. In a more basic level, computing radii of convergence or getting global information on the domain of definition of such functions pertaining to Mirror Symmetry is a current research problem; see for results on certain families of mirror maps. The following phenomenon provides further evidence that condition (i) is overly restrictive. Let f be the holomorphic function on D given by f(q) = qJ(q) and set $\theta \defeq q\frac{d}{dq}$. Since $\QQ(J,\theta J, \theta^2J) = \QQ(E\_2,E\_4,E\_6)$ (see, for instance, the explicit formulas in Paragraph 1), it follows from Theorem [thmnes] that $$\begin{aligned} \trdeg\_{\QQ}\QQ(z,f(z), \theta f(z), \theta^2f(z))\ge 3\end{aligned}$$ for any $z \in D\minus \{0\}$. However, Nesterenko’s method cannot be directly applied to the system (f, θf, θ2f) since the sequence c(j) does not grows polynomialy in j.[2](#fn2) All the other good properties are nevertheless satisfied: f, θf, and θ2f have integral Taylor coefficients, the Ramanujan equations imply that f satisfies a third order algebraic differential equation with rational coefficients, and a condition similar to (ii) also holds. This paper grew from an observation of J.-B. Bost and H. Randriambololona that the growth condition (i) in Nesterenko’s method could be replaced by a geometric notion of moderate growth formulated in terms of characteristic functions à la Nevanlinna Theory. Besides being weaker than the growth condition in (i), which in principle enlarges the domain of application of Nesterenko’s method, this geometric growth condition is preserved under some algebraic manipulations on the input functions, thereby eliminating the odd phenomenon explained in last paragraph. We next explain our main results. Further directions and open problems are indicated below. Our main results: a geometric approach -------------------------------------- Our main theorem is a general geometric formulation of Nesterenko’s method valid for arbitrary rings of algebraic integers and more general quasi-projective ambient spaces. Let us first informally introduce the geometric notions which will replace conditions (i) and (ii) above. ### Moderate growth Let X be a smooth projective variety over $\CC$, and h be a C∞ Hermitian metric on the complex manifold $X(\CC)$. Let $\omega \defeq -\Im h$ be the positive real (1,1)-form on $X(\CC)$ associated to h. To fix ideas, the reader may consider the example $X = \PP\_{\CC}^n$ endowed with the Fubini-Study metric, for which ω is given in homogeneous coordinates z = (z0 : ⋯ : zn) by $$\begin{aligned} \omega = \frac{i}{2\pi}\partial \overline{\partial}\log \|\mathbf{z}\|^2 = \frac{i}{2\pi} \left(\frac{\sum\_{j=0}^ndz\_j\wedge d\overline{z}\_j}{\|\mathbf{z}\|^2} - \frac{\sum\_{j,k=0}^n\overline{z}\_jz\_kdz\_j\wedge d\overline{z}\_k}{\|\mathbf{z}\|^4} \right)\text{,}\end{aligned}$$ where ∣z∣2 = ∑j = 0n∣zj∣2. Let R > 0 be a real number and denote by $D\_R \defeq \{z \in \CC \mid \|z\|<R\}$ the complex disk of radius R centered at the origin. One may measure the growth of an analytic map $\varphi: D\_R \to X(\CC)$ as follows. For each t ∈ (0, R), the area of the “disk” φ(Dt) in $X(\CC)$ with respect to the metric h is given by $$\begin{aligned} A\_{\varphi}(t) \defeq \int\_{D\_t}\varphi^\\omega\text{.}\end{aligned}$$ We may then form the characteristic function* $$\begin{aligned} T\_{\varphi} : (0,R) &\to \RR\_{\ge 0}\\ r &\mapsto T\_{\varphi}(r) \defeq \int\_{0}^r A\_{\varphi}(t)d\log t\text{,}\end{aligned}$$ and we say that φ has moderate growth in X if $$\begin{aligned} \limsup\_{r \to R-} \frac{T\_{\varphi}(r)}{\log \frac{1}{1-\frac{r}{R}}} < +\infty\end{aligned}$$ By the compactness of $X(\CC)$, moderate growth does not depend on the choice of Hermitian metric. When R = 1, any analytic map $\varphi: D \to \CC^n \subset \PP^n(\CC)$ whose coordinates have Taylor coefficients of polynomial growth has moderate growth in $\PP\_{\CC}^n$ (see Example [polgrowth] below). Therefore, moderate growth generalizes the growth condition (i) in Nesterenko’s method. Moderate growth is nonetheless more flexible than polynomial growth on Taylor coefficients. For instance, as long as the image of $\varphi : D\_R \to X(\CC)$ is Zariski-dense in X, moderate growth is a birational invariant in the following sense: if f : X → Y is a birational morphism between smooth projective varieties over $\CC$, then φ has moderate growth in X if and only if f ∘ φ has moderate growth in Y (see Theorem [funcfield] and Corollary [corobirinv] below). In particular, this allows us to define, via compactifications, an unambiguous notion of moderate growth in smooth quasi-projective varieties. ### ZL-density Let k be a field, X a projective variety over k of dimension n, and L an ample line bundle on X. Consider a parametrized formal curve $\hat{\varphi} : \Spf k[\![q]\!] \to X$ in X, i.e., φ̂ is a morphism of formal k-schemes. We say that φ̂ is ZL-dense in X if there exists a constant C > 0 such that, for every integer d ≥ 1 and every non-zero global section $s \in \Gamma(X,L^{\tensor d})\minus\{0\}$, we have $$\begin{aligned} \ord\_{q=0}\hat{\varphi}^\s \le C d^n\text{.}\end{aligned}$$ The exponent n* = dimX in the above polynomial bound is the smallest possible (see Proposition [propsmallest] below). Moreover, since L is ample, a ZL-dense formal curve has Zariski-dense image. Thus, ZL-density may be regarded as a strong form of Zariski-density. For a formal curve $\hat{\varphi}: \Spf k[\![q]\!] \to \AA^n\_k \subset \PP^n\_k$ the above notion boils down to a classical Zero Lemma property — here, $L=\mathcal{O}\_{\PP^n\_k}(1)$. For instance, taking $k=\CC$, n = 4, and φ̂ defined by the system of formal series (q, E2(q), E4(q), E6(q)), the Zero Lemma in condition (ii) above amounts to asserting that φ̂ is ZL-dense in $\PP^4\_{\CC}$. We shall prove that ZL-density does not depend on the choice of L (see Proposition [amplediv] below). Actually, if X is only quasi-projective and the closed point φ̂(0) in the image of φ̂ is a regular point of X, then we shall prove that ZL-density does not depend on the choice of a projective compactification of X (see Corollary [corostandard] below). ### Statement of our main theorem and proof method Let K be a number field and OK be its ring of integers. By an arithmetic scheme over OK we mean an integral scheme X endowed with a separated and flat morphism of finite type $\mathcal{X} \to \Spec \mathcal{O}\_K$. The following theorem formalizes and generalizes Nesterenko’s method. [lethm] Let X be a quasi-projective arithmetic scheme over OK of relative dimension n ≥ 2, with smooth generic fiber XK, and let $\hat{\varphi}: \Spf \mathcal{O}\_K[\![q]\!] \to \mathcal{X}$ be a morphism of formal OK-schemes such that, for every field embedding $\sigma : K \hookrightarrow \CC$, the formal curve $\hat{\varphi}\_{\sigma} : \Spf \CC[\![q]\!] \to \mathcal{X}\_{\sigma}$, obtained from φ̂ by base change, lifts to an analytic curve $\varphi\_{\sigma}: D\_{R\_{\sigma}}\subset \CC \to \mathcal{X}\_{\sigma}^{\an}$ defined on a disk of radius Rσ > 0 centered at the origin. Assume that $$\begin{aligned} \prod\_{\sigma: K \into \CC} R\_{\sigma}=1 \end{aligned}$$ and that there exists a vector field $v \in \Gamma(\mathcal{X}\_K,T\_{\mathcal{X}\_K/K})\minus\{0\}$ on the generic fiber of X such that $\hat{\varphi}\_K : \Spf K[\![q]\!] \to \mathcal{X}\_K$ satisfies the differential equation $$\begin{aligned} q\frac{d\hat{\varphi}\_K}{dq} = v\circ \hat{\varphi}\_K\text{.} \end{aligned}$$ If, moreover, 1. the formal curve φ̂K is ZL-dense in XK, and 2. for each field embedding $\sigma: K \hookrightarrow \CC$, the analytic curve $\varphi\_{\sigma}: D\_{R\_{\sigma}} \to \mathcal{X}\_{\sigma}^{\an}$ has moderate growth, then, for every $\sigma : K \hookrightarrow \CC$, and every $z \in D\_{R\_{\sigma}}\smallsetminus\{0\}$, the field of definition K(φσ(z)) of the complex point φσ(z) in XK satisfies $$\begin{aligned} \trdeg\_{\QQ} K(\varphi\_{\sigma}(z))\ge n-1\text{.} \end{aligned}$$ Let us remark that the conditions of ZL-density and of moderate growth, corresponding to conditions (i) and (ii) in Nesterenko’s method, are actually very mild hypotheses. For instance, ZL-density is automatic whenever φ̂ is a smooth integral curve of some vector field satisfying Nesterenko’s D-property. When the ambient space is an affine space, this is also a theorem of Nesterenko ( Theorem 6), which was recently extended to a geometric framework by Binyamini. In Appendix [appzl] we explain how to slightly modify Binyamini’s arguments to prove a similar statement for any smooth quasi-projective variety. Moderate growth, in turn, is satisfied for curves having uniformly bounded derivative on the disk (endowed with the Poincaré metric; see Example [bd] for a precise statement). In particular, a theorem of Brunella ( Theorem 16; see also Theorem 15) implies that, for a generic one dimensional holomorphic foliation (with singularities) F on $\PP^n(\CC)$, any integral curve to F parametrized by a disk has moderate growth. In the broader context of Transcendental Number Theory, our result may be regarded as complementary to the Siegel-Shidlovsky and Schneider-Lang theories, which also deal with algebraic independence or transcendence of values of integral curves of algebraic vector fields (see,, and for general geometric formulations). Indeed, while the Siegel-Shidlovsky and Schneider-Lang criteria handle curves parametrized by parabolic Riemann surfaces, our theorem deals with the hyperbolic case. Our proof of Theorem [lethm] bears the same general structure of Nesterenko’s method. We also start by reducing it to a diophantine approximation statement: Theorem [lethm’] below. This is done via the same algebraic independence criterion of Philippon; we explain in Appendix [genphil] how to generalize it to arbitrary quasi-projective varieties. The first step in the method, concerning the construction of “auxiliary polynomials”, is replaced in our geometric framework by a construction of “auxiliary sections” given by Theorem [auxsec] below, the proof of which makes essential use of Bost’s method of slopes in Arakelov Theory (; cf., ). The second step also involves estimating some higher order derivative — here, our main tool is a general result comparing, for a section of a Hermitian line bundle on a disk, norms of jets at two distinct points; see Proposition [proptaylor] below and its corollaries. The third and last step is essentially the same trick using the differential equation as explained above. Further directions and open problems ------------------------------------ With Theorem [lethm] in hand, we may turn the puzzling remark explained above into a precise mathematical question: is there any example of application of Theorem [lethm] whose resulting transcendence statement is not contained in Theorem [thmnes]? As promising as the potential candidates from the theory of Mirror Symmetry may seem, one must face, given our current state of knowledge, the logical possibility of a negative answer. However, let us remark that a proof of this fact would be as remarkable as the discovery of a new example, since it would imply that modular functions and their derivatives are the only ones satisfying the (quite general) hypotheses of Theorem [lethm], which make no explict reference to their geometric nature in terms of moduli of elliptic curves. This paper, specially our geometric formulation of Theorem [lethm], should also be seen as a first step in a larger program aiming to adapt and apply Nesterenko’s method to the study of algebraic independence of abelian periods of genus g > 1. Since values of quasimodular forms can be expressed in terms of periods of elliptic curves, the theorem of Nesterenko recovers in particular a famous result of Chudnovsky establishing Grothendieck’s Period Conjecture for complex multiplication elliptic curves. A family of analogs of (E2, E4, E6) related to the Siegel moduli spaces Ag was constructed in and ; in this setting involving analytic functions of several variables, in addition to being easier to read the relevant properties required by Nesterenko’s method from geometry, the appearance of truly geometric phenomena — such as positive dimensional ``special subvarieties" of Ag — strongly suggests a geometric approach. Organization of this article ---------------------------- A great effort has been done to isolate all the different techniques intervening in Nesterenko’s method and to place them in their natural generality. This distillation process is aimed not only at improving the readability of our paper, but also at making these techniques suitable for other applications in Diophantine Approximation. Section [strongzd] contains the definition of ZL-density and some of its basic properties; here we use elementary Intersection Theory. Section [mgje] defines characteristic functions and moderate growth for 2-forms on a disk and contains basic versions of the jet estimates we shall need later; our main result here is Proposition [proptaylor] (see also Corollary [corotaylor2]). Both Sections [strongzd] and [mgje] are self-contained and are of independent interest. Section [mgqp] treats the special case of moderate growth for analytic curves (as explained in this introduction) and it depends only on the beginning of Section [mgje]. Its main objective is to prove that, under a non-degeneracy hypothesis, this concept is a birational invariant of the target space (Theorem [funcfield]); this is essentially classical material on Nevanlinna Theory. Section [secauxsec] is devoted to the construction of “auxiliary sections” in a geometric context. Here, we combine the concepts of moderate growth developed in Sections [mgje] and [mgqp] with Bost’s slope inequality to obtain Theorem [auxsec]. This section contains a review of the prerequisites in Arakelov Theory. In Section [derivesections] we explain how vector fields induce derivations on global sections of line bundles and we provide some L∞ estimates. This section is also self-contained and of independent interest. Section [finalproof] contains a proof of Theorem [lethm]. The reader will recognize, in Lemmas [lemma1], [lemma2], and [lemma3], natural generalizations of the three steps of Nesterenko’s method explained above. Finally, Appendices [genphil] and [appzl] concern geometric generalizations of results of Philippon and Binyamini originally stated only for affine (or projective) spaces, as explained above. In Appendix [appzl] we make use of the basic constructions of Section [derivesections]. Acknowledgments --------------- This work was supported by a public grant as part of the FMJH project, and is part of my PhD thesis at Université Paris-Sud, Orsay, under the supervision of Jean-Benoît Bost. I thank him and Hugues Randriam for allowing me to use their preliminary non-published notes on moderate growth as a starting point for this paper. I am also grateful to Dinh Tuan Huynh for a fruitful discussion on Nevanlinna Theory and to Daniel Bertrand for pointing me out a missing reference. Finally, I thank the Max-Planck-Institut für Mathematik, Bonn, for the excellent working conditions in which this paper was finished. Terminology and notation ------------------------ ### By an (algebraic) variety over a field k we mean a separated integral scheme of finite type over k. ### Recall that a line bundle L on a scheme X is semiample if there exists an integer m ≥ 1 such that $L^{\tensor m}$ is generated by its global sections. Observe that ample line bundles are semiample, and that semiampleness is preserved under pullbacks. ### A real (1,1)-form ω on a complex manifold M can always be written, in local coordinates (z1, …, zn) on M, as $$\begin{aligned} \omega = \frac{i}{2}\sum\_{k,l=1}^nh\_{kl}dz\_k\wedge d\overline{z}\_l\end{aligned}$$ where $H\defeq (h\_{kl})\_{1\le k,l\le n}$ is a Hermitian matrix. We say that ω is positive (resp. semipositive) if the matrix H is positive-definite (resp. positive-semidefinite). Note that semipositive (1,1)-forms are stable under pullbacks. ### By a Hermitian line bundle $\overline{L}=(L,\\| \ \\|)$ on a complex manifold M, we mean a holomorphic line bundle L on M endowed with a C∞ Hermitian metric ∥ ∥. If Θ denotes the curvature of the Chern connection on L associated to ∥ ∥ (locally, $\Theta = - \partial \overline{\partial}\log \\|e\\|^2$ where e is some trivialization of L), then we define the Chern curvature of $\overline{L}$ by $$\begin{aligned} c\_1(\overline{L}) = \frac{i}{2\pi} \Theta\text{.}\end{aligned}$$ This is a closed real C∞ (1, 1)-form on M whose class in $H^2(M,\RR)$ coincides with the first Chern class c1(L). We say that $\overline{L}$ is positive (resp. semipositive) if $c\_1(\overline{L})$ is positive (resp. semipositive). ### We use the standard notation $$d^c = \frac{i}{4\pi}(\overline{\partial} - \partial)= \frac{1}{4\pi} \left(r\frac{\partial}{\partial r}\tensor d\theta -\frac{1}{r}\frac{\partial}{\partial \theta}\tensor dr \right)\text{,}$$ so that $dd^c = \frac{i}{2\pi}\partial \overline{\partial}$. ### The continuous function $\log^+: \RR \to \RR$ is defined by $$\begin{aligned} \log^+x = \begin{cases} \log x & \text{ if }x\ge 1\\ 0 & \text{ otherwhise}. \end{cases}\end{aligned}$$ ZL-dense formal curves in quasi-projective varieties ==================================================== In this section we introduce the purely algebraic concept of ZL-dense formal curves, and we prove some of its basic properties. This notion refines the property of being Zariski-dense and isolates the content of the Zero Lemma necessary in Nesterenko’s method; that is, a formal curve satisfies the Zero Lemma if and only if it is ZL-dense. Degree of a divisor with respect to a line bundle ------------------------------------------------- Let k be a field and X be a variety over k. Recall from 2.5 that (the isomorphism class of) a line bundle L on X defines an additive operator $$\begin{aligned} \alpha \mapsto c\_1(L)\cap \alpha\end{aligned}$$ on the abelian group of algebraic cycles in X modulo rational equivalence; if α is the class of a subvariety V of X, then c1(L) ∩ α is by definition the class of the cycle in V associated to any Cartier divisor D of V for which L∣V ≅ OV(D). The r-fold composition of this operator with itself is denoted by α ↦ c1(L)r ∩ α. We say that a cycle class α in X is semipositive if there exists an integer m ≥ 1 such that mα can be represented by a non-negative cycle in X (i.e., a cycle of the form ∑imi[Vi] with each mi ≥ 0). For instance, the cycle class of a Cartier divisor D is semipositive if and only if some positive multiple of D is linearly equivalent to an effective divisor. [lemmasemiample] Let L be a semiample line bundle on X (see [defsemiample] for a definition). Then, for any semipositive cycle class α in X, c1(L) ∩ α is semipositive. Let m ≥ 1 be an integer such that mα is represented by the cycle ∑imi[Vi], with each mi ≥ 0. As L is semiample, there exists an integer n ≥ 1 such that $L^{\tensor n}$ is generated by global sections. In particular, for any subvariety V of X, the line bundle $L^{\tensor n}\|\_V$ on V admits a non-zero global section sV. For every i, $c\_1(L^{\tensor n})\cap [V\_i]$ is the cycle class induced by the effective Cartier divisor ÷ (sVi) on Vi, so that $nm\, c\_1(L)\cap \alpha = c\_1(L^{\tensor n})\cap m\alpha$ is represented by the non-negative cycle ∑imi[ ÷ (sVi)]. Still following the terminology of, for any line bundle L on X, and any r-cycle class α in X, the *L-degree* of α is defined by $$\begin{aligned} \deg\_L\alpha = \deg( c\_1(L)^{r}\cap \alpha)\text{,}\end{aligned}$$ where deg denotes the degree function on zero-cycle classes. If D is a Cartier divisor on X, then we denote by $$\begin{aligned} \deg\_LD = \deg(c\_1(L)^{\dim X - 1}\cap [D])\end{aligned}$$ the L-degree of the cycle class [D] induced by D. Observe that the degree of a semipositive zero-cycle class is non-negative. In the next result, we use the following easy consequence of Lemma [lemmasemiample]: if L is semiample and α is semipositive, then degLα ≥ 0 (cf. Lemma 12.1). [propsemiample] Let X be a projective variety over a field k and L (resp. M) be an ample (resp. semiample) line bundle on X. Then, there exists an integer m ≥ 1 such that, for any semipositive r-cycle class α in X, we have $$\begin{aligned} 0 \le \deg\_M\alpha \le m^r\deg\_L\alpha\text{.} \end{aligned}$$ Let m ≥ 1 be an integer such that $N\defeq L^{\tensor m}\tensor M^{\vee}$ is semiample. For any r-cycle class α in X, we have $$\begin{aligned} m^r\deg\_L\alpha = \deg\_{L^{\tensor m}}\alpha = \deg\_{N\tensor M}\alpha = \sum\_{s=0}^r \binom{r}{s}\deg\_M(c\_1(N)^{s}\cap \alpha)\text{.} \end{aligned}$$ Since N is semiample and α is semipositive, it follows from Lemma [lemmasemiample] that each c1(N)s ∩ α is semipositive. As M is also semiample, we conclude that each term in the right-hand side of the above equation is non-negative, so that mrdegLα ≥ degMα ≥ 0. By combining the above proposition with an induction argument in r, one can actually prove the following generalization. Let X be a projective variety over a field k, L be an ample line bundle on X, and M be any line bundle on X. Then there exists a constant C > 0 such that ∣degMα∣ ≤ CdegLα for every semipositive cycle class α in X. [comparable] Let X be a projective variety over a field k. If L and M are ample line bundles on X, then there exist constants C1, C2 > 0 such that $$\begin{aligned} C\_1\deg\_MD \le \deg\_LD \le C\_2\deg\_MD \end{aligned}$$ for any effective Cartier divisor D in X. $\blacksquare$ ZL-dense formal curves in projective varieties ---------------------------------------------- Let X be an algebraic variety over a field k and consider the ring of formal power series k[ [q] ]. By a formal curve in X we mean a morphism of k-schemes $\hat{\varphi} : \Spec k[\![q]\!] \to X$, or, equivalently, a morphism of formal k-schemes $\hat{\varphi} : \Spf k[\![q]\!] \to X$. The k-point of X obtained by composing the k-point of $\Spec k[\![q]\!]$ given by the ideal (q) ⊂ k[ [q] ] with φ̂ is denoted by φ̂(0). Let $\hat{\varphi} : \Spec k[\![q]\!] \to X$ be a formal curve in X, and D be an effective Cartier divisor in X. We define the intersection multiplicity of D with φ̂ (at φ̂(0)) by $$\begin{aligned} \mult\_{\hat{\varphi}}D \defeq \ord\_{0} \hat{\varphi}^\f\text{,}\end{aligned}$$ where f* ∈ OX, φ̂(0) is any local equation for D around φ̂(0). This clearly does not depend on the choice of f. The multiplicity function $\mult\_{\hat{\varphi}}$ is additive and takes values in $\NN \cup\{+\infty\}$. Let X be a projective variety of dimension n over a field k and let L be any ample line bundle on X. We say that a formal curve $\hat{\varphi}: \Spec k[\![q]\!] \to X$ in X is ZL-dense if there exists a constant C > 0 such that $$\begin{aligned} \label{boundZLdense} \mult\_{\hat{\varphi}}D \le C (\deg\_LD)^n \end{aligned}$$ for every effective Cartier divisor D in X. Observe that the choice of L in the above definition is irrelevant by Corollary [comparable]. Let us remark that the exponent n = dimX intervening in the polynomial bound ([boundZLdense]) is the smallest possible one: [propsmallest] Let X be a projective variety of dimension n over k endowed with an ample line bundle L, and φ̂ be a formal curve in X. Then there exists ɛ > 0 and sequence of effective Cartier divisors (Di)i ≥ 1 on X satisfying $$\begin{aligned} \lim\_{i\rightarrow +\infty}\deg\_L D\_i = +\infty \ \ \text{ and }\ \ \mult\_{\hat{\varphi}}D\_i \ge \varepsilon(\deg\_LD\_i)^n\text{ for every }i\ge 1\text{.} \end{aligned}$$ In the above statement we allow the possibility that $\mult\_{\hat{\varphi}}D\_i = +\infty$ (i.e. the divisor Di vanishes identically along the formal curve φ̂) by adopting the standard convention that + ∞ ≥ t for every $t\in\RR$. Assume first that $X=\PP^n\_k$ and L = O(1). Consider the natural projection $\AA^n\_k\minus\{0\} \to \PP^n\_k$ and lift φ̂ to some $\hat{\psi}: \Spec k[\![q]\!] \to \AA^n\_k\minus\{0\}$. Let i ≥ 1 be an integer. If P = ∑∣I∣ = iaIXI ∈ k[X0, …, Xn] is a homogeneous polynomial of degree i, seen as a regular function on $\AA^n\_k\minus\{0\}$, then we may write ψ̂\*P = ∑j = 0∞(∑∣I∣ = iaIbI, j)qj ∈ k[ [q] ], for some bI, j ∈ k depending on the coefficients of the n + 1 formal series defining ψ̂. Since $$\begin{aligned} \text{card}\{I \in \NN^{n+1}\mid \|I\|=i\} = \binom{i+n}{n}> \frac{1}{n!}i^n\text{,} \end{aligned}$$ it follows from elementary linear algebra that there exists a non-zero homogeneous polynomial Pi of degree i such that $\ord\_0 \hat{\psi}^\P\_i\ge \frac{1}{n!}i^n$. By considering the Cartier divisors D*i on $\PP^n\_k$ induced by Pi, we see that we may take ɛ = 1/n! in this case. The general case follows from the above one by considering a finite surjective morphism $f:X \to \PP^n\_k$ satisfying f\O(1) ≅ L*m for some m ≥ 1. It follows from the above proof that the Cartier divisors Di can actually be taken in the linear system $\|L^{\tensor m i}\|$ for some fixed integer m ≥ 1. Any ZL-dense formal curve φ̂ in a projective variety X has a dense image in the Zariski topology. Indeed, since X is projective, any Zariski-closed subset of X is contained in the support of some effective Cartier divisor of X; then, one simply remarks that ZL-density implies that $\mult\_{\hat{\varphi}}D < + \infty$ for any effective Cartier divisor D in X, so that the image of φ̂ is not contained in the. The following example shows that the converse is not true in general. [Lacunary series] Let k be a field and (ni)i ≥ 0 be an increasing sequence of natural numbers satisfying $\lim\_{i\rightarrow +\infty}\frac{n\_{i+1}}{n\_i^2} = +\infty$. If h ∈ k[ [q] ] is any formal series of the form $$\begin{aligned} h(q) = \sum\_{i\ge 0}a\_i q^{n\_i}\text{,} \ \ \ \ a\_i\neq 0 \end{aligned}$$ then the formal curve $\hat{\varphi} : \Spec k[\![q]\!] \to \PP^2\_k$, given in homogeneous coordinates by, is not ZL-dense. Indeed, for any integer d ≥ 0, we may consider the homogeneous polynomial of degree nd $$\begin{aligned} P\_d = X\_0^{n\_d-1}X\_2 - \sum\_{i=0}^d a\_i X\_0^{n\_d-n\_i}X\_1^{n\_i} \end{aligned}$$ so that $$\begin{aligned} \frac{\mult\_{\hat{\varphi}} \div(P\_d)}{n\_d^2} = \frac{n\_{d+1}}{n\_d^2} \to +\infty \end{aligned}$$ as d → + ∞. Observe that the image of φ̂ is indeed Zariski-dense. By contradiction, if $C\subset \PP^2\_k$ is an irreducible curve containing the image of φ̂, then, for any effective Cartier divisor D in $\PP^2\_k$ whose support does not contains C, we have $\mult\_{\hat{\varphi}}D = i(\hat{\varphi}(0),C\cdot D) \le \deg C \cdot \deg D$. By construction, this is absurd for D = ÷ (Pd) and d sufficiently large. For natural examples of ZL-dense formal curves we refer to Appendix [appzl]. Reformulation in terms of sections of an ample line bundle ---------------------------------------------------------- Let X be an algebraic variety over a field k and $\hat{\varphi}: \Spec k[\![q]\!] \to X$ be a formal curve. If L is any line bundle on X, and s is a section of L on a neighborhood of φ̂(0), we may consider the vanishing order of $\hat{\varphi}^\s \in \Gamma(\Spec k[\![q]\!], \hat{\varphi}^\L)$ at q = 0, which coincides with the intersection multiplicity of the effective Cartier divisor ÷ (s) with φ̂: $$\begin{aligned} \ord\_{0}\hat{\varphi}^\s = \mult\_{\hat{\varphi}} \div(s)\text{.} \end{aligned}$$ The next proposition shows that ZL-density is a condition that has to be checked only for Cartier divisors arising from sections of powers of some fixed ample line bundle. [amplediv] Let X* be a projective variety of dimension n over a field k and L be an ample line bundle on X. A formal curve $\hat{\varphi}: \Spec k[\![q]\!] \to X$ is ZL-dense if and only if there exists a constant C > 0 such that $$\begin{aligned} \ord\_{0}\hat{\varphi}^\s \le C d^n \end{aligned}$$ for any integer d* ≥ 1, and any $s \in \Gamma(X, L^{\tensor d}) \minus \{0\}$. The necessity follows from the fact that, for any $s \in \Gamma(X,L^{\tensor d})\minus \{0\}$, degL ÷ (s) = (degLX)d. To prove the sufficiency, fix any finite surjective morphism $f: X \to \PP^n\_k$ such that f\O(1) is isomorphic to $L^{\tensor m}$ for some m* ≥ 1. If E is an effective Cartier divisor in $\PP\_k^n$, then there exists a section $s \in \Gamma(\PP^n\_k, \mathcal{O}(\deg\_{\mathcal{O}(1)} E))$ satisfying E = ÷ (s), so that $$\begin{aligned} \label{aux1} \mult\_{f\circ \hat{\varphi}} E = \ord\_{0}(f\circ \hat{\varphi})^\s = \ord\_{0}\hat{\varphi}^\(f^\s) \le C (m\deg\_{\mathcal{O}(1)}E)^n = Cm^n(\deg\_{\mathcal{O}(1)}E)^n\text{.} \end{aligned}$$ Let D* be an effective Cartier divisor in X. Since f is finite and $\PP^n\_k$ is normal, we may define the pushforward f\*D by taking norms: there is an open affine covering (Ui)i of $\PP^n\_k$ such that D admits a local equation hi on each f− 1(Ui), and we define f\*D = [(Normf(hi), Ui)i]. As f\*f\*D − D is effective, we obtain $$\begin{aligned} \mult\_{\hat{\varphi}}D \le \mult\_{\hat{\varphi}}f^\f\_\D = \mult\_{f\circ \hat{\varphi}}f\_\D\text{.} \end{aligned}$$ Note that the Weil divisor associated to f\D coincides with the pushforward (of cycles) of the Weil divisor associated to D (cf. Proposition 1.4). In particular, the projection formula gives $$\begin{aligned} \deg\_{\mathcal{O}(1)}f\_\D = \deg\_{L^{\tensor m}}D = m^{n-1}\deg\_{L}D \end{aligned}$$ so that, by ([aux1]), $$\begin{aligned} \mult\_{\hat{\varphi}}D \le Cm^n (\deg\_{\mathcal{O}(1)}f\_\D)^n = Cm^{n^2}(\deg\_LD)^n\text{.} \end{aligned}$$ An advantage of considering the above equivalent form of ZL-density stems from the vector space structure of the sets $\Gamma(X,L^{\tensor d})$, d ≥ 1. In general, a formal curve $\hat{\varphi}: \Spec k[\![q]\!] \to X$ induces, for every integer d ≥ 1, a decreasing filtration by linear subspaces (Edi)i ≥ 0 on the k-vector space $E\_d \defeq \Gamma(X,L^{\tensor d})$ defined by $E\_d^i\defeq \{s \in E\_d \mid \ord\_0\hat{\varphi}^\s\ge i\}$. Since E*d is finite dimensional, there exists id ≥ 1 such that Edid = ⋂i ≥ 0Edi = {s ∈ Ed ∣ φ̂\*s = 0}. In other words, for every s ∈ Ed such that φ̂\*s ≠ 0, we have $\ord\_0\hat{\varphi}^\s <i\_d$. In particular, this shows that one may replace in Proposition [amplediv] the condition ``for any integer d* ≥ 1“ by the weaker ``for any sufficiently large integer d”. As a first application of Proposition [amplediv], we use the filtration (Edi)i ≥ 0 to show that ZL-density is a geometric property. Let X be a geometrically integral projective variety over a field k and $\hat{\varphi} : \Spf k[\![q]\!] \to X$ be a formal curve. Then, for any field extension K of k, the formal curve $\hat{\varphi}\_K : \Spf K[\![q]\!] \to X\_K$, obtained from φ̂ by base change, is ZL-dense in XK if and only if φ̂ is ZL-dense in X. Let d ≥ 1 and i ≥ 0 be integers. Note that $E\_d\tensor\_kK$ may be canonically identified with $\Gamma(X\_K, L\_K^{\tensor d})$. Moreover, since Edi is the kernel of the k-linear map $$\begin{aligned} E\_d &\to \Gamma(\Spec k[\![q]\!],\hat{\varphi}^\L^{\tensor d})\tensor\_k k[\![q]\!]/(q^{i})\\ s &\mapsto \hat{\varphi}^\s \mod q^{i}\text{,} \end{aligned}$$ we conclude that $E\_d^{i}\tensor\_k K = \{t \in E\_d\tensor\_kK \mid \ord\_0\hat{\varphi}\_K^\t \ge i\}$. In particular, for every integer d* ≥ 1, and any real number κ > 0, $\ord\_0\hat{\varphi}^\s\le \kappa$ for every $s \in \Gamma(X,L^{\tensor d})\minus\{0\}$ if and only if $\ord\_0\hat{\varphi}^\\_Kt\le \kappa$ for every $t \in \Gamma(X\_K,L\_K^{\tensor d})\minus\{0\}$. As another application of Proposition [amplediv], we prove the following result which will be used in our proof of Proposition [propmod]. [surj] Let f : X → Y be a surjective morphism between projective varieties of dimension n over a field k and let $\hat{\varphi}:\Spec k[\![q]\!] \to X$ be a formal curve in X. If φ̂ is ZL-dense in X, then f ∘ φ̂ is ZL-dense in Y. Let L be an ample line bundle on X admitting a global section $s \in \Gamma(X,L)\minus \{0\}$, and M be any ample line bundle on Y. Since f\*M is semiample, $N\defeq L \tensor f^{\}M$ is ample ( Proposition 4.5.6 (ii)). Let d* ≥ 1 be an integer, and $t\in \Gamma(Y,M^{\tensor d})\minus\{0\}$. Since f is surjective, f\*t ≠ 0. Thus $s^{\tensor d}\tensor f^\t$ is a non-zero global section of $N^{\tensor d}$ and, since φ̂* is ZL-dense in X, there exists a constant C > 0 independent of d such that $$\begin{aligned} \ord\_{0}\hat{\varphi}^\(s^{\tensor d}\tensor f^\t) \le Cd^n\text{.} \end{aligned}$$ To complete the proof, it is sufficient to remark that $$\begin{aligned} \ord\_{0}(f\circ \hat{\varphi})^\(t) = \ord\_{0}\hat{\varphi}^\(f^\t) \le \ord\_{0}\hat{\varphi}^\(s^{\tensor d}\tensor f^\t)\text{.} \end{aligned}$$ The above proposition, combined with the arguments in the proof of Proposition [amplediv], actually shows that for any finite surjective morphism f* : X → Y between projective varieties over a field k, with Y normal, a formal curve $\hat{\varphi}: \Spec k[\![q]\!] \to X$ is ZL-dense in X if and only if f ∘ φ̂ is ZL-dense in Y. ZL-density in quasi-projective varieties ---------------------------------------- We defined a notion of ZL-density for formal curves in projective varieties. In this paragraph, under a mild technical condition, we extend this notion, via compactification, to formal curves in quasi-projective varieties. To assure that we obtain a well defined notion, we must show that this does not depend on the choice of compactification. We start by reformulating ZL-density in terms of Weil divisors in place of Cartier divisors. Let X be an algebraic variety over a field k, and $\hat{\varphi}: \Spec k[\![q]\!] \to X$ be a formal curve. If φ̂(0) is a regular point of X, then we may define the intersection multiplicity of a Weil divisor with φ̂: if U is a regular open neighborhood of φ̂(0) and Z is a Weil divisor on X, then Z ∩ U is induced by some Cartier divisor D on U, and we define $$\begin{aligned} \mult\_{\hat{\varphi}}Z = \mult\_{\hat{\varphi}}D\text{.}\end{aligned}$$ We may thus mimic the proof of Proposition [amplediv] to obtain the following result. Let X be a projective variety of dimension n over a field k, L be an ample line bundle on X, and $\hat{\varphi} : \Spec k[\![q]\!] \to X$ be a formal curve such that φ̂(0) is a regular point of X. Then, φ̂ is ZL-dense in X if and only if there exists a constant C > 0 such that, for every effective Weil divisor Z on X, $$\begin{aligned} \mult\_{\hat{\varphi}}Z \le C (\deg\_LZ)^n \end{aligned}$$ We are now in position to prove that a modification away from φ̂ does not affect ZL-density. [propmod] Let f : X → Y be a proper morphism between projective varieties over a field k, and U be an open subset of Y such that f induces an isomorphism $f^{-1}(U) \stackrel{\sim}{\to} U$. If $\hat{\varphi}: \Spec k[\![q]\!] \to f^{-1}(U)\subset X$ is a formal curve such that φ̂(0) is a regular point of X, then φ̂ is ZL-dense in X if and only if f ∘ φ̂ is ZL-dense in Y. Since f : X → Y is a proper birational morphism, and Y is irreducible, f is surjective. By Proposition [surj], if φ̂ is ZL-dense in X, then f ∘ φ̂ is ZL-dense in Y. Conversely, suppose that f ∘ φ̂ is ZL-dense in Y. Fix an ample line bundle L (resp. M) on X (resp. Y), and let Z be an effective Weil divisor on X. Since f is an isomorphism over U and φ̂ factors through f− 1(U), we have $$\begin{aligned} \mult\_{\hat{\varphi}}Z = \mult\_{f\circ \hat{\varphi}}f\_\Z\text{.} \end{aligned}$$ As f* ∘ φ̂ is ZL-dense in Y, there is a constant C1 > 0 (not depending on Z) such that $$\begin{aligned} \mult\_{f\circ\hat{\varphi}}f\_\Z \le C\_1 (\deg\_M f\_\Z)^n\text{,} \end{aligned}$$ where n = dimY = dimX. By the projection formula, degMf\*Z = degf\MZ. Since f\M is semiample, it follows from Proposition [propsemiample] that there exists a constant C2 > 0 such that degf\MZ* ≤ C2degLZ. We conclude that $$\begin{aligned} \mult\_{\hat{\varphi}}Z \le C\_1C^n\_2 (\deg\_LZ)^n\text{.} \end{aligned}$$ Let us remark that it is essential in our proof above to consider the pushforward of a Weil divisor on X under the proper morphism f. This explains why we reformulated ZL-density in terms of Weil divisors as the pushforward of a Cartier divisor under a proper morphism is not well defined in general. [corostandard] Let X be a quasi-projective variety over a field k, and $\hat{\varphi}: \Spec k[\![q]\!] \to X$ be a formal curve such that φ̂(0) is a regular point of X. If $j\_i:X \hookrightarrow \overline{X}\_i$, i = 1, 2, are two projective compactifications of X, then j1 ∘ φ̂ is ZL-dense in $\overline{X}\_1$ if and only if j2 ∘ φ̂ is ZL-dense in $\overline{X}\_2$. Consider the scheme theoretic image $\overline{X}$ of $(j\_1,j\_2): X \to \overline{X}\_1\times\_k\overline{X}\_2$ and apply Proposition [propmod] to the natural projections $\overline{X} \to \overline{X}\_i$, i = 1, 2. This enables us to define a good notion of ZL-density in a quasi-projective variety. Let X be a quasi-projective variety over a field k, and $\hat{\varphi}: \Spec k[\![q]\!] \to X$ be a formal curve such that φ̂(0) is a regular point of X. We say that φ̂ is ZL-dense in X if there exists a projective compactification $j: X \hookrightarrow \overline{X}$ of X such that j ∘ φ̂ is ZL-dense in $\overline{X}$. Moderate growth and jet estimates on complex disks ================================================== In this section we introduce characteristic functions and moderate growth of certain 2-forms on a complex disk; these are purely analytic notions. We then proceed to establishing natural estimates on jets of sections of holomorphic line bundles on disks. In a sense, our exposition is more basic than the usual accounts on Value Distribution Theory, since characteristic functions of analytic curves will be a special case of our construction. The kind of jet estimates we consider here play a central role in Diophantine Approximation and transcendence proofs. They notably appear in such proofs using the formalism of Arakelov Geometry, to estimate the height of evaluation maps, when applying Bost’s method of slopes (see, for instance,,,,,, ). Characteristic functions ------------------------ Let r > 0 be a real number, and $p\in D\_r \defeq \{z \in \CC \mid \|z\|<r\}$. Recall that the Green’s function of Dr at p is defined by $$\begin{aligned} g\_{D\_r,p}: \CC &\to (-\infty,+\infty]\\ z&\mapsto \log^+ \left\|\frac{r^2 -\overline{p}z}{r(z-p)}\right\|\text{.}\end{aligned}$$ This is a superharmonic (thus locally integrable) function on $\CC$, real-valued and continuous on $\CC\minus\{p\}$, strictly positive and harmonic (thus C∞) on $D\_r\minus \{p\}$, and vanishing identically on $\CC \minus D\_r$. For any locally bounded 2-form α defined on an open neighborhood of Dr, we denote $$\begin{aligned} T\_{\alpha, p}(r) \defeq \int\_{\CC}g\_{D\_r,p}\alpha\text{.}\end{aligned}$$ [intparts] An integration by parts with u(t) = ∫Dtα and v(t) = logt shows that $$\begin{aligned} T\_{\alpha,0}(r) = \int\_0^r \left( \int\_{D\_t}\alpha\right)\frac{dt}{t}\text{.}\end{aligned}$$ Since gDr, p is the composition of gDr, 0 with the automorphism of Dr given by $$\begin{aligned} \sigma\_{r,p}(z) = \frac{r^2(z-p)}{r^2-\overline{p}z}\text{,}\end{aligned}$$ we obtain $$\begin{aligned} T\_{\alpha,p}(r) = \int\_0^r \left( \int\_{D\_t}(\sigma\_{r,p}^{-1})^\\alpha\right)\frac{dt}{t}\text{.}\end{aligned}$$ Let R* > 0 be a real number, and α be a locally bounded semipositive (1, 1)-form on the disk DR. The non-decreasing function $$\begin{aligned} T\_{\alpha} : (0,R) &\to \RR\_{\ge 0}\\ r &\mapsto T\_{\alpha,0}(r)\end{aligned}$$ is the characteristic function of α in DR. We shall be particularly interested in the following special case. Let $\overline{L} = (L, \\| \ \\|)$ be a semipositive Hermitian line bundle on DR. The characteristic function of $\overline{L}$ in DR is defined by $$\begin{aligned} T\_{\overline{L}} \defeq T\_{c\_1(\overline{L})}\text{.}\end{aligned}$$ We also denote $T\_{\overline{L},p}(r) = T\_{c\_1(\overline{L}),p}(r)$ for p ∈ Dr ⊂ DR. Let d ≥ 1 be an integer. As $c\_1(\overline{L}^{\tensor d}) = d\cdot c\_1(\overline{L})$, we have $T\_{\overline{L}^{\tensor d},p}(r) = d\cdot T\_{\overline{L},p}(r)$. Forms of moderate growth ------------------------ Let R > 0 be a real number, and α be a locally bounded semipositive (1,1)-form on DR. [defmodgrowth] We say that α has moderate growth if there exist constants a, b > 0 such that $$\begin{aligned} T\_{\alpha}(r) \le a + b \log \frac{1}{1-\frac{r}{R}} \end{aligned}$$ for any r ∈ (0, R). When $\alpha = c\_1(\overline{L})$ for some semipositive Hermitian line bundle $\overline{L}$ on DR, we rather say that $\overline{L}$ has moderate growth. The motivating example of a form of moderate growth is the following one. [Poincaré form][poincareform] Let $$\begin{aligned} d\mu\_R \defeq \frac{i}{2}\left(\frac{R}{R^2-\|z\|^2} \right)^2 dz \wedge d\overline{z}\end{aligned}$$ be the 2-form associated to the surface element of the Poincaré metric $\frac{R}{R^2-\|z\|^2}\|dz\|$ on DR. A direct computation shows that, for any r ∈ (0, R), $$\begin{aligned} T\_{d\mu\_R}(r) = \frac{\pi}{2}\log \frac{1}{1+\frac{r}{R}} + \frac{\pi}{2}\log \frac{1}{1-\frac{r}{R}}\text{.} \end{aligned}$$ Thus, the 2-form dμR on DR has moderate growth. We have defined moderate growth for a form α as a growth condition on Tα, 0(r) with respect to r. Our next result shows that a similar growth condition for Tα, p(r) holds uniformly for p varying in a fixed compact subset. [lemmagreenfunc] Let 0 < r0 < r1 < R1 < R be real numbers. Then there exists a constant C > 0 such that, for every p ∈ Dr0, and every r ∈ [R1, R), we have $$\begin{aligned} g\_{D\_r,p} \le g\_{D\_{r\_1},p} + Cg\_{D\_r,0}\text{.} \end{aligned}$$ Let r ∈ [R1, R) and p ∈ Dr0. We set $$\begin{aligned} C\_{r,p} \defeq \max\_{z\in \partial D\_{r\_1}}\frac{g\_{D\_r,p}(z)}{g\_{D\_r,0}(z)} \end{aligned}$$ and $$\begin{aligned} f\_{r,p} \defeq g\_{D\_{r},p} - g\_{D\_{r\_1},p} - C\_{r,p}g\_{D\_r,0}\text{.} \end{aligned}$$ By definition of Cr, p, we see that fr, p ≤ 0 over ∂Dr1. Moreover, fr, p vanishes identically on ∂Dr. Since fr, p is subharmonic over the domains Dr1 and $D\_r \minus \overline{D}\_{r\_1}$, by the Maximum Principle, we conclude that fr, p ≤ 0 everywhere. To finish the proof, it is sufficient to remark that Cr, p is uniformly bounded for r ∈ [R1, R) and p ∈ Dr0. Indeed, for z ∈ ∂Dr1, we have $$\begin{aligned} \frac{g\_{D\_r,p}(z)}{g\_{D\_r,0}(z)} = \frac{\log\left\|\frac{r^2-\overline{p}z}{r(z-p)} \right\|}{\log\left\|\frac{r}{z} \right\|} \le \frac{\log \frac{R^2 + r\_0r\_1}{R\_1(r\_1-r\_0)}}{\log \frac{R\_1}{r\_1}}\text{.} \end{aligned}$$ [lemmacompact] Let R > 0 be a real number, K ⊂ DR be a compact subset, and α be a locally bounded semipositive (1,1)-form on DR. Fix R1 ∈ (0, R) such that K ⊂ DR1. If α has moderate growth, then there exist real numbers a, b > 0 such that $$\begin{aligned} {\sup}\_{p\in K}T\_{\alpha,p}(r) \le a + b \log \frac{1}{1-\frac{r}{R}} \end{aligned}$$ for every r ∈ [R1, R). Let 0 < r0 < r1 be real numbers such that K ⊂ Dr0 and r1 < R1. By Lemma [lemmagreenfunc], there exists a constant C > 0 such that $$\begin{aligned} g\_{D\_r,p} \le g\_{D\_{r\_1},p} + C g\_{D\_r,0} \end{aligned}$$ for every r ∈ [R1, R) and every p ∈ K, so that $$\begin{aligned} T\_{\alpha,p}(r) \le T\_{\alpha,p}(r\_1) + C T\_{\alpha,0}(r)\text{.} \end{aligned}$$ Since α has moderate growth, to conclude it is sufficient to remark that the function p ↦ Tα, p(r1) is continuous, thus bounded on the compact K. Jets and characteristic functions --------------------------------- Let r > 0 be a real number and p ∈ Dr. We define a probability measure πr, p supported on ∂Dr by $$\begin{aligned} \int \psi \pi\_{r,p} = \frac{1}{2\pi}\int\_{0}^{2\pi}\psi(\sigma\_{r,p}^{-1}(re^{i\theta}))d\theta\text{,} \end{aligned}$$ where σr, p is the function defined in Remark [intparts]. For the next proposition, we shall need the following classical result. [greenfunc] As an equality of distributions on $\CC$, we have $$\begin{aligned} -2dd^cg\_{D\_r,p} = \delta\_p - \pi\_{r,p} \text{.} \end{aligned}$$ Apply Remark [intparts] and Stokes’ Theorem (see also [dc]). Let U be an open subset of $\CC$ and $\overline{L} =(L,\\| \ \\|)$ be a Hermitian line bundle over U. If s ∈ Γ(U, L) and z ∈ U, the mt*h jet of s at z* is denoted by jzms. When s has vanishing order at least m at z, jzms is simply an element of the fiber of $L\tensor (\Omega^1\_{U})^{\tensor m}$ at z. In this case, if r is a real number strictly greater than ∣z∣, we denote by ∥jzms∥r the norm of jzms with respect to the metric ∥ ∥ on L and the norm on ΩDr, z1 given by the dual of the Poincaré metric $\frac{r}{r^2-\|z\|^2}\|dz\|$ on Dr. The following result, relating jets of sections with characteristic functions, is a basic tool in Nevanlinna Theory (see, for instance, Section 2.3); variants of it were used in the context of Diophantine Approximation in Proposition 4.14, Section 3, and Theorem 5.13. [propmain] Let R > 0 be a real number, $\overline{L} = (L, \\| \ \\|)$ be a semipositive Hermitian line bundle on DR, and p ∈ DR. For every real number r ∈ (∣p∣, R) and every global section $s \in \Gamma(D\_R,L)\minus \{0\}$, if $m \defeq \ord\_{p}s$ denotes the vanishing order of s at p, we have $$\begin{aligned} \label{mainformula} \log \\|j\_p^ms\\|\_r =T\_{\overline{L},p}(r) + \int\log \\|s\\| \pi\_{r,p} - \int g\_{D\_r,p}\delta\_{\div(s)-m[p]}\text{.} \end{aligned}$$ We start with a lemma that follows immediately from the explicit formula for the Green’s functions on disks (cf. Paragraph [charfunc]). [lemmamainprop] With the above notation, if E ⊂ DR denotes the support of the divisor ÷ (s), then the distribution log∥s∥ + mgDr, p on DR defines a C∞ function over $(D\_r \minus E)\cup \{p\}$ and a continuous function over $\partial D\_r \minus E$. Moreover, $$\begin{aligned} \lim\_{z\rightarrow p}(\log \\|s(z)\\| + m g\_{D\_r,p}(z)) = \log \\|j\_p^ms\\|\_r\text{.} \end{aligned}$$ $\blacksquare$ Observe now that, for a fixed s, both sides in formula ([mainformula]) are continuous with respect to r. Since E is discrete, we may thus assume that ∂Dr ∩ E = ∅. [Proof of Proposition [propmain]] The Poincaré-Lelong formula yields the identity of currents $$\begin{aligned} dd^c \log \\|s\\|^2 = \delta\_{\div(s)} -c\_1(\overline{L})\text{.} \end{aligned}$$ Thus, by Lemma [greenfunc], $$\begin{aligned} c\_1(\overline{L})= -2dd^c(\log \\|s\\| + mg\_{D\_r,p})+\delta\_{\div(s)-m[p]} + m\pi\_{r,p}\text{.} \end{aligned}$$ Since πr, p is supported on ∂Dr, and gDr, p vanishes identically on ∂Dr, we obtain $$\begin{aligned} T\_{\overline{L},p}(r) = \int g\_{D\_r,p}\cdot(-2dd^c(\log \\|s\\| + mg\_{D\_r,p})) + \int g\_{D\_r,p}\delta\_{\div(s)-m[p]}\text{.} \end{aligned}$$ Note that, by our choice of r and by Lemma [lemmamainprop], the above products of distributions are well-defined. By another application of Lemma [greenfunc], $$\begin{aligned} T\_{\overline{L},p}(r) &= \int (\log \\|s\\| + mg\_{D\_r,p})(\delta\_p -\pi\_{r,p}) + \int g\_{D\_r,p}\delta\_{\div(s)-m[p]}\\ & =\int (\log \\|s\\| + mg\_{D\_r,p})\delta\_p - \int \log\\|s\\| \pi\_{r,p} + \int g\_{D\_r,p}\delta\_{\div(s)-m[p]} \end{aligned}$$ where in the second equality we used once again that gDr, p vanishes identically on ∂Dr. To conclude, we apply once more Lemma [lemmamainprop], which ensures that the function log∥s∥ + mgDr, p tends to log∥jpms∥r at p. Let us keep the notation of Proposition [propmain]. Then, $$\label{jet1} \log \\| j\_p^m s \\|\_r \le T\_{\overline{L},p}(r) + \log \\| s \\|\_{L^{\infty}(\partial D\_r)} \text{.}$$ If, moreover, pʹ is another point of Dr, and mʹ denotes the vanishing order of s at pʹ, then $$\label{jet2} \log \\| j\_p^m s \\|\_r \le T\_{\overline{L},p}(r) + \log \\| s \\|\_{L^{\infty}(\partial D\_r)} -m'g\_{D\_r,p}(p')\text{.}$$ Since πr, p is a probability measure over ∂Dr, we have ∫log∥s∥πr, p ≤ log∥s∥L∞(∂Dr). Thus, the estimate ([jet1]) (resp. ([jet2])) follows immediately from the non-negativity both of the function gDr, p and of the distribution δ÷ (s) − m[p] (resp. δ÷ (s) − m[p] − mʹ[pʹ]). A first application of moderate growth -------------------------------------- We shall need the following elementary inequality. [trick] Let A, B, and R be positive real numbers. Set $$\begin{aligned} r\defeq R \frac{B}{A+B}\text{.} \end{aligned}$$ If log(B/A) ≥ 2, then $$\begin{aligned} A \log \left(\frac{1}{1-\frac{r}{R}}\right) - B \log r \le 2A \log \left(\frac{B}{A}\right) - B \log R\text{.} \end{aligned}$$ By homogeneity, we may assume that A = R = 1, so that r = B/(1 + B) and our statement is equivalent to: $$\begin{aligned} \log (1+B) - B \log (B/(1+B)) \le 2 \log B \end{aligned}$$ when logB ≥ 2. By subtracting logB from both sides, we see that this is yet equivalent to: $$\begin{aligned} (1+B)\log (1 + 1/B) \le \log B \end{aligned}$$ when logB ≥ 2. Now, this last inequality follows trivially from the fact that log(1 + 1/B) ≤ 1/B. Note that r as above is the minimum of the real function $t \mapsto A \log \left(\frac{1}{1-\frac{t}{R}}\right) - B \log t$ defined on the open interval (0, R). [prop1mg] Let R > 0 be a real number and $\overline{L}=(L, \\| \ \\|)$ be a semipositive Hermitian line bundle on DR. If $\overline{L}$ has moderate growth, then there exist constants κ1, κ2 > 0 such that for every integer d ≥ 1 and every bounded global section $s \in \Gamma(D\_R,L^{\tensor d})\minus \{0\}$, if we denote $m \defeq \ord\_0s$, then $$\begin{aligned} \log \\|j\_0^ms\\|\_R \le \kappa\_1 d + \kappa\_2 d\log^+ m + \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.} \end{aligned}$$ Since $\overline{L}$ has moderate growth, there exist real numbers a, b > 0 such that $$\begin{aligned} T\_{\overline{L}}(r) \le a + b \log\left( \frac{1}{1-\frac{r}{R}}\right)\end{aligned}$$ for any r ∈ (0, R). We may assume that b > 1. By the jet estimate ([jet1]), for any r ∈ (0, R), we have $$\begin{aligned} \log \\|j\_0^ms\\|\_r\le d\cdot T\_{\overline{L}}(r) + \log \\|s\\|\_{L^{\infty}(\partial D\_r)} \le d\cdot T\_{\overline{L}}(r) + \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.}\end{aligned}$$ As $$\begin{aligned} \\|j\_0^ms\\|\_r = \\|j\_0^ms\\|\_R\left(\frac{r}{R} \right)^m\text{,}\end{aligned}$$ we obtain $$\begin{aligned} \label{bound} \log \\|j\_0^ms\\|\_R \le ad + bd\log \left(\frac{1}{1-\frac{r}{R}} \right) - m \log r + m\log R + \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.}\end{aligned}$$ The result being trivial for m = 0, we may assume that m > 0. We now consider two cases. If log(m/bd) < 2, then we may take $r \defeq R/2$ in ([bound]) to obtain $$\begin{aligned} \log \\|j\_0^ms\\|\_R \le (a + (1+e^2)b\log 2)d + \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.}\end{aligned}$$ If log(m/bd) ≥ 2, we apply Lemma [trick] for A = bd and B = m: $$\begin{aligned} \log \\|j\_0^ms\\|\_R \le ad + 2bd\log \left(\frac{m}{bd} \right) + \log \\|s\\|\_{L^{\infty}(D\_R)}\le ad + 2bd\log m + \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.}\end{aligned}$$ Bounding jets via Taylor coefficients at another point ------------------------------------------------------ In this paragraph, we compare Taylor coefficients at different points. We start with a general result, and next we explain how moderate growth improves the estimate. [proptaylor] Let R > 0 be a real number, $\overline{L} = (L,\\| \ \\|)$ be a semipositive Hermitian line bundle on DR, and $q \in D\_R\minus \{0\}$. Fix a real number R0 satisfying ∣q∣ < R0 < R, and a global holomorphic section s0 ∈ Γ(DR, L) such that s0(q) ≠ 0. Then there exists a real number κ > 1 such that, for every integer J ≥ 1, every integer d ≥ 1, and every global section $s \in \Gamma(D\_R,L^{\tensor d})\minus\{0\}$, if f denotes the germ of holomorphic function at q such that $s=fs^{\tensor d}\_0$ in a neighborhood of q, and if $m \defeq \ord\_0s$, then $$\begin{aligned} \log \\|j\_0^ms\\|\_R \le \log \left(\left(\frac{R\_0}{\|q\|} \right)^{-J}\left(\frac{R}{R\_0}\right)^m\\|s\\|\_{L^{\infty}(\partial D\_{R\_0})} + \kappa^{d+m+J}\max\_{0\le j < J}\frac{\|f^{(j)}(q)\|}{j!} \right) + d\cdot T\_{\overline{L}}(R\_0)\end{aligned}$$ Let us first remark that if such a constant κ > 0 exists for s0 ∈ Γ(DR, L) trivializing L at q, then an analogous constant κ̄ > 0 will exist for any other trivialization s̄0 of L in a neighborhood of q — we do not require s̄0 to be a global section. Indeed, if we write $s = \bar{f}\bar{s}^{\tensor d}\_0$ and s̄0 = us0 in a neighborhood of q, then f = f̄ud and $$\begin{aligned} \max\_{0\le j<J}\frac{\|f^{(j)}(q)\|}{j!}\le \max\_{0\le j < J}\sum\_{k+ l=j}\frac{\|\bar{f}^{(k)}(q)\|}{k!}\frac{\|(u^d)^{(l)}(q)\|}{l!}\le \left(J \max\_{0\le j < J}\frac{\|(u^d)^{(j)}(q)\|}{j!}\right) \max\_{0\le j < J}\frac{\|\bar{f}^{(j)}(q)\|}{j!}\text{.} \end{aligned}$$ We conclude by the Cauchy inequalities, which ensure that $\max\_{0\le j < J}\frac{\|(u^{d})^{(j)}(q)\|}{j!}$ grows at most exponentially in d + J. By the above remark, up to replacing s0 by $z^{-\ord\_0(s\_0)}s\_0$, we can assume that s0 trivializes L both at q and at 0. Let $a\_j \in \CC$ be defined by the expansion $$\begin{aligned} f(z) = z^m \sum\_{j=0}^{\infty}a\_j(z-q)^j \end{aligned}$$ in a neighborhood of q, and set $$\begin{aligned} g(z)\defeq z^m\sum\_{j=0}^{J-1}a\_j(z-q)^j\text{.} \end{aligned}$$ Note that g extends uniquely to a holomorphic function on DR. Let $s\_1,s\_2 \in \Gamma(D\_R,L^{\tensor d})$ be given by $s\_1 \defeq gs^{\tensor d}\_0$ and $s\_2\defeq s-s\_1$. Observe that both s1 and s2 have vanishing order at least m at 0. Next, we estimate ∥j0msi∥R0, i = 1, 2; for this, we shall first assume that each j0msi ≠ 0. By the jet estimate ([jet1]) for p = 0, we have $$\begin{aligned} \log \\|j\_0^ms\_1\\|\_{R\_0} \le d\cdot T\_{\overline{L}}(R\_0) + \log \\|s\_1\\|\_{L^{\infty}(\partial D\_{R\_0})}\text{.} \end{aligned}$$ Since $\ord\_qs\_2\ge J$, by the jet estimate ([jet2]) for p = 0 and pʹ = q, we have $$\begin{aligned} \log \\|j\_0^ms\_2\\|\_{R\_0} \le d\cdot T\_{\overline{L}}(R\_0) + \log \\|s\_2\\|\_{L^{\infty}(\partial D\_{R\_0})} -\log \left(\frac{R\_0}{\|q\|} \right)J\text{.} \end{aligned}$$ Thus $$\begin{aligned} \\|j\_0^ms\\|\_R &= \\|j\_0^ms\\|\_{R\_0}\left(\frac{R}{R\_0}\right)^m \le (\\|j\_0^ms\_1\\|\_{R\_0} + \\|j\_0^ms\_2\\|\_{R\_0})\left(\frac{R}{R\_0}\right)^m\\ &\le \left (\left(\frac{R}{R\_0}\right)^m\\|s\_1\\|\_{L^{\infty}(\partial D\_{R\_0})} + \left(\frac{R\_0}{\|q\|} \right)^{-J}\left(\frac{R}{R\_0}\right)^m\\|s\_2\\|\_{L^{\infty}(\partial D\_{R\_0})}\right)\exp(d\cdot T\_{\overline{L}}(R\_0))\text{.} \end{aligned}$$ Using that ∥s2∥L∞(∂DR0) ≤ ∥s1∥L∞(∂DR0) + ∥s∥L∞(∂DR0), we get $$\begin{aligned} \\|j\_0^ms\\|\_R \le \left(\left(\frac{R\_0}{\|q\|} \right)^{-J}\left(\frac{R}{R\_0}\right)^m\\|s\\|\_{L^{\infty}(\partial D\_{R\_0})} + \left(1 + \left(\frac{R\_0}{\|q\|} \right)^{-J}\right)\left(\frac{R}{R\_0}\right)^m\\|s\_1\\|\_{L^{\infty}(\partial D\_{R\_0})}\right)\exp(d\cdot T\_{\overline{L}}(R\_0))\text{.} \end{aligned}$$ It should be clear at this point that the same estimate holds if j0ms1 = 0 or j0ms2 = 0. We now estimate ∥s1∥L∞(∂DR0). For any z ∈ ∂DR0, we have $$\begin{aligned} \|g(z)\| &= R\_0^m \left\|\sum\_{j=0}^{J-1}a\_j(z-q)^j \right\|\le R\_0^m\left(\sum\_{j=0}^{J-1}(2R\_0)^j\right)\max\_{0\le j < J}\|a\_j\| \le JR\_0^m\max\{1,(2R\_0)^J\}\max\_{0\le j < J}\|a\_j\|\text{,} \end{aligned}$$ so that $$\begin{aligned} \\|s\_1\\|\_{L^{\infty}(\partial D\_{R\_0})} = \sup\_{z \in \partial D\_{R\_0}}\|g(z)\|\\|s\_0(z)\\|^d\le \\|s\_0\\|^d\_{L^{\infty}(\partial D\_{R\_0})}JR\_0^m\max\{1,(2R\_0)^J\}\max\_{0\le j < J}\|a\_j\|. \end{aligned}$$ To finish, we must bound the coefficients aj. By definition, for any $j\in \NN$, $$\begin{aligned} a\_j = \frac{1}{j!}\left.\frac{d^j}{dz^j}\right\|\_{z=q}\left(\frac{f(z)}{z^m}\right) = \sum\_{k=0}^j\left(\frac{(-1)^k}{q^{m+k}}\binom{k+m-1}{k}\frac{f^{(j-k)}(q)}{(j-k)!} \right)\text{.} \end{aligned}$$ If j < J, then, for any 0 ≤ k ≤ j, we have the (crude but sufficient) estimate $$\begin{aligned} \binom{k+m-1}{k} < \binom{J+m -1}{J} < 2^{m+J-1} < 2^{m+J}\text{,} \end{aligned}$$ so that $$\begin{aligned} \|a\_j\| \le \left(\sum\_{k=0}^j\frac{1}{\|q\|^{m+k}}\right)2^{m+J}\max\_{0\le k \le j}\frac{\|f^{(k)}(q)\|}{k!}\text{.} \end{aligned}$$ Thus, $$\begin{aligned} \max\_{0\le j < J}\|a\_j\| \le J (2\max\{1,\|q\|^{-1}\})^{m+J}\max\_{0\le j <J}\frac{\|f^{(j)}(q)\|}{j!}\text{.} \end{aligned}$$ [boundcompact] Let R > 0 be a real number, $\overline{L} = (L,\\| \ \\|)$ be a semipositive Hermitian line bundle over DR, and let K ⊂ DR be a compact subset. If $\overline{L}$ has moderate growth, then there exist real numbers κ0, κ1 > 1 and an integer d0 ≥ 1 such that, for any integer d ≥ d0 and any bounded section $s \in \Gamma(D\_R,L^{\tensor d})\minus \{0\}$ for which $m \defeq \ord\_0s$ satisfies m ≥ κ0d, we have $$\begin{aligned} \\|s(z)\\| \le m^{\kappa\_1d}\left(\frac{\|z\|}{R} \right)^m\\|s\\|\_{L^{\infty}(D\_R)} \end{aligned}$$ for every $z \in K\minus \{0\}$. Fix any R1 ∈ (0, R) such that K ⊂ DR1. By Proposition [lemmacompact], there exist real numbers a, b > 0 such that $$\begin{aligned} T\_{\overline{L}, z}(r) \le a + b \log \frac{1}{1-\frac{r}{R}} \end{aligned}$$ for any z ∈ K and every r ∈ [R1, R). Let $s \in \Gamma(D\_R,L^{\tensor d})\minus\{0\}$ be a bounded section, and $z \in K\minus \{0\}$. We may assume that $\ord\_zs = 0$. By the jet estimate ([jet2]) for p = z and pʹ = 0, we have, for every r ∈ [R1, R), $$\begin{aligned} \log \\|s(z)\\| &\le d\cdot T\_{\overline{L},z}(r) + \log \\|s\\|\_{L^{\infty}(\partial D\_r)} -m\log \frac{r}{\|z\|}\\ & \le ad + bd \log \frac{1}{1-\frac{r}{R}} - m \log r + m \log \|z\| + \log \\|s\\|\_{L^{\infty}(D\_{R})}\text{.} \end{aligned}$$ Assume that m ≥ e2bd (i.e. log(m/bd) ≥ 2). It follows from Lemma [trick] for A = bd and B = m that, if $$\begin{aligned} r\defeq R \frac{m}{bd+m}\text{,}\end{aligned}$$ then $$\begin{aligned} bd \log \frac{1}{1-\frac{r}{R}} -m\log r \le 2bd\log \frac{m}{bd} - m\log R\text{.}\end{aligned}$$ If we also require that $m\ge b\frac{R\_1}{R-R\_1}d$, then r ≥ R1, so that $$\begin{aligned} \log \\|s(z)\\| \le 2bd\log m + (ad -2bd\log bd) + m\log \frac{\|z\|}{R}+ \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.} \end{aligned}$$ Now, for every integer $d\ge e^{\frac{a}{b}}/b$, we have ad − 2bdlogbd ≤ 0, and we get $$\begin{aligned} \log \\|s(z)\\| \le 2bd\log m +m\log \frac{\|z\|}{R} + \log \\|s\\|\_{L^{\infty}(D\_R)}\text{.}\end{aligned}$$ We may thus take $\kappa\_0 \defeq b \max\left\{e^2, \frac{R\_1}{R-R\_1}\right\}$, $\kappa\_1 \defeq 2b$, and $d\_0 \defeq \lceil e^{\frac{a}{b}}/b\rceil$. The following result is a combination of Proposition [proptaylor] together with the existence of a non-zero global section of L (see the remark following the statement; actually, L is holomorphically trivial on DR) and Proposition [boundcompact]. [corotaylor] Let R > 0 be a real number, $\overline{L} = (L,\\| \ \\|)$ be a semipositive Hermitian line bundle on DR, $q \in D\_R\minus \{0\}$, and s0 be a holomorphic trivialization of L in a neighborhood of q. Assume moreover that $\overline{L}$ has moderate growth. Then there exist real numbers κi > 1, i = 0, …, 4, and an integer d0 ≥ 1, such that, for any integer J ≥ 1, any integer d ≥ d0, and every bounded section $s \in \Gamma(D\_R,L^{\tensor d})\minus \{0\}$ for which $m\defeq \ord\_0 s$ satisfies m ≥ κ0d, if f denotes the germ of holomorphic function at q such that $s = fs\_0^{\tensor d}$ in a neighborhood of q, we have $$\begin{aligned} \log \\|j\_0^ms\\|\_R \le \log \left(\frac{m^{\kappa\_1d}}{\kappa\_2^J}\\|s\\|\_{L^{\infty}(D\_R)} + \kappa\_3^{d+m+J} \max\_{0\le j < J}\frac{\|f^{(j)}(q)\|}{j!}\right) + \kappa\_4d\text{.} \end{aligned}$$ Fix any real number R0 ∈ (∣q∣, R). We take $\kappa\_2 \defeq R\_0/\|q\|$, $\kappa\_3 \defeq \kappa$ given by Proposition [proptaylor], and $\kappa\_4 \defeq T\_{\overline{L}}(R\_0)$. Since $\overline{L}$ has moderate growth, we may apply Proposition [boundcompact] to the compact K = ∂DR0 to obtain real numbers κ0, κ1 > 0 and an integer d0 ≥ 1 such that $$\begin{aligned} \\|s\\|\_{L^{\infty}(\partial D\_{R\_0})} \le m^{\kappa\_1 d}\left(\frac{R\_0}{R} \right)^m \\|s\\|\_{L^{\infty}(D\_R)}\end{aligned}$$ for any integer d ≥ d0 and any bounded section $s \in \Gamma(D\_R,L^{\tensor d})\minus\{0\}$ such that $m\defeq \ord\_0s\ge \kappa\_0d$. We conclude by combining this bound with the estimate given by Proposition [proptaylor]. In practice, we shall be concerned with the following particular situation. [corotaylor2] Let R > 0 be a real number, $\overline{L} = (L,\\| \ \\|)$ be a semipositive Hermitian line bundle of moderate growth on DR, and $q \in D\_R\minus \{0\}$. Fix a holomorphic trivialization s0 of L in a neighborhood of q, real constants c0, c1, c2 > 0, with c0 < c1, and an integer n ≥ 2. For any real number C > 0, there exist real numbers γ0, γ1 > 0 such that, for any sufficiently large integer d, and any bounded section $s\in \Gamma(D\_R,L^{\tensor d})$ satisfying $$\begin{aligned} c\_0d^n \le m\defeq \ord\_0s \le c\_1d^n\text{, }\ \ \ \log \\|s\\|\_{L^{\infty}(D\_R)}\le c\_2d\log d\text{,} \end{aligned}$$ and $$\begin{aligned} \max\_{0\le j< \lceil\gamma\_0d\log d\rceil}\log \frac{\|f^{(j)}(q)\|}{j!} \le -\gamma\_1d^n\text{,} \end{aligned}$$ where $s = fs\_0^{\tensor d}$ on a neighborhood of q, we have $$\begin{aligned} \log \\|j\_0^ms\\|\_R \le -C d\log d \text{.} \end{aligned}$$ Let κi > 1, i = 0, …, 4, be the constants given by Corollary [corotaylor]. We claim that it suffices to take γ0 > (logκ2)− 1(nκ1 + c2 + C) and γ1 > c1logκ3. Indeed, let $s \in \Gamma(D\_R,L^{\tensor d})$ be as in the statement. Since n ≥ 2 and m ≥ c0dn, if d is sufficiently large, we have m ≥ κ0d, so that the conclusion of Corollary [corotaylor] for $J\defeq \lceil \gamma\_0d\log d \rceil$ applies: $$\begin{aligned} \label{ct1-1} \log \\|j\_0^ms\\|\_R \le \log \left(\frac{m^{\kappa\_1d}}{\kappa\_2^J}\\|s\\|\_{L^{\infty}(D\_R)} + \kappa\_3^{d+m+J} \max\_{0\le j < J}\frac{\|f^{(j)}(q)\|}{j!}\right) + \kappa\_4d\text{.} \end{aligned}$$ Since m ≤ c1dn, log∥s∥L∞(DR) ≤ c2dlogd, and $J \ge \gamma\_0\d\log d$, we obtain $$\begin{aligned} \log \left(\frac{m^{\kappa\_1d}}{\kappa\_2^J}\\|s\\|\_{L^{\infty}(D\_R)}\right)\le (n\kappa\_1+c\_2 - (\log\kappa\_2)\gamma\_0)d\log d + \kappa\_1(\log c\_1) d\text{.} \end{aligned}$$ Thus, by our choice of γ0, if d is sufficiently large, we get $$\begin{aligned} \label{ct1-2} \log \left(\frac{m^{\kappa\_1d}}{\kappa\_2^J}\\|s\\|\_{L^{\infty}(D\_R)}\right) \le -(C+\varepsilon\_1)d\log d\text{,} \end{aligned}$$ for some ɛ1 > 0. Since m ≤ c1dn, J ≤ γ0dlogd + 1, and $\max\_{0\le j< J}\log \frac{\|f^{(j)}(q)\|}{j!} \le -\gamma\_1d^n$, we have $$\begin{aligned} \log \left(\kappa\_3^{d+m+J} \max\_{0\le j < J}\frac{\|f^{(j)}(q)\|}{j!}\right) \le (c\_1\log \kappa\_3 - \gamma\_1)d^n + \gamma\_0(\log\kappa\_3)d\log d + (\log \kappa\_3)(d+1)\text{.} \end{aligned}$$ Thus, as n ≥ 2, and by our choice of γ1, if d is sufficiently large, we obtain $$\begin{aligned} \label{ct1-3} \log \left(\kappa\_3^{d+m+J} \max\_{0\le j < J}\frac{\|f^{(j)}(q)\|}{j!}\right) \le -\varepsilon\_2 d^n \end{aligned}$$ for some ɛ2 > 0. We conclude by applying ([ct1-2]) and ([ct1-3]) in ([ct1-1]), and by taking d to be sufficiently large. Analytic curves of moderate growth in quasi-projective varieties ================================================================ This section contains mostly well-known techniques and results in Nevanlinna Theory. These are nevertheless written in the literature in a form not suitable for our purposes. Although our proofs may vary, many of the theory concerning growth of entire analytic maps (“parabolic case”) easily translate into our hyperbolic situation; we refer the reader to the recent monograph for a thorough exposition of the general parabolic theory (in several variables). Analytic curves of moderate growth in compact complex manifolds --------------------------------------------------------------- Let R > 0 be a real number, M be a compact complex manifold, and φ : DR → M be an analytic map. Fix any Hermitian metric h on M, and let $\omega \defeq -\Im h$ be the positive (1, 1)-form associated to h; in other words, if $h = \sum\_{k,l=1}^n h\_{kl}dz\_k\tensor d\bar{z}\_l$ in a local chart (z1, …, zn) of M, then $\omega = \frac{i}{2}\sum\_{k,l=1}^n h\_{kl}dz\_k\wedge d\bar{z}\_l$. [defmodgrowthcurve] We say that φ : DR → M has moderate growth if the semipositive (1,1)-form φ\*ω on DR has moderate growth (see Definition [defmodgrowth]). This notion does not depend on the choice of the Hermitian metric h. Indeed, since M is compact, any two Hermitian metrics on M are “comparable”: if h0 is another Hermitian metric on M, then there exist real numbers α, β > 0 such that α∥ ∥h0 ≤ ∥ ∥h ≤ β∥ ∥h0. It follows from Remark [intparts] that Tφ\*ω(r) = ∫0r(∫Dtφ\*ω)dlogt can be thought of as a logarithmic integral of the areas of the disks φ(Dt) in M for 0 < t < r. We next consider a simple example of curves of moderate growth. [Bounded derivative][bd] Let φ : DR → M be an analytic map, and h be a Hermitian metric on M. Then we can write $$\begin{aligned} \varphi^\\omega = \\|\varphi'(z)\\|\_{R,h}^2 d\mu\_R\text{,}\end{aligned}$$ where dμR* is the Poincaré form defined in Example [poincareform], and ∥φʹ(z)∥R, h denotes the norm of the tangent map Dzφ : TzDR → Tφ(z)M with respect to the Poincaré metric on DR, and the Hermitian metric h on M. Since dμR has moderate growth, the analytic curve φ has moderate growth in M whenever the function z ↦ ∥φʹ(z)∥R, h is bounded on DR (e.g., φ extends continuously to $\overline{D\_R}\subset \CC$). Nevanlinna’s characteristic function ------------------------------------ Let M be a complex manifold, $\overline{L}=(L,\\| \ \\|)$ be a semipositive Hermitian line bundle on M, and $s\_0 \in \Gamma(M,L)\minus\{0\}$ be a non-zero global section. Let R > 0 be a real number and φ : DR → M be an analytic map whose image is not contained in the support of ÷ (s0). We define, for every r ∈ (0, R), $$\begin{aligned} m\_{\varphi,\overline{L},s\_0}(r) \defeq \frac{1}{2\pi}\int\_0^{2\pi} \log \frac{1}{\\|s\_0(\varphi(re^{i\theta}))\\|}d\theta\end{aligned}$$ and $$\begin{aligned} N\_{\varphi,\overline{L},s\_0}(r) \defeq (\ord\_0\varphi^\s\_0)\log r + \sum\_{0<\|z\|< r} (\ord\_z\varphi^\s\_0) \log \frac{r}{\|z\|}\text{.}\end{aligned}$$ Then we can form the Nevanlinna characteristic function on the interval (0, R) $$\begin{aligned} T\_{\varphi,\overline{L},s\_0} \defeq m\_{\varphi,\overline{L},s\_0} + N\_{\varphi,\overline{L},s\_0}\text{.}\end{aligned}$$ For the next proposition, we introduce a temporary notation. If s is a global section of φ\*L and $m=\ord\_0s$, we denote by ℓ(s) the unique element of the fiber of φ\*L at 0 ∈ DR such that $j\_0^ms = \ell(s)\tensor dz^{\tensor m}$ (the “leading coefficient” of s). The following classical identity (cf. Theorem 2.3.31) is an immediate corollary of Proposition [propmain] applied to the section s = φ\*s0 and the point p = 0. [Nevanlinna’s First Fundamental Theorem][fft] For every r ∈ (0, R), we have $$\begin{aligned} T\_{\varphi^\\overline{L}}(r) = T\_{\varphi,\overline{L},s\_0}(r) + \log \\|\ell(\varphi^\s\_0)\\|\text{.}\end{aligned}$$ $\blacksquare$ As an application we show that, when R = 1, polynomial growth of Taylor coefficients implies moderate growth. [polgrowth] Let $\varphi=(\varphi\_1,\ldots,\varphi\_n): D\to \CC^n$ be an analytic map with coordinates φi(z) = ∑j = 0∞aijzj. Assume that there exist a real number C > 1 and an integer d ≥ 1 such that $$\begin{aligned} \|a\_{ij}\|\le C j^d \end{aligned}$$ for every 1 ≤ i ≤ n and j ≥ 0. Then, when identifying $\CC^n$ with the open affine subset $U\_0 = \{(p\_0:\cdots:p\_n) \in \PP^n(\CC) \mid p\_0\neq 0\}$ of $\PP^n(\CC)$ via (z1, …, zn) ↦ (1 : z1 : ⋯ : zn), the analytic curve $\varphi: D \to \PP^n(\CC)$ has moderate growth. Indeed, let $\overline{\mathcal{O}(1)}$ denote the line bundle O(1) on $\PP^n(\CC)$ endowed with the Fubini-Study metric; that is, $$\begin{aligned} \\|X\_i(p)\\| = \frac{\|p\_i\|}{\sqrt{\|p\_0\|^2 + \cdots + \|p\_n\|^2}}\end{aligned}$$ for every 0 ≤ i ≤ n, and $p = (p\_0:\cdots:p\_n)\in \PP^n(\CC)$. Since $N\_{\varphi,\overline{\mathcal{O}(1)},X\_0}$ vanishes identically, by Proposition [fft], it is sufficient to prove that there exist a, b > 0 such that $$\begin{aligned} m\_{\varphi,\overline{\mathcal{O}(1)},X\_0}(r) \le a + b \log \frac{1}{1-r}\end{aligned}$$ for every r ∈ (0, 1). For any real numbers t1, …, tm ≥ 0, we have log+(∑i = 1mti) ≤ ∑i = 1mlog+ti + logm, so that $$\begin{aligned} \log \frac{1}{\\|X\_0(\varphi(re^{i\theta}))\\|} = \log \sqrt{1 + \sum\_{i=1}^n\|\varphi\_i(re^{i\theta})\|^2} \le \sum\_{i=1}^n\log^+\|\varphi\_i(re^{i\theta})\| + \log(\sqrt{1+n})\text{.}\end{aligned}$$ Since $$\begin{aligned} \|\varphi\_i(re^{i\theta})\| \le C\sum\_{j=0}^{\infty}j^dr^j \le Cd!\left(\frac{1}{1-r} \right)^{d+1}\text{,}\end{aligned}$$ we may take $a = \log (\sqrt{1+n}) + n\log(Cd!)$ and b = n(d + 1). The field of moderate functions on a disk ----------------------------------------- In this paragraph we study more closely the case $M=\PP^1(\CC)$. We refer to Chapters V-VII for a survey on the classical work on this subject. Let R > 0 be a real number and f be meromorphic function on DR, i.e., an analytic map $f:D\_R \to \PP^1(\CC)$ which is not constant equal to ∞ = (0 : 1). We say that f is a moderate function on DR if the analytic map $f:D\_R \to \PP^1(\CC)$ has moderated growth. If $\overline{\mathcal{O}(1)}$ denotes the line bundle O(1) on $\PP^1(\CC)$ endowed with the Fubini-Study metric (see Example [polgrowth]), we denote $$\begin{aligned} T\_{f}\defeq T\_{f^\\overline{\mathcal{O}(1)}}\text{.}\end{aligned}$$ By Proposition [fft], we have $$\begin{aligned} T\_{f} = m\_{f,\overline{\mathcal{O}(1)},X\_0} + N\_{f,\overline{\mathcal{O}(1)},X\_0} + O(1)\text{,}\end{aligned}$$ where O(1) denotes a constant. To lighten the notation, we shall write $m\_f = m\_{f,\overline{\mathcal{O}(1)},X\_0}$ (resp. $N\_f = N\_{f,\overline{\mathcal{O}(1)},X\_0}$). Let KD*R denote the field of meromorphic functions on DR. It is classical (and easy to prove) that characteristic functions are compatible with the algebraic structure of KDR in the following sense: for $f,g\in \mathcal{K}\_{D\_R}\minus\{0\}$ and $n \in \ZZ\minus\{0\}$, we have $$\begin{aligned} \label{algop} T\_{f+g} \le T\_f + T\_g + O(1)\text{, }\ \ T\_{fg} \le T\_f + T\_g + O(1)\text{, }\ \ T\_{f^{n}} = \|n\|T\_f +O(1)\end{aligned}$$ It follows from the above relations that the subset KDRm of KDR consisting of moderate meromorphic functions is a field. [cf. Lemma 2.5.15][algcomp] Let f, f1, …, fn be meromorphic functions on DR. If f is algebraic over the field $\CC(f\_1,\ldots,f\_n)\subset \mathcal{K}\_{D\_R}$, then there exist real numbers a, b > 0 such that $$\begin{aligned} T\_f \le a + b \sum\_{i=1}^nT\_{f\_i}\text{.} \end{aligned}$$ Let d be the degree of f over $\CC(f\_1,\ldots,f\_n)$. If d = 0, then the result follows immediately from formulas ([algop]). Assume that d ≥ 1, and let $P = X^d - g\_{d-1}X^{d-1} - \cdots - g\_0 \in \CC(f\_1,\ldots,f\_n)[X]$ be the minimal polynomial of f. Since each $g\_i \in \CC(f\_1,\ldots,f\_n)$, it suffices to prove that Tf ≤ ∑i = 0d − 1Tgi + O(1). By formulas ([algop]), we have $$\begin{aligned} T\_{f^{d}} = T\_{(g\_{d-1}f^{d-2} + \cdots + g\_1)f + g\_0} \le T\_{g\_{d-1}f^{d-2} + \cdots + g\_1} + T\_f + T\_{g\_0} + O(1)\text{.} \end{aligned}$$ By descending induction, we get $$\begin{aligned} T\_{f^d} \le (d-1)T\_f + \sum\_{i=0}^{d-1}T\_{g\_i} + O(1)\text{.} \end{aligned}$$ As Tfd = d ⋅ Tf + O(1), we obtain $$\begin{aligned} T\_f \le \sum\_{i=0}^{d-1}T\_{g\_i}+ O(1)\text{.} \end{aligned}$$ The field of moderate functions KDRm is algebraically closed in KDR. $\blacksquare$ In particular, since the inclusion $D\_R \to \PP^1(\CC)$ is easily seen to be an analytic map of moderate growth (see Example [bd]), the field KDRm contains the field of (univalued) algebraic meromorphic functions on DR. Birational invariance and moderate growth in quasi-projective varieties ----------------------------------------------------------------------- In this paragraph, we establish the birational invariance of moderate growth under a non-degeneracy hypothesis. Our arguments follow closely those of 2.5; we claim no originality here. In what follows, if f is a meromorphic function on DR, we denote the divisor of zeros (resp. poles) of f by ÷ 0(f) (resp. ÷ ∞(f)), so that ÷ (f) = ÷ 0(f) − ÷ ∞(f). [cf. Theorem 2.5.7][lemmacomp1] Let M be a compact complex manifold endowed with a semipositive Hermitian line bundle $\overline{L} = (L,\\|\ \\|)$. Fix global sections s0, s1 ∈ Γ(M, L), with s0 ≠ 0. Then, for any analytic map φ : DR → M whose image is not contained in the support of ÷ (s0), if we denote by f the unique meromorphic function on DR such that fφ\*s0 = φ\*s1, we have $$\begin{aligned} T\_{f} \le T\_{\varphi, \overline{L}, s\_0} + O(1)\text{.} \end{aligned}$$ Let H be the support of ÷ (s0). For $p\in M \minus H$, we have $$\begin{aligned} \log \sqrt{1 + \frac{\\|s\_1(p)\\|^2}{\\|s\_0(p)\\|^2}} = \log \frac{1}{\\|s\_0(p)\\|} + \log \sqrt{\\|s\_0(p)\\|^2 + \\|s\_1(p)\\|^2}\text{.} \end{aligned}$$ Since M is compact, the functions ∥si∥ on M are bounded, so that $$\begin{aligned} \log \sqrt{1 + \frac{\\|s\_1\\|^2}{\\|s\_0\\|^2}} \le \log \frac{1}{\\|s\_0\\|} + O(1)\end{aligned}$$ over $M \minus H$. In particular, we get $$\begin{aligned} m\_{f} \le m\_{\varphi, \overline{L},s\_0} +O(1)\text{.}\end{aligned}$$ Since ÷ ∞(f) ≤ ÷ (φ\*s0), the bound $$\begin{aligned} N\_{f} \le N\_{\varphi,\overline{L},s\_0}\end{aligned}$$ is trivial. For the next lemma, we endow the line bundle O(1) over $\PP^n(\CC)$ with the Fubini-Study metric as in Example [polgrowth]. Moreover, if E = ∑z ∈ DRnz[z] is a divisor in DR, we denote $i(z,E) \defeq n\_z$. [lemmacomp2] Let $\varphi: D\_R \to \PP^n(\CC)$ be an analytic map whose image is not contained in the support of ÷ (X0). For 1 ≤ j ≤ n, let us denote by fj the unique meromorphic function on DR such that fjφ\*X0 = φ\Xj. Then $$\begin{aligned} T\_{\varphi, \overline{\mathcal{O}(1)},X\_0} \le \sum\_{j=1}^n T\_{f\_j} + O(1)\text{.} \end{aligned}$$ We first prove that $m\_{\varphi,\overline{\mathcal{O}(1)},X\_0} \le \sum\_{j=1}^n m\_{f\_j}$. For any real numbers t1, …, t*n ≥ 0, we have $$\begin{aligned} \log \left(1+ \sum\_{j=1}^nt\_j \right) \le \sum\_{j=1}^n \log(1+t\_j)\text{.} \end{aligned}$$ Thus, for any r ∈ (0, R), $$\begin{aligned} m\_{\varphi,\overline{\mathcal{O}(1)},X\_0}(r) &= \frac{1}{2\pi} \int\_0^{2\pi}\log \sqrt{1 + \sum\_{j=1}^{n}\|f\_j(re^{i\theta})\|^2}d\theta \\ &\le \sum\_{j=1}^n \frac{1}{2\pi}\int\_0^{2\pi}\log \sqrt{1 + \|f\_j(re^{i\theta})\|^2}d\theta = \sum\_{j=1}^nm\_{f\_j}(r) \end{aligned}$$ Next, observe that to prove that $N\_{\varphi,\overline{\mathcal{O}(1)},X\_0}\le \sum\_{j=1}^n N\_{f\_j}$ it suffices to show that $$\begin{aligned} \div(\varphi^\X\_0) \le \sum\_{j=1}^n{\div}\_{\infty}(f\_j)\text{.}\end{aligned}$$ Since each ÷ ∞(f*j) is an effective divisor, it is sufficient to prove that, for every z ∈ DR, there exists 1 ≤ j ≤ n such that i(z, ÷ (φ\*X0)) ≤ i(z, ÷ ∞(fj)). Now, for any 1 ≤ j ≤ n, since fjφ\*X0 = φ\Xj, we may write $$\begin{aligned} {\div}\_{\infty}(f\_j) = \div (\varphi^\X\_0) + {\div}\_0(f\_j) - \div(\varphi	arxiv_0000454
Sub-millimeter galaxies as progenitors of compact quiescent galaxies ==================================================================== Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, UK Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9, 9000 Gent, Belgium National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903 ETH Zurich,Institute for Astronomy, Department of Physics, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland European Southern Observatory, Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago 19, Chile MPE, Postfach 1312, 85741 Garching, Germany Institute for Astronomy, 2680 Woodlawn Drive, University of Hawaii, Honolulu, HI 96822 The Henry A. Rowland Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA Institut dAstrophysique de Paris, UMR7095 CNRS, Universite Pierre et Marie Curie, 98 bis Boulevard Arago, 75014 Paris, France Department of Astronomy, Cornell University, 220 Space Sciences Building, Ithaca, NY 14853, USA Astronomy Department, California Institute of Technology, MC 249-17, 1200 East California Boulevard, Pasadena, CA 91125, USA Three billion years after the big bang (at redshift z = 2), half of the most massive galaxies were already old, quiescent systems with little to no residual star formation and extremely compact with stellar mass densities at least an order of magnitude larger than in low redshift ellipticals, their descendants. Little is known about how they formed, but their evolved, dense stellar populations suggest formation within intense, compact starbursts 1-2 Gyr earlier (at 3 < z < 6). Simulations show that gas-rich major mergers can give rise to such starbursts which produce dense remnants. Sub-millimeter selected galaxies (SMGs) are prime examples of intense, gas-rich, starbursts. With a new, representative spectroscopic sample of compact quiescent galaxies at z = 2 and a statistically well-understood sample of SMGs, we show that z = 3 − 6 SMGs are consistent with being the progenitors of z = 2 quiescent galaxies, matching their formation redshifts and their distributions of sizes, stellar masses and internal velocities. Assuming an evolutionary connection, their space densities also match if the mean duty cycle of SMG starbursts is 42− 29+ 40 Myr (consistent with independent estimates), which indicates that the bulk of stars in these massive galaxies were formed in a major, early surge of star-formation. These results suggests a coherent picture of the formation history of the most massive galaxies in the universe, from their initial burst of violent star-formation through their appearance as high stellar-density galaxy cores and to their ultimate fate as giant ellipticals. Introduction ============ One of the most remarkable discoveries in galaxy evolution studies in the past years is that up to half of the most massive galaxies ($\mathrm{log(M\_{\}/M\_{\odot})>11}$) at z\approx2 are old quiescent systems with extremely compact structure, corresponding to stellar densities orders of magnitudes higher than seen in local elliptical galaxies. Much effort has gone into confirming their extreme properties and investigating their evolutionary path to the local universe. Virial arguments and simulations indicate that the most important process is likely to be minor dry merging, but observations suggest that other processes are likely also important, e.g. the continuous addition of increasingly larger newly quenched galaxies to the quenched population with decreasing redshift. The formation path of these extreme systems is largely unknown. Simulations indicate that highly dissipational interactions on short timescales provide plausible mechanisms for creating compact stellar populations, either through major mergers, or dynamical instabilities fed by cold gas accretion. A possible scenario is major gas-rich mergers at high redshift, in which the gas is driven to the center, igniting a massive nuclear starburst, followed by an AGN/QSO phase that quenches the star formation, and leaves behind a compact remnant. This is consistent with local stellar archaeology studies which imply that massive ellipticals must have short formation timescales of less than 1 Gyr,. Several authors have pointed out that sub-millimeter galaxies (SMGs) may be examples of the above scenario, but see for a counter example. The SMG population is dominated by galaxies undergoing intense, dust enshrouded starbursts. A large fraction of SMGs with measured CO profiles, show double peaked profiles, evident of ongoing major mergers or rotation. The auto-correlation length of SMGs is similar to that of optically selected QSOs, suggesting that SMGs and QSOs live in similar mass haloes and that the ignition of a QSO could be the event that quenches the star formation in SMGs. This is consistent with observations suggesting that the hosts of the most luminous QSOs, i.e those likely associated with the formation of massive quiescent galaxies, are found to be primarily major mergers, a result which is cooborated by who find that luminous AGN in massive z* ∼ 2 galaxies must be triggered by external processes. Interestingly, also finds evidence for low luminosity AGN in the vast majority of massive quiescent galaxies at z ∼ 2, suggesting that AGN play an active role in the quenching their star-formation. The correlation length of SMGs is similar to that of z\sim2 galaxies with $\mathrm{M\_\>5 \times 10^{10}\, M\_{\odot}}$ (r\_0=7.66\pm0.78), while z\sim2 galaxies with $\mathrm{M\_\>10^{11}\, M\_{\odot}}$ cluster more strongly. Recent advances in near infrared (NIR) spectroscopy have made it possible for the first time to accurately constrain the age, dust content and past star formation history of the brightest z\sim2 quiescent galaxies through absorption line diagnostics and spectral fitting in the rest frame-optical. These galaxies have spectra typical of post starburst galaxies, with no detected emission lines, but with strong Balmer absorption lines, suggesting that they underwent major starbursts which were quenched 1-2 Gyr prior to the time of observation (i.e. at 3<z<6). Several of these galaxies show evidence of significant dust abundance (with A\_V’s up to ∼ 1 magnitude), and they are baryon dominated, as is the case for local post starburst galaxies. In combination with their extremely compact stellar populations these observations suggest that the majority of the stars in z\sim2 quiescent galaxies formed in intense, possibly dust enshrouded nuclear starbursts, a scenario very similar to what is observed in z\sim2 SMGs. Velocity dispersions of z\sim2 quiescent galaxies measured from the width of absorption lines are in the range 300 − 500 km\, s− 1, significantly higher than in local ellipticals of similar stellar mass, but comparable to the FWHM of molecular lines in 2<z<3 SMGs. The line emitting gas of SMGs, as traced by high-J CO lines, is found to be spatially very compact, with a mean size of $\mathrm{\left < R\_{e}\right >=2.0\pm0.3 kpc}$, comparable to the mean spatial extent $\mathrm{\left < R\_{e}\right >=1.96\pm0.8 kpc}$, of the stellar populations in the quiescent z\sim2 galaxies. We note however that studies of lower-J CO lines suggest that some SMGs may have more extended CO-disks. The median dynamical masses measured from CO(1-0) for z\sim2 SMGs, $\mathrm{\left < M\_{dyn}\right >=(2.3 \pm1.4)\times 10^{11}\, M\_{\odot}}$, is similar to that measured for z\sim2 quiescent galaxies $\mathrm{\left < M\_{dyn}\right>} =(2.5 \pm 1.3)\times 10^{11}\, \mathrm{M\_{\odot}}$. Despite the many similarities between SMGs and z\sim2 quiescent galaxies, a major obstacle in establishing an evolutionary link between the two galaxy types is their similar redshift distribution. While the quiescent nature and derived ages for z\sim2 quiescent galaxies suggest they formed at z\gtrsim3 the peak of the known SMG population was until recently found to be at z\sim2 with very few examples known at z\gtrsim 3, rendering an evolutionary link between the two populations unlikely. Recently, improved selection techniques have however uncovered a substantial tail stretching out to redshifts of z\sim6. In this paper we present evidence for a direct evolutionary link between the two extreme galaxy populations, by comparing the properties of two unique samples in the COSMOS field: (i) a spectroscopically confirmed, representative sample of compact z\sim2 quiescent galaxies with high resolution HST/WFC3 imaging, and (ii) a statistical sample of z\gtrsim 3 SMGs. In Section [section.samples] we introduce the samples, and in Section [section.results] we present our results. In particular, in Section [section.redshiftdistributions] we show that the distribution of formation redshifts for the z\sim2 quiescent galaxies is similar to the observed redshift distribution of z\gtrsim 3 SMGs, and in Section [section.numberdensities] we compare the co-moving number densities of the two populations. In Section [section.structure] we derive structural properties of the z\gtrsim 3 SMGs and in Section [section.masssize] show that their stellar mass-size relation is similar to those of z\sim2 quiescent galaxies. In Section [section.dutycycle] and [section.eddington] we show that the duty cycle of the z > 3 SMG starburst (derived assuming they are progenitors of z\sim2 quiescent galaxies), is consistent with independent estimates, and with the formation time scale derived for z ∼ 2 quiescent galaxies, assuming they formed in Eddington limited starbursts. In Section [section.discussion] we summarize and discuss the results. Throughout this paper we assume a standard flat universe with $\mathrm{\Omega\_{\lambda}=0.73}$, $\mathrm{\Omega\_{m}=0.27}$ and $\mathrm{H\_0=71 km s^{-1} Mpc^{-3}}$. All stellar masses are derived assuming a IMF. llccc AzTEC 1 & $4.64\tablenotemark{a}$ & < 2.6 & unresolved & 1.3 − 2.7 AzTEC 3 & $5.299\tablenotemark{a}$ & < 2.4& unresolved & < 3 ± 2 AzTEC 4 & 4.93− 1.11+ 0.43 & < 2.5 & unresolved &- AzTEC 5 & $3.971\tablenotemark{a}$ & 0.5 ± 0.4& HST/WFC3 &- AzTEC 8 & $3.179\tablenotemark{a}$ & < 3.0 & unresolved &- AzTEC 10 & 2.79− 1.29+ 1.86 & 0.7 ± 0.1& - & - AzTEC 11-S & $>2.58\tablenotemark{b}$ & - & not detected & - AzTEC 13 & $> 3.59\tablenotemark{b}$ & - & not detected&- AzTEC 14-E & $> 3.03\tablenotemark{b}$ & - &not detected&- AzTEC 15 & 3.17− 0.37+ 0.29 & 5.0 ± 0.8 & very faint&- J1000+0234 & $4.542\tablenotemark{a}$ & 3.7 ± 0.2 & - &- Vd-17871 & $4.622\tablenotemark{a}$ & 1.3 ± 1.4 & - &- GISMO-AK03 & $4.757\tablenotemark{a}$ & 1.6 ± 0.6 & HST/WFC3 & - [table.sample] Samples ======= lclccccrccc AzTEC 1 &10.9− 0.1+ 0.1 & 0.47% & 0.070 & 25.0 & 9.1 ± 0.1 & 13.36 ± 0.09 &2291 ± 528&11.7 ± 0.1&secure & AzTEC 3 & 11.2− 0.1+ 0.1 & 0.47% & 0.290 & 25.0 & 9.3 ± 0.1 & 13.37 ± 0.04 &2344 ± 226&11.3 ± 0.1&secure & 10.7 AzTEC 4 &11.2− 0.1+ 0.1& 4.58% & 0.080 & 25.0 & 9.6 ± 0.2 & 13.25 ± 0.15& 1778 ± 733&11.6 ± 0.2 &tentative & AzTEC 5 & 10.9− 0.5+ 0.5 & 0.47% & 0.190 & 25.0 & 9.4 ± 0.1 &13.43 ± 0.02&2692 ± 127 &11.4 ± 0.1&secure & AzTEC 8 & 11.5− 0.1+ 0.1& 0.47% & 0.090 & 25.0 & 9.7 ± 0.1 & 13.45 ± 0.01&2818 ± 66 &11.7 ± 0.1 &secure& AzTEC 10 & 10.5− 0.1+ 0.1& 2.50% & 0.060 & 5.00 & 9.6 ± 0.1 & 12.58 ± 0.10&380 ± 98 &11.8 ± 0.1 &secure & AzTEC 11-S & - & 0.47% & 0.040 & 25.0 & 9.6 ± 0.1&13.30 ± 0.01&1995 ± 46 &- &secure & AzTEC 13 & - & 4.58% & 0.160 & 2.00 & 9.9 ± 0.3 & 12.70 ± 0.20&501 ± 293 &- &upper limits& AzTEC 14-E & - & 0.47% & 0.290 & 0.70 & 9.8 ± 0.2 & 12.48 ± 0.18&302 ± 155 &- &upper limits& AzTEC 15 & 11.2− 0.1+ 0.1& 3.19% & 0.010 & 20.0 & 9.3 ± 0.1&12.73 ± 0.08& 537 ± 108&11.4 ± 0.1 &secure& J1000+0234 & 10.7− 0.1+ 0.1& 1.77% & 0.150 & 25.0 & 9.3 ± 0.1 &13.17 ± 0.09&575 ± 275&11.8 ± 0.4 &tentative& 10.4 Vd-17871 &10.9− 0.1+ 0.1& 4.58% & 0.250 & 25.0 & 9.1 ± 0.1 &13.09 ± 0.06&1230 ± 182 &11.2 ± 0.2&secure & GISMO-AK03 &12.1− 0.1+ 0.1& 4.58% & 0.290 & 3.00 & 9.5 ± 0.2 &12.66 ± 0.19&457 ± 250 &11.5 ± 0.2 &secure &. “Secure”: The source is relatively isolated and detected at S/N>3. “Tentative”: The source is detected at S/N>3, but the flux density estimates may be affected by bright closeby objects. “Upper limit”: The source is not detected at S/N>3. [table.mdust] Sample of $z\gtrsim 3$ SMGs --------------------------- Based on dedicated follow-up studies with sub-mm-interferometers (PdBI, SMA, CARMA) and optical/mm-spectroscopy (with Keck/DEIMOS, EVLA, PdBI) towards 1.1 mm and $870~\micron$ selected sources in the COSMOS field presented the redshift distribution of SMGs. This sample shows a tail of z\gtrsim 3 SMGs, corresponding to a significantly larger number density at these high redshifts than found in previous surveys. A possible reason for the difference is that previous surveys did not have (sub)-mm followup interferometry, and therefore may be subject to identification biases. E.g. show that many of the galaxies in the sample break up into multiple sources when studied at high resolution, which enevitably lead to mis-identifications for some of the sources. Here we use the sample to estimate the comoving number density and other properties of z\gtrsim 3 SMGs. Our starting point is a 1.1 mm-selected sample, drawn from the AzTEC/JCMT 0.15 square degree survey of the COSMOS field (Scott et al. 2008), and observed with the SMA at 890 μm and \sim2 \arcsec angular resolution to unambiguously associate multi-wavelength counterparts. The 17 SMGs identified by the SMA follow-up form a statistical sample as they are drawn from a signal-to-noise limited ($\mathrm{S/N\_{1.1mm}}>4.5$), and flux-limited ($F\_\mathrm{1.1mm}\gtrsim4.2$ mJy), 1.1 mm-selected sample, drawn from a contiguous area of 0.15 square degrees. We include one more SMG in this sample, J1000+0234 (F1.1mm = 4.8 ± 1.5 mJy, $\mathrm{S/N\_{1.1mm}}\sim3$), which is confirmed to be at z=4.542. Nine out of these 18 interferometrically detected galaxies have spectroscopic redshifts (four are confirmed to be at z\gtrsim 3; ; Karim et al., in prep), while for the remainder precise photometric redshifts ($\sigma\_\mathrm{{\Delta z / (1+z\_\mathrm{spec})}}=0.09$) were computed by. The z\gtrsim 3 SMGs from this sample are listed in Table [table.sample]. The top 11 objects constitute our statistical sample. We will use these in the following sections to estimate the redshift distribution and comoving number density of $z\gtrsim 3$ SMGs. The bottom two objects, are additional spectroscopically confirmed z\gtrsim 3 SMGs in the COSMOS field which we add to the sample for structural analysis only. The flux-limited sub-mm selection ensures a relatively homogenous sample of the most intensely starforming dust obscured galaxies at ${z\gtrsim3}$: Due to the negative k-correction, the sub-mm flux detection limit corresponds roughly to a cut in SFR over the considered redshift range. Note that while a fraction of single dish detected SMGs break up into multiple components when studied with interferometry at $\le 2\arcsec$ resolution, this is only the case for two of the galaxies studied here (AzTEC 11 and 14). In the present study we assume that the close individual components are related and count them as one in the number density calculations (thus assuming they will eventually merge into one galaxy). As the galaxies are not resolved in the MIR-mm photometry, derived properties (infrared luminosities, star formation rates, dust masses etc) are for the combined systems. Neither of the two galaxies are detected in the optical-NIR so the derived sizes and stellar masses for the sample are not affected. Far-infrared emission of the z\gtrsim 3 SMGs -------------------------------------------- In order to directly constrain the SFRs, dust and gas masses of the z\gtrsim 3 SMGs, we made use of the (sub)-mm (AzTEC, LABOCA, MAMBO, SMA, CARMA, PdBI) and FIR (Spitzer MIPS, Herschel PACS and SPIRE) observations of the COSMOS field. The Herschel data consist of deep PACS 100 and 160 μm observations, taken as part of the PACS Evolutionary Probe guaranteed time key programme, and SPIRE 250, 350 and 500 μm observations taken as part of the Herschel Multi-tiered Extragalactic Survey. PACS and SPIRE flux densities were measured using a PSF fitting analysis, guided by the position of sources detected in the deep COSMOS 24 μm observations from the Multiband Imaging Photometer onboard the Spitzer Space Observatory. We cross-matched our $z\gtrsim 3$ SMGs sample with this MIPS-PACS-SPIRE catalogue using a matching radius of 2. Results of these matches were all visually checked. For $z\gtrsim3$ SMGs not included in the MIPS-PACS-SPIRE catalogue because of a lack of MIPS counterpart, we compute their PACS and SPIRE flux densities using a PSF fitting analysis guided by their positions. Further details of the FIR photometry are presented in Smolčić et al. (in prep.). Among the 13 z\gtrsim 3 SMGs, 9 have secure mid/far-infrared detections, 2 have tentative mid/far-infrared detections and 2 are undetected at infrared wavelengths. From the FIR–mm SED of the z\gtrsim 3 SMGs, we infer their infrared luminosities and dust masses using the dust model of as described in detail in. The infrared luminosity ($L\_{{\rm IR}}$) is derived by integrating the best fitting normalized SED templates from the DL07 library from rest-frame 8 to 1000 μm. From these we can accurately estimate the star-formation activity of the z\gtrsim 3 SMGs, using the standard $L\_{\rm IR}$-to-SFR conversion of, assuming a Chabrier IMF : $$\label{equation.sfr} {\rm SFR\,[M\_{\odot}\,yr^{-1}]\,=\,10^{-10}\,}L\_{{\rm IR}}\,{\rm [L\_{\odot}]}.$$ Finally we estimate the gas masses of the sample through $\mathrm{log(M\_{gas}/M\_{dust})=-0.85\Z+9.4}$, where $\mathrm{Z=2.18\times log(M\_{\star})}$$\mathrm{-0.0896\log(M\_{\star})^2-4.51}$ [1](#fn1). This method has been used successfully in the local Universe as well as at high redshift. Assumptions and limitations of this method in the case of high redshift galaxies are extensively discussed in. Results of the FIR SED fits and derived quantities are summarized in Table [table.mdust], and used in the following analysis to establish an evolutionary link between ${z\gtrsim 3}$ SMGs and quiescent z ∼ 2 galaxies. The derived gas masses are comparable to or larger than the derived stellar masses: $\mathrm{\left< f\_g \right > = \left< M\_{gas}/(M\_{\star}+M\_{gas})\right>=0.71\pm0.03}$, in agreement with the high gas fractions found in previous studies of high redshift SMGs. We caution however, that gas masses estimated from FIR SED fits are relatively uncertain (potentially up to a factor of 5 -10). E.g. in Table [table.mdust] we list gas masses for two objects in our sample which have independent estimates derived from CO line emission. These are significantly different from our SED estimates. The main factors contributing to the uncertainties in the SED estimates are that the sub-mm measurements dont trace cold gas very well, in which the (sub)mm/CO flux ratio is much lower than in the starburst nucleus, but where a lot of the gas mass may reside. The others are the metallicity correction (which has a large scatter) and the assumption about the gas-to-dust ratio. The main factors contributing to the uncertainty of the CO measurements is the assumed αCO which can be uncertain by a factor > 2, and the excitation corrections, which can be uncertain by a factor of ∼ 2 − 4. Stellar Mass Estimates for the $z\gtrsim 3$ SMGs ------------------------------------------------ We estimate stellar masses of the $z\gtrsim 3$ SMGs from their UV-MIR ($8\micron$) broad band photometry as described in Smolčić et al. (in prep.). Briefly, stellar masses were derived by fitting the observed broadband UV-MIR spectral energy distributions with the MAGPHYS code. The stellar component in the MAGPHYS models is based on stellar population synthesis models assuming various star formation histories (exponentially-declining SFHs (with random timescales) + superimposed stochastic bursts) and a IMF. The stellar masses for the SMGs, and their formal uncertainties drawn from the probability distribution function (generated from the χ2 fit values by MAGPHYS) are given in Tab. [table.mdust]. We note however, that stellar masses for SMGs are strongly dependent on the assumed star formation histories, and may lead to systematic discrepancies of ± 0.5 dex given different assumptions and stellar population synthesis models, and whether or not emission lines are included in the templates. For example, using the double SFHs implemented in GRASIL, we find systematically higher stellar masses, consistent with the results from. On the other hand, dynamical mass considerations based on CO line observations for two objects in our sample suggest lower stellar masses than inferred by MAGPHYS. Hence, here we adopt the middle values, i.e. the stellar masses computed by MAGPHYS+BC03, noting that these may be subject to systematic uncertainties. Sample of z ∼ 2 Compact Quiescent Galaxies -------------------------------------------- It is well established from deep multi-waveband photometric surveys, that a substantial population of quiescent massive galaxies with extremely compact structure exists at z ∼ 2. Samples of spectroscopically confirmed, z ∼ 2 quiescent galaxies with accurate stellar population model fits and high angular resolution space based NIR imaging are much more sparse. As a high quality comparison set to the z > 3 SMGs we use the sample of. This sample consist of 16 spectroscopically confirmed massive quiescent galaxies, selected from the 3DHST survey in the COSMOS field by requiring strong 4000Å breaks in the grism observations. As shown in K13 this effectively selects a representative sample of massive (logM > 10.9) evolved quiescent galaxies around z ∼ 2. The high S/N grism spectra around the break in combination with multi-waveband photometry from the COSMOS survey allows for strong constraints on the stellar populations including stellar masses, dust contents, mean stellar ages, i.e the time elapsed since the last major episode of star formation, as well as formation redshifts (derived from the stellar ages). The sample is also covered by high resolution NIR imaging with HST/WFC3 from the CANDELS survey yielding accurate constraints on the rest-frame optical surface brightness profiles, and effective radii (r\_e). Results ======= Redshift Distributions ---------------------- From the spectroscopic redshifts and mean stellar ages available for the quiescent z ∼ 2 galaxy sample described in Section [section.quiescent] we can estimate the distribution of their formation redshifts. In Figure [figure.zform] this distribution is compared to the observed redshift distribution of the sample of ${z\gtrsim 3}$ SMGs described in Section [section.z3smg]. Due to the small number of galaxies in both samples, a one to one correspondence is not expected. However, we stress that the two distributions are similar, with a peak at z ∼ 3 and a tail towards higher redshifts. A two-sample Kolmogorov-Smirnov (K-S) test yields a statistic of 0.29 with a p-value of 54%, and is thus not inconsistent with the two redshift distributions being drawn from the same parent distribution. [figure.zform] Co-moving Number Densities -------------------------- The next step in establishing an evolutionary connection between ${z\gtrsim 3}$ SMGs and quiescent galaxies at z ∼ 2 is to compare their co-moving number densities. The comoving number density of massive quiescent galaxies as a function of redshift is well constrained from photometric redshift and stellar population model fits to deep multi-waveband photometry. studied the number density evolution of star forming and quiescent galaxies (separated using their rest-frame UVJ colors) in a sample complete at stellar masses $\mathrm{log(M/ M\_{\odot})>11}$ out to z ∼ 2.2. Here we estimate a comoving number density of $\mathrm{6.0 \pm 2.1 \times 10^{-5} Mpc^{-3}}$ for quiescent galaxies at z ∼ 2 with $\mathrm{log(M/ M\_{\odot})>11}$ as the mean of the densities measured at z = 1.9 and z = 2.1 by. The error includes a contribution of cosmic variance of 12%. To derive the surface number density of ${z\gtrsim 3}$ SMGs we take all SMGs from the 1.1 mm-selected COSMOS sample that could lie at z > 3 given their lower or upper 99% confidence levels of the photometric redshift. We then derive an average value of the surface density by taking the most probable photometric redshift (or spectroscopic redshift where available)[2](#fn2), and the lower[3](#fn3) and upper[4](#fn4) surface density values by taking the limiting redshifts corresponding to the 99% confidence intervals of the photometric redshifts. This yields a surface density of $z\gtrsim 3$, bright ($F\_\mathrm{1.1mm}\gtrsim4.2$ mJy) SMGs of 60 ± 10 deg− 2. Note that conservatively excluding from the analysis all 3 SMGs in the sample that are not significantly detected at other wavelengths (AzTEC11S, AzTEC13, AzTEC14E), and thus only have lower redshift limits, yields a surface density of 40 deg− 2. The derived surface density values for ${z\gtrsim 3}$ SMGs may be subject to systematic effects. The completeness of the AzTEC/JCMT COSMOS survey, shown in Fig. 8 in, is roughly 50%, 70%, and 90% at F1.1mm = 4.2, 5, and 6 mJy. Taking this into account in combination with the deboosted 1.1 mm fluxes of the SMGs yields that the derived surface densities could be roughly a factor of 1.5 higher than that reported above. On the other hand, the AzTEC/JCMT COSMOS field may be overdense, which would imply that the true ${z \gtrsim 3}$ SMG surface density averaged over larger area would be lower. Our best estimate of the co-moving number density of 3 < z < 6 SMGs is $\mathrm{2.1 \pm 0.4\times 10^{-6}~Mpc^{-3}}$, significantly lower than the space density of z ∼ 2 quiescent galaxies. This is expected as ${z\gtrsim 3}$ galaxies only enter the mm-selection criterion during their intense starburst phase. In Section [section.dutycycle] we use the observed difference in co-moving number densities to constrain the duty cycle of the SMG starbursts. [figure.gallery] Rest-frame UV-Optical Structure of z > 3 SMGs: Disks, Spheroids or Mergers? --------------------------------------------------------------------------- The high redshifts and large amounts of dust in the ${z\gtrsim 3}$ SMGs renders them extremely faint in the rest-frame UV and optical, despite their high stellar masses and star formation rates. This makes it challenging to constrain their structure. To achieve the least biased estimates of the distribution of stellar mass one would need to study the surface brightness distributions in the rest-frame optical/NIR, or as close to these wavebands as possible. Ideally, the observations would be done in the observed mid infrared, but at the low spatial resolution of current facilities (e.g. Spitzer) the galaxies remain unresolved. Until JWST becomes operational the best that can be achieved is to study the galaxies in the observed NIR. For most of the galaxies in the sample this wavelength range probes rest-frame wavelengths around the 4000 Å break, and thus should be a relatively good tracer of the stellar mass distribution. For two galaxies (AzTEC5 and GISMO-AK03) we use space based NIR imaging with HST/WFC3, which is available from the CANDELS survey. This is preferable to groundbased imaging given the higher resolution (FWHM$\sim0.2\arcsec$). For the remaining galaxies we use deep NIR imaging provided by the UltraVista survey (5σ AB depths range from 23.7 in the K-band to 24.6 in the Y-band, McCracken et al 2012). The resolution of these observations is lower (FWHM ∼ 0.8), but it has been demonstrated that relatively unbiased sizes (down to a fraction of the FWHMPSF) can be derived from such data when the S/N is high and the PSF is well known. To increase the S/N we stack the Y, J, H and K band images. Postage stamp images of the galaxies are shown in the top panel of Figure [figure.gallery]. NIR counterparts of 10 of the 13 sources are detected, 8 of which have relatively high signal to noise (the faintest ones have S/N ∼ 10). We fit 2D Sersic models to their surface brightness distributions with, using similarly stacked images of nearby stars as PSF models. We find the sersic n to be relatively poorly constrained from the data. Leaving it free in the fits in all cases results in low values n < 2, with a median value of \left<n\right>=0.6\pm0.1, but with relatively large errors. To limit the degrees of freedom in the fits we therefore fix it to n = 1. The reduced χ2 of these fits are in all cases similar to those with n free, and better than fits with n fixed to 4. The best fitting effective radius encompassing half the light of the model, are reported in Table [table.sample]. Half of the detected galaxies (5) have close companions. In these cases we model both components simultaneously and report the parameters for the main component (closest to the center of the mm-emission). Also listed in Table [table.sample] are restframe FIR sizes for two galaxies in our sample derived from interferometric sub-mm imaging observations. These agree with the sizes derived from the NIR data. The restframe FIR sizes directly measures the extend of the starforming regions which we hypothesize evolves into the compact stellar populations at z=2, so the agreement is encouraging. Our analysis shows, that apart from being very compact, the z > 3 SMGs are not isolated, smooth single component galaxies. All the detected galaxies show evidence of close companions or clumpy sub-structure (see Figure [figure.gallery]). From these observations alone, it is not possible to deduce whether this is due to chance projections, ongoing minor/major mergers, or perhaps multiple star forming regions in individual galaxies, as resolved photometry and spectroscopy is not available. We note however that the two galaxies with HST/WFC3 data appear to have well separated individual components of comparable brightness, favoring the merger interpretation. This is consistent with direct observational evidence for SMGs being major mergers, i.e. having multiple close components at the same redshift. Simulations suggest that the timescale for major mergers are typically 0.39 ± 0.30 Gyr. The cosmic time available between the observed epoch of the SMGs at z=3-6 and their proposed remnants at z=2 is 1-2 Gyr. If (some of) the SMGs are major mergers, there is thus sufficient time available for them to coalesce to a single quiescent remnant at z = 2. In the local universe most star forming galaxies are well fit by exponential disk profiles corresponding to n=1, while irregular galaxies and (pre-coalescence) mergers are often best fit by models with lower n-values (n < 1). At the S/N and resolution of the galaxies in the Ultravista data the confidence in derived Sersic parameters are limited. However, the persisting low values found for the whole z > 3 SMG sample, including the two galaxies with the higher resolution HST/WFC3 data, suggests that the galaxies are more consistent with disks or mergers than spheroids. A similar conclusion was found for a sample of 22 SMGs at 1 < z < 3 with HST/WFC3 data for which the majority were best fit by low n Sersic models, with a mean \left< n \right>=1.2 \pm 0.1. If z > 3 SMGs are progenitors of z ∼ 2 quiescent galaxies, then their evolution must include a transformation of their surface brightness profiles which increase their Sersic indices, as surface brightness profiles of quiescent galaxies at z ∼ 2 are more centrally concentrated, e.g. the sample of K13 has \left < n \right >=4.0 \pm0.1. We discuss a possible mechanism for this transformation in Section [section.discussion]. Mass-Size relation ------------------ Combining the derived stellar masses and effective radii of the 3 < z < 6 SMGs, in Figure [figure.masssize] we compare their stellar mass-size distribution to that of z = 2 quiescent galaxies and of massive early type galaxies in the local universe. Two of the ten NIR-detected SMGs are relatively extended with effective radii comparable to those in local galaxies of similar mass. Both of these (AzTEC 10 and AzTEC 15) appear from their NIR images to be ongoing mergers. The remaining 8 galaxies are extremely compact, with r\_e \lesssim 2.5~kpc. Four are unresolved in the Ultravista data. For these we adopt upper limits on their effective radii corresponding to $\mathrm{0.5\times FWHM\_{PSF}}$. The stellar mass-size distribution of the 3 < z < 6 SMGs is similar to that of z ∼ 2 quiescent galaxies. Both populations are smaller by an average factor of ∼ 3 than local galaxies of similar mass. From the derived quantities we can infer the mean internal stellar mass surface densities within the effective radius ($\mathrm{\Sigma=0.5M\_{\star}/\pi r\_{e}^2}$) of the z > 3 SMGs and z = 2 compact quiescent galaxies (cQGs) which we find to be similar $\mathrm{\left<log(\Sigma)\right >\_{SMGs} \sim9.9 \pm 0.1 M\_{\odot}{kpc}^{-2}}$, $\mathrm{\left<log(\Sigma)\right >\_{cQGs} \sim9.8 \pm 0.1 M\_{\odot}{kpc}^{-2}}$ more than an order of magnitude higher than in local early type galaxies of similar mass. This is consistent with a picture where the SMGs passively evolve into compact quiescent galaxies after their starbursts are quenched. [figure.masssize] Duty Cycle of SMG Starbursts ---------------------------- The observed space density of $z\gtrsim 3$ SMGs is a factor of \sim30 lower than the space density of z ∼ 2 quiescent galaxies (see Section [section.numberdensities]). However, the SMGs only enter the sub-mm selected ($F\_\mathrm{1.1mm}\gtrsim4.2$ mJy) sample during their intense starburst phase where they have very high star formation rates. The duration of this phase, i.e. the duty cycle $\mathrm{t\_{burst}}$, which ends when the supply of gas is depleted or the star formation is quenched, e.g. by feedback from supernovae or active galactic nuclei, has been estimated to be in the range 40-200 Myr, based on gas depletion timescales and clustering analysis. If we assume that all the ${z\gtrsim 3}$ SMGs evolve into z ∼ 2 quiescent galaxies, and that they only undergo one SMG phase we can estimate the average duty cycle of their starbursts from the observed comoving number densities of the two populations, as $$\mathrm{t\_{burst}=t\_{obs}\times (n\_{SMG,z\gtrsim 3}/n\_{q,z=2})},$$ where $\mathrm{t\_{obs}}$ is the cosmic epoch corresponding to the redshift interval 3 < z < 6 from which the ${z\gtrsim 3}$ SMGs are selected. Using the comoving number densities we can thus constrain the duty cycle of the SMGs to tburst = 42− 15+ 33Myr. This number however does not include possible systematics uncertainties on the number density of SMGs discussed in Section [section.numberdensities]. If we conservatively assume the two extreme cases where (i) the SMG sample is 100% complete, and the field is three times overdense, and (ii) the sample is a factor of 1.5 incomplete but not overdense, the derived timescale are in the range $\mathrm{14<t\_{SMG}<62\, Myr}$. The systematic uncertainties on the timescale is thus of the order of 24 Myr. Therefore our constraints on the average dutycycle in ${z\gtrsim 3}$ SMGs is $\mathrm{t\_{burst}=42\_{-29}^{+40}Myr}$, where the errors have been added in quadrature. This value is consistent with the independently estimated duty cycles based on gas depletion time scales, thus affirming the idea that ${z\gtrsim 3}$ SMGs are progenitors of z ∼ 2 quiescent galaxies. The derived timescale does not depend strongly on the z = 6 upper limit adopted for the SMG redshift distribution. Adopting limits of z=5.5 or z=7 instead, leads to timescales of 44 and 37 Myr, respectively. We note that the validity of the timescale calculation presented here, relies on the assumption of a direct evolutionary connection between the two populations, implying that all z = 2 quiescent galaxies were once z > 3 SMGs, and all z > 3 SMGs evolve into z = 2 quiescent galaxies. Star formation Rate and Timescale of z=2 Quiescent Galaxies During their Formation ---------------------------------------------------------------------------------- We can infer a lower limit on the star formation rate of the z=2 quiescent galaxies during their formation by assuming that they started forming stars at z = 10 and did so at a constant rate until their inferred formation redshifts. The minimum average SFR needed to acquire their observed stellar masses at z=2 calculated in this way is $\langle\mbox{SFR}\_{\rm min}\rangle = 115 \pm 5\, M\_\odot\,\mbox{yr}^{-1}$. This is a factor of > 3 larger than the observed average SFR in star-forming Lyman break galaxies (LBGs) at z > 3. Furthermore, the space density of z ∼ 2 quiescent galaxies with logM\/M⊙ > 11, is 5, 10 and > 100 times larger than that of similar mass LBGs at z=4,5 and 6 respectively. Their progenitors must therefore have had much larger SFRs and are missing from LBG samples. This suggests that they must be dust obscured starburst galaxies. Based on the observed line widths and compact spatial extent of molecular line emitting regions, SMGs are often argued to be maximum starbursts, i.e to be forming stars at a rate close to the Eddington limit. Assuming a spherical symmetric geometry, an isothermal sphere density structure, a small volume filling factor for molecular gas and a Chabrier IMF, based on, approximate this “maximum star formation rate” as $$\label{equation.eddington} \mathrm{SFR\_{MAX}}=480\, \sigma\_{400}^2 \, D\_\mathrm{kpc} \, \kappa\_{100}^{-1}\, \, \, \,\, \, \, \mathrm{[M\_{\odot}\, yr^{-1}]}$$ where σ400 is the line-of-sight gas velocity dispersion in units of 400 km s− 1, κ100 is the opacity in units of cm2g− 1, and Dkpc is the characteristic physical scale of the starburst (usually approximated as the Gaussian FWHM of the line emitting region). In Figure [figure.eddington], the blue curves show probability distributions for a 1000 realizations of ongoing SFRs in the ${z\gtrsim 3}$ SMGs, estimated from their total infrared luminosity and associated errors, through Equation [equation.sfr]. The SMGs are forming stars at high rates $\mathrm{500-3000 M\_{\odot}yr^{-1}}$, close to the Eddington limit. E.g. estimated the maximum starformation rate of AzTEC4 and AzTEC8 to be in the range 1900 − 3800M⊙y*r− 1, comparable to the values derived here (see Tab.[table.mdust]). In the following we investigate if the observed properties of z ∼ 2 quiescent galaxies are consistent with having formed under such conditions. Assuming that z ∼ 2 quiescent galaxies formed in Eddington limited maximum starbursts, we can estimate the maximum SFR and the duration of this burst, from the observed size, velocity dispersion and stellar mass of the quiescent remnants. In Figure [figure.eddington] the red curve shows the distribution of $\mathrm{SFR\_{MAX}}$ for the sample of z ∼ 2 quiescent galaxies described in Section [section.quiescent], calculated from Equation [equation.eddington], assuming κ100 = 1, $\mathrm{D\_\mathrm{kpc}=2\,r\_{e,c}}$ (where $\mathrm{r\_{e,c}}$ are the effective radii measured for the individual galaxies) and $\sigma\_{400}=\left<\sigma\right>/400\, \mathrm{km\, s^{-1}}$, where $\left<\sigma\right>=363\pm100 \, \mathrm{km \, s^{-1}}$ is the mean velocity dispersion measured for z ∼ 2 quiescent galaxies in the litterature. We use this mean value as measured velocity dispersions for the K13 sample are not available. There is a good general correspondence between the $\mathrm{SFR\_{MAX}}$ distribution of quiescent z ∼ 2 galaxies, and the SFR distribution of ${z\gtrsim 3}$ SMGs. The $\mathrm{SFR\_{MAX}}$ distribution peaks at higher SFRs than the observed distribution in ${z\gtrsim 3}$ SMGs, indicating that some of the z ∼ 2 quiescent galaxies may have formed in starbursts with sub-Eddington SFRs. Also plotted in Figure [figure.eddington] (b) is the duration of this “maximum starburst” $$\label{equation.tburst-eddington} \mathrm{t\_{burst}=\Delta M\_{\star}/SFR\_{MAX}}$$ assuming a constant star fomation rate $\mathrm{SFR}=\mathrm{SFR\_{max}}$ during the burst, and that all the stellar mass of the z\sim2 quiescent galaxies was created during this burst, i.e. $\mathrm{\Delta M\_{\star} = M\_{\star}}$. While consistent within the errors, the mean derived timescales for Eddington limited starbursts is about a factor of two longer than the starburst timescale derived from comparing comoving number densities. This can be accounted for by changing some of the assumptions, e.g., if the SMG starbursts are triggered by major mergers, a fraction of the stellar mass must have been formed in the progenitor galaxies, prior to the merger. In Figure [figure.eddington] (c) we show that if we assume that only half of the observed stellar mass in z ∼ 2 quiescent galaxies was created in a ${z\gtrsim 3}$ Eddington limited starburst, i.e. $\mathrm{\Delta M\_{\star} = 0.5 \,M\_{\star}}$, there is excellent agreement between the derived timescales, consistent with the idea that ${z\gtrsim 3}$ SMGs are the progenitors of z ∼ 2 quiescent galaxies. Interestingly this is consistent with the results of who found that on average ∼ 45% of the stellar mass in a sample of z > 4 SMGs was formed in their ongoing starbursts. If half the stellar mass formed prior to the merger that ignite the SMG starburst, an implication is that the merger progenitors must have been gas rich star forming galaxies, in agreement with the high gas fractions found in high redshift star forming galaxies [figure.eddington] Additional Stellar Mass Growth and Quenching of the $z\gtrsim 3$ SMGs --------------------------------------------------------------------- The similar mass-size distribution of the ${z\gtrsim 3}$ SMGs and z ∼ 2 quiescent galaxies is in agreement with what one would expect if the ${z\gtrsim 3}$ SMGs evolve passively into z ∼ 2 quiescent galaxies, after they have been quenched. Prior to the quenching however the ongoing starburst will increase the stellar masses of the galaxies. In Figure [figure.deltam] we show that the distribution of stellar masses in the ${z\gtrsim 3}$ SMGs is broader than that of $\mathrm{log(M/M\_{\odot}) >11}$ quiescent galaxies at z ∼ 2. We can estimate the growth of stellar mass in the individual $z\gtrsim 3$ SMGs from their gas masses, inferred from the FIR SED fits (see Table [table.mdust]). From these we can estimate the final stellar masses of the ${z\gtrsim 3}$ SMGs if we assume a star formation effeciency, i.e the fraction of gas that is turned into stars during the starburst. In the simulations of the gas fraction decrease from 45% to 40% in isolated disks and from 17.5% to 15% in merging galaxies, from the peak of the starburst to when it ends, corresponding to a decrease in gas mass of 5% and 15% during this time. If we assume that this gas is turned into stars, and that we are observing the SMGs at the peak of their starburst, the models thus indicate that ∼ 10 ± 5% of the observed gas mass in the $z\gtrsim 3$ SMGs will be turned in to stars during the remainder of the burst. In Figure [figure.deltam] we compare the final stellar mass distribution of the ${z\gtrsim 3}$ SMGs with that of quiescent z ∼ 2 quiescent galaxies, assuming that 10% of the derived gas mass in the $z\gtrsim 3$ SMGs are turned into stars before the starbursts are quenched. The two distributions are similar, with a K-S test statistic of 0.33 and a probability of 67%, in agreement with a direct evolutionary link between the two populations. The mass increase from the time the SMGs are observed up to the end of the starburst will likely not significantly increase the effective radii, since the process is highly dissipative, resulting in a slight horizontal shift in the $\mathrm{M\_{\star}-r\_e}$ plane (blue points in Figure [figure.masssize]). The continued starbursts and subsequent quenching, may also provide the mechanism needed to transform the observed low-n disk-like surface brightness profiles observed in SMGs to the higher n bulge-like profiles observed in quiescent galaxies at z ∼ 2. Most of the stellar mass will be added in the nuclear regions of the SMGs, which is likely highly obscured by dust. Once the quenching sets in and most of the dust is destroyed or blown away, a more centrally concentrated surface brightness distribution could be revealed. Note that if, as assumed here, only 10% of the large derived gas masses in the $z\gtrsim 3$ SMGs is turned into stars during the remainder of the burst, the following quenching mechanism must be highly effecient at heating or expelling the substantial amounts of leftover gas. A possible mechanism for expelling the gas is through outflows, driven by strong winds associsted with the maximum starbursts. Tentative evidence for such outflows have recently been observed in the 163 OH line profile in an SMG at z=6.3. We stress that the large systematic uncertainties on the derived stellar masses for the SMGs could potentially influence our conclusions. [figure.deltam] Summary and Discussion ====================== The Link between $z\gtrsim 3$ SMGs and z ∼ 2 Compact, Quiescent Galaxies -------------------------------------------------------------------------- In this paper we presented evidence for a direct evolutionary connection between two of the most extreme galaxy types in the universe, the highest redshift (${z\gtrsim 3}$) SMGs which host some of the most intense starbursts known, and quiescent galaxies at z ∼ 2 which host the densest conglomerations of stellar mass known. The comparison was motivated by the recent discovery of a significant population of SMG at 3 < z < 6 and high resolution imaging and spectroscopic studies of z ∼ 2 quiescent galaxies which show that the majority of their stars likely formed in massive nuclear, possibly dust eshrouded starbursts in this redshift range. From a unique flux-limited statistical sample of ${z\gtrsim 3}$ SMGs in the COSMOS field, we have put robust constraints on their co-moving number density, which we then put in context of the comoving number densities of quiescent galaxies of similar mass at z ∼ 2. If ${z\gtrsim 3}$ SMGs are progenitors of z ∼ 2 quiescent galaxies, then our data implies that the SMG duty cycle must be $\mathrm{t\_{burst}=42\_{-29}^{+40}Myr}$, where the errorbars include our best estimates of the effects of cosmic variance, photometric redshift errors and incompleteness. This timescale is independent from, but in good agreement with estimates based on SMG gas depletion timescales $\mathrm{t\_{burst}\sim 40-200\, \mathrm{Myr}}$, estimates from hydrodynamical merger simulations $\mathrm{t\_{burst}\sim50\, \mathrm{Myr}}$, and estimates based on the time, compact starburst galaxies spend above the main sequence of star formation $\mathrm{t\_{burst}<70}$ Myr. Importantly, as our estimate of the SMG starburst timescale is based only on number density arguments, it is relatively independent on assumptions of the underlying stellar inital mass function (IMF), which is a large potential systematic uncertainty, e.g. in depletion timescale estimates. Based on stellar masses derived from UV-MIR photometry and sizes derived from deep NIR imaging, we have shown that the mass-size distribution of the ${z\gtrsim 3}$ galaxies is remarkably similar to that observed for compact quiescent massive galaxies at z ∼ 2, with similar mean internal stellar mass surface densities $\mathrm{\left<log(\Sigma)\right > \sim9.8 M\_{\odot}{kpc}^{-2}}$. The surface brightness distributions of the ${z\gtrsim 3}$ SMGs are best fit by Sersic models with low Sersic n parameters, typical of local star forming disk galaxies or mergers, and the majority show multiple components or irregularities indicative of ongoing merging and/or clumpy structure. Many similarities between z ∼ 2 quiescent galaxies and SMGs exist: they have similar stellar masses, characteristic internal velocities, dynamical masses, sizes, correlation lengths etc. Millimeter measurements of ${z\gtrsim 3}$ SMGs in continuum and CO show signatures of merging or rotation, with molecular emission line widths in the range 300 − 700 km s− 1 (with a few outliers), and a mean $\left< \mathrm{FWHM} \right >=456 \pm 253$ km s− 1 similar to stellar velocity dispersions σ = 300 − 500 km s− 1 measured in z ∼ 2 quiescent galaxies. For example, for AzTEC 3, at z = 5.3, measured a CO linewith of $\mathrm{487 km~s^{-1}}$, and a gas depletion timescale of 30 Myr, similar to the SMG starburst timescale derived here. At the depth and resolution of the present data, it is impossible to make strong claims about how many $z\gtrsim 3$ SMGs are in the process of merging. However all the detected galaxies show evidence of close companions, multiple components or clumpy structure, and have low derived Sersic indices, consistent with expectations for merging galaxies. In particular the two galaxies with HST/WFC3 data appears to be major mergers. [figure.sequence] The evidence presented in this paper is in support of a direct evolutionary connection between z\gtrsim 3 SMGs, through compact quiescent galaxies at z ∼ 2, to giant elliptical galaxies in the local universe. In this scenario (illustrated in Figure [figure.sequence]) gas rich, major mergers in the early universe, trigger nuclear dust enshrouded starbursts[5](#fn5), which on average last 42− 29+ 40 Myr, followed by star formation quenching, either due to gas exhaustion, feedback from the starburst or the ignition of an AGN, leaving behind compact stellar remnants to evolve passivly for about a Gyr into the compact quiescent galaxies we observe at z ∼ 2. Over the next 10 Gyr, these then grow gradually, primarily through minor merging, into local elliptical galaxies. Connection to Compact Star Forming Galaxies at 2.5 < z < 3 ------------------------------------------------------------ found a population of relatively massive (log(M/M⊙ > 10) compact star forming galaxies (cSFGs) at 1.4 < z < 3, which show evidence of quenching beginning to set-in (lower specific star formation rates than typical star forming galaxies and increased AGN fractions). Their masses, sizes and number densities (which increase with decreasing redshift, at the same time the number density of quiescent galaxies increase), suggest that the highest redshift examples of these may be progenitors of compact quiescent z ∼ 2 galaxies. These galaxies are thus good candidates for transition objects in the evolutionary sequence suggested here between the ${z\gtrsim 3}$ SMGs and the z ∼ 2 quiescent galaxies. The comoving number density of the most massive cSFGs ($\mathrm{log(M/M\_{\odot} > 10.8}$) at 2.5 < z < 3 is $\mathrm{\sim 5.4 \pm 2.5 \times 10^{-5}\,Mpc^{-3}}$, comparable to the number density for z ∼ 2 quiescent galaxies. However, the cSFGs are not massive enough to be descendants of the brightest ${z\gtrsim 3}$ SMGs, or progenitors of most of the massive z ∼ 2 quiescent galaxies considered here, as none of the cSFGs have $\mathrm{log(M/M\_{\odot}) > 11}$ (Barro, private communication), but are likely decendents of less intense starbursts at ${z\gtrsim 3}$ and progenitors of slightly lower mass quiescent z = 2 galaxies. Caveats and Outlook ------------------- One of the largest uncertainty in the derivables for the ${z\gtrsim 3}$ SMG sample are associated with their stellar masses. As extensively discusses in stellar masses for SMGs are highly dependent on the assumed star formation history, and may differ by up to ± 0.5 dex given different assumptions and models. Dynamical mass considerations may set an upper limit to stellar masses, however the $z\gtrsim 3$ SMGs samples with available dynamical mass estimates are still sparse, as well as subject of their own biases. The sample of ${z\gtrsim 3}$ SMGs is still small, and only partially spectroscopically confirmed. Future, larger and deeper mm surveys, over multiple fields, will allow for better constraints on the evolution of the co-moving number density of starburst galaxies, to the highest redshifts, and to study the effects of cosmic variance. This will allow for more detailed tests and modeling of the proposed scenario in different redshift and mass bins, rather that in the single mass bin and two redshift ranges as possible with the present data. E.g., the proposed scenario implies that the significant population of z ∼ 2 SMGs should evolve into compact, ∼ 1 Gyr old, massive post starburst galaxies at z ∼ 1.5. Interestingly recently published a spectroscopic sample of galaxies with exactly these properties. Similarly, if compact quiescent galaxies at $z\gtrsim3$ are found in the future, the properties of these should match those of the highest redshift z > 5 SMGs. With deeper data it will also be possible to push to lower star formation rates, and not only consider the most extreme starbursts. This will likely provide a way of fitting the 2.5 < z < 3 cSFG discussed in Section [section.csfg] into the evolutionary picture. Cosmological surface brightnes dimming and the large amounts (and unknown distribution) of dust in SMGs make them extremely faint in the rest-frame UV and optical, and likely bias the sizes measured, even in very deep NIR imaging data. However, we do note that one of the galaxy in our sample (AzTEC1), has been resolved in high resolution submillimeter imaging, with a derived extend of $0.1-0.2\arcsec$, corresponding to physical size of 1.3 − 2.7 kpc, consistent with the constraints on the effective radius we measure from the UltraVISTA data (r\_e<2.6 kpc, see Table [table.sample]). ALMA will greatly improve estimates of the sizes of high redshift SMGs, through high resolution observations of the restframe FIR dust continuum. We have argued in this paper that the observed structural properties are consistent with the SMGs being disks or mergers, but the constraints are uncertain, due to the relatively low S/N and spatial resolution of the images, e.g. the Sersic n parameters and effective radii could be underestimated, due to obscuration by dust and cosmological surface brightness dimming. With ALMA it will be straightforward to determine redshifts from molecular lines, and constrain the internal dynamics of the galaxies, e.g. estimate velocity dispersions, rotational velocities and search for evidence of merging. This will provide powerful diagnostics to help map the transformation of the most massive galaxies in the universe from enigmatic starburst at cosmic dawn to dead remnants, a few gigayears later. ST acknowledges the support of Lundbeck foundation and is grateful for the hospitality and support of the Institute for Astronomy, University of Hawaii, during the visit where this work was initiated. The research leading to these results has received funding from the European Union’s Seventh Framework programme under grant agreement 229517. KS gratefully acknowledges support from Swiss National Science Foundation Grant PP00P2\_138979/1. MJM acknowledges the support of the FWO-Vlaanderen and the Science and Technology Facilities Council. 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Taking the most probably photometric redshift yields that 9 SMGs (AzTEC1, AzTEC3, AzTEC4, AzTEC5, AzTEC8, AzTEC13, AzTEC14E, AzTEC15 & J1000+0234), are at z > 3[↩](#fnref2) 3. In this case 8 SMGs (AzTEC1, AzTEC3, AzTEC4, AzTEC5, AzTEC8, AzTEC13, AzTEC14E & J1000+0234), are at z > 3[↩](#fnref3) 4. In this case 10 SMGs (AzTEC1, AzTEC3, AzTEC4, AzTEC5, AzTEC8, AzTEC10, AzTEC11S, AzTEC13, AzTEC14E & J1000+0234), are at z > 3[↩](#fnref4) 5. The SMG image in the figure is adopted from Targett et al, 2013[↩](#fnref5) Sub-millimeter galaxies as progenitors of compact quiescent galaxies ==================================================================== Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, UK Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9, 9000 Gent, Belgium National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903 ETH Zurich,Institute for Astronomy, Department of Physics, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland European Southern Observatory, Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago 19, Chile MPE, Postfach 1312, 85741 Garching, Germany Institute for Astronomy, 2680 Woodlawn Drive, University of Hawaii, Honolulu, HI 96822 The Henry A. Rowland Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA Institut dAstrophysique de Paris, UMR7095 CNRS, Universite Pierre et Marie Curie, 98 bis Boulevard Arago, 75014 Paris, France Department of Astronomy, Cornell University, 220 Space Sciences Building, Ithaca, NY 14853, USA Astronomy Department, California Institute of Technology, MC 249-17, 1200 East California Boulevard, Pasadena, CA 91125, USA Three billion years after the big bang (at redshift z = 2), half of the most massive galaxies were already old, quiescent systems with little to no residual star formation and extremely compact with stellar mass densities at least an order of magnitude larger than in low redshift ellipticals, their descendants. Little is known about how they formed, but their evolved, dense stellar populations suggest formation within intense, compact starbursts 1-2 Gyr earlier (at 3 < z < 6). Simulations show that gas-rich major mergers can give rise to such starbursts which produce dense remnants. Sub-millimeter selected galaxies (SMGs) are prime examples of intense, gas-rich, starbursts. With a new, representative spectroscopic sample of compact quiescent galaxies at z = 2 and a statistically well-understood sample of SMGs, we show that z = 3 − 6 SMGs are consistent with being the progenitors of z = 2 quiescent galaxies, matching their formation redshifts and their distributions of sizes, stellar masses and internal velocities. Assuming an evolutionary connection, their space densities also match if the mean duty cycle of SMG starbursts is 42− 29+ 40 Myr (consistent with independent estimates), which indicates that the bulk of stars in these massive galaxies were formed in a major, early surge of star-formation. These results suggests a coherent picture of the formation history of the most massive galaxies in the universe, from their initial burst of violent star-formation through their appearance as high stellar-density galaxy cores and to their ultimate fate as giant ellipticals. Introduction ============ One of the most remarkable discoveries in galaxy evolution studies in the past years is that up to half of the most massive galaxies ($\mathrm{log(M\_{\}/M\_{\odot})>11}$) at z\approx2 are old quiescent systems with extremely compact structure, corresponding to stellar densities orders of magnitudes higher than seen in local elliptical galaxies. Much effort has gone into confirming their extreme properties and investigating their evolutionary path to the local universe. Virial arguments and simulations indicate that the most important process is likely to be minor dry merging, but observations suggest that other processes are likely also important, e.g. the continuous addition of increasingly larger newly quenched galaxies to the quenched population with decreasing redshift. The formation path of these extreme systems is largely unknown. Simulations indicate that highly dissipational interactions on short timescales provide plausible mechanisms for creating compact stellar populations, either through major mergers, or dynamical instabilities fed by cold gas accretion. A possible scenario is major gas-rich mergers at high redshift, in which the gas is driven to the center, igniting a massive nuclear starburst, followed by an AGN/QSO phase that quenches the star formation, and leaves behind a compact remnant. This is consistent with local stellar archaeology studies which imply that massive ellipticals must have short formation timescales of less than 1 Gyr,. Several authors have pointed out that sub-millimeter galaxies (SMGs) may be examples of the above scenario, but see for a counter example. The SMG population is dominated by galaxies undergoing intense, dust enshrouded starbursts. A large fraction of SMGs with measured CO profiles, show double peaked profiles, evident of ongoing major mergers or rotation. The auto-correlation length of SMGs is similar to that of optically selected QSOs, suggesting that SMGs and QSOs live in similar mass haloes and that the ignition of a QSO could be the event that quenches the star formation in SMGs. This is consistent with observations suggesting that the hosts of the most luminous QSOs, i.e those likely associated with the formation of massive quiescent galaxies, are found to be primarily major mergers, a result which is cooborated by who find that luminous AGN in massive z* ∼ 2 galaxies must be triggered by external processes. Interestingly, also finds evidence for low luminosity AGN in the vast majority of massive quiescent galaxies at z ∼ 2, suggesting that AGN play an active role in the quenching their star-formation. The correlation length of SMGs is similar to that of z\sim2 galaxies with $\mathrm{M\_\>5 \times 10^{10}\, M\_{\odot}}$ (r\_0=7.66\pm0.78), while z\sim2 galaxies with $\mathrm{M\_\>10^{11}\, M\_{\odot}}$ cluster more strongly. Recent advances in near infrared (NIR) spectroscopy have made it possible for the first time to accurately constrain the age, dust content and past star formation history of the brightest z\sim2 quiescent galaxies through absorption line diagnostics and spectral fitting in the rest frame-optical. These galaxies have spectra typical of post starburst galaxies, with no detected emission lines, but with strong Balmer absorption lines, suggesting that they underwent major starbursts which were quenched 1-2 Gyr prior to the time of observation (i.e. at 3<z<6). Several of these galaxies show evidence of significant dust abundance (with A\_V’s up to ∼ 1 magnitude), and they are baryon dominated, as is the case for local post starburst galaxies. In combination with their extremely compact stellar populations these observations suggest that the majority of the stars in z\sim2 quiescent galaxies formed in intense, possibly dust enshrouded nuclear starbursts, a scenario very similar to what is observed in z\sim2 SMGs. Velocity dispersions of z\sim2 quiescent galaxies measured from the width of absorption lines are in the range 300 − 500 km\, s− 1, significantly higher than in local ellipticals of similar stellar mass, but comparable to the FWHM of molecular lines in 2<z<3 SMGs. The line emitting gas of SMGs, as traced by high-J CO lines, is found to be spatially very compact, with a mean size of $\mathrm{\left < R\_{e}\right >=2.0\pm0.3 kpc}$, comparable to the mean spatial extent $\mathrm{\left < R\_{e}\right >=1.96\pm0.8 kpc}$, of the stellar populations in the quiescent z\sim2 galaxies. We note however that studies of lower-J CO lines suggest that some SMGs may have more extended CO-disks. The median dynamical masses measured from CO(1-0) for z\sim2 SMGs, $\mathrm{\left < M\_{dyn}\right >=(2.3 \pm1.4)\times 10^{11}\, M\_{\odot}}$, is similar to that measured for z\sim2 quiescent galaxies $\mathrm{\left < M\_{dyn}\right>} =(2.5 \pm 1.3)\times 10^{11}\, \mathrm{M\_{\odot}}$. Despite the many similarities between SMGs and z\sim2 quiescent galaxies, a major obstacle in establishing an evolutionary link between the two galaxy types is their similar redshift distribution. While the quiescent nature and derived ages for z\sim2 quiescent galaxies suggest they formed at z\gtrsim3 the peak of the known SMG population was until recently found to be at z\sim2 with very few examples known at z\gtrsim 3, rendering an evolutionary link between the two populations unlikely. Recently, improved selection techniques have however uncovered a substantial tail stretching out to redshifts of z\sim6. In this paper we present evidence for a direct evolutionary link between the two extreme galaxy populations, by comparing the properties of two unique samples in the COSMOS field: (i) a spectroscopically confirmed, representative sample of compact z\sim2 quiescent galaxies with high resolution HST/WFC3 imaging, and (ii) a statistical sample of z\gtrsim 3 SMGs. In Section [section.samples] we introduce the samples, and in Section [section.results] we present our results. In particular, in Section [section.redshiftdistributions] we show that the distribution of formation redshifts for the z\sim2 quiescent galaxies is similar to the observed redshift distribution of z\gtrsim 3 SMGs, and in Section [section.numberdensities] we compare the co-moving number densities of the two populations. In Section [section.structure] we derive structural properties of the z\gtrsim 3 SMGs and in Section [section.masssize] show that their stellar mass-size relation is similar to those of z\sim2 quiescent galaxies. In Section [section.dutycycle] and [section.eddington] we show that the duty cycle of the z > 3 SMG starburst (derived assuming they are progenitors of z\sim2 quiescent galaxies), is consistent with independent estimates, and with the formation time scale derived for z ∼ 2 quiescent galaxies, assuming they formed in Eddington limited starbursts. In Section [section.discussion] we summarize and discuss the results. Throughout this paper we assume a standard flat universe with $\mathrm{\Omega\_{\lambda}=0.73}$, $\mathrm{\Omega\_{m}=0.27}$ and $\mathrm{H\_0=71 km s^{-1} Mpc^{-3}}$. All stellar masses are derived assuming a IMF. llccc AzTEC 1 & $4.64\tablenotemark{a}$ & < 2.6 & unresolved & 1.3 − 2.7 AzTEC 3 & $5.299\tablenotemark{a}$ & < 2.4& unresolved & < 3 ± 2 AzTEC 4 & 4.93− 1.11+ 0.43 & < 2.5 & unresolved &- AzTEC 5 & $3.971\tablenotemark{a}$ & 0.5 ± 0.4& HST/WFC3 &- AzTEC 8 & $3.179\tablenotemark{a}$ & < 3.0 & unresolved &- AzTEC 10 & 2.79− 1.29+ 1.86 & 0.7 ± 0.1& - & - AzTEC 11-S & $>2.58\tablenotemark{b}$ & - & not detected & - AzTEC 13 & $> 3.59\tablenotemark{b}$ & - & not detected&- AzTEC 14-E & $> 3.03\tablenotemark{b}$ & - &not detected&- AzTEC 15 & 3.17− 0.37+ 0.29 & 5.0 ± 0.8 & very faint&- J1000+0234 & $4.542\tablenotemark{a}$ & 3.7 ± 0.2 & - &- Vd-17871 & $4.622\tablenotemark{a}$ & 1.3 ± 1.4 & - &- GISMO-AK03 & $4.757\tablenotemark{a}$ & 1.6 ± 0.6 & HST/WFC3 & - [table.sample] Samples ======= lclccccrccc AzTEC 1 &10.9− 0.1+ 0.1 & 0.47% & 0.070 & 25.0 & 9.1 ± 0.1 & 13.36 ± 0.09 &2291 ± 528&11.7 ± 0.1&secure & AzTEC 3 & 11.2− 0.1+ 0.1 & 0.47% & 0.290 & 25.0 & 9.3 ± 0.1 & 13.37 ± 0.04 &2344 ± 226&11.3 ± 0.1&secure & 10.7 AzTEC 4 &11.2− 0.1+ 0.1& 4.58% & 0.080 & 25.0 & 9.6 ± 0.2 & 13.25 ± 0.15& 1778 ± 733&11.6 ± 0.2 &tentative & AzTEC 5 & 10.9− 0.5+ 0.5 & 0.47% & 0.190 & 25.0 & 9.4 ± 0.1 &13.43 ± 0.02&2692 ± 127 &11.4 ± 0.1&secure & AzTEC 8 & 11.5− 0.1+ 0.1& 0.47% & 0.090 & 25.0 & 9.7 ± 0.1 & 13.45 ± 0.01&2818 ± 66 &11.7 ± 0.1 &secure& AzTEC 10 & 10.5− 0.1+ 0.1& 2.50% & 0.060 & 5.00 & 9.6 ± 0.1 & 12.58 ± 0.10&380 ± 98 &11.8 ± 0.1 &secure & AzTEC 11-S & - & 0.47% & 0.040 & 25.0 & 9.6 ± 0.1&13.30 ± 0.01&1995 ± 46 &- &secure & AzTEC 13 & - & 4.58% & 0.160 & 2.00 & 9.9 ± 0.3 & 12.70 ± 0.20&501 ± 293 &- &upper limits& AzTEC 14-E & - & 0.47% & 0.290 & 0.70 & 9.8 ± 0.2 & 12.48 ± 0.18&302 ± 155 &- &upper limits& AzTEC 15 & 11.2− 0.1+ 0.1& 3.19% & 0.010 & 20.0 & 9.3 ± 0.1&12.73 ± 0.08& 537 ± 108&11.4 ± 0.1 &secure& J1000+0234 & 10.7− 0.1+ 0.1& 1.77% & 0.150 & 25.0 & 9.3 ± 0.1 &13.17 ± 0.09&575 ± 275&11.8 ± 0.4 &tentative& 10.4 Vd-17871 &10.9− 0.1+ 0.1& 4.58% & 0.250 & 25.0 & 9.1 ± 0.1 &13.09 ± 0.06&1230 ± 182 &11.2 ± 0.2&secure & GISMO-AK03 &12.1− 0.1+ 0.1& 4.58% & 0.290 & 3.00 & 9.5 ± 0.2 &12.66 ± 0.19&457 ± 250 &11.5 ± 0.2 &secure &. “Secure”: The source is relatively isolated and detected at S/N>3. “Tentative”: The source is detected at S/N>3, but the flux density estimates may be affected by bright closeby objects. “Upper limit”: The source is not detected at S/N>3. [table.mdust] Sample of $z\gtrsim 3$ SMGs --------------------------- Based on dedicated follow-up studies with sub-mm-interferometers (PdBI, SMA, CARMA) and optical/mm-spectroscopy (with Keck/DEIMOS, EVLA, PdBI) towards 1.1 mm and $870~\micron$ selected sources in the COSMOS field presented the redshift distribution of SMGs. This sample shows a tail of z\gtrsim 3 SMGs, corresponding to a significantly larger number density at these high redshifts than found in previous surveys. A possible reason for the difference is that previous surveys did not have (sub)-mm followup interferometry, and therefore may be subject to identification biases. E.g. show that many of the galaxies in the sample break up into multiple sources when studied at high resolution, which enevitably lead to mis-identifications for some of the sources. Here we use the sample to estimate the comoving number density and other properties of z\gtrsim 3 SMGs. Our starting point is a 1.1 mm-selected sample, drawn from the AzTEC/JCMT 0.15 square degree survey of the COSMOS field (Scott et al. 2008), and observed with the SMA at 890 μm and \sim2 \arcsec angular resolution to unambiguously associate multi-wavelength counterparts. The 17 SMGs identified by the SMA follow-up form a statistical sample as they are drawn from a signal-to-noise limited ($\mathrm{S/N\_{1.1mm}}>4.5$), and flux-limited ($F\_\mathrm{1.1mm}\gtrsim4.2$ mJy), 1.1 mm-selected sample, drawn from a contiguous area of 0.15 square degrees. We include one more SMG in this sample, J1000+0234 (F1.1mm = 4.8 ± 1.5 mJy, $\mathrm{S/N\_{1.1mm}}\sim3$), which is confirmed to be at z=4.542. Nine out of these 18 interferometrically detected galaxies have spectroscopic redshifts (four are confirmed to be at z\gtrsim 3; ; Karim et al., in prep), while for the remainder precise photometric redshifts ($\sigma\_\mathrm{{\Delta z / (1+z\_\mathrm{spec})}}=0.09$) were computed by. The z\gtrsim 3 SMGs from this sample are listed in Table [table.sample]. The top 11 objects constitute our statistical sample. We will use these in the following sections to estimate the redshift distribution and comoving number density of $z\gtrsim 3$ SMGs. The bottom two objects, are additional spectroscopically confirmed z\gtrsim 3 SMGs in the COSMOS field which we add to the sample for structural analysis only. The flux-limited sub-mm selection ensures a relatively homogenous sample of the most intensely starforming dust obscured galaxies at ${z\gtrsim3}$: Due to the negative k-correction, the sub-mm flux detection limit corresponds roughly to a cut in SFR over the considered redshift range. Note that while a fraction of single dish detected SMGs break up into multiple components when studied with interferometry at $\le 2\arcsec$ resolution, this is only the case for two of the galaxies studied here (AzTEC 11 and 14). In the present study we assume that the close individual components are related and count them as one in the number density calculations (thus assuming they will eventually merge into one galaxy). As the galaxies are not resolved in the MIR-mm photometry, derived properties (infrared luminosities, star formation rates, dust masses etc) are for the combined systems. Neither of the two galaxies are detected in the optical-NIR so the derived sizes and stellar masses for the sample are not affected. Far-infrared emission of the z\gtrsim 3 SMGs -------------------------------------------- In order to directly constrain the SFRs, dust and gas masses of the z\gtrsim 3 SMGs, we made use of the (sub)-mm (AzTEC, LABOCA, MAMBO, SMA, CARMA, PdBI) and FIR (Spitzer MIPS, Herschel PACS and SPIRE) observations of the COSMOS field. The Herschel data consist of deep PACS 100 and 160 μm observations, taken as part of the PACS Evolutionary Probe guaranteed time key programme, and SPIRE 250, 350 and 500 μm observations taken as part of the Herschel Multi-tiered Extragalactic Survey. PACS and SPIRE flux densities were measured using a PSF fitting analysis, guided by the position of sources detected in the deep COSMOS 24 μm observations from the Multiband Imaging Photometer onboard the Spitzer Space Observatory. We cross-matched our $z\gtrsim 3$ SMGs sample with this MIPS-PACS-SPIRE catalogue using a matching radius of 2. Results of these matches were all visually checked. For $z\gtrsim3$ SMGs not included in the MIPS-PACS-SPIRE catalogue because of a lack of MIPS counterpart, we compute their PACS and SPIRE flux densities using a PSF fitting analysis guided by their positions. Further details of the FIR photometry are presented in Smolčić et al. (in prep.). Among the 13 z\gtrsim 3 SMGs, 9 have secure mid/far-infrared detections, 2 have tentative mid/far-infrared detections and 2 are undetected at infrared wavelengths. From the FIR–mm SED of the z\gtrsim 3 SMGs, we infer their infrared luminosities and dust masses using the dust model of as described in detail in. The infrared luminosity ($L\_{{\rm IR}}$) is derived by integrating the best fitting normalized SED templates from the DL07 library from rest-frame 8 to 1000 μm. From these we can accurately estimate the star-formation activity of the z\gtrsim 3 SMGs, using the standard $L\_{\rm IR}$-to-SFR conversion of, assuming a Chabrier IMF : $$\label{equation.sfr} {\rm SFR\,[M\_{\odot}\,yr^{-1}]\,=\,10^{-10}\,}L\_{{\rm IR}}\,{\rm [L\_{\odot}]}.$$ Finally we estimate the gas masses of the sample through $\mathrm{log(M\_{gas}/M\_{dust})=-0.85\Z+9.4}$, where $\mathrm{Z=2.18\times log(M\_{\star})}$$\mathrm{-0.0896\log(M\_{\star})^2-4.51}$ [1](#fn1). This method has been used successfully in the local Universe as well as at high redshift. Assumptions and limitations of this method in the case of high redshift galaxies are extensively discussed in. Results of the FIR SED fits and derived quantities are summarized in Table [table.mdust], and used in the following analysis to establish an evolutionary link between ${z\gtrsim 3}$ SMGs and quiescent z ∼ 2 galaxies. The derived gas masses are comparable to or larger than the derived stellar masses: $\mathrm{\left< f\_g \right > = \left< M\_{gas}/(M\_{\star}+M\_{gas})\right>=0.71\pm0.03}$, in agreement with the high gas fractions found in previous studies of high redshift SMGs. We caution however, that gas masses estimated from FIR SED fits are relatively uncertain (potentially up to a factor of 5 -10). E.g. in Table [table.mdust] we list gas masses for two objects in our sample which have independent estimates derived from CO line emission. These are significantly different from our SED estimates. The main factors contributing to the uncertainties in the SED estimates are that the sub-mm measurements dont trace cold gas very well, in which the (sub)mm/CO flux ratio is much lower than in the starburst nucleus, but where a lot of the gas mass may reside. The others are the metallicity correction (which has a large scatter) and the assumption about the gas-to-dust ratio. The main factors contributing to the uncertainty of the CO measurements is the assumed αCO which can be uncertain by a factor > 2, and the excitation corrections, which can be uncertain by a factor of ∼ 2 − 4. Stellar Mass Estimates for the $z\gtrsim 3$ SMGs ------------------------------------------------ We estimate stellar masses of the $z\gtrsim 3$ SMGs from their UV-MIR ($8\micron$) broad band photometry as described in Smolčić et al. (in prep.). Briefly, stellar masses were derived by fitting the observed broadband UV-MIR spectral energy distributions with the MAGPHYS code. The stellar component in the MAGPHYS models is based on stellar population synthesis models assuming various star formation histories (exponentially-declining SFHs (with random timescales) + superimposed stochastic bursts) and a IMF. The stellar masses for the SMGs, and their formal uncertainties drawn from the probability distribution function (generated from the χ2 fit values by MAGPHYS) are given in Tab. [table.mdust]. We note however, that stellar masses for SMGs are strongly dependent on the assumed star formation histories, and may lead to systematic discrepancies of ± 0.5 dex given different assumptions and stellar population synthesis models, and whether or not emission lines are included in the templates. For example, using the double SFHs implemented in GRASIL, we find systematically higher stellar masses, consistent with the results from. On the other hand, dynamical mass considerations based on CO line observations for two objects in our sample suggest lower stellar masses than inferred by MAGPHYS. Hence, here we adopt the middle values, i.e. the stellar masses computed by MAGPHYS+BC03, noting that these may be subject to systematic uncertainties. Sample of z ∼ 2 Compact Quiescent Galaxies -------------------------------------------- It is well established from deep multi-waveband photometric surveys, that a substantial population of quiescent massive galaxies with extremely compact structure exists at z ∼ 2. Samples of spectroscopically confirmed, z ∼ 2 quiescent galaxies with accurate stellar population model fits and high angular resolution space based NIR imaging are much more sparse. As a high quality comparison set to the z > 3 SMGs we use the sample of. This sample consist of 16 spectroscopically confirmed massive quiescent galaxies, selected from the 3DHST survey in the COSMOS field by requiring strong 4000Å breaks in the grism observations. As shown in K13 this effectively selects a representative sample of massive (logM > 10.9) evolved quiescent galaxies around z ∼ 2. The high S/N grism spectra around the break in combination with multi-waveband photometry from the COSMOS survey allows for strong constraints on the stellar populations including stellar masses, dust contents, mean stellar ages, i.e the time elapsed since the last major episode of star formation, as well as formation redshifts (derived from the stellar ages). The sample is also covered by high resolution NIR imaging with HST/WFC3 from the CANDELS survey yielding accurate constraints on the rest-frame optical surface brightness profiles, and effective radii (r\_e). Results ======= Redshift Distributions ---------------------- From the spectroscopic redshifts and mean stellar ages available for the quiescent z ∼ 2 galaxy sample described in Section [section.quiescent] we can estimate the distribution of their formation redshifts. In Figure [figure.zform] this distribution is compared to the observed redshift distribution of the sample of ${z\gtrsim 3}$ SMGs described in Section [section.z3smg]. Due to the small number of galaxies in both samples, a one to one correspondence is not expected. However, we stress that the two distributions are similar, with a peak at z ∼ 3 and a tail towards higher redshifts. A two-sample Kolmogorov-Smirnov (K-S) test yields a statistic of 0.29 with a p-value of 54%, and is thus not inconsistent with the two redshift distributions being drawn from the same parent distribution. [figure.zform] Co-moving Number Densities -------------------------- The next step in establishing an evolutionary connection between ${z\gtrsim 3}$ SMGs and quiescent galaxies at z ∼ 2 is to compare their co-moving number densities. The comoving number density of massive quiescent galaxies as a function of redshift is well constrained from photometric redshift and stellar population model fits to deep multi-waveband photometry. studied the number density evolution of star forming and quiescent galaxies (separated using their rest-frame UVJ colors) in a sample complete at stellar masses $\mathrm{log(M/ M\_{\odot})>11}$ out to z ∼ 2.2. Here we estimate a comoving number density of $\mathrm{6.0 \pm 2.1 \times 10^{-5} Mpc^{-3}}$ for quiescent galaxies at z ∼ 2 with $\mathrm{log(M/ M\_{\odot})>11}$ as the mean of the densities measured at z = 1.9 and z = 2.1 by. The error includes a contribution of cosmic variance of 12%. To derive the surface number density of ${z\gtrsim 3}$ SMGs we take all SMGs from the 1.1 mm-selected COSMOS sample that could lie at z > 3 given their lower or upper 99% confidence levels of the photometric redshift. We then derive an average value of the surface density by taking the most probable photometric redshift (or spectroscopic redshift where available)[2](#fn2), and the lower[3](#fn3) and upper[4](#fn4) surface density values by taking the limiting redshifts corresponding to the 99% confidence intervals of the photometric redshifts. This yields a surface density of $z\gtrsim 3$, bright ($F\_\mathrm{1.1mm}\gtrsim4.2$ mJy) SMGs of 60 ± 10 deg− 2. Note that conservatively excluding from the analysis all 3 SMGs in the sample that are not significantly detected at other wavelengths (AzTEC11S, AzTEC13, AzTEC14E), and thus only have lower redshift limits, yields a surface density of 40 deg− 2. The derived surface density values for ${z\gtrsim 3}$ SMGs may be subject to systematic effects. The completeness of the AzTEC/JCMT COSMOS survey, shown in Fig. 8 in, is roughly 50%, 70%, and 90% at F1.1mm = 4.2, 5, and 6 mJy. Taking this into account in combination with the deboosted 1.1 mm fluxes of the SMGs yields that the derived surface densities could be roughly a factor of 1.5 higher than that reported above. On the other hand, the AzTEC/JCMT COSMOS field may be overdense, which would imply that the true ${z \gtrsim 3}$ SMG surface density averaged over larger area would be lower. Our best estimate of the co-moving number density of 3 < z < 6 SMGs is $\mathrm{2.1 \pm 0.4\times 10^{-6}~Mpc^{-3}}$, significantly lower than the space density of z ∼ 2 quiescent galaxies. This is expected as ${z\gtrsim 3}$ galaxies only enter the mm-selection criterion during their intense starburst phase. In Section [section.dutycycle] we use the observed difference in co-moving number densities to constrain the duty cycle of the SMG starbursts. [figure.gallery] Rest-frame UV-Optical Structure of z > 3 SMGs: Disks, Spheroids or Mergers? --------------------------------------------------------------------------- The high redshifts and large amounts of dust in the ${z\gtrsim 3}$ SMGs renders them extremely faint in the rest-frame UV and optical, despite their high stellar masses and star formation rates. This makes it challenging to constrain their structure. To achieve the least biased estimates of the distribution of stellar mass one would need to study the surface brightness distributions in the rest-frame optical/NIR, or as close to these wavebands as possible. Ideally, the observations would be done in the observed mid infrared, but at the low spatial resolution of current facilities (e.g. Spitzer) the galaxies remain unresolved. Until JWST becomes operational the best that can be achieved is to study the galaxies in the observed NIR. For most of the galaxies in the sample this wavelength range probes rest-frame wavelengths around the 4000 Å break, and thus should be a relatively good tracer of the stellar mass distribution. For two galaxies (AzTEC5 and GISMO-AK03) we use space based NIR imaging with HST/WFC3, which is available from the CANDELS survey. This is preferable to groundbased imaging given the higher resolution (FWHM$\sim0.2\arcsec$). For the remaining galaxies we use deep NIR imaging provided by the UltraVista survey (5σ AB depths range from 23.7 in the K-band to 24.6 in the Y-band, McCracken et al 2012). The resolution of these observations is lower (FWHM ∼ 0.8), but it has been demonstrated that relatively unbiased sizes (down to a fraction of the FWHMPSF) can be derived from such data when the S/N is high and the PSF is well known. To increase the S/N we stack the Y, J, H and K band images. Postage stamp images of the galaxies are shown in the top panel of Figure [figure.gallery]. NIR counterparts of 10 of the 13 sources are detected, 8 of which have relatively high signal to noise (the faintest ones have S/N ∼ 10). We fit 2D Sersic models to their surface brightness distributions with, using similarly stacked images of nearby stars as PSF models. We find the sersic n to be relatively poorly constrained from the data. Leaving it free in the fits in all cases results in low values n < 2, with a median value of \left<n\right>=0.6\pm0.1, but with relatively large errors. To limit the degrees of freedom in the fits we therefore fix it to n = 1. The reduced χ2 of these fits are in all cases similar to those with n free, and better than fits with n fixed to 4. The best fitting effective radius encompassing half the light of the model, are reported in Table [table.sample]. Half of the detected galaxies (5) have close companions. In these cases we model both components simultaneously and report the parameters for the main component (closest to the center of the mm-emission). Also listed in Table [table.sample] are restframe FIR sizes for two galaxies in our sample derived from interferometric sub-mm imaging observations. These agree with the sizes derived from the NIR data. The restframe FIR sizes directly measures the extend of the starforming regions which we hypothesize evolves into the compact stellar populations at z=2, so the agreement is encouraging. Our analysis shows, that apart from being very compact, the z > 3 SMGs are not isolated, smooth single component galaxies. All the detected galaxies show evidence of close companions or clumpy sub-structure (see Figure [figure.gallery]). From these observations alone, it is not possible to deduce whether this is due to chance projections, ongoing minor/major mergers, or perhaps multiple star forming regions in individual galaxies, as resolved photometry and spectroscopy is not available. We note however that the two galaxies with HST/WFC3 data appear to have well separated individual components of comparable brightness, favoring the merger interpretation. This is consistent with direct observational evidence for SMGs being major mergers, i.e. having multiple close components at the same redshift. Simulations suggest that the timescale for major mergers are typically 0.39 ± 0.30 Gyr. The cosmic time available between the observed epoch of the SMGs at z=3-6 and their proposed remnants at z=2 is 1-2 Gyr. If (some of) the SMGs are major mergers, there is thus sufficient time available for them to coalesce to a single quiescent remnant at z = 2. In the local universe most star forming galaxies are well fit by exponential disk profiles corresponding to n=1, while irregular galaxies and (pre-coalescence) mergers are often best fit by models with lower n-values (n < 1). At the S/N and resolution of the galaxies in the Ultravista data the confidence in derived Sersic parameters are limited. However, the persisting low values found for the whole z > 3 SMG sample, including the two galaxies with the higher resolution HST/WFC3 data, suggests that the galaxies are more consistent with disks or mergers than spheroids. A similar conclusion was found for a sample of 22 SMGs at 1 < z < 3 with HST/WFC3 data for which the majority were best fit by low n Sersic models, with a mean \left< n \right>=1.2 \pm 0.1. If z > 3 SMGs are progenitors of z ∼ 2 quiescent galaxies, then their evolution must include a transformation of their surface brightness profiles which increase their Sersic indices, as surface brightness profiles of quiescent galaxies at z ∼ 2 are more centrally concentrated, e.g. the sample of K13 has \left < n \right >=4.0 \pm0.1. We discuss a possible mechanism for this transformation in Section [section.discussion]. Mass-Size relation ------------------ Combining the derived stellar masses and effective radii of the 3 < z < 6 SMGs, in Figure [figure.masssize] we compare their stellar mass-size distribution to that of z = 2 quiescent galaxies and of massive early type galaxies in the local universe. Two of the ten NIR-detected SMGs are relatively extended with effective radii comparable to those in local galaxies of similar mass. Both of these (AzTEC 10 and AzTEC 15) appear from their NIR images to be ongoing mergers. The remaining 8 galaxies are extremely compact, with r\_e \lesssim 2.5~kpc. Four are unresolved in the Ultravista data. For these we adopt upper limits on their effective radii corresponding to $\mathrm{0.5\times FWHM\_{PSF}}$. The stellar mass-size distribution of the 3 < z < 6 SMGs is similar to that of z ∼ 2 quiescent galaxies. Both populations are smaller by an average factor of ∼ 3 than local galaxies of similar mass. From the derived quantities we can infer the mean internal stellar mass surface densities within the effective radius ($\mathrm{\Sigma=0.5M\_{\star}/\pi r\_{e}^2}$) of the z > 3 SMGs and z = 2 compact quiescent galaxies (cQGs) which we find to be similar $\mathrm{\left<log(\Sigma)\right >\_{SMGs} \sim9.9 \pm 0.1 M\_{\odot}{kpc}^{-2}}$, $\mathrm{\left<log(\Sigma)\right >\_{cQGs} \sim9.8 \pm 0.1 M\_{\odot}{kpc}^{-2}}$ more than an order of magnitude higher than in local early type galaxies of similar mass. This is consistent with a picture where the SMGs passively evolve into compact quiescent galaxies after their starbursts are quenched. [figure.masssize] Duty Cycle of SMG Starbursts ---------------------------- The observed space density of $z\gtrsim 3$ SMGs is a factor of \sim30 lower than the space density of z ∼ 2 quiescent galaxies (see Section [section.numberdensities]). However, the SMGs only enter the sub-mm selected ($F\_\mathrm{1.1mm}\gtrsim4.2$ mJy) sample during their intense starburst phase where they have very high star formation rates. The duration of this phase, i.e. the duty cycle $\mathrm{t\_{burst}}$, which ends when the supply of gas is depleted or the star formation is quenched, e.g. by feedback from supernovae or active galactic nuclei, has been estimated to be in the range 40-200 Myr, based on gas depletion timescales and clustering analysis. If we assume that all the ${z\gtrsim 3}$ SMGs evolve into z ∼ 2 quiescent galaxies, and that they only undergo one SMG phase we can estimate the average duty cycle of their starbursts from the observed comoving number densities of the two populations, as $$\mathrm{t\_{burst}=t\_{obs}\times (n\_{SMG,z\gtrsim 3}/n\_{q,z=2})},$$ where $\mathrm{t\_{obs}}$ is the cosmic epoch corresponding to the redshift interval 3 < z < 6 from which the ${z\gtrsim 3}$ SMGs are selected. Using the comoving number densities we can thus constrain the duty cycle of the SMGs to tburst = 42− 15+ 33Myr. This number however does not include possible systematics uncertainties on the number density of SMGs discussed in Section [section.numberdensities]. If we conservatively assume the two extreme cases where (i) the SMG sample is 100% complete, and the field is three times overdense, and (ii) the sample is a factor of 1.5 incomplete but not overdense, the derived timescale are in the range $\mathrm{14<t\_{SMG}<62\, Myr}$. The systematic uncertainties on the timescale is thus of the order of 24 Myr. Therefore our constraints on the average dutycycle in ${z\gtrsim 3}$ SMGs is $\mathrm{t\_{burst}=42\_{-29}^{+40}Myr}$, where the errors have been added in quadrature. This value is consistent with the independently estimated duty cycles based on gas depletion time scales, thus affirming the idea that ${z\gtrsim 3}$ SMGs are progenitors of z ∼ 2 quiescent galaxies. The derived timescale does not depend strongly on the z = 6 upper limit adopted for the SMG redshift distribution. Adopting limits of z=5.5 or z=7 instead, leads to timescales of 44 and 37 Myr, respectively. We note that the validity of the timescale calculation presented here, relies on the assumption of a direct evolutionary connection between the two populations, implying that all z = 2 quiescent galaxies were once z > 3 SMGs, and all z > 3 SMGs evolve into z = 2 quiescent galaxies. Star formation Rate and Timescale of z=2 Quiescent Galaxies During their Formation ---------------------------------------------------------------------------------- We can infer a lower limit on the star formation rate of the z=2 quiescent galaxies during their formation by assuming that they started forming stars at z = 10 and did so at a constant rate until their inferred formation redshifts. The minimum average SFR needed to acquire their observed stellar masses at z=2 calculated in this way is $\langle\mbox{SFR}\_{\rm min}\rangle = 115 \pm 5\, M\_\odot\,\mbox{yr}^{-1}$. This is a factor of > 3 larger than the observed average SFR in star-forming Lyman break galaxies (LBGs) at z > 3. Furthermore, the space density of z ∼ 2 quiescent galaxies with logM\/M⊙ > 11, is 5, 10 and > 100 times larger than that of similar mass LBGs at z=4,5 and 6 respectively. Their progenitors must therefore have had much larger SFRs and are missing from LBG samples. This suggests that they must be dust obscured starburst galaxies. Based on the observed line widths and compact spatial extent of molecular line emitting regions, SMGs are often argued to be maximum starbursts, i.e to be forming stars at a rate close to the Eddington limit. Assuming a spherical symmetric geometry, an isothermal sphere density structure, a small volume filling factor for molecular gas and a Chabrier IMF, based on, approximate this “maximum star formation rate” as $$\label{equation.eddington} \mathrm{SFR\_{MAX}}=480\, \sigma\_{400}^2 \, D\_\mathrm{kpc} \, \kappa\_{100}^{-1}\, \, \, \,\, \, \, \mathrm{[M\_{\odot}\, yr^{-1}]}$$ where σ400 is the line-of-sight gas velocity dispersion in units of 400 km s− 1, κ100 is the opacity in units of cm2g− 1, and Dkpc is the characteristic physical scale of the starburst (usually approximated as the Gaussian FWHM of the line emitting region). In Figure [figure.eddington], the blue curves show probability distributions for a 1000 realizations of ongoing SFRs in the ${z\gtrsim 3}$ SMGs, estimated from their total infrared luminosity and associated errors, through Equation [equation.sfr]. The SMGs are forming stars at high rates $\mathrm{500-3000 M\_{\odot}yr^{-1}}$, close to the Eddington limit. E.g. estimated the maximum starformation rate of AzTEC4 and AzTEC8 to be in the range 1900 − 3800M⊙y*r− 1, comparable to the values derived here (see Tab.[table.mdust]). In the following we investigate if the observed properties of z ∼ 2 quiescent galaxies are consistent with having formed under such conditions. Assuming that z ∼ 2 quiescent galaxies formed in Eddington limited maximum starbursts, we can estimate the maximum SFR and the duration of this burst, from the observed size, velocity dispersion and stellar mass of the quiescent remnants. In Figure [figure.eddington] the red curve shows the distribution of $\mathrm{SFR\_{MAX}}$ for the sample of z ∼ 2 quiescent galaxies described in Section [section.quiescent], calculated from Equation [equation.eddington], assuming κ100 = 1, $\mathrm{D\_\mathrm{kpc}=2\,r\_{e,c}}$ (where $\mathrm{r\_{e,c}}$ are the effective radii measured for the individual galaxies) and $\sigma\_{400}=\left<\sigma\right>/400\, \mathrm{km\, s^{-1}}$, where $\left<\sigma\right>=363\pm100 \, \mathrm{km \, s^{-1}}$ is the mean velocity dispersion measured for z ∼ 2 quiescent galaxies in the litterature. We use this mean value as measured velocity dispersions for the K13 sample are not available. There is a good general correspondence between the $\mathrm{SFR\_{MAX}}$ distribution of quiescent z ∼ 2 galaxies, and the SFR distribution of ${z\gtrsim 3}$ SMGs. The $\mathrm{SFR\_{MAX}}$ distribution peaks at higher SFRs than the observed distribution in ${z\gtrsim 3}$ SMGs, indicating that some of the z ∼ 2 quiescent galaxies may have formed in starbursts with sub-Eddington SFRs. Also plotted in Figure [figure.eddington] (b) is the duration of this “maximum starburst” $$\label{equation.tburst-eddington} \mathrm{t\_{burst}=\Delta M\_{\star}/SFR\_{MAX}}$$ assuming a constant star fomation rate $\mathrm{SFR}=\mathrm{SFR\_{max}}$ during the burst, and that all the stellar mass of the z\sim2 quiescent galaxies was created during this burst, i.e. $\mathrm{\Delta M\_{\star} = M\_{\star}}$. While consistent within the errors, the mean derived timescales for Eddington limited starbursts is about a factor of two longer than the starburst timescale derived from comparing comoving number densities. This can be accounted for by changing some of the assumptions, e.g., if the SMG starbursts are triggered by major mergers, a fraction of the stellar mass must have been formed in the progenitor galaxies, prior to the merger. In Figure [figure.eddington] (c) we show that if we assume that only half of the observed stellar mass in z ∼ 2 quiescent galaxies was created in a ${z\gtrsim 3}$ Eddington limited starburst, i.e. $\mathrm{\Delta M\_{\star} = 0.5 \,M\_{\star}}$, there is excellent agreement between the derived timescales, consistent with the idea that ${z\gtrsim 3}$ SMGs are the progenitors of z ∼ 2 quiescent galaxies. Interestingly this is consistent with the results of who found that on average ∼ 45% of the stellar mass in a sample of z > 4 SMGs was formed in their ongoing starbursts. If half the stellar mass formed prior to the merger that ignite the SMG starburst, an implication is that the merger progenitors must have been gas rich star forming galaxies, in agreement with the high gas fractions found in high redshift star forming galaxies [figure.eddington] Additional Stellar Mass Growth and Quenching of the $z\gtrsim 3$ SMGs --------------------------------------------------------------------- The similar mass-size distribution of the ${z\gtrsim 3}$ SMGs and z ∼ 2 quiescent galaxies is in agreement with what one would expect if the ${z\gtrsim 3}$ SMGs evolve passively into z ∼ 2 quiescent galaxies, after they have been quenched. Prior to the quenching however the ongoing starburst will increase the stellar masses of the galaxies. In Figure [figure.deltam] we show that the distribution of stellar masses in the ${z\gtrsim 3}$ SMGs is broader than that of $\mathrm{log(M/M\_{\odot}) >11}$ quiescent galaxies at z ∼ 2. We can estimate the growth of stellar mass in the individual $z\gtrsim 3$ SMGs from their gas masses, inferred from the FIR SED fits (see Table [table.mdust]). From these we can estimate the final stellar masses of the ${z\gtrsim 3}$ SMGs if we assume a star formation effeciency, i.e the fraction of gas that is turned into stars during the starburst. In the simulations of the gas fraction decrease from 45% to 40% in isolated disks and from 17.5% to 15% in merging galaxies, from the peak of the starburst to when it ends, corresponding to a decrease in gas mass of 5% and 15% during this time. If we assume that this gas is turned into stars, and that we are observing the SMGs at the peak of their starburst, the models thus indicate that ∼ 10 ± 5% of the observed gas mass in the $z\gtrsim 3$ SMGs will be turned in to stars during the remainder of the burst. In Figure [figure.deltam] we compare the final stellar mass distribution of the ${z\gtrsim 3}$ SMGs with that of quiescent z ∼ 2 quiescent galaxies, assuming that 10% of the derived gas mass in the $z\gtrsim 3$ SMGs are turned into stars before the starbursts are quenched. The two distributions are similar, with a K-S test statistic of 0.33 and a probability of 67%, in agreement with a direct evolutionary link between the two populations. The mass increase from the time the SMGs are observed up to the end of the starburst will likely not significantly increase the effective radii, since the process is highly dissipative, resulting in a slight horizontal shift in the $\mathrm{M\_{\star}-r\_e}$ plane (blue points in Figure [figure.masssize]). The continued starbursts and subsequent quenching, may also provide the mechanism needed to transform the observed low-n disk-like surface brightness profiles observed in SMGs to the higher n bulge-like profiles observed in quiescent galaxies at z ∼ 2. Most of the stellar mass will be added in the nuclear regions of the SMGs, which is likely highly obscured by dust. Once the quenching sets in and most of the dust is destroyed or blown away, a more centrally concentrated surface brightness distribution could be revealed. Note that if, as assumed here, only 10% of the large derived gas masses in the $z\gtrsim 3$ SMGs is turned into stars during the remainder of the burst, the following quenching mechanism must be highly effecient at heating or expelling the substantial amounts of leftover gas. A possible mechanism for expelling the gas is through outflows, driven by strong winds associsted with the maximum starbursts. Tentative evidence for such outflows have recently been observed in the 163 OH line profile in an SMG at z=6.3. We stress that the large systematic uncertainties on the derived stellar masses for the SMGs could potentially influence our conclusions. [figure.deltam] Summary and Discussion ====================== The Link between $z\gtrsim 3$ SMGs and z ∼ 2 Compact, Quiescent Galaxies -------------------------------------------------------------------------- In this paper we presented evidence for a direct evolutionary connection between two of the most extreme galaxy types in the universe, the highest redshift (${z	arxiv_0000457
){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\doteq}M\_1^A {\cong}M\_2^A$, where M1A and M2A are the LTSs in Lemma [L:trdiff]. By choosing f(L) = L ∣∣ M2A we get $L {\equiv}L {\:\|\|\:}{{\mathsf}{Run}(A)} {\cong}L {\:\|\|\:}M\_2^A = f(L)$, so L ≅ f(L). Because M2A lacks stable failures and blocks all visible actions of L in L ∣∣ M2A, we have ${{\mathit}{Sf}}(f(L)) = \emptyset$, ${{\mathit}{Div}}(f(L)) = \{\varepsilon\}$, and ${{\mathit}{Inf}}(f(L)) = \emptyset$. They are constants, so Lemma [L:nocongr] yields the claim if we choose k = 0 in it.It is widely known that the equivalence induced by Σ and ${{\mathit}{Tr}}$ is a congruence. The next theorem says that climbing up the ladder, ${{\mathit}{Inf}}$ has to be preserved. (187,26)(4,13)(-4,8)(-0.9,1.8) (4,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(10,7)τ (6,13)(30,13)(18,16)b1 (32,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (32,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(26,7)τ (46,15)⋮ (60,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(66,7)τ (62,13)(86,13)(74,16)b2 (88,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (88,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(82,7)τ (102,15)⋮ (116,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(122,7)τ (118,13)(142,13)(130,16)b3 (144,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (144,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(138,7)τ (158,15)⋮ (182,13)⋯ (4,13)2(32,13)2(60,13)2(88,13)2(116,13)2 (144,13)2 [F:testinftr] [T:Trace] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, “ ≅ ” preserves ${{\mathit}{Tr}}$ but not ${{\mathit}{Inf}}$, and ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$, then “ ≅ ” is the equivalence induced by Σ and ${{\mathit}{Tr}}$. There are M1, M2, and ξ such that M1 ≅ M2 and $\xi \in {{\mathit}{Inf}}(M\_1) \setminus {{\mathit}{Inf}}(M\_2)$. Because “ ≅ ” preserves ${{\mathit}{Tr}}$, Theorem [T:noSigma] implies that it also preserves Σ, so we may let ΣM = Σ(M1) = Σ(M2). Let b1b2⋯ = ξ[1]. Let A be any set such that $\tau \notin A$. Let $\{a\_1, a\_2, \ldots{\gray, a\_m}\} = {A^{[2]}}$. When i ∈ {1, 2}, let MiA = ⌊ (Tξ ∣∣ ⌈Mi⌉[1]) \ ΣM[1] ⌋[2], where Σ(Tξ) = ΣM[1] ∪ A[2] and otherwise Tξ is like in Fig. [F:testinftr]. Because X[i] and Y[j] are disjoint whenever i ≠ j, we have Σ(M1A) = Σ(M2A) = A. Thanks to the τ-loops in Fig. [F:testinftr], ${{\mathit}{Sf}}(M\_1^A) = {{\mathit}{Sf}}(M\_2^A) = \emptyset$. By ([E:Inf]), M1 can execute any finite prefix of ξ. This yields ${{\mathit}{Tr}}(M\_1^A) = {{\mathit}{Div}}(M\_1^A) = A^\$. By the congruence property M1A* ≅ M2A. Because “ ≅ ” preserves ${{\mathit}{Tr}}$, also ${{\mathit}{Tr}}(M\_2^A) = {{\mathit}{Div}}(M\_2^A) = A^\$. Since M1 can but M2 cannot execute ξ* completely, we get ${{\mathit}{Inf}}(M\_1^A) = A^\omega$ and ${{\mathit}{Inf}}(M\_2^A) = \emptyset$. Let L be any LTS and A = Σ(L). We can reason ${{\mathsf}{Run}(A)} {\equiv}{{\mathsf}{Run}(A)} {\:\|\|\:}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{{\mathsf}{Run}(A)} {\:\|\|\:}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\doteq}M\_1^A {\cong}M\_2^A$, and $L {\equiv}L {\:\|\|\:}{{\mathsf}{Run}(A)} {\cong}L {\:\|\|\:}M\_2^A$. Lemma [L:nocongr] gives the claim if we choose k = 1, $X\_1 = {{\mathit}{Tr}}$, and f(L) = L ∣∣ M2A, because then L ≅ f(L), ${{\mathit}{Sf}}(f(L)) = \emptyset$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Tr}}(L)$, and ${{\mathit}{Inf}}(f(L)) = \emptyset$. The above proof constructed a function f(L) that throws away all information (modulo “ ≐ ”) except Σ and Tr, while preserving “ ≅ ”. Information on ${{\mathit}{Sf}}$ and ${{\mathit}{Div}}$ was thrown away using the assumption that ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$. Information on ${{\mathit}{Inf}}$ was thrown away by starting with an arbitrary difference on ${{\mathit}{Inf}}$, and amplifying it to a function fʹ(L, M) = L ∣∣ ⌊ (Tξ ∣∣ ⌈M⌉[1]) \ ΣM[1] ⌋[2] so that fʹ(L, M1) preserves ${{\mathit}{Inf}}(L)$ while fʹ(L, M2) wipes it out. The permission to also throw away all information on ${{\mathit}{Sf}}$ and ${{\mathit}{Div}}$ simplified the design. We have L ≅ fʹ(L, M1) ≅ fʹ(L, M2) = f(L), where the first “ ≅ ” takes care of ${{\mathit}{Sf}}$ and ${{\mathit}{Div}}$, and the second of ${{\mathit}{Inf}}$. In the construction of f, despite the use of notation defined in this section, ultimately only operators from Section [S:def] were used. By Theorem [T:Tr], there are no more congruences in this section. In conclusion, altogether precisely four abstract linear-time congruences satisfy ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$: those induced by the first zero, one, two, or three of Σ, ${{\mathit}{Tr}}$, and ${{\mathit}{Inf}}$. That also the last one is a congruence is widely known and proven, e.g., in . When Deadlock Is Bothlock Is Not Livelock ========================================= In this section we show that only three congruences that are implied by “ ≐ ” satisfy ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$. We also introduce an “internal choice” operator L ⊓ Lʹ that will be used in this and later sections. It can be built from the four operators in Section [S:def], so any equivalence that is a congruence with respect to them also is a congruence with respect to internal choice. The next theorem tells that all congruences in this section preserve ${{\mathit}{Sf}}$. By Theorem [T:noSigma], they also preserve Σ. [T:Sf] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and ${\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$, then “ ≅ ” preserves ${{\mathit}{Sf}}$. If “ ≅ ” does not preserve ${{\mathit}{Sf}}$, then there are M1, M2, σ = b1⋯bn, and $A = \{a\_1, \ldots{\gray, a\_m}\}$ such that M1 ≅ M2 and $(\sigma, A) \in {{\mathit}{Sf}}(M\_1) \setminus {{\mathit}{Sf}}(M\_2)$. Let ΣM = Σ(M1). If Σ(M2) ≠ ΣM, then Theorem [T:noSigma] yields ${\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$. Otherwise, if TσA is the LTS in Fig. [F:testDl] with Σ(TσA) = ΣM, we have $(M\_2 {\:\|\|\:}T\_\sigma^A) \setminus \Sigma\_M {\doteq}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$ and $(M\_1 {\:\|\|\:}T\_\sigma^A) \setminus \Sigma\_M {\doteq}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$. In both cases, ${\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$, contrary to our assumption. Thus “ ≅ ” preserves ${{\mathit}{Sf}}$. The equivalence induced by Σ and ${{\mathit}{Sf}}$ is a congruence . However, if the so-called interrupt operator found in CSP or Lotos is employed, then it is no longer a congruence . To prove the next result, the “internal choice” operator of CSP would be handy. It is equivalent to the CCS expression τ.P + τ.Q. Fortunately, it can be built from our operators. $$\begin{aligned} L\_1 \sqcap L\_2 & := & \big((\,L\_C {\:\|\|\:}c\_1.{\lceilL\_1\rceil^{[1]}} {\:\|\|\:}c\_2.{\lceilL\_2\rceil^{[2]}}\,) \setminus \{c\_1, c\_2\}\big)\Phi \textrm{,}\nonumber\end{aligned}$$ where c1 = 1[0], c2 = 2[0], Φ = {(a[1], a) ∣ a ∈ Σ1} ∪ {(a[2], a) ∣ a ∈ Σ2}, and LC has $S\_C = \{\hat s\_C, s\_C\}$, ΣC = {c1, c2}, $\Delta\_C = \{ (\hat s\_C,c\_1,s\_C), (\hat s\_C,c\_2,s\_C) \}$, and $\hat s\_C \neq s\_C$ (please see Fig. [F:simpleLTSs]). (Here c1 and c2 could be any distinct symbols that are not in Σ1[1] ∪ Σ2[2].) The CFFD-semantics of this operator is simple: $$\begin{aligned} \Sigma(L \sqcap L') & = & \Sigma(L) \cup \Sigma(L')\nonumber\\ {{\mathit}{Sf}}(L \sqcap L') & = & {{\mathit}{Sf}}(L) \cup {{\mathit}{Sf}}(L')\nonumber\\ {{\mathit}{Div}}(L \sqcap L') & = & {{\mathit}{Div}}(L) \cup {{\mathit}{Div}}(L')\nonumber\\ {{\mathit}{Inf}}(L \sqcap L') & = & {{\mathit}{Inf}}(L) \cup {{\mathit}{Inf}}(L')\nonumber\end{aligned}$$ The next congruence in this section also preserves ${{\mathit}{Tr}}$. [T:SfTr] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, “ ≅ ” preserves ${{\mathit}{Sf}}$ but not ${{\mathit}{Tr}}$, and ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$, then “ ≅ ” is the equivalence induced by Σ and ${{\mathit}{Sf}}$. Let L be any LTS and A = Σ(L). By Theorem [T:noSigma], “ ≅ ” preserves Σ. The assumptions of Lemma [L:trdiff] hold, so we can use its M1A and M2A. Let $f(L) = (L {\:\|\|\:}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}) \sqcap M\_1^A$. We have $L {\equiv}L {\:\|\|\:}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}L {\:\|\|\:}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}{\doteq}(L {\:\|\|\:}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}) \sqcap M\_2^A$, so L ≅ f(L). Furthermore, ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = A^\* = \Sigma(L)^\$, and ${{\mathit}{Inf}}(f(L)) = A^\omega = \Sigma(L)^\omega$. With k* = 1 and $X\_1 = {{\mathit}{Sf}}$, Lemma [L:nocongr] gives the claim.The equivalence induced by Σ, ${{\mathit}{Tr}}$, and ${{\mathit}{Sf}}$ is a congruence . At the next level, also ${{\mathit}{Inf}}$ has to be preserved. To prove this, we need a more complicated construction than in the proof of Theorem [T:Trace], because this time ${{\mathit}{Sf}}$ has to be preserved. (40,49)(-13,0)(10,41)(-4,8)(-0.9,1.8)(0,49)R1A (10,41)(-2,0)(-10,0)(-7,-7)(-1.4,-1.4)(3,33)τ (10,39)(10,15)(11,30.5)τ (10,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (24,15)⋮ (38,15)(38,41)(12,41)(38,40)τ (10,41)2(10,13)2(38,13)2 (137,49)(-10,0)(12,41)(-4,8)(-0.9,1.8)(3,49)R2A (14,41)(38,41)(26,43)τ (42,41)(66,41)(54,43)τ (40,39)(40,15)(41,31)τ (70,41)(94,41)(82,43)τ (68,39)(68,15)(69,31)τ (98,41)(122,41)(110,43)τ (96,39)(96,15)(97,31)τ (132,41)⋯(132,27)⋯ (12,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(6,7)τ (40,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (26,15)⋮ (68,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (54,15)⋮ (96,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (82,15)⋮ (124,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (110,15)⋮ (132,13)⋯ (12,41)2(40,41)2(68,41)2(96,41)2 (12,13)2(40,13)2(68,13)2(96,13)2 [F:Inf2] [T:TrSf] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, “ ≅ ” preserves ${{\mathit}{Tr}}$ and ${{\mathit}{Sf}}$ but not ${{\mathit}{Inf}}$, and ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$, then “ ≅ ” is the equivalence induced by Σ, ${{\mathit}{Tr}}$, and ${{\mathit}{Sf}}$. Let M1 ≅ M2, $\xi \in {{\mathit}{Inf}}(M\_1) \setminus {{\mathit}{Inf}}(M\_2)$, b1b2⋯ = ξ[1], and A be any set such that $\tau \notin A$. By Theorem [T:noSigma], “ ≅ ” preserves Σ. Let ΣM = Σ(M1) = Σ(M2). When i ∈ {1, 2}, let MiA = ⌊ (Tξ ∣∣ ⌈Mi⌉[1]) \ ΣM[1] ⌋[2], where Σ(Tξ) = ΣM[1] ∪ A[2] and otherwise Tξ is like in Fig. [F:testinftr]. Because Tξ does not have stable states, we have ${{\mathit}{Sf}}(M\_1^A) = {{\mathit}{Sf}}(M\_2^A) = \emptyset$. Because ⌈M2⌉[1] lacks the infinite trace b1b2⋯, we have ${{\mathit}{Inf}}(M\_2^A) = \emptyset$. Let R1A and R2A be the LTSs in Fig. [F:Inf2]. We have ${{\mathit}{Div}}(R\_2^A) = A^\$. These imply M2A* ⊓ R2A ≐ R2A. On the other hand, ${{\mathit}{Inf}}(M\_1^A) = {{\mathit}{Inf}}(R\_1^A) = A^\omega$, ${{\mathit}{Sf}}(R\_1^A) = {{\mathit}{Sf}}(R\_2^A)$, and also ${{\mathit}{Div}}(R\_1^A) = A^\$, so M1A* ⊓ R2A ≐ R1A. As a consequence, R1A ≐ M1A ⊓ R2A ≅ M2A ⊓ R2A ≐ R2A. By choosing A = Σ(L) and f(L) = L ∣∣ R2A we get $L {\equiv}L {\:\|\|\:}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}L {\:\|\|\:}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}{\doteq}L {\:\|\|\:}R\_1^A {\cong}L {\:\|\|\:}R\_2^A$, so L ≅ f(L). We have ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Tr}}(f(L)) = {{\mathit}{Tr}}(L)$, and ${{\mathit}{Inf}}(f(L)) = \emptyset$. With k = 2, $X\_1 = {{\mathit}{Tr}}$, and $X\_2 = {{\mathit}{Sf}}$, Lemma [L:nocongr] gives the claim. The equivalence induced by Σ, ${{\mathit}{Tr}}$, ${{\mathit}{Sf}}$, and ${{\mathit}{Inf}}$ is the intersection of the equivalences induced by $(\Sigma, {{\mathit}{Tr}}, {{\mathit}{Sf}})$ and $(\Sigma, {{\mathit}{Tr}}, {{\mathit}{Inf}})$. So it is the intersection of two congruences and thus a congruence. We now show that it is the last one in this section. [T:TrSfail] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, “ ≅ ” preserves ${{\mathit}{Sf}}$ and ${{\mathit}{Inf}}$, and ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$, then “ ≅ ” is the equivalence induced by Σ, ${{\mathit}{Tr}}$, ${{\mathit}{Sf}}$, and ${{\mathit}{Inf}}$. By Lemma [L:Inf=>Tr], “ ≅ ” preserves Σ and ${{\mathit}{Tr}}$. Let $f(L) = L {\:\|\|\:}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$. We have $L {\equiv}L {\:\|\|\:}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}L {\:\|\|\:}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}= f(L)$. Clearly ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Tr}}(L)$, and ${{\mathit}{Inf}}(f(L)) = {{\mathit}{Inf}}(L)$. Letting k = 3, $X\_1 = {{\mathit}{Tr}}$, $X\_2 = {{\mathit}{Sf}}$, and $X\_3 = {{\mathit}{Inf}}$, Lemma [L:nocongr] gives the claim. To summarize, precisely three abstract linear-time congruences satisfy ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$: those induced by $(\Sigma, {{\mathit}{Sf}})$, $(\Sigma, {{\mathit}{Tr}}, {{\mathit}{Sf}})$, and $(\Sigma, {{\mathit}{Tr}}, {{\mathit}{Sf}}, {{\mathit}{Inf}})$. When All Three Are Non-equivalent ================================= (-12,-12)(301,221) (-10,80)(-10,113)(120,40)(120,-10)(100,-10) (123,16)Section [S:DLisLL]: ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$ (22,110)(22,140)(152,67)(152,37) (155,54)Section [S:BLnotLL]: ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$ (22,154)(22,225)(152,152)(152,81) (155,118)Section [S:ALLnot]: ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$ (22,170)(86,134)(88,156)(152,120) (30,30)(108,32)(-30,-30)(0,0) (-30,-30)(0,0)(30,30)(-32,18)(108,32)(-30,-30)(0,0) (-30,-30)(0,0)(108,32)(32,-18)(0,0) (32,-18)(0,0)(30,30)(-96,54)(108,32)(-30,-30)(0,0) (-30,-30)(0,0)(108,32)(64,-36)(0,0) (64,-36)(0,0) (30,30)(0,55)(108,32)(0,-55)(0,0) (0,-55)(0,0)(30,30)(0,55)(-32,18)(108,32)(0,-55)(0,0) (0,-55)(0,0)(108,32)(32,-18)(0,0) (32,-18)(0,0)(30,30)(0,55)(-96,54)(108,32)(0,-55)(0,0) (0,-55)(0,0)(108,32)(64,-36)(0,0) (64,-36)(0,0) (30,30)(0,85)(-32,18)(108,32)(0,-30)(0,0) (0,-30)(0,0)(30,30)(0,85)(-64,36)(108,32)(32,-18)(0,0) (32,-18)(0,0)(30,30)(0,85)(-96,54)(108,32)(0,-30)(0,0) (0,-30)(0,0)(108,32)(32,-18)(0,0) (32,-18)(0,0) (108,32)(0,-30)(0,0) (0,-30)(0,0)(-32,18)(108,32)(32,-18)(0,0) (32,-18)(0,0)(-96,54)(108,32)(64,-36)(0,0) (64,-36)(0,0) (0,-30)(108,32)(0,0)2 (-3,-2)(108,32)(0,0)2 (-3,-2)(-32,18)(108,32)(0,0)2 (-3,-2)(-96,54)(108,32)(0,0)2 (-3,-2) (30,30)(108,32)(0,0)2 (-3,-2)(30,30)(-32,18)(108,32)(0,0)2 (-3,-2)(30,30)(-96,54)(108,32)(0,0)2 (-3,-2) (30,30)(0,55)(108,32)(0,0)2 (-3,-2)(108,32)(-3,-8)(108,32)(-3,-14) (30,30)(0,55)(-32,18)(108,32)(0,0)2 (-3,-2)(30,30)(0,55)(-96,54)(108,32)(0,0)2 (-3,-2) (30,30)(0,85)(-32,18)(108,32)(0,0)2 (-3,-2)(108,32)(-3,-8)(30,30)(0,85)(-64,36)(108,32)(0,0)2 (-3,-2) (30,30)(0,85)(-96,54)(108,32)(0,0)2 (-3,-2) [F:ALLnot] Figure [F:ALLnot] shows the results of the previous two sections and this section. In this section we survey the region where ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$. We need new semantic sets. They are defined in Subsection [S:minD]. Many proofs in this and the next section treat end states of divergence traces differently from end states of other traces. For this to be possible, no state must be simultaneously the end state of both a divergence trace and a nondivergent trace. Subsection [S:Una] presents a construct with which LTSs can be transformed into such a form, while preserving bisimilarity. The theorems that there are no other congruences are presented in Subsections [S:lower-ALLnot] and [S:upper-ALLnot]. New kinds of divergence and infinite traces ------------------------------------------- In this subsection we define new semantic sets that are related to divergence traces or infinite traces, and briefly study their properties. Minimal divergence traces ${{\mathit}{minD}}$ are divergence traces whose proper prefixes are not divergence traces. Finite extensions of minimal divergence traces ${{\mathit}{extT}}$ are an alternative representation for the same information (assuming that Σ is available). Also infinite extensions ${{\mathit}{extI}}$ can be derived from ${{\mathit}{minD}}$. Always-nondivergent traces ${{\mathit}{anT}}$ are traces which and whose proper prefixes are not divergence traces, and similarly with always-nondivergent infinite traces ${{\mathit}{anI}}$. Eventually-always-nondivergent infinite traces ${{\mathit}{eanI}}$ may have a finite number of divergence traces as prefixes. Always-eventually-nondivergent infinite traces ${{\mathit}{aenI}}$ have an infinite number of prefixes that are not divergence traces. $$\begin{aligned} {{\mathit}{minD}}(L) & := & \{ a\_1 \cdots a\_n \in {{\mathit}{Div}}(L) \mid \forall i; 0 \leq i < n: a\_1 \cdots a\_i \notin {{\mathit}{Div}}(L) \}\nonumber\\ {{\mathit}{extT}}(L) & := & \{ a\_1 \cdots a\_n \in \Sigma(L)^\* \mid \exists i; 0 \leq i \leq n: a\_1 \cdots a\_i \in {{\mathit}{minD}}(L) \}\nonumber\\ {{\mathit}{extI}}(L) & := & \{ a\_1 a\_2 \cdots \in \Sigma(L)^\omega \mid \exists i; i \geq 0: a\_1 \cdots a\_i \in {{\mathit}{minD}}(L) \}\nonumber\\ {{\mathit}{anT}}(L) & := & {{\mathit}{Tr}}(L) \setminus {{\mathit}{extT}}(L)\nonumber\\ {{\mathit}{anI}}(L) & := & {{\mathit}{Inf}}(L) \setminus {{\mathit}{extI}}(L)\nonumber\\ {{\mathit}{eanI}}(L) & := & \{ a\_1 a\_2 \cdots \in {{\mathit}{Inf}}(L) \mid \exists n; n \geq 0: \forall i; i \geq n: a\_1 \cdots a\_i \notin {{\mathit}{Div}}(L) \}\nonumber\\ {{\mathit}{aenI}}(L) & := & \{ a\_1 a\_2 \cdots \in {{\mathit}{Inf}}(L) \mid \forall n; n \geq 0: \exists i; i \geq n: a\_1 \cdots a\_i \notin {{\mathit}{Div}}(L) \}\nonumber\end{aligned}$$ We have $$\begin{array}{rcll} {{\mathit}{minD}}(L) & = & \{ a\_1 \cdots a\_n \in {{\mathit}{extT}}(L) \mid n=0 \vee a\_1 \cdots a\_{n-1} \notin {{\mathit}{extT}}(L) \} & \textrm{,}\\ {{\mathit}{anT}}(L) & = & {{\mathit}{Sf}}^{{\mathit}{Tr}}(L) \setminus {{\mathit}{extT}}(L) & \textrm{, and}\\ {{\mathit}{anI}}(L) & \subseteq & \begin{array}{@{}ccccc@{}} {{\mathit}{eanI}}(L) & \subseteq & {{\mathit}{aenI}}(L) & \subseteq & {{\mathit}{Inf}}(L) \end{array} & \textrm{.} \end{array}$$ [L:minD=>anT] Any congruence that preserves ${{\mathit}{minD}}$ also preserves Σ and ${{\mathit}{anT}}$. By Theorem [T:noSigma], it preserves Σ. Let L ≅ Lʹ and $b \notin \Sigma(L) \cup \{\tau\}$. For each σ = a1⋯an ∈ Σ(L)\, let T*σ be the LTS whose graph is ${ \begin{pspicture}(99,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}} {\psline[linearc=2]{->}}(6,2)(18,2)\rput[B](12,5.5){$a\_1$} {\psline[linearc=2]{->}}(22,2)(34,2)\rput[B](28,5.5){$a\_2$} \rput(44,2){$\cdots$} {\psline[linearc=2]{->}}(54,2)(66,2)\rput[B](60,5.5){$a\_n$} {\psline[linearc=2]{->}}(70,2)(82,2)\rput[B](76,4){$b$} {\rput[lb](84,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[r](99,5){$\tau$} {\pscircle[fillcolor=white]}(4,2){2}{\pscircle[fillcolor=white]}(20,2){2}{\pscircle[fillcolor=white]}(68,2){2}{\pscircle[fillcolor=white]}(84,2){2} \end{pspicture}}$ and whose alphabet is Σ(L) ∪ {b}. We have $\sigma \in {{\mathit}{Tr}}(L)$ if and only if $\sigma b \in {{\mathit}{Div}}(L {\:\|\|\:}T\_\sigma)$. If 0 ≤ i ≤ n, then $a\_1 \cdots a\_i \in {{\mathit}{Div}}(L)$ if and only if $a\_1 \cdots a\_i \in {{\mathit}{Div}}(L {\:\|\|\:}T\_\sigma)$. Therefore, $\sigma \in {{\mathit}{anT}}(L)$ if and only if $\sigma b \in {{\mathit}{minD}}(L {\:\|\|\:}T\_\sigma)$ if and only if $\sigma b \in {{\mathit}{minD}}(L' {\:\|\|\:}T\_\sigma)$ if and only if $\sigma \in {{\mathit}{anT}}(L')$. [L:minD=>anI] Any congruence that preserves ${{\mathit}{minD}}$ also preserves Σ and ${{\mathit}{anI}}$. By Theorem [T:noSigma], it preserves Σ. Let L ≅ Lʹ, $a\_1 a\_2 \cdots \in {{\mathit}{anI}}(L)$, and $T = {\begin{pspicture}(49,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}} {\psline[linearc=2]{->}}(6,2)(18,2)\rput[B](12,5.5){$a\_1$} {\psline[linearc=2]{->}}(22,2)(34,2)\rput[B](28,5.5){$a\_2$} \rput(44,2){$\cdots$} {\pscircle[fillcolor=white]}(4,2){2}{\pscircle[fillcolor=white]}(20,2){2} \end{pspicture}}$ with Σ(T) = Σ(L). None of a1⋯ai is in ${{\mathit}{minD}}(L) = {{\mathit}{minD}}(L')$, yielding ${{\mathit}{minD}}(L' {\:\|\|\:}T) = \emptyset$. On the other hand, $\varepsilon \in {{\mathit}{minD}}(\,(L {\:\|\|\:}T) \setminus \Sigma(L)\,) = {{\mathit}{minD}}(\,(L' {\:\|\|\:}T) \setminus \Sigma(L')\,)$. So $a\_1 \cdots a\_i \notin {{\mathit}{Div}}(L' {\:\|\|\:}T)$, $a\_1 a\_2 \cdots \in {{\mathit}{Inf}}(L' {\:\|\|\:}T)$, $a\_1 a\_2 \cdots \in {{\mathit}{Inf}}(L')$, and $a\_1 a\_2 \cdots \in {{\mathit}{anI}}(L')$. [L:Div=>Tr] Any congruence that preserves ${{\mathit}{Div}}$ also preserves ${{\mathit}{Tr}}$. $\sigma \in {{\mathit}{Tr}}(L) \Leftrightarrow \sigma \in {{\mathit}{Div}}(L {\:\|\|\:}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}})$. [L:Div=>eanI] Any congruence that preserves ${{\mathit}{Div}}$ also preserves Σ and ${{\mathit}{eanI}}$. By Theorem [T:noSigma], it preserves Σ. Let L ≅ Lʹ and $\xi \in {{\mathit}{eanI}}(L)$. If no prefix of ξ is in ${{\mathit}{Div}}(L)$, then let i = 1, and otherwise let i be 2 plus the length of the longest prefix of ξ that is in ${{\mathit}{Div}}(L)$. Let $a\_i \notin \Sigma(L) \cup \{\tau\}$ and, when 1 ≤ j ≠ i, let aj be such that ξ = a1⋯ai − 1ai + 1⋯. When j ≥ 0, none of a1⋯ai − 1ai + 1⋯ai + j is in ${{\mathit}{Div}}(L) = {{\mathit}{Div}}(L')$. Let T be the LTS whose alphabet is Σ(L) ∪ {ai} and whose graph is. We have $a\_1 \cdots a\_{i+j} \notin {{\mathit}{Div}}(L' {\:\|\|\:}T)$ but $a\_i \in {{\mathit}{Div}}(\,(L {\:\|\|\:}T) \setminus \Sigma(L)\,) = {{\mathit}{Div}}(\,(L' {\:\|\|\:}T) \setminus \Sigma(L')\,)$. As a consequence, $a\_1 a\_2 \cdots \in {{\mathit}{Inf}}(L' {\:\|\|\:}T)$, $\xi \in {{\mathit}{Inf}}(L')$, and $\xi \in {{\mathit}{eanI}}(L')$. Lower sub-region ---------------- In this subsection we survey the part of the current region that is below the dashed grey line in Fig. [F:ALLnot]. Thanks to the next theorem, all congruences in this and the next section preserve ${{\mathit}{minD}}$. [T:minD] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$, then “ ≅ ” preserves ${{\mathit}{minD}}$. To derive a contradiction, let L ≅ Lʹ and $\sigma = a\_1 \cdots a\_n \in {{\mathit}{minD}}(L) \setminus {{\mathit}{minD}}(L')$. If there is an i < n such that $a\_1 \cdots a\_i \in {{\mathit}{minD}}(L')$, then swap the roles of L and Lʹ, and use a1⋯ai instead of σ. Now no prefix of σ is in ${{\mathit}{minD}}(L')$. If Σ(L) ≠ Σ(Lʹ), then Theorem [T:noSigma] yields ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$. Otherwise, let Tσ be the LTS whose graph is ${\begin{pspicture}(70,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}} {\psline[linearc=2]{->}}(6,2)(18,2)\rput[B](12,5.5){$a\_1$} {\psline[linearc=2]{->}}(22,2)(34,2)\rput[B](28,5.5){$a\_2$} \rput(44,2){$\cdots$} {\psline[linearc=2]{->}}(54,2)(66,2)\rput[B](60,5.5){$a\_n$} {\pscircle[fillcolor=white]}(4,2){2}{\pscircle[fillcolor=white]}(20,2){2}{\pscircle[fillcolor=white]}(68,2){2} \end{pspicture}}$ and alphabet is Σ(L). We have $(L {\:\|\|\:}T\_\sigma) \setminus \Sigma(L) {\doteq}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$ or $(L {\:\|\|\:}T\_\sigma) \setminus \Sigma(L) {\doteq}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$. Furthermore, $(L' {\:\|\|\:}T\_\sigma) \setminus \Sigma(L) {\doteq}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$. These imply ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$ or ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$. If ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$, then ${\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\doteq}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}{\:\|\|\:}{\begin{pspicture}(20,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\rput[lb](4,2){{\psline[linearc=2]{->}}(2,0)(10,0)(7,7)(1.4,1.4)}}\rput[l](15,5){$\tau$}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}{\:\|\|\:}{\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\equiv}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$, so ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$. All cases contradict the assumption ${\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}\not{\cong}{\begin{pspicture}(33,10)(0,2){\psline[linearc=2]{->}}(15,10)(15,4) {\rput[lb](15,2){{\psline[linearc=2]{->}}(-1.4,1.4)(-7,7)(-10,0)(-2,0)}}\rput[r](5,5){$\tau$} {\psline[linearc=2]{->}}(17,2)(29,2)\rput[B](23,4){$\tau$} {\pscircle[fillcolor=white]}(15,2){2}{\pscircle[fillcolor=white]}(31,2){2} \end{pspicture}}$. By Lemmas [L:minD=>anT] and [L:minD=>anI], all congruences in this and the next section also preserve Σ, ${{\mathit}{anT}}$, and ${{\mathit}{anI}}$. Furthermore, in this section also Theorem [T:Sf] is applicable. So ${{\mathit}{Sf}}$ must be added to the semantics. Doing so yields a congruence (proof skipped). After adding ${{\mathit}{Sf}}$, ${{\mathit}{anT}}$ can be removed because ${{\mathit}{anT}}(L) = {{\mathit}{Sf}}^{{\mathit}{Tr}}(L) \setminus {{\mathit}{extT}}(L)$. Thus the weakest congruence in this section is induced by Σ, ${{\mathit}{Sf}}$, ${{\mathit}{minD}}$, and ${{\mathit}{anI}}$. Adding ${{\mathit}{Tr}}$ to this also yields a congruence. The next theorem says that it is the next congruence. We will need the construction in the proof of the theorem also in Section [S:BLisLL], so we isolate it in a lemma. [L:noTr] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{minD}}$ but not ${{\mathit}{Tr}}$, then for every LTS L there is an LTS f(L) such that L ≅ f(L), ${{\mathit}{Tr}}(f(L)) = {{\mathit}{anT}}(L) \cup {{\mathit}{extT}}(L)$, ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{extT}}(L)$, and ${{\mathit}{Inf}}(f(L)) = {{\mathit}{anI}}(L) \cup {{\mathit}{extI}}(L)$. By Theorem [T:noSigma], “ ≅ ” preserves Σ. Let M1 ≅ M2, $\sigma \in {{\mathit}{Tr}}(M\_1) \setminus {{\mathit}{Tr}}(M\_2)$, b1⋯bn = σ[1], and c = 1[0]. Let ΣM = Σ(M1) = Σ(M2). For any LTS L, let ΣL = Σ(L), and let g(L) be the LTS that is obtained as follows: the label of every visible transition is transformed from a to a[2], and a c-transition to the initial state of Fig. [F:testTr] is added to every divergent state. In Fig. [F:testTr], $\{a\_1, \ldots{\gray, a\_m}\} = {\Sigma\_L^{[2]}}$. The alphabet of g(L) is {c} ∪ ΣM[1] ∪ ΣL[2]. When i ∈ {1, 2}, let fi(L) = ⌊ (g(L) ∣∣ c.⌈Mi⌉[1]) \ ({c} ∪ ΣM[1]) ⌋[2]. By construction, fi(L) can do everything that L can do, and also try to hiddenly execute cσ[1]. Attempts to execute cσ[1] start at divergent states and, thanks to the τ-loops in Fig. [F:testTr], do not lead to stable states. Thus ${{\mathit}{Sf}}(f\_1(L)) = {{\mathit}{Sf}}(f\_2(L)) = {{\mathit}{Sf}}(L)$. Because f2(L) cannot execute cσ[1] completely, f2(L) ≐ L. On the other hand, f1(L) can, so ${{\mathit}{Tr}}(f\_1(L)) = {{\mathit}{anT}}(L) \cup {{\mathit}{extT}}(L)$, ${{\mathit}{Div}}(f\_1(L)) = {{\mathit}{extT}}(L)$, and ${{\mathit}{Inf}}(f\_1(L)) = {{\mathit}{anI}}(L) \cup {{\mathit}{extI}}(L)$. We have L ≐ f2(L) ≅ f1(L). Therefore, f1 qualifies as the f of the claim. [T:minDTr] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Sf}}$ and ${{\mathit}{minD}}$ but not ${{\mathit}{Tr}}$, then “ ≅ ” is the equivalence induced by Σ, ${{\mathit}{Sf}}$, ${{\mathit}{minD}}$, and ${{\mathit}{anI}}$. Lemma [L:minD=>anI] implies that “ ≅ ” preserves Σ and ${{\mathit}{anI}}$. Because ${{\mathit}{extT}}(L)$ and ${{\mathit}{extI}}(L)$ are functions of Σ(L) and ${{\mathit}{minD}}(L)$, the f of Lemma [L:noTr] qualifies as the f of Lemma [L:nocongr] with k = 3, $X\_1 = {{\mathit}{Sf}}$, $X\_2 = {{\mathit}{minD}}$, and $X\_3 = {{\mathit}{anI}}$. Unambiguation of LTSs --------------------- In this subsection we motivate and present two functions, called ${{\mathsf}{Una}}$ and ${{\mathsf}{PD}}$, that transform any LTS to a bisimilar LTS that has some useful property. (81,51)(41,30)(-4,8)(-0.9,1.8)(38,42)L (39,30)(15,30)(27,32)b (41,30)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a (41,30)(1.4,-1.4)(7,-7)(21,-7)(26.6,-1.4)(14,-8.5)b (13,30)(0,2)(0,10)(-7,7)(-1.4,1.4)(6,33)τ (69,30)(0,2)(0,10)(7,7)(1.4,1.4)(76,33)a (13,30)2(41,30)2(69,30)2 (12,0)(81,51)(41,30)(-4,8)(-0.9,1.8)(38,40)${{\mathsf}{Det}}(L)$ (43,30)(67,30)(55,32)a (69,30)(0,2)(0,10)(7,7)(1.4,1.4)(76,33)a (41,28)(41,4)(42,16)b (42.4,3.4)(67.6,28.6)(60,15)a (41,30)2(69,30)2(41,2)2 (81,51)(41,30)(-4,8)(-0.9,1.8)(38,40)${{\mathsf}{Una}}(L)$ (39,30)(15,30)(27,32)b(43,30)(67,30)(55,32)a (13,30)(0,2)(0,10)(-7,7)(-1.4,1.4)(6,33)τ (69,30)(0,2)(0,10)(7,7)(1.4,1.4)(76,33)a (41,28)(41,4)(42,16)b (42.4,3.4)(67.6,28.6)(60,15)a (13,30)2(41,30)2(69,30)2 (41,2)2 (81,51)(41,30)(-4,8)(-0.9,1.8)(38,40)${{\mathsf}{PD}}(L)$ (39,30)(15,30)(27,32)b(43,30)(67,30)(55,32)a (13,30)(0,2)(0,10)(-7,7)(-1.4,1.4)(6,33)τ (69,30)(0,2)(0,10)(7,7)(1.4,1.4)(76,33)a (41,28)(41,4)(42,16)b(43,2)(67,2)(55,4)a (69,2)(0,2)(0,10)(7,7)(1.4,1.4)(76,5)a (13,30)2(41,30)2(69,30)2 (41,2)2(69,2)2 [F:DetUna] To continue the survey, we need a construction that preserves ${{\mathit}{anI}}$ but not ${{\mathit}{Inf}}$. It will block infinite traces after a minimal divergence trace, while not affecting them before a minimal divergence trace. Blocking does not have the desired effect unless all executions of each minimal divergence trace switch it on. Forcing the execution of the switch at every divergent state does not suffice, because the same trace may have two executions, one leading to a divergent and the other to a nondivergent state. This is exemplified by the trace b of the L in Fig. [F:DetUna]. Even if we knew that this is the case with some nondivergent state, we could not blindly implement the switch there, because it may also be reachable via another, always-nondivergent trace. An example is the trace a in the figure. To cope with this problem, we define a function ${{\mathsf}{Una}}$ that, given an LTS, yields a bisimilar LTS where different traces lead to the same state only if they have the same futures. This is obtained by keeping track, in a new component of the state, of the set of original states that can be reached via the trace that has been executed so far. To do that, we first define the determinization of L as the LTS $$\begin{aligned} {{\mathsf}{Det}}(L) & := & (S{\_{\mathsf}{D}}, \Sigma, \Delta{\_{\mathsf}{D}}, \hat s{\_{\mathsf}{D}})\textrm{, where}\nonumber\\ S\_\sigma & = & \{ s \mid \hat s {\,{=}\sigma{\Rightarrow}\,} s \}\nonumber\\ S{\_{\mathsf}{D}}& = & \{ S\_\sigma \mid \sigma \in {{\mathit}{Tr}}(L) \}\nonumber\\ \Delta{\_{\mathsf}{D}}& = & \{ (S\_\sigma, a, S\_{\sigma a}) \mid a \neq \tau \wedge \sigma a \in {{\mathit}{Tr}}(L) \}\nonumber\\ \hat s{\_{\mathsf}{D}}& = & S\_\varepsilon\nonumber\end{aligned}$$ [L:Det] If $\sigma \in {{\mathit}{Tr}}(L)$, then $\hat s{\_{\mathsf}{D}}{\,{=}\sigma{\Rightarrow}\,} S\_\sigma$. If $\hat s{\_{\mathsf}{D}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{D}}$, then $s{\_{\mathsf}{D}}= S\_\sigma$ and $\sigma \in {{\mathit}{Tr}}(L)$. We prove the first claim by induction. Clearly $\hat s{\_{\mathsf}{D}}{\,{=}\varepsilon{\Rightarrow}\,} \hat s{\_{\mathsf}{D}}= S\_\varepsilon$. If $\sigma a \in {{\mathit}{Tr}}(L)$, then a ≠ τ and $(S\_\sigma, a, S\_{\sigma a}) \in \Delta{\_{\mathsf}{D}}$. By the induction assumption $\hat s{\_{\mathsf}{D}}{\,{=}\sigma{\Rightarrow}\,} S\_\sigma$, yielding $\hat s{\_{\mathsf}{D}}{\,{=}\sigma a{\Rightarrow}\,} S\_{\sigma a}$. Also the second claim is proven by induction. The definition of $\Delta{\_{\mathsf}{D}}$ constructs no τ-transitions, so if $\hat s{\_{\mathsf}{D}}{\,{=}\varepsilon{\Rightarrow}\,} s{\_{\mathsf}{D}}$, then $s{\_{\mathsf}{D}}= \hat s{\_{\mathsf}{D}}= S\_\varepsilon$. Trivially $\varepsilon \in {{\mathit}{Tr}}(L)$. If $\hat s{\_{\mathsf}{D}}{\,{=}\sigma a{\Rightarrow}\,} s{\_{\mathsf}{D}}$, then consider the last transition along the path. By the definition of $\Delta{\_{\mathsf}{D}}$, it is of the form (Sρ, a, Sρa), where $\rho a \in {{\mathit}{Tr}}(L)$, $\hat s{\_{\mathsf}{D}}{\,{=}\sigma{\Rightarrow}\,} S\_\rho$, and $S\_{\rho a} = s{\_{\mathsf}{D}}$. By the induction assumption Sρ = Sσ. We get $s{\_{\mathsf}{D}}= S\_{\rho a} = \{ s \mid \exists s' \in S\_\rho: s' {\,{=}a{\Rightarrow}\,} s \} = \{ s \mid \exists s' \in S\_\sigma: s' {\,{=}a{\Rightarrow}\,} s \} = S\_{\sigma a}$. Because $\rho a \in {{\mathit}{Tr}}(L)$, we have ∅ ≠ Sρa = Sσa, so $\sigma a \in {{\mathit}{Tr}}(L)$. Then we define the unambiguation of L as $$\begin{aligned} {{\mathsf}{Una}}(L) & := & L {\:\|\|\:}{{\mathsf}{Det}}(L)\textrm{.}\nonumber\end{aligned}$$ [L:Una-bs] ${{\mathsf}{Una}}(L) {\equiv}L$, that is, ${{\mathsf}{Una}}(L)$ is bisimilar with L. Let ${{\mathsf}{Una}}(L) = (S{\_{\mathsf}{U}}, \Sigma{\_{\mathsf}{U}}, \Delta{\_{\mathsf}{U}}, \hat s{\_{\mathsf}{U}})$. The states of ${{\mathsf}{Una}}(L)$ are of the form $s{\_{\mathsf}{U}}= (s,S\_\sigma)$. Let “ ∼ ” ⊆ $S \times S{\_{\mathsf}{U}}$ be defined by s ∼ (sʹ, Sσ) if and only if $\hat s {\,{=}\sigma{\Rightarrow}\,} s = s'$. We now show that “ ∼ ” is a bisimulation. “(1)”, etc., refer to the numbers in the definition on p. . * Clearly $\Sigma({{\mathsf}{Una}}(L)) = \Sigma{\_{\mathsf}{U}}= \Sigma \cup \Sigma = \Sigma = \Sigma(L)$. * We have $\hat s{\_{\mathsf}{U}}= (\hat s, \hat s{\_{\mathsf}{D}}) = (\hat s, S\_\varepsilon)$ and $\hat s {\,{=}\varepsilon{\Rightarrow}\,} \hat s$, so $\hat s \sim \hat s{\_{\mathsf}{U}}$. * Let s ∼ (s, Sσ), that is, $\hat s {\,{=}\sigma{\Rightarrow}\,} s$. * If (s, τ, sʹ) ∈ Δ, then $((s,S\_\sigma), \tau, (s',S\_\sigma)) \in \Delta{\_{\mathsf}{U}}$ and $\hat s {\,{=}\sigma{\Rightarrow}\,} s'$, yielding sʹ ∼ (sʹ, Sσ). If (s, a, sʹ) ∈ Δ where a ∈ Σ, then $\hat s {\,{=}\sigma a{\Rightarrow}\,} s'$. The definition of $\Delta{\_{\mathsf}{D}}$ yields $(S\_\sigma, a, S\_{\sigma a}) \in \Delta{\_{\mathsf}{D}}$, implying $((s,S\_\sigma), a, (s',S\_{\sigma a})) \in \Delta{\_{\mathsf}{U}}$. We have sʹ ∼ (sʹ, Sσa). * If $((s,S\_\sigma), \tau, (s',s'{\_{\mathsf}{D}})) \in \Delta{\_{\mathsf}{U}}$, then by the definitions of “∣∣” and $\Delta{\_{\mathsf}{D}}$ we have (s, τ, sʹ) ∈ Δ and $s'{\_{\mathsf}{D}}= S\_\sigma$. Furthermore, $\hat s {\,{=}\sigma{\Rightarrow}\,} s'$. So $s' \sim (s',s'{\_{\mathsf}{D}})$. If $((s,S\_\sigma), a, (s',s'{\_{\mathsf}{D}})) \in \Delta{\_{\mathsf}{U}}$ where a ∈ Σ, then (s, a, sʹ) ∈ Δ. It implies $\hat s {\,{=}\sigma a{\Rightarrow}\,} s'$. We also have $(S\_\sigma, a, s'{\_{\mathsf}{D}}) \in \Delta{\_{\mathsf}{D}}$, yielding $\sigma a \in {{\mathit}{Tr}}({{\mathsf}{Det}}(L))$ and $s'{\_{\mathsf}{D}}= S\_{\sigma a}$ by Lemma [L:Det]. Again $s' \sim (s',s'{\_{\mathsf}{D}})$. We say that a state of ${{\mathsf}{Una}}(L)$ is potentially divergent if it can be reached via a divergence trace, and certainly nondivergent otherwise. These phrases do not actually refer to the properties of the state but to the properties of the traces that lead to it. The essential useful property of ${{\mathsf}{Una}}(L)$ is stated in the following lemma. [L:Una-div] If state $s{\_{\mathsf}{U}}$ of ${{\mathsf}{Una}}(L)$ is potentially divergent, then all traces that lead to it belong to ${{\mathit}{Div}}(L)$. If state $s{\_{\mathsf}{U}}$ of ${{\mathsf}{Una}}(L)$ is certainly nondivergent, then no trace that leads to it belongs to ${{\mathit}{Div}}(L)$. If $\hat s{\_{\mathsf}{U}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{U}}$, then $s{\_{\mathsf}{U}}$ is of the form $(s, s{\_{\mathsf}{D}})$, where $\hat s {\,{=}\sigma{\Rightarrow}\,} s$ and $\hat s{\_{\mathsf}{D}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{D}}$. By Lemma [L:Det], $s{\_{\mathsf}{D}}= S\_\sigma$. If also $\hat s{\_{\mathsf}{U}}{\,{=}\rho{\Rightarrow}\,} s{\_{\mathsf}{U}}$, then $S\_\rho = s{\_{\mathsf}{D}}= S\_\sigma$. If $\sigma \in {{\mathit}{Div}}(L)$, then there is an sʹ ∈ Sσ = Sρ such that sʹ − τω → , implying $\rho \in {{\mathit}{Div}}(L)$. Therefore, either none or all of the traces that lead to $s{\_{\mathsf}{U}}$ are divergence traces. In Fig. [F:DetUna], the rightmost state of L has been split to two states in ${{\mathsf}{Una}}(L)$, a certainly nondivergent one led to by a and a potentially divergent one led to by b. Then we define a function ${{\mathsf}{PD}}$ that makes the following property hold while preserving bisimilarity: for every state s, either no or all traces that lead to s have a divergence trace as a prefix. This is obtained by adding a component to ${{\mathsf}{Una}}(L)$ that remembers if the execution has gone through a divergence trace. Formally, by ${{\mathsf}{PD}}(L)$ we mean the LTS $(S{\_{\mathsf}{P}}, \Sigma, \Delta{\_{\mathsf}{P}}, \hat s{\_{\mathsf}{P}})$ that is obtained as follows. Let $[\sigma] = {\mathsf}{pre}$ if $\sigma \in {{\mathit}{anT}}(L)$ and $[\sigma] = {\mathsf}{post}$ otherwise. Let στ = σ and σa = σa if a ∈ Σ. First L is replaced by ${{\mathsf}{Una}}(L) = (S{\_{\mathsf}{U}}, \Sigma, \Delta{\_{\mathsf}{U}}, \hat s{\_{\mathsf}{U}})$. Then let $$\begin{aligned} S{\_{\mathsf}{P}}& = & \{ (s{\_{\mathsf}{U}}, [\sigma]) \mid \hat s{\_{\mathsf}{U}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{U}}\}\nonumber\\ \Delta{\_{\mathsf}{P}}& = & \{ ((s{\_{\mathsf}{U}}, [\sigma]), a, (s'{\_{\mathsf}{U}}, [\sigma\_a])) \mid \hat s{\_{\mathsf}{U}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{U}}\wedge (s{\_{\mathsf}{U}}, a, s'{\_{\mathsf}{U}}) \in \Delta{\_{\mathsf}{U}}\}\nonumber\\ \hat s{\_{\mathsf}{P}}& = & (\hat s{\_{\mathsf}{U}}, [\varepsilon])\nonumber\end{aligned}$$ We say that $(s{\_{\mathsf}{U}}, x)$ is pre-divergent if $x = {\mathsf}{pre}$ and post-divergent otherwise. [L:PPD] We have ${{\mathsf}{PD}}(L) {\equiv}L$. If state $s{\_{\mathsf}{P}}$ of ${{\mathsf}{PD}}(L)$ is pre-divergent, then all traces that lead to it belong to ${{\mathit}{anT}}(L)$. If state $s{\_{\mathsf}{P}}$ of ${{\mathsf}{PD}}(L)$ is post-divergent, then no trace that leads to it belongs to ${{\mathit}{anT}}(L)$. We have ${{\mathsf}{PD}}(L) {\equiv}{{\mathsf}{Una}}(L) {\equiv}L$, because the relation $(s{\_{\mathsf}{U}}, [\sigma]) \sim s'{\_{\mathsf}{U}}\Leftrightarrow s{\_{\mathsf}{U}}= s'{\_{\mathsf}{U}}$ is a bisimulation between $S{\_{\mathsf}{P}}$ and $S{\_{\mathsf}{U}}$. If $[\sigma\_a] = {\mathsf}{pre}$, then $\sigma\_a \in {{\mathit}{anT}}(L)$, implying $\sigma \in {{\mathit}{anT}}(L)$ and $[\sigma] = {\mathsf}{pre}$. Thus ${{\mathsf}{PD}}(L)$ has no transitions from post-divergent to pre-divergent states. Let $\hat s{\_{\mathsf}{P}}{\,{=}\rho{\Rightarrow}\,} (s{\_{\mathsf}{U}},x)$ and $\rho \in {{\mathit}{Div}}(L)$. Because $(s{\_{\mathsf}{U}},x) \in S{\_{\mathsf}{P}}$, there is a σ such that $\hat s{\_{\mathsf}{U}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{U}}$ and x = [σ]. Because $\rho \in {{\mathit}{Div}}(L)$, $s{\_{\mathsf}{U}}$ is potentially divergent. By Lemma [L:Una-div], all traces that lead to it are divergence traces. That includes σ. Thus $x = {\mathsf}{post}$. As a consequence, each divergence trace only leads to post-divergent states. By the first result in this proof, the same holds for each trace that has a divergence trace as a prefix. If an execution of ${{\mathsf}{PD}}(L)$ leads to a post-divergent state, then $\hat s{\_{\mathsf}{P}}$ is post-divergent or the execution contains a transition of the form $((s{\_{\mathsf}{U}}, {\mathsf}{pre}), a, (s'{\_{\mathsf}{U}}, {\mathsf}{post}))$. In the first case, $[\varepsilon] = {\mathsf}{post}$, so $\varepsilon \in {{\mathit}{Div}}(L)$. In the second case, by the definition of $\Delta{\_{\mathsf}{P}}$, there is a σ such that $\hat s{\_{\mathsf}{U}}{\,{=}\sigma{\Rightarrow}\,} s{\_{\mathsf}{U}}$, $\sigma \in {{\mathit}{anT}}(L)$, and $\sigma\_a \notin {{\mathit}{anT}}(L)$. This implies $\sigma a \in {{\mathit}{Div}}(L)$. So $s'{\_{\mathsf}{U}}$ is potentially divergent and all traces that lead to it are divergence traces. As a consequence, each post-divergent state has a divergence trace in each of its histories. In Fig. [F:DetUna], the rightmost state of ${{\mathsf}{Una}}(L)$ has been split to two states in ${{\mathsf}{PD}}(L)$, one such that all traces leading to it start with the only divergence trace b, and another such that no trace leading to it starts with b. Upper sub-region ---------------- In this subsection we survey the rest of the current region. Armed with ${{\mathsf}{PD}}$, we can attack the case where ${{\mathit}{Tr}}$, ${{\mathit}{Sf}}$, and ${{\mathit}{minD}}$ are preserved, but ${{\mathit}{Div}}$ and ${{\mathit}{Inf}}$ are not. This time there is no unique next congruence, but two. Therefore, the proof consists of two parts, where the first throws away information on divergence traces that are not minimal, and the second on infinite traces that are not always-nondivergent. Again, to reuse the construction in Section [S:BLisLL], we present it as a lemma that does not assume that ${{\mathit}{Sf}}$ is preserved. [L:noDiv] Assume that “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Tr}}$ and ${{\mathit}{minD}}$ but not ${{\mathit}{Div}}$. 1. For every LTS L there is an LTS f(L) such that L ≅ f(L), ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Tr}}(L) \cap {{\mathit}{extT}}(L)$, and ${{\mathit}{Inf}}(f(L)) = {{\mathit}{Inf}}(L)$. 2. If “ ≅ ” does not preserve ${{\mathit}{Inf}}$, then for every LTS L there is an LTS f(L) such that L ≅ f(L), ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Tr}}(L) \cap {{\mathit}{extT}}(L)$, and ${{\mathit}{Inf}}(f(L)) = {{\mathit}{anI}}(L)$. (180,20)(0,10)(24,10)(12,13)a (28,10)(52,10)(40,13)c (56,10)(80,10)(68,13)b1 (54,10)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(53,4)τ (84,10)(108,10)(96,13)b2 (118,10)⋯ (118,3)⋯ (82,10)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(81,4)τ (128,10)(152,10)(140,13)bn − 1 (154,10)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(153,4)τ (156,10)(180,10)(168,13)bn (26,10)2(54,10)2(82,10)2(154,10)2 [F:testdiv] Let M1 ≅ M2, $\sigma \in {{\mathit}{Div}}(M\_1) \setminus {{\mathit}{Div}}(M\_2)$, b1⋯bn = σ[1], c = 1[0], and d = 2[0]. By Theorem [T:noSigma], “ ≅ ” preserves Σ, so we may let ΣM = Σ(M1) = Σ(M2). For any LTS L, let ΣL = Σ(L) and let g(L) be the LTS that is obtained as follows. First L is replaced by ${{\mathsf}{PD}}({\lceilL\rceil^{[2]}})$. If $\hat s{\_{\mathsf}{P}}$ is pre-divergent, then it is the new initial state, and each transition (s, a, sʹ) where s is pre-divergent and sʹ is post-divergent is replaced by a copy of the LTS fragment shown in Fig. [F:testdiv]. Otherwise a copy of Fig. [F:testdiv] is added such that its a-transition is left out, the start state of the c-transition is the new initial state, and the LTS fragment leads to $\hat s{\_{\mathsf}{P}}$. The alphabet of g(L) is {c} ∪ ΣM[1] ∪ ΣL[2]. When completing a minimal divergence trace of ⌈L⌉[2], g(L) executes cσ[1] before continuing, but otherwise it behaves like ⌈L⌉[2]. Later, in the proof of claim (b), we will introduce ΣN, Nʹ1, and Nʹ2. To have a place for them in our construction, we now let $N'\_0 = {\begin{pspicture}(6,10)(0,2){\rput[lb](4,2){{\psline[linearc=2]{->}}(-4,8)(-0.9,1.8)}}{\pscircle[fillcolor=white]}(4,2){2} \end{pspicture}}$ (with Σ(Nʹ0) = ∅). When i ∈ {1, 2} and j ∈ {0, 1, 2}, let Mʹi = c.⌈Mi ⊓ M2⌉[1] and fi, j(L) = ⌊ ( g(L) ∣∣ Mʹi ∣∣ Nʹj ) \ ({c, d} ∪ ΣM[1] ∪ ΣN[3]) ⌋[2]. Clearly Nʹ0 has no effect on the behaviour. With Nʹ0, independently of what ΣN is, also the hiding with ΣN[3] has no effect. We show now that L ≐ f2, 0(L). Before completing any minimal divergence trace, f2, 0(L) behaves like L. When g(L) executes c, one of the two copies of M2 in Mʹ2 is switched on. Then g(L) tries to execute σ[1]. If it fails because M2 blocks it, then f2, 0(L) diverges due to the τ-loops in Fig. [F:testdiv]. That is still equivalent to L, because the trace that has been executed is a minimal divergence trace. For the same reason it is okay if M2 diverges before completing σ. The execution of σ may also succeed, because $\sigma \in {{\mathit}{Div}}(M\_1) \subseteq {{\mathit}{Tr}}(M\_1) = {{\mathit}{Tr}}(M\_2)$. In that case, because $\sigma \notin {{\mathit}{Div}}(M\_2)$, M2 is left in a nondivergent state, having no effect on the further behaviour. So f2, 0(L) continues like L. Because Mʹ1 has a copy of both M1 and M2, f1, 0(L) behaves otherwise like f2, 0(L), but it has additional behaviour caused by M1 starting in Mʹ1, executing σ completely, and diverging. In that case, every subsequent state of f1, 0(L) is divergent. Thus L ≐ f2, 0(L) ≅ f1, 0(L), ${{\mathit}{Tr}}(f\_{1,0}(L)) = {{\mathit}{Tr}}(L)$, ${{\mathit}{Sf}}(f\_{1,0}(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{minD}}(f\_{1,0}(L)) = {{\mathit}{minD}}(L)$, ${{\mathit}{Div}}(f\_{1,0}(L)) = {{\mathit}{Tr}}(L) \cap {{\mathit}{extT}}(L)$, ${{\mathit}{anI}}(f\_{1,0}(L)) = {{\mathit}{anI}}(L)$, and ${{\mathit}{Inf}}(f\_{1,0}(L)) = {{\mathit}{Inf}}(L)$. As a consequence, f1, 0 qualifies as the f of claim (a). (259,89)(19,57)(19,50)(19,48)(-1.4,1.4)(-7,7)(-10,0)(-2,0)(10,52)a1 (18,37)am (7,47)⋱ (20.2,49.6)(45.8,81.4)(33,71)c (20.2,46.4)(45.8,14.6)(33,36)c (35,42) (14,41)(38,41)(26,43)τ (42,41)(66,41)(54,43)τ (40,39)(40,15)(41,31)τ (70,41)(94,41)(82,43)τ (68,39)(68,15)(69,31)τ (98,41)(122,41)(110,43)τ (96,39)(96,15)(97,31)τ (132,41)⋯(132,27)⋯ (12,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(6,7)τ (40,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (26,15)⋮ (68,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (54,15)⋮ (96,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (82,15)⋮ (124,13)(-1.4,1.4)(-7,7)(-21,7)(-26.6,1.4)(-14,8.5)a1 (110,15)⋮ (132,13)⋯ (12,41)2(40,41)2(68,41)2(96,41)2 (12,13)2(40,13)2(68,13)2(96,13)2 (47,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(54,7)τ (49,13)(73,13)(61,16)d (75,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(82,7)τ (77,13)(101,13)(89,16)e1 (103,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(97,7)τ (103,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (117,15)⋮ (131,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(138,7)τ (133,13)(157,13)(145,16)e2 (159,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(153,7)τ (159,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (173,15)⋮ (187,13)(0,-2)(0,-10)(7,-7)(1.4,-1.4)(194,7)τ (189,13)(213,13)(201,16)e3 (215,13)(0,-2)(0,-10)(-7,-7)(-1.4,-1.4)(209,7)τ (215,13)(1.4,1.4)(7,7)(21,7)(26.6,1.4)(14,8.5)a1 (229,15)⋮ (253,13)⋯ (19,48)2(47,13)2(75,13)2(103,13)2(131,13)2 (159,13)2(187,13)2(215,13)2 [F:testinftr2] In the case of claim (b), there are N1, N2, and ξ such that N1 ≅ N2 and $\xi \in {{\mathit}{Inf}}(N\_1) \setminus {{\mathit}{Inf}}(N\_2)$. Let e1e2⋯ = ξ[3], ΣN = Σ(N1) = Σ(N2), and $\{a\_1, a\_2, \ldots{\gray, a\_m}\} = {\Sigma\_L^{[2]}}$. When j ∈ {1, 2}, let Nʹj = Tξ ∣∣ d.⌈Nj⌉[3], where Tξ is the LTS in Fig. [F:testinftr2] with the alphabet {c, d} ∪ ΣL[2] ∪ ΣN[3]. If j ∈ {1, 2}, c makes Tξ enter one of its two branches. Its initial state and upper branch can parallel any finite execution of g(L). Because Tξ never refuses any other subset of ΣL[2] than ∅, and because of the stable states initially and in the upper branch, ${{\mathit}{Sf}}(f\_{1,j}(L)) = {{\mathit}{Sf}}(f\_{1,0}(L))$. Furthermore, ${{\mathit}{Div}}(f\_{1,j}(L)) = {{\mathit}{Div}}(f\_{1,0}(L))$, because Tξ cannot diverge before executing c, and all traces that involve the execution of c are in ${{\mathit}{Div}}(f\_{1,0}(L))$. The upper branch of Tξ does not yield infinite traces. In its lower branch Tξ switches Nj on by executing d. Thanks to the initial state of Tξ and because N2 cannot execute ξ, we have ${{\mathit}{Inf}}(f\_{1,2}(L)) = {{\mathit}{anI}}(f\_{1,0}(L))$. Because N1 can execute ξ, we have ${{\mathit}{Inf}}(f\_{1,1}(L)) = {{\mathit}{Inf}}(f\_{1,0}(L))$. We get f1, 0(L) ≐ f1, 1(L) ≅ f1, 2(L). So f1, 2 qualifies as the f of claim (b). [T:Tr-minD-anI] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Tr}}$, ${{\mathit}{Sf}}$, and ${{\mathit}{minD}}$ but neither ${{\mathit}{Div}}$ nor ${{\mathit}{Inf}}$, then “ ≅ ” is the equivalence induced by Σ, ${{\mathit}{Tr}}$, ${{\mathit}{Sf}}$, ${{\mathit}{minD}}$, and ${{\mathit}{anI}}$. By Lemma [L:minD=>anI], “ ≅ ” preserves Σ and ${{\mathit}{anI}}$. The f of Lemma [L:noDiv](b) qualifies as the f of Lemma [L:nocongr]. We have now two directions to go: one where ${{\mathit}{Inf}}$ is preserved and another where ${{\mathit}{Div}}$ is preserved. Given the work we have done already, the former is easy. [T:Tr-minD-Inf] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Sf}}$, ${{\mathit}{minD}}$, and ${{\mathit}{Inf}}$ but not ${{\mathit}{Div}}$, then “ ≅ ” is the equivalence induced by Σ, ${{\mathit}{Tr}}$, ${{\mathit}{Sf}}$, ${{\mathit}{minD}}$, and ${{\mathit}{Inf}}$. By Lemma [L:Inf=>Tr], “ ≅ ” preserves Σ and ${{\mathit}{Tr}}$. The f of Lemma [L:noDiv](a) qualifies as the f of Lemma [L:nocongr]. We still have the case where ${{\mathit}{Div}}$ is preserved but ${{\mathit}{Inf}}$ is not. [L:noaenI] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Div}}$ but not ${{\mathit}{aenI}}$, then for every LTS L there is an LTS f(L) such that L ≅ f(L), ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Div}}(L)$, and ${{\mathit}{Inf}}(f(L)) = {{\mathit}{eanI}}(L)$. Let M1 ≅ M2 and $\xi \in {{\mathit}{aenI}}(M\_1) \setminus {{\mathit}{aenI}}(M\_2)$. By Theorem [T:noSigma], “ ≅ ” preserves Σ. Let ΣM = Σ(M1) = Σ(M2), c = 0[0], c1 = 1[0], and c2 = 2[0]. Because “ ≅ ” preserves ${{\mathit}{Div}}$, M1 and M2 agree on which prefixes of ξ are divergence traces. Infinitely many of them are not, by the definition of ${{\mathit}{aenI}}$. So non-empty σ1, σ2, σ3, … exist such that σ1σ2σ3⋯ = ξ[1] and σ1, σ1σ2, σ1σ2σ3, … are not divergence traces. Let Tξ be the LTS whose alphabet is {c, c1, c2} ∪ ΣM[1] and whose graph is (257,10)(4,2)(-4,8)(-0.9,1.8) (6,2)(22,2)(14,5)c1 (26,2)(42,2)(34,5)c (46,2)(0,0)(3,3)(6,0)(9,-3)(12,0)(15,3)(18,0)(21,-3)(24,0)(27,3)(30,0)(36,0)(64,5)σ1 (86,2)(102,2)(94,5)c2 (106,2)(122,2)(114,5)c1 (126,2)(0,0)(3,3)(6,0)(9,-3)(12,0)(15,3)(18,0)(21,-3)(24,0)(27,3)(30,0)(36,0)(144,5)σ2 (166,2)(182,2)(174,5)c2 (186,2)(202,2)(194,5)c1 (206,2)(0,0)(3,3)(6,0)(9,-3)(12,0)(15,3)(18,0)(21,-3)(24,0)(27,3)(30,0)(36,0)(224,5)σ3 (252,2)⋯ (4,2)2(24,2)2(44,2)2(84,2)2(104,2)2 (124,2)2(164,2)2(184,2)2(204,2)2 . For any LTS L, let g(L) be the LTS that is obtained as follows. First L is replaced by ${{\mathsf}{Una}}({\lceilL\rceil^{[2]}})$. Then each transition whose label a is visible and which ends in a potentially divergent state is replaced by (68,12)(0,2)(0,2)(20,2)(10,5)a (24,2)(44,2)(32,5)c1 (46,2)(0,2)(0,10)(7,7)(1.4,1.4)(45,8)τ (48,2)(68,2)(60,5)c2 (22,2)2(46,2)2 . The alphabet of the result is {c1, c2} ∪ ΣL[2], where ΣL = Σ(L). When i ∈ {1, 2}, let fi(L) = ⌊ ( g(L) ∣∣ Tξ ∣∣ c.⌈Mi⌉[1] ) \ ({c, c1, c2} ∪ ΣM[1]) ⌋[2]. Each time when g(L) is about to enter a potentially divergent state, it executes c1. This makes Tξ move one step and then let c.⌈Mi⌉[1] try to execute up to a nondivergent state. If it succeeds, Tξ lets g(L) continue by executing c2. In the opposite case, g(L) is trapped in the τ-loop between c1 and c2. The LTS M1 has every prefix of ξ as its trace. By Lemma [L:Div=>Tr], “ ≅ ” preserves ${{\mathit}{Tr}}$. So both ⌈M1⌉[1] and ⌈M2⌉[1] may succeed in executing σ1σ2⋯σi for any i. This implies ${{\mathit}{Tr}}(f\_1(L)) = {{\mathit}{Tr}}(f\_2(L)) = {{\mathit}{Tr}}(L)$. Clearly g(L) mimics the divergence traces of L. When M1 or M2 diverges, g(L) is in a τ-loop and the trace that has been executed is a divergence trace. Thus ${{\mathit}{Div}}(f\_1(L)) = {{\mathit}{Div}}(f\_2(L)) = {{\mathit}{Div}}(L)$. When g(L) is in a stable state (other than the start states of c1), then c.⌈M1⌉[1] and c.⌈M2⌉[1] do not diverge, so ${{\mathit}{Sf}}(f\_1(L)) = {{\mathit}{Sf}}(f\_2(L)) = {{\mathit}{Sf}}(L)$. Because M2 does but M1 does not necessarily prevent g(L) from infinitely many times continuing with c2 after a divergence trace, we have ${{\mathit}{Inf}}(f\_1(L)) = {{\mathit}{Inf}}(L)$ but ${{\mathit}{Inf}}(f\_2(L)) = {{\mathit}{eanI}}(L)$. So L ≐ f1(L) ≅ f2(L). [T:Tr-Div-eanI] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Sf}}$ and ${{\mathit}{Div}}$ but not ${{\mathit}{aenI}}$, then “ ≅ ” is the equivalence induced by Σ, ${{\mathit}{Sf}}$, ${{\mathit}{Div}}$, and ${{\mathit}{eanI}}$. By Lemma [L:Div=>eanI], “ ≅ ” preserves Σ and ${{\mathit}{eanI}}$. The f of Lemma [L:noaenI] qualifies as the f of Lemma [L:nocongr]. [L:noInf] If “ ≅ ” is a congruence, “ ≐ ” implies “ ≅ ”, and “ ≅ ” preserves ${{\mathit}{Div}}$ and ${{\mathit}{aenI}}$ but not ${{\mathit}{Inf}}$, then for every LTS L there is an LTS f(L) such that L ≅ f(L), ${{\mathit}{Sf}}(f(L)) = {{\mathit}{Sf}}(L)$, ${{\mathit}{Div}}(f(L)) = {{\mathit}{Div}}(L)$, and ${{\mathit}{Inf}}(f(L)) = {{\mathit}{aenI}}(L)$. For the purpose of this proof, we define eventually-always-divergent infinite traces as ${\mathit{eadI}}(L) = {{\mathit}{Inf}}(L) \setminus {{\mathit}{aenI}}	arxiv_0000473
Double robust estimation of partially adaptive treatment strategies =================================================================== Precision medicine aims to tailor treatment decisions according to patients’ characteristics. G-estimation and dynamic weighted ordinary least squares (dWOLS) are double robust statistical methods that can be used to identify optimal adaptive treatment strategies. They require both a model for the outcome and a model for the treatment and are consistent if at least one of these models is correctly specified. It is underappreciated that these methods additionally require modeling all existing treatment-confounder interactions to yield consistent estimators. Identifying partially adaptive treatment strategies that tailor treatments according to only a few covariates, ignoring some interactions, may be preferable in practice. It has been proposed to combine inverse probability weighting and G-estimation to address this issue, but we argue that the resulting estimator is not expected to be double robust. Building on G-estimation and dWOLS, we propose alternative estimators of partially adaptive strategies and demonstrate their double robustness. We investigate and compare the empirical performance of six estimators in a simulation study. As expected, estimators combining inverse probability weighting with either G-estimation or dWOLS are biased when the treatment model is incorrectly specified. The other estimators are unbiased if either the treatment or the outcome model are correctly specified and have similar standard errors. Using data maintained by the Centre des Maladies du Sein, the methods are illustrated to estimate a partially adaptive treatment strategy for tailoring hormonal therapy use in breast cancer patients according to their estrogen receptor status and body mass index. R software implementing our estimators is provided. Keywords: Causal inference; Double robustness; Dynamic treatment regimens; Inverse probability weighting; Personalized medicine; Precision medicine Introduction ============ Precision medicine, sometimes also called personalized medicine, is a medical approach that aims to improve patients’ outcomes by tailoring treatment decisions taking into account their genes, environments and lifestyles (United States Food and Drug Administration, 2018). This approach has garnered increasing attention over recent years. Notably, multiple countries have put forward initiatives to accelerate research on precision medicine, including the United States, Canada and various European countries. Adaptive treatment strategies (ATSs), or dynamic treatment regimes, is a type of precision medicine. ATSs aim to determine time-dependent treatment decision rules that use prior patients’ information to optimize a clinical outcome among patients sharing similar characteristics. Multiple statistical techniques have been proposed for estimating ATSs (for example, Watkins, 1989; Murphy, 2003; Robins, 2004; Wallace and Moodie, 2015). G-estimation (Robins, 2004) and dynamic weighted ordinary least squares (dWOLS; Wallace and Moodie, 2015) have the advantage of offering some robustness to statistical modeling errors. These methods require the specification of two models: a treatment model that relates potential confounders to the treatment decisions, and an outcome model that relates potential confounders and treatments to the outcome. The outcome model is further divided in a treatment-free component that includes only potential confounders, and a “blip” component that features terms related to the treatment effect and possible effect modification according to tailoring variables. The blip is the statistical quantity of interest for determining optimal decision rules. G-estimation and dWOLS yield consistent estimators of the blip’s parameters if either the treatment model or the treatment-free component is correctly specified, but not necessarily both, a property known as double robustness. The blip must, however, be correctly specified. Other double robust methods for estimating ATSs have been proposed (for example, Petersen et al, 2004; van der Laan and Petersen, 2007), but we focus on G-estimation and dWOLS because of their relative ease of implementation and interpretation within a regression-like framework. G-estimation and dWOLS do not formally require including the same covariates in the blip component as in the treatment-free component. For dWOLS, any covariate interacting with the treatment in the blip must be included as main effect in the treatment-free component (that is, the “hierarchy” of the model must be preserved), while G-estimation has no specific requirement. In practice, it is common to include fewer covariates in the blip than in the treatment-free model. For example, when estimating an ATS for the treatment of type 2 diabetes, Simoneau et al (2020) included multiple potential confounders in the treatment-free component, but only glycemic control, body mass index, previous treatment and history of hypoglycemia in the blip component. Multiple reasons may motivate tailoring treatment decisions according to only a few variables. For instance, some variables may be confounders in the context of a given dataset, but are expected to be unavailable for treatment tailoring in the intended clinical context. Simplifying decision rules to facilitate their use in clinical practice or to improve statistical power are other examples of reasons to exclude some variables from the blip. A perhaps underappreciated possible consequence of excluding some potential confounders from the blip is that the blip may be incorrectly specified if some true treatment effect modification remains unmodeled. As mentioned previously, both G-estimation and dWOLS require the blip to be correctly specified for consistent estimation of its parameters. A solution to this problem has been proposed by van der Laan and Robins (2003). It combines G-estimation with inverse probability of treatment weighting (IPTW), where the IPTW is used to control the residual confounding bias attributable to the (voluntary) possible misspecification of the blip. However, we expect this solution is not fully double robust, since it relies on the correct specification of a treatment model to account for the blip misspecification. In this paper, we propose alternative double robust estimators of partially adaptive treatment strategies (PATSs), where the treatment is tailored according to only some covariates, excluding potential true effect modifiers. In the next section, we introduce the notation and briefly review dWOLS estimation of an ATS.. Analogous G-estimation methods are available in Appendix C. In Section [s:simulation], we employ simulation studies to illustrate the potential bias of using standard ATSs estimators for estimating PATSs as well as the double robustness of our proposed estimators. In Section [s:application], we estimate a PATS that aims to tailor hormonal therapy for treating breast cancer as a function of estrogen receptor status and body mass index (BMI) in data maintained by the Centre des Maladies du Sein. We conclude in Section [s:discussion] with a discussion of the results and perspectives for future research. Notation and review of adaptive treatment strategy estimation ============================================================= Notation -------- Let Aj be the exposure at time j (j = 1, ..., K), Xj the pre-treatment covariates at time j, and Y the final outcome. Without loss of generality, we assume that a greater value of Y represents a better clinical outcome and that Aj = 0 represents a reference value for the treatment (for example, a placebo or standard care). We use over- and underbars to represent the set of past and future values of a variable, respectively, both including the present. For example, Āj = {A1, ..., Aj} and $\underline{A}\_j = \{A\_{j}, A\_{j+1},..., A\_K\}$. We further denote by Hj the observed history before treatment j, including previous treatments; hence, Hj = {Āj − 1, X̄j}. Let dj denote a treatment decision rule, which is a function of the observed history. The goal of an ATS is to determine the optimal treatment decision rule djopt at each time j in order to maximize the final outcome Y. To define the causal parameter of interest more formally, we make use of the counterfactual framework. For example, Yd̄Kopt is the outcome that would have been observed if, possibly contrary to the fact, the treatment strategy had been optimal throughout the entire follow-up. Using this notation, the causal parameter of interest in an ATS is $$\begin{aligned} \gamma\_j(a\_j, h\_j) = \mathbb{E}\left[Y^{\bar{a}\_{j-1}, a\_j, \underline{d}\_{j+1}^{opt}} - Y^{\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt}}\|H\_j = h\_j\right] = \mathbb{E}\left[Y^{\bar{a}\_j,\underline{d}\_{j+1}^{opt}} - Y^{\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt}}\|H\_j = h\_j\right].\end{aligned}$$ This is the blip to 0 function. It represents the effect of treatment Aj = aj compared to Aj = 0 for a given history hj, assuming that the future treatment strategy is optimal ($\underline{a}\_{j+1} = \underline{d}\_{j+1}^{opt}$).. Note that the optimal future treatment strategy may not be the same under treatment history āj as under treatment history (āj − 1, 0). In other words, $\underline{d}\_{j+1}^{opt}$ may not represent the same treatment in $(\bar{a}\_j,\underline{d}\_{j+1}^{opt})$ as in $(\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt})$, however in both cases the same treatment rule (or function) is applied; only the arguments to that function differ. Non-parametric identification of γj(aj, hj) is possible from the observed data under the following assumptions: * Consistency: If Ā = ā then Y = Yā. This assumption entails that the outcome of a given subject is not affected by the treatment of other subjects (absence of interference) and that there are not multiple versions of each treatment level a. * Sequential exchangeability (no unmeasured confounders): $(Y^{\bar{a}}, \underline{X}^{\bar{a}}\_{j+1}) \coprod A\_j \| \bar{A}\_{j-1}, \bar{X}\_j$ for all ā and j = 1, ..., K. * Positivity: P(Aj = aj∣hj) > 0 for all j = 1, ..., K, all aj, and all hj such that f(hj) > 0. If the positivity assumption fails to hold, it is still possible to estimate an optimal ATS among feasible strategies, that is, among strategies that have been observed in the data. In practice, a parametric model is assumed for the blip function, such that we can write γj(aj, hj; ψj). Estimation of adaptive treatment strategies ------------------------------------------- We now briefly present the dWOLS algorithm for estimating the blip γj(aj, hj; ψj). More details can be found elsewhere (for example, see Wallace and Moodie, 2015) and a description of the G-estimation algorithm can be found in Appendix A, as well as in Robins (2004) or Moodie, Richardson and Stephens (2007). Estimation of ψj using dWOLS is performed recursively. First, balancing weights such that Aj∐Hj in the weighted data must be specified. Constructing such weights generally involves specifying a model for E[Aj∣Hj];. Wallace and Moodie (2015) give a few examples of such balancing weights in the case of a binary treatment and observed that $w(A\_j, H\_j) = \|A\_j - \hat{\mathbb{E}}[A\_j\|H\_j]\|$ performs particularly well. These weights correspond to the overlap weights, which have been shown to yield a minimal variance weighted estimator of a certain average treatment effect under some assumptions (Li, Morgan and Zaslavsky, 2018; Li and Li, 2019). Schulz and Moodie (2020) propose various balancing weights for the multilevel and continuous treatment cases. Define the pseudo-outcome ỸiK = Yi and Ỹij = Yi + ∑k = j + 1K[γk(dkopt, hik; ψ̂k) − γk(aik, hik; ψ̂k)] if j ≠ K. Finally, a model for E[Ỹj∣Aj, Hj] = fj(hj; βj) + γj(aj, hj; ψj) is specified, where fj(hj; βj) is a function relating Hj to the outcome, and its parameters are estimated using the weighted data. Estimation of partially adaptive treatment strategies ===================================================== We now turn our attention to the estimation of PATSs, that is, where the objective is to tailor treatment decisions only according to a subset of the measured history. To formally define the causal parameter of interest of a PATS, we partition the covariates history at each time-point in two disjoint subsets Hj = (Hj\* ∪ HjC), where Hj\* are the covariates that are intended to be used for tailoring the treatment at time j and HjC are the other covariates. The parameter of interest is defined as Nonparametric identification ---------------------------- Combining dWOLS and Inverse probability of treatment weighting -------------------------------------------------------------- A first estimator of γj\(a*j, hj\; ψ*j\) is based on the solution proposed by van der Laan and Robins (2003), Sections 6.5.2 and 6.5.3, in the more general context where the causal parameter of interest is expressed as a function of the density of the counterfactual outcomes. Their proposal consists of combining G-estimation with IPTW, where the IPTW is employed to account for the possible misspecification of the blip due to unmodeled true effect modification. We detail an analogous estimator combining IPTW with dWOLS, which we denote “IPTW+dWOLS.” Following Wallace and Moodie (2015), we show in Appendix D that the solution to the IPTW+dWOLS estimator is also a solution the “IPTW+G-estimation” estimating equations in certain circumstances. Next, balancing weights w\(Aj, Hj\) such that A*j∐Hj\* in the weighted data must be specified. In addition, weights $\varpi(A\_j, H\_j, H\_j^\) = \frac{P(A\_j\|H\_j^\)}{P(A\_j\|H\_j)}$ are computed.. Finally, specify a model for E[Ỹj\∣A*j, Hj] = fj(hj; βj) + γj\(a*j, hj\; ψ*j\) and estimate its parameters using the data weighted according to $w^\(A\_j, H\_j^\) \times \varpi(A\_j, H\_j, H\_j^\)$. Alternative estimators ---------------------- The last remaining step is to compute the outer expectation of ([eq.PATS2]). A first approach for doing this is to estimate the density fHjC∣Hj\* and integrate γj†(aj, hj; ψj†) over HjC. In simple situations, the nonparametric empirical estimator $\hat{f}\_{H\_j^C\|H\_j^\}(h\_j^c\|h\_j^\) = \frac{\sum\_{i=1}^n I(H\_{ij}^C = h\_j^c, H\_{ij}^\* = h\_j^\)}{\sum\_{i=1}^n I(H\_{ij}^\ = h\_j^\)}$, where I( ⋅ ) is the usual indicator function, could be used. Other density estimators would generally be required. We henceforth call this approach “integrate dWOLS” and the analogous G-estimation approach as “integrate G-estimation.” An alternative implementation consists in computing the expectation through a linear model. More precisely, we calculate Q(a, hij) = γ*j†(a, hij; ψ̂\) for all observations, that is, the predicted value of γ*j†(Aj, hij; ψj†) when setting Aj = a. Next, we regress the predicted γj†(Aj, hij; ψj†) on covariates Hj\, thus fitting a model E[Q(a, H*j)∣Hj\] = γ\(a, hj\; ψ*j\). A clear advantage of this implementation is that it does not require estimating densities or computing integrals. This approach is called “CE dWOLS” and the analogous G-estimation approach is called “CE G-estimation” moving forward. Double robustness ----------------- The intuition for the double robustness of “integrate dWOLS” and “CE dWOLS” is the same as that of dWOLS presented at the end of Section 2.2. More formally, the double robustness of our estimators can be demonstrated recursively. First, if considering the usual IPTW w(A*j, Hj) = Aj/P(Aj = 1∣Hj) + (1 − Aj)/P(Aj = 0∣Hj), note that the double robustness of ψ̂K\* and ψ̂K (i.e. the treatment rule parameters from the final stage) is a direct consequence of the results of Kang and Schafer (2007) and Robins et al (2007) who demonstrate the double robustness of the ordinary least square estimator weighted according to IPTW, as long as an intercept term is included in the outcome (treatment-free) model. In Appendix B, we provide a proof of the double robustness of ψ̂K for the case where the weights w(Aj, Hj) = ∣Aj − E[Aj∣Hj]∣ are used. The double robustness of ψ̂K\* follows directly. To demonstrate the double robustness of ψ̂j\* for j < K, assume that γk\(a*k, hk\; ψ*k\) and γ*k(ak, hk; ψk) for k > j are consistently estimated. As a result, the mean of Ỹj\* for each (aj, hj) is also consistent for $\mathbb{E}\left[Y^{\bar{a}\_j,\underline{d}\_{j+1}^{opt\}}\|h\_j\right]$. Appealing once more to the previous double robustness results, it follows that γ*j†(aj, hj; ψ̂j†) is a double robust estimator for $\mathbb{E}\left[Y^{\bar{a}\_j,\underline{d}\_{j+1}^{opt\}} - Y^{\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt\}}\|h\_j\right]$. Hence, ψ̂j\* is also double robust for ψj\. We conclude that consistent estimation of the parameters of γ*j\(a*j, hj\; ψ*j\) at each time-point requires either the treatment model or the treatment-free model to be correctly specified, in addition to correct specification of the blips γ*j\(a*j, hj\; ψ*j\), γ*j(aj, hj; ψj) and γj†(aj, hj; ψj†) with respect to included covariates (which may, by virtue of the PATS model, be a subset of all variables that interact with treatment), and the consistent estimation of the parameters of later blips. Inferences ---------- Analytical calculation of confidence intervals for PATS is challenging because of the form of the parameter of interest. Alternatively, nonparametric bootstrap may be employed. However, this approach, as well as usual asymptotic variance estimators, would not be valid in cases where some γj\(a*j, hj\; ψ*j\) = 0 when K* > 1 for j ≠ K and some hj\. Indeed, the optimal treatment strategy $\underline{d}\_{j}^{opt\}$ is not uniquely defined in such situations, which yield estimators with non-regular limiting distribution (Robins, 2004). Even when γj\(a*j, hj\; ψ*j\) is small relative to the sample size but nonzero, poor inferences may be produced by standard bootstrap because of the near non-regularity of the estimator (Moodie and Richardson, 2010). However, the m-out-of-n* bootstrap has been found to perform well in such settings (Chakraborty, Laber and Zhao, 2013; Simoneau et al, 2018). This type of bootstrap is similar to regular nonparametric bootstrap, except that m < n observations are sampled at each replication. While the validity of this type of bootstrap relies on an appropriate choice of m, Chakraborty et al (2013) proposed a data-adaptive method for this choice that has been found to perform well. The data-adaptive choice determines m based, in part, on an estimate of the extent to which the estimator is non-regular, which is determined from the data by the estimated proportion of subjects for whom there is no unique optimal choice of treatment (see Appendix F for details). We therefore recommend utilizing this data-adaptive m-out-of-n bootstrap in general, although the regular nonparametric bootstrap may also be considered when nonregularity is not expected and the additional computational burden of the adaptive m-out-of-n bootstrap is prohibitive. Simulation study ================ Scenarios --------- Additional simulations in a two time-points setting are available in Appendix E. The simulation scenarios are inspired by those presented in Wallace and Moodie (2015) and Wallace, Moodie and Stephens (2016). Three different scenarios are considered. In all scenarios, H = X = (X1, X2) are two pre-treatment covariates and both are effect modifiers. However, only X1 is intended to be used to tailor treatment (H\* = X1, HC = X2). In all three scenarios X1 ∼ Bernoulli(p = 0.5) and X2 ∼ Bernoulli(p = 0.5). In Scenarios 1 and 3, A ∼ Bernoulli(p = expit{ − 0.5 + X1 + 0.5X2}); in Scenario 2, A ∼ Bernoulli(p = expit{ − 0.5 + X1 + 0.5X2 + X1X2}). In Scenario 1 and 2, Y ∼ N(0.25X1 + X2 + A{0.5 − X1 + 1.5X2}, 1); in Scenario 3, Y ∼ N(0.25X1 + X2 + X1X2 + A{0.5 − X1 + 1.5X2}, 1). The true γ1\(A, X1) is A(ψ0\ + ψ1\*X1) = A(0.5 + 1.5E[X2∣X1] − X1) = A(1.25 − X1). The optimal PATS is thus dopt \* = 1, whether X1 = 0 or X1 = 1. Note that the optimal PATS varies according to covariates’ values in the two time-points setting considered in Appendix E. The parameters ψ0\* and ψ1\* are estimated with “IPTW+G-estimation”, “IPTW+dWOLS”, “integrate G-estimation”, “integrate dWOLS”, “CE G-estimation”, “CE dWOLS” in addition to usual implementations of G-estimation and dWOLS for ATS. For all estimators, the blip model includes only X1, and the treatment and the treatment-free models only include main terms (no interactions). As such, in Scenario 1, both the treatment and the treatment-free models are correctly specified. In Scenario 2, only the treatment-free model is correctly specified. In Scenario 3, only the treatment model is correctly specified. A total of 1000 replications of each simulation scenario were generated for each sample sizes of n = 100, 1000 and 10 000. We report below the estimated relative bias ((average estimate - true value)/true value × 100%; Rel. bias) and standard deviation (SD) over the 1000 replications for each estimator in each combination of scenarios and sample size. The proportion of observations for which the optimal PATS was correctly identified and the expected loss (difference between the expected outcome under the true optimal PATS and the estimated optimal PATS) were also calculated within the same simulated data. In addition, we have explored the coverage of confidence intervals using the adaptive m-out-of-n bootstrap proposed by Chakraborty et al (2013) in a two-time-points scenario with near non-regularity (see Appendix F for details). Recall that non-regularity is not of concern in the single-stage setting, and so m-out-of-n results are not presented for the simulations involving only a single treatment decision. Results ------- The results of the main simulations are presented in Tables 1 and 2. Standard G-estimation and dWOLS, i.e. the approaches that use the incorrect (reduced) blip model with no adjustment for the partial nature of the ATS, produced estimates with non-negligible bias in all scenarios. On the other hand, all PATS estimators we introduced achieved unbiased estimation when the treatment model was correctly specified (Scenario 1 and 3). When the treatment model was misspecified, some bias remained for IPTW+G-estimation and IPTW+dWOLS, but not for the other PATS estimators. The standard deviation of the estimates were almost identical for all PATS estimators. The true optimal PATS was identified less often by standard G-estimation and dWOLS than the PATS estimators for n = 100 and n = 1000 in Scenarios 1 and 3, and for all sample sizes in Scenario 2. IPTW+dWOLS and IPTW+G-estimation also identified the optimal PATS less often than the other PATS estimators in Scenario 2 when n = 100 and n = 1000. For observations (“individuals”) whose true optimal PATS failed to be identified, the expected loss was always 0.25. Similar results were observed in the two time-point simulations, except that the difference in the ability to identify the optimal PATS between standard ATS estimators and the PATS estimators was more pronounced (see Appendix E). The coverage of the confidence intervals produced using the adaptive m-out-of-n bootstrap was approximately 95% (between 95.2% and 95.7% depending on the parameter), as desired (Appendix F). [h!] [tab1] [!htbp] [tab4] Application =========== Context ------- Breast cancer is the most common cancer among women and also the leading cause of cancer deaths among them (Bray et al, 2018). Various treatments can be used to treat patients who receive a breast cancer diagnosis including surgery, chemotherapy, radiotherapy and hormonal therapy. Hormonal therapy seeks stop hormone production or to interfere with the ability of hormones to attach to cancer cells and thus prevent their growth. This treatment is currently recommended for patients whose tumors are hormone receptor positive. Unfortunately, not all cancers that are hormone receptor positive respond adequately to hormonal therapy. Even when hormonal therapy helps treating the cancer itself, it can have multiple undesirable side effects, including hot flashes, sexual problems, weight gain, nausea, fatigue, high cholesterol and osteoporosis (Canadian Cancer Society, 2021). As such, it is important to appropriately tailor hormonal therapy so that it is prescribed only to patients who will benefit from it. Obesity is associated with poorer outcomes among patients with breast cancer (Chan et al, 2014). It is also known to affect the circulating levels of estrogen in the body (Calle and Kaaks, 2004). We thus hypothesized that obesity may modify the effect of hormonal therapy on survival in breast cancer patients. More precisely, we believed that hormonal therapy would only be beneficial in hormone receptor positive women with normal weight. While a breast cancer is generally considered to be hormone receptor positive if it is either estrogen receptor positive (ER+), progesterone receptor positive (PR+) or both, the relevance of testing PR is somewhat controversial. Indeed, it is already recommended to provide hormonal therapy when the cancer is ER+, regardless of PR status, and the profile ER-PR+ is very rare, representing only 3% of all breast cancers (Dunnwald, Rossing and Li, 2007). In fact, this profile is so rare that some have argued that testing PR has very little therapeutic impact and is unlikely to be cost-effective (Olivotto et al, 2004). Consequently, while effect modification by PR status is expected, tailoring hormonal therapy treatment according to PR status may not be necessary. To illustrate our method, we thus consider the estimation of a single-stage PATS that tailor hormonal therapy decisions according to both ER status and obesity, while neglecting the known interaction with PR status. Data and analysis ----------------- We analysed data concerning women diagnosed with a non-metastatic breast cancer between 1987 and 2009 that are part of the breast cancer registry maintained by the Centre des Maladies du Sein, in Québec, Canada. We compared the (log transformed) number of years of survival since breast cancer diagnosis (Y) between women who received hormonal therapy (A = 1) and those who did not (A = 0). Data on age, body mass index (BMI), menopause, cancer’s grade and stage, ER status, PR status, type of surgery, first degree familial history of breast cancer, hormone replacement therapy, chemotherapy, radiotherapy, trastuzumab and year of diagnosis were considered as potential confounders (X). While ER status, PR status and BMI were all considered as potential effect modifiers (H), only ER status and BMI were used as tailoring variables (H\). Before proceeding with the analysis, we treated the missing data as follows. When menopause status was missing, it was either imputed to “no” when age < 50 and to “yes” when age ≥ 50. Based on clinical and contextual knowledge, missing data on chemotherapy and radiotherapy were imputed to their most likely value (“no” for chemotherapy and “yes” for radiotherapy). Missing data on progesterone status and cancer’s grade were considered as an “unknown” category. Observations with missing data on other variables were deleted. A total of 5444 individuals were thus included in the analysis. To analyze the data, we used both the naive dWOLS estimator and the CE dWOLS estimator we have introduced. For both estimators, the treatment model was fitted using a logistic regression of the treatment according to all potential confounders. The outcome model was a linear regression of the natural logarithm of the survival time according to all potential confounders and including interaction terms between treatment and ER-BMI categories (ER+ and BMI < 25, ER+ and BMI ≥ 25, ER- and BMI < 25, ER- and BMI ≥ 25). The results are of this regression model are interpreted as differences in expected log years of survival. Except for years, which was modeled using a restricted cubic splines with five knots, all other covariates were entered in the same categories as presented in Table [TabAppli]. Because the survival time was right censored for 4151 patients, the outcome model was fitted only among observations that were not censored using an inverse probability of censoring weight as proposed by Simoneau et al (2020). Again, the censoring model was a logistic regression including all potential confounders. Censoring weights were truncated at their 99.9t*h percentile to reduce influence of large weights and to achieve a better representativeness of the weighted uncensored observations to the complete sample. Standard nonparametric bootstrap with 5000 replicates was used to produce confidence intervals. [!htbp] [TabAppli] @ lrrr & No hormonal therapy & Hormonal therapy & SMD & 1800 (33.0) & 3644 (67.0) Age & & & 0.419 ≤ 39 & 186 (10.3) & 119 ( 3.3) & 40–49 & 496 (27.6) & 625 (17.2) & 50–59 & 477 (26.5) & 1189 (32.6) & 60–69 & 353 (19.6) & 1009 (27.7) & ≥ 70 & 288 (16.0) & 702 (19.3) & BMI ≥ 25 & 698 (38.8) & 1698 (46.6) & 0.159 Menopause & 658 (36.5) & 875 (24.0) & 0.276 Grade & & & 0.791 1 & 225 (12.5) & 1055 (29.0) & 2 & 343 (19.1) & 1406 (38.6) & 3 & 994 (55.2) & 838 (23.0) & Unknown & 238 (13.2) & 345 ( 9.5) & Stage & & & 0.077 I & 809 (44.9) & 1671 (45.9) & II & 746 (41.4) & 1569 (43.1) & III & 245 (13.6) & 404 (11.1) & Estrogen + & 819 (45.5) & 3517 (96.5) & 1.360 Progesterone & & & 0.840 + & 604 (33.6) & 2627 (72.1) & - & 999 (55.5) & 805 (22.1) & Unknown & 197 (10.9) & 212 ( 5.8) & Surgery & & & 0.076 Mastectomy & 482 (26.8) & 867 (23.8) & Breast-conserving & 1289 (71.6) & 2730 (74.9) & None & 29 ( 1.6) & 47 ( 1.3) & 1st degree familial history & 399 (22.2) & 980 (26.9) & 0.110 Hormone replacement therapy & 1170 (65.0) & 1920 (52.7) & 0.252 Chemotherapy & 977 (54.3) & 1527 (41.9) & 0.250 Radiotherapy & 1307 (72.6) & 2875 (78.9) & 0.147 Trastuzumab & 70 ( 3.9) & 113 ( 3.1) & 0.043 Year & & & 0.729 1985–1989 & 269 (14.9) & 119 ( 3.3) & 1990–1994 & 384 (21.3) & 328 ( 9.0) & 1995–1999 & 402 (22.3) & 513 (14.1) & 2000–2004 & 328 (18.2) & 1164 (31.9) & 2005–2009 & 417 (23.2) & 1520 (41.7) & [-1.8ex] All numbers are n (%). SMD = standardized mean difference. [!htbp] [TabAppli2] Results ------- Table [TabAppli] presents the characteristics of the 3644 (67.0%) patients who received and 1800 (33.0%) who did not receive hormonal therapy. Among others, women who received hormonal therapy were overall older, had a greater BMI and a higher cancer’s grade. Unsurprisingly, there is a large imbalance between treated and untreated patients according to ER and PR statuses, since these markers are used for tailoring treatment in clinical practice. Hormonal therapy has also become more frequent in time. Results of the PATS analysis are presented in Table [TabAppli2]. In this specific application, only minor differences were observed between the results of the naive dWOLS and CE dWOLS estimators. In both analyses, hormonal therapy was associated with a lower survival time among ER- patients with a BMI ≥ 25 (CE dWOLS: -0.49; 95% CI: -0.83 to -0.12). Among the ER- participants with BMI < 25, a negative association was also observed, but the results were also compatible with an absence of difference (CE dWOLS: -0.28, 95%CI: -0.68 to 0.10). Among ER+ participants with BMI < 25, the observed difference in log survival times was close to zero with 95% confidence intervals covering equally small beneficial and detrimental associations (CE dWOLS: -0.01, 95% CI: -0.14 to 0.13). Finally, among ER+ participants with BMI ≥ 25, a small positive association was observed, but the confidence intervals indicated the data were compatible with a range of associations going from a slightly negative to moderately positive (CE dWOLS: 0.13, 95% CI: -0.10 to 0.34). Discussion ========== Better evidence-based tailoring of treatment decisions according to patients’ characteristics and to their evolving condition has become a priority. Robust statistical methods for identifying optimal treatment strategies are important tools for reaching this objective. Although it may be theoretically interesting to find the optimal treatment strategy according to multiple, or even all, relevant characteristics, this is often too ambitious or impractical. Indeed, achieving this would require infeasibly large datasets and may render treatment decisions overly complicated. Identifying optimal partially adaptive treatment strategies that aim to tailor treatment decisions according to a few selected characteristics may be a more reasonable goal in practice. In this paper, we proposed methods for identifying optimal PATSs that benefit from the double robustness property. Methods for producing inferences were additionally discussed. To facilitate the use of these methods, we supply functions in `R` as supplementary material. All our proposed estimators had similar performance in the simulation study in scenarios we investigated. As such, because CE dWOLS is simpler to implement, it may be preferred in practice. The simulation study results also supports the validity of the data-adaptive m-out-of-n bootstrap for producing inferences. It is noteworthy that the algorithm for implementing the m-out-of-n bootstrap for ATSs (or PATSs) has been developed for the two time-point setting. Extending its application to a multiple time-point setting isn’t trivial and should be explored in future studies. Furthermore, while double robustness is an interesting property that offers two chances at correctly specifying models to control confounding bias, it may often be the case in practice that both models are incorrectly specified. There are some situations where double robust methods perform worse than non double robust ones when both models are incorrectly specified, particularly in settings where positivity may be violated (Kang and Schafer, 2007). We have also illustrated the usefulness of our method for estimating an optimal PATS concerning the use of hormonal therapy in treating breast cancer patients. In this illustration, we were interested in tailoring treatment only according to ER status and BMI, but not PR status, although effect modification by PR is expected. Neglecting PR status was motivated by the fact that the additional clinical value of PR, above ER alone, is controversial. Although similar results were obtained using our proposed PATS estimator and a naive ATS estimator in this example, this needs not always be the case as illustrated in our simulation study. We believe the lack of difference between ATS and PATS estimators in this specific application might be due to the very low prevalence of the ER-PR+ status. From a substantive point of view, hormonal therapy was associated with poorer survival among ER- women, especially those with a greater BMI. The results were more imprecise regarding ER+ breast cancer patients, but provide some evidence suggesting that ER+ breast cancer patients with a greater BMI may benefit more from hormonal therapy than ER+ patients with a lower BMI. Given the large widths of confidence intervals, more studies are required to further explore this unexpected result. It is possible that residual confounding or selection bias is present although a very rich set of covariates were adjusted for. Moreover, in an analysis where only effect modification according to ER status was considered (not presented), the results were in the expected direction: hormonal therapy was associated with a better survival among ER+ participants, but not among ER- participants. In addition to its practical importance, we believe the current work also opens the door to further important methodological developments. First, it seems reasonable to assume that tailoring treatment decisions according to only a few variables may increase the precision of the estimators of the selected interaction terms, since fewer treatment-covariate interaction parameters need to be estimated. It would be worthy to verify this hypothesis and investigate if this reduces the risk of identifying as “optimal” treatment strategies that truly have low expected value. PATS estimators could also provide a framework for data-driven selection of the most important tailoring features. Acknowledgements ================ This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada. DT is also supported by a career award from the Fonds de recherche du Québec - Santé. Part of this work was conducted while DT was a visiting professor at McGill University. DT would like thank Université Laval for funding his visit at McGill University, and McGill University for welcoming him. 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Appendix A - G-estimation of adaptive treatment strategies ========================================================== The G-estimation of the parameters ψj is performed recursively, for j = K, ..., 1. First, a vector-valued function S(Aj) of the same length as ψj containing effect modifiers of the treatment effect must be specified. The optimal form for S(Aj) is known (Robins, 2004), but is generally complex and S(aj) = ∂γj(aj, hj; ψj)/∂ψj is thus commonly used. For example, if γj(aj, hj; ψj) = aj(ψ0j + ψ1jhj) then S(aj) could be (aj, ajhj). Next, a model for E[S(Aj)∣Hj] is specified. Define GiK = Yi − γK(aiK, hiK; ψK) and Gij = Yi − γj(aij, hij; ψj) + ∑k = j + 1K[γk(dkopt, hik; ψ̂k) − γk(aik, hik; ψ̂k)] if j ≠ K, where ψ̂k is the estimated value of ψk. A model for E[Gj∣Hj] is then specified. The G-estimate of ψj is the solution ψ̂j to the estimating equations 0 = ∑i = 1n{S(Aij) − E[S(Aij)∣Hij]}{Gij − E[Gij∣Hij]}. This equation has a closed-form solution for continuous Y when linear models are used for the treatment-free (E[Gij∣Hij]) and blip components. Appendix B - Double robustness of weighted ordinary least squares ================================================================= Consider the case where the outcome model may be misspecified, but the treatment model is correctly specified. We consider the weights w = ∣A − E[A∣X]∣, which benefit from the balancing property A∐X in the weighted data. We show that E[ψ̂] converges in probability to ψ. We first consider the following outcome model, for which the treatment-free component (Xiβ) may be misspecified: Yi = Xiβ + AiXiψ + εi. The weighted least squares estimating equations are $$\begin{aligned} 0 &= \sum w\_i X^\top\_i (Y\_i - X\_i \hat{\beta} - A\_i X\_i \hat{\psi}) \\ 0 &= \sum w\_i A\_i X^\top\_i (Y\_i - X\_i \hat{\beta} - A\_i X\_i \hat{\psi}).\end{aligned}$$ Isolating β̂ in both equations $$\begin{aligned} \hat{\beta} &= (\sum w\_i X\_i^\top X\_i)^{-1}[(\sum w\_i X\_i^\top Y\_i) - (\sum w\_i A\_i X\_i^\top X\_i)\psi] \\ \hat{\beta} &= (\sum w\_i A\_i X\_i^\top X\_i)^{-1}[(\sum w\_i A\_i X\_i^\top Y\_i) - (\sum w\_i A\_i X\_i^\top X\_i)\psi], \end{aligned}$$ then equating both equations, the estimator for ψ is obtained $$\begin{aligned} \hat{\psi} &= [(\sum w\_i X\_i^\top X\_i)^{-1} (\sum w\_i A\_i X\_i^\top X\_i) - I\_p]^{-1} \times \\ & [(\sum w\_i X\_i^\top X\_i)^{-1}(\sum w\_i X\_i^\top Y\_i) - (\sum w\_i A\_i X\_i^\top X\_i)^{-1}(\sum w\_i A\_i X\_i^\top Y\_i)] \\ & = [(\sum w\_i (A\_i + (1 - A\_i)) X\_i^\top X\_i)^{-1} (\sum w\_i A\_i X\_i^\top X\_i) - I\_p]^{-1} \times \\ & [(\sum (A\_i + (1 - A\_i)) w\_i X\_i^\top X\_i)^{-1}(\sum w\_i X\_i^\top Y\_i) - (\sum w\_i A\_i X\_i^\top X\_i)^{-1}(\sum w\_i A\_i X\_i^\top Y\_i)]. \\\end{aligned}$$ Since E[wAη(X)] = E[w(1 − A)η(X)] = E[π(x)η(X)], where η(X) is any function of X and π(X) = P(A = 1∣X)P(A = 0∣X) (Li and Li 2019), we get $$\begin{aligned} \hat{\psi} &\stackrel{p}{\rightarrow} [(2\sum \pi(X\_i) X\_i^\top X\_i)^{-1} (\sum \pi(X\_i) X\_i^\top X\_i) - I\_p]^{-1} \times \\ & [(2\sum \pi(X\_i) X\_i^\top X\_i)^{-1}(\sum w\_i X\_i^\top Y\_i) - (\sum \pi(X\_i) X\_i^\top X\_i)^{-1}(\sum w\_i A\_i X\_i^\top Y\_i)] \\ &= -[(\sum \pi(X\_i) X\_i^\top X\_i)^{-1}(\sum w\_i X\_i^\top Y\_i) - 2(\sum \pi(X\_i) X\_i^\top X\_i)^{-1}(\sum w\_i A\_i X\_i^\top Y\_i)] \\ &= -(\sum \pi(X\_i) X\_i^\top X\_i)^{-1}[\sum (w\_i X\_i^\top Y\_i - 2 w\_i A\_i X\_i^\top Y\_i)] \\ &= -(\sum \pi(X\_i) X\_i^\top X\_i)^{-1}[\sum (w\_i (A\_i + (1 - A\_i)) X\_i^\top Y\_i - 2 w\_i A\_i X\_i^\top Y\_i)] \\ &= (\sum \pi(X\_i) X\_i^\top X\_i)^{-1}[\sum w\_i A\_i X\_i^\top Y\_i - w\_i (1 - A\_i) X\_i^\top Y\_i)] \\\end{aligned}$$ Taking the expectation on both sides $$\begin{aligned} \mathbb{E}\left[\hat{\psi}\right] &\stackrel{p}{\rightarrow} \mathbb{E}\left\{(\sum \pi(X\_i) X\_i^\top X\_i)^{-1}[\sum w\_i A\_i X\_i^\top Y\_i - w\_i (1 - A\_i) X\_i^\top Y\_i]\right\} \nonumber \\ &= \mathbb{E}\left[\mathbb{E}\left\{(\sum \pi(X\_i) X\_i^\top X\_i)^{-1}[\sum w\_i A\_i X\_i^\top Y\_i - w\_i (1 - A\_i) X\_i^\top Y\_i]\|X\_i \right\} \right] \nonumber \\ &= \mathbb{E}\left[(\sum \pi(X\_i) X\_i^\top X\_i)^{-1} \sum X\_i^\top \left\{\mathbb{E}\left[w\_i A\_i Y\_i\|X\_i\right] - \mathbb{E}\left[w\_i(1 - A\_i) Y\_i\|X\_i\right]\right\}\right]. \label{eq1}\end{aligned}$$ Consider E[wiAiYi∣Xi]: $$\begin{aligned} \mathbb{E}\left[w\_i A\_i Y\_i\|X\_i\right] &= \mathbb{E}\left[w\_i Y\_i^1\|A\_i = 1, X\_i\right]P(A\_i = 1\|X\_i) \\ &= \mathbb{E}[w\_i\|A\_i = 1, X\_i] \mathbb{E}\left[Y\_i^1\|X\_i\right]P(A\_i = 1\|X\_i) \\ &= P(A\_i = 0\|X\_i) \mathbb{E}\left[Y\_i^1\|X\_i\right]P(A\_i = 1\|X\_i) \\ & = \pi(X\_i) \mathbb{E}\left[Y\_i^1\|X\_i\right],\end{aligned}$$ where the first equality is obtained using the consistency assumption and the second using the exchangeability assumption. Similarly, we can show that E[wiAiYi∣Xi] = π(Xi)E[Yi0∣Xi]. Inserting these results in ([eq1]) $$\begin{aligned} \mathbb{E}\left[\hat{\psi}\right] &\stackrel{p}{\rightarrow} \mathbb{E}\left\{(\sum \pi(X\_i) X\_i^\top X\_i^\top)^{-1} \sum \pi(X\_i) X\_i^\top \mathbb{E}\left[Y\_i^1 - Y\_i^0\|X\_i\right]\right\} \\ &= \mathbb{E}\left\{(\sum \pi(X\_i) X\_i^\top X\_i^\top)^{-1} \sum \pi(X\_i) X\_i^\top X\_i \psi \right\} = \psi. \end{aligned}$$ Now consider the case where the outcome model is correctly specified, but the treatment model may be misspecified. Denote Z = (X, AX) and θ = (β, ψ)⊤, we now show that E[θ̂] = θ. $$\begin{aligned} \mathbb{E}[\hat{\theta}] &= \mathbb{E}\left[(Z^\top W Z)^{-1} Z^\top W Y \right] \\ &= \mathbb{E}\left[(Z^\top W Z)^{-1} Z^\top W (Z\theta + \varepsilon) \right] \\ &= \mathbb{E}\left[(Z^\top W Z)^{-1} X^\top W Z\theta + (Z^\top W Z)^{-1} Z^\top W \varepsilon \right] \\ &= \theta + \mathbb{E}\left[(Z^\top W Z)^{-1} Z^\top W \varepsilon \right] \\ &= \theta\end{aligned}$$ because Z⊤Wɛ = 0 by construction. In the multiple time-point setting, we can first note that the consistency of the dWOLS estimator for the parameters of the blip at the last time-point (j = K) follows directly from the previous results. For j < K, we assume that the later blips were estimated consistently. As a consequence, E[Ỹj] = E{Y + ∑k = j + 1K[γk(dkopt, hk; ψ̂k) − γk(aik, hk; ψ̂k)]} is consistent for E{Y + ∑k = j + 1K[γk(dkopt, hk; ψk) − γk(aik, hk; ψk)]}. Under our causal assumptions, it can be verified that $\mathbb{E}\{Y + \sum\_{k=j+1}^K[\gamma\_k(d\_k^{opt}, h\_{k}; \psi\_k) - \gamma\_k(a\_{k}, h\_k; \psi\_k)]\|A\_j, H\_j\} = \mathbb{E}[Y^{\bar{a}\_{j},\underline{d}\_{j+1}^{opt}}\|H\_j]$. Together, these results imply that E[Ỹj∣Aj, Hj] is a consistent estimator for $\mathbb{E}[Y^{\bar{a}\_{j},\underline{d}\_{j+1}^{opt}}\|H\_j]$. The rest of the proof follows the same steps as above, replacing Y by Ỹ, X by H and adding back the subscript j. Appendix C - G-estimation estimators of partially adaptive treatment strategies =============================================================================== This Appendix details the steps of the G-estimation analogues to the dWOLS estimators of PATS presented in the main manuscript. IPTW+G-estimation ----------------- 1. Estimate ψj with the G-estimation algorithm. 2. Specify a vector valued function S\(A*j) of the same length as ψj\, containing the effect modifiers H*j\. 3. Define GiK\ = Yi − γj\(aiK, hiK\; ψK\) and Gij\ = Yi − γj\(aij, hij\; ψj\) + ∑k* = j + 1K[γk\(dkopt* \, hik; ψ̂*k\) − γ*k(aik, hik; ψ̂k)] if j ≠ K. 4. Specify a model for E[Gj\∣H*j]. 5. Specify a model for E[S\(A*j)∣Hj\]. 6. Compute weights $\varpi(A\_j, H\_j, H\_j^\) = \frac{P(A\_j\|H\_j^\)}{P(A\_j\|H\_j)}$. 7. ψ̂*j\* is the solution ψj\* to the estimating equations ${0 = \sum\_{i=1}^n \varpi(A\_{ij}, H\_{ij}, H\_{ij}^\) \{S^\(A\_{ij}) - \mathbb{E}[S^\(A\_{ij})\|H\_{ij}^\]\}\{G\_{ij}^\* - \mathbb{E}[G\_{ij}^\\|H\_{ij}]\}}$. Integrate G-estimation ---------------------- 1. Estimate ψ*j with the G-estimation algorithm. 2. Specify a vector valued function S(Aj) of the same length as ψj, containing the effect modifiers among Hj. 3. Define GiK\* = Yi − γj†(aiK, hiK) and Gij\* = Yi − γj†(aij, hij) + ∑k = j + 1K[γk\(dkopt* \, hik; ψ̂*k\) − γ*k(aik, hik; ψ̂k)] if j ≠ K. 4. Specify a model for E[Gj\∣H*j]. 5. Specify a model for E[S(Aj)∣Hj]. 6. Estimate γj†(aj, hj) as the solution to 0 = ∑i = 1n{S(Aij) − E[S(Aij)∣Hij]}{Gij\* − E[Gij\∣Hij]}. 7. Estimate fHj*C∣Hj\* and compute $\hat{\gamma}\_j^\(a\_j, h\_j^\) = \int\_{H\_j^C} \hat{\gamma\_j}^\dagger(a\_j, h\_j) \hat{f}\_{H\_j^C\|H\_j^\} \mu(H\_j^C)$. CE G-estimation --------------- All steps except the last one are the same as in the algorithm of “integrate G-estimation.” This step is replaced by the two following steps 7. Compute Q(a, hij) = γ̂†(a, hij) for all observations. 8. Run a regression of E[Q(a, H*j)∣Hj\] = γ\(a, hj\). Appendix D - Connection between the IPTW+dWOLS and IPTW+G-estimation estimators =============================================================================== We show that the estimate of the IPTW+dWOLS is also a solution to the IPTW+G-estimation estimating equations in certain circumstances to illustrate the close connection between IPTW+dWOLS and IPTW+G-estimation. Our presentation closely follows the connection between G-estimation and dWOLS for estimating ATSs made by Wallace and Moodie (2015). We consider the single time point setting where γ\(a, h\) = A*H\*ψ\, $\mathbb{E}[G\|H^\] = f(h^\; \bm{\beta}) = \bm{\beta} \bm{H}^\$ and S\(A) = ∂γ*j\(a*j, hj\; ψ*j\)/∂ψ*j\* = AH\, where H\ is assumed to include a column of 1 for the intercept. The estimating equations of IPTW+G-estimation can be written as: $$\begin{aligned} 0 = \sum\_{a\_i = 0}& \varpi(a\_i, h\_i, h\_i^\) h^\\_i \left(-\mathbb{E}[A\|H^\\_{i}] \right)(y\_i - \bm{\beta} h\_i) \\ +& \sum\_{a\_i = 1} \varpi(a\_i, h\_i, h\_i^\) h^\\_i \left(1-\mathbb{E}[A\|H^\\_i] \right)(y\_i - \psi^\h^\\_i - \bm{\beta}h\_i). \end{aligned} \label{eq0}$$ The dWOLS estimating equations can be written in a similar form: $$\begin{aligned} 0 = \sum\_{a\_i = 0}& \varpi(a\_i, h\_i, h\_i^\) \times w^\(a\_i, h\_i^\) h\_i (y\_i - \bm{\beta} h\_i) \\ +& \sum\_{a\_i = 1} \varpi(a\_i, h\_i, h\_i^\) \times w^\(a\_i, h\_i^\) h\_i (y\_i - \psi^\h^\\_i - \bm{\beta} h\_i)\\ 0 = \sum\_{a\_i =1} & \varpi(a\_i, h\_i, h\_i^\) \times w^\(a\_i, h\_i^\) h^\\_i (y\_i - \bm{\psi}^\h^\\_i - \bm{\beta} h\_i).\end{aligned}$$ Let w(ai, hi\) = ∣A*i − E[A∣hi\]∣. The dWOLS estimating equations are: $$\begin{aligned} 0 = \sum\_{a\_i = 0}& \varpi(a\_i, h\_i, h\_i^\) h\_i \left(\mathbb{E}[A\|h\_i^\] \right) (y\_i - \bm{\beta} h\_i) \nonumber \\ +& \sum\_{a\_i = 1} \varpi(a\_i, h\_i, h\_i^\) h\_i \left(1-\mathbb{E}[A\|h\_i^\] \right) (y\_i - \psi^\h^\\_i - \bm{\beta} h\_i) \label{eq1.1} \\ 0 = \sum\_{a\_i = 1}& \varpi(a\_i, h\_i, h\_i^\) h^\\_i \left(1-\mathbb{E}[A\|h\_i^\] \right) (y\_i - \psi^\h^\\_i - \bm{\beta} h\_i). \label{eq2}\end{aligned}$$ Remarking that hi = (hi\, hiC), then writing the system of equations {([eq2]) - ([eq1.1]), ([eq2])}, we get $$\begin{aligned} 0 & = \sum\_{a\_i = 0} \varpi(a\_i, h\_i, h\_i^\) h^\\_i \left(-\mathbb{E}[A\|h\_i^\] \right) (y\_i - \bm{\beta} h\_i) \label{eq4}\\ 0 & = \sum \varpi(a\_i, h\_i, h\_i^\) h\_i^C \left(a\_i -\mathbb{E}[A\|h\_i^\] \right) (y\_i - \bm{\beta} h\_i - a\_i \bm{\psi}^\h^\\_i) \label{eq5} \\ 0 & = \sum\_{a\_i = 1} \varpi(a\_i, h\_i, h\_i^\) h^\\_i \left(1-\mathbb{E}[A\|h\_i^\] \right) (y\_i - \bm{\psi}^\h^\\_i - \bm{\beta} h\_i). \label{eq6} \end{aligned}$$ Since the sum of the first and last equations from this IPTW+dWOLS is equal to the estimating equations from IPTW+G-estimation [([eq4]) + ([eq6]) = ([eq0])], the solution of IPTW+dWOLS is also a solution to the IPTW+G-estimation estimating equations. Appendix E - Additional simulation results with two time-points =============================================================== We consider three additional scenarios with two time-points. At each time-point, two pre-treatment covariates are measured, X1 and X2. Both are confounders and effect modifiers for the treatment at their respective time-point. However, only X1 is intended to be used for tailoring treatment. Using the notation introduced in the paper, we have H1 = (X11, X12), H2 = (X11, X12, A1, X21, X22), H1\ = X11, H2\* = X21. In all scenarios $$\begin{aligned} X\_{11} \sim Bernoulli(p = 0.5), \\ X\_{12} \sim Bernoulli(p = 0.5). \\\end{aligned}$$ In Scenarios 1 and 3, A1 ∼ Bernoulli[p = expit( − 1 + X11 + X12)], whereas in Scenario 2, A1 ∼ Bernoulli[p = expit( − 1 + X11 + X12 + X11X12)]. In all scenarios $$\begin{aligned} X\_{21} \sim Bernoulli[p = expit(-1 + X\_{11} + A\_1)], \\ X\_{22} \sim Bernoulli[p = expit(-1 + X\_{12} + A\_1)]. \\\end{aligned}$$ In Scenarios 1 and 3, A2 ∼ Bernoulli[p = expit( − 1 + X21 + X22)], and in Scenario 2 A2 ∼ Bernoulli[p = expit( − 1 + X21 + X22 + X21X22)]. In all scenarios $$\begin{aligned} A\_1^{opt} = I(\bm{X}\_1\psi\_1 > 0),\\ \mu\_1 = (A\_1^{opt} - A\_1)\bm{X}\_1 \psi\_1,\\ A\_2^{opt} = I(\bm{X}\_2\psi\_2 > 0),\\ \mu\_2 = (A\_2^{opt} - A\_2)\bm{X}\_2 \psi\_2, \end{aligned}$$ where I( ⋅ ) is the usual indicator function, $\bm{X}\_j = (1, X\_{j1}, X\_{j2}), j = 1,2$, and ψ1 = ψ2 = ( − 0.5, 1, − 1). Finally, in Scenario 1 and 2, Y ∼ N(X11 + X12 − μ1 − μ2, 1), and in Scenario 3, Y ∼ N(X11 + X12 + X11X12 − μ1 − μ2, 1). The true value of the causal effect at the second time-point is γ2\(A2, X12) = A2(ψ20\ + ψ21\*X21) = A2{(ψ20 + E[X22∣X21 = 0]ψ22) + [ψ21 + (E[X22∣X21 = 1] − E[X22∣X21 = 0])ψ22]X21} ≈ A2( − 1.01 + 1.03X21). The true value of the causal effects at the first time-point were estimated through Monte Carlo simulations of the counterfactuals with a sample size of 1 000 000, γ1\(A1, X11) ≈ A1( − 0.96 + 0.91X11). As in the main simulation, the treatment and the treatment-free models only include main terms. Hence, in Scenario 1, both the treatment score and the treatment-free models are correctly specified. In Scenario 2, only the treatment-free model is correctly specified. In Scenario 3, only the treatment model is correctly specified. A total of 1000 replications of each scenario were generated for n* = 100, 1000 and 10 000. The results are summarized in Tables 1-4. As in the main simulations, standard G-estimation and dWOLS were biased in all three scenarios. In Scenarios 1 and 3, where the treatment model was correctly specified, the PATS estimators we proposed produced estimates with very low bias, especially for larger sample sizes. When the treatment model was misspecified, some bias remained for IPTW+G-estimation and IPTW+dWOLS. The bias for the other methods was somewhat larger when the treatment model was misspecified then when it was correctly specified, but could be considered as negligible (<5% for n = 100, and <3% for n = 1000 and n = 10 000). Standard ATS estimators identified the optimal PATS less often than the PATS estimators under all Scenarios, especially at larger sample sizes. The expected loss when the optimal PATS was incorrectly estimated was similar across estimators and varied from 0.076 to 1.102 with a mean of 0.128. The expected loss was slightly larger in Scenario 2 than in Scenarios 1 and 3 with means varying between ≈ 0.13 and 0.17 depending on the estimator and sample size. [h!] [!htbp] [!htbp] [!htbp] Appendix F - Additional simulation to investigate confidence intervals ====================================================================== Scenario -------- We performed additional simulations to explore the ability of the m-out-of-n bootstrap to yield adequate confidence intervals for the parameters of a PATS. Because of the important computational burden of this type of bootstrap, we considered a single simulation scenario with two time-points, Scenario 1 of Appendix E, and a single estimator, “CE dWOLS.” We generated 1000 datasets of size n = 300. Confidence intervals -------------------- Confidence intervals were constructed using the m-out-of-n where m was chosen data-adaptively using the procedure proposed by Chakraborty et al (2013). Briefly, a first standard non-parameteric (n-out-of-n) bootstrap was first performed with B1 = 500 replicates. Using these, we estimated the variance-covariance matrix of the estimators of the parameters of the blip at the second time-point (ψ̂2\). We then constructed 95% confidence intervals for the linear predictor a2h2ψ̂2\ for all observations. We estimated a non-regularity parameter p as the proportion of the observations for which the optimal treatment at the second time-point was not uniquely defined, that is, the proportion of the observations for which the preceding confidence intervals included the null value. Next, we took B2 = 500 samples with replacement of size $m = n^{\frac{1+\alpha(1 - \hat{p})}{1 + \alpha}}$ within each first-stage bootstrap sample, where α = 0.025 and p̂ is the estimated value of p. For each of the B1 first stage replicate, 95% confidence intervals for the parameters of the blip were computed as the 2.5th and 97.5th percentiles of the B2 = 500 second-stage bootstrap replicates. As such, B1 = 500 confidence intervals were computed for each parameter. We computed the proportion of these confidence intervals that included their respective first-stage bootstrap estimate. If this proportion was greater than 95% then α̂ = α and m̂ = m. Otherwise we increased the value of α by 0.025 and restarted the second-stage bootstrap with the new corresponding value of m. This procedure was repeated until at least 95% of the confidence intervals produced at the second stage included the first stage estimates. Note that if p̂ = 0, then m̂ = n and no second-stage bootstrap was performed. Once the procedure stopped, we performed one last non-parametric bootstrap with B1 = 500 replicates of size m̂ and 95% confidence intervals were based on the 2.5th and 97.5th percentiles of the replicates. Results ------- The value of p̂ varied between 0 and 1, with a mean of 0.47. When p̂ ≠ 0, α̂ was either 0.025 or 0.05 in 81.5% of the simulation replicates. Its mean value was 0.04 and the maximum was 0.125. When p̂ ≠ 0, each α value explored took about 50 minutes of computation time. The sample size of the m-out-of-n bootstrap varied between 217 and 300, with a mean of 269. The coverage for all four PATS parameters was approximately 95%: 95.7% for ψ10\, 95.2% for ψ11\, 95.2% for ψ20\* and 95.1% for ψ21\. Double robust estimation of partially adaptive treatment strategies =================================================================== Precision medicine aims to tailor treatment decisions according to patients’ characteristics. G-estimation and dynamic weighted ordinary least squares (dWOLS) are double robust statistical methods that can be used to identify optimal adaptive treatment strategies. They require both a model for the outcome and a model for the treatment and are consistent if at least one of these models is correctly specified. It is underappreciated that these methods additionally require modeling all existing treatment-confounder interactions to yield consistent estimators. Identifying partially adaptive treatment strategies that tailor treatments according to only a few covariates, ignoring some interactions, may be preferable in practice. It has been proposed to combine inverse probability weighting and G-estimation to address this issue, but we argue that the resulting estimator is not expected to be double robust. Building on G-estimation and dWOLS, we propose alternative estimators of partially adaptive strategies and demonstrate their double robustness. We investigate and compare the empirical performance of six estimators in a simulation study. As expected, estimators combining inverse probability weighting with either G-estimation or dWOLS are biased when the treatment model is incorrectly specified. The other estimators are unbiased if either the treatment or the outcome model are correctly specified and have similar standard errors. Using data maintained by the Centre des Maladies du Sein, the methods are illustrated to estimate a partially adaptive treatment strategy for tailoring hormonal therapy use in breast cancer patients according to their estrogen receptor status and body mass index. R software implementing our estimators is provided. Keywords: Causal inference; Double robustness; Dynamic treatment regimens; Inverse probability weighting; Personalized medicine; Precision medicine Introduction ============ Precision medicine, sometimes also called personalized medicine, is a medical approach that aims to improve patients’ outcomes by tailoring treatment decisions taking into account their genes, environments and lifestyles (United States Food and Drug Administration, 2018). This approach has garnered increasing attention over recent years. Notably, multiple countries have put forward initiatives to accelerate research on precision medicine, including the United States, Canada and various European countries. Adaptive treatment strategies (ATSs), or dynamic treatment regimes, is a type of precision medicine. ATSs aim to determine time-dependent treatment decision rules that use prior patients’ information to optimize a clinical outcome among patients sharing similar characteristics. Multiple statistical techniques have been proposed for estimating ATSs (for example, Watkins, 1989; Murphy, 2003; Robins, 2004; Wallace and Moodie, 2015). G-estimation (Robins, 2004) and dynamic weighted ordinary least squares (dWOLS; Wallace and Moodie, 2015) have the advantage of offering some robustness to statistical modeling errors. These methods require the specification of two models: a treatment model that relates potential confounders to the treatment decisions, and an outcome model that relates potential confounders and treatments to the outcome. The outcome model is further divided in a treatment-free component that includes only potential confounders, and a “blip” component that features terms related to the treatment effect and possible effect modification according to tailoring variables. The blip is the statistical quantity of interest for determining optimal decision rules. G-estimation and dWOLS yield consistent estimators of the blip’s parameters if either the treatment model or the treatment-free component is correctly specified, but not necessarily both, a property known as double robustness. The blip must, however, be correctly specified. Other double robust methods for estimating ATSs have been proposed (for example, Petersen et al, 2004; van der Laan and Petersen, 2007), but we focus on G-estimation and dWOLS because of their relative ease of implementation and interpretation within a regression-like framework. G-estimation and dWOLS do not formally require including the same covariates in the blip component as in the treatment-free component. For dWOLS, any covariate interacting with the treatment in the blip must be included as main effect in the treatment-free component (that is, the “hierarchy” of the model must be preserved), while G-estimation has no specific requirement. In practice, it is common to include fewer covariates in the blip than in the treatment-free model. For example, when estimating an ATS for the treatment of type 2 diabetes, Simoneau et al (2020) included multiple potential confounders in the treatment-free component, but only glycemic control, body mass index, previous treatment and history of hypoglycemia in the blip component. Multiple reasons may motivate tailoring treatment decisions according to only a few variables. For instance, some variables may be confounders in the context of a given dataset, but are expected to be unavailable for treatment tailoring in the intended clinical context. Simplifying decision rules to facilitate their use in clinical practice or to improve statistical power are other examples of reasons to exclude some variables from the blip. A perhaps underappreciated possible consequence of excluding some potential confounders from the blip is that the blip may be incorrectly specified if some true treatment effect modification remains unmodeled. As mentioned previously, both G-estimation and dWOLS require the blip to be correctly specified for consistent estimation of its parameters. A solution to this problem has been proposed by van der Laan and Robins (2003). It combines G-estimation with inverse probability of treatment weighting (IPTW), where the IPTW is used to control the residual confounding bias attributable to the (voluntary) possible misspecification of the blip. However, we expect this solution is not fully double robust, since it relies on the correct specification of a treatment model to account for the blip misspecification. In this paper, we propose alternative double robust estimators of partially adaptive treatment strategies (PATSs), where the treatment is tailored according to only some covariates, excluding potential true effect modifiers. In the next section, we introduce the notation and briefly review dWOLS estimation of an ATS.. Analogous G-estimation methods are available in Appendix C. In Section [s:simulation], we employ simulation studies to illustrate the potential bias of using standard ATSs estimators for estimating PATSs as well as the double robustness of our proposed estimators. In Section [s:application], we estimate a PATS that aims to tailor hormonal therapy for treating breast cancer as a function of estrogen receptor status and body mass index (BMI) in data maintained by the Centre des Maladies du Sein. We conclude in Section [s:discussion] with a discussion of the results and perspectives for future research. Notation and review of adaptive treatment strategy estimation ============================================================= Notation -------- Let A*j be the exposure at time j (j = 1, ..., K), Xj the pre-treatment covariates at time j, and Y the final outcome. Without loss of generality, we assume that a greater value of Y represents a better clinical outcome and that Aj = 0 represents a reference value for the treatment (for example, a placebo or standard care). We use over- and underbars to represent the set of past and future values of a variable, respectively, both including the present. For example, Āj = {A1, ..., Aj} and $\underline{A}\_j = \{A\_{j}, A\_{j+1},..., A\_K\}$. We further denote by Hj the observed history before treatment j, including previous treatments; hence, Hj = {Āj − 1, X̄j}. Let dj denote a treatment decision rule, which is a function of the observed history. The goal of an ATS is to determine the optimal treatment decision rule djopt at each time j in order to maximize the final outcome Y. To define the causal parameter of interest more formally, we make use of the counterfactual framework. For example, Yd̄Kopt is the outcome that would have been observed if, possibly contrary to the fact, the treatment strategy had been optimal throughout the entire follow-up. Using this notation, the causal parameter of interest in an ATS is $$\begin{aligned} \gamma\_j(a\_j, h\_j) = \mathbb{E}\left[Y^{\bar{a}\_{j-1}, a\_j, \underline{d}\_{j+1}^{opt}} - Y^{\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt}}\|H\_j = h\_j\right] = \mathbb{E}\left[Y^{\bar{a}\_j,\underline{d}\_{j+1}^{opt}} - Y^{\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt}}\|H\_j = h\_j\right].\end{aligned}$$ This is the blip to 0 function. It represents the effect of treatment Aj = aj compared to Aj = 0 for a given history hj, assuming that the future treatment strategy is optimal ($\underline{a}\_{j+1} = \underline{d}\_{j+1}^{opt}$).. Note that the optimal future treatment strategy may not be the same under treatment history āj as under treatment history (āj − 1, 0). In other words, $\underline{d}\_{j+1}^{opt}$ may not represent the same treatment in $(\bar{a}\_j,\underline{d}\_{j+1}^{opt})$ as in $(\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt})$, however in both cases the same treatment rule (or function) is applied; only the arguments to that function differ. Non-parametric identification of γj(aj, hj) is possible from the observed data under the following assumptions: * Consistency: If Ā = ā then Y = Yā. This assumption entails that the outcome of a given subject is not affected by the treatment of other subjects (absence of interference) and that there are not multiple versions of each treatment level a. * Sequential exchangeability (no unmeasured confounders): $(Y^{\bar{a}}, \underline{X}^{\bar{a}}\_{j+1}) \coprod A\_j \| \bar{A}\_{j-1}, \bar{X}\_j$ for all ā and j = 1, ..., K. * Positivity: P(Aj = aj∣hj) > 0 for all j = 1, ..., K, all aj, and all hj such that f(hj) > 0. If the positivity assumption fails to hold, it is still possible to estimate an optimal ATS among feasible strategies, that is, among strategies that have been observed in the data. In practice, a parametric model is assumed for the blip function, such that we can write γj(aj, hj; ψj). Estimation of adaptive treatment strategies ------------------------------------------- We now briefly present the dWOLS algorithm for estimating the blip γj(aj, hj; ψj). More details can be found elsewhere (for example, see Wallace and Moodie, 2015) and a description of the G-estimation algorithm can be found in Appendix A, as well as in Robins (2004) or Moodie, Richardson and Stephens (2007). Estimation of ψj using dWOLS is performed recursively. First, balancing weights such that Aj∐Hj in the weighted data must be specified. Constructing such weights generally involves specifying a model for E[Aj∣Hj];. Wallace and Moodie (2015) give a few examples of such balancing weights in the case of a binary treatment and observed that $w(A\_j, H\_j) = \|A\_j - \hat{\mathbb{E}}[A\_j\|H\_j]\|$ performs particularly well. These weights correspond to the overlap weights, which have been shown to yield a minimal variance weighted estimator of a certain average treatment effect under some assumptions (Li, Morgan and Zaslavsky, 2018; Li and Li, 2019). Schulz and Moodie (2020) propose various balancing weights for the multilevel and continuous treatment cases. Define the pseudo-outcome ỸiK = Yi and Ỹij = Yi + ∑k = j + 1K[γk(dkopt, hik; ψ̂k) − γk(aik, hik; ψ̂k)] if j ≠ K. Finally, a model for E[Ỹj∣Aj, Hj] = fj(hj; βj) + γj(aj, hj; ψj) is specified, where fj(hj; βj) is a function relating Hj to the outcome, and its parameters are estimated using the weighted data. Estimation of partially adaptive treatment strategies ===================================================== We now turn our attention to the estimation of PATSs, that is, where the objective is to tailor treatment decisions only according to a subset of the measured history. To formally define the causal parameter of interest of a PATS, we partition the covariates history at each time-point in two disjoint subsets Hj = (Hj\* ∪ HjC), where Hj\* are the covariates that are intended to be used for tailoring the treatment at time j and HjC are the other covariates. The parameter of interest is defined as Nonparametric identification ---------------------------- Combining dWOLS and Inverse probability of treatment weighting -------------------------------------------------------------- A first estimator of γj\(a*j, hj\; ψ*j\) is based on the solution proposed by van der Laan and Robins (2003), Sections 6.5.2 and 6.5.3, in the more general context where the causal parameter of interest is expressed as a function of the density of the counterfactual outcomes. Their proposal consists of combining G-estimation with IPTW, where the IPTW is employed to account for the possible misspecification of the blip due to unmodeled true effect modification. We detail an analogous estimator combining IPTW with dWOLS, which we denote “IPTW+dWOLS.” Following Wallace and Moodie (2015), we show in Appendix D that the solution to the IPTW+dWOLS estimator is also a solution the “IPTW+G-estimation” estimating equations in certain circumstances. Next, balancing weights w\(Aj, Hj\) such that A*j∐Hj\* in the weighted data must be specified. In addition, weights $\varpi(A\_j, H\_j, H\_j^\) = \frac{P(A\_j\|H\_j^\)}{P(A\_j\|H\_j)}$ are computed.. Finally, specify a model for E[Ỹj\∣A*j, Hj] = fj(hj; βj) + γj\(a*j, hj\; ψ*j\) and estimate its parameters using the data weighted according to $w^\(A\_j, H\_j^\) \times \varpi(A\_j, H\_j, H\_j^\)$. Alternative estimators ---------------------- The last remaining step is to compute the outer expectation of ([eq.PATS2]). A first approach for doing this is to estimate the density fHjC∣Hj\* and integrate γj†(aj, hj; ψj†) over HjC. In simple situations, the nonparametric empirical estimator $\hat{f}\_{H\_j^C\|H\_j^\}(h\_j^c\|h\_j^\) = \frac{\sum\_{i=1}^n I(H\_{ij}^C = h\_j^c, H\_{ij}^\* = h\_j^\)}{\sum\_{i=1}^n I(H\_{ij}^\ = h\_j^\)}$, where I( ⋅ ) is the usual indicator function, could be used. Other density estimators would generally be required. We henceforth call this approach “integrate dWOLS” and the analogous G-estimation approach as “integrate G-estimation.” An alternative implementation consists in computing the expectation through a linear model. More precisely, we calculate Q(a, hij) = γ*j†(a, hij; ψ̂\) for all observations, that is, the predicted value of γ*j†(Aj, hij; ψj†) when setting Aj = a. Next, we regress the predicted γj†(Aj, hij; ψj†) on covariates Hj\, thus fitting a model E[Q(a, H*j)∣Hj\] = γ\(a, hj\; ψ*j\). A clear advantage of this implementation is that it does not require estimating densities or computing integrals. This approach is called “CE dWOLS” and the analogous G-estimation approach is called “CE G-estimation” moving forward. Double robustness ----------------- The intuition for the double robustness of “integrate dWOLS” and “CE dWOLS” is the same as that of dWOLS presented at the end of Section 2.2. More formally, the double robustness of our estimators can be demonstrated recursively. First, if considering the usual IPTW w(A*j, Hj) = Aj/P(Aj = 1∣Hj) + (1 − Aj)/P(Aj = 0∣Hj), note that the double robustness of ψ̂K\* and ψ̂K (i.e. the treatment rule parameters from the final stage) is a direct consequence of the results of Kang and Schafer (2007) and Robins et al (2007) who demonstrate the double robustness of the ordinary least square estimator weighted according to IPTW, as long as an intercept term is included in the outcome (treatment-free) model. In Appendix B, we provide a proof of the double robustness of ψ̂K for the case where the weights w(Aj, Hj) = ∣Aj − E[Aj∣Hj]∣ are used. The double robustness of ψ̂K\* follows directly. To demonstrate the double robustness of ψ̂j\* for j < K, assume that γk\(a*k, hk\; ψ*k\) and γ*k(ak, hk; ψk) for k > j are consistently estimated. As a result, the mean of Ỹj\* for each (aj, hj) is also consistent for $\mathbb{E}\left[Y^{\bar{a}\_j,\underline{d}\_{j+1}^{opt\}}\|h\_j\right]$. Appealing once more to the previous double robustness results, it follows that γ*j†(aj, hj; ψ̂j†) is a double robust estimator for $\mathbb{E}\left[Y^{\bar{a}\_j,\underline{d}\_{j+1}^{opt\}} - Y^{\bar{a}\_{j-1},0,\underline{d}\_{j+1}^{opt\}}\|h\_j\right]$. Hence, ψ̂j\* is also double robust for ψj\. We conclude that consistent estimation of the parameters of γ*j\(a*j, hj\; ψ*j\) at each time-point requires either the treatment model or the treatment-free model to be correctly specified, in addition to correct specification of the blips γ*j\(a*j, hj\; ψ*j\), γ*j(aj, hj; ψj) and γj†(aj, hj; ψj†) with respect to included covariates (which may, by virtue of the PATS model, be a subset of all variables that interact with treatment), and the consistent estimation of the parameters of later blips. Inferences ---------- Analytical calculation of confidence intervals for PATS is challenging because of the form of the parameter of interest. Alternatively, nonparametric bootstrap may be employed. However, this approach, as well as usual asymptotic variance estimators, would not be valid in cases where some γj\(a*j, hj\; ψ*j\) = 0 when K* > 1 for j ≠ K and some hj\. Indeed, the optimal treatment strategy $\underline{d}\_{j}^{opt\}$ is not uniquely defined in such situations, which yield estimators with non-regular limiting distribution (Robins, 2004). Even when γj\(a*j, hj\; ψ*j\) is small relative to the sample size but nonzero, poor inferences may be produced by standard bootstrap because of the near non-regularity of the estimator (Moodie and Richardson, 2010). However, the m-out-of-n* bootstrap has been found to perform well in such settings (Chakraborty, Laber and Zhao, 2013; Simoneau et al, 2018). This type of bootstrap is similar to regular nonparametric bootstrap, except that m < n observations are sampled at each replication. While the validity of this type of bootstrap relies on an appropriate choice of m, Chakraborty et al (2013) proposed a data-adaptive method for this choice that has been found to perform well. The data-adaptive choice determines m based, in part, on an estimate of the extent to which the estimator is non-regular, which is determined from the data by the estimated proportion of subjects for whom there is no unique optimal choice of treatment (see Appendix F for details). We therefore recommend utilizing this data-adaptive m-out-of-n bootstrap in general, although the regular nonparametric bootstrap may also be considered when nonregularity is not expected and the additional computational burden of the adaptive m-out-of-n bootstrap is prohibitive. Simulation study ================ Scenarios --------- Additional simulations in a two time-points setting are available in Appendix E. The simulation scenarios are inspired by those presented in Wallace and Moodie (2015) and Wallace, Moodie and Stephens (2016). Three different scenarios are considered. In all scenarios, H = X = (X1, X2) are two pre-treatment covariates and both are effect modifiers. However, only X1 is intended to be used to tailor treatment (H\* = X1, HC = X2). In all three scenarios X1 ∼ Bernoulli(p = 0.5) and X2 ∼ Bernoulli(p = 0.5). In Scenarios 1 and 3, A ∼ Bernoulli(p = expit{ − 0.5 + X1 + 0.5X2}); in Scenario 2, A ∼ Bernoulli(p = expit{ − 0.5 + X1 + 0.5X2 + X1X2}). In Scenario 1 and 2, Y ∼ N(0.25X1 + X2 + A{0.5 − X1 + 1.5X2}, 1); in Scenario 3, Y ∼ N(0.25X1 + X2 + X1X2 + A{0.5 − X1 + 1.5X2}, 1). The true γ1\(A, X1) is A(ψ0\ + ψ1\*X1) = A(0.5 + 1.5E[X2∣X1] − X1) = A(1.25 − X1). The optimal PATS is thus dopt \* = 1, whether X1 = 0 or X1 = 1. Note that the optimal PATS varies according to covariates’ values in the two time-points setting considered in Appendix E. The parameters ψ0\* and ψ1\* are estimated with “IPTW+G-estimation”, “IPTW+dWOLS”, “integrate G-estimation”, “integrate dWOLS”, “CE G-estimation”, “CE dWOLS” in addition to usual implementations of G-estimation and dWOLS for ATS. For all estimators, the blip model includes only X1, and the treatment and the treatment-free models only include main terms (no interactions). As such, in Scenario 1, both the treatment and the treatment-free models are correctly specified. In Scenario 2, only the treatment-free model is correctly specified. In Scenario 3, only the treatment model is correctly specified. A total of 1000 replications of each simulation scenario were generated for each sample sizes of n = 100, 1000 and 10 000. We report below the estimated relative bias ((average estimate - true value)/true value × 100%; Rel. bias) and standard deviation (SD) over the 1000 replications for each estimator in each combination of scenarios and sample size. The proportion of observations for which the optimal PATS was correctly identified and the expected loss (difference between the expected outcome under the true optimal PATS and the estimated optimal PATS) were also calculated within the same simulated data. In addition, we have explored the coverage of confidence intervals using the adaptive m-out-of-n bootstrap proposed by Chakraborty et al (2013) in a two-time-points scenario with near non-regularity (see Appendix F for details). Recall that non-regularity is not of concern in the single-stage setting, and so m-out-of-n results are not presented for the simulations involving only a single treatment decision. Results ------- The results of the main simulations are presented in Tables 1 and 2. Standard G-estimation and dWOLS, i.e. the approaches that use the incorrect (reduced) blip model with no adjustment for the partial nature of the ATS, produced estimates with non-negligible bias in all scenarios. On the other hand, all PATS estimators we introduced achieved unbiased estimation when the treatment model was correctly specified (Scenario 1 and 3). When the treatment model was misspecified, some bias remained for IPTW+G-estimation and IPTW+dWOLS, but not for the other PATS estimators. The standard deviation of the estimates were almost identical for all PATS estimators. The true optimal PATS was identified less often by standard G-estimation and dWOLS than the PATS estimators for n = 100 and n = 1000 in Scenarios 1 and 3, and for all sample sizes in Scenario 2. IPTW+dWOLS and IPTW+G-estimation also identified the optimal PATS less often than the other PATS estimators in Scenario 2 when n = 100 and n = 1000. For observations (“individuals”) whose true optimal PATS failed to be identified, the expected loss was always 0.25. Similar results were observed in the two time-point simulations, except that the difference in the ability to identify the optimal PATS between standard ATS estimators and the PATS estimators was more pronounced (see Appendix E). The coverage of the confidence intervals produced using the adaptive m-out-of-n bootstrap was approximately 95% (between 95.2% and 95.7% depending on the parameter), as desired (Appendix F). [h!] [tab1] [!htbp] [tab4] Application =========== Context ------- Breast cancer is the most common cancer among women and also the leading cause of cancer deaths among them (Bray et al, 2018). Various treatments can be used to treat patients who receive a breast cancer diagnosis including surgery, chemotherapy, radiotherapy and hormonal therapy. Hormonal therapy seeks stop hormone production or to interfere with the ability of hormones to attach to cancer cells and thus prevent their growth. This treatment is currently recommended for patients whose tumors are hormone receptor positive. Unfortunately, not all cancers that are hormone receptor positive respond adequately to hormonal therapy. Even when hormonal therapy helps treating the cancer itself, it can have multiple undesirable side effects, including hot flashes, sexual problems, weight gain, nausea, fatigue, high cholesterol and osteoporosis (Canadian Cancer Society, 2021). As such, it is important to appropriately tailor hormonal therapy so that it is prescribed only to patients who will benefit from it. Obesity is associated with poorer outcomes among patients with breast cancer (Chan et al, 2014). It is also known to affect the circulating levels of estrogen in the body (Calle and Kaaks, 2004). We thus hypothesized that obesity may modify the effect of hormonal therapy on survival in breast cancer patients. More precisely, we believed that hormonal therapy would only be beneficial in hormone receptor positive women with normal weight. While a breast cancer is generally considered to be hormone receptor positive if it is either estrogen receptor positive (ER+), progesterone receptor positive (PR+) or both, the relevance of testing PR is somewhat controversial. Indeed, it is already recommended to provide hormonal therapy when the cancer is ER+, regardless of PR status, and the profile ER-PR+ is very rare, representing only 3% of all breast cancers (Dunnwald, Rossing and Li, 2007). In fact, this profile is so rare that some have argued that testing PR has very little therapeutic impact and is unlikely to be cost-effective (Olivotto et al, 2004). Consequently, while effect modification by PR status is expected, tailoring hormonal therapy treatment according to PR status may not be necessary. To illustrate our method, we thus consider the estimation of a single-stage PATS that tailor hormonal therapy decisions according to both ER status and obesity, while neglecting the known interaction with PR status. Data and analysis ----------------- We analysed data concerning women diagnosed with a non-metastatic breast cancer between 1987 and 2009 that are part of the breast cancer registry maintained by the Centre des Maladies du Sein, in Québec, Canada. We compared the (log transformed) number of years of survival since breast cancer diagnosis (Y) between women who received hormonal therapy (A = 1) and those who did not (A = 0). Data on age, body mass index (BMI), menopause, cancer’s grade and stage, ER status, PR status, type of surgery, first degree familial history of breast cancer, hormone replacement therapy, chemotherapy, radiotherapy, trastuzumab and year of diagnosis were considered as potential confounders (X). While ER status, PR status and BMI were all considered as potential effect modifiers (H), only ER status and BMI were used as tailoring variables (H\). Before proceeding with the analysis, we treated the missing data as follows. When menopause status was missing, it was either imputed to “no” when age < 50 and to “yes” when age ≥ 50. Based on clinical and contextual knowledge, missing data on chemotherapy and radiotherapy were imputed to their most likely value (“no” for chemotherapy and “yes” for radiotherapy). Missing data on progesterone status and cancer’s grade were considered as an “unknown” category. Observations with missing data on other variables were deleted. A total of 5444 individuals were thus included in the analysis. To analyze the data, we used both the naive dWOLS estimator and the CE dWOLS estimator we have introduced. For both estimators, the treatment model was fitted using a logistic regression of the treatment according to all potential confounders. The outcome model was a linear regression of the natural logarithm of the survival time according to all potential confounders and including interaction terms between treatment and ER-BMI categories (ER+ and BMI < 25, ER+ and BMI ≥ 25, ER- and BMI < 25, ER- and BMI ≥ 25). The results are of this regression model are interpreted as differences in expected log years of survival. Except for years, which was modeled using a restricted cubic splines with five knots, all other covariates were entered in the same categories as presented in Table [TabAppli]. Because the survival time was right censored for 4151 patients, the outcome model was fitted only among observations that were not censored using an inverse probability of censoring weight as proposed by Simoneau et al (2020). Again, the censoring model was a logistic regression including all potential confounders. Censoring weights were truncated at their 99.9t*h percentile to reduce influence of large weights and to achieve a better representativeness of the weighted uncensored observations to the complete sample. Standard nonparametric bootstrap with 5000 replicates was used to produce confidence intervals. [!htbp] [TabAppli] @ lrrr & No hormonal therapy & Hormonal therapy & SMD & 1800 (33.0) & 3644 (67.0) Age & & & 0.419 ≤ 39 & 186 (10.3) & 119 ( 3.3) & 40–49 & 496 (27.6) & 625 (17.2) & 50–59 & 477 (26.5) & 1189 (32.6) & 60–69 & 353 (19.6) & 1009 (27.7) & ≥ 70 & 288 (16.0) & 702 (19.3) & BMI ≥ 25 & 698 (38.8) & 1698 (46.6) & 0.159 Menopause & 658 (36.5) & 875 (24.0) & 0.276 Grade & & & 0.791 1 & 225 (12.5) & 1055 (29.0) & 2 & 343 (19.1) & 1406 (38.6) & 3 & 994 (55.2) & 838 (23.0) & Unknown & 238 (13.2) & 345 ( 9.5) & Stage & & & 0.077 I & 809 (44.9) & 1671 (45.9) & II & 746 (41.4) & 1569 (43.1) & III & 245 (13.6) & 404 (11.1) & Estrogen + & 819 (45.5) & 3517 (96.5) & 1.360 Progesterone & & & 0.840 + & 604 (33.6) & 2627 (72.1) & - & 999 (55.5) & 805 (22.1) & Unknown & 197 (10.9) & 212 ( 5.8) & Surgery & & & 0.076 Mastectomy & 482 (26.8) & 867 (23.8) & Breast-conserving & 1289 (71.6) & 2730 (74.9) & None & 29 ( 1.6) & 47 ( 1.3) & 1st degree familial history & 399 (22.2) & 980 (26.9) & 0.110 Hormone replacement therapy & 1170 (65.0) & 1920 (52.7) & 0.252 Chemotherapy & 977 (54.3) & 1527 (41.9) & 0.250 Radiotherapy & 1307 (72.6) & 2875 (78.9) & 0.147 Trastuzumab & 70 ( 3.9) & 113 ( 3.1) & 0.043 Year & & & 0.729 1985–1989 & 269 (14.9) & 119 ( 3.3) & 1990–1994 & 384 (21.3) & 328 ( 9.0) & 1995–1999 & 402 (22.3) & 513 (14.1) & 2000–2004 & 328 (18.2) & 1164 (31.9) & 2005–2009 & 417 (23.2) & 1520 (41.7) & [-1.8ex] All numbers are n (%). SMD = standardized mean difference. [!htbp] [TabAppli2] Results ------- Table [TabAppli] presents the characteristics of the 3644 (67.0%) patients who received and 1800 (33.0%) who did not receive hormonal therapy. Among others, women who received hormonal therapy were overall older, had a greater BMI and a higher cancer’s grade. Unsurprisingly, there is a large imbalance between treated and untreated patients according to ER and PR statuses, since these markers are used for tailoring treatment in clinical practice. Hormonal therapy has also become more frequent in time. Results of the PATS analysis are presented in Table [TabAppli2]. In this specific application, only minor differences were observed between the results of the naive dWOLS and CE dWOLS estimators. In both analyses, hormonal therapy was associated with a lower survival time among ER- patients with a BMI ≥ 25 (CE dWOLS: -0.49; 95% CI: -0.83 to -0.12). Among the ER- participants with BMI < 25, a negative association was also observed, but the results were also compatible with an absence of difference (CE dWOLS: -0.28, 95%CI: -0.68 to 0.10). Among ER+ participants with BMI < 25, the observed difference in log survival times was close to zero with 95% confidence intervals covering equally small beneficial and detrimental associations (CE dWOLS: -0.01, 95% CI: -0.14 to 0.13). Finally, among ER+ participants with BMI ≥ 25, a small positive association was observed, but the confidence intervals indicated the data were compatible with a range of associations going from a slightly negative to moderately positive (CE dWOLS: 0.13, 95% CI: -0.10 to 0.34). Discussion ========== Better evidence-based tailoring of treatment decisions according to patients’ characteristics and to their evolving condition has become a priority. Robust statistical methods for identifying optimal treatment strategies are important tools for reaching this objective. Although it may be theoretically interesting to find the optimal treatment strategy according to multiple, or even all, relevant characteristics, this is often too ambitious or impractical. Indeed, achieving this would require infeasibly large datasets and may render treatment decisions overly complicated. Identifying optimal partially adaptive treatment strategies that aim to tailor treatment decisions according to a few selected characteristics may be a more reasonable goal in practice. In this paper, we proposed methods for identifying optimal PATSs that benefit from the double robustness property. Methods for producing inferences were additionally discussed. To facilitate the use of these methods, we supply functions in `R` as supplementary material. All our proposed estimators had similar performance in the simulation study in scenarios we investigated. As such, because CE dWOLS is simpler to implement, it may be preferred in practice. The simulation study results also supports the validity of the data-adaptive m-out-of-n bootstrap for producing inferences. It is noteworthy that the algorithm for implementing the m-out-of-n bootstrap for ATSs (or PATSs) has been developed for the two time-point setting. Extending its application to a multiple time-point setting isn’t trivial and should be explored in future studies. Furthermore, while double robustness is an interesting property that offers two chances at correctly specifying models to control confounding bias, it may often be the case in practice that both models are incorrectly specified. There are some situations where double robust methods perform worse than non double robust ones when both models are incorrectly specified, particularly in settings where positivity may be violated (Kang and Schafer, 2007). We have also illustrated the usefulness of our method for estimating an optimal PATS concerning the use of hormonal therapy in treating breast cancer patients. In this illustration, we were interested in tailoring treatment only according to ER status and BMI, but not PR status, although effect modification by PR is expected. Neglecting PR status was motivated by the fact that the additional clinical value of PR, above ER alone, is controversial. Although similar results were obtained using our proposed PATS estimator and a naive ATS estimator in this example, this needs not always be the case as illustrated in our simulation study. We believe the lack of difference between ATS and PATS estimators in this specific application might be due to the very low prevalence of the ER-PR+ status. From a substantive point of view, hormonal therapy was associated with poorer survival among ER- women, especially those with a greater BMI. The results were more imprecise regarding ER+ breast cancer patients, but provide some evidence suggesting that ER+ breast cancer patients with a greater BMI may benefit more from hormonal therapy than ER+ patients with a lower BMI. Given the large widths of confidence intervals, more studies are required to further explore this unexpected result. It is possible that residual confounding or selection bias is present although a very rich set of covariates were adjusted for. Moreover, in an analysis where only effect modification according to ER status was considered (not presented), the results were in the expected direction: hormonal therapy was associated with a better survival among ER+ participants, but not among ER- participants. In addition to its practical importance, we believe the current work also opens the door to further important methodological developments. First, it seems reasonable to assume that tailoring treatment decisions according to only a few variables may increase the precision of the estimators of the selected interaction terms, since fewer treatment-covariate interaction parameters need to be estimated. It would be worthy to verify this hypothesis and investigate if this reduces the risk of identifying as “optimal” treatment strategies that truly have low expected value. PATS estimators could also provide a framework for data-driven selection of the most important tailoring features. Acknowledgements ================ This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada. DT is also supported by a career award from the Fonds de recherche du Québec - Santé. Part of this work was conducted while DT was a visiting professor at McGill University. DT would like thank Université Laval for funding his visit at McGill University, and McGill University for welcoming him. 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Appendix A - G-estimation of adaptive treatment strategies ========================================================== The G-estimation of the parameters ψj is performed recursively, for j = K, ..., 1. First, a vector-valued function S(Aj) of the same length as ψj containing effect modifiers of the treatment effect must be specified. The optimal form for S(Aj) is known (Robins, 2004), but is generally complex and S(aj) = ∂γj(aj, hj; ψj)/∂ψj is thus commonly used. For example, if γj(aj, hj; ψj) = aj(ψ0j + ψ1jhj) then S(aj) could be (aj, ajhj). Next, a model for E[S(Aj)∣Hj] is specified. Define GiK = Yi − γK(aiK, hiK; ψK) and Gij = Yi − γj(aij, hij; ψj) + ∑k = j + 1K[γk(dkopt, hik; ψ̂k) − γk(aik, hik; ψ̂k)] if j ≠ K, where ψ̂k is the estimated value of ψk. A model for E[Gj∣Hj] is then specified. The G-estimate of ψj is the solution ψ̂j to the estimating equations 0 = ∑i = 1n{S(Aij) − E[S(Aij)∣Hij]}{Gij − E[Gij∣Hij]}. This equation has a closed-form solution for continuous Y when linear models are used for the treatment-free (E[Gij∣Hij]) and blip components. Appendix B - Double robustness of weighted ordinary least squares ================================================================= Consider the case where the outcome model may be misspecified, but the treatment model is correctly specified. We consider the weights w = ∣A − E[A∣X]∣, which benefit from the balancing property A∐X in the weighted data. We show that E[ψ̂] converges in probability to ψ. We first consider the following outcome model, for which the treatment-free component (Xiβ) may be misspecified: Yi = Xiβ + AiXiψ + εi. The weighted least squares estimating equations are $$\begin{aligned} 0 &= \sum w	arxiv_0000480
; elle basée sur des manipulations algébriques des anneaux de séries du même type que $\Z\_{1^-}[\![T]\!]$. Celle que nous proposons est, en revanche, purement géométrique. La seule difficulté réside dans le fait que le cadre adapté à ce problème est celui, fort naturel mais sans doute encore un peu exotique, de la droite de Berkovich sur un anneau d’entiers de corps de nombres (la construction et les propriétés de cet espace font l’objet de l’annexe [sectionberko]). Signalons, pour finir, que, contrairement à l’habitude, nous construisons un revêtement d’un ouvert d’un espace affine ; les résultats de type GAGA y tombent donc en défaut et nous utiliserons, pour pallier ce manque, le fait que l’ouvert en question soit un espace de Stein. Notations Nous désignerons par $\N$ l’ensemble des nombres entiers positifs et par $\N^\$ le sous-ensemble formé de ceux qui ne sont pas nuls. Remerciements* La dernière partie de cet article a été rédigée au cours de l’année que j’ai passée à l’université de Ratisbonne. Je souhaite remercier Klaus Künnemann, qui m’a permis d’y séjourner, pour son accueil et ses encouragements. Ma gratitude va également à Antoine Chambert-Loir dont les conseils concernant la structure de cet texte m’ont permis, je l’espère, d’en accroître l’intérêt et la clarté. Merci également à Antoine Ducros de m’avoir communiqué ses notes sur les théorèmes GAGA. Stratégie de raccord ==================== Nous rappelons ici une démonstration classique du fait que tout groupe fini est groupe de Galois d’un revêtement de la variété algébrique $\P^1\_{\C}$. Ce n’est qu’un prétexte pour présenter la stratégie de raccord que nous utiliserons constamment par la suite. Considérons tout d’abord le cas des groupes cycliques. Pour disposer de plus de souplesse, nous allons commencer par construire des revêtements de la variété analytique $\P^1(\C)$, et même des revêtements de petits ouverts de cette variété. Soit $m\in\N^\$. Choisissons un point P* de $\P^1(\C)$, une coordonnée locale z au voisinage de ce point, un disque ouvert DP sur lequel elle est définie et un disque fermé EP de rayon strictement positif contenu dans DP. Soient a et b deux points distincts de EP. Considérons le revêtement connexe et lisse XP du disque DP donné par l’équation um = (z − a)(z − b)m − 1. C’est un revêtement galoisien de groupe $\Z/m\Z$. En outre, il est trivial au-dessus du complémentaire du disque EP. Remarquons que pour déterminer le groupe de Galois nous avons utilisé le fait que le corps $\C$ contienne une racine primitive $m\eme$ de l’unité. Nous allons maintenant recoller des revêtements du type précédent afin d’en construire qui possèdent des groupes de Galois finis arbitraires. Fixons un groupe fini G. Notons n son ordre et choisissons-en des générateurs g1, …, gt, avec $t\in\N^\$. Soit $i\in\cn{1}{t}$. Notons n*i l’ordre de l’élément gi dans le groupe G et posons di = n/ni. Choisissons un point Pi de $\P^1(\C)$ et construisons, par la méthode du paragraphe précédent, un $\Z/n\_{i}\Z$-revêtement XPi au-dessus d’un disque ouvert DPi et trivial hors d’un disque fermé EPi ⊂ DPi. Indexons les feuillets de ce revêtement par les entiers compris entre 0 et ni − 1 de façon compatible avec l’action du groupe $\Z/n\_{i}\Z\simeq \la g\_{i} \ra$. Considérons maintenant $\textrm{Ind}\_{\la g\_{i} \ra}^G(X\_{P\_{i}})$, le G-revêtement induit par le $\la g\_{i} \ra$-revêtement XPi. Rappelons qu’il est constitué topologiquement de di copies de XPi. Nous pouvons envoyer, de façon bijective, les feuillets de ce revêtement sur les éléments du groupe. Pour ce faire, choisissons, dans G, des représentants ai, 0, …, ai, di − 1 des éléments du quotient $G/\la g\_{i} \ra$. L’application qui envoie le feuillet indexé par k de la copie indexée par l de XPi sur l’élément ai, lgik de G est bijective. Nous pouvons alors décrire l’action du groupe G sur le revêtement $\textrm{Ind}\_{\la g\_{i} \ra}^G(X\_{P\_{i}})$ de la façon suivante : l’élément g de G envoie le feuillet associé à l’élément h de G sur le feuillet associé à l’élément hg. Notons Dʹ le complémentaire de la réunion des disques EP1, …, EPt dans $\P^1(\C)$. Considérons le G-revêtement $\textrm{Ind}\_{\la e \ra}^G(D')$ induit par le revêtement trivial de Dʹ et indexons ses feuillets par les éléments de G de façon compatible avec l’action de ce groupe. Raccordons, maintenant, les revêtements que nous venons de construire. Nous supposerons que les disques DPi, avec $i\in\cn{1}{t}$, sont deux à deux disjoints. Nous pouvons facilement nous ramener à ce cas en les réduisant, si besoin est. Pour tout élément i de $\cn{1}{t}$, nous recollons alors, au-dessus de l’intersection DPi ∩ Dʹ, les feuillets associés aux mêmes éléments du groupe G (cf. figure ). Nous obtenons ainsi un revêtement Y de $\P^1(\C)$ dont on vérifie facilement qu’il est connexe, lisse et galoisien de groupe G. [figurerecollement] (-4,-9)(19,7) (-3,-7)(-1,-7) (-1,-7)(0,-7) (0,-7)(2,-7) (13,-7)(15,-7) (15,-7)(16,-7) (16,-7)(18,-7) (-3,-8)(-1,-8) (0,-8)(15,-8) (16,-8)(18,-8) (15,6)(15,3) (15,3)(16,3) (16,3)(16,6) (16,6)(15,6) (15,1)(15,-2) (15,-2)(16,-2) (16,-2)(16,1) (16,1)(15,1) (15,5)(13,5) (15,4.5)(13,4.5) (15,5.5)(13,5.5) (15,0)(13,0) (15,0.5)(13,0.5) (-1,4)(-1,0) (-1,0)(0,0) (0,0)(0,4) (0,4)(-1,4) (0,2.5)(2,2.5) (0,3.5)(2,3.5) (-1,3.5)(-3,3.5) (-1,2.5)(-3,2.5) (16,5.5)(18,5.5) (16,5)(18,5) (16,4.5)(18,4.5) (16,0.5)(18,0.5) (16,0)(18,0) (0,1.5)(2,1.5) (0,0.5)(2,0.5) (15,-1.5)(13,-1.5) (-1,1.5)(-3,1.5) (-1,0.5)(-3,0.5) (16,-1.5)(18,-1.5) (1,3.6)⋮ (1,2.6)⋮ (1,1.6)⋮ (-2,3.6)⋮ (-2,2.6)⋮ (-2,1.6)⋮ (14,4.3)⋮ (17,4.3)⋮ (14,-0.1)⋮ (17,-0.1)⋮ (15.5,2.87)⋮ (-0.5,5.85)⋮ (-0.5,-0.69)⋮ (15.5,2.19)⋮ (-0.5,5.17)⋮ (-0.5,0)⋮ (7.7,-2.82)Y (7.3,-5.5)P1(C) (7.96,-3.6)(7.98,-5.2) (14.8,-0.5)dj − 1 (15.41,5.2)1 (7.79,-7.17)Dʹ (-0.9,-6.3)EPi (-2.93,-6.3)DPi (15.1,-6.3)EPj (16.9,-6.3)DPj (-0.63,3)l (2.39,3.81)0 (2.5,2.7)k (2.5,1.76)kʹ (2.5,0.74)ni − 1 (12.2,5.75)0 (12.2,5.27)1 (12.2,4.76)2 (12.2,0.77)0 (12.2,0.26)1 (12.2,-1.78)nj − 1 (6.77,4.59)ai, lgik = aj, 1gj2 (5.5,0.1)ai, lgikʹ = aj, dj − 1gj− 1 Ainsi avons-nous obtenu une variété analytique complexe Y vérifiant les propriétés requises. Il nous reste à montrer que c’est, en réalité, une variété algébrique. Ce résultat découle du théorème d’existence de Riemann ou, si l’on veut, des théorèmes GAGA de J.-P. Serre. Nous avons finalement obtenu le résultat suivant : [surC] Tout groupe fini est groupe de Galois d’une extension du corps $\C(T)$. Pour résumer, rappelons en quelques mots la stratégie de la preuve : 1. Construire des revêtements cycliques sur de petits ouverts, triviaux au voisinage du bord. 2. Raccorder ces revêtements. 3. Montrer que le revêtement obtenu est algébrique. D. Harbater l’a développée dans plusieurs contextes et utilisée pour démontrer de nombreux résultats. Nous renvoyons le lecteur désireux d’en savoir plus au texte . La simplicité de cette stratégie de raccord (patching chez D. Harbater) invite à l’appliquer dans de nombreux contextes géométriques, pour peu que l’on dispose d’une bonne notion de petits ouverts et de théorèmes d’algébricité. Ce n’est pas le cas de la géométrie algébrique, où deux ouverts non vides de la droite projective se coupent toujours, mais ce devrait l’être pour toute géométrie analytique. Dans la suite de ce texte, nous illustrerons cette idée en appliquant la stratégie indiquée dans le cadre des espaces de Berkovich sur un corps ultramétrique complet, à la section [sectiondemoberko], puis sur un anneau d’entiers de corps de nombres, à la section [sectiondemo]. Problème inverse de Galois sur une droite relative ================================================== Soient k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet et X un espace k-analytique géométriquement irréductible. Nous noterons $\Os$ le faisceau structural sur cet espace. Soit B une partie de l’espace X. Notons Y le produit fibré $X\times\_{k}\P^{1,\an}\_{k}$, π : Y → X et $\lambda : Y \to \P^{1,\an}\_{k}$ les morphismes naturels de projection. Signalons que, d’après , théorèmes 7.16 et 8.4, l’espace Y est géométriquement irréductible. Notons Y(B) l’image réciproque de B par le morphisme π. Dans les numéros [constructionlocaleberko], [numerorecollementberko] et [conclusion], nous supposerons que la partie B est compacte (pour les applications, dans la preuve des corollaires [corcomplet] et [corfertile], elle sera même réduite à un point). Nous la munirons alors du faisceau des fonctions surconvergentes, c’est-à-dire du faisceau $j^{-1}\Os$, où $j : B \hookrightarrow X$ désigne l’inclusion. De façon générale, nous utiliserons, sans plus le préciser, le faisceau des fonctions surconvergentes pour toute partie compacte. En particulier, l’espace localement annelé associé à l’espace Y(B) que nous obtiendrons ainsi ne sera autre que le germe (Y, π− 1(B)) au sens de V. Berkovich (cf. , §3.4 ou , §2). Construction locale de revêtements cycliques -------------------------------------------- Dans le cas complexe, la construction locale était particulièrement simple, car nous disposions de racines primitives de l’unité de tout ordre. Lorsque nous cherchons à construire un revêtement cyclique dont l’ordre n est premier à l’exposant caractéristique du corps de base, la situation n’est guère plus compliquée, car nous disposons encore de racines primitives n\`emes de l’unité sur certains ouverts. Nous utiliserons alors une extension de Kummer bien choisie. Dans les autres cas, nous ferons appel à la théorie d’Artin-Schreier-Witt. Nous noterons p l’exposant caractéristique du corps k. Fixons une extension finie et séparable K de k. Soit P ∈ k[T] le polynôme minimal unitaire d’un élément primitif de cette extension. Notons t le point de $\P^{1,\an}\_{k}$ défini par l’annulation de ce polynôme. ### Extensions de Kummer Soit n un entier supérieur ou égal à 2 et premier à p. Supposons que le corps K contient une racine primitive $n\eme$ de l’unité. Puisque l’anneau local au point t est hensélien, il contient une racine primitive $n\eme$ de l’unité, que nous noterons ζ. Cette racine est définie sur un voisinage du point t, que nous pouvons supposer de la forme $$V\_{u} = \left\{\left.y\in\P^{1,\an}\_{k}\,\right\|\, \|P(y)\|\le u\right\},$$ avec u > 0. Posons V = λ− 1(Vu). Nous noterons encore P et ζ les rétrotirettes des éléments P et ζ par le morphisme λ. Remarquons que l’élément ζ de $\Os(V)$ est encore une racine primitive $n\eme$ de l’unité. Soient U un voisinage de B dans X et α un élément de $\Os(U)$ nul en tout point de B. Posons $$Q(S) = S^n-P^n-\alpha \in \Os(\pi^{-1}(U))[S].$$ Définissons un préfaisceau $\Fs$ sur π− 1(U) en posant, pour toute partie ouverte W de π− 1(U), $$\Fs(W) = \Os(W)[S]/(Q(S))$$ et en utilisant les morphismes de restriction induits par ceux du faisceau $\Os$. Le caractère unitaire du polynôme Q assure que $\Fs$ est un faisceau de $\Os\_{\pi^{-1}(U)}$-algèbres cohérent. Le faisceau $\Fs$ est l’image directe du faisceau structural d’une courbe analytique sur Y. Celle-ci nous est donnée comme un revêtement ramifié de degré n de π− 1(U). Soit $v\in\of{]}{0,u}{]}$. Posons $$V(B) = \left\{y\in Y(B)\, \big\|\, \|P(y)\|\le v\right\}$$ et Vʹ(B) = {y ∈ V(B) ∣ P(y) ≠ 0}. Remarquons que le complémentaire de Vʹ(B) dans V(B) est fermé dans Y(B). [nonramifieberko] Il existe un isomorphisme de $\Os\_{V'(B)}$-algèbres $$\varphi : \Fs \to \Os^n$$ tel que, pour tout ouvert W de Vʹ(B) et tout élément s de $\Fs(W)$, nous ayons φ(ζs) = τ(φ(s)), où τ désigne l’automorphisme du faisceau $\Os^n$ qui consiste à faire agir la permutation cyclique (1 2 ⋯ n) sur les coordonnées. Le rayon de convergence de la série (1 + T)1/n = ∑i = 0+ ∞C1/ni Ti ∈ k[ [T] ] est strictement positif. La série $$P\, \sum\_{i=0}^{+\infty} C\_{1/n}^i\, \left(\frac{\alpha}{P^n}\right)^{i}$$ définit donc un élément ω de $\Os(V'(B))$, qui est une racine du polynôme Q. Nous avons alors l’égalité $$Q(S) = S^n-P^n-\alpha = \prod\_{j=0}^n (S - \zeta^{j} \omega) \textrm{ dans } \Os(V'(B)).$$ Par conséquent, le morphisme $$\begin{array}{ccc} \Fs & \to & \Os^n\\ R(S) & \mapsto & \left(R(\omega), R(\zeta^{-1} \omega),\ldots, R(\zeta^{-(n-1)} \omega)\right) \end{array}$$ est un isomorphisme. On vérifie immédiatement qu’il satisfait la condition requise. La première partie du résultat signifie que le revêtement associé au faisceau $\Fs$ est trivial au-dessus de Vʹ(B). La seconde assure que le groupe $\of{\la}{\zeta}{\ra} \simeq \Z/n\Z$ agit sur le revêtement par une permutation cyclique des feuillets du lieu trivial. Nous allons maintenant imposer des conditions sur les données B et α de façon que le faisceau $\Fs$ soit associé à un revêtement irréductible. Nous dirons que la partie B de X satisfait la condition (CGI) si elle est compacte et possède un système fondamental de voisinages affinoïdes géométriquement intègres. Sous les conditions de cette définition, l’anneau $\Os(B)$ est intègre et la partie B connexe. En particulier, le principe du prolongement analytique vaut sur B et les anneaux locaux en tous les points de B sont intègres. [Vxirr] Supposons que la partie B de X satisfait la condition (CGI). Alors, la partie V(B) de Y possède un système fondamental de voisinages affinoïdes irréductibles. En particulier, pour tout point z de V(B), le morphisme naturel $$\Os(V(B)) \to \Os\_{z}$$ est injectif. Soient Uʹ un voisinage affinoïde géométriquement irréductible de B dans U et vʹ un nombre réel strictement supérieur à v. La partie de Y définie par $$\{z\in \pi^{-1}(U')\, \big\|\, \|P(z)\|\le v'\}$$ est alors irréductible, d’après , théorème 8.4. En effet, ce n’est autre que le produit, au-dessus de k, de l’espace géométriquement irréductible Uʹ par l’espace $$\left\{\left. x\in\P^{1,\an}\_{k}\, \right\|\, \|P(x)\|\le v'\right\},$$ qui est irréductible, car le polynôme P est irréductible sur k. La condition (CGI) assurent que l’ensemble des parties de la forme précédente est un système fondamental de voisinages de V(B) dans Y. Nous supposerons désormais que la partie compacte $\boldsymbol{B}$ de $\boldsymbol{X}$ satisfait la condition (CGI). Nous dirons que l’élément α de $\Os(B)$ satisfait la condition (In, K) s’il existe un point x de B qui vérifie les deux conditions suivantes : 1. les corps K et $\Frac(\Os\_{x})$ sont linéairement disjoints sur k ; 2. le polynôme Sn − α est irréductible sur le corps $\Frac(\Os\_{x})\otimes\_{k} K$. [lemintlocalberko] Supposons que l’élément α de $\Os(B)$ satisfait la condition (In, K). Alors, le polynôme Q(S) = Sn − Pn − α est irréductible sur le corps Frac$(\Os(V(B)))$. En particulier, l’anneau $\Fs(V(B))$ est intègre. Soit x un point de B satisfaisant les propriétés énoncées dans la définition de la condition (In, K). Puisque les corps K et $\Frac(\Os\_{x})$ sont linéairement disjoints sur k, le polynôme P est irréductible sur $\Os\_{x}$. Il l’est donc encore dans $\Hs(x)[T]$, puisque le corps κ(x) et l’anneau local $\Os\_{x}$ sont henséliens. Notons Z(B) l’ensemble des points de Y(B) en lesquels la fonction P est nulle. D’après le raisonnement précédent, la trace de la fibre π− 1(x) sur Z comporte un seul point, que nous noterons z. Notons $\Os\_{Z}$ le faisceau structural sur Z. Nous avons alors un isomorphisme $$\Os\_{x}[T]/(P(T)) \xrightarrow[]{\sim} \Os\_{Z,z}.$$ Le polynôme Sn − α est donc irréductible sur le corps $\Frac(\Os\_{Z,z})$, isomorphe à $\Frac(\Os\_{x})\otimes\_{k} K$. Par conséquent, le polynôme Q(S) est irréductible sur le corps $\Frac(\Os\_{z})$. D’après le lemme [Vxirr], le morphisme naturel $\Os(V(B))\to \Os\_{z}$ est injectif. Par conséquent, le corps $\Frac(\Os(V(B)))$ est un sous-corps de $\Frac(\Os\_{z})$. On en déduit que le polynôme Q(S) est irréductible sur le corps $\Frac(\Os(V(B)))$. Le polynôme Q(S) étant unitaire, l’unicité de la division euclidienne assure que le morphisme $$\Os(V(B))[S]/(Q(S)) \to \Frac(\Os(V(B)))[S]/(Q(S))$$ est injectif. Puisque l’anneau au but est intègre, celui à la source, qui n’est autre que l’anneau $\Fs(V(B))$, l’est également. Ce résultat signifie que la courbe associée au faisceau $\Fs$ est intègre, c’est-à-dire réduite et irréductible. ### Extensions d’Artin-Schreier-Witt Il nous reste à traiter le cas des groupes cycliques dont l’ordre n’est pas premier à l’exposant caractéristique p du corps k. Nous supposons désormais que p est un nombre premier et chercherons à construire un revêtement cyclique d’ordre n = pr, où r est un entier supérieur à 1. Il s’agit essentiellement de remplacer, dans le numéro précédent, la théorie de Kummer par celle d’Artin-Schreier-Witt. Nous nous contenterons d’indiquer les grandes lignes de la preuve. Dans ce numéro, comme dans le précédent, nous supposerons que la partie compacte $\boldsymbol{B}$ de $\boldsymbol{X}$ satisfait la condition (CGI). Soit α un élément de $\Os(B)$ nul en tout point de B. Nous dirons que l’élément α de $\Os(B)$ satisfait la condition (Ip, K) s’il existe un point x de B qui vérifie les deux conditions suivantes : 1. les corps K et $\Frac(\Os\_{x})$ sont linéairement disjoints sur k ; 2. le polynôme Sp − α est irréductible sur le corps $\Frac(\Os\_{x})\otimes\_{k} K$. Soit v > 0. Posons $$V(B) = \left\{y\in Y(B)\, \big\|\, \|P(y)\|\le v\right\}$$ et Vʹ(B) = {y ∈ V(B) ∣ P(y) ≠ 0}. Notons Wr l’anneau des vecteurs de Witt de longueur r sur $\Os(V(B))[S\_{0},\ldots,S\_{r-1}]$. Posons S = (S0, …, Sr − 1) ∈ Wr et, pour tout élément a de $\Os(V(B))[S\_{0},\ldots,S\_{r-1}]$, {a} = (a, 0, …, 0) ∈ Wr. Pour tout $i\in\cn{0}{r-1}$, définissons un polynôme Qi(S0, …, Sr − 1) à coefficients dans $\Os(V(B))$ par la formule (Q0, …, Qr − 1) = F(S) − {P}p − 1S − {α}dans Wr. Le groupe $\Z/p^r\Z$ agit sur l’anneau $\Os(V(B))[S\_{0},\ldots,S\_{r-1}]/(Q\_{0},\ldots,Q\_{r-1})$ en laissant stable $\Os(V(B))$ et en envoyant Si, pour $i\in\cn{0}{r-1}$, sur la (i + 1)\`eme coordonnée du vecteur S + {P} dans Wr. Par analogie avec la construction précédente, nous définissons un faisceau sur V(B) par $$\Fs = \Os[S\_{0},\ldots,S\_{r-1}]/(Q\_{0},\ldots,Q\_{r-1}).$$ Les propriétés des vecteurs de Witt montrent que, pour tout $i\in\cn{0}{r-1}$, nous avons $$Q\_{i} = S\_{i}^p - P^{(p-1)p^i}S\_{i} \mod (\alpha, Q\_{0},\ldots,Q\_{i-1}).$$ Soit y ∈ Vʹ(B). L’image S0p − Pp − 1S0 du polynôme Q0(S0) dans le corps résiduel κ(y) de l’anneau local $\Os\_{y}$ est scindé à racines simples. Puisque cet anneau local est hensélien, le polynôme P0(S0) possède p racines simples dans $\Os\_{y}$. En raisonnant par récurrence sur le nombre de variables, on montre ainsi que le système d’équations polynomiales donné par Q0, …, Qr − 1 possède exactement pr racines ω1, …, ωpr dans $\Os\_{y}^r$ et que le morphisme $$\begin{array}{cccc} \psi : & \Os\_{y}[S\_{0},\ldots,S\_{r-1}]/(Q\_{0},\ldots,Q\_{r-1}) & \to & \Os\_{y}^{p^r}\\ & R(S\_{0},\ldots,S\_{r-1}) & \mapsto & (R(\omega\_{i}))\_{1\le i\le p^r} \end{array}$$ est un isomorphisme. On en déduit que le revêtement est trivial sur Vʹ(B), comme à la proposition [nonramifieberko]. Supposons que l’élément α de $\Os(B)$ satisfait la condition (Ip, K). L’énoncé du lemme [lemintlocalberko] vaut alors encore. Pour le démontrer, l’on remplace simplement les arguments de la théorie de Kummer par ceux de la théorie d’Artin-Schreier-Witt. L’argument-clé consiste à utiliser le fait que le polynôme Sp − α est irréductible sur $\Os\_{x}$, où x est un point de B satisfaisant les propriétés énoncées dans la définition de la condition (Ip, K), et à en déduire que le polynôme Sp − Pp − 1S − α est irréductible sur $\Os\_{z}$, où z désigne l’unique point de la fibre π− 1(x) en lequel P s’annule. Raccord et retour à l’algèbre ----------------------------- Soit G un groupe fini. Soient g1, …, gt, avec $t\in\N^\$, des générateurs du groupe G. Nous pouvons les choisir de façon que, pour tout élément i* de $\cn{1}{t}$, il existe un nombre premier pi et un entier ri ≥ 1 tels que le sous-groupe de G engendré par gi soit cyclique d’ordre piri. Nous supposerons qu’il existe $s\in\cn{1}{t}$ tel que, pour tout i ≤ s, pi ≠ p et, pour tout i ≥ s + 1, pi = p. Nous nous plaçons sous les hypothèses suivantes : * la partie B de X satisfait la condition (CGI) ; * pour tout élément i de $\cn{1}{s}$, il existe une extension séparable Ki de k contenant une racine primitive $(p\_{i}^{r\_{i}})\eme$ de l’unité et un élément αi de $\Os(B)$ qui satisfait la condition Ipiri, Ki ; * pour tout élément i de $\cn{s+1}{t}$, il existe une extension séparable Ki de k et un élément αi de $\Os(B)$ qui satisfait la condition Ip, Ki ; * les corps K1, …, Kt sont deux à deux non isomorphes. Supposons que les élément α1, …, αt sont nuls en tout point de B. Soit i un élément de $\cn{1}{t}$. Construisons par la méthode du numéro [Kummer] ou [Artin-Schreier-Witt] un revêtement galoisien de groupe $\Z/p\_{i}^{r\_{i}}\Z$. Il est défini au-dessus d’une partie Vi et trivial au-dessus d’une partie Vʹi. Notons $\textrm{Ind}\_{\la g\_{i} \ra}^G(V\_{i})$ le G-revêtement induit. Puisque les corps K1, …, Kt sont deux à deux non isomorphes, nous pouvons choisir les parties V1, …, Vt de façon qu’elles soient deux à deux disjointes (il suffit de choisir des éléments v assez petits). Pour $i\in\cn{1}{t}$, posons Vʺi = Vi \ Vʹi. C’est une partie fermée de Y(B). Posons Yʹ(B) = Y(B) \ ⋃1 ≤ i ≤ tVʺi et considérons le G-revêtement $\textrm{Ind}\_{\la e \ra}^G(Y'(B))$ induit par le revêtement trivial au-dessus de Yʹ(B). Raccordons, à présent, les différents revêtements selon les relations entre les éléments du groupe G, par la méthode décrite au numéro [sectioncomplexe]. Nous obtenons un revêtement de Y(B), galoisien de groupe G. On montre à l’aide du lemme [lemintlocalberko] et de son analogue dans le cas des revêtements d’Artin-Schreier-Witt, qu’il est intègre. Un théorème du type GAGA (cf. corollaire [GAGAvoiscompact]) assure qu’il est algébrique. En passant aux corps de fonctions, nous obtenons donc finalement une extension finie et galoisienne de groupe G $$\Ms(Y(B)) = \Frac(\Os(B))(T) \to L,$$ où $\Ms$ désigne le faisceau des fonctions méromorphes. La construction que nous avons menée étant purement géométrique, on se convainc aisément que cette extension est régulière, c’est-à-dire que le corps $\Frac(\Os(B))$ est algébriquement fermé dans le corps L. [galoisberko] Il existe une extension finie du corps $\Frac(\Os(B))(T)$ qui est régulière et galoisienne de groupe de Galois G. Conclusion ---------- Regroupons les résultats que nous avons obtenus jusqu’ici. [galoisinverse] Soient k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet. Soient X un espace k-analytique et B une partie compacte de X qui possède un système fondamental de voisinages affinoïdes géométriquement intègres. Supposons que pour tout nombre premier q différent de la caractéristique du corps k et tout entier $r\in\N^\$, il existe une famille infinie $\Ks\_{q^r}$ de corps deux à deux non isomorphes satisfaisant les propriétés suivantes : 1. tout élément de $\Ks\_{q^r}$ est une extension finie et séparable de k* contenant une racine primitive $(q^r)\eme$ de l’unité ; 2. pour tout élément K de $\Ks\_{q^r}$, il existe un élément x de B et un élément α de $\Os(B)$ nul en tout point de B tels que les corps K et $\Frac(\Os\_{x})$ soient linéairement disjoints et le polynôme Sqr − α soit irréductible sur leur compositum. Si la caractéristique du corps k est un nombre premier p, supposons en outre qu’il existe une famille infinie $\Ks\_{p}$ de corps deux à deux non isomorphes satisfaisant les propriétés suivantes : 1. tout élément de $\Ks\_{p}$ est une extension finie et séparable de k ; 2. pour tout élément K de $\Ks\_{p}$, il existe un élément x de B et un élément α de $\Os(B)$ nul en tout point de B tels que les corps K et $\Frac(\Os\_{x})$ soient linéairement disjoints et le polynôme Sp − α soit irréductible sur leur compositum. Alors, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(\Os(B))(T)$. Ce théorème ne contient malheureusement aucun résultat qui ne soit déjà connu. En effet, le corps $\Frac(\Os(B))$ contient toujours un corps complet pour une valeur absolue non triviale (un corps de séries de Laurent engendré par l’un des éléments α lorsque k est trivialement valué) et, sur un tel corps, le résultat est dû à D. Harbater (cf. , corollary 2.4). Nous allons, à présent, appliquer ce résultat général dans des cas particuliers. [corcomplet] Soit k un corps muni d’une valeur absolue ultramétrique non triviale pour laquelle il est complet. Alors, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps k(T). Appliquons le théorème [galoisinverse] en choisissant pour espace X la droite analytique $\E{1}{k}$, dont nous noterons U la variable, et pour partie B le point 0. Soit K une extension finie du corps k. Choisissons pour point x le point 0 : l’anneau local $\Os\_{0}$ est l’anneau des séries en une variable à coefficients dans k de rayon de convergence strictement positif. Par conséquent, les corps K et $\Frac(\Os\_{0})$ sont linéairement disjoints sur k. Choisissons pour fonction α la fonction U : pour tout entier n ≥ 1, le polynôme Sn − U est irréductible sur le corps $\Frac(\Os\_{0})\otimes\_{k} K$, qui est un sous-corps de K( (U) ), par le théorème d’Eisenstein. Les hypothèses du théorème [galoisinverse] sont donc satisfaites. Soit G un groupe fini. Il existe une extension finie et régulière du corps $\Frac(\Os(B))(T)=\Frac(\Os\_{0})(T)$ qui est galoisienne de groupe G. Puisque l’anneau local $\Os\_{0}$ est composé de séries convergentes et que le corps k n’est pas trivialement valué, toute variété qui possède un point sur $\Frac(\Os\_{0})$ en possède un sur k. En utilisant le théorème de Bertini-Noether (cf. , proposition 10.4.2), on montre alors que l’on peut spécialiser l’extension précédente en une extension finie et régulière du corps k(T) qui est galoisienne de groupe G. Rappelons qu’un corps k est dit fertile[2](#fn2) si tout k-schéma de type fini qui possède un point sur k( (U) ) en possède un sur k (l’on peut démontrer que cela équivaut à demander que toute k-courbe lisse qui possède un point sur k en possède une infinité). F. Pop a démontré que, si k est un corps fertile, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps k(T) (cf. , main theorem A). [corfertile] Soit k un corps. Alors, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps k( (U) )(T). En particulier, si le corps k est fertile ou contient un corps fertile, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps k(T). Munissons le corps k de la valuation triviale et considérons la même situation que dans la preuve précédente. Nous avons alors $\Os(B)=\Os\_{0}=k[\![U]\!]$ et le théorème [galoisinverse] fournit le résultat annoncé. Lorsque le corps k est fertile, le théorème de Bertini-Noether permet de spécialiser l’extension précédente en une extension de k(T) possédant les mêmes propriétés. La régularité de l’extension de k(T) permet d’obtenir, par produit tensoriel, pour tout corps L contenant k, une extension de L(T) possédant encore les propriétés requises. Le résultat du second corollaire contient le résultat du premier, puisque tout corps complet pour une valeur absolue ultramétrique non triviale est fertile. Cependant, la preuve de ce dernier énoncé étant assez difficile (on peut, par exemple, le démontrer en utilisant l’approximation d’Artin), nous avons choisi de proposer une preuve directe du corollaire [corcomplet]. Cas de la valuation triviale ---------------------------- Lorsque le corps k est trivialement valué, nous pouvons obtenir des résultats plus généraux. Nous indiquons simplement ici les modifications à apporter au raisonnement qui précède. Supposons que le corps k est muni de la valeur absolue triviale. Pour tout $n\in\N$, nous pouvons alors définir une application, appelée flot (cf. , 1.3), de $\E{n}{k}\times\R\_{+}^\$ dans $\E{n}{k}$ de la façon suivante. Soient x* un point de $\E{n}{k}$ – il est associé à une semi-norme multiplicative ∣.∣x sur k[T1, …, Tn] qui induit la valeur absolue triviale sur k – et $\eps$ un nombre réel strictement positif. L’image du couple $(x,\eps)$ est le point de $\E{n}{k}$ associée à la semi-norme multiplicative $\|.\|\_{x}^\eps$. Par restriction à la source et au but, nous pouvons encore définir le flot sur tout fermé de Zariski d’un espace affine analytique. Signalons qu’une fonction définie au voisinage d’un point se prolonge, et ce de façon unique, à un voisinage de sa trajectoire sous le flot (ibid., proposition 1.3.10). Nous considérerons désormais un espace analytique X qui est un fermé de Zariski d’un espace affine analytique et une partie ouverte B de X. Nous définissons comme précédemment Y, π et λ. Expliquons comment adapter les constructions locales du numéro [constructionlocaleberko]. Comme alors, choisissons une extension finie et séparable K de k. Soit P ∈ k[T] le polynôme minimal unitaire d’un élément primitif de cette extension et notons t le point de $\P^{1,\an}\_{k}$ défini par l’annulation de ce polynôme. Reprenons, à présent, le raisonnement du numéro [Kummer]. À cet effet, choisissons un entier n supérieur ou égal à 2 et premier à p et supposons que le corps K contient une racine primitive $n\eme$ de l’unité. Par hensélianité, elle se relève, dans l’anneau local au point t, en une racine primitive $n\eme$ de l’unité, que nous noterons ζ. Les propriétés du flot assurent qu’elle est définie sur l’ouvert $$\left\{\left.y\in\P^{1,\an}\_{k}\,\right\|\, \|P(y)\|<1\right\}.$$ Soit α un élément de $\Os(B)$. Insistons sur le fait que nous ne supposons plus qu’il soit nul en tout point de B. Nous définissons alors un faisceau $\Fs$, comme précédemment, au-dessus de l’ouvert $$V\_{t}(B) = \left\{y\in Y(B)\, \big\|\, \|P(y)\|<1\right\}.$$ Puisque le corps k est trivialement valué, le rayon de convergence de la série (1 + T)1/n est égal à 1 et le revêtement associé à $\Fs$ est trivial au-dessus de l’ouvert $$V\_{t}'(B) = \left\{y\in V(B)\, \big\|\, \|\alpha(y)\| < \|P(y)\|^n\right\}.$$ Supposons, en outre, que l’élément α est de valeur absolue strictement inférieure à 1 en tout point de B. Alors, le complémentaire de la partie Vtʹ(B) dans Vt(B) est fermé dans Y(B). Pour assurer l’irréductibilité du revêtement associé au faisceau $\Fs$, nous remplaçons la condition (CGI) par la condition suivante : l’ouvert B est limite inductive d’espaces affinoïdes géométriquement intègres. Le résultat du lemme [lemintlocalberko] vaut alors encore. Passons aux résultats du numéro [Artin-Schreier-Witt]. Supposons donc que p est un nombre premier et considérons un entier n de la forme pr, avec $r\in\N^\$. Soit α* un élément de $\Os(B)$ et posons, de nouveau, $$V\_{t}(B) = \left\{y\in Y(B)\, \big\|\, \|P(y)\|<1\right\}.$$ Les propriétés du flot permettent de préciser le domaine de définition des racines du polynôme P0(S0). Notons B0 le lieu d’annulation de α dans B. Supposons qu’il ne soit pas vide et que pour tout point b de B vérifiant ∣α(b)∣ < 1 et tout voisinage B+ de B0, le flot joigne le point b à un point de B+ (c’est en particulier le cas dès que la partie B est stable par le flot, que l’élément α est de valeur absolue strictement inférieure à 1 en tout point de B et s’annule sur B). Posons $$V'\_{t}(B) = \left\{y\in V(B)\, \big\|\, \|\alpha(y)\|<1 \textrm{ et } \|P(y)\| < 1\right\}.$$ Les propriétés du flot assurent que le revêtement associé à $\Fs$ est encore trivial sur une partie Vʹt(B) de Vt(B) dont le complémentaire dans Vt(B) est fermé. Si l’élément α est de valeur absolue strictement inférieure à 1 en tout point de B, nous pouvons même choisir la partie Vʹt(B) de façon que son complémentaire dans Vt(B) soit fermé dans Y(B). Dans la preuve du théorème [galoisberko], il faut finalement remplacer le résultat de type GAGA du corollaire [GAGAvoiscompact] par celui du corollaire [GAGAsurouvert]. Nous obtenons finalement le résultat suivant : [galoisinversetriv] Soient k un corps. Munissons-le de la valeur absolue triviale. Soient X un espace de Zariski d’un espace affine k-analytique et B une partie ouverte de X stable par le flot qui soit limite inductive d’espaces affinoïdes géométriquement intègres. Supposons que pour tout nombre premier q différent de la caractéristique du corps k et tout entier $r\in\N^\$, il existe une famille infinie $\Ks\_{q^r}$ de corps deux à deux non isomorphes satisfaisant les propriétés suivantes : 1. tout élément de $\Ks\_{q^r}$ est une extension finie et séparable de k* contenant une racine primitive $(q^r)\eme$ de l’unité ; 2. pour tout élément K de $\Ks\_{q^r}$, il existe un élément x de B et un élément α de $\Os(B)$ de valeur absolue strictement inférieure à 1 en tout point de B et qui s’annule sur B tels que les corps K et $\Frac(\Os\_{x})$ soient linéairement disjoints et le polynôme Sqr − α soit irréductible sur leur compositum. Si la caractéristique du corps k est un nombre premier p, supposons en outre qu’il existe une famille infinie $\Ks\_{p}$ de corps deux à deux non isomorphes satisfaisant les propriétés suivantes : 1. tout élément de $\Ks\_{p}$ est une extension finie et séparable de k ; 2. pour tout élément K de $\Ks\_{p}$, il existe un élément x de B et un élément α de $\Os(B)$ de valeur absolue strictement inférieure à 1 en tout point de B et qui s’annule sur B tels que les corps K et $\Frac(\Os\_{x})$ soient linéairement disjoints et le polynôme Sp − α soit irréductible sur leur compositum. Alors, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(\Os(B))(T)$. De nouveau, le résultat de ce théorème est connu, puisque le corps $\Frac(\Os(B))$ contient un corps de séries de Laurent sur k (engendré par l’un des éléments α). [corvaltriv] Soit k un corps. Notons k+ ∞, 1−[ [U1, U2] ] le sous-anneau de k[U1][ [U2] ] composé des séries de la forme ∑n ≥ 0an(U1)U2n qui vérifient la condition $$\forall r>0,\forall s\in\of{[}{0,1}{[},\ \lim\_{n\to +\infty} (r^{\deg(a\_{n})}\, s^n) = 0.$$ Alors, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(k\_{+\infty,1^-}[\![U\_{1},U\_{2}]\!])(T)$. Appliquons le théorème [galoisinverse] en choisissant pour espace X l’espace analytique de dimension deux $\E{2}{k}$, dont nous noterons U1 et U2 les variables, et pour partie B le disque ouvert relatif de rayon 1 au-dessus de $\E{1}{k}$ : $$B = \left\{\left.x\in\E{2}{k}\,\right\|\, \|U\_{2}(x)\|<1\right\}.$$ Cette partie est stable par le flot et nous avons $\Os(B) = k\_{+\infty,1^-}[\![U\_{1},U\_{2}]\!]$. Soit K une extension finie du corps k. Choisissons pour point x le point de coordonnées (0, 0) : l’anneau local $\Os\_{x}$ est isomorphe à k[ [U1, U2] ]. Par conséquent, les corps K et $\Frac(\Os\_{x})$ sont linéairement disjoints sur k. Choisissons pour fonction α la fonction U2 : pour tout entier n ≥ 1, le polynôme Sn − U2 est irréductible sur le corps $\Frac(\Os\_{x})\otimes\_{k} K \simeq K[\![U\_{1},U\_{2}]\!]$. Les hypothèses du théorème [galoisinverse] sont donc satisfaites. On en déduit le résultat attendu. Ce dernier énoncé peut surprendre, puisqu’il découle directement du corollaire [corfertile]. En effet, le corps $\Frac(k\_{r^-,1^-}[\![U\_{1},U\_{2}]\!])$ contient le corps k( (U2) ). Il présente cependant un intérêt dans le cadre de l’analogie entre corps de fonctions et corps de nombres. La droite analytique sur un corps trivialement valué est proche, à bien des égards, du spectre analytique – espace analytique de dimension 0 – d’un anneau d’entiers de corps de nombres (cf. annexe [sectionberko]). Il semble donc raisonnable d’envisager que le résultat du corollaire précédent reste vrai en remplaçant l’anneau kr−, 1−[ [U1, U2] ] par l’anneau du disque ouvert de rayon 1 au-dessus du spectre d’un anneau d’entiers de corps de nombres. Signalons que les constructions effectuées peuvent effectivement être menées dans ce cadre. Pour conclure, seuls manquent encore les théorèmes du type GAGA sur les espaces de Berkovich au-dessus des anneaux d’entiers de corps de nombres. Soient K un corps de nombres, A l’anneau de ses entiers et Σ∞ l’ensemble des plongements de K dans $\C$. Notons A1−[ [X] ] le sous-anneau de A[ [X] ] composé des séries f telles que, pour tout σ ∈ Σ∞, la série à coefficients complexes σ(f) a un rayon de convergence supérieur ou égal à 1. Alors, tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(A\_{1^-}[\![X]\!])(T)$. Nous ignorons si le corps $\Frac(A\_{1^-}[\![X]\!])$ est fertile, même lorsque l’anneau A est l’anneau $\Z$ des entiers. Problème inverse de Galois sur un disque relatif ================================================ Soit A un anneau d’entiers de corps de nombres. Nous allons maintenant appliquer la stratégie de raccord décrite à la section [sectioncomplexe] dans le cadre des espaces de Berkovich sur A (cf. annexe [sectionberko]). Plus précisément, un groupe fini G étant donné, nous allons construire un revêtement galoisien de groupe G du disque $$\D=\left\{\left. x\in\E{1}{A}\, \right\|\, \|T(x)\|<1\right\}.$$ Cela indique que la troisième étape de notre démonstration différera fondamentalement de la troisième étape de la démonstration du cas complexe. En effet, le disque $\D$ est un espace affine et non plus projectif comme l’était $\P^1(\C)$. En particulier, les théorèmes GAGA n’y sont pas valables. Nous utiliserons, pour les remplacer, le caractère Stein du disque $\D$. Nous noterons $X=\E{1}{A}$ et, pour tout idéal maximal $\m$ de A, $$\D\_{\m} = \D \cap \pi^{-1}(\of{[}{a\_{0},\tilde{a}\_{\m}}{]}) \textrm{ et } \D'\_{\m} = \D \cap \pi^{-1}(\of{]}{a\_{0},\tilde{a}\_{\m}}{]}).$$ Ces deux parties sont connexes. Construction locale de revêtements cycliques -------------------------------------------- Dans le cas complexe, la construction locale était particulièrement simple car nous disposions de racines primitives de l’unité de tout ordre. Elle ne sera guère plus difficile ici puisque, comme nous allons l’expliquer, un entier n ≥ 1 étant donné, il existe toujours une branche de $\Mc(A)$, et même une infinité, dont l’anneau des fonctions contient une racine primitive n\`eme de l’unité. Soient n ≥ 1 un entier, p un nombre premier congru à 1 modulo n et $\m$ un idéal maximal de A contenant p. Notons $\hat{A}\_{\m}$ le complété de l’anneau A pour la topologie $\m$-adique. Soit $\pi\_{\m}$ une uniformisante de l’anneau $\hat{A}\_{\m}$. Posons $$Q(S) = S^n-\pi\_{\m}^n-T \in \Os(\D'\_{\m})[S].$$ Définissons un préfaisceau $\Fs$ sur $D'\_{\m}$ en posant, pour toute partie ouverte W de $D'\_{\m}$, $$\Fs(W) = \Os(W)[S]/(Q(S))$$ et en utilisant les morphismes de restriction induits par ceux du faisceau $\Os$. Le caractère unitaire du polynôme Q assure que $\Fs$ est un faisceau de $\Os\_{D'\_{\m}}$-algèbres cohérent. Nous considérons ce faisceau comme l’image directe du faisceau d’un revêtement fini de $\D'\_{\m}$. Le résultat classique qui suit explique le choix des entiers n et p. L’anneau $\Z\_{p}$ contient une racine primitive $n\eme$ de l’unité et, pour tout $i\in\N$, nous avons $C\_{1/n}^i \in \Z\_{p}$. Soit $\zeta\in\hat{A}\_{\m}$ une racine primitive n\`eme de l’unité. Posons $$U = \left\{x\in \D'\_{p}\, \big\|\, \|T(x)\|< \|\pi\_{\m}(x)\|^n \right\}.$$ Le résultat suivant affirme que le revêtement de $\D'\_{\m}$ associé au faiseau $\Fs$ est trivial au-dessus de l’ouvert U. [nonramifie] Il existe un isomorphisme de $\Os\_{U}$-algèbres $$\varphi : \Fs \to \Os^n$$ tel que, pour tout ouvert V de U et tout élément s de $\Fs(V)$, nous ayons φ(ζs) = τ(φ(s)), où τ désigne l’automorphisme du faisceau $\Os^n$ qui consiste à faire agir la permutation cyclique (1 2 ⋯ n) sur les coordonnées. En utilisant le lemme précédent, on montre que la fonction $\pi\_{\m}^{-n}\, T$ possède une racine $n\eme$ dans $\Os(U)$. Nous la noterons ω. On en déduit l’égalité $$Q(S) = S^n-\pi\_{\m}^n-T = \prod\_{j=0}^n (S - \pi\_{\m}\, \zeta^{j}\, \omega) \textrm{ dans } \Os(U)[S].$$ Par conséquent, le morphisme $$\begin{array}{ccc} \Fs & \to & \Os^n\\ R(S) & \mapsto & \left(R(\pi\_{\m}\, \omega), R(\pi\_{\m}\, \zeta^{-1}\, \omega),\ldots, R(\pi\_{\m}\, \zeta^{-(n-1)}\, \omega)\right) \end{array}$$ est un isomorphisme. On vérifie immédiatement qu’il satisfait la condition requise. Démontrons, à présent, que le revêtement est irréductible. [lemintlocal] Le polynôme $Q(S)=S^n-\pi\_{\m}^n-T$ est irréductible sur le corps Frac$(\Os(\D'\_{\m}))$. En particulier, l’anneau $\Fs(\D'\_{\m})$ est intègre. Notons $z\_{\m}$ le point 0 de la fibre $\pi^{-1}(\tilde{a}\_{\m})$. D’après la discussion menée à la fin de la section [sectionberko], l’anneau local en ce point est isomorphe à l’anneau $\hat{A}\_{\m}[\![T]\!]$. Commençons par montrer que le polynôme Q(S) est irréductible sur le corps $\Frac(\Os\_{z\_{\m}})$. Pour des raisons de valuation T-adique, l’élément de $\hat{A}\_{\m}[\![T]\!]$ n’est racine d\`eme dans $\hat{A}\_{\m}[\![T]\!]$ pour aucun diviseur de n. D’après la théorie de Kummer, cela impose au polynôme Q(S) d’être irréductible sur $\Frac(\Os\_{z\_{\m}})$. Les mêmes arguments que dans la preuve du lemme [lemintlocalberko] permettent alors de conclure. Nous pouvons même être plus précis et démontrer un principe du prolongement analytique. [integreunebranche] Soient x un point de U et i un élément de $\cn{1}{n}$. Le morphisme $$\rho\_{i,x} : \Fs(\D'\_{\m}) \to \Fs\_{x} \xrightarrow[\sim]{\varphi\_{x}} \Os\_{x}^n \xrightarrow[]{p\_{i}} \Os\_{x},$$ où pi est la projection sur le i\`eme facteur, est injectif. Soit s un élément de l’anneau $\Fs(\D'\_{\m})=\Os(\D'\_{\m})[S]/(Q(S))$ dont l’image par le morphisme ρi, x est nulle. Choisissons un élément R(S) de $\Os(\D'\_{\m})[S]$ qui représente la section s. Reprenons les notations de la preuve de la proposition [nonramifie]. Par hypothèse, nous avons $$R(\pi\_{\m}\, \zeta^{-i}\, \omega)=0 \textrm{ dans } \Os\_{x}.$$ Pour montrer que l’élément s est nul, il suffit de montrer que le polynôme Q(S) est le polynôme minimal de l’élément $\pi\_{\m}\, \zeta^{-i}\, \omega$ sur le corps $\Frac(\Os(\D'\_{\m}))$. C’est bien le cas, puisque le lemme précédent assure que le polynôme Q est irréductible sur le corps $\Frac(\Os(\D'\_{\m}))$. Terminons par un résultat topologique. [Festferme] La partie $$F = \D'\_{\m} \setminus U = \left\{x\in \D'\_{\m}\, \big\|\, \|T(x)\|\ge \|\pi\_{\m}(x)\|^n \right\}$$ est fermée dans le disque $\D$. Il suffit de montrer que F est fermée dans $\D\_{\m}$ puisque cette dernière partie est elle-même fermée dans $\D$. En d’autres termes, nous souhaitons montrer que la partie $$V = U \cup (\D\cap \pi^{-1}(a\_{0}))$$ est ouverte dans $\D\_{\m}$. Puisque U est une partie ouverte de $\D\_{\m}$, il suffit de montrer que V est voisinage dans $\D\_{\m}$ de chacun des points de $\D\cap \pi^{-1}(a\_{0})$. Soit x un point de $\D\cap \pi^{-1}(a\_{0})$. Posons r = ∣T(x)∣. C’est un élément de l’intervalle $\of{]}{0,1}{[}$. Soient s un élément de $\of{]}{r,1}{[}$ et $\eps$ un élément de $\of{]}{0,1}{[}$ tels que l’on ait $\|\pi\_{\m}\|\_{\m}^{n\eps} > s$. La partie $$\left\{y\in \pi^{-1}({[}{a\_{0},a\_{\m}^\eps}{[}) \, \big\|\, \|T(y)\|<s \right\}$$ est un voisinage ouvert du point x dans $\D\_{\m}$ qui est contenu dans V. Raccord et retour à l’algèbre ----------------------------- Soit G un groupe fini. Notons $n\in\N^\$ son ordre. Soient g1, …, g*t, avec $t\in\N^\$, des générateurs du groupe G. Pour tout élément i* de $\cn{1}{t}$, notons ni l’ordre de l’élément gi, choisissons un nombre premier pi congru à 1 modulo ni et un idéal maximal $\m\_{i}$ de A contenant pi. Nous pouvons supposer que les $\m\_{i}$ sont distincts. Soit i un élément de $\cn{1}{t}$. Construisons par la méthode du numéro [constructionlocale] un revêtement galoisien de groupe $\Z/n\_{i}\Z$. Il est défini au-dessus de $\D'\_{\m\_{i}}$ et trivial au-dessus de $$U\_{i} = \left\{x\in \D'\_{\m\_{i}}\, \big\|\, \|T(x)\| < \|\pi\_{\m\_{i}}(x)\|^{n\_{i}} \right\}.$$ Notons $\textrm{Ind}\_{\la g\_{i} \ra}^G(\D'\_{\m\_{i}})$ le G-revêtement induit (cf. section [sectioncomplexe]). D’après le lemme [Festferme], pour tout élément i de $\cn{1}{t}$, la partie $F\_{i}=\D'\_{\m\_{i}}\setminus U\_{i}$ est fermée dans $\D$. Définissons une partie ouverte de $\D$ par $$U\_{0} = \D\setminus \bigcup\_{1\le i\le t} F\_{i}.$$ On se convainc aisément qu’elle est connexe. Considérons le G-revêtement $\textrm{Ind}\_{\la e \ra}^G(U\_{0})$ induit par le revêtement trivial au-dessus de U0. Recollons ces différents revêtements par la méthode décrite au numéro [sectioncomplexe]. Nous obtenons un revêtement de $\D$, galoisien de groupe G, associé à un faisceau $\Gs$. On montre à l’aide du lemme [integreunebranche] qu’il est intègre, c’est-à-dire que l’anneau $\Gs(\D)$ est intègre. Nous disposons, à présent, d’un revêtement du disque $\D$ possédant le groupe de Galois désiré G. Il nous reste à montrer que l’extension induite entre les corps de fonctions est galoisienne de même groupe. Nous utiliserons, pour ce faire, le caractère Stein du disque $\D$ (cf. théorème [lemniscateStein]). [GinjAutGsD] Le groupe des automorphismes de $\Os(\D)$-algèbres du faisceau $\Gs(\D)$ est isomorphe à G. Soient $\As$ et $\Bs$ deux faisceaux de $\Os\_{\D}$-algèbres cohérents. Considérons l’application $$\textrm{Mor}\_{\Os}(\As,\Bs) \to \textrm{Mor}\_{\Os(\D)}(\As(\D),\Bs(\D)).$$ Elle est bijective car les faisceaux $\As$ et $\Bs$ satisfont le théorème A sur $\D$. Par construction, le groupe des automorphismes de $\Os\_{\D}$-algèbres du faisceau $\Gs$ est isomorphe à G. On en déduit le résultat attendu. Il reste à montrer que l’extension $\Os(\D) \to \Gs(\D)$ est entière. Puisque les théorèmes du type GAGA ne sont pas valables dans ce cadre, nous utiliserons un raisonnement direct. [racinedegreinfn] Tout élément de $\Gs(\D)$ annule un polynôme unitaire à coefficients dans $\Os(\D)$ de degré inférieur à n. Soit s un élément de $\Gs(\D)$. Nous supposerons, tout d’abord, qu’il existe un point x0 de U0 tel que toutes les coordonnées de son image sx0 dans $\Gs\_{x\_{0}}=\Os\_{X,x\_{0}}^n$ soient distinctes. Puisque l’ouvert U0 est connexe, le principe du prolongement analytique (cf. théorème [prolan]) assure qu’en tout point x de U0, toutes les coordonnées du germe sx sont distinctes. Notons a1, …, an les coordonnées de l’image de s dans $\Gs(U\_{0})=\Os(U\_{0})^n$. Posons $$M(Z) = \prod\_{l=1}^n (Z-a\_{l}) \in \Os(U\_{0})[Z].$$ En tout point x de U0, l’image du polynôme M est l’unique polynôme unitaire de degré inférieur à n à coefficients dans $\Os\_{x}$ qui annule le germe sx. Pour tout élément j de $\cn{0}{t}$, posons $V\_{j} = U\_{0} \cup \bigcup\_{1\le i\le j} \D'\_{\m\_{i}}$. Montrons, par récurrence, que pour tout élément j de $\cn{0}{t}$, il existe un polynôme unitaire Nj de degré n à coefficients dans $\Os(V\_{j})$ qui annule l’élément s∣Vj de $\Gs(V\_{j})$. Nous avons déjà traité le cas j = 0. Soit maintenant un élément j de $\cn{0}{t-1}$ pour lequel l’hypothèse de récurrence est vérifiée. L’élément $s\_{\|\D'\_{\m\_{j+1}}}$ de l’anneau $\Gs(\D'\_{\m\_{j+1}})=\Os(\D'\_{\m\_{j+1}})[S]/(S^{n\_{j+1}}-p\_{j+1}^{n\_{j+1}}-T)$ est annulé par un polynôme unitaire Mj + 1 de degré inférieur à n à coefficients dans le corps $\Os(\D'\_{\m\_{j+1}})$. Soit x un élément de $U\_{j+1} = \D'\_{\m\_{j+1}}\cap U\_{0}$. Nous avons démontré qu’il existe un unique polynôme unitaire de degré inférieur à n à coefficients dans $\Os\_{x}$ qui annule le germe sx. On en déduit que les images des polynômes Nj et Mj + 1 dans $\Os\_{x}[Z]$ coïncident. L’ouvert Uj + 1 étant connexe, d’après le théorème [prolan], les images de ces polynômes dans $\Os(U\_{j+1})[Z]$ coïncident. On en déduit que le polynôme Nj se prolonge en un polynôme unitaire Nj + 1 de degré inférieur à n à coefficients dans $\Os(V\_{j+1})$ qui annule l’élément s∣Vj + 1 de $\Gs(V\_{j+1})$. On déduit finalement le résultat attendu du cas j = t. Soit x0 un point de l’ouvert U0. La fibre du faisceau $\Gs$ au point x0 est isomorphe à l’algèbre $\Os\_{x\_{0}}^n$. D’après le théorème [lemniscateStein], le faisceau $\Gs$ vérifie le théorème A sur le disque $\D$. On en déduit qu’il existe un élément s0 de $\Gs(\D)$ dont toutes les coordonnées de l’image dans la fibre $\Gs\_{x\_{0}}=\Os\_{x\_{0}}^n$ sont distinctes. Soit s un élément de $\Gs(\D)$. Il existe un élément λ de $\Os(\D)$ tel que toutes les coordonnées du germe de la section s1 = s + λs0 au point x0 soient distinctes. Le raisonnement qui précède montre qu’il existe deux polynômes unitaires P0 et P1 de degré inférieur à n à coefficients dans $\Os(\D)$ qui annulent respectivement les sections s0 et s1. On en déduit qu’il existe un polynôme unitaire P de degré inférieur à n à coefficients dans $\Os(\D)$ qui annule la section s. [AalgfermeGsD] L’anneau A est algébriquement fermé dans l’anneau $\Gs(\D)$. Soit P un polynôme unitaire à coefficients dans A sans racines dans A. Supposons, par l’absurde, qu’il existe une section s de $\Gs(\D)$ qui est racine du polynôme P. Notons z0 le point 0 de la fibre π− 1(a0) de l’espace X. C’est un point de l’ouvert U0. Notons a la première coordonnée de l’image du germe sz0 par l’isomorphisme $\Gs\_{z\_{0}} \xrightarrow[]{\sim} \Os\_{z\_{0}}^t$. C’est un élément de $\Os\_{z\_{0}}$ qui vérifie l’égalité P(a) = 0. D’après la discussion menée à la fin de la section [sectionberko], l’anneau local $\Os\_{z\_{0}}$ se plonge dans l’anneau K[ [T] ]. On en déduit que le polynôme P possède une racine dans l’anneau K[ [T] ] et donc dans le corps K. Puisque l’anneau A est algébriquement fermé dans le corps K, cette racine doit appartenir à A. Nous avons abouti à une contradiction. On en déduit le résultat annoncé. Introduisons une définition correspondant à cette propriété. Une extension L du corps $\Frac(\Os(\D))$ est dite régulière si le corps K est algébriquement fermé dans L. Regroupons, à présent, les résultats obtenus. L’extension de corps $$\Frac(\Os(\D)) \to \Frac(\Gs(\D))$$ est finie de degré n, régulière et galoisienne de groupe G. L’extension $\Frac(\Os(\D)) \to \Frac(\Gs(\D))$ est finie et de degré inférieur à n d’après le lemme [racinedegreinfn]. Elle est régulière d’après le lemme [AalgfermeGsD]. On déduit de la proposition [GinjAutGsD] qu’il existe un morphisme injectif du groupe G dans le groupe des $\Frac(\Os(\D))$-automorphismes du corps Frac($\Gs(\D)$). Or le groupe G a pour cardinal n. On en déduit que l’extension $\Frac(\Os(\D)) \to \Frac(\Gs(\D))$ est exactement de degré n, qu’elle est galoisienne et que son groupe de Galois est isomorphe au groupe G. Puisque le disque $\D$ est connexe, les théorèmes [prolan] et [lemniscateStein] assurent que l’anneau des sections méromorphes globales $\Ms(\D)$ est un corps isomorphe à $\Frac(\Os(\D))$. L’extension $\Frac(\Os(\D)) \to \Frac(\Gs(\D))$ est donc bien l’extension obtenue à partir du revêtement du disque $\D$ associé au faisceau $\Gs$ en passant aux corps de fonctions. Conclusion et généralisations ----------------------------- Regroupons, à présent, les résultats que nous avons obtenus. Puisque nous sommes partis d’un groupe fini G arbitraire, nous avons finalement démontré que tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(\Os(\D))$. D’après la description de l’anneau $\Os(\D)$ donnée à la fin de la section [sectionberko], ce dernier est isomorphe au corps $\Frac(\Aun)$, où $\Aun$ désigne l’anneau des séries en une variable à coefficients dans A de rayon de convergence complexe supérieur ou égal à 1 en toute place infinie. Lorsque $A=\Z$, nous retrouvons bien ainsi le résultat de D. Harbater (cf. , corollary 3.8) énoncé en introduction. [thZun] Soit A un anneau d’entiers de corps de nombres. Tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(\Aun)$. Pour tout r > 1, l’anneau Ar−[ [T] ] des séries en une variable à coefficients dans A de rayon de convergence complexe supérieur ou égal à r en toute place infinie (une seule suffirait) est réduit à l’anneau de polynômes A[T]. Si nous disposions du théorème précédent pour un certain nombre réel r > 1, nous aurions donc résolu le problème inverse de Galois géométrique sur K. Pour finir, nous regroupons plusieurs résultats proches de celui du théorème [thZun]. Les démonstrations en sont fort similaires et nous n’indiquerons que les modifications à effectuer. Ainsi que nous l’avons déjà signalé, le spectres analytique d’un anneau d’entiers de corps de nombres présente de nombreuses similitudes avec la droite analytique sur un corps trivialement valué. C’est donc, tout d’abord, dans ce cadre que nous allons nous placer. Soit k un corps. Munissons-le de la valeur absolue triviale afin d’en faire un corps ultramétrique complet. Considérons, maintenant, l’espace $\E{2}{k}$, analogue de $\E{1}{A}$. Nous noterons U et T les coordonnées sur cet espace. Lorsque la caractéristique du corps k est nulle, pour tout entier n ≥ 1, il existe une infinité de branches de la droite $\E{1}{k}$ dont l’anneau des fonctions contient une racine primitive $n\eme$ de l’unité. Posons $$\D\_{k} = \left\{x\in\E{2}{k}\, \big\|\, \|T(x)\|<1\right\}.$$ En appliquant le raisonnement suivi dans cette section, nous démontrons que tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(\Os(\D\_{k}))$. Par régulière, nous entendons ici que le corps k(U) est algébriquement fermé dans l’extension en question. Supposons, à présent, que le corps k est de caractéristique p, où p est un nombre premier. Dans ce cas, la construction des revêtements cycliques locaux est plus complexe. Cependant, il est possible de la mener à bien en faisant appel aux extensions d’Artin-Schreier-Witt, comme nous l’avons déjà fait au numéro [constructionlocaleberko]. Il est plus difficile de montrer qu’un tel revêtement est trivial sur une partie dont le complémentaire est fermé dans $\D\_{k}$, mais les propriétés du flot nous permettent d’y parvenir. En utilisant une description explicite de l’anneau $\Os(\D\_{k})$, nous obtenons finalement le résultat suivant : [GaloisDk] Soit k un corps. Notons k+ ∞, 1−[ [U, T] ] le sous-anneau de k[U][ [T] ] composé des séries de la forme ∑n ≥ 0an(U)Tn qui vérifient la condition $$\forall r>0,\forall s\in\of{[}{0,1}{[},\ \lim\_{n\to +\infty} (r^{\deg(a\_{n})}\, s^n) = 0.$$ Tout groupe fini est groupe de Galois d’une extension finie et régulière du corps $\Frac(k\_{+\infty,1^-}[\![U,T]\!])$. Le théorème précédent nous permet, en particulier, de réaliser, pour tout corps k, tout groupe fini comme groupe de Galois sur le corps des fractions de k[U][ [T] ]. Nous étendons ainsi des résultats de D. Harbater (cf. , corollary 1.4 et corollary 1.5). Théorèmes GAGA relatifs sur un affinoïde ======================================== Soit k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet, $\As$ une algèbre k-affinoïde et X un schéma localement de type fini sur $\As$. Dans , 2.6, V. Berkovich a défini, de manière fonctorielle, l’analytifié $X^\an$ du schéma X. Il vient avec un morphisme d’espaces localement annelés $X^\an \to X$, qui est plat et surjectif (cette dernière propriété tombe évidemment en défaut dans le cas complexe). À tout faisceau de $\Os\_{X}$-modules $\Fs$, nous pouvons associer, par rétrotirette, un faisceau de $\Os\_{X^\an}$-modules, que nous noterons $\Fs^\an$. Remarquons que l’analytifié d’un faisceau cohérent est encore cohérent. Dans la lignée des théorèmes GAGA de J.-P. Serre (cf. et , exposé XII), nous allons nous intéresser aux propriétés du foncteur d’analytification lorsque l’espace X est propre. Précisément, nous allons démontrer le théorème suivant : [GAGAsuraffinoide] Soit k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet, $\As$ une algèbre k-affinoïde et X un $\As$-schéma propre. Alors 1. pour tout faisceau de $\Os\_{X}$-modules cohérent $\Fs$ et tout entier $q\in\N$, le morphisme $$H^q(X,\Fs) \to H^q(X^\an,\Fs^\an)$$ est un isomorphisme ; 2. le foncteur d’analytification $$\Fs \to \Fs^\an$$ induit une équivalence entre la catégorie des $\Os\_{X}$-modules cohérents et celle des $\Os\_{X^\an}$-modules cohérents. La preuve originale de J.-P. Serre, qui concerne l’analytification complexe, peut être adaptée à notre contexte sans difficultés majeures. Signalons que le théorème précédent a d’ailleurs déjà été obtenu par U. Köpf dans le cadre de la géométrie rigide (cf. ) et par A. Ducros en général, dans un texte inédit. Nous en rédigeons cependant une démonstration pour la commodité du lecteur, sans prétendre aucunement à l’originalité. Comme dans le cas complexe, on se ramène à démontrer le théorème pour un espace X de la forme $\P^r\_{\As}$ et on utilise les résultats classiques concernant les faisceaux cohérents sur un tel espace. Deux propriétés joueront un rôle essentiel dans la preuve : la platitude du morphisme $X^\an \to X$ et la finitude cohomologique des morphismes propres. Ce dernier point prend la forme du théorème suivant : [kiehlfini] Soit k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet, $\As$ une algèbre k-affinoïde et X un $\As$-espace analytique propre. Pour tout faisceau de $\Os\_{X}$-modules cohérent et tout entier naturel q, le $\As$-module $H^q(X,\Fs)$ est un $\As$-module de Banach de type fini. Ce résultat a été démontré par R. Kiehl dans le cas d’un corps de valuation non triviale et d’objets strictement affinoïdes (cf. , Theorem 3.3). Il a été étendu au cas général par V. Berkovich (cf. , proposition 3.3.5). Indiquons pour finir le seul véritable ajout que nous avons dû faire à la preuve de (et qui figure chez A. Ducros) : il s’agit du lemme [chgtbasekL], un résultat de changement de base, utilisé pour pallier le fait qu’un point de l’espace k-analytique $\P^{r,\an}\_{k}$ n’est pas toujours situé sur un hyperplan. Démonstration ------------- Commençons par quelques réductions classiques. En utilisant le lemme de Chow (cf., théorème 5.6.1), on montre qu’il suffit de prouver le théorème dans le cas où X est un schéma projectif sur $\As$. Dans le cas complexe, les détails de l’argument figurent dans l’exposé XII de ; un raisonnement en tout point semblable vaut ici. Si X est un schéma projectif sur $\As$, il existe une immersion fermée $\varphi : X \to \P^r\_{\As}$, pour un certain entier naturel r. Pour tout faisceau de $\Os\_{X}$-modules cohérent $\Fs$, le faisceau de $\Os\_{\P^r\_{\As}}$-modules $\varphi\_{\}(\Fs)$, qui n’est autre que le prolongement de $\Fs$ par zéro, est encore cohérent. On vérifie que cette opération de prolongement commute à l’analytification et préserve la cohomologie. En outre, elle possède un inverse à gauche : la restriction à Y. En utilisant ces propriétés, on montre qu’il suffit de prouver le théorème dans le cas où X* est un espace projectif sur $\As$. C’est ce que nous supposerons désormais. ### Assertion i) lorsque $\Fs=\Os(n)$ Nous allons démontrer, par récurrence sur r, que, pour tout entier naturel r et tout entier relatif n, l’assertion i) du théorème est vraie lorsque $X = \P^r\_{\As}$ et $\Fs = \Os\_{X}(n)$. Pour r = 0, le résultat découle du théorème d’acyclicité de Tate. Soit $r\in\N$ tel que le résultat soit vrai pour $\P^r\_{\As}$. Posons $X=\P^{r+1}\_{\As}$. Soit t une section non nulle du fibré $\Os\_{X}(1)$ et Y l’hyperplan de X (isomorphe à $\P^r\_{\As}$) qu’elle définit. Nous noterons $\Os\_{Y}$ à la fois le faisceau structural sur Y et son prolongement par zéro à X. D’après l’hypothèse de récurrence, pour tout entier $q\in\N$, le morphisme $$H^q(X,\Os\_{Y}(n)) = H^q(Y,\Os\_{Y}(n)) \to H^q(Y^\an,\Os\_{Y^\an}(n)) = H^q(X^\an,\Os\_{Y^\an}(n))$$ est un isomorphisme. Pour tout $n\in\Z$, la multiplication par t définit une suite exacte courte $$0 \to \Os\_{X}(n-1) \to \Os\_{X}(n) \to \Os\_{Y}(n) \to 0.$$ En écrivant la suite exacte longue associée et en utilisant le lemme des cinq, on montre que l’on a un isomorphisme $H^q(X,\Os\_{X}(n)) \simeq H^q(X^\an,\Os\_{X^\an}(n))$ pour tout $q\in\N$ si, et seulement si, on a un isomorphisme $H^q(X,\Os\_{X}(n-1)) \simeq H^q(X^\an,\Os\_{X^\an}(n-1))$ pour tout $q\in\N$. Un calcul explicite montre que l’on a $H^q(X,\Os\_{X}) \simeq H^q(X^\an,\Os\_{X^\an})$ pour tout $q\in\N$. On en déduit le résultat annoncé. ### Assertion i) en général Soit $r\in\N$. Posons $X=\P^r\_{\As}$. Nous allons démontrer, par une récurrence descendante sur q, que, pour tout entier naturel q, l’assertion i) du théorème est vraie pour Hq. Si q > r, pour tout faisceau de $\Os\_{X}$-modules cohérent $\Fs$, les groupes $H^q(X,\Fs)$ et $H^q(X^\an,\Fs^\an)$ sont tous deux nuls, et le résultat est vrai. Soit $q\in\N^\$ tel que le résultat soit vrai pour H*q. Soit $\Fs$ un faisceau de $\Os\_{X}$-modules cohérent. Nous pouvons l’insérer dans une suite exacte de faisceaux de $\Os\_{X}$-modules cohérents $$0 \to \Rs \to \Ls \to \Fs \to 0,$$ où $\Ls$ est somme directe de faisceaux isomorphes à $\Os(n)$, avec $n\in\Z$. Insérons le faisceau $\Rs$ dans une suite exacte courte du même type $$0 \to \Rs' \to \Ls' \to \Rs \to 0.$$ D’après l’hypothèse de récurrence et le numéro [iOn], le résultat vaut pour le faisceau $\Rs'$ en rang q et pour le faisceau $\Ls'$ en rangs q et q − 1. Le lemme des cinq assure alors que le morphisme $$H^{q-1}(X,\Rs) \to H^{q-1}(X^\an,\Rs^\an)$$ est surjectif. Pour les mêmes raisons que précédemment, nous savons en outre que le résultat vaut pour le faisceau $\Rs$ en rang q et pour le faisceau $\Ls$ en rangs q et q − 1. Une nouvelle application du lemme des cinq assure alors que le morphisme $$H^{q-1}(X,\Fs) \to H^{q-1}(X^\an,\Fs^\an)$$ est bijectif. ### Pleine fidélité du foncteur $\Fs \to \Fs^\an$ Soit X un espace projectif sur $\As$. Soient $\Fs$ et $\Gs$ deux faisceaux de $\Os\_{X}$-modules cohérents. Soit $x^\an$ un point de $X^\an$. Notons x son image dans X. Nous disposons des isomorphismes $$\Hom(\Fs,\Gs)^\an\_{x^\an} \simeq \textrm{Hom}(\Fs,\Gs)\_{x} \otimes\_{\Os\_{X,x}} \Os\_{X^\an,x^\an}$$ et $$\Hom(\Fs^\an,\Gs^\an)\_{x^\an} \simeq \textrm{Hom}(\Fs\_{x} \otimes\_{\Os\_{X,x}} \Os\_{X^\an,x^\an}, \Gs\_{x} \otimes\_{\Os\_{X,x}} \Os\_{X^\an,x^\an}).$$ La platitude du morphisme naturel $X^\an \to X$ entraîne que le morphisme $$\Hom(\Fs,\Gs)^\an \to \Hom(\Fs^\an,\Gs^\an)$$ est un isomorphisme. On conclut en appliquant le résultat de l’assertion i) du théorème au faisceau cohérent $\Hom(\Fs,\Gs)$ et à l’entier q = 0. ### Surjectivité essentielle du foncteur $\Fs \to \Fs^\an$ Nous allons démontrer, par récurrence sur r, que, pour tout entier naturel r, le foncteur $\Fs \to \Fs^\an$ est essentiellement surjectif lorsque $X = \P^r\_{\As}$. Lorsque r = 0, le résultat est classique (cf. , proposition 2.3.1). Soit $r\in\N$ tel que le résultat soit vrai pour le schéma $\P^r\_{\As}$. Posons $X=\P^{r+1}\_{\As}$. Commençons par une série de lemmes. [lemhyp] Pour tout hyperplan projectif Y de X et tout faisceau de $\Os\_{X^\an}$-modules cohérent $\Ns$, il existe un entier n0 tel que $$\forall n\ge n\_{0}, \forall q\ge 1,\ H^q(Y^\an,\Ns\_{\|Y^\an}(n))=0.$$ On démontre ce résultat en appliquant l’hypothèse de récurrence au faisceau $\Ns\_{\|Y^\an}$, puis en utilisant le résultat analogue pour les schémas projectifs et les isomorphismes fournis par l’assertion i) du théorème. [chgtbasekL] Soit L une extension valuée complète de k. Notons $\pi : X^\an\_{L} \to X^\an$ le morphisme de projection. Soient x un point de $X^\an$ et xL l’un de ses antécédents par le morphisme π. Soit $\Fs$ un faisceau de $\Os\_{X^\an}$-modules cohérent. Supposons que la fibre $\pi^\(\Fs)\_{x\_{L}}$ soit engendrée par l’ensemble des sections globales de $\pi^\(\Fs)$. Alors, la fibre $\Fs\_{x}$ est engendrée par l’ensemble des sections globales de $\Fs$. D’après le théorème [kiehlfini], $\Fs(X^\an)$ est un $\As$-module de Banach fini. Considérons une famille (f1, …, fr), avec $r\in\N$, qui l’engendre. Notons $\Gs$ le conoyau du morphisme $\Os\_{X^\an}^r \to \Fs$ défini par cette famille. Puisque le produit tensoriel est exact à droite, le faisceau $\pi^\(\Gs)$ est le conoyau du morphisme $\Os\_{X^\an\_{L}}^r \to \pi^\(\Fs)$ défini par la famille (π\(f1), …, π\(fr)). L’exactitude du foncteur $\cdot \hat{\otimes}\_{k} L$ assure que les $(\As\hat{\otimes}\_{k} L)$-modules $\pi^\(\Fs)(X^\an\_{L})$ et $\Fs(X^\an) \hat{\otimes}\_{k} L$ sont isomorphes. En particulier, la famille (π\(f1), …, π\(f*r)) engendre $\pi^\(\Fs)(X^\an\_{L})$. Puisque, par hypothèse, cet ensemble engendre $\pi^\(\Fs)\_{x\_{L}}$, la fibre $\pi^\(\Gs)\_{x\_{L}} \simeq \Gs\_{x} \otimes\_{\Os\_{X^\an,x}} \Os\_{X^\an\_{L},x\_{L}}$ est nulle. A fortiori, nous avons $\Gs\_{x} \otimes\_{\Os\_{X^\an,x}} \kappa(x\_{L})=0$. Puisque $\Gs\_{x} \otimes\_{\Os\_{X^\an,x}} \kappa(x\_{L}) \simeq \Gs\_{x} \otimes\_{\Os\_{X^\an,x}} \kappa(x) \otimes\_{\kappa(x)} \kappa(x\_{L})$, nous avons même $\Gs\_{x} \otimes\_{\Os\_{X^\an,x}} \kappa(x)=0$, d’où l’on déduit que $\Gs\_{x}=0$, par le lemme de Nakayama. [lemgen] Soient $\Ns$ un faisceau de $\Os\_{X^\an}$-modules cohérent et x* un point de $X^\an$. Il existe un entier naturel n0 tel que, pour tout n ≥ n0, la fibre $\Ns(n)\_{x}$ soit engendrée par l’ensemble des sections globales de $\Ns(n)$. D’après le lemme [chgtbasekL], quitte à effectuer un changement de corps de base de k à $\Hs(x)$ (et à modifier les autres données en conséquence), nous pouvons supposer que le point x est k-rationnel. Il est alors situé sur l’analytifié $Y^\an$ d’un certain hyperplan projectif Y de X. Soit t une section de $\Os\_{X}(1)$ de lieu des zéros Y. La multiplication par t définit une suite exacte $$0 \to \Ns' \to \Ns(-1) \to \Ns \to \Ns\_{\|Y^\an} \to 0,$$ où $\Ns'$ est un faisceau de $\Os\_{X^\an}$-modules cohérent supporté par $Y^\an$. Soit $n\in\Z$. En tensorisant la suite précédente par $\Os\_{X^\an}(n)$ puis en la scindant, nous obtenons les deux suites exactes courtes $$0 \to \Ns'(n) \to \Ns(n-1) \to \Ps\_{n} \to 0$$ et $$0 \to \Ps\_{n} \to \Ns(n) \to \Ns\_{\|Y^\an}(n) \to 0,$$ qui donnent naissance aux deux suites exactes de cohomologie $$H^1(X^\an,\Ns(n-1)) \to H^1(X^\an,\Ps(n)) \to H^2(X^\an,\Ns'(n))$$ et $$H^1(X^\an,\Ps\_{n}) \to H^1(X^\an,\Ns(n)) \to H^1(X^\an,\Ns\_{\|Y^\an}(n)).$$ D’après le lemme [lemhyp], il existe un entier n1 tel que, pour tout n ≥ n1, les groupes de cohomologie $$H^2(X^\an,\Ns'(n)) \textrm{ et } H^1(X^\an,\Ns\_{\|Y^\an}(n))$$ soient nuls et, par conséquent, le morphisme composé $$H^1(X^\an,\Ns(n-1)) \to H^1(X^\an,\Ps\_{n}) \to H^1(X^\an,\Ns(n))$$ soit surjectif. Le morphisme $X^\an \to \Mc(\As)$ étant propre, le théorème [kiehlfini], assure que le $\As$-module $H^1(X^\an,\Ns(n\_{1}-1))$ est de type fini, et donc noethérien. On en déduit qu’il existe un entier n2 ≥ n1 tel que, pour tout n ≥ n2, le morphisme $$H^1(X^\an,\Ns(n-1)) \to H^1(X^\an,\Ns(n))$$ soit un isomorphisme. Par conséquent, pour tout n ≥ n2, le morphisme surjectif $$H^1(X^\an,\Ps\_{n}) \to H^1(X^\an,\Ns(n))$$ est bijectif, d’où l’on déduit, en considérant la suite exacte longue associé à la seconde suite exacte courte, que le morphisme $$H^0(X^\an,\Ns(n)) \to H^0(X^\an,\Ns\_{\|Y^\an}(n))$$ est surjectif. D’après l’hypothèse de récurrence, le faisceau de $\Os\_{Y^\an}$-modules cohérent $\Ns\_{\|Y^\an}$ est l’analytifié d’un faisceau de $\Os\_{Y}$-modules cohérent $\Gs$. Notons $x^\alg$ l’image du point x dans Y. Les résultats classiques sur les schémas projectifs assurent qu’il existe un entier n0 ≥ n2 tel que, pour tout n ≥ n0, la fibre $\Gs(n)\_{x^\alg}$ soit engendrée, en tant que $\Os\_{Y,x^\alg}$-module, par l’ensemble des sections globales $H^0(Y,\Gs(n))$. En utilisant l’assertion i) du théorème, on en déduit que le résultat vaut encore en remplaçant respectivement Y par $Y^\an$, $\Gs$ par $\Ns\_{\|Y^\an}$ et $x^\alg$ par x. Notons $\Is$ le faisceau d’idéaux qui définit $Y^\an$ dans $X^\an$. Remarquons que sa fibre $\Is\_{x}$ est contenue dans l’idéal maximal $\m\_{x}$ de $\Os\_{X^\an,x}$. Soit n ≥ n0. Nous venons de montrer que $\Ns(n)\_{x}\otimes\_{\Os\_{X^\an,x}} (\Os\_{X^\an,x}/\Is\_{x})$ est engendré par $H^0(X^\an,\Ns\_{\|Y^\an}(n))$. On en déduit que $\Ns(n)\_{x}\otimes\_{\Os\_{X^\an,x}} (\Os\_{X^\an,x}/\m\_{x})$ est engendré par $H^0(X^\an,\Ns\_{\|Y^\an}(n))$, et donc par $H^0(X^\an,\Ns(n))$. On conclut par le lemme de Nakayama. Terminons, à présent, la démonstration. Soit $\Ns$ un faisceau de $\Os\_{X^\an}$-modules cohérent. En utilisant le résultat du lemme précédent et la compacité de $X^\an$, on montre qu’il existe un entier n tel qu’en tout point x de $X^\an$, la fibre $\Ns(n)\_{x}$ soit engendrée par $H^0(X^\an,\Ns(n))$. On en déduit l’existence d’un entier naturel p, d’un faisceau de $\Os\_{X^\an}$-modules cohérent $\Rs$ et d’une suite exacte $$0 \to \Rs \to \Os\_{X^\an}(-n)^p \to \Ns \to 0.$$ En appliquant le même raisonnement au faisceau $\Rs$, nous parvenons finalement à écrire le faisceau $\Ns$ comme le conoyau d’un morphisme $\varphi : \Os\_{X^\an}(-m)^q \to \Os\_{X^\an}(-n)^p$, avec $m\in\Z$ et $q\in\N$. Puisque le foncteur d’analytification est pleinement fidèle, le morphisme φ est l’analytifié d’un morphisme $\varphi^\alg : \Os\_{X}(-m)^q \to \Os\_{X}(-n)^p$. L’exactitude à droite du foncteur d’analytification assure alors que le faisceau $\Ns$ est isomorphe à l’analytifié du conoyau du morphisme $\varphi^\alg$, qui est un faisceau de $\Os\_{X}$-modules cohérent. Corollaires ----------- Nous énonçons ici deux corollaires du théorème [GAGAsuraffinoide]. Ils ont également pour objet des résultats de type GAGA, mais sur des bases qui ne sont plus nécessairement affinoïdes. Nous sommes convaincu qu’il est possible de les étendre à une base quelconque, de façon à obtenir un analogue parfait des théorèmes obtenus par M. Hakim dans le cadre de la géométrie analytique complexe (cf. , chapitre VIII, théorèmes 3.2 et 3.5). Cependant, pour éviter d’avoir à utiliser le formalisme un peu lourd des schémas relatifs sur un espace analytique, nous nous contenterons d’énoncer les deux cas particuliers que nous utilisons dans cet article. Soit k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet, $\As$ une algèbre k-affinoïde et X un $\As$-schéma propre. Soit B une partie compacte de $\Mc(\As)$ possédant un système fondamental de voisinages affinoïdes. Rappelons que la notation $\Os(B)$ désigne l’anneau des germes de fonctions analytiques au voisinage de B. Notons $Y = X\times\_{\Spec(\As)}\Spec(\Os(B))$ et $Y^\an$ l’image réciproque de B dans $X^\an$. Munissons $Y^\an$ du faisceau des fonctions surconvergentes. En utilisant le morphisme d’analytification au-dessus d’un espace affinoïde défini par V. Berkovich, on montre qu’il existe un morphisme d’espaces localement annelés $Y^\an \to Y$. La rétrotirette d’un faisceau de $\Os\_{Y}$-modules cohérent $\Fs$ est un faisceau de $\Os\_{Y^\an}$-modules cohérent, que nous noterons $\Fs^\an$. [GAGAvoiscompact] Supposons que nous nous trouvons dans la situation décrite ci-dessus. Alors 1. pour tout faisceau de $\Os\_{Y}$-modules cohérent $\Fs$ et tout entier $q\in\N$, le morphisme $$H^q(Y,\Fs) \to H^q(Y^\an,\Fs^\an)$$ est un isomorphisme ; 2. le foncteur d’analytification $$\Fs \to \Fs^\an$$ induit une équivalence entre la catégorie des $\Os\_{Y}$-modules cohérents et celle des $\Os\_{Y^\an}$-modules cohérents. Il faut tout d’abord remarquer que l’espace $Y^\an$ est compact. En reprenant le raisonnement de la preuve de la proposition 1 de , on en déduit que tout faisceau cohérent sur $Y^\an$ se prolonge en un faisceau cohérent sur un voisinage de $Y^\an$, et donc sur une partie de la forme $Y^\an\times\_{\Mc(\As)} V$, où V est un voisinage affinoïde de B. En utilisant ce raisonnement et le théorème [GAGAsuraffinoide], on obtient le résultat attendu. Soit k un corps muni d’une valeur absolue ultramétrique pour laquelle il est complet, $\As$ une algèbre k-affinoïde et X un $\As$-schéma propre. Soit B un espace $\As$-analytique qui soit limite inductive d’espaces affinoïdes. Notons $Z = X\times\_{\Spec(\As)}\Spec(\Os(B))$. En utilisant le morphisme d’analytification au-dessus d’un espace affinoïde défini par V. Berkovich, on construit, par limite inductive, un espace analytique $Z^\an$ et un morphisme d’espaces localement annelés $Z^\an \to Z$. Comme précédemment, la rétrotirette d’un faisceau de $\Os\_{Z}$-modules cohérent $\Fs$ est un faisceau de $\Os\_{Z^\an}$-modules cohérent, que nous noterons $\Fs^\an$. Le résultat suivant se déduit alors aisément du théorème [GAGAsuraffinoide]. [GAGAsurouvert] Supposons que nous nous trouvons dans la situation décrite ci-dessus. Alors 1. pour tout faisceau de $\Os\_{Z}$-modules cohérent $\Fs$ et tout entier $q\in\N$, le morphisme $$H^q(Z,\Fs) \to H^q(Z^\an,\Fs^\an)$$ est un isomorphisme ; 2. le foncteur d’analytification $$\Fs \to \Fs^\an$$ induit une équivalence entre la catégorie des $\Os\_{Z}$-modules cohérents et celle des $\Os\_{Z^\an}$-modules cohérents. La droite de Berkovich sur un anneau d’entiers de corps de nombres ================================================================== Dans cette annexe, nous présentons succintement la droite de Berkovich sur un anneau d’entiers de corps de nombres. Nous invitons le lecteur dont ces prémices auront éveillé la curiosité à parcourir l’ouvrage pour approfondir ce sujet. Définitions ----------- Soit K un corps de nombres. Notons A l’anneau de ses entiers. Commençons par rappeler la définition d’espace affine analytique sur A. Elle est due à V. Berkovich (cf. , §1.5). Soit $n\in\N$. L’espace affine analytique de dimension n sur A, noté $\E{n}{A}$, est l’ensemble des semi-normes multiplicatives sur A[T1, …, Tn], c’est-à-dire l’ensemble des applications $$\|.\| : A[T\_{1},\ldots,T\_{n}] \to \R\_{+}$$ qui vérifient les propriétés suivantes : 1. ∣0∣ = 0 et ∣1∣ = 1 ; 2. ∀P, Q ∈ A[T1, …, Tn], ∣P + Q∣ ≤ ∣P∣ + ∣Q∣ ; 3. ∀P, Q ∈ A[T1, …, Tn], ∣PQ∣ = ∣P∣ ∣Q∣. Dans la définition proposée par V. Berkovich figure une condition supplémentaire qui fait intervenir une norme sur l’anneau A. Pour a ∈ A, posons ∥a∥ = maxσ ∈ Σ∞(∣σ(a)∣∞), où Σ∞ désigne l’ensemble des plongements complexes du corps K et ∣.∣∞ la valeur absolue usuelle sur $\C$. La fonction $\l.\l : A\to \R\_{+}$ définit une norme sur A et, lorsque l’on munit l’anneau A de cette norme, la définition de Berkovich coïncide avec la nôtre. Signalons que, quelle que soit la norme dont on munit A (sous réserve tout de même qu’elle soit sous-multiplicative et fasse de A un espace complet), on obtient un espace contenu dans celui que nous avons défini. Soit x un point de $\E{n}{A}$. Il lui est associé une semi-norme multiplicative ∣.∣x sur A[T1, …, Tn]. L’ensemble $\p\_{x}$ des éléments sur lesquels elle s’annule est un idéal premier de A[T1, …, Tn]. Le quotient est un anneau intègre sur lequel la semi-norme ∣.∣x induit une valeur absolue. Nous noterons $\Hs(x)$ le complété du corps des fractions de cet anneau pour cette valeur absolue. Nous noterons simplement ∣.∣ la valeur absolue sur le corps $\Hs(x)$, cela n’entraînant pas de confusion. La construction fournit un morphisme $$A[T\_{1},\ldots,T\_{n}] \to \Hs(x).$$ L’image d’un élément P de A[T1, …, Tn] par ce morphisme sera notée P(x). Avec ces notations, nous avons donc ∣P(x)∣ = ∣P∣x. Munissons, à présent, l’espace analytique $\E{n}{A}$ d’une topologie : celle engendrée par les ensembles de la forme $$\{x \in \E{n}{A}\, \|\, r < \|P(x)\| < s\},$$ pour P ∈ A[T1, …, Tn] et $r,s\in\R$. Pour finir, nous définissons un faisceau d’anneaux $\Os$ sur $\E{n}{A}$ de la façon suivante : pour tout ouvert U de $\E{n}{A}$, l’anneau $\Os(U)$ est constitué des applications $$f : U\to \bigsqcup\_{x\in U} \Hs(x)$$ qui vérifient les deux conditions suivantes : 1. ∀x ∈ U, $f(x)\in\Hs(x)$ ; 2. f est localement limite uniforme de fractions rationnelles sans pôles. Dimension 0 ----------- Afin de rendre plus palpables les définitions précédentes, nous allons décrire explicitement $\E{0}{A}$, l’espace affine analytique de dimension 0 sur A, que nous noterons plus volontiers $\Mc(A)$. Nous noterons ∣.∣∞ la valeur absolue usuelle sur $\C$ et, pour tout idéal maximal $\m$ de A, nous noterons $\|.\|\_{\m}$ la valeur absolue $\m$-adique normalisée. Du théorème d’Ostrowski, l’on déduit que les points de $\Mc(A)$ sont exactement 1. la valeur absolue triviale ∣.∣0 (nous noterons a0 le point associé) ; 2. la valeur absolue archimédienne $\|\sigma(.)\|\_{\infty}^\eps$ (nous noterons $a\_{\sigma}^\eps$ le point associé) pour tout toute classe de conjugaison de plongements complexes σ de K et tout élément $\eps$ de $\of{]}{0,1}{]}$ ; 3. la valeur absolue $\m$-adique $\|.\|\_{\m}^\eps$ (nous noterons $a\_{\m}^\eps$ le point associé) pour tout idéal maximal $\m$ de A et tout élément $\eps$ de $\of{]}{0,+\infty}{[}$ ; 4. la semi-norme $\|.\|\_{\m}^{+\infty}$ (nous noterons $\tilde{a}\_{\m}$ le point associé) induite par la valeur absolue triviale sur le corps fini $A/\m$ pour tout idéal maximal $\m$ de A. Nous pouvons également décrire la topologie de l’espace $\Mc(A)$ (cf. figure , tracée dans le cas où $K=\Q$ pour simplifier les notations, mais aisément généralisable). Pour cela, il suffit d’indiquer que chacune des branches tracée sur la figure est homéomorphe à un segment réel et qu’un voisinage du point central a0 est une partie qui contient entièrement toutes les branches à l’exception d’un nombre fini, et qui contient un voisinage de a0 dans chacune des branches restantes. Si l’on préfère, l’espace $\Mc(A)$ possède la topologie du compactifié d’Alexandrov de la réunion disjointe de ses branches privées de a0, le point a0 jouant le rôle du point à l’infini. Nous pouvons également décrire explicitement les sections du faisceau structural sur les ouverts de $\Mc(A)$. Nous avons représenté les différents cas à la figure , de nouveau dans le cas où $K=\Q$. Dimension 1 ----------- Venons-en, à présent, à l’espace affine analytique de dimension 1 sur A. Nous noterons T la coordonnée sur cet espace. Remarquons, tout d’abord, que le morphisme A → A[T] induit un morphisme de projection $$\pi : \E{1}{A} \to \Mc(A).$$ Cela permet d’obtenir une description topologique de la droite de Berkovich sur A : la fibre de π au-dessus d’un point x de $\Mc(A)$ est isomorphe à la droite de Berkovich sur le corps $\Hs(x)$. Si $\Hs(x)=\C$, cette droite est isomorphe à l’espace $\C$ et, si $\Hs(x)=\R$, elle est isomorphe à son quotient par la conjugaison complexe. Nous ne chercherons pas à obtenir de description plus précise et nous contenterons d’indiquer quelques propriétés (cf. , théorèmes 4.4.1 et 4.5.5). [resume] 1. L’espace $\E{1}{A}$ est localement compact, métrisable et de dimension topologique 3. 2. L’espace $\E{1}{A}$ est localement connexe par arcs. 3. Le morphisme de projection $\pi : \E{1}{A}\to \Mc(A)$ est ouvert. 4. En tout point x de $\E{1}{A}$, l’anneau local $\Os\_{x}$ est hensélien, noethérien, régulier, de dimension inférieure à 2 et le corps résiduel κ(x) est hensélien. 5. Le faisceau structural $\Os$ est cohérent. Signalons encore que la droite de Berkovich sur A satisfait au principe du prolongement analytique (ibid., théorèmes 4.4.2 et 7.1.9, corollaire 4.4.5). [prolan] Soit U une partie connexe de $\E{1}{A}$. 1. Le principe du prolongement analytique vaut sur U. En particulier, l’anneau $\Os(U)$ est intègre. 2. L’anneau des sections méromorphes $\Ms(U)$ est un corps. 3. Si U est de Stein, le morphisme naturel $\Frac(\Os(U))\to \Ms(U)$ est un isomorphisme. Rappelons ici ce que nous entendons par espace de Stein. Nous dirons qu’un espace localement annelé $(X,\Os\_{X})$ est de Stein s’il satisfait les conclusions des théorèmes A et B de H. Cartan : 1. pour tout faisceau de $\Os\_{X}$-modules cohérent $\Fs$ et tout point x de X, la fibre $\Fs\_{x}$ est engendrée par l’ensemble des sections globales $\Fs(X)$ ; 2. pour tout faisceau de $\Os\_{X}$-modules cohérent $\Fs$ et tout entier $q\in\N^\$, nous avons $H^q(X,\Fs)=0$. Donnons quelques exemples de sous-espaces de la droite analytique $\AZ$ qui sont des espaces de Stein (ibid., théorème 6.6.29). [lemniscateStein] Soient V* une partie ouverte et connexe de l’espace $\Mc(A)$ et. Soit P(T) un polynôme unitaire à coefficients dans $\Os(V)$. Les parties suivantes de la droite analytique $\E{1}{A}$ sont des espaces de Stein : 1. $\left\{x\in \pi^{-1}(V)\, \big\|\, s < \|P(T)(x)\| < t\right\}$ ; 2. $\left\{x\in \pi^{-1}(V)\, \big\|\, \|P(T)(x)\| > s\right\}$. Pour terminer, disons quelques mots des sections globales sur les parties de la droite analytique $\E{1}{A}$. Sur les disques, elles s’expriment essentiellement en termes de séries dont les coefficients sont des fonctions sur $\Mc(A)$. Considérons, par exemple, le disque ouvert relatif de rayon 1 : $$\D=\left\{\left. x\in\E{1}{A}\, \right\|\, \|T(x)\|<1\right\}.$$ Le morphisme naturel $A[T]\to \Os(\D)$ induit un isomorphisme $$A\_{1^-}[\![T]\!] \xrightarrow[]{\sim} \Os(\D),$$ où A1−[ [T] ] désigne l’anneau constitué des séries de la forme ∑n ≥ 0an Tn ∈ A[ [T] ] telles que le rayon de convergence de la série à coefficients complexes ∑n ≥ 0σ(an) Tn soit supérieur ou égal à 1, pour tout plongement complexe σ de K. On déduit cette description du théorème 3.2.16 de ibid. À partir de la description des anneaux de sections sur les disques, nous pouvons déduire celle des anneaux locaux en certains points. Nous nous contenterons de deux exemples. Soit $\m$ un idéal maximal de A. Notons $z\_{\m}$ le point 0 de la fibre de π au-dessus du point $\tilde{a}\_{\m}$. D’après le corollaire 3.2.5 de ibid., le morphisme naturel $A[T] \to \Os\_{z\_{p}}$ induit un isomorphisme $$\hat{A}\_{\m}[\![T]\!] \xrightarrow[]{\sim} \Os\_{z\_{p}}.$$ Notons z0 le point 0 de la fibre de π au-dessus du point a0. D’après le corollaire 3.2.8 de ibid., le morphisme naturel $A[T] \to \Os\_{z\_{0}}$ induit un isomorphisme $$E \xrightarrow[]{\sim} \Os\_{z\_{0}},$$ où E désigne l’anneau constitué des éléments f de K[ [T] ] qui vérifient les propriétés suivantes : 1. ∃a ∈ A \ {0}, f(aT) ∈ A[ [T] ] ; 2. pour tout plongement complexe σ de K, le rayon de convergence complexe de la série σ(f) est strictement positif ; 3. pour tout idéal maximal $\m$ de A, le rayon de convergence $\m$-adique de la série f est strictement positif (il suffit d’imposer cette condition pour les idéaux maximaux qui contiennent un élément a possédant les propriétés décrites en i). --- 1. L’auteur est membre du projet jeunes chercheurs Espaces de Berkovich de l’agence nationale de la recherche.[↩](#fnref1) 2. Nous empruntons ce terme à L. Moret-Bailly (cf. ). Les corps fertiles sont connus sous beaucoup d’autres noms. Ils ont été introduits par F. Pop dans sous l’appellation de large fields.[↩](#fnref2)	arxiv_0000481
mathcal{P} \vartriangleright\mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$ are equal; and there are no idempotents of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright \mathbf{R}$ apart from the Mean, nor any of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash\mathcal{R}$. Coincident randomizations ------------------------- Condition ([eq:coincond]) must hold; $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright \mathcal{R}$; use the decomposition $(\mathcal{P} \vartriangleright \mathcal {Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$, whose idempotents have the form P, $\mathbf{P} \vartriangleright \mathbf{Q}$, $\mathbf{P} \vartriangleright\mathbf{R}$, P ⊢ Q or P ⊢ R; if condition ([eq:coin]) holds, this is the same as the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$, which is done from left to right; otherwise, there may be a refinement of R giving a left-to-right decomposition. Double randomizations --------------------- We require that ∣Υ∣ = ∣Γ∣ and that R be structure balanced in relation to both Q and P, so that the decomposition is $ \mathcal{P} \vartriangleright\mathcal {R} = \mathcal{P} \vartriangleright(\mathcal{Q} \vartriangleright\mathcal {R})$, which is done from right to left. It appears that they can also be formulated as randomized-inclusive randomizations using pseudofactors to refine Q to Q2 for which the left-to-right decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}\_2) \vartriangleright\mathcal{R}$ is correct. Structure-balanced experiments with four or more tiers ====================================================== Each experiment in Sections [s:uincl]–[s:double] involves only one type of multiple randomization, and so involves three tiers and three structures. However, multitiered experiments are not limited to this configuration. Examples 12–14 in each have four tiers and involve more than one type of multiple randomization. In general, there is the set of observational units, Ω, and each randomization adds another set of objects with its associated tier. Section 7 in shows how to deal with three or more randomizations which follow each other in a chain. Mixtures of other types of multiple randomization should be amenable to successive decompositions of the sort summarized in Section [s:summary2], so long as they are handled in the correct order. Thus we can use a recursive procedure in which each new structure refines the decomposition of VΩ obtained using structures accounted for previously. All that is required is that each successive structure should be structure balanced in relation to the previous decomposition. [fig:corn] One class of experiments with both two–one randomizations and chain randomizations consists of multiphase experiments in which different treatment factors are applied in different phases, as the following example demonstrates. [(A two-phase corn seed germination experiment)] [eg:corn] Example 12 of has the four tiers shown in Figure [fig:corn]. Here we have taken the opportunity to correct the diagram given in. The 36 Lots of grain within each Plot should be completely randomized to Plates ∧ Containers within each Interval. This will not be achieved by permuting Containers within Intervals and Plates within Intervals ∧ Containers, as implied in the rightmost panel of Figure [fig:corn]. We introduce pseudofactors L1 and L2 for Lots, with nine and four levels, respectively, like the pseudofactors for Seedlings in Example [eg:PlantExp]. The 36 Lots must be randomly allocated to the combinations of levels of L1 and L2, independently within each level of Sites ∧ Blocks ∧ Plots, so that neither pseudofactor corresponds to any inherent source of variation. At each randomization, an orthogonal design is used, so there is no difficulty in constructing the decomposition in Table [tab:corn]. Here L1[P ∧ B ∧ S] is the part of the source Lots[P ∧ B ∧ S] which is confounded with Containers[I]. The source Lots[P ∧ B ∧ S]⊢ is the part of Lots[P ∧ B ∧ S] which is orthogonal to L1[P ∧ B ∧ S]: it is confounded with Plates[C ∧ I]. [tab:corn] @lrlrlrlr@ & & & & & & source & & & & & & & Mean & 1 & Mean & 1 & Mean & 1 & Mean & 1 Intervals& 17 & Sites & 2 && & & Blocks[S] & 3 && & & Plots[B ∧ S] & 12 & Harvesters &2 & & & & Residual & 10 Containers[I] & 144 & L1[P ∧ B ∧ S] & 144 & & &Temperature & 2 & & & & & & Moistures& 2 & & & & & & T # M & 4 & & & & & & Residual & 136 Plates[C ∧ I] & 486 & Lots[P ∧ B ∧ S]⊢ & 486 Bailey suggests an analysis for this example which we reproduce in the first three columns of Table [tab:Baileycorn](a). In this, the 3-level factors Temperature and Moisture have been combined into a single 9-level Treatment factor, the intertier interactions of Sites, Harvesters and Treatments have been included, and the notation × is used in place of #. We cannot be sure, but it is plausible that he based this decomposition on the crossing and nesting relationships summarized in the formula $$\label{eq:tbb} \bigl(\mathrm{T} \\mathrm{H} \(\mathrm{S}/\mathrm{B})\bigr) /\mathrm{Q},$$ where T, H, S, B and Q represent factors for Treatments, Harvesters, Sites, Blocks and Plates, with 9, 3, 3, 2 and 4 levels, respectively. The sources derived from this are in the final column of Table [tab:Baileycorn](a), with degrees of freedom matching those in the preceding column. [tab:Baileycorn] @llrl@ (a) & source & d.f. & source from ([eq:tbb]) Phase I: field study & Site & 2 & S & Experimental error (a) & 3 & B[S] & Harvester & 2 & H &Harvester × Site & 4 & H # S & Experimental error (b) & 6 & H # B[S] [4pt] Phase II: laboratory study & Treatment & 8 & T & Treatment × Site & 16 & T # S & Experimental error (c) & 24 & T # B[S] & Treatment × Harvester & 16 & T # H & Treatment × Harvester × Site & 32 & T # H # S &Experimental error (d) & 48 & T # H # B[S] & Residual & 486& Q[T ∧ H ∧ S ∧ B] @lrlrlrlr@ (b) & & & & & & source & d.f. & source & d.f. & source & d.f. & source & d.f. Mean & 1 & Mean & 1 & Mean & 1 & Mean & 1 Intervals& 17 & Sites & 2 & & Blocks[S] & 3 && & & Plots[B ∧ S] & 12 & Harvesters & 2 & & & & H # S & 4 & & & & Residual & 6 Containers[I] & 144 & Lots[P ∧ B ∧ S]1 & 144 & & & Treatments & 8 & & & & & & T # S & 16 & & & & & & T # H & 16 & & & & & & T # H # S & 32 & & & & & & Residual & 72 Plates[C ∧ I] & 486 & Lots[P ∧ B ∧ S]⊢ & 486 Revision of Table [tab:corn] along similar lines, and with pseudosources replaced with actual sources, yields the skeleton analysis-of-variance table in Table [tab:Baileycorn](b). Note that, given Step 4 in Table 1 of, an intertier interaction will generally occur in the right-most tier that contains a main effect in the interaction. Table [tab:Baileycorn](b) differs from Table [tab:Baileycorn](a) in the following ways. 1. The rationale for the sources in Table [tab:Baileycorn](a) is unclear. We had to reverse-engineer it by producing formula ([eq:tbb]). On the other hand, the sources in Table [tab:Baileycorn](b) are based on the relationships between factors within each tier and on the confounding between sources from different tiers. 2. Table [tab:Baileycorn](a) does not show, as Table [tab:Baileycorn](b) does, the successive decomposition of the vector space indexed by the observational units. The impression given is that there is a set of sources that arise from the field phase and another set that arises from the laboratory phase. 3. Table [tab:Baileycorn](a) has four sources called “experimental error” and does not mention plates, containers, intervals, blocks, plots or lots. Hence, there is no indication of the sources of error variation. By contrast, each source called “Residual” in Table [tab:Baileycorn](b) is unambiguously identified; and the labelling shows that all terms are affected by variation from both phases. For example, the Residual for Plots[B ∧ S], labelled Experimental error (b) in Table [tab:Baileycorn](a), clearly arises from variability associated with Plots within the Sites-Blocks combinations and variability associated with Intervals. Similarly, it can be seen from Table [tab:Baileycorn](b) that the Residual in Table [tab:Baileycorn](a) arises from variability associated with Plates and Lots. 4. As discussed in, Section 7.1, the usual default is that there are no intertier interactions because such inclusions would mean that the analysis cannot be justified by the randomization used. It parallels the assumption of unit-treatment additivity in single-randomization experiments. The approach using Table [tab:corn] forces the statistician to to consult the researcher about whether intertier interactions should be included, and, if so, to justify them. Tables [tab:Baileycorn](a) and (b) include the intertier interactions of Sites, Harvesters and Treatments, which suggests that it is anticipated that Harvesters and Treatments will perform differently at different Sites. 5. Even with the addition of intertier interactions, the decompositions in Tables [tab:Baileycorn](a) and (b) are not equivalent, and so neither are the mixed models underlying them. Experimental errors (c) and (d) from Table [tab:Baileycorn](a) are combined into the Residual with 72 degrees of freedom for Lots[P ∧ B ∧ S]1 in Table [tab:Baileycorn](b). To justify an analysis based on Table [tab:Baileycorn](a), one would need to argue that unit-treatment interaction of Treatments with Blocks within Sites can be anticipated in this experiment. Discussion ========== Implications of incoherent unrandomized-inclusive randomizations ---------------------------------------------------------------- The phenomenon of incoherent unrandomized-inclusive randomizations is described in , Section 5.2.1. Essentially, when there has been a randomization to factors that are crossed, one or more of these factors become nested in the second randomization. Consider the cherry rootstock experiment in Example [eg:SuperStruct]. The trees tier gives an orthogonal decomposition of VΩ into sources Mean, Blocks and Trees[Blocks] of dimensions 1, 2 and 27, respectively, in the left-hand column of Table [tab:ANOVASuper]. Similarly, the rootstocks tier decomposes VΥ into sources Mean and Rootstocks of dimensions 1 and 9. The result of the first randomization is to make the Mean sources equal and to place the Rootstocks source inside Trees[Blocks], thus giving the finer decomposition of VΩ shown in the middle column of Table [tab:ANOVASuper]. The result of the unrandomized-inclusive randomization should be to further decompose the decomposition resulting from the first two tiers. In the extended Youden square, the source Viruses is orthogonal to Blocks but partially confounded with Rootstocks, so the Viruses source defines the decomposition in the right-hand column in Table [tab:ANOVASuper]. That is, the source Viruses further decomposes the sources Rootstocks and the Residual for Trees[Blocks], as required. In we discussed the possibility that the designer of the superimposed experiment ignores the inherent crossing of the factors Blocks and Rootstocks and randomizes Viruses to Blocks in Rootstocks in a balancedincomplete-block design. Then the randomizations are incoherent. The permutation group for the second randomization does not preserve the structure arising from the first two tiers, exhibited by the two left-most columns in Table [tab:ANOVASuper]. We can see immediately that this randomization is senseless because it destroys the Blocks subspace preserved by the first randomization. This randomization might have some appeal if no block effects had been detected during the 20 years of the original experiment, but then the analysis of the second experiment would be based on an assumed model rather than on the intratier structures. [fig:cc] [b] [tab:cc] @lrlrlrclr@ & & & & & & source & d.f. & source & d.f. & source & d.f. & eff. & source & d.f. Mean & 1 & Mean & 1 & Mean & 1 && Mean & 1 Benches & 2 & & & B[S]B & 2 Plants[B] & 9 & Soils & 3 & Soils &3 && & &Residual &6 & B[S]⊢ & 6 Altitudes & 2 & & &A[S]A & 2 A # B & 4 & && A # B[S]A # B & 4 &? & Viruses & 2 & & & & & &? & V # S & 2 A # P[B] & 18 && & A[S]⊢ & 6 && & & & &A # B[S]⊢ & 12 &?& Viruses & 2 & & & & & &?& V # S & 6 & & & & & & & Residual & 4 Other examples of incoherent unrandomized-inclusive randomizations are more complicated, and perhaps less easily detected. One is the design proposed by several authors for a split-plot experiment in which the subplot treatments are to be assigned using a row-column design. Example [eg:cc] illustrates how consideration of the decomposition table for the proposed design facilitates the design process and helps the detection of incoherence. [(Split-plots in a row–column design)][eg:cc] Example 11 in is based on the design with split-plots in a row–column design given in Cochran and Cox , Section 7.33. Diagrams for the two randomizations are given in Figure [fig:cc], with leaf treatments named as viruses for clarity, soil treatments designated as different soils for brevity, and Altitude substituted for Layer so that no two factors begin with the same letter. Two diagrams are needed, because the assumed structure on leaves changes between the randomizations, as shown in the two right-hand panels. At first sight, this experiment seems to involve unrandomized-inclusive randomizations, because soils are randomized to leaves in the first randomization, and then viruses are randomized to leaves, taking into account the location of the soils. However, the change in the assumed structure on the leaves between the two randomizations makes them incoherent rather than unrandomized-inclusive. Table [tab:cc] shows an attempt to build up a decomposition table for this design. The first two columns follow directly from the randomization in the top half of Figure [fig:cc]. The third column corresponds to the leaves tier in the bottom half of Figure [fig:cc]. When we use it to refine the decomposition given by the first two tiers, we find that the Soils source occurs in two tiers. Although this can happen in special circumstances like those in Example [eg:RotateGraze], this is already a signal that something may be wrong. We also find that the nesting, in this tier, of Benches within Soils gives a source B[S] with 8 degrees of freedom. This is the sum of the previous sources Benches and Residual in Plants[B]; these two parts are denoted B[S]B and B[S]⊢ in Table [tab:cc]. Similarly, the nesting of Altitudes within Soils, in this tier, gives sources A[S] and A # B[S] which are each the sum of two previous sources. [b] [tab:SPLwLS] @lcccccccccccc@ & & & & & & & & & & & & & & & & & & Altitude Top & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 Middle & 2 & 2 & 1 & 1 & 0 & 0 & 2 & 2 & 1 & 1 & 0 & 0 Bottom & 1 & 1 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 1 & 1 The real difficulties come when we try to incorporate the column for the viruses tier, because the location of the Viruses source depends on the outcome of the randomizations. For the outcome given in, Section 7.33, and, Example 11, the Viruses source does not have first-order balance in relation to either Altitudes # Benches or Altitudes # Plants[Benches]. The interaction Viruses # Soils has the same problem. If Altitudes # Benches is merged with Altitudes # Plants[Benches] in the decomposition table, then the analysis is orthogonal and is equivalent to that given in. However, this does not allow for consistent Altitude differences across Plants, so it removes six spurious degrees of freedom from what Cochran and Cox call “Error (b)” in, page 310. The problem is that the design for the Viruses does not respect the factor relationships established in applying the Soils. As Yates showed in, if the randomization respects Benches and Altitudes then a randomization-based model must include their interaction. What is needed is a design for a two-tiered experiment in which the twelve treatments (combinations of levels of Soils and Viruses) are randomized to leaves1 in such a way that there is a refinement of the natural decomposition of the treatments space which is structure balanced in relation to Altitudes # Benches. For example, one might choose the systematic design in Table [tab:SPLwLS] and then randomize benches, altitudes, and plants within benches. In this design the twelve treatments are arranged in a (3 × 3)/4 semi-Latin square constructed from a pair of mutually orthogonal Latin squares of order 3. The Viruses are arranged according to one square for soils s0 and s1, and according to the other square for s2 and s3. Theorem 5.4 in shows that this design is the most efficient with respect to Altitudes # Benches. Let S1 be a pseudofactor for Soils whose two levels distinguish between the first two and the last two levels of Soils. The design is structure balanced: Viruses and Viruses # S1 have efficiency factor 1/2 in Altitudes # Benches, while the rest of the interaction Viruses # Soils is orthogonal to Altitudes # Benches. It has the advantage of having 10 degrees of freedom for the Residual for Altitudes # Plants[Benches], two more than for the Cochran and Cox design. Thus construction of the decomposition table when designing the experiment can help to detect problems with a proposed design. In this case, it helped to draw attention to the incoherent randomizations, to highlight the associated problems and to give insight into how they might be redressed. Other structures ---------------- All the examples in and this paper are poset block structures, being defined by some factors and their nesting relationships, as explained in. More generally, a structure may be a Tjur structure that is defined by a family of mutually orthogonal partitions or generalized factors (see or ). Again, the generalized factors are summarized in the Hasse diagram that depicts their marginality relations. There is one projector P for each generalized factor F, obtained from the Hasse diagram just as in Section 3 in, so that the effect of P on any vector is still achieved by a straightforward sequence of averaging operations and subtractions. It is possible for some of these projectors to be zero. Structures derived from tiers belong to this class. Another common source of structure is an association scheme : for example, the triangular scheme for all unordered pairs from a set of parental types, which is appropriate in a diallel experiment with no self-crosses when the cross (i, j) is regarded as the same as the cross (j, i). Then the matrices P are the minimal idempotents of the association algebra, and the corresponding subspaces are its common eigenspaces, Chapter 2. The effect of P is a linear combination of the operations of taking sums over associate classes. In the case of the triangular association scheme with n parental types, the subspaces have dimensions 1, n − 1 and n(n − 3)/2; they correspond to the Mean, differences between parental types and differences orthogonal to parental types, respectively. The decomposition R3 in Example 5 in comes from an association scheme with two associate classes. The set of treatments in a rectangular lattice design exhibits yet another kind of structure. Although this structure derives neither from partitions nor from an association scheme, the effect of each P is achieved by averaging and subtracting. The results here and in apply to any structure that is an orthogonal decomposition of the relevant vector space, so long as each structure can be regarded as a decomposition of VΩ. For a Tjur structure Q on a set Υ randomized to Ω, condition (4.1) in must hold in order for Q to be regarded as an orthogonal decomposition of VΩ. For structures not defined by partitions, it seems that we need QiX'XQj to be zero whenever Qi ≠ Qj, where X is the Ω × Υ design matrix. For an association scheme, this implies that the design must be equireplicate. The analogue of Theorem 5.1(a) in for association schemes is given in, Section 7.7. We admit that there are relevant experimental structures, such as neighbour relations in a field or increasing quantities of dose, that are not adequately described by an orthogonal decomposition of the space. Nonetheless, a theory which covers designed experiments where all the structures are orthogonal decompositions has wide applicability, and we limit ourselves to such structures here and in. Multiphase experiments ---------------------- Multiphase experiments are one of the commoner types of multitiered experiment. As outlined in, Section 8.1, two-phase experiments may involve almost any of the different types of multiple randomizations and, as is evident from Section [s:summary2], these differ in their assumptions. If treatments are introduced only in the first phase, then the randomizations form a chain, as in. In, Wood, Williams and Speed consider a class of such two-phase designs for which R is orthogonal in relation to the natural structure Q1 on the middle tier, and there is a refinement Q2 of Q1 such that $\mathcal{Q}\_2 \vartriangleright\mathcal{R}$ is structure balanced in relation to P. The results there are less general than ours. First, the assumptions for the second phase are in the nature of those for randomized-inclusive randomizations only. Second, the designs are restricted to those for which the design for the first phase is orthogonal. If treatments are introduced after the first phase, as in Example [eg:corn], then some form of two-to-one randomization is needed. Similarly, Brien andDemétrio describe a three-phase experiment involving composed and coincident randomizations. Further work ------------ While obtaining mixed model analyses of multitiered experiments has been described in, Section 7, and, it remains to establish their randomization analysis. The effects of intertier interactions on the analysis need to be investigated. We would like to establish conditions under which closed-form expressions are available for the Residual or Restricted Maximum Likelihood (REML) estimates of the variance components and Estimated Generalized Least Squares (EGLS) estimates of the fixed effects. Also required is a derivation of the extended algorithm described in for obtaining the ANOVA for a multitiered experiment. Furthermore, we have provided the basis for assessing a particular design for a multitiered experiment, yet general principles for designing them are still needed. 99 Bailey, R. A. (1992). Efficient semi-Latin squares. Statist. Sinica 2 413–437. Bailey, R. A. (1996). Orthogonal partitions in designed experiments. Des. Codes Cryptogr. 8 45–77. Bailey, R. A. (2004). Association Schemes: Designed Experiments, Algebra and Combinatorics. Cambridge Studies in Advanced Mathematics 84. Cambridge Univ. Press, Cambridge. Bailey, R. A. and Speed, T. P. (1986). Rectangular lattice designs: Efficiency factors and analysis. Ann. Statist. 14 874–895. Bailey, T. B. (2006). Contribution to the discussion of “Multiple randomizations” by C. J. Brien and R. A. Bailey. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 604. Bose, R. C. and Mesner, D. M. (1959). On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 30 21–38. Bose, R. C. and Shimamoto, T. (1952). Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Amer. Statist. Assoc. 47 151–284. Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics 39 53–59. Brien, C. J. and Bailey, R. A. (2006). Multiple randomizations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 571–609. Brien, C. J. and Bailey, R. A. (2009). Decomposition tables for experiments I. A chain of randomizations. Ann. Statist. 37 4184–4213. Brien, C. J. and Demétrio, C. G. B. (2009). Formulating mixed models for experiments, including longitudinal experiments. J. Agric. Biol. Environ. Stat. 14 253–280. Brien, C. J. and Payne, R. W. (1999). Tiers, structure formulae and the analysis of complicated experiments. The Statistician 48 41–52. Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York. Eccleston, J. and Russell, K. (1975). Connectedness and orthogonality in multi-factor designs. Biometrika 62 341–345. James, A. T. and Wilkinson, G. N. (1971). Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. Biometrika 58 279–294. Patterson, H. D. and Thompson, R. (1971). Recovery of inter-block information when block sizes are unequal. Biometrika 58 545–554. Tjur, T. (1984). Analysis of variance models in orthogonal designs (with discussion). Internat. Statist. Rev. 52 33–81. Wood, J. T., Williams, E. R. and Speed, T. P. (1988). Non-orthogonal block structure in two-phase designs. Aust. J. Stat. 30A 225–237. Yates, F. (1935). Complex experiments (with discussion). J. R. Statist. Soc. Suppl. 2 181–247. Decomposition tables for experiments. II. Two–one randomizations ================================================================ and We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomized-inclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used. [class=AMS]. . Introduction ============ The purpose of this paper, and its prequel, is to establish the orthogonal decomposition of the data space for experiments that involve multiple randomizations, so that the properties of proposed designs can be evaluated. In, this was done for randomizations that follow each other in a chain, as in Figure [fig:intro](a). Here, analogous results to those in are obtained for experiments in which the randomizations are two-to-one, as in Figure [fig:intro](b). In such randomizations, two different sets of objects are directly randomized to a third, as in Figures [fig:cherry], [fig:indep] and [fig:plant]. The unrandomized-inclusive, independent and coincident randomizations from are of this type. Also covered are experiments in which the randomization is two-from-one in that two different sets of objects have a single set of objects randomized to them; that is, experiments with double randomizations [see Figure [fig:intro](c)]. As in, we always denote the set of observational units by Ω, so that the data space is the set VΩ of all real vectors indexed by Ω. This data space has an orthogonal decomposition into subspaces defined by inherent factors and managerial constraints. We call this decomposition the “structure” on Ω, and identify it with the set P of mutually orthogonal idempotent matrices which project onto those subspaces. Thus if P ∈ P then P is an Ω × Ω matrix, because its rows and columns are labelled by the elements of Ω. In the setting of Figure [fig:intro](b), there are two other sets, Υ and Γ, which typically contain treatments of different types to be randomized to Ω. For example, in Figure [fig:cherry], the set of treatments (Γ) and the set of rootstocks (Υ) are randomized to the set of trees (Ω). Then VΥ is the space of all real vectors indexed by Υ, and VΓ is defined similarly. Each of the sets Υ and Γ also has a structure defined on it, the structures being orthogonal decompositions of VΥ and VΓ, respectively. These are identified with complete sets Q and R of mutually orthogonal idempotent matrices. [fig:intro] There is an immediate technical difficulty. As first defined, a matrix Q in Q is not the same size as a matrix P in P. However, the outcome of the randomization of Υ to Ω is a function f which allocates element f(ω) of Υ to observational unit ω. This function defines a subspace VΥf of VΩ isomorphic to VΥ. Similarly, the outcome of the randomization of Γ to Ω is a function g which allocates element g(ω) of Γ to observational unit ω. Thus we have a subspace VΓg of VΩ isomorphic to VΓ. From now on, we identify VΥf with VΥ, and VΓg with VΓ. We also assume that equation (4.1) in holds for both f and g, so that we may regard each matrix Q in Q and each matrix R in R as an Ω × Ω matrix without losing orthogonality or idempotence. This condition is satisfied for all equi-replicate allocations, and for many others. In it was seen that a standard two-tiered experiment has just two sets of objects, Ω and Υ say, typically observational units and treatments. To evaluate the design for such an experiment, one needs the decomposition of the data space VΩ that takes into account both P and Q. Brien and Bailey introduced the notation $\mathcal{P} \vartriangleright\mathcal{Q}$ for the set of idempotents for this decomposition, and established expressions for its elements under the assumption that Q is structure balanced in relation to P. They exhibited the decomposition in decomposition tables based on sources corresponding to the elements of P and Q. For the idempotents for two sources from different tiers, such as P in P and Q in Q, we follow James and Wilkinson in defining Q to have first-order balance in relation to P if there is a scalar λPQ such that QPQ = λPQQ. If this is satisfied and λPQ ≠ 0, then $\mathbf{P} \vartriangleright\mathbf{Q}$ is defined in to be λPQ− 1PQP, which is the matrix of orthogonal projection onto ImPQ, the part of the source P pertaining to the source Q. The scalar λPQ is called the efficiency factor; it lies in [0, 1] and indicates the proportion of the information pertaining to the source Q that is (partially) confounded with the source P. Furthermore, a structure Q is defined in to be structure balanced in relation to another structure P if (i) all idempotents from Q have first-order balance in relation to all idempotents from P; (ii) all pairs of distinct elements of Q remain orthogonal when projected onto an element of P, that is, for all P in P and all pairs of distinct Q1 and Q2 in Q, the product. If Q is structure balanced in relation to P, and P ∈ P, then the residual subspace for Q in ImP is just the orthogonal complement in ImP of all the spaces ImPQ: its matrix of orthogonal projection P ⊢ Q is given by $$\label{eq:resid} \mathbf{P} \vdash\mathcal{Q} = \mathbf{P} - {\sum\_{\mathbf{Q}\in\mathcal{Q}}\hspace\{-2pt}}' \mathbf{P} \vartriangleright\mathbf{Q},$$ where ∑Q* ∈ Qʹ means summation over all Q in Q with λPQ ≠ 0. This notation was extended in to describe the decomposition for three-tiered experiments where the two randomizations follow each other in a chain, as in composed and randomized-inclusive randomizations [see Figure [fig:intro](a)]. This involved combining the three structures P, Q and R defined on three sets of objects to yield the two equivalent decompositions $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$ and $\mathcal{P} \vartriangleright(\mathcal{Q} \vartriangleright \mathcal{R})$. It was seen that the idempotents of these decompositions could be any of the following forms: $(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright\mathbf{R}$, $\mathbf{P} \vartriangleright(\mathbf{Q} \vartriangleright\mathbf{R})$, $(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash\mathcal{R}$, $\mathbf{P} \vartriangleright(\mathbf{Q} \vdash\mathcal{R})$, and P ⊢ Q, where P, Q and R are idempotents in P, Q, R, respectively. In some cases, some idempotents in $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$ may reduce to idempotents of the form P, $\mathbf{P} \vartriangleright\mathbf{Q}$, Q, R or $\mathbf{Q} \vartriangleright\mathbf{R}$. In Sections [s:uincl]–[s:onone] of this paper, corresponding results are obtained for the two-to-one randomizations: unrandomized-inclusive, independent and coincident randomizations. It is shown that, in addition to the decompositions above, the following decompositions occur: $\mathcal{P} \vartriangleright\mathcal{R}$, $(\mathcal{P} \vartriangleright\mathcal{R}) \vartriangleright \mathcal{Q}$ and $(\mathcal{P} \vartriangleright\mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$, where “▫” denotes “the combination of compatible decompositions” in a sense defined in Section [s:onone]. Also, the list of forms of idempotents is expanded to include: $\mathbf{P} \vartriangleright\mathbf{R}$, P ⊢ R, $(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf{R}$ and (P ⊢ Q) ⊢ R. Section [s:double] deals with experiments having the only two-from-one randomization: double randomizations. There are differences between different types of multiple randomization in the reduced forms for the above idempotents and in the efficiency factors. Section [s:summary2] gives recommendations for when the different types of multiple randomization should be used. How the results might be applied to experiments with more than three tiers is outlined in Section [s:fourII]. We finish in Section [s:discuss] with a discussion of a number of issues that arise in the decompositions for multitiered experiments. Unrandomized-inclusive randomizations ===================================== In an experiment with unrandomized-inclusive randomizations, Υ is randomized to Ω in an initial two-tiered experiment. The unrandomized-inclusive randomization involves a third set, Γ, which is randomized to Ω taking account of the result of the first randomization. As for randomized-inclusive randomizations, the order of the two randomizations is fixed. Two functions are required to encapsulate the results of these randomizations, say f: Ω → Υ and g: Ω → Γ. For ω in Ω, f(ω) is the element of Υ assigned to ω by the first randomization, and g(ω) is the element of Γ assigned to ω by the second randomization. The set-up is represented diagrammatically in Figure [f:uInclusive]. [f:uInclusive] We consider experiments in which the structure Q on Υ is structure balanced in relation to the structure P on Ω, so that the first randomization gives the combined decomposition $\mathcal{P} \vartriangleright\mathcal{Q}$ of VΩ described in. The second randomization takes account of $\mathcal{P} \vartriangleright \mathcal{Q}$, both in the choice of systematic design and in restricting the permutations of Ω to preserve $\mathcal{P} \vartriangleright\mathcal{Q}$, so we assume that the structure R on Γ is structure balanced in relation to $\mathcal{P} \vartriangleright \mathcal{Q}$. Put IQ = ∑Q ∈ QQ, which is the matrix of orthogonal projection onto VΥ. The condition for Q to be structure balanced in relation to P can be written as IQPQ = λPQQ for all P in P and all Q in Q. Similarly, put IR = ∑R ∈ RR, which is the matrix of orthogonal projection onto VΓ. [th:RbalancedP] Let P, Q and R be orthogonal decompositions of the spaces VΩ, VΥ and VΓ, respectively, with VΥ ≤ VΩ and VΓ ≤ VΩ. If Q is structure balanced in relation to P with efficiency factors λPQ, and R is structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{Q}$ with efficiency factors $\lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}}$ and λP ⊢ Q, R, then: 1. R is structure balanced in relation to P with efficiency matrix ΛPR whose entries are $\lambda\_{\mathbf{P} \mathbf{R}} = ( \lambda\_{\mathbf{P} \vdash \mathcal{Q}, \mathbf{R}} + \sum\_{\mathbf{Q} \in\mathcal{Q}}' \lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} )$; 2. the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$ is $$\begin{aligned} & & \{(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright\mathbf{R}\dvtx\mathbf{P} \in\mathcal{P}, \mathbf {Q} \in\mathcal{Q}, \mathbf{R} \in\mathcal{R}, \lambda \_{\mathbf{PQ}} \neq0, \lambda\_{\mathbf{P} \vartriangleright \mathbf{Q}, \mathbf{R}} \neq0 \} \\ &&\qquad{}\cup \{(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash\mathcal {R}\dvtx\mathbf{P} \in\mathcal{P}, \mathbf{Q} \in\mathcal{Q}, \lambda\_{\mathbf{PQ}} \neq0 \} \\ &&\qquad{}\cup \{(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf {R}\dvtx\mathbf{P} \in\mathcal{P}, \mathbf{R} \in\mathcal{R}, \lambda\_{\mathbf{P} \vdash\mathcal{Q}, \mathbf{R}} \neq0 \} \\ &&\qquad{}\cup \{(\mathbf{P} \vdash\mathcal{Q}) \vdash\mathcal{R}\dvtx\mathbf {P} \in\mathcal{P} \}.\end{aligned}$$ (a) Because R is structure balanced in relation to the decomposition $\mathcal{P} \vartriangleright\mathcal{Q}$, we have $\mathbf{I}\_\mathcal{R} (\mathbf{P} \vartriangleright\mathbf {Q})\mathbf{R} = \lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} \mathbf{R}$ and IR(P ⊢ Q)R = λP ⊢ Q, RR, for all P in P, all Q in Q with λPQ ≠ 0, and all R in R. Now, $\mathbf{P} = (\mathbf{P} \vdash\mathcal{Q}) + \sum\_{\mathbf{Q} \in\mathcal{Q}}' \mathbf{P} \vartriangleright \mathbf{Q}$, so $$\mathbf{I}\_\mathcal{R}\mathbf{P}\mathbf{R} = \mathbf{I}\_\mathcal{R} (\mathbf{P} \vdash\mathcal{Q}) \mathbf{R} + {\sum\_{\mathbf{Q} \in\mathcal{Q}}\hspace\{-2pt}}' \mathbf{I}\_\mathcal{R} (\mathbf{P} \vartriangleright\mathbf{Q}) \mathbf{R} = \biggl( \lambda\_{\mathbf{P} \vdash\mathcal{Q}, \mathbf{R}} + {\sum\_{\mathbf{Q} \in\mathcal{Q}}\hspace\{-2pt}}' \lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} \biggr) \mathbf{R}.$$ This proves that R is structure balanced in relation to P with the given efficiency matrix. (b) Since R is structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{Q}$, we may apply the “$\vartriangleright$” operator to elements of $\mathcal{P} \vartriangleright\mathcal{Q}$ and R, to obtain $$(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright\mathbf{R} = \lambda^{-1}\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} (\mathbf{P} \vartriangleright\mathbf{Q}) \mathbf{R} (\mathbf{P} \vartriangleright\mathbf{Q}) = \lambda^{-1}\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} (\lambda^{-1}\_{\mathbf{PQ}} \mathbf{PQP}) \mathbf{R} (\lambda^{-1}\_{\mathbf{PQ}} \mathbf{PQP}).$$ Moreover, writing ∑R ∈ R\* to mean summation over R ∈ R with $\lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} \neq0$, applying equation ([eq:resid]) to $\mathbf{P} \vartriangleright \mathbf{Q}$ and R gives $$(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash\mathcal{R} = \mathbf{P} \vartriangleright\mathbf{Q} - {\sum\_{\mathbf{R} \in \mathcal{R}}\hspace\{-2pt}}^\ (\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright\mathbf{R}.$$ Similarly, $$(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf{R} = \lambda^{-1}\_{\mathbf{P} \vdash\mathcal{Q}, \mathbf{R}} (\mathbf{P} \vdash\mathcal{Q}) \mathbf{R} (\mathbf{P} \vdash \mathcal{Q})$$ and $$(\mathbf{P} \vdash\mathcal{Q}) \vdash\mathcal{R} = \mathbf{P} \vdash\mathcal{Q} - {\sum\_{\mathbf{R} \in\mathcal{R}}\hspace\{-2pt}}^\ (\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf{R}.$$ Thus, using Definition 4 in, the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$ is as given. The expression for $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$ in Theorem [th:RbalancedP](b) differs from that in equation (5.1) of because $(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf{R}$ is zero for composed and randomized-inclusive randomizations, but may not be zero for unrandomized-inclusive randomizations. For simplicity, we write the one-dimensional space for the Mean as V0, with projector P0 = Q0 = R0 = n− 1J, where n = ∣Ω∣ and J is the n × n all-1 matrix. As Brien and Bailey show, unrandomized-inclusive randomizations are common in superimposed experiments. In such an experiment, it may well be the case that VΓ ∩ V0⊥ is orthogonal to every $\mathbf{P} \vartriangleright \mathbf{Q}$ of the decomposition $\mathcal{P} \vartriangleright \mathcal{Q}$. In this case, the decomposition has the simpler form given by Corollary [eq:u2dec]. [eq:u2dec] Suppose that Q is structure balanced in relation to P and that R is structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{Q}$. If $(\mathbf{P} \vartriangleright\mathbf{Q})\mathbf{R} = \mathbf{0}$ for all P in P, all Q in Q and all R in R \ {R0}, then $$\begin{aligned} (\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R} & = & \{(\mathbf{P} \vartriangleright\mathbf{Q})\dvtx\mathbf{P} \in \mathcal{P}, \mathbf{Q} \in\mathcal{Q}, \lambda\_{\mathbf{PQ}} \neq0 \} \\ & & {}\cup \{(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf {R}\dvtx\mathbf{P} \in\mathcal{P}, \mathbf{R} \in\mathcal{R}, \lambda\_{\mathbf{P} \mathbf{R}} \neq0 \} \\ & & {}\cup \{(\mathbf{P} \vdash\mathcal{Q}) \vdash\mathcal{R}\dvtx\mathbf {P} \in\mathcal{P} \}.\end{aligned}$$ If $\lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} = 0$ for all Q ∈ Q, then λPR = λP ⊢ Q, R. If this is true for all R, then $(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash\mathcal{R} = \mathbf{P} \vartriangleright\mathbf{Q}$. The result follows. [th:qrzero] Suppose that Q is structure balanced in relation to P, and let P ∈ P \ {P0}. The following conditions are equivalent. $ (\mathbf{P} \vartriangleright\mathbf{Q}) \mathbf{I}\_{\mathcal {R}}={\mathbf 0}$ for all Q in Q with λPQ ≠ 0. QPR = 0 for all Q in Q and all R in R. IQPIR = 0. If these are satsified for all P in P \ {P0}, then VΥ ∩ V0⊥ is orthogonal to VΓ ∩ V0⊥, and all combinations of elements of Υ with elements of Γ occur on Ω. If λPQ = 0 then QP = 0 so QPR = 0. If λPQ ≠ 0 then $\mathbf{QPR} = \lambda^{-1}\_{\mathbf{PQ}}\times \mathbf{I}\_{\mathcal{Q}} \mathbf{PQP} \mathbf{I}\_{\mathcal{R}}\mathbf{R} = \mathbf{I}\_{\mathcal{Q}} (\mathbf{P}\vartriangleright\mathbf{Q}) \mathbf{I}\_{\mathcal{R}}\mathbf{R}$. Condition (i) implies that all these terms are zero, which implies condition (ii). Summing QPR over all Q and all R gives IQPIR, so condition (ii) implies condition (iii). Finally, if λPQ ≠ 0 then $(\mathbf{P} \vartriangleright \mathbf{Q}) \mathbf{I}\_\mathcal{R} = \lambda\_{\mathbf{PQ}}^{-1} \mathbf{PQP} \mathbf{I}\_\mathcal{R} = \lambda\_{\mathbf{PQ}}^{-1} \mathbf{PQ} (\mathbf{I}\_\mathcal{Q} \mathbf{P} \mathbf{I}\_\mathcal{R})$, so condition (iii) implies condition (i). Summing condition (iii) over all P in P \ {P0} gives 0 = IQ(IP − P0)IR = IQIR − IQP0IR = (IQ − Q0)(IR − R0), since P0 = Q0 = R0. This shows that VΥ ∩ V0⊥ is orthogonal to VΓ ∩ V0⊥. This implies that VΥ ∩ VΓ = V0, so Proposition 2 of shows that the Universe is the only partition marginal to both Υ and Γ considered as factors on Ω. Then orthogonality and Proposition 3 of show that all combinations of Υ and Γ occur on Ω. The conditions in Lemma [th:qrzero] are a general form of adjusted orthogonality. [(Superimposed experiment in a row-column design)] [eg:SuperStruct] The initial experiment in Example 10 in is a randomized complete-block design to investigate cherry rootstocks: there are three blocks of ten trees each, and there are ten types of rootstock. Many years later, a set of virus treatments is superimposed on this, using the extended Youden square in Table [tab:cherry]. [b] [tab:cherry] @lccccccccccc@ & & & & & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 Blocks & I & A & B & A & C & D & C & B & E & E & D & II & D & E & B & D & E & A & C & C & A & B & III & E & A & C & E & B & D & D & B & C & A This “square” is a 3 × 10 rectangle whose rows correspond to Blocks and columns to Rootstocks. Each of the five treatments occurs twice in each Block (row), while their disposition in Rootstocks (columns) is that of a balanced incomplete-block design. The sets of objects for this experiment are trees, rootstocks and treatments. Figure [fig:cherry] shows both randomizations. [fig:cherry] For this example, using the notation for sources in, but writing PMean as P0, the three structures are P = {P0, PB, PT[B]}, Q = {Q0, QR} and R = {R0, RV}. We have $\mathcal{P} \vartriangleright\mathcal{Q} = \{\mathbf{P}\_0 \vartriangleright\mathbf{Q}\_0, \mathbf {P}\_{\mathrm{B}}, \mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf{Q}\_{\mathrm{R}}, \mathbf{P}\_{\mathrm {T}[\mathrm{B}]} \vdash\mathcal{Q} \}$, with $\mathbf{P}\_0 \vartriangleright\mathbf{Q}\_0 = \mathbf{P}\_0$, $\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright \mathbf{Q}\_{\mathrm{R}} = \mathbf{Q}\_{\mathrm{R}}$ and PT[B] ⊢ Q = PT[B] − QR. See the first two columns of Table [tab:ANOVASuper]. [b] [tab:ANOVASuper] @lrclrcllr@ && && && && & && & && & source & Mean & 1 && Mean & 1 & & & Mean & 1 Blocks & 2 Trees[Blocks] & 27 && Rootstocks& 9 && $\frac{1}{6}$ & Viruses & 4 & & & & &&&Residual & 5 & & & & && Residual & 18 && $\frac{5}{6}$ &Viruses & 4 & & & & &&& Residual & 14 The efficiency factors for the structure on treatments in relation to the joint decomposition of trees and rootstocks are derived from the extended Youden square. Viruses are orthogonal to Blocks, which means that PBRV = 0, and hence $\mathbf{P}\_{\mathrm{T[\mathrm{B}]}} \mathbf {R}\_{\mathrm{V}} = \mathbf{R}\_{\mathrm{V}}$. Viruses have first-order balance in relation to Rootstocks, with λR, V = 1/6. Hence $\mathbf{R}\_{\mathrm{V}}(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright \mathbf{Q}\_{\mathrm{R}}) \mathbf{R}\_{\mathrm{V}} = \mathbf{R}\_{\mathrm{V}}\mathbf{Q}\_{\mathrm{R}} \mathbf{R}\_{\mathrm {V}} = (1/6) \mathbf{R}\_{\mathrm{V}}$, and so $\lambda\_{\mathrm{T}[\mathrm{B}]\vartriangleright\mathrm{R}, \mathrm{V}} = 1/6$. Similarly, λT[B] ⊢ Q, V = 5/6. Theorem [th:RbalancedP] shows that the structure on treatments is orthogonal in relation to the structure on trees since $$\lambda\_{\mathrm{T} [\mathrm{B}], \mathrm{V}} = \lambda\_{\mathrm{T} [\mathrm{B}] \vartriangleright\mathrm{R}, \mathrm{V}} + \lambda\_{\mathrm{T}[\mathrm{B}] \vdash \mathcal{Q}, \mathrm{V}} = \tfrac{1}{6} + \tfrac{5}{6} = 1.$$ To obtain the full decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$, take $\mathcal{P} \vartriangleright\mathcal{Q}$ and refine it by R. In this experiment VΓ ∩ V0⊥ is not orthogonal to VΥ ∩ V0⊥, because the Viruses source is not orthogonal to Rootstocks. This leads to nonorthogonality between R and $\mathcal{P} \vartriangleright\mathcal{Q}$. In particular, the Viruses source is not orthogonal to $\mathrm{Trees} [\mathrm{Blocks}] \vartriangleright\mathrm{Rootstocks}$. Consequently, the decomposition is given by Theorem [th:RbalancedP](b) rather than Corollary [eq:u2dec]. The full decomposition of Vtrees, that contains six elements, one for each line in the decomposition table, is in Table [tab:ANOVASuper]: $$(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R} = \left\{ \begin{array}{l} (\mathbf{P}\_0 \vartriangleright\mathbf{Q}\_0) \vartriangleright\mathbf{R}\_0, \mathbf{P}\_{\mathrm{B}}, \\ \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr) \vartriangleright\mathbf{R}\_{\mathrm{V}}, \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr) \vdash\mathcal{R}, \\ \bigl(\mathbf{P}\_{\mathrm {T}[\mathrm{B}]} \vdash\mathcal{Q}\bigr) \vartriangleright\mathbf {R}\_{\mathrm{V}}, \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash \mathcal{Q}\bigr) \vdash\mathcal{R} \end{array} \right \}$$ with $$\begin{aligned} (\mathbf{P}\_0 \vartriangleright\mathbf{Q}\_0) \vartriangleright \mathbf{R}\_0 &=& \mathbf{P}\_0 = \mathbf{Q}\_0 = \mathbf{R}\_0, \\ \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr) \vartriangleright\mathbf{R}\_{\mathrm{V}} &=& 6\bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]}\vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr)\mathbf{R}\_{\mathrm{V}} \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]}\vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr) \\ &=& 6\mathbf{Q}\_{\mathrm{R}} \mathbf{R}\_{\mathrm{V}} \mathbf {Q}\_{\mathrm{R}} = \mathbf{Q}\_{\mathrm{R}} \vartriangleright\mathbf{R}\_{\mathrm {V}}, \\ \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr) \vdash\mathcal{R} &=& \mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{R}} - \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{R}}\bigr) \vartriangleright\mathbf{R}\_{\mathrm{V}} \\ &=& \mathbf{Q}\_{\mathrm{R}} - \mathbf{Q}\_{\mathrm{R}} \vartriangleright \mathbf{R}\_{\mathrm{V}},\\ \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash\mathcal{Q}\bigr) \vartriangleright\mathbf{R}\_{\mathrm{V}} &=& \lambda^{-1}\_{\mathrm{T}[\mathrm{B}]\vdash\mathcal {Q},\mathrm{V}} \bigl( \mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash\mathcal {Q}\bigr)\mathbf{R}\_{\mathrm{V}} \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash\mathcal{Q}\bigr), \\ \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash\mathcal{Q}\bigr) \vdash\mathcal{R} &=& \mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash\mathcal{Q} - \bigl(\mathbf{P}\_{\mathrm{T}[\mathrm{B}]} \vdash\mathcal{Q}\bigr) \vartriangleright\mathbf{R}\_{\mathrm{V}}.\end{aligned}$$ As expected, this decomposition does contain a nontrivial idempotent of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright \mathbf{R}$. Also, unlike the chain randomizations in, it contains an idempotent of the form $(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright\mathbf{R}$. The efficiency factors are recorded in the decomposition in Table [tab:ANOVASuper], which shows that the Viruses source is partly confounded with both Rootstocks and the part of Trees[Blocks] that is orthogonal to Rootstocks. A consequence of this is that four Rootstocks degrees of freedom cannot be separated from Virus differences. However, there are five Rootstocks degrees of freedom that are orthogonal to Virus differences. Further, while the Viruses source has first-order balance in relation to Rootstocks, the reverse is not true. Independent or coincident randomizations ======================================== For independent or coincident randomizations, two sets of objects are randomized to the third; thus we could have Γ and Υ randomized to Ω. Two functions are needed to encapsulate the results of these randomizations, say f: Ω → Γ and g: Ω → Υ. The set-up is represented diagrammatically in Figure [f:Coincide]. A particular feature of these randomizations is that there is no intrinsic ordering of Γ and Υ, because neither randomization takes account of the outcome of the other. Associated with Ω, Υ and Γ are the decompositions P, Q and R. We assume that Q and R are both structure balanced in relation to P. [f:Coincide] [fig:indep] The difference between coincident and independent randomizations is that, for coincident randomizations, there are sources from the two randomized tiers which are both (partly) confounded with the same source in the unrandomized tier. For independent randomizations this does not occur (apart from the Mean). Independent randomizations -------------------------- For a pair of independent randomizations, the two functions are randomized by two permutations chosen independently from the same group of permutations of Ω. The precise definition of independence, which we were unable to give in, is that the conditions in Lemma [th:qrzero] are satisfied, for all P in P \ {P0}, for all possible outcomes of the two randomizations. If λPQλPR ≠ 0 then some outcomes will have QPR ≠ 0, violating these conditions. Hence independent randomizations require that λPQλPR = 0 for all Q in Q and all R in R unless P = P0. Lemma [th:qrzero] shows that, if Q and R are both structure balanced in relation to P, then they are also structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{R}$ and $\mathcal{P} \vartriangleright\mathcal{Q}$, respectively, with $\lambda\_{\mathbf{P} \vartriangleright\mathbf{Q}, \mathbf{R}} = \lambda\_{\mathbf{P} \vartriangleright\mathbf{R}, \mathbf{Q}} = 0$ unless P = P0, Q = Q0 and R = R0. Therefore $$\begin{aligned} \label{eq:decindep} (\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R} &=& (\mathcal{P} \vartriangleright\mathcal{R}) \vartriangleright \mathcal{Q}\nonumber\\ &=& \{\mathbf{P} \vartriangleright\mathbf{Q}\dvtx\mathbf{P}\in \mathcal{P}, \mathbf{Q}\in\mathcal{Q}, \lambda\_{\mathbf {PQ}}\ne0 \} \nonumber\\ && {} \cup \{\mathbf{P} \vdash\mathcal{Q}\dvtx\mathbf {P} \in\mathcal{P}, \mathbf{P}\mathbf{I}\_\mathcal{Q} \ne\mathbf {0} \} \nonumber\\[-9pt]\\[-9pt] && {} \cup \{\mathbf{P} \vartriangleright\mathbf {R}\dvtx\mathbf{P}\in\mathcal{P}, \mathbf{R}\in\mathcal{R}, \lambda\_{\mathbf{PR}}\ne0 \} \nonumber\\ && {} \cup \{\mathbf{P} \vdash\mathcal{R}\dvtx\mathbf {P} \in\mathcal{P}, \mathbf{P}\mathbf{I}\_\mathcal{R} \ne\mathbf {0} \} \nonumber\\ && {} \cup \{\mathbf{P}\dvtx\mathbf{P} \in\mathcal {P}, \mathbf{P}\mathbf{I}\_\mathcal{Q} = \mathbf{P}\mathbf {I}\_\mathcal{R} = \mathbf{0} \}.\nonumber\end{aligned}$$ As outlined in, Section 8.5, wherever possible we reduce two independent randomizations to a single randomization. However, as noted in, Section 4.3, this is not always possible—for example, when it is not physically possible to do them simultaneously. [(Superimposed experiment using split plots)] [eg:SuperSplit] Example 6 in is a superimposed experiment in which the second set of treatments (fertilizers) is randomized to subunits (trees) of the original experimental units (plots). The randomizations are independent, being carried out at different times and with the later one taking no account of the earlier one except to force fertilizers to be orthogonal to rootstocks. See Figure [fig:indep]. Table [tab:indep] shows the decomposition. [tab:indep] @lclclc@ & & & & source & d.f. & source & d.f. & source & d.f. Mean & 1 & Mean & 1 & Mean & 1 Blocks & b − 1 Plots[B] &b(r − 1) & Rootstocks & r − 1 & & Residual & (b − 1)(r − 1) Trees[P ∧ B] & br(t − 1) & & & Fertilizers & t − 1 & & & & Residual & (br − 1)(t − 1) In this example the independence of the randomizations implies that $(\mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \vartriangleright \mathbf{Q}\_{\mathrm{R}}) \vartriangleright\mathbf{R}\_{\mathrm{F}} = \mathbf{0}$ and so the conditions in Lemma [th:qrzero] are satisfied. Coincident randomizations ------------------------- For coincident randomizations, there are idempotents P in P \ {P0}, Q in Q \ {Q0} and R in R \ {R0} such that PQ and PR are both nonzero. If ImPQ and ImPR are both proper subspaces of ImP, then the relationship between Q and R depends on the choice of the two independent permutations used in randomizing Υ and Γ to Ω; restricting one of the randomizations to preserve the relationship would make the multiple randomizations unrandomized inclusive rather than coincident. On the other hand, if ImPQ = ImP then ImPR is always contained in ImPQ. If Q is structure balanced in relation to P and ImPQ = ImP, then $\mathbf{P}\vartriangleright\mathbf{Q} = \mathbf{P}$ and the two sources corresponding to Q and P have the same number of degrees of freedom. The condition for coincident randomizations hinted at in, Section 4.2, is precisely that $$\label{eq:coincond} \begin{tabular}{p{325pt}} for all $\mathbf{P}$ in $\mathcal{P}$, $\mathbf{Q}$ in $\mathcal{Q}$ and $\mathbf{R}$ in $\mathcal{R}$, if $\mathbf{PQ}$ and $\mathbf{PR}$ are both nonzero then one of $\mathbf{P} \vartriangleright \mathbf{Q}$ and $\mathbf{P} \vartriangleright\mathbf{R}$ is equal to $\mathbf{P}$. \end{tabular}\hspace\{-35pt}$$ A special, commonly occurring, case arises when Q and R can be assigned to the two randomized sets of objects such that the following condition is satisfied: $$\label{eq:coin} \begin{tabular}{p{310pt}} for all $\mathbf{P}$ in $\mathcal{P}$ and $\mathbf{Q}$ in $\mathcal{Q}$, if $\mathbf{PQ}$ and $\mathbf{P}\mathbf{I}\_{\mathcal{R}}$ are both nonzero then\break $\mathbf{P}\vartriangleright\mathbf{Q} = \mathbf{P}$. \end{tabular}\hspace\{-30pt}$$ [th:RbalancedPQ] If Q and R are both structure balanced in relation to P and condition ([eq:coin]) is satisfied then R is structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{Q}$, with $\lambda\_{\mathbf {P}\vartriangleright \mathbf{Q}, \mathbf{R}} = \lambda\_{\mathbf{PR}}$ if λPQ ≠ 0 and λP ⊢ Q, R = λPR if P ⊢ Q ≠ 0. Moreover, the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright\mathcal{R}$ is $$\begin{aligned} \label{eq:cdec} & & \{\mathbf{P} \vartriangleright\mathbf{R}\dvtx\mathbf{P} \in \mathcal{P}, \mathbf{R} \in\mathcal{R}, \lambda\_{\mathbf{PR}} \neq0 \}\nonumber\\ &&\qquad{}\cup \{(\mathbf{P} \vdash\mathcal{R})\dvtx\mathbf{P} \in\mathcal {P}, \mathbf{PI}\_{\mathcal{R}}\ne0 \} \nonumber\\[-8pt]\\[-8pt] &&\qquad{}\cup \{(\mathbf{P} \vartriangleright\mathbf{Q}) \dvtx\mathbf{P} \in \mathcal{P}, \mathbf{Q} \in\mathcal{Q}, \lambda\_{\mathbf{PQ}} \neq0, \mathbf{PI}\_{\mathcal{R}}=0 \} \nonumber\\ &&\qquad{} \cup \{(\mathbf{P} \vdash\mathcal{Q})\dvtx\mathbf{P} \in\mathcal {P}, \mathbf{PI}\_{\mathcal{R}}=0 \}. \nonumber\end{aligned}$$ If λPR = 0 then PR = 0, so $(\mathbf{P} \vartriangleright\mathbf{Q}) \mathbf{R} = \mathbf {0}$ for all Q with λPQ ≠ 0, and hence (P ⊢ Q)R = 0. Suppose that PIR ≠ 0. Then either PIQ = 0 or there is a unique Q in Q with λPQ ≠ 0, which satisfies $\mathbf{P} = \mathbf{P} \vartriangleright\mathbf{Q}$. In the first case, P = P ⊢ Q: therefore IR(P ⊢ Q)R = IRPR = λPRR, and $(\mathbf{P}\vdash\mathcal{Q}) \vartriangleright\mathbf{R} = \mathbf{P} \vartriangleright\mathbf{R}$. In the second case, $\mathbf{I}\_\mathcal{R} (\mathbf{P} \vartriangleright\mathbf{Q}) \mathbf {R} = \mathbf{I}\_\mathcal{R} \mathbf{P} \mathbf{R} = \lambda\_{\mathbf{PR}} \mathbf{R}$ and $(\mathbf{P}\vartriangleright\mathbf{Q}) \vartriangleright \mathbf{R} = \mathbf{P} \vartriangleright\mathbf{R}$. [(A plant experiment)] [eg:PlantExp] Example 5 in is an experiment to investigate five varieties and two spray regimes. Each bench has one spray regime and two seedlings of each variety. See Figure [fig:plant]. The sets are positions, seedlings and regimes. The diagram includes the pseudofactor S1 for Seedlings[Varieties], which indexes the groups of seedlings randomized to the different benches. Although the factor Seedlings is nested in Varieties, S1 is not, because each of its levels is taken across all levels of Varieties. [fig:plant] The Hasse diagrams displaying the structures for this experiment are in Figure [f:HassePlant]. The decomposition is in Table [tab:ANOVAPlant], where the source Seedlings[Varieties] ⊢ S1 is the part of Seedlings[Varieties] which is orthogonal to the source S1. [f:HassePlant] [b] [tab:ANOVAPlant] @lrlrlr@ & & & & source & & & & & Mean & 1 & Mean & 1 & Mean & 1 Benches & 5 & S1 & 5 & Regimes &1 & & & & Residual &4 Positions[Benches] & 54 & Varieties & 4 & & Seedlings[Varieties] ⊢ S1 & 50 The full decomposition of Vpositions in this case contains five elements and is $$(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R} = \left\{ \begin{array}{l} (\mathbf{P}\_0 \vartriangleright\mathbf{Q}\_0) \vartriangleright\mathbf{R}\_0, \\ (\mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf{Q}\_{\mathrm{S}\_1}) \vartriangleright \mathbf{R}\_{\mathrm{R}}, (\mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf{Q}\_{\mathrm{S}\_1}) \vdash\mathcal{R}, \\ \mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{V}}, \mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \vartriangleright\mathbf{Q}\_{\mathrm{S}[\mathrm{V}] \vdash\mathrm {S}\_1} \end{array} \right\} .$$ This experiment clearly meets condition ([eq:coin]), because the only source for positions which is nonorthogonal to sources from both of the randomized tiers is the Benches source, and the five-dimensional pseudosource S1 is equal to Benches. That is, $\mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf {Q}\_{\mathrm {S}\_1} = \mathbf{P}\_{\mathrm{B}} = \mathbf{Q}\_{\mathrm{S}\_1}$. The other source nonorthogonal to Benches is the one-dimensional source Regimes, which is a proper subspace of the Benches source, and so $(\mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf {Q}\_{\mathrm {S}\_1}) \vartriangleright\mathbf{R}\_{\mathrm{R}} = \mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf{R}\_{\mathrm{R}} = \mathbf{R}\_{\mathrm{R}}$. Consequently, the elements of the full decomposition can be written as follows: $$(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R} = \left\{ \begin{array}{l} \mathbf{P}\_0, \mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf{R}\_{\mathrm{R}}, \mathbf{P}\_{\mathrm {B}} \vdash\mathcal{R}, \\ \mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \vartriangleright\mathbf{Q}\_{\mathrm{V}}, \mathbf{P}\_{\mathrm {P}[\mathrm{B}]} \vartriangleright\mathbf{Q}\_{\mathrm{S}[\mathrm {V}] \vdash\mathrm{S}\_1} \end{array} \right\} .$$ On noting that PBRR = RR, PP[B]QV = QV and PP[B]QS[V] ⊢ S1 = QS[V] ⊢ S1, the decomposition further reduces to $$(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R} = \bigl\{\mathbf{P}\_0, \mathbf{R}\_{\mathrm{R}}, \mathbf {P}\_{\mathrm{B}} - \mathbf{R}\_{\mathrm{R}}, \mathbf{Q}\_{\mathrm {V}}, \mathbf{Q}\_{\mathrm{S}[\mathrm{V}] \vdash\mathrm {S}\_1} \bigr\}.$$ Decomposition ([eq:cdec]) is convenient for algorithms, because it is $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$, like the decompositions in Theorem 5.1(d) in, Theorem [th:RbalancedP](b), Corollary [eq:u2dec] and equation ([eq:decindep]). However, it gives the false impression that the decomposition of VΩ must have P refined by Q, then $\mathcal{P} \vartriangleright\mathcal{Q}$ refined by R, suggesting that Q and R have different roles. Moreover, condition ([eq:coin]) does not hold for all pairs of coincident randomizations. We therefore introduce another joint decomposition that emphasizes the symmetry between Q and R. Let B and C be orthogonal decompositions of the same space VΩ. Then B is compatible with C if BC = CB for all B in B and all C in C. Lemma 2.4 in shows that if B and C are compatible then the nonzero products BC, for B in B and C in C, give another orthogonal decomposition of VΩ, which is a refinement of both B and C. If B and C are orthogonal decompositions of VΩ which are compatible with each other, then the decomposition B ▫ C of VΩ is defined to be $$\mathcal{B} {\,\square\,}\mathcal{C} = \{\mathbf{BC}\dvtx\mathbf{B} \in\mathcal{B}, \mathbf{C} \in\mathcal{C}, \mathbf{BC} \ne \mathbf{0} \}.$$ Thus B ▫ C = C ▫ B. Moreover, if B and C are also both compatible with D, then B ▫ C is compatible with D, B is compatible with C ▫ D, and (B ▫ C) ▫ D = B ▫ (C ▫ D). Hence if B1, …, Bm are pairwise compatible then there is no need for parentheses in defining B1 ▫ B2 ▫ ⋯ ▫ Bm. This decomposition could be referred to as “B1 combined with B2 combined with ⋯ combined with Bm.” [th:commute] If PQPRP is symmetric for all P in P, all Q in Q and all R in R, then $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright\mathcal{R}$. If PQPRP is symmetric then PQPRP = PRPQP. Hence if λPQ × λPR ≠ 0 then ${(\mathbf{P} \vartriangleright\mathbf{Q})}{ (\mathbf{P} \vartriangleright\mathbf{R})} = \lambda\_{\mathbf{PQ}}^{-1} \lambda\_{\mathbf{PR}}^{-1} \mathbf{PQPPRP} =\break \lambda\_{\mathbf{PQ}}^{-1} \lambda\_{\mathbf{PR}}^{-1} \mathbf{PRPPQP} = (\mathbf{P} \vartriangleright\mathbf{R}) (\mathbf{P} \vartriangleright \mathbf{Q})$. Thus if $\mathbf{P} \vartriangleright\mathbf{Q}$ is defined then it commutes with P and with every $\mathbf{P} \vartriangleright \mathbf{R}$, so it commutes with P ⊢ R. Similarly, if $\mathbf{P} \vartriangleright\mathbf{R}$ is defined then it commutes with P ⊢ Q. Now the same argument shows that P ⊢ Q commutes with P ⊢ R. If Pi and Pj are different elements of P then $\mathbf{P}\_i \vartriangleright\mathbf{Q}$ and Pi ⊢ Q commute with $\mathbf{P}\_j \vartriangleright\mathbf{R}$ and Pj ⊢ R, because all products are zero. Hence $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright\mathcal{R}$. [th:compat] If the conditions in Lemma [th:qrzero] are, or condition ([eq:coincond]) is, satisfied, then $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright\mathcal{R}$. The first conditions imply that QPR = 0 or P = Q = R = P0. The second implies that QPR = 0 or PQP = λPQP or PRP = λPRP. In each case, PQPRP is symmetric, so Lemma [th:commute] completes the proof. Thus the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$, which is symmetric in Q and R, can be used for coincident or independent randomizations, or for unrandomized-inclusive randomizations which satisfy the conditions in Lemma [th:qrzero]. It is the same as decomposition ([eq:cdec]) for coincident randomizations when condition ([eq:coin]) holds, the same as the decomposition in Corollary [eq:u2dec] for unrandomized-inclusive randomizations when the conditions in Lemma [th:qrzero] hold, and the same as decomposition ([eq:decindep]) for independent randomizations. Condition ([eq:coincond]) shows that, for a pair of coincident randomizations, each idempotent in $(\mathcal{P} \vartriangleright \mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$ has one of the following forms: P, $\mathbf{P} \vartriangleright\mathbf{Q}$, $\mathbf{P} \vartriangleright \mathbf{R}$, P ⊢ Q or P ⊢ R. If a pair of coincident randomizations does not satisfy condition ([eq:coin]), then it may be possible to refine R to, say, R2 in such a way that R2 is structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{Q}$, so that the decomposition in Theorem [th:RbalancedP](b) can be used. It is possible if R = P whenever $\mathbf{P} \vartriangleright\mathbf{R} = \mathbf{P}$. [(Continued)] As already noted, this example satisfies condition ([eq:coin]), so $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright\mathcal{R}$. Here $$\begin{aligned} \mathcal{P} \vartriangleright\mathcal{Q} & = & \bigl\{\mathbf{P}\_0 \vartriangleright\mathbf{Q}\_0, \mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf{Q}\_{\mathrm{S}\_1}, \mathbf{P}\_{\mathrm {P}[\mathrm{B}]} \vartriangleright\mathbf{Q}\_{\mathrm{V}}, \mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \vartriangleright\mathbf {Q}\_{\mathrm{S}[\mathrm{V}] \vdash\mathrm{S}\_1} \bigr\}\\ & = & \bigl\{\mathbf{P}\_0, \mathbf{P}\_{\mathrm{B}}, \mathbf {Q}\_{\mathrm{V}}, \mathbf{Q}\_{\mathrm{S}[\mathrm{V}] \vdash \mathrm{S}\_1} \bigr\}\end{aligned}$$ and $$\begin{aligned} \mathcal{P} \vartriangleright\mathcal{R} &=& \bigl\{\mathbf{P}\_0 \vartriangleright\mathbf{R}\_0, \mathbf{P}\_{\mathrm{B}} \vartriangleright\mathbf{R}\_{\mathrm{R}}, \mathbf{P}\_{\mathrm {B}} \vdash\mathcal{R}, \mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \bigr\}\\ & = & \bigl\{\mathbf{P}\_0, \mathbf{R}\_{\mathrm{R}}, \mathbf {P}\_{\mathrm{B}} - \mathbf{R}\_{\mathrm{R}}, \mathbf{P}\_{\mathrm {P}[\mathrm{B}]} \bigr\}.\end{aligned}$$ Then $$\begin{aligned} (\mathcal{P} \vartriangleright\mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright \mathcal{R}) & =& \bigl\{\mathbf{P}\_0^2, \mathbf{P}\_{\mathrm{B}}\mathbf {R}\_{\mathrm{R}}, \mathbf{P}\_{\mathrm{B}}(\mathbf{P}\_{\mathrm {B}} - \mathbf{R}\_{\mathrm{R}}), \mathbf{Q}\_{\mathrm{V}}\mathbf {P}\_{\mathrm{P}[\mathrm{B}]}, \mathbf{Q}\_{\mathrm{S}[\mathrm {V}] \vdash\mathrm{S}\_1}\mathbf{P}\_{\mathrm{P}[\mathrm{B}]} \bigr\} \nonumber\\ &=& \bigl\{\mathbf{P}\_0, \mathbf{R}\_{\mathrm{R}}, \mathbf {P}\_{\mathrm{B}} - \mathbf{R}\_{\mathrm{R}}, \mathbf{Q}\_{\mathrm {V}}, \mathbf{Q}\_{\mathrm{S}[\mathrm{V}] \vdash\mathrm {S}\_1} \bigr\} \\ & = & (\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}.\end{aligned}$$ Double randomizations ===================== Double randomization is the one known type of two-from-one randomizations. In an experiment with double randomization, one set of objects is randomized to two others; thus we could have Γ randomized to Υ and to Ω. We follow the convention that the set of observational units is designated as Ω. Two functions are needed to encapsulate the results of these randomizations, say f: Ω → Γ and g: Υ → Γ. These two functions are randomized independently using two different groups of permutations. The set-up is shown in Figure [f:Double]. [f:Double] Now we obtain a subspace VΓf of VΩ and a subspace VΓg of VΥ, both isomorphic to VΓ. If ∣Υ∣ = ∣Γ∣ then VΥ = VΓg, so we may effectively identify VΥ, VΓ and VΓf. If ∣Υ∣ > ∣Γ∣ then we cannot identify VΥ with a subspace of VΩ without further information explicitly assigning an element of Υ to each observational unit in Ω. This may not be possible (see, e.g., Figure 28 in ). Thus we shall assume that ∣Υ∣ = ∣Γ∣. Associated with Ω, Υ and Γ are the decompositions P, Q and R. If R is structure balanced in relation to Q and ∣Υ∣ = ∣Γ∣, then Lemma 4.2 in shows that $\mathcal{Q}\vartriangleright\mathcal {R} = \mathcal{R}$. Therefore it suffices to have R structure balanced in relation to P. Then the overall decomposition is $\mathcal{P} \vartriangleright\mathcal{R} = \mathcal{P} \vartriangleright (\mathcal{Q} \vartriangleright \mathcal{R})$, which must be done from right to left. [(An improperly replicated rotational grazing experiment)] [eg:RotateGraze] Example 8 in is the rotational grazing trial shown in Figure [fig:improper], with Cows substituted for Animals. The double randomization of Availability results in the assignment of Cows to Paddocks, the Cows assigned to an Availability forming a single herd that is used to graze all Paddocks with the same level of Availability. The sets of objects are observational units, paddocks and treatments, and the numbers of paddocks and treatments are equal, as required. The Hasse diagrams for treatments and observational units are like the middle diagram in Figure [f:HassePlant]; that for paddocks is trivial. [fig:improper] The structures on observational units, paddocks and treatments are P = {P0, PC, PR, PC#R}, Q = {Q0, QP} and R = {R0, RA, RR, RA#R}, respectively. This leads to the decomposition $\mathcal{P} \vartriangleright(\mathcal{Q} \vartriangleright \mathcal{R})$ in Table [tab:ANOVARotateGraze]. It shows that there are no residual degrees of freedom for testing any treatment differences—hence the experiment being dubbed improperly replicated. In this case, $$\mathcal{Q} \vartriangleright\mathcal{R} = \mathcal{R} = \{\mathbf{Q}\_0\vartriangleright\mathbf{R}\_0, \mathbf {Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A}}, \mathbf {Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{R}}, \mathbf {Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A} \# \mathrm {R}} \}$$ with $$\begin{aligned} \mathbf{Q}\_0\vartriangleright\mathbf{R}\_0 &=& \mathbf{R}\_0,\qquad \mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A}} = \mathbf {R}\_{\mathrm{A}},\\ \mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{R}} &=& \mathbf {R}\_{\mathrm{R}},\qquad \mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A} \# \mathrm{R}} = \mathbf{R}\_{\mathrm{A} \#\mathrm{R}}.\end{aligned}$$ Also, PR is equal to either R or 0 for all P ∈ P and all R ∈ R. That is, R is orthogonal in relation to P. Therefore the complete decomposition for the experiment is $$\mathcal{P} \vartriangleright(\mathcal{Q} \vartriangleright\mathcal {R}) = \left\{ \begin{array}{l} \mathbf{P}\_0\vartriangleright(\mathbf {Q}\_0\vartriangleright\mathbf{R}\_0), \mathbf{P}\_{\mathrm{C}} \vartriangleright(\mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf {R}\_{\mathrm{A}}), \mathbf{P}\_{\mathrm{C}} \vdash(\mathcal{Q} \vartriangleright\mathcal{R}), \\ \mathbf{P}\_{\mathrm{R}} \vartriangleright(\mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf {R}\_{\mathrm{R}}), \mathbf{P}\_{\mathrm{C} \#\mathrm{R}} \vartriangleright(\mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf {R}\_{\mathrm{A} \# \mathrm{R}}), \mathbf{P}\_{\mathrm{C} \#\mathrm {R}} \vdash(\mathcal{Q} \vartriangleright\mathcal{R}) \end{array} \right\}$$ with $$\begin{aligned} \mathbf{P}\_0\vartriangleright\mathbf{Q}\_0\vartriangleright\mathbf {R}\_0 &=& \mathbf{P}\_0,\qquad \mathbf{P}\_{\mathrm{C}} \vartriangleright(\mathbf{Q}\_{\mathrm{P}} \vartriangleright \mathbf{R}\_{\mathrm{A}}) = \mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A}} = \mathbf {R}\_{\mathrm{A}}, \\ \mathbf{P}\_{\mathrm{C}} \vdash(\mathcal{Q} \vartriangleright \mathcal{R}) &=& \mathbf{P}\_{\mathrm{C}} -(\mathbf{Q}\_{\mathrm{P}} \vartriangleright \mathbf {R}\_{\mathrm{A}}) = \mathbf{P}\_{\mathrm{C}}- \mathbf{R}\_{\mathrm{A}}, \\ \mathbf{P}\_{\mathrm{R}} \vartriangleright(\mathbf{Q}\_{\mathrm{P}} \vartriangleright \mathbf{R}\_{\mathrm{R}}) &=& \mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{R}} = \mathbf {R}\_{\mathrm{R}}, \\ \mathbf{P}\_{\mathrm{C} \#\mathrm{R}} \vartriangleright (\mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A} \# \mathrm{R}}) &=& \mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A} \# \mathrm{R}} = \mathbf{R}\_{\mathrm{A} \#\mathrm{R}}, \\ \mathbf{P}\_{\mathrm{C} \#\mathrm{R}} \vdash(\mathcal{Q} \vartriangleright\mathcal{R}) &=& \mathbf{P}\_{\mathrm{C} \#\mathrm{R}} - (\mathbf{Q}\_{\mathrm{P}} \vartriangleright\mathbf{R}\_{\mathrm{A} \# \mathrm{R}}) = \mathbf{P}\_{\mathrm{C} \#\mathrm{R}} - \mathbf{R}\_{\mathrm{A} \#\mathrm{R}}.\end{aligned}$$ [tab:ANOVARotateGraze] @lrclrclr@ && && && && source & d.f. && source & d.f. && source & d.f. Mean & 1 && Mean & 1 && Mean & 1 Cows & 14 && Paddocks & 2 && Availability & 2 &&& & & & Residual & 12 Rotations & 3 && Paddocks & 3 && Rotations & 3 Cows # Rotations & 42 && Paddocks & 6 && Availability # Rotations &6 [-4pt] &&& & & & Residual & 36 In this example was redone as a case of randomized-inclusive randomization, using two pseudofactors PA and PR for Paddocks, aliased with Availability and Rotations, respectively. These are required if Q itself is to be structure balanced in relation to P, giving a decomposition from left to right like the one in Section 6 in. Summary ======= We have shown in and here that, under structure balance, the six different types of multiple randomization identified in all lead to orthogonal decompositions of VΩ using some of the following idempotents: P, $\mathbf{P} \vartriangleright\mathbf{Q}$, $\mathbf{P} \vartriangleright\mathbf{R}$, $(\mathbf{P} \vartriangleright \mathbf{Q}) \vartriangleright\mathbf{R}$, $\mathbf{P} \vartriangleright(\mathbf{Q} \vartriangleright\mathbf{R})$, P ⊢ Q, P ⊢ R, $(\mathbf{P} \vartriangleright \mathbf{Q}) \vdash\mathcal{R}$, $\mathbf{P} \vartriangleright(\mathbf{Q} \vdash \mathcal{R})$, $(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright \mathbf{R}$ and (P ⊢ Q) ⊢ R. The differences between the different multiple randomizations lead to differences in the reduced forms for these elements and in the efficiency factors. Composed randomizations ----------------------- If each design is structure balanced then so is the composite; the decompositions $\mathcal{P} \vartriangleright(\mathcal{Q} \vartriangleright \mathcal{R})$ and $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$ are equal, and so the decomposition may be done in either order; and there are no idempotents of the form $(\mathbf{P} \vdash\mathcal{Q}) \vartriangleright \mathbf{R}$ or (P ⊢ Q) ⊢ R. Randomized-inclusive randomizations ----------------------------------- The structures Q1 and R1 for design 1 are refined to Q and R using the pseudofactors that are necessary for the second randomization, and then the results are the same as for composed randomizations. Unrandomized-inclusive randomizations ------------------------------------- We must have R structure balanced in relation to $\mathcal{P} \vartriangleright\mathcal{Q}$; use the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$, which is done from left to right; if the conditions in Lemma [th:qrzero] hold then there are no idempotents of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright \mathbf{R}$ apart from the Mean, nor any of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash \mathcal{R}$, the decomposition $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright\mathcal{R}$, and $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright\mathcal{R} = (\mathcal{P} \vartriangleright \mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$. Independent randomizations -------------------------- The conditions in Lemma [th:qrzero] must hold; if both designs are structure balanced then each remains structure balanced after the other has been taken into account; $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright\mathcal{R}$; the decompositions $(\mathcal{P} \vartriangleright \mathcal{Q}) \vartriangleright\mathcal{R}$, $(\mathcal{P} \vartriangleright \mathcal{R}) \vartriangleright \mathcal{Q}$ and $(\mathcal{P} \vartriangleright\mathcal{Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$ are equal; and there are no idempotents of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vartriangleright \mathbf{R}$ apart from the Mean, nor any of the form $(\mathbf{P} \vartriangleright\mathbf{Q}) \vdash\mathcal{R}$. Coincident randomizations ------------------------- Condition ([eq:coincond]) must hold; $\mathcal{P} \vartriangleright\mathcal{Q}$ is compatible with $\mathcal{P} \vartriangleright \mathcal{R}$; use the decomposition $(\mathcal{P} \vartriangleright \mathcal {Q}) {\,\square\,}(\mathcal{P} \vartriangleright\mathcal{R})$, whose idempotents have the form P, $\mathbf{P} \vartriangleright \mathbf{Q}$, $\mathbf{P} \vartriangleright\mathbf{R}$, P ⊢ Q or P ⊢ R; if condition ([eq:coin]) holds, this is the same as the decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}) \vartriangleright \mathcal{R}$, which is done from left to right; otherwise, there may be a refinement of R giving a left-to-right decomposition. Double randomizations --------------------- We require that ∣Υ∣ = ∣Γ∣ and that R be structure balanced in relation to both Q and P, so that the decomposition is $ \mathcal{P} \vartriangleright\mathcal {R} = \mathcal{P} \vartriangleright(\mathcal{Q} \vartriangleright\mathcal {R})$, which is done from right to left. It appears that they can also be formulated as randomized-inclusive randomizations using pseudofactors to refine Q to Q2 for which the left-to-right decomposition $(\mathcal{P} \vartriangleright\mathcal{Q}\_2) \vartriangleright\mathcal{R}$ is correct. Structure-balanced experiments with four or more tiers ====================================================== Each experiment in Sections [s:uincl]–[s:double] involves only one type of multiple randomization, and so involves three tiers and three structures. However, multitiered experiments are not limited to this configuration. Examples 12–14 in each have four tiers and involve more than one type of multiple randomization. In general, there is the set of observational units, Ω, and each randomization adds another set of objects with its associated tier. Section 7 in shows how to deal with three or more randomizations which follow each other in a chain. Mixtures of other types of multiple randomization should be amenable to successive decompositions of the sort summarized in Section [s:summary2], so long as they are handled in the correct order. Thus we can use a recursive procedure in which each new structure refines the decomposition of VΩ obtained using structures accounted for previously. All that is required is that each successive structure should be structure balanced in relation to the previous decomposition. [fig:corn] One class of experiments with both two–one randomizations and chain randomizations consists of multiphase experiments in which different treatment factors are applied in different phases, as the following example demonstrates. [(A two-phase corn seed germination experiment)] [eg:corn] Example 12 of has the four tiers shown in Figure [fig:corn]. Here we have taken the opportunity to correct the diagram given in. The 36 Lots of grain within each Plot should be completely randomized to Plates ∧ Containers within each Interval. This will not be achieved by permuting Containers within Intervals and Plates within Intervals ∧ Containers, as implied in the rightmost panel of Figure [fig:corn]. We introduce pseudofactors L1 and L2 for Lots, with nine and four levels, respectively, like the pseudofactors for Seedlings in Example [eg:PlantExp]. The 36 Lots must be randomly allocated to the combinations of levels of L1 and L2, independently within each level of Sites ∧ Blocks ∧ Plots, so that neither pseudofactor corresponds to any inherent source of variation. At each randomization, an orthogonal design is used, so there is no difficulty in constructing the decomposition in Table [tab:corn]. Here L1[P ∧ B ∧ S] is the part of the source Lots[P ∧ B ∧ S] which is confounded with Containers[I]. The source Lots[P ∧ B ∧ S]⊢ is the part of Lots[P ∧ B ∧ S] which is orthogonal to L1[P ∧ B ∧ S]: it is confounded with Plates[C ∧ I]. [tab:corn] @lrlrlrlr@ & & & & & & source & & & & & & & Mean & 1 & Mean & 1 & Mean & 1 & Mean & 1 Intervals& 17 & Sites & 2 && & & Blocks[S] & 3 && & & Plots[B ∧ S] & 12 & Harvesters &2 & & & & Residual & 10 Containers[I] & 144 & L1[P ∧ B ∧ S] & 144 & & &Temperature & 2 & & & & & & Moistures& 2 & & & & & & T # M & 4 & & & & & & Residual & 136 Plates[C ∧ I] & 486 & Lots[P ∧ B ∧ S]⊢ & 486 Bailey suggests an analysis for this example which we reproduce in the first three columns of Table [tab:Baileycorn](a). In this, the 3-level factors Temperature and Moisture have been combined into a single 9-level Treatment factor, the intertier interactions of Sites, Harvesters and Treatments have been included, and the notation × is used in place of #. We cannot be sure, but it is plausible that he based this decomposition on the crossing and nesting relationships summarized in the formula $$\label{eq:tbb} \bigl(\mathrm{T} \\mathrm{H} \(\mathrm{S}/\mathrm{B})\bigr) /\mathrm{Q},$$ where T, H, S, B and Q represent factors for Treatments, Harvesters, Sites, Blocks and Plates, with 9, 3, 3, 2 and 4 levels, respectively. The sources derived from this are in the final column of Table [tab:Baileycorn](a), with degrees of freedom matching those in the preceding column. [tab:Baileycorn] @llrl@ (a) & source & d.f. & source from ([eq:tbb]) Phase I: field study & Site & 2 & S & Experimental error (a) & 3 & B[S] & Harvester & 2 & H &Harvester × Site & 4 & H # S & Experimental error (b) & 6 & H # B[S] [4pt] Phase II: laboratory study & Treatment & 8 & T & Treatment × Site & 16 & T # S & Experimental error (c) & 24 & T # B[S] & Treatment × Harvester & 16 & T # H & Treatment × Harvester × Site & 32 & T # H # S &Experimental error (d) & 48 & T # H # B[S] & Residual & 486& Q[T ∧ H ∧ S ∧ B] @lrlrlrlr@ (b) & & & & & & source & d.f. & source & d.f. & source & d.f. & source & d.f. Mean & 1 & Mean & 1 & Mean & 1 & Mean & 1 Intervals& 17 & Sites & 2 & & Blocks[S] & 3 && & & Plots[B ∧ S] & 12 & Harvesters & 2 & & & & H # S & 4 & & & & Residual & 6 Containers[I] & 144 & Lots[P ∧ B ∧ S]1 & 144 & & & Treatments & 8 & & & & & & T # S & 16 & & & & & & T # H & 16 & & & & & & T # H # S & 32 & & & & & & Residual & 72 Plates[C ∧ I] & 486 & Lots[P ∧ B ∧ S]⊢ & 486 Revision of Table [tab:corn] along similar lines, and with pseudosources replaced with actual sources, yields the skeleton analysis-of-variance table in Table [tab:Baileycorn](b). Note that, given Step 4 in Table 1 of, an intertier interaction will generally occur in the right-most tier that contains a main effect in the interaction. Table [tab:Baileycorn](b) differs from Table [tab:Baileycorn](a) in the following ways. 1. The rationale for the sources in Table [tab:Baileycorn](a) is unclear. We had to reverse-engineer it by producing formula ([eq:tbb]). On the other hand, the sources in Table [tab:Baileycorn](b) are based on the relationships between factors within each tier and on the confounding between sources from different tiers. 2. Table [tab:Baileycorn](a) does not show, as Table [tab:Baileycorn](b) does, the successive decomposition of the vector space indexed by the observational units. The impression given is that there is a set of sources that arise from the field phase and another set that arises from the laboratory phase. 3. Table [tab:Baileycorn](a) has four sources called “experimental error” and does not mention plates, containers, intervals, blocks, plots or lots. Hence, there is no indication of the sources of error variation. By contrast, each source called “Residual” in Table [tab:Baileycorn](b) is unambiguously identified; and the labelling shows that all terms are affected by variation from both phases. For example, the Residual for Plots[B ∧ S], labelled Experimental error (b) in Table [tab:Baileycorn](a), clearly arises from variability associated with Plots within the Sites-Blocks combinations and variability associated with Intervals. Similarly, it can be seen from Table [tab:Baileycorn](b) that the Residual in Table [tab:Baileycorn](a) arises from variability associated with Plates and Lots. 4. As discussed in, Section 7.1, the usual default is that there are no intertier interactions because such inclusions would mean that the analysis cannot be justified by the randomization used. It parallels the assumption of unit-treatment additivity in single-randomization experiments. The approach using Table [tab:corn] forces the statistician to to consult the researcher about whether intertier interactions should be included, and, if so, to justify them. Tables [tab:Baileycorn](a) and (b) include the intertier interactions of Sites, Harvesters and Treatments, which suggests that it is anticipated that Harvesters and Treatments will perform differently at different Sites. 5. Even with the addition of intertier interactions, the decompositions in Tables [tab:Baileycorn](a) and (b) are not equivalent, and so neither are the mixed models underlying them. Experimental errors (c) and (d) from Table [tab:Baileycorn](a) are combined into the Residual with 72 degrees of freedom for Lots[P ∧ B ∧ S]1 in Table [tab:Baileycorn](b). To justify an analysis based on Table [tab:Baileycorn](a), one would need to argue that unit-treatment interaction of Treatments with Blocks within Sites can be anticipated in this experiment. Discussion ========== Implications of incoherent unrandomized-inclusive randomizations ---------------------------------------------------------------- The phenomenon of incoherent unrandomized-inclusive randomizations is described in , Section 5.2.1. Essentially, when there has been a randomization to factors that are crossed, one or more of these factors become nested in the second randomization. Consider the cherry rootstock experiment in Example [eg:SuperStruct]. The trees tier gives an orthogonal decomposition of VΩ into sources Mean, Blocks and Trees[Blocks] of dimensions 1, 2 and 27, respectively, in the left-hand column of Table [tab:ANOVASuper]. Similarly, the rootstocks tier decomposes VΥ into sources Mean and Rootstocks of dimensions 1 and 9. The result of the first randomization is to make the Mean sources equal and to place the Rootstocks source inside Trees[Blocks], thus giving the finer decomposition of VΩ shown in the middle column of Table [tab:ANOVASuper]. The result of the unrandomized-inclusive randomization should be to further decompose the decomposition resulting from the first two tiers. In the extended Youden square, the source Viruses is orthogonal to Blocks but partially confounded with Rootstocks, so the Viruses source defines the decomposition in the right-hand column in Table [tab:ANOVASuper]. That is, the source Viruses further decomposes the sources Rootstocks and the Residual for Trees[Blocks], as required. In we discussed the possibility that the designer of the superimposed experiment ignores the inherent crossing of the factors Blocks and Rootstocks and randomizes Viruses to Blocks in Rootstocks in a balancedincomplete-block design. Then the randomizations are incoherent. The permutation group for the second randomization does not preserve the structure arising from the first two tiers, exhibited by the two left-most columns in Table [tab:ANOVASuper]. We can see immediately that this randomization is senseless because it destroys the Blocks subspace preserved by the first randomization. This randomization might have some appeal if no block effects had been detected during the 20 years of the original experiment, but then the analysis of the second experiment would be based on an assumed model rather than on the intratier structures. [fig:cc] [b] [tab:cc] @lrlrlrclr@ & & & & & & source & d.f. & source & d.f. & source & d.f. & eff. & source & d.f. Mean & 1 & Mean & 1 & Mean & 1 && Mean & 1 Benches & 2 & & & B[S]B & 2 Plants[B] & 9 & Soils & 3 & Soils &3 && & &Residual &6 & B[S]⊢ & 6 Altitudes & 2 & & &A[S]A & 2 A # B & 4 & && A # B[S]A # B & 4 &? & Viruses & 2 & & & & & &? & V # S & 2 A # P[B] & 18 && & A[S]⊢ & 6 && & & & &A # B[S]⊢ & 12 &?& Viruses & 2 & & & & & &?& V # S & 6 & & & & & & & Residual & 4 Other examples of incoherent unrandomized-inclusive randomizations are more complicated, and perhaps less easily detected. One is the design proposed by several authors for a split-plot experiment in which the subplot treatments are to be assigned using a row-column design. Example [eg:cc] illustrates how consideration of the decomposition table for the proposed design facilitates the design process and helps the detection of incoherence. [(Split-plots in a row–column design)][eg:cc] Example 11 in is based on the design with split-plots in a row–column design given in Cochran and Cox , Section 7.33. Diagrams for the two randomizations are given in Figure [fig:cc], with leaf treatments named as viruses for clarity, soil treatments designated as different soils for brevity, and Altitude substituted for Layer so that no two factors begin with the same letter. Two diagrams are needed, because the assumed structure on leaves changes between the randomizations, as shown in the two right-hand panels. At first sight, this experiment seems to involve unrandomized-inclusive randomizations, because soils are randomized to leaves in the first randomization, and then viruses are randomized to leaves, taking into account the location of the soils. However, the change in the assumed structure on the leaves between the two randomizations makes them incoherent rather than unrandomized-inclusive. Table [tab:cc] shows an attempt to build up a decomposition table for this design. The first two columns follow directly from the randomization in the top half of Figure [fig:cc]. The third column corresponds to the leaves tier in the bottom half of Figure [fig:cc]. When we use it to refine the decomposition given by the first two tiers, we find that the Soils source occurs in two tiers. Although this can happen in special circumstances like those in Example [eg:RotateGraze], this is already a signal that something may be wrong. We also find that the nesting, in this tier, of Benches within Soils gives a source B[S] with 8 degrees of freedom. This is the sum of the previous sources Benches and Residual in Plants[B]; these two parts are denoted B[S]B and B[S]⊢ in Table [tab:cc]. Similarly, the nesting of Altitudes within Soils, in this tier, gives sources A[S] and A # B[S] which are each the sum of two previous sources. [b] [tab:SPLwLS] @lcccccccccccc@ & & & & & & & & & & & & & & & & & & Altitude Top & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 Middle & 2 & 2 & 1 & 1 & 0 & 0 & 2 & 2 & 1 & 1 & 0 & 0 Bottom & 1 & 1 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 1 & 1 The real difficulties come when we try to incorporate the column for the viruses tier, because the location of the Viruses source depends on the outcome of the randomizations. For the outcome given in, Section 7.33, and, Example 11, the Viruses source does not have first-order balance in relation to either Altitudes # Benches or Altitudes # Plants[Benches]. The interaction Viruses # Soils has the same problem. If Altitudes # Benches is merged with Altitudes # Plants[Benches] in the decomposition table, then the analysis is orthogonal and is equivalent to that given in. However, this does not allow for consistent Altitude differences across Plants, so it removes six spurious degrees of freedom from what Cochran and Cox call “Error (b)” in, page 310. The problem is that the design for the Viruses does not respect the factor relationships established in applying the Soils. As Yates showed in, if the randomization respects Benches and Altitudes then a randomization-based model must include their interaction. What is needed is a design for a two-tiered experiment in which the twelve treatments (combinations of levels of Soils and Viruses) are randomized to leaves1 in such a way that there is a refinement of the natural decomposition of the treatments space which is structure balanced in relation to Altitudes # Benches. For example, one might choose the systematic design in Table [tab:SPLwLS] and then randomize benches, altitudes, and plants within benches. In this design the twelve treatments are arranged in a (3 × 3)/4 semi-Latin square constructed from a pair of mutually orthogonal Latin squares of order 3. The Viruses are arranged according to one square for soils s0 and s1, and according to the other square for s2 and s3. Theorem 5.4 in shows that this design is the most efficient with respect to Altitudes # Benches. Let S1 be a pseudofactor for Soils whose two levels distinguish between the first two and the last two levels of Soils. The design is structure balanced: Viruses and Viruses # S1 have efficiency factor 1/2 in Altitudes # Benches, while the rest of the interaction Viruses # Soils is orthogonal to Altitudes # Benches. It has the advantage of having 10 degrees of freedom for the Residual for Altitudes # Plants[Benches], two more than for the Cochran and Cox design. Thus construction of the decomposition table when designing the experiment can help to detect problems with a proposed design. In this case, it helped to draw attention to the incoherent randomizations, to highlight the associated problems and to give insight into how they might be redressed. Other structures ---------------- All the examples in and this paper are poset block structures, being defined by some factors and their nesting relationships, as explained in. More generally, a structure may be a Tjur structure that is defined by a family of mutually orthogonal partitions or generalized factors (see or ). Again, the generalized factors are summarized in the Hasse diagram that depicts their marginality relations. There is one projector P for each generalized factor F, obtained from the Hasse diagram just as in Section 3 in, so that the effect of P on any vector is still achieved by a straightforward sequence of averaging operations and subtractions. It is possible for some of these projectors to be zero. Structures derived from tiers belong to this class. Another common source of structure is an association scheme : for example, the triangular scheme for all unordered pairs from a set of parental types, which is appropriate in a diallel experiment with no self-crosses when the cross (i, j) is regarded as the same as the cross (j, i). Then the matrices P are the minimal idempotents of the association algebra, and the corresponding subspaces are its common eigenspaces, Chapter 2. The effect of P is a linear combination of the operations of taking sums over associate classes. In the case of the triangular association scheme with n parental types, the subspaces have dimensions 1, n − 1 and n(n − 3)/2; they correspond to the Mean, differences between parental types and differences orthogonal to parental types, respectively. The decomposition R3 in Example 5 in comes from an association scheme with two associate classes. The set of treatments in a rectangular lattice design exhibits yet another kind of structure. Although this structure derives neither from partitions nor from an association scheme, the effect of each P is achieved by averaging and subtracting. The results here and in apply to any structure that is an orthogonal decomposition of the relevant vector space, so long as each structure can be regarded as a decomposition of VΩ. For a Tjur structure Q on a set Υ randomized to Ω, condition (4.1) in must hold in order for Q to be regarded as an orthogonal decomposition of VΩ. For structures not defined by partitions, it seems that we need QiX'XQj to be zero whenever Qi ≠ Qj, where X is the Ω × Υ design matrix. For an association scheme, this implies that the design must be equireplicate. The analogue of Theorem 5.1(a) in for association schemes is given in, Section 7.7. We admit that there are relevant experimental structures, such as neighbour relations in a field or increasing quantities of dose, that are not adequately described by an orthogonal decomposition of the space. Nonetheless, a theory which covers designed experiments where all the structures are orthogonal decompositions has wide applicability, and we limit ourselves to such structures here and in. Multiphase experiments ---------------------- Multiphase experiments are one of the commoner types of multitiered experiment. As outlined in, Section 8.1, two-phase experiments may involve almost any of the different types of multiple randomizations and, as is evident from Section [s:summary2], these differ in their assumptions. If treatments are introduced only in the first phase, then the randomizations form a chain, as in. In, Wood, Williams and Speed consider a class of such two-phase designs for which R is orthogonal in relation to the natural structure Q1 on the middle tier, and there is a refinement Q2 of Q1 such that $\mathcal{Q}\_2 \vartriangleright\mathcal{R}$ is structure balanced in relation to P. The results there are less general than ours. First, the assumptions for the second phase are in the nature of those for randomized-inclusive randomizations only. Second, the designs are restricted to those for which the design for the first phase is orthogonal. If treatments are introduced after the first phase, as in Example [eg:corn], then some form of two-to-one randomization is needed. Similarly, Brien andDemétrio describe a three-phase experiment involving composed and coincident randomizations. Further work ------------ While obtaining mixed model analyses of multitiered experiments has been described in, Section 7, and, it remains to establish their randomization analysis. The effects of intertier interactions on the analysis need to be investigated. We would like to establish conditions under which closed-form expressions are available for the Residual or Restricted Maximum Likelihood (REML) estimates of the variance components and Estimated Generalized Least Squares (EGLS) estimates of the fixed effects. Also required is a derivation of the extended algorithm described in for obtaining the ANOVA for a multitiered experiment. Furthermore, we have provided the basis for assessing a particular design for a multitiered experiment, yet general principles for designing them are still needed. 99 Bailey, R. A. (1992). Efficient semi-Latin squares. Statist. Sinica 2 413–437. Bailey, R. A. (1996). Orthogonal partitions in designed experiments. Des. Codes Cryptogr. 8 45–77. Bailey, R. A. (2004). Association Schemes: Designed Experiments, Algebra and Combinatorics. Cambridge Studies in Advanced Mathematics 84. Cambridge Univ. Press, Cambridge. Bailey, R. A. and Speed, T. P. (1986). Rectangular lattice designs: Efficiency factors and analysis. Ann. Statist. 14 874–895. Bailey, T. B. (2006). Contribution to the discussion of “Multiple randomizations” by C. J. Brien and R. A. Bailey. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 604. Bose, R. C. and Mesner, D. M. (1959). On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 30 21–38. Bose, R. C. and Shimamoto, T. (1952). Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Amer. Statist. Assoc. 47 151–284. Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics 39 53–59. Brien, C. J. and Bailey, R. A. (2006). Multiple randomizations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 571–609. Brien, C. J. and Bailey, R. A. (2009). Decomposition tables for experiments I. A chain of randomizations. Ann. Statist. 37 4184–4213. Brien, C. J. and Demétrio, C. G. B. (2009). Formulating mixed models for experiments, including longitudinal experiments. J. Agric. Biol. Environ. Stat. 14 253–280. Brien, C. J. and Payne, R. W. (1999). Tiers, structure formulae and the analysis of complicated experiments. The Statistician 48 41–52. Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York. Eccleston, J. and Russell, K. (1975). Connectedness and orthogonality in multi-factor designs. Biometrika 62 341–345. James, A. T. and Wilkinson, G. N. (1971). Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. Biometrika 58 279–294. Patterson, H. D. and Thompson, R. (1971). Recovery of inter-block information when block sizes are unequal. Biometrika 58 545–554. Tjur, T. (1984). Analysis of variance models in orthogonal designs (with discussion). Internat. Statist. Rev. 52 33–81. Wood, J. T., Williams, E. R. and Speed, T. P. (1988). Non-orthogonal block structure in two-phase designs. Aust. J. Stat. 30A 225–237. Yates, F. (1935). Complex experiments (with discussion). J. R. Statist. Soc. Suppl. 2 181–247.	arxiv_0000482
the existing literature on MNL bandits, it is commonly assumed that v0 = 1 . On the other hand, assume that v0 = K [1](#fn1) to induce a tighter lower bound in terms of K. Later, we will explore how these differing assumptions create fundamentally different problems, leading to different regret lower bounds (Subsection [subsec:mainlower]). The choice response for each item i ∈ St ∪ {0} is defined as $y\_{ti} := \mathbbm{1}(c\_t = i) \in \{0,1\}$. Hence, the choice feedback variable $\yb\_t := (y\_{t0}, y\_{ti\_1}, \dots y\_{ti\_{l}}) $ is sampled from the following multinomial (MNL) distribution: $ \yb\_{t} \sim \operatorname{MNL} \{ 1, ( p\_t(0 \| S\_t, \wb^\star), \dots, p\_t(i\_l \| S\_t, \wb^\star) )\}$, where the parameter 1 indicates that $\yb\_t$ is a single-trial sample, i.e., yt0 + ∑k = 1lytik = 1. For each i ∈ St ∪ {0}, we define the noise $\epsilon\_{ti}:= y\_{ti} - p\_{t}(i \| S\_t, \wb^\star)$. Since each εti is a bounded random variable in [0, 1], εti is 1/4-sub-Gaussian. At every round t, the reward rti for each item i is also given. Then, we define the expected revenue of the assortment S as $$\begin{aligned} R\_{t}(S, \wb^\star):= \sum\_{i \in S} p\_t(i \| S, \wb^\star) r\_{ti} = \frac{ \sum\_{i \in S }\exp(x\_{ti}^\top \wb^\star )r\_{ti} }{ v\_0 \!+\!\sum\_{j \in S }\exp( x\_{tj}^\top \wb^\star )}\end{aligned}$$ and define St⋆ as the offline optimal assortment at time t when $\wb^\star$ is known a prior, i.e., $S\_{t}^\star = \argmax\_{S \in \mathcal{S}} \sum\_{i \in S} R\_{t}(S, \wb^\star)$. Our objective is to minimize the cumulative regret over the T periods: $$\begin{aligned} \Regret (\wb^\star ) = \sum\_{t=1}^T R\_{t}(S\_{t}^\star, \wb^\star) - R\_{t}(S\_{t}, \wb^\star). \end{aligned}$$ When K = 1, rt1 = 1, and v0 = 1, the MNL bandit recovers the binary logistic bandit with $R\_t(S=\{x\}, \wb^\star) = \sigma\left( x^\top \wb^\star \right) = 1/ (1 + \exp (-x^\top \wb^\star) )$, where σ( ⋅ ) is the sigmoid function. Consistent with previous works on MNL bandits , we make the following assumptions: [Bounded assumption] [assum:boundedassumption] We assume that $\\| \wb^\star \\|\_2 \leq 1$, and for all t ≥ 1, i ∈ [N], ∥xti∥2 ≤ 1 and rti ∈ [0, 1]. [Problem-dependent constant] [assum:kappa] There exist κ > 0 such that for every item i ∈ S and any S ∈ S, and all round t, $\min\_{\wb \in \mathcal{W}} p\_t(i \| S, \wb) p\_t(0 \| S, \wb) \geq \kappa$, where $\mathcal{W} = \{ \wb \in \RR^d \mid \\| \wb \\|\_2 \leq 1 \}$. In Assumption [assum:boundedassumption], we assume that the reward for each item i is bounded by a constant, allowing the norm of the reward vector to depend on K, e.g., $\\| \bm{\rho}\_{t} \\|\_2 \leq \sqrt{K}$. In contrast, assume that the norm of the reward vector $\bm{\rho}\_{t} = [r\_{t1}, \dots r\_{t\|S\_t\|}]^\top \in \RR^{\|S\_t\|}$ is bounded by a constant, independent of K, e.g., $\\| \bm{\rho}\_{t} \\|\_2 \leq 1$. Thus, our assumption regarding rewards is weaker than theirs. Assumption [assum:kappa] is common in contextual MNL bandits . Note that 1/κ depends on the maximum size of the assortment K, i.e., $1/\kappa = \BigO(K^{2} )$. One of the primary goals of this paper is to show that as the assortment size K increases, we can achieve an improved (or at least not worsened) regret bound. To this end, we design a dynamic assortment policy that enjoys improved dependence on κ. Note that our algorithm does not need to know κ a priori, whereas do. Algorithms and Main Results =========================== In this section, we begin by proving the tightest regret lower bound under uniform rewards (Subsection [subsec:mainlower]), explicitly showing the dependence on the utility for the outside option v0. We then introduce, an algorithm that achieves minimax optimality, up to logarithmic factors under uniform rewards (Subsection [subsec:mainupper]). Notably, is designed for efficiency, requiring only an O(Kd3) computation cost per iteration and an O(d2) storage cost. Finally, we establish the tightest regret lower bound and a matching minimax optimal regret upper bound (up to logarithmic factors) under non-uniform rewards (Subsection [subsec:mainnonuniform]). Regret Lower Bound under Uniform Rewards ---------------------------------------- In this subsection, we present a lower bound for the worst-case expected regret in the uniform reward setting (rti = 1). This covers all applications where the objective is to maximize the appropriate “click-through rate” by offering the assortment. [Regret lower bound, Uniform rewards] [thm:lowerbound] Let d be divisible by 4 and let Assumption [assum:boundedassumption] hold true. Suppose T ≥ C ⋅ d4(v0 + K)/K for some constant C > 0. Then, in the uniform reward setting, for any policy π, there exists a worst-case problem instance with N = Θ(K ⋅ 2d) items such that the worst-case expected regret of π is lower bounded as follows: $$\begin{aligned} \sup\_{\wb} \EE^\pi\_{\wb} \left[ \Regret(\wb) \right] = \Omega\left( \frac{\sqrt{v\_0 K}}{ v\_0 + K } \cdot d\sqrt{T} \right). \end{aligned}$$ Discussion of Theorem [thm:lowerbound]. If v0 = Θ(1), Theorem [thm:lowerbound] demonstrates a regret lower bound of $\Omega(d \sqrt{\smash[b]{T/K}})$. This indicates that, under uniform rewards, increasing the assortment size K leads to an improvement in regret. Compared to the lower bound $\Omega(d \sqrt{T}/K)$ proposed by , our lower bound is improved by a factor of $\sqrt{K}$. This improvement is mainly due to the establishment of a tighter upper bound for the KL divergence (Lemma [lemma:lowerKL]). Notably, also considered uniform rewards with v0 = Θ(1). On the other hand, and established regret lower bounds of $\Omega(\sqrt{NT})$ and $\Omega(\sqrt{NT/K})$, respectively, in non-contextual MNL bandits with uniform rewards, by setting v0 = K to achieve these regrets. Theorem [thm:lowerbound] shows that if v0 = Θ(K), we can obtain a regret lower bound of $\Omega(d \sqrt{T})$, which is consistent with the K-independent regret in . To the best of our knowledge, this result is the first to explicitly show the dependency of regret on the utility for the outside option v0. Minimax Optimal Regret Upper Bound under Uniform Rewards -------------------------------------------------------- In this subsection, we propose a new algorithm , which enjoys minimax optimal regret up to logarithmic factors in the case of uniform rewards. Note that, since the revenue is an increasing function when rewards are uniform, maximizing the expected revenue $ R\_{t}(S, \wb)$ over all S ∈ S always yields exactly K items, i.e., ∣St∣ = ∣St⋆∣ = K. Our first step involves constructing the confidence set for the online parameter. Online parameter estimation. Instead of performing MLE as in previous works works , inspired by , we use the mirror descent algorithm to estimate parameter. We first define the multinomial logistic loss function at round t as: $$\begin{aligned} \label{eq:loss} \ell\_t(\wb) := - \sum\_{i \in S\_t} y\_{ti} \log p\_t(i \| S\_t, \wb).\end{aligned}$$ In Proposition [prop:selfconcordant], we will show that the loss function has the constant parameter self-concordant-like property. We estimate the true parameter $\wb^\star$ as follows: $$\label{eq:online\_update} \wb\_{t+1} = \argmin\_{\wb \in \mathcal{W}} \langle \nabla \ell\_{t}(\wb\_t), \wb \rangle + \frac{1}{2 \eta} \\| \wb - \wb\_{t} \\|\_{\tilde{H}\_{t}}^2 \,, \quad \forall t \geq 1$$ where η > 0 is the step-size parameter to be specified later, and $\tilde{H}\_{t} := H\_t + \eta \Gcal\_t(\wb\_t)$, where $$\begin{aligned} \Gcal\_t(\wb) := \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb) x\_{ti} x\_{ti}^\top - \sum\_{i \in S\_t} \sum\_{j \in S\_t} p\_t(i \| S\_t, \wb) p\_t(j \| S\_t, \wb) x\_{ti} x\_{tj}^\top,\end{aligned}$$ and $H\_t := \lambda \Ib\_d + \sum\_{s=1}^{t-1} \Gcal\_s(\wb\_{s+1})$. Note that $\Gcal\_t(\wb) = \nabla^2 \ell\_{t}(\wb)$. This online estimator is efficient in terms of both computation and storage. By a standard online mirror descent formulation , can be solved using a single projected gradient step through the following equivalent formula: $$\begin{aligned} \wb^{\prime}\_{t+1} = \wb\_t - \eta \tilde{H}\_t^{-1} \nabla \ell\_t(\wb\_t), \quad \text{and} \quad \wb\_{t+1} = \argmin\_{\wb \in \mathcal{W}} \\| \wb - \wb^{\prime}\_{t+1} \\|\_{\tilde{H}\_t},\end{aligned}$$ which enjoys a computational cost of only O(Kd3), completely independent of t . Regarding storage costs, the estimator does not need to store all historical data because both H̃t and Ht can be updated incrementally, requiring only O(d2) storage. Furthermore, the estimator allows for a κ-independent confidence set, leading to an improved regret. [Online parameter confidence set] [lemma:onlineconfidenceset] Let δ ∈ (0, 1]. Under Assumption [assum:boundedassumption], with $\eta = \frac{1}{2}\log (K+1) + 2$ and $\lambda = 84 \sqrt{2}d \eta$, we define the following confidence set $$\begin{aligned} \Ccal\_t(\delta) := \{ \wb \in \mathcal{W} \mid \\| \wb\_t - \wb \\|\_{H\_t} \leq \beta\_{t}(\delta) \}, \end{aligned}$$ where $\beta\_{t}(\delta) =\BigO \left( \sqrt{d} \log t \log K \right)$. Then, we have $\textup{Pr}[\forall t \geq 1, \wb^\star \in \Ccal\_t(\delta)] \geq 1- \delta$. Armed with the online estimator, we construct the computationally efficient optimistic revenue. [t!] [alg:minimax] [1] Inputs: regularization parameter λ, probability δ, confidence radius βt(δ),step size η. Initialize: H1 = λId and $\wb\_1$ as any point in W, Compute $\alpha\_{ti} = x\_{ti}^\top \wb\_t + \beta\_{t}(\delta) \\|x\_{ti}\\|\_{H\_t^{-1}}$ for all i ∈ [N]. Offer $S\_t = \argmax\_{S \in \mathcal{S}} \tilde{R}\_t (S)$ and observe $\yb\_t$. Update $\tilde{H}\_{t} = H\_t + \eta \Gcal\_t(\wb\_t)$, and update the estimator $\wb\_{t+1}$ by . Update $H\_{t+1} = H\_t + \Gcal\_t(\wb\_{t+1})$. Computationally efficient optimistic expected revenue. To balance the exploration and exploitation trade-off, we use the upper confidence bounds (UCB) technique, which have been widely studied in many bandit problems, including K-arm bandits and linear bandits . At each time t, given the confidence set in Lemma [lemma:onlineconfidenceset], we first calculate the optimistic utility αti for each item as follows: $$\begin{aligned} \alpha\_{ti} := x\_{ti}^\top \wb\_t + \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}}. \label{eq:optimistic\_utility}\end{aligned}$$ The optimistic utility αti is composed of two parts: the mean utility estimate $x\_{ti}^\top \wb\_t$ and the standard deviation βt(δ)∥xti∥Ht− 1. In the proof of the regret upper bound, we show that αti serves as an upper bound for $x\_{ti}^\top \wb^\star$, assuming that the true parameter $\wb^\star$ falls within the confidence set $\Ccal\_t(\delta)$. Based on αti, we construct the optimistic expected revenue for the assortment S, defined as follows: $$\begin{aligned} \tilde{R}\_{t}(S) := \frac{\sum\_{i \in S} \exp( \alpha\_{ti} ) r\_{ti} }{v\_0 + \sum\_{j \in S} \exp(\alpha\_{ti})}, \label{eq:opt\_revenue}\end{aligned}$$ where rti = 1. Then, we offer the set St that maximizes the optimistic expected revenue, $S\_t = \argmax\_{S \in \mathcal{S}} \tilde{R}\_{t}(S) $. Given our assumption that all rewards are of unit value, the optimization problem is equivalent to selecting the K items with the highest optimistic utility αti. Consequently, solving the optimization problem incurs a constant computational cost of O(N). [Comparison to ] In, the MNL choice model is outlined with a shared context xt and distinct parameters $\wb\_{1}^\star, \dots, \wb\_K^\star$ for each choice. Conversely, our model employs a single parameter $\wb^\star$ across all choices and has varying contexts for each item in the assortment S, xt1, …xti∣S∣. Due to this discrepancy in the choice model, directly applying Proposition 1 from , which constructs the optimistic revenue by adding bonus terms to the estimated revenue, incurs an exponential computational cost in our problem setting. This complexity arises because the optimistic revenue must be calculated for every possible assortment S ∈ S; therefore, it is necessary to enumerate all potential assortments (N choose K) to identify the one that maximizes the optimistic revenue As a result, extending the approach in to our setting is non-trivial, requiring a different analysis. We now present the regret upper bound of in the uniform reward setting. [Regret upper bound of , Uniform rewards] [thm:upperbound] Let δ ∈ (0, 1] and Assumptions [assum:boundedassumption] and [assum:kappa] hold. In the uniform reward setting, by setting $\eta = \frac{1}{2}\log (K+1) + 2$ and $\lambda = 84 \sqrt{2}d \eta$, with probability at least 1 − δ, the cumulative regret of is upper-bounded by $$\begin{aligned} \Regret (\wb^\star ) = \BigOTilde \left( \frac{\sqrt{ v\_0K}}{ v\_0 + K } \cdot d\sqrt{T} + \frac{1}{\kappa} d^2 \right). \end{aligned}$$ Discussion of Theorem [thm:upperbound]. If $T \geq \BigO (d^2 (v\_0 + K)^2/(\kappa^2 v\_0 K))$, Theorem [thm:upperbound] shows that our algorithm achieves minimax optimal regret (up to logarithmic factor) in terms of all d, T, K, and even v0. To the best of our knowledge, ignoring logarithmic factors, our proposed algorithm is the first computationally efficient, minimax optimal algorithm in (adversarial) contextual MNL bandits. When v0 = Θ(1), which is the convention in existing MNL bandit literature , obtains $\BigOTilde(d \sqrt{T/K})$ regret. This represents an improvement over the previous upper bound of [2](#fn2), which is $\BigOTilde(d K \sqrt{\kappa^{\prime} T } + d^2K^4/\kappa )$, where $\kappa^{\prime}= \BigO(1/K)$, by a factor of K. This improvement can be attributed to two key factors: an improved, constant, self-concordant-like property of the loss function (Proposition [prop:selfconcordant]) and a K-free elliptical potential (Lemma [lemma:elliptical]). Furthermore, by employing an improved bound for the second derivative of the revenue (Lemma [lemma:revenuesecondpd]), we achieve an enhancement in the regret for the second term, d2/κ, by a factor of K4, in comparison to . Unless v0 = Θ(K), Theorem [thm:upperbound] indicates that the regret decreases as the assortment size K increases. To the best of our knowledge, this is the first algorithm in MNL bandits to show that increasing K results in a reduction in regret. Moreover, when reduced to the logistic bandit, i.e., K = 1, rt1 = 1, and v0 = 1, our algorithm can also achieve a regret of $\BigOTilde(d\sqrt{\kappa T})$ by Corollary 1 in , which is consistent with the results in . [Efficiency of ] [remark:uppercomputationcost] The proposed algorithm is computationally efficient in both parameter updates and assortment selections. Since we employ online parameter estimation, akin to , our algorithm demonstrates computational efficiency in parameter estimation, incurring only incurring O(Kd3) computation cost and O(d2) storage cost, which are completely independent of t. Furthermore, a naive approach to selecting the optimistic assortment requires enumerating all possible (N choose K) assortments, resulting in exponential computational cost . However, by constructing the optimistic expected revenue according to (inspired by ), our algorithm needs only $\BigO(N)$ computational cost. Regret Upper & Lower Bounds under Non-Uniform Rewards ----------------------------------------------------- In this subsection, we propose regret upper and lower bounds in the non-uniform reward setting. In this scenario, the sizes of both the chosen assortment St, and the optimal assortment, St⋆ are not fixed at K. Therefore, we cannot guarantee an improvement in regret even as K increases. We first prove the regret lower bound in the non-uniform reward setting. [Regret lower bound, Non-uniform rewards] [theorem:lowerboundnonuniform] Under the same conditions as Theorem [thm:lowerbound], let the rewards be non-uniform and v0 = Θ(1). Then, for any policy π, there exists a worst-case problem instance such that the worst-case expected regret of π is lower bounded as follows: $$\begin{aligned} \sup\_{\wb} \EE^\pi\_{\wb} \left[ \Regret(\wb) \right] = \Omega \left( d\sqrt{T} \right). \end{aligned}$$ Discussion of Theorem [theorem:lowerboundnonuniform]. In contrast to Theorem [thm:lowerbound], which considers uniform rewards, the regret lower bound is independent of the assortment size K. Note that Theorem [theorem:lowerboundnonuniform] does not claim that non-uniform rewards inherently make the problem more difficult. Rather, it implies that there exists an instance with adversarial non-uniform rewards, where regret does not improve even with an increase in K. Moreover, the assumption that v0 = Θ(1) is common in the existing literature on contextual MNL bandits (refer Appendix [appsubsec:outside]). To the best of our knowledge, this is the first established lower bound for non-uniform rewards in MNL bandits. We also prove a matching upper bound up to logarithmic factors. The algorithm is also applicable in the case of non-uniform rewards. However, because the optimistic expected revenue R̃t(S) is no longer an increasing function of αti, optimizing for $S\_t = \argmax\_{S \in \mathcal{S}} \tilde{R}\_t (S)$ no longer equates to simply selecting the top K items with the highest optimistic utility. Instead, we employ assortment optimization methods introduced in , which are efficient polynomial-time (independent of t) [3](#fn3) algorithms available for solving this optimization problem. Therefore, our algorithm is also computationally efficient under non-uniform rewards. [Regret upper bound of , Non-uniform rewards] [theorem:upperboundnonuniform] Under the same assumptions and parameter settings as Theorem [thm:upperbound], if the rewards are non-uniform and v0 = Θ(1), then with a probability of at least 1 − δ, the cumulative regret of is upper-bounded by $$\begin{aligned} \Regret (\wb^\star ) &= \BigOTilde \left( d\sqrt{T} + \frac{1}{\kappa}d^2 \right). \end{aligned}$$ Discussion of Theorem [theorem:upperboundnonuniform]. If $T \geq \BigO (d^2 / \kappa^2)$, our algorithm achieves a regret of $\BigOTilde(d \sqrt{T})$ when the reward for each item is non-uniform, demonstrating that is minimax optimal up to a logarithmic factor. Recall that we relax the bounded assumption on the reward compared to (refer Assumption [assum:boundedassumption]); thus, we allow the sum of the squared rewards in the assortment to scale with K. Consequently, we need a novel approach to achieve the regret that does not scale with K. To this end, we centralize the features and propose a novel elliptical potential lemma for them, as detailed in Lemma [lemma:ellipticalxtilde]. Note that our algorithm is capable of achieving 1/κ-free regret (in the leading term) under both uniform and non-uniform rewards. In contrast, the algorithm in is limited to achieving this only in the uniform reward setting. Numerical Experiments ===================== [t] [b]0.245 [fig:N=100K=5d=5dist=0] [b]0.245 [fig:N=100K=10d=5dist=0] [b]0.245 [fig:N=100K=15d=5dist=0] [b]0.245 [fig:N=100K=15d=5dist=0rmain] [b]0.245 [fig:N=100K=5d=5dist=0nu] [b]0.245 [fig:N=100K=10d=5dist=0nu] [b]0.245 [fig:N=100K=15d=5dist=0nu] [b]0.245 [fig:N=100K=15d=5dist=0rnumain] [fig:regretmain] In this section, we empirically evaluate the performance of our algorithm . We measure cumulative regret over T = 3000 rounds. For each experimental setup, we run the algorithms across 20 independent instances and report the average performance. In each instance, the underlying parameter $\wb^\star$ is randomly sampled from a d-dimensional uniform distribution, where each element of $\wb^\star$ lies within the range $[-1/\sqrt{d},1/\sqrt{d}]$ and is not known to the algorithms. Additionally, the context features xti are drawn from a d-dimensional multivariate Gaussian distribution, with each element of xti clipped to the range $[-1/\sqrt{d},1/\sqrt{d}]$. This setup ensures compliance with Assumption [assum:boundedassumption]. In the uniform reward setting (first row of Figure [fig:regretmain]), the combinatorial optimization step to choose the assortment reduces to sorting items by their utility estimate. In the non-uniform reward setting (second row of Figure [fig:regretmain]), the rewards are sampled from a uniform distribution in each round, i.e., rti ∼ Unif(0, 1). Refer Appendix [appsec:experimatdetails] for more details. We compare the performance of with those of the practical and state-of-the-art algorithms: the Upper Confidence Bound-based algorithm, `UCB-MNL` , and the Thompson Sampling-based algorithm, `TS-MNL` . Figure [fig:regretmain] demonstrates that our algorithm significantly outperforms other baseline algorithms. In the uniform reward setting, as K increases, the cumulative regrets of all algorithms tend to decrease. In contrast, this trend is not observed in the non-uniform reward setting. Furthermore, the results also show that our algorithm maintains a constant computation cost per round, while the other algorithms exhibit a linear dependence on t. In Appendix [appsec:experimatdetails], we present the additional runtime curves (Figure [fig:runtime]) as well as the regret curves of the other configuration where v0 = Θ(K) (Figure [fig:varyingv0]). All of these empirical results align with our theoretical results. Conclusion ========== In this paper, we propose minimax optimal lower and upper bounds for both uniform and non-uniform reward settings. We propose a computationally efficient algorithm, , that achieves a regret of $\BigOTilde(d\sqrt{\smash[b]{T/K}})$ under uniform rewards and $\BigOTilde(d \sqrt{T})$ under non-uniform rewards. We also prove matching lower bounds of $\Omega(d\sqrt{\smash[b]{T/K}})$ and $\Omega(d \sqrt{T})$ for each setting, respectively. Moreover, our empirical results support our theoretical findings, demonstrating that is not only provably but also experimentally efficient. Further Related Work ==================== In this section, we discuss additional related works that complement Section [sec:Related]. For simplicity, we consider only the dependence on the number of rounds t for a computation cost in big-$\BigO$ notation. Logistic Bandits. The logistic bandit model focuses on environments with binary rewards and explores the impact of non-linearity on the exploration-exploitation trade-off for parametrized bandits. The main research interest has been the algorithms’ dependence on the degree of non-linearity κ, which can grow exponentially in terms of the diameter of the decision domain W. introduced the first efficient algorithm for binary logistic bandits with a $\BigO(1)$ computation cost, achieving a regret of $\BigOTilde(d\sqrt{T}/\kappa)$. enhanced the regret to $\BigOTilde(d\sqrt{\smash[b]{T/\kappa}})$ with a $\BigO(t)$ computation cost. However, their regret bounds still suffered from a harmful dependence on 1/κ. addressed this by achieving the tightest regret upper bound of $\BigOTilde(d\sqrt{\kappa T})$ with a $\BigO( t)$ computation cost, while achieved the same regret with an improved computation cost of $\BigO(\log t)$. More recently, proposed a jointly efficient algorithm that achieves the optimal regret with a constant $\BigO(1)$ computation cost. Note that the logistic bandit is a special case of the multinomial logistic (MNL) bandit. When the maximum assortment size is one (K = 1), rewards are uniform (rt1 = 1), and the utility for the outside option is one v0 = 1, the MNL bandit reduces to the logistic bandit. In this logistic bandit setting, our proposed algorithm,, can achieve a regret upper bound of $\BigOTilde(d\sqrt{\kappa T})$ with a constant $\BigO(1)$ computation cost, consistent with the result in . Multinomial Logistic (MNL) Bandits. There are two main approaches to multinomial logistic (MNL) bandits: the multiple-parameter choice model and the single-parameter choice model. In the multiple-parameter choice model, the learner estimates parameters for each choice in the assortment ($\wb\_1^\star, \dots, \wb\_K^\star$) with a shared context xt. In this setting, proposed a feasible algorithm that achieves a regret upper bound of $\BigOTilde(dK\sqrt{\kappa T})$ with a $\BigO(t)$ computation cost. They also proposed an intractable algorithm that achieves an improved regret of $\BigOTilde(dK^{3/2}\sqrt{ T})$. introduced a computationally and statistically efficient algorithm that obtains a regret of $\BigOTilde(dK\sqrt{ T})$. Recently, further improved the regret by a factor of $\sqrt{K}$, achieving $\BigOTilde(d \sqrt{KT})$ regret. In the multiple-parameter case, the regret’s dependence on K is unavoidable since the number of unknown parameters depends on K. On the other hand, the single-parameter choice model, closely related to ours, shares the parameter $\wb^\star$ cross the choices, with varying contexts for each choice. The learner offers a set of items St, with ∣St∣ ≤ K at each round. This setting involves a combinatorial optimization to choose the assortment St, making it more challenging to devise a tractable algorithm. As extensively discussed in Section [sec:Related], no previous studies have definitively confirmed whether the existing lower or upper bounds are tight. As shown in Table [tab:regrets], many studies have presented their results in inconsistent settings with varying reward structures and values of v0, adding to the ambiguity about the bounds’ optimality. In this paper, we address these issues by bridging the gap between the lower and upper bounds of regret through a careful categorization of the settings. We propose an algorithm that is both provably optimal, up to logarithmic factors, and computationally efficient, significantly enhancing the theoretical and practical understanding of MNL bandits. Generalized Linear Bandits. In generalized linear bandits , the expected rewards are modeled using a generalized linear model. These problems generalize logistic bandits by incorporating a general exponential family link function instead of the logistic link function. The algorithms proposed for generalized linear bandits also exhibit a dependence on the nonlinear term κ. However, our problem setting (single-parameter MNL bandits) considers a more complex state space where multiple arms are pulled simultaneously. Combinatorial Bandits. Another related stream of literature is combinatorial bandits , particularly top-k combinatorial bandits . In top-k combinatorial bandits, the decision set includes all subsets of size k out of n arms, and the reward for each action is the sum of the rewards of the k selected arms. In this framework, the rewards are assumed to be independent of the entire set of arms played in round t. In contrast, in our setting, the reward for each individual arm depends on the whole set of arms played. Notation ======== We denote T as the total number of rounds and t ∈ [T] as the current round. We denote N as the total number of items, K as the maximum size of assortments, and d as the dimension of feature vectors. For notational simplicity, we define the loss function in two different forms throughout the proof: $$\begin{aligned} \ell\_t(\wb) &= - \sum\_{i \in S\_t} y\_{ti} \log p\_t(i \| S\_t, \wb) =- \sum\_{i \in S\_t} y\_{ti} \log \left( \frac{\exp(x\_{ti}^\top \wb )}{ v\_0 \!+\!\sum\_{j \in S\_t }\exp( x\_{tj}^\top \wb )}\right), \\ \ell(\zb\_t, \yb\_t) &= - \sum\_{i \in S\_t} y\_{ti} \log \left( \frac{\exp( z\_{ti} )}{ v\_0 \!+\!\sum\_{j \in S\_t }\exp( z\_{tj} )}\right),\end{aligned}$$ where $z\_{ti} = x\_{ti}^\top \wb$, $\zb\_t = (z\_{ti})\_{i \in S\_t} \in \RR^{\|S\_t\|}$, and $\yb\_t = (y\_{ti})\_{i \in S\_t} \in \RR^{\|S\_t\|}$. Thus, $\ell\_t(\wb) = \ell(\zb\_t, \yb\_t)$. We offer a Table [tablesymbols] for convenient reference. [htp!] [tablesymbols] ll xti & feature vector for item i given at round t rti & reward for item i given at round t St & assortment chosen by an algorithm at round t 0 & outside option yti & choice response for each item i ∈ St ∪ {0} at round t $ R\_{t}(S, \wb^\star)$ & $:= \sum\_{i \in S} p\_t(i \| S, \wb^\star) r\_{ti} $, expected revenue of the assortment S at round t $\ell\_t(\wb) $ & $:= - \sum\_{i \in S\_t} y\_{ti} \log \left( \frac{\exp(x\_{ti}^\top \wb )}{ v\_0 \!+\!\sum\_{j \in S\_t }\exp( x\_{tj}^\top \wb )}\right)$, loss function at round t $\ell(\zb\_t, \yb\_t)$ & $:= - \sum\_{i \in S\_t} y\_{ti} \log \left( \frac{\exp( z\_{ti} )}{ v\_0 \!+\!\sum\_{j \in S\_t }\exp( z\_{tj} )}\right)$, loss function at round t, $z\_{ti} = x\_{ti}^\top \wb$ λ & regularization parameter $\Gcal\_t(\wb)$ & $:= \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb) x\_{ti} x\_{ti}^\top - \sum\_{i \in S\_t} \sum\_{j \in S\_t} p\_t(i \| S\_t, \wb) p\_t(j \| S\_t, \wb) x\_{ti} x\_{tj}^\top$ Ht & $:= \lambda \Ib\_d + \sum\_{s=1}^{t-1} \Gcal\_s(\wb\_{s+1})$ H̃t & $:= H\_t + \eta \Gcal\_t(\wb\_t)$ αti & $:= x\_{ti}^\top \wb\_t + \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}}$, optimistic utility for item i at round t βt(δ) & $:= \BigO \left( \sqrt{d} \log t \log K \right)$, confidence radius at round t R̃t(S) & $:= \frac{\sum\_{i \in S} \exp( \alpha\_{ti} ) r\_{ti} }{v\_0 + \sum\_{j \in S} \exp(\alpha\_{ti})}$, optimistic expected revenue for the assortment S at round t Properties of MNL function ========================== In this section, we present key properties of the MNL function and its associated loss, which are used throughout the paper. Utility for Outside Option: is Common in Contextual MNL Bandits --------------------------------------------------------------- In this subsection, we explain why the assumption that v0 = Θ(1) is made without loss of generality. Let the original feature vectors be $x'\_{ti} \in \RR^d$ for every item i ∈ [N]. Suppose that a context for the outside option xt0ʹ is given and the probability of choosing any item i ∈ St ∪ {0} is defined as $$\begin{aligned} p\_t(i \| S\_t, \wb^\star) = \frac{\exp((x'\_{ti})^\top \wb^\star )}{\sum\_{j \in S\_t \cup \{0\} }\exp( (x'\_{tj})^\top \wb^\star )}.\end{aligned}$$ Then, by dividing the denominator and numerator by $\exp\left( (x'\_{t0})^\top \wb^\star \right)$, and defining xti : = xʹti − xʹt0, we obtain the MNL probability in the form presented in with v0 = exp(0) = 1. Note that this division does not change the probability. Therefore, v0 = Θ(1) is natural and common in contextual MNL bandit literature. Self-concordant-like Function ----------------------------- [Self-concordant-like function, ] [def:self-concordant-like] A convex function $f \in \mathcal{C}^3(\RR^m)$ is M-self-concordant-like function with constant M if: $$\begin{aligned} \|\phi^{\prime\prime\prime} (s) \| \leq M \\|\bb \\|\_2 \phi^{\prime\prime}(s). \end{aligned}$$ for $s \in \RR$ and M > 0, where $\phi(s) := f(\ab + s\bb)$ for any $\ab, \bb \in \RR^m$. Then, the MNL loss defined in is $3\sqrt{2}$-self-concordant-like function. [prop:selfconcordant] For any t ∈ [T], the multinomial logistic loss $\ell\_{t}(\wb)$, defined in , is $3\sqrt{2}$-self-concordant-like. Consider the function ϕ(s) : = log(∑i = 0neais + bi), where $\ab = [a\_0, \dots, a\_{n}]^\top \in \RR^{n+1}$ and $\bb = [b\_0, \dots, b\_{n}]^\top \in \RR^{n+1}$. Then, by simple calculus, we have $$\begin{aligned} \phi^{\prime\prime} (s) = \frac{\sum\_{i<j} (a\_i - a\_j)^2 e^{a\_is + b\_i} e^{a\_js + b\_j} }{\left(\sum\_{i=0}^n e^{a\_is + b\_i} \right)^2} \geq 0, \end{aligned}$$ and $$\begin{aligned} \phi^{\prime\prime\prime} (s) = \frac{\sum\_{i<j} (a\_i - a\_j)^2 e^{a\_is + b\_i} e^{a\_js + b\_j} \left[ \sum\_{k=0}^n (a\_i + a\_j -2a\_k)e^{a\_ks + b\_k} \right]}{\left(\sum\_{i=0}^n e^{a\_is + b\_i} \right)^3}. \numberthis \label{eq:self\_concordant\_third\_derivative} \end{aligned}$$ Note that for all i, j, k = 0, …, n, $$\begin{aligned} \|a\_i + a\_j -2a\_k\| \leq \sqrt{6}\sqrt{a\_i^2 + a\_j^2 + a\_k^2} \leq 3\sqrt{2} \max\_{i=0, \dots, n} \| a\_i\|. \label{eq:self\_concordant\_improved\_norm} \end{aligned}$$ Therefore, we have $$\begin{aligned} \left\| \sum\_{k=0}^n (a\_i + a\_j -2a\_k)e^{a\_ks + b\_k} \right\| \leq \sum\_{k=0}^n \left\|a\_i + a\_j -2a\_k\right\|e^{a\_ks + b\_k} \leq 3\sqrt{2} \max\_{i=0, \dots, n} \| a\_i \| \sum\_{k=0}^n e^{a\_ks + b\_k}. \numberthis \label{eq:self\_concordant\_improved\_norm2} \end{aligned}$$ Plugging in into , we obtain $$\begin{aligned} \phi^{\prime\prime\prime}(s) \leq 3\sqrt{2} \max\_{i=0, \dots, n} \|a\_i \| \phi^{\prime\prime}(s). \label{eq:self\_concordant\_like\_improved} \end{aligned}$$ Now, we are ready to prove the proposition. For any t ∈ [T], let n = ∣St∣ and c1 = xti1, c2 = xti2, …, cn = xtin. Define a function $f \in \mathcal{C}^3: \RR^d \rightarrow\ \RR$ as $f(\thetab) := \log \left( v\_0 + \sum\_{i=1}^n e^{c\_i^\top \thetab } \right)$. Let $\deltab \in \RR^d$ and let $f(\thetab + s \deltab) = \log \left( v\_0 + \sum\_{i=1}^n e^{c\_i^\top \thetab + s c\_i^\top \deltab} \right) = \log \left( \sum\_{i =0}^n e^{a\_i s + b\_i} \right) = \phi(s)$, where $a\_i = c\_i^\top \deltab$, $b\_i = c\_i^\top \thetab$ for i = 1, …, n, and ai = 0 and bi = logv0 for i = 0. Then, by , we get $$\begin{aligned} \|\phi^{\prime\prime\prime}(s)\| &\leq 3\sqrt{2} \max\_{i=0, \dots, n} \|a\_i \| \phi^{\prime \prime}(s) = 3\sqrt{2} \max\_{i=1, \dots, n} \| c\_i^\top \deltab \| \phi^{\prime \prime}(s) \\ &\leq 3\sqrt{2} \max\_{i=1, \dots, n} \\| c\_i \\|\_2 \\|\deltab \\|\_2 \phi^{\prime \prime}(s) \leq 3\sqrt{2} \\|\deltab \\|\_2 \phi^{\prime \prime}(s) , \end{aligned}$$ where the last inequality holds due to Assumption [assum:boundedassumption] that ∥ci∥2 = ∥xtji∥2 ≤ 1. Then, by Definition [def:self-concordant-like], f is $3\sqrt{2}$-self-concordant-like. Since ℓt is the sum of f and a linear operator, which has third derivatives equal to zero, it follows that ℓt is also $3\sqrt{2}$-self-concordant-like function. [remark:selfconcordant] Contrary to the findings of , which suggest that the MNL loss function $\sqrt{6K}$-self-concordant-like, our loss function is $3\sqrt{2}$-self-concordant-like. This yields an improved regret bound on the order of $\mathcal{O}(\sqrt{K})$. The improvement arises due to a K-independent self-concordant-like property of ℓt, as shown in Proposition [prop:selfconcordant]. In , Lemma 4 from is used, which describes a $\sqrt{6} \\|\ab\\|\_2$ self-concordant-like property. However, in the analysis of [eq:selfconcordantimprovednorm], we show that their analysis is not tight because they bound the term $\sqrt{\smash[b]{a\_i^2 + a\_j^2 + a\_k^2}}$ by $\\|\ab\\|\_2 = \sqrt{\sum\_{i=0}^n a\_i^2}$, thus making its upper bound dependent on K, i.e., n = ∣St∣ ≤ K. In contrast, we bound the same term by a constant, maxi = 1, …, n∥ai∥2, which allows our loss function to exhibit a constant $3\sqrt{2}$-self-concordant-like property. This key difference accounts for the $\sqrt{K}$-improved regret. [Theorem 3 of ] [lemma:tranthm3] A convex function $\ell \in \mathcal{C}^3: \RR^d \rightarrow \RR$ is M-self-concordant-like if and only if for any $\vb, \ub\_1, \ub\_2, \ub\_3 \in \RR^d$, we have $$\begin{aligned} \| \langle D^3 \ell(\vb)[\ub\_1]\ub\_2, \ub\_3 \rangle \| \leq M \\|\ub\_1 \\|\_2 \\| \ub\_2 \\|\_{ \nabla^2 \ell(\vb) } \\| \ub\_3 \\|\_{ \nabla^2 \ell(\vb) }. \end{aligned}$$ Proof of Theorem [thm:lowerbound] ================================= In this section, we provide the proof of Theorem [thm:lowerbound]. The proof structure is similar to the one presented in . However, unlike their approach, we explicitly derive a bound that includes v0. Furthermore, by establishing a tighter upper bound for the KL divergence (Lemma [lemma:lowerKL]), we derive a bound that is tighter than the one provided by . Adversarial Construction and Bayes Risk --------------------------------------- Let $\epsilon \in ( 0, 1/d\sqrt{d} )$ be a small positive parameter to be specified later. For every subset V ⊆ [d], we define the corresponding parameter $\wb\_{V} \in \RR^d$ as $[\wb\_{V}]\_{j} = \epsilon$ for all j ∈ V, and $[\wb\_{V}]\_j = 0$ for all $j \notin V$. Then, we consider the following parameter set $$\begin{aligned} \wb \in \mathcal{W} := \{ \wb\_{V} : V \in \mathcal{V}\_{d/4} \} := \{ \wb\_{V}: V \subseteq [d], \|V\| = d/4 \},\end{aligned}$$ where Vk denotes the class of all subsets of [d] whose size is k. Moreover, note that d/4 is a positive integer, as d is divisible by 4 by construction. The context vectors {xti} are constructed to be invariant across rounds t. For each t and U ∈ Vd/4, K identical context vectors [4](#fn4) xU are constructed as follows: $$\begin{aligned} [x\_{U}]\_{j} = 1/\sqrt{d} \quad \text{for} \,\, j \in U; \quad [x\_{U}]\_{j} = 0 \quad \text{for} \,\, j \notin U.\end{aligned}$$ For all V, U ∈ Vd/4, it can be verified that $\wb\_V$ and xU satisfy the requirements of a bounded assumption [assum:boundedassumption] as follows: $$\begin{aligned} \\| \wb\_V \\|\_{2} \leq \sqrt{d \epsilon^2} \leq 1, \quad \\| x\_{U} \\|\_2 \leq \sqrt{d \cdot 1/d } = 1.\end{aligned}$$ Therefore, the worst-case expected regret of any policy π can be lower bounded by the worst-case expected regret of parameters belonging to W, which can be further lower bounded by the “average” regret over a uniform prior over W as follows: $$\begin{aligned} \sup\_{\wb} \EE\_{\wb}^{\pi} \left[ \Regret(\wb) \right] &= \sup\_{\wb} \EE\_{\wb}^{\pi} \sum\_{t=1}^T R(S^\star, \wb) - R(S\_{t}, \wb) \\ &\geq \max\_{\wb\_V \in \mathcal{W}} \EE\_{\wb\_V}^{\pi} \sum\_{t=1}^T R(S^\star, \wb\_V) - R(S\_{t}, \wb\_V) \\ &\geq \frac{1}{\|\mathcal{V}\_{d/4}\|} \sum\_{V \in \mathcal{V}\_{d/4}} \EE\_{\wb\_V}^{\pi} \sum\_{t=1}^T R(S^\star, \wb\_V) - R(S\_{t}, \wb\_V) \\ &= \frac{1}{\|\mathcal{V}\_{d/4}\|} \sum\_{V \in \mathcal{V}\_{d/4}} \EE\_{\wb\_V}^{\pi} \sum\_{t=1}^T \left[ \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in S\_t} p(i \| S\_t, \wb\_V) \right] . \numberthis \label{eq:lower\_bayes\_risk}\end{aligned}$$ This reduces the task of lower bounding the worst-case regret of any policy to the task of lower bounding the Bayes risk of the constructed parameter set. Main Proof of Theorem [thm:lowerbound] -------------------------------------- [Proof of Theorem [thm:lowerbound]] For any sequence of assortments {St}t = 1T produced by policy π, we denote an alternative sequence {S̃t}t = 1T that provably enjoys less regret under parameterization $\wb\_{V}$. Let xU1, …, xUL be the distinct feature vectors contained in assortments St (if St = ∅, then one may choose an arbitrary feature xU) with U1, …, UL ∈ Vd/4. Let U⋆ be the subset among U1, …, UL that maximizes $x\_U^\top \wb\_V$, i.e., $U^\star \in \argmax\_{U \in \{U\_1, \dots, U\_L \}} x\_U^\top \wb\_V$, where $\wb\_V$ is the underlying parameter. Then, we define S̃t as the assortment consisting of all K items corresponding to feature xU⋆, i.e., $\tilde{S}\_t = \{ \underbrace{x\_{U^\star}, \dots, x\_{U^\star}}\_{K} \}$. Since the expected revenue is an increasing function, we have the following observation: [Proposition 1 in ] [prop:lowerStilde] $$\begin{aligned} \sum\_{i \in S\_t} p(i \| S\_t, \wb\_V) \leq \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V). \end{aligned}$$ Proposition [prop:lowerStilde] implies that $\sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in S\_t} p(i \| S\_t, \wb\_V) \geq \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V)$. Hence, it is sufficient to bound $\sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) $ instead of $\sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in S\_t} p(i \| S\_t, \wb\_V)$. To simplify notation, we denote Ũt as the unique U⋆ ∈ Vd/4 in S̃t. We also use $\EE\_{V}$ and PV to denote the expected value and probability, respectively, as governed by the law parameterized by $\wb\_V$ and under policy π. Then, we can establish a lower bound for $\sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) $ as follows: [lemma:lowerrevenuegap] Suppose $\epsilon \in (0, 1/d\sqrt{d})$ and define δ : = d/4 − ∣Ũt ∩ V∣. Then, we have $$\begin{aligned} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) \geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\delta \epsilon}{2 \sqrt{d}} . \end{aligned}$$ For any j ∈ V, define random variables M̃j : = ∑t = 1T1{j ∈ Ũt}. Then, by Lemma [lemma:lowerrevenuegap], for all V ∈ Vd/4, we have $$\begin{aligned} \EE\_{V} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) \geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{4} - \sum\_{j \in V} \EE\_{V}[\tilde{M}\_j] \right). \numberthis \label{eq:lower\_expected\_revenue\_gap}\end{aligned}$$ Furthermore, we define Vd/4(j) : = {V ∈ Vd/4 : j ∈ V} and Vd/4 − 1 : = {V ⊆ [d] : ∣V∣ = d/4 − 1}. By taking the average of both sides of Equation with respect to all V ∈ Vd/4, we obtain $$\begin{aligned} &\frac{1}{\left\|\mathcal{V}\_{d/4} \right\|} \sum\_{V \in \mathcal{V}\_{d/4}} \EE\_{V} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) \\ &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \cdot \frac{1}{\left\|\mathcal{V}\_{d/4} \right\|} \sum\_{V \in \mathcal{V}\_{d/4}} \left( \frac{dT}{4} - \sum\_{j \in V} \EE\_{V}[\tilde{M}\_j] \right) \\ &= \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{4} - \frac{1}{\left\|\mathcal{V}\_{d/4} \right\|} \sum\_{j=1}^d \sum\_{V \in \mathcal{V}\_{d/4}^{(j)}} \EE\_{V}[ \tilde{M}\_j] \right) \\ &= \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left(\frac{dT}{4} - \frac{1}{\left\|\mathcal{V}\_{d/4} \right\|} \sum\_{V \in \mathcal{V}\_{d/4 - 1}} \sum\_{j \notin V} \EE\_{V \cup \{j\}}[ \tilde{M}\_j] \right) \\ &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left(\frac{dT}{4} - \frac{ \left\| \mathcal{V}\_{d/4 -1} \right\| }{\left\|\mathcal{V}\_{d/4} \right\|} \max\_{V \in \mathcal{V}\_{d/4 -1}} \sum\_{j \notin V} \EE\_{V \cup \{j\}}[ \tilde{M}\_j] \right) \\ &= \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left(\frac{dT}{4} - \frac{ \left\| \mathcal{V}\_{d/4 -1} \right\| }{\left\|\mathcal{V}\_{d/4} \right\|} \max\_{V \in \mathcal{V}\_{d/4 -1}} \sum\_{j \notin V} \EE\_{V} [ \tilde{M}\_j] +\EE\_{V \cup \{j\}}[ \tilde{M}\_j] - \EE\_{V} [ \tilde{M}\_j] \right).\end{aligned}$$ For any fixed V, we get $\sum\_{j \notin V} \EE\_{V} [ \tilde{M}\_j] \leq \sum\_{j=1}^d \EE\_{V} [ \tilde{M}\_j ] \leq dT/4$. Also, we have $\frac{ \left\| \mathcal{V}\_{d/4 -1} \right\| }{\left\|\mathcal{V}\_{d/4} \right\|} = \binom{d}{d/4 - 1} / \binom{d}{d/4} = \frac{d/4}{3d/4 +1} \leq \frac{1}{3}$. Consequently, we derive that $$\begin{aligned} \frac{1}{\left\|\mathcal{V}\_{d/4} \right\|} &\sum\_{V \in \mathcal{V}\_{d/4}} \EE\_{V} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) \\ &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{6} - \max\_{V \in \mathcal{V}\_{d/4 -1}} \sum\_{j \notin V} \left\| \EE\_{V \cup \{j\}}[ \tilde{M}\_j] - \EE\_{V} [ \tilde{M}\_j] \right\| \right). \numberthis \label{eq:lower\_expected\_revenue\_gap\_before\_pinsker}\end{aligned}$$ Now we bound the term $\left\| \EE\_{V \cup \{j\}}[ \tilde{M}\_j] - \EE\_{V} [ \tilde{M}\_j] \right\|$ in for any V ∈ Vd/4 − 1. For simplicity, let P = PV and Q = PV ∪ {j} denote the laws under $\wb\_V$ and $\wb\_{V \cup j}$, respectively. Then, we have $$\begin{aligned} \left\| \EE\_{P}[ \tilde{M}\_j] - \EE\_{Q} [ \tilde{M}\_j\right] \| &\leq \sum\_{t=0}^T t \cdot \left\| P[\tilde{M}\_j =t ] - Q [\tilde{M}\_j = t] \right\| \\ &\leq T \cdot \sum\_{t=0}^T \left\| P[\tilde{M}\_j =t ] - Q [\tilde{M}\_j = t] \right\| \\ &\leq T \cdot \\| P-Q \\|\_{\operatorname{TV}} \leq T \cdot \sqrt{\frac{1}{2} \operatorname{KL}(P \\| Q)}, \numberthis \label{eq:lower\_pinskers}\end{aligned}$$ where ∥P − Q∥TV = supA∣P(A) − Q(A)∣ \| is the total variation distance between P and Q, $\operatorname{KL}(P \\| Q) = \int (\log \dd P / \dd Q)\dd P$ is s the Kullback-Leibler (KL) divergence between P and Q, and the last inequality holds by Pinsker’s inequality. Now, we bound the KL divergence term using the following Lemma. [lemma:lowerKL] For any V ∈ Vd/4 − 1 and j ∈ [d], there exists a positive constant CKL > 0 such that $$\begin{aligned} \operatorname{KL}(P\_{V} \\| Q\_{V \cup \{j\}}) \leq C\_{\operatorname{KL}} \cdot \frac{v\_0 K}{(v\_0 + K)^2} \cdot \frac{ \EE\_{V}[ \tilde{M}\_j]\epsilon^2}{d}. \end{aligned}$$ Therefore, combining , , and Lemma [lemma:lowerKL], we have $$\begin{aligned} \frac{1}{\left\|\mathcal{V}\_{d/4} \right\|} &\sum\_{V \in \mathcal{V}\_{d/4}} \EE\_{V} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) \\ &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{6} - T \sum\_{j=1}^d \sqrt{ C\_{\operatorname{KL}} \cdot \frac{v\_0 K}{(v\_0 + K)^2} \cdot \frac{ \EE\_{V}[ \tilde{M}\_j]\epsilon^2}{d} } \right) \\ &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{6} - T \sqrt{d} \cdot \sqrt{ \sum\_{j=1}^d C\_{\operatorname{KL}} \cdot \frac{v\_0 K}{(v\_0 + K)^2} \cdot \frac{ \EE\_{V}[ \tilde{M}\_j]\epsilon^2}{d} } \right) \\ &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{6} - T \sqrt{d} \cdot \sqrt{ \frac{C\_{\operatorname{KL}} }{4} \cdot \frac{v\_0 K}{(v\_0 + K)^2} \cdot T \epsilon^2 } \right) ,\end{aligned}$$ where the second inequality is due to the Cauchy-Schwartz inequality and the last inequality holds because $\sum\_{j=1}^d \EE\_{V}[ \tilde{M}\_j] \leq dT/4$. Let CKLʹ = CKL/4. By setting $\epsilon = \sqrt{\frac{d}{144 C\_{\operatorname{KL}}' T } \cdot \frac{(v\_0 + K)^2}{v\_0 K}}$, we have $$\begin{aligned} \sup\_{\wb} \EE\_{\wb}^{\pi} \left[ \Regret(\wb) \right] &\geq \frac{v\_0 K}{(v\_0 + K e)^2} \cdot \frac{\epsilon}{2 \sqrt{d}} \left( \frac{dT}{6} - \sqrt{C\_{\operatorname{KL}}'\cdot \frac{v\_0 K}{(v\_0 + K)^2} d T \epsilon^2 } \right) \\ &= \frac{v\_0 K }{(v\_0 + K e)^2} \cdot \sqrt{\frac{(v\_0 + K)^2}{v\_0 K} } \cdot \frac{1}{288 \sqrt{C\_{\operatorname{KL}}'} } d\sqrt{T} \\ &= \Omega \left( \frac{\sqrt{v\_0 K}}{v\_0 + K} \cdot d\sqrt{T} \right).\end{aligned}$$ This concludes the proof of Theorem [thm:lowerbound]. Proofs of Lemmas for Theorem [thm:lowerbound] --------------------------------------------- ### Proof of Lemma [lemma:lowerrevenuegap] [Proof of Lemma [lemma:lowerrevenuegap]] Let x = xV and x̂ = xŨt be the corresponding context vectors. Then, we have $$\begin{aligned} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) &= \frac{K \exp \left( x^\top \wb\_{V} \right)}{v\_0 + K \exp \left( x^\top \wb\_{V} \right)} - \frac{K \exp \left( \hat{x}^\top \wb\_{V} \right)}{v\_0 + K \exp \left( \hat{x}^\top \wb\_{V} \right)} \\ &= \frac{v\_0 K \left( \exp \left( x^\top \wb\_{V} \right) - \exp \left( \hat{x}^\top \wb\_{V} \right) \right)}{\left( v\_0 + K \exp \left( x^\top \wb\_{V} \right)\right) \left(v\_0 + \exp \left( \hat{x}^\top \wb\_{V} \right) \right)} \\ &\geq \frac{v\_0 K \left( \exp \left( x^\top \wb\_{V} \right) - \exp \left( \hat{x}^\top \wb\_{V} \right) \right)}{(v\_0 + K e)^2}, \numberthis \label{eq:lemma:lower\_revenue\_gap\_intermediate} \end{aligned}$$ where the inequality holds since $\max\left\{\exp \left( x^\top \wb\_{V} \right), \exp \left( \hat{x}^\top \wb\_{V} \right) \right\} \leq e$. To further bound the right-hand side of , we use the fact that 1 + a ≤ ea ≤ 1 + a + a2/2 for all a ∈ [0, 1], which can be easily shown by Taylor expansion. Thus, we get $$\begin{aligned} \sum\_{i \in S^\star} p(i \| S^\star, \wb\_V) - \sum\_{i \in \tilde{S}\_t} p(i \| \tilde{S}\_t, \wb\_V) &\geq \frac{v\_0 K \left( (x - \hat{x})^\top \wb\_V - (\hat{x}^\top \wb\_V)^2/2 \right)}{(v\_0 + K e)^2} \\ &\geq \frac{v\_0 K \left( \delta \epsilon/ \sqrt{d} - (\sqrt{d} \epsilon)^2/2 \right) }{(v\_0 + K e)^2} \\ &\geq \frac{v\_0 K \delta \epsilon }{2 \sqrt{d}(v\_0 + K e)^2}, \end{aligned}$$ where the last inequality holds because $(\sqrt{d} \epsilon)^2 \leq \delta \epsilon/\sqrt{d}$ when $\epsilon \in (0, 1/d\sqrt{d})$. This concludes the proof. ### Proof of Lemma [lemma:lowerKL] [Proof of Lemma [lemma:lowerKL]] Fix a round t, an assortment S̃t, and Ũt. Let U = Ũt. Define mj(S̃t) : = ∑xU ∈ S̃t1{j ∈ U}/K. Let {pi}i ∈ S̃t ∪ {0} and {qi}i ∈ S̃t ∪ {0} be the probabilities of choosing item i under parameterization $\wb\_{V}$ and $\wb\_{V \cup \{j\}}$, respectively. Then, we have $$\begin{aligned} \operatorname{KL}\left( P\_{V} (\cdot \| \tilde{S}\_t ) \\| P\_{V \cup \{j\}} (\cdot \| \tilde{S}\_t) \right) = \sum\_{i \in \tilde{S}\_t \cup \{0\} } p\_i \log \frac{p\_i}{q\_i} \leq \sum\_{i \in \tilde{S}\_t \cup \{0\} } p\_i \frac{p\_i - q\_i}{q\_i} \leq \sum\_{i \in \tilde{S}\_t \cup \{0\} } \frac{(p\_i - q\_i)^2}{q\_i}, \end{aligned}$$ where the first inequality holds because log(1 + x) ≤ x for all x > − 1. Let x̂ = xU. Now, we separately upper bound (pi − qi)2/qi, by analyzing the following three cases: Case 1. The outside option, i = 0. For i = 0, $q\_i \geq \frac{v\_0}{v\_0 + Ke}$. Thus, we have $$\begin{aligned} \|p\_i - q\_i\| &= \left\| \frac{v\_0}{v\_0 + \sum\_{i \in \tilde{S}\_t}\exp\left( x\_i^\top \wb\_V \right) } - \frac{v\_0}{v\_0 + \sum\_{i \in \tilde{S}\_t} \exp\left( x\_i^\top \wb\_{V \cup \{j\} } \right) } \right\| \\ &= \left\| \frac{v\_0}{v\_0 + K\exp\left( \hat{x}^\top \wb\_V \right) } - \frac{v\_0}{v\_0 + K \exp\left( \hat{x}^\top \wb\_{V \cup \{j\} } \right) } \right\| \\ &\leq \frac{v\_0 K }{(v\_0 + K/e)^2} \left\| \exp\left( \hat{x}^\top \wb\_V \right) - \exp\left( \hat{x}^\top \wb\_{V \cup \{j\} } \right) \right\| \\ &= \frac{v\_0 K}{(v\_0 + K/e)^2} \left\| e^{\bar{c}\_1} (\hat{x}^\top \wb\_V - \hat{x}^\top \wb\_{V \cup \{j\} }) \right\| \\ &\leq \frac{v\_0 K e}{(v\_0 + K/e)^2} \left\| \hat{x}^\top \left(\wb\_V - \wb\_{V \cup \{j\} } \right) \right\| \leq \frac{v\_0 K e}{(v\_0 + K/e)^2} \cdot \frac{m\_j(\tilde{S}\_t) \epsilon}{\sqrt{d}} , \end{aligned}$$ where the third equality holds by applying the mean value theorem for the exponential function, with $\bar{c}\_1 := (1-u)(\hat{x}^\top \wb\_V) +u (\hat{x}^\top \wb\_{V \cup \{j\} })$ for some u ∈ (0, 1). Then, there exist an absolute constant C0 such that $$\begin{aligned} \frac{(p\_0 - q\_0)^2}{q\_0} &\leq \frac{v\_0^2 K^2 e^2}{(v\_0 + K/e)^4} \cdot \frac{ \left(m\_j(\tilde{S}\_t)\right)^2 \epsilon^2}{d} \cdot \frac{v\_0 + Ke}{v\_0} \\ &\leq C\_0 \cdot \frac{v\_0 K^2}{(v\_0 + K)^3 } \cdot \frac{ m\_j(\tilde{S}\_t) \epsilon^2}{ d }, \numberthis \label{eq:lower\_pinsker\_case1} \end{aligned}$$ where the last inequality holds since mj(S̃t) ≤ 1. Case 2. i ∈ S̃t and $j \notin U$. Then, for any i ∈ S̃t corresponding to xi = x̂ and $j \notin U$, we have $$\begin{aligned} \|p\_i - q\_i\| &= \left\| \frac{\exp\left( \hat{x}^\top \wb\_{V} \right) }{v\_0 + K \exp\left( \hat{x}^\top \wb\_V \right) } - \frac{\exp\left( \hat{x}^\top \wb\_{V \cup \{j \}} \right)}{v\_0 + K \exp\left( \hat{x}^\top \wb\_{V \cup \{j\} } \right) } \right\| =0 , \end{aligned}$$ where the last equality holds because $\exp\left( \hat{x}^\top \wb\_{V} \right) = \exp\left( \hat{x}^\top \wb\_{V \cup \{j \}} \right)$, given that $j \notin U$. Thus, we get $$\begin{aligned} \sum\_{i \in \tilde{S}\_t, j \notin U} \frac{(p\_i - q\_i)^2}{q\_i} =0, \numberthis \label{eq:lower\_pinsker\_case2} \end{aligned}$$ Case 3. i ∈ S̃t and j ∈ U. Recall that for any i ∈ S̃t, $q\_i \geq \frac{e^{-1}}{v\_0 + Ke}$. Then, for any i ∈ S̃t corresponding to xi = x̂ and j ∈ U, we have $$\begin{aligned} \|p\_i - q\_i\| &= \left\| \frac{\exp\left( \hat{x}^\top \wb\_{V} \right) }{v\_0 + K \exp\left( \hat{x}^\top \wb\_V \right) } - \frac{\exp\left( \hat{x}^\top \wb\_{V \cup \{j \}} \right)}{v\_0 + K \exp\left( \hat{x}^\top \wb\_{V \cup \{j\} } \right) } \right\| \\ &= \left\| \frac{ \exp\left(\bar{c}\_2 \right) }{v\_0 + K \exp\left(\bar{c}\_2 \right) } \cdot \hat{x}^\top \left(\wb\_V - \wb\_{V \cup \{j\} } \right) - \frac{ K \exp\left(2\bar{c}\_2 \right) }{ \left(v\_0 + K \exp\left(\bar{c}\_2 \right) \right)^2} \cdot \hat{x}^\top \left(\wb\_V - \wb\_{V \cup \{j\} } \right) \right\| \\ &= \frac{ \exp\left(\bar{c}\_2 \right) v\_0 }{\left(v\_0 + K \exp\left(\bar{c}\_2 \right) \right)^2} \left\| \hat{x}^\top \left(\wb\_V - \wb\_{V \cup \{j\} } \right) \right\| \\ &\leq \frac{ v\_0 e }{(v\_0 + K/e)^2} \left\| \hat{x}^\top \left(\wb\_V - \wb\_{V \cup \{j\} } \right) \right\| \leq \frac{ v\_0 e }{(v\_0 + K/e)^2} \cdot \frac{m\_j(\tilde{S}\_t) \epsilon}{\sqrt{d}} , \end{aligned}$$ the second equality holds by applying the mean value theorem, with $\bar{c}\_2 := (1-u)(\hat{x}^\top \wb\_V) +u (\hat{x}^\top \wb\_{V \cup \{j\} })$ for some u ∈ (0, 1). Then, there exist an absolute constant C1 such that $$\begin{aligned} \sum\_{i \in \tilde{S}\_t, j \in U} \frac{(p\_i - q\_i)^2}{q\_i} &\leq K m\_j(\tilde{S}\_t) \cdot \frac{ v\_0^2 e^2 }{(v\_0 + K/e)^4} \cdot \frac{\left(m\_j(\tilde{S}\_t)\right)^2 \epsilon^2}{d} \cdot \frac{v\_0 + Ke}{e^{-1}} \\ &\leq C\_1 \cdot \frac{v\_0^2 K}{(v\_0 + K)^3} \cdot \frac{ m\_j(\tilde{S}\_t) \epsilon^2}{ d } , \numberthis \label{eq:lower\_pinsker\_case3} \end{aligned}$$ where the last inequality holds since mj(S̃t) ≤ 1. Combining , , and , we derive that $$\begin{aligned} \sum\_{i \in \tilde{S}\_t \cup \{0\} } \frac{(p\_i - q\_i)^2}{q\_i} &\leq \left( C\_0 \cdot \frac{v\_0 K^2}{(v\_0 + K)^3 } + C\_1 \cdot \frac{v\_0^2 K}{(v\_0 + K)^3} \right) \cdot \frac{ m\_j(\tilde{S}\_t) \epsilon^2}{d} \\ &\leq \max\{C\_0, C\_1\} \cdot \frac{v\_0K}{(v\_0 + K)^2} \cdot \frac{ m\_j(\tilde{S}\_t) \epsilon^2}{d} \\ &= C\_{\operatorname{KL}} \cdot \frac{v\_0K}{(v\_0 + K)^2} \cdot \frac{ m\_j(\tilde{S}\_t) \epsilon^2}{d}, \end{aligned}$$ where CKL = max{C0, C1}. Since M̃j = ∑t = 1Tmj(S̃t) by definition, and subsequently summing over all t = 1 to T, we have $$\begin{aligned} \operatorname{KL}\left( P\_V \\| Q\_{V \cup \{j\} } \right) &= \sum\_{t=1}^T \EE\_{V} \left[ \operatorname{KL}\left( P\_{V} (\cdot \| \tilde{S}\_t ) \\| P\_{V \cup \{j\}} (\cdot \| \tilde{S}\_t) \right) \right] \\ &\leq C\_{\operatorname{KL}} \cdot \frac{v\_0K}{(v\_0 + K)^2} \cdot \frac{ \EE\_{V}[ \tilde{M}\_j]\epsilon^2}{d}, \end{aligned}$$ where the equality holds by the chain rule of relative entropy (cf. Exercise 14.11 of ). This concludes the proof. Proof of Theorem [thm:upperbound] ================================= In this section, we present the proof of Theorem [thm:upperbound]. Note that when the rewards are uniform, the revenue increases as a function of the assortment size. Therefore, maximizing the expected revenue $ R\_{t}(S, \wb)$ across all possible assortments S ∈ S always contains exactly K items. In other words, the size of the chosen assortment St and the size of the optimal assortment St⋆ both equal to K. Main Proof of Theorem [thm:upperbound] -------------------------------------- Before presenting the proof, we introduce useful lemmas, whose proof can be found in Appendix [appsubsec:usefullemmasthm1]. Lemma [lemma:utility] shows the optimistic utility for the context vectors. [lemma:utility] Let $\alpha\_{ti} = x\_{ti}^\top \wb\_{t} + \beta\_t \\| x\_{ti}\\|\_{H\_t^{-1}}$. If $\wb^\star \in \mathcal{C}\_t(\delta)$, then we have $$\begin{aligned} 0 \leq \alpha\_{ti} - x\_{ti}^\top \wb^\star \leq 2 \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}}. \end{aligned}$$ Lemma [lemma:revenuesecondpd] is a K-free elliptical potential lemma that improves upon the one presented in Lemma 10 of in terms of K. Lemma 10 of states: $\sum\_{s=1}^{t} \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb^\star) p\_s(0 \| S\_s, \wb^\star) \\| x\_{si}\\|\_{H\_s(\wb^\star)^{-1} }^2 \leq 2d K \log \left( \lambda\_{t+1} + \frac{2tK}{d} \right)$ and $\sum\_{s=1}^{t} \max\_{i \in S\_s} \\| x\_{si}\\|\_{H\_s(\wb^\star)^{-1} }^2 \leq 2d\left( K + \frac{1}{\kappa} \right) \log \left(\lambda\_{t+1} + \frac{2tK}{d} \right)$, where $H\_t(\wb) = \sum\_{s=1}^{t-1} \Gcal\_s(\wb) + \lambda\_t \Ib\_d$. [lemma:elliptical] Let $H\_t = \lambda \Ib\_d + \sum\_{s=1}^{t-1} \Gcal\_s(\wb\_{s+1})$, where $\Gcal\_s(\wb) = \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb) x\_{si} x\_{si}^\top - \sum\_{i \in S\_s} \sum\_{j \in S\_s} p\_s(i \| S\_s, \wb) p\_s(j \| S\_s, \wb) x\_{si} x\_{sj}^\top$. Suppose λ ≥ 1. Then the following statements hold true: 1. $\sum\_{s=1}^{t} \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(0 \| S\_s, \wb\_{s+1}) \\| x\_{si}\\|\_{H\_s^{-1}}^2 \leq 2d \log \left( 1+ \frac{t}{d \lambda} \right)$, 2. $\sum\_{s=1}^{t} \max\_{i \in S\_s} \\| x\_{si}\\|\_{H\_s^{-1}}^2 \leq \frac{2}{\kappa} d \log \left( 1+ \frac{t}{d \lambda} \right)$. Moreover, we provide a tighter bound for the second derivative of the expected revenue than that presented in Lemma 12 of . Lemma 12 of states: $\left\| \frac{\partial^2 Q}{\partial i \partial j} \right\| \leq 5$. [lemma:revenuesecondpd] Define $Q:\RR^K \rightarrow \RR$, such that for any $\ub = (u\_1, \dots, u\_K) \in \RR^K$, $Q(\ub) = \sum\_{i=1}^K \frac{\exp(u\_i)}{v\_0 + \sum\_{k=1}^K \exp(u\_k)}$. Let $p\_i(\ub) = \frac{\exp(u\_i)}{v\_0 + \sum\_{k=1}^K \exp(u\_k)}$. Then, for all i ∈ [K], we have $$\begin{aligned} \left\| \frac{\partial^2 Q}{\partial i \partial j} \right\| \leq \begin{cases} 3 p\_i(\ub) & \text{if} \,\,\, i=j, \\ 2p\_i(\ub) p\_j(\ub) & \text{if} \,\,\, i \neq j. \end{cases}\end{aligned}$$ Now, we are ready to prove Theorem [thm:upperbound]. [Proof of Theorem [thm:upperbound]] First, we bound the regret as follows: $$\begin{aligned} \Regret (\wb^\star) &= \sum\_{t=1}^T R\_{t}(S\_{t}^\star, \wb^\star) - R\_{t}(S\_{t}, \wb^\star) = \sum\_{t=1}^T \left[ \sum\_{i \in S\_t^\star} p\_t(i \| S\_t^\star, \wb^\star) - \sum\_{i \in S\_t} p\_t (i \| S\_t, \wb^\star) \right] \\ &= \sum\_{t=1}^T \left[ \frac{\sum\_{i \in S\_t^\star} \exp(x\_{ti}^\top \wb^\star)}{v\_0 + \sum\_{j \in S\_t^\star} \exp(x\_{tj}^\top \wb^\star) } - \frac{\sum\_{i \in S\_t} \exp(x\_{ti}^\top \wb^\star)}{v\_0 + \sum\_{j \in S\_t} \exp(x\_{tj}^\top \wb^\star)} \right] \\ &\leq \sum\_{t=1}^T \left[ \frac{\sum\_{i \in S\_t^\star} \exp(\alpha\_{ti})}{v\_0 + \sum\_{j \in S\_t^\star} \exp(\alpha\_{tj}) } - \frac{\sum\_{i \in S\_t} \exp(x\_{ti}^\top \wb^\star)}{v\_0 + \sum\_{j \in S\_t} \exp(x\_{tj}^\top \wb^\star)} \right] \\ &\leq \sum\_{t=1}^T \left[ \frac{\sum\_{i \in S\_t} \exp(\alpha\_{ti})}{v\_0 + \sum\_{j \in S\_t} \exp(\alpha\_{tj}) } - \frac{\sum\_{i \in S\_t} \exp(x\_{ti}^\top \wb^\star)}{v\_0 + \sum\_{j \in S\_t} \exp(x\_{tj}^\top \wb^\star)} \right] \\ &= \sum\_{t=1}^T \tilde{R}\_t(S\_{t}) - R\_{t}(S\_{t}, \wb^\star) , \end{aligned}$$ where the first inequality holds by Lemma [lemma:utility], and the last inequality holds by the assortment selection of Algorithm [alg:minimax]. Now, we define $Q:\RR^K \rightarrow \RR$, such that for all $\ub = (u\_1, \dots, u\_K) \in \RR^K$, $Q(\ub) = \sum\_{i=1}^K \frac{\exp(u\_i)}{v\_0 + \sum\_{j=1}^K \exp(u\_j)}$. Noting that St always contains K elements since the expected revenue is an increasing function in the uniform reward setting, we can write St = {i1, …, iK}. Moreover, for all t ≥ 1, let $\ub\_t = (u\_{ti\_1}, \dots u\_{ti\_K} )^\top = (\alpha\_{ti\_1}, \dots, \alpha\_{ti\_{K}} )^\top$ and $\ub^\star\_t = (u\_{ti\_1}^\star, \dots u\_{ti\_K}^\star )^\top = (x\_{ti\_1}^\top \wb^\star, \dots, x\_{ti\_{K}}^\top \wb^\star)^\top$. Then, by a second order Taylor expansion, we have $$\begin{aligned} \sum\_{t=1}^T \tilde{R}\_t(S\_{t}) - R\_{t}(S\_{t}, \wb^\star) &= \sum\_{t=1}^T Q(\ub\_t) - Q(\ub^\star\_t) \\ &= \underbrace{ \sum\_{t=1}^T \nabla Q(\ub\_t^\star)^\top (\ub\_t - \ub^\star\_t) }\_{\texttt{(A)}} + \underbrace{\frac{1}{2} \sum\_{t=1}^T (\ub\_t - \ub^\star\_t)^\top \nabla^2 Q(\bar{\ub}\_t) (\ub\_t - \ub^\star\_t)}\_{\texttt{(B)}}, \end{aligned}$$ where $\bar{\ub}\_t = (\bar{u}\_{ti\_1}, \dots, \bar{u}\_{ti\_K})^\top \in \RR^K$ is the convex combination of $\ub\_t$ and $\ub^\star\_t$. First, we bound the term `(A)`. $$\begin{aligned} & \sum\_{t=1}^T \nabla Q(\ub\_t^\star)^\top (\ub\_t - \ub^\star\_t) \\ &= \sum\_{t=1}^T \sum\_{i \in S\_t} \frac{\exp(x\_{ti}^\top \wb^\star )}{v\_0 + \sum\_{k\in S\_t} \exp(x\_{tk}^\top \wb^\star ) } (u\_{ti} - u\_{ti}^\star ) - \sum\_{i \in S\_t} \sum\_{j \in S\_t} \frac{\exp(x\_{ti}^\top \wb^\star ) \exp(x\_{tj}^\top \wb^\star )}{(v\_0 + \sum\_{k\in S\_t} \exp(x\_{tk}^\top \wb^\star ))^2 } (u\_{ti} - u\_{ti}^\star ) \\ &= \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) (u\_{ti} - u\_{ti}^\star ) - \sum\_{i \in S\_t} \sum\_{j \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(j \| S\_t, \wb^\star) (u\_{ti} - u\_{ti}^\star ) \\ &= \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \left( 1 - \sum\_{j \in S\_t} p\_t(j \| S\_t, \wb^\star) \right) (u\_{ti} - u\_{ti}^\star ) \\ &= \sum\_{t=1}^T \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \wb^\star) (u\_{ti} - u\_{ti}^\star ) \\ &\leq \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \wb^\star) 2 \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}} \\ &\leq 2 \beta\_T(\delta) \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \wb^\star) \\| x\_{ti} \\|\_{H\_t^{-1}}, \end{aligned}$$ where the first inequality holds by Lemma [lemma:utility], and the last inequality holds because βt is increasing for t ∈ [T]. Now we bound the term `(B)`. Let $p\_i(\bar{\ub}\_t) = \frac{\exp(\bar{u}\_{ti})}{v\_0 + \sum\_{k=1}^K \exp(\bar{u}\_{tk})}$. Then, we have $$\begin{aligned} &\frac{1}{2} \sum\_{t=1}^T (\ub\_t - \ub^\star\_t)^\top \nabla^2 Q(\bar{\ub}\_t) (\ub\_t - \ub^\star\_t) \\ &= \frac{1}{2} \sum\_{t=1}^T \sum\_{i \in S\_t} \sum\_{j \in S\_t}(u\_{ti} - u\_{ti}^\star ) \frac{\partial^2 Q}{\partial i \partial j} (u\_{tj} - u\_{tj}^\star ) \\ &= \frac{1}{2} \sum\_{t=1}^T \sum\_{i \in S\_t} \sum\_{j \in S\_t, j \neq i} (u\_{ti} - u\_{ti}^\star ) \frac{\partial^2 Q}{\partial i \partial j} (u\_{tj} - u\_{tj}^\star ) + \frac{1}{2} \sum\_{t=1}^T \sum\_{i \in S\_t } (u\_{ti} - u\_{ti}^\star ) \frac{\partial^2 Q}{\partial i \partial i} (u\_{ti} - u\_{ti}^\star ) \\ &\leq \sum\_{t=1}^T \sum\_{i \in S\_t} \sum\_{j \in S\_t, j \neq i} \|u\_{ti} - u\_{ti}^\star\| p\_i(\bar{\ub}\_t) p\_j(\bar{\ub}\_t) \|u\_{tj} - u\_{tj}^\star\| + \frac{3}{2} \sum\_{t=1}^T \sum\_{i \in S\_t } (u\_{ti} - u\_{ti}^\star)^2 p\_i(\bar{\ub}\_t), \numberthis \label{eq:term\_B} \end{aligned}$$ where the inequality is by Lemma [lemma:revenuesecondpd]. To bound the first term in , by applying the AM-GM inequality, we get $$\begin{aligned} \sum\_{t=1}^T & \sum\_{i \in S\_t} \sum\_{j \in S\_t, j \neq i} \|u\_{ti} - u\_{ti}^\star\| p\_i(\bar{\ub}\_t) p\_j(\bar{\ub}\_t) \|u\_{tj} - u\_{tj}^\star\| \\ &\leq \sum\_{t=1}^T\sum\_{i \in S\_t} \sum\_{j \in S\_t} \|u\_{ti} - u\_{ti}^\star\| p\_i(\bar{\ub}\_t) p\_j(\bar{\ub}\_t) \|u\_{tj} - u\_{tj}^\star\| \\ &\leq \frac{1}{2} \sum\_{t=1}^T\sum\_{i \in S\_t} \sum\_{j \in S\_t} (u\_{ti} - u\_{ti}^\star)^2 p\_i(\bar{\ub}\_t) p\_j(\bar{\ub}\_t) + \frac{1}{2} \sum\_{i \in S\_t} \sum\_{j \in S\_t} (u\_{tj} - u\_{tj}^\star)^2 p\_i(\bar{\ub}\_t) p\_j(\bar{\ub}\_t) \\ &\leq \sum\_{t=1}^T\sum\_{i \in S\_t} (u\_{ti} - u\_{ti}^\star)^2 p\_i(\bar{\ub}\_t) . \numberthis \label{eq:term\_B\_first} \end{aligned}$$ By plugging into , we have $$\begin{aligned} \frac{1}{2} \sum\_{t=1}^T (\ub\_t - \ub^\star\_t)^\top \nabla^2 Q(\bar{\ub}\_t) (\ub\_t - \ub^\star\_t) &\leq \frac{5}{2} \sum\_{t=1}^T \sum\_{i \in S\_t } (u\_{ti} - u\_{ti}^\star)^2 p\_i(\bar{\ub}\_t) \\ &\leq 10 \sum\_{t=1}^T\sum\_{i \in S\_t } p\_i(\bar{\ub}\_t) \beta\_t(\delta)^2 \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\ &\leq 10 \sum\_{t=1}^T \max\_{i \in S\_t } \beta\_t(\delta)^2 \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\ &\leq 10 \beta\_T(\delta)^2 \sum\_{t=1}^T \max\_{i \in S\_t } \\| x\_{ti} \\|\_{H\_t^{-1}}^2 , \end{aligned}$$ where the second inequality holds by Lemma [lemma:utility]. Combining the upper bound for the terms `(A)` and `(B)`, with probability at least 1 − δ, we have $$\begin{aligned} \sum\_{t=1}^T \tilde{R}\_t(S\_{t}) - R\_{t}(S\_{t}, \wb^\star) &\leq 2 \beta\_T(\delta) \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \wb^\star) \\| x\_{ti} \\|\_{H\_t^{-1}} \\ &+ 10 \beta\_T(\delta)^2 \sum\_{t=1}^T \max\_{i \in S\_t } \\| x\_{ti} \\|\_{H\_t^{-1}}^2. \numberthis \label{eq:regret\_decomposition} \end{aligned}$$ Now, we bound each term of respectively. For the first term, we decompose it as follows: $$\begin{aligned} \sum\_{t=1}^T \sum\_{i \in S\_t} &p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \wb^\star) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &= \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &+ \sum\_{t=1}^T \sum\_{i \in S\_t} \left( p\_t(i \| S\_t, \wb^\star) - p\_t(i \| S\_t, \wb\_{t+1}) \right) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &+ \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \left( p\_t(0 \| S\_t, \wb^\star) - p\_t(0 \| S\_t, \wb\_{t+1}) \right) \\|x\_{ti} \\|\_{H\_t^{-1}}. \numberthis \label{eq:bouns\_first\_decomposition} \end{aligned}$$ To bound the first term on the right-hand side of , we apply the Cauchy-Schwarz inequality. $$\begin{aligned} &\sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &\leq \sqrt{ \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1})} \sqrt{\sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}}^2 } \\ &\leq \frac{\sqrt{v\_0 K}}{(v\_0 + K e^{-1})} \sqrt{T} \sqrt{\sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}}^2 } \\ &\leq \frac{\sqrt{v\_0 K}}{(v\_0 + K e^{-1})} \sqrt{ T \cdot 2d \log \left( 1+ \frac{T}{d \lambda} \right)}, \numberthis \label{eq:bouns\_first\_decomposition\_1} \end{aligned}$$ where the last inequality holds by Lemma [lemma:elliptical]. Now, we bound the second term on the right-hand side of . Let the virtual context for the outside option be xt0 = 0. Then, by the mean value theorem, there exists $\xib\_t = (1-c) \wb^\star + c \wb\_{t+1}$ for some c ∈ (0, 1) such that $$\begin{aligned} & \sum\_{i \in S\_t} \left( p\_t(i \| S\_t, \wb^\star) - p\_t(i \| S\_t, \wb\_{t+1}) \right) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &= \sum\_{i \in S\_t} \nabla p\_t(i \| S\_t, \xib\_t)^\top (\wb^\star - \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1}) \\| x\_{ti} \\|\_{H\_t^{-1}} \\ &= \sum\_{i \in S\_t} \left( p\_t(i \| S\_t, \xib\_t) x\_{ti} - p\_t(i \| S\_t, \xib\_t) \sum\_{j \in S\_t} p\_t(j \| S\_t, \xib\_t) x\_{tj} \right)^\top \!\!(\wb^\star - \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1})\\| x\_{ti} \\|\_{H\_t^{-1}} \\ &\leq \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \left\|x\_{ti}^\top (\wb^\star - \wb\_{t+1}) \right\| p\_t(0 \| S\_t, \wb\_{t+1})\\| x\_{ti} \\|\_{H\_t^{-1}} \\ &+ \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}} \sum\_{j \in S\_t} p\_t(j \| S\_t, \xib\_t) \left\| x\_{tj}^\top (\wb^\star - \wb\_{t+1}) \right\| p\_t(0 \| S\_t, \wb\_{t+1}) \\ &\leq \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} + \left( \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}} \right)^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t}. \\ \end{aligned}$$ Then, since xt0 = 0, we can further bound the right-hand side as: $$\begin{aligned} &\sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} + \left( \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}} \right)^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} \\ &= \sum\_{i \in S\_t } p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} + \left( \sum\_{i \in S\_t \cup \{0\} } p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}} \right)^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} \\ &\leq \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} + \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} \\ &\leq 2\sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} \\ &\leq 2 \beta\_t(\delta) \sum\_{i \in S\_t} p\_t(i \| S\_t, \xib\_t) \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \leq 2 \beta\_t(\delta) \max\_{i \in S\_t} \\| x\_{ti} \\|\_{H\_t^{-1}}^2 , \end{aligned}$$ where the first inequality holds due to Jensen’s inequality and the second-to-last inequality holds by Lemma [lemma:onlineconfidenceset]. Hence, we get $$\begin{aligned} \sum\_{t=1}^T \sum\_{i \in S\_t} \left( p\_t(i \| S\_t, \wb^\star) - p\_t(i \| S\_t, \wb\_{t+1}) \right) p\_t(0 \| S\_t, \wb\_{t+1}) \\|x\_{ti} \\|\_{H\_t^{-1}} &\leq 2 \beta\_T(\delta) \sum\_{t=1}^T \max\_{i \in S\_t} \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\ &\leq \frac{4d }{\kappa} \beta\_T(\delta) \log \left( 1+ \frac{T}{d \lambda} \right) \numberthis \label{eq:bouns\_first\_decomposition\_2} \end{aligned}$$ where the last inequality holds by Lemma [lemma:elliptical]. Finally, we bound the third term on the right-hand side of . By the mean value theorem, there exists $\xib'\_t = (1-c') \wb^\star + c' \wb\_{t+1}$ for some cʹ ∈ (0, 1) such that $$\begin{aligned} &\sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \left( p\_t(0 \| S\_t, \wb^\star) - p\_t(0 \| S\_t, \wb\_{t+1}) \right) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &= \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \nabla p\_t(0 \| S\_t, \xib'\_t)^\top (\wb^\star - \wb\_{t+1}) \\| x\_{ti} \\|\_{H\_t^{-1}} \\ &= - \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \xib'\_t) \sum\_{j \in S\_t} p\_t(j \| S\_t, \xib'\_t) x\_{tj}^\top (\wb^\star - \wb\_{t+1}) \\| x\_{ti} \\|\_{H\_t^{-1}} \\ &\leq \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \\| x\_{ti} \\|\_{H\_t^{-1}} p\_t(0 \| S\_t, \xib'\_t) \sum\_{j \in S\_t} p\_t(j \| S\_t, \xib'\_t) \\|x\_{tj}\\|\_{H\_t^{-1}} \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} \\ &\leq \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \\| x\_{ti} \\|\_{H\_t^{-1}} \sum\_{j \in S\_t} p\_t(j \| S\_t, \xib'\_t) \\|x\_{tj}\\|\_{H\_t^{-1}} \\|\wb^\star - \wb\_{t+1}\\|\_{H\_t} \\ &\leq \beta\_t(\delta) \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \\| x\_{ti} \\|\_{H\_t^{-1}} \sum\_{j \in S\_t} p\_t(j \| S\_t, \xib'\_t) \\|x\_{tj}\\|\_{H\_t^{-1}} \\ &\leq \beta\_t(\delta) \left( \max\_{i \in S\_t} \\| x\_{ti} \\|\_{H\_t^{-1}}\right)^2 = \beta\_t(\delta) \max\_{i \in S\_t} \\| x\_{ti} \\|\_{H\_t^{-1}}^2, \end{aligned}$$ where the third inequality holds by Lemma [lemma:onlineconfidenceset], and the last inequality holds since (maxiai)2 = maxiai2 for any ai ≥ 0. Therefore, we have $$\begin{aligned} \sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) \left( p\_t(0 \| S\_t, \wb^\star) - p\_t(0 \| S\_t, \wb\_{t+1}) \right) \\|x\_{ti} \\|\_{H\_t^{-1}} &\leq \beta\_T(\delta) \sum\_{t=1}^T \max\_{i \in S\_t} \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \\ &\leq \frac{2d}{\kappa} \beta\_T(\delta) \log \left( 1+ \frac{T}{d \lambda} \right), \numberthis \label{eq:bouns\_first\_decomposition\_3} \end{aligned}$$ where the last inequality holds by Lemma [lemma:elliptical]. By plugging , , and into and multiplying 2βT(δ), we get $$\begin{aligned} 2 \beta\_T(\delta) &\sum\_{t=1}^T \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb^\star) p\_t(0 \| S\_t, \wb^\star) \\|x\_{ti} \\|\_{H\_t^{-1}} \\ &\leq 2\sqrt{2} \frac{\sqrt{v\_0 K}}{(v\_0 + K e^{-1})} \beta\_T(\delta) \sqrt{dT} \sqrt{ \log \left( 1+ \frac{T}{d \lambda} \right) } + \frac{12 d}{\kappa} \beta\_T(\delta)^2 \log \left( 1+ \frac{T}{d \lambda} \right) . \numberthis \label{eq:regret\_decomposition\_1} \end{aligned}$$ Moreover, by applying Lemma [lemma:elliptical], we can directly bound the second term of . $$\begin{aligned} 10 \beta\_T(\delta)^2 \sum\_{t=1}^T \max\_{i \in S\_t } \\| x\_{ti} \\|\_{H\_t^{-1}}^2 \leq 10 \beta\_T(\delta)^2 \cdot \frac{2}{\kappa} d \log \left( 1+ \frac{T}{d \lambda} \right). \numberthis \label{eq:regret\_decomposition\_2} \end{aligned}$$ Finally, plugging and into , we obtain $$\begin{aligned} \Regret(\wb^\star) &\leq 2\sqrt{2} \frac{\sqrt{v\_0 K}}{(v\_0 + K e^{-1})} \beta\_T(\delta) \sqrt{dT} \sqrt{ \log \left( 1+ \frac{T}{d \lambda} \right) } + \frac{32d}{\kappa} \beta\_T(\delta)^2 \log \left( 1+ \frac{T}{d \lambda} \right) \\ &= \BigOTilde \left ( \frac{\sqrt{v\_0 K}}{v\_0 + K } d\sqrt{T} + \frac{1}{\kappa}d^2 \right), \end{aligned}$$ where $\beta\_{T}(\delta) =\BigO \left( \sqrt{d} \log T \log K \right)$. This concludes the proof of Theorem [thm:upperbound]. Useful Lemmas for Theorem [thm:upperbound] ------------------------------------------ ### Proof of Lemma [lemma:utility] [Proof of Lemma [lemma:utility]] Under the condition $\wb^\star \in \mathcal{C}\_t(\delta)$, we have $$\begin{aligned} \left\| x\_{ti}^\top \wb\_{t} - x\_{ti}^\top \wb^\star \right\| \leq \\| x\_{ti} \\|\_{H\_t^{-1}} \\| \wb\_{t}- \wb^\star \\|\_{H\_2} \leq \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}}, \end{aligned}$$ where the first inequality is by Hölder’s inequality, and the last inequality holds by Lemma [lemma:onlineconfidenceset]. Hence, it follows that $$\begin{aligned} \alpha\_{ti} - x\_{ti}^\top \wb^\star = x\_{ti}^\top \wb\_{t} - x\_{ti}^\top \wb^\star + \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}} \leq 2 \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}}. \end{aligned}$$ Moreover, from $ x\_{ti}^\top \wb\_{t} - x\_{ti}^\top \wb^\star \geq -\beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}}$, we also have $$\begin{aligned} \alpha\_{ti} - x\_{ti}^\top \wb^\star = x\_{ti}^\top \wb\_{t} - x\_{ti}^\top \wb^\star + \beta\_t(\delta) \\| x\_{ti} \\|\_{H\_t^{-1}} \geq 0. \end{aligned}$$ This concludes the proof. ### Proof of Lemma [lemma:elliptical] [Proof of Lemma [lemma:elliptical]] Since xx⊤ + yy⊤ ≽ xy⊤ + yx⊤ for any $x,y \in \RR^d$, it follows that $$\begin{aligned} &\Gcal\_s(\wb\_{s+1}) \\ &= \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top - \sum\_{i \in S\_s} \sum\_{j \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(j \| S\_s, \wb\_{s+1}) x\_{si} x\_{sj}^\top \\ &= \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top - \frac{1}{2} \sum\_{i \in S\_s} \sum\_{j \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(j \| S\_s, \wb\_{s+1}) (x\_{si} x\_{sj}^\top + x\_{sj} x\_{si}^\top ) \\ &\succeq \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top - \frac{1}{2} \sum\_{i \in S\_s} \sum\_{j \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(j \| S\_s, \wb\_{s+1}) (x\_{si} x\_{si}^\top + x\_{sj} x\_{sj}^\top ) \\ &= \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top - \sum\_{i \in S\_s} \sum\_{j \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(j \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top. \end{aligned}$$ Hence, we have $$\begin{aligned} \Gcal\_s(\wb\_{s+1}) &\succeq \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) \left( 1 - \sum\_{j \in S\_s}p\_s(j \| S\_s, \wb\_{s+1}) \right) x\_{si} x\_{si}^\top \\ &=\sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(0 \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top \numberthis \label{eq:hessian\_for\_elliptical\_lowerbound}, \end{aligned}$$ which implies that $$\begin{aligned} H\_{t+1} \succeq H\_t + \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_t(0 \| S\_t, \wb\_{t+1}) x\_{ti} x\_{ti}^\top. \end{aligned}$$ Then, we get $$\begin{aligned} \det \left( H\_{t+1} \right) \geq \det \left( H\_t \right) \left( 1 + \sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_0(i \| S\_t, \wb\_{t+1}) \\| x\_{ti} \\|\_{H\_{t}^{-1}}^2 \right). \end{aligned}$$ Since λ ≥ 1, for all t ≥ 1, we have $\sum\_{i \in S\_t} p\_t(i \| S\_t, \wb\_{t+1}) p\_0(i \| S\_t, \wb\_{t+1}) \\| x\_{ti} \\|\_{H\_{t}^{-1}}^2 \leq 1$. Then, using the fact that z ≤ 2log(1 + z) for any z ∈ [0, 1], we get $$\begin{aligned} \sum\_{s=1}^{t} \sum\_{i \in S\_s} & p\_s(i \| S\_s, \wb\_{s+1}) p\_s(0 \| S\_s, \wb\_{s+1}) \\| x\_{si}\\|\_{H\_s^{-1}}^2 \\ &\leq 2 \sum\_{s=1}^{t} \log \left( 1 + p\_s(i \| S\_s, \wb\_{s+1}) p\_s(0 \| S\_s, \wb\_{s+1}) \\| x\_{si}\\|\_{H\_s^{-1}}^2 \right) \\ &\leq 2 \sum\_{s=1}^{t} \log \left( \frac{\det (H\_{s+1})}{\det(H\_s)} \right) \\ &\leq 2 d \log \left( \frac{\operatorname{tr}(H\_{t+1})}{d \lambda} \right) \leq 2d \log \left(1 + \frac{t}{d \lambda} \right). \end{aligned}$$ This proves the first inequality. To establish the proof for the second inequality, we return to : $$\begin{aligned} \Gcal\_s(\wb\_{s+1}) &\succeq \sum\_{i \in S\_s} p\_s(i \| S\_s, \wb\_{s+1}) p\_s(0 \| S\_s, \wb\_{s+1}) x\_{si} x\_{si}^\top \succeq \kappa \sum\_{i \in S\_s} x\_{si} x\_{si}^\top, \end{aligned}$$ which implies that $$\begin{aligned} H\_{t+1} = H\_t + \Gcal\_t(\wb\_{t+1}) \succeq H\_t + \kappa \sum\_{i \in S\_t} x\_{ti}x\_{ti}^\top . \end{aligned}$$ Since λ ≥ 1, for all t ≥ 1, we have κmaxi ∈ St∥xti∥Ht− 12 ≤ κ. We then conclude on the same way: $$\begin{aligned} \sum\_{s=1}^{t} \max\_{i \in S\_s} \\| x\_{si}\\|\_{H\_s^{-1}}^2 &\leq \frac{2}{\kappa} \sum\_{s=1}^{t} \log \left( 1 + \kappa\max\_{i \in S\_s} \\| x\_{si}\\|\_{H\_s^{-1}}^2 \right) \\ &\leq \frac{2}{\kappa} \sum\_{s=1}^{t} \log \left( \frac{\det (H\_{s+1})}{\det(H\_s)} \right) \\ &\leq \frac{2}{\kappa} d \log \left( 1+ \frac{t}{d \lambda} \right), \end{aligned}$$ which proves the second inequality. ### Proof of Lemma [lemma:revenuesecondpd] [Proof of Lemma [lemma:revenuesecondpd]] Let i, j ∈ [K]. We first have $$\begin{aligned} \frac{\partial Q}{\partial i} = \frac{e^{u\_i}}{v\_0 + \sum\_{k=1}^K e^{u\_k}} - \frac{e^{u\_i} \left( \sum\_{k=1}^K e^{u\_k} \right) }{(v\_0 + \sum\_{k=1}^K e^{u\_k})^2} \end{aligned}$$ Then, we get $$\begin{aligned} &\frac{\partial^2 Q}{\partial i \partial j} \\ &= \frac{\mathbbm{1}\_{i=j}e^{u\_i}}{v\_0 + \sum\_{k=1}^K e^{u\_k}} - \frac{e^{u\_i} e^{u\_j}}{(v\_0 + \sum\_{k=1}^K e^{u\_k})^2} - \frac{\mathbbm{1}\_{i=j}e^{u\_i} \left( \sum\_{k=1}^K e^{u\_k} \right) + e^{u\_i}e^{u\_j} }{(v\_0 + \sum\_{k=1}^K e^{u\_k})^2} \\ &+ \frac{e^{u\_i} \left( \sum\_{k=1}^K e^{u\_k} \right) 2e^{u\_j} \left(v\_0 + \sum\_{k=1}^K e^{u\_k} \right) }{(v\_0 + \sum\_{k=1}^K e^{u\_k})^4} \\ &= \frac{\mathbbm{1}\_{i=j}e^{u\_i}}{v\_0 + \sum\_{k=1}^K e^{u\_k}} - \frac{e^{u\_i} e^{u\_j}}{(v\_0 + \sum\_{k=1}^K e^{u	arxiv_0000486
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JR was partially supported by NSF Grants DMS-0504212 and DMS-0805003, and also thanks the Department of Pure Mathematics at The University of Adelaide for its hospitality during visits in August 2007 and March 2009.[↩](#fnref1) A noncommutative sigma-model ============================ [1](#fn1) We begin to study a sigma-model in which both the spacetime manifold and the two-dimensional string world-sheet are made noncommutative. We focus on the case where both the spacetime manifold and the two-dimensional string world-sheet are replaced by noncommutative 2-tori. In this situation, we are able to determine when maps between such noncommutative tori exist, to derive the Euler-Lagrange equations, to classify many of the critical points of the Lagrangian, and to study the associated partition function. Introduction ============ Noncommutative geometry is playing an increasingly important role in physical field theories, especially quantum field theory and string theory. Connes proposed a general formulation of action functionals in noncommutative spacetime, and there is now a large literature on noncommutative field theories (surveyed in part in and ). Thus it seems appropriate now to study fully noncommutative sigma-models. In our previous work, we argued that a consistent approach to T-duality for spacetimes X which are principal torus bundles over another space Z, with X possibly equipped with a non-trivial H-flux, forces the consideration of “noncommutative” T-duals in some situations. A special case of this phenomenon was also previously noted by Lowe, Nastase and Ramgoolam. However, this work left open the question of what sort of sigma-model should apply in the situation where the “target space” is no longer a space at all but a noncommutative C\-algebra, and in particular (as this is the simplest interesting case), a noncommutative torus. In classical sigma-models in string theory, the fields are maps g: Σ → X, where Σ is closed and 2-dimensional, and the target space X* is 10-dimensional spacetime. The leading term in the action is S(g) = ∫Σ‖∇g(x)‖2dσ(x), where the gradient and norm are computed with respect to suitable Riemannian (or pseudo-Riemannian) metrics on Σ and X, σ is volume measure on Σ, and critical points of the action are just harmonic maps Σ → X. Usually one adds to a Wess-Zumino term, related to the H-flux, an Einstein term, corresponding to general relativity on X, and various other terms, but here we will focus on (except in Section [sec:WZ], where the Wess-Zumino term will also come up). The question we want to treat here is what should replace maps g: Σ → X and the action when X becomes noncommutative. More precisely, we will be interested in the case where we replace C0(X), the algebra of continuous functions on X vanishing at infinity, by a noncommutative torus. At the end of the paper, we will also comment on what happens in the more complicated case, considered in,, and, where A = Γ0(Z, E) is the algebra of sections vanishing at infinity of a continuous field E of noncommutative 2-tori over a space Z, which plays the role of reduced or “physically observable” spacetime. (In other words, we think of X as a bundle over Z with noncommutative 2-torus fibers.) Naively, since a map g: Σ → X is equivalent to a C\-algebra morphism C0(X) → C(Σ), one’s first guess would be to consider \ -homomorphisms A → C(Σ), where Σ is still an ordinary 2-manifold. The problem with this approach when A is complicated is that often there are no such maps. For example, if A = C0(Z) ⊗ Aθ with θ irrational (this is Γ0(Z, E) for a trivial field E of noncommutative tori over Z), then simplicity of Aθ implies there are no non-zero \* -homomorphisms A → C(Σ). Thus the first thing we see is that once spacetime becomes noncommutative, it is necessary to allow the world-sheet Σ to become noncommutative as well. In most of this paper, we consider a sigma-model based on \* -homomorphisms between noncommutative 2-tori. The first problem is to determine when such maps exist, and this is studied in Section [sec:morphisms]. The main result here is Theorem [thm:homostoirrat], which determines necessary and sufficient conditions for existence of a non-zero \* -homomorphism from AΘ to Mn(Aθ), when Θ and θ are irrational and n ≥ 1. The main section of the paper is Section [sec:harmonic], which studies an energy functional on such \* -homomorphisms. The critical points of the energy are called harmonic maps, and we classify many of them when Θ = θ. We also determine the Euler-Lagrange equations for harmonic maps (Proposition [prop:ELeq]), which are considerably more complicated than in the commutative case. Subsection [sec:rational] deals in more detail with the special case of maps from $C({{\mathbb T}}^2)$ to a rational noncommutative torus. Even this case is remarkably complicated, and we discover interesting connections with the field equations studied in. Section [sec:refinements] deals with various variations on the theory, such as how to incorporate general metrics and the Wess-Zumino term, and what happens when spacetime is a “bundle” of noncommutative tori and not just a single noncommutative torus. Finally, Section [sec:physics] discusses what the partition function for our sigma-model may look like. The authors are very grateful to Joachim Cuntz, Hanfeng Li, and the referee of this paper for several helpful comments. They are especially grateful to Hanfeng Li for writing the appendix, which resolves two problems which were unsolved when the first draft of this paper was written. Classification of morphisms between irrational rotation algebras ================================================================ In principle one should allow replacement of Σ by general noncommutative Riemann surfaces, as defined for example in (in the case of genus 0) and (in the case of genus > 1), but since here we take our spacetimes to be noncommutative tori, it is natural to consider the “genus one” case and to replace C(Σ) by Aθ for some θ. This case was already discussed and studied in, but only in the case of exceptionally simple target spaces X. In fact, in and, X was taken to be S0, i.e., the algebra A was taken to be ${\mathbb C}\oplus{\mathbb C}$. (Or alternatively, one could say that they took $A={\mathbb C}$, but allowed non-unital maps.) We begin by classifying \* -homomorphisms. We begin with the (easy) case of unital maps. [thm:unitalhomostoirrat] Fix Θ and θ in (0, 1), both irrational. There is a unital \* -homomorphism φ: AΘ → Aθ if and only if Θ = cθ + d for some $c,\,d\in{\mathbb Z}$, c ≠ 0. Such a \* -homomorphism φ can be chosen to be an isomorphism onto its image if and only if c = ± 1. Remember from that projections in irrational rotation algebras are determined up to unitary equivalence by their traces, that K0(Aθ) is mapped isomorphically to the ordered group ${\mathbb Z}+ \theta{\mathbb Z}\subset {\mathbb R}$ by the unique normalized trace Tr on Aθ, and that the range of the trace Tr on projections from Aθ itself is precisely $({\mathbb Z}+ \theta{\mathbb Z})\cap [0,\,1]$. Now a unital \* -homomorphism φ: AΘ → Aθ must induce an order-preserving map φ\* of K0 groups sending the class of the identity to the class of the identity. Since both K0 groups are identified with dense subgroups of ${\mathbb R}$, with the induced order and with the class of the identity represented by the number 1, this map can be identified with the inclusion of a subgroup, with 1 going to 1. So Θ, identified with a generator of K0(AΘ), must lie in ${\mathbb Z}+ \theta{\mathbb Z}$, say, Θ = c θ + d for some $c,\,d\in{\mathbb Z}$. That proves necessity of the condition, but sufficiency is easy, since Acθ + d ≅ Acθ is the universal C\-algebra on two unitaries U* and V satisfying UV = e2πicθVU, while Aθ is the universal C\-algebra on two unitaries u* and v satisfying uv = e2πiθvu. So define φ by φ(U) = uc, φ(V) = v, and the required condition is satisfied. Note of course that if c = ± 1, then the images of U and V generate Aθ and φ is surjective, whereas if ∣c∣ ≠ 1, then φ\* is not surjective (and so φ can’t be, either). With notation as in Theorem [thm:unitalhomostoirrat], if c = ± 1, it is natural to ask if it follows that any φ inducing the isomorphism on K0 is a \* -isomorphism. The answer is definitely “no.” In fact, by, which applies because of, for any given possible map K0(AΘ) → K0(Aθ), there is a \* -homomorphism AΘ → Aθ inducing any desired group homomorphism ${{\mathbb Z}}^2 \cong K\_1(A\_\Theta) \to K\_1(A\_\theta)\cong {{\mathbb Z}}^2$, including the 0-map. In particular, Aθ always has proper (i.e., non-invertible) unital \* -endomorphisms. (To prove this, take Θ = θ, and observe that if the induced map on K1 is not invertible, then the endomorphism of Aθ cannot be invertible.) It is not clear, however, whether or not such endomorphisms constructed using the inductive limit structure of can be chosen to be smooth. But Kodaka has constructed smooth unital \* -endomorphisms Φ of Aθ, whose image has nontrivial relative commutant, but only when θ is a quadratic irrational of a certain type. For a slight improvement on his result, see Theorem [thm:endos] below. Note that the de la Harpe-Skandalis determinant Δ, with the defining property $$\Delta(e^y) = \frac{{\operatorname{Tr}}(y)}{2\pi i}\mod {\mathbb Z}+\theta{\mathbb Z},$$ maps the abelianization of the connected component of the identity in the unitary group of Aθ to ${\mathbb C}^\times/({\mathbb Z}+\theta{\mathbb Z})$. Thomsen has proved that everything in the kernel of Δ is a finite product of commutators. But for the element e2πiθ ∈ kerΔ, we get a stronger result. Since (by ) Aθ has a proper \* -endomorphism φ inducing the 0-map on K1, that means there are two unitaries in Aθ (namely, φ(U) and φ(V)) in the connected component of the identity in the unitary group with commutator e2πiθ. As far as \* -automorphisms of Aθ are concerned, some structural facts have been obtained by Elliott, Kodaka, and Elliott-Rørdam. Elliott and Rørdam showed that ${\overline{\operatorname{Inn}}}(A\_\theta)$, the closure of the inner automorphisms, is topologically simple, and that ${\operatorname{Aut}}(A\_\theta)/{\overline{\operatorname{Inn}}}(A\_\theta) \cong GL(2,{\mathbb Z})$. However, if one looks instead at smooth automorphisms, what one can call diffeomorphisms, one sees a different picture. For θ satisfying a certain Diophantine condition, Aut(Aθ∞) is an iterated semidirect product, $(U(A\_\theta^\infty)\_0/{\mathbb T}) \rtimes ({\mathbb T}^2\rtimes SL(2,{\mathbb Z}))$. This is not true without the Diophantine condition, but it may still be that ${\operatorname{Aut}}(A\_\theta^\infty) ={\overline{\operatorname{Inn}}}(A\_\theta^\infty)\rtimes SL(2,{\mathbb Z})$ for all θ. (See Elliott’s review of in MathSciNet.) Next we consider \* -homomorphisms that are not necessarily unital. We can attack the problem in two steps. If there is a non-zero \* -homomorphism φ: AΘ → Mℓ(Aθ), not necessarily unital, then φ(1AΘ) = p is a self-adjoint projection, and imφ ⊆ pMℓ(Aθ)p, which is an algebra strongly Morita-equivalent to Aθ. By, pMℓ(Aθ)p must be isomorphic to Mn(Aβ) for some β in the orbit of θ under the action of $GL(2,{\mathbb Z})$ on ${\mathbb R}$ by linear fractional transformations. So we are essentially reduced to the unital case covered in Theorem [thm:unitalhomostoirrat], except that we have to allow for the possibility of passage to matrix algebras. (This would be the case even if ℓ = 1, since there is not necessarily any relationship between n and ℓ.) This modification is covered in the following: [thm:unitalhomostomatirrat] Fix Θ and θ in (0, 1), both irrational, and $n\in {\mathbb N}$, n ≥ 1. There is a unital \* -homomorphism φ: AΘ → Mn(Aθ) if and only if nΘ = cθ + d for some $c,\,d\in{\mathbb Z}$, c ≠ 0. Such a \* -homomorphism φ can be chosen to be an isomorphism onto its image if and only if n = 1 and c = ± 1. The argument is similar to that for Theorem [thm:unitalhomostoirrat], since K0(Mn(Aθ)) is again isomorphic (as an ordered group) to ${\mathbb Z}+\theta{\mathbb Z}$, but this time the class of the identity is represented by n, so that if both K0 groups are identified with subgroups of ${\mathbb R}$ in the usual way, φ\* must be multiplication by n. Hence if φ exists, $n\Theta \in {\mathbb Z}+\theta{\mathbb Z}$. For the other direction, suppose we know that nΘ = cθ + d. We need to construct an embedding of AΘ into a matrix algebra over Aθ. By, AΘ = A(cθ + d)/n is strongly Morita equivalent to An/(cθ + d), which embeds unitally into A1/(cθ + d) as in the proof of Theorem [thm:unitalhomostoirrat], and A1/(cθ + d) is Morita equivalent to Acθ + d ≅ Acθ, which embeds unitally into Aθ. Stringing things together, we get an embedding of AΘ into a matrix algebra over Aθ. (By, when two unital C\-algebras are Morita equivalent, each one embeds as a corner into a matrix algebra over the other.) So we get a non-zero \ -homomorphism AΘ → Mℓ(Aθ) (not necessarily unital), possibly with ℓ ≠ n. The induced map φ\* on K0 can be identified with an order-preserving homomorphism from ${\mathbb Z}+ \left(\frac{c\theta+d}{n}\right){\mathbb Z}$ to ${\mathbb Z}+ {\theta}{\mathbb Z}$. But in fact we can determine this map precisely, using the fact that the Morita equivalence from A(cθ + d)/n to An/(cθ + d) is associated to multiplication by n/(cθ + d), and the Morita equivalence from A1/(cθ + d) to Acθ + d is associated to multiplication by cθ + d. Thus the composite map φ\* is multiplication by n, and sends the class of 1AΘ to n, which is the class of 1n in K0(Mℓ(Aθ)), where necessarily ℓ ≥ n. Since (by ) projections are determined up to unitary equivalence by their classes in K0, we can conjugate by a unitary and arrange for φ to map AΘ unitally to Mn(Aθ). For the last statement we use, which says that AΘ can be isomorphic to Mn(Aθ) only if n = 1. We can now reorganize our conclusions in a way that is algebraically more appealing. First, it’s helpful in terms of motivation to point out the following purely algebraic lemma, which we suspect is known, though we don’t know where to look it up. [lem:GL2Q] Let M be the submonoid not a subgroup of $GL(2,{\mathbb Q})$ consisting of matrices in $M\_2({\mathbb Z})$ with non-zero determinant, i.e., of integral matrices having inverses that are not necessarily integral. Then M is generated by $GL(2,{\mathbb Z})$ and by the matrices of the form $\begin{pmatrix} r & 0 \\ 0 & 1\end{pmatrix}$, $r\in{\mathbb Z}\smallsetminus \{0\}$. First we recall that applying an elementary row or column operation to a matrix is the same as pre- or post-multiplying by an elementary matrix of the form $\begin{pmatrix} 1 & \star \\ 0 & 1\end{pmatrix}$ or $\begin{pmatrix} 1 & 0 \\ \star & 1\end{pmatrix}$. So it will suffice to show that, given any matrix B ∈ M, we can write it as a product of matrices that reduce via elementary row or column operations (over ${\mathbb Z}$) to things of the form $\begin{pmatrix} \star & 0 \\ 0 & 1\end{pmatrix}$. The proof of this is almost the same as for. Write $B = \begin{pmatrix} b\_{11} & b\_{12} \\ b\_{21} & b\_{22}\end{pmatrix}$. Since B is nonsingular, b11 and b21 can’t both be 0. Suppose bj1 is the smaller of the two in absolute value (or if the absolute values are the same, choose j = 1). Subtracting an integral multiple of the j-th row from the other row, we can arrange to decrease the minimal absolute value of the elements in the first column. Proceeding this way and using the Euclidean algorithm, we can reduce the first column to either $\begin{pmatrix} r\\ 0\end{pmatrix}$ or $\begin{pmatrix} 0\\ r\end{pmatrix}$ (with r the greatest common divisor of the original b11 and b21). Since we can, if necessary, left multiply by the elementary matrix $\begin{pmatrix} 0 & 1 \\ -1& 0\end{pmatrix}$, we can assume the first column has been reduced to $\begin{pmatrix} r\\ 0\end{pmatrix}$, and thus that B has been reduced to the form $$\begin{pmatrix} b\_{11} & b\_{12} \\ 0 & b\_{22}\end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & b\_{22}\end{pmatrix} \begin{pmatrix} 1 & b\_{12} \\ 0 & 1\end{pmatrix} \begin{pmatrix} b\_{11} & 0 \\ 0 & 1\end{pmatrix}.$$ And finally, $\begin{pmatrix} 1 & 0 \\ 0 & b\_{22}\end{pmatrix}$ is conjugate to $\begin{pmatrix} b\_{22} & 0 \\ 0 & 1\end{pmatrix}$ under the elementary matrix $\begin{pmatrix} 0 & 1 \\ -1& 0\end{pmatrix}$. [rem:Mgen] We can relate this back to the proofs of Theorems [thm:unitalhomostoirrat] and [thm:unitalhomostomatirrat]. Elements of M lying in $GL(2,{\mathbb Z})$ correspond to Morita equivalences of irrational rotation algebras. Elements of the form $\begin{pmatrix} r & 0 \\ 0 & 1\end{pmatrix}$ act on θ by multiplication by r, and correspond to inclusions $A\_{r\theta} \hookrightarrow A\_{\theta}$. Lemma [lem:GL2Q] says that general elements of M are built out of these two cases. This motivates the following Theorem [thm:homostoirrat]. [rem:Hecke] The appearance of the monoid $GL(2,{\mathbb Z})\subset M\subset GL(2,{\mathbb Q})$, and also the statement of Lemma [lem:GL2Q], are somewhat reminiscent of the theory of Hecke operators in the theory of modular forms, which also involve the action of the same monoid M (on $GL(2,{\mathbb R})/GL(2,{\mathbb Z})$). [thm:homostoirrat] Fix Θ and θ in (0, 1), both irrational. Then there is a non-zero \* -homomorphism φ: AΘ → Mn(Aθ) for some n, not necessarily unital, if and only if Θ lies in the orbit of θ under the action of the monoid M on ${\mathbb R}$ by linear fractional transformations. The possibilities for Tr(φ(1AΘ)) are precisely the numbers t = cθ + d > 0, $c,\,d\in {\mathbb Z}$ such that $t\Theta\in {\mathbb Z}+ \theta{\mathbb Z}$. Once t is chosen, n can be taken to be any integer ≥ t. First suppose φ exists, and let p = φ(1AΘ). Then φ\: K0(AΘ) → K0(M*n(Aθ)) = K0(Aθ) must be an injection of ordered groups sending 1 ∈ K0(AΘ) to $t={\operatorname{Tr}}(p)=c\theta + d\in {\mathbb Z}+\theta{\mathbb Z}$. Since both groups are dense subgroups of ${\mathbb R}$, this map must be multiplication by t and must send Θ to something in ${\mathbb Z}+ \theta{\mathbb Z}$. So we have tΘ = aθ + b for some $a,\,b\in{\mathbb Z}$, and $$\Theta = \frac{a\theta + b}{c\theta + d}= \begin{pmatrix}a&b\\c&d\end{pmatrix}\cdot \theta.$$ The matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ has integer entries, and can’t be singular since the numerator and denominator are both non-zero (being Tr(φ(q)) and Tr(φ(1)), respectively, where q is a Rieffel projection in AΘ with trace Θ), and (a b) and (c d) can’t be rational multiples of each other (as that would imply Θ is rational). Hence $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ lies in M, and t is as required. And since p ≤ 1n, t ≤ n. To prove the converse, suppose $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in M$ and (cθ + d)Θ = aθ + b. Let t = cθ + d and choose any integer n ≥ t. Since the range of the trace on projections in Mn(Aθ) is $[0,\,n]\cap ({\mathbb Z}+ \theta{\mathbb Z})$, we can choose a self-adjoint projection p ∈ Mn(Aθ) with Tr(p) = t. The subalgebra pMn(Aθ)p of Mn(Aθ) is a full corner (since Aθ is simple), hence is strongly Morita equivalent to Aθ, hence is \* -isomorphic to Mk(Aβ) for some β in the orbit of $GL(2,{\mathbb Z})$ acting on θ. In fact, we can compute k and β; k is the (positive) greatest common divisor of c and d, and β is obtained by completing the row vector $\bigl(\frac{c}{k}\ \frac{d}{k}\bigr)$ to a matrix $$\begin{pmatrix}a'&b'\\ \frac{c}{k} & \frac{d}{k}\end{pmatrix} \in GL(2,{\mathbb Z})$$ and then letting this act on θ. By Theorem [thm:unitalhomostomatirrat], there is a \* -homomorphism φ: AΘ → pMn(Aθ)p ≅ Mk(Aβ) with φ(1AΘ) = p if and only if $k\Theta\in{\mathbb Z}+\beta{\mathbb Z}$. But, by assumption, $$k\Theta = k \frac{a\theta + b}{c\theta + d} = \frac{a\theta + b}{\frac{c}{k}\theta +\frac{d}{k}}$$ while $$\beta = \frac{a'\theta + b'}{\frac{c}{k}\theta +\frac{d}{k}} \quad \text{with} \quad \begin{pmatrix}a'&b'\\ \frac{c}{k} & \frac{d}{k}\end{pmatrix} \in GL(2,{\mathbb Z}).$$ Note that the transpose matrix $$\begin{pmatrix}a'& \frac{c}{k} \\ b' & \frac{d}{k}\end{pmatrix}$$ also lies in $GL(2,{\mathbb Z})$. So we can we can solve for integers r and s such that $$\begin{pmatrix}a'& \frac{c}{k} \\ b' & \frac{d}{k}\end{pmatrix} \begin{pmatrix}r \\ s\end{pmatrix} = \begin{pmatrix}a \\ b\end{pmatrix}.$$ That says exactly that $$\begin{aligned} r\beta + s &= r\cdot \frac{a'\theta + b'}{\frac{c}{k}\theta +\frac{d}{k}} + s\\ &= \frac{r\bigl(a'\theta + b'\bigr) + s\bigl(\frac{c}{k}\theta +\frac{d}{k}\bigr)}{\frac{c}{k}\theta +\frac{d}{k}} \\ &= \frac{\bigl(a'r+\frac{c}{k}s\bigr)\theta + \bigl(b'r+\frac{d}{k}s\bigr)}{\frac{c}{k}\theta +\frac{d}{k}} \\ &= \frac{a\theta + b}{\frac{c}{k}\theta +\frac{d}{k}}\\ &= k\Theta, \end{aligned}$$ as required. Harmonic maps between noncommutative tori ========================================= The action and some of its minima for maps between noncommutative tori ---------------------------------------------------------------------- In this section we consider the analogue of the action functional in the context of the \* -homomorphisms classified in the last section. For simplicity, consider first of all a unital \* -homomorphism φ: AΘ → Aθ as in Theorem [thm:unitalhomostoirrat]. As before, denote the canonical generators of AΘ and Aθ by U and V, u and v, respectively. The natural analogue of S(g) in our situation is $$\label{eq:qlag} \begin{aligned} {\mathcal{L}}(\varphi) &= {\operatorname{Tr}}\Bigl(\delta\_1(\varphi(U))^\\delta\_1(\varphi(U)) + \delta\_2(\varphi(U))^\\delta\_2(\varphi(U)) \\ &\qquad + \delta\_1(\varphi(V))^\\delta\_1(\varphi(V)) + \delta\_2(\varphi(V))^\\delta\_2(\varphi(V))\Bigr). \end{aligned}$$ (Except for a factor of two, this is the same as the sum of the “energies” of the unitaries φ(U) and φ(V) in Aθ, as defined in.) Here δ1 and δ2 are the infinitesimal generators for the “gauge action” of the group ${\mathbb T}^2$ on Aθ. More precisely, δ1 and δ2 are defined on the smooth subalgebra Aθ∞ by the formulas δ1(u) = 2πiu, δ2(u) = 0, δ1(v) = 0, δ2(v) = 2πiv. The derivations δ1 and δ2 play the role of measuring partial derivatives in the two coordinate directions in Aθ (which, we recall, plays the role of the worldsheet Σ), the product of an operator with its adjoint has replaced the norm squared, and integration over Σ has been replaced by the trace. Note for example that if Θ = θ and φ = Id, the identity map, then we obtain $${\mathcal{L}}({\operatorname{Id}}) = {\operatorname{Tr}}\Bigl(\delta\_1(u)^\\delta\_1(u) + 0 + 0 + \delta\_2(v)^\\delta\_2(v)\Bigr) = 8\pi^2.$$ More generally, for the \* -automorphism φA: u ↦ upvq, v ↦ urvs, with $A=\begin{pmatrix}p&q\\r&s\end{pmatrix}\in SL(2,{\mathbb Z})$, we obtain $$\label{eq:minenergy} \begin{aligned} {\mathcal{L}}( \varphi\_A) &= {\operatorname{Tr}}\Bigl(\delta\_1(u^pv^q)^\\delta\_1(u^pv^q) + \delta\_2(u^pv^q)^\\delta\_2(u^pv^q) \\ &\quad + \delta\_1(u^rv^s)^\\delta\_1(u^rv^s) + \delta\_2(u^rv^s)^\\delta\_2(u^rv^s)\Bigr)\\ &= 4\pi^2\Bigl(p^2 + q^2 + r^2 + s^2\Bigr). \end{aligned}$$ [conj:minenergy] The value of L(φA) is minimal among all L(φ), $\varphi{\colon\,}A\_\theta^\infty\circlearrowleft$ a \* -endomorphism inducing the matrix $A\in SL(2,{\mathbb Z})$ on $K\_1(A\_\theta) \cong {\mathbb Z}^2$. Note that this conjecture is a close relative of, which deals with maps C(S1) → Aθ instead of maps AΘ → Aθ. That conjecture said that the multiples of umvn minimize the energy of the unitaries in their connected components. Since L(φ) is twice the sum of the energies of φ(U) and φ(V), immediately implies the present conjecture. The following results provide support for Conjecture [conj:minenergy]. Conjecture is true if $\varphi{\colon\,}A\_\theta^\infty\circlearrowleft$ maps u to a scalar multiple of itself. In this case, p = s = 1 and q = 0. The minimum is achieved precisely when φ(v) = λurv, $\lambda\in {\mathbb T}$. Let φ(u) = μu and φ(v) = w, where $\mu \in {\mathbb T}$, w is unitary and smooth, and (necessarily) uw = e2πiθwu. Since also uv = e2πiθvu, it follows that wv\* is a unitary commuting with u. Since the \* -subalgebra generated by u is maximal abelian, that implies that w = f(u)v, where $f{\colon\,}{\mathbb T}\to {\mathbb T}$ is continuous, and the parameter r is the winding number of f. Now we compute that δ1(f(u)v) = 2πifʹ(u)uv, δ2(f(u)v) = 2πif(u)v, and hence: $$\label{eq:enofphi} \begin{aligned} {\mathcal{L}}(\varphi) &= {\operatorname{Tr}}\Bigl(\delta\_1(u)^\\delta\_1(u) + \delta\_2(u)^\\delta\_2(u) \\ &\qquad + \delta\_1(f(u)v)^\\delta\_1(f(u)v) + \delta\_2(f(u)v)^\\delta\_2(f(u)v)\Bigr)\\ &= 4\pi^2{\operatorname{Tr}}\Bigl(2 + v^\u^\f'(u)^\f'(u)uv\Bigr)\\ &= 4\pi^2{\operatorname{Tr}}\Bigl(2 + f'(u)^\f'(u)\Bigr). \end{aligned}$$ We can pull $f{\colon\,}{\mathbb T}\to {\mathbb T}$ back to a function $[0,\, 1] \to {\mathbb R}$ via the covering map z = e2πit, and then the winding number of f (as a self-map of ${\mathbb T}$) translates into the difference f(1) − f(0) (for f defined on [0, 1]). The problem of minimizing is thus the same as that of minimizing ∫01\|fʹ(t)\|2 dt in the class of smooth functions $f{\colon\,}[0,\,1] \to {\mathbb R}$ with f(1) − f(0) = r. Since such a function can be written as f(t) = f(0) + tr + g(t), with g(0) = g(1) = 0, and fʹ(t) = r + gʹ(t), we have $$\int\_0^1\vert f'(t)\vert^2 \,dt = \int\_0^1 \bigl( r^2 + 2 r g'(t) + g'(t)^2 \bigr) \,dt = r^2 + \Vert g'\Vert^2\_{L^2} \ge r^2,$$ with equality exactly when gʹ ≡ 0, i.e., g constant, and thus g ≡ 0 since g(0) = 0. Thus equality occurs when (going back to the original notation) f(u) = λur, i.e., φ(v) = λurv, for some constant $\lambda\in{\mathbb T}$. We now give a complete proof of Conjecture [conj:minenergy] for \* -automorphisms, in the case where the Diophantine condition of is satisfied. The same proof works in general modulo a technical point which we will discuss below. [thm:varhomphi] Conjecture is true for \* -automorphisms, assuming the Diophantine condition of is satisfied. In other words, if φ is an \* -automorphism of Aθ∞ inducing the map given by $A\in SL(2,{\mathbb Z})$ on K1(Aθ), and if θ satisfies the Diophantine condition of, then L(φ) ≥ L(φA), with equality if and only if φ(u) = λφA(u), φ(v) = μφA(v), for some $\lambda,\mu\in{\mathbb T}$. What we use from is that the hypothesis on θ ensures that we can write φ(u) = λwφA(u)w\, φ(v) = μwφ*A(v)w\, for some $\lambda,\mu\in{\mathbb T}$ and for some unitary w* ∈ Aθ∞. Suppose $A = \begin{pmatrix}p&q\\r&s\end{pmatrix}\in SL(2,{\mathbb Z})$. Since L(φ) is a sum of four terms, all of which have basically the same form, it will be enough to estimate the first term; the estimate for the other three is precisely analogous. We find that $$\begin{gathered} \delta\_1(\varphi(u))=\delta\_1(\lambda w u^p v^q w^\) \\ = \lambda\bigl(\delta\_1(w) u^p v^q w^\ + w 2 \pi i p u^p v^q w^\* + w u^p v^q \delta\_1(w)^\* \bigr),\end{gathered}$$ so the first term in L(φ), ${\operatorname{Tr}}\bigl(\delta\_1(\varphi(u))^\* \delta\_1(\varphi(u)) \bigr)$ is a sum of nine terms, three “principal” terms and six “cross” terms. Note that $\overline\lambda$ in δ1(φ(u))\* cancels the λ in δ1(φ(u)), so we can ignore the λ altogether. The three principal terms are $$\label{eq:prin} \begin{aligned} &{\operatorname{Tr}}\Bigl( \bigl(\delta\_1(w) u^p v^q w^\* \bigr)^\* \bigl(\delta\_1(w) u^p v^q w^\* \bigr) \\ & \qquad + \bigl( w 2 \pi i p u^p v^q w^\\bigr)^\ \bigl( w 2 \pi i p u^p v^q w^\\bigr) \\ & \qquad + \bigl( w u^p v^q \delta\_1(w)^\ \bigr)^\\bigl( w u^p v^q \delta\_1(w)^\ \bigr) \Bigr)\\ &= 2 {\operatorname{Tr}}\bigl( \delta\_1(w)^\* \delta\_1(w) \bigr) + 4\pi^2 p^2, \end{aligned}$$ where in the last step we have used (several times) the invariance of the trace under inner automorphisms. Now consider the six cross-terms. These are $$\label{eq:cross} \begin{aligned} &{\operatorname{Tr}}\Bigl( \bigl(\delta\_1(w) u^p v^q w^\* \bigr)^\* \bigl( w 2 \pi i p \,u^p v^q w^\\bigr) \\ & \qquad + \bigl(\delta\_1(w) u^p v^q w^\ \bigr)^\* \bigl( w u^p v^q \delta\_1(w)^\* \bigr) \\ & \qquad + \bigl( w 2 \pi i p\, u^p v^q w^\\bigr)^\ \bigl(\delta\_1(w) u^p v^q w^\* \bigr) \\ & \qquad + \bigl( w 2 \pi i p \, u^p v^q w^\\bigr)^\ \bigl( w u^p v^q \delta\_1(w)^\* \bigr) \\ & \qquad + \bigl( w u^p v^q \delta\_1(w)^\* \bigr)^\* \bigl(\delta\_1(w) u^p v^q w^\* \bigr) \\ & \qquad + \bigl( w u^p v^q \delta\_1(w)^\* \bigr)^\* \bigl( w 2 \pi i p\, u^p v^q w^\\bigr) \Bigr) \\ & = {\operatorname{Tr}}\Bigl( 2 \pi i p \,\delta\_1(w)^\ w + w \bigl( u^p v^q \bigr)^\* \delta\_1(w)^\* w u^p v^q \delta\_1(w)^\* \\ & \qquad - 2 \pi i p \, w^\\delta\_1(w) - 2 \pi i p \, w \, \delta\_1(w)^\ \\ & \qquad + \delta\_1(w)\bigl( u^p v^q \bigr)^\* w^\* \delta\_1(w) u^p v^q w^\* + 2 \pi i p \, \delta\_1(w) w^\* \Bigr)\\ &= {\operatorname{Tr}}\Bigl( w \bigl( u^p v^q \bigr)^\* \delta\_1(w)^\* w u^p v^q \delta\_1(w)^\* \\ & \qquad + \delta\_1(w)\bigl( u^p v^q \bigr)^\* w^\* \delta\_1(w) u^p v^q w^\* \Bigr). \end{aligned}$$ (Note the use of “integration by parts,”.) Now we put and together. We obtain $$\begin{aligned} {\operatorname{Tr}}\bigl(\delta\_1(\varphi(u))^\* \delta\_1(\varphi(u)) \bigr) &= 4\pi^2 p^2 + {\operatorname{Tr}}\Bigl( 2\delta\_1(w)^\* \delta\_1(w)\\ & \qquad \qquad + w \bigl( u^p v^q \bigr)^\* \delta\_1(w)^\* w u^p v^q \delta\_1(w)^\* \\ & \qquad \qquad + \delta\_1(w)\bigl( u^p v^q \bigr)^\* w^\* \delta\_1(w) u^p v^q w^\* \Bigr). \end{aligned}$$ We make the substitutions T = δ1(w)\*w and W = upvq. Note that W is unitary. We obtain $$\begin{aligned} {\operatorname{Tr}}\bigl(\delta\_1(\varphi(u))^\* \delta\_1(\varphi(u)) \bigr) &= 4\pi^2 p^2 + {\operatorname{Tr}}\Bigl( T T^\* + T^\* T + W^\* T W T + T^\* W^\* T^\* W\Bigr) \\ &\text{(using invariance of the trace under cyclic permutations)}\\ &= 4\pi^2 p^2 + {\operatorname{Tr}}\Bigl( T^\* W W^\* T + T^\* W T^\* W^\* + W T W^\* T \\ & \qquad\qquad \qquad + W T T^\* W^\\Bigr)\\ &= 4\pi^2 p^2 + {\operatorname{Tr}}\Bigl( \bigl( W^\ T + T^\* W^\* \bigr)^\* \bigl( W^\* T + T^\* W^\* \bigr) \Bigr)\\ & \ge 4\pi^2 p^2. \end{aligned}$$ Furthermore, equality holds only if W\*T + T\*W\* = 0, i.e., T = − WT\*W\. Similar estimates with the other three terms in the energy show that ${\mathcal{L}}(\varphi) \ge {\mathcal{L}}(\varphi\_A) = 4\pi^2 \bigl( p^2 + q^2 + r^2 + s^2 \bigr)$, with equality only if δ*j(w)\*w = − Ww\δj(w)W\ and δj(w)\*w = − W1w\δj(w)W1\, where W1 = urvs. (The conditions involving W1 come from the analysis of the last two terms in L(φ), which use the second row of the matrix A.) So if equality holds, W and W1 both conjugate w\δj(w) to the negative of its adjoint. In particular, w\δj(w) commutes with W\*W1. But this unitary generates a maximal abelian subalgebra, so w\δj(w) is a function f* of W\*W1. So w\δj(w) = f(W\W1) with Wf(W\*W1)W\* = − f(W\*W1)\. One can check that these equations can be satisfied only if f* = 0. Indeed, we have the commutation relation WW1 = e2πiθW1W, so W(W\*W1)nW\* = (W1W\)n* = e2πinθ(W\*W1)n. If we expand f in a Fourier series, f(W\*W1) = ∑ncn(W\*W1)n, then we must have $$\begin{aligned} - f(W^\W\_1)^\ &= - \sum\_n \overline{c\_n} (W^\* W\_1)^{-n} = - \sum\_n \overline{c\_{-n}} (W^\* W\_1)^{n} \\ & = \sum\_n c\_n W (W^\* W\_1)^n W^\* = \sum\_n c\_n e^{2\pi i n \theta} (W^\* W\_1)^n. \end{aligned}$$ Equating coefficients gives $$- \overline{c\_{-n}} = c\_n e^{2\pi i n \theta}, \quad \text{and replacing $n$ by $-n$,}\quad - \overline{c\_n} = c\_{-n} e^{-2\pi i n \theta}.$$ These give $$-c\_{-n} = \overline{c\_n} e^{- 2\pi i n \theta} = - c\_{-n} e^{-4\pi i n \theta},$$ so all cn must vanish for n ≠ 0. Thus f is a constant equal to its negative, i.e., f = 0, so δ1(w) = 0 and δ2(w) = 0, w is a scalar, and φ differs from φA only by a gauge transformation. That completes the proof. [rem:nonDio] Note that the same proof always shows that L(φA) ≤ L(φ) for any φ in the orbit of φA under gauge automorphisms and inner automorphisms, and thus, by continuity, under automorphisms in the closure (in the topology of pointwise C∞ convergence) of the inner automorphisms. So if the conjecture of Elliott that ${\operatorname{Aut}}(A\_\theta^\infty) ={\overline{\operatorname{Inn}}}(A\_\theta^\infty)\rtimes SL(2,{\mathbb Z})$ mentioned earlier is true, the Diophantine condition in Theorem [thm:varhomphi] is unnecessary. After the first draft of this paper was written, Hanfeng Li succeeded in proving and Conjecture [conj:minenergy] (in complete generality). His solution is given in the appendix. [rem:enofendo] Of course, so far we have neglected smooth proper \* -endomorphisms of Aθ, which by certainly exist at least for certain quadratic irrational values of θ. We do not know if one can construct such endomorphisms to be energy-minimizing. But we can slightly improve the result of as follows. [thm:endos] Suppose θ is irrational. Then there is a necessarily injective unital \* -endomorphism Φ: Aθ → Aθ, with image $B \subsetneq A\_\theta$ having non-trivial relative commutant and with a conditional expectation of index-finite type from Aθ onto B, if and only if θ is a quadratic irrational number. When this is the case, Φ can be chosen to be smooth. The “only if” direction and the idea behind the “if” direction are both in. We just need to modify his construction as follows. Suppose θ is a quadratic irrational. Thus there exist $a,\,b,\,c\in{{\mathbb Z}}$ with aθ2 + bθ + c = 0, a ≠ 0. Choose $d\in {{\mathbb Z}}$ with 0 < aθ + d < 1, and let e be an orthogonal projection in Aθ with trace aθ + d. Since $$(a\theta + d)\theta = a\theta^2 + d\theta = (d-b)\theta - c \in {{\mathbb Z}}+\theta{{\mathbb Z}},$$ by Theorem [thm:homostoirrat], there is an injective \* -homomorphism φ1: Aθ → Aθ with image eAθe. Let e⊥ = 1 − e. Since Tr(1 − e) = − aθ + 1 − d and $$(-a\theta + 1 - d)\theta = -a\theta^2 + (1-d)\theta = (1+b-d)\theta + c \in {{\mathbb Z}}+\theta{{\mathbb Z}},$$ there is also an injective \* -homomorphism φ2: Aθ → Aθ with image e⊥Aθe⊥. Since eAθe and e⊥Aθe⊥ are orthogonal, Φ = φ1 + φ2 is a unital \* -endomorphism of Aθ whose image has e in its relative commutant. It is clear (since e can be chosen smooth) that Φ can be chosen to be smooth. The last part of the argument can be taken more-or-less verbatim from. Let $$\Psi(x)= \frac12\Bigl( exe + {e^\perp}x{e^\perp}+ \varphi\_2(\varphi\_1^{-1}(exe)) + \varphi\_1(\varphi\_2^{-1}({e^\perp}x{e^\perp})) \Bigr).$$ Then Ψ is a faithful conditional expectation onto the image of Φ, and it has index-finite type as shown in. [rem:end0] As pointed out earlier by Kodaka, the endomorphisms constructed in Theorem [thm:endos] can be constructed to implement a wide variety of maps on K1. In fact, one can even choose Φ so that Φ\* = 0 on K1, with Φ taking both u and v to the connected component of the identity in the unitary group. One can see this as follows. The map Φ constructed in Theorem [thm:endos] can be written as ι ∘ Δ, where Δ: Aθ → Aθ × Aθ is the diagonal map and ι is an inclusion of Aθ × Aθ into Aθ (which exists for θ a quadratic irrational). Since “block direct sum” agrees with the addition in K1, it follows that (in the notation of the proof above) Φ\* = (φ1)\* + (φ2)\* on K1. One can easily arrange to have (φ1)\* = (φ2)\* = Id, which would make Φ\* = multiplication by 2. But if φ3 is the automorphism of Aθ with u ↦ u− 1, v ↦ v− 1 and we replace Φ = ι ∘ Δ by Φʹ = ι ∘ (Id × φ3) ∘ Δ, then since (φ3)\* = − 1 on K1, we get an endomorphism Φʹ inducing the 0-map on K1. In fact, one can modify the construction so that Φ\* is any desired endomorphism of K1. So far we have seen how to get Φ\* = 2 or Φ\* = 0. To get Φ\* = 1, use a construction with three blocks. In other words, choose mutually orthogonal projections e and f in Aθ so that there exist \* -isomorphisms φ1, φ2, and φ3 from Aθ onto each of eAθe, fAθf, and (1 − e − f)Aθ(1 − e − f), respectively. (With a, b, c, d as above, this can be done by choosing Tre = (aθ + d)2 and Trf = (aθ + d)(1 − d − aθ).) As above, one can arrange to have (φ1)\* = (φ2)\* = 1 on K1 and (φ3)\* = − 1. So if Φ = φ1 + φ2 + φ3, Φ is a unital \* -endomorphism inducing multiplication by 1 + 1 − 1 = 1 on K1. Other cases can be done similarly. Euler-Lagrange equations ------------------------ In Proposition [prop:ELeq] below, we determine the Euler-Lagrange equations for the energy functional, L(φ) in. One striking difference with the classical commutative case, is that one cannot get rid of the “integral” Tr in the Euler-Lagrange equations whenever θ is irrational. In Corollary [cor:satisfiesEL], we construct explicit harmonic maps with respect to L. [prop:ELeq] Let L(φ) denote the energy functional for a unital \* -endomorphism φ of Aθ. Then the Euler-Lagrange equations for φ to be a harmonic map, that is, a critical point of L, are: $$0 = \sum\_{j=1}^2\Big\{{\operatorname{Tr}}\left(A\,\delta\_j\left[\varphi(u)^\* \delta\_j(\varphi(u))\right]\right) + {\operatorname{Tr}}\left(B\,\delta\_j\left[\varphi(v)^\\delta\_j (\varphi(v)) \right]\right) \Big\}$$ where A, B* are self-adjoint elements in Aθ, constrained to satisfy the equation, A − φ(v)\Aφ(v) = B* − φ(u)\Bφ(u). Consider the 1-parameter family of \ -endomorphisms of Aθ defined by $$\begin{aligned} \varphi\_t(u) &= \varphi(u) e^{ih\_1(t)} \\ & = \varphi(u)[ 1 + ith\_1'(0) + O(t^2)],\\ \varphi\_t(v) &= \varphi(v) e^{ih\_2(t)} \\ & = \varphi(u)[ 1 + ith\_2'(0) + O(t^2)],\end{aligned}$$ where hj(t), j = 1, 2 are 1-parameter families of self-adjoint operators with h1(0) = 0 = h2(0). Therefore δj(φt(u)) = δj(φ(u)) + itδj(φ(u))h1ʹ(0) + itφ(u)δj(h1ʹ(0)) + O(t2), and taking adjoints, δj(φt(u))\* = δj(φ(u))\* − ith1ʹ(0)δj(φ(u))\* − itδj(h1ʹ(0))φ(u)\* + O(t2), and similarly with v in place of u, h2 in place of h1. Using this, the term of order t in Tr(δj(φt(u))\δj(φ*t(u))) equals $$\label{eqn:der1} \begin{aligned} & i{\operatorname{Tr}}\left(\delta\_j( h\_1'(0))\left( \delta\_j(\varphi(u))^\\varphi(u) - \varphi (u)^\ \delta\_j(\varphi(u))\right)\right)\\ &= -2i {\operatorname{Tr}}\left(\delta\_j( h\_1'(0))\varphi(u)^\* \delta\_j(\varphi(u))\right). \end{aligned}$$ (Here we used the fact that since φ(u) is unitary, δj(φ(u))\*φ(u) + φ(u)\δj(φ(u)) = 0.) Because of “integration by parts”, equation equals 2iTr(h1ʹ(0) δ*j[φ(u)\δj(φ(u))]). Similarly, we calculate the term of order t* in Tr(δj(φt(v))\δj(φ*t(v))) to be 2iTr(h2ʹ(0) δj[φ(v)\δj(φ(v))]). Setting A* = h1ʹ(0), B = h2ʹ(0), we deduce that the Euler-Lagrange equations for L, defined by $0=\frac{d}{dt}{\mathcal{L}}(\varphi\_t)\Big\|\_{t=0}$, are given as in the Proposition. We next differentiate the constraint equations, $$\begin{aligned} 0&= \frac{d}{dt}\left(\varphi\_t(u)\varphi\_t(v) - e^{2\pi i\theta} \varphi\_t(v) \varphi\_t(u)\right)\Big\|\_{t=0} \\ & = \varphi(u)h\_1'(0)\varphi(v) + \varphi(u)\varphi(v) h\_2'(0) - e^{2 \pi i\theta} \left[\varphi(v)h\_2'(0)\varphi(u) + \varphi(v)\varphi(u)h\_1'(0)\right].\end{aligned}$$ Using the fact that φ is a unital \* -endomorphism of Aθ, that is, φ satisfies φ(u)φ(v) = e2πiθφ(v)φ(u) we easily see that the constraint equations of the Proposition are also valid. The following is not especially interesting since it is already implied by the stronger result in, but it illustrates how one might check this condition in some cases. [cor:satisfiesEL] If φA is the \* -automorphism of Aθ∞ defined by φA(u) = upvq and φA(v) = urvs, with $A=\begin{pmatrix}p&q\\r&s\end{pmatrix} \in SL(2,{\mathbb Z})$, then φA is a critical point of L(φ). We compute: $$\begin{aligned} \delta\_1(\varphi\_A(u)) & = 2\pi i p \varphi\_A(u), \quad \delta\_2 (\varphi\_A(u)) = 2\pi i q\varphi\_A(u), \\ \delta\_1(\varphi\_A(v)) &= 2\pi i r \varphi\_A(v), \quad \delta\_2 (\varphi\_A(v)) = 2\pi i s \varphi\_A(v).\end{aligned}$$ Therefore $$\begin{aligned} \varphi\_A(u)^\\delta\_1(\varphi\_A(u)) & = 2\pi i p, \\ \varphi\_A(u)^\\delta\_2(\varphi\_A(u)) & = 2\pi i q, \\ \varphi\_A(v)^\\delta\_1(\varphi\_A(v)) & = 2\pi i r, \\ \varphi\_A(v)^\\delta\_2(\varphi\_A(v)) & = 2\pi i s, \end{aligned}$$ Applying any derivation δj, j = 1, 2, to any of the terms above gives zero, since they are all constants. Therefore φA is a critical point of L, by the Euler-Lagrange equations in Proposition [prop:ELeq]. Of course, a major question is to determine how many critical points there are for L aside from those of the special form φA, $A\in SL(2,{{\mathbb Z}})$. Certain maps between rational noncommutative tori ------------------------------------------------- In this subsection we investigate certain harmonic maps between rational noncommutative tori. This is an exception to our general focus on irrational rotation algebras, but it might shed some light on what seems to be the most difficult case, of (possibly nonunital) maps φ: AΘ → Mm(Aθ) implementing a Morita equivalence when $$\Theta = 1/\theta = \begin{pmatrix}0&1\\1&0 \end{pmatrix}\cdot \theta.$$ In effect, we consider this same situation, but in the case where Θ = n > 1 is a positive integer, so that $A\_\Theta=C({{\mathbb T}}^2)$, the universal C\-algebra generated by two commuting unitaries U* and V. In this case, Aθ = A1/n is an algebra of sections of a bundle over ${{\mathbb T}}^2$ with fibers $M\_n({{\mathbb C}})$. This bundle is in fact the endomorphism bundle of a complex vector bundle V over ${{\mathbb T}}^2$, with Chern class c1(V) ≡ 1 (mod n). (More generally, Ak/n is the algebra of sections of the endomorphism bundle of a vector bundle of Chern class ≡ k (mod n); one can see this, for instance, from the explicit description of the algebra in.) If u and v are the canonical unitary generators of A1/n, then un and vn are both central, and generate the center of A1/n, which is isomorphic to C(T2), the copy of ${{\mathbb T}}^2$ here being identified with the spectrum of the algebra A1/n. Since the normalized trace on A1/n sends 1 to 1, it takes the value $\frac{1}{n}$ on rank-one projections e, which exist in abundance. (The fact that there are lots of global rank-one projections is due to the fact that the Dixmier-Douady invariant of the algebra vanishes.) A choice of e determines a \* -isomorphism φe from $A\_0=A\_n=C({{\mathbb T}}^2)$ to eA1/ne, sending U to eun and V to evn. Let us compute the action functional on φe. [prop:ratLandi] With notation as above, i.e., with e a self-adjoint projection in A1/n and $$\varphi\_e{\colon\,}C({{\mathbb T}}^2)\xrightarrow{\cong} e A\_{1/n}e,\qquad \varphi\_e(U)=eu^n,\,\varphi\_e(V)=ev^n,$$ we have L(φe) = 2Tr(δ1(e)2 + δ2(e)2 + 4π2n2) . Thus, up to a renormalization, this is the same as the action functional on e as defined in. Thus φe is harmonic exactly when e is harmonic. We have $$\delta\_1(eu^n)=\delta\_1(e)\,u^n + 2\pi i n\, e\,u^n = \bigl(\delta\_1(e)+2\pi i n\, e\bigr)u^n \text{ and } \delta\_2(eu^n)=\delta\_2(e)\,u^n\,,$$ and similarly for evn (with the roles of δ1 and δ2 reversed). Since un and vn are central, they cancel out when we compute $\bigl(\delta\_1(eu^n)\bigr)^\\delta\_1(eu^n)$, etc., and we obtain $$\begin{aligned} \bigl(\delta\_1(eu^n)\bigr)^\\delta\_1(eu^n) &= \bigl(\delta\_1(e)+2\pi i n\,e\bigr)^\* \bigl(\delta\_1(e)+2\pi i n\,e\bigr) \\ & = \bigl(\delta\_1(e)\bigr)^2 + 2\pi i n \bigl(\delta\_1(e)e-e\delta\_1(e)\bigr) +4\pi^2 n^2,\\ \bigl(\delta\_2(eu^n)\bigr)^\\delta\_2(eu^n) &= \bigl(\delta\_2(e)\bigr)^2,\\ \bigl(\delta\_1(ev^n)\bigr)^\\delta\_1(ev^n) &= \bigl(\delta\_1(e)\bigr)^2,\\ \bigl(\delta\_2(ev^n)\bigr)^\\delta\_2(ev^n) &= \bigl(\delta\_2(e)\bigr)^2 + 2\pi i n \bigl(\delta\_2(e)e-e\delta\_2(e)\bigr) +4\pi^2 n^2,\\ \end{aligned}$$ and the result follows since the “cross-terms” have vanishing trace. While a complete classification seems difficult, we at least have an existence theorem. [thm:harmonicrat] There exist harmonic nonunital \ -isomorphisms $\varphi\_e{\colon\,}C({{\mathbb T}}^2) \to A\_{1/n}$. By Proposition [prop:ratLandi], it suffices to show that A1/n contains harmonic rank-1 projections. In terms of the realization of A1/n as Γ(T2, End(V)), the sections of the endomorphism bundle of the complex vector bundle V, this is equivalent to showing that ${{\mathbb P}}(V)$, the ${{\mathbb C}}{{\mathbb P}}^{n-1}$-bundle over ${{\mathbb T}}^2$ whose fiber at a point x is the projective space of 1-dimensional subspaces of Vx, has harmonic sections for its natural connection. One way to prove this is by using holomorphic geometry. Realize ${{\mathbb T}}^2$ as an elliptic curve $E={{\mathbb C}}/({{\mathbb Z}}+i{{\mathbb Z}})$ and V as a holomorphic bundle. Then a holomorphic section of ${{\mathbb P}}(V)$ is certainly harmonic. But a holomorphic section of ${{\mathbb P}}(V)$ will exist provided V has an everywhere non-vanishing holomorphic section s, since the line through s(z) is a point of ${{\mathbb P}}(V\_z)$ varying holomorphically with z. Since n = rankV > dimE = 1, this is possible by, assuming that V has “sufficient holomorphic sections,” i.e., that there is a holomorphic section through any point in any fiber. The condition of having sufficient sections is weaker than being ample, which we can arrange by changing c1(V) to be sufficiently positive (recall that only c1(V) mod n is fixed, so we have this flexibility). In preparation for Example [ex:rat2] below, it will be useful to give a concrete model for the algebra A1/n. [prop:A1n] Let n > 1, and let ζ = e2πi/n. Fix the n × n matrices $$u\_0 = \begin{pmatrix} 0 & 1 & 0& \cdots\\ 0 & 0 & 1 &\cdots\\ 0 & 0 & 0 &\ddots\\ 1& 0 & 0 &\cdots \end{pmatrix},\qquad v\_0 = \begin{pmatrix} 1 & 0 & 0& \cdots\\ 0 & \zeta & 0 &\cdots\\ 0 & 0 & \zeta^2 &\cdots\\ 0& 0 & 0 &\ddots \end{pmatrix},$$ or in other words $v\_0 = \text{\textup{diag}}(1,\zeta,\zeta^2,\cdots,\zeta^{n-1})$. Then A1/n can be identified with the algebra of continuous functions $f{\colon\,}{{\mathbb T}}^2 \to M\_n({{\mathbb C}})$ satisfying the transformation rules $$\begin{cases} f(\zeta\lambda,\,\mu) = v\_0^{-1} f(\lambda,\,\mu) v\_0,\\ f(\lambda,\,\zeta\mu) = u\_0 f(\lambda,\,\mu) u\_0^{-1}. \end {cases}$$ Observe that u0n = v0n = 1 and that u0v0 = ζv0u0. It is then easy to see that the most general irreducible representation of A1/n is equivalent to one of the form πμ, λ: u ↦ μu0, v ↦ λv0 for some $(\mu,\,\lambda) \in {{\mathbb T}}^2$. However, we are “overcounting,” because it is clear that v0− 1 conjugates πμ, λ to πζμ, λ, and u0 conjugates πμ, λ to πμ, ζλ. The spectrum of the algebra A1/n can thus be identified with the quotient of ${{\mathbb T}}^2$ by the action by multiplication by n-th roots of unity in both coordinates. The result easily follows. [ex:rat2] We now give a specific example of this situation in which one can write down an explicit harmonic map. We suspect one can do something similar in general, but to make the calculations easier, we restrict to the case n = 2. Proposition [prop:A1n] describes A1/2 as the algebra of continuous functions $f{\colon\,}{{\mathbb T}}^2\to M\_2({{\mathbb C}})$ satisfying $$\label{eq:fconds} \begin{aligned} f(-\lambda,\mu) &= \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix} f(\lambda,\mu) \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix},\\ f(\lambda,-\mu) &= \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix} f(\lambda,\mu) \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}. \end{aligned}$$ If we write λ = eiθ1 and μ = eiθ2, we can rewrite by thinking of $$f=\begin{pmatrix} f\_{11}&f\_{12}\\f\_{21}&f\_{22} \end{pmatrix}$$ as defined on [0, π] × [0, π], subject to boundary conditions $$\label{eq:fboundconds} \begin{alignedat}{2} f\_{11}(\pi,\theta\_2)&=f\_{11}(0,\theta\_2), &\qquad f\_{22}(\pi,\theta\_2)&=f\_{22}(0,\theta\_2), \\ f\_{12}(\pi,\theta\_2)&=-f\_{12}(0,\theta\_2), &\qquad f\_{21}(\pi,\theta\_2)&=-f\_{21}(0,\theta\_2), \\ f\_{11}(\theta\_1,\pi)&=f\_{22}(\theta\_1,0), &\qquad f\_{22}(\theta\_1,\pi)&=f\_{11}(\theta\_1,0), \\ f\_{12}(\theta\_1,\pi)&=f\_{21}(\theta\_1,0), &\qquad f\_{21}(\theta\_1,\pi)&=f\_{12}(\theta\_1,0). \end{alignedat}$$ To get a nonunital harmonic map inducing an isomorphism from $C({{\mathbb T}})$ to a nonunital subalgebra of A1/2, we need by Proposition [prop:ratLandi] to choose f satisfying so that for all θ1 and θ2, f(θ1, θ2) is self-adjoint with trace 1 and determinant 0, and so that f is harmonic. The conditions as well as the conditions for f to be a rank-one projection will be satisfied provided that f is of the form: $$\label{eq:f} f\left({\theta\_1},\, {\theta\_2}\right) = \frac{1}{2}\begin{pmatrix} 1 +\cos(g(\theta\_1))\,\cos\theta\_2 & \sin(g(\theta\_1)) - i \cos(g(\theta\_1))\,\sin\theta\_2\\ \sin(g(\theta\_1)) + i \cos(g(\theta\_1))\,\sin\theta\_2 & 1-\cos(g(\theta\_1))\,\cos\theta\_2 \end{pmatrix}$$ with g real-valued and satisfying the conditions $$\label{eq:gh} g(0)=-\frac{\pi}{2},\qquad g(\pi)=\frac{\pi}{2}.$$ For f to be harmonic, we need to make sure it satisfies the Euler-Lagrange equation f(Δf) = (Δf)f, which is derived in. In the realization of Proposition [prop:A1n], the canonical generators of A1/2 are given by u(eiθ1, eiθ2) = eiθ1u0, v(eiθ1, eiθ2) = eiθ2v0, so that δ1 and δ2 act by $2\pi\frac{\partial}{\partial \theta\_1}$ and $2\pi\frac{\partial}{\partial \theta\_2}$, respectively. Thus up to a factor of 4π2, Δ can be identified with the usual Laplacian in the variables θ1 and θ2. A messy calculation, which we performed with $\text{\textsl{Mathematica}}^{\text{\textregistered}}$, though one can check it by hand, shows that the commutator of f and Δf vanishes exactly when the function g in satisfies the nonlinear (pendulum) differential equation 2gʺ(θ) + sin(2g(θ)) = 0. Subject to the boundary conditions, this has a unique solution, which Mathematica plots as in Figure [fig:gtheta]. [fig:gtheta] Plot of g(θ) as computed by Mathematica Note incidentally that Mathematica calculations show that this solution is neither self-dual nor anti-self-dual, in the sense of. In fact, writing out the self-duality and anti-self-duality equations for a projection of the form shows that they reduce to gʹ(θ) = ± cos(g(θ)), so the only self-dual or anti-self-dual projections of this form satisfying the initial condition g(0) = − π/2 are constant (and thus don’t satisfy the other boundary condition in ). It may be of interest to compute the value of L for this example. The normalized trace TrA on A1/2 for matrix-valued functions f satisfying is $${\operatorname{Tr}}\_A f = \frac{1}{2\pi^2}\int\_0^\pi\int\_0^\pi {\operatorname{Tr}}f(\theta\_1,\theta\_2)\,d\theta\_1\, d\theta\_2,$$ so by Proposition [prop:ratLandi], $$\begin{aligned} {\mathcal{L}}(\varphi\_f) & = 2 {\operatorname{Tr}}\_A \left(\delta\_1(f)^2 + \delta\_2(f)^2 + 4\pi^2n^2\right)\text{ with } n=2\\ &= 2 {\operatorname{Tr}}\_A \left(4\pi^2 \left(\frac{\partial f}{\partial \theta\_1}\right)^2+ 4\pi^2 \left(\frac{\partial f}{\partial \theta\_2}\right)^2+ 4\pi^2\cdot 4\right) \\ &= 8\pi^2 \left(4 + \frac{1}{2\pi^2}\int\_0^\pi\int\_0^\pi {\operatorname{Tr}}\left( \left(\frac{\partial f}{\partial \theta\_1}\right)^2 + \left(\frac{\partial f}{\partial \theta\_2}\right)^2 \right) \,d\theta\_1\, d\theta\_2\right) \\ &= 8\pi^2 \left(4 + 0.1116\right) \approx 32.89 \pi^2. \end{aligned}$$ (The integral was computed numerically with Mathematica.) Variations and Refinements ========================== One can argue that what we have done up till now was somewhat special, in that we took a very special form for the metric on the “worldsheet,” and ignored the Wess-Zumino term in the action. In this section, we discuss how to generalize the results given earlier in the paper. The modifications to the proofs given in the earlier sections are routine, and most arguments will not be repeated. Spectral triples and sigma-models --------------------------------- In this subsection, we write a general sigma-model energy functional for spectral triples, that specializes to the cases existing in the literature, including what was discussed earlier in the paper. It is an explicit variant of the discussion in and. Recall that a spectral triple (A, H, D) is given by an involutive unital algebra A represented as bounded operators on a Hilbert space H and a self-adjoint operator D with compact resolvent such that the commutators [D, a] are bounded for all a ∈ A. A spectral triple (A, H, D) is said to be even if the Hilbert space H is endowed with a ${\mathbb Z}\_2$-grading γ which commutes with all a ∈ A and anti-commutes with D. Suppose in addition that (A, H, D) is (2, ∞)-summable, which means (assuming for simplicity that D has no nullspace) that Trω(a∣D∣− 2) < ∞, where Trω denotes the Dixmier trace. We recall from VI.3 in that ψ2(a0, a1, a2) = Tr((1 + γ)a0[D, a1][D, a2]) defines a positive Hochschild 2-cocycle on A, where $\gamma = \left(\begin{array}{cc}1 & 0 \\0 & -1\end{array}\right)$ is the grading operator on H, and where Tr denotes the Dixmier trace composed with D− 2. In this paper, although we consider the canonical trace Tr instead of the above trace, all the properties go through with either choice. Using the Dixmier trace Trω composed with D− 2 has the advantage of scale invariance, i.e., it is invariant under the replacement of D by λD for any nonzero $\lambda \in {\mathbb C}$, which becomes relevant when one varies the metric, although for special classes of metrics, the scale invariance can be obtained by other means also. The positivity of ψ2 means that ⟨a0 ⊗ a1, b0 ⊗ b1⟩ = ψ2(b0\*a0, a1, b1\) defines a positive sesquilinear form on A ⊗ A. We now give a prescription for energy functionals in the sigma-model consisting of homomorphisms φ: B → A, from a smooth subalgebra of a C\-algebra B with target the given even (2, ∞)-summable spectral triple (A, H, D). Observing that φ\(ψ2) is a positive Hochschild 2-cocycle on B, we need to choose a formal “metric” on B, which is a positive element G* ∈ Ω2(B) in the space of universal 2-forms on B. Then evaluation LG, D(φ) = φ\(ψ2)(G) ≥ 0 defines a general sigma-model action. Summarizing, the data for a general sigma-model action consists of 1. A (2, ∞)-summable spectral triple (A, H, D); 2. A positive element G* ∈ Ω2(B) in the space of universal 2-forms on B, known as a metric on B. Consider a unital C\-algebra generated by the n* unitaries {Uj : i = 1, …n}, with finitely many relations as in, and let B be a suitable subalgebra consisting of rapidly vanishing series whose terms are (noncommutative) monomials in the Ui’s. Then a choice of positive element G ∈ Ω2(B) (or metric on B) is given by G = ∑j, k = 1nGjk(dUj)\dU*k, where the matrix (Gjk) is symmetric, real-valued, and positive definite. Then we compute the energy functional in this case, LG, D(φ) = φ\(ψ2)(G) = ∑j, k* = 1nGjkTr((1 + γ)[D, φ(Uj)\][D, φ(U*k)]) ≥ 0. The Euler-Lagrange equations for φ to be a critical point of LD can be derived as in Proposition [prop:ELeq], but since the equations are long, we omit them. We next give several examples of this sigma-model energy functional. In all of these cases, the target algebra A will be Aθ∞. The first example is the Dabrowski-Krajewski-Landi model, consisting of non-unital \* -homomorphisms $\varphi\colon {\mathbb C}\longrightarrow A\_\theta^\infty$. Note that φ(1) = e is a projection in the noncommutative torus Aθ, and for any (2, ∞)-summable spectral triple (Aθ∞, H, D) on the noncommutative torus, our sigma-model energy functional is LD(φ) = Tr[(1 + γ)[D, e][D, e]]. Choosing the even spectral triple given by ${\mathcal{H}}= L^2(A\_\theta) \otimes {\mathbb C}^2$ consisting of the Hilbert space closure of Aθ in the canonical scalar product coming from the trace, tensored with the 2-dimensional representation space of spinors, and D = γ1δ1 + γ2δ2, where $$\gamma\_1 = \left(\begin{array}{cc}0 & 1 \\1 & 0\end{array}\right), \quad \gamma\_2 = \left(\begin{array}{cc}0 & -i \\ i & 0\end{array}\right)$$ are the Pauli matrices, we calculate that LD(φ) = ∑j = 12Tr[(δje)2], recovering the action in and the Euler-Lagrange equation (Δe)e = e(Δe) there. Next, we consider the model in Rosenberg, consisting of unital \* -homomorphisms φ: C(S1) → Aθ∞. Let U be the unitary given by multiplication by the coordinate function z on S1 (considered as the unit circle ${{\mathbb T}}$ in $ {\mathbb C}$). Then φ(U) is a unitary in the noncommutative torus Aθ, and for any (2, ∞)-summable spectral triple (Aθ∞, H, D) on the noncommutative torus, our sigma-model energy functional is LD(φ) = Tr[(1 + γ)[D, φ(U)\][D, φ(U)]]. Choosing the particular spectral triple on the noncommutative torus as above, we calculate that LD(φ) = ∑j* = 12Tr[(δj(φ(U)))\δj(φ(U))], recovering the action in and the Euler-Lagrange equation φ(U)\Δ(φ(U)) + (δ1(φ(U)))\*δ1(φ(U)) + (δ2(φ(U)))\*δ2(φ(U)) = 0 there. The final example is the one treated in this paper. For any (smooth) homomorphism φ: AΘ → Aθ and any (2, ∞)-summable spectral triple (Aθ∞, H, D), and any positive element G ∈ Ω2(AΘ) (or metric on AΘ) given by G = ∑j, k = 12Gij(dUj)\dU*k, the energy of φ is LG, D(φ) = φ\(ψ2)(G) = ∑j, k* = 12GjkTr((1 + γ)[D, φ(Uj)\][D, φ(U*k)]) ≥ 0. where U, V are the canonical generators of AΘ. Choosing the particular spectral triple on the noncommutative torus as above, we obtain the action and Euler-Lagrange equation considered in §3. One can consider other choices of spectral triples on Aθ defined as follows. For instance, let $g = \left(\begin{array}{cc} g\_{11} & g\_{12} \\g\_{21} & g\_{22}\end{array}\right) \in M\_2({\mathbb R})$ be a symmetric real-valued positive definite matrix. Then one can consider the 2-dimensional complexified Clifford algebra, with self-adjoint generators $\gamma\_\mu \in M\_2({\mathbb C})$ and relations γμγν + γνγμ = gμν, μ, ν = 1, 2, where (gμν) denotes the matrix g− 1. Then with H as before, define D = ∑μ = 12γμδμ. The energy in this more general case is LG, D(φ) = φ\(ψ2)(G) = ∑j, k* = 12∑μ, ν = 12GjkgμνTr(δμ(φ(Uj))\δν(φ(U*k)) ≥ 0. In this case, the trace Tr is either the Dixmier trace composed with D− 2, or the canonical trace on Aθ multiplied by the factor $\sqrt{\det(g)}$, to make the energy scale invariant. The Euler-Lagrange equations in this case are an easy modification of those in Proposition [prop:ELeq]. The Wess-Zumino term -------------------- There is a rather large literature on “noncommutative Wess-Zumino theory” or “noncommutative WZW theory,” referred to in and summarized in part in the survey articles and. Most of this literature seems to deal with the Wess-Zumino-Witten model (where spacetime is a compact group) or with the Moyal product, but we have been unable to find anything that applies to our situation where both spacetime and the worldsheet are represented by noncommutative C\-algebras (or dense subalgebras thereof). For that reason, we will attempt here to reformulate the theory from scratch. The classical Wess-Zumino term is associated to a closed 3-form H* with integral periods on X (the spacetime manifold). If Σ2 is the boundary of a 3-manifold W3, and if φ: Σ → X extends to $\widetilde\varphi{\colon\,}W\to X$, the Wess-Zumino term is $${\mathcal{L}}\_{WZ}(\varphi)=\int\_W (\widetilde\varphi)^\(H).$$ The fact that H* has integral periods guarantees that e2πiLWZ(φ) is well-defined, i.e., independent of the choice of W and the extension $\widetilde\varphi$ of φ. To generalize this to the noncommutative world, we need to dualize all spaces and maps. We replace X by B (which in the classical case would be C0(X)), Σ by A, and W by ${\mathcal C}$. Since H classically was a cochain on X (for de Rham cohomology), it becomes an odd cyclic cycle on B. The integral period condition can be replaced by requiring $$\label{eq:intper} \langle H, u\rangle \in {{\mathbb Z}}$$ for all classes u ∈ K1(B) (dual K-theory, defined via spectral triples or some similar theory). The inclusion $\Sigma\hookrightarrow W$ dualizes to a map $q{\colon\,}{\mathcal C}\to {\mathcal{A}}$, and we suppose φ: B → A has a factorization $$\xymatrix{& {\mathcal C}\ar[d]^q \\ {\mathcal{B}}\ar[r]^\varphi \ar@{.>}[ru]^{\widetilde\varphi} & \,{\mathcal{A}}. }$$ The noncommutative Wess-Zumino term then becomes $${\mathcal{L}}\_{WZ}(\varphi)=\langle {\widetilde\varphi}\_\(H), [{\mathcal C}] \rangle,$$ with $[{\mathcal C}]$ a cyclic cochain corresponding to integration over W. The integral period condition is relevant for the same reason as in the classical case—if we have another “boundary” map $q'{\colon\,}{\mathcal C}'\to {\mathcal{A}}$ and corresponding $\widetilde\varphi'{\colon\,}{\mathcal{B}}\to {\mathcal C}'$, and if ${\mathcal C}\oplus\_{\mathcal{A}}{\mathcal C}'$ is “closed,” so that $[{\mathcal C}]-[{\mathcal C}']$ corresponds to a class $u\in K^1({\mathcal C}\oplus\_{\mathcal{A}}{\mathcal C}')$, then $$\langle {\widetilde\varphi}\_\(H), [{\mathcal C}]\rangle -\langle {\widetilde\varphi'}\_\(H), [{\mathcal C}']\rangle = \langle H, (\widetilde\varphi\oplus \widetilde\varphi')^\(u) \rangle \in {{\mathbb Z}},$$ and thus e2πiLWZ(φ) is the same whether computed via $[{\mathcal C}]$ or via $[{\mathcal C}']$. Now we want to apply this theory when A = Aθ (or a suitable smooth subalgebra, say Aθ∞). If we realize Aθ as the crossed product $C^\infty(S^1)\rtimes\_ \theta {\mathbb Z}$, we can view Aθ∞ as the “boundary” of ${\mathcal C}= C^\infty(D^2)\rtimes\_\theta {\mathbb Z}$, where D2 denotes the unit disk in ${\mathbb C}$. The natural element $[{\mathcal C}]$ is the trace on ${\mathcal C}$ coming from normalized Lebesgue measure on D2. To summarize, it is possible to enhance the sigma-model action on a spacetime algebra B with the addition of a Wess-Zumino term LWZ(φ), depending on a choice of a “flux” H. More general spacetimes ----------------------- In references such as, T-duality considerations suggested that very often one should consider spacetimes which are not just noncommutative tori, but “bundles” of noncommutative tori over some base space, such as the C\-algebra of the discrete Heisenberg group, called the “rotation algebra” in. A theory of some of these bundles was developed in. For present purposes, the following definition will suffice: [def:ATheta] Let Z* be a compact space and let $\Theta{\colon\,}Z\to{\mathbb T}$ be a continuous function from Z to the circle group. We define the noncommutative torus bundle algebra associated to (Z, Θ) to be the universal C\-algebra A* = A(Z, Θ) generated over a central copy of C(Z) (continuous functions vanishing on the base space, Z) by two unitaries u and v, which can be thought of as continuous functions from Z to the unitaries on a fixed Hilbert space H, satisfying the commutation rule u(z)v(z) = Θ(z)v(z)u(z). Note that A is the algebra Γ(Z, E) of sections of a continuous field E of rotation algebras, with fiber AlogΘ(z)/(2πi) over z ∈ Z. [ex:ATheta] The reader should keep in mind three key examples of Definition [def:ATheta]. If Z = {z} is a point, A(Z, Θ) is just the rotation algebra AlogΘ(z)/(2πi). More generally, if Θ is a constant function with constant value e2πiθ, then A(Z, Θ) = C(Z) ⊗ Aθ. And finally, there is a key example with a nontrivial function Θ, that already came up in from T-dualization of ${\mathbb T}^3$ (viewed as a principal ${\mathbb T}^2$-bundle over ${\mathbb T}$) with a nontrivial H-flux, namely the group C\-algebra of the integral Heisenberg group. In this example, $Z=S^1={\mathbb T}$ and $\Theta{\colon\,}{\mathbb T}\to {\mathbb T}$ is the identity map. If w* is the canonical unitary generator of C(Z), then in this case the commutation rule becomes simply uv = wvu (with w central), so as explained in, A is the universal C\-algebra on three unitaries u, v, w, satisfying this commutation rule. [rem:AThetatoAtheta] Let A* = A(Z, Θ) be as in Definition [def:ATheta], and fix θ irrational. Then homomorphisms A → Aθ, not assumed necessarily to be unital, can be identified with triples consisting of the following: 1. a projection p ∈ Aθ which represents the image of 1 ∈ A, 2. a unital \* -homomorphism ρ from C(Z) to pAθp, and 3. a unitary representation of the Heisenberg commutation relations into the unital C\-algebra pAθ*p, with the images of u and v commuting with ρ(C(Z)). Even in the case discussed above with A = C\(u, v, w* ∣ uv = wvu) and in the special case of unital maps, the classification of maps φ: A → Aθ is remarkably intricate. For example, choose any n mutually orthogonal self-adjoint projections p1, ⋯, pn in Aθ with p1 + ⋯ + pn = 1. Each pjAθpj is Morita equivalent to Aθ, and is thus isomorphic to a matrix algebra $M\_{n\_j}\bigl(A\_{\theta\_j}\bigr)$, $\theta\_j \in GL(2,{{\mathbb Z}})\cdot \theta$. For each j, there is a unital map $\varphi\_j{\colon\,}A\to M\_{n\_j}\bigl(A\_{\theta\_j}\bigr)$ sending the central unitary w to e2πiθj. Then φ1 ⊕ ⋯ ⊕ φn is a unital \* -homomorphism from A to Aθ sending w to ∑e2πiθjpj. Since n can be chosen arbitrarily large, one sees that there are quite a lot of inequivalent maps. In this particular example, K1(A) is a free abelian group on 3 generators, u, v, and an additional generator W ∈ M2(A) (w does not give an independent element since it is the commutator of u and v). A notion of “energy” for such maps φ may be obtained by summing the energies of the three unitaries φ(u), φ(v), and φ(W) (for the last of these, one needs to extend φ to matrices over A in the usual way). Estimates for the energy can again be obtained using the results and methods of. A physical model ================ To write the partition function for the sigma-model studied in this paper, recall the expression for the energy from equation, $${\mathcal{L}}\_{G,D}(\varphi) = \varphi^\(\psi\_2)(G) = \sqrt{\det(g)} \sum\_{j,k=1}^2 \sum\_{\mu,\nu=1}^2 G\_{ij} g^{\mu\nu} {\operatorname{Tr}}(\delta\_ \mu(\varphi(U\_j))^\\delta\_\nu(\varphi(U\_k)).$$ It is possible to parametrize the metrics (gμν) by a complex parameter τ, $$g(\tau)= (g\_{\mu\nu}(\tau)) = \left(\begin{array}{cc} 1 & \tau\_1 \\ \tau\_1 & \|\tau\|^2\end{array}\right)$$ where $\tau= \tau\_1 + i \tau\_2 \in {\mathbb C}$ is such that τ2 > 0. Note that g is invertible with inverse given by $$g^{-1}(\tau) = (g^{\mu\nu}(\tau)) = \tau\_2^{-2} \left(\begin{array} {cc} \|\tau\|^2 & -\tau\_1 \\ -\tau\_1 & 1 \end{array}\right)$$ and $\sqrt{\det(g)} = \tau\_2$. The “genus 1” partition function is $$Z(G,z) = \int\_{\tau\in {\mathbb C}, \tau\_2>0} \frac{d\tau\wedge d\bar\tau} {{\tau\_2}^2} Z(G, \tau, z)$$ where $$Z(G, \tau, z) = \int {\mathcal D}[\varphi] e^{-z{\mathcal{L}}\_{G,\tau}(\varphi)}/ \int {\mathcal D}[\varphi].$$ is the renormalized integral. Here LG, τ = LG, D, where we emphasize the dependence of the energy on τ. This integral is much too difficult to deal with even in the commutative case, so we oversimplify by considering the semiclassical approximation, which is a sum over the critical points. Even this turns out to be highly nontrivial, and we discuss it below. In the special case when Θ = θ and is not a quadratic irrational, then the semiclassical approximation to the partition function above is Z(G, τ, z) ≈ ∑m ∈ M/{ ± 1}∑Ae− zLG, τ(φA), up to a normalizing factor, in the notation as explained later in this section. In this approximation, $$Z(G,z) \approx \int\_{\tau\in {\mathbb C}, \tau\_2>0} \frac{d\tau\wedge d\bar\tau} {{\tau\_2}^2} \sum\_{m\in M/\{\pm 1\}} \sum\_{A} e^{-z{\mathcal{L}}\_{G,\tau} (\varphi\_A)}.$$ We expect Z(G) and Z(G− 1) to be related as in the classical case, as a manifestation of T-duality. In the rest of this section we specialize to a (rather oversimplified) special case based on the results of Section [sec:action-tori]. As explained before, we basically take our spacetime to be a noncommutative 2-torus, and for simplicity, we ignore the integral over τ (the parameter for the metric on the worldsheet) and take τ = i. As pointed out by Schwarz, changing a noncommutative torus to a Morita equivalent noncommutative torus in many cases amounts to an application of T-duality, and should not change the underlying physics. For that reason, it is perhaps appropriate to stabilize and take our spacetime to be represented by the algebra AΘ ⊗ K (K as usual denoting the algebra of compact operators), which encodes all noncommutative tori Morita equivalent to AΘ at once. (Recall AΘʹ is Morita equivalent to AΘ if and only if they become isomorphic after tensoring with K, by the Brown-Green-Rieffel theorem.) Since this algebra is stable, to obtain maps into the worldsheet algebras we should take the latter to be stable also, and thus we consider a sigma-model based on maps φ: AΘ ⊗ K → Aθ ⊗ K, where θ is allowed to vary (but Θ remains fixed). Via the results of Section [sec:morphisms], such maps exist precisely when there is a morphism of ordered abelian subgroups of ${{\mathbb R}}$, from ${\mathbb Z}+ {\mathbb Z}\Theta$ to ${\mathbb Z}+ {\mathbb Z}\theta$, or when there exists $c\theta + d\in {\mathbb Z}+ {\mathbb Z}\theta$, cθ + d > 0, such that $(c\theta + d)\Theta \in {\mathbb Z}+ {\mathbb Z}\theta$, i.e., when there exists $m\in M = GL(2,{\mathbb Q})\cap M\_2({\mathbb Z})$ (satisfying the sign condition cθ + d > 0) such that Θ = m ⋅ θ or θ = m− 1 ⋅ Θ for the action of $GL(2,{\mathbb Q})$ on ${\mathbb R}$ by linear fractional transformations. Given that Θ = m ⋅ θ for some m ∈ M, the matrix m determines the map φ\: K0(AΘ ⊗ K) → K0(A*θ ⊗ K), which turns out to be multiplication by $$\label{eq:denom} {\mathcal D}\left(m,\,\theta\right) = \left\|c\theta + d\right\| \text{ if } m= \begin{pmatrix}a&b\\c&d\end{pmatrix}.$$ (Note the similarity with the factor that appears in the transformation law for modular forms. Also note that if Θ = m ⋅ θ, then also Θ = mʹ ⋅ θ for many other matrices mʹ, since one can multiply both rows by the same positive constant factor.) However, m does not determine the map induced by φ on K1. A natural generalization of Conjecture [conj:minenergy] would suggest that if θ = Θ and m = 1, at least if θ is not a quadratic irrational (so as to exclude the Kodaka-like maps), then the induced map φ\* on K1 has a matrix $$A=\begin{pmatrix}p&q\\r&s\end{pmatrix}$$ in $SL(2,{{\mathbb Z}})$, and there should be (up to gauge equivalence) a unique energy-minimizing map φ: AΘ ⊗ K → Aθ ⊗ K with energy 4π2 (p2 + q2 + r2 + s2). Note that p2 + q2 + r2 + s2 is the squared Hilbert-Schmidt norm of A (i.e., the sum of the squares of the entries). We want to generalize this to the case of other values of m. Unfortunately, the calculation in Section [sec:rational] suggests that there may not be a good formula for the energy of a harmonic map just in terms of the induced maps on K0 and K1. But a rough approximation to the partition function might be something like $$\label{eq:partition} Z(z) \approx \sum\_{m\in M/\{\pm 1\}} \sum\_{A} e^{- 4\pi^2 {\mathcal D}(m, \theta) \Vert A\Vert\_{HS}^2 z}\,.$$ The formula $ 4\pi^2 {\mathcal D}(m, \theta) \Vert A\Vert\_{HS}^2$ for the energy is valid not just for the automorphisms φA but also for the map U ↦ upvq, V ↦ urvs with $$A = \begin{pmatrix}p&q\\r&s\end{pmatrix},\quad \det A = n$$ from Anθ to Aθ, which one can check to be harmonic, just as in Corollary [cor:satisfiesEL]. The associated map on K0 corresponds to the matrix $$m=\begin{pmatrix}n&0\\0&1\end{pmatrix}$$ with ${\mathcal D}(m,\theta)=1$. 10 J. Anderson and W. Paschke, The rotation algebra, Houston J. Math. 15 (1989), no. 1, 1–26. M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. M. D. Brabanter, The classification of rational rotation algebras, Arch. Math. (Basel) 43 (1984), 79–83. L. G. Brown, P. Green and M. A. 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Evans, The structure of the irrational rotation C\-algebra, Ann. of Math. (2) 138* (1993), no. 3, 477–501. G. A. Elliott and M. Rørdam, The automorphism group of the irrational rotation $C\sp \$-algebra, Comm. Math. Phys. 155* (1993), no. 1, 3–26. K. Kodaka, Endomorphisms of certain irrational rotation $C\sp \$-algebras, Illinois J. Math. 36* (1992), no. 4, 643–658. K. Kodaka, A note on endomorphisms of irrational rotation $C\sp \$-algebras, Proc. Amer. Math. Soc. 122* (1994), no. 4, 1171–1172. K. Kodaka, Diffeomorphism of irrational rotation $C\sp \$-algebras by nongeneric rotations, II, J. Operator Theory 27* (1992), no. 2, 341–345. H. Li, Appendix to V. Mathai and J. Rosenberg’s paper ``A noncommutative sigma-model", arXiv:0909.4978. T. Loring, Lifting solutions to perturbing problems in $C\sp \$-algebras, Fields Institute Monographs, 8, Amer. Math. Soc., Providence, RI, 1997. D. A. Lowe, H. Nastase and S. 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R. J. Szabo, Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), no. 4, 207–299, arXiv:hep-th/0109062. K. Thomsen, Finite sums and products of commutators in inductive limit $C\sp \$-algebras, Ann. Inst. Fourier (Grenoble) 43* (1993), no. 1, 225–249. --- 1. Both authors thank the Erwin Schrödinger International Institute for Mathematical Physics for its hospitality and support under the program in Gerbes, Groupoids, and Quantum Field Theory in Spring 2006, which made the beginning of this work possible. VM was partially supported by the Australian Research Council. JR was partially supported by NSF Grants DMS-0504212 and DMS-0805003, and also thanks the Department of Pure Mathematics at The University of Adelaide for its hospitality during visits in August 2007 and March 2009.[↩](#fnref1)	arxiv_0000493
use the induction hypothesis and conclude $\SequentCLJ(e\_2){}$. Using the rule $\LJImplL$, we obtain $\SequentCLJ(e\_1 \impl e\_2){}$. Inductive Case 4. Suppose e = ∀x.e1, and therefore we have $\tproduct{\forall x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the minimum expected value of $\lambda y. \tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. It means that $\tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for some input w. But then, by the induction hypothesis, we have $\SequentCLJ(e\_1[y / x]){}$. We therefore can deduce $\SequentCLJ(\forall x. e\_1){}$. Inductive Case 5. Suppose e = ∃x.e1, and therefore we have $\tproduct{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the maximum expected value of $\lambda y. \tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. Seeing that 0 is the bottom value, it means that $\tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for all inputs. But then, by the induction hypothesis, we have $\SequentCLJ(e\_1){}$. We therefore can deduce $\SequentCLJ(\exists x. e\_1){}$, ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Its corollary is: [Soundness of Product DL] Given a formula e, for any contexts N, Γ, Q if$\tproduct{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tproduct{\top}^{\hNetCtx,\hRandCtx, \hVarCtx} = 1$ then $\SequentCLJ(){e}$. Proof of Soundness for DL2 -------------------------- We now provide the proof for Theorem [th:d1]. It is important to remember that DL2 translation does not include a stand-alone negation operator or implication - therefore this proof is done for a limited version of. We start with a helper lemma. [Adequacy of intervals and arithmetic operations in DL2][lem:eval1] If $\validForm{e\_1 \bowtie e\_2} : \RealType$ and $\tDLtwo{e\_1 \bowtie e\_2}^{\semCtx} = 0$ then $\tempty{e\_1}^{\semCtx}\_{DL2} \bowtie \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. Since e1 ⋈ e2 is well-typed each of e1, e2 has type $\RealType$. If ⋈ is = = , then $\tDLtwo{e\_1 == e\_2}^{\semCtx} = 0$ means $\tDLtwo{e\_1}^{\semCtx} - \tDLtwo{e\_2}^{\semCtx} = 0$, that is, $\tDLtwo{e\_1}^{\semCtx} = \tDLtwo{e\_2}^{\semCtx}$. But then, $\tempty{e\_1}^{\semCtx}\_{DL2} == \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. If ⋈ is ≤ , then $\tDLtwo{e\_1 \bowtie e\_2}^{\semCtx} = 0$ means $\tDLtwo{e\_1}^{\semCtx} - \tDLtwo{e\_2}^{\semCtx} \leq 0$, that is, $\tDLtwo{e\_1}^{\semCtx} \leq \tDLtwo{e\_2}^{\semCtx}$. But then, $\tempty{e\_1}^{\semCtx}\_{DL2} \leq \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. If ⋈ is ≠ , then $\tDLtwo{e\_1 \neq e\_2}^{\semCtx} = 0$ means $[\tDLtwo{e\_1}^{\semCtx} == \tDLtwo{e\_2}^{\semCtx}] = 0$, that is, $\tDLtwo{e\_1}^{\semCtx} \neq \tDLtwo{e\_2}^{\semCtx}$. But then, $\tempty{e\_1}^{\semCtx}\_{DL2} \neq \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. The remaining comparisons in ⋈ are all defined using the already proven comparisons. We now move on to the proof of Theorem [th:d1]. [Soundness of DL2] Given a formula e, taking DL2 with just connectives ∧ , ∨ and quantifiers, for any contexts N, Γ, Q: if $\tDLtwo{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tDLtwo{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}$ then $\SequentCLJ(){e}$. Base Case 1. Suppose e = ⊤. But we know $\SequentCLJ(){\top}$ by the rule (⊤). Base Case 2. Suppose e = e1 ⋈ e2, where ⋈ is a comparison operator, and e1, e2 are real numbers. Moreover, $\tDLtwo { e\_1 \bowtie e\_2}^{\semCtx} = 0$. By Lemma [lem:eval1], we have $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{DL2} \bowtie^\* \tempty{e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{DL2} = \top$. But then we obtain a proof for $\SequentCLJ{e\_1 \bowtie e\_2}$ by the rules $ \LJArithR $ and (⊤). Inductive Case 1. Suppose e = e1 ∧ e2, and therefore ${tDLtwo{e}^{\semCtx} = \tDLtwo{e\_1}^{\semCtx} + \tDLtwo{e\_2}^{\semCtx} = 0}$.That means we have $\tDLtwo{e\_1}^{\semCtx} = \tDLtwo{e\_2}^{\semCtx} = 0 $ And, by the induction hypothesis, we have that $\SequentCLJ(){e\_1}$ and $\SequentCLJ(){e\_2}$. But then, by the rule $\LJConjR$, we have $\SequentCLJ(){e\_1 \land e\_2}$. Inductive Case 2. Suppose e = ∀x.e1, and therefore we have $\tDLtwo{\forall x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the minimum expected value of $\lambda y. \tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. Seeing that 0 is the top value, it means that $\tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for all inputs y. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1}$. We therefore can deduce $\SequentCLJ(){\forall x. e\_1}$ ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Inductive Case 3. Suppose e = ∃x.e1, and therefore we have $\tDLtwo{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the maximum expected value of $\lambda y. \tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. It means that $\tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for at least one input w. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1[y / x]}$. We therefore can deduce $\SequentCLJ(){\exists x. e\_1}$. Further Discussion of Logical and Geometric Properties ====================================================== The DLs defined as part of LDL have the following properties: * $\tDLtwo{}$ is commutative, scale-invariant, associative, sound (for a limited ) and has shadow-lifting. It is not idempotent and does not have quantifier commutativity. While its semantics for logical connectives is weakly smooth, $\tDLtwo{}$ is not weakly smooth due to the presence of min and max in translation of comparisons. While no proofs about the properties were provided in the translation of conjunction in DL2 and product based fuzzy logic is standard addition. The more common properties including idempotence, commutativity, associativity and min-max boundedness in given domain are known and proven. Considering the semantics of conjunction is addition it is simple enough to reason about its partial derivatives and therefore shadow-lifting. * $\tGodel{}$ is idempotent, commutative, scale-invariant, associative, sound and has quantifier commutativity. It does not have shadow-lifting and it is not weakly smooth. The semantics of conjunction is a minimum between the two elements which prohibits shadow-lifting as well as smoothness by definition. The simpler properties are well known and proved in fuzzy logic literature as it is an established fuzzy logic. * $\tlukasiewicz{}$ is commutative, sound (for a syntax excluding either negation or implication) and associative. It is not idempotent, scale-invariant, weakly smooth and does not have shadow-lifting or quantifier commutativity. The presence of maxima in the semantics of conjunction however naturally prohibits shadow lifting and smoothness. Similarly to Gödel it is an established fuzzy logic and many of the properties of its semantics of conjunction have been proven. * $\tyager{}$ is commutative, sound (for a limited syntax) and associative. It is not idempotent, scale-invariant, weakly smooth and does not have shadow-lifting or quantifier commutativity. Similarly to the previous two the presence of maxima prohibits shadow lifting and smoothness. This is another one of the well established fuzzy logics for which majority of the properties have been already investigated by the community. * $\tproduct{}$ is commutative, associative, sound and has shadow-lifting. It is not idempotent, scale-invariant, does not have quantifier commutativity. While its semantics of logical connectives are weakly smooth $\tproduct{}$ is not weakly smooth due to the semantics of comparisons. Furthermore as its semantics of conjunction is the arithmetic operation of multiplication it is easy to reason about its partial derivates and therefore shadow lifting. Since it is a well-established fuzzy logic its properties the more common logical properties have been extensively studied already. * $\tSTL{}$ is idempotent, commutative, scale-invariant, weakly smooth and has shadow-lifting. It is not associative or sound and does not have quantifier commutativity. The properties of this translation have been proven in the paper it originates from as indicated in Table [tab:properties], aside from quantifier commutativity which has been added in this paper. The translation has been changed for the purposes of LDL however the proofs would remain analogous. Furthermore it is the only DL for which the soundness does not hold, for either full or limited. * Only $\tGodel{}$’s quantifiers commute with connectives, as its connectives are defined via max and min (commuting with maxima and minima used in quantifier interpretation). Logic of Differentiable Logics: Towards a Uniform Semantics of DL ================================================================= Differentiable logics (DL) have recently been proposed as a method of training neural networks to satisfy logical specifications. A DL consists of a syntax in which specifications are stated and an interpretation function that translates expressions in the syntax into loss functions. These loss functions can then be used during training with standard gradient descent algorithms. The variety of existing DLs and the differing levels of formality with which they are treated makes a systematic comparative study of their properties and implementations difficult. This paper remedies this problem by suggesting a meta-language for defining DLs that we call the Logic of Differentiable Logics, or LDL. Syntactically, it generalises the syntax of existing DLs to FOL, and for the first time introduces the formalism for reasoning about vectors and learners. Semantically, it introduces a general interpretation function that can be instantiated to define loss functions arising from different existing DLs. We use LDL to establish several theoretical properties of existing DLs and to conduct their empirical study in neural network verification. Keywords: Differentiable Logic, Fuzzy Logic, Probabilistic Logic, Machine Learning, Training with Constraints, Types. Introduction ============ Logics for reasoning with uncertainty and probability have long been known and used in programming: e.g. fuzzy logic , probabilistic logic and variants of thereof in logic programming domain . Recently, rising awareness of the problems related to machine learning verification opened a novel application area for those ideas. Differentiable Logics (DLs) is a family of methods that applies key insights from fuzzy logic and probability theory to enhance this domain with property-driven learning . As a motivating example, consider the problem of verification of neural networks. A neural network is a function $f: \Real^n \rightarrow \Real^m$ parametrised by a set of weights $\ws$. A training dataset $\dataset$ is a set of pairs $(\x,\y)$ consisting of an input $\x \in \Real^n$ and the desired output $\y \in \Real^m$. It is assumed that the outputs $\y$ are generated from $\x$ by some function $\hypothesis : \Real^n \rightarrow \Real^m$ and that $\x$ is drawn from some probability distribution over $\Real^n$. The goal of training is to use the dataset $\dataset$ to find weights $\ws$ such that f approximates $\hypothesis$ well over input regions with high probability density. The standard approach is to use a loss function $\lossfn$, that given a pair $(\x, \y)$ calculates how much $f(\x)$ differs from the desired output $\y$. Gradients of $\lossfn$ with respect to the network’s weights can then be used to update the weights during training. However in addition to the dataset $\dataset$, in certain problem domains we may have additional information about $\hypothesis$ in the form of a mathematical property $\property$ that we know $\hypothesis$ must satisfy. Given that $\hypothesis$ satisfies property $\property$, we would like to ensure that f also satisfies $\property$. A common example of such a property is that $\hypothesis$ should be robust, i.e. small perturbations to the input only result in small perturbations to the output. For example, in image classification tasks changing a single pixel should not significantly alter what the image is classified as . [ε-δ-robustness] [def:robustness] Given constants $\epsilon, \delta \in \Real$, a function f is *ε-δ-robust* around a point $ \hat{\x} \in \Real^n$ if: $$\forall \x \in \Real^n : \|\|\x - \hat{\x}\|\| \leq \epsilon \implies \|\| f(\x) - f(\hat{\x})\|\| \leq \delta$$ formulate several possible notions of robustness. Throughout this paper, we will use the term ``robustness property" to refer to the above definition. The problem of verifying the robustness of neural networks has received significant attention from the verification community , and it is known to be difficult both theoretically and in practice . However, even leaving aside the challenges of undecidability of non-linear real arithmetic , and scalability , the biggest obstacle to verification is that the majority of neural networks do not succeed in learning $\property$ from the training dataset $\dataset$ . The concept of a differentiable logic (DL) was introduced to address this challenge by verification-motivated training. This idea is sometimes referred to as continuous verification , referring to the loop between training and verification. The key idea in differentiable logic is to use $\property$ to generate an alternative logical loss function $\lossfn\_\property$, that calculates a penalty depending on how much f deviates from satisfying $\property$. When combined with the standard data-driven loss function L, the network is trained to both fit the data and satisfy $\property$. A DL therefore has two components: a suitable language for expressing the properties and an interpretation function that can translate expressions in the language into a suitable loss function. Although the idea sounds simple, developing good principles of DL design has been surprisingly challenging. The machine-learning community has proposed several DLs such as DL2 for supervised learning , and a signal temporal logic based-DL for reinforcement learning (STL) . However, both approaches had shortcomings from the perspective of formal logic: the former fell short of introducing quantifiers as part of the language (and thus modelled robustness semi-formally in all experiments), and the latter only covered propositional fragment (which is not sufficient to reasoning about robustness). Solutions were offered from the perspective of formal logic. In it was shown that one can use various propositional fuzzy logics to create loss functions. However, these fuzzy DLs did not stretch to cover the features of the DLs that came from machine learning community, and in particular could not stretch to express the above robustness property (Definition [def:robustness]), which needs not just quantification over infinite domains (quantifiers for finite domains were given in ), but also a formalism to express properties concerning vectors and functions over vectors. The first problem is thus: There does not exist a DL that formally covers a sufficient fragment of first-order logic to express key properties in machine learning verification, such as robustness. The second problem has to do with formalisation of different DLs. In many of the existing DL approaches syntax, semantics and pragmatics are not well-separated, which inhibits their formal analysis. We have already given an example of DL2 treating quantifiers empirically outside of the language. But the problem runs deeper than modelling quantifiers. To illustrate, let us take a fragment of syntax on which all DLs are supposed to agree. It will give us a toy propositional language a : = p ∣ a ∧ a They assume that each propositional variable p is interpreted in a domain $\domainSymbol \subseteq \Real$. The domains vary vastly across the DLs (from fuzzy set [0, 1] to (∞, ∞) in STL) and the choice of a domain can have important ramifications for both the syntax and the semantics. For example DL2’s domain [0, ∞] severely restricts the translation of negation compared to other DLs. Conjunction is interpreted by + in DL2 , by min, × and other more complex operations in different fuzzy DLs in. In STL , in order to make loss functions satisfy a shadow-lifting property, the authors propose a complex formula computing conjunction, that includes natural exponents alongside other operations. But this forces them to redefine the syntax for conjunction and thus obtain a different language a : = p∣⋀M(a1, …, aM) where ⋀M denotes a (non-associative!) conjunction for M elements. [t] \|p5cm\|c\|c\| Translation & Domain $\domainSymbol$ & Translation of conjunction $\translation (\val\_1 \wedge \val\_2)$ DL2 () & [0, ∞] & $\translation (\val\_1) + \translation (\val\_2)$ Gödel () & [0, 1] & $\min(\translation(\val\_1), \translation(\val\_2))$ STL () & ( − ∞, ∞) &$ \translation (\bigwedge\_M) = \begin{cases} \dfrac{\sum\_i \val\_{\min} e^{\tilde{\val\_i}} e^{\nu \tilde{\val\_i}}}{\sum\_i e^{\nu \tilde{A\_i}}} & \text{if}\ \val\_{\min} < 0 \\ \dfrac{\sum\_i \val\_{\min} e^{-\nu \tilde{\val\_i}}}{\sum\_i e^{-\nu \tilde{\val\_i}}} & \text{if}\ \val\_{\min} > 0 \\ 0 & \text{if}\ \val\_{\min} = 0 \\ \end{cases} $ && $\nu \in \Real^+$ (constant) &&$ \val\_{\min} = \min (\translation(\val\_1), \ldots, \translation(\val\_M)) $ &&$ \tilde{\val\_i} = \dfrac{\translation(\val\_i) - \val\_{\min}}{\val\_{\min}} $ [table:original-dl] As consequence of the above two problems, the third problem is lack of unified, general syntax and semantics able to express multiple different DLs and modular on the choice of DL, that would make it possible to choose one best suited for concrete task or design new DLs in an easy way. In this paper, we propose a solution to all of these problems. The solution comes in a form of a meta-DL, which we call a logic of differentiable logics (LDL). On the syntactic side (Section [sec:syntax]), it is a typed first-order language with negation and universal and existential quantification that can express properties of functions and vectors. On the semantic side (Section [sec:semantics]), interpretation is defined to be parametric on the choice of the interpretation domain $\domainSymbol$ or a particular choice of logical connectives. This parametric nature of interpretation simplifies both the theoretical study and implementations that compare different DLs. Moreover, the language has an implicit formal treatment of neural networks via a special kind of context – a solution that we found necessary in achieving a sufficient level of generality in the semantics. For the first time the semantics formally introduces the notion of a probability distribution that corresponds to the data from which the data is assumed to be drawn. We demonstrate the power of this approach by using LDL to prove soundness of various DLs in Sections [subsec:soundness], [subsec:soundness2] and systematically compare their geometric properties in Section [sec:properties]. In Section [sec:eval] we use LDL to provide a uniform empirical comparison of performance of all mentioned DLs on improving robustness. Background ========== In the previous section, we informally introduced the notions of probability distributions and loss functions. We now formally define these. In what follows, quantities such as $\x$ which are written using a bold font refer to vectors and the notation xi refers to the ith element of $\x$. Probability Distributions and Expectations ------------------------------------------ Following the standard definitions , we start by considering a random variable X that ranges over some domain D. The probability distribution for X characterises how probable it is for a sample from it to take a given value in the domain D. Depending on whether D is discrete or continuous, the variable is called discrete or continuous random variable, respectively. Formally, given a continuous random variable X with domain D, the probability distribution function (PDF) $ \distr\_X : D \rightarrow [0,1]$ is a function that satisfies: $ \int\_{D} \distr\_{X}(x) dx = 1$. This definition can be generalised to random vector variables $\X$ over the domain $\D = D\_1 \times \ldots \times D\_n$ as follows. Each element of the vector is assumed to be drawn from a random variable Xi over domain Di. The joint PDF is a function $\distr\_{X\_1, \ldots, X\_n} : D\_1 \times \ldots \times D\_n \rightarrow [0,1]$ that satisfies: $$\int\_{D\_1} \ldots \int\_{D\_n} \distr\_{X\_1, \ldots, X\_n}(x\_1, \ldots, x\_n) dx\_n \ldots dx\_1 = 1$$ For brevity we will write $\distr\_\X$ instead of $\distr\_{X\_1, \ldots, X\_n}$ and $\int\_\D$ and $d\x$ rather than the full integral above. Given a function $g: \Real^n \rightarrow \Real$, we can calculate an average value that g takes on $\X$, given a probability distribution $\distr\_{\X}$. Formally $\mathbb{E}[g(\X)]$, the expected value for $g: \Real^n \rightarrow \Real $ over the random variable $\X$, is defined as: $$\mathbb{E}[g(\X)] = \int\_\D \distr\_{\X}(\x) g(\x) d\x. \label{eq:vector-integral}$$ Throughout the paper we assume that different random variables are independent. Loss Functions -------------- In standard machine learning training, given a neural network f, a loss function $\losssymbol: \Real^n \times \Real^m \rightarrow \Real$ computes a penalty proportional to the difference between the output of f on a training input x and a desired output y. There has been a large number of well performing loss functions proposed, with the most popular in classification tasks being cross-entropy loss : Given a classifier ${f: \Real^n \rightarrow [0,1]^m}$, the cross-entropy loss $\losssymbol\_{ce}$ is defined as $$\label{eq:ce} \losssymbol\_{ce}(\x, \y) = - \sum\_{i=1}^{m} \y\_i \log(f(\x)\_i)$$ where $\y\_i$ is the true probability of $\x$ belonging to class i and $f(\x)\_i$ the probability for class i as predicted by f when applied to $\x$. However, this notion of a loss function is not expressive enough to capture loss functions generated by DLs. Firstly, the property $\property$ may depend on other parameters apart from a labelled input/output pair. For example, the definition of robustness in Definition [def:robustness], depends not only on some input x̂ but also on parameters ε and δ. Secondly, properties may relate more than one neural network, for example, in student-teacher scenarios . Therefore we generalise our notion of a loss function as follows: [Generalised loss function] [eq:loss-function] Given a set of neural networks $\hNetCtx$ and parameters $\hVarCtx$, a loss function $\losssymbol^{\hNetCtx, \hVarCtx}$ is any function of type $\hNetCtx \times \hVarCtx \rightarrow \Real$. Note that we recover cross-entropy loss from this definition, by setting $\hNetCtx = \Real^n \rightarrow \Real^m$ and $\hVarCtx = {\Real^n \times \Real^m}$, whereas in a teacher-student scenario it might be something like $\hNetCtx = (\Real^n \rightarrow \Real^m) \times (\Real^n \rightarrow \Real^m)$ and $\hVarCtx = {\Real^n \times \Real}$. We distinguish between the neural networks and the other parameters, because during training we will differentiate the loss function with respect to the former but not the latter. Syntax and Type-system of LDL ============================= Figure [fig:syntax] formally defines the syntax of LDL. The first line of the definition of the set ⟨expr⟩ of expressions includes: bound variables x, neural network variables f and constants of types $\RealType$, Index and $\BoolType$. The second line of ⟨expr⟩ defines the standard syntax for lambda functions, applications and let bindings to facilitate language modularity. In this aspect the syntax is richer then any of the existing DLs. The third and fourth lines contain standard operations on Booleans (,,, ) and Reals (,,,, = = ,...). The fifth line contains vector operations: is used to construct vectors, and the lookup operation $\AtSymbol{}$ retrieves the value of a vector at the provided index, i.e. v! i accesses the value at the ith position in the vector v. The final line contains quantifiers. LDL is a typed language. In the first line of the block defining the set ⟨type⟩, there is the function type constructor $\FunType{\typeClass\_1}{\typeClass\_2}$, which represents the type of functions which take inputs of type $\typeClass\_1$ and produce outputs of type $\typeClass\_2$. Next there are the standard Bool and Real types. Finally, there are: Vec n, the type of vectors of length n, and Index n, the type of indices into vectors of length n. The fact that these two types are parametrised by the size of the Vector will allow the type-system to statically eliminate specifications with out-of-bounds errors. [t]0.45 <expr> ∋ e ::= $\identClass$ \| $\netClass$ \| $\: \in \Real$ \| $\: \in \Nat$ \| $\: \in \Bool$ \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| [t]0.5 <type> ∋ τ ::= \| s <simple type> ∋ s ::= \| for $n \in \Nat$ [fig:syntax] To illustrate LDL in use, we now present one possible encoding of the robustness property from Definition [def:robustness] for a network f of size 784 (28 × 28 pixel images). [Encoding of robustness property in LDL] [ex:wte] This example illustrates how LDL has already fulfilled several of our goals defined in Section [sec:introduction]: [a)] While other DLs work with different fragments of propositional and first-order logics, LDL covers the whole subset of FOL, and in particular quantifiers are first-class constructs in the language. LDL allows one to express properties of neural networks concisely and at a high level of abstraction. The presence of the Vec type makes explicit that the logic is intended to reason over vectors of reals. LDL is strongly typed, which will allow us to define a type-system, and subsequently a rigorous notion of an interpretation function over well-typed terms. In our presentation, the types have to be written explicitly, but it would be simple to remove this constraint by using a standard type-inference algorithm to infer most of them. $$\begin{gathered} \infer[(networkVar)]{\hcTypeRel{f}{\FunType{\VecType{m}}{\VecType{n}}}} {\Xi[f] = (m, n)} \semSpace \infer[(boundVar)]{\hcTypeRel{\id}{\tau}}{\Delta[\id] = \tau} \\ \\ \infer[(real)]{\hcTypeRel{\elReal}{\RealType}}{\elReal \in \Real} \semSpace \infer[(index)]{\hcTypeRel{\elNat}{\FinType{n}}}{\elNat \in \{0,...,n-1\}} \semSpace \infer[(bool)]{\hcTypeRel{\elBool}{\BoolType}}{\elBool \in \ttype{\BoolType}} \\ \\ \infer[(app)]{\hcTypeRel{\App{e\_1}{e\_2}}{\tau\_2}}{\hcTypeRel{e\_1}{\FunType{\tau\_1}{\tau\_2}} & \hcTypeRel{e\_2}{\tau\_1}} \\ \\ \infer[(lam)]{\hcTypeRel{\Lam{\id}{\tau\_1}{e}}{\FunType{\tau\_1}{\tau\_2}}}{\hTypeRel{\hcNetCtx,\consNew{x\rightarrow \tau}{\hcVarCtx}}{e}{\tau\_2}} \semSpace \infer[(let)]{\hcTypeRel{\Let{\id}{\tau\_1}{e\_1}{e\_2}}{\tau\_2}}{\hcTypeRel{e\_1}{\tau\_1} & \hTypeRel{\hcNetCtx,\consNew{x\rightarrow \tau}{\hcVarCtx}}{e\_2}{\tau\_2}} \\ \\ \infer[(and) (or) (implies)]{\hcTypeRel{\wedge,\vee, \Rightarrow}{\FunType{\BoolType}{\FunType{\BoolType}{\BoolType}}}}{} \semSpace \infer[(not)]{\hcTypeRel{\text{\NotText{}}}{\FunType{\BoolType}{\BoolType}}}{} \\ \\ \infer[(add) (mul)]{\hcTypeRel{\text{\AddSymbol{}}, \text{\MulSymbol{}}}{\FunType{\RealType}{\FunType{\RealType}{\RealType}}}}{} \semSpace \infer[(\bowtie)]{\hcTypeRel{\bowtie}{\FunType{\RealType}{\FunType{\RealType}{\BoolType}}}}{} \\ \\ \infer[(vec)]{\hcTypeRel{\Seq{e\_1}{e\_n}}{\VecType{n}}}{\hcTypeRel{e\_1}{\RealType} & \ldots & \hcTypeRel{e\_n}{\RealType}} \semSpace \infer[(lookup)]{\hcTypeRel{\:\AtSymbol{}}{\FunType{\VecType{n}}{\FunType{\FinType{n}}{\RealType}}}}{} \\ \\ \infer[(forall)]{\hcTypeRel{\Forall{\id}{\tau}{e}}{\BoolType}}{\hTypeRel{\hcNetCtx,\consNew{x\rightarrow \tau}{\hcVarCtx}}{e}{\BoolType} & \tau \neq \FunType{\tau\_1}{\tau\_2}} \quad \infer[(exists)]{\hcTypeRel{\Exists{\id}{\tau}{e}}{\BoolType}}{\hTypeRel{\hcNetCtx,\consNew{x\rightarrow \tau}{\hcVarCtx}}{e}{\BoolType} & \tau \neq \FunType{\tau\_1}{\tau\_2}} \end{gathered}$$ [fig:types] We now define a typing relation for well-typed expressions in LDL. A bound variable context, Δ, is a partial function that assigns each bound variable x currently in scope a type τ. We will use the notation Δ[x → τ] to represent updating Δ with a new mapping from x to τ. A network context, Ξ, is a function that assigns each network variable f a pair of natural numbers (m, n) such that m is the number of inputs to the network and n is the number of outputs. We will use the notation Ξ[f → (m, n)] to represent updating Ξ with a new mapping from f to (m, n). The set of well-typed expressions is the collection of all e for which there exists contexts Ξ and Δ and type τ such that $\haTypeRel$, which is defined inductively in Figure [fig:types], holds. When $\Forall{\id}{\tau}{e}$ is well typed and $\tau = \RealType$ or $\tau = \VecType{n}$, we will say that the quantifier ∀ is infinite, otherwise the quantifier is finite; similarly for ∃. Note that for simplicity, we assume all bound syntactic variable names for quantifiers are unique. In practice this can be achieved by applying typical binding techniques such as de Brujin indices . Loss Function Semantics of LDL ============================== We translate the LDL syntax into loss functions in a manner that is parametric on the choice of the concrete DL. Firstly, in Section [subsec:types], we interpret LDL types as sets of values. Secondly, in Section [subsec:expressions] we interpret expressions, splitting the definitions in two parts: expressions whose interpretation is independent of the choice of DL (e.g. application, lambda, vectors) and expressions whose interpretation is dependent on the choice of DL (e.g. logical connectives, comparison operators). In Section [subsec:quantifiers] we define semantics of quantifiers uniformly for all logics. Semantics of LDL Types ---------------------- We first define a mapping $\ttype{\cdot}$ from LDL types to the set of values that expressions of that type will be mapped to: Note that the type Bool is absent as it is dependent on the choice of DL, given in Figure [fig:expr]. Semantics of LDL Expressions ---------------------------- The next step is to interpret well-typed expressions, $\hcTypeRel{e}{\tau}$, which will be dependent on the semantic context of the interpretation. An expression’s semantic context is comprised of three parts: * A semantic network context $\hNetCtx$ is a function that maps each network variable f ∈ Ξ, such that Ξ[f] = (m, n), to a function $\mathtt{f} : \Real^m \rightarrow \Real^n$, the actual (external) implementation of the neural network. We refer to the set of such functions as $\ttype{\Xi}$. * Let q(e) be the set of infinitely quantified syntactic variables within expression e. A semantic quantifier context $\hRandCtx$ is a function that maps each variable x in q(e) to a random variable X, from which values for x are sampled from (by discussion of Section [sec:datasets-and-classifiers] this also extends to cover random vector variables). We refer to the set of such functions as $\ttype{q(e)}$. * A semantic bound context $\hVarCtx$ is a partial function that assigns each bound variable x ∈ Δ a semantic value in $\ttype{\Delta(x)}$. We refer to the set of such functions as $\ttype{\Delta}$. [Semantic context] Consider the LDL expression in Example [ex:wte]: $\robustexpr$ contains a single network variable f, and therefore the network context will be of the form: $\hNetCtx = [ f \mapsto (\mathtt{f}: \Real^{784} \rightarrow \Real^{10}) ]$ where `f` is the actual neural network implementation (e.g. the Tensorflow/PyTorch object). The expression contains a single infinitely quantified variable $\x$, and therefore the quantifier context $\hRandCtx$ will be of the form $[\x \mapsto \X]$, where $\X$ is random vector variable of size 784, and represents the underlying distribution that the input images are being drawn from. At the top-level of the expression, no bound variables in scope and therefore $\hVarCtx$ is empty. However, when interpreting the subexpression $bounded\ \x\ \hat{\x}\ \epsilon$,the bound context $\hVarCtx$ will be of the form: $[bounded \mapsto v\_1, \: \epsilon \mapsto v\_2,\: \: \hat{\x} \mapsto v\_3, \: \x \mapsto v\_4]$ where v1 is the interpretation of the let bound expression bounded, $v\_2 \in \Real$ is the concrete value of ε, and $v\_3, v\_4 \in \Real^{784}$ are the value passed in for $\hat{\x}$ and the current value of the quantified variable $\x$ respectively. In general the interpretation of an expression will also be dependent on some differentiable logic $\logic$. In this paper, $\logic$ will stand for either of: DL2 , STL , or Fuzzy DLs (G, L, Y, p) . In this section, we present the formalisation for the first three (DL2, STL, G) while the remaining fuzzy logic variants (L, Y, p) can be found in Appendix [appendix:additional-dls]. Therefore, in general we will use the notation $\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}$ to represent the mapping of LDL expression e to the corresponding value in the loss function semantics using logic $\logic$ in the semantic context $(\hNetCtx, \hRandCtx, \hVarCtx)$. Figure [fig:generic-semantics] shows the definition of $\texpr{\cdot}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}$ for the LDL expressions whose semantics are independent of the choice of $\logic$. Note that, with the exception of network variables, the semantics is standard and could belong to any functional language. Figure [fig:expr] interprets the expressions that depend on the choice of DL, as follows. $$\begin{gathered} \texpr{\id}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \hVarCtx[\id] \semSpace \texpr{f}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \hNetCtx[f] \semSpace \texpr{\elReal}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \elReal \semSpace \texpr{\elNat}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \elNat \\ \\ \texpr{\Lam{\id}{\tau}{e}}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \lambda (y : \ttype{\tau}). \texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx[x\rightarrow y]}{\logic} \semSpace \texpr{\App{e\_1}{e\_2}}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \texpr{e\_1}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}(\texpr{e\_2}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}) \\ \\ \texpr{\Let{\id}{\tau}{e\_1}{e\_2}}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \texpr{e\_2}{\hNetCtx}{\hRandCtx}{\hVarCtx[x \rightarrow \texpr{e\_1}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}]}{\logic} \qquad\: \texpr{[e\_1,..., e\_n]}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = <{ \texpr{e\_1}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}, \ldots, \texpr{e\_n}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}}> \\ \\ \texpr{!}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \lam{(\val\_1: \ttype{\VecType{n}}), (\val\_2:\ttype{\FinType{n}})}{{\val\_1}\_{\val\_2}} \\ \\ \texpr{+}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \lam{(\val\_1, \val\_2: \ttype{\RealType})}{\val\_1 + \val\_2} \semSpace \texpr{\times}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic} = \lam{(\val\_1, \val\_2: \ttype{\RealType})}{\val\_1 \times \val\_2} \end{gathered}$$ [fig:generic-semantics] \| Syntax \| DL2 interpretation \| Gödel interpretation \| STL interpretation \| \| --- \| --- \| --- \| --- \| \| $\tempty{\BoolType}\_{L}$ \| [ − ∞, 0] \| [0, 1] \| [ − ∞, ∞] \| \| $\tempty{\top}\_{L}$ \| 0 \| 1 \| \| \| $\tempty{\bot}\_{L}$ \| \| 0 \| \| \| $\tempty{\neg}\_{L}$ \| - \| $\lam{\val}1-\val$ \| $\lam{\val}- \val$ \| \| $\tempty{\wedge}\_{L}$ \| $ \lam{\val\_1,\val\_2}{\val\_1 + \val\_2} $ \| $ \lam{\val\_1,\val\_2} \min (\val\_1, \val\_2)$ \| andS \| \| $\tempty{\vee}\_{L}$ \| $ \lam{\val\_1,\val\_2}{-\val\_1 \times \val\_2} $ \| $ \lam{\val\_1,\val\_2} \max (\val\_1, \val\_2)$ \| orS \| \| $\tempty{\impl}\_{L}$ \| - \| $ \lam{\val\_1,\val\_2} \max (1 - \val\_1, \val\_2)$ \| - \| \| $ \tempty{==}\_{L} $ \| $ \lam{\val\_1, \val\_2}{- \| \val\_1 - \val\_2 \| } $ \| \| \| \| $ \tempty{\leq}\_{L} $ \| $\lam{\val\_1, \val\_2}{- \max(\val\_1 - \val\_2, 0)}$ \| \| \| lcc $ and\_S = \lam{\val\_1,\ldots,\val\_M} \begin{cases} \dfrac{\sum\_i \val\_{\min} e^{\tilde{\val\_i}} e^{\nu \tilde{\val\_i}}}{\sum\_i e^{\nu \tilde{\val\_i}}} & \text{if}\ \val\_{\min} < 0 \\ \dfrac{\sum\_i \val\_i e^{-\nu \tilde{\val\_i}}}{\sum\_i e^{-\nu \tilde{\val\_i}}} & \text{if}\ \val\_{\min} > 0 \\ 0 & \text{if}\ \val\_{\min} = 0 \\ \end{cases} $ where \| \| \| --- \| \| $\nu \in \Real^+$ (constant) \| \| $ \val\_{\min} = \min (\val\_1, \ldots, \val\_M) $ \| \| $ \tilde{\val\_i} = \dfrac{\val\_i - \val\_{\min}}{\val\_{\min}} $ \| orS = analogous to andS & & [fig:expr] Booleans. When giving the semantics of LDL types in Section [subsec:types], we intentionally omitted the interpretation of $\BoolType{}$, as it is dependent on the DL. In DL2, $\BoolType{}$ was originally mapped to [0, ∞], where 0 is interpreted as true, and other values were corresponding to the degree of falsity. Thus, ⊤ is interpreted as 0, but ⊥ did not exist in DL2. In Fuzzy DLs with the domain [0, 1], 1 stood for absolute truth, and other values – for degrees of (partial) truth; finally, in STL interval [ − ∞, ∞], all values but 0 stood for degrees of falsity and truth. Note that we swap the domain of DL2 for [∞, 0] in order to fit with the ordering of truth values in other logics. We will now see how these choices determine interpretation for predicates and connectives. Predicates. The predicates are given by comparisons in LDL. Here we interpret just ≤ and = = , the remaining comparisons are given in Appendix [app:comparisons]. Originally, only DL2 had these comparisons. Inequality of two terms a1 and a2 can be interpreted there by a measure of how different they are. We slightly modify the DL2 translation in order to adapt it in a similar way to other DLs, mapping the difference between $\val\_1$ and $\val\_2$ inside of the chosen domain. Logical Connectives. The interpretation of ∧ and ∨ follows the definitions given in the literature. As Figure [fig:expr] makes it clear, the interpretation of negation for DL2 is not defined. By design, DL2 negation is pushed inwards to the level of comparisons between terms, see Appendix [ap:neg]. STL translation did not have a defined interpretation for implication. Semantics of LDL Quantifiers ---------------------------- So far, we have given an interpretation for quantifier-free formulae. However, none of works studied so far have included infinite quantifiers as first class constructs in the DL syntax. In this section, we therefore propose novel semantics for $\texpr{\forall x: \tau.\ e}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}$ and $\texpr{\exists x: \tau.\ e}{\hNetCtx}{\hRandCtx}{\hVarCtx}{\logic}$. Finite quantifiers were introduced in via finitely composed conjunction and disjunction. We extend this idea to other DLs. Given an expression ∀x : τ. e with a finite quantifier over the variable x, and given the interpretation $\ttype{\tau}{} = \{d\_1, \ldots, d\_n\}$, $\texpr{\forall x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L} = \texpr{\ e [x/d\_1] \land \ldots \land e [x/ d\_n]}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}$; analogously for ∃ and ∨ . Note that we have only two finite types, $ \FinType{n} $ or $\BoolType$, and for the latter we take $\ttype{\BoolType}{} = \{\texpr{\top}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}, \texpr{\bot}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}\}$ to interpret the quantifiers. To proceed with infinite quantifiers, recall that context Q gives us a probability distribution for every first-order variable. This gives us a way to use the definition of an expectation for a function from Section [sec:datasets-and-classifiers] to interpret quantifiers. Recall that, for $g: \Real^n \rightarrow \Real $ we had: $$\mathbb{E}[g(\X)] = \int\_{- \infty}^{\infty} \distr\_{\X}(\x) g(\x) d \x.$$ were the first to interpret universally quantified formulae via expectation maximisation methods, in which case the optimised parameters were neural networks weights, and the quantified properties concerned robustness of the given neural network. Our goal is to propose a unifying approach to both universal and existential quantification that will fit with all DLs that we study here, and will not make any restricting assumptions about the universal properties that the language can express. For example, we defined LDL to admit expressions that may or may not refer to neural networks (or weights), and may express properties more general than robustness. With this in mind, we introduce the following notation. For a function $g: \Real^n \rightarrow \Real $, we say that $\x\_{\min}$ is the global minimum (resp. global maximum) if $g(\x\_{\min}) \leq g(\y)$ (resp. $g(\x\_{\max}) \geq g(\y)$) for any $\y$ on which g is defined. We define a γ-ball around a point $\x$ as follows: $\mathbb{B}\_{\x}^{\gamma} = \{\y \ \mid \ \|\| \x - \y \|\| \leq \gamma \}$. We call the expectation $$\mathbb{E}\_{\min}[g(\X)] = \lim\_{\gamma \rightarrow 0} \int\_{\x \in \mathbb{B}\_{\x\_{\min}}^{\gamma}} \distr\_{\X}(\x) g(\x) d \x$$ minimised expected value for g* (over the random variable $\X$). Taking $\x \in \mathbb{B}\_{\x\_{\max}}^{\gamma}$ in the above formula will give the definition of $\mathbb{E}\_{\max}[g(\X)]$, the maximised expected value for g (over the random variable $\X$). The key insight when applying this is that given a logic $\logic$ and the context $(\semCtx)$, the loss for the body e* of the quantified expressions ∀x : τ. e or ∃x : τ. e can be calculated for any particular given semantic value for the quantified variable x. Therefore we can construct a function $\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx} {\hVarCtx[x\rightarrow y]}{\logic}$ of type $\ttype{\tau} \rightarrow \Real$ that takes in the value and interprets the body of the quantifier with respect to it. This gives us interpretation of universal and existential quantifiers as minimised (or maximised) expected values for the interpretation of their body: $$\begin{aligned} \texpr{\forall x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L} = \mathbb{E}\_{\min}[(\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx[x\rightarrow y]}{L})(Q[x])] \\ \texpr{\exists x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L} = \mathbb{E}\_{\max}[(\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx[x\rightarrow y]}{L})(Q[x])] \end{aligned}$$ The formulae above refer to the minimised (resp. maximised) expectations for the function $\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx[x\rightarrow y]}{L}$ over the random variable Q[x]. Note that this is the first time we use the semantic quantifier context Q to map a syntactic variable x to a random variable. Another notable feature of the resulting interpretation is that the interpretation for quantifiers is parametric on the choice of the logic $\logic$, and that it has the advantage of being compositional which opens a new degree of generality: it allows for arbitrary nesting of quantifiers (in well-formed expressions). This will greatly simplify the proof of soundness for LDL. With this in mind, we introduce the following notation. Assume we are given a small $\gamma \in \Real$. For a function $g: \Real^n \rightarrow \Real $, we say that $\x$ is the global minimum (resp. global maximum) if $g(\x) \leq g(\y)$ (resp. $g(\x) \geq g(\y)$) for any $\y$ on which g is defined. In this case, we say that an interval [a, b], such that $\|\mathbf{a}-\x\| = \|\mathbf{b}-\x\| =\gamma$, is a minimum (resp. maximum) interval for g*. We call the expectation $$\mathbb{E}^{\gamma}[g(\X)] = \int\_{a}^{b} \distr\_{\X}(\x) g(\x) d \x$$ minimised (resp. maximised) expected value, depending on whether [a, b] is a minimum (resp. maximum) interval for g. We will use the notation $ \mathbb{E}^{\gamma}\_{min}[g(\X)] $ and $ \mathbb{E}^{\gamma}\_{max}[g(\X)]$ to refer to minimised and maximised expected values, respectively. Note that, given γ, a* and b are not uniquely defined, but they always exist, and can be obtained by sampling in $ \Real^n$. The key insight when applying this is that given a logic $\logic$ and the context $(\semCtx)$, the loss for the body e of the quantified expressions ∀x : τ. e or ∃x : τ. e can be calculated for any particular given semantic value for the quantified variable x. Therefore we can construct a function $\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx} {\hVarCtx[x\rightarrow y]}{\logic}$ of type $\ttype{\tau} \rightarrow \Real$ that takes in the value and interprets the body of the quantifier with respect to it. This gives us interpretation of universal and existential quantifiers as minimised (or maximised) expected values for the interpretation of their body: $$\begin{aligned} \texpr{\forall x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L} = \mathbb{E}\_{\min}^{\gamma}[(\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx[x\rightarrow y]}{L})(X)] \text{ where } Q[x] = (X, \gamma) \\ \texpr{\exists x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L} = \mathbb{E}\_{\max}^{\gamma}[(\lambda y.\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx[x\rightarrow y]}{L})(X)] \text{ where } Q[x] = (X, \gamma)\end{aligned}$$ Note that the interpretation for quantifiers is parametric on the choice of the logic $\logic$, and that it has the advantage of being compositional which opens a new degree of generality: it allows for arbitrary nesting of quantifiers (in well-formed expressions). This will greatly simplify the proof of soundness for LDL. We can now see that DL2’s interpretation of quantifiers can be expressed as a special case of this definition. In particular, for a quantified formula ∀x.P(x), where P is a robustness property, take $\|\|\x - \hat{\x}\|\| \leq \epsilon$ and empirically compute the ``worst perturbation“ in the ε-interval from x̂ using a ``PGD adversarial attack” projected within that interval. This worst perturbation is the global minimum within the chosen ε-ball. And the loss function that optimises the neural network weights minimises the expectation. This key example motivates our use of expectation terminology in the interpretation of quantifiers. An alternative would be to use just the global minimum (or maximum) of a function directly when defining $\texpr{\forall x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}$ and $\texpr{\exists x: \tau.\ e }{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}$. But such a choice will not generalise over the DL2 implementation (or any other optimisation method). Another attempt to model quantification over real domains was made by within the framework of ``Logic Tensor Networks". There, all variables were mapped to finite sequences of real numbers, by-passing explicit use of the notions of random variables and probability distributions over random variables. Intuitively, each given data set has only a finite number of objects, thus giving a finite domain to map to. This solution would not be satisfactory for a DL, that must take into consideration the fact that loss functions are ultimately designed to compute approximations of the unknown probability distribution, from which the given data is sampled (cf. also the discussion given in Introduction). [Semantics of Quantified Expressions] We calculate a loss function for the robustness property in Example [ex:wte] with respect to DL2 logic in the context $(\semCtx)$ where $\hNetCtx = [ f \mapsto (\mathtt{f}: \Real^{784} \rightarrow \Real^{10}) ]$, $\hRandCtx = [x \mapsto \X]$ and $\hVarCtx$ is empty. Since DL2 has no separate definition of implication we translate the implication in robustness property in a standard manner using negation and disjunction and pushing the negation inwards to the level of comparisons: $$\begin{aligned} \texpr{e^\}{\hNetCtx}{\hRandCtx}{\hVarCtx}{DL2} = \lambda\epsilon.\lambda\delta.\lambda\hat{\x}.\ \mathbb{E}\_{\min}[(\lambda \x. -(\tempty{\text{bounded}}\ \x\ \hat{\x}\ \epsilon) \times (\tempty{\text{bounded}}\ f(\x)\ f(\hat{\x})\ \delta))(\X)] \\ \quad \text{where} \ \tempty{\text{bounded}} = \lambda \x. \lambda \y. \lambda v. \sum\_{i=0}^{783} -\max (-v-(\x\_i - \y\_i),0) - \max(v-(\x\_i - \y\_i),0) \end{aligned}$$ The final problem to consider is whether the interpretation of infinite quantifiers via global minima and maxima commutes with the interpretation for ∧ and ∨ : in general it does not. Only the Gödel interpretation for these connectives (based on min* and max operators) commutes with quantifiers interpreted as minimised or maximised expected values. Section [sec:properties] considers this problem in detail. Using LDL to explore properties of DLs ====================================== Soundness and Completeness of DLs --------------------------------- In all existing work, soundness and completeness of DLs was proved for propositional fragments of those logics. In the proofs the standard Boolean semantics of propositional classical logic is taken as a decidable procedure for membership of the set of all true formulae $\texpr{\top}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}$. The main soundness result establishes that if $\texpr{e}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L} \in \texpr{\top}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}$ then e is true in the propositional logic. Completeness shows that the implication holds in the other direction as well. However, as LDL is equipped with quantifiers such a procedure is no longer decidable, we may no longer rely on this method. For a typed FOL, one could define a Kripke-style semantics for characterising the set of all true formulae, or take provable FOL formulae to characterise the set of true FOL formulae. In this section we take the latter approach. Specifically, we take the set of FOL formulae provable in logic by Gentzen . We will say that a DL is sound, if the set of formulae that it interprets as $\texpr{\top}{\hNetCtx}{\hRandCtx}{\hVarCtx}{L}$ is a subset of formulae provable in. ### Type Soundness of Interpretation We start by proving soundness of the typing relation in LDL. The following result, proven by induction on the typing judgement, will be useful in proving soundness of LDL: [Type Soundness of LDL][th:type-sound] For all differentiable logics L and well typed expressions $\hcTypeRel{e}{\tau}$, then for all $\hNetCtx \in \ttype{\Xi}$, $\hRandCtx \in \ttype{q(e)}$ and $\hVarCtx \in \ttype{\Delta}$ we have $\tempty{e}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \ttype{\tau}$. See Appendix [ap:type-soundness]. We will use the fact that all expressions of type $\RealType$ can be interpreted as real numbers in Section [subsec:soundness]. [Arithmetic evaluation of real expressions][lem:expr-real] If $\validForm{e: \RealType}$ then $\tempty{e}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \Real$. Obtained as a corollary of Theorem [th:type-sound] by instantiating τ with $\RealType$. ### Sequent Calculus Figure [fig:terms] gives a formal definition of well-formed FOL formulae in LDL. This definition is a basis for defining sequents in. 7pt gather\* [fig:terms] [fig:rules-CLJ] We define the set of FOL formulae by induction on the formula shape, in a standard way (see Figure [fig:terms]). FOL formulae are a subset of well-typed expressions of LDL. We use ΓT to denote a set of FOL formulae. The rules in Figure [fig:rules-CLJ] follow the standard formulation of (including notation ΓT, ψ for ΓT ∪ {ψ}). The equivalence closure of the reduction relation (convertibility) is denoted by $\conv$. We will assume that for each comparison relation ⋈ in the language, we have a corresponding oracle ⋈ \* that decides, for any pair r1, r2 of real numbers whether r1 ⋈ \*r2 holds. Formally, r1 ⋈ \*r2 returns ⊤ if the relation holds, and ⊥ otherwise. It is standard in intuitionistic logic to model negation ¬e as $e \impl \bot$, but we use an equivalent formulation from that introduces intuitionistic negation explicitly in the rules. The only additional rule we need here is (Arith). We assume the standard structural rules for which can be found in Appendix [appendix:LJ-struct-rules]. Given this we can now define what it means for a differentiable logic L to be sound and complete with respect to a set of formulae provable in. [Soundness][df:sound] A logic L is sound if for any well-typed formula $\validForm{e}$, and any contexts $\hNetCtx \in \ttype{\Xi}$, $\hRandCtx \in \ttype{q(e)}$, $\hVarCtx \in \ttype{\Delta}$ if $\tempty{e}\_L^{\hNetCtx,\hRandCtx, \hVarCtx} = \tempty{\top}\_L^{\hNetCtx,\hRandCtx, \hVarCtx}$ then $\SequentCLJ(){e}$. [Completeness][df:compl] A logic L is complete if for any well-typed formula $\validForm{e}$, and any contexts $\hNetCtx \in \ttype{\Xi}$, $\hRandCtx \in \ttype{q(e)}$, $\hVarCtx \in \ttype{\Delta}$ if $\SequentCLJ(){e}$ then $\tempty{e}\_L^{\hNetCtx,\hRandCtx, \hVarCtx} = \tempty{\top}\_L^{\hNetCtx,\hRandCtx, \hVarCtx}$. ### Most Fuzzy DLs are sound but incomplete We start with showing that fuzzy DL comparison operators are valid. [Soundness of FDL comparisons][lem:eval] If $\validForm{e\_1 \bowtie e\_2}$ and any contexts $\hNetCtx \in \ttype{\Xi}$, $\hRandCtx \in \ttype{q(e)}$, $\hVarCtx \in \ttype{\Delta}$, then for all L in fuzzy differentiable logics (FDL) the following holds: If $\tempty{e\_1 \bowtie e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \tempty{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L}$ then $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} \bowtie^\* \tempty{e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \top$. If $\tempty{e\_1 \bowtie e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \tempty{\bot}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L}$ then $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} \bowtie^\* \tempty{e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \bot$. See Appendix [app:lemma]. This result uses Lemma [lem:expr-real]. [l:s1] Given a well-typed formula e, for any any contexts $\hNetCtx \in \ttype{\Xi}$, $\hRandCtx \in \ttype{q(e)}$, $\hVarCtx \in \ttype{\Delta}$ the following hold: 1. If $\tGodel{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tGodel{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}$ then $\SequentCLJ(){e}$. 2. If $\tGodel{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tGodel{\bot}^{\hNetCtx,\hRandCtx, \hVarCtx}$ then $\SequentCLJ(e){}$. The auxiliary lemma is proven by case analysis on e, mutual induction, and relies on Lemma [lem:eval], see Appendix [app:proof-godel]. [Soundness of Gödel DL][th:s] Gödel DL is sound. Soundness is obtained as a corollary of Lemma [l:s1]. Other fuzzy DLs, with the exception of Łukasiewicz and Yager are also sound with respect to, with proofs following the same scheme. For the Łukasiewicz DL, we cannot prove the equivalent of Lemma [l:s1]. Intuitively, this is because Łukasiewicz implication is not strong enough: e.g. $ \tempty{e\_1 \impl e\_2} $ evaluates to 1 as long as $\tempty{e\_1} \leq \tempty{e\_2}$. Because $\tempty{e\_1}$ and $\tempty{e\_2}$ can take any values in the interval [0, 1] (as long as $\tempty{e\_1} \leq \tempty{e\_2}$), it is impossible to apply an inductive argument on $\tempty{e\_1}$ and $\tempty{e\_2}$. The Yager DL is a generalisation of the Łukasiewicz DL, in particular, when p = 1 in the Yager DL translation, it coincides with the Łukasiewicz DL. Thus, the same problem stands for Yager. Proper study of the question of (alternative approaches to) soundess of the Łukasiewicz and Yager DLs is left for future work. We note that the recent work by has shed light on logical properties of Łukasiewicz logic, and we intend to build upon those results. Theorem [th:s] does not hold for the opposite direction of implication (and thus completeness fails). To see this, consider the following derivation for $\SequentCLJ{\neg(5 \leq 3)}$ in, which we would have liked to be equivalent to having $\tGodel{\neg(5 \leq 3)} \in \tGodel{\top}$. However, taking $\tGodel{5 \leq 3} = 1 - 0.25 = 0.75$, we get $\tGodel{\neg(5 \leq 3)} = 1 - 0.75 = 0.25 \notin \tGodel{\top}$. This is the problem that will be common for all Fuzzy DLs. ### DL2 is sound and incomplete LDL helps to establish a generic approach to proving soundness results for a variety of DLs. Firstly, a variant of Lemma [lem:eval] holds for DL2. Next we obtain its soundness for DL2, where DL2 has just the connectives ∧ , ∨ and quantifiers, as $\impl$ and ¬ are not defined. [Soundness of DL2][th:d1] DL2 is sound. See Appendix [app:dl2-adequacy]. The proof follows the structure of similar proofs for FDLs. This is a welcome simplification, and it will be useful in future computer formalisations of these logics. Seeing the result is given for an incomplete set of connectives, we omit discussion of completeness for DL2. ### STL is neither sound nor complete Incompleteness of STL is discussed in §[subsec:soundness]. In addition, note that its connectives and quantifiers lack properties that are provable in : e.g. associativity of conjunction, or commutativity of a universal quantifier with respect to conjunction. The definition for soundness relies on the interpretation of $\tSTL{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}$. do not formally define $\tSTL{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}$, and for that reason alone soundness is unattainable for the original STL. We show an attempt to propose an interpretation $\tSTL{\top}^{\hNetCtx,\hRandCtx, \hVarCtx} = \infty$ and show how we use LDL’s generic approach to analyse the result. Informally, their intuition is that any positive number in [0, ∞) belongs to $\tSTL{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}$. This motivates, for example, $\tSTL{ 3 \leq 5}^{\hNetCtx,\hRandCtx, \hVarCtx} = 2$ or $\tSTL{ 3 == 3}^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. But none of these evaluates to ∞. Thus, although the soundness proof goes through, the set of expressions that a variant of Lemma [lem:eval] for STL covers will be empty: such soundness would say nothing about the FOL fragment of LDL. A solution would be to re-define interpretation for all predicates in a binary fashion: e.g. for a1 = = a2, return ∞ if a1 = a2 and return − ∞ otherwise. But this would sacrifice continuity and smoothness of the resulting loss functions, – key properties for STL design. Logical and Geometric Properties of DLs --------------------------------------- Both and suggest a selection of desirable properties for their DLs. Some are logical properties (e.g. associativity or commutativity) and some have a geometric origin, such as (weak) smoothness or shadow-lifting. As some of the DLs we consider do not have associative connectives, we will use $\conjM$ as a notation for conjunction of exactly M conjuncts. Similar definition can be made for disjunction, which we omit. In all of the following definitions we will omit the contexts for clarity as they do not change. We will denote a well-typed formula by e. Starting with geometric properties, smoothness is generally desirable from optimisation perspective as it aids gradient based methods used in neural network training : [Weak smoothness] The $\tempty{e}\_L$ is weakly smooth if it is continuous everywhere and its gradient is continuous at all points in the interval where there is a unique minimum. [Scale invariance for $\conjM$] The $\tempty{e}\_L$ is scale-invariant if, for any real α ≤ 0 $$\alpha \tempty{\conjM (A\_1,..., A\_M)}\_L = \tbig{\conjM}\_L (\alpha \tempty{A\_1}\_L,..., \alpha \tempty{A\_M}\_L)$$ Shadow-lifting is a property original to and motivates the STL conjunction. It characterises gradual improvement when training the neural network: if one conjunct increases, the value of entire conjunction should increase as well. [Shadow-lifting property for $\conjM$] The $\tempty{e}\_L$ satisfies the shadow-lifting property if, $\forall i. \tempty{A\_i}\_L \neq 0$: $$\left. \dfrac{\partial \tempty{\conjM(A\_1,..., A\_M)}\_L}{\partial \tempty{A\_i}\_L}\right\rvert\_{A\_1,..., A\_M} >0$$ where ∂ denotes partial differentiation. Coming from the logic perspective, we have the following desirable properties: [Commutativity, idempotence and associativity of $\conjM$] The $\tempty{e}\_L$ is commutative if for any permutation π of the integers i ∈ 1, ..., M $$\tempty{\conjM (A\_1,..., A\_M)}\_L = \tempty{\conjM (A\_{k\_{\pi(1)}},..., A\_{k\_{\pi(M)}})}\_L$$ it is idempotent and associative if $$\begin{aligned} \tempty{\conjM(A,..., A)}\_L &= \tempty{A}\_L\\ \tempty{\conjL{2}(\conjL{2}(A\_1, A\_2), A\_3)}\_L &= \tempty{\conjL{2}(A\_1, \conjL{2}(A\_2, A\_3))}\_L \end{aligned}$$ The below property has not appeared in the literature before, but as previous sections have shown, it proves to be important when generalising DLs to FOL: [Quantifier commutativity] The $\tempty{e}\_L$ satisfies the quantifier commutativity if $$\tempty{\forall x. \conjM(A\_1,..., A\_M)}\_L = \tempty{\conjM(\forall x. A\_1, \ldots, \forall x. A\_M)}\_L$$ Similarly for ∃ and ∨ . [t] [tab:properties] center \|p0.28\|c\|c\|c\|c\|c\|c\| Properties:& DL2 & Gödel & Łukasiewicz& Yager & Product & STL Weak Smoothness & yes\* & no &no &no &yes\* & yes Shadow-lifting & yes & no &no&no&yes& yes Scale invariance & yes & yes &no&no&no& yes Idempotence & no & yes &no &no &no & yes Commutativity & yes & yes &yes &yes &yes & yes Associativity &yes &yes &yes &yes &yes &no Quantifier commutativity & no & yes & no & no& no & no Soundness & yes & yes & no & no & yes & no Table [tab:properties] summarises how different DLs compare and contains results already established in literature (as cited) as well as provides new results. While we do not provide proofs, the reader can fine an in-depth explanation of the reasons behind the mentioned properties in Appendix [app:properties]. The table inspires the following observations. [i)] Starting with geometric properties, smoothness is traditionally important for differentiation , and connectives for DL2 and Product DL satisfy the property. However, in the FOL extension only STL remains weakly smooth, see also discussion of § [subsec:STL-S]. For the remaining geometric properties, DL2 and STL satisfy shadow-lifting and scale invariance, but Fuzzy logics disagree on these. Thus, only STL satisfies all geometric properties. For logical properties, suggested that a DL cannot both satisfy shadow-lifting, idempotence and associativity. However, their proof of this fact fails within LDL. We leave formal investigation of this conjecture for future work. Since the semantics of quantifiers are defined as global minima and maxima, only connectives of Gödel DL commute with respect to them. This is a highly desirable property, that has not been previously discussed in the literature. It impacts completeness of DLs (cf. § [subsec:soundness] and [subsec:STL-S]). Only Gödel DL satisfies all logical properties, but it fails smoothness and shadow-lifting properties. On the opposite side of the spectrum, STL behaves well geometrically, but fails associativity and quantifier commutativity, and has unresolved issues with soundness (§ [subsec:STL-S]). It warrants future research whether a DL with optimal combination of logical and geometric properties may be formulated. Empirical Evaluation -------------------- LDL offers a unified framework to evaluate different DLs. We have implemented the syntax and semantics as an extension to the Vehicle tool which supports the general syntax of LDL. The implementation is highly modular due to the semantics of each DL sharing the same semantics for the core of the syntax as explained in Section [sec:semantics]. We explain the workflow for our running example – the robustness property. Taking the specification of Example [ex:wte], it is compiled by Vehicle to a loss function in Python where it can be used to train a chosen neural network. The loss function resulting from the LDL semantics is typically used in combination with a standard loss function such as cross-entropy. If we denote cross-entropy loss together with its inputs as LCE and the LDL loss together with its inputs and parameters as LDL then the final loss will be of the form: L = α ⋅ LCE + β ⋅ LDL where the weights $\alpha, \beta \in \Real$ are parameters. For the preliminary tests we used a simple 2-layer network to classify MNIST images . As seen in Figure [fig:experiments] we have tested the effect of varying the $\alpha, \beta \in \Real$ parameters and therefore the weight of the custom loss in training on accuracy and constraint satisfaction; the results are broadly consistent with trends reported in the literature previously. We note that formulating heuristics for finding minima/maxima of loss functions, which are necessary for the implementation of quantifier interpretation, can be very complex, and are beyond the scope of this paper. Figure [fig:experiments] shows results implemented using random sampling instead of sampling around global maxima or minima. We leave implementation of heuristics of finding global minima/maxima, as well as thorough experimental study, for future work. .4![image](figures/all-DLs-accuracy.png) .4![image](figures/all-DLs-constraint-satisfaction-1.png) [fig:experiments] Conclusions, Related and Future Work ==================================== #### Conclusions. We have presented a general language (LTL) for expressing properties of neural networks. Our contributions are: 1) LDL generalises other known DLs and provides a language that is rich enough to express complex verification properties; 2) with LDL we achieved the level of rigour that allows to formally separate the formal language from its semantics, and thus opens a way for systematic analysis of properties of different DLs; 3) by defining different DLs in LDL, we proved properties concerning their soundness and resulting loss functions, and opened the way for uniform empirical evaluation of DLs. We now discuss related work. #### Learning Logical Properties. There are many methods of passing external knowledge in the form of logical constraints to neural networks. The survey by discusses multitude of approaches including methods based on loss functions , to which LDL belongs, but also others such as tailoring neural network architectures , or guaranteeing constraint satisfaction at the level of neural network outputs . As establish, the majority of approaches are tailored to specific problems, and only and go as far as to include quantifiers. LDL generalises both. Analysis of properties of loss functions. Property analysis, especially of smoothness or bilateral properties , is a prominent field . One of LDL’s achievements is to expose trade-offs between satisfying desired geometric and logic properties of a loss functions. Neural Network Verification. While this work does not attempt to verify neural networks, we draw our motivation from this area of research. It has been observed in verification literature that neural networks often fail to satisfy logical constraints . One of proposed solutions is training the NN to satisfy a constraint prior to verifying them . This belongs to an approach referred to as continuous verification which focuses on the cycle between training and verification. LDL fits into this trend. Indeed, the tool Vehicle that implements LDL is also built to work with NN verifiers . Logics for Uncertainty and Probabilistic Logics. LDLs have a strong connection to fuzzy logic . Via the use of probability distributions and expectations, we draw our connection to Probabilistic Prolog and similar languages . We differ as none of those approaches can be used to formulate loss functions, which is the main goal of LDL. Adversarial training. Starting with the seminal paper by , thousands of papers in machine learning literature have been devoted to adversarial attacks and robust training of neural networks. Majority of those papers does not use a formal logical language for attack or loss function generation. LDL opens new avenues for this community, as allows one to formulate other properties of interest apart from robustness, see e.g. . #### Future work. We intend to use LDL to further study the questions of a suitable semantics for DL, both proof-theoretic and denotational, possibly taking inspiration from . For proof-theoretic semantics, we need to find new DLs with tighter correspondence to ; and moreover new calculi can be developed on the basis of to suit the purpose. Since loss functions are used in computation, we conjecture that constructive logics (or semi-constructive logics as in ) will be useful in this domain. For denotational semantics, we intend to explore Kripke frame semantics. Future work could also include finding the best combination of logical and geometric properties in a DL and formulating new DLs that satisfy them. Thorough evaluation of performance of all the DLs is also left to future work: it includes formulating heuristics for finding local/global maxima for quantifier interpretation, and evaluation of LDL effectiveness within the continuous verification cycle. Finally, finding novel ways of defining quantifiers that commute with DL connectives is an interesting challenge. Acknowledgements ================ This work was supported by the EPSRC grant EP/T026952/1, AISEC: AI Secure and Explainable by Construction and the EPSRC DTP Scholarship for N. Ślusarz. We thank anonymous referees, James McKinna, Wen Kokke, Bob Atkey and Emile Van Krieken for valuable comments on the early versions of this paper, and Marco Casadio for contributions to the Vehicle implementation. Supplementary Definitions for Different DLs =========================================== Additional DLs based on Fuzzy logic ----------------------------------- LDL can be used to express other DLs based on fuzzy logic aside from ones defined in Section [sec:semantics]. Figure [fig:expr1] gives a few more examples originally from . The semantics of comparison operators are omitted as they are identical to those of the logic. \| Syntax \| Łukasiewicz \| Yager \| product \| \| --- \| --- \| --- \| --- \| \| $\tempty{\BoolType}\_L$ \| [0, 1] \| [0, 1] \| [0, 1] \| \| $\tempty{\top}\_L$ \| 1 \| 1 \| 1 \| \| $\tempty{\bot}\_L$ \| 0 \| 0 \| 0 \| \| $\tempty{\neg}\_L$ \| $\lam{\val}1 - \val$ \| $\lam{\val}1 - \val$ \| $\lam{\val}1 - \val$ \| \| $\tempty{\wedge}\_L$ \| $ \lam{\val\_1,\val\_2} \max(\val\_1+\val\_2-1,0)) $ \| $ \lam{\val\_1,\val\_2}\max (1 - ((1-\val\_1)^p + (1-\val\_2)^p)^{1/p},0)$ \| $\lam{\val\_1,\val\_2} \val\_1 \cdot \val\_2$ \| \| $\tempty{\vee}\_L$ \| $ \lam{\val\_1,\val\_2} \min (\val\_1+\val\_2,1) $ \| $ \lam{\val\_1,\val\_2}\min ((\val\_1^p + \val\_2^p)^{1/p},1)$ \| $\lam{\val\_1,\val\_2} \val\_1 + \val\_2 - \val\_1 \times \val\_2$ \| \| $\tempty{\impl}\_L$ \| $ \lam{\val\_1,\val\_2} \min (1 -\val\_1+\val\_2,1) $ \| - \| $ \lam{\val\_1,\val\_2} 1 - \val\_1 + \val\_1 \times \val\_2 $ \| \| $ \tempty{==}\_L $ \| \| \| \| \| $ \tempty{\leq}\_L $ \| \| \| \| [fig:expr1] Remaining comparisons --------------------- We have defined the semantics of both ≤ and = = in Section [subsec:expressions]. In many logics the semantics of other comparisons could be expressed in terms of the semantics of ≤ , = = and logical connectives. However, as not all DLs presented have negation, this approach is not feasible in general and therefore we define all the remaining comparisons in Figure [fig:more-comparisons]. \|c\|c\|c\|c\| Syntax & DL2 & Fuzzy Logics & STL $\tempty{\neq}\_{L}$ & $ \lam{\val\_1,\val\_2}{-\xi[\val\_1=\val\_2]} $& & $ \tempty{<}\_{L} $ & $\lam{\val\_1, \val\_2} \tDLtwo{(\val\_1\leq\val\_2)\land (\val\_1 \neq \val\_2)}$ & & $ \tempty{\geq}\_{L} $ & $\lam{\val\_1, \val\_2}{\val\_2 \leq \val\_1}$ & & $ \tempty{>}\_{L} $ & $\lam{\val\_1, \val\_2} \tDLtwo{(\val\_1\geq\val\_2)\land (\val\_1 \neq \val\_2)}$& & [fig:more-comparisons] Negation in DL2 --------------- DL2 does not have an interpretation for negation - instead negation of a term is pushed inwards syntactically to the level of comparisons between terms. While this cannot be expressed as a single lambda function, it is possible to express it as a partial function that applies the same operation on LDL syntax, as defined in Figure [eq:lowerNot]. It is however not possible to define a total function as it is not possible to push the negation through parts of syntax such as a lambda. $$\begin{aligned} \tDLtwo{\neg(a\_1 \wedge a\_2)} & = \tDLtwo{\neg a\_1 \vee \neg a\_2}\\ \tDLtwo{\neg(a\_1 \vee a\_2)} & = \tDLtwo{\neg a\_1 \wedge \neg a\_2}\\ \tDLtwo{\neg(a\_1 \leq a\_2)} & = \tDLtwo{a\_1 > a\_2 }\\ \tDLtwo{\neg(a\_1 < a\_2)} & = \tDLtwo{a\_1 \geq a\_2 }\\ \tDLtwo{\neg(a\_1 \geq a\_2)} & = \tDLtwo{a\_1 < a\_2 }\\ \tDLtwo{\neg(a\_1 > a\_2)} & = \tDLtwo{a\_1 \leq a\_2 }\\ \tDLtwo{\neg(a\_1 == a\_2)} & = \tDLtwo{a\_1 \neq a\_2 }\\ \tDLtwo{\neg(a\_1 \neq a\_2)} & = \tDLtwo{a\_1 == a\_2 } \end{aligned}$$ [eq:lowerNot] Supplementary Definitions for. =============================== Structural Rules for LJ ----------------------- has four structural rules: weakening on the left, exchange on the left, contraction on the left, and weakening on the right. $$\begin{gathered} \AxiomC{$\SequentCLJ(\ThAssumsEnv){e}$} \UnaryInfC{$\SequentCLJ(\ThAssumsEnv, e\_1){e}$} \bottomAlignProof \DisplayProof \qquad \AxiomC{$\SequentCLJ{e}$} \UnaryInfC{$\SequentCLJ{e, e\_1}$} \bottomAlignProof \DisplayProof \qquad \AxiomC{$\SequentCLJ(\ThAssumsEnv, e\_1, e\_2, \Gamma\_T'){e}$} \UnaryInfC{$\SequentCLJ(\ThAssumsEnv, e\_2, e\_1, \Gamma\_T'){e}$} \bottomAlignProof \DisplayProof \qquad \AxiomC{$\SequentCLJ(\ThAssumsEnv, e\_1, e\_1){e}$} \UnaryInfC{$\SequentCLJ(\ThAssumsEnv, e\_1){e}$} \bottomAlignProof \DisplayProof \end{gathered}$$ Proof of Type Soundness of LDL ============================== We prove by induction on the typing judgment the Theorem [th:type-sound] which reads as follows. [Type Soundness of LDL] For all differentiable logics L and well typed expressions $\hcTypeRel{e}{\tau}$, then for all $\hNetCtx \in \ttype{\Xi}$, $\hRandCtx \in \ttype{q(e)}$ and $\hVarCtx \in \ttype{\Delta}$ we have $\tempty{e}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \ttype{\tau}$. Base Case 1. Suppose we have $\hcTypeRel{e}{\VecType{m} \to \VecType{n}}$ (e is a network variable). Then we have $e: \VecType{m} \to \VecType{n} \in \Xi$. But then we have $N[e] \in (\Real^n)^{\Real^m}$ by assumption. Base Case 2. Suppose we have $\hcTypeRel{e}{\tau}$ (e is a bound variable). Then we have e : τ ∈ Δ. But then we have $\Gamma[e] \in \ttype{\tau}$ by the definition of Γ. Base Case 3. Suppose we have $\hcTypeRel{\elReal}{\RealType}$. By definition in Figure [fig:types] we also have $r \in \Real$. Base Case 4. Suppose we have $\hcTypeRel{i}{\FinType}$. By definition in Figure [fig:types] we also have i ∈ Index n. Base Case 5. Suppose we have $\hcTypeRel{b}{\BoolType}$. By definition in Figure [fig:types] we have $b \in \ttype{\BoolType}$. The interpretation of $\ttype{\BoolType}$ depends on the DL in question but is always a subset of $\Real$ and does not impact the proof. Base Case 6. Suppose we have $\hcTypeRel{e}{+ :\RealType \to \RealType \to \RealType}$. This follows by definition in Figure [fig:types]. Base Case 7. Suppose we have $\hcTypeRel{e}{\times :\RealType \to \RealType \to \RealType}$. This follows by definition in Figure [fig:types]. Base Case 8. Suppose we have $\hcTypeRel{e}{\wedge,\vee,\implies :\BoolType \to \BoolType \to \BoolType}$. This follows by definition in Figure [fig:types]. Base Case 9. $\hcTypeRel{e}{\not :\BoolType \to \BoolType}$ This follows by definition in Figure [fig:types]. Base Case 9. $\hcTypeRel{e}{\bowtie :\RealType \to \RealType \to \BoolType}$ This follows by definition in Table [fig:types]. Inductive Case 1. Suppose e is an application and we have $\hcTypeRel{\App{e\_1}{e\_2}}{\tau\_2}$. By definition in Figure [fig:types] we have $ \hcTypeRel{e\_1}{\FunType{\tau\_1}{\tau\_2}} $ and $ \hcTypeRel{e\_2}{\tau\_1} $. Then by the induction hypothesis we have $\tempty{e\_1}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \ttype{\tau\_2}^{\ttype{\tau\_1}}$ and $\tempty{e\_2}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \ttype{\tau\_1}$. By application we directly have that $ \tempty{e\_1}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \tempty{e\_2}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \ttype{\tau\_2}$. Inductive Case 2. Suppose we have $\hcTypeRel{\Lam{\id}{\tau\_1}{e}}{\FunType{\tau\_1}{\tau\_2}}$ (e is a lambda). Then by definition in Figure [fig:types] we have $ \hTypeRel{\hcNetCtx,\consNew{\id \rightarrow \tau\_1}{\hcVarCtx}}{e}{\tau\_2} $. We want to show that $\lambda y. \tempty{e}^{\hNetCtx,\hRandCtx,\hVarCtx[x\rightarrow y]}\_L \in \ttype{\tau\_2}^{\ttype{\tau\_1}}$. Assuming $y \in \ttype{\tau\_1}$ we therefore need to show that $\tempty{e}^{\hNetCtx,\hRandCtx,\hVarCtx[x\rightarrow y]}\_L \in \ttype{\tau\_2}$. By the induction hypothesis we can now show that $\forall z:\tau \in \hcVarCtx[(x\rightarrow\tau\_1)]\.\ (\hVarCtx[x\rightarrow \tempty{e\_1}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L])[z] \in \ttype{\tau}$. Inductive Case 3. Suppose we have $\hcTypeRel{\text{let} (x\ :\ \tau\_1) = e\_1\ \text{in}\ e\_2}{\tau\_2}$ (e1 is a let). Then by definition in Figure [fig:types] and the induction hypothesis we have $\tempty{e\_1}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L \in \ttype{\tau\_1}$ and that $\hTypeRel{\hcNetCtx,\consNew{\id,\tau\_1}{\hcVarCtx}}{e\_2}{\tau\_2}$. We want to show that $\tempty{e\_2}^{\hNetCtx,\hRandCtx,\hVarCtx[x\rightarrow \tempty{e\_1}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L]}\_L \in \ttype{\tau\_2}$. By induction hypothesis we have that $\forall z. \tau \in \consNew{\id \rightarrow \tau\_1}{\hcVarCtx}. (\hVarCtx[x\rightarrow \tempty{e\_1}^{\hNetCtx,\hRandCtx,\hVarCtx}\_L])[z] \in \ttype{\tau} $ which follows from other induction hypothesis. Inductive Case 4. Suppose e is a vector. By definition in Figure [fig:types] we have $\hcTypeRel{e\_1}{\RealType} \ldots \hcTypeRel{e\_n}{\RealType}$. This follows from induction hypothesis on each ei. Proofs of Soundness of LDL Relative to ====================================== Proof of Lemma [lem:eval] ------------------------- We start with a lemma needed for base case of adequacy proofs for all FDLs. In the proofs, we reduce ⋈ to { = = , ≤ , ≠ }, in the cases of other comparison operators can be proved analogously. [Soundness of FDL comparisons] If $\validForm{e\_1 \bowtie e\_2}$, then for all L in fuzzy differentiable logics (FDL) the following holds: If $\tempty{e\_1 \bowtie e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \tempty{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L}$ then $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} \bowtie \tempty{e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \top$. If $\tempty{e\_1 \bowtie e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \tempty{\bot}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L}$ then $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} \bowtie \tempty{e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{L} = \bot$. The proof of Lemma [lem:eval] proceeds by case-reasoning on the comparison operators. Since e1 ⋈ e2 is well-typed, each of e1, e2 has type $\RealType$. In that case, $\tempty{e\_i}^{\semCtx} \in \Real$ by Lemma [lem:expr-real]. Case 1. If ⋈ is = = , then $\tempty{e\_1 == e\_2}^{\semCtx}\_{FDL} = 1$ means $\tempty{e\_1}^{\semCtx}\_{FDL} - \tempty{e\_2}^{\semCtx}\_{FDL} = 0$, that is, $\tempty{e\_1}^{\semCtx}\_{FDL} = \tempty{e\_2}^{\semCtx}\_{FDL}$. But then, $\tempty{e\_1}^{\semCtx}\_{FDL} ==^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \top$. Similarly, $\tempty{e\_1 == e\_2}\_{FDL} = 0$ means $\tempty{e\_1}^{\semCtx}\_{FDL} - \tempty{e\_2}^{\semCtx}\_{FDL} \neq 0$, that is, $\tempty{e\_1}^{\semCtx} \neq \tempty{e\_2}^{\semCtx}\_{FDL}$. But then, $\tempty{e\_1}^{\semCtx}\_{FDL} ==^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \bot$. Case 2. If ⋈ is ≤ , then $\tempty{e\_1 \leq e\_2}^{\semCtx}\_{FDL} = 1$ means $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{FDL} - \tempty{e\_2}^{\semCtx}\_{FDL} \leq 0$, that is, $\tempty{e\_1}^{\semCtx}\_{FDL} \leq \tempty{e\_2}^{\semCtx}\_{FDL}$. But then, $\tempty{e\_1}^{\semCtx}\_{FDL} \leq^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \top$. Similarly, $\tempty{e\_1 \leq e\_2}^{\semCtx}\_{FDL} = 0$ means $\tempty{e\_1}^{\semCtx}\_{FDL} - \tempty{e\_2}^{\semCtx}\_{FDL} > 0$, that is, $\tempty{e\_1}^{\semCtx}\_{FDL} > \tempty{e\_2}^{\semCtx}\_{FDL}$. But then, $\tempty{e\_1}^{\semCtx}\_{FDL} \leq^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \bot$. Case 3. If ⋈ is ≠ , then $\tempty{e\_1 \neq e\_2}^{\semCtx}\_{FDL} = 1$ means $[\tempty{e\_1}^{\semCtx}\_{FDL} = \tempty{e\_2}^{\semCtx}\_{FDL}] = 0$, that is, $\tempty{e\_1}^{\semCtx}\_{FDL} \neq \tempty{e\_2}^{\semCtx}\_{FDL}$. But then, $\tempty{e\_1}^{\semCtx}\_{FDL} \neq^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \top$. Similarly, $\tempty{e\_1 \leq e\_2}^{\semCtx}\_{FDL} = 0$ means $[\tempty{e\_1}^{\semCtx}\_{FDL} = \tempty{e\_2}^{\semCtx}\_{FDL}] = 1$, that is, $\tempty{e\_1}^{\semCtx}\_{FDL} = \tempty{e\_2}^{\semCtx}\_{FDL}$. But then, $\tempty{e\_1}^{\semCtx}\_{FDL} \neq^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \bot$. The remaining comparisons in ⋈ are all defined using the already proven comparisons. Proof of Soundness for Gödel DL ------------------------------- The below proof is for Lemma [l:s1]. The mutually inductive proof proceeds by case-reasoning on the shape of the formula e, and by induction on the structure of $\tGodel{}^{\semCtx}$. As finite quantifiers are defined via conjunctions, we only cover the case of infinite quantifiers explicitly. As the proof is mutually inductive between two parts of Lemma [l:s1], we will refer to them accordingly as Lemma [l:s1] (Part 1) and Lemma [l:s1] (Part 2). We start with Lemma [l:s1] (Part 1). The case of e = ⊥ is automatically excluded, so our first base case is: Base Case 1. Suppose e = ⊤. But we know $\SequentCLJ(){\top}$ by the rule (⊤). Base Case 2. Suppose e = e1 ⋈ e2. As $\tGodel{e\_1 \bowtie e\_2}^{\semCtx} = 1$, by Lemma [lem:eval], we have $\tempty{e\_1}^{\semCtx}\_{FDL} \bowtie^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \top$. But then, we can derive $\SequentCLJ(){ e\_1 \bowtie e\_2}$ by the rules $ \LJArithR$ and (⊤). Inductive Case 1. Suppose e = ¬e1, and therefore $\tGodel{e}^{\semCtx} = 1-\tGodel{e\_1}^{\semCtx} = 1$. This is only possible when $\tGodel{e\_1}^{\semCtx} = 0$. Then, by Lemma [l:s1] (Part 2), we have $\SequentCLJ(e\_1){}$. But then, by the rule $\LJNegR$, we can derive $\SequentCLJ(){\neg e\_1}$. Inductive Case 2. Suppose e = e1 ∧ e2, and therefore $\tGodel{e}^{\semCtx} = \min (\tGodel{e\_1}^{\semCtx}, \tGodel{e\_2}^{\semCtx}) = 1$. This means that $\tGodel{e\_1}^{\semCtx} = \tGodel{e\_2}^{\semCtx} = 1 $. And, by the induction hypothesis, we have that $\SequentCLJ(){e\_1}$ and $\SequentCLJ(){e\_2}$. But then, by the rule $\LJConjR$, we have $\SequentCLJ(){e\_1 \land e\_2}$. Inductive Case 3. Suppose $e = e\_1 \impl e\_2$, and therefore $\tGodel{e}^{\semCtx} = \max (1-\tGodel{e\_1}^{\semCtx}, \tGodel{e\_2}^{\semCtx}) = 1$. This means that at least one of the following is true: $1-\tGodel{e\_1}^{\semCtx} = 1$ or $\tGodel{e\_2}^{\semCtx} = 1$. In the first case we have $\tGodel{e\_1}^{\semCtx} = 0$. Then, by Lemma [l:s1] (Part 2), we have $\SequentCLJ(e\_1){}$. In the second case, by the induction hypothesis, we have $\SequentCLJ(){e\_2}$. In either case, using one of the weakening rules, we can obtain $\SequentCLJ(\Gamma\_T, e\_1){e\_2}$. This allows us to use the rule $\LJImplR$ to derive $\SequentCLJ{e\_1 \impl e\_2}$. Inductive Case 4. Suppose e = ∀x : τ.e1, and therefore we have $\tGodel{\forall x: \tau.\ e\_1 }^{\semCtx} = 1$. This means that the minimum expected value of $\lambda y. \tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 1. Seeing that 1 is the top value, it means that $\tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 1$ for all inputs y. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1}$. We therefore can deduce $\SequentCLJ(){\forall x. e\_1}$ ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Inductive Case 5. Suppose e = ∃x : τ.e1, and therefore we have $\tGodel{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 1$. This means that the maximum expected value of $\lambda y. \tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 1. Tt means that $\tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 1$ for at least one input w. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1[y / x]}$. We therefore can deduce $\SequentCLJ(){\exists x. e\_1}$. We now move on to the proof of Lemma [l:s1] (Part 2). The case of e = ⊤ is automatically excluded, so our first base case is: Base Case 1. Suppose e = ⊥. But we know $\SequentCLJ(\bot){}$ by the (⊥). Base Case 2. Suppose e = e1 ⋈ e2, where ⋈ is a comparison operator. Moreover, $\tGodel { e\_1 \bowtie e\_2}^{\semCtx} = 0$. By Lemma [lem:eval], we have $\tempty{e\_1}^{\semCtx}\_{FDL} \bowtie^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \bot$. But then we obtain a proof for $\SequentCLJ(\Gamma\_T,e\_1 \leq e\_2){ }$ by the rules $ \LJArithL $ and (⊥). Inductive Case 1. Suppose e = ¬e1, and therefore $\tGodel{e}^{\semCtx} = 1-\tGodel{e\_1}^{\semCtx} = 0$. This is only possible when $\tGodel{e\_1}^{\semCtx} = 1$. Then $\tGodel{e\_1}^{\semCtx} \in \tGodel{\top}^{\semCtx}$. Then by Lemma [l:s1] (Part 1) we have $\SequentCLJ(){e\_1 }$. But then by $\LJNegL$ we can derive $\SequentCLJ(\neg e\_1){ }$. Inductive Case 2. Suppose e = e1 ∧ e2, and therefore $\tGodel{e}^{\semCtx} = \min (\tGodel{e\_1}^{\semCtx}, \tGodel{e\_2}^{\semCtx}) = 0$. This means that at least one of the following is true: $\tGodel{e\_1}^{\semCtx} = 0$ or $ \tGodel{e\_2}^{\semCtx} = 0 $. And, by the induction hypothesis, we have that $\SequentCLJ(e\_1){}$ or $\SequentCLJ(e\_2){}$. But then, by the rule $\LJConjL$, we have $\SequentCLJ(e\_1 \land e\_2){}$. Inductive Case 3. Suppose $e = e\_1 \impl e\_2$, and therefore ${\tGodel{e}^{\semCtx} = \max (1-\tGodel{e\_1}^{\semCtx}, \tGodel{e\_2}^{\semCtx}) = 0}$. This means that both $1-\tGodel{e\_1}^{\semCtx} = 0$ and $\tGodel{e\_2}^{\semCtx} = 0$. This gives us $\tGodel{e\_1}^{\semCtx} = 1$, therefore by Lemma [l:s1] (Part 1) we obtain $\SequentCLJ(){e\_1}$. For $\tGodel{e\_2}^{\semCtx} = 0$, we use the induction hypothesis and conclude $\SequentCLJ(e\_2){}$. Using the rule $\LJImplL$, we obtain $\SequentCLJ(e\_1 \impl e\_2){}$. Inductive Case 4. Suppose e = ∀x.e1, and therefore we have $\tGodel{\forall x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the minimum expected value of $\lambda y. \tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. It means that $\tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for some input w. But then, by the induction hypothesis, we have $\SequentCLJ(e\_1[y / w]){}$. We therefore can deduce $\SequentCLJ(\forall x. e\_1){}$. Inductive Case 5. Suppose e = ∃x.e1, and therefore we have $\tGodel{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the maximum expected value of $\lambda y. \tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. Seeing that 0 is the bottom value, it means that $\tGodel{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for all inputs. But then, by the induction hypothesis, we have $\SequentCLJ(e\_1){}$. We therefore can deduce $\SequentCLJ(\exists x. e\_1){}$, ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Proof of Soundness for Product DL --------------------------------- We first define analogous theorem for another fuzzy logic - product, denoted $\tproduct{}$. [th:p1] Given a formula e, for any contexts N, Γ, Q the following hold: 1. if $\tproduct{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tproduct{\top}^{\hNetCtx,\hRandCtx, \hVarCtx} $ then $\SequentCLJ(){e}$. 2. if $\tproduct{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tproduct{\bot}^{\hNetCtx,\hRandCtx, \hVarCtx}$ then $\SequentCLJ(e){}$. The mutually inductive proof proceeds by case-reasoning on the shape of the formula e, and by induction on the structure of $\tproduct{}^{\semCtx}$, a choice of one of the DLs. As the proof is mutually inductive between two parts of Lemma [th:p1], we will refer to them accordingly as Lemma [th:p1] (Part 1) and Lemma [th:p1] (Part 2). We start with the first part of Lemma [th:p1]. The case of e = ⊥ is automatically excluded, so our first base case is: Base Case 1. Suppose e = ⊤. But we know $\SequentCLJ(){\top}$ by the rule (⊤). Base Case 2. Suppose e = e1 ⋈ e2. Moreover, $\tproduct {e\_1 \bowtie e\_2}^{\semCtx} = 1$. By Lemma [lem:eval], we have $\tempty{e\_1}^{\semCtx}\_{FDL} \bowtie^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \top$. But then, we can derive $\SequentCLJ(){ e\_1 \bowtie e\_2}$ by the rules $ \LJArithR$ and (⊤). Inductive Case 1. Suppose e = ¬e1, and therefore $\tproduct{e}^{\semCtx} = 1-\tproduct{e\_1}^{\semCtx} = 1$. This is only possible when $\tproduct{e\_1} = 0$. Then, by Lemma [th:p1] (Part 2), we have $\SequentCLJ(e\_1){}$. But then, by the rule $\LJNegR$, we can derive $\SequentCLJ(){\neg e\_1}$. Inductive Case 2. Suppose e = e1 ∧ e2, and therefore $\tproduct{e}^{\semCtx} = \tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx} = 1$. Since 1 is the top value that means we have $\tproduct{e\_1}^{\semCtx} = \tproduct{e\_2}^{\semCtx} = 1 $ And, by the induction hypothesis, we have that $\SequentCLJ(){e\_1}$ and $\SequentCLJ(){e\_2}$. But then, by the rule $\LJConjR$, we have $\SequentCLJ(){e\_1 \land e\_2}$. Inductive Case 3. Suppose $e = e\_1 \impl e\_2$, and therefore $\tproduct{e}^{\semCtx} = 1-\tproduct{e\_1}^{\semCtx}+ \tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx} = 1$. This means that $\tproduct{e\_1}^{\semCtx}= \tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx}$. From this at least one of the following is true $\tproduct{e\_1}^{\semCtx} = 0$ or $\tproduct{e\_2}^{\semCtx} = 1 $. In the first case by Lemma [th:p1] (Part 2) we have $\SequentCLJ(e\_1){}$. In the second case, by induction hypothesis we have $\SequentCLJ(){e\_2}$. Now using one of the weakening rules, we can obtain $\SequentCLJ(e\_1){e\_2}$. This allows us to use the rule $\LJImplR$ to derive $\SequentCLJ(){e\_1 \impl e\_2}$. Inductive Case 4. Suppose e = ∀x.e1, and therefore we have $\tproduct{\forall x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 1$. This means that the minimum expected value of $\lambda y. \tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 1. Seeing that 1 is the top value, it means that $\tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 1$ for all inputs y. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1}$. We therefore can deduce $\SequentCLJ(){\forall x. e\_1}$ ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Inductive Case 5. Suppose e = ∃x.e1, and therefore we have $\tproduct{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 1$. This means that the maximum expected value of $\lambda y. \tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 1. It means that $\tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 1$ for at least one input w. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1[y / x]}$. We therefore can deduce $\SequentCLJ(){\exists x. e\_1}$. We now move on to the proof of the second part of Lemma [th:p1]. The case of e = ⊤ is automatically excluded, so our first base case is: Base Case 1. Suppose e = ⊥. But we know $\SequentCLJ(\bot){}$ by the (⊥). Base Case 2. Suppose e = e1 ⋈ e2, where ⋈ is a comparison operator, and e1, e2 are real numbers. Moreover, $\tproduct { e\_1 \bowtie e\_2}^{\semCtx} = 0$. By Lemma [lem:eval], we have $\tempty{e\_1}^{\semCtx}\_{FDL} \bowtie^\* \tempty{e\_2}^{\semCtx}\_{FDL} = \bot$. But then we obtain a proof for $\SequentCLJ(\Gamma\_T,e\_1 \leq e\_2){ }$ by the rules $ \LJArithL $ and (⊥). Inductive Case 1. Suppose e = ¬e1, and therefore $\tproduct{e}^{\semCtx} = 1-\tproduct{e\_1}^{\semCtx} = 0$. This is only possible when $\tproduct{e\_1}^{\semCtx} = 1$. Then $\tproduct{e\_1}^{\semCtx} \in \tproduct{\top}^{\semCtx}$. Then by Lemma [th:p1] (Part 1) we have $\SequentCLJ(){e\_1 }$. But then by $\LJNegL$ we can derive $\SequentCLJ(\neg e\_1){ }$. Inductive Case 2. Suppose e = e1 ∧ e2, and therefore $\tproduct{e}^{\semCtx} = \tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx} = 0$. This means that at east one of the following holds - $ \tproduct{e\_1}^{\semCtx} = 0 $ or $ \tproduct{e\_2}^{\semCtx} = 0 $. In both cases respectively by induction hypothesis we have $\SequentCLJ(e\_1){}$ or $\SequentCLJ(e\_2){}$. But then, by the rule $(\LJConjL)$, we have $\SequentCLJ(e\_1 \land e\_2){}$. Inductive Case 3. Suppose $e = e\_1 \impl e\_2$, and therefore $\tproduct{e}^{\semCtx} = 1-\tproduct{e\_1}^{\semCtx}+ \tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx} = 0$. This means that $\tproduct{e\_1}^{\semCtx}-1= \tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx}$. However as $\tproduct{e\_1}^{\semCtx} \times \tproduct{e\_2}^{\semCtx} \in [0,1]$ that means that $\tproduct{e\_1}^{\semCtx}-1 \in [0,1]$ and therefore $\tproduct{e\_1}^{\semCtx} = 1$ and by Lemma [th:p1] (Part 1) we obtain $\SequentCLJ(){e\_1}$. From this we have $\tproduct{e\_2}^{\semCtx} = 0$ and we use the induction hypothesis and conclude $\SequentCLJ(e\_2){}$. Using the rule $\LJImplL$, we obtain $\SequentCLJ(e\_1 \impl e\_2){}$. Inductive Case 4. Suppose e = ∀x.e1, and therefore we have $\tproduct{\forall x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the minimum expected value of $\lambda y. \tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. It means that $\tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for some input w. But then, by the induction hypothesis, we have $\SequentCLJ(e\_1[y / x]){}$. We therefore can deduce $\SequentCLJ(\forall x. e\_1){}$. Inductive Case 5. Suppose e = ∃x.e1, and therefore we have $\tproduct{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the maximum expected value of $\lambda y. \tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. Seeing that 0 is the bottom value, it means that $\tproduct{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for all inputs. But then, by the induction hypothesis, we have $\SequentCLJ(e\_1){}$. We therefore can deduce $\SequentCLJ(\exists x. e\_1){}$, ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Its corollary is: [Soundness of Product DL] Given a formula e, for any contexts N, Γ, Q if$\tproduct{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tproduct{\top}^{\hNetCtx,\hRandCtx, \hVarCtx} = 1$ then $\SequentCLJ(){e}$. Proof of Soundness for DL2 -------------------------- We now provide the proof for Theorem [th:d1]. It is important to remember that DL2 translation does not include a stand-alone negation operator or implication - therefore this proof is done for a limited version of. We start with a helper lemma. [Adequacy of intervals and arithmetic operations in DL2][lem:eval1] If $\validForm{e\_1 \bowtie e\_2} : \RealType$ and $\tDLtwo{e\_1 \bowtie e\_2}^{\semCtx} = 0$ then $\tempty{e\_1}^{\semCtx}\_{DL2} \bowtie \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. Since e1 ⋈ e2 is well-typed each of e1, e2 has type $\RealType$. If ⋈ is = = , then $\tDLtwo{e\_1 == e\_2}^{\semCtx} = 0$ means $\tDLtwo{e\_1}^{\semCtx} - \tDLtwo{e\_2}^{\semCtx} = 0$, that is, $\tDLtwo{e\_1}^{\semCtx} = \tDLtwo{e\_2}^{\semCtx}$. But then, $\tempty{e\_1}^{\semCtx}\_{DL2} == \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. If ⋈ is ≤ , then $\tDLtwo{e\_1 \bowtie e\_2}^{\semCtx} = 0$ means $\tDLtwo{e\_1}^{\semCtx} - \tDLtwo{e\_2}^{\semCtx} \leq 0$, that is, $\tDLtwo{e\_1}^{\semCtx} \leq \tDLtwo{e\_2}^{\semCtx}$. But then, $\tempty{e\_1}^{\semCtx}\_{DL2} \leq \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. If ⋈ is ≠ , then $\tDLtwo{e\_1 \neq e\_2}^{\semCtx} = 0$ means $[\tDLtwo{e\_1}^{\semCtx} == \tDLtwo{e\_2}^{\semCtx}] = 0$, that is, $\tDLtwo{e\_1}^{\semCtx} \neq \tDLtwo{e\_2}^{\semCtx}$. But then, $\tempty{e\_1}^{\semCtx}\_{DL2} \neq \tempty{e\_2}^{\semCtx}\_{DL2} = \top$. The remaining comparisons in ⋈ are all defined using the already proven comparisons. We now move on to the proof of Theorem [th:d1]. [Soundness of DL2] Given a formula e, taking DL2 with just connectives ∧ , ∨ and quantifiers, for any contexts N, Γ, Q: if $\tDLtwo{e}^{\hNetCtx,\hRandCtx, \hVarCtx} = \tDLtwo{\top}^{\hNetCtx,\hRandCtx, \hVarCtx}$ then $\SequentCLJ(){e}$. Base Case 1. Suppose e = ⊤. But we know $\SequentCLJ(){\top}$ by the rule (⊤). Base Case 2. Suppose e = e1 ⋈ e2, where ⋈ is a comparison operator, and e1, e2 are real numbers. Moreover, $\tDLtwo { e\_1 \bowtie e\_2}^{\semCtx} = 0$. By Lemma [lem:eval1], we have $\tempty{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{DL2} \bowtie^\* \tempty{e\_2}^{\hNetCtx,\hRandCtx, \hVarCtx}\_{DL2} = \top$. But then we obtain a proof for $\SequentCLJ{e\_1 \bowtie e\_2}$ by the rules $ \LJArithR $ and (⊤). Inductive Case 1. Suppose e = e1 ∧ e2, and therefore ${tDLtwo{e}^{\semCtx} = \tDLtwo{e\_1}^{\semCtx} + \tDLtwo{e\_2}^{\semCtx} = 0}$.That means we have $\tDLtwo{e\_1}^{\semCtx} = \tDLtwo{e\_2}^{\semCtx} = 0 $ And, by the induction hypothesis, we have that $\SequentCLJ(){e\_1}$ and $\SequentCLJ(){e\_2}$. But then, by the rule $\LJConjR$, we have $\SequentCLJ(){e\_1 \land e\_2}$. Inductive Case 2. Suppose e = ∀x.e1, and therefore we have $\tDLtwo{\forall x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the minimum expected value of $\lambda y. \tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. Seeing that 0 is the top value, it means that $\tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for all inputs y. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1}$. We therefore can deduce $\SequentCLJ(){\forall x. e\_1}$ ($x \notin FV(\Gamma\_T)$, as ΓT is empty). Inductive Case 3. Suppose e = ∃x.e1, and therefore we have $\tDLtwo{\exists x: \tau.\ e\_1 }^{\hNetCtx,\hRandCtx, \hVarCtx} = 0$. This means that the maximum expected value of $\lambda y. \tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]}$ is 0. It means that $\tDLtwo{e\_1}^{\hNetCtx,\hRandCtx, \hVarCtx[x\rightarrow y]} = 0$ for at least one input w. But then, by the induction hypothesis, we have $\SequentCLJ(){e\_1[y / x]}$. We therefore can deduce $\SequentCLJ(){\exists x. e\_1}$. Further Discussion of Logical and Geometric Properties ====================================================== The DLs defined as part of LDL have the following properties: * $\tDLtwo{}$ is commutative, scale-invariant, associative, sound (for a limited ) and has shadow-lifting. It is not idempotent and does not have quantifier commutativity. While its semantics for logical connectives is weakly smooth, $\tDLtwo{}$ is not weakly smooth due to the presence of min and max in translation of comparisons. While no proofs about the properties were provided in the translation of conjunction in DL2 and product based fuzzy logic is standard addition. The more common properties including idempotence, commutativity, associativity and min-max boundedness in given domain are known and proven. Considering the semantics of conjunction is addition it is simple enough to reason about its partial derivatives and therefore shadow-lifting. * $\tGodel{}$ is idempotent, commutative, scale-invariant, associative, sound and has quantifier commutativity. It does not have shadow-lifting and it is not weakly smooth. The semantics of conjunction is a minimum between the two elements which prohibits shadow-lifting as well as smoothness by definition. The simpler properties are well known and proved in fuzzy logic literature as it is an established fuzzy logic. * $\tlukasiewicz{}$ is commutative, sound (for a syntax excluding either negation or implication) and associative. It is not idempotent, scale-invariant, weakly smooth and does not have shadow-lifting or quantifier commutativity. The presence of maxima in the semantics of conjunction however naturally prohibits shadow lifting and smoothness. Similarly to Gödel it is an established fuzzy logic and many of the properties of its semantics of conjunction have been proven. * $\tyager{}$ is commutative, sound (for a limited syntax) and associative. It is not idempotent, scale-invariant, weakly smooth and does not have shadow-lifting or quantifier commutativity. Similarly to the previous two the presence of maxima prohibits shadow lifting and smoothness. This is another one of the well established fuzzy logics for which majority of the properties have been already investigated by the community. * $\tproduct{}$ is commutative, associative, sound and has shadow-lifting. It is not idempotent, scale-invariant, does not have quantifier commutativity. While its semantics of logical connectives are weakly smooth $\tproduct{}$ is not weakly smooth due to the semantics of comparisons. Furthermore as its semantics of conjunction is the arithmetic operation of multiplication it is easy to reason about its partial derivates and therefore shadow lifting. Since it is a well-established fuzzy logic its properties the more common logical properties have been extensively studied already. * $\tSTL{}$ is idempotent, commutative, scale-invariant, weakly smooth and has shadow-lifting. It is not associative or sound and does not have quantifier commutativity. The properties of this translation have been proven in the paper it originates from as indicated in Table [tab:properties], aside from quantifier commutativity which has been added in this paper. The translation has been changed for the purposes of LDL however the proofs would remain analogous. Furthermore it is the only DL for which the soundness does not hold, for either full or limited. * Only $\tGodel{}$’s quantifiers commute with connectives, as its connectives are defined via max and min (commuting with maxima and minima used in quantifier interpretation).	arxiv_0000512
Integrated Sachs-Wolfe map reconstruction in the presence of systematic errors ============================================================================== The decay of gravitational potentials in the presence of dark energy leads to an additional, late-time contribution to anisotropies in the cosmic microwave background (CMB) at large angular scales. The imprint of this so-called integrated Sachs-Wolfe (ISW) effect to the CMB angular power spectrum has been detected and studied in detail, but reconstructing its spatial contributions to the CMB map, which would offer the tantalizing possibility of separating the early- from the late-time contributions to CMB temperature fluctuations, is more challenging. Here we study the technique for reconstructing the ISW map based on information from galaxy surveys and focus in particular on how its accuracy is impacted by the presence of photometric calibration errors in input galaxy maps, which were previously found to be a dominant contaminant for ISW signal estimation. We find that both including tomographic information from a single survey and using data from multiple, complementary galaxy surveys improve the reconstruction by mitigating the impact of spurious power contributions from calibration errors. A high-fidelity reconstruction further requires one to account for the contribution of calibration errors to the observed galaxy power spectrum in the model used to construct the ISW estimator. We find that if the photometric calibration errors in galaxy surveys can be independently controlled at the level required to obtain unbiased dark energy constraints, then it is possible to reconstruct ISW maps with excellent accuracy using a combination of maps from two galaxy surveys with properties similar to Euclid and SPHEREx. Introduction ============ Cosmic microwave background (CMB) photons undergo a frequency shift as they travel to us from the last scattering surface. On top of the redshift due to the expansion of the Universe, an additional contribution to the temperature anisotropy is introduced whenever the universe is not matter dominated—for example, right after recombination when radiation contributes non-negligibly, or at late times when dark energy becomes important. This so-called integrated Sachs-Wolfe (ISW) effect is given by $$\label{ISWorigexpr} \left.\frac{\Delta T}{\bar{T}}\right\|\_{ISW}({{\bf \hat n}}) = \frac{2}{c^2}\int\_{t\_\}^{t\_0}dt\,\frac{\partial \Phi(\mathbf{r},t)}{\partial t},$$ where t0 is the present time, t⋆ is that of recombination, c* is the speed of light, r is the position in comoving coordinates, and Φ is the gravitational potential. The late-time ISW signal (referred to hereafter simply as ‘ISW’) has been statistically detected via measurements of the cross-correlation of CMB temperature maps with galaxy maps and, more recently, with maps of CMB lensing convergence. These detections serve as an important consistency test of the standard model of cosmology, and can help constrain the properties of dark energy. The ISW can provide additional information beyond its power spectrum if its map can be reconstructed with sufficient signal-to-noise. Since the total large-angle CMB temperature anisotropy is the sum of early- (hereafter ‘primordial’) and late-time contributions, $$\left.\frac{\Delta T}{\bar{T}}\right\|({{\bf \hat n}}) = \left.\frac{\Delta T}{\bar{T}}\right\|\_\mathrm{prim}({{\bf \hat n}}) + \left.\frac{\Delta T}{\bar{T}}\right\|\_\mathrm{ISW}({{\bf \hat n}}),$$ reconstructing the ISW map would allow us to isolate the primordial-only anisotropy. This separation of the CMB into early- and late-time contributions can also be useful for a variety of cosmological tests. For example, one could study the temporal origin of the large-angle CMB anomalies reported in, e.g., Ref . One could also subtract the realization-specific contaminating ISW contribution to estimation of primordial non-Gaussianity, something that is currently done using theoretical templates for the ISW-lensing bispectrum. Motivated by these considerations, reconstruction of the ISW map has been the focus of a number of recent analyses . In this paper we study how ISW map reconstruction is affected by a class of observational and astrophysical systematic errors which we will refer to broadly as photometric calibration errors or, for conciseness, calibration errors. These systematics afflict all galaxy surveys at large angular scales, contributing to the significant excess of power at large scales found in many recent surveys, including the Sloan Digital Sky Survey (SDSS), MegaZ, WISE-AGN and WISE-GAL, and NVSS. Calibration errors are thus already established as one of the most significant systematics impacting large-angle measurements of galaxy surveys, a fact that has broad implications, such as for measuring scale-dependent bias as a signal of primordial non-Gaussianity. As the statistical power of galaxy surveys continues to grow, the control and understanding of systematics like calibration errors is becoming even more important. There is a variety of ways in which modern photometric surveys assess and mitigate contamination from systematics, many of which rely on cross-correlating galaxy maps with known systematics templates. This can be used to identify contaminated regions, which are then masked or excluded from the analysis (as in Ref. ). Alternatively, one can use these templates to subtract or marginalize over systematics-induced spatial variations in the calculation of, for example, the two-point clustering signal. Such an approach was taken in Ref. to study the overall detection significance of the ISW effect in SDSS data. The authors found results similar to Ref. , the authors of which instead accounted for excess power by adding a low-redshift spike in the source distributions. Most of these correlation corrections are perturbative, however, and therefore require fairly clean maps in which systematic effects are minor to begin with. Additionally, while corrections to the two-point statistics are important for the inference of cosmological parameters, they do not remove the systematics from the maps themselves. propose an alternative approach, wherein measurement biases are characterized by injecting fake objects into Dark Energy Survey images. This neatly avoids the reliance on having small levels of contamination in the input maps, but it still cannot account for certain systematics, such as dust or flux calibration. Whatever the approach taken, some level of residual calibration error will remain. Some of us previously showed that at levels of calibration control consistent with current and near-future surveys, residual calibration errors are by far the dominant systematic for ISW signal reconstruction . This motivates us to study their impact in more detail. Namely, we would like to study whether the presence of residual calibration errors can be mitigated by combining information from multiple input maps or through better modeling of the contributions of systematics to observed galaxy power. We also wish to investigate to what extent residual calibration errors similarly impact the signal-to-noise ratio of galaxy-CMB cross-correlation and, in turn, the significance of ISW detection. With this aim, we use ensembles of simulated maps to characterize the performance of ISW reconstruction based on surveys like Euclid and SPHEREx, two proposed wide-angle surveys of which the properties are expected to be good for ISW detection and reconstruction. We also consider the benefits of including Planck-like simulations of CMB intensity in the reconstruction effort. We begin in Sec. [sec:method] by describing our model for calibration errors, how we reconstruct the ISW map and evaluate its quality, and which input data sets we use. In Sec. [sec:results], we compare the performance of ISW reconstruction when using one versus multiple surveys and investigate the impact different assumptions have on the results. In Sec. [sec:SNcal], we relate map reconstruction to the total signal-to-noise ratio of ISW detection, and we conclude in Sec. [sec:conclusion]. [fig:plancknvssplot] [fig:planckmphgplot] Methodology =========== Modeling calibration errors --------------------------- Photometric calibration is a challenge faced by all photometric galaxy surveys. It refers to the adjustments required to establish a consistent spatial and temporal measurement of flux of the target objects in different observation bands. A number of different systematics must be accounted for in calibration, including but not limited to detector sensitivity variation on the focal plane, variation in observing conditions, the presence of foreground stars (as galaxies near them are less likely to be detected), and extinction by dust. Calibration errors are introduced if these systematics are incompletely or inaccurately accounted for. Our focus is on how calibration errors affect galaxy number counts. To illustrate this, imagine that a perfectly uniform screen (of e.g. dust) blocks some light from all galaxies. This pushes the faintest galaxies below the survey’s flux limit, and leads to observation of fewer galaxies in all directions. A pure monopole change such as this increases shot noise but does not affect the angular clustering signal of galaxies. In contrast, in a more realistic scenario where the opacity of this “screen” depends on direction, it affects the observed galaxy clustering signal by adding spurious power and by coupling different multipoles of the measured power spectrum . The presence of unaccounted-for calibration errors can thus introduce biases in cosmological parameter estimates from Large Scale Structure (LSS) surveys. These are particularly severe for the ISW effect and other measurements based on signals that, like calibration errors, enter primarily at large angular scales. In keeping with the picture of calibration errors as a direction-dependent screen, we model them as a modulation of the true galaxy number counts $N(\hat{\bf n})$, where ${{\bf \hat n}}$ is the direction on the sky. The observed, modulated counts are $${N\_{\rm obs}}({\hat{\bf n}}) = [1+c({\hat{\bf n}})]\, N({\hat{\bf n}}), \label{eq:nobs}$$ where the field $c({\hat{\bf n}})$ describes the screening effect of calibration errors. While we will generally refer to this kind of modulation as the result of ``calibration errors," as Eq. ([eq:nobs]) makes clear, this formalism can describe any residual effect that modulates a survey’s selection function. Though the expression in Eq. ([eq:nobs]) will necessarily couple different multipoles, at low ℓ, the impact of calibration errors on the observed galaxy power spectrum is well approximated by CobsCgal + Ccal, [eq:clobs] neglecting multiplicative terms. Following, we model the calibration error field $c({{\bf \hat n}})$ as a Gaussian random field with power spectrum ${C\_{\ell}^{\rm cal}}$ and quantify the level of residual calibration errors using its variance, $${{\sigma^2\_{\rm cal}}}\equiv {\rm Var}(c(\hat{\bf n})) = \sum\_{\ell=0}^\infty {2\ell+1\over 4\pi}{C\_{\ell}^{\rm cal}}. \label{eq:calib\_var}$$ While the conversion between $\sigma\_{\rm cal}$ and the rms magnitude error depends on the faint-end slope of the luminosity function of tracers in the survey, they can be related roughly as $(\delta m)^{\rm rms}\simeq \sigma\_{\rm cal}$. Thus a survey with residual calibration errors of ${{\sigma^2\_{\rm cal}}}=10^{-6}$ has been calibrated to roughly a milimagnitude. Motivated by power spectrum estimates for maps of dust extinction corrections and magnitude limit variations in existing surveys (see Fig. 5 and 6 in Ref. ), we choose the fiducial calibration error power spectrum to be $$\label{eq:clcalmodel} {C\_\ell}^{\rm cal} = \alpha^{\rm cal} \exp{\left[-(\ell/w^{\rm cal})^2\right]},$$ with $w^{\rm cal}=10$. The normalization constant $\alpha^{\rm cal}$ is varied to achieve the desired ${{\sigma^2\_{\rm cal}}}$. Figures [fig:plancknvssplot] and [fig:planckmphgplot] show the impact of calibration errors of this form on the angular power spectrum of the NVSS and SDSS MphG galaxy surveys, which have been used to reconstruct the ISW map in previous studies For our fiducial model, we assume that calibration error maps for different redshift bins and surveys are uncorrelated with one another. We briefly examine the impact of relaxing such an assumption in Sec. [sec:caldetails]. ISW estimator ------------- Similarly to , we work with the optimal estimator derived by . It takes as input n maps, which can include any tracers that carry information about the ISW signal, namely LSS, CMB, or lensing convergence maps. Letting gℓmi represent the spherical components of the ith input map, where i ∈ {1, …, n}, the optimal estimator for the spherical component of the ISW signal is [eq:iswestsimple] mISW = i=1nRigmi. The operator $$R\_{\ell}^i\equiv -N\_{\ell}[D\_{\ell}^{-1}]\_{{\rm ISW-}i} \label{eq:R\_def}$$ is a reconstruction filter derived from the covariance matrix, [eq:Dl] D = ( cccc CISW,ISW&CISW,1 &&C n C1,ISW&C1,1 &&C1,n &&& Cn,ISW&Cn,1&&Cn,n ). In this expression, superscript numbers label the input maps and $N\_{\ell} \equiv 1/[D\_{\ell}^{-1}]\_{\rm ISW-ISW}$ estimates the variance of the reconstruction at multipole ℓ. When a single input map A, is used, this expression reduces to a simple Wiener filter, mISW gmA. [eq:wiener] We demonstrate in Appendix [app:otherest] that Eq. ([eq:iswestsimple]) is equivalent to the estimator of, where the CMB temperature map is treated separately from LSS maps, and show that it reduces to the Linear Covariance Based (LCB) filter first proposed by . In constructing this ISW estimator, one must make a choice about how to obtain the necessary angular power and cross-power spectra in the covariance matrix. The Cℓ’s can either be extracted from observations (as in Refs. ) or computed analytically for an assumed cosmology (as in Refs. ). Analytic calculation is straightforward but introduces a model dependence which can potentially bias results if, for example, calibration error contributions are not modeled correctly . Measuring Cℓ from observations produces a model-independent estimator and so can help in the case where the theory spectra are inaccurate, but at the expense of limited precision due to sample variance, especially at large scales, scales with low power, or for map combinations that have little correlation.[1](#fn1) Hybrid methods can also be used, as in Ref. , where account for observed excesses in the autopower of NVSS data by using a smoothed fit to data to get the galaxy map’s autopower, but analytically compute its cross-correlation with the ISW signal. We therefore consider two limiting cases of constructing the estimator in order to investigate how calibration errors impact the ISW reconstruction: 1. a ‘worst’ case estimator filter, ${{R\_\ell({C\_\ell}^{\rm th})}}$, where we use the fiducial theory Cℓ’s in the estimator, in which calibration errors’ power contributions are not modeled at all, and 2. a ‘best’ case estimator filter, ${{R\_\ell({C\_\ell}^{\rm sky})}}$, in which calibration error power contributions are modeled perfectly (i.e. the covariance matrix is known). This case may be approximated by, e.g. a smoothed fit of the observed LSS power. The theoretical spectra are related simply through the expression [eq:clcalrel] Csky= Cth+ Ccal. where ${C\_{\ell}^{\rm cal}}$ is the power spectrum of the calibration error field described in Sec. [sec:calmodel]. We consider these cases in Secs. [sec:euconlyresults] and [sec:eucpluscmb] respectively. Quality statistic ----------------- To quantify the accuracy of a given reconstruction, we use the correlation coefficient between the temperature maps of the true [$T^{\rm ISW}({{\bf \hat n}})$] and reconstructed [$T^{\rm rec}({{\bf \hat n}})$] ISW signal, = ISWrec, [eq:rhofrommap] where $\bar{T^{X}}$ and σX2 are the mean and variance of map $T^{X}({{\bf \hat n}})$, respectively.[2](#fn2) We do not include pixel weights in our calculation of ρ, as is done to account for masking effects in Ref. . This is because we work with only full-sky maps, as will be discussed in the next section. The correlation coefficient can be rewritten in terms of the cross-power between the true ISW map realization and the input tracers, = ISWrec, [eq:rhoraw] where the tilde denotes pseduo-Cℓ measured from a given map realization, and we have used Eq. ([eq:iswestsimple]) to write $$\begin{aligned} \tilde{C}\_{\ell}^{\rm ISW-rec}&=\sum\_i\frac{1}{2\ell+1}\sum\_{m} [{a\_{\ell m}}^{\rm ISW}]^\R\_{\ell}^i g\_{\ell m}^i \\ &= \sum\_iR\_{\ell}^i\tilde{C}\_{\ell}^{{\rm ISW}-i}.\end{aligned}$$ Because the measured correlation coefficient depends on the specific realization, we assess reconstruction accuracy for a given set of input map properties as follows. We simulate a large number of realizations of correlated maps, then apply the ISW estimator to obtain associated reconstructed ISW maps, and by comparing these with the true ISW maps we obtain a sample distribution for ρ. Its mean value ρ̄, which in the limit of an infinitely large ensemble will approach an expectation value ⟨ρ⟩, provides a statistical measure of how accurately the estimator can reproduce the true ISW signal. Studying how ρ̄* changes in response to variations in survey properties and modeling choices therefore allows us to understand which factors are most important for obtaining an accurate ISW reconstruction. We can avoid the computational cost of generating many simulation ensembles by noting that we can obtain a good estimate for the expectation value of ρ if we make the approximation $$\begin{aligned} \langle\rho\rangle &= \left< {\frac{\frac{1}{4\pi} \sum\_{\ell, i}(2\ell+1)R\_{\ell}^i\tilde{{C\_\ell}}^{{\rm ISW}-i}} {\sigma\_{\rm ISW}\sigma\_{\rm rec} }} \right> \label{eq:rhoexp}\\ &\approx \frac{\frac{1}{4\pi} \sum\_{\ell, i}(2\ell+1)R\_{\ell}^i{C\_\ell}^{{\rm ISW}-i}}{\hat \sigma\_{\rm ISW}\hat \sigma\_{\rm rec}}, \label{eq:rhoest}\end{aligned}$$ that is, we replace the pseudo-Cℓ’s with their expectation value across realizations, $\tilde{{C\_\ell}}~\rightarrow~{{C\_\ell}}$. We will refer to the quantity in Eq. ([eq:rhoest]) as ρ̂, defining $$\begin{aligned} \hat \sigma\_{\rm ISW} &= \sqrt{ \frac{1}{4\pi}\sum\_{\ell}\,(2\ell+1)\,{C\_\ell}^{\rm ISW}} \\ \hat \sigma\_{\rm rec} &=\sqrt{\frac{1}{4\pi} \sum\_{\ell, i, j}\,(2\ell+1)\,R\_{\ell}^iR\_{\ell}^j{C\_\ell}^{ij}} \,, \label{eq:sigrecest}\end{aligned}$$ to approximate the rms fluctuations in the true and reconstructed ISW maps. Here the indices i and j label the input tracer maps and the sum over ℓ runs over the multipoles ℓ ∈ [2, 95], a range chosen to conservatively to include all scales where the ISW signal is important. We have tested the approximation ρ̂ ≈ ⟨ρ⟩ in Eq. ([eq:rhoest]) extensively and found it works well when the estimator filter Rℓ is built from analytically computed spectra but can break down if Rℓ is composed of $\tilde{C\_\ell}$’s extracted from map realizations. This behavior is related to the way in which using measured Cℓ’s makes ρ depend on $\tilde{C\_\ell}$, such that $\bar{\rho} = \rho(\langle\tilde{C\_\ell}\rangle)$ is no longer a good approximation of $\langle\rho(\tilde{C\_\ell})\rangle$. Appendix [sec:appapprox] discusses this in more detail. Simulated surveys ----------------- By working with simulated maps, we are able to study in detail how calibration error levels and modeling choices affect ISW signal reconstruction. Since we are concerned only with large scales, we model the ISW signal, total CMB temperature anisotropy, and galaxy number density fluctuations as correlated Gaussian fields. We use `HEALPY` to generate map realizations based on input auto- and cross-power spectra which we compute analytically following the standard expressions given e.g. in Ref. . We use the Limber approximation for ℓ ≥ 20, having verified that this affects ρ at the level of 0.1% or less for the surveys and range of ${{\sigma^2\_{\rm cal}}}$ considered here. We compute Cℓ for multipoles with ℓ ≤ 95, as this range contains almost all of the ISW signal . Accordingly, our simulations are sets of `HEALPIX` maps of resolution ${\rm NSIDE}=(\ell\_{\rm max}+1)/3=32$. We refer the reader to Ref. for a more detailed description of the reconstruction pipeline. Because our goal is to study the impact of calibration errors and not survey geometry, we assume full-sky coverage in all of our analyses. found that in overlapping regions of partial sky LSS surveys, ISW reconstruction quality degrades only slightly compared to the full-sky case. Therefore, the performance of a given estimator using full-sky maps should be indicative of its performance using maps with only partial sky coverage. Our fiducial cosmological model is ΛCDM, with the best-fit cosmological parameter values from Planck 2015, {Ωch2, Ωbh2, Ωνh2, h, ns} = {0.1188, 0.0223, 0, 0.6774, 0.9667}. Unless otherwise stated, ISW reconstructions are performed on 2000 map realizations for each analysis and include multipole information down to $\ell\_{\rm min}=2$. Within this framework, four pieces of information are required to model a LSS survey: the distribution of its sources along the line of sight n(z), a prescription for how they are binned in redshift, their linear bias b(z), and their projected number density per steradian n̄. Below we describe how our choices for these characteristics are based on the properties of promising future probes of the ISW effect. ### Euclid-like LSS survey Our fiducial survey is modeled on Euclid, a future LSS survey with large sky coverage and a deep redshift distribution , which is expected to be an excellent probe of the ISW effect . We assume the redshift distribution used by, [eq:fiddndz] = z2, which has a maximum at $z\_{\rm peak}\simeq 1.21z\_0$. We choose z0 = 0.7 and n̄ = 3.5 × 108, with a photo-z redshift uncertainty of σ(z) = 0.05(1 + z) which smoothes the edges of redshift bins. For simplicity, we assume a constant galaxy bias of b(z) = 1. Our results are qualitatively insensitive to this choice as long as the bias is reasonably well approximated for the input maps. This is because the bias term cancels in the estimate of the ISW signal, so that fractional differences between true and modeled bias have little impact on ρ. We refer the reader to Ref. for further details on both fitting for bias and the impact that mismodeling can have on reconstruction. In Sec. [sec:euconlyresults] we investigate the improvement in ISW map reconstruction when the fiducial Euclid-like survey is split into six redshift bins with edges at z ∈ {0.01, 0.4, 0.8, 1.2, 1.6, 2, 3.5} (see inset of Fig. [fig:euconlyplot]), as compared to the unbinned case. We subsequently use the six-binned Euclid survey as our fiducial case. ### SPHEREx-like LSS survey We model a second survey on the SPHEREx All-Sky Spectral Survey (SPHEREx), a proposed survey that has been optimized to study LSS in the low-redshift universe. One of its goals is to place stringent limits on primordial non-Gaussianity , which will require rigorous control of calibration errors. Given this, SPHEREx will provide excellent input map candidates for ISW map reconstruction. Its shallower reach makes it complementary to the deeper mapping of the LSS provided by Euclid. SPHEREx will identify galaxies with varying levels of redshift uncertainty, ranging from σz < 0.003(1 + z) up to σz > 0.1(1 + z). Grouping these into catalogs with different levels of precision provides collections of galaxies useful for different science goals. The σz < 0.1(1 + z) catalog with a projected ∼ 300 million galaxies was identified in Ref. as the best subsample for $f\_{\rm NL}^{\rm loc}$ detection. Our investigations confirm it to be the best for ISW detection as well. We therefore fit its projected redshift distribution given in Ref. to the functional form for dn/dz given in Eq. ([eq:fiddndz]). We select z0 = 0.46, which results in a peak dn/dz of $z\_{\rm peak} \simeq 0.56$. We have confirmed that our results are not strongly sensitive to changes in this redshift distribution, in agreement with the findings of Ref. . We use a projected number density of n̄ = 6.6 × 107 and consider the case where the survey is split into six redshift bins. We choose their edges by scaling the Euclid-like survey’s binned redshift distribution to the SPHEREx median redshift, resulting in redshift bin edges at z ∈ {0.01, 0.26, 0.53, 0.79, 1.05, 1.31, 2.30}. This still provides sufficient sampling of the field in each bin to ensure that shot noise is subdominant to the galaxy signal power. ### Planck-like CMB survey CMB data have frequently been used in conjunction with LSS data for ISW map reconstruction. Recent examples include Ref. , which used NVSS radio data, the Planck lensing convergence map, and Planck temperature data. That analysis was subsequently extended to include more LSS tracers in Ref. . However, in both of these cases, residual systematics limit the usefulness of lensing data to scales of ℓ ≥ 10 and ℓ ≥ 8, respectively. investigated the usefulness of CMB data for ISW reconstruction using a simulation pipeline similar to ours, finding that both CMB temperature and polarization data only modestly improve reconstruction quality but carry a greater benefit when the LSS tracers themselves contain less information (due to e.g. noise or other properties of the survey). It is then natural to ask whether CMB data can help mitigate the impact of calibration errors in LSS maps. We therefore consider CMB temperature as an additional input map. To compute the total CMB temperature power spectrum, CℓTT, we compute the primordial-only contributions using a modified version of `CAMB` and add them to our calculations for ${C\_\ell}^{\rm ISW}$. As the CMB power spectrum is determined within the limits of cosmic variance at low ℓ and the ISW signal is already dominated by the primary (that is, non-ISW) CMB anisotropies, we do not include calibration errors in the generation of CMB temperature maps. Though CMB polarization and lensing could provide additional information, residual systematics remain at large scales for each (see Refs. and, respectively), so for simplicity we do not include them in this analysis. Results ======= To characterize the impact of calibration errors in LSS surveys on the ISW map reconstruction, and the potential to mitigate these impacts, we look at multiple combinations of input maps with different properties. Specifically, we consider the impact of binning in redshift, of adding CMB intensity data, and of including additional LSS information from another survey. For each of these studies, we examine two limiting cases for the estimator. The best case scenario, which we will reference as ${{R\_\ell({C\_\ell}^{\rm sky})}}$, is when one perfectly models all contributions to the galaxy power, including residual calibration errors. The worst case, referenced by ${{R\_\ell({C\_\ell}^{\rm th})}}$, is when the estimator is built out of theoretical spectra with no power from calibration errors. The power spectra in these two cases are related by Eq. ([eq:clcalrel]). We use the analytical $\hat\rho$ to estimate the mean reconstruction quality across a wide range of ${{\sigma^2\_{\rm cal}}}$, while performing reconstruction on simulated maps for selected values, to both verify the accuracy of $\hat\rho$ and to generate error bars for the spread of ρ across simulations. One survey: Binning in redshift ------------------------------- [fig:euconlyplot] [fig:eucpluscmb] We first consider the Euclid-like survey alone and investigate the impact of binning in redshift on the quality of reconstructions in the presence of calibration errors. We model calibration errors in the binned case by adding the calibration field’s power to the autopowerpower spectrum of each bin i per Eq. ([eq:clobs]): ${C\_\ell}^{i,i} \rightarrow {C\_\ell}^{i,i} + {C\_\ell}^{\rm cal}$. We do not add any power to the cross-spectra, though we test the impact of contamination in the cross-spectra in Sec. [sec:caldetails].[3](#fn3) The results of this study are shown in Fig. [fig:euconlyplot]. For reference, we use a vertical shaded band to mark the level of calibration corresponding to current surveys, defined roughly as the range bounded by the residual SDSS DR8 limiting magnitude variations and the SDSS über-calibration. (‘Future’ levels are defined roughly as those between that required to limit bias on cosmological parameters to below their projected uncertainties and an intermediate level before bias becomes unacceptable; see Refs. and for details.) As shown in Fig. [fig:euconlyplot], splitting the survey into six redshift bins results in significant improvement in reconstruction at all levels of calibration error. This improvement is comparable to reducing ${{\sigma^2\_{\rm cal}}}$ of the single-bin case by a factor of 10 at ‘current’ levels. Tomographic information is useful because it allows galaxy counts to be weighted more optimally, taking advantage of the fact that the ISW effect becomes stronger at lower redshift as dark energy becomes more dominant and structure growth slows. For instance, considering the expected ISW reconstruction power from each bin when using optimal weights (i.e. the squared contribution of each term in Eq. [eq:iswestsimple], using ${{R\_\ell({C\_\ell}^{\rm sky})}}$), we find that with no calibration error, the first redshift bin contributes 87% as much power as the second bin, with subsequent bins contributing 58%, 31%, 15% and 10% as much power, respectively. There is additional benefit to binning when calibration errors are considered. Since the low-redshift bins have a higher clustering signal than the high-redshift bins, they are less impacted by the same level of calibration error. Thus, the optimal weighting changes depending on the level of calibration error. When calibration errors are increased to ${{\sigma^2\_{\rm cal}}}=10^{-4}$, for example, the first bin contributes the most power to the reconstruction, with bins 2 − 6 only contributing 39%, 12%, 4%, 2%, and 1% as much power. As we will show later, this error-level-dependent weighting will mean adding information from a shallower survey such as SPHEREx makes reconstruction more robust against calibration errors. The importance of accounting for calibration errors in the estimator is apparent in the difference between the dashed and solid curves, where doing so improves $\bar\rho$ for ${{\sigma^2\_{\rm cal}}}\gtrsim 10^{-6}$, with $\Delta \bar\rho \approx 0.1-0.2$ at current levels of calibration. This improvement is roughly comparable to the improvement seen from binning in redshift. Though for clarity we do not include this case in the Figure, we additionally studied the effect of using the observed, unsmoothed galaxy-galaxy power in the estimator (that is, $\tilde{C\_\ell}$, power spectra extracted from map realizations rather than computed analytically). We find that in this case $\bar\rho$ converges to the same value as the ${{R\_\ell({C\_\ell}^{\rm sky})}}$ case when calibration errors are very large, but is greatly reduced from $\bar\rho$ found using either ${{R\_\ell({C\_\ell}^{\rm sky})}}$ or ${{R\_\ell({C\_\ell}^{\rm th})}}$ when calibration errors are small (${{\sigma^2\_{\rm cal}}}\ll 10^{-5}$). For example, for a single input map in the limit of no calibration errors, quality reduces from ρ̄ = 0.93 to 0.83 when we switch to using observed $\tilde{C\_\ell}$’s. If we also use the observed (unsmoothed) cross-correlation between the LSS map and the CMB for the galaxy-ISW term in the estimator, reconstruction quality is further degraded to ρ̄ = 0.74 in the absence of calibration errors. This is because primary CMB anisotropies are large compared to ISW contributions, causing the measured galaxy-CMB correlation to receive relatively large noise contributions from chance correlations between LSS maps and the primordial CMB. Given the significant improvement in reconstruction that binning provides, from here forward we adopt the configuration with six tomographic bins as our fiducial Euclid-like survey. Effect of adding Planck TT data ----------------------------------- We now consider adding information from the Planck-like CMB temperature map described in Sec. [sec:modsurveys]. When used as the only input map, the reconstruction is considerably worse than that found using the ideal Euclid-like survey (Table [table:rhotable]). We include it in our study, however, because any realistic study attempting to reconstruct the ISW signal will likely include CMB temperature data. Additionally, the reconstruction quality attainable with CMB temperature data alone provides a useful baseline against which to compare the performance of estimators based on LSS maps. With CMB temperature data alone, we find an average reconstruction quality of ρ̄ = 0.46, in good agreement with Ref. . To put this into proper context, however, it is important to note that there is a large scatter around that mean; while the average reconstruction quality is indicative of performance, any single realization, such as that of our own Universe, can vary substantially in fidelity. The purple band in Fig. [fig:eucpluscmb] shows the extent of this scatter for ISW reconstruction based on just the CMB map. When CMB temperature information is combined with that from LSS maps, it significantly improves reconstruction quality, but only if the true galaxy power spectrum ${C\_\ell}^{\rm sky}$ (including calibration error contributions) is used in the estimator, as can been seen by the behavior of the solid curves on the right-hand side of Fig. [fig:eucpluscmb]. The blue $\rho({{\sigma^2\_{\rm cal}}})$ curve describing the CMB+LSS reconstruction tracks the maximum of the curves corresponding to reconstructions using the CMB and LSS input maps separately, shown by the purple and red curves, respectively. This occurs because the estimator down-weights the LSS survey the more it is affected by calibration errors, converging to the TT-only reconstruction quality in the limit of large calibration errors. If one does not model calibration error power contributions, however, then any improvement from combining multiple input maps is marginal at best and can in fact result in a worse reconstruction than just using the CMB data alone. This demonstrates the importance of ensuring that the LSS Cℓ’s used in the ISW estimator are a good fit to the observed spectra. Effect of an additional LSS survey: SPHEREx-like ------------------------------------------------ We now consider the addition of our fiducial six-bin SPHEREx-like survey described in Sec. [sec:spherexintro], assuming for simplicity that it has the same level of calibration errors as the Euclid-like survey. Results are shown in Fig. [fig:multsurveyplot]. In the limit of no calibration errors, the SPHEREx-like survey offers little additional information. In fact, adding both SPHEREx and CMB TT results in negligible improvement over the Euclid-like only case ($\Delta\bar\rho<0.003$ compared to a spread of $\sigma\_{\rm Euc+Spx+TT}=0.019$). However, by comparing the black and blue curves we see that including the SPHEREx-like survey does make the reconstruction somewhat more robust against calibration errors. The reason for this is similar to why binning in redshift is helpful: recall that, in the case of binning, having narrow, low-redshift bins means having some bins with higher galaxy autopower than the unbinned case, which then have less susceptibility to a given level of calibration error. Similarly, SPHEREx has a shallower redshift distribution, and thus an intrinsically higher clustering signal, so that it can actually provide a better reconstruction than the Euclid-like survey at moderate levels of calibration error. We would expect to see similarly increased robustness to calibration errors for any tracer with a larger clustering signal, including tracers with a larger bias. Finally, just as for Euclid, we find that if calibration errors are not accounted for in the estimator, then adding LSS data can actually result in a worse reconstruction than that from using CMB temperature data alone. \|c \|c \|c c \|c c \| & Rℓ & & ${{\sigma^2\_{\rm cal}}}$ & 0 & 10− 6 & 10− 4 & 10− 6 & 10− 4 [0.5ex] TT & 0.46 & - & -& -& - Euclid (1 bin) & 0.92 & 0.83 & 0.19 & 0.84& 0.29 Euclid (6 bin) & 0.95 & 0.91 & 0.41& 0.92& 0.57 SPHEREx (6 bin) & 0.89 & 0.88 & 0.52 & 0.88& 0.62 Euc + Spx + TT & 0.96 & 0.92 & 0.47 & 0.93& 0.73 [.5ex] [table:rhotable] [fig:multsurveyplot] Effect of varying calibration error properties ---------------------------------------------- We now test how sensitive the results in the previous sections are to our assumptions about calibration errors, showing the results in Fig. [fig:calprops]. First, the left panel shows what happens when we vary the level of cross-correlation between the calibration errors of different LSS maps. It is conceivable that residual calibration errors can be correlated across different bins of a single survey, or even across different surveys, especially if the error has an astronomical origin. To model such correlation, we set the level of cross-correlation between the calibration errors of maps i and j using a parameter $r\_{\rm cc}$, where Ccal, ij = rcc, ij [eq:crosscorr] As we only consider cases where calibration errors in all maps are characterized by the same ${C\_\ell}^{\rm cal}$, this reduces to Ccal, ij = rccCcal,ij. [eq:crosscorr-simp] [fig:calprops] We consider the six-bin fiducial Euclid-like survey and find that this kind of correlated error results in mild degradation of the reconstruction for ${{\sigma^2\_{\rm cal}}}\lesssim10^{-4}$, but otherwise it has little effect as long as calibration errors are correctly modeled in the estimator [that is, ${{R\_\ell({C\_\ell}^{\rm sky})}}$ is used]. If calibration errors are not accounted for [${{R\_\ell({C\_\ell}^{\rm th})}}$ is used], reconstruction suffers considerably, as shown by the dotted curve. We also use a dashed curve [labeled ${R\_\ell}({C\_\ell}^{\rm XY,th})$] for the case where the estimator filter correctly accounts for the autopower contributions of calibration errors but neglects the cross-power contributions. As seen by comparing the solid, dashed, and dotted orange curves in Fig. [fig:calprops], reconstruction quality is far more sensitive to accurate modeling of the calibration error contribution to the autopower than to the cross-power. Thus, fitting the observed autopower for each map but using theoretical cross-powers, as is done in Ref. , should harm the reconstruction relatively little, depending on the fitting scheme; we find $\Delta \hat\rho \approx -0.03$ at ${{\sigma^2\_{\rm cal}}}= 10^{-4}$ for $r\_{\rm cc}=0.2$, far less than the typical variation over realizations shown in Fig. [fig:euconlyplot]. Additionally, we study the impact of changing the shape of the calibration error power spectrum ${C\_{\ell}^{\rm cal}}$, showing the results in the right panel of Fig. [fig:calprops]. We first vary the width parameter $w^{\rm cal}$ of the calibration error power spectrum ${C\_{\ell}^{\rm cal}}$ given in Eq. ([eq:clcalmodel]). Results for different values of $w^{\rm cal}$ are qualitatively similar, though for fixed ${{\sigma^2\_{\rm cal}}}$, the reconstruction is less sensitive to calibration errors when $w^{\rm cal}$ is larger. The reason for this is that ρ is most sensitive to contamination at the lowest multipoles, as will be discussed in Sec. [sec:SNcal]. Using a power law ${C\_{\ell}^{\rm cal}}\propto(\ell+1)^{-2}$ gives results similar to our fiducial Gaussian form with $w^{\rm cal}=10$. [fig:SN2l] Impact of calibration errors on S/N estimates ============================================= Given the extent to which calibration errors degrade the quality of ISW signal reconstruction, it makes sense to ask whether they also impact the signal-to-noise (S/N) of ISW detection. Detection of the ISW signal via the cross-correlation between the CMB and LSS maps has been the subject of considerable study, as it serves as an important consistency test for the presence of dark energy. The significance of detection varies considerably depending on the LSS tracer sample and the statistical methods used, as well as how systematics in the LSS data are treated . used systematics templates to correct the observed power spectra for SDSS galaxies, finding a S/N loss of ∼ 0.5 if such corrections are neglected. introduced a low-redshift spike in the source distributions in order to reproduce the observed excess autopower in NVSS and SDSS catalogs and estimate that such systematics result in an uncertainty of ΔS/N ± 0.4. The most recent results come from the Planck Collaboration, which found ∼ 4σ evidence for the ISW effect, with most of the signal coming from cross-correlation of the CMB temperature with the NVSS radio catalog and CMB lensing. The maximum achievable signal-to-noise can be obtained by considering an ideal survey that perfectly traces the ISW (i.e. ${C\_\ell}^{gg}={C\_\ell}^{Tg}={C\_\ell}^{\rm ISW}$), resulting in a maximum S/N ∼ 6 − 10 for ΛCDM cosmology. Our goal is to study how calibration errors impact the significance of ISW detection. There are multiple ways one can quantify detection of the ISW effect, including correlation detection between LSS and the CMB, template matching to an assumed model, or model comparison. Each of these methods relies on different assumptions and tests different statistical questions (see Ref. for a detailed review). Here we adopt the simple correlation detection statistic which quantifies the expected deviation from a null hypothesis of no correlation between LSS (g) and CMB temperature (T). In this formalism the S/N for ISW detection is ()2 (Tg) (Ccov)-1 Tg, [eq:SNdef] where we have assumed the multipoles contribute independently to the S/N. Here $\mathbf{{C\_\ell}}^{Tg}$ is a vector of the ISW-LSS cross-spectra, and the covariance matrix elements corresponding to LSS maps i and j can be written as $$\begin{aligned} {C\_\ell}^{{\rm cov},ij} &= \left<\Delta {C\_\ell}^{Ti} \Delta {C\_\ell}^{Tj}\right> \\ &\simeq \frac{{C\_\ell}^{Ti}{C\_\ell}^{Tj} + {C\_\ell}^{TT} ({C\_\ell}^{ij} + {C\_\ell}^{{\rm cal},ij} + \delta\_{ij} \frac{1}{\bar{n}\_{ij}})} {f\_{sky}(2\ell+1)}, \label{eq:covar}\end{aligned}$$ where the last term in the numerator is due to shot noise and δij is the Kronecker delta.[4](#fn4) Equations ([eq:SNdef]) and ([eq:covar]) demonstrate that all cosmological tests using LSS-CMB cross-correlation are limited in their constraining power due to sample variance and the relatively large amplitude of the primordial CMB fluctuations. They also make it clear that calibration errors will reduce the significance of ISW detection. We assume calibration errors to be uncorrelated between maps, so ${C\_\ell}^{{\rm cal},ij}~\rightarrow~\delta\_{ij}{C\_\ell}^{{\rm cal},ij}$. For a single LSS map, Eq. ([eq:SNdef]) reduces to the form ()2 fsky (CTg)2 + CTT (Cgg + Ccal + 1/\|ng). [eq:SNonemap] If there are no calibration errors, we find S/N = 6.6 for our Euclid-like survey, which is near the maximum[5](#fn5) for this cosmology, S/N = 6.7. As ${{\sigma^2\_{\rm cal}}}$ increases from 0 to current levels, the total S/N reduces to 4.9 − 5.7, a drop of only ∼ 15% − 30%. This can be seen in the S/N values listed for various ${{\sigma^2\_{\rm cal}}}$ in the legend of Fig. [fig:SN2l]. In contrast, for the same level of error, average reconstruction quality $\hat\rho$ is reduced by 40% − 60%. Clearly, ISW signal reconstruction is substantially more affected by calibration errors than is ISW detection significance. The greater robustness of the total S/N to calibration errors is due to the fact that it has support at higher multipoles. This is most easily illustrated in the single-map case, where the contribution per multipole to the total signal-to-noise is ()2= (2+1). [eq:SNell] Figure [fig:SN2l] shows how the contribution per multipole responds to different levels of calibration error. As ${{\sigma^2\_{\rm cal}}}$ increases, the signal-to-noise decreases at lower multipoles, but contributions at higher multipoles remain unchanged. These higher-multipole contributions are thus still available to contribute to the overall S/N. Map reconstruction is more sensitive to the largest scales. For the single-map case, this can be illustrated analytically as follows. Using the single-map estimator from Eq. ([eq:wiener]), we can write the estimated reconstruction quality statistic as $$\begin{aligned} \hat{\rho} &= \frac{\frac{1}{4\pi} \sum\_{\ell}(2\ell+1)\left(\frac{{C\_\ell}^{Tg}}{{C\_\ell}^{gg}}\right){C\_\ell}^{Tg}} {\sigma\_{\rm ISW} \sqrt{\frac{1}{4\pi}\sum\_{\ell}(2\ell+1)\left(\frac{{C\_\ell}^{Tg}}{{C\_\ell}^{gg}}\right)^2{C\_\ell}^{gg}}} \nonumber\\ &= \frac{1} {\sigma\_{\rm ISW}} \sqrt{\frac{1}{4\pi}\sum\_{\ell}(2\ell+1)\frac{({C\_\ell}^{Tg})^2}{{C\_\ell}^{gg}}} \nonumber\\ &= \frac{1} {\sigma\_{\rm ISW}}\sqrt{\frac{1}{4\pi}\sum\_{\ell} \left(\frac{S}{N}\right)^2\_\ell {C\_\ell}^{TT}\left(1+\frac{({C\_\ell}^{Tg})^2}{{C\_\ell}^{gg}{C\_\ell}^{TT}}\right)} \nonumber\\ &\approx \frac{1} {\sigma\_{\rm ISW}}\sqrt{\frac{1}{4\pi}\sum\_{\ell}\left(\frac{S}{N}\right)^2\_\ell {C\_\ell}^{TT}}. \label{eq:rho\_from\_SN}\end{aligned}$$ Here, (S/N)ℓ2 is the quantity given by Eq. ([eq:SNell]) which, when summed over ℓ, gives (S/N)2. Thus, we see from Eq. ([eq:rhofromSN]) that $\hat\rho$ is proportional to a total (S/N) of which the terms are weighted by CℓTT. Since CℓTT drops sharply as ∼ ℓ− 2, the quality of map reconstruction $\hat\rho$ is more impacted by large-angle calibration errors than the overall S/N is. This is also a primary cause for the degradation in reconstruction quality seen when ${{\sigma^2\_{\rm cal}}}$ was concentrated at lower multipoles in Sec. [sec:caldetails]. Conclusions =========== Reconstruction of the integrated Sachs-Wolfe signal would allow, for the first time, a clean separation of the CMB temperature anisotropies into contributions from 300,000 years after the big bang and those from some ∼ 10 billion years later. This, in turn, would allow for a more informed assessment of the origin of the “large-angle CMB anomalies” and a more complete elimination of ISW contaminants to CMB-based measurements of primordial non-Gaussianity. Accurate ISW reconstruction requires wide-angle large-scale structure maps from which the gravitational potential evolution can be inferred, but in practice, these maps are plagued by photometric calibration errors due to a host of atmospheric, instrumental, and selection-induced systematics. In our previous work, we found that the realistic levels of residual calibration error severely degrade the accuracy of the reconstructed ISW map. In this paper, we investigated how the effects of residual photometric calibration errors on the ISW map reconstruction can be mitigated by using tomographic information and by combining data from multiple surveys. To quantify the amount of residual calibration errors, we use their variance ${{\sigma^2\_{\rm cal}}}$, the square root of which is roughly equal to the rms magnitude fluctuations induced by these systematics. We find that for a Euclid-like survey with a single redshift bin, to achieve a reconstruction comparable in quality to that derived from the CMB temperature map alone (with an average correlation between the true and reconstructed ISW maps of only ρ̄ ≈ 0.46), one must limit the variance of the calibration error field to ${{\sigma^2\_{\rm cal}}}\lesssim 10^{-5}$. This can be improved significantly if we exploit the tomographic information available by binning the LSS data in redshift (Fig. [fig:euconlyplot]). We also show that if the model spectra in the estimator differ substantially from those used to generate the input maps e.g. by using theoretical power spectra that do not account for the observed excess autopower in the LSS survey from calibration errors, then reconstruction quality is substantially degraded. It is therefore crucial to verify that the theoretical spectra in the estimator are a good fit to those observed or to otherwise use smoothed fits. We next consider how using multiple input maps, probing different tracers and redshift ranges, improves ISW signal reconstruction. We find that as long as the excess power contributed by calibration errors is adequately modeled in the estimator, the resulting reconstruction is always better than that from either of the input maps individually. If the excess power from calibration errors is not modeled, however, adding a map can actually degrade reconstruction (Fig. [fig:eucpluscmb]). The CMB temperature map adds information to the reconstruction at all levels of calibration error, but is especially valuable if the LSS maps are subject to calibration errors with ${{\sigma^2\_{\rm cal}}}\gtrsim 10^{-5}$. Using a six-bin SPHEREx-like survey provides qualitatively similar results to the Euclid-like one, but because it is shallower, the reconstruction is less accurate in the limit of no calibration errors ($\rho\_{\rm Spx} =0.89\pm0.04$ vs. $\rho\_{\rm Euc}=0.95\pm0.02$, where errors indicate the 68% spread across realizations). However SPHEREx’s shallower depth and therefore intrinsically higher clustering signal becomes an asset in the presence of calibration errors, making the survey more robust against calibration errors and leading to a better reconstruction for ${{\sigma^2\_{\rm cal}}}\gtrsim 6\times10^{-6}$. (Similarly, we would expect the increased clustering of tracers with larger bias to help mitigate the effects of calibration error as well.) Therefore, a combination of a shallower and a deeper LSS survey provides complementarity useful for separation of calibration errors from the ISW signal and necessary for a good ISW map reconstruction in the presence of such errors. Using all three simulated surveys as input—Euclid, SPHEREx, and CMB temperature—significantly improves reconstruction compared to using a single survey with current levels of residual calibration errors, or CMB temperature data alone. We find that if the calibration errors can be controlled to the level of ${{\sigma^2\_{\rm cal}}}\lesssim 10^{-6}$ (${{\sigma^2\_{\rm cal}}}\lesssim 10^{-5}$), then the combination of Euclid, SPHEREx, and CMB temperature maps can produce the ISW map reconstruction to an excellent accuracy of ρ = 0.93 ± 0.03 (ρ = 0.87 ± 0.05). This is roughly the same level of calibration control required for future LSS surveys to avoid biasing measurements of cosmological parameters like the non-Gaussianity parameter fNL and the dark energy equation of state. Thus, high-quality ISW reconstructions will, in a sense, “come for free” with the developments needed for cosmology constraints with next-generation surveys. We additionally test the robustness of our results against changes to the properties of the calibration error field, looking at cross-correlations between calibration errors in different maps as well as the shape of their spectrum. We found that cross-correlation between the calibration errors of different tracer maps degrades the reconstruction most for $10^{-6} \lesssim {{\sigma^2\_{\rm cal}}}\lesssim 10^{-4}$, but that this effect is relatively minor, provided the auto-correlation contribution is accounted for in the estimator (Fig. [fig:calprops], left). We also find that, compared to map reconstruction accuracy, the overall significance of ISW detection is less strongly affected by calibration errors. This is because its signal is distributed more broadly in multipole space than that of the map reconstruction quality statistic. To clarify this, we analytically relate ρ̂ to the commonly used ISW detection S/N statistic in the case of a single input LSS map and show that ρ̂ is weighted by an additional factor of CℓTT, demonstrating map reconstruction’s greater sensitivity to the largest scales [Eq. ([eq:rhofromSN])]. As an extension to this work, one could study how the inclusion of CMB lensing and polarization data can improve ISW map reconstruction, provided the systematics present in those data sets could be sufficiently accurately modeled. The results of indicate that the use of lensing as input can contribute significantly to reconstruction quality, but they also show that current noise levels limit its effectiveness. Notably, the residual lensing systematics at ℓ ≤ 8 present a challenge, as this is where the ISW signal is strongest, and we expect these systematics to affect reconstruction with CMB lensing and polarization in a manner broadly similar to unaccounted for calibration errors in LSS maps at those scales. Further work could also be performed to concretely explore how to best approximate the ‘best case’ reconstruction scenario, wherein calibration errors are fully accounted for, by using real input data. Here we have only characterized the limiting cases where the residual calibration error contribution to the LSS power is fully known or fully unknown, and have not addressed intermediate cases where they are partially accounted for. Finally, we have only worked in the full-sky case whereas real data will necessarily have only partial sky coverage. Others have already shown that incomplete sky coverage only very minorly degrades reconstruction quality for areas covered by the input data sets, and we do not expect this to change in the presence of calibration errors. Nevertheless, a full analysis that attempts to make predictions for real surveys should take the actual sky coverage and survey-specific systematics into account. Even with these considerations, it is clear that accounting for the presence of residual calibration errors in LSS surveys is a critical step for any reconstruction of the ISW map, as their presence and treatment impact both the survey characteristics and set of input maps that produce the optimal reconstruction. The authors have been supported by DOE under Contract No. DE-FG02-95ER40899. D. H. has also been supported by NASA under Contract No. 14-ATP14-0005. J. M. has been supported by the Rackham Graduate School through a Predoctoral Fellowship. Equivalence with Other Estimators ================================= We now demonstrate the equivalence between our estimator for the ISW map coefficients $\hat{a}\_{\ell m}^{\rm ISW}$, and the estimators proposed by and. Our estimator in Eq. ([eq:iswestsimple]) is based on a version of the likelihood from that has been reformulated to handle observed CMB maps like any other input map. derive their estimator using the likelihood $$\begin{aligned} \label{eq:mdlike} \mathcal{L}(T^{\rm ISW})&\propto \frac{1}{\sqrt{{\rm det}(CD)}} \times \exp\left\{-\frac{1}{2}{\boldmath{d}}^TD^{-1}{\boldmath{d}}\right\} \\ &\times \exp\left\{-\frac{1}{2}\left(T^{\rm obs} - T^{\rm ISW}\right)C^{-1}\left(T^{\rm obs} - T^{\rm ISW}\right)\right\}, \nonumber\end{aligned}$$ where C ≡ Cp + Cn is the angular power spectrum of the primordial Cp and noise Cn contributions to CMB temperature fluctuations, ${\boldmath{d}}$ is a vector of ISW and LSS tracer maps, and D is the covariance matrix between the ISW and LSS tracers (see Eqs. (4-6)), with ISW maps associated with the first (1) index. This likelihood is a product of the independent likelihoods for $(T^{\rm obs}-T^{\rm ISW})$ and for the input maps in d. Instead of explicitly including independent terms for the primordial CMB and LSS tracers (which are assumed to have no cross-correlation), we include the total observed CMB temperature, Tobs = Tp + TISW, where Tp includes both the primordial CMB temperature as well as any instrumental noise terms. We then expand the data vector to include $T^{\rm obs}$: m = (amISW,gm1,…,gmn) (amISW,gm1,…,gmn, amobs), with aℓm indicating spherical components of ISW and CMB temperature fluctuations and gℓm indicating components of LSS overdensity. The covariance matrix is similarly expanded to account for the cross-correlation of Tobs with the ISW and LSS tracers $$\begin{aligned} D\_{\ell} &\rightarrow \left(\begin{array}{ccccc} C\_{\ell}^{\rm{ISW,ISW}}&C\_{\ell}^{\rm{ISW, }1} &\cdots &C\_{\ell}^{\rm{ISW, }\textit{n}}&C\_{\ell}^{\rm{ISW,obs}}\\ C\_{\ell}^{1\rm{,ISW}}&C\_{\ell}^{1,1} &\cdots &C\_{\ell}^{1,n}&C\_{\ell}^{1\rm{,obs}}\\ \vdots&\vdots &\ddots &\vdots\\ C\_{\ell}^{n\rm{,ISW}}&C\_{\ell}^{n,1}&\cdots &C\_{\ell}^{n,n}&C\_{\ell}^{n\rm{,obs}}\\ C\_{\ell}^{\rm{obs, ISW}}&C\_{\ell}^{1,\rm{obs}}&\cdots &C\_{\ell}^{n,\rm{obs}}&C\_{\ell}^{\rm{obs,obs}}\\ \end{array}\right).\end{aligned}$$ Assuming that at the scales we consider the observed CMB is cross-correlated with other LSS tracers only through the ISW, we have $$\begin{aligned} C\_{\ell}^{\rm{obs, ISW}} &= C\_{\ell}^{\rm{ISW, ISW}} \nonumber\\ C\_{\ell}^{\rm{obs, LSS}\_i} &= C\_{\ell}^{\rm{ISW, LSS}\_i}, \\ C\_{\ell}^{\rm{obs,obs}} &= C\_{\ell}^{p,p} + C\_{\ell}^{\rm{ISW, ISW}},\nonumber\end{aligned}$$ assuming there is no residual cross-correlation between the primordial and late-time CMB. Maximizing the resulting likelihood (TISW) {-TD-1}, [eq:uslike] gives the optimal estimator given in Sec. [sec:iswest]. To show that this is equivalent to the estimator derived from Eq. ([eq:mdlike]), we focus on the case of using CMB temperature and a single LSS tracer as input maps. For compactness, and to make the connections with other ISW estimators in the literature more apparent, we adopt notation from , where s, g, and T indicate the ISW, LSS tracer, and observed CMB temperature signals, respectively. We then have m = (ams,gmg, amT), and $$\begin{aligned} D\_{\ell} &= \left(\begin{array}{ccc} {{C\_\ell}^{{s}{s}}}&{{C\_\ell}^{{s}{g}}}&{{C\_\ell}^{{s}{s}}}\\ {{C\_\ell}^{{s}{g}}}&{{C\_\ell}^{{g}{g}}}&{{C\_\ell}^{{s}{g}}}\\ {{C\_\ell}^{{s}{s}}}&{{C\_\ell}^{{s}{g}}}&{{C\_\ell}^{{T}{T}}}\\ \end{array}\right).\end{aligned}$$ From Eqs. ([eq:iswestsimple]) and ([eq:Rdef]) our estimator gives $$\begin{aligned} \hat{a}\_{\ell m}^{{s}} &= \frac{-1}{[{D\_\ell}^{-1}]\_{11}}\left([{D\_\ell}^{-1}]\_{12}~{g\_{\ell m}}^{g}+ [{D\_\ell}^{-1}]\_{13}~{a\_{\ell m}}^{T}\right)\nonumber\\ &= \left(\frac{{{C\_\ell}^{{s}{g}}}({{C\_\ell}^{{T}{T}}}-{{C\_\ell}^{{s}{s}}})}{{{C\_\ell}^{{g}{g}}}{{C\_\ell}^{{T}{T}}}- ({{C\_\ell}^{{s}{g}}})^2}\right){g\_{\ell m}}^{g}\\ &+ \left(\frac{{{C\_\ell}^{{s}{s}}}{{C\_\ell}^{{g}{g}}}-({{C\_\ell}^{{s}{g}}})^2}{{{C\_\ell}^{{g}{g}}}{{C\_\ell}^{{T}{T}}}- ({{C\_\ell}^{{s}{g}}})^2}\right){a\_{\ell m}}^{T}. \nonumber \label{eq:2binestus}\end{aligned}$$ We now calculate the estimator of . Denoting their covariance matrix without the CMB as Dℓʹ, we use their Eq. (9) $$\begin{aligned} \hat{a}\_{\ell m}^{{s}, \mathrm{MD}} &= \left([C\_{\ell}^{pp}]^{-1} + [D'^{-1}]\_{11}\right)^{-1}\nonumber\\ &\times \left(-[{D\_\ell}'^{-1}]\_{12}~{g\_{\ell m}}^{g}+ [C\_{\ell}^{pp}]^{-1}{a\_{\ell m}}^{T}\right)\\ &= \left(\frac{1}{{{C\_\ell}^{pp}}} + \frac{{{C\_\ell}^{{g}{g}}}}{\mathrm{det}\|{D\_\ell}'\|}\right)^{-1} \left[\frac{{{C\_\ell}^{{s}{g}}}}{\mathrm{det}\|{D\_\ell}'\|}{g\_{\ell m}}^{g}+ \frac{1}{{{C\_\ell}^{pp}}}{a\_{\ell m}}^{T}\right] \nonumber \\ &= \left(\frac{1}{\mathrm{det}\|{D\_\ell}'\| + {{C\_\ell}^{{g}{g}}}{{C\_\ell}^{pp}}}\right) \left[{{C\_\ell}^{pp}}{{C\_\ell}^{{s}{g}}}{g\_{\ell m}}^{g}+ (\mathrm{det}\|{D\_\ell}'\|){a\_{\ell m}}^{T}\right] \nonumber\end{aligned}$$ Expanding the determinant, we find $$\begin{aligned} \hat{a}\_{\ell m}^{{s}, \mathrm{MD}} &= \left(\frac{{{C\_\ell}^{{s}{g}}}{{C\_\ell}^{pp}}}{{{{C\_\ell}^{{g}{g}}}({{C\_\ell}^{{s}{s}}}+{{C\_\ell}^{pp}}) - ({{C\_\ell}^{{s}{g}}})^2}}\right){g\_{\ell m}}^{g}\\ &+ \left(\frac{{{C\_\ell}^{{s}{s}}}{{C\_\ell}^{{g}{g}}}-({{C\_\ell}^{{s}{g}}})^2}{{{C\_\ell}^{{g}{g}}}({{C\_\ell}^{{s}{s}}}+{{C\_\ell}^{pp}}) - ({{C\_\ell}^{{s}{g}}})^2}\right) {a\_{\ell m}}^{T}, \end{aligned}$$ which, using the relation CℓTT = Cℓpp + Cℓss, is equivalent to the estimator given by Eq. ([eq:2binestus]). This is also equivalent to the estimator proposed in , which uses the Cholesky decomposition (L) of the covariance matrix Dℓʺ (denoted Cℓ in Ref. ). $$\begin{aligned} {D\_\ell}'' &= \begin{bmatrix} {{C\_\ell}^{{g}{g}}}& {{C\_\ell}^{{s}{g}}}\\ {{C\_\ell}^{{s}{g}}}& {{C\_\ell}^{{s}{s}}}\\ \end{bmatrix} = L\_\ell L\_\ell^T.\end{aligned}$$ Note that here the ISW index is last instead of first, in contrast to the covariance matrices Dℓ and Dℓʹ used previously. The Cholesky decomposition is written as L= & 0 & . [eq:cholesky] Eqs. (8) and (9) from Ref. give the ISW estimate (their ŝℓm) as $$\begin{aligned} \hat{a}\_{\ell m}^{{s}, \mathrm{B08}} &= \frac{L\_{12}}{L\_{11}}\left(1-\frac{L\_{22}^2}{L\_{22}^2 + {{C\_\ell}^{pp}}}\right) {g\_{\ell m}}^{g}+ \frac{L\_{22}^2}{L\_{22}^2 + {{C\_\ell}^{pp}}} {a\_{\ell m}}^{T}\\\end{aligned}$$ where we have suppressed the ℓ-dependence of L and combined their observed ISW signal (sℓm) and noise (nℓm) terms into the single term aℓmT. We use Cℓpp to denote the combined power of noise and the primordial CMB, in keeping with the notation above. Plugging this into Eq. ([eq:cholesky]), we obtain $$\begin{aligned} \hat{a}\_{\ell m}^{{s}, \mathrm{B08}} &= \frac{{{C\_\ell}^{{s}{g}}}}{{{C\_\ell}^{{g}{g}}}}\left(\frac{{{C\_\ell}^{pp}}}{({{C\_\ell}^{{s}{s}}}+ {{C\_\ell}^{pp}})-({{C\_\ell}^{{s}{g}}})^2/{{C\_\ell}^{{g}{g}}}}\right) {g\_{\ell m}}^{g}\nonumber \\ &+ \frac{{{C\_\ell}^{{s}{s}}}-({{C\_\ell}^{{s}{g}}})^2/{{C\_\ell}^{{g}{g}}}}{({{C\_\ell}^{{s}{s}}}+{{C\_\ell}^{pp}}) - ({{C\_\ell}^{{s}{g}}})^2/{{C\_\ell}^{{g}{g}}}} {a\_{\ell m}}^{T}\nonumber\\[0.2cm] &= \left(\frac{{{C\_\ell}^{{s}{g}}}({{C\_\ell}^{{T}{T}}}-{{C\_\ell}^{{s}{s}}})}{{{C\_\ell}^{{g}{g}}}{{C\_\ell}^{{T}{T}}}- ({{C\_\ell}^{{s}{g}}})^2}\right){g\_{\ell m}}^{g}\nonumber \\ &+ \left(\frac{{{C\_\ell}^{{s}{s}}}{{C\_\ell}^{{g}{g}}}-({{C\_\ell}^{{s}{g}}})^2}{{{C\_\ell}^{{g}{g}}}{{C\_\ell}^{{T}{T}}}- ({{C\_\ell}^{{s}{g}}})^2}\right){a\_{\ell m}}^{T},\end{aligned}$$ which is the same as Eq. ([eq:2binestus]). Estimating ρ with ${R\_\ell}(\tilde{C\_\ell})$ ================================================ Here we show why using raw pseudo-Cℓ’s ($\tilde{C\_\ell}$) in the estimator results in a degraded reconstruction, for which ρ̄ is not well approximated by ρ̂ (Eq. ([eq:rhoest])). For a given realization, ρ is constructed from the covariance between the true and reconstructed ISW maps (Cov$(T^{\rm ISW},T^{\rm rec})$, i.e. the numerator in Eq. ([eq:rhoraw])) normalized by the square root of the individual variances of the true and reconstructed ISW maps ($\sigma\_{\rm True}^2$ and $\sigma\_{\rm rec}^2$, respectively). We therefore focus on how using realization-specific $\tilde{C\_\ell}$’s in the estimator filter Rℓ affects the individual Cℓ contributions to $\sigma\_{\rm rec}^2$ and Cov$(T^{\rm ISW},T^{\rm rec})$ ($\sigma\_{\rm ISW}^2$ is unaffected by our choice of Rℓ). For simplicity, we work with a single input map. If the ISW estimator filter Rℓ is constructed from analytically computed model Cℓ’s, the power spectrum of the ISW map for a given realization will be $$\begin{aligned} \tilde{C\_\ell}^{\rm rec-rec, th} &= {R\_\ell}^2({C\_{\ell}^{\rm th}})\tilde{C\_\ell}^{\rm gal-gal},\\ &= \left(\frac{{C\_\ell}^{\rm ISW-gal}}{{C\_\ell}^{\rm gal-gal}}\right)^2\,\tilde{C\_\ell}^{\rm gal-gal}. \end{aligned}$$ We add the superscript “th” to distinguish this reconstructed ISW power spectrum from the one where the filter Rℓ is built from $\tilde{C\_\ell}$’s, which will be discussed shortly. The expectation value for this over many realizations is $$\begin{aligned} \left\langle \tilde{C\_\ell}^{\rm rec-rec, th}\right\rangle &= \left(\frac{{C\_\ell}^{\rm ISW-gal}}{{C\_\ell}^{\rm gal-gal}}\right)^2\,\left\langle\tilde{C\_\ell}^{\rm gal-gal}\right\rangle,\\ &=\frac{\left({C\_\ell}^{\rm ISW-gal}\right)^2}{{C\_\ell}^{\rm gal-gal}}. \end{aligned}$$ Now let us look at the behavior of the reconstructed ISW power when the galaxy autopower spectra in the estimator filter are extracted from the observed maps. Denoting this version of the filter by R, we write $$\begin{aligned} {\tilde {C\_\ell}^{\rm rec-rec}}&= \tilde{R\_\ell}^2\tilde{C\_\ell}^{\rm gal-gal},\\ &=\left({C\_\ell}^{\rm ISW-gal}\right)^2\left(\frac{1}{\tilde{C\_\ell}^{\rm gal-gal}}\right). \label{eq:Rlsingle} \end{aligned}$$ Because the measured $\tilde{C\_\ell}^{\rm gal-gal}$ appears in the denominator of this expression, taking its expectation value over many realizations is somewhat more complicated. To do so we use the fact that $(2\ell+1)\tilde{C\_\ell}^{\rm gal-gal}$ is χ2-distributed with 2ℓ + 1 degrees of freedom. This means ${\tilde {C\_\ell}^{\rm rec-rec}}/(2\ell+1)$ follows an inverse-χ2 distribution, with an expectation value[6](#fn6) < > =. Therefore the average reconstructed power is $$\begin{aligned} \left<\tilde{C\_\ell}^{\rm rec-rec}\right> &= \frac{2\ell+1}{2\ell-1}\left<\tilde{C\_\ell}^{\rm rec-rec, th}\right> \label{eq:recrecanal} \\ &= \left<\tilde{C\_\ell}^{\rm rec-rec, th}\right>\left(1 + \frac{2}{2\ell-1}\right). \end{aligned}$$ Because $\langle{\tilde {C\_\ell}^{\rm rec-rec}}\rangle$ is strictly positive, this increased power results in an increase in the total variance of the reconstruction map $\tilde \sigma\_{\rm rec}^2$ compared to that from the theory-only filter reconstruction $\sigma\_{\rm rec, th}^2$ $$\begin{aligned} \langle\tilde \sigma\_{\rm rec}^2\rangle &=\frac{1}{4\pi} \sum\_{\ell}\,(2\ell+1)\,\left<{\tilde {C\_\ell}^{\rm rec-rec}}\right> \\ &= \sigma\_{\rm rec, th}^2 + \frac{1}{4\pi} \sum\_{\ell}\,(2\ell+1)\,\frac{\left<\tilde{C\_\ell}^{\rm rec-rec, th}\right>}{\ell-1/2}, \end{aligned}$$ In contrast, we find the average cross-power $\langle\tilde{C\_\ell}^{\rm ISW-rec}\rangle$ between reconstructed and true ISW maps remains unchanged. The increased power of the reconstruction thus results in a net decrease in $\left<\rho\right>$, per Eq. ([eq:rhoraw]), and hence is not well approximated by simply substituting the theory Cℓ, as is done to compute $\hat \rho$. Additionally, this suggests that a simple scaling of Rℓ in order to ``debias" the reconstruction will not improve ρ. To understand why the cross-power does not increase, we again use the observed galaxy autopower in the estimator and approximate its expectation value. The cross-power is given by $$\begin{aligned} \tilde{C\_\ell}^{\rm ISW-rec} &= \tilde{R\_\ell}\tilde{C\_\ell}^{\rm ISW-gal}\\ &= \left(\frac{{C\_\ell}^{\rm ISW-gal}}{\tilde{C\_\ell}^{\rm gal-gal}}\right) \tilde{C\_\ell}^{\rm ISW-gal}. \label{eq:singleClrecisw} \end{aligned}$$ Here we have a quotient of two non-independent χ2 random variables. Generically, we can approximate the average of a function of two random variables X and Y through a second-order Taylor expansion about the mean of each (μX, μY): $$\begin{aligned} \left< f(X,Y)\right> &\approx f(\mu\_X,\mu\_Y) + \frac{1}{2}f''\_{XX}(\mu\_X,\mu\_Y)\left<(X-\mu\_X)^2\right> \\ &+ f''\_{XY}(\mu\_X,\mu\_Y)\left<(X-\mu\_X)(Y-\mu\_Y)\right> \\ &+ \frac{1}{2}f''\_{YY}(\mu\_X,\mu\_Y)\left<(Y-\mu\_Y)^2\right>,\end{aligned}$$ where a prime indicates a derivative with respect to the respective subscripted variable. By taking f(X, Y) to be $\tilde{C\_\ell}^{\rm ISW-rec}$, of the form X/Y, then from Eq. ([eq:singleClrecisw]) we can approximate the mean cross-power to be $$\begin{aligned} & \left<\tilde{C\_\ell}^{\rm ISW-rec}\right> \approx {C\_\ell}^{\rm ISW-gal}\frac{\left<\tilde{C\_\ell}^{\rm ISW-gal}\right>}{\left<\tilde{C\_\ell}^{\rm gal-gal}\right>}\\ &\times \left(1 - \frac{\text{Cov}(\tilde{C\_\ell}^{\rm ISW-gal}, \tilde{C\_\ell}^{\rm gal-gal})}{\left<\tilde{C\_\ell}^{\rm ISW-gal}\right>\left<\tilde{C\_\ell}^{\rm gal-gal}\right>} + \frac{\text{Var}(\tilde{C\_\ell}^{\rm gal-gal})}{\left<\tilde{C\_\ell}^{\rm gal-gal}\right>^2}\right)\nonumber\\[0.2cm] &= {C\_\ell}^{\rm ISW-gal}\frac{\left<\tilde{C\_\ell}^{\rm ISW-gal}\right>}{\left<\tilde{C\_\ell}^{\rm gal-gal}\right>}\left(1 - \frac{2}{2\ell+1} + \frac{2}{2\ell+1}\right),\end{aligned}$$ where we used $$\begin{aligned} \mathrm{Var}\left(\tilde{C\_\ell}^{\rm gal-gal}\right) &= \frac{2}{2\ell+1}{C\_\ell}^{\rm gal-gal}, \\ \mathrm{Cov}\left(\tilde{C\_\ell}^{\rm gal-gal},\tilde{C\_\ell}^{\rm ISW-gal}\right) &= \frac{2}{2\ell+1}{C\_\ell}^{\rm gal-gal}{C\_\ell}^{\rm ISW-gal}.\end{aligned}$$ The corrective terms vanish and we find $$\begin{aligned} \left<\tilde{C\_\ell}^{\rm ISW-rec}\right> &\approx {C\_\ell}^{\rm ISW-gal}\frac{\left<\tilde{C\_\ell}^{\rm ISW-gal}\right>}{\left<\tilde{C\_\ell}^{\rm gal-gal}\right>} \\ &=\left<\tilde{C\_\ell}^{\mathrm{ISW-rec, th}}\right>. \end{aligned}$$ Then on average, the cross-power between the true and reconstructed ISW maps is unchanged from the theory case. Since the multipoles are independent, this means the total covariance between the true and reconstructed ISW maps is unchanged as well: $$\begin{aligned} \left<\tilde{\mathrm{Cov}}\left(T^{\rm ISW},T^{\rm rec}\right)\right> &=\frac{1}{4\pi} \sum\_{\ell}\,(2\ell+1)\,\left<\tilde{C\_\ell}^{\mathrm{ISW-rec}}\right> \\ &=\frac{1}{4\pi} \sum\_{\ell}\,(2\ell+1)\,\left<\tilde{C\_\ell}^{\mathrm{ISW-rec, th}}\right> \end{aligned}$$ While for the autopower $\tilde{C\_\ell}^{\rm rec-rec}$ we were able to derive an analytical result, a similar Taylor expansion treatment to the same order as the cross-power results in an additive correction of 2/(2ℓ + 1), or <Crec-rec>()<C>, which is a good approximation to the analytical result found in Eq. ([eq:recrecanal]). --- 1. Using the observed spectra also violates an assumption in the maximum likelihood derivation of the estimator, in which the covariance is assumed to be known (i.e. independent of the measured signal).[↩](#fnref1) 2. We also considered s, which measures the rms error between true and reconstructed ISW maps as a complementary quality statistic, but found that for the cases studied here, the information it provided was largely redundant to that given by ρ.[↩](#fnref2) 3. In reality, the power contribution from calibration errors will also vary somewhat across bins, depending on the redshift dependence of the faint-end slope of the luminosity function for the tracer population. We have assumed here for simplicity that the power contribution is independent of redshift.[↩](#fnref3) 4. Strictly speaking, this will result in a slight underestimate of the significance, as technically the null hypothesis covariance, with CℓTi = CℓTj = 0 in Eq. ([eq:covar]), should be used. However, as the galaxy-ISW cross-power terms are small compared to the galaxy autopower, we follow the practice in most of the literature of keeping them in the S/N calculation.[↩](#fnref4) 5. This limit can in principle be increased, e.g., through the method of in which the observed LSS map is used to reduce the local variance and which in our case brings the maximum possible S/N to 7.2, or through the inclusion of polarization data as in the work by.[↩](#fnref5) 6. We refer the reader to Refs. and for discussions of the bias introduced when inverting an estimator, with implications specifically for estimating the inverse covariance matrix.[↩](#fnref6) Integrated Sachs-Wolfe map reconstruction in the presence of systematic errors ============================================================================== The decay of gravitational potentials in the presence of dark energy leads to an additional, late-time contribution to anisotropies in the cosmic microwave background (CMB) at large angular scales. The imprint of this so-called integrated Sachs-Wolfe (ISW) effect to the CMB angular power spectrum has been detected and studied in detail, but reconstructing its spatial contributions to the CMB map, which would offer the tantalizing possibility of separating the early- from the late-time contributions to CMB temperature fluctuations, is more challenging. Here we study the technique for reconstructing the ISW map based on information from galaxy surveys and focus in particular on how its accuracy is impacted by the presence of photometric calibration errors in input galaxy maps, which were previously found to be a dominant contaminant for ISW signal estimation. We find that both including tomographic information from a single survey and using data from multiple, complementary galaxy surveys improve the reconstruction by mitigating the impact of spurious power contributions from calibration errors. A high-fidelity reconstruction further requires one to account for the contribution of calibration errors to the observed galaxy power spectrum in the model used to construct the ISW estimator. We find that if the photometric calibration errors in galaxy surveys can be independently controlled at the level required to obtain unbiased dark energy constraints, then it is possible to reconstruct ISW maps with excellent accuracy using a combination of maps from two galaxy surveys with properties similar to Euclid and SPHEREx. Introduction ============ Cosmic microwave background (CMB) photons undergo a frequency shift as they travel to us from the last scattering surface. On top of the redshift due to the expansion of the Universe, an additional contribution to the temperature anisotropy is introduced whenever the universe is not matter dominated—for example, right after recombination when radiation contributes non-negligibly, or at late times when dark energy becomes important. This so-called integrated Sachs-Wolfe (ISW) effect is given by $$\label{ISWorigexpr} \left.\frac{\Delta T}{\bar{T}}\right\|\_{ISW}({{\bf \hat n}}) = \frac{2}{c^2}\int\_{t\_\}^{t\_0}dt\,\frac{\partial \Phi(\mathbf{r},t)}{\partial t},$$ where t0 is the present time, t⋆ is that of recombination, c* is the speed of light, r is the position in comoving coordinates, and Φ is the gravitational potential. The late-time ISW signal (referred to hereafter simply as ‘ISW’) has been statistically detected via measurements of the cross-correlation of CMB temperature maps with galaxy maps and, more recently, with maps of CMB lensing convergence. These detections serve as an important consistency test of the standard model of cosmology, and can help constrain the properties of dark energy. The ISW can provide additional information beyond its power spectrum if its map can be reconstructed with sufficient signal-to-noise. Since the total large-angle CMB temperature anisotropy is the sum of early- (hereafter ‘primordial’) and late-time contributions, $$\left.\frac{\Delta T}{\bar{T}}\right\|({{\bf \hat n}}) = \left.\frac{\Delta T}{\bar{T}}\right\|\_\mathrm{prim}({{\bf \hat n}}) + \left.\frac{\Delta T}{\bar{T}}\right\|\_\mathrm{ISW}({{\bf \hat n}}),$$ reconstructing the ISW map would allow us to isolate the primordial-only anisotropy. This separation of the CMB into early- and late-time contributions can also be useful for a variety of cosmological tests. For example, one could study the temporal origin of the large-angle CMB anomalies reported in, e.g., Ref . One could also subtract the realization-specific contaminating ISW contribution to estimation of primordial non-Gaussianity, something that is currently done using theoretical templates for the ISW-lensing bispectrum. Motivated by these considerations, reconstruction of the ISW map has been the focus of a number of recent analyses . In this paper we study how ISW map reconstruction is affected by a class of observational and astrophysical systematic errors which we will refer to broadly as photometric calibration errors or, for conciseness, calibration errors. These systematics afflict all galaxy surveys at large angular scales, contributing to the significant excess of power at large scales found in many recent surveys, including the Sloan Digital Sky Survey (SDSS), MegaZ, WISE-AGN and WISE-GAL, and NVSS. Calibration errors are thus already established as one of the most significant systematics impacting large-angle measurements of galaxy surveys, a fact that has broad implications, such as for measuring scale-dependent bias as a signal of primordial non-Gaussianity. As the statistical power of galaxy surveys continues to grow, the control and understanding of systematics like calibration errors is becoming even more important. There is a variety of ways in which modern photometric surveys assess and mitigate contamination from systematics, many of which rely on cross-correlating galaxy maps with known systematics templates. This can be used to identify contaminated regions, which are then masked or excluded from the analysis (as in Ref. ). Alternatively, one can use these templates to subtract or marginalize over systematics-induced spatial variations in the calculation of, for example, the two-point clustering signal. Such an approach was taken in Ref. to study the overall detection significance of the ISW effect in SDSS data. The authors found results similar to Ref. , the authors of which instead accounted for excess power by adding a low-redshift spike in the source distributions. Most of these correlation corrections are perturbative, however, and therefore require fairly clean maps in which systematic effects are minor to begin with. Additionally, while corrections to the two-point statistics are important for the inference of cosmological parameters, they do not remove the systematics from the maps themselves. propose an alternative approach, wherein measurement biases are characterized by injecting fake objects into Dark Energy Survey images. This neatly avoids the reliance on having small levels of contamination in the input maps, but it still cannot account for certain systematics, such as dust or flux calibration. Whatever the approach taken, some level of residual calibration error will remain. Some of us previously showed that at levels of calibration control consistent with current and near-future surveys, residual calibration errors are by far the dominant systematic for ISW signal reconstruction . This motivates us to study their impact in more detail. Namely, we would like to study whether the presence of residual calibration errors can be mitigated by combining information from multiple input maps or through better modeling of the contributions of systematics to observed galaxy power. We also wish to investigate to what extent residual calibration errors similarly impact the signal-to-noise ratio of galaxy-CMB cross-correlation and, in turn, the significance of ISW detection. With this aim, we use ensembles of simulated maps to characterize the performance of ISW reconstruction based on surveys like Euclid and SPHEREx, two proposed wide-angle surveys of which the properties are expected to be good for ISW detection and reconstruction. We also consider the benefits of including Planck-like simulations of CMB intensity in the reconstruction effort. We begin in Sec. [sec:method] by describing our model for calibration errors, how we reconstruct the ISW map and evaluate its quality, and which input data sets we use. In Sec. [sec:results], we compare the performance of ISW reconstruction when using one versus multiple surveys and investigate the impact different assumptions have on the results. In Sec. [sec:SNcal], we relate map reconstruction to the total signal-to-noise ratio of ISW detection, and we conclude in Sec. [sec:conclusion]. [fig:plancknvssplot] [fig:planckmphgplot] Methodology =========== Modeling calibration errors --------------------------- Photometric calibration is a challenge faced by all photometric galaxy surveys. It refers to the adjustments required to establish a consistent spatial and temporal measurement of flux of the target objects in different observation bands. A number of different systematics must be accounted for in calibration, including but not limited to detector sensitivity variation on the focal plane, variation in observing conditions, the presence of foreground stars (as galaxies near them are less likely to be detected), and extinction by dust. Calibration errors are introduced if these systematics are incompletely or inaccurately accounted for. Our focus is on how calibration errors affect galaxy number counts. To illustrate this, imagine that a perfectly uniform screen (of e.g. dust) blocks some light from all galaxies. This pushes the faintest galaxies below the survey’s flux limit, and leads to observation of fewer galaxies in all directions. A pure monopole change such as this increases shot noise but does not affect the angular clustering signal of galaxies. In contrast, in a more realistic scenario where the opacity of this “screen” depends on direction, it affects the observed galaxy clustering signal by adding spurious power and by coupling different multipoles of the measured power spectrum . The presence of unaccounted-for calibration errors can thus introduce biases in cosmological parameter estimates from Large Scale Structure (LSS) surveys. These are particularly severe for the ISW effect and other measurements based on signals that, like calibration errors, enter primarily at large angular scales. In keeping with the picture of calibration errors as a direction-dependent screen, we model them as a modulation of the true galaxy number counts $N(\hat{\bf n})$, where ${{\bf \hat n}}$ is the direction on the sky. The observed, modulated counts are $${N\_{\rm obs}}({\hat{\bf n}}) = [1+c({\hat{\bf n}})]\, N({\hat{\bf n}}), \label{eq:nobs}$$ where the field $c({\hat{\bf n}})$ describes the screening effect of calibration errors. While we will generally refer to this kind of modulation as the result of ``calibration errors," as Eq. ([eq:nobs]) makes clear, this formalism can describe any residual effect that modulates a survey’s selection function. Though the expression in Eq. ([eq:nobs]) will necessarily couple different multipoles, at low ℓ, the impact of calibration errors on the observed galaxy power spectrum is well approximated by CobsCgal + Ccal, [eq:clobs] neglecting multiplicative terms. Following, we model the calibration error field $c({{\bf \hat n}})$ as a Gaussian random field with power spectrum ${C\_{\ell}^{\rm cal}}$ and quantify the level of residual calibration errors using its variance, $${{\sigma^2\_{\rm cal}}}\equiv {\rm Var}(c(\hat{\bf n})) = \sum\_{\ell=0}^\infty {2\ell+1\over 4\pi}{C\_{\ell}^{\rm cal}}. \label{eq:calib\_var}$$ While the conversion between $\sigma\_{\rm cal}$ and the rms magnitude error depends on the faint-end slope of the luminosity function of tracers in the survey, they can be related roughly as $(\delta m)^{\rm rms}\simeq \sigma\_{\rm cal}$. Thus a survey with residual calibration errors of ${{\sigma^2\_{\rm cal}}}=10^{-6}$ has been calibrated to roughly a milimagnitude. Motivated by power spectrum estimates for maps of dust extinction corrections and magnitude limit variations in existing surveys (see Fig. 5 and 6 in Ref. ), we choose the fiducial calibration error power spectrum to be $$\label{eq:clcalmodel} {C\_\ell}^{\rm cal} = \alpha^{\rm cal} \exp{\left[-(\ell/w^{\rm cal})^2\right]},$$ with $w^{\rm cal}=10$. The normalization constant $\alpha^{\rm cal}$ is varied to achieve the desired ${{\sigma^2\_{\rm cal}}}$. Figures [fig:plancknvssplot] and [fig:planckmphgplot] show the impact of calibration errors of this form on the angular power spectrum of the NVSS and SDSS MphG galaxy surveys, which have been used to reconstruct the ISW map in previous studies For our fiducial model, we assume that calibration error maps for different redshift bins and surveys are uncorrelated with one another. We briefly examine the impact of relaxing such an assumption in Sec. [sec:caldetails]. ISW estimator ------------- Similarly to , we work with the optimal estimator derived by . It takes as input n maps, which can include any tracers that carry information about the ISW signal, namely LSS, CMB, or lensing convergence maps. Letting gℓmi represent the spherical components of the ith input map, where i ∈ {1, …, n}, the optimal estimator for the spherical component of the ISW signal is [eq:iswestsimple] mISW = i=1nRigmi. The operator $$R\_{\ell}^i\equiv -N\_{\ell}[D\_{\ell}^{-1}]\_{{\rm ISW-}i} \label{eq:R\_def}$$ is a reconstruction filter derived from the covariance matrix, [eq:Dl] D = ( cccc CISW,ISW&CISW,1 &&C n C1,ISW&C1,1 &&C1,n &&& Cn,ISW&Cn,1&&Cn,n ). In this expression, superscript numbers label the input maps and $N\_{\ell} \equiv 1/[D\_{\ell}^{-1}]\_{\rm ISW-ISW}$ estimates the variance of the reconstruction at multipole ℓ. When a single input map A, is used, this expression reduces to a simple Wiener filter, mISW gmA. [eq:wiener] We demonstrate in Appendix [app:otherest] that Eq. ([eq:iswestsimple]) is equivalent to the estimator of, where the CMB temperature map is treated separately from LSS maps, and show that it reduces to the Linear Covariance Based (LCB) filter first proposed by . In constructing this ISW estimator, one must make a choice about how to obtain the necessary angular power and cross-power spectra in the covariance matrix. The Cℓ’s can either be extracted from observations (as in Refs. ) or computed analytically for an assumed cosmology (as in Refs. ). Analytic calculation is straightforward but introduces a model dependence which can potentially bias results if, for example, calibration error contributions are not modeled correctly . Measuring Cℓ from observations produces a model-independent estimator and so can help in the case where the theory spectra are inaccurate, but at the expense of limited precision due to sample variance, especially at large scales, scales with low power, or for map combinations that have little correlation.[1](#fn1) Hybrid methods can also be used, as in Ref. , where account for observed excesses in the autopower of NVSS data by using a smoothed fit to data to get the galaxy map’s autopower, but analytically compute its cross-correlation with the ISW signal. We therefore consider two limiting cases of constructing the estimator in order to investigate how calibration errors impact the ISW reconstruction: 1. a ‘worst’ case estimator filter, ${{R\_\ell({C\_\ell}^{\rm th})}}$, where we use the fiducial theory Cℓ’s in the estimator, in which calibration errors’ power contributions are not modeled at all, and 2. a ‘best’ case estimator filter, ${{R\_\ell({C\_\ell}^{\rm sky})}}$, in which calibration error power contributions are modeled perfectly (i.e. the covariance matrix is known). This case may be approximated by, e.g. a smoothed fit of the observed LSS power. The theoretical spectra are related simply through the expression [eq:clcalrel] Csky= Cth+ Ccal. where ${C\_{\ell}^{\rm cal}}$ is the power spectrum of the calibration error field described in Sec. [sec:calmodel]. We consider these cases in Secs. [sec:euconlyresults] and [sec:eucpluscmb] respectively. Quality statistic ----------------- To quantify the accuracy of a given reconstruction, we use the correlation coefficient between the temperature maps of the true [$T^{\rm ISW}({{\bf \hat n}})$] and reconstructed [$T^{\rm rec}({{\bf \hat n}})$] ISW signal, = ISWrec, [eq:rhofrommap] where $\bar{T^{X}}$ and σX2 are the mean and variance of map $T^{X}({{\bf \hat n}})$, respectively.[2](#fn2) We do not include pixel weights in our calculation of ρ, as is done to account for masking effects in Ref. . This is because we work with only full-sky maps, as will be discussed in the next section. The correlation coefficient can be rewritten in terms of the cross-power between the true ISW map realization and the input tracers, = ISWrec, [eq:rhoraw] where the tilde denotes pseduo-Cℓ measured from a given map realization, and we have used Eq. ([eq:iswestsimple]) to write $$\begin{aligned} \tilde{C}\_{\ell}^{\rm ISW-rec}&=\sum\_i\frac{1}{2\ell+1}\sum\_{m} [{a\_{\ell m}}^{\rm ISW}]^\R\_{\ell}^i g\_{\ell m}^i \\ &= \sum\_iR\_{\ell}^i\tilde{C}\_{\ell}^{{\rm ISW}-i}.\end{aligned}$$ Because the measured correlation coefficient depends on the specific realization, we assess reconstruction accuracy for a given set of input map properties as follows. We simulate a large number of realizations of correlated maps, then apply the ISW estimator to obtain associated reconstructed ISW maps, and by comparing these with the true ISW maps we obtain a sample distribution for ρ. Its mean value ρ̄, which in the limit of an infinitely large ensemble will approach an expectation value ⟨ρ⟩, provides a statistical measure of how accurately the estimator can reproduce the true ISW signal. Studying how ρ̄* changes in response to variations in survey properties and modeling choices therefore allows us to understand which factors are most important for obtaining an accurate ISW reconstruction. We can avoid the computational cost of generating many simulation ensembles by noting that we can obtain a good estimate for the expectation value of ρ if we make the approximation $$\begin{aligned} \langle\rho\rangle &= \left< {\frac{\frac{1}{4\pi} \sum\_{\ell, i}(2\ell+1)R\_{\ell}^i\tilde{{C\_\ell}}^{{\rm ISW}-i}} {\sigma\_{\rm ISW}\sigma\_{\rm rec} }} \right> \label{eq:rhoexp}\\ &\approx \frac{\frac{1}{4\pi} \sum\_{\ell, i}(2\ell+1)R\_{\ell}^i{C\_\ell}^{{\rm ISW}-i}}{\hat \sigma\_{\rm ISW}\hat \sigma\_{\rm rec}}, \label{eq:rhoest}\end{aligned}$$ that is, we replace the pseudo-Cℓ’s with their expectation value across realizations, $\tilde{{C\_\ell}}~\rightarrow~{{C\_\ell}}$. We will refer to the quantity in Eq. ([eq:rhoest]) as ρ̂, defining $$\begin{aligned} \hat \sigma\_{\rm ISW} &= \sqrt{ \frac{1}{4\pi}\sum\_{\ell}\,(2\ell+1)\,{C\_\ell}^{\rm ISW}} \\ \hat \sigma\_{\rm rec} &=\sqrt{\frac{1}{4\pi} \sum\_{\ell, i, j}\,(2\ell+1)\,R\_{\ell}^iR\_{\ell}^j{C\_\ell}^{ij}} \,, \label{eq:sigrecest}\end{aligned}$$ to approximate the rms fluctuations in the true and reconstructed ISW maps. Here the indices i and j label the input tracer maps and the sum over ℓ runs over the multipoles ℓ ∈ [2, 95], a range chosen to conservatively to include all scales where the ISW signal is important. We have tested the approximation ρ̂ ≈ ⟨ρ⟩ in Eq. ([eq:rhoest]) extensively and found it works well when the estimator filter Rℓ is built from analytically computed spectra but can break down if Rℓ is composed of $\tilde{C\_\ell}$’s extracted from map realizations. This behavior is related to the way in which using measured Cℓ’s makes ρ depend on $\tilde{C\_\ell}$, such that $\bar{\rho} = \rho(\langle\tilde{C\_\ell}\rangle)$ is no longer a good approximation of $\langle\rho(\tilde{C\_\ell})\rangle$. Appendix [sec:appapprox] discusses this in more detail. Simulated surveys ----------------- By working with simulated maps, we are able to study in detail how calibration error levels and modeling choices affect ISW signal reconstruction. Since we are concerned only with large scales, we model the ISW signal, total CMB temperature anisotropy, and galaxy number density fluctuations as correlated Gaussian fields. We use `HEALPY` to generate map realizations based on input auto- and cross-power spectra which we compute analytically following the standard expressions given e.g. in Ref. . We use the Limber approximation for ℓ ≥ 20, having verified that this affects ρ at the level of 0.1% or less for the surveys and range of ${{\sigma^2\_{\rm cal}}}$ considered here. We compute Cℓ for multipoles with ℓ ≤ 95, as this range contains almost all of the ISW signal . Accordingly, our simulations are sets of `HEALPIX` maps of resolution ${\rm NSIDE}=(\ell\_{\rm max}+1)/3=32$. We refer the reader to Ref. for a more detailed description of the reconstruction pipeline. Because our goal is to study the impact of calibration errors and not survey geometry, we assume full-sky coverage in all of our analyses. found that in overlapping regions of partial sky LSS surveys, ISW reconstruction quality degrades only slightly compared to the full-sky case. Therefore, the performance of a given estimator using full-sky maps should be indicative of its performance using maps with only partial sky coverage. Our fiducial cosmological model is ΛCDM, with the best-fit cosmological parameter values from Planck 2015, {Ωch2, Ωbh2, Ωνh2, h, ns} = {0.1188, 0.0223, 0, 0.6774, 0.9667}. Unless otherwise stated, ISW reconstructions are performed on 2000 map realizations for each analysis and include multipole information down to $\ell\_{\rm min}=2$. Within this framework, four pieces of information are required to model a LSS survey: the distribution of its sources along the line of sight n(z), a prescription for how they are binned in redshift, their linear bias b(z), and their projected number density per steradian n̄. Below we describe how our choices for these characteristics are based on the properties of promising future probes of the ISW effect. ### Euclid-like LSS survey Our fiducial survey is modeled on Euclid, a future LSS survey with large sky coverage and a deep redshift distribution , which is expected to be an excellent probe of the ISW effect . We assume the redshift distribution used by, [eq:fiddndz] = z2, which has a maximum at $z\_{\rm peak}\simeq 1.21z\_0$. We choose z0 = 0.7 and n̄ = 3.5 × 108, with a photo-z redshift uncertainty of σ(z) = 0.05(1 + z) which smoothes the edges of redshift bins. For simplicity, we assume a constant galaxy bias of b(z) = 1. Our results are qualitatively insensitive to this choice as long as the bias is reasonably well approximated for the input maps. This is because the bias term cancels in the estimate of the ISW signal, so that fractional differences between true and modeled bias have little impact on ρ. We refer the reader to Ref. for further details on both fitting for bias and the impact that mismodeling can have on reconstruction. In Sec. [sec:euconlyresults] we investigate the improvement in ISW map reconstruction when the fiducial Euclid-like survey is split into six redshift bins with edges at z ∈ {0.01, 0.4, 0.8, 1.2, 1.6, 2, 3.5} (see inset of Fig. [fig:euconlyplot]), as compared to the unbinned case. We subsequently use the six-binned Euclid survey as our fiducial case. ### SPHEREx-like LSS survey We model a second survey on the SPHEREx All-Sky Spectral Survey (SPHEREx), a proposed survey that has been optimized to study LSS in the low-redshift universe. One of its goals is to place stringent limits on primordial non-Gaussianity , which will require rigorous control of calibration errors. Given this, SPHEREx will provide excellent input map candidates for ISW map reconstruction. Its shallower reach makes it complementary to the deeper mapping of the LSS provided by Euclid. SPHEREx will identify galaxies with varying levels of redshift uncertainty, ranging from σz < 0.003(1 + z) up to σz > 0.1(1 + z). Grouping these into catalogs with different levels of precision provides collections of galaxies useful for different science goals. The σz < 0.1(1 + z) catalog with a projected ∼ 300 million galaxies was identified in Ref. as the best subsample for $f\_{\rm NL}^{\rm loc}$ detection. Our investigations confirm it to be the best for ISW detection as well. We therefore fit its projected redshift distribution given in Ref. to the functional form for dn/dz given in Eq. ([eq:fiddndz]). We select z0 = 0.46, which results in a peak dn/dz of $z\_{\rm peak} \simeq 0.56$. We have confirmed that our results are not strongly sensitive to changes in this redshift distribution, in agreement with the findings of Ref. . We use a projected number density of n̄ = 6.6 × 107 and consider the case where the survey is split into six redshift bins. We choose their edges by scaling the Euclid-like survey’s binned redshift distribution to the SPHEREx median redshift, resulting in redshift bin edges at z ∈ {0.01, 0.26, 0.53, 0.79, 1.05, 1.31, 2.30}. This still provides sufficient sampling of the field in each bin to ensure that shot noise is subdominant to the galaxy signal power. ### Planck-like CMB survey CMB data have frequently been used in conjunction with LSS data for ISW map reconstruction. Recent examples include Ref. , which used NVSS radio data, the Planck lensing convergence map, and Planck temperature data. That analysis was subsequently extended to include more LSS tracers in Ref. . However, in both of these cases, residual systematics limit the usefulness of lensing data to scales of ℓ ≥ 10 and ℓ ≥ 8, respectively. investigated the usefulness of CMB data for ISW reconstruction using a simulation pipeline similar to ours, finding that both CMB temperature and polarization data only modestly improve reconstruction quality but carry a greater benefit when the LSS tracers themselves contain less information (due to e.g. noise or other properties of the survey). It is then natural to ask whether CMB data can help mitigate the impact of calibration errors in LSS maps. We therefore consider CMB temperature as an additional input map. To compute the total CMB temperature power spectrum, CℓTT, we compute the primordial-only contributions using a modified version of `CAMB` and add them to our calculations for ${C\_\ell}^{\rm ISW}$. As the CMB power spectrum is determined within the limits of cosmic variance at low ℓ and the ISW signal is already dominated by the primary (that is, non-ISW) CMB anisotropies, we do not include calibration errors in the generation of CMB temperature maps. Though CMB polarization and lensing could provide additional information, residual systematics remain at large scales for each (see Refs. and, respectively), so for simplicity we do not include them in this analysis. Results ======= To characterize the impact of calibration errors in LSS surveys on the ISW map reconstruction, and the potential to mitigate these impacts, we look at multiple combinations of input maps with different properties. Specifically, we consider the impact of binning in redshift, of adding CMB intensity data, and of including additional LSS information from another survey. For each of these studies, we examine two limiting cases for the estimator. The best case scenario, which we will reference as ${{R\_\ell({C\_\ell}^{\rm sky})}}$, is when one perfectly models all contributions to the galaxy power, including residual calibration errors. The worst case, referenced by ${{R\_\ell({C\_\ell}^{\rm th})}}$, is when the estimator is built out of theoretical spectra with no power from calibration errors. The power spectra in these two cases are related by Eq. ([eq:clcalrel]). We use the analytical $\hat\rho$ to estimate the mean reconstruction quality across a wide range of ${{\sigma^2\_{\rm cal}}}$, while performing reconstruction on simulated maps for selected values, to both verify the accuracy of $\hat\rho$ and to generate error bars for the spread of ρ across simulations. One survey: Binning in redshift ------------------------------- [fig:euconlyplot] [fig:eucpluscmb] We first consider the Euclid-like survey alone and investigate the impact of binning in redshift on the quality of reconstructions in the presence of calibration errors. We model calibration errors in the binned case by adding the calibration field’s power to the autopowerpower spectrum of each bin i per Eq. ([eq:clobs]): ${C\_\ell}^{i,i} \rightarrow {C\_\ell}^{i,i} + {C\_\ell}^{\rm cal}$. We do not add any power to the cross-spectra, though we test the impact of contamination in the cross-spectra in Sec. [sec:caldetails].[3](#fn3) The results of this study are shown in Fig. [fig:euconlyplot]. For reference, we use a vertical shaded band to mark the level of calibration corresponding to current surveys, defined roughly as the range bounded by the residual SDSS DR8 limiting magnitude variations and the SDSS über-calibration. (‘Future’ levels are defined roughly as those between that required to limit bias on cosmological parameters to below their projected uncertainties and an intermediate level before bias becomes unacceptable; see Refs. and for details.) As shown in Fig. [fig:euconlyplot], splitting the survey into six redshift bins results in significant improvement in reconstruction at all levels of calibration error. This improvement is comparable to reducing ${{\sigma^2\_{\rm cal}}}$ of the single-bin case by a factor of 10 at ‘current’ levels. Tomographic information is useful because it allows galaxy counts to be weighted more optimally, taking advantage of the fact that the ISW effect becomes stronger at lower redshift as dark energy becomes more dominant and structure growth slows. For instance, considering the expected ISW reconstruction power from each bin when using optimal weights (i.e. the squared contribution of each term in Eq. [eq:iswestsimple], using ${{R\_\ell({C\_\ell}^{\rm sky})}}$), we find that with no calibration error, the first redshift bin contributes 87% as much power as the second bin, with subsequent bins contributing 58%, 31%, 15% and 10% as much power, respectively. There is additional benefit to binning when calibration errors are considered. Since the low-redshift bins have a higher clustering signal than the high-redshift bins, they are less impacted by the same level of calibration error. Thus, the optimal weighting changes depending on the level of calibration error. When calibration errors are increased to ${{\sigma^2\_{\rm cal}}}=10^{-4}$, for example, the first bin contributes the most power to the reconstruction, with bins 2 − 6 only contributing 39%, 12%, 4%, 2%, and 1% as much power. As we will show later, this error-level-dependent weighting will mean adding information from a shallower survey such as SPHEREx makes reconstruction more robust against calibration errors. The importance of accounting for calibration errors in the estimator is apparent in the difference between the dashed and solid curves, where doing so improves $\bar\rho$ for ${{\sigma^2\_{\rm cal}}}\gtrsim 10^{-6}$, with $\Delta \bar\rho \approx 0.1-0.2$ at current levels of calibration. This improvement is roughly comparable to the improvement seen from binning in redshift. Though for clarity we do not include this case in the Figure, we additionally studied the effect of using the observed, unsmoothed galaxy-galaxy power in the estimator (that is, $\tilde{C\_\ell}$, power spectra extracted from map realizations rather than computed analytically). We find that in this case $\bar\rho$ converges to the same value as the ${{R\_\ell({C\_\ell}^{\rm sky})}}$ case when calibration errors are very large, but is greatly reduced from $\bar\rho$ found using either ${{R\_\ell({C\_\ell}^{\rm sky})}}$ or ${{R\_\ell({C\_\ell}^{\rm th})}}$ when calibration errors are small (${{\sigma^2\_{\rm cal}}}\ll 10^{-5}$). For example, for a single input map in the limit of no calibration errors, quality reduces from ρ̄ = 0.93 to 0.83 when we switch to using observed $\tilde{C\_\ell}$’s. If we also use the observed (unsmoothed) cross-correlation between the LSS map and the CMB for the galaxy-ISW term in the estimator, reconstruction quality is further degraded to ρ̄ = 0.74 in the absence of calibration errors. This is because primary CMB anisotropies are large compared to ISW contributions, causing the measured galaxy-CMB correlation to receive relatively large noise contributions from chance correlations between LSS maps and the primordial CMB. Given the significant improvement in reconstruction that binning provides, from here forward we adopt the configuration with six tomographic bins as our fiducial Euclid-like survey. Effect of adding Planck TT data ----------------------------------- We now consider adding information from the Planck-like CMB temperature map described in Sec. [sec:modsurveys]. When used as the only input map, the reconstruction is considerably worse than that found using the ideal Euclid-like survey (Table [table:rhotable]). We include it in our study, however, because any realistic study attempting to reconstruct the ISW signal will likely include CMB temperature data. Additionally, the reconstruction quality attainable with CMB temperature data alone provides a useful baseline against which to compare the performance of estimators based on LSS maps. With CMB temperature data alone, we find an average reconstruction quality of ρ̄ = 0.46, in good agreement with Ref. . To put this into proper context, however, it is important to note that there is a large scatter around that mean; while the average reconstruction quality is indicative of performance, any single realization, such as that of our own Universe, can vary substantially in fidelity. The purple band in Fig. [fig:eucpluscmb] shows the extent of this scatter for ISW reconstruction based on just the CMB map. When CMB temperature information is combined with that from LSS maps, it significantly improves reconstruction quality, but only if the true galaxy power spectrum ${C\_\ell}^{\rm sky}$ (including calibration error contributions) is used in the estimator, as can been seen by the behavior of the solid curves on the right-hand side of Fig. [fig:eucpluscmb]. The blue $\rho({{\sigma^2\_{\rm cal}}})$ curve describing the CMB+LSS reconstruction tracks the maximum of the curves corresponding to reconstructions using the CMB and LSS input maps separately, shown by the purple and red curves, respectively. This occurs because the estimator down-weights the LSS survey the more it is affected by calibration errors, converging to the TT-only reconstruction quality in the limit of large calibration errors. If one does not model calibration error power contributions, however, then any improvement from combining multiple input maps is marginal at best and can in fact result in a worse reconstruction than just using the CMB data alone. This demonstrates the importance of ensuring that the LSS Cℓ’s used in the ISW estimator are a good fit to the observed spectra. Effect of an additional LSS survey: SPHEREx-like ------------------------------------------------ We now consider the addition of our fiducial six-bin SPHEREx-like survey described in Sec. [sec:spherexintro], assuming for simplicity that it has the same level of calibration errors as the Euclid-like survey. Results are shown in Fig. [fig:multsurveyplot]. In the limit of no calibration errors, the SPHEREx-like survey offers little additional information. In fact, adding both SPHEREx and CMB TT results in negligible improvement over the Euclid-like only case ($\Delta\bar\rho<0.003$ compared to a spread of $\sigma\_{\rm Euc+Spx+TT}=0.019$). However, by comparing the black and blue curves we see that including the SPHEREx-like survey does make the reconstruction somewhat more robust against calibration errors. The reason for this is similar to why binning in redshift is helpful: recall that, in the case of binning, having narrow, low-redshift bins means having some bins with higher galaxy autopower than the unbinned case, which then have less susceptibility to a given level of calibration error. Similarly, SPHEREx has a shallower redshift distribution, and thus an intrinsically higher clustering signal, so that it can actually provide a better reconstruction than the Euclid-like survey at moderate levels of calibration error. We would expect to see similarly increased robustness to calibration errors for any tracer with a larger clustering signal, including tracers with a larger bias. Finally, just as for Euclid, we find that if calibration errors are not accounted for in the estimator, then adding LSS data can actually result in a worse reconstruction than that from using CMB temperature data alone. \|c \|c \|c c \|c c \| & Rℓ & & ${{\sigma^2\_{\rm cal}}}$ & 0 & 10− 6 & 10− 4 & 10− 6 & 10− 4 [0.5ex] TT & 0.46 & - & -& -& - Euclid (1 bin) & 0.92 & 0.83 & 0.19 & 0.84& 0.29 Euclid (6 bin) & 0.95 & 0.91 & 0.41& 0.92& 0.57 SPHEREx (6 bin) & 0.89 & 0.88 & 0.52 & 0.88& 0.62 Euc + Spx + TT & 0.96 & 0.92 & 0.47 & 0.93& 0.73 [.5ex] [table:rhotable] [fig:multsurveyplot] Effect of varying calibration error properties ---------------------------------------------- We now test how sensitive the results in the previous sections are to our assumptions about calibration errors, showing the results in Fig. [fig:calprops]. First, the left panel shows what happens when we vary the level of cross-correlation between the calibration errors of different LSS maps. It is conceivable that residual calibration errors can be correlated across different bins of a single survey, or even across different surveys, especially if the error has an astronomical origin. To model such correlation, we set the level of cross-correlation between the calibration errors of maps i and j using a parameter $r\_{\rm cc}$, where Ccal, ij = rcc, ij [eq:crosscorr] As we only consider cases where calibration errors in all maps are characterized by the same ${C\_\ell}^{\rm cal}$, this reduces to Ccal, ij = rccCcal,ij. [eq:crosscorr-simp] [fig:calprops] We consider the six-bin fiducial Euclid-like survey and find that this kind of correlated error results in mild degradation of the reconstruction for ${{\sigma^2\_{\rm cal}}}\lesssim10^{-4}$, but otherwise it has little effect as long as calibration errors are correctly modeled in the estimator [that is, ${{R\_\ell({C\_\ell}^{\rm sky})}}$ is used]. If calibration errors are not accounted for [${{R\_\ell({C\_\ell}^{\rm th})}}$ is used], reconstruction suffers considerably, as shown by the dotted curve. We also use a dashed curve [labeled ${R\_\ell}({C\_\ell}^{\rm XY,th})$] for the case where the estimator filter correctly accounts for the autopower contributions of calibration errors but neglects the cross-power contributions. As seen by comparing the solid, dashed, and dotted orange curves in Fig. [fig:calprops], reconstruction quality is far more sensitive to accurate modeling of the calibration error contribution to the autopower than to the cross-power. Thus, fitting the observed autopower for each map but using theoretical cross-powers, as is done in Ref. , should harm the reconstruction relatively little, depending on the fitting scheme; we find $\Delta \hat\rho \approx -0.03$ at ${{\sigma^2\_{\rm cal}}}= 10^{-4}$ for $r\_{\rm cc}=0.2$, far less than the typical variation over realizations shown in Fig. [fig:euconlyplot]. Additionally, we study the impact of changing the shape of the calibration error power spectrum ${C\_{\ell}^{\rm cal}}$, showing the results in the right panel of Fig. [fig:calprops]. We first vary the width parameter $w^{\rm cal}$ of the calibration error power spectrum ${C\_{\ell}^{\rm cal}}$ given in Eq. ([eq:clcalmodel]). Results for different values of $w^{\rm cal}$ are qualitatively similar, though for fixed ${{\sigma^2\_{\rm cal}}}$, the reconstruction is less sensitive to calibration errors when $w^{\rm cal}$ is larger. The reason for this is that ρ is most sensitive to contamination at the lowest multipoles, as will be discussed in Sec. [sec:SNcal]. Using a power law ${C\_{\ell}^{\rm cal}}\propto(\ell+1)^{-2}$ gives results similar to our fiducial Gaussian form with $w^{\rm cal}=10$. [fig:SN2l] Impact of calibration errors on S/N estimates ============================================= Given the extent to which calibration errors degrade the quality of ISW signal reconstruction, it makes sense to ask whether they also impact the signal-to-noise (S/N) of ISW detection. Detection of the ISW signal via the cross-correlation between the CMB and LSS maps has been the subject of considerable study, as it serves as an important consistency test for the presence of dark energy. The significance of detection varies considerably depending on the LSS tracer sample and the statistical methods used, as well as how systematics in the LSS data are treated . used systematics templates to correct the observed power spectra for SDSS galaxies, finding a S/N loss of ∼ 0.5 if such corrections are neglected. introduced a low-redshift spike in the source distributions in order to reproduce the observed excess autopower in NVSS and SDSS catalogs and estimate that such systematics result in an uncertainty of ΔS/N ± 0.4. The most recent results come from the Planck Collaboration, which found ∼ 4σ evidence for the ISW effect, with most of the signal coming from cross-correlation of the CMB temperature with the NVSS radio catalog and CMB lensing. The maximum achievable signal-to-noise can be obtained by considering an ideal survey that perfectly traces the ISW (i.e. ${C\_\ell}^{gg}={C\_\ell}^{Tg}={C\_\ell}^{\rm ISW}$), resulting in a maximum S/N ∼ 6 − 10 for ΛCDM cosmology. Our goal is to study how calibration errors impact the significance of ISW detection. There are multiple ways one can quantify detection of the ISW effect, including correlation detection between LSS and the CMB, template matching to an assumed model, or model comparison. Each of these methods relies on different assumptions and tests different statistical questions (see Ref. for a detailed review). Here we adopt the simple correlation detection statistic which quantifies the expected deviation from a null hypothesis of no correlation between LSS (g) and CMB temperature (T). In this formalism the S/N for ISW detection is ()2 (Tg) (Ccov)-1 Tg, [eq:SNdef] where we have assumed the multipoles contribute independently to the S/N. Here $\mathbf{{C\_\ell}}^{Tg}$ is a vector of the ISW-LSS cross-spectra, and the covariance matrix elements corresponding to LSS maps i and j can be written as $$\begin{aligned} {C\_\ell}^{{\rm cov},ij} &= \left<\Delta {C\_\ell}^{Ti} \Delta {C\_\ell}^{Tj}\right> \\ &\simeq \frac{{C\_\ell}^{Ti}{C\_\ell}^{Tj} + {C\_\ell}^{TT} ({C\_\ell}^{ij} + {C\_\ell}^{{\rm cal},ij} + \delta\_{ij} \frac{1}{\bar{n}\_{ij}})} {f\_{sky}(2\ell+1)}, \label{eq:covar}\end{aligned}$$ where the last term in the numerator is due to shot noise and δij is the Kronecker delta.[4](#fn4) Equations ([eq:SNdef]) and ([eq:covar]) demonstrate that all cosmological tests using LSS-CMB cross-correlation are limited in their constraining power due to sample variance and the relatively large amplitude of the primordial CMB fluctuations. They also make it clear that calibration errors will reduce the significance of ISW detection. We assume calibration errors to be uncorrelated between maps, so ${C\_\ell}^{{\rm cal},ij}~\rightarrow~\delta\_{ij}{C\_\ell}^{{\rm cal},ij}$. For a single LSS map, Eq. ([eq:SNdef]) reduces to the form ()2 fsky (CTg)2 + CTT (Cgg + Ccal + 1/\|ng). [eq:SNonemap] If there are no calibration errors, we find S/N = 6.6 for our Euclid-like survey, which is near the maximum[5](#fn5) for this cosmology, S/N = 6.7. As ${{\sigma^2\_{\rm cal}}}$ increases from 0 to current levels, the total S/N reduces to 4.9 − 5.7, a drop of only ∼ 15% − 30%. This can be seen in the S/N values listed for various ${{\sigma^2\_{\rm cal}}}$ in the legend of Fig. [fig:SN2l]. In contrast, for the same level of error, average reconstruction quality $\hat\rho$ is reduced by 40% − 60%. Clearly, ISW signal reconstruction is substantially more affected by calibration errors than is ISW detection significance. The greater robustness of the total S/N to calibration errors is due to the fact that it has support at higher multipoles. This is most easily illustrated in the single-map case, where the contribution per multipole to the total signal-to-noise is ()2= (2+1). [eq:SNell] Figure [fig:SN2l] shows how the contribution per multipole responds to different levels of calibration error. As ${{\sigma^2\_{\rm cal}}}$ increases, the signal-to-noise decreases at lower multipoles, but contributions at higher multipoles remain unchanged. These higher-multipole contributions are thus still available to contribute to the overall S/N. Map reconstruction is more sensitive to the largest scales. For the single-map case, this can be illustrated analytically as follows. Using the single-map estimator from Eq. ([eq:wiener]), we can write the estimated reconstruction quality statistic as $$\begin{aligned} \hat{\rho} &= \frac{\frac{1}{4\pi} \sum\_{\ell}(2\ell+1)\left(\frac{{C\_\ell}^{Tg}}{{C\_\ell}^{gg}}\right){C\_\ell}^{Tg}} {\sigma\_{\rm ISW} \sqrt{\frac{1}{4\pi}\sum\_{\ell}(2\ell+1)\left(\frac{{C\_\ell}^{Tg}}{{C\_\ell}^{gg}}\right)^2{C\_\ell}^{gg}}} \nonumber\\ &= \frac{1} {\sigma\_{\rm ISW}} \sqrt{\frac{1}{4\pi}\sum\_{\ell}(2\ell+1)\frac{({C\_\ell}^{Tg})^2}{{C\_\ell}^{gg}}} \nonumber\\ &= \frac{1} {\sigma\_{\rm ISW}}\sqrt{\frac{1}{4\pi}\sum\_{\ell} \left(\frac{S}{N}\right)^2\_\ell {C\_\ell}^{TT}\left(1+\frac{({C\_\ell}^{Tg})^2}{{C\_\ell}^{gg}{C\_\ell}^{TT}}\right)} \nonumber\\ &\approx \frac{1} {\sigma\_{\rm ISW}}\sqrt{\frac{1}{4\pi}\sum\_{\ell}\left(\frac{S}{N}\right)^2\_\ell {C\_\ell}^{TT}}. \label{eq:rho\_from\_SN}\end{aligned}$$ Here, (S/N)ℓ2 is the quantity given by Eq. ([eq:SNell]) which, when summed over ℓ, gives (S/N)2. Thus, we see from Eq. ([eq:rhofromSN]) that $\hat\rho$ is proportional to a total (S/N) of which the terms are weighted by CℓTT. Since CℓTT drops sharply as ∼ ℓ− 2, the quality of map reconstruction $\hat\rho$ is more impacted by large-angle calibration errors than the overall S/N is. This is also a primary cause for the degradation in reconstruction quality seen when ${{\sigma^2\_{\rm cal}}}$ was concentrated at lower multipoles in Sec. [sec:caldetails]. Conclusions =========== Reconstruction of the integrated Sachs-Wolfe signal would allow, for the first time, a clean separation of the CMB temperature anisotropies into contributions from 300,000 years after the big bang and those from some ∼ 10 billion years later. This, in turn, would allow for a more informed assessment of the origin of the “large-angle CMB anomalies” and a more complete elimination of ISW contaminants to CMB-based measurements of primordial non-Gaussianity. Accurate ISW reconstruction requires wide-angle large-scale structure maps from which the gravitational potential evolution can be inferred, but in practice, these maps are plagued by photometric calibration errors due to a host of atmospheric, instrumental, and selection-induced systematics. In our previous work, we found that the realistic levels of residual calibration error severely degrade the accuracy of the reconstructed ISW map. In this paper, we investigated how the effects of residual photometric calibration errors on the ISW map reconstruction can be mitigated by using tomographic information and by combining data from multiple surveys. To quantify the amount of residual calibration errors, we use their variance ${{\sigma^2\_{\rm cal}}}$, the square root of which is roughly equal to the rms magnitude fluctuations induced by these systematics. We find that for a Euclid-like survey with a single redshift bin, to achieve a reconstruction comparable in quality to that derived from the CMB temperature map alone (with an average correlation between the true and reconstructed ISW maps of only ρ̄ ≈ 0.46), one must limit the variance of the calibration error field to ${{\sigma^2\_{\rm cal}}}\lesssim 10^{-5}$. This can be improved significantly if we exploit the tomographic information available by binning the LSS data in redshift (Fig. [fig:euconlyplot]). We also show that if the model spectra in the estimator differ substantially from those used to generate the input maps e.g. by using theoretical power spectra that do not account for the observed excess autopower in the LSS survey from calibration errors, then reconstruction quality is substantially degraded. It is therefore crucial to verify that the theoretical spectra in the estimator are a good fit to those observed or to otherwise use smoothed fits. We next consider how using multiple input maps, probing different tracers and redshift ranges, improves ISW signal reconstruction. We find that as long as the excess power contributed by calibration errors is adequately modeled in the estimator, the resulting reconstruction is always better than that from either of the input maps individually. If the excess power from calibration errors is not modeled, however, adding a map can actually degrade reconstruction (Fig. [fig:eucpluscmb]). The CMB temperature map adds information to the reconstruction at all levels of calibration error, but is especially valuable if the LSS maps are subject to calibration errors with ${{\sigma^2\_{\rm cal}}}\gtrsim 10^{-5}$. Using a six-bin SPHEREx-like survey provides qualitatively similar results to the Euclid-like one, but because it is shallower, the reconstruction is less accurate in the limit of no calibration errors ($\rho\_{\rm Spx} =0.89\pm0.04$ vs. $\rho\_{\rm Euc}=0.95\pm0.02$, where errors indicate the 68% spread across realizations). However SPHEREx’s shallower depth and therefore intrinsically higher clustering signal becomes an asset in the presence of calibration errors, making the survey more robust against calibration errors and leading to a better reconstruction for ${{\sigma^2\_{\rm cal}}}\gtrsim 6\times10^{-6}$. (Similarly, we would expect the increased clustering of tracers with larger bias to help mitigate the effects of calibration error as well.) Therefore, a combination of a shallower and a deeper LSS survey provides complementarity useful for separation of calibration errors from the ISW signal and necessary for a good ISW map reconstruction in the presence of such errors. Using all three simulated surveys as input—Euclid, SPHEREx	arxiv_0000515
Cluster algebras and Jones polynomials ====================================== [1](#fn1) We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by continued fractions, and it has recently been shown that one can associate to each continued fraction a snake graph, and hence a cluster variable in a cluster algebra. We show that up to normalization by the leading term the Jones polynomial of the 2-bridge link is equal to the specialization of this cluster variable obtained by setting all initial cluster variables to 1 and specializing the initial principal coefficients of the cluster algebra as follows y1 = t− 2 and yi = − t− 1, for all i > 1. As a consequence we obtain a direct formula for the Jones polynomial of a 2-bridge link as the numerator of a continued fraction of Laurent polynomials in q = − t− 1. We also obtain formulas for the degree and the width of the Jones polynomial, as well as for the first three and the last three coefficients. Along the way, we also develop some basic facts about even continued fractions and construct their snake graphs. We show that the snake graph of an even continued fraction is ismorphic to the snake graph of a positive continued fraction if the continued fractions have the same value. We also give recursive formulas for the Jones polynomials. Introduction ============ Cluster algebras were introduced by Fomin and Zelevinsky in 2002 in the context of canonical bases in Lie theory. Since then many people have worked on cluster algebras and the subject has developed into a theory of its own with deep connections to a number of research areas including representation theory of associative algebras and Lie algebras, combinatorics, number theory, dynamical systems, hyperbolic geometry, algebraic geometry and string theory. In this paper, we develop a new connection between cluster algebras and knot theory. The key to this relation is the use of continued fractions in both areas. To every continued fraction, one can associate a knot or a 2-component link which is built up as a sequence of braids each of which corresponding to one entry of the continued fraction. The class of knots (and links) obtained in this way are called 2-bridge knots. They were first studied by Schubert in 1956 in and the relation to continued fractions goes back to Conway in 1970 in. A significant part of knot theory is concerned with knot invariants, and one of the most important knot invariants is the Jones polynomial, introduced by Jones in 1984 in. The Jones polynomial of an oriented link is a Laurent polynomial in one variable t± 1/2 with integer coefficients. A cluster algebra is a Z-subalgebra of a field of rational functions in several variables. To define a cluster algebra, one constructs a possibly infinite set of generators called cluster variables, by a recursive method called mutation. Each cluster variable is a polynomial in two types of variables; the initial cluster variables x1± 1, x2± 1, …, xN± 1 and the initial principal coefficients y1, y2, …, yN, see, and with positive integer coefficients, see. The *F-polynomial* of the cluster variable is obtained by setting all initial cluster variables equal to 1. The relation between cluster algebras and continued fractions was established recently by Çanakçi and the second author in. They showed that the set of positive continued fractions is in bijection with the set of abstract snake graphs. These are planar graphs that appeared naturally in the study of cluster algebras from surfaces in (also for the special case of the once-punctured torus). The bijection is such that the numerator of the continued fraction is equal to the number of perfect matchings of the snake graph. In, the authors gave a combinatorial formula for the cluster variables for cluster algebras from surfaces, and in this formula was used to construct canonical bases for these cluster algebras in the case where the surface has no punctures. For each cluster variable, the authors construct a weighted snake graph from geometric information provided by the surface. The combinatorial formula expresses the cluster variables as a sum over all perfect matchings of that snake graph. In, the authors used their new approach to snake graphs via continued fractions to give another formula for the cluster variables of a cluster algebra of surface type (with trivial coefficients) which expresses the cluster variable as the numerator of a continued fraction of Laurent polynomials. This formula was generalized by Rabideau to include cluster algebras with principal coefficients. Our main result in this paper is that these cluster variables specialize to the Jones polynomials of 2-bridge links. In order to make this statement precise, we must first ensure that the 2-bridge link as well as its orientation is uniquely determined by the continued fraction. We found that the most natural way to do so is to work with even continued fractions [b1, b2, …, bm], where each entry bi is an even non-zero integer (possibly negative). This is no restriction, since every 2-bridge link is associated to an even continued fraction. Moreover, if [b1, b2, …, bm] is an even continued fraction and [a1, a2, …, an] is a positive continued fraction such that both have the same value [b1, b2, …, bm] = p/q = [a1, a2, …, an], then the associated 2-bridge links are isotopic and the associated snake graphs are isomorphic. To state our main theorem, recall from above that each cluster variable is a polynomial in variables x1± 1, x2± 1, …, xN± 1, y1, y2, …, yN, and that the Jones polynomial is a polynomial in one variable t± 1/2. Denote by Fb1, b2, …, bm the specialization of the cluster variable of the continued fraction [b1, b2, …, bm] obtained by setting x1 = x2 = ⋯ = xN = 1, y2 = y3 = ⋯ = yN = − t− 1 and y1 = t− 2. Then our main result is the following. [thm main intro] (Theorem [thm main]) Let [b1, b2, …, bm] be an even continued fraction, let V[b1, b2, …, bm] be the Jones polynomial of the corresponding 2-bridge link, and let Fb1, b2, …, bm be the specialized cluster variable. Then V[b1, b2, …, bm] = δ tj Fb1, b2, …, bm, where δ = ± 1 and $j= \sum\_{i=1}^m \max\left( (-1)^{i+1} b\_i + \frac{\textup{sign}(b\_ib\_{i-1})}{2}\,\ -\frac{1}{2}\right).$ We also give a simple formula for the sign δ. Thus the Jones polynomial is completely determined by the cluster variable, and hence this theorem establishes a direct connection between knot theory on the one hand and cluster algebras on the other. This also shows that the Jones polynomial of a 2-bridge knot is an alternating sum, which is not true for arbitrary knots. Another direct consequence of the theorem is that the difference between the highest and the lowest degree in the Jones polynomial is equal to the sum of the entries in the positive continued fraction a1 + a2 + ⋯ + an. We also obtain the following direct formula for the Jones polynomial in terms of the continued fraction. In the statement, we use the notation [a]q = 1 + q + q2 + … + qa − 1 for the q-analogue of the positive integer a, and we let q = − t− 1, and ℓi = a1 + a2 + ⋯ai. (Theorem [thm main2]) Let [a1, a2, …, an] be a positive continued fraction. Then up to normalization by its leading term, V[a1, a2, …, an] is equal to the numerator of the following continued fraction * If n is odd, $$\Big[\, [a\_1+1]\_q -q\,\ [a\_2]\_q \,q^{-\ell\_2},\ [a\_3]\_q\,q^{\ell\_2+1},\ldots, [a\_{2i}]\_q\, q^{-\ell\_{2i}},\ [a\_{2i+1}]\_q\, q^{\ell\_{2i}+1},\ldots, \ [a\_{n}]\_q\, q^{\ell\_{n-1}+1}\Big].$$ * If n is even, $$q^{\ell\_n}\,\Big[\, [a\_1+1]\_q -q\,\ [a\_2]\_q \,q^{-\ell\_2},\ [a\_3]\_q\,q^{\ell\_2+1},\ldots, [a\_{2i}]\_q\, q^{-\ell\_{2i}},\ [a\_{2i+1}]\_q\, q^{\ell\_{2i}+1},\ldots, \ [a\_{n}]\_q\, q^{-\ell\_{n}}\Big].$$ This result is very useful for computations. We give several examples, the biggest being the Jones polynomial of a link with 20 crossings, corresponding to the continued fraction [2, 3, 4, 5, 6]. The theorem also allows us to obtain direct formulas for the first 3 and the last 3 co coefficients of the Jones polynomials in Theorem [thm coeff]. As mentioned above, a main ingredient in the proofs is the use of even continued fractions. These were used in knot theory previously, see for example, but not in cluster algebras. We also did not find an algebraic treatment of the topic in the literature, so we include the basic algebraic foundation of even continued fractions in section [sect cf], and we apply it to the study of snake graphs in section [sect sg]. Our main result in section [sect sg] is the following. (Theorem [thm 1]) Given a positive continued fraction [a1, a2, …, an] and an even continued fraction [b1, b2, …, bm] such that both have the same value p/q, then the corresponding snake graphs are isomorphic. We also study the Jones polynomials of 2-bridge links and develop recursive formulas which we need to prove our main results. Let us point out that Hikami and Inoue have obtained a realization of knots in a cluster algebra of a disc with several punctures using the braid presentation of the knot in. Their approach as well as their results are very different from ours. Let us also mention the work of Shende, Treumann and Williams, who relate cluster algebras to Legendrian knots. Also their work is very different from ours. The paper is organized as follows. In section [sect cf], we review positive continued fractions and develop basic results for even continued fractions. We introduce snake graphs of even continued fractions in section [sect sg] and compare them to snake graphs of positive continued fractions. Section [sect knots] contains a short overview of 2-bridge links. We also fix conventions for the links and their orientations in that section. In section [sect Jones], we define the Jones polynomial and develop recursive formulas for the Jones polynomial of 2-bridge links from the definition. We also compute the degree of the Jones polynomial and the sign of the leading coefficient. Section [sect F] contains a brief review of cluster algebras, especially the formula for the (specialized) cluster variables in terms of continued fractions. We also prove recursive formulas for the F-polynomials in this section. Our main results are stated and proved in section [sect main]. This section also contains several examples. We would like to thank Michael Shapiro for helpful comments. Positive continued fractions vs even continued fractions ======================================================== In this section, we list a few results on continued fractions that we will need later. For a standard introduction, we refer to. For a more extensive treatment of continued fractions see. For the relation between continued fractions and cluster algebras see. In this paper a continued fraction is an expression of the form $$[a\_1,a\_2,\ldots,a\_n]= a\_1+\cfrac{1}{a\_2+\cfrac{1}{a\_3+\cfrac{1}{\ddots +\cfrac{1}{a\_n}}}}$$ where the ai are integers (unless stated otherwise) and an ≠ 0. We will be mainly concerned with the following two types of continued fractions. A continued fraction is called positive if each ai is a positive integer, and it is called even if each ai is a nonzero even integer. For example, the rational number $\frac{27}{10}$ can be represented as a positive continued fraction as well as an even continued fraction as follows. $$\frac{27}{10}=[2,1,2,3]=[2,2,-2,4].$$ Indeed, the first expression is obtained by the Euclidean division algorithm $$\begin{array}{rcl} 27&=&2\cdot 10 +7 \\ 10&=&1\cdot 7 +3 \\ 7&=&2\cdot 3 +1 \\ 3&=&3\cdot 1,\\ \end{array}$$ and the second expression is obtained by the following division algorithm $$\begin{array}{rcl} 27&=&2\cdot 10 +7 \\ 10&=&2\cdot 7 +(-4) \\ 7&=&(-2) \cdot (-4) + (-1) \\ -4&=&4\cdot (-1). \\ \end{array}$$ We often use the notation N[a1, a2, …, an] for the numerator of the continued fraction. Thus N[2, 1, 2, 3] = 27. [lem 0] Let p > q > 0 be integers. * [(a)] There exist unique integers a, r such that p = aq + r, with a > 0 and 0 ≤ r < q. * [(b)] There exist unique integers b, s such that p = bq + s, with b ∈ 2Z \ {0} and − q ≤ s < q. Moreover, if pq is even then so is qs. Part (a) is the Euclidean division algorithm in Z. To prove (b), we let b = a and s = r if a is even, and we let b = a + 1 and s = r − q if a is odd. Then p = bq + s and the inequalities 0 ≤ r < q imply the inequalities − q ≤ s < q. To prove uniqueness, suppose we have another pair bʹ, sʹ satisfying the required property. Then, since bq + s = bʹq + sʹ, we have ∣b − bʹ∣q = ∣sʹ − s∣ < 2q, and since b − bʹ is even, it follows that b = bʹ and s = sʹ. [ex 1] The even continued fraction of p/(p − 1) is [2, − 2, 2, − 2, …, ± 2] with a total of m = p − 1 coefficients. Thus 2/1 = [2], 3/2 = [2, − 2], 4/3 = [2, − 2, 2], 5/4 = [2, − 2, 2, − 2]. [prop 01] Let p, q be relatively prime integers with p > q > 0. Then * [(a)] p/q has a positive continued fraction expansion p/q = [a1, a2, …, an] which is unique up to replacing the coefficient an by the two coefficients an − 1, 1. * [(b)] if p or q is even, then p/q has a unique even continued fraction expansion. * [(c)] if p and q are both odd then p/q does not have an even continued fraction expansion. Part (a) is well known, see for example. We prove part (b) by induction on p. If p = 2 then q = 1 and p/q = 2 = [2]. To show that this expansion is unique, suppose that 2 = [b1, b2, …, bm] with m > 1. Then 2 = b1 + 1/[b2, …, bm], which implies that − 1 ≤ 2 − b1 ≤ 1. Since b1 is even, we conclude b1 = 2 and thus 1/[b2, …, bm] = 0 which is impossible. Now suppose that p ≥ 2. Lemma [lem 0] implies the existence of unique b1, s with p = b1q + s with b1 even and − q ≤ s < q, and qs is even. By induction, we may assume that q/s has a unique even continued fraction expansion q/s = [b2, …, bm]. Thus p/q = b1 + s/q = [b1, b2, …, bm]. To show uniqueness, suppose p/q = [b1ʹ, b2ʹ, …, bmʹ] = b1ʹ + 1/[b2ʹ, …, bmʹ]. Then p = b1ʹq + q/[b2ʹ, …, bmʹ] with − q < q/[b2ʹ, …, bmʹ] < q, and then the uniqueness in Lemma [lem 0] implies that b1ʹ = b1 and q/[b2ʹ, …, bmʹ] = s. Now the statement follows from the uniqueness of the expansion q/s = [b2, …, bm]. To show part (c), we prove by induction that if [b1, b2, …, bm] = p/q with bi even, then one of p or q is even. Clearly [b1] = b1 is even. For the induction step, suppose that p/q = [b1, b2, …, bm] = b1 + 1/[b2, …, bm], and let [b2, …, bm] = q/r. Then one of q or r is even, by induction. If q is even we are done, and if r is even then p/q = b1 + r/q implies that p is even. [prop 1] Let p > q > 1 be relatively prime integers such that p or q is even. Denote by p/q = [a1, a2, …, an] a positive continued fraction expansion and by p/q = [b1, b2, …, bm] the even continued fraction expansion. Let q/r = [a2, …, an]. Then $$[b\_2,\ldots,b\_m] = \left\{\begin{array}{ll} q/r &\textup{if $a\_1 $ is even;}\\ -q/(q-r) &\textup{if $a\_1 $ is odd.} \end{array}\right.$$ By Lemma [lem 0], we have p = a1q + r = b1q + s and [b2, …, bm] = q/s. If a1 is even, then a1 = b1, r = s. If a1 is odd, then b1 = a1 + 1 and s = r − q. [prop 2] Let p/q = [b1, b2, …, bm] with nonzero even integers bi. Then p is odd if and only if m is even. If m = 1 then p = b1 is even. If m = 2 then p = b1b2 + 1 is odd. Now suppose m > 2. Then the recursion relation for the convergents of the continued fraction gives [b1, b2, …, bm] = bmN[b1, b2, …, bm − 1] + N[b1, …, bm − 2]. The first summand on the right hand side is always even since bm is even. Thus the parity of N[b1, b2, …, bm] is the same as the parity of N[b1, b2, …bm − 2], and the result follows by induction. The following definition will be crucial in the whole paper. [defsignb] Let [b1, b2, …, bm] be an even continued fraction. * The sign sequence ${\mathop{sgn}}{[b\_1,b\_2,\ldots,b\_m]}$ of [b1, b2, …, bm] is the sequence $$\begin{array}{cccccccc} ( \underbrace{ {\mathop{sgn}}(b\_1),\ldots,{\mathop{sgn}}(b\_1)},& \underbrace{ -{\mathop{sgn}}(b\_2),\ldots,-{\mathop{sgn}}(b\_2)},& \ldots & \underbrace{(-1)^{m+1}{\mathop{sgn}}(b\_m),\ldots,(-1)^{m+1}{\mathop{sgn}}(b\_m))}. \\ \| b\_1\| & \|b\_2\| &\ldots&\|b\_m\| \\ \end{array}$$ Thus the first ∣b1∣ entries of ${\mathop{sgn}}{[b\_1,b\_2,\ldots,b\_m]}$ are equal to the sign of b1, the next ∣b2∣ entries are equal to the sign of − b2 and so on. * The type sequence ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}$ of [b1, b2, …, b*m]* is the sequence $$({\mathop{sgn}}(b\_1),-{\mathop{sgn}}(b\_2),\ldots,(-1)^{m+1}{\mathop{sgn}}(b\_m)).$$ For example $$\begin{array} {rclrrcl}{\mathop{sgn}}[2,4,2]&=&(+,+,\,-,-,-,-,\,+,+),&\quad&{\mathop{type}}[2,4,2]& =& (+,-,+) \\ {\mathop{sgn}}[2,2,-2,4]&=&(+,+,\,-,-,\,-,-,-,-,\,-,-), & &{\mathop{type}}[2,2,-2,4]&=&(+,-,-,-). \end{array}$$ Snake graphs ============ In this section, we review the construction of a snake graph from a positive continued fraction, and we introduce the new construction of a snake graph from an even continued fraction. We show that the two constructions are compatible in Theorem [thm 1]. Abstract snake graphs --------------------- Abstract snake graphs have been introduced and studied motivated by the snake graphs appearing in the combinatorial formulas for elements in cluster algebras of surface type in. In this section we review the definition. Throughout we fix the standard orthonormal basis of the plane. A tile G is a square in the plane whose sides are parallel or orthogonal to the elements in the fixed basis. All tiles considered will have the same side length. We consider a tile G as a graph with four vertices and four edges in the obvious way. A snake graph G is a connected planar graph consisting of a finite sequence of tiles G1, G2, …, Gd with d ≥ 1, such that Gi and Gi + 1 share exactly one edge ei and this edge is either the north edge of Gi and the south edge of Gi + 1 or the east edge of Gi and the west edge of Gi + 1, for each i = 1, …, d − 1. An example is given in Figure [signfigure]. [signfigure] The graph consisting of two vertices and one edge joining them is also considered a snake graph. The d − 1 edges e1, e2, …, ed − 1 which are contained in two tiles are called interior edges of G and the other edges are called boundary edges. We will always use the natural ordering of the set of interior edges, so that ei is the edge shared by the tiles Gi and Gi + 1. We denote by SWG the 2 element set containing the south and the west edge of the first tile of G and by GN E the 2 element set containing the north and the east edge of the last tile of G. If G is a single edge, we let SWG = ∅ and GN E = ∅. A snake graph G is called straight if all its tiles lie in one column or one row, and a snake graph is called zigzag if no three consecutive tiles are straight. We say that two snake graphs are isomorphic if they are isomorphic as graphs. A sign function f on a snake graph G is a map f from the set of edges of G to { + , − } such that on every tile in G the north and the west edge have the same sign, the south and the east edge have the same sign and the sign on the north edge is opposite to the sign on the south edge. See Figure [signfigure] for an example. Note that on every snake graph there are exactly two sign functions. A snake graph is determined up to symmetry by its sequence of tiles together with a sign function on its interior edges. Snake graphs from positive continued fractions ---------------------------------------------- In this section, we recall a construction from which associates a snake graph to a positive continued fraction. Given a snake graph G with d tiles, we consider the set of interior edges {e1, e2, …, ed − 1} and we let e0 be one of the two boundary edges in SWG, and we let ed be one of the two boundary edges in GN E. Let f be a sign function on G and consider the sign sequence (f(e0), f(e1), …, f(ed)). Let − ε ∈ { ± } be the first sign f(e0) in this sequence, and define a continued fraction [a1, a2, …, an] from the sign sequence as follows $$\label{eqsign1} \begin{array}{cccccccc} ( \underbrace{ -{\epsilon},\ldots,-{\epsilon}},& \underbrace{ {\epsilon},\ldots,{\epsilon}},& \underbrace{ -{\epsilon},\ldots,-{\epsilon}},& \ldots,& \underbrace{\pm{\epsilon},\ldots,\pm{\epsilon}}). \\ a\_1 & a\_2 & a\_3&\ldots&a\_n \end{array}$$ Thus a1 is the number of entries before the first sign change, a2 is the number of entries between the first and the second sign change and so on. See the top row of Figure [match4] for examples. Conversely, to every positive integer ai, we associate the snake graph G[ai] consisting of ai − 1 tiles and ai − 2 interior edges all of which have the same sign. Thus G[ai] is a zigzag snake graph, meaning that no three consecutive tiles lie in one row or in one column. If [a1, a2, …, an] is a positive continued fraction, its snake graph G[a1, a2, …, an] is defined as the unique snake graph with d = a1 + a2 + ⋯ + an − 1 tiles such that the signs of the d − 1 interior edges form the following sign sequence. $$\label{eqsign} \begin{array}{cccccccc} ( \underbrace{ -{\epsilon},\ldots,-{\epsilon}},& \underbrace{ {\epsilon},\ldots,{\epsilon}},& \underbrace{ -{\epsilon},\ldots,-{\epsilon}},& \ldots,& \underbrace{\pm{\epsilon},\ldots,\pm{\epsilon}}). \\ a\_1-1 & a\_2 & a\_3&\ldots&a\_n-1 \end{array}$$ Furthermore the continued fraction determines a choice of edges e0 in SWG and ed in GN E via the sign condition. For example, the snake graph of 27/10 = [2, 1, 2, 3] is shown in Figure [fig2710]. Note that between any two consecutive zigzag graphs G[ai] and G[ai + 1] there is exactly one tile that is not part of the zigzag graphs. [fig2710] Snake graphs were introduced in to describe cluster variables in cluster algebra of surface type. The cluster variable is given as a sum over all perfect matchings of the snake graph. Therefore the main interest in the construction of snake graphs from continued fractions stems from the following result. $${[a\_1,a\_2,\ldots,a\_n]}=\frac{m({\mathcal{G}}{[a\_1,a\_2,\ldots,a\_n]})}{m({\mathcal{G}}[a\_2,\ldots,a\_n])},$$ where m(G) is the number of perfect matchings of the graph G. Moreover the fraction on the right hand side is reduced. In particular, the numerator N[a1, a2, …, an] of the continued fraction is the number of perfect matchings of the snake graph G[a1, a2, …, an]. Snake graphs from even continued fractions ------------------------------------------ In this subsection, we extend the construction of to even continued fractions. Let [b1, b2, …, bm] be an even continued fraction. Again we form the zigzag graphs G[ ∣b1∣ ], …, G[ ∣bm∣ ] of the absolute values of the bi, but now we join them according to the following rule, see Figure [figgluing]. [figgluing] Without loss of generality, assume that the last tile of G[ ∣bi∣ ] is north of the second to last tile, then the first tile of G[ ∣bi + 1∣ ] is south of its second tile. If bi and bi + 1 have the same sign then we join the two graphs by drawing two horizontal edges that form a connecting tile together with the east edge of the last tile of G[ ∣bi∣ ] and the west edge of the first tile of G[ ∣bi + 1∣ ]. This case is illustrated in the left picture in Figure [figgluing]. If bi and bi + 1 have opposite signs then we glue the two graphs by identifying the north edge of the last tile of G[ ∣bi∣ ] with the south edge of the first tile of G[ ∣bi + 1∣ ]. This case is illustrated in the right picture in Figure [figgluing]. Thus if bi and bi + 1 have the same sign then the sign function on the interior edges changes from the subgraph G[ ∣bi∣ ] to G[ ∣bi + 1∣ ]. On the other hand, if bi and bi + 1 have opposite signs then the sign function stays the same from G[ ∣bi∣ ] to G[ ∣bi + 1∣ ]. The following lemma follows directly from the construction. [lem extra] * G[ − b1, − b2, …, − bm] ≅ G[b1, b2, …, bm]. * ${\mathcal{G}}[b\_1,b\_2]\cong\left\{\begin{array}{ll} {\mathcal{G}}[\,\|b\_1\|\,,\,\|b\_2\|\,] &\textup{if $b1b2>0$;}\\[8pt] {\mathcal{G}}[\,\|b\_1\|-1,1,\,\|b\_2\|-1] &\textup{if $b1b2<0$.} \end{array}\right.$ * G[2, − 2, 2, − 2, …] is a zigzag snake graph. * G[2, 2, − 2, − 2, 2, 2, − 2, − 2, …] is a straight snake graph. ### The sign sequence of a snake graph of an even continued fraction We choose our sign function f such that the south edge of the first tile e0 has the same sign as b1. Thus $f(e\_0)={\mathop{sgn}}(b\_1)$. We want to define a sequence of edges by e0ʹ = e0, e1ʹ, …, eʹβ in G[b1, b2, …, bm] that realize the sign sequence ${\mathop{sgn}}{[b\_1,b\_2,\ldots,b\_m]}$ for [b1, b2, …, bm] introduced in Definition [defsignb]; here β = ∣b1∣ + ⋯∣bm∣. The first edge in the sequence is e0. On the subgraph G[ ∣b1∣ ], we choose the interior edges eʹ1, …, eʹ∣b1∣ − 1 and the unique edge eʹ∣b1∣ ∈ G[ ∣b1∣ ]N E such that $f(e'\_{\|b\_1\|})={\mathop{sgn}}(b\_1)$. Similarly, on the subgraph G[ ∣bi∣ ], we choose the interior edges and the two unique edges in SWG[ ∣bi∣ ] and G[ ∣bi∣ ]N E whose signs are $(-1)^{i+1}{\mathop{sgn}}(b\_i).$ See Figure [fig signb] and the bottom row of Figure [match4] for examples. Thus we have the following equality of sign sequences $$\label{eqsignb} (f(e\_0'),f(e\_2'),\ldots,f(e'\_{{\beta}}))={\mathop{sgn}}{[b\_1,b\_2,\ldots,b\_m]}.$$ [fig signb] [match4] Correspondence between the two constructions -------------------------------------------- We are ready for the main result of this section. [thm 1] Let [a1, a2, …, an] = p/q be a positive continued fraction with a1 > 1, and let [b1, b2, …, bm] be an even continued fraction such that [b1, b2, …, bm] = p/q or p/(p − q). Then the following snake graphs are isomorphic G[a1, a2, …, an] ≅ G[1, a1 − 1, a2, …, an] ≅ G[b1, b2, …, bm]. In particular, the number of perfect matchings of G[b1, b2, …, bm] is equal to p. The first isomorphism follows directly from the construction of the snake graphs. Indeed the difference between the two snake graphs consists only in the choice of the boundary edge e0. To prove the second isomorphism, we proceed by induction on n. If n = 1 then G[a1] is a zigzag graph with a1 − 1 tiles. If a1 is even then b1 = a1 and we are done. If a1 is odd, then the even continued fraction cannot be equal to a1/1 and therefore we must have [b1, b2, …, bm] = a1/(a1 − 1). We have seen in Example [ex 1] that [b1, b2, …, bm] = [2, − 2, 2, − 2, ... ± 2] with m = a1 − 1. Since the signs in this continued fraction are alternating, the corresponding snake graph G[b1, b2, …, bm] is a zigzag snake graph with a1 − 1 tiles, thus G[a1] ≅ G[b1, b2, …, bm]. Suppose now that n > 1. Since G[a1, a2, …, an] ≅ G[1, a1 − 1, a2, …an] and the continued fraction [1, a1 − 1, a2, …an] = p/(p − q), we may assume without loss of generality that [a1, a2, …, an] = [b1, b2, …, bm] = p/q. Let q/r denote the value of [a2, …, an]. From Proposition [prop 1] we know that [b2, …, bm] = q/r or − q/(q − r). Thus by induction, we may assume that G[a2, …, an] ≅ G[b2, …, bm]. The graph G[a1, a2, …, an] is obtained from the subgraphs G[a1] and G[a2, …, an] as described above by inserting two horizontal edges to form the connecting tile Ga1. Since G[a1] has a1 − 1 tiles, the connecting tile Ga1 is the a1-th tile of the snake graph G[a1, a2, …, an]. If a1 is even, then a1 = b1, and then b1 and b2 have the same sign by the division algorithm. Thus G[a2, …, an] ≅ G[b2, …, bm] and the gluing with G[a1] = G[b1] which yields G[a1, a2, …, an] respectively G[b1, b2, …, bm] is the same. This shows the result if a1 is even. Suppose now that a1 is odd. Then b1 = a1 + 1 > 0 and b2 < 0. Now, G[b1] has one more tile than G[a1], however, since b1 and b2 have opposite signs, the procedure by which we glue G[b1] with G[b2, …, bm] yields the same snake graph as the gluing of G[a1] with G[a2, …, an]. Thus G[a1, a2, …, an] is isomorphic to G[b1, b2, …, bm]. [cor 3.3] The number of tiles of G[b1, b2, …, bm] is equal to $$\sum\_{i=1}^m\|b\_i\| - 1- \textup{(number of sign changes in $b\_1,b\_2,\ldots,b\_m$)}.$$ If m = 1 then G[b1] has b1 − 1 tiles. For m > 1, we have seen that G[b1, b2, …, bm] is obtained by gluing G[b1] to G[b2, …, bm]. By induction we may assume that the number of tiles in G[b2, …, bm] is ∑i = 2m∣bi∣ − 1 − (number of sign changes in b2, …, bm), and gluing G[b1] to it will increase this number by b1, if b1 and b2 have the same sign, and by b1 − 1, if the signs of b1 and b2 are opposite. This completes the proof. Theorem [thm 1] and its corollary also hold more generally for continued fractions with entries bi ∈ Z \ {0, − 1}, but we will not need this here. Two-bridge knots from continued fractions ========================================= In this section, we briefly review two-bridge knots. These knots have been studied first by Schubert in 1956. For a detailed introduction see for example. A knot is a subset of R3 that is homeomorphic to a circle. A link with r* components* is a subset of R3 that is homemorphic to a disjoint union of r circles. Thus a knot is a link with one component. A knot is alternating if the crossings alternate between over and under when traveling along a strand. To every rational number p/q ≥ 1 one can associate a link C(p/q) or C[a1, a2, …, an] using the positive continued fraction expansion p/q = [a1, a2, …, an]. 27/10 = [2, 1, 2, 3] corresponds to the knot shown on the left in Figure [knot2123]. [knot2123] The link consists of n pieces, each of which is a 2-strand braid with ai crossings, where i = 1, 2, …, n. These pieces are joined in such a way that the link is alternating, see Figure [knotschema]. [knotschema] It is well known that the link C(p/q) is a knot if p is odd and it is a link with exactly two connected components if p is even. Moreover, in the case where p is odd, the two knots C(p/q) and C(p/(p − q)) are isotopic. Indeed this follows simply from the fact that the continued fractions are related as follows p/q = [a1, a2, …, an] and p/(p − q) = [1, a1 − 1, a2, …, an]. Thus when we are considering the knot or link C[a1, a2, …, an], we may always assume that a1 > 1. On the other hand, by Proposition [prop 01], at least one of the rational numbers p/q and p/(p − q) can be represented by an even continued fraction [b1, b2, …, bm]. We can also construct a link C[b1, b2, …, bm] from this even continued fraction, essentially in the same way, except that a sign change now means that the diagram is not alternating at that point. See for example the knot C[2, 2, − 2, 4] in Figure [knot2123]. It is known that if p/q is equal to the positive continued fraction [a1, a2, …, an] and to the even continued fraction [b1, b2, …, bm] then the corresponding links C[a1, a2, …, an] and C[b1, b2, …, bm] are isotopic. See for example. Orientation ----------- One can orient the strand of a knot in one of two ways. In a link, one can orient each conponente in one of two ways. Fixing an orientation of the strands induces a sign on each crossing according to the following cases. $$\begin{array} {cccc} \scalebox{0.6}{} & \qquad \textup{and}\qquad& \scalebox{0.6}{} \\ \textup{positive}&& \textup{negative} \end{array}$$ We shall always use the following convention for the orientations of the strands. If [b1, b2, …, bm] is an even continued fraction then the first crossing in the b1-braid is if b1 > 0 and it is if b1 < 0. In particular, the sign of the first crossing is equal to the sign of b1. The ∣bi∣ crossings in the i-th braid all have the same sign. We call the braid positive if these signs are + and negative if they are − . [lem 3] With the conventions above, if [b1, b2, …, bm] is an even continued fraction then the last crossing is of the following form. $$\begin{array}{clccl} \scalebox{0.4}{} &\textup{if $m$ is even and $b\_m>0$;} &\qquad& \scalebox{0.4}{} &\textup{if $m$ is odd and $b\_m<0$};\\ \scalebox{0.4}{} &\textup{if $m$ is even and $b\_m<0$;}& & \scalebox{0.4}{} &\textup{if $m$ is odd and $b\_m>0$}. \end{array}$$ Using our schematic illustration of the link, we see that the strand that enters the bi-braid from southwest (respectively southeast) must leave the bi-braid towards southeast (respectively southwest), because bi is even. Similarly the strand entering the bi-braid from the northeast (respectively northwest) will exit the braid to the northwest (respectively northeast). If m is even, the orientation is as shown in Figure [orientation]. [orientation] Thus if m is even then, since bm is even, the last crossing is if bm > 0 and if bm < 0. The odd case is proved in a similar way. [cor 2] The signs of the crossings in the bi-braid are $(-1)^{i+1}{\mathop{sgn}}(b\_i)$. In particular, * the sign sequence ${\mathop{sgn}}{[b\_1,b\_2,\ldots,b\_m]}$ of the even continued fraction is equal to the sequence of the signs of the crossings in the link diagram C[b1, b2, …, bm]. * the type sequence ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}$ of the even continued fraction is equal to the sequence of the signs of the braids in C[b1, b2, …, bm]. This follows directly from Lemma [lem 3]. Jones polynomial ================ The Jones polynomial of an oriented link is an important invariant. For 2-bridge links, the Jones polynomial has been computed in. For general facts about the Jones polynomial see for example the book by Lickorish. The Jones polynomial V(L) of an oriented link L can be defined recursively as follows. The Jones polynomial of the unknot is 1, and whenever three oriented links L−, L+ and L0 are the same except in the neighborhood of a point where they are as shown in Figure [crossing5] then V(L−) = t− 2 V(L+) + ε V(L0), where ε = t− 1(t− 1/2 − t1/2) = t− 3/2 − t− 1/2. [crossing5] Equivalently, V(L+) = t2 V(L+) + ε̄ V(L0), where ε̄ = t2( − ε) = t(t1/2 − t− 1/2). Usually the defining identity is stated as t− 1V(L+) − t V(L−) + (t− 1/2 − t1/2)V(L0) = 0. The Jones polynomial V ∘ ∘ of two disjoint copies of the unknot can be computed using the defining relation as shown on the top left of Figure [fig Jonex]. Since L− and L+ are both the unknot we see that V ∘ ∘ = (1 − t− 2)/ε = − t− 1/2 − t1/2. The Jones polynomial of the Hopf link is computed in the top right of Figure [fig Jonex]. We have $$V\_{\textup{Hopf link}}=t^{-2} (-t^{-{1}/{2}}-t^{{1}/{2}})+{\epsilon}=-t^{-5/2}-t^{-1/2}.$$ The Jones polynomial of the trefoil knot is computed in the bottom left of Figure [fig Jonex]. We have $$V\_{\textup{trefoil}} =t^{-2} +{\epsilon}V\_{\textup{Hopf link}}=t^{-2} +( t^{-3/2}-t^{-1/2})(-t^{-5/2}-t^{-1/2})=-t^{-4}+t^{-3}+t^{-1}.$$ [fig Jonex] The Jones polynomial is a Laurent polynomial in t1/2. Let $\overline{(\ )}\colon \mathbb{Z}[t^{1/2},t^{-1/2}]\to \mathbb{Z}[t^{1/2},t^{-1/2}]$ be the algebra automorphism of order two that sends t1/2 to t− 1/2. This is consistent with our definition of ε̄. If $\overline L$ is the mirror image of the link L then $V(\overline{L})=\overline{V(L)}$. If L is a 2-bridge link, L = C[b1, b2, …, bm] = C(p/q) with [b1, b2, …, bm] an even continued fraction, then the following are equivalent $$\textup{$L$ is a knot} \ \Longleftrightarrow\ \textup{$p$ is odd}\ \Longleftrightarrow\ \textup{$m$ is even}\ \Longleftrightarrow\ V(L)\in \mathbb{Z}[t,t^{-1}].$$ If L is not a knot then it is a 2-component link and V(L) ∈ t1/2Z[t, t− 1]. If L is a knot and Lʹ is the same knot with reversed orientation (running through L in the opposite direction) then V(L) = V(Lʹ). Thus the Jones polynomial of a knot does not depend on the orientation of the knot. If L is a link with components L1, L2 and Lʹ is obtained from L by reversing the orientation of one component, then V(Lʹ) = t− 3 lk(L1, L2)V(L), where $lk(L\_1,L\_2) = 1/2 (\textup{sum of signs of crossings between $L1andL2$})$, see. [ex 4] Let L = C[4] with first crossing and let Lʹ be the same link with first crossing. Then V(L) = − t1/2 + t3/2 − t5/2 − t9/2 and V(Lʹ) = − t− 11/2 + t− 9/2 − t− 7/2 − t− 3/2. On the other hand, the number lk(L1, L2) is $\frac{1}{2}(-4)=-2$, and thus V(Lʹ) = t− 6 V(L). It is important to keep this small subtlety in mind when working with data bases, especially if it is not immediately obvious which conventions were used for the orientations of the link components. In this paper, the orientation is fixed. The Jones polynomial of a 2-bridge knot --------------------------------------- In this subsection, we compute recursion formulas for the Jones polynomials of 2-bridge knots and links. We also give a direct formula for the degree and compare the signs of the leading coefficients. Recall that because of our conventions (see section [sect knots]), the orientation of the links is fixed and therefore the Jones polynomials are well defined. In what follows, we will work with recursive formulas for the Jones polynomials, and we will use the conventions that the expressions C[b1, b2, …, b0] and C[ ] both denote the unknot and hence V[b1, b2, …, b0] = V[ ] = 1, and the expressions C[b1, b2, …, b− 1] and C[0] both denote two disjoint copies of the unknot and hence V[b1, b2, …, b− 1] = V[0] = − t− 1/2 − t1/2. Recall the definition of the type sequence ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}$ in Definition [defsignb]. We will use the notation ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}=(\ldots,-)$ to indicate that the last entry of the type sequence is a minus sign. [lem 4] Let [a1, a2, …, an] = p/q be a positive continued fraction and [b1, b2, …, bm] = p/q or p/(p − q) an even continued fraction and suppose that m ≥ 1. So the two 2-bridge links C[a1, a2, …, an] and C[b1, b2, …, bm] are isotopic. Let V[a1, a2, …, an] or V[b1, b2, …, bm] denote their Jones polynomials. Thus V[a1, a2, …, an] = V[b1, b2, …, bm]. Then $$V\_{{[b\_1,b\_2,\ldots,b\_m]}}=\left\{\begin{array}{ll} t^{-2}\,V\_{[b\_1,b\_2,\ldots, \,{\mathop{sgn}}(b\_{m})(\|b\_{m}\|-2)\,]} +{\epsilon}\,V\_{[b\_1,b\_2,\ldots,b\_{m-1}]} &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m}]=(\ldots,-)$};\\[8pt] t^{\,2}\ V\_{[b\_1,b\_2,\ldots,{ \,{\mathop{sgn}}(b\_{m})(\|b\_{m}\|-2)\,}]} +\overline{{\epsilon}}\,V\_{[b\_1,b\_2,\ldots,b\_{m-1}]}&\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m}]=(\ldots,+)$.} \end{array}\right.$$ In particular, $$V\_{[b\_1]}=\left\{\begin{array}{ll} t^{-2}\,V\_{[b\_1+2]}+{\epsilon}&\textup{ if $b\_1<0$;}\\[8pt] t^{\,2}\ V\_{[b\_1-2]}+\overline{\epsilon}&\textup{ if $b\_1>0$.} \end{array}\right.$$ In type (…, − ), the sign of the last crossing in C[b1, b2, …, bm] is negative. Therefore the relation V(L−) = t− 2 V(L+) + ε V(L0) applied to the last crossing yields the identity $$V\_{{[b\_1,b\_2,\ldots,b\_m]}} = t^{-2} \, V\_{[b\_1,b\_2,\ldots,{ \,\textup{sign}(b\_{m})(\|b\_{m}\|-2)\,}]} + {\epsilon}\,V\_ {[b\_1,b\_2,\ldots,b\_{m-1}]}.$$ Indeed, by Lemma [lem 3] the last crossing is or. Therefore after the smoothing of this crossing, which leads to L0, the last braid becomes trivial, see Figure [smoothing]. [smoothing] In type (…, + ), the last crossing has positive sign and we use the analogous argument with the smoothing relation $V(L\_+)= t^{2}\,V(L\_-) +\overline{\epsilon}\,V(L\_0)$. Applying the lemma $\frac{1}{2}\|b\_m\|$ times, we obtain the recursive formula of the following theorem. In its statement, we will use the notation [b]q = 1 + q + q2 + ⋯ + qb − 1 = (1 − qb)/(1 − q) for the q-analogue of a positive integer b. Setting q = − t− 1 and $\overline{q}=-t$, this gives the short hands $$[b]\_q=1-t^{-1}+t^{-2} -t^{-3}+\cdots \pm t^{-b+1} \qquad\textup{ and }\qquad [b]\_{\overline{q}}=1-t^{1}+t^{2} -t^{3}+\cdots \pm t^{b-1}.$$ [thm 5] Let [b1, b2, …, bm] be an even continued fraction with m ≥ 1. Then $$V\_{{[b\_1,b\_2,\ldots,b\_m]}}= \left\{ \begin{array} {ll} t^{-\|b\_m\|}\,V\_{[b\_1,b\_2,\ldots,b\_{m-2}]} -t^{-\frac{1}{2}}[\,\|b\_m\|\,]\_q\,V\_{[b\_1,b\_2,\ldots,b\_{m-1}]} &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m}]=(\ldots,-)$:}\\[8pt] t^{\|b\_m\|}\ V\_{[b\_1,b\_2,\ldots,b\_{m-2}]} -\ t^{\frac{1}{2}}\ [\,\|b\_m\|\,]\_{\overline {q}}\,V\_{[b\_1,b\_2,\ldots,b\_{m-1}]} &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m}]=(\ldots,+)$.}\\ \end{array}\right.$$ In particular, $$V\_{[b\_1]}= \left\{ \begin{array} {ll} t^{b\_1}(-t^{-1/2}-t^{1/2})-t^{-\frac{1}{2}}\,[-b\_1]\_q, &\textup{if $b\_1<0$;}\\ t^{b\_1}(-t^{-1/2}-t^{1/2})-t^{\frac{1}{2}}\,[b\_1]\_{\overline q} &\textup{if $b\_1>0$.}\\ \end{array}\right.$$ This follows simply by applying Lemma [lem 4] exactly bm/2 times and using the facts that $${\epsilon}(1+t^{-2}+t^{-4}+\cdots+t^{-\|b\_m\| +2}) = -t^{-\frac{1}{2}}[\,\|b\_m\|\,]\_q,$$ $$\overline{{\epsilon}}(1+t^{2}+t^{4}+\cdots+t^{\|b\_m\| -2}) = -t^{\frac{1}{2}}[\,\|b\_m\|\,]\_{\overline{q}},$$ and C[b1, b2, …, bm − 1, 0] = C[b1, b2, …, bm − 2]. For the case where m = 1, we also need to observe that C[0] is a disjoint union of two unknots, and thus V[0] = − t− 1/2 − t1/2, and C[ ] is the unknot, hence V[ ] = 1 In the remainder of the section, we compute the degree and the sign of the leading term of the Jones polynomial. The degree of a Laurent polynomial in t1/2 is the highest exponent of t that appears, and the leading term is the term that realizes the degree. For example, the degree of t− 3/2 − t− 1/2 is − 1/2 and the leading term is − t− 1/2. For simplicity, we shall use the following notation for i = 0, 1, 2, …, m − 1. $$\begin{array} {cccccc} V\_i=V\_{[b\_1,b\_2,\ldots,b\_{m-i}]} & \quad & j\_i= \deg V\_i& \quad & {\delta}\_i t^{j\_i}= \textup{leading term of } V\_i. \end{array}$$ [cor 6] With the notation above $$j\_0 \, \le \left\{ \begin{array} {ll} \max (j\_2 -\|b\_m\| \, \ j\_1 -\textstyle\frac{1}{2} ) &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m}]=(\ldots,-)$; }\\ \\ \max (j\_2 +\|b\_m \| \, \ j\_1 -\textstyle\frac{1}{2}+\|b\_m\|) &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m}]=(\ldots,+)$.} \end{array}\right.$$ with equality if $j\_2 -\|b\_m\| \ne j\_1 -\textstyle\frac{1}{2}$, respectively $j\_2 +\|b\_m \| \ne j\_1 -\textstyle\frac{1}{2}+\|b\_m\|$. This follows directly from Theorem [thm 5] and the fact that deg[b]q = 1 and $\deg [b]\_{\overline {q}}=b-1$, for b > 0. [cor 7] The degrees j0, j1 and j2 compare as follows accoding to the type of [b1, b2, …, bm]. $$\begin{array} {rcl} j\_0 &=&j\_1+ \left\{ \begin{array} {ll} -\frac{1}{2} &\textup{in type }(\ldots,-);\\[8pt] \|b\_m\|+\frac{1}{2}\qquad&\textup{in type }(\ldots,-,+);\\[8pt] \|b\_m\|-\frac{1}{2}&\textup{in type }(\ldots,+,+);\\[8pt] \end{array} \right.\\ \\ j\_0&=&j\_2+\left\{ \begin{array} {ll} -1 &\textup{in type }(\ldots,-,-);\\[8pt] \|b\_{m-1}\| &\textup{in type }(\ldots,-,+,-);\\[8pt] \|b\_{m-1}\| - 1 &\textup{in type }(\ldots,+,+,-);\\[8pt] \|b\_{m}\| &\textup{in type }(\ldots,-,+);\\[8pt] \|b\_{m}\| + \|b\_{m-1}\| &\textup{in type }(\dots,-,+,+);\\[8pt] \|b\_{m}\| + \|b\_{m-1}\| -1 &\textup{in type }(\dots,+,+,+).\\[8pt] \end{array} \right. \end{array}$$ We use induction on m. If m = 1, we have $j\_1=\deg V\_{\textup{unknot}}=0$ and j2 = deg( − t− 1/2 − t1/2) = 1/2. On the other hand, Theorem [thm 5] implies $$j\_0=\left\{ \begin{array} {ll} -\frac{1}{2} &\textup{if $b\_1<0$};\\ b\_1+\frac{1}{2} &\textup{if $b\_1>0$}. \end{array}\right.$$ Therefore $$j\_0=\left\{ \begin{array} {ll} j\_1-\frac{1}{2} &\textup{if $b\_1<0$};\\ j\_1+b\_1+\frac{1}{2} &\textup{if $b\_1>0$}, \end{array}\right. \qquad \textup{and} \qquad j\_0=\left\{ \begin{array} {ll} j\_2-1 &\textup{if $b\_1<0$};\\ j\_2+b\_1 &\textup{if $b\_1>0$}. \end{array}\right.$$ This shows the result for m = 1. Suppose now m > 1. By induction, we have $$\label{eq 58} j\_1=j\_2+\left\{ \begin{array} {ll} -\frac{1}{2} &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m-1}]=(\ldots,-)$};\\[5pt] \|b\_{m-1}\|+\frac{1}{2} &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m-1}]=(\ldots,-,+)$};\\[5pt] \|b\_{m-1}\|-\frac{1}{2} &\textup{if ${\mathop{type}}[b\_1,\ldots,b\_{m-1}]=(\ldots,+,+)$}.\\ \end{array}\right.$$ In particular, j1 ≥ j2 − 1/2, whence j1 − 1/2 > j2 − ∣bm∣, since bm is a nonzero even integer. Therefore Corollary [cor 6] implies that, if ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}=(\ldots,-)$, then j0 = j1 − 1/2, and the three cases in equation ([eq 58]) prove the three cases of statement where the type ends in a minus sign. Next suppose that ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}=(\ldots,+,+)$. In this case, equation ([eq 58]) implies that j1 = j2 + ∣bm − 1∣ ± 1/2 > j2 + 1. Then Corollary [cor 6] yields $j\_0=j\_1-1/2+\|b\_m\|=j\_2+\|b\_{m}\|+\|b\_{m-1}]-\frac{1}{2}\pm\frac{1}{2}.$ Finally, suppose that ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}=(\ldots,-,+)$. Then equation ([eq 58]) implies that j1 = j2 − 1/2 and thus j1 − 1/2 < j2. In this case, Corollary [cor 6] yields j0 = j2 + ∣bm∣ = j1 + ∣bm∣ + 1/2. Corollary [cor 7] allows us to determine which of the two polynomials on the right hand side of the equations in Theorem [thm 5] contains the leading term. Indeed, if [b1, b2, …, bm] is of type (…, − ), the equation is $$V\_0= t^{-\|b\_m\|}\,V\_2 -t^{-\frac{1}{2}}[\,\|b\_m\|\,]\_q\,V\_1.$$ The leading term of the first polynomial on the right hand side has sign δ2 and its degree j2 − ∣bm∣ is strictly smaller than j2 − 1 ≤ j0, by Corollary [cor 7]. The leading term of the second polynomial has sign − δ1 and its degree is $j\_1-\frac{1}{2}=j\_0$. Therefore the leading term of V0 is ${\delta}\,t^{j\_0}=-{\delta}\_1\,t^{j\_1-\frac{1}{2}}$. On the other hand, if [b1, b2, …, bm] is of type (…, + ), the equation in Theorem [thm 5] is $$V\_0= t^{\|b\_m\|}\,V\_2 -t^{\frac{1}{2}}[\,\|b\_m\|\,]\_{\overline q}\,V\_1.$$ The first term on the right hand side has sign δ2 and its degree j2 + ∣bm∣ is equal to j0 in type (…, − , + ). The second term has sign δ1 (since bm is even) and its degree is $j\_1+ \|b\_m\| -\frac{1}{2}$ which is equal to j0 in type (…, + , + ). This leads us to the following corollary. [cor 8] The leading terms satisfy $$\label{eq 59} {\delta}\_0\,t^{j\_0}=\left\{\begin{array}{ll} -{\delta}\_1\,t^{j\_1-\frac{1}{2}} &\textup{ in type $(\ldots,-)$;}\\[5pt] {\delta}\_2\,t^{j\_2+\|b\_m\|} &\textup{ in type $(\ldots,-,+)$;}\\[5pt] {\delta}\_1\,t^{j\_1+\|b\_m\|-\frac{1}{2} } &\textup{ in type $(\ldots,+,+)$.}\end{array}\right.$$ Moreover, the coefficients δ0, δ1 and δ2 compare according to the type of [b1, b2, …, bm] as follows. $${\delta}\_0=\left\{\begin{array}{ll} -{\delta}\_1 &\textup{ in type $(\ldots,-)$;}\\ -{\delta}\_1 &\textup{ in type $(\ldots,-,+)$;}\\ {\delta}\_1 &\textup{ in type $(\ldots,+,+)$;}\\ \end{array}\right.\qquad \textup{and}\qquad {\delta}\_0=\left\{\begin{array}{ll} {\delta}\_2 &\textup{ in type $(\ldots,-,-)$;}\\ {\delta}\_2 &\textup{ in type $(\ldots,-,+,-)$;}\\ -{\delta}\_2 &\textup{ in type $(\ldots,+,+,-)$;}\\ {\delta}\_2 &\textup{ in type $(\ldots,-,+)$;}\\ -{\delta}\_2 &\textup{ in type $(\ldots,-,+,+)$;}\\ {\delta}\_2 &\textup{ in type $(\ldots,+,+,+)$.}\\ \end{array}\right.$$ It only remains to show the last equations comparing δ0 with δ1 and δ2. Two of the three equations comparing δ0 and δ1 follow directly from equation ([eq 59]). In the remaining equation, the type is (…, − , + ) and equation ([eq 59]) yields δ0 = δ2 as well as δ1 = − δ2, because ${\mathop{type}}[b\_1,\ldots,b\_{m-1}]=(\ldots,-)$. Thus δ0 = − δ1. Now consider the equations comparing δ0 and δ2. In type (…, − , − ) we have δ0 = − δ1 and δ1 = − δ2, thus δ0 = δ2. In type (…, + , − ) we have δ0 = − δ1, wheras δ1 = − δ2 in type (…, − , + , − ) and δ1 = δ2 in type (…, + , + , − ). Thus δ0 = δ2 in the former case and δ0 = − δ2 in the latter. In type (…, − , + ) we have δ0 = − δ1 and δ1 = − δ2, thus δ0 = δ2. In type (…, + , + ) we have δ0 = δ1, wheras δ1 = − δ2 in type (…, − , + , + ) and δ1 = δ2 in type (…, + , + , + ). Thus δ0 = − δ2 in the former case and δ0 = δ2 in the latter. This completes the proof. We close this section with a direct formula for the degree of the Jones polynomial. [thm degree] Let [b1, b2, …, bm] be an even continued fraction. Then the Jones polynomial V[b1, b2, …, bm] of the associated 2-bridge knot has degree $$\sum\_{i=1}^m \max\left( (-1)^{i+1} b\_i + \frac{\textup{sign}(b\_ib\_{i-1})}{2}\,\ -\frac{1}{2}\right),$$ where we use the convention that $\textup{sign}(b\_0)=1$. Moreover, the sign of its leading term is equal to ( − 1)m − τ, where τ is the number of times the subsequence + , + appears in the type sequence of [b1, b2, …, bm]. If m = 1, then Theorem [thm 5] implies that the leading term of Vb1 is $$\left\{ \begin{array} {ll}-t^{ b\_1+\frac{1}{2}} &\textup{if $b\_1>0$;}\\ -t^{-\frac{1}{2}} &\textup{if $b\_1<0$,} \end{array}\right.$$ so the degree is equal to $ \max( b\_1 + \frac{\textup{sign}(b\_1)}{2}\,\ -\frac{1}{2})$ and the sign is equal to ( − 1) = ( − 1)m. Now suppose that m > 1. Corollary [cor 7] implies $$\deg V\_{{[b\_1,b\_2,\ldots,b\_m]}}=j\_0=j\_1 +\left\{ \begin{array} {ll} -\frac{1}{2} &\textup{in type $(\ldots,-)$;}\\ \|b\_m\| +\frac{\textup{sign}(b\_m b\_{m-1})}{2} &\textup{in type $(\ldots,+)$.}\\ \end{array}\right.$$ By induction, we may assume that j1 is equal to the sum of the first m − 1 terms in the theorem. Thus we must show that the m-th term satisfies $$\max\left( (-1)^{m+1} b\_m + \frac{\textup{sign}(b\_mb\_{m-1})}{2}\,\ -\frac{1}{2}\right) = \left\{ \begin{array} {ll} -\frac{1}{2} &\textup{in type $(\ldots,-)$;}\\ \|b\_m\|+\frac{\textup{sign}(b\_m b\_{m-1})}{2} &\textup{in type $(\ldots,+)$.}\\ \end{array}\right.$$ Recall that ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}=(\ldots,-)$ if m is even and bm > 0, or if m is odd and bm < 0. Therefore the maximum on the left hand side is equal to − 1/2 in this case. On the other hand, if ${\mathop{type}}{[b\_1,b\_2,\ldots,b\_m]}=(\ldots,+)$, then the maximum equals $\|b\_m\| +\frac{\textup{sign}(b\_m b\_{m-1})}{2} $. This proves the statement about the degree. To determine the sign, we use Corollary [cor 8], which shows how the sign changes in terms of the entries at positions 2, 3, …, m in the type sequence of [b1, b2, …, bm]. Namely, the sign changes for every − sign in these positions of the type sequence and for each + sign that is a direct successor of a − sign. In other words, the number of sign changes is precisely m − 1 − τ. Now the result follows since the sign is − in the case m = 1. For each i, the maximum in the theorem is equal to − 1/2 if the crossings in the bi braid are negative and it is ∣bi∣ ± 1/2 if the crossings are positive. Thus we can express the degree in terms of the crossings of the link as follows $$\begin{array}{rcl}\deg V\_{{[b\_1,b\_2,\ldots,b\_m]}}&=&-\frac{1}{2}\#\,\textup{negative braids\,} +\#\,\textup{positive crossings} \\[8pt] &&+\frac{1}{2} \#\,\textup{consecutive pairs of braids with sign } (+,-)\\[8pt] &&-\frac{1}{2}\#\,\textup{consecutive pairs of braids with sign }(+,+). \end{array}$$ For the continued fraction [2, 2, − 2, 4], the formula of the theorem gives $$\deg V\_{[2,2,-2,4]}=\left(2+\frac{1}{2}\right)-\frac{1}{2}-\frac{1}{2}-\frac{1}{2}=1,$$ and the formula from the remark gives $$\deg V\_{[2,2,-2,4]}=-\frac{3}{2}+2+\frac{1}{2}=1.$$ The type sequence of [2, 2, − 2, 4] is ( + , − , − , − ), hence τ = 0 and thus the sign of the leading term is ( − 1)4 = + 1. The Jones polynomial is V[2, 2, − 2, 4] = t − 2 + 4t− 1 − 4t− 2 + 5t− 3 − 5t− 4 + 3t− 5 − 2t− 6 + t− 7. Cluster algebras and specialized F-polynomials ================================================ Cluster algebras ---------------- We recall a few facts about cluster algebras with principal coefficients. For more detailed information, we refer to the original paper or the lecture notes. Let Q be a quiver without loops and oriented 2-cycles, and let N denote the number of vertices of Q. Let ZP denote the ring of Laurent polynomials in variables y1, y2, …, yN and let QP be its field of fractions. The cluster algebra A(Q) of Q with principal coefficients is a ZP-subalgebra of the field of rational functions QP(x1, x2, …, xN). To define the cluster algebra one constructs a set of generators, the cluster variables, by a recursive method called mutation. It is known that the cluster variables are elements of the ring Z[x1± 1, x2± 1, …, xN± 1, y1, y2, …, yN] with positive coefficients. The *F-polynomial* is the polynomial in Z[y1, y2, …, yN] obtained from the cluster variable by setting all xi equal to 1. If the quiver Q is the adjacency quiver of a triangulation of a surface with marked points then the cluster algebra is said to be of surface type, see. In this case, each cluster variable is given as a sum over all perfect matchings of the weighted snake graph associated to the cluster variable. In another formula was given, that computes the cluster variables as continued fractions of Laurent polynomials. Let Q be the quiver $\xymatrix{1&\ar[l]2}$. Then the ‘largest’ cluster variable in A(Q) is equal to (x2 + y1 + x1y1y2)/x1x2 and its F-polynomial is 1 + y1 + y1y2. Let Q be the quiver $\xymatrix{1&\ar[l]2\ar[r]&3}$. Then the ‘largest’ cluster variable in A(Q) is equal to (x22 + x2y1 + x2y3 + y1y3 + x1x3y1y2y3)/x1x2x3 and its F-polynomial is 1 + y1 + y3 + y1y3 + y1y2y3. Specialized F-polynomials --------------------------- We shall show that the Jones polynomial of a 2-bridge link is equal (up to normalization by its leading term) to the specialization of a corresponding cluster variable at xi = 1, y1 = t− 2 and yi = − t− 1 if i ≠ 1. To make this statement precise, we need to fix our notation. Let [a1, a2, …, an] be a positive continued fraction with a1 ≥ 2. As we have seen in section [sect knots], this is not a restriction from the point of view of 2-bridge knots. Let G[a1, a2, …, an] be the snake graph of the continued fraction, let d = a1 + a2 + ⋯ + an − 1 be the number of tiles of this graph and label the tiles 1, 2, …, d. Let A be any cluster algebra with principal coefficients in which we can realize the snake graph G[a1, a2, …, an] as the snake graph of a cluster variable, with the sole condition that the first tile of the snake graph corresponds to the initial cluster coefficient y1 and no other tile of the graph corresponds to the same coefficient y1. Denote the initial seed (with principal coefficients) by ((x1, x2, …, xN), (y1, y2, …, yN), Q) where (x1, x2, …, xN) is the initial cluster, (y1, y2, …, yN) is the initial coefficient tuple and Q is the initial quiver. For example, one can choose A to be of type Ad and let the i-th tile correspond to the i-th coefficient and choose the initial seed such that its quiver Q is the acyclic quiver corresponding to the snake graph G[a1, a2, …, an]. That means that Q is of the form $$\xymatrix@C15pt{1&\ar[l]2&\ar[l]\cdots&\ar[l]\ell\_1\ar[r]&\ell\_1+1\ar[r]&\cdots\ar[r]&\ell\_2&\ell\_2+1\ar[l]&\ar[l]\cdots&\ar[l]\ell\_3\ar[r]&\cdots}$$ where ℓi = a1 + a2 + ⋯ai. In this cluster algebra, the cluster variable corresponding to the continued fraction [a1, a2, …, an] is the one whose denominator is the product of all initial cluster variables x1x2…xd, or equivalently, the cluster variable that, under the Caldero-Chapoton map, corresponds to the largest indecomposable representation of Q, the one with dimension 1 at every vertex. In the chosen cluster algebra A, let x[a1, a2, …, an] denote the cluster variable whose snake graph is G[a1, a2, …, an] and let F[a1, a2, …, an] be its F-polynomial. Recall from that F[a1, a2, …, an] is obtained from x[a1, a2, …, an] by setting all initial cluster variables x1, x2, …xN equal to 1. It has been shown in that x[a1, a2, …, an] and F[a1, a2, …, an] can be written as the numerator of a continued fraction of Laurent polynomials. We will use a specialization of the F-polynomial by setting y1 = q2 and yi = q, for all i = 2, 3, …, N, where q = − t− 1. In other words $$y\_1=t^{-2} \qquad\textup{ and } \qquad y\_2=y\_3=\ldots=y\_N=-t^{-1}.$$ We denote this specialization by Fa1, a2, …, an. Thus $${F\_{a\_1,a\_2,\ldots,a\_n}}=F{[a\_1,a\_2,\ldots,a\_n]}\Big\|\_{y\_1=t^{-2};\ y\_i=-t^{-1},\ i>1}.$$ We do not yet know an intrinsic reason why y1 is different from y2, …, yN. Using the main result of, we have the following formula. Recall that [b]q = 1 + q + q2 + ⋯ + qb − 1 and ℓi = a1 + a2 + ⋯ai. Also recall that the notation N[L1, L2, …, Ln] is defined recursively in equation ([def N]), where N[L1] = L1. [prop F] * If n is odd, the specialized F-polynomial Fa1, a2, …, an is equal to $$N\Big[\, [a\_1+1]\_q -q\,\ [a\_2]\_q \,q^{-\ell\_2},\ [a\_3]\_q\,q^{\ell\_2+1},\ [a\_4]\_q\, q^{-\ell\_4},\ldots, [a\_{2i}]\_q\, q^{-\ell\_{2i}},\ [a\_{2i+1}]\_q\, q^{\ell\_{2i}+1},\ldots, \ [a\_{n}]\_q\, q^{\ell\_{n-1}+1}\Big]$$ * If n is even, Fa1, a2, …, an is equal to the result in (a) multiplied by qℓn. In her formula, Rabideau uses the following notation. $$\begin{array} {rclcrcl} C\_i&=& \left\{ \begin{array}{ll} \displaystyle \prod\_{j=1}^{\ell\_{i-1}} y\_j & \textup{if $ i $ is odd},\\ \\ \displaystyle \prod\_{j=1}^{(\ell\_i)-1}y\_j^{-1} & \textup{if $i$ is even,}\ \end{array} \right. &\qquad& \varphi\_i&=& \left\{\begin{array}{ll} \displaystyle \sum\_{k=\ell\_{i-1}}^{(\ell\_i) -1}\ \prod\_{j=(\ell\_{i-1})+1}^{k} y\_j & \textup{if $i$ is odd},\\ \\ \displaystyle \sum\_{k=(\ell\_{i-1})+1}^{\ell\_i}\ \prod\_{j=k}^{(\ell\_i)-1} y\_j & \textup{if $i$ is even,} \end{array} \right. \end{array}$$ and Li = Ciφi. With this notation $$F{[a\_1,a\_2,\ldots,a\_n]}=\left\{ \begin{array} {ll} N[{\mathcal{L}}\_1,{\mathcal{L}}\_2,\ldots,{\mathcal{L}}\_n] & \textup{if $n $ is odd},\\ \\ C\_n^{-1}N[{\mathcal{L}}\_1,{\mathcal{L}}\_2,\ldots,{\mathcal{L}}\_n] & \textup{if $n $ is even}. \end{array}\right.$$ Under our specialization y1 ↦ q2 and yi ↦ q (i > 1), the quantities above transform as follows. $$\begin{array} {rclcrcl} C\_i &\mapsto& \left\{ \begin{array}{ll} q^{\ell\_{i-1}+1} & \textup{if $i $ is odd};\\ \\ q^{-\ell\_{i}} & \textup{if $i$ is even;} \end{array} \right. \\ \\ \displaystyle\varphi\_1=\sum\_{k=0}^{a\_1 -1}\ \prod\_{j=1}^{k} y\_j & \mapsto& 1+q^2+q^3+\cdots +q^{a\_1}\ =\ [a\_1+1]\_q-q;\\ \varphi\_i&\mapsto & \displaystyle \sum\_{k=0}^{\ell\_i-\ell\_{i-1} -1}\ q^k \ =\ [a\_i]\_q & (i>1). \end{array}$$ Thus $${\mathcal{L}}\_1\mapsto [a\_1+1]\_q-q \quad \textup { and } \quad {\mathcal{L}}\_i \mapsto \left\{ \begin{array}{ll} [a\_i]\_q \,q^{-\ell\_i} & \textup{if $i $ is odd};\\ \\ {} [a\_i]\_q \,q^{\ell\_{i-1}+1} & \textup{if $i$ is even,} \end{array} \right.$$ and the proof is complete. If the ai are positive integers then N[a1, a2, …, an] is the numerator of the continued fraction, or to be more precise, the positve numerator, since numerators are only defined up to multiplication by units ± 1 in Z. Since the Li ∈ Z[t, t− 1] are Laurent polynomials then N[L1, L2, …, Ln] is also a representative of the numerator of the continued fraction. Note however that the powers of t are also units in this ring. This is the reason why we define the notation N by the recursion in equation ([def N]). F-polynomials of even continued fractions ------------------------------------------- We now want to define specialized F-polynomials for even continued fractions. If the value of the even continued fraction is equal to a positive rational number r/s, then its F-polynomial is the F-polynomial of the positive continued fraction of r/s. However, if the value of the even continued fraction is negative, our definition involves the bar automorphism defined in section [sect knots]. Note that r/s > 0 if and only if b1 > 0. [def F] Let [b1, b2, …, bm] be an even continued fraction with value r/s ∈ Q. Let [a1, a2, …, an] be the positive continued fraction expansion of the absolute value ∣r/s∣ of r/s and let d = a1 + ⋯ + an − 1. We define the specialized F-polynomial of [b1, b2, …, bm] as follows. $${F\_{b\_1,b\_2,\ldots,b\_m}}= \left\{\begin{array}{rl} {F\_{a\_1,a\_2,\ldots,a\_n}}&\textup{if $b\_1>0$;}\\[8pt] (-t^{-1})^{d+1}\,\overline{{F\_{a\_1,a\_2,\ldots,a\_n}}} &\textup{if $b\_1<0$.} \end{array}\right.\\$$ [ex 6.4] The even continued fraction [2, − 2] is equal to 3/2. Thus F2, − 2 = F1, 2 which is equal q3N[ [2]q − q, [2]q q− 3 ] = ([2]q − q)[2]q + q3 = (1)(1 + q) + q3 = 1 + q + q3 = 1 − t− 1 − t− 3. Thus F2, − 2 = 1 − t− 1 − t− 3. On the other hand, [ − 2, 2] = − 3/2. Thus $$F\_{-2,2}=(-t^{-1})^3\,\overline{F\_{1,2}} =-t^{-3}(1-t^{1}-t^{3})=-t^{-3}+t^{-2}+1.$$ Note that we also have F− 2, 2 = F3. [ex 6.5] The even continued fraction [4] is equal to 4/1. Thus F4 = [5]q − q = 1 + t− 2 − t− 3 + t− 4. On the other hand, $$F\_{-4}=(-t^{-1})^4\,\overline{F\_{4}} =t^{-4}+t^{-2}-t^{-1}+1.$$ Note that we also have F− 4 = F1, 3. [ex 6.6] The even continued fraction [4, − 2] is equal to 7/2 which is equal to the positive continued fraction [3, 2]. Thus $$\begin{array}{rcl}F\_{4,-2}&=&F\_{3,2}= q^5N[ \,[4]\_q -q\,,\,[2]\_q\,q^{-5}\,]= ([4]\_q-q)[2]\_q+q^5\\ &=& 1-t^{-1}+t^{-2}-2t^{-3}+t^{-4}-t^{-5}.\end{array}$$ On the other hand, $$F\_{-4,2}=(-t^{-1})^5\, \overline{F\_{3,2}} =- t^{-5}+t^{-4}-t^{-3}+2t^{-2}-t^{-1}+1.$$ Note that F− 4, 2 = F1, 2, 2. [lemF] Let yij denote the coefficient variable in the cluster algebra that corresponds to the j-th tile in the snake graph G[a1, a2, …, an] ≅ G[b1, b2, …, bm], where j = 1, …, d. * The F-polynomial F[a1, a2, …, an] is a polynomial in Z≥ 0[y1, y2…, yN] of the form F[a1, a2, …, an] = ∑χα1, …, αd yi1α1⋯yidαd, where αj ∈ {0, 1}, with constant term 1 and highest degree term yi1yi2⋯yid. * The specialized F-polynomial Fb1, b2, …, bm is a polynomial in Z[t− 1] of the form Fb1, b2, …, bm = ∑i = 0d + 1( − 1)iσi t− i, where σi ∈ Z≥ 0, with constant term 1 and lowest degree term ( − 1)d + 1 t− d − 1. In particular, the degree of Fb1, b2, …, bm is 1. Part (a) was conjectured in and proved in except for positivity which is shown in for surface type and in for arbitrary quivers. The sum in (b) is alternating, since the sum in (a) is positive. The lowest degree in (b) is − d − 1, since y1 is specialized to t− 2 and every other yi is specialized to − t− 1. The following proposition has already been observed in the examples above. In the proof, we shall work with the minimal matching P− of the snake graph G[a1, a2, …, an]. The snake graph has precisely two perfect matchings P− and P+ which contain only boundary edges of the snake graph. By convention, the minimal matching P− is the one that contains the edge e0, the south edge of the first tile. The minimal matching has trivial coefficient y(P−) = 1. The matching P+ is called the maximal matching of the snake graph, and its coefficient y(P+) = yi1yi2⋯yid is the product of the y-coefficients of all tiles in the snake graph. The term y(P−) is the constant term in the polynomial F[a1, a2, …, an] and y(P+) is the term of highest degree. [prop Fbar] Let [a1, a2, …, an] be a positive continued fraction and let d = a1 + a2 + ⋯ + an − 1. Then $$\begin{array}{rcl} F{[a\_1,a\_2,\ldots,a\_n]}&=& y\_{i\_1} y\_{i\_2}\cdots y\_{i\_d} \ \overline{F[1,a\_1-1,a\_2,\ldots,a\_n]}.\\ \\ {F\_{a\_1,a\_2,\ldots,a\_n}}&=&q^{d+1} \ \overline{F\_{1,a\_1-1,a\_2,\ldots,a\_n}}. \end{array}$$ The snake graph G[1, a1 − 1, a2, …, an] is obtained from the snake graph G[a1, a2, …, an] by a reflection along the line containing the diagonal of the first tile. For example, $${\LARGE\scalebox{0.55}{}}$$ This reflection induces a bijection φ between the sets of perfect matchings of the snake graphs. Under this bijection, the minimal matching of G[a1, a2, …, an] is mapped to the maximal matching of G[1, a1 − 1, a2, …, an], and this implies that for an arbitrary perfect matching P of G[a1, a2, …, an] with height function y(P) = yi1α1⋯yidαd, the corresponding perfect matching φ(P) of G[1, a1 − 1, a2, …, an] has the complementary height function, that is, $$y(\varphi(P))= y\_{i\_1}^{1-{\alpha}\_1}y\_{i\_2}^{1-{\alpha}\_2}\cdots y\_{i\_d}^{1-{\alpha}\_d}=y\_{i\_1}y\_{i\_2}\cdots y\_{i\_d} \, \overline{y(P)}.$$ This proves the first identity, and the second follows by specialization. Our next lemma gives a recursive formula for the F-polynomial of an even continued fraction. The result mainly follows from results in, however, we are using even continued fractions here instead of positive continued fractions in loc.cit. [lem 6] Let [b1, b2, …, bm] be an even continued fraction. Then we have the following identity of F-polynomials depending on the type sequence of [b1, b2, …, bm]. If b1 > 0 then F[b1, b2, …, bm] is equal to $$\left\{\begin{array} {ll} - F[b\_1,b\_2,\ldots,b\_{m-2}] \prod\_{{\mathcal{G}}[b\_m]} y\_i\ +\ F[b\_1,b\_2,\ldots,b\_{m-1}] F[b\_m] &\textup{in type $(\ldots,-,-)$};\\ \\ F[b\_1,b\_2,\ldots,b\_{m-2}] \prod\_{{\mathcal{G}}\setminus{\mathcal{G}}\_2} y\_i\ +\ F[b\_1,b\_2,\ldots,b\_{m-1}] F[b\_m] &\textup{in type $(\ldots,+,-)$};\\ \\ F[b\_1,b\_2,\ldots,b\_{m-2}]\ +\ F[b\_1,b\_2,\ldots,b\_{m-1}] F[b\_m] y\_\ell&\textup{in type $(\ldots,-,+)$};\\ \\ - F[b\_1,b\_2,\ldots,b\_{m-2}] \prod\_{{\mathcal{G}}\_1\setminus{\mathcal{G}}\_2} y\_i \ +\ F[b\_1,b\_2,\ldots,b\_{m-1}] F[b\_m]&\textup{in type $(\ldots,+,+)$}, \end{array}\right.$$ where * ∏G[bm] runs over all tiles Gi ∈ G[bm]; * ∏G \ G2 runs over all tiles Gi that lie in G[b1, b2, …, bm] but not in G[b1, b2, …, bm − 2]; * yℓ is the coefficient of the tile Gℓ that connects G[b1, …, bm − 1] and G[bm] in G[b1, b2, …, bm]; and * ∏G1 \ G2 runs over all tiles Gi that lie in G[b1, …, bm − 1] but not in G[b1, b2, …, bm − 2]. If b1 < 0, the above formulas hold if we replace the types with their negatives. One reason for the separate cases is that the construction of G[b1, b2, …, bm] is different if bm and bm − 1 have the same sign or not, see section [sect sgb]. Suppose first that bm and bm − 1 have the same sign, thus the type sequence has a sign change in the last two positions. In this case, the grafting with a single edge formula gives the following two results. If the minimal matching P− restricts to a perfect matching of G[bm] then G[b1, b2, …, bm] = G[b1, b2, …, bm − 2]∏G \ G2yi + G[b1, b2, …, bm − 1] G[bm]. If the P− does not restrict to a perfect matching of G[bm] then G[b1, b2, …, bm] = G[b1, b2, …, bm − 2] + G[b1, b2, …, bm − 1] G[bm] yℓ. Suppose now that bm and bm − 1 have opposite signs, thus the last two entries in the type sequence are equal. In this case, yields the following two results. If the P− restricts to a perfect matching of G[bm] then G[b1, b2, …, bm] = − G[b1, b2, …, bm − 2]∏G1 \ G2yi + G[b1, b2, …, bm − 1] G[bm], If the P− does not restrict to a perfect matching of G[bm] then G[b1, b2, …, bm] = − G[b1, b2, …, bm − 2]∏G[bm]yi + G[b1, b2, …, bm − 1] G[bm]. Thus we need to determine when P− restricts to a perfect matching of G[bm]. See Figure [match5] for examples that illustrate the argument. Suppose first that b1 > 0. Then the first edge of the minimal matching e0 has sign f(e0) = + . Moreover e0 is a south edge, which implies that all south edges and all west edges of the minimal matching P− have sign + and all north and all east edges in P− have sign − , see. In particular, the last edge e ∈ GN E of the minimal matching has sign − . On the other hand, the last edge eʹβ ∈ GN E of the sign sequence of the even continued fraction [b1, b2, …, bm] has sign − if and only if [b1, b2, …, bm] is of type (…, − ). Thus eʹβ is in the minimal matching P− if and only if the type sequence of [b1, b2, …, bm] ends in a minus sign. Since G[bm] is a zigzag graph, if P− contains eʹβ, then it contains two boundary edges of the last tile and thus restricts to a matching of the last tile. Moreover, P− contains 3 boundary edges of the last two tiles and thus restricts to a matching of the last two tiles. Continuing this way, we see that P− restricts to a matching of the last bm − 1 tiles. The question whether P− restricts to a matching of the last bm tiles only depends on how G[bm] is glued to G[b1, …, bm − 1]. Thus, the minimal matching restricts to G[bm] in type (…, + , − ); these cases are marked (…, + , − ) in Figure [match5]. On the other hand, the minimal matching does not restrict to G[bm] in type (…, − , − ). Now suppose [b1, b2, …, bm] is of type (…, + ). Then eʹβ is not in P−. Thus P− contains only one edge of the last tile, only two edges of the last two tiles and only bm − 2 edges of the last bm − 1 tiles. Again, the question whether P− restricts to a matching of the last bm tiles only depends on how G[bm] is glued to G[b1, …, bm − 1]. Thus, the minimal matching restricts to G[bm] in type (…, + , + ); these cases are marked (…, + , + ) in Figure [match5]. On the other hand, the minimal matching does not restrict to G[bm] in type (…, − , + ). This proves the statement in the case where b1 > 0. If b1 < 0 the above argument is still valid, except that in this case, the sign of the edge e0 is − , and therefore the roles of the signs are reversed. [match5] We now apply this result to the specialization of the F-polynomial. [cor 8bis] Let [b1, b2, …, bm] be an even continued fraction with b1 > 0. Then Fb1, b2, …, bm = μ(t) Fb1, b2, …, bm − 2 + ν(t) [ ∣bm∣ ]q Fb1, b2, …, bm − 1 where the functions ν(t) and μ(t) depend on the type sequence of [b1, b2, …, bm] as follows \|c \| c \| r \| ν(t)&μ(t)&${\mathop{type}}[b\_1,\ldots,b\_m]$ && 1&t− ∣bm∣ + 1&(…, − , − ) 1&t− ∣bm∣ − ∣bm − 1∣&(…, − , + , − ) 1& − t− ∣bm∣ − ∣bm − 1∣ + 1 &(…, + , + , − ) && − t− 1 &1&(…, − , + ) 1& − t− ∣bm − 1∣&(…, − , + , + ) 1&t− ∣bm − 1∣ + 1&(…, + , + , + ) According to Lemma [lem 6], we only need to check that μ(t), ν(t) are the correct functions. Moreover it is clear from the lemma that ν(t) = 1 in all cases except in the case (…, − , + ) where ν(t) = − t− 1. In type (…, − , − ), Lemma [lem 6] implies that μ(t) = − ( − t− 1)α, where α is the number of tiles in G[bm]. Since bm and bm − 1 have opposite signs we have α = ∣bm∣ − 1, and since bm is even, we see that μ(t) = t− ∣bm∣ + 1. In type (…, + , − ), Lemma [lem 6] implies that μ(t) = ( − t− 1)α, where α is the number of tiles in G[b1, b2, …, bm] \ G[b1, b2, …, bm − 2]. According to Corollary [cor 3.3], this number is $\|b\_m\|+\|b\_{m-1}\| - \textup{(number of sign changes in $bm-2,bm-1)$}$. Thus α = ∣bm∣ + ∣bm − 1∣ in type (…, − , + , − ); and α = ∣bm∣ + ∣bm − 1∣ − 1 in type (…, + , + , − ). Since bm − 1 and bm − 2 are both even, it follows that μ(t) = t− ∣bm∣ − ∣bm − 1∣ in type (…, − , + , − ), and μ(t) = − t− ∣bm∣ − ∣bm − 1∣ + 1 in type (…, + , + , − ). In type (⋯, − , + ), Lemma [lem 6] implies μ(t) = 1. In type (⋯, + , + ), Lemma [lem 6] implies that μ(t) = − ( − t− 1)α, where α is the number of tiles in G[b1, b2, …, bm − 1] \ G[b1, b2, …, bm − 2]. Similar to the previous case, we see that α = ∣bm − 1∣ in type (⋯, − , + , + ) ; and α = ∣bm − 1∣ − 1 in type (⋯, + , + , + ), and this completes the proof. Main results ============ In this section, we state and prove our main theorem. Recall that we have fixed the orientation of all knots and links in section [sect knots]. [thm main] Let [b1, b2, …, bm] be an even continued fraction, let V[b1, b2, …, bm] be the Jones polynomial of the corresponding 2-bridge link, and let Fb1, b2, …, bm be the specialized F-polynomial of the corresponding cluster variable. Then V[b1, b2, …, bm] = ( − 1)m − τ tj Fb1, b2, …, bm where τ is the number of subsequences + , + in the type sequence of [b1, b2, …, bm] and $$\textstyle j= \sum\_{i=1}^m \max\left( (-1)^{i+1} b\_i + \frac{\textup{sign}(b\_ib\_{i-1})}{2}\,\ -\frac{1}{2}\right).$$ We proceed by induction on m. For m = 1, we will show that V[b1] = − tjFb1. Suppose first that b1 > 0. Then $j=b\_1+\frac{1}{2}$ and according to Definition [def F] we have Fb1 = [b1 + 1]q − q, with q = − t− 1. On the other hand, Theorem [thm 5] implies $$\begin{array} {rcl} V\_{[b\_1]}&=& t^{b\_1}(-t^{-\frac{1}{2}}-t^{\frac{1}{2}})-t^{\frac{1}{2}}\,[b\_1]\_{\overline{q}}\\ &=&-t^{b\_1+\frac{1}{2}}(t^{-1}+1+t^{-b\_1} (1-t+t^2-\cdots -t^{b\_1-1}))\\ &=&-t^{b\_1+\frac{1}{2}}([b\_1+1]\_q-q), \end{array}$$ and the result follows. Now suppose that b1 < 0. Then $j=-\frac{1}{2}$ and Definition [def F] implies $$F\_{b\_1}= (-t^{-1})^{\|b\_1\|} \overline{F\_{\|b\_1\|}} =t^{-\|b\_1\|} [b\_1+1]\_{\overline{q}}\,-\overline{q} = t^{-\|b\_1\|}(1+t^2-t^3+t^4-\cdots+t^{\|b\_1\|} ) = [\|b\_1\|+1]\_q	arxiv_0000522
A formula for F-Polynomials in terms of C-Vectors and Stabilization of F-Polynomials ========================================================================================== Given a quiver associated to a cluster algebra and a sequence of vertices, iterative mutation leads to F-Polynomials which appear in numerous places in the cluster algebraic literature. The coefficients of the monomials in these F-Polynomials are difficult to understand and have been an area of study for many years. In this paper, we present a general closed-form formula for these coefficients in terms of elementary manipulations with C-matrices. We then demonstrate the effectiveness of the formula by using it to derive simple explicit formulas for F-Polynomials of specific classes of quivers and mutation sequences. Work has been done to do these cases in ad-hoc combinatorial ways, but our formula recovers and improves known formulas with a general method. Secondly, we investigate convergence of F-polynomials. In themselves, they do not converge, but by changing bases using the C-matrix, they conjecturally do. Since our formula relates C-matrix entries to coefficients, we are able to apply it to make considerable progress on the conjecture. Specifically, we show stability for green mutation sequences. Finally, we look at exact formulas for these stable deformed F-polynomials in instances where they illustrate properties that the F-polynomials themselves do not. Introduction ============ Cluster algebras, introduced by Fomin and Zelevinsky in 2002, are commutative algebras with a distinguished set of generators, known as cluster variables. Their original motivation came from an algebraic framework for total positivity and canonical bases in Lie Theory. In recent years, connections have been made to other areas, such as physics, quiver representations, Calabi-Yau categories, and Poisson geometry. The cluster variables are grouped into sets of constant cardinality v, the clusters, and the integer v is called the rank of the cluster algebra. Start with an initial cluster ${\textup{\textbf{x}}}$ (together with a skew symmetrizable integer v × v matrix B and a coefficient vector ${\textup{\textbf{p}}} = \left<(p\_1^+,p\_1^-),(p\_2^+,p\_2^-),\ldots,(p\_v^+,p\_v^-)\right>$ whose entries are elements of a torsion-free abelian group. We denote the entry in the kth row and ith column of B as B[k, i]. The set of cluster variables is obtained by repeated application of so called mutations. To perform a mutation at k, replace xk in the cluster with $$x'\_k=\frac{1}{x\_k}\left(p\_k^{+}\prod\_{B[k,i]>0}x\_i^{B[k,i]}+p\_k^{-}\prod\_{B[k,i]<0}x\_i^{-B[k,i]}\right)$$ This creates a new cluster variable. By a result of Fomin and Zelevinsky known as the Laurent phenomenon, this new cluster variable will always be a Laurent polynomial. There is also a corresponding change to the matrix B and vector p, which will be defined properly in the next section. When one sets each initial xi to 1 and initial p vector as $\left<y\_1,\ldots,y\_v\right>$, each cluster variable is a polynomial with indeterminates $\left<y\_1,\ldots,y\_v\right>$. These polynomials are called F-polynomials. It turns out that the coefficients of monomials in F-polynomials are the same as those we would have gotten by setting each initial yi to 1 and leaving the initial x vector as indeterminates $\left<x\_1,\ldots,x\_v\right>$. As such, results about coefficients of F-polynomials also apply to cluster variables. F-Polynomials have been an interesting area of study since the inception of the field of cluster algebras. Fomin and Zelevinsky conjectured in that every F-polynomial has all positive coefficients. This conjecture was later proven in the skew-symmetric case in and then the entire skew-symmetrizable case in. Nonetheless, we are far from completely understanding them. For one, not very many explicit formulas exist for the coefficients of F-polynomials. Since they are defined through recurrence relations, it is not clear that we can actually get a closed form formulation instead. The main result of our paper is to derive a closed form based only on the C-matrix entries, which are related to the p-vectors defined above. Some of the most general previous results regarding F-polynomial explicit forms are the following: * A formula in terms of the Euler-Poincare characteristic of quiver Grassmannians obtained in. * A combinatorial interpretation for cluster algebras from surfaces given in. * A formula for cluster variables corresponding to string modules as a product of 2 × 2 matrices obtained in. * A formula using broken lines and reflections in. The last result is the only one that is completely general; the others are restrictive in the cluster algebras they account for. The first is mostly general, restricting only in that it only applies to skew-symmetric cluster algebras. However, the first and last have limited computational power, as the coefficients involve objects that are hard to understand and compute. As said earlier, the main result of this paper is to present a formula for the F-polynomial that is completely general and completely explicit. This is in itself interesting, because it shows that the recurrence relation for F-polynomials can be solved out. Moreover, we show that our formula is useful. We demonstrate that this formula has computational power, meaning its terms are easy to understand, unlike the formula using the Euler-Poincare characteristic of quiver Grassmannians. We demonstrate this by using it to easily obtain algebraic formulas for F-polynomials of specific classes of quivers, that are as simple or simpler than previously known. Though our formula is general, our examples and applications of it restrict to the skew-symmetric case, because that is where patterns are most evident. Furthermore, the formula has an application to geometry in this case, where the cluster algebra can be interpreted with quivers. The formula for the F-polynomial in terms of the Euler-Poincare characteristic of quiver Grassmannians mentioned earlier in says the following: Fn = ∑eχ(Gre(ϕn))∏i = 1vyiei Here, e is a vector in Nv and ϕn is a representation of the original quiver defined by certain mutation rules. As such, our expression for the coefficient of a given monomial in an F-Polynomial also gives a formula for the corresponding Euler-Poincare characteristic of quiver Grassmannians associated to dimension vectors parameterized by e. This is interesting, because these are often very difficult to understand and compute. Finally, it is well known that F-polynomial coefficients for specific quivers represent various combinatorial objects; for example pyramid partitions, T-paths in surfaces with marked points, and snake graphs. Therefore, our F-polynomial formula also provides a way to count these combinatorial objects. Secondly, we use our formula to make progress on a conjecture made by Eager and Franco in Section 9.5 of. They explore convergence of F-polynomials in the skew symmetric case. In themselves, F-polynomials do not converge, but when one changes basis; i.e. acts with an appropriate linear operator on the exponent vector of each monomial, they appear to converge. This phenomenon and precise linear operator was conjectured in and formalized by Grace Zhang in. This bears a striking resemblance to our formula for the coefficients of F-Polynomials, and we use our results to prove Eager and Franco’s conjecture for a certain class of quivers known as green quivers. This includes the quiver they originally conjectured stabilization for, the dP1 quiver, known in combinatorics for giving rise to the Somos-4 sequence. This also extends results of, which prove stabilization for some specific green quivers. Definitions =========== We start with the general definitions, and later state the simplified definitions for the skew-symmetric case, since that is the case we study most closely. The way we define F-polynomials is purely combinatorial; we do not use the cluster algebra notation directly. In both cases, we would start with a matrix B and mutation sequence v1, …, vn to obtain the F-polynomial Fn. In our definitions, the matrix B is essentially replaced by the notion of a generalized quiver defined below in Section [generaldefs]. An interested reader should look at and to understand why the cluster algebra definition of an F-polynomial is the same. Furthermore, the reader interested in only the skew-symmetric case can ignore the concept of “attached” edges, and instead look at Section [simpledefs] for the quiver and mutation definitions. From there, the reader can come back to this section, and ignore any occurrence of the word “attached” and assume that all edges are attached to both incident vertices. In this case, we also have the matrix A is the same as E and C the same as D. General Definitions ------------------- [Generalized Quiver] A generalized quiver Q is a directed graph on vertices 1, …, ∣Q∣, where every edge is “attached” to one of its incident vertices, along with a Laurent polynomial label at every vertex. The number of vertices ∣Q∣ will be denoted v. However the edges must satisfy certain properties: * There are no 2-cycles involving edges both attached to the same vertex. * When written in a matrix B with entry B[i, j] being the number of edges attached to j going from j → i (negative if edges go in), B is skew-symmetrizable. Intuitively, a generalized quiver is a generalized version of a quiver, reflecting the fact that the “adjacency matrix” B need not be skew-symmetric. [Generalized Quiver Mutation] An generalized quiver mutation at vertex i is defined by changing both the quiver around vertex i and the Laurent polynomial at vertex i. The quiver changes are defined as follows. * For every path j → i → k with the first edge attached to i and the second to k, draw a new edge from j to k attached to k. * For every path j → i → k the first edge attached to j and the second to i, draw a new edge from j to k attached to k. Reverse the direction of every edge pointing with i as a vertex. * Delete all 2-cycles involving edges both attached to the same vertex. The Laurent polynomial is replaced as follows. Let Vk denote the Laurent polynomial currently on vertex k. Replace Vi with the following function P. $$P=\frac{\prod\_{j \in S\_1}V\_j + \prod\_{j \in S\_2}V\_j}{V\_i}$$ where S1 is the set of vertices with multiplicity pointing into i attached to i and S2 is the set of vertices with multiplicity of edges pointing out of i attached to i. Notice that the vertex i may no longer have a Laurent polynomial label. However, the following theorem by Fomin and Zelevinsky rectifies that. [Laurent Phenomenon ] Take a generalized quiver Q with starting polynomials x1, …, xv at each corresponding vertex. For any mutation sequence, the function P as defined above is actually a Laurent polynomial. [Framed Generalized Quiver] The framed generalized quiver of a generalized quiver Q with labels (polynomials) 1 at every vertex is the quiver formed by adding a vertex iʹ for every vertex i with an edge pointing from iʹ to i attached to i. The new vertices iʹ have the label yi. A base vertex of the framed generalized quiver is a vertex that is also a vertex of the original quiver. [F-Polynomial] The nth $\bm{F}$-Polynomial of a framed generalized quiver Q accompanied with a mutation sequence v1, …, vn of vertices is the polynomial at vertex vn after performing mutations at v1, …, vn sequentially. [Vn(i) or Vi] Given a quiver Q and a sequence of mutations v1, …, vn, $\bm{V\_n(i)}$ is the F-polynomial on vertex i after the n mutations. We will usually denote this F-polynomial more simply as Vi if we are otherwise clear about the termination point n for the sequence of mutations. [C-matrix] The $\bm{C}$-matrix of the framed generalized quiver Q and mutation sequence v1, …, vn with v base vertices is the v × v matrix with entry (i, j) as the number of edges from frozen vertex iʹ to base vertex j attached to j. It is denoted Cn. [Ai] Given a framed generalized quiver Q and a sequence of mutations v1, …, vi, let $\bm{A\_i^g}$ be the identity matrix with entry (vi, vi) replaced with − 1 and for every arrow pointing from vi to another base quiver vertex u, add 1 to the entry (vi, u). $\bm{A\_i^r}$ is defined analogously with arrows pointing into vi. We will just refer to the relevant Aig or Air by Ai, where the relevant one is what corresponds to the color of ri as will be defined soon. The other one will be Ai\. [C-Matrix Facts, Theorem 5.6 ] Given a framed quiver Q* and a sequence of mutations v1, …, vn, Cn = A1A2⋯An, where for each i you choose g if all the arrows are pointing into vi before the ith mutation, and r otherwise. Also, Cn is invertible. [Ei] Given a framed generalized quiver Q and a sequence of mutations v1, …, vi, let $\bm{E\_i^g}$ be the identity matrix with entry (vi, vi) replaced with − 1 and for every arrow pointing from vi to another base quiver vertex u attached to vi, add 1 to the entry (vi, u). $\bm{E\_i^r}$ is defined analogously with arrows pointing into vi. We will just refer to the relevant Eig or Eir by Ei, where the relevant one is what corresponds to the color of ri as will be defined soon. The other one will be Ei\. [D-Matrix] The $\bm{D}$-matrix* is similar to the C-matrix, defined slightly differently though: Dn = E1⋯En. Note that Dn is also invertible. The D-matrix between two steps is defined as follows. For two indices m and n, let Dm, n be Dm− 1Dn. By extension, Dn = D0, n. [ri] [ri] Given the quiver formed by a framed quiver Q and a mutation sequence v1, …, vi, let $\bm{r\_i}$ be the monomial that is the product of the frozen nodes pointing into/from vertex vi. More precisely, we have ri = ∏j = 1vyjδCi[j, vi]∣ where δ ∈ { ± 1} chosen so that the exponents are nonnegative. The ri is called green if the frozen nodes are pointing in before the final mutation (out afterwards) and red if they are all pointing out. [Sign Coherence ] Given the quiver formed by a framed quiver Q and a mutation sequence v1, …, vi, all the frozen vertices pointing into any vertex u at the end of the process are in the same direction (all in or all out), and there will be at least one arrow. Thus, every vertex will be green or red. [δi, j] Let δi, j be 1 if ri and rj are the same color, and − 1 otherwise. Mutation 0 counts as red for the purposes of assigning δ0, i. The sign δ0, i agrees with the sign δ from Definition [ri]. We call a mutation sequence green if all the mutations are green. We often just say the quiver is green if it is clear what mutation sequence we are referring to. The next two definitions bare no obvious significance, but simplify expressions in future sections. Loosely, the particular values ai, j and bi, j are important because they are the building blocks of coefficients of monomials in the F-polynomials. [ai, j and bi, j] Given a framed quiver Q and a mutation sequence v1, v2, …, , for two indices i ≤ j, let $\bm{a\_{i,j}} = D\_{i,j}^{-1}[v\_j,v\_i]$. Similarly, let $\bm{b\_{i,j}} = E^\\_jE\_jD\_{i,j}^{-1}[v\_j,v\_i]$, or equal to 0 when i* = j. [W] Given a framed quiver Q and a mutation sequence v1, …, vn, and a sequence of indices w1, …, wk where the wi’s are all between 1 and n inclusive, let $$\bm{W(n, w\_1,\ldots,w\_k)}=\prod\_{i=1}^{k} \left( a\_{w\_i,n} + \sum\_{j=i+1}^{k} (-a\_{w\_i,w\_j}+b\_{w\_i,w\_j})\right).$$ [ϕ] We define $\bm{\phi({\textup{\textbf{w}}})}$ to be the number of distinct permutations of ${\textup{\textbf{w}}}=w\_1,\ldots,w\_k$ divided by k!. Note that this is just $\frac{1}{\prod{a\_i!}}$ where the ai are the number of occurrences of each of the distinct values taken on by w1, …, wk. Simplifications in the Skew-symmetric Case ------------------------------------------ In the case where B is skew-symmetric, for any edge between two base vertices, the number of edges attached to each vertex is the same. As such, we can define the quiver, rather than generalized quiver, so that only one set of edges is included, and attached can be disregarded. The analogous definitions are straightforward, but we redefine for completeness. To understand that all proofs completed in the general case also apply here, one should check that all the variables defined, C, A, ai, j, bi, j, ri, etc. are all the same when returning the quiver to its generalized quiver form. [Quiver] A quiver Q is a directed graph, along with a Laurent polynomial label at every vertex. The variable v denotes the number of vertices in Q, or ∣Q∣. The directed graph must have no 2-cycles. Any quiver can be viewed, instead, as a generalized quiver by replacing each edge with two edges, one attached to each incident vertex. [Quiver Mutation] An quiver mutation at vertex i is defined by changing both the quiver around vertex i and the Laurent polynomial at vertex i. The quiver changes are defined as follows. * For every path j → i → k, draw a new edge from j to k. * For every path j → i → k, draw a new edge from j to k. Reverse the direction of every edge pointing with i as a vertex. * Delete all 2-cycles. The Laurent polynomial is replaced as follows. Let Vk denote the Laurent polynomial currently on vertex k. Replace Vi with the following polynomial P. $$P=\frac{\prod\_{j \in S\_1}V\_j + \prod\_{j \in S\_2}V\_j}{V\_i}$$ where S1 is the multiset of edges pointing into i attached to i and S2 is the multiset of edges pointing out of i. [Framed Quiver] The framed quiver of a quiver Q with labels 1 at every vertex is the quiver formed by adding a vertex iʹ for every vertex i with an edge pointing from iʹ to i. The new vertices iʹ have the label yi. A base vertex of the framed quiver is a quiver that is also a vertex of the original quiver. [C-matrix] The $\bm{C}$-matrix of the quiver formed by a framed quiver Q and mutation sequence v1, …, vn with v base vertices is the v × v matrix with entry (i, j) as the number of edges from frozen vertex iʹ to base vertex j. It is denoted Cn. The C-matrix between two steps is defined as follows. For two indices m and n, let $\bm{C\_{m,n}}$ be Cm− 1Cn. It is worth noting that Cn = C0, n. [Ai] Given a framed quiver Q and a sequence of mutations v1, …, vi, let $\bm{A\_i^g}$ be the identity matrix with entry (vi, vi) replaced with − 1 and for every arrow pointing from vi to another base quiver vertex u, add 1 to the entry (vi, u). $\bm{A\_i^r}$ is defined analogously with arrows pointing into vi. We will just refer to the relevant Aig or Air by Ai, where the relevant one is what corresponds to the color of ri as will be defined soon. The other one will be Ai\. The A-matrix is the same as the E-Matrix. Both will be denoted by A. The C-matrix is the same as the D-matrix. Both will be denoted by C. Also note the following: [a*i, j and bi, j] Given a framed quiver Q and a mutation sequence v1, v2, …, for two indices i ≤ j, let $\bm{a\_{i,j}} = C\_{i,j}^{-1}[v\_j,v\_i]$. Similarly, let, $\bm{b\_{i,j}} = A^\\_jA\_jC\_{i,j}^{-1}[v\_j,v\_i]$, or equal to 0 when i* = j. [W] Given a framed quiver Q and a mutation sequence v1, …, vn, and a sequence of indices w1, …, wk where the wi’s are all between 1 and n inclusive, let $$\bm{W(n, w\_1,\ldots,w\_k)}=\prod\_{i=1}^{k} \left( a\_{w\_i,n} + \sum\_{j=i+1}^{k} (-a\_{w\_i,w\_j}+b\_{w\_i,w\_j}) \right).$$ We will use ai, j, bi, j and W regardless of whether we are dealing with a quiver or generalized quiver. Definitions for Stabilization ----------------------------- [Matrix Action on Polynomial] Given an v × v matrix M and a polynomial P in x1, …, xv, we let $\bm{M(P)}$ denote the polynomial formed when M acts on the vector of exponents for each monomial in P. [Deformed F-Polynomial] The nth $\bm{S}$-polynomial (deformed F-polynomial) of a framed quiver Q with a mutation sequence v1, …, vn is defined as − C− 1(Fn). [Fundamental ri] An ri of a framed quiver Q and mutation sequence v1, v2, … is called fundamental if it is not expressible as the product of at least 2 other rj’s with j < i. Let the set Mn denote the set of fundamental ri for a framed quiver Q and sequence of mutations v1, …, vn. M is the set of fundamental ri for the entire mutation sequence v1, v2, …. [Basic] Given a framed quiver Q and a mutation sequence v1, …, vn, a monomial m is called basic if its coefficient is nonzero in a polynomial that appears on one of the vertices after n mutations. In other words, its coefficient is nonzero in some element of the cluster. Let Pn denote the set of basic monomials after n mutations. $\bm{F}$-Polynomial Formula =========================== Derivation ---------- [formula] Given a framed quiver or a framed generalized quiver Q and a mutation sequence v1, …, vn, Fn can be calculated as follows. Let W be the set of (possibly empty) sequences ${\textup{\textbf{w}}}$ with 1 ≤ w1 ≤ … ≤ wk ≤ n. $$F\_n = \sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) W(n, w\_1,\ldots,w\_k) \prod\_{i=1}^{k}r\_{w\_i}$$ Consequently, the coefficient of a given monomial p is the sum of all sequences 1 ≤ w1 ≤ … ≤ wk ≤ n so that p = ∏i = 1krwi of $$\phi({\textup{\textbf{w}}}) W(n, w\_1,\ldots,w\_k).$$ (Recall that rwi is a monomial in y1, …, yv as in Definition [ri].) It suffices to prove the statement in the case of framed generalized quivers. Consider the steps v1, …, vn of the mutation process on the quiver Q that results in Fn. At each step i, define ≻ i and ≺ i as follows. If ri is green, any edge in the quiver before the ith mutation attached to a denoted a → b is now denoted a ≻ ib. Similarly, a ← b is denoted a ≺ ib. If ri is red, the reverse occurs: ≻ i replaces ← , and ≺ i replaces → . Thus, ≻ replaces → or ← corresponding to the direction of the frozen vertices into vi. At each step i, the following occurs to calculate Fi. We have FiVvi = ∏vi ≻ jVj + ri∏vi ≺ jVj. We modify this mutation process: instead of plugging in the monomial ri, replace ri with the variable zi. Define the analog of Fi as Zi and the analog of Vj as ZVj, so ZVj is the Z power series at the jth vertex. Thus, Zn will be some rational function in the zi’s, computed by the following recurrence relation: ZiZVvi = ∏vi ≻ jZVj + zi∏vi ≺ jZVj and …, F− 1, F0 = 1. When we plug in ri for each zi, we get back Fn. As such, to get the coefficient of a given monomial p, do the following: express Zn in its MacLaurin series form, i.e. an element of C[[z1, …, zn]]. Then, add the coefficients over all sequences w1, …, wk so that p = rw1⋯rwk. Now, we compute the coefficient of a given zw1⋯zwk. We show it is equal to ϕ(w1, …, wk)W(n, w1, …, wk). Encode the sequence ${\textup{\textbf{w}}}$ instead as two sequences ${\textup{\textbf{q}}}$ and ${\textup{\textbf{m}}}$ where ${\textup{\textbf{q}}}$ consists of the distinct terms of ${\textup{\textbf{w}}}$ in increasing order, and ${\textup{\textbf{m}}}$ is the corresponding multiplicity of each. Let $l=\|{\textup{\textbf{q}}}\|$. The coefficient of the term zw1⋯zwk = zq1m1⋯zqlml in Zn is the same as the one in the power series computed as follows: look at the entire modified mutation process, except on a mutation step i not corresponding to a qj, replace zi with 0. This is because replacing all irrelevant zi’s with 0 does not affect the coefficient of the desired term. Call this new power series Yn. YVvi is defined analogously as well. Consider the following power series, L1, …, Ll. $$\begin{aligned} L\_1&=(1+z\_{q\_1}) \\ L\_2&=\left(1+z\_{q\_2}(1+z\_{q\_1})^{-a\_{q\_1,q\_2}+b\_{q\_1,q\_2}}\right) \\ &. \\ &. \\ &. \\ L\_l&=\left(1+z\_{q\_l}(L\_{l-1}^{-a\_{q\_{l-1},q\_l}+b\_{q\_{l-1},q\_l}}\cdots L\_1^{-a\_{q\_1,q\_l}+b\_{q\_1,q\_l}})\right) \end{aligned}$$ Let Hi : = ∏j = 1rLjaqj, i. where r corresponds to the final qr preceding i (so Hn would be ∏j = 1lLjaqj, i). We show that at any given step i, Yi = Hi. We do so by induction. #### Base Case: $\bm{i\leq q\_1}$: For this case, we have Yq1 = Hq1 = (1 + zq1) and for i < q1, we have Yi = Hi = 1. Yi is 1 because all the relevant zi are 0, and Hi is 1 because it is the product over no Lj and therefore just 1. We have the following recurrence relation. YiYVvi = ∏vi ≻ jYVj + zi∏vi ≺ jYVj I claim that the relation HiHVvi = ∏vi ≻ jHVj + zi∏vi ≺ jHVj holds as well. Because of the inductive hypothesis, proving this recurrence relation would suffice to show the power series are equal. #### Case 1: $\bm{i\neq q\_r}$: We have zi = 0, so we want to show $$\begin{aligned} H\_iHV\_{v\_i}&=\prod\_{v\_i \succ j}HV\_j+z\_i\prod\_{v\_i \prec j}HV\_j \\ \iff H\_i&=\frac{\prod\_{v\_i \succ j}HV\_j}{HV\_{v\_i}}. \end{aligned}$$ We look at the exponent for each Ls in Hi and show that it is, in fact, the same on both sides of the equation. The exponent on the right hand side, denoted $e\_s\left( \frac{\prod\_{v\_i \succ j}HV\_j}{HV\_{v\_i}} \right)$ is calculated as follows: ∑vi ≻ jes(HVj) − es(HVvi). Let ij be the last index of a mutation at vertex j. By the inductive hypothesis, $$e\_s\left( \frac{\prod\_{v\_i \succ j}HV\_j}{HV\_{v\_i}} \right)=\sum\_{v\_i \succ j}a\_{q\_s,i\_j}-a\_{q\_s,i\_{v\_i}}.$$ Recall that $$D\_{i,j}=E\_{i+1}\cdots E\_j \implies D^{-1}\_{i,j}=E\_j\cdots E\_{i+1}$$ since the Ei are involutions. Multiplication by any Ei only changes the corresponding row, so since ai, j = Di, j− 1[vj, vi], we can replace terms of the form aqs, ij with Dqs, i − 1− 1[j, vqs], since no entry in the jth row of the corresponding matrix changes between step ij and step i − 1. It’s useful to note this also stays 0 when Ls corresponds to a term that appears after one of our HV terms is calculated. Thus, we have as our expression: $$e\_s\left( \frac{\prod\_{v\_i \succ j}HV\_j}{HV\_{v\_i}} \right)=\sum\_{v\_i \succ j}D^{-1}\_{q\_s,i-1}[j,v\_{q\_s}] -D^{-1}\_{q\_s,i-1}[v\_i,v\_{q\_s}].$$ However, the right hand side is exactly the operation corresponding to obtaining the entry Dqs, i− 1[vi, vqs] = aqs, i from how Ei is defined: add the contributions from all the vertices pointing opposite of the frozen vertices, and subtract the current vertex’s contribution. This exactly proves our desired statement that $$e\_s(H\_i)=e\_s\left( \frac{\prod\_{v\_i \succ j}HV\_j}{HV\_{v\_i}} \right).$$ #### Case 2: $\bm{i=q\_r}$: Just as in the last case, we can try to compute $$\frac{\prod\_{v\_i \succ j}HV\_j+z\_i\prod\_{v\_i \prec j}HV\_j}{HV\_{v\_i}}.$$ By the first case, $$\frac{\prod\_{v\_i \succ j}HV\_j}{HV\_{v\_i}}$$ is Hi without the contribution from Lr. We now have to show that adding $\frac{z\_i\prod\_{v\_i \prec j}HV\_j}{HV\_{v\_i}}$ gives the same contribution as multiplying by Lr. By the same logic as the proof from the first case, we find the term $$\frac{\prod\_{v\_i \prec j}HV\_j}{HV\_{v\_i}}.$$ Again we find es of the expression for each k. We get $$e\_s\left(\frac{\prod\_{v\_i \prec j}HV\_j}{HV\_{v\_i}}\right)=\sum\_{j \succ v\_i}D^{-1}\_{q\_s,i-1}[j,v\_{q\_s}] -D^{-1}\_{q\_s,i-1}[v\_i,v\_{q\_s}].$$ This is exactly the operation of Ei\, so we get $$e\_s\left(\frac{\prod\_{j \succ v\_i}HV\_j}{HV\_{v\_i}}\right)=b\_{q\_j,i}.$$ Now that we have these identities, we can show that our recurrence relation in fact gives H*i. $$\begin{aligned} &\frac{\prod\_{v\_i \succ j}HV\_j+z\_i\prod\_{v\_i \prec j}HV\_j}{HV\_{v\_i}} \\ =&\prod\_{j=1}^{r-1}L\_j^{a\_{q\_j,i}}+z\_i\prod\_{j=1}^{r-1}L\_j^{b\_{q\_j,i}} \\ =&(1+z\_i\prod\_{j=1}^{r-1}L\_j^{b\_{q\_j,i}-a\_{q\_j,i}})\prod\_{j=1}^{r-1}L\_j^{a\_{q\_j,i}} \\ =&L\_r\prod\_{j=1}^{r-1}L\_j^{a\_{q\_j,i}} \\ =&H\_i \end{aligned}$$ which is what we desired. Now we must actually compute the coefficient of zw1⋯zwk in Hn = ∏j = 1kLjaqj, n. We prove that it is ϕ(w1, …, wk)W(n, w1, …, wk) by induction on $l=\|{\textup{\textbf{q}}}\|$. #### Base Case: $\|{\textup{\textbf{q}}}\|=1$: Then Hn = L1aq1, n. By the extended binomial theorem, the coefficient of zq1m1 is in fact $$\frac{(a\_{q\_1,i})(a\_{q\_1,i}-1),\ldots,(a\_{q\_1,i}-m\_1+1)}{m\_1!}.$$ #### Inductive step: $\bm{\|}{\textup{\textbf{q}}}\bm{\|=l}$: If we instead compute the coefficient of the term zq2m2⋯zqlml, then we can apply the inductive hypothesis to get the desired result. The corresponding Li, which we call Liʹ, we have are defined as follows. $$\begin{aligned} L\_2'&=(1+z\_{q\_2}) \\ L\_3'&=\left(1+z\_{q\_3}(1+z\_{q\_2})^{-a\_{q\_1,q\_2}+b\_{q\_1,q\_2}}\right) \\ &. \\ &. \\ &. \\ L\_l'&=\left(1+z\_{q\_l}(L\_{l-1}'^{-a\_{q\_{l-1},q\_l}+b\_{q\_{l-1},q\_l}}\cdots L\_1'^{-a\_{q\_1,q\_l}+b\_{q\_1,q\_l}})\right) \end{aligned}$$ This gives Hʹn = ∏j = 1lLjʹaqj, n. By inspecting the way the Li’s are formed, we see that we can form Hn by replacing every instance of zqi with zqi(1 + zq1)− aq1, qi + bq1, qi, and multiplying the entire expression by (1 + zq1)aq1, n. If we expand only q2, …, ql, leaving the (1 + zq1) expressions, which corresponds to just expanding Hʹn, the term zq2m2⋯zqlml corresponds to the term (1 + zq1)aq1, n + ∑i = 2l − miaq1, qi + mibq1, qizq2m2⋯zqlml. Any other terms cannot have a monomial that has the correct number of factors of all of zq2, …, zql. Thus, we only need to find the coefficient of zw1⋯zwk in this monomial. The corresponding term in Hʹn already has coefficient ϕ(wm1 + 1, wk)W(n, wm1 + 1, …, wk), so the monomial itself is that times (1 + zq1)aq1, n + ∑i = 2l − miaq1, qi + mibq1, qizq2m2⋯zqlml. To find the total coefficient, we multiply this by the coefficient of zq1m1 in the expansion of (1 + zq1)aq1, n + ∑i = 2l − miaq1, qi + mibq1, qi which by extended binomial theorem is $$\frac{a\_{q\_1,n}+\sum\_{i=2}^{l}-a\_{q\_1,q\_i}+b\_{q\_1,q\_i} \cdots a\_{q\_1,n}+\sum\_{i=2}^{l}-m\_ia\_{q\_1,q\_i}+m\_ib\_{q\_1,q\_i}-m\_1+1}{m\_1!}.$$ Since ai, i is 1 and bi, i is 0, this can be rewritten as $$\frac{\prod\_{j=1}^{m\_1}\left(a\_{w\_1,n}+\sum\_{i=m\_1+1}^{k}-a\_{w\_j,w\_i}+b\_{w\_j,w\_i} \right)}{m\_1!}.$$ This, in turn, is exactly ϕ(w1, …, wk)W(n, w1, …, wk), thus proving our inductive step. Formula Discussion ------------------ In this section, we discuss the formula, its terms, and implications. The formula has the unfortunate property that it is summing over an infinite set of sequences. However, this is easily remedied; it is fairly simple to find an upper bound on the degree of Fn. For example, take the maximum number of edges in the quiver at any stage m and raise this value to the nth power. This must be an upper bound, because at each step the maximum degree of a label on the quiver must be less than m times the previous maximum value. Then, simply sum only over sequences where the expression ∏i = 1krwi has degree less than our upper bound. This is clearly finite, since each rwi is positive. Consequently, this at least limits us to only checking sequences with length less than the upper bound on the degree. In the next few sections, we apply our formula to get simpler formulas for specific cases where we know what the relevant entries of the C-matrix are. In many of the cases, we do not limit the sequences we are summing over, if there isn’t a specifically interesting way to do so. It is therefore valuable to keep in mind that this procedure would work. Secondly, let us look at the full expanded version of the formula and understand what the terms mean. We have for sequences ${\textup{\textbf{w}}}$ with 1 ≤ w1 ≤ … ≤ wk ≤ n, $$\label{formulafull} \begin{aligned} F\_n = \sum\_{{\textup{\textbf{w}}}\in W} & \phi(w\_1,\ldots,w\_k) \prod\_{i=1}^{k} \left( D\_{w\_i,n}^{-1}[v\_n,v\_{w\_i}] + \sum\_{j=i+1}^{k} (-D\_{w\_i,w\_j}^{-1}[v\_{w\_j},v\_{w\_i}]+A^\\_{w\_j}A\_{w\_j}D\_{w\_i,w\_j}^{-1}[v\_{w\_j},v\_{w\_i}])\right) \\ &\prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}\delta\_{0,w\_j}C\_{0,w\_j}[i,v\_{w\_j}]}. \end{aligned}$$ First of all, we should understand which sequences ${\textup{\textbf{w}}}$ contribute to a given monomial term. These would be the terms that have the same ∏i* = 1krwi, or align for each i on ∑j = 1kδ0, wjC0, wj[i, vwj]. Though this appears complicated to work with, C-matrix entries are often easy to understand, so it becomes simple to categorize which sequences contribute to a given monomial term. Now, let us understand what the D-matrix entries mean in the skew-symmetric case, where they are the same as the C-matrix entries. Recall that Cn[i, j] is the number of arrows pointing into vertex j from vertex iʹ after n mutation steps on the mutation sequence used to obtain Cn. Similarly, Cm, n[i, j] is the number of arrows pointing into vertex j from vertex iʹ after n − m mutation steps where the initial base quiver is actually the original quiver after m mutation steps. There is a caveat though: the color sequence must be the same as it was on the original quiver. If not, we can force it to be the same, loosely by turning a red mutation into a negative analog of a green mutation and vice versa. Cm, n[i, j] also makes sense for n < m (simply do the mutations from step n to m in reverse), so this lets us understand what the entries of inverse matrices mean, since Cm, n[i, j] = Cn, m− 1[i, j]. Another thing to understand is the following. The formula does not do a good job of highlighting which sequences of ri’s are going to contribute negatively and positively to the coefficient of a given monomial. There might be a way to group the sequences, so each group of sequences will be a nonnegative contributor. It would be very cool if we could recover positivity using this formula. It could also help simplify the formula, by reducing the set of sequences we have to sum over, if we know certain groups give 0. Now we state the analog of the F-polynomial formula for deformed F-polynomials. Given a framed quiver Q and a mutation sequence v1, …, vn, we have that Sn can be calculated as follows. Let W be the set of (possibly empty) sequences ${\textup{\textbf{w}}}$ with 0 ≤ w1 ≤ … ≤ wk ≤ n − 1. $$S\_n = \sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}})W(n,n-w\_k,\ldots,n-w\_1) \prod\_{i=1}^{k}C^{-1}\_n(r\_{w\_i})$$ Consequently, for a given monomial p, over all sequences 0 ≤ w1 ≤ … ≤ wk ≤ n − 1 so that p = ∏i = 1vyi∑j = 1kδn − wj, nCn − wj, n− 1[i, vn − wj] the value of the coefficient is the sum over all the sequences of $$\phi({\textup{\textbf{w}}}) W(n,n-w\_k,\ldots,n-w\_1).$$ This is Theorem [formula], with the yi exponents transformed according to the stabilization rule. This is interesting when we consider the full form of the formula in the skew symmetric case, namely $$\begin{aligned}S\_n = &\sum\_{0\leq w\_1\ldots w\_k\leq n-1} \phi(w\_1,\ldots,w\_k)\prod\_{i=1}^{k} \bigg((a\_{n-w\_i,n}) + \sum\_{j=1}^{i-1} (-C^{-1}\_{n-w\_j,n-w\_i}[v\_{n-w\_i},v\_{n-w\_j}]+ \\ &A^\\_{n-w\_i}A\_{n-w\_i}C^{-1}\_{n-w\_j,n-w\_i}[v\_{n-w\_i},v\_{n-w\_j}])\bigg) \prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}\delta\_{n-w\_j,n}C^{-1}\_{n-w\_j,n}[i,v\_{n-w\_j}]}. \end{aligned}$$ The terms C*n − wj, n− 1[i, vn − wj] appear both in the exponents of the yi terms and as a part of the coefficient calculation. This alludes to why Cn− 1 is a natural operation, and also why it might cause convergence. Finally, if we look at the proof of the theorem, recall that we had a definition of Li and Hn : = ∏j = 1rLjaqj, n where q1...ql represent the sequence of zi (the ri) that are not replaced by 0. This gives a polynomial that aligns with the F-polynomial on all terms where only the rqj are considered. In the case where q1 = 1…qn = n is our sequence, when there are n mutation steps, this process gives the F-polynomial exactly. More specifically, we obtain $$\begin{aligned} L\_1&=(1+z\_1) \\ L\_2&=\left(1+z\_2(1+z\_1)^{-a\_{1,2}+b\_{1,2}}\right) \\ &. \\ &. \\ &. \\ L\_n&=\left(1+z\_n(L\_{n-1}^{-a\_{n-1,n}+b\_{n-1,n}}\cdots L\_1^{-a\_{1,n}+b\_{1,n}})\right) \end{aligned}$$ and Fn = ∏j = 1nLjaj, n The author was informed by Dr. Bernhard Keller that for the skew-symmetric case, this is analogous to a formula found by Nagao in. Finally, we demonstrate an example of our main formula in theorem [formula]. [k2] Let K2 be the quiver with vertices 1 and 2, and two arrows from vertex 1 to vertex 2. Consider the mutation sequence 1, 2, 1. We aspire to find F3 given this sequence. Our formula states $$F\_3 = \sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) W(3, w\_1,\ldots,w\_k) \prod\_{i=1}^{k}r\_{w\_i}$$ where W consists of nondecreasing sequences of 1,2,3’s. As alluded earlier, we can simplify this set by only considering sequences such that ∏i = 1krwi has degree at most the degree of F3 which is 3 in y1 and 2 in y2. Noting that r1 = y1, r2 = y12y2, r3 = y13y22, the sequences in W are [1], [1, 1], [1, 1, 1], [1, 2], [2], [3]. Thus, we obtain (shortening the input to W fixing n = 3), F3 = ϕ(1)W(1)r1 + ϕ(1, 1)W(1, 1)r12 + ϕ(1, 1, 1)W(1, 1, 1)r13 + ϕ(1, 2)W(1, 2)r1r2 + ϕ(2)W(2)r2 + ϕ(3)W(3)r3. To compute the W values, we need to know the relevant ai, j and bi, j values. We know ai, i and bi, i to be 1 and 0 respectively, so we compute the others. These are computed in terms of C-matrix entires. Importantly, we have Agi&= -1 & 2 0 & 1 Agi&= 1 & 0 2 & -1 Ari&= -1 & 0 2 & 1 Ari&= 1 & 2 0 & -1 . Recall ai, j = Ci, j− 1[vj, vi]. This yields a1, 2 = 2, a1, 3 = 3, a2, 3 = 2. Also recall bi, j = Aj\AjCi, j− 1[v*j, vi], which yields b1, 2 = 0, b1, 3 = 1, b2, 3 = 0. Recall that W(n, w1, …, wk) = ∏i = 1k((awi, n) + ∑j = i + 1k( − awi, wj + bwi, wj)). Plugging in, we get W(1) = 3, W(1, 1) = 6, W(1, 1, 1) = 6, W(1, 2) = 2, W(2) = 2, and W(3) = 1. Noting ϕ values, this yields $$F\_3 = 3y\_1 + \frac{1}{2}6y\_1^2 + \frac{1}{6}\cdot 6 y\_1^3 + 2y\_1^3y\_2 + 2y\_1^2y\_2 + y\_1^3y\_2^2.$$ This simplifies to the correct expression of F3 = 3y1 + 3y12 + y13 + 2y13y2 + 2y12y2 + y13y22. Exact F-Polynomials ===================== Symmetric Quivers ----------------- We aspire to compute Fn for graphs with useful symmetry properties. We then use this to directly do the next two cases. Let a Symmetric Quiver Q be one with the following properties. Take a quiver Q with base quiver B and the periodic mutation sequence 1, …, v, 1, …, with the following properties. * The base quiver B is reversible; that is reversing all the arrows and relabeling vertex i with v + 1 − i returns the same quiver. * The sequence of mutations is cyclic; that is mutating B at 1 returns the original quiver upon relabeling vertex i with i − 1, where 1 is relabeled with v. * The base quiver has a symmetrical structure; that is swapping every vertex 2 ≤ i ≤ v with v + 2 − i and fixing vertex 1 returns a quiver with the same arrows pointing in and out of vertex 1. * The quiver is entirely green. For the quiver Q define f(xv, …, x1) = − x1 + ∑edge in B1 → ixi and g(xv, …, x1) = − x1 + ∑edge in Bi → 1xi. Define the recurrence relation si = f(si − 1, …, si − v) and …, s− 2 = s− 1 = 0, s0 = 1. Let sʹi = g(si − 1, …, si − v) [symr] Take a symmetric quiver Q and mutation sequence 1, …, v, 1, …. with the definitions above. Then, rk = ∏i = 1vyisk + 1 − i. We prove the statement by induction. #### Base Case: step $\bm{k=1}$: We have ri = y1 and ∏i = 1vyis1 − i = y1. #### Inductive Step: step $\bm{k \neq 1}$: We are mutating at mutation step k. We need to find the contributors to the frozen vertices pointing into k. Consider the previous cycle of mutation steps, and possible arrows pointing into or out of vertex k when the vertex was mutated (thus contributing to vertex k) Due to symmetry and cyclicity, the things that were pointing in must be precisely what is pointing out of vertex k now, and vice versa. This easily gives the relation ei = f(ei − 1, …, ei − v) where ei is the exponent vector of a given ri. It is worth verifying that this logic also holds when k < v (vertices that have not yet been mutated will have contribution 0). In turn, the relation proves the desired result for k by induction, and so we have proven the lemma. [sym] Take a symmetric quiver Q and mutation sequence 1, …, v, 1, …. Let W be the set of sequences ${\textup{\textbf{w}}}$ with 1 ≤ w1 ≤ … ≤ wk ≤ n. Then $$F\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_j+1-i}}.$$ We have $$F\_n = \sum\_{1\leq w\_1,\ldots,w\_k\leq n} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( (a\_{w\_i,n}) + \sum\_{j=i+1}^{k} (-a\_{w\_i,w\_j}+b\_{w\_i,w\_j}) \right) \prod\_{i=1}^{k}r\_{w\_i}.$$ Since the quiver is reversible, note that Ck− 1 = Ck after swapping indices 0 and v − 1, 1 and v − 2 etc. Furthermore, Ci, k = Ck − i after cyclic shifting the indices. With a little work, one can therefore see that ai, k = sk − i and similarly bi, k = sʹk − i. Further, one can see that a0, i = si and b0, i = sʹi because of the lemma. Therefore, $$F\_n = \sum\_{1\leq w\_1,\ldots,w\_k\leq n} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( (s\_{n-w\_i}) + \sum\_{j=i+1}^{k} (-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i}) \right) \prod\_{i=1}^{k}r\_{w\_i}$$ By lemma [symr], we get $$F\_n = \sum\_{1\leq w\_1\ldots w\_k\leq n} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( (s\_{n-w\_i}) + \sum\_{j=i+1}^{k} (-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i}) \right) \prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_j+1-i}}$$ which is the desired expression. Kr ------ The framed quiver $\bm{K\_r}$ is the quiver with base quiver having two vertices, 0 and 1, and r arrows from 0 to 1. K3 is illustrated below. (1) at (0,0) 1; (2) at (4,0) 2; (1) to [bend right=20] (2); (1) to (2); (1) to [bend left=20] (2); (1’) at (0,2) 1’; (2’) at (4,2) 2’; (1’) to (1); (2’) to (2); [sdef] Define the sequence ${\textup{\textbf{s}}}$ as …, s− 2 = s− 1 = 0, s0 = 1 and sk + 2 = rsk + 1 − sk for k ≥ − 1. The main focus of the papers and are to find explicit formulas for the F-Polynomials of the quivers Kr. The former derives an algebraic expression that appears to be more complicated than ours. The latter interprets the coefficients as the count of a combinatorial object, and thus is incomparable to our work. It is interesting that an explicit formula follows as a simple corollary of our general formula. We can compute Fn for Kr as follows. Let W be the set of (possibly empty) sequences ${\textup{\textbf{w}}}$ with 1 ≤ w1 ≤ … ≤ wk ≤ n. $$F\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( s\_{n-w\_i} - \sum\_{j=i+1}^{k} s\_{w\_j-w\_i} + s\_{w\_j-w\_i-2} \right)y\_1^{\sum\_{i=1}^{k}s\_{w\_i-1}}y\_2^{\sum\_{i=1}^{k}s\_{w\_i-2}}$$ Notice that Kr is a symmetric quiver when r ≥ 2. Recall Theorem [sym] which states that $$F\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_j+1-i}}.$$ Here, ${\textup{\textbf{s}}}$ is defined exactly as in [sdef], and sʹi = − si − 2. Plugging in therefore gives $$F\_n = \sum\_{1\leq w\_1\ldots w\_k\leq n} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( (s\_{n-w\_i}) - \sum\_{j=i+1}^{k} (s\_{w\_j-w\_i}+s\_{w\_j-w\_i-2}) \right) y\_1^{\sum\_{j=1}^{k}s\_{w\_j-1}}y\_2^{\sum\_{j=1}^{k}s\_{w\_j-2}}$$ This is exactly our desired expression. Note that the degree in y1 of Fn is sn − 1, so we only have to sum over sequences with ∑i = 1kswi − 1 ≤ sn − 1. Another thing to note is that this language greatly simplifies Example [k2]. Gale Robinson Quivers --------------------- We now aspire to compute Fn for Gale Robinson Quivers specifically. Define the Gale Robinson Quiver Gv, r, t as follows. * There are v vertices labeled 1, …, v. * For all 1 ≤ i ≤ v − r, draw an arrow i → i + r, and for all 1 ≤ j ≤ r, draw an arrow j → v − r + j. * For all 1 ≤ i ≤ v − t, draw an arrow t + i → i, and for all 1 ≤ j ≤ t, draw an arrow v − s + j → j. * For all 1 ≤ i ≤ v − r − t, draw an arrow from r + i → t + i and for all 1 ≤ j ≤ t − r, draw an arrow r + j → v − t + j. * Delete any 2-cycles created in the above process. We illustrate G7, 2, 3 below. (1) at (0,3) 1; (2) at (2.34,1.86) 2; (3) at (2.91,-0.66) 3; (4) at (1.29,-2.70) 4; (5) at (-1.29,-2.70) 5; (6) at (-2.91,-0.66) 6; (7) at (-2.34,1.86) 7; (1) to (3); (1) to (6); (2) to (4); (2) to (7); (3) to (4); (3) to [bend right=10] (5); (3) to [bend left=10] (5); (4) to (1); (4) to (5); (4) to (6); (5) to (1); (5) to (2); (5) to (7); (6) to (2); (6) to (3); (7) to (3); (7) to (4); (1’) at (0,4) 1’; (2’) at (3.12,2.48) 2’; (3’) at (3.88,-0.88) 3’; (4’) at (1.72,-3.60) 4’; (5’) at (-1.72,-3.60) 5’; (6’) at (-3.88,-0.88) 6’; (7’) at (-3.12,2.48) 7’; (1’) to (1); (2’) to (2); (3’) to (3); (4’) to (4); (5’) to (5); (6’) to (6); (7’) to (7); Gv, r, t has the following properties. * The base quiver B is reversible. * The sequence of mutations is cyclic. * The quiver is entirely green. * Only 1 + r and v + 1 − r point out of vertex 1 and 1 + t and v + 1 − t point into 0 (this implies it is symmetric). The first statement is easy to check, since the arrows are explicitly stated. The second and third are well-known. The fourth, again, is easy to check since the arrows are explicitly stated. Take the quiver Gv, r, t. Define a sequence ${\textup{\textbf{s}}}$ with si as the number of partitions of i into parts r, v − r (0 for negative values of i, 1 for 0). Let W be the set of sequences with 1 ≤ w1 ≤ … ≤ wk ≤ n. Then $$F\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}-s\_{w\_j-w\_i-v}+s\_{w\_j-w\_i-t}+s\_{w\_j-w\_i-v+t} \right)\prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_j-i}}$$ A Gale Robinson Quiver is in fact a symmetric quiver. Therefore, we can just plug into the formula in the previous section. Define si as in the previous section. It is not difficult to see that si is the number of partitions of i into parts r, v − r (0 for negative values of i, 1 for 0). We also have siʹ = si − t + si − v + t − si − v. Thus, literally plugging in to Theorem [sym] gives the desired equation of $$F\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}-s\_{w\_j-w\_i-v}+s\_{w\_j-w\_i-t}+s\_{w\_j-w\_i-n+t} \right)\prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_j-i}}$$ Gale Robinson quivers have been specifically studied in depth already. A paper by Jeong, Musiker and Zhang has a combinatorial interpretation for the coefficients in terms of brane tilings. As graphs admissible on surfaces, they also have combinatorial interpretations in terms of snake graphs and paths on marked surfaces. Furthermore a paper by Glick and Weyman has an explicit formula for the coefficients formed by summing a function over order ideals of a poset. An advantage of our formula over the ones listed is that it is more explicit; it does not abstractly summing over a combinatorial object we do not know how to count, which is a pitfall of all the above formulas. Ours only involves understanding the partition function, which is simple when there are only two possible parts the number can be divided into. Positivity of Exponents for Deformed F-Polynomials ==================================================== In this section, we begin our study of deformed F-polynomials, aspiring to prove that the deformed F-polynomial is, in fact, a polynomial. Take a quiver Q and a sequence of mutations v1, …, vn. Every monomial in any Fn is expressible as the product of not necessarily distinct ri with i ≤ n. Assume there is a framed quiver Q and sequence of mutations v1, v2, … such that some Fn has a monomial not expressible as the product of the ri’s. Take the smallest n for which this is true, and the let the monomial of smallest degree in Fn for which this is true be m. Let Vvn be the old label of vertex vn. Let S1 be the vertices corresponding to the set of base quiver vertices pointing in the same direction as the edges from frozen vertices. Let S2 be the vertices corresponding to the other edges. Then, we have FnVvn = rn∏i ∈ S1Vi + ∏i ∈ S2Vi It is easy to see that each Fi has a positive constant term of 1; namely Vvn does, so then m is a term on the left hand side. Since there are no negative coefficients, the coefficient of m on the left hand side must be positive. However, every term on the right hand side is expressible as a product of ri’s since every term in every Fi is, which is what the Vi’s are, and so is rn. Recall our set Mn for quiver Q and mutations v1, …, vn which is the set of fundamental ri (ri not expressible as the product of other rj). Every monomial in any Fn is expressible as the product of not necessarily distinct elements of Mn. In fact, it’s expressible as the product of not necessarily distinct elements of Mm for any m ≥ n, including M itself. By the previous lemma, we know every monomial is the product of ri’s. Now, if an ri is not fundamental, express it as a product of other ri’s. Repeating this process must result in a product of fundamental ri’s, since degrees are decreasing. Given a framed quiver Q and a sequence of mutations v1, …, vn, m ∈ Mn, its coefficient can be computed as follows: Let g be the number of j ≤ n such that m = rj and rj is green, and let r be the number of such j ≤ n so that rj is red. Let cf(m, Vi) be the coefficient of m in Vi. Then $$-(g-r)C\_n^{-1}(m)=\left<cf(m,V\_1),cf(m,V\_2),\ldots,cf(m,V\_{v})\right>$$ First, we compute the coefficient of m in Ft for any t. If m is fundamental, the only possible sequences of ri’s producting to m consists of the individual ri’s themselves that equal m. For all such ri, plugging into our formula for F-polynomials gives that the inner expression $\phi({\textup{\textbf{w}}}) W(n, w\_1,\ldots,w\_k)$ simplifies to − Ci, t− 1[vn, vi]. Since − Ci, t− 1[vn, vi] = δ0, iCt− 1(m)[vt], summing over all the ri equaling m gives − (g − r)Ct− 1(m)[vt]. Now, we compute Vi. Take t so that t is the last step less than or equal to n where vertex vt is mutated. If no occurrences of m as an ri are between t + 1 and n inclusive, then the coefficient of m is in fact − (g − r)Ct− 1(m)[vt], where the g and r correspond to number of occurrences of m preceding n. Note that Cn− 1 = An⋯At + 1Ct− 1 where each mutation Ai for t + 1 ≤ i ≤ n corresponds to a vertex vi. It is easy to see that left multiplication by this matrix only affects the vith row of the matrix. As such, no mutation affects row vt, so Ct− 1(m)[vt] = Cn− 1(m)[vt]. Now we have the second case, where some occurrence of m proceeds spot t in the mutation sequence. Take a such occurrence of this to be at index s. Then, by the way the C matrix is defined, Cs− 1(m) is the identity vector with a 1 at the spot vm ≠ vt. Then, recalling that the vtth row of the C− 1-matrix does not change between spot t and s and m, we have Ct− 1(m)[vt] = Cs− 1(m)[vt] = Cn− 1(m)[vt] = 0. As such, the coefficient of m, which is − (g − r)Ct− 1(m)[vt] is 0, but this is also equal to − (g − r)Cn− 1(m)[vt] since this is also 0. Thus, in both cases, the expression simplifies to − (g − r)Cn− 1(m)[vt], and we are done. [posexps] Take a framed quiver Q and a mutation sequence v1, …, vn, with the following property: for any fundamental m ∈ Mn, the number of occurrences of i ≤ n such that ri = m and ri is green is greater than the number where ri is red. Then, Sn is a polynomial; that is none of the monomial terms have negative exponents if the rn satisfy this property. Express any monomial in Sn as a product of fundamental terms. Choose a sequence of fundamentals m1, …, mk and express as ∏mi. Since matrix operations are linear, we have that − Cn− 1(∏mi) = ∏ − Cn− 1(mi). We know that each monomial − Cn− 1(mi) has all positive exponents because of the following. Due to positivity of coefficients of Fn, every m ∈ Mn is a valid monomial when acted on by − Cn− 1, since the coefficient vector multiplied by (g − r) is the same as the exponent vector of − Cn− 1(mi), and the multiplication by (g − r) preserves sign. Given a framed quiver Q and a mutation sequence v1, v2, … with every ri green, Sn is a polynomial. This is clear; for any ri the number of green occurrences must be at least the number of red occurrences since there are only green occurrences. Thus, this follows from Theorem [posexps]. Almost conversely to Theorem [posexps], we have the following. [rgfails] Take a framed quiver Q and a mutation sequence v1, …, vn, with the following property: there exists a fundamental m so that the number of occurrences of i ≤ n such that ri = m and ri is red is greater than the number where ri is green. Then, Sn is not a polynomial. Consider the monomial corresponding to m: − Cn− 1(m). We claim it will have all negative exponents. Its coefficient vector in the original cluster of F-polynomials is − (g − r)Cn− 1(m). This has to be always positive due to positivity of coefficients. Since r > g, this gives us that − Cn− 1(m) has all negative entries, as desired. Since we expect that the deformed F-polynomial is always a polynomial, we expect that the implied statement is always true. Given a framed quiver Q and a mutation sequence v1, v2, …, it is true that for any m ∈ M, for any n the number of occurrences of i ≤ n such that ri = m and ri is green is at least the number where ri is red. The reason we cannot get an if and only if statement with this method directly is that the case where g = r for a fundamental monomial m is tricky to deal with. Replacing basic with fundamental provides an approach that circumvents this, though we do not know how to carry it through. [basiccon] Given a framed quiver Q and a sequence of mutations v1, …, vn, p ∈ Pn, the coefficient of p is the following: $$-C\_n^{-1}(p)=\left< cf(p,V\_1),cf(p,V\_2),\ldots,cf(p,V\_{v})\right>$$ If Conjecture [basiccon] is true, then for any framed quiver Q and mutation sequence v1, …, vn, Sn is a polynomial. For any basic monomial, the coefficient vector is the same as the exponent vector, by the conjecture. Any monomial can be rewritten as the product of basics. Each of these primes has positive coefficient vector, and therefore positive exponent vector, so the product will have positive exponent vector. [primecon] Take a quiver Q and a sequence of mutations v1, …, vn. Let ϕn, i be the representation on the base quiver corresponding to the last step of the mutation process up to and including the nth step that occurred at vertex i. Let a monomial p be prime if any subrepresentation of ϕn, i with dimensions corresponding to the degree vector of p is indecomposable. Then the space of subrepresentations of ϕn, i with dimensions corresponding to the degree vector of p has Euler-Poincare characteristic of the quiver Grassmannian equal to − Cn− 1(p)i If Conjecture [primecon] is true, then for any framed quiver Q and mutation sequence v1, …, vn, Sn is a polynomial. For any prime monomial, the coefficient vector is the same as the exponent vector, by the conjecture. Any monomial can be rewritten as the product of primes; rewrite it as the direct sum of two representations, and inductively decompose it into primes. Each of these primes has positive coefficient vector, and therefore positive exponent vector, so the product will have positive exponent vector. Convergence in Periodic Case for Green Quivers ============================================== Take a framed quiver Q and a green mutation sequence v1, …, vn. For any ri, we have that − Cn− 1(ri) is a monomial with positive exponents. We show each ri is the product of fundamentals. Assume ri is not a fundamental. Then, by definition, it is the product of other rj’s. Repeating recursively, it is the product of fundamentals. Then, by a similar argument as before, − Cn− 1(ri) has to have all positive components, since − Cn− 1 of all its components in the decomposition into fundamentals is positive. Given a green quiver Q and a mutation sequence v1, v2, …, all the ri’s are distinct. Assume monomial m is both ri and rj for j > i. Consider the quiver after j mutation steps. First we show m is fundamental. If not, we will have − Cj− 1(m) = ∏ − Cj− 1(ra). However, the left side is just the 0 vector with a 1 at vj, and this has degree 1 so it cannot be the product of other terms. Now we know that the coefficient vector of a fundamental ri is simply − (g − r)Cj− 1(ri) which is 0 everywhere and 2 at vi. In order for this to be true, at the previous step, the coefficient vector had to be 0 everywhere and − 1 at vi. This is because nothing other than vi changed; the new ri contributes 1, so the old term must contribute 1, and it was negated so must have been − 1 before. This contradicts positivity of coefficients. Take a green quiver Q, and a periodic mutation sequence v1, …, vp, …, vn. For any monomial m in the stabilized polynomial, there exists a constant c so that for any sequence w1, …, wk with ( − Cn− 1(rw1))⋯( − Cn− 1(rwk)) = m, then n − c ≤ w1, …, wk, independent of n. Consider instead the monomials after acted on by − Cn− 1. Let the sequence of mutations be 1, …, p, 1, …, ending in a. We have − Cn− 1 = An⋯A1 = Aa⋯A1Ap⋯A1⋯A1. For fixed a, consider each of the monomials Aa⋯A1⋯(eva), at each possible termination point of the matrix product, and assume two of them are equal. Say the length of the sequences at these two termination points are b and c. Then take k larger than both b and c and equal to $a \mod p$;. Then − Ck − b, k− 1(eva) = − Ck − c, k− 1(eva) However then $$C(-C^{-1}\_{k-b,k}(e\_{v\_a}))=C(-C^{-1}\_{k-c,n}(e\_{v\_a})) \implies C\_{k-b}(e\_{v\_{k-b}})=C\_{k-c}(e\_{v\_{k-c}})$$ This means both rvk − b and rvk − c are equal. However, we already showed that all the ri are distinct in a green quiver. Thus, there is some k for any monomial p so that the only occurrence of p has to be at mutation n − k after stabilizing the corresponding ri, assuming $n\equiv a \mod p$. Now, take the max of this value k (call it max(k) for all monomials p∣m over all the congruence classes $\mod p$ (hence over all n)). Since only monomials p∣m can contribute to m, we get that all contributors must be after n − max(k), as desired. Given a green quiver Q, and a periodic mutation sequence v1, …, vp, …, for any i, Si, Si + p, … converges as a formal power series. Consider any monomial m and we show its coefficient converges. Take n larger than the c value for monomial m from the previous lemma. Thus, we have our equation: Sn = ∑0 ≤ w1…wk ≤ n − 1ϕ(w1, …, wk)W(n, n − wk, …, n − w1)∏i = 1vyi∑j = 1kδn − wi, nCn − wi, n− 1[j, vn − wi] which can be rewritten as Sn = ∑0 ≤ w1…wk ≤ cϕ(w1, …, wk)W(n, n − wk, …, n − w1)∏i = 1vyi∑j = 1kδn − wj, nCn − wj, n− 1[j, vn − wj] It would be clear these expressions were equal for sets of n where we could show Cn − w, n− 1 was only dependent on w; i.e. for a fixed w, the same regardless of n. However, this is a property true of all n in a congruence class $\mod p$. Thus, the expression is in fact equal for all n > c. Recall from before that stability was originally conjectured by Eager and Franco in their paper, using the dP1 quiver. As a green quiver, our paper proves this case. We also write an explicit formula for the deformed F-polynomial in the proceeding section. Exact Formulas for Deformed F-Polynomials =========================================== In this section, we present some exact formulas for these limit deformed F-polynomials we have now shown to exist. We restrict to the green quiver case, since that’s where we have shown convergence. Symmetric Quivers ----------------- For this subsection, we write the exact deformed F-polynomials for symmetric quivers. It does not illustrate much, but allows for simpler proofs in proceeding sections. [syms] Let Sn be the deformed F-polynomial for the symmetric quiver Q. Let W be the set of (possibly empty) sequences ${\textup{\textbf{w}}}$ with 0 ≤ w1 ≤ … ≤ wk ≤ n − 1. Then $$S\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_{v\_n+1-i}^{\sum\_{j=1}^{k}s\_{n-w\_j+1-i}}.$$ For this proof, consider vertices by their value $\mod v$. Recall the formula from Theorem [sym] which states $$F\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_j+1-i}}.$$ We simply need to deform the exponents of the monomials. By Lemma [symr], rk = ∏i = 1vyisk + 1 − i so the latter part of the expression can be rewritten as ∏j = 1krwj. Now we deform the exponents to get ∏j = 1k − Cn− 1(rwj) which is ∏j = 1kCwj, n− 1(evwj). Now since the quiver is symmetric, each Ai is the same, with the vertices cyclic shifted according to vi. Thus Cwj, n− 1 will just be AvnAvn − 1⋯Awj where there can be many periods in between. Mapping vn to vertex 1, vn − 1 to vertex 2 and so on, we get that Cwj, n− 1(evwj) is C0, n − wj(evn + 1 − vwj) under the new labeling. However, this is just rn − wj. Then we can use Lemma [symr] to see that this is rn − wj = ∏i = 1vyisn − wj + 1 − i. Returning the vertices to their original labels, we get that each original rk deforms to ∏i = 1vyvn + 1 − isn − k + 1 − i and thus $$S\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_{v\_n+1-i}^{\sum\_{j=1}^{k}s\_{n-w\_j+1-i}}.$$ ${\widetilde}{A\_{1,r}}$ ------------------------ The quiver ${\widetilde}{A\_{1,r}}$ is on vertices 1, …, r + 1 with the following edges: * An edge from i → i + 1 for 1 ≤ i ≤ r. * An edge from r + 1 → 1. In this section, we look for explicit formulas for deformed F-polynomials for the quiver ${\widetilde}{A\_{1,r}}$. In her paper, Grace Zhang looks at the quiver ${\widetilde}{A\_{1,1}}$. She shows the following. Let S∞ be the deformed F-polynomial for the quiver ${\widetilde}{A\_{1,2}}$. Then, up to swapping y1 and y2, we have $$S\_{\infty}=1+\frac{y\_2}{(1-y\_1y\_2)^2}.$$ Our theorem generalizes this result. Let S∞ be the deformed F-polynomial for the quiver ${\widetilde}{A\_{1,r}}$. Then $$S\_{\infty}=1+\frac{(y\_{r+1}+y\_{r+1}y\_{r}+\ldots+y\_{r+1}\cdots y\_2)}{(1-y\_1\cdots y\_{r+1})^2}$$ up to a cyclic shift of variables. Let W be the set of sequences 0 ≤ w1 ≤ … ≤ n − 1. By Theorem [syms] looking at indices modulo r we know that $$S\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_{v\_n+1-i}^{\sum\_{j=1}^{k}s\_{n-w\_j+1-i}}$$ where $s\_i={\left\lceil}\frac{i}{r} {\right\rceil}$. Consider any monomial m that we are trying to find the coefficient of in the stabilized polynomial which has nonzero coefficient. We claim that the exponent of yvn is at most one greater than the exponent of yvn + 1 in any term m with nonzero coefficient. Because ${\textup{\textbf{s}}}$ increases every r terms and there are r + 1 vertices, the difference between the exponent of yvn and yvn + 1 is going to be k. This also means that all sequences creating a given monomial are all the same length. Thus, we just have to show that the only terms with nonzero coefficient in the limit are the ones with all corresponding sequences ${\textup{\textbf{w}}}$ having length 1. Assume some sequence ${\textup{\textbf{w}}}$ with length greater than 1 corresponds to some monomial which has nonzero coefficient in the limit, specifically for Sn and all future Sn + k(r + 1). In Sn it corresponds to w1, …, wk. Then consider the polynomial Sn + n(r + 1). The sequences w1 + n(r + 1), …, wk + n(r + 1) would contribute to this monomial in Sn + n(r + 1). However, in the original un-deformed polynomial, this would mean the corresponding sequence rw1 + n(r + 1)⋯rwk + n(r + 1) products to the corresponding term. However, this terms clearly has degree too large in the original polynomial, so must have 0 coefficient. As such, any term of that form will eventually drop out of the limit. Therefore, it suffices to sum over all sequences ${\textup{\textbf{w}}}$ with length 0 or 1. The sequences with length 0 only create the term 1. Letting Sʹn be the deformed polynomial summing over only length 0 and 1 sequences, we get Sʹn = 1 + ∑1 ≤ w ≤ nsn − w∏i = 1vyvn + 1 − isn − w + 1 − i where $s\_i={\left\lceil}\frac{i}{r} {\right\rceil}$. Since we only care about the yi up to cyclic shift, and instead summing over the length 1 sequences ${\textup{\textbf{q}}}$ with 1 ≤ q1 ≤ n mapping ${\textup{\textbf{w}}}$ to $n-{\textup{\textbf{q}}}$, we can rewrite the above as Sʹn = ∑1 ≤ q ≤ nsq∏i = 1vyv + 1 − isq + 1 − i up to cyclic shift of variables, and taking the limit as n → ∞ gives Sʹ∞ = ∑1 ≤ qsq∏i = 1vyv + 1 − isq + 1 − i. A sequence of exponents occurs if and only if the exponent of yv is one more than the exponent of y1 and the exponent sequence is nonstrictly decreasing. Thus, we get Sʹ∞ = ∑0 ≤ e(e + 1)(yr + 1e + 1yre⋯y1e + yr + 1e + 1 + yre + 1⋯yre + … + yr + 1e + 1⋯y2e + 1y1e). Because the limits of the Sʹ and the S are the same, this simplifies to $$S\_{\infty}=1+\frac{y\_{r+1}(1+y\_{r}+y\_{r}y\_{r-1}+\ldots+y\_{r}\cdots y\_2)}{(1-y\_1\cdots y\_{r+1})^2}$$ as desired. Kr ------ In this section, we investigate the deformed F-polynomial for the Kr quiver. It is worthwhile to note that $K\_2={\widetilde}{A\_{1,1}}$. As such, this case can be viewed as a different generalization of the Zhang’s result for the ${\widetilde}{A\_{1,1}}$ quiver. We redefine the relevant sequences. Define the sequence s as …, s− 2 = s− 1 = 0, s0 = 1 and sk + 2 = rsk + 1 − sk for k ≥ 0. Let $p=\frac{r+\sqrt{r^2-4}}{2}$ Let the norm at b of a sequence 1 ≤ w1 ≤ … ≤ wk be $$N\_b({\textup{\textbf{w}}})=\sum\_{i=1}^{k} b^{w\_i}.$$ The explicit value of any given si for i ≥ 0 can be calculated as follows: $$s\_i = \frac{1}{\sqrt{r^2-4}}p^i - \frac{1}{\sqrt{r^2-4}}p^{-i}.$$ The deformed F-polynomial S∞ is calculated as follows. Let Q be the set of (possibly empty) sequences ${\textup{\textbf{w}}}$ with 0 ≤ w1 ≤ … ≤ wk with $N\_{1/p}({\textup{\textbf{w}}})$ at most 1. $$S\_{\infty}=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( s\_{w\_i} - \sum\_{j=1}^{i} s\_{w\_i-w\_j} + s\_{w\_i-w\_j-2} \right)y\_1^{\sum\_{i=1}^{k}s\_{w\_i}}y\_1^{\sum\_{i=1}^{k}s\_{w\_i-1}}$$ Alternately, $$S\_{\infty}=\sum\_{{\textup{\textbf{w}}}\in W} \phi({\textup{\textbf{w}}}) \prod\_{i=1}^{k} \left( s\_{w\_i} - \sum\_{j=1}^{i} s\_{w\_i-w\_j} + s\_{w\_i-w\_j-2} \right)y\_1^{{\left\lfloor}\frac{1}{\sqrt{r^2-4}}pN\_p(q){\right\rfloor}}y\_2^{{\left\lfloor}\frac{1}{\sqrt{r^2-4}}N\_p({\textup{\textbf{w}}}){\right\rfloor}}$$ up to a permutation of variables. Recall the statement of our formula from Theorem [syms]: $$S\_n=\sum\_{{\textup{\textbf{w}}}\in W} \phi(w\_1,\ldots,w\_k) \left( \prod\_{i=1}^{k}s\_{n-w\_i}+\sum\_{j=i}^{k}-s\_{w\_j-w\_i}+s'\_{w\_j-w\_i} \right)\prod\_{i=1}^{v}y\_{v\_n+1-i}^{\sum\_{j=1}^{k}s\_{n-w\_j+1-i}}.$$ Noting that there are only two vertices, this simplifies to Sn = ∑1 ≤ w1…wk ≤ nϕ(w1, …, wk)∏i = 1k((swi) − ∑j = 1i − 1(swj − wi + swj − wi − 2))y1∑j = 1kswiy2∑j = 1kswi − 1 where y1 and y2 are swapped if n is even. We ignore this distinction in the proof that follows. This is exactly our desired expression, except we need to limit the set of sequences we are summing over, using the norm condition. We take a monomial m with degree in y1 being d. Then, all possible sequences w that allow for this monomial have at most d terms and all wi < d (d is a very crude bound, but suffices). Take n extremely large. Now, we look at the degree of the term corresponding to w in the original F-polynomial. We want to show both that if the norm is at most 1, the degree will be this must be less than or equal to sn − 1, and if norm is greater than 1 it will be larger. The degree is $$\sum\_{i=1}^{k}s\_{n-w\_i} = \frac{1}{\sqrt{r^2-4}}(p^nN\_{1/p}(w) - \sum\_{i=1}(1/p)^{n-w\_i})$$ by using the explicit form of the si. sn − 1 can be expressed as $$\frac{1}{\sqrt{r^2-4}}p^n- \frac{1}{\sqrt{r^2-4}}p^{n}$$ When n goes to infinity, the first term of each expression dominates, and this easily proves the desired result; the first is smaller when norm is less than 1, and larger otherwise. If the norm is exactly 1, the first terms of each expression are equal, so we compare the second term. The first expression clearly has a smaller first term, so the result holds in this case as well. Now we prove the second part; that $$\sum\_{j=1}^{k}s\_{w\_i}={\left\lfloor}\frac{1}{\sqrt{r^2-4}}N\_p(q){\right\rfloor}$$ and $$\sum\_{j=1}^{k}s\_{w\_i-1}={\left\lfloor}\frac{1}{\sqrt{r^2-4}}pN\_p(q){\right\rfloor}$$ for a sequence with norm at most 1. For the first part, first rewrite $$\sum\_{j=1}^{k}s\_{w\_i}=\frac{1}{\sqrt{r^2-4}}(pN\_p(w)-p^{-1}N\_{1/p}(w))$$ The second term (after expanding) is strictly greater than 0 and less than 1, and since the expression must be an integer, we have proven our desired result. We have $$\sum\_{j=1}^{k}s\_{w\_i}=\frac{1}{p\sqrt{r^2-4}}(N\_p(w)-N\_{1/p}(w))$$ Again the second term after expanding has to be greater than 0 and less than 1, proving the desired result. We now find another formulation for the deformed F-polynomial of the quiver Kr highlighting exactly which monomials have nonzero coefficient. The coefficient of the term y1⌊pa⌋ + 1y2a for integers a ≥ 0 are the only ones that are nonzero, and are expressed as follows. Let W be the set of sequences ${\textup{\textbf{w}}}$ with 0 ≤ w1 ≤ … ≤ wk and with $\sum\_{{\textup{\textbf{w}}}\in W}s\_{w\_i-1}=a$. Then, the coefficient corresponding to that value of a is ϕ(w1, …, wk)∏i = 1k((swi) − ∑j = 1i − 1(swj − wi + swj − wi − 2)) We need to find $${\left\lfloor}p{\left\lfloor}\frac{1}{\sqrt{r^2-4}}N\_p(q){\right\rfloor}{\right\rfloor}+1$$ Recall from the previous proof that our expression of ${\left\lfloor}\frac{1}{\sqrt{r^2-4}}N\_p(q){\right\rfloor}$ is also $$\frac{1}{\sqrt{r^2-4}}N\_p(q)-\frac{1}{\sqrt{r^2-4}}N\_{1/p}(q)$$ Multiplying by p gives $$\frac{p}{\sqrt{r^2-4}}N\_p(q)+\frac{p}{\sqrt{r^2-4}}N\_{1/p}(q)$$ which is $$\sum\_{i=1}^{\|q\|}s\_{w\_i}+\frac{1}{p\sqrt{r^2-4}}N\_{1/p}({\textup{\textbf{w}}})-\frac{1}{\sqrt{r^2-4}}N\_{1/p}({\textup{\textbf{w}}})$$ which is $$\sum\_{i=1}^{\|q\|}s\_{w\_i}+\frac{p-1}{p}\frac{1}{\sqrt{r^2-4}}N\_{1/p}(q)$$ The second term is non-strictly less than 1, but greater than 0, so taking the floor and adding 1 gives ∑i = 1∣q∣swi as desired for the first part. For the second part, we know that we get a given term by summing the expression ϕ(w1, …, wk)∏i = 1k((swi) − ∑j = 1i − 1(swj − wi + swj − wi − 2)) over sequences ${\textup{\textbf{w}}}$ with 0 ≤ w1 ≤ … ≤ wk with $\sum\_{{\textup{\textbf{w}}}\in W}s\_{w\_i-1}=a$. Thus, it suffices to prove that $a={\left\lfloor}\frac{p}{\sqrt{r^2-4}}N\_p(q) {\right\rfloor}$. Note that $$s\_{w\_i-1}=\frac{1}{p\sqrt{r^2-4}}N\_p({\textup{\textbf{w}}})-\frac{p}{\sqrt{r^2-4}}N\_{1/p}({\textup{\textbf{w}}}).$$ The second term is less than 1, so we get that the overall expression is just the floor of the first part, as desired. The deformed F-polynomial is interesting for two reasons: it allows for the convenient norm property to determine which monomials are included. Further, the renormalization has y1 and y2 uniquely determine each other. Gale Robinson Quivers --------------------- We now look at the explicit deformed F-polynomial for the Gale Robinson Quivers. Essentially we just plug into our explicit formula and nothing extremely interesting happens, but it allows us to completely close the Eager and Franco’s original conjecture data around the dP1 quiver. Take the quiver Gv, r, t. Define a sequence ${\textup{\textbf{s}}}$ with si as the number of partitions of i into parts r, v − r (0 for negative values of i, 1 for 0). Let W be the set of sequences ${\textup{\textbf{w}}}$ with 0 ≤ w1 ≤ … ≤ wk. $$S\_{\infty}=\sum\_{{\textup{\textbf{w}}} \in W} \phi(w\_1,\ldots,w\_k)\prod\_{i=1}^{k} \left( s\_{w\_i} + \sum\_{j=1}^{i-1} (-s\_{w\_j-w\_i}-s\_{w\_j-w\_i-v}+s\_{w\_j-w\_i-t}+s\_{q\_j-q\_i-n+t}) \right) \prod\_{i=1}^{v}y\_i^{\sum\_{j=1}^{k}s\_{w\_i-j+1}}$$ We have by Theorem [syms] that Sn = ∑0 ≤ w1…wk ≤ n − 1ϕ(w1, …, wk)∏i = 1k(swi + ∑j = 1i − 1( − swj − wi − swj − wi − v + swj − wi − t + swj − qi − n + t))∏i = 1vyi∑j = 1kswi − j + 1 since that is how to calculate the corresponding sequences, and so taking the limit gives S∞ = ∑0 ≤ w1…wkϕ(w1, …, wk)∏i = 1k(swi + ∑j = 1i − 1( − swj − wi − swj − wi − v + swj − wi − t + swj − wi − n + t))∏i = 1vyi∑j = 1kswi − j + 1 Now we find the exact deformed F-polynomial for the quiver dP1 = G4, 2, 1. If we can, it is always more interesting to calculate the coefficient of a given monomial directly, rather than summing up a number of terms that could possibly be contributing to the same polynomial. We will do this for the dP1 quiver, since the sequence ${\textup{\textbf{s}}}$ is easy to understand. Take the quiver dP1 and mutation sequence 1, 2, 3, 4, 1, 2, … which has n terms. The coefficient of the term y1ay2by3cy4d is calculated as follows. Let W be the set of sequences of nonnegative integers and half-integers so that * a − c terms are integers. * b − d terms are half-integers. * The sum of all the integer terms is c. * The sum of the floors of all the half integer terms is d. Then the desired coefficient is $$\sum\_{{\textup{\textbf{w}}}\in W}\phi(w\_1,\ldots,w\_k)\prod\_{i=1}^{k} \left( \frac{(-1)^{2w\_i}+1}{2}(w\_i+1) - \sum\_{j=1}^{i-1} (-1)^{2w\_j-2w\_i}(2w\_j-2w\_i + 4) \right)$$ Let s(k) be k if k is a positive integer, and 0 otherwise. Noting that sk as defined in the Gale Robinson quiver is $\frac{k}{2}$ if k is even and 0 otherwise, plugging in gives: $$S\_{\infty}=\sum\_{0\leq w\_1 \ldots w\_k, w\_i} \phi(w\_1,\ldots,w\_k)\prod\_{i=1}^{k} \left( \frac{((-1)^{w\_i}+1)w\_i}{4} - \sum\_{j=1}^{i-1} (-1)^{w\_j-w\_i}(w\_j-w\_i + 4) \right) \prod\_{i=1}^{4}y\_i^{\sum\_{j=1}^{k}\frac{(-1)^{w\_i-j+1}+1}{4}}$$ Letting $q\_i=\frac{w\_i}{2}$ gives the following expression. $$S\_{\infty}=\sum\_{q\in Q} \phi(q\_1,\ldots,q\_k)\prod\_{i=1}^{k} \left( q(w\_i) - \sum\_{j=1}^{i-1} (-1)^{2q\_j-2q\_i}(2q\_j-2q\_i + 4) \right) \prod\_{i=1}^{4}y\_i^{\sum\_{j=1}^{k}s(q\_j-\frac{1}{2}i)}$$ For any term y1ay2by3cy4d it is contributed to by the sequences described in the corollary statement. Adding the contributions proves the desired result. Future Research =============== The Formula ----------- One important avenue of future research involving our formula in Theorem [formula] consists of finding more use cases for the formula. For example, it might be possible to group terms and use the formula to recover positivity of coefficients, at least in certain cases. We might be able to do this by grouping terms of the formula. Furthermore, we know that the Laurent Phenomenon proves that the F-polynomials are in fact polynomials. However, our formula has nontrivial expansion for the coefficient of all monomials. Thus, for monomials that are too large, or obviously otherwise have coefficient 0, we get an expression involving the C-matrix entries that must be zero. We have not yet investigated these; it might be interesting. It might even help with positivity; one might be able to say that if large terms have coefficient 0, the smaller terms must have coefficient positive. The following provides intuition for why such a proof might work. [posex] Take a quiver Q and a mutation sequence v1, …, vn. If each rik can only be constructed as a product of ri’s, and not any of the other terms, then the coefficient of ri must be positive. The coefficient of ri by our formula is ai, n, and the coefficient of rik is $$\frac{1}{k!}\prod\_{i=1}^{k}\left(a\_{i,n}-\sum\_{j=1}^{k-1}(a\_{i,i}-b\_{i,i})\right)$$ which is $$\frac{1}{k!}\prod\_{i=1}^{k}( a\_{i,n}-k+1)$$ This is only 0 for large k when ai, n is positive. This idea is complicated to generalize, since in large enough monomials, every part-contribution (the particular sequence we are summing over) need not be 0. However, there might be a clever way to group terms to deal with this. Another thing we want to do with our formula is re-express it without the ϕ(w1, …, wk) term. Our formula currently reads $$F\_n = \sum\_{{\textup{\textbf{w}}}\in W} \phi(w\_1,\ldots,w\_k) W(n, w\_1,\ldots,w\_k) \prod\_{i=1}^{k}r\_{w\_i}$$ over all sequences ${\textup{\textbf{w}}}$ with 1 ≤ w1 ≤ … ≤ wk ≤ n. This can be rewritten as $$F\_n = \sum\_{{\textup{\textbf{q}}}\in Q} \frac{1}{k!} W(n, \sigma(q\_1,\ldots,q\_k)) \prod\_{i=1}^{k}r\_{q\_i}$$ over all sequences ${\textup{\textbf{q}}}$ with 1 ≤ q1, …, qk ≤ n (not increasing order), where σ(q1, …, qk) is the sequence permuted to be in increasing order. This is because the extra occurrences of each sequence ${\textup{\textbf{w}}}$ exactly account for the ϕ term. However, this formula isn’t any more insightful than the the original one, because the W term still involves permuting the sequence to be in increasing order. Our goal would be to find an explicit expression Wʹ(n, q1, …, qk) so that $$F\_n = \sum\_{{\textup{\textbf{q}}}\in Q} \frac{1}{k!} W'(n, q\_1,\ldots,q\_k) \prod\_{i=1}^{k}r\_{q\_i}.$$ In particular, we want an explicit formula for Wʹ(n, q1, …, qk) = W(n, σ(q1, …, qk)) that does not involve first permuting the sequence ${\textup{\textbf{q}}}$ into increasing order. Another possible avenue of research would be to see how our work relates to dilogarithm identities. Keller and Nakanashi discuss these identities, and they appear to be an area where explicit formulas may allow for better results. Deformed F-Polynomials ------------------------ Regarding deformed F-polynomials, or S-polynomials we have the following three conjectures. Given a framed quiver Q and a mutation sequence v1, v2, …, Sn is actually a polynomial; that is none of the monomial terms have negative exponents. Given a framed quiver Q and a mutation sequence v1, v2, …, Sn is bounded; i.e. for each monomial term ∏xiai, there is some c(a1, …, an) dependent on Q and the mutation sequence so that the coefficient of ∏xiai in Sn is less than c(a1, …, an). Given a framed quiver Q and a periodic mutation sequence v1, …, vp, v1, …, vp, …, such that performing the mutations v1, …, vp on the base quiver returns the quiver back to the original, for every i, the sequence Si, Si + p, S*	arxiv_0000523
Towards the Erds-Gallai Cycle Decomposition Conjecture ====================================================== In the 1960’s, Erds and Gallai conjectured that the edges of any n-vertex graph can be decomposed into O(n) cycles and edges. We improve upon the previous best bound of O(nloglogn) cycles and edges due to Conlon, Fox and Sudakov, by showing an n-vertex graph can always be decomposed into O(nlog⋆n) cycles and edges, where log⋆n is the iterated logarithm function. Introduction ============ When is it possible to decompose a graph into edge disjoint subgraphs with certain properties? Many classical problems in extremal combinatorics fall within this framework and its natural hypergraph generalisation, while decomposition problems have strong links to many other fields, including the design of experiments, coding theory, complexity theory and distributed computing (see, for example, ). The particular case where we seek to decompose a graph into cycles has a long history, dating back to the 18th century and Euler’s result on the existence of Euler tours. As Veblen observed for his algebraic approach to the Four-Colour Theorem, Euler’s result immediately implies that any graph with even vertex degrees (i.e., any Eulerian graph) has a decomposition into cycles. As it is immediate that any graph with a vertex of odd degree cannot be decomposed into cycles, this exactly characterises which graphs have cycle decompositions. Another very classical cycle decomposition result is due to Walecki from 1892, who showed it is possible to decompose any complete graph with an odd number of vertices into Hamilton cycles. This gives a cycle decomposition into few cycles, indeed, into optimally few cycles. This raises a very natural question of whether every Eulerian graph has a cycle decomposition into few cycles? That only O(n) cycles might be needed to decompose any n-vertex Eulerian graph is easily seen to be equivalent to the following classical conjecture of Erds and Gallai dating back to the 1960’s, which is one of the major open problems on graph decompositions. [conj:EG] Any n-vertex graph can be decomposed into O(n) cycles and edges. While is equivalent to conjecturing that every n-vertex Eulerian graph can be decomposed into O(n) cycles, as noted above, if they both hold then the optimal implicit constants in these conjectures seem likely to be different. For the Eulerian problem, Hajós conjectured that $\frac n2$ cycles should be sufficient (see also ), while the best known lower bound for the number of cycles and edges required in is $(\frac{3}{2}-o(1))n$, as observed by Erds in 1983, improving on a previous construction of Gallai (see ). Since its formulation, the Erds-Gallai Conjecture has often been highlighted (see, for example, ), with Erds himself mentioning it in many of his open problem collections. Despite this attention, and a lot of work on related problems over the years, direct progress towards the Erds-Gallai Conjecture has only been made within the last decade. The previous related results, which we discuss first, are mostly on the analogous path decomposition problem and the covering version of the Erds-Gallai conjecture. Path decompositions. In the 1960’s, Gallai posed the analogous path decomposition version of. In particular, he conjectured that any connected n-vertex graph can be decomposed into at most $\frac{n+1}{2}$ paths. Lovász in 1968 proved that any graph can be decomposed into at most n − 1 paths. This follows easily from his complete solution to the problem of how many paths or cycles one needs to decompose an n-vertex graph, to which the answer is ${ \left\lfloor \frac n2 \right\rfloor }$. Currently, the best general bound in the path decomposition problem is due independently to Dean and Kouider and Yan , who showed that any graph can be decomposed into at most $\lfloor \frac23 n\rfloor$ paths. Gallai’s path decomposition conjecture is known to hold for quite a few special classes of graphs, with connected planar graphs being the most recent addition to the list. This latest result is due to Blanché, Bonamy and Bonichon in, where a more exhaustive list of partial results can also be found. Covering problems. Another interesting direction which has attracted a lot of attention is the covering version of, in which we do not insist that the cycles we find should be disjoint, only that together they contain all the edges of the host graph. In 1985, Pyber proved the covering version of the Erds-Gallai conjecture, showing that the edges of any n-vertex graph can be covered with n − 1 cycles and edges. The analogous covering version of Gallai’s conjecture, raised by Chung in 1980, has been settled first approximately by Pyber in 1996 and then completely by Fan in 2002, who showed that the edges of any connected graph can be covered by ${ \left\lceil \frac{n}{2} \right\rceil }$ paths. The covering version of Hajós’s conjecture was also solved by Fan , who showed that any n-vertex Eulerian graph can be covered by at most $\lfloor \frac{n-1}{2}\rfloor$ cycles, settling another conjecture of Chung. As with the other two covering results above, this bound is best possible. Results on the Erds-Gallai conjecture. In more recent years, the Erds-Gallai conjecture (along with more accurate results on the implicit bounds) has been shown to hold for two large specific classes of graphs – random graphs and graphs with linear minimum degree. The conjecture was first established for a typical binomial random graph G(n, p) (for any p = p(n)) by Conlon, Fox and Sudakov. Korándi, Krivelevich, and Sudakov found the correct leading constant here, showing that $(\frac{1}{4}+\frac{p}{2}+o(1))n$ cycles and edges are typically sufficient to decompose G(n, p). For constant edge probability p, Glock, Kühn, and Osthus were even able to determine with high probability the exact minimum number of cycles and edges required to decompose a (quasi)random graph. On the other hand, the Erds-Gallai conjecture was first shown to hold for graphs with linear minimum degree again by Conlon, Fox and Sudakov. Very recently, the asymptotically correct bound of $(\frac32+o(1)) n$ cycles and edges has been proved by Girão, Granet, Kühn, and Osthus for large graphs with linear minimum degree. A fundamental challenge towards establishing the Erds-Gallai conjecture is its generality, and indeed these previous results make progress only by imposing a fairly strong constraint on the structure or randomness of the graph. For almost 50 years, the best known bound in the general case of the Erds-Gallai conjecture (as observed by Erds and Gallai) came from a simple argument involving the iterative removal of a longest cycle, which shows that an n-vertex graph can always be decomposed into O(nlogn) cycles and edges. In 2014, Fox, Conlon and Sudakov made the first major breakthrough on this problem, showing that such a decomposition with only O(nloglogn) cycles and edges always exists. Here we will give the following improvement on this bound, where log⋆n is the iterated logarithm function. theoremlogstar [thm:logstar] Any n-vertex graph can be decomposed into O(nlog⋆n) cycles and edges. Key to the decompositions used by Conlon, Fox and Sudakov was to show that a) graphs H with certain expansion properties can be decomposed into O(∣H∣) cycles and few edges and b) any n-vertex graph G can be decomposed nicely into such ‘expanders’ H and a small number of leftover edges. Combined, this gives a decomposition of G into O(n) cycles and some leftover edges, and it can be shown that iterating this on the leftover edges while removing any particularly long cycles causes the average degree of the leftover edges to drop significantly each time, so that after loglogn iterations the decomposition given by is achieved. To prove, essentially, we need the average degree of the leftover edges to drop much faster, and so at b) we have to take a much weaker condition on the ‘expanders’ H. Effectively we replace the strong expansion used in, with a very weak sublinear expansion (as introduced by Komlós and Szemerédi ), in particular using a robust sublinear expansion where sets expand sublinearly despite the additional removal of a possibly-superlinear set of edges (see for a discussion of these forms of expansion and their background). Using this much weaker form of expansion introduces a raft of issues when we decompose an expander into few cycles and edges (for a) above), resulting in a very different approach to that used in. In order to do this, we introduce a range of new tools, which we hope will find further applications. In particular, we would highlight a new approach to robust sublinear expansion (see ) and the (surprisingly difficult) result that randomly sampling the vertices of an expander is likely to induce a subgraph with a (somewhat weaker) expansion property (see ). Additional new tools include a similar result but while randomly sampling edges, the (almost) decomposition of any graph into robust sublinear expanders, and the finding of a sparse ‘connective skeleton’ in expanders to connect vertex pairs with paths. These, and other tools, and how they come together to prove Theorem [thm:logstar], are discussed in. As discussed in, sublinear expansion has been useful in many different settings in which our tools may also be useful (see for some examples). In particular, robust sublinear expansion (specifically considering the deletion of superlinearly many edges) is a very recent concept and we hope our new perspective and tools will contribute to its development and use. Several of our intermediate results and tools might also ultimately prove useful towards proving the Erds-Gallai conjecture in full as they often decompose any n-vertex graph into O(n) cycles and a graph with some other structure imposed. This is discussed further in our concluding remarks in. Preliminaries ============= After we introduce our notation, we give a detailed sketch of our methods before outlining the rest of the paper. Notation -------- Given a graph G we will denote by V(G) and E(G) its vertex and edge set, respectively. Given a vertex v ∈ V(G), we denote its degree by dG(v) and the set of its neighbours by NG(v). We write Δ(G) for the maximum degree of a vertex in a graph G. Given a subset of vertices U ⊆ V(G) we denote by NG(U) the set of vertices in V(G) \ U which have a neighbour in U. Given U ⊆ V(G) we define BGi(U) as the set of vertices at distance at most i from a vertex of U in the graph G, i.e. the ball of radius i around U in G, and write simply BG(U) = BG1(U). Given V ⊆ V(G) we write G[V] for the subgraph of G induced by the vertex set V, and write G \ V for G[V(G) \ V]. Given F ⊆ E(G) we write G − F for the subgraph of G obtained by deleting all the edges in F. Given multiple (hyper)graphs H1, …, Ht we write H1 ∪ … ∪ Ht for the (hyper)graph with vertex set ⋃i ∈ [t]V(Hi) and edge set ⋃i ∈ [t]E(Hi). Given vertices v and u, by a vu-path/walk we refer to a path/walk joining v and u. We write X ∼ Bin(n, p) to mean that X is a random variable distributed according to the binomial distribution with parameters n and p. We denote by G(n, p) the binomial random graph defined as the graph with vertex set [n] in which we sample every edge with probability p independently from all other edges. We write G ∼ G(n, p) to mean that G is sampled according to G(n, p). For each n ≥ 1, [n] = {1, …, n}. All our logarithms have base two. For each k ≥ 1, let $\log^{[k]}(n)=\underbrace{\log \log \ldots \log}\_{k \text{ times}} n$, and let log[0]n = n. The iterated logarithm function log⋆n is the minimum number of times we need to apply the logarithm function to n until it becomes at most one, that is, the least k ≥ 0 such that log[k]n ≤ 1. Throughout the paper, we make no attempt to optimise constants and logarithmic factors; often, we are wasteful to improve readability. With the same goal, we also omit floor and ceiling signs wherever they are not crucial. Proof sketch ------------ Our methods to find an (edge) decomposition into cycles and edges is iterative, where a single iteration, applied to an n-vertex graph G with average degree d, performs the following steps for some appropriately large constant C. * Repeatedly remove any cycle of length at least d and add it to the decomposition, giving at most $\frac n2$ new cycles. * Decompose G into edge disjoint subgraphs Ri with a certain expansion property which, combined with the lack of long cycles, guarantees that ∣Ri∣ = O(dlog4d). These subgraphs are almost vertex disjoint, so that ∑i∣Ri∣ ≤ 2n. * Decompose each of the subgraphs Ri into cycles and edges, using in total O(n) cycles and O(nlogCd) edges. This finds a decomposition of G into O(n) cycles and a subgraph consisting of leftover edges of average degree at most O(logCd). We now iterate by applying the same argument to our much sparser graph consisting of leftover edges. After at most O(log⋆d) iterations we will be left with a graph of constant average degree. We then simply make all its edges part of our decomposition, which together with the O(n) cycles we found at each of the O(log⋆d) iterations gives our desired decomposition for – in fact, using only O(nlog⋆d) edges for any n-vertex graph with average degree d. The majority of our work lies in carrying out the final step in this iteration, where the key part of this step is to show the following intermediate result. theoremdecompcyclelinear [thm:decompexander] There exists C > 0 such that any r-vertex graph decomposes into O(r) cycles and O(rlogCr) edges. Note that this intermediate result is applied to graphs Ri, each of which has order at most O(dlog4d), so that, as ∑i∣Ri∣ ≤ 2n, we get in total O(n) cycles and O(∑i∣Ri∣logC∣Ri∣) = O(∑i∣Ri∣logCd) = O(nlogCd) edges in total from the decomposition in the final step of the iteration. Though we use a slightly modified iteration argument, till now our approach has the same structure as the one taken by Conlon, Fox and Sudakov in, where they prove a weaker version of in which they allow, instead of O(rlogCr), up to O(r2 − 1/10) edges in the decomposition. This much weaker bound leads them to iterate O(loglogd) times and thus use a decomposition using O(nloglogd) cycles and edges in total. The key difference is that we are able to replace the extremely strong expansion properties used in with a very weak form of expansion. To make this change successfully, we need to carefully develop the weak expansion property we use (which originates with Komlós and Szemerédi ) as well as solve the variety of problems caused by working with this weak expansion. For this development, the main insight is the new perspective we bring to robust sublinear expansion. That we can decompose an arbitrary graph into edges and expanders follows relatively naturally from the definition of robust sublinear expansion (see for more details on the expander partitioning lemma), and the same decomposition result allows us to carry out the second step of the iteration mentioned above. This leaves the difficult task of decomposing an expander graph into few cycles and edges, which we now discuss. Decomposing expanders into few cycles and edges. For some constant C > 0, we now assume that we wish to decompose an n-vertex graph G with the following (slightly simplified) expansion condition (see for the full condition we use): for each U ⊆ V(G) and F ⊆ E(G) with $\|U\|\leq \frac23 n$ and ∣F∣ ≤ ∣U∣logCn, we have $$\label{expandprop} \|N\_{G-F}(U)\|\geq \frac{1}{\log^2n}\|U\|.$$ As the size of the neighbourhood guaranteed in is smaller than ∣U∣, this type of expansion is known as sublinear expansion. Roughly speaking, our main strategy is to set aside a ‘sparse connecting skeleton’ H ⊆ G, before initially decomposing the edges of G − H into O(n) paths and cycles using a result of Lovász stated in the introduction. We then use short paths from the connecting skeleton H to connect up each initial path into a cycle, before simply taking each unused edge of H as part of our decomposition. In order for this to produce a correct decomposition we need H to be very sparse, in particular with at most nlogO(1)n edges. To aid the connection of the paths from the initial path/cycle decomposition, we need that each vertex does not appear too often as an endvertex of these paths. This we ensure by proving a simple, but crucial, corollary of Lovász’s result (see ), which will allow us to decompose G − H into a collection P of O(n) paths in which each vertex appears as an endvertex at most twice, so that the endvertices of the paths are well spread across the graph. More problematically, note that in order to get an actual cycle we need to connect the endvertices of each path P ∈ P using a path in H which is internally vertex disjoint from P. To deal with this, we change this outline slightly as follows. We partition V(G) = V1 ∪ V2 ∪ V3 by placing each vertex independently into a set Vi uniformly at random, and show that G contains sparse subgraphs H1, H2, H3 (each with nlogO(1)n edges) with the following property for each i ∈ [3], where a path through Vi is one whose interior vertices are all in Vi. We place no restriction on the endvertices themselves, so in particular a single edge path is a path through any set since it contains no interior vertices. 1. For any set $\mathcal{P}\subseteq \binom{V(G)}{2}$ such that each vertex appears in at most 2 pairs in P, there are edge disjoint paths Pxy, {x, y} ∈ P, such that, for each {x, y} ∈ P, Pxy is an xy-path through Vi in Hi with length O(log7n).[prop:sketch] We then split the edges of G − H1 − H2 − H3 into three subgraphs G1, G2, G3 in such a way that V(Gi) = Vi + 1 ∪ Vi + 2, for each i ∈ [3], with indices taken modulo 3. Applying our path decomposition corollary to each Gi then gives a decomposition of all the edges outside of H1 ∪ H2 ∪ H3 into paths whose endvertices are well spread across the graph. Note that the paths decomposing Gi completely avoid Vi so by using the property [prop:sketch] we can connect each of these paths into actual cycles using edges of Hi. In total we find O(n) edge disjoint cycles which use all the edges of G − H1 − H2 − H3 so the total number of uncovered edges, which all belong to H1 ∪ H2 ∪ H3, is small. With such a weak expansion property as that at, whether we can do this is initially far from clear. Building up to this, we ask the following three questions. 1. Can we connect pairs of vertices with edge disjoint paths using the whole of G? I.e., does property [prop:sketch] hold if Hi = G and Vi = V(G)? 2. If so, can we do this using only a random subset of vertices Vi for the interior vertices of the paths? I.e., does property [prop:sketch] hold if Hi = G? 3. If so, can we do this using only a sparse subgraph Hi of G? I.e., can we find a sparse Hi ⊆ G so that property [prop:sketch] holds for Hi? *i) Finding edge disjoint paths in G.* The expansion condition on G at is sufficient to imply that any pair of vertices in H are connected by a path of length O(log3n) by expanding the neighbourhoods around each of the two vertices until they become large enough that they must overlap. Moreover, the robustness of our condition at (i.e., that this expansion can avoid using an arbitrary, but not too large, set of edges F) allows us, with only a bit more work, to show that, for any collection P of pairs of vertices as in property [prop:sketch], we could find at least Θ(log4n) ⋅ ∣P∣ edge disjoint paths in G which each connect some vertex pair in P and have length O(log3n). As this holds in fact for any subset Pʹ ⊆ P in place of P, this allows us to use the Aharoni-Haxell hypergraph matching theorem (see ) to select, for each pair {x, y} ∈ P, an xy-path in G, so that all these paths are edge disjoint. In total this allows us to answer question i) positively. *ii) Connecting through the random vertex subset V*i.** With i ∈ [3] and Vi a random subset of V with size approximately $\frac{n}3$ as chosen above, unfortunately it seems it does not follow easily that H[Vi] likely satisfies a similar expansion property to . Indeed, firstly, if the inequality is tight for a set U, then with probability exp( − Θ(∣U∣/log2n)) we have that U has no neighbours selected into our random subset Vi, too high a probability to naively take a union bound to avoid this event over all sets U with any fixed size. Secondly, the robustness condition we need in order to find multiple edge disjoint paths (to then apply the Aharoni-Haxell hypergraph matching theorem) requires us to avoid an arbitrary set of ∣U∣logO(1)n edges, and again there are too many choices to just take a union bound. The first problem here is the most difficult to overcome. The second problem can be overcome by splitting the edges of G randomly into t subgraphs Gi, i ∈ [t], for some t = logO(1)n, and showing then that (with high probability) each of these has some (slightly weaker) expansion property (see ). When we look for edge disjoint collections of paths in each graph Gi separately for the application of the Aharoni-Haxell hypergraph matching theorem, by finding $\frac 1t$ fraction of the required paths in each graph Gi we need to avoid fewer edges by expanding in each Gi separately rather than G. Thus, we have fewer sets of edges over which to take a union bound, solving the second problem above. The splitting into subgraphs is done in Section [sec:edge-partition-of-expanders] after a brief discussion of the parameters (see also the discussion at the start of Section [sec:connectinexpander]), but effectively this approach works because the splitting is very efficient: we do not need to reduce the lower bound in by more than a factor of 4 despite splitting into polylogarithmically many subgraphs (the upper bound on ∣F∣ for which this holds, though, will decrease quite a bit more). Solving the first problem to get some weak expansion into Vi is the crux of this paper, and is where our new perspective on robust sublinear expansion is critical. We will show that it follows from this new perspective that (as stated more precisely in ), for such sets U and F, we have either 1. NG − F(U) is actually much larger than guaranteed by, or 2. there is a set Uʹ ⊆ U which is much smaller than U but whose neighbourhood alone contains at least (essentially) $\frac{\|U\|}{3\log^2n}$ vertices in NG − F(U). Given this, a natural approach is to take a union bound over all ‘well-expanding’ sets U, meaning that they fall under condition 1. above, to guarantee a constant fraction of their neighbourhood gets sampled into our random set Vi. Since the subset Uʹ from condition 2. is well-expanding this will guarantee us that it expands inside Vi. We would now like to use the fact that Uʹ expands inside Vi to conclude the same happens for the original set U from condition 2. above containing Uʹ. However, a major issue here is that in order to achieve this we would need a bound on ∣(NG − F(Uʹ) \ U) ∩ Vi∣ to get the expansion for any relevant set U for which we use the well-expanding subset Uʹ – having to bound a random variable depending on Uʹ and U spoils our union bound approach over the smaller sets Uʹ. For the first expansion of U into Vi this is avoidable (by only considering such sets disjoint from Vi), but we need to expand multiple times to reach most of the vertices in Vi and, after the first expansion, avoiding expanding sets that contain vertices in Vi is unavoidable. To get around this, when we identify the well-expanding set Uʹ, we look at its successive neighbourhoods in G − F and show that enough of these vertices are chosen to be in Vi together with all the vertices along a path going back to Uʹ so that Uʹ expands via such short paths to reach more than half of the vertices of Vi. In an early draft of this work, this was shown by carefully analysing an intricate random process. Fortunately, however, we will instead give here a much easier proof by combining our new perspective on robust sublinear expansion with an adaptation of a clever application of the sprinkling method appearing in a very recent work of Tomon . We defer a more detailed sketch for this part of the argument to, in particular until after we have introduced in full our new perspective on robust sublinear expansion, which remains crucial for this new approach. In total, though, this will allow us to answer question ii) positively. I.e., property [prop:sketch] holds if we are allowed to use all the edges of G to make connections through Vi. Let us also stress an important point, which already played a role at various points in the above arguments and that is that the paths Pxy we find will always be short, namely of length logO(1)n. This again plays an important role in answering the next question, namely finding an appropriate sparse subgraph Hi ⊆ G with the same property, which we turn to next. iii) Finding sparse connecting skeletons. Before we look for a subgraph Hi ⊆ G with the property [prop:sketch] and nlogO(1)n edges, can we even find any graph with these properties? A binomial random graph is a natural candidate for such a graph, and, indeed, if H is a binomial random graph with vertex set V(G) and edge probability $p=\omega\left(\frac{\log n}n\right)$ then it will have, with high probability, the property [prop:sketch] if we replace Hi with H and choose Vi to be any fixed set of linear size. (We prove this as, with a larger than optimal value of p for simplicity.) We then use H as a template to construct the sparse expanding skeleton Hi. We first sample our large random subset of vertices Vi, and use our answer to question ii) to guarantee that property [prop:sketch] holds with high probability in a slightly stronger form where every vertex is allowed to appear in O(log5n) pairs in P. In particular, we will use it with P being the set of pairs of vertices making an edge of our template graph H, which we choose to be sparse and well-connected through Vi, as well as have maximum degree O(log5n). For each edge xy ∈ E(H), we find an xy-path Pxy through Vi with length logO(1)n in G so that all these paths are edge disjoint. We then let Hi be the union of all these paths, noting that, as H is sparse and the paths Pxy are relatively short, Hi is also relatively sparse. Then, given an arbitrary collection P of pairs to connect, we first find edge disjoint paths connecting them through Vi in H, before replacing each edge xy on one of these paths in H with the corresponding path Pxy through Vi. This creates a set of edge disjoint xy-walks through Vi in Hi — as each such walk contains an xy-path, we can find the paths required by property [prop:sketch]. Note that, when we do this for each i ∈ [3], we need to ensure that the graphs Hi we find are edge disjoint, but this is easy to do by reusing some of our previous work, splitting G into a union of edge disjoint expanders G1, G2, G3, before finding each subgraph Hi in the respective subgraph Gi. Organisation of the paper ------------------------- In the rest of this section we will introduce some general preliminary results, including some concentration results in and the Aharoni-Haxell hypergraph matching theorem in, before showing strongly expanding graphs (namely G(n, p)) satisfy a certain strong connectivity property in. In, we introduce robust sublinear expansion and prove a number of useful properties of this type of expansion. In, we establish that our weaker expansion implies a similar (though weaker) connectivity property as that used in. In, we use the machinery we developed to prove our main result,. Finally, in, we make some concluding remarks. Concentration inequalities -------------------------- We will often use a basic version of Chernoff’s inequality for the binomial random variable (see, for example, ). [Chernoff’s bound][chernoff] Let n be an integer and 0 ≤ δ, p ≤ 1. If X ∼ Bin(n, p), then, setting μ = EX = np, we have P(X > (1 + δ)μ) ≤ e− δ2μ/3, and P(X < (1 − δ)μ) ≤ e− δ2μ/2. We will also make use of the following well-known martingale concentration result (see Chapter 7 of ). [lem:mcd] Suppose that X : ∏i = 1NΩi → R is k-Lipschitz. Then, for each t > 0, $$\mathbb{P}(\|X-\mathbb{E} X\|>t)\leq 2\exp\left(\frac{-2t^2}{k^2N}\right).$$ The Aharoni-Haxell hypergraph matching theorem and edge disjoint paths ---------------------------------------------------------------------- We will use the following hypergraph version of Hall’s theorem due to Aharoni and Haxell, which is an immediate consequence of Corollary 1.2 in (noting that we can add new, unique, vertices to each edge in the theorem to make the hypergraphs ℓ-uniform). A matching in a hypergraph is a collection of pairwise vertex disjoint edges. [thm:hyperhall] Let r ∈ N, and let H1, …, Hr be a collection of hypergraphs with at most ℓ vertices in each edge. Suppose that, for each I ⊆ [r], there is a matching in ⋃i ∈ IHi containing more than ℓ(∣I∣ − 1) edges. Then, there is an injective function f : [r] → ⋃i ∈ [r]E(Hi) such that f(i) ∈ E(Hi) for each i ∈ [r] and {f(i) : i ∈ [r]} is a matching of r edges. We will use to find edge disjoint paths between vertex pairs, to show that a graph is well-connected under the following definition, recalling that a path through V is a path with all its internal vertices in V. [defn:path-connected] A graph G is (ℓ, t)-path connected through a vertex subset V ⊆ V(G) if, for any $\mathcal{P} \subseteq \binom{V(G)}{2}$ in which every vertex appears in at most t pairs in P, there are edge disjoint paths P{x, y}, {x, y} ∈ P, such that, for each {x, y} ∈ P, P{x, y} is an xy-path through V with length at most ℓ. We denote by $\binom{V(G)}{2}$ the multiset of pairs of distinct vertices of G, so in particular the same pair may appear multiple times in the collection P. Typically, t will be a small constant and ℓ will be at most polylogarithmic in the number of vertices. Given a graph G, a vertex set V ⊆ V(G) and a collection $\mathcal{P}\subseteq \binom{V(G)}{2}$, we translate the pair connectivity property into a hypergraph matching problem as follows. For some ℓ ∈ N, and each {x, y} ∈ P, let H{x, y} be the hypergraph with vertex set E(G) and add as an edge the set E(P) for each xy-path P in G with interior vertices in V and length at most ℓ. If there is an injective function f : P → ⋃{x, y} ∈ PE(H{x, y}) such that f({x, y}) ∈ E(H{x, y}) for each {x, y} ∈ P and {f({x, y}) : {x, y} ∈ P} is a matching, then, for each {x, y} ∈ P, let P{x, y} be the path in G with edge set f({x, y}). By the definition of H{x, y}, each path P{x, y} is an xy-path in G with length at most ℓ and interior vertices in V, and, as {f({x, y}) : {x, y} ∈ P} is a matching, these paths are all edge disjoint. Therefore, in order to prove that G is (ℓ, t)-path connected through V, it suffices to take a general such collection P, define the relevant hypergraphs H{x, y}, and prove that the associated condition holds for an application of. Existence of well-connected sparse graphs ----------------------------------------- Random graphs typically present a natural candidate for a sparse, well-connected, graph, and we use this to prove the existence of our template in (using the connectivity property in ). Similar properties have been studied before in random graphs for various applications (see e.g. ), and our main lemma (Lemma [lem:template]) can be obtained as a (not quite immediate) corollary of Lemma 3.4. from combined with a multi-round exposure argument like the one we use below. We include a different proof of for completeness but also use this to introduce two intermediate results ( and ) that we later use in the same manner to prove our key technical result,. We first remind the reader that a path through a subset of vertices V is a path whose internal vertices are all in V. Let us also introduce, given U, V ⊆ V(G), the ball of radius i* around U within V* which we will denote as BGi(U, V), namely it is the set of vertices in V which can be reached by a path through V of length at most i starting from a vertex in U. The starting vertex in U is not required to be in V itself, as is usual with our definition of paths through a set. We do, however, only consider reachable vertices within V, so that BGi(U, V) ⊆ V. For, we take an expansion property of sets of size t ∈ N and use this to connect a pair of vertices from a set of 2t − 1 pairs (c.f. the collection P in ). [prop:connectonepair] Let 1 ≤ ℓ, t ≤ n. Let G be an n-vertex graph and let V ⊆ V(G) be of size ∣V∣ ≥ 4t − 2 such that, for every U ⊆ V(G) with size ∣U∣ = t, we have $\|B^\ell\_{G}(U,V)\|>\frac{\|V\|}2$. Let x1, …, x2t − 1, y1, …, y2t − 1 be distinct vertices of G. Then, for some j ∈ [2t − 1], there is an xjyj-path in G through V with length at most 4ℓlogn. Let Ix be the set of i ∈ [2t − 1] for which $\|B^{2\ell\log n}\_G(x\_i,V)\|\leq \frac{\|V\|}2$ and let Iy be the set of i ∈ [2t − 1] for which $\|B^{2\ell\log n}\_G(y\_i,V)\|\leq \frac{\|V\|}2$. Note that the required path can be found if there exists some j ∈ [2t − 1] with $j\notin I\_x$ and $j\notin I\_y$, for then BG2ℓlogn(xj, V) and BG2ℓlogn(yj, V) each have size larger than $\frac{\|V\|}2$ and must therefore intersect. Thus, we can assume that there is no such j, and, therefore, without loss of generality, that ∣Ix∣ ≥ t. Let r ≥ 0 be the largest integer for which there is a set X ⊆ {xi : i ∈ Ix} for which $\|X\|\leq t\left(\frac23\right)^r$ and $\|B^{(r+1)\ell}\_G(X,V)\|>\frac{\|V\|}2$, and let X be any such set. Note that this is possible as ∣Ix∣ ≥ t and any subset of {xi : i ∈ Ix} with size t satisfies these conditions for r = 0 by the assumption of the proposition. Now, as X ≠ ∅, we have that $t\left(\frac23\right)^r\geq 1$, so that $r\leq 2\log t\leq 2\log \frac{n}2$ (using $\left(\frac23\right)^2<\frac12$ and n ≥ ∣V∣ ≥ 4t − 2 ≥ 2t) and thus r < (2logn) − 1. Thus, by the definition of Ix, $\|B^{(r+1)\ell}\_G(x\_i,V)\|\leq \|B^{2\ell\log n}\_G(x\_i,V)\|\leq \frac{\|V\|}2$ for each i ∈ Ix, and hence ∣X∣ ≥ 2. This allows us to partition X = X0 ∪ X1 with $\|X\_0\|,\|X\_1\|\leq \frac23 \|X\|\leq t\left(\frac23\right)^{r+1}$. As $$\|B^{(r+1)\ell}\_G(X\_0,V)\|+\|B^{(r+1)\ell}\_G(X\_1,V)\|\geq \|B^{(r+1)\ell}\_G(X\_0\cup X\_1,V)\|=\|B^{(r+1)\ell}\_G(X,V)\|>\frac{\|V\|}2\geq 2t-1,$$ we can pick j ∈ [2] such that ∣BG(r + 1)ℓ(Xj, V)∣ ≥ t. Therefore, using the expansion of t-sets into V, we have $\|B^{(r+2)\ell}\_G(X\_j,V)\|>\frac{\|V\|}2$, contradicting the maximality of r as $\|X\_j\|\leq t\left(\frac23\right)^{r+1}$. We now take a stronger expansion property and use in combination with the Aharoni-Haxell hypergraph matching theorem () to find edge disjoint paths connecting a set of vertex pairs (see also the discussion after ), proving. [lem:connectmanypairs] Let n ≥ 2 and 1 ≤ ℓ, k ≤ n. Let G be an n-vertex graph and let V ⊆ V(G) so that $\|V\|\ge \frac n8+1$ and suppose, for each U ⊆ V(G) and F ⊆ E(G) with U ≠ ∅ and ∣F∣ ≤ 29k∣U∣(ℓlogn)2, we have $\|B^\ell\_{G-F}(U,V)\|>\frac{\|V\|}2$. Then, G is (4ℓlogn, k)-path connected through V. To show that G is (4ℓlogn, k)-path connected through V, let $\mathcal{P}\subseteq \binom{V(G)}{2}$ be an arbitrary collection of vertex pairs such that each vertex appears in at most k pairs in P. Let r = ∣P∣, and order the pairs in P. For each i ∈ [r], let Hi be the hypergraph with vertex set E(G) and edge set corresponding to the edge sets of paths through V with length at most 4ℓlogn connecting the i-th pair of vertices in P, noting that the size of any edge is at most 4ℓlogn. Now, for each I ⊆ [r], let MI be a maximal matching in ⋃i ∈ IHi. We will show that ∣MI∣ ≥ 4ℓlogn ⋅ ∣I∣ for each I ⊆ [r]. Towards a contradiction, suppose that, for some I ⊆ [r], we have ∣MI∣ < 4ℓlogn ⋅ ∣I∣, noting that we must have I ≠ ∅. Let F be the set of edges of G in any path corresponding to an edge in MI, so that F = ⋃e ∈ MIV(e). Note that ∣F∣ < 4ℓlogn ⋅ 4ℓlogn ⋅ ∣I∣ = (4ℓlogn)2∣I∣. Let Iʹ be a maximal subset of I such that no vertex appears in a pair in Iʹ more than once, so that 2k∣Iʹ∣ ≥ ∣I∣. Note also that this ensures that $\|I'\|\le \frac n2$. Now, let t = ⌈∣Iʹ∣/16⌉ ≥ 1, so that 2t − 1 ≤ 16t − 15 ≤ ∣Iʹ∣ ≤ 16t and $4t-2 \le { \left\lceil \frac n8 \right\rceil }+1\le\|V\|$. Note that $\|F\|< (4\ell\log n)^2\|I\|\leq k\frac{\|I'\|}2(8\ell\log n)^2\leq 8kt(8\ell\log n)^2$, and therefore any set U ⊆ V(G) with size t satisfies $\|B^\ell\_{G-F}(U,V)\|>\frac{\|V\|}2$. Then, by applied to G − F, for some j ∈ Iʹ there is a path in G − F between the j-th pair in P with interior vertices in V and length at most 4ℓlogn. Such a path corresponds to an edge of Hj with no vertices in F, a contradiction to the maximality of MI. Thus, we must have ∣MI∣ ≥ 4ℓlogn ⋅ ∣I∣. Therefore, by, there is a set of paths Pi, i ∈ [r], in G with E(Pi) ∈ Hi for each i ∈ [r], such that E(Pi), i ∈ [r], form edge disjoint sets. Thus, G is (4ℓlogn, k)-path connected through V. We now prove the existence of our template graph, by showing an appropriate expansion condition is likely in a certain binomial random graph and applying. [lem:template] For any large enough n, there exists an n-vertex graph G with Δ(G) ≤ 28log5n and a set V ⊆ V(G) with $\|V\|=\frac n6$ such that G is $\left(\frac14\log^2 n,2\right)$-path connected through V. Let $p=\frac{150\log^5 n}{n}$, let V ⊆ [n] be a set of size $\frac n6$ and let G ∼ G(n, p). As $p=\omega\left(\frac{\log n}{n}\right)$, a standard application of Chernoff’s inequality () shows that, with high probability, $\Delta(G)\leq \frac32pn= 225\log^5 n$. Therefore, it is sufficient to show that, with high probability, G is $\left(\frac14\log^2 n,2\right)$-path connected through V. With high probability, for each U ⊆ V(G) and each F ⊆ E(G) with U ≠ ∅ and ∣F∣ ≤ 4∣U∣log4n we have $$\label{eqn:U} \|B\_{G-F}(U,V)\| > \min\left\{2^{16}\|U\|,\frac {\|V\|}{2}\right\}.$$ For large n, we will show for each U ⊆ V(G) and F ⊆ E(G) with U ≠ ∅ and ∣F∣ ≤ 4∣U∣log4n that holds with probability at least 1 − 2− 10∣U∣log5n, so that holds for all U ⊆ V(G) and F ⊆ E(G) with ∣F∣ ≤ 4∣U∣log4n with probability at least $$1-\sum\_{u=1}^{n} \binom{n}{u}\sum\_{f=0}^{4u\log^4n}\binom{\binom{n}{2}}{f} \cdot 2^{-10u\log^5 n}\geq 1-\sum\_{u=1}^nn^u\cdot n^2\cdot n^{8u\log^4n}\cdot 2^{-10u\log^5 n}\geq 1-\sum\_{u=1}^nn^{-2}= 1-n^{-1},$$ and the claim holds. Let then U ⊆ V(G) with u = ∣U∣ ≥ 1 and F ⊆ E(G) with f = ∣F∣ ≤ 4ulog4n. Note that if $\|B\_{G-F}(U,V)\| \leq \min\left\{2^{16}u,\frac {\|V\|}{2}\right\}$, then there is some set X = BG − F(U, V) with size at most 216u such that $\|V\setminus X\|\geq \frac{\|V\|}2$ and there are no edges between U and V \ X other than in F. The probability of such a set X existing is at most $$\sum\_{i=0}^{2^{16}u}\binom{n}{i}\cdot (1-p)^{u\cdot \frac{\|V\|}{2}-\|F\|}\leq n^{2^{16}u+1}\cdot e^{-p(un/12-f)}\leq e^{-pun/15}\leq 2^{-10u\log^5n},$$ as required. With high probability then, we have that the conclusion of the claim holds for G. For any U ⊆ V(G) and F ⊆ E(G) with U ≠ ∅ and ∣F∣ ≤ 4∣U∣log4n, we have then by induction that, for each i ≥ 0, $$\|B\_{G-F}^i(U,V)\| > \min\left\{2^{16i}\|U\|,\frac{\|V\|}2\right\}.$$ Setting $i=\ell=\frac{\log n}{16}$, we thus have $\|B^{\ell}\_{G-F}(U, V)\|> \frac{\|V\|}2.$ Thus, by with $\ell=\frac{\log n}{16}$ and k = 2, we have that G is $\left(\frac14\log^2 n,2\right)$-connected through V. Robust sublinear expansion ========================== In this section, we will explain the expansion we use and its background, our new perspective on this type of expansion, and prove some key results using the expansion. Before doing so, for convenience, we state the definition of expansion that we use, as follows. [defn:robust-sublinear-expansion] An n-vertex graph G is an (ɛ, s)-expander if, for every U ⊆ V(G) and F ⊆ E(G) with $1\le \|U\|\leq \frac23n$ and ∣F∣ ≤ s∣U∣, we have $$\|N\_{G-F}(U)\|\geq \frac{{\varepsilon}\|U\|}{\log^2 n}.\label{eqn:expands}$$ As the bound on the size of the neighbourhood guaranteed at is o(∣U∣) as n → ∞, we consider this to be sublinear expansion. We often use Definition [defn:robust-sublinear-expansion] when s is polylogarithmic in n, so that the set of edges F may be of size ω(∣U∣) as n → ∞, and call this robust sublinear expansion. Note that, for each v ∈ V(G), setting U = {v} and setting F to be the set of edges incident to v, we have that does not hold, so we must have ∣F∣ > s. Thus, δ(G) > s for any (ɛ, s)-expander G, and therefore the expanders we will work with will always have minimum degree at least polylogarithmic in n. We discuss this notion of expansion in more detail with the relevant background in Section [sec:generalexpansionchat]. In Section [subsec:alternative], we then introduce an alternative perspective of this expansion. In, we prove a lemma which almost decomposes an arbitrary graph into expanders. Finally, in, we prove sublinear expanders can be (edge) partitioned into expanders (with only slightly weaker expansion parameters). Expansion --------- Classical graph expansion is an immensely powerful idea in graph theory and computer science that has seen a very wide variety of applications (see, for example, the survey ). A typical such property in a graph G says that ∣NG(U)∣ ≥ λ∣U∣ for any set U ⊆ V(G) which is not too large, where λ is some strictly positive absolute constant not dependent on the graphs G considered, though other notions have been considered instead of requiring a large ∣NG(U)∣, for example bounding the number of edges in G between U and V(G) \ U (as indeed used by Conlon, Fox and Sudakov ). Sublinear expansion is a weaker notion of this classical expansion introduced by Komlós and Szemerédi , where we take a much smaller value of λ, but which is significant as every graph contains a sublinear expander H with λ = Θ(1/log2∣H∣) (and even has a nice decomposition into sublinear expanders, as we will prove and use). Komlós and Szemerédi used sublinear expansion to find minors in sparse graphs, and more recently, sublinear expansion has found a host of other applications (see, for example, ). Such sublinear expansion in a graph G has some very weak robustness properties, in that if λ∣U∣/2 vertices in V(G) \ U are removed from the graph then the set U will still expand (with the neighbourhood of U still having at least λ∣U∣/2 vertices), and this property is used in many of the applications of sublinear expansion cited above. However, we will distinguish robust sublinear expansion to be that where U expands despite the removal of any set F of at most s∣U∣ edges in G, where s grows with ∣G∣ so that this bound is superlinear in ∣U∣, as in. Such robust sublinear expansion has recently been developed essentially independently by groups of different authors, appearing in some form in work by Haslegrave, Kim, and Liu and by Sudakov and Tomon (with the parallel clearer in the expansion used in subsequent work by Jiang, Methuku and Yepremyan and by Tomon ). Roughly speaking, the expansion we use, as given in Definition [defn:robust-sublinear-expansion], is a slightly weaker version of that used by Haslegrave, Kim, and Liu (so that we can find an almost-decomposition into such expanders) and a stronger version than subsequent developments of the expansion used by Sudakov and Tomon (which makes it more powerful when we use it). An alternative notion of robustness ----------------------------------- In, we consider the expansion to be robust as sets expand despite an arbitrary removal of a small number of edges. This can be alternatively encoded by recording that every vertex subset U either expands very well (by some factor greater than 1) or that its ‘robust neighbourhood’ of vertices with plenty of edges towards U expands well, though perhaps sublinearly (as in Proposition [prop:expansion-red-blue] below). It will be convenient to define this robust neighbourhood for any parameter d as NG, d(U) : = {v ∈ V(G) \ U : ∣NG(v) ∩ U∣ ≥ d}, that is, the set of vertices in a graph G, outside of a subset of vertices U, which have degree at least d towards U. [prop:expansion-red-blue] Let G be an n-vertex (ɛ, s)-expander, $U\subseteq V(G), \|U\|\le \frac23 n$ and F a set of at most s∣U∣/2 edges. Then, for any 0 < d ≤ s, either $$\text{\textbf{\emph{a)}} }\;\;\|N\_{G-F}(U)\| \ge \frac{s\|U\|}{2d},\;\;\;\text{ or }\;\;\;\text{\textbf{\emph{b)}}}\;\; \|N\_{G-F,d}(U)\|\ge \frac{{\varepsilon}\|U\|}{\log^{2} n}.$$ Suppose a) is not satisfied, so that $\|N\_{G-F}(U)\|<\frac{s\|U\|}{2d}$. Let X = NG − F(U) \ NG − F, d(U), so that $\|X\|<\frac{s\|U\|}{2d}$. Let Fʹ be the edges of G − F between U and X, so that ∣Fʹ∣ < ∣X∣d ≤ s∣U∣/2, and hence ∣F∣ + ∣Fʹ∣ ≤ s∣U∣. Note that, by the definition of Fʹ, we have NG − F, d(U) = NG − F − Fʹ(U). As G is an (ɛ, s)-expander, we thus have $$\|N\_{G-F,d}(U)\|=\|N\_{G-F-F'}(U)\|\geq \frac{{\varepsilon}\|U\|}{\log^{2} n},$$ and therefore b) holds, as required. The following proposition shows more structure can be found in both outcomes of the above proposition. Though a more general variant follows easily, the parameters are tailored for our intended application. [prop:red-blue-expansion-robust-both] There is an n0 such that the following holds for each n ≥ n0, 1 ≥ ɛ ≥ 2− 9 and s ≥ 8log13n. Let G be an n-vertex (ɛ, s)-expander, let U ⊆ V(G) have size $\|U\|\le \frac23 n$ and let F be a set of at most s∣U∣/4 edges. Then, in G − F we can find either 1. $\frac{\|U\|}{\log^{7} n}$ vertex disjoint stars, each with log9n leaves, centre in U and all its leaves in V(G) \ U, or 2. a bipartite subgraph H with vertex classes U and X ⊆ V(G) \ U such that * $\|X\| \ge \frac{{\varepsilon}\|U\|}{2\log^2 n}$ and * every vertex in X has degree at least log4n in H and every vertex of U has degree at most 2log9n in H. Take a maximal collection of vertex disjoint stars in G − F with log9n leaves and centre in U and leaves outside of U. Let C ⊆ U be the set of centres of these stars and L ⊆ V(G) \ U be the set consisting of all their leaves. Assuming a) does not hold, we can thus assume that $\|C\| \le \frac{\|U\|}{\log^{7} n}$, ∣L∣ ≤ ∣U∣log2n, and, by the maximality, that there is no vertex in U \ C with at least log9n neighbours in G − F in V(G) \ (U ∪ L). Thus, $$\label{eqn:NGW} \|N\_{G-F}(U\setminus C)\|\leq \|C\|+\|L\|+\|U\setminus C\|\cdot \log^9n\leq \frac{\|U\|}{\log^{7} n}+\|U\| \log^2 n+\|U\|\log^9n< 2\|U\|\log^9n.$$ Let d = log4n and Δ = 2log9n. We now construct the set X ⊆ V(G) \ U and the bipartite subgraph H through the following process, starting with X0 = ∅ and setting H0 to be the graph with vertex set U ∪ X0 and no edges. Let r = ∣V(G) \ U∣ and label the vertices of V(G) \ U arbitrarily as v1, …, vr. For each i ≥ 1, if possible pick a star Si in G − F with centre vi and d leaves in U such that Hi − 1 ∪ Si has maximum degree at most Δ, and let Hi = Hi − 1 ∪ Si and Xi = Xi − 1 ∪ {vi}, while otherwise we set Hi = Hi − 1 and Xi = Xi − 1. Finally, let H = Hr and X = Xr = V(Hr) \ U. We will show that b) holds for this choice of H with bipartition (U, X). Firstly, observe that Δ(Hi) ≤ Δ for each i ∈ [r] by construction, and that every vertex vi in X has degree exactly d in H, so the second condition in b) holds. Thus, we only need to show that $\|X\| \ge \frac{{\varepsilon}\|U\|}{2\log^2 n}$ holds, which will follow as no vertex in U \ C has $\frac{\Delta}2=\log^9n$ neighbours in G − F in X \ L due to the maximality of our family of disjoint stars defining C and L. Indeed, let Uʹ be the set of vertices in U \ C with degree exactly Δ in H. As each vertex in Uʹ ⊆ U \ C has fewer than $\frac{\Delta}2$ neighbours in G − F in X \ L, it must have at least $\frac{\Delta}2$ neighbours in H in X ∩ L. As each vertex in X ∩ L has d neighbours in H, we have $$\|U'\|\leq \frac{d\|X \cap L\|}{\Delta/2}\le \frac{2d\|L\|}{\Delta}\leq \frac{2d\cdot \|U\|\log^2n}{\Delta}=\frac{2\log^4 n\cdot \|U\|\log^2n}{2\log^9n}\le \frac{{\varepsilon}\|U\|}{8\log^2 n},$$ where the last inequality follows for sufficiently large n. Let B = C ∪ Uʹ, so that $$\label{eqn:Bbound} \|B\|\leq \frac{\|U\|}{\log^{7} n}+\frac{{\varepsilon}\|U\|}{8\log^2 n}\le \frac{{\varepsilon}\|U\|}{6\log^2 n},$$ and, in particular, $\|U\setminus B\|\geq \frac{2}3 \|U\|$. Then, by applied to U \ B and F with d, using that ∣F∣ ≤ s∣U∣/4 ≤ s∣U \ B∣/2, we have either $\|N\_{G-F}(U\setminus B)\|\geq \frac{s\|U\setminus B\|}{2d}$ or $\|N\_{G-F,d}(U\setminus B)\|\geq \frac{{\varepsilon}\|U\setminus B\|}{\log^2 n}$. As $$\frac{s\|U\setminus B\|}{2d}\geq \frac{s\|U\|}{3d}\geq \|U\|+2\|U\|\log^9 n,$$ the former contradicts, since ∣NG − F(U \ C)∣ ≥ ∣NG − F(U \ B)∣ − ∣U∣. Therefore, we must have that $\|N\_{G-F,d}(U\setminus B)\|\geq \frac{{\varepsilon}\|U\setminus B\|}{\log^2n}$. Every vertex vi in NG − F, d(U \ B) \ B has at least d neighbours in G − F in U \ B which, not being in B = Uʹ ∪ C, by definition of Uʹ must all have degree strictly less than Δ in H. This implies vi ∈ X, since we could add it together with some d of these neighbours. Hence, we must have NG − F, d(U \ B) \ B ⊆ X, and $$\|X\|\geq \|N\_{G-F,d}(U\setminus B)\|-\|B\|\overset{\eqref{eqn:Bbound}}{\geq} \frac{{\varepsilon}\|U\setminus B\|}{\log^2 n} - \frac{{\varepsilon}\|U\|}{6\log^2 n} \geq \frac{{\varepsilon}\|U\|}{2\log^2 n},$$ as required, where we have used that $\|U\setminus B\|\geq \frac{2}3 \|U\|$. Almost decomposing an arbitrary graph into expanders ---------------------------------------------------- The following lemma almost decomposes a graph into robust sublinear expanders with, on average, very little overlap between their vertex sets. Setting s = 0 in the below lemma (as we do in one application) obtains a full decomposition, although without any robustness. [splitting-into-expanders] Given an n-vertex graph G, a non-negative integer s and ɛ ≤ 2− 5 we can delete up to 4snlogn edges from G so that the remaining edges may be partitioned into graphs G1, …, Gr such that ∑i = 1r∣Gi∣ ≤ 2n and each Gi is an (ɛ, s)-expander. We prove this by induction on n, under the stronger condition that the graphs G1, …, Gr in the partition satisfy $\sum\_{i=1}^r \|G\_i\| \le 2n-\frac{2n}{2+\log n}$. Since $2n-\frac{2n}{2+\log n}\ge n,$ and any 1-vertex graph G is trivially an (ɛ, s)-expander, the lemma holds for n = 1 with G1 = G. Let us then assume n ≥ 2 and that the claim holds for all graphs with at most n − 1 vertices. Letting G be an n-vertex graph, note that, as $2n-\frac{2n}{2+\log n}\ge n,$ if G is an (ɛ, s)-expander then the trivial partition of G1 = G demonstrates the claim holds for G. Thus, we can assume G is not an (ɛ, s)-expander, and in particular, that there exists a non-empty set of vertices U ⊆ V(G) with $\|U\| \le \frac23n$ and a set F of at most s∣U∣ edges such that $\|N\_{G-F}(U)\| < \frac{{\varepsilon}\|U\|}{\log^2 n}.$ Let G1 = G[U ∪ NG − F(U)] − F and let G2 = G \ U − E(G1) − F, so that G1 and G2 form an edge partition of G − F and, setting n1 = ∣G1∣ and n2 = ∣G2∣, we have $$\label{eqn:n1n2} n\_1+n\_2=\|G\_1\|+\|G\_2\|=\|G\|+\|N\_{G-F}(U)\| < n+\frac{{\varepsilon}\|U\|}{\log^2 n}\le n+\frac{{\varepsilon}n\_1}{\log^2 n}.$$ Now, n2 = n − ∣U∣ < n and $$\label{eqn:n1} n\_1\leq \|U\|+\frac{{\varepsilon}\|U\|}{\log^2 n}\leq \frac23n+{\varepsilon}n\leq \frac34n<n,$$ so there exist sets E1 ⊆ E(G1) and E2 ⊆ E(G2) and partitions G1, 1, …, G1, r1 and G2, 1, …, G2, r2 of G1 − E1 and G2 − E2 into edge disjoint (ɛ, s)-expanders so that, for each i ∈ [2], ∣Ei∣ ≤ 4snilogni, and $$\sum\_{j=1}^{r\_i}\|G\_{i,j}\|\leq 2n\_i-\frac{2n\_i}{2+\log n\_i}.$$ Therefore, we can remove F ∪ E1 ∪ E2 from G and decompose the remaining edges into (ɛ, s)-expanders G1, 1, …, G1, r1, G2, 1, …, G2, r2. We need then only check that ∣F ∪ E1 ∪ E2∣ ≤ 4snlogn and that the sum of the vertices of the expanders in this decomposition is at most $2n-\frac{2n}{2+\log n}$. Firstly, note that, from, we have $\log n\_1\leq \log \frac34n<\log n-\frac 25$, so that $$\begin{aligned} \frac{1}{s}(\|F\|+\|E\_1\|+\|E\_2\|)&\leq \|U\|+4n\_1\log n\_1+4n\_2\log n\_2 \leq n\_1+4n\_1\left(\log n-\frac{2}{5}\right)+4n\_2\log n \nonumber\\ & = 4(n\_1+n\_2)\log n -\frac{3}{5}n\_1 \overset{\eqref{eqn:n1n2}}{\leq} 4\left(n+\frac{{\varepsilon}n\_1}{\log^2 n}\right)\log n -\frac{3}{5}n\_1 \nonumber\\ & \leq 4n\log n.\label{eqn:EEE}\end{aligned}$$ Secondly, again as $ \log \frac34n<\log n-\frac 25$, we have $$\label{eqn:n1logs} \frac{2n\_1}{2+\log n\_1} \overset{\eqref{eqn:n1}}{\ge} \frac{2n\_1}{2+\log (3n/4)} \ge \frac{2n\_1}{8/5+\log n}=\frac{2n\_1}{2+\log n}+\frac{2n\_1\cdot 2/5}{(8/5+\log n)(2+\log n)} > \frac{2n\_1}{2+\log n}+\frac{n\_1}{10\log^2 n},$$ so that $$\begin{aligned} \sum\_{i=1}^2\sum\_{j=1}^{r\_i}\|G\_{i,j}\|&\leq 2n\_1+2n\_2-\frac{2n\_1}{2+\log n\_1}-\frac{2n\_2}{2+\log n\_2}\overset{\eqref{eqn:n1logs}}{<} (n\_1+n\_2)\left(2-\frac{2}{2+\log n}\right)-\frac{n\_1}{10\log^2 n} \\ &\overset{\eqref{eqn:n1n2}}{\leq} \left(n+\frac{{\varepsilon}n\_1}{\log^2 n}\right)\left(2-\frac{2}{2+\log n}\right)-\frac{n\_1}{10\log^2 n}\leq 2n-\frac{2n}{2+\log n}.\end{aligned}$$ In combination with, this shows that G has the required decomposition, completing the inductive step and hence the proof. Decomposing an expander into many expanders ------------------------------------------- The following lemma partitions the edges of an expander into a chosen number of expanders with the same vertex set (and a slightly weaker expansion condition). Our expanders have two parameters, and it is key that this splitting is particularly efficient with the first parameter (as further discussed at the start of Section [sec:connectinexpander]) – that is, we will split an (ɛ, s)-expander into polylogarithmically many (ɛ/4, sʹ)-expanders for some appropriate sʹ, so that, while s will be reduced polylogarithmically to get sʹ, the first parameter, ɛ, is only reduced to ɛ/4. [lem:partitionedgesintoexpanders] Let n, k, s ∈ N and 0 < ɛ ≤ 1. Suppose that G is an n-vertex (ɛ, s)-expander and s ≥ 212ɛ− 1k2log4n. Then, there are edge disjoint graphs G1, …, Gk such that E(G) = ⋃i ∈ [k]E(Gi) and, for each i ∈ [k], Gi is an $\left(\frac{{\varepsilon}}4,\frac{\sqrt{s{\varepsilon}}}{ 8k \log n}\right)$-expander with vertex set V(G). If n = 1, then the claim is trivially true, so let us assume that n ≥ 2. Furthermore, observe that as ɛ > 0, we must have s ≤ δ(G) for otherwise, we can remove all the neighbours of a vertex with minimum degree by removing at most s edges, contradicting that G is an (ɛ, s)-expander, so certainly s ≤ n. Let H be a random subgraph of G with vertex set V(G), which contains every edge independently with probability $\frac 1k$. Then, assign every edge of G to one of the graphs G1, …, Gk uniformly and independently at random, so that each Gi is a random subgraph with the same distribution as H. Letting $s'=\frac{\sqrt{s{\varepsilon}}}{ 8k \log n}\geq 8\log n$, we will show that the probability H is not an $\left(\frac{{\varepsilon}}4,s'\right)$-expander is strictly less than $\frac 1k$. Thus, by a union bound, the probability that each Gi is an $\left(\frac{{\varepsilon}}4,s'\right)$-expander is strictly positive, so some decomposition as required by the lemma must exist. To show that H is not an $\left(\frac{{\varepsilon}}4,s'\right)$-expander with probability less than $\frac 1k$, we will take a union bound over all subsets U of V(G) for the event that U fails the conditions of $(\frac{{\varepsilon}}{4},s')$-expansion in H. For this, set $d=\sqrt{s/{\varepsilon}} \log n$ and note that $\frac{s}{d}=8ks'$, $s'=\frac{{\varepsilon}d}{8k \log^2 n}$ and $s\geq \sqrt{s}\cdot \sqrt{{\varepsilon}^{-1}\log^2n}=d\geq 64 k$. Let U ⊆ V(G) and $u=\|U\|\leq \frac{2n}{3}$. By with d, U and F = ∅, we have either a) $\|N\_G(U)\|\geq \frac{su}{2d}=4ks'u$ or b) $\|N\_{G,d}(U)\|\geq \frac{{\varepsilon}u}{\log^2 n}$. If a) holds, then ∣NH(U)∣ is dominated by Bin(4ksʹu, 1/k), so that P(∣NH(U)∣ ≥ 2sʹu) ≥ 1 − e− sʹu/2 by a Chernoff bound (). Note that if ∣NH(U)∣ ≥ 2sʹu then for any F ⊆ E(H) with ∣F∣ ≤ sʹu we have $\|N\_{H-F}(U)\|\geq s'u\geq \frac{{\varepsilon}u}{4\log^2n}$. Thus, when a) holds, the probability U fails the $\left(\frac{{\varepsilon}}4,s'\right)$-expansion condition is at most e− sʹu/2 ≤ e− 4ulogn as sʹ ≥ 8logn. If b) holds, then we have NG, d(U) ≠ ∅, so in particular u = ∣U∣ ≥ d, which in turn implies $\frac{{\varepsilon}u}{\log^2n}\ge \frac{\sqrt{s{\varepsilon}}}{\log n}\ge 64k\log n$. Note now that the probability that any vertex v ∈ NG, d(U) is not in NH, d/2k(U) is at most $p:={\mathbb{P}}\left(\text{Bin}\left(d,\frac1k\right)< \frac{d}{2k}\right)$, where we have once again by a Chernoff bound that $$\label{eqn:p} p\leq e^{-d/8k}\le \frac 14.$$ Therefore, if we write $t=\frac{{\varepsilon}u}{2\log^2 n}$, as $\|N\_{G,d}(U)\|\geq \frac{{\varepsilon}u}{\log^2 n}=2t$, we have $$\begin{aligned} \label{eqn:NHU} {\mathbb{P}}\left(\|N\_{H,d/2k}(U)\|< \frac{{\varepsilon}u}{2\log^2n}\right)&\le \binom{{ \left\lceil 2t \right\rceil }}{{ \left\lceil 2t \right\rceil }-{ \left\lfloor t \right\rfloor }}\cdot p^{{ \left\lceil 2t \right\rceil }-{ \left\lfloor t \right\rfloor }} \le 2^{{ \left\lceil 2t \right\rceil }}\cdot p^{{ \left\lceil 2t \right\rceil }-{ \left\lfloor t \right\rfloor }}\le (4p)^t \overset{\eqref{eqn:p}}{\le} e^{\frac{-{\varepsilon}u d}{32k\log^2 n}}= e^{-s'u/4}\notag\\&\le e^{-2u\log n}.\end{aligned}$$ Note that, if $\|N\_{H,d/2k}(U)\|\geq \frac{{\varepsilon}u}{2\log^2n}$, then, for any F ⊆ E(H) with ∣F∣ ≤ sʹu, we have $$\|N\_{H-F}(U)\|\geq \|N\_{H,d/2k}(U)\|-\frac{\|F\|}{d/2k}\geq \frac{{\varepsilon}u}{2\log^2n}-\frac{2ks'u}{d}=\frac{{\varepsilon}u}{2\log^2n}-\frac{su}{4d^2}= \frac{{\varepsilon}u}{4\log^2n}.$$ Thus, implies that the probability U does not satisfy the (ɛ/4, sʹ)-expansion condition is at most e− 2ulogn. Therefore, whichever of a) or b) holds, the probability U does not satisfy the (ɛ/4, sʹ)-expansion condition is at most e− 2ulogn. Hence, the probability that H is not an (ɛ/4, sʹ)-expander is at most $$\sum\_{u=1}^{2n/3}\binom{n}{u}e^{-2u\log n} \le \sum\_{u=1}^{2n/3} n^u\cdot n^{-2u} \le n^{-1}+\sum\_{u=2}^{2n/3} n^{-2}\leq \frac{2}{n}< \frac{1}{k},$$ as required, where in the last inequality we make use of the observation that n ≥ s ≥ 212ɛ− 1k2log4n > 2k. Finding edge disjoint paths through random vertex sets ====================================================== We will now show that a robust sublinear expander is not only well-connected (in the sense of Definition [defn:path-connected], and as used in Section [sec:template]), but is likely to be well-connected through any large random vertex subset. That is, we prove the following result. [thm:pathconnect] Let G be an n-vertex (ɛ, s)-expander with 1 ≥ ɛ ≥ 2− 7 and s ≥ log135n. Let V ⊆ V(G) be a random subset chosen by including each vertex independently at random with probability $\frac13$. Then, with high probability, G is (4log5n, 28log5n)-path connected through V. The challenge of proving Theorem [thm:pathconnect] is discussed in Section [subsec:sketch], and in particular in the answer to question ii) there. Having since then proved Lemma [lem:connectmanypairs], let us note that, for suitable polylogarithmic parameters s̄ and ℓ, by this lemma, to prove Theorem [thm:pathconnect] it is sufficient to show that, with high probability, 1. $\|B^\ell\_{G-F}(U,V)\|> \frac{\|V\|}2$ for each U ⊆ V(G) and F ⊆ E(G) with ∣F∣ ≤ s̄∣U∣.[youdag] Here, let us remind the reader that given U, V ⊆ V(G), we denote by BGi(U, V) the set of vertices in V which can be reached by a path through V (with all internal vertices in V) of length at most i starting from a vertex in U. To further motivate our approach in this section, let us recap some key aspects discussed in Section [subsec:sketch] with the parameters we have now introduced. As our condition is that G is an n-vertex (ɛ, s)-expander, we could have sets U ⊆ V(G) with ∣NG(U)∣ = ɛ∣U∣/log2n, so that, with probability (2/3)ɛ∣U∣/log2n, we may have NG(U) ∩ V = ∅, so that [youdag] does not even hold for U with F = ∅. Thus, if we take a union bound approach to [youdag], we could reasonably only hope to bound this over exp(O(ɛu/log2n)) pairs (U, F) with ∣U∣ = u (for any relevant u). As discussed in Section [subsec:sketch], we solve this in two ways. Firstly, for our union bound, we consider for [youdag] only sets U, which expand particularly well, much more than the lower bound from the expansion condition. By showing that a general set contains a well-expanding set that is not much smaller, we will then be able to deduce that all sets satisfy at least some weaker form of [youdag]. Secondly, we randomly partition the edges of G into expanders G1, …, Gk, for some appropriate polylogarithmic value of k, each of which will be an (ɛ/4, sʹ)-expander for some appropriate sʹ. As the first parameter in the expansion (ɛ) has only reduced by a factor of 4, and this is the parameter that appears in the number of pairs for which we can hope to take a union bound, there is little cost to looking at expansion only within any given subgraph Gi. Then, when we want to expand, avoiding some larger set of edges F, with ∣F∣ = s̄∣U∣ as in [youdag], we can find the smallest set of edges F ∩ E(Gi), and expand U in Gi while only avoiding a relatively much smaller set of edges F ∩ E(Gi). Thus, for our union bound, we only need to consider these smaller sets of edges, and this plays a crucial role in reducing the number of pairs (U, F) that we need to consider when applying the union bound. We do this all in three stages. Firstly, in Section [sec:expand1], we show that for any set U ⊆ V(G) with ∣NG(U)∣ ≥ ∣U∣log24n and any F ⊆ E(G) with ∣F∣ ≤ ∣U∣ we have $\|B^\ell\_{G-F}(U,V)\| > \frac{\|V\|}2$ holds with probability 1 − exp( − Ω(∣U∣log2n)). That is, we show the desired property holds for a well-expanding set U with high enough probability that we can take a union bound over all well-expanding sets U (and any edge set F with, say, ∣F∣ ≤ ∣U∣) to get that this property holds for all well-expanding sets with probability 1 − o(1/n). As discussed in Section [subsec:sketch], if the set U does not expand well, then we can not guarantee it has the property we want with high enough probability for a union bound over all sets U, so we only consider well-expanding sets U here. Then, in Section [sec:expand2], we show that every set U in our expander contains a well-expanding set Uʹ which is not that much smaller than U (see ), indeed, we will find such a Uʹ satisfying ∣Uʹ∣ ≥ μ∣U∣, where μ ≥ 1/log27n. Thus, for any edge set F with ∣F∣ ≤ μ∣U∣ ≤ ∣Uʹ∣ we have $\|B^\ell\_{G-F}(U,V)\|\geq \|B^\ell\_{G-F}(U',V)\| > \frac{\|V\|}2$. This is almost the condition we need to apply Lemma [lem:connectmanypairs]. However, μ is too small, namely by a polylogarithmic factor smaller than the value of s̄ that we need. Therefore, in Section [subsec:well-connectivity-weak-expander], we find the stronger expansion property we need by first splitting an (ɛ, s)-expander into polylogarithmically many edge disjoint expanders via, before showing that with high probability each one of these has the above-mentioned weaker expansion property into V. Finally, we combine these properties to show that, indeed, G has the desired stronger expansion property into V. This allows us to apply Lemma [lem:connectmanypairs], completing the proof of Theorem [thm:pathconnect]. Expansion of well-expanding sets into a random vertex set --------------------------------------------------------- In an n-vertex (ɛ, s)-expander G, given U ⊆ V(G) such that ∣NG(U)∣ ≥ ∣U∣log24n and F ⊆ E(G) with ∣F∣ ≤ s∣U∣/4, when V ⊆ V(G) is chosen by selecting each vertex independently at random with probability $\frac 13$, we wish to show that, with some large probability, we have $\|B^\ell\_{G-F}(U,V)\|>\frac{\|V\|}2$, for some appropriate parameters ɛ, s and ℓ (thus proving Lemma [lem:expandintorandom] below). To prove this we will adapt a ‘sprinkling’ argument from a very recent work of Tomon , thus avoiding a much more complex argument from initial versions of this work. To prove this we will reveal the vertices in V in ℓ batches, using the so-called sprinkling method, by partitioning V randomly into sets V1 ∪ … ∪ Vℓ, weighted so that most of the vertices in V are likely to be in Vℓ. A natural approach here would be to prove a likely bound on BG − Fi(U, V1 ∪ … ∪ Vi) for each i ∈ [ℓ], resulting in a bound on BG − Fℓ(U, V1 ∪ … ∪ Vℓ) = BG − Fℓ(U, V), so let us emphasise that this is not what we do. Instead, for 0 ≤ i ≤ ℓ we track the size of sets Bi which are defined as the set of vertices v ∈ V(G) which can be reached from U by a path all of whose internal vertices are in V1 ∪ … ∪ Vi − 1 (i.e., a path through this set) of length at most i. It is crucial here that we do not insist v belongs to either U or V1 ∪ … ∪ Vi. One can think of vertices in Bi as having potential of making all their neighbours in G − F reachable in the same way (so being in Bi + 1) if they get sampled into our next random subset Vi. In particular, every vertex with a neighbour in Bi which gets sampled into Vi will join Bi + 1. This combined with our notion of robust expansion (as discussed further below) allows us to show that it is likely that Bi + 1 will increase in size compared to Bi until for some i ≤ ℓ − 1 its size is at least $\frac23 n$. In particular, we will have $B\_{\ell} \ge \frac23 n$. The final stage is slightly different, here since Bℓ is independent of our final random set Vℓ we will likely have almost $\frac23$ of the vertices of Vℓ belonging to Bℓ. As our random sets are weighted heavily towards Vℓ, it is likely to contain more than $\frac34$ of the vertices of V so that we will likely have $\|V\_\ell\cap B\_\ell\|>\frac{\|V\|}2$, so that, finally, we have $$\|B^\ell\_{G-F}(U,V)\|\geq \|V\_\ell\cap B\_\ell\|>\frac{\|V\|}2,$$ as required. That the sets Bi, 1 ≤ i ≤ ℓ − 1, are very likely to increase notably in size will follow from our notion of robust expansion (as proved in the claim below). In particular, at step i, tells us that one of two cases a) or b) may occur. a) Bi has many large vertex disjoint stars extending from Bi. In this case we use that, for each centre sampled into Vi, the (many) corresponding leaves are added to Bi + 1. We will have that many more leaves are added for each successful centre than the sampling probability for Vi, so that this is a good increase in size. b) Bi has a large robust neighbourhood whose vertices have many neighbours in G − F in Bi. Each vertex in this robust neighbourhood is likely to have at least one of these neighbours in Bi sampled into Vi, whereupon it will then be in Bi + 1. (In fact, we need a slightly stronger property to hold so that the sampling of each vertex in Bi does not have too strong an influence on the size of Bi + 1 \ Bi, which is why we use the subgraph H provided by case b) of.) Thus, in either case, ∣Bi∣ is likely to increase. [lem:expandintorandom] Let n ≥ 2. Suppose that G is an n-vertex (ɛ, s)-expander with 2− 9 ≤ ɛ ≤ 1 and s ≥ 8log13n. Let U ⊆ V(G) satisfy ∣NG(U)∣ ≥ ∣U∣log24n and let F ⊆ E(G) satisfy ∣F∣ ≤ ∣U∣. Let V ⊆ V(G) be a random subset chosen by including each vertex independently at random with probability $\frac13$. Then, with probability 1 − e− Ω(∣U∣log2n), $$\label{expand} \|B^{\log^4n}\_{G-F}(U, V)\|>\frac{\|V\|}{2}.$$ Let ℓ = log4n, $q=\frac3{11}$ and let p be such that $1-(1-p)^{\ell-1}(1-q)=\frac{1}{3}$, i.e., that $(1-p)^{\ell-1}=\frac{11}{12}$, so that $$\label{eqn:p15} p\ge \frac{1}{15\log ^4 n}.$$ Let G be an n-vertex (ɛ, s)-expander, U ⊆ V(G) with ∣NG(U)∣ ≥ ∣U∣log24n and F ⊆ E(G) with ∣F∣ ≤ ∣U∣. Independently, for each i ∈ {1, …, ℓ}, let Vi be a random subset of V(G) with each vertex included independently at random with probability p if i ≤ ℓ − 1 and with probability q if i = ℓ. Set V = V1 ∪ … ∪ Vℓ, and note that each vertex is included in V independently at random with probability $\frac13$. Thus, we wish to show that, with probability at least 1 − e− Ω(∣U∣log2n) we have $\|B^{\ell}\_{G-F}(U, V)\|>\frac{\|V\|}{2}$. For each 0 ≤ i ≤ ℓ, let Bi be the set of vertices of G which can be reached via a path in G − F which starts in U and has length at most i and whose internal vertices (if there are any) are in V1 ∪ … ∪ Vi − 1. In particular, then, we have B0 = U and B1 = BG − F(U). Observe also that B0 ⊆ B1 ⊆ … ⊆ Bℓ. We emphasise that the vertices of Bi do not have to themselves be inside V1 ∪ … ∪ Vi − 1, only the interior vertices of some path from U to the vertex in Bi are required to be inside V1 ∪ … ∪ Vi − 1. An important property of Bi is that it is completely determined by the sets U, V1, …, Vi − 1, so is in particular independent of Vi. Note also that any vertex in NG − F(Bi) with a neighbour in Bi that gets sampled into Vi belongs to Bi + 1. These two observations will be the key behind why the sets Bi + 1 will grow in size until they occupy most of the set V(G). In particular, finally, observe that Bℓ ∩ Vℓ ⊆ BG − Fℓ(U, V). We now show that indeed, for each 1 ≤ i ≤ ℓ − 1, that, unless Bi is already very large, Bi + 1 is likely to be larger than Bi, as follows. For each 1 ≤ i ≤ ℓ − 1, with probability 1 − e− Ω(∣U∣log2n), either $\|B\_i\| \ge \frac23 n$, or $$\|B\_{i+1}\setminus B\_i\| \ge \frac{{\varepsilon}\|B\_i\|}{2^6\log^{2} n}.$$ For each v ∈ NG − F(Bi), v is in Bi + 1 if at least one of its neighbours in G − F in Bi gets sampled into Vi. That is, {v ∈ NG − F(Bi) : NG − F(v, Bi) ∩ Vi ≠ ∅} ⊆ Bi + 1 \ Bi. We will show that, for any set W ⊆ V(G) with $\|W\|\leq \frac23 n$ and B1 ⊆ W $$\label{eqn:overkill2} {\mathbb{P}}\left(\|\{v\in N\_{G-F}(W):N\_{G-F}(v,W)\cap V\_i\neq\emptyset\}\|\geq \frac{{\varepsilon}\|W\|}{2^6\log^{2} n}\right)\ge 1-e^{-\Omega\left({\|B\_1\|}/{\log^{22} n}\right)}.$$ Thus, we will have for all 1 ≤ i ≤ ℓ − 1 $$\begin{aligned} {\mathbb{P}}\left(\|B\_i\| \ge \frac23 n \: \text{ or } \: \|B\_{i+1}\setminus B\_i\| \ge \frac{{\varepsilon}\|B\_i\|}{2^6\log^{2} n}\right) &\overset{\textcolor{white}{\eqref{eqn:overkill1}}}{\geq} {\mathbb{P}}\left(\|B\_{i+1}\setminus B\_i\| \ge \frac{{\varepsilon}\|B\_i\|}{2^6\log^{2} n}\: \big\| \: \|B\_i\|\leq \frac{2}{3}n\right) \\ &\overset{\eqref{eqn:overkill1}}{\geq} {\mathbb{P}}\left(\|\{v\in N\_{G-F}(B\_i):N\_{G-F}(v,B\_i)\cap V\_i\neq\emptyset\}\|\ge \frac{{\varepsilon}\|B\_i\|}{2^6\log^{2} n}\: \big\| \: \|B\_i\|\leq \frac{2}{3}n\right)\\ &\overset{\eqref{eqn:overkill2}}{\ge} 1-e^{-\Omega\left({\|B\_1\|}/{\log^{22} n}\right)}\ge 1-e^{-\Omega\left({\|U\|}{\log^{2} n}\right)},\end{aligned}$$ where in the last inequality we used $\|B\_1\|=\|B\_{G-F}(U)\|\ge \|U\|\log^{24} n-\|F\|\geq \frac12\|U\|\log^{24}n$. Let then W ⊆ V(G) with $\|W\|\leq \frac23n$ and B1 ⊆ W. As $\|W\|\leq \frac23 n$, and ∣F∣ ≤ ∣U∣ ≤ ∣B1∣ ≤ ∣W∣ ≤ s∣W∣/4 we can apply to W and F to show one of two cases a) or b) holds and we will show that holds in either case. a) Suppose G − F contains $\frac{\|W\|}{\log^{7} n}$ vertex disjoint stars with log9n leaves, with the centre in W and all leaves in NG − F(W). Let C ⊆ W be the set of centres of such a collection of stars, and note that ∣{v ∈ NG − F(W) : NG − F(v, W) ∩ Vi ≠ ∅}∣ ≥ ∣C ∩ Vi∣log9n. By a Chernoff bound () and, with probability at least 1 − e− p∣C∣/8 = 1 − e− Ω(∣W∣/log11n), we have $\|C\cap V\_i\|\geq \frac{p\|C\|}2\geq \frac{\|W\|}{2^6\log^{11}n}$. Thus, in combination with, we have that holds as ɛ ≤ 1. b) Suppose instead that there is a bipartite subgraph H ⊆ G − F with vertex classes W and X such that * $\|X\| \ge \frac{{\varepsilon}\|W\|}{2\log^2 n}$ and * every vertex in X has degree at least log4n in H and every vertex of U has degree at most Δ : = 2log9n in H. For each v ∈ X, the probability that v has no neighbours in H in Vi is at most $$(1-p)^{\log^4n}\le e^{-p\log^4n}\overset{\eqref{eqn:p15}}{\leq} e^{-1/15}\leq \frac{15}{16}.$$ Let Y be the random variable counting the number of vertices of X having a neighbour in Vi in H, so that ${\mathbb{E}}Y \ge \frac{\|X\|}{16}$. Observe also that Y is Δ-Lipschitz since for each v ∈ W the event {v ∈ Vi} affects Y by at most dH(v) ≤ Δ. Hence, by with k = Δ, $t=\frac{\|X\|}{32}$ and N = ∣W∣, we have $${\mathbb{P}}\left(Y < \frac{\|X\|}{32}\right)\le {\mathbb{P}}\left(Y < {\mathbb{E}}Y - \frac{\|X\|}{32} \right)\le2\exp\left(-\frac{2^{-9}\|X\|^2}{\Delta^2\|W\|}\right)= e^{-\Omega\left({\|W\|}/{\log^{22} n}\right)}.$$ Each vertex in X with a neighbour in Vi in H lies in {v ∈ NG − F(W) : NG − F(v, W) ∩ Vi ≠ ∅}, so therefore, with probability at least 1 − e− Ω(∣W∣/log22n), we have $\|\{v\in N\_{G-F}(W):N\_{G-F}(v,W)\cap V\_i\neq\emptyset\}\|\geq Y\geq \frac{\|X\|}{32}\geq \frac{{\varepsilon}\|W\|}{2^6\log^2 n}$ and thus holds as well in case b), completing the proof. As Bℓ and Vℓ are independent and $q=\frac3{11}$ (so that $\frac{2q}{3}=\frac{2}{11}>\frac{4}{23}$), by Chernoff’s bound (), we have that $${\mathbb{P}}\left(\|B\_{\ell}\cap V\_\ell\|\le\frac{4}{23}n \: \big\| \: \|B\_{\ell}\|\geq \frac{2}3n\right)\leq {\mathbb{P}}\left(\operatorname{Bin}\left(\frac{2}3n,q\right)\leq \frac{4}{23}n\right)\le e^{-\Theta(n)},$$ and, similarly, as $\frac13<\frac8{23}$ we have ${\mathbb{P}}\left(\|V\|\geq \frac{8}{23}n\right)\le e^{-\Theta(n)}$. Thus, by the claim, we have in total that 1. [itm1] for each i ∈ [ℓ − 1], $\|B\_i\| \ge \frac23 n$ or $\|B\_{i+1}\setminus B\_i\| \ge \frac{{\varepsilon}\|B\_i\|}{2^6\log^{2} n}$, and 2. $\|B\_\ell\|<\frac23 n$ or $\|B\_\ell\cap V\_\ell\|>\frac{4}{23}n$, and 3. [itm3] $\|V\|\leq \frac{8}{23}n$ with probability at least 1 − log4n ⋅ e− Ω(∣U∣log2n) − 2− Θ(n) ≥ 1 − e− Ω(∣U∣log2n). However, if [itm1]–[itm3] all hold, then, for each i ∈ [ℓ − 1], we have $$\|B\_i\| \ge \min\left\{\frac23 n,\left(1+\frac{{\varepsilon}}{2^6\log^2 n}\right)^{i}\|U\|\right\}\geq \min\left\{\frac23 n,\exp\left(\frac{{\varepsilon}i}{2^7\log^2 n}\right)\right\},$$ so that, setting i = ℓ = log4n, we conclude $\|B\_{\ell}\|\geq \frac23 n$, and hence, by ii) and iii), that $\|B\_\ell\cap V\_\ell\|>\frac{\|V\|}2$. Thus, by, we have that $\|B^{\ell}\_{G-F}(U,V)\|>\frac{\|V\|}2$ with probability at least 1 − e− Ω(∣U∣log2n). Expansion into a random vertex set ---------------------------------- Having picked V ⊆ V(G) with vertex probability $\frac13$ in an n-vertex (ɛ, s)-expander G, tells us that for any fixed, well-expanding subset of vertices U and small set F of edges we can reach more than one half of the vertices of V by short paths through V in G − F with pretty high probability. We now want to use this to show that a similar property holds simultaneously for all vertex subsets U. As we cannot directly take a union bound over all subsets U, we first show that any vertex subset U in an (ɛ, s)-expander contains a subset Uʹ ⊆ U which expands particularly well (relative to its own size), which is not much smaller than U, and so that Uʹ captures much of the guaranteed expansion of U (as it will easily follow that $\|N\_G(U')\|\geq \frac{{\varepsilon}\|U\|}{3\log^2n}$). This follows easily from the definition of expansion but is perhaps easier to immediately see why it is true from the perspective introduced in. [prop:well-expanding-core] Let n ≥ 2, 0 < ɛ ≤ 1 and s ≥ log24n. Let G be an n-vertex (ɛ, s)-expander and let U ⊆ V(G) have size $\|U\|\le \frac{2}3n$. Then, there is a set Uʹ ⊆ U with ∣NG(Uʹ)∣ ≥ ∣Uʹ∣log24n and $\|U'\|\geq \frac{{\varepsilon}\|U\|}{3\log^{26}n}$. If U = ∅, then Uʹ = ∅ easily satisfies the required conditions, so we can assume that ∣U∣ ≥ 1. Let, then, Uʹ ⊆ U be maximal subject to ∣NG(Uʹ)∣ ≥ ∣Uʹ∣log24n, noting this is possible as Uʹ = ∅ satisfies these conditions. Suppose that U ≠ Uʹ, for otherwise U satisfies the conditions itself. Then ∣NG(Uʹ)∣ < (∣Uʹ∣ + 1)log24n + 1 or we could add an arbitrary vertex to Uʹ and contradict maximality. Similarly we know that, for every vertex v ∈ U \ Uʹ, v has at most log24n neighbours outside of Uʹ ∪ NG(Uʹ), for otherwise Uʹ ∪ {v} contradicts the maximality. Let F be the set consisting of edges uv with u ∈ U \ Uʹ and v ∈ V(G) \ (Uʹ ∪ NG(Uʹ)), so that ∣F∣ ≤ ∣U \ Uʹ∣log24n ≤ s∣U∣. Thus, we have, by the definition of expansion (and the choice of F), that $$\label{eqn:rain} \frac{{\varepsilon}\|U\|}{\log^2n} \le \|N\_{G-F}(U)\| \le \|N\_{G}(U')\| \le (\|U'\|+1)\log^{24}n+1\leq 3\|U'\|\log^{24}n,$$ so that $\|U'\|\geq \frac{{\varepsilon}\|U\|}{3\log^{26}n}$, as required. Note that in the sequence of inequalities at we gain that Uʹ ≠ ∅ (as ∣U∣ ≥ 1) in time to use this for the last inequality. We now show that we can ensure the conclusion of holds for all well-expanding sets simultaneously by taking a union bound, and then use Proposition [prop:well-expanding-core] to deduce an expansion property for all sets, as follows. [lem:expandintorandom2] Suppose that G is an n-vertex (ɛ, s)-expander with 2− 9 ≤ ɛ ≤ 1 and s ≥ 2log24n. Let V ⊆ V(G) be a random subset chosen by including each vertex independently at random with probability $\frac13$. Then, with probability at least 1 − o(1/n), for every U ⊆ V(G) and every set F ⊆ E(G) with $\|F\|\leq \frac{\|U\|}{\log^{27}n}$ $$\label{eqn:B1UV} \|B\_{G-F}^{\log^4n}(U,V)\|> \frac{\|V\|}{2}.$$ Say a set Uʹ ⊆ V(G) expands well in G if ∣NG(Uʹ)∣ ≥ ∣Uʹ∣log24n. Given a non-empty well-expanding set Uʹ ⊆ V(G) and a set of edges F of size at most ∣Uʹ∣, applied to Uʹ implies that $$\label{eq:1} \|B\_{G-F}^{\log^4n}(U', V)\|> \frac{\|V\|}2$$ fails with probability at most e− Ω(∣Uʹ∣log2n). Now a union bound over all pairs (Uʹ, F) such that Uʹ is a well-expanding set in G and F is a set of at most ∣Uʹ∣ edges tells us that some such pair (Uʹ, F) fails with probability at most $$\begin{aligned} \sum\_{(U',F)} e^{-\Omega(\|U'\|\log^2n)} &\le \sum\_{u=1}^n\sum\_{f=1}^{u} \binom{n}{u}\binom{n^2}{f} \cdot e^{-\Omega(u\log^2n)} \\ & \le \sum\_{u=1}^n u\cdot n^{3u}\cdot e^{-\Omega(u\log^2n)} \le \sum\_{u=1}^n e^{-\Omega(u\log^2n)}=o(1/n).\end{aligned}$$ Thus, with probability 1 − o(1/n), we can assume that holds for every well-expanding set Uʹ and set F ⊆ E(G) with ∣F∣ ≤ ∣Uʹ∣. We will now show that this implies holds for all U ⊆ V(G) and F ⊆ E(G) with $\|F\|\leq \frac{\|U\|}{\log^{27}n}$, completing the proof. Let then U ⊆ V(G) with $\|U\|\leq \frac23n$ and let F ⊆ E(G) satisfy the (slightly weaker) condition $\|F\|\leq \frac{2 \|U\|}{\log^{27}n}$. Then, by, there is a set Uʹ ⊆ U which is well-expanding for which $\|U'\|\geq \frac{{\varepsilon}\|U\|}{3\log^{26}n}$. Noting that ∣F∣ ≤ ∣Uʹ∣ (as we may assume n is large with probability 1 − o(1/n)), we therefore have that $$\|B\_{G-F}^{\log^4n}(U,V)\|\geq \|B\_{G-F}^{\log^4n}(U',V)\|> \frac{\|V\|}{2}.$$ Finally, consider U ⊆ V(G) with $\|U\|> \frac23n$ and let F ⊆ E(G) satisfy $\|F\|\leq \frac{\|U\|}{\log^{27}n}$. Let Ū ⊆ U be an arbitrary subset with $\frac{n}2\leq \|\bar{U}\|\leq \frac23 n$, so that we have $\|F\|\leq \frac{2\|\bar{U}\|}{\log^{27}n}$, and hence, from what we have just shown, $$\|B\_{G-F}^{\log^4n}(U,V)\|\geq \|B\_{G-F}^{\log^4n}(\bar{U},V)\|> \frac{\|V\|}{2},$$ as required. Path connectedness through a random subset in expanders ------------------------------------------------------- We are now ready to prove Theorem [thm:pathconnect]. As discussed at the start of this section, we first split the edges of the graph G into expanders, before applying Lemma [lem:expandintorandom2] to each of these, to get (with high probability), a strong enough expansion condition to apply. [ of Theorem [thm:pathconnect]] To recap: we have an n-vertex (ɛ, s)-expander, G, with 2− 7 ≤ ɛ ≤ 1 and s ≥ log135n, and a random subset V ⊆ V(G) where each vertex is included independently at random with probability $\frac13$. To prove Theorem [thm:pathconnect], we need to show that, with high probability, G is (4log5n, 28log5n)-path connected through V. Let k = 217log42n, so that s ≥ 212ɛ− 1k2log4n, and let $s'=\frac{\sqrt{s{\varepsilon}}}{ 8k \log n} \ge 2\log ^{24} n$. Using Lemma [lem:partitionedgesintoexpanders], take edge disjoint graphs G1, …, Gk such that E(G) = ⋃i ∈ [k]E(Gi) and, for each i ∈ [k], Gi is an $\left(\frac{{\varepsilon}}4,s'\right)$-expander. Then, by and a union bound over the k graphs Gi, with high probability we can assume that, for each i ∈ [k] and every U ⊆ V(Gi) and F ⊆ E(Gi) with $\|F\|\leq \frac{\|U\|}{\log^{27}n}$, $$\|B^{\log^4n}\_{G\_i-F}(U,V)\|> \frac{\|V\|}{2}.$$ Now, let U ⊆ V(G) and F ⊆ E(G) with ∣F∣ ≤ 217∣U∣log15n. As the graphs Gi, i ∈ [k], are edge disjoint, there must be some i ∈ [k] with $\|F\cap E(G\_i)\|\leq \frac{2^{17}\|U\|\log^{15}n}k=\frac{\|U\|}{\log^{27}n}$, and therefore $$\|B^{\log^4n}\_{G-F}(U,V)\|\geq \|B^{\log^4n}\_{G\_i-F}(U,V)\|> \frac{\|V\|}{2}.$$ We also know by Chernoff’s inequality () that with high probability $\|V\| \ge \frac n8 +1$, and thus, by, applied with k = 28log5n and ℓ = log4n we conclude that G is (4log5n, 28log5n)-connected, as desired. Cycle decompositions ==================== In this section, we will prove our main results,. Before doing this we need to put together a few final ingredients. In Section [sec:connectinexpander], we established a very robust connectivity property of expanders. In we will show that in an expander one can find a subgraph with few edges and yet (effectively) the same connectivity property through a random subset V, with high probability. We will refer to this subgraph as a skeleton of our graph, divide the vertex set into three as V(G) = V1 ∪ V2 ∪ V3, and find three matching skeletons. In we show that any graph can be decomposed into few paths in such a way that no vertex is used as an endvertex many times. In we combine these results to decompose any expander into linearly many cycles and a few leftover edges. As outlined in Section [subsec:sketch], we achieve this by setting aside the skeletons, then decomposing the remainder of the graph into three sets of paths and finally using the connection properties of the skeletons to join the endvertices of these sets of paths, where the sets of paths are matched to the skeletons to ensure this creates edge disjoint cycles. These cycles decompose all the edges in the graph which are not in the skeleton and since the skeleton is chosen to be sparse this gives us the result. The final ingredient in the proof of, given in, is to decompose an arbitrary graph into expanders and a few leftover edges via, to each of which we can apply our expander decomposition result. All that will remain, then, is to iteratively apply in, while removing some additional cycles, to deduce. Finding the skeletons --------------------- To find the skeletons, we will use to embed a sparse well-connected ‘template’ graph (from ) with its edges replaced by relatively short edge disjoint paths, and show that the image of this embedding has the properties we need of a skeleton, as follows. [lem:sparseconnect] Let G be an n-vertex graph which is an (ɛ, s)-expander with 2− 7 ≤ ɛ ≤ 1 and s ≥ log135n. Let V ⊆ V(G) be chosen by including each vertex independently at random with probability $\frac13$. Then, with high probability, there is a subgraph of G with at most 29nlog10n edges which is (log7n, 2)-path connected through V. By Theorem [thm:pathconnect] applied to G and V, G is with high probability (4log5n, 28log5n)-path connected through V. Note that, by Chernoff’s inequality (), we can, in addition, ensure with high probability that $\|V\|\geq \frac n6$, and since our goal is to show a statement with high probability, we may assume that n is large enough to apply. That is, by that lemma, we may assume	arxiv_0000530
Negative modes of Coleman-De Luccia bounces =========================================== We investigate the negative modes about Coleman-De Luccia bounces governing vacuum transitions in de Sitter space, with the goal of gaining physical insight into the various anomalous results associated with these that have been reported in the literature. For the case of bounces with radii much less that the horizon distance H− 1 we find two distinct regimes, distinguished by the magnitude of the bubble nucleation rate Γ. If Γ/H4 ≫ 1, then the behavior of the modes contributing to the determinant factors in Γ is much as it is in flat spacetime, and the calculation of Γ goes over smoothly to the flat spacetime calculation as the gravitational coupling is taken to zero. This is not the case if Γ/H4 ≪ 1. These two regimes correspond to the two possible outcomes of de Sitter vacuum decay — either a rapidly completed transition or non-percolation and eternal inflation. For bounces with radii comparable to the horizon length, we confirm previous results concerning anomalous negative modes with support on the bounce wall. We also find further evidence supporting previous claims, based on thin-wall arguments, of the absence of expected negative modes for a class of bounces that arises when the initial and final vacua are nearly degenerate. Introduction ============ The process in which a false vacuum of a scalar field theory decays by the nucleation of true vacuum bubbles was first studied in detail some time ago . The bubble nucleation rate has the exponential suppression typical of tunneling processes, with the exponent being twice the WKB barrier penetration integral or, equivalently, the Euclidean action of the bounce solution to the Euclidean field equations . By using path integral methods to calculate the energy density of the false vacuum, the pre-exponential factor can also be obtained . This turns out to contain a factor of $[\det S\_E''(\phi\_{\rm bounce})]^{-1/2}$, where SEʺ denotes the second functional variation of the Euclidean action. An essential ingredient of this calculation is the fact that $S\_E''(\phi\_{\rm bounce})$ has a single negative eigenvalue, which arises from a mode corresponding to expansion or contraction of the bounce. Because of the square root, this negative mode leads to a factor of i that makes the false vacuum energy complex, with its imaginary part interpreted in terms of a rate of decay by bubble nucleation. It is crucial here that there is only a single negative eigenvalue and thus one factor of i; if, for example, there were an even number of such eigenvalues, the contribution to the false vacuum energy would be purely real. Indeed, it has been shown that the bounce with lowest action has one, and only one, negative mode . Coleman and De Luccia (CDL) proposed that this formalism could be extended to include gravitational effects by adding a Euclidean Einstein-Hilbert term to the action. Although they did not address the issue of the prefactor, it seems natural to assume that it should be analogous to that for the non-gravitational case, and that the dominant bounce should again have a single negative mode. The matter is complicated by the fact that with gravity included the fluctuations about the bounce enjoy a gauge freedom, corresponding to the invariance under coordinate transformations of the underlying theory. A consequence of this gauge freedom is the existence of constraints that must be imposed on the possible fluctuations. A number of authors have studied the negative mode issue in the context of bounces corresponding to bubble nucleation in a de Sitter spacetime . Some curious, troubling, and sometimes contradictory, results, often depending on the choice of gauge, have been obtained. Among these are: 1) An action for fluctuations about a bounce with large radius that is unbounded from below, suggesting an infinite family of negative modes with support near the bubble wall. 2) Indications of a similar phenomenon for all de Sitter CDL bounces, but with the additional negative modes having support on a region defined by the horizon. 3) Arguments for the existence of a single negative mode, but subject to restrictions on the form of the bounce solution. 4) Claims that there are no negative modes, and that the factor of i comes from a proper rotation of contours in the Hamiltonian path integral. 5) The absence of a negative mode in the thin-wall approximation for a certain class of bounces. In our view, these works only address one part of the problem, namely how to develop an algorithm that, at least in principle[1](#fn1), gives an unambiguous result for the bubble nucleation rate. There is another aspect that should be addressed. When anomalous or apparently meaningless mathematical results are encountered, it is often the case that they are indicative of a particular physical aspect of the problem being studied. Thus, one should look for a physical understanding as to why these negative modes arise (or do not) in particular cases, even if they can ultimately be eliminated from the calculation. Also, one should expect to find a smooth transition from the curved spacetime calculation to the non-gravitational one in an appropriate limit. Ideally, this should include not just a continuous evolution of the bounce and its action, but also a gradual disappearance of the potential pathologies as one approaches the limit in which κ ≡ 8πG = 0. As we will see, this limit is rather subtle. Our goal in this paper is to make some progress toward this end by exploring the regions where the quadratic fluctuation action becomes negative. We work in a purely Lagrangian framework. Rather than fixing a gauge, we impose the requirement that the fluctuations obey the constraint, and then write the quadratic fluctuation Lagrangian in terms of manifestly gauge-invariant combinations of fields. For fluctuations about a bounce with O(4)-symmetry, which we assume, the objects of primary interest are spherically symmetric scalar fluctuations, which can be described by a single field χ. We examine the situations in which χ can have negative modes, and study the form of those modes as κ is varied. The remainder of this paper is organized as follows. In Sec. [tunnelSec] we review the formalism for treating vacuum decay in both flat spacetime (i.e., with gravity ignored) and with gravitational effects included. In Sec. [negativeSec] we discuss negative modes in flat spacetime. In the course of this discussion we describe an example where bounces with many negative modes turn out to not only be physically meaningful, but to actually be dominant. Section [perturb-sec] describes the perturbative expansion of the scalar field and metric about an O(4)-symmetric CDL bounce and gives the expression for the quadratic part of the fluctuation Lagrangian. Next, in Sec. [negmode-sec], we explore the various types of negative modes that can arise from this Lagrangian. Many of the results that we present here are based on numerical explorations of particular scalar field models. In Sec. [multibounceSec] we discuss the negative mode problem for multibounce solutions. This is almost a trivial aside in flat spacetime, but becomes more complex with gravity included. In fact, we will see that it is key to understanding the limit in which gravitational effects disappear. Finally, in Sec. [conclusionSec] we summarize our results and discuss how they reflect the various modes of vacuum decay in de Sitter spacetime. There are two appendices containing some technical details. Review of tunneling =================== Tunneling in flat spacetime --------------------------- The WKB approximation leads to an expression of the form γ = C e− B for the decay rate of a state that decays by tunneling through a potential energy barrier. For the case of a single particle in one dimension with a standard kinetic energy, $$B = 2 \int\_{x\_1}^{x\_2} dx \, \sqrt{2m[V(x) -E]} \,, \label{tunnelInt}$$ where x1 and x2 are the classical turning points that bound the particle’s path through the barrier. With multiple degrees of freedom, the WKB approximation requires that we consider all paths through the barrier and pick out the one along which the one-dimensional tunneling exponent is a minimum . This minimization problem is equivalent to the problem of finding a stationary point of the Euclidean action, which in turn is equivalent to finding a solution of the Euclidean equations of motion. The solution relevant for tunneling, known as the bounce, starts at an initial configuration at Euclidean time $\tau\_{\rm in}$, runs through the barrier to an exit point at some τ0, and then returns, in a “τ-reversed” fashion, to $\tau\_{\rm fin}$. The intermediate Euclidean times give a parameterization of the tunneling path through configuration space. If the initial configuration is a local minimum of the potential energy (i.e., a false vacuum), then $\tau\_{\rm in}=-\infty$ and $\tau\_{\rm fin}=\infty$. For a scalar field theory with a standard Lagrangian the tunneling exponent B is equal to the difference between the Euclidean actions of the bounce solution ϕ(x) and the uniform false vacuum. Even if these actions are divergent, their difference, $$\begin{aligned} B &=& S\_E(\phi) - S\_E(\phi\_{\rm fv}) \cr &=& \int\_{-\infty}^\infty d\tau \int d^3x\, \left[ \frac12 \left({d\phi \over d\tau}\right)^2 + \frac12 ({\bm \nabla}\phi)^2 + U(\phi) - U(\phi\_{\rm fv}) \right] \,,\end{aligned}$$ is finite and well-defined. Note that the tunneling integral and the Euclidean action differ for paths that are not solutions of the equations of motion. In fact, although the bounce is a local minimum of the integral through the potential energy barrier, it only corresponds to a saddle point of SE. In a scalar field theory, with thermal and gravitational effects ignored, the bounce describing tunneling from a false vacuum is a solution on Euclidean R4 that approaches the false vacuum value $\phi\_{\rm fv}$ as either $\|{\bf x}\|$ or τ tends to infinity. With only minimal restrictions on the potential U(ϕ), it can be shown that the bounce with the smallest action is O(4)-symmetric . Figure [flat-bounce] illustrates such a bounce. [flat-bounce] The prefactor C can be obtained by a path integral argument . Consider the matrix element $$\begin{aligned} \langle \phi\_{\rm fv}\| e^{-HT} \| \phi\_{\rm fv}\rangle &=& \int[d\phi({\bf x},\tau)]\, e^{-S\_E[\phi]} \cr &=& \sum\_n e^{-E\_nT} \langle \phi\_{\rm fv}\| n\rangle \langle n\| \phi\_{\rm fv} \rangle \,.\end{aligned}$$ Here the path integral is restricted to paths that begin and end on the false vacuum configuration. In the limit T → ∞, the sum over energy eigenstates in the second expression is dominated by the state with the lowest energy among those that contribute to the matrix element. Identifying this state as the false vacuum, we have $$E\_{\rm fv} = -\lim\_{T\to \infty} {1\over T} \ln \, \langle \phi\_{\rm fv}\| e^{-HT} \| \phi\_{\rm fv}\rangle \,. \label{EfvFromLog}$$ The path integral is evaluated by expanding about its stationary points. The first of these is the constant solution with $\phi(x) = \phi\_{\rm fv}$ everywhere. A Gaussian integral about this gives $$I\_0 = [\det S\_E''(\phi\_{\rm fv})]^{-1/2} \, e^{-S\_E(\phi\_{\rm fv})} \,,$$ where SʺE denotes the second functional derivative of the Euclidean action. It is convenient to consider space to have a finite volume Ω which is to be taken to infinity at the end of the calculation, so that $S\_E(\phi\_{\rm fv}) = \Omega T \,U(\phi\_{\rm fv})$. Next, we have the bounce solution. This would give a similar Gaussian integral were it not for two factors. First, the spectrum of $S''\_E(\phi\_{\rm bounce})$ includes four zero modes, corresponding to the freedom to translate the bounce in Euclidean space and time. These are handled by introducing collective coordinates specifying the location of the center of the bounce. Second, the spectrum also contains a mode with negative eigenvalue, corresponding roughly (and exactly, in the thin-wall limit) to expansion or contraction of the bounce. This negative mode leads to a factor of i when the square root of the determinant is taken. The contribution to the path integral can be written as $$\begin{aligned} I\_1 &=& \frac{i}{2}\,\Omega T \left\|{\det' S\_E''(\phi\_{\rm bounce})] \over \det S\_E''(\phi\_{\rm fv})}\right\|^{-1/2} \, J \, e^{-[S\_E(\phi\_{\rm bounce}) - S\_E(\phi\_{\rm fv})]}\, I\_0 \cr &\equiv & i \Omega T K e^{-B} \, I\_0 \,. \label{I1formula}\end{aligned}$$ Here the factor of ΩT arises from integrating over the four collective coordinates, J contains the Jacobian factors associated with the introduction of the collective coordinates, and the factor of 1/2 comes from a careful treatment of the negative mode. Finally, the prime on the bounce determinant indicates that the product of eigenvalues is to be taken only over the nonzero eigenvalues. There are also approximate stationary points corresponding to many well-separated bounces. The n-bounce solution has an action $$S\_E(\phi\_{n\rm -bounce}) = S\_E(\phi\_{\rm fv}) + n \left[S\_E(\phi\_{\rm bounce}) - S\_E(\phi\_{\rm fv}) \right] = S\_E(\phi\_{\rm fv}) + n B \,.$$ The integration over the collective coordinates gives a factor of (ΩT)n/n!, with the n! entering because the bounces are indistinguishable. Similarly, the factor of J becomes Jn. Only the determinant factors remain to be considered. Outside the wall region the bounce solution rapidly approaches the pure false vacuum, with $\|\phi\_{\rm bounce}(x) - \phi\_{\rm fv}\|$ decreasing exponentially with distance. Thus, we can imagine evaluating the determinant for the one-bounce case by dividing Euclidean space into a large (compared to the bounce radius) region enclosing the bounce, and the remainder. The full determinant is then the product of the contributions of the two regions. In the latter region the bounce is exponentially close to the false vacuum and the contributions to the bounce and the false vacuum determinants are essentially equal. In the region containing the bounce the determinants corresponding to $\phi\_{\rm bounce}$ and $\phi\_{\rm fv}$ differ precisely by the ratio that appears in Eq. ([I1formula]). For an n-bounce configuration, with the bounces all well separated, the Euclidean spacetime can be divided into n regions, each containing one bounce, and the remainder, in which the field is essentially equal to its false vacuum region. The full determinant is the product of the contributions from each of these regions. The net result is that the contribution to the path integral from n-bounce configurations is $$I\_n = {1\over n!} \left( i \Omega T K e^{-B} \right)^n I\_0 \,.$$ Summing the contributions from all values of n gives $$\begin{aligned} I &=& I\_0 \sum\_{n=0}^\infty {1\over n!} \left( i \Omega T K e^{-B} \right)^n \cr &=& I\_0 \exp\left[ i \Omega T K e^{-B} \right] \,. \label{Isum}\end{aligned}$$ Taking the logarithm and using Eq. ([EfvFromLog]) gives $$E\_{\rm fv} = -\lim\_{T\to \infty} \left( {\ln \,I\_0\over T}\right) - i\Omega K e^{-B} \,.$$ The first term on the right-hand side is real, but the second is imaginary, making the false vacuum energy complex. As usual, we interpret this as a signal that the false vacuum is unstable, with a decay rate $$\gamma = - 2 \,{\rm Im}\, E\_{\rm fv} = 2 \Omega K e^{-B} \,.$$ This is proportional to Ω, corresponding to the fact that a bubble can nucleate anywhere in space. The bubble nucleation rate per unit volume is Γ = 2K e− B . This calculation implicitly assumes that in the multibounce solutions the individual bounces are separated by distances large relative to the bounce four-volume ${\cal V}\_b$. To test this dilute-gas approximation, we note that the sum in Eq. ([Isum]) is dominated by the terms with n ≈ ΩTKe− B . For the approximation to be valid, we must require that the volume occupied by these n bounces be much less than the total Euclidean spacetime volume ΩT; i.e., $${\cal V}\_4 K e^{-B} \ll 1 \,,$$ where ${\cal V}\_4$ is the four-dimensional volume of the bounce. Dimensional arguments suggest that ${\cal V}\_4 K$ is typically of order unity, in which case this condition reduces to a lower bound on B. A useful illustrative example is given by the thin-wall approximation, which applies in the limit where the difference between the false and true vacuum values of the potential, $\epsilon \equiv U(\phi\_{\rm fv}) - U(\phi\_{\rm tv})$, is sufficiently small relative to the surface tension σ of the bubble . The bounce can then be obtained by considering O(4)-symmetric configurations in which a true vacuum region of radius R is separated from the exterior false vacuum by a thin wall with action per unit area σ. The total Euclidean action, less that of the pure false vacuum, is $$S\_E(R) = 2\pi^2 \sigma R^3 - {\pi^2 \over 2}\, \epsilon R^4 \,.$$ The stationary point of this action at $\bar R \equiv 3 \sigma/\epsilon$ gives the bounce radius. Both the bounce radius and the tunneling exponent $$B = {27 \pi^2\over 2}\, {\sigma^4 \over \epsilon^3}$$ tend to infinity in the ε → 0 limit of degenerate vacua. The fact that $$\left. {d^2 S\_E \over d R^2}\right\|\_{\bar R} = -{18 \pi^2 \sigma^2 \over \epsilon} \label{TWAsecondDeriv}$$ is negative shows that the bounce is a maximum of the action among the one-parameter family of thin-wall configurations, and thus a saddle point of the action, with at least one negative eigenvalue, on the full configuration space. Finally, recall that the bounce is supposed to describe a family of configurations interpolating between the initial pure vacuum state, at τ = − ∞, and a state of equal potential energy containing a bubble, on a constant τ slice through the center of the bounce. In the thin-wall limit the difference in the static potential energy between these two configurations is $$\Delta E = 4\pi R^2 \sigma - \frac43 \pi R^3 \epsilon \,.$$ This does indeed vanish if $R = \bar R$. The above discussion readily generalizes to the case of nonzero temperature T . In this case Γ is obtained from the imaginary part of the free energy, rather than the energy, of the metastable false vacuum. The Euclidean path integral is then over paths that are periodic in Euclidean time with period β = 1/T. Furthermore, the classical Euclidean equations leading to the bounce must be obtained using the finite-temperature effective potential. Just as at T = 0, the path integral includes contributions not just from a single bounce, but also from all multibounce solutions. Summing over these exponentiates the single bounce contribution, and the nucleation rate can be read off from the exponent. Including gravitational effects ------------------------------- Coleman and De Luccia argued that gravitational effects could be incorporated by adding a Euclidean Einstein-Hilbert term to the action, so that $$S\_E = \int d^4x \sqrt{g} \left[-{1 \over 2\kappa}R +\frac12\,g^{ab} \partial\_a\phi\,\partial\_b \phi +U(\phi) \right] + S\_{\rm bdy} \,,$$ where κ = 8πG and $S\_{\rm bdy}$ is the Euclidean version of the Gibbons-Hawking boundary term . Although it has not been proven that the bounce of minimum action continues to be O(4)-symmetric when gravitational effects are included, this is widely believed to be true. If one assumes this to be the case, the Euclidean metric can be written as ds2 = N(ξ)dξ2 + ρ(ξ)2dΩ32 , where dΩ32 is the usual metric on the unit three-sphere. It is convenient to choose the origin of ξ to be a zero of ρ, so that ρ(0) = 0. The curvature scalar is $$R = {6 \over N \rho^2}\left(N-{\dot\rho}^2 - \rho\ddot\rho\right) + {3 \dot\rho\dot N\over \rho N^2} \,,$$ with overdots denoting derivatives with respect to ξ. With the scalar field ϕ depending only on ξ, the Euclidean action takes the form $$S\_E = 2\pi^2 \int d\xi \sqrt{N} \left\{ \rho^3\ \left[{1\over 2N}\,{\dot\phi}^2 + U(\phi)\right] +\frac3\kappa \left[{1\over N}\left(\rho^2\ddot\rho + \rho{\dot\rho}^2\right) -\rho -{\rho^2\dot\rho \dot N \over 2 N^2} \right] \right\} + S\_{\rm bdy} \,.$$ An integration by parts to remove the second derivative term recasts this as $$S\_E = 2\pi^2 \int d\xi\sqrt{N} \left\{ \rho^3\left[{1 \over 2N}\,{\dot\phi}^2 + U(\phi)\right] -\frac3\kappa \left( {\rho{\dot\rho}^2 \over N} +\rho\right) \right\} \,,$$ with the boundary terms from the integration by parts precisely canceling $S\_{\rm bdy}$. We note that the action does not contain any derivatives of N. Consequently, variation with respect to N yields a constraint equation (which is in fact the Gξξ Einstein equation), $$0= \rho^3\left({{\dot\phi}^2\over 2N} - U \right) -{3\over \kappa}\left({\rho{\dot\rho}^2 \over N} -\rho \right) \,. \label{rhoN-eq}$$ The existence of this constraint is related to the freedom to make a coordinate transformation to redefine ξ. We make use of this gauge freedom to set N(ξ) = 1. Having done so, we can write the constraint equation as $${\dot\rho}^2 = 1 + {\kappa \over 3} \, \rho^2 \left(\frac12 {\dot\phi}^2 - U\right) \,. \label{rho-eq}$$ Varying the action with respect to ϕ gives the scalar field equation of motion $$\ddot\phi + {3 \dot\rho\over \rho}\dot\phi = {dU \over d\phi} \,. \label{phi-eq}$$ Equations ([rho-eq]) and ([phi-eq]) are a complete set of field equations. Variation of the action with respect to ρ does not yield an independent equation, reflecting the fact that in spherically symmetric configurations the gravitational dynamics is determined completely by the matter distribution. However, differentiating Eq. ([rho-eq]) and then using Eq. ([phi-eq]) yields the useful identity $$\ddot \rho = - \frac{\kappa}{3} \, \rho\left(\dot\phi^2 + U \right) \,.$$ For bubble nucleation in a Minkowski or anti-de Sitter false vacuum the bounce solution has the topology of R4, just as in the non-gravitational case. De Sitter spacetime, the case on which we will focus, is different. In this case ρ always has a second zero, at a value $\xi=\xi\_{\rm max}$. The bounce is then topologically a four-sphere, with a finite four-volume of order H− 4, where $$H = \sqrt{{\kappa \over 3}\, U(\phi\_{\rm fv}) } \label{HubbleDef}$$ is the Hubble parameter of the false vacuum.[2](#fn2) In particular, for the solution of Eqs. ([rho-eq]) and ([phi-eq]) describing the pure false vacuum, with $\phi= \phi\_{\rm fv}$ everywhere, the metric is the standard round metric on the four-sphere, with ρ = H− 1sin(Hξ) . Its Euclidean action, which must be subtracted from that of the bounce to obtain B, is $$S\_E = - {24\pi^2 \over \kappa^2 \, U(\phi\_{\rm fv})} = - {8\pi^2 \over \kappa} \, H^{-2} \,.$$ In our discussions it will be useful to distinguish two classes of bounce solutions: a) Small bubble bounces: The difference between the values of the potential at the false vacuum and at the top of the barrier defines a mass scale μ via $$U\_{\rm top} - U\_{\rm fv} \equiv \mu^4 \,. \label{mudef}$$ If μ ≪ MP and the difference between the true and false vacuum values of U is not too small, then the bounce describes the nucleation of bubbles whose radius $\bar \rho$ at nucleation is much smaller than H− 1. If ξ ≪ H− 1, then ρ(ξ) ≈ ξ and ϕ(ξ) differs only slightly from the corresponding flat-space bounce. For larger ξ, on the other hand, ρ = H− 1sin[H(ξ + Δ)] , where Δ = O(μ/MP2). The scalar field in this large-ξ region is exponentially close to its false vacuum value, with $\|\phi - \phi\_{\rm fv}\| \sim \mu e^{-\mu(\xi - \bar\xi)}$. b) Large bubble bounces: Bubbles of size comparable to H− 1 can arise, even with μ ≪ MP, if ε is sufficiently small. The flat-space thin-wall results are replaced by $${1 \over \bar\rho^2} = H^2 + \left( {\epsilon \over 3 \sigma} - {\kappa \sigma \over 4} \right)^2 \,. \label{TWArhoBar}$$ As ε is decreased, the bubble radius increases until it reaches a maximum value, $\bar\rho = H^{-1}$, when the quantity in parentheses vanishes. Until this point, the false vacuum region is larger than a hemisphere, while the true vacuum occupies less than a hemisphere. With a further decrease in ε, $\bar\rho$ begins to decrease, and the true and false vacuum regions are both less than hemispheres. In contrast to the flat-space thin-wall bounce, whose radius and action tend to infinity in the ε → 0 limit, both $\bar\rho$ and the bounce action remain finite in this limit. To distinguish between the two regimes, we will denote bounces as being type A or type B, according to whether the false vacuum region is greater than or less than a hemisphere. We saw in Eq. ([TWAsecondDeriv]) that with gravity ignored the bounce was a maximum, as a function of R, of the thin-wall-approximation action. Similarly, for type A bounces the $\bar\rho$ of Eq. ([TWArhoBar]) gives a maximum of the thin-wall action, and therefore a saddle point of the full action functional . However, for type B thin-wall solutions the second derivative of the action with respect to $\bar\rho$ is positive, corresponding to a minimum. We can generalize the distinction between type A and type B solutions beyond the thin-wall limit. In the thin-wall limit the maximum of ρ in a type B bounce occurs at the bubble wall. More generally, we will say that a bounce is type B if the maximum of ρ occurs within the wall region, and type A otherwise. (Because the beginning and end of the wall are not precisely defined, there will still be some ambiguity concerning borderline cases.) [CDL-bounce] The fact that the bounce topology is compact raises questions: Why is the volume finite? How can the bounce be understood as a sequence of configurations forming a tunneling path through a potential energy barrier? It was argued in that the bounce should be understood as mediating a transition in a de Sitter horizon volume (giving a finite three-volume ∼ H− 3) at a finite de Sitter temperature $T\_{\rm dS} = H/2\pi$ (giving periodicity ∼ H− 1 in a fourth Euclidean coordinate). In this picture the path through configuration space is given by a series of radial slices, each of which represents a three-dimensional ball, as shown in Fig. [CDL-bounce] (see also ). The boundaries of the various three-balls, which are two-spheres at the horizon, all meet and are identified. The initial and final slices are the two horizontal slices in this figure. In the non-gravitational case, the bounce connects initial and final configurations that are each turning points, with vanishing derivatives with respect to the Euclidean time, that can be continued to real time configurations with vanishing time derivatives. With gravitational effects included, the initial and final slices each correspond to a configuration on a horizon volume with instantaneously static metric and scalar field. In Appendix [horizonApp] we verify explicitly that these limiting configurations are indeed bounded by horizons. Negative modes ============== The occurrence of a negative eigenvalue in the spectrum of fluctuations about the bounce was an essential ingredient in the path integral derivation of the flat-space nucleation rate. It gave rise to the factor of i in Eq. ([I1formula]), which led to the bounce terms being a contribution to the imaginary part of the false vacuum energy. The existence of the negative mode is readily established in flat spacetime. An O(4)-symmetric bounce obeys $${d^2\phi \over ds^2} + {3 \over s}{d\phi \over ds} = U'(\phi) \,, \label{flatScalarEq}$$ where $s = \sqrt{\tau^2 + {\bf r}^2}$. The eigenmodes about this bounce satisfy [ − □E + Uʺ(ϕb)]η = λη . The spherical symmetry of the underlying bounce allows us to decompose the modes as products of a radial function and a four-dimensional spherical harmonic, and to recast Eq. ([BoxEeq]) into the form of a Schroedinger equation, with the angular derivatives giving rise to an angular momentum barrier term. We know that the breaking of translational invariance gives rise to four zero modes that transform as the components of a four-dimensional vector. Standard arguments then show that there must be a rotationally invariant scalar mode with a lower, and therefore negative, eigenvalue λ that obeys $$- {d^2\eta \over ds^2} - {3 \over s}{d\eta \over ds} + U''(\phi\_b) \eta = \lambda \eta \,. \label{flat-mode-eq}$$ This is readily verified for the case of a thin-wall bounce with radius $\bar R$. Differentiating Eq. ([flatScalarEq]) with respect to s gives $$\left[ -{d^2 \over ds^2}- {3 \over s}{d \over ds} + U''(\phi\_b) \right] {d\phi\_b \over ds} = -{3\over s^2} {d\phi\_b \over ds} \,. \label{flat-neg-mode}$$ In the thin-wall limit dϕ/ds vanishes except in a narrow region where $s \approx \bar R$, and so we see that there is an eigenfunction η(s) ≈ dϕ/ds with negative eigenvalue $\lambda \approx -3/\bar R^2$. What about multiple negative modes? The determinant factor would acquire a factor of i for each such mode, and so would be real if there were an even number of negative modes. Three such modes would give an imaginary determinant factor, but with the wrong sign. Although five modes, or any larger number equal to $1 \mod 4$, would give an imaginary part of the right sign, this would seem to be an unusual constraint on the bounce. The situation was clarified by Coleman , who showed that if a bounce has multiple negative modes then the corresponding tunneling path through configuration space is a saddle point, rather than a minimum, of the barrier penetration integral. As an explicit illustration of this, consider a theory with a single scalar field whose potential has three minima, at ϕA < ϕB < ϕC, of increasing depth, UA > UB > UC . It is convenient to choose the zero of energy so that UA = 0. Let us assume that the energy density differences $$\begin{aligned} \epsilon\_1 &\equiv& U\_A - U\_B \,, \\ \epsilon\_2 &\equiv& U\_B - U\_C \,,\end{aligned}$$ and the corresponding surface tensions σ1 and σ2 are such that the thin-wall approximation is valid for both the A-to-B and the B-to-C transitions. The corresponding thin-wall bubble radii are $$\bar R\_j = {3 \sigma\_j \over \epsilon\_j}\,, \qquad j=1,2 \,.$$ Now consider a “double-bounce” configuration in which a spherical C-vacuum region of radius R2 is centered within a spherical B-vacuum region of radius R1, which itself is surrounded by an A-vacuum background. This is clearly a solution of the Euclidean field equations if $R\_1 =\bar R\_1$ and $R\_2 =\bar R\_2$. It is also clear that there are two negative modes, corresponding to independent variations of R1 and R2. With a single spherical thin-wall configuration $\bar R$ is the unique radius consistent with having the four-dimensional fields interpolate between the false vacuum, at τ = − ∞, and a three-dimensional configuration of equal energy containing a single bubble, at τ = 0. With two nested three-spheres there is a one-parameter family of configurations satisfying this equal energy condition, with their radii constrained to obey $$0= 4\pi \sigma\_1 R\_1^2 -{4\pi \over 3} \epsilon\_1 R\_1^3 + 4\pi \sigma\_2 R\_2^2 -{4\pi \over 3} \epsilon\_2 R\_2^3 \,.$$ For each choice of R1 and R2 obeying this condition one can calculate the potential energy[3](#fn3) $$V(\tau) = \int d^3x \left[\frac12 ({\bm \nabla} \phi)^2 + U(\phi)\right]$$ as a function of τ and then calculate the field theory generalization of the barrier penetration integral of Eq. ([tunnelInt]). As expected, this has a minimum when R2 = 0 and $R\_1= \bar R\_1$ (i.e. the ordinary A-to-B bounce). On the other hand, the choice $R\_1 =\bar R\_1$ and $R\_2 =\bar R\_2$ that gives the double-bounce solution turns out to maximize this tunneling integral. Thus, we would not expect this double-bounce solution to correspond, even with a reduced rate, to a new mode of bubble nucleation. However, suppose that the parameters are such that the preferred radius, $\bar R\_2$, of the B-to-C bounce is much less than $\bar R\_1$. There will then be solutions in which a single A-to-B bounce contains many B-to-C bounces. In evaluating the path integral, we would then have to sum not only over all numbers of A-to-B bounces, as in Eq. ([Isum]), but also over the numbers of B-to-C bounces within each of these bubbles. This would lead to a double sum formula $$\begin{aligned} I &=& I\_0 \sum\_{n=0}^\infty {1\over n!}\, \left[i\Omega TK\_1e^{-B\_1} \sum\_{k=0} {1 \over k!}\left(i {\cal V}\_4 K\_2 e^{-B\_2}\right)^k \right]^n \cr &=& I\_0 \sum\_{n=0}^\infty {1\over n!}\, \left[i\Omega TK\_1e^{-B\_1} \sum\_{k=0} {1 \over k!}\left({i\over 2}\, {\cal V}\_4 \Gamma\_2\right)^k \right]^n \,,\end{aligned}$$ where $${\cal V}\_4 = {81 \pi^2 \over 2} \, \left(\sigma \over \epsilon \right)^4$$ is the four-volume of the A-to-B bounce and Γ2 is the bubble nucleation rate for the B-to-C transition. Because ΩT is understood to be eventually taken to infinity, the upper limit on the sum over n can be taken to be infinity, allowing us to convert the sum to an exponential. By contrast, ${\cal V}\_4$ is fixed and finite, leading us to consider two regimes. If ${\cal V}\_4 \Gamma\_2$ is much less than unity, the sum over k is dominated by the k = 0 term, the A-to-B bounces containing B-to-C sub-bounces can be ignored, and there is no significant correction to Γ1. On the other hand, if ${\cal V}\_4 \Gamma\_2$ is much greater than unity, the sum over k can be approximated by an exponential, which has the effect of replacing B1 by $$\tilde B\_1 = B\_1 - {i\over 2} {\cal V}\_4 \Gamma\_2 = {27 \pi^2 \over 2} \, {\sigma\_1^4 \over \epsilon\_1^3} \left( 1 - {3i\over 2} \, {\Gamma\_2 \over \epsilon\_1}\right) \,.$$ To leading order, this is the same as making the replacement ε1 → ε1 + iΓ2/2 or, in other words, as if we had given the energy density of vacuum B an imaginary part Γ2/2. This describes a situation in which vacuum B is sufficiently unstable that bubbles of B already contain tiny bubbles of C at the time that they nucleate. Thus, although though the double-bounce with two negative modes has little physical relevance, the solutions with many sub-bounces and hence many negative modes are physically meaningful.[4](#fn4) Although they have a somewhat higher action than the unadorned single bounce, this is outweighed by the fact that they are more numerous. In essence, it is a case of “entropy” overcoming “energy”. Perturbative expansion about a CDL bounce ========================================= In this section we consider the expansion of the Euclidean action about an O(4)-symmetric bounce solution of Eqs. ([rho-eq]) and ([phi-eq]), and obtain the contribution from the terms quadratic in fluctuations about the bounce. We require that these fluctuations obey the constraints that follow from the freedom to make coordinate transformations. To exclude fluctuations that are purely coordinate transformations, we write the quadratic action in terms of explicitly gauge-invariant quantities. It is convenient to exploit the O(4) symmetry by expanding the fluctuations in terms of O(4) harmonics. The normal modes can be classified as scalar, vector, or tensor. Because the only matter source is a scalar field, it suffices for us to focus on the scalar modes. We begin by considering the most important case, the O(4)-symmetric zero angular momentum modes. O(4)-symmetric fluctuations --------------------------- With O(4) symmetry retained, we can write the metric as ds2 = [1 + 2A(ξ)]dξ2 + ρ(ξ)2[1 + 2Ψ(ξ)]dΩ32 , corresponding to the perturbations $$\begin{aligned} N(\xi) &\rightarrow& 1 + A(\xi) \,, \cr \rho(\xi) &\rightarrow& \rho(\xi)[1 + \Psi(\xi)] \,, \label{zeroEllPert}\end{aligned}$$ and define the perturbed scalar field ϕ(ξ) → ϕ(ξ) + Φ(ξ) . Next we expand the total action about the background solution. The first-order correction to the action vanishes by the background equations of motion. The quadratic terms are given by $$S^{(2)}\_E=2\pi^2 \int L^{(2)}\_E(\Phi,\Psi,A;\dot{\Phi},\dot{\Psi}) d\xi \,,$$ where $$\begin{aligned} L^{(2)}\_E(\Phi,\Psi,A;\dot{\Phi},\dot{\Psi})= -\frac{3}{\kappa}\rho^3 \dot{\Psi}^2+ \frac{3}{\kappa}\rho\Psi^2+\frac{1}{2}\rho^3\dot{\Phi}^2 +\frac{1}{2}\rho^3 U''\Phi^2-3\rho^3 \dot{\phi}\dot{\Psi}\Phi \,\cr +\left(-\rho^3\dot{\phi}\dot{\Phi}+\rho^3 U' \Phi +\frac{6}{\kappa}\dot{\rho}\rho^2\dot{\Psi} +\frac{6}{\kappa}\rho \Psi\right)A-\frac{3}{\kappa}\rho Q A^2 \,. \label{zeroEllL2}\end{aligned}$$ Here we have defined[5](#fn5) $$Q= \left(1 - {\kappa \rho^2 U \over 3}\right) = \dot\rho^2 -\frac{\kappa \rho^2\dot{\phi}^2}{6} \,, \label{Qdef}$$ with the two expressions being equal because of Eq. ([rho-eq]). The theory is invariant under coordinate transformations. For perturbations about a background solution, the coordinate transformation ξ → ξ + α(ξ) gives the infinitesimal gauge transformation $$\delta\_G \Phi = \dot{\phi}\alpha, \quad \delta\_G \Psi=\frac{\dot{\rho}}{\rho}\alpha, \quad \delta\_G A = \dot{\alpha} \,. \label{O4gaugeTrans}$$ This leaves LE(2) unchanged, up to a total derivative (see Appendix [mode-app]), and so is an invariance of the action. Closely related to this gauge invariance is the existence of a constraint that follows from the fact that LE(2) does not contain derivatives of A. Requiring that δLE(2)/δA vanish gives $$0 = {\cal C}^{(1)} \equiv \frac{\kappa \rho^2}{6} \left(\dot{\phi}\dot{\Phi}-U'\Phi\right) -\left(\rho \dot{\rho}\dot{\Psi}+\Psi\right)+ Q A \,.$$ This is nothing more than the linear term in the expansion of the constraint arising from the variation N(ξ), and could have been obtained by substituting Eqs. ([zeroEllPert]) and ([PhiPert]) into Eq. ([rhoN-eq]). Using this constraint we can eliminate A from Eq. ([zeroEllL2]). After some tedious calculations, described in Appendix [mode-app], we obtain the remarkably simple expression $$L^{(2)}\_E(\chi;\dot{\chi})=\frac{\rho^3}{2Q}\dot{\chi}^2+ \frac{\rho^3}{2Q}f(\rho,\phi)\chi^2 \,, \label{g-inv-L2}$$ where we have defined the gauge-invariant quantity $$\chi \equiv \dot\rho \Phi-\rho\dot\phi\Psi \label{ZeroEllChi}$$ and $$f(\rho,\phi) = U''+\frac{\kappa \rho^2 U'^2}{3 Q} +\frac{\kappa \rho\dot{\phi} U'}{3\dot{\rho} Q} +\frac{2\kappa \dot\phi^2}{3} - \frac{\kappa\rho\dot\phi U'}{\dot\rho} -\frac{4\kappa U}{3} -{\ddot\rho \dot Q\over \dot\rho Q} \,. \label{fForL2}$$ Our expression for the second-order Lagrangian LE(2) is now manifestly gauge-invariant. One can check that the constraint ${\cal C}^{(1)}$ is likewise gauge-invariant. Non-spherically-symmetric perturbations --------------------------------------- The O(4)-symmetric perturbations of the metric are completely described by the two functions A and Ψ. Less symmetric perturbations require additional fields. For arbitrary angular momentum the perturbed metric can be written as $$\begin{aligned} ds^2 &=& [1+2 A\_l(\xi)Y\_l(\Omega)]\, d\xi^2 + B\_l(\xi) \nabla\_a Y\_l(\Omega)\, d\xi \,dz^a \, \cr &&+\rho(\xi)^2\left\{\bar{g}\_{ab} [1+2\Psi\_l(\xi)Y\_l(\Omega)] +2C\_l(\xi)k^{-2}\left(\nabla\_a \nabla\_b +\frac{k^2}3\bar{g}\_{ab}\right) Y\_l(\Omega)\right\}dz^a dz^b \,. \cr && \label{generalPertMetric}\end{aligned}$$ Here ḡab is the standard round metric on the three-sphere with coordinates za and ∇a is the corresponding covariant derivative. The spherical harmonics Yl(Ω) are eigenfunctions of the Laplacian on the three-sphere, with ΔYl(Ω) = − l(l + 2) Yl(Ω) = − k2 Yl(Ω) . For simplicity of notation we have defined k2 = l(l + 2) and have suppressed the indices corresponding to the (l + 1)2 degeneracy of the modes and spherical harmonics of angular momentum l. In the following we will also omit the subscript l on the harmonics and the coefficient functions. For l = k = 0, the quantities multiplying B and C vanish, and Eq. ([generalPertMetric]) reduces to our previous expression, Eq. ([ZeroEllMetric]). If l = 1, and thus k2 = 3, the quantity multiplying C vanishes as a result of the identity ∂a∂bY1m(Ω) = − ḡabY1m(Ω) and there are only three metric coefficients, A, B, and Ψ. Expanding to second order in these perturbations gives the quadratic Lagrangian $$\begin{aligned} L^{(2)}\_E&=&-\frac{3}{\kappa}\rho^3\dot{\Psi}^2 -\frac{\rho}{\kappa}(k^2-3)\Psi^2 +\frac{1}{2}\rho^3\dot{\Phi}^2 +\frac{1}{2}\left(k^2\rho+\rho^3 U''\right)\Phi^2 -3\rho^3\dot{\phi}\dot{\Psi}\Phi\,\cr &&+\frac{\rho^3}{3\kappa}\left(\frac{k^2-3}{k^2}\right)\dot{C}^2 -\frac{\rho}{9\kappa}(k^2-3)C^2-\frac{2\rho}{3\kappa}(k^2-3)\Psi C \,\cr &&+\left[-\rho^3\dot{\phi}\dot{\Phi}+\rho^3 U'\Phi +\frac{6}{\kappa}\dot{\rho}\rho^2\dot{\Psi} +\frac{6}{\kappa}(1-k^2/3)\rho\Psi+ \frac{2}{\kappa}(1-k^2/3)\rho C\right]A -\frac{3}{\kappa}\rho Q A^2 \,\cr &&+\frac{2\dot{\rho} k^2}{\kappa} AB -\left[\frac{2\rho}{\kappa}\dot{\Psi} +\frac{2\rho}{3\kappa}\left(\frac{k^2-3}{k^2}\right)\dot{C} +\rho\dot{\phi}\Phi\right]k^2 B -\frac{k^2}{\kappa \rho}B^2 \,. \label{GeneralL2}\end{aligned}$$ This Lagrangian is invariant under the two-parameter coordinate transformation $$\begin{aligned} \xi&\rightarrow& \xi+\alpha(\xi)Y(\Omega) \,, \\ z\_a &\rightarrow& z\_a + \beta(\xi)\partial\_a Y(\Omega) \,,\end{aligned}$$ under which the perturbations transform as $$\delta\_G \Phi = \dot{\phi}\alpha,\quad \delta\_G \Psi=\frac{\dot{\rho}}{\rho}\alpha-\frac{k^2}{3}\beta, \quad \delta\_G A = \dot{\alpha}, \quad \delta\_G B=\alpha+\rho^2 \dot{\beta},\quad \delta\_G C = k^2\beta \,.$$ With two independent gauge parameters α and β we expect to have two constraints. Indeed, we see that no derivatives of A or B appear in L(2). Differentiating with respect to these quantities leads to the constraints $$0 ={\cal C}^{(1)}\_A \equiv \frac{\kappa \rho^2}{6} \left(\dot{\phi}\dot{\Phi}-U'\Phi\right) -\left[\rho \dot{\rho}\dot{\Psi}+\left(1 -\frac{k^2}{3}\right)\left(\Psi+\frac{C}{3}\right) +\frac{\dot{\rho}k^2}{3\rho}B\right]+ Q A$$ and $$0 = {\cal C}^{(1)}\_B \equiv \dot{\Psi}+\frac{k^2-3}{3k^2}\dot{C} +\frac{\kappa\dot{\phi}}{2}\Phi-\frac{\dot{\rho}}{\rho}A+\frac{1}{\rho^2}B=0 \,.$$ Both of these constraints are gauge-invariant, leading us to again seek a manifestly gauge-independent form of the Lagrangian. Using the constraints to eliminate A and B and proceeding as before, we obtain $$L^{(2)}\_E=\frac{\rho^3(1-k^2/3)}{2(Q-\dot\rho^2k^2/3)} \dot{\chi}^2 +\frac{\rho^3(1-k^2/3)}{2(Q-\dot\rho^2k^2/3)}f(\rho,\phi)\chi^2 \,, \label{L2forAllEll}$$ where now χ is the only possible gauge-invariant generalization of Eq. ([ZeroEllChi]), $$\chi\equiv \dot\rho\Phi- \rho\dot\phi\Psi -\frac{\rho\dot{\phi}}{3}C \,,$$ and $$\begin{aligned} f &=&U''+ {k^2 \over \rho^2} +\frac{\kappa}{3\dot\rho(Q-\dot\rho^2 k^2/3)}\left\{\rho\dot{\phi}U' + \rho^2 \dot\rho U'^2 -k^2\left[2 \rho \dot{\rho}^2 \dot{\phi}U' -3\dot{\rho}\dot{\phi}^2Q +\dot{\rho}\dot{\phi}^2\right]\right\} \cr &&\qquad +\frac{2\kappa \dot\phi^2}{3} - \frac{\kappa\rho \dot\phi U'}{\dot\rho} -\frac{4\kappa U}{3} -{\ddot \rho \over \dot \rho} \left({\dot Q - 2\dot\rho\ddot\rho k^2/3 \over Q - \dot\rho^2 k^2/3}\right) \,.\end{aligned}$$ If l = k = 0, Eq. ([L2forAllEll]) reduces to the result of the previous subsection. For l = 1 and k2 = 3, LE(2) vanishes identically. Finally, note that the coefficient of $\dot\chi^2$ is positive for all l ≥ 2. Perturbations about a homogeneous solution ------------------------------------------ The analysis of the previous two subsections is modified somewhat if the unperturbed solution is homogeneous, with ϕ constant and equal to an extremum of U(ϕ) and the metric being the usual round metric on the four-sphere. This could be the false vacuum solution given by Eqs. ([HubbleDef]) and ([SphereRho]) or the analogous true vacuum solution, in either case with Uʺ > 0. Alternatively, it could be the Hawking-Moss solution with $\phi=\phi\_{\rm HM}$ at a local maximum of U. If we examine the quadratic Lagrangians in Eqs. ([zeroEllL2]) and ([GeneralL2]), we see that the terms coupling Φ with a metric perturbation are all absent if $\dot\phi$ and Uʹ both vanish. Furthermore, Φ drops out of the constraints, and Q is identically equal to unity. Finally, χ is simply Φ. Redoing the analysis, we find that the metric terms are removed by the constraints, and we obtain simply $$L^{(2)}\_E = \frac12 \rho^3 \left[ \dot\Phi^2 + \left( U'' +{k^2\over \rho^2}\right)\Phi^2 \right] \,.$$ (Note that the l = 1 modes do not drop out, in contrast with the perturbations about inhomogeneous bounces.) The eigenmodes of this Lagrangian are just the five-dimensional spherical harmonics. If $$\frac{U''}{H\_{\rm top}^2} < -N(N+3)$$ with N = 0, 1, …, then there are N + 1 negative eigenvalues. If these are numbered n = 0, 1, …, eigenvalue n has a degeneracy (2n + 3)(n + 2)(n + 1)/6. Negative modes with gravity =========================== In this section we investigate the χ-field modes with negative eigenvalues that arise from the quadratic Euclidean actions that were obtained in Sec. [perturb-sec]. These actions, given in Eqs. ([g-inv-L2]) and ([L2forAllEll]), are comprised of a “kinetic energy” term quadratic in $\dot \chi$ and a “potential energy” term quadratic in χ. For l = 0 the kinetic energy can be either positive or negative, depending on the sign of Q, while the sign of the potential energy is that of f/Q. For l ≥ 2 the kinetic energy is always positive and the sign of the potential energy is that of f. (As already noted, there are no gauge-invariant modes for l = 1.) We can distinguish two classes of negative modes: a) Slowly varying or “standard” negative modes: These are associated with a positive kinetic energy and a negative potential energy. They are the analogues of the negative mode about the flat-space bounce. We will restrict our investigation to the l = 0 case; although we do not expect negative modes with higher l (there are none in flat spacetime), we have not been able to prove that these are impossible. b) Rapidly oscillating modes: These arise if l = 0 and Q < 0, giving a negative kinetic energy. Low-amplitude, short-wavelength oscillations of the form χ ∼ sin(ωξ)/ω in the region of negative Q can yield modes with negative eigenvalues whose magnitudes grow without bound as ω tends toward infinity; this is a manifestation of the conformal mode problem of Euclidean gravity . We will consider these two cases separately. Slowly varying negative modes ----------------------------- In flat spacetime these are associated with the variation of the bubble radius R. Indeed, in the thin-wall approximation, they are signaled by the fact that d2SE/dR2 is negative at $R=\bar R$, as noted in Eq. ([TWAsecondDeriv]). As has already been noted, when the thin-wall approximation is applied to the case with gravitational effects included, one finds that for a bubble with a wall at $\rho=\bar\rho$ the second derivative $d^2 S\_E/d\bar\rho^2$ is negative for a type A de Sitter bounce, but positive for a type B bounce. This suggests that there is a slowly-varying negative mode in the former case, but not in the latter. To study this issue further, and going beyond the thin-wall limit, we took as an example a theory with a scalar field potential, U(ϕ) = (ϕ − 3)2ϕ2 + 0.5ϕ2 + 1.5 , whose false and true vacua are at ϕ = 2.81 and ϕ = 0, respectively. We studied the behavior of the bounce solutions as the strength of gravity was progressively increased. We worked with a dimensionless ϕ and κ; the translation to physical quantities is obtained by noting that for this potential the scalar field mass scale μ, defined in Eq. ([mudef]), is related to the Planck mass by $${\mu \over M\_{\rm Pl}} = 0.24 \sqrt{\kappa} \,.$$ cc .25in (a) 3.0in (b) .25in (c) 3.0in (d) [kappa=01fig] cc .25in (a) 3.0in (b) .25in (c) 3.0in (d) [kappa=055fig] To make the comparison with the flat-space case clearer we define $$y = \chi/\sqrt{Q}$$ and rewrite Eq. ([g-inv-L2]) as $$L^{(2)}\_E = {\rho^3\over 2}\left( \dot y^2 + {\cal V} y^2 \right) \,, \label{rewriteL}$$ with $${\cal V} = U'' +{\kappa \rho^2 {U'}^2 \over 3 Q} +{\kappa \rho \dot\phi U' \over 3 \dot\rho Q} -{3\dot\rho \dot Q\over 2\rho Q} - {\ddot Q \over 2Q} + {3 \dot Q^2 \over 4 Q^2} +{3\ddot\rho \over \rho} + {\dddot \rho \over \dot \rho} -{\ddot\rho\dot Q\over \dot\rho Q} \,. \label{calVdef}$$ Figures [kappa=01fig] - [kappa=09fig] show the evolution of the bounce and the lowest eigenmode as the gravitational coupling κ is increased. In Fig. [kappa=01fig] we show the bounce fields ϕ(ξ) and ρ(ξ) for the weak gravity case κ = 0.01 (i.e., $\mu = 0.024 M\_{\rm Pl}$). The bubble wall is located well before the maximum of ρ, so this is clearly a type A bounce, and might almost be classified as a small-bubble bounce. The dominant term in ${\cal V}$ is Uʺ, as in flat space. The negative mode is concentrated on the wall region of the bounce. Its eigenvalue is λ0 = − (0.40μ)2, which can be compared with the flat-space (κ = 0) value − (0.39μ)2. There is a second minimum of Q at the value of ξ where ρ reaches its maximum. From Eq. ([Qdef]) we see that Q is negative there, although the factor of $\dot\phi^2$ makes its absolute value exponentially small. We will return to this minimum and the narrow region of negative Q surrounding it in the next subsection. Increasing the strength of gravity to κ = 0.055 (i.e., $\mu = 0.056 M\_{\rm Pl}$) yields the results shown in Fig. [kappa=055fig]. The maximum of ρ is outside the bubble wall (although just barely so), and so this is a large type A bounce. Q remains positive throughout the wall region, although it reaches a minimum value of 0.015 near the center of the bubble wall. Gravitational effects have significantly modified ${\cal V}$, whose negative region is much narrower. The negative mode has become somewhat narrower, but as before it is centered about the minimum of Q in the wall. The eigenvalue has increased to − (0.21μ)2. cc .25in (a) 3.0in (b) .25in (c) 3.0in (d) [kappa=057fig] The effect of a further increase in κ, to 0.057 (and, by a numerical coincidence, $\mu = 0.057 M\_{\rm Pl}$), is shown in Fig. [kappa=057fig]. The profiles of ρ and ϕ show little change, and the bounce remains type A. The minimum value of Q in the wall remains positive, at 0.00027. The dip in ${\cal V}$ is now much deeper and much narrower. Most importantly, the lowest eigenvalue is now positive, at (0.23μ)2, even though this is still a type A bounce. With even a slight further increase of κ, the minimum value of Q on the bubble wall becomes negative. This is illustrated in Fig. [kappa=07fig] for the case κ = 0.07 ($\mu= 0.063 M\_{\rm Pl}$). Because the change of variables from χ to y can no longer be carried out, $\cal V$ ceases to be a useful quantity, and there is no analogue of the lowest eigenmodes of the previous examples. If there is a slowly varying negative mode, it is not simply related to the ones found in type A solutions with weaker gravity and smaller bounces. With κ increased to 0.09 ($\mu= 0.072 M\_{\rm Pl}$), we have a bounce, shown in Fig. [kappa=09fig], that can be viewed as being on the borderline between type A to type B. We see that the regions of negative Q at the wall and at $\rho\_{\rm max}$ have merged to form a single negative-Q region. cc .25in (a) 3.0in (b) [kappa=07fig] cc .25in (a) 3.0in (b) [kappa=09fig] Finally, in Fig. [typeBfig] we illustrate a clear-cut type B bounce. Instead of Eq. ([firstPotential]), the potential is now given by U = 10(ϕ2 − 0.25)2 + 0.1(ϕ + 1) . The plots are for κ = 1, corresponding to $\mu/M\_{\rm Pl} = 0.17$. The potential ${\cal W} = f/Q$ is everywhere positive, so there is no possibility of a standard slowly varying negative mode. cc .25in (a) 3.0in (b) (c) [typeBfig] Rapidly oscillating modes ------------------------- These arise whenever Q, defined by Eq. ([Qdef]), is negative; i.e., whenever $${\kappa \rho^2 \dot\phi^2 \over 6 \dot\rho^2} > 1 \,. \label{negQcondition}$$ This can happen in two different situations. For some bounces, Q becomes negative in a portion of the bubble wall, leading to what we will call “wall modes”; we saw an example of this in Fig. [kappa=07fig]. For all bounces arising from de Sitter space, Q becomes negative in a region about the maximum of ρ; the resulting negative modes may be called “$\rho\_{\rm max}$ modes”. Of course, the distinction between these classes only applies for type A bounces; for type B bounces, where $\rho\_{\rm max}$ is reached in the bubble wall, the distinction disappears. Indeed, Fig. [kappa=09fig] illustrates how the two classes merge as type A crosses over to type B. We start by considering a small bubble bounce, with $\mu \ll M\_{\rm Pl}$ and the bubble radius $\bar \rho \ll H^{-1}$. In the wall region $\dot\phi^2 \sim \mu^4$, while $\dot\rho \approx 1$, so $${\kappa \rho^2 \dot\phi^2 \over 6 \dot\rho^2} \sim { \mu^4 \bar\rho^2 \over M\_{\rm Pl}^2 } \,.$$ For a typical small bubble bounce this is much less than unity, and so there are no wall modes. There will, however, be $\rho\_{\rm max}$ modes. These will be centered about ξ = (π/2H) − Δ ≡ ξ0, where ρ reaches its maximum. From Eq. ([rhoInFV]) we find that in this region ρ ≈ H− 1 and $$\dot\rho \approx \sin[H(\xi\_0-\xi)] \approx H(\xi\_0-\xi) \,,$$ while $\dot\phi$ is exponentially small, with $\|\dot \phi\| \sim \mu^2 e^{-\mu\pi/2H}$. It follows that Q is only negative in a region with an exponentially small width, $$\Delta \xi \approx {\mu^2 \over H^2 M\_{\rm Pl}} \, e^{-\mu \pi/2H} \,,$$ that is much less than a Planck length. As a result, the eigenvalues of the negative modes will all be super-Planckian. We turn now to the case of a large type A bounce. In the thin-wall approximation this has ρ ≈ Ht− 1sin(Htξ) in the true vacuum region. In order to have a region of negative Q, we need that at the wall $${\kappa \rho^2 \dot\phi^2 \over 6 \dot\rho^2} \sim {\mu^4 \over H\_t^2 M\_{\rm Pl}^2} \, \tan^2(H\_t \bar\xi) \sim {\mu^4 \over U\_{\rm tv}} \, \tan^2(H\_t \bar\xi)$$ be greater than unity. As we have already seen in Fig. [kappa=07fig], it is easy to construct examples where this is true without invoking any Planckian mass scales . These type A bounces will also have $\rho\_{\rm max}$ modes, with support in a region of width $$\Delta \xi \approx {\mu^2 M\_{\rm Pl}\over U\_{\rm tv}} \, e^{-\mu (\xi\_0 - \xi)} \,.$$ Even for a large type A bounce, such as those illustrated in Figs. [kappa=055fig]-[kappa=07fig], where the wall is only a few e-foldings away from $\rho\_{\rm max}$, this width remains small, although not necessarily sub-Planckian, almost until the point where the type A bounce goes over to type B. For type B bounces, where ρ reaches its maximum inside the bubble wall, the distinction between the two types of rapidly oscillating negative modes disappears; a typical example is illustrated in Fig. [typeBfig]. Table [negmode-table] summarizes the results of this section. \| \| Slowly varying \| Wall oscillating \| $\rho\_{\rm max}$ oscillating \| \| --- \| --- \| --- \| --- \| \| Small type A \| Yes \| No \| Planckian \| \| Large type A \| Usually \| Possible \| Yes \| \| Type B \| No \| — \| Yes \| [negmode-table] Negative modes about multibounce configurations =============================================== In discussions of tunneling in flat spacetime, one usually focuses on the single-bounce solution and its associated determinant factor and normal modes. In the usual dilute gas approximation, the deviation of the determinant from its value in the pure false vacuum can be approximated as being local to the bounce, so that for multibounce quasi-stationary points the determinant term, including its single factor of i, is simply repeated for each additional bounce. Summing over all numbers of bounces leads to an exponential, with the factor of i promoted to the exponent. The situation is more subtle with gravitational effects included. For large type A bounces (and of course for all type B bounces), there may not even be room on the Euclidean sphere to have several well-separated bounces. On the other hand, if the scalar field mass scale μ is far below the Planck mass and the bounce radius is much less than H− 1, there is no problem at all with configurations containing large numbers of component bounces. However, one wonders what becomes of the $\rho\_{\rm max}$ negative modes, which are not localized about the bounce, when there are multiple bounces. To start, consider an O(4)-symmetric solution with two bounces centered at antipodal points[6](#fn6), which we may take to be ξ = 0 and $\xi=\xi\_{\rm max}$. There is potentially a negative-Q region about the “equator” at $\xi=\xi\_{\rm max}/2$. By symmetry, $\dot\phi=0$ here, so both terms in $$Q = \dot\rho^2 - {\kappa \rho^2 \dot\phi^2 \over 6}$$ vanish at the equator. If the bounces have radii much less than the horizon length, $\dot\phi$ will be exponentially small, Q will be positive, and the infinite set of oscillating negative modes will be absent. On the other hand, with larger bounces one can easily find parameters that would make the first term smaller than the second in the region close to the equator, giving a negative Q and a family of oscillating negative modes. Having seen that two-bounce solutions may or may not have negative-Q regions, let us turn to the more generic multibounce case, with many component small bounces. A schematic view of such a solution is shown in Fig. [manybubblesfig]. Here the small circles represent the individual bounces, with ϕ close to its true vacuum value near their centers. As one moves away from one of these circles ϕ rapidly approaches $\phi\_{\rm fv}$. Thus, in most of the space $\|\phi - \phi\_{\rm fv}\|$ is exponentially small. The magnitude of this exponential tail decreases as the distance to the nearest bounce increases. Roughly speaking, this tail decreases in magnitude until one reaches the dotted lines in the figure, which schematically represent the boundaries that separate the “domains” of the individual bounces. [manybubblesfig] Although we have no overall O(4) symmetry to guide us, we can look to our analysis of the single bounce case for guidance. We begin by remarking that the fields in the interior of each domain are, to a good approximation, the same as those in a region of comparable size about the bounce in a one-bounce solution. Since the rapidly oscillating negative modes in the one-bounce solution have support far from this region, we should not expect to find them in the domain interiors. Hence, if rapidly oscillating negative modes exist, they should be located near the domain boundaries. However, even that seems quite unlikely. With many component bounces, the typical domain size is much smaller than the curvature radius of the underlying four-sphere. Thus, viewed on the scale of a few domains, the solution is barely distinguishable from the corresponding flat-space multibounce configuration, which we know has no oscillating negative modes. To phrase this differently, in the one-bounce solution the negative-Q region is a thin three-sphere shell with a curvature radius equal to that of the four-sphere background. In the multibounce configuration the domain boundaries approximate thin three-sphere shells with curvature radii much smaller than that of the underlying background. We conclude that while rapidly oscillating negative modes exist for the one-bounce solution, and possibly even for quasi-stationary points with a handful of bounces, they will be absent for configurations with many bounces. We need to ask ourselves which of these cases is the one of relevance for us. In the discussion of tunneling in flat spacetime in Sec. [flatTunnelSec] we argued that the path integral was dominated by configurations with n bounces, with n given by Eq. ([dominantN]). With a total four-volume of ΩT, we had n ∼ (ΩT)Γ. Of course, the four-volume was understood to be taken to infinity at the end of the calculation, so the conclusion was that the path integral was dominated by configurations in which the bounces were infinite in number but with a finite density of the order of Γ. If we ignore for the moment any complications from rapidly varying negative modes (which will, in any case, be absent if n is large), then analogous reasoning leads to the conclusion that the dominant configurations for de Sitter tunneling should be ones with $$n \approx \left({8\pi^2 \over 3}\, H^{-4}\right) \left(K e^{-B}\right)$$ component bounces. The first factor, ${8\pi^2 \over 3}\, H^{-4}$, is the four-volume of the Euclidean de Sitter space. The second factor is (apart from a factor of 2) the usual expression for the nucleation rate per unit volume, Γ. Dimensional analysis suggests that $$r \equiv {\Gamma \over H^4} \sim \left({M\_{\rm Pl}\over \mu} \right)^4 \, e^{-B} \,.$$ Depending on the parameters of the scalar field theory, this ratio could easily be much less than or much greater than unity; given the exponential dependence on the bounce action, obtaining a value very close to unity, while possible, is unlikely. If r ≫ 1 the path integral is overwhelmingly dominated by configurations with many bounces. These configurations have only the usual slowly varying negative modes, one for each bounce. The anomalous rapidly varying negative modes are absent. On the other hand, if r ≪ 1 the contributions from multibounce configurations are negligible, and the single-bounce solution, with its infinite set of negative modes, makes the important contribution to the path integral.[7](#fn7) Discussion and summary ====================== In examining various CDL bounces, we have found patterns of negative modes that are rather different from those encountered in the absence of gravity. It seems natural to ask why matters aren’t the same as in flat spacetime. However, one might instead ask why they should be the same. The issue is not so much the formal difference between the two cases, with constraints and gauge issues involved in one case and not the other, but rather the fact that different physical processes come into play when gravitational effects are included. This can be seen already in the case of small bounces, with all mass scales far below the Planck mass and the bounce radius much less than the horizon distance. In the absence of gravity we have a standard first-order transition. Once the transition begins (e.g., by cooling below a critical temperature) small bubbles start to nucleate in the initial false vacuum. These bubbles then expand, collide, and coalesce to form a homogeneous true vacuum. Varying the magnitude of Γ changes the time scale for completion of the transition, but makes no qualitative change in the process. With gravity brought into play, and a de Sitter false vacuum, matters are more complicated. If r = Γ/H4 ≫ 1, the time scale for bubble nucleation, expansion, and coalescence is short compared to the Hubble time and the transition to a homogeneous new phase is completed very much as in flat spacetime. On the other hand, if r ≪ 1, the cosmic expansion outpaces the bubble nucleation and the transition to the true vacuum is never truly completed . More precisely, if r is less than a critical value rc (which can be shown to lie in the range 1.1 × 10− 6 < rc < 0.24), then the new phase never percolates . Instead, false vacuum regions always remain, continuing to nucleate new bubbles of the true vacuum, and eternal inflation ensues. The underlying explanation for the difference between the two cases is the manner in which the bubble grows in de Sitter space. After nucleating with a small initial radius, the bubble expands at a speed that quickly approaches the speed of light, with the bubble wall approximately tracing out a light cone. On time scales much less than H− 1, there is very little difference from the flat-space case; if Γ is large enough for the transition to have been completed by this time, nothing unusual is found. However, once t ∼ H− 1 and the bubble radius approaches the horizon length, the cosmic expansion dominates and the bubble radius, as measured in comoving coordinates, becomes essentially constant. Hence, bubbles that are initially separated by a distance greater than 2H− 1 never meet. Bubbles that nucleate later have smaller asymptotic comoving sizes; they nucleate and expand in the spaces left between the older bubbles, but never quite fill out these false vacuum regions. The bounce solution itself shows no distinction between the regimes of large and small r. In both cases the CDL bounce approaches the flat-space solution as κ → 0. However, differences appear when we look at the pre-exponential factor. The two regimes correspond to the two cases found in Sec. [multibounceSec]. If r ≫ 1 the path integral is dominated by configurations with many bounces. These have the usual expansion/contraction negative modes, one for each component bounce, but no indication at all of any anomalous rapidly varying negative modes. If instead r ≪ 1, only the single-bounce configuration, with its rapidly oscillating negative modes near $\rho\_{\rm max}$, is relevant. While these $\rho\_{\rm max}$ modes do not appear in some Hamiltonian choices of gauge and, because their eigenvalues are super-Planckian, may well disappear with a proper treatment of quantum gravity, there is another class of modes that also distinguishes between the two regimes. As we saw in Eq. ([L2forAllEll]), the l = 1 modes make no contribution to the quadratic fluctuation Lagrangian about a single bounce. The zero-eigenvalue l = 1 modes simply correspond to translation of the bounce, and must be replaced by collective coordinates even in flat spacetime. However, the flat-space l = 1 modes with nonzero eigenvalue are true fluctuation modes, and it is puzzling that they do not have direct analogues about the single de Sitter CDL bounce. With the multibounce configurations matters are different. The low-lying fluctuations in the neighborhood of each component bounce, including the ones that locally look like l = 1 modes, closely approximate their flat-space counterparts. Thus, for relatively rapid nucleation, with r ≫ 1, the physical processes proceed very much as in flat spacetime, and the calculation of Γ, including the pre-exponential factor, smoothly transitions between the gravitational and the non-gravitational cases. If instead r ≪ 1, the physical process in the presence of gravity is qualitatively different from when κ = 0, and this difference is reflected in the calculation of the pre-exponential factor. Note that $\rho\_{\rm max}$, where the potential pathologies in the calculation arise, corresponds to the location of the horizon in the initial and final states. This fits nicely with the fact that it is the existence of a horizon that is responsible for the failure of the transition to truly complete. It is instructive to recall that the basis of the path integral calculation was the interpretation of the imaginary part of the energy of the false vacuum in terms of the lifetime of an unstable false vacuum state. On relatively small scales, with or without gravity, we have a region of space that is initially in a homogeneous false vacuum state and then tunnels to a state in which a true vacuum bubble is embedded in a surrounding false vacuum region. On a larger scale, matters are more subtle. Without gravity, or with gravity and r ≫ 1, the continued bubble nucleation leads to a complete transition to the true vacuum; the false vacuum unambiguously decays. On the other hand, if r ≪ 1, an eternal inflation scenario results and the false vacuum never fully disappears; in a sense, its lifetime is infinite. Let us turn now to the case of large bounces, but, as always, with all mass scales well below the Planck scale. Several physical processes are possible. First, we can have nucleation of a single bubble that, even initially, occupies a large fraction of a horizon volume. Given the size of the bubble, the possibility of multibounce configurations is severely constrained. Second, there can be bounces that occupy a full horizon volume and that are naturally interpreted as mediating transitions that occur over the entire horizon volume. These include type B bounces, with a horizon volume of one vacuum tunneling into a horizon volume of a different vacuum, and Hawking-Moss bounces, with an entire horizon volume thermally fluctuating to the top of the potential barrier. We might also include here the oscillating bounce solutions. Although it has been argued that these are not relevant for vacuum transitions because they have multiple negative modes even when the gravitational background is held fixed , it may be that this conclusion should be revisited. Finally, there can be “up-tunneling”, with a bubble of a higher (false) vacuum forming within a region of lower (true) vacuum . Issues of time-slicing arise when considering these bubbles. In discussions of vacuum tunneling one often speaks as if the bubble nucleates suddenly and simultaneously over a hypersurface of constant time, with the classical evolution and expansion of the bubble beginning sharply at some t = t0. Even ignoring the neglect of the quantum fuzziness, already in flat spacetime there is an ambiguity due to the possibility of Lorentz boosting the nucleation hypersurface, although this is known to not affect the large time evolution of the bubble . This issue becomes more acute with de Sitter bubble nucleation, especially for the case of large bounces. For example, if the de Sitter spacetime is described in global, closed-universe coordinates, then one can always find a de Sitter transformation that brings the center of the bounce to a point on the t = 0 “waist” of the de Sitter hyperboloid and the nucleation hypersurface to this hypersurface. Indeed, this is the simplest choice for continuing the Euclidean solution to a Lorentzian one. From this viewpoint, the horizon volume occupies half of the full t = 0 de Sitter space, and the nucleated bubble a large fraction of that half. Alternatively, one could use flat-universe coordinates (which cover half of the fully extended de Sitter spacetime). With this choice a spacelike hypersurface of fixed time is infinite in extent and the initial horizon volume, and the nucleated bubble within it, appear as just small regions within a larger universe. This latter viewpoint fits much better with the idea that the vacuum transition should not be sensitive to the global structure of spacetime. Even with a given choice of slicing, there are ambiguities in defining particles, and thus the vacuum state, for the true vacuum, and perhaps even more so for the false vacuum. The analysis of Ref. derived the CDL bounce by working within the static coordinate patch of a horizon volume. The vacuum naturally corresponding to these coordinates is not the same as, e.g, the Bunch-Davies vacuum. Does the choice of one or the other affect the value of the imaginary part of the vacuum energy? Focusing now on the negative modes, we have both unanticipated modes that are present, and an expected mode that is absent. The former include the formally infinite family of $\rho\_{\rm max}$ oscillating modes that, unlike those associated with small bounces, can have sub-Planckian eigenvalues. As with the corresponding small bounce modes, the fact that $\rho\_{\rm max}$ corresponds to the horizon fits well with the fact that the existence of the horizon plays a significant role in determining the nature of the transition governed by these bounces. Also unanticipated are the oscillating negative modes with support on the bubble wall. In our analysis these appear when $\kappa \rho^2 \dot\phi^2/6 > \dot\rho^2$; with the Hamiltonian gauge choice used in Refs. , they require the only slightly stronger condition that $\kappa \rho^2 \dot\phi^2/6 > 1$. We find these modes puzzling. The condition for their existence does not entail any unusually large local energy density. Rather, the large circumference of the bounce, a nonlocal property, seems to be crucial. The expected mode that goes missing is the standard slowly varying mode corresponding to radial expansion or contraction of the bounce. This is absent from the thin-wall analysis for all type B bounces (but see Ref. ). Going beyond the thin-wall approximation, our numerical results showed the mode deviating more and more from its expected form, eventually merging with the rapidly oscillating negative modes, as the bounce moved from type A toward type B. Is this a problem? Is it an indication that type B bounces do not correspond to vacuum transitions? We think not. Recall from the discussion below Eq. ([TWArhoBar]) that the type B regime is the one where ε, the energy density difference of the two vacua, is small or even zero. Bubble nucleation certainly occurs for large positive ε, and we know that the thermal nature of de Sitter spacetime allows the very same bounces to mediate tunneling upward; i.e., with large negative ε . Continuity arguments then strongly suggest that tunneling with ε ≈ 0 should also be possible. However, this tunneling need not be a decay. A single particle in a double-well potential can tunnel back and forth between two degenerate minima. The corresponding Euclidean solution has no negative mode. Perhaps the type B bounces should be understood as describing a finite system — the fields within a horizon volume — tunneling between two degenerate states. Let us sum up briefly. We have investigated the negative mode problem of de Sitter CDL bounces in a manner somewhat complementary to previous studies, with the goal of gaining physical insight into the various anomalies and pathologies associated with these modes. For the case of small bounces, we have found that the presence or absence of potential oscillating negative modes, as well as of the l = 1 modes, depends on the value of Γ/H4. There are two regimes, which correspond to the two possible outcomes of de Sitter vacuum decay — either a rapidly completed transition or eternal inflation. Only the former goes over smoothly to the flat spacetime case, showing that the true weak gravity limit of bubble nucleation requires not only that the mass scales be well below the Planck mass and the initial bubble radius much less than the horizon distance, but also that the characteristic time scale for nucleation be small compared to H− 1. For large bounces, we found further evidence, confirming the indications from thin-wall arguments, that type B bounces do not have the usual expansion/contraction negative mode. We argued that, nevertheless, these bounces correspond to vacuum transitions. On the other hand, the underlying physical origin of the oscillating negative wall modes that arise when the bounce is sufficiently large remain somewhat obscure. Elucidating this issue is among the problems that remain to be resolved by future investigations. We are grateful for the hospitality of the Korea Institute for Advanced Study, where part of this research was performed. This work was supported in part by U.S. Department of Energy grants DE-FG02-92ER40699 and DE-SC0011941. Horizon volumes and the bounce solution ======================================= In this appendix we show that the the slices of the Coleman-De Luccia bounce that we identified in Fig. [CDL-bounce] as giving the configurations at the beginning and end of tunneling are indeed horizon volumes. The O(4)-invariant Euclidean metric can be written as ds2 = dξ2 + ρ(ξ)2(dχ2 + cos2χ dΩ22) . The three-dimensional slices in which we are interested are given by χ = 0, and either 0 ≤ ξ < ξ0 or $ \xi\_0 < \xi < \xi\_{\rm max}$, where ξ0 is the location of the maximum of ρ. This Euclidean metric can be continued to a Lorentzian one along the hypersurface χ = 0 by writing χ = it and going to real t. This gives the metric $$\begin{aligned} ds^2 &=& - \rho^2 \, dt^2 + d\xi^2 + \rho^2 \cosh^2t \,d\Omega\_2^2 \cr &=& -\rho^2 \, dt^2 + {1 \over {\dot\rho}^2 }\, d\rho^2 + \rho^2 \cosh^2t \,d\Omega\_2^2 \,.\end{aligned}$$ We see that on the t = 0 hypersurface ρ coincides with the usual choice for radial coordinate. The vanishing of $g^{\rho\rho} = \dot\rho^2$ at ξ = ξ0 indicates the presence of a horizon, as claimed, and verifies that the two three-dimensional slices each correspond to a horizon volume in an instantaneously static metric. In fact, because the bounce is a solution of the Euclidean equations, we can use Eq. ([rho-eq]) and write $$\begin{aligned} g^{\rho\rho} &=& 1 +{\kappa \over 3} \rho^2 \left[ \frac12 \dot\phi^2 -U \right] \cr &=& 1 - {\kappa {\cal M}(\rho) \over 4\pi \rho} \,.\end{aligned}$$ Here we have defined $${\cal M}(\rho) = {4\pi \over 3} \, \rho^3 \left[ \frac12 \dot\phi^2 -U \right] \,.$$ Differentiating it with respect to ρ, we obtain $$\begin{aligned} {d{\cal M} \over d\rho} &=& 4\pi \left\{\rho^2 \left[ \frac12 \dot\phi^2 - U\right] +\frac13\rho^3 \left[\dot\phi\ddot\phi - {dU\over d\phi}\dot\phi\right]{1 \over \dot\rho} \right\} \cr &=& -4\pi \rho^2 \left[ \frac12 \dot\phi^2 + U\right] \,,\end{aligned}$$ where the last line is obtained with the help of Eq. ([phi-eq]). This, together with the fact that ${\cal M}$ vanishes at the zeros of ρ, allows us to write $${\cal M}(\rho) = 4\pi \int\_0^\rho d\rho \, \rho^2 \left[\frac12 g^{\rho\rho} \left({d\phi \over d\rho}\right)^2 + U \right] \,,$$ which is the familiar result for a static spherically symmetric field configuration. A similar result follows for the slice with $\xi\_0 < \xi <\xi\_{\rm max}$. Perturbation Lagrangian ======================= In this appendix we outline the steps leading to Eq. ([g-inv-L2]) for the quadratic Lagrangian governing O(4)-invariant fluctuations about the bounce. Simple substitution of the perturbations defined in Eqs. ([zeroEllL2]) and ([PhiPert]) into the unperturbed Lagrangian gives the expression in Eq. ([ZeroEllMetric]), which can be written as $$L^{(2)}\_E = -\frac{3}{\kappa}\rho^3 \dot{\Psi}^2+ \frac{3}{\kappa}\rho\Psi^2+\frac{1}{2}\rho^3\dot{\Phi}^2 +\frac{1}{2}\rho^3 U''\Phi^2-3\rho^3 \dot{\phi}\dot{\Psi}\Phi + FA -\frac{3}{\kappa}\rho Q A^2 \,, \label{FullAction}$$ where $$F= -\rho^3\dot{\phi}\dot{\Phi}+\rho^3 U' \Phi +\frac{6}{\kappa}\dot{\rho}\rho^2\dot{\Psi} +\frac{6}{\kappa}\rho \Psi \,.$$ Varying LE(2) with respect to A leads to the constraint equation $$F = {6\over \kappa} \, \rho Q A \,.$$ Using this to eliminate A, we can rewrite the Lagrangian in a form, $$L\_E^{(2)} = -\frac{3}{\kappa}\rho^3 \dot{\Psi}^2+ \frac{3}{\kappa}\rho\Psi^2+\frac{1}{2}\rho^3\dot{\Phi}^2 +\frac{1}{2}\rho^3 U''\Phi^2-3\rho^3 \dot{\phi}\dot{\Psi}\Phi + {\kappa \over 12}\, {F^2 \over \rho Q} \,,$$ that involves only Φ and Ψ. Now let us define a field $$Y = \Phi - {\rho \dot\phi \over \dot\rho} \, \Psi$$ that is invariant under the infinitesimal gauge transformation of Eq. ([O4gaugeTrans]). Substituting this into our previous expression gives $$\begin{aligned} L\_E^{(2)} &=& {\rho^3 \over 2Q} \left[\dot\rho^2 \dot Y^2 + \left(U'' Q +{\kappa\over 6}\rho^2 U'^2 \right)Y^2 - {\kappa\over 3}\rho^2 \dot\phi \,U' \dot Y Y\right] +\cdots \cr &=& {\rho^3\over 2Q} \left[\dot\rho^2 \dot Y^2 + \left(U'' + {\kappa \rho^2 U'^2\over 3Q} + {\kappa\rho \dot\phi U'\over 3\dot\rho Q} \right)\dot\rho^2 Y^2 \right] +\cdots \,. \label{actionWithDots}\end{aligned}$$ In the first line the ellipsis denotes terms containing Ψ or $\dot\Psi$, while in the second it also includes total derivative terms arising from the integration by parts. Now recall that the action is gauge-invariant. In particular, one can verify that under an infinitesimal gauge transformation the expression in Eq. ([FullAction]) only changes by a total derivative, $$\delta\_G L\_E^{(2)} = {d \over d\xi} \left[\left( \rho^3 \ddot\phi \Phi +{6\over \kappa}\rho\Psi\right)\alpha\right] \,,$$ that makes no contribution to the total action. When the action is rewritten in terms of Y and Ψ, the terms shown explicitly in Eq. ([actionWithDots]) are manifestly gauge-invariant. By contrast, the omitted terms involving Ψ or $\dot\Psi$ cannot possibly be gauge-invariant, and so must in fact vanish. Finally, noting that Y is potentially singular when $\dot\rho=0$, let us define $$\chi = \dot\rho Y = \dot\rho \Phi - \rho \dot\phi \Psi \,.$$ Substituting this into Eq. ([actionWithDots]), performing an integration by parts to eliminate the terms proportional to $\chi \dot\chi$, and dropping total derivatives and terms involving Ψ then leads to the expression given in Eqs. ([g-inv-L2]) and ([fForL2]). 99 M. B. Voloshin, I. Y. Kobzarev, and L. B. Okun, Sov. J. Nucl. Phys. 20, 644 (1975) [Yad. 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Weinberg, Phys. Rev. D 36, 1088 (1987). J. Garriga, S. Kanno, M. Sasaki, J. Soda and A. Vilenkin, JCAP 1212, 006 (2012). M. B. Fröb, J. Garriga, S. Kanno, M. Sasaki, J. Soda, T. Tanaka and A. Vilenkin, JCAP 1404, 009 (2014). I-S. Yang, Phys. Rev. D 87, 084026 (2013). --- 1. Of course, even if the issues associated with negative modes are resolved, a complete calculation of the functional determinant would require an understanding of how to renormalize gravity.[↩](#fnref1) 2. To simplify notation we will omit the subscript on H when it refers to the false vacuum; when we need to refer to the true vacuum Hubble parameter this will be indicated by a subscript.[↩](#fnref2) 3. For some choices of parameters and of the Rj there is a range of τ in which the potential energy is less than that of the original false vacuum. The cases with such classically allowed regions were the focus of , but are not of relevance for the present discussion.[↩](#fnref3) 4. In a two-field model a similar mechanism can result in the wall tension acquiring a small imaginary part .[↩](#fnref4) 5. This definition agrees with that in Refs. and, but differs from the quantity denoted by Q in Refs. and . For l = 0 modes the latter two references define $Q=1 - \kappa \rho^2 \dot\phi^2/6$.[↩](#fnref5) 6. This is the simplest example of an oscillating bounce solution .[↩](#fnref6) 7. The situation here is somewhat akin to that with the “interior” B-to-C bounces discussed in Sec. 3. In that case, the finite Euclidean volume of the A-to-B bounce led to two limiting regimes, one with no B-to-C bounces and one with many. Here the finite volume of the Euclidean spacetime implies limiting cases with one and with many CDL bounces.[↩](#fnref7) Negative modes of Coleman-De Luccia bounces =========================================== We investigate the negative modes about Coleman-De Luccia bounces governing vacuum transitions in de Sitter space, with the goal of gaining physical insight into the various anomalous results associated with these that have been reported in the literature. For the case of bounces with radii much less that the horizon distance H− 1 we find two distinct regimes, distinguished by the magnitude of the bubble nucleation rate Γ. If Γ/H4 ≫ 1, then the behavior of the modes contributing to the determinant factors in Γ is much as it is in flat spacetime, and the calculation of Γ goes over smoothly to the flat spacetime calculation as the gravitational coupling is taken to zero. This is not the case if Γ/H4 ≪ 1. These two regimes correspond to the two possible outcomes of de Sitter vacuum decay — either a rapidly completed transition or non-percolation and eternal inflation. For bounces with radii comparable to the horizon length, we confirm previous results concerning anomalous negative modes with support on the bounce wall. We also find further evidence supporting previous claims, based on thin-wall arguments, of the absence of expected negative modes for a class of bounces that arises when the initial and final vacua are nearly degenerate. Introduction ============ The process in which a false vacuum of a scalar field theory decays by the nucleation of true vacuum bubbles was first studied in detail some time ago . The bubble nucleation rate has the exponential suppression typical of tunneling processes, with the exponent being twice the WKB barrier penetration integral or, equivalently, the Euclidean action of the bounce solution to the Euclidean field equations . By using path integral methods to calculate the energy density of the false vacuum, the pre-exponential factor can also be obtained . This turns out to contain a factor of $[\det S\_E''(\phi\_{\rm bounce})]^{-1/2}$, where SEʺ denotes the second functional variation of the Euclidean action. An essential ingredient of this calculation is the fact that $S\_E''(\phi\_{\rm bounce})$ has a single negative eigenvalue, which arises from a mode corresponding to expansion or contraction of the bounce. Because of the square root, this negative mode leads to a factor of i that makes the false vacuum energy complex, with its imaginary part interpreted in terms of a rate of decay by bubble nucleation. It is crucial here that there is only a single negative eigenvalue and thus one factor of i; if, for example, there were an even number of such eigenvalues, the contribution to the false vacuum energy would be purely real. Indeed, it has been shown that the bounce with lowest action has one, and only one, negative mode . Coleman and De Luccia (CDL) proposed that this formalism could be extended to include gravitational effects by adding a Euclidean Einstein-Hilbert term to the action. Although they did not address the issue of the prefactor, it seems natural to assume that it should be analogous to that for the non-gravitational case, and that the dominant bounce should again have a single negative mode. The matter is complicated by the fact that with gravity included the fluctuations about the bounce enjoy a gauge freedom, corresponding to the invariance under coordinate transformations of the underlying theory. A consequence of this gauge freedom is the existence of constraints that must be imposed on the possible fluctuations. A number of authors have studied the negative mode issue in the context of bounces corresponding to bubble nucleation in a de Sitter spacetime . Some curious, troubling, and sometimes contradictory, results, often depending on the choice of gauge, have been obtained. Among these are: 1) An action for fluctuations about a bounce with large radius that is unbounded from below, suggesting an infinite family of negative modes with support near the bubble wall. 2) Indications of a similar phenomenon for all de Sitter CDL bounces, but with the additional negative modes having support on a region defined by the horizon. 3) Arguments for the existence of a single negative mode, but subject to restrictions on the form of the bounce solution. 4) Claims that there are no negative modes, and that the factor of i comes from a proper rotation of contours in the Hamiltonian path integral. 5) The absence of a negative mode in the thin-wall approximation for a certain class of bounces. In our view, these works only address one part of the problem, namely how to develop an algorithm that, at least in principle[1](#fn1), gives an unambiguous result for the bubble nucleation rate. There is another aspect that should be addressed. When anomalous or apparently meaningless mathematical results are encountered, it is often the case that they are indicative of a particular physical aspect of the problem being studied. Thus, one should look for a physical understanding as to why these negative modes arise (or do not) in particular cases, even if they can ultimately be eliminated from the calculation. Also, one should expect to find a smooth transition from the curved spacetime calculation to the non-gravitational one in an appropriate limit. Ideally, this should include not just a continuous evolution of the bounce and its action, but also a gradual disappearance of the potential pathologies as one approaches the limit in which κ ≡ 8πG = 0. As we will see, this limit is rather subtle. Our goal in this paper is to make some progress toward this end by exploring the regions where the quadratic fluctuation action becomes negative. We work in a purely Lagrangian framework. Rather than fixing a gauge, we impose the requirement that the fluctuations obey the constraint, and then write the quadratic fluctuation Lagrangian in terms of manifestly gauge-invariant combinations of fields. For fluctuations about a bounce with O(4)-symmetry, which we assume, the objects of primary interest are spherically symmetric scalar fluctuations, which can be described by a single field χ. We examine the situations in which χ can have negative modes, and study the form of those modes as κ is varied. The remainder of this paper is organized as follows. In Sec. [tunnelSec] we review the formalism for treating vacuum decay in both flat spacetime (i.e., with gravity ignored) and with gravitational effects included. In Sec. [negativeSec] we discuss negative modes in flat spacetime. In the course of this discussion we describe an example where bounces with many negative modes turn out to not only be physically meaningful, but to actually be dominant. Section [perturb-sec] describes the perturbative expansion of the scalar field and metric about an O(4)-symmetric CDL bounce and gives the expression for the quadratic part of the fluctuation Lagrangian. Next, in Sec. [negmode-sec], we explore the various types of negative modes that can arise from this Lagrangian. Many of the results that we present here are based on numerical explorations of particular scalar field models. In Sec. [multibounceSec] we discuss the negative mode problem for multibounce solutions. This is almost a trivial aside in flat spacetime, but becomes more complex with gravity included. In fact, we will see that it is key to understanding the limit in which gravitational effects disappear. Finally, in Sec. [conclusionSec] we summarize our results and discuss how they reflect the various modes of vacuum decay in de Sitter spacetime. There are two appendices containing some technical details. Review of tunneling =================== Tunneling in flat spacetime --------------------------- The WKB approximation leads to an expression of the form γ = C e− B for the decay rate of a state that decays by tunneling through a potential energy barrier. For the case of a single particle in one dimension with a standard kinetic energy, $$B = 2 \int\_{x\_1}^{x\_2} dx \, \sqrt{2m[V(x) -E]} \,, \label{tunnelInt}$$ where x1 and x2 are the classical turning points that bound the particle’s path through the barrier. With multiple degrees of freedom, the WKB approximation requires that we consider all paths through the barrier and pick out the one along which the one-dimensional tunneling exponent is a minimum . This minimization problem is equivalent to the problem of finding a stationary point of the Euclidean action, which in turn is equivalent to finding a solution of the Euclidean equations of motion. The solution relevant for tunneling, known as the bounce, starts at an initial configuration at Euclidean time $\tau\_{\rm in}$, runs through the barrier to an exit point at some τ0, and then returns, in a “τ-reversed” fashion, to $\tau\_{\rm fin}$. The intermediate Euclidean times give a parameterization of the tunneling path through configuration space. If the initial configuration is a local minimum of the potential energy (i.e., a false vacuum), then $\tau\_{\rm in}=-\infty$ and $\tau\_{\rm fin}=\infty$. For a scalar field theory with a standard Lagrangian the tunneling exponent B is equal to the difference between the Euclidean actions of the bounce solution ϕ(x) and the uniform false vacuum. Even if these actions are divergent, their difference, $$\begin{aligned} B &=& S\_E(\phi) - S\_E(\phi\_{\rm fv}) \cr &=& \int\_{-\infty}^\infty d\tau \int d^3x\, \left[ \frac12 \left({d\phi \over d\tau}\right)^2 + \frac12 ({\bm \nabla}\phi)^2 + U(\phi) - U(\phi\_{\rm fv}) \right] \,,\end{aligned}$$ is finite and well-defined. Note that the tunneling integral and the Euclidean action differ for paths that are not solutions of the equations of motion. In fact, although the bounce is a local minimum of the integral through the potential energy barrier, it only corresponds to a saddle point of SE. In a scalar field theory, with thermal and gravitational effects ignored, the bounce describing tunneling from a false vacuum is a solution on Euclidean R4 that approaches the false vacuum value $\phi\_{\rm fv}$ as either $\|{\bf x}\|$ or τ tends to infinity. With only minimal restrictions on the potential U(ϕ), it can be shown that the bounce with the smallest action is O(4)-symmetric . Figure [flat-bounce] illustrates such a bounce. [flat-bounce] The prefactor C can be obtained by a path integral argument . Consider the matrix element $$\begin{aligned} \langle \phi\_{\rm fv}\| e^{-HT} \| \phi\_{\rm fv}\rangle &=& \int[d\phi({\bf x},\tau)]\, e^{-S\_E[\phi]} \cr &=& \sum\_n e^{-E\_nT} \langle \phi\_{\rm fv}\| n\rangle \langle n\| \phi\_{\rm fv} \rangle \,.\end{aligned}$$ Here the path integral is restricted to paths that begin and end on the false vacuum configuration. In the limit T → ∞, the sum over energy eigenstates in the second expression is dominated by the state with the lowest energy among those that contribute to the matrix element. Identifying this state as the false vacuum, we have $$E\_{\rm fv} = -\lim\_{T\to \infty} {1\over T} \ln \, \langle \phi\_{\rm fv}\| e^{-HT} \| \phi\_{\rm fv}\rangle \,. \label{EfvFromLog}$$ The path integral is evaluated by expanding about its stationary points. The first of these is the constant solution with $\phi(x) = \phi\_{\rm fv}$ everywhere. A Gaussian integral about this gives $$I\_0 = [\det S\_E''(\phi\_{\rm fv})]^{-1/2} \, e^{-S\_E(\phi\_{\rm fv})} \,,$$ where SʺE denotes the second functional derivative of the Euclidean action. It is convenient to consider space to have a finite volume Ω which is to be taken to infinity at the end of the calculation, so that $S\_E(\phi\_{\rm fv}) = \Omega T \,U(\phi\_{\rm fv})$. Next, we have the bounce solution. This would give a similar Gaussian integral were it not for two factors. First, the spectrum of $S''\_E(\phi\_{\rm bounce})$ includes four zero modes, corresponding to the freedom to translate the bounce in Euclidean space and time. These are handled by introducing collective coordinates specifying the location of the center of the bounce. Second, the spectrum also contains a mode with negative eigenvalue, corresponding roughly (and exactly, in the thin-wall limit) to expansion or contraction of the bounce. This negative mode leads to a factor of i when the square root of the determinant is taken. The contribution to the path integral can be written as $$\begin{aligned} I\_1 &=& \frac{i}{2}\,\Omega T \left\|{\det' S\_E''(\phi\_{\rm bounce})] \over \det S\_E''(\phi\_{\rm fv})}\right\|^{-1/2} \, J \, e^{-[S\_E(\phi\_{\rm bounce}) - S\_E(\phi\_{\rm fv})]}\, I\_0 \cr &\equiv & i \Omega T K e^{-B} \, I\_0 \,. \label{I1formula}\end{aligned}$$ Here the factor of ΩT arises from integrating over the four collective coordinates, J contains the Jacobian factors associated with the introduction of the collective coordinates, and the factor of 1/2 comes from a careful treatment of the negative mode. Finally, the prime on the bounce determinant indicates that the product of eigenvalues is to be taken only over the nonzero eigenvalues. There are also approximate stationary points corresponding to many well-separated bounces. The n-bounce solution has an action $$S\_E(\phi\_{n\rm -bounce}) = S\_E(\phi\_{\rm fv}) + n \left[S\_E(\phi\_{\rm bounce}) - S\_E(\phi\_{\rm fv}) \right] = S\_E(\phi\_{\rm fv}) + n B \,.$$ The integration over the collective coordinates gives a factor of (ΩT)n/n!, with the n! entering because the bounces are indistinguishable. Similarly, the factor of J becomes Jn. Only the determinant factors remain to be considered. Outside the wall region the bounce solution rapidly approaches the pure false vacuum, with $\|\phi\_{\rm bounce}(x) - \phi\_{\rm fv}\|$ decreasing exponentially with distance. Thus, we can imagine evaluating the determinant for the one-bounce case by dividing Euclidean space into a large (compared to the bounce radius) region enclosing the bounce, and the remainder. The full determinant is then the product of the contributions of the two regions. In the latter region the bounce is exponentially close to the false vacuum and the contributions to the bounce and the false vacuum determinants are essentially equal. In the region containing the bounce the determinants corresponding to $\phi\_{\rm bounce}$ and $\phi\_{\rm fv}$ differ precisely by the ratio that appears in Eq. ([I1formula]). For an n-bounce configuration, with the bounces all well separated, the Euclidean spacetime can be divided into n regions, each containing one bounce, and the remainder, in which the field is essentially equal to its false vacuum region. The full determinant is the product of the contributions from each of these regions. The net result is that the contribution to the path integral from n-bounce configurations is $$I\_n = {1\over n!} \left( i \Omega T K e^{-B} \right)^n I\_0 \,.$$ Summing the contributions from all values of n gives $$\begin{aligned} I &=& I\_0 \sum\_{n=0}^\infty {1\over n!} \left( i \Omega T K e^{-B} \right)^n \cr &=& I\_0 \exp\left[ i \Omega T K e^{-B} \right] \,. \label{Isum}\end{aligned}$$ Taking the logarithm and using Eq. ([EfvFromLog]) gives $$E\_{\rm fv} = -\lim\_{T\to \infty} \left( {\ln \,I\_0\over T}\right) - i\Omega K e^{-B} \,.$$ The first term on the right-hand side is real, but the second is imaginary, making the false vacuum energy complex. As usual, we interpret this as a signal that the false vacuum is unstable, with a decay rate $$\gamma = - 2 \,{\rm Im}\, E\_{\rm fv} = 2 \Omega K e^{-B} \,.$$ This is proportional to Ω, corresponding to the fact that a bubble can nucleate anywhere in space. The bubble nucleation rate per unit volume is Γ = 2K e− B . This calculation implicitly assumes that in the multibounce solutions the individual bounces are separated by distances large relative to the bounce four-volume ${\cal V}\_b$. To test this dilute-gas approximation, we note that the sum in Eq. ([Isum]) is dominated by the terms with n ≈ ΩTKe− B . For the approximation to be valid, we must require that the volume occupied by these n bounces be much less than the total Euclidean spacetime volume ΩT; i.e., $${\cal V}\_4 K e^{-B} \ll 1 \,,$$ where ${\cal V}\_4$ is the four-dimensional volume of the bounce. Dimensional arguments suggest that ${\cal V}\_4 K$ is typically of order unity, in which case this condition reduces to a lower bound on B. A useful illustrative example is given by the thin-wall approximation, which applies in the limit where the difference between the false and true vacuum values of the potential, $\epsilon \equiv U(\phi\_{\rm fv}) - U(\phi\_{\rm tv})$, is sufficiently small relative to the surface tension σ of the bubble . The bounce can then be obtained by considering O(4)-symmetric configurations in which a true vacuum region of radius R is separated from the exterior false vacuum by a thin wall with action per unit area σ. The total Euclidean action, less that of the pure false vacuum, is $$S\_E(R) = 2\pi^2 \sigma R^3 - {\pi^2 \over 2}\, \epsilon R^4 \,.$$ The stationary point of this action at $\bar R \equiv 3 \sigma/\epsilon$ gives the bounce radius. Both the bounce radius and the tunneling exponent $$B = {27 \pi^2\over 2}\, {\sigma^4 \over \epsilon^3}$$ tend to infinity in the ε → 0 limit of degenerate vacua. The fact that $$\left. {d^2 S\_E \over d R^2}\right\|\_{\bar R} = -{18 \pi^2 \sigma^2 \over \epsilon} \label{TWAsecondDeriv}$$ is negative shows that the bounce is a maximum of the action among the one-parameter family of thin-wall configurations, and thus a saddle point of the action, with at least one negative eigenvalue, on the full configuration space. Finally, recall that the bounce is supposed to describe a family of configurations interpolating between the initial pure vacuum state, at τ = − ∞, and a state of equal potential energy containing a bubble, on a constant τ slice through the center of the bounce. In the thin-wall limit the difference in the static potential energy between these two configurations is $$\Delta E = 4\pi R^2 \sigma - \frac43 \pi R^3 \epsilon \,.$$ This does indeed vanish if $R = \bar R$. The above discussion readily generalizes to the case of nonzero temperature T . In this case Γ is obtained from the imaginary part of the free energy, rather than the energy, of the metastable false vacuum. The Euclidean path integral is then over paths that are periodic in Euclidean time with period β = 1/T. Furthermore, the classical Euclidean equations leading to the bounce must be obtained using the finite-temperature effective potential. Just as at T = 0, the path integral includes contributions not just from a single bounce, but also from all multibounce solutions. Summing over these exponentiates the single bounce contribution, and the nucleation rate can be read off from the exponent. Including gravitational effects ------------------------------- Coleman and De Luccia argued that gravitational effects could be incorporated by adding a Euclidean Einstein-Hilbert term to the action, so that $$S\_E = \int d^4x \sqrt{g} \left[-{1 \over 2\kappa}R +\frac12\,g^{ab} \partial\_a\phi\,\partial\_b \phi +U(\phi) \right] + S\_{\rm bdy} \,,$$ where κ = 8πG and $S\_{\rm bdy}$ is the Euclidean version of the Gibbons-Hawking boundary term . Although it has not been proven that the bounce of minimum action continues to be O(4)-symmetric when gravitational effects are included, this is widely believed to be true. If one assumes this to be the case, the Euclidean metric can be written as ds2 = N(ξ)dξ2 + ρ(ξ)2dΩ32 , where dΩ32 is the usual metric on the unit three-sphere. It is convenient to choose the origin of ξ to be a zero of ρ, so that ρ(0) = 0. The curvature scalar is $$R = {6 \over N \rho^2}\left(N-{\dot\rho}^2 - \rho\ddot\rho\right) + {3 \dot\rho\dot N\over \rho N^2} \,,$$ with overdots denoting derivatives with respect to ξ. With the scalar field ϕ depending only on ξ, the Euclidean action takes the form $$S\_E = 2\pi^2 \int d\xi \sqrt{N} \left\{ \rho^3\ \left[{1\over 2N}\,{\dot\phi}^2 + U(\phi)\right] +\frac3\kappa \left[{1\over N}\left(\rho^2\ddot\rho + \rho{\dot\rho}^2\right) -\rho -{\rho^2\dot\rho \dot N \over 2 N^2} \right] \right\} + S\_{\rm bdy} \,.$$ An integration by parts to remove the second derivative term recasts this as $$S\_E = 2\pi^2 \int d\xi\sqrt{N} \left\{ \rho^3\left[{1 \over 2N}\,{\dot\phi}^2 + U(\phi)\right] -\frac3\kappa \left( {\rho{\dot\rho}^2 \over N} +\rho\right) \right\} \,,$$ with the boundary terms from the integration by parts precisely canceling $S\_{\rm bdy}$. We note that the action does not contain any derivatives of N. Consequently, variation with respect to N yields a constraint equation (which is in fact the Gξξ Einstein equation), $$0= \rho^3\left({{\dot\phi}^2\over 2N} - U \right) -{3\over \kappa}\left({\rho{\dot\rho}^2 \over N} -\rho \right) \,. \label{rhoN-eq}$$ The existence of this constraint is related to the freedom to make a coordinate transformation to redefine ξ. We make use of this gauge freedom to set N(ξ) = 1. Having done so, we can write the constraint equation as $${\dot\rho}^2 = 1 + {\kappa \over 3} \, \rho^2 \left(\frac12 {\dot\phi}^2 - U\right) \,. \label{rho-eq}$$ Varying the action with respect to ϕ gives the scalar field equation of motion $$\ddot\phi + {3 \dot\rho\over \rho}\dot\phi = {dU \over d\phi} \,. \label{phi-eq}$$ Equations ([rho-eq]) and ([phi-eq]) are a complete set of field equations. Variation of the action with respect to ρ does not yield an independent equation, reflecting the fact that in spherically symmetric configurations the gravitational dynamics is determined completely by the matter distribution. However, differentiating Eq. ([rho-eq]) and then using Eq. ([phi-eq]) yields the useful identity $$\ddot \rho = - \frac{\kappa}{3} \, \rho\left(\dot\phi^2 + U \right) \,.$$ For bubble nucleation in a Minkowski or anti-de Sitter false vacuum the bounce solution has the topology of R4, just as in the non-gravitational case. De Sitter spacetime, the case on which we will focus, is different. In this case ρ always has a second zero, at a value $\xi=\xi\_{\rm max}$. The bounce is then topologically a four-sphere, with a finite four-volume of order H− 4, where $$H = \sqrt{{\kappa \over 3}\, U(\phi\_{\rm fv}) } \label{HubbleDef}$$ is the Hubble parameter of the false vacuum.[2](#fn2) In particular, for the solution of Eqs. ([rho-eq]) and ([phi-eq]) describing the pure false vacuum, with $\phi= \phi\_{\rm fv}$ everywhere, the metric is the standard round metric on the four-sphere, with ρ = H− 1sin(Hξ) . Its Euclidean action, which must be subtracted from that of the bounce to obtain B, is $$S\_E = - {24\pi^2 \over \kappa^2 \, U(\phi\_{\rm fv})} = - {8\pi^2 \over \kappa} \, H^{-2} \,.$$ In our discussions it will be useful to distinguish two classes of bounce solutions: a) Small bubble bounces: The difference between the values of the potential at the false vacuum and at the top of the barrier defines a mass scale μ via $$U\_{\rm top} - U\_{\rm fv} \equiv \mu^4 \,. \label{mudef}$$ If μ ≪ MP and the difference between the true and false vacuum values of U is not too small, then the bounce describes the nucleation of bubbles whose radius $\bar \rho$ at nucleation is much smaller than H− 1. If ξ ≪ H− 1, then ρ(ξ) ≈ ξ and ϕ(ξ) differs only slightly from the corresponding flat-space bounce. For larger ξ, on the other hand, ρ = H− 1sin[H(ξ + Δ)] , where Δ = O(μ/MP2). The scalar field in this large-ξ region is exponentially close to its false vacuum value, with $\|\phi - \phi\_{\rm fv}\| \sim \mu e^{-\mu(\xi - \bar\xi)}$. b) Large bubble bounces: Bubbles of size comparable to H− 1 can arise, even with μ ≪ MP, if ε is sufficiently small. The flat-space thin-wall results are replaced by $${1 \over \bar\rho^2} = H^2 + \left( {\epsilon \over 3 \sigma} - {\kappa \sigma \over 4} \right)^2 \,. \label{TWArhoBar}$$ As ε is decreased, the bubble radius increases until it reaches a maximum value, $\bar\rho = H^{-1}$, when the quantity in parentheses vanishes. Until this point, the false vacuum region is larger than a hemisphere, while the true vacuum occupies less than a hemisphere. With a further decrease in ε, $\bar\rho$ begins to decrease, and the true and false vacuum regions are both less than hemispheres. In contrast to the flat-space thin-wall bounce, whose radius and action tend to infinity in the ε → 0 limit, both $\bar\rho$ and the bounce action remain finite in this limit. To distinguish between the two regimes, we will denote bounces as being type A or type B, according to whether the false vacuum region is greater than or less than a hemisphere. We saw in Eq. ([TWAsecondDeriv]) that with gravity ignored the bounce was a maximum, as a function of R, of the thin-wall-approximation action. Similarly, for type A bounces the $\bar\rho$ of Eq. ([TWArhoBar]) gives a maximum of the thin-wall action, and therefore a saddle point of the full action functional . However, for type B thin-wall solutions the second derivative of the action with respect to $\bar\rho$ is positive, corresponding to a minimum. We can generalize the distinction between type A and type B solutions beyond the thin-wall limit. In the thin-wall limit the maximum of ρ in a type B bounce occurs at the bubble wall. More generally, we will say that a bounce is type B if the maximum of ρ occurs within the wall region, and type A otherwise. (Because the beginning and end of the wall are not precisely defined, there will still be some ambiguity concerning borderline cases.) [CDL-bounce] The fact that the bounce topology is compact raises questions: Why is the volume finite? How can the bounce be understood as a sequence of configurations forming a tunneling path through a potential energy barrier? It was argued in that the bounce should be understood as mediating a transition in a de Sitter horizon volume (giving a finite three-volume ∼ H− 3) at a finite de Sitter temperature $T\_{\rm dS} = H/2\pi$ (giving periodicity ∼ H− 1 in a fourth Euclidean coordinate). In this picture the path through configuration space is given by a series of radial slices, each of which represents a three-dimensional ball, as shown in Fig. [CDL-bounce] (see also ). The boundaries of the various three-balls, which are two-spheres at the horizon, all meet and are identified. The initial and final slices are the two horizontal slices in this figure. In the non-gravitational case, the bounce connects initial and final configurations that are each turning points, with vanishing derivatives with respect to the Euclidean time, that can be continued to real time configurations with vanishing time derivatives. With gravitational effects included, the initial and final slices each correspond to a configuration on a horizon volume with instantaneously static metric and scalar field. In Appendix [horizonApp] we verify explicitly that these limiting configurations are indeed bounded by horizons. Negative modes ============== The occurrence of a negative eigenvalue in the spectrum of fluctuations about the bounce was an essential ingredient in the path integral derivation of the flat-space nucleation rate. It gave rise to the factor of i in Eq. ([I1formula]), which led to the bounce terms being a contribution to the imaginary part of the false vacuum energy. The existence of the negative mode is readily established in flat spacetime. An O(4)-symmetric bounce obeys $${d^2\phi \over ds^2} + {3 \over s}{d\phi \over ds} = U'(\phi) \,, \label{flatScalarEq}$$ where $s = \sqrt{\tau^2 + {\bf r}^2}$. The eigenmodes about this bounce satisfy [ − □E + Uʺ(ϕb)]η = λη . The spherical symmetry of the underlying bounce allows us to decompose the modes as products of a radial function and a four-dimensional spherical harmonic, and to recast Eq. ([BoxEeq]) into the form of a Schroedinger equation, with the angular derivatives giving rise to an angular momentum barrier term. We know that the breaking of translational invariance gives rise to four zero modes that transform as the components of a four-dimensional vector. Standard arguments then show that there must be a rotationally invariant scalar mode with a lower, and therefore negative, eigenvalue λ that obeys $$- {d^2\eta \over ds^2} - {3 \over s}{d\eta \over ds} + U''(\phi\_b) \eta = \lambda \eta \,. \label{flat-mode-eq}$$ This is readily verified for the case of a thin-wall bounce with radius $\bar R$. Differentiating Eq. ([flatScalarEq]) with respect to s gives $$\left[ -{d^2 \over ds^2}- {3 \over s}{d \over ds} + U''(\phi\_b) \right] {d\phi\_b \over ds} = -{3\over s^2} {d\phi\_b \over ds} \,. \label{flat-neg-mode}$$ In the thin-wall limit dϕ/ds vanishes except in a narrow region where $s \approx \bar R$, and so we see that there is an eigenfunction η(s) ≈ dϕ/ds with negative eigenvalue $\lambda \approx -3/\bar R^2$. What about multiple negative modes? The determinant factor would acquire a factor of i for each such mode, and so would be real if there were an even number of negative modes. Three such modes would give an imaginary determinant factor, but with the wrong sign. Although five modes, or any larger number equal to $1 \mod 4$, would give an imaginary part of the right sign, this would seem to be an unusual constraint on the bounce. The situation was clarified by Coleman , who showed that if a bounce has multiple negative modes then the corresponding tunneling path through configuration space is a saddle point, rather than a minimum, of the barrier penetration integral. As an explicit illustration of this, consider a theory with a single scalar field whose potential has three minima, at ϕA < ϕB < ϕC, of increasing depth, UA > UB > UC . It is convenient	arxiv_0000534
Evolving black hole horizons in General Relativity and alternative gravity ========================================================================== From the microscopic point of view, realistic black holes are time-dependent and the teleological concept of event horizon fails. At present, the apparent or the trapping horizon seem its best replacements in various areas of black hole physics. We discuss the known phenomenology of apparent and trapping horizons for analytical solutions of General Relativity and alternative theories of gravity. These specific examples (we focus on spherically symmetric inhomogeneities in a background cosmological spacetime) are useful as toy models for research on various aspects of black hole physics. Keywords: black holes, evolving horizons, apparent horizons. Introduction ============ In the literature on gravitation and quantum field theory in curved space, several kinds of horizon are studied, including Rindler horizons for accelerated observers in Minkowski space, black hole horizons, and cosmological horizons. Research on classical and semiclassical black hole physics has unveiled inner, outer, Cauchy, and extremal horizons. The early literature on black holes and the 1970s development of black hole thermodynamics focused on stationary (or even static) black holes and on event horizons (see, e.g., Refs. ). However, highly dynamical situations are of great interest for theorists, including gravitational collapse, the merger of a black hole with a compact object, evaporation of a small black hole due to Hawking radiation, and black holes interacting with non-trivial environments. Examples of nontrivial environments occur in the case of black holes accreting or emitting gravitating matter such as Vaidya spacetimes (which recur in mathematical studies of horizon dynamics); black holes immersed in a cosmological background other than de Sitter space; black holes emitting (and possibly accreting) Hawking radiation in the final stages of their evolution when backreaction is significant; or black holes with variable mass due to other conceivable processes. In all these situations the concept of event horizon must be generalized. Moreover, if the black hole is located in a non-Minkowskian background, its mass-energy (which is usually the internal energy appearing in the first law of black hole thermodynamics) needs to be defined carefully through some notion of quasi-local energy, and is related to the notion of horizon. In Rindler’s words, an horizon is “a frontier between things observable and things unobservable”. The horizon concept, which is the product of strong gravity, is perhaps the most impressive feature of a black hole spacetime and, traditionally, the one that best characterizes the black hole concept itself. Various notions of black hole horizon studied in the technical literature include event, Killing, inner, outer, Cauchy, apparent, trapping, quasi-local, isolated, dynamical, and slowly evolving horizons (for reviews see Refs. ). The notions of event, apparent, trapping, and dynamical horizon usually coincide for stationary black holes but they are quite different from each other in the case of dynamical black holes with masses evolving in time. The usual definition of black hole event horizon turns out to be pretty much useless for practical purposes in highly dynamical situations because it requires the knowledge of the entire causal structure of spacetime (including future null infinity I+), which is physically impossible. Time-evolving black holes are not just a theoretician’s playground but they are also important for astrophysics. The remarkable improvements in astronomical techniques in recent years and their projected developments in the near future have stressed the important roles that stellar mass and supermassive black holes play in the modelling of astrophysical systems. The improvement of ground-based gravitational wave detectors (presumably nearing their first detection in the next few years) and the development of space-based detectors prompt enormous theoretical efforts to predict in detail the gravitational waves emitted by astrophysical black holes and build template banks for interferometric detectors. Progress in this theoretical programme went hand-in-hand with the improvement in computing power; however, for numerical calculations on black hole systems, the notion of event horizon is again of little use in the highly dynamical situations involving gravitational collapse or the evolution or merger of a close binary system with a black hole component. In practice, “black holes” are identified with outermost marginally trapped surfaces and apparent horizons in numerical studies. The concept of horizon is not limited to black holes: there are also Rindler horizons for uniformly accelerated observers in Minkowski space and cosmological horizons. In cosmology, in addition to the particle and event horizons familiar from the standard literature on inflation, cosmological apparent and trapping horizons have also been studied more and more intensely in the recent past. Shortly after the discovery of Hawking radiation from stationary black holes and the completion of black hole thermodynamics, it was pointed out that the event horizon of de Sitter space should also be attributed a temperature and an entropy. The region of de Sitter space below this horizon is static and the de Sitter horizon does not evolve, therefore the latter can be considered, to a certain extent, as a cosmological analogue of the Schwarzschild event and Killing horizon. In this logic, the analogue of time-dependent black hole horizons would be the time-dependent apparent and trapping horizons of Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes. Many theoretical efforts went into generalizing the “standard” black hole thermodynamics for event horizons to moving horizons or horizon constructs different from event horizons. Thermodynamical studies of FLRW apparent horizons have also appeared. In principle, while it is reasonable that “slowly moving” horizons are meaningful from the thermodynamical point of view in some adiabatic approximation, it is not at all clear that fast-evolving horizons constitute thermodynamical systems and, if they do, they would most likely require non-equilibrium thermodynamics, as opposed to equilibrium thermodynamics, for their description. This feature would clearly complicate the study of highly time-dependent horizons. The (now classical) thermodynamics of stationary black hole horizons does not make reference to the field equations of the gravitational theory and, therefore, black holes in theories of gravity alternative to General Relativity (GR) can be usefully studied as well (it is true, however, that some dependence on the action remains, for example the entropy of a stationary black hole horizon of area A in scalar-tensor gravity with a Brans-Dicke-like scalar field ϕ is not S = A/4G in units c = ℏ = 1, but rather S = ϕA/4, which can be understood naively by noting that ϕ ∼ 1/G plays the role of the inverse of the effective gravitational coupling in these theories — see Ref. for details). In recent years there has been a renewed interest in alternative theories of gravity with various motivations. First of all, the search for a quantum theory of gravity has generated considerable interest in low-energy effective actions, which invariably contain ingredients foreign to Einstein’s theory such as scalar fields coupled non-minimally to the curvature (which give a scalar-tensor nature to the gravitational theory), or higher derivative terms, or non-local terms. From this point of view, the question is not if but when gravity will deviate from GR. Further motivation for extending GR comes from attempts (see for reviews) to explain the current acceleration of the universe discovered with type Ia supernovae without invoking an ad hoc dark energy. One should also mention attempts to replace the concept of dark matter at galactic and cluster scales by modifying not just relativistic, but even Newtonian gravity, given that dark matter particles seem to elude direct detection (there is yet no agreement from the various groups on reports of possible signals). From the astrophysical point of view, if primordial black holes formed in the early universe, they would have had a scale comparable to the Hubble scale and very dynamical horizons evolving on the Hubble time scale, and an important question is how fast these black holes could accrete and grow. To summarize, there are currently many avenues of research in theoretical gravitational physics in which evolving horizons play some role. Here we review the main definitions and properties of horizons and we focus on apparent and trapping black hole horizons. While it is questionable that apparent horizons are the “correct” notion of horizon in the dynamical case (and there are indications that they may not satisfy a quantum generalized second law ), they are the ones that are used in practice in numerical relativity and there seems to be no better candidate at the moment for the concept of “horizon” when time-dependence and interaction with the environment are allowed. Since only a few exact solutions of GR (and even less of other theories of gravity) are known for which the horizons are explicitly time-dependent, here we focus on solutions describing black holes embedded in cosmological backgrounds, which have been studied in some detail. Fig. [Generic] shows how a conformal diagram of an hypothetical cosmological black hole may look like. The conformal diagram of an hypothetical cosmological black hole. The bottom horizontal (dashed) line represents a Big Bang singularity, the top horizontal line (dashed) is a spacelike black hole singularity, and an apparent horizon (marked AH) can change from timelike, to null, to spacelike, and it can be located inside or outside the event horizon (forty-five degrees line marked EH) according to whether the energy conditions are satisfied or not. If Rablalb ≥ 0 for all null vectors la, then the apparent horizon lies inside the event horizon (, p. 311). [Generic] There are many additional reasons for pursuing the study of analytical solutions of GR and of gravitational theories representing a central inhomogeneity in a cosmological background. The non-linearity of the field equations prevents the splitting of solutions into a “background” and a “deviation” from it in general, but we drop the quotation marks and from now on the term “background” refers to the asymptotic structure of spacetime. We have already mentioned the use of alternative theories of gravity, and of f(Rcc) gravity in particular (where Rcc is the Ricci curvature of spacetime and f is the Lagrangian density of the gravitational field), to explain the present acceleration of the universe without dark energy: since these theories are designed to produce a time-varying effective cosmological “constant”, black hole spacetimes in these theories are naturally asymptotically FLRW, not asymptotically flat, and are dynamical. It is also of interest to study the spatial variation of fundamental constants throughout the universe, and scalar-tensor theories of gravity embody the variation of the gravitational “constant”, hence the search for analytical solutions describing condensations in cosmological spacetimes in these theories. Overall, very few such solutions are known in alternative gravity. But then one realizes that also analytical solutions of GR interpretable as central objects in cosmological backgrounds are quite interesting. The first solution of this kind is the McVittie spacetime, which was invented to address the problem of whether, or how, the expansion of the universe affects local systems (see Ref. for a recent review on this subject). The McVittie solution of GR has a complex structure and is not yet completely understood. Relatively few other solutions of GR with similar features have been discovered, including Swiss-cheese and other models. Recent interest in cosmological condensations in GR arises also from a different attempt to explain the present cosmic acceleration without dark energy and without modifying gravity. The idea is that the backreaction of inhomogeneities on the expansion of the universe could be sufficient to produce the observed acceleration. However, the implementation of this idea has several formal problems and its proponents have not yet shown convincingly that this idea explains the magnitude or even the sign of the cosmic acceleration (more mathematically oriented work puts this proposed solution to the cosmic acceleration problem in jeopardy ). The study of exact inhomogeneous universes has also been pursued in yet another attempt to explain the current acceleration of the universe, the dominant idea being that we live inside a giant void which mimics an accelerated expansion; some of the analytical GR solutions considered are related to black holes in expanding universes (see for a review). Independent motivation for the study of evolving horizons, and one not insignificant for astrophysics, comes from the renewed interest in exact models of spherical accretion by black holes, in particular the accretion of dark or phantom energy. This issue may again be relevant for primordial black holes which need to grow fast if they are to survive until the present era. The plan of this review paper is the following: first, we review basic material in the next section. The following section discusses analytical solutions of GR, and is followed by a section on spacetimes with similar features in other theories of gravity. Due to space limitations it is not possible to discuss all the known solutions and their details, or to review all the works on cosmological black holes, but we do provide more detail for a few solutions to illustrate the techniques used, and the selection made is no doubt biased. Moreover, we do not discuss here the more mathematical approaches to time-evolving horizons and the various existence and uniqueness theorems for horizons. The metric signature used is − + + + and we follow the conventions of Ref. (the speed of light c and Newton’s constant G are set to unity except where, occasionally, we restore them explicitly). Various notions of horizon ========================== Let us review briefly the geometry of the congruences of null geodesics crossing a horizon, which are used in the definition of non-stationary horizons. Null geodesic congruences and trapped surfaces ---------------------------------------------- Consider a congruence of null geodesics with tangent la = dxa/dλ, where λ is an affine parameter along each geodesic and lala = 0, lc∇cla = 0. The metric hab in the 2-space orthogonal to la is determined by the following : pick another null vector field na such that ncnc = 0 and lcnc = − 1, then we have hab gab+la nb +lb na. hab is purely spatial and hab is a projection operator on the 2-space orthogonal to la, i.e., hab la = hab lb = 0 , haa = 2 , hac hcb = hab. The choice of na is not unique but the geometric quantities of interest to us do not depend on it once la is fixed. Let ηa be the geodesic deviation vector (it corresponds to a one parameter subfamily of the congruence since its choice is not unique) and define the tensor field Bab b la, which satisfies lb∇b ηa = Bab ηb and is orthogonal to the null geodesics, Babla = Bablb = 0. The transverse part of the deviation vector is a b b=a+( ncc) la and the orthogonal component of lc∇cηa, denoted by a tilde, is = hab hcd Bbc d d d. The transverse tensor $ \widetilde{B}\_{ab}$ is decomposed into its symmetric and antisymmetric parts, and the symmetric part is further decomposed into its trace and trace-free parts as ab=(ab) + [ab]= ( hab+ab ) + ab, where the trace = gab ab= gab Bab =c lc is the expansion of the affinely parametrized congruence, ab= hab is the expansion tensor, ab= (ab) - hab is the shear tensor, and ab= [ab] is the vorticity tensor. The expansion, shear, and vorticity tensors are purely transversal (i.e., orthogonal to la) and the shear and vorticity are trace-free. The shear scalar and vorticity scalar 2 ab ab, 2 ab ab are non-negative. The expansion propagates along a null geodesic according to the celebrated Raychaudhuri equation, which was the main tool in the proof of the singularity theorems of Hawking and Penrose, = - -2+2 -Rabla lb ; similar propagation equations hold for σab and ωab. If the congruence of null geodesics with tangent la is not affinely parametrized, the geodesic equation assumes the form lc c la = la, where the quantity κ which measures the failure of la to be affinely parametrized is sometimes used, on a horizon, as a possible definition of surface gravity (there are various inequivalent definitions of surface gravity in the literature). The expansion is now =c lc- or [expansion] l = habalb = alb. Eq. ([expansion]) is independent of the field equations of the theory and can be applied when lc and nc are not normalized to satisfy lcnc = − 1 as usual. With non-affine parametrization, the Raychaudhuri equation picks up an extra term, = - -2+2 -Rabla lb. A compact and orientable 2-surface embedded in 4-space has two independent directions orthogonal to it, corresponding to ingoing and outgoing null rays. One is naturally led to study congruences of ingoing and outgoing null geodesics with tangent fields la and na, respectively, and the way they propagate in strong gravity. Let us provide now some basic definitions for closed 2-surfaces (usually it is assumed that these 2-surfaces are spacelike but we will not impose this requirement here): * A normal surface corresponds to θl > 0 and θn < 0 (for example, a 2-sphere in Minkowski space satisfies this property). * A trapped surface corresponds to θl < 0 and θn < 0. The outgoing, in addition to the ingoing, future-directed null rays converge here instead of diverging and outward-propagating light is dragged back by strong gravity. * A marginally outer trapped (or marginal) surface (MOTS) corresponds to θl = 0 (where la is the outgoing null normal to the surface) and θn < 0. * An untrapped surface is one with θlθn < 0. * An antitrapped surface corresponds to θl > 0 and θn > 0 (both outgoing and ingoing future-directed null rays are diverging). * A marginally outer trapped tube (MOTT) is a 3-dimensional surface which can be foliated entirely by marginally outer trapped (2-dimensional) surfaces. In GR Penrose has proved that, if a spacetime contains a trapped surface, the null energy condition holds, and there is a non-compact Cauchy surface for the spacetime, then this spacetime contains a singularity. Trapped surfaces are probably essential features in the concept of black hole and notions of “horizon” of practical utility will be identified with boundaries of spacetime regions containing trapped surfaces. At present, the mathematical conditions for the existence and uniqueness of MOTSs are not completely clear. It is known that, in general, a MOTT can be distorted smoothly, hence MOTTs are non-unique. Let us review the various kinds of horizons appearing in the literature on black holes, cosmology, quantum field theory in curved spaces, and the corresponding thermodynamics. Event horizons -------------- The event horizon is the traditional notion of horizon for stationary black holes in GR. An event horizon is a connected component of the boundary ∂(J−(I+)) of the causal past J−(I+) of future null infinity I+. This is the most peculiar feature of a black hole: the horizon is a causal boundary separating a region from which nothing can come out to reach a distant observer from a region in which signals can be sent out and eventually arrive to this observer. An event horizon is generated by the null geodesics which fail to reach future null infinity and, therefore (provided that it is smooth) is always a null hypersurface. In astrophysics the concept of event horizon is implicitly taken as a synonym of black hole. However, since to define and locate an event horizon one must know all the future history of spacetime (one must know all the geodesics which do reach future null infinity and, tracing them back, the boundary of the region from which they originate), an event horizon is a globally defined concept. For an observer to state that a black hole event horizon has formed requires knowledge of the spacetime outside his or her future light cone, which is impossible to achieve unless the spacetime is stationary, the black hole has existed forever, and nothing changes (a common expression is that the event horizon has a teleological nature). It has been shown that, in a collapsing Vaidya spacetime, an event horizon forms and grows starting from the centre and an observer can cross it and be unaware of it even though his or her causal past consists entirely of a portion of Minkowski space: the event horizon cannot be detected by this observer with a physical experiment. In other words, the event horizon “knows” about events belonging to a spacetime region very far away and in its future but not causally connected to it (“clarvoyance” ). Due to its global nature, an event horizon is not a practical concept and it is nearly impossible to locate precisely an event horizon in a dynamical situation. Realistic astrophysical black holes have not existed forever but are formed by gravitational collapse. In numerical relativity, codes designed to follow a collapse situation, a binary system merger, or other dynamical situations generating black holes, eventually crash and it is impossible to follow the evolution of a system to future null infinity. It is routine in numerical relativity to employ marginally trapped surfaces as proxies for event horizons (e.g., ). The event horizon ${\cal H}$ is a tube in spacetime; a very common abuse of terminology consists of referring to the intersections of ${\cal H}$ with surfaces of constant time (which produce 2-surfaces) as “event horizons” (this abuse of terminology extends to all the notions of horizon that we define below). Killing horizons ---------------- When present, a Killing vector field ka satisfying the Killing equation ∇akb + ∇bka = 0 defines a Killing horizon ${\cal H}$ of the spacetime (M, gab), which is a null hypersurface which is everywhere tangent to a Killing vector field k*a which becomes null, kckc = 0, on ${\cal H}$. This Killing vector field is timelike, kck*c < 0, in a spacetime region which has ${\cal H}$ as boundary. Stationary event horizons in GR are Killing horizons, for example in the Schwarzschild geometry the event horizon R = 2M is also a Killing horizon and the timelike Killing vector ka = (∂/∂t)a in the R > 2M region outside the event horizon becomes null at R = 2M and spacelike for R < 2M. An event horizon in a locally static spacetime is also a Killing horizon for the Killing vector ka = (∂/∂t)a associated with the time symmetry. If the spacetime is stationary and asymptotically flat (but not necessarily static), it must be axisymmetric and an event horizon is a Killing horizon for the Killing vector ka =( /t )a +H ( /)a, which is a linear combination of the vectors associated with time and rotational symmetries, and where ΩH is the angular velocity at the horizon (this statement requires the assumption that the Einstein-Maxwell equations hold and some assumption on the matter stress-energy tensor ). When present, a Killing horizon defines a notion of surface gravity $\kappa\_{{\small {\sf Killing}}}$, as we will see below. Of course, the concept of Killing horizon is useless in spacetimes which do not admit timelike Killing vectors. Attempts to use conformal Killing horizons in spacetimes conformal to the Schwarzschild one (, see also ) have not been fruitful. However, the introduction of the Kodama vector, which is defined in spacetimes without Killing vectors, in place of a Killing field is much more useful to introduce a surface gravity. Apparent horizons ----------------- A future apparent horizon is the closure of a surface (usually a 3-surface) which is foliated by marginal surfaces; it is defined by the conditions on the time slicings $$\begin{aligned} & \theta\_{l} = 0 \,,& \label{AHcondition1} \\ & \theta\_{n} < 0 \,,& \label{AHcondition2} \end{aligned}$$ where θl and θn are the expansions of the future-directed outgoing and ingoing null geodesic congruences, respectively (this more practical definition differs from that of Hawking and Ellis, which is rather impractical ). Eq. ([AHcondition1]) expresses the fact that the congruence of future-pointing outgoing null rays momentarily stops expanding and, presumably, these rays turn around at the horizon, while the condition ([AHcondition2]) originally served the purpose of distinguishing between black holes and white holes. Apparent horizons are defined quasi-locally and are independent of the global causal structure of spacetime, contrary to event horizons. However, apparent horizons (and also trapping horizons, see below) depend on the choice of the foliation of the 3-surface with marginal surfaces and, of course, also the ingoing and outgoing null geodesics orthogonal to these surfaces do, as well as their expansions θl and θn. While the expansions are scalars, and are therefore independent of the coordinate system chosen, sometimes a coordinate choice makes it easier to specify locally the foliation (for example by choosing spacelike surfaces of constant time coordinate — different time coordinates identify different families of hypersurfaces of constant time), which is a geometric object and is coordinate-independent. Congruences of outgoing and ingoing null geodesics orthogonal to these surfaces will, of course, change when changing the foliation. The dependence of apparent horizons on the spacetime slicing is illustrated by the fact that non-symmetric slicings of the Schwarzschild spacetime can be found for which no apparent horizons exist. Apparent horizons are, in general, quite distinct from event horizons: for example, event and apparent horizons do not coincide in the Reissner-Nordström black hole (inner horizon) and in the Vaidya spacetime. Also in static black holes which are perturbed, the apparent and the event horizons do not coincide. During the spherical collapse of uncharged matter an event horizon forms before the apparent horizon does and the two come closer and closer until they eventually coincide asymptotically as the final static state is reached. In GR, a black hole apparent horizon lies inside the event horizon provided that the null curvature condition Rab lalb ≥ 0 ∀ null vector la is satisfied. This requirement coincides with the null energy condition $ T\_{ab}\l^al^b \geq 0 $ ∀ null vector la if the Einstein equations are imposed, and in this case it is believed to be a reasonable condition on classical matter. However, Hawking radiation itself violates the weak and the null energy conditions, as does quantum matter. A simple scalar field non-minimally coupled to the curvature can also violate all of the energy conditions. The null curvature condition is easily violated also in alternative theories of gravity (for example, Brans-Dicke and scalar-tensor gravity) and the black hole apparent horizon has been observed to lie outside of the event horizon during spherical collapse in Brans-Dicke theory, although it eventually settles inside of it when the static Schwarzschild state is achieved (note that the GR black hole is the endpoint of collapse in general scalar-tensor gravity for asymptotically flat black holes without matter other than the Brans-Dicke scalar field outside the horizon ). To summarize, the cherished notion of event horizon seems rather useless in general dynamical situations and apparent horizons appear to be more practical in spite of their fundamental limitations of depending on the spacetime slicing and of possibly being timelike surfaces (this last drawback is probably the most puzzling one ). Trapping horizons ----------------- A future outer trapping horizon (FOTH) is the closure of a surface (usually a 3-surface) foliated by marginal surfaces such that on its 2-dimensional “time slicings” (, see also and references therein) $$\begin{aligned} & \theta\_{l} = 0 \,,& \label{THcondition1} \\ &&\nonumber\\ & \theta\_{n} < 0 \,,& \label{THcondition2} \\ &&\nonumber\\ & {\cal L}\_n \, \theta\_{l} = n^{a}\nabla\_{a} \, \theta\_{l} < 0 \,, & \label{THcondition3} \end{aligned}$$ where θl and θn are the expansions of the future-directed outgoing and ingoing null geodesic congruences, respectively. The inequality ([THcondition3]) serves the purpose of distinguishing between inner and outer horizons, e.g., in the non-extremal Reissner-Nordström solution, and also distinguishes between apparent horizons and trapping horizons (it is not imposed for apparent horizons but it is required for trapping ones), and its sign distinguishes between future and past horizons. The definition of a past inner trapping horizon (PITH) is obtained by exchanging la with na and reversing the signs of the inequalities, $$\begin{aligned} & \theta\_{n} = 0 \,,& \label{PITHcondition1} \\ &&\nonumber\\ & \theta\_{l} > 0 \,,& \label{PITHcondition2} \\ &&\nonumber\\ & {\cal L}\_l \theta\_n = l^{a}\nabla\_{a} \, \theta\_{n} > 0 \,.& \label{PITHcondition3} \end{aligned}$$ The past inner trapping horizon identifies a white hole or a cosmological horizon. As one moves just inside an outer trapping horizon, one encounters trapped surfaces, while trapped surfaces are encountered as as one moves just outside an inner trapping horizon. As an example, consider the static Reissner-Nordström black hole with the natural spherically symmetric foliation: the event horizon r = r+ is a future outer trapping horizon (FOTH), the inner (Cauchy) horizon r = r− is a future inner trapping horizon (FITH), while the white hole horizons are past trapping horizons (PTHs). Black hole trapping horizons have been associated with thermodynamics, and it is claimed that it is the trapping horizon area and not the area of the event horizon which should be associated with entropy in black hole thermodynamics. This claim is controversial. The Parikh-Wilczek “tunneling” approach is in principle applicable also to apparent and trapping horizons, not only to event horizons but also this aspect is not entirely free of controversy. In general, trapping horizons do not coincide with event horizons. Dramatic examples are spacetimes which possess trapping horizons but not event horizons. The difference between the areas of the trapping and the event horizon in particular spacetimes have been studied in Ref. . Isolated, dynamical, and slowly evolving horizons ------------------------------------------------- Isolated horizons correspond to isolated systems in thermal equilibrium not interacting with their surroundings, which are described by a stress-energy tensor Tab. The concept of isolated horizon has been introduced in relation with loop quantum gravity and, in a general perspective, it is too restrictive when one wants to allow mass-energy to cross the “horizon” (whichever way the latter is defined) in one direction or the other. A weakly isolated horizon is a null surface ${\cal H} $ with null normal l*a such that θl = 0, − Tabla is a future-oriented and causal vector, and ${\cal L}\_l \left( n\_b \nabla\_a l^b \right) =0$. In this context l*a is a Killing vector for the intrinsic geometry on ${\cal H}$, without reference to the surroundings, and can therefore be used to define a “completely local Killing horizon” when there are no energy flows across ${\cal H}$. The vector field la generates a congruence of null geodesics on ${\cal H}$, which can be used to define a surface gravity κ via the (non-affinely parametrized) geodesic equation laa lb= lb, which gives =-nb laa lb using nblb = − 1. This surface gravity κ is constant on the weakly isolated horizon ${\cal H}$, which corresponds to the zeroth law of thermodynamics. Since the vector field na is not unique also this surface gravity is not unique. A Hamiltonian analysis of the phase space of isolated horizons, identifying boundary terms with the energies of these boundaries, leads to a first law of thermodynamics for isolated horizons with rotational symmetry, H H = A + H J, where J is the angular momentum, $H\_{{\cal H}}$ the Hamiltonian, A the area of the 2-dimensional cross-sections of ${\cal H}$, and $\Omega\_{ {\cal H}} $ the angular velocity of the horizon. A dynamical horizon is a spacelike marginally trapped tube foliated by marginally trapped 2-surfaces (MTT). This definition allows for energy fluxes across the dynamical horizon. A set of flux laws describing the related changes in the area of the dynamical horizons have been formulated. An apparent horizon which is everywhere spacelike coincides with a dynamical horizon, but an apparent horizon is not required to be spacelike. Being spacelike, dynamical horizons can be crossed only in one direction by causal curves, while this is not the case for apparent horizons which can be partially or entirely timelike. Finally, slowly evolving horizons have also been introduced and studied : these are “almost isolated” FOTHs and they are intended to represent black hole horizons which evolve slowly in time, as is expected in many astrophysical processes but not, for example, in the final stages of black hole evaporation. They are analogous to thermodynamical systems in quasi-equilibrium. Kodama vector ------------- In the literature one finds several notions of surface gravity associated with horizons. In stationary situations, in which a timelike Killing vector field outside the horizon becomes null on it, these notions of surface gravity coincide. In dynamical situations there is no timelike Killing vector and these surface gravities turn out to be inequivalent. In spherical symmetry, the Kodama vector mimics the properties of a Killing vector and originates a (miracolously) conserved current and a surface gravity. The Kodama vector is a generalization of the notion of Killing vector field to spacetimes which do not admit one, and has been used in place of a Killing vector in the thermodynamics of dynamically evolving horizons. The Kodama vector is defined only for spherically symmetric spacetimes (see for an attempt to introduce a Kodama-like vector in non-spherical spacetimes). Let the metric be [hab] ds2=habdxa dxb +R2 d(2)2, where (a, b) = (t, R), R is the areal radius, and dΩ(2)2 = dθ2 + sin2θ dφ2 is the line element on the unit 2-sphere. Let εab be the volume form associated with the 2-metric hab ; then the Kodama vector is Ka - ab b R (with Kθ = Kφ = 0). The Kodama vector satisfies Ka∇aR = − εab∇aR∇bR = 0. In a static spacetime the Kodama vector is parallel (in general, not equal) to the timelike Killing vector. In the region in which it is timelike, the Kodama vector defines a class of preferred observers with four-velocity $ u^a \equiv K^a/ \sqrt{ \| K^c K\_c \| } $ (the Kodama vector is timelike in asymptotically flat regions). The Kodama vector is divergence-free, [Kodamadivergence] a Ka=0, which has the consequence that the Kodama energy current Ja GabKb (where Gab is the Einstein tensor) is covariantly conserved, ∇aJa = 0, even if there is no timelike Killing vector (a property referred to as the “Kodama miracle” ). By writing the spherical metric in Schwarzschild-like coordinates, [ABgauge] ds2=-A (t,R ) dt2+B(t,R ) dR2+R2 d(2)2, the Kodama vector assumes the simple form (e.g., ) [KodamaAB] Ka= ( )a. The Noether charge associated with the Kodama conserved current is the Misner-Sharp-Hernandez energy of spacetime (which, again, is defined only for spherically symmetric spacetimes). Surface gravities ----------------- Traditionally, surface gravity is defined in terms of geometric properties of the metric tensor and it also shows up in black hole thermodynamics as the proportionality factor between the variation of the black hole mass (which plays the role of internal energy) dM and the variation of the event horizon area (proportional to the entropy) dA. Since it is unclear which definition of black hole mass is appropriate in non-trivial backgrounds (see the review ), also the definition of surface gravity suffers from the same ambiguities. Surface gravity is also a semiclassical quantity since for stationary black holes it coincides, up to a constant, with the Hawking temperature of a black hole. The textbook definition of surface gravity is given on a Killing horizon. Given that Killing fields are not available in non-stationary situations, a different concept of surface gravity is necessary there. The recurrent definitions are reviewed in Ref. and are briefly recalled here. a Killing horizon defines the surface gravity $\kappa\_{{\small {\sf Killing}}}$ as follows : on the Killing horizon the Killing vector ka satisfies (e.g., ) [Killingsurfacegravity] kaa kb kb, so $\kappa\_{ {\sf Killing}}$ measures the failure of the geodesic Killing vector ka to be affinely parametrized on the Killing horizon. Another property of the Killing surface gravity is [Killingsurfacegravity2] 2=- ( a kb ) ( a kb ). In static spacetimes, $\kappa\_{{\small {\sf Killing}}}$ is interpreted as the limiting force required at spatial infinity to hold in place a unit test mass just above the event horizon by means of an infinitely long massless string (which shows the non-local nature of the notion of Killing surface gravity). Since the Killing equation ∇akb + ∇bka = 0 determines the Killing vector ka only up to an overall normalization, there is freedom to rescale ka and the value of the surface gravity depends on the non-affine parametrization chosen for ka. However, in static/stationary situations one has the freedom of imposing that kckc = − 1 at spatial infinity. The Killing surface gravity can be generalized to any event horizon that is not a Killing horizon by replacing the Killing vector ka with the null generator of the event horizon. let la and na be the outgoing and ingoing null normals to a marginally trapped (spacelike compact 2-dimensional) surface, with the expansion of la vanishing, and assume that la and na are normalized so that lcnc = − 1. In general, la is not a horizon generator but is a non-affinely parametrized geodesic vector on the trapping horizon, which allows one to define a surface gravity κ as laa lb lb, or = -nb la a lb. The value of κ depends on the parametrization of la and there are various proposals for this. In general, writing la as the tangent to a null curve xμ(λ) with parameter λ, a parameter change (dependent on the spacetime point) λ → λʹ means that the components of la change according to l = l’ = = l (x) l and $$\begin{aligned} && l^{\nu '} \nabla\_{\nu '} l^{\mu '} = \kappa'l^{\mu '} \,, \\ &&\nonumber\\ && \Omega l^{\nu}\nabla\_{\nu} \left( \Omega l^{\mu} \right)=\kappa' \Omega \, l^{\mu} \,, \end{aligned}$$ and finally ’ = +lcc. The Hayward proposal (for spherical symmetry) is based on the Kodama vector Ka. In spherical symmetry the Kodama vector satisfies $$\begin{aligned} && \nabla\_b \left( K\_a T^{ab} \right) \propto \nabla\_b J^b =0 \,, \\ && \nonumber\\ && K^c K\_c=-1 \;\;\;\;\mbox{ at spatial infinity},\end{aligned}$$ and it is taken to be future-directed. The ensuing surface gravity for a trapping horizon is given by [xvbcquesta] gab Kc ( c Ka -a Kc ) = Kodama Kb. This definition is unique because the Kodama vector is unique. $\kappa\_{{\small {\sf Kodama}}}$ agrees with the surface gravity on the horizon of a Reissner-Nordström black hole but not with other definitions of dynamical surface gravity. An expression equivalent to ([xvbcquesta]) is [kappaKodamaHayward] = (h)R= ( h R ), where R is the areal radius and h is the determinant of the metric hab in the 2-space orthogonal to ∇aR. The Hamilton-Jacobi approach, a variant of the Parikh-Wilczek method, employs the Kodama-Hayward definition of surface gravity (for a review of tunneling methods see Ref. ). this proposal for spherically symmetric asymptotically flat spacetimes is based on the ingoing null normal na being normalized so that nata = − 1, where ta is the asymptotic time-translational Killing vector at spatial infinity. na is affinely parametrized everywhere and at spatial infinity is parametrized by the proper time of static observers. Requiring that lcnc = − 1 fixes the parametrization of la, yielding = -nb la a lb. this proposal of Ashtekar, Beetle, and Fairhurst applies to an isolated horizon. na is normalized so that its expansion agrees with that of the Reissner-Nordström case and with lana = − 1. This choice fixes a unique surface gravity as a function of the horizon parameters. However, this concept appears to be limited, for example it cannot be extended to the Einstein-Yang-Mills case. this definition extends the previous proposal. On the isolated horizon the normal is τa = Bla + Cna, with B and C scalar fields defined there, which weight the contributions of la and na (for an isolated horizon it is B = 1 , C = 0). The surface gravity is BF -B na lb b la -C la nb b na. other proposals for surface gravity include Hayward’s trapping gravity and the Mukohyama and Hayward proposal. The surface gravities listed here are computed in Ref. for a general spherically symmetric metric in Eddington-Finkelstein coordinates and in terms of the Misner-Sharp-Hernandez mass. Ref. compares these definitions for spherical black holes in Painlevé-Gullstrand coordinates. Spherical symmetry ------------------ Assuming spherical symmetry greatly simplifies the solution of the field equations and the study of horizons. Although this is an unrealistic assumption for rotating astrophysical black holes and for universes with realistic inhomogeneities, it is important for the fundamental theory. In spherically symmetric spacetimes a useful tool is the Misner-Sharp-Hernandez mass M, which here coincides with the Hawking-Hayward quasi-local mass. The Misner-Sharp-Hernandez mass is defined in GR and for spherical symmetry. Using the areal radius R and angular coordinates (θ, φ), a spherical line element can be written as [2normal] ds2=habdxa dxb +R2 d(2)2 (a,b=1,2). The Misner-Sharp-Hernandez mass M is defined by [MisnerSharpmass] 1- c R c R or M= ( 1-haba R b R ). [MSHz2] Horizons in spherical symmetry are discussed in a clear way in the formalism of Nielsen and Visser. These authors consider the most general spherically symmetric metric (not necessarily stationary or asymptotically flat) with a spherically symmetric spacetime slicing, which assumes the form [generalsphericalmetric] ds2=--2(t, R) dt2 + +R2d(2)2 in Schwarzschild-like coordinates, where M(t, R) a posteriori turns out to be the Misner-Sharp-Hernandez mass. The line element ([generalsphericalmetric]) can be recast in Painlevé-Gullstrand coordinates as [generalsphericalPG] ds2=- ( 1- ) d2 + ddR +dR2 +R2d(2)2, where ϕ(τ, R) and M(τ, R) are implicit functions of (τ, R) and the spacelike hypersurfaces τ = constant are flat. Using the implicit functions of (τ, R) $$\begin{aligned} c \left(\tau, R \right) & \equiv & \frac{ \mbox{e}^{-\phi(t,R)} }{\left( \partial\tau/ \partial t \right)} \,,\\ &&\nonumber\\ v \left(\tau, R \right) & \equiv & \sqrt{\frac{2M(t,R)}{R} } \, \frac{ \mbox{e}^{-\phi(t,R)} }{\partial\tau/\partial t} =c \, \sqrt{\frac{2M}{R} } \,,\end{aligned}$$ the line element becomes [vvvucci] ds2= -d2 +2v (, R ) ddR +dR2 +R2 d(2)2. A number of practical results are then obtained. The outgoing radial null geodesic congruence has tangent field with components (in Painlevé-Gullstrand coordinates (τ, R, θ, φ)) l= ( 1, c(, R)-v(, R), 0,0 ), while the ingoing radial null geodesics have tangent field n= ( 1, -c(, R)-v(, R), 0,0 ), where the normalization gabla nb=-2 is adopted. The expansions of these radial null geodesic congruences are $$\begin{aligned} \theta\_l &=& \frac{2}{R} \left( 1-\sqrt{ \frac{2M}{R} } \right) \,,\\ &&\nonumber\\ \theta\_n &=& - \, \frac{2}{R} \left( 1 +\sqrt{ \frac{2M}{R} } \right) \,. \label{2.159}\end{aligned}$$ A sphere of radius R is trapped if R < 2M, marginal if R = 2M, and untrapped if R > 2M. The apparent horizon corresponding to θl = 0 and θn < 0 is given by [AH] =1 c Rc R. \|AH =0 gRR. \|AH=0, where the last equation holds in both Painlevé-Gullstrand coordinates and in the gauge ([generalsphericalmetric]) and is obtained by using the fact that the inverse of the metric ([vvvucci]) has components ( g )= ( cccc 1 & -v &0 &0 &&& -v & -(c2-v2) &0 &0 &&& 0 & 0 & &0 &&& 0 & 0 & 0 & ). In practice, the condition gRR = 0 is a very convenient recipe to locate the apparent horizons in the presence of spherical symmetry when the areal radius R is used as a coordinate and it is often convenient to perform a coordinate transformation to this radial coordinate and to rewrite the line element using R. The gradient of the areal radius R and the normal na = ∇aR to the surfaces R = const. become null at the apparent horizon; this recipe is reminiscent of the change in the causal character of the Schwarzschild radial coordinate on the Schwarzschild event horizon. However the apparent horizon is not, in general, a null surface. We have also n l. \|AH=na a AH =- ( 1+ ), where a prime and an overdot denote partial differentiation with respect to R and τ, respectively, and the subscript AH identifies quantities evaluated on the apparent horizon. 1 − 2MʹAH > 0 is required for the horizon to be outer in a spacetime with regular asymptotic region, hence the condition for the apparent horizon to be also a trapping horizon is AH >-2cAH. If matter satisfies the null energy condition, and assuming the Einstein equations, the area of the apparent horizon cannot decrease. Various energy fluxes across the apparent horizon are also discussed and computed in. The Nielsen-Visser surface gravity at the horizon is computed from lb∇bla = κl la, which gives l ( ) =. An extremal horizon will be one with vanishing surface gravity, 1-2M’(, RH() )=0. The fact that the Misner-Sharp-Hernandez mass can be used to define and locate apparent horizons in spherically symmetric spacetimes shows that the apparent horizon is a quasi-local concept and is independent of the global causal structure. However, it does not appear to be a completely local notion (it depends on a surface, not only on the spacetime point). Evolving horizons, cosmological black holes, and naked singularities in GR ========================================================================== Let us review briefly some dynamical and spherically symmetric solutions of the Einstein equations of particular significance, paying attention to the structure and dynamics of their apparent horizons. The Schwarzschild-de Sitter-Kottler spacetime --------------------------------------------- The Schwarzschild-de Sitter-Kottler spacetime is locally static but it is useful to review it in order to understand the apparent horizons of more complicated dynamical solutions. It has line element ds2=-( 1--H2R2 ) dt2+ ( 1--H2R2 )-1 dR2+ R2 d (2)2,[staticSdS] where the constant $H=\sqrt{\Lambda/3} $ is the Hubble parameter of the de Sitter background, Λ > 0 is the cosmological constant, and m > 0 is a second parameter related to the mass of the central inhomogeneity (e.g., ). The static coordinates (t, R, θ, φ) cover the region R1 < R < R2. The apparent horizons are located by gRR = 0, which is equivalent to the cubic equation [3] 1- -H2R2 =0, with roots $$\begin{aligned} R\_1&=&\frac{2}{\sqrt{3}H}\sin\psi \,,\nonumber\\ &&\nonumber\\ R\_2&=&\frac{1}{H}\cos\psi -\frac{1}{\sqrt{3}H}\sin\psi \,,\nonumber\\ &&\nonumber\\ R\_3&=&-\frac{1}{H}\cos\psi -\frac{1}{\sqrt{3}H} \sin\psi \,, \end{aligned}$$ with $\sin (3\psi )=3\sqrt{3} \, mH$. m and H are both necessarily positive in an expanding universe, then R3 is negative and there are at most two apparent horizons. When R1 and R2 are real, R1 is a black hole apparent horizon which reduces to the R = 2m Schwarzschild horizon in the limit H → 0, while R2 is a cosmological apparent horizon which reduces to the R = 1/H de Sitter horizon in the limit m → 0. The metric ([staticSdS]) is static in the region between these two horizons. Both apparent horizons exist only if 0 < sin(3ψ) < 1 and, since the metric is locally static, the apparent black hole and cosmological horizons are also event horizons. If sin(3ψ) = 1 these horizons coincide (extremal Nariai black hole). For sin(3ψ) > 1 the roots are complex-valued and there is a naked singularity. To summarize: if $ mH<1/(3\sqrt{3}) $ there are two horizons of radii R1 and R2; if $ mH=1/(3\sqrt{3}) $ the two horizons coincide, R1 = R2; if $ mH>1/(3\sqrt{3})$ there are no apparent horizons. The interpretation seems to be that the would-be black hole horizon would become larger than the cosmological one but, strictly speaking, the roots corresponding to the apparent horizons are complex in this case. The black hole horizon has area $ {\cal A} = 4 \pi R\_1^2$ which is, of course, time-independent. In the non-extremal case the central singularity is eternal and spacelike (, see this reference also for a conformal diagram) and is surrounded by the black hole event horizon at all times for the parameter values for which this horizon exists. A sphere of radius R has the Misner-Sharp-Hernandez mass MMSH=m+ =m+ R3, where $\rho=\frac{\Lambda}{8\pi} $. The Schwarzschild-de Sitter-Kottler black hole has been studied extensively in relation to its thermodynamics. Here we do not discuss anti-de Sitter black holes corresponding to Λ < 0, which are the subject of much recent interest due to the fluid-gravity duality. The McVittie solution --------------------- The 1933 McVittie solution of the Einstein equations is a generalization of the Schwarzschild-de Sitter-Kottler solution and represents a central object embedded in a FLRW (not necessarily a locally static de Sitter) background. Even after many works, this solution is not completely understood. The McVittie solution with negative cosmological constant was analyzed in Ref. and an electrically charged version of the McVittie spacetime was found in Ref. . In this subsection we restrict to a spatially flat FLRW background and to zero electric charge. A simplifying assumption of McVittie consists of the no-accretion condition Gtr̄ = 0 (in spherical coordinates, where Gμν is the Einstein tensor) which forbids any mass-energy flow (which, in spherical symmetry, could only be radial), Ttr̄ = 0. Generalizations of the McVittie solution allowing radial energy fluxes are more complicated and will be considered later. McVittie was led to his solution by the problem of the effect of the cosmological expansion on local systems. Different approaches to this problem generated other solutions, such as the Swiss-cheese model (this problem has seen an extensive literature devoted to it but is not completely solved ). Unlike the Schwarzschild-de Sitter-Kottler spacetime, black holes in more general FLRW backgrounds are dynamical. The McVittie line element in isotropic coordinates is $$\label{McVittieisotropic} ds^2=-\frac{ \left(1-\frac{m(t)}{2\bar{r}} \right)^2}{ \left(1+\frac{m(t)}{2\bar{r}} \right)^2} \, dt^2+ a^2(t) \left( 1+\frac{m(t)}{2\bar{r}} \right)^4 \left( d\bar{r}^2 +\bar{r}^2 d\Omega\_{(2)}^2 \right) \,,$$ where the function m(t) is required to satisfy the McVittie no-accretion condition Ttr̄ = 0 on the stress-energy tensor Tab, which becomes $$\label{35} \frac{\dot{m}}{m}+\frac{\dot{a}}{a}=0$$ with solution $$\label{36} m(t)=\frac{m\_0}{a(t)} \,,$$ where m0 is a constant, therefore, $$ds^2=-\frac{ \left[ 1-\frac{m\_0}{2\bar{r}a(t)} \right]^2}{ \left[ 1+\frac{m\_0}{2\bar{r}a(t)} \right]^2} \, dt^2+ a^2(t) \left[ 1+\frac{m\_0}{2\bar{r}a(t)} \right]^4 \left( d\bar{r}^2 +\bar{r}^2 d\Omega\_{(2)}^2 \right) \,.$$ The McVittie metric reduces to the Schwarzschild one in isotropic coordinates if a ≡ 1 and to the FLRW metric if m0 = 0 and is singular on the 2-sphere r̄ = m0/2 (which reduces to the Schwarzschild horizon if a ≡ 1). This singularity is spacelike (and is represented as a horizontal line in conformal diagrams). There is another spacetime singularity at r̄ = 0. McVittie’s original interpretation of the line element ([McVittieisotropic]) as describing a point mass at r̄ = 0 is made untenable by the fact that this point mass would be surrounded by the r̄ = m0/2 singularity. We will only consider the region r̄ > 2m0 here. The energy density of the source fluid is finite but its pressure $$\label{pressure} P \left( t, \bar{r} \right) =-\, \frac{1}{8\pi} \left[ 3H^2+\frac{2\dot{H}\left( 1+\frac{m\_0}{2\bar{r}} \right) }{1-\frac{m\_0}{2\bar{r}} }\right]$$ diverges at r̄ = m0/2 with the Ricci scalar Raa = 8π(3P − ρ), with the exception of a de Sitter background with $\dot{H}=0$. The apparent horizons were studied in Refs. and interpreted in, which we follow here. We rewrite the line element ([McVittieisotropic]) in terms of the areal radius R( t, \|r ) a(t) \|r ( 1+ )2 ; the differentials dr̄ and dR are related by $$\begin{aligned} dR &=& \left( 1+\frac{m}{2\bar{r}} \right) a\bar{r} \left[ H \left( 1+\frac{m}{2\bar{r}} \right) +\frac{\dot{m}}{\bar{r}} \right] dt + a \left( 1+\frac{m}{2\bar{r}} \right) \left( 1 - \frac{m}{2\bar{r}} \right) d\bar{r} \nonumber\\ &&\nonumber\\ &=& a \left( 1+\frac{m}{2\bar{r}}\right) \left( 1-\frac{m}{2\bar{r}}\right) \left( H\bar{r}dt + d\bar{r} \right) \,,\end{aligned}$$ where the relation ([35]), which gives H ( 1+ ) + = H ( 1 - ), has been used and d \|r= -H\|r dt. Using this relation in ([McVittieisotropic]) and noting that ( )2= 1- where m/r̄ = ma/R = m0/R (here ma is constant because of eq. ([35])) leads to [nondiagonalMcVittie] ds2 = -( 1--H2R2 ) dt2 + - dtdR +R2 d(2)2, where $H\equiv \dot{a}/a$. The cross-term in dtdR is then eliminated by defining a new time T(t, R) such that [formdT] dT= ( dt+dR ), with the integrating factor F(t, R) and function β(t, R) to be determined. dT is an exact differential if the 1-form ([formdT]) is closed, or = F -. Now replace dt with FdT − βdR in eq. ([nondiagonalMcVittie]), obtaining $$\begin{aligned} ds^2 & = & -\left( 1-\frac{2m\_0}{R}-H^2R^2 \right) F^2 dT^2\nonumber\\ &&\nonumber\\ &\, & + \left[ -\left( 1-\frac{2m\_0}{R} -H^2R^2 \right)\beta^2 + \frac{1}{ 1-\frac{2m\_0}{R} } +\frac{2\beta HR}{ \sqrt{ 1-\frac{2m\_0}{R} }} \right] dR^2 \nonumber\\ &&\nonumber\\ &\,& + 2F \left[ \left( 1-\frac{2m\_0}{R} -H^2R^2 \right) \beta - \frac{HR}{ \sqrt{1-\frac{2m\_0}{R}}} \right] dT dR +R^2 d\Omega\_{(2)}^2 \,.\nonumber\\ &&\end{aligned}$$ Imposing now that (t, R) =, the line element is diagonalized, [diagonalMcVittie] ds2 = -( 1--H2R2 ) F2 dT2 + +R2 d(2)2. The singularity r̄ = m/2 corresponds to the proper radius R = 2m a(t) = 2m0 and does not expand. Let us study now the apparent horizons of the McVittie spacetime. For simplicity, we restrict ourselves to a spatially flat FLRW background. The Einstein equations provide the density of the fluid (t)= H2(t), The McVittie metric admits arbitrary FLRW backgrounds generated by cosmic fluids satisfying any constant equation of state. For brevity, we restrict to a cosmic fluid which reduces to dust at spatial infinity and corresponding to an equation of state parameter w = 0. Then the pressure is P(t,R )=(t) ( -1 ) [5]. The apparent horizons are located by gRR = 0 or, using eq. ([diagonalMcVittie]) [8] 1--H2(t) R2=0. This cubic in R is the same as the Schwarzschild-de Sitter-Kottler horizon condition ([3]) but with a time-dependent Hubble parameter. The resulting time-dependent apparent horizons R1(t) and R2(t) are again the solutions R1, 2 of eq. ([3]) but now with time-dependent coefficient H(t). The location of the apparent horizons of the McVittie spacetime depends on the cosmic time. Again, the condition for both horizons to exist is 0 < sin(3ψ) < 1, which corresponds to $m\_0H(t)<1/(3\sqrt{3})$. However, unlike the Schwarschild-de Sitter-Kottler case with constant H, this inequality is only satisfied at certain times. The critical time at which $m\_0 H(t)=1/(3\sqrt{3})$ is unique for a dust-dominated background with H(t) = 2/(3t) and is $t\_\* =2\sqrt{3} \, m\_0$. Three possibilities arise: 1. for t < t\* it is $m\_0>\frac{1}{3\sqrt{3} \,H(t)}$ and both R1(t) and R2(t) are complex. There are no apparent horizons. 2. The critical time t = t\* corresponds to $m\_0=\frac{1}{3\sqrt{3}\,H(t)}$. R1(t) and R2(t) coincide at a real value and there is a single apparent horizon at $R\_\=\frac{1}{\sqrt{3}\,H(t\_\)}$. 3. For t > t\* it is $ m\_0 < \frac{1}{3\sqrt{3}\,H(t)}$ and there are two apparent horizons of real positive radii R1(t) and R2(t). The behaviour of the apparent horizons is described in fig. [McV1]. The McVittie cosmological (dashed) and black hole (solid) apparent horizons in a dust-dominated background universe. Time t (on the horizontal axis) and radius R (on the vertical axis) are in units of m0 and we arbitrarily fix m0 = 1. [McV1] At times t < t\* there is a naked singularity at R = 2m0: while the Hubble parameter H(t) diverges near the Big Bang, the mass coefficient m0 stays supercritical at $m\_0>\frac{1}{3\sqrt{3}\,H(t)}$. As the Schwarzschild-de Sitter-Kottler experience teaches us, a black hole horizon cannot be accommodated in this small universe and the singularity is naked: the putative black hole is too large to fit in the observable universe (varying speed of light cosmologies have a related phenomenology — the radii of primordial black holes and the Compton wavelengths of massive particle states can become larger than the Hubble radius ). At the critical time t\* a black hole apparent horizon and a cosmological apparent horizon appear together at radius $R\_1(t\_\)=R\_2(t\_\) =\frac{1}{\sqrt{3}\,H(t\_\)}$, in analogy with the Nariai black hole of the Schwarzschild-de Sitter-Kottler solution. This critical black hole is instantaneous. As time progresses to t* > t\, this single horizon splits into an evolving black hole apparent horizon surrounded by an evolving cosmological horizon. The black hole apparent horizon shrinks, asymptoting to the spacetime singularity at 2m0 from above as t* → + ∞, while the cosmological apparent horizon expands monotonically, tending to 1/H(t) in the same limit. The well known singularity R = 2m0, where the Ricci scalar a=-8 T=8( -3P )=8(t) ( 4- ) diverges, separates the two disconnected spacetime regions R < 2m0 and R > 2m0 and is spacelike. One can compare the rate of change of the apparent horizon radii with respect to that of the cosmic substratum, obtaining -H= -H( 1+ ) :[eq:rateofexp] the apparent horizons are not comoving except for trivial cases. The sum of the areas of the two apparent horizons of the McVittie spacetime is a non-decreasing function of time but undergoes a discontinuous jump from zero at the critical time t\. ### A phantom background A background FLRW universe dominated by a phantom fluid with equation of state parameter w* ≡ P/ρ < − 1 which violates the weak energy condition can be considered. Phantom fluids, studied in conjunction with the present cosmic acceleration, cause a Big Rip singularity at a finite future trip. A phantom background universe for the McVittie solution was studied in Ref.. The scale factor of a phantom-dominated spatially flat FLRW universe is a(t)=, where A is a constant. The Hubble parameter H(t)= is qualitatively the time-reverse of that of a dust-dominated universe H(t) = 2/(3t). The Hubble parameter for a phantom fluid is finite at t = 0 and increases until the Big Rip, at which it diverges. The apparent horizons around McVittie black holes embedded in a phantom fluid behave in the opposite way to those in a background with w > − 1 (fig. [McV2]). [McV2] The radii of the McVittie apparent horizons (vertical axis) versus time (horizontal axis) in a phantom-dominated universe (here w = − 1.5 and trip = 0). An idealized interior solution for the McVittie metric describing a relativistic star of uniform density in a FLRW background was found by Nolan and it generalizes the Schwarzschild interior solution with a Minkowski background, to which it reduces when a = const. It belongs to the Kustaanheimo family of shear-free solutions. The star surface is comoving with the cosmic substratum. The generalization of the Tolman-Oppenheimer-Volkoff equation for this crude star model was written down in . Recent works on the McVittie spacetime study its conformal structure, which means integrating numerically the null geodesics or deriving general analytical results upon assuming something on the expansion. Lake and Abdelqader find that null geodesics asymptote to the singularity without entering it. Depending on the form of the scale factor, a bifurcation surface may appear which splits the spacetime boundary into a black hole horizon in the future and a white hole horizon in the past. This behaviour seems a reflection of the McVittie no-accretion condition which applies to a timelike dust and, in the limit, also to a null dust. da Silva et al. find that the presence or absence of this white hole horizon depends crucially on the expansion history of the universe a(t), and prove a theorem in this regard for McVittie spacetimes for which the background is non-superaccelerating (i.e., $\dot{H} \leq 0$) and de Sitter at late times. It would be desirable to extend the result to backgrounds which at late times asymptote to any FLRW space, not just de Sitter. See Refs. for the corresponding conformal diagrams. Area quantization and McVittie solutions as toy models ------------------------------------------------------ As an example of the use of cosmological black holes as toy models to exemplify unintuitive physics, we quote the current issue of the quantization of black hole areas. Inspired by certain stringy black holes, there has been excitement in the string community about the fact that the areas A± of black hole inner ( − ) and outer ( + ) horizons satisfy the relation [expression1] A = 8lpl2 ( ), N1, N2, or [expression2] A+ A- =( 8lpl2)2 N, N, where lpl is the Planck length. These area-quantizing relations have somehow come to be seen as universal. While certain stringy black holes remarkably do satisfy these relations, this property is certainly not universal, as shown by Visser using 4-dimensional GR black holes. The McVittie solutions provide further, and even more convincing examples: if eq. ([expression1]) or eq. ([expression2]) is satisfied at an instant of time, it fails at subsequent times due to the dynamical character of the horizons, and realistic black holes are dynamical if nothing else because of Hawking radiation and of quantum fluctuations. Generalized McVittie solutions ------------------------------ Generalized McVittie solutions with spacetime metric of the form ([McVittieisotropic] ) but without the no-accretion restriction ([35]) were introduced in Ref. . In principle such metrics could be meaningless: in the “Synge approach” one can always impose that an invented metric solves the Einstein equations and run these equations from left to right to compute the corresponding formal stress-energy tensor Tab. This Tab is usually found to be completely unphysical and violates all reasonable energy conditions, beginning with the positivity of the energy density. Rather surprisingly, generalized McVittie solutions with reasonable matter sources exist. In isotropic coordinates, generalized McVittie solutions can be presented as $$ds^2= -\frac{B^2\left(t, \bar{r} \right)}{ A^2\left(t, \bar{r} \right)}\, dt^2 +a^2(t) A^4 \left(t, \bar{r} \right) \left( d\bar{r}^2+\bar{r}^2 d\Omega^2\_{(2)} \right) \,,$$ where m(t) ≥ 0 and $$A \left(t, \bar{r} \right) = 1+\frac{m(t)}{2\bar{r}} \,, \;\;\;\;\;\;\; B \left(t, \bar{r} \right) = 1-\frac{m(t)}{2\bar{r}} \,.$$ The only non-vanishing components of the mixed Einstein tensor are $$\begin{aligned} G\_t^t &=& -\, \frac{3A^2}{B^2}\left( \frac{\dot{a}}{a} +\frac{\dot{m}}{\bar{r}A} \right)^2 \,, \label{einst1} \\ &&\nonumber \\ G\_t^{\bar{r}} &=& \frac{2m}{ \bar{r}^2 a^2 A^5 B} \left( \frac{\dot{m}}{m} + \frac{\dot{a}}{a} \right) \,, \\ &&\label{einst2} \nonumber \\ G\_{\bar{r}} ^{\bar{r}} &=& G\_{\theta}^{\theta} =G\_{\varphi}^{\varphi} = - \frac{A^2}{B^2}\left\{ 2 \frac{d}{dt} \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right) + \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right) \right.\nonumber \\ && \nonumber \\ && \left. \cdot \left[ 3 \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right) + \frac{2\dot{m}}{\bar{r}AB} \right]\right\} \label{einst3}\end{aligned}$$ (the unusual feature that Gr̄r̄ = Gθθ is named “spatial Ricci isotropy” in Ref. ). The quantity $$C\equiv \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} = \frac{\dot{M}}{M}-\frac{\dot{m}}{m} \frac{B}{A}$$ appearing in the Einstein tensor reduces to $ \dot{M}/M $ where M(t) m(t)a(t) for the special subclass of solutions with m = constant. This subclass will be called “comoving mass” solutions. On the surface r̄ = m/2, C reduces to $$C\_{\Sigma}= \frac{\dot{a}}{a}+\frac{\dot{m}}{m} = \frac{\dot{M}}{M}$$ for any function m(t). McVittie solutions correspond to CΣ = 0, while comoving mass solutions have C = CΣ = H everywhere. The Ricci scalar $${R^a}\_a = \frac{3A^2}{B^2}\left( 2\dot{C} +4C^2 +\frac{ 2\dot{m}C}{\bar{r}AB} \right)$$ diverges on the surface r̄ = m/2 unless m is a constant. Imperfect fluids can be contemplated as matter sources for this metric. ### Single perfect fluid If the matter source of the generalized McVittie metric is a single perfect fluid with stress energy tensor Tab = (P + ρ)uaub + Pgab and a radial fluid flow described by the fluid four-velocity uμ = (u0, u, 0, 0), is allowed, then the only possible solution of the Einstein equations is the Schwarzschild-de Sitter-Kottler black hole. This is easily seen, since the normalization ucuc = − 1 yields $$u^t=\frac{A}{B} \, \sqrt{ 1+a^2 A^4 u^2}$$ and, using eqs. ([einst1])-([einst3]), the Einstein equations imply that $$\label{delta0} \dot{M}=- B^2 au \left( P+\rho \right) {\cal A} \sqrt{ 1+a^2A^4 u^2} \,,$$ where =dd =4a2 A4 \|r2 is the area of a sphere of isotropic radius r̄ and $$\begin{aligned} && 3\left( \frac{AC}{B} \right)^2=8\pi \left[ \left( P+\rho \right)a^2A^4 u^2 +\rho \right] \,, \\ &&\nonumber \\ && -\left( \frac{A}{B} \right)^2 \left( 2\dot{C}+3C^2 +\frac{2\dot{m}C}{\bar{r}AB} \right)= 8\pi \left[ \left( P+\rho \right)a^2A^4 u^2 +P \right] \,, \label{delta1} \\ && \nonumber \\ && -\left( \frac{A}{B} \right)^2 \left( 2\dot{C}+3C^2 + \frac{2\dot{m}C}{\bar{r}AB} \right)=8\pi P \,. \label{delta2} \end{aligned}$$ Eqs. ([delta1]) and ([delta2]) combined give P = − ρ: only the de Sitter equation of state is allowed and then eq. ([delta0]) implies that $\dot{M}=0$. ### Imperfect fluid and no radial mass flow Consider now the imperfect fluid stress-energy tensor Tab = (P + ρ)uaub + Pgab + qaub + qbua , as a source for the generalized McVittie solutions, where the purely spatial vector qc describes a radial energy flow, $$u^{\mu}=\left( \frac{A}{B}, 0,0,0 \right) \,, \;\;\;\;\; q^{\alpha}=\left( 0, q, 0,0 \right) \;, \;\;\;\;\; q^cu\_c=0 \,,$$ and ucuc = − 1 (in principle one could take qc to be spacelike instead of purely spatial ). The (t, r̄) component of the Einstein equations yields $$\frac{\dot{m}}{m}+\frac{\dot{a}}{a}= -\frac{4\pi G}{m}\, \bar{r}^2 a^2 A^4 B^2 q \,.$$ Furthermore, it is $$\frac{\dot{M}}{M}=\frac{\dot{m}}{m}+\frac{\dot{a}}{a}$$ and the area of a sphere Σ of constant time and constant isotropic radius r̄ is =dd =4a2 A4 \|r2, then energy flow, area ${\cal A}$, and accretion rate are related by $$\label{accretionrate} \dot{M} (t)=- a B^2 {\cal A} q \,.$$ In the case of inflow (q < 0), this condition can be written on a sphere of radius r̄ ≫ m as $\dot{M} \simeq a {\cal A} \left\| q \right\| $; for a 2-sphere, M increases due to the inflow of matter alone (but it receives another contribution from the evolution of the cosmological fluid contained in it). The energy density and pressure obtained from the Einstein equations are $$\begin{aligned} \rho \left( t, \bar{r}\right) &=& \frac{1}{8\pi } \, \frac{3A^2}{B^2} \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right)^2 \,, \label{newdensity} \\ &&\nonumber \\ P \left( t, \bar{r}\right) &=& \frac{- A^2}{8\pi B^2} \left\{ 2 \frac{d}{dt} \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right) + \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right)\left[ 3 \left( \frac{\dot{a}}{a}+\frac{\dot{m}}{\bar{r}A} \right)+\frac{2\dot{m}}{\bar{r}AB} \right]\right\}\,; \label{newpressure}\end{aligned}$$ clearly the energy density is always non-negative. In terms of the quantity C, eq. ([newpressure]) becomes the generalization of the Raychaudhuri equation of FLRW space $$\label{generalizedRaychaudhuri} \dot{C}=-\,\frac{3C^2}{2}-\frac{\dot{m}}{\bar{r}AB}\, C -4\pi \, \frac{B^2}{A^2} \, P \,.$$ It reduces to the usual Raychaudhuri equation of FLRW cosmology in the limit m → 0, =--4P, and then the Hamiltonian constraint H2 = 8πρ/3 yields $$\label{FLRWHdot} \dot{H}=-4\pi \left(P+\rho \right) \,.$$ When m ≠ 0, instead, eq. ([newdensity]) yields the generalization $$\dot{C}=-4\pi \, \frac{B^2}{A^2}\left( P+\rho \right) - \frac{\dot{m}C}{\bar{r}AB} \,.$$ ### Imperfect fluid and radial mass flow Let us consider now an imperfect fluid with stress-energy tensor of the form ([imperfect]) with both radial mass flow and energy current present and of the form $$u^{\mu}=\left( \frac{A}{B}\sqrt{1+a^2A^4u^2}, u, 0, 0 \right)\,, \;\;\;\;\;\;\;\;\; q^{\mu}=\left( 0,q,0,0 \right) \,.$$ By using the components ([einst1])-([einst3]) of the Einstein tensor, the field equations become $$\label{deltadelta1} \dot{M} =- aB^2 {\cal A} \sqrt{1+a^2A^4u^2} \left[ \left( P+\rho \right)u+q \right] \,,$$ $$-3\left( \frac{AC}{B} \right)^2 =-8\pi \left[ \left( P+\rho \right)a^2 A^4 u^2 +\rho \right] \,,$$ $$-\left( \frac{A}{B} \right)^2 \left( 2\dot{C}+3C^2 +\frac{2\dot{m}C}{\bar{r}AB} \right)=8\pi \left[ \left( P+\rho \right)a^2 A^4 u^2 +P+2a^2 A^4 qu \right] \,,$$ $$-\left( \frac{A}{B} \right)^2 \left( 2\dot{C}+3C^2 +\frac{2\dot{m}C}{\bar{r}AB} \right)=8\pi P \,.$$ Adding the last two equations yields $$\label{460} q=-\left(P+\rho \right)\frac{u}{2}$$ (equivalently, this equation can be seen as a consequence of the spatial Ricci isotropy Gr̄r̄ = Gθθ), i.e., to an ingoing radial mass flow there corresponds an outgoing radial heat current if P > − ρ. By substituting eq. ([460]) into eq. ([deltadelta1]), one obtains the accretion rate $$\dot{M} =-\frac{1}{2} aB^2 \sqrt{1+a^2A^4 u^2} \left( P+\rho \right){\cal A}u \,,$$ where $\left( P+\rho \right) {\cal A} u $ can be seen as the flux of gravitating energy through the surface of area ${\cal A}$ (remember that u < 0). The energy density is given by $$8\pi \rho= \frac{A^2}{B^2}\left[ 3C^2 +\left( \dot{C}+\frac{\dot{m}C}{\bar{r}AB} \right) \frac{2 a^2A^4u^2}{1+a^2A^4 u^2} \right] \,.$$ ### The “comoving mass” solution In the class of generalized McVittie solutions of GR, the choice M(t) = m0 a(t) where m0 is a constant, selects a special one which is a late-time attractor within this class. The corresponding line element in isotropic coordinates is [eq:ds-ph] ds2 = - dt2 +a2(t)(1+ )4 (dr2+r2d2(2) ). The apparent horizons of this metric were studied in Ref. by transforming to areal radius. A nice feature of this solution is that the apparent horizons are given analytically by $$\begin{aligned} R\_{c}&=&{\frac{1}{2H}}{\left(1+\sqrt{1-8m\_0\dot{a}} \, \right)} \,, \\ && \nonumber\\ R\_{b}&=&{\frac{1}{2H}}{\left(1-\sqrt{1-8m\_0\dot{a}} \, \right)} \,. \label{Rb}\end{aligned}$$ Rc is a cosmological and Rb is a black hole apparent horizon. The surface r = m0/2 (or r̃ = 2m0, or R = 2m0a = 2M(t)) is a spacetime singularity contained inside the black hole apparent horizon when the latter exists, since Rc, b > 2m0a = 2M. The black hole and cosmological apparent horizons have the qualitative behaviour already discussed for all the McVittie and generalized McVittie solutions. ### The general class of solutions For the wider class of generalized McVittie solutions with arbitrary dependence m(t) ≥ 0, an analysis of the apparent horizons using the areal radius identifies them as the roots of the equation $$\label{addhorizons} HR+\dot{m}a\sqrt{ \frac{\tilde{r}}{r} } =\pm \left( 1-\frac{2M}{R} \right) \,,$$ where r̃ ≡ R/a. Since M(t) = m(t)a(t), the left hand side can be written as $$HR+ M\left( 1+\frac{m}{2r} \right)\left( \frac{\dot{M}}{M}-H \right) \,$$ where the factor $M\left( 1+\frac{m}{2r} \right)$ quantifies the deviation of the radius from 2M (r > m/2 corresponds to R > 2M and to $M\left( 1+\frac{2m}{r} \right) >2M$), while the factor $ \left( \frac{\dot{M}}{M}-H \right) $ is the difference between the percent rate of change of M and that of the scale factor of the substratum. The vanishing of this factor corresponds to an analog of stationary accretion for a time-dependent background. Then, the special solution with M(t) = m0a(t) corresponds to stationary accretion relative to the FLRW background. Eq. ([addhorizons]), which becomes $$\label{addquadratic} HR^2 +\left[ M\left( 1+\frac{m}{2r} \right)\left( \frac{\dot{M}}{M}-H \right)-1 \right]R +2M=0 \,,$$ is not a quadratic algebraic equation, but it can be treated formally as such, providing the formal roots Rc,b= { 1-M( 1+ ) }. Since r = r(R), this is really an implicit equation for the radii Rc, b of the cosmological and black hole apparent horizons. When the argument of the square root is positive there are a cosmological apparent horizon at Rc and a black hole apparent horizon at Rb. When this argument vanishes, these two apparent horizons coincide at $ \sqrt{\frac{2M}{H}}$. If this argument becomes negative, the apparent horizons disappear leaving behind a naked singularity. ### Attractor behaviour of the “comoving mass” solution “Comoving mass” solutions are generic under certain assumptions, in the sense that all other generalized McVittie solutions approach them at late times. In fact, assume that the universe always expands, that m(t) ≥ 0, and that the function m(t) is continuous with its first derivative. Then, using r̃ ≡ R/a, one obtains [200] H + = -( 1+ ) +. Since m ≥ 0 the left-hand side is always non-negative and $\dot{m} \left( 1+\frac{m}{2r}\right)<\frac{1}{a}$. Then, given that $1+\frac{m}{2r}>0$, in an expanding universe in which a → + ∞, one has $ \dot{m}\_{\infty} \equiv \lim\_{t\rightarrow +\infty} \dot{m}(t) \leq 0$. If $\dot{m}\_{\infty}=0$, the quantity m(t) becomes asymptotically comoving. The other possibility is $\dot{m}\_{\infty}<0$. In this case, there is a time t̄ such that ∀ t > t̄ it is $\dot{m}(t)<0$. Then there are only two options: since m(t) ≥ 0, either m(t) reaches the value zero at a finite time t\* with derivative $\dot{m}\_\\equiv \dot{m}(t\_\) < 0$, or else m(t) → m0 = const. with $\dot{m}(t)\rightarrow 0$, i.e., m(t) has a horizontal asymptote. In the first case one has, at t = t\, $ HR =\left\| \dot{m}\_\\right\| a +1 $, which yields the radius of the black hole apparent horizon at t\* r\* rhorizon(t\)= (\|\\| + ). Late in the history of the universe we have a black hole of zero mass M(t\) = a(t\)m(t\) but finite radius r\. As time evolution continues, one would have negative mass M and finite radius of the black hole apparent horizon. This unphysical situation for m(t\) = 0 with m(t* > t\) < 0 is discarded. In the second case $\dot{m}(t) \rightarrow 0$ at late times and t* → + ∞ if the cosmic expansion continues forever or t → trip if a Big Rip occurs at trip. The physical meaning of $\dot{m} \rightarrow 0$ is that, at late times, the rate of increase of the black hole mass is at most the Hubble rate and becomes comoving. The Sultana-Dyer solution ------------------------- The Sultana-Dyer solution of GR is a Petrov type D metric interpreted as a black hole embedded in a spatially flat FLRW universe. This solution was generated by extending a metric resulting from the conformal transformation of the Schwarzschild metric gab(S) → Ω2 gab(S) with conformal factor Ω = a(t) = η2 equal to the scale factor of a dust-filled k = 0 FLRW universe in conformal time η. That is, this spacetime is conformally static and admits a conformal Killing vector ξa (fig. [SultanaDyerReview]). Conformal diagram of the Sultana-Dyer spacetime. [SultanaDyerReview] The authors of aimed at changing the Schwarzschild timelike Killing field ξc into a conformal Killing field defined for ξc∇cΩ ≠ 0, thus generating a conformal Killing horizon (which, however, seems of little relevance in modern studies of time-evolving horizons). The Sultana-Dyer metric is $$\label{SultanaDyeroriginal} ds^2=a^2(\eta) \left[ - \left(1-\frac{2m\_0}{r} \right)d\eta^2 +\frac{4m\_0}{r} \, d\eta dr + \left(1+\frac{2m\_0}{r} \right)dr^2 + r^2 d\Omega^2\_{(2)} \right] \,,$$ where m0 is a constant and a(η) = η2. The coordinate transformation (t, r) =t+2m0 \| -1 \|, turns the line element into the form $$ds^2=a^2(t, r) \left[ - \left(1-\frac{2m\_0}{r}\right) dt^2 + \frac{dr^2}{ 1 - \frac{2m\_0}{r} } + r^2 d\Omega^2\_{(2)} \right] \,,$$ which is explicitly conformal to the Schwarzschild metric with conformal factor = a(t, r)=2 (t,r)= ( t+2m0 \| -1 \| )2. The matter source of the Sultana-Dyer spacetime is a mixture of two non-interacting perfect fluids with stress-energy tensor Tab= Tab(I)+Tab(II), where Tab(I) = ρua ub describes an ordinary dust with timelike 4-velocity uc and Tab(II) = ρn ka kb describes a null dust with density ρn and kckc = 0. A problem of this solution is that the cosmological fluid becomes tachyonic with negative energy density at late times near r̄ = m0/2. Let us use, in the rest of this subsection, the quantity [bigM] M(\|t) m0 a( \|t), which is not constant in the Sultana-Dyer solution. The locus r = 2m0 is not a singularity, but the conformal factor Ω vanishes there. The metric ([SultanaDyeroriginal]), however, is not singular there. The Ricci curvature is a= ( 1- + ), [ssscurvature] and is not singular at r = 2m0 (where η → − ∞) but is singular at r = 0 (central singularity) and for η = 0 (Big Bang singularity). The problem of Hawking emission from the Sultana-Dyer black hole was approached in Ref.. These authors considered quantum radiation from a massless conformally coupled scalar field ϕ and computed the renormalized stress-energy tensor ⟨Tab⟩ of ϕ taking advantage of the simplifications introduced by the fact that the Sultana-Dyer spacetime is conformal to the Schwarzschild one and taking into account the conformal anomaly and particle creation by the FLRW background. Discarding complicated corrections which are small if the black hole is evolving slowly, the effective Hawking temperature from the Sultana-Dyer black hole was computed as Teff= =, where TSchw = (8πm0)− 1 is the Hawking temperature of the Schwarzschild black hole which seeds the Sultana-Dyer metric. The more general relation [Tconformal] T= for spacetimes conformally related to the Schwarzschild spacetime by a trasformation with conformal factor Ω is conjectured in. Independent support for eq. ([Tconformal]) comes from dimensional considerations related to the use of conformal transformations. The Husain-Martinez-Nuñez solution ---------------------------------- In the 1994 Husain-Martinez-Nuñez solution of GR a new phenomenology of the apparent horizons appears. This spacetime describes an inhomogeneous universe with a spatially flat FLRW background sourced by a free, minimally coupled, scalar field. The coupled Einstein-Klein-Gordon equations reduce to $$\begin{aligned} && R\_{ab}=8\pi \nabla\_a \phi \nabla\_b \phi \,. \,, \label{HMNfieldeq1}\\ &&\nonumber\\ && \Box \phi=0 \,,\end{aligned}$$ and the Husain-Martinez-Nuñez solution to them is $$\begin{aligned} ds^2 &=& \left( A\_0 \eta +B\_0 \right) \left[ - \left( 1-\frac{2C}{r}\right)^{\alpha} d\eta^2 +\frac{dr^2}{\left( 1-\frac{2C}{r}\right)^{\alpha} } \right. \nonumber\\ &&\nonumber\\ &\, & \left. + r^2 \left( 1-\frac{2C}{r}\right)^{1-\alpha} d\Omega\_{(2)}^2 \right] \,, \label{HMNconformaltime}\\ &&\nonumber\\ \phi(\eta, r ) &=& \pm \frac{1}{4\sqrt{\pi}} \, \ln \left[ D\left( 1-\frac{2C}{r}\right)^{\alpha/\sqrt{3}} \left( A\_0 \eta +B\_0 \right)^{\sqrt{3}} \right] \,,\label{HMNscalar}\end{aligned}$$ where A0, B0, C, and D are non-negative constants, $\alpha =\pm \sqrt{3}/2$, and η > 0. The additive constant B0 becomes irrelevant and can be dropped whenever A0 ≠ 0. When A0 = 0, the Husain-Martinez-Nuñez metric degenerates into the static Fisher spacetime [Fisher] ds2=-V(r) d2 + +r2 V1-(r) d(2)2, where V(r) = 1 − 2μ/r, μ and ν are parameters, and the Fisher scalar field is (r)=0 V(r). The Fisher solution of the coupled Einstein-Klein-Gordon equations, also referred to as the Janis-Newman-Winicour-Wyman solution, has been rediscovered many times. Its features are a naked singularity at r = 2C and its asymptotic flatness. It is claimed that this solution is the most general static and spherically symmetric solution of the Einstein equations with zero cosmological constant and a massless, minimally coupled, scalar field, but it is unstable. The general Husain-Martinez-Nuñez metric is conformal to the Fisher metric with conformal factor $\Omega=\sqrt{A\_0 \eta+B\_0}$ equal to the scale factor of the background FLRW space and with only two possible values of the parameter ν. From now on, we set the constant B0 to zero by labelling the Big Bang by η = 0. The sign in eq. ([HMNscalar]) is not associated with the sign of α. The full metric is asymptotically FLRW for r → + ∞ and is FLRW if C = 0 (in which case the constant A0 can be eliminated by rescaling the time coordinate η). The Ricci scalar a = 8cc = -, immediately identifies a spacetime singularity at r = 2C (for both values of the parameter α). The scalar ϕ also diverges there, and a Big Bang singularity is present at η = 0. Only the coordinate range 2C < r < + ∞ is physical and the lower limit r = 2C corresponds to zero areal radius [HMNarealradius] R(, r)= r ( 1-). Let us introduce the comoving time t defined by dt = adη (where $a(\eta)=\sqrt{A\_0\eta}$ is the FLRW scale factor) in place of the conformal time η, then it is t=d a()= 3/2 by choosing η = 0 at t = 0, or =( t )2/3 and a(t)== a0 t1/3, a0=( )1/3. This power law for the scale factor is consistent with the stiff equation of state P = ρ/3 of a free massless scalar field in a FLRW universe and with the general solution $ a(t)= \mbox{const.} \, t^{\frac{2}{3(w+1)}} $ (where w ≡ P/ρ). The Husain-Martinez-Nuñez solution in comoving time reads ds2 = - ( 1-) dt2 +a2(t) [HMNcomoving] with ( t, r ) =. The areal radius ([HMNarealradius]) increases with r for r > 2C. It is useful to rewrite the line element in terms of the areal radius R. By setting A(r) 1-, B(r) 1-, we have $R(r)=a(t)r A^{\frac{1-\alpha}{2}}(r) $ and dr=. The metric is then $$\begin{aligned} ds^2&=& -A^{\alpha} \left[ 1- \frac{ H^2R^2 A^{2(1-\alpha)} }{B^2(r)} \right] dt^2 +\frac{H^2R^2A^{2-\alpha}(r) }{B^2(r)}\, dR^2 \nonumber\\ &&\nonumber\\ &\, & - \, \frac{2HR A^{\frac{3-\alpha}{2}}}{B^2(r)} \, dt \, dR +R^2 d\Omega\_{(2)}^2 \,.\label{HSMcross}\end{aligned}$$ The time-radius cross-term is eliminated by introducing a new time T with differential dT= ( dt+dR ), where β(t, R) is a function to be determined and F(t, R) is an integrating factor which must satisfy ( )= ( ) in order for dT to be an exact differential. Using dt = FdT − βdR in eq. ([HSMcross]) and choosing (t,R)=, the line element becomes $$\begin{aligned} ds^2 &=&- A^{\alpha}(r) \left[ 1- \frac{H^2R^2 A^{2(1-\alpha)}(r) }{B^2(r)} \right] F^2dt^2 \nonumber\\ &&\nonumber\\ &\, & +\frac{H^2 R^2 A^{2-\alpha}(r)}{B^2(r)} \left[ 1+ \frac{ A^{1-\alpha}(r)} { B^2(r) -H^2R^2 A^{2(1-\alpha)}(r)} \right] dR^2 +R^2 d\Omega\_{(2)}^2 \,.\nonumber\\ &&\end{aligned}$$ The apparent horizons, located by gRR = 0, must satisfy [HMNAH] B(r)=H(t)RA1-(r), where now r = r(t, R), or = ( 1- )-1 [HMNoriginalAH] using the original coordinates (η, r). For r → + ∞ (corresponding to R → + ∞), this equation reduces to R ≃ H− 1, the radius of the cosmological apparent horizon in spatially flat FLRW space. Eq. ([HMNAH]) must be solved numerically. Let x ≡ C/r, then the equation locating the apparent horizons is [TeKaPo] HR=( 1- )-1. The left hand side can be written as HR= ( 1-2x ), which expresses the radius of the apparent horizons in units of H− 1 (the radius of the cosmological apparent horizon of the FLRW background if it did not have the central inhomogeneity). The right hand side is [1 − (α + 1)x](1 − 2x)α − 1. Eq. ([TeKaPo]) and the equation defining the areal radius give $$\begin{aligned} t(x) &=& \left\{ \frac{2Ca\_0}{3} \, \frac{ (1-2x)^{3(1-\alpha)}}{x\left[ 1-(\alpha+1)x\right]} \right\}^{3/2} \,,\label{HMNparametricR}\\ &&\nonumber\\ R(x) &=& a\_0 \, t^{1/3}(x) \, \frac{2C}{x}\left( 1-2x \right)^{ \frac{1-\alpha}{2}} \,.\end{aligned}$$ This is a parametric representation of the function R(t) and can be used to plot this function. The result is illustrated in figs. [HMNfigure1] and [HMNfigure2]. The radii of the apparent horizons of the Husain-Martinez-Nuñez spacetime (vertical axis) versus comoving time (horizontal axis) for $\alpha=\sqrt{3}/2$ (t and R are measured in arbitrary units of length and the parameter values are chosen so that (Ca0)3/2 = 103 in eq. ([HMNparametricR])). [HMNfigure1] If $\alpha=\sqrt{3}/2$, between the Big Bang and a critical time t\* there is only one expanding apparent horizon, then two other apparent horizons are created at t\. One is a cosmological apparent horizon which expands forever and the other is a black hole horizon which contracts until it meets the first (expanding) black hole apparent horizon. When they meet, these two annihilate and a naked singularity appears at R* = 0 in a FLRW universe. This phenomenology of apparent horizons differs from that of the McVittie and generalized McVittie solutions. The “S-curve” phenomenology of fig. [HMNfigure1] appears also in Lemaître-Tolman-Bondi spacetimes already for a dust fluid much simpler than a scalar field (multiple “S”s are possible, for example see fig. 9 of Ref. ) and in analytical solutions of Brans-Dicke and f(Rcc) gravity. The scalar field is regular on the apparent horizons. The radius of the Husain-Martinez-Nuñez apparent horizon (vertical axis) versus comoving time (horizontal axis) for $\alpha=-\sqrt{3}/2$. There is always only one, expanding, cosmological apparent horizon and there is a naked singularity at R = 0. [HMNfigure2] For $\alpha=-\sqrt{3}/2$ there is only one, forever expanding, cosmological apparent horizon and the universe contains a naked singularity at R = 0 (fig. [HMNfigure2]), with the usual Big Bang singularity at t = 0. The apparent horizons are spacelike, as can be seen by studying the normal vector to these surfaces and checking that it always lies inside the light cone in an (η, r) diagram. Eq. ([HMNoriginalAH]) yields = along the apparent horizons. Differentiate this relation with respect to R to obtain, r \|AH = ( 1- )- { 1- }. Along radial null geodesics it is, r\|light cone = ( 1- )-, which follows from ds2 = 0 with dθ = dφ = 0. Therefore, it is \| \| =1- 1 and the normal to the apparent horizons is always pointing inside the light cone, except at the spacetime points at which this vector becomes tangent to the light cone and is null, which occurs when a pair of apparent horizons is created or destroyed. This occurrence is in agreement with a general result of Ref. stating that a trapping horizon created by a massless scalar field must be spacelike (however, even simple potentials V(ϕ) can make the trapping horizon be non-spacelike). The nature of the singularity at r = 2C (or R = 0) is easily established: all surfaces R = const. have equation Φ(R) ≡ R − const. = 0 and gradient Nμ ≡ ∇μΦ = δμR in coordinates (t, R, θ, φ). The norm squared is Nc Nc= gRR= and, because $B(r) \rightarrow \frac{1-\alpha}{2}$ and A(r) → 0+ as r → 2C+, it is NcNc > 0 and NcNc → + ∞ as r → 2C+. The singularity at R = 0 is timelike for both values of the parameter α. For $\alpha=+\sqrt{3}/2$, the Husain-Martinez-Nuñez spacetime is interpreted as describing the creation and annihilation of pairs of black hole apparent horizons. The central singularity at R = 0 is created with the universe in the Big Bang and does not result from a collapse process (this is also the case for $\alpha=-\sqrt{3}/2$). The Fonarev and generalized Fonarev solutions --------------------------------------------- The Fonarev solution of the Einstein equations with a minimally coupled scalar field in an exponential potential as the matter source generalizes the Husain-Martinez-Nuñez solution. It describes a central inhomogeneity embedded in a scalar field FLRW universe. The action is $$\begin{aligned} S=\frac{1}{2\kappa} \int d^4x\sqrt{-g}\left[ {R^a}\_a -\frac{1}{2} \nabla\_{a}\phi\nabla^{a}\phi -V\left(\phi \right)\right] \,,\end{aligned}$$ where κ ≡ 8πG and $$\begin{aligned} V\left(\phi\right)=V\_0 \, \mbox{e}^{-\lambda \phi} \,,\end{aligned}$$ and V0 and λ are two positive constants (this potential has been investigated in great detail in cosmology). The coupled Einstein-Klein-Gordon equations simplify to $$\begin{aligned} && R\_{ab}=8\pi \left( \nabla\_a\phi \nabla\_b \phi +g\_{ab}V \right) \,, \label{FonarevFieldEq1}\\ &&\nonumber\\ &&\Box \phi-\frac{d V}{d\phi}=0 \,.\end{aligned}$$ The spherically symmetric Fonarev line element and scalar field are $$\begin{aligned} \label{eq:fonarev} ds^2 & = & a^2 \left(\eta\right) \left[ -f^2\left(r\right) d\eta^2+\frac{dr^2}{f^2\left(r\right)} +S^2 \left(r\right) d\Omega^2\_{(2)} \right] \,,\\ &&\nonumber \\ \phi \left( \eta, r \right) &=& \frac{1}{\sqrt{\lambda^2+2}}\ln \left(1-\frac{2w}{r}\right) +\lambda\ln a + \frac{1}{\lambda} \ln \left[ \frac{V\_0\left(\lambda^2-2\right)^2 }{2A\_0^2 \left(6-\lambda^2\right)} \right] \,,\nonumber\\ &&\end{aligned}$$ where $$\begin{aligned} f(r) &=&\left(1-\frac{2w}{r}\right)^{\frac{\alpha}{2}} \,,\ \ \ \ \alpha =\frac{\lambda}{\sqrt{\lambda^2+2}} \,, \\ S(r) & = & r \left(1-\frac{2w}{r}\right)^{\frac{1-\alpha}{2} } \,, \ \ \ a(\eta) =A\_0 \|\eta\|^{\frac{2}{\lambda^2-2}} \,,\end{aligned}$$ with w and A0 constants and η is the conformal time. For simplicity we choose A0 = 1. When w = 0 the metric ([eq:fonarev]) reduces to a spatially flat FLRW one while, when a ≡ 1 and α = 1, it degenerates into the Schwarzschild solution (however, the value α = 1 is not possible if $\alpha=\frac{\lambda}{\sqrt{\lambda^2+2}}$). The line element becomes asymptotically that of spatially flat FLRW space as r → + ∞. The Husain-Martinez-Nuñez class of solutions ([HMNconformaltime]) is recovered by setting $\lambda=\pm \sqrt{6}$ and V0 = 0. See Refs. for the corresponding conformal diagrams. ### A generalized Fonarev solution A generalized Fonarev solution corresponding to a dynamical phantom scalar field solution of GR is known. It is obtained from the Fonarev solution via the transformation $$\begin{aligned} \phi\rightarrow i\phi \,, \ \ \ \ \lambda\rightarrow -i\lambda \,.\end{aligned}$$ The corresponding action is $$\begin{aligned} S=\frac{1}{2\kappa} \int d^4x\sqrt{-g}\left[ {R^a}\_a +\frac{1}{2}\nabla\_a \phi\nabla^a\phi- V\left(\phi \right)\right] \end{aligned}$$ and it contains a phantom field endowed with the “wrong” sign of the kinetic term. The generalized Fonarev line element representing a dynamical black hole immersed in a phantom FLRW background is $$\begin{aligned} ds^2&=& a^2 \left(\eta\right) \left[ -f^2\left(r\right) d\eta^2 +\frac{dr^2 }{f\left(r\right)^2} +S^2 \left(r\right) d\Omega^2\_{(2)}\right] \,,\\ &&\nonumber \\ \phi \left( \eta, r \right)&=&\frac{1}{\lambda} \ln \left[ \frac{V\_0\left(\lambda^2 +2\right)^2}{2\left(\lambda^2+6\right)} \right] -\lambda\ln a -\frac{1}{\sqrt{\lambda^2-2}}\ln \left(1-\frac{2w}{r}\right) \,,\nonumber\\ &&\end{aligned}$$ where $$\begin{aligned} f(r) &=&\left(1-\frac{2w}{r}\right)^{\alpha/2} \,,\ \ \ \ \alpha=-\frac{\lambda}{\sqrt{\lambda^2-2}} \,, \label{xinoceros}\\ S(r) & = & r \left(1-\frac{2w}{r}\right)^{\frac{1-\alpha}{2}} \,, \ \ \ a(\eta) =\eta^{-\frac{2}{\lambda^2+2}} \,.\end{aligned}$$ Assuming that $\lambda>\sqrt{2}$, it is of interest to understand the physical meaning of the constant w. When $\lambda \gg \sqrt{2}$ it is a ≈ 1 and α ≈ − 1 and the metric approximates to [eqq:metric] ds2 -(1-)-1 d2+(1-)dr2 +r2(1-)2d2(2). The coordinate transformation $$\begin{aligned} \label{eq:coordi}y=r\left(1-\frac{2w}{r}\right) \,,\end{aligned}$$ transforms the line element ([eqq:metric]) into [eq:metricsch] ds2 = -(1+)d2+ (1+)-1dy2 +y2d2(2) ; this is the Schwarzschild spacetime with mass − w. The parameter w corresponds to the negative of the mass in this limit and from now on we will use − M instead of w. Let us locate the apparent horizons as the parameters M and α vary. This phantom black hole solution can be cast in the form $$\begin{aligned} ds^2&=&\frac{1}{\eta^{\frac{2\alpha^2-2}{2\alpha^2-1}}} \left[-\left(1+\frac{2M}{r}\right)^{\alpha}d\eta^2 +\left(1+\frac{2M}{r}\right)^{-\alpha}dr^2 \right. \nonumber\\ &&\nonumber\\ &\, & \left.+r^2\left( 1+\frac{2M}{r}\right)^{1+\alpha}d\Omega^2\_{(2)}\right] \,;\end{aligned}$$ the replacement of the conformal time η with the comoving time t leads to $$\begin{aligned} \label{eq:cosmic time} ds^2&=&-\left(1+\frac{2M}{r}\right) ^{\alpha}dt^2 \nonumber\\ &&\nonumber\\ &\, & +a^2 \left(t\right) \left[ \left(1+\frac{2M}{r}\right)^{-\alpha}dr^2 +r^2\left(1+\frac{2M}{r}\right)^{1 +\alpha}d\Omega^2\_{(2)}\right] \,,\\ &&\nonumber \\ a\left(t\right)&=& \left(t\_0-t\right)^{ -\frac{2\left( \alpha^2-1\right)}{\alpha^2}} \,,\nonumber\\ && \end{aligned}$$ where the integration constant t0 marks the time of the Big Rip and it is α < − 1 since $\lambda>\sqrt{2}$. The exponent α is determined by the slope of the potential according to eq. ([xinoceros]). When M = 0 the spacetime ([eq:cosmic time]) reduces to a phantom-dominated FLRW cosmos. By setting, for simplicity, α = − 3 or λ = 3/2, the line element ([eq:cosmic time]) reduces to $$\begin{aligned} ds^2&=&-\left(1+\frac{2M}{r}\right) ^{-3}dt^2 \nonumber\\ &&\nonumber\\ &\, & +a^2 \left(t\right) \left[\left( 1+\frac{2M}{r}\right)^{3}dr^2 +r^2\left(1+ \frac{2M}{r}\right)^{-2}d\Omega^2\_{(2)} \right] \,,\nonumber\\ &&\nonumber \\ a\left(t\right) &=&\left(t\_0-t\right)^{-16/9} \,.\end{aligned}$$ In terms of the areal radius R = ar(1 + 2M/r)− 1, the equation locating the apparent horizons is 1+( 1+ )-1 -( 1+ )5=0, where $H\equiv \dot{a}/a$ is the Hubble parameter of the background. Further setting $x\equiv 1+\sqrt{1+\frac{8Ma}{R}}$ yields aMHx4 − 4x2 + 12x − 8 = 0 . This quartic equation has only two real positive roots corresponding to a cosmological apparent horizon Rc and a black hole apparent horizon Rb. The qualitative behaviour of the apparent horizons is the same as that of the McVittie and generalized McVittie classes of solutions with a phantom FLRW substratum: a black hole apparent horizon inflates while a cosmological apparent horizon shrinks. At a critical time these two apparent horizons meet and disappear leaving behind a naked singularity. The Swiss-Cheese model ---------------------- In 1945, apparently unaware of McVittie’s work from a decade earlier, Einstein and Straus constructed the solution of GR now called “Einstein-Straus vacuole” or “Swiss-cheese model” by pasting a Schwarzschild-like region of spacetime onto a dust-dominated FLRW universe across a timelike hypersurface (for reviews see, which we partially follow here). There is a black hole event horizon in this spacetime and the usual energy conditions are satisfied. Let the interior Schwarzschild region be denoted with ${\cal M}^{-}$ and the exterior FLRW region with ${\cal M}^+$ and let Σ be a spacelike 2-sphere of constant comoving radius rΣ. The coordinate charts covering Σ are the FLRW {t, θ, φ} and the Schwarzschild chart {T(t), θ, φ}. The metric in the two regions is given by $$\begin{aligned} ds^2\_{(-)} & = & -\left( 1-\frac{2m}{R} \right)dT^2 +\frac{dR^2}{1-\frac{2m}{R}} +R^2 d\Omega\_{(2)}^2 \,, \nonumber\\ &&\nonumber\\ ds^2\_{(+)} & = & -dt^2 +a^2(t) \left( \frac{dr^2}{1-kr^2} +r^2 d\Omega\_{(2)}^2 \right) \,.\end{aligned}$$ Choose on Σ the triad of orthonormal vectors { e(t), e( ), e( ) }= { t,, }, where α, β = r, θ, φ. The equation of Σ is Φ(r) ≡ r − rΣ = 0 and the gradient of Φ is Na ≡ ∇aΦ = δar, with norm squared $N\_a N^a= g^{rr}=\frac{1-kr^2}{a^2} $. The unit normal to Σ, therefore, has components n==( 0,, 0, 0 ). The extrinsic curvature of Σ is given by the usual formula K= e(a)e(b) anb which is used to compute Kαβ in ${\cal M}^{-}$ and ${\cal M}^{+}$. The continuity of the first and second fundamental forms on Σ requires $$\begin{aligned} && R\_{\Sigma}(t)=a(t) r\_{\Sigma} \,,\\ &&\nonumber\\ && \left( 1-\frac{2m}{R\_{\Sigma}} \right) \left( \frac{dT}{dt} \right)^2 - \left( \frac{dR}{dt} \right)^2 \frac{1}{1-\frac{2m}{R\_{\Sigma}} }=1 \,,\\\end{aligned}$$ the combination of which yields =( 1- )-1. Using the Hamiltonian constraint of FLRW space $H^2=\frac{8\pi}{3} \, \rho -\frac{k}{a^2}$, one obtains 1- +H2 R2 = 1-kr2 + ( R2 - ). [mmmmmatch] In the absence of surface distributions of mass-energy on Σ, the stress-energy tensor of the matter source of this solution of the Einstein equations must also be continuous across Σ. Since the interior is vacuum, the pressure on the outside is forced to vanish, P( + ) = P( − ) = 0, which implies that only a dust-dominated FLRW background can match the Schwarzschild solution. Moreover, the energy density must be continuous at Σ, implying that =, which means that the mass of the black hole inside the vacuole must equal the mass that a sphere of volume 4πRΣ3/3 would have in the FLRW background (note that this volume is not the proper volume of such a sphere unless the FLRW curvature index k vanishes). This condition yields $ \frac{8\pi }{3} \, \rho R\_{\Sigma}^2 = \frac{2m}{ar\_{\Sigma}} $. Eq. ([mmmmmatch]) then reduces to 1- +H2 R2 = 1-kr2. The continuity of the matter distribution across Σ can be seen as the continuity of the Misner-Sharp-Hernandez mass MMSH( + ) = MMSH( − ) (see for a detailed discussion). The Einstein-Straus model has no accretion onto the central inhomogeneity. The interior Schwarzschild region is shielded from the cosmological expansion (and also the exterior FLRW region sees no effect from the central hole) and is static and, because of this fact, the Swiss-cheese model is often used as supporting evidence that the cosmological expansion does not affect local systems. However, the boundary of the vacuole is expanding and perfectly comoving; if the vacuole is regarded as the “local object” (instead of the black hole in it, which is insulated by a vacuum region), this argument fails. The Einstein-Straus vacuole has few drawbacks: it is unable to describe the Solar System and is unstable with respect to non-spherical perturbations and to perturbations of the matching condition MMSH( + ) = MMSH( − ). The Einstein-Straus vacuole was generalized to include a cosmological constant, obtaining a Schwarschild-(anti-)de Sitter instead of Schwarzschild interior, or to include a fluid with pressure in the interior region. Also the generalization obtained by matching a Schwarschild interior with an inhomogeneous Lemaître-Tolman-Bondi exterior has been studied. The Hawking radiation emitted by the Einstein-Straus black hole has been studied in. It is found that such a black hole in an expanding universe is excited to a non-equilibrium state and emits with stronger intensity than a thermal one. Other GR solutions ------------------ There are several other analytical solutions of the Einstein equations describing central inhomogeneities in FLRW backgrounds. While one has to be careful as many of them do not have reasonable matter sources, they are of some interest. They cannot be included here due to space limitations (for a more general and rigorous treatment of inhomogeneous cosmologies see the book by Krasiński ). They include, among others, the well known Lemaître-Tolman-Bondi and Szekeres solutions (e.g., ), the Oppenheimer-Snyder solution, members of the large Barnes family, the solutions of Dyer, McClure, and collaborators, the Roberts solution with a scalar field, Patel and Trivedi’s and Vaidya’s Kerr-FLRW solutions, Balbinot’s evaporating black hole, and other solutions can be obtained from the previous ones with cut-and-paste techniques, possibly to excise regions in which the energy conditions are violated. Also asymptotically flat metrics describing transient and time-dependent horizons have been studied and other solutions are of potential interest. Some cosmological black holes and naked singularities in alternative gravity ============================================================================ Few solutions of theories of gravity alternative to GR and representing cosmological black holes at least part of the time are known, most of them in scalar-tensor gravity. Here we review a few. The simplest scalar-tensor theory, Brans-Dicke gravity, is described by the action SBD=d4x, where ${\cal L}^{(m)}$ is the matter Lagrangian, ϕ is the Brans-Dicke scalar field (roughly speaking, the inverse of the effective gravitational coupling strength), and ω is a parameter (“Brans-Dicke coupling”). In more general scalar-tensor theories, the Brans-Dicke coupling is promoted to a function of ϕ, ω = ω(ϕ). The conformal cousin of the Husain-Martinez-Nuñez solution ---------------------------------------------------------- A solution of Brans-Dicke gravity was generated, but not interpreted, by Clifton, Mota, and Barrow by conformally transforming the Husain-Martinez-Nuñez solution, gμν(HMN) → Ω2 gμν(HMN) = ϕgμν(HMN) with $ \phi \longrightarrow \tilde{\phi}=\sqrt{\frac{2\omega+3}{16\pi}} \, \ln \phi $. This is the inverse of the usual transformation from the Jordan frame to the Einstein frame which turns gravity with a scalar field non-minimally coupled to the Ricci scalar into GR with a scalar field with canonical kinetic energy but non-minimally coupled to matter. The Clifton-Mota-Barrow solution is $$\begin{aligned} ds^2 &=& -A^{\alpha \left( 1-\frac{1}{\sqrt{3}\, \beta}\right)} (r) \, dt^2 \\ &\,& +A^{-\alpha \left( 1+\frac{1}{\sqrt{3}\, \beta}\right)} (r) \, t^{\frac{2 \left( \beta-\sqrt{3}\right)}{3\beta - \sqrt{3}} } \left[ dr^2 + r^2 A(r) d\Omega\_{(2)}^2 \right] \,,\\ &&\nonumber\\ \phi( t,r) &=& A^{\frac{\pm 1}{2\beta}} (r)\, t^{\frac{2}{\sqrt{3}\, \beta -1}} \,, \end{aligned}$$ where A(r) = 1-, =, >-3/2, =/2. There are singularities at r = 2C and at t = 0 (here it must be 2C < r < + ∞ and t > 0). The scale factor of the spatially flat FLRW background is a(t)= tt. The solution was interpreted in Ref. . We rewrite the 2-parameter line element as ds2=-A(r) dt2 +A(r) a2(t)dr2 +R2(t,r)d(2)2, [lineelementq] where = ( 1- ), = -( 1+ ), and R(t,r)=A(r) a(t) r is the areal radius. It is useful to study the area of the 2-spheres of symmetry: we have $ \partial R/\partial r = a(t)A^{\frac{\Theta-1}{2}}(r) \left( 1- r\_0/r \right) $ where r0 = (1 − Θ)C or R0(t)=( ) (1-) a(t) C. The critical value r0 lies in the physical spacetime region r0 > 2C if Θ < − 1. R has the limit R(t, r)= + r2C+ For Θ < − 1, the areal radius R(r) has a minimum at r0, the area 4πR2 of the 2-spheres of symmetry is minimum there, and there is a wormhole throat joining two spacetime regions. Since = ( 1+ ) for $\alpha=\pm \sqrt{3}/2$, the condition Θ < − 1 requires $\alpha =+\sqrt{3}/2$ (this is a necessary but not sufficient condition for the throat to exist). The sufficient condition Θ < − 1 constrains the Brans-Dicke parameter as < 5.46 0. For − 3/2 < ω < ω0 the solution can be interpreted as a cosmological Brans-Dicke wormhole. The region 2C < r < r0 is not a FLRW region and the scalar field is finite and non-zero at r0: (t, r0 )= t ( ). The wormhole throat is exactly comoving with the cosmic substratum, which is relevant for the problem of cosmic expansion versus local systems and disappears in the limit C → 0. Let us study the existence and location of the apparent horizons of this spacetime. The relation between differentials dr=, turns the line element into ds2 = -Adt2 ++R2 d(2)2, where D1(r)=A(r) 2. Starightforward manipulations yield ds2 = - dt2 - dtdR + + R2 d(2)2 where $H \equiv \dot{a}/a$. The inverse metric in coordinates (t, R, θ, φ) is ( g )=( cccc - & - & 0 & 0 &&& - & & 0 & 0 &&& 0 & 0 & R-2 & 0 &&& 0 & 0 & 0 &R-2-2 ). The apparent horizons are located by the roots of *g	arxiv_0000537
Weak vorticity formulation for the incompressible 2D Euler equations in domains with boundary ============================================================================================= In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data. We formulate our results for flows outside a finite number of smooth obstacles. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulations around connected components of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Our first result is existence of a solution for the weak velocity formulation with vortex sheet initial data for flow outside a finite number of smooth obstacles. The proof is a straightforward adaptation of Delort’s original existence result and requires the usual sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Next, we focus on weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove that the circulations around each boundary component cannot be smaller than the initial data circulation, so that nonnegative vorticity may be absorbed by the boundary, but not produced by the boundary. In addition, we prove that if the weak solution conserves circulation at the boundary components it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation at the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. In addition, we discuss the definition of the mechanical force which the flow exerts on material boundary components and its relation with conservation of circulation. Finally, we describe the corresponding results for a bounded domain with holes, and the adaptations required in the proofs. Introduction ============ For two-dimensional incompressible fluid flow, a vortex sheet is a curve where the velocity of the fluid has a tangential discontinuity. Vortex sheets are an idealized model of a thin region where the fluid is subjected to intense, strongly localized shear. Flows with vortex sheets are of critical physical interest in fluid mechanics for several reasons, specially because such flows are common in situations of practical interest, such as in the wake of an airfoil. Thus, the mathematical description of vortex sheet motion is a classical topic in fluid dynamics. In 1990, Delort proved global-in-time existence of weak solutions for the incompressible Euler equations having, as initial vorticity, a compactly supported, bounded Radon measure with distinguished sign in H− 1, plus an arbitrary, compactly supported, Lp function, with p > 1, see. This includes a large class of examples in which the initial vorticity is actually supported on a curve (classical vortex sheets). This result was later extended to certain symmetric configurations of vorticity with sign change, see. Very little is known regarding Delort’s weak solutions beyond their existence. Some interesting open questions are the conservation of kinetic energy, conservation of the total variation of vorticity, and the behavior of the support of vorticity. The original work by Delort included flows in the full plane, in bounded domains and in compact manifolds without boundary. The proof was based on a compensated-compactness argument for certain quadratic expressions in the components of velocity in order to pass to the limit in the weak formulation of the momentum equations along an approximate solution sequence obtained by mollifying initial data and exactly solving the equations. There is a large literature directly associated with Delort’s Theorem. The convergence to a weak solution was extended to approximations obtained by vanishing viscosity, see, numerical approximations, see and Euler-α, see. The initial data class was extended to the limiting case p = 1, see, and an alternative proof using harmonic analysis was produced, see. In 1995, S. Schochet presented a simplified proof of the full-plane case by introducing the weak vorticity formulation of the Euler equations, where the compensated compactness argument at the heart of the original result becomes an elementary algebraic trick, see. This “weak vorticity formulation” of the vortex sheet initial data problem is the focus of the present work. Originally, this weak formulation of the vortex sheet initial data problem was of interest for the simplification it provided of Delort’s original argument, for enabling the extensions to some symmetric, sign-changing initial data in and for the general physical relevance of the vortex dynamics point of view in incompressible fluid dynamics. Recently, with the discovery and rapid development of the theory of wild solutions of the Euler equations by De Lellis and Szekelyhidi, see and references therein, it became clear that the weak form of the momentum formulation of the Euler equations is severely incomplete. However, the extension of the theory of wild solutions to weak vorticity formulations of the Euler equations is an important open problem, which suggests that there may be additional information encoded on the vortex dynamics which would be very interesting to uncover. In the present article, we are interested in ideal flows in domains with boundary. We observe first that such flows, in themselves, are unphysical. Indeed, all ideal flows are, in fact, slightly viscous, and it has been known since Prandtl, see, that, in the presence of rigid boundaries, ideal flows and slightly viscous flows behave very differently. However, the mathematical study of ideal flows in domains with boundary has some physical relevance, first because the inviscid flow serves as a natural scaffolding for constructing slightly viscous flow perturbatively, and, second, because the vanishing viscosity limit for incompressible flows in a domain with boundary is an important open problem, which could, in principle, lead to some very irregular solutions of the inviscid equations. We refer the reader to for a broad discussion of this problem and its relation to turbulence modelling. Perhaps the main qualitative feature of the way in which slightly viscous flow interacts with a rigid boundary is vorticity production and shedding. In general, ideal flows cannot exchange vorticity with a wall, but, as we will see, at the level of regularity of vortex sheets, this becomes an interesting possibility. The weak vorticity formulation, as originally proposed by Schochet in, applied to full plane flows only. In this article we adapt this notion to flows in domains with boundary in a way which provides a quantitative accounting of the possible vorticity production through the interaction of irregular inviscid flow with a rigid wall. In, Lopes Filho, Nussenzveig Lopes and Xin introduced the weak vorticity formulation for half-plane flows. The weak vorticity formulation makes use of the Biot-Savart law, which is explicit in the full plane and in the half-plane. In the notion of boundary coupled weak solution was also introduced, as a necessary and sufficient condition for the validity of the method of images. This work was later extended to compactly supported perturbations of the half-plane in. The point of departure for the present work is to look for weak solutions of the incompressible Euler equations in exterior domains with vortex sheet initial data. Here, an exterior domain is the complement of a finite number of smooth, disjoint, obstacles. Existence of such weak solutions can be easily established - we show that Delort’s original proof can be immediately adapted to the exterior domain. We then ask ourselves - what about a weak vorticity formulation? The key new issue in the exterior domain is the topology. A vortex dynamical description of ideal flow in domains with topology requires us to keep track of velocity circulation in the 1-homology classes of the domain, a family of conserved quantities for smooth flow, due to Kelvin’s Circulation Theorem. We prove that one can make sense and keep track of these velocity circulations for flows with vortex sheet regularity. Even for smooth flows, reducing ideal fluid dynamics to vortex dynamics in a domain with holes is a rather recent development. A full description, for the case of bounded domains, was developed by Flucher and Gustafsson in, see also for a more explicit account. Part of the work of the present paper is to adapt and extend the vortex dynamics formulation of to exterior domains and recast it as a weak solution theory which includes vortex sheet flows. The remainder of this paper is divided as follows. In Sections 2 and 3 we put together the basic notation and description of smooth vortex dynamics in an exterior domain. In Section 4 we adapt this machinery to the weak solution context. The main point is that circulation of velocity around connected components of the boundary is well-defined, as long as velocity is at least locally integrable and vorticity is a bounded Radon measure. Still in Section 4, we develop a suitable approximation result. In Section 5 we state and prove an adaptation of Delort’s existence result to exterior domain flow. In Section 6 we state and prove the main result of this article – the equivalence between the weak velocity and weak vorticity formulations of the 2D Euler equations. In Section 7 we derive some properties of weak solutions which arise as limits of exact solutions with mollified initial data, we discuss the connection of boundary-coupled weak solutions and conservation of circulation on connected components of the boundary, and we prove existence of boundary-coupled weak solutions with integrable initial vorticity. In Section 8 we extend the equivalence between weak velocity and weak vorticity formulations to an equivalence between their boundary-coupled versions. This implies that the mechanical coupling between connected components of the boundary and the fluid flow is well-defined if and only if the solution is boundary-coupled. In Section 9 we describe the adaptation of our results to bounded domains with holes, a simpler situation than the exterior domain, we derive conclusions and we propose some open problems. Finally, in the Appendix we prove estimates for the Green’s function and the Biot-Savart kernel on bounded domains with holes which, although not new, do not appear to be easily available in the literature. Harmonic vector fields and the Green’s function =============================================== In this section we introduce basic notation and collect a few estimates which will be needed throughout the paper. Let Ω ⊂ R2 be a smooth open set such that the complement of $\overline\Om$ is the union of k ∈ N\* disjoint, simply connected, smooth, open sets Ω1, …, Ωk. Let Γi be the boundary of $\Om\_i$ so that $\partial\Om=\Gamma\_1\cup\dots\cup\Gamma\_k$. We will assume, for convenience, throughout this paper, that one of the obstacles, say Ω1, contains the unit ball B(0; 1); this can be done without loss of generality, incorporating a translation and dilation in all the proofs, if needed. We also introduce notation which will be used hereafter: i = i(x) = (x1, − x2)/∣x∣2, for x = (x1, x2). Let $i(\Om)\cup\{0\} = {\widetilde}{\Om}$. It is easy to see that $i({\mathbb{R}}^2 \setminus \overline{\Omega}\_1) \cup \{0\}$ is a simply connected subset of B(0; 1) which contains i(Ωj) for all j = 1, …, k. Therefore ${\widetilde}{\Om}$ is a subset of $i({\mathbb{R}}^2 \setminus \overline{\Omega}\_1)\cup\{0\}$ with k − 1 holes; the outer boundary of ${\widetilde}{\Omega}$ is i(Γ1). We denote by G = G(x, y) the Green’s function of the Laplacian in $\Om$ with Dirichlet boundary conditions. We also introduce the function K = K(x, y) ≡ ∇x⊥G(x, y), where ∇x⊥ = ( − ∂x2, ∂x1), known as the kernel of the Biot-Savart law. We will denote by $\yinf$ the space of functions $f\in C^\infty(\overline\Om)$ with bounded support and such that f is constant in a neighborhood of each Γj (with a constant depending on j). We also denote by $\yinfb$ the space of functions $f\in C^\infty(\overline\Om)$ with bounded support and such that f is constant on each Γj (with a constant depending on j). The space of smooth divergence free vector fields compactly supported in $\Om$ is denoted by $C^\infty\_{c,\sigma}(\Om)$. Next, we introduce the harmonic measures $\w\_j$, j = 1, …, k, in $\Om$. These are solutions of the boundary-value problem: $$\left\{ \begin{array}{ll} \Delta \w \_j = 0, & \mbox{ in } \Omega, \\ \w \_j = \delta\_{j\ell},& \mbox{ on } \Gamma \_{\ell}, \ \ell = 1, \ldots, k.\\ \w \_j \text{ has a finite limit at }\infty. \end{array} \right.$$ Observe that this system can be viewed as a Dirichlet problem for the Laplacian in $\Om\,\cup\,\{\infty\}$. Existence and uniqueness of the harmonic measures is well-known, and they can be expressed by means of an explicit formula in terms of the Green’s function: $$\w \_j(y)=-\frac1{2\pi}\int\_{\Gamma\_j}\frac{\partial G(x,y)}{\partial \hat n\_x}\d S\_x$$ where $\hat n\_x$ is the exterior unit normal vector at $\partial\Om$ (see, for example, Chapter 1, Section 10 of ). In what follows we adopt the convention (a, b)⊥ = ( − b, a). We denote by τ̂ the unit tangent vector to $\partial\Om$ oriented in the counterclockwise direction, i.e. τ̂ = − n̂⊥. In K. Kikuchi constructed a special basis for the harmonic vector fields in Ω (i.e., vector fields which are both solenoidal and irrotational and which are tangent to the boundary), generators of the homology of Ω, $\bX\_j, \, j = 1, \ldots k.$ It was shown in that there exist harmonic functions Ψj, j = 1, …ℓ, such that $$\label{propsPsij} \begin{array}{l} \mathrm{(i)} \;\;\bX\_j = \nabla^{\perp}\Psi\_j,\\ \\ \mathrm{(ii)}\;\; \Psi\_j (x) = \frac{1}{2\pi}\log\|x\| + {\mathscr{O}}(1), \text{ as } \|x\| \to \infty,\\ \\ \mathrm{(iii)}\;\; \int\_{\Gamma \_{\ell}} \bX\_j \cdot \hat{\tau} \d S = \delta\_{j\ell},\\ \\ \mathrm{(iv)} \;\;\Psi\_j\big\|\_{\Gamma \_{\ell}} = c\_{j\ell}, \text{ where the $c\_{j\ell}$ are constants},\\ \\ \mathrm{(v)} \;\;\|\bX\_j(x)\| {\leqslant}{\displaystyle}{\frac{C}{\|x\|}}. \end{array}$$ That these harmonic vector fields are uniquely defined was proved in. In addition, since the harmonic vector fields can be represented as holomorphic functions (if X = (X1, X2) is a harmonic vector field, then X2 + iX1 is holomorphic), we may use a Laurent expansion at infinity to deduce that $$\label{harumph1} X\_j (x) = \frac{x^{\perp}}{2 \pi \|x\|^2} + \mathcal{O}(\|x\|^{-2}), \mbox{ as } \|x\| \to \infty.$$ We will denote by $\xspace=\langle \X\_1,\dots,\X\_k\rangle$ the vector space spanned by the harmonic vector fields $\X\_1,\dots,\X\_k$. We end this section with two estimates involving the kernel of the Biot-Savart law and the Green’s function. [propK] There exists a constant M1 > 0 depending only on $\Om$ such that $$\label{estk} \|K(x,y)\|{\leqslant}M\_1\frac{\|y\|}{\|x\|\|x-y\|}$$ and $$\label{estg} \|G(x,y)\|{\leqslant}M\_1+M\_1\left\| \log\frac{\|x\|\|y\|}{\|x-y\|}\right\|.$$ Let $G\_{{\widetilde}{\Om}}$ denote the Green’s function for ${\widetilde}{\Om}$. With complex notation we have that i(z) = 1/z, so i is an holomorphic function. It is well known that the Green’s functions of two conformally equivalent domains are obtained via composition with the conformal mapping between the two domains. Therefore the Green’s function for $\Om$ is $G(x,y)= G\_{{\widetilde}{\Om}}(i(x),i(y))$. By construction, $$K(x,y) = \nabla^{\perp}\_x G(x,y)= -Di(x)K\_{{\widetilde}{\Om}}(i(x),i(y)),$$ where $K\_{{\widetilde}{\Om}}$ denotes the kernel of the Biot-Savart law for ${\widetilde}{\Om}$. We will use the following estimate: $$\label{thisisnotgood} \|K\_{{\widetilde}{\Om}}({\widetilde}{x},{\widetilde}{y})\| \leqslant \frac{C}{\|{\widetilde}{x}-{\widetilde}{y}\|},$$ valid for ${\widetilde}{x}$, ${\widetilde}{y}\in{\widetilde}{\Om}$ and some constant C > 0 which depends only on ${\widetilde}{\Om}$. The proof of this estimate is included in the Appendix, see Proposition [appGreenAndBS]. We deduce, from, the following pointwise estimate for the Biot-Savart kernel: $$\|K(x,y)\| \leqslant \frac{C}{\|x\|^2\|i(x)-i(y)\|}=\frac{C\|y\|}{\|x\|\|x-y\|}.$$ Similarly, estimate follows from the bound $$\|G\_{{\widetilde}{\Om}}({\widetilde}{x},{\widetilde}{y})\| \leqslant C(1+\|\log(\|{\widetilde}{x}-{\widetilde}{y}\|)\|)$$ which is also included in Proposition [appGreenAndBS]. [propestK] There exists a constant M2 depending only on $\Om$ such that $$\label{estK} \|f(x)\cdot K(x,y)+f(y)\cdot K(y,x)\|{\leqslant}M\_2\\|f\\|\_{W^{1,\infty}(\Om)} \quad\forall x,y\in\Omega,\ x\neq y.$$ for every vector valued function $f\in W^{1,\infty}(\overline\Om;\R^2)$ whose restriction to $\partial\Om$ is normal to the boundary. In particular we may take f = ∇φ, $\varphi \in \yinfb$. The case when f vanishes on the boundary $\partial\Om$ was considered in. We observe first from that there exists a constant $M\_3=M\_3(\Om)$ such that $$\label{estK1} \|K(x,y)\|{\leqslant}M\_3+\frac{M\_3}{\|x-y\|} \quad\forall x,y\in\Omega,\ x\neq y.$$ Define $$H(x,y)=G(x,y)-\frac1{2\pi}\log\|x-y\|.$$ We have that the function H is harmonic in both its arguments on Ω × Ω. Therefore K(x, y) + K(y, x) = ∇x⊥G(x, y) + ∇y⊥G(y, x) = ∇x⊥H(x, y) + ∇y⊥H(y, x) is also harmonic and smooth on $\overline\Omega\times\overline\Omega\setminus\{(x,x);\ x\in\partial\Omega\}$. Moreover, given that G(x, y) vanishes when x or y belongs to $\partial\Om$, we also have that $$\label{Kbord} K(x,y)=0\quad \forall x\in\Om,\ y\in\partial\Om \qquad\text{and}\qquad K(x,y)\text{ tangent to }\partial\Om \quad \forall x\in\partial\Om,\ y\in\overline\Om\setminus\{x\}.$$ We bound $$\label{estK2} \begin{split} \|f(x)\cdot K(x,y)+f(y)\cdot &K(y,x)\| {\leqslant}\|f(x)\cdot [K(x,y)+K(y,x)]\|+\|[f(x)-f(y)]\cdot K(y,x)\|\\ &{\leqslant}\|f(x)\cdot [K(x,y)+K(y,x)]\|+2M\_3\\|f\\|\_{L^\infty(\Om)}+M\_3\\|\nabla f\\|\_{L^\infty(\Om)} \end{split}$$ where we used. Next, let us fix $x\in\Om$. The function y ↦ f(x) ⋅ [K(x, y) + K(y, x)] is harmonic in $\Om$ and smooth on $\overline\Om$. By the maximum principle we therefore have that $$\begin{gathered} \label{estK3} \sup\_{y\in\Om} \|f(x)\cdot [K(x,y)+K(y,x)]\|\\ {\leqslant}\max\bigl(\limsup\_{\|y\|\to\infty} \|f(x)\cdot [K(x,y)+K(y,x)]\|, \max\_{y\in\partial\Om}\|f(x)\cdot [K(x,y)+K(y,x)]\|\bigr).\end{gathered}$$ Using we have that $$\label{estK4} \limsup\_{\|y\|\to\infty} \|f(x)\cdot [K(x,y)+K(y,x)]\|{\leqslant}2M\_3\\|f\\|\_{L^\infty(\Om)}.$$ Moreover, using that f is normal to the boundary $\partial\Om$ we have from that for $y\in\partial\Om$ $$\begin{gathered} \label{estK5} \|f(x)\cdot [K(x,y)+K(y,x)]\| =\|f(x)\cdot K(y,x)\| =\|[f(x)-f(y)]\cdot K(y,x)\|\\ {\leqslant}2M\_3\\|f\\|\_{L^\infty(\Om)}+M\_3\\|\nabla f\\|\_{L^\infty(\Om)}.\end{gathered}$$ We conclude from relations – that $$\|f(x)\cdot K(x,y)+f(y)\cdot K(y,x)\| {\leqslant}4M\_3\\|f\\|\_{L^\infty(\Om)}+2M\_3\\|\nabla f\\|\_{L^\infty(\Om)}.$$ This completes the proof of the proposition. We end this section with the following convergence result. [weakconvmeas] Let X be a metric space, locally compact and σ-compact. Let $(\mu\_n)\_{n\in\N}$ be a tight sequence of bounded measures converging weakly to a measure μ. Suppose that $(\|\mu\_n\|)\_{n\in\N}$ converges weakly to another measure ν. Then for any bounded borelian function f continuous outside a ν-negligible set, we have that limn → ∞∫f dμn = ∫f dμ. This lemma is almost the same as. The difference here is that we have the additional assumption of tightness for the sequence of measures while allowing for test functions f that does not necessarily vanish at infinity as required in. Recall that a sequence of measures μn is called tight if, for any $\ep>0$, there is a compact subset $K\_\ep$ of X such that $\|\mu\_n\|(X\setminus K\_\ep)<\ep$ for all $n\in\N$. Let $\ep>0$ and $K\_\ep$ a compact set of X such that $\|\mu\_n\|(X\setminus K\_\ep)<\ep$ for all $n\in\N$. Since X is σ-compact and μ is bounded, we can moreover assume that $\|\mu\|(X\setminus K\_\ep)<\ep$. Let $g\_\ep:X\to[0,1]$ a compactly supported continuous function such that $g\_\ep\bigl\|\_{K\_\ep}\equiv1$. Since the function $fg\_\ep$ is continuous outside a ν-negligible set and vanishing at infinity, we can apply to deduce that $$\lim\_{n\to\infty}\int fg\_\ep\,d\mu\_n=\int fg\_\ep\,d\mu.$$ We infer that $$\begin{gathered} \limsup\_{n\to\infty}\Bigl\|\int f\,d\mu\_n-\int f\,d\mu\Bigr\| =\limsup\_{n\to\infty}\Bigl\|\int f(1-g\_\ep)\,d\mu\_n-\int f(1-g\_\ep)\,d\mu\Bigr\|\\ {\leqslant}\\|f\\|\_{L^\infty}(\|\mu\_n\|(X\setminus K\_\ep)+\|\mu\|(X\setminus K\_\ep)){\leqslant}2\ep \\|f\\|\_{L^\infty}.\end{gathered}$$ Letting $\ep\to0$ completes the proof. Velocity, vorticity and circulation: classical setting ====================================================== One important dynamic variable for incompressible flow, specially in 2D, is the vorticity, the curl of velocity. If u = (u1, u2) is the velocity then the vorticity is $\om=\partial\_{x\_1}u\_2-\partial\_{x\_2}u\_1 \equiv \mbox{ curl } u$. In bounded simply connected domains the velocity can be easily recovered from the vorticity by means of the regularizing linear operator ∇⊥Δ− 1, where Δ− 1 is the inverse Dirichlet Laplacian. In our case, the fluid domain is neither simply connected nor bounded. The purpose of this section is to discuss reconstruction of the velocity from the vorticity for exterior domains, for smooth flows. The fact that $\Om$ is not simply connected implies that we need to assign extra conditions in order to recover velocity from vorticity, for instance, the circulation of the velocity around each obstacle. For a given vector field u, we define γj the circulation around Γj as follows $$\gamma\_j=\int\_{\Gamma\_j}u\cdot\hat\tau\d S.$$ As in the previous section, we denote by K the Biot-Savart operator ∇⊥Δ− 1, where Δ is the Dirichlet Laplacian in $\Om$, and we abuse notation, denoting also by K = K(x, y) its singular kernel. The fact that $\Om$ is unbounded implies that the properties of the Biot-Savart operator are a delicate issue. We discuss how to express the velocity field u from the vorticity $\om$ and the circulations γ1, …, γk. We start by studying the Biot-Savart operator on $\Om$. For $\om\in C^\infty\_c (\Om)$ we denote by $K[\om]$ the value of the operator K on $\om$, given by $$K[\om](x)=\int\_{\Om}K(x,y)\om(y)\d y.$$ We will also use the analogous notation for the Green’s function $$\Delta^{-1} \omega = G[\omega](x)=\int\_{\Om}G(x,y)\omega(y)\d y.$$ We have the following [propdefK] Let $\om\in C^\infty\_c (\Om)$. We have that $K[\om]$ is smooth, divergence free, tangent to the boundary, square integrable on $\Om$ and such that $\operatorname{curl}K[\om]=\om$. Moreover, the circulation of $K[\om]$ on Γj is given by $$\label{circK} \int\_{ \Gamma \_j} K [ \omega] \cdot \hat{\tau} \d S = - \int\_{\Omega } \w \_{j} \omega\d x.$$ Denoting $\psi=G[\om]$ we have that $$\label{psi} \left\{ \begin{array}{ll} \Delta \psi = \omega, & \text{ in } \Om,\\ \psi=0, &\mbox{ on } \partial\Om,\\ K[\om]=\nabla^{\perp}\psi, & \text{ in } \overline\Om. \end{array} \right.$$ We infer that $K[\om]$ is smooth, divergence free, tangent to the boundary and such that $\operatorname{curl}K[\om]=\om$. Moreover, since ω has compact support, from we deduce that $K[\om]$ decays like O(1/∣x∣2) as ∣x∣ → ∞ so $K[\om]\in L^2(\Om)$. It remains to prove the formula for the circulation. We have $$\begin{aligned} \int\_{ \Gamma \_j} K [ \omega] \cdot \hat{\tau}\d S &=\int\_{ \partial \Omega } (K [ \omega])^\perp \w \_{j}\cdot \hat{ n} \d S(\sigma) \\ &= \int\_{\Omega \cap B(0;M)} \text{div} \ [ (K [ \omega])^\perp\w \_{j} ] \d x - \int\_{ \partial B(0;M) } (K [ \omega])^\perp \w \_{j}\cdot\frac x{\|x\|}\d S(\sigma)\\ &= \int\_{\Omega \cap B(0;M) } \w \_{j} \text{div} \ [ (K [ \omega])^\perp] \d x+ \int\_{\Omega \cap B(0;M) } \nabla \w \_{j} \cdot (K [ \omega])^\perp \d x \\ &\hskip 6cm - \int\_{ \partial B(0;M) } (K [ \omega])^\perp \w \_{j}\cdot \frac x{\|x\|} \d S(\sigma) \\ &\equiv I\_1 + I\_2 + I\_3.\end{aligned}$$ Let us examine each of these three terms. We have, for the first term, $$\begin{gathered} I\_1 = \int\_{\Omega \cap B(0;M) } \w \_{j} \text{div} \ (K [ \omega])^\perp \d x = - \int\_{\Omega \cap B(0;M) } \w \_{j} \text{ curl } (K [ \omega])\d x\\ = - \int\_{\Omega \cap B(0;M) } \w \_{j} \omega \d x = - \int\_{\Omega } \w \_{j} \omega\d x,\end{gathered}$$ if M is sufficiently large. Using and integrating by parts the second term then gives $$\begin{gathered} I\_2 = \int\_{\Omega \cap B(0;M) } \nabla \w \_{j} \cdot (K [ \omega])^\perp \d x = \int\_{\Omega \cap B(0;M) } ( \Delta \w \_{j}) \,( \psi) \d x \\ - \int\_{ \partial B(0;M) } \psi \nabla \w \_{j}\cdot\frac x{\|x\|} \d S(\sigma) - \int\_{ \partial \Omega } \psi \nabla \w \_{j} \cdot\hat{n} \d S(\sigma) = - \int\_{ \partial B(0;M) } \psi \, \nabla \w \_{j}\cdot\frac x{\|x\|} \d S(\sigma).\end{gathered}$$ We have used here that $\Delta \w \_j = 0$ in Ω and that ψ = 0 on ∂Ω, see. Now, the same argument used to show the pointwise estimates for K[ω] in Proposition [propK] can be used to show that $\|\nabla \w \_j\| = {\mathscr{O}}(M^{-2})$ for ∣x∣ = M. Furthermore, from we deduce that $\psi=G[\om]$ is bounded. Thus we obtain ∣I2∣ → 0 as M → ∞. Finally, we note that $$I\_3 = \int\_{ \partial B(0;M) } \w \_{j} (K [ \omega])^\perp \cdot \frac x{\|x\|} \d S(\sigma) \rightarrow 0$$ as M → ∞, since $\w \_j$ is bounded and ∣K[ω]∣ = O(M− 2) for ∣x∣ = M (which is a consequence of ). This establishes and completes the proof of the proposition. A result analogous to Proposition [propdefK] for bounded domains was explicitly stated and proved in, and it was implicit in the analysis contained in. As an easy consequence of this proposition we show how to recover velocity from vorticity in an exterior domain with k holes. Denote by $C^1\_b(\overline{\Omega})$ the set of C1-functions which are bounded on $\overline{\Omega}$. [milton] Let ω ∈ Cc∞(Ω) and let γj ∈ R, j = 1, …, k be given constants. Then there exists one and only one divergence free vector field $u \in C^1\_b ( \overline{\Omega} )$, tangent to ∂Ω, vanishing at infinity, such that curl u = ω and $$\int\_{\Gamma \_j} u \cdot \hat{\tau} \d S = \gamma\_j, \;\; j=1,\ldots,k.$$ This vector field is given by the formula: $$\label{recover} u = K [ \omega] + \sum\_{j=1}^k \, \left( \int\_{\Omega } \w \_j \omega \d x + \gamma\_j \right) \, \bX\_j.$$ The existence part is trivial since, from and Proposition [propdefK], it follows that the vector field defined in has all the required properties. Now, if $v\in C^1\_b(\overline{\Omega})$ is another divergence free vector field, tangent to the boundary, vanishing at infinity, whose curl is ω and whose circulations around each Ωj are γj, then consider the difference between v and u, ū ≐ v − u. We find that ū will be divergence free, curl-free, tangent to the boundary ∂Ω; ū will vanish at infinity and ū will have vanishing circulation around each hole Ωj. Thus ū satisfies the hypothesis of, which implies that ū = 0. This concludes the proof. Velocity, vorticity and circulation: weak setting ================================================= We need to concern ourselves with the technical issue of reconstructing velocity from vorticity and circulations in a less regular setting. This is the subject of the present section. We will work in the remainder of this paper with velocity fields of the form when $\om$ has weak regularity. Given that $K[\om]$ is expected to belong to L2, the natural function space for the velocity field is $L^2(\Om)+\xspace$. Given that all harmonic vector fields $\X\_j$ can be written as $\frac{x^\perp}{2\pi\|x\|^2}+{\mathscr{O}}(1/\|x\|^2)$ when ∣x∣ → ∞ we observe that $L^2(\Om)+\xspace=L^2(\Om)+\langle\frac{x^\perp}{\|x\|^2}\rangle$. Next, if u can be decomposed as in then $\operatorname{curl}u=\operatorname{curl}K[\om]$, so we expect the vorticity to be the curl of a square integrable velocity field. It is then natural to introduce the space $$\H=\{\om\in \mathscr{D}'(\Om);\ \exists v\in L^2(\Om)\text{ such that } \operatorname{div}v=0,\ \operatorname{curl}v=\om,\ v\cdot\hat{n}\bigl\|\_{\partial\Om}=0\}.$$ endowed with the norm $$\label{defnormh} \\|\om\\|\_{\H}=\inf\{\\|v\\|\_{L^2(\Om)}\ ;\ v\in L^2(\Om), \ \operatorname{div}v=0,\ \operatorname{curl}v=\om,\ v\cdot\hat{n}\bigl\|\_{\partial\Om}=0\}.$$ Clearly $\H$ can be identified with the quotient space $\H=Y/M$, where $Y=\{v\in L^2(\Om), \ \operatorname{div}v=0,\ v\cdot\hat{n}\bigl\|\_{\partial\Om}=0\}$ is endowed with the L2 norm, and where M = {v ∈ Y, curlv = 0}. The space Y is a classical space in mathematical fluid mechanics and is well-known to be a Hilbert space. We have that M is a closed subspace of Y. Indeed, the convergence in L2 implies the convergence in the sense of distributions which, in turn, implies the convergence of the curl in the sense of distributions. Therefore the L2 limit of a sequence of curl-free vector fields is a curl-free vector field. We conclude that $\H$ is a Banach space as the quotient of a Banach space by a closed subspace. It is reasonable to ask whether $\H$ is the same as H− 1(Ω). This is obviously true if the underlying domain is smooth and bounded, but problematic in unbounded domains. In fact, if we take ω such that its Fourier transform ω̂(ξ) is given by φ(ξ)(log∣ξ∣)− 1/2, where φ is a cutoff, identically 1 for ∣ξ∣ < 1/2, vanishing for ∣ξ∣ > 2/3, then ω ∈ H− 1(R2) but $\omega \notin \mathscr{H}^{-1}({\mathbb{R}}^2)$. Nevertheless, it is not hard to show that for vorticity with compact support in Ω, the two notions $\H$ and H− 1(Ω) are the same. In this paper we are interested in flows with vortex sheet regularity, and, to this end, we will consider velocities in $\L$ and vorticities in $\H\cap\bm$. Next we show that, for any locally integrable velocity field whose vorticity is in $\bm$, one can define the circulation on each connected component of the boundary. More precisely, we have the following lemma. [defcirc] Let $v\in L^1\loc(\Om)$ be a vector field such that $\om\equiv\operatorname{curl}v\in \bm $. Then the circulations γ1, …, γk of v on the connected components of the boundary Γ1, …, Γk are well defined through the following formula: $$\label{defcirceq} \int\_\Omega\varphi\d \om + \int\_\Omega v\cdot\nabla^\perp\varphi=-\sum\_{j=1}^k\gamma\_j\;\varphi\bigl\|\_{\Gamma \_j}$$ for all $\varphi\in \yinf$. For consistency, we note that, if v is smooth, then the Stokes formula implies that holds true with γj equal to the circulation of v along Γj given by the usual formula γj : = − ∫Γju ⋅ n̂⊥dS. It is enough to show existence and uniqueness of a set of constants {γj} for which holds. We begin with existence. Let us fix j ∈ {1, …, k} and consider some $\varphi\_j\in C^\infty\_c(\overline\Omega)$ such that φj is equal to 1 in a neighborhood of Γj and vanishes in a neighborhood of ∂Ω \ Γj. We set $$\gamma\_j\equiv -\int\_\Omega\varphi\_j\d\om -\int\_\Omega v\cdot\nabla^\perp\varphi\_j.$$ For φ as in the statement of the lemma, we define $L\_j=\varphi\bigl\|\_{\Gamma \_j}$ and $\overline\varphi\equiv\varphi-\sum\_{j=1}^kL\_j\varphi\_j$. Clearly $\overline\varphi\in C\_c ^\infty(\Omega)$, so we have, in the sense of distributions, that $$\int\_\Omega\overline\varphi\d \om =\langle \om,\overline\varphi\rangle\_{\mathscr{D}',\mathscr{D}} =\langle \operatorname{curl}v,\overline\varphi\rangle\_{\mathscr{D}',\mathscr{D}} =-\langle v,\nabla^\perp\overline\varphi\rangle\_{\mathscr{D}',\mathscr{D}} =-\int\_\Omega v\cdot\nabla^\perp\overline\varphi.$$ We infer that $$\int\_\Omega\varphi\d \om + \int\_\Omega v\cdot\nabla^\perp\varphi =\sum\_{j=1}^k L\_j\bigl(\int\_\Omega\varphi\_j\d \om + \int\_\Omega v\cdot\nabla^\perp\varphi\_j\bigr) = -\sum\_{j=1}^k L\_j\gamma\_j$$ so that holds true. This proves existence of the constants γ1, …, γk in. Uniqueness is clear since, if γʹ1, …, γʹk were to verify as well, then we would have $$\sum\_{j=1}^k(\gamma\_j-\gamma'\_j)\;\varphi\bigl\|\_{\Gamma \_j} =0$$ for all $\varphi\in \yinf$. Choosing φ = φj above implies that γj = γʹj. [defcircremark] If we assume that v is L1 up to the boundary, i.e. $v\in L^1\_{loc}(\overline\Om)$, then we can allow for the more general class of test functions $\varphi\in\yinfb$ in. Let us prove this. Suppose first that $\varphi\in\yinfb$ is vanishing on the boundary and use relation with test function φk = φχk where χk is the cutoff function defined on page. Because φ vanishes on the boundary, we have that φk and ∇φk are uniformly bounded. By the dominated convergence theorem we infer that $\int\_\Om \varphi^k\,d\om\to\int\_\Om \varphi\,d\om$ and $\int\_\Om v\cdot\nabla^\perp\varphi^k\to \int\_\Om v\cdot\nabla^\perp\varphi$ as k → ∞ implying that relation holds true for all $\varphi\in\yinfb$ vanishing on the boundary. If $\varphi\in\yinfb$ does not vanish on the boundary, then $\varphi-\sum\_{j=1}^k\varphi\bigl\|\_{\Gamma\_j}\varphi\_j$ does. Since $\varphi-\sum\_{j=1}^k\varphi\bigl\|\_{\Gamma\_j}\varphi\_j$ and φj can be used as test functions in, so does φ. Our next objective is to define the Biot-Savart operator for vorticities in $\H\cap\bm$. First we prove that divergence free vector fields in $\L$ which are tangent to the boundary are uniquely determined by their curl, together with the circulations around each boundary component. [unicdec] Let $u\in\L$ be divergence free, curl free, tangent to the boundary and with vanishing circulation on each of the connected components of the boundary. Then u = 0. We show first that $u\in L^2(\Om)$. We have that u can be decomposed as follows: $$u=v+\sum\_{j=1}^k\al\_j \X\_j,$$ where the vector field $v\in L^2(\Om)$ is divergence free, curl free, tangent to the boundary and has circulation $-\al\_j$ on Γj, for all j. Let R be such that $\Om^c\subset B(0;R)$ and let $\kappa\in C\_c(\overline\Om;[0,1])$ be such that suppκ ⊂ B(0; 2R) and κ ≡ 1 on B(0; R). Using we have that, for any n ≥ 1, $$\begin{gathered} \left\| \sum\_{j=1}^k\al\_j \right\|=\left\|\int\_\Om v\cdot\nabla^\perp[\kappa(x/n)] \right\| =\left\|\frac1n \int\_\Om v\cdot\nabla^\perp\kappa(x/n) \right\| {\leqslant}\frac1n \\|v\\|\_{L^2(Rn<\|x\|<2Rn)}\\|\nabla^\perp\kappa(x/n) \\|\_{L^2}\\ {\leqslant}\\|v\\|\_{L^2(Rn<\|x\|<2Rn)}\\|\nabla^\perp\kappa\\|\_{L^2} \stackrel{n\to\infty}{\xrightarrow{\hspace\{1cm}}}0.\end{gathered}$$ Therefore we have that $\sum\_{j=1}^k\al\_j=0$. Since, for all j, we have $$\X\_j=\frac{x^\perp}{2\pi\|x\|^2}+{\mathscr{O}}(1/\|x\|^2)\in \frac{x^\perp}{2\pi\|x\|^2}+L^2(\Om),$$ we infer that $$\sum\_{j=1}^k\al\_j \X\_j \in L^2(\Om),$$ so $u\in L^2(\Om)$. Recall that the closure of $C^\infty\_{c,\sigma}(\Om)$ in $L^2(\Om)$ is the space of square integrable, divergence free vector fields tangent to the boundary. Therefore there exists a sequence $u\_j\in C^\infty\_{c,\sigma}(\Om)$ such that u*j → u in $L^2(\Om)$ as j → ∞. Since $u\_j\in C^\infty\_{c,\sigma}(\Om)$ there exists some $\psi\_j\in\yinf$ such that uj = ∇⊥ψj. We now use relation together with the fact that u is curl free and has vanishing circulation on each of the connected components of the boundary, to deduce that $$\int\_\Om u\cdot u\_j= \int\_\Om u\cdot \nabla^\perp\psi\_j=0.$$ Letting j → ∞, we infer that $\int\_\Om\|u\|^2=0$, so that u = 0. This completes the proof. Using Proposition [unicdec], we can define an extension of the Biot-Savart operator K to $\H\cap\bm$. [defKfield] Let $\om\in\H\cap\bm$. We define $K[\om]$ as the unique vector field in $\L$ which is divergence free, tangent to the boundary, with curl equal to $\om$ and such that, for any j, its circulation on Γj is $-\int\_\Om\w\_j\d\om$. Note that, by definition, if $\omega \in \H$ there exists a vector field $v \in \L$ with curlv = ω, divergence free and tangent to the boundary. If, in addition, ω is assumed to be a bounded measure, then the circulations of v around boundary components are well-defined; this is the content of Lemma [defcirc]. All that is needed to find K[ω] is to adjust v by adding a suitable linear combination of harmonic vector fields, so that the resulting field has the required circulations. Uniqueness follows from Proposition [unicdec]. We will need, in the sequel, an approximation result. We introduce some additional notation. Let χn be a cutoff function at distance $\frac1n$ from the boundary and for ∣x∣ < n. More precisely, we assume that [chin] $$\begin{gathered} \chi\_n\in C\_c^\infty(\Omega;[0,1]), \quad \chi\_n\equiv1 \text{ in }\Sigma\_{\frac2n}^c\cap B\_n, \quad \chi\_n\equiv0 \text{ in }\Sigma\_{\frac1n}\cap B\_{2n}^c\\ \\|\nabla\chi\_n\\|\_{L^\infty(\Sigma\_{\frac2n}\setminus\Sigma\_{\frac1n})}{\leqslant}Cn,\quad \\|\nabla\chi\_n\\|\_{L^\infty(B\_{2n}\setminus B\_n)}{\leqslant}\frac Cn,\end{gathered}$$ where Σa = {x ∈ Ω ; d(x, ∂Ω) ≤ a} and Ba = B(0; a). Fix $$\eta\in C^\infty\_c(\R^2;\R\_+),\quad \operatorname{supp}\eta\subset B\_{\frac12},\quad \int\eta=1,$$ and set ηn(x) = n2η(nx). Before we state the main result of this section we need to prove the following Poincaré inequality. [Poincare-lemma] There exists a constant C = C(Ω) such that for any R > 0 and for any function $f\in H\_{loc}^1(\overline\Om)$ which vanishes on $\partial\Om$ and such that $\nabla f\in L^2(\Om)$, we have the following inequality: $$\label{Poincare} \\|f\\|\_{L^2(\Sigma\_R)}{\leqslant}CR \\|\nabla f\\|\_{L^2(\Om)}.$$ Relation with some constant C(R) instead of CR is well-known. We only need to see how the constant depends on R. Therefore, we only need to consider the cases R small and R large. We infer that it suffices to assume that the exterior domain $\Om$ have only one hole. Indeed, we can extend f with zero values on all holes except one of them. The resulting extension is still in Hloc1 and it suffices to work with this extension. Suppose now that $\Om=\{x\ ;\ \|x\|>1\}$. By density, we may assume that f is smooth. Using the polar coordinates (x, y) = r(cosθ, sinθ) we may write f(r, θ) = ∫1r∂τf(τ, θ) dτ so that ∣f(r, θ)∣2 ≤ (r − 1)∫1r∣∂τf(τ, θ)∣2 dτ Then $$\begin{aligned} \\|f\\|\_{L^2(\Sigma\_R)}^2 &=\int\_{1<\|x\|<R+1}\|f(x)\|^2\,dx\\ &=\int\_{1}^{1+R}\int\_0^{2\pi}\|f(r,\theta)\|^2\,dr\,d\theta\\ &{\leqslant}\iint\_{1<\tau<r<1+R}\int\_0^{2\pi}(r-1)\|\partial\_\tau f(\tau,\theta)\|^2\,d\tau \,dr\,d\theta\\ &= \int\_{1}^{1+R}\int\_0^{2\pi}\bigl[\frac{R^2}2-\frac{(\tau-1)^2}2\bigr]\|\partial\_\tau f(\tau,\theta)\|^2\,d\tau \,d\theta\\ &{\leqslant}\frac{R^2}2 \\|\partial\_r f\\|\_{L^2(\Sigma\_R)}^2.\end{aligned}$$ Suppose now that $\R^2\setminus\Om$ is not a disk. By the Kellogg-Warschawski theorem (see for example ) there exists a biholomorphism T : {∣x∣ > 1} → Ω which is smooth up to the boundary. In fact, is stated for simply connected domains, but using the inversion z ↦ 1/z it holds true for the exterior of a simply connected obstacle too. Moreover, there exist C1, C2 > 0 such that C1 < ∣Tʹ(z)∣ < C2 for all z. In particular, we have that ΣR ⊂ T(ΣC3R(Dc)) for some constant C3 > 0 (here ΣR(Dc) denotes the set ΣR associated to Dc = {∣x∣ > 1}). Therefore ∥f∥L2(ΣR) ≤ C2∥f ∘ T∥L2(ΣC3R(Dc)). Moreover, we also have that $\\|\nabla (f\circ T)\\|\_{L^2(\|x\|>1)}{\leqslant}C\_4 \\|\nabla f\\|\_{L^2(\Om)}$ for some constant C4 > 0, so relation in the general case follows from the particular case of the exterior of the unit disk applied to the function f ∘ T. We now prove the following approximation result. [aproxom] Let $\om\in\H\cap\bm$. We have that $K[\om]\in L^2(\Om)$. Moreover, let us define ωn = (χnω) \* ηn. Then we have that $$\begin{gathered} \omega^n\text{ is bounded in } L^1(\Omega),\label{conv2}\\ \int\_\Omega \varphi\omega^n\stackrel{n\to\infty}\longrightarrow \int\_\Omega\varphi\d\omega \text{ and } \quad\forall\varphi\in C^0\_b(\overline\Omega),\label{conv1}\\ K[\om^n] \stackrel{n\to\infty}\longrightarrow K[\om] \text{ strongly in } L^2(\Om)\label{conv123}.\end{gathered}$$ Moreover, any weak limit in the sense of measures of any subsequence of $\|\om^n\|$ is a continuous measure. Above, we denoted by $C^0\_b(\overline\Omega)$ the space of bounded functions on Ω, continuous up to the boundary and with a finite limit as ∣x∣ → ∞. Claim is obvious: $$\int\_\Omega\|\omega^n\|=\int \_\Omega \| (\chi\_n\omega)\ast\eta\_n\| {\leqslant}\int\_\Omega \chi\_n \d\|\omega\| {\leqslant}\int\_\Omega1\d\|\omega\|=\|\omega\|(\Omega)<\infty.$$ Next we prove. Let $\varphi\in C^0\_b(\overline\Omega)$. We extend it to a continuous function on $\R^2$, again denoted by φ. For f smooth, denote $\check f = \check f (z) = f(-z)$. Then we have: $$\begin{gathered} \int\_\Omega \varphi\omega^n- \int\_\Omega\varphi\d\omega =\int\_\Omega \varphi(\chi\_n\omega)\ast\eta\_n- \int\_\Omega\varphi\d\omega =\int\_\Omega\chi\_n(\varphi\ast\check\eta\_n)\d\omega - \int\_\Omega\varphi\d\omega\\ =\int\_\Omega\chi\_n(\varphi\ast\check\eta\_n-\varphi)\d\omega +\int\_\Omega(\chi\_n-1)\varphi\d\omega\equiv I\_1+I\_2.\end{gathered}$$ We first bound I2: $$\|I\_2\|{\leqslant}\int\_{\Sigma\_{\frac2n}\cup B\_n^c} \|(\chi\_n-1)\varphi\|\d\|\omega\|{\leqslant}\\|\varphi\\|\_{L^\infty}(\|\omega\|(\Sigma\_{\frac2n})+\|\omega\|(B\_n^c))\stackrel{n\to\infty}\longrightarrow 0.$$ Next, we know by classical results that $\varphi\ast\check\eta\_n\to\varphi$ uniformly in $\R^2$. Therefore $$\|I\_1\|{\leqslant}\\|\varphi\ast\check\eta\_n-\varphi\\|\_{L^\infty(\R^2)}\|\omega\|(\Omega)\stackrel{n\to\infty}\longrightarrow 0.$$ This completes the proof of. We prove now that any weak limit in the sense of measures of any subsequence of $\|\om^n\|$ is a continuous measure. Let $\om\_+$, respectively $\om\_-$, be the positive part, respectively the negative part, of the measure $\om$. Since $\om\in\H\subset H^{-1}(\Omega)$ we infer that $\om$ is a continuous measure (see ). Since $\om=\om\_++\om\_-$ and $\om\_+$, $\om\_-$ are orthogonal measures, we have that $\om\_+$ and $\om\_-$ are also continuous measures. Let $\om^n\_+=(\chi\_n\om\_+)\ast\eta\_n$ and $\om^n\_-=(\chi\_n\om\_-)\ast\eta\_n$. Then $\om^n\_+$ and $\om\_-^n$ are single-signed, although they are not necessarily the positive and negative parts of $\om^n$. Nevertheless, we have the bound $\|\om^n\|{\leqslant}\om^n\_+-\om^n\_-$. We can apply the result proved in relation to the measures $\om\_+$ and $\om\_-$. Indeed, we did not use the hypothesis that $\om\in\H$ to prove that result. We therefore have that $\om^n\_+\rightharpoonup\om\_+$ and $\om^n\_-\rightharpoonup\om\_-$ weakly in the sense of measures. Therefore $\om^n\_+-\om^n\_-\rightharpoonup\om\_+-\om\_-=\|\om\|$ weakly in the sense of measures. From the bound $\|\om^n\|{\leqslant}\om^n\_+-\om^n\_-$ we infer that any weak limit in the sense of measures of any subsequence of $\|\om^n\|$ must be bounded by $\|\om\|$. Since $\|\om\|$ is a continuous measure, we deduce that any weak limit in the sense of measures of any subsequence of $\|\om^n\|$ must be a continuous measure. The proof that $K[\om]\in L^2(\Omega)$ and the proof of is done in three steps. First we show that $K[\om^n]$ is bounded in $L^2(\Om)$. Second, we prove that $K[\om^n]$ converges weakly to $K[\om]$ in L2(Ω) (showing in particular that $K[\om]\in L^2(\Omega)$). Third, we show the strong convergence of $K[\om^n]$ to $K[\om]$ in L2(Ω). We prove now that $K[\om^n]$ is bounded in $L^2(\Om)$. Let $f\in C^\infty\_c(\Om)$ be a divergence free vector field and let us define F = G[curlf]. Clearly $F\in C^\infty(\overline\Om)$ is bounded and vanishes on $\partial\Om$. We extend it to $\R^2$ by setting F to vanish on $\Om^c$. We have that $$\begin{gathered} \label{idistr} \int\_\Om K[\om^n]\cdot f =\int\_\Om\nabla^\perp G[\om^n]\cdot f = -\int\_\Om G[\om^n] \operatorname{curl}f =- \int\_\Om G[\om^n]\triangle F =- \lim\_{R\to\infty}\int\_{\Om\cap B\_R} G[\om^n]\triangle F\\ =- \lim\_{R\to\infty}\Bigl(\int\_{\Om\cap B\_R} \triangle G[\om^n] F+\int\_{\|x\|=R} G[\om^n]\ \frac x{\|x\|}\cdot \nabla G[\operatorname{curl}f]-\int\_{\|x\|=R} \frac x{\|x\|}\cdot\nabla G[\om^n]\cdot G[\operatorname{curl}f]\Bigr) =-\int\_\Om \om^nF.\end{gathered}$$ We used above that, for a compactly supported smooth function h, we have that G[h](x) = O(1) and ∇G[h](x) = O(1/∣x∣2), as ∣x∣ → ∞. We recall now that $\om\in\H$ and we choose some $v\in L^2(\Om)$ such that divv = 0, $\operatorname{curl}v=\om$ and v is tangent to the boundary. Next, we observe that $$\begin{gathered} \label{jdistr} \int\_\Om \om^nF=\int\_\Om (\chi\_n\om)\ast\eta\_n\ F =\int\_\Om \chi\_n(F\ast\check\eta\_n)\d\om =\langle\om, \chi\_n(F\ast\check\eta\_n)\rangle\_{\mathcal{D}'(\Om),\mathcal{D}(\Om)}\\ =\langle\operatorname{curl}v, \chi\_n(F\ast\check\eta\_n)\rangle\_{\mathcal{D}'(\Om),\mathcal{D}(\Om)} =-\langle v, \nabla^\perp[\chi\_n(F\ast\check\eta\_n)]\rangle\_{\mathcal{D}'(\Om),\mathcal{D}(\Om)}\\ =-\int\_\Om v \nabla^\perp\chi\_n(F\ast\check\eta\_n) -\int\_\Om v \chi\_n(\nabla^\perp F\ast\check\eta\_n).\end{gathered}$$ We estimate each of the two last terms above: $$\bigl\|\int\_\Om v \chi\_n(\nabla^\perp F\ast\check\eta\_n)\bigr\|{\leqslant}\\|v \\|\_{L^2(\Om)}\\|\nabla^\perp F\ast\check\eta\_n\\|\_{L^2(\Sigma\_{\frac1n}^c\cap B\_{2n})}{\leqslant}\\|v \\|\_{L^2(\Om)}\\|\nabla^\perp F\\|\_{L^2(\Om)}$$ and $$\begin{aligned} \bigl\| \int\_\Om v \nabla^\perp\chi\_n(F\ast\check\eta\_n) \bigl\| &=\bigl\| \int\_{(\Sigma\_{\frac2n}\setminus\Sigma\_{\frac1n})\cup(B\_{2n}\setminus B\_n)} v \nabla^\perp\chi\_n(F\ast\check\eta\_n) \bigl\|\\ &{\leqslant}\\|v \\|\_{L^2(\Om)}\bigl(Cn\\|F\\|\_{L^2(\Sigma\_{\frac5{2n}}\setminus\Sigma\_{\frac1{2n}})}+\frac Cn\\|F\\|\_{L^2(B\_{2n+1}\setminus B\_{n-1})}\bigr)\\ &{\leqslant}C\\|v \\|\_{L^2(\Om)}\\|\nabla F\\|\_{L^2(\Om)}\end{aligned}$$ where we used Lemma [Poincare-lemma] and the fact that F vanishes on $\partial\Om$. We infer from and, together with the estimates performed above, that the following inequality holds true: $$\label{conv8} \bigl\| \int\_\Om K[\om^n]\cdot f\bigr\|{\leqslant}C \\|v \\|\_{L^2(\Om)}\\|\nabla F\\|\_{L^2(\Om)}.$$ Next we observe that $$\label{conv7} \\|\nabla F\\|\_{L^2(\Om)}{\leqslant}\\|f\\|\_{L^2(\Om)}.$$ Indeed, we have that △F = curlf on $\Om$ so $$\label{Fstuff1} -\int\_\Om\triangle F\ F= -\int\_\Om\operatorname{curl}f\ F=\int\_\Om f\cdot\nabla^\perp F{\leqslant}\\|f\\|\_{L^2(\Om)}\\|\nabla F\\|\_{L^2(\Om)}.$$ On the other hand, $$\label{Fstuff2} -\int\_\Om\triangle F\ F =\int\_\Om\|\nabla F\|^2-\lim\_{R\to\infty}\int\_{\|x\|=R}\frac x{\|x\|}\cdot\nabla F\ F= \\|\nabla F\\|\_{L^2(\Om)}^2.$$ Putting together and, we get. Using in we obtain that $$\left\| \int\_\Om K[\om^n]\cdot f\right\|{\leqslant}C \\|v \\|\_{L^2(\Om)}\\|f\\|\_{L^2(\Om)},$$ for all divergence free vector fields $f\in C^\infty\_c(\Om)$. This implies that $$\\|K[\om^n]\\|\_{L^2(\Om)} {\leqslant}C \\|v \\|\_{L^2(\Om)}.$$ We have, hence, established that $K[\om^n]$ is bounded in L2. Therefore, there exists a subsequence $K[\om^{n\_k}]$ which converges weakly to some vector field $w\in L^2(\Om)$. Given that $K[\om^{n\_k}]$ is divergence free and tangent to the boundary, the same holds true for the weak limit w. Moreover, we have that $K[\om^{n\_k}]\to w$ in $\mathscr{D}'(\Om)$ so $\om^{n\_k}=\operatorname{curl}K[\om^{n\_k}]\to \operatorname{curl}w$ in $\mathscr{D}'(\Om)$. Using we have that $ \om^{n\_k}\to\om$ in $\mathscr{D}'(\Om)$, hence $\operatorname{curl}w=\om$. We use Proposition [propdefK] and Lemma [defcirc] with $\om$ replaced by $\om^{n\_k}$ and w replaced by $K[\om^{n\_k}]$ to deduce the following identity: $$\int\_\Om\varphi \om^{n\_k}+\int\_\Om K[\om^{n\_k}]\cdot\nabla^\perp\varphi=\sum\_{j=1}^k\varphi\bigl\|\_{\Gamma\_j}\int\_\Om\w\_j \om^{n\_k},$$ for all $\varphi\in \yinf$. Letting nk → ∞ and using together with the fact that $K[\om^{n\_k}]\rightharpoonup w$ weakly in L2 implies that $$\int\_\Om\varphi \d\om+\int\_\Om w\cdot\nabla^\perp\varphi=\sum\_{j=1}^k\varphi\bigl\|\_{\Gamma\_j}\int\_\Om\w\_j \d\om.$$ From Lemma [defcirc] we infer that w and $K[\om]$ both have circulation around Γj equal to $-\int\_\Om\w\_j\d\om$, so $K[\om]=w$. In particular, we have that $K[\om]\in L^2(\Om)$. We proved that the limit of any weakly convergent subsequence of $K[\om^n]$ in L2 must necessarily be $K[\om]$. We conclude that the whole sequence $K[\om^n]$ converges to $K[\om]$ weakly in $L^2(\Om)$. Finally, we show the strong convergence of $K[\om^n]$ to $K[\om]$ in $L^2(\Om)$. Given that we already know the weak convergence, it suffices to show the convergence of the norms. We use relations and with $f=f\_n=K[\om^n]$, $F=F\_n=G[\operatorname{curl}f\_n]=G[\om^n]$ and $v=K[\om]$ to obtain that $$\int\_\Omega\|K[\om^n]\|^2= \int\_\Omega K[\om^n]\cdot f\_n =\int\_\Omega K[\om]\nabla^\perp\chi\_n(F\_n\ast\check{\eta}\_n) +\int\_\Omega K[\om]\chi\_n(\nabla^\perp F\_n \ast\check\eta\_n) \equiv J\_1+J\_2.$$ In fact, we assumed in relations and that f ∈ Cc∞(Ω) while $K[\om^n]$ is not compactly supported. However, when we look at the proof of these relations, we observe that we only require sufficient decay at infinity for f. Since $K[\om^n]={\mathscr{O}}(\|x\|^{-2})$ as ∣x∣ → ∞ is sufficient decay, we see that the proofs of and go through for the choice $f=K[\om^n]$. Next, since fn = ∇⊥Fn, we have that $$J\_2=\int\_\Omega \eta\_n\ast(\chi\_n K[\om])\cdot f\_n.$$ By classical results, we know that $\eta\_n\ast(\chi\_n K[\om])\to K[\om]$ strongly in L2(Ω). On the other hand, we have that $f\_n=K[\om^n]\to K[\om]$ weakly in L2(Ω). Therefore, we can pass to the limit in the term J2: $$J\_2 \stackrel{n\to\infty}\longrightarrow \int\_\Omega\|K[\om]\|^2.$$ To bound the term J1, recall that $\operatorname{supp}\nabla\chi\_n\subset \Sigma\_{\frac2n}\cup B\_n^c$. We have that $$\|J\_1\|{\leqslant}\\|K[\om]\\|\_{L^2( \Sigma\_{\frac2n}\cup B\_n^c)}\\|\nabla^\perp\chi\_n(F\_n\ast\check{\eta}\_n)\\|\_{L^2(\Omega)}.$$ As in the estimates that follow relation, we can bound $$\\|\nabla^\perp\chi\_n(F\_n\ast\check{\eta}\_n)\\|\_{L^2(\Omega)}{\leqslant}C \\|\nabla F\_n\\|\_{L^2(\Omega)}=C \\|f\_n\\|\_{L^2(\Omega)}=C \\|K[\om^n]\\|\_{L^2(\Omega)}{\leqslant}C'$$ independently of n. But clearly $$\\|K[\om]\\|\_{L^2( \Sigma\_{\frac2n}\cup B\_n^c)} \stackrel{n\to\infty}\longrightarrow 0$$ so $$J\_1 \stackrel{n\to\infty}\longrightarrow 0.$$ We conclude from the above relations $\\|K[\om^n]\\|\_{L^2(\Omega)}\to \\|K[\om]\\|\_{L^2(\Omega)}$ as n → ∞. This completes the proof of Proposition [aproxom]. We proved in relation the following identity: $$\int\_\Om K[\om]\cdot f=-\int\_\Om \om \, G[\operatorname{curl}f]$$ for all $\om \in C^\infty\_c(\Om)$ and $f\in \bigl(C^\infty\_{c}(\Om)\bigr)^2$. This would allow us to define $K[\om]$ in the sense of distributions for all measures $\om\in \bm$, without requiring $\omega \in \H$. For the sake of convenience we chose, in Definition [defKfield], to define $K[\om]$ in the subset of measures in $\H$. Given a vector field $u\in\L$ which is divergence free and tangent to the boundary such that $\omega=\operatorname{curl}u\in\bm $, one can compute the circulations γ1, …, γk on each of the connected components of the boundary Γ1, …, Γk by using Lemma [defcirc]. Conversely, given vorticity and circulations, one can recover the velocity in this weak setting. The next proposition is the analogue of Proposition [milton] in the weak setting. [uomega] Let $\omega\in\bm\cap \H $ and consider some arbitrary real numbers γ1, …, γk. There exists a unique vector field $u\in \L$ such that curlu = ω, divu = 0, u is tangent to the boundary and with circulations given by γ1, …, γk. Moreover, we have that $$\label{defucirc} u=K[\omega]+\sum\_{j=1}^k\bigl(\gamma\_j+\int\_\Omega\w \_j\d\om \bigr) \bX\_j.$$ The uniqueness part is proved in Proposition [unicdec]. To prove the existence part, we observe that the vector field defined in has all the required properties. Indeed, since $K[\om]\in L^2(\Om)$ we have that $u\in\L$. It is obvious that $\operatorname{curl}u=\om$, divu = 0 and that u is tangent to the boundary. We conclude the proof by recalling that the circulation of $K[\om]$ around Γj is given by $-\int\_\Omega\w \_j\d\omega$. We end this section with the following characterization of the $\H$ norm. [charH-1norm] For any $\omega\in\bm\cap \H $, we have that $\\|\om\\|\_{\H}=\\|K[\om]\\|\_{L^2(\Om)}.$ Given that $K[\om]$ is square-integrable, divergence free, tangent to the boundary and with curl $\om$, we see from the definition of $\\|\om\\|\_{\H}$ given in that it suffices to show that $ \\|K[\om]\\|\_{L^2(\Om)}{\leqslant}\\|v\\|\_{L^2(\Om)}$ for every square integrable vector field v which is divergence free, tangent to the boundary and of curl $\om$. Let v be such a vector field. Then $v-K[\om]$ is square integrable, divergence and curl free and tangent to the boundary. Therefore it must be a linear combination of the harmonic vector fields X1, …, Xn: $$v-K[\om]=\sum\_{j=1}^k\al\_j X\_j.$$ From we see that $$\sum\_{j=1}^k\al\_j X\_j= \left(\sum\_{j=1}^k\al\_j \right)\frac{x^\perp}{2\pi\|x\|^2}+{\mathscr{O}}(\|x\|^{-2})$$ at infinity. Since the left-hand side is square integrable, we deduce that $\sum\_{j=1}^k\al\_j =0$. Clearly $$\\|v\\|\_{L^2(\Om)}^2= \\|K[\om]\\|\_{L^2(\Om)}^2+\\|\sum\_{j=1}^k\al\_j X\_j\\|\_{L^2(\Om)}^2+2\int\_\Om K[\om]\cdot \sum\_{j=1}^k\al\_j X\_j.$$ It suffices to show that the last term on the right-hand side vanishes. Let $\om^n$ be defined as in Proposition [aproxom]. Then, since $K[\om^n]\to K[\om]$ in $L^2(\Om)$, we have that $$\int\_\Om K[\om]\cdot \sum\_{j=1}^k\al\_j X\_j=\lim\_{n\to\infty}\int\_\Om K[\om^n]\cdot \sum\_{j=1}^k\al\_j X\_j.$$ But using that $G[\om^n]$ is bounded and that $\sum\_{j=1}^k\al\_j X\_j$ is curl free and O(∣x∣− 2) at infinity, we have that $$\int\_\Om K[\om^n]\cdot \sum\_{j=1}^k\al\_j X\_j =\int\_\Om \nabla^\perp G[\om^n]\cdot \sum\_{j=1}^k\al\_j X\_j =-\int\_\Om G[\om^n]\operatorname{curl}\sum\_{j=1}^k\al\_j X\_j =0$$ since the harmonic fields Xj are curl free. This completes the proof. An adaptation of Delort’s theorem to exterior domains ===================================================== In this section we provide a precise statement and a proof of Delort’s Theorem in the exterior domain $\Om$. We consider incompressible ideal fluid flow in the exterior domain $\Om \equiv {\mathbb{R}}^2 \setminus \Omega$, modeled by the Euler equations. Recall the initial-boundary-value problem for the 2-D incompressible Euler equations: $$\label{2DincEuler} \left\{ \begin{array}{ll} \partial\_t u + u \cdot \nabla u + \nabla p = 0, & \mbox{ in } {\mathbb{R}}\_+ \times \Om,\\ \text{div } u =0, & \mbox{ in }\overline{{\mathbb{R}}}\_+\times\Om,\\ u \cdot \hat{n} = 0, & \text{ on } \overline{{\mathbb{R}}}\_+ \times \partial \Om,\\ \| u(t,x)\| \rightarrow 0, & \text{ as } \| x\| \rightarrow \infty, \;\; t\in{\mathbb{R}}\_+,\\ u(0,x)=u\_0(x), & \text{ on } \{t=0\}\times\Om. \end{array} \right.$$ Here, u = (u1, u2) is the velocity field and p is the scalar pressure. The evolution equation for ω = curl u is known as the vorticity equation: $$\left\{ \begin{array}{ll} \partial\_t \omega + u \cdot \nabla \omega = 0, &\mbox{ in } {\mathbb{R}}\_+ \times \Omega\\ \mbox{div } u = 0, \;\; \mbox{ curl } u = \omega, & \mbox{ in } \overline{\R}\_+\times\Omega\\ u \cdot \hat{n} = 0, & \text{ on } \overline{\R}\_+\times \partial \Omega,\\ \| u(t,x)\| \rightarrow 0, & \text{ as } \| x\| \rightarrow \infty, \;\; t\in{\mathbb{R}}\_+,\\ \omega(0,x)=\omega\_0(x) = \mbox{ curl } u\_0, & \text{ on } \{t=0\}\times\Omega. \end{array} \right.$$ We begin with a precise formulation of what it means to be a weak solution of. [eulerweaksol] Let $u\_0 \in L^2\loc (\overline\Om)$ and $u = u(x,t) \in L^{\infty}\loc([0,\infty); L^2\loc (\overline\Om)) \cap C^0([0,\infty); \mathscr{D}'(\Om))$ be a vector field. We will say that u is a weak solution of the 2D incompressible Euler equations in $\Om$ with initial data u0 if, for any divergence free test vector field $\phi \in C^{\infty}\_c([0,\infty)\times\Om)$ we have: 1. We have the integral identity $$\label{defveleq} \int\_0^\infty \int\_{\Om}\partial\_t \phi \cdot u \d x \d t + \int\_0^\infty \int\_{\Om} \nabla\phi : u \otimes u \d x\d t + \int\_{\Om} u\_0 \cdot \phi(0,\cdot) \d x = 0.$$ 2. For each time t ∈ [0, ∞), the vector field u( ⋅ , t) is divergence free in the sense of distributions in $\Om$. 3. The boundary condition u( ⋅ , t) ⋅ n = 0 is satisfied in the trace sense in $\partial \Om$, for every time. The condition that $u\in C^0([0,\infty); \mathscr{D}'(\Om))$ is a consequence of $u \in L^{\infty}\loc([0,\infty); L^2\loc (\overline\Om))$ and of the integral relation in (a). We chose to require it explicitly in Definition [eulerweaksol] only to make sense of parts (b) and (c). Moreover, the fact that u is $L^{\infty}\loc([0,\infty);L^2\loc )$ and C([0, ∞); Dʹ) implies that u is continuous into weak $L^2\loc $, which, in particular, implies that it belongs to $L^2\loc $ pointwise in time. This, together with the (pointwise in time) divergence free condition implies that the normal component of u has a trace at the boundary for each fixed time, so that the boundary data can be imposed pointwise in time. In, Delort proved his existence result for bounded domains, full space and compact manifolds without boundaries. However, his proof of the bounded domain case is local, because it is based on the local behavior of certain quadratic expressions in the components of velocity. To extend Delort’s result to an exterior domain we first require a strategy to obtain an approximate solution sequence. This can be done by mollifying initial data and using Kikuchi’s existence result, see. We give now a precise statement of a version of Delort’s existence result for flows in an exterior domain. [theodelort] Assume that $u\_0\in \L $ is divergence free, tangent to the boundary and that $\om\_0 = \mbox{ curl } u\_0 \in\bmp + L^1(\Om)$. Then there exists a global weak solution $u\in L^\infty([0,\infty);\L)$ of the incompressible Euler equations with initial velocity u0. We start by observing that $\om\_0\in\H$. Indeed, we write u0 = v0 + Y, with v0 ∈ L2(Ω) and Y ∈ X. Since both u0 and Y are divergence free and tangent to ∂Ω, so is v0 and, since curl Y = 0, it follows that $\om\_0 = \mbox{ curl } v\_0$. Hence, $\om\_0$ satisfies the requirements in the definition of $\H$. By Lemma [defcirc], the circulations of u0 on each of Γ1, …, Γk are well defined: γ1, 0, …, γk, 0. From Proposition [uomega] we know that $$\label{defuzero} u\_0=K[\omega\_0]+\sum\_{j=1}^k\bigl(\gamma\_{j,0}+\int\_\Omega\w \_j\d\om\_0 \bigr) \bX\_j=K[\omega\_0]+\Y,\quad\text{where } \Y=\sum\_{j=1}^k\bigl(\gamma\_{j,0}+\int\_\Omega\w \_j\d\om\_0 \bigr) \bX\_j.$$ Next, we smooth out $\om\_0$ by convolution with ηn (defined in ). We smooth out the initial vorticity as in Proposition [aproxom] by writing $\om\_0^n=\eta\_n\ast(\chi\_n\om\_0)$ and we define $$\label{defuzerobis} u^n\_0=K[\omega^n\_0]+\Y.$$ From we have that u0 − u0n goes to 0 strongly in $L^2(\Om)$. Since u0n is smooth, one can use Kikuchi’s result to construct a global smooth solution un of the incompressible Euler equations in $\Om$ with initial velocity u0n. Let us define $$\om^{n}=\operatorname{curl}u^{n}\quad\text{and}\quad v^{n}=K[\om^{n}].$$ From Proposition [milton], we know that the circulation of u0n on Γj is given by $$\gamma\_{j,0}+\int\_\Om \w \_j\d\om\_0-\int\_\Om \w \_j\om\_0^n.$$ Since un is smooth, the Kelvin circulation theorem applies so the circulation of un on Γj is given by the same quantity. Using again Proposition [milton] we obtain that $$\label{unprop} u^n=K[\omega^n]+\sum\_{j=1}^k\bigl(\gamma\_{j,0}+\int\_\Om \w \_j\d\om\_0-\int\_\Om \w \_j\om\_0^n+\int\_\Omega\w \_j\om^n \bigr) \bX\_j.$$ Observe now that $u^n-\Y=\mathcal{O}(\|x\|^{-2})$ as ∣x∣ → ∞. Indeed, we know that K[ωn] = O(∣x∣− 2). Using we have that $$u^n-\Y=\frac{x^\perp}{2\pi\|x\|^2}\sum\_{j=1}^k\Bigl( \int\_\Omega\w \_j\om^n -\int\_\Om \w \_j\om\_0^n\Bigr)+\mathcal{O}(\|x\|^{-2}) =\frac{x^\perp}{2\pi\|x\|^2}\Bigl( \int\_\Omega\om^n -\int\_\Om \om\_0^n\Bigr)+\mathcal{O}(\|x\|^{-2})=\mathcal{O}(\|x\|^{-2})$$ as ∣x∣ → ∞. We used above that $\sum\_{j=1}^k\w \_j$ is a harmonic function, equal to 1 on ∂Ω and with a finite limit at ∞ so, by uniqueness, it must be identically equal to 1. Similarly $\partial\_t(u^n-\Y)=\mathcal{O}(\|x\|^{-2})$ as ∣x∣ → ∞. Since $\Y$ is time independent, we have that $$\partial\_t(u^n-\Y)+u^n\cdot\nabla u^n=-\nabla p\_n.$$ In the relation above, the first term on the left-hand side is O(∣x∣− 2) as ∣x∣ → ∞. Thanks to and to the fact that ∇un is uniformly bounded we also have that the second term on the left-hand side is O(∣x∣− 1) as ∣x∣ → ∞. We conclude that the left-hand side above is O(∣x∣− 1) as ∣x∣ → ∞, so it belongs to Lp(Ω) for all p > 2. Since the Leray projector P is well-defined and continuous on every Lp(Ω), 1 < p < ∞ (see for example ), we can apply it to the relation above to obtain that $$\label{equn} \partial\_t(u^n-\Y)+\mathbb{P}(u^n\cdot\nabla u^n)=0.$$ Since the term above belongs to any Lp(Ω), p > 2, and since $u^n-\Y\in L^p(\Omega)$ for all p > 1, we can multiply by $u^n-\Y$ to obtain, after integrating by parts, that $$\begin{aligned} \frac12\dt\\| u^n-\Y\\|\_{L^2(\Om)}^2 &=-\int\_\Om u^n\cdot\nabla u^n\cdot (u^n-\Y) = -\int\_\Om u^n\cdot\nabla Y\cdot (u^n-\Y)\\ &= -\int\_\Om (u^n-\Y)\cdot\nabla \Y\cdot (u^n-\Y) -\int\_\Om \Y\cdot\nabla \Y\cdot (u^n-\Y)\\ &{\leqslant}\\| u^n-\Y\\|\_{L^2(\Om)}^2 \\| \nabla\Y\\|\_{L^\infty(\Om)}+\\| u^n-\Y\\|\_{L^2(\Om)}\\| \Y\cdot\nabla\Y\\|\_{L^\infty(\Om)}.\end{aligned}$$ The Gronwall lemma implies that un − Y is uniformly bounded in $L^\infty\loc([0,\infty);L^2(\Om))$. From the vorticity equation we have that $\om^n$ is bounded in $L^\infty(\R\_+;L^1(\Om))$. Therefore, there exists a subsequence unp of un and some $u\in L^\infty\loc([0,\infty);\Y+L^2(\Om))$ such that unp − u converges to 0 weak \* in $L^\infty\loc([0,\infty);L^2(\Om))$. Passing to further subsequences as needed, we also have that there exists $\om\in L^\infty(\R\_+;\bmp + L^1(\Om))$ so that $\om^{n\_p}\to\om$ weak \* in $L^\infty(\R\_+;\bmp + L^1(\Om))$. To see this, we split $\om\_0$ into its positive and negative parts, so that $\om\_0^{+} \in \bmp$ and $\om\_0^{-}\in L^1(\Om)$. Then, $\om\_0^{n,-} = \eta\_n \ast (\chi\_n\om\_0^-)$ converges weakly (even strongly) in $L^1(\Om)$ to $\om\_0^-$. Let Φn = Φn(t, x) denote the flow map associated to un. Write the corresponding smooth $\om^n$ as $\om^n = \om^{n,+} + \om^{n,-}$, where $\om^{n,+} = \om^{n,+}(t,x) = \om^{n,+}\_0 ((\Phi^n)^{-1}(t,x))$ and $\om^{n,-} = \om^{n,-}(t,x) = \om^{n,-}\_0 ((\Phi^n)^{-1}(t,x))$. Then, since $\om^{n,-}(t,\cdot) $ is a rearrangement of $\om^{n,-}\_0(\cdot)$ and since, by the Dunford-Pettis theorem, $\om\_0^{n,-}$ is uniformly integrable, so is $\om^{n,-}$, uniformly in time. Furthermore, $\om^{n,+} {\geqslant}0$, while $\om^{n,-}{\leqslant}0$. It follows that, passing to subsequences as needed, $\om^{n\_p,+}$ converges weak- \* in $L^\infty\loc(\R\_+;\bmp)$, to some $\om^+ \in L^\infty\loc(\R\_+;\bmp)$, while, by the Dunford-Pettis theorem, $\om^{n\_p,-}$ converges weak- \* in $L^\infty\loc(\R\_+;L^1(\Om))$ to some $\om^- \in L^\infty\loc(\R\_+;L^1(\Om))$, $\om^- {\leqslant}0$. With this notation we find $\om = \om^+ + \om^-$. We can further assume that $\|\om^{n\_p}\|$ converges weak \* in $L^\infty(\R\_+;\bmp)$ to some μ. We prove now that μ is a continuous measure. Since $\om^{n\_p}$ is bounded in $L^\infty(\R\_+;H^{-1}\_{loc}(\Om))$, we have that $\om\in L^\infty(\R\_+;H^{-1}\_{loc}(\Om))$. Therefore $\om$ is a continuous measure. But $\om^-\in L^\infty\_{loc}(\R\_+;L^1(\Om))$ so $\om^-$ is continuous as a measure. Since $\om=\om^++\om^-$ is a continuous measure, we infer that $\om^+$ is a continuous measure too. Finally, from the bound $\|\om^{n\_p}\|{\leqslant}\om^{n,+}-\om^{n,-}$ we deduce that $\|\mu\|{\leqslant}\om^+-\om^-$ so μ must be continuous. We claim now that u1npu2np → u1u2 and (u1np)2 − (u2np)2 → (u1)2 − (u2)2 in the sense of distributions of $\R\_+\times\Om$. To show this, let $\B$ be a ball whose closure is included in $\Om$ and let $f\in C^\infty\_c (\B;\R\_+)$. We denote by ψn the unique function defined on $\B$ such that un = ∇⊥ψn and $\int\_{\B}\psi\_n=0$. Let us define $$\tu^n=\nabla^\perp(f\psi^n)$$ and extend it with zero values on $\R^2\setminus\B$. We define in the same manner ψ and $\tu$ associated to u. Given that un is bounded in $L^\infty\loc([0,\infty);L^2(\B))$, the Poincaré inequality implies that ψn is bounded in $L^\infty\loc([0,\infty);H^1(\B))$. Given that unp − u converges to 0 weak \* in $L^\infty\loc([0,\infty);L^2(\Om))$, every weak limit in $L^\infty\loc([0,\infty);H^1(\B))$ of ψnp is a function of vanishing mean on $\B$ and whose ∇⊥ is u. So such a weak limit is necessarily ψ. Having established that ψnp → ψ weak \* in $L^\infty\loc([0,\infty);H^1(\B))$, it is trivial to see that $\tu^{n\_p}\to\tu$ weak \* in $L^\infty\loc([0,\infty);L^2(\R^2))$. Moreover, we have that $$\operatorname{curl}\tu^n=\operatorname{curl}(fu^n+\psi^n\nabla^\perp f) =f\om^n+2u^n\cdot\nabla^\perp f+\psi^n\triangle f.$$ Clearly $f\om^n$ is bounded in $L^\infty\loc\bigl([0,\infty);L^1(\R^2)\bigr)$. Moreover, given that un and ψn are bounded in $L^2(\B)$ we infer that $$2u^n\cdot\nabla^\perp f+\psi^n\triangle f \quad\text{is bounded in}\quad L^\infty\loc\bigl([0,\infty);L^1(\R^2)\cap L^2(\R^2)\bigr).$$ In particular, $\operatorname{curl}\tu^n$ is bounded in $L^\infty\loc\bigl([0,\infty);L^1(\R^2)\bigr)$. Given that the weak limit of $\|\om^{n\_p}\|$ is a continuous measure, the same holds true for $\|\operatorname{curl}\tu^{n\_p}\|$. Given the informations we have on $\tu^{n\_p}$ we can now apply, to deduce that $$\tu^{n\_p}\_1\tu^{n\_p}\_2\to \tu\_1\tu\_2 \quad\text{and}\quad (\tu^{n\_p}\_1)^2-(\tu^{n\_p}\_2)^2\to (\tu\_1)^2-(\tu\_2)^2$$ in the sense of distributions of $\R\_+\times\R^2$. Given that $f\in C^\infty\_c (\B;\R\_+)$ is arbitrary we infer that holds true in the sense of distributions of $\R\_+\times\B$. Since $\B$ is an arbitrary ball relatively compact in $\Om$, we infer that holds true in the sense of distributions of $\R\_+\times\Om$. Given, it is trivial to pass to the limit in the equation of unp. Indeed, let us consider $\phi\in C^\infty\_c ([0,\infty)\times\Om)$ be a divergence free test vector field. We multiply the equation of unp by ϕ and integrate to obtain $$\int\_\Om u^{n\_p}\_0 \cdot \phi(0,\cdot)+\int\_0^\infty\int\_\Om u^{n\_p}\cdot \partial\_t\phi+\int\_0^\infty\int\_\Om (u^{n\_p}\otimes u^{n\_p})\cdot\nabla\phi=0.$$ When we send np → ∞, the first two terms on the left-hand side converge trivially to the desired limit. The second term passes to the limit thanks to, since (unp ⊗ unp) ⋅ ∇ϕ = u1npu2np(∂1ϕ2 + ∂2ϕ1) + [(u1np)2 − (u2np)2]∂1ϕ1. We obtain that $$\int\_\Om u\_0 \cdot \phi(0,\cdot)+\int\_0^\infty\int\_\Om u\cdot \partial\_t\phi+\int\_0^\infty\int\_\Om (u\otimes u)\cdot\nabla\phi=0$$ which is the formulation in the sense of distributions of the incompressible Euler equations in velocity form. Weak vorticity formulation ========================== The aim of this section is to give a vorticity formulation which is equivalent to the velocity formulation. This is an extension of a result due to S. Schochet, see. More precisely, we will prove the following theorem. [equivth] Let $u\_0\in \L $ be divergence free, tangent to the boundary and assume that $\om\_0 = \mbox{ curl } u\_0 \in\bm$. Let $u\in L^\infty\loc([0,\infty);\L)$ be a divergence free vector field, tangent to the boundary, and such that $\om=\operatorname{curl}u\in L^\infty\loc([0,\infty);\bm)$. Let γj(t) be the circulation of u(t) on Γj (defined a.e. in t thanks to Lemma [defcirc]). Assume that u is a weak solution of the 2D incompressible Euler equations with initial data u0. Then $\gamma\_j\in L^\infty\_{\loc}([0,\infty))$ for all j. Moreover, $\om$ and γj, j ∈ {1, …, k}, verify the following identity: $$\begin{gathered} \label{weakvort} \int\_0^\infty\int\_\Om\partial\_t\varphi\d \om+ \sum\_{j=1}^k\int\_0^\infty \gamma\_j(t) \partial\_t\varphi(t,\cdot)\bigl\|\_{\Gamma\_j}\d t+\sum\_{j=1}^k\int\_0^\infty\bigl(\gamma\_j+\int\_\Omega\w \_j\d\omega\bigr)\int\_\Omega \bX\_j \cdot\nabla\varphi \d\omega\\ +\int\_0^\infty\iint\_{\Om\times\Om}\frac{1}{2} \Big( \nabla\_x \varphi (x) \cdot K (x,y) + \nabla\_y\varphi(y)\cdot K(y,x)\Big)\d\omega(x)\d\omega(y)\\ +\int\_\Om\varphi(0,\cdot)\d \om\_0+\sum\_{j=1}^k\gamma\_{j}(0)\varphi(0,\cdot)\bigl\|\_{\Gamma\_j}=0 \end{gathered}$$ for all test functions $\varphi\in C^\infty\_c([0,\infty);\yinf)$. Conversely, let $\om\_0 \in \bm \cap \H$ and consider real numbers γj, 0, j = 1, …, k. Suppose that $\om\in L^\infty\loc([0,\infty);\bm \cap \H)$ and $\gamma\_j \in L^{\infty}\_{\loc}([0,\infty))$ verify identity for all test functions $\varphi\in C^\infty\_c([0,\infty);\yinf)$, with γj(0) = γj, 0. Set u = u(t) to be the vector field given by Proposition [uomega] in terms of ω = ω(t) and γj = γj(t). Then $u \in L^{\infty}\_{\loc}([0,\infty));\L)$ and u is a weak solution of the 2D incompressible Euler equations with initial data u0 = u(0). We observe from Proposition [propestK] that the function ∇xφ(x) ⋅ K(x, y) + ∇yφ(y) ⋅ K(y, x) is bounded and continuous except on the diagonal. On the other hand, since $\om=\operatorname{curl}u\in L^\infty\loc([0,\infty);\bmp\cap\H)$ the measure $\om\otimes\om$ attaches no mass to the diagonal. We infer that all the terms appearing in are well defined. We remark that can be written as an equation in $\mathscr{D}'(\R\_+)$: $$\begin{gathered} \label{weakvortr} \partial\_t\int\_\Om\theta\d\om+\sum\_{j=1}^k\gamma'\_j \theta\bigl\|\_{\Gamma\_j} =\sum\_{j=1}^k\bigl(\gamma\_j+\int\_\Omega\w \_j\d\omega\bigr)\int\_\Omega \bX\_j \cdot\nabla\theta \d\omega\\ +\iint\_{\Om\times\Om}\frac{1}{2} \Big( \nabla\_x \theta (x) \cdot K (x,y) + \nabla\_y\theta(y)\cdot K(y,x)\Big)\d\omega(x)\d\omega(y)\end{gathered}$$ for all test functions $\theta\in \yinf$. In fact, is equivalent to plus the initial conditions $\om\bigl\|\_{t=0}=\om\_0$ and γj(0) given. To show Theorem [equivth], we observe first that the map φ ↦ ∇⊥φ = ϕ gives a one to one correspondence from $\yinf$ into $C^\infty\_{c,\sigma} (\Om)$. Therefore, the velocity formulation in the sense of distributions (see relation ) can be written as follows $$\label{weakvel} \int\_\Om u\_0 \cdot \nabla^\perp\varphi(0,\cdot)+\int\_0^\infty\int\_\Om u\cdot \partial\_t\nabla^\perp\varphi+\int\_0^\infty\int\_\Om (u\otimes u)\cdot\nabla\nabla^\perp\varphi=0$$ for all test functions $\varphi\in C^\infty\_c ([0,\infty);\yinf)$. Using Lemma [defcirc], we have that $$\label{ident1} \int\_\Om u\_0 \cdot \nabla^\perp\varphi(0,\cdot)= -\int\_\Om \varphi(0,\cdot)\d\om\_0-\sum\_{j=1}^k\gamma\_{j,0} \varphi(0,\cdot)\bigl\|\_{\Gamma\_j}$$ and $$\label{ident2} \int\_0^\infty\int\_\Om u\cdot \partial\_t\nabla^\perp\varphi = -\int\_0^\infty\int\_\Om \partial\_t\varphi\d\om-\sum\_{j=1}^k\int\_0^\infty\gamma\_j \partial\_t \varphi(t,\cdot)\bigl\|\_{\Gamma\_j}.$$ In order to show Theorem [equivth] it suffices to prove the following proposition. [propident] Let $u\in \L$ be a divergence free vector field tangent to the boundary such that $\omega=\operatorname{curl}u\in\bm\cap\H$. Let γ1, …, γk be the circulations of u on each of the connected components of the boundary Γ1, …, Γk. Let $\varphi\in \yinfb$. The following identity holds true: $$\label{identity} \int\_\Omega (u\otimes u):\nabla\nabla^\perp\varphi =-\iint\_{\Omega\times\Omega} H\_\varphi(x,y) \d\omega(x)\d\omega(y)-\sum\_{j=1}^k\al\_j\int\_\Omega \bX\_j \cdot\nabla\varphi \d\omega$$ where $$H\_\varphi(x,y) = \frac{1}{2} \Big( \nabla\_x \varphi (x) \cdot K (x,y) + \nabla\_y\varphi(y)\cdot K(y,x)\Big)\quad\text{and}\quad \al\_j=\gamma\_j+\int\_\Omega\w \_j\d\omega.$$ Indeed, assume that this proposition is proved. Let us first suppose that $u\_0\in \L $ is divergence free, tangent to the boundary and such that $\om\_0 = \mbox{ curl } u\_0 \in\bm$. Let $u\in L^\infty\loc([0,\infty);\L)$ be a divergence free vector field, tangent to the boundary, and such that $\om=\operatorname{curl}u\in L^\infty\loc([0,\infty);\bm)$. Let γj(t) be the circulation of u(t) on Γj. From the definition of the circulation, see relation, we immediately see that $\gamma\_j\in L^\infty\_{\loc}([0,\infty))$. Assume that u is a weak solution of the 2D incompressible Euler equations with initial data u0. Given that $u\in L^\infty\loc([0,\infty);\L)$ and $\om\in L^\infty\loc([0,\infty);\bm\cap\H)$ we have that $u(t)\in \L$ and $\om(t)=\operatorname{curl}u(t)\in \bm\cap\H$ for almost all times t. For those times t we can apply Proposition [propident] to obtain relation for u(t) and $\om(t)$. Integrating this relation in time implies that $$\begin{gathered} \label{ident3} \int\_0^\infty\int\_\Om (u\otimes u)\cdot\nabla\nabla^\perp\varphi = -\int\_0^\infty\iint\_{\Om\times\Om}\frac{1}{2} \Big( \nabla\_x \varphi (x) \cdot K (x,y) + \nabla\_y\varphi(y)\cdot K(y,x)\Big)\d\omega(x)\d\omega(y)\\ -\sum\_{j=1}^k\int\_0^\infty\bigl(\gamma\_j+\int\_\Omega\w \_j\d\omega\bigr)\int\_\Omega \bX\_j \cdot\nabla\varphi \d\omega\d t.\end{gathered}$$ Combining relations, and shows that the left-hand side of is equal up to a sign to the left-hand side of. Conversely, let $\om\_0 \in \bm \cap \H$ and consider γj, 0 ∈ R, j = 1, …, k. Suppose that $\om\in L^\infty\loc([0,\infty);\bm \cap \H)$ and $\gamma\_j \in L^{\infty}\_{\loc}([0,\infty))$ verify identity for all test functions $\varphi\in C^\infty\_c([0,\infty);\yinf)$, with γj(0) = γj, 0. Set u = u(t) to be the vector field given by Proposition [uomega] in terms of ω = ω(t) and γj = γj(t). We begin by noting that, by Proposition [charH-1norm], together with the hypothesis $\om\in L^\infty\loc([0,\infty); \H)$, it follows that $K[\om] \in L^\infty\loc([0,\infty);L^2(\Om))$. Furthermore, since $\om\in L^\infty\loc([0,\infty);\bm )$ and $\gamma\_j \in L^{\infty}\_{\loc}([0,\infty))$, we find that u given by Proposition [uomega], $$u = K[\om] + \sum\_{i=1}^k (\gamma\_j + \int\_{\Om} \mathbf{w}\_j\,\mathrm{d}\om) \mathbf{X}\_j,$$ belongs to $ L^{\infty}\_{\loc}([0,\infty));\L)$. Next, let $\phi \in C^{\infty}\_c([0,\infty)\times\Om)$ be a divergence free test vector field. Clearly, there exists $\varphi \in C^{\infty}\_c([0,\infty);\yinf)$ such that ϕ = ∇⊥φ. With this notation the identities, hold true, and Proposition [propident] implies, as before, that is valid as well. This trivially implies that u is a weak solution of the 2D incompressible Euler equations with initial data u0 = u(0). This completes the proof of Theorem [equivth], once we establish Proposition [propident]. We prove now Proposition [propident]. [Proof of Proposition [propident]] We show first when u and $\om$ are smooth. More precisely, we assume that $\om\in C^\infty\_c (\Om)$. Then, by Proposition [milton] we have that $$u=K[\om]+\sum\_{j=1}^k \al\_j\X\_j,\qquad \al\_j=\gamma\_j+\int\_\Om\w\_j\om.$$ We integrate by parts the left-hand side of : $$\int\_\Om (u\otimes u):\nabla\nabla^\perp\varphi =- \int\_\Om \operatorname{div}(u\otimes u)\cdot\nabla^\perp\varphi =- \int\_\Om u\cdot\nabla u\cdot\nabla^\perp\varphi =\int\_\Om \operatorname{curl}(u\cdot\nabla u)\varphi -\int\_{\partial\Om} u\cdot\nabla u\cdot\hat n^\perp\varphi.$$ We claim that the boundary integral above vanishes. Indeed, let $C\_j=\varphi\bigl\|\_{\Gamma\_j}$ and write $u\bigl\|\_{\Gamma\_j}=\beta\hat n^\perp$ with β a scalar function defined on Γj. Then $$\int\_{\Gamma\_j} u\cdot\nabla u\cdot\hat n^\perp\varphi = C\_j \int\_{\Gamma\_j} \beta \hat n^\perp\cdot\nabla u\cdot\hat n^\perp = C\_j \int\_{\Gamma\_j} \hat n^\perp\cdot\nabla u\cdot u = \frac{C\_j}{2} \int\_{\Gamma\_j} \hat n^\perp\cdot\nabla (\|u\|^2) =0$$ since Γj is a closed curve. Since $\operatorname{curl}(u\cdot\nabla u)=u\cdot\nabla\om$ we infer that $$\begin{gathered} \int\_\Om (u\otimes u):\nabla\nabla^\perp\varphi =\int\_\Om u\cdot\nabla\om\varphi =-\int\_\Om u\cdot\nabla\varphi\om =-\int\_\Om K[\om]\cdot\nabla\varphi\om -\sum\_{j=1}^k\al\_j\int\_\Om \X\_j\cdot\nabla\varphi\om \\ =-\iint\_{\Om\times\Om}K(x,y)\cdot\nabla\varphi(x)\om(x)\om(y)\d x\d y -\sum\_{j=1}^k\al\_j\int\_\Om \X\_j\cdot\nabla\varphi\om. \end{gathered}$$ Symmetrizing the term involving the kernel of the Biot-Savart law K(x, y) implies. We prove now the general case. Let $\om^n$ be defined like in Proposition [aproxom] and let us introduce $$u^n= K[\omega^n]+\sum\_{j=1}^k\bigl(\gamma\_j+\int\_\Omega\w \_j\omega^n\bigr) \bX\_j.$$ From Proposition [uomega], we know that $$u=K[\omega]+\sum\_{j=1}^k\bigl(\gamma\_j+\int\_\Omega\w \_j\d\om \bigr) \bX\_j.$$ Since ωn and un are smooth, we have that $$\int\_\Omega (u^n\otimes u^n):\nabla\nabla^\perp\varphi =-\iint\_{\Omega\times\Omega} H\_\varphi(x,y) \omega^n(x)\omega^n(y)\d x\d y-\sum\_{j=1}^k\al^n\_j\int\_\Omega \bX\_j \cdot\nabla\varphi \ \omega^n$$ where $$\al^n\_j=\gamma\_j+\int\_\Omega\w \_j\omega^n.$$ Given relations,, and it not difficult to pass to the limit in the above relation and obtain. First, from we infer that $\al^n\_j\to \al\_j$ as n → ∞ for all j ∈ {1, …, k}. Next, from we infer that $$u^n\to u \quad\text{strongly in }L^2\loc(\overline\Om).$$ so that $$\int\_\Omega (u^n\otimes u^n):\nabla\nabla^\perp\varphi \stackrel{n\to\infty}\longrightarrow \int\_\Omega (u\otimes u):\nabla\nabla^\perp\varphi.$$ It remains to show that $$\label{convH} \iint\_{\Omega\times\Omega} H\_\varphi(x,y) \omega^n(x)\omega^n(y)\d x\d y \stackrel{n\to\infty}\longrightarrow \iint\_{\Omega\times\Omega} H\_\varphi(x,y) \d\omega(x)\d\omega(y).$$ By Proposition [propestK] we have that the function Hφ(x, y) is bounded and continuous outside the diagonal. Since $\om^n$ is bounded in $L^1(\Om)$, there is a subsequence $\om^{n\_p}$ such that $\|\om^{n\_p}\|$ converges weakly in the sense of measures to some positive bounded measure μ. From Proposition [aproxom], we know that the measure μ is continuous. Therefore, the mass that μ(x) ⊗ μ(y) attaches to the diagonal $D=\{(x,x);\ x\in\Om\}$ vanishes since $$\mu(x)\otimes\mu(y)(D) =\iint\_D1\d\mu(x)\d\mu(y) =\int\_\Om\mu(\{x\})\d\mu(x) =0.$$ From the definition of $\om^n$ it is easy to observe that $$\int\_{\Sigma\_\ep}\|\om^n\|{\leqslant}\|\omega\|(\Sigma\_{\ep+\frac1{2n}}) \quad\text{and}\quad \int\_{B\_R^c}\|\om^n\|{\leqslant}\|\omega\|(B\_{R-1}).$$ This implies that the sequence of measures $\om^n$ is tight. We conclude that the convergence stated in relation holds true for the subsequence $\om^{n\_p}$ as a consequence of Lemma [weakconvmeas]. This completes the proof of Proposition [propident]. [weakvortformul] Let $\om\_0 \in \bm \cap \H$ and consider real numbers γj, 0, j = 1, …, k. We say that the (k + 1)-tuple $(\om, \gamma\_1,\ldots,\gamma\_k)$, is a solution of the weak vorticity formulation of the incompressible 2D Euler equations in $\Om$, with initial vorticity $\om\_0$ and initial circulations γj, 0, if $\om\in L^\infty\loc([0,\infty);\bm \cap \H)$, if $\gamma\_j \in L^{\infty}\_{\loc}([0,\infty))$, j = 1, …, k, and if the identity, with γj(0) = γj, 0, holds true for every test function $\varphi\in C^\infty\_c([0,\infty);\yinf)$. We have shown, in Theorem [equivth], that there is a one-to-one correspondence between weak solutions of the incompressible 2D Euler equations in $\Om$ and solutions of the weak vorticity formulation. Notice that the weak vorticity formulation allows for weak solutions for which the Kelvin circulation theorem is no longer valid along boundary components. Next, our aim is to collect as much information as possible on the circulations γj of a weak solution. Suppose that $(\om, \gamma\_1,\ldots,\gamma\_k)$ is a solution of the weak vorticity formulation as in Definition [weakvortformul]. Then the circulations γj, j = 1, …, k can be expressed uniquely in terms of $\om$. Indeed, take in successive test functions $\theta\_{\ell} \in \yinf$, vanishing in a neighborhood of the boundary except in the neighborhood of one of the Γℓ, where it equals 1. We then obtain the following linear system of ODEs for the circulations: $$\begin{gathered} \partial\_t\int\_\Om\theta\_{\ell}\d\om+ \gamma'\_{\ell} =\sum\_{j=1}^k\bigl(\gamma\_j+\int\_\Omega\w \_j\d\omega\bigr)\int\_\Omega \bX\_j \cdot\nabla\theta\_{\ell} \d\omega\\ +\iint\_{\Om\times\Om}\frac{1}{2} \Big( \nabla\_x \theta\_{\ell} (x) \cdot K (x,y) + \nabla\_y\theta\_{\ell}(y)\cdot K(y,x)\Big)\d\omega(x)\d\omega(y),\end{gathered}$$ ℓ = 1, …, k. Clearly, this system of ODEs for the circulations has a unique solution. So we can express the circulations γj in terms of ω. Plugging the formula for the circulations in γj in and taking a test function θ ∈ Cc∞(Ω), it is possible to obtain an equation in the sense of distributions for ω only. However, this equation is not very enlightening. [en-e-thm] Let $(\om, \gamma\_1,\ldots,\gamma\_k)$ be a solution of the weak vorticity formulation as in Definition [weakvortformul]. Then the following conservation law holds true: $$\label{en-e} \int\_\Om 1\d\om+\sum\_{j=1}^k\gamma\_j(t)=\int\_\Om1\d\om\_0+\sum\_{j=1}^k\gamma\_{j,0},$$ for all t ≥ 0. Let $\rho\in C^\infty\_c (\R\_+)$ and $\theta\in C^\infty\_c (\R^2;[0,1]$ such that θ(x) = 1 if ∣x∣ ≤ 1 and θ(x) = 0 if ∣x∣ ≥ 2. Set θn(x) = θ(x/n). We use with φ = ρ(t)θn(x) and n sufficiently large to obtain $$\begin{gathered} \label{conservation} \int\_0^\infty\rho'\int\_\Om\theta\_n\d \om+ \sum\_{j=1}^k\int\_0^\infty \gamma\_j \rho' +\sum\_{j=1}^k\int\_0^\infty\rho\bigl(\gamma\_j+\int\_\Omega\w \_j\d\omega\bigr)\int\_\Omega \bX\_j \cdot\nabla\theta\_n\d\omega\\ +\int\_0^\infty\rho\iint\_{\Om\times\Om}\frac{1}{2} \Big( \nabla\_x \theta\_n (x) \cdot K (x,y) + \nabla\_y\theta\_n(y)\cdot K(y,x)\Big)\d\omega(x)\d\omega(y)\\ +\rho(0)\int\_\Om\theta\_n\d \om\_0+\rho(0)\sum\_{j=1}^k\gamma\_{j,0}=0.\end{gathered}$$ We let now n → ∞. By the dominated convergence theorem we have that $$\int\_0^\infty\rho'\int\_\Om\theta\_n\d \om\to \int\_0^\infty\rho'\int\_\Om1\d \om \quad\text{and}\quad \int\_\Om\theta\_n\d \om\_0\to \int\_\Om1\d \om\_0.$$ Moreover $$\Bigl\|\int\_\Omega \bX\_j \cdot\nabla\theta\_n\d\omega\Bigr\| {\leqslant}\\| \bX\_j\\|\_{L^\infty}\\| \nabla\theta\_n\\|\_{L^\infty}\int\_\Om1\d\om={\mathscr{O}}(1/n)$$ uniformly in time. Using relation we also have that $$\| \nabla\_x \theta\_n (x) \cdot K (x,y) + \nabla\_y\theta\_n(y)\cdot K(y,x)\| {\leqslant}M\_2\\|\nabla\theta\_n\\|\_{W^{1,\infty}(\Om)}={\mathscr{O}}(1/n)$$ uniformly in t and x. We conclude that after passing to the limit n → ∞ in we get $$\int\_0^\infty\rho'(\int\_\Om1\d \om+ \sum\_{j=1}^k\gamma\_j) +\rho(0)(\int\_\Om1\d \om\_0+\sum\_{j=1}^k\gamma\_{j,0})=0.$$ This completes the proof of. Weak solutions obtained by mollifying initial data ================================================== In this section we will specify those properties enjoyed by weak solutions which are weak limits of smooth solutions, obtained by smoothing out initial data. We begin by recalling a stronger notion of weak solution, introduced by two of the authors in and, called boundary-coupled weak solution for flows in a domain with a single, infinite, connected boundary. In that context a boundary-coupled weak solution is a solution satisfying the weak vorticity formulation for the wider class of admissible test functions vanishing on the boundary, but not necessarily their derivatives (instead of vanishing in a neighborhood of the boundary as assumed in relation ). Let $\om\_0 \in \bm \cap \H$ and consider real numbers γj, 0, j = 1, …, k. We say that the (k + 1)-tuple $(\om, \gamma\_1,\ldots,\gamma\_k)$, is a boundary-coupled weak solution of the incompressible 2D Euler equations in $\Om$, with initial vorticity $\om\_0$ and initial circulations γj, 0, if $\om\in L^\infty\loc([0,\infty);\bm \cap \H)$, if $\gamma\_j \in L^{\infty}\_{\loc}([0,\infty))$, j = 1, …, k, and if the identity, with γj(0) = γj, 0, holds true for every test function $\varphi\in C^\infty\_c([0,\infty);\yinfb)$. We will see that a sufficient condition for a weak limit of smooth solutions to satisfy this stronger notion is that the circulations γj be conserved. [thmsomething] Let $\om\_0 \in (\bmp + L^1(\Om))\cap \H$, and consider γ0j ∈ R, j = 1, …, k. Let $(\om,\gamma\_1,\ldots,\gamma\_k)$ be a solution of the weak vorticity formulation as constructed in Theorem [theodelort], i.e. by smoothing out the initial vorticity and passing to the limit using the argument given by Delort. Then we have that 1. [en-a] γj(t) ≥ γj, 0 for almost every t ≥ 0. 2. [en-c] If γj(t) ≡ γj, 0 for almost all t > 0 then the solution is a boundary-coupled weak solution. To prove, we go back to the notation and proof of Theorem [theodelort]. We have that $\om^n$ is bounded in $L^\infty\loc(\R\_+;\bmp + L^1(\Om))$, so there exists some $\omb\in L^\infty\loc(\R\_+;\bmb)$ and a subsequence $\om^{n\_p}$ such that $\om^{n\_p}\rightharpoonup\omb$ in $L^\infty\loc(\R\_+;\bmb)$ weak \* . We also recall that $\om^n = \om^{n,+} + \om^{n,-}$, with $\om^{n,+} {\geqslant}0$ and $\om^{n,-} {\leqslant}0$, such that $\om^{n,+} $ is bounded in $L^\infty\loc({\mathbb{R}}\_+,\bmpb)$ and, passing to subsequences if needed, such that $\om^{n\_p,-} \rightharpoonup \om^-$ weak- \* in $L^\infty\loc({\mathbb{R}}\_+,L^1(\Om))$. Hence, passing to further subsequences if necessary, there exists $\omb^+ \in L^\infty\loc(\R\_+;\bmpb)$ such that $\om^{n\_p,+}\rightharpoonup\omb^+$ weak \* in $L^\infty\loc(\R\_+;\bmpb)$. Thus $\omb - \om^- = \omb^+ \in L^\infty\loc({\mathbb{R}}\_+,\bmpb)$. We can assume without loss of generality that the subsequence np is the same as the one considered in the proof of Theorem [theodelort]. Since the solution un is smooth, the Kelvin circulation theorem holds true so the circulation of un on Γj is constant in time, denoted by γj, 0n. We get from relations, and that $$\gamma\_{j,0}^n=\gamma\_{j,0}+\int\_\Om\w\_j\d\om\_0-\int\_\Om\w\_j \om\_0^n.$$ We clearly have that γj, 0n → γj, 0 as n → ∞. Using we have that $$\label{needthisforfinal1} \int\_0^\infty\int\_\Om \varphi\om^{n\_p}+ \int\_0^\infty\int\_\Om u^{n\_p}\cdot\nabla^\perp\varphi=-\sum\_{j=1}^k \gamma\_{j,0}^n \int\_0^\infty\varphi\bigl\|\_{\Gamma\_j}$$ for all $\varphi\in C^\infty\_c ([0,\infty);\yinf)$. Passing to the limit p → ∞ above yields $$\label{needthisforfinal2} \int\_0^\infty\int\_{\overline\Om} \varphi\d\omb+ \int\_0^\infty\int\_\Om u\cdot\nabla^\perp\varphi=-\sum\_{j=1}^k \gamma\_{j,0} \int\_0^\infty\varphi\bigl\|\_{\Gamma\_j}.$$ Using again Lemma [defcirc] we have that $$\int\_0^\infty\int\_{\Om} \varphi\d\om+ \int\_0^\infty\int\_\Om u\cdot\nabla^\perp\varphi=-\sum\_{j=1}^k \int\_0^\infty\gamma\_j\varphi\bigl\|\_{\Gamma\_j}.$$ Subtracting the two previous relations we get that $$\sum\_{j=1}^k \int\_0^\infty\bigl[\gamma\_j- \gamma\_{j,0}-\omb(\Gamma\_j)\bigr]\varphi	arxiv_0000538
is large. In any case, this shows that in the regime m ≫ n/s the speed of change might have a non-trivial effect on the problem difficulty. Exploring this is out of the scope of this work, but might be an interesting topic of future research. Final remarks ============= In this paper we studied the problem of the detection of signals that evolve dynamically over time. We introduced a simple model for the evolution of the signal that allowed us to explicitly characterize the difficulty of the problem with a special regard to the effect of the speed of change. We also showed the potential advantages that adaptively collecting the observations bring to the table and showed that these are more and more pronounced as the speed of change decreases, which is in line with previous results dealing with signal detection using adaptive sensing. The lower bounds derived in this paper provide a clear picture of the role of the rate of change parameter p, but unfortunately still do not span the entire range of problems we would like to consider (e.g. Theorem [thm:nalower] applies only to p = 0, 1 and part (ii) of Theorem [thm:1sparselower] applies only to s = 1). The latter difficulties appear to be mostly technical and the authors suspect these might be possible to address with carefully chosen reductions. Our contributions merely scratch the surface of this interesting problem, and below we highlight a few interesting directions for future work in this regard. Large vs. small sample regimes: in this work we focus primarily on the case m ≈ n/s, which may be deemed as the small sample regime. When the number of measurements m is significantly larger the type of tests and performance tradeoffs will likely be different, even under the non-adaptive sensing paradigm. For instance, we expect the signal dynamics to have an effect on performance, meaning that it is easier to detect signals non-adaptively when p is smaller. Other interesting questions arise in that setting as well — what is the optimal non-adaptive sensing design? These questions become even more intriguing when one considers adaptive sensing. Restricted dynamics: in the model considered in this paper when signal components change they can move to any unoccupied location in the signal vector. This assumption simplifies the setup, but in some applications might be too unrestrictive. For instance, if signal components can only move to adjacent locations at each time step the effect of the speed of change will likely be less pronounced in the difficulty of detection (at least for adaptive sensing). Understanding the effect of such restrictions could prove valuable in certain applications, such as detection of a disease outbreak in a network, besides being interesting from a theoretical point of view. Structures: in certain situations the signal support can be assumed to have structure to it, for instance all anomalous items might be consecutive or have some other pattern. In some cases the structure of the support has a huge effect on the difficulty of the problems of detection and recovery (see for instance ). How structural restrictions affect these tasks for dynamically evolving signals could be a fruitful avenue of research. Support recovery: another common question in such settings is how well can we estimate the support of a signal. That is, instead of deciding only if there are anomalous items or not, we need to determine which of the items are anomalous. This is also an interesting problem to study for dynamically evolving signals, although a precise formulation of the objective and performance metric for such estimators is less immediate than for static signals. Acknowledgements ================ This work was partially supported by a grant from the Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO 613.001.114). We are very grateful for the comments of the two anonymous referees, which helped improving the presentation. Appendix ======== [Proof of Lemma [lem:event]] We write $$\begin{aligned} {\mathbb{P}}(\Omega\_c ) & \geq {\mathbb{E}}\left( \{ N-1 >mp/2\} \cap \{ \forall j:\ l\_j \leq cm/N \} \right) \\ & = {\mathbb{E}}\left[ {\mathbf{1}}\{ N-1 > mp/2 \} {\mathbb{E}}\left[ {\mathbf{1}}\{ \forall j:\ l\_j\leq cm/N \} \ \big\| N \right] \right] \,\end{aligned}$$ We first lower bound the inner conditional probability. Note that if N ≤ c this probability is one (since cm/N ≥ m and lj ≤ m by definition). When N > c, we will upper bound the probability of the complementary event. Note that given N the distribution of \theta is uniform from the set of 0 − 1 sequences of length m containing exactly N ones, and for which also θm = 1. Hence, to upper bound P(∃j : lj > cm/N), we simply need to count the number of sequences described above for which we have a long block. We can get an upper bound on this count in the following way. First note that since the last element of the sequence is always one, we can simply think of sequences of length [m − 1] containing N − 1 ones. Consider an interval of length cm/N in the set [m − 1]. Now consider the sequences containing N − 1 ones, and for which there are no ones in the aforementioned interval. Note that for all such sequences the existence of at least one long interval is guaranteed. We can simply count how many 0 − 1 sequences can be generated like this. This number is an upper bound on the number of 0 − 1 sequences that have N ones, the last element of the sequence is one and for which ∃j : lj > cm/N. We thus have $$\begin{aligned} {\mathbb{P}}(\exists j:\ l\_j> cm/N \|N) & \leq (m-cm/N) \frac{{m-cm/N \choose N-1}}{{m-1 \choose N-1}} \\ & = (m-cm/N) \frac{(m-cm/N)(m-cm/N -1)\dots (m-cm/N -N+2)}{(m-1)(m-2)\dots (m-N+1)} \\ & \leq \frac{m}{m-1}(1-c/N) \left( \frac{m-cm/N}{m-2} \right)^{N-2} \\ & < \left( \frac{m-cm/N}{m-2} \right)^{N-2} \.\end{aligned}$$ Now consider the logarithm of the expression above. Using log(1 + x) ≤ x, we get $$\begin{aligned} \log {\mathbb{P}}(\exists j:\ l\_j> cm/N \|N) & < (N-2) \left( \log \frac{m}{m-2} + \log (1-c/N) \right) \\ & \leq (N-2) \left( \frac{2}{m-2} - \frac{c}{N} \right) \\ & \leq -\log{2} \,\end{aligned}$$ whenever c ≥ 6 + 3log2, using the fact that 3 ≤ c ≤ N ≤ m. Hence P(Ωc) ≥ P(N − 1 > mp/2)/2. All that remains is to use the fact that N − 1 ∼ Bin(m − 1, p). For instance Chebyshev’s inequality yields $${\mathbb{P}}(N-1 \leq mp/2) \leq \frac{4(m-1)p(1-p)}{(mp)^2} \leq 1/2 \,$$ when p ≥ 8/m and so the claim is proved. --- 1. In principle one could ensure these are sampled without replacement from [n], but this would only unnecessarily complicate the analysis without yielding significant performance gains.[↩](#fnref1) 2. In this informal discussion, the notations ≈ and $\gtrsim$ hide constant factors and/or log(1/ɛ) terms.[↩](#fnref2) 3. In principle one can recall that N − 1 ∼ Bin(m − 1, p) and proceed from there, although it will overcomplicate the derivation. In any case, this will at most allow us to replace the term p2 by p inside the logarithm in the statement of the theorem, which is not very relevant.[↩](#fnref3) Are there needles in a moving haystack? Adaptive sensing for detection of dynamically evolving signals ====================================================================================================== In this paper we investigate the problem of detecting dynamically evolving signals. We model the signal as an n dimensional vector that is either zero or has s non-zero components. At each time step t ∈ N the non-zero components change their location independently with probability p. The statistical problem is to decide whether the signal is a zero vector or in fact it has non-zero components. This decision is based on m noisy observations of individual signal components collected at times t = 1, …, m. We consider two different sensing paradigms, namely adaptive and non-adaptive sensing. For non-adaptive sensing the choice of components to measure has to be decided before the data collection process started, while for adaptive sensing one can adjust the sensing process based on observations collected earlier. We characterize the difficulty of this detection problem in both sensing paradigms in terms of the aforementioned parameters, with special interest to the speed of change of the active components. In addition we provide an adaptive sensing algorithm for this problem and contrast its performance to that of non-adaptive detection algorithms. Introduction ============ Detection of sparse signals is a problem that has been studied with great attention in the past. The usual setting of this problem involves a (potentially) very large number of items, of which a (typically) much smaller number may be exhibiting anomalous behavior. A natural question one can ask if it is possible to reliably detect if there are indeed some items showing anomalous behavior? Questions like this are encountered in a number of research fields. Some examples include epidemiology where one wishes to quickly detect an outbreak or the environmental risk factors of a disease, identifying changes between multiple images, and microarray data studies to name a few. A common point in the examples above is that even though it is not known which items are anomalous, their identity remains fixed throughout the sampling/measurement process. However, in certain situations the identity of these items may change over time. Consider for instance a signal intelligence setting where one wishes to detect covert communications. Suppose that our task is to survey a signal spectrum, a small fraction of which may be used for communication, meaning that some frequencies would exhibit increased power. On one hand we do not know beforehand which frequencies are used, but also the other parties may change the frequencies they communicate through over time. This means we will be chasing a moving target. This introduces a further hindrance in our ability to detect whether someone is using the surveyed signal spectrum for covert communications. Other motivating examples for such a problem include spectrum scanning in a cognitive radio system, detection of hot spots of a rapidly spreading disease, detection of momentary astronomical events or intrusions into computer systems. The main question that we aim to answer in this paper is how the dynamical aspects of the signal affect the difficulty of the detection problem. In the more classical framework of the signal detection problem, inference is based on observations that are collected non-adaptively. However, dealing with time-dependent signals naturally leads to a setting where measurements can be obtained in a sequential and adaptive manner, using information gleaned in the past to guide subsequent sensing actions. Furthermore, in certain situations it is impossible to monitor the entire system at once, but instead one can only partially observe the system at any given time. It is known that, in certain situations, adaptive sensing procedures can very significantly outperform non-adaptive ones in signal detection tasks. Hence our goal is to understand the differences between adaptive and non-adaptive sensing procedures when used for detecting dynamically evolving signals, in situations where the system can only be partially monitored. #### Contributions: In this paper we introduce a simple framework for studying the detection problem of time-evolving signals. Our signal of interest is an n-dimensional vector xt ∈ Rn, where t ∈ N denotes the time index. We take a hypothesis testing point of view. Under the null the signal is static and equal to the zero vector for all t, while under the alternative the signal is a time-evolving s-sparse vector. At each time step t ∈ N we flip a biased coin independently for each non-zero signal component to decide if these will “move” to a different location. Thus, the coin bias p encodes the speed of change of the signal support in some sense. At each time step we are allowed to select one component of the signal to observe through additive standard normal noise, and we are allowed to collect up to m measurements. Our goal is to decide whether the signal is zero or not, based on the collected observations. We present an adaptive sensing algorithm that addresses the above problem, and show it is near-optimal by deriving the fundamental performance limits of any sensing and detection procedure. We do this in both the adaptive sensing and non-adaptive sensing settings for a range of parameter values p and s. It is easy to see that the above problem can not be solved reliably unless we are allowed to collect on the order of n/s measurements. When the number of measurements is of this order, we can reliably detect the presence of the signal when the smallest non-zero component scales roughly like $\sqrt{p \log (n/s)}$ in the adaptive sensing setting (Theorems [thm:upper] and [thm:1sparselower]). In the non-adaptive sensing setting detection is possible only when the smallest non-zero component scales like $\sqrt{\log (n/s)}$ (Theorem [thm:nalower]). Hence, under the adaptive sensing paradigm the speed of change influences the difficulty of the detection problem, with slowly changing signals being easier to detect. Contrasting this, in the non-adaptive sensing setting the speed of change appears to have no strong effect in the problem difficulty when m is of the order n/s. When the number of measurements m is significantly larger than n/s the picture changes quite a bit, and a theoretical analysis of that case is beyond the contribution of this paper. Nevertheless we provide some simulation results indicating that, in the non-adaptive sensing setting, the signal dynamics will then influence the detection ability. Despite its simplicity, the setting introduced in this paper provides a good starting point to understand the problem of detecting dynamically evolving signals. Although we provide several answers in this setting many questions remain (both technical and conceptual). We hope that this work opens the door for many interesting and exciting extensions and developments, some of which are highlighted in Section [sec:end]. #### Related work: The setting where the identity of the anomalous items is fixed over time has been widely studied in the literature. Classically this problem has been addressed when each entry of the vector is observed exactly once. In this context both the fundamental limits of the detection problem and the optimal tests are well understood (see and references therein). The same problem has been investigated in the adaptive sensing setting as well. In the authors provide an efficient adaptive sensing algorithm for identifying a few anomalous items among a large number of items. These results were generalized in to cope with a wide variety of distributions. The algorithms outlined in these works can in principle also be used to solve the detection problem, that is where only the presence or absence of anomalous items needs to be decided. In and bounds on the fundamental difficulty of the estimation problem were derived, whereas in bounds for the detection problems were provided as well. Our work here has a similar flavor to all the above, but tackling the problem when the anomalous items may change positions while the measurement process is taking place. This brings a new temporal dimension to the signal detection problems referenced above. Statistical inference problems pertaining time-dependent signals have been investigated in various settings in the past. However, the papers referenced below only have varying degrees of connection to the problem we are considering, as despite our best efforts, we were only able to find a few instances that resemble our setting. A setting that has some degree of temporal dependence is the monitoring of multi-channel systems. This problem was introduced in and later revisited in and. In this setting each channel of a multi-channel system contains a Wiener process, a few of which are anomalous and have a deterministic drift. The observer is allowed to monitor one channel at a time with the goal to localize the anomalous channels as quickly as possible. Although there is a clear temporal aspect to these problems, the anomalous channels identity is unchanged during the process. Another prototypical example of inference concerning temporal data is change-point detection in a system involving multiple processes. In this problem we have multiple sensors observing stochastic processes. After some unknown time a change occurs in the statistical behavior of some of the processes, and our goal is to detect when such a change occurs as quickly as possible. This setting has been studied in, a Bayesian version of the problem was investigated in, while the authors of deal with a version of the above problem where only one of the sensors is compromised. This setting shares similarities to ours, but there are some key differences. In the change-point detection setting, once a process becomes anomalous it remains so indefinitely. Since some processes are bound to exhibit anomalous behavior, the goal is to minimize the detection delay. Contrasting this, in the setting we consider an anomalous process can revert back to the nominal state, and there is a possibility that none of the processes are anomalous at any time. Hence our goal is to decide between the presence or absence of any anomalous processes over the measurement horizon. A set of more closely related work is concerned with the spectrum scanning of multichannel cognitive radio systems. Here the aim is to quickly and accurately determine the availability of each spectrum band of a multi-band system where the occupancy status changes over time. Alternatively one might only aim to quickly find a single band that is available. This problem has been studied in and, in which the authors provide efficient algorithms for the problem at hand. A very similar problem was investigated in, where one observes multiple ON/OFF processes and wishes to catch one in the ON state. Although the underlying models of these problems come very close to the one we consider, these works are also change-point detection problems in spirit. Hence a similar comment applies here as well, namely that the goal of the algorithms of and is to detect a change-point while minimizing some notion of regret (such as detection delay or sampling cost), which is somewhat different to the problem we are aiming to tackle. #### Organization: Section [sec:setup] introduces the problem setup, including the signal and observation models and the inference goals. In Section [sec:upper] we introduce an adaptive sensing algorithm and analyze its performance. Section [sec:lower] is dedicated to the characterization of the difficulty of the detection of dynamically evolving signals. In particular we show that the algorithm presented in Section [sec:upper] is near-optimal, and examine the difference between adaptive and non-adaptive sensing procedures. In Section [sec:sim] we present numerical evidence supporting a conjecture on the non-adaptive sensing performance limit in the regime when m is of the order n/s. Concluding remarks and avenues for future research are provided in Section [sec:end]. Problem setup ============= For notational convenience let [k] = {1, …, k} where k ∈ N. In our setting the underlying (unobserved) signal at time t is a n-dimensional vector, where time t ∈ N is discrete. Let μ > 0 and denote the unknown signal at time t ∈ N by ${{\boldsymbol{x}}}^{(t)}\equiv\left(x\_1^{(t)},\ldots,x\_n^{(t)}\right)\in{\mathbb{R}}^n$, where $$x^{(t)}\_i = \left\{ \begin{array}{ll} \mu & \text{ if }i \in S^{(t)}\\ 0 & \text{ if } i \notin S^{(t)} \end{array} \right. \,$$ and S(t) ⊂ [n] is the support of the signal at time t. We refer to the components of ${{\boldsymbol{x}}}^{(t)}$ corresponding to the support S(t) as the active components of the signal at time t. In Section [subsec:signalmodel] we model the signal as a random process with the property that, at any time, the number of active components is much smaller than n. In this idealized model the active components of ${{\boldsymbol{x}}}^{(t)}$ have all same value, which might seem restrictive at first. However, when the active components have different signs and magnitudes, the arguments of all the proofs hold throughout the paper with μ playing the role of the minimum absolute value of the active components. Although a more refined analysis is likely possible, where the minimum is replaced by a suitable function of the magnitudes of active components, we choose to sacrifice generality for the sake of clarity (see also Remark [rem:minimumvsaverage] below). The signal is only observable through m noisy coordinate-wise measurements of the form Yt = xAt(t) + Wt, t ∈ [m], where At ∈ [n] is the index of the entry of the signal measured at time t and Wt are independent and identically distributed (i.i.d.) standard normal random variables. In the general adaptive sensing setting At is a (possibly random) measurable function of {Yj, Aj}j ∈ [t − 1] and Wt is independent of $\{{{\boldsymbol{x}}}^{(j)},A\_j \}\_{j\in [t]}$ and {Yj}j ∈ [t − 1]. This means the choice of signal component to be measured can depend on the past observations. A more restrictive setting is that of non-adaptive sensing, where the choice of components to be measured has to be made before any data is collected. Formally At is independent from {Yj, Aj}j ∈ [t − 1] for all t ∈ [m]. This measurement model is very similar to that of, and, where measurements are of the form Yt = xAt + Γt− 1Wt, t = 1, 2, …, when x is a (time-independent) signal, At are as above, and Γt ∈ R represent the precision of the measurements (that can be also chosen adaptively). In those papers the authors impose a restriction on the total precision used (and not on the number of measurements). However, since often the precision is related to the amount of time we have for an observation it is somewhat more appealing to consider fixed precision measurements instead. See also Remark [rem:discretevscontinuous] for an alternative model closer in spirit to that of the above papers. Recently considered an extension of the classical sparse signal detection problem in which the measurements are heteroscedastic, and derived the asymptotic constants of the detection boundary. In principle, a model similar in spirit to the one presented in that work could also be considered here as well, by assuming that measurements on active components not only have elevated means, but also variance different to 1. The ideas of can be used to modify our detection procedure (in particular the Sequential Thresholding Test – see Algorithm [STT]) to craft a procedure that can deal with measurements of different variances. However, the question of heteroscedasticity for dynamically evolving signals is too rich to be dealt with in the present work. Signal dynamics --------------- We consider what might be the simplest non-trivial stochastic model for the evolution of the signal. Our goal is to model situations where the signal support S(t) changes “slowly” over time. For concreteness consider first a particular situation, where we assume that at any time t there is a single active component (so ∣S(t)∣ = 1 for all t ∈ N). We model the support evolution as a Markov process: the support S(1) is chosen uniformly at random over the set [n] (that is, the active component is equally likely to be any of the [n] components); for t ≥ 1 we flip a biased coin with heads probability p ∈ [0, 1] independent of all the past, and if the outcome is heads then S(t + 1) is chosen uniformly at random over [n], otherwise S(t + 1) = S(t). In words, at each time instant the active component stays in place with probability 1 − p and “jumps” to another location with probability p. Thus when p = 1 the signal has a new support drawn uniformly at random at each time t ∈ N, whereas in case p = 0 the support is chosen randomly at the beginning and stays the same over time. In general, the parameter p can be interpreted as the speed of change of the support, with larger values corresponding to a faster rate of change. This basic model of signal dynamics can be easily generalized to multiple active components model as follows. Let s ∈ [n] be the sparsity of our signal. We enforce that ∣S(t)∣ = s for t ∈ N, meaning the signal sparsity does not change over time. For t = 1, S(t) is chosen uniformly at random from the set {S ⊆ [n] : ∣S∣ = s}. For time t ≥ 1, we flip s independent biased coins, each corresponding to an active component, to decide which components move and which stay in the same place. Formally take p ∈ [0, 1] and let θi(t) ∼ Ber(p) be independent for every i ∈ [s], t ∈ N. Consider an enumeration of S(t) as S(t) ≡ {Si(t)}i ∈ [s]. If θi(t) = 0 component Si(t) will also be included in S(t + 1), otherwise it will move. The support set S(t + 1) is chosen uniformly at random from the set $$\bigg\{ S \subset [n]:\ \|S\|=s,\ S\cap S^{(t)} = \{ S^{(t)}\_i: \theta^{(t)}\_i =0 \} \bigg\} \.$$ For illustration purposes we provide some simulated results in Figure [fig:support] (n is chosen quite small for visual clarity only). [fig:support] [rem:discretevscontinuous] Although we consider time to be discrete, continuous-time counterparts of this model are certainly possible (e.g., by taking the transition times to be generated by a Poisson process). A realistic measurement model in this case would require the variance of the observation noise to be inversely proportional to the time between consecutive measurements, effectively playing a similar role to the precision parameter as in. Testing if a signal is present ------------------------------ In the setting described one can envision several inference goals. One might try to “track” the active components of the signal, attempting to minimize the total number of errors over time. A somewhat different and in a sense statistically easier goal is to detect the presence of a signal, attempting to answer the question: are there any needles in this moving haystack? This is the question we pursue in this paper, and it can be naturally formulated as a binary hypothesis test. Under the null hypothesis there is no signal present, that is S(t) = ∅ for every t ∈ N. Under the alternative hypothesis there is a signal support evolving according to the model described above, for some s ∈ [n] and p ∈ [0, 1]. Ultimately, after we collected m observations we have to decide whether or not to reject the null hypothesis. Formally, let Ψ : {At, Yt}t ∈ [m] → {0, 1} be a test function where the outcome 1 indicates the null hypothesis should be rejected. We evaluate the performance of any test Ψ ≡ Ψ({At, Yt}t ∈ [m]) in terms of the maximum of the type I and type II error probabilities, which we call the risk of a test R(Ψ). Namely we require R(Ψ) ≡ maxi = 0, 1Pi(Ψ ≠ i) ≤ ɛ, with some fixed ɛ ∈ (0, 1/2), where P0 and P1 denote the probability measure of the observations and the null and alternative hypothesis, respectively. Later on we also use the notation Ei, i ∈ {0, 1} to denote the expectation operator under the null and alternative hypothesis respectively. Note that both the null and alternative hypothesis are simple in the current setup (as we assume p and μ to be known). In particular, the density of the observations y = (y1, …, ym) under the alternative can be written as the following mixture: $${\ensuremath {\,\mathrm{d}}}{\mathbb{P}}\_1 ({\mathbf{y}}) = {\mathbb{E}}\left[ \prod\_{t\in [m]} g \left( A\_t \| \{ y\_j,A\_j \}\_{j\in [t-1]} \right) \left( {\mathbf{1}}\{ A\_t \in S^{(t)} \} f\_\mu (y\_t) + {\mathbf{1}}\{ A\_t \notin S^{(t)} \} f\_0 (y\_t) \right) \right] \,$$ where fμ is the density of a normal distribution with mean μ and variance 1, {S(t)}t ∈ [m] are the supports evolving as defined in Section [sec:setup], and g(At∣{yj, Aj}j ∈ [t − 1]) is the density of the sensing action at time t. Note, however, that our detection procedures in Section [sec:upper] do not require knowledge of μ or p. The main goal of this work is to understand how large the signal strength μ needs to be, as a function of n, m, s, p and ɛ to ensure is satisfied. To this end we first propose a specific adaptive sensing algorithm and evaluate its performance in Section [sec:upper]. Furthermore in Section [sec:lower] we prove that, in several settings, this algorithm is essentially optimal, by showing lower bounds on μ that are necessary for detection by any sensing and testing strategy. In the subsequent sections we will see that there is a complex interplay between the parameters n, m, s and p in how they affect the minimum signal strength required for reliable detection. It is noteworthy to stress that even when we restrict ourselves to the case p = 1 the nature of the optimal test changes radically depending on the interplay between the remaining parameters n, m and s. In this case, the signal support is reset at every time t ∈ N, which means that regardless of the sampling strategy (the choice of At) we are in the situation akin to a so-called sparse mixture model. These models are now well understood (see,, and references therein). We know that in the case of mixture models, for very sparse signals a type of scan test (which is essentially a generalized likelihood-ratio test) performs optimally, whereas for less sparse signals a global test based on the sum of all the observations is optimal. In our case the interplay between the parameters n, s and m determines the level of sparsity of the sample under the alternative. This in turn means that when p = 1 the optimal test and the scaling required for μ, depends on the relation between m and s/n. The above phenomenon becomes even more complex when p < 1. Note, however, that unless m is at least of the order of n/s reliable detection is impossible (regardless of the value of p). The reason behind this is that no sampling strategy will sample an active component under the alternative in fewer measurements with sufficiently large probability. To see this consider the case p = 0 and suppose there is no observation noise. Let the sampling strategy be arbitrary and let Ω denote the event that the algorithm does not sample an active component. When m ≤ n/s we have $$\begin{aligned} {\mathbb{P}}\_1 (\Omega ) & \geq \frac{{n-s \choose m}}{{n \choose m}} = \frac{(n-s)(n-s-1)\dots (n-s-m+1)}{n(n-1)\dots (n-m+1)} \\ & \geq \left( 1-\frac{s}{n-m} \right)^m \geq \left( 1-\frac{2s}{n} \right)^{n/s} \.\end{aligned}$$ The expression on the right is bounded away from zero when n/s is large enough. Hence regardless of the sampling strategy, there is a strictly positive probability that no active components are sampled under the alternative, which shows that can not hold for ɛ smaller than $\left( 1-\frac{2s}{n} \right)^{n/s}$. When p > 0, sampling an active component becomes even harder, hence the same rationale holds. In this paper we focus primarily on the regime where the number of measurements m is only slightly larger than n/s (what might be deemed to be the “small sample” regime). If we are interested in scenarios where one needs a detection outcome as soon as possible this is the interesting regime to consider. Interestingly, when m is significantly larger than n/s the optimal sensing and testing strategies, as well as the fundamental difficulty of the problem appears to be quite different than that of the small sample regime, and is an interesting and likely fruitful direction for future work. In Section [sec:sim] we conducted a small numerical experiment illustrating how the fundamental performance behavior changes in that regime. [rem:minimumvsaverage] The results in this paper can be very naturally generalized for signals with different signs and magnitudes, by considering the class of signals characterized by the minimum signal magnitude. In the regime where m is of the order of n/s this is essentially the most natural characterization, since only a very small number of active components will actually be observed (so a very low magnitude component will hinder the performance of any method). When m is significantly larger the picture changes quite significantly and pursuing these results is an interesting avenue for future research beyond the scope of this paper. A detection procedure ===================== In this section we present an adaptive sensing detection algorithm for the setting in Section [sec:setup] and analyze its performance. To devise such a procedure we use a similar approach as taken by — first devise a sensible procedure that works when there is no observation noise (i.e., when Wt ≡ 0), and then make it robust to noise by using sequential testing ideas. Consider a setting where there is no measurement noise, that is, when measuring a component of ${{\boldsymbol{x}}}^{(t)}$ we know for sure whether that component is zero or not. In such a setting if we find an active component we can immediately stop and deem Ψ = 1. Note that it is wasteful to make more than one measurement per component, and that, before hitting an active component, we have absolutely no prior knowledge on the location of active components. Therefore an optimal adaptive sensing design is random component sampling without replacement. If we look at a large enough number of randomly chosen components and only observe zeros, it becomes reasonable to conclude that there are no active components and so we deem Ψ = 0. Bear in mind though that in case we did not observe any active components we might have simply been unlucky, and missed them even though they are present. Hence, there is always a possibility for a false negative decision regardless of how many components we observe, unless p = 0 and m ≥ n − s. The procedure that we propose is a “robustified” version of the one explained above, so that it can deal with measurement noise. This is done by performing a simple sequential test to gauge the identity of the component that we are observing. A natural candidate for this is the Sequential Likelihood-Ratio Test (SLRT), introduced in. However, the dynamical nature of the signal causes some difficulties. In particular the identity/activity of the component that we are observing might change while performing the test, creating many analytic hinderances in the study of the SLRT performance. We instead use a simplified testing/stopping criteria that is easier to analyze in such a scenario. The basic detection algorithm, presented in Algorithm [basicalgorithm], queries components uniformly at random one after another and tests their identity (whether they are active or not during the subsequent time period) using the sequential test to be described later. Once a component is deemed to have been active we set Ψ = 1 and stop collecting data. If after examining T components or exhausting our measurement budget no components are deemed active we set Ψ = 0. Formally, let {Qj}j ∈ [T] denote the components queried by Algorithm [basicalgorithm]. We choose Qj, j ∈ [T] to be independent Unif([n]) random variables.[1](#fn1) The appropriate number of queries T ≤ m will be chosen later. For each Qj we run a sequential test to determine the identity of that component. We refer to our sequential test as Sequential Thresholding Test (STT). To gauge the identity of Qj, j ∈ [T], the STT algorithm makes multiple measurements at that coordinate. The exact number of measurements depends on the observed values (in a way we describe in detail later), and hence it is random. We denote the number of observations collected by STT at coordinate Qj by Nj. Formally, this means that At = Qj for $t\in \big[ 1+ {\sum\_{i=1}^{j-1}} N\_i, {\sum\_{i=1}^j} N\_i \big]$. At the end of the jth run of STT (j = 1, 2, …, T), the STT returns either that an active component was present at coordinate Qj, or that no active component was present at that location. In the former case there is no need to collect any more samples: Algorithm [basicalgorithm] stops and declares Ψ = 1. Otherwise we continue with applying STT to coordinate Qj + 1. If all T runs of STT found no signal, or we exhaust our measurement budget, Algorithm [basicalgorithm] stops and returns Ψ = 0. [h][basicalgorithm] Number of queries T ∈ N Queries $Q\_1,\dots,Q\_T \displaystyle{\mathop{\sim}^{iid}} {\operatorname{Unif}}([n])$ Perform a STT for the component indexed by Qj If the STT returns ``Signal": set Ψ = 1 and break If measurement budget is exhausted: set Ψ = 0 and break The sequential test that we use to examine the identity of a queried component is based on the ideas of distilled sensing introduced and analyzed in and the Sequential Thresholding procedure of. The distilled sensing algorithm is designed to recover the support of a sparse signal (whose active components remain the same during the sampling process). The main idea there is to use the fact that the signal is sparse and try to measure active components as often as possible, while not wasting too many measurements on components that are not part of the support. Our aim here is somewhat similar: on one hand we wish to quickly identify when the component that we are sampling is non-active so that we can move on to probe a different location of the signal. On the other hand in case we are sampling an active component we wish to keep sampling it as long as it is active to collect as much evidence as possible. However, unlike in the original setting of distilled sensing, we need to be able to quickly detect that we are sampling an active component, as it will eventually move away because of the dynamics. To address the last point the STT algorithm in Algorithm [STT] uses an evolving threshold for detection depending on the number of observations collected. We present STT in a way that emphasizes that it is a stand-alone routine plugged into the detection algorithm above, and not necessarily specific to the problem at hand. Hence, when discussing STT, the observations the STT makes are denoted by X(1), X(2), …. In the context of Algorithm [basicalgorithm], for the jth call of STT we have X(1), X(2), … to be independent normal random variables with variance one and means respectively xQj(Tj)xQj(Tj + 1), …, where Tj = 1 + ∑i = 1j − 1Ni. [h][STT] k ∈ N, t1 > t2 > … > tk > 0 STT sequentially observe X(1), X(2), …, X(k) Observe X(j) and compute $\overline{X}^{(j)} = \sum\_{i=1}^j X^{(i)} /j$ If $\overline{X}^{(j)} \leq t\_k$: break and declare No signal If $\overline{X}^{(j)} > t\_j$: break and declare Signal In words, STT collects at most k measurements sequentially and keeps track of the running average until one of the stopping conditions is met. The first stopping condition says that once the running average drops below the threshold tk we stop and declare that there is no signal present. The second says that if the running average at step j exceeds a threshold tj, we stop and conclude that a signal component is present. Note that after each measurement the upper threshold decreases, eventually reaching tk, hence the procedure necessarily terminates after at most k measurements. Key to the performance of the STT is a good choice of k and {tj}j ∈ [k], which is informed by the following heuristic argument: the sample collected by the detection algorithm consists of T blocks of measurements, where each block corresponds to an application of STT. Let the block lengths be denoted by {Nj}j ∈ [T]. Suppose for a moment that blocks entirely consist of either zero mean or non-zero mean measurements. In this case we can simply think of each block j as a single measurement with mean multiplied by $\sqrt{N\_j}$ for all j ∈ [T]. This would reduce the problem to a detection problem in a T-dimensional vector, each component being normally distributed and having unit variance. This is a well-understood setting, and we know that in this case the signal strength needs to scale as $\sqrt{\log T}$ when there are not too many active components (see for instance and the references therein). Recall that we are concerned with the case where the number of measurements we are allowed to make is of the order n/s. Hence we do not expect to encounter active components too many times. This heuristic shows that we should calibrate STT in a way that when it encounters j consecutive measurements with elevated mean, it should be able to detect it when $\mu \approx \sqrt{\tfrac{1}{j} \log T}$[2](#fn2). Furthermore, considering the tail properties of the Gaussian distribution, it is easy to see that we also need $\mu \gtrsim \sqrt{\log \tfrac{1}{\varepsilon}}$ for reliable detection. Recalling that j ≤ k, this shows that choosing k greater than logT does not buy us anything. Informed by the above heuristic argument we choose the parameters of STT so that the following result holds. [lem:STT] Let ɛ ∈ (0, 1) and define the parameters of STT as $$\begin{aligned} k & = \lfloor \log (T/2) \rfloor \, \\ t\_j & = \sqrt{\frac{c(2\varepsilon /T)}{j} \log \frac{2T}{\varepsilon}},\ j\in [k] \,\end{aligned}$$ where $$c(x)= 2\left( 1+ \frac{\log \log (1/x)}{\log (1/x)} \right).$$ Denote the observations available to the STT by X(1), …, X(k) (note that the STT may terminate without observing all the variables). Then the following holds: 1. If $X^{(i)} \displaystyle{\mathop{\sim}^{\text{i.i.d.}}} \mathcal{N}(0,1)$ for i ∈ [k], then STT declares ``Signal" with probability at most ɛ/T. 2. For any j ∈ [k], if the $X^{(i)} \displaystyle{\mathop{\sim}^{\text{i.i.d.}}} \mathcal{N}(\mu,1)$ for i ∈ [j] with $$\mu \geq \sqrt{\frac{c(2\varepsilon /T )}{j} \log \frac{2T}{\varepsilon}} + \sqrt{2 \log \frac{4}{\varepsilon}} \,$$ then STT declares ``No Signal" with probability at most ɛ/3. Note that, for (ii) it suffices for the first j observations to have elevated mean to guarantee the good performance of the STT. [Proof of Lemma [lem:STT]] For the first part suppose note that the STT declares ``Signal" if at any time step j ∈ [k] the running average $\overline{X}\_j$ exceeds the threshold tj. $$\begin{aligned} {\mathbb{P}}\left( \exists j \in [k]: \overline{X}^{(j)} \geq t\_j \right) & \leq \sum\_{j=1}^{k} {\mathbb{P}}(\overline{X}^{(j)} \geq t\_j) \\ & \leq \sum\_{j=1}^{k} \frac{1}{2} \exp \left( - \frac{j t\_j^2}{2} \right) \\ & = \sum\_{j=1}^{\lfloor \log (T/2) \rfloor} \frac{1}{2} \exp \left( -\frac{c(2\varepsilon /T)}{2} \log \frac{T}{2\varepsilon} \right) \\ & \leq \frac{1}{2}\log (T/2) \cdot \left( \frac{2\varepsilon}{T} \right)^{c(2\varepsilon /T)/2} \,\end{aligned}$$ where the first inequality follows by a union bound, and the second inequality is follows by a tail bound on Gaussian random variables noting that $\overline X\_j\sim\mathcal{N}(0,1/j)$. The last expression above is at most ɛ/T, which can be checked by taking the logarithm: $$\begin{aligned} \log \left( \frac{1}{2} \log (T/2) \cdot \left( \frac{2\varepsilon}{T} \right)^{c(2\varepsilon /T)/2} \right) &= \log \log (T/2) + \left( 1- \frac{\log \log (T/(2\varepsilon))}{\log (2\varepsilon/T)} \right) \log (2\varepsilon /T) -\log 2\\ & = \log \log (T/2) + \log (2\varepsilon/T) - \log \log (T/(2\varepsilon)) -\log 2\\ & \leq \log \frac{\varepsilon}{T} \.\end{aligned}$$ For the second part assume the conditions in (ii) hold for μ as given in the lemma. Define the event $$\Omega = \left\{ \exists i\in [j-1]: \overline{X}^{(i)} \leq t\_k \right\} \.$$ Note that if this event happens, we stop and declare ``No signal" in one of the first j − 1 steps. $$\begin{aligned} {\mathbb{P}}(\textrm{Declare ``No signal"}) & = {\mathbb{P}}(\Omega ) + {\mathbb{P}}(\textrm{Declare ``No signal"})\|\overline \Omega ){\mathbb{P}}( \overline{\Omega} ) \\ & \leq {\mathbb{P}}(\Omega ) + {\mathbb{P}}( \overline{X}^{(j)} \leq t\_j\|\overline\Omega ){\mathbb{P}}( \overline{\Omega} ) \\ & \leq {\mathbb{P}}(\Omega ) + {\mathbb{P}}(\overline{X}^{(j)} \leq t\_j ) \.\end{aligned}$$ Using a union bound and the same Gaussian tail bound as before, the last expression can be upper bounded by $$\label{eqn:miss} \sum\_{i=1}^{j-1} \frac{1}{2} \exp \left( -\frac{i (\mu -t\_k)^2}{2} \right) + \frac{1}{2} \exp \left( - \frac{j(\mu -t\_j)^2}{2} \right) \.$$ Considering the first term above, note that $$\mu - t\_k \geq t\_j + \sqrt{2 \log \frac{4}{\varepsilon}} -t\_k \geq \sqrt{2 \log \frac{4}{\varepsilon}} \,$$ since tj ≥ tk (recall that j ≤ k). Hence the first term can be upper bounded as $$\sum\_{i=1}^{j-1} \frac{1}{2} \exp \left( -\frac{i (\mu -t\_k)^2}{2} \right) \leq \frac{1}{2} \sum\_{i=1}^{j-1} (\varepsilon /4)^i \leq \frac{\varepsilon}{2} \frac{1}{4-\varepsilon} \leq \varepsilon /6 \.$$ On the other hand, when μ satisfies the inequality above, the second term is simply upper bounded by (ɛ/4)j, and so the left-hand-side of is less than ɛ/6 + ɛ/8 < ɛ/3. Using Lemma [lem:STT], we can establish a performance guarantee for our detection algorithm. Though it is possible to derive a result for fixed n and s it is more transparent to state a result for large n instead, better highlighting the impact of parameter p. Keeping this comment in mind, note that $2 \leq c(x) \leq 2(1+1/e)\leq 2\sqrt{2}$ and c(x) → 2 as x → 0. Thus, keeping ɛ fixed and letting T → ∞, we see that if there exists a τ > 1 for which $$\mu \geq \tau \sqrt{\frac{2}{j} \log T} + \sqrt{2 \log \frac{4}{\varepsilon}} \,$$ then for T large enough the condition on μ in Lemma [lem:STT] is satisfied. Furthermore, recall that our main interest is how the algorithm performs when the time horizon (number of measurements) is only slightly larger than n/s. [thm:upper] Fix ɛ ∈ (0, 1/3) and assume s ≡ sn = o(n/(logn)2) as n → ∞. The parameter p ≡ pn is also allowed to depend on n. Set $T= \tfrac{9n}{2s} \log\_2 \tfrac{3}{\varepsilon}$ and the parameters of STT according to Lemma [lem:STT]. If the measurement budget is m ≥ 2T the detection algorithm satisfies R(Ψ) = maxi = 0, 1Pi(Ψ ≠ i) ≤ ɛ, whenever $$\mu \geq \tau \sqrt{2 \max \{ 2p,\tfrac{1}{\log (n/s)} \} \log (n/s)} + \sqrt{2 \log \frac{4}{\varepsilon}} \,$$ for n large enough and τ > 1 fixed (but arbitrary). Before we move on to the proof of this result, let us discuss its message. First note that the detection algorithm is agnostic about the speed of change p and the signal strength μ, though it does require knowledge of the sparsity s to set the parameter T. The number of measurements that we require is a multiple of n/s, which is the minimum amount necessary to be able to solve the problem (see Section [sec:testing]). Furthermore, when p < 1/(2log(n/s)) the signal strength needs to scale as $\sqrt{\log (1/\varepsilon )}$, and when p ≥ 2/log(n/s) it needs to scale as $\sqrt{p \log (n/s)}$. This matches the intuition that the speed of change p affects the problem difficulty in a monotonic fashion. We will show in Section [sec:lower] that in the regime m ≈ n/s this scaling of μ is necessary to reliably solve this detection problem. In Figure [fig:STTdetection] we present an illustration of the above detection algorithm. We can clearly see the “random” exploration (in red) and the “tracking” of active components (in green). Note that in this case the algorithm missed that an active component was hit at time 8, so more exploration was needed. [fig:STTdetection] [rem:mregime] As we have mentioned in Section [sec:testing], for now we are interested in the case where the number of observations we can make is roughly n/s. Note that Theorem [thm:upper] claims the same performance guarantee for every m that is at least of order n/s. In fact, it is not hard to see that the performance of this algorithm does not improve as m increases, hinting that it is suboptimal for large m. Actually this algorithm completely ignores the fact that a component might have multiple periods of activity over time, and that activity evidence from multiple components might be combined for detection, in a more global fashion. Consider the following simple algorithm: sample components uniformly at random in each step t ∈ [m]. Then in each step we hit an active component with probability s/n. We then roughly have ms/n active components in our sample under the alternative. Consider the standardized sum of our observations. Under the null this follows a standard normal distribution, whereas under the alternative it is distributed as $N(\sqrt{m} s \mu/n,1)$. Thus reliable detection using this simple global algorithm is possible when μ is of the order $n/(\sqrt{m} s)$. Hence this algorithm clearly outperforms the one above when m is large enough (compared to n/s). This phenomena is not unlike that present in sparse mixture detection problems (e.g. as in ) where depending on the sparsity a global test might be optimal. [Proof of Theorem [thm:upper]] In light of Lemma [lem:STT], the type I error probability is at most ɛ by a union bound. Hence we are left with studying the alternative. There are two ways that our algorithm can make a type II error. Either the measurement budget is exhausted, or we fail to identify an active component in T runs of STT. We bound the probability of the first event by ɛ/3, and of the second event by 2ɛ/3 ensuring that under the alternative the probability of error is bounded by ɛ. We start with upper bounding the probability of exhausting our measurement budget. Let Nj denote the number of measurements that STT makes when called for the jth time, for j ∈ [T]. Note that these random variables are independent and identically distributed, because the components to query are selected uniformly at random independently from the past, the dynamic evolution of the model is memoryless, and the observation noise is independent. First we upper bound E1(N1). Note that 1 ≤ N1 ≤ k, where k = ⌊log(T/2)⌋ by Lemma [lem:STT]. Let Ω denote the event that a non-zero mean observation appears at location A1 in any of the first k steps. By the law of total expectation we have $${\mathbb{E}}\_1 (N\_1) \leq k {\mathbb{P}}\_1 (\Omega ) + {\mathbb{E}}\_1 (N\_1 \|\overline{\Omega} ) \.$$ Note that $$\begin{aligned} {\mathbb{P}}\_1 (\Omega ) & = {\mathbb{P}}\_1 (\exists t\in[k]: A\_1 \in S^{(t)} ) \leq \sum\_{t=1}^k {\mathbb{P}}\_1 (A\_1 \in S^{(t)} ) \\ & \leq \frac{s}{n} + (k-1)\frac{s}{n-s} \leq \frac{ks}{n-s} \,\end{aligned}$$ since the choice of A1 (and S(1)) is random, and in each subsequent step the probability that a signal component moves to location A1 is at most s/(n − s) regardless of p. On the other hand, recalling that $t\_k = \sqrt{\tfrac{c(2\varepsilon /T)}{k} \log \tfrac{T}{2\varepsilon}} \geq \sqrt{2}$ is the lower stopping boundary of STT, $$\begin{aligned} {\mathbb{E}}\_1 (N\_1\|\overline{\Omega}) & = 1+\sum\_{t=2}^k {\mathbb{P}}\_0 (N\_1 \geq t)\\ & \leq 1+\sum\_{t=2}^k {\mathbb{P}}\_0 (\overline{X}\_{t-1} > t\_k) \leq 1+\sum\_{t=2}^k {\mathbb{P}}\_0 (\overline{X}\_{t-1} > \sqrt{2}) \\ & \leq 1+ \frac{1}{2} \sum\_{t=1}^{k-1} e^{-t} \leq 1+ \frac{1}{2(e-1)} < 3/2 \.\end{aligned}$$ Hence $${\mathbb{E}}\_1 (N\_1) \leq 1+ \frac{1}{2(e-1)} + \frac{k^2s}{n-s} < 3/2 \,$$ for large enough n, since the last term can be made arbitrarily small by the definition of T, and the assumption on s. Since N1 is also a bounded random variable, an easy (but crude) way of proceeding is to use Hoeffding’s inequality to get $$\begin{aligned} {\mathbb{P}}\_1 \left( \sum\_{j=1}^T N\_j > m \right) & = {\mathbb{P}}\_1 \left( \sum\_{j=1}^T N\_j - {\mathbb{E}}\_1 \Big( \sum\_{j=1}^T N\_j \Big) > m - {\mathbb{E}}\_1 \Big( \sum\_{j=1}^T N\_j \Big) \right) \\ & \leq {\mathbb{P}}\_1 \left( \sum\_{i=1}^T N\_i - {\mathbb{E}}\_1 \Big( \sum\_{i=1}^T N\_i \Big) > T/2 \right) \\ & \leq \exp \left( - \frac{T}{2k^2} \right)=\exp \left( - \frac{T}{2\lfloor\log(T/2)\rfloor^2} \right)\leq \varepsilon/3 \,\end{aligned}$$ provided T is large enough, which is the case if n is large enough. This shows that the probability that the measurement budget is exhausted is bounded by ɛ/3. The final step in the proof is to guarantee that the algorithm identifies an active component in one of the T tests with high probability. To show this, we first guarantee that there will be an instance in the repeated application of STT where the first 1/(2p) observations that the procedure has access to have elevated mean (when p = 0 we only need that the STT probes an active component at least once). Then we can apply Lemma [lem:STT] together with a union bound to conclude the proof. Let Tj = 1 + ∑i = 1j − 1Ni denote the time when STT starts for the jth time. Let N = ∑j = 1T1{Qj ∈ S(Tj)} denote the number of times an active component is sampled at the start of an STT. Note that N ∼ Bin(T, s/n). In these situations the STT has access to a sequence of active measurements (of random length). Denote the number of consecutive active observations these STTs have access to by {ηi}i ∈ [N], and for now assume p > 0. Note that ηi ∼ Geom(p) and {ηi}i ∈ [N] are independent. We have $$\begin{aligned} {\mathbb{P}}(\forall i \in [N]: \eta\_i < 1/(2p) ) & \leq {\mathbb{P}}\left( \forall i\in [N]: \eta\_i <1/(2p)\ \| N\geq \log\_2 \tfrac{3}{\varepsilon} \right) \\ & \qquad + {\mathbb{P}}\left( N < \log\_2 \tfrac{3}{\varepsilon} \right) \.\end{aligned}$$ On one hand, note that the median of ηi is ⌈ − 1/log2(1 − p)⌉ which is greater than 1/(2p). This can be easily checked by considering the cases p ≥ 1/2 and p < 1/2 separately. Hence the first term above can be upper bounded as $${\mathbb{P}}\left( \forall i\in [N]: \eta\_i <\lceil -1/\log\_2 (1-p) \rceil\ \| N\geq \log \tfrac{3}{\varepsilon} \right) \leq 2^{-\log\_2 \tfrac{3}{\varepsilon}} = \varepsilon /3 \.$$ On the other hand, N ∼ Bin(T, s/n) and so by Bernstein’s inequality, $${\mathbb{P}}\left( N < (1-\delta ) \frac{Ts}{n} \right) \leq \exp \left( -\frac{3 \delta^2}{8} \frac{Ts}{n} \right) \,$$ for any δ ∈ (0, 1). However, note that plugging in the value of T together with δ = 2/3 yields $${\mathbb{P}}\left( N<\log\_2 \tfrac{3}{\varepsilon} \right) = {\mathbb{P}}\left( N < (1-\delta)\frac{Ts}{n}\right)\leq \exp \left( -\frac{49}{48} \log\_2 \tfrac{3}{\varepsilon} \right) < \varepsilon /3 \,$$ since log2x > logx for x > 1. So we conclude that the probability that there is no block (out of T) with the first 1/(2p) observations active is bounded by 2ɛ/3. When p = 0, we only need to control P(N = 0), for which we can simply use the inequality above since $\log\_2 \tfrac{3}{\varepsilon} >0$. Finally, if such a block is present the probability STT will not detect it is bounded by ɛ/3 via part (ii) of Lemma [lem:STT], provided $$\mu\geq \sqrt{\frac{c(2\varepsilon/T)}{\min \{ 1/(2p),\lfloor \log (T/2) \rfloor \} }\log\left(\frac{T}{2\varepsilon}\right)}+\sqrt{2\log\frac{4}{\varepsilon}}\,$$ where one should note that the blocks sampled by the STT are never larger than ⌊log(T/2)⌋. It is easily checked that the above condition is met for the choices in the theorem, provided n is large enough, concluding the proof. Lower bounds ============ In this section we identify conditions for the signal strength that are necessary for the existence of a sensing procedure to have small risk, namely R(Ψ) = maxi = 0, 1Pi(Ψ ≠ i) ≤ ɛ. We consider first the non-adaptive sensing setting. This is done both for comparison purposes (to highlight the gains of sensing adaptivity) but also illustrates some of the interesting features of this problem. In this case the sensing procedure is simply the choice of when and where to measure a component, before any data is collected. Then we consider the adaptive sensing setting to show the near-optimality of the algorithm proposed in Section [sec:upper]. In both cases our primary interests in on the regime m ≈ n/s, as highlighted in Section [sec:testing]. Non-adaptive sensing -------------------- In the non-adaptive sensing setting, the sampling strategy {At}t ∈ [m] needs to be specified before any observations are made. Note that this does not exclude the possibility of having a random design of the sensing actions. Common sense tells us that supports that are changing fast are harder to detect than those that are changing slowly, provided all other parameters are fixed. In other words, the problem difficulty should be increasing in the parameter p, meaning the signal magnitude μ needed to ensure should grow monotonically in p. Formalizing this heuristic in general turns out to be technically challenging with the methodologies we are aware of. Because of this we focus on the two extreme cases: when the signal is static (p = 0), and when the entire signal resets at each time instance (p = 1). Note that in the case s = 1 it is relatively easy to formalize that the problem difficulty is non-decreasing in p. Suppose there exists an algorithm (denoted by Alg) that performs accurate detection for some p > 0, and suppose we need to perform the detection task of a static signal. The idea is to transform the signal into one that has the same distribution as if it were generated according to the model of Section [subsec:signalmodel] with parameter p, and apply Alg to the modified signal. If such a transformation is possible than the existence of Alg implies the existence of an accurate detection procedure – in other words, the problem difficulty is non-decreasing in p. Such a transformation is easy to construct for s = 1, in fact one can almost follow the description of the signal model of Section [subsec:signalmodel] word-by-word. Let {θt}t ∈ [m − 1] be i.i.d. Ber(p) variables and w.l.o.g. θm = 1 — these represent the coin flips in the description of Section [subsec:signalmodel]. Let N = ∑t ∈ [m]1{θt = 1} be the number of times the coin came up heads and τ0 = 0 and τj = inf{t > τj − 1 : θt = 1}, j ∈ [N] be the instances when the coin came up heads. Finally, let {πi}i ∈ [N] be permutations of [n] drawn independently and uniformly at random (from the set of possible permutations). It is clear that a static support that is permuted by πi on the time intervals [τi − 1 + 1, τi] will “look” like a support sequence evolving with parameter p. Formally, one can show that if S ≡ {S(t)}t ∈ [m] is a static support sequence (chosen uniformly at random) then ${\widetilde}{{\mathbf{S}}} \equiv \{ {\widetilde}{S}^{(t)} \}\_{t\in [m]}$ defined as $${\widetilde}{S}^{(t)} = \sum\_{i\in [N]} {\mathbf{1}}\{ t\in [\tau\_{i-1}+1,\tau\_i ] \pi\_i (S^{(t)})$$ is distributed as a support sequence generated according to the model described in Section [subsec:signalmodel] with parameter p. Hence for s = 1 the problem difficulty is indeed non-decreasing in p. Nonetheless the authors did not find an obvious way to extend this argument to general sparsities, because the signal components change their locations at possibly different times. We note at this point that if one considered a more restrictive model where the entire support of the signal would reset simultaneously (a setting perhaps not vastly different to the one we are considering) would enable an argument similar to the above. We have the following result for these two extreme cases, which we prove at the end of the section. Note that these are not asymptotic, and hold for any n, m and s satisfying the assumptions in the statement. [thm:nalower] Let n, s, m ∈ N be fixed (with s ≤ n), consider a setup described in Section [sec:setup], and suppose there is a non-adaptive sensing design and a test Ψ satisfying R(Ψ) = maxi = 0, 1Pi(Ψ ≠ i) ≤ ɛ. * If p = 0, s ≤ n/2, n/s ≤ m and ɛ ≤ 1/(2e) then necessarily $$\mu \geq \sqrt{\frac{n}{2ms} \log \left( \frac{2 n}{s^2} \log \left( \tfrac{1}{e}-4\varepsilon \right) +1 \right)} \.$$ * If p = 1 and ɛ < 1/2 then necessarily $$\mu \geq \sqrt{\log \left( \frac{n^2}{s^2 m} \log \left( 4(1-2\varepsilon )^2 +1 \right) +1 \right)} \.$$ Considering the case p = 1, the result above tells us that when m scales like n/s, the signal strength needs to scale as $\sqrt{\log (n/s)}$ for detection to be possible. This is the same scaling that is guaranteed by Theorem [thm:upper]. This should come as no surprise, since when p = 1 we have 1{At ∈ S(t)} ∼ Ber(s/n) independently for every t ∈ [m], regardless of the choice of At. Hence the resulting measurements {Yt}t ∈ [m] follow the same mixture distribution under the alternative, no matter what sampling strategy we use. Although settings like these have been studied extensively (see and references therein), those works consider asymptotic results. As such we find it useful to prove a non-asymptotic result for our particular problem, though we point out that this can be simply established by following the steps of the referenced proofs. Contrasting with this one has the (arguably) more interesting case when the signal is static (p = 0). Although the problem of detecting static signals have been the focus of much work (see for instance ), a key difference in our setting is that the sensing actions of the experimenter are not fixed, but are free to choose. This results in a qualitatively different statement, as the following remark attests. [rem:NAlower] In particular, the first part of the theorem above is interesting in its own right. It tells us that, for static signals, if the experimenter is free to choose the sensing actions, the signal magnitude needs to scale at least as $\sqrt{\tfrac{n}{sm} \log \tfrac{n}{s^2}}$ for detection to be possible. It is easy to see that this rate can (almost) be achieved using a sub-sampling scheme: select roughly n/s components at random and collect an equal number of samples of each. Average the observations for each component separately, and declare a signal if any of these averages is above the threshold $\sqrt{\tfrac{n}{sm} \log \tfrac{n}{s}}$. Basic calculations show that this procedure has low probability of error. Contrasting this, the lower bounds of, which pertain the situation where we measure each component of the vector exactly once, scale as $\sqrt{\log \tfrac{n}{s^2}}$. Hence, the additional flexibility of where to sample buys us a multiplicative factor of $\sqrt{\tfrac{n}{sm}}$, even though no feedback from the observations is used. If we can use this feedback, we can also get rid of the log-factor, as shown in. In light of the previous remark, the authors suspect the lower bound in part (i) of the Theorem is slightly loose. Namely, the term s2 appears to be due to slack in the second moment method in Equation [eqn:slack], and it might be possible to replace it by s via a more sophisticated truncation argument. The result above tells us that in the regime m ≈ n/s, the signal strength needs to scale as $\sqrt{\log (n/s^2)}$ for detection to be possible — approximately the same magnitude as required for p = 1. On the other hand Theorem [thm:upper] guarantees the existence of an adaptive sensing procedure that reliably detects static signals of constant magnitude (in terms of the parameters n and s) using roughly n/s measurements. This shows that adaptive sensing gains over non-adaptive sensing become more pronounced as the speed of change decreases. Finally we point out once more that the requirements for the signal strength of Theorem [thm:nalower] are essentially the same for p = 0 and p = 1. Although we did not succeed in proving a result that holds for any value of p due to technical difficulties, we conjecture that the lower bound or general values of p should interpolate between these two extremes. In other words, we suspect that the problem difficulty is essentially independent of p in the non-adaptive case when m is of the order (or slightly larger than) n/s. This conjecture is further supported by numerical simulations of testing error probability presented in Section [sec:sim]. [Proof of Theorem [thm:nalower]] (i): To prove the claim above for p = 0 we use the truncated second moment method, an approach suggested by to address problems in the regular second moment method when the distribution of the likelihood ratio under the null has tails that are too heavy (and therefore too large of a second moment). First, note that $$\label{eqn:TVbound} \max\_{i=0,1} {\mathbb{P}}\_i (\Psi \neq i) \geq \frac{1}{2} \sum\_{i=0}^1 {\mathbb{P}}\_i (\Psi \neq i) = \frac{1}{2} \left( 1-\frac{1}{2} {\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}})-1\|\right) \right) \,$$ where $L({{\boldsymbol{Y}}})$ denotes the likelihood-ratio of the observations ${{\boldsymbol{Y}}}=(Y\_1,\ldots,Y\_m)$, and E0 is the expectation taken with respect to the distribution of the observations ${{\boldsymbol{Y}}}$ under the null. The second equality is well known (see for instance ), and can be easily checked using simple algebraic manipulations. A common way to proceed is to use either Cauchy-Schwarz’s or Jensen’s inequality to get $${\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}})-1\|\right) \leq \sqrt{{\mathbb{E}}\_0 \left( (L({{\boldsymbol{Y}}})-1)^2 \right)} = \sqrt{{\operatorname{Var}}\_0 (L({{\boldsymbol{Y}}}))} \.$$ Therefore, to get a lower bound on the risk we need to get a good upper bound on the variance of the likelihood ratio. This is often referred to as the second moment method. However, in some cases there is a lot of slack in the bound and the variance is too large to yield interesting results — so a modification of the above argument is needed. Let Y denote the sample space, and let ${\widetilde}{L}({{\boldsymbol{y}}}): \mathcal{Y} \to {\mathbb{R}}$ be an arbitrary function. Instead of using the Cauchy-Schwarz inequality right away, let us continue the first chain of inequalities as $$\begin{aligned} {\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}})-1\|\right) & = {\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}}) -{\widetilde}{L}({{\boldsymbol{Y}}}) + {\widetilde}{L}({{\boldsymbol{Y}}}) -1\| \right) \\ & \leq {\mathbb{E}}\_0 \left( \|{\widetilde}{L}({{\boldsymbol{Y}}}) -1\| \right) + {\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}}) -{\widetilde}{L}({{\boldsymbol{Y}}})\| \right) \\ & \leq \sqrt{{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}})^2 \right) -2{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}})\right) +1} + {\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}}) -{\widetilde}{L}({{\boldsymbol{Y}}})\| \right) \.\end{aligned}$$ Furthermore, if ${\widetilde}{L}({{\boldsymbol{y}}})\leq L({{\boldsymbol{y}}})$ for every ${{\boldsymbol{y}}}\in \mathcal{Y}$, then we have $$\label{eqn:truncated\_method} {\mathbb{E}}\_0 \left( \|L({{\boldsymbol{Y}}})-1\|\right) \leq \sqrt{{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}})^2 \right) -2{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}})\right) +1} +1 -{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}}) \right) \.$$ In order to proceed, we need to lower bound ${\mathbb{E}}\_0 ({\widetilde}{L}({{\boldsymbol{Y}}}))$ and upper bound ${\mathbb{E}}\_0 ({\widetilde}{L}({{\boldsymbol{Y}}})^2)$. To get a sharp lower bound with this method, we need a good choice for ${\widetilde}{L}({{\boldsymbol{y}}})$. This is often achieved by truncating the original likelihood-ratio by multiplying with the indicator of a well chosen event. In our setting the likelihood-ratio can be expressed in a convenient way. Note that under the null the observations are independent standard normal regardless of the sensing actions, hence $${\ensuremath {\,\mathrm{d}}}{\mathbb{P}}\_0 ({{\boldsymbol{y}}}) = \prod\_{t\in [m]} f\_0(y\_t) \,$$ where fμ( ⋅ ) is the density of a normal random variable with mean μ and variance 1. Under the alternative, the density of the observations is a mixture. Recall that we are considering the case p = 0 therefore the signal support S(t) does not change over time, namely S(t) = S for all t ∈ [m]. The conditional density of the observations given the sensing actions A = (A1, …, Am) and the support S can be written as $${\ensuremath {\,\mathrm{d}}}{\mathbb{P}}\_1 ({{\boldsymbol{y}}}\|A,S) = \prod\_{t\in [m]} \left( {\mathbf{1}}\{ A\_t \in S\} f\_\mu (y\_t) + {\mathbf{1}}\{ A\_t \notin S\} f\_0 (y\_t) \right) \.$$ Hence the likelihood-ratio can be expressed as $$L({{\boldsymbol{y}}}) = {\mathbb{E}}\left( \exp \left( \sum\_{t\in [m]} {\mathbf{1}}\{ A\_t \in S\} \log \frac{f\_\mu (y\_t)}{f\_0 (y\_t)} \right) \right) \.$$ Using the second moment method without truncation, one would need to upper bound the second moment of the likelihood ratio above. Unfortunately, this yields a loose bound on μ. The reason is that the second moment will be extremely large when the signal is sampled often, even if this event is relatively rare. In other words, if ∑t ∈ [m]1{At ∈ S} is large one will face problems. Note that, since the support is chosen uniformly at random, E(∑t ∈ [m]1{At ∈ S}) = ms/n. However, for certain choices of design ∑t ∈ [m]1{At ∈ S} can be very far from the mean (e.g., if A1 = ⋯ = Am then ∑t ∈ [m]1{At ∈ S} is equal to m with probability s/n and zero otherwise). This causes the second moment of the likelihood ratio to be extremely large. To resolve this issue we truncate the likelihood-ratio to exclude these somewhat troublesome instances. Begin by defining the sets $$A\_{\rm big} = \{ i:\ \sum\_{t\in [m]} {\mathbf{1}}\{ A\_t =i \} > 2ms/n \} \quad\text{ and }\quad A\_{\rm small} = [n] \setminus A\_{\rm big}\.$$ In words, for a given sensing design the signal components are divided in two disjoint subsets: one subset contains signal components that are sampled often, whereas the other contains the remaining components. A simple pigeon hole principle shows that $\|A\_{\rm big}\|\leq n/(2s)$. Now define $${\widetilde}{L}({{\boldsymbol{Y}}}) = {\mathbb{E}}\left[\left. {\mathbf{1}}\{S\subseteq A\_{\rm small}\} \exp \left( \sum\_{t\in [m]} {\mathbf{1}}\{ A\_t \in S\} \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)}\right)\right\| {{\boldsymbol{Y}}}\right] \.$$ Clearly ${\widetilde}{L}({{\boldsymbol{y}}}) \leq L({{\boldsymbol{y}}})$ for all ${{\boldsymbol{y}}}\in \mathcal{Y}$, and so we can apply by controlling the first and second moments of ${\widetilde}L({{\boldsymbol{Y}}})$. First note that, since the event $S\subseteq A\_{\rm small}$ does not involve the observations ${{\boldsymbol{Y}}}$ we can easily conclude that $${\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}}) \right) = {\mathbb{P}}(S\subseteq A\_{\rm small}) = {\mathbb{E}}\left( {\mathbb{P}}\left(S\subseteq A\_{\rm small}\|{{\boldsymbol{A}}} \right) \right) \,$$ where ${{\boldsymbol{A}}}\equiv (A\_1,\dots,A\_m )$. The conditional probability on the right can be lower bounded as $$\begin{aligned} {\mathbb{P}}\left(S\subseteq A\_{\rm small}\|{{\boldsymbol{A}}} \right) & = \frac{{\|A\_{\rm small}\| \choose s}}{{n \choose s}} = \frac{\|A\_{\rm small}\| (\|A\_{\rm small}\|-1)\dots (\|A\_{\rm small}\|-s+1)}{n(n-1)\dots (n-s+1)} \\ & \geq \left( \frac{\|A\_{\rm small}\|-s+1}{n-s+1} \right)^s \geq \left( \frac{n\left(1-\tfrac{1}{2s}\right)-s+1}{n-s+1} \right)^s \\ & = \left( 1-\frac{n}{2s(n-s+1)} \right)^s \geq \left( 1-\frac{1}{s} \right)^s\\ & \geq \frac{1}{e}\,\end{aligned}$$ where we used $\|A\_{\rm small}\|\geq n\left(1-\frac{1}{2s}\right)$ and 1 ≤ s ≤ n/2. We are left with upper bounding the second moment of ${\widetilde}{L}({{\boldsymbol{Y}}})$. First, note that in the non-adaptive sensing setting A = (A1, …, Am) and S are independent. The proof proceeds by careful conditioning on these random quantities. We use Jensen’s inequality to write $$\begin{aligned} {\mathbb{E}}\_0 ({\widetilde}{L}(Y)^2) & = {\mathbb{E}}\_0 \Bigg[ \Bigg({\mathbb{E}}\Bigg[{\mathbf{1}}\{S\subseteq A\_{\rm small}\} \exp \Bigg( \sum\_{t\in [m]} {\mathbf{1}}\{ A\_t \in S\} \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)}\Bigg)\Bigg\| {{\boldsymbol{Y}}}\Bigg] \Bigg)^2\Bigg]\\ & \leq {\mathbb{E}}\_0\Bigg[{\mathbb{E}}\Bigg[\Bigg({\mathbb{E}}\Bigg[\underbrace{{\mathbf{1}}\{S\subseteq A\_{\rm small}\} \exp \Bigg( \sum\_{t\in [m]} {\mathbf{1}}\{ A\_t \in S\} \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)}\Bigg)}\_{h(S,{{\boldsymbol{Y}}},{{\boldsymbol{A}}})}\Bigg\| {{\boldsymbol{Y}}},{{\boldsymbol{A}}}\Bigg]\Bigg)^2\Bigg\|{{\boldsymbol{Y}}}\Bigg]\Bigg]\.\end{aligned}$$ At this point it is convenient to introduce an extra random variable Sʹ, independent from S and identically distributed. Then $$\begin{aligned} \left({\mathbb{E}}\left[\left. h(S,{{\boldsymbol{Y}}},{{\boldsymbol{A}}})\right\| {{\boldsymbol{Y}}},{{\boldsymbol{A}}}\right]\right)^2 &= {\mathbb{E}}\left[\left. h(S,{{\boldsymbol{Y}}},{{\boldsymbol{A}}})\right\| {{\boldsymbol{Y}}},{{\boldsymbol{A}}}\right]\ {\mathbb{E}}\left[\left. h(S',{{\boldsymbol{Y}}},{{\boldsymbol{A}}})\right\| {{\boldsymbol{Y}}},{{\boldsymbol{A}}}\right]\\ &= {\mathbb{E}}\left[\left. h(S,{{\boldsymbol{Y}}},{{\boldsymbol{A}}})h(S',{{\boldsymbol{Y}}},{{\boldsymbol{A}}})\right\| {{\boldsymbol{Y}}},{{\boldsymbol{A}}}\right]\.\end{aligned}$$ Therefore we conclude that $$\begin{aligned} {\mathbb{E}}\_0\left[{\widetilde}L({{\boldsymbol{Y}}})^2\right] &\leq {\mathbb{E}}\_0 \left[{\mathbf{1}}\{S\subseteq A\_{\rm small}\} {\mathbf{1}}\{S'\subseteq A\_{\rm small}\} \exp \left( \sum\_{t\in [m]} \left({\mathbf{1}}\{ A\_t \in S\}+{\mathbf{1}}\{ A\_t \in S'\}\right) \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)}\right)\right]\\ &= {\mathbb{E}}\left[ {\mathbb{E}}\_0\left[\left. {\mathbf{1}}\{S,S'\subseteq A\_{\rm small}\} \exp \left( \sum\_{t\in [m]} \left({\mathbf{1}}\{ A\_t \in S\}+{\mathbf{1}}\{ A\_t \in S'\}\right) \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)}\right)\right\|{{\boldsymbol{A}}},S,S'\right]\right]\\ &= {\mathbb{E}}\left[{\mathbf{1}}\{S,S'\subseteq A\_{\rm small}\}\prod\_{t\in [m]} {\mathbb{E}}\_0\left[\left. \exp \left(\left({\mathbf{1}}\{ A\_t \in S\}+{\mathbf{1}}\{ A\_t \in S'\}\right) \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)}\right)\right\|{{\boldsymbol{A}}},S,S'\right]\right]\\ &= {\mathbb{E}}\left[{\mathbf{1}}\{S,S'\subseteq A\_{\rm small}\} \exp\left(\mu^2 \sum\_{t\in [m]} {\mathbf{1}}\{A\_t\in S\cap S'\}\right)\right]\.\end{aligned}$$ We are now in a good position to finish the bound. Note that, when $S,S'\subseteq A\_{\rm small}$ we have ∑t ∈ [m]1{At = i} ≤ 2ms/n. It follows that $$\begin{aligned} {\mathbb{E}}\_0\left[{\widetilde}L({{\boldsymbol{Y}}})^2\right] &\leq {\mathbb{E}}\left[{\mathbb{E}}\left[\left.{\mathbf{1}}\{S,S'\subseteq A\_{\rm small}\} \exp\left(\mu^2 \sum\_{i\in[n]} {\mathbf{1}}\{i\in S\cap S'\}\sum\_{t\in[m]} {\mathbf{1}}\{A\_t=i\}\right) \right\|{{\boldsymbol{A}}}\right]\right]\\ &\leq {\mathbb{E}}\left[{\mathbb{E}}\left[\left. \exp\left(\frac{2ms\mu^2}{n} \sum\_{i\in[n]} {\mathbf{1}}\{i\in S\cap S'\}\right) \right\|{{\boldsymbol{A}}}\right]\right]\\ &= {\mathbb{E}}\left[ \exp \left( \lambda \sum\_{i\in[n]} {\mathbf{1}}\{i\in S\cap S'\} \right) \right]\,\end{aligned}$$ where $\lambda=\frac{2ms\mu^2}{n}$. The beauty of the last expression is that it no longer involves the sensing actions or the observations, and depends only on the support. Using the negative association property of 1{i ∈ S ∩ Sʹ} as introduced in we can finally bound the second moment of the truncated likelihood as $$\begin{aligned} {\mathbb{E}}\_0 \left({\widetilde}{L}({{\boldsymbol{Y}}})^2\right) & \leq {\mathbb{E}}\left[\exp\left(\lambda\sum\_{i\in[n]} {\mathbf{1}}\{i\in S\cap S'\}]\right)\right]\nonumber\\ & = {\mathbb{E}}\left[\prod\_{i\in[n]} e^{\lambda {\mathbf{1}}\{i\in S\cap S'\}}\right] \leq \prod\_{i\in[n]} {\mathbb{E}}\left[ e^{\lambda {\mathbf{1}}\{i\in S\cap S'\}}\right]\nonumber\\ & = \left( 1+ \frac{s^2}{n^2} \left( e^\lambda -1 \right) \right)^n = \left( 1+ \frac{s^2}{n^2} \left( e^{2\mu^2 ms/n} -1 \right) \right)^n \label{eqn:slack}\.\end{aligned}$$ We have now all the ingredients needed to complete the proof. Note that, on one hand, if maxi = 0, 1Pi(Ψ ≠ i) ≤ ɛ then necessarily ${\mathbb{E}}\_0[\|L({{\boldsymbol{Y}}})-1\|]\geq 2-4\varepsilon$. On the other hand, from we know that $$\begin{aligned} {\mathbb{E}}\_0[\|L({{\boldsymbol{Y}}})-1\|] &\leq \sqrt{{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}})^2 \right) -2{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}})\right) +1} +1 -{\mathbb{E}}\_0 \left( {\widetilde}{L}({{\boldsymbol{Y}}}) \right)\\ &< \left(1+ \frac{s^2}{n^2} \left( e^{2\mu^2 ms/n} -1\right) \right)^{n/2} +\frac{1}{2}\.\end{aligned}$$ This means that $$\frac{s^2}{n^2}\left(e^{2\mu^2 ms/n} -1\right) > \left(\frac{3}{2}-4\varepsilon\right)^{2/n}-1\geq \frac{2}{n}\log\left(\frac{3}{2}-4\varepsilon\right)\,$$ where the last inequality uses the fact that x − 1 ≥ logx. The final result ensues by simple algebraic manipulation. (ii): Proving the claim for p = 1 requires considerably less technical effort. In particular we can use the original second moment method, without truncation. Therefore, we simply need to upper bound the second moment of the likelihood-ratio. Using essentially the same calculations as before, we get E0(L(Y)2) = E[exp(μ2∑t ∈ [m]1{At ∈ S(t) ∩ Sʹ(t)})]. When p = 1 we have that 1{At ∈ S(t) ∩ Sʹ(t)} ∼ Ber(s2/n2) and these random variables are independent, so we can simply evaluate the above expression and get $${\mathbb{E}}\_0 \left( L(Y)^2 \right) = \left( 1+ \frac{s^2}{n^2} \left( e^{\mu^2} -1 \right) \right)^m \.$$ Plugging this into the inequalities above (not using the truncation), we get $$\mu \geq \sqrt{\log \left( \frac{n^2}{s^2} \left( \sqrt[m]{4(1-2\varepsilon )^2}-1 \right) +1 \right)} \.$$ The desired result follows by using x − 1 ≥ logx. Adaptive sensing ---------------- In the adaptive sensing setting, the decision where to sample at time t can depend on information gleaned up to that point. For the static case (p = 0) the fundamental limits of the detection problem using adaptive sensing have been studied in. Those lower bounds are derived for a slightly more general setting than the one considered here, in that the total precision of the measurements is constrained, but not the total number of measurements. Nevertheless, this bound is still valid in our setting, and states that for any adaptive sensing and testing procedure Ψ if max{P0(Ψ ≠ 0), P1(Ψ ≠ 1)} ≤ ɛ then necessarily $$\mu \geq \sqrt{\frac{2 n}{sm} \log \frac{1}{2\varepsilon}} \.$$ In the regime m ≈ n/s the bound states that the signal strength needs scale as $\sqrt{\log (1/\varepsilon )}$. This coincides (up to constants) with the bound of Theorem [thm:upper] when p ≤ 2/log(n/s). This tells us that when the signal changes slowly enough, the problem is essentially non-dynamic in nature. On the other extreme end of the spectrum is the case p = 1. We have seen previously that in this case the non-adaptive and adaptive sensing settings are identical, by virtue of the fact that 1{At ∈ S(t)} ∼ Ber(s/n) for every t ∈ [m] and independent, regardless of the choice of At. What remains to be understood are the fundamental limits for the intermediate regime. ### Non-extreme dynamics (p ∈ (0, 1)) For general values of p we start by considering the case s = 1, which we call the 1-sparse case. This case is considerably simpler to analyze than the general s-sparse setting, as now whenever the active component changes the entire signal resets. This effectively creates a number of independent static signals on the time horizon. [thm:1sparselower] Consider the setup in Section [sec:setup] and suppose there exists a test Ψ such that maxi = 0, 1Pi(Ψ ≠ i) ≤ ɛ. * The signal strength must satisfy $$\mu \geq \sqrt{\frac{2n}{sm} \log \frac{1}{4\varepsilon}} \.$$ * When s = 1 and p ≥ 8/m, then necessarily $$\mu \geq \sqrt{\frac{p}{2c} \log \left( \log \left( \left( \tfrac{5}{4} -4\varepsilon \right)^2 +\tfrac{1}{2} \right) \frac{p^2 n^2}{4 c^2 m} +1 \right)} \,$$ with c = 6 + 3log2. We provide the proof of Theorem [thm:1sparselower] at the end of the section. Part (i) holds regardless of the values of p and s, so it is necessarily loose when p is large. On the other hand part (ii) already captures the role of the rate of change p, and it is the main contribution in this result. Let us compare the above bound on μ with the guarantees for Algorithm [basicalgorithm] proved in Theorem [thm:upper]. Note that c and ɛ are constants. Thus the bound on the signal strength in the above result scales as $\sqrt{p \log(p^2 n^2/m)}$. Recall that we are interested in the regime m ≈ n/s and that s = 1, as we are considering the 1-sparse case. In that setting the bound above scales as $\sqrt{p \log (p^2 n)}$. Also note that the scaling of the performance guarantee of Theorem [thm:upper] matches that of the lower bound from when p < 1/logn. Hence we only need to assess the result of Theorem [thm:1sparselower] when p ≥ 1/logn. In this case, the scaling of that bound is at least as big as $\sqrt{p(\log n - 2\log \log n)} \approx \sqrt{p \log n}$. This shows near-optimality of the algorithm proposed in Section [sec:upper], in terms of its scaling in the parameters n and p. Due to technical reasons we were unable to generalize the result for signals of sparsity greater than one. As noted above, a key feature of the 1-sparse case is that the signal decouples into independent static signals over time. This key property is lost when we consider signals with sparsity greater than one, and this proves to be a major obstacle to obtain a rigorous formal proof. However, we conjecture that a similar result to the one above holds for s-sparse signals, with n replaced by n/s. The heuristic behind this is that a general s-sparse signal of dimension n should behave very much like an s-fold concatenation of an 1-sparse signal of dimension n/s, when viewed through the lens of one measurement per time-index (one expects this to actually be a statistical reduction, and this problem should be statistically easier than the original one). For such a signal the result above would follow directly with the signal dimension n replaced by n/s. When p ≥ 8/m, if the risk of an adaptive sensing and test procedure is less or equal to ɛ then necessarily $$\mu \geq \sqrt{\frac{p}{2c} \log \left( \log \left( \left( \tfrac{5}{4} -4\varepsilon \right)^2 +\tfrac{1}{2} \right) \frac{p^2 n^2}{4 c^2 s^2 m} +1 \right)} \,$$ with c = 6 + 3log2. [Proof of Theorem [thm:1sparselower]] We prove the two parts of the statement separately. (i): The proof is very similar to that of Theorem 3.1 in, with small modifications to be able to deal with dynamically evolving signals (which actually simplify the argument). By Theorem 2.2 of we have $$\label{eqn:tsybakov} \inf\_{\Psi} \max\_{i=0,1} {\mathbb{P}}\_i (\Psi \neq i) \geq \frac{1}{4} e^{- {\operatorname{KL}}({\mathbb{P}}\_0 \\| {\mathbb{P}}\_1 )} \,$$ where KL(P0∥P1) denotes the Kullback-Leibler divergence between the distribution of the data Y under the null and alternative respectively. This divergence can be simply upper bounded using Jensen’s inequality as $$\begin{aligned} {\operatorname{KL}}({\mathbb{P}}\_0 \\| {\mathbb{P}}\_1 ) & = {\mathbb{E}}\_0 \left[ -\log L({\mathbf{Y}}) \right] \\ & \leq {\mathbb{E}}\_0 \left[ {\mathbb{E}}\left[ \left. -\sum\_{t\in [m]} {\mathbf{1}}\{ A\_t \in S^{(t)} \} \log \frac{f\_\mu (Y\_t)}{f\_0 (Y\_t)} \right\| {\mathbf{Y}}\right] \right] \.\end{aligned}$$ Changing the order of integration and expanding the densities fμ( ⋅ ) and f0( ⋅ ) we get $${\operatorname{KL}}({\mathbb{P}}\_0 \\| {\mathbb{P}}\_1 ) \leq \frac{\mu^2}{2} {\mathbb{E}}\left[ \sum\_{t\in [m]} {\mathbf{1}}\{ A\_t \in S^{(t)} \} \right] = \frac{\mu^2}{2}\frac{sm}{n} \,$$ where the last step follows from the symmetry of the supports. In particular note that E[1{At ∈ S(t)}∣At] = s/n for every t ∈ [m]. Plugging this bound into the right side of, using that the left side of is at most ɛ due to our assumption, and rearranging concludes the proof of the first claim. (ii): We use the truncated second moment method, as in the proof of Theorem [thm:nalower]. Recall that, from and, we have $$2 \max\_{i=0,1} {\mathbb{P}}\_i (\Psi \neq i) \geq 1- \frac{1}{2} \left( \sqrt{{\mathbb{E}}\_0 \left( {\widetilde}{L}(Y)^2 \right) -2{\mathbb{E}}\_0 \left( {\widetilde}{L}(Y)\right) +1} +1 -{\mathbb{E}}\_0 \left( {\widetilde}{L}(Y) \right) \right) \,$$ for any function ${\widetilde}{L}(\cdot )$ satisfying ${\widetilde}{L}({\mathbf{y}}) \leq L({\mathbf{y}}),\ \forall {\mathbf{y}}\in \mathcal{Y}$, where L( ⋅ ) is the likelihood function. ![Illustration of the notation introduced for the proof part (ii) of Theorem [thm:1sparselower].](Notation.png "fig:") [fig:notation] To aid the presentation we begin by introducing some convenient notation, illustrated in Figure [fig:notation]. Recall that the variables $\theta^{(t)}\_{i} \displaystyle{\mathop{\sim}^{\text{i.i.d.}}}\ {\operatorname{Ber}}(p)$, t ∈ [m], i ∈ [s] identify the change points of the signal. Since now we are dealing with the 1-sparse case we have one variable per time index, so in what follows we drop the subscript from the previous notation. Furthermore, note that our time horizon is m, so we enforce θ(m) = 1 as this does not change the model and it is convenient for the presentation. Let the total number of change points over the time horizon be N = ∑t ∈ [m]1{θ(t) = 1}. Note that N − 1 ∼ Bin(m − 1, p). Let τ0 = 0 and for j ∈ N let τj = min{t > τj − 1 : θ(t) = 1} denote the time instances when the signal changes (so τN = m), as illustrated in Figure [fig:notation]. Note that on the time intervals [τj + 1, τj + 1] the signal is static. Let lj = τj + 1 − τj denote the length of these intervals, and Sj, j ∈ [N] denote the correspoding signal support. Finally, for any t ∈ [m] let the number of change points up to time t be N(t) = max{j : τj ≤ t}. It is important to note that the random variables θ(t) completely determine the variables τj, N(t) and N. Let us first explicitly write the likelihood of the observations in the model under consideration. We use the shorthand notation y = {yt}t ∈ [m], A = {At}t ∈ [m], S = {S(t)}t ∈ [m], \theta = {θt}t ∈ [m]. As before, the density of y under the alternative is a mixture. In particular, denoting the density of N(μ, 1) by fμ, the conditional density of y can be written as $$\begin{aligned} {\ensuremath {\,\mathrm{d}}}{\mathbb{P}}\_1 ({\mathbf{y}}\|{\mathbf{A}},{\mathbf{S}}) & = \prod\_{t\in [m]} \left( {\mathbf{1}}\{ A\_t \in S^{(t)} \} f\_\mu (y\_t) + {\mathbf{1}}\{ A\_t \notin S^{(t)} \} f\_0 (y\_t) \right) \\ & = \prod\_{j\in [N]} \prod\_{t= \tau\_{j-1}+1}^{\tau\_j} \left( {\mathbf{1}}\{ A\_t \in S\_j \} f\_\mu (y\_t) + {\mathbf{1}}\{ A\_t \notin S\_j \} f\_0 (y\_t) \right) \.\end{aligned}$$ Hence, the likelihood ratio is $$L({\mathbf{y}}) = {\mathbb{E}}\left[ {\mathbb{E}}\left[ \left. \exp \left( \sum\_{j\in [N]} \sum\_{t=\tau\_{j-1}+1}^{\tau\_j} {\mathbf{1}}\{ A\_t\in S\_j \} \log \frac{f\_\mu (y\_t)}{f\_0 (y\_t)} \right) \right\| {\mathbf{\theta}},{\mathbf{A}}\right] \right] \,$$ where conditioning on \theta and A is done in order to conveniently define ${\widetilde}L({\mathbf{y}})$. Consider the event Ωc = {∀j : lj ≤ 2c/p}, with some fixed c > 0. This event says that the signal is never static for a time longer than 2c/p. Note that this event is determined exclusively by the variables {θt}t ∈ [m]. We define the truncated likelihood as $${\widetilde}{L}({\mathbf{y}}) = {\mathbb{E}}\left[ {\mathbf{1}}\{\Omega\_c\} {\mathbb{E}}\left[ \left. \exp \left( \sum\_{j\in [N]} \sum\_{t=\tau\_{j-1}+1}^{\tau\_j} {\mathbf{1}}\{ A\_t\in S\_j \} \log \frac{f\_\mu (y\_t)}{f\_0 (y\_t)} \right) \right\| {\mathbf{\theta}},{\mathbf{A}}\right] \right] \.$$ As in the proof of Theorem [thm:nalower], we need to upper bound ${\mathbb{E}}\_0 \left( {\widetilde}{L}({\mathbf{Y}})^2 \right)$ and lower bound ${\mathbb{E}}\_0 \left( {\widetilde}{L}({\mathbf{Y}})\right)$. We start with the latter. Since the event Ωc only involves the variables \theta, we have $${\mathbb{E}}\_0 \left( {\widetilde}{L}({\mathbf{Y}})\right) = {\mathbb{P}}(\Omega\_c ) \.$$ We have the following result, the proof of which is presented in the Appendix. [lem:event] Consider the event Ωc = {∀j : lj ≤ 2c/p}. In the model described above P(Ωc) > 1/4 whenever c ≥ 6 + 3log2 and p ≥ 8/m. According to Lemma [lem:event], we have an appropriate bound for ${\mathbb{E}}\_0 \left( {\widetilde}{L}(Y)\right)$ when c ≥ 6 + 3log2. All that remains is to derive an upper bound on the truncated second moment. This can be done much the same way as in the proof of Theorem [thm:nalower]. Using Jensen’s inequality, we have $${\mathbb{E}}\_0 \left[ {\widetilde}{L}({\mathbf{Y}})^2 \right] \leq {\mathbb{E}}\_0 \left[ {\mathbb{E}}\left[ {\mathbf{1}}\{\Omega\_c\} {\mathbb{E}}\left[ \left. \exp \left( \sum\_{j\in [N]} \sum\_{t=\tau\_{j-1}+1}^{\tau\_j} {\mathbf{1}}\{ A\_t\in S\_j \} \log \frac{f\_\mu (y\_t)}{f\_0 (y\_t)} \right) \right\| {\mathbf{\theta}},{\mathbf{A}}\right]^2 \right] \right] \.$$ Note that given \theta, the Sj ∼ Unif([n]) and independent for j ∈ [N]. Let {Sʹj}j ∈ [N] be an independent copy of {Sj}j ∈ [N]. Following the same reasoning as in Theorem [thm:nalower] we can write the square of the conditional expectation above as the product of two expectations using the random variables {Sj, Sʹj}j ∈ [N], and change the order of the expectations to get $${\mathbb{E}}\_0 \left[ {\widetilde}{L}({\mathbf{Y}})^2 \right] \leq {\mathbb{E}}\left[ {\mathbf{1}}\{\Omega\_c\} {\mathbb{E}}\left[ \left. \exp \left( \mu^2 \sum\_{j\in [N]} \sum\_{t=\tau\_{j-1}+1}^{\tau\_j} {\mathbf{1}}\{A\_t \in S\_j \cap S'\_j \} \right) \right\| {\mathbf{\theta}},{\mathbf{A}}\right] \right] \.$$ So far we have not taken into account the fact that we are allowed an adaptive design. This is captured by the crude bound below. ∑t = τj − 1 + 1τj1{At ∈ Sj ∩ Sʹj} ≤ lj1{∃t ∈ [τj − 1 + 1, τj] : At ∈ Sj ∩ Sʹj}. Informally this means that, if the used design “hits” the signal at any place in the interval [τj − 1 + 1, τj] it is assumed the design hit the signal in the entire interval (capturing more information). Furthermore P(∃t ∈ [τj − 1 + 1, τj] : At ∈ Sj ∩ Sʹj∣A, \theta) = P(Sj ∈ {At : t ∈ [τj − 1 + 1, τj]}∣A, \theta)2. However, ∣{At : τj − 1 + 1 ≤ t ≤ τj}∣ ≤ τj − τj − 1 : = lj thus the probability above is bounded from above by lj2/n2. Using all this yields $$\begin{aligned} {\mathbb{E}}\_0 \left[ {\widetilde}{L}({\mathbf{Y}})^2 \right] & \leq {\mathbb{E}}\left[ {\mathbf{1}}\{\Omega\_c\} \prod\_{j\in [N]} {\mathbb{E}}\left[ \left. \exp \left( l\_j \mu^2 {\mathbf{1}}\{\exists t\in [\tau\_{j-1}+1,\tau\_j ]:\ A\_t \in S\_j \cap S'\_j \} \mu^2 \right) \right\| {\mathbf{\theta}},{\mathbf{A}}\right] \right] \\ & \leq {\mathbb{E}}\left[ {\mathbf{1}}\{\Omega\_c\} \prod\_{j\in [N]} \left( 1+ \frac{l\_j^2}{n^2} \left( e^{l\_j \mu^2} -1 \right) \right) \right] \.\end{aligned}$$ The last expression is readily upper bounded by the fact that N ≤ m. Although this is a crude bound[3](#fn3) it is enough for our purposes. Also, on the event Ωc we have the upper bound lj ≤ 2c/p for every j ∈ [N]. We conclude that $${\mathbb{E}}\_0 \left( {\widetilde}{L}({\mathbf{Y}})^2 \right) \leq \left( 1+ \frac{4c^2}{p^2 n^2} \left( e^{2c\mu^2 /p} -1 \right) \right)^m \.$$ Combining our results yields that if there exists a test for which maxi = 0, 1P(Ψ ≠ i) ≤ ɛ, we must have $$\sqrt{\left( 1+ \frac{4c^2}{p^2 n^2} \left( e^{2c\mu^2 /p}-1 \right) \right)^m -\frac{1}{2}} +\frac{3}{4} \geq 2-4\varepsilon \.$$ Rearranging gives $$\frac{4c^2}{p^2 n^2} \left( e^{2c\mu^2 /p}-1 \right) \geq \sqrt[m]{\left( \frac{5}{4} - 4\varepsilon \right)^2 + \frac{1}{2}}-1 \.$$ Using the inequality logx ≤ x − 1 on the right hand side, and rearranging concludes the proof. Numerical evaluation of the non-adaptive lower bound ==================================================== Although the lower bound in Theorem [thm:nalower] only deals with the extreme cases p ∈ {0, 1}, we conjecture that in the regime m ≈ n/s the same scaling of μ is necessary for reliable detection, regardless of the value of p. To corroborate this conjecture we provide a brief section of numerical experiments. We numerically estimate the right hand side of, which is a lower bound on the maximal probability of error. We do so for several values of p ∈ [0, 1], and for each p we plot the value of the lower bound as a function of μ. Note that the sampling strategy has a large impact on the value in question. We know that when p = 0 a sub-sampling scheme is near-optimal (see Remark [rem:NAlower]), and so it should also be reasonable for small values of p. On the other hand, the sampling strategy is irrelevant for p = 1, and probably essentially irrelevant for large p. This motivates using a sub-sampling scheme in all the experiments. Furthermore, note that unless we sample c ⋅ n/s different components, the probability ${\mathbb{P}}\_1 (\forall t\in [m]:\ A\_t \notin S^{(t)})$ can not be small. To ensure an upper bound of ɛ on the previous probability, we need to choose c ≡ c(ɛ) = log(1/ɛ). Considering all the above, we set up our experiment as follows. We set n = 5000, s = ⌈n1/4⌉ = 9 and m = c(ɛ)n/s with ɛ = 0.05. In this case, sub-sampling reduces to measuring m randomly selected components (one measurement each). We note that we experimented using multiple values of s across a wide range of sparsity levels, but found qualitatively the same result in all cases. Based on previous work concerning the sparse-mixture model (e.g. ) we expect the lower bound to reach the value ɛ when $\mu \approx \sqrt{2\log (n/s)}$. Hence, we set $\mu\_t \approx t\cdot \sqrt{2\log (n/s)}$, and plot the r.h.s. of as a function of t. The left panel of Figure [fig:NAlower] seems to support our conjecture that the problem difficulty is independent of p in the regime m ≈ n/s, as all the curves are on top of each other. Furthermore, since there is always a non-negligible chance of not sampling a signal component, the lower bound is bounded away from zero, even as μt grows large. [fig:NAlower] To contrast this, we present another simulation with the same setup, except that the number of measurements m ≫ n/s. In particular, we set m = n, but otherwise use the same parameters. Note that in this case, sub-sampling amounts to sampling c(ɛ)n/s randomly chosen components, but now we sample each of these m/(c(ɛ)n/s) consecutive times. To keep the two plots on the same horizontal scale, we set $\mu\_t = t\cdot \sqrt{ (2c(\varepsilon )n /sm) \log (n/s)}$ in the right panel of Figure [fig:NAlower]. It seems that in this case, the curves are no longer on top of each other, suggesting that the value of p has an impact on the problem difficulty. Surprisingly, the curve corresponding to p = 1 is the one that descends the fastest, though the difference is only marginal. Though the cause of this is unclear, a possible reason might be that for faster signals the chance of not sampling active components at all is diminished, an effect that is more pronounced when m is large. In any case, this shows that in the regime m ≫ n/s the speed of change might have a non-trivial effect on the problem difficulty. Exploring this is out of the scope of this work, but might be an interesting topic of future research. Final remarks ============= In this paper we studied the problem of the detection of signals that evolve dynamically over time. We introduced a simple model for the evolution of the signal that allowed us to explicitly characterize the difficulty of the problem with a special regard to the effect of the speed of change. We also showed the potential advantages that adaptively collecting the observations bring to the table and showed that these are more and more pronounced as the speed of change decreases, which is in line with previous results dealing with signal detection using adaptive sensing. The lower bounds derived in this paper provide a clear picture of the role of the rate of change parameter p, but unfortunately still do not span the entire range of problems we would like to consider (e.g. Theorem [thm:nalower] applies only to p = 0, 1 and part (ii) of Theorem [thm:1sparselower] applies only to s = 1). The latter difficulties appear to be mostly technical and the authors suspect these might be possible to address with carefully chosen reductions. Our contributions merely scratch the surface of this interesting problem, and below we highlight a few interesting directions for future work in this regard. Large vs. small sample regimes: in this work we focus primarily on the case m ≈ n/s, which may be deemed as the small sample regime. When the number of measurements m is significantly larger the type of tests and performance tradeoffs will likely be different, even under the non-adaptive sensing paradigm. For instance, we expect the signal dynamics to have an effect on performance, meaning that it is easier to detect signals non-adaptively when p is smaller. Other interesting questions arise in that setting as well — what is the optimal non-adaptive sensing design? These questions become even more intriguing when one considers adaptive sensing. Restricted dynamics: in the model considered in this paper when signal components change they can move to any unoccupied location in the signal vector. This assumption simplifies the setup, but in some applications might be too unrestrictive. For instance, if signal components can only move to adjacent locations at each time step the effect of the speed of change will likely be less pronounced in the difficulty of detection (at least for adaptive sensing). Understanding the effect of such restrictions could prove valuable in certain applications, such as detection of a disease outbreak in a network, besides being interesting from a theoretical point of view. Structures: in certain situations the signal support can be assumed to have structure to it, for instance all anomalous items might be consecutive or have some other pattern. In some cases the structure of the support has a huge effect on the difficulty of the problems of detection and recovery (see for instance ). How structural restrictions affect these tasks for dynamically evolving signals could be a fruitful avenue of research. Support recovery: another common question in such settings is how well can we estimate the support of a signal. That is, instead of deciding only if there are anomalous items or not, we need to determine which of the items are anomalous. This is also an interesting problem to study for dynamically evolving signals, although a precise formulation of the objective and performance metric for such estimators is less immediate than for static signals. Acknowledgements ================ This work was partially supported by a grant from the Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO 613.001.114). We are very grateful for the comments of the two anonymous referees, which helped improving the presentation. Appendix ======== [Proof of Lemma [lem:event]] We write $$\begin{aligned} {\mathbb{P}}(\Omega\_c ) & \geq {\mathbb{E}}\left( \{ N-1 >mp/2\} \cap \{ \forall j:\ l\_j \leq cm/N \} \right) \\ & = {\mathbb{E}}\left[ {\mathbf{1}}\{ N-1 > mp/2 \} {\mathbb{E}}\left[ {\mathbf{1}}\{ \forall j:\ l\_j\leq cm/N \} \ \big\| N \right] \right] \,\end{aligned}$$ We first lower bound the inner conditional probability. Note that if N ≤ c this probability is one (since cm/N ≥ m and lj ≤ m by definition). When N > c, we will upper bound the probability of the complementary event. Note that given N the distribution of \theta is uniform from the set of 0 − 1 sequences of length m containing exactly N ones, and for which also θm = 1. Hence, to upper bound P(∃j : lj > cm/N), we simply need to count the number of sequences described above for which we have a long block. We can get an upper bound on this count in the following way. First note that since the last element of the sequence is always one, we can simply think of sequences of length [m − 1] containing N − 1 ones. Consider an interval of length cm/N in the set [m − 1]. Now consider the sequences containing N − 1 ones, and for which there are no ones in the aforementioned interval. Note that for all such sequences the existence of at least one long interval is guaranteed. We can simply count how many 0 − 1 sequences can be generated like this. This number is an upper bound on the number of 0 − 1 sequences that have N ones, the last element of the sequence is one and for which ∃j : lj > cm/N. We thus have $$\begin{aligned} {\mathbb{P}}(\exists j:\ l\_j> cm/N \|N) & \leq (m-cm/N) \frac{{m-cm/N \choose N-1}}{{m-1 \choose N-1}} \\ & = (m-cm/N) \frac{(m-cm/N)(m-cm/N -1)\dots (m-cm/N -N+2)}{(m-1)(m-2)\dots (m-N+1)} \\ & \leq \frac{m}{m-1}(1-c/N) \left( \frac{m-cm/N}{m-2} \right)^{N-2} \\ & < \left( \frac{m-cm/N}{m-2} \right)^{N-2} \.\end{aligned}$$ Now consider the logarithm of the expression above. Using log(1 + x) ≤ x, we get $$\begin{aligned} \log {\mathbb{P}}(\exists j:\ l\_j> cm/N \|N) & < (N-2) \left( \log \frac{m}{m-2} + \log (1-c/N) \right) \\ & \leq (N-2) \left( \frac{2}{m-2} - \frac{c}{N} \right) \\ & \leq -\log{2} \,\end{aligned}$$ whenever c ≥ 6 + 3log2, using the fact that 3 ≤ c ≤ N ≤ m. Hence P(Ωc) ≥ P(N − 1 > mp/2)/2. All that remains is to use the fact that N − 1 ∼ Bin(m − 1, p). For instance Chebyshev’s inequality yields $${\mathbb{P}}(N-1 \leq mp/2) \leq \frac{4(m-1)p(1-p)}{(mp)^2} \leq 1/2 \,$$ when p ≥ 8/m and so the claim is proved. --- 1. In principle one could ensure these are sampled without replacement from [n], but this would only unnecessarily complicate the analysis without yielding significant performance gains.[↩](#fnref1) 2. In this informal discussion, the notations ≈ and $\gtrsim$ hide constant factors and/or log(1/ɛ) terms.[↩](#fnref2) 3. In principle one can recall that N − 1 ∼ Bin(m − 1, p) and proceed from there, although it will overcomplicate the derivation. In any case, this will at most allow us to replace the term p2 by p inside the logarithm in the statement of the theorem, which is not very relevant.[↩](#fnref3)	arxiv_0000539
Inertial waves in rotating bodies: a WKBJ formalism for inertial modes and a comparison with numerical results ============================================================================================================== ### Accepted Received ; in original form [firstpage] Inertial waves governed by Coriolis forces may play an important role in several astrophysical settings, such as eg. tidal interactions, which may occur in extrasolar planetary systems and close binary systems, or in rotating compact objects emitting gravitational waves. Additionally, they are of interest in other research fields, eg. in geophysics. However, their analysis is complicated by the fact that in the inviscid case the normal mode spectrum is either everywhere dense or continuous in any frequency interval contained within the inertial range. Moreover, the equations governing the corresponding eigenproblem are, in general, non-separable. In this paper we develop a consistent WKBJ formalism, together with a formal first order perturbation theory for calculating the properties of the normal modes of a uniformly rotating coreless body (modelled as a polytrope and referred hereafter to as a planet) under the assumption of a spherically symmetric structure. The eigenfrequencies, spatial form of the associated eigenfunctions and other properties we obtained analytically using the WKBJ eigenfunctions are in good agreement with corresponding results obtained by numerical means for a variety of planet models even for global modes with a large scale distribution of perturbed quantities. This indicates that even though they are embedded in a dense spectrum, such modes can be identified and followed as model parameters changed and that first order perturbation theory can be applied. This is used to estimate corrections to the eigenfrequencies as a consequence of the anelastic approximation, which we argue here to be small when the rotation frequency is small. These are compared with simulation results in an accompanying paper with a good agreement between theoretical and numerical results. The results reported here may provide a basis of theoretical investigations of inertial waves in many astrophysical and other applications, where a rotating body can be modelled as a uniformly rotating barotropic object, for which the density has, close to its surface, an approximately power law dependence on distance from the surface. hydrodynamics; stars: oscillations, binaries, rotation; planetary systems: formation Introduction ============ In astrophysical applications inertial waves that can exist in rotating bodies may be excited by several different physical mechanisms, most notably through tidal perturbation by a companion (eg. Papaloizou & Pringle 1981, hereafter PP) or in the case of compact objects through secular instability arising through gravitational wave losses (eg. Chandrasekhar 1970, Friedman & Schutz 1978, Andersson 1998, Friedman & Morsink 1998). They also can play a role in other physical systems. For example, they can also be excited by several mechanisms in the Earth’s fluid core with possible detection being announced (Aldridge & Lumb 1987). For rotating planets and stars that have a barotropic equation of state these wave modes are governed by Coriolis forces and so have oscillation periods that are comparable to the rotation period. They are accordingly readily excited by tidal interaction with a perturbing body when the characteristic time associated with the orbit is comparable to the rotation period, which is expected naturally when the rotation period and orbit become tidally coupled. They may then play an important role in governing the secular orbital evolution of the system. Inertial modes excited in close binary systems in circular orbit were considered by PP and Savonije & Papaloizou (1997). Wu (2005)a,b considered the excitation of inertial modes in Jupiter as a result of tidal interaction with a satellite and excitation as a result of a parabolic encounter of a planet or star with a central star was studied by Papaloizou & Ivanov (2005), hereafter referred to as PI and Ivanov & Papaloizou (2007), hereafter referred to as IP. The latter work was applied to the problem of circularisation of extrasolar giant planets starting with high eccentricity. In that work the planet was assumed coreless. Ogilvie & Lin (2004) and Ogilvie (2009) have considered the case of a cored planet in circular orbit around a central star and found that inertial waves play an important role. The importance of the role played by inertial waves in the transfer of the rotational energy of a rotating neutron star to gravitational waves via the Chandrasekhar-Friedman-Schutz (CFS) instability was pointed out by Andersson (1998). Later studies mainly concentrated on physical mechanisms of dissipation of energy stored in these modes that limit amplitudes of the modes, and, consequently, the strength of the gravitational wave signal. In these studies either numerical methods or simple local estimates of properties of inertial modes were mainly used, see eg. Kokkotas (2008) for a recent review and references. An analytical treatment of problems related to inertial waves, such as eg. finding normal mode spectra and eigenfunctions, and coupling them to other physical fields, etc., is difficult due to a number of principal complicating technical issues. In particular, the dynamical equations governing the perturbations of a rotating body (called planet later on) are, in general, non-separable, for compressible fluids. When such fluids are considered and rotation is assumed to be small, a low frequency anelastic approximation that filters out the high frequency modes is often used (see eg. PP). This simplifies the problem to finding solutions to leading order in the small parameter R\3Ω2/(G*M), where Ω is the rotation frequency, G is the constant of gravity and M\, R\ are the mass and radius of the planet. In this approximation eigenfrequencies of inertial modes are proportional to Ω, while the form of the spatial distribution of perturbed quantities does not depend on the rotation rate. However, even when this approximation is adopted, the problem is, in general, non-separable apart from models with a special form of density distribution, see Arras et al (2003), Wu (2005)a and below. Additionally, the problem of calculating the inertial mode spectrum and its response to tidal forcing is complicated by the fact that in the inviscid case the spectrum is either everywhere dense or continuous in any frequency interval it spans (Papaloizou & Pringle, 1982). This is in contrast to the situation of, for example, high frequency p modes, which are discrete with well separated eigenvalues. When the anelastic approximation is adopted the singular ill posed nature of the inviscid eigenvalue problem is seen to come from the fact that the governing equation is hyperbolic and the nature of the spectrum is determined by the properties of the characteristics (eg. Wood 1977). A discrete spectrum is believed to occur when there are no such trajectories that define periodic attractors. Otherwise the inviscid spectrum is continuous. Then, when a small viscosity is introduced the spectrum becomes discrete but normal modes have energy focused onto wave attractors (see eg. Ogilvie & Lin 2004). Given these complexities it is desirable to work with and compare a variety of analytical and numerical approaches. Coreless inviscid rotating planets with an assumed spherical or ellipsoidal shape have a discrete but everywhere dense spectrum that makes difficulties for example with mode identification and application of standard perturbation theory. However, numerical work indicates that there are well defined global modes that can be identified and followed through a sequence of models (eg. Lockitch & Friedman, 1999, hereafter LF, and PI). In this paper we investigate the inertial mode spectrum of a uniformly rotating coreless barotropic planet or star and its tidal response by a WKBJ approach coupled with first order perturbation theory and compare its eigenvalue predictions with numerical results obtained by a variety of authors and find good agreement apart from some unidentified WKBJ modes that are near the limits of the spectrum and for which the perturbation theory appears not to work. For the identified modes we also find remarkably good agreement for the form of the eigenfunctions. This indicates that they can be represented at low resolution with small scale phenomena being unimportant, meaningful mode identification (in that the modes can be followed from one model to another) and at least first order perturbation theory works for these modes. This is also confirmed in a following paper (hereafter referred to as PIN) where we investigate the inertial mode spectrum and its tidal response by numerical solution of an initial value problem without the anelastic approximation. We are able to confirm the validity of the anelastic approximation and the applicability of the first order perturbation theory developed here for demonstrating this as well as estimating eigenvalues. Thus a suggestion of Goodman & Lackner (2009) that tidal interaction might be seriously overestimated by use of the anelastic approximation is not confirmed. A WKBJ approach to the same problem was also considered by Arras et al (2003) and Wu (2005)a. However, in this work only terms of leading order in an expansion in inverse powers of a large WKBJ parameter λ (see the text below for its definition) were taken into account and treatment of perturbations near the surface and close to the rotational axis were oversimplified. As a consequence, although their results are correct in the formal limit λ → ∞, they cannot be used to make a correspondence between WKBJ modes and those obtained numerically, or an approximate description of modes with a scale that is not very small. In this paper we treat the problem in a more extended way, considering terms of the next O(λ− 1) order together with an accurate treatment of perturbations near the surface and close to the rotation axis. Additionally, we consider a frequency correction of the next order, O(λ− 2), for modes having non-zero azimuthal number, m. We checked results obtained with use of the WKBJ formalism against practically all numerical data existing in the literature finding good agreement in practically all cases. Therefore, we can assume that our formalism may be applied to provide an approximate analytic description of inertial modes, including those with large scale variations, where the WKBJ approach might be expected to be invalid. Also, different quantities associated with the modes may be described within the framework of our formalism or its natural extension, such as the tidal overlap integrals (see PI and IP), quantities determining the growth rate due to the CFS instability and decay of inertial waves due different processes, eg. by non-linear mode-mode interactions (see eg. Schenk et al 2002, Arras et al 2003). Thus, the formalism developed here may provide a basis for the analytic treatment of inertial waves in many different astrophysical applications. The plan of the paper is as follows. In section [sec2] we briefly review the basic equations and their linearised form for a uniformly rotating barotropic planet or star. In section [anelastic] we go on to consider these in the anelastic approximation which is appropriate when the rotation frequency of the star is very much less than the critical or break up rotation frequency. We give a simple physical argument why we expect this approximation to be valid in this limit even when the sound speed tends to a small value or possibly zero at the surface of the configuration. In section [sec2.5] we give a brief discussion about when discrete normal modes may be expected to occur such as in the case of a coreless slowly rotating planet with surface boundary assumed to be either spherical or ellipsoidal. We then present a formal first order perturbation theory that can be used to estimate corrections to eigenfrequencies occurring as either a consequence of terms neglected in the WKBJ approximation or the anelastic approximation. The latter application is tested by a direct comparison with the results of numerical simulations in PIN. Section [sec2.6] concludes with a brief account of the form of the anelastic equations in pseudo-spheroidal coordinates in which they become separable for density profiles of the form ρ ∝ (1 − r2/R\2)β, where r* is the local radius, R\* is the surface radius and β is a constant. (Arras et al 2003, Wu 2005a). In section [sec3] we develop a WKBJ approximation for calculating the normal modes which is based on the idea that in the short wavelength limit these modes coincide with those appropriate to separable cases which include the homogeneous incompressible sphere as a well known example. Solutions of a general WKBJ form appropriate to the interior of the sphere are matched to solutions appropriate to the surface regions where they become separable which is the case when the density vanishes as a power of the distance to the boundary as is expected for a polytropic equation of state. This matching results in an expression for the eigenfrequencies given in section [sec3.5]. In section [surface] we go on to develop expressions for the eigenfunctions appropriate to any location in the planet including the rotation axis and the critical latitude region where one of the inertial mode characteristics is tangential to the planet surface. These solutions are then used to obtain corrections to the eigenfrequencies resulting from density gradient terms neglected in the initial WKBJ approximation in section [sec3.9]. In section [sec4] we compare the corrected eigenfrequencies obtained from the WKBJ approximation with those obtained numerically by several different authors who used differing numerical approaches and find good agreement even for global modes. A similar comparison with the results of numerical simulations for a polytropic model with positive results is reported in PIN. We also compare the forms of the eigenfunctions with those obtained in Ivanov & Papaloizou (2007) and find a good agreement even for global modes. Finally in section [sec5.1] we discuss our results in the context of the evaluation of the overlap integrals that occur in evaluating the response to tidal forcing. We show that these vanish smoothly in the limit that the polytropic index tends to zero and we indicate that they vanish at the lowest WKBJ order and are thus expected to vanish rapidly as the order of the mode increases. We go on to summarize our conclusions in section [Conclu]. Basic definitions and equations =============================== In this section we review the formalism and equations we adopt in this paper. As much of this has been presented in previous work (PI, IP) only a brief review is given here. In what follows we continue to investigate oscillations of a uniformly rotating fully convective body referred hereafter to as a planet, focusing on the low frequency branch associated with inertial waves. Framework for linear perturbation analysis ------------------------------------------ The planet is characterised by its mass M\, radius R\ and the associated characteristic frequency $$\Omega\_{\}=\sqrt{GM\_{\}\over R\_{\}^{3}}, \label{eqn p2}$$ where G* is the gravitational constant. We adopt a cylindrical coordinate system $(\varpi, \phi, z)$ and associated spherical coordinate system (r, ϕ, θ) with origin at the centre of mass of the planet. In this paper we make use of the Fourier transform of a general perturbation quantity, say Q, with respect to the azimuthal angle ϕ and the time t in the form $$Q = \sum\_{m}\left( \exp({im\phi})\int^{+\infty}\_{-\infty}d\sigma \tilde Q\_{m}\exp({-i\sigma t}) + cc \right ), \label{eq p1}$$ where the sum is over m = 0 and 2 and cc denotes the complex conjugate of the preceding quantity hereafter. The reality of Q implies that the Fourier transform, indicated by tilde satisfies $\tilde Q\_{m}(\sigma) = \tilde Q\_{-m}^\(-\sigma).$ The inner products of two complex scalars Y1, Y2, that are functions of $\varpi,$ z* and t are defined as $$(Y\_{1}\|Y\_{2})=\int \_S\varpi d\varpi dz (Y\_{1}^{\}Y\_{2}), \label{eq p2}$$ where \ denotes the complex conjugate. Note that the definition of the inner product differs from what is given in IP where the planet’s density ρ was used as a weight function. Integrals of this type are always taken over the section S of the unperturbed planet for which ϕ = 0. Linearised equations of motion governing the response to tidal perturbation --------------------------------------------------------------------------- We assume that the planet is rotating with uniform angular velocity ${\mbox{\boldmath${\Omega }$}}$. The hydrodynamic equations for the perturbed quantities take the simplest form in the rotating frame with z axis along the direction of rotation. Since the planet is fully convective, the entropy per unit of mass of the planetary gas remains approximately the same over the volume of the planet, and the pressure P can be considered as a function of density ρ only, thus P = P(ρ). As the characteristic oscillation periods associated with inertial modes are in general significantly shorter than the global thermal timescale we may adopt the approximation that perturbations of the planet can be assumed to be adiabatic. Then the relation P = P(ρ) holds during perturbation as well leading to a barotropic equation of state. In the barotropic approximation the linearised Euler equations take the form (see PI) $${D^{2} {{\mbox{\boldmath${ \xi}$}}} \over Dt^{2}}+2{{\mbox{\boldmath${\Omega}$}}}\times {D {\mbox{\boldmath${\xi}$}}\over Dt}=-\nabla W, \label{eq p3}$$ where W = cs2ρʹ/ρ + Ψint + Ψext, ${{\mbox{\boldmath${ \xi}$}}}$ is the Lagrangian displacement vector, ρʹ is the density perturbation, cs is the adiabatic sound speed, Ψint is the stellar gravitational potential arising from the perturbations and Ψext is an external forcing potential, say, the tidal potential in the problem of excitation of inertial waves by tides, see PI and IP. The linearised continuity equation is $$\rho^{'}=-\nabla \cdot (\rho {{\mbox{\boldmath${\xi}$}}}). \label{eq p6}$$ Note that the centrifugal term is absent in equation $(\ref{eq p3})$ being formally incorporated into the potential governing the static equilibrium of the unperturbed star. The convective derivative ${D\over Dt} \equiv {\partial \over \partial t}$ as there is no unperturbed motion in the rotating frame. Although incorporation of the perturbation to the internal gravitational potential presents no principal difficulty, in this paper, for simplicity we neglect it, setting Ψint = 0. This procedure known as ’the Cowling approximation’ can be formally justified in the case when perturbations of small spatial scale in the WKBJ limit are considered. However, it turns out that when low frequency inertial modes are considered the Cowling approximation has been found to lead to results which are in qualitative and quantitative agreement with those obtained numerically for global modes obtained with a proper treatment of perturbations to the gravitational potential (see below). Therefore, we do not expect that the use of the Cowling approximation can significantly influence our main conclusions. Provided that the expressions for the density and sound speed are specified for some unperturbed model of the planet, the set of equations $(\ref{eq p3}-\ref{eq p6})$ is complete. Now we express the Lagrangian displacement vector and the density perturbation in terms of W with help of equations $(\ref{eq p3})$ and $(\ref{eq p4})$, and substitute the result into the continuity equation from which we obtain an equation for its Fourier transform in the form $$\sigma^{2} {{\mbox{\boldmath${A}$}}} \tilde W\_{m} -\sigma {{\mbox{\boldmath${B}$}}} \tilde W\_{m} -{{\mbox{\boldmath${C}$}}}\tilde W\_{m} = \sigma^{2}d{\rho \over c\_{s}^{2}} (\tilde \Psi^{ext}\_{m}-\tilde W\_{m}), \label{eq 4}$$ where d = 4Ω2 − σ2, and $${{\mbox{\boldmath${A}$}}}=-{1\over \varpi}{\partial \over \partial \varpi} \left (\varpi \rho{\partial\over \partial \varpi}\right )-{\partial \over \partial z}\left (\rho {\partial \over \partial z}\right )+ {m^{2} \rho \over \varpi^{2}}, \label{eq 5}$$ $${{\mbox{\boldmath${B}$}}} =-{2m\Omega \over \varpi}{\partial \rho \over \partial \varpi}, \quad {{\mbox{\boldmath${C}$}}}=-4\Omega^{2}{\partial \over \partial z}\left(\rho {\partial \over \partial z}\right). \label{eq 6}$$ It is very important to note that the operators ${{\mbox{\boldmath${A }$}}}$, ${{\mbox{\boldmath${B}$}}}$ and ${{\mbox{\boldmath${C}$}}}$ are self-adjoint when the inner product $$(W\_{1}\|W\_{2})=\int\_{V} dz \varpi d\varpi W^{\}\_{1}W\_{2}, \label{eq 7}$$ with V* denoting the volume of the star. Here these operators are assumed to act on well behaved functions and the density is taken to vanish at the surface boundary. Also when m ≠ 0 ${{\mbox{\boldmath${A}$}}}$ and ${{\mbox{\boldmath${B}$}}}$ are positive definite and ${{\mbox{\boldmath${C}$}}}$ is non negative. When m = 0, ${{\mbox{\boldmath${A}$}}}$ remains positive definite if consideration is restricted to the physically acceptable variations W that conserve mass, this constraint eliminating the possibility that W is constant. When the Cowling approximation is adopted equation ([eq 4]) fully specifies solutions to the problem of forced linear perturbations of a rotating barotropic planet. In the general case, a complete set of equations is described in PI. The anelastic approximation --------------------------- When $\tilde \Psi^{ext}\_{m} = 0,$ equation ([eq 4]) leads to an eigenvalue problem describing the free oscillations of a rotating star in the form $$\sigma^{2} {{\mbox{\boldmath${A}$}}} \tilde W\_{m} -\sigma {{\mbox{\boldmath${B}$}}} \tilde W\_{m} -{{\mbox{\boldmath${C}$}}}\tilde W\_{m} = -\sigma^{2}d{\rho \over c\_{s}^{2}} \tilde W\_{m}, \label{eq 41}$$ Assuming that rotation of the planet is relatively slow such that the angular velocity Ω ≪ Ω\, these may be classified as f* or p modes with eigenfrequencies such that ∣σ∣ > Ω\* or inertial modes with eigenfrequencies ∣σ∣ ∼ Ω. The f and p modes exist in non rotating stars and can be treated in a framework of perturbation theory taking advantage of the small parameter α = ∣Ω/σ∣ (see eg. Ivanov & Papaloizou 2004 and references therein). On the other hand for inertial waves α is of order unity, such a perturbation approach cannot be used. Since, in general, equation ([eq 41]) is rather complicated even for numerical solution, in order to make it more tractable the so-called ’anelastic approximation’ has been frequently used (see eg. PP, Lockitch & Friedman 1999 and Dintrans & Ouyed 2001) for which the right hand side of ([eq 41]) is neglected. In order to justify this approximation we note that for eigenfunctions that are non singular everywhere in the planet, we can crudely estimate the derivatives entering equation ([eq 41]) as ∣∇W∣ ∼ kW/R\* and ∣∇2W∣ ∼ k2W/R\2, where the parameter k* > 1. Consider first the interior region of the planet where we approximately have cs2 ∼ GM\/R\. It follows from equation ([eq 41]) that the left hand side and the right hand side can be respectively estimated as $\sim \Omega^{2}\rho k^{2}\tilde W\_{m}/R\_{\}^{2}, \quad {\rm and} \quad \Omega^{4}\rho \tilde W\_{m}/ c\_{s}^{2}.$ The ratio of these is of order $$\hspace{6cm} {\Omega^{2} R\_{\}^{2}\over c\_{s}^{2}k^{2}} \sim {\Omega^{2}\over\Omega\_\^2 k^{2}} \ll 1. \label{eqn3}$$ This estimate is, however, not valid near the boundary of the planet where c*s2 → 0 and the left hand side of the inequality ([eqn3]) diverges. However, in the same limit the terms containing the density gradient on the left hand side of ([eq 41]) will dominate terms involving the second derivatives of $\tilde W\_{n}$. Thus in this limit the magnitude of the contribution from terms on the left hand side of ([eq 4]) may be estimated to be $$\hspace{6cm} {\Omega^{2}}\left\|{d\rho\over dr}\right\| k\tilde W\_{m}/ R\_{\} \sim \Omega^{2}\Omega\_{\}^{2}{\rho k\tilde W\_{m}/ c^{2}\_{s}}, \label{eqn4}$$ where we remark that it follows from hydrostatic equilibrium that close to the surface ∣dρ/dr∣ ∼ ρGM\/(c*s2R\2). Accordingly, when r* → R\* the ratio of the terms on the right and left hand sides of equation ([eq 41]) can be estimated as $$\hspace{6cm} \sim {\Omega^{2}\over\Omega\_\^2 k} \ll 1 \label{eqn5}$$ From equations ([eqn3]) and ([eqn5]) it follows that when Ω ≪ Ω\ the terms determining deviation from the anelastic approximation are small compared to the leading terms everywhere in the planet. Accordingly, in the slow rotation regime, we can use this approximation to find the leading order solutions for eigenfrequencies and eigenfunctions and then proceed to regard the terms on the right hand side of ([eq 41]) as a perturbation. The validity of the anelastic approximation in the context of the tidal excitation of inertial modes has been recently questioned in a recent paper by Goodman & Lackner (2009) on account of the divergence of the terms on the right hand side of equation([eq 41]) as r → R\* although an actual demonstration of its failure was not given. In fact the above discussion, which also applies to equation ([eq 4]) as this differs only by the addition of a forcing term, indicates that these terms are never important provided Ω/Ω\* is sufficiently small. This is to be expected because as Ω is reduced, the structure of the modes remains unaffected in the anelastic approximation whereas the radial width of the region where terms on the right hand side of equation([eq 41]) might become comparable to any other terms shrinks to zero. We also note that the vanishing of the normal velocity at the boundary in the anelastic approximation is correct in the limit Ω/Ω\* → 0 as the ratio of the horizontal to normal components there can be shown using the above arguments to also be on the order of (Ω/Ω\)2. Finally in PIN, we find by comparing the results of tidal forcing calculations using a spectral approach with the anelastic approximation, to those obtained using direct numerical solution of the initial value problem, that it gives good results even when Ω/Ω\ is not very small. Self-adjoint formalism ---------------------- It was shown by PI and IP that both quite generally and also when the anelastic approximation is used equations ([eq 4]) and ([eq 41]) can be brought to the standard form leading to an eigenvalue problem for a self-adjoint operator. Here we describe the approach, which leads to the self-adjoint formulation of the problem in the anelastic approximation. The self-adjoint and non negative character of the operators ${{\mbox{\boldmath${A}$}}}$, ${{\mbox{\boldmath${B}$}}}$ and ${{\mbox{\boldmath${C}$}}}$ is made use of to formally introduce their square roots, eg. ${{\mbox{\boldmath${A}$}}}^{1/2}$, defined by condition ${{\mbox{\boldmath${A}$}}}={{\mbox{\boldmath${A}$}}}^{1/2} {{\mbox{\boldmath${A}$}}}^{1/2}$, etc. As is standard, the requirement of non negativity, makes the definitions of these square roots unique. The positive definiteness of ${{\mbox{\boldmath${A}$}}}$ (see above discussion) also allows definition of the inverse of ${{\mbox{\boldmath${A}$}}}^{1/2}$, ${{\mbox{\boldmath${A}$}}}^{-1/2}$. Let us consider a new generalised two dimensional vector $\vec Z$ with components such that $ \vec Z = (Z\_1, Z\_2)$ and the straightforward generalisation of the inner product given by equation ([eq 7]). It is now easy to see that equation ([eq 4]) is equivalent to $$\sigma \vec Z ={{\mbox{\boldmath${H}$}}} \vec Z +\vec S, \label{eq 12}$$ where $${{\mbox{\boldmath${H}$}}}= \left( \begin{array}{cc} {{\mbox{\boldmath${A}$}}}^{-1/2}{{\mbox{\boldmath${B}$}}} {{\mbox{\boldmath${A}$}}}^{-1/2} & {{\mbox{\boldmath${A}$}}}^{-1/2} {{\mbox{\boldmath${C}$}}}^{1/2} \\ {{\mbox{\boldmath${C}$}}}^{1/2} {{\mbox{\boldmath${A}$}}}^{-1/2} & 0 \end{array}\right), \label{eq 13}$$ and the vector $\vec S\_{}$ has the components $$({{\mbox{\boldmath${A}$}}}^{-1/2}\sigma^{2}d{\rho \over c\_{s}^{2}} (\tilde \Psi^{ext}\_{m}-\tilde W\_{m}), \quad 0) \label{eqn6}.$$ Note that as follows from ([eq 12]) the relation between the components of $\vec Z$ and $\tilde W\_{m}$ can be taken to be given by $$Z\_1 = \sigma {{\mbox{\boldmath${A}$}}}^{1/2} \tilde W\_{m}, \qquad Z\_2 = {{\mbox{\boldmath${C}$}}}^{1/2} \tilde W\_{m}. \label{eq 13n}$$ Since the off diagonal elements in the matrix ([eq 13]) are adjoint of each other and the diagonal elements are self adjoint, it is clear that the operator ${{\mbox{\boldmath${ H}$}}}$ is self-adjoint. Equation ([eq 12]) can be formally solved using the spectral decomposition of ${{\mbox{\boldmath${ H}$}}}.$ We now make a few remarks concerning the spectrum. The oscillation spectrum of a rotating fluid contained within an axisymmetric domain ------------------------------------------------------------------------------------ It has been known for many years (see eg. Greenspan 1968, Stewartson & Rickard 1969) that the eigenvalue problem we consider is not well posed in the inertial mode range − 2Ω < σ < 2Ω. This is because in this spectral range the eigenvalue equation ([eq 41]) becomes a hyperbolic partial differential equation with boundary conditions specified on the planet boundary. The form of the spectrum depends on the behaviour of the characteristics, which correspond to localised inertial waves, under successive reflections from the boundary. Note that these reflections maintain a constant angle with the rotation axis rather than the normal to the boundary. The situation was conveniently summarised by Wood (1977) (see also Fokin 1994a,b and references therein). There are three types of behaviour of the characteristic paths for frequencies in the inertial mode range. They may all close forming periodic trajectories, they may be ergodic, or there may be a finite number of periodic trajectories that form attractors. The first two types of behaviour are believed to be associated with discrete normal modes while the third type leads to wave attractors and a continuous spectrum. The homogeneous sphere within a spherical or ellipsoidal boundary exhibits the first two kinds of behaviour and has discrete normal modes which form a dense spectrum (eg. Bryan 1889) while the same system with a solid core has wave attractors (eg. Ogilvie & Lin 2004, Ogilvie 2009). Note the characteristics behave in the same way for all spheres or ellipsoids with a continuous density distribution so that these should have normal modes. Note too that in the limit of very short wavelength modes only the second derivative terms matter in equation ([eq 41]) and the system becomes equivalent to the two dimensional case studied by Ralston (1973) and Schaeffer (1975). In that case the normal modes are associated with the frequencies for which all characteristic paths are periodic. They form a dense spectrum and are infinitely degenerate. From this discussion we expect the modes of a system with a continuous density distribution to approach the same form as those of the homogeneous sphere, an aspect upon which we build our later WKBJ approach. ### Formal solution of ([eq 12]) in the anelastic approximation From the above discussion we expect the normal modes for the cases of interest to form a discrete but dense spectrum. The anelastic approximation can be implemented by setting $\tilde W\_{m}=0 $ in equation ([eqn6]). In this case we can look for a solution to ([eq 12]) in the form $$\vec Z=\sum\_{k} \alpha\_{k} \vec Z\_{k}, \label{eq 13a}$$ where $\vec Z\_{k}$ are the real eigenfunctions of ${{\mbox{\boldmath${ H}$}}}$ satisfying $$\sigma\_{k} \vec Z\_{k} ={{\mbox{\boldmath${ H}$}}} \vec Z\_{k}, \label{eq 14}$$ the associated necessarily real eigenfrequencies being σk. Substituting ([eq 14]) into ([eq 12]) we obtain $$\alpha\_{k} ={<\vec Z\_{k}\| \vec S>\over <\vec Z\_{k}\|\vec Z\_{k}>(\sigma-\sigma\_{k})}. \label{eq 15}$$ The operator ${{\mbox{\boldmath${ H}$}}}$ induces the inner product and associated orthogonality relation for eigenfunctions according to the rule $$<\vec Z\_{k}\| \vec Z\_{l}>=\sigma\_{k}\sigma\_{l}(W\_{k}\| {{\mbox{\boldmath${A}$}}} W\_{l})+(W\_{k}\|{{\mbox{\boldmath${C}$}}} W\_{l}) = \delta\_{kl},\label{eqn7}$$ where $$W\_{k}={{\mbox{\boldmath${C}$}}}^{-1/2}Z\_{2}^{k}=\sigma\_{k}^{-1}{{\mbox{\boldmath${A}$}}}^{-1/2}Z\_{1}^{k}. \label{eqn8}$$ Using ([eqn6]) and ([eq 15]) we explicitly obtain $$\tilde W\_{m}= \sigma^{2}d\sum\_{k} {\sigma\_{k}\over N\_{k}(\sigma-\sigma\_{k})} (W\_{k}\| {\rho \over c\_{s}^{2}}\tilde \Psi^{ext}\_{m}) W\_{k}, \label{eq 16}$$ where $$N\_{k}=\sigma^{2}\_{k}(W\_{k}\| {{\mbox{\boldmath${A}$}}} W\_{k})+(W\_{k}\|{{\mbox{\boldmath${C}$}}} W\_{k})\label{eq 16a}$$ is the norm. The decomposition ([eq 13a]) should be valid for any vector $\vec F$ with components (F, 0), where F is any function of the spatial coordinates. The second component of this equality shows that in order for this to be valid an identity $$\sum\_{k} {<\vec F\|\vec Z\_{k}> \over N\_{k}}Z\_{2}^{k}=0$$ must be hold (IP). This identity allows us to represent the relation ([eq 16]) in a different form (PI): $$\tilde W\_{m}= \sigma d\sum\_{k} {\sigma\_{k}^{2}\over N\_{k}(\sigma-\sigma\_{k})} (W\_{k}\| {\rho \over c\_{s}^{2}}\tilde \Psi^{ext}\_{m}) W\_{k}. \label{eq 16b}$$ Note that in response problems such as the problem of excitation of the inertial waves during the periastron flyby, in order to take account of causality issues correctly when extending to the complex σ plane, one should add a small imaginary part in the resonance denominator in ([eq 16b]) according to the Landau prescription: (σ − σk) → (σ + iν − σk), where ν > 0 is a small real quantity. ### Corrections to the anelastic approximation When external forces are absent and the potential $\tilde \Psi^{ext}\_{m}$ is set to zero, equation ([eq 4]) (or, alternatively, equation ([eq 12])) defines the full eigenvalue problem. Under very general assumptions it was shown by IP that this problem can be formally solved in an analogous manner. However, it is rather difficult to use the general expressions obtained by IP without making further approximations. Here we note that, given that the spectrum is discrete, we may find conditions satisfied by the eigenfunctions and eigenvalues by replacing $\tilde \Psi^{ext}\_{m}$ by $-\tilde W\_{m}$ in equations ([eq 16]) and ([eq 16b]). These conditions relate any eigenfunction, now equated to $\tilde W\_{m}$ and its associated eigenvalue σ to the eigenfunctions and eigenvalues of the anelastic problem. Proceeding in this way we go on to form the quantity $$\sigma\sigma\_{l}(W\_{l}\| {{\mbox{\boldmath${A}$}}} \tilde W\_{m})+(W\_{l}\|{{\mbox{\boldmath${C}$}}} \tilde W\_{m}) = - {\sigma^2 d \sigma\_{l}\over (\sigma-\sigma\_{l})} (W\_{l}\| {\rho \over c\_{s}^{2}}\tilde W\_{m}), \label{eqn16c}$$ where Wl is an anelastic eigenfunction and we have made use of the orthogonality relation ([eqn7]). As argued in section [anelastic], the quantity on the right hand side can be regarded as a perturbation where the small parameter is ε = (Ω/Ω\)2. Provided an eigenfunction can be identified as ε* → 0 and is non degenerate with $\tilde W\_{m}\rightarrow W\_{l},$ it follows from ([eqn16c]) that in this limit $$(\sigma-\sigma\_{l}) = -{ d\_l\sigma^3\_{l}\over N\_{l}} (W\_{l}\| {\rho \over c\_{s}^{2}} W\_{l}), \label{eqn16d}$$ where d0 = 4Ω2 − σl2. The spectrum of inertial modes is dense. This may lead to a potential difficulties in identifying and following modes as parameters change as we discussed above. However, it is possible to argue that this problem can be alleviated for large scale global modes by for example modifying the eigenvalue problem by adding terms that have a very small effect on the global modes but spectrally separate close by short scale modes. Dintrans and Ouyed (2001) adopt such a procedure by adding a viscosity and this enables them to identify and follow global modes. Note that a similar situation would result if conservative high order derivative terms were added that preserved the self-adjoint form of the problem. Numerical work presented below and in PIN also confirms that global modes have a clear identity and can be followed as parameters change provided that the angular frequency is sufficiently small. Thus we both expect and verify the validity of the expression ([eqn16d]) in this limit. For larger values of Ω one should take into account a possibility of mixing between two neighbouring large scale global modes to explain results of numerical calculations, see PIN. In this case expression ([eqn16d]) should be modified in an appropriate way. ### Eigenvalues corresponding to opposite signs of m In the next section we find solutions of the eigenvalue problem in the WKBJ approximation. It will be shown that the corresponding eigenvalues and eigenmodes are independent of the sign of m to two leading orders. This is explained by the fact that to that order solutions are determined only by operators containing second and first derivatives in equation ([eq 4]). On the other hand it follows from the same equation that the only dependence on sign of m is determined by the operator ${{\mbox{\boldmath${B}$}}}$ which does not contain any derivatives of W. In order to find the first correction to the WKBJ eigenfrequencies that depends on sign of m, we treat the operator ${{\mbox{\boldmath${B}$}}}$ as a perturbation. This leads to a change in the eigenfrequency that can be found by using the same formalism that lead to equation ([eqn16d]) but then simply replacing σldlρWl/cs2 in that equation by $- {{\mbox{\boldmath${B}$}}} W\_{l}.$ Equation ([eqn16d]) then gives $$\sigma-\sigma\_{l} =-2m\Omega \sigma\_l^{2}\left(\int d\varpi dz {\partial \rho \over \partial \varpi} W\_{l}^{2}\right )/N\_{l}. \label{eqnn2}$$ Note that since ${\partial \rho \over \partial \varpi} < 0$ it follows from equation ([eqnn2]) that when Ω > 0 the sign of σ − σl is proportional to the sign of m. A form of equation ([eq 4]) valid for a spherical planet -------------------------------------------------------- In what follows we assume that an object experiencing tidal interactions can be approximated as having a spherically symmetric structure. In this case it is appropriate to use another form of ([eq 4]) with $\tilde \Psi^{ext}\_{m}=0$, which is especially convenient for an analysis of WKBJ solutions. We can obtain this from ([eq 4]) using the fact that for a spherical star ${\partial \over \partial \varpi}\rho ={\varpi \over r} {d\over dr}\rho $ and ${\partial \over \partial z}\rho ={z \over r} {d\over dr}\rho $. We obtain $$\left[\sigma^{2}\Delta -4\Omega^{2}{\partial^{2} \over \partial z^{2}}\right] W ={1\over rH}\left(\left[\sigma^{2} \varpi {\partial W \over \partial \varpi} -d z {\partial W \over \partial z }\right] -\left[2m\sigma \Omega -{d (\sigma/ \Omega\_{K}(r))^{2}}\right]W\right), \label{eqn12}$$ where we set, for simplicity, $W\equiv \tilde W\_{m}$, Δ is the Laplace operator, $H=-{d r\over d \ln\rho}$ is a characteristic density scale height and $\Omega\_{K}(r)=\sqrt {GM(r)\over r^{3}}$, where M(r) is the mass enclosed within a radius r. Note that we use the hydrostatic balance equation $-c\_{s}^{2}{d\rho \over dr}={GM(r)\over r^{3}}$ to obtain equation ([eqn12]) from equation ([eq 4]). The last term in the second square braces on the right hand side describes correction to the anelastic approximation. It is discarded when the WKBJ approximation is used. ### Pseudo-spheroidal coordinates When the density approaches a constant value, H tends to infinity and the right hand side of equation ([eqn12]) vanishes. In this case it describes an incompressible fluid, see eg. Greenspan (1968). It was shown by Bryan (1889) that in this case this equation is separable in special ’pseudo-spheroidal’ orthogonal coordinates defined by the relations $$\varpi =R\_{\}\sqrt{{(1-x\_{1}^{2})(1-x\_{2}^{2})\over (1-\mu^{2})}}, \quad z={R\_\x\_{1}x\_{2}\over \mu}, {\rm \ \ where \ \ the\ \ constant\ \ parameter\ \ } \mu ={\sigma \over 2\Omega}. \label{eqn13}$$ Since the governing equations are invariant to the mapping (σ, m) → ( − σ, − m), without loss of generality we assume from now on that μ > 0 for all modes while m can have either sign. Also, from equation ([eqn12]) it follows that the modes should be either even or odd with respect to the reflection in the equatorial plane z → − z. Therefore, it is sufficient to consider only the upper hemisphere $(z >0, \sqrt{\varpi^{2}+z^{2}} \le R\_{\})$. In this region we can assume that the variables x1 and x2 are contained within the intervals [μ, 1] and [0, μ], respectively. A detailed description of this coordinate system can be found in eg. Arras et al (2003), Wu (2005)a. Using the new variables equation ([eqn12]) takes the form $$(\hat D\_{2} -\hat D\_{1})W={1\over \mu^{2}(1-\mu^{2})Hr} \left( \hat AW -(x\_{1}^{2}-x\_{2}^{2})(m\mu-4\mu^{2}(1-\mu^{2})({\Omega /\Omega\_{K}(r)})^{2})W\right) \label{eqn15}$$ where $$\hat D\_{i}= -{d\over d x\_{i}}(1-x\_{i}^{2}){d\over d x\_{i}} +{m^{2}\over (1-x\_{i}^{2})}, \label{eqn16}$$ $$\hat A = x\_{1}(1-x\_{1}^{2})(x\_{2}^{2}-\mu^{2}){\partial \over \partial x\_{1}} -x\_{2}(1-x\_{2}^{2})(x\_{1}^{2}-\mu^{2}){\partial \over \partial x\_{2}}, \label{eqn17}$$ and the quantities r, H, Ω\(r) are understood to be functions of the variables x1 and x2. It is easy to see that the eigenfunctions of the operators $\hat D\_{i}$ are the associated Legendre functions, Pνm(xi), and we have DiPνm(xi) = λ2Pνm(xi), where $\lambda=\sqrt{\nu(\nu+1)}$. Let us stress that as the domains of x1 and x2 are not [ − 1, 1], ν is not necessarily an integer. In some important cases equation ([eqn15]) is separable. Firstly, when the gas is incompressible, the right hand side of ([eqn15]) is zero. In this case it follows from equation ([eqn18]) that the solution can be represented as product of two associated Legendre functions. Secondly, as was mentioned by Arras et al (2003) and later explored in detail by Wu (2005)a,b when equation ([eqn15]) is considered in the anelastic approximation it is separable for planetary models with density profiles of the form $$\rho=C\left(1-\left({r\over R\_{\}}\right)^{2}\right)^{\beta}, \label{eqn19}$$ where C* is a constant. These models include the incompressible one which corresponds to β = 0. It was also noted by Arras (2003) and Wu (2005)a,b that for polytropic models with equation of state $$\hspace{6cm} p=k\rho^{\gamma}, \label{eqn20}$$ close to surface the density distribution has the form $$\hspace{6cm} \rho=Dx^{n}, \label{eqn21}$$ where x = 1 − r/R\, n* = 1/(γ − 1) is the polytropic index, and D is a constant. In the asymptotic limit r → R\* this expression coincides with what is obtained from equation ([eqn19]) with β = n. This proves that when polytropic models are considered equation ([eqn15]) is separable in a plane parallel approximation often adopted close to the surface. Here we would like to note that this is valid even when the anelastic approximation is relaxed since close to the surface we have ΩK(r) ≈ Ω\* = const and the additional term appearing in this case in the braces on the right hand side of ([eqn15]) has the same spatial structure of a term already present in the anelastic approximation. WKBJ solutions for the normal modes =================================== In general equation ([eqn15]) should be solved numerically. We can, however, look for analytical solutions to ([eqn15]) in the WKBJ approximation assuming that solutions are fast oscillating functions in the planet’s interior. The first and second derivatives of these functions are assumed to be proportional to first and second power of a large parameter λ, the value of which is specified below. This problem has been analysed before by Arras (2003) and Wu (2005)a who obtained expressions for eigenvalues and eigenfunctions for the problem of free oscillations in the inertial mode spectral range. Here we revisit the problem, taking into account terms that appear at the next order in an asymptotic expansion of the quantities of interest in a power series in 1/λ. This will allow us to obtain analytic expressions which agree with numerical results, even for the rather small values of λ, appropriate to global modes (see below). Natural units ------------- In what follows in order to simplify notation we express all dimensional quantities in natural units. These are such that the spatial coordinates, density, angular velocity and sound speed are expressed in units of R\, the mean density ${\bar \rho}=3M\_{\}/ (4\pi R\_{\}^{3})$,$\sqrt {GM\_{\}/ R\_{\}^3}$ and $\sqrt {GM\_{\}/ R\_{\}}$ respectively. All other quantities of interest are expressed in terms of powers of these basic units. WKBJ solutions -------------- It is easy to find from either ([eqn12]) or ([eqn15]) that in the planet’s interior far from the rotational axis, the WKBJ solution should have the form $$W\_{WKBJ}={1\over \sqrt{\rho \varpi}}(C\_{1}e^{i\lambda \phi\_{+}(u\_{+})} + C\_{2}e^{i\lambda \phi\_{-}(u\_{-})}+cc) +O({1\over \lambda}), \label{eqn22}$$ where ϕ±(u±) are arbitrary functions of $$u\_{\pm}=\varpi \pm {\mu \over \sqrt{1-\mu^{2}}}z, \label{eqn23}$$ the constancy of which defines the characteristics of equation ([eqn15]). Acceptable forms for the functions ϕ± have to be determined by matching the solution ([eqn22]) to approximate solutions valid near the surface boundary and near the rotational axis. It turns out that this matching is possible if the WKBJ solution has the form $$W\_{WKBJ}={1\over \sqrt{\rho \varpi}}(\cos(\lambda y\_{1}+\phi\_{1})\cos(\lambda y\_{2} + \phi\_{2})), \label{eqn24}$$ where y1, 2 = arccos(x1, 2) and ϕ1, 2 are constants to be determined. One can readily check with help of the coordinate transformations ([eqn13]) that this form agrees with the general expression ([eqn22]), see equation ([eqn50]) below. Since y1, 2 are multivalued functions of x1, 2 we should specify a one-valued branch of these. Taking into account that our calculations will be done for positive values of x1, 2, we assume below that values of y1, 2 are in the range [0, π/2]. For simplicity, in the main text we are going to consider the modes even with respect to reflection z* → − z, called hereafter ‘the even modes’. For example, such modes are excited by tidal interactions since tidal potential is an even function of z. The case of the modes odd with respect to this reflection (’the odd modes’) can be dealt with in a similar way. This case is considered in Appendix A. From equation ([eqn13]) it follows that reflection of the coordinate z leads to the reflection of the coordinate x2 such that x2 → − x2, while the coordinate y2 changes according to the rule y2 → π − y2. We readily find that ([eqn24]) is unchanged under this transformation provided the phase ϕ2 = − πλ/2, (see also eg. Wu 2005a). We remark that the same result is obtained by requiring that the derivative of ([eqn24]) with respect to y2 vanish on the equator where y2 = π/2. Matching near the rotation axis ------------------------------- In the WKBJ approximation sufficiently far from the rotational axis all terms proportional to W give small corrections to the solution ([eqn22]) and are formally discarded. However, when $\varpi \rightarrow 0$ and, accordingly, x1 → 1, it follows from equation ([eqn16]) that the term proportional to m2W in the expression for the operator $\hat D\_{1}$ diverges in this limit and should be retained. When this is done the phase ϕ1 can be found from condition of regularity of W close to the rotation axis $\varpi=0$. We begin by using the WKBJ solution already found to develop an approximate expression for W that is appropriate for small values of δ = 1 − x1 and which can be matched at large distances from the rotation axis. An appropriate expression for W which can be matched to the correct WKBJ limit sufficiently far from the rotational axis is $$W\propto {1\over \sqrt{\rho} {(1-x\_{2}^{2})^{1/4}}}\cos \lambda (y\_{2}-\pi/2)W\_{a}(\delta), \label{eqn25}$$ where we take into account that the factor $\sqrt{\varpi}$ entering ([eqn22]) is proportional to the product (1 − x12)1/4(1 − x22)1/4 ∝ δ1/4(1 − x22)1/4, see equation ([eqn13]), and the factor δ1/4 is formally incorporated in the definition of Wa(δ) which is to be found by imposing the condition of regularity on the rotation axis. In order to do this we obtain an equation for Wa from equation ([eqn12]) (or ([eqn15])) that retains terms containing the derivatives and terms that potentially diverge in the limit $\varpi \rightarrow 0$ while other terms can be discarded. From equations ([eqn12]) and ([eqn15]) it follows that Wa(δ) satisfies equation ([eqn18]) in the limit of small δ $$\delta^{2}{d^{2}\over d\delta^{2}}W\_{a}+\delta {d\over d\delta}W\_{a} +({\lambda^{2}\over 2}\delta -{m^{2}\over 4})W\_{a}=0. \label{eqn26}$$ The solution to ([eqn26]) regular at the point δ = 0 can be expressed in terms of the Bessel function $$W\_a\propto J\_{\|m\|}(\sqrt{2}\lambda \delta^{1/2}), \label{eqn27}$$ where we assume from now on that λ is positive[2](#fn2). In the limit of large (λδ) the asymptotic form of the expression ([eqn27]) is $$W\_a\propto \delta^{-1/4}\cos\left (\sqrt{2}\lambda \delta^{1/2} -\|m\|{\pi \over 2}-{\pi\over 4}\right). \label{eqn28}$$ It is easy to see that when δ is small $y\_{1}\approx \sqrt{2\delta}$. Therefore, from equations ([eqn25]) and ([eqn28]) it follows that the solution has the required form ([eqn24]) provided that $$\phi\_{1}=-\|m\|{\pi\over 2}-{\pi\over 4}, \label{eqn29}$$ and we have, accordingly, $$W\_{WKBJ}={1\over \sqrt{\rho \varpi}}F, \quad F=\cos\left(\lambda y\_{1}-\|m\|{\pi\over 2}-{\pi\over 4}\right )\cos\left (\lambda y\_{2} -\lambda{\pi\over 2}\right). \label{eqn30}$$ Note that the phase ([eqn29]), which can in fact be verified with reference to the incompressible sphere, differs from that given in Arras et al (2003) and Wu (2005)a. This disagreement is due to an oversimplified treatment of the WKBJ solution close to the rotational axis in these papers. Matching at the planet surface ------------------------------- The eigenvalues appropriate to the problem of free oscillations can be found by matching the solution ([eqn30]) to approximate solutions valid near the surface of the planet. In pseudo-spheroidal coordinates ([eqn13]) the equation determining the upper hemispherical surface of the planet (r = R\, z* > 0) has two branches: 1) x1 = μ, 0 < x2 < μ and 2) μ < x1 < 1, x2 = μ. In order to simultaneously consider solutions to equation ([eqn15]) that can be close to either of these branches, we introduce two new coordinates δi, with i = 1 corresponding to the first branch and i = 2 corresponding to the second branch, that are defined by the relation xi = μ ± δi, where the sign + ( − ) corresponds to the 1st (2nd) branch, and assume later on that the δi are small. The form of the solutions close to the surface depends on the density profile. In what follows we the consider the planet models with a polytropic equation of state for which the density profile close to the surface is given by equation ([eqn21]). The variable x = 1 − r/R\* entering equation ([eqn21]) can be expressed through xj and δi as $$x=\mp {(x\_{j}^{2}-\mu^{2})\delta\_{i}\over \mu (1-\mu^{2})}, \label{eqn31}$$ where we assume from now on that the upper (lower) sign corresponds to the 1st (2nd) branch and the index i takes on the values 1 ( first branch) and 2 (2nd branch) with j ≠ i. We now look for solutions close to the surface that have λδj large but λδi small. This is possible because in the WKBJ theory λ is a large parameter. The domain for which λδi is small for both i = 1 and i = 2 is called the critical latitude domain and will be considered separately below. Solutions valid in all of these domains must match correctly on to a solution of the form ([eqn30]) in order to produce a valid eigenfunction. Using equations ([eqn21]) and ([eqn24]) we can look for a solution close to the surface in the form W ∝ ∣xj2 − μ2∣− n/2(1 − xj2)− 1/4cos(λyj + ϕj)Wi(δi), where we also use equation ([eqn13]) in order to express the factor $\varpi^{-1/2}$ in terms xj setting δi = 0 there. Substituting this expression in equation ([eqn15]) and taking the limit δi → 0 we obtain $$\delta {d^{2}\over d\delta^{2}}W+n{d\over d\delta }W + {1\over (1-\mu^{2})}(\lambda^{2}\delta \pm nm\_{\})W=0, \label{eqn33}$$ where, for simplicity, we omit the index (i) in the quantities δ*i and Wi, and $$m\_{\} =m-4\mu(1-\mu^{2})\frac{\Omega^{2}}{\Omega\_\^2}. \label{eqn34}$$ We recall that the term proportional to Ω/Ω\* gives the correction to the anelastic approximation. Since in the low frequency limit Ω/Ω\* is assumed to be much smaller than unity, this term is small and we approximately have m\* ≈ m. Equation ([eqn33]) can be brought into a standard form by the change of variables $$\zeta=2i\kappa \delta, \quad \Psi= e^{i\kappa \delta}W=e^{\zeta/2}W, \ \ {\rm where} \ \ \kappa={\lambda \over \sqrt{1-\mu^{2}}}. \label{eqn35}$$ Adopting these we obtain $$\zeta \Psi^{\prime \prime}+(n-\zeta)\Psi^{\prime}-{1\over 2}(n\pm i\chi)\Psi=0, \label{eqn36}$$ where a prime denotes differentiation with respect to ζ and $\chi=nm\_{\}/ (\lambda \sqrt{1-\mu^{2}}).$ This is the confluent hyper-geometric equation. Its solution that is regular at the surface is expressed in terms of the confluent hyper-geometric function Φ(a, b, z) as W* ∝ e− ζ/2Φ((n ± iχ)/2, n, ζ). Note that this solution is similar to solutions of the Schrodinger equation with the Coulomb potential describing wave functions belonging to continuous part of its spectrum, ( see eg. Landau & Lifshitz 1977). In the limit of ∣κδ∣ ≫ 1 we obtain from ([eqn37]) $$W\propto \zeta^{-n/2}\left ({1\over \Gamma ({1\over 2}(n\mp i\chi))}\exp \left( i\left(-{\|\zeta\|\over 2}\mp {\chi \over 2}\ln\|\zeta\|+{\pi n\over 4}\right)\right)+cc\right), \label{eqn39}$$ where G(z) is the gamma function. Since the quantity χ is assumed to be small we can approximately write $${1/ \Gamma ({1\over 2}(n\mp i\chi))}\approx (1\pm i\psi({n\over 2}){\chi \over 2})/ \Gamma ({n\over 2}) \approx \exp\left(i\psi({n\over 2}){\chi \over 2}\right)/ \Gamma ({n\over 2}), \label{eqn40}$$ where $\psi(z)\equiv {d\over dz}\Gamma(z)$ is the psi function. In the same approximation equation ([eqn39]) can be rewritten in the form $$W\propto \zeta^{-n/2}\cos \left ({\|\zeta\|\over 2}\pm {\chi \over 2}\left(\ln\|\zeta\|-\psi({n\over 2})\right)-{\pi n\over 4}\right ). \label{eqn41}$$ After substituting the result expressed by equation ([eqn41]) into ([eqn32]) the resulting expression should be of the general` form (given by [eqn24]) evaluated close to the surface. This, however, cannot be realised on account of the presence of the factor ln∣ζ∣ in ([eqn41]). This term, having a coordinate dependence of order of 1/λ after removing a constant phase would formally require terms of that order that are not accounted for in the expressions ([eqn22]) and ([eqn24]) to enable matching, therefore to the order we are currently working, it is discarded. Since only this term depends on the sign of m and on the correction to the anelastic approximation, both dependencies are absent in the resulting approximation. Another way of obtaining solutions to ([eqn33]) compatible with the form ([eqn24]) inside the planet is to set to zero the small quantity χ ∝ 1/λ in equation ([eqn37]). In this case the solution can be expressed in terms of a Bessel function such that [3](#fn3) W(δ) ∝ δ(1 − n)/2J1/2(n − 1)(κδ). Note that this expression is equivalent to ([eqn37]) when the anelastic approximation is adopted and m = 0. When κδ → ∞ we get $$W\_{i}\propto \delta\_{i}^{-n/2}\cos \left({\lambda \over \sqrt{1-\mu^{2}}} \delta\_{i}-{\pi n\over 4}\right), \label{eqn42}$$ where we the index (i) has been restored and we use the explicit expression for κ. Substituting ([eqn42]) into equation ([eqn32]), taking into account that the factor $(\|x\_{j}^{2}-\mu^{2}\|\delta\_{i})^{-n/2} \propto \sqrt \rho $, and that close to the surface we have $$y\_{i}\approx \arccos \mu \mp {\delta\_{i}\over \sqrt{1-\mu^{2}}}. \label{eqn43}$$ Determination of the eigenfrequencies ------------------------------------- It can now be seen that the expression ([eqn32]) has the required form ([eqn24]) provided that the phases ϕi satisfy appropriate appropriate conditions. However, these phases have already been determined from the requirements of regularity on the rotation axis and symmetry with respect to reflection in the equatorial plane and are accordingly specified through equation ([eqn30]) which equation([eqn32]) must match. It is readily seen that the expressions ([eqn30]) and ([eqn32]) can be compatible only for particular choices of λ and μ. These compatibility conditions determine the eigenspectrum of the problem in the WKBJ approximation. They are easily found from equations ([eqn30]), ([eqn42]) and ([eqn43])to be given by $${\pi \over 4}n+\pi k\_{1}=\lambda \arccos (\mu)-{\pi\over 2}\|m\|-{\pi \over 4} \quad {\rm and} \quad {\pi \over 4}n+\pi k\_{2}=\lambda \left({\pi \over 2}-\arccos (\mu)\right). \label{eqn44}$$ Here k1 and k2 are positive or negative integers that must be chosen in a way which ensures that the angle arccos(μ) belongs to the branch $0 < \arccos(\mu) < {\pi\over 2}$. Adding the above relations we obtain $$\lambda =2l+n+\|m\|+{1\over 2}, \label{eqn45}$$ where l = k1 + k2. Substituting ([eqn45]) into the first expression in ([eqn44]) we obtain an expression for the eigenfrequency $$\mu =\frac{\sigma}{2\Omega}= \cos\left( \pi {k+{\|m\|/ 2}+(n+1)/4\over \lambda}\right), \label{eqn46}$$ where we set k ≡ k1 from now on. As shown in Appendix A the modes with different symmetry with respect to reflection z → − z (both the ’even’ and the ‘odd’ modes) can be described by the same expression ([eqn46]) provided that the expression for λ changes to $$\lambda=p+n+\|m\|+{1\over 2}, \label{eqn45n}$$ where the integer p is even for the even modes while for those with odd symmetry p is odd. For the WKBJ approximation to be valid λ should be large, and, accordingly, l ≫ 1. We would like, however, to consider all values of l and k allowed by our assumption that λ is positive and arccos(μ) belongs to the interval (0, π/2). These conditions imply that l is positive and lead to inequality: $$-\left[{(1+n)\over 4}+{\|m\|\over 2}\right] \le k \le \left[l+{n\over 4}\right], \label{eqn47}$$ where [Q] means that integer part of Q is taken. When l and k are sufficiently large one may neglect other quantities in the argument of the cosine in equation ([eqn46]). In this case one gets μ = cos(πk/(2l)) - an expression obtained in previous papers (see Arras 2003 and Wu 2005a). One may also consider the limit of an incompressible fluid by setting n = 0 in ([eqn46]). In this case the expression ([eqn46]) gives the correct asymptotic eigenfrequencies appropriate to the high order modes of pulsation of an incompressible fluid in a rotating spherical container, see Appendix B for details. A general expression for eigenfunctions close to the surface of the planet -------------------------------------------------------------------------- The purpose here is to establish an expression for W that is approximately valid in the whole region close to the surface where the separation of variables is possible and the eigenfunction can be written as the product of functions of x1 alone and x2 alone. Also, this expression should approach the WKBJ expression ([eqn30]) in the limit of sufficiently large λx in order to have the norm that will be given by equation ([eqn57]) below. Close to the planet surface we have x ≡ 1 − r ≪ 1, the density profile can be represented in the form ([eqn21]), and equation ([eq 4]) becomes separable in the pseudo-spheroidal coordinates (x1, x2) (Arras et al 2003, Wu 2005a). As described already in section [Surfmatch], in these coordinates the region close to the surface is described by two branches xi = μ ± δi, (i = 1, 2) see equation ([eqn30a]). We denote these branches as the ( + ) branch for which i = 1 and the ( − ) branch for which i = 2, respectively. When one of the coordinates, xj, say, is sufficiently far from the value xj = μ the eigenfunction is proportional to the expression given by equation ([eqn32]). In practice the requirement that xj is far from μ is that ∣λδj∣ be large. When this parameter is of order unity or less, xj is considered to be close to μ. When both coordinates are close to μ in this sense, the eigenfunction is proportional, with, in the limit of large λ, proportionality factor being slowly varying, to the product W1(δ1)W2(δ2), where Wi(xi) can be found from equation ([eqn38]). From equation ([eqn13]) it follows that when x1 ≈ x2 ≈ μ the spherical polar angle θ is close to the critical latitude defined by θ = arccos(μ). Accordingly, we describe this region as the region near the critical latitude. ### An expression for the eigenfunction near the critical latitude In order to obtain an expression for the eigenfunction that is valid near the critical latitude we proceed as follows. At first we consider a region of the planet sufficiently far from rotational axis, where λ(1 − x1) ≫ 1. We start from the form of solution given by equation([eqn32]) in the form $$W\propto (\pm(\mu^2-x\_{2,1}^{2})^{-n/2}(1-x\_{2,1}^{2})^{-1/4}\cos (\lambda y\_{2,1} +\phi\_{2,1})\bar W(\delta\_{1,2}), \label{eqn320}$$ where here and below the first index and upper sign (second index and lower sign) correspond to the ( + ) branch (( − ) branch). The function $\bar W(y) $ satisfies equation ([eqn33]) with m\* set to zero. As we discussed above the term proportional to m\* gives a correction which will be calculated below. The fact that the function $\bar W(y) $ is normalised in order to have the appropriate limit in the case of large y is stressed by the overbar. We have $$\bar W(y) =\sqrt {\pi\kappa \over 2}y^{(1-n)/2}J\_{(n-1)/2}(\kappa y), \label{eqn60}$$ where we recall that $\kappa =\lambda/\sqrt{1-\mu^{2}}.$ Equation ([eqn320]) is not valid near the critical latitude where both x1 and x2 are close to μ. To obtain a modified form that is valid, the function cos(λy2, 1 + ϕ2, 1) must be replaced by a function that matches this when λ∣y2, 1 − arccos(μ)∣ is large but which takes has the correct form to result in the proportionality of the eigenfunction, as indicated above to $\bar W(\delta\_1)\bar W(\delta\_2),$ when this is small. It can be seen that an expression for W having the required properties can be written in the form $$W\_{\pm}=D\_{\}^{-1/2}(1-x^{2}\_{2,1})^{-1/4}(\pm(\mu^{2}-x\_{2,1}^{2}))^{-n/2} \bar W(\delta\_{1,2})\Delta\_{2,1}^{n/2}\bar W(\Delta\_{2,1}), \label{eqn58} \hspace {3mm} {\rm where}$$ $$\Delta\_{2,1}=\pm\sqrt{1-\mu^{2}}(y\_{2,1}-\arccos (\mu) ). \label{eqn59}$$ Here the quantity D\ = D((1 − μ2)μ)− n, where D is defined in equation ([eqn21]) [4](#fn4). We remark that in the limit κδ1, 2 → ∞ together with the limit κΔ1, 2 → ∞ we can use the asymptotic expansion of Bessel functions $$J\_{(n-1)/2}(\kappa y)=\sqrt{{2\over \pi \kappa y}}[\cos{(\kappa y -{\pi n\over 4})} (\sum\_{k=0}^{\infty} {A\_{k}\over (\kappa y)^{2k}}) - \sin{(\kappa y -{\pi n\over 4})} (\sum\_{k=0}^{\infty} {B\_{k}\over (\kappa y)^{(2k+1)}})], \label{eqn61aa}$$ where $$A\_{k}={(-1)^{k}\over 2^{2k} (2k)!}{\Gamma ({n\over 2}+2k)\over \Gamma ({n\over 2}-2k)}, \quad B\_{k}={(-1)^{k}\over 2^{(2k+1)} (2k+1)!}{\Gamma ({n\over 2}+2k+1)\over \Gamma ({n\over 2}-2k-1)}, \label{eqn61bb}$$ together with equation([eqn44]) to show that expression ([eqn58]) attains the required form specified in equations $(\ref{eqn24})$ and $(\ref{eqn30})$. ### An expression for the eigenfunction near the pole and surface In the region close to the pole of the planet we have x2 ∼ μ and δ = 1 − x1 ≪ 1. The discussion given above excluded consideration of this domain which needs to be considered separately. Close to the pole but away from the surface the solution is given by equation ([eqn25]). From very similar considerations to those above, close to the surface where ∣λδ2∣ is of order unity or less and to the pole where 1 − x1 ≪ 1, the solution is proportional, to within, in the limit of large λ, a slowly varying proportionality factor, to the product of $\bar W{(\delta\_2)}$ and the solution given by equation ([eqn27]) which is valid near the rotation axis. Thus in this domain $W\propto W\_{a}(\delta )\bar W{(\delta\_2)}.$ An an approximate solution for Wpole that satisfies the required matching conditions and also reduces to the form ([eqn58]) in the limit λ(1 − x1) ≫ 1 can be written as $$W\_{pole}=(-1)^{k\_{1}}D\_{\}^{-1/2}(1-x\_{1}^{2})^{-1/4}(x\_{1}^{2}-\mu^{2})^{-n/2} \sqrt{{\pi \lambda y\_{1}\over 2}}J\_{\|m\|}(\lambda y\_{1})\bar W(\delta\_{2}),\label{eqn61n}$$ where the factor ( − 1)k1 takes into account that when k1 is odd W*a and WWKBJ differ by sign in the matching region, see equation ([eqn44]). An approximate expression valid over the whole surface domain ------------------------------------------------------------- We now use an interpolation procedure to combine expressions derived above, that are valid in separate domains inside the planet, to form single expressions that can be used over the whole domain. To do this we introduce a function η[x] that is defined for x ∈ [0, 1] and belongs to C(∞). We stipulate that η[x] decreases monotonically with x and is such that η[x] ≡ 1 when x ∈ [0, 1 − x\] and η[x] ≡ 0 when x* ∈ [x\, 1] The quantity x\ is a parameter such that 1/2 < x\* < 1. An explicit form of η[x] as well as the value of x\* are not important for our purposes. This is because all representations in the different domains attain a matching asymptotic form in the planet interior and elsewhere. ### An expression for W− We may now write down an approximate solution valid close to the surface for all x1 ∈ [μ, 1] for W− which we denote as W−\* as W−\* = η[z1]W− + (1 − η[z1])Wpole, where z1 = (x1 − μ)/(1 − μ) and we recall that W− in the above is to be obtained from equation ([eqn58]). The expression ([eqn62n]) can be rewritten in another useful form, which explicitly shows that the solution is separable close to the surface: $$W\_{-}^{\}=D\_{\}^{-1/2}(1-x\_{1}^{2})^{-1/4}(x\_{1}^{2}-\mu^{2})^{-n/2}\tilde W\_{1}(x\_{1})\bar W(\delta\_{2}), \label{eqn62na}$$ where $$\tilde W\_{1}=\eta [z\_{1}] \Delta\_{1}^{n/2}\bar W (\Delta\_{1})+(1-\eta [z\_{1}]) \sqrt{\pi \lambda y\_{1}\over 2}J\_{\|m\|}(\lambda y\_{1}). \label{eqn62nb}$$ ### An expression for W+ In this case we formulate an expression valid close to the surface and for all x2 ∈ (0, μ). Close to the equatorial plane and away from the critical latitude, and κ(μ − x2) = κΔ2 ≫ 1 it is convenient to represent the function W+ in terms of an asymptotic series in ascending powers of (κΔ2)− 1. Substituting the series ([eqn61aa]) in ([eqn58]) for W+ we obtain: $$W\_{eq}=D\_{\}^{-1/2}(1-x\_{2}^{2})^{-1/4}(\mu^{2}-x\_{2}^{2})^{-n/2} \tilde W\_{eq}(x\_{2})\bar W(\delta\_{1}), \label{eqn62nc}$$ where $$\tilde W\_{eq}(x\_{2})=(-1)^{k\_{2}}\cos \lambda (y\_{2}-{\pi\over 2}) \sum\_{k=0}^{\infty} {A\_{k}\over (\kappa \Delta\_{2})^{2k}} +(-1)^{(k\_{2}+1)} \sin \lambda (y\_{2}- \pi/ 2) \sum\_{k=0}^{\infty} {B\_{k}\over (\kappa \Delta\_{2})^{(2k+1)}}, \label{eqn62nd}$$ where the coefficients A*k and Bk are given in equation ([eqn61bb]) and we take into account that κΔ2 − nπ/4 = λ(y2 − π/2) + πk2, according to equations ([eqn44]) and ([eqn59]). On the other hand it is convenient to use the expression ([eqn58]) directly in the region close to the critical latitude. The expressions ([eqn58]) and ([eqn62nc]) can be combined with help of the function η[x] to provide a single function W+\, which is analogous to the function W−\ discussed above and can be used for x2 ∈ (0, μ): $$W\_{+}^{\}=D\_{\}^{-1/2}(1-x\_{2}^{2})^{-1/4}(\mu^{2}-x\_{2}^{2})^{-n/2} \tilde W\_{2}(x\_{2})\bar W(\delta\_{1}), \label{eqn62nf}$$ where $$W\_{2}(x\_{2})=\eta [z\_{2}] \Delta\_{2}^{n/2}\bar W(\Delta\_{2}) + (1-\eta [z\_{2}]) \tilde W\_{eq}(x\_{2}), \label{eqn62ng}$$ and z2 = 1 − x2/μ. Calculation of the mode norm ---------------------------- In order to calculate different quantities related to a particular mode we need an expression for the mode norm given by equation ([eq 16a]). One can show that when λ is large enough the interior of the planet gives the dominant contribution to the integrals determining the norm. Thus the expression ([eqn30]) for the eigenfunction is appropriate. In addition one can simplify the expression ([eq 16a]) by noting that eigenfunctions satisfy equation ([eq 4]) with the right hand side set to zero. Furthermore, the term involving the operator ${\mbox{\boldmath${B}$}}$ in equation ([eq 4]) may be neglected. This is because it does not contain second derivatives and therefore contributes a higher order correction to the WKBJ approximation as discussed above. Additionally, for the same reason, only the terms proportional to the second derivatives in the operators ${\mbox{\boldmath${A}$}}$ and ${\mbox{\boldmath${C}$}}$ need to be retained since these terms give the leading contributions to the norm being proportional to λ2. Thus we have $$\sigma^{2}{{\mbox{\boldmath${A}$}}}W\_{WKBJ}\approx {{\mbox{\boldmath${C}$}}}W\_{WKBJ}, \label{eqn48}$$ and $$N\approx 2(W\_{WKBJ}\|{{\mbox{\boldmath${C}$}}}W\_{WKBJ})=8\Omega^{2}\int \_Vd\varpi dz\left ({\partial \over \partial z}F\right)^{2}, \label{eqn49}$$ where in order to evaluate the norm ([eq 16a]), we use the explicit form of ${\mbox{\boldmath${C}$}}$ given by equation ([eq 6]) and, after an integration by parts, adopt equation ([eqn30]) for the eigenfunction. The quantity F can be expressed in the form $$F={1\over 2}(\cos (\lambda y\_{+} -\Psi\_{+})+\cos(\lambda y\_{-}-\Psi\_{-})), \label{eqn50}$$ where $$y\_{\pm}=y\_{2}\pm y\_{1}, \quad \Psi\_{\pm}={\pi \over 2}\|m\|+{\pi \over 4}\pm {\pi \over 2}\lambda. \label{eqn51}$$ Note that y± can be readily expressed in terms of coordinates (x1, x2) and $(\varpi,z)$: $$\cos y\_{\pm} =x\_{1}x\_{2}\mp \sqrt{1-x\_{1}^{2}}\sqrt{1-x\_{2}^{2}} =\mu z\mp (\sqrt{1-\mu^{2}})\varpi, \label{eqn52}$$ where we use equations ([eqn13]). From equation ([eqn52]) it follows that the quantities y± are constant on characteristics of equation ([eq 4]), see equation ([eqn23]). By differentiating ([eqn50]) we then obtain $${\partial F \over \partial z}=- {\mu \lambda\over 2} \left({\sin (\lambda y\_{+} -\Psi\_{+})\over \sin y\_{+}} +{\sin (\lambda y\_{-} -\Psi\_{-})\over \sin y\_{-}}\right). \label{eqn54}$$ In the limit of large λ the integral of square of the derivative ([eqn54]) in ([eqn49]) can be approximately calculated taking into account that only average values of sin2(λy± − Ψ±) ≈ 1/2 give a significant contribution to the integral. In this way we obtain $$\int \_V d\varpi dz \left({\partial F\over \partial z}\right)^{2}={\mu^{2}\lambda^{2}\over 8}I, \label{eqn55}$$ where $$I\equiv \int \_V rdr d\theta \lbrace {1\over \sin^{2}y\_{+}}+{1\over \sin^{2}y\_{-}}\rbrace=2\pi. \label{eqn56}$$ Note that in order to evaluate the integral ([eqn56]) we use the fact that from ([eqn52]) we have sin2y± = 1 − r2cos2(θ ± arccos(μ)). The integral is elementary and most easily done by noting that it is easily shown to be independent of μ, and can accordingly be evaluated setting μ = 1. Substituting ([eqn56]) into ([eqn49]) we obtain a very simple expression for the norm: N = 2π(μλΩ)2. Calculation of the frequency correction σ1m ----------------------------------------------- As follows from equation ([eqn46]) the eigenfrequencies of the modes are degenerate with respect to change of sign of m in the approximation we use. The correction to the eigenfrequencies accounting for the dependence on sign of m, σ1m, is calculated above, see equation ([eqnn2]). As follows from this equation, σ1m is determined by the integral $$I=-\int \_V d\varpi dz {\partial \rho \over \partial \varpi} W\_{k}^{2}. \label{eqn62}$$ The integral ([eqn62]) has contributions from the interior of the planet where the mode eigenfunction is given by ([eqn30]) and also close to the surface where equations ([eqn58]) and ([eqn61n]) apply. Contributions to the integral close to the surface arise from both the ( + ) and ( − ) branches. Accordingly, we have I = IWKBJ + I+ + I−. At first let us evaluate the contribution from the inner region, IWKBJ. In order to do this we substitute ([eqn30]) into ([eqn62]), thus obtaining $$I^{WKBJ}=-\int dr d\theta F^{2}{d \over d r}\ln \rho. \label{eqn63}$$ Since the quantity F is rapidly oscillating we use the average value of F2, < F2 > = 1/4 in ([eqn63]). Hence $$I^{WKBJ}=-{\pi\over 4}\int\_0^{1-\epsilon}dr {d \over d r}\ln \rho ={\pi\over 4}\ln {\rho\_{c}\over \rho\_{\}}, \label{eqn64}$$ where ρ*c is the value of the central density of the planet and ρ\* ≪ ρc is a value of density close to the surface of the planet (r = r\* = 1 − ε) above which the contribution to I has to be determined by a separate treatment of the surface region. With help of equation ([eqn21]) equation ([eqn63]) can be rewritten in the form $$I^{WKBJ}={\pi\over 4}(\ln {\rho\_{c}\over D} -n\ln x\_{\}), \label{eqn65}$$ where x\ = 1 − r\* = ε is the dimensionless distance from the surface corresponding to the density ρ\: ρ\ = ρ(x\). Now let us evaluate the contribution to integral from the region close to the surface. For definiteness, let us consider the ( + ) branch where δ1 = x1 − μ* is assumed to be small. An analysis of contribution of the region close to the critical latitude to the integral ([eqn62]) shows that this contribution is small and can be neglected. Thus, we can use an expression of the form ([eqn32]), choosing i = 1 and j = 2 there, that correctly matches ([eqn30]) in the interior. We substitute this in ([eqn62]) and change the integration variables from cylindrical coordinates to pseudo-spheroidal coordinates, using the fact that $$\varpi d\varpi dz = {x\_{1}^{2}-x\_{2}^{2}\over (1-\mu^{2})\mu}dx\_{1}dx\_{2}\approx {\mu^{2}-x\_{2}^{2}\over (1-\mu^{2})\mu}d\delta\_{1}dx\_{2}, \label{eqn66}$$ (see eg. Wu 2005a). Also, we use the approximate density profile given by equation ([eqn21]) and express the radial variable x there in terms of δ1 and x2 with help of ([eqn31]). In this way we obtain $$I^{+}=n\int dx\_{2}d\delta\_{1} (1-x\_{2}^{2})^{-1/2}F\_{2}^{2}\delta\_{1}^{n-1}\bar W^{2}(\delta\_{1}), \label{eqn67}$$ where F2 = cos(λy2 + ϕ2). The integral ([eqn67]) should be evaluated in a volume bounded by the surfaces x = 0 and x = x\. First let us evaluate the integral $I\_{n}=\int d\delta\_{1} \delta\_{1}^{n-1}\bar W(\delta\_{1})^{2}$. Using the explicit expression ([eqn60]) for $\bar W(y)$, we obtain $$I\_{n}={\pi \over 2}\int^{y\_{\}}\_{0}dy J^{2}\_{(n-1)/2}(y). \label{eqn68}$$ Here we note that the integral converges only for the case n > 0. The integration variable y = κδ1 with y\* = κδ\. The condition y* = y\* defines the surface x = x\. From equation ([eqn31]) we have $$\delta\_{\}= {(1-\mu^{2})\mu x\_{\}\over \mu^{2}-x\_{2}^{2}}. \label{eqn69}$$ The integral ([eqn68]) logarithmically diverges when y\ → ∞. When y\* is sufficiently large it can be represented in the form $$I\_{n}={1\over 2}\ln (B\_{n}y\_{\}), \label{eqn70}$$ where the constant B*n can be calculated numerically for a general value of n[5](#fn5). Substituting ([eqn70]) to ([eqn67]), using the averaged value of F22, < F22 > = 1/2, we obtain $$I^{+}={n\over 2}\int^{\mu}\_{0}dx\_{2}(1-x\_{2}^{2})^{-1/2}\ln (B\_{n}y\_{\}), \label{eqn71}$$ where a multiplicative factor of two has been applied in order to account for the contributions from both the upper and lower hemispheres. Using equation ([eqn69]) we can bring ([eqn71]) to the form $$I^{+}={n\over 2}(\ln (B\_{n}\kappa (1-\mu^{2})\mu x\_{\}) \int^{\mu}\_{0}dx\_{2}(1-x\_{2}^{2})^{-1/2}-\Phi^{+}(\mu)), \label{eqn72}$$ where Φ+(μ) = ∫0μdx(1 − x2)− 1/2ln(μ2 − x2). The same approach can be used to evaluate the integral corresponding to the ( − ) branch with the result: $$I^{-}={n\over 2}(\ln (B\_{n}\kappa (1-\mu^{2})\mu x\_{\}) \int^{1}\_{\mu}dx\_{1}(1-x\_{1}^{2})^{-1/2}-\Phi^{-}(\mu)), \label{eqn74}$$ where Φ−(μ) = ∫μ1d*x(1 − x2)− 1/2ln(x2 − μ2). Thus, the integral corresponding the the region close to the surface, Is = I+ + I− can be evaluated as $$I^{s}={n\over 2}({\pi \over 2}\ln (B\_{n}\kappa (1-\mu^{2})\mu x\_{\}) -\Phi\_{tot}), \label{eqn76}$$ where Φtot* = ∫01dx(1 − x2)− 1/2ln∣x2 − μ2∣. It can be shown (eg. Prudnikov et al 1986) that the last integral does not depend on μ: Φtot = − πln2. Substituting this value to ([eqn76]), remembering that the total integral I = IWKBJ + Is and adding ([eqn76]) to ([eqn65]), we obtain the final expression for the integral $$I={\pi \over 4}\ln \left[{\rho\_{c}\over D}(4B\_{n}\lambda \mu\sqrt{ 1-\mu^{2}})^{n}\right]. \label{eqn78}$$ Note that the integral ([eqn78]) does not depend on position of the matching point x\. Substituting it and the norm ([eqn57]) to the expression for the frequency correction ([eqnn2]) we have $$\sigma\_{1}^{m}= {m\Omega \over \lambda^{2}}\ln \left[{\rho\_{c}\over D}(4B\_{n}\lambda \mu\sqrt{ 1-\mu^{2}})^{n}\right]. \label{eqn79}$$ In Appendix B we show that the expression ([eqn79]) has a correct form in the limiting case of an incompressible fluid n* = 0. The expression ([eqn79]) is not valid when ∣μ∣ is sufficiently close to 1. We recall that one can prove that the absolute value of any eigenfrequency must be less than or equal to 2Ω,( see eg. PP). This condition may be violated when Δμ = 1 − μ ≪ 1 and the correction ([eqn79]) is added to the unperturbed frequency ([eqn46]). A similar constraint may be obtained from consideration of the assumptions leading to ([eqn70]). Indeed, the matching radius should obviously be smaller than the radius of the star, and, accordingly, x\* < 1. For small values of Δμ this condition together with equation ([eqn69]) leads to δ\* < μ, where we assume that a ’typical’ value of x2 entering ([eqn69]) is of the order of ∼ μ ≈ 1. On the other hand, for the validity of the asymptotic expression ([eqn67]) we should have κδ\* > 1, and therefore $$\delta\_{\} < {\sqrt{\Delta \mu}\over \lambda}. \label{eqn81}$$ Combining inequalities ([eqn80]) and ([eqn81]) we obtain $$\Delta \mu > {C \over \lambda^{2}}, \label{eqn82}$$ where a coefficient C* > 1 can be obtained from a more accurate analysis. Thus, our simple approach is likely to be invalid for eigenfrequencies with absolute values sufficiently close to 2Ω. Therefore, we discard unperturbed values ([eqn46]) that lead to eigenfrequencies with absolute values larger than 2Ω when the correction ([eqn79]) is added. The same analysis can be used to calculate the quantity σ1an determining the correction to the anelastic approximation. As follows from equation ([eqn16d]) the correction is proportional to an integral very similar in form to that given by equation ([eqn62]). This integral also has contributions from the surface and the interior where the standard WKBJ solutions may be used. These contributions are comparable and so they should be matched at some radius. In fact, it may be shown that the surface contribution is equal to the surface contribution to ([eqn62]) given by equation ([eqn76]). The internal WKBJ contribution is more complicated and should, in general, be evaluated numerically. Therefore, for simplicity, we adopt a different approach when dealing with the correction to the anelastic approximation. We calculate σ1an numerically using the expression ([eqn16d]) for several ’global’ eigenmodes (i.e. modes with a large scale distribution of perturbed quantities). Such modes are of especial importance in applications of the formalism. For example, as discussed in PI, IP and in PIN, those mainly determine the dynamic tidal response of the planet. Since σ1an ∝ λ− 2, we expect that corrections to small scale WKBJ modes are less significant. They are, therefore, neglected. . A comparison of analytical and numerical results ================================================= [f1] In this section we compare the frequencies obtained from the approach described above with those obtained by a number of authors who have employed a variety numerical methods. To obtain the values from the above analysis we use equation ([eqn46]) for the eigenfrequencies, and, in the case of non zero m, add the expression for the correction to the frequency given by ([eqn79]). We compare the results for polytropes with polytropic indices n = 1, and 1.5. The quantities Bn entering ([eqn79]) for these cases were obtained numerically with help of equation ([eqn70]). We found B1 ≈ 14.4 and B3/2 ≈ 5.97. The range of allowed values of k is found from equation ([eqn47]). Additionally, we shall discard modes, which have ∣μ∣ close to 0 or 1 (see discussion below). We comment here that it follows from equation ([eqn79]) that the frequency correction σ1m becomes undefined for such values of μ. [f2] [f3] [f1n] [f2n] Most of the eigenvalues considered are for the model with n = 1, which has been extensively discussed, (see eg. LF and Dintrans & Ouyed, 2001). The amount of attention paid to this model is due to the fact that it approximately reproduces the density distribution of a cold coreless Jupiter mass sufficiently below the planet’s surface. Some eigenvalues for planetary models (mostly the so-called ’global modes’ with m = 2 and l = 1) were calculated by IP. The properties of the global modes in this case are very similar to those of a polytrope with n = 1. Since the different numerical approaches essentially agree with each other for the case n = 1 with non zero values of m, we compare our results with the numerical values obtained by LF. LF consider both even and odd modes and classify the mode order by an integer l0, which is related to integer p defined in equation ([eqn45n]) through l0 = p + ∣m∣ − 1. Let us remind that when the even modes are considered p = 2l, see equation ([eqn45]), and for the odd modes p is related to the integer lo classifying the odd modes as p = 2lo + 1, see equation ([AA3]) of Appendix A. Using this definition, it is easy to see that LF give eigenfrequencies of modes having even symmetry with l = lmin = 1 and l = 2 for the case ∣m∣ > 0 and l = lmin = 2 and l = 3 for the case m = 0. For the case of m = 1, we also considered the next order even modes with l = 4 and compared eigenfrequencies with what was obtained by Dintrans & Ouyed (2001). We also compare the WKBJ odd modes for m = 1, 2, 3 and 4 with the results of LS. The integer lo, in this case, takes the values lo = lmino = 0, 1 and 2. For the case n = 1.5 the comparison of the WKBJ eigenfrequencies is made with the numerical results obtained by spectral methods in PI. The results of the comparisons for n = 1 are shown in Figs. [f1]- [f2n], where positions of the eigenfrequencies within the allowed range − 2 < σ/Ω < 2 are shown. The WKBJ eigenfrequencies, σ are found from σ = σ\* + σ1m, where σ\* = 2μ, and μ, σ1m are given by equations ([eqn46]), ([eqn79]), respectively. In our analytical investigation we assumed that the quantity μ, and, accordingly, σ\, is positive, but allowed the sign of the azimuthal number m* to be either positive or negative. For the purpose of this section it is convenient to take a different but equivalent point of view and assume that the sign of m is fixed: m > 1 and allow the quantities σ\* to have either sign. The positive and negative signs correspond to prograde and retrograde mode propagation with respect to direction of rotation of the planet, respectively. Let us first discuss the result of comparison of the even modes, which is presented in Figs. [f1]-[f3]. In Fig. [f1] illustrates the case with m = 2. This is the most important value when one considers the problem of dynamical tides excited by a perturbing companion. The WKBJ eigenfrequencies are shown as open circles for l = 1 and squares for l = 2, respectively. The corresponding numerical values are indicated by crosses and stars. As seen in Fig. [f1], the numerical and analytical values show quite good agreement with each other. This agreement is good even for the modes with the smallest possible value of l = 1, which have a global distribution of perturbed quantities over the volume of the star and therefore might not be expected to be in any kind of agreement with the results of a WKBJ theory. This might be explained by the rather large value of the parameter λ for these modes, being equal to 5.5. From the discussion above, its inverse λ− 1 ≈ 0.18 is assumed to be a small parameter in our WKBJ expansions. There are, however, three unidentified modes for the case l = 1 as well as for l = 2. These modes have frequencies concentrating near the borders of the allowed frequency range with σ ≈ ± 2 as well as in the region close to σ = 0. It is possible that although these correspond to global modes for the incompressible (n = 0) model, these do not retain their character when n is increased to 1 and beyond through a strong coupling to short wavelength modes nearby in eigenfrequency (see discussion in section [sec2.5]). Discarding these modes, hereafter referred to as unidentified modes, the relative difference between the analytical and numerical results is of order of or smaller than 10 per cent. In Fig. [f2] we illustrate the case with m = 0 in the left panel and the case with m = 1 in the right panel. For m = 0 the eigenfrequency distribution is symmetric with respect to the reflection σ → − σ, and, therefore, only positive values of σ are shown. As for the previous case, when m = 0 the agreement between the WKBJ values classified as ’physical’ and the results of the numerical study is quite good, even when the ’global’ mode with σ ≈ 1.3 is considered, the relative difference being of the order of 3 per cent for this mode. Again, this may be accounted for by the relatively large value of λ = 5.5 for m = 0 and lmin = 2. Additionally, on this plot we show the results of a calculation by Dintrans & Ouyed (2001). They calculated five modes for m = 0. Three of them have locations nearly the same as those obtained by LF. They are, therefore, not shown in the plot. As seen from the plot, the other two modes may be identified with with the analytical modes having l = 4. Note that the agreement gets better with increasing WKBJ order, l, as expected. When ∣m∣ = 1 the agreement is similar to the case with ∣m∣ = 2 apart from the mode with l = 2 and k = 2, where the WKBJ value σ ≈ 0.33 is approximately twice as small as the numerical one. The reason for this disagreement is unclear to us. Note, however, that there seems to be a disagreement between numerical methods in this case. Only one of the modes with l = 2 and ∣m∣ = 1 obtained by LF can be reliably identified with an eigenfrequency given by Dintrans & Ouyed (2001). However, the ’global’ modes with ∣m∣ = 1 and l = 1 as well as all LF modes obtained for the case m = 0 have their counterparts in the results of Dintrans & Ouyed 2001. In Fig. [f3] we show the cases of relatively large values of ∣m∣ = 3 ( left panel) and ∣m∣ = 4 (right panel). The agreement gets worse with increasing ∣m∣, and in the case of the ’global’ mode with ∣m∣ = 4, l = 1 and the numerical value σ/Ω ≈ 0.6, the disagreement is of the order of 30 per cent. This may be explained by the fact that in our theoretical scheme the value of ∣m∣ is assumed to be much smaller than the value of λ. However, for the mode with the largest disagreement the ratio ∣m∣/λ ≈ 0.53 is not very small. In Figs. [f1n] and [f2n] the comparison of the odd modes is made. The results are similar to the previous case. Apart from the presence of unidentified WKBJ modes, all numerically obtained frequencies have well identified WKBJ counterparts. The agreement is getting better with increase of the WKBJ order lo and is getting worse with increase of the azimuthal number m. Note a rather good agreement between the ’global’ WKBJ and numerical modes corresponding to lo = 0. In fact, as was shown by PP (see also Papaloizou & Pringle 1977), eigenfrequencies and eigenfunctions of these modes can be calculated analytically giving very simple results $$\sigma=-{2m\Omega \over m+1}, \quad W=z\varpi^{m}. \label{ne1}$$ Note that neither eigenfrequencies nor eigenfunctions depend on the planet’s density distribution in this case. In summary, we point out that agreement between the numerical and WKBJ frequencies is unexpectedly good taking into account the fact that the WKBJ theory should not, strictly speaking, be applied to modes with such small values of λ. Apart from the existence of the unidentified modes in the WKBJ scheme and the two ’physical’ even modes and the global odd mode corresponding to m = 4 with the rather large disagreements alluded to above the agreement between analytical and numerical results is of the order of or smaller than 15 − 20 per cent for all the remaining identified 27 even and 35 odd modes. As we shall see below, good agreement is also found when the spatial distribution of the modes is compared. Properties of eigenfunctions ---------------------------- In order to calculate distributions of the quantity W over the volume of the planet we use equation ([eqn30]) in the bulk of the planet and equation ([eqn58]-[eqn62na]) close to the surface and smoothly interpolate between the two regions. Since the function η[x] defined in section [dom] is inconvenient for a numerical implementation we consider instead of it a function, which is zero in the regions [0, 1 − x\] and [x\, 1] and represented as a ratio of two polynomials of x in the intermediate region, which are chosen in such a way to ensure that several first derivatives are equal to zero in both points x = 1 − x\* and x = x\. Let us stress that for self-consistency we use the frequency σ\ (or μ) as given by equation ([eqn46]) in those equations even when the frequency correction is non zero. As above, the numerical results for the n = 1.5 polytrope are taken from PI. The WKBJ results for n = 1 are compared with those obtained for a realistic model of a planet of one Jupiter mass. This model has a first order phase transition between metallic and molecular hydrogen, which has been discussed, eg., in IP. [ff1] [ff2] [ff3] A comparison of the different results is shown in Figs. [ff1] and [ff2]. Note that in all cases shown in this section the same contour levels are used for the numerical and analytical data. In Fig. [ff1] we show the distribution of W over the planet’s volume for the m = 2 modes and n = 1.5. Numerical results are presented in the upper plots, which are taken from PI. These are for modes with σ = − 1.06 ( upper left plot), σ = 0.67 (upper middle plot) and σ = 0.67 (upper right plot). The modes with σ = − 1.06 and σ = 0.67 are the so-called two main global modes, according to PI. They mainly determine transfer of energy and angular momentum through dynamic tides induced by a parabolic encounter. The respective WKBJ counterparts have the smallest possible WKBJ order l = 1. The corresponding analytic eigenfrequencies are σ = − 0.99 (σ\* = − 1.32) and σ = 0.64 (σ\* = 0.39). The distribution shown on the upper right plot may be identified with a next order mode having σ = 0.435 and σ\* = 0.29. One can see that there is a surprisingly good agreement between the analytical and numerical results. In particular, the retrograde mode represented on the left hand side plots has a ’spot’ in distribution at the angle ≈ π/4 with respect to the rotational axis. This agrees with position of the critical latitude since arccosμ = arccos∣σ\/2∣ ≈ 0.27π. The distributions shown on the middle and right plots correspond to prograde modes. They have a well pronounced approximately vertical isolines. The main global mode may be distinguished from the mode corresponding to the next order by the number of nodes in the horizontal direction, this being one in the case of the global mode and two for the next order mode. We have checked that similar agreement exists between the WKBJ and numerical results corresponding to n* = 1. Since the distributions are quite similar they are not shown here. For m = 0 we compare the WKBJ results with calculations done by a spectral method for a model of a planet of Jupiter size and mass in Fig. [ff2]. As in the previous case the upper plots correspond to the numerical results. From left to right the numerical values of the eigenfrequencies are 1.35 (the main global mode), 1.01 and 0.79. Their analytical counterparts have l = 2, k = 1, σ = 1.33; l = 3, k = 2, σ = 1.03 and l = 4, k = 3, σ = 0.83 respectively. Note that a more pronounced disagreement in eigenfrequencies corresponding to the mode represented on the right hand side plot is mainly determined by the fact that this mode has a distribution concentrated near the surface of the planet, where the equation of state differs from that of a n = 1 polytrope. One can see that again there is very good agreement between the results. This is especially good for the main global mode represented in the plots on the left hand side. The agreement gets somewhat worse moving from right to left. This may be explained by a number of factors such as inaccuracies of the numerical and analytical methods as well as the physical effects determined by changes of the equation of state in the outer layers of the planet and the presence of the phase transition. These factors mainly influence modes with a small spatial structure while the large scale main global mode is hardly affected by them. Finally we consider the global odd modes and compare the analytic distributions given by equation ([ne1]) with the corresponding WKBJ distributions for m = 1, 2 and 3 in Fig. [ff3]. Although there is a disagreement in position of the spot close to the critical latitude, which is situated on the planet’s surface in the case of the exact analytic solutions and slightly interior to the surface of the planet in the case of the WKBJ distributions, there is a similarity in the distributions in the planet’s interior. This is quite surprising since in this case the analytic distributions do not depend on the planet’s structure at all while the WKBJ distributions are determined by the density distribution close to the planet’s surface. Discussion ========== Overlap integrals ----------------- As we pointed out in the Introduction, integrals of the form $$\hat Q\_{k}={Q\_{k}\over \sqrt N\_{k}} \quad {\rm with}\quad Q\_{k}= ({\rho \over c\_{s}^{2}}W\_{k}\|\Phi), \label{eqn84}$$ where Wk corresponds to a particular eigenmode and Φ is some smooth function, appear in astrophysical applications of the theory developed in this paper. In particular, as was discussed in PI and IP, integrals of this type enter in expressions for the transfer of energy and angular momentum transferred during the periastron passage of a massive perturber. These apply to the case when the spectrum of normal modes is discrete and they involve integrals of form ([eqn84]), where Φ = P2mr2 with P2m being the associated Legendre function. Assuming that Wk varies on a small spatial scale while the function Φ is smoothly varying such integrals may, in principal, be evaluated using our formalism with help of a theory of asymptotic evaluation of multidimensional integrals, see eg. Fedoryuk (1987), Wong (1989). However, some important integrals of form ([eqn84]) require an extension of our formalism, which can provide a smooth matching of the solution close to the surface to the WKBJ solution in the inner part of the planet that is valid at the next orders in inverse powers of λ. This is due to cancellations of leading terms in corresponding asymptotic series. Since this problem appears to be a rather generic one we would like to discuss it here in more detail for the important case when $\Phi =P^{2}\_{2}r^2=3\varpi^{2}$. The overlap integral of this type determines excitation of the m = 2 modes which are the most important for the tidal problem (eg. PI, IP and see also PIN). Explicitly, we have in this case $$Q\_{k}= 3\int dV \left(\varpi^2 {\rho \over c\_{s}^{2}}W\_{k}\right), \label{eqn84a}$$ where $dV=dz\varpi d\varpi $. Note that this integral must converge to zero in the incompressible limit n → 0 as in this case it is well known that inertial modes are not excited in the anelastic approximation. This fact, however, is not obvious for the integral written in the form ([eqn84a]) because close to the surface we have $${\rho \over c\_{s}^{2}}\approx nCx^{(n-1)}, \label{eqn85}$$ with the constant C converging to a nonzero value as n → 0. Therefore, as the eigenfunctions are regular, the integral Qk/n has a logarithmic divergence at the surface of the planet as n → 0. This raises the possibility that the overlap integral might converge to a nonzero value or behave pathologically as the incompressible limit is approached. In order to show that, in fact, this is not so and Qk(n → 0) → 0 in a smooth manner, let us consider some fiducial models having the property that the quantity $$\omega\_0^2 = -\frac{c^2\_s}{r \rho }\frac{d\rho }{dr}\label{Good1}$$ is constant. For models in hydrostatic equilibrium under their own gravity, constancy of ω02 implies that ratio M(r)/r3, where M(r) is the mass interior to the radius r, is constant. The model must accordingly be incompressible. Goodman & Lackner (2009) obtained a wider class of models in hydrostatic equilibrium under a fixed quadratic gravitational potential. Because the potential is fixed independently of the mass distribution and there are no constraints on the equation of state, such models may be constructed for an arbitrary density distribution. Now let use consider the integral $$Q\_{fud}= 3\int dV \left (\varpi^2 {\rho \over c\_{s}^{2}}\omega\_0^2W\_{k}\right). \label{eqn84b}$$ For the fiducial models described above this is identical to the overlap integral ([eqn84a]) where we note that we may adopt natural units such that the constant ω02, which should be identified with the surface value of GM(r)/r3 is equal to unity in that case. More generally the integrand in ([eqn84b]) can be transformed using equation of hydrostatic equilibrium ([Good1]) such that $$Q\_{fud}= -3\int dV \left ({\varpi^2\over r} {d\rho \over dr}W\_{k}\right)=-3\int dV \left (\varpi {\partial \rho \over \partial \varpi}W\_{k}\right). \label{eqn84c}$$ Now let us consider equation ([eq 41]) for free normal modes in the anelastic approximation by setting σ = σk and the right hand side of this equation to zero. Then we multiply it by $\varpi^2$, set m = 2, and integrate over dV. After removing derivatives of Wk by integrating by parts, assuming that the density vanishes at the surface boundary, it is easy to see that it follows from ([eq 41]) that Qfud = 0 in the anelastic approximation provided σk ≠ 2Ω. This means, in particular, that m = 2 inertial waves cannot be excited in the Goodman & Lackner (2009) models in this approximation as was found by these authors when compressibility was fully taken into account (see PIN for additional discussion). Using the fact that Qfud = 0 we may rewrite ([eqn84a]) for models under their own self-gravity quite generally, adopting natural units, as $$Q\_{k}=Q\_{k}-Q\_{fud}= 3\int dV \varpi^2 {\rho \over c\_{s}^{2}}\left(1-\frac{M(r)}{r^3}\right)W\_{k}. \label{eqn84d}$$ Taking into account that the factor in the brackets is proportional to x for small x, and, accordingly, in this limit the integrand is proportional to nxn we see that now the logarithmic divergence of Qk/n disappears and, therefore, it is clear from the representation ([eqn84d]) that the overlap integral indeed smoothly tends to zero in the limit n → 0. The theory of asymptotic evaluation of integrals of the form ([eqn84d]) tells that the values of such integrals are determined either by inner stationary points, where gradient of the WKBJ phase vanishes or contributions close to the surface or other parts of the integration domain, where the WKBJ approximation is not valid. From the expression of the Wk in the WKBJ regime ([eqn30]) it follows that there are no stationary points in the inner region of the planet. Considering the regions close to the surface it appears to be reasonable to assume that the leading contribution is determined by the region close to the critical latitude, where a ’hot spot’ is observed in distributions of Wk, see the previous section. In this region the quantities δ1 = x1 − μ and δ2 = μ − x2 are small. We can use them as new integration variables in ([eqn84d]) with help of ([eqn66]), separate the contribution of this region to the integral by introduction of the functions η[δ1, 2] defined in section 3.7 in the integrand, and decompose the quantities in front of Wk in powers of δ1, 2 taking into account that x ∝ δ1δ2 and $\varpi^{2}\approx (1-\mu^{2})$ in the leading order. Assuming that $W\_{k}\approx \bar W(\delta\_{1}) \bar W(\delta\_{2})$, where $\bar W(y)$ is given by equation ([eqn60]), it is easy to see that the leading contribution to ([eqn84]) is given by a symmetric combination of two integrals involving Bessel functions Qk ∝ I1(δ1)I2(δ2) + I2(δ1)I1(δ2), where I1(y) = ∫y = 0dyη[y]ynJ(n − 1)/2(κy), I2(y) = ∫y = 0dyη[y]y(n + 1)J(n − 1)/2(κy), where it is assumed that η[y > y\] = 0, y\ lies within the range of integration and 1/κ ≪ y\* ≪ 1. As was shown by Larichev (1973) the integral I1(y) = 0 for any particular form of the function η[y]. Thus, the leading order contribution to the overlap integral from the surface region close to the critical latitude is equal to zero. In principal, one can look for the next order terms. However, in this case our simple approach to the problem seems to be inadequate since eg. the assumption that Wk can be represented as a product of two functions separately depending on the coordinates may be broken at this level, etc.. A more accurate approach is left for a future work. We note, however, that this cancellation means that the overlap integral should decay rapidly with increasing λ, possibly being inversely proportional to a large power of λ. This may qualitatively explain why a small number of relatively large scale modes are significantly excited by dynamic tides, see PI, IP and PIN. Conclusions ----------- In this paper we have developed a WKBJ approximation, together with a formal first order perturbation approach for calculating the normal modes of a uniformly rotating coreless planet under the assumption of a spherically symmetric structure. Matching of the general WKBJ form valid in the interior to separable solutions valid near the surface resulted in expressions for eigenfunctions that were valid at any location within the planet together with an expression for the associated eigenfrequencies given in section [sec3.5]. Corrections as a result of density gradient terms neglected in the initial WKBJ approach were also obtained from formal first order perturbation theory. The corrected WKBJ eigenfrequencies obtained using the WKBJ eigenfunctions were compared with results obtained numerically by several different authors and found to be in good agreement, away from the limits of the inertial mode spectrum where identifications could be made, even for modes with a global structure. We also compared the spatial forms of the eigenfunctions with those obtained using the spectral method described in IP finding similar good agreement. This is consistent with the idea that these global modes can be identified and that first order perturbation theory works even though they are embedded in a dense spectrum. In further support of this, the formal first order perturbation theory developed here is subsequently used to estimate corrections to the eigenfrequencies as a consequence of the anelastic approximation and is then compared with simulation results for a polytropic model with n = 1 in PIN. These different approaches are found to be in agreement for small enough rotation frequencies and also indicates that, as implied by the simplified discussion in section [anelastic] of this paper, that corrections as a result of the anelastic approximation are never very significant for the models adopted. Our results show that the problem of finding eigenfrequencies and eigenvalues of inertial modes allows for an approximate analytical treatment, even in the case of modes having a large scale distribution of perturbed quantities. Although we consider only the case of a polytropic planet, our formalism can be applied to a much wider context. Indeed, the approach developed here is mainly determined by the form of the density close to planet’s surface, where we assume that it is proportional to a power of distance from the surface. Thus, we expect that our main results remain unchanged for any density distribution, which is approximately power-law close to the surface. In particular, according to our results, all models of type having approximately the same behaviour of the density distribution close to the surface should have approximately the same eigenspectrum. The formalism developed here can be extended for an approximate analytic evaluation of different quantities associated with inertial modes, such as overlap integrals characterising interaction of inertial waves with different physical fields, growth rates due to the CFS instability and decay rates due to various viscous interactions and non-linear mode-mode interaction. It may provide a basis for a perturbative analytic analysis of more complicated models, such as realistic models of star and planets flattened by rotation or models of relativistic stars. As we discussed above, for a given value of WKBJ order, l, some modes are identified with modes obtained numerically while others remain unidentified. Eigenfrequencies of the unidentified modes are always either situated close to the boundaries of the frequency range allowed for inertial modes, σ = ± 2Ω or situated close to the origin σ = 0. We believe that our theory is not applicable to these modes, and they develop a small scale contribution controlled by the closeness of the positions of their eigenfrequencies to σ = 0 and ± 2Ω, and thus effectively move to higher order than allowed for. Accordingly we de not consider these modes when comparing our results with results of direct numerical calculations of the excitation of inertial waves due to a tidal encounter reported in PIN. Acknowledgements ================ We are grateful to the referee, Jeremy Goodman, for his comments, which led to improvement of the paper. PBI was supported in part by RFBR grant 08-02-00159-a, by the governmental grant NSh-2469.2008.2 and by the Dynasty Foundation. This paper was prepared to the press when both P.B.I. and J. C. B. P. took part in the Isaac Newton programme ’Dynamics of Discs and Planets’. 99 Aldridge, K. D., Lumb, L. I., 1987, Nature, 325, 421 Andersson, N., 1998, ApJ, 502, 708 Arras, P., Flanagan, E. E., Morsink, S. 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− 2.0 ± 0.1 down to 0.38 pc. This is however not appropriate within the BH sphere of influence, if dynamically relaxed, where the equilibrium distribution (radius independent mass and energy flow) is the famous cusp of ρ ∝ r− 7/4 (where only one mass species is present, cf., ). Unrelaxed clusters around adiabatically growing BHs are expected to have shallower slopes or steeper slopes in the case of rotating systems and non-isothermal clusters. In the case of a minor merger (equivalent to q = 0.1), the structure of the more massive galaxy does not change significantly and equation ([eq:isothermal-density]) likely represents the stellar environment seen by the secondary BH after its parent galaxy is absorbed. This picture is somewhat naïve in the case of a major merger (q = 1), but violent relaxation due to the rapidly varying potential in a newly merged galaxy causes widening of the stellar energy distribution and is analogous to relaxation by collisions in a gas. This process tends to drive galaxies towards a universal steady state. A nice demonstration of this appears in the N-body merger simulations of, who find an r− 2 density profile at the time the equal mass binary becomes hard, which extends down to the scale of the binary separation. Throughout this work we assume that the BHs are ‘naked’, that is, do not carry clusters of bound stars. In the equal mass case, the stellar mass bound to a single BH once the binary becomes hard does not exceed 10 per cent of its mass (assuming that the cluster also has a power law density profile with index of -2 and normalization based on the M–σ relation). This will most likely not affect the late time evolution, but will surely affect the early inspiral phase, and the exact inspiral time, but will probably not have a major effect. Algorithm ========= Equal masses ------------ Below we provide a technical description of the application of the technique described in Section [sec:math] for an equal mass binary simulation. The slightly different procedure for the 10:1 mass ratio case is discussed in the following Section. First, a realization of a singular isothermal sphere is produced up to a cutoff radius of ${R\_{\rm max}}$, with two equal mass BHs placed on the x-axis at x = ± R0; at this initial distance the BHs are still unbound to each other and their inspiraling orbits are governed by the bulge’s gravity and dynamical friction. The simulation duration is divided into (equal) intervals Δt, after each the stellar force acting on the BHs is updated. The actual time steps of the ODE solver are smaller if necessary, so close encounters can be resolved. Within each interval we do the following: 1. Symmetrically advance[2](#fn2) the two BHs from ti to ti + 1, under their mutual gravity, and the effective forces exerted by the stars. The calculated antiparallel f and perpendicular f̃ embody the stars’ pull on the BHs, so the bulge potential (equation [eq:bulge-potential]) should not be considered additionally. This gives a short orbital segment which is stored in memory. 2. Advance each star from ti to ti + 1, under the influence of the two BHs and the bulge potential. The motion of the BHs in this interval is already set from the previous stage, so this is essentially a short scattering experiment. This is the most computationally demanding stage of each iteration, but easily parallelized. 3. Sum up the energies and angular momenta of all stars and subtract the values from the previous iteration to obtain ΔE⋆ and ΔL⋆. 4. Use the BHs’ path to obtain S, P and Q. 5. Calculate f and f̃ from equations ([eq:force-sym-par]) and ([eq:force-sym-per]), to be used in the next Δt interval. Unless otherwise indicated, our simulations end at t = 10 000 time units (or 107 years for a $10^8~{\rm M}\_\odot$ primary); this is equivalent to ∼ 1 000 revolutions after the binary becomes hard in most equal mass simulations. Integration of a single star is terminated prematurely in three cases: if it reaches a distance of ${R\_{\rm max}}+10$ length units from the centre of the system, reaches $r\_{\rm tidal}=10^{-3}$ from either BH or takes more than 35 integration time steps to complete the interval of Δt = 0.1 (see Section [sec:qa]). In the first case the star is considered to have escaped the system (or diverged); the extra 10 length units beyond ${R\_{\rm max}}$ are an arbitrary ‘padding’ required for technical reasons. The second case represents tidal disruption of the star (the orbit is then said to have crashed). The true tidal disruption radius is up to two orders of magnitude smaller than our $r\_{\rm tidal}$, but this choice has a negligible effect on the BHs because the rate of crashing stars is negligible compared to the rate of diverging stars. The choice of ${R\_{\rm max}}$ is rather arbitrary, and is chosen to be large enough to minimize its effect on the results, as discussed in Section [sec:Rmax]. Notably, the forces on the BHs are one interval retarded. Thus, the interval duration must be short enough in order not to break conservation of energy and angular momentum. In all simulations we chose to use equal intervals of Δt = 0.1, and since simulations done with this choice both ran reasonably quickly and performed well in terms of conservation, we did not thoroughly investigate changing Δt. Certainly making the interval length longer or adaptive can significantly speed up the simulations. The code outlined above is very simple in terms of decision making; the bulk of CPU effort is made to individually advance stars (scattering experiments) with only a small non-parallelizable overhead between intervals. A typical simulation with a large realization (N = 5 × 106) would run for ∼ few days on a medium strength personal desktop computer. Mass ratio ---------- As noted in Section [sec:forces], the force acting on the BHs can be uniquely found from the energy and angular momentum differences only in the cases of equal masses and high mass ratios (where it is possible to assume that the primary BH is fixed at the centre). However, when the velocity is perpendicular to the radius vector (i.e. at pericentre or apocentre), the values of Qi approach zero, and thus f̃i are left out of the coupled equations ([eq:forces-L]) and ([eq:forces-E]), and cannot be solved for. At the same time, fi are overdetermined as they have to compensate for both energy and angular momentum changes. Even in intervals that contain an apsis, there is usually a solution unless the orbital segment happens to be symmetric about the apsis. Since only a few segments are affected, the effect on the BBH orbits is not significant. In the case of equal masses, no special treatment of these segments was necessary; but in the case of very high mass ratios (i.e. q ≪ 1) the finite numerical fluctuations are larger due to the fact that the secondary BH is less massive. In these cases the total angular momentum would discontinuously drop at an apsis, and the otherwise smooth path would suddenly break at this segment. We compromised on simulating a 10:1 binary, and utilized two techniques to improve accuracy. First, each step was performed twice: after calculating the frictional force, the stars and BH were reset to their original positions, and advanced again with the newly calculated force on the BH. Second, we attempted to compensate for the accumulated error: instead of calculating the force components using purely ΔE⋆ and ΔL⋆ of the last Δt interval, we added their respective accumulated errors (the correction term had weight of 0.1 per cent). Those adjustments dramatically improved the accuracy of the 10:1 simulations, but unfortunately we did not yet overcome all the technical problems associated with reliably simulating a higher mass ratio inspiral from ∼ 50 pc down to stalling separation. Quality Assurance ----------------- If the routine that advances the stars has a bug or is just not accurate enough, the values of ΔE⋆ and ΔL⋆ obtained in each interval will be faulty; but since the frictional force is calculated in such a way that would compensate for any change of E and L in the stars, the bug might remain undetected. Thus, an important validity test of E and L conservation in the code is inherently not available. We can, however, test the ODE solver for a similar problem, and infer that our stellar orbits are at least well behaved in the original problem. Assume a BBH with a circular orbit of constant radius R; the BBH orbit is now decoupled from the stars and each stellar orbit can be integrated separately. This is the restricted 3-body problem plus a spherically symmetric potential; as in, the energy in the rest frame of the BHs (the frame which rotates with the same angular frequency) is the only conserved quantity. The Jacobi integral is a constant of motion related to the rest frame energy by CJ = − 2E. We performed a number of tests where the BBH was in a circular orbit with a radius in the range of 1 ≤ R ≤ 40 length units. The stellar model was the same as described in Section [sec:bulge] and the orbits were evolved for t = 10 000 time units. It was found that CJ is well conserved for the great majority of stars: only 0.2 to 1.6 per cent of orbits were cast off as rogue orbits, exceeding 35 steps per Δt = 0.1 interval; the rest had an average ∣ΔCJ∣ of (4 to 11) × 10− 5 energy units (CJ is typically of order unity). When applying a short softening length of 0.04 (corresponding to ), 0.2 to 1.7 per cent of orbits were eliminated by the same criterion and the rest had an average ∣ΔCJ∣ of (5 to 17) × 10− 5 energy units. Thus, we did not apply softening in the actual simulations. This type of number of steps filter against rogue orbits was found to work better than putting a lower limit on the allowed step size. Accuracy may be improved simply by integrating close encounters with smaller error tolerance, or by employing more elaborate techniques such as regularization. The small number of stars which are lost due to the ODE solver’s inability to handle them does not have a significant systematic effect on the BBH orbital evolution. Conservation of E and L --------------------------- As noted previously, total energy and angular momentum conservation is not a built-in requirement of the method, but rather indicates a successful transfer of energy and angular from the BBH to the stellar population. In the upper panels of Figure [fig:conservation] we show energy of the BBH (solid black line), the stellar population (solid red line) and their sum (dashed blue line); the lower panels is the same but for the angular momenta. The left panel is for an equal mass simulation while the right panel is for a 10:1 mass ratio simulation. The quality of energy conservation is attested by the absolute difference of total energy between the beginning and the end of the simulation; it is also useful to look at the fluctuations in total energy, a very crude estimate of which is the amplitude of the largest fluctuation (this is not a very good measure since fluctuation amplitude can be high in some parts of the orbit mild in others). These two quantities have dimensions of energy, and it is most sensible to normalize them with respect to the absolute difference in the energy of stars (or the BHs). Thus, for a simulation ending at t = T: $$\begin{aligned} \epsilon\_1^E(T) &= [E\_{\rm tot}(T) - E\_{\rm tot}(0)] / \|E\_\star(T) - E\_\star(0)\|,\\ \epsilon\_2^E(T) &= {\rm max}(\|E\_{\rm tot}(t) - \langle E\_{\rm tot} \rangle\|) / \|E\_\star(T) - E\_\star(0)\|.\end{aligned}$$ We similarly define ε1L and ε2L for the angular momentum. All these ε parameters must be very small. If the quantity that is supposed to be conserved has some trend, then usually ε1 = ε2, otherwise it only fluctuates around its mean value and ε1 < ε2. For an equal mass simulation with N = 5 × 106 (discussed in detail in Section [sec:N]) case at T = 43 200, we get ε1E, ε1L < 10− 6 and ε2E = ε2L = 3 × 10− 4. For a 10:1 mass ratio simulation (discussed in detail in Section [sec:e-growth]) at T = 21 800, we get ε1L = 4 × 10− 6 and ε2L = 6 × 10− 5, with a decreasing trend in total energy: ε1E = ε2E = 0.008. The trend in total energy begins at t ∼ 14 000 and is probably due to the fact that there is an accumulated inaccuracy in the solution of the BHs’ equations when they are so close together at pericentre. At t = 14 000 we get ε1E = 8 × 10− 5 and ε2E = 4 × 10− 4 with values for the angular momentum similar to the end of the simulation. Thus, the numerical effect previously discussed is unrelated the eccentricity growth observed in this simulation. [fig:conservation] Results ======= We performed a total of 58 simulations of an equal mass binary and a 10:1 mass ratio binary, varying N, R0, ${R\_{\rm max}}$ and the initial eccentricity e0. In Figure [fig:examples] we shows r(t), for selected simulations. In the 10:1 simulations, the primary BH is fixed at the centre, so r is the BBH separation; in the equal mass simulations r is half the separation. As can be seen in the figure, the semi-major axis stalls in all simulations but eccentricity (indicated by the thickness of the lines) does not reach a steady state in the unequal mass cases. These results are discussed in more detail below. [fig:examples] Number of Particles ------------------- Here we show two things: how the results converge with increasing number of stars N, and how the results depend on the specific realization of the stellar distribution. We present seven pairs of simulations with N between 50 000 and 350 000; for each N the two simulations have a different random seed, so that the stars have different initial positions and velocities, but are drawn from the same distribution. All simulations are of an equal mass binary starting at R0 = 60 with the local circular velocity; the cutoff radius of the stellar sphere is $R\_{\rm max}=200$. In Figure [fig:N] we show the ‘final state’ (i.e. at t = 10 000) semi-major axes a (equivalent to separation, in the equal mass case), and eccentricities e as functions of N; these two numbers are the best way to appreciate differences between similar simulations. From this small sample, it is apparent that the effect of changing N in this range is comparable in magnitude to that of changing the realization. In this set of simulations, the range of semi-major axis values is 0.07 length units or ∼ 5 per cent of the sample’s average a; the eccentricities are small and in the range 0.03 < e < 0.1. Thus, increasing N beyond 100 000 (within $R\_{\rm max}=200$) is unnecessary for this level of accuracy and following tests are made using this number. The average of a in these simulations is 1.43 (in physical units, for $10^8 ~{\rm M}\_\odot$ BHs, this is equivalent to 1.1 pc); this is approximately 30 per cent below the hard binary separation of 2. suggested the following formula for the stalling separation: $$\frac{a\_{\rm stall}}{r\_h^\prime} = 0.2 \frac{q}{(1+q)^2}\label{eq:stall},$$ where rh′ is the radius containing a mass in stars equal to twice the combined mass, or M(rh′) = 2M•(1 + q), at the time of stalling. We measured the accumulated mass in the above simulations and got that in all of them rh′ = 37.5 with a very small spread. By substituting this information and q = 1 into equation ([eq:stall]), one gets $a\_{\rm stall}=1.88$. It is important to note that in the simulation there was no actual stalling of the BBH, and $a\_{\rm stall}$ was estimated as the value of a in which a clear change in the hardening rate took place; rh′ was determined at the time when this change occurred. In our simulations both rh′ and $a\_{\rm stall}$ were determined at t = 10 000, which is well after the hardening rate has dropped. Also, equation ([eq:stall]) was calibrated by using simulations with mass ratios 0.025 ≤ q ≤ 0.5. To test the convergence of the results, we also performed a single simulation with significantly more particles (N = 5 × 106) and longer duration (t = 43 200); we show the inverse semi-major axis as a function of time in Figure [fig:long]. As seen in the figure, there is still some slow evolution of the semi-major after t = 10 000. At t = 40 000 the value of a is 1.35, which is 5.6 per cent lower than the value at t = 10 000 and only one per cent lower than the value at t = 30 000; the decay rate at the end of this simulation is $\dot{a}=10^{-6}$ velocity units, equivalent to $\sim 10^{-9}~{\rm pc\ yr^{-1}}$ for a $10^8~{\rm M}\_\odot$ primary. In this specific run, eccentricity was especially low at e < 0.01. The orbits produced in the other simulations in this set (with N ≤ 3.5 × 105) are very similar to the orbit shown in Fig. [fig:long], regardless of the number of particle (cf., spherical and triaxial cases). Note that a large N is required only to minimize the statistical fluctuations, our scheme is not subject to an artificial stellar relaxation mechanism, which requires a very large N to overcome. [fig:N] [fig:long] *R${}\_{\bf max}$ and R${}\_{\bf 0}$ ----------------------------------------------- Since we only simulate the spherical component and not a full merger, the initial and boundary conditions need be assumed: the initial distance of the BHs from the centre, R0, and the cutoff radius of the stellar sphere, ${R\_{\rm max}}$. Here we present six trios of simulations with ${R\_{\rm max}}$ between 120 and 520, each three simulations are of an equal mass binary starting at R0 = 30, 60 and 90 with the local circular velocity. The number of particles is $N = 500 \times {R\_{\rm max}}$, so as to keep the particle density profiles of equal normalizations (but different cutoff radii) in all simulations; this is due to the fact that the number of particles (or mass) grows linearly with radius in an isothermal sphere. In Figure [fig:R] we show the final state semi-major axes a* (stalling separation) as a function of ${R\_{\rm max}}$. From the results discussed in Section [sec:N], a characteristic error of ∼ 0.04 units on the semi-major axis can be attributed to a specific realization and number-of-particles statistical fluctuation. The simulations with R0 = 30 give systematically lower values for a; the values for R0 = 60 do appear to be systematically lower than the R0 = 90 simulations, but the two sets are within the errors of each other. There appears to be a trend of decreasing values of a with increasing ${R\_{\rm max}}$ up to ∼ 300, which is to be expected both because there is a larger supply of particles that can interact with the BHs, and because the potential well is deeper and stars that have already interacted with the BHs have a larger probability to fall back to the centre and interact again. Nevertheless, we see from this small sample that these effects are weak and comparable in magnitude to those discussed in the Section [sec:N]. In this set of simulations, the eccentricities are also small and in the range 0.02 < e < 0.11; no correlation of eccentricity was observed with either R0 or ${R\_{\rm max}}$. [fig:R] Mass Ratios ----------- As noted in section [sec:alg-ratio], the simulation of an unequal mass binary is somewhat different in nature. Thus, when studying the evolution of a 10:1 binary, we followed the analysis of Section [sec:N] and performed a number of different tests with increasing number of particles and different realizations. Here we present four pairs of simulations with N between 100 000 and 400 000; for each N the two simulations have a different random seed, so that the stars have different initial positions and velocities, but are drawn from the same distribution. All simulations are of a binary with a 10:1 mass ratio starting at R0 = 60 with the local circular velocity; the cutoff radius of the stellar sphere is ${R\_{\rm max}}= 200$. In Figure [fig:M] we show the final state semi-major axes a and eccentricities e as functions of N. As in Section [sec:N], changing N in this range produces no apparent effect on the stalling radius. Within this set of simulations, the semi-major axes are consistent; the sample’s average a is 0.241 ± 0.005. We measured the accumulated mass in the above simulations and got that in this case rh′ = 16.8 with a very small spread. By substituting this information and q = 0.1 into equation ([eq:stall]), one gets $a\_{\rm stall}=0.28$. The caveats of using this equation were explained in Section [sec:N]. If we recalibrate equation ([eq:stall]) using our two values of q, the prefactor is lowered from 0.2 to approximately 0.16; it is even somewhat smaller considering the fact that the true value of $a\_{\rm stall}$ can be ∼ 6 per cent lower than its measured value at t = 10 000 (see Figure [fig:long]). In contrast with the equal mass case, the eccentricities do not reach a steady state value. In one of the N = 105 simulations, the eccentricity increases very rapidly after the binary becomes hard. At t = 9 400 the pericentre distance reached 10− 3 and the simulation is terminated. In the rest of the simulations, The eccentricity values range between 0.05 and 0.28 at t = 10 000, however, in all but one of these simulations eccentricity is slowly increasing. We calculate Te → 1, a very rough estimation for the time of gravitational wave regime, by fitting e(t) with a linear function in the range 6 000 < t < 10 000 and continue it to e = 1. At this time span the rise in eccentricity is approximately linear, however this trend breaks at approximately e = 0.85, so in fact Te → 1 can underestimate the time to the gravitational wave regime by some ≈ 20 per cent. In Table [tab:e] we show the results for Te → 1 for this simulation set and also for the initially eccentric runs. The values range between approximately 4 × 104 and 3 × 105 time units, or on the order of 108 years for a $10^7~{\rm M\_\odot}$ secondary around a $10^8~{\rm M\_\odot}$ primary. [fig:M] [tab:e] \| N [105] \| e0 \| $e\_{\rm f}$ \| Te → 1 \| \| --- \| --- \| --- \| --- \| \| 1 \| 0 \| 0.05 \| 2.7 × 105 \| \| 1 \| 0 \| ∼ 1 \| $\lesssim 10^4$ \| \| 2 \| 0 \| 0.21 \| 4.8 × 104 \| \| 2 \| 0 \| 0.13 \| 2.2 × 105 \| \| 3 \| 0 \| 0.15 \| 1.4 × 105 \| \| 3 \| 0 \| 0.18 \| 5.9 × 104 \| \| 4 \| 0 \| 0.05 \| 4.4 × 107 \| \| 4 \| 0 \| 0.28 \| 3.5 × 104 \| \| 2 \| 0.1 \| 0.42 \| 2.0 × 104 \| \| 2 \| 0.1 \| 0.33 \| 2.5 × 104 \| \| 2 \| 0.2 \| 0.47 \| 2.1 × 104 \| \| 2 \| 0.2 \| 0.59 \| 1.7 × 104 \| \| 50 \| 0.2 \| 0.52 \| 1.8 × 104 \| \| 2 \| 0.3 \| 0.21 \| 9.2 × 104 \| \| 2 \| 0.3 \| ∼ 1 \| $\lesssim 10^4$ \| \| 2 \| 0.4 \| 0.96 \| ≈ 104 \| \| 2 \| 0.4 \| ∼ 1 \| $\lesssim 10^4$ \| Eccentricity growth ------------------- Here we tested how the initial eccentricity affects the results. We performed simulations with four different initial eccentricities between 0.1 and 0.4 with semi-major axes as in the circular simulations of Sections [sec:N] and [sec:mass-ratios]. The initial eccentricity e0 is that of a BBH with the same initial conditions and that moves in the initial potential (equation [eq:bulge-potential]) with no friction; the orbit is not really an ellipse since the potential in not Keplerian, so e0 corresponds to the mean orbital eccentricity, defined as the difference between the maximum and minimum separations divided by the major axis. For each e0, there are two simulations (with two different realizations) for an equal mass binary and two for a 10:1 binary. The number of particles is N = 2 × 105 with cutoff radius ${R\_{\rm max}}=200$; we performed a single simulation with significantly more particles (N = 5 × 106) for the e0 = 0.2 case. In Figure [fig:E0] we show the final state semi-major axes a and eccentricities e as functions of e0. This figure also includes four data points with e0 = 0 that have already been presented in Figs. [fig:N] & [fig:M]. The stalling separation is independent of initial eccentricity in the tested range, the sample’s average a is 1.42 ± 0.04 for the equal mass simulations and 0.246 ± 0.010 for the unequal mass simulations. As in the other tests we performed for equal mass binaries, the final orbits are very much circular; the eccentricities are in the range 0.01 < e < 0.1 despite the initially significant eccentricity. However, the final eccentricities in the 10:1 cases do appear to be generally correlated with e0. As in Section [sec:mass-ratios], here too, eccentricities are still increasing when the simulations terminate at t = 10 000; the eccentricity timescales Te → 1 for the 10:1 simulations are shown in Table [tab:e], which shows that the eccentricity growth rate is related to e0. [fig:E0] found that eccentricity growth is generally mild for equal mass binaries with very small initial eccentricities, but also that initial eccentricity e0 > 0.3 leads to very high peak eccentricity almost regardless of the other system parameters. Although all our equal mass simulations end up in nearly circular orbit, this is not inconsistent with : while in the latter work the BHs are launched from within their radius of influence, in our simulations the BHs are launched from much further out, in the dynamical friction regime. In our simulations, an equal mass binary becomes less eccentric as it inspirals from r > 60. For one of the e0 = 0.4 simulations, the rapid eccentricity decrease ceased at about r = 1.4 (approximately twice the stalling radius) where the value was e = 0.035. At r = 4 (equivalent to the initial radius of ’s equal mass binaries) the eccentricity was 0.12; no significant eccentricity growth occurred in ’s equal mass simulation with initial eccentricity of 0.1. While it has been shown by that BBHs lose memory of their initial eccentricity if they corotate with a massive gaseous disc, studies of eccentric orbits of hard binaries is motivated for the purely stellar dynamical case by the theory of linear response for dynamical friction. However, this theory is derived from a first order perturbative expansion and is not applicable when close encounters dominate the evolution, and the system is not well described by an analytical approximation. It is important to note another major difference between our work and which greatly affects the evolution of the binary orbital parameters, mostly the semi-major axis: while in our work the loss cone empties, implicitly assumes that the loss cone is always full at $r > r\_{\rm inf}$. This leads to a very rapid decay of the binary separation and quick coalescence due to gravitational wave emission. Similarly, calculated the eccentricity evolution in the case of a very small secondary BH that does not affect the stellar density profile. There, eccentricity grows because the orbit passes in and out of a flat core, where the star are fast and the drag force is much less efficient at pericentre than at apocentre. In our simulations, however, the much more massive secondary forms a cavity slightly larger than its apocentre, and eccentricity grows where the density is essentially zero. In Figure [fig:e-evolution] we show the evolution of the semi-major axis for all the runs with e0 = 0.2 and 10:1 mass ratio. The dotted green lines represent the two realizations with 2 × 105 particles while the solid blue line represents the larger N = 5 × 106 realization. The first two simulations are arbitrarily terminated at t = 10 000 while the latter is stopped only when the eccentricity reaches 0.99. If we scale to physical units for a primary BH of $10^8~{\rm M\_\odot}$, the end of the simulation is 22 Myr from its beginning. For this mass scaling, using the formula for orbital decay due to gravitational waves, the timescale for coalescence at the end of the simulation is just 1 Myr. The rapid growth of the eccentricity while the semi-major axis remains fairly constant indicates a high value for the the dimensionless eccentricity growth, defined as: $$K = \frac{{\rm d}e}{{\rm d}\ln (1/a)}.$$ For single scattering of unbound stars from a fixed background, derived a maximal value of K of about 0.3 for mass ratio of 16:1, consistent with previous scattering experiments (; ). The value of K for the simulation presented in Fig. [fig:e-evolution] is at least an order of magnitude larger (we only roughly estimated the value from the results). The difference is probably due to the very different nature of the orbits in the restricted 3-body problem versus the more realistic model used here. We refer an in-depth study of the physical mechanism behind the eccentricity growth to a subsequent work. It is interesting to compare our results to those of, who also got ‘runaway’ growth of eccentricity while the semi-major axis stalled, but one must note the critical differences between the two studies. Most notably, used a mass ratio of 100:1 while we took only 10:1. Additionally, they used an initially very shallow central density profile, ρ ∝ r− 3/4 while our bulge model (see Section [sec:bulge]) was an isothermal sphere, ρ ∝ r− 2. More importantly, the mass of their entire stellar population was just $1.25\times 10^9~{\rm M\_\odot}$, which is 8 times less than the primary BH’s mass and only 12.5 times more than the secondary’s mass. By comparison, the total stellar mass in our model is 25 times the mass of the primary and 250 times the mass of the secondary. Thus, their entire simulation is deep within the primary’s sphere of influence, where its gravity dominates, while our simulations started with the secondary well outside the primary’s radius of influence. Let us scale our work to theirs by setting M8 = 100 in the scaling equations of Section [sec:units]. start their A32k simulations (with N = 32 768) at $R\_0 = 20~{\rm pc}$ within which there are less than 2 500 particles, their BH stalls at $a = 3.9~{\rm pc}$. We initiate the secondary BH at $r=637~{\rm pc}$ and get stalling at 2.1 pc; there are initially 40 000 particles enclosed within $r = 20~{\rm pc}$ in our large N = 5 × 106 simulation (marked with a star in Fig. [fig:E0]). The secondary BH is 20 000 times more massive than a field star in our simulation, versus 2 600 in the simulation. Their stellar bulge model is very small compared to ours, with 90 per cent of their mass is within $R\_0 = 190~{\rm pc}$, while our model is truncated at ${R\_{\rm max}}= 1\,800~{\rm pc}$. The semi-major axes in the two studies evolve at very different rates: while the binary takes 19 million years (Myr) to sink from $a\_0 = 20~{\rm pc}$ to 10 pc in their shallow cusp, our binary does the same journey in only 0.18 Myr. However, when applying equation (6) of (derive from their numerical results) for our physical parameters, the timescale for significant eccentricity growth is extremely short at 0.78 Myr; the lifetime of the system from our simulation is approximately 85 Myr (scaled with M8 = 100), which is two orders of magnitude longer. The difference might be due to the mass scaling assumed in their formula. A final note about precession: in the unequal mass case, the BHs exhibit very small precession during the hard phase. In the large simulation of Fig. [fig:e-evolution], between t1 = 10 000 and t2 = 11 000 the semi-major axis precesses by 0.311∘. During this period of 1 Myr (scaled with M8 = 1), the average semi-major axis is 0.18 pc, and it drops by 0.002 pc; the average eccentricity is 0.559 and the growth is by 0.072. This precession can be produced, for example, by perturbing the Keplerian potential with a uniform density field of 20,000 solar masses per parsec cubed. Even with our large number of particles, this density corresponds to only four particles enclosed in the sphere with radius equals to the apocentre. In the snapshot taken at t1, there was one particle inside this region, and it was probably a transient since there are no stable orbits there except those tightly bound to one of the BHs. In principle, a small flux of particles to this region can produce the measured precession, but torques due to the anisotropy of the potential at larger distances are more likely to cause the precession. For comparison, general relativistic precession of the orbit (not reproduced in the simulation) would be $56^\circ~\rm{Myr}^{-1}$. We cannot yet say whether this precession compromises resonances that possibly induce the eccentricity growth, but will refer to this point in a future paper. [fig:e-evolution] Dynamical friction ================== The orbital decay of a massive object within a galaxy down to its centre is well approximated by Chandrasekhar’s dynamical friction formula. However, the assumption of a uniform background in his classical treatment does not hold in real galaxies. Thus, corrections to the Coulomb logarithm are necessary to account for the changing background with radius. performed a detailed theoretical investigation of Chandrasekhar’s formula in the presence of a density gradient and gave an improved analytical formula for the Coulomb logarithm. took into account also self consistent velocity distribution functions, and made a comprehensive examination of the applicability of the new formula to sinking massive objects in a number of galaxy models, using high resolution N-body and particle-mesh codes. Their results suggest a delay in the orbital decay with respect to the standard formula, which quantitatively varies according to the studied case. They notably give an explicit solution for the decay of a massive object moving on a ‘circular’ orbit. Their formula (equation 25 in their paper) is very general and holds for an arbitrary power law density profile (it does not hold for very flat cores where fast moving stars contribute to most of the frictional drag; see ). Here we bring their formula in our model’s units and adjusted the parameters for an isothermal sphere; the radial evolution is given implicitly by: $$\textstyle t = 52.918 \times \left\{{\rm Ei}\left[2\ln\left(\frac{1}{8}R\_0\right)\right]-{\rm Ei}\left[2\ln\left(\frac{1}{8}r\right)\right]\right\}\label{eq:Just}$$ The prefactor is an exactly calculable number. The special function ${\rm Ei}(x)$ is called the exponential integral, it has real values only for x > 0. Thus, the domain of definition of equation ([eq:Just]) is r > 8. However, the assumption of a nearly circular orbit breaks well above r = 8. The angle between the velocity vector and the tangent can be derived by finding the radial velocity from equation ([eq:Just]): $$\theta = 195.99^\circ \times \frac{1}{r}\ln\left({\textstyle\frac{1}{8}}r\right),\label{eq:spiral-angle}$$ where this approximation hold only for small angles or large r. It is generally difficult to simulate a full infall of a compact object into a galaxy centre due to the collisional nature of the interactions and the low number density of particles which can generally be obtained at the outskirts of galaxy models. used the particle-mesh code Superbox, which is collisionless and uses fixed time steps; This type of code, unlike direct N-body codes, allows a large particle number to be simulated in a relatively short time. Using the code we designed for BBHs, we can also study the early part of the inspiral, which is dominated by dynamical friction. Since our code resolves collisions between the BHs and stars, this study is complimentary to that of Just et al.. In Figure [fig:df] we show a simulation of an equal mass binary initiated from R0 = 500 (solid blue line) and the theoretical curve (dashed black line), equation ([eq:Just]). The initial velocity is the local circular velocity, but the initial velocity vector is tilted by 1.62∘; this angle is obtained be substituting r = R0 in equation ([eq:spiral-angle]). If the initial velocity is purely tangential, then the spiral becomes slightly ‘eccentric’. A second run with a different realization (not shown) gave very similar results, including the position of the wiggles. It should be pointed out that this simulation is just a proof of concept; we do not expect the stellar distribution to be spherically symmetric at the early stages of a major merger. The orbital decay in our simulation is initially very well described by equation ([eq:Just]); deviations become significant below r ∼ 300, where mutual gravity of the two BHs is still negligible compared to the gravity of the bulge. These deviations are possibly due to the fact that the phase space distribution in the actual simulation (at the time and radius where the deviations occur) is no longer consistent with the assumption of isothermal sphere used to derive equation ([eq:Just]), in particular the velocity distribution might not be described well by a Gaussian. [fig:df] Summary ======= Using a conservation-based scheme, we were able to follow the evolution of a BBH from a wide separation (enclosed stellar mass greater than the combined BH mass) down to sub-parsec scale. Our code resolves star-BH collisions and can run with N > 106 stars on a desktop computer. We verified that our scheme yields convergent results which are independent of the number of particles, and the initial and boundary conditions. By performing scattering experiments on the inspiraling BBH, we will be able to extend and calculate the signature of the inspiral on the background stellar phase space distribution as a function of projected position. This calculation improves on N-body simulation by reducing statistical fluctuations and having no spurious relaxation (and thus no loss cone refilling). We performed calculations for both an equal mass binary and a 10:1 mass ratio. Our calculations reveal: 1. The inspiral from a radius scale of tens of parsecs to the hard binary radius occurs on a time scale of a few million years for a $10^8 ~{\rm M}\_\odot$ primary, with only a weak dependence of the timescale on the mass ( ∝ M80.29, equation [eq:units-T]). 2. 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E-mail: <!-- h='physics.technion.ac.il';a='@';n='ym';e=n+a+h; document.write('<a h'+'ref'+'="ma'+'ilto'+':'+e+'">'+e+'<\/'+'a'+'>'); // --> ym at physics dot technion dot ac dot il (YM); <!-- h='physics.technion.ac.il';a='@';n='laor';e=n+a+h; document.write('<a h'+'ref'+'="ma'+'ilto'+':'+e+'">'+e+'<\/'+'a'+'>'); // --> laor at physics dot technion dot ac dot il (AL)[↩](#fnref1) 2. The word ‘advance’ in this context means solve the equation of motion by means of Runge–Kutta method of order five with adaptive step size control (e.g. ).[↩](#fnref2) A conservation-based method for simulating the inspiral of binary black holes ============================================================================= ### Accepted 2012 January 17. Received 2012 January 16; in original form 2011 October 28 [firstpage] We present a new approach to studying the evolution of massive black hole binaries in a stellar environment. By imposing conservation of total energy and angular momentum in scattering experiments, we find the dissipation forces that are exerted on the black holes by the stars, and thus obtain the decaying path of the binary from the classical dynamical friction regime down to subparsec scales. Our scheme lies between scattering experiments and N-body simulations. While still resolving collisions between stars and black holes, it is fast enough and allows to use a large enough number of particles to reach a smooth and convergent result. We studied both an equal mass and a 10:1 mass ratio binaries under various initial conditions. We show that while an equal mass binary stalls at a nearly circular orbit, a runaway growth of eccentricity occurs in the unequal mass case. This effect reduces the timescale for black hole coalescence through gravitational radiation to well below the Hubble time, even in spherical and gasless systems formed by dry mergers. black hole physics – stars: kinematics and dynamics – galaxies: nuclei. Introduction ============ Supermassive black hole (BH) binaries are formed as a result of galactic mergers. The two BHs sink towards the bottom of the potential well and form a bound pair at the centre of the stellar distribution. According to the classical picture the binary hardens until the loss cone is depleted (i.e. depleted of stars on low angular momentum orbits that pass close enough to the binary for a significant interaction), and stalls at a separation too large for the emission of gravitational radiation to cause rapid inspiral and coalescence (the ‘final parsec problem’). This picture has been tested extensively using N-body simulations and scattering experiments. Early numerical studies concentrated on the simple case of an equal mass binary on a circular orbit within a spherically symmetric stellar distribution. Those studies (e.g. ; ; ; ), with a relatively small number of particles, were somewhat inconsistent with each other and could not confirm loss cone depletion due to spurious numerical relaxation over long time scales. In recent years, more detailed investigations have been performed. Apart from the increased number of particles and longer integration times, various complications have been considered such as triaxiality and rotation (e.g. ; ), post Newtonian corrections (e.g. ) massive perturbers (e.g. ) and gas discs (e.g. ). These studies found that binary coalescence in less than the Hubble time is feasible even if only one of these factors is present. Thus, the final parsec is potentially not a problem in a realistic merger remnant. More recent work (e.g. ; ) has focused on the evolution of the binary eccentricity rather than just the binary separation. For high mass ratios ( ∼ 100:1), these authors showed that even in a spherically symmetric environment, eccentricity can increase to almost unity. Very high eccentricity means that in pericentre the two BHs can be close enough together that gravitational wave emission becomes efficient. This enhanced energy loss to gravitational radiation (as compared to the circular case with the same semi-major axis) can reduce coalescence timescale to well below the Hubble time. Direct N-body simulations are the most accurate way to study binary BH (BBH) evolution, but they are computationally expensive. It is therefore difficult to perform diverse enough tests to cover the problem’s parameter space. By compromising for an unrealistically small number of particles, one introduces spurious relaxation. In spherical galaxies, this process drives unrealistic loss cone repopulation (; ) and causes the hardening rate to be highly N-dependent (; ). followed the evolution of a BBH in triaxial and rotating galaxy models and found that the hardening rate was N-independent, implying a collisionless mode of loss cone repopulation. Our previous work dealt with a possible stellar kinematical signature of stars around a BBH. In we produced kinematical maps (projected maps of mean stellar velocity, velocity dispersion and higher velocity moments); this was achieved by scattering stars on a BBH on a fixed circular orbit. To produce more realistic kinematical maps, we looked for a way to calculate a BBH path fast and accurately so it could be used for following scattering experiments. Good kinematical maps require a very large number ($N\lesssim 10^8$) of stars on small scales, otherwise the high moments of the line of sight velocity distribution are poorly resolved. N-body simulations cannot be made yet with such a large number of particles on small enough scales, but this is not a problem for scattering experiments which are performed on a precalculated BBH path. Since we were already working with scattering experiments, we derived a way to use the existing code base and adapt it to work more like an actual N-body simulation, where the effect of scatterings on the BBH orbit is taken into account, giving the orbital evolution. In this paper we present a new scheme we developed to simulate the inspiral of the BBH from the galactic scale to the hard binary scale. Our method is based on imposing conservation of total energy and angular momentum, instead of directly summing the forces of individual stars on the BHs, and lies between scattering experiments and N-body simulations. Studying this type of systems with scattering experiments is significantly faster than a full N-body treatment and is appropriate when the bulge has relaxed. The phase space stellar distribution is followed accurately and a more realistic number of stars can be included. Thus, we are able to run many simulations and probe a large range of parameters under reasonable physical assumptions, as well as the convergence of the solution, and compare the inspiral timescale, the stalling radius, and the eccentricity evolution to earlier calculations. Since in our method we obtain the forces on the BHs, it is also possible to test analytical expressions for dynamical friction. The original treatment by is for a homogeneous background, therefore we compare our results to who investigate dynamical friction in power law cusps. Our solution extends beyond the range of validity of their formula and into the hard binary regime of the BHs. We studied the evolution of both an equal mass binary and a 10:1 mass ratio binary under different initial conditions. We found that both cases presented stalling of the semi-major axis, but in the unequal mass case, eccentricity tended to grow towards unity on timescales well below the Hubble time. In Section [sec:math] we give a general mathematical formulation of our simulation scheme. The model for the specific simulations we performed in this work is described in Section [sec:model] while a technical description of the scheme appears in Section [sec:algorithm]. In Section [sec:results] we present the results of all BBH simulations and discuss their physical implication. In Section [sec:df] we discuss dynamical friction, compare to earlier results, and show that our code is applicable in very large radii, thus potentially helpful in future studies of phenomena related to dynamical friction. Finally, we give a short summary in Section [sec:summary]. Mathematical Formulation ======================== Motivation ---------- The two basic ideas of our solution are the separation of timescales and the balance of energy and angular momentum between the BHs and the stars. There are three timescales in the system corresponding to changes in: 1. stellar orbits following close encounters 2. the BHs’ orbits 3. the background stellar potential In more detail: a star significantly changes its original orbit during a close encounter with a BH, but the BH’s path is only slightly perturbed as the force exerted on it is due to many small ‘scattering’ events; the change of the background stellar potential is due to the collective response of stars to the perturbation, which evolves on the dynamical timescale. A yet longer timescale would be of 2-body relaxation, which is longer by a factor of ∼ 0.1N/lnN and is expected to be well above the Hubble time for a real galaxy core. However, relaxation time is potentially significantly shorter if the dominant relaxation mode is not 2-body relaxation. The most basic kind of an N-body simulation uses very small steps, after each the vector forces exerted by all the field stars (particles) are summed up and applied to the BHs in the next step; each particle is propagated the same way. Energy and angular momentum are globally conserved (within a given numerical error tolerance) since it is a closed system. If N is not large enough, noisy potential and unrealistically massive stars lead to spurious relaxation. If the time step is too large, close encounters cannot be resolved. Our scheme uses a ‘large’ Δt, which we call an interval, within which there is one or more actual steps of the ODE solver. Let BH number i (where i ∈ {1, 2}) propagate one interval between times t0 and t1 = t0 + Δt from some vector position ${\bf s}\_i(t\_0)$ to ${\bf s}\_i(t\_1)$; we use the term segment for the path length. After each interval, the energies and angular momenta of all the particles are summed up. If no work was done and no torque exerted due to the background stellar potential, then the energy ΔE⋆ (angular momentum ΔL⋆) gained by the stars during some interval must be equal and opposite to the energy $\Delta E\_{{\rm BH}}$ (angular momentum $\Delta L\_{{\rm BH}}$) lost to the BHs during this interval. For simplicity, we use a static and spherically symmetric model for the stellar potential, so these requirements are automatically fulfilled. The scheme can accommodate more complicated models as well: the demand for spherical (or more generally, axial) symmetry, which is necessary if one assumes that change of a star’s angular momentum is only due to interaction with the BHs, can be relaxed if the torque component due to the stellar bulge is considered separately; a slowly varying potential can also be taken into account if the proper adjustments are made to the code. The evolution of a BBH is dominated by close and fast encounters with the field stars and the evolution of the stellar gravity field is only a secular effect. Thus, the part of ΔE⋆ due only to the change in stellar potential is negligible after each interval, justifying the use of a static model. However, at late times, the initial stellar potential is no longer consistent with the actual spatial distribution. The static potential assumption is still reasonable as far as the BHs are concerned, since by the time that any significant evolution of the background potential has taken place, the BHs will have fallen deep enough in the potential well, where the dominant force is the other BH’s gravity rather than the background stellar potential (so that the exact shape of the potential well does not matter). Stars further out orbit in a ‘wrong’ potential, but as long as approximate spherical symmetry is preserved, their interaction rate with the BHs should not be significantly influenced by this (and during a close encounter, the background stellar potential is of course unimportant). Assuming additionally that the interval is short enough so that the forces on the BHs due to the background stars do not vary significantly, we perform scattering experiments in each interval to find ΔE⋆ and ΔL⋆. Using the simple algebra described below, we find the average forces on the BHs over this interval. We also assume a purely planar motion of the BHs. Thus, unless stated otherwise, by ‘angular momentum’ and ‘torque’ we mean only the z component of these vectors. Conservation laws ----------------- The basic conservation equations are $$\begin{aligned} \Delta E\_{{\rm BH}} &= -\Delta E\_{\star}=\int\_{{\bf s}\_1(t\_{0})}^{{\bf s}\_1(t\_1)}{\bf F}\_1\cdot{\bf ds}\_1 + \int\_{{\bf s}\_2(t\_{0})}^{{\bf s}\_2(t\_1)}{\bf F}\_2\cdot{\bf ds}\_2,\label{eq:cons-E}\\ \Delta L\_{{\rm BH}} &= -\Delta L\_{\star}=\int\_{t\_{0}}^{t\_1}\tau\_1{\rm d}t + \int\_{t\_{0}}^{t\_1}\tau\_2{\rm d}t,\label{eq:cons-L}\end{aligned}$$ where ${\bf F}\_i$ and $\tau\_i\equiv\left({\bf r}\_i\times{\bf F}\_i\right)\cdot\hat{{\bf z}}$ are the force and torque, respectively, exerted on BH number i by the stellar population. Let us write ${\bf F}\_i$ in the following form: $${\bf F}\_i=-f\_i\hat{{\bf v}}\_i+\tilde{f}\_i\hat{{\bf u}}\_i,$$ where $\hat{{\bf v}}\_i$ and $\hat{{\bf u}}\_i$ are the unit vectors parallel and perpendicular to the velocity of the BH. Note that f is ‘drag like’ and directed opposite to the velocity vector. If an object moves through a uniform background, symmetry dictates that the mean force would be antiparallel to the velocity vector. However in realistic environments there must be also a force due to the inhomogeneities of the background, so a perpendicular force component is required. In Cartesian coordinates: $$\begin{aligned} \hat{{\bf v}}\_i &= \frac{v\_{ix}}{v\_i}\hat{{\bf x}}+\frac{v\_{iy}}{v\_i}\hat{{\bf y}},\\ \hat{{\bf u}}\_i &= \frac{v\_{iy}}{v\_i}\hat{{\bf x}}-\frac{v\_{ix}}{v\_i}\hat{{\bf y}}.\end{aligned}$$ To simplify equation ([eq:cons-E]) we write ${\bf ds}\_i=\hat{{\bf v}}\_i{\rm d}s\_i$ and thus $$\begin{aligned} \int\_{{\bf s}\_i(t\_{0})}^{{\bf s}\_i(t\_1)}{\bf F}\_i\cdot{\bf ds}\_i &= \int\_{{\bf s}\_i(t\_{0})}^{{\bf s}\_i(t\_1)}\left(-f\_i\hat{{\bf v}}\_i+\tilde{f}\_i\hat{{\bf u}}\_i\right)\cdot\hat{{\bf v}}\_i{\rm d}s\_i\notag\\ {} &=-f\_i\int\_{{\bf s}\_i(t\_{0})}^{{\bf s}\_i(t\_1)}{\rm d}s\_i\equiv-f\_i\mathcal{S}\_i.\label{eq:def-S}\end{aligned}$$ The perpendicular force component f̃ disappears due to the dot product, while f is assumed constant along the path and can be taken out of the integral, which defines S (which is simply the path’s length). Finally, equation ([eq:cons-E]) for the energy becomes ΔE⋆ = f1S1 + f2S2. To simplify equation ([eq:cons-L]) we write τ1 in Cartesian coordinates: $$\tau\_i = -\frac{f\_i}{v\_i}(x\_iv\_{iy}-y\_iv\_{ix}) - \frac{\tilde{f}\_i}{v\_i}(x\_iv\_{ix}+y\_iv\_{iy}),$$ and thus $$\begin{aligned} \int\_{t\_{0}}^{t\_1}\tau\_i{\rm d}t &= -f\_i\int\_{t\_{0}}^{t\_1}\frac{x\_iv\_{iy}-y\_iv\_{ix}}{v\_i}{\rm d}t-\tilde{f\_i}\int\_{t\_{0}}^{t\_1}\frac{x\_iv\_{ix}+y\_iv\_{iy}}{v\_i}{\rm d}t \notag \\ {} &\equiv -f\_i\mathcal{P}\_i-\tilde{f\_i}\mathcal{Q}\_i.\end{aligned}$$ The above integrals define P and Q. Equation ([eq:cons-L]) for the angular momentum becomes $$\Delta L\_{\star}=f\_1\mathcal{P}\_1+\tilde{f\_1}\mathcal{Q}\_1+f\_2\mathcal{P}\_2+\tilde{f\_2}\mathcal{Q}\_2.\label{eq:forces-L}$$ If we evolve the BHs between t0 and t1 under their mutual gravity alone, energy (and angular momentum) will be conserved along the produced orbital segment: $\Delta E\_{\rm BH}=0$. Stars scattered on this orbital segment will not their conserve energy: ΔE⋆ ≠ 0; so energy is also globally not conserved between t0 and t1. By solving equations ([eq:forces-E]) and ([eq:forces-L]) for the (non conserving) forces, the BHs can be evolved again in this time interval, under the additional forces, producing an orbital segment for which $\Delta E\_{\rm BH}=-\Delta E\_\star$. The revised orbital segment is only slightly different from the original, since the additional forces are much smaller than the forces exerted by the other BH and the background potential. Stars scattered on the revised orbit will have a slightly different ΔE⋆. This process of alternatingly evolving the BHs and the scattering of stars can be repeated until converges is achieved. Since ΔE⋆ and ΔL⋆ are directly obtained from the scattering experiments and the six integrals (calligraphic letters) are calculated from the orbital segments, equations ([eq:forces-E]) and ([eq:forces-L]) are a linear system of two equations with four variables: f1, f2, f̃1 and f̃2. It is worth noting that without the perpendicular force, which is expected to be negligible in the standard dynamical friction formalism, it is impossible to conserve energy and angular momentum simultaneously. In particular cases, additional constraints give an exact solution as explained in the following section. Finding the forces ------------------ ### Symmetric motion In this case, the masses are equal and the initial conditions are symmetric with respect to the centre of the system. The motion of one BH mirrors that of its companion, so that the path integrals are equal for the two BHs (the index is therefore dropped) and the forces acting on them must also be equal due to symmetry. The solution is $$\begin{aligned} f &= \frac{1}{2\mathcal{S}}\Delta E\_\star,\label{eq:force-sym-par}\\ \tilde{f} &= \frac{1}{2\mathcal{SQ}}(\Delta L\_\star \mathcal{S}-\Delta E\_\star \mathcal{P}).\label{eq:force-sym-per}\end{aligned}$$ There is no solution for Q = 0 (but S ≠ 0 is guaranteed by equation [eq:def-S]). In points where the BHs’ velocity is purely tangential, $\tilde f$ is parallel to the radius vector and thus exerts no torque (and work is never done by $\tilde f$). At these points, $\tilde f$ is free but fi is overdetermined (must change both $E\_{\rm BH}$ and $L\_{\rm BH}$ by the specified amounts). Segments which are symmetric about such points have Q = 0. ### High mass ratio In this case one mass is much larger that the other, and is assumed to sit motionless at the centre of the system. The forces f and f̃ act on the secondary BH only. The integrals S, P and Q are also calculated for the secondary only. The solution for Q ≠ 0 is $$\begin{aligned} f &= \frac{1}{\mathcal{S}}\Delta E\_\star,\\ \tilde{f} &= \frac{1}{\mathcal{SQ}}(\Delta L\_\star \mathcal{S}-\Delta E\_\star \mathcal{P}).\end{aligned}$$ ### General solution The results for the limiting cases described above motivate us to look for solutions of the form $$\begin{aligned} f\_1 & =\frac{\alpha}{\mathcal{S}\_1}\Delta E\_{\star}, & \tilde{f}\_1 & =\frac{\alpha}{\mathcal{S}\_1\mathcal{Q}\_1}\left(\Delta L\_{\star}\mathcal{S}\_1-\Delta E\_{\star}\mathcal{P}\_1\right),\\ f\_2 & =\frac{1-\alpha}{\mathcal{S}\_2}\Delta E\_{\star}, & \tilde{f}\_2 & =\frac{1-\alpha}{\mathcal{S}\_2\mathcal{Q}\_2}\left(\Delta L\_{\star}\mathcal{S}\_2-\Delta E\_{\star}\mathcal{P}\_2\right).\end{aligned}$$ There is a mathematical solution for every α, but we know from the limiting cases that $0\leq\alpha\leq\frac{1}{2}$. The value of α is a function of the mass ratio q, but it may also be dependent on other factors, such as the local stellar densities at the instantaneous position of either BH. In the case of a high mass ratio, one BH is almost stationary; therefore α must approach zero faster or at least as fast as S1. The force f1 does not have to approach zero, but the acceleration f1/M1 does. Model ===== Units ----- For reasons of consistency with our previous work, we use a unit system in which mass is measured in units of the primary BH’s mass, velocity is measured in units of 4σ and G = 1 (where σ and G are the stellar velocity dispersion and the gravitational constant, respectively). The hard binary separation is defined as $$a\_{\rm h} \equiv \frac{q}{1+q} \frac{GM\_\bullet}{4\sigma^2},$$ where q ≤ 1 is the mass ratio of the secondary and primary BHs and M• is the mass of the primary. In our units, the hard binary separation of an equal mass binary (q = 1) is 2. The base units are therefore scalable by M• and σ. Only one parameter is required if we also use the M–σ relation (e.g. ). The units of length, time and velocity units and their scaling, using M• and σ, are $$\begin{aligned} [{\rm L}] &= {\textstyle\frac{1}{16}} GM\_\bullet\sigma^{-2} = 0.77\ M\_8^{0.53} ~{\rm pc},\\ [{\rm T}] &= {\textstyle\frac{1}{64}} GM\_\bullet\sigma^{-3} = 1\,000\ M\_8^{0.29} ~{\rm yr},\label{eq:units-T}\\ [{\rm V}] &= 4\sigma = 750\ M\_8^{0.24} ~{\rm km~s^{-1}},\end{aligned}$$ where M8 is the physical mass of the primary BH in units of $10^8 ~{\rm M}\_\odot$. Note that in all simulations we used a stellar velocity dispersion of 0.25 velocity units. Bulge Properties ---------------- In all our simulations, stars are initially distributed in a singular isothermal sphere and follow a Maxwell–Boltzmann distribution with 1D velocity dispersion σ. To avoid the non-physical divergence of the potential, we assume a core structure: $$\rho(r)= \begin{cases} \rho\_0 & r\leq h\\ \rho\_0\left(\frac{h}{r}\right)^{2} & r>h \end{cases}\label{eq:isothermal-density}$$ where h is an arbitrary break radius set to 1 and ρ0 = σ2/2πGh2. The expression for the gravitational potential (or the bulge potential) derived from the above density is $$\Phi\_{\rm bulge}(r)= \begin{cases} \frac{\sigma^2}{3Gh^2}r^2 & r\leq h\\ \frac{\sigma^{2}}{G}\left[\frac{4h}{3r}+2\ln\left(\frac{r}{h}\right)-1\right] & r>h \end{cases}\label{eq:bulge-potential}$$ found that massive elliptical galaxies, within their effective radii, are well approximated by a power law ellipsoid with an index of − 2. also found that the density of the nuclear star cluster of the Milky way can be described by a broken power law with index of − 2.0 ± 0.1 down to 0.38 pc. This is however not appropriate within the BH sphere of influence, if dynamically relaxed, where the equilibrium distribution (radius independent mass and energy flow) is the famous cusp of ρ ∝ r− 7/4 (where only one mass species is present, cf., ). Unrelaxed clusters around adiabatically growing BHs are expected to have shallower slopes or steeper slopes in the case of rotating systems and non-isothermal clusters. In the case of a minor merger (equivalent to q = 0.1), the structure of the more massive galaxy does not change significantly and equation ([eq:isothermal-density]) likely represents the stellar environment seen by the secondary BH after its parent galaxy is absorbed. This picture is somewhat naïve in the case of a major merger (q = 1), but violent relaxation due to the rapidly varying potential in a newly merged galaxy causes widening of the stellar energy distribution and is analogous to relaxation by collisions in a gas. This process tends to drive galaxies towards a universal steady state. A nice demonstration of this appears in the N-body merger simulations of, who find an r− 2 density profile at the time the equal mass binary becomes hard, which extends down to the scale of the binary separation. Throughout this work we assume that the BHs are ‘naked’, that is, do not carry clusters of bound stars. In the equal mass case, the stellar mass bound to a single BH once the binary becomes hard does not exceed 10 per cent of its mass (assuming that the cluster also has a power law density profile with index of -2 and normalization based on the M–σ relation). This will most likely not affect the late time evolution, but will surely affect the early inspiral phase, and the exact inspiral time, but will probably not have a major effect. Algorithm ========= Equal masses ------------ Below we provide a technical description of the application of the technique described in Section [sec:math] for an equal mass binary simulation. The slightly different procedure for the 10:1 mass ratio case is discussed in the following Section. First, a realization of a singular isothermal sphere is produced up to a cutoff radius of ${R\_{\rm max}}$, with two equal mass BHs placed on the x-axis at x = ± R0; at this initial distance the BHs are still unbound to each other and their inspiraling orbits are governed by the bulge’s gravity and dynamical friction. The simulation duration is divided into (equal) intervals Δt, after each the stellar force acting on the BHs is updated. The actual time steps of the ODE solver are smaller if necessary, so close encounters can be resolved. Within each interval we do the following: 1. Symmetrically advance[2](#fn2) the two BHs from ti to ti + 1, under their mutual gravity, and the effective forces exerted by the stars. The calculated antiparallel f and perpendicular f̃ embody the stars’ pull on the BHs, so the bulge potential (equation [eq:bulge-potential]) should not be considered additionally. This gives a short orbital segment which is stored in memory. 2. Advance each star from ti to ti + 1, under the influence of the two BHs and the bulge potential. The motion of the BHs in this interval is already set from the previous stage, so this is essentially a short scattering experiment. This is the most computationally demanding stage of each iteration, but easily parallelized. 3. Sum up the energies and angular momenta of all stars and subtract the values from the previous iteration to obtain ΔE⋆ and ΔL⋆. 4. Use the BHs’ path to obtain S, P and Q. 5. Calculate f and f̃ from equations ([eq:force-sym-par]) and ([eq:force-sym-per]), to be used in the next Δt interval. Unless otherwise indicated, our simulations end at t = 10 000 time units (or 107 years for a $10^8~{\rm M}\_\odot$ primary); this is equivalent to ∼ 1 000 revolutions after the binary becomes hard in most equal mass simulations. Integration of a single star is terminated prematurely in three cases: if it reaches a distance of ${R\_{\rm max}}+10$ length units from the centre of the system, reaches $r\_{\rm tidal}=10^{-3}$ from either BH or takes more than 35 integration time steps to complete the interval of Δt = 0.1 (see Section [sec:qa]). In the first case the star is considered to have escaped the system (or diverged); the extra 10 length units beyond ${R\_{\rm max}}$ are an arbitrary ‘padding’ required for technical reasons. The second case represents tidal disruption of the star (the orbit is then said to have crashed). The true tidal disruption radius is up to two orders of magnitude smaller than our $r\_{\rm tidal}$, but this choice has a negligible effect on the BHs because the rate of crashing stars is negligible compared to the rate of diverging stars. The choice of ${R\_{\rm max}}$ is rather arbitrary, and is chosen to be large enough to minimize its effect on the results, as discussed in Section [sec:Rmax]. Notably, the forces on the BHs are one interval retarded. Thus, the interval duration must be short enough in order not to break conservation of energy and angular momentum. In all simulations we chose to use equal intervals of Δt = 0.1, and since simulations done with this choice both ran reasonably quickly and performed well in terms of conservation, we did not thoroughly investigate changing Δt. Certainly making the interval length longer or adaptive can significantly speed up the simulations. The code outlined above is very simple in terms of decision making; the bulk of CPU effort is made to individually advance stars (scattering experiments) with only a small non-parallelizable overhead between intervals. A typical simulation with a large realization (N = 5 × 106) would run for ∼ few days on a medium strength personal desktop computer. Mass ratio ---------- As noted in Section [sec:forces], the force acting on the BHs can be uniquely found from the energy and angular momentum differences only in the cases of equal masses and high mass ratios (where it is possible to assume that the primary BH is fixed at the centre). However, when the velocity is perpendicular to the radius vector (i.e. at pericentre or apocentre), the values of Qi approach zero, and thus f̃i are left out of the coupled equations ([eq:forces-L]) and ([eq:forces-E]), and cannot be solved for. At the same time, fi are overdetermined as they have to compensate for both energy and angular momentum changes. Even in intervals that contain an apsis, there is usually a solution unless the orbital segment happens to be symmetric about the apsis. Since only a few segments are affected, the effect on the BBH orbits is not significant. In the case of equal masses, no special treatment of these segments was necessary; but in the case of very high mass ratios (i.e. q ≪ 1) the finite numerical fluctuations are larger due to the fact that the secondary BH is less massive. In these cases the total angular momentum would discontinuously drop at an apsis, and the otherwise smooth path would suddenly break at this segment. We compromised on simulating a 10:1 binary, and utilized two techniques to improve accuracy. First, each step was performed twice: after calculating the frictional force, the stars and BH were reset to their original positions, and advanced again with the newly calculated force on the BH. Second, we attempted to compensate for the accumulated error: instead of calculating the force components using purely ΔE⋆ and ΔL⋆ of the last Δt interval, we added their respective accumulated errors (the correction term had weight of 0.1 per cent). Those adjustments dramatically improved the accuracy of the 10:1 simulations, but unfortunately we did not yet overcome all the technical problems associated with reliably simulating a higher mass ratio inspiral from ∼ 50 pc down to stalling separation. Quality Assurance ----------------- If the routine that advances the stars has a bug or is just not accurate enough, the values of ΔE⋆ and ΔL⋆ obtained in each interval will be faulty; but since the frictional force is calculated in such a way that would compensate for any change of E and L in the stars, the bug might remain undetected. Thus, an important validity test of E and L conservation in the code is inherently not available. We can, however, test the ODE solver for a similar problem, and infer that our stellar orbits are at least well behaved in the original problem. Assume a BBH with a circular orbit of constant radius R; the BBH orbit is now decoupled from the stars and each stellar orbit can be integrated separately. This is the restricted 3-body problem plus a spherically symmetric potential; as in, the energy in the rest frame of the BHs (the frame which rotates with the same angular frequency) is the only conserved quantity. The Jacobi integral is a constant of motion related to the rest frame energy by CJ = − 2E. We performed a number of tests where the BBH was in a circular orbit with a radius in the range of 1 ≤ R ≤ 40 length units. The stellar model was the same as described in Section [sec:bulge] and the orbits were evolved for t = 10 000 time units. It was found that CJ is well conserved for the great majority of stars: only 0.2 to 1.6 per cent of orbits were cast off as rogue orbits, exceeding 35 steps per Δt = 0.1 interval; the rest had an average ∣ΔCJ∣ of (4 to 11) × 10− 5 energy units (CJ is typically of order unity). When applying a short softening length of 0.04 (corresponding to ), 0.2 to 1.7 per cent of orbits were eliminated by the same criterion and the rest had an average ∣ΔCJ∣ of (5 to 17) × 10− 5 energy units. Thus, we did not apply softening in the actual simulations. This type of number of steps filter against rogue orbits was found to work better than putting a lower limit on the allowed step size. Accuracy may be improved simply by integrating close encounters with smaller error tolerance, or by employing more elaborate techniques such as regularization. The small number of stars which are lost due to the ODE solver’s inability to handle them does not have a significant systematic effect on the BBH orbital evolution. Conservation of E and L --------------------------- As noted previously, total energy and angular momentum conservation is not a built-in requirement of the method, but rather indicates a successful transfer of energy and angular from the BBH to the stellar population. In the upper panels of Figure [fig:conservation] we show energy of the BBH (solid black line), the stellar population (solid red line) and their sum (dashed blue line); the lower panels is the same but for the angular momenta. The left panel is for an equal mass simulation while the right panel is for a 10:1 mass ratio simulation. The quality of energy conservation is attested by the absolute difference of total energy between the beginning and the end of the simulation; it is also useful to look at the fluctuations in total energy, a very crude estimate of which is the amplitude of the largest fluctuation (this is not a very good measure since fluctuation amplitude can be high in some parts of the orbit mild in others). These two quantities have dimensions of energy, and it is most sensible to normalize them with respect to the absolute difference in the energy of stars (or the BHs). Thus, for a simulation ending at t = T: $$\begin{aligned} \epsilon\_1^E(T) &= [E\_{\rm tot}(T) - E\_{\rm tot}(0)] / \|E\_\star(T) - E\_\star(0)\|,\\ \epsilon\_2^E(T) &= {\rm max}(\|E\_{\rm tot}(t) - \langle E\_{\rm tot} \rangle\|) / \|E\_\star(T) - E\_\star(0)\|.\end{aligned}$$ We similarly define ε1L and ε2L for the angular momentum. All these ε parameters must be very small. If the quantity that is supposed to be conserved has some trend, then usually ε1 = ε2, otherwise it only fluctuates around its mean value and ε1 < ε2. For an equal mass simulation with N = 5 × 106 (discussed in detail in Section [sec:N]) case at T = 43 200, we get ε1E, ε1L < 10− 6 and ε2E = ε2L = 3 × 10− 4. For a 10:1 mass ratio simulation (discussed in detail in Section [sec:e-growth]) at T = 21 800, we get ε1L = 4 × 10− 6 and ε2L = 6 × 10− 5, with a decreasing trend in total energy: ε1E = ε2E = 0.008. The trend in total energy begins at t ∼ 14 000 and is probably due to the fact that there is an accumulated inaccuracy in the solution of the BHs’ equations when they are so close together at pericentre. At t = 14 000 we get ε1E = 8 × 10− 5 and ε2E = 4 × 10− 4 with values for the angular momentum similar to the end of the simulation. Thus, the numerical effect previously discussed is unrelated the eccentricity growth observed in this simulation. [fig:conservation] Results ======= We performed a total of 58 simulations of an equal mass binary and a 10:1 mass ratio binary, varying N, R0, ${R\_{\rm max}}$ and the initial eccentricity e0. In Figure [fig:examples] we shows r(t), for selected simulations. In the 10:1 simulations, the primary BH is fixed at the centre, so r is the BBH separation; in the equal mass simulations r is half the separation. As can be seen in the figure, the semi-major axis stalls in all simulations but eccentricity (indicated by the thickness of the lines) does not reach a steady state in the unequal mass cases. These results are discussed in more detail below. [fig:examples] Number of Particles ------------------- Here we show two things: how the results converge with increasing number of stars N, and how the results depend on the specific realization of the stellar distribution. We present seven pairs of simulations with N between 50 000 and 350 000; for each N the two simulations have a different random seed, so that the stars have different initial positions and velocities, but are drawn from the same distribution. All simulations are of an equal mass binary starting at R0 = 60 with the local circular velocity; the cutoff radius of the stellar sphere is $R\_{\rm max}=200$. In Figure [fig:N] we show the ‘final state’ (i.e. at t = 10 000) semi-major axes a (equivalent to separation, in the equal mass case), and eccentricities e as functions of N; these two numbers are the best way to appreciate differences between similar simulations. From this small sample, it is apparent that the effect of changing N in this range is comparable in magnitude to that of changing the realization. In this set of simulations, the range of semi-major axis values is 0.07 length units or ∼ 5 per cent of the sample’s average a; the eccentricities are small and in the range 0.03 < e < 0.1. Thus, increasing N beyond 100 000 (within $R\_{\rm max}=200$) is unnecessary for this level of accuracy and following tests are made using this number. The average of a in these simulations is 1.43 (in physical units, for $10^8 ~{\rm M}\_\odot$ BHs, this is equivalent to 1.1 pc); this is approximately 30 per cent below the hard binary separation of 2. suggested the following formula for the stalling separation: $$\frac{a\_{\rm stall}}{r\_h^\prime} = 0.2 \frac{q}{(1+q)^2}\label{eq:stall},$$ where rh′ is the radius containing a mass in stars equal to twice the combined mass, or M(rh′) = 2M•(1 + q), at the time of stalling. We measured the accumulated mass in the above simulations and got that in all of them rh′ = 37.5 with a very small spread. By substituting this information and q = 1 into equation ([eq:stall]), one gets $a\_{\rm stall}=1.88$. It is important to note that in the simulation there was no actual stalling of the BBH, and $a\_{\rm stall}$ was estimated as the value of a in which a clear change in the hardening rate took place; rh′ was determined at the time when this change occurred. In our simulations both rh′ and $a\_{\rm stall}$ were determined at t = 10 000, which is well after the hardening rate has dropped. Also, equation ([eq:stall]) was calibrated by using simulations with mass ratios 0.025 ≤ q ≤ 0.5. To test the convergence of the results, we also performed a single simulation with significantly more particles (N = 5 × 106) and longer duration (t = 43 200); we show the inverse semi-major axis as a function of time in Figure [fig:long]. As seen in the figure, there is still some slow evolution of the semi-major after t = 10 000. At t = 40 000 the value of a is 1.35, which is 5.6 per cent lower than the value at t = 10 000 and only one per cent lower than the value at t = 30 000; the decay rate at the end of this simulation is $\dot{a}=10^{-6}$ velocity units, equivalent to $\sim 10^{-9}~{\rm pc\ yr^{-1}}$ for a $10^8~{\rm M}\_\odot$ primary. In this specific run, eccentricity was especially low at e < 0.01. The orbits produced in the other simulations in this set (with N ≤ 3.5 × 105) are very similar to the orbit shown in Fig. [fig:long], regardless of the number of particle (cf., spherical and triaxial cases). Note that a large N is required only to minimize the statistical fluctuations, our scheme is not subject to an artificial stellar relaxation mechanism, which requires a very large N to overcome. [fig:N] [fig:long] *R${}\_{\bf max}$ and R${}\_{\bf 0}$ ----------------------------------------------- Since we only simulate the spherical component and not a full merger, the initial and boundary conditions need be assumed: the initial distance of the BHs from the centre, R0, and the cutoff radius of the stellar sphere, ${R\_{\rm max}}$. Here we present six trios of simulations with ${R\_{\rm max}}$ between 120 and 520, each three simulations are of an equal mass binary starting at R0 = 30, 60 and 90 with the local circular velocity. The number of particles is $N = 500 \times {R\_{\rm max}}$, so as to keep the particle density profiles of equal normalizations (but different cutoff radii) in all simulations; this is due to the fact that the number of particles (or mass) grows linearly with radius in an isothermal sphere. In Figure [fig:R] we show the final state semi-major axes a* (stalling separation) as a function of ${R\_{\rm max}}$. From the results discussed in Section [sec:N], a characteristic error of ∼ 0.04 units on the semi-major axis can be attributed to a specific realization and number-of-particles statistical fluctuation. The simulations with R0 = 30 give systematically lower values for a; the values for R0 = 60 do appear to be systematically lower than the R0 = 90 simulations, but the two sets are within the errors of each other. There appears to be a trend of decreasing values of a with increasing ${R\_{\rm max}}$ up to ∼ 300, which is to be expected both because there is a larger supply of particles that can interact with the BHs, and because the potential well is deeper and stars that have already interacted with the BHs have a larger probability to fall back to the centre and interact again. Nevertheless, we see from this small sample that these effects are weak and comparable in magnitude to those discussed in the Section [sec:N]. In this set of simulations, the eccentricities are also small and in the range 0.02 < e < 0.11; no correlation of eccentricity was observed with either R0 or ${R\_{\rm max}}$. [fig:R] Mass Ratios ----------- As noted in section [sec:alg-ratio], the simulation of an unequal mass binary is somewhat different in nature. Thus, when studying the evolution of a 10:1 binary, we followed the analysis of Section [sec:N] and performed a number of different tests with increasing number of particles and different realizations. Here we present four pairs of simulations with N between 100 000 and 400 000; for each N the two simulations have a different random seed, so that the stars have different initial positions and velocities, but are drawn from the same distribution. All simulations are of a binary with a 10:1 mass ratio starting at R0 = 60 with the local circular velocity; the cutoff radius of the stellar sphere is ${R\_{\rm max}}= 200$. In Figure [fig:M] we show the final state semi-major axes a and eccentricities e as functions of N. As in Section [sec:N], changing N in this range produces no apparent effect on the stalling radius. Within this set of simulations, the semi-major axes are consistent; the sample’s average a is 0.241 ± 0.005. We measured the accumulated mass in the above simulations and got that in this case rh′ = 16.8 with a very small spread. By substituting this information and q = 0.1 into equation ([eq:stall]), one gets $a\_{\rm stall}=0.28$. The caveats of using this equation were explained in Section [sec:N]. If we recalibrate equation ([eq:stall]) using our two values of q, the prefactor is lowered from 0.2 to approximately 0.16; it is even somewhat smaller considering the fact that the true value of $a\_{\rm stall}$ can be ∼ 6 per cent lower than its measured value at t = 10 000 (see Figure [fig:long]). In contrast with the equal mass case, the eccentricities do not reach a steady state value. In one of the N = 105 simulations, the eccentricity increases very rapidly after the binary becomes hard. At t = 9 400 the pericentre distance reached 10− 3 and the simulation is terminated. In the rest of the simulations, The eccentricity values range between 0.05 and 0.28 at t = 10 000, however, in all but one of these simulations eccentricity is slowly increasing. We calculate Te → 1, a very rough estimation for the time of gravitational wave regime, by fitting e(t) with a linear function in the range 6 000 < t < 10 000 and continue it to e = 1. At this time span the rise in eccentricity is approximately linear, however this trend breaks at approximately e = 0.85, so in fact Te → 1 can underestimate the time to the gravitational wave regime by some ≈ 20 per cent. In Table [tab:e] we show the results for Te → 1 for this simulation set and also for the initially eccentric runs. The values range between approximately 4 × 104 and 3 × 105 time units, or on the order of 108 years for a $10^7~{\rm M\_\odot}$ secondary around a $10^8~{\rm M\_\odot}$ primary. [fig:M] [tab:e] \| N [105] \| e0 \| $e\_{\rm f}$ \| Te → 1 \| \| --- \| --- \| --- \| --- \| \| 1 \| 0 \| 0.05 \| 2.7 × 105 \| \| 1 \| 0 \| ∼ 1 \| $\lesssim 10^4$ \| \| 2 \| 0 \| 0.21 \| 4.8 × 104 \| \| 2 \| 0 \| 0.13 \| 2.2 × 105 \| \| 3 \| 0 \| 0.15 \| 1.4 × 105 \| \| 3 \| 0 \| 0.18 \| 5.9 × 104 \| \| 4 \| 0 \| 0.05 \| 4.4 × 107 \| \| 4 \| 0 \| 0.28 \| 3.5 × 104 \| \| 2 \| 0.1 \| 0.42 \| 2.0 × 104 \| \| 2 \| 0.1 \| 0.33 \| 2.5 × 104 \| \| 2 \| 0.2 \| 0.47 \| 2.1 × 104 \| \| 2 \| 0.2 \| 0.59 \| 1.7 × 104 \| \| 50 \| 0.2 \| 0.52 \| 1.8 × 104 \| \| 2 \| 0.3 \| 0.21 \| 9.2 × 104 \| \| 2 \| 0.3 \| ∼ 1 \| $\lesssim 10^4$ \| \| 2 \| 0.4 \| 0.96 \| ≈ 104 \| \| 2 \| 0.4 \| ∼ 1 \| $\lesssim 10^4$ \| Eccentricity growth ------------------- Here we tested how the initial eccentricity affects the results. We performed simulations with four different initial eccentricities between 0.1 and 0.4 with semi-major axes as in the circular simulations of Sections [sec:N] and [sec:mass-ratios]. The initial eccentricity e0 is that of a BBH with the same initial conditions and that moves in the initial potential (equation [eq:bulge-potential]) with no friction; the orbit is not really an ellipse since the potential in not Keplerian, so e0 corresponds to the mean orbital eccentricity, defined as the difference between the maximum and minimum separations divided by the major axis. For each e0, there are two simulations (with two different realizations) for an equal mass binary and two for a 10:1 binary. The number of particles is N = 2 × 105 with cutoff radius ${R\_{\rm max}}=200$; we performed a single simulation with significantly more particles (N = 5 × 106) for the e0 = 0.2 case. In Figure [fig:E0] we show the final state semi-major axes a and eccentricities e as functions of e0. This figure also includes four data points with e0 = 0 that have already been presented in Figs. [fig:N] & [fig:M]. The stalling separation is independent of initial eccentricity in the tested range, the sample’s average a is 1.42 ± 0.04 for the equal mass simulations and 0.246 ± 0.010 for the unequal mass simulations. As in the other tests we performed for equal mass binaries, the final orbits are very much circular; the eccentricities are in the range 0.01 < e < 0.1 despite the initially significant eccentricity. However, the final eccentricities in the 10:1 cases do appear to be generally correlated with e0. As in Section [sec:mass-ratios], here too, eccentricities are still increasing when the simulations terminate at t = 10 000; the eccentricity timescales Te → 1 for the 10:1 simulations are shown in Table [tab:e], which shows that the eccentricity growth rate is related to e0. [fig:E0] found that eccentricity growth is generally mild for equal mass binaries with very small initial eccentricities, but also that initial eccentricity e0 > 0.3 leads to very high peak eccentricity almost regardless of the other system parameters. Although all our equal mass simulations end up in nearly circular orbit, this is not inconsistent with : while in the latter work the BHs are launched from within their radius of influence, in our simulations the BHs are launched from much further out, in the dynamical friction regime. In our simulations, an equal mass binary becomes less eccentric as it inspirals from r > 60. For one of the e0 = 0.4 simulations, the rapid eccentricity decrease ceased at about r = 1.4 (approximately twice the stalling radius) where the value was e = 0.035. At r = 4 (equivalent to the initial radius of ’s equal mass binaries) the eccentricity was 0.12; no significant eccentricity growth occurred in ’s equal mass simulation with initial eccentricity of 0.1. While it has been shown by that BBHs lose memory of their initial eccentricity if they corotate with a massive gaseous disc, studies of eccentric orbits of hard binaries is motivated for the purely stellar dynamical case by the theory of linear response for dynamical friction. However, this theory is derived from a first order perturbative expansion and is not applicable when close encounters dominate the evolution, and the system is not well described by an analytical approximation. It is important to note another major difference between our work and which greatly affects the evolution of the binary orbital parameters, mostly the semi-major axis: while in our work the loss cone empties, implicitly assumes that the loss cone is always full at $r > r\_{\rm inf}$. This leads to a very rapid decay of the binary separation and quick coalescence due to gravitational wave emission. Similarly, calculated the eccentricity evolution in the case of a very small secondary BH that does not affect the stellar density profile. There, eccentricity grows because the orbit passes in and out of a flat core, where the star are fast and the drag force is much less efficient at pericentre than at apocentre. In our simulations, however, the much more massive secondary forms a cavity slightly larger than its apocentre, and eccentricity grows where the density is essentially zero. In Figure [fig:e-evolution] we show the evolution of the semi-major axis for all the runs with e0 = 0.2 and 10:1 mass ratio. The dotted green lines represent the two realizations with 2 × 105 particles while the solid blue line represents the larger N = 5 × 106 realization. The first two simulations are arbitrarily terminated at t = 10 000 while the latter is stopped only when the eccentricity reaches 0.99. If we scale to physical units for a primary BH of $10^8~{\rm M\_\odot}$, the end of the simulation is 22 Myr from its beginning. For this mass scaling, using the formula for orbital decay due to gravitational waves, the timescale for coalescence at the end of the simulation is just 1 Myr. The rapid growth of the eccentricity while the semi-major axis remains fairly constant indicates a high value for the the dimensionless eccentricity growth, defined as: $$K = \frac{{\rm d}e}{{\rm d}\ln (1/a)}.$$ For single scattering of unbound stars from a fixed background, derived a maximal value of K of about 0.3 for mass ratio of 16:1, consistent with previous scattering experiments (; ). The value of K for the simulation presented in Fig. [fig:e-evolution] is at least an order of magnitude larger (we only roughly estimated the value from the results). The difference is probably due to the very different nature of the orbits in the restricted 3-body problem versus the more realistic model used here. We refer an in-depth study of the physical mechanism behind the eccentricity growth to a subsequent work. It is interesting to compare our results to those of, who also got ‘runaway’ growth of eccentricity while the semi-major axis stalled, but one must note the critical differences between the two studies. Most notably, used a mass ratio of 100:1 while we took only 10:1. Additionally, they used an initially very shallow central density profile, ρ ∝ r− 3/4 while our bulge model (see Section [sec:bulge]) was an isothermal sphere, ρ ∝ r− 2. More importantly, the mass of their entire stellar population was just $1.25\times 10^9~{\rm M\_\odot}$, which is 8 times less than the primary BH’s mass and only 12.5 times more than the secondary’s mass. By comparison, the total stellar mass in our model is 25 times the mass of the primary and 250 times the mass of the secondary. Thus, their entire simulation is deep within the primary’s sphere of influence, where its gravity dominates, while our simulations started with the secondary well outside the primary’s radius of influence. Let us scale our work to theirs by setting M8 = 100 in the scaling equations of Section [sec:units]. start their A32k simulations (with N = 32 768) at $R\_0 = 20~{\rm pc}$ within which there are less than 2 500 particles, their BH stalls at $a = 3.9~{\rm pc}$. We initiate the secondary BH at $r=637~{\rm pc}$ and get stalling at 2.1 pc; there are initially 40 000 particles enclosed within $r = 20~{\rm pc}$ in our large N = 5 × 106 simulation (marked with a star in Fig. [fig:E0]). The secondary BH is 20 000 times more massive than a field star in our simulation, versus 2 600 in the simulation. Their stellar bulge model is very small compared to ours, with 90 per cent of their mass is within $R\_0 = 190~{\rm pc}$, while our model is truncated at ${R\_{\rm max}}= 1\,800~{\rm pc}$. The semi-major axes in the two studies evolve at very different rates: while the binary takes 19 million years (Myr) to sink from $a\_0 = 20~{\rm pc}$ to 10 pc in their shallow cusp, our binary does the same journey in only 0.18 Myr. However, when applying equation (6) of (derive from their numerical results) for our physical parameters, the timescale for significant eccentricity growth is extremely short at 0.78 Myr; the lifetime of the system from our simulation is approximately 85 Myr (scaled with M8 = 100), which is two orders of magnitude longer. The difference might be due to the mass scaling assumed in their formula. A final note about precession: in the unequal mass case, the BHs exhibit very small precession during the hard phase. In the large simulation of Fig. [fig:e-evolution], between t1 = 10 000 and t2 = 11 000 the semi-major axis precesses by 0.311∘. During this period of 1 Myr (scaled with M8 = 1), the average semi-major axis is 0.18 pc, and it drops by 0.002 pc; the average eccentricity is 0.559 and the growth is by 0.072. This precession can be produced, for example, by perturbing the Keplerian potential with a uniform density field of 20,000 solar masses per parsec cubed. Even with our large number of particles, this density corresponds to only four particles enclosed in the sphere with radius equals to the apocentre. In the snapshot taken at t1, there was one particle inside this region, and it was probably a transient since there are no stable orbits there except those tightly bound to one of the BHs. In principle, a small flux of particles to this region can produce the measured precession, but torques due to the anisotropy of the potential at larger distances are more likely to cause the precession. For comparison, general relativistic precession of the orbit (not reproduced in the simulation) would be $56^\circ~\rm{Myr}^{-1}$. We cannot yet say whether this precession compromises resonances that possibly induce the eccentricity growth, but will refer to this point in a future paper. [fig:e-evolution] Dynamical friction ================== The orbital decay of a massive object within a galaxy down to its centre is well approximated by Chandrasekhar’s dynamical friction formula. However, the assumption of a uniform background in his classical treatment does not hold in real galaxies. Thus, corrections to the Coulomb logarithm are necessary to account for the changing background with radius. performed a detailed theoretical investigation of Chandrasekhar’s formula in the presence of a density gradient and gave an improved analytical formula for the Coulomb logarithm. took into account also self consistent velocity distribution functions, and made a comprehensive examination of the applicability of the new formula to sinking massive objects in a number of galaxy models, using high resolution N-body and particle-mesh codes. Their results suggest a delay in the orbital decay with respect to the standard formula, which quantitatively varies according to the studied case. They notably give an explicit solution for the decay of a massive object moving on a ‘circular’ orbit. Their formula (equation 25 in their paper) is very general and holds for an arbitrary power law density profile (it does not hold for very flat cores where fast moving stars contribute to most of the frictional drag; see ). Here we bring their formula in our model’s units and adjusted the parameters for an isothermal sphere; the radial evolution is given implicitly by: $$\textstyle t = 52.918 \times \left\{{\rm Ei}\left[2\ln\left(\frac{1}{8}R\_0\right)\right]-{\rm Ei}\left[2\ln\left(\frac{1}{8}r\right)\right]\right\}\label{eq:Just}$$ The prefactor is an exactly calculable number. The special function ${\rm Ei}(x)$ is called the exponential integral, it has real values only for x > 0. Thus, the domain of definition of equation ([eq:Just]) is r > 8. However, the assumption of a nearly circular orbit breaks well above r = 8. The angle between the velocity vector and the tangent can be derived by finding the radial velocity from equation ([eq:Just]): $$\theta = 195.99^\circ \times \frac{1}{r}\ln\left({\textstyle\frac{1}{8}}r\right),\label{eq:spiral-angle}$$ where this approximation hold only for small angles or large r. It is generally difficult to simulate a full infall of a compact object into a galaxy centre due to the collisional nature of the interactions and the low number density of particles which can generally be obtained at the outskirts of galaxy models. used the particle-mesh code Superbox, which is collisionless and uses fixed time steps; This type of code, unlike direct N-body codes, allows a large particle number to be simulated in a relatively short time. Using the code we designed for BBHs, we can also study the early part of the inspiral, which is dominated by dynamical friction. Since our code resolves collisions between the BHs and stars, this study is complimentary to that of Just et al.. In Figure [fig:df] we show a simulation of an equal mass binary initiated from R0 = 500 (solid blue line) and the theoretical curve (dashed black line), equation ([eq:Just]). The initial velocity is the local circular velocity, but the initial velocity vector is tilted by 1.62∘; this angle is obtained be substituting r = R0 in equation ([eq:spiral-angle]). If the initial velocity is purely tangential, then the spiral becomes slightly ‘eccentric’. A second run with a different realization (not shown) gave very similar results, including the position of the wiggles. It should be pointed out that this simulation is just a proof of concept; we do not expect the stellar distribution to be spherically symmetric at the early stages of a major merger. The orbital decay in our simulation is initially very well described by equation ([eq:Just]); deviations become significant below r ∼ 300, where mutual gravity of the two BHs is still negligible compared to the gravity of the bulge. These deviations are possibly due to the fact that the phase space distribution in the actual simulation (at the time and radius where the deviations occur) is no longer consistent with the assumption of isothermal sphere used to derive equation ([eq:Just]), in particular the velocity distribution might not be described well by a Gaussian. [fig:df] Summary ======= Using a conservation-based scheme, we were able to follow the evolution of a BBH from a wide separation (enclosed stellar mass greater than the combined BH mass) down to sub-parsec scale. Our code resolves star-BH collisions and can run with N > 106 stars on a desktop computer. We verified that our scheme yields convergent results which are independent of the number of particles, and the initial and boundary conditions. By performing scattering experiments on the inspiraling BBH, we will be able to extend and calculate the signature of the inspiral on the background stellar phase space distribution as a function of projected position. This calculation improves on N-body simulation by reducing statistical fluctuations and having no spurious relaxation (and thus no loss cone refilling). We performed calculations for both an equal mass binary and a 10:1 mass ratio. Our calculations reveal: 1. The inspiral from a radius scale of tens of parsecs to the hard binary radius occurs on a time scale of a few million years for a $10^8 ~{\rm M}\_\odot$ primary, with only a weak dependence of the timescale on the mass ( ∝ M80.29, equation [eq:units-T]). 2. The inspiral ends at a radius which is ∼ 30 per cent smaller than the simple analytical estimate for the hard binary radius, and consistent with. 3. An equal mass binary inspiral leads to a nearly circular final orbit, regardless of the initial eccentricity. 4. Eccentricity increases and coalescence due to gravitational wave emission will occur for a binary with a mass ratio of 10:1 in less than 108 years ( × M80.29). If the stellar distribution is triaxial or rotating the lifetime of such systems is potentially shorter. While we used a static, spherically symmetric background potential to account for star-star interactions, it is straightforward to extended this method to treat more complicated cases such as an adiabatically evolving potential (e.g. due to core depletion during the BBH inspiral), non symmetric stellar models and perturbers within the scattering method. 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E-mail: <!-- h='physics.technion.ac.il';a='@';n='ym';e=n+a+h; document.write('<a h'+'ref'+'="ma'+'ilto'+':'+e+'">'+e+'<\/'+'a'+'>'); // --> ym at physics dot technion dot ac dot il (YM); <!-- h='physics.technion.ac.il';a='@';n='laor';e=n+a+h; document.write('<a h'+'ref'+'="ma'+'ilto'+':'+e+'">'+e+'<\/'+'a'+'>'); // --> laor at physics dot technion dot ac dot il (AL)[↩](#fnref1) 2. The word ‘advance’ in this context means solve the equation of motion by means of Runge–Kutta method of order five with adaptive step size control (e.g. ).[↩](#fnref2)	arxiv_0000541
Generalized Pareto Regression Trees for extreme events analysis =============================================================== 1 Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, LPSM, 4 place Jussieu, F-75005 Paris, France, 2 Mission Risques Naturels, 1 rue Jules Lefebvre 75009 Paris, France E-mails : [email protected], [email protected], [email protected], olivier.lopez@sorbonne- universite.fr In this paper, we provide finite sample results to assess the consistency of Generalized Pareto regression trees, as tools to perform extreme value regression. The results that we provide are obtained from concentration inequalities, and are valid for a finite sample size, taking into account a misspecification bias that arises from the use of a “Peaks over Threshold” approach. The properties that we derive also legitimate the pruning strategies (i.e. the model selection rules) used to select a proper tree that achieves compromise between bias and variance. The methodology is illustrated through a simulation study, and a real data application in insurance against natural disasters. Extreme value theory; Regression trees; Concentration Inequalities; Generalized Pareto Distribution. Introduction ============ Extreme value theory (EVT) is the branch of statistics which has been developed and broadly used to handle extreme events, such as extreme floods, heat waves episodes or extreme financial losses. One of the key results behind the success of this approach was proved by Balkema and de Haan in : they show that the tail of the distribution of a series of observations can be approximated by a parametric family of distributions, namely Generalized Pareto (GP) distributions. This property allows the statistician to find information from the largest observations of a random sample to extrapolate the tail. This yields the so-called Peaks over Threshold (PoT) method introduced in which consists in fitting a GP distribution to the excesses above some (high) suitably chosen threshold. In a regression framework, the parameters of this GP distribution depend on covariates reflecting the fact that different values of these covariates may result in a different tail behavior of the response variable. In this paper, we study the use of regression trees to perform GP regression on the excesses. This ensemble method, introduced by, determines clusters of similar tail behaviors depending on the value of the covariates, based on a recursive partition of the sample and simple model selection rules. In the present work, we provide theoretical results and empirical evidence on the consistency of such a procedure and of these selection rules. The result we provide are based on concentration inequalities, in order to hold for finite sample sizes. The main difficulty stands in the misspecification of the model and on handling the fact that the distributions are heavy tailed. Tail regression is a challenging task. Several papers have been interested in extreme quantile regression, to name a few, in 2005, Chernozhukov and, in 2012, Wang et al. derive extreme quantile estimators assuming a linear form for the conditional quantile. In 2019, Gardes and Stupfler and Velthoen et al. use conditional intermediate-level quantiles to extrapolate above the threshold and deduce estimators for extreme conditional quantiles. Another approach is to model the parameters of the GP distribution of functions of the covariates e.g. as local polynomials or as generalized additive models. Very recently, in 2021, Velthoen et al. proposed to a gradient boosting procedure to estimate conditional GP distribution. Let us note that the nonparametric approaches rely on regularity assumptions on the way the tail of the distribution evolves with the covariates (which are required to be continuous through the use of kernel smoothing). A nice feature of the regression tree approach we consider in the present paper is its ability to handle several covariates which components may be either discrete or continuous. Moreover, this method is adapted to situations where the tail behavior is supposed to be significantly different depending on the characteristics, as for example it is the case in an application to cyber-insurance considered in a former paper, see. Regression trees, introduced by Breiman along with the CART algorithm (for Clustering And Regression Trees), are flexible tools to perform a regression and clustering task simultaneously. They have been used in various fields, including industry, geology (see e.g. ), ecology (see e.g. ), claim reserving in insurance. Through the iterative splitting algorithm used in CART, nonlinearities are introduced in the way the distribution is modeled, while furnishing an intelligible interpretation of the final classification of response variables. The splitting criterion—used to iteratively separate observations into clusters of similar behaviors—depends on the type of problems one is considering. While the standard CART algorithm relies on mean-squared criterion to perform mean-regression, alternative loss functions have been considered as in for quantile regression, or in who used a log-likelihood based loss. Loh provide detailed descriptions of regression trees procedures and a review of their variants. In this paper, building on the Balkema and de Haan result, we use a GP log-likelihood loss, as in, to perform extreme value regression. The rest of the paper is organized as follows. In Section [secmethods], we introduce notations and describe the GP regression tree algorithm. Section [secmain] lists the main results of this paper, that is deviation bounds for the regression tree estimator for finite sample size, and consistency of the “pruning” (that is model selection) strategy. Empirical results are gathered in Section [secsimu], which provides a simulation study, and a real data analysis in natural disaster insurance. Detailed proofs of the technical results are shown in the Appendix. Regression trees for extreme value analysis =========================================== This section describes the estimation method (GP regression trees) that is considered in the paper. Some classical results in EVT are given in Section [secevtreg] to motivate the GP approximation. Regression trees adapted to this context are described in Section [secGPDtrees]. A short discussion on the advantage of this technique compared to competing approaches is developed in Section [subsec:compete]. Extreme value theory and regression ----------------------------------- Let us consider independent and identically distributed observations Y1, Y2, … with an unknown survival function $\overline{F}$ (that is $\overline{F}(y)=P(Y\_1>y)$). A natural way to define extreme events is to consider the values of Yi which have exceeded some high threshold u. The excesses above u are then defined as the variables Yi − u given that Yi > u. The asymptotic behavior of extreme events is characterized by the distribution of the excesses which is given by $$\overline{F}\_u(z)= P[Y\_1 - u > z \mid Y\_1> u] = \frac{\overline F(u+z)}{\overline F(u)}, \ z>0 \,.$$ In 1975, Pickands showed that, if $\overline F$ satisfies the following property $$\label{eq:rv} \lim\_{t \to \infty} \frac{\overline{F}(ty)}{\overline{F}(y)} = y^{-1/\gamma\_0}, \, \forall y >0,$$ with γ0 > 0, then $$\label{eqplus} \lim\_{u \to \infty} \sup\_{z>0} \|\overline F\_u(z) - \overline{H}\_{\sigma\_{0u},\gamma\_0}(z) \| = 0$$ for some σ0u > 0 and $\overline{H}\_{\sigma\_{0u},\gamma\_0}$ necessarily belongs to the Generalized Pareto (GP) distributions family which distribution function is of the form $$\overline{H}\_{\sigma\_{0u},\gamma\_0}(z) = \left(1+ \gamma\_0 \frac{z}{\sigma\_{0u}}\right)^{-1/\gamma\_0}, \ z>0,$$ where σ0u > 0 is a scale parameter and γ0 > 0 is a shape parameter, which reflects the heaviness of the tail distribution. Especially, if γ0 ∈ ]0; 1[, the expectation of Y1 is finite whereas if γ0 ≥ 1 the expectation of Y1 is infinite. More details on these results can be found in e.g.. In practice, the so-called Peaks over Threshold (PoT) method is widely used, see. It consists in choosing a high threshold u and fitting a GP distribution on the excesses above that threshold u. The estimation of the parameters σ0u and γ0 may be done by maximizing the GP likelihood. The choice of the threshold u can be understood as a compromise between bias and variance: the smaller the threshold, the less valid the asymptotic approximation, leading to bias; on the other hand, a too high threshold will generate few excesses to fit the model, leading to high variance. The existing methods are mostly graphical, up to our knowledge, no automatic data-driven selection procedure is available. In the present paper, we consider a regression framework, that is that our goal to know the impact of some random covariates X on the tail of the distribution of a response variable Y. The previous convergence results hold, but for quantities σ0u, γ0 and u that may depend on X. More precisely, this means that, if we assume that γ0(x) > 0 for all x (which is the assumption that we will make throughout this paper), then ([eq:rv]) becomes $$\label{eq:rv2} \lim\_{t \to \infty} \frac{\overline{F}(ty\|\mathbf{x})}{\overline{F}(y\|\mathbf{x})} = y^{-1/\gamma\_0(\mathbf{x})}, \, \forall y >0,$$ where $\overline F(y\|\mathbf{x})=\mathbb{P}(Y\geq y \|\mathbf{X}=\mathbf{x}),$ see and references therein, and ([eqplus]) becomes $$\label{eqplus2} \lim\_{u(\mathbf{x}) \to \infty} \sup\_{z>0} \|\overline F\_{u(\mathbf{x})}(z \mid \mathbf{x}) - \overline{H}\_{\sigma\_{0u(\mathbf x)}(\mathbf{x}),\gamma\_0 (\mathbf x)}(z) \| = 0.$$ where $\overline F\_{u(\mathbf{x})}(z \mid \mathbf{x})=P[Y-u(\mathbf{x}) > z \mid Y > u(\mathbf{x}),\mathbf{X}=\mathbf{x}].$ Suppose that we observe (Yi, Xi)1 ≤ i ≤ n a sample of (Y, X), where X belongs to a compact set X. Following the PoT approach, the estimation of the function γ0(X) and σ0(X) = σ0u(X) can typically be done by fitting a regression model on the data points (Yi, Xi) such that Yi exceeds a proper threshold u(Xi). More precisely, let us define $$\begin{aligned} \theta^\(\mathbf x)&=&\arg \max\_{ \theta\in \Theta} \mathbb {\mathbb{E}}[\phi(Y-u(\mathbf{X}),\theta)\|\mathbf{X} =\mathbf{x},Y\geq u(\mathbf{x})] \nonumber \\ &=& \arg \max\_{ \theta\in \Theta}{\mathbb{E}}[\phi(Y-u(\mathbf{X}),\theta)\mathbf{1}\_{Y\geq u(\mathbf{X})}\|\mathbf{X}=\mathbf{x}], \label{m\_est} \end{aligned}$$ where θ* = (σ, γ)τ (where aτ denotes the transpose of a vector a) and ϕ is the GP log-likelihood function, that is $$\phi(z,\theta) = -\log(\sigma)-\left(\frac{1}{\gamma}+1\right)\log\left(1+\frac{\gamma z}{\sigma} \right).$$ From ([eqplus2]), θ\(x) should be close to θ0(x) = (σ0(x), γ0(x))τ* for u(x) large enough. Based on this idea, Beirlant (2004) proposed a nonparametric approximation of the loss function maximized by θ\. This technique, based on local polynomials, requires continuity of the covariates and some smoothness assumptions on θ0. On the other hand, parametric methods have also been proposed, but relying on a stronger assumption on the shape of θ0. In the next section, we introduce a regression tree approach which is adapted to both continuous and discrete variables, and that relies on few assumptions (since the estimated regression function θ0 does not need to be smooth). GPD regression trees -------------------- Regression Trees are a convenient tool to capture heterogeneous behaviors in the data, see. These models aim at constituting classes of observations which have a relatively similar behavior in terms of the response variable Y. These classes are defined by “rules”, which affect an observation to one of these classes according to the values of its covariates X. These rules are obtained from the data through the CART (Clustering And Regression Tree) algorithm, and the non-linearity of the procedure allows for an adaptation to the estimation of large classes of regression functions. Fitting regression trees relies on a so-called “growing phase”, described in our context in Section [subsec:buildingProc], which corresponds to the determination of these splitting rules. Section [sec5] shows how an estimator of the regression function θ0 can be deduced from such a tree. The “pruning step”, which can be understood as a model selection procedure, is described in Section [subsec:pruning]. ### Growing step: construction of the maximal tree The CART algorithm consists in determining iteratively a set of “rules” x* = (x(1), …, x(d)) → Rj(x) to split the data, aiming at optimizing some objective function (also referred to as splitting criterion). In our case, we want to approximate the criterion ([mest]), that is we are searching for a regression function θ̂(X) among some class such that $\sum\_{i=1}^n \phi(Y\_i-u(\mathbf{X}\_i),\widehat{\theta}(\mathbf X\_i))\mathbf{1}\_{Y\_i\geq u(\mathbf{X}\_i)}$ is maximal. To shorten the notation, let φ(Yi, θ) = ϕ(Yi − u(Xi), θ)1Yi ≥ u(Xi). A set of rules (Rj)j ∈ J is a set of maps such that Rj(x) = 1 or 0 depending on whether some conditions are satisfied by x, with Rj(x)Rjʹ(x) = 0 for j ≠ jʹ and ∑jRj(x) = 1. In case of regression trees, these partitioning rules have a particular structure, since they can be written, for quantitative covariates (the case of x containing qualitative variables is described in Remark [rqual] below), as Rj(x) = 1x1 ≤ x < x2 for some x1 ∈ Rd and x2 ∈ Rd, with comparison symbols to be understood as component-wise comparisons. In other terms, if d = 1, rules can be identified as partitioning segments, if d = 2 they are rectangles (hyper-rectangles in the general case). The determination of these rules from one step to another can be represented as a binary tree, since each rule Rj at step k generates two rules Rj1 and Rj2 (with Rj1(x) + Rj2(x) = 0 if Rj(x) = 0) at step k + 1. The algorithm can be summarized as follows: Step 1: R1(x) = 1 for all x, and n1 = 1 (corresponds to the root of the tree). Step k+1: Let (R1, ...Rnk) denote the rules obtained at step k. For j = 1, …, nk, * if all observations such that Rj(Xi) = 1 have the same characteristics, then keep rule j as it is no longer possible to segment the population; * else, rule Rj is replaced by two new rules Rj1 and Rj2 determined in the following way: for each component X(ℓ) of X = (X(1), …, X(d)), define the best threshold xj ⋆(ℓ) to split the data, such that xj ⋆(ℓ) = argmaxx(ℓ)Φ(Rj, x(ℓ)), with $$\begin{aligned} \Phi(R\_j,x^{(\ell)}) & = & \sum\_{i=1}^n \varphi(Y\_i,\theta\_{\ell-}(\mathbf{X}\_i,R\_j)) \mathbf{1}\_{X\_{i}^{(\ell)}\leq x^{(\ell)}} R\_j(\mathbf{x})\\ & + & \sum\_{i=1}^n \varphi(Y\_i,\theta\_{\ell+}(\mathbf{X}\_i,R\_j)) \mathbf{1}\_{X\_{i}^{(\ell)} > x^{(\ell)}} R\_j(\mathbf{x}), \end{aligned}$$ where $$\begin{aligned} \widehat{\theta}(R\_j)&=&\arg \max\_{\theta\in \Theta}\sum\_{i=1}^n \varphi(Y\_i,\theta(\mathbf{X}\_i))R\_j(\mathbf{X}\_i), \\ \theta\_{\ell-}(x,R\_j) &=& \arg \max\_{\theta\in \Theta} \sum\_{i=1}^n \varphi(Y\_i,\theta(\mathbf{X}\_i))\mathbf{1}\_{X\_{i}^{(\ell)}\leq x}R\_j(\mathbf{X}\_i), \\ \theta\_{\ell+}(x,R\_j) &=& \arg \max\_{\theta\in \Theta} \sum\_{i=1}^n \varphi(Y\_i,\theta(\mathbf{X}\_i))\mathbf{1}\_{X\_{i}^{(\ell)} > x}R\_j(\mathbf{X}\_i). \end{aligned}$$ Then, select the best component index to consider: $\widehat{\ell} = \arg\max\_{\ell} \Phi(R\_j,x^{(\ell)}\_{j\star})$. Define the two new rules $R\_{j1}(\mathbf{x})=R\_j(\mathbf{x})\mathbf{1}\_{x^{(\widehat{\ell})} \leq x^{(\widehat{\ell})}\_{j\star}},$ and $R\_{j2}(\mathbf{x})=R\_j(\mathbf{x})\mathbf{1}\_{x^{(\widehat{\ell})} > x^{(\widehat{\ell})}\_{j\star}}.$ * Let nk + 1 denote the new number of rules. Stopping rule: stop if nk + 1 = nk. 0.3cm This algorithm has a binary tree structure. The list of rules (Rj)1 ≤ j ≤ nk are identified with the leaves of the tree at step k, and the number of leaves of the tree is increasing from step k to step k + 1. The stopping rule can also be slightly modified to ensure that there is a minimal number of points of the original data in each leaf of the tree at each step. [rqual] In this version of the CART algorithm, all covariates are continuous or {0, 1} − valued. For qualitative variables with more than two modalities, they must be transformed into binary variables, or the algorithm must be slightly modified so that the splitting step of each Rj should be done by finding the best partition into two groups on the values of the modalities that minimizes the loss function. This can be done by ordering the modalities with respect to the average value—or the median value—of the response for observations associated with this modality. ### From the tree to the parameter estimation From a given set of rules R = (Rj)j = 1, …, s, let Tj = {x : Rj(x) = 1}, the jth leaf of the corresponding tree. The estimator θ̂ associated with a tree T = (Tℓ)ℓ = 1, ..., K (where K is the total number of leaves) is obtained as θ̂(x) = ∑ℓ = 1Kθ̂(Rj)Rj(x) = ∑ℓ = 1Kθ̂ℓ1x ∈ Tℓ. The maximal tree is the Tmax obtained once the previous algorithm stops. It corresponds to a trivial estimator of m, since either the number of observations in a leaf is one, or all observations in this leaf have the same characteristics x. The pruning step, presented in the next section, consists in extracting from the maximal tree a subtree that achieves a compromise between simplicity and good fit. ### Selection of a subtree: pruning step For the pruning step, a standard way to proceed is to use a penalized approach to select the appropriate subtree, see. For a given tree TK with K leaves (Tℓ)ℓ = 1, ..., K, associated with the corresponding estimator θ̂, the performance of this tree is measured through the following criterion $$\label{c\_pen} \frac{1}{k\_n}\sum\_{\ell=1}^K \sum\_{i=1}^n \varphi(Y\_i-u,\widehat{\theta}(\mathbf{X}\_i))\mathbf{1}\_{\mathbf X\_i \in \mathcal T\_\ell}-\lambda K.$$ For a given level of penalty λ, the selected tree is the one that maximizes criterion ([cpen]), achieving a compromise between good fit and simplicity. To determine this optimal tree, it is not necessary to compute all the subtrees from the maximal tree. It suffices to determine, for all K ≥ 0, the subtree TK which maximizes the criterion ([cpen]) among all subtrees with K leaves, and then to determine the final tree among a list of Kmax trees (where Kmax is the number of leaves of the maximal tree). The trees TK are easy to determine, since TK is obtained by removing one leaf to TK + 1, see p.284–290 in. The penalization constant λ can be chosen using a test sample or k − fold cross-validation. In the first case, data are split into two parts before making the tree grow (a training data of size n and a test sample which is not used in computing the tree). In the second case, the dataset is randomly split into k parts which successively act as a training or a test sample, see for example. Comparison with competing approaches ------------------------------------ Compared to competing approaches in extreme value regression, the advantage of the procedure is to introduce discontinuities in the regression function while parametric approaches suppose a form of linearity, e.g.. The more flexible non-parametric approaches, as, rely on smoothing techniques that require the covariates to be continuous. Chavez-Demoulin et al. propose a semi-parametric framework to separate the continuous covariates from the discrete ones. Smoothing splines are used to estimate non-parametrically the continuous part, while the influence of discrete covariates is captured by a parametric function. Main results ============ [secmain] In this section, we show that the GP regression tree procedure defined in Section [secGPDtrees] is consistent. Notations and assumptions used throughout this section are listed in Section [subsec:not]. We then state our first main results on the consistency of a fixed tree with K leaves, by separating the stochastic part of the error (Section [subsec:bounds:estimate]) from the misspecification part (Section [subsec:misspec]) caused by the GP approximation. The consistency of the pruning methodology is studied in Section [subsec:consistency:pruning]. Notations --------- Let us recall that the PoT approach consists in considering observations such that Yi ≥ u(Xi). Below, we will restrain ourselves to the case where u(x) = u. Our results easily extend to the case where u(x) = ∑j = 1muj1x ∈ Xj, where (Xj)1 ≤ j ≤ m are subsets of the space of covariates. Another possible extension would be to assume that u(x) = f(β, x) for some parameter β and f a known function. Nevertheless, a choice of such a particular threshold function seems hard to justify. Hence, we restrain ourselves to the simplest case. Moreover, the result we provide holds uniformly for u ∈ [umin, umax] to cover adaptive choice of this parameter. Conditions on umin and umax are given in Assumption [au]. [au] If n denote the number of observations, let kn be an intermediate sequence, that is kn → ∞ and kn/n → 0, as n → ∞. Then, let kn/n denote the average proportion of Y larger than umin, that is P(Y ≥ umin) = knn− 1. Moreover, assume that $$\mathbb{P}(Y\geq u\_{\max})=\frac{u\_0 k\_n}{n},$$ for some constant u0 ≥ 1. Here, kn will denote the average number (up to some constant) of observations on which the model is fitted. It is hence related to the rate of convergence of the procedure. The following assumption introduces conditions on this rate kn and on the space of parameters. [arate] We assume the parameter space to be Θ = S × Γ where * S = [σmin, σn], with σn = O(na1), with a1 > 0, * Γ is a compact set [γmin, γmax], with γmin > 0. Moreover, assume that kn = O(na2), with a2 > 0, and that the number of leaves of the maximal tree Kmax satisfies Kmax ≤ κkn, with κ > 0. Next, let us introduce some notations regarding the trees. Consider a tree T(u) with K leaves denoted $\mathcal T\_\ell$, ℓ = 1, …, K. Introducing the (normalized) contribution of the log-likelihood to the ℓth leaf, say $$L\_n^\ell (\theta,u) = \frac{1}{k\_n} \sum\_{i=1}^n \phi(Y\_i - u,\theta) \mathbf{1}\_{Y\_i >u} \mathbf{1}\_{\mathbf X\_i \in \mathcal T\_\ell},$$ let θ̂ℓ(u) = argmaxθLnℓ(θ, u), the estimated value of the parameter in the leaf $\mathcal T\_\ell.$ This estimator is expected to be close to θℓ\(u) = argmaxθ*Lℓ(θ, u), introducing Lℓ(θ, u) = knn− 1E[Lnℓ(θ, u)]. We denote by T\(u∣T) the tree with same leaves as T, but with parameters θℓ\(u). This quantity is not exactly our target: ideally, we would like to estimate θ0, ℓ(u) = (σ0(Tℓ, u), γ0(Tℓ)), such that $$\lim\_{t \to \infty} \sup\_{z>0} \|\overline F\_{t}(z \mid \mathcal{T}\_{\ell}) - \overline{H}\_{\sigma\_0(\mathcal{T}\_{\ell},t),\gamma\_0(\mathcal{T}\_{\ell})}(z) \| = 0,$$ where $\overline F\_{t}(z \mid \mathcal{T}\_{\ell}) =\mathbb{P}(Y-t\geq z \| \mathbf{X}\in \mathcal{T}\_{\ell},Y\geq t).$ We denote T0(u∣T) the tree with same leaves as T but with parameters θ0, ℓ(u). If θ = (θℓ)ℓ = 1, …, K denotes the set of parameters of a tree with K leaves $(\mathcal T\_\ell)\_{\ell=1,\ldots,K}$, we will denote $\theta(\mathbf x)$ the function defined by $$\theta(\mathbf x) = \sum\_{\ell = 1}^K \theta\_\ell \mathbf 1\_{\mathbf x \in \mathcal T\_\ell}.$$ We will first focus on the difference T(u) and T\(u∣T) in Section [subsec:bounds:estimate], which is the stochastic part of the error. On the other hand, the difference between T\(u∣T) and T0(u∣T) (and ultimately the difference between θ̂(x) and θ0(x)) is studied in Section [subsec:misspec] and can be understood as a misspecification term, caused by the fact that the excesses above the threshold are not exactly GP distributed. For ℓ = 1, …, K, let ∇θLℓ(θ, u) denote the gradient of Lℓ(θ, u), denoting $$\nabla\_{\theta} L^\ell (\theta,u) = {\mathbb{E}}\left( \begin{array}{cc} g\_{\theta,\ell}(Y\_i - u) \\ h\_{\theta,\ell}(Y\_i - u) \end{array} \mathbf{1}\_{\mathbf X\_i \in \mathcal T\_\ell} \mathbf{1}\_{Y\_i > u} \right)$$ with, for z > 0, $$\begin{aligned} g\_{\theta}(z) &=& \partial\_\sigma \phi(z,\theta) = \left(-\frac1 \sigma + \left(1 + \frac 1 \gamma \right)\frac{\gamma z}{\sigma^2(1+ \frac{\gamma z}{\sigma})}\right), \\ h\_{\theta}(z) &=& \partial\_\gamma \phi(z,\theta) = \left(-\frac{1}{\gamma^2} \log\left(1+\frac{\gamma z}{\sigma}\right) + \left(1+\frac 1 \gamma \right)\frac{ z}{\sigma+\gamma z}\right). \end{aligned}$$ To handle the stochastic part, we shall add a few assumptions. We first need a domination condition on the class of the derivatives of the functions y → ϕ(y − u, θ). These derivatives are uniformly bounded by Φ(y) = C(1 + log(1 + wy)), where C is a constant (not depending on n), and w = γmax/σmin. [aphi] Assume that, for some ρ0 > 0, mρ0 = E[exp(ρ0Φ(Y))] < ∞. In fact, this assumption is automatically satisfied if Assumption [arate] holds: since γ(x) ≥ γmin > 0, E[∣Y∣1/γ − ɛ] < ∞, for any ɛ > 0. Additionally, we need some regularity assumptions on the criterion Lℓ. [amoche] Let $$M^\ell\_{\theta\_1,\theta\_2,\theta\_3,\theta\_4}(u)=\mathbb{E}\left[\left(\begin{array}{cc} \partial\_\sigma g\_{\theta\_1}(Y-u) & \partial\_{\gamma} g\_{\theta\_2}(Y-u) \\ \partial\_\sigma h\_{\theta\_3}(Y-u) & \partial\_{\gamma} h\_{\theta\_4}(Y-u)\end{array}\right)\mathbf{1}\_{Y\geq u}\mid \mathbf X \in \mathcal T\_{\ell}\right].$$ Assume that there exists a constant $\mathfrak C\_1>0$ such that $$\inf\_{a,b\in \mathbb{R}}\inf\_{\theta\_1,\theta\_2,\theta\_3,\theta\_4\in \Theta}\inf\_{u \in[u\_{\min}, u\_{\max}]}\inf\_{\ell=,\ldots,K} \left\|M^\ell\_{\theta\_1,\theta\_2,\theta\_3,\theta\_4}(u)\left(\begin{array}{c} a \\ b \end{array}\right)\right\|\geq \mathfrak C\_1\max(\|a\|,\|b\|).$$ The condition on the infimum can be relaxed: Assumption [amoche] comes naturally in using a Taylor expansion. Hence, the infimum with respect of θ1, …, θ4 can be restricted to θ2 to θ3 belonging to a small neighborhood of θ1 (and not to the whole set Θ). Deviation bounds for our estimator ---------------------------------- In this section, we study the consistency of a fitted tree T(u), a subtree of the maximal tree Tmax(u), with K leaves $(\mathcal T\_\ell)\_{\ell=1,\ldots, K}$,. We compare this fitted tree to T\(u∣T), which is the tree based on the same subdivision, but where, in each leaf ℓ, the parameter is θℓ\(u) (instead of θ̂ℓ(u) in T(u)). The first step is to define a distance between trees. Let us define ∥(a, b)∥∞ = max(∣a∣, ∣b∣), and for two trees T and S, ∥T − S∥2 = (∫∥T(x) − S(x)∥∞2dP(x))1/2. The main result of this section is a deviation bound for ∥T(u) − T\(u∣T)∥2, which is Theorem [thm:deviation:bounds] below. [thm:deviation:bounds] Under Assumptions [au] to [amoche], and let β* > 0 such that βa2 ≥ 10/ρ0 (with ρ0 defined in Assumption [aphi]) and for t ≥ c1K(logkn)kn− 1, with c1 > 0, $$\begin{aligned} \lefteqn{\mathbb{P}\left(\sup\_{u\_{\min}\leq u\leq u\_{\max}}\\| T(u)- T^\(u\|T)\\|\_2^2\geq t\right) }\\ &\leq & 2\left(\exp\left(- \frac{{ \mathcal C\_1} k\_n t}{K\beta^2 (\log k\_n)^{2}} \right) + \exp\left(-\frac{{ \mathcal C\_2} k\_n t^{1/2}}{K^{1/2}\beta \log k\_n} \right)\right) +\frac{{ \mathcal C\_3}K}{k\_n^{5/2} t^{3/2}},\end{aligned}$$ where ${ \mathcal C\_1}$, ${ \mathcal C\_2}$ and ${ \mathcal C\_3}$ are positive constants. The proof of Theorem [thm:deviation:bounds] is postponed to the appendix section (Section [sec1]). The exponential terms on the right-hand side come from concentration inequalities proved by Einmahl and Mason, while the polynomially decreasing term is related to the fact that the log-likelihood is an unbounded quantity, but that can still controlled when considering its expectation. As a by-product, we obtain the following Corollary [corexp] (by integration of the bound of Theorem [thm:deviation:bounds]). [corexp] $$\mathbb{E}\left[{ \sup\_{u\_{\min}\leq u\leq u\_{\max}}\\|T(u)- T^\(u\|T)\\|\_2^2}\right]\leq { \mathcal C\_4} \frac{K\beta ^2(\log k\_n)^2}{k\_n}.$$ From Corollary [corexp], one can see that the L2 − norm of the stochastic part of the error, E[supumin ≤ u ≤ umax∥T(u) − T\(u* ∣ T)∥22]1/2, is proportional to K1/2, and, as expected, increases with the complexity of the tree. On the other hand, the error decreases almost at rate kn1/2 (up to some logarithmic factor), which is the convergence rate of standard estimators used to estimate the tail parameter in absence of covariates. The proof is again postponed to the appendix (Section [sec2]). Misspecification bias --------------------- For $\mathbf X = \mathbf x$, the ultimate goal is to estimate the tail index parameter $\theta\_0(\mathbf x)=(\sigma\_{0u}(\mathbf x),\gamma\_0(\mathbf x))$, introduced in, by maximization of the GP likelihood. The difference between $\theta\_0(\mathbf x)$ and $\theta^\(\mathbf x)$ can be understood as a misspecification term due to the fact that the observations above the threshold are not exactly distributed according to a GP distribution. This bias term can be controlled under second order conditions which are standard in Extreme Value Analysis. Indeed, recall that assuming that the underlying distribution $\overline{F}(\cdot\|\mathbf x)$ satisfies Condition guarantees that asymptotically the associate excesses above the threshold u* are GP distributed. For finite samples, the excesses are thus not exactly GP distributed which introduces some bias term. In order to control this bias term, a second-order condition is needed, that is a condition to control the rate of convergence in Condition. There exist numerous ways to express this second-order condition. Here, we consider the same condition as Condition C.6 in. First, Condition can be translated into $$\overline{F} (y\mid \mathbf x) = y^{-1/\gamma\_0(\mathbf x)} \eta(y \mid \mathbf x)\,, \forall y>0,$$ where η is a slow-varying function, that is $\eta(ty\mid \mathbf x)/\eta(t\mid \mathbf x) \rightarrow 1$ as t → ∞, for all y > 0. [assum:rv:second] Assume that for all $\mathbf x$, there exist a constant c and a function ψ such that $$\eta(ty \mid \mathbf x)/\eta(t\mid \mathbf x) = 1 + c\psi(t)\int\_1^t v^{\rho-1} {\mathrm{d}}v + o(\psi(t ))$$ as t → ∞ for each y > 0 with ψ(t) > 0 and ψ(t) → 0 as t → ∞ and ρ ≤ 0. Let us note that we could also consider the case of c, ψ and ρ depending on x, and then assume some uniform bound over x of these quantities. We chose this more restrictive formulation to simplify the notations. The next result guarantees that the bias term tends to 0 as u → ∞. [prp:bias] There exists a constant c and a function ψ such that ψ(u) > 0 and ψ(u) → 0 as u → ∞, and such that, for $\mathbf X = \mathbf x$, $$\begin{aligned} \\|\theta\_{0}(\mathbf x) - \theta^\(\mathbf x)\\|\_\infty &\leq& \mathfrak C\_2 (u)\frac{k\_n}{n} \left( 1 + c\gamma\_{\max} \psi(u) + o(\psi(u)) \right) \,,\end{aligned}$$ where $\mathfrak C\_2 (u)$ is a constant depending on u, γmin and γmax. Consistency of the pruning step ------------------------------- The previous results cover the case of a tree with fixed number of leaves K. In practice, the question is to select the proper subtree of Tmax(u), the maximal tree obtained once the previous step of the CART procedure has stopped, with some “optimal” number of leaves, which is the objective of the pruning step described in Section [subsec:pruning]. As seen in Corollary [corexp], the stochastic part of the error put to the square increases proportionally to K. This is closely related to the natural inflation of the log-likelihood (which is locally quadratic) when the number of leaves increases, justifying a penalty proportional to K, as in. The aim of Theorem [thselection] is to corroborate this choice. First of all, for a decomposition (TℓK)ℓ = 1, ..., K* of K leaves, let us define TK(u) the tree with parameters θ̂ℓK(u) estimated with the CART procedure, TK\(u) the tree with parameters $$\theta^{\K}\_{\ell}(u) =\arg \max\_{\theta\in \Theta} {\mathbb{E}}\left[\phi(Y - u,\theta) \mathbf{1}\_{Y >u} \mathbf{1}\_{\mathbf X\_i \in \mathcal T\_\ell^K}\right],$$ and $\mathbf{x}\rightarrow \theta^{\K}(\mathbf{x})=\sum\_{\ell = 1}^K\theta^{\K}\_{\ell}(u) \mathbf{1}\_{\mathbf x \in \mathcal T\_\ell^K}$ the corresponding regression function. Moreover, let K0(u) = argmaxK = 1, ..., KmaxE[ϕ(Y − u, θ\* K(X))1Y > u]. In words, T\(u) = T*K0(u)\(u) is the subtree of Tmax(u) that achieves the closest proximity to x* → θ\* K(x) in the sense that it maximizes the expectation of the (pseudo)-log-likelihood. Second of all, we denote, as explained in, the selected number of leaves $$\widehat{K}(u) = \arg\max\_{K=1,\ldots, K\_{\max}} \left\{\frac{1}{k\_n}\sum\_{\ell=1}^K\sum\_{i=1}^n \phi(Y\_i-u, \widehat{\theta}^{K}(\mathbf X\_i)) \mathbf 1\_{Y\_i>u} \mathbf 1\_{\mathbf X\_i \in \mathcal T\_\ell} - \alpha K \right\} \,,$$ and $\widehat{T}(u)=T\_{\widehat K(u)}(u)$ the corresponding selected tree. Define the log-likelihood Ln(TK, u) associated with a tree TK(u) with K leaves $(\mathcal T\_{\ell}^K)\_{\ell = 1,\ldots, K}$ with parameters $\widehat \theta^K(u)=\left(\widehat\theta^K\_{\ell}(u)\right)\_{\ell=1,\ldots,K}$ $$L\_n(T\_K,u) = \sum\_{\ell = 1}^K L\_n^{\ell}(\widehat \theta^K\_\ell,u) \,.$$ Then L(TK, u) = E[Ln(TK, u)]. Finally, for two trees T and S, ΔLn(T, S) = Ln(T, u) − Ln(S, u) and similarly, ΔL(T, S) = L(T, u) − L(S, u). The following Theorem [thselection] shows that the pruning methodology selects a tree $\widehat T(u)$ which approximately achieves the same rate as TK0(u), even if K0(u) is unknown, provided that the penalty constant λ belongs to some reasonable interval. [thselection] Let $\mathfrak D = \inf\_u \inf\_{K < K\_0(u)} \Delta L(T^\(u),T^\\_K(u))$ and suppose that there exists a constant c2 > 0 sauch that the penalization constant λ satisfies c2{logkn}1/2kn− 1/2 ≤ λ ≤ (D − 2c2{log(kn)}1/2kn− 1/2)kn− 1, then, for all u ∈ [umin, umax], $$\mathbb{E}\left[\\|\widehat{T}(u)-T^\(u)\\|^2\_2\right]\leq \frac{\mathcal{C}\_5 K\_0(u) (\log k\_n)^2 }{k\_n},$$ where C5 is a constant depending on T\(u). The proof is given in Section [sec3]. Simulation study and real data analysis ======================================= This section is devoted to the illustration of the GP regression procedure on simulated data (Section [subsec:simus]) and on a real dataset (Section [subseb:realdata]). Simulations ----------- In this section, we assess the performance of the GP regression procedure on simulated data and compare it with the competing approach proposed by. We first describe the simulation framework and then discuss the experiments results. We consider the following regression framework: X is a one-dimensional variable uniformly distributed on [0, 1], and the response variable Y, conditionally on X = x, is distributed according to a Burr distribution of parameters (σ, γ0(x)) which survival function is given by $$\label{eq\_burr} \overline{F}(y \mid x) =\frac{1}{1+ \left(y/\sigma\right)^{1/\gamma\_0(x)}} \,,$$ with σ > 0 and γ0(x) for all x. Note that $\overline{F}(\cdot \mid x)$ satisfies the property. We consider two cases: (i) γ0(x) as a step-wise function and (ii) γ0(x) as smooth function. In both cases, the scale parameter σ was fixed equal to 1. #### (i) step-wise function: In this case, the function γ0 is taken as $$\gamma\_0(x) = \begin{cases} 0.5 & \quad \text{if $0 \leq x <0.25$}\\ 1 & \quad \text{if $0.25 \leq x <0.75$}\\ 1.5 & \quad \text{if $0.75 \leq x \leq 1$}. \end{cases}$$ #### (ii) smooth function: In this case, the function γ0 is taken as, for x ∈ [0, 1], $$\label{eq\_simu\_model} \gamma\_0(x) = 1 + \frac{\tanh(10(x-1/4))}{4} + \frac{\tanh(10(x-3/4))}{4} \,.$$ We simulate 1 000 replications for different sizes of the observation sample (n = 1000, 2500, 5000, 10 000 and 25 000) according to the described framework for both cases (i) and (ii). For each sample, we consider the excesses above the 0.90-empirical quantile, which corresponds to kn = 100, 250, 500, 1 000 and 2 500. For each simulated sample, we compute the regression tree procedure (CART), and the method based on generalized additive model (GAM) proposed by. Next we compute ∫01(γ̂(x) − γ0(x))2dx for each estimator. The empirical mean squared error is then obtained by averaging these errors over the 1 000 replications. Results are shown in Table [tab:mse]. The boxplots of the empirical quadratic errors are shown in the supplementary material (Section A). [!ht] c \| kn \| 100 \| 250 \| 500 \| 1 000 \| 2 500 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| GP CART \| 0.290 \| 0.129 \| 0.107 \| 0.080 \| 0.050 \| \| GAM \| 0.313 \| 0.196 \| 0.122 \| 0.081 \| 0.048 \| a) \| kn \| 100 \| 250 \| 500 \| 1 000 \| 2 500 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| GP CART \| 0.227 \| 0.108 \| 0.079 \| 0.059 \| 0.043 \| \| GAM \| 0.233 \| 0.144 \| 0.068 \| 0.034 \| 0.016 \| b) [tab:mse] Let us note that the GAM approach is not designed to capture non-smooth functions like in the step-wise case. Nevertheless, we see that this technique manages to fit relatively correctly even in this case when the sample size is large. For kn = 1 000 and 2 500, the results of the GAM approach are similar or even slightly better than the regression tree method. On the other hand, we observe that regression trees lead to a better fit for small sample sizes, even in the smooth case where it is not designed to take into account the regularity of γ0(x). Prediction of the cost of flooding events in France --------------------------------------------------- In order to improve the knowledge and the management of natural catastrophes, the French Federation of Insurance (FFA) is interested in the prediction of the cost of such events, especially of the most severe ones, shortly after their occurrence. These catastrophic events present some heterogeneity in their intensity depending on their characteristics, such as the affected meteorological region or the number of individual houses in flood risk area. The prediction of their cost thus becomes a challenging task. In this section, we illustrate how the GP regression tree procedure can be used to gain further insight in this heterogeneity. The ability of the procedure to design classes of events that are more homogeneous (in view of analyzing the tail of their distribution) is an appealing property in view of operation applications in insurance. The database we consider was obtained through a partnership with the FFA, in particular with one of its dedicated technical body, the association of French insurance undertaking for natural risk knowledge and reduction (Mission Risques Naturels, MRN). It consists of all 3 100 flooding events that have been granted the status of natural catastrophe in France from 1999 to 2019 (let us note that the status “natural catastrophe” is a French specificity, with some legal consequences when an event receives this label, see ).This database is fed by 13 contributors including the major French insurance companies, allowing this database to cover 70% of French non-life insurance market. The database gathers information regarding each flooding event (its cost, the meteorological region, the season, the number of affected hydrological regions, the number of individual houses and the number of professional business premises in flood-risk area). Note that, since the purpose of this database is the fast prediction of the cost of a flooding event (as soon as possible after its occurrence), the variables that are registered correspond to quantities that are available before the event, or soon after it. The variable of interest, the total cost of a flooding event, is highly volatile. Indeed, it ranges between 0 and 394 376 000 euros with an empirical variance equal to 1.77e + 14. Figure [fig:map:floods] shows the average of the costs of the 10% most onerous flooding events within each meteorological region. This highlights the heterogeneity of the severity of the most severe events. Furthermore, the top ten most onerous events represent 43% of the total cost of this database and the top hundred 80%. ![image](CARTE_CART.png) [fig:map:floods] Now, let us recall that our goal is to understand the heterogeneity of the total cost of the most severe flooding events, that is of extreme flooding events. As explained in Section [secevtreg], the definition of extreme events consists in choosing a threshold u, which should be chosen as a bias-variance trade-off. We chose a value of u = 100 000 based practical considerations and validated by sensitivity analyses (shown in the supplementary material, Section B). This yields 1 100 extreme events, that is for which the cost is larger than u. The GP regression tree was performed on the database corresponding to the flooding events extracted from the original database for which the total cost is larger than u (=100 000 euros). The variables of this database and their characteristics are summarized in Table [tab:var:data]. Again, it can be noticed that the cost, the variable of interest, is highly volatile. [!ht] c m3cm\|c\|c\|c\|c\|c\|c Variable & Min & 1st Q & Median & Mean & 3rd Q & Max Cost (in euros) & 100 005 & 183 901 & 390 761 & 4 949 576 & 1 339 936 & 394 376 166 Number of affected hydrological regions & 1 & 3 & 5 & 6.53 & 8 & 35 Number of individual houses in flood risk area & 0 & 48 504 & 141 512 & 345 826 & 415 488 & 5 705 590 Number of professional business premises in flood risk area & 0 & 17 525 & 54 921 & 168 950 & 185 772 & 2 431 039 a) \| Variable \| Category \| Number of observations \| \| --- \| --- \| --- \| \| \| Center \| 89 \| \| \| North West \| 111 \| \| \| North \| 166 \| \| \| North-East \| 99 \| \| \| East \| 135 \| \| \| South \| 281 \| \| \| West \| 49 \| \| \| South West \| 158 \| \| \| Spring \| 358 \| \| \| Summer \| 336 \| \| \| Autumn \| 251 \| \| \| Winter \| 143 \| b) [tab:var:data] The tree obtained from GP regression procedure is shown in Figure [fig:tree:floods] (the quantile-quantile plots of the GP fit in each leaf are shown in the supplementary material, Section C). The tree is composed of 6 leaves, with three splits according to only 3 covariates: the number of individual houses, the number of professional business premises in flood-risk area and the number of affected meteorological regions. This seems reasonable since the first two covariates represent the exposure to floods, but also the population density of the affected area and the third one the extent of the flood. In each leaf, are given the shape and scale parameters. The worst case scenario corresponds to the leaf on the far right, with a shape parameter equal to 1 and containing 9% of all flooding events. This leaf corresponds to events for which more than 9 meteorological regions are affected and more than 597 518 professional business premises are in flood-risk area. The least severe case corresponds to the third leaf from the left, with a shape parameter equal to 0.24 and containing only 3% of the events. Table [tab:var:leaves] presents for each leaf the empirical median and mean of the costs and the theoretical median and mean of the corresponding GP distribution. Let us recall that for a GP distribution with a scale parameter σ and a shape parameter γ, the theoretical median is given by σ(2γ − 1)/γ and the empirical mean by σ/(1 − γ) for γ < 1 and ∞ for γ ≥ 1. First of all, for every leaf, the median is much smaller than the mean suggesting that we are indeed dealing with extreme events. Then, the empirical and theoretical medians are of the same order for each leaf while the empirical and theoretical (when it exits) means are only comparable for the leaves 3 and 5 for which the shape parameter is significantly different from 1. [fig:tree:floods] \| Leaf \| Shape parameter \| Empirical Median \| Theoretical Median \| Empirical Mean \| Theoretical Mean \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 1 \| 0.91 \| 207 044 \| 104 793 \| 711 740 \| 1 366 968 \| \| 2 \| 0.97 \| 364 513 \| 276 879 \| 1 325 493 \| 13 168 585 \| \| 3 \| 0.24 \| 900 945 \| 1 045 203 \| 1 929 512 \| 1 938 357 \| \| 4 \| 0.99 \| 578 437 \| 529 377 \| 3 868 125 \| 807 158 756 \| \| 5 \| 0.29 \| 2 974 918 \| 3 339 911 \| 6 086 955 \| 6 245 812 \| \| 6 \| 1.00 \| 9 980 686 \| 9 152 030 \| 37 335 807 \| ∞ \| [tab:var:leaves] Conclusion ========== In this paper, we investigated the consistency of Generalized Pareto regression trees, applied to extreme value regression. The results that we derive are non-asymptotic, and allow to justify the consistency of the pruning methodology used to select a proper subtree. Let us note that the conditions under which our results hold are relatively weak, in the sense that they hold even if the tail index γ is arbitrary close to zero (the special case γ = 0 is excluded) or large. Moreover, no regularity assumptions on the target parameters is required, due to the flexibility of the regression tree procedure. Through the simulation study and the real data analysis, we investigated the practical performances of the methodology. The regression tree approach can be applied in various situations, and still provides interpretability of the results. On the other hand, regression trees may be unstable, since quite sensitive to some changes on the data that have been used to fit them. Hence, this work is a first step into the direction of studying other relied methodologies, like random forests (see for example ) in this field of extreme value regression. Proofs ====== In this Section, we present in details the proof of the results presented throughout the paper. Concentration inequalities required to obtain the results are presented in Section [seccons]. These inequalities are used to obtain deviation bounds in Section [secdev], which are the key ingredients of the proof of Theorem [thm:deviation:bounds] (Section [sec1]), Corollary [corexp] (Section [sec2]), and Theorem [thselection] (Section [sec3]). Section [seccovering] shows some results on covering numbers that are required to control the complexity of some classes of functions considered in the proofs. Some technical lemmas are gathered in Section [sec4]. Concentration inequalities -------------------------- The proofs of the main results are mostly based on concentration inequalities. The following inequality was proved initially by Talagrand, see also. [talagrand] Let $(\mathbf V\_i)\_{1\leq i \leq n}$ denote i.i.d. replications of a random vector $\mathbf V,$ and let (ɛi)1 ≤ i ≤ n denote a vector of i.i.d. Rademacher variables (that is, P(ɛi = − 1) = P(ɛi = 1) = 1/2) independent from $(\mathbf V\_i)\_{1\leq i \leq n}.$ Let F be a pointwise measurable class of functions bounded by a finite constant M0. Then, for all t, $$\begin{aligned} \mathbb{P}\left(\sup\_{\varphi \in \mathfrak{F}}\left\\|\sum\_{i=1}^n \{\varphi(\mathbf V\_i)-\mathbb {\mathbb{E}}[\varphi(\mathbf V)]\}\right\\|\_{\infty}>A\_1\left\{\mathbb \mathbb{E}\left[\sup\_{\varphi\in \mathfrak{F}}\left\\|\sum\_{i=1}^n \varphi(\mathbf V\_i)\varepsilon\_i\right\\|\_{\infty}\right]+t\right\}\right) \\ \leq 2\left\{\exp\left(-\frac{A\_2t^2}{nv\_{\mathfrak{F}}}\right)+\exp\left(-\frac{A\_2t}{M\_0}\right)\right\},\end{aligned}$$ with $v\_{\mathfrak{F}}=\sup\_{\varphi \in \mathfrak{F}}\mathrm{Var}(\\|\varphi(\mathbf V)\\|\_{\infty}),$ and where A1 and A2 are universal constants. The difficulty in using Proposition [talagrand] comes from the need to control the symmetrized quantity $\mathbb{E}\left[\sup\_{\varphi\in \mathfrak{F}}\left\\|\sum\_{i=1}^n \varphi(\mathbf V\_i)\varepsilon\_i\right\\|\right].$ Proposition [emason] is due to Einmahl and Mason and allows this control via some assumptions on the considered class of functions F. We first need to introduce some notations regarding covering numbers of a class of functions. More details can be found for example in Chapter 2.6 of. Let us consider a class of functions F with envelope Φ (which means that for (almost) all v, f ∈ F, ∣f(v)∣ ≤ Φ(v)). Then, for any probability measure Q, introduce N(ɛ, F, Q) the minimum number of L2(Q) balls of radius ɛ to cover the class F. Then, define NΦ(ɛ, F) = supQ : Q(Φ2) < ∞N(ɛ(Q(Φ2)1/2), F, Q). [emason] Let F be a point-wise measurable class of functions bounded by M0 with envelope Φ such that, for some constants A3, α ≥ 1, and $0\leq \sqrt{v} \leq M\_0,$ we have 1. NΦ(ɛ, F) ≤ A3ɛ− α, for 0 < ɛ < 1, 2. $\sup\_{\varphi \in \mathfrak{F}}\mathbb{E}\left[\varphi(\mathbf V)^2\right]\leq v,$ 3. $M\_0\leq \frac{1}{4\alpha^{1/2}}\sqrt{nv/\log(A\_4M\_0/\sqrt{v}) },$ with A4 = max(e, A31/α). Then, for some absolute constant A5, $$\mathbb{E}\left[\sup\_{\varphi\in \mathfrak{F}}\left\\|\sum\_{i=1}^n \varphi(\mathbf V\_i)\varepsilon\_i\right\\|\right]\leq A\_5\sqrt{\alpha n v \log(A\_4M\_0/\sqrt{v})}.$$ Deviation results ----------------- We first introduce some notations that will be used throughout Sections [secdev] to [seccovering]. In the following, fθ is a function indexed by θ = (σ, γ)τ denoting either ϕ( ⋅ , θ) or gθ = ∂σϕ( ⋅ , θ), or hθ = ∂γϕ( ⋅ , θ). Let us note that the functions y ↦ gθ(y − u) and y ↦ hθ(y − u) are uniformly bounded (eventually up to some multiplication by a constant) by Φ(y) = log(1 + wy), where w = γmax/σmin (see Assumption [aphi]). On the other hand, y ↦ ϕ(y − u, θ) is bounded by logσn + Φ(y) = O(log(kn)) + Φ(y). We consider in the following a class of functions F defined as $$\mathfrak F =\left\{y \mapsto f\_{ \theta}(y-u)\mathbf{1}\_{y\geq u}\mathbf{1}\_{\mathbf{x}\in \mathcal{T}\_\ell}, \; \theta \in \Theta,\; u\in [u\_{\min};u\_{\max}],\ell=1,...,K\right\}. \label{eq\_frakF}$$ Next, recall that for ℓ = 1, …, K $$L\_n^{\ell}(\theta,u)=\frac{1}{k\_n}\sum\_{i=1}^n \phi(Y\_i-u,\theta)\mathbf{1}\_{Y\_i>u}\mathbf{1}\_{\mathbf{X}\_i\in \mathcal{T}\_{\ell}},$$ is the (normalized) GP log-likelihood in the leaf ℓ of the tree $T(u)=(\mathcal T\_\ell)\_{\ell=1,\ldots,K}$. The key results behind Theorems [thm:deviation:bounds] and [thselection] relies on studying the deviation of the processes $$\begin{aligned} \nonumber \mathcal{W}\_0^{\ell}(\theta,u) &=& L\_n^{\ell}(\theta,u)-L^{\ell}(\theta,u), \\ \mathcal{W}\_1^\ell (\theta,u) &=& \nabla\_\theta L\_n^\ell (\theta,u)-\nabla\_\theta L^\ell (\theta,u), \nonumber\end{aligned}$$ indexed by θ, u and ℓ. We study these deviations by decomposing Wiℓ(θ, u), for i = 0, 1, (which is a sum of i.i.d. observations) into two sums: * the first one gathers observations smaller than some bound (more precisely, such that Φ(Yi) ≤ Mn), which is considered in Theorem [th1]. Since these observations are bounded (even if this bound in fact depends on n and can tend to infinity when n grows), we can apply a concentration inequality such as the one of Section [seccons]; * in the second one, we consider the observations larger than this bound, and control them through the fact that the function Φ is assumed to have a finite exponential moment (see Assumption [aphi]). Corollary [corthetal], which provides deviation bounds for estimation errors in the leaves of the tree, is then a direct consequence. [th1] Let Mn = βlogkn, with β > 0 and $$\underline{\mathcal Z}(M\_n) = \sup\_{f\in \mathfrak{F}} \left\| \frac{1}{k\_n} \sum\_{i=1}^n \left(f(Y\_i) \mathbf{1}\_{\Phi(Y\_i)\leq M\_n} - {\mathbb{E}}\left[ f(Y\_i) \mathbf{1}\_{\Phi(Y\_i) \leq M\_n}\right]\right) \right\|$$ Then, under Assumptions [au], [arate] and [amoche], $$\label{Zmoins} \mathbb{P}\left(\underline{\mathcal Z}(M\_n) \geq t\right) \leq 2\left(\exp\left(- \frac{{ C\_1} k\_n t^2}{M\_n^2} \right) + \exp\left(-\frac{{ C\_2} k\_n t}{M\_n} \right)\right),$$ for $t \geq {\mathfrak c\_1} (\log k\_n)^{1/2} k\_n^{-1/2}$. Let us stress that $\sup\_{f \in \mathfrak F} \\|f(y)\mathbf{1}\_{\Phi(y)\leq M\_n}\\|\_{\infty} \leq M\_n.$ From Proposition [talagrand], $$\begin{aligned} \label{Zint} &&\mathbb{P} \left(\underline{\mathcal{Z}}(M\_n) \geq A\_1 \left\{\mathbb{E} \left[\sup\_{f\in \mathfrak{F}}\frac{1}{k\_n}\left\|\sum\_{i=1}^n f(Y\_i)\mathbf{1}\_{\Phi(Y\_i)\leq M\_n}\varepsilon\_i\right\|\right] +t\right\} \right) \\ &\leq& 2 \left( \exp\left(- \frac{A\_2 k\_n^2 t^2}{nv\_{\mathfrak F}} \right) + \exp\left(-\frac{A\_2 k\_n t}{M\_n} \right)\right) \,. \notag\end{aligned}$$ From Lemma [lemmavphi], vF ≤ Mn2knn− 1, which shows that the first exponential term on the right-hand side of ([Zint]) is smaller than $$\label{t1} \exp\left(- \frac{A\_2 k\_n t^2}{M\_n^2} \right).$$ We can now apply Proposition [emason] (combined with Lemma [lem:cov:numbers]) to this class of functions with v = Mn2knn− 1 and M0 = Mn. Hence, $$\mathbb{E} \left[\sup\_{f\in \mathfrak{F}}\frac{1}{k\_n}\left\|\sum\_{i=1}^n f(Y\_i)\mathbf{1}\_{\Phi(Y\_i)\leq M\_n} \varepsilon\_i\right\|\right] \leq \frac{{ A\_6}}{k\_n}\sqrt{nv \mathfrak{s}\_n}={ A\_6} \frac{ \mathfrak{s}^{1/2}\_n}{k\_n^{1/2}} \;,$$ where Aʹ6 > 0 and sn = log(σnαK4(d + 1)(d + 2)n/kn) (α > 0 being defined in Lemma [lem:cov:numbers]). From Assumption [arate], we see that sn = O(log(kn)) (let us recall that K is necessarily less than n). Whence, if ${ \mathfrak c\_1}= 2A\_1{A'\_6}$, for $t\geq {\mathfrak c\_1} \left\{\log\left(k\_n\right)\right\}^{1/2} k\_n^{-1/2}$, $$\mathbb{P}\left(\underline{\mathcal Z}(M\_n) \geq t\right) \leq \mathbb{P} \left(\underline{\mathcal Z}(M\_n) \geq A\_1 \left\{\mathbb{E} \left[\sup\_{f\in \mathfrak{F}}\frac{1}{k\_n}\left\|\sum\_{i=1}^n f(Y\_i)\mathbf{1}\_{\Phi(Y\_i)\leq M\_n}\varepsilon\_i\right\|\right] + \frac{t}{2A\_1}\right\} \right) \;.$$ Equation ([Zmoins]) follows from ([Zint]) and ([t1]) with C1 = A2A1− 2/4 and C2 = A2A1− 1/2. [th2] Define $$\overline{\mathcal Z}(M\_n)=\sup\_{f\in \mathfrak{F}} \left\|\frac{1}{k\_n} \sum\_{i=1}^n \left( f(Y\_i )\mathbf{1}\_{\Phi(Y\_i) > M\_n}\right)- {\mathbb{E}}\left[ f(Y\_i)\mathbf{1}\_{\Phi(Y\_i) > M\_n}\right] \right\|.$$ Then, under Assumptions [au], [arate] and [aphi], for Mn = βlogkn = βa2logn and βa2 ≥ 10/ρ0, and $t\geq {\mathfrak c\_2} k\_n^{-1/2},$ $$\label{res:thm2} \mathbb{P}\left(\overline{\mathcal Z}(M\_n) \geq t\right) \leq \frac{{ C\_{3}}}{k\_n^{5/2} t^3}.$$ Let βʹ = βa2. $\overline{\mathcal Z}(M\_n)$ is upper-bounded by $$\frac{1}{k\_n} \sum\_{i=1}^n \left\{\Phi(Y\_i) \mathbf{1}\_{\Phi(Y\_i) \geq M\_n}\mathbf{1}\_{Y\_i\geq u\_{\min}} + \mathbb{E} \left[ \Phi(Y) \mathbf{1}\_{\Phi(Y) \geq M\_n}\mathbf{1}\_{Y\geq u\_{\min}}\right]\right\} \,.$$ A bound for E1, n = E[Φ(Y)1Φ(Y) ≥ Mn1Y ≥ umin] is obtained from Lemma [lemmamoments], and nE1, n/kn ≤ e1kn− 1/2 if βʹ ≥ 2/ρ0. Next, from Markov inequality, $$\begin{aligned} t^3\mathbb{P}\left(\frac{1}{k\_n} \sum\_{i=1}^n \Phi(Y\_i) \mathbf{1}\_{\Phi(Y\_i) \geq M\_n}\mathbf{1}\_{Y\_i\geq u\_{\min}}\geq t\right)&\leq & \frac{n E\_{3,n}}{k\_n^3}+\frac{n(n-1)E\_{2,n}E\_{1,n}}{k\_n^3}\\ &&+\frac{n(n-1)(n-2)E\_{1,n}^3}{k\_n^3}.\end{aligned}$$ From Lemma [lemmamoments], we get $$\begin{aligned} \frac{n E\_{3,n}}{k\_n^3} &\leq & \frac{\mathfrak{e}\_3 n^{-(\rho\_0\beta'/4-1/2)}}{k\_n^{5/2}}, \\ \frac{n(n-1)E\_{2,n}E\_{1,n}}{k\_n^3} &\leq & \frac{\mathfrak{e}\_2\mathfrak{e}\_1n^{-(\rho\_0\beta'/2-3/2)}}{k\_n^{5/2}}, \\ \frac{n(n-1)(n-2)E\_{1,n}^3}{k\_n^3} &\leq & \frac{\mathfrak{e}\_1^3n^{-(\rho\_0\beta'/4-5/2)}}{k\_n^{5/2}}.\end{aligned}$$ Each of these terms is bounded by max(e3, e\_2e1, e13)kn− 5/2 for βʹ ≥ 10/ρ0. Thus, for t ≥ 2e1kn− 1/2 and βʹ ≥ 10/ρ0, $$\begin{aligned} \lefteqn{ \mathbb{P}\left(\overline{\mathcal Z}\_n \geq t \right) }\\ &\leq & \mathbb{P} \left(\frac{1}{k\_n} \sum\_{i=1}^n \Phi(Y\_i) \mathbf{1}\_{\Phi(Y\_i) \geq M\_n}\mathbf{1}\_{Y\_i\geq u\_{\min}} \geq \frac{t}{2}\right) + \mathbb{P}\left( \mathbb{E} \left[ \Phi(Y) \mathbf{1}\_{\Phi(Y) \geq M\_n}\mathbf{1}\_{Y\geq u\_{\min}}\right] \geq \frac{t}{2}\right)\\ &\leq & \frac{8\max(\mathfrak{e}\_3,\mathfrak{e\_2}\mathfrak{c}\_1,\mathfrak{e}\_1^3)}{t^3k\_n^{5/2}}\end{aligned}$$ We now apply these results to deduce deviation bounds on the estimators θ̂ℓ in the leaves of the tree. [corthetal] Under the assumptions of Theorem [th1] and [th2] and Assumption [amoche], for $t\geq \mathfrak c\_3 (\log k\_n)^{1/2}k\_n^{-1/2},$ $$\begin{aligned} \mathbb{P}\left(\sup\_{\substack{\ell=1,\ldots, K,\\ { u\_{\min} \leq u \leq u\_{\max}}}} \\|\widehat{\theta}\_\ell(u)-\theta^\\_\ell(u)\\|\_{\infty}\geq t\right)\leq 2\left(\exp\left(- \frac{{ C\_4} k\_n t^2}{\beta^2 (\log k\_n)^{2}} \right) + \exp\left(-\frac{{ C\_5} k\_n t}{\beta \log k\_n} \right)\right)+\frac{{C\_6}}{k\_n^{5/2} t^3}.\end{aligned}$$ For 1 ≤ ℓ ≤ K* and umin ≤ u ≤ umax, write θ = (s, γ)τ and θℓ\(u) = (sℓ\(u), γℓ\(u))τ, and let m*u, ℓ(θ) = ∇θLℓ(θ, u). From a Taylor expansion, $$m\_{u,\ell}(\theta)=\mathbb{E}\left[\left(\begin{array}{cc} \partial\_s g\_{\tilde{s}\_1,\gamma}(Y-u) & \partial\_{\gamma} g\_{s,\tilde{\gamma}\_1}(Y-u) \\ \partial\_s h\_{\tilde{s}\_2,\gamma}(Y-u) & \partial\_{\gamma} h\_{s,\tilde{\gamma}\_2}(Y-u)\end{array}\right)\mathbf{1}\_{\mathbf{X}\in \mathcal T\_\ell}\mathbf{1}\_{Y\geq u}\right](\theta-\theta^\\_\ell(u))^\tau,$$ for some parameters γ̃*j (resp. s̃j) between γ and γℓ\(u) (resp. s* and sℓ\(u)). From Assumption [amoche], we get, for all ℓ = 1, …, K, $$\frac{n}{k\_n}\\|m\_{u,\ell}(\theta)\\|\_{\infty}\geq \mathfrak C\_1\\|\theta-\theta^\\_{\ell}(u)\\|\_{\infty}.$$ Hence, for all ℓ = 1, …, K, $$\mathbb{P}\left(\\|\widehat{\theta}\_\ell(u)-\theta^\\_\ell(u)\\|\_{\infty}\geq t\right)\leq \mathbb{P}\left(\frac{n}{k\_n}\\|m\_{u,\ell}(\widehat{\theta})\\|\_{\infty}\geq \mathfrak C\_1t\right).$$ Since for all ℓ = 1, …, K, ∇θLnℓ(θ̂) = 0, $\mathcal{W}\_1^\ell(\widehat{\theta}(u),u)=-\frac{n}{k\_n}m\_{u,\ell}(\widehat{\theta}).$ Hence, $$\mathbb{P}\left(\sup\_{\substack{\ell=1,\ldots, K, \\{ u\_{\min} \leq u \leq u\_{\max}}}} \\|\widehat{\theta}\_\ell(u)-\theta^\\_l(u)\\|\_{\infty}\geq t\right)\leq \mathbb{P}\left(\sup\_{\substack{\ell=1,\ldots,K,\\{ u\_{\min} \leq u \leq u\_{\max}}}}\\|\mathcal{W}\_1^\ell(\widehat{\theta}(u),u)\\|\_{\infty}\geq \mathfrak C\_1t\right),$$ and the right-hand side is bounded by $$\mathbb{P}\left(\overline{\mathcal Z}(M\_n)\geq \frac{\mathfrak C\_1t}{2}\right)+\mathbb{P}\left(\underline{\mathcal Z}(M\_n)\geq \frac{\mathfrak C\_1t}{2}\right).$$ The result follows from Theorem [th1] and [th2]. Proof of Theorem [thm:deviation:bounds] --------------------------------------- The proof of Theorem [thm:deviation:bounds] then consists in gathering the results on the leaves obtained in Corollary [corthetal]. Let umin ≤ u ≤ umax, ∥T(u) − T\(u∣T)∥22 ≤ ∑ℓ = 1K∥θ̂ℓ(u) − θℓ\(u)∥∞2 ≤ Ksupℓ = 1, ..., K∥θ̂ℓ(u) − θℓ\(u)∥∞2. Hence $$\begin{aligned} \mathbb{P}\left(\sup\_{u\_{\min} \leq u \leq u\_{\max}}\\|T(u)-T^\(u\|T)\\|\_2^2\geq t\right) \leq \mathbb{P}\left(\sup\_{\substack{\ell=1,\ldots,K,\\{ u\_{\min} \leq u \leq u\_{\max}}}}\\|\widehat{\theta}\_\ell(u)-\theta^\\_\ell(u)\\|\_{\infty}\geq t^{1/2}K^{-1/2}\right).\end{aligned}$$ The results follows from Corollary [corthetal], and from the assumption on K* ≤ Kmax = O(kn3). Proof of Corollary [corexp] --------------------------- Write E[supumin ≤ u ≤ umax∥T(u) − T\(u∣T)∥22] = ∫0∞P(supumin ≤ u* ≤ umax∥T(u) − T\(u∣T)∥22 ≥ t)d*t. Let tn = c1K(logkn)kn− 1, then ∫0∞P(supumin ≤ u ≤ umax∥T(u) − T\(u∣T)∥22 ≥ t)d*t ≤ tn + ∫tn∞P(supumin ≤ u ≤ umax∥T(u) − T\(u∣T)∥22 ≥ t)d*t. We now use Theorem [thm:deviation:bounds] to bound the integral on the right-hand side. Since $\int\_0^{\infty}\exp(-a t)dt=\frac{1}{a},$ $\int\_0^{\infty} \exp(-a^{1/2} t^{1/2})dt=\frac 2 a,$ and ∫1∞t− 3/2dt = 2, we get $$\begin{aligned} \mathbb{E}\left[{ \sup\_{u\_{\min} \leq u \leq u\_{\max}}}\\|T(u)-T^\(u\|T)\\|\_2^2\right]&\leq &{ t\_n+ \frac{2K\beta^2 (\log k\_n)^2}{\mathcal C\_1 k\_n }+\frac{4K \beta^2 (\log k\_n)^2}{\mathcal C\_2^2 k\_n } + \frac{2 \mathcal C\_3K}{k\_n^{5/2} }}\\ & \leq & \frac{c\_1 K \log k\_n}{k\_n}+ \frac{2 K \beta^2 (\log k\_n)^2}{\mathcal C\_1 k\_n }+\frac{4K \beta^2(\log k\_n)^2}{\mathcal C\_2^2 k\_n } + \frac{2 \mathcal C\_3K}{k\_n^{5/2} }\\ & \leq & \frac{\mathcal C\_4 K (\log k\_n)^2}{k\_n}. \end{aligned}$$ Proof of Proposition [prp:bias] ------------------------------- Let $\mathbf x$ fixed, then, $$\\|\theta^\(\mathbf x) - \theta\_0(\mathbf x)\\|\_\infty = \\|\sum\_{\ell=1}^{K\_{\max}}\left(\theta^\\_\ell - \theta\_{0}(\mathbf x) \right) \mathbf 1\_{\mathbf x \in \mathcal T\_\ell}\\|\_\infty \leq \sum\_{\ell=1}^{K\_{\max}}\\|\theta^\\_\ell - \theta\_{0}(\mathbf x) \\|\_\infty \mathbf 1\_{\mathbf x \in \mathcal T\_\ell} \,.$$ Now, from Taylor expansion, for ℓ = 1, …, K, conditionally on $\mathbf X \in \mathcal T\_\ell$, $$\begin{aligned} \nabla\_\theta L^\ell(\theta\_{0}(\mathbf x),u) &= & \nabla\_\theta L^\ell(\theta^\\_\ell,u) + \nabla^2\_\theta L^\ell (\widetilde{\theta}\_\ell)(\theta\_{0}(\mathbf x) - \theta^\\_\ell)^\tau \\ &=& 0 + \mathbb{E}\left[\left(\begin{array}{cc} \partial\_\sigma g\_{\tilde{\sigma}\_1,\gamma}(Y-u) & \partial\_{\gamma} g\_{\sigma,\tilde{\gamma}\_1}(Y-u) \\ \partial\_\sigma h\_{\tilde{\sigma}\_2,\gamma}(Y-u) & \partial\_{\gamma} h\_{\sigma,\tilde{\gamma}\_2}(Y-u)\end{array}\right)\mathbf{1}\_{Y\geq u} \mid \mathbf X \in \mathcal T\_\ell \right](\theta\_{0}(\mathbf x) - \theta^\\_\ell)^\tau \\\end{aligned}$$ for some parameters $\tilde \gamma\_j$ (resp. $\tilde \sigma\_j$) between $\gamma\_{0}(\mathbf x)$ and γℓ\ (resp. $\sigma\_{0}(\mathbf x)$ and σℓ\). Thus, under Assumption [amoche], $$\begin{aligned} \\|\theta\_{0}(\mathbf x) - \theta^\\_\ell\\|\_\infty &\leq& \frac{1}{\mathfrak C\_1} \\|\nabla\_\theta L^\ell(\theta\_{0}(\mathbf x),u) \\|\_\infty \\ &\leq& \frac{1}{\mathfrak C\_1}\frac{k\_n}{n} \max\left(\| {\mathbb{E}}\left[g\_{\theta\_{0}(\mathbf x)}(Z) \mid \mathbf X \in \mathcal T\_\ell\right]\|,{\mathbb{E}}\left[h\_{\theta\_{0}(\mathbf x)}(Z) \mid \mathbf X \in \mathcal T\_\ell\right] \right) \,,\end{aligned}$$ where Z is a random variable distributed according to the distribution Fu defined in Section [secevtreg] with $\sigma\_0(\mathbf x) =u\gamma\_0(\mathbf x)$ and with $$\begin{aligned} {\mathbb{E}}\left[g\_{\theta\_{0}(\mathbf x)}(Z) \mid \mathbf X \in \mathcal T\_\ell\right] &=& -\frac{1}{u \gamma\_{0}(\mathbf x)} + \frac{1}{u^2\gamma\_{0}(\mathbf x)} \left(1+\frac{1}{\gamma\_{0}(\mathbf x)} \right) {\mathbb{E}}\left[ \frac{Z}{1+Z/u}\mid \mathbf X \in \mathcal T\_\ell \right] \\ {\mathbb{E}}\left[h\_{\theta\_{0}(\mathbf x)}(Z) \mid \mathbf X \in \mathcal T\_\ell\right] &=&-\frac{1}{\gamma\_{0}(\mathbf x)^2}{\mathbb{E}}\left[ \log(1+Z/u) \mid \mathbf X \in \mathcal T\_\ell \right] \\ &&+ \frac{1}{u\gamma\_{0}(\mathbf x)}\left(1+\frac{1}{\gamma\_{0}(\mathbf x)} \right) {\mathbb{E}}\left[ \frac{Z}{1+Z/u}\mid \mathbf X \in \mathcal T\_\ell\right] \,. \end{aligned}$$ Under Assumption [assum:rv:second], we have $$\overline{F}\_{u}(z) = \left( 1+\frac{z}{u}\right)^{-1/\gamma\_{0}(\mathbf x)} \left \{1 + c\psi(u) \int\_1^{1+z/u} v^{\rho-1} {\mathrm{d}}v + o(\psi(u))\right\} \,.$$ $$\begin{aligned} {\mathbb{E}}\left[ \frac{Z}{1+Z/u}\mid \mathbf X \in \mathcal T\_\ell\right] &=& \int\_0^u \overline{F}\_u \left(\frac{t}{1-t/u} \right) {\mathrm{d}}t\\ &=& \frac{u}{1+1/\gamma\_0(\mathbf x)} \left(1 + \frac{ c\psi(u)}{1+1/\gamma\_{0}(\mathbf x)-\rho} + o(\psi(u)) \right)\\ &\leq& u \left( 1 + c\gamma\_{0}(\mathbf x)\psi(u) + o(\psi(u)) \right)\end{aligned}$$ and then $$\begin{aligned} {\mathbb{E}}\left[ \log(1+Z/u) \mid \mathbf X \in \mathcal T\_\ell \right] &=& \int\_0^u \mathbb{P}\left[Z \geq u({\mathrm{e}}^t-1) \mid \mathbf X \in \mathcal T\_\ell \right] {\mathrm{d}}t \\ &=& \gamma\_{0}(\mathbf x)\left(1 + \frac{c\psi(u)}{1/\gamma\_{0}(\mathbf x)-\rho} + o(\psi(u)) \right)\\ &\leq& \gamma\_{0}(\mathbf x) \left(1+c\gamma\_{0}(\mathbf x)\psi(\mathbf x)(u) + o(\psi(u)) \right) \,.\end{aligned}$$ Consequently, $$\begin{aligned} \|{\mathbb{E}}\left[g\_{\theta\_0(\mathbf x)}(Z) \mid \mathbf X \in \mathcal T\_\ell \right] \| \leq \frac{1}{\gamma\_{\min}}\left(1+\frac{1}{u}\left(1+\frac{1}{\gamma\_{\min}}\right) \right) \left( 1 + c\gamma\_{0}(\mathbf x) \psi(u) + o(\psi(u)) \right)\end{aligned}$$ and $$\begin{aligned} \|{\mathbb{E}}\left[h\_{\theta\_0(\mathbf x)}(Z) \mid \mathbf X = \mathbf x\right]\| \leq \frac{1}{\gamma\_{\min}}\left(1 +\frac{1}{\gamma\_{\min}}+\frac{\gamma\_{\max}}{\gamma\_{\min}}\right) \left( 1 + c\gamma\_0(\mathbf x) \psi(u ) + o(\psi(u)) \right) \,. \end{aligned}$$ Hence, $$\begin{aligned} \\|\theta\_{0}(\mathbf x) - \theta^\\_\ell\\|\_\infty &\leq& \mathfrak C\_2(u)\frac{k\_n}{n} \left( 1 + c\gamma\_{\max} \psi(u) + o(\psi(u)) \right) \,,\end{aligned}$$ where $\mathfrak C\_2(u)=\frac{1}{\mathfrak C\_1}\frac{1}{\gamma\_{\min}}\max\left(1+\frac{1}{u}+\frac{1}{u\gamma\_{\min}},1+\frac{1}{\gamma\_{\min}}+\frac{\gamma\_{\max}}{\gamma\_{\min}} \right)$. Finally, $$\begin{aligned} \\|\theta^\(\mathbf x) - \theta\_0(\mathbf x)\\|\_\infty &\leq& \sum\_{\ell=1}^{K\_{\max}}\\|\theta^\\_\ell - \theta\_{0}(\mathbf x) \\|\_\infty \mathbf 1\_{\mathbf x \in \mathcal T\_\ell}\\ &\leq& \mathfrak C\_2(u)\frac{k\_n}{n} \left( 1 + c\gamma\_{\max} \psi(u) + o(\psi(u)) \right)\sum\_{\ell=1}^{K\_{\max}} \mathbf 1\_{\mathbf x \in \mathcal T\_\ell}\\ &\leq& \mathfrak C\_2(u)\frac{k\_n}{n} \left( 1 + c\gamma\_{\max} \psi(u) + o(\psi(u)) \right) \,.\end{aligned}$$ Proof of Theorem [thselection] ------------------------------ The following lemma will be needed to prove Theorem [thselection]. [pk] Let $\mathfrak D = \inf\_u\inf\_{K < K\_0(u)} \Delta L(T^\(u),T^\\_K(u))$ and u* ∈ [umin, umax] fixed. Suppose that there exists a constant c2 > 0 such that the penalization constant λ satisfies c2{logkn}1/2kn− 1/2 ≤ λ ≤ (D − 2c2{log(kn)}1/2kn− 1/2)kn− 1, then, for K > K0(u), $$\begin{aligned} \mathbb{P}(\widehat{K}(u)=K) &\leq & 2\left(\exp\left(- \frac{ { C\_1} k\_n \lambda^2(K-K\_0(u))^2}{\beta^2 (\log k\_n)^{2}} \right) + \exp\left(-\frac{{ C\_2} k\_n \lambda(K-K\_0(u)))}{\beta \log k\_n} \right)\right) \\ && +\frac{{ C\_3}}{k\_n^{5/2} \lambda^3(K-K\_0(u))^3},\end{aligned}$$ and, for K < K0(u), $$\begin{aligned} \mathbb{P}(\widehat{K}(u)=K) &\leq & 4\exp\left(- \frac{ C\_1 k\_n \{\mathfrak{D}-\lambda(K\_0(u)-K)\}^2}{\beta^2 (\log k\_n)^{2}} \right) \\ &&+4 \exp\left(-\frac{C\_2 k\_n \{\mathfrak{D}-\lambda(K\_0(u)-K)\}}{\beta \log k\_n} \right)\\ && +\frac{2C\_3}{k\_n^{5/2} \{\mathfrak{D}-\lambda(K\_0(u)-K)\}^3}.\end{aligned}$$ Let u ∈ [umin, umax] fixed. If K̂(u) = K, this means that ΔLn(TK(u), TK0(u)(u)) : = Ln(TK, u) − Ln(TK0(u), u) > λ(K − K0(u)). Decompose $$\begin{aligned} \Delta L\_n(T\_K(u),T\_{K\_0}(u))&=&\{L\_n(T\_K,u)-L\_n(T^\\_K,u)\}+\{L\_n(T^\\_K,u)-L\_n(T^\,u)\}\\ &&+\{L\_n(T^\,u)-L\_n(T\_{K\_0(u)},u)\}.\end{aligned}$$ Since Ln(T\, u) − L*n(TK0(u), u) < 0, ΔLn(TK(u), TK0(u)(u)) ≤ {Ln(TK, u) − Ln(TK\, u)} + {L*n(TK\, u) − L*n(T\, u)}. For K* > K0(u), TK\(u) = T\(u), hence, $$\begin{aligned} \mathbb{P}(\widehat{K}(u)=K)&\leq &\mathbb{P}\left(\Delta L\_n(T\_K(u), T^\\_K(u))>\lambda (K-K\_0(u))\right)\\ &\leq &\mathbb{P}\left( \left\|\Delta L\_n(T\_K(u), T^\\_K(u)) - \Delta L( T\_K(u), T^\\_K(u))\right\|>\lambda (K-K\_0(u))\right).\end{aligned}$$ For K* > K0(u), a bound is then obtained from Theorems [th1] and [th2] if λ(K − K0(u)) ≥ c1{log(kn)}1/2kn− 1/2, that is λ ≥ c1{logkn}1/2kn− 1/2. Now, for K < K0(u), $$\begin{aligned} \Delta L\_n(T^\\_K(u),T^\(u)) &\leq& \|\Delta L\_n(T^\\_{K}(u),T^\(u))-\Delta L(T^\\_{K}(u),T^\(u))\|+\Delta L(T^\\_{K}(u),T^\(u))\\ & \leq & \|\Delta L\_n(T^\(u),T^\\_{K}(u))-\Delta L(T^\(u),T^\\_K(u))\| - \mathfrak D(K\_0(u),K).\end{aligned}$$ where $\mathfrak D = \inf\_{K<K\_0(u), u\in[u\_{\min}, u\_{\max}]} \mathfrak D(K\_0(u),K),$ Hence, $$\begin{aligned} \lefteqn{\mathbb{P}(\widehat{K}(u)=K)}\\ &\leq &\mathbb{P}\left(\Delta L\_n(T\_K(u), T^\\_K(u))\geq \frac{\mathfrak D- \lambda(K\_0(u)-K)}{2}\right) \\ && + \mathbb{P}\left(\|\Delta L\_n(T^\(u),T^\\_K(u))-\Delta L(T^\(u),T^\\_K(u))\|\geq \frac{\mathfrak D - \lambda(K\_0(u)-K)}{2}\right) \\ &\leq &\mathbb{P}\left(\left\|\Delta L\_n(T\_K(u), T^\\_K(u))-\Delta L( T\_K(u), T^\\_K(u))\right\|\geq \frac{\mathfrak D - \lambda(K\_0(u)-K)}{2}\right) \\ && + \mathbb{P}\left(\|\Delta L\_n(T^\(u),T^\\_K(u))-\Delta L(T^\(u),T^\\_K(u))\|\geq \frac{\mathfrak D - \lambda(K\_0(u)-K)}{2}\right).\end{aligned}$$ These two probabilities can be bounded using Theorems [th1] and [th2] provided that, for all K* < K0(u), $$\frac{\mathfrak D - \lambda(K\_0(u)-K)}{2} \geq \mathfrak c\_1 \{\log(k\_n)\}^{1/2} k\_n^{-1/2},$$ that is, $$\lambda \leq \mathfrak D - 2{\mathfrak c\_1} \{\log(k\_n)\}^{1/2} k\_n^{-1/2}.$$ We are now ready to prove Theorem [thselection]. Let u ∈ [umin, umax] fixed. $$\begin{aligned} {\mathbb{E}}\left[ \\|\widehat{T}(u)-T^\(u)\\|\_{2}^2\right]&=&\sum\_{K=1}^{K\_{\max}}{\mathbb{E}}\left[\\|T\_K(u)-T^\(u)\\|\_2^2\mathbf{1}\_{\widehat{K}(u)=K}\right]\\ &\leq& {\mathbb{E}}\left[\\|T\_{K\_0(u)}(u)-T^\(u)\\|\_2^2\right] + \sum\_{K=1, K\neq K\_0(u)}^{K\_{\max}}K \mathbb{P}(\widehat{K}(u)=K) \\ && +\sum\_{K=1, K\neq K\_0(u)}^{K\_{\max}}\mathbb{E}\left[\\|T\_K(u)-T^\(u)\\|\_2^2 \mathbf{1}\_{\\|T\_K(u)-T^\(u)\\|\_2^2> K}\mathbf{1}\_{\widehat{K}(u)=K}\right]\\ &\leq & {\mathbb{E}}\left[\\|T\_{K\_0(u)}(u)-T^\(u)\\|\_2^2\right] + \sum\_{K=1}^{K\_0(u)-1}K \mathbb{P}(\widehat{K}(u)=K)\\ && + \sum\_{K=K\_0(u)+1}^{K\_{\max}}K \mathbb{P}(\widehat{K}(u)=K)\\ && +2\sum\_{K=1, K\neq K\_0(u)}^{K\_{\max}}\mathbb{E}\left[\\|T\_K(u)-T^\\_K(u)\\|\_2^2\mathbf{1}\_{\\|T\_K(u)-T^\(u)\\|\_2^2> K}\right]\\ &&+2\sum\_{K=1, K\neq K\_0(u)}^{K\_{\max}}\mathbb{P}(\widehat{K}(u)=K) \\|T^\(u)-T\_K^\(u)\\|\_2^2.\end{aligned}$$ Firstly, from Theorem [thm:deviation:bounds], $$\begin{aligned} \lefteqn{\mathbb{E}\left[\\|T\_K(u)-T\_K^\(u)\\|\_2^2 \mathbf{1}\_{\\|T\_K(u)-T^\(u)\\|\_2^2> K}\right]}\\ & = & K \mathbb{P}\left(\\|T\_K(u)-T\_K^\(u)\\|\_2^2 > K\right) + \int\_K^{\infty} \mathbb{P}\left(\\|T\_K(u)-T\_K^\(u)\\|\_2^2 > t\right) {\mathrm{d}}t\\ &\leq& 2 K\left(1+\frac{\beta^2 (\log k\_n)^2}{\mathcal C\_1 k\_n}\right)\exp\left(-\frac{\mathcal C\_1 k\_n }{ \beta^2 (\log k\_n)^2}\right)\\ && +2K\left( 1+ \frac{2\beta (\log k\_n)}{\mathcal C\_2 k\_n}+\frac{2\beta^2 (\log k\_n)^2}{\mathcal C\_2^2 k\_n^2}\right)\exp\left(-\frac{\mathcal C\_2 k\_n}{\beta (\log k\_n)}\right) + \frac{2\mathcal C\_3 K^{1/2}}{k\_n^{5/2} }\,.\end{aligned}$$ Secondly, recall that $$\\|T^\\_K(u)-T^\(u)\\|^2\_2 = \int \\|\theta^{K\}(\mathbf x)-\theta^{\}(\mathbf x)\\|^2\_{\infty}{\mathrm{d}}\mathbb P(\mathbf x) \leq K\_{\max} \sum\_{\ell=1}^{K\_{\max}}\\|\mu(\mathcal T\_\ell)\theta^{K\}\_\ell-\theta^{\}\_\ell\\|\_\infty^2 \,,$$ where $\mu(\mathcal T\_\ell) = \mathbb{P}(\mathbf X \in \mathcal T\_\ell)$. Following the same idea as in the proof of Proposition [prp:bias], from Taylor’s expansion, under Assumptions [amoche] and [assum:rv:second], $$\begin{aligned} \\|\theta^{K\}\_{\ell} - \theta^\\_\ell\\|\_\infty^2 & \leq & \mathfrak C^2\_2(u)\frac{k\_n^2}{n^2} \left( 1 + c\gamma\_{\max} \psi(u) + o(\psi(u)) \right)^2 \,.\end{aligned}$$ Hence, $$\begin{aligned} \\|T^\\_K(u)-T^\(u)\\|\_2^2 &\leq& \mathfrak C^2\_2(u)\frac{k\_n^2}{n^2} (1 + c\gamma\_{\max} \psi(u) + o(\psi(u)))^2\sum\_{\ell=1}^{K\_{\max}} \mathbf 1\_{\mathbf x \in \mathcal T\_\ell}\\ &\leq& \mathfrak C\_3(u)\frac{k\_n^2}{n^2} \,.\end{aligned}$$ Finally, $$\begin{aligned} {\mathbb{E}}\left[\\|\widehat{T}(u)-T^\(u)\\|\_{2}^2 \right] &\leq & \frac{\mathcal{C}\_5 K\_0(u) (\log k\_n)^2 }{k\_n},\end{aligned}$$ for some constant C5.. Covering numbers ================ [lem:cov:numbers] Following the notations of the proof of Theorem [th1], the class of functions F satisfies $$\mathcal{N}\_{\Phi}(\varepsilon,\mathfrak{F})\leq \frac{\mathfrak C\_4K^{4(d+1)(d+2)}\\|\Phi \\|\_2^{\alpha\_1}\sigma\_{n}^{\alpha}}{\varepsilon^{\alpha}},$$ for some constants $\mathfrak C\_4>0$ and α* > 0 (not depending on n nor K). Let $$\begin{aligned} g\_{\theta}(z) &=& -\frac{1}{\sigma}+\left(\frac{1}{\gamma}+1\right)\frac{\gamma z}{\sigma^2(1+\frac{z\gamma}{\sigma})}, \\ h\_{\theta}(z) &=& -\frac{1}{\gamma^2}\log\left(1+\frac{z\gamma}{\sigma} \right)+\frac{\left(\frac{1}{\gamma}+1\right)z}{\sigma+z\gamma},\end{aligned}$$ for z > 0. For θ and θʹ in S × Γ, we have (from a straightforward Taylor expansion), ∣gθ(y − u) − gθʹ(y − u)∣ ≤ C∣γ − γʹ∣ + Cʹ∣σ − σʹ∣, for some constants C and Cʹ. More precisely, one can take $$\begin{aligned} C &=& \frac{6}{\gamma\_{\min}^2\sigma\_{\min}},\\ C' &=& \frac{1}{\sigma\_{\min}^2}\left(1+3\left\{1+\frac{1}{\gamma\_{\min}}\right\}\right).\end{aligned}$$ Next, observe that ∣gθʹ(y − u) − gθʹ(y − uʹ)∣ ≤ Cʺ∣u − uʹ∣, where Cʺ = 4γmax2/[γminσ3]. Which leads to ∣gθ(y − u) − gθʹ(y − uʹ)∣ ≤ Cgmax(∥θ − θʹ∥∞, ∣u − uʹ∣), for some constant Cg > 0. Similarly, ∣hθ(y − u) − hθʹ(y − u)∣ ≤ C1(4 + log(1 + wy))∣γ − γʹ∣ + C2∣σ − σʹ∣, Next, ∣hθʹ(y − u) − hθʹ(y − uʹ)∣ ≤ C7∣u − uʹ∣, where C7 = 5/(γminσmin), leading to, for some Ch > 0, ∣hθ(y − u) − hθʹ(y − uʹ)∣ ≤ Chmax(∥θ − θʹ∥∞, ∣u − uʹ∣). On the other hand, $$\|\phi(y-u,\theta)-\phi(y-u,\theta')\|\leq\frac{1}{\gamma\_{\min}^2}(2+\log(1+wy))\|\gamma-\gamma'\|+ \frac{3}{\gamma\_{\min}\sigma\_{\min}}\|\sigma-\sigma'\|,$$ and $$\|\phi(y-u,\theta')-\phi(y-u',\theta')\|\leq \frac{1}{\sigma\_{\min}}\|u-u'\|.$$ Define F1 = {gθ( ⋅ − u) : θ ∈ S × Γ, u ∈ [umin, umax]}, F2 = {hθ( ⋅ − u) : θ ∈ S × Γ, u ∈ [umin, umax]}, and F3 = {ϕ( ⋅ − u, θ) : θ ∈ S × Γ, u ∈ [umin, umax]}. From Example 19.7 in, we get, for i = 1, ..., 3, N(ɛ, Fi) ≤ Fi∥Φ∥2α1σnα1ɛ− α1, for some α > 0 and constants Fi. On the other hand, let $$\mathfrak{F}\_4=\left\{\mathbf x \mapsto \mathbf{1}\_{\mathbf x \in \mathcal T\_\ell} \colon \ell =1,\ldots,K \right\},$$ and $$\mathfrak{F}\_5=\left\{y \mapsto \mathbf{1}\_{y>u} \colon u \in \mathcal U \right \}.$$ From Lemma 4 in, we have N(ɛ, F4) ≤ mkKα2ɛ− α2, where α2 = 4(d + 1)(d + 2), and where k is the number of discrete components taking at most m modalities. On the other hand, from Example 19.6 in, N(ɛ, F5) ≤ 2ɛ− 2. From Lemma A.1 in, we get, for i = 1, …, 3, $$N(\varepsilon,\mathfrak{F}\_i\mathfrak{F}\_4\mathfrak{F}\_5)\leq \frac{4 m^kK^{\alpha\_2}\max(C\_g,C\_h)\\|\Phi \\|\_2^{\alpha\_1}\sigma\_{n}^{\alpha\_1}}{\varepsilon^{\alpha\_1+\alpha\_2+\alpha\_3}}.$$ Multiplying FiF4F5 by a single indicator function 1Φ(Yi) ≤ Mn does not change the covering number, and the result follows. Technical Lemmas ================ [lemmavphi] With vF defined in Proposition [talagrand], $$v\_{\mathfrak{F}}\leq \frac{M\_n^2k\_n}{n}.$$ We have $$\begin{aligned} v\_{\mathfrak{F}} &\leq& \mathbb{E}\left[\Phi(Y)^2 \mathbf{1}\_{Y\geq u\_{\min}}\mathbf{1}\_{\Phi(Y)\leq M\_n}\right] \\ &\leq & M\_n^2 \mathbb{P}(Y\geq u\_{\min})=\frac{M\_n^2k\_n}{n}.\end{aligned}$$ [lemmamoments] Define, for j = 1, 2, 3, Ej, n = E[Φ(Y)j1Φ(Y) ≥ Mn1Y ≥ umin]. Under the assumptions of Theorem [th2], $$E\_{j,n}\leq \frac{\mathfrak{e}\_j k\_n^{1/2}}{n^{1/2}n^{\rho\_0\beta a\_2/4}}.$$ Applying twice Cauchy-Schwarz inequality leads to $$E\_{j,n}\leq \mathbb{P}(Y\geq u\_{\min})^{1/2}{\mathbb{E}}[\Phi(Y)^{2j}\mathbf{1}\_{\Phi(Y)\geq M\_n}]^{1/2}\leq \frac{k\_n^{1/2}}{n^{1/2}}{\mathbb{E}}[\Phi(Y)^{4j}]^{1/4}\mathbb{P}(\Phi(Y)\geq M\_n)^{1/4}.$$ Next, from Chernoff inequality, $$\nonumber \mathbb{P}(\Phi(Y)\geq M\_n)\leq \exp(-\rho\_0 M\_n){\mathbb{E}}[\exp(\rho\_0 \Phi(Y))]\leq \frac{m\_{\rho\_0}}{n^{\rho\_0\beta a\_2}}.$$ Generalized Pareto Regression Trees for extreme events analysis =============================================================== 1 Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, LPSM, 4 place Jussieu, F-75005 Paris, France, 2 Mission Risques Naturels, 1 rue Jules Lefebvre 75009 Paris, France E-mails : [email protected], [email protected], [email protected], olivier.lopez@sorbonne- universite.fr In this paper, we provide finite sample results to assess the consistency of Generalized Pareto regression trees, as tools to perform extreme value regression. The results that we provide are obtained from concentration inequalities, and are valid for a finite sample size, taking into account a misspecification bias that arises from the use of a “Peaks over Threshold” approach. The properties that we derive also legitimate the pruning strategies (i.e. the model selection rules) used to select a proper tree that achieves compromise between bias and variance. The methodology is illustrated through a simulation study, and a real data application in insurance against natural disasters. Extreme value theory; Regression trees; Concentration Inequalities; Generalized Pareto Distribution. Introduction ============ Extreme value theory (EVT) is the branch of statistics which has been developed and broadly used to handle extreme events, such as extreme floods, heat waves episodes or extreme financial losses. One of the key results behind the success of this approach was proved by Balkema and de Haan in : they show that the tail of the distribution of a series of observations can be approximated by a parametric family of distributions, namely Generalized Pareto (GP) distributions. This property allows the statistician to find information from the largest observations of a random sample to extrapolate the tail. This yields the so-called Peaks over Threshold (PoT) method introduced in which consists in fitting a GP distribution to the excesses above some (high) suitably chosen threshold. In a regression framework, the parameters of this GP distribution depend on covariates reflecting the fact that different values of these covariates may result in a different tail behavior of the response variable. In this paper, we study the use of regression trees to perform GP regression on the excesses. This ensemble method, introduced by, determines clusters of similar tail behaviors depending on the value of the covariates, based on a recursive partition of the sample and simple model selection rules. In the present work, we provide theoretical results and empirical evidence on the consistency of such a procedure and of these selection rules. The result we provide are based on concentration inequalities, in order to hold for finite sample sizes. The main difficulty stands in the misspecification of the model and on handling the fact that the distributions are heavy tailed. Tail regression is a challenging task. Several papers have been interested in extreme quantile regression, to name a few, in 2005, Chernozhukov and, in 2012, Wang et al. derive extreme quantile estimators assuming a linear form for the conditional quantile. In 2019, Gardes and Stupfler and Velthoen et al. use conditional intermediate-level quantiles to extrapolate above the threshold and deduce estimators for extreme conditional quantiles. Another approach is to model the parameters of the GP distribution of functions of the covariates e.g. as local polynomials or as generalized additive models. Very recently, in 2021, Velthoen et al. proposed to a gradient boosting procedure to estimate conditional GP distribution. Let us note that the nonparametric approaches rely on regularity assumptions on the way the tail of the distribution evolves with the covariates (which are required to be continuous through the use of kernel smoothing). A nice feature of the regression tree approach we consider in the present paper is its ability to handle several covariates which components may be either discrete or continuous. Moreover, this method is adapted to situations where the tail behavior is supposed to be significantly different depending on the characteristics, as for example it is the case in an application to cyber-insurance considered in a former paper, see. Regression trees, introduced by Breiman along with the CART algorithm (for Clustering And Regression Trees), are flexible tools to perform a regression and clustering task simultaneously. They have been used in various fields, including industry, geology (see e.g. ), ecology (see e.g. ), claim reserving in insurance. Through the iterative splitting algorithm used in CART, nonlinearities are introduced in the way the distribution is modeled, while furnishing an intelligible interpretation of the final classification of response variables. The splitting criterion—used to iteratively separate observations into clusters of similar behaviors—depends on the type of problems one is considering. While the standard CART algorithm relies on mean-squared criterion to perform mean-regression, alternative loss functions have been considered as in for quantile regression, or in who used a log-likelihood based loss. Loh provide detailed descriptions of regression trees procedures and a review of their variants. In this paper, building on the Balkema and de Haan result, we use a GP log-likelihood loss, as in, to perform extreme value regression. The rest of the paper is organized as follows. In Section [secmethods], we introduce notations and describe the GP regression tree algorithm. Section [secmain] lists the main results of this paper, that is deviation bounds for the regression tree estimator for finite sample size, and consistency of the “pruning” (that is model selection) strategy. Empirical results are gathered in Section [secsimu], which provides a simulation study, and a real data analysis in natural disaster insurance. Detailed proofs of the technical results are shown in the Appendix. Regression trees for extreme value analysis =========================================== This section describes the estimation method (GP regression trees) that is considered in the paper. Some classical results in EVT are given in Section [secevtreg] to motivate the GP approximation. Regression trees adapted to this context are described in Section [secGPDtrees]. A short discussion on the advantage of this technique compared to competing approaches is developed in Section [subsec:compete]. Extreme value theory and regression ----------------------------------- Let us consider independent and identically distributed observations Y1, Y2, … with an unknown survival function $\overline{F}$ (that is $\overline{F}(y)=P(Y\_1>y)$). A natural way to define extreme events is to consider the values of Yi which have exceeded some high threshold u. The excesses above u are then defined as the variables Yi − u given that Yi > u. The asymptotic behavior of extreme events is characterized by the distribution of the excesses which is given by $$\overline{F}\_u(z)= P[Y\_1 - u > z \mid Y\_1> u] = \frac{\overline F(u+z)}{\overline F(u)}, \ z>0 \,.$$ In 1975, Pickands showed that, if $\overline F$ satisfies the following property $$\label{eq:rv} \lim\_{t \to \infty} \frac{\overline{F}(ty)}{\overline{F}(y)} = y^{-1/\gamma\_0}, \, \forall y >0,$$ with γ0 > 0, then $$\label{eqplus} \lim\_{u \to \infty} \sup\_{z>0} \|\overline F\_u(z) - \overline{H}\_{\sigma\_{0u},\gamma\_0}(z) \| = 0$$ for some σ0u > 0 and $\overline{H}\_{\sigma\_{0u},\gamma\_0}$ necessarily belongs to the Generalized Pareto (GP) distributions family which distribution function is of the form $$\overline{H}\_{\sigma\_{0u},\gamma\_0}(z) = \left(1+ \gamma\_0 \frac{z}{\sigma\_{0u}}\right)^{-1/\gamma\_0}, \ z>0,$$ where σ0u > 0 is a scale parameter and γ0 > 0 is a shape parameter, which reflects the heaviness of the tail distribution. Especially, if γ0 ∈ ]0; 1[, the expectation of Y1 is finite whereas if γ0 ≥ 1 the expectation of Y1 is infinite. More details on these results can be found in e.g.. In practice, the so-called Peaks over Threshold (PoT) method is widely used, see. It consists in choosing a high threshold u and fitting a GP distribution on the excesses above that threshold u. The estimation of the parameters σ0u and γ0 may be done by maximizing the GP likelihood. The choice of the threshold u can be understood as a compromise between bias and variance: the smaller the threshold, the less valid the asymptotic approximation, leading to bias; on the other hand, a too high threshold will generate few excesses to fit the model, leading to high variance. The existing methods are mostly graphical, up to our knowledge, no automatic data-driven selection procedure is available. In the present paper, we consider a regression framework, that is that our goal to know the impact of some random covariates X on the tail of the distribution of a response variable Y. The previous convergence results hold, but for quantities σ0u, γ0 and u that may depend on X. More precisely, this means that, if we assume that γ0(x) > 0 for all x (which is the assumption that we will make throughout this paper), then ([eq:rv]) becomes $$\label{eq:rv2} \lim\_{t \to \infty} \frac{\overline{F}(ty\|\mathbf{x})}{\overline{F}(y\|\mathbf{x})} = y^{-1/\gamma\_0(\mathbf{x})}, \, \forall y >0,$$ where $\overline F(y\|\mathbf{x})=\mathbb{P}(Y\geq y \|\mathbf{X}=\mathbf{x}),$ see and references therein, and ([eqplus]) becomes $$\label{eqplus2} \lim\_{u(\mathbf{x}) \to \infty} \sup\_{z>0} \|\overline F\_{u(\mathbf{x})}(z \mid \mathbf{x}) - \overline{H}\_{\sigma\_{0u(\mathbf x)}(\mathbf{x}),\gamma\_0 (\mathbf x)}(z) \| = 0.$$ where $\overline F\_{u(\mathbf{x})}(z \mid \mathbf{x})=P[Y-u(\mathbf{x}) > z \mid Y > u(\mathbf{x}),\mathbf{X}=\mathbf{x}].$ Suppose that we observe (Yi, Xi)1 ≤ i ≤ n a sample of (Y, X), where X belongs to a compact set X. Following the PoT approach, the estimation of the function γ0(X) and σ0(X) = σ0u(X) can typically be done by fitting a regression model on the data points (Yi, Xi) such that Yi exceeds a proper threshold u(Xi). More precisely, let us define $$\begin{aligned} \theta^\(\mathbf x)&=&\arg \max\_{ \theta\in \Theta} \mathbb {\mathbb{E}}[\phi(Y-u(\mathbf{X}),\theta)\|\mathbf{X} =\mathbf{x},Y\geq u(\mathbf{x})] \nonumber \\ &=& \arg \max\_{ \theta\in \Theta}{\mathbb{E}}[\phi(Y-u(\mathbf{X}),\theta)\mathbf{1}\_{Y\geq u(\mathbf{X})}\|\mathbf{X}=\mathbf{x}], \label{m\_est} \end{aligned}$$ where θ* = (σ, γ)τ (where aτ	arxiv_0000544
Topological Twists of Supersymmetric Algebras of Observables ============================================================ We explain how to perform topological twisting of supersymmetric field theories in the language of factorization algebras. Namely, given a supersymmetric factorization algebra with a choice of a topological supercharge we construct an algebra over the operad of little disks. We also explain the role of the twisting homomorphism allowing us to construct an algebra over the operad of framed little disks. Finally, we give a complete classification of topological supercharges and twisting homomorphisms in dimensions 1 through 10. Introduction ============ Quantum field theories contain a large amount of information, part of which has been formalized mathematically. For instance, given a quantum field theory on a manifold M, we may consider the set of observables $\obs(U)$ supported on an open subset $U\sub M$. Given a disjoint collection of open subsets $U\_1,\dots, U\_n\sub V\sub M$ we may combine observables supported on each open set to get an operator product map $\obs(U\_1)\otimes \dots \otimes \obs(U\_n)\rightarrow \obs(V)$. In this way, Costello and Gwilliam gave a formalization of the structure of observables in perturbative quantum field theories as a factorization algebra, see. Topological field theories do not depend on the precise shape of the manifold, so we expect that the maps $\obs(U)\rightarrow \obs(V)$ are equivalences for an embedding of open subsets $U\sub V$ which is a homotopy equivalence. In other words, observables in topological field theories are described by locally-constant factorization algebras. According to, in the case where $M=\RR^n$ locally-constant factorization algebras are the same as $\bb E\_n$-algebras, i.e. algebras over the operad of little n-disks. Most quantum field theories are not topological, since they explicitly depend on a metric on M. To obtain a topological quantum field theory from a more general quantum field theory, and as a first example, to obtain a topological field theoretic description for Donaldson invariants of 4-manifolds, Witten introduced the idea of a topological twisting. We may start with a non-topological quantum field theory and consider its deformations to a topological quantum field theory (note that local constancy is an open condition). Deformations of factorization algebras are parametrized by their derivations of degree 1 and a large source of such derivations is given by quantum field theories which in addition carry an action of a super Poincaré algebra. More explicitly, consider a translation-invariant factorization algebra $\obs$ on $\RR^n$ together with a degree 1 derivation Q such that the generators of translations are Q-exact. Then we expect that the twisted factorization algebra $\obs^Q$, i.e. $\obs$ equipped with the differential $\d + Q$, corresponds to a topological field theory. In particular, one should be able to extract an $\bb E\_n$-algebra from it. In this paper we will formalize this idea, explaining in what sense we can recover $\bb E\_n$-algebras and more refined structures from topologically twisted factorization algebras. We begin with an explanatory outline of our results, followed by a more detailed summary in which we will make more precise statements. * We can construct $\bb E\_n$-algebras from topologically twisted factorization algebras subject to a single condition: that the map $\obs(B\_r(0)) \to \obs(B\_R(0))$ on concentric balls of radii R > r is an equivalence. * This condition is automatically satisfied in many natural examples, including those coming from superconformal field theories. * In many of these natural examples these $\bb E\_n$-algebras are additionally compatible with the action of $\SO(n)$. This means the topological field theories can be defined on any oriented n-manifold. We do not know whether the twisted factorization algebra $\obs^Q$ is locally-constant, so our arguments will not use Lurie’s result comparing locally-constant factorization algebras and $\bb E\_n$-algebras. Instead we will equip the local observables of the topologically twisted factorization algebra directly with an action of a specific model for the $\bb E\_n$-operad. In addition to understanding twisted factorization algebras abstractly we will give a complete classification of the possible twists coming from super Poincaré algebra actions in dimensions 1 to 10. Topological twists ------------------ Let us summarize our results in more precise terms. Suppose $\obs$ is a translation-invariant factorization algebra on $\RR^n$ where translations are homotopically trivial (we say $\obs$ is a de Rham translation-invariant factorization algebra: we define this precisely in Definition [dRtranslationinvariancedef]). There is a coloured operad $\disk\_n^{\col}$ with the set of colours $\RR\_{>0}$ which is defined in the same way as the operad of little n-disks, but where the colours correspond to the radii of the disks. Let $B\_r(0)\sub \RR^n$ be the standard open disk of radius r centered at the origin. [See Theorem [dRtranslationinvariancethm]] Let $\obs$ be a de Rham translation-invariant factorization algebra on $\RR^n$. Then the collection $\{\obs(B\_r(0))\}\_{r>0}$ becomes an algebra over $\disk\_n^{\col}$. It is then natural to ask when a $\disk\_n^{\col}$-algebra {A(r)}r > 0 is an $\bb E\_n$-algebra, i.e. when it is possible to forget the parameterization by radii of disks. The unary operations in $\disk\_n^{\col}$ give maps A(r) → A(R) for R ≥ r. Moreover, we have a natural forgetful functor from $\bb E\_n$-algebras to $\disk\_n^{\col}$-algebras where the family {A(r)} is constant. [See Theorem [Entheorem]] The forgetful functor of ∞-categories ${\mathcal{A}\mathrm{lg}}\_{\bb E\_n}\rightarrow {\mathcal{A}\mathrm{lg}}\_{\disk\_n^{\col}}$ is fully faithful. Its essential image is given by those families {A(r)} where the map A(r) → A(R) for R ≥ r is an equivalence. Supersymmetric field theories provide a natural family of examples of de Rham translation-invariant factorization algebras. We review the definitions of supersymmetry algebras and supersymmetric field theories in Section [twistsection]. In particular, a supertranslation algebra $\mf{A}$ is a super Lie algebra with even part $V=\CC^n$, the Lie algebra of translations. The group of R-symmetries is a subgroup of the group of automorphisms of $\mf{A}$ which preserve the even part. A topological supercharge is a square-zero odd element $Q\in\mf{A}$ such that the bracket [Q, − ] is surjective onto the even part of $\mf{A}$. We then have the following statement. [See Proposition [topologicallytwistedtheoriesaredRtranslationinvtprop]] Let $\obs$ be an $\mf{A}$-supersymmetric factorization algebra on $\RR^n$ and $Q\in\mf{A}$ a topological supercharge with an abelian section. Moreover, suppose $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ is a quasi-isomorphism for R ≥ r. Then the collection $\{\obs^Q(B\_r(0))\}$ forms an $\bb E\_n$-algebra. In particular this justifies the term “topological supercharge”: by twisting a supersymmetric factorization algebra by a topological supercharge we produce a topological field theory. Note that in the previous claim we had to choose an abelian section for Q, i.e. an abelian subalgebra of $\mf{A}\_{\mr{odd}}$ which maps under [Q, − ] isomorphically onto the even part. We show in Proposition [commutingpotentialprop] that in all examples of interest such a choice can be made. Let us also remark that the factorization algebra $\obs$ we start with has a $\ZZ\times \ZZ/2$-grading corresponding to the ghost number and the fermionic charge. If we can extend a topological supercharge Q to a twisting datum (meaning we choose a circle action on $\obs$ compatible with Q, see Definition [def:twistingdatum]), then the twisted factorization algebra $\obs^Q$ will be $\ZZ$-graded. Without this choice $\obs^Q$ is merely $\ZZ/2$-graded. When we classify square-zero supercharges in Section [twistedtheoriessection], we will specify for each supercharge whether we can extend it to a twisting datum. We may also consider square-zero supercharges which are not topological, i.e. when the image of [Q, − ] is a proper subspace of V. We show in Proposition [atleastholoprop] that the image is at least half-dimensional. We call square-zero supercharges with a half-dimensional image holomorphic. For instance, in dimension 2 factorization algebras twisted by holomorphic supercharges give rise to vertex algebras as explained in. There are also intermediate cases between holomorphic and topological supercharges which we call holomorphic-topological supercharges. One would like to think of an n-dimensional field theory twisted by a holomorphic-topological supercharge whose image has codimension k as a field theory on $\CC^k \times \RR^{n-2k}$ which is holomorphic along $\CC^k$ and topological along $\RR^{n-2k}$. Such theories were originally considered by Kapustin, and the term “holomorphic-topological” first appeared in. It is natural to ask if there are ways to ensure that $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ is a quasi-isomorphism. If $\obs^Q$ admits a homotopically trivial dilation action, the map $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ is a quasi-isomorphism by Proposition [dRdilationprop]. In particular, we show in Lemma [dilationpotentiallemma] that such a homotopically trivial dilation action exists in many superconformal field theories. Moreover, it is enough to have this action only on the classical level (see Remark [remark:classicaldilationaction]). Oriented field theories ----------------------- If M is a framed n-manifold and A is an $\bb E\_n$-algebra, we may consider its factorization homology ∫MA (see e.g. ) which is a cochain complex. In particular, this allows us to define observables on any framed manifold. If we want to define factorization homology for a manifold M which is only oriented, we need to extend A to a framed $\bb E\_n$-algebra, i.e. an algebra over the operad of framed little n-disks. In other words, $\bb E\_n$ is an operad in $\SO(n)$-spaces and a framed $\bb E\_n$-algebra is an $\bb E\_n$-algebra equipped with a compatible action of the chains ${\mathrm{C}}\_\bullet(\SO(n))$ (see e.g. ). More generally, for a Lie group G with a homomorphism G → O(n) we may define the factorization homology ∫MA over M when M carries a G-reduction of the frame bundle and A is an $\bb E\_n^G$-algebra. We recover the oriented case for $G=\SO(n)$ and the framed case for $G=\pt$. We may ask when it is possible to extend the twisted factorization algebra $\obs^Q$ described above to a framed $\bb E\_n$-algebra. In general, we consider supersymmetric quantum field theories, so the untwisted factorization algebra $\obs$ carries a compatible action of rotations. However, the supercharge Q is never preserved by this action, so this action of rotations does not survive the twist. To correct this problem, Witten modifies the action of rotations by means of a twisting homomorphism. A twisting homomorphism is a homomorphism $\phi\colon \SO(n)\rightarrow G\_R$ from the group of rotations to the group of R-symmetries which allows us to define a new action of rotations on the factorization algebra $\obs$ using the embedding $(\id, \phi)\colon \SO(n)\rightarrow \SO(n)\times G\_R$. If the supercharge Q is scalar under the ϕ-twisted $\SO(n)$-action, we say that Q is compatible with the twisting homomorphism ϕ. When we classify square-zero supercharges in Section [twistedtheoriessection], we will also classify possible twisting homomorphisms and say which supercharges are compatible. Note that if Q is any square-zero supercharge (not necessarily topological) in dimension n ≥ 3 compatible with a twisting homomorphism ϕ, then Q is automatically a topological supercharge by Theorem [compatibletwistinghomthm] (this is usually taken as the definition of a topological supercharge). To summarize, given a topological supercharge Q compatible with a twisting homomorphism ϕ we get an $\SO(n)$-action on $\obs^Q$. However, to define a framed $\bb E\_n$-algebra we need to have an action of the chains ${\mathrm{C}}\_\bullet(\SO(n))$ on $\obs^Q$. Explicitly, we are looking for $\SO(n)$-actions in the de Rham sense (see Definition [dRactiondef]), i.e. where the Lie algebra action is homotopically trivial. Then we get the following statement (see Theorem [twistedtheorywithtwistinghomthm]). Let $\obs$ be an $\mf{A}$-supersymmetric factorization algebra on $\RR^n$, Q a topological supercharge compatible with a twisting homomorphism $\phi\colon \SO(n)\rightarrow G\_R$. Suppose that the following two conditions hold. * The factorization map $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ is a quasi-isomorphism. * The ϕ-twisted $\SO(n)$-action on $\obs^Q$ extends to a de Rham $\SO(n)$-action. Then the collection $\{\obs^Q(B\_r(0))\}$ forms a framed $\bb E\_n$-algebra. Note that if we did not have an extension of the ϕ-twisted $\SO(n)$-action to a de Rham $\SO(n)$-action, we would get the structure of a $\bb E\_n^{\SO(n)\_\delta}$-algebra, where $\SO(n)\_\delta$ is the Lie group $\SO(n)$ considered with the discrete topology. To such an algebra we can apply factorization homology over an oriented manifold M if the manifold carries a flat connection on its tangent bundle (for instance, it is an affine manifold). There are also versions of this theorem for any Lie group G with a homomorphism G → O(n). One might then wonder how one can construct an extension of the ϕ-twisted $\SO(n)$-action to a de Rham $\SO(n)$-action. There are two natural mechanisms for that: * In the case of superconformal theories in dimension 2, 3 or 4, the homotopy trivialization may be obtained from the generators of the superconformal algebra as shown in Propositions [2dsuperconformalpotential], [3dsuperconformalpotential] and [4dsuperconformalpotential] (in the case of dimension 4, this works only for certain twisting homomorphisms and topological supercharges). * If $\obs$ is an ℏ-family of factorization algebras obtained by deformation quantization of a classical field theory and the ϕ-twisted $\SO(n)$-action on $\obs^Q$ is inner, then the ϕ-twisted $\SO(n)$-action on the quantum observables $\obs^Q[\hbar^{-1}]$ is automatically homotopically trivial (See Remark [remark:inneractionhomotopytrivial]). One expects that a topological field theory should assign diffeomorphism invariants to manifolds. For observables this means that the complex of observables on $\RR^n$ carries a $\mr{Diff}(\RR^n)$-action in the de Rham sense. Given a supersymmetric factorization algebra $\obs$, it usually only carries a Poincaré action (i.e. an action of Killing vector fields) since the action of the theory involves the choice of a metric. The obstruction to extending this action to an action of all vector fields is given by the gravitational stress-energy tensor (see Section [stressenergysection]). If Q is a topological supercharge compatible with a twisting homomorphism ϕ, then one still has an action of Killing vector fields on $\obs^Q$. Moreover, in many supersymmetric field theories the gravitational stress-energy tensor is Q-exact which shows that there is an action of all vector fields on $\obs^Q$ (Lemma [exactSEtensorlemma]). Note, however, that we do not expect this to be a de Rham $\mr{Diff}(\RR^n)$-action. Structure of the paper ---------------------- Section [Factorizationsection] takes place in the world of homological and homotopical algebra – the abstract mathematical machinery that we will use is concentrated entirely in this section. We remind the reader the basics of factorization algebras (in Section [perturbativesection]) and group actions thereon (in Section [groupactionsection]). In particular, we explain what it means to have a group action in the de Rham sense. We study factorization algebras with an action of the group of translations (Section [translationinvariantsection]), and investigate how this condition can be reformulated operadically. We then study de Rham translation-invariant factorization algebras (Section [Ensection]) and show how to obtain the data $\disk\_n^{\col}$-algebras from them. Analyzing the natural pushout of coloured operads, we show that $\bb E\_n$-algebras sit fully faithfully in the ∞-category of $\disk\_n^{\col}$-algebras (Theorem [Entheorem]). Finally, in Section [Gstructuresection] we study analogues of these results for theories with a G-structure. Section [twistsection] is devoted to supersymmetric field theories. This section is much less high-tech than the previous section: other than the definition of the twist we will perform calculations relying only on ordinary Lie theory. We begin in Section [susyalgebrasection] with a recollection on (complex) supersymmetry algebras in various dimensions and define what it means for a factorization algebra on $\RR^n$ to be supersymmetric. We then define topological twists of supersymmetric factorization algebras by topological supercharges (Section [twistdefsection]) and explain when we get a de Rham translation-invariant theory (Proposition [topologicallytwistedtheoriesaredRtranslationinvtprop]). When we additionally have the data of a twisting homomorphism from a group G, we explain how we obtain de Rham translation-invariant theories with G-structure (Section [twistinghomsection]). In Section [purespinorsection] we provide a recollection on pure spinors which will be later used in the classification of twists in dimensions 7 through 10. In Section [stressenergysection] we explain what the gravitational stress-energy tensor is and why supersymmetric field theories carry an action of all vector fields (which is possibly not homotopically trivial). In Section [dilationsection] we explain how the additional data of an $\bb E\_n$-algebra arises from a twisted theory where we additionally have a compatible action of the group of dilations. Finally, Section [superconformalsection] is devoted to superconformal algebras which are certain extensions of supersymmetry algebras. We explain there how superconformal symmetry can be used to construct nullhomotopies for rotations and to show that the factorization maps $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ are quasi-isomorphisms. In Section [twistedtheoriessection] we provide a classification of possible twists of supersymmetric theories. This section engages most closely with the techniques used in the physics literature on topological twists. We give a classification of twists in dimensions 1 through 10: * In each dimension we classify orbits of square-zero supercharges under the complexified Lorentz group, R-symmetry and scaling. For each class of square-zero supercharges we specify the dimension of its image, i.e. we explain whether it is topological, holomorphic, or one of the intermediate holomorphic-topological possibilities. * We also classify possible twisting homomorphisms from $\SO(n)$ and their compatible topological supercharges. It turns out that these exist only in dimensions ≤ 5 and in higher dimensions we list some twisting homomorphisms from subgroups of $\SO(n)$ which might be of interest. We also explain whether a square-zero supercharge can be promoted to a twisting datum and whether this can be done compatibly with the twisting homomorphism. * Finally, we provide references where twisted supersymmetric theories for a given square-zero supercharge have been studied. Let us note that there are several cases (in particular, in dimensions above 5) which have not been analyzed from either the mathematical or physical perspective to our best knowledge. Let us mention a recent article of Eager, Saberi and Walcher appearing subsequent to the first version of this paper that also studies the possible square zero supercharges in supersymmetric field theories, and includes discussion of the geometry of the varieties of such supercharges. Acknowledgements ---------------- We would like to thank Kevin Costello, Owen Gwilliam, Vasily Pestun and Brian Williams for helpful discussions and the anonymous referees for many comments and suggestions. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. CE acknowledges the support of IHÉS. The research of CE on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (QUASIFT grant agreement 677368). PS was supported by the NCCR SwissMAP grant of the Swiss National Science Foundation. Factorization Algebras and $\bb E\_n$-Algebras ============================================== Throughout the paper we work over the ground field $\CC$ of complex numbers. Thus, all complexes, Lie algebras and so on are over $\CC$. The chain complexes of observables will be differentiable chain complexes, see. Models for Perturbative Field Theory ------------------------------------ We begin by outlining the theory of factorization algebras as a model for classical and quantum field theory on $\RR^n$. One can model a classical perturbative field theory and its quantization starting from either of two “dual” points of view. 1. The BV–BRST complex of fields: a sheaf of L∞-algebras on $\RR^n$ that gives a derived model for the space of gauge-equivalence classes of solutions to the equations of motion. 2. The factorization algebra of observables: a multiplicative version of a cosheaf of cochain complexes on $\RR^n$ built from the BV–BRST complex by taking Chevalley-Eilenberg cochains. Our focus for this article will be the second point of view, that of factorization algebras, but we will conclude this section with some remarks on the BV–BRST complex. Let us begin by explaining what a factorization algebra is. The idea is that we can model observables in a field theory on each open subset of the spacetime (i.e. of $\RR^n$), along with the data of how to extend observables from a smaller open set to a larger one. That is, we will describe a kind of precosheaf. Fix a symmetric monoidal category C, for instance the category of vector spaces. A prefactorization algebra on $\RR^n$ is a “multiplicative” precosheaf $\obs$ with values in C on $\RR^n$. That is, an assignment of $\obs(U)\in{\mathcal{C}}$ to each open set, along with structure maps $\bigotimes\_{i=1}^n \obs(U\_i) \to \obs(V)$ for each pairwise disjoint collection of open subsets $U\_1, \ldots, U\_n \sub V$ satisfying the natural compatibility conditions. A factorization algebra on $\RR^n$ is a prefactorization algebra which satisfies descent for covers {Ui} of open sets U satisfying the condition that for any sequence of points x1, …, xn in U there is an element Ui of the cover so that $\{x\_1, \ldots, x\_n\} \sub U\_i$. Such covers are known as Weiss covers (see for a precise definition). The idea here is that the local sections $\obs(U)$ on an open set U model the (classical or quantum) $\CC$-valued observables of a theory that depend only on measurements inside of U, so in particular there is a canonical extension (our structure maps) $\obs(U) \to \obs(V)$ for every pair of open sets $U \sub V$. The sheaf condition is intended to convey the idea that all observables are determined by their values on arbitrarily small neighbourhoods of finite sets of points. The observables in a classical field theory carry an additional structure: a degree 1 Poisson bracket. A $\bb P\_0$-algebra is a commutative dg algebra A together with a Poisson bracket of cohomological degree 1: that is, a Lie bracket { − , − }: A ⊗ A → A[1] which is a derivation for the product. Let $\alg\_{\bb P\_0}$ denote the symmetric monoidal category of $\bb P\_0$-algebras with tensor product given by the tensor product of the underlying complexes. Let us briefly comment on the passage from classical to quantum observables which is given by BV quantization introduced in. We will follow the modern account given in. A BD0-algebra is a complex of flat $\CC[[\hbar]]$-modules together with a commutative multiplication and a Poisson bracket of degree 1 satisfying the relation $$\d(ab) = \d(a) b + (-1)^{\|a\|} a \d(b) + \hbar\{a, b\}.$$ Note that if Ã is a BD0-algebra, Ã/ℏ becomes a $\bb P\_0$-algebra. Conversely, given a $\bb P\_0$-algebra A, a deformation quantization is a BD0-algebra Ã together with an isomorphism Ã/ℏ ≅ A of $\bb P\_0$-algebras. Similarly, given a prefactorization algebra $\obs$ representing the classical observables in a classical field theory, a quantization is a prefactorization algebra $\obs^q$ valued in BD0-algebras together with an isomorphism $\obs^q/\hbar\cong \obs$. There is a comprehensive theory governing the process of quantizing factorization algebras of classical observables via renormalization. Throughout the paper our results will be valid in all of the following settings: * (Classical). We consider prefactorization algebras valued in $\bb P\_0$-algebras. * (Quantum). We consider prefactorization algebras valued in complexes. * (Deformation quantization). We consider prefactorization algebras valued in BD0-algebras. We will conclude this section with a comment on the dual point of view, where we model a field theory starting from its space of classical fields and action functional. We represent classical field theory from this point of view using the Batalin-Vilkovisky formalism – we refer to Costello and Costello and Gwilliam for a more detailed account. [classicalfieldtheorydef] A classical field theory on $\RR^n$ is a sheaf L of L∞-algebras on $\RR^n$ along with an invariant antisymmetric pairing of degree − 3 (sometimes called the antibracket): $$\langle -,-\rangle \colon L \otimes L \to \dens\_{\RR^n}[-3].$$ In order to match up with the physics terminology we will refer to this sheaf of Lie algebras as the BV–BRST complex of the theory. These BV–BRST complexes can be constructed algorithmically from the more traditional physical data of a space of fields with a gauge action and an action functional; it models the tangent complex to the derived critical locus of the action functional. In brief, one models the space of infinitesimal gauge transformations, with its action on the space of physical fields, by a sheaf Φ of L∞-algebras (the “BRST complex”). The BV-BRST complex is then constructed as the sum Φ ⊕ Φ![ − 3] – where Φ! is a topological dual space to Φ tensored by the sheaf of densities on $\RR^n$ – with L∞ structure deformed using the classical action functional. For details, see. Recall that the Chevalley-Eilenberg cochain complex of an L∞ algebra ≫ is the cochain complex $C^\bullet(\gg) = \sym(\gg[1])^\$ with differential coming from the L∞ structure on ≫ . An invariant symmetric pairing of degree − 3 on ≫ induces a $\bb P\_0$-algebra structure on C•( ≫ ). Let L* be an L∞ algebra describing a classical field theory on $\RR^n$. Its algebra of classical observables is the prefactorization algebra $\obs$ valued in $\alg\_{\bb P\_0}$ that assigns to an open set V the $\bb P\_0$-algebra $$\obs\_L(V) := C^\bullet(L(V)).$$ In fact, in reasonable circumstances this prefactorization algebra is a factorization algebra (see for the precise statement). Smooth Group Actions -------------------- There are several senses in which a field theory can be thought of as equivariant with respect to the action of a Lie group G. We will mostly consider group actions in a strong sense, where we have a smooth action of the Lie group relating observables on different open sets along with a compatible infinitesimal action of its Lie algebra by derivations on each open set. We will consider derivations of algebras as follows: * If A is a $\bb P\_0$- or a BD0-algebra, a derivation F: A → A is an endomorphism which is a derivation of the multiplication and the bracket. * If A is a complex, a derivation is merely an endomorphism. [] A degree k derivation of a prefactorization algebra $\obs$ on $\RR^n$ is a cohomological degree k derivation $F\_U \colon \obs(U) \to \obs(U)$ for each open set U in $\RR^n$ that collectively satisfy a Leibniz rule. That is, if $U\_1, \ldots, U\_m \sub V$ are disjoint open subsets of an open set, and mU1, …, UmV is the associated factorization map, we require that $$\begin{aligned} F\_V \circ &m\_{U\_1, \ldots, U\_m}^V(\OO\_1, \ldots, \OO\_m) \\ &= \sum\_{i=1}^m (-1)^{k(\|\OO\_1\| + \cdots + \|\OO\_{i-1}\|)} m\_{U\_1, \ldots, U\_m}^V(\OO\_1, \ldots, F\_{U\_i}(\OO\_i), \ldots, \OO\_m).\end{aligned}$$ The complex $\der(\obs)$ of derivations of arbitrary degree is naturally a dg Lie algebra, with the bracket defined on each open set. [actionofliealgdef] If ≫ is a dg Lie algebra and $\obs$ is a prefactorization algebra on $\RR^n$, an action of ≫ on $\obs$ is a morphism $\gg \to \der(\obs)$ of dg Lie algebras. We may also consider inner actions of dg Lie algebras on the observables: * If $\obs$ is a prefactorization algebra valued in $\bb P\_0$ or BD0-algebras, then there is a morphism of dg Lie algebras $\obs(\RR^n)[-1] \to \der^\bullet(\obs)$ defined by $\OO \mapsto \{\OO, -\}$. * If $\obs$ is a prefactorization algebra valued in complexes, we equip $\obs(\RR^n)[-1]$ with the trivial bracket and let $\obs(\RR^n)[-1]\to \der^\bullet(\obs)$ be the zero map. A ≫ -action $\gg \to \der^\bullet(\obs)$ is inner if it factors through a morphism $\gg \to \obs(\RR^n)[-1]$ of dg Lie algebras. Note that by definition an inner action on a prefactorization algebra valued in complexes is trivial. Similarly, an inner action on a prefactorization algebra $\obs$ valued in BD0-algebras is homotopically trivial on $\obs[\hbar^{-1}]$ (see e.g. ). [remark:inneractionhomotopytrivial] The following definition generalises the definition of a smooth translation action in. Let G be a Lie group with an action $\rho\colon G\rightarrow \mr{Diff}(\RR^n)$ on $\RR^n$ by diffeomorphisms. We say G acts smoothly on a prefactorization algebra $\obs$ on $\RR^n$ if we have an action of the complexified Lie algebra ≫ on $\obs$ by derivations, which we denote by $v \mapsto \d / \d v$, along with isomorphisms $\alpha\_g \colon \obs(U) \to \obs(\rho(g)(U))$ for each g ∈ G and each open subset $U \sub \RR^n$ which satisfy the following conditions. * For each g1, g2 ∈ G, αg1 ∘ αg2 = αg1g2. * If $U\_1,U\_2\sub V$ are disjoint open subsets such that ρ(g)(U1) and ρ(g)(U2) are disjoint for g ∈ G, then the square $$\xymatrix{ \obs(U\_1) \otimes \obs(U\_2) \ar[r]^-{\alpha\_g} \ar[d] &\obs(\rho(g)(U\_1)) \otimes \obs(\rho(g)(U\_2)) \ar[d] \\ \obs(V) \ar[r]^{\alpha\_g} &\obs(\rho(g)(V)) }$$ commutes. * For each collection $U\_1, \ldots, U\_k\sub V$ of disjoint open subsets, the composite map $$\begin{aligned} m\_{g\_1, \ldots, g\_k} \colon \obs(U\_1) \otimes \cdots \otimes \obs(U\_k) &\to \obs(\rho(g\_1)(U\_1)) \otimes \cdots \otimes \obs(\rho(g\_k)(U\_k)) \\ &\to \obs(V)\end{aligned}$$ varies smoothly in the space {(g1, …, gk) ∈ Gk : the ρ(gk)(Uk)are disjoint subsets of V}. * For each v ∈ ≫ we have the following compatibility between the Lie group and Lie algebra actions: $$\left( \frac \partial {\partial v} \right)\_i m\_{g\_1, \ldots, g\_k}(\OO\_1, \ldots, \OO\_k) = m\_{g\_1, \ldots, g\_k}(\OO\_1, \ldots, \frac \partial {\partial v} \OO\_i, \ldots, \OO\_k)$$ for every i, where on the left we are taking the directional derivative with respect to the element (0, …, 0, Lgi(v), 0, …, 0) ∈ Tg1, …, gkGk with v in the ith slot. Translation-invariant Prefactorization Algebras ----------------------------------------------- Consider the action of $\RR^n$ on itself by translations. A translation-invariant prefactorization algebra is a prefactorization algebra $\obs$ on $\RR^n$ equipped with a smooth action of the translation group $\RR^n$. Let us give an operadic reformulation of this structure. Let $B\_r(x)\sub \RR^n$ be the open n-disk of radius r with center at $x\in\RR^n$ and let $P\_r(x)\sub \CC^n$ be the open n-polydisk of radius r with center at $x\in\CC^n$. Denote by $\overline{B}\_r(x)$ and $\overline{P}\_r(x)$ their closures. We consider the following four operads: * Let $\disk^{\man}\_n$ be the $\RR\_{>0}$-coloured operad in smooth manifolds defined as follows. Define $$\disk^{\man}\_n(r\_1, \dots, r\_k \| R)\sub \RR^{nk}$$ to be the subset of vectors $x\_1, \dots, x\_k\in\RR^n$ such that the map $\overline{B}\_{r\_1}(x\_1)\sqcup \dots \sqcup \overline{B}\_{r\_k}(x\_k)\rightarrow \overline{B}\_R(0)$ is injective. The composition in $\disk^{\man}\_n$ is defined by embedding of disks. * Let $\pdisk^{\man}\_n$ be the $\RR\_{>0}$-coloured operad in complex manifolds where $$\pdisk^{\man}\_n(r\_1, \dots, r\_k \| R)\sub \CC^{nk}$$ is the subset of vectors $x\_1, \dots, x\_k\in \CC^n$ such that the map $\overline{P}\_{r\_1}(x\_1)\sqcup \dots \sqcup \overline{P}\_{r\_k}(x\_k)\rightarrow \overline{P}\_R(0)$ is injective. * Let $\disk^{\col}\_n$ be the $\RR\_{>0}$-coloured operad in simplicial sets given by taking the singular set of $\disk^{\man}\_n$. * Let $\disk\_n$ be the operad in simplicial sets defined as follows. $\disk\_n(k)$ has a map to $\RR\_{>0}^k$ whose fiber at $(r\_1, \dots, r\_k)\in\RR\_{>0}^k$ is $\disk^{\col}\_n(r\_1, \dots, r\_k \| 1)$. The composition is given by rescaling and embedding the disks. $\disk\_n$ is a model for the $\bb E\_n$-operad of little disks (see e.g. ) and $\disk\_n$-algebras are $\bb E\_n$-algebras. We have a morphism of coloured operads $\disk^{\col}\_n\rightarrow \disk\_n$ given by collapsing all colours so that the map $$\disk^{\col}\_n(r\_1, \dots, r\_k \| R)\longrightarrow \disk\_n(k)$$ is given by sending an embedding Br1(x1) ⊔ … ⊔ Brk(xk) → BR(0) to the rescaled embedding Br1/R(x1/R) ⊔ … ⊔ Brk/R(xk/R) → B1(0). The difference between the coloured operad $\disk^{\col}\_n$ and the operad $\disk\_n$ is that in the former we retain the information about the radii of the disks. As will be explained in Theorem [Entheorem], $\disk\_n$-algebras are $\disk^{\col}\_n$-algebras which are locally-constant in the radial direction. Let O be an operad in manifolds. The functor C∞ from manifolds to convenient vector spaces is symmetric monoidal, so C∞(O) forms a cooperad in convenient vector spaces. Similarly, Ω•(O) forms a cooperad in complexes of convenient vector spaces (see ). We may then talk about algebras over C∞(O) or Ω•(O) in differentiable chain complexes. The singular chain complexes C•(O) form a dg operad and we may talk about algebras over C•(O) in chain complexes. For a manifold M we have an integration map ${\mathrm{C}}\_\bullet(M)\otimes \Omega^\bullet(M)\rightarrow \RR$, so an algebra in differentiable chain complexes over the cooperad Ω•(O) gives rise to an algebra in chain complexes over the operad C•(O). For a prefactorization algebra $\obs$ on $\RR^n$ we denote by ${\mathcal{F}}\_r = \obs(B\_r(0))$ the observables on the standard disk of radius r. The following statement is shown in. [translationinvarianceprop] A translation-invariant prefactorization algebra $\obs$ on $\RR^n$ gives rise to an algebra $\{{\mathcal{F}}\_r\}\_{r\in\RR\_{>0}}$ over the coloured cooperad ${\mathrm{C}}^\infty(\disk^{\man}\_n)$ in the category of differentiable chain complexes. If we have additional structure on a translation-invariant prefactorization algebra, we may enhance this collection to an algebra structure over either the coloured operad $\pdisk^{\man}\_{n/2}$ or the coloured operad $\disk^{\col}\_n$. We begin with the holomorphic case. Recall that if $\obs$ is a translation-invariant prefactorization algebra on $\RR^{2n}\cong \CC^n$ we get derivations $\frac{\partial}{\partial \overline{z}\_i}$ of $\obs$ for every i = 1, …, n. [holtranslationinvariancedef] A holomorphic translation-invariant prefactorization algebra is a translation-invariant prefactorization algebra on $\CC^n$ equipped with the following data: for every $v\in\CC^n$ we have derivations ηi of $\obs$ of degree − 1 satisfying the following equations: $$\begin{aligned} \d \eta\_i &= \frac{\partial}{\partial \overline{z}\_i} \\ \left[\frac{\partial}{\partial \overline{z}\_i}, \eta\_j\right] &= 0 \\ [\eta\_i, \eta\_j] &= 0\end{aligned}$$ for every i, j = 1, …, n. The following statement is shown in. [holtranslationinvarianceprop] A holomorphic translation-invariant prefactorization algebra $\obs$ on $\CC^n$ gives rise to an algebra $\{\obs(P\_r(0))\}\_{r\in\RR\_{>0}}$ over the coloured cooperad $\Omega^{0, \bullet}(\pdisk^{\man}\_n)$ in the category of differentiable chain complexes. An algebra over the cooperad $\Omega^{0, \bullet}(\pdisk^{\man}\_1)$ is closely related to the notion of a vertex algebra. We refer to and for more details. De Rham Translation Invariance ------------------------------ In this section we explain how to produce $\bb E\_n$ algebras from translation-invariant prefactorization algebras where the translation action is homotopically trivialized. The arguments in this section (but only in this section and the subsequent Section [Gstructuresection]) will use techniques from homotopical algebra. At a first pass, in order to understand the results of this section as they will be applied in the rest of the paper it will be sufficient to read the initial definitions, up to Definition [dRtranslationinvariancedef] and then the statement of Corollary [Ensummarycor]. We will be interested in actions of the homotopy type of a Lie group on a prefactorization algebra. Such an action can be defined in the following way. [htpicallytrivialactiondef] Let G be a Lie group which acts on $\RR^n$ with an ideal $\gg\_0\sub \gg$ in its complexified Lie algebra which is stable under the G-action. Suppose $\obs$ is a prefactorization algebra on $\RR^n$ with a smooth G-action. We say the ≫ 0-action is homotopically trivial if it is equipped with a G-equivariant map $\eta\colon \gg\_0\rightarrow \der(\obs)[-1]$ of graded vector spaces satisfying the following equations: $$\begin{aligned} \d\eta(v) &= \frac{\partial}{\partial v} \\ [\eta(v), \eta(w)] &= 0.\end{aligned}$$ Note that if G is a connected Lie group, we may rephrase G-equivariance of $\eta\colon \gg\_0\rightarrow \der(\obs)[-1]$ via the equation $$\eta([v, w]) = \left[\eta(v), \frac{\partial}{\partial w}\right]$$ for any v ∈ ≫ 0 and w ∈ ≫ . [holotranslationinvtexample] Consider $G=\RR^{2n}\cong \CC^n$ acting on $\RR^{2n}$ by translations. We have $\gg=\CC^n\otimes\_\RR\CC$ and let $$\gg\_0 = \mathrm{span}\left\{\frac{\partial}{\partial \overline{z}\_i}\right\}\_i\sub \gg.$$ Then a translation-invariant prefactorization algebra on $\CC^n$ for which the ≫ 0-action is homotopically trivial is exactly a holomorphic translation-invariant prefactorization algebra (see Definition [holtranslationinvariancedef]). [dRactiondef] Let G be a Lie group which acts on $\RR^n$. A $G\_{\dR}$-action on a prefactorization algebra $\obs$ on $\RR^n$ is a smooth action of G on $\obs$ for which the ≫ -action is homotopically trivial. We can now give the following topological analogue of Definition [holtranslationinvariancedef]. [dRtranslationinvariancedef] A translation-invariant prefactorization algebra $\obs$ on $\RR^n$ is de Rham translation-invariant if the smooth $\RR^n$-action on $\obs$ is extended to a $\RR^n\_{\dR}$-action. De Rham translation invariance allow us to enhance the action of $\disk^{\man}\_n$ to a locally-constant action as follows. [dRtranslationinvariancethm] A de Rham translation-invariant prefactorization algebra $\obs$ on $\RR^n$ gives rise to an algebra $\{{\mathcal{F}}\_r\}\_{r\in\RR\_{>0}}$ over the coloured cooperad $\Omega^\bullet(\disk^{\man}\_n)$ in the category of differentiable chain complexes. Let us first state a preliminary result. Let ≫ be a finite-dimensional dg Lie algebra with a representation $f\colon \gg\rightarrow \eend(V)$ and a map $\eta\colon \gg\rightarrow \eend(V)[-1]$ satisfying the following equations: $$\begin{aligned} f(v) &= \d \eta(v) + \eta(\d v) \\ \eta([v, w]) &= [\eta(v), f(w)] \\ [\eta(v), \eta(w)] &= 0\end{aligned}$$ Then the map V → C•( ≫ , V) given by v ↦ exp(∑ieiη(ei))v is a chain map, where {ei} is a basis of g and {ei} is the dual basis of g\. [lm:homotopysection] [Proof of Theorem [dRtranslationinvariancethm]] We will give a proof in a slightly more general setting so that it will be amenable to generalizations. Let $\widetilde{G} = \RR^n$ be the group of translations and let $\widetilde{\gg}=\RR^n$ be its Lie algebra. By definition the space $\disk^{\man}\_n(r\_1, \dots, r\_k \| R)$ is an open subset of $\widetilde{G}^k$, so we can identify $$\Omega^\bullet(\disk^{\man}\_n(r\_1, \dots, r\_k \| R)) \cong {\mathrm{C}}^\bullet(\widetilde{\gg}, C^\infty(\disk^{\man}\_n(r\_1, \dots, r\_k \| R))).$$ We have maps $$\mu^0\colon {\mathcal{F}}\_{r\_1}\otimes \dots \otimes {\mathcal{F}}\_{r\_k}\longrightarrow C^\infty(\disk^{\man}\_n(r\_1, \dots, r\_k \| R), {\mathcal{F}}\_R)$$ provided by Proposition [translationinvarianceprop] which are $\widetilde{\gg}^{\oplus k}$-equivariant. We define $$\mu\in \Omega^\bullet(\disk^{\man}\_n(r\_1, \dots, r\_k \| R), \hom({\mathcal{F}}\_{r\_1}\otimes \dots\otimes {\mathcal{F}}\_{r\_k}, {\mathcal{F}}\_R))$$ to be the composite $$\begin{aligned} {\mathcal{F}}\_{r\_1}\otimes \dots\otimes {\mathcal{F}}\_{r\_k} &\rightarrow {\mathrm{C}}^\bullet(\widetilde{\gg}^{\oplus k}, {\mathcal{F}}\_{r\_1}\otimes \dots\otimes {\mathcal{F}}\_{r\_k}) \\ &\xrightarrow{\mu^0} {\mathrm{C}}^\bullet(\widetilde{\gg}^{\oplus k}, C^\infty(\disk^{\man}\_n(r\_1, \dots, r\_k \| R), {\mathcal{F}}\_R))\end{aligned}$$ where the map in the first line is given by Lemma [lm:homotopysection]. By construction it is a chain map and it is straightforward to check the operadic identities for μ* using those for μ0. Thus, by the above proposition we obtain an algebra over the coloured operad ${\mathrm{C}}\_\bullet(\disk^{\man}\_n)$ in chain complexes. In the rest of this section we explain a precise relationship between such algebras over the coloured operad ${\mathrm{C}}\_\bullet(\disk^{\man}\_n)$ and $\disk\_n$-algebras (i.e. $\bb E\_n$-algebras). Fix a presentable symmetric monoidal ∞-category C and let ${\mathcal{A}\mathrm{lg}}\_{\disk^{\col}\_n}({\mathcal{C}})$ be the ∞-category of algebras over $\disk^{\col}\_n$ considered as an ∞-operad (see ). if C is the ∞-category of chain complexes, by we may identify ${\mathcal{A}\mathrm{lg}}\_{\disk^{\col}\_n}({\mathcal{C}})$ as the localization of the category of ${\mathrm{C}}\_\bullet(\disk^{\col}\_n)$-algebras in chain complexes with respect to quasi-isomorphisms. Given a simplicial category D we denote by π0(D) the category obtained by applying π0 to the Hom-sets. Let F: D → E be a functor of simplicial categories. It is a Dwyer–Kan equivalence if F: π0(D) → π0(E) is essentially surjective and homD(x, y) → homE(F(x), F(y)) is a weak equivalence of simplicial sets for every x, y ∈ D. Given a simplicial coloured operad O, consdering only operations of arity 1 we obtain a simplicial category O1. Recall from that there is a model structure on simplicial categories with weak equivalences given by Dwyer–Kan equivalences. Also recall from that the category of simplicial coloured operads carries a model structure where F: O → P is a weak equivalence if O1 → P1 is a Dwyer–Kan equivalence and for every collection of colours, O(c1, …, ck∣d) → P(F(c1), …, F(ck)∣F(d)) is an equivalence of simplicial sets. We will only consider simplicial coloured operads with O(0) = \* . We may view $\RR\_{>0}$ as a poset, hence as a category. In particular, we have the associated $\RR\_{>0}$-coloured operad concentrated in arity 1 that we denote by the same symbol. The diagram of simplicial coloured operads $$\xymatrix{ \disk\_n & \disk^{\col}\_n \ar[l] \\ 1 \ar[u] & \RR\_{>0} \ar[l] \ar[u] }$$ is homotopy coCartesian. [prop:diskcoCartesian] By, the homotopy pushout $\disk^{\col}\_n\sqcup^h\_{\RR\_{>0}} 1$ is equivalent to the strict pushout $${\mathcal{P}}= \disk^{\col}\_n \underset {\RR\_{>0}} \sqcup I,$$ where $\RR\_{>0}\rightarrow I$ is a cofibrant replacement of $\RR\_{>0}\rightarrow 1$. The inclusion of simplicial categories into simplicial coloured operads is left Quillen, so we may assume that I is concentrated in arity 1 and has the set of colours $\RR\_{>0}$. Moreover, since I → 1 is an acyclic fibration, I(r∣R) is contractible for any $r, R\in\RR\_{>0}$. By construction the set of colours of P is $\RR\_{>0}$ and we can identify $${\mathcal{P}}(r\_1, \dots, r\_k \| R)\cong \underset{a\_1, \dots, a\_k\rightarrow 0}{\colim} \underset{b\rightarrow\infty}{\colim}\ \disk^{\col}\_n(a\_1, \dots, a\_k \| b)\times I(b\|R)\times \prod\_i I(r\_i\|a\_i).$$ The functors P1 → I1 → \* are both Dwyer–Kan equivalences, so ${\mathcal{P}}\_{\textbf{1}}\rightarrow (\disk\_n)\_{\textbf{1}}\cong \ast$ is a Dwyer–Kan equivalence as well. Let ${\mathrm{Conf}}\_k(\RR^n)\sub \RR^{nk}$ be the space of configurations of k ordered points on $\RR^n$. Then we have maps $${\mathcal{P}}(r\_1, \dots, r\_k \| R)\longrightarrow \disk\_n(k)\longrightarrow {\mathrm{Conf}}\_k(\RR^n),$$ where $\disk\_n(k)\rightarrow {\mathrm{Conf}}\_k(\RR^n)$ is a weak equivalence. Since the simplicial sets I( − ∣ − ) are contractible, the map $${\mathcal{P}}(r\_1, \dots, r\_k \| R)\longrightarrow \underset{a\_1, \dots, a\_k\rightarrow 0}{\colim} \underset{b\rightarrow\infty}{\colim}\ \disk^{\col}\_n(a\_1, \dots, a\_k \| b)$$ is a weak equivalence since weak equivalences in simplicial sets are stable under filtered colimits. Hence the map ${\mathcal{P}}(r\_1, \dots, r\_k \| R)\rightarrow {\mathrm{Conf}}\_k(\RR^n)$ is also a weak equivalence. Therefore, the map ${\mathcal{P}}(r\_1, \dots, r\_k \| R)\rightarrow \disk\_n(k)$ described above is a weak equivalence as well and we conclude that ${\mathcal{P}}\rightarrow \disk\_n$ is a weak equivalence of simplicial coloured operads. Let $$\xymatrix{ {\mathcal{D}}\ar^{\tilde{F}}[r] \ar^{\tilde{G}}[d] & {\mathcal{D}}\_1 \ar^{G}[d] \\ {\mathcal{D}}\_2 \ar^{F}[r] & {\mathcal{D}}\_0 }$$ be a Cartesian square of ∞-categories. Then: 1. The essential image of F̃ consists of objects x ∈ D1 such that G(x) is in the essential image of F. 2. If F is fully faithful, so is F̃. [prop:Cartesiansquarecategories] The embedding S → Cat∞ of ∞-groupoids into ∞-categories has a right adjoint ( − )∼: Cat∞ → S given by taking the maximal ∞-groupoid D∼ contained in D ∈ Cat∞. Since it is a right adjoint, it preserves fiber products, so we obtain a Cartesian square of ∞-groupoids of objects $$\xymatrix{ {\mathcal{D}}^\sim \ar[r] \ar[d] & {\mathcal{D}}\_1^\sim \ar[d] \\ {\mathcal{D}}\_2^\sim \ar[r] & {\mathcal{D}}\_0^\sim }$$ which proves the first claim. We also have a Cartesian square of ∞-categories $$\xymatrix{ {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}) \ar[r] \ar[d] & {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}\_1) \ar[d] \\ {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}\_2) \ar[r] & {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}\_0) }$$ and hence a Cartesian square of ∞-groupoids $$\xymatrix{ {\mathrm{Fun}}(\Delta^1, {\mathcal{D}})^\sim \ar[r] \ar[d] & {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}\_1)^\sim \ar[d] \\ {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}\_2)^\sim \ar[r] & {\mathrm{Fun}}(\Delta^1, {\mathcal{D}}\_0)^\sim }$$ For any two objects x, y ∈ D taking the fiber of the natural map Fun(Δ1, D)∼ → D∼ × D∼ in the above diagram we obtain a Cartesian square of ∞-groupoids $$\xymatrix{ \hom\_{\mathcal{D}}(x, y) \ar^{\tilde{F}}[r] \ar^{\tilde{G}}[d] & \hom\_{{\mathcal{D}}\_1}(\tilde{F}(x), \tilde{F}(y)) \ar^{G}[d] \\ \hom\_{{\mathcal{D}}\_2}(\tilde{G}(x), \tilde{G}(y)) \ar^{F}[r] & \hom\_{{\mathcal{D}}\_0}(G\tilde{F}(x), G\tilde{F}(y)) }$$ If F is fully faithful, the bottom map is an equivalence. Since the square is Cartesian, the top map is also an equivalence. In other words, in this case F̃: D → D1 is fully faithful. [Entheorem] The forgetful functor ${\mathcal{A}\mathrm{lg}}\_{\disk\_n}({\mathcal{C}})\rightarrow {\mathcal{A}\mathrm{lg}}\_{\disk^{\col}\_n}({\mathcal{C}})$ is fully faithful with essential image given by algebras $\{A(r)\}\_{r\in\RR\_{>0}}$ where the natural map A(r) → A(R) for r ≤ R is an equivalence. From Proposition [prop:diskcoCartesian] we obtain a homotopy coCartesian square of ∞-operads $$\xymatrix{ \disk\_n & \disk^{\col}\_n \ar[l]\\ 1 \ar[u] & \RR\_{>0} \ar[l] \ar[u] }$$ Therefore, we get a Cartesian square of ∞-categories of algebras $$\xymatrix{ {\mathcal{A}\mathrm{lg}}\_{\disk\_n}({\mathcal{C}}) \ar^{\tilde{F}}[r] \ar^{\tilde{G}}[d] & {\mathcal{A}\mathrm{lg}}\_{\disk^{\col}\_n}({\mathcal{C}}) \ar^{G}[d] \\ {\mathcal{C}}\ar^{F}[r] & {\mathcal{A}\mathrm{lg}}\_{\RR\_{>0}}({\mathcal{C}}) }$$ The functor $F\colon {\mathcal{C}}\rightarrow {\mathcal{A}\mathrm{lg}}\_{\RR\_{>0}}({\mathcal{C}})$ sends an object V to a sequence $\{V\}\_{r\in\RR\_{>0}}$ with all maps the identities. This functor has a left adjoint $\colim\colon {\mathcal{A}\mathrm{lg}}\_{\RR\_{>0}}({\mathcal{C}})\rightarrow {\mathcal{C}}$ given by evaluating the colimit of a coloured collection over the poset $\RR\_{>0}$. The unit of the adjunction $\colim\circ F\rightarrow \id$ is an equivalence, so F is fully faithful. The essential image is contained in coloured collections $\{V(r)\}\_{r\in\RR\_{>0}}$ where V(r) → V(R) is an equivalence for every pair r ≤ R. But the unit $\id\rightarrow F\circ \colim$ is an equivalence on such coloured collections, so the essential image consists of precisely such coloured collections. Therefore, by Proposition [prop:Cartesiansquarecategories] we can conclude that the forgetful functor $$\tilde{F}\colon {\mathcal{A}\mathrm{lg}}\_{\disk\_n}({\mathcal{C}})\longrightarrow {\mathcal{A}\mathrm{lg}}\_{\disk^{\col}\_n}({\mathcal{C}})$$ is fully faithful with essential image given by those algebras $\{A(r)\}\_{r\in\RR\_{>0}}$ where the morphism A(r) → A(R) is an equivalence. We can summarise the results of this section in the following way: the following is just a restating of the combination of Theorem [dRtranslationinvariancethm] and Theorem [Entheorem]. [Ensummarycor] If $\obs$ is a de Rham translation-invariant prefactorization algebra on $\RR^n$ such that the factorization map $\obs(B\_r(0)) \to \obs(B\_R(0))$ is a quasi-isomorphism for all r ≤ R, then $\obs(B\_1(0))$ can be canonically equipped with the structure of a $\disk\_n$-algebra. Suppose $\obs$ arises from a classical field theory, i.e. $\obs$ is a prefactorization algebra on $\RR^n$ valued in the category of $\bb P\_0$-algebras. Then $\obs(B\_1(0))$ becomes an $\bb E\_n$-algebra in the ∞-category of $\bb P\_0$-algebras. By we may identify $\bb E\_n$-algebras in $\bb P\_0$-algebras with $\bb P\_n$-algebras, i.e. commutative dg algebras with a Poisson bracket of degree 1 − n. G-Structures -------------- Fix a finite-dimensional Lie group G with a homomorphism G → O(n). A translation-invariant prefactorization algebra $\obs$ on $\RR^n$ has a *G-structure* if the smooth action of $\RR^n$ is extended to a smooth action of the group $G \ltimes \RR^n$. Let us now explain operadic consequences of a G-structure on a prefactorization algebra. Recall from that if O is a G-operad, then we can construct a semidirect product operad $G\ltimes {\mathcal{O}}$ such that a $G\ltimes {\mathcal{O}}$-algebra in chain complexes is the same as an O-algebra in chain complexes equipped with a G-action. Observe that the (coloured) operads $\disk^{\man}\_n, \disk^{\col}\_n, \disk\_n$ by construction carry a natural O(n)-action. Thus, we may consider their G-equivariant versions: * $\disk^{\man, G}\_n$ is the $\RR\_{>0}$-coloured operad in smooth manifolds $G\ltimes \disk^{\man}\_n$. * $\disk^{\col, G}\_n$ is the $\RR\_{>0}$-coloured operad in simplicial sets $G\ltimes \disk^{\man}\_n$ where we consider G as a simplicial group by taking its singular complex. * $\disk^G\_n$ is the operad in simplicial sets $G\ltimes \disk\_n$ where we again treat G as a simplicial group. The operad $\disk^{\SO(n)}\_n$ is the operad of framed little n-disks and so $\disk^{\SO(n)}\_n$-algebras are framed $\bb E\_n$-algebras. We have the following variant of Proposition [translationinvarianceprop] with G-structures. [translationinvarianceGprop] A translation-invariant prefactorization algebra $\obs$ on $\RR^n$ with a G-structure gives rise to an algebra $\{{\mathcal{F}}\_r\}\_{r\in\RR\_{>0}}$ over the coloured cooperad $C^\infty(\disk^{\man, G}\_n)$. We may also consider de Rham translation-invariant versions. A prefactorization algebra $\obs$ on $\RR^n$ is de Rham translation-invariant with a G-structure if it carries a smooth $G\ltimes \RR^n$-action for which the Lie algebra action $\RR^n\sub \gg\ltimes \RR^n$ is homotopically trivial. A prefactorization algebra $\obs$ on $\RR^n$ is de Rham translation-invariant with a $G\_{\dR}$-structure if it carries a $(G\ltimes \RR^n)\_{\dR}$-action. For G a Lie group we denote by Gδ the Lie group with the discrete topology. Then we have the following operadic interpretations of G- and $G\_{\dR}$-actions. [G-structuredoperadalgebrathm] * A de Rham translation-invariant prefactorization algebra $\obs$ on $\RR^n$ with a G-structure gives rise to an algebra $\{{\mathcal{F}}\_r\}\_{r\in\RR\_{>0}}$ in differentiable chain complexes over the coloured cooperad $\Omega^\bullet(\disk^{\man, G\_\delta}\_n)$. * A de Rham translation-invariant prefactorization algebra $\obs$ on $\RR^n$ with a $G\_{\dR}$-structure gives rise to an algebra $\{{\mathcal{F}}\_r\}\_{r\in\RR\_{>0}}$ in differentiable chain complexes over the coloured cooperad $\Omega^\bullet(\disk^{\man, G}\_n)$. The proof is identical to the proof of Theorem [dRtranslationinvariancethm] where we perform the following modifications: * If $\obs$ has a G-structure, we let $\widetilde{G} = G\ltimes \RR^n$ and $\widetilde{\gg} = \RR^n$. * If $\obs$ has a $G\_{\dR}$-structure, we let $\widetilde{G} = G\ltimes \RR^n$ and $\widetilde{\gg} = \gg\ltimes \RR^n$. As before, an algebra in differentiable chain complexes over $\Omega^\bullet(\disk^{\man, G}\_n)$ gives rise to an algebra in chain complexes over ${\mathrm{C}}\_\bullet(\disk^{\col, G}\_n)$. Finally, we have an analogue of Theorem [Entheorem] for prefactorization algebras with a $G\_{\dR}$-structure. [G-structuredEntheorem] The forgetful functor ${\mathcal{A}\mathrm{lg}}\_{\disk^G\_n}({\mathcal{C}})\rightarrow {\mathcal{A}\mathrm{lg}}\_{\disk^{\col, G}\_n}({\mathcal{C}})$ is fully faithful with essential image given by algebras $\{A(r)\}\_{r\in\RR\_{>0}}$ where the natural map A(r) → A(R) for r ≤ R is a quasi-isomorphism. We can summarize Theorem [G-structuredoperadalgebrathm] and Theorem [G-structuredEntheorem] in the following way. [Efrnsummarycor] If $\obs$ is a de Rham translation-invariant prefactorization algebra on $\RR^n$ with an $\SO(n)\_{\dR}$-structure such that the factorization map $\obs(B\_r(0)) \to \obs(B\_R(0))$ is a quasi-isomorphism for all r ≤ R, then $\obs(B\_1(0))$ can be canonically equipped with the structure of a framed $\bb E\_n$-algebra. Supersymmetry and Topological Twists ==================================== At this point we will change gears and begin to apply the results of the previous section to our main focus: the theory of twists of supersymmetric field theories. We will begin by explaining what supersymmetric theories are in the factorization algebra context, giving the definition of twisting, then apply our results to explain the circumstances under which the observables in topologically twisted supersymmetric field theories admit $\bb E\_n$-structures and either G- or $G\_{\dR}$-structures. Supersymmetry Algebras ---------------------- We refer to for some details on the material presented here. Let $V\_\RR=\RR^n$ endowed with a nondegenerate symmetric bilinear form. Denote by $V=V\_\RR\otimes \CC$ its complexification and consider the complex Lie algebra $\so(V)$. Let us recall that if n = 2q + 1, $\so(V)$ has a distinguished fundamental representation S called the spin representation. If n = 2q, $\so(V)$ has a pair of distinguished fundamental representations S+ and S− called the semi-spin representations which are exchanged under the outer automorphism group (recall that it is $\ZZ/2$ for q ≥ 5). Elements of S (in odd dimensions) or S+ ⊕ S− (in even dimensions) are called Dirac spinors, and elements of S+ or S− (in even dimensions) are called Weyl spinors. A representation Σ of $\so(V)$ is spinorial if it splits as a sum of semi-spin (if n = 2q) or spin (if n = 2q + 1) representations. Note that under the standard inclusion $\so(n-1)\sub \so(n)$ a spinorial representation restricts to a spinorial representation. We will use certain canonical $\so(V)$-equivariant nondegenerate pairings Γ from spin representations to V. The precise form of the pairing depends on dim(V) modulo 8: * If $\dim(V)\equiv 0, 4\pmod 8$, we have a pairing Γ: S+ ⊗ S− → V. * If $\dim(V)\equiv 1, 3\pmod 8$, we have a pairing $\Gamma\colon \sym^2(S)\rightarrow V$. * If $\dim(V)\equiv 2\pmod 8$, we have pairings $\Gamma\_{\pm}\colon \sym^2(S\_{\pm})\rightarrow V$. * If $\dim(V)\equiv 5, 7\pmod 8$, we have a pairing Γ: ∧ 2(S) → V. * If $\dim(V)\equiv 6\pmod 8$, we have pairings Γ±: ∧ 2(S±) → V. We then have the following proposition characterizing possible spinorial representations. [prop:spinorialreps] Suppose Σ is a spinorial representation of $\so(V)$ equipped with a nondegenerate $\so(V)$-equivariant pairing $\sym^2(\Sigma)\rightarrow V$. Then Σ has the following form. * Suppose $\dim(V)\equiv 0, 4\pmod 8$. There is a vector space W such that Σ ≅ S+ ⊗ W ⊕ S− ⊗ W\. Suppose $\dim(V)\equiv 1,3\pmod 8$. There is a vector space W equipped with a nondegenerate symmetric bilinear pairing such that Σ ≅ S ⊗ W. * Suppose $\dim(V)\equiv 2 \pmod 8$. There is a pair of vector spaces W+ and W− equipped with nondegenerate symmetric bilinear pairings such that Σ ≅ S+ ⊗ W+ ⊕ S− ⊗ W−. * Suppose $\dim(V)\equiv 5, 7\pmod 8$. There is a symplectic vector space W such that Σ ≅ S ⊗ W. * Suppose $\dim(V)\equiv 6\pmod 8$. There is a pair of symplectic vector spaces W+ and W− such that Σ ≅ S+ ⊗ W+ ⊕ S− ⊗ W−. Using Proposition [prop:spinorialreps] we can define spinorial representations using a single number or a pair of numbers as follows: * If $\dim(V)\equiv 0, 1, 3, 4\pmod 8$, we let $\mc{N} = \dim(W)$. * If $\dim(V)\equiv 2 \pmod 8$, we let $\mc{N}\_{\pm}=\dim(W\_{\pm})$. * If $\dim(V)\equiv 5, 7\pmod 8$, we let $2\mc{N} = \dim(W)$. * If $\dim(V)\equiv 6\pmod 8$, we let $2\mc{N}\_{\pm} = \dim(W\_{\pm})$. We are now ready to define supertranslation Lie algebras associated to spinorial representations Σ equipped with a nondegenerate pairing $\Gamma \colon \sym^2(\Sigma)\rightarrow V$. The supertranslation Lie algebra associated to Σ and Γ is the $\so(V)$-equivariant super Lie algebra T = V ⊕ ΠΣ with the only nontrivial bracket given by the pairing $\Gamma \colon \sym^2(\Sigma)\rightarrow V$. We can also define super Poincaré algebra as follows. Fix a nondegenerate symmetric bilinear pairing on $V\_\RR$. Recall that the Poincaré group is $\ISO(V\_\RR) = \Spin(V\_\RR)\ltimes V\_\RR$, so that its complexified Lie algebra, the Poincaré algebra, is $\mf{iso}(V) = \so(V)\ltimes V$. The super Poincaré algebra associated to Σ and Γ is the super Lie algebra $\so(V)\ltimes T$ where T is the supertranslation Lie algebra. Note that the super Poincaré algebra carries a large outer automorphism group corresponding to automorphisms of the auxiliary space W. Many field theories of interest carry an action of a subgroup of this outer automorphism group. So, we will define a supersymmetry algebra to incorporate those symmetries as well. Fix the following data: 1. A spinorial representation Σ of $\so(V)$. 2. An $\so(V)$-equivariant nondegenerate pairing $\Gamma\colon \sym^2(\Sigma)\to V$. 3. A complex Lie group GR, the group of R-symmetries, which is a subgroup of the group of outer automorphisms of the super Poincaré algebra that act trivially on the even part. The supersymmetry algebra associated to a choice of Σ, Γ and GR is the super Lie algebra $$\mf A = (\mf{iso}(V) \oplus \gg\_R) \ltimes \Pi \Sigma.$$ With this discussion in hand, we can define what it means for a prefactorization algebra to be supersymmetric. From now on all prefactorization algebras are $\ZZ/2$-graded in addition to the original cohomological grading. In the symmetric monoidal structure on $\ZZ/2$-graded complexes that we consider, there is a Koszul sign coming from both gradings. We say a prefactorization algebra $\obs$ on $\RR^n$ is $\mf A$-supersymmetric for the supersymmetry algebra $\mf A$ if it admits a smooth $\ISO(p,q) \times G\_R$-action, where $\ISO(p,q)$ acts on $\RR^n$ by isometries and GR acts trivially, so that the action of $\mf{iso}(n) \oplus \gg\_R$ extends to an action $\nu\colon \mf A\to\der(\obs)$ of $\mf A$. Topological Twists of Supersymmetric Theories --------------------------------------------- We now proceed to give a definition of the topological twist of an $\mf A$-supersymmetric factorization algebra $\obs$ on $\RR^n$ with respect to an odd element $Q \in \mf A$. We refer to and for more details on these ideas. A square-zero supercharge is an odd element $Q\in \mf A$ satisfying [Q, Q] = 0. We say Q is topological if the map $[Q,-] \colon \Sigma \to \RR^n$ is surjective. We say Q is holomorphic if n is even and the image of the map [Q, − ] has dimension n/2. Note that all supercharges are at least holomorphic, i.e. the image of [Q, − ] has dimension at least n/2 (see Proposition [atleastholoprop]). A twisting datum is a pair (α, Q) where Q is a square-zero supercharge and α: U(1) → GR is a homomorphism so that Q has α-weight 1. [def:twistingdatum] Given a twisting datum, we can construct the twisted theory in the following way. A graded mixed complex is a cochain complex M equipped with an additional $(\ZZ \times \ZZ/2)$-grading and a square-zero endomorphism Q of $(\ZZ \times \ZZ/2)$-degree (1, 1) and cohomological degree 0. Given a graded mixed complex M, we construct a $\ZZ/2$-graded complex MQ as $$M^Q = \underset{m\rightarrow\infty}{\colim}\prod\_{n\geq -m} (\Pi^n M(n)[-n])$$ with the differential $\d\_Q = \d + Q$, where $\d$ is the original differential on M and M(n) is the weight n (i.e. degree n with respect to the auxiliary $\ZZ$-grading) part of M. The shifts have been arranged so that $\d\_Q$ has cohomological degree 1 and even $\ZZ/2$-grading. This defines a lax symmetric monoidal functor from graded mixed complexes to $\ZZ/2$-complexes. This functor is analogous to the Tate realization functor introduced in. Given a twisting datum (α, Q), $\obs(U)$ becomes a graded mixed complex for any open subset $U\sub \RR^n$ where the extra grading is given by α and the mixed structure by ν(Q). Thus, we may construct the twisted complex $\obs(U)^Q$. Since the functor ( − )Q is lax symmetric monoidal, it will send prefactorization algebras to prefactorization algebras and will preserve $\bb P\_0$- or BD0-structures. We call $\obs^Q$ the twisted prefactorization algebra. Suppose we did not choose the homomorphism α, so that $\obs$ did not have an extra grading. Then we may still construct $\obs^Q$, but the cohomological $\ZZ\times \ZZ/2$-grading will collapse to a cohomological $\ZZ/2$-grading. [rmk:twistedgrading] [Twisting the BV–BRST Complex] If the factorization algebra $\obs$ is constructed as the Chevalley-Eilenberg complex of a classical field theory L as in Definition [classicalfieldtheorydef] then we could instead define the twisted factorization algebra by twisting the sheaf L of L∞-algebras with respect to the twisting datum (α, Q), and then taking the Chevalley-Eilenberg complex of the result. This is the point of view taken in and. [topologicallytwistedtheoriesaredRtranslationinvtprop] Suppose a prefactorization algebra $\obs$ on $\RR^n$ is $\mf{A}$-supersymmetric Let (α, Q) be a twisting datum where Q is topological and fix an abelian section $\mf{a}\sub \mf A$, i.e. an odd abelian subalgebra such that $[Q, -]\colon \mf{a}\rightarrow V$ is an isomorphism. Then $\obs^Q$ is de Rham translation-invariant. In particular, if $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ is a quasi-isomorphism for R ≥ r then the collection $\{\obs^Q(B\_r(0))\}$ forms an $\bb E\_n$-algebra. Recall from Definition [htpicallytrivialactiondef] that in order to extend the V-action on $\obs^Q$ to a $V\_\dR$-action we need to specify a linear map $\eta \colon V \to \der(\obs)[-1]$ so that [η(v), η(w)] = 0, [ν(v), η(w)] = 0 for all v, w ∈ V and $\d\_Q \eta(v) = \frac {\partial}{\partial v}$. We construct η using the action of $\mf{a}\sub \mf A$. Since Q has α-weight 1 and V has α-weight 0, the subalgebra $\mf a$ has α-weight − 1, therefore the action $\nu \colon \mf a \to \der(\obs)$ becomes a map $\mf a \to \der(\obs^Q)[-1]$ of the desired degree. We define $\eta \colon V \to \der(\obs^Q)[-1]$ by composing this map with the inverse to the isomorphism $[Q, -]\colon \mf{a}\rightarrow V$. The fact that ν was a Lie map implies that [η(v), η(w)] = 0 and [ν(v), η(w)] = 0 for all v, w ∈ V. Moreover, we have $$\begin{aligned} \d\_Q \eta(v) &= \d \eta(v) + [\nu(Q), \eta(v)] \\ &= 0 + \nu(v) = \frac{\partial}{\partial v}\end{aligned}$$ since η(v) is a cocycle in $\der(\obs)$. The final sentence of the statement is a direct application of Corollary [Ensummarycor]. We can make an analogous statement in the case where Q is a holomorphic supercharge, where instead of a de Rham translation invariant factorization algebra the twist becomes holomorphic translation invariant as in Example [holotranslationinvtexample]. Suppose a prefactorization algebra $\obs$ on $\RR^{2n}$ is $\mf{A}$-supersymmetric. Let (α, Q) be a twisting datum where Q isholomorphic, and let $V\_{\mr{hol}} \sub V$ be the n-dimensional image of the map [Q, − ]. Fix $\mf{a}\sub \mf A$, an odd abelian subalgebra such that $[Q, -]\colon \mf{a}\rightarrow V\_{\mr{hol}}$ is an isomorphism. Then $\obs^Q$ is holomorphic translation-invariant. The proof is identical to that of Proposition [topologicallytwistedtheoriesaredRtranslationinvtprop]. Let us now show that we can find the abelian section $\mf{a}\sub \mf{A}$ in the examples of interest to us. [commutingpotentialprop] Let $Q\in\mf{A}$ be a square-zero supercharge and let $S\_Q \sub \CC^n$ be the image of the bracket $[Q,-] \colon \Sigma \to \CC^n$. If either 1. $n \equiv 5, 6, 7 \pmod 8$, 2. $n \equiv 0, 4 \pmod 8$ and Q lies in $S\_+\otimes W\sub \Sigma$ or $S\_-\otimes W^\\sub \Sigma$, 3. $n \equiv 1, 3 \pmod 8$ and $\mc N \geq 2$, or 4. $n \equiv 2 \pmod 8$ and $\mc N\_+$ or $\mc N\_- \geq 2$, then there is a section σ: S*Q → Σ of [Q, − ]: Σ → SQ with an abelian image. Let Σ be a spinorial representation of $\so(n)$, and let $\Gamma\_n \colon \sym^2(\Sigma) \to \CC^n$ be a nondegenerate $\so(n)$-equivariant pairing. If we consider the subalgebra $\so(n-1)\sub \so(n)$, we obtain a nondegenerate $\so(n-1)$-equivariant pairing $$\Gamma\_{n-1}\colon \sym^2(\Sigma)\xrightarrow{\Gamma\_n} \CC^n\rightarrow \CC^{n-1}.$$ In fact, every such $\so(n-1)$-equivariant $\CC^{n-1}$-valued pairing on Σ arises in this way. As such, if we choose Q in Σ and an abelian section σ: SQ → Σ, in dimension n, we obtain an abelian section in dimension n − 1. In general, Γn(Q, Q) ≠ 0, so we will prove the statement for all supercharges which will not necessarily square to zero. We can now build abelian subalgebras on a case-by-case basis. * Suppose $n \equiv 0, 4 \pmod 8$, so $\Sigma \iso S\_+\otimes W \oplus S\_-\otimes W^\$. Suppose Q* ∈ S+ ⊗ W. Then any section σ: SQ → S− ⊗ W\* has abelian image. * Suppose $n \equiv 7 \pmod 8$. Given a spinorial representation Σ of $\so(n)$, we can write it as a restriction of the spinorial representation S+ ⊗ W ⊕ S− ⊗ W\* of $\so(n+1)$ where both S+ and S− restrict to the spin representation of $\so(n)$. We may always arrange Q ∈ Σ to lie in S+ ⊗ W, so by the previous point we get an abelian section. The same analysis applies to the case $n\equiv 3\pmod 8$ with $\mc{N} \geq 2$. * Suppose $n \equiv 2, 6 \pmod 8$. In this case Σ = S+ ⊗ W+ ⊕ S− ⊗ W−. Suppose dim(W+) ≥ dim(W−) and consider the $\so(n)$-equivariant pairing Γn on S+ ⊗ W+ ⊕ S− ⊗ W+. It arises by restriction from an $\so(n+1)$-equivariant pairing Γn + 1 on (S+ ⊕ S−) ⊗ W+, where S+ ⊕ S− is the spin representation of $\so(n+1)$. We can choose a surjective map W+ → W− compatible with the pairings and lift Q to an element $\widetilde{Q}\in (S\_+\oplus S\_-)\otimes W\_+$. If $n\equiv 6 \pmod 8$ or $n\equiv 2\pmod 8$ and $\mc{N}\_+\geq 2$, then by the previous results we may find an abelian section $S\_{\widetilde{Q}}\rightarrow (S\_+\oplus S\_-)\otimes W\_+$. Post-composing it with the projection W+ → W−, we obtain an abelian section for Q. * Suppose $n \equiv 1, 5 \pmod 8$. In this case Σ = S ⊗ W which arises as a restriction of the spinorial representation S+ ⊗ W with its pairing in dimension n + 1. Thus, if $n\equiv 1\pmod 8$ and $\mc{N}\geq 2$ or $n\equiv 5\pmod 8$, we obtain an abelian section. This argument includes all the square-zero supercharges in dimensions less than 8 with the exception of supercharges in dimension 4 which contain components in both S+ and S−. In the latter case it is easy to construct an abelian section by hand. All the topological supercharges we will discuss in Section [twistedtheoriessection] are included in this result. The only square-zero supercharges that are not included are non-topological supercharges in dimensions 8, 9 and 10. We expect that the hypotheses in our statement can be weakened; we are not aware of any counter-examples to the existence of an abelian section to [Q, − ] in any dimension. Twisting Homomorphisms ---------------------- Supersymmetric theories in particular include an action of the spin group $\Spin(V\_\RR)$. However, this spin action does not survive the twisting procedure. An action of a Lie group G on the observables of a supersymmetric field theory is only well-defined after twisting by Q only if it preserves Q, i.e. we have [ν(Q), ν(X)] = 0 for every X ∈ ≫ . We can see the following. [twistinghomgivesGactionprop] If $\obs$ is an $\mf A$-supersymmetric factorization algebra on $\RR^n$ and Q is a square-zero supercharge, the twisted factorization algebra $\obs^Q$ admits a smooth action of $\stab(Q) \sub \Spin(V\_\RR) \times G\_R$. Since the representation Σ does not contain a trivial subrepresentation, a non-zero Q is never preserved under $\Spin(V\_\RR)$. Nevertheless, one can define a $\Spin(V\_\RR)$-action on $\obs^Q$ by means of a twisting homomorphism. Consider a Lie group G equipped with a homomorphism $G\rightarrow \Spin(V\_\RR)$. In particular, G acts on supersymmetric field theories on $\RR^n$. Let $\obs$ be an $\mf A$-supersymmetric factorization algebra on $\RR^n$. A twisting homomorphism is a homomorphism ϕ: G → GR. A square-zero supercharge Q is compatible with the twisting homomorphism ϕ if the action of the subgroup $(\id,\phi) \colon G \to G \times G\_R$ on Q is trivial. If a supercharge Q is compatible with a twisting homomorphism, then ϕ defines a smooth G-action on the twisted factorization algebra $\obs^Q$. To put this in context, in the physics literature a topological twist is often defined to be a pair consisting of a twisting homomorphism ϕ and a compatible topological supercharge Q (as in Witten’s original construction ). In fact, this is sometimes taken as the definition of a topological supercharge. This is justified by the following result. [compatibletwistinghomthm] Let Q be a square-zero supercharge in a supersymmetry algebra in dimension n ≥ 3 which is compatible with a twisting homomorphism $\phi\colon \Spin(V\_\RR)\to G\_R$. Then Q is a topological supercharge. By construction Σ carries an action of $\so(n)\oplus \gg\_R$. In particular, we may consider a modified $\so(n)$-action on Σ via the twisting homomorphism $(\id, \phi)\colon \so(n)\to\so(n)\oplus \gg\_R$. We denote this subalgebra by $\so(n)'$. By assumption $\CC Q\sub \Sigma$ is a trivial $\so(n)'$-subrepresentation. Therefore, $(\CC Q)\otimes \Sigma\sub \Sigma\otimes \Sigma$ is also a subrepresentation. The map $\Gamma\colon\Sigma\otimes \Sigma\to \CC^n$ is $\so(n)'$-equivariant since GR acts by automorphisms of the Lie structure. Therefore, the image of $[Q, -]\colon \Sigma\rightarrow \CC^n$ is an $\so(n)$-subrepresentation. By nondegeneracy of the pairing Γ, the image of [Q, − ] is nonzero. But for n > 2 the representation $\CC^n$ of $\so(n)$ is irreducible, so [Q, − ] must be surjective. The theorem is false in dimension 2. For instance, choose the spinorial representation Σ = S+⊕ 2 and the R-symmetry group $G\_R = \SO(2)$. If one takes the twisting homomorphism $\phi \colon \Spin(2) \to \SO(2)$ to be the standard cover, then there is a compatible supercharge. However in this supersymmetry algebra no supercharges are topological. We will discuss this supersymmetry algebra more completely in Section [twistedtheoriessection]. The formalism of twisting homomorphisms allows us to apply Corollary [Efrnsummarycor] in the context of topologically twisted theories. [twistedtheorywithtwistinghomthm] Let $\obs$ be an $\mf{A}$-supersymmetric factorization algebra on $\RR^n$, Q a square-zero supercharge compatible with a twisting homomorphism $\phi\colon \SO(n)\rightarrow G\_R$. Suppose that the following two conditions hold. * The factorization map $\obs^Q(B\_r(0))\rightarrow \obs^Q(B\_R(0))$ is a quasi-isomorphism. * The ϕ-twisted $\SO(n)$-action on $\obs^Q$ extends to a de Rham $\SO(n)$-action. Then the collection $\{\obs^Q(B\_r(0))\}$ forms a framed $\bb E\_n$-algebra. This is a direct application of Corollary [Efrnsummarycor]. By Theorem [compatibletwistinghomthm] Q is a topological supercharge. Then by Proposition [topologicallytwistedtheoriesaredRtranslationinvtprop] and Proposition [commutingpotentialprop] the Q-twisted factorization algebra is de Rham translation invariant. The hypotheses in the statement of the theorem ensure that we can apply Corollary [Efrnsummarycor]. Pure Spinors ------------ The subspace of square-zero supercharges in any dimension is closely related to the space of pure spinors introduced by Cartan and Chevalley. We will make reference to this theory when we discuss $\mc N=1$ or $\mc N=(1,0)$ square-zero supercharges in high dimensions. We refer to, and for a reference on pure spinors. Let S be an irreducible representation of the Clifford algebra $\mr{Cl}(V)$. Here S coincides with the Dirac spinor representation of $\so(V)$: that is, in odd dimensions S is the spin representation of $\so(V)$ and in even dimensions S ≅ S+ ⊕ S− is the sum of the semi-spin representations of $\so(V)$. Let $L\sub V$ be a maximal isotropic subspace and let $\SO(V)\_L\sub \SO(V)$ be the subgroup preserving $L\sub V$. We can identify $\SO(V)\_L\cong \GL(L)\ltimes G\_L$ where GL has the following description: * If n is even, GL = ∧ 2L. * If n is odd, GL is a central extension 0 → ∧ 2L → GL → L → 0. Explicitly, GL is the group of pairs (v, ω) ∈ (L ⊕ ∧ 2L) with the multiplication rule (v1, ω1)(v2, ω2) = (v1 + v2, ω1 + ω2 + v1 ∧ v2). The preimage $\Spin(V)\_L\sub \Spin(V)$ of $\SO(V)\_L\sub \SO(V)$ under $\Spin(V)\rightarrow \SO(V)$ is identified with ${\mathrm{ML}}(L)\ltimes G\_L$, where ML(L) is the metalinear group. Then we can identify the restriction of the spinor representation to the subgroup $\Spin(V)\_L$ as S ≅ ∧ •L ⊗ det(L)− 1/2 where v ∈ L acts by multiplication by 1 + v and ω ∈ ∧ 2L acts by multiplication by eω. Pick a nondegenerate pairing $\langle -, -\rangle\colon S\otimes S\rightarrow \CC$ such that ⟨ρ(v)Q1, ρ(v)Q2⟩ = ⟨Q1, Q2⟩ for v a unit vector, which is unique up to rescaling. Here we write ρ(v) to denote the Clifford multiplication by the vector v ∈ V. Then we define a nondegenerate pairing Γ: S ⊗ S → V by (v, Γ(Q1, Q2)) = ⟨ρ(v)Q1, Q2⟩ for v ∈ V and Q1, Q2 ∈ S, where ( − , − ) denotes the inner product on V. These pairings coincide with the symmetric and anti-symmetric pairings we described at the beginning of Section [susyalgebrasection]. For a spinor Q ∈ S, its nullspace $T\_Q\sub V$ is the space of vectors v ∈ V such that ρ(v)Q = 0. It is easy to see that for any nonzero spinor Q ∈ S, the nullspace $T\_Q\sub V$ is isotropic. This motivates the following definition. The spinor is called pure if TQ is a maximal isotropic subspace, i.e. $\dim T\_Q = \lfloor \frac n2 \rfloor$. From the description of the spinor module S as a $\Spin(V)\_L$-representation we get the following statement (see also ). [prop:purespinorstabilizer] Let $L\sub V$ be a maximal isotropic subspace. 1. The subspace $\{Q\in S\colon \rho(L)Q=0\}\sub S$ is one-dimensional and isomorphic to det(L)1/2 as a $\Spin(V)\_L$-representation. 2. Suppose Q ∈ S is a pure spinor such that ρ(L)Q = 0. Then its stabilizer in $\Spin(V)$ is isomorphic to $(\SL(L)\times \ZZ/2\ZZ)\ltimes G\_L$. Now, let us investigate the image of the Γ pairing with a pure spinor, i.e. the subspace $$S\_Q = \mr{Im}(\Gamma(Q,-)) \sub V.$$ [purespinorimage] Let Q be a spinor. Then SQ = TQ⊥. In particular, if Q is a pure spinor, * If n is even, then SQ = TQ. * If n is odd, then $T\_Q \sub S\_Q = T\_Q^\perp$ is a codimension 1 subspace. Let v ∈ SQ⊥, i.e. (v, w) = 0 for all w ∈ SQ. That is for every Qʹ ∈ S we have 0 = (v, Γ(Q, Qʹ)) = ⟨ρ(v)Q, Qʹ⟩. Since this is true for every Qʹ, we get that ρ(v)Q = 0, i.e. SQ⊥ = TQ. Taking the orthogonal complement again, we get SQ = TQ⊥. If Q is a pure spinor, TQ is maximal isotropic, so in even dimensions SQ = TQ and in odd dimensions $T\_Q\sub S\_Q$ has codimension 1. Let us return to prove the statement we claimed in Section [twistsection]: that all supercharges are at least holomorphic. [atleastholoprop] Let Q ∈ Σ be an element of an arbitrary spinorial representation in dimension n equipped with a non-degenerate pairing. Then the image space SQ has dimension at least n/2. First, if Q ∈ S, then by Proposition [purespinorimage] SQ has dimension ≥ n/2, since TQ has dimension ≤ n/2. If Σ = S ⊗ W, then if we choose a basis {w1, …, wk} for W, we can write Q = Q1 ⊗ w1 + ⋯ + Qk ⊗ wk, and then $S\_Q = \mr{span}(S\_{Q\_1},\ldots, S\_{Q\_k})$ which then also has dimension ≥ n/2. More generally, suppose Σ = S+ ⊗ W+ ⊕ S− ⊕ W− where there is a strict inclusion $W\_- \subsetneq W\_+$. Then S+ and S− are paired with themselves. For an element Q1 ⊗ w1 ∈ S+ ⊗ W+ we can identify the image of Γ(Q1 ⊗ w1, − ) in the spaces S+ ⊗ W+ ⊕ S− ⊕ W− and in (S+ ⊕ S−) ⊗ W+. So we still have that $S\_{Q\_1} \sub S\_Q$, so SQ has dimension ≥ n/2. Now, what is the relationship between pure spinors and those satisfying Γ(Q, Q) = 0 (i.e. that square to zero in the supersymmetry algebra)? To answer this question, we will use an interesting decomposition. Every spinor Q induces a matrix element $$Q \otimes Q \in S \otimes S\cong S\otimes S^\* \iso \eend(S)$$ using the invariant pairing on S. Since S is irreducible, we can identify endomorphisms with elements of the Clifford algebra, and thus decompose the spinor bilinear as $$Q \otimes Q = \sum\_{p=0}^n F\_p \in \eend(S)\cong \mr{Cl}(V)\cong \wedge^\bullet(V),$$ where Fp ∈ ∧ pV. We can explicitly identify these elements Fp as, in index notation, the Clifford algebra element Qiγija1⋯apQj. There is a nice characterisation of pure spinors in this language. [Chevalley] [ChevalleyCriterion] A non-zero spinor Q is pure if and only if it is Weyl (in even dimensions), and Fp = 0 for all p < ⌊n/2⌋. For instance, if n ≥ 4 and Q is a pure spinor, we have F1 = Γ(Q, Q) = 0. In other words, any pure spinor in these dimensions is square-zero. We have the following observation (see ). [WeylCriterion] If Q is a Weyl spinor in dimension n = 2m, the tensor Fp is zero unless $p\equiv m\pmod 4$. Finally, we can characterize pure spinors in odd dimensions in terms of those in even dimensions. Indeed, suppose n is odd and S is the spinor module. If we denote by S+ the semi-spin representation of $\Spin(V\oplus {\mathrm{C}})$, then we can identify S ≅ S+ as $\Spin(V)$-representations. [][prop:oddimensionpurespinor] A spinor Q ∈ S in an odd dimension n is pure if and only if it is pure as a Weyl spinor Q ∈ S+ in dimension n + 1. Now, let us use these facts to investigate pure spinors in various dimensions. * If n = 2, 4, 6 every non-zero Weyl spinor is pure. * If n = 1, 3, 5 every non-zero spinor is pure. * If n = 8 we have a single constraint for a Weyl spinor to be pure, that F0 = 0. That is, the spinor satisfies ⟨Q, Q⟩ = 0. All Weyl spinors satisfy Γ(Q, Q) = 0 however, so in this dimension squaring to zero is strictly weaker than being pure. As we will see in Section [dim8section] all impure square-zero spinors in dimension 8 are in fact topological. * If n = 7 we can use Proposition [prop:oddimensionpurespinor] to conclude that a spinor Q is pure if and only if ⟨Q, Q⟩8 = 0, where ⟨ − , − ⟩8 is the scalar-valued spinor pairing in 8 dimensions. * If n = 10 the constraint for a Weyl spinor to be pure is that F1 = 0. That is, a Weyl spinor Q in dimension 10 is pure if and only if Γ(Q, Q) = 0. * If n = 9 we can again use Proposition [prop:oddimensionpurespinor] to conclude that a spinor Q is pure if and only if Γ10(Q, Q) = 0. To summarise, there are non-trivial pure spinor constraints only in dimensions at least 7. In the classification of twists we will refer to purity only in those large dimensions where it is a non-trivial condition. The Stress-Energy Tensor ------------------------ Often topologically twisted theories carry a natural action of all vector fields extending the action of the Poincaré algebra. In this section we explain how this action can be constructed. The results in this section will not be used in the rest of the paper, but instead help to motivate the concept of the topological twist by means of the following structure: topologically twisted field theories are often automatically generally covariant, meaning they carry an action of the group of diffeomorphisms of spacetime. Let g0 be the standard metric on $\RR^n$ of signature (p, q) with constant coefficients. The local Lie algebra of Killing vector fields is the sheaf of dg Lie algebras given by $$L\_{\mr{Kill}} = \big(T\_{\RR^n} \to \sym^2 T\_{\RR^n}\big)$$ placed in degrees 0 and 1, where the differential sends a vector field X to the Lie derivative $\LL\_X(g\_0)$, and where the Lie bracket is likewise given by the Lie derivative. Note that the Poincaré algebra $\mf{iso}(n)$ acts by Killing vector fields on $\RR^n$, so we have a natural morphism $\mf{iso}(n) \to L\_{\mr{Kill}}(U)$ for each open set $U\sub \RR^n$. A prefactorization algebra on $\RR^n$ is isometry-invariant if carries a smooth $\ISO(p, q)$-action and the infinitesimal Poincaré algebra action extends to an inner action $\nu\colon L\_{\mr{Kill}}\rightarrow \obs(\RR^n)[-1]$ of $L\_{\mr{Kill}}$. The gravitational stress-energy tensor $T\_{\mr{grav}}$ of an isometry-invariant factorization algebra $\obs$ is the composite $$T\_{\mr{grav}}\colon \sym^2 T\_{\RR\_n}\longrightarrow L\_{\mr{Kill}}[1]\longrightarrow \obs(\RR^n).$$ The gravitational stress-energy tensor of a field theory models infinitesimal deformations of the background metric g0. Note that this is not the same as the “canonical” stress-energy tensor modelling the conserved currents of the global translation action. These two tensors are related by a correction term involving the conserved currents for rotations: this is known as the Belinfante-Rosenfeld formula (see for instance for a rigorous account of this relationship). Given a quantum field theory arising from a classical Lagrangian description where the fields are tensors, i.e. sections of $T\_{\RR^n}^{\otimes a}\otimes (T^\\_{\RR^n})^{\otimes b}$, we have a natural action of the Lie algebra of vector fields on $\RR^n$ on the space of fields. This gives a isometry invariant structure on the factorization algebra of observables $\obs$ where the gravitational stress-energy tensor is $$T\_{\mr{grav}}(\delta g) = \int \left.\frac{\delta S}{\delta (g^{\mu\nu}(x))}\right\|\_{g=g\_0} \delta g^{\mu\nu}(x),$$ the variation of the action with respect to the metric. Typical supersymmetric field theories involve fields which are sections of spinor bundles, so they are not locally isometry invariant in this sense. However, consider a twisting homomorphism $\phi\colon \SO(p, q)\rightarrow G\_R$ so that under the induced $\SO(p, q)$-action the fields have integer spins (i.e. are tensors). Then we obtain that the prefactorization algebra of observables $\obs$ is locally isometry invariant with respect to the twisted $\SO(p, q)$ action. If we moreover assume that the supercharge Q* is compatible with the twisting homomorphism ϕ, the twisted prefactorization algebra $\obs^Q$ is also locally isometry invariant. Let us fix a supersymmetric factorization algebra $\obs$ on $\RR^n$ with a twisting homomorphism $\phi\colon \SO(p, q)\rightarrow G\_R$ and a compatible supercharge Q. Moreover, suppose that the twisted factorization algebra $\obs^Q$ is locally isometry invariant. [exactSEtensorlemma] Suppose $\obs$ is equipped with a map $$\Lambda\colon \Gamma(\RR^n, \sym^2 T\_{\RR^n})\rightarrow \obs(\RR^n)[-1]$$ satisfying the following equations: 1. $\d\_Q \Lambda(\delta g) = T\_{\mr{grav}}(\delta g)$. 2. $\Lambda(\LL\_{[v, w]} g\_0) = [\nu(v), \Lambda(\LL\_w(g\_0))] + [\Lambda(\LL\_v(g\_0)), \nu(w)] - [\Lambda(\LL\_v g\_0), \Lambda(\LL\_w g\_0)]$. Then the Poincaré action on $\obs^Q$ extends to an infinitesimal action of the tangent sheaf $T\_{\RR^n}$ as a sheaf of Lie algebras. Define a map $$\nu\_Q\colon \Gamma(\RR^n, T\_{\RR^n})\rightarrow \obs^Q(\RR^n)[-1]$$ by $$\nu\_Q(v) = \nu(v) - \Lambda(\LL\_v(g\_0)).$$ The fact that it is a morphism of dg Lie algebras follows from the equations on Λ. The Lie algebra of vector fields on $\RR^n$ is a Lie algebra of the group $\mr{Diff}(\RR^n)$ of diffeomorphisms, so we have shown that topologically twisted theories with null-homotopic stress-energy tensor are generally covariant (at least infinitesimally – the infinitesimal $T\_{\RR^n}$-action need not necessarily exponentiate). Witten discusses topological twisting from this perspective in his original paper on topological quantum field theories. In physical examples where one has a Lagrangian description for a classical field theory with an explicit metric dependence one can check that the action functional consists of a Q-exact term plus a metric independent term. This implies that the variation of the action with respect to the metric, the gravitational stress-energy tensor, is Q-exact. Dilation Actions ---------------- When can we apply the result of Section [Ensection] to topologically twisted field theories? Since topological twists are automatically de Rham translation-invariant, according to Corollary [Ensummarycor] the factorization algebra $\obs^Q$ of twisted observables admits a canonical $\bb E\_n$-algebra structure whenever the factorization map $\mr{fact}\_{r,R} \colon \obs^Q(B\_r(0)) \to \obs^Q(B\_R(0))$ associated to concentric balls is a quasi-isomorphism. When we have a direct description of the local twisted observables, we can check this condition by hand. Alternatively, we can abstractly describe quasi-inverses to the factorization maps on concentric balls by means of a dilation action. A prefactorization algebra $\obs$ on $\RR^n$ is dilation-invariant if it admits a smooth action of the group $\RR\_{>0}$ acting on $\RR^n$ by dilations. Such a smooth dilation action defines a rescaling map $\alpha\_{R,r} \colon \obs(B\_R(0)) \to \obs(B\_r(0))$ for any pair of radii r, R. This rescaling map makes a de Rham translation-invariant factorization algebra into an $\bb E\_n$-algebra as long as the dilation action is itself homotopically trivial. [dRdilationdef] A dilation-invariant prefactorization algebra $\obs$ on $\RR^n$ is de Rham dilation-invariant if the smooth $\RR\_{>0}$-action extends to an action of $(\RR\_{>0})\_{\dR}$. [dRdilationprop] Let $\obs$ be a de Rham dilation-invariant prefactorization algebra on $\RR^n$. The rescaling map αR, r provides a quasi-inverse to the factorization map $\mr{fact}\_{r,R}$. Fix r > 0 and consider the family of endomorphisms $m\_s = \mr{fact}\_{sr,r} \circ \alpha\_{r,sr}$ of the local observables $\obs(B\_r(0))$ parameterised by rescalings 0 < s ≤ 1. By the definition of a smooth action, ms is a smooth family of endomorphisms of $\obs(B\_r(0))$ which is the identity at s = 1. Thus, to show that it is homotopic to the identity, it will be enough to show that $\frac{\d}{\d s} m\_s$ is null-homotopic for any s. If we denote by $$\epsilon = \d \eta + \eta \d$$ the derivation generating the dilation action, then we have $$\begin{aligned} \frac{\d}{\d s} m\_s(\OO) &= m\_s(\epsilon(\OO)) \\ &= \d m\_s(\eta(\OO)) + m\_s(\eta(\d\OO)).\end{aligned}$$ Thus, ms ∘ η provides a nullhomotopy for $\frac{\d}{\d s} m\_s$. Suppose $\obs$ is a prefactorization algebra on $\RR^n$ valued in BD0-algebras. Then the factorization maps $\mr{fact}\_{r, R}$ are quasi-isomorphisms if and only if they are quasi-isomorphisms at ℏ = 0. This can be proved by a spectral sequence argument similar to. There are de Rham dilation-invariant field theories on the classical level where the dilation action is anomalous on the quantum level; this argument tells us that $\mr{fact}\_{r, R}$ is nevertheless still a quasi-isomorphism on the quantum level. [remark:classicaldilationaction] Combining Proposition [dRdilationprop] and Corollary [Ensummarycor] we get the following. Let $\obs$ be a de Rham dilation-invariant prefactorization algebra on $\RR^n$. Then $\obs(B\_1(0))$ can be canonically equipped with the structure of a $\disk\_n$-algebra. Superconformal Theories ----------------------- We can build twisted theories which are de Rham dilation invariant as well as de Rham translation-invariant starting from not just a supersymmetry action, but a superconformal action. We refer to Kac’s survey article for a comprehensive account of the theory of Lie superalgebras and superconformal algebras. Fix an inner product on $\RR^n$ of signature (p, q). The conformal group $\mr{CO}(p,q)$ of signature (p, q) is the group of oriented conformal transformations of $\RR^n$. We always have an inclusion $\SO(p+1, q+1)\sub \mr{CO}(p, q)$. If (p, q) = (1, 1), (1, 0) or (0, 1), the conformal group is infinite-dimensional. In the other cases this inclusion is an isomorphism. Let us denote by $\mf{co}(p, q)$ the complexified Lie algebra of $\mr{CO}(p, q)$. A superconformal algebra is a simple super Lie algebra whose bosonic part has form $\mf{co}(p,q) \oplus \gg\_R$ for some Lie algebra ≫ R, and where the action of $\mf{co}(p,q)$ (if n ≤ 2, the action of its subalgebra isomorphic to $\so(p+1,q+1)$) on the fermionic part defines a spinorial representation. The possible superconformal algebras in dimensions ≥ 3 were classified by Nahm and Shnider, who showed that they only exist in dimensions up to 6 [1](#fn1). A factorization algebra on $\RR^n$ is superconformal if it admits a smooth action of $\mr{CO}(p, q)\times G\_R$, so that the action of $\mf{co}(p, q)\oplus \gg\_R$ extends to an action of a superconformal algebra $\mf{A}$. We can explicitly describe the possible superconformal algebras starting from dimension 6 and working down using Kac’s classification of finite-dimensional semisimple super Lie algebras. That is, identifying those super Lie algebras where the bosonic part is equivalent to $\so(n+2) \oplus \gg\_R$ and where the action on the fermionic part is spinorial. Let $\mf{A}$ be a superconformal algebra in dimension n. Then its even part is $$\mf{A}\_{\mr{even}} = \so(n)\oplus V\oplus V\oplus \CC\cdot D\oplus \gg\_R,$$ where the two copies of V correspond to translations and special conformal transformations respectively and where D is the generator of dilations. Its odd part is $$\mf{A}\_{\mr{odd}} = \Sigma\oplus \Sigma^\$$ where Σ is a spinorial representation of $\so(n)$. The superconformal algebra $\mf{A}$ contains a subalgebra spanned by Σ and translations; it also contains an isomorphic subalgebra spanned by Σ\ and special conformal transformations. We also have a bracket $$\Sigma\otimes \Sigma^\\longrightarrow \so(n)\oplus \CC\cdot D\oplus \gg\_R$$ whose D* component corresponds to the natural pairing. This bracket has the property that if Σ = S ⊗ W the projection $\Sigma \otimes \Sigma^\* \to \so(n) \oplus \CC \cdot D$ factors through the natural evaluation pairing on W: $$\Sigma\otimes \Sigma^\* \to S \otimes S^\* \to \so(n)\oplus \CC\cdot D.$$ Likewise if Σ = S+ ⊗ W+ ⊕ S− ⊗ W− it factors through $$\Sigma\otimes \Sigma^\* \to S\_+ \otimes S\_+^\* \oplus S\_- \otimes S\_-^\* \to \so(n)\oplus \CC\cdot D.$$ We refer to for explicit formulas for brackets. * In dimension 6 there are superconformal algebras with $\mc N=(k,0)$ supersymmetry for all k isomorphic to $\osp(8\|k)$. * In dimension 5 there is a unique superconformal algebra: the exceptional super Lie algebra $\mf f(4)$ with bosonic part $\so(7) \oplus \sl(2)$. This superconformal algebra contains an $\mc N=1$ supersymmetry algebra. * In dimension 4 there is a superconformal algebra for all $\mc N$ given by $\sl(4\|\mc N)$ unless $\mc N=4$ in which case we should quotient by the center to get $\mf{psl}(4\|4)$. * In dimension 3 there is again a superconformal algebra for all $\mc N$ isomorphic to $\osp(\mc N\|4)$. * In dimension 2 the superconformal algebras are infinite-dimensional. For a classification of the possible superconformal algebras see Fattori-Kac. The most notable 2-dimensional superconformal algebras are the supersymmetric extensions of the Virasoro algebra such as the Ramond, Neveu–Schwarz and $\mc N=2$ superconformal algebras. * In dimension 1 there are many possible superconformal algebras, where we restrict attention to supersymmetric extensions of the small conformal algebra $\so(3) \iso \sl(2)$. We refer to for more details. Since superconformal factorization algebras are in particular supersymmetric it makes sense to form their twist by a square-zero supercharge Q. These twisted theories admit a smooth action of the subgroup of the conformal group stabilizing Q just like we discussed in Section [twistinghomsection]. As such, we can make a Q-twisted factorization algebra $\obs^Q$ dilation invariant whenever we have a twisting homomorphism for the dilation action of the form $$\psi \colon \RR\_{>0} \to G\_R$$ compatible with Q. Note that since $\RR\_{>0}$ is contractible, it is enough to specify the twisting homomorphism on the level of Lie algebras. In fact, this is always possible. [dilationpotentiallemma] If Q is a square-zero supercharge in a superconformal algebra $\mf A$ as follows: * The $\mc N=(2,2)$ superconformal algebra in dimension 2 with Q a topological supercharge. * The $\mc N=k$ superconformal algebra in dimension 3 with Q of rank 2. * The $\mc N=k$ superconformal algebra in dimension 4 with Q of rank at least 2. * The $\mc N=1$ superconformal algebra in dimension 5 with Q of rank 2. * The $\mc N=(k,0)$ superconformal algebra in dimension 6 with Q of rank 4. Then there exists a twisting homomorphism ψ for the dilation group so that Q is invariant for the twisted dilation action. Furthermore, this twisted dilation action is de Rham. We can characterize the data of a dilation twisting homomorphism as an element J of the R-symmetry algebra $\mf g\_R$ so that [J, Q] = [D, Q] in the superconformal algebra, where D is the infinitesimal dilation. This twisted dilation action actually makes the twisted theory de Rham dilation invariant if J − D is actually Q-exact, i.e. if there exists an odd element Z of the superconformal algebra so that [Z, Q] = J − D. That is, it suffices to find Z so that the $\so(n)$ component of [Z, Q] vanishes but the D component of [Z, Q] does not vanish. We can check that such elements Z and therefore J exist by considering the classification of superconformal algebras. That is, we’ll consider Q ⊗ Z as an element of Σ ⊗ Σ\, and choose Z* so that Q ⊗ Z vanishes when we project onto the irreducible summands of the $\so(n)$-module Σ ⊗ Σ\* isomorphic to the adjoint representation, but does not vanish when we project those summands corresponding to the trivial representation. Decompose Q = ∑si ⊗ wi as a sum of pure tensors in Σ = S ⊗ W or S+ ⊗ W+ ⊕ S− ⊗ W−. Recall that the odd part of the superconformal algebra splits as Σ ⊕ Σ\, and recall that the $\so(n) \oplus \CC \cdot D$ part of the bracket between Σ and Σ\ factors through the evaluation pairing on W, or the separate evaluation pairings on W+ and W−. We will construct Z ∈ Σ\* in each dimension in turn. * In dimension 2 the $\mc{N}=2$ Neveu–Schwarz algebra is spanned by the Virasoro generators $\{L\_n\}\_{n\in\ZZ}$, odd elements $\{G^+\_i,G^-\_i\}\_{i\in \frac 12+\ZZ}$, R-symmetry generators $\{J\_n\}\_{n\in\ZZ}$ and the central element 1 with brackets of the form $$[G^+\_i, G^-\_j] = L\_{i+j} + \frac{1}{2} (i-j) J\_{i+j} + \delta\_{i,-j} \left(\frac{i^2}6 - \frac 1{24}\right) c \textbf{1},$$ where c is the central charge. We consider the $\mc{N}=(2, 2)$ superconformal algebra which is given by the sum of two $\mc{N}=2$ Neveu–Schwarz algebras with generators we denote by {Gi+, Gi−, Ln, Jn, 1} and $\{\overline{G}^+\_i, \overline{G}^-\_i, \overline{L}\_n, \overline{J}\_n, \overline{\textbf{1}}\}$. In particular, for the natural $\mc N=(2,2)$ topological supercharge $Q = G^-\_{-1/2} + \overline{G}^-\_{-1/2}$ we have $$[Q, G^+\_{1/2} + \overline{G}^+\_{1/2}] = L\_0 + \overline{L}\_0 + \frac{1}{2} (J\_0 + \overline{J}\_0),$$ where $L\_0 + \overline{L}\_0$ is the generator of dilations. In particular, there is a twisting homomorphism making Q dilation-invariant. * In dimension 3 $S \iso S^\$ and the tensor product of representations decomposes as $S \otimes S^\ \iso \sym^2(S) \oplus \wedge^2(S) \iso \so(3) \oplus \CC$. Since the bracket between Σ and Σ\* is $\so(3)$-equivariant, the $\so(3) \oplus \CC \cdot D$ part of the bracket in the superconformal algebra is uniquely determined up to rescaling. Choose f1, f2 ∈ W\* so that fi(wj) = δij for j = 1, ⋯, 4. Then if Z = s2 ⊗ f1 − s1 ⊗ f2 ∈ S ⊗ W\* then, since Q had rank at least 2, the image of Q ⊗ Z in $\sym^2(S)$ vanishes, and so [Q, Z] = D + J for some J as required. * In dimension 4 $S\_\pm \iso S^\\_\pm$ and the tensor product of representations decomposes as $S\_+ \otimes S\_+^\ \oplus S\_- \otimes S\_-^\* \iso \sym^2(S\_+) \oplus \sym^2(S\_-) \oplus \wedge^2(S\_+) \oplus \wedge^2(S\_-) \iso \so(4) \oplus \CC \oplus \CC$. Therefore as in dimension 3 $\so(4)$-equivariant of the bracket, along with the $\CC \cdot D$ term being symmetrical in + and − , means the $\so(4) \oplus \CC \cdot D$ part of the bracket in the superconformal algebra is uniquely determined up to rescaling. Choose f1, f2 ∈ W+\* ⊕ W−\* so that fi(wj) = δij for j = 1, ⋯, 4. Then if Z = s2 ⊗ f1 − s1 ⊗ f2 ∈ Σ\* then, since Q had rank at least 2, the image of Q ⊗ Z in $\sym^2(S\_+) \oplus \sym^2(S\_-)$ vanishes, and so [Q, Z] = D + J for some J. * In dimension 5 $S \iso S^\$ and the tensor product of representations decomposes as $S \otimes S^\ \iso \sym^2(S) \oplus \wedge^2(S) \iso \so(5) \oplus (V \oplus \CC)$. Therefore as above the $\so(5) \oplus \CC \cdot D$ part of the bracket in the superconformal algebra is uniquely determined up to rescaling. Here W has a symplectic structure which induces an isomorphism W ⊗ W\, so let Z* = Q = s1 ⊗ w1 + s2 ⊗ w2. Then, since Q had rank at least 2, [Z, Q] = ω(w1, w2)(s1 ⊗ s2 − s2 ⊗ s1), which vanishes when projected to $\sym^2(S)$. It also vanishes when projected to V because Q squares to zero in the supersymmetry algebra. However since s1 ≠ s2 it does not vanish when projected to $\CC \cdot D$, so [Q, Z] = λD + J for some J and some non zero $\lambda \in \CC$. * In dimension 6 S+ ⊗ S+\* decomposes into irreducible summands as $\so(6) \oplus \CC$, where we view S as the defining representation of $\sl(4)$, so $S\_+ \otimes S\_+^\* \iso \gl(4)$ and the projection onto $\CC$ is given by the trace. Again this uniquely determined the $\so(6) \oplus \CC \cdot D$ part of the bracket in the superconformal algebra is uniquely determined up to rescaling. Choose Z ∈ Σ\* of rank 4 so that the image of Q ⊗ Z under the evaluation pairing Σ ⊗ Σ → S+ ⊗ S+\* is represented by the matrix $\lambda \cdot \mr{id}\_4$ for λ ≠ 0. This vanishes under the projection to $\so(6) \iso \sl(4)$, but not under the projection to $\CC \cdot D$. Therefore [Q, Z] = λD + J for some J and some non zero $\lambda \in \CC$. In dimension 3 and larger this result is best possible: if Q has lower rank then there is never a twisting homomorphism for the dilation group coming from the superconformal algebra. However as we will see explicitly in Section [twistedtheoriessection]	arxiv_0000545
A Graphical Approach to Treatment Effect Estimation with Observational Network Data =================================================================================== We propose an easy-to-use adjustment estimator for the effect of a treatment based on observational data from a single (social) network of units. The approach allows for interactions among units within the network, called interference, and for observed confounding. We define a simplified causal graph that does not differentiate between units, called generic graph. Using valid adjustment sets determined in the generic graph, we can identify the treatment effect and build a corresponding estimator. We establish the estimator’s consistency and its convergence to a Gaussian limiting distribution at the parametric rate under certain regularity conditions that restrict the growth of dependencies among units. We empirically verify the theoretical properties of our estimator through a simulation study and apply it to estimate the effect of a strict facial-mask policy on the spread of COVID-19 in Switzerland. Keywords: causality, graphical model, interference, valid adjustment. Introduction ============ One common assumptions in causal inference is the stable unit-treatment value assumption (SUTVA). Part of SUTVA is the no-interference assumption, that is, the assumption that the treatment status of a unit may only influence the outcome of that unit and not the outcome of any other unit. In practical applications, however, interference is common as units can interact. For example the vaccination of a person against an infectious disease also helps protect the health of that person’s social contacts. Another example are students interacting in their class at school, such that a child’s test score at the end of the year is not only affected by the student’s math instruction type, but also the instruction type other students in the class received. Ignoring interference can lead to faulty conclusions. It is therefore important to account for interference when estimating treatment effects in networks, but there are three major difficulties in doing so. First, in the classical i.i.d. setting with a binary treatment and N independent units, there is one counterfactual treatment for each of the N units, namely the treatment that was not assigned to that unit. In the interference setting with N dependent units, there are 2N − 1 counterfactual treatments for each unit, namely one for every possible treatment assignment of the N units except the observed one. As a result, it is less clear how to define causal effects such as the average treatment effect (ATE). One standard target effect in the literature is the difference between the average expected unit-specific outcome of two different hypothetical stochastic treatment interventions that assign treatments to units independently with a user-specified treatment probability. We call this class of effects global treatment effects. A special case is the average total treatment effect, also called the global average treatment effect (GATE), which contrasts the hypothetical interventions of treating all units versus treating none. Second, to account for interference, it is generally necessary to model it by making assumptions on the specific structure and pathways of the interference. A common assumption in the literature, called partial interference, is that interference takes place in arbitrary form but only within nonoverlapping groups of units and not across these groups. Another is to describe the dependencies among units via a known interaction network graph, in which the nodes represent the units and the edges indicate relations between units that facilitate interaction, such as geographical proximity. Given a network graph, it is possible to model interference by summarizing a unit’s dependence on the treatment of other units through a finite set of functions that are common to all the units in the population and depend on the network graph. In the structural equation model (SEM) framework these functions are generally called interference features. Third, in many applications only observational data may be available. In such settings, it is important to account for confounding when estimating treatment effects in networks. This is a difficult problem, but one that has been extensively studied in the i.i.d. setting. For example, given knowledge of the underlying causal structure in the form of a causal graph, the class of adjustment sets that correct for confounding has been fully characterized. The members of this class are called valid adjustment sets. It is, however, unclear under what conditions we can apply these graphical results from the i.i.d. setting to settings with interference. In this paper we consider the estimation of treatment effects based on observational data from networks with interference and within-unit confounding, that is, confounding between a unit’s treatment and its outcome. The target effects are global treatment effects and we work in the framework of SEMs. Concretely, we assume a class of SEMs Se on explicit variables, that is, covariates Ci, treatments Wi and outcomes Yi, for all units i = 1, …, N. With such explicit SEMs we can represent the simultaneous presence of within-unit confounding and interference. Based on the explicit directed acyclic graph (DAG) Ge corresponding to Se, we define the generic graph G on the variables $\boldsymbol{C}$, W and Y by stacking the subgraphs for each unit i of Ge. While the generic graph is not as informative as the original explicit DAG, we show that for the class of explicit SEMs we consider, the generic graph can be used to identify a class of causal effects we call unit-specific effects. Global treatment effects, however, do not belong to the class of unit-specific effects. To obtain an identification result for global treatment effects, we therefore adopt the approach of modelling interference via interference features, a finite set of known functions of the known interaction network graph and the treatment vector of the entire population. In addition, we assume a linear outcome model. Based on these two assumptions we show that we can rewrite the target global treatment effect as the weighted average of unit-specific effects, where the weights can be explicitly computed or approximated, and the unit-specific effects can be identified from the generic graph G, using tools from causal graphical models. In particular, we will use graphical criteria for valid adjustment sets. Based on this identification result we then propose an adjustment estimator for global average treatment effects. Under some regularity conditions that limit the growth of dependencies between units, we prove that this estimator is consistent and converges at the parametric N− 1/2-rate to a Gaussian limiting distribution with finite variance that can be consistently estimated. Methodologically, our work is most similar to the work of and, with whom we share the assumption of a linear outcome model., however, does not allow for confounding and are interested in the bias of estimating the ATE if the units were isolated. Conceptually, our work is also related to the semi-parametric estimation of treatment effects in networks. This literature, however, either makes simplifying assumptions under which graphical identifiability results are trivial and/or estimate other treatment effects. Finally, there exists literature on identifying treatment effects in networks using explicit DAGs. However, the number of nodes in these graphs grows with the number of units N and as a result these graphs become difficult to use for larger sample sizes. The paper is organized as follows. In Section [sec:prelim] we introduce the set-up and the target effects. In Section [sec:identification] we introduce the generic graph and interference features and discuss the identification of treatment effects using the generic graph. In Section [sec:consistency] we showcase the use of the generic graph by proposing an adjustment estimator. In Section [sec:empirical] we perform a simulation study to verify the properties of the adjustment estimator and apply our methods to estimate the effect of a strict facial-mask policy on the spread of COVID-19 in Switzerland. The code for the simulation study and the facial-mask policy analysis is available at [github.com/henckell/InterferenceCode](https://github.com/henckell/InterferenceCode) and proofs are provided in the appendix. Preliminaries ============= We consider a population of N units. For each unit i we observe a binary treatment Wi ∈ {0, 1}, a possibly multivariate vector of covariates Ci ∈ RDC, and a continuous outcome Yi ∈ R. We aim to estimate a causal effect of the treatment on the outcome accounting for the presence of within-unit confounding and interference. We illustrate the problem in Example [example1]. [example1] We consider people interacting in their social network. Given a person i, the severity of a viral disease is the outcome Yi and the vaccination against the disease is the treatment Wi. Each person chooses whether to take the vaccination or not. This decision is governed by the variable Ci, representing the severity of previous infections with the disease. The variable Ci also affects the outcome, that is, the severity of a new infection with the disease. Thus, Ci constitutes within-unit confounding through the confounding path Wi ← Ci → Yi. In addition, the treatment status Wj of person j affects person j’s viral load. If person i is in close contact with person j, person j’s viral load may in turn affect the severity of disease Yi of person i. The fact that the treatment of person j affects the outcome of person i constitutes interference. Throughout the paper, we consider two types of random variables. Variables that distinguish between units, called explicit variables, and variables that do not, called generic variables. For example, we use Wi to denote the explicit treatment variable for unit i and W to denote the generic treatment variable that does not distinguish between units. We use $\boldsymbol{\bar{W}}=(W\_1,\dots,W\_N)^T \in {\ensuremath{{\mathbb R}}}^N$ to denote the treatment vector for all units. To ease notation we use $\boldsymbol{\bar{W}}\_{-i}=(W\_1,\dots,W\_{i-1},W\_{i+1},\dots,W\_{N})^T \in {\ensuremath{{\mathbb R}}}^{(N-1)}$ to denote the treatment vector for all units but i. We use the same notation for random vectors, e.g., $\boldsymbol{C}\_i \in {\ensuremath{{\mathbb R}}}^{D\_C}, \boldsymbol{C} \in {\ensuremath{{\mathbb R}}}^{D\_C},\boldsymbol{\bar{C}} \in {\ensuremath{{\mathbb R}}}^{N \times D\_C}$ and $\boldsymbol{\bar{C}}\_{-i} \in {\ensuremath{{\mathbb R}}}^{(N-1) \times D\_C}$ are the explicit vector of covariates for unit i, the generic vector of covariates, the matrix of covariates for all units and the matrix of covariates for all units but i, respectively. We now introduce our set-up and the treatment effects that are the targets of inference. Please refer to Appendix [appendix:prelims] for the graphical notions used throughout, such as the definition of a DAG or the latent projection. Explicit Models with Confounding and Interference ------------------------------------------------- [t]0.35 [>=stealth, shorten >=1pt, node distance=2cm, on grid, scale=0.6, transform shape ] (w1) at (0,0) W1; (c1) at ((w1) + ( − 50 : 2)) C1; (y1) at ((w1) + (0 : 2.5)) Y1; (w3) at (2.5,3) W3; (c3) at ((w3) + ( − 50 : 2)) C3; (y3) at ((w3) + (0 : 2.5)) Y3; (w2) at (-2.5,3) W2; (c2) at ((w2) + ( − 50 : 2)) C2; (y2) at ((w2) + (0 : 2.5)) Y2; (w2) edge (y2) (c1) edge (y1) (c2) edge (w2) (c2) edge (y2) (c1) edge (w1) (c3) edge (w3) (c3) edge (y3) (w1) edge (y1) (w3) edge (y3); [subfig: explicit DAG isolated] [t]0.35 [>=stealth, shorten >=1pt, node distance=2cm, on grid, scale=0.6, transform shape ] (w1) at (0,0) W1; (c1) at ((w1) + ( − 50 : 2)) C1; (y1) at ((w1) + (0 : 2.5)) Y1; (w3) at (2.5,3) W3; (c3) at ((w3) + ( − 50 : 2)) C3; (y3) at ((w3) + (0 : 2.5)) Y3; (w2) at (-2.5,3) W2; (c2) at ((w2) + ( − 50 : 2)) C2; (y2) at ((w2) + (0 : 2.5)) Y2; (w2) edge (y2) (c1) edge (y1) (c2) edge (w2) (c2) edge (y2) (c1) edge (w1) (c3) edge (w3) (c3) edge (y3) (w1) edge (y1) (w3) edge[] (y2) (w2) edge[bend left=20] (y3) (w3) edge[] (y1) (w1) edge[] (y2) (w3) edge (y3); [subfig: explicit DAG interference] [t]0.25 [>=stealth, shorten >=1pt, node distance=2cm, on grid, scale=0.6, transform shape ] (w) at (0,0) W; (y) at (2.5,0) Y; (c) at (1.25,-1.5) C; (w) edge (y) (c) edge (w) (c) edge (y); [subfig: generic DAG] [fig: no features] In the classical setting where units do not interact with each other, it is common to write structural equations which do not specify or differentiate between units. This implicitly assumes (1) that the structural equations and therefore the causal relationships between variables of a unit are the same for all units and (2) that there are no causal effects between units. To make these assumptions explicit, we consider structural equations on the explicit variables Ci, Wi and Yi, for i = 1, …, N. We define an explicit SEM Se as a SEM on explicit variables, and call the DAG Ge corresponding to Se an explicit DAG. An example of an explicit DAG Ge on N = 3 units is shown in Figure [subfig: explicit DAG isolated]. It represents the classical case with no interference between the three units. Explicit SEMs allow us to characterize settings where the assumptions (1) and/or (2) are violated. An example of an explicit DAG Ge on N = 3 units with interference between all three units is shown in Figure [subfig: explicit DAG interference]. We limit our considerations to a specific class of explicit SEMs Se with interference, defined in the following assumption. For simplicity we restrict ourselves to recursive SEMs, that is, we do not allow cycles. [def:sem0] The explicit recursive SEM Se with within-unit confounding and interference is given by $$\begin{aligned} &\boldsymbol{C}\_i \leftarrow g\_{\boldsymbol{C}}(\boldsymbol{C}\_i, \boldsymbol{\epsilon}\_{C\_i}), \ W\_i \leftarrow g\_W(\boldsymbol{C}\_i,\epsilon\_{W\_i}) \text{ and } Y\_i \leftarrow g\_{Y,i}(\boldsymbol{C}\_i, W\_i, \boldsymbol{\bar{W}}\_{-i}, \epsilon\_{Y\_i}), \end{aligned}$$ for each unit i = 1, …, N. We assume that $\boldsymbol{\epsilon}\_{C\_i},\epsilon\_{W\_i}$ and εYi are jointly independent error terms with expectation zero, and that their distribution does not depend on i. Under Assumption [def:sem0], a unit i may depend on another unit j solely through interference. In the explicit DAG, this means that we allow edges from Wj to Yi for j ≠ i, but no other between-unit edges. Furthermore, $g\_{\boldsymbol{C}}(\cdot)$ and gW( ⋅ ) do not depend on i, that is, they are functions common to all units. Target Treatment Effects ------------------------ We consider hypothetical stochastic interventions or policies, where the treatments are assigned independently to each unit with some fixed probability π ∈ [0, 1]. We denote such a stochastic intervention with $\mathrm{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))$ using the do-notation by. Due to interference between the units, ${\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]$ may differ for i = 1, …, N, and we therefore consider their average. The causal effect of interest is the contrast under two different stochastic interventions: $$\tau\_N(\pi,\eta):= \frac{1}{N}\sum\_{i=1}^N\left( {\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]- {\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\eta))] \right).$$ We call the effect τN(π, η) a global treatment effect, as it considers a simultaneous intervention on all units. The GATE, τN(1, 0), is a special case. Identification of the Target Treatment Effects ============================================== While explicit DAGs can be used for causal inference, they become complex for even a moderate number of units N, since the number of nodes is increasing in the number of units. In the classical setting, where there are no causal effects between different units, we overcome this difficulty by implicitly stacking the induced subgraphs for each unit in the explicit DAG Ge to obtain the conventional DAG G on variables that are not indexed by i. In this section, we first generalize this stacking approach to any explicit DAG Ge. We refer to the resulting graph as a generic graph G. While the generic graph is not as informative as the explicit DAG, we show that for the class of explicit SEMs satisfying Assumption [def:sem0], the generic graph can be used to identify a class of causal effects we call unit-specific effects. However, the global treatment effect τN(π, η) does not belong to this class. We overcome this problem by modelling interference via interference features and showing that τN(π, η) can be decomposed into a weighted average of unit-specific effects. The weights in this decomposition only depend on our choice of interference features and can be explicitly computed or approximated. The unit-specific effects, on the other hand, can be identified with graphical criteria for valid adjustment sets, applied to the generic graph. Thus, this approach allows us to identify the target treatment effect τN(π, η) in the presence of within-unit confounding and interference. Generic Graphs -------------- [Generic graph] [def:genericDAG] Consider an explicit DAG Ge on explicit variables Vi, i = 1, …, N. The corresponding generic graph $\mathcal{G}=(\boldsymbol{V},\boldsymbol{E})$ is defined as follows: if Ai → Bi for any i in Ge, then add A → B to the edge set $\boldsymbol{E}$. Definition [def:genericDAG] is similar to the isolated interaction model from Definition 2 of, in that it only considers within-unit edges. For explicit SEMs satisfying Assumption [def:sem0], the generic graph is guaranteed to be a DAG, since the induced subgraph on Vi of the explicit DAG is the same for all units. For illustration, consider the two explicit DAGs Ge in Figures [subfig: explicit DAG isolated] and [subfig: explicit DAG interference]. Both have the same generic graph G, shown in Figure [subfig: generic DAG]. For explicit SEMs satisfying Assumption [def:sem0], the generic graph G is also the latent projection as defined by of Ge over $\boldsymbol{\bar{V}}\_{-i}$. This implies that interventional distributions $f(b \mid {{\text{do}}}( \boldsymbol{A}=\boldsymbol{a}))$ for {B} ∪ A ⊆ Vi, that is, belonging to the same unit i, factorize according to G. In other words, G is a causal DAG for each $\boldsymbol{V}\_i$. We also provide an explicit proof of this fact in Proposition [lemma:strucpreserv] of Appendix [sec:identnofeat]. Thus, we can use the generic graph G corresponding to an explicit SEM satisfying Assumption [def:sem0] to identify the following class of causal effects. [Unit-specific effects] Consider an explicit SEM Se on explicit variables $\boldsymbol{\bar{V}}=\bigcup\_{i=1}^N \boldsymbol{V}\_i$. Let $\boldsymbol{A} \subset \boldsymbol{\bar{V}}$ and $B \in \boldsymbol{\bar{V}} \setminus \mathbf{A}$ and consider causal effects of $\boldsymbol{A}$ on B of the form $\frac{\partial}{\partial \boldsymbol{a}}{\ensuremath{{\mathbb E}}}[B \mid \text{do}(\mathbf{A}=\boldsymbol{a})]$ or ${\ensuremath{{\mathbb E}}}[B \mid \text{do}(\mathbf{A}=\boldsymbol{a})]- {\ensuremath{{\mathbb E}}}[B \mid \text{do}(\mathbf{A}=\boldsymbol{a'})]$ for some $\boldsymbol{a}\neq \boldsymbol{a'}$ in the support of $\boldsymbol{A}$. We say that the causal effect is unit-specific if A ∪ {B} ⊂ Vi for some unit i. An example of an average of unit-specific effects is the expected average treatment effect (EATE), given by $$\frac{1}{N}\sum\_{i=1}^N \big({\ensuremath{{\mathbb E}}}[Y\_i\mid {{\text{do}}}(W\_i=1)]- \allowbreak {\ensuremath{{\mathbb E}}}[Y\_i\mid {{\text{do}}}(W\_i=0)]\big).$$ It captures how, on average, the outcome of a unit is affected by its own treatment. We are, however, interested in the global treatment effect τN(π, η), which involves interventions on $\boldsymbol{\bar{W}}$. Since τN(π, η) is not unit-specific, it may not be identifiable using G. In the following section we show that we can overcome this problem if we impose additional structure on the interference mechanism by introducing interference features. Interference Features --------------------- We refine Assumption [def:sem0] on the explicit SEM Se, by assuming that the outcome model takes the form $ Y\_i \leftarrow g\_Y'(\boldsymbol{C}\_i, W\_i, \boldsymbol{X}\_i,\epsilon\_{Y\_i})$, where $\boldsymbol{X}\_i$ are possibly multivariate interference features capturing the effect of $\boldsymbol{\bar{W}}\_{-i}$ on Yi, and the function gYʹ( ⋅ ) does not depend on i. Specifically, we assume that for each unit i the effect of $\boldsymbol{\bar{W}}\_{-i}$ on Yi is modulated by a known and nonrandom interaction network graph IN, with nodes i = 1, …, N representing the units and edges representing interaction between the respective units, such as, for example, friendship ties in a social group or geographical adjacency between agricultural fields. We use the convention that all edges in IN are directed, with an edge i → j indicating that there is an interaction from i to j. If the interaction is bi-directional, we add the edge j → i in IN. We also use IN to denote the corresponding adjacency matrix IN ∈ {0, 1}N × N, where IijN = 1 if there is an edge i → j. Similarly to and, we assume that for each unit i = 1, …, N, the interference features $\boldsymbol{X}\_i=(X\_{i1},\ldots, X\_{iP})^T$ are functions of IN and the treatment vector $\boldsymbol{\bar{W}}\_{-i}$, that is, $X\_{ik} = h^k(\boldsymbol{\bar{W}}\_{-i},I^N)$ for k = 1, …, P, where the functions hk( ⋅ ) : R(N − 1) × (N × N) ↦ R do not depend on i. [t]0.3 [>=stealth, shorten >=1pt, node distance=2cm, on grid, scale=0.7, transform shape, align=center,minimum size=3em] (1) at (0,0) 1; (2) [right =5em of 1]2; (3) [above= 5em of 2]3; (1) edge (2) (3) edge (1) (2) edge[bend right=10] (3) (3) edge[bend right=10] (2); [subfig: interaction graph small] [t]0.3 [>=stealth, shorten >=1pt, node distance=2cm, on grid, auto, scale=0.7, transform shape, align=center,minimum size=3em] [xshift=-5cm] (2) at (0,0) 2; (1) [left =5em of 2] 1; (3) [right = 5em of 2] 3; (4) [above = 5em of 3] 4; (5) [above = 5em of 2] 5; (6) [above = 5em of 1] 6; (5) edge (1) (2) edge[bend right=10] (5) (5) edge[bend right=10] (2) (6) edge (5) (2) edge (3); [subfig: interaction graph big] [t]0.3 [>=stealth, shorten >=1pt, node distance=2cm, on grid, auto, scale=0.7, transform shape, align=center,minimum size=3em] (2) at (0,0) 2; (1) [left =5em of 2] 1; (3) [right = 5em of 2] 3; (4) [above = 5em of 3] 4; (5) [above = 5em of 2] 5; (6) [above = 5em of 1] 6; (1) edge (2) (1) edge (6) (2) edge (6) (5) edge (3); [subfig: dependency graph] [fig:network] [example1revisited] A natural interference feature is the fraction of treated parents $$\begin{aligned} X\_{i}^1 \coloneqq \frac{1}{\left\|\mathcal{N}\_i^{1}\right\|}\sum\_{j \in \mathcal{N}\_i^{1}}W\_j,\label{formula:feat1}\end{aligned}$$ where $\mathcal{N}\_i^{1} \coloneqq \left\{ j\in \{1,\ldots,N\}: I^N\_{ji}=1\right\}$ denotes the parents of i in IN. Another possible interference feature is the indicator that at least 50% of the parents of i are treated. To model interference beyond the parents in IN, we may, for example, consider the fraction of treated parents of parents $$\begin{aligned} X\_{i}^2 &\coloneqq \frac{1}{\left\| \mathcal{N}\_i^{(2)}\right\|}\sum\_{j \in \mathcal{N}\_i^{(2)}}W\_j, \label{formula:feat3}\end{aligned}$$ where $\mathcal{N}\_i^{(2)} \coloneqq \left\{ j \in \{1,\ldots,N\} \setminus \{i\}: \text{there exists l such that } I^N\_{jl}I^N\_{li}=1\right\}$. We further assume that the outcome equation is linear and may differ for treated units (Wi = 1) and untreated units (Wi = 0), but is common to units within these two treatment groups. Specifically, we assume that $$Y\_i \leftarrow (1-W\_i) (1,\boldsymbol{X}\_i^T) \boldsymbol{\beta}\_{0} + W\_i (1,\boldsymbol{X}\_i^T) \boldsymbol{\beta}\_{1} + \boldsymbol{C}\_i^T\boldsymbol{\gamma} + \epsilon\_{Y\_i}, \label{eq:outcomewithbeta}$$ where $\boldsymbol{X}\_i \coloneqq (X\_{i1}, X\_{i2}, \ldots, X\_{iP})^T \in \mathbb{R}^{P}$, and $\boldsymbol{\beta}\_{0}, \boldsymbol{\beta}\_{1} \in \mathbb{R}^{P+1}$. We summarize our assumptions on the model in Assumption [def:sem], where we also reparameterize the outcome model with coefficients $\boldsymbol{\alpha}\_{0} = \boldsymbol{\beta}\_{0}$ and $\boldsymbol{\alpha}\_{1} = \boldsymbol{\beta}\_{1}-\boldsymbol{\beta}\_{0}$ as these are the parameters we will estimate. [def:sem] The explicit recursive SEM Se with interference features and linear outcome model is given by $$\begin{aligned} &\boldsymbol{C}\_i \leftarrow g\_{\boldsymbol{C}}(\boldsymbol{C}\_i, \boldsymbol{\epsilon}\_{C\_i}), \ W\_i \leftarrow g\_W(\boldsymbol{C}\_i,\epsilon\_{W\_i}),\notag\\ &\boldsymbol{X}\_i \leftarrow h(\boldsymbol{\bar{W}}\_{-i},I^N), \ \boldsymbol{O}\_i \leftarrow W\_i \boldsymbol{X}\_i \text{ and} \notag\\ &Y\_i \leftarrow (1,\boldsymbol{X}\_i^T) \boldsymbol{\alpha}\_{0} + (W\_i,\boldsymbol{O}^T\_i) \boldsymbol{\alpha}\_{1} + \boldsymbol{C}\_i^T \boldsymbol{\gamma}+ \epsilon\_{Y\_i},\label{eq:outcomeX} \end{aligned}$$ for each unit i = 1, …, N. We assume that the interaction network graph IN and the functions h( ⋅ ) : = (h1( ⋅ ), …, hP( ⋅ ))T are known. Further, we assume that $\boldsymbol{\epsilon}\_{C\_i},\epsilon\_{W\_i}$ and εYi are jointly independent error terms with expectation zero, and that their distributions does not depend on i. [t]0.6 [>=stealth, shorten >=1pt, node distance=2cm, on grid, scale=0.5, transform shape ] (y1) at (0,0) Y2; (o1) [above =6em of y1] $\boldsymbol{O}\_{2}$; (w1) [left =6em of o1] W2; (x1) [right =6em of o1] $\boldsymbol{X}\_{2}$; (c11) [below =6em of x1] C21; (c12) [left =6em of y1] C22; (c13) [above =6em of w1] C23; (y2) at (-7,0) Y1; (o2) [above =6em of y2] $\boldsymbol{O}\_{1}$; (w2) [left =6em of o2] W1; (x2) [right =6em of o2] $\boldsymbol{X}\_{1}$; (c21) [below =6em of x2] C11; (c22) [left =6em of y2] C12; (c23) [above =6em of w2] C13; (y3) at (7,0) Y3; (o3) [above =6em of y3] $\boldsymbol{O}\_{3}$; (w3) [left =6em of o3] W3; (x3) [right =6em of o3] $\boldsymbol{X}\_{3}$; (c31) [below =6em of x3] C31; (c32) [left =6em of y3] C32; (c33) [above =6em of w3] C33; (w1) edge (y1) (c12) edge (w1) (x1) edge (y1) (o1) edge (y1) (x1) edge (o1) (w1) edge (o1) (c11) edge[bend left] (c12) (c11) edge (y1) (c13) edge (w1) (w2) edge (y2) (c22) edge (w2) (x2) edge (y2) (o2) edge (y2) (x2) edge (o2) (w2) edge (o2) (c21) edge[bend left] (c22) (c21) edge (y2) (c23) edge (w2) (w3) edge (y3) (c32) edge (w3) (x3) edge (y3) (o3) edge (y3) (x3) edge (o3) (w3) edge (o3) (c31) edge[bend left] (c32) (c31) edge (y3) (c33) edge (w3) (w3) edge[bend right] (x2) (w3) edge[] (x1) (w1) edge[bend right=20] (x3) (w2) edge[bend right=20] (x1); [subfig: explicit features] [t]0.35 [>=stealth, shorten >=1pt, node distance=2cm, on grid, scale=0.5, transform shape] (y) at (0,0) Y; (o) [above =6em of y] $\boldsymbol{O}$; (w) [left =6em of o] W; (x) [right =6em of o] $\boldsymbol{X}$; (c1) [right =6em of y] C1; (c2) [left =6em of y] C2; (c3) [above =6em of w] C3; (w) edge (y) (c2) edge (w) (x1) edge (y) (o) edge (y) (x) edge (o) (w) edge (o) (c1) edge[bend left] (c2) (c1) edge (y) (c3) edge (w); [subfig: generic features] [fig: feature DAGs] The interference features are a tool to model the interference mechanisms and are not unique. We also only need to know the features up to shift and scale (see Lemma [lemma:invariance] in Appendix [app:proofssecidentwithfeat]). The feature model is flexible, in that we allow for arbitrary features and arbitrary combinations of them, as long as the explicit SEM Se respects Assumption [def:sem]. We do, however, impose further conditions on the asymptotic behaviour of the features in Section [sec:consistency]. Given an explicit SEM respecting Assumption [def:sem], we consider a corresponding explicit DAG Ge with possibly multivariate nodes for $\boldsymbol{X}\_i$ and $\boldsymbol{O}\_i$. We interpret the structural equation of a multivariate node in Ge as the vector of structural equations of each of the variables in the node. An intervention ${{\text{do}}}(\boldsymbol{X}\_i =\boldsymbol{x})$ on a multivariate node is given by simultaneously replacing all structural equations of the vector of structural equations by the vector $\boldsymbol{x}$. Treating $\boldsymbol{X}\_i$ and $\boldsymbol{O}\_i$ as single nodes in Ge implies that the corresponding generic graph G also contains single nodes for $\boldsymbol{X}$ and $\boldsymbol{O}$. The generic graph coincides with the latent projection of Ge over $\boldsymbol{\bar{V}}\_{-i} $ for a given unit i. Therefore, G can again be interpreted causally for each Vi, i = 1, …, N in the sense that interventional distributions $f(b \mid {{\text{do}}}( \boldsymbol{A}=\boldsymbol{a}))$ for {B} ∪ A ⊆ Vi for some i factorize according to G for all units i. We also provide a proof of this fact in Proposition [lemma:strucpreservwithfeat] in Appendix [sec:identfeat] and note that it does not hold if we treat each Xik as an individual node, that is, allow for interventions on proper subsets of Xi. [example:explicitDAG] Consider a model on three units and suppose that for each unit the explicit SEM takes the form $$\begin{aligned} \begin{split} &C\_{i1} \leftarrow -2 + \epsilon\_{C\_{i1}}, \ C\_{i2} \leftarrow 2C\_{i1} + \epsilon\_{C\_{i2}}, \ C\_{i3} \leftarrow 0.5 + \epsilon\_{C\_{i3}}, \\ &W\_i \sim \mathrm{Bern}\left( \frac{1}{1+ \exp{(-C\_{i2}- 5 C\_{i3})}} \right), \\ &\boldsymbol{X}\_i \leftarrow h(\boldsymbol{\bar{W}}\_{-i},I^N), \ \boldsymbol{O}\_i \leftarrow W\_i \boldsymbol{X}\_i \text{ and}\\ &Y\_i \leftarrow (1,\boldsymbol{X}\_i^T)\boldsymbol{\alpha}\_{0} + (W\_i,\boldsymbol{O}^T\_i) \boldsymbol{\alpha}\_{1} + 1.5 C\_{i1}+ \epsilon\_{Y\_i}, \end{split}\end{aligned}$$ where IN is the interaction graph given in Figure [subfig: interaction graph small] and $h(\boldsymbol{\bar{W}}\_{-i},I^N)$ is the fraction of treated parents as defined in equation. Clearly, this explicit SEM satisfies Assumption [def:sem]. The corresponding explicit and generic DAGs are shown shown in Figure [fig: feature DAGs]. Based on Assumption [def:sem] we can decompose the global treatment effect τN(π, η) into a weighted linear combination of unit-specific effects that we can identify using the interpretation of the generic graph G as a causal DAG. The decomposition is analogous to the decomposition result by for the setting without confounding. [Decomposition of global treatment effects]proplemmataureformulation [lemma:reformulation] Let Se be an explicit SEM satisfying Assumption [def:sem]. Then $$\begin{aligned} \label{eq: weighted} \tau\_N(\pi, \eta) &= \boldsymbol{\omega}^N\_{0}(\pi,\eta)^T \boldsymbol{\alpha}\_{0} + \boldsymbol{\omega}^N\_{1}(\pi,\eta)^T(\boldsymbol{\alpha}\_{0} +\boldsymbol{\alpha}\_{1} ),\end{aligned}$$ where $$\begin{aligned} \begin{split} \boldsymbol{\omega}^N\_{0}(\pi,\eta)^T = \frac{1}{N}\sum\_{i=1}^N & \left( (1-\pi) {\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}^T\_i)\mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \right. \\ &\quad\quad \left. - (1-\eta) {\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}^T\_i)\mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\eta))] \right) \text{ and} \\ \boldsymbol{\omega}^N\_{1}(\pi,\eta)^T = \frac{1}{N}\sum\_{i=1}^N &\left( \pi {\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}^T\_i)\mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \right. \\ &\quad\quad \left.- \eta {\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}^T\_i) \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\eta))] \right). \end{split}\end{aligned}$$ The weights $\boldsymbol{\omega}^N\_{0}(\pi,\eta)$ and $\boldsymbol{\omega}^N\_{1}(\pi,\eta)$ are functions of the expected value of the interference features Xi under the stochastic interventions on $\boldsymbol{\bar{W}}$ with probabilities π and η, respectively. Even though the effect of $\boldsymbol{\bar{W}}$ on $\boldsymbol{X}\_i$ is not unit-specific, we can exploit our knowledge of the interaction network graph IN and the interference function h( ⋅ ) to either compute ${\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}^T\_i) \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]$ and ${\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}^T\_i) \mid \text{do}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim}\text{Bern}(\eta))]$ in closed-form or approximate them with a simulation, holding IN and h( ⋅ ) fixed and randomly drawing $\boldsymbol{\bar{W}}$ with probability π or η, respectively. Effectively, we absorb the nonunit-specific part of our target effect τN(π, η) in the computable weights $\boldsymbol{\omega}^N\_{1}(\pi,\eta)$ and $\boldsymbol{\omega}^N\_{0}(\pi,\eta)$, and as a result we only need to estimate $\boldsymbol{\alpha}\_{0}$ and $\boldsymbol{\alpha}\_{1}$. We now show that $(\boldsymbol{\alpha}^T\_{0},\boldsymbol{\alpha}^T\_{1})$ is the unit-specific joint total effect of $(1,\boldsymbol{X}^T\_i, W\_i, \boldsymbol{O}^T\_i)$ on Yi for all units i = 1, …, N. Here we treat the intercept term in equation as an additional nonrandom cause of Yi that we may intervene on. We do so for notational convenience, since the intercept’s causal effect cancels in equation and is therefore irrelevant for computing τN(π, η). [Total joint effect]lemmalemmaisolatedeffectfeat [lemma:totaljointeffet] Let Se be an explicit SEM satisfying Assumption [def:sem]. Then $(\boldsymbol{\alpha}^T\_{0},\boldsymbol{\alpha}^T\_{1})$ is the total joint effect of $(1,\boldsymbol{X}^T\_i, W\_i, \boldsymbol{O}^T\_i)$ on Yi. Since $(\boldsymbol{\alpha}^T\_{0},\boldsymbol{\alpha}^T\_{1})$ is a unit-specific effect we can identify it using the generic graph G employing the graphical characterization of valid adjustment sets from the causal graphical models literature. The following theorem summarizes our main identification result. [Identification]thmtheoremident [theorem:identificantion] Let Se be an explicit SEM satisfying Assumption [def:sem]. Then $\tau\_N(\pi,\eta) = \boldsymbol{\omega}^N\_{0}(\pi,\eta)^T \boldsymbol{\alpha}\_{0}+ \boldsymbol{\omega}^N\_{1}(\pi,\eta)^T(\boldsymbol{\alpha}\_{0} +\boldsymbol{\alpha}\_{1} )$, where the weights $ \boldsymbol{\omega}^N\_{0}(\pi,\eta)$ and $ \boldsymbol{\omega}^N\_{1}(\pi,\eta)$ are computable, and $(\boldsymbol{\alpha}^T\_{0},\boldsymbol{\alpha}^T\_{1})$ is the total joint effect of $(1,\boldsymbol{X}^T\_i, W\_i, \boldsymbol{O}^T\_i)$ on Yi in Se for all i = 1, …, N, and can be identified via adjustment in the generic graph G. The proof of Theorem [theorem:identificantion] uses that we can interpret the generic graph causally, in the sense that the truncated factorization formula holds for unit-specific effects (see Proposition [lemma:strucpreservwithfeat] in Appendix [app:proofssecidentwithfeat]). This also implies that identification of effects is possible through other graphical tools for causal DAGs such as the frontdoor-criterion or instrumental variables. We focus on adjustment for simplicity and leave these alternatives for future research. Estimation of Target Treatment Effects ====================================== Based on the identification result in Theorem [theorem:identificantion], we propose an adjustment estimator for the causal effect τN(π, η). In order to derive asymptotic properties for this estimator, we need to make restrictions on the behavior of the interaction network graph and the feature functions. As a tool to make these restrictions, we first introduce the interference dependency graph. Interference Dependency Graph ----------------------------- As discussed before, we consider settings where the units exhibit interference via interference features $\boldsymbol{X}\_i$ that are functions of the other units’ treatment vector $\boldsymbol{\bar{W}}\_{-i}$ and the interaction network graph IN. Since we do not restrict the interference functions to be local in IN, the absence of an edge i ← j or i → j in IN does not necessarily indicate independence between any variable Vi ∈ Vi and any $V\_j \in \boldsymbol{V}\_j$. We use an additional undirected graph called the interference dependency graph in which the absence of an edge i − j does imply independence between Vi and $\boldsymbol{V}\_j$. Dependency graphs are a standard approach to characterize dependencies between random variables. We use a specific version, namely the interference dependency graph on networks as proposed by, which is a function of the interaction network graph IN and the feature functions hk( ⋅ ), k = 1, …, P. The following definition is written for general $U\_{ik} \leftarrow h^k(\boldsymbol{\bar{W}}\_{-i},I^N)$, but we mostly consider the case Uik = Xik. [Interference Dependency Graph] [def:depgraph] Consider a treatment vector $\boldsymbol{\bar{W}}$ and an interaction network graph IN. Given P functions h1( ⋅ ), …, hP( ⋅ ), let $\boldsymbol{\bar{U}}$ be the matrix with entries $U\_{ik} \leftarrow h^k(\boldsymbol{\bar{W}}\_{-i},I^N)$ for i = 1, …, N and k = 1, …, P, and let $\boldsymbol{U}\_j$ denote the jth row of $\boldsymbol{\bar{U}}$. We characterize the interference dependency graph by its adjacency matrix $D\_{ij}(\boldsymbol{\bar{U}}, \boldsymbol{\bar{W}}) \in \{0,1\}^{N \times N}$, where for two units i ≠ j it holds that $D\_{ij}(\boldsymbol{\bar{U}}, \boldsymbol{\bar{W}}) = D\_{ji}(\boldsymbol{\bar{U}}, \boldsymbol{\bar{W}}) =1$, if one of the following conditions holds: (a) Wi affects $\boldsymbol{U}\_j$, (b) Wj affects $\boldsymbol{U}\_i$ or (c) $\boldsymbol{U}\_i$ and $\boldsymbol{U}\_j$ are affected by some Wl, l ∈ {1, …, N} \ {i, j}. Here, affect means that Wi appears in the generating equation of Uj. By definition, the interference dependency graph $D(\boldsymbol{\bar{U}}, \boldsymbol{\bar{W}})$ is undirected, that is, it does not reflect whether the dependence between units i and j arises because Wi causally affects $\boldsymbol{U}\_j$ or Wj causally affects $\boldsymbol{U}\_i$. Consider $D(\boldsymbol{\bar{X}}, \boldsymbol{\bar{W}})$, the interference dependency graph for the interference features $\boldsymbol{\bar{X}}$. By Assumption [def:sem], interference between units may only occur via the features $\boldsymbol{X}\_i$ and therefore the absence of an edge i − j in $D(\boldsymbol{\bar{X}}, \boldsymbol{\bar{W}})$ implies that $\boldsymbol{V}\_i$ and $\boldsymbol{V}\_j$ are independent. [ex:depgraph] Consider the interaction network graph IN in Figure [subfig: interaction graph big]. Suppose we choose as interference feature the fraction of treated parents of parents as defined in equation. The resulting dependency graph $D(\boldsymbol{\bar{X}}, \boldsymbol{\bar{W}})$ is given in Figure [subfig: dependency graph]. Estimating Treatments Effects via Adjustment -------------------------------------------- We propose an estimator of τN(π, η) based on an adjustment estimator for $\boldsymbol{\alpha}=(\boldsymbol{\alpha}^T\_{0},\boldsymbol{\alpha}^T\_{1})^T$. Let Z be a valid adjustment set relative to $(\{\boldsymbol{X}, W, \boldsymbol{O}\},Y)$ in the generic graph G. For each unit i, let $ \mathbf{M}\_i:= (1,\boldsymbol{X}^T\_i, W\_i, \boldsymbol{O}^T\_i, \mathbf{Z}^T\_i)^T$ and consider the OLS-estimator $$\label{alphahatols} \boldsymbol{\hat{\alpha}}^{\text{full}}=(\mathbf{\bar{M}}^{T}\mathbf{\bar{M}})^{-1}\mathbf{\bar{M}}^{T}\boldsymbol{\bar{Y}}.$$ We denote the components of $\boldsymbol{\hat{\alpha}}^{\text{full}}$ corresponding to $(1,\boldsymbol{X}^T)$ by $\boldsymbol{\hat{\alpha}}\_{0}$ and those corresponding to $(W,\boldsymbol{O}^T)$ by $\boldsymbol{\hat{\alpha}}\_{1}$. Given $\boldsymbol{\hat{\alpha}}\_{0}$ and $\boldsymbol{\hat{\alpha}}\_{1}$, we estimate τN(π, η) by $$\label{pluginest} \hat{\tau}\_N(\pi, \eta) = \boldsymbol{\omega}^N\_{0}(\pi,\eta)^T \boldsymbol{\hat\alpha}\_{0} + \boldsymbol{\omega}^N\_{1}(\pi, \eta)^T(\boldsymbol{\hat\alpha}\_{0} + \boldsymbol{\hat\alpha}\_{1}),$$ where $\boldsymbol{\omega}^N\_{0}(\pi,\eta)$ and $\boldsymbol{\omega}^N\_{1}(\pi,\eta)$ can either be computed in closed-form or can be approximated through simulation. The following theorem shows that under mild assumptions on the interference features and their dependency graph, the estimator τ̂N(π, η) is consistent for τN(π, η). [Consistency]thmpropconsistencyols [prop:consistency-ols] Consider a sequence of explicit SEMs SeN and corresponding interaction network graphs IN, satisfying Assumption [def:sem] such that the SeN only differ in IN and N. Let GeN be the corresponding explicit DAGs, let Z be a valid adjustment set relative to $( \{\boldsymbol{X},W,\boldsymbol{O}\}, Y)$ in the generic graph G common to all GeN, let $\mathbf{M}\_i= (1,\boldsymbol{X}^T\_i, W\_i, \boldsymbol{O}^T\_i, \mathbf{Z}^T\_i)^T$ and let τ̂N(π, η) be as defined in equation. Then, $\hat{\tau}\_N(\pi, \eta) -\tau\_N(\pi,\eta) \xrightarrow[]{P} 0$, given that 1. the limits $\lim\_{N\rightarrow \infty}\frac{1}{N}\sum\_{i = 1}^N {\ensuremath{{\mathbb E}}}[\boldsymbol{X}\_i \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\theta))]$ for θ = π and θ = η exist, 2. dmax(N) ∈ o(N), where $d\_{\text{max}}(N) := \max\_{i \in \{1,\dots, N\}} \sum\_{j=1}^ND\_{ij}(\boldsymbol{\bar{X}}, \boldsymbol{\bar{W}})$ is the maximal degree in the interference dependency graph, holds and in addition the following regularity conditions hold: 1. ${\ensuremath{{\mathbb E}}}[Y\_i^4] < \infty$ and ${\ensuremath{{\mathbb E}}}\left[{\lVert \mathbf{M}\_i \rVert}^4\right] < \infty$ for i = 1, …, N, where ∥ ⋅ ∥ denotes the Euclidean norm, 2. $ {\ensuremath{{\mathbb E}}}\left[\mathbf{M}\_i\mathbf{M}\_i^T\right] < \infty$ is invertible for i = 1, …, N, 3. $ \lim\_{N\rightarrow \infty}\frac{1}{N}\sum\_{i=1}^N {\ensuremath{{\mathbb E}}}\left[\mathbf{M}\_i\mathbf{M}\_i^T\right] = \Sigma\_{\mathbf{M} \mathbf{M}} < \infty$ elementwise, where ΣMM is invertible and 4. ${\ensuremath{{\mathbb E}}}[\mathbf{P\_i} \mid \mathbf{Z}\_i]=\delta^T \mathbf{Z}\_i$ for i = 1, …, N, and some matrix δ, where $\mathbf{P}\_i = \mathrm{pa}(Y\_i,G\_e) \setminus \{\boldsymbol{X}\_i,W\_i,\boldsymbol{O}\_i\}$. We require Condition i) to ensure that the limit of the target effect τN(π, τ) exists. We require Condition ii) to ensure that a weak law of large number holds for the estimator $\boldsymbol{\hat{\alpha}}^{\text{full}}$. Both conditions are implicit restriction on the sequence of IN and the interference functions hk( ⋅ ) and allow us to avoid explicitly modelling them. For example, if the feature function is the number of treated parents, than Condition i) implies that the average number of parents in IN converges. We discuss Condition ii) more thoroughly in Example [example: maximal degree local]. The other four conditions are more standard statistical regularity conditions. The next theorem shows that under a stricter set of assumptions the estimator τ̂N(π, η) is also asymptotically normal. [Asymptotic Normality]thmpropnormalityols [prop:normality-ols] Consider a sequence of explicit SEMs SeN and corresponding interaction network graphs IN, satisfying Assumption [def:sem] such that the SeN only differ in IN and N. Let GeN be the corresponding explicit DAGs, let Z be a valid adjustment set relative to $( \{\boldsymbol{X},W,\boldsymbol{O}\}, Y)$ in the generic graph G common to all GeN, let $\boldsymbol{M}= \{\boldsymbol{X},W,\boldsymbol{O},\boldsymbol{Z}\}$ and let τ̂N(π, η) be as defined in equation. Then, $\sqrt{N}\Big(\hat{\tau}\_N(\pi, \eta) -\tau\_N(\pi,\eta)\Big)\xrightarrow[]{d} \mathcal{N}(0, \sigma^2),$ given that the conditions from Theorem [prop:consistency-ols] hold, 1. dmax(N) ∈ o(N1/4), where $d\_{\text{max}}(N) := \max\_{i \in \{1,\dots, N\}} \sum\_{j=1}^ND\_{ij}(\boldsymbol{\bar{X}}, \boldsymbol{\bar{W}})$ is the maximal degree in the interference dependency graph, holds and in addition the following regularity conditions hold: 1. ${\ensuremath{{\mathbb E}}}[Y\_i^8] < \infty$ and ${\ensuremath{{\mathbb E}}}\left[{\lVert \mathbf{M}\_i \rVert}^8\right] < \infty$ for i = 1, …, N and 2. $\lim\_{N\rightarrow \infty}\frac{1}{N}\sum\_{i=1}^N{\ensuremath{{\mathbb E}}}\left[ \epsilon\_i^2\mathbf{M}\_i \mathbf{M}\_i ^T\right] =\Sigma\_{\epsilon^2\mathbf{M} \mathbf{M}} < \infty$, where $\epsilon\_i :=Y\_i - \mathbf{M}\_i^T\boldsymbol{\alpha}^{\text{full}}$, with population level regression coefficients $\boldsymbol{\alpha}^{\text{full}}$ from the regression of Yi on Mi. The asymptotic variance σ2 is finite and given by $$\sigma^2 = \begin{pmatrix} \boldsymbol{\omega}\_{0}(\pi,\eta) + \boldsymbol{\omega}\_{1}(\pi,\eta ) \vspace{0.1cm} \\ \boldsymbol{\omega}\_{1}(\pi,\eta ) \vspace{0.1cm}\\ \boldsymbol{0} \vspace{0.1cm} \end{pmatrix}^T\Sigma\_{\mathbf{M} \mathbf{M}}^{-1}\Sigma\_{\epsilon^2\mathbf{M} \mathbf{M}} \Sigma\_{\mathbf{M}\mathbf{M}}^{-1}\begin{pmatrix} \boldsymbol{\omega}\_{0}(\pi,\eta) + \boldsymbol{\omega}\_{1}(\pi,\eta ) \vspace{0.1cm} \\ \boldsymbol{\omega}\_{1}(\pi,\eta ) \vspace{0.1cm}\\ \boldsymbol{0} \vspace{0.1cm} \end{pmatrix},$$ where $\boldsymbol{\omega}\_{0}(\pi,\eta ) = \lim\_{N \rightarrow \infty} \boldsymbol{\omega}^N\_{0}(\pi,\eta )$,$\boldsymbol{\omega}\_{1}(\pi,\eta )= \lim\_{N \rightarrow \infty} \boldsymbol{\omega}^N\_{1}(\pi,\eta )$, and $ \boldsymbol{0}$ denotes a vector of zeros in ${\ensuremath{{\mathbb R}}}^{\|\mathbf{Z}\|}$. We propose a plug-in estimator for the asymptotic variance σ2 and show that it is consistent in the Appendix (Lemma [lemma:varest]). The asymptotic normality and the consistent variance estimator, provide the asymptotically valid confidence interval $$CI\_{1-\alpha} := \hat{\tau}\_N(\pi,\eta) \pm z\_{1-\alpha/2}\sqrt{\frac{\hat{\sigma}\_N^2}{N}}, \label{formula:CI}$$ where z1 − α/2 is the (1 − α/2)-quantile of a standard normal distribution. We now provide two examples. The first illustrates how the growth of the maximal degree of the interference dependency graph depends on both the interaction network graph IN and the features $\boldsymbol{X}\_i$. The second illustrates how we can use the generic graph to find valid adjustment sets. Let Se be an explicit SEM satisfying Assumption [def:sem], with features $\boldsymbol{X}^1$ as per equation and $\boldsymbol{X}^2$ as per equation that depend on some given interaction network graph IN. Here, the interference dependency graph (see Definition [def:depgraph]) contains an edge between any two units i ≠ j in the following three cases: (i) IijN = 1, (ii) IjiN = 1 and (iii) PijN ≥ 1 or PjiN ≥ 1, where PN = (IN)TIN is the Gram matrix of IN. The maximal degree dmax(N) of the interference dependency graph $D(\boldsymbol{\bar{X}}, \boldsymbol{\bar{W}})$ is therefore given by $$d\_{\text{max}}(N) = \max\_{i \in \{1,\ldots N\}} \sum\_{j\in \{1,\ldots N\}\setminus i} \mathds{1} \left\{(I^N + P^N)\_{ij}\geq 1\right\},$$ where $ \mathds{1} \{\cdot\}$ denotes the indicator function. Whether dmax(N) satisfies dmax(N) ∈ o(N) or dmax(N) ∈ o(N1/4) depends on the specific sequence of IN. It will, for example, hold if the IN have bounded maximal degree. Suppose an interference feature is non-local in IN, for example $\boldsymbol{\bar{X}}= \boldsymbol{T} \boldsymbol{\bar{W}}$, where $\boldsymbol{T}=(T\_{ij})$ with Tij = ∣∣Ni1∣ − ∣Nj1∣∣ the difference in in-degree centrality between nodes i and j in IN. Then dmax(N) ∈ o(N) will only hold for very specific sequences of IN such as the sequence of empty graphs. [example: maximal degree local] [example:ols] Consider the explicit SEM Se from Example [example:explicitDAG] and the corresponding generic graph G given in Figure [subfig: generic features]. By the adjustment criterion (see Appendix [appendix:prelims]), the valid adjustment sets relative to $( \{\boldsymbol{X},W,O\}, Y)$ in the generic graph G are {C1}, {C2}, {C1, C2}, {C1, C3}, {C2, C3}, and {C1, C2, C3}. Based on research from the i.i.d. setting it is likely that using {C1} results in a smaller asymptotic variance estimator than using the alternative adjustment sets. Empirical Validation ==================== In a simulation study we validate the performance and theoretical properties of our adjustment estimator and compare it to alternative estimators that do not control either for within-unit confounding and/or interference. In addition, we apply our adjustment estimator to a real data example, where we estimate the effect of a strict facial-mask policy on the spread of COVID-19 in the early phase of the pandemic in Switzerland. Simulation Study ---------------- We consider three different structures for the interaction network graphs IN: First, Erds–Rényi networks I(N, pN), where for each pair of units i ≠ j ∈ {1, …, N}, we either draw both edges {i → j, i ← j} or neither of them with probability pN. Second, family networks of disjoint families, where within a family all members are pairwise connected and the family sizes are randomly sampled between 1 and 6. Third, directed square 2-dimensional lattices with at most one edge between two units. [t!] c\|c\|c\|c & Erds–Rényi & Family network & 2d-lattice Features & (X1) & (X1, X2) & (X1, X2) $\boldsymbol{\alpha}\_0$ & (2, 1)T & (2, 1)T & (2, 1, 0.5)T $\boldsymbol{\alpha}\_1$ & (0.4, 1.1)T & (0.4, 1.1)T & (0.4, 1.1, 0.5)T Target effect & τN(0.7, 0.2) & τN(1, 0) & τN(0.5, 0.1) Sample sizes & @c@ 300, 600, 1200, 2400, 4800 & @c@ 300, 600, 1200, 2400, 4800 & @c@ 289, 576, 1225, 2401, 4761 [table: simulation settings] Throughout we consider explicit SEMs of the form given in Example [example:explicitDAG], where the error terms εCi1, εCi2, εCi3, and εYi are mean zero Gaussian random variables with variance 1, except εYi which is uniformly distributed. We assume that we do not observe Ci1 for some or all units i. For the Erds–Rényi and family networks we choose $h(\boldsymbol{\bar{W}}\_{-i},I^N) = X\_i^1$ and for the 2-d lattices we choose $h(\boldsymbol{\bar{W}}\_{-i},I^N) = (X\_i^1, X\_{i}^2)$, where Xi1 is the fraction of treated parents in IN as per equation, and Xi3 is the fraction of treated parents of parents in IN as per equation. We summarize the features, $\boldsymbol{\alpha}$ vectors, sample sizes and target effects we consider in Table [table: simulation settings]. For each graph-type and sample size we draw `nrep.graph` = 50 interaction network graphs IN. For each of the `nrep.graph` network graphs IN we draw the data `nrep.data` = 100 times according to the explicit SEM Se from Example [example:explicitDAG]. We sample different IN to investigate how the interaction network graph affects the estimator’s performance but emphasize that Theorems [prop:consistency-ols] and [prop:normality-ols] hold for a fixed sequence of IN. To estimate the target global treatment effects, we use the valid adjustment set {C2} determined graphically in the generic graph G, shown in Figure [subfig: generic features]. For comparison, we consider, in addition to the estimator we propose (called fully adjusted estimator in the following), three additional OLS-based estimators (called naive, confounding adjusted, and interference adjusted estimator) according to equation, where we choose Mi as follows: 1. Mi = (1, Wi)T, 2. Mi = (1, Wi, Ci2)T, 3. $ \mathbf{M}\_i = (1,W\_i,\boldsymbol{X}\_i^T,\boldsymbol{O}\_{i}^T )^T$, 4. $ \mathbf{M}\_i = (1,W\_i,\boldsymbol{X}\_i^T,\boldsymbol{O}\_i^T, C\_{i2})^T$. For each IN we use the four estimators to estimate the target effect across the `nrep.data` data-sets and use the results to compute the root mean-squared-error (RMSE), the empirical bias and the logarithm of the empirical variance. Before discussing the results, we first discuss the maximal degree dmax(N) of the interference dependency graphs. For the family networks and the 2-d lattices it is clear that the maximal degree of the interaction network graph IN does not increase with N, and therefore it naturally holds that dmax(N) ∈ o(N1/4) for any sequence IN (see Example [example: maximal degree local]). Thus Theorems [prop:consistency-ols] and [prop:normality-ols] hold in these cases. For the Erds–Rényi networks I(N, pN) we performed a simulation to observe how dmax(N) grows with N for three different regimes: pN = N/10, pN = N− 2/3 and pN = 0.2. Specifically, we drew 100 interaction network graphs for each N and computed the average maximal degree of the corresponding interference dependency graphs. A plot of the logarithm of the average maximal degree, d̄max(N), against the logarithm of N is shown in Figure [fig:slopes]. For I(N, 10/N) the slope is 0.17 < 0.25, that is, dmax(N) empirically satisfies dmax(N) ∈ o(N1/4). Based on this, we expect Theorems [prop:consistency-ols] and [prop:normality-ols] to hold. For I(N, N− 2/3) the slope is 0.64 > 0.25, that is, dmax(N) empirically satisfies dmax(N) ∈ o(N) but not dmax(N) ∈ o(N1/4). Based on this, we expect that Theorem [prop:consistency-ols] holds. For I(N, 0.2) the slope is 1, that is, dmax(N) does not satisfy dmax(N) ∈ o(N). Therefore, neither Theorem [prop:consistency-ols] nor [prop:normality-ols] can be applied in this case. [fig:slopes] We present the results with three plots, showing (i) the average root mean-squared-error (RMSE), (ii) the average empirical bias and (iii) the average logarithm of the empirical variance of τ̂N(π, η) against the logarithm of N for these four estimators, with the average taken over the `nrep.graph` = 50 network graphs IN. We assess the asymptotic normality and the consistency of the variance estimator (Lemma [lemma:varest]) in Appendix [app:normality]. [b]0.9 [res:const10] [b]0.9 [res:random23] [b]0.9 [res:random02] [figure: simulation study] The results for the Erds–Rényi networks are shown in Figure [figure: simulation study]. The results for the family networks and for the 2-d lattices are shown and discussed in Appendix [app:furtherres]. The empirical bias plots show that the naive and the confounding adjusted estimator underestimate τN(π, η), while the interference adjusted estimator overestimates τN(π, η). In contrast the fully adjusted estimator appears to be close to unbiased even for small N. The variance plots also corroborate our results: for I(N, 10/N), the only case where we expect Theorem [prop:normality-ols] to hold, the variance of the fully adjusted estimator converges to zero with rate N− 1/2. We also verified that the fully adjusted estimator converges, when properly scaled, to a normal distribution (see Appendix [app:normality]). For I(N, N− 2/3) we observe that, while the fully adjusted estimator still seems consistent, the convergence rate is slower than N− 1/2. For I(N, 0.2), the variance for the fully adjusted estimator does not appear to converge to zero, indicating inconsistency. Strict Facial-Mask Policy Data Analysis --------------------------------------- We now apply our estimator to study the effect of introducing a strict facial-mask policy on the spread of COVID-19 in Switzerland between July 2020 and December 2020. During several weeks in this early phase of the pandemic, the cantons of Switzerland could choose to adopt the government-determined facial-mask policy (mandatory facial-mask wearing on public transport) or a strict facial-mask policy (mandatory facial-mask wearing on public transport and in all public or shared spaces where social distancing was not possible). This data set was gathered and analysed by and we closely follow their approach, including the causal assumptions. The key difference is that they estimate the causal effect of the strict facial-mask policy on the spread of COVID-19, without considering interference between neighboring cantons. Since people commute between neighboring cantons, the facial-mask policy of neighboring cantons might have had an effect on the spread of COVID-19 in a given canton. Here, we estimate the GATE τN(1, 0), contrasting the hypothetical intervention of introducing the strict facial-mask policy nationally as compared to not introducing it in any canton. We assume the following explicit SEM satisfying Assumption [def:sem], $$\begin{aligned} &\boldsymbol{C}\_{i,t} \leftarrow g\_{\boldsymbol{C}}(\boldsymbol{C}\_{i,t},\boldsymbol{\epsilon}\_{C\_{i,t}}), \ W\_{i,t} \leftarrow g\_W(\boldsymbol{C}\_{i,t},\epsilon\_{W\_{i,t}}),\notag\\ &X\_{i,t} \leftarrow \frac{1}{\left\|\mathcal{N}\_i^{1}\right\|}\sum\_{j \in \mathcal{N}\_i^{1}}W\_{j,t}, \ O\_{i,t} \leftarrow W\_{i,t} X\_{i,t} \text{ and}\notag\\ & Y\_{i,t} =(1,X\_{i,t}) \boldsymbol{\alpha}\_{0} + ( W\_{i,t}, O\_{i,t}) \boldsymbol{\alpha}\_{1} +\boldsymbol{C}\_{i,t}^T \boldsymbol{\gamma} +\epsilon\_{Y\_{i,t}},\label{eq:real} \end{aligned}$$ for each canton i = 1, …, N = 26 and week t = 1, …, T = 24. Here, a unit is given by a tuple (i, t). We assume that $(\boldsymbol{\epsilon}\_{C\_{i,t}},\epsilon\_{W\_{i,t}},\epsilon\_{Y\_{i,t}})$ are jointly independent error terms with expectation zero, and that their distributions do not depend on i or t. Here, Ni1 denotes the neighbors of canton i in $I^N \in {\ensuremath{{\mathbb R}}}^{N \times N}$, where IN is the geographical adjacency matrix. 0.5 [>=stealth, shorten >=1pt, node distance=2cm, on grid, auto, scale=0.6, transform shape, align=center, minimum size=3em ] (m) at (0,0) W; (y) [right=6cm of m] Y; (ylag)[above =3cm of m] J; (meteo)[above =3cm of y ]$\boldsymbol{M}$; (p) [below= 3cm of m] $\boldsymbol{P}$; (w)[right= 3cm of ylag]D; (g)[left= 2.5cm of p]$\boldsymbol{E}$; (h)[below =3cm of y] H; (o)[below left=9cm and 1.5cm of w,diamond] O; (x)[right =3cm of o, diamond] X; (w) edge (m) (w) edge (p) (w) edge (ylag) (w) edge (y) (h) edge (y) (g) edge (m) (g) edge (p) (ylag) edge (m) (ylag) edge[bend right = 20] (p) (ylag) edge (y) (meteo) edge (y) (p) edge (y) (h) edge (m) (m) edge (o) (x) edge (o) (h) edge(p); (x) edge (y) (m) edge (y) (o) edge (y); [subfig: real data generic DAG] 0.5 [subfig: real data results] [fig:dagface] We now describe the response variable, the treatment variable and the covariates we consider. * To specify the response variables, let Gi, t = ln(Ai, t/Ai, t − 1), where Ai, t is the number of reported new cases in canton i in week t. Due to the delay between the time of infection and the reporting of a new case, Gi, t reflects the pandemic situation of a time period before t. Therefore, as response variable we use a future value of Gi, t. Specifically, Yi, t = Gi, t + 2. * Treatment variable, given by the strict facial-mask policy indicator, where 0 denotes the baseline government-determined policy and 1 the strict facial-mask policy. * Indicators reflecting policies on the closing of workplaces, restrictions on gatherings and cancellations of public events. * Unobserved factors that determine the policy variables Wi, t and $\boldsymbol{P}\_{i,t}$. * Canton-specific demographic variables, given by population size, people of age ≥ 80 years in %, and people per km2. * Holiday indicator, where 1 denotes public school holiday. * Meteorological variables, given by sunshine in minutes per day, air temperature in ∘C, and mean relative humidity in %. * Information about the pandemic available to the public in week t, given by the lagged response variable Ji, t = Yi, t − 2. * and Oi, t: Interference feature and its product with the treatment Wi, t. We use weekly data to remove weekly patterns and refer to for more details on the variables and the origin of the data. The assumed generic graph is shown in Figure [subfig: real data generic DAG]. In our analysis, we adjust for $\{\mathbf{D}, H, \boldsymbol{M}, \boldsymbol{P},J\}$ which according to the generic graph is a valid adjustment set. Note that we cannot adjust for $\boldsymbol{E}$ as it is unobserved. In addition to the fully adjusted estimator we again consider the naive, confounding adjusted and interference adjusted estimators described in Section [sec:simul]. The results in Figure [subfig: real data results] show the point estimates τ̂N(1, 0) with their 95%-confidence intervals, computed using equation. All four estimates are significantly negative, indicating that introducing the strict facial-mask policy nationally would have reduced the spread of COVID-19. The fully adjusted estimator provides the smallest estimate, indicating the presence of interference and illustrating the importance of taking it into account. As is always the case with observational data, the results need to be treated with care, as we assume, among other things, to know a valid adjustment set. Discussion ========== There are two natural avenues to generalize the results of this paper. First, in Section [sec:identification] we show that for an explicit SEM following Assumption [def:sem] the generic graph is a causal DAG. It is possible to derive similar results under weaker assumptions. For example, we do not allow for within-unit paths between Wi on Yi that are mediated by some Ci, that is Wi → Ci → Yi, but the results generalize to explicit DAGs with such paths. For valid adjustment we can, however, assume that no such path exists without loss of generality. Second, by the identifiability results from Section [sec:identification], adjustment is only one possible strategy to estimate τN(π, η). Possible alternatives include the front-door criterion and instrumental variables. We restrict ourselves to models satisfying Assumptions [def:sem] as well as adjustment to keep the presentation concise and focused on the crucial insight that we can adapt causal graphical model tools from the i.i.d. setting to network effects. There are three important caveats to our results. First, we require that the interference features be known. In practice, this will generally not be the case. There is, however, novel research on learning the interference mechanism. Second, we assume a linear outcome model. This is needed for the important decomposition result in Proposition [lemma:reformulation]. It may be possible to generalize our results to more flexible outcome models, as long as they admit a decomposition similar to Proposition [lemma:reformulation]. A natural candidate is the class of partially linear models. Third, the constraints on the maximal degree of the interference dependency graph in Theorems [prop:consistency-ols] and [prop:normality-ols] are hard to formally verify, even in relatively simple examples. Graphical Preliminaries ======================= We now give an overview of the graphical terminology used throughout the paper. Graphs and Paths: A graph $G=(\boldsymbol{V}, \boldsymbol{E})$ is a tuple consisting of node-set $\boldsymbol{V}$ and edge-set $\boldsymbol{E}$. Edges may be directed ( → ), bi-directed ( ↔ ), or undirected ( − ). Two edges are adjacent if they have a common node. A path is a sequence of adjacent edges without repetition of a node. A path may consist of just a single edge. We call the first and the final node on a path the endpoint nodes and all remaining nodes on the path nonendpoint nodes. DAGs: A path from node A to node B, where all edges on the path point towards B, together with an edge B → A forms a directed cycle. A directed graph without directed cycles is called a directed acyclic graph (DAG). Proper and Causal Paths: Let $G = (\boldsymbol{V}, \boldsymbol{E})$ be a DAG. A path from a set of nodes $\boldsymbol{A}$ to a set of nodes $\boldsymbol{B}$ in G is a path from a node $V \in \boldsymbol{A}$ to a node $V' \in \boldsymbol{B}$. A path from $\boldsymbol{A}$ to $\boldsymbol{B}$ is called a proper path if only the first node is in $\boldsymbol{A}$. A path from node A to node B in G is called a causal path if all edges on the path point towards B. Otherwise, we call the path noncausal. Parents and Descendants: Let G be a DAG. We define the parents of node B in G as all the nodes A such that the edge A → B exists in G and denote them pa(B, G). We define the descendants of A in G as all the nodes B, such that there exists a causal path from A to B in G and denote them by de(A, G). We use the convention that A ∈ de(A, G). For a set $\boldsymbol{A}$, let $\text{de}(\boldsymbol{A},G)=\bigcup\_{A\in \boldsymbol{A}} \text{de}(A,G)$. Colliders: A nonendpoint node V on a path p in a DAG G is a collider if p contains a subpath of the form U → V ← W. Otherwise, V is called a noncollider on p. Blocking and d-Separation: (Definition 1.2.3 in and Section 2.1 in ) Let $\boldsymbol{A}$ be a set of nodes in a DAG G. A path p is blocked by $\boldsymbol{A}$ if i) p contains a noncollider that is in $\boldsymbol{A}$, or ii) p contains a collider B such that no descendant of B is in $\boldsymbol{A}$. If $\boldsymbol{A}$, $\boldsymbol{B}$ and Z are three pairwise disjoint sets of nodes in G, then Z d-separates $\boldsymbol{A}$ from $\boldsymbol{B}$ if Z blocks every path between $\boldsymbol{A}$ and $\boldsymbol{B}$ in G. We then write $\boldsymbol{A} \ {\ensuremath{{ \perp\_{G}}}}\ \boldsymbol{B} \mid \mathbf{Z}$. Otherwise, we write $\boldsymbol{A} \ {\ensuremath{{ \not\! \perp\_{G}}}}\ \boldsymbol{B} \mid \mathbf{Z}$. (Recursive) Structural Equation Model (SEM): Let $G = (\boldsymbol{V}, \boldsymbol{E})$ be a DAG. The random vector $\boldsymbol{V}= (V\_1, \ldots, V\_k)^T$ is generated from a structural equation model (SEM) compatible with G if each Vj, j ∈ {1, …, k}, is generated by a structural equation, $$V\_j \leftarrow f\_j(\boldsymbol{V}\_{\text{pa}(V\_j, G)}, \epsilon\_j),$$ where fj are functions and εj are independent error terms with expectation 0. Each structural equation is interpreted as the generating mechanism, denoted by the assignment operator ← . Each structural equation is assumed to be invariant to possible changes in the other structural equations. A SEM is called recursive if there exists an ordering such that fj( ⋅ , εj) only depends on variables Vs with s < j for all j = 1, …, k. do-Intervention: A do-intervention do(Vj = Aj) in a SEM is modeled by replacing the structural equation $$\begin{aligned} V\_j \leftarrow h\_j(\boldsymbol{V}\_{\text{pa}(V\_j, G)}, \epsilon\_j) \ \ \text{by} \ \ V\_j \leftarrow A\_j,\end{aligned}$$ where Aj may be deterministic or random. Total Joint Effect: The total joint effect of a set of random variables $\boldsymbol{A} = (A\_1,\ldots, A\_k)$ on a random variable B is given by $ \boldsymbol{\theta}\_{b\boldsymbol{a}} := (\theta\_{b a\_1},\ldots,\theta\_{b a\_k})^T,$ where $$\theta\_{b a\_i}:= \frac{\partial}{\partial a\_i}{\ensuremath{{\mathbb E}}}[B \mid \text{do}(\boldsymbol{A}= \boldsymbol{a}) ], \ \text{for} \ i = 1,\ldots, k.$$ Causal and Forbidden Nodes: Let G be a DAG. We define the causal nodes with respect to $(\boldsymbol{A},\boldsymbol{B})$ in G as all nodes on proper causal paths from $\boldsymbol{A}$ to $\boldsymbol{B}$ excluding $\boldsymbol{A}$ and denote them by $\text{cn}(\boldsymbol{A}, \boldsymbol{B}, G)$. We define the forbidden nodes relative to $(\boldsymbol{A},\boldsymbol{B})$ in G as the descendants of the causal nodes as well as $\boldsymbol{A}$ and denote them by $\text{forb}(\boldsymbol{A},\boldsymbol{B},G)$. Valid Adjustment Sets: Consider disjoint node sets $\boldsymbol{A},\{B\}$ and Z in a DAG $G=(\boldsymbol{V},\boldsymbol{E})$ such that $\boldsymbol{V}$ is generated from a SEM compatible with G. We refer to Z as a valid adjustment set relative to $(\boldsymbol{A}, B)$ in G if * $\mathbf{Z} \cap \mathrm{forb}(\boldsymbol{A},B,G) = \emptyset$, and * Z blocks all proper noncausal paths from $\boldsymbol{A}$ to B. Latent Projection: Let G be a DAG with node set $\boldsymbol{A} \cup \boldsymbol{B}$ where $\boldsymbol{A} \cap \boldsymbol{B}= \emptyset.$ The latent projection of G over $\boldsymbol{B}$ is a graph denoted $G^{\boldsymbol{B}}$ with node set $\boldsymbol{A}$ and edge-set defined as follows: For distinct nodes $A\_i, A\_j \in \boldsymbol{A}$, * $G^{\boldsymbol{B}}$ contains a directed edge Ai → Aj if G contains a directed path Ai → ⋯ → Aj on which all nonendpoint nodes are in $\boldsymbol{B}$, * $G^{\boldsymbol{B}}$ contains a bi-directed edge Ai ↔ Aj if G contains a path of the form Ai ← ⋯ → Aj on which all nonendpoint nodes are noncolliders and in $\boldsymbol{B}$. Proofs for Section [sec:identification] ======================================= Proofs for Section [sec:identnofeat] ------------------------------------ The following definition formalizes what the generic graph G can be interpreted causally means. [Truncated factorization preserving generic graph] [def:structrpreserv] Consider an explicit DAG Ge with a compatible explicit SEM Se on explicit variables $\boldsymbol{V}\_i$, i = 1, …, N. We say that the generic graph $\mathcal{G}=(\boldsymbol{V},\boldsymbol{E})$ is truncated factorization preserving for Ge if it holds for all i = 1, …, N and for all A ⊂ V that $$f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\boldsymbol{A}\_i = \mathbf{a}\_i))= \begin{cases} \prod\_{V \in \mathbf{V}\_i \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,\mathcal{G})), & \text{if }\mathbf{A}\_i =\mathbf{a}\_i, \\ 0, & \text{otherwise,} \end{cases}$$ where for any node $N\_i \in \boldsymbol{V}\_i$ we define pa(Ni, G) = pa(N, G), that is, the parent set of the node N in G corresponding to Ni according to Definition [def:genericDAG]. []proplemmastrucpreservnofeat [lemma:strucpreserv] Let Se be an explicit SEM satisfying Assumption [def:sem0] and let Ge be the corresponding explicit DAG. Then the generic graph G of Ge is truncated factorization preserving. Let $\boldsymbol{A} \subset \boldsymbol{V}$, where $\boldsymbol{V}$ is the node-set of the generic graph $\mathcal{G}=(\boldsymbol{V}, \boldsymbol{E})$. Note first that since the explicit SEM Se is compatible with the explicit DAG Ge, the truncated factorization formula holds with respect to Ge, that is, $$f(\boldsymbol{\bar{v}} \setminus \boldsymbol{a}\_i \mid \text{do}(\mathbf{A}\_i =\mathbf{a}\_i))= \begin{cases} \prod\_{V \in \boldsymbol{\bar{V}} \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,G\_e)), & \text{if }\mathbf{A}\_i =\mathbf{a}\_i, \\ 0, & \text{otherwise,} \end{cases} \label{eq11}$$ where $\boldsymbol{\bar{V}} = \bigcup\_{i=1}^N\boldsymbol{V}\_i$. Further, let $\boldsymbol{\bar{V}}\_{-i} = \boldsymbol{\bar{V}} \setminus \mathbf{V}\_i$ and $\boldsymbol{\bar{Y}}= \bigcup\_{i=1}^N Y\_i$. We distinguish two cases. The first case is $Y\_i \in \boldsymbol{A}\_i$. In the case that Ai = ai, integrating out all variables in $\boldsymbol{\bar{V}}\_{-i}$ we obtain $$\begin{aligned} &f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i =\mathbf{a}\_i)) \notag =\int\_{\boldsymbol{\bar{v}}\_{-i}}\prod\_{V \in \boldsymbol{\bar{V}} \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,G\_e))d\boldsymbol{\bar{v}}\_{-i} \notag\\ &= \prod\_{V \in \mathbf{V}\_i \setminus \mathbf{A}\_i } f(v \mid \mathrm{pa}(V,G\_e)) \int\_{\boldsymbol{\bar{v}}\_{-i}} \prod\_{V \in \boldsymbol{\bar{V}}\_{-i}} f(y \mid \mathrm{pa}(Y,G\_e)) d\boldsymbol{\bar{v}}\_{-i} \\ &= \prod\_{V \in \mathbf{V}\_i \setminus \mathbf{A}\_i } f(v \mid \mathrm{pa}(V,G\_e)) = \prod\_{V \in \mathbf{V}\_i \setminus \mathbf{A}\_i } f(v \mid \mathrm{pa}(V,\mathcal{G})), \notag\end{aligned}$$ since the parents of any node $V \in \mathbf{V}\_i \setminus \boldsymbol{A}\_i $ are in $\mathbf{V}\_i \setminus \boldsymbol{A}\_i $, as $Y\_i \in \boldsymbol{A}\_i$ and Yi is the only variable with parents indexed by other units j. Thus pa(V, Ge) = pa(V, G), where pa(V, G) is defined in Definition [def:structrpreserv], for all nodes in Vi \ Ai, i = 1, …, N. This concludes the proof of the first case. The second case is $Y\_i \notin \boldsymbol{A}\_i$. In the case that Ai = ai, integrating out all variables in $\boldsymbol{\bar{V}}\_{-i}$ we obtain that $$\begin{aligned} &f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i =\mathbf{a}\_i)) \notag \\ &=\int\_{\boldsymbol{\bar{v}}\_{-i}}\prod\_{V \in \boldsymbol{\bar{V}} \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,G\_e))d\boldsymbol{\bar{v}}\_{-i} \notag\\ &= \prod\_{V \in \mathbf{V}\_i \setminus (\mathbf{A}\_i \cup \{Y\_i\}) } f(v \mid \mathrm{pa}(V,G\_e)) \int\_{\boldsymbol{\bar{v}}\_{-i}} \prod\_{Y \in \boldsymbol{\bar{Y}}} f(y \mid \mathrm{pa}(Y,G\_e)) \notag \\ & \quad \quad \prod\_{V \in \boldsymbol{\bar{V}}\_{-i} \setminus \boldsymbol{\bar{Y}}} f(v \mid \mathrm{pa}(V,G\_e))d\boldsymbol{\bar{v}}\_{-i}, \label{prop1:eq1}\end{aligned}$$ where we use that all parents of nodes in $\mathbf{V}\_i \setminus (\boldsymbol{A}\_i \cup\{Y\_i\}) $ are themselves in $\mathbf{V}\_i \setminus (\boldsymbol{A}\_i \cup\{Y\_i\})$. Furthermore, considering the integral in equation, we get $$\begin{aligned} &\int\_{\boldsymbol{\bar{v}}\_{-i}} \prod\_{Y \in \boldsymbol{\bar{Y}}} f(y \mid \mathrm{pa}(Y,G\_e)) \prod\_{V \in \boldsymbol{\bar{V}}\_{-i} \setminus \boldsymbol{\bar{Y}}} f(v \mid \mathrm{pa}(V,G\_e))d\boldsymbol{\bar{v}}\_{-i}\\ &=\int\_{\boldsymbol{\bar{v}}\_{-i}} f(y\_i \mid \mathrm{pa}(Y\_i,G\_e)) \prod\_{j \neq i} f(y\_i \mid \mathrm{pa}(Y\_i,G\_e)) f(\boldsymbol{\bar{v}}\_{-i} \setminus \boldsymbol{\bar{y}}\_{-i}) d\boldsymbol{\bar{v}}\_{-i}\\ &= \int\_{\boldsymbol{\bar{v}}\_{-i}} f(y\_i\mid w\_i,\boldsymbol{c}\_i,\boldsymbol{\bar{w}}\_{-i}) \prod\_{j \neq i} f(y\_j\mid w\_j,\boldsymbol{c}\_j,\boldsymbol{\bar{w}}\_{-j}) f(\boldsymbol{\bar{v}}\_{-i} \setminus \boldsymbol{\bar{y}}\_{-i}) d\boldsymbol{\bar{v}}\_{-i}\\ &= \int\_{\boldsymbol{\bar{v}}\_{-i}} f(y\_i\mid w\_i,\boldsymbol{c}\_i,\boldsymbol{\bar{w}}\_{-i}) \prod\_{j \neq i} f(y\_j\mid w\_i,\boldsymbol{c}\_i, \boldsymbol{c}\_j, \boldsymbol{\bar{w}}\_{-i}) f(\boldsymbol{\bar{v}}\_{-i} \setminus \boldsymbol{\bar{y}}\_{-i} \mid w\_i,\boldsymbol{c}\_i)d\boldsymbol{\bar{v}}\_{-i}\\ &= \int\_{\boldsymbol{\bar{v}}\_{-i}} f(y\_i\mid w\_i,\boldsymbol{c}\_i,\boldsymbol{\bar{v}}\_{-i}) f(\boldsymbol{\bar{v}}\_{-i} \mid w\_i,\boldsymbol{c}\_i)d\boldsymbol{\bar{v}}\_{-i} = f(y\_i \mid w\_i,\boldsymbol{c}\_i)= f(y\_i \mid \mathrm{pa}(y\_i,\mathcal{G})),\end{aligned}$$ where in the first equality we used that $\boldsymbol{\bar{V}}\_{-i} \setminus \boldsymbol{\bar{Y}}\_{-i}$ is an ancestral set, and in the third equality that $Y\_j {\perp \!\!\! \perp}\boldsymbol{C}\_i \mid \boldsymbol{C}\_j, \boldsymbol{\bar{W}}$ and $\boldsymbol{\bar{V}}\_{-i} \setminus \boldsymbol{\bar{Y}}\_{-i} {\perp \!\!\! \perp}W\_i,\boldsymbol{C}\_i$, which follow from Assumption [def:sem0] and the local Markov property, that is, for all $V \in \boldsymbol{\bar{V}}$ it holds that $V {\perp \!\!\! \perp}\boldsymbol{\bar{V}} \setminus \{\text{de}(V, G\_e) \cup \text{pa}(V, G\_e) \} \mid \text{pa}(V, G\_e)$. Thus, combining the above we get $$\begin{aligned} f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i &=\mathbf{a}\_i))= f(y\_i \mid w\_i,\boldsymbol{c}\_i) \prod\_{V \in \mathbf{V}\_i \setminus (\mathbf{A}\_i \cup \{Y\_i\}) } f(v \mid \mathrm{pa}(V,G\_e)) \\ & = \prod\_{V \in \mathbf{V}\_i \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,\mathcal{G})), \end{aligned}$$ since the parents of any node V ∈ Vi \ (Ai ∪ {Yi}) are in Vi and thus pa(V, Ge) = pa(V, G), where pa(V, G) is defined in Definition [def:structrpreserv]. Proofs for Section [sec:identfeat] ---------------------------------- [Invariance of τN(π, η) to linear transformations of features]lemmalemmainvariance [lemma:invariance] Consider an explicit SEM Se satisfying Assumption [def:sem] with features $\boldsymbol{X}\_i =(X\_{i1}, X\_{i2}, \ldots, X\_{iP})^T$. Let τN(π, η) be the treatment effect obtained by using $\boldsymbol{X}\_i$ as features and τ̃N(Pπ, Pη) the treatment effect obtained by replacing $\boldsymbol{X}\_i$ with $\boldsymbol{\tilde{X}}\_i = (l\_1(X\_{i1}), l\_2(X\_{i2}), \ldots, l\_P(X\_{iP}))^T$, where lk(x) : = akx + bk, for k = 1, …, P, with ak, bk ∈ R. It then holds that τN(π, η) = τ̃N(Pπ, Pη). Let unit i be fixed and let us focus on the outcome equation in. Let us look at the case Wi = 1. We reformulate the generating equation of the outcome Yi in Se as $$\begin{aligned} Y\_i &= (1,\boldsymbol{X}\_i^T)\boldsymbol{\beta}\_{1} + \boldsymbol{C}\_i^T\boldsymbol{\gamma} + \epsilon\_{Y\_i} \\ &=\beta^{1}\_0 + \beta^{1}\_1 X\_{i1} + \ldots + \beta^{1}\_P X\_{iP} + \boldsymbol{C}\_i^T\boldsymbol{\gamma} + \epsilon\_{Y\_i} \\ &= \beta^{1}\_0 - \frac{\beta^{1}\_1}{a\_1} b\_1 - \ldots -\frac{\beta^{1}\_P}{a\_P} b\_P + \frac{\beta^{1}\_1}{a\_1} (a\_1X\_{i1} +b\_1) + \ldots + \frac{\beta^{1}\_P}{a\_P} (a\_PX\_{iP} +b\_P) + \boldsymbol{C}\_i^T\boldsymbol{\gamma} + \epsilon\_{Y\_i} \\ &= \tilde{\beta}^1\_0 + \tilde{\beta}^1\_1 l\_1(X\_{i1}) + \ldots + \tilde{\beta}^1\_P l\_P(X\_{iP}) +\boldsymbol{C}\_i^T \boldsymbol{\gamma} + \epsilon\_{Y\_i},\end{aligned}$$ where $$\begin{aligned} \tilde{\beta}^1\_0 &:= \beta^{1}\_0 - \frac{\beta^{1}\_1}{a\_1} b\_1 - \ldots -\frac{\beta^{1}\_P}{a\_P} b\_P, \\ \tilde{\beta}^1\_j &: = \frac{\beta^{1}\_j}{a\_j} \text{ for } j = 1,\ldots, P\end{aligned}$$ and $\boldsymbol{\beta}\_1 = (\beta^1\_0,\dots,\beta^1\_P)$. In the following, we write $\boldsymbol{\omega}\_{1}$ instead of $\boldsymbol{\omega}^N\_{1}(\pi,\eta)$ and $\boldsymbol{\omega}\_{0}$ instead of $\boldsymbol{\omega}^N\_{0}(\pi,\eta)$ to ease notation. We also write $\tilde{\boldsymbol{\omega}}\_{1}$ instead of $\tilde{\boldsymbol{\omega}}^N\_{1}(\pi, \eta)$ and $\tilde{\boldsymbol{\omega}}\_{0}$ instead of $\tilde{\boldsymbol{\omega}}^N\_{0}(\pi, \eta)$, where $\tilde{\boldsymbol{\omega}}\_{1} = (1,l\_1(\omega^{1}\_1), \ldots,l\_P(\omega^{1}\_P))$ are the weights obtained if the linearly transformed features lk(Xik) are used to compute the weights per the equations in Proposition [lemma:reformulation]. It then follows that $\boldsymbol{\omega}^{T}\_1\boldsymbol{\beta}\_{1} = \tilde{\boldsymbol{\omega}}^{T}\_{1}\tilde{\boldsymbol{\beta}}\_{1}$, where $\tilde{\boldsymbol{\beta}}\_{1} = (\tilde{\beta}^{1}\_0,\ldots,\tilde{\beta}^{1}\_P) $, and by the same arguments $\boldsymbol{\omega}^{T}\_{0}\boldsymbol{\beta}\_{0} = \tilde{\boldsymbol{\omega}}^{T}\_{0}\tilde{\boldsymbol{\beta}}\_{0}$, which proves the result. []proplemmastrucpreservwithfeat [lemma:strucpreservwithfeat] Let Se be an explicit SEM satisfying Assumption [def:sem] and let Ge be the corresponding explicit DAG. Then the generic graph G of Ge is truncated factorization preserving, if there is one multivariate node for the features $\boldsymbol{X}\_i$ in Ge. Let $\boldsymbol{A} \subset \boldsymbol{V}$, where $\boldsymbol{V}$ is the node-set of the generic graph $\mathcal{G}=(\boldsymbol{V}, \boldsymbol{E})$. Note first that since the explicit SEM Se is compatible with the explicit DAG Ge, the truncated factorization formula holds with respect to Ge, that is, $$f(\boldsymbol{\bar{v}} \setminus \boldsymbol{a}\_i \mid \text{do}(\mathbf{A}\_i =\mathbf{a}\_i))= \begin{cases} \prod\_{V \in \boldsymbol{\bar{V}} \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,G\_e)), & \text{if }\mathbf{A}\_i =\mathbf{a}\_i, \\ 0, & \text{otherwise,} \end{cases}$$ where $\boldsymbol{\bar{V}} = \bigcup\_{i=1}^N \boldsymbol{V}\_i$. Further, let $\boldsymbol{\bar{V}}\_{-i} = \boldsymbol{\bar{V}} \setminus \mathbf{V}\_i$. We distinguish two cases. The first case is $\boldsymbol{X}\_i \subseteq \boldsymbol{A}\_i$. In the case that Ai = ai, integrating out all variables in $\boldsymbol{\bar{V}}\_{-i}$ we obtain that $$\begin{aligned} &f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i =\mathbf{a}\_i)) \notag =\int\_{\boldsymbol{\bar{v}}\_{-i}}\prod\_{V \in \boldsymbol{\bar{V}} \setminus \mathbf{A}\_i}f(v \mid \mathrm{pa}(V,G\_e))d\boldsymbol{\bar{v}}\_{-i} \notag\\ &= \prod\_{V \in \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i} f(v \mid \mathrm{pa}(V,G\_e)) \int\_{\boldsymbol{\bar{v}}\_{-i}} \prod\_{V \in \boldsymbol{\bar{V}}\_{-i}} f(v\mid \mathrm{pa}(V,G\_e)) d\boldsymbol{\bar{v}}\_{-i}, \label{prop:eq1}\end{aligned}$$ since the parents of any node $ V \in \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i$ are in $ \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i$, since $\boldsymbol{X}\_i \subseteq \boldsymbol{A}\_i$ and $\boldsymbol{X}\_i$ are the only variables with parents indexed by other units j. In the following, we show that the integral in equation equals 1. Consider the product of densities in equation, $$\begin{aligned} &\prod\_{V \in \boldsymbol{\bar{V}}\_{-i}} f(v\mid \mathrm{pa}(V,G\_e)) \notag \\ &= \prod\_{j \neq i} f(\boldsymbol{c}\_j) f(w\_j \mid \boldsymbol{c}\_j) f(\boldsymbol{o}\_j \mid w\_j, \boldsymbol{x}\_j) f(\boldsymbol{x}\_j \mid \boldsymbol{\bar{w}}\_{-j}) f( y\_j \mid w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j, \boldsymbol{x}\_j) \notag \\ &= \prod\_{j \neq i} f(w\_j,\boldsymbol{c}\_j) f(\boldsymbol{o}\_j \mid w\_j, \boldsymbol{x}\_j, \boldsymbol{\bar{w}}\_{-j}) f(\boldsymbol{x}\_j \mid w\_j, \boldsymbol{\bar{w}}\_{-j}) f( y\_j \mid w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j, \boldsymbol{x}\_j) \notag\\ &= \prod\_{j \neq i} f(w\_j,\boldsymbol{c}\_j) f(\boldsymbol{o}\_j,\boldsymbol{x}\_j \mid w\_j, \boldsymbol{\bar{w}}\_{-j}) f( y\_j \mid w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j, \boldsymbol{x}\_j) \notag\\ &=\prod\_{j \neq i} f(w\_j,\boldsymbol{c}\_j \mid \boldsymbol{\bar{w}}\_{-j}) f(\boldsymbol{o}\_j,\boldsymbol{x}\_j \mid w\_j, \boldsymbol{\bar{w}}\_{-j}, \boldsymbol{c}\_j) f( y\_j \mid w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j, \boldsymbol{x}\_j, \boldsymbol{\bar{w}}\_{-j}) \notag\\ &= \prod\_{j \neq i} f(w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid \boldsymbol{\bar{w}}\_{-j}), \label{dens}\end{aligned}$$ where in the second and fourth equality we used Assumption [def:sem] and the local Markov property in Ge, that is, for all $V \in \boldsymbol{\bar{V}}$ it holds that $V {\perp \!\!\! \perp}\boldsymbol{\bar{V}} \setminus \{\text{de}(V, G\_e) \cup \text{pa}(V, G\_e) \} \mid \text{pa}(V, G\_e)$. Especially, we used that $f(\boldsymbol{x}\_j \mid \mathrm{pa}(\boldsymbol{X}\_j,G\_e)) = f(\boldsymbol{x}\_j \mid \boldsymbol{\bar{w}}\_{-j})$ by the local Markov property, even though it does not necessarily hold that $\mathrm{pa}(\boldsymbol{X}\_j,G\_e) = \boldsymbol{\bar{W}}\_{-j}$, that is, not all Wj for j ≠ i need to be in $\mathrm{pa}(\boldsymbol{X}\_j,G\_e)$. We now consider the density in equation for a given j ≠ i, $$\begin{aligned} f(w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid \boldsymbol{\bar{w}}\_{-j}) &= f(w\_j \mid \boldsymbol{\bar{w}}\_{-j}) f( \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid w\_j, \boldsymbol{\bar{w}}\_{-j})\\ &= f(w\_j ) f( \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid w\_i, \boldsymbol{\bar{w}}\_{-i})\\ &= f(w\_j) \frac{f( \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j,w\_i\mid \boldsymbol{\bar{w}}\_{-i})}{f(w\_i)},\end{aligned}$$ using that Wi ⊥ ⊥ Wj for j ≠ i by d-separation in Ge. Using this reformulation of the density and considering the whole integral in equation leads to $$\begin{aligned} &\int\_{\boldsymbol{\bar{v}}\_{-i}} \prod\_{j \neq i} f(w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid \boldsymbol{\bar{w}}\_{-j}) d\boldsymbol{\bar{v}}\_{-i} \\ &= \int\_{\boldsymbol{\bar{v}}\_{-i}} \prod\_{j \neq i} \frac{f(w\_j)}{f(w\_i)}f( \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j,y\_j,w\_i\mid \boldsymbol{\bar{w}}\_{-i}) d\boldsymbol{\bar{v}}\_{-i}.\end{aligned}$$ We now fix j ≠ i. Using Fubini we integrate out all variables indexed by j and obtain, $$\begin{aligned} & \int\_{\boldsymbol{v}\_j}\frac{f(w\_j)}{f(w\_i)} f( \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j,w\_i\mid \boldsymbol{\bar{w}}\_{-i}) d\boldsymbol{v}\_j\\ &=\int\_{w\_j}\frac{f(w\_j)}{f(w\_i)} \left(\int\_{\boldsymbol{c}\_j} \int\_{\boldsymbol{o}\_j} \int\_{\boldsymbol{x}\_j} \int\_{y\_j}f( \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j,w\_i\mid \boldsymbol{\bar{w}}\_{-i}) dy\_j d\boldsymbol{x}\_j d\boldsymbol{o}\_j d\boldsymbol{c}\_j\right) dw\_j\\ &= \int\_{w\_j}\frac{f(w\_j)}{f(w\_i)} f(w\_i \mid \boldsymbol{\bar{w}}\_{-i}) dw\_j = \int\_{w\_j}\frac{f(w\_j)}{f(w\_i)} f(w\_i ) dw\_j= 1,\end{aligned}$$ using in the third equality again that Wi ⊥ ⊥ Wj for j ≠ i by d-separation in Ge. Thus, combining the above we get in the case that $\boldsymbol{A}\_i = \boldsymbol{a}\_i$ that $$f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i = \mathbf{a}\_i)) =\prod\_{V \in \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i} f(v \mid \mathrm{pa}(V,G\_e)) = \prod\_{V \in \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i} f(v \mid \mathrm{pa}(V,\mathcal{G})),$$ since the parents of any node $V \in\ \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i$ are in Vi and thus pa(V, Ge) = pa(V, G), where pa(V, G) are defined in Definition [def:structrpreserv]. This concludes the proof of the first case. The second case is $\boldsymbol{X}\_i \cap \boldsymbol{A}\_i = \emptyset$. In the case that Ai = ai, integrating out all variables in $\boldsymbol{\bar{V}}\_{-i}$ we obtain that $$\begin{aligned} &f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i =\mathbf{a}\_i)) =\int\_{\boldsymbol{\bar{v}}\_{-i}}\prod\_{V \in \boldsymbol{\bar{V}} \setminus \boldsymbol{A}\_i}f(v \mid \mathrm{pa}(V,G\_e))d\boldsymbol{\bar{v}}\_{-i} \notag\\ &= \prod\_{V \in \boldsymbol{V}\_i \setminus \{\mathbf{A}\_i \cup \boldsymbol{X}\_i\}} f(v \mid \mathrm{pa}(V,G\_e)) \int\_{\boldsymbol{\bar{v}}\_{-i}} f(\boldsymbol{x}\_i \mid \mathrm{pa}(\boldsymbol{X}\_i,G\_e)) \prod\_{V \in \boldsymbol{\bar{V}}\_{-i}} f(v\mid \mathrm{pa}(V,G\_e)) d\boldsymbol{\bar{v}}\_{-i}, \label{proppart2}\end{aligned}$$ where we use again that the parents of any node $V \in \boldsymbol{V}\_i \setminus \{\boldsymbol{A}\_i \cup \boldsymbol{X}\_i\}$ are in $ \boldsymbol{V}\_i \setminus \{\boldsymbol{A}\_i \cup \boldsymbol{X}\_i\}$, since $\boldsymbol{X}\_i \cap \boldsymbol{A}\_i = \emptyset$ and $\boldsymbol{X}\_i$ are the only variables with parents indexed by other units j. We now consider the integral in equation, $$\begin{aligned} &\int\_{\boldsymbol{\bar{v}}\_{-i}} f(\boldsymbol{x}\_i \mid \boldsymbol{\bar{w}}\_{-i} ) \prod\_{V \in \boldsymbol{\bar{V}}\_{-i}} f(v\mid \mathrm{pa}(V,G\_e)) d\boldsymbol{\bar{v}}\_{-i} \notag \\ &= \int\_{\boldsymbol{\bar{v}}\_{-i}} f(\boldsymbol{x}\_i \mid \boldsymbol{\bar{w}}\_{-i} )\prod\_{j \neq i} f(w\_j, \boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid \boldsymbol{\bar{w}}\_{-j}) d\boldsymbol{\bar{v}}\_{-i}\\ &= \int\_{\boldsymbol{\bar{v}}\_{-i}} f(\boldsymbol{x}\_i \mid \boldsymbol{\bar{w}}\_{-i} )\prod\_{j \neq i} f(w\_j \mid \boldsymbol{\bar{w}}\_{-j}) f(\boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid w\_i, \boldsymbol{\bar{w}}\_{-i}) d\boldsymbol{\bar{v}}\_{-i} \\ &= \int\_{w\_j, j \neq i} f(\boldsymbol{x}\_i \mid \boldsymbol{\bar{w}}\_{-i} ) \\ &\quad\quad \left(\prod\_{j \neq i} f(w\_j) \int\_{\boldsymbol{c}\_j} \int\_{\boldsymbol{o}\_j} \int\_{\boldsymbol{x}\_j} \int\_{y\_j} f(\boldsymbol{c}\_j, \boldsymbol{o}\_j,\boldsymbol{x}\_j, y\_j \mid w\_i, \boldsymbol{\bar{w}}\_{-i}) dy\_j d\boldsymbol{x}\_j d\boldsymbol{o}\_j d\boldsymbol{c}\_j \right) d \boldsymbol{\bar{w}}\_{-i}\\ &= \int\_{w\_j, j \neq i} f(\boldsymbol{x}\_i \mid \boldsymbol{\bar{w}}\_{-i} ) \left(\prod\_{j \neq i} f(w\_j) \right) d \boldsymbol{\bar{w}}\_{-i}= \int\_{w\_j, j \neq i} f(\boldsymbol{x}\_i \mid \boldsymbol{\bar{w}}\_{-i} ) f(\boldsymbol{\bar{w}}\_{-i}) d \boldsymbol{\bar{w}}\_{-i}\\ &= \int\_{w\_j, j \neq i} f(\boldsymbol{x}\_i,\boldsymbol{\bar{w}}\_{-i} ) d \boldsymbol{\bar{w}}\_{-i} = f(\boldsymbol{x}\_i),\end{aligned}$$ using in the second equality again equation and that Wi ⊥ ⊥ Wj for j ≠ i by d-separation in Ge. Thus, combining the above we get in the case that $\boldsymbol{A}\_i = \boldsymbol{a}\_i$ that $$f(\boldsymbol{v}\_i \setminus \boldsymbol{a}\_i \mid {{\text{do}}}(\mathbf{A}\_i = \mathbf{a}\_i)) =f(\boldsymbol{x}\_i)\prod\_{V \in \boldsymbol{V}\_i \setminus \{\boldsymbol{A}\_i \cup \boldsymbol{X}\_i\}} f(v \mid \mathrm{pa}(V,G\_e)) = \prod\_{V \in \boldsymbol{V}\_i \setminus \boldsymbol{A}\_i} f(v \mid \mathrm{pa}(V,\mathcal{G})),$$ since the parents of any node $V \in \boldsymbol{V}\_i \setminus \{\boldsymbol{A}\_i \cup \boldsymbol{X}\_i\}$ are in $ \boldsymbol{V}\_i \setminus \{\boldsymbol{A}\_i \cup \boldsymbol{X}\_i\}$, and thus pa(V, Ge) = pa(V, G), where pa(V, G) are defined in Definition [def:structrpreserv]. In addition, the parent set of $\boldsymbol{X}\_i$ in G is the empty set. This concludes the proof of the second case. Let us consider first the term ${\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]$ for a fixed unit i. Plugging in the outcome equation, we obtain $$\begin{aligned} &{\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \\ &\quad= (1-{\ensuremath{{\mathbb E}}}[W\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]){\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}\_i^T) \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \boldsymbol{\beta}\_{0} \\ &\quad\quad+{\ensuremath{{\mathbb E}}}[W\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]{\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}\_i^T) \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \boldsymbol{\beta}\_{1} \\ &\quad\quad+ {\ensuremath{{\mathbb E}}}[\boldsymbol{C}\_i^T \mid {{\text{do}}}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \boldsymbol{\gamma} + {\ensuremath{{\mathbb E}}}[\epsilon\_{Y\_i} \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi)) ]\\ &\quad= (1-\pi){\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}\_i^T) \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \boldsymbol{\beta}\_{0} \\ &\quad\quad+ \pi{\ensuremath{{\mathbb E}}}[(1,\boldsymbol{X}\_i^T) \mid {{\text{do}}}(\boldsymbol{\bar{W}}\_{-i} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))] \boldsymbol{\beta}\_{1} + {\ensuremath{{\mathbb E}}}[\boldsymbol{C}\_i^T] \boldsymbol{\gamma},\end{aligned}$$ where the first equality holds because $W\_i {\perp \!\!\! \perp}\boldsymbol{X}\_i$ by d-separation in Ge. The second equality holds because Wi and $\boldsymbol{\bar{W}}\_{-i}$ are d-separated in the explicit graph obtained by removing all incoming edges into the nodes in $\boldsymbol{\bar{W}}\_{-i}$ (do-calculus Rule 1 ). Similarly, Ci and $\boldsymbol{\bar{W}}\_{-i}$ are d-separated in the explicit graph obtained by removing all incoming edges into the nodes in $\boldsymbol{\bar{W}}\_{-i}$. This yields $$\begin{aligned} \tau\_N(\pi,\eta) &:= \frac{1}{N}\sum\_{i=1}^N\left( {\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\pi))]- {\ensuremath{{\mathbb E}}}[Y\_i \mid \text{do}(\boldsymbol{\bar{W}} \overset{\text{i.i.d.}}{\sim} \text{Bern}(\eta))] \right)\\ &= \boldsymbol{\omega}^N\_{0}(\pi,\eta)^T\boldsymbol{\beta}\_{0} + \boldsymbol{\omega}^N\_{1}(\pi,\eta)^T \boldsymbol{\beta}\_{1}\\ &= \boldsymbol{\omega}^N\_{0}(\pi,\eta)^T \boldsymbol{\alpha}\_{0} + \boldsymbol{\omega}^N\_{1}(\pi,\eta)^T(\boldsymbol{\alpha}\_{0} +\boldsymbol{\alpha}\_{1} ),\end{aligned}$$ where $\boldsymbol{\beta}\_{1} = \boldsymbol{\alpha}\_{0}+ \boldsymbol{\alpha}\_{1}$, $\boldsymbol{\beta}\_{0} =\boldsymbol{\alpha}\_{0}$, and the weights $\boldsymbol{\omega}^N\_{1}(\pi,\eta)$ and $\boldsymbol{\omega}^N\_{0}(\pi,\eta)$ are as defined in the statement of Proposition [lemma:reformulation]. Recall the outcome equation, $$\begin{aligned} Y\_i \leftarrow (1,\boldsymbol{X}\_i^T)\boldsymbol{\alpha}\_{0} + (W\_i,\boldsymbol{O}^T\_i) \boldsymbol{\alpha}\_{1} + \boldsymbol{C}\_i^T \boldsymbol{\gamma}+ \epsilon\_{Y\_i},\ \ i = 1,\ldots, N\end{aligned}$$ with $\boldsymbol{X}\_i \coloneqq (X\_{i1}, X\_{i2}, \ldots, X\_{iP})^T \in \mathbb{R}^{P+1}$	arxiv_0000546
Hyperbolization of affine Lie algebras ====================================== In 1983, Feingold and Frenkel posed a question about possible relations between affine Lie algebras, hyperbolic Kac–Moody algebras and Siegel modular forms. In this paper we give an automorphic answer to this question and its generalization. We classify hyperbolic Borcherds–Kac–Moody superalgebras whose super-denominators define reflective automorphic products of singular weight on lattices of type 2U ⊕ L. As a consequence, we prove that there are exactly 81 affine Lie algebras $\hat{\mathfrak{g}}$ which have nice extensions to hyperbolic BKM superalgebras for which the leading Fourier–Jacobi coefficients of super-denominators coincide with the denominators of $\hat{\mathfrak{g}}$. We find that 69 of them appear in Schellekens’ list of semi-simple V1 structures of holomorphic CFT of central charge 24, while 8 of them correspond to the N = 1 structures of holomorphic SCFT of central charge 12 composed of 24 chiral fermions. The last 4 cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. This clarifies the relationship of affine Lie algebras, vertex algebras and hyperbolic BKM superalgebras at the level of modular forms. Introduction ============ Affine Lie algebras are the simplest class of infinite dimensional Kac–Moody Lie algebras, and they have numerous connections with other areas of mathematics and theoretical physics. The next simplest class of Kac–Moody algebras after the affine algebras are the hyperbolic Lie algebras. In 1983, as an extension of the affine Lie algebra of type A1, Feingold and Frenkel investigated the hyperbolic Kac–Moody algebra with Cartan matrix $$\left( \begin{array}{ccc} 2 & -2 & 0 \\ -2 & 2 & -1 \\ 0 & -1 & 2 \end{array} \right)$$ and found a connection with the Igusa cusp form of weight 35 on ${\mathop{\mathrm {Sp}}\nolimits}\_2({\mathbb Z})$. This suggests exploring more general relationships between hyperbolic Kac–Moody Lie algebras and modular forms of several variables. In 1988, Borcherds introduced generalized Kac–Moody algebras, which are now usually called Borcherds–Kac–Moody or simply BKM algebras. These infinite-dimensional Lie algebras are also defined in terms of Chevalley–Serre generators and relations that are encoded in a generalized Cartan matrix, and they differ from Kac–Moody algebra mainly by allowing the diagonal entries of the Cartan matrix to be non-positive. In other words, simple roots are allowed to be imaginary, whereas simple roots in a Kac–Moody algebra are always real. The best-known example of a BKM algebra is the monster Lie algebra. In 1992, Borcherds constructed this algebra as the BRST cohomology related to the monster vertex algebra by means of the no-ghost theorem from string theory. By considering the action of the monster group on the denominator identity of the monster Lie algebra, Borcherds proved the celebrated monstrous moonshine conjecture. Furthermore, he observed that the denominator functions of some BKM algebras are modular forms on orthogonal groups of signature (l, 2). In 1995 and 1998 Borcherds developed the theory of singular theta lift to construct modular forms for arithmetic subgroups of ${\mathop{\null\mathrm {O}}\nolimits}(l,2)$ which have infinite product expansions. These are called automorphic products, or Borcherds products, and they are natural candidates for the denominator functions of BKM algebras. Similarly to affine Lie algebras, BKM algebras and automorphic products also have many applications in physics. For example, Harvey and Moore proposed that BKM algebras should play as the underlying organizing structure of BPS states in string compactifications; in particular, the denominators of BKM algebras might be generating functions for BPS states. In 1996 Gritsenko and Nikulin constructed an automorphic correction of the rank-three hyperbolic Lie algebra considered previously by Feingold and Frenkel. More precisely, they extended this hyperbolic Lie algebra to a (hyperbolic) BKM algebra by adding infinitely many imaginary simple roots, such that the denominator of the resulting BKM algebra is exactly the Igusa cusp form of weight 35. Later, they constructed automorphic corrections of other hyperbolic Lie algebras in a series of papers. A common feature of these corrections is that the resulting BKM algbera only has finitely many real simple roots, a Weyl chamber of finite volume and a Weyl vector of negative norm. Moreover, its denominator usually defines a cuspidal automorphic product on ${\mathop{\null\mathrm {O}}\nolimits}(l,2)$. These corrections extend affine Lie algebras in a nice way, but they have several features that are not preferred from our perspective: * for the automorphic correction G of an affine Lie algebra $\hat{\mathfrak{g}}$, the multiplicity of an imaginary root of $\hat{\mathfrak{g}}$ in G is strictly greater than its multiplicity in $\hat{\mathfrak{g}}$; * the set of imaginary simple roots is very complicated, although the set of real simple roots is easy to describe; * the role of the vertex algebra is unclear. In this paper we extend affine Lie algebras to (hyperbolic) BKM algebras in a different way. Certain affine Kac–Moody algebras $\hat{\mathfrak{g}}$ will be extended to BKM algebras Gg which have infinitely many real simple roots and satisfy: * for any root α of $\hat{\mathfrak{g}}$, the root multiplicities of α in $\hat{\mathfrak{g}}$ and Gg are the same; * the imaginary simple roots of Gg are negative integral multiples of the Weyl vector; * the Lie algebras $\hat{\mathfrak{g}}$ and Gg are closely related to some exceptional vertex algebras. Our main results are about the classification and construction of such extensions, which are connected with various types of modular forms. In the rest of the introduction, we will explain the setup and state the main theorems. Denominators of affine Lie algebras and Jacobi forms ---------------------------------------------------- Let g be a finite-dimensional simple Lie algebra of rank r and let Δg+ be a set of positive roots. The product side of the denominator identity of the associated affine Lie algebra $\hat{\mathfrak{g}}$ is the holomorphic function $$\vartheta\_{\mathfrak{g}}(\tau,\mathfrak{z})=\eta(\tau)^r \prod\_{\alpha\in \Delta\_\mathfrak{g}^+}\frac{\vartheta(\tau, {{\langle{\alpha,\mathfrak{z}}\rangle}})}{\eta(\tau)},$$ where η and ϑ are the Dedekind eta function and the odd Jacobi theta function, respectively: $$\begin{aligned} \eta(\tau)&=q^{\frac{1}{24}}\prod\_{n=1}^\infty(1-q^n), \quad \tau\in {\mathbb H}, \; q=e^{2\pi i\tau},\\ \vartheta(\tau,z)&=-q^{\frac{1}{8}}\zeta^{-\frac{1}{2}}\prod\_{n=1}^\infty(1-q^{n-1}\zeta)(1-q^n\zeta^{-1})(1-q^n), \quad z\in {\mathbb C}, \; \zeta= e^{2\pi iz}.\end{aligned}$$ The function ϑg has nice modular properties and is an example of a Jacobi form in the sense of Eichler and Zagier. Let L be an even positive definite lattice. A Jacobi form of integral weight k and index L is a holomorphic function $\varphi : {\mathbb H}\times (L \otimes {\mathbb C}) \rightarrow {\mathbb C}$ that is modular under ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$ and doubly quasi-periodic, namely $$\begin{aligned} \varphi \left( \frac{a\tau +b}{c\tau + d},\frac{\mathfrak{z}}{c\tau + d} \right)& = (c\tau + d)^k \exp{\left(i \pi t \frac{c(\mathfrak{z},\mathfrak{z})}{c \tau + d}\right)} \varphi ( \tau, \mathfrak{z} ), \quad A=\begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix} \in {\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z}),\\ \varphi (\tau, \mathfrak{z}+ x \tau + y)&= \exp{\big(-i \pi t \big( (x,x)\tau +2(x,\mathfrak{z})\big)\big)} \varphi (\tau, \mathfrak{z} ), \quad x,y\in L,\end{aligned}$$ and whose Fourier expansion satisfies a boundary condition. The function ϑg is a Jacobi form of weight r/2 and index Pg∨(hg∨) for some character, where Pg∨ is the dual of the root lattice and hg∨ is the dual Coxeter number. Note that Jacobi forms defined by an expression similar to ϑg are called theta blocks following Gritsenko–Skoruppa–Zagier. Automorphic products of singular weight --------------------------------------- A modular form of integral weight k and trivial character for an arithmetic subgroup $\Gamma<{\mathop{\null\mathrm {O}}\nolimits}(l,2)$ is a holomorphic function on the associated type IV symmetric domain which satisfies $$\begin{aligned} F(t\mathcal{Z})&=t^{-k} F(\mathcal{Z}), \quad t\in {\mathbb C}^\times,\\ F(g\mathcal{Z})&=F(\mathcal{Z}), \qquad g \in \Gamma. \end{aligned}$$ Let M be an even lattice of signature (l, 2). The input into Borcherds’ theta lift is a vector-valued modular form of weight 1 − l/2 with integral Fourier expansion for the Weil representation of ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$ attached to the discriminant form Mʹ/M, and the output is a meromorphic modular form for a certain subgroup of ${\mathop{\null\mathrm {O}}\nolimits}(M)$ which has an infinite product expansion at any 0-dimensional cusp and whose divisors are linear combinations of hyperplanes. Since the denominators of affine Lie algebras satisfy modularity, it is natural to focus on hyperbolic BKM algebras whose denominators are modular. Let G be a BKM algebra whose denominator function defines the Fourier expansion of an ${\mathop{\null\mathrm {O}}\nolimits}(l,2)$-modular form F at some 0-dimensional cusp. Since F has an infinite product expansion, by Bruinier’s converse theorem one expects that it can be constructed by the Borcherds lift. In the case, the roots and their multiplicities are encoded in the Fourier expansion of the input. When F has singular weight, i.e. weight l/2 − 1, the Fourier expansion is supported only on isotropic vectors, which often forces the imaginary simple roots of G to be negative integral multiples of the Weyl vector. Moreover, it is conjectured in this particular case that G can be constructed as the BRST cohomology related to some vertex algebra, similarly to the monster algebra. This suggests focusing on BKM algebras whose denominators are automorphic products of singular weight. The zeros of F that contain the cusp are actually hyperplanes orthogonal to real roots of G, hence F is anti-invariant under the reflections through these hyperplanes. It is natural to expect that F is anti-invariant under all reflections associated with zeros of F. This has been proven by the last two named authors. It follows that F is a reflective modular form. A non-constant modular form on $\Gamma < {\mathop{\null\mathrm {O}}\nolimits}(M)$ is called reflective if it vanishes only on mirrors of reflections fixing the lattice M. Reflective modular forms were introduced in 1998 by Borcherds and Gritsenko–Nikulin, and their classification has been an active project for the past thirty years (see ), because they have nice applications to hyperbolic reflection groups, algebraic geometry and free algebras of modular forms in addition to infinite-dimensional Lie algebras. Main results ------------ BKM algebras whose denominator functions are reflective automorphic products of singular weight are exceptional. The main examples are the fake monster algebra and their twists by the Conway group Co0. There are conjecturally only finitely many such algebras and constructing and classifying them is an open problem. Many partial results have been proved towards such a classification. In this paper we contribute some new results in this direction. Let U be an even unimodular lattice of signature (1, 1) and let L be an even, positive-definite lattice. The input of the Borcherds lift on 2U ⊕ L can be identified with Jacobi forms of weight 0 and index L. We will identify affine Lie algebras, which naturally extends to BKM algebras or superalgebras whose denominators or super-denominators are reflective Borcherds product of singular weight on lattices of type 2U ⊕ L. The setting is inspired by the following result. [MTH1] Let F be a reflective Borcherds product of singular weight on 2U ⊕ L whose Jacobi form input has Fourier expansion $$\phi(\tau,\mathfrak{z}) = \sum\_{n\in {\mathbb Z}} \sum\_{\ell \in L'} f(n,\ell) q^n \zeta^\ell$$ satisfying that f(0, ℓ) ≥ 0 for all ℓ ∈ Lʹ. If L is the Leech lattice, then F is the denominator of the fake monster algebra and ϕ is the full character of the Leech lattice vertex operator algebra. Otherwise, the set R = {ℓ ∈ Lʹ : ℓ ≠ 0, f(0, ℓ) ≠ 0} determines a finite-dimensional semi-simple Lie algebra g = ⨁j = 1sgj, kj with the same rank as L such that the identity $$\label{MEQ} C:= \frac{\dim \mathfrak{g}}{24} - a = \frac{h\_j^\vee}{k\_j}$$ holds for any 1 ≤ j ≤ s and such that the leading Fourier–Jacobi coefficient of F at the 1-dimensional cusp determined by 2U coincides with the denominator of the associated affine Lie algebra $\hat{\mathfrak{g}}$. For any 1 ≤ j ≤ s, gj is a simple ideal of g, kj is a positive integer indicating the level of gj and hj∨ is the dual Coxeter number of gj. The number a equals f( − 1, 0), which has to be 0 or 1. When a = 0, kj > 1 for any 1 ≤ j ≤ s. The cases a = 0 and a = 1 are called symmetric and anti-symmetric, respectively. If the q0-term of ϕ has negative Fourier coefficients, then R determines a semi-simple Lie superalgebra and the corresponding BKM superalgebra will have odd real roots. We will consider this case in a separate paper. If the Fourier expansion of F defines the (super)-denominator of a BKM (super)-algebra G, then the (super)-denominator has the form $$e^\rho \prod\_{\alpha>0} \big( 1 - e^{-\alpha} \big)^{f(nm,\ell)},$$ where ρ is the Weyl vector of F, and where (n, ℓ, m) are coordinates of positive roots α ∈ U ⊕ Lʹ with $n\in{\mathbb Z}$, $m\in {\mathbb N}$, ℓ ∈ Lʹ and α2 = ℓ2 − 2nm. The above $\hat{\mathfrak{g}}$ is embedded into G as the sum of the root spaces associated with roots of type ± (n, ℓ, 0). In this way, G can be regarded as a graded module over $\hat{\mathfrak{g}}$ graded by $m\in {\mathbb N}$. This leads us to define G as a hyperbolization of $\hat{\mathfrak{g}}$ and F as a hyperbolization of the denominator of $\hat{\mathfrak{g}}$. It turns out that there are only 81 affine Lie algebras with a hyperbolization: [MTH2] There are 81 possibilities for the semi-simple Lie algebra g in Theorem [MTH1] and they fall into three categories: 1. 69 make up Schellekens’ list of semi-simple V1 structures of holomorphic vertex operator algebras of central charge 24; 2. 8 are the N = 1 structures of holomorphic vertex operator superalgebras F24 of central charge 12 composed of 24 fermions; 3. The remaining 4 cases A1, 16, A1, 82, A1, 44 and A2, 9 possess an exceptional modular invariant that comes from a nontrivial automorphism of the fusion algebra. Case (1) is anti-symmetric, while Cases (2) and (3) are symmetric. Schellekens’ list was established using the solutions of Equation with a = 1. Holomorphic vertex operator superalgebras of central charge 12 were classified by Creutzig, Duncan and Riedler, and the N = 1 structures of F24 were determined in, corresponding to solutions of Equation with a = 0 and C = 1. The exceptional modular invariants mentioned in (3) were discovered around the 1990s by Moore and Seiberg, Verstegen and Gannon. The 4 exotic cases satisfy Equation with a = 0 and C < 1. We now present hyperbolizations of these affine Lie algebras. [MTH3] For any g in Theorem [MTH2] there exists a singular-weight reflective Borcherds product Ψg on some lattice 2U ⊕ Lg whose leading Fourier–Jacobi coefficient equals the denominator of $\hat{\mathfrak{g}}$. Moreover, the Jacobi form input ϕg can be expressed as a ${\mathbb Z}$-linear combination of full characters of the affine vertex operator algebra generated by $\hat{\mathfrak{g}}$. The construction will be briefly summarized here. If g is in Schellekens’ list, Lg is the orbit lattice in Höhn’s construction of the holomorphic VOA of central charge 24 with V1 = g, and ϕg is the full character of the VOA. If g is of symmetric type, Lg is the maximal even sublattice of the coweight lattice of g. If g defines an N = 1 structure of F24, then the Jacobi form input can be expressed in terms of characters of F24 as $$\phi\_{\mathfrak{g}}=(\chi\_{\mathrm{NS}} - \chi\_{\widetilde{\mathrm{NS}}} - \chi\_{\mathrm{R}})/2.$$ Finally, we will explain the relation between Jacobi form inputs and exceptional modular invariants for the remaining four g. The Jacobi form input for g = A1, 16 can be written in terms of affine characters as $$\phi\_{A\_{1,16}}=\chi^{A\_{1,16}}\_{2,\frac19}+\chi^{A\_{1,16}}\_{14,\frac{28}{9}}-\chi^{A\_{1,16}}\_{8,\frac{10}{9}},$$ and we find that the difference between the simple current modular invariant and the exceptional modular invariant is given by ∣ϕA1, 16∣2. Similar relations hold for the other three g. Note that these Lg are chosen so that the resulting BKM superalgebra has root lattice U ⊕ Lgʹ. Clearly, the Borcherds products Ψg in Theorem [MTH3] are closely related to vertex algebras. We therefore expect that the BRST cohomology related to these vertex algebras define the BKM (super)-algebras with Ψg as the (super)-denominators. This type of realization has been achieved in under some technical assumptions for g from Schellekens’ list and the N = 1 structure of F24. However, such a realization is completely open for g related to the four exceptional modular invariants. Affine Lie algebras, vertex algebras and BKM (super)-algebras are therefore closely connected from the point of view of the attached modular forms. The connections are illustrated in Figure [fig:hyperbolization]. [node distance=3cm, auto] (A) Affine Lie algebra; (AA) [right of=A] ; (B) [right of=AA] Vertex algebra; (AAA) [right of=B] ; (C) [right of=AAA] BKM superalgebra; (D) [below of=A] Theta block; (E) [below of=B] Weight 0 Jacobi form; (F) [below of=C] Borcherds product; (A) to node (B); (B) to node (C); (D) to node (E); (E) to node Lift (F); (A) to node (D); (B) to node (E); (C) to node (F); (F) to node (D); (A) to node Hyperbolization (C); (A) to node (E); [fig:hyperbolization] Outline of the proof -------------------- The proof of Theorem [MTH1] relies on some previous results. We know from that singular-weight reflective Borcherds products have only simple zeros and are anti-invariant under reflections associated with their zeros. Therefore, the nonzero coefficients f(0, ℓ) of the Jacobi form input have to be 1 if ℓ ≠ 0, and f(0, 0) equals the rank of g by singular-weight. The theorem follows by applying an argument used by the second named author to classify reflective modular forms. Theorem [MTH1] further shows that the central charge of the affine vertex operator algebra generated by $\hat{\mathfrak{g}}$ is $$c\_\mathfrak{g}= \frac{24(C+a)}{C+1}.$$ In particular, cg = 24 if a = 1, and cg = 12 if a = 0 and C = 1. This motivates the three groupings in Theorem [MTH2]. To prove Theorem [MTH2] we first solve equations of type. In the anti-symmetric case, Equation was first derived by Schellekens in the context of conformal field theories. Schellekens found 221 solutions to this equation and eliminated 152 of them to arrive at his list. We also have the same extra solutions to eliminate, but we have to use a completely different approach. In the symmetric case, there are 17 solutions to Equation and we have to rule out 5 of them. In the setting of Theorem [MTH1], we can prove that L(C) is an integral lattice and that L is bounded by Qg < L < Pg, where Qg and Pg are the coroot lattice and coweight lattice of g, respectively. On one hand, for every extraneous g, we will be able to find an even overlattice K of L for which there is no reflective Borcherds product on 2U ⊕ K satisfying certain constraints. On the other hand, we prove that if 2U ⊕ L has a singular-weight reflective Borcherds product then 2U ⊕ K has also a reflective Borcherds product of the same type. Taken together, this allows us to rule out all 157 extraneous solutions of Equation. We will now sketch the proof of Theorem [MTH3], beginning with the anti-symmetric case. Let V be a holomorphic VOA of central charge 24 with semi-simple V1 = g. The full character χV of V is known to be a Jacobi form of weight 0 and lattice index Qg with non-negative integral Fourier expansion. This immediately implies that the singular theta lift of χV, denoted $\Borch(\chi\_V)$, defines a holomorphic Borcherds product of singular weight on 2U ⊕ Qg. It remains to find an extension L of Qg for which $\Borch(\chi\_V)$ is reflective on 2U ⊕ L. We recognize that L should be the Höhn’ orbit lattice Lg. Höhn proposed a construction of V as the simple current extension of the tensor product of the lattice VOA VLg and a certain VOA Wg of central charge $24-{\mathop{\mathrm {rk}}\nolimits}(\mathfrak{g})$ with trivial weight-one subspace. This construction corresponds to the theta decomposition of χV as a Jacobi form of index Lg. It is possible to prove directly that $\Borch(\chi\_V)$ is reflective on 2U ⊕ Lg by computing the Fourier expansion of χV; however, this is only feasible for certain specific g. We carry out the calculation for g = B12, 2, A2, 2F4, 6 and C4, 10, in which case [Lg : Qg] ≤ 2. To complete the proof for the remaining g, we relate $\Borch(\chi\_V)$ to a twisted denominator of the fake monster algebra. By, there exists a conjugacy class [g] of the Conway group Co0 such that Wg is isomorphic to the orbifold VΛgĝ, where Λg is the coinvariant sublattice of the Leech lattice Λ. Moreover, $\Borch(\chi\_V)$ for distinct V define the same modular form on type IV symmetric domain if V1 = g correspond to the same [g]. Combining this fact with our previous calculation of χV, we prove that $\Borch(\chi\_V)$ is reflective if V1 corresponds to a class [g] whose order is distinct from its level. The last two named authors proved that the g-twisted denominator of the fake monster algebra defines a reflective Borcherds product Φg of singular weight on U(ng) ⊕ U ⊕ Λg if g has the same level and order ng, where Λg is the sublattice of Λ fixed by g. We identify $\Borch(\chi\_V)$ with the associated Φg and thus prove that $\Borch(\chi\_V)$ is reflective if V1 corresponds to some [g] with the equal level and order. As a by-product, we find that the BKM algebra constructed as the BRST cohomology related to V and the g-twist of the fake monster algebra are isomorphic if g has equal order and level, but they are not isomorphic if the order and level of g are distinct. Then we consider the symmetric case with C = 1. For any N = 1 structure of F24, we show that there exists a Co0-conjugacy class [g] with level equal to its order such that the Borcherds product $\Borch(\phi\_\mathfrak{g})$ is the g-twisted denominator of the fake monster algebra. This is proven by identifying their Jacobi form inputs, and relies on the construction of these g-twisted denominators as Gritsenko (additive) lifts due to Dittmann and the second named author. We then confirm that $\Borch(\phi\_\mathfrak{g})$ is a singular-weight reflective Borcherds product and that the BKM superalgebra constructed in as the BRST cohomology related to F24 is isomorphic to the g-twist of the fake monster algebra. In the final case, we determine the Jacobi form inputs in terms of affine characters and prove that their singular theta lifts are reflective by direct calculation. Outline of the paper -------------------- In Section [sec:preliminaries], we quickly introduce Jacobi forms, affine Kac–Moody algebras, reflective modular forms, automorphic products and Borcherds–Kac–Moody algebras. We also fix some notation that will be used frequently later on. In Section [sec:hyperbolization] we define the hyperbolization of an affine Lie algebra and explain its motivation. In Section [sec:root-system] we present the proof of Theorem [MTH1] and the solutions of Equation. In Section [sec:main-results] we state the full version of the main results. In Section [sec:construct-anti], we first introduce holomorphic VOA of central charge 24 and Höhn’s construction, as well as their relations with twisted denominators of the fake monster algebra. We then construct hyperbolizations of the affine Lie algebras on Schellekens’ list. In Section [sec:construct-symmetric-C=1], we begin by introducing holomorphic SVOA of central charge 12 and type F24, and then construct hyperbolizations for the eight N = 1 structures of F24. As an application, we construct many exceptional modular invariants by considering the conformal embedding from the affine VOA generated by the N = 1 structure to F24 and considering an automorphism of the D12, 1 fusion algebra. In Section [sec:construct-symmetric-C<1], we construct hyperbolizations of the remaining affine Lie algebras and explain their connection with some exceptional modular invariants. Combining Sections [sec:construct-anti]-[sec:construct-symmetric-C<1] completes the proof of Theorem [MTH3]. In Section [sec:additive lifts], we present a uniform construction of the 12 symmetric singular-weight reflective Borcherds products as additive lifts. In Section [sec:Fourier-expansion], we compute the Fourier expansions of these singular-weight reflective Borcherds products at the 0-dimensional cusp determined by one copy of U. Section [sec:exclude-anti-root] is devoted to the proof of the anti-symmetric case of Theorem [MTH2]. Section [sec:exclude-root] contains the proof of the symmetric case of Theorem [MTH2]. In Section [sec:paramodular] we give an application of our main results. We use the pull-back to construct a new infinite series of anti-symmetric Siegel paramodular forms of weight 3. In Section [sec:problems] we raise some related questions and conjectures. This article ends with several long tables. Acknowledgements KS thanks Thomas Creutzig, Zhihao Duan, Kimyeong Lee and Hiraku Nakajima for useful discussions. HW thanks Valery Gritsenko, Xingjun Lin and Nils Scheithauer for fruitful discussions. KS and HW thank the hospitality of IBS Center for Geometry and Physics where part of this work was done. KS would also like to thank KIAS (grant QP081001, QP081002), MPIM Bonn, Inter-University Centre in Dubrovnik and Isaac Newton Institute for Mathematical Sciences in Cambridge (during the program Black holes: bridges between number theory and holographic quantum information, by EPSRC grant no EP/R014604/1) for support and hospitality where work on this paper was undertaken. Preliminaries ============= In this section, we will first review Jacobi forms and denominator identities of affine Lie algebras. We then define reflective modular forms on orthogonal groups ${\mathop{\null\mathrm {O}}\nolimits}(n,2)$ and review Borcherds’ theory of automorphic products, as well as the denominator identities of Borcherds–Kac–Moody algebras. This background is necessary to state and prove the main theorems. Jacobi forms of lattice index ----------------------------- Let ${\mathbb Z}$ and ${\mathbb N}$ denote the sets of integers and non-negative integers, respectively. Let L be an even integral positive-definite lattice with bilinear form ( − , − ) and dual lattice L’={ vLQ: (x,y)Z, }. For any nonzero integer a, the lattice with abelian group L and bilinear form a( − , − ) is denoted by L(a). Let ${\mathbb H}$ be the complex upper half plane. The Dedekind eta function $$\eta(\tau)=q^{1/24}\prod\_{j=1}^\infty(1-q^j), \quad q=e^{2\pi i\tau}, \; \tau\in {\mathbb H}$$ is a modular form of weight $\frac{1}{2}$ on ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$ with a multiplier system of order 24. We denote this multiplier system by υη. Let D be a positive integer, $k\in\frac{1}{2}{\mathbb Z}$ and $t\in \frac{1}{2}{\mathbb N}$. A holomorphic function $\varphi : {\mathbb H}\times (L \otimes {\mathbb C}) \rightarrow {\mathbb C}$ is called a weakly holomorphic Jacobi form of weight k, index t and character (or multiplier system) υηD for L if it satisfies $$\begin{aligned} \varphi \left( \frac{a\tau +b}{c\tau + d},\frac{\mathfrak{z}}{c\tau + d} \right)& = \upsilon\_\eta^D(A) (c\tau + d)^k \exp{\left(i \pi t \frac{c(\mathfrak{z},\mathfrak{z})}{c \tau + d}\right)} \varphi ( \tau, \mathfrak{z} ),\\ \varphi (\tau, \mathfrak{z}+ x \tau + y)&= (-1)^{t\big((x,x)+(y,y)\big)}\exp{\big(-i \pi t \big( (x,x)\tau +2(x,\mathfrak{z})\big)\big)} \varphi (\tau, \mathfrak{z} ),\end{aligned}$$ for $A=\begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix} \in {\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$ and x, y ∈ L, and if its Fourier expansion takes the form $$\varphi ( \tau, \mathfrak{z} )=\sum\_{ \substack{-\infty \ll n\in {\mathbb Z}-\frac{D}{24} \\ \ell \in \frac{1}{2}L' } }f(n,\ell)q^n\zeta^\ell, \quad \zeta^\ell=e^{2\pi i (\ell, \mathfrak{z})},$$ where n ≫ − ∞ means that n is bounded from below. If f(n, ℓ) = 0 whenever n < 0, then φ is called a weak Jacobi form. If f(n, ℓ) = 0 whenever 2nt − (ℓ, ℓ) < 0, then φ is called a holomorphic Jacobi form. We denote the vector spaces of weakly holomorphic, weak and holomorphic Jacobi forms of weight k, index t and character υηD for L by $$J^{!}\_{k,L,t}(\upsilon\_\eta^D) \supset J^{\w}\_{k,L,t}(\upsilon\_\eta^D) \supset J\_{k,L,t}(\upsilon\_\eta^D),$$ respectively. We simply write $J^{!}\_{k,L,t}\supset J^{\w}\_{k,L,t} \supset J\_{k,L,t}$ if the character is trivial. The spaces Jk, t of classical Jacobi forms introduced by Eichler–Zagier are simply Jk, A1, t for the lattice $A\_1={\mathbb Z}$ with (x, x) = 2x2. If φ ∈ Jk, L, t!(υηD) is nonzero for some t > 0 then L(t) is necessarily an integral lattice. Jacobi forms of index t for L are also called Jacobi forms of (lattice) index L(t) and we sometimes write $$J^!\_{k,L(t)}=J^!\_{k,L,t}, \quad J^{\w}\_{k,L(t)}=J^{\w}\_{k,L,t}, \quad J\_{k,L(t)}=J\_{k,L,t}.$$ We will use the following Hecke operators to construct Jacobi forms later. [Proposition 3.1 in ][prop:Hecke] Let φ ∈ Jk, L, t!(υηD). Assume that $k\in {\mathbb Z}$ and D is an even divisor of 24. If Q = 24/D is odd, we further assume that $t\in {\mathbb Z}$. Then for any positive integer m coprime to Q we have $$\varphi \lvert\_{k,t}T\_{-}^{(Q)}(m)(\tau,\mathfrak{z})=m^{-1}\sum\_{\substack{ad=m,\,a>0\\ 0\leq b <d}}a^k \upsilon\_{\eta}^D(\sigma\_a)\varphi\left(\frac{a\tau+bQ}{d},a\mathfrak{z}\right)\in J\_{k,L,mt}^!(\upsilon\_{\eta}^{D x}),$$ where $x,y\in {\mathbb Z}$ such that mx + Qy = 1, and $$\sigma\_a= \left(\begin{array}{cc} dx+Qdxy & -Qy \\ Qy & a \end{array} \right) \in {\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z}).$$ Moreover, the Fourier coefficients fm( − , − ) of φ\|k, tT−(Q)(m) are linear combinations of the Fourier coefficients f( − , − ) of φ, i.e. $$f\_m(n,\ell)=\sum\_{a \mid (n,\ell,m)}a^{k-1} \upsilon\_{\eta}^D(\sigma\_a)f\left( \frac{nm}{a^2},\frac{\ell}{a}\right),$$ where a ∣ (n, ℓ, m) means that a ∣ nQ, $a^{-1}\ell\in \frac{1}{2}L'$ and a ∣ m. Affine Kac–Moody algebras and theta blocks ------------------------------------------ We review untwisted affine Kac–Moody algebras following and identify their denominators as important examples of Jacobi forms. Let g be a simple Lie algebra of rank r with Cartan subalgebra h and root system Δg. We fix a set of simple roots {α1, ..., αr} ⊂ h\* and denote the set of positive roots by Δg+ and the highest root by θ. Note that dimg = r + ∣Δg∣. The invariant symmetric bilinear form ⟨ − , − ⟩ on h\* is normalized such that long roots have (square) norm two; in particular, ⟨θ, θ⟩ = 2. We identify h with h\* and define the coroot of a root α as α∨ = 2α/⟨α, α⟩. The fundamental weights wi ∈ h\* are defined by ⟨wi, αi∨⟩ = δi, j, 1 ≤ i, j ≤ r, where δi, j = 1 if i = j and 0 otherwise. The Weyl vector ρg is defined as $$\rho\_\mathfrak{g}=\frac{1}{2}\sum\_{\alpha\in \Delta\_\mathfrak{g}^+} \alpha = \sum\_{j=1}^r w\_j.$$ Let Qg be the rational lattice generated by the roots of g and let Qg∨ be the even integral lattice generated by the coroots, or equivalently by the long roots of g. The weight lattice Pg, generated by the fundamental weights, is the dual of the coroot lattice: $$P\_\mathfrak{g}=(Q\_\mathfrak{g}^\vee)'=\{ x\in Q\_\mathfrak{g}\otimes{\mathbb Q}: {{\langle{x, \alpha^\vee}\rangle}}\in {\mathbb Z}, \; \alpha\in \Delta\_\mathfrak{g} \}\supset Q\_\mathfrak{g}.$$ The coweight lattice Pg∨ is the dual of the root lattice: $$P\_\mathfrak{g}^\vee=Q\_\mathfrak{g}'=\{ x\in Q\_\mathfrak{g}\otimes{\mathbb Q}: {{\langle{x, \alpha}\rangle}}\in {\mathbb Z}, \; \alpha\in \Delta\_\mathfrak{g} \}\supset Q\_\mathfrak{g}^\vee.$$ The reflection associated with a root α is defined as $$\sigma\_{\alpha}(x)= x - {{\langle{x,\alpha^\vee}\rangle}}\alpha, \quad x\in Q\_\mathfrak{g}\otimes {\mathbb R}.$$ The reflections associated with simple roots generate the Weyl group Wg. The coroot of θ can be written as an ${\mathbb N}$-linear combination of the coroots of simple roots, θ = θ∨ = ∑j = 1raj∨αj∨. The integers aj∨ are called comarks. The number $$h\_\mathfrak{g}^\vee=1+\sum\_{j=1}^r a\_j^\vee = \frac{1}{r}\sum\_{\alpha\in \Delta\_\mathfrak{g}^+} {{\langle{\alpha, \alpha}\rangle}}$$ is the dual Coxeter number and it satisfies the identity $$\sum\_{\alpha\in \Delta\_\mathfrak{g}^+} {{\langle{\alpha, \mathfrak{z}}\rangle}}^2 = h\_\mathfrak{g}^\vee{{\langle{\mathfrak{z},\mathfrak{z}}\rangle}}, \quad \mathfrak{z}\in Q\_\mathfrak{g}\otimes{\mathbb C}.$$ The above identity implies that the rescaled lattice Pg∨(hg∨) is integral. The classification of irreducible root systems into types An for n ≥ 1, Bn for n ≥ 2, Cn for n ≥ 3, Dn for n ≥ 4, and the exceptional systems E6, E7, E8, F4 and G2 is well-known. We use the same symbol to stand for the corresponding root lattice with its normalized bilinear form. Some useful data is summarized in Table [tab:data] for convenience; for the coordinates of the fundamental weights and the values of the comarks, see. [ht] $$\begin{array}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline \Delta\_\mathfrak{g} & A\_n & B\_n & C\_n & D\_n & E\_6 & E\_7 & E\_8 & G\_2 & F\_4 \\ \hline \|\Delta\_\mathfrak{g}\| & n(n+1) & 2n^2 & 2n^2 & 2n(n-1) & 72 & 126 & 240 & 12 & 48 \\ \hline h^\vee\_{\mathfrak{g}} & n+1 & 2n-1 & n+1 & 2(n-1)& 12 & 18 & 30 & 4 & 9 \\ \hline Q^\vee\_\mathfrak{g} & A\_n & D\_n & nA\_1 & D\_n & E\_6 & E\_7 & E\_8 & A\_2 & D\_4 \\ \hline P^\vee\_\mathfrak{g} & A\_n' & {\mathbb Z}^n & D\_n'(2) & D\_n' & E\_6' & E\_7' & E\_8 & A\_2 & D\_4 \\ \hline \end{array}$$ [tab:data] The untwisted affine Kac–Moody algebra $\hat{\mathfrak{g}}$ is an extension of g defined by $$\hat{\mathfrak{g}} = {\mathbb C}[t,t^{-1}]\otimes \mathfrak{g} \oplus {\mathbb C}K \oplus {\mathbb C}d,$$ where K is a central element and d is a derivative. The algebra $\hat{\mathfrak{g}}$ is an infinite-dimensional Lie algebra with affine Cartan subalgebra $\hat{\mathfrak{h}}=\mathfrak{h}\oplus {\mathbb C}K \oplus {\mathbb C}d$. We write $\hat{\mathfrak{h}}^\=\mathfrak{h}^\\oplus {\mathbb C}\hat{w}\_0 \oplus {\mathbb C}\delta$ with $$\begin{aligned} &\hat{w}\_0(\mathfrak{h}\oplus {\mathbb C}d)=0, \qquad\hat{w}\_0(K)=1,\\ &\delta(\mathfrak{h}\oplus {\mathbb C}K)=0, \qquad\ \delta(d)=1.\end{aligned}$$ We embed h\* into $\hat{\mathfrak{h}}^\$ and define α0 = δ* − θ and α0∨ = K − θ∨. Then {α0, α1, ⋯, αr} is a set of simple roots of $\hat{\mathfrak{g}}$ and {α0∨, α1∨, ⋯, αr∨} is the corresponding set of coroots. Setting ŵi = wi + ai∨ŵ0, 1 ≤ i ≤ r, {ŵ0, ŵ1, ⋯, ŵr} is a set of fundamental weights of $\hat{\mathfrak{g}}$. The Weyl vector of $\hat{\mathfrak{g}}$ is defined by ρ̂g = ρg + hg∨ŵ0 − (dimg/24)δ, such that the norm of ρ̂g is zero; indeed, by the “strange formula” of Freudenthal–de Vries, hg∨ ⋅ dimg = 12⟨ρg, ρg⟩. The integrable highest weight representations of $\hat{\mathfrak{g}}$ are indexed by dominant integral weights, which are elements of $$\hat{P}\_+ = \sum\_{j=0}^r {\mathbb N}\hat{w}\_j + {\mathbb C}\delta.$$ The level of λ = ∑j = 0rxjŵj + cδ ∈ P̂+ is the integer λ(K) = λ0 + ∑j = 1rxjaj∨. The character of the irreducible highest weight module labeled by a weight λ ∈ P̂+ of level k is given by the Weyl–Kac character formula, $$\chi\_\lambda^{\hat{\mathfrak{g}}} = \frac{\sum\_{\sigma\in W\_{\hat{\mathfrak{g}}}} (-1)^{\ell(\sigma)} e^{\sigma(\lambda+\rho\_{\hat{\mathfrak{g}}})} }{e^{\rho\_{\hat{\mathfrak{g}}}} \prod\_{\alpha \in \Delta\_{\hat{\mathfrak{g}}}^+}(1-e^{-\alpha})^{\mathrm{mult}(\alpha)}},$$ where $W\_{\hat{\mathfrak{g}}}$ is the Weyl group of $\hat{\mathfrak{g}}$, a semi-direct product of Wg by a certain group of translations, ℓ(σ) is the length of σ, $\Delta\_{\hat{\mathfrak{g}}}^+$ is the set of positive roots of $\hat{\mathfrak{g}}$, and mult(α) is the multiplicity of α, i.e. the dimension of the root space $\hat{\mathfrak{g}}\_\alpha$. When λ = 0, we have $\chi\_\lambda^{\hat{\mathfrak{g}}}=1$, which gives the Macdonald–Weyl denominator identity: $$e^{\rho\_{\hat{\mathfrak{g}}}} \prod\_{\alpha \in \Delta\_{\hat{\mathfrak{g}}}^+}(1-e^{-\alpha})^{\mathrm{mult}(\alpha)} = \sum\_{\sigma\in W\_{\hat{\mathfrak{g}}}} (-1)^{\ell(\sigma)}e^{\sigma(\rho\_{\hat{\mathfrak{g}}})}.$$ The algebra $\hat{\mathfrak{g}}$ has real roots (with norm > 0) and imaginary roots (with norm = 0). The set of positive real roots is $$\Delta\_{\hat{\mathfrak{g}}}^{+,\mathrm{re}} = \{ \alpha + n\delta : 0 < n \in {\mathbb Z}, \alpha \in \Delta\_\mathfrak{g} \} \cup \Delta\_\mathfrak{g}^+.$$ The set of positive imaginary roots is $$\Delta\_{\hat{\mathfrak{g}}}^{+,\mathrm{im}} = \{ n\delta : 0< n \in {\mathbb Z}\}.$$ Every real root has multiplicity one, but every imaginary root has multiplicity r. Let $(\tau, \mathfrak{z})\in {\mathbb H}\times \mathfrak{h}$. By interpreting the formal variable eα as e2πiα(z − τd), the character $\chi\_\lambda^{\hat{\mathfrak{g}}}(\tau, \mathfrak{z})$ defines a holomorphic function on ${\mathbb H}\times {\mathbb C}^r$. The product side of the denominator identity can be written as a theta block (introduced by Gritsenko–Skoruppa–Zagier ) $$\vartheta\_{\mathfrak{g}}(\tau,\mathfrak{z})=\eta(\tau)^r \prod\_{\alpha\in \Delta\_\mathfrak{g}^+}\frac{\vartheta(\tau, {{\langle{\alpha,\mathfrak{z}}\rangle}})}{\eta(\tau)},$$ where the odd Jacobi theta function $$\vartheta(\tau,z)=\sum\_{n\in{\mathbb Z}}\left(\frac{-4}{n}\right)q^{\frac{n^2}{8}}\zeta^{\frac{n}{2}} =-q^{\frac{1}{8}}\zeta^{-\frac{1}{2}}\prod\_{n=1}^\infty(1-q^{n-1}\zeta)(1-q^n\zeta^{-1})(1-q^n), \quad z\in {\mathbb C}, \; \zeta= e^{2\pi iz}$$ is the denominator of $\chi\_\lambda^{\hat{\mathfrak{g}}}$ for the A1 Lie algebra g. Note that ϑg is a holomorphic Jacobi form of (singular) weight r/2 and lattice index Pg∨(hg∨) with multiplier system υηdimg. By, all characters $\chi\_\lambda^{\hat{\mathfrak{g}}}$ of fixed level k form a vector-valued weakly holomorphic Jacobi form of weight 0 and lattice index Qg∨(k) which is invariant under the action of the Weyl group Wg on z. We refer to for the precise transformation laws with respect to the generators of ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$. Let k be a positive integer. The irreducible $\hat{\mathfrak{g}}$-module $L\_{\hat{\mathfrak{g}}}(k,0)$ associated with kŵ0 has a canonical structure as a simple rational vertex operator algebra. This is called the affine VOA generated by $\hat{\mathfrak{g}}$ at level k. Such VOAs are physically realized as the well-known Wess–Zumino–Witten (WZW) models, which are nonlinear sigma models describing mapping fields from Riemann surfaces to Lie group manifolds. The central charge of $L\_{\hat{\mathfrak{g}}}(k,0)$ is $$\label{cc} c\_\mathfrak{g}=\frac{k \dim \mathfrak{g}}{k + h\_\mathfrak{g}^\vee}.$$ The irreducible modules of $L\_{\hat{\mathfrak{g}}}(k,0)$ are the irreducible $\hat{\mathfrak{g}}$-modules $L\_{\hat{\mathfrak{g}}}(k,\lambda)$ associated with level k dominant integral weights kŵ0 + λ ∈ P̂+ for any $\lambda\in \sum\_{j=1}^r {\mathbb N}w\_j$ that satisfies ⟨λ, θ∨⟩ = ∑j = 1raj∨λj ≤ k, where λ = ∑j = 1rλjwj. The conformal weight of $L\_{\hat{\mathfrak{g}}}(k,\lambda)$ is $$\label{conformalw} h\_\lambda=\frac{{{\langle{\lambda, \lambda + 2\rho\_\mathfrak{g}}\rangle}}}{2(k+h^\vee\_\mathfrak{g})}.$$ The full character of $L\_{\hat{\mathfrak{g}}}(k,\lambda)$ is actually $\chi\_{k\hat{w}\_0+\lambda}^{\hat{\mathfrak{g}}}(\tau,\mathfrak{z})$ as defined above. When the root system of g is Rr as in Table [tab:data] and λ = ∑j = 1rλjwj, we often write $\chi\_{k\hat{w}\_0+\lambda}^{\hat{\mathfrak{g}}}$ as χλ1⋯λr, hλRr, k. [notation] Let g = ⊕ j = 1sgj be a semi-simple Lie algebra. We will often have to associate to each simple ideal gj a positive integer kj called the level. In this case we indicate the levels by writing g = ⊕ j = 1sgj, kj. For simplicity we write hj∨ = hg\_j∨, Qj = Qgj, Qj∨ = Qgj∨, Pj = Pgj, Pj∨ = Pgj∨. We further fix two lattices Qg = ⨁j = 1sQj∨(kj) and Pg = ⨁j = 1sPj∨(kj). Let Vg denote the affine vertex operator algebra $$\bigotimes\_{j=1}^s L\_{\hat{\mathfrak{g}}\_j}(k\_j,0).$$ Clearly, the denominator of $\hat{\mathfrak{g}}$ is given by the theta block ϑg : = ϑg1 ⊗ ϑg2 ⊗ ⋯ ⊗ ϑgs. Some remarks on the denominator identities are in order: 1. Gritsenko, Skoruppa and Zagier found a direct proof of the denominator identity based on their theory of theta blocks. 2. The function ϑg(τ, z)/η(τ)dimg equals the modular Jacobian of the generators of the ring of weak Jacobi forms for Qg∨ invariant under Wg (see ). 3. If a theta block defines a holomorphic Jacobi form of singular weight, then it has to be the denominator of an affine Lie algebra (). Therefore, there is an one-to-one correspondence between affine Lie algebras and singular-weight theta blocks. Reflective modular forms on orthogonal groups --------------------------------------------- Let M be an even integral lattice of signature (l, 2) with l ≥ 3. We choose one of the two connected components of $$\{\mathcal{Z} \in M\otimes {\mathbb C}: (\mathcal{Z}, \mathcal{Z})=0, (\mathcal{Z},\bar{\mathcal{Z}}) < 0\}$$ and label it ${\mathcal A}(M)$. The symmetric domain of type IV attached to M is $${\mathcal D}(M):={\mathcal A}(M) / \mathbb{C}^{\times}=\{[\mathcal{Z}] \in {\mathbb P}(M\otimes {\mathbb C}): \mathcal{Z} \in {\mathcal A}(M) \}.$$ Let ${\mathop{\null\mathrm {O}}\nolimits}^+(M)$ denote the subgroup of ${\mathop{\null\mathrm {O}}\nolimits}(M\otimes{\mathbb R})$ which preserves M and ${\mathcal A}(M)$. Let Γ be a finite-index subgroup of ${\mathop{\null\mathrm {O}}\nolimits}^+(M)$. The most important example of Γ will be the discriminant kernel +(M)={ g +(M): g(v) - v M, }, where Mʹ is the dual lattice of M. Let $k\in {\mathbb Z}$ and $\chi: \Gamma\to {\mathbb C}^\times$ be a character. A holomorphic function $F: {\mathcal A}(M)\to {\mathbb C}$ is called a modular form of weight k and character χ on Γ if it satisfies $$\begin{aligned} F(t\mathcal{Z})&=t^{-k}F(\mathcal{Z}), \quad\ \text{for all $t \in {\mathbb C}^\times$},\\ F(g\mathcal{Z})&=\chi(g)F(\mathcal{Z}), \quad \text{ for all $g\in \Gamma$}.\end{aligned}$$ Modular forms can be represented by their Fourier expansions. Let c be a primitive isotropic vector of M and choose cʹ ∈ Mʹ satisfying (c, cʹ) = 1. Then Mc, cʹ = M ∩ c⊥ ∩ (cʹ)⊥ is an even lattice of signature (l − 1, 1). Around the cusp c, one can identify the symmetric domain ${\mathcal D}(M)$ with a tube domain ${\mathbb H}\_{c,c'}$, a connected component of $$\{ Z = X + iY :\; X, Y \in M\_{c,c'}\otimes{\mathbb R}, \; (Y,Y)<0 \}.$$ This induces an action of ${\mathop{\null\mathrm {O}}\nolimits}^+(M)$ on ${\mathbb H}\_{c,c'}$ and an automorphy factor, which allows us to define modular forms of half-integral weight. A modular form of trivial character on $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(M)$ can be expanded on ${\mathbb H}\_{c,c'}$ as F(Z) = ∑λ ∈ Mc, cʹʹ, (λ, λ) ≤ 0c(λ)e2πi(λ, Z). Modular forms F on general Γ have similar expansions with Mc, cʹ replaced by a finite-index sublattice. If F is nonzero, then either k = 0 in which case F is constant, or k ≥ l/2 − 1. The smallest possible positive weight l/2 − 1 is called the singular weight. When F has singular weight, its Fourier coefficients c(λ) are zero whenever (λ, λ) ≠ 0. Let $\lambda \in M\otimes{\mathbb Q}$ be a vector of positive norm. The rational quadratic divisor associated with λ is $$\lambda^\perp=\{ [\mathcal{Z}] \in {\mathcal D}(M) : (\mathcal{Z}, \lambda)=0 \}.$$ We define the associated reflection $\sigma\_\lambda \in {\mathop{\null\mathrm {O}}\nolimits}^+(M\otimes{\mathbb Q})$ as $$\sigma\_\lambda(x)=x-\frac{2(x,\lambda)}{(\lambda,\lambda)}\lambda, \quad x\in M\otimes{\mathbb Q}.$$ The divisor λ⊥ is called reflective if σλ fixes M, i.e. $\sigma\_\lambda \in {\mathop{\null\mathrm {O}}\nolimits}^+(M)$. If λ is a primitive vector in Mʹ, then λ⊥ is reflective if and only if there exists a positive integer d such that (λ, λ) = 2/d and dλ ∈ M. More precisely, the order of λ in Mʹ/M is either d, or d/2 in which case d/2 is necessarily even. A non-constant (holomorphic) modular form on Γ is called reflective if its zero divisor is a linear combination of reflective rational quadratic divisors. Borcherds products ------------------ Let M be an even lattice of signature (b+, b−) with discriminant form DM : = (Mʹ/M, q), where q(x) = (x, x)/2 is the induced quadratic form. Let Mp2(Z) be the metaplectic group, which consists of pairs A = (A, ϕA), where $A = \begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix} \in \mathrm{SL}\_2(\mathbb{Z})$ and ϕA is a holomorphic square root of τ ↦ cτ + d on H, with the standard generators $$T = \big(\begin{psmallmatrix} 1 & 1 \\ 0 & 1 \end{psmallmatrix}, 1\big) \quad \text{and} \quad S = \big(\begin{psmallmatrix} 0 & -1 \\ 1 & 0 \end{psmallmatrix}, \sqrt{\tau}\big).$$ The Weil representation ρM is the unitary representation of Mp2(Z) on the group ring C[DM] = span(ex : x ∈ DM) defined by $$\rho\_M(T) e\_x = \mathbf{e}(-\mathfrak{q}(x)) e\_x \quad \text{and} \quad \rho\_M(S) e\_x = \frac{\mathbf{e}( \mathrm{sign}(M) / 8)}{\sqrt{\|D\_M\|}} \sum\_{y \in D\_M} \mathbf{e}(( x,y)) e\_y,$$ where e(t) = e2πit for $t\in {\mathbb C}$, and $\mathrm{sign}(M)=b^+ - b^- \mod 8$. The dual representation of ρM is the complex conjugate of ρM; moreover, ρ̄M = ρM( − 1). Let $k \in \frac{1}{2}\mathbb{Z}$. A holomorphic function f : H → C[DM] is called a weakly holomorphic modular form of weight k if f satisfies A()-2k f(A ) = M(A) f(), and if f is represented by a Fourier series of the form $$f(\tau) = \sum\_{x \in D\_M} \sum\_{\substack{n \in \mathbb{Z} - \mathfrak{q}(x)\\ n \gg -\infty}} c\_x(n) q^n e\_x.$$ The finite sum cx(n)qnex with n < 0 is called the principal part of f. If f is holomorphic at infinity, i.e. its principal part is zero, then it is a holomorphic modular form. Note that $k+\mathrm{sign}(M)/2 \in {\mathbb Z}$ if non-zero f exist, and that if sign(M) is even then ρM factors through a representation of ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$. We denote the spaces of weakly holomorphic and holomorphic modular forms of weight k for ρM respectively by Mk!(ρM) ⊃ Mk(ρM). There is a natural homomorphism ${\mathop{\null\mathrm {O}}\nolimits}(M) \to {\mathop{\null\mathrm {O}}\nolimits}(D\_M)$ with kernel $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}(M)$. We define an action of ${\mathop{\null\mathrm {O}}\nolimits}(D\_M)$ on modular forms f = ∑cx(n)qnex by $$\sigma(f)=\sum c\_x(n)q^n e\_{\sigma(x)}, \quad \sigma \in {\mathop{\null\mathrm {O}}\nolimits}(D\_M).$$ We now assume that M has signature (l, 2) with l ≥ 3. Let f be a weakly holomorphic modular form of weight 1 − l/2 for ρM with integral principal part. The Borcherds singular theta lift produces a meromorphic modular form $\Borch(f)$ of weight c0(0)/2 and some character (or multiplier system) on $${\mathop{\null\mathrm {O}}\nolimits}^+(M,f)=\{ \sigma \in {\mathop{\null\mathrm {O}}\nolimits}^+(M): \sigma(f)=f\} \supset \widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(M)$$ which has an infinite product expansion at any 0-dimensional cusp involving the Fourier coefficients of f (see below). Moreover, the divisor of $\Borch(f)$ is a linear combination of rational quadratic divisors λ⊥, each with multiplicity ∑d = 1∞cdλ( − d2λ2/2), where λ ∈ Mʹ are primitive vectors of positive norm. There is an identification between modular forms for the Weil representation and Jacobi forms. Let L be an even positive-definite lattice of rank ${\mathop{\mathrm {rk}}\nolimits}(L)$. Then one has the isomorphism $$\begin{split} M\_{k-\frac{1}{2}{\mathop{\mathrm {rk}}\nolimits}(L)}^!(\rho\_L) &\stackrel{\sim}{\longrightarrow} J\_{k,L}^! \\ f(\tau)=\sum\_{\gamma\in L'/L} f\_\gamma(\tau)e\_\gamma &\longmapsto \sum\_{\gamma\in L'/L} f\_\gamma(\tau) \Theta\_{L,\gamma}(\tau,\mathfrak{z}), \end{split}$$ where ΘL, γ is the Jacobi theta function of L + γ, defined as $$\Theta\_{L,\gamma}(\tau,\mathfrak{z}) = \sum\_{\ell \in L+\gamma} e^{\pi i(\ell,\ell)\tau + 2\pi i(\ell, \mathfrak{z})}, \quad (\tau,\mathfrak{z})\in{\mathbb H}\times (L\otimes{\mathbb C}).$$ This map can be extended to modular forms with characters and it induces an isomorphism between holomorphic modular forms. It follows that a holomorphic Jacobi form of weight $\frac{1}{2}{\mathop{\mathrm {rk}}\nolimits}(L)$ and index L is a ${\mathbb C}$-linear combination of ΘL, γ. We call $\frac{1}{2}{\mathop{\mathrm {rk}}\nolimits}(L)$ the singular weight for Jacobi forms. It will be convenient to represent Borcherds products in terms of Jacobi forms. Let ϕ ∈ J0, L, 1! be a weakly holomorphic Jacobi form of trivial character with Fourier expansion $$\phi(\tau,\mathfrak{z})=\sum\_{n\in {\mathbb Z}}\sum\_{\ell\in L'}f(n,\ell)q^n\zeta^\ell.$$ The terms f(n, ℓ)qnζℓ with 2n − (ℓ, ℓ) < 0 are called singular Fourier coefficients and they correspond to the principal part of the preimage of ϕ under the above isomorphism. [rem:singular] The Fourier expansion of ϕ satisfies f(n, ℓ) = f(n, − ℓ) and f(n1, ℓ1) = f(n2, ℓ2) if 2n1 − (ℓ1, ℓ1) = 2n2 − (ℓ2, ℓ2) and if ℓ1 − ℓ2 ∈ L. Therefore, any singular Fourier coefficient of ϕ already appears as a summand f(n, ℓ)qnζℓ where n ≤ δ̂L, ℓ ∈ Lʹ and 2n < (ℓ, ℓ), where δ̂L is the largest integer less than δL/2 and $$\delta\_L := \max\big\{ \min\{(y,y): y\in L + x \} : x \in L' \big\}.$$ Let U be a hyperbolic plane, i.e. an even unimodular lattice of signature (1, 1). We write $$U={\mathbb Z}e + {\mathbb Z}f, \quad \text{where} \quad e^2=f^2=0 \quad \text{and} \quad (e,f)=-1.$$ Let $U\_1={\mathbb Z}e\_1 +{\mathbb Z}f\_1$ be a second hyperbolic plane and define M = U1 ⊕ U ⊕ L. We fix coordinates on the tube domain about U1 by writing $${\mathbb H}(L)=\{ Z= \tau f + \mathfrak{z} +\omega e : \tau, \omega \in {\mathbb H},\; \mathfrak{z}\in L\otimes{\mathbb C}, \; 2{\mathop{\mathrm {Im}}\nolimits}(\tau){\mathop{\mathrm {Im}}\nolimits}(\omega) -(\mathfrak{z},\mathfrak{z})>0 \},$$ such that − (α, Z) = nτ − (ℓ, z) + mω for α = ne + ℓ + mf ∈ U ⊕ Lʹ. These coordinates are chosen such that the infinite expansions below match the denominators of Borcherds–Kac–Moody algebras. [notation2] We write v = x1e1 + xe + ℓ + yf + y1f1 ∈ U1 ⊕ U ⊕ Lʹ in the coordinate (x1, x, ℓ, y, y1) and α = xe + ℓ + yf ∈ U ⊕ Lʹ in the coordinate (x, ℓ, y). In particular, v2 = (ℓ, ℓ) − 2(xy + x1y1). [][th:product] Assume the above ϕ has integral singular Fourier coefficients. Then the Borcherds theta lift of ϕ is a meromorphic modular form of weight f(0, 0)/2 on $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(M)$ and it can be expanded on an open subset of the tube domain ${\mathbb H}(L)$ as $$\Borch(\phi)(Z)=q^A \zeta^{\vec{B}} \xi^C\prod\_{\substack{n,m\in{\mathbb Z},\; \ell \in L'\\ (n,\ell,m)>0}}\Big(1-q^n \zeta^{-\ell} \xi^m\Big)^{f(nm,\ell)},$$ where q = exp(2πiτ), ζℓ = exp(2πi(ℓ, z)), ξ = exp(2πiω), where the positivity condition (n, ℓ, m) > 0 means that either m > 0, or m = 0 and n > 0, or m = n = 0 and ℓ > 0, and where the Weyl vector ρ = ( − A, B⃗, − C) of $\Borch(\phi)$ is defined by $$A=\frac{1}{24}\sum\_{\ell\in L'}f(0,\ell),\quad \vec{B}=\frac{1}{2}\sum\_{\ell>0} f(0,\ell)\ell, \quad C=\frac{1}{2{\mathop{\mathrm {rk}}\nolimits}(L)}\sum\_{\ell\in L'}f(0,\ell)(\ell,\ell).$$ The Fourier–Jacobi expansion of $\Borch(\phi)$ on ${\mathbb H}(L)$ is $$\Borch(\phi)(Z)= \biggl(\Theta\_{f(0,\ast)} (\tau,\mathfrak{z})\cdot \xi^C \biggr) \exp \left(-\sum\_{m=1}^\infty \Big(\phi \|\_0 T\_{-}^{(1)}(m)\Big) (\tau, \mathfrak{z}) \cdot \xi^m \right),$$ where the leading Fourier–Jacobi coefficient is given by a generalized theta block $$\Theta\_{f(0,\ast)}(\tau,\mathfrak{z}) =\eta(\tau)^{f(0,0)}\prod\_{\ell >0} \biggl(\frac{\vartheta(\tau,(\ell,\mathfrak{z}))}{\eta(\tau)} \biggr)^{f(0,\ell)}.$$ We also need the following additive lifts with characters to construct orthogonal modular forms. [Theorem 3.2 in ][th:additive] Let D be an even divisor of 24, $k\in{\mathbb N}$, $t\in \frac{1}{2}{\mathbb N}$. If Q = 24/D is odd, we further assume that $t\in{\mathbb N}$. Let φ ∈ Jk, L, t(υηD). Let Gk(τ) be the normalized Eisenstein series of weight k on ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$ whose Fourier coefficient at q is 1. Then the function $$\Grit(\varphi)(Z)=f(0,0)G\_k(\tau)+\sum\_{0<m\in 1+Q{\mathbb Z}}\Big(\varphi \lvert\_{k,t}T\_{-}^{(Q)}(m)\Big)(\tau,\mathfrak{z})\cdot \xi^{m/Q}$$ is a (holomorphic) modular form of weight k and some character on $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(U\_1\oplus U\oplus L(Qt))$. The form $\Grit(\varphi)$ is always invariant under the involution (ω, z, τ) ↦ (τ, z, ω). For a given lattice M, we will often need to determine all holomorphic, reflective Borcherds products and in some cases prove that no such products exist. To do this, we apply Borcherds’s obstruction criterion which states that a formal sum ∑n < 0∑x ∈ DMcx(n)qnex occurs as the principal part of a weakly-holomorphic modular form of weight κ for ρM if and only if the identity ∑n < 0∑x ∈ DMcx(n)ax( − n) = 0 holds for every cusp form ∑n > 0∑x ∈ DMax(n)qnex of weight 2 − κ for the dual representation ρM( − 1). We construct a basis of cusp forms for ρM( − 1) following and realize the problem of computing holomorphic reflective products as the problem of enumerating integral lattice points in a polyhedral cone defined by finitely many inequalities (non-negative order along reflective divisors) and finitely many linear equations. The solution can be conveniently expressed by the notion of a Hilbert basis, i.e. a minimal system of reflective products F1, ..., Fr such that every holomorphic reflective product can be written in the form F = F1n1 ⋅ ... ⋅ Frnr with non-negative integers n1, ..., nr. Given the polyhedral cone, we used the software Normaliz to find a Hilbert basis. Borcherds–Kac–Moody superalgebras --------------------------------- Borcherds–Kac–Moody superalgebras are infinite dimensional Lie superalgebras introduced by Borcherds in 1988 which generalize affine Kac–Moody algebras. They can be defined in terms of Chevalley–Serre generators and relations which are encoded in a generalized Cartan matrix. The restrictions on the generalized Cartan matrix are weaker; in particular, simple roots are allowed to have non-positive norm (i.e. they can be imaginary roots). BKM superalgebras also have a character formula for highest-weight modules and a denominator identity. We will review the denominator identity following because it contains valuable information about the roots, root multiplicities and the Weyl group. Let G = G0 ⊕ G1 be a BKM superalgebra with even and odd components G0 and G1, respectively. If G1 is trivial, G is a BKM algebra. Let H be the generalized Cartan subalgebra of G. Let Δ+ be the set of positive roots of G. We have the root space decomposition G = ⨁α ∈ Δ+Gα ⊕ H ⊕ ⨁α ∈ Δ+G− α. For α ∈ Δ+, we define mult0(α) = dim(Gα ∩ G0), mult1(α) = dim(Gα ∩ G1), mult(α) = dim(Gα), and define the super-multiplicity s-mult(α) as mult0(α) − mult1(α). The sets of positive even roots and odd roots are respectively Δ0+ = {α ∈ Δ+ : mult0(α) > 0} and Δ1+ = {α ∈ Δ+ : mult1(α) > 0}. Let I be a countable set indexing the simple roots αi and S be the subset of I indexing odd roots. By expanding a root as α = ∑i ∈ Ikiαi, we define the height and even height of α respectively as ht(α) = ∑i ∈ Iki and ht0(α) = ∑i ∈ I ∖ Ski. We further define two formal sums T = eρ∑μ( − 1)ht(μ)e− μ, Tʹ = eρ∑μ( − 1)ht0(μ)e− μ, where ρ is the Weyl vector of G and the sums are taken over all sets μ of distinct pairwise orthogonal imaginary simple roots. Let W be the Weyl group generated by reflections associated with real simple roots. In this paper we assume that G has no odd real roots. Under this assumption, one has the denominator identity $$e^\rho \cdot \frac{\prod\_{\alpha \in \Delta\_0^+}(1-e^{-\alpha})^{\mathrm{mult}\_0(\alpha)}}{\prod\_{\alpha \in \Delta\_1^+}(1+e^{-\alpha})^{\mathrm{mult}\_1(\alpha)}} = \sum\_{\sigma\in W} \det(\sigma) \sigma(T)$$ and the super-denominator identity $$e^\rho \cdot \frac{\prod\_{\alpha \in \Delta\_0^+}(1-e^{-\alpha})^{\mathrm{mult}\_0(\alpha)}}{\prod\_{\alpha \in \Delta\_1^+}(1-e^{-\alpha})^{\mathrm{mult}\_1(\alpha)}} = \sum\_{\sigma\in W} \det(\sigma) \sigma(T').$$ An affine Kac–Moody algebra G is a BKM algebra with no imaginary simple roots. Therefore, T = eρ, and we recover the denominator identity for affine Kac–Moody algebras. Following Borcherds, we study BKM superalgebras that have a hyperbolic root lattice (i.e. its signature is of type (l, 1)) and whose denominator or super-denominator is the Fourier expansion of a holomorphic modular form F on ${\mathop{\null\mathrm {O}}\nolimits}(l+1,2)$ at some 0-dimensional cusp. The product side of the denominator identity suggests that F can usually be constructed as a Borcherds product, in which case the roots and their multiplicities are encoded in a weakly holomorphic modular form for the Weil representation of ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$. The real roots of G correspond to the zeros of F through the cusp, so F is locally reflective. Usually, F is also globally reflective. This is particularly interesting when F is of singular weight. In this case, imaginary simple roots are easy to describe, and in many cases they are just the multiples of the Weyl vector. Moreover, one can write the sum side of the denominator identity explicitly as in, and it is expected that such a BKM superalgebra has a natural construction as the BRST cohomology of a suitable vertex algebra. In this paper, we will classify this type of BKM superalgebra under some mild conditions and work out the corresponding vertex algebras. Our classification is closely related to the fake monster algebra and its twists, so we will review that theory as well. In 1990 Borcherds constructed the fake monster algebra G as the BRST cohomology related to the Leech lattice VOA. The root lattice of the BKM algebra G is U ⊕ Λ, where Λ is the Leech lattice. The Weyl vector is ρ = ( − 1, 0, 0) (see Notation [notation2]) and the real roots are α ∈ U ⊕ Λ with α2 = 2. The real simple roots are characterized by the equality (ρ, α) = 1. The imaginary simple roots are nρ for negative integers n, each with multiplicity 24. The denominator of G defines a reflective Borcherds product Φ12 of singular weight on 2U ⊕ Λ whose input is the full character of the Leech lattice VOA (see ). Let $\mathrm{Co}\_0={\mathop{\null\mathrm {O}}\nolimits}(\Lambda)$ be the Conway group. Borcherds proved that the denominator identity of G is a cohomological identity and that Co0 acts on G naturally. In this way, he obtained a twisted denominator identity for each conjugacy class [g] of Co0 and proved that it is the (untwisted) super-denominator identity of a BKM superalgebra, denoted Gg. The root lattice of Gg is Lg = U ⊕ Λg, where Λg = {v ∈ Λ : g(v) = v} is the fixed-point sublattice. The real simple roots are the roots α of Lg satisfying (ρ, α) = α2/2 and the imaginary simple roots are mρ with super-multiplicity ∑k∣(m, ng)bk for all negative integers m, where ng is the order of g and where bk describe the cycle shape ∏bk ≠ 0kbk of g. The last two named authors proved that the twisted denominator function associated with g defines a holomorphic Borcherds product (denoted Φg) of singular weight on U(Ng) ⊕ U ⊕ Λg, where Ng is the level of g, and that Φg is reflective on U(Ng) ⊕ U ⊕ Λg if Ng = ng, confirming a conjecture of Borcherds. The hyperbolization of affine Kac–Moody algebras ================================================ It is well-known how to realize affine Kac–Moody algebras as extensions of semi-simple Lie algebras. In this section, we introduce extensions of affine Kac–Moody algebras to BKM superalgebras that we call their hyperbolizations. Let L be an even positive-definite lattice of rank ${\mathop{\mathrm {rk}}\nolimits}(L)$, let U and U1 be two hyperbolic planes, and define M = U1 ⊕ U ⊕ L. Let G be a BKM superalgebra with root lattice U ⊕ Lʹ and generalized Cartan subalgebra H with no odd real roots. We assume that the super-denominator of G is given by the Fourier expansion of a Borcherds product F for $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(M)$ at the 0-dimensional cusp determined by U1. Every divisor λ⊥ of F lies in the $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(M)$-orbit of the rational quadratic divisor associated with some primitive vector α ∈ U ⊕ Lʹ. If α⊥ appears in the divisor of F, then α is a real root. Since we have assumed that G has no odd real roots, the divisor α⊥ is always a zero and therefore F is holomorphic. Moreover, the associated reflection σα lies in the Weyl group of G. Therefore, σα fixes the root lattice U ⊕ Lʹ, so it also fixes Mʹ = U1 ⊕ U ⊕ Lʹ, which implies that $\sigma\_\alpha \in {\mathop{\null\mathrm {O}}\nolimits}^+(M')={\mathop{\null\mathrm {O}}\nolimits}^+(M)$ and finally that F is a reflective modular form. We denote by ϕ the Jacobi form input of F and write its expansion at ∞ as $$\phi(\tau,\mathfrak{z}) = \sum\_{n\in{\mathbb Z},\; \ell\in L'} f(n,\ell)q^n \zeta^\ell \in J\_{0,L}^!.$$ By Theorem [th:product], the product side of the denominator identity is (see Notation [notation2]) eρ∏α > 0(1 − e− α)f(nm, ℓ), α = (n, ℓ, m) ∈ U ⊕ Lʹ, ρ = ( − A, B⃗, − C). In particular, if f(nm, ℓ) ≠ 0 then α is a root of G and mult0(α) − mult1(α) = f(nm, ℓ). The weight of F is given by f(0, 0)/2. The root α is positive if and only if either m > 0, or m = 0 and n > 0, or m = n = 0 and ℓ > 0. Let R+ be the set of positive roots of G of type (n, ℓ, 0). For any positive integer n, (n, 0, 0) is an imaginary root in R+. All other roots in R+ are (even) real and have multiplicity f(0, ℓ). Since the multiplicity of a real root is always 1 (see e.g. ), we have f(0, ℓ) = 1. Assume further that none of (n, 0, 0) is odd. Then the even roots (n, 0, 0) always have multiplicity f(0, 0). Now consider the sum of root spaces G0 : = ⨁α ∈ R+Gα ⊕ H ⊕ ⨁α ∈ R+G− α. If F is of singular weight, i.e. $f(0,0)={\mathop{\mathrm {rk}}\nolimits}(L)$, then G0 defines an affine Kac–Moody algebra $\hat{\mathfrak{g}}$ for which g is a semi-simple Lie algebra whose positive roots are the roots of type (0, ℓ, 0) in R+ (here, we can view K = δ = (1, 0, 0) and d = ŵ0 = (0, 0, − 1)). The leading Fourier–Jacobi coefficient of F at the 1-dimensional cusp determined by U1 ⊕ U is a holomorphic Jacobi form of singular weight given by the theta block $$\eta(\tau)^{{\mathop{\mathrm {rk}}\nolimits}(L)} \prod\_{(0,\ell,0)\in \mathcal{R}^+} \frac{\vartheta(\tau, (\ell, \mathfrak{z}))}{\eta(\tau)}, \quad \mathfrak{z}\in L\otimes{\mathbb C},$$ which equals the denominator ϑg of $\hat{\mathfrak{g}}$. In particular, G is an extension of $\hat{\mathfrak{g}}$. For m > 0 we define Gm to be the sum of root spaces associated with roots of type ± ( \* , \* , m). Then G = ⨁m = 0∞Gm is a graded module over $\hat{\mathfrak{g}}=\mathcal{G}^0$. From this point of view, it is natural to regard G as a hyperbolization of $\hat{\mathfrak{g}}$. Motivated by the discussion above, we introduce the following definition. [def:hyperbolization] Let $\hat{\mathfrak{g}}$ be an affine Kac–Moody algebra. A BKM superalgebra G without odd real roots is called a hyperbolization of $\hat{\mathfrak{g}}$ if there exists an even positive-definite lattice L such that the root lattice of G is U ⊕ Lʹ and the super-denominator of G defines a holomorphic Borcherds product F of (singular) weight ${\mathop{\mathrm {rk}}\nolimits}(L)/2$ on 2U ⊕ L whose leading Fourier–Jacobi coefficient is the denominator ϑg of $\hat{\mathfrak{g}}$. We also call F a hyperbolization of ϑg. The notion of hyperbolization was first introduced by Gritsenko and the second named author in, and. In this paper we will classify affine Kac–Moody algebras that admit hyperbolizations and construct a hyperbolization for every such affine Kac–Moody algebra. Moreover, we express every Jacobi form input ϕ as a ${\mathbb Z}$-linear combination of full characters of an associated vertex operator (super)algebra. Root systems associated with singular-weight reflective modular forms ===================================================================== In this section, we classify reflective automorphic products of singular weight on lattices of type 2U ⊕ L whose Jacobi form inputs have non-negative q0-terms. Although Jacobi form inputs that have negative coefficients in their q0-terms can lift to holomorphic Borcherds products, the corresponding BKM superalgebras in these cases will have odd real roots. By an argument similar to and, we will show that any such Borcherds product belongs to a root system satisfying certain strong restrictions. These restrictions yield a finite list of root systems and a finite list of candidates for affine Kac–Moody algebras that admit hyperbolizations. [th:root-system] Let L be an even positive-definite lattice of rank ${\mathop{\mathrm {rk}}\nolimits}(L)$ and let U be a hyperbolic plane. Suppose there is a reflective Borcherds product F of singular weight on $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(2U\oplus L)$ whose Jacobi form input has non-negative q0-term. Then either (a) or (b) below holds: * The lattice L is the Leech lattice, F is the denominator Φ12 of the fake monster algebra, and the Jacobi form input is the full character of the Leech lattice VOA. * There exists a semi-simple Lie algebra g = ⊕ j = 1sgj, kj of rank ${\mathop{\mathrm {rk}}\nolimits}(L)$ satisfying restrictions (1) or (1ʹ) and (2) and (3), where gj are simple Lie algebras and kj are positive integers for 1 ≤ j ≤ s. We refer to Notation [notation] for the symbols below. Let λ ∈ U with λ2 = 2. + If F vanishes on λ⊥, then $$\label{antiC} C:=\frac{\dim \mathfrak{g}}{24} -1 = \frac{h\_j^\vee}{k\_j}, \quad \text{for $1\leq j\leq s$}.$$ + If F does not vanish on λ⊥, then $$\label{symC} C:=\frac{\dim \mathfrak{g}}{24} = \frac{h\_j^\vee}{k\_j} \quad \text{and} \quad k\_j>1, \quad \text{for $1\leq j\leq s$}.$$ + The rescaled lattice L(C) is integral and L is bounded by Qg < L < Pg. + The leading Fourier–Jacobi coefficient of F at the 1-dimensional cusp determined by 2U coincides with the denominator of the affine Kac–Moody algebra $\hat{\mathfrak{g}}$. Let cg denote the central charge of the affine VOA generated by $\hat{\mathfrak{g}}=\oplus\_{j=1}^s\hat{\mathfrak{g}}\_{j,k\_j}$. Then cg is always 24 in case (1) and in case (1ʹ) we have $$c\_\mathfrak{g} = \frac{24C}{C+1}.$$ We call case (1) (resp. (1ʹ)) the anti-symmetric (resp. symmetric) case, since F is anti-invariant (resp. invariant) under the involution (ω, z, τ) ↦ (τ, z, ω). This is a more precise version of in the specific case of singular weight. We provide a detailed proof here as it combines the proof of with some recent results proved in, and because it explains the construction of the Lie algebra g. Let ϕ ∈ J0, L! be the Jacobi form input of F, and write its Fourier expansion as $$\phi(\tau,\mathfrak{z}) = \sum\_{n\in {\mathbb Z},\; \ell\in L'} f(n,\ell) q^n \zeta^\ell.$$ By assumption, f(0, ℓ) are non-negative integers. Each vector v = (n, ℓ, 1) ∈ U ⊕ Lʹ is primitive in M : = 2U ⊕ L and satisfies v2 = (ℓ, ℓ) − 2n. The multiplicity of v⊥ in the divisor of F is ∑d = 1∞f(d2n, dℓ). By, the Fourier expansion of F begins with ϕ(τ, z) = f( − 1, 0)q− 1 + ∑ℓ ∈ Lʹf(0, ℓ)ζℓ + O(q). Since F is of singular weight, $f(0,0)={\mathop{\mathrm {rk}}\nolimits}(L)$. We know from that a holomorphic Borcherds product of singular weight has only simple zeros. Therefore, f( − 1, 0) and f(0, ℓ) can only be 0 or 1. By definition, f( − 1, 0) = 1 in the anti-symmetric case and f( − 1, 0) = 0 in the symmetric case. We define R = {ℓ ∈ Lʹ ∖ {0} : f(0, ℓ) = 1}. By, we have $$\begin{aligned} \label{eq:A} \frac{\|\mathcal{R}\|+{\mathop{\mathrm {rk}}\nolimits}(L)}{24} -f(-1,0) &= \frac{1}{2{\mathop{\mathrm {rk}}\nolimits}(L)}\sum\_{\ell \in \mathcal{R}} (\ell, \ell) =:C, \\ \label{eq:B} \sum\_{\ell \in \mathcal{R}} (\ell, \mathfrak{z})^2 &= 2C (\mathfrak{z}, \mathfrak{z}).\end{aligned}$$ If R is empty, then C = 0, f( − 1, 0) = 1 and ${\mathop{\mathrm {rk}}\nolimits}(L)=24$. In this case, shows that M is isomorphic to the even unimodular lattice of signature (26, 2), which then forces L to be the Leech lattice Λ and F to be the Borcherds form Φ12 (see ). The Jacobi form input of Φ12 is given by ΘΛ, 0(τ, z)/η24(τ) = q− 1 + 24 + O(q), which equals the full character of the Leech lattice VOA. Now assume that R is non-empty. Equation implies that C > 0 and that $L\otimes{\mathbb Q}$ is spanned by R over ${\mathbb Q}$. We will now show that R is a rescaled root system. Let ℓ1, ℓ2 ∈ R. Then − ℓ1 ∈ R because f(0, ℓ) = f(0, − ℓ). For any integer a > 1, aℓ1 ∉ R, as otherwise (0, ℓ1, 1)⊥ would have multiplicity at least 2 in the divisor of F, contradicting the fact that all zeros of F are simple. Since F is reflective, the reflection σ(0, 0, ℓ1, 1, 0) fixes M (see Notation [notation2]). From $$\sigma\_{(0,0,\ell\_1,1,0)}\big((0,0,\ell\_2,1,0)\big)= \big(0,0,\sigma\_{\ell\_1}(\ell\_2),1-2(\ell\_1,\ell\_2)/(\ell\_1,\ell\_1),0\big)=:\lambda \in M$$ we obtain $$\sigma\_{\ell\_1}(\ell\_2)\in L' \quad \text{and} \quad 2(\ell\_1,\ell\_2)/(\ell\_1,\ell\_1) \in {\mathbb Z}.$$ We know from that F is anti-invariant under σ(0, 0, ℓ1, 1, 0) and therefore vanishes on λ⊥. Since (0, 0, σℓ1(ℓ2), 1, 0) is primitive in Mʹ, by the Eichler criterion (see e.g. ) there exists $g\in \widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(M)$ such that g(λ) = (0, 0, σℓ1(ℓ2), 1, 0). Therefore, F also vanishes on (0, 0, σℓ1(ℓ2), 1, 0)⊥, from which it follows that f(0, σℓ1(ℓ2)) = 1 and that σℓ1(ℓ2) ∈ R. This proves that R is a (rescaled) root system. Now we can decompose R into rescaled irreducible root systems R = ⨁j = 1sRj(aj), where Rj are standard irreducible root systems and aj are rational numbers by which the (squared) norms of the roots in Rj are rescaled. Since (0, 0, ℓ, 1, 0)⊥ is reflective for any ℓ ∈ R, there exists a positive integer d such that ℓ2 = 2/d and dℓ ∈ L. It follows that aj = 1/kj for some positive integers kj. In the symmetric case, there are no 2-roots in R, so every kj is larger than 1. Let gj, kj be the simple Lie algebra of type Rj with level kj. We see from and that the semi-simple Lie algebra g = ⊕ j = 1sgj, kj satisfies condition (1) or (1ʹ) of the theorem. By, L(C) is an integral lattice. The set R = ⊕ j = 1sRj(1/kj) generates a sublattice of Lʹ over ${\mathbb Z}$, from which it follows that ⨁j = 1sQj(1/kj) < Lʹ and therefore L < ⨁j = 1sPj∨(kj). The vectors ${\mathop{\mathrm {ord}}\nolimits}(\ell) \ell$ for ℓ ∈ R generate a sublattice of L over ${\mathbb Z}$, where ${\mathop{\mathrm {ord}}\nolimits}(\ell)$ is the order of ℓ in Lʹ/L. It is easy to check that this lattice contains the lattice generated by long roots of ⊕ j = 1sRj(kj), and therefore ⊕ j = 1sQj∨(kj) < L. This proves claim (2). Finally, let Δj+ be the set of positive roots of Rj and let ⟨ − , − ⟩ be the normalized bilinear form on Rj that was fixed in Section [subsec:theta-blocks]. For α ∈ Δj+, we have $\alpha / k\_j \in \mathcal{R} \subsetneq L'$ and (α/kj, z) = kj⟨α/kj, z⟩ = ⟨α, z⟩. Therefore, the leading Fourier–Jacobi coefficient of F is a Jacobi form of lattice index L(C) given by $$\eta(\tau)^{{\mathop{\mathrm {rk}}\nolimits}(L)} \prod\_{0<\ell \in \mathcal{R}} \frac{\vartheta(\tau, (\ell, \mathfrak{z}))}{\eta(\tau)} = \prod\_{j=1}^s \Big( \eta(\tau)^{{\mathop{\mathrm {rk}}\nolimits}(\mathfrak{g}\_{j})} \prod\_{\alpha \in \Delta\_j^+}\frac{\vartheta(\tau, {{\langle{\alpha, \mathfrak{z}}\rangle}})}{\eta(\tau)} \Big),$$ which is actually the denominator of $\hat{\mathfrak{g}}$. Note that ⨁j = 1sQj∨(hj∨) < L(C) < ⨁j = 1sPj∨(hj∨), which matches the index L(C) of the above leading Fourier–Jacobi coefficient of F. The last assertion follows from the central charge formula $$\begin{aligned} c\_\mathfrak{g} = \sum\_{j=1}^s \frac{k\_j \dim \mathfrak{g}\_{j}}{ k\_j+h\_j^\vee } = \sum\_{j=1}^s \frac{\dim \mathfrak{g}\_{j}}{1 + C} = \frac{\dim \mathfrak{g}}{1+C}\end{aligned}$$ and because dimg = 24(C + 1) in the anti-symmetric case and dimg = 24C in the symmetric case. In the above theorem, when Rj is of type E8, F4 or G2, we have Qj∨ = Pj∨. Let J denote the subset of such j. Then we have (see ) L = K ⊕ ⨁j ∈ JPj∨(kj), where ⨁j ≠ JQj∨(kj) < K < ⨁j ≠ JPj∨(kj). [cor:solutions] Equation has 221 solutions, and they are listed in Table [table:anti1]. Equation has 17 solutions, and they are listed in Table [table:sym]. Equation was first derived by Schellekens in 1993 in the context of holomorphic vertex operator algebras of central charge 24, and he determined the solutions that are listed in Table [table:anti1]. The solutions of Equation can be found in a similar way. The classification of affine Lie algebras with hyperbolizations =============================================================== In this section we state the main theorems and explain the ideas behind the proofs. Our first main result is the classification of root systems associated with reflective Borcherds products of singular weight on lattices of type 2U ⊕ L, which was introduced in Theorem [th:root-system]. [th:non-existence] Suppose 2U ⊕ L has a reflective Borcherds product of singular weight whose Jacobi form input has non-negative q0-term. Then the associated semi-simple Lie algebra g defined in Theorem [th:root-system] is one of the 81 Lie algebras colored blue in Tables [table:anti1] and [table:sym]. To prove this, we have to rule out most of the solutions of Equation and Equation. There are 238 root systems in Tables [table:anti1] and [table:sym] altogether. A root system will be called extraneous if it is not one of the 81 root systems described in Theorem [th:non-existence]; there are 152 extraneous root systems of anti-symmetric type and 5 extraneous root systems of symmetric type. Let g be a semi-simple Lie algebra associated with a singular-weight reflective Borcherds product on 2U ⊕ L that is extraneous. By Theorem [th:root-system] (2), the underlying lattice L satisfies the bounds Qg < L < Pg. We will determine an explicit even overlattice K of L with the property that any singular-weight reflective Borcherds product on 2U ⊕ L lifts to a reflective Borcherds product satisfying certain conditions on 2U ⊕ K, but for which no such reflective products on 2U ⊕ K exist. This leads to a contradiction. The complete proof appears in Section [sec:exclude-anti-root] for anti-symmetric root systems and in Section [sec:exclude-root] for symmetric root systems. The second main result is the construction of hyperbolizations for the 81 semi-simple Lie algebras in the theorem above. [th:construction] For each semi-simple Lie algebra g in Theorem [th:root-system], there exists an even positive definite lattice Lg with the same rank as g that satisfies the following conditions: 1. There is a singular-weight reflective Borcherds product Ψg on 2U ⊕ Lg; 2. The leading Fourier–Jacobi coefficient of Ψg at the 1-dimensional cusp determined by 2U is the denominator of the affine Lie algbera $\hat{\mathfrak{g}}$; 3. The Jacobi form input ϕg of Ψg is a ${\mathbb Z}$-linear combination of full characters of the affine vertex operator algebra generated by $\hat{\mathfrak{g}}$; 4. The lattice generated by dρ and by those λ = (n, ℓ, m) ∈ U ⊕ Lgʹ for which Ψg vanishes on λ⊥ is U ⊕ Lgʹ; that is, U ⊕ Lgʹ is the root lattice of the BKM (super)algebra with (super)-denominator Ψg. Here, ρ is the Weyl vector of Ψg and d is the denominator of C as defined in Theorem [th:root-system]. 5. When g is of symmetric type, the Borcherds product Ψg coincides with the Gritsenko (additive) lift of the denominator of $\hat{\mathfrak{g}}$. The above lattices Lg and Jacobi forms ϕg are constructed as follows. * Let g be one of the 69 semi-simple Lie algebras of anti-symmetric type. In this case, g appears as the semi-simple V1 structure of a holomorphic vertex operator algebra V of central charge 24 in Schellekens’s list. Then Lg is the orbit lattice in Höhn’s construction of V and ϕg is the unique full character of V. * Let g be one of the 8 semi-simple Lie algebras of symmetric type with C = 1. In this case, g appears as the N = 1 structure of a holomorphic vertex operator superalgebra of central charge 12 composed of 24 chiral fermions. Then Lg is the maximal even sublattice of Pg, and ϕg can be expressed as a ${\mathbb Z}$-linear combination of the full NS-, $\widetilde{\mathrm{NS}}$- and R-characters of the associated SVOA: $$\phi\_{\mathfrak{g}}=(\chi\_{\mathrm{NS}} - \chi\_{\widetilde{\mathrm{NS}}} - \chi\_{\mathrm{R}})/2.$$ * Let g = A1, 16, A1, 82, A1, 44 or A2, 9. These semi-simple Lie algebras are of symmetric type with C < 1. In these cases, the affine Lie algebras $\hat{\mathfrak{g}}$ admit exceptional modular invariants coming from a nontrivial automorphism of the fusion algebra. Then Lg = Pg, the expressions of ϕg in terms of affine characters are given in Theorem [th:input-character], and the relationship between ϕg and the exceptional modular invariants is explained in Remark [rem:exceptional-modular-invariants]. The proof of Theorem [th:construction] is divided into four sections. In Sections [sec:construct-anti]-[sec:construct-symmetric-C<1] we prove parts (a), (b) and (c) respectively. In Section [sec:additive lifts] we prove property (5). The connections between our construction and the twists of the fake monster algebra are also explained in Sections [sec:construct-anti] and [sec:construct-symmetric-C=1]. Our last main result is the complete classification of affine Lie algebras with hyperbolizations. This is a direct consequence of Theorem [th:non-existence] and Theorem [th:construction]. [th:classification] There are exactly 81 affine Kac–Moody algebras which have a hyperbolization in the sense of Definition [def:hyperbolization]. The 81 algebras are colored blue in Tables [table:anti1] and [table:sym]. Combining Theorem [th:non-existence] and Theorem [th:construction], we obtain the following constraint on the existence of singular automorphic products: There are reflective Borcherds products of singular weight on lattices of type 2U ⊕ L whose input has non-negative principal part if and only if $${\mathop{\mathrm {rk}}\nolimits}(L) \in \{1, 2, 4, 6, 8, 10, 12, 16, 24\}.$$ The antisymmetric case: holomorphic CFT of central charge 24 ============================================================ The 69 semi-simple Lie algebras g in the anti-symmetric case of Theorem [th:non-existence] coincide with the semi-simple V1 structures of holomorphic vertex operator algebras (VOA) of central charge 24 in Schellekens’ list. In 2017, Höhn found a uniform construction of the holomorphic VOAs. For each g, we will take Höhn’s orbit lattice Lg as the underlying lattice L and construct the reflective Borcherds product of singular weight as the Borcherds lift of the full character of the holomorphic VOA. We also explain the connection between these hyperbolizations and the twisted denominators of the fake monster algebra. Holomorphic CFTs of central charge 24 and Schellekens’ list ----------------------------------------------------------- Let V be a holomorphic vertex operator algebra of central charge 24. The weight-one subspace V1 has a natural Lie algebra structure and by, V1 is either trivial, abelian of dimension 24, or semi-simple. In the first case it was conjectured in that V is isomorphic to the monster VOA, which was proved under some conditions in. In the second case V is isomorphic to the Leech lattice VOA. We focus on the third case where V1 is a semi-simple Lie algebra g. Let ( − , − ) be the unique symmetric, non-degenerate, invariant bilinear form on V normalized such that (1, 1) = − 1, where 1 is the vacuum vector. The restriction of ( − , − ) to a simple ideal gj of g satisfies ( − , − ) = kj⟨ − , − ⟩ for some positive integer kj, where ⟨ − , − ⟩ is the normalized bilinear form of gj (see ). We indicate these integers by writing g = g1, k1 ⊕ ⋯ ⊕ gs, ks. Then the affine vertex operator algebra $$V\_\mathfrak{g}\cong L\_{\hat{\mathfrak{g}}\_1}(k\_1,0)\otimes \cdots \otimes L\_{\hat{\mathfrak{g}}\_s}(k\_s,0)$$ generated by V1 is a full vertex operator subalgebra of V. As a Vg-module, one can decompose V into finitely many irreducible Vg-modules $$\label{eq:V-decomposition} V=\bigoplus\_{\lambda\_1,...,\lambda\_s} m\_{\lambda\_1,...,\lambda\_s} L\_{\hat{\mathfrak{g}}\_1}(k\_1,\lambda\_1)\otimes \cdots \otimes L\_{\hat{\mathfrak{g}}\_s}(k\_s,\lambda\_s),$$ where the sum runs over the dominant integral weights λj of gj that satisfy ⟨λj, θj∨⟩ ≤ kj, where θj is the highest root of gj. In 1993 Schellekens established the equation (see Notation [notation]) -1 =, and showed that it has exactly 221 solutions. By excluding 152 of them, Schellekens proved that V1 is isomorphic to one of the 69 semi-simple Lie algebras in (named Schellekens’s list) and further determined the induced decompositions. This result was reproved in using similar arguments and in by means of the ``very strange formula". Schellekens also conjectured that there exists a unique holomorphic VOA of central charge 24 with V1 = g for every g on his list. By the work of many authors over the past three decades, this celebrated conjecture was finally proved (see for the existence and for the uniqueness). There are at least three uniform proofs of the conjecture: the Leech lattice method of Höhn (), the generalized deep hole method of Möller–Scheithauer, and the Niemeier lattice method of Höhn–Möller. Höhn’s construction of holomorphic CFT of central charge 24 ----------------------------------------------------------- Our construction of hyperbolizations relies heavily on Höhn’s argument, which is reviewed below. Let Qg be the rescaled coroot lattice (see Notation [notation]). Then V contains the lattice VOA associated with Qg denoted VQg. If we let Wg = ComV(VQg) be the commutant (or centralizer), then the double commutant ComV(Wg) = ComV(ComV(VQg)) is a lattice VOA extending VQg. Therefore, there exists an even positive definite lattice Lg ⊃ Qg such that ComV(Wg) is isomorphic to the lattice VOA VLg. It is well known that VLg has group-like fusion: all irreducible VLg-modules are simple current modules. In this case, the set of all irreducible modules R(VLg) forms an abelian group with respect to the fusion product, and it also carries the quadratic form $\mathfrak{q}: R(V\_{L\_\mathfrak{g}})\cong L\_\mathfrak{g}'/L\_\mathfrak{g} \to {\mathbb Q}/{\mathbb Z}$ defined via $$\mathfrak{q}(V\_{\alpha + L\_\mathfrak{g}}) = \mathrm{wt}(V\_{\alpha + L\_\mathfrak{g}}) = (\alpha,\alpha)/2 \mod {\mathbb Z},$$ where wt( − ) denotes the conformal weight of the module. The full character of the irreducible module labeled by α + Lg ∈ Lgʹ/Lg can be expressed in terms of the Jacobi theta function and η-function: $$\chi\_{V\_{\alpha+L\_\mathfrak{g}}}(\tau, \mathfrak{z}) = \frac{\Theta\_{L\_\mathfrak{g},\alpha}(\tau,\mathfrak{z})}{\eta(\tau)^{{\mathop{\mathrm {rk}}\nolimits}(L\_\mathfrak{g})}}, \quad (\tau,\mathfrak{z})\in {\mathbb H}\times (L\_\mathfrak{g}\otimes{\mathbb C}).$$ It is also known that Wg is strongly rational and also has group-like fusion (see e.g. ). The set R(Wg) of irreducible modules forms a quadratic space isomorphic to (R(VLg), − q), where the quadratic form is defined by reducing the conformal weight modulo ${\mathbb Z}$. Therefore, V is a simple current extension of Wg ⊗ VLg, i.e. there is an isometry ι : Lgʹ/Lg → R(Wg) such that V ≅ ⨁α + Lg ∈ Lgʹ/LgWι(α + Lg) ⊗ Vα + Lg. Note that VLg has central charge ${\mathop{\mathrm {rk}}\nolimits}(L\_\mathfrak{g})={\mathop{\mathrm {rk}}\nolimits}(V\_1)={\mathop{\mathrm {rk}}\nolimits}(\mathfrak{g})$, that Wg has central charge $24-{\mathop{\mathrm {rk}}\nolimits}(\mathfrak{g})$, and that the weight-one subspace of Wg is zero. Höhn computed the lattice Lg for each of the 69 semi-simple g and called it the orbit lattice of g. Let h be the Cartan subalgebra of g. Recall that the full character of V, defined by $$\chi\_{V}(\tau, \mathfrak{z}) = \mathrm{Tr}\_V (e^{2\pi i\mathfrak{z}} q^{L\_0-1}), \quad q=e^{2\pi i\tau}, \quad (\tau, \mathfrak{z})\in{\mathbb H}\times \mathfrak{h},$$ is a weakly holomorphic Jacobi form of weight 0 and lattice index Qg (with trivial character on ${\mathop{\mathrm {SL}}\nolimits}\_2({\mathbb Z})$) (see e.g. ). The above construction implies that χV also defines a weakly holomorphic Jacobi form of weight 0 and lattice index Lg. Moreover, the character decomposition χV(τ, z) = ∑α + Lg ∈ Lgʹ/LgχVα + Lg(τ, z) ⋅ χWι(α + Lg)(τ) determines the theta decomposition of χV as a Jacobi form of lattice index Lg. In particular, the vector-valued function $$\label{eq:character-orbifold} f\_\mathfrak{g}(\tau):= \sum\_{\alpha \in L\_\mathfrak{g}'/L\_\mathfrak{g}} \eta(\tau)^{-{\mathop{\mathrm {rk}}\nolimits}(\mathfrak{g})}\cdot\chi\_{W\_{\iota(\alpha+L\_\mathfrak{g})}}(\tau) e\_\alpha$$ is a weakly holomorphic modular form of weight $-{\mathop{\mathrm {rk}}\nolimits}(\mathfrak{g})/2$ for the Weil representation ρLg. Höhn described Wg in an elegant way. Let Λ continue to be the Leech lattice and let g:=(g)={ x: (x,y)=0 } be the coinvariant sublattice for $g\in \mathrm{Co}\_0={\mathop{\null\mathrm {O}}\nolimits}(\Lambda)$. Let ĝ ∈ Aut(VΛ) be a standard lift of g; that is, ĝ is a lift of g to the Leech lattice VOA that acts trivially on Λg. Höhn conjectured that Wg is isomorphic to an orbifold VOA VΛgĝ where [g] is one of 11 particular conjugacy classes [g] of Co0 (see Tables [table:8cycleshapes] and [table:3cycleshapes]). This would imply that V is isomorphic to a simple-current extension of VΛgĝ ⊗ VLg. Lam proved that the orbifold VΛgĝ with Λg ≠ {0} has group-like fusion and described the corresponding quadratic form explicitly, and in this way was able to confirm Höhn’s conjecture. The simple-current extension of VΛgĝ ⊗ VLg depends on the double cosets $$\label{eq:double-coset} {\mathop{\null\mathrm {O}}\nolimits}(L\_\mathfrak{g}) \backslash {\mathop{\null\mathrm {O}}\nolimits}(R(W\_\mathfrak{g}),-\mathfrak{q}) / \mathrm{Aut}(W\_\mathfrak{g}).$$ By counting the numbers of the above double cosets denoted n(Lg), Betsumiya, Lam and Shimakura supplemented Höhn’s proof of the uniqueness of holomorphic VOAs of central charge 24 with semi-simple V1. We will now describe the 69 simple current extensions explicitly. The construction involves 11 distinct Co0-conjugacy classes [g], which are characterized by the property that the discriminant form (R(VΛgĝ), − q) of signature ${\mathop{\mathrm {rk}}\nolimits}(\Lambda^g) \mod 8$ can be realized by an even positive definite lattice of rank ${\mathop{\mathrm {rk}}\nolimits}(\Lambda^g)$. The orbit lattices Lg lie in the 11 associated genera. 1. Let [g] be one of the 8 conjugacy classes in Table [table:8cycleshapes]. For each orbit lattice of genus [g], the number of double cosets is always 1, so the number of inequivalent simple-current extensions equals to the number of classes in the genus. There are in total 58 classes (including the Leech lattice) in the 8 genera, so the 8 conjugacy classes [g] induce 57 holomorphic VOAs of central charge 24 with semi-simple V1 = g. 2. Let [g] be the conjugacy class of cycle shape 212. The associated (lattice) genus has 2 classes D12(2) and E8(2) ⊕ D4(2) and the corresponding numbers of double cosets are respectively 6 and 3. Therefore, this [g] induces 9 simple-current extensions. 3. Let [g] be the	arxiv_0000547
Orthogonal root numbers of tempered parameters ============================================== ### 5 July 2021 We show that an orthogonal root number of a tempered L-parameter φ decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a certain involution of Langlands’s central character for φ. The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil-Deligne representations arising in the work of Hiraga, Ichino, and Ikeda on the Plancherel measure. Introduction ============ To every representation of the absolute Galois group Γ k of a local field k, or more generally, of its Weil-Deligne group WDk, we can attach various local constants, the γ-, L-, and ɛ-factors, each a function of a complex parameter s. Of these, the ɛ-factor has the simplest description as a function of s: it is an exponential function, s ↦ a ⋅ bs − 1/2. Understanding this local constant thus amounts to understanding the base b and constant term a. The base of the ɛ-factor (or depending on one’s choice of terminology, its logarithm) is known as the Artin conductor, taken as 1 for k archimedean. This quantity measures the ramification of the representation. Although the Artin conductor has its own subtleties, especially in the presence of wild ramification, there are good formulas to compute it, formulas that make precise the sense in which the conductor measures ramification. The constant term a is known as the root number. We denote it here by ω, an additive function of Weil-Deligne representations. The choice of exponent s − 1/2 ensures that the root number is a complex number of modulus one, so we can refer to it informally as the sign of the ɛ-factor. This sign is much more subtle than the Artin conductor. Here is one example of its subtlety. The problem of computing a root number generalizes the problem of computing the sign of a Gauss sum. For quadratic Gauss sums, the sign is determined by a congruence condition on the modulus, a classical result of Gauss and the most difficult of the basic properties of these sums. For cubic Gauss sums, the situation is much more complicated: there is no congruence condition to describe the sign, and in fact, the sign is randomly and uniformly distributed on the unit circle. In other words, for as simple a class of Galois representations as the cubic characters, the root number exhibits great complexity. Fortunately, there is a special class of Galois representations whose root numbers are more amenable to computation: the orthogonal representations. After all, the orthogonal characters are precisely the quadratic characters. Using formal properties of root numbers, it is easy to see that the root number of an orthogonal representation is a fourth root of unity, and the square of the root number can be described in terms of the determinant of the representation. Ultimately, then, computing an orthogonal root number is a matter of distinguishing between two square roots. Deligne calculated the distinction by expressing the root number of an orthogonal representation in terms of the second Stiefel-Whitney class of the representation. Stiefel-Whitney classes are a notion from algebraic topology, but they can be reinterpreted purely in terms of group cohomology, as the pullback of a certain universal cohomology class. In this way, Deligne’s formula reduces computing the root number of an orthogonal representation to a problem in group cohomology. In the Langlands program we generalize the study of Weil-Deligne representations to the study of L-parameters for a quasi-split reductive k-group G. To extend the definition of the local constants to this setting we reduce to the case of the general linear group by composing an L-parameter WDk → LG with a complex representation r: LG → GL(V) and computing the local constants of the representation r ∘ φ. Deligne’s formula suggests the possibility of computing the root numbers $\omega(\varphi,{{r}}){\stackrel{{\textnormal}{def}}{=}}\omega({{r}}\circ\varphi)$ for orthogonal r, in other words, the orthogonal root numbers of φ. The hope is a formula for ω(φ, r) that incorporates information about the L-parameter φ and the complex orthogonal representation r. In a 2010 paper, the direct inspiration for this article, Gross and Reeder gave a conjectural formula for a particular class of orthogonal root numbers and proved the formula when G is split. Motivated in part by a conjecture of Hiraga, Ichino, and Ikeda on the formal degree of discrete series, they took the adjoint representation $\operatorname{Ad}\colon{}^LG\to\operatorname{GL}(\hat{{\mathfrak}g})$, an orthogonal representation, and set about computing $\omega(\varphi,{\textnormal}{Ad})$. Their conjectural answer has three factors. First, there is a recipe, due to Langlands, that constructs from the L-parameter φ a character χφ of the center of G. At the same time, we can conjecturally assign to φ a finite set Πφ of smooth irreducible representations of G(k). The set Πφ is called the *L-packet* of φ and the assignment φ ↦ Πφ is called the local Langlands correspondence. It is expected that all elements of Πφ have the same central character, and further, that this character is χφ. Second, Gross and Reeder evaluated this central character on a certain canonical involution $z\_{\textnormal}{Ad}$ in the center of G, defined as the value of the sum of the positive coroots on − 1. Third, when k is nonarchimedean the L-group admits a particular distinguished parameter called the principal parameter $\varphi\_{\textnormal}{prin}\colon\operatorname{WD}\_k\to{}^LG$. This parameter is trivial on the Weil group, and its restriction to the Deligne SL2 corresponds, via the Jacobson-Morozov theorem, to the sum of elements in a pinning of $\widehat G$, a nilpotent element of the Lie algebra. The L-packet of the principal parameter captures the Steinberg representation. When k is archimedean, we define the principal parameter to be the trivial crossed homomorphism $W\_k\to\widehat G$. [] [thm2] If k is nonarchimdean of characteristic zero, the center of G is anisotropic, and the parameter φ: WDk → LG is discrete then $$\frac{\omega(\varphi,{\textnormal}{Ad})}{\omega(\varphi\_{\textnormal}{prin},{\textnormal}{Ad})} = \chi\_\varphi(z\_{\textnormal}{Ad}).$$ The goal of this article is to verify Gross and Reeder’s conjecture. Actually, we will prove a more general version that relaxes the base field to an arbitrary local field, the adjoint representation to an arbitrary orthogonal representation, and discreteness of the parameter to temperedness. We generalize Gross and Reeder’s canonical involution $z\_{\textnormal}{Ad}$ to an involution zr (see ) that depends only on the restriction of r to $\widehat G$. The original motivation for our generalization was to compute the adjoint ɛ-factor arising in a conjecture of Hiraga, Ichino, and Ikeda describing the Plancherel measure, of which the formal degree conjecture that motivated Gross and Reeder is a special case. In the more general conjecture, the adjoint representation of LG is replaced by a certain “relative adjoint representation” of the L-group of a Levi. From this perspective it is natural, and in the end costs little, to relax the adjoint representation to an arbitrary orthogonal representation. [thm1] Let r be an orthogonal representation of LG and φ: WDk → LG a tempered parameter. Then $$\frac{\omega(\varphi,{{r}})}{\omega(\varphi\_{\textnormal}{prin},{{r}})} = \chi\_\varphi(z\_{{r}}).$$ The does not fully compute the orthogonal root number ω(φ, r): instead, it disentangles the root number into an automorphic factor, χφ(zr), and a Galois factor, $\omega(\varphi\_{\textnormal}{prin},{{r}})$. If one wanted to use to pin down an orthogonal root number precisely, it seems that the main challenge would be to compute the orthogonal root number of the principal parameter. In general I expect no better resolution to this problem than the rough answer provided by Clifford theory, though for special classes of representations, such as the adjoint representation, it might be possible to say more. Although our work is informed by the local Langlands correspondence, everything here takes place on the Galois side. A stronger version of the theorem would assert that χφ(zr) is actually the value on zr of the central character of the L-packet of φ; this is much more difficult because it requires some knowledge of L-packets. Lapid proved the stronger version of for generic irreducible representations of certain classical groups. Gross and Reeder proved for split G by an argument in group cohomology. This article is an outgrowth of an observation that their argument can be generalized in various directions. To relax the split assumption in Gross and Reeder’s proof, we generalize, in of, the basic lemma from group cohomology that underlies their proof, taking into account the Galois action on $\widehat G$. Since the pin extension of a complex orthogonal group is not topologically split, we must use Borel cohomology here instead of continuous (group) cohomology. With this modification, proves equal two particular Borel cohomology classes in ${\textnormal}H\_{\textnormal}{Borel}^2({}^LG,\{\pm1\})$. then follows from the equation by pullback along the parameter φ, once we properly identify the factors of the pulled-back equation. Most of this article is devoted to identifying these factors. Two of the factors are root numbers, and their recognition as such passes through Deligne’s theorem. Gross and Reeder already reformulated the theorem in the language of group cohomology for determinant-one orthogonal representations of Galois type, and it is mostly a matter of collecting definitions from the algebraic topology literature, in, to extend their reformulation to tempered orthogonal representations. The third factor of the pulled-back equation is the value of the central character χφ on the involution zr. Identifying this factor requires several detours that are surely well known to experts in the field. Any spin representation of a complex reductive group gives rise to a character of its topological fundamental group. The first detour, in, is to compute this character. Here we offer a small correction to an exercise of Bourbaki. The second detour, in, is to compute the second cohomology group of the Weil group with coefficients in the character lattice of a k-torus, generalizing a standard computation for the absolute Galois group. It turns out that this cohomology group is the character group of the Harish-Chandra subgroup of the torus. In we extend these results, for k nonarchimedean of characteristic zero, to any finite type k-group of multiplicative type, using a generalization of Tate duality due to Karpuk. Finally, in the brief we weave together these disparate threads to prove, and then explain the connection with the conjectures of Hiraga, Ichino, and Ikeda. Notational conventions ---------------------- Let k be a nonarchimedean local field of residue characteristic p and characteristic different from 2. The latter assumption is no great loss because is trivially satisfied if + 1 = − 1. Let Γ k be the absolute Galois group of k, let Wk be the Weil group of k, and let $$\operatorname{WD}\_k {\stackrel{{\textnormal}{def}}{=}}\begin{cases} W\_k &{\textnormal}{if $k$ is archimedean} \\ W\_k\times\operatorname{SL}\_2({\mathbb C}) &{\textnormal}{if not} \end{cases}$$ be the Weil-Deligne group of k. In practice we can immediately forget WDk and work with Wk because root numbers of orthogonal Weil-Deligne representations are unaffected by restriction to the Weil group. For k nonarchimedean, let Ik ⊂ Wk be the inertia subgroup. We reserve the letter G for a group, of two kinds: either a reductive group or a topological group. When G is a topological group, we assume it to be Hausdorff. When G is a reductive group, we assume it to be connected and quasi-split. To best align with the statement of the key we work with the Weil form of the L-group, ${}^LG{\stackrel{{\textnormal}{def}}{=}}\widehat G\rtimes W\_k$. The choice of the Weil form over the Galois form is not essential because in practice, we can replace the Weil-group factor of this semidirect product with the Galois group of any extension of k that is large enough to contain the splitting field of G and to trivialize the L-parameter relevant to the problem at hand. For us, a representation of the L-group LG is a representation in the sense of Borel’s Corvallis article, that is, a finite-dimensional complex vector space V together with a continuous homomorphism LG → GL(V) whose restriction to $\widehat G$ is a morphism of complex algebraic groups. Similarly, a representation of a complex reductive group is assumed algebraic. In general, the ɛ-factor, and hence the root number, depends not only on a Weil-Deligne representation but also on a nontrivial additive character of k and a Haar measure on k. Since there are formulas that explain the dependence of these factors on the character and the measure, nothing is lost in fixing them. We follow Gross and Reeder in computing ɛ-factors with respect to the Haar measure that assigns ${\mathcal}O\_k$ measure one and an additive character that is trivial on ${\mathcal}O\_k$ and nontrivial on the inverse of some uniformizer. In this article we use three kinds of cohomology: the singular cohomology of a topological space, denoted by ${\textnormal}H^\bullet\_{\textnormal}{sing}$; the continuous cohomology of a topological group, denoted by ${\textnormal}H^\bullet$; and the Borel cohomology of a topological group, denoted by ${\textnormal}H\_{\textnormal}{Borel}^\bullet$, whose definition we review in. Given any group A of order two, let sgn: A → { ± 1} denote the canonical isomorphism to the group { ± 1} of order two. Acknowledgments --------------- I am grateful to Jack Carlisle for discussing the cohomology of classifying spaces, to Peter Dillery for discussing the cohomology of the Weil group, to my advisor, Tasho Kaletha, for discussing the contents of this article and providing detailed feedback on it, and to Karol Koziol for pointing me to Flach’s article. This research was supported by the National Science Foundation RTG grant DMS 1840234. Group cohomology ================ This section collects general facts on group cohomology that inform the rest of this article. In, we review the relationship between extensions and cohomology for discrete groups. In we explain how the relationship behaves in the presence of topology. And in we state and prove the main cohomological lemma,, underlying the proof of. The remaining sections of the article will flesh out the connection between and. Group extensions ---------------- In this subsection we work in the category of discrete groups. An extension of the group G by the group A is an exact sequence 1 & A & E & G & 1, permitting one to identify A with a subgroup of E and G with the quotient of G by this subgroup. We call A, E, and G, the first, second, and third terms of the extension, respectively. We always assume that the extension is abelian, in other words, that A is an abelian group. In fact, in our application all extensions are central, that is, having A as a central subgroup of G. Since A is abelian, the conjugation action of G on E descends to an action of G on A: that is, A is a G-module. Conversely, starting from a G-module A, an extension of G by A is an extension of G by the abelian group A with the property that the G-action on A induced by the extension agrees with the given action. A morphism of extensions is a commutative diagram 1 & A & E & G & 1 1 & A’ & E’ & G’ & 1. As for extensions, we can speak of the first, second, and third terms of a morphism, namely, the maps A → Aʹ, E → Eʹ, and G → Gʹ, respectively. The isomorphisms in this category are precisely the morphisms whose terms are all isomorphisms. An easy diagram chase shows that a morphism is an isomorphism as soon as its first and third terms are isomorphisms. For this reason, we often restrict attention to the isomorphisms whose first and third terms are equalities; we call such an isomorphism an equivalence. Besides the isomorphisms, there are two kinds of morphisms of interest. First, given two G-modules A and Aʹ and a G-equivariant homomorphism α: A → Aʹ, we can construct the morphism of extensions 1 & A & E & G & 1 1 & A’ & \(E) & G & 1 in which α\(E) is the cokernel of the map $A\to A'\rtimes E$ sending a to α(a)a. We call the bottom extension (or sometimes, its second term) the pushout of the top extension.[1](#fn1) The pushout morphism is universal in the sense that any morphism with first term α from the top extension to an extension of G by Aʹ factors through the pushout morphism. Second, given a homomorphism γ: Gʹ → G, we can construct the morphism of extensions 1 & A & \(E) & G’ & 1 1 & A & E & G & 1 in which γ\(E) is the pullback (of sets, hence of groups) of the right square. We call the top extension (or sometimes, its second term) the pullback of the bottom extension. The pullback morphism is universal in the sense that any morphism with third term γ from an extension of A by Gʹ to the bottom extension factors through the pullback morphism. The theory of group extensions is relevant to us because of its connection to group cohomology, which relates to root numbers, in turn, by Deligne’s theorem. Given an extension 1 & A & E & G & 1, let s: G → E be a set-theoretic section of E → G. The formula zs(g, gʹ) = s(g)s(gʹ)s(ggʹ)− 1 defines a 2-cocyle $z\_s\in{\textnormal}Z^2(G,A)$. The cohomology class of the resulting cocycle is independent of the choice of section: any two such sections differ by a function G → A, and the coboundary of this function exhibits a cohomology between the cocycles. Conversely, given a 2-cocycle $z\in{\textnormal}Z^2(G,A)$, define the group extension 1 & A & AzG & G & 1 in which $A\boxtimes\_zG=A\times G$ with multiplication (a, g) ⋅ (aʹ, gʹ) = (a ⋅ ga ⋅ z(g, gʹ), ggʹ). The equivalence class of the resulting extension depends only on the cohomology class of the cocycle: any function G → A whose coboundary exhibits a cohomology between two different cocycles gives rise to an equivalence of the corresponding extensions. In summary, given a group G and a G-module A, we have constructed a canonical bijection between equivalence classes of extensions of G by A and the cohomology group ${\textnormal}H^2(G,A)$. The bijection is compatible with pushforward and pullback of cohomology classes and extensions. The dictionary between extensions and cohomology classes nicely answers, or rather, reformulates, a natural question in the theory of group extensions: when does the diagram 1 & A & E & G & 1 1 & A’ & E’ & G’ & 1 extend to a morphism of extensions? The answer to the question is a special case of our key lemma,, and is also used in the proof of the lemma. To answer the question, use the universal properties of pullback and pushforward to extend the candidate morphism to the diagram 1 & A & E & G & 1 1 & A’ & \(E) & G & 1 1 & A’ & \(E’) & G & 1 1 & A’ & E’ & G’ & 1 It follows that the original diagram can be extended to a morphism if and only if the middle extensions in the diagram are isomorphic. In other words, letting $c\in{\textnormal}H^2(G,A)$ and $c'\in{\textnormal}H^2(G',A')$ denote the cohomology classes classifying the top and bottom extensions, the original diagram can be extended to a morphism if and only if $$\alpha\_\(c) = \gamma^\(c') \qquad{\textnormal}{in ${\textnormal}H^2(G,A')$.}$$ Borel cohomology ---------------- A sequence of topological groups 1 & A & E & G & 1 is a topological extension of G by A if A → E is a closed subgroup and the induced map from the cokernel of A → E to G, where the source has the quotient topology, is an isomorphism. The pullback and pushout constructions from work just as well in this setting, provided all maps in question are continuous. Classifying topological extensions is more subtle than classifying extensions of discrete groups, however, because not every continuous surjection of topological spaces admits a continuous section. If we were to carry out the work of in the category of topological groups, where all maps are required to be continuous, we would find that the continuous cohomology ${\textnormal}H^2(G,A)$ classifies extensions of G by A that are topologically split, that is, whose second term is a direct product G × A as a topological space. The collection of topologically split extensions is much too small for most purposes. For instance, the universal-cover group extension of a Lie group is never topologically split, but we need a theory that can see the spin extension of the special orthogonal group because of its great relevance to the study of Stiefel-Whitney classes. To capture the topological extensions that are not topologically split one must enlarge the continuous cohomology group. The correct enlargement is known as Borel cohomology,[2](#fn2) a variant of discrete group cohomology in which one requires that cochains be Borel measurable. I refer the reader to Stasheff’s survey article as well as Moore’s papers on the subject for more information on Borel cohomology. [thm22] Let G and A be separable locally compact groups with A abelian. There is a canonical natural isomorphism between ${\textnormal}H\_{\textnormal}{Borel}^2(G,A)$ and the set of equivalence classes of topological extensions of G by A. Mackey proved this result in his thesis. The same construction as in yields the bijection, with the following additional argument. Given an extension of G by A, one must find a section whose associated cocycle is Borel measurable. This is Mackey’s Théorème 3, and its proof shows that the section need only be Borel-measurable. Conversely, given a Borel-measurable cocycle z : G × G → A, one must show endow the extension $A\boxtimes\_z G$ with a locally compact topology making it an extension of topological groups. This is Mackey’s Théorème 2. A uniqueness statement in that theorem ensures that the topology on the extension can be recovered from any measurable cocycle classifying it. [thm31] Using the fact that ${\mathbb C}^\times$ is connected and { ± 1} is discrete, it is easy to show that the continuous cohomology group ${\textnormal}H^2({\mathbb C}^\times,\{\pm1\})$ is trivial. On the other hand, since ${\mathbb C}^\times=W\_{\mathbb C}$, we know that ${\textnormal}H^2\_{\textnormal}{Borel}({\mathbb C}^\times,\{\pm1\})\simeq\{\pm1\}$. The nontrivial cohomology class here represents the topologically non-split squaring extension 1 & {1} & C & C& 1. Since Borel cohomology is not widely used in the Langlands program, we point out several relevant properties. First, in the nonarchimedean case, the pullback of a Borel cocycle along an L-parameter is automatically continuous. Let W and G be topological groups, let A be a continuous G-module, and let φ : W → G be a continuous homomorphism. If φ factors through a discrete quotient of W then pullback by φ induces a map $$\varphi^\:{\textnormal}H\_{\textnormal}{Borel}^i(G,A) \to {\textnormal}H^i(W,A).$$ Pullback along φ* takes an arbitrary cocycle Gn → A to a cocycle Wn → A that is constant on cosets of an open subgroups. So the pulled-back cocycle is continuous. Second, in the archimedean case, Borel ${\textnormal}H^1$ works just as well as continuous ${\textnormal}H^1$ in the local Langlands correspondence. [thm32] Let G and A be separable completely metrizable topological groups, in other words, Polish groups, and let A carry a continuous G-module structure. The natural map $${\textnormal}H^1(G,A) \to {\textnormal}H^1\_{\textnormal}{Borel}(G,A)$$ is an isomorphism. Results of this kind are known as automatic continuity. A theorem of Banach and Pettis, nicely explained in Rosendal’s overview of automatic continuity, proves the result when the action of G on A is trivial, in which case ${\textnormal}H^1(G,A) = \operatorname{Hom}\_{\textnormal}{cts}(G,A)$. In general, use their theorem together with the fact that a crossed homomorphism G → A is the same as a homomorphism $A\rtimes G\to G$ that restricts to the identity on G. It would be interesting to see if Clausen and Scholze’s condensed mathematics offers a replacement to Borel cohomology. We will not pursue this idea here. The key lemma ------------- The following is the main technical tool underpinning the proof of. There is nothing essential in its use of Borel cohomology. [thm4] Let A, Aʹ, E, G, Gʹ, and W be topological groups with A and Aʹ abelian. Assume that either all groups are discrete or all groups are separable and locally compact. Let the group W act continuously on the groups G and E by group automorphisms, let E → G be a continuous W-equivariant homomorphism, and let 1 & A & E & G & 1 1 & A’ & E’ & G’ & 1 be a morphism of topological extensions. Let f: W → Gʹ be a continuous homomorphism such that the map $\gamma f\colon G\rtimes W\to G'$ is a homomorphism. Consider the diagram 1 & A & EW & GW & 1 & (c) 1 & A’ & E’ & G’ & 1 & (c’). with top and bottom extensions classified by the cohomology classes c and cʹ, respectively, and let $p\colon G\rtimes W\to W$ denote the canonical projection. Then (γf)\(cʹ) = α\(c) ⋅ p\*f\(cʹ). The assumption on the topologies of the groups involved implies, by in the locally compact case and the discussion of in the discrete case, that the relevant extensions are classified by Borel cohomology classes. The proof rests on an understanding of ${\textnormal}H\_{\textnormal}{Borel}^2(G\rtimes W,A)$. The semidirect product fits into a split short exact sequence 1 & G & G W & W & 1 in which $i\colon G\to G\rtimes W$ is the canonical inclusion, $p\colon G\rtimes W\to W$ is the canonical projection, and $s\colon W\to G\rtimes W$ is the canonical inclusion. This sequence dualizes to a split exact sequence of abelian groups 1 & HBorel2(W,A) & HBorel2(GW,A) & HBorel2(G,A). The map p\ realizes ${\textnormal}H\_{\textnormal}{Borel}^2(W,A)$ as a direct summand of ${\textnormal}H\_{\textnormal}{Borel}^2(G\rtimes W,A)$ with a canonical complement, the kernel of s\. The map p\s\* is projection onto the summand, and the map i\* identifies its complement, the kernel of s\, with a subgroup of ${\textnormal}H\_{\textnormal}{Borel}^2(G,A)$. To prove the lemma, it therefore suffices to show that s\(γf)\(cʹ) = f\(cʹ) and that i\(γ*f)\(cʹ) = i\α\(c). The first equation follows from the identity f* = (γf) ∘ s. The second equation amounts to showing that γ\(cʹ) = α\i\(c), and this is a consequence of the existence of ɛ. Later, in, we need the following compatibility condition. [thm16] In the setting of, a Borel-measurable crossed homomorphism φ: W* → G maps under the coboundary ${\textnormal}H\_{\textnormal}{Borel}^1(W,G) \to{\textnormal}H\_{\textnormal}{Borel}^2(W,A)$ to the pullback of c along the homomorphism $\varphi\cdot{\textnormal}{id} \colon W\to G\rtimes W$. By, we could have written “continuous” here instead of “Borel-measurable”. Let s: G → E be a Borel-measurable section of E → G and let $\widetilde\varphi=s\circ\varphi$ be a lift of φ to E. The coboundary of φ is given by the formula $$(w,w') \mapsto \widetilde\varphi(w) \cdot{}^w\widetilde\varphi(w') \cdot\widetilde\varphi(ww')^{-1}.$$ On the other hand, the class c is represented by the 2-cocycle (g, w; gʹ, wʹ) ↦ s(g)w ⋅ s(gʹ)wʹ ⋅ (s(g ⋅ wgʹ)wwʹ)− 1 = s(g) ⋅ ws(gʹ) ⋅ s(g ⋅ wgʹ)− 1 corresponding to the section $(s,{\textnormal}{id})\colon G\rtimes W\to E\rtimes W$. Pulling back this function along φ replaces g by φ(w) and recovers the coboundary of φ. For our application of to the proof of, the group W is the Weil group, the top extension is the universal cover of the dual group, the bottom extension is the universal cover of a complex orthogonal group, and the morphism between them arises from the given orthogonal representation ${{r}}\colon {}^LG\to{\textnormal}O(V)$: $$\label{thm11} \begin{tikzcd} 1 \rar & \pi\_1(\widehat G) \rar\dar{e\_{{r}}} & \widehat G\_{\textnormal}{univ} \rar\dar & \widehat G \rar\dar{{{r}}\|\_{\widehat G}} & 1 \\ 1 \rar & \{\pm1\} \rar & \operatorname{Pin}(V) \rar & {\textnormal}O(V) \rar & 1 & (c\_{\textnormal}{pin}). \end{tikzcd}$$ Here er is the “spin character”, which we study in, and the class $c\_{\textnormal}{pin}$ classifies the bottom extension. Let $c\_G\in{\textnormal}H^2\_{\textnormal}{Borel}({}^LG,\pi\_1(\widehat G))$ classify the extension $\widehat G\_{\textnormal}{univ}\rtimes W\_k$ of LG by $\pi\_1(\widehat G)$, as in the. With this setup, we prove the theorem by pulling back the conclusion of along the given L-parameter. After pullback, the quotient of the cohomology classes ${{r}}^\(c\_{\textnormal}{pin})$ and $p^\{{r}}\|\_{W\_k}^\(c\_{\textnormal}{pin})$ becomes a quotient of root numbers and the cohomology class e*r, \(c*G) becomes the value of a central character on an involution. The goal of the remainder of the article is to explain these identifications, thereby proving. Stiefel-Whitney classes ======================= Deligne’s formula for orthogonal root numbers is a key technical tool supporting the main results of this article. To use his formula effectively, we need a workable definition of the second Stiefel-Whitney class. The goal of this largely expository section is to explain how to interpret in terms of group cohomology the second Stiefel-Whitney class of a bounded complex orthogonal representation of a countable discrete group. In the end, the class is the pullback of a certain pin-group extension. This interpretation is surely well known to the experts, and special cases have already appeared in the literature, for instance, in a paper of Gunarwardena, Kahn, and Thomas on real orthogonal representations of finite groups. We generalize their results by working with countable discrete groups instead of finite groups and complex representations instead of real representations. Classifying spaces ------------------ One way to construct characteristic classes is to pull them back from universal cohomology classes of a certain classifying space. In this subsection, we review the theory of classifying spaces, loosely following Mitchell’s notes on classifying spaces and Section 6 of Stasheff’s survey article. Let G be a Lie group. We assume G to be second countable but we do not assume G to be connected. So G could be a complex reductive group or a countable discrete group. Let P → B be a principal G-bundle. Although G is a manifold we do not impose any smoothness or differentiability assumption on bundles: they are simply continuous. If P is weakly contractible, that is, having trivial homotopy groups, then we call B a classifying space for G and P a universal G-bundle. The bundle is universal in the following sense: for every CW-complex X, the canonical map from the set of homotopy classes of maps X → B to the set of equivalence classes of principal G-bundles over X, defined by pulling back the universal G-bundle, is a bijection. As the universal property makes reference only to homotopy classes of maps, a classifying space for G is defined uniquely only up to homotopy equivalence. It turns out, though this is not clear from the definition, that classifying spaces exist for every G. Often we can construct the classifying space by hand. For example, the classifying space of a compact orthogonal group of rank n is the Grassmannian of n-planes in ${\mathbb R}^{\oplus{\mathbb N}}$. But for a general group such a geometric construction is difficult, and we can instead construct the classifying space by simplicial methods. The simplicial construction of classifying spaces makes clear their functoriality: a homomorphism r: G → H of topological groups gives rise to a map Br: BG → BH. Functoriality also follows from Yoneda’s Lemma, without needing to choose a specific model for the classifying space: the map Br represents the balanced product functor P ↦ H × GP from principal G-bundles to principal H-bundles. The homotopy class of the classifying space BG depends only on homotopy type of the group G in the following sense: any group homomorphism G → H that is a homotopy equivalence induces a homotopy equivalence of classifying spaces. At the same time, a theorem of Iwasawa and Malcev implies that the inclusion into G of a maximal compact subgroup K is a homotopy equivalence. Hence the map BK → BG is a homotopy equivalence. Universal characteristic classes live in the singular cohomology of classifying spaces. To translate characteristic classes into the language of group cohomology, therefore, we should strive to interpret the singular cohomology of a classifying space in terms of group cohomology. In his thesis, Wigner gave such an interpretation using Borel cohomology. [thm3] Let A be a discrete abelian group. There is a canonical natural isomorphism $${\textnormal}H\_{\textnormal}{sing}^i(B(\cdot),A) \simeq H\_{\textnormal}{Borel}^i(\cdot,A)$$ of contravariant functors from the category of Lie groups to the category of abelian groups. Let G be a Lie group. Wigner defined cohomology groups ${\textnormal}H\_{\textnormal}{Wig}^\bullet(G,A)$ which he showed to agree with the Borel cohomology groups ${\textnormal}H\_{\textnormal}{Borel}^\bullet(G,A)$. Wigner’s Theorem 4 shows that ${\textnormal}H^\bullet(G,A)$ agrees with the cohomology of the constant sheaf ${\underline}A$ on the classifying space BG. Since G is a Lie group, BG is sufficiently nice (homotopy equivalent to a CW complex, say) that this sheaf cohomology agrees with the singular cohomology ${\textnormal}H\_{\textnormal}{sing}^i(BG,A)$. Definition for vector bundles ----------------------------- Let X be a CW complex. The theory of Stiefel-Whitney classes assigns to a rank-n real vector bundle V over X a family of cohomology classes $$w\_i(V)\in{\textnormal}H^i\_{\textnormal}{sing}(X,{\mathbb F}\_2), \qquad i = 0,1,\dots,n.$$ These characteristic classes, along with others like the Chern classes, provide a powerful algebraic framework for computations with vector bundles. The standard source for the subject is Milnor and Stasheff’s book on characteristic classes ; for our application, Chapters 4 to 9 and 14 are especially relevant. Stiefel-Whitney classes are characterized abstractly, via a series of axioms: unitality, naturality, multiplicativity of the total class, and nontriviality. To show that these axioms do indeed define a collection of cohomology classes, and that this collection is unique, one defines the classes as the pullback of certain universal cohomology classes on a classifying space. The naturality axiom forces such a description. The classifying space $B\!\operatorname{GL}\_n({\mathbb R})$ is the Grassmannian of n-planes in ${\mathbb R}^{\oplus{\mathbb N}}$, topologized as a direct limit, and the universal bundle over $B\!\operatorname{GL}\_n({\mathbb R})$ is just the tautological bundle over the Grassmannian, whose fiber over a point is the n-plane the point represents. The ${\mathbb F}\_2$ singular cohomology ring of this infinite Grassmannian is a graded polynomial ring with one generator for each i = 1, 2, …, n. We call the generator in degree i the ith universal singular Stiefel-Whitney class. Given a rank-n real vector bundle V on X classified by the map $f\_V\colon X\to B\!\operatorname{GL}\_n({\mathbb R})$, we define the ith Stiefel-Whitney class of V as the image under the pullback $$f\_V^\: {\textnormal}H^i\_{\textnormal}{sing}(B\!\operatorname{GL}\_n({\mathbb R}),{\mathbb F}\_2) \to {\textnormal}H^i\_{\textnormal}{sing}(X,{\mathbb F}\_2)$$ of the ith universal singular Stiefel-Whitney class. Since the compact orthogonal group ${\textnormal}O\_n$ of rank n* is a maximal compact subgroup of $\operatorname{GL}\_n({\mathbb R})$, the classifying spaces of the two groups are homotopy equivalent to each other and real vector bundles are classified by maps to $B{\textnormal}O\_n$. In the literature one often works with the classifying spaces of this maximal compact subgroup instead of the ambient general linear group, as we do here. Definition for representations ------------------------------ It is now clear how to define the Stiefel-Whitney classes of a real representation. Let $w\_{i,{\textnormal}{univ}}\in {\textnormal}H^i\_{\textnormal}{Borel}(\operatorname{GL}\_n({\mathbb R}),\{\pm1\})$ denote the Borel cohomology class corresponding, via, to the ith universal singular Stiefel-Whitney class. We call $w\_{i,{\textnormal}{univ}}$ the *ith universal Stiefel-Whitney class. Set $w\_{i,{\textnormal}{univ}} {\stackrel{{\textnormal}{def}}{=}}0$ if i* < 0 or i > n. Let G be a Lie group. The *ith Stiefel-Whitney class* of a real representation ${{r}}:G\to\operatorname{GL}\_n({\mathbb R})$ is the cohomology class $$w\_i({{r}}) {\stackrel{{\textnormal}{def}}{=}}{{r}}^\(w\_{i,{\textnormal}{univ}}) \in {\textnormal}H^i\_{\textnormal}{Borel}(G,\{\pm1\}).$$ This definition is not enough for us, however. Since the relevant representations of L-groups are complex, not real, we need to define the Stiefel-Whitney classes of a complex representation, the goal of this subsection. In outline, to make the definition we simply require that base change from ${\mathbb R}$ to ${\mathbb C}$ leave Stiefel-Whitney classes unchanged. This stipulation defines Stiefel-Whitney classes for all complex representations that descend to ${\mathbb R}$, in particular, the bounded orthogonal representations. An ${\mathbb R}$-structure* on a complex vector space V is a real subspace V0 ⊆ V such that the map $V\_0\otimes\_{\mathbb R}{\mathbb C}\to V$ is an isomorphism. An ${\mathbb R}$-structure on a complex representation (r, V) is a real representation (r0, V0) such that V0 is an ${\mathbb R}$-structure on V and r factors through r0. An isomorphism of ${\mathbb R}$-structures on a complex representation is an isomorphism of the corresponding real representations. It is not the case that every complex orthogonal representation admits an ${\mathbb R}$-structure: for instance, the two-dimensional representation of ${\mathbb Z}$ that sends 1 to the diagonal matrix with entries (2i, − i/2) is orthogonal but does not admit an ${\mathbb R}$-structure because its character takes imaginary values. However, if the representation is in addition bounded, meaning the closure of its image is compact, then it does admit an ${\mathbb R}$-structure. [thm5] Every bounded complex orthogonal representation admits an ${\mathbb R}$-structure. An exercise in point-set topology shows that the closure of the image of the representation is again a group, and by hypothesis the closure is compact. Hence the representation factors through some compact orthogonal group ${\textnormal}O\_n$, where n is the complex dimension of the representation, because ${\textnormal}O\_n$ is a maximal compact subgroup of the complex orthogonal group. The base change problem appears in many other settings than representation theory. For example, we can study the base change of varieties from ${\mathbb R}$ to ${\mathbb C}$. In that setting, it can happen that ${\mathbb R}$-varieties are nonisomorphic but become isomorphic upon base change to ${\mathbb C}$. In our setting the story is simpler: any two real forms of a (semisimple) representation must be isomorphic. [thm35] Let G be a group and r : G → GL(V) a finite-dimensional complex representation. If r is semisimple then any two of its ${\mathbb R}$-structures are isomorphic. The isomorphism classes of ${\mathbb R}$-structures on r are classified by the Galois cohomology set ${\textnormal}H^1(\Gamma\!\_{\mathbb R},\operatorname{Aut}\_G({{r}}))$. Since r is semisimple, AutG(r) is a product of complex general linear groups. Hence the cohomology set is trivial by Hilbert’s Theorem 90. By the, the following definition does not depend on the choice of ${\mathbb R}$-structure. Let G be a Lie group and let (r, V) be a semisimple complex representation of G that admits an ${\mathbb R}$-structure (r0, V0). The *ith Stiefel-Whitney class* of (r, V) is $$w\_i({{r}}) {\stackrel{{\textnormal}{def}}{=}}w\_i({{r}}\_0) \in {\textnormal}H^i\_{\textnormal}{Borel}(G,\{\pm1\}).$$ We are most interested in the second Stiefel-Whitney class because of its appearance in Deligne’s theorem on root numbers. [thm36] Let V0 be a real anisotropic quadratic space and let $V {\stackrel{{\textnormal}{def}}{=}}V\_0\otimes\_{\mathbb R}{\mathbb C}$, a complex quadratic space. There exists an extension Pin(V) of ${\textnormal}O(V)$ by { ± 1}, called the (complex) pin group, with the following property: the class $c\_{\textnormal}{pin}\in{\textnormal}H^2\_{\textnormal}{Borel}({\textnormal}O(V),\{\pm1\})$ that classifies Pin(V) is the pullback of the second universal Stiefel-Whitney class $$w\_{2,{\textnormal}{univ}}\in {\textnormal}H^2\_{\textnormal}{Borel}({\textnormal}O(V\_0),\{\pm1\}) \simeq {\textnormal}H^2\_{\textnormal}{Borel}(\operatorname{GL}(V\_0),\{\pm1\}).$$ We call the class $c\_{\textnormal}{pin}\in{\textnormal}H^2\_{\textnormal}{Borel}({\textnormal}O(V),\{\pm1\})$ the pin class. Let Pin(V0) be the extension of ${\textnormal}O(V\_0)$ by { ± 1} classified by $w\_{2,{\textnormal}{univ}}$. It is well-known that Pin(V0) is (the rational points of) the standard (real) algebraic pin group defined using the Clifford algebra Cl(V0); see, for instance, Appendix I of or the introduction to. To be specific, the group Pin(V0) is the stabilizer of the subspace V0 ⊆ Cl(V0) under the conjugation action of the units group Cl(V0)× on Cl(V0). The main anti-involution of Cl(V0) is the automorphism α induced by the order-reversing automorphism v1 ⊗ ⋯ ⊗ vd ↦ vd ⊗ ⋯ ⊗ v1 of the tensor algebra on V0. The spinor norm is the homomorphism $\operatorname{Cl}(V\_0)\to{\mathbb C}^\times$ sending x ∈ Cl(V0) to xα(x). Finally, Pin(V0) is the kernel of the spinor norm. It is now clear that we may take as Pin(V) the complex algebraic group obtained from (the algebraic group underlying) Pin(V0) by base change from ${\mathbb R}$ to ${\mathbb C}$. It follows from the that the second Stiefel-Whitney class of a bounded complex orthogonal representation $r:G\to{\textnormal}O(V)$ is classified by the pullback of group extensions r\Pin(V). This conclusion is our final reformulation of the second Stiefel-Whitney class in the language of group cohomology. There are two variant and non-isomorphic definitions of the pin group, stemming from the fact that the elements $w\_{2,{\textnormal}{univ}}$ and $w\_{1,{\textnormal}{univ}}^2$ of ${\textnormal}H^2\_{\textnormal}{Borel}({\textnormal}O(V),\{\pm1\})$ are distinct. Conrad’s SGA 3 article, whose notation for the Pin group agrees with ours, nicely explains the difference; Remarks C.4.9 and C.5.1 are especially relevant. The other variant of the pin group, which we do not use here, is due to Atiyah, Bott, and Shapiro, and is denoted by Pin− in Conrad’s article. The definition is similar to ours but one modifies the spinor norm by a sign twist. Deligne’s theorem ----------------- We start by defining the Stiefel-Whitney classes of a complex orthogonal representation of the Weil group. When the field k* is archimedean, this task is already complete because $W\_{\mathbb R}$ and $W\_{\mathbb C}$ are Lie groups. When k is nonarchimedean, we use the fact that every complex representation of Wk factors through a discrete quotient. [thm33] Let k be nonarchimedean, let V be a complex quadratic space, and let ${{r}}\colon W\_k\to{\textnormal}O(V)$ be a complex orthogonal representation of Wk. The ith Stiefel-Whitney class of (r, V) is the image of wi(r) under the inflation map $${\textnormal}H^i(W\_k/\ker{{r}},\{\pm1\}) \to {\textnormal}H^i(W\_k,\{\pm1\}).$$ The evident compatibility of Stiefel-Whitney classes with inflation shows that we are free to replace kerr by any open subgroup of Wk on which r is trivial. We can now state our reformulation of Deligne’s theorem on root numbers. [thm19] Let ${{r}}\colon W\_k\to{\textnormal}O(V)$ be a bounded complex orthogonal representation and let $c\_{\textnormal}{pin}\in{\textnormal}H\_{\textnormal}{Borel}^2({\textnormal}O(V),\{\pm1\})$ be the pin class of. Then $$\frac{\omega({{r}})}{\omega(\det{{r}})} = \operatorname{sgn}{{r}}^\(c\_{\textnormal}{pin}).$$ The statement of the uses that the group ${\textnormal}H^2(W\_k,\{\pm1\})$ is cyclic of order two. The function sgn was defined in ; it uniquely identifies this group with the group { ± 1}. Applying Deligne’s theorem to the virtual representation $${{r}}- \det{{r}}- (\dim{{r}}-1)\cdot{\textnormal}{triv}$$ of dimension zero and determinant shows that the lefthand side of the equation equals sgn(w2(r)). And by, $w\_2(r) = r^\(c\_{\textnormal}{pin})$. Spin lifting ============ Let G be a complex reductive group, identified with its set of ${\mathbb C}$-points. In our application G will be the Langlands dual of a reductive k-group, but in this section we restrict attention to complex groups so there is no need to decorate G with a hat. Consider an (algebraic, complex) orthogonal representation ${{r}}\colon G\to{\textnormal}O(V)$. As G is (by assumption) connected, the representation r factors through the identity component SO(V) of ${\textnormal}O(V)$, and we may without loss of generality replace ${\textnormal}O(V)$ by SO(V) as the target of r. The special orthogonal group SO(V) is not simply connected. Supposing that dimV > 2, its universal cover p: Spin(V) → SO(V), a double cover, is an algebraic group called the spin group. We can construct the spin group as a subgroup of the units group of a Clifford algebra, though for our purposes, we can understand the group using the combinatorics of root systems alone. When dimV = 2, so that $\operatorname{SO}(V)={\mathbb G}\_{{\textnormal}m}$ is a one-dimensional torus, we define $\operatorname{Spin}(V)={\mathbb G}\_{{\textnormal}m}$ with double cover p: Spin(V) → SO(V) the squaring map. The existence of the spin group creates a dichotomy in the orthogonal representations r: G → SO(V) of G: either the representation lifts to the spin group or it does not. We can study and refine this lifting question by passing to the universal cover $G\_{\textnormal}{univ}$ of G: $$G\_{\textnormal}{univ} = {\mathfrak}z\times G\_{\textnormal}{sc}$$ where ${\mathfrak}z$ is the Lie algebra of the center Z of G and $G\_{\textnormal}{sc}$ is the simply-connected cover of the derived subgroup of G. Standard facts about covering spaces imply that r lifts to a homomorphism ${{r}}\_{\textnormal}{univ}\colon G\_{\textnormal}{univ}\to\operatorname{Spin}(V)$. Restricting ${{r}}\_{\textnormal}{univ}$ to the kernels of the projections yields the following commutative diagram, which is essentially equivalent to : $$\begin{tikzcd} 1 \rar & \pi\_1(G) \rar\dar{e\_{{r}}} & G\_{\textnormal}{univ} \rar\dar{{{r}}\_{\textnormal}{univ}} & G \rar\dar{{{r}}} & 1 \\ 1 \rar & \{\pm1\} \rar & \operatorname{Spin}(V) \rar & \operatorname{SO}(V) \rar & 1. \end{tikzcd}$$ Then r lifts to the spin group if and only if the character er: π1(G) → { ± 1}, which we call the spin character of r, is trivial. Our goal in this section is to give a formula for the spin character of a representation in terms of its weights. We start in with a criterion for r to lift to the spin group. From this criterion we then deduce, in, a formula for the spin character. All that is new here is our exposition: this calculation forms part of the canon of the representation theory of compact Lie groups. With that said, there is a small discrepancy between our answer and Bourbaki’s, which I believe to be an error on Bourbaki’s part. This discrepancy is discussed in. Given a representation (r, V) of G and a maximal torus T, let Φ(V) ⊆ X\(T) denote the set of weights of T* on V. Lifting criterion ----------------- The goal of this subsection is to describe which orthogonal representations of G lift to the spin group. We start with the essential case where G = T is a torus; the general case reduces easily to this one. Let S be a maximal torus of SO(V) and let $\widetilde S\subseteq\operatorname{Spin}(V)$ be its preimage in the spin group, again a maximal torus. Any homomorphism T → SO(V) can be conjugated to take values in S, after which point the lift to the spin group, if it exists, factors through $\widetilde S$. Passing to character lattices, the lifting question reduces to the algebra question of whether the homomorphism X\(S) → X\(T) can be extended to the group $X^\(\widetilde S)$, which contains X\(S) as an index-two subgroup because $\widetilde S\to S$ is a double-cover. & S T & S & X\(S) X\(T) & X\(S) To solve this extension problem, we need to understand the double cover $\widetilde S\to S$ as well as the relationship between the homomorphism T* → S and the weights of the original orthogonal representation. Let n be the rank of Spin(V), so that dimV = 2n or 2n + 1. Fix a set I of cardinality dimV equipped with an involution i ↦ − i that fixes no element of I when dimV is even and exactly one element of I, which we denote by 0, when dimV is odd. First, the double cover. Choose an I-indexed basis e = (ei)i ∈ I of V for which ⟨ei, ej⟩ = [i = − j];[3](#fn3) we call such a basis a Witt basis. Let Se denote the group of $s=(s\_i)\_{i\in I}\in{\mathbb G}\_{{\textnormal}m}^I$ such that si ⋅ s− i = 1 and such that, when dimV is odd, s0 = 1. The rank-n torus Se acts on V by $$s\cdot e\_i {\stackrel{{\textnormal}{def}}{=}}s\_ie\_i,$$ realizing Se as a subgroup of SO(V). Evidently Se is a maximal torus, and every other maximal torus of SO(V) arises from a Witt basis e by this construction. Passing to the character lattice, describing a basis of X\(S*e) requires a choice of gauge p: I \ {0} → { ± 1}, that is, a negation-equivariant function. Given p, say i > 0 if p(i) = + 1 and i < 0 if p(i) = − 1, for i ∈ I. The characters fi: s ↦ si for i > 0 give a basis for X\(S). We can use the Clifford-algebra description of Spin(V), or even easier, the Bourbaki root-system tables, to work out the character lattice of $X^\(\widetilde S)$. Taking fp = (fi)i > 0 as a basis for $X^\(S)\_{\mathbb Q}$, the character lattice of $X^\(\widetilde S)$ is the set of elements of $\tfrac12 X^\(S)$ whose coordinates are all integers or all half-integers. In particular, $X^\(\widetilde S)$ is generated by X\(S) and the vector $\tilde f\_p = \tfrac12\sum\_{i>0} f\_i$. It follows that for A* an abelian group, a homomorphism X\(S) → A* extends to a homomorphism $X^\(\widetilde S)\to A$ if and only if, letting a*i denote the image of fi, the sum ∑i > 0ai lies in 2A. If this property is satisfied then $\tilde f\_p$ can map to any element whose double is ∑i > 0ai. When A is 2-torsionfree there is at most one such element, hence at most one extension. Consequently, $\widetilde S$ can be described as the quotient S/B where B is the group of (ɛi)i ∈ I ∈ { ± 1}I with ɛi = ɛ− i, ɛ0 = 1, and ∏i > 0ɛi = 1. In this description the twofold cover $\widetilde S\to S$ is induced by the squaring map on S. Next, the weights. Since V is an orthogonal representation of T, its set of weights is negation-invariant. Let p: Φ(V) \ {0} → { ± 1} be a gauge. Using the gauge we can write down an orthogonal decomposition V = V0 ⊕ ⨁α > 0(Vα ⊕ V− α) in which Vα is the orthogonal complement of V− α when α ≠ 0. At this point the gauge is only a notational convenience since the summands in the decomposition do not depend on it. Choose a Witt basis (ei)i ∈ I for V consisting of weight vectors of S and let αi denote the weight of ei. If dimV is odd then e0 is of weight 0. The map X\(S) → X\(T) dual to the homomorphism T → S sends the basis vector fi to the character αi. These two analyses combine to a criterion for the map T → S to lift to $\widetilde S$. Choose a gauge p on X\(T) and define the element $$\label{thm8} \rho\_{{r}}{\stackrel{{\textnormal}{def}}{=}}\frac12\sum\_{\alpha>0} (\dim V\_\alpha)\alpha \in \frac12 X^\(T).$$ Although ρr depends on the choice of gauge, we will only ever use it in a way that is independent of the choice of gauge. Now our criterion is this: the representation r lifts to the spin group if and only if ρr ∈ X\(T). When G* is no longer abelian, we can reduce the spin-lifting problem to the abelian case using the observation that all the obstructions to lifting lie in a maximal torus. [thm9] Let f: G → H be a homomorphism of complex reductive groups, T ⊆ G a split maximal torus of G, and $\widetilde H\to H$ an isogeny. Then f lifts to $\widetilde H$ if and only if f∣T lifts to $\widetilde H$. The is true over much more general bases than the complex numbers. Counter to the spirit of SGA 3, we prove it using the analytic topology on G. Isogenies are covering spaces. This claim follows from the lifting criterion for covering spaces together with the surjectivity of the map π1(T) → π1(G). Combining with our analysis of the abelian case completely solves the problem of lifting an orthogonal representation to the spin group. [thm12] Let G be a reductive group, T ⊆ G a maximal torus, and (r, V) an orthogonal representation. Then r lifts to the spin group if and only if ρr ∈ X\(T). Comparison with Bourbaki ------------------------ Our differs from at least one work in the canon of Lie groups, Chapter 9 of Bourbaki’s Groupes et algèbres de Lie. I believe there is a small error in Bourbaki’s account of the lifting criterion. In light of the famous scrupulousness with which the Bourbaki group prepared their treatises, a few words are in order to explain the discrepancy. Chapter 9 of Bourbaki’s book studies compact connected Lie groups. Although this setting is different from ours, the algebraic setting, there is a standard dictionary between the compact and algebraic settings, and this dictionary gives a comparison between our work and Bourbaki’s. Exercise 7a of Section 7 of concerns the spin lifting question. The difference between Bourbaki’s answer and ours, the quantity ρ*r defined in, is that we take into account the multiplicity of the weights, in other words, the dimensions of the weight spaces, while Bourbaki’s analogue of ρr, $$\frac12\sum\_{0<\alpha\in\Phi(V)} \alpha,$$ does not weight the sum by multiplicity. In this subsection we give an example explaining why Bourbaki’s criterion is incorrect. Our theory predicts that the double of an orthogonal representation lifts to the spin group. Here is an independent proof of this prediction. It shows that Bourbaki’s exercise cannot be correct, as we explain after the proof. [thm10] Let r: G → SO(V) be an orthogonal representation of a reductive group G. Then r ⊕ r lifts to the spin group. It suffices to prove the claim in the case where r is the tautological (identity) orthogonal representation of G = SO(V). Consider the following commutative diagram, in which the left horizontal arrows are diagonal inclusions, the right horizontal arrows are multiplication maps, and the vertical arrows are the canonical projection from the spin group to the special orthogonal group. (V) & (V0)(0V) & (VV) (V) & (V0)(0V) & (VV). The claim amounts to showing that the bottom horizontal composite arrow lifts to Spin(V ⊕ V). Recall that the kernel of Spin(V) → SO(V) is negation in the Clifford algebra, which we denote by − 1. The claim is equivalent to the statement that − 1 lies in the kernel of the horizontal top composite arrow. And this statement follows from the fact that for any isometric embedding W → V of quadratic spaces, the induced map Spin(W) → Spin(V) sends the center to the center. This fact is a consequence of the Clifford-algebra definition of the spin group: the induced spin-group map is the restriction of the induced Clifford algebra map Cl(W) → Cl(V), and this map is the identity on the copy of ${\mathbb C}$ inside both algebras. It is clear that the tautological orthogonal representation of SO(V) does not lift to the spin group: otherwise, the spin cover of SO(V) would split and Spin(V) would be disconnected. One the other hand, since Φ(V) = Φ(V ⊕ V) and Bourbaki’s criterion is sensitive only to the set of weights, that criterion would imply, along with, that the tautological representation does lift. Spin character -------------- In this subsection we build on our work from to give a formula for the spin character of an orthogonal representation r: G → SO(V). First, let’s review the construction of the universal covering projection of a complex torus T. The universal cover of T can be identified with the Lie algebra ${\mathfrak}t$ and the universal covering map ${\mathfrak}t\to T$ is then the exponential map from the theory of Lie groups. The universal cover is functorial because formation of Lie algebras is functorial: the morphism on universal covers induced by a morphism of tori is the differential of the morphism on the Lie algebras. For our purposes it is more useful, however, to describe the universal cover ${\mathfrak}t$ using cocharacter lattices. The evaluation map $X\_\(T)\otimes{\mathbb C}^\times\to T$ is an isomorphism, forming the tensor product over ${\mathbb Z}$. It is conventional to use exponential notation for these tensors, writing a*λ for λ ⊗ a. Similarly, evaluation of the derivative at 1 gives a canonical isomorphism $X\_\(T)\otimes{\mathbb C}\to{\mathfrak}t$. In these coordinates, the exponential cover ${\mathfrak}t\to T$ is simply the map $X\_\(T)\otimes{\mathbb C}\to X\_\(T)\otimes{\mathbb C}^\times$ induced by the exponential function λ* ⊗ a ↦ exp(2πia)λ. The universal cover map identifies its kernel, X\(T), with the fundamental group of T. More concretely, the identification X\(T) ≃ π1(T) is restriction of cocharacters to the unit circle of ${\mathbb C}^\times$. In this tensor product model, functoriality of the universal cover follows from functoriality of the cocharacter lattice: a homomorphism f: T → S of tori induces a map f\: X\(T) → X\(S), and the induced map $f\_{\textnormal}{univ}\colon T\_{\textnormal}{univ}\to S\_{\textnormal}{univ}$ is simply $$f\_{\textnormal}{univ}\colon \lambda\otimes a\mapsto f\_\(\lambda)\otimes a.$$ Of particular importance are the isogenies f: T → S, for which $f\_{\textnormal}{univ}$ relates two different descriptions of the same universal cover. We first work out the character er in the case where G = T is a torus; the general case follows immediately from this special case. Choose a gauge on X\(T) \ {0} so that we may speak of positive and negative characters. Retain the notation from, so that I* indexes a Witt basis (ei)i ∈ I of V of weight vectors, S is the maximal torus of SO(V) corresponding to the basis, and $\widetilde S$ is its double cover in Spin(V). Here $X\_\(S)=\ker({\mathbb Z}^I\to{\mathbb Z}^{I/\pm})$, a basis for X\(S) is (fi − f− i)i > 0 where (fi)∈ I is the standard basis of ${\mathbb Z}^I$, and $X\_\(\widetilde S)$ is the set of elements of $\tfrac12 X\_\(S)$ whose fi-coefficients are either all integers or all half-integers. Now consider the following diagram, where Sʹ = S and the dashed arrow is squaring. Tuniv &&& T Suniv & S’ & S & S The map X\(Sʹ) = X\(S) → X\(S) induced by squaring is multiplication by 2. Therefore, under the identification $S\_{\textnormal}{univ}=X\_\(S)\otimes{\mathbb C}$, the map $S\_{\textnormal}{univ}\to X\_\(S')\otimes{\mathbb C}^\times=S'$ is λ* ⊗ a ↦ exp(2πia/2)λ λ ∈ X\(S). At the same time, our work in shows that the map r\: X\(T) → X\(S) is r\: λ* ↦ ∑i > 0⟨λ, αi⟩(fi − f− i). As α− i = αi− 1, it follows that the map $X\_\(T)\to\widetilde S$ is given by the formula $$\lambda\mapsto \bigl(\exp(\pi i\langle\lambda, \alpha\_i\rangle)\bigr)\_{i\in I},$$ where the target element is interpreted as a coset in S* following the discussion in. Belying the notational complexity of this formula, every component of the tuple is ± 1. We know that the image in $\widetilde S$ of this tuple lies in the center of Spin(V), an order-two subgroup of $\widetilde S$. To identify the image as + 1 or − 1, we take the product of the elements of the tuple with positive index i. The final formula, therefore, is $$e\_{{r}}(\lambda) = \prod\_{i>0} \exp\bigl(\pi i\langle\lambda, \alpha\_i\rangle\bigr) = \exp\bigl(\pi i\langle\lambda,2\rho\_{{r}}\rangle\bigr) = (-1)^{\langle\lambda,2\rho\_{{r}}\rangle}, \qquad \lambda\in X\_\(T)\simeq\pi\_1(T).$$ When G* is not a torus, we choose a split maximal torus T in G and use the fact that the inclusion i: T → G induces a surjection π1(T) → π1(G). A diagram chase shows that the spin character er ∘ i for the restriction of r to T factors through er. In this way we reduce to the case where G is a torus. [thm24] Let G be a complex reductive group, let T ⊆ G be a maximal torus, and let r: G → SO(V) be an orthogonal representation. The spin character er: π1(G) → { ± 1} induced by r is given by the formula er(λ) = ( − 1)⟨λ, 2ρr⟩, where λ ∈ X\(T) ≃ π1(G) and ρ*r is defined in. Central characters and Weil cohomology ====================================== To apply to prove, we need to interpret the image of the class φ\(c*G) under the map $$e\_{{{r}},\}: {\textnormal}H^2(W\_k,\pi\_1(\widehat G)) \to {\textnormal}H^2(W\_k,\{\pm1\}).$$ It turns out that φ\(cG) corresponds to Langlands’s central character χφ – conjecturally, the central character of the L-packet of φ – and that the map er, \* corresponds to evaluation of the character on a certain involution zr. The goal of this section is to justify and explain these claims, and ultimately, in, to show that er, \*φ\(c*G) = χφ(zr). Let Z be the center of the quasi-split reductive group G. The main difficulty is to interpret the cohomology group $${\textnormal}H^2(W\_k,X^\(Z)).$$ After reviewing the general setting in which the cohomology of the Weil group is computed, in, we show in that when Z* is connected, this group is the character group of the Harish-Chandra subgroup Z(k)1 of Z(k). When chark = 0 and k is nonarchimedean this identification exists even for Z disconnected, as we explain in. Although the center of G need not be connected in general, Langlands’s definition of χφ permits us to reduce, in, to the connected-center case. We conclude in by defining the involution zr and proving. In what follows, we use continuous cohomology for the Weil group of a nonarchimedean field and Borel cohomology for the Weil group of an archimedean field. The difference is sometimes elided in the notation to avoid overburdening the reader. Cohomology of the Weil group ---------------------------- What kind of group cohomology should we use to study the Weil group? Over an archimedean field the right answer is Borel cohomology, as shows. Over a nonarchimedean field, one[4](#fn4) right answer is continuous cohomology. Some care is required, however, because the Weil group is not profinite, only locally profinite. For profinite groups we can easily reduce most foundational problems to the setting of finite groups, where topology is irrelevant, but this is no longer the case for locally profinite groups. Fortunately, Flach has written a nice article that addresses technical concerns in the continuous cohomology of the Weil group. Flach first situates this cohomology in a general topos-theoretic setting, using theory from SGA 4, and then shows that this general theory recovers the usual definition of continuous cohomology by continuous cochains. One particularly useful consequence of his work is the existence of the usual long exact sequence for any short exact sequence of topological modules whose quotient map locally admits continuous sections. Another useful consequence is the following. [thm23] Let V be a discrete Wk-module whose underlying abelian group is uniquely divisible, in other words, a ${\mathbb Q}$-vector space. 1. If k is nonarchimedean then ${\textnormal}H^i(W\_k,V) = 0$ for i ≥ 2. 2. If k is archimedean then ${\textnormal}H\_{\textnormal}{Borel}^i(W\_k,V) = 0$ for i odd. First, assume k is nonarchimedean. There is a Hochschild-Serre spectral sequence $${\textnormal}H^i({\mathbb Z},{\textnormal}H^j(I\_k,V)) \implies {\textnormal}H^{i+j}(W\_k,V)$$ coming from the short exact sequence $1 \to I\_k \to W\_k \to {\mathbb Z}\to 1$. This spectral sequence is an instance of a more general spectral sequence that Flach constructed. In the more general sequence, the group ${\textnormal}H^j(I\_k,V)$ is replaced by a sheaf which may or may not be representable. But since V is discrete and Wk is locally profinite, this sheaf is representable and there are no technical problems. Since V is uniquely divisible and Ik is profinite, ${\textnormal}H^j(I\_k,V) = 0$ for j ≥ 1. Since ${\mathbb Z}$ has cohomological dimension one, ${\textnormal}H^i({\mathbb Z},-)$ vanishes for i ≥ 2. So all entries on the starting page of the spectral sequence vanish except possibly those in positions (0, 0) and (1, 0). Next, suppose k is archimedean. If $k={\mathbb C}$ then $W\_{\mathbb C}={\mathbb C}^\times= S^1\times{\mathbb R}\_{>0}$ has classifying space the infinite-dimensional complex projective space. Its integral, hence rational, cohomology is known to be concentrated in even degrees. If $k={\mathbb R}$ then use the Hochschild-Serre spectral sequence in Borel cohomology for the short exact sequence $1\to{\mathbb C}^\times\to W\_{\mathbb R}\to\Gamma\!\_{\mathbb R}\to1$ together with the vanishing of ${\textnormal}H^{\geq1}(\Gamma\!\_{\mathbb R},-)$ on uniquely divisible groups to show that the map ${\textnormal}H\_{\textnormal}{Borel}^i(W\_{\mathbb R},V) \to{\textnormal}H\_{\textnormal}{Borel}^i({\mathbb C}^\times,V)^{\Gamma\!\_{\mathbb R}}$ is an isomorphism for i ≥ 2. Harish-Chandra subgroup ----------------------- Recall that G is a quasi-split reductive k-group. Define the pairing $G(k)\otimes \operatorname{Hom}\_{\textnormal}{$k$-gp}(G,{\mathbb G}\_{{\textnormal}m})\to{\mathbb R}$ by ⟨g, α⟩ = ordk(α(g)), where $\operatorname{ord}\_k\colon k^\times\to{\mathbb R}$ is the absolute value when k is archimedean and the discrete valuation when k is nonarchimedean. Currying yields a map $$\theta\_G: G(k)\to \operatorname{Hom}(\operatorname{Hom}\_{\textnormal}{$k$-gp}(G,{\mathbb G}\_{{\textnormal}m}),{\mathbb R}).$$ In particular, if G = T is a torus then the target of θG is $X\_\(T)^{\Gamma\!\_k}\_{\mathbb R}$. A character of G(k) is unramified* if it is inflated along θG. Let G be a k-group. The Harish-Chandra subgroup G(k)1 of G(k) is the kernel of the homomorphism θG. For example, if G is finite then G(k)1 = G(k) because roots of unity have valuation zero. If G = T is a torus then the group T(k)1 is compact, hence profinite. The Harish-Chandra subgroup is relevant because its character group is exactly the cohomology group of interest to us, as we now explain. [thm30] Let T be a k-torus. The image of the composite map (taken in Borel cohomology if k is archimedean) H1(Wk,X\(T)C) & H1(Wk,T) & cts(T(k),C), in which the second map is the local Langlands correspondence for tori, is the group of unramified characters of T(k). When $T={\mathbb G}\_{{\textnormal}m}$ the claim is clear, and Shapiro’s lemma then proves the claim for any induced torus. In general, embed T* in an induced torus S. Using the diagram H1(Wk,X\(S)C) & H1(Wk,S) & cts(S(k),C) H1(Wk,X\(T)C) & H1(Wk,T) & cts(T(k),C) and the fact that the left vertical arrow is a surjection by, it suffices to show that the unramified characters of T(k) are precisely the restrictions of the unramified characters of S(k). Clearly restriction preserves unramification, and every unramified character of T(k) extends to an unramified character of S(k) because X\(T)Γ k* is a summand of X\(S)Γ k. The universal cover exact sequence for $\widehat T$ expands, by, to the exact sequence H1(Wk,X\(T)C) & H1(Wk,T) & H2(Wk,X\(T)) & 0. At the same time, there is a canonical identification $$\operatorname{coker}\bigl(\operatorname{Hom}(X\_\(T)\_{\mathbb R}^{\Gamma\!\_k},{\mathbb C}^\times) \to\operatorname{Hom}\_{\textnormal}{cts}(T(k),{\mathbb C}^\times)\bigr) \simeq \operatorname{Hom}\_{\textnormal}{cts}(T(k)^{1},{\mathbb C}^\times).$$ Combining these two facts yields a cohomological description of the character group of T(k)1. Let T be a k-torus. The local Langlands correspondence and universal cover coboundary furnish an isomorphism (in Borel cohomology if k is archimedean) $${\textnormal}H^2(W\_k,X^\(T)) \simeq \operatorname{Hom}\_{\textnormal}{cts}(T(k)^{1},{\mathbb C}^\times).$$ Evaluation at an involution --------------------------- Let T* be a k-torus. Every Galois-equivariant homomorphism X\(T) → { ± 1} induces a map $${\textnormal}H^2(W\_k,X^\(T)) \to {\textnormal}H^2(W\_k,\{\pm1\}).$$ The source is the character group of T(k)1 and the target is a group of order two. It follows by Pontryagin duality that this map is evaluation of characters at an involution. The goal of this subsection is to show that when $T={\mathbb G}\_{{\textnormal}m}$ and the map $X^\(T)={\mathbb Z}\to\{\pm1\}$ is nontrivial, this involution is nontrivial. Although the claim is quite weak, it is all that is needed for our computation of e*r, \*φ\(c*G). When k is nonarchimedean of characteristic zero we can describe ${\textnormal}H^2(W\_k,X^\(T))$ using the cup-product pairing of Tate duality, as in, and the claim follows immediately from the naturality of that pairing. But in general there is no such naturality statement compatible with our coboundary description of ${\textnormal}H^2(W\_k,X^\(T))$. Instead, we prove nontriviality using an argument with long exact sequences that rests on the following computation. [thm27] Let A be a Wk-module which is finitely generated as an abelian group. 1. If k is archimedean then ${\textnormal}H\_{\textnormal}{Borel}^3(W\_k,A)$ is torsionfree. 2. If k is nonarchimedean then ${\textnormal}H^3(W\_k,A)$ is a p-group. A (possibly infinite) group is a *p-group* if each of its elements has order a power of p. When k is archimedean, we can argue as in the proof of. For $k={\mathbb C}$ we conclude that ${\textnormal}H\_{\textnormal}{Borel}^3(W\_{\mathbb C},A)=0$ and for $k={\mathbb R}$ we conclude, using the calculation on page 310 of, that ${\textnormal}H\_{\textnormal}{Borel}^3(W\_{\mathbb R},A)$ is a free ${\mathbb Z}$-module of finite type. Next, assume k is nonarchimedean. When chark = 0, Karpuk proved the stronger statement that ${\textnormal}H^i(W\_k,A) = 0$ for all i ≥ 3. A slight modification of his argument, which we outline below, proves the. The Hochschild-Serre spectral sequence, mentioned in the proof of, gives the vanishing for i ≥ 4 and can be used to show, for i = 3, that ${\textnormal}H^2(W\_k,P) \simeq {\textnormal}H^3(W\_k,M)$, where $P{\stackrel{{\textnormal}{def}}{=}}M\_{\mathbb Q}/M$. Let P[n] denote the n-torsion subgroup of P. The cohomology of Wk in P can be computed as a direct limit of the cohomologies in the torsion subgroups: $${\textnormal}H^2(W\_k,P) \simeq \varinjlim\_n {\textnormal}H^2(W\_k,P[n]).$$ Since P[n] decomposes as a direct sum of its Sylow subgroups, we can rewrite the direct limit as $${\textnormal}H^2(W\_k,P) \simeq {\textnormal}H^2(W\_k,P[p^\infty]) \oplus \varinjlim\_{p\nmid n} {\textnormal}H^2(W\_k,P[n])$$ where $P[p^\infty]{\stackrel{{\textnormal}{def}}{=}}\bigcup\_{n\geq0} P[p^n]$. Since the group P[p∞] is a p-group, the cohomology group ${\textnormal}H^2(W\_k,P[p^\infty])$ is a p-group as well. It therefore suffices to show that the second summand above vanishes. By Tate duality (see ), if n is prime to p then ${\textnormal}H^2(W\_k,P[n])$ is dual to HomWk(P[n], μ) where μ is the group of roots of unity in $\bar k$. It follows that the second term above, $\varinjlim\_{p\nmid n} {\textnormal}H^2(W\_k,P[n])$, is dual to HomWk(P/P[p∞], μ). Since P[p∞] is a summand of P, it suffices to show that HomWk(P, μ) = 0. Karpuk has an argument for the vanishing that works just as well when chark > 0. together with a long exact sequence argument proves our desired nontriviality. [thm28] Let ℓ be a finite separable extension of k, let $M = \operatorname{Ind}\_{\ell/k}{\mathbb Z}$, and let f: M → { ± 1} be the unique nontrivial homomorphism. If chark ≠ 2 then the induced map $$f\_\: {\textnormal}H^2(W\_k,M) \to {\textnormal}H^2(W\_k,\{\pm1\})$$ (taken in Borel cohomology if k* is archimedean) is nontrivial. Since f factors through the augmentation map $M\to{\mathbb Z}$, it suffices to prove the in the case where ℓ = k and $M={\mathbb Z}$. The long exact sequence for the short exact sequence $1\to{\mathbb Z}{\xrightarrow}2{\mathbb Z}\to\{\pm1\}\to1$ yields a coboundary map $${\textnormal}H^2(W\_k,\{\pm1\})\to{\textnormal}H^3(W\_k,{\mathbb Z})$$ whose cokernel measures the failure of surjectivity. But by the coboundary map must be trivial: there are no nontrivial homomorphisms between { ± 1} and a torsionfree group or a p-group with p ≠ 2. Involutions from representations of -groups ------------------------------------------- In this subsection we explain how to construct from an orthogonal representation r of LG a central involution zr ∈ Z(k), generalizing Gross and Reeder’s canonical involution. We then relate this involution to our description of the character group of Z(k)1. The involution zr depends only on the restriction of r to $\widehat G$. This restriction is an algebraic representation of a complex reductive group and can thus be understood through its weights. We compute the weights of the representation with respect to a fixed, Galois-stable maximal torus $\widehat T$ of $\widehat G$, which is dual to a minimal Levi T of the quasi-split group G. Since ${{r}}\|\_{\widehat G}$ is the restriction of a representation of the L-group, the multiset of its weights is Galois-stable. Moreover, each weight can be interpreted as a coweight of T. Let $m\colon X\_\(T)\to{\mathbb N}$ be a Galois-invariant multiset of weights. In our application m* will be the multiset of weights of a representation of LG, but the greater generality is convenient for some proofs. As in, choose a gauge $X^\(\widehat T)\setminus\{0\} \to\{\pm1\}$. This time we require the gauge to be Galois-invariant, and the requirement can be met because the Galois action preserves some pinning containing $\widehat T$. Define $$z\_m {\stackrel{{\textnormal}{def}}{=}}\sum\_{0<\varpi\in X\_\(T)} \varpi(-1)^{m(\varpi)}.$$ The involution zm is independent of the choice of gauge. For r a representation of LG we set $z\_{{r}}{\stackrel{{\textnormal}{def}}{=}}z\_m$ where $m(\varpi)$ is the multiplicity of $\varpi$ in ${{r}}\|\_{\widehat T}$. In this setting, the generalization of the spin character of is the character $e\_m\colon\pi\_1(\widehat G)\simeq X^\(T)\to\{\pm1\}$ defined by the formula $$e\_m(\lambda) = \prod\_{0<\varpi\in X\_\(T)} (-1)^{m(\varpi)\langle\lambda,\varpi\rangle}.$$ [thm25] Let $m\colon X\_\(T)\to{\mathbb N}$ be a Galois-invariant multiset of weights. If G* has connected center then the composite map cts(Z(k)1,C) & H2(Wk,1(G)) & H2(Wk,{1}) & {1} (taken in Borel cohomology if k is archimedean) is evaluation at zm. By additivity it suffices to consider the case where m is a (multiplicity-one) Galois orbit Ω. After choosing a gauge, we can factor em: X\(T) → { ± 1} as the composition X\(T) & Z[] & {1} of the map $$\lambda\mapsto\sum\_{\varpi\in\Omega} \langle\lambda,\varpi\rangle[\varpi]$$ and the mod-two augmentation map ${\mathbb Z}[\Omega]\to\{\pm1\}$. Apply the functor ${\textnormal}H^2(W\_k,-)$ to this composition. By naturality of the coboundary, the image of the first map is isomorphic to the restriction map $$\operatorname{Hom}\_{\textnormal}{cts}(T(k)^{1},{\mathbb C}^\times) \to \operatorname{Hom}\_{\textnormal}{cts}(S(k)^{1},{\mathbb C}^\times)$$ where S is the k-torus with character group ${\mathbb Z}[\Omega]$. The image of the second map is a homomorphism $$\operatorname{Hom}\_{\textnormal}{cts}(S(k)^{1},{\mathbb C}^\times) \to\{\pm1\}$$ which is nontrivial by. And by Pontryagin duality this map must be evaluation at an element of order two. Since S(k) ≃ ℓ× where ℓ is the fixed field of any element of Ω, the group S(k) has a unique element of order two, namely − 1. The follows. Let χφ denote Langlands’s central character, the construction of which is summarized in Borel’s Corvallis article, [thm29] Let φ: Wk → LG be an L-parameter, let ${{r}}\colon {}^LG\to{\textnormal}O(V)$ be a complex representation of LG, and let $c\_G\in{\textnormal}H\_{\textnormal}{Borel}^2({}^LG,\pi\_1(\widehat G))$ classify the extension $\widehat G\_{\textnormal}{univ}\rtimes W\_k$. Then sgn(er, \*φ\(c*G)) = χφ(zr). First, assume the center of G is connected. Let $\varphi\_0\colon W\_k\to\widehat G$ denote the 1-cocycle corresponding to φ, so that φ(w) = φ0(w)w. The morphism of short exact sequences 1 & 1(G) & Guniv & G & 1 1 & X\(Z) & & Z & 1 gives rise to a commutative square H1(Wk,G) & H1(Wk,Z) H2(Wk,1(G)) & H2(Wk, X\(Z)). Pass φ0 around this square. By, the left arrow maps φ0 to φ\(c*G). The top arrow maps φ0 to the cocycle representing Langlands’s character $\chi\_\varphi\colon Z(k)\to{\mathbb C}^\times$, and the image of this cocycle under the right arrow corresponds to the restriction of χφ to Z(k)1. Hence this restriction corresponds to φ\(c*G). The now follows from. For a general G, whose center need not be connected, Langlands’s construction of χφ starts with a choice of embedding of G into a group G1 with connected center. Let Z1 be the center of G1. It turns out that every L-parameter φ: Wk → LG lifts to an L-parameter φ1: Wk → LG1. One then defines χφ as the restriction to Z(k) of the character χφ1 of Z1(k). The resulting character χφ is independent of the choice of lift and the choice of G1. The projection $\widehat G\_1\to\widehat G$ induces the diagram [row sep=small] & 1(G1) & G1,univWk & LG1 & & (c1 cG1) {1} & & & & Wk & 1(G) & GunivWk & LG & & (cG) with exact rows. Let $c\_1\in{\textnormal}H^2({}^LG\_1,\pi\_1(G\_1))$ classify the upper extension. On the one hand, the existence of g implies that h\(c*G) = f\(c1), from which we conclude that e*r, \*φ\(c*G) = er, \*φ1\* h\(c*G) = er, \*φ1\* f\(c1) = e*r ∘ h, \*φ1\(c1). On the other hand, its easy to see from the definition of z*r that χφ(zr) = χφ1(zr ∘ h). These calculations reduce the general case to the previous one. Synthesis ========= In this section we complete the proof of and explain its relationship to conjectures of Hiraga, Ichino, and Ikeda. Proof of Theorem A ------------------ Let G be a quasi-split reductive k-group, let φ: WDk → LG be a tempered L-parameter, and let ${{r}}\colon{}^LG\to{\textnormal}O(V)$ be a complex orthogonal representation. Since root numbers of orthogonal Weil-Deligne representations are unaffected by semisimplification, we replace φ by its restriction φ: Wk → LG to the Weil group. The natural action of the Weil group on $\widehat G$ lifts to an action on its universal cover. Using this action, apply to the diagram 1 & 1(G) & Guniv & G & 1 1 & {1} & (V) & O(V) & 1 & (cpin), taking W = Wk. Here $c\_{\textnormal}{pin}\in{\textnormal}H\_{\textnormal}{Borel}^2({\textnormal}O(V),\{\pm1\})$ classifies the (bottom) pin extension and $c\_G\in{\textnormal}H\_{\textnormal}{Borel}^2({}^LG,\pi\_1(\widehat G))$ classifies the extension $\widehat G\_{\textnormal}{univ}\rtimes W\_k$. We conclude from the that $$\label{thm20} {{r}}^\(c\_{\textnormal}{pin}) = e\_{{{r}},\}(c\_G)\cdot p^\{{r}}\|\_{W\_k}^\(c\_{\textnormal}{pin})$$ where p: LG → W is the projection. Hence $$\varphi^\{{r}}^\(c\_{\textnormal}{pin}) = \varphi^\e\_{{{r}},\}(c\_G)\cdot \varphi^\p^\{{r}}\|\_{W\_k}^\(c\_{\textnormal}{pin}),$$ valued in ${\textnormal}H^2(W\_k,\{\pm1\})$ for k* nonarchimedean and in ${\textnormal}H\_{\textnormal}{Borel}^2(W\_k,\{\pm1\})$ for k archimedean. By our formulation of Deligne’s theorem,, $$\frac{\omega(\varphi,{{r}})}{\omega(\varphi,\det{{r}})} = \operatorname{sgn}\bigl(\varphi^\{{r}}^\(c\_{\textnormal}{pin})\bigr).$$ At the same time, letting $\varphi\_{\textnormal}{prin}\colon\operatorname{WD}\_k\to{}^LG$ denote the principal parameter, the composition r∣Wk ∘ p ∘ φ is the restriction of ${{r}}\circ\varphi\_{\textnormal}{prin}$ to the Weil group. Deligne’s theorem again implies that $$\frac{\omega(\varphi\_{\textnormal}{prin},{{r}})}{\omega(\varphi\_{\textnormal}{prin},\det{{r}})} = \operatorname{sgn}\bigl(\varphi^\p^\{{r}}\|\_{W\_k}^\(c\_{\textnormal}{pin})\bigr).$$ Since $\widehat G$ is connected and r* is orthogonal, detr restricts trivially to $\widehat G$. Hence $$\omega(\varphi\_{\textnormal}{prin},\det{{r}}) = \omega(\varphi,\det{{r}}).$$ All in all, then, simplifies to $$\frac{\omega(\varphi,{{r}})}{\omega(\varphi\_{\textnormal}{prin},{{r}})} = \operatorname{sgn}\bigl(\varphi^\e\_{{{r}},\}(c\_G)\bigr).$$ Finally, identifies the righthand side with χφ(zr). Application to Plancherel measure --------------------------------- Let G be a reductive k-group and let M be a Levi subgroup of G. From this data we can form an orthogonal representation rM of LM which one might call the relative adjoint representation for G ⊇ M, namely, the representation $$\operatorname{Lie}(\widehat G)/\operatorname{Lie}(Z(\widehat M)^{\Gamma\!\_k}).$$ The nonzero weights of this representation on $\widehat M$ are the elements of the root system of $\widehat G$. Now let φ: WDk → LM be a tempered discrete L-parameter for M and π a representation of G(k) in the L-packet of φ. Hiraga, Ichino, and Ikeda predicted a relationship between the gamma factor of the Weil-Deligne representation rM ∘ φ and the Plancherel measure on the part of the tempered dual of G(k) coming from parabolic inductions of unramified twists of π. largely computes the root number of this Weil-Deligne representation: $$\omega(\varphi,r\_M) = \omega(\varphi\_{\textnormal}{prin},r\_M) \cdot \pi(z\_{\textnormal}{Ad})$$ where $z\_{\textnormal}{Ad}$ is Gross and Reeder’s canonical involution for G (not M). Characters via Weil-Tate duality ================================ Let T be a finite type k-group of multiplicative type. Our description of ${\textnormal}H^2(W\_k,X^\(T))$ in when T* is a torus brings to mind a related and more classical result, Tate duality, which in this case describes ${\textnormal}H^2(\Gamma\!\_k,X^\(T))$ as the character group of the profinite completion $T(k)\_{\textnormal}{pro}$ of T(k). This conclusion holds more generally whenever T* is reduced. By analogy, one would hope that our description of ${\textnormal}H^2(W\_k,X^\(T))$ from as the character group of T(k)1 could be strengthened by relaxing the hypothesis that T* is a torus to the hypothesis that T is reduced. In this appendix we provide partial evidence, in, for this hope by giving such a description in two cases, both for nonarchimedean k: when chark is arbitrary and T is finite and reduced, and when chark = 0 and T is arbitrary. Our starting point is Karpuk’s thesis, which constructs a cup-product pairing generalizing Tate duality to the Weil group, provided that chark = 0. After explaining how this works, we will have two identifications of ${\textnormal}H^2(W\_k,X^\(T))$ with a character group when T* is a torus, one by coboundary and one by cup product. We then show that the two identifications are inverses of each other. To compare the Harish-Chandra subgroup with Karpuk’s formulation of Weil-Tate duality, we need a small lemma on (metric) completions. Let $\ell=\breve k$ be the completion of the maximal unramified extension of k and let $\bar\ell$ be a separable closure of ℓ. By Krasner’s lemma, Γℓ = Ik. [thm15] An element $a\in\bar\ell$ is algebraic over k if and only if its Galois orbit is finite. The forward implication is clear. For the reverse implication, consider the polynomial $f(x)\in\bar\ell[x]$ whose set of roots is the Galois orbit of a. Since the coefficients of f are Galois-invariant, it suffices to show that k is the set of Galois-invariant elements of $\bar\ell$. This claim is a theorem of Tate which holds also when chark > 0. [thm14] Let T be a finite type k-group of multiplicative type. If chark = 0 or T is finite and reduced then the cup-product pairing $${\textnormal}H^2(W\_k,X^\(T))\otimes {\textnormal}H^0\bigl(W\_k,\operatorname{Hom}(X^\(T),{\mathcal}O\_{\bar\ell}^\times)\bigr) \to{\mathbb Q}/{\mathbb Z}$$ induces a canonical identification $${\textnormal}H^2(W\_k,X^\(T))\simeq \operatorname{Hom}\_{\textnormal}{cts}(T(k)^{1},{\mathbb C}^\times).$$ Here we interpret ${\mathbb Q}/{\mathbb Z}$ as a subgroup of ${\mathbb C}^\times$ via the exponential map t* ↦ exp(2πit). Let $\bar k$ be the separable closure of k in $\bar\ell$. First, assume chark = 0. Karpuk showed that the topological groups ${\textnormal}H^2(W\_k,X^\(T))$ and $\operatorname{Hom}(X^\(T),{\mathcal}O\_{\bar\ell}^\times)^{W\_k}$ are Pontryagin dual to each other. The split short exact sequence for the valuation on $\bar\ell^\times$ gives rise to an exact sequence describing Karpuk’s group as a kernel: 1 & (X\(T),O\|)Wk & (X\(T),\|)Wk & (X\(T),Z)Wk. Since the action of W*k on X\(T) has finite orbits, any W*k-equivariant homomorphism $X^\(T)\to\bar\ell^\times$ factors through $\bar k^\times$ by. Hence we may rewrite the sequence as 1 & (X\(T),O\|)Wk & T(k) & X\(T). So the lefthand group is the Harish-Chandra subgroup of T(k). When T* is finite and reduced, the fact that the inflation map $${\textnormal}H^2(\Gamma\!\_k,X^\(T)) \to {\textnormal}H^2(W\_k,X^\(T))$$ is an isomorphism reduces the problem to Tate duality for finite modules. Our next goal is to compare the description of with our earlier coboundary description when T is a torus. Tate duality is the intermediary in the comparison. Let $T(k)\_{\textnormal}{pro}$ denote the profinite completion of T(k). Since T(k)1 is compact, it is a subgroup of $T(k)\_{\textnormal}{pro}$. [thm17] Let T be a k-torus. The following square commutes, where the horizontal arrows are the cup-product pairings and the vertical arrows are restriction. H2(k,X\(T)) & cts(T(k)pro,C) H2(Wk,X\(T)) & cts(T(k)1,C). It suffices to observe that the subgroup inclusion $T(k)^{1}{\hookrightarrow}T(k)\_{\textnormal}{pro}$ can be identified with the map $${\textnormal}H^0(W\_k,\operatorname{Hom}(X^\(T),{\mathcal}O\_{\bar\ell}^\times)) \to{\textnormal}H^0(\Gamma\!\_k,\operatorname{Hom}(X^\(T),\bar\ell^\times))$$ induced by inclusion ${\mathcal}O\_{\bar\ell}^\times {\hookrightarrow}\bar\ell^\times$. identifies the righthand ${\textnormal}H^0$ group with T(k). Tate duality describes ${\textnormal}H^2(\Gamma\!\_k,X^\(T))$ as the character group of $T(k)\_{\textnormal}{pro}$. Moreover, the extension of Artin reciprocity to tori describes ${\textnormal}H^1(\Gamma\!\_k,\widehat T)$ as the same character group. The two comparisons are related by the coboundary map for the exponential exact sequence 1 & X\(T) & X\(T)C& T & 1. There is a twist, however: the composite isomorphism below is inversion. $$\operatorname{Hom}\_{\textnormal}{cts}(T(k)\_{\textnormal}{pro}) \simeq {\textnormal}H^1(\Gamma\!\_k,\widehat T) \to {\textnormal}H^2(\Gamma\!\_k,X^\(T)) \simeq \operatorname{Hom}\_{\textnormal}{cts}(T(k)\_{\textnormal}{pro})$$ [thm18] Let T be a k-torus. The following square anticommutes, where the top arrow is the (profinitely completed) Langlands correspondence, the bottom arrow is from, the left arrow is the exponential coboundary, and the right arrow is restriction. H1(Wk,T) & cts(T(k),C) H2(Wk,X\(T)) & cts(T(k)1,C). Consider the following extension of the given diagram in which the left square commutes by the compatibility of inflation and coboundary. H1(k,T) & H1(Wk,T) & cts(T(k),C) H2(k,X\(T)) & H2(Wk,X\(T)) & cts(T(k)1,C). By and the compatibility between Tate duality and the local Langlands correspondence, the right square becomes commutative after restriction along the upper left arrow. We’ll use this to deduce commutativity of the right square by a diagram chase. The bottom left arrow is surjective by and exactness of Pontryagin duality. Starting with an element $c\in{\textnormal}H^1(W\_k,\widehat T)$, move it clockwise around the left square, choosing a lift along the bottom left arrow. We thereby produce an element $c'\in{\textnormal}H^1(W\_k,\widehat T)$ in the image of ${\textnormal}H^1(\Gamma\!\_k,\widehat T)$. The elements c* and cʹ have the same image in ${\textnormal}H^2(W\_k,X^\(T))$, hence differ by an element in the kernel of the middle vertical arrow, or in other words, by an element in the image of ${\textnormal}H^1(W\_k,X^\(T)\_{\mathbb C})$. To prove commutativity of the right square, it therefore suffices to show that the following composition $f\_T\colon{\textnormal}H^1(W\_k,X^\(T)) \to\operatorname{Hom}\_{\textnormal}{cts}(T(k)^{1},{\mathbb C}^\times)$ vanishes: H1(Wk,X\(T)C) & H1(Wk,T) & cts(T(k),C) & cts(T(k)1,C). For this, we use the same proof strategy as in the proof of. When $T={\mathbb G}\_{{\textnormal}m}$, the image of the first map in $\operatorname{Hom}\_{\textnormal}{cts}(k^\times,{\mathbb C}^\times)$ is the group of unramified characters and the claim is clear. It follows from Shapiro’s lemma that fT vanishes for any torus T that is induced, in other words, a product of Weil restrictions of split tori. For a general torus T, choose an embedding $T{\hookrightarrow}S$ into an induced torus S and let R denote the cokernel, a third torus. This embedding yields the commutative square H1(Wk,X\(S)C) & cts(S(k)1,C) H1(Wk,X\(T)C) & cts(T(k)1,C). Since ${\textnormal}H^2(W\_k,X^\(R)\_{\mathbb C}) = 0$ by, the left arrow is surjective. We can thus deduce the vanishing of f*T from the vanishing of fS. --- 1. Our pushout is usually not isomorphic to the pushout in the category of groups. The category-theoretic pushout is the amalgamated free product.[↩](#fnref1) 2. Borel cohomology is named after Émile, not Armand. It is sometimes also called Moore cohomology.[↩](#fnref2) 3. Here $[{\mathcal}P]$ is the Iverson bracket popularized by Knuth : it equals 0 if the property ${\mathcal}P$ is false and 1 if ${\mathcal}P$ is true.[↩](#fnref3) 4. Lichtenbaum remarked that Borel and continuous cohomology agree in many cases, in particular, when the coefficient group is discrete and countable. We will not use this result, though it would slightly simplify the exposition.[↩](#fnref4) Orthogonal root numbers of tempered parameters ============================================== ### 5 July 2021 We show that an orthogonal root number of a tempered L-parameter φ decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a certain involution of Langlands’s central character for φ. The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil-Deligne representations arising in the work of Hiraga, Ichino, and Ikeda on the Plancherel measure. Introduction ============ To every representation of the absolute Galois group Γ k of a local field k, or more generally, of its Weil-Deligne group WDk, we can attach various local constants, the γ-, L-, and ɛ-factors, each a function of a complex parameter s. Of these, the ɛ-factor has the simplest description as a function of s: it is an exponential function, s ↦ a ⋅ bs − 1/2. Understanding this local constant thus amounts to understanding the base b and constant term a. The base of the ɛ-factor (or depending on one’s choice of terminology, its logarithm) is known as the Artin conductor, taken as 1 for k archimedean. This quantity measures the ramification of the representation. Although the Artin conductor has its own subtleties, especially in the presence of wild ramification, there are good formulas to compute it, formulas that make precise the sense in which the conductor measures ramification. The constant term a is known as the root number. We denote it here by ω, an additive function of Weil-Deligne representations. The choice of exponent s − 1/2 ensures that the root number is a complex number of modulus one, so we can refer to it informally as the sign of the ɛ-factor. This sign is much more subtle than the Artin conductor. Here is one example of its subtlety. The problem of computing a root number generalizes the problem of computing the sign of a Gauss sum. For quadratic Gauss sums, the sign is determined by a congruence condition on the modulus, a classical result of Gauss and the most difficult of the basic properties of these sums. For cubic Gauss sums, the situation is much more complicated: there is no congruence condition to describe the sign, and in fact, the sign is randomly and uniformly distributed on the unit circle. In other words, for as simple a class of Galois representations as the cubic characters, the root number exhibits great complexity. Fortunately, there is a special class of Galois representations whose root numbers are more amenable to computation: the orthogonal representations. After all, the orthogonal characters are precisely the quadratic characters. Using formal properties of root numbers, it is easy to see that the root number of an orthogonal representation is a fourth root of unity, and the square of the root number can be described in terms of the determinant of the representation. Ultimately, then, computing an orthogonal root number is a matter of distinguishing between two square roots. Deligne calculated the distinction by expressing the root number of an orthogonal representation in terms of the second Stiefel-Whitney class of the representation. Stiefel-Whitney classes are a notion from algebraic topology, but they can be reinterpreted purely in terms of group cohomology, as the pullback of a certain universal cohomology class. In this way, Deligne’s formula reduces computing the root number of an orthogonal representation to a problem in group cohomology. In the Langlands program we generalize the study of Weil-Deligne representations to the study of L-parameters for a quasi-split reductive k-group G. To extend the definition of the local constants to this setting we reduce to the case of the general linear group by composing an L-parameter WDk → LG with a complex representation r: LG → GL(V) and computing the local constants of the representation r ∘ φ. Deligne’s formula suggests the possibility of computing the root numbers $\omega(\varphi,{{r}}){\stackrel{{\textnormal}{def}}{=}}\omega({{r}}\circ\varphi)$ for orthogonal r, in other words, the orthogonal root numbers of φ. The hope is a formula for ω(φ, r) that incorporates information about the L-parameter φ and the complex orthogonal representation r. In a 2010 paper, the direct inspiration for this article, Gross and Reeder gave a conjectural formula for a particular class of orthogonal root numbers and proved the formula when G is split. Motivated in part by a conjecture of Hiraga, Ichino, and Ikeda on the formal degree of discrete series, they took the adjoint representation $\operatorname{Ad}\colon{}^LG\to\operatorname{GL}(\hat{{\mathfrak}g})$, an orthogonal representation, and set about computing $\omega(\varphi,{\textnormal}{Ad})$. Their conjectural answer has three factors. First, there is a recipe, due to Langlands, that constructs from the L-parameter φ a character χφ of the center of G. At the same time, we can conjecturally assign to φ a finite set Πφ of smooth irreducible representations of G(k). The set Πφ is called the *L-packet* of φ and the assignment φ ↦ Πφ is called the local Langlands correspondence. It is expected that all elements of Πφ have the same central character, and further, that this character is χφ. Second, Gross and Reeder evaluated this central character on a certain canonical involution $z\_{\textnormal}{Ad}$ in the center of G, defined as the value of the sum of the positive coroots on − 1. Third, when k is nonarchimedean the L-group admits a particular distinguished parameter called the principal parameter $\varphi\_{\textnormal}{prin}\colon\operatorname{WD}\_k\to{}^LG$. This parameter is trivial on the Weil group, and its restriction to the Deligne SL2 corresponds, via the Jacobson-Morozov theorem, to the sum of elements in a pinning of $\widehat G$, a nilpotent element of the Lie algebra. The L-packet of the principal parameter captures the Steinberg representation. When k is archimedean, we define the principal parameter to be the trivial crossed homomorphism $W\_k\to\widehat G$. [] [thm2] If k is nonarchimdean of characteristic zero, the center of G is anisotropic, and the parameter φ: WDk → LG is discrete then $$\frac{\omega(\varphi,{\textnormal}{Ad})}{\omega(\varphi\_{\textnormal}{prin},{\textnormal}{Ad})} = \chi\_\varphi(z\_{\textnormal}{Ad}).$$ The goal of this article is to verify Gross and Reeder’s conjecture. Actually, we will prove a more general version that relaxes the base field to an arbitrary local field, the adjoint representation to an arbitrary orthogonal representation, and discreteness of the parameter to temperedness. We generalize Gross and Reeder’s canonical involution $z\_{\textnormal}{Ad}$ to an involution zr (see ) that depends only on the restriction of r to $\widehat G$. The original motivation for our generalization was to compute the adjoint ɛ-factor arising in a conjecture of Hiraga, Ichino, and Ikeda describing the Plancherel measure, of which the formal degree conjecture that motivated Gross and Reeder is a special case. In the more general conjecture, the adjoint representation of LG is replaced by a certain “relative adjoint representation” of the L-group of a Levi. From this perspective it is natural, and in the end costs little, to relax the adjoint representation to an arbitrary orthogonal representation. [thm1] Let r be an orthogonal representation of LG and φ: WDk → LG a tempered parameter. Then $$\frac{\omega(\varphi,{{r}})}{\omega(\varphi\_{\textnormal}{prin},{{r}})} = \chi\_\varphi(z\_{{r}}).$$ The does not fully compute the orthogonal root number ω(φ, r): instead, it disentangles the root number into an automorphic factor, χφ(zr), and a Galois factor, $\omega(\varphi\_{\textnormal}{prin},{{r}})$. If one wanted to use to pin down an orthogonal root number precisely, it seems that the main challenge would be to compute the orthogonal root number of the principal parameter. In general I expect no better resolution to this problem than the rough answer provided by Clifford theory, though for special classes of representations, such as the adjoint representation, it might be possible to say more. Although our work is informed by the local Langlands correspondence, everything here takes place on the Galois side. A stronger version of the theorem would assert that χφ(zr) is actually the value on zr of the central character of the L-packet of φ; this is much more difficult because it requires some knowledge of L-packets. Lapid proved the stronger version of for generic irreducible representations of certain classical groups. Gross and Reeder proved for split G by an argument in group cohomology. This article is an outgrowth of an observation that their argument can be generalized in various directions. To relax the split assumption in Gross and Reeder’s proof, we generalize, in of, the basic lemma from group cohomology that underlies their proof, taking into account the Galois action on $\widehat G$. Since the pin extension of a complex orthogonal group is not topologically split, we must use Borel cohomology here instead of continuous (group) cohomology. With this modification, proves equal two particular Borel cohomology classes in ${\textnormal}H\_{\textnormal}{Borel}^2({}^LG,\{\pm1\})$. then follows from the equation by pullback along the parameter φ, once we properly identify the factors of the pulled-back equation. Most of this article is devoted to identifying these factors. Two of the factors are root numbers, and their recognition as such passes through Deligne’s theorem. Gross and Reeder already reformulated the theorem in the language of group cohomology for determinant-one orthogonal representations of Galois type, and it is mostly a matter of collecting definitions from the algebraic topology literature, in, to extend their reformulation to tempered orthogonal representations. The third factor of the pulled-back equation is the value of the central character χφ on the involution zr. Identifying this factor requires several detours that are surely well known to experts in the field. Any spin representation of a complex reductive group gives rise to a character of its topological fundamental group. The first detour, in, is to compute this character. Here we offer a small correction to an exercise of Bourbaki. The second detour, in, is to compute the second cohomology group of the Weil group with coefficients in the character lattice of a k-torus, generalizing a standard computation for the absolute Galois group. It turns out that this cohomology group is the character group of the Harish-Chandra subgroup of the torus. In we extend these results, for k nonarchimedean of characteristic zero, to any finite type k-group of multiplicative type, using a generalization of Tate duality due to Karpuk. Finally, in the brief we weave together these disparate threads to prove, and then explain the connection with the conjectures of Hiraga, Ichino, and Ikeda. Notational conventions ---------------------- Let k be a nonarchimedean local field of residue characteristic p and characteristic different from 2. The latter assumption is no great loss because is trivially satisfied if + 1 = − 1. Let Γ k be the absolute Galois group of k, let Wk be the Weil group of k, and let $$\operatorname{WD}\_k {\stackrel{{\textnormal}{def}}{=}}\begin{cases} W\_k &{\textnormal}{if $k$ is archimedean} \\ W\_k\times\operatorname{SL}\_2({\mathbb C}) &{\textnormal}{if not} \end{cases}$$ be the Weil-Deligne group of k. In practice we can immediately forget WDk and work with Wk because root numbers of orthogonal Weil-Deligne representations are unaffected by restriction to the Weil group. For k nonarchimedean, let Ik ⊂ Wk be the inertia subgroup. We reserve the letter G for a group, of two kinds: either a reductive group or a topological group. When G is a topological group, we assume it to be Hausdorff. When G is a reductive group, we assume it to be connected and quasi-split. To best align with the statement of the key we work with the Weil form of the L-group, ${}^LG{\stackrel{{\textnormal}{def}}{=}}\widehat G\rtimes W\_k$. The choice of the Weil form over the Galois form is not essential because in practice, we can replace the Weil-group factor of this semidirect product with the Galois group of any extension of k that is large enough to contain the splitting field of G and to trivialize the L-parameter relevant to the problem at hand. For us, a representation of the L-group LG is a representation in the sense of Borel’s Corvallis article, that is, a finite-dimensional complex vector space V together with a continuous homomorphism LG → GL(V) whose restriction to $\widehat G$ is a morphism of complex algebraic groups. Similarly, a representation of a complex reductive group is assumed algebraic. In general, the ɛ-factor, and hence the root number, depends not only on a Weil-Deligne representation but also on a nontrivial additive character of k and a Haar measure on k. Since there are formulas that explain the dependence of these factors on the character and the measure, nothing is lost in fixing them. We follow Gross and Reeder in computing ɛ-factors with respect to the Haar measure that assigns ${\mathcal}O\_k$ measure one and an additive character that is trivial on ${\mathcal}O\_k$ and nontrivial on the inverse of some uniformizer. In this article we use three kinds of cohomology: the singular cohomology of a topological space, denoted by ${\textnormal}H^\bullet\_{\textnormal}{sing}$; the continuous cohomology of a topological group, denoted by ${\textnormal}H^\bullet$; and the Borel cohomology of a topological group, denoted by ${\textnormal}H\_{\textnormal}{Borel}^\bullet$, whose definition we review in. Given any group A of order two, let sgn: A → { ± 1} denote the canonical isomorphism to the group { ± 1} of order two. Acknowledgments --------------- I am grateful to Jack Carlisle for discussing the cohomology of classifying spaces,	arxiv_0000548
Quantum Mechanics on a background modulo observation ==================================================== Abstract In this work we will answer the following question: What remains of Quantum Mechanics when we transform the background space-time into a space modularized by observation or measurement regions? This new moduli space is constructed by identifying regions of space-time where quantum phase comparison (observation, measurement) is implied. We call it Observation Modular space (OM-space). In addition we replace in QM statements the Plank constant (h) by the quantity ζ04π2 (where ζ0 is the Plank Length) or otherwise, replacing P0 (the Planck Momentum) by 4π2. This maps Quantum Mechanics into a very rich dual Number Theory which we call Observation Modular Quantum Mechanics (OM-QM). We find the OM-dual to the Dirac Equation, the quantum Wave Function and a free particle’s mass. The OM-QM counterparts of the Energy turns out to be a simple function of the zeroes of the Riemann zeta function. We also find the OM-QM correspondents to the electron spin, the electron charge, the Electric Field and the Fine Structure Constant. We also find the OM-QM correspondents of the Heisenberg uncertainty relation and Einstein’s General Relativity Field equation emerging as certain limits of a unique OM-QM equation. We also get the OM-QM correspondents of the Gravitational Constant and the Cosmological Constant. We find the analog of holography in the OM-QM side and we get an interpretation of spin as a high dimensional curvature. An interpretation of the OM-QM correspondence is proposed as giving the part of QM information which is not measurement or observation dependent. Some potential future applications of this correspondence are discussed. Introduction ============ In this work we will ask the following question: What remains of Quantum Mechanics when we identify (modularize) regions of the space-time background where observation or measurement is implied? To be precise we are not talking here about the Observer itself as this is a more complex topic which in general is not well understood and difficult to model inside Quantum Mechanics (QM). Neither we are talking here about the Measurement or Observation process itself which is also a very hard problem in QM. One would think (naively) that according to the Copenhagen interpretation of QM the answer to this question would be : Nothing at all remains except from Mathematics disconnected from QM physical content. According to this interpretation (the most standard and accepted one) QM is only talking about the information one can get in measurements involving an observed system. Thus, if we strip QM of observation (represented by a concrete physical mechanism like quantum wave function phase comparison between different regions of space-time) there should be no physically meaningful information content left in the residual Theory. That is, naively we could think that if we reduce QM over a moduli space defined modulo those regions where that observation or quantum phase comparison is happening, the resulting Theory should have no structure left other than probably pure Mathematics. Here we will prove that, contrary to that naive conclusion, we can construct a very specific such modulli space for which the reduction of QM on it is highly structured and non-trivial. We will call this moduli space Observation Modular space (OM-space ). Moreover, this resulting Theory while being purely Mathematical heritages (by design) a lot of the interconnections between Physical observables and makes evident surprising new interconnections and insights coming from the mathematical structures in it. Although one could define such modularization in apparently infinite ways, we use a very specific parametrization of the target theory which gives very suggestive and meaningful interconnections between Mathematical objects matching the ones in the Physical side. We call the residual Theory resulting after this parametrization and modularization Observation Modular Quantum Mechanics (OM-QM ). The intuition which takes us to this special parametrization is the following: The number 4π2 (a transcendental number) and the Physical quantity h (the Quantum of Action or Planck’s constant) play a similar role in Number Theory and in Physics respectively. 4π2 is connected to the distribution of ensembles of prime numbers. It is the result of the multiplication of all prime numbers “alla Euler”, that is, via Analytical Continuation of that infinite product. It’s in that sense that we should understand the statement that 4π2 “knows” about the distribution of ensembles of prime numbers. By the other hand the Planck constant encodes the information about the distribution of ensembles of quantum states in an arbitrary physical system. In both cases we are talking about statistical distributions of certain irreducible elements of the theory. In the case of h that relationship with the distribution of quantum states is more “by design”. From a mathematical point of view the Dirac axiom of QM is a statement about the relative distribution of eignenvalues of certain non-commuting, Hermitean operators (the physical observables) acting on a Hilbert space. h is the only parameter in such distribution. We argue here that there is more than an analogy between the roles of those two constants. We conjecture that every statement in Quantum Mechanics can be mapped into a corresponding Number Theoretical statement which results from replacing h with the quantity ζ04π2 (where ζ0 is the Planck Length) while at the same time modularizing the Space-Time background by identifying regions where quantum phase comparison is implied. Another way to say the same is that there is a correspondence between a Quantum Mechanical statements and a certain Number Theory statement which results from replacing the Planck’s momentum, defined as : $$\label{Eq1} P\_0 = \frac{h}{\zeta\_0}$$ with the number 4π2 and transforming Space-Time in the aforementioned way. We will call this reparametrization the P0-4π2 correspondence. We will call the full process of transforming a QM statement into its OM-QM dual the OM-correspondence. This process amounts to reducing Quantum Mechanics over a moduli-space background where the modularity is given by identifying regions of space-time where observation or measurement is implicit. We will define and explore this OM-space in the rest of this work. The effects of such OM-correspondence on QM seems to be fine tuned to give us a dual highly non-trivial Number Theory surprisingly full of physical and mathematical interconnections. We will analyze its internal consistency as well as potential connections with several similar ideas which have appeared in the past in the literature. Indeed, monduli spaces in relation to quantum based theories are not a new topic. The subject of Topological Quantum Field Theories (QFT) develops similar ideas but in a Field Theory context. In QFT the interest is on finding the effects on a Quantum Field Theory when one replaces normal space-time with a moduli-space constructed via some kind of modularity related to the Gauge Symmetries in the Fields . In QFT the starting point is an action which is independent on the space-time metric. Those actions are dependent only in certain functions of the field potential which give raise to Feynman amplitudes which in turn are topolocical invariants (Knot invariants sometimes) of the moduli space. A reference which is even more relevant for our work is Gao’s work on AdS/CFT correspondence over a quotient space. In it a moduli (quotient) space version of the bulk AdS space is introduced. The modularization in this case is via some large, general quotient Group. Gao proceeds then to find the effects of this modularization in the boundary quantum CFT Theory using the AdS/CFT correspondence. This is very similar to what we will attempt in our work. What is new in our approach is that we are in general interested in the effects of that modularization in Quantum Mechanics iteslf in a more realistic setup. We are not going to base on a toy model bulk space-time background as AdS but instead base in the usual 4-Dimensional space time and in full Quantum Mechanics. Neither we will use a large group for producing a quotient space but instead we will use observation regions of space as our quotient criteria. Also it is new the P0-4π2 substitution we use in our OM-correspondence. The main conclusion we will be lead to is that the OM-QM construction reduces the degrees of freedom available in Quantum Mechanical information in a very special way. It leaves us with a Physics independent core of that information which itself contains a rich structure. We can say that that residual Number Theory core we are left with is the Observation independent content of Quantum Mechanical information. We will see that topics which are difficult or obscure in the usual QM framework become natural and easy to deal with in the OM-QM side. Topics in QM do get mapped to very concrete and clear Mathematical statements which often throw some light on their nature in the QM side. In Mathematics we often see similar simplification or insight when we pass some problems to a modular framework. OM-QM can be seen as the counterpart of that Mathematical modular simplification or insight in QM. We will first apply the OM-correspondence to the Quantum Mechanics of a free electron. That is, we are going to find the OM-dual of the Dirac Equation. This will gives as a by-result that the OM-QM counterparts of the particle mass and energy are simple functions of the zeroes of the Riemann Zeta function. Then we will use this correspondence for finding a fist principles derivation of the value of the Fine Structure Constant (it precise numeric value!) In this work we will adopt a Physics approach for developing the topics at hand. We will not delve too deep in the Mathematics aspects although we will be dealing with deep Mathematical concepts. For example when talking about moduli-spaces one could adopt a very strict Mathematical approach as that is in reality a very complex Mathematical subject. We will use here the wording moduli-spaces as a placeholder for an eventual future formalization of the Physical ideas we will propose. We will go as deep as we need in the Mathematics but not so deep that we lose focus on our main goal, finding a precise definition of the OM-correspondence and proving that it produces an interesting and very non trivial effect in QM. The Theory we will develop below seems to be a small common playground where Physics and Mathematics can play together for the sake of finding new or known interconnections. We propose that OM-QM is in fact a toy model of a Quantum Gravity Theory. In the final section of this work we will briefly talk about potential future applications o this correspondence. The OM correspondence ===================== The correspondence we are postulating in this paper can be summarized in the following procedure: * [*step 1] Take a valid Quantum Mechanical statement (an equation, a definition, a postulate). Let’s call this the P0 side of the correspondence. The OM corresponding statement can be constructed via the following operations:* * [*step 2] Replace any instance of the Planck constant h by the quantity ζ04π2 (where ζ0 is the Planck Length). Or equivalently, replace P0 (the Planck Momentum) in the QM side by the number 4π2* * [step 3] Identify all regions in the background space-time where a quantum phase comparison (that is measurement or observation) is implicit in the quantum statement (up to homotopy classes). Keep the form of the QM equations covariant in the transformation. Then find the effect of such transformation in all elements in the quantum statement. We will make a unique and precise definition of this modularization in next section. In the rest of the paper we will notate with a tilde the OM-QM correspondent of any quantum quantity. Then by definition we have $$\label{Eq2} \tilde{P\_0} = 4 \pi^2$$ We will also make use of terms like “OM-Energy” or “OM-mass” for referring to the OM correspondents of certain Physical observables. OM-space: Space-Time modulo regions with implied Quantum wave function phase comparison ======================================================================================= Let’s analyze the step 3 in the above mentioned procedure. As one of the elements of that step we impose covariance of the Quantum equations under the space-time transformation we are performing. That is, we are imposing that the form of the dual Number Theoretic statement should be the same as the form of the Quantum Mechanical statement. We know which are the space-time transformations under which Quantum Mechanics is covariant. Ignoring (by now) the presence of any gauge fields in the Theory, in the case of non-relativistic QM those are the transformations from the Galileo Group. In the case of relativistic QM those are the Lorentz group transformations multiplied by the internal spin symmetry group. These internal spin symmetry groups are representations of the SU(2) group acting on 4 components spinors. The internal symmetry groups can include Gauge groups if we are talking about the QM of particles interacting with Gauge fields . This covariance enforces the elements of the QM equations to transforms in very specific ways. For example mass must transform as scalar under Lorentz transformations. Gauge charges (electric, color, lepton flavors) will transform as vectors under the internal gauge groups. The particle states will in turn transform as spinors in the relativistic case and under the internal gauge groups if gauge charges and fields are considered. The space-time transformation we use in Step-3 of our correspondence is obviously none of those transformations under which QM is covariant. Therefore if we want to impose covariance under our space-time transformation the way the QM elements transform must be different than in the way QM transforms under its covariance transformations. We must assume that all elements in the QM equation must transform in general in order to keep covariance. Elements as Mass, Gauge Charges (electric of other), Spin and the state vectors will all change in these transformations in ways we will find in the rest of this work. In fact Step-3 is all about the fact that even when we are imposing that the form of the equation should not change then all its elements should change to keep that covariance. Finding out how those elements change in order to keep covariance is the core of this work. In this work we are indeed talking about a type of covariance which is not Lorentz or Gauge covariance. We argue that this covariance is related to the observation itself. We are imposing that the QM laws should keep their form even when we take out observation or measurements out from the picture. Or otherwise, what we are defining here is QM Modulo Observation. Although Step-3 in our OM correspondence is a simple definition from the physical point of view it is not very practical if one wants to apply it in an operational way to real quantum mechanical statements of any kind. If we want to apply this correspondence in practice we need to describe in a more precise and operational way what we mean by modularizing the Phase Space by identifying regions where quantum phase comparison is implicit. We will use a similar reasoning to the one Maldacena and Susskind employed in their ER=EPR correspondence . They needed to make more quantitative their principle of identifying regions of space-time connected via entanglement. In their case they studied the correlations between states in entangled Space-Time regions and compared them to the correlations of events in Space-Time regions connected via an Einstein-Rosen bridge. They mapped the “gluing” of those regions into a more operational notion: probability correlation of states in those regions. For the OM correspondence we need a similar more operational form of Step 3. We will not try to find at this point an explicit form of the Phase Space transformation implied in that step (we will do that later in this work). We will define instead our moduli Space-Time in a totally equivalent way from the Quantum Mechanical point of view. It is enough describing how such transformation of the space-time acts on Integrals over paths in Phase Space. In Feynman’s interpretation of QM, the central object of the Theory is the quantum Amplitude given by the Feynman path integral $$\label{Eq3} \mathcal{A}\_{\alpha \rightarrow \beta} = \int\_\mathbf{C} ( \exp{ ( \frac{i\S}{\hbar} ) } ) ds$$ where C* is the set of all possible paths in Space-Time taking us from an initial point α to a final point β and S is the Classical Action for the system. All the information about results of measurements is contained in that amplitude. According to the Copenhagen interpretation of QM that is all the information we can expect to know about quantum systems. Therefore, if we find what is the effect of the modularization of Phase Space described in Step 3 on that Amplitude we will have an equivalent and more operational form of that Step 3 in the OM correspondence. ![image](Fiure1) Here we make a crucial observation about the end points of the set of paths C (the start classical space-time point α and end point β ). There is a quantum phase comparison assumed in each of them. We motivate this by saying that the fact we are talking about those two extreme points for all the paths carries the assumption that there is a measurement implicit at both ends. Nevertheless, the value of the Feynman amplitude does not depend on a measurement or observation of one concrete observer over the system. According to the Copenhagen interpretation of QM the only thing this Amplitude is saying is that if one performs the experiment or observation for measuring the probability of transition of the system from a state centered (according to the uncertainty dispersion) in the space-time point α to a state centered in space-time point β then the value of such probability is given by Eq.[Eq3]. In this Amplitude formula we are always speaking about information from experiments or observations. There is observation implicit in it. The set of paths C is in general infinite and other than the experimental constraints there is no limitation in how long or convoluted they can be. In fact QM says we have to integrate over all of the possible paths. The potential observational information in the set of paths comes from two sources: from the common end-points and from some experimental or topological constraints implicit in the set up of the Classical system. For example if we are performing the double slit experiment, there are obvious topological/geometrical constraints implicit in the measurement and in the amplitude. In addition we have the constraints coming from the extreme points of the the paths. In Step 3 of the OM correspondence we explicitly mention that the identification of regions of the space-time is up to homotopy classes. This topological reduction or identification of paths accounts for the observation of topological constraints in the system. But we still have to take into account the part of the amplitude implicit in the end points of all the paths. We argue here that this localization of the end points are equivalent to a measurement. In fact we argue that any kind of constraint on the system should be considered as assumed measurement and therefore a quantum phase comparison. In any measurement or constraint a quantum phase comparison is implicit. Therefore following Step 3 we can identify both end points α and β up to a cell of minimal size in space-time (a pixel given by Heisenberg’s uncertainty relation). Going forward, we will consider that minimal size pixel to be a Planck Length-Time radius ball in 4 dimensions. The net result of this reduction of number of paths due to topological equivalence and to the identification of the end points as described above is that the set of paths C gets mapped to a finite set of closed loops united in a fuzzy way on that minimal size pixel of space-time. In Figure 1 we describe this transformation in a pictorial way in 2-D for the sake of visualization. First thing one can notice in Figure 1.c is the way the transformed end parts of the paths “approach” the minimal size cell. They enter the region close to the minimal cell in an almost parallel way. Not totally parallel but the closer they get to the minimum cell the more parallel they become. The reason for this is the following: If this transformation is a proper modularization then regions near the extreme path points should map to regions close together. Regions far from the path’s end points are free of this constraint. That means the paths will be mapped into sort of “petal loops” (Going forward we will call these loops the OM-petals) meeting at the central minimal cell. This fact will be important at the moment of interpreting the meaning of this spatial transformation. In Figure 1.d we can also see that under the transformation above the winding of the resulting loop around the minimal cell can be a Natural Number (n) larger or equal to 1. In order to differentiate these winding loops from OM-petals we will call them contractible OM-loops. In a 2-D visualization as Figure 1.d this winding around the minimum cell is partially dependent on the projection of the 3-D spacial paths to a 2-D view. However the maximum winding number is not dependent of this projection. In at least one of the ways to perform this 2-D projection one gets a maximum possible such winding number. In any case the resulting general topology will be always the one of a Knot. Going forward we will call it the OM-Knot. This OM-Knot is composed of one (or more) OM-petals an a integer number n of contractible OM-loops. We can make at this point another important observation which we will use in the rest of this paper. The length of the paths connecting the extreme points gets transformed into a integer multiple of the minimal cell space size (the Planck Length) Nζ0 This is due to the fact that the loop can be homotopycally deformed to a loop curled directly over the minimal cell. Therefore we get the following OM correspondence for one position coordinate $$\label{Eq5} \tilde{x}= \frac{x}{\zeta\_0} = N$$ Let’s note that this variable N and the number n of contractible loops are in general different natural numbers. n is partially dependent on the geometry of the system we are considering but it can also depend on symmetries or other physical constraints. In this work we will almost ignore the variable n and use it only for visualization. This variable hides in reality a very deep quantum baggage and will be the topic of a future work. In this paper we will focus mainly in the variable N which has a pure geometric origin. The details of what the OM-Knot “petals” do far from the central minimal size cell is not influencing paths integrals of holomorphic (or branch-holomorphic) functions. Only the path integration around the central cell contributes to the end result. We see here the first hint that the Theory resulting from the applying the OM correspondence to QM will lead us to a Mathematical Theory, more precisely a Number Theoretical framework. We will use the number N loosely in this paper. We won’t consider it to be strictly a Natural number. For us it will be just a numeric parameter which arises due to the background transformation introduced in the OM correspondence. We can allow it to vary as a real parameter for the purpose of the rest of this paper. However, we will see later that most of the OM-QM counterparts of the quantum statements adopt the form of Number Theoretical statements and N will naturally be identifiable as a Natural. From the discussion above we get that the Space-Time transformation defined above allow us to give an operational form to Step 3 in our correspondence as the following way to transform a general path integral of an arbitrary phase-space function f(x,p) ∫C[α − β]f(x, p) → ∫L[α − β]f̃(x̃, p̃) where C[α − β] is a path (or a set of them) connecting states centered at Space-Time points α and β, L[α − β] is a “petal-like” Loop (or a set of them) connecting at a minimal cell of the transformed space at points α − ε and α + ε. f̃, x̃ and p̃ are the OM-QM corresponding objects to the integrated function, position and momentum. Here x should be understood as a vector in 3 dimensions though below we will work over one of its components. We should keep in mind that for Knots the number of spatial dimensions is very relevant but we will see we will be able to work with each coordinate separately for the sake of visualization and for reducing complexity. x̃ is given by Eq.[Eq4]. p̃ can be found easily knowing x̃ and remembering the expression of the quantum momentum operator $$\label{Eq7} p=i\hbar \dfrac{d \Psi(x)}{dx}$$ with Ψ(x) the Wave Function of the quantum system. Applying the OM correspondence we get $$\label{Eq8} \tilde{p}=i 2 \pi \dfrac{d \tilde{\Psi}(N) }{dN}$$ where $ \tilde{\Psi}(N) $ is the OM-QM counterpart of the Wave Function. Later we will find a concrete expression for this $ \tilde{\Psi}(N) $. By now we only need to know that it is a function of N through its dependency on x̃. We need one more element to be able to make progress in finding what corresponds to QM in the OM-QM side. Although we could start with a non relativistic version of QM we will instead work in a Relativistic quantum framework from the start. To that end, we will need to find the OM counterpart of the speed of light ( c̃ ) as this is something which also gets mapped to certain mathematical object under the OM-correspondence. We know how distances get mapped according to Eq.[Eq4]. In order to know how a speed gets transformed we need to understand how time gets mapped in the OM-QM side. In the definition of the OM correspondence we mentioned that this is essentially an holographic theory where we are reducing degrees of freedom via the identification of certain regions of space-time. In every holographic transformation time gets mapped to an spacial radial coordinate tangent to the target spacial dimensions . In Figure 1.2 we visualize how this looks for the case of our modularized space-time. ![image](Figure2.jpg) We can see time can be identified with the radius r in that picture and we should assume the transformed time flows in the direction of the central minimal cell. The length of the spacial dimensions is as we know Nζ0. Therefore we have $$\label{Eq9} \|r\|= \frac{N \zeta\_0}{2 \pi}$$ this can be assumed to be a minimum time similar as Nζ0 is a minimum length. Then we can write $$\label{Eq10} \|\tilde{c}\| = \frac{N \zeta\_0}{\|r\|}=2 \pi$$ and finally to take into account the direction of this transformed radial coordinate to point inwards towards the central cell (r⃗ = i∣r∣), we finally find c̃ = − 2iπ The micro and medium scale structure of OM-space ================================================ In Figure 1 and in the previous section we got a visualization of the micro structure of the OM-space. Each loop of the fuzzy OM-knot leaves and enters the central minimal size cell in an almost parallel fashion. In general the loop can wind around the fuzzy center cell for a large number of full rounds. In a two dimensional visualization all those loops can be thought as a ring structure made of loops circling the central fuzzy center of the OM-Knot. However, we must notice that as we do not allow self crossing of the loop into itself the re-injection of the loop into the central region must occur in a direction above or below that ring structure. In the following figure we see a visualization of this medium scale structure. ![image](Figure3) It is very similar to an object known in the area of Dynamical Systems and Chaos as a Rosller attractor . To be precise this is just and analogy we are proposing. The real target object has a very close resemblance of the Rosller attractor but its real structure can have some variations as we will see later. Nevertheless, going forward we are going to use the name Rosller-like geometry to describe the micro structure of the OM-pace. We propose that this mathematical object is in effect a good representation of the medium scale structure of OM-space. It also shows a flow of loops around a central region in an almost two dimensional ring with re-injection flow along a perpendicular direction. In Chaos Theory the Rosller attractor araises in the phase space orbits of dynamical systems with dissipation due to interaction or coupling of two dimension in the system. The coupling between two of the system’s dimensions is mapped into the re-injection of the flow of the orbits in one of those coupled dimensions into the general flow of orbits in the other dimensions. This causes a dimensional reduction in the resulting attractor of the orbits flow. The resulting object has a dimension between 2 and 3 with a fractal geometry. Locally this kind of strange attractors behave as a Baker Map. That is, each loop around central region stretches phase space volumes in one of the coupled dimensions while expanding them in the non coupled directions. The net effect if a volume preserving transformation for each cycle of orbits flow around the central region. A known characteristic of this kind of Baker flows is that the separation between distances (dN) between points in the flow increases in an universal exponential fashion involving the Lyapunov (λ) coefficients of the loops flow $$\label{Eq12} d\_{N+1} \propto d\_{N+1} \exp{(\sqrt{\pi \lambda })}$$ where for a Baker type map (locally a logistic like map), the Lyapunonv coeficient is equal to the Feigenbaund Constant (δ = 4, 66920160910299.... ) The proportionality constant depends on the square of the Hausdorff dimension of the Rosller fractal. If the attractor is embedded in a three dimensional space then this proportionality constant will be the square of a number between 2 and 3 and rather close to 3 (the dimension of a locally logistic fractal embedded in 3D space). This will be important in another section later in this paper. It is worth noting that every observable of the system depending only on the scale we observe the system follows the same scaling universal behavior. For example, in many systems one of those properties depending only on the observation scale is Entropy. We will make use of this property later in the context of the OM correspondence. The OM-QM counterpart of the free particle Dirac Equation ========================================================= Having found expressions both for x̃, p̃ and c̃ we can now try to find the OM-QM counterpart of the Quantum Wave Function, Physical Observables such as Energy and parameters such as the mass and spin. All of them get transformed to Number Theoretical objects under the OM correspondence. We work with all physical units in full display as in the OM correspondence we want to understand how all physical elements transform in the OM-QM side. The free-particle Dirac equation has the following form (iℏγμδμ − mc)Ψ(x, t) = 0 where the γμ are the 4 by 4 Dirac Matrices, m is the particle’s mass, c is the speed of light and Ψ(x, t) is a 4 components bi-spinor describing the particle quantum state. Plane waves solution to this equation make necessary the emergence of the particle spin. Step 3 of the OM correspondence in the case of the free-particle gives the simplest possible modulrization of the space-time as there is no topological or experimental constraint other than the ones implicit in the extremes of paths connecting the particle’s initial position and end position. The fuzzy OM-Knot resulting from the identification of initial and end points produces a simple knot with only one petal with a minimum cell center. Let’s apply our OM-correspondence to Eq.[Eq13]. We obtain the following differential equation $$\label{Eq14} (i 2 \pi (\gamma^0 \dfrac{\delta}{\delta N\_0} ) - i 2 \pi (\gamma^{1,2,3} \dfrac{\delta}{\delta N\_{1,2,3}} ) ) \tilde{\Psi}(N\_{0,1,2,3}) = \tilde{m}\tilde{c} \tilde{\Psi } (N\_{0,1,2,3})$$ Here the numeric parameters N0, 1, 2, 3 are the OM-QM side of each of the space time coordinates. We have also substituted the derivation in terms of the space-time coordinates by their OM-QM counterparts and m̃ also transforms to a Number Theoretical expression as well as we saw the speed of light also does. We will see later that m̃ is in fact a function of the numbers N0, 1, 2, 3. The particle’s state transforms to a 4-components vector where each of its components are functions of the numbers N0, 1, 2, 3. Eq.[Eq14] is the full OM-QM correspondent of the Dirac Equation. At this point we could try to find a plane wave solution and understand what is the OM-QM version of $ \tilde{\Psi } (N\_{0,1,2,3} ) $ that is, of what is the OM-QM counterpart of the free particle state. The γ matrices do mix the derivatives on the four Ni parameters. Therefore, finding a solution for one of the components of $ \tilde{\Psi } (N\_{0,1,2,3} ) $ involve derivatives in all four Ni parameters. We will make here a simplification in order to find of a solution to Eq.[Eq14]. Let’s recall the topological meaning of the parameters Ni as winding of paths in space-time around a minimum cell. We can always find a topology of those “petal loops” where the projection of a 3-dimensional such knot according to any of the dimensions onto any of the 3 possible 2-dimensional views has the same winding around the central minimum cell (irrespective of the dimension we decide to project). That is, for the kind of knots we are dealing with the three “directions” Ni are essentially equivalent. We can then simplify Eq.[Eq14] considering that all four derivatives are equal. In a geometric context this would not be possible even when we consider all coordinates as equivalent. However, here we are dealing with pure mathematical objects, numbers in fact. They cannot be labeled as different if they are equal in value. For them we can then simplify Eq.[Eq14] to this form for the 4 components of $ \tilde{\Psi } (N\_{0,1,2,3} ) $ $$\label{Eq15} (i 2 \pi (\gamma^\mu ) \dfrac{\delta}{\delta N} ) ) \tilde{\Psi}(N) = - 2i \pi \tilde{m} \tilde{\Psi } (N)$$ Substituting the Dirac γ matrices Eq.[Eq15] is equal to $$\label{Eq16} \begin{pmatrix} i-1 \\ 1-i \\ 1-i \\ i-1 \end{pmatrix} \dfrac{\delta}{\delta N} \tilde{\Psi } (N) = \tilde{m} \tilde{\Psi } (N)$$ By similarity with the Dirac equation solution, the numbers that appear in the r.h.s. before the derivative are the OM-QM counterparts of the spin. We see that the possible values of s̃ are s̃ = ± (i − 1) The values of s̃ are independent of N as they come from the gamma matrices. Therefore the general solution to Eq.[Eq15] is $$\label{Eq18} \tilde{\Psi } (N) = e^{\frac{\int^{N\_f}\_{N\_i} \tilde{m}(t) dt}{\tilde{s}}}$$ We can propose a reasonable guess for a solution of Eq.[Eq15]. Let’s start by assuming a solution of this form $$\label{Eq19} \tilde{\Psi } (N) = e^{\phi(N)}$$ If we assume an almost plane-wave solution type then ϕ(N) has the meaning of a total phase difference after parallel transport over the fuzzy Knot we discussed in previous section. This is due to the fact that this phase difference corresponds to the comparison of the free particle states at the initial space-time point and at the final space-time point. As we saw before this phase comparison is done after parallel transport over the “petals” of the fuzzy OM-Knot generated by the OM correspondence. Comparison after parallel transport is equivalent to Curvature. In this case it is total curvature integrated over the whole OM-Knot “petals”. If we call this curvature as R(N) then we have $$\label{Eq20} \tilde{\Psi } (N) = e^{\phi(N)} = e^{\int^{N\_f}\_{N\_i} R(t) dt} = e^{\frac{\int^{N\_f}\_{N\_i} \tilde{m}(t) dt}{\tilde{s}}}$$ Essentially what we are saying is that a very reasonable guess for the OM-QM counterpart of the free particle quantum state is a topological invariant of the OM-Knot and concretely, the exponential of the total Knot curvature. This invariant is in general a function counting the statistics of possible crossing at the fuzzy center of the OM-Knot. From Eq.[Eq20] we get $$\label{Eq21} \phi'(N) = R(N) = \frac{\tilde{m}(N) }{\tilde{s}}$$ where $ \phi'(N) = \dfrac{d \phi}{d N}$ We see here a first hint that the OM-QM statements gives very non trivial and interesting connections between elements coming from the QM side. In Eq.[Eq21] we see an explicit relationship between the OM-QM counterparts of spin, mass and curvature. We cannot see that direct connection in the QM side but in the OM-QM side this arises explicitly. Using the presence of the total OM-Knot curvature in the formula for $ \tilde{\Psi } (N) $ we can find a more explicit and meaningful form of that solution. From the work of Mazur it is known that the crossing statistics of fuzzy Knots are given by logarithms of the Prime Numbers . To each fuzzy Knot one can assign a prime and to each prime one can assign a fuzzy Knot. Mazur gave the following correspondence log(p) = vol(K) where p is a prime number and vol(K) is the volume of the hyperbolic complement of a Knot. When talking about a fuzzy Knot that prime p counts the crossing statistics at the fuzzy center. By the other side that volume is equal to the total curvature integrated over the Knot. Therefore, we have ∫NiNfR(t)dt = log(p) At this point we make contact with Number Theory by recognizing that in the formula above we can use the von Mangoldt function Λ(p) instead of log(p). The von Mangoldt function is defined by $$\label{Eq24} \Lambda (\textit{p}) = \bigl\{ \begin{array}{cll} \log(\textit{p})& if & N = \textit{p} ^k \\ \\ 0 & &otherwise \end{array}$$ This is not a simple substitution as it takes into account the topological meaning of Eq.[Eq20]. Substituting this into Eq.[Eq20] we get the following possible solution for (each of the 4 components of) the OM-QM transformed Dirac Equation $$\label{Eq25} \tilde{\Psi } (N) = e^{(\sum\_{q={\textit{p}^r}} \Lambda (\textit{N}) \delta (N-q))}$$ As a side note, the exponent in this solution is equal to the derivative of the Chevyshev function (ψc). The Chevyshev function is in turn equal to the logarithm of the Least Common Multiplier function of all numbers from 1 to N ψc = log(l.c.m.(1, ..., N)) Finally we find the following alternative form of $ \tilde{\Psi } (N) $ $$\label{Eq27} \tilde{\Psi } (N) = exp[ \frac{l.c.m.'(1,...,N)}{l.c.m.(1,...,N)}]$$ This gives an interesting interpretation to the solution of the OM-QM transformed Dirac Equation as a kind of focus or convolution formula for all scales from 1 to N. Let’s remember that N in our framework represent scales in terms of the Planck Length. Therefore, $ \tilde{\Psi } (N) $ is in fact the exponential of the logarithmic derivative of a scale function which is able to resolve at the same time all the scales from 1 to N. The OM-QM counterpart of Mass and Energy ======================================== We will focus at this point in using Eq.[Eq25] for finding the OM-QM counterparts of other Quantum Mechanical observables or parameters. By substituting Eq.[Eq25] into Eq.[Eq16] we get the following form for the OM-QM counterpart of the ]mass parameter in the Dirac equation m̃(N) = s̃∑q = prΛ(q)δ(N − q) Let’s find the OM-QM version of the other equation satisfied by each one of the 4 components of the bi-spinor solution to the Dirac equation, that is the Klein-Gordon equation. $$\label{Eq29} (\frac{1}{c^2}\dfrac{\delta^2}{\delta t^2} - \Delta^2+\frac{m^2 c^2}{\hbar^2} ) \Psi(\vec{r}, t) =0$$ We will consider the plane waves type solutions which allows us to substitute the time derivatives side by an energy eigenvalues term. If we apply the OM correspondence and use the transformed Dirac Equation to this equation we get the following OM-QM form of the Klein-Gordon Equation $$\label{Eq30} \frac{\tilde{E} ^2 (N)}{(4 \pi^2)^2} \tilde{\Psi}(N) = (\tilde{s}^2 \dfrac{d^2}{dN^2} + \tilde{m}^2(N) ) \tilde{\Psi}(N)$$ where Ẽ is the OM-QM counterpart of Energy which as well mass becomes a function of the parameter N. To get to this equation we have made use of the same approximation we used in the case of the Dirac equation which allowed us to remove the dependency on the three spatial coordinates and work in one dimension. Using Eq.[Eq20] we obtain this alternative version of the transformed Klein-Gordon equation $$\label{Eq31} \frac{\tilde{E} ^2 }{(4 \pi^2)^2} \tilde{\Psi}(N) = (\tilde{s} \tilde{m}' + 2 \tilde{m}^2 ) \tilde{\Psi}(N)$$ where we used the notation m̃ʹ as the derivative of m̃ respect to N. To find an explicit formula for m̃ʹ we can use Eq.[Eq28] and the method of derivation under the sum sign $$\label{Eq32} \tilde{m}' = \tilde{s} \sum\_{q={\textit{p}^r}} ( \dfrac{d(\Lambda (\textit{q}^b) \delta (N-q^b) ) }{db})\|\_{b=1} = \tilde{s} \sum\_{q={\textit{p}^r}} (\dfrac{d(\Lambda (\textit{q})) }{dq} \log(q)q + \Lambda (\textit{q}) ) \delta (N-q)$$ Let’s see how the derivative of the von-Mangoldt function is related to the Riemann ζ-function. A known identity for the logarythmic derivative of the Riemann ζ-function says that $$\label{Eq33} \frac{\zeta'}{\zeta} = \sum\_{q={\textit{p}^r}} \frac{\Lambda(q)}{q} \delta (N-q)$$ Otherwise we can write $$\label{Eq34} \frac{\delta \ln (\zeta)}{\delta N} = \sum\_{q={\textit{p}^r}} \frac{\delta \Lambda(q)}{\delta q} \delta (N-q)$$ That means that Λ(q) is the Moebius inverse of (ln(ζ(q)))ʹ. Then Λʹ(q) is the Moebius inverse of ζ. This implies a duality relationship between those two functions. From this we conclude that the peaks of the r.h.s. in Eq.[Eq32] will occur on the zeroes of the ζ-function. Then Eq.[Eq32] can be considered an statistical average of those zeroes. We finally arrive to the following expression for m̃ʹ m̃ʹ = m + s̃∑q = pr(σqlog(q)q)δ(N − q) where σq are the imaginary part of the zeroes of the Riemann ζ-function. The statistical average of the real part of those zeroes (equal to 1/2 if the Riemann hypothesis is true) is zero. Substituting this into Eq.[Eq31] we get the following formula for Ẽ $$\label{Eq36} \frac{\tilde{E} ^2 }{(4 \pi^2)^2} = \tilde{s}^2 \sum\_{q={\textit{p}^r}} \sigma\_q \log(q)q ) \delta (N-q) +\tilde{m} + 2 \tilde{m}^2$$ That means that the OM-QM correspondent of Energy for the free particle is a simple function of the imaginary part of the zeroes of the Riemann ζ-function. It has been long theorized that the imaginary part of the zeroes of the Riemann ζ-function are the eigenvalues of some non-hermitean Hamiltonian . Many promissing Hamiltonians have been tried but a proof of such conjecture still is out of reach. Here we have arrived to a similar relationship in the context of the OM correspondence. This shows the Number-Theory we arrive at with the OM correspondence is highly non trivial and can give us very valuable information about the Quantum side. We see that problems which seem to be hard or mysterious in a pure Quantum realm seems to be mapped to Mathematical relationships which are much simpler to treat. We see here a first hint that the OM-QM side of the correspondence seems to be a common playground for both Quantum Physics and Mathematics where the interplay of concepts from both sides can give raise to unexpected relationships (or conjectured as in the case above). The values of m̃ and Ẽ =========================== Let’s explore the formulas for m̃ and Ẽ to see which values they can possibly have. From Eq.[Eq28] we see that only the following values are possible for m̃ $$\label{Eq37} \tilde{m}(N) = \bigl\{ \begin{array}{ccl} 0 & if & N \neq p^k, p=prime \\ \\ \frac{\tilde{s}}{N} & if & N= p^k \end{array}$$ Thus, only for N which is a power of a prime number m̃ is not null. The values it take are coherent with it the already used formula relating OM-Knot curvature to m̃ $$\label{Eq38} \tilde{R} (N)=\frac{\tilde{m}}{\tilde{s}}$$ as R̃ is precisely of the form 1/N. Inserting those values of m̃ into Eq.[Eq36] we get the following possible values for Ẽ2 $$\label{Eq39} \tilde{E}^2(N) = \bigl\{ \begin{array}{lcl} 0 & if & N \neq p^k, p=prime \\ \\ \tilde{P}\_0^2 (\tilde{s}^2 \sum\_{q={\textit{p}^r}} \sigma\_q \log(q)q ) \delta (N-q) + \frac{\tilde{s}}{N} + 2 (\frac{\tilde{s}}{N})^2) & if & N= p^k \end{array}$$ We cam rewrite the values in the case N = pk in terms of the OM-Knot curvature in this way $$\label{Eq40} \tilde{E}^2(N) = \bigl\{ \begin{array}{lcl} 0 & if & N \neq p^k, p=prime \\ \tilde{P}\_0^2 (\tilde{s}^2 \sum\_{q={\textit{p}^r}} \sigma\_q \log(q)q ) \delta (N-q) + \tilde{s} \tilde{R}(N) + 2 \tilde{s}^2 \tilde{R}^2(N)) & if & N= p^k \end{array}$$ We can reduce further this expression noting that the sum in it selects only one of the possible summing elements Ẽ2(pk) = P̃02(s̃2σpplog(p))Δp + s̃R̃(pk) + 2s̃2R̃2(pk)) where Δp is a distribution version of the Dirac delta centered at p. We will return to this important expression later in this work. The value of the Fine Structure Constant ======================================== As an application of our correspondence let’s find find the OM-QM correspondent of the Fine Structure Constant. The value of this fundamental constant cannot be calculated from first principles in Quantum Mechanics even when is form in terms of other physical constants appears in the spin-orbit contribution to the energy eigenvalues in atomic Physics. Its is a non-dimensional number whose inverse is close to the rational $ \frac{1}{137} $. One of the ways to define formally this constant is through the following limit $$\label{Eq42} \alpha^{-2} = \lim\_{n \rightarrow \infty} \frac{{E^2}\_{n-1}-{E^2}\_{n-2}}{{E^2}\_{n}-{E^2}\_{n-1}}$$ where En are the atomic energy eigenvalues. By similarity we can define the OM-QM correspondent of the Fine Structure Constant by the following number $$\label{Eq43} \tilde{\alpha}^{-2} = \lim\_{i \rightarrow \infty} \frac{\tilde{{E^2}}\_{i-1}-\tilde{{E^2}}\_{i-2}}{\tilde{{E^2}}\_{i}-\tilde{{E^2}}\_{i-1}}$$ where Ẽn are the OM-QM correspondent of Energy defined in Eq.[Eq39]. Let’s calculate explicitly this number α̃. Looking at Eq.$ \ref{Eq36} $ and Eq.$ \ref{Eq28} $ we notice that terms consecutive in N related to mass are the same for N → ∞. That means they will cancel in α̃. We only need to consider differences of consecutive terms of the form γ(N) = s̃2∑q = prσqlog(q)qδ(N − q) Thus, we get the following formula for α̃− 2 $$\label{Eq45} \tilde{\alpha}^{-2} = \lim\_{n \rightarrow \infty} \frac{\gamma\_{n-1}-\gamma\_{n-2}}{\gamma\_{n}-\gamma\_{n-1}}$$ We know the difference σn − σn − 1 tends to zero in the infinite. Thus we get the following limit for α̃ $$\label{Eq46} \tilde{\alpha}^{-2} = \lim\_{N \rightarrow \infty} K \frac{\log(N-1) (N-1)-\log(N-2) (N-2)}{\log(N) N -\log(N-1) (N-1)}$$ where K is the proportionality constant already mentioned in Section 4. Taking into account that N in our setup corresponds to a scale in terms of the Plank Length, all terms in α̃− 2 depend only on the scale and also have the form of an entropy term. We found in section 4 that such functions in the moduli space we are working on do scale in a very particular way (Eq.12). Specially entropy like functions do scale in that way. Therefore we get the following result $$\label{Eq47} \tilde{\alpha}^{-2} = K \exp{(\sqrt{\pi \delta})}$$ where δ is the Feingenbaum constant. We get finally the following valued for α̃− 1 $$\label{Eq48} \tilde{\alpha}^{-1} = \sqrt{K} \sqrt{\exp{(\sqrt{\pi \delta})}}$$ $$\label{Eq49} \sqrt{\exp{(\sqrt{\pi \delta})}} \approx 46.061512666666666666666666666667....$$ From our discussion in Section 4, we know the constant K is related to the square of the Hausdorff dimension of the local structure of OM-space. The dimension of a Rosller fractal embedded in 3D is a number between 2 and 3 and rather close to 3. A value of $ \sqrt{K}=3 $ gives a value of α̃− 1 = 138, 184538...... which is already close to the experimental value. However if we consider K to be the square of the Rosller fractal dimension (D=2.974283562752.....) we get the value α̃− 1 ≈ 137. $$\label{Eq50} \tilde{\alpha}^{-1} = D \sqrt{\exp{(\sqrt{\pi \delta})}}$$ In essence we have proven that assuming covariance of the form of α− 1 under our correspondence allows us to deduce its experimental numeric value and its value does not change from the Quantum Mechanical value side. Then the Fine Structure Constant is an invariant in our correspondence α̃ = α Let’s notice that α̃− 1 is equal to the Lyapunov coefficient of the Rosller-like structure of OM-space times a coefficient dependent on its dimension. The Lyapunov coefficient generates in the Rosller internal flow the conservation of volume in the space complementary to the Rosller geometry. We propose that the conservation of this dual or complementary volume is the OM-QM correspondent to conservation of probability during QM evolution. By the other hand, α is a quantum Amplitude. More precisely, the probability of absorption or emission of a photon by a charged particle. Quantum Amplitudes are always volumes of certain geometrical objects embedded in certain spaces. Then, we can think of α̃− 1 as related to the volume of the hyperbolic complement of the OM-Knot. Then it makes sense that its value is almost exactly the coefficient of expansion-contraction in the OM-knot which secures conservation of that volume. The OM-QM correspondent of the electron’s charge and Electric Field =================================================================== As all physical quantities in the QM side of our correspondence have a OM-QM counterpart it is natural to try to find the OM-QM version of the electron’s electric charge. This is related to α via the formula $$\label{Eq52} \alpha = \frac{e^2}{\hbar 4 \pi \epsilon}$$ where e is the electron charge and ε is the vacumm electric permeability. This expresion of α comes from atomic physics and it contains the constant ε (the vacuum electric permitivity) which cannot be determined from first principles and comes from experimental measurement. If we invert this expression and apply the OM correspondence we find $$\label{Eq53} \tilde{\alpha}^{-1} = \frac{ 8 \pi^2 \tilde{\zeta\_0} \tilde{\epsilon}}{\tilde{e}^2}$$ where ẽ, ε and $ \tilde{\zeta\_0} $ are the OM-QM version of e, ε and the Planck Length. As The Planck Length is 1 in the scales transformation implied in our correspondence we can assume $ \tilde{\zeta\_0} = 1 $ Then from Eq.[Eq50] we get $$\label{Eq54} \tilde{\alpha}^{-1} = \frac{ 8 \pi^2 \tilde{\zeta\_0} \tilde{\epsilon}}{\tilde{e}^2} = D \sqrt{\exp{(\sqrt{\pi \delta})}}$$ which after rearrangement give us the following formula for the quotient $$\label{Eq55} \frac{\tilde{e}^2}{\tilde{\epsilon}} = \frac{8 \pi^2}{D \sqrt{ \exp{(\sqrt{\pi \delta})} }}$$ we will disentangle this quotient in the l.h.s. making an educated guess of which part of the r.h.s. corresponds to denominator and numerator. We arrive to the following OM-QM versions for e and ε ẽ = 2π $$\label{Eq57} \tilde{\epsilon} = \frac{D}{2} \sqrt{ \exp{(\sqrt{\pi \delta})} } = \frac{\tilde{\alpha}^{-1}}{2}$$ The value of ẽ can be interpreted as the residue of a potential of the form 1/z integrated around a loop around the minimum cell. What is more surprising is that the OM-QM counterpart of ε turns out to be half the OM-QM counterpart of α− 1. As we saw previously α− 1 is related to the Lyapunov coefficient of the Rosller-like microstucture of OM-space. Then ε̃ also is. This seems to indicate a totally unexpected connection between the OM-QM counterpart of ε̃ and the conservation of volume in the OM-speace Rosller-like geometry. Following we will see two arguments why this connection makes sense. First we can think in the following physical argument. In the previous section we showed that α̃− 1 is also related to the same dual volume conservation. Also, Physically, α̃− 1 and ε̃ are talking about the same property, the amplitude of the coupling of photons to particles and to the vacuum respectively. We see that their OM-QM correspondents fall into the exact same numerical value. We conjecture that this is an example of a general behavior of the OM-correspondence. Similar properties in the QM realm which differ only on whether they are talking about particles or about vacuum space will be mapped into the same mathematical object in the OM-QM side. We propose this is fact a central principle in the OM-correspondence, an analog to the Equivalence Principle in General Relativity. Now let’s see a geometric argument on why this connection is plausible. Let’s analyze the behavior of the (classical) Electric Field under the OM correspondence. In the following figure we show how the Electric Field of the electron changes after the OM correspondence ![image](figure4) We start with a charged particle in normal space-time with classical Electric Field pointing outwards the particle’s position both in the initial and final positions in the evolution path. When we modularize this space all the space-time regions near the extreme of the path are mapped to the region near the minimal cell and then in the OM-space all the transformed Electric Field lines will come out radially from the region near the minimal cell. As our modularization implies all lines have to enter this region in almost parallel fashion (and pointing in the same direction!) that means that one set of lines will be inwards to this region an another will be outwards to it. Moreover, due to the gluing of this central region we will have a mirror symmetry between the transformed fields entering and leaving this region. Thus, the transformed Electric Field is not spherical but rather hyperbolic in symmetry as the point in the infinite also has to be identified in the modularization. The transformed Electric Field looks like a Magnetic Field in an Hyperbolic Volume around the central region of the OM-Knot. If we identify this hyperbolic region where the Electric Fields lands with the hyperbolic complement of the OM-Knot itself then we have a very satisfying visualization of why the the OM-QM versions of the vacuum electric permeability is also so intimately and exactly related to the flow of volumes in the Rosller-like OM-space. It is worth mentioning that the emergence of a sort of the Electric-Magnetic duality described above is a common feature of Field Theories formulated on moduli spaces. Thus it is not strange seeing this also appearing in our correspondence. What is new here is that we can identify the target “Magnetic lines” as occupying a very meaningful volume in the moduli space: the hyperbolic complement of the OM-Knot. The OM-QM correspondent of Heisenberg’s Uncertainity relation ============================================================= Let’s recall the form of the only non null possible values of Ẽ(N) Ẽ2(pk) = P̃02(s̃2σpplog(p))Δp + s̃R̃(pk) + 2s̃2R̃2(pk)) where Δp is a distribution version of the Dirac delta centered at p. Let’s analyze this expression in two limits. First let’s see it in the limit p → ∞ limp → ∞Ẽ2(pk) → P̃02(s̃2σpplog(p))Δp We can notice that the expression plog(p))Δp is dependent only of the scale p and we know how those quantities transform under change of p (Eq.[Eq12]). Then if we calculate the difference of limp → ∞Ẽ2(pk) between two consecutive values of p we get the following formula $$\label{Eq60} (\Delta (\tilde{E}(p^k)))^2 = \tilde{P}\_0^2 \tilde{s}^2 D^2 \exp{(\sqrt{\pi \delta})} (\sigma\_{p+1} - \sigma\_{p})$$ Recalling the relationship between $ D^2 \exp{(\sqrt{\pi \delta})} $ and α̃− 2 we get the following in this limit $$\label{Eq61} \Delta (\tilde{E}(p^k)) (\frac{\tilde{\alpha}}{(\Delta(\sigma))^{1/2}}) = \tilde{P}\_0 \tilde{s}$$ We argue that this is the OM-QM correspondent to the Heisenberg Uncertainty relation for Energy and Time. In the r.h.s. of this equation we have the correspondent of the Plank constant. and in the l.h.s. a product of variations. One of them is the variation of the OM-QM correspondent of Energy. As we are assuming covariance under our correspondence then the quantity $ \frac{\tilde{\alpha}}{(\Delta(\sigma))^{1/2}} $ must be the OM-QM correspondent of variations of certain times. Thus, (Δ(σ))1/2 must be the OM-QM correspondent of variations of certain frequencies. This give us as a bonus the interpretation of the imaginary parts of the zeroes of the Riemann ζ-function as the OM-QM correspondent of frequencies associated to Ẽ. The OM correspondent Einstein’s Gravitational field equation ============================================================ Equation Eq.[Eq36] is the OM-QM correspondent to Klein-Gordon’s equation for a free particle, a second order relationship between Energy, momentum and mass. In the OM-QM side this becomes a relationship between the OM-QM correspondent of Energy, one term related to the OM-Knot curvature and an additional term which we will discuss in more detail below. In this section we will work with the values of this equation for finite values of the prime p. It is known that getting a first order relationship between Energy observable, momentum and mass was the starting point for Dirac for finding the quantum state Equation for the free particle in a relativistic context. The prize to pay for getting a first order relationship was that the coefficients in the equation no longer could be scalars but needed to be matrices, the Dirac matrices. We will see that in the context of our correspondence we also can find a first order relationship starting with Eq.[Eq41]. Let’s start by taking the difference of Ẽ2(p) between two consecutive but finite primes p. We get the following relation Δ(Ẽ) = P̃0(s̃Δ(R̃) + 2s̃2Δ(R̃2) + s̃2Δ(σpplog(p)Δp))1/2 where we have approximated Δ(Ẽ2) by [Δ(Ẽ)]2 The quadratic term Δ(R̃2) dominates over the linear one Δ(R̃) for finite p. It can also be approximated by [Δ(R̃)]2. We can then write Δ(Ẽ) = P̃0s̃(2[Δ(R̃)]2 + Δ(σpplog(p)Δp))1/2 The only way we can reduce this equation to a first order equation is if the factor under the square root is a square, that is whether there exists a Pythagorean relationship between 2[Δ(R̃)]2 and Δ(σpplog(p)Δp). From Number Theory we know that this question is related to the possibility to have unique complex prime factorization in a Number Field made up of elements of the form $$\label{Eq64} \sqrt{2} \Delta (\tilde{R}) + i \sqrt{\Delta (\sigma\_p p \log(p) \Delta\_p)})$$ And that is tied to the existense of a number close to an integer Number and of the form $ \gamma^2 \exp{(\sqrt{\pi \delta})} $ whith γ and δ positive numbers. The two squares in the Pytagorean relationship should sum up to precisely this number. In the case of the elements inside the square root in Eq.[Eq64] we know that both elements depend only of the scale and under changes in the number p they change according to the Logistic relationship $ D^2 \exp{(\sqrt{\pi \delta})} $. And we saw in previous sections that this number is very close to the square of the inverse of the Fine Structure Constant. It s also the factor by which volumes in the OM-space transforms under changes in p. This proves that we can write Eq.[Eq63] as the product of two mutually conjugate complex equations of the form $$\label{Eq65} \Delta\_+ (\tilde{E}) = \tilde{P}\_0 \tilde{s} (\sqrt{2} \Delta (\tilde{R}) + i \sqrt{ \Delta (\sigma\_p p \log(p) \Delta\_p)} )$$ We want to emphasize how remarkable this relationship is. Its existence is dictated Mathematically by the way OM-space transforms under changes in p which in turn determines the value of the inverse of the Fine Structure Constant. It is precisely that value (so close to an integer) what makes this relationship a valid one. It is a first order relationship between Ẽ, Δ(R̃) and another term. Similar to the case in the original Dirac’s result we have here a price to pay for the passing to a first order relation. In the Dirac equation there was the appearance of Dirac gamma matrices. In our case it is the occurrence of an imaginary unit in front of that last term $ \sqrt{\Delta (\sigma\_p p \log(p) \Delta\_p)} $. At this point we make connection with Einstein’s Field Equation in General Relativity as they are also a relationship between Energy and Curvature. They also include an extra term historically called the Cosmological Constant term, a term which modernly we associate to the action of Dark Energy. We are proposing that Eq.[Eq65] is indeed the OM-QM correspondent to Einstein Field Equation for the Gravitation Field in their following form $$\label{Eq66} T = \frac{c^4}{8 \pi G} ( (1-\frac{D}{2}) R + D \Lambda )$$ where T is the net value of the Mass-Energy tensor, G is the Gravitational Constant, D is the space Dimension and Λ is the Cosmological Constant term. Applying the OM correspondence to this equation we get $$\label{Eq67} \tilde{T} = \frac{\tilde{c}^4}{8 \pi \tilde{G}} ( (1-\frac{D}{2}) \tilde{R} + D \tilde{\Lambda} )$$ substituting c̃4 = 8π2 we arrive to the following formula $$\label{Eq68} \tilde{T} = \frac{2 \pi^3 }{ \tilde{G}} ( (1-\frac{D}{2}) \tilde{R} + D \tilde{\Lambda} )$$ Comparing Eq.[Eq68] and Eq.[Eq65] we get the following relationship $$\label{Eq69} \frac{2 \pi^3 }{ \tilde{G}} (1-\frac{D}{2}) = \sqrt{2} \tilde{s} \tilde{P}\_0$$ from which we get the following expression for the OM-QM correspondent to the Gravitational Constant $$\label{Eq70} \tilde{G} = \frac{\pi (1-\frac{D}{2})}{2^{2/3} \tilde{s}}$$ We also get the following from the comparison $$\label{Eq71} \frac{2 \pi^3 }{\tilde{G}} D \tilde{\Lambda} = i \tilde{s} \tilde{P\_0} \sqrt{\Delta (\sigma\_p p \log(p) \Delta\_p)}$$ from which we get the following form of the OM-QM correspondent to the Cosmological Constant $$\label{Eq72} \tilde{\Lambda} = \frac{i \pi }{\sqrt{2}} \frac{(1-\frac{D}{2})}{D}\sqrt{\Delta (\sigma\_p p \log(p) \Delta\_p) }$$ The OM-QM correspondent of Holography ===================================== Holography is one of the most exiting discoveries in Physics in the recent decades. Starting with the original idea from Gerard T’Hooft and followed by the insights of Susskind and Maldacena it has already got experimental applications in several areas of Physics . In essence Holography states that every Classical Physical Theory with Gravity in a certain D+1 volume (in a certain space) can be mapped to Quantum Theory without Gravity on the D dimensional boundary of that same D+1 volume. Quantum Mechanics is an holographic Theory, so is General Relativity. That is why it is generally believed that Holography will be an essential component of any future Quantum Gravity Theory. One explicit toy model of such Theory is the so called AdS/CFT correspondence by Maldacena. Here AdS is a classical gravity theory living in a finite hyperbolic space (AdS) and CFT is the holographic quantum dual theory living in the boundary of AdS. This correspondence gives a very convenient toy-model for visualization of the details of the holographic mapping to the boundary and a how the quantities inside the gravitational bulk do get mapped to quantum observables in the boundary. Gravity for example must become null in the boundary. More recently Maldacena and Susskind have proposed a physical interpretation of entanglement as the holographic dual of an Einstein-Rossen bridge, the so called ER=EPR Theory. Entanglement being the the source of so many of the apparently counter intuitive properties of QM would then be the shadow of an object in a higher dimensional gravitational dual. Let’s see how holography gets mapped under the OM correspondence. We start by analyzing the formula for G̃ $$\label{Eq73} \tilde{G} = \frac{\pi (1-\frac{D}{2})}{2^{2/3} \tilde{s} }$$ Let’s remember that D is the dimension of the Rossler-like micro-structure of the OM-space. It’s a fractal dimension close to 2.9. Let’s see how is the behavior of G̃ when we vary this dimension D towards 2. It indeed becomes null as it should if we were dealing with a proper holograhic limit. Let’s now analyze the denominator. Here we have s̃, which for us is just a complex number by now arising in the our transformed Dirac equation. Using that equation we can rewrite the following $$\label{Eq74} \tilde{G} = \frac{\pi (1-\frac{D}{2}) \tilde{R}}{2^{2/3} \tilde{m} }$$ where we have substituted s̃ by the quotient of the OM-Knot curvature and m̃. We can interpret then G̃ as how much D-1 dimensional curvature is generated by m̃. Here we are off course using the fact that we have imposed covariance of the meaning of the mathematical quantities we have in our equations respect to the physical quantities in the quantum side of our correspondence. That is, the purely mathematical equations in the OM-QM side can shed some meaningful information about the observables in the P0 side and vice versa. Rearranging Eq.[Eq73] we get $$\label{Eq75} \tilde{s} \tilde{G} = \frac{\pi (1-\frac{D}{2})}{2^{2/3} }$$ in the r.h.s. of this relation we have a pure number. That tell us that m̃ has a duality relation with G̃. m̃ must represent something dual to a curvature in D-1 dimensions. Eq.[Eq75] is an holographic relation too. Therefore we propose this give us an interpretation of s̃ as a curvature in higher dimensional OM-space. If we believe in the covariance imposed into our correspondence then this give us the following interpretation of the spin observable in QM The quantum mechanical observable we call spin in the usual QM is the D-1 shadow of the curvature of the OM-Knot in a dimension perpendicular to all those D-1 dimensions we can observe. The spin in the OM correspondence can be interpreted as a curvature in a higher dimension we cannot observe directly. We just see its shadow as quantum mechanical effects. This dimension is coupled to the dimensions we can observe Going forward we will adopt the following notation. We will call R̃H this curvature in this higher dimension we cannot observe directly and we will use it instead of s̃ in our equations. We will call R̃L the curvature in the D-1 dimensions which we can observe directly. Thence, we can rewrite the transformed Dirac equation in this suggestive way R̃HR̃L = m̃ Let’s visualize this in the Rosller-like micro structure of the OM-space. In the following figure we can see the parts of the OM-Knot in the D-1 dimensions we can observe and the “reinsertion flow” of the knotted paths which must necessarely go outside of the D-1 volume in order to achieve the Rosller reinsertion. ![image](Figure5) The curvature of this “reinsertion flow” is the spin. This also explains the quantized values of spin we observe in QM. The reinsertion flow can flip only a integer multiple of 1/2 times in its trajectory in the high dimension. That is why we see a total shadow effect in the lower dimensional in the form of the appearence of a quantity which can only have values which are multiples of 1/2. As we see, if we work in a QM setup we are really not seeing the full picture but only the holographic projection of the D dimensional Rosller OM-space into D-1 dimensions. As we saw in the case of the spin, quantization in D-1 dimensions is an effect of the coupling of that extra dimension to the D-1 ones we can observe directly. The spin and other quantized quantities can be then viewed as a window to that higher dimension we cannot observe. Let’s then try to generate a framework where we see an unified picture in all the D dimensions of the Rosller OM-space. We start by noting that the total curvature in the full OM-space can be written in this way R̃ = R̃H + R̃L Let’s take the square of this total curvature R̃2 = R̃H2 + R̃L2 + 2R̃HR̃L and using Eq.[Eq76] we get R̃2 = R̃H2 + R̃L2 + 2m̃ We saw before that the possible values of R are of the form $ \frac{1}{N} $. Then R2 = 1/N2. As R is curvature in the hyperbolic complement of the OM-Knot then the inverse of the square of the curvature radius is equal the variation of D-1 dimensional Area in that hyperbolic complement. Visualization of this fact is difficult in OM-space. However an analogy with the AdS/CFT visualization of the relations between Area in AdS space and length in the boundary of that space is a good analogy for any holographic theory. We can then write R̃L2 = ΔÃL where ΔÃL is a variation of Area in the hyperbolic complement of the OM-Knot in the D-1 projection. Using this we can rewrite Eq.[Eq79] in this form ΔÃL = R̃H2 − R̃2 + 2m̃ The difference R̃H2 − R̃2 is equal to minus the difference of OM-Area after one single flow of the OM-Knot in the higher dimension. We will note this in this by ΔÃH. Thus we arrive to this result ΔÃL = − ΔÃH + 2m̃ *In itself this is a remarkable relationship between OM-mass and variation of the total OM-Area. It provides an holographic interpretation of m̃* as a variation of the OM-QM correspondent of Area of the hyperbolic complement of the OM-Knot.** We can give a more explicitly holographic form of this formula recalling that in any holographic theory there is a universal relationship between the Area in the D dimensional Theory and the entanglement entropy in the quantum theory living in the boundary D-1 space. This is the so called Area Law of holography. This relationship first appeared in Black Hole physics in the works of Jacobson and Hawkings. In its pure Quantum Mechanical form we can write it like this ΔAL = − ΔSL4G Where SL is the entanglement entropy in the boundary of the D dimensional space and G is the Gravitation constant. Its OM-QM form can be written as ΔÃL = − ΔS̃L4G̃ Substituting it in Eq.[Eq82] we get the following relationship $$\label{Eq85} \Delta \tilde{S}\_L = \frac{\Delta\tilde{A}\_H}{4 \tilde{G}} - \frac{\tilde{m}}{2 \tilde{G}}$$ This, we propose, is the OM-QM version of another universal holographic relation, the Ryu-Takayanagi relation which (in its pure Quantum Mechanical version) it states the following relationship between Area, space related Entanglement Entropy and matter related entropy $$\label{Eq86} \Delta S\_L = \frac{\Delta A\_H}{4 G} - (matter-entropy-term)$$ The emergence of such a relationship strongly supports the idea that OM-QM is also a proper holographic Theory and holography is covariantly preserved in the OM-correspondence. Conclusions and outlook ======================= In this work we have proven that we can construct a very special spatial background by identifying regions of normal space where observation (or measurement) are implied (OM-space). We have shown that Quantum Mechanics over this background reduces to a highly non trivial and interesting Number Theory (OM-QM) if we simultaneously make use of a reparametrization replacing the Planck constant by the quantity 4π2ζ0 (where zeta0 is the Plank Length) or equivalently replacing the Plank Momentum (P0) by the number 4π2. Although we are transforming QM statements into pure Mathematical ones we are demanding this transformation to preserve the relationships among the original physical objects into the corresponding relationships among the target mathematical objects. This requirement of covariance causes that all objects in the QM side do transform under this correspondence. Elements like mass, energy, wave function, electric charge, spin, position, time, speed of light and all their derived objects become Number Theoretical objects in the OM-QM side. The functions and values they adopt as well as the transformed QM equations give a lot of new and non trivial insight about the objects the Physical side. We found the OM-QM equivalent to the Dirac and Klein-Gordon equations. From them we have learned that the OM-QM correspondent to Energy of a free particle are simple functions of the zeroes of the Riemann ζ-function. Similar relation between those zeroes and quantum energy eigenvalues been suggested in the past by several authors and here it appears naturally. The exploration of the meso and micro structure of OM-space lead us to a toy model local representation via a geometrical object similar to a Rolller attractor which exposes the dimensional reduction implicit in the modularization of OM-space. One of the main conclusions of this work is that topics which seems to be hard or of unknown origins in the QM side become easy to treat or to interpret in the OM-QM side. The mathematical insight we can get in the OM-QM side can shed some light on the nature or relationship among physical objects in the QM side (and vice versa!). This Rosller like structure allowed us to propose a first principles deduction to the exact numerical value of the Fine Structure Constant. Based on this toy model we were also able to find the OM-QM analogues of the Heisenberg relationship for Energy and Time an the Einstein Gravitation Field Equations emerging as two limits from the OM-QM analog to the Klein Gordon equation. Although the intention in this work has been to stick to basic Quantum Mechanics not mixing when possible into the hard topics o Quantum Gravity we think that OM-QM is in fact some sort of Quantum Gravity Theory. It is so in the same sense that for example Maldacena’s AdS/CFT is a full Quantum Gravity toy model. In our case we were not attempting to insert Gravity in the discussion at all but it appeared naturally into OM-QM through the implications of the OM-space Rosller micro structure into equations relating Energy and curvature of mathematical objects in the OM-QM side. It is exactly the internal geometrical flow inside this Rosller like structure which allows us to pass from a second order Klein-Gordon OM-QM equation to a first order Einstein Gravitation field OM-QM equation. This is a highly non-trivial result and unique to the a Rosller like structure of OM-space. No other kind of geometrical structure would allow for such a result. The fact that holography is preserved and manifest in the OM-correspondence is also a sign that OM-QM is a type of Quantum Gravity toy model. It not only proves good internal consistency but it also provides new insights like an interpretation of spin of a free particle as higher dimensional curvature of the Rosller OM-space micro structure. Also the OM-QM mass correspondent gets an interpretation as variations of the OM-Area of a geometrical object in the OM-QM side. We demonstrated that this in disguise the OM-QM version of the Ryu-Takayanagi universal relation for holographic theories. Another way to see OM-QM as a Quantum Gravity Theory is that based on analysis of the OM correspondents of similar QM quantities we were able to conjecture an OM-version of the Equivalence Principle in General Relativity. In essence we found that statements or quantities in the QM side which differ only in that one is speaking about particles and the other about vacuum space are mapped in exactly the same mathematical object in the OM-QM side. This important conjecture, its application and meaning is an obvious future venue of investigation for the OM-correspondence. We think that it could play a similar central role in the OM-correspondence as the Equivalence Principle plays in General Relativity. Our intention in the paper was introducing the OM-correspondence with an absolute minimum of assumptions or without entering in side discussions on topics which are hard to grasp in physical or mathematical terms. Nevertheless, we see that the richness and cross connected nature of OM-QM is so large that one can speculate with the potential applications of it to some of those harder problems. As we mentioned above, exploring the value of OM-QM as a valuable toy mode for Quantum Gravity would be one of the potential future directions. In addition in as a work following up to this one we will apply the OM-correspondence to the following topics which in standard QM are hard or have some mysterious aspects to them : * The State reduction principle (or Wave function collapse) * The measurement problem in QM * The EPR experiment interpretation As potential future work venues we will explore the possibility of an experimentally observable consequence of OM-QM in the Cosmic Microwave Background Map or in the observed matter clustering in the Early Universe. Applying the OM-correspondence to Black Hole physics could be also a potentially great area for exploration. Finally, we want to bring attention on what is probably the most striking and new aspect of the OM-correspondence and that is the reparametrization we have called the P0 − 4π2 correspondence. We have explicitly avoided entering in any deep interpretation on why this works so well together with the OM-space structure. This reparametrization seems to balance the effect in QM of the OM-space modularization in such a way that by requiring covariance in the form of equations we get in the mathematical side objects which have very desirable mathematical relationships if they would be correspondents to their physical counterparts. This is most evident in the case of a pure numeric quantity which is also crucial in QM, the Fine Structure Constant. The OM-correspondence maps it into something which is also very central to OM-QM. Also the relationship between the OM-QM values of the free particle Energy and the zeroes of the Riemann ζ-function is possible only through this parametrization. We suspect the P0 − 4π2 correspondence says something deep about the relationship between QM and Mathematics. OM-QM itself seems to be a sort of common playground where both QM and parts of Mahematics can be mapped one to each other. The search for this kind of relationships between QM and several areas of Mathematics has been a very active area of research in last decades. The Langlands Program is one of those ideas in Mathematics trying to build bridges with QM at a fundamental level. We think that the OM-correspondence may be studied from a more formal mathematical point of view which could shed some light on the deeper meaning of the P0 − 4π2 correspondence. We propose that studying the potential relevance of the OM-correspondence for the Langlands Program in Mathematics could be a high value potential future research venue. Quantum Mechanics on a background modulo observation ==================================================== Abstract In this work we will answer the following question: What remains of Quantum Mechanics when we transform the background space-time into a space modularized by observation or measurement regions? This new moduli space is constructed by identifying regions of space-time where quantum phase comparison (observation, measurement) is implied. We call it Observation Modular space (OM-space). In addition we replace in QM statements the Plank constant (h) by the quantity ζ04π2 (where ζ0 is the Plank Length) or otherwise, replacing P0 (the Planck Momentum) by 4π2. This maps Quantum Mechanics into a very rich dual Number Theory which we call Observation Modular Quantum Mechanics (OM-QM). We find the OM-dual to the Dirac Equation, the quantum Wave Function and a free particle’s mass. The OM-QM counterparts of the Energy turns out to be a simple function of the zeroes of the Riemann zeta function. We also find the OM-QM correspondents to the electron spin, the electron charge, the Electric Field and the Fine Structure Constant. We also find the OM-QM correspondents of the Heisenberg uncertainty relation and Einstein’s General Relativity Field equation emerging as certain limits of a unique OM-QM equation. We also get the OM-QM correspondents of the Gravitational Constant and the Cosmological Constant. We find the analog of holography in the OM-QM side and we get an interpretation of spin as a high dimensional curvature. An interpretation of the OM-QM correspondence is proposed as giving the part of QM information which is not measurement or observation dependent. Some potential future applications of this correspondence are discussed. Introduction ============ In this work we will ask the following question: What remains of Quantum Mechanics when we identify (modularize) regions of the space-time background where observation or measurement is implied? To be precise we are not talking here about the Observer itself as this is a more complex topic which in general is not well understood and difficult to model inside Quantum Mechanics (QM). Neither we are talking here about the Measurement or Observation process itself which is also a very hard problem in QM. One would think (naively) that according to the Copenhagen interpretation of QM the answer to this question would be : Nothing at all remains except from Mathematics disconnected from QM physical content. According to this interpretation (the most standard and accepted one) QM is only talking about the information one can get in measurements involving an observed system. Thus, if we strip QM of observation (represented by a concrete physical mechanism like quantum wave function phase comparison between different regions of space-time) there should be no physically meaningful information content left in the residual Theory. That is, naively we could think that if we reduce QM over a moduli space defined modulo those regions where that observation or quantum phase comparison is happening, the resulting Theory should have no structure left other than probably pure Mathematics. Here we will prove that, contrary to that naive conclusion, we can construct a very specific such modulli space for which the reduction of QM on it is highly structured and non-trivial. We will call this moduli space Observation Modular space (OM-space ). Moreover, this resulting Theory while being purely Mathematical heritages (by design) a lot of the interconnections between Physical observables and makes evident surprising new interconnections and insights coming from the mathematical structures in it. Although one could define such modularization in apparently infinite ways, we use a very specific parametrization of the target theory which gives very suggestive and meaningful interconnections between Mathematical objects matching the ones in the Physical side. We call the residual Theory resulting after this parametrization and modularization Observation Modular Quantum Mechanics (OM-QM ). The intuition which takes us to this special parametrization is the following: The number 4π2 (a transcendental number) and the Physical quantity h (the Quantum of Action or Planck’s constant) play a similar role in Number Theory and in Physics respectively. 4π2 is connected to the distribution of ensembles of prime numbers. It is the result of the multiplication of all prime numbers “alla Euler”, that is, via Analytical Continuation of that infinite product. It’s in that sense that we should understand the statement that 4π2 “knows” about the distribution of ensembles of prime numbers. By the other hand the Planck constant encodes the information about the distribution of ensembles of quantum states in an arbitrary physical system. In both cases we are talking about statistical distributions of certain irreducible elements of the theory. In the case of h that relationship with the distribution of quantum states is more “by design”. From a mathematical point of view the Dirac axiom of QM is a statement about the relative distribution of eignenvalues of certain non-commuting, Hermitean operators (the physical observables) acting on a Hilbert space. h is the only parameter in such distribution. We argue here that there is more than an analogy between the roles of those two constants. We conjecture that every statement in Quantum Mechanics can be mapped into a corresponding Number Theoretical statement which results from replacing h with the quantity ζ04π2 (where ζ0 is the Planck Length) while at the same time modularizing the Space-Time background by identifying regions where quantum phase comparison is implied. Another way to say the same is that there is a correspondence between a Quantum Mechanical statements and a certain Number Theory statement which results from replacing the Planck’s momentum, defined as : $$\label{Eq1} P\_0 = \frac{h}{\zeta\_0}$$ with the number 4π2 and transforming Space-Time in the aforementioned way. We will call this reparametrization the P0-4π2 correspondence. We will call the full process of transforming a QM statement into its OM-QM dual the OM-correspondence. This process amounts to reducing Quantum Mechanics over a moduli-space background where the modularity is given by identifying regions of space-time where observation or measurement is implicit. We will define and explore this OM-space in the rest of this work. The effects of such OM-correspondence on QM seems to be fine tuned to give us a dual highly non-trivial Number Theory surprisingly full of physical and mathematical interconnections. We will analyze its internal consistency as well as potential connections with several similar ideas which have appeared in the past in the literature. Indeed, monduli spaces in relation to quantum based theories are not a new topic. The subject of Topological Quantum Field Theories (QFT) develops similar ideas but in a Field Theory context. In QFT the interest is on finding the effects on a Quantum Field Theory when one replaces normal space-time with a moduli-space constructed via some kind of modularity related to the Gauge Symmetries in the Fields . In QFT the starting point is an action which is independent on the space-time metric. Those actions are dependent only in certain functions of the field potential which give raise to Feynman amplitudes which in turn are topolocical invariants (Knot invariants sometimes) of the moduli space. A reference which is even more relevant for our work is Gao’s work on AdS/CFT correspondence over a quotient space. In it a moduli (quotient) space version of the bulk AdS space is introduced. The modularization in this case is via some large, general quotient Group. Gao proceeds then to find the effects of this modularization in the boundary quantum CFT Theory using the AdS/CFT correspondence. This is very similar to what we will attempt in our work. What is new in our approach is that we are in general interested in the effects of that modularization in Quantum Mechanics iteslf in a more realistic setup. We are not going to base on a toy model bulk space-time background as AdS but instead base in the usual 4-Dimensional space time and in full Quantum Mechanics. Neither we will use a large group for producing a quotient space but instead we will use observation regions of space as our quotient criteria. Also it is new the P0-4π2 substitution we use in our OM-correspondence. The main conclusion we will be lead to is that the OM-QM construction reduces the degrees of freedom available in Quantum Mechanical information in a very special way. It leaves us with a Physics independent core of that information which itself contains a rich structure. We can say that that residual Number Theory core we are left with is the Observation independent content of Quantum Mechanical information. We will see that topics which are difficult or obscure in the usual QM framework become natural and easy to deal with in the OM-QM side. Topics in QM do get mapped to very concrete and clear Mathematical statements which often throw some light on their nature in the QM side. In Mathematics we often see similar simplification or insight when we pass some problems to a modular framework. OM-QM can be seen as the counterpart of that Mathematical modular simplification or insight in QM. We will first apply the OM-correspondence to the Quantum Mechanics of a free electron. That is, we are going to find the OM-dual of the Dirac Equation. This will gives as a by-result that the OM-QM counterparts of the particle mass and energy are simple functions of the zeroes of the Riemann Zeta function. Then we will use this correspondence for finding a fist principles derivation of the value of the Fine Structure Constant (it precise numeric value!) In this work we will adopt a Physics approach for developing the topics at hand. We will not delve too deep in the Mathematics aspects although we will be dealing with deep Mathematical concepts. For example when talking about moduli-spaces one could adopt a very strict Mathematical approach as that is in reality a very complex Mathematical subject. We will use here the wording moduli-spaces as a placeholder for an eventual future formalization of the Physical ideas we will propose. We will go as deep as we need in the Mathematics but not so deep that we lose focus on our main goal, finding a precise definition of the OM-correspondence and proving that it produces an interesting and very non trivial effect in QM. The Theory we will develop below seems to be a small common playground where Physics and Mathematics can play together for the sake of finding new or known interconnections. We propose that OM-QM is in fact a toy model of a Quantum Gravity Theory. In the final section of this work we will briefly talk about potential future applications o this correspondence. The OM correspondence ===================== The correspondence we are postulating in this paper can be summarized in the following procedure: * [*step 1] Take a valid Quantum Mechanical statement (an equation, a definition, a postulate). Let’s call this the P0 side of the correspondence. The OM corresponding statement can be constructed via the following operations:* * [*step 2] Replace any instance of the Planck constant h by the quantity ζ04π2 (where ζ0 is the Planck Length). Or equivalently, replace P0 (the Planck Momentum) in the QM side by the number 4π2* * [step 3] Identify all regions in the background space-time where a quantum phase comparison (that is measurement or observation) is implicit in the quantum statement (up to homotopy classes). Keep the form of the QM equations covariant in the transformation. Then find the effect of such transformation in all elements in the quantum statement. We will make a unique and precise definition of this modularization in next section. In the rest of the paper we will notate with a tilde the OM-QM correspondent of any quantum quantity. Then by definition we have $$\label{Eq2} \tilde{P\_0} = 4 \pi^2$$ We will also make use of terms like “OM-Energy” or “OM-mass” for referring to the OM correspondents of certain Physical observables. OM-space: Space-Time modulo regions with implied Quantum wave function phase comparison ======================================================================================= Let’s analyze the step 3 in the above mentioned procedure. As one of the elements of that step we impose covariance of the Quantum equations under the space-time transformation we are performing. That is, we are imposing that the form of the dual Number Theoretic statement should be the same as the form of the Quantum Mechanical statement. We know which are the space-time transformations under which Quantum Mechanics is covariant. Ignoring (by now) the presence of any gauge fields in the Theory, in the case of non-relativistic QM those are the transformations from the Galileo Group. In the case of relativistic QM those are the Lorentz group transformations multiplied by the internal spin symmetry group. These internal spin symmetry groups are representations of the SU(2) group acting on 4 components spinors. The internal symmetry groups can include Gauge groups if we are talking about the QM of particles interacting with Gauge fields . This covariance enforces the elements of the QM equations to transforms in very specific ways. For example mass must transform as scalar under Lorentz transformations. Gauge charges (electric, color, lepton flavors) will transform as vectors under the internal gauge groups. The particle states will in turn transform as spinors in the relativistic case and under the internal gauge groups if gauge charges and fields are considered. The space-time transformation we use in Step-3 of our correspondence is obviously none of those transformations under which QM is covariant. Therefore if we want to impose covariance under our space-time transformation the way the QM elements transform must be different than in the way QM transforms under its covariance transformations. We must assume that all elements in the QM equation must transform in general in order to keep covariance. Elements as Mass, Gauge Charges (electric of other), Spin and the state vectors will all change in these transformations in ways we will find in the rest of this work. In fact Step-3 is all about the fact that even when we are imposing that the form of the equation should not change then all its elements should change to keep that covariance. Finding out how those elements change in order to keep covariance is the core of this work. In this work we are indeed talking about a type of covariance which is not Lorentz or Gauge covariance. We argue that this covariance is related to the observation itself. We are imposing that the QM laws should keep their form even when we take out observation or measurements out from the picture. Or otherwise, what we are defining here is QM Modulo Observation. Although Step-3 in our OM correspondence is a simple definition from the physical point of view it is not very practical if one wants to apply it in an operational way to real quantum mechanical statements of any kind. If we want to apply this correspondence in practice we need to describe in a more precise and operational way what we mean by modularizing the Phase Space by identifying regions where quantum phase comparison is implicit. We will use a similar reasoning to the one Maldacena and Susskind employed in their ER=EPR correspondence . They needed to make more quantitative their principle of identifying regions of space-time connected via entanglement. In their case they studied the correlations between states in entangled Space-Time regions and compared them to the correlations of events in Space-Time regions connected via an Einstein-Rosen bridge. They mapped the “gluing” of those regions into a more operational notion: probability correlation of states in those regions. For the OM correspondence we need a similar more operational form of Step 3. We will not try to find at this point an explicit form of the Phase Space transformation implied in that step (we will do that later in this work). We will define instead our moduli Space-Time in a totally equivalent way from the Quantum Mechanical point of view. It is enough describing how such transformation of the space-time acts on Integrals over paths in Phase Space. In Feynman’s interpretation of QM, the central object of the Theory is the quantum Amplitude given by the Feynman path integral $$\label{Eq3} \mathcal{A}\_{\alpha \rightarrow \beta} = \int\_\mathbf{C} ( \exp{ ( \frac{i\S}{\hbar} ) } ) ds$$ where C* is the set of all possible paths in Space-Time taking us from an initial point α to a final point β and S is the Classical Action for the system. All the information about results of measurements is contained in that amplitude. According to the Copenhagen interpretation of QM that is all the information we can expect to know about quantum systems. Therefore, if we find what is the effect of the modularization of Phase Space described in Step 3 on that Amplitude we will have an equivalent and more operational form of that Step 3 in the OM correspondence. ![image](Fiure1) Here we make a crucial observation about the end points of the set of paths C (the start classical space-time point α and end point β ). There is a quantum phase comparison assumed in each of them. We motivate this by saying that the fact we are talking about those two extreme points for all the paths carries the assumption that there is a measurement implicit at both ends. Nevertheless, the value of the Feynman amplitude does not depend on a measurement or observation of one concrete observer over the system. According to the Copenhagen interpretation of QM the only thing this Amplitude is saying is that if one performs the experiment or observation for measuring the probability of transition of the system from a state centered (according to the uncertainty dispersion) in the space-time point α to a state centered in space-time point β then the value of such probability is given by Eq.[Eq3]. In this Amplitude formula we are always speaking about information from experiments or observations. There is observation implicit in it. The set of paths C is in general infinite and other than the experimental constraints there is no limitation in how long or convoluted they can be. In fact QM says we have to integrate over all of the possible paths. The potential observational information in the set of paths comes from two sources: from the common end-points and from some experimental or topological constraints implicit in the set up of the Classical system. For example if we are performing the double slit experiment, there are obvious topological/geometrical constraints implicit in the measurement and in the amplitude. In addition we have the constraints coming from the extreme points of the the paths. In Step 3 of the OM correspondence we explicitly mention that the identification of regions of the space-time is up to homotopy classes. This topological reduction or identification of paths accounts for the observation of topological constraints in the system. But we still have to take into account the part of the amplitude implicit in the end points of all the paths. We argue here that this localization of the end points are equivalent to a measurement. In fact we argue that any kind of constraint on the system should be considered as assumed measurement and therefore a quantum phase comparison. In any measurement or constraint a quantum phase comparison is implicit. Therefore following Step 3 we can identify both end points α and β up to a cell of minimal size in space-time (a pixel given by Heisenberg’s uncertainty relation). Going forward, we will consider that minimal size pixel to be a Planck Length-Time radius ball in 4 dimensions. The net result of this reduction of number of paths due to topological equivalence and to the identification of the end points as described above is that the set of paths C gets mapped to a finite set of closed loops united in a fuzzy way on that minimal size pixel of space-time. In Figure 1 we describe this transformation in a pictorial way in 2-D for the sake of visualization. First thing one can notice in Figure 1.c is the way the transformed end parts of the paths “approach” the minimal size cell. They enter the region close to the minimal cell in an almost parallel way. Not totally parallel but the closer they get to the minimum cell the more parallel they become. The reason for this is the following: If this transformation is a proper modularization then regions near the extreme path points should map to regions close together. Regions far from the path’s end points are free of this constraint. That means the paths will be mapped into sort of “petal loops” (Going forward we will call these loops the OM-petals) meeting at the central minimal cell. This fact will be important at the moment of interpreting the meaning of this spatial transformation. In Figure 1.d we can also see that under the transformation above the winding of the resulting loop around the minimal cell can be a Natural Number (n) larger or equal to 1. In order to differentiate these winding loops from OM-petals we will call them contractible OM-loops. In a 2-D visualization as Figure 1.d this winding around the minimum cell is partially dependent on the projection of the 3-D spacial paths to a 2-D view. However the maximum winding number is not dependent of this projection. In at least one of the ways to perform this 2-D projection one gets a maximum possible such winding number. In any case the resulting general topology will be always the one of a Knot. Going forward we will call it the OM-Knot. This OM-Knot is composed of one (or more) OM-petals an a integer number n of contractible OM-loops. We can make at this point another important observation which we will use in the rest of this paper. The length of the paths connecting the extreme points gets transformed into a integer multiple of the minimal cell space size (the Planck Length) Nζ0 This is due to the fact that the loop can be homotopycally deformed to a loop curled directly over the minimal cell. Therefore we get the following OM correspondence for one position coordinate $$\label{Eq5} \tilde{x}= \frac{x}{\zeta\_0} = N$$ Let’s note that this variable N and the number n of contractible loops are in general different natural numbers. n is partially dependent on the geometry of the system we are considering but it can also depend on symmetries or other physical constraints. In this work we will almost ignore the variable n and use it only for visualization. This variable hides in reality a very deep quantum baggage and will be the topic of a future work. In this paper we will focus mainly in the variable N which has a pure geometric origin. The details of what the OM-Knot “petals” do far from the central minimal size cell is not influencing paths integrals of holomorphic (or branch-holomorphic) functions. Only the path integration around the central cell contributes to the end result. We see here the first hint that the Theory resulting from the applying the OM correspondence to QM will lead us to a Mathematical Theory, more precisely a Number Theoretical framework. We will use the number N loosely in this paper. We won’t consider it to be strictly a Natural number. For us it will be just a numeric parameter which arises due to the background transformation introduced in the OM correspondence. We can allow it to vary as a real parameter for the purpose of the rest of this paper. However, we will see later that most of the OM-QM counterparts of the quantum statements adopt the form of Number Theoretical statements and N will naturally be identifiable as a Natural. From the discussion above we get that the Space-Time transformation defined above allow us to give an operational form to Step 3 in our correspondence as the following way to transform a general path integral of an arbitrary phase-space function f(x,p) ∫C[α − β]f(x, p) → ∫L[α − β]f̃(x̃, p̃) where C[α − β] is a path (or a set of them) connecting states centered at Space-Time points α and β, L[α − β] is a “petal-like” Loop (or a set of them) connecting at a minimal cell of the transformed space at points α − ε and α + ε. f̃, x̃ and p̃ are the OM-QM corresponding objects to the integrated function, position and momentum. Here x should be understood as a vector in 3 dimensions though below we will work over one of its components. We should keep in mind that for Knots the number of spatial dimensions is very relevant but we will see we will be able to work with each coordinate separately for the sake of visualization and for reducing complexity. x̃ is given by Eq.[Eq4]. p̃ can be found easily knowing x̃ and remembering the expression of the quantum momentum operator $$\label{Eq7} p=i\hbar \dfrac{d \Psi(x)}{dx}$$ with Ψ(x) the Wave Function of the quantum system. Applying the OM correspondence we get $$\label{Eq8} \tilde{p}=i 2 \pi \dfrac{d \tilde{\Psi}(N) }{dN}$$ where $ \tilde{\Psi}(N) $ is the OM-QM counterpart of the Wave Function. Later we will find a concrete expression for this $ \tilde{\Psi}(N) $. By now we only need to know that it is a function of N through its dependency on x̃. We need one more element to be able to make progress in finding what corresponds to QM in the OM-QM side. Although we could start with a non relativistic version of QM we will instead work in a Relativistic quantum framework from the start. To that end, we will need to find the OM counterpart of the speed of light ( c̃ ) as this is something which also gets mapped to certain mathematical object under the OM-correspondence. We know how distances get mapped according to Eq.[Eq4]. In order to know how a speed gets transformed we need to understand how time gets mapped in the OM-QM side. In the definition of the OM correspondence we mentioned that this is essentially an holographic theory where we are reducing degrees of freedom via the identification of certain regions of space-time. In every holographic transformation time gets mapped to an spacial radial coordinate tangent to the target spacial dimensions . In Figure 1.2 we visualize how this looks for the case of our modularized space-time. ![image](Figure2.jpg) We can see time can be identified with the radius r in that picture and we should assume the transformed time flows in the direction of the central minimal cell. The length of the spacial dimensions is as we know Nζ0. Therefore we have $$\label{Eq9} \|r\|= \frac{N \zeta\_0}{2 \pi}$$ this can be assumed to be a minimum time similar as Nζ0 is a minimum length. Then we can write $$\label{Eq10} \|\tilde{c}\| = \frac{N \zeta\_0}{\|r\|}=2 \pi$$ and finally to take into account the direction of this transformed radial coordinate to point inwards towards the central cell (r⃗ = i∣r∣), we finally find c̃ = − 2iπ The micro and medium scale structure of OM-space ================================================ In Figure 1 and in the previous section we got a visualization of the micro structure of the OM-space. Each loop of the fuzzy OM-knot leaves and enters the central minimal size cell in an almost parallel fashion. In general the loop can wind around the fuzzy center cell for a large number of full rounds. In a two dimensional visualization all those loops can be thought as a ring structure made of loops circling the central fuzzy center of the OM-Knot. However, we must notice that as we do not allow self crossing of the loop into itself the re-injection of the loop into the central region must occur in a direction above or below that ring structure. In the following figure we see a visualization of this medium scale structure. ![image](Figure3) It is very similar to an object known in the area of Dynamical Systems and Chaos as a Rosller attractor . To be precise this is just and analogy we are proposing. The real target object has a very close resemblance of the Rosller attractor but its real structure can have some variations as we will see later. Nevertheless, going forward we are going to use the name Rosller-like geometry to describe the micro structure of the OM-pace. We propose that this mathematical object is in effect a good representation of the medium scale structure of OM-space. It also shows a flow of loops around a central region in an almost two dimensional ring with re-injection flow along a perpendicular direction. In Chaos Theory the Rosller attractor araises in the phase space orbits of dynamical systems with dissipation due to interaction or coupling of two dimension in the system. The coupling between two of the system’s dimensions is mapped into the re-injection of the flow of the orbits in one of those coupled dimensions into the general flow of orbits in the other dimensions. This causes a dimensional reduction in the resulting attractor of the orbits flow. The resulting object has a dimension between 2 and 3 with a fractal geometry. Locally this kind of strange attractors behave as a Baker Map. That is, each loop around central region stretches phase space volumes in one of the coupled dimensions while expanding them in the non coupled directions. The net effect if a volume preserving transformation for each cycle of orbits flow around the central region. A known characteristic of this kind of Baker flows is that the separation between distances (dN) between points in the flow increases in an universal exponential fashion involving the Lyapunov (λ) coefficients of the loops flow $$\label{Eq12} d\_{N+1} \propto d\_{N+1} \exp{(\sqrt{\pi \lambda })}$$ where for a Baker type map (locally a logistic like map), the Lyapunonv coeficient is equal to the Feigenbaund Constant (δ = 4, 66920160910299.... ) The proportionality constant depends on the square of the Hausdorff dimension of the Rosller fractal. If the attractor is embedded in a three dimensional space then this proportionality constant will be the square of a number between 2 and 3 and rather close to 3 (the dimension of a locally logistic fractal embedded in 3D space). This will be important in another section later in this paper. It is worth noting that every observable of the system depending only on the scale we observe the system follows the same scaling universal behavior. For example, in many systems one of those properties depending only on the observation scale is Entropy. We will make use of this property later in the context of the OM correspondence. The OM-QM counterpart of the free particle Dirac Equation ========================================================= Having found expressions both for x̃, p̃ and c̃ we can now try to find the OM-QM counterpart of the Quantum Wave Function, Physical Observables such as Energy and parameters such as the mass and spin. All of them get transformed to Number Theoretical objects under the OM correspondence. We work with all physical units in full display as in the OM correspondence we want to understand how all physical elements transform in the OM-QM side. The free-particle Dirac equation has the following form (iℏγμδμ − mc)Ψ(x, t) = 0 where the γμ are the 4 by 4 Dirac Matrices, m is the particle’s mass, c is the speed of light and Ψ(x, t) is a 4 components bi-spinor describing the particle quantum state. Plane waves solution to this equation make necessary the emergence of the particle spin. Step 3 of the OM correspondence in the case of the free-particle gives the simplest possible modulrization of the space-time as there is no topological or experimental constraint other than the ones implicit in the extremes of paths connecting the particle’s initial position and end position. The fuzzy OM-Knot resulting from the identification of initial and end points produces a simple knot with only one petal with a minimum cell center. Let’s apply our OM-correspondence to Eq.[Eq13]. We obtain the following differential equation $$\label{Eq14} (i 2 \pi (\gamma^0 \dfrac{\delta}{\delta N\_0} ) - i 2 \pi (\gamma^{1,2,3} \dfrac{\delta}{\delta N\_{1,2,3}} ) ) \tilde{\Psi}(N\_{0,1,2,3}) = \tilde{m}\tilde{c} \tilde{\Psi } (N\_{0,1,2,3})$$ Here the numeric parameters N0, 1, 2, 3 are the OM-QM side of each of the space time coordinates. We have also substituted the derivation in terms of the space-time coordinates by their OM-QM counterparts and m̃ also transforms to a Number Theoretical expression as well as we saw the speed of light also does. We will see later that m̃ is in fact a function of the numbers N0, 1, 2, 3. The particle’s state transforms to a 4-components vector where each of its components are functions of the numbers N0, 1, 2, 3. Eq.[Eq14] is the full OM-QM correspondent of the Dirac Equation. At this point we could try to find a plane wave solution and understand what is the OM-QM version of $ \tilde{\Psi } (N\_{0,1,2,3} ) $ that is, of what is the OM-QM counterpart of the free particle state. The γ matrices do mix the derivatives on the four Ni parameters. Therefore, finding a solution for one of the components of $ \tilde{\Psi } (N\_{0,1,2,3} ) $ involve derivatives in all four Ni parameters. We will make here a simplification in order to find of a solution to Eq.[Eq14]. Let’s recall the topological meaning of the parameters Ni as winding of paths in space-time around a minimum cell. We can always find a topology of those “petal loops” where the projection of a 3-dimensional such knot according to any of the dimensions onto any of the 3 possible 2-dimensional views has the same winding around the central minimum cell (irrespective of the dimension we decide to project). That is, for the kind of knots we are dealing with the three “directions” Ni are essentially equivalent. We can then simplify Eq.[Eq14] considering that all four derivatives are equal. In a geometric context this would not be possible even when we consider all coordinates as equivalent. However, here we are dealing with pure mathematical objects, numbers in fact. They cannot be labeled as different if they are equal in value. For them we can then simplify Eq.[Eq14] to this form for the 4 components of $ \tilde{\Psi } (N\_{0,1,2,3} ) $ $$\label{Eq15} (i 2 \pi (\gamma^\mu ) \dfrac{\delta}{\delta N} ) ) \tilde{\Psi}(N) = - 2i \pi \tilde{m} \tilde{\Psi } (N)$$ Substituting the Dirac γ matrices Eq.[Eq15] is equal to $$\label{Eq16} \begin{pmatrix} i-1 \\ 1-i \\ 1-i \\ i-1 \end{pmatrix} \dfrac{\delta}{\delta N} \tilde{\Psi } (N) = \tilde{m} \tilde{\Psi } (N)$$ By similarity with the Dirac equation solution, the numbers that appear in the r.h.s. before the derivative are the OM-QM counterparts of the spin. We see that the possible values of s̃ are s̃ = ± (i − 1) The values of s̃ are independent of N as they come from the gamma matrices. Therefore the general solution to Eq.[Eq15] is $$\label{Eq18} \tilde{\Psi } (N) = e^{\frac{\int^{N\_f}\_{N\_i} \tilde{m}(t) dt}{\tilde{s}}}$$ We can propose a reasonable guess for a solution of Eq.[Eq15]. Let’s start by assuming a solution of this form $$\label{Eq19} \tilde{\Psi } (N) = e^{\phi(N)}$$ If we assume an almost plane-wave solution type then ϕ(N) has the meaning of a total phase difference after parallel transport over the fuzzy Knot we discussed in previous section. This is due to the fact that this phase difference corresponds to the comparison of the free particle states at the initial space-time point and at the final space-time point. As we saw before this phase comparison is done after parallel transport over the “petals” of the fuzzy OM-Knot generated by the OM correspondence. Comparison after parallel transport is equivalent to Curvature. In this case it is total curvature integrated over the whole OM-Knot “petals”. If we call this curvature as R(N) then we have $$\label{Eq20} \tilde{\Psi } (N) = e^{\phi(N)} = e^{\int^{N\_f}\_{N\_i} R(t) dt} = e^{\frac{\int^{N\_f}\_{N\_i} \tilde{m}(t) dt}{\tilde{s}}}$$ Essentially what we are saying is that a very reasonable guess for the OM-QM counterpart of the free particle quantum state is a topological invariant of the OM-Knot and concretely, the exponential of the total Knot curvature. This invariant is in general a function counting the statistics of possible crossing at the fuzzy center of the OM-Knot. From Eq.[Eq20] we get $$\label{Eq21} \phi'(N) = R(N) = \frac{\tilde{m}(N) }{\tilde{s}}$$ where $ \phi'(N) = \dfrac{d \phi}{d N}$ We see here a first hint that the OM-QM statements gives very non trivial and interesting connections between elements coming from the QM side. In Eq.[Eq21] we see an explicit relationship between the OM-QM counterparts of spin, mass and curvature. We cannot see that direct connection in the QM side but in the OM-QM side this arises explicitly. Using the presence of the total OM-Knot curvature in the formula for $ \tilde{\Psi } (N) $ we can find a more explicit and meaningful form of that solution. From the work of Mazur it is known that the crossing statistics of fuzzy Knots are given by logarithms of the Prime Numbers . To each fuzzy Knot one can assign a prime and to each prime one can assign a fuzzy Knot. Mazur gave the following correspondence log(p) = vol(K) where p is a prime number and vol(K) is the volume of the hyperbolic complement of a Knot. When talking about a fuzzy Knot that prime p counts the crossing statistics at the fuzzy center. By the other side that volume is equal to the total curvature integrated over the Knot. Therefore, we have ∫NiNfR(t)dt = log(p) At this point we make contact with Number Theory by recognizing that in the formula above we can use the von Mangoldt function Λ(p) instead of log(p). The von Mangoldt function is defined by $$\label{Eq24} \Lambda (\textit{p}) = \bigl\{ \begin{array}{cll} \log(\textit{p})& if & N = \textit{p} ^k \\ \\ 0 & &otherwise \end{array}$$ This is not a simple substitution as it takes into account the topological meaning of Eq.[Eq20]. Substituting this into Eq.[Eq20] we get the following possible solution for (each of the 4 components of) the OM-QM transformed Dirac Equation $$\label{Eq25} \tilde{\Psi } (N) = e^{(\sum\_{q={\textit{p}^r}} \Lambda (\textit{N}) \delta (N-q))}$$ As a side note, the exponent in this solution is equal to the derivative of the Chevyshev function (ψc). The Chevyshev function is in turn equal to the logarithm of the Least Common Multiplier function of all numbers from 1 to N ψc = log(l.c.m.(1, ..., N)) Finally we find the following alternative form of $ \tilde{\Psi } (N) $ $$\label{Eq27} \tilde{\Psi } (N) = exp[ \frac{l.c.m.'(1,...,N)}{l.c.m.(1,...,N)}]$$ This gives an interesting interpretation to the solution of the OM-QM transformed Dirac Equation as a kind of focus or convolution formula for all scales from 1 to N. Let’s remember that N in our framework represent scales in terms of the Planck Length. Therefore, $ \tilde{\Psi } (N) $ is in fact the exponential of the logarithmic derivative of a scale function which is able to resolve at the same time all the scales from 1 to N. The OM-QM counterpart of Mass and Energy ======================================== We will focus at this point in using Eq.[Eq25] for finding the OM-QM counterparts of other Quantum Mechanical observables or parameters. By substituting Eq.[Eq25] into Eq.[Eq16] we get the following form for the OM-QM counterpart of the ]mass parameter in the Dirac equation m̃(N) = s̃∑q = prΛ(q)δ(N − q) Let’s find the OM-QM version of the other equation satisfied by each one of the 4 components of the bi-spinor solution to the Dirac equation, that is the Klein-Gordon equation. $$\label{Eq29} (\frac{1}{c^2}\dfrac{\delta^2}{\delta t^2} - \Delta^2+\frac{m^2 c^2}{\hbar^2} ) \Psi(\vec{r}, t) =0$$ We will consider the plane waves type solutions which allows us to substitute the time derivatives side by an energy eigenvalues term. If we apply the OM correspondence and use the transformed Dirac Equation to this equation we get the following OM-QM form of the Klein-Gordon Equation $$\label{Eq30} \frac{\tilde{E} ^2 (N)}{(4 \pi^2)^2} \tilde{\Psi}(N) = (\tilde{s}^2 \dfrac{d^2}{dN^2} + \tilde{m}^2(N) ) \tilde{\Psi}(N)$$ where Ẽ is the OM-QM counterpart of Energy which as well mass becomes a function of the parameter N. To get to this equation we have made use of the same approximation we used in the case of the Dirac equation which allowed us to remove the dependency on the three spatial coordinates and work in one dimension. Using Eq.[Eq20] we obtain this alternative version of the transformed Klein-Gordon equation $$\label{Eq31} \frac{\tilde{E} ^2 }{(4 \pi^2)^2} \tilde{\Psi}(N) = (\tilde{s} \tilde{m}' + 2 \tilde{m}^2 ) \tilde{\Psi}(N)$$ where we used the notation m̃ʹ as the derivative of m̃ respect to N. To find an explicit formula for m̃ʹ we can use Eq.[Eq28] and the method of derivation under the sum sign $$\label{Eq32} \tilde{m}' = \tilde{s} \sum\_{q={\textit{p}^r}} ( \dfrac{d(\Lambda (\textit{q}^b) \delta (N-q^b) ) }{db})\|\_{b=1} = \tilde{s} \sum\_{q={\textit{p}^r}} (\dfrac{d(\Lambda (\textit{q})) }{dq} \log(q)q + \Lambda (\textit{q}) ) \delta (N-q)$$ Let’s see how the derivative of the von-Mangoldt function is related to the Riemann ζ-function. A known identity for the logarythmic derivative of the Riemann ζ-function says that $$\label{Eq33} \frac{\zeta'}{\zeta} = \sum\_{q={\textit{p}^r}} \frac{\Lambda(q)}{q} \delta (N-q)$$ Otherwise we can write $$\label{Eq34} \frac{\delta \ln (\zeta)}{\delta N} = \sum\_{q={\textit{p}^r}} \frac{\delta \Lambda(q)}{\delta q} \delta (N-q)$$ That means that Λ(q) is the Moebius inverse of (ln(ζ(q)))ʹ. Then Λʹ(q) is the Moebius inverse of ζ. This implies a duality relationship between those two functions. From this we conclude that the peaks of the r.h.s. in Eq.[Eq32] will occur on the zeroes of the ζ-function. Then Eq.[Eq32] can be considered an statistical average of those zeroes. We finally arrive to the following expression for m̃ʹ m̃ʹ = m + s̃∑q = pr(σqlog(q)q)δ(N − q) where σq are the imaginary part of the zeroes of the Riemann ζ-function. The statistical average of the real part of those zeroes (equal to 1/2 if the Riemann hypothesis is true) is zero. Substituting this into Eq.[Eq31] we get the following formula for Ẽ $$\label{Eq36} \frac{\tilde{E} ^2 }{(4 \pi^2)^2} = \tilde{s}^2 \sum\_{q={\textit{p}^r}} \sigma\_q \log(q)q ) \delta (N-q) +\tilde{m} + 2 \tilde{m}^2$$ That means that the OM-QM correspondent of Energy for the free particle is a simple function of the imaginary part of the zeroes of the Riemann ζ-function. It has been long theorized that the imaginary part of the zeroes of the Riemann ζ-function are the eigenvalues of some non-hermitean Hamiltonian . Many promissing Hamiltonians have been tried but a proof of such conjecture still is out of reach. Here we have arrived to a similar relationship in the context of the OM correspondence. This shows the Number-Theory we arrive at with the OM correspondence is highly non trivial and can give us very valuable information about the Quantum side. We see that problems which seem to be hard or mysterious in a pure Quantum realm seems to be mapped to Mathematical relationships which are much simpler to treat. We see here a first hint that the OM-QM side of the correspondence seems to be a common playground for both Quantum Physics and Mathematics where the interplay of concepts from both sides can give raise to unexpected relationships (or conjectured as in the case above). The values of m̃ and Ẽ =========================== Let’s explore the formulas for m̃ and Ẽ to see which values they can possibly have. From Eq.[Eq28] we see that only the following values are possible for m̃ $$\label{Eq37} \tilde{m}(N) = \bigl\{ \begin{array}{ccl} 0 & if & N \neq p^k, p=prime \\ \\ \frac{\tilde{s}}{N} & if & N= p^k \end{array}$$ Thus, only for N which is a power of a prime number m̃ is not null. The values it take are coherent with it the already used formula relating OM-Knot curvature to m̃ $$\label{Eq38} \tilde{R} (N)=\frac{\tilde{m}}{\tilde{s}}$$ as R̃ is precisely of the form 1/N. Inserting those values of m̃ into Eq.[Eq36] we get the following possible values for Ẽ2 $$\label{Eq39} \tilde{E}^2(N) = \bigl\{ \begin{array}{lcl} 0 & if & N \neq p^k, p=prime \\ \\ \tilde{P}\_0^2 (\tilde{s}^2 \sum\_{q={\textit{p}^r}} \sigma\_q \log(q)q ) \delta (N-q) + \frac{\tilde{s}}{N} + 2 (\frac{\tilde{s}}{N})^2) & if & N= p^k \end{array}$$ We cam rewrite the values in the case N = pk in terms of the OM-Knot curvature in this way $$\label{Eq40} \tilde{E}^2(N) = \bigl\{ \begin{array}{lcl} 0 & if & N \neq p^k, p=prime \\ \tilde{P}\_0^2 (\tilde{s}^2 \sum\_{q={\textit{p}^r}} \sigma\_q \log(q)q ) \delta (N-q) + \tilde{s} \tilde{R}(N) + 2 \tilde{s}^2 \tilde{R}^2(N)) & if & N= p^k \end{array}$$ We can reduce further this expression noting that the sum in it selects only one of the possible summing elements Ẽ2(pk) = P̃02(s̃2σpplog(p))Δp + s̃R̃(pk) + 2s̃2R̃2(pk)) where Δp is a distribution version of the Dirac delta centered at p. We will return to this important expression later in this work. The value of the Fine Structure Constant ======================================== As an application of our correspondence let’s find find the OM-QM correspondent of the Fine Structure Constant. The value of this fundamental constant cannot be calculated from first principles in Quantum Mechanics even when is form in terms of other physical constants appears in the spin-orbit contribution to the energy eigenvalues in atomic Physics. Its is a non-dimensional number whose inverse is close to the rational $ \frac{1}{137} $. One of the ways to define formally this constant is through the following limit $$\label{Eq42} \alpha^{-2} = \lim\_{n \rightarrow \infty} \frac{{E^2}\_{n-1}-{E^2}\_{n-2}}{{E^2}\_{n}-{E^2}\_{n-1}}$$ where En are the atomic energy eigenvalues. By similarity we can define the OM-QM correspondent of the Fine Structure Constant by the following number $$\label{Eq43} \tilde{\alpha}^{-2} = \lim\_{i \rightarrow \infty} \frac{\tilde{{E^2}}\_{i-1}-\tilde{{E^2}}\_{i-2}}{\tilde{{E^2}}\_{i}-\tilde{{E^2}}\_{i-1}}$$ where Ẽn are the OM-QM correspondent of Energy defined in Eq.[Eq39]. Let’s calculate explicitly this number α̃. Looking at Eq.$ \ref{Eq36} $ and Eq.$ \ref{Eq28} $ we notice that terms consecutive in N related to mass are the same for N → ∞. That means they will cancel in α̃. We only need to consider differences of consecutive terms of the form γ(N) = s̃2∑q = prσqlog(q)qδ(N − q) Thus, we get the following formula for α̃− 2 $$\label{Eq45} \tilde{\alpha}^{-2} = \lim\_{n \rightarrow \infty} \frac{\gamma\_{n-1}-\gamma\_{n-2}}{\gamma\_{n}-\gamma\_{n-1}}$$ We know the difference σn − σn − 1 tends to zero in the infinite. Thus we get the following limit for α̃ $$\label{Eq46} \tilde{\alpha}^{-2} = \lim\_{N \rightarrow \infty} K \frac{\log(N-1) (N-1)-\log(N-2) (N-2)}{\log(N) N -\log(N-1) (N-1)}$$ where K is the proportionality constant already mentioned in Section 4. Taking into account that N in our setup corresponds to a scale in terms of the Plank Length, all terms in α̃− 2 depend only on the scale and also have the form of an entropy term. We found in section 4 that such functions in the moduli space we are working on do scale in a very particular way (Eq.12). Specially entropy like functions do scale in that way. Therefore we get the following result $$\label{Eq47} \tilde{\alpha}^{-2} = K \exp{(\sqrt{\pi \delta})}$$ where δ is the Feingenbaum constant. We get finally the following valued for α̃− 1 $$\label{Eq48} \tilde{\alpha}^{-1} = \sqrt{K} \sqrt{\exp{(\sqrt{\pi \delta})}}$$ $$\label{Eq49} \sqrt{\exp{(\sqrt{\pi \delta})}} \approx 46.061512666666666666666666666667....$$ From our discussion in Section 4, we know the constant K is related to the square of the Hausdorff dimension of the local structure of OM-space. The dimension of a Rosller fractal embedded in 3D is a number between 2 and 3 and rather close to 3. A value of $ \sqrt{K}=3 $ gives a value of α̃− 1 = 138, 184538...... which is already close to the experimental value. However if we consider K to be the square of the Rosller fractal dimension (D=2.974283562752.....) we get the value α̃− 1 ≈ 137. $$\label{Eq50} \tilde{\alpha}^{-1} = D \sqrt{\exp{(\sqrt{\pi \delta})}}$$ In essence we have proven that assuming covariance of the form of α− 1 under our correspondence allows us to deduce its experimental numeric value and its value does not change from the Quantum Mechanical value side. Then the Fine Structure Constant is an invariant in our correspondence α̃ = α Let’s notice that α̃− 1 is equal to the Lyapunov coefficient of the Rosller-like structure of OM-space times a coefficient dependent on its dimension. The Lyapunov coefficient generates in the Rosller internal flow the conservation of volume in the space complementary to the Rosller geometry. We propose that the conservation of this dual or complementary volume is the OM-QM correspondent to conservation of probability during QM evolution. By the other hand, α is a quantum Amplitude. More precisely, the probability of absorption or emission of a photon by a charged particle. Quantum Amplitudes are always volumes of certain geometrical objects embedded in certain spaces. Then, we can think of α̃− 1 as related to the volume of the hyperbolic complement of the OM-Knot. Then it makes sense that its value is almost exactly the coefficient of expansion-contraction in the OM-knot which secures conservation of that volume. The OM-QM correspondent of the electron’s charge and Electric Field =================================================================== As all physical quantities in the QM side of our correspondence have a OM-QM counterpart it is natural to try to find the OM-QM version of the electron’s electric charge. This is related to α via the formula $$\label{Eq52} \alpha = \frac{e^2}{\hbar 4 \pi \epsilon}$$ where e is the electron charge and ε is the vacumm electric permeability. This expresion of α comes from atomic physics and it contains the constant ε (the vacuum electric permitivity) which cannot be determined from first principles and comes from experimental measurement. If we invert this expression and apply the OM correspondence we find $$\label{Eq53} \tilde{\alpha}^{-1} = \frac{ 8 \pi^2 \tilde{\zeta\_0} \tilde{\epsilon}}{\tilde{e}^2}$$ where ẽ, ε and $ \tilde{\zeta\_0} $ are the OM-QM version of e, ε and the Planck Length. As The Planck Length is 1 in the scales transformation implied in our correspondence we can assume $ \tilde{\zeta\_0} = 1 $ Then from Eq.[Eq50] we get $$\label{Eq54} \tilde{\alpha}^{-1} = \frac{ 8 \pi^2 \tilde{\zeta\_0} \tilde{\epsilon}}{\tilde{e}^2} = D \sqrt{\exp{(\sqrt{\pi \delta})}}$$ which after rearrangement give us the following formula for the quotient $$\label{Eq55} \frac{\tilde{e}^2}{\tilde{\epsilon}} = \frac{8 \pi^2}{D \sqrt{ \exp{(\sqrt{\pi \delta})} }}$$ we will disentangle this quotient in the l.h.s. making an educated guess of which part of the r.h.s. corresponds to denominator and numerator. We arrive to the following OM-QM versions for e and ε ẽ = 2π $$\label{Eq57} \tilde{\epsilon} = \frac{D}{2} \sqrt{ \exp{(\sqrt{\pi \delta})} } = \frac{\tilde{\alpha}^{-1}}{2}$$ The value of ẽ can be interpreted as the residue of a potential of the form 1/z integrated around a loop around the minimum cell. What is more surprising is that the OM-QM counterpart of ε turns out to be half the OM-QM counterpart of α−	arxiv_0000554
Hot UV-bright stars of galactic globular clusters[1](#fn1)[2](#fn2) =================================================================== ### Received April 15, 2019 / Accepted May 7, 2019 We wish to derive effective temperatures, surface gravities, and helium abundances of the luminous hot UV-bright stars in these clusters to determine their evolutionary status and compare the observed numbers to predictions from evolutionary theory. We obtained FORS2 spectroscopy of eleven of these UV-selected objects (covering a range of − 2.3 < [Fe/H] < − 1.0), which we (re-)analysed together with previously observed data. We used model atmospheres of different metallicities, including super-solar ones. Where possible, we verified our atmospheric parameters using UV spectrophotometry and searched for metal lines in the optical spectra. We calculated evolutionary sequences for four metallicity regimes and used them together with information about the HB morphology of the globular clusters to estimate the expected numbers of post-AGB stars. We find that metal-rich model spectra are required to analyse stars hotter than 40000K. Seven of the eleven new luminous UV-bright stars are post-AGB or post-early AGB stars, while two are evolving away from the HB, one is a foreground white dwarf, and another is a white dwarf merger. Taking into account published information on other hot UV-bright stars in globular clusters, we find that the number of observed hot post-AGB stars generally agrees with the predicted values, although the numbers are still low. Spectroscopy is clearly required to identify the evolutionary status of hot UV-bright stars. For hotter stars, metal-rich model spectra are required to reproduce their optical and UV spectra, which may affect the flux contribution of hot post-AGB stars to the UV spectra of evolved populations. While the observed numbers of post-AGB and post-early AGB stars roughly agree with the predictions, our current comparison is affected by low number statistics. Introduction ============ Ultraviolet images of globular clusters are often dominated by one or two hot, luminous, UV-bright stars that are more than a magnitude brighter than the horizontal branch (HB) and hotter than 7000K. Such stars are in various evolutionary stages after helium core burning (HeCB). These evolutionary stages are the most uncertain phases of the evolution of low-mass single stars, caused by uncertainties in the size of the C/O-core due to our poor understanding of convective boundary mixing, and our lack of understanding of winds in red giant branch (RGB) stars. The previous mass loss on the RGB plays an important role here because it determines whether a star even ascends the asymptotic giant branch (AGB): HB stars with very low hydrogen envelope masses evolve directly from the extreme horizontal branch (EHB, T$\_{\rm eff}\gtrsim$ 20000K) to the white dwarf stage, whereas stars with hydrogen envelope masses of more than 0.02M⊙ will at least partly ascend the AGB. Owing to the previous uncertainties, both the masses of the C/O-core and the H-rich envelope of post-HeCB are loosely constrained. In particular, theoretical post-AGB tracks have only been very little tested for old low-mass stars because suitable observations are scarce. A similar situation holds for post-EHB stars that evolve directly to the white dwarf phase and avoid the AGB. As an example, found far fewer hot post-AGB stars than expected in their Space Telescope Imaging Spectrograph (STIS) UV imagery of M32, while reported a similar lack of hot post-AGB stars in the Galactic halo. The status of hot post-AGB stars in globular clusters remains uncertain: with only about one hot post-AGB star per cluster, many clusters must be observed in order to compare number counts, luminosities, and surface gravities with evolutionary tracks. The detection of hot post-AGB stars in optical colour-magnitude diagrams is limited by selection effects that are caused by crowding in the cluster cores and by the large bolometric corrections for these hot stars. More complete searches are possible for hot post-AGB stars in planetary nebulae, for example, by using Oiii imaging, as performed by and. The meaningfulness of such searches, however, is limited by the long evolutionary timescales of low-mass post-AGB stars, which make the appearance of a planetary nebula unlikely. Only four planetary nebulae have indeed thus far been discovered in the Galactic globular cluster system. and first pointed out the advantages of wide-field UV imagery to obtain a complete sample of hot post-AGB stars in a large number of globular clusters. The UV suppresses the dominant cool-star population and emphasizes the hot stars, ensuring that all hot post-AGB stars in the cluster are detected. Fourteen globular clusters were observed with Ultraviolet Imaging Telescope (UIT, ) in 1990 and 1995 at 1620Å with a 40ʹ field of view (FOV). used ground-based spectroscopy to derive temperatures and gravities of the newly discovered hot UV-bright stars. Their results clearly illustrated the points made above, namely that the HB morphology has a strong influence on the evolutionary status of hot UV-bright stars and that optical selections suffer from a strong bias towards the most luminous hot stars. Subsequently, the Galaxy Evolution Explorer (GALEX) satellite was used to obtain wide-field UV images of 41 globular clusters at 1520Å with a 1.2∘ diameter FOV. Because GALEX observations are limited by its brightness constraints to mostly high Galactic latitude fields, we obtained archival UV images of an additional 31 clusters using the Swift Ultraviolet-Optical Telescope (UVOT, ). The UVOT images are less satisfactory because its FOV is smaller (17ʹx17ʹ) and its wavelength is longer. The solar-blind UVM2 filter on the UVOT has a central wavelength of 2250Å, while the UVW2 filter has a central wavelength of 1930Å. The UV imagery from all three wide-field telescopes was complemented by UV imagery with the Hubble Space Telescope (HST, ), which provides much higher spatial resolution and photometric precision, but rarely encompasses the entire globular cluster within its FOV. This brings the total number of globular clusters with UV imaging data to 78. Among these, we recovered all previously known hot post-AGB star candidates and found 19 new hot post-AGB star candidates and 16 other new hot stars (mainly apparent EHB or post-EHB stars in globular clusters where only red HB stars have been known so far). Because each cluster contains only a few post-AGB stars, a large sample of clusters is needed to test stellar evolution predictions. To this aim, we started a project to study the whole sample of globular clusters with UV imagery and compare it with stellar evolution predictions. We are currently working on a paper describing the results of this greatly enlarged sample and the new hot post-AGB candidates. As a first step, we describe here the optical spectroscopic observations of 10 of the brightest hot post-AGB star candidates in the Southern Hemisphere and one post-EHB star candidate in M4. The paper is organised as follows. First we describe in Sect.[sec:obsproc] the new observations as well as unpublished data from, and in Sects.[sec:rv] and [sec:param] we derive stellar parameters for all the stars in our sample and discuss some particular cases. This sample is enlarged in Section [sec:lit] by including all UV-bright stars that have been studied in previous publications. In Sect.[sec:masses] we derive masses from the atmospheric parameters and the distances of the globular cluster, and we discuss the effect of Gaia results for some of the closest clusters. In Sect.[sec:evol] we then present stellar evolution models of post-HeCB stars at different metallicities and for different masses on the HB and compare simple estimates of the number of luminous hot UV-bright stars predicted by the models with those observed in our sample. Finally, we end the paper in Sect.[sec:conclusions] with some preliminary conclusions and discuss what needs to be done to improve the comparison. [!h] llllrclc globular & star & α2000 & δ2000 & mUV & UV Tel. & V & Ref cluster & & [h:m:s] & [∘::] & [mag] & &[mag] & NGC5139 & UIT151 & 13:27:05.06 & − 47:21:56.6 & 15.29& UIT & 16.577& (1) & UIT644 & 13:26:44.96 & − 47:27:09.5 & 13.12& UIT & 12.713 & (1) & UIT1275 & 13:26:52.77 & − 47:29:44.1 & 14.75& UIT & 14.306 & (1) & UIT1425 & 13:26:11.56 & − 47:30:49.2 & 15.31& UIT & 15.660 & (1) & ROA5342 & 13:25:45.42 & − 47:24:02.0 & 14.73& UIT & 15.944 & (1) & Dk3873 & 13:26:13.95 & − 47:25:30.1 & 15.29& UIT & 16.531 & (1) NGC6121 & UVBS2 & 16:23:26.25 & − 26:31:27.0 & 15.07 & UIT & 16.388& (2) NGC6656 & UVBS2 & 18:36:22.85 & − 23:55:19.3 & 14.88 & Swift UVOT & 14.533 & (3) NGC6712 & C49 & 18:53:03.45 & − 08:42:30.5 & 17.00 & Swift UVOT & 16.72 & (4) NGC6779 & ZNG2 & 19:16:41.28 & + 30 12 48.5 & 14.40 & Swift UVOT & 15.13 & (5) NGC7099 & UVBS2 & 21:40:18.08 & − 23:13:22.3 & 15.54 & GALEX & 16.75 & (6) NGC5139 & ROA542 & 13:25:49.91 & − 47:22:59.7 & 15.331 & UIT & 12.876 & (1) & ROA3596 & 13:27:46.09 & − 47:30:57.6 & 14.731 & UIT & 14.179 & (1) Observations and data reduction =============================== The coordinates and brightness of our targets are listed in Table[tab:target]. The data were taken with the FOcal Reducer and low-dispersion Spectrograph 2 (FORS2, ) at the ESO Very Large Telescope (VLT) UT1. We used the grism GRIS\_1200B with a slit width of 05 and verified the resolution from the arc lamp images. The stars in ωCen (except for ωCen − UIT151) were observed in multi-object spectroscopy (MOS) mode with movable slitlets of about 20 length and have a spectral resolution of about 1.7Å. All other stars (including ωCen − UIT151) were observed with the long slit with a length of 68 and have a spectral resolution of about 1.4Å. We always took two exposures of each target for consistency checks. The data were observed between April 1 and July 22, 2012 (date at the beginning of the night). They were processed soon afterwards with the [ESO FORS2 pipeline](http://www.eso.org/sci/software/pipelines/) (version fors-4.12.8), except for the flux calibration (see Appendix[app:flux]). For the bias correction the bias applied to science and calibration data was scaled with the prescan level of these data. In order to take possible structure in the bias into account, the full (prescan-corrected) master bias was subtracted and not just a number. The flat fields were summed per setup and date and normalized using a smoothing with a radius of 10 pixels along both axes. The wavelength calibration used 15 arc lines for the long-slit spectra and 15–17 arc lines for the MOS spectra (depending on the slitlet location). The average accuracy achieved in this way was 0.1pixel for the long-slit spectra and 0.24pixel for the MOS data. We used local sky subtraction, that is, fitting of the sky spectrum per slit on the two-dimensional spectra (before rectification and rebinning). Because of the brightness of our targets and the blue wavelength range, the sky level was rather low, even though most of the data were observed while the Moon was above the horizon. To verify the quality of the sky subtraction, we checked our spectra for the presence of G-band features, which should not be present in these hot stars, and found none. The details of the response determination for this volume-phase holographic grism are described in Appendix[app:flux]. The extracted spectra showed that NGC6656 − UVBS2 has a close cool fainter neighbour, whose spectrum overlaps with that of the intended target. We did not see any significant lines in the neighbour’s spectrum, therefore we smoothed it with a median filter of 14Å half-width and subtracted it from the spectrum of NGC6656 − UVBS2 (see Fig.[fig:M22 − UVBS1]). In another attempt to avoid the contamination from the cool neighbour we extracted only the part of the spectrum away from the neighbour (marked in red in Fig.[fig:spec]). The different slopes of the two spectra point towards possible over- and/or undercorrection of the flux from the cool neighbour. The ratio of the two spectra shows no noticeable line residuals. Considering the experience with crowded field spectroscopy recorded in , we prefer to use the narrowly extracted spectrum for further analysis. [fig:M22 − UVBS1] [fig:spec] Unpublished data from --------------------- After discovering two unpublished spectra (NGC5139 − ROA542 and NGC5139 − ROA3596) from the same observing run as, we decided to analyse all the data from that observing run again. Based on comments from V. Dixon and P. Chayer, we checked the resolution of these data and found that the line width used in was too high. A pixel scale of 0.336/pixel and dispersion of 100Å/mm yield a resolution of 5.6Å and not 6.7Å, as used by. The lower value explains most of the differences between the results reported here and those from. Radial velocities ================= To correct the observed spectra to laboratory wavelengths, we first corrected to the heliocentric system using the midas command compute/barycor. Then we determined the radial velocities by fitting Gaussian profiles to the cores of strong lines using the midas command center/gauss. We selected the lines and their fit range manually to ensure that no surviving cosmics or noise peaks distort the results. We did not use lines at wavelengths below 3800Å as the bluest arc line in our calibration data is at 3888Å. The resulting radial velocities are listed in Table[tab:params], together with the rms error of the individual measurements and the velocities of the corresponding clusters from the literature. The one glaring discrepancy is the radial velocity for NGC7099 − UVBS2 whose value suggests that this star is not a cluster member. This is also supported by the parameters derived in Sect.[sec:param], which argue in favour of a foreground white dwarf. Two stars with very small formal errors ($\lesssim$10kms− 1) show velocity differences well above these errors: NGC6656 − UVBS2 ( + 33.6kms− 1) and NGC5139 − UIT1275 ( + 40.5kms− 1). We recall, however, that one pixel in our data corresponds to 48kms− 1 and the resolution of our data corresponds to 90–120kms− 1. Therefore we decided to consider all stars except for NGC7099 − UVBS2 as members of their respective globular clusters. [tab:params] llrrrr\|c\|rrrl cluster & star & $T\_{\rm eff}$ & logg & $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ & M & status & $v\_{\rm hel}$ & nv & $v\_{\rm cluster}$ & Ref. & & [K] & [cm s− 1] & & [M⊙] & & [kms− 1] & & [kms− 1] & NGC5139 & UIT151 & 62600 ± 2400 & 5.11 ± 0.10& − 0.99 ± 0.08& 0.24 & peAGB & + 231.1 ± 12 & 9 & + 231.8 & (1) & UIT644 & 18000 ± 800& 2.82 ± 0.10& − 0.35 ± 0.06& 0.29 & pAGB & + 232.3 ± 13 & 26 & & & UIT1275 & 17200 ± 1100& 3.34 ± 0.18& − 1.27 ± 0.18& 0.25 & pHB & + 272.3 ± 6& 14& & & UIT1425 & 11700 ± 200& 3.57 ± 0.08& − 1.72 ± 0.36& 0.23 & pHB & + 213.6 ± 17 & 9& & & Dk3873 & 49700 ± 1800& 4.99 ± 0.12& − 1.21 ± 0.12& 0.28 & peAGB & + 231.2 ± 26 & 11& & & ROA5342 & 52000 ± 2000 & 4.79 ± 0.10& − 1.17 ± 0.10& 0.29 & peAGB & + 227.6 ± 10 & 10& & NGC6121 & UVBS2 & 46600 ± 600& 5.82 ± 0.12& + 0.89 ± 0.18&0.85& WDM & + 74.6 ± 15 & 17& + 71.5 & (2) NGC6656 & UVBS2 & 78800 ± 9900 & 4.58 ± 0.20& + 0.36 ± 0.16&0.18& pAGB & − 111.3 ± 4& 3& − 144.9 & (2) NGC6712 & C49 & 53300 ± 4000 & 4.50 ± 0.12& − 1.04 ± 0.12&0.31& pAGB & − 108.0 ± 8& 8& − 109.0 & (3) NGC6779 & ZNG2 & 21500 ± 1800 & 3.07 ± 0.16& − 0.85 ± 0.08&0.19& pAGB & − 123.5 ± 9& 15& − 138.1 & (4) NGC7099 & UVBS2 & 65000 ± 2000 & 7.22 ± 0.12& − 3.29 ± 0.12& & WD & + 11.4 ± 8& 3& − 184.4 & (2) NGC5139 & ROA542 & 10600 ± 200 & 2.55 ± 0.10 & − 1.00 &0.34& pHB & & & & & ROA3596 & 17800 ± 1000 & 3.56 ± 0.16 & − 1.26 ± 0.18&0.41& pHB & & & & NGC2808 & C2946 & 24900 ± 1800 & 4.78 ± 0.20 & − 1.88 ± 0.14 &0.68& pHB & & & & & C2947 & 15000 ± 1200 & 3.92 ± 0.24 & − 1.45 ± 0.44 &0.41& pHB & & & & & C4594 & 19900 ± 1600 & 3.79 ± 0.22 & − 1.63 ± 0.26 &0.34& pHB & & & & NGC6121 & Y453 & 54900 ± 2000 & 5.62 ± 0.14 & − 1.25 ± 0.10 &0.59& peAGB & & & & NGC6723 & III60 & 43000 ± 1400 & 4.72 ± 0.14 & − 1.19 ± 0.14 &1.02& peAGB & & & & & IV9 & 25900 ± 2000 & 4.02 ± 0.22 & − 1.06 ± 0.10 &0.88& peAGB & & & & NGC6752 & B2004 & 34500 ± 800 & 5.18 ± 0.14 & − 2.45 ± 0.24 &0.35& pHB & & & & Atmospheric parameters ====================== Following and, we analysed the spectra with grids of non-local thermal equilibrium (NLTE) line-blanketed model atmospheres and NLTE model spectra that were calculated with the NLTE model atmosphere code TLUSTY and the companion spectrum synthesis code SYNSPEC. The model atmosphere grids were tailored to the studied sample with the parameter space delimited in Table[tab:models]. Over 1100 new model atmospheres have been produced with steps in the model grids of 2000K in effective temperature, 0.25dex in surface gravity, and 0.5dex in helium-to-hydrogen abundance ratio by number, so that the observed spectra can be matched with model spectra that were interpolated from the grids. Scaled-solar abundances based on at the cluster’s mean metallicities have been assumed, and kept the same for models with various He/H abundance ratios. The implied change in the total mass fraction of the heavy elements has very limited consequences on our analysis (see ). The model atmospheres at higher temperature ($T\_{\rm eff} \geq 40\,000$K) allow for departures from LTE for 1132 explicit levels and superlevels of 52 ions (H, He, C, N, O, Ne, Mg, Al, Si, P, S, and Fe), as in. A detailed description of the model atoms and the source of the atomic data can be found in. Microturbulent velocity is assumed to be $v\_{\rm t} = 5\,{\rm km\,s}^{-1}$. At lower temperatures ($T\_{\rm eff} \leq 30\,000$K), we naturally considered lower ionization stages and excluded some highest ions. The model atmospheres allow for departures from LTE for 1127 explicit levels and superlevels of 46 ions (H, He, C, N, O, Ne, Mg, Al, Si, S, and Fe), with the same sources for model atoms. Microturbulent velocity is assumed to be $v\_{\rm t} = 2\,{\rm km\,s}^{-1}$. Recently, have proposed that setting microturbulence to zero provides a better fit to the spectra of the iron lines in the 21400K post-AGB star Barnard29 in NGC6205. However, the choice of microturbulence should have little effect on the broad hydrogen and helium lines studied here. [tab:models] \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Clusters \| [M/H] \| $T\_{\rm eff}$ \| logg \| $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ \| $v\_{\rm t}$ \| Number of \| \| \| \| [kK] \| \| \| [kms− 1] \| models \| \| NGC6121 & NGC6712 \| − 1.0 \| 40–70 \| 4.00–6.00 \| − 1.5… + 1.0 \| 5. \| 256 \| \| NGC5139 & NGC6656 \| − 1.5 \| 40–82 \| 4.25–6.00 \| − 3.0… + 0.5 \| 5. \| 426 \| \| \| \| 16–28 \| 2.75–5.75 \| − 3.0…0.0 \| 2. \| 332 \| \| NGC6779 \| − 2.0 \| 18–24 \| 2.75–3.25 \| − 1.0… − 0.5 \| 2. \| 24 \| \| NGC7099 \| − 2.3 \| 50–60 \| 5.50–6.75 \| − 3.0… − 2.0 \| 5. \| 108 \| For the cool stars NGC5139 − ROA3596, NGC5139 − UIT1275, and NGC5139 − UIT1425, which are most likely post-HB stars, we used the solar metallicity LTE model spectra described in to simulate the effects of radiative levitation. For the re-analysis of the stars cooler than 35000K from and for NGC5139 − ROA542 we used LTE model spectra like those from for metallicities [M/H] = − 1.0 and − 1.5. For NGC7099 − UVBS2 we used the metal-free NLTE model spectra described in. To establish the best fit to the observed spectra, we used the routines developed by and, as modified by, which employ a χ2 test. The σ necessary for the calculation of χ2 is estimated from the noise in the continuum regions of the spectra. The fit program normalizes model spectra observed spectra using the same points for the continuum definition. We fitted the Balmer (and corresponding Heii) lines H10 to Hβ (excluding Hε due to the interstellar Caii absorption line), the Hei lines λλ4026.29Å, 4120.81Å, 4387.59Å, 4471.60Å, and 4921.94Å, and the Heii lines λλ4025.60Å, 4199.83Å (only for stars hotter than 30000K), 4541.59Å, and 4685.68Å. These fit routines underestimate the errors by at least a factor of 2 (Napiwotzki priv. comm.). We therefore provide formal errors multiplied by 2 to account for this effect. In addition, the errors provided by the fit routine do not include possible systematic errors due to flat-field inaccuracies or imperfect sky subtraction, for instance. The results of the line profile fits are listed in Table[tab:params]. For some of the spectra (NGC5139 − UIT151, NGC5139 − Dk3873, NGC5139 − ROA5342, and NGC6712 − C49) we could not fit the H, Hei, and Heii lines simultaneously because the observed Hei lines were weaker than predicted, while the H and Heii lines were stronger than predicted. We saw similar effects for NGC6121 − Y453 and NGC6723 − III60 from. All these stars have effective temperatures between about 50000K and 63000K and roughly solar helium abundances. A similar behaviour was observed by for BD + 28∘4211. They suspected missing opacities to be the cause of the problem. They showed that the effects of increasing metallicity on the hydrogen and helium lines saturate at [M/H] = + 1 and therefore used model atmospheres with [M/H] = + 1 for their analysis. With these model atmospheres, they were able to reconcile the parameters derived from optical spectra with those derived from UV spectra. We therefore decided to follow the same path and analysed our hotter stars with very metal-rich model spectra. For the synthetic spectra we included only hydrogen and helium lines, as the high metallicity may only be a proxy for missing opacities (see, however, Sect.[sec:UVphot] and Fig.[fig:UV] for further information). The new results are listed in Table[tab:metal-rich]. For NGC6121 − Y453 we note that obtained similar parameters from the same optical spectrum, but a much higher temperature of 72000K (with logg of 5.7) from far-ultraviolet (FUV) data. For reasons of consistency, we use the value derived here for further discussion. The effective temperatures clearly decrease and the surface gravities increase with the metal-rich model atmospheres. This effect is also known for cooler stars (e.g. ). [tab:metal-rich] llrrrr cluster & star & $T\_{\rm eff}$ & logg & $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ & M & & [K] & [cm s− 2] & & [M⊙] NGC5139 & UIT151 & 57400 ± 2200 & 5.37 ± 0.08& − 1.07 ± 0.08 & 0.51 & Dk3873 & 47900 ± 1200& 5.15 ± 0.08& − 1.19 ± 0.08 & 0.42 & ROA5342 & 49800 ± 1400& 4.98 ± 0.08& − 1.15 ± 0.06 & 0.46 NGC6121 & UVBS2 & 46200 ± 400& 5.92 ± 0.12& + 0.96 ± 0.14 & 1.06 NGC6656 & UVBS2 & 62300 ± 9600 & 4.59 ± 0.16& + 0.40 ± 0.18 & 0.31 NGC6712 & C49 & 47800 ± 1800& 4.64 ± 0.08& − 1.10 ± 0.08 & 0.51 NGC6121 & Y453 & 56500 ± 1800 & 5.71 ± 0.12 & − 1.16 ± 0.10 & 0.79 NGC6723 & III60 & 42300 ± 1400 & 4.80 ± 0.12 & − 1.13 ± 0.12 & 1.19 NGC6656 − UVBS2 --------------- The analysis of NGC6656 − UVBS2 was difficult for two reasons: It has a cool neighbour very close by, and it has a planetary nebula. The first results using model spectra with the cluster abundance pointed towards a temperature of almost 80000K, while the results obtained with metal-rich model spectra suggest a much cooler temperature of 62000K. Based on a low-resolution spectrum, suggested an effective temperature of 75000K from the similarity to the sdO star KS292, which had been analysed by. The spectrum shows absorption lines of Civ and Nv similar to the star SDSSJ155610.40 + 254640.3, which has a temperature of about 100000K. The deeper Heii lines of NGC6656 − UVBS2 are consistent with a lower temperature, which unfortunately does not allow us to distinguish between the two temperatures mentioned before. suggested that NGC6656 − UVBS2 is the result of a stellar merger. Interestingly enough, SDSSJ155610.40 + 254640.3 was classified by as a PG1159 star, a class of H-deficient post-AGB stars. In any case, it seems that NGC6656 − UVBS2 is closely related to the star ZNG1 in NGC5904 (see Table[tab:lit]), which has also been connected to both a merger and PG1159 stars. While NGC5904 − ZNG1 shows a high rotational velocity of about 170km s− 1, we see no evidence for fast rotation in our spectrum of NGC6656 − UVBS2. We recall, however, that the resolution of our data is about 100km s− 1. Metal lines ----------- We took care to verify that potential metal lines were present in both individual spectra for each star. We found potential metal lines only in the stars listed below and compared their strength to that predicted by the best-fitting model spectrum. The number of lines for each ion are given in parentheses, and the stars are sorted by evolutionary stage. A detailed abundance analysis is beyond the scope of this paper. NGC5139 − UIT1425 (pHB) Mgii (1) significantly weaker than predicted by the metal-rich model spectrum with metal lines, which is usual for HB stars in this temperature range. NGC5139 − UIT151 (peAGB) Nv (2) in agreement with the metal-rich model spectrum with metal lines, which also predicts strong Ciii and Oiv lines that are not observed, however. NGC5139 − UIT644 (pAGB) Nii (9), Cii (1) Mgii (1), and Siiii (2) much stronger than predicted by the metal-poor model spectrum used for the analysis, especially Nii. NGC6656 − UVBS2 (pAGB) Civ (3), Nv (2), much stronger than predicted by the model spectrum with metal lines for [M/H] = + 1. NGC6121 − UVBS2 (WDM) Ciii (7), much stronger than predicted by the model spectrum with metal lines for [M/H] = + 1. UV spectrophotometry -------------------- Several of our targets have archival UV spectra from either the International Ultraviolet Explorer (IUE), the Far Ultraviolet Spectroscopic Explorer (FUSE), or the HST. We defer a complete discussion of the UV spectra to a subsequent paper, but discuss here some of the IUE and HST Faint Object Spectrograph (FOS) data in ωCen (Table[tab:uvspectra]), which (1) provide additional justification for our use of metal-rich atmospheres for the sdO stars, and (2) provide evidence of binarity for the star UIT1425. The data were obtained from the [Mikulski Archive for Space Telescope (MAST)](http://archive.stsci.edu/iue/), and the IUE spectra were converted into the HST absolute calibration scale using the transformation in. We restricted use of the IUE data to the short-wavelength ( < 1950 Å) prime (SWP) camera because the IUE large (10 × 20) aperture includes light from background red stars at longer wavelengths. We expect the SWP images to be free of background contamination because examination of the UIT 1620Å image of ωCen showed that no UV sources are included within the IUE aperture for any of the target stars. In addition, the FOS and IUE spectra of UIT151 show fair agreement in Fig.[fig:UV], even though the FOS data use a much smaller (1 circular) aperture. [tab:uvspectra] llrllrrlrr Star & Telescope & Mode & ID & Date & Exp & & & & & [s] UIT151 & HST/FOS & G160L & Y2SS402T & 1996-04-28 & 2400 UIT151 & IUE & SWP & 54154 & 1995-03-16 & 23400 Dk3873 & IUE & SWP & 48271 & 1993-07-31 & 19200 ROA5342 & IUE & SWP & 54333 & 1995-04-08 & 25500 UIT1425 & IUE & SWP & 54804 & 1995-05-31 & 26820 The high-resolution study of field sdO stars by showed that the UV spectrum of these stars is dominated by numerous lines of iron and nickel. Even though our low-resolution UV spectra cannot individually resolve these lines, the lines are sufficiently numerous at high metallicity to affect the apparent continuum. Figure[fig:UV] shows surprisingly good agreement between the low-resolution UV spectra of the three sdO stars and models with [M/H] = + 1.0. The Civ doublet near 1550Å appears much weaker than predicted by the metal-rich models, suggesting that the light elements do not follow the abundance enhancement seen in iron and nickel, an effect also seen in some field sdO stars. We are able to use the known metallicity spread of ω Cen to set an upper limit of [M/H] = − 1.0 for the metallicity of the progenitors of these sdO stars. The factor of ≈ 100 enhancement in iron and nickel must then be attributed to radiative levitation. For NGC5139 − UIT1425 a model spectrum with the parameters derived from our optical spectra cannot reproduce the UV data (see Fig.[fig:UIT1425]). This star is most likely a binary, as its optical colours (B − V= − 0.01, V − I= + 0.20) are appropriate for a blue HB star, whereas its UV − V colour and steep IUE spectrum is more indicative of an sdO star. The optical spectrum, however, shows no trace of Heii, which would be a clear sign of the hotter star. Unfortunately, there is no HST imagery in this region of ωCen, so that we cannot rule out that the binary is a chance alignment. We checked the spatial (line by line) IUE image, but found no presence of more than one star within the large aperture. [fig:UV] [fig:UIT1425] Data from the literature ======================== In Table[tab:lit] we list the physical parameters published for hot UV-bright stars other than those discussed here. For NGC6656 − II-81 we note that its position in the $T\_{\rm eff}, \log g$ diagram is consistent with being a post-AGB instead of a post-HB star. [tab:lit] llrllrrlrr cluster & star & $T\_{\rm eff}$ & logg & $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ & status & Ref. & V & Ref. & M & & [K] & [cgs] & & & & [mag] & & [M⊙] NGC104 & BS & 11000 & 2.5 & − 1 & pAGB & (1) & 10.73 & (13)& 1.25 NGC1851 & UV5 & 16000 & 2.5 & − 1 & pAGB & (2) & 13.26& (14)& 0.41 NGC5139 & ROA5701 & 25000 & 3.3 & − 1 & pAGB & (3) & 13.13 & (24) & 0.32 NGC5272 & vZ1128 & 36600 & 3.95 & − 0.84 & peAGB & (4) & 15.03 & (15)& 0.41 NGC5904 & ZNG1 & 44300 & 4.3 & +0.52 & pAGB & (5) & 14.54 & (16)& 0.72 NGC6205 & B29 & 21400 & 3.10 & − 0.89 & pAGB & (6) & 13.116& (23) & 0.38 NGC6254 & ZNG1 & 27000 & 3.6 & − 1.33 & pAGB & (7) & 13.23 & (17)& 0.59 NGC6397 & ROB162 & 51000 & 4.5 & − 1.0 & pAGB & (8) & 13.1 & (9)& 0.50 NGC6656 & II-81 & 38000 & 4.2 & — & pHB & (9) & 14.0 & (18) & 0.47 NGC6712 & ZNG1 & 11000 & 2.1 & − 1.19 & pAGB & (7) & 13.33 & (19)& 0.35 NGC6752 & B852 & 39000 & 5.2 & − 2.0 & pHB & (10) & 15.91 & (20)& 0.52 & B4380 & 32000 & 5.3 & − 2.3 & pHB & (10) & 15.93 &(20)& 0.79 & B1754 & 40000 & 5.0 & − 1.52 & pHB & (10) & 15.99 & (20)& 0.30 NGC7078 & K648 & 39000 & 3.9 & − 1.10 & pAGB & (11) & 14.73 & (11)& 0.62 & K996 & 11500 & 2.5 & — & pHB & (12) & 14.31 & (21)& 0.26 & ZNG1 & 28000 & 3.7 & − 1.22 & pAGB & (7) & 14.8 & (17)& 0.55 We wish to compare results between different globular clusters, therefore we decided to use homogeneous data sets for their ages, metallicities, and integrated V magnitudes. Unfortunately, we found no such data set for the distances and reddenings. Therefore we used the values from that are at least created in a defined way from the various sources. The values we used in our analysis are listed in Table[tab:par-gc]. [tab:par-gc] l\|rlrrrr\|rrr Cluster & [M/H] & age & Vt & (m − M)V & EB − V & BCV & & & [109 yr] & [mag] & [mag] & [mag] & [mag] & $f\_{\rm RHB}$ & $f\_{\rm BHB}$ & $f\_{\rm EHB}$ NGC104 & − 0.76 & 13.1 & 4.01 & 13.37 & 0.04 & − 0.53 & 1.00 & 0.00 & 0.00 NGC1851 & − 1.18 & 10.0 & 7.16 & 15.47 & 0.02 & − 0.41 & 0.64 & 0.36 & 0.00 NGC5139 & − 1.64 & 11.5 & 3.85 & 13.94 & 0.12 & − 0.38 & 0.06 & 0.71 & 0.23 NGC5272 & − 1.50 & 11.4 & 6.36 & 15.07 & 0.01 & − 0.38 & 0.32 & 0.68 & 0.00 NGC5904 & − 1.33 & 10.6 & 5.58 & 14.46 & 0.03 & − 0.40 & 0.21 & 0.79 & 0.00 NGC5986 & − 1.63 & 12.2 & 7.46 & 15.96 & 0.28 & − 0.38 & 0.00 & 0.91 & 0.09 NGC6121 & − 1.18 & 12.5 & 5.77 & 12.82 & 0.35 & − 0.43 & 0.39 & 0.61 & 0.00 NGC6205 & − 1.58 & 11.6 & 5.82 & 14.33 & 0.02 & − 0.38 & 0.00 & 0.70 & 0.30 NGC6254 & − 1.57 & 11.4 & 6.60 & 14.08 & 0.28 & − 0.38 & 0.00 & 1.00 & 0.00 NGC6397 & − 1.99 & 12.7 & 6.20 & 12.37 & 0.18 & − 0.37 & 0.00 & 1.00 & 0.00 NGC6656 & − 1.70 & 12.7 & 5.08 & 13.60 & 0.34 & − 0.38 & 0.00 & 0.73 & 0.27 NGC6712 & − 1.02 & 12.0 & 8.04 & 15.60 & 0.45 & − 0.44 & 0.85 & 0.15 & 0.00 NGC6723 & − 1.10 & 13.1 & 7.03 & 14.84 & 0.05 & − 0.44 & 0.58 & 0.42 & 0.00 NGC6752 & − 1.55 & 11.8 & 5.32 & 13.13 & 0.04 & − 0.38 & 0.00 & 0.72 & 0.28 NGC6779 & − 2.00 & 13.7 & 8.17 & 15.68 & 0.26 & − 0.37 & 0.00 & 1.00 & 0.00 NGC7078 & − 2.33 & 12.9 & 6.32 & 15.39 & 0.10 & − 0.35 & 0.10 & 0.90 & 0.00 NGC7099 & − 2.33 & 12.9 & 7.35 & 14.64 & 0.03 & − 0.35 & 0.00 & 1.00 & 0.00 Masses and distances ==================== Using the atmospheric parameters together with the observed brightness of the stars and reddening and distances to their parent globular cluster, we determined masses using Eq.[eq:massbc], $$\label{eq:mass\_bc} \log {\frac{M}{M\_\odot}} = \log{\frac{g\_\ast}{g\_\odot}} - 4\cdot \log {\frac{T\_\ast}{T\_\odot}} - {\frac{M\_V+BC-4.74}{2.5}} ,$$ which can be rewritten to $$\label{eq:mass\_bc2} \log {\frac{M}{M\_\odot}} = \log g\_\ast + 0.4 \cdot [(m-M)\_0-V\_\ast+A\_V - BC] -4 \cdot \log T\_\ast +C3 ,$$ with $$\label{eq:C3} C3 = -\log g\_\odot + 4\cdot \log T\_\odot + {\frac{4.74}{2.5}} ,$$ with the bolometric corrections from. The results are listed in Tables[tab:params] and [tab:lit] and plotted in Fig.[fig:alltm]. [fig:alltm] From stellar evolution we would generally expect masses of 0.5M⊙ to 0.6M⊙. Figure [fig:alltm] shows that the masses derived for stars below 25000K tend to be too low, while the masses derived for hotter stars scatter towards too high masses[3](#fn3). Some of the globular clusters discussed in this paper are close enough to have parallax estimates from the Gaia DR2. The Gaia collaboration compared the parallaxes derived from the Gaia measurements to those listed in and found a systematic difference of 0.029mas, with the Gaia parallaxes being smaller. We therefore used the reported Gaia parallaxes only for those clusters that had parallax values of more than 0.23mas, that is, about eight times the systematic offset. To derive distances, we applied the reported systematic offset to the parallaxes. We then added interstellar extinction as 3.2 ⋅ EB − V. The apparent distance moduli derived that way are 0.29mag smaller for NGC6121 and 0.16mag, 0.18mag, and 0.08mag larger for NGC6397, NGC6656, and NGC6752, respectively, than those from. It is interesting to note that we find higher-than-expected masses for the two stars in NGC6121, which would be reduced by 30% with the Gaia distance to 0.81M⊙ and 0.65M⊙, respectively. For NGC6397, using the Gaia distance would increase the masses by 16%, moving them from about 0.5M⊙ to about 0.6M⊙. For NGC6656, using the Gaia distance would increase the mass of NGC6656 − UVBS2 from 0.31M⊙ to 0.38M⊙ (without the additional reddening correction) and from 0.55M⊙ to 0.65M⊙ (with the additional reddening correction). The change of 8% for the masses of the stars in NGC6752 is negligible. Stellar evolution models and evolutionary fluxes ================================================ [tab:par-gc-model] [tab:evolini] \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| $M\_{\rm ZAMS}$ \| $Z\_{\rm ZAMS}$ \| $Y\_{\rm ZAMS}$ \| $X\_{\rm ZAMS}$ \| \| \| [M⊙] \| \| \| \| \| − 1.0 \| 0.85 \| 0.00172 \| 0.24844 \| 0.74984 \| \| − 1.5 \| 0.83 \| 0.000548 \| 0.246096 \| 0.753356 \| \| − 2.0 \| 0.82 \| 0.000174 \| 0.245348 \| 0.754478 \| \| − 2.3 \| 0.82 \| 0.000087 \| 0.245174 \| 0.754739 \| [tab:result-gc] lc\|ccc Cluster & N$^{\rm post-AGB}\_{\rm obs}$ & [M/H]$\_{\rm tracks}$ & N$^{\rm post-AGB}\_{\rm exp}$ NGC104 & 1 & − 1.00 & 0.08 – 0.65 NGC1851 & 1 & − 1.00 & 0.15 – 1.04 NGC5139 & 5 & − 1.50 & 1.31 – 13.58 NGC5272 & 1 & − 1.50 & 0.27 – 1.48 NGC5904 & 1 & − 1.50 & 0.38 – 2.11 NGC5986 & 1 & − 1.50 & 0.61 – 4.29 NGC6121 & 1 & − 1.00 & 0.19 – 1.32 NGC6205 & 1 & − 1.50 & 0.23 – 2.71 NGC6254 & 1 & − 1.50 & 0.26 – 1.44 NGC6397 & 1 & − 2.00 & 0.07 – 0.26 NGC6656 & 2 & − 1.50 & 0.59 – 6.65 NGC6712 & 2 & − 1.00 & 0.14 – 0.98 NGC6723 & 2 & − 1.00 & 0.12 – 0.83 NGC6752 & 0 & − 1.50 & 0.13 – 1.48 NGC6779 & 1 & − 2.00 & 0.29 – 1.13 NGC7078 & 2 & − 2.30 & 1.04 – 3.71 NGC7099 & 0 & − 2.30 & 0.17 – 0.64 [fig:alltg] The hot UV-bright phase contains stars in different evolutionary stages. The most luminous of these stars ((L/L⊙) $\gtrsim$ 3.1) are believed to be post-AGB stars, which go through a luminous UV-bright phase as they leave the AGB and move rapidly toward their final white dwarf state. Despite their short lifetimes ($\lesssim 10^{5}$yr), hot post-AGB stars ($T\_{\rm eff} >$ 7000K) can dominate the total UV flux of an old stellar population. UV-bright stars with luminosities 2.65 $\lesssim \log$ (L/L⊙)$\lesssim$ 3.1 are a mixture of low-mass post-AGB stars, which are stars that departed from the AGB before the beginning of the thermal pulses (post-early AGB stars), and post-EHB stars that managed to ignite the H-burning shell after the departure from the HB, but never reached the AGB. The post-early AGB population arises from hot HB stars with sufficient envelope mass to return to the AGB, but which peel off the AGB prior to the thermally pulsing phase. Less luminous hot UV-bright stars (1.8 $\lesssim \log$ (L/L⊙)$\lesssim$ 2.65) are mostly evolving from the EHB to the white dwarf domain (post-EHB stars), but some may also evolve from the hot end of the blue horizontal branch (BHB) to the AGB phase. The population of post-EHB stars is expected to be about 15–20% of the population of EHB stars. We now wish to compare the number of observed post-AGB stars with expectations from stellar evolution models. In the absence of a detailed study of the distribution of stars in the HB of each cluster it is not possible to perform detailed synthetic population simulations for each cluster. However, using the available information from the HST UV Legacy Survey of Galactic Globular Clusters we estimated the fractions of stars populating the red HB (RHB), BHB, and EHB, based on the HB morphology in the V, V − I colour-magnitude diagram (see Table[tab:par-gc] for results). Using these numbers, we estimated how many post-AGB stars should evolve from the RHB, BHB, and EHB in each cluster. The post-AGB evolutionary models computed for this work are an extension of the models presented by. The models were computed for values of [M/H] = − 1, − 1.5, − 2 and − 2.3 under the assumption of a scaled-solar mixture. The helium-metallicity relation was taken as in, that is, Y = 0.245 + 2 × Z. The initial masses of the models were chosen to correspond to ages between 11.5 to 12 Gyr. The higher or lower ages of some of the globular clusters (see Table[tab:par-gc]) are no problem because age does not influence the mass of the helium core at the time of the helium core flash, nor the following evolution. It might affect the mass of the hydrogen-rich envelope, but this mass is varied artifically in our models anyway to populate the whole horizontal branch. The corresponding initial parameters for the stellar evolution sequences are shown in Table[tab:evolini]. Different mass loss was applied to each model before the zero-age horizontal branch (ZAHB) in order to obtain different masses at the ZAHB and a complete coverage of the horizontal branch (from the RHB to the EHB) and post-HB evolution (see Table[tab:sequences] for details). Convective boundary mixing at all evolutionary stages and mass loss on the AGB were also adopted as in. Figure[fig:alltg] shows the atmospheric parameters derived in Sect.[sec:param] compared to the evolutionary tracks. The evolutionary stage derived from this comparison can be found in Table[tab:params]. The stars are colour-coded by their helium abundance because this helps to identify the evolutionary status of a star. Significantly sub-solar helium abundances due to diffusion are known for HB stars hotter than about 11000K. The diffusion patterns are erased once convection starts in the atmosphere as the star evolves towards cooler temperatures. A low helium abundance is therefore unlikely for stars that ascended the AGB (even if only partially). Fig.[fig:alltg] shows that helium-poor stars are found along post-(E)HB tracks and not along post-(e)AGB tracks. Helium-rich stars, on the other hand, are found only along the post-AGB tracks (except for the white dwarf merger). In order to compare the stellar evolution models with the observed number of stars, we adopted the evolutionary flux method (see for details). This method assumes that the late evolutionary stages are much shorter than the main-sequence evolution and that the number of “dying stars” can be equated to the number of stars leaving the main sequence. Under these assumptions, the number of stars Nk in a simple stellar population at each evolutionary stage k is given by $$N\_k= B\times L\_{\rm total}\times t\_k, \label{eq: evolflux}$$ where tk is the duration of the evolutionary stage k, $L\_{\rm total}$ is the total luminosity of the stellar population, and B stands for the specific evolutionary flux. For old stellar populations, like the globular clusters studied in this work, we can approximate B ≃ 2 × 10− 11 stars per year and per solar luminosity (see for details). In our case, we are interested in the time spent by the models in the region of the Hertzprung-Rusell (HR) diagram where we would observe them as hot UV-bright post-AGB stars. We define the region in the HR diagram corresponding to hot UV-bright post-AGB stars detected in optical colours as that defined by logL/L⊙ > 2.65 and $4.9<\log T\_{\rm eff}<3.845$ (grey zone in Fig.[fig:tracks]). Each computed track casts a timescale t corresponding to the time spent in that part of the HR diagram. With the parameters shown in Table[tab:par-gc] and with Eq.[eq: evolflux], we can compute for each cluster the expected numbers Nt predicted by each track t of a similar metallicity. It should be emphasized that Eq.[eq: evolflux] assumes that the whole cluster population evolved through that specific track k. Each post-HB track computed for the cluster metallicity gives a different expected number Nk (Eq.[eq: evolflux]), and the actual number of expected post-HB stars will depend on the frequency with which each specific track is followed in each cluster. Lacking a better characterization of the HB demographics, we can split the HB into the RHB, BHB, and EHB and then estimate the number of stars evolving from the RHB, BHB, or EHB using the relative populations of the RHB, BHB, and EHB in each cluster[4](#fn4) ($f\_{\rm RHB}$, $f\_{\rm BHB}$, and $f\_{\rm EHB}$ respectively, see Table[tab:par-gc]). Using all sequences evolving from the RHB, BHB, or EHB, we obtain a range of expected numbers for hot post-AGB stars evolving from each part of the HB ($N^{\rm RHB}$, $N^{\rm BHB}$, or $N^{\rm EHB}$, respectively). The actual range of expected UV-bright stars in the post-AGB region of the HR diagram (N) of a given cluster is then estimated as $$N= f\_{\rm EHB} \cdot N^{\rm EHB} + f\_{\rm BHB} \cdot N^{\rm BHB} + f\_{\rm RHB} \cdot N^{\rm RHB}. \label{eq: number}$$ [fig:tracks] The resulting ranges of expected numbers for each cluster are shown in Table[tab:result-gc]. We excluded NGC2808 from this investigation because we know that several luminous hot UV-bright stars are not included in our study. [fig:result-gc] Overall, the agreement between the number of post-AGB stars estimated (N$^{\rm post-AGB}\_{\rm exp}$, fourth column in Table[tab:result-gc]) and the observed number of post-AGB stars (N$^{\rm post-AGB}\_{\rm obs}$, second column in Table[tab:result-gc]) in each cluster is good (see also Fig.[fig:result-gc]). In 11 out of 17 cases ( ≈ 65%) the observed number of stars lies within the computed range, while in 3 other cases (NGC104, NGC6752, and NGC7099) the observed number is just an integer number next to the expected range. A more significant discrepancy is observed for NGC6712 and NGC6723, where the upper boundary of the predicted ranges is a factor ≈ 2 lower than the observed number. An even larger discrepancy is observed for NGC6397, for which the largest predicted number is about four times smaller than the observed one. In these three cases, the observed value is higher than the predicted ones, but a word of caution is in order. We have preselected for our sample those clusters that do harbour hot UV-bright stars. It is therefore possible that our clusters are biased to larger post-AGB numbers. Given the low number of the expected post-AGB stars and the fact that the observed number is necessarily an integer, it is difficult to point to this discrepancy as a discrepancy with the stellar evolution models. A full comparison of all clusters searched for hot UV-bright stars is required for firmer conclusions. Conclusions =========== Our results first confirm that spectroscopic observations of UV-bright stars are required to verify their evolutionary status, as is shown by the cases of NGC7099 − UVBS2 and NGC6121 − UVBS2. The need for metal-rich model spectra in the analysis of hot evolved stars has been reported elsewhere , and we can confirm that additional opacities are required to reproduce the Hei and Heii lines simultaneously and the UV spectra. This should be kept in mind when the contribution of hot post-AGB stars to the UV flux of evolved populations is estimated. We find general agreement for the number of observed hot post-AGB stars compared to predictions from evolutionary theory, although the numbers per cluster are low (at most two, except for ωCen, which has five). Some discrepancies are observed in the clusters NGC6397, NGC6712, and NGC6723, which show somewhat larger numbers than expected from the models. Because of the small-number statistics nature of the comparison, it is unclear, however, whether this is due to a problem with the models or a result of the way the sample was preselected. In order to improve our constraints on low-mass post-AGB stellar evolution models, the number of studied globular clusters needs to be increased. When the full set of 78 globular clusters with UV-imaging is studied, it will be possible to combine clusters with similar metallicities and HB morphologies to improve the number statistics in the comparisons. We are grateful to Nicole Reindl, who made her co-added spectrum of SDSSJ155610.40 + 254640.3 and its model spectrum available to us. We thank Simon Jeffery for the merger track and his explanations of the merger behaviour. We appreciate the anonymous referee’s comments that improved the readability and clarity of this paper substantially. M.M.M.B. is partially supported through ANPCyT grant PICT-2016-0053 and MinCyT-DAAD bilateral cooperation program through grant DA/16/07. This research has made use of NASA’s Astrophysics Data System Bibliographic Services and of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Flux calibration ================ GRIS\_1200B is a volume-phase holographic grism. Its response therefore depends on the position of the slit along the dispersion axis. The best solution for a flux calibration is to take the flux standard star at the same place on the detector as the science spectrum. This is done for data taken with the long slit by creating a 5 wide long slit with the slit blades of the MOS slitlets. For MOS observations with distributed slitlets, this would require taking several flux standard star observations, which is not feasible in service mode. To correct the response derived from standard stars taken at different positions than the science targets, we made use of the fact that the varying response is seen also in the spectral energy distribution of the screen flat fields taken for the MOS data (see and for a more detailed description). Therefore we took the ratio of the wavelength-calibrated master screen flat and normalized master flat and averaged it along the slit. Then we divided the extracted standard star spectra and the extracted science spectra by their flat-field spectra. This procedure will of course introduce the lamp spectrum into these spectra, but as long as the lamp spectrum does not change, this can be corrected for with the response curve. During this exercise we noted three potential problems: 1. The narrow-slit flat fields taken until June 22, 2012, show two emission lines at about 3944.2Å and 3961.8Å. The lines are not visible in the 5 flat fields taken for the standard stars. The smoothing of the master screen flat along the dispersion axis will smear out these lines to a radius of at least ± 10 pixels or ± 7Å (relative to the peak position). We therefore did not use these regions in our later analysis. 2. Between June 22 and July 22, 2012, the spectral energy distribution of the flat-field lamps changed. 3. Even between flat fields taken within 30 minutes of each other, the spectral energy distribution changed by some 20% (see Fig.[fig:specratio]). [fig:specratio] All extracted spectra (science and standard stars) were corrected for atmospheric extinction using the extinction coefficients of. In order to have a finely sampled response, we used observations of EG274 (April 20, 2012) and Feige110 (June 22, 2012), for which finely sampled reference spectra exist. We determined the radial velocity of the observed spectrum using the Hδ line, aligned the noise-free reference spectrum to the same radial velocity, and resampled it to the same wavelength steps as the observed spectrum. Then we computed the ratio of the observed extinction-corrected standard star spectrum and the resampled reference spectrum. This ratio showed small residuals at the positions of strong stellar lines, which we masked when we fit a smoothed spline to the ratio. This fit was then used to flux-calibrate the extinction-corrected science data. As noted above, the spectral energy distribution of the flat fields changed between June 22 and July 22, 2012. Only long-slit spectra were taken after June 22. We therefore decided to use the flat-field spectrum taken for the long slit on June 14, 2012, to correct all long-slit spectra. This procedure allowed us to use the standard star observed on June 22, 2012 also for data from July. In principle, the long-slit spectra do not even need to be corrected if a standard star was observed at the same position (as is the case for Feige110). However, the smoothed flat-field spectra show some variations in detector response at the blue end that are hard to fit without a high risk of fitting noise in the response. The division by the flat-field spectra is therefore useful beyond correcting the position dependency of the response. In order to deal with the short-term variations of the flat-field lamp we used the same flat-field spectrum for the standard star taken at the position of the LSS 05 slit and all science data taken with the long slit. For the MOS data we could not use the same flat field for science data and standard star because they were observed at different positions on the CCD. Details of the stellar evolution sequences ========================================== [tab:sequences] cccccc $M\_{\rm ZAHB}$ & $T\_{\rm eff, ZAHB}$ & $\log g\_{\rm ZAHB}$ & HB location & post-HB behavior & $M\_{\rm final}$ [M⊙] & [K] & [cm s− 2] & & &[M⊙] 0.490 & 30138 & 5.74 & EHB & post-EHB, no thermal pulses & 0.490 0.494 & 26706 & 5.52 & EHB & 1 thermal pulse (like a LTP) & 0.493 0.495 & 26152 & 5.49 & EHB & 1 thermal pulse (like a LTP) & 0.495 0.500 & 24040 & 5.34 & EHB & 2 thermal pulses (like a LTP)& 0.496 0.530 & 17770 & 4.78 & BHB & post-EAGB, 2 thermal pulses (like a LTP)& 0.499 0.550 & 15264 & 4.48 & BHB & TP-AGB + LTP & 0.504 0.580 & 11097 & 3.80 & BHB & TP-AGB + LTP & 0.513 0.600 & 8815 & 3.35 & BHB & TP-AGB & 0.518 0.650 & 5724 & 2.56 & RHB & TP-AGB +LTP & 0.528 ‡ 0.700 & 5484 & 2.50 & RHB & TP-AGB & 0.537 0.750 & 5392 & 2.47 & RHB & TP-AGB & 0.545 0.850 & 5315 & 2.46 & RHB & TP-AGB & 0.555 0.495 & 29013 & 5.66 & EHB & post-EHB, no thermal pulses & 0.495 0.500 & 26116 & 5.47 & EHB & 1 thermal pulse (like a LTP) & 0.499 0.510 & 22942 & 5.23 & EHB & post-EAGB, 1 thermal pulse (like a LTP) & 0.500 0.530 & 19297 & 4.91 & BHB & post-EAGB, 1 thermal pulse (like a LTP) & 0.501 0.550 & 16897 & 4.66 & BHB & TP-AGB + LTP & 0.505 0.600 & 12354 & 3.99 & BHB & TP-AGB + LTP & 0.519 0.650 & 8670 & 3.31 & BHB & TP-AGB + VLTP & 0.525 ‡ 0.700 & 6105 & 2.67 & RRLyr & TP-AGB + LTP & 0.540 ‡ 0.750 & 5700 & 2.54 & RRLyr & TP-AGB + LTP & 0.550 ‡ 0.830 & 5542 & 2.50 & RHB & TP-AGB & 0.557 0.505 & 29455 & 5.65 & EHB & post-EHB, no thermal pulses & 0.505 0.506 & 28773 & 5.61 & EHB & post-EHB, no thermal pulses & 0.506 0.510 & 26760 & 5.48 & EHB & 1 thermal pulse (like a LTP) & 0.509 0.520 & 23648 & 5.26 & EHB & post-EAGB, 1 thermal pulse (like a LTP) & 0.510 0.550 & 18774 & 4.83 & BHB & post-EAGB, 1 thermal pulse (like a LTP) & 0.513 0.600 & 14431 & 4.31 & BHB & TP-AGB & 0.525 0.700 & 9064 & 3.40 & BHB & TP-AGB & 0.547 0.820 & 6164 & 2.69 & RRLyr & TP-AGB & 0.557 0.505 & 30346 & 5.72 & EHB & post-EHB, no thermal pulses & 0.505 0.515 & 25384 & 5.39 & EHB & 1 thermal pulse (like a LTP) & 0.511 0.520 & 24034 & 5.29 & EHB & post-EAGB, 1 thermal pulse (like a LTP) & 0.512 0.550 & 19090 & 4.86 & BHB & TP-AGB & 0.513 0.600 & 14787 & 4.35 & BHB & TP-AGB+VLTP & 0.510 0.700 & 9681 & 3.51 & BHB & TP-AGB & 0.546 0.820 & 7967 & 2.92 & RRLyr & TP-AGB & 0.570 --- 1. Based on observations with the ESO Very Large Telescope at Paranal Observatory, Chile (proposal ID 089.C-0210)[↩](#fnref1) 2. The extracted spectra, their best-fitting model spectra, and the evolutionary tracks used in this paper will be available at CDS.[↩](#fnref2) 3. The dust in the planetary nebula IRAS18333 − 2357 surrounding NGC6656 − UVBS2 (at about 62000K) probably provides additional extinction so that the derived mass is definitely a lower limit. estimated a reddening of EB − V = 0.54 compared to a global reddening of 0.34 used for the mass determination. This would increase the mass by some 80%, leading to a mass of ∼ 0.55 M⊙, which is typical of low-mass central stars of planetary nebulae.[↩](#fnref3) 4. The lifetimes on the HB are not strongly affected by the effective temperature of the models, therefore this is a decent approximation.[↩](#fnref4) Hot UV-bright stars of galactic globular clusters[1](#fn1)[2](#fn2) =================================================================== ### Received April 15, 2019 / Accepted May 7, 2019 We wish to derive effective temperatures, surface gravities, and helium abundances of the luminous hot UV-bright stars in these clusters to determine their evolutionary status and compare the observed numbers to predictions from evolutionary theory. We obtained FORS2 spectroscopy of eleven of these UV-selected objects (covering a range of − 2.3 < [Fe/H] < − 1.0), which we (re-)analysed together with previously observed data. We used model atmospheres of different metallicities, including super-solar ones. Where possible, we verified our atmospheric parameters using UV spectrophotometry and searched for metal lines in the optical spectra. We calculated evolutionary sequences for four metallicity regimes and used them together with information about the HB morphology of the globular clusters to estimate the expected numbers of post-AGB stars. We find that metal-rich model spectra are required to analyse stars hotter than 40000K. Seven of the eleven new luminous UV-bright stars are post-AGB or post-early AGB stars, while two are evolving away from the HB, one is a foreground white dwarf, and another is a white dwarf merger. Taking into account published information on other hot UV-bright stars in globular clusters, we find that the number of observed hot post-AGB stars generally agrees with the predicted values, although the numbers are still low. Spectroscopy is clearly required to identify the evolutionary status of hot UV-bright stars. For hotter stars, metal-rich model spectra are required to reproduce their optical and UV spectra, which may affect the flux contribution of hot post-AGB stars to the UV spectra of evolved populations. While the observed numbers of post-AGB and post-early AGB stars roughly agree with the predictions, our current comparison is affected by low number statistics. Introduction ============ Ultraviolet images of globular clusters are often dominated by one or two hot, luminous, UV-bright stars that are more than a magnitude brighter than the horizontal branch (HB) and hotter than 7000K. Such stars are in various evolutionary stages after helium core burning (HeCB). These evolutionary stages are the most uncertain phases of the evolution of low-mass single stars, caused by uncertainties in the size of the C/O-core due to our poor understanding of convective boundary mixing, and our lack of understanding of winds in red giant branch (RGB) stars. The previous mass loss on the RGB plays an important role here because it determines whether a star even ascends the asymptotic giant branch (AGB): HB stars with very low hydrogen envelope masses evolve directly from the extreme horizontal branch (EHB, T$\_{\rm eff}\gtrsim$ 20000K) to the white dwarf stage, whereas stars with hydrogen envelope masses of more than 0.02M⊙ will at least partly ascend the AGB. Owing to the previous uncertainties, both the masses of the C/O-core and the H-rich envelope of post-HeCB are loosely constrained. In particular, theoretical post-AGB tracks have only been very little tested for old low-mass stars because suitable observations are scarce. A similar situation holds for post-EHB stars that evolve directly to the white dwarf phase and avoid the AGB. As an example, found far fewer hot post-AGB stars than expected in their Space Telescope Imaging Spectrograph (STIS) UV imagery of M32, while reported a similar lack of hot post-AGB stars in the Galactic halo. The status of hot post-AGB stars in globular clusters remains uncertain: with only about one hot post-AGB star per cluster, many clusters must be observed in order to compare number counts, luminosities, and surface gravities with evolutionary tracks. The detection of hot post-AGB stars in optical colour-magnitude diagrams is limited by selection effects that are caused by crowding in the cluster cores and by the large bolometric corrections for these hot stars. More complete searches are possible for hot post-AGB stars in planetary nebulae, for example, by using Oiii imaging, as performed by and. The meaningfulness of such searches, however, is limited by the long evolutionary timescales of low-mass post-AGB stars, which make the appearance of a planetary nebula unlikely. Only four planetary nebulae have indeed thus far been discovered in the Galactic globular cluster system. and first pointed out the advantages of wide-field UV imagery to obtain a complete sample of hot post-AGB stars in a large number of globular clusters. The UV suppresses the dominant cool-star population and emphasizes the hot stars, ensuring that all hot post-AGB stars in the cluster are detected. Fourteen globular clusters were observed with Ultraviolet Imaging Telescope (UIT, ) in 1990 and 1995 at 1620Å with a 40ʹ field of view (FOV). used ground-based spectroscopy to derive temperatures and gravities of the newly discovered hot UV-bright stars. Their results clearly illustrated the points made above, namely that the HB morphology has a strong influence on the evolutionary status of hot UV-bright stars and that optical selections suffer from a strong bias towards the most luminous hot stars. Subsequently, the Galaxy Evolution Explorer (GALEX) satellite was used to obtain wide-field UV images of 41 globular clusters at 1520Å with a 1.2∘ diameter FOV. Because GALEX observations are limited by its brightness constraints to mostly high Galactic latitude fields, we obtained archival UV images of an additional 31 clusters using the Swift Ultraviolet-Optical Telescope (UVOT, ). The UVOT images are less satisfactory because its FOV is smaller (17ʹx17ʹ) and its wavelength is longer. The solar-blind UVM2 filter on the UVOT has a central wavelength of 2250Å, while the UVW2 filter has a central wavelength of 1930Å. The UV imagery from all three wide-field telescopes was complemented by UV imagery with the Hubble Space Telescope (HST, ), which provides much higher spatial resolution and photometric precision, but rarely encompasses the entire globular cluster within its FOV. This brings the total number of globular clusters with UV imaging data to 78. Among these, we recovered all previously known hot post-AGB star candidates and found 19 new hot post-AGB star candidates and 16 other new hot stars (mainly apparent EHB or post-EHB stars in globular clusters where only red HB stars have been known so far). Because each cluster contains only a few post-AGB stars, a large sample of clusters is needed to test stellar evolution predictions. To this aim, we started a project to study the whole sample of globular clusters with UV imagery and compare it with stellar evolution predictions. We are currently working on a paper describing the results of this greatly enlarged sample and the new hot post-AGB candidates. As a first step, we describe here the optical spectroscopic observations of 10 of the brightest hot post-AGB star candidates in the Southern Hemisphere and one post-EHB star candidate in M4. The paper is organised as follows. First we describe in Sect.[sec:obsproc] the new observations as well as unpublished data from, and in Sects.[sec:rv] and [sec:param] we derive stellar parameters for all the stars in our sample and discuss some particular cases. This sample is enlarged in Section [sec:lit] by including all UV-bright stars that have been studied in previous publications. In Sect.[sec:masses] we derive masses from the atmospheric parameters and the distances of the globular cluster, and we discuss the effect of Gaia results for some of the closest clusters. In Sect.[sec:evol] we then present stellar evolution models of post-HeCB stars at different metallicities and for different masses on the HB and compare simple estimates of the number of luminous hot UV-bright stars predicted by the models with those observed in our sample. Finally, we end the paper in Sect.[sec:conclusions] with some preliminary conclusions and discuss what needs to be done to improve the comparison. [!h] llllrclc globular & star & α2000 & δ2000 & mUV & UV Tel. & V & Ref cluster & & [h:m:s] & [∘::] & [mag] & &[mag] & NGC5139 & UIT151 & 13:27:05.06 & − 47:21:56.6 & 15.29& UIT & 16.577& (1) & UIT644 & 13:26:44.96 & − 47:27:09.5 & 13.12& UIT & 12.713 & (1) & UIT1275 & 13:26:52.77 & − 47:29:44.1 & 14.75& UIT & 14.306 & (1) & UIT1425 & 13:26:11.56 & − 47:30:49.2 & 15.31& UIT & 15.660 & (1) & ROA5342 & 13:25:45.42 & − 47:24:02.0 & 14.73& UIT & 15.944 & (1) & Dk3873 & 13:26:13.95 & − 47:25:30.1 & 15.29& UIT & 16.531 & (1) NGC6121 & UVBS2 & 16:23:26.25 & − 26:31:27.0 & 15.07 & UIT & 16.388& (2) NGC6656 & UVBS2 & 18:36:22.85 & − 23:55:19.3 & 14.88 & Swift UVOT & 14.533 & (3) NGC6712 & C49 & 18:53:03.45 & − 08:42:30.5 & 17.00 & Swift UVOT & 16.72 & (4) NGC6779 & ZNG2 & 19:16:41.28 & + 30 12 48.5 & 14.40 & Swift UVOT & 15.13 & (5) NGC7099 & UVBS2 & 21:40:18.08 & − 23:13:22.3 & 15.54 & GALEX & 16.75 & (6) NGC5139 & ROA542 & 13:25:49.91 & − 47:22:59.7 & 15.331 & UIT & 12.876 & (1) & ROA3596 & 13:27:46.09 & − 47:30:57.6 & 14.731 & UIT & 14.179 & (1) Observations and data reduction =============================== The coordinates and brightness of our targets are listed in Table[tab:target]. The data were taken with the FOcal Reducer and low-dispersion Spectrograph 2 (FORS2, ) at the ESO Very Large Telescope (VLT) UT1. We used the grism GRIS\_1200B with a slit width of 05 and verified the resolution from the arc lamp images. The stars in ωCen (except for ωCen − UIT151) were observed in multi-object spectroscopy (MOS) mode with movable slitlets of about 20 length and have a spectral resolution of about 1.7Å. All other stars (including ωCen − UIT151) were observed with the long slit with a length of 68 and have a spectral resolution of about 1.4Å. We always took two exposures of each target for consistency checks. The data were observed between April 1 and July 22, 2012 (date at the beginning of the night). They were processed soon afterwards with the [ESO FORS2 pipeline](http://www.eso.org/sci/software/pipelines/) (version fors-4.12.8), except for the flux calibration (see Appendix[app:flux]). For the bias correction the bias applied to science and calibration data was scaled with the prescan level of these data. In order to take possible structure in the bias into account, the full (prescan-corrected) master bias was subtracted and not just a number. The flat fields were summed per setup and date and normalized using a smoothing with a radius of 10 pixels along both axes. The wavelength calibration used 15 arc lines for the long-slit spectra and 15–17 arc lines for the MOS spectra (depending on the slitlet location). The average accuracy achieved in this way was 0.1pixel for the long-slit spectra and 0.24pixel for the MOS data. We used local sky subtraction, that is, fitting of the sky spectrum per slit on the two-dimensional spectra (before rectification and rebinning). Because of the brightness of our targets and the blue wavelength range, the sky level was rather low, even though most of the data were observed while the Moon was above the horizon. To verify the quality of the sky subtraction, we checked our spectra for the presence of G-band features, which should not be present in these hot stars, and found none. The details of the response determination for this volume-phase holographic grism are described in Appendix[app:flux]. The extracted spectra showed that NGC6656 − UVBS2 has a close cool fainter neighbour, whose spectrum overlaps with that of the intended target. We did not see any significant lines in the neighbour’s spectrum, therefore we smoothed it with a median filter of 14Å half-width and subtracted it from the spectrum of NGC6656 − UVBS2 (see Fig.[fig:M22 − UVBS1]). In another attempt to avoid the contamination from the cool neighbour we extracted only the part of the spectrum away from the neighbour (marked in red in Fig.[fig:spec]). The different slopes of the two spectra point towards possible over- and/or undercorrection of the flux from the cool neighbour. The ratio of the two spectra shows no noticeable line residuals. Considering the experience with crowded field spectroscopy recorded in , we prefer to use the narrowly extracted spectrum for further analysis. [fig:M22 − UVBS1] [fig:spec] Unpublished data from --------------------- After discovering two unpublished spectra (NGC5139 − ROA542 and NGC5139 − ROA3596) from the same observing run as, we decided to analyse all the data from that observing run again. Based on comments from V. Dixon and P. Chayer, we checked the resolution of these data and found that the line width used in was too high. A pixel scale of 0.336/pixel and dispersion of 100Å/mm yield a resolution of 5.6Å and not 6.7Å, as used by. The lower value explains most of the differences between the results reported here and those from. Radial velocities ================= To correct the observed spectra to laboratory wavelengths, we first corrected to the heliocentric system using the midas command compute/barycor. Then we determined the radial velocities by fitting Gaussian profiles to the cores of strong lines using the midas command center/gauss. We selected the lines and their fit range manually to ensure that no surviving cosmics or noise peaks distort the results. We did not use lines at wavelengths below 3800Å as the bluest arc line in our calibration data is at 3888Å. The resulting radial velocities are listed in Table[tab:params], together with the rms error of the individual measurements and the velocities of the corresponding clusters from the literature. The one glaring discrepancy is the radial velocity for NGC7099 − UVBS2 whose value suggests that this star is not a cluster member. This is also supported by the parameters derived in Sect.[sec:param], which argue in favour of a foreground white dwarf. Two stars with very small formal errors ($\lesssim$10kms− 1) show velocity differences well above these errors: NGC6656 − UVBS2 ( + 33.6kms− 1) and NGC5139 − UIT1275 ( + 40.5kms− 1). We recall, however, that one pixel in our data corresponds to 48kms− 1 and the resolution of our data corresponds to 90–120kms− 1. Therefore we decided to consider all stars except for NGC7099 − UVBS2 as members of their respective globular clusters. [tab:params] llrrrr\|c\|rrrl cluster & star & $T\_{\rm eff}$ & logg & $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ & M & status & $v\_{\rm hel}$ & nv & $v\_{\rm cluster}$ & Ref. & & [K] & [cm s− 1] & & [M⊙] & & [kms− 1] & & [kms− 1] & NGC5139 & UIT151 & 62600 ± 2400 & 5.11 ± 0.10& − 0.99 ± 0.08& 0.24 & peAGB & + 231.1 ± 12 & 9 & + 231.8 & (1) & UIT644 & 18000 ± 800& 2.82 ± 0.10& − 0.35 ± 0.06& 0.29 & pAGB & + 232.3 ± 13 & 26 & & & UIT1275 & 17200 ± 1100& 3.34 ± 0.18& − 1.27 ± 0.18& 0.25 & pHB & + 272.3 ± 6& 14& & & UIT1425 & 11700 ± 200& 3.57 ± 0.08& − 1.72 ± 0.36& 0.23 & pHB & + 213.6 ± 17 & 9& & & Dk3873 & 49700 ± 1800& 4.99 ± 0.12& − 1.21 ± 0.12& 0.28 & peAGB & + 231.2 ± 26 & 11& & & ROA5342 & 52000 ± 2000 & 4.79 ± 0.10& − 1.17 ± 0.10& 0.29 & peAGB & + 227.6 ± 10 & 10& & NGC6121 & UVBS2 & 46600 ± 600& 5.82 ± 0.12& + 0.89 ± 0.18&0.85& WDM & + 74.6 ± 15 & 17& + 71.5 & (2) NGC6656 & UVBS2 & 78800 ± 9900 & 4.58 ± 0.20& + 0.36 ± 0.16&0.18& pAGB & − 111.3 ± 4& 3& − 144.9 & (2) NGC6712 & C49 & 53300 ± 4000 & 4.50 ± 0.12& − 1.04 ± 0.12&0.31& pAGB & − 108.0 ± 8& 8& − 109.0 & (3) NGC6779 & ZNG2 & 21500 ± 1800 & 3.07 ± 0.16& − 0.85 ± 0.08&0.19& pAGB & − 123.5 ± 9& 15& − 138.1 & (4) NGC7099 & UVBS2 & 65000 ± 2000 & 7.22 ± 0.12& − 3.29 ± 0.12& & WD & + 11.4 ± 8& 3& − 184.4 & (2) NGC5139 & ROA542 & 10600 ± 200 & 2.55 ± 0.10 & − 1.00 &0.34& pHB & & & & & ROA3596 & 17800 ± 1000 & 3.56 ± 0.16 & − 1.26 ± 0.18&0.41& pHB & & & & NGC2808 & C2946 & 24900 ± 1800 & 4.78 ± 0.20 & − 1.88 ± 0.14 &0.68& pHB & & & & & C2947 & 15000 ± 1200 & 3.92 ± 0.24 & − 1.45 ± 0.44 &0.41& pHB & & & & & C4594 & 19900 ± 1600 & 3.79 ± 0.22 & − 1.63 ± 0.26 &0.34& pHB & & & & NGC6121 & Y453 & 54900 ± 2000 & 5.62 ± 0.14 & − 1.25 ± 0.10 &0.59& peAGB & & & & NGC6723 & III60 & 43000 ± 1400 & 4.72 ± 0.14 & − 1.19 ± 0.14 &1.02& peAGB & & & & & IV9 & 25900 ± 2000 & 4.02 ± 0.22 & − 1.06 ± 0.10 &0.88& peAGB & & & & NGC6752 & B2004 & 34500 ± 800 & 5.18 ± 0.14 & − 2.45 ± 0.24 &0.35& pHB & & & & Atmospheric parameters ====================== Following and, we analysed the spectra with grids of non-local thermal equilibrium (NLTE) line-blanketed model atmospheres and NLTE model spectra that were calculated with the NLTE model atmosphere code TLUSTY and the companion spectrum synthesis code SYNSPEC. The model atmosphere grids were tailored to the studied sample with the parameter space delimited in Table[tab:models]. Over 1100 new model atmospheres have been produced with steps in the model grids of 2000K in effective temperature, 0.25dex in surface gravity, and 0.5dex in helium-to-hydrogen abundance ratio by number, so that the observed spectra can be matched with model spectra that were interpolated from the grids. Scaled-solar abundances based on at the cluster’s mean metallicities have been assumed, and kept the same for models with various He/H abundance ratios. The implied change in the total mass fraction of the heavy elements has very limited consequences on our analysis (see ). The model atmospheres at higher temperature ($T\_{\rm eff} \geq 40\,000$K) allow for departures from LTE for 1132 explicit levels and superlevels of 52 ions (H, He, C, N, O, Ne, Mg, Al, Si, P, S, and Fe), as in. A detailed description of the model atoms and the source of the atomic data can be found in. Microturbulent velocity is assumed to be $v\_{\rm t} = 5\,{\rm km\,s}^{-1}$. At lower temperatures ($T\_{\rm eff} \leq 30\,000$K), we naturally considered lower ionization stages and excluded some highest ions. The model atmospheres allow for departures from LTE for 1127 explicit levels and superlevels of 46 ions (H, He, C, N, O, Ne, Mg, Al, Si, S, and Fe), with the same sources for model atoms. Microturbulent velocity is assumed to be $v\_{\rm t} = 2\,{\rm km\,s}^{-1}$. Recently, have proposed that setting microturbulence to zero provides a better fit to the spectra of the iron lines in the 21400K post-AGB star Barnard29 in NGC6205. However, the choice of microturbulence should have little effect on the broad hydrogen and helium lines studied here. [tab:models] \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Clusters \| [M/H] \| $T\_{\rm eff}$ \| logg \| $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ \| $v\_{\rm t}$ \| Number of \| \| \| \| [kK] \| \| \| [kms− 1] \| models \| \| NGC6121 & NGC6712 \| − 1.0 \| 40–70 \| 4.00–6.00 \| − 1.5… + 1.0 \| 5. \| 256 \| \| NGC5139 & NGC6656 \| − 1.5 \| 40–82 \| 4.25–6.00 \| − 3.0… + 0.5 \| 5. \| 426 \| \| \| \| 16–28 \| 2.75–5.75 \| − 3.0…0.0 \| 2. \| 332 \| \| NGC6779 \| − 2.0 \| 18–24 \| 2.75–3.25 \| − 1.0… − 0.5 \| 2. \| 24 \| \| NGC7099 \| − 2.3 \| 50–60 \| 5.50–6.75 \| − 3.0… − 2.0 \| 5. \| 108 \| For the cool stars NGC5139 − ROA3596, NGC5139 − UIT1275, and NGC5139 − UIT1425, which are most likely post-HB stars, we used the solar metallicity LTE model spectra described in to simulate the effects of radiative levitation. For the re-analysis of the stars cooler than 35000K from and for NGC5139 − ROA542 we used LTE model spectra like those from for metallicities [M/H] = − 1.0 and − 1.5. For NGC7099 − UVBS2 we used the metal-free NLTE model spectra described in. To establish the best fit to the observed spectra, we used the routines developed by and, as modified by, which employ a χ2 test. The σ necessary for the calculation of χ2 is estimated from the noise in the continuum regions of the spectra. The fit program normalizes model spectra observed spectra using the same points for the continuum definition. We fitted the Balmer (and corresponding Heii) lines H10 to Hβ (excluding Hε due to the interstellar Caii absorption line), the Hei lines λλ4026.29Å, 4120.81Å, 4387.59Å, 4471.60Å, and 4921.94Å, and the Heii lines λλ4025.60Å, 4199.83Å (only for stars hotter than 30000K), 4541.59Å, and 4685.68Å. These fit routines underestimate the errors by at least a factor of 2 (Napiwotzki priv. comm.). We therefore provide formal errors multiplied by 2 to account for this effect. In addition, the errors provided by the fit routine do not include possible systematic errors due to flat-field inaccuracies or imperfect sky subtraction, for instance. The results of the line profile fits are listed in Table[tab:params]. For some of the spectra (NGC5139 − UIT151, NGC5139 − Dk3873, NGC5139 − ROA5342, and NGC6712 − C49) we could not fit the H, Hei, and Heii lines simultaneously because the observed Hei lines were weaker than predicted, while the H and Heii lines were stronger than predicted. We saw similar effects for NGC6121 − Y453 and NGC6723 − III60 from. All these stars have effective temperatures between about 50000K and 63000K and roughly solar helium abundances. A similar behaviour was observed by for BD + 28∘4211. They suspected missing opacities to be the cause of the problem. They showed that the effects of increasing metallicity on the hydrogen and helium lines saturate at [M/H] = + 1 and therefore used model atmospheres with [M/H] = + 1 for their analysis. With these model atmospheres, they were able to reconcile the parameters derived from optical spectra with those derived from UV spectra. We therefore decided to follow the same path and analysed our hotter stars with very metal-rich model spectra. For the synthetic spectra we included only hydrogen and helium lines, as the high metallicity may only be a proxy for missing opacities (see, however, Sect.[sec:UVphot] and Fig.[fig:UV] for further information). The new results are listed in Table[tab:metal-rich]. For NGC6121 − Y453 we note that obtained similar parameters from the same optical spectrum, but a much higher temperature of 72000K (with logg of 5.7) from far-ultraviolet (FUV) data. For reasons of consistency, we use the value derived here for further discussion. The effective temperatures clearly decrease and the surface gravities increase with the metal-rich model atmospheres. This effect is also known for cooler stars (e.g. ). [tab:metal-rich] llrrrr cluster & star & $T\_{\rm eff}$ & logg & $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ & M & & [K] & [cm s− 2] & & [M⊙] NGC5139 & UIT151 & 57400 ± 2200 & 5.37 ± 0.08& − 1.07 ± 0.08 & 0.51 & Dk3873 & 47900 ± 1200& 5.15 ± 0.08& − 1.19 ± 0.08 & 0.42 & ROA5342 & 49800 ± 1400& 4.98 ± 0.08& − 1.15 ± 0.06 & 0.46 NGC6121 & UVBS2 & 46200 ± 400& 5.92 ± 0.12& + 0.96 ± 0.14 & 1.06 NGC6656 & UVBS2 & 62300 ± 9600 & 4.59 ± 0.16& + 0.40 ± 0.18 & 0.31 NGC6712 & C49 & 47800 ± 1800& 4.64 ± 0.08& − 1.10 ± 0.08 & 0.51 NGC6121 & Y453 & 56500 ± 1800 & 5.71 ± 0.12 & − 1.16 ± 0.10 & 0.79 NGC6723 & III60 & 42300 ± 1400 & 4.80 ± 0.12 & − 1.13 ± 0.12 & 1.19 NGC6656 − UVBS2 --------------- The analysis of NGC6656 − UVBS2 was difficult for two reasons: It has a cool neighbour very close by, and it has a planetary nebula. The first results using model spectra with the cluster abundance pointed towards a temperature of almost 80000K, while the results obtained with metal-rich model spectra suggest a much cooler temperature of 62000K. Based on a low-resolution spectrum, suggested an effective temperature of 75000K from the similarity to the sdO star KS292, which had been analysed by. The spectrum shows absorption lines of Civ and Nv similar to the star SDSSJ155610.40 + 254640.3, which has a temperature of about 100000K. The deeper Heii lines of NGC6656 − UVBS2 are consistent with a lower temperature, which unfortunately does not allow us to distinguish between the two temperatures mentioned before. suggested that NGC6656 − UVBS2 is the result of a stellar merger. Interestingly enough, SDSSJ155610.40 + 254640.3 was classified by as a PG1159 star, a class of H-deficient post-AGB stars. In any case, it seems that NGC6656 − UVBS2 is closely related to the star ZNG1 in NGC5904 (see Table[tab:lit]), which has also been connected to both a merger and PG1159 stars. While NGC5904 − ZNG1 shows a high rotational velocity of about 170km s− 1, we see no evidence for fast rotation in our spectrum of NGC6656 − UVBS2. We recall, however, that the resolution of our data is about 100km s− 1. Metal lines ----------- We took care to verify that potential metal lines were present in both individual spectra for each star. We found potential metal lines only in the stars listed below and compared their strength to that predicted by the best-fitting model spectrum. The number of lines for each ion are given in parentheses, and the stars are sorted by evolutionary stage. A detailed abundance analysis is beyond the scope of this paper. NGC5139 − UIT1425 (pHB) Mgii (1) significantly weaker than predicted by the metal-rich model spectrum with metal lines, which is usual for HB stars in this temperature range. NGC5139 − UIT151 (peAGB) Nv (2) in agreement with the metal-rich model spectrum with metal lines, which also predicts strong Ciii and Oiv lines that are not observed, however. NGC5139 − UIT644 (pAGB) Nii (9), Cii (1) Mgii (1), and Siiii (2) much stronger than predicted by the metal-poor model spectrum used for the analysis, especially Nii. NGC6656 − UVBS2 (pAGB) Civ (3), Nv (2), much stronger than predicted by the model spectrum with metal lines for [M/H] = + 1. NGC6121 − UVBS2 (WDM) Ciii (7), much stronger than predicted by the model spectrum with metal lines for [M/H] = + 1. UV spectrophotometry -------------------- Several of our targets have archival UV spectra from either the International Ultraviolet Explorer (IUE), the Far Ultraviolet Spectroscopic Explorer (FUSE), or the HST. We defer a complete discussion of the UV spectra to a subsequent paper, but discuss here some of the IUE and HST Faint Object Spectrograph (FOS) data in ωCen (Table[tab:uvspectra]), which (1) provide additional justification for our use of metal-rich atmospheres for the sdO stars, and (2) provide evidence of binarity for the star UIT1425. The data were obtained from the [Mikulski Archive for Space Telescope (MAST)](http://archive.stsci.edu/iue/), and the IUE spectra were converted into the HST absolute calibration scale using the transformation in. We restricted use of the IUE data to the short-wavelength ( < 1950 Å) prime (SWP) camera because the IUE large (10 × 20) aperture includes light from background red stars at longer wavelengths. We expect the SWP images to be free of background contamination because examination of the UIT 1620Å image of ωCen showed that no UV sources are included within the IUE aperture for any of the target stars. In addition, the FOS and IUE spectra of UIT151 show fair agreement in Fig.[fig:UV], even though the FOS data use a much smaller (1 circular) aperture. [tab:uvspectra] llrllrrlrr Star & Telescope & Mode & ID & Date & Exp & & & & & [s] UIT151 & HST/FOS & G160L & Y2SS402T & 1996-04-28 & 2400 UIT151 & IUE & SWP & 54154 & 1995-03-16 & 23400 Dk3873 & IUE & SWP & 48271 & 1993-07-31 & 19200 ROA5342 & IUE & SWP & 54333 & 1995-04-08 & 25500 UIT1425 & IUE & SWP & 54804 & 1995-05-31 & 26820 The high-resolution study of field sdO stars by showed that the UV spectrum of these stars is dominated by numerous lines of iron and nickel. Even though our low-resolution UV spectra cannot individually resolve these lines, the lines are sufficiently numerous at high metallicity to affect the apparent continuum. Figure[fig:UV] shows surprisingly good agreement between the low-resolution UV spectra of the three sdO stars and models with [M/H] = + 1.0. The Civ doublet near 1550Å appears much weaker than predicted by the metal-rich models, suggesting that the light elements do not follow the abundance enhancement seen in iron and nickel, an effect also seen in some field sdO stars. We are able to use the known metallicity spread of ω Cen to set an upper limit of [M/H] = − 1.0 for the metallicity of the progenitors of these sdO stars. The factor of ≈ 100 enhancement in iron and nickel must then be attributed to radiative levitation. For NGC5139 − UIT1425 a model spectrum with the parameters derived from our optical spectra cannot reproduce the UV data (see Fig.[fig:UIT1425]). This star is most likely a binary, as its optical colours (B − V= − 0.01, V − I= + 0.20) are appropriate for a blue HB star, whereas its UV − V colour and steep IUE spectrum is more indicative of an sdO star. The optical spectrum, however, shows no trace of Heii, which would be a clear sign of the hotter star. Unfortunately, there is no HST imagery in this region of ωCen, so that we cannot rule out that the binary is a chance alignment. We checked the spatial (line by line) IUE image, but found no presence of more than one star within the large aperture. [fig:UV] [fig:UIT1425] Data from the literature ======================== In Table[tab:lit] we list the physical parameters published for hot UV-bright stars other than those discussed here. For NGC6656 − II-81 we note that its position in the $T\_{\rm eff}, \log g$ diagram is consistent with being a post-AGB instead of a post-HB star. [tab:lit] llrllrrlrr cluster & star & $T\_{\rm eff}$ & logg & $\log {\frac{n\_{\rm He}}{n\_{\rm H}}}$ & status & Ref. & V & Ref. & M & & [K] & [cgs] & & & & [mag] & & [M⊙] NGC104 & BS & 11000 & 2.5 & − 1 & pAGB & (1) & 10.73 & (13)& 1.25 NGC1851 & UV5 & 16000 & 2.5 & − 1 & pAGB & (2) & 13.26& (14)& 0.41 NGC5139 & ROA5701 & 25000 & 3.3 & − 1 & pAGB & (3) & 13.13 & (24) & 0.32 NGC5272 & vZ1128 & 36600 & 3.95 & − 0.84 & peAGB & (4) & 15.03 & (15)& 0.41 NGC5904 & ZNG1 & 44300 & 4.3 & +0.52 & pAGB & (5) & 14.54 & (16)& 0.72 NGC6205 & B29 & 21400 & 3.10 & − 0.89 & pAGB & (6) & 13.116& (23) & 0.38 NGC6254 & ZNG1 & 27000 & 3.6 & − 1.33 & pAGB & (7) & 13.23 & (17)& 0.59 NGC6397 & ROB162 & 51000 & 4.5 & − 1.0 & pAGB & (8) & 13.1 & (9)& 0.50 NGC6656 & II-81 & 38000 & 4.2 & — & pHB & (9) & 14.0 & (18) & 0.47 NGC6712 & ZNG1 & 11000 & 2.1 & − 1.19 & pAGB & (7) & 13.33 & (19)& 0.35 NGC6752 & B852 & 39000 & 5.2 & − 2.0 & pHB & (10) & 15.91 & (20)& 0.52 & B4380 & 32000 & 5.3 & − 2.3 & pHB & (10) & 15.93 &(20)& 0.79 & B1754 & 40000 & 5.0 & − 1.52 & pHB & (10) & 15.99 & (20)& 0.30 NGC7078 & K648 & 39000 & 3.9 & − 1.10 & pAGB & (11) & 14.73 & (11)& 0.62 & K996 & 11500 & 2.5 & — & pHB & (12) & 14.31 & (21)& 0.26 & ZNG1 & 28000 & 3.7 & − 1.22 & pAGB & (7) & 14.8 & (17)& 0.55 We wish to compare results between different globular clusters, therefore we decided to use homogeneous data sets for their ages, metallicities, and integrated V magnitudes. Unfortunately, we found no such data set for the distances and reddenings. Therefore we used the values from that are at least created in a defined way from the various sources. The values we used in our analysis are listed in Table[tab:par-gc]. [tab:par-gc] l\|rlrrrr\|rrr Cluster & [M/H] & age & Vt & (m − M)V & EB − V & BCV & & & [109 yr] & [mag] & [mag] & [mag] & [mag] & $f\_{\rm RHB}$ & $f\_{\rm BHB}$ & $f\_{\rm EHB}$ NGC104 & − 0.76 & 13.1 & 4.01 & 13.37 & 0.04 & − 0.53 & 1.00 & 0.00 & 0.00 NGC1851 & − 1.18 & 10.0 & 7.16 & 15.47 & 0.02 & − 0.41 & 0.64 & 0.36 & 0.00 NGC5139 & − 1.64 & 11.5 & 3.85 & 13.94 & 0.12 & − 0.38 & 0.06 & 0.71 & 0.23 NGC5272 & − 1.50 & 11.4 & 6.36 & 15.07 & 0.01 & − 0.38 & 0.32 & 0.68 & 0.00 NGC5904 & − 1.33 & 10.6 & 5.58 & 14.46 & 0.03 & − 0.40 & 0.21 & 0.79 & 0.00 NGC5986 & − 1.63 & 12.2 & 7.46 & 15.96 & 0.28 & − 0.38 & 0.00 & 0.91 & 0.09 NGC6121 & − 1.18 & 12.5 & 5.77 & 12.82 & 0.35 & − 0.43 & 0.39 & 0.61 & 0.00 NGC6205 & − 1.58 & 11.6 & 5.82 & 14.33 & 0.02 & − 0.38 & 0.00 & 0.70 & 0.30 NGC6254 & − 1.57 & 11.4 & 6.60 & 14.08 & 0.28 & − 0.38 & 0.00 & 1.00 & 0.00 NGC6397 & − 1.99 & 12.7 & 6.20 & 12.37 & 0.18 & − 0.37 & 0.00 & 1.00 & 0.00 NGC6656 & − 1.70 & 12.7 & 5.08 & 13.60 & 0.34 & − 0.38 & 0.00 & 0.73 & 0.27 NGC6712 & − 1.02 & 12.0 & 8.04 & 15.60 & 0.45 & − 0.44 & 0.85 & 0.15 & 0.00 NGC6723 & − 1.10 & 13.1 & 7.03 & 14.84 & 0.05 & − 0.44 & 0.58 & 0.42 & 0.00 NGC6752 & − 1.55 & 11.8 & 5.32 & 13.13 & 0.04 & − 0.38 & 0.00 & 0.72 & 0.28 NGC6779 & − 2.00 & 13.7 & 8.17 & 15.68 & 0.26 & − 0.37 & 0.00 & 1.00 & 0.00 NGC7078 & − 2.33 & 12.9 & 6.32 & 15.39 & 0.10 & − 0.35 & 0.10 & 0.90 & 0.00 NGC7099 & − 2.33 & 12.9 & 7.35 & 14.64 & 0.03 & − 0.35 & 0.00 & 1.00 & 0.00 Masses and distances ==================== Using the atmospheric parameters together with the observed brightness of the stars and reddening and distances to their parent globular cluster, we determined masses using Eq.[eq:massbc], $$\label{eq:mass\_bc} \log {\frac{M}{M\_\odot}} = \log{\frac{g\_\ast}{g\_\odot}} - 4\cdot \log {\frac{T\_\ast}{T\_\odot}} - {\frac{M\_V+BC-4.74}{2.5}} ,$$ which can be rewritten to $$\label{eq:mass\_bc2} \log {\frac{M}{M\_\odot}} = \log g\_\ast + 0.4 \cdot [(m-M)\_0-V\_\ast+A\_V - BC] -4 \cdot \log T\_\ast +C3 ,$$ with $$\label{eq:C3} C3 = -\log g\_\odot + 4\cdot \log T\_\odot + {\frac{4.74}{2.5}} ,$$ with the bolometric corrections from. The results are listed in Tables[tab:params] and [tab:lit] and plotted in Fig.[fig:alltm]. [fig:alltm] From stellar evolution we would generally expect masses of 0.5M⊙ to 0.6M⊙. Figure [fig:alltm] shows that the masses derived for stars below 25000K tend to be too low, while the masses derived for hotter stars scatter towards too high masses[3](#fn3). Some of the globular clusters discussed in this paper are close enough to have parallax estimates from the Gaia DR2. The Gaia collaboration compared the parallaxes derived from the Gaia measurements to those listed in and found a systematic difference of 0.029mas, with the Gaia parallaxes being smaller. We therefore used the reported Gaia parallaxes only for those clusters that had parallax values of more than 0.23mas, that is, about eight times the systematic offset. To derive distances, we applied the reported systematic offset to the parallaxes. We then added interstellar extinction as 3.2 ⋅ EB − V. The apparent distance moduli derived that way are 0.29mag smaller for NGC6121 and 0.16mag, 0.18mag, and 0.08mag larger for NGC6397, NGC6656, and NGC6752, respectively, than those from. It is interesting to note that we find higher-than-expected masses for the two stars in NGC6121, which would be reduced by 30% with the Gaia distance to 0.81M⊙ and 0.65M⊙, respectively. For NGC6397, using the Gaia distance would increase the masses by 16%, moving them from about 0.5M⊙ to about 0.6M⊙. For NGC6656, using the Gaia distance would increase the mass of NGC6656 − UVBS2 from 0.31M⊙ to 0.38M⊙ (without the additional reddening correction) and from 0.55M⊙ to 0.65M⊙ (with the additional reddening correction). The change of 8% for the masses of the stars in NGC6752 is negligible. Stellar evolution models and evolutionary fluxes ================================================ [tab:par-gc-model] [tab:evolini] \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| $M\_{\rm ZAMS}$ \| $Z\_{\rm ZAMS}$ \| $Y\_{\rm ZAMS}$ \| $X\_{\rm ZAMS}$ \| \| \| [M⊙] \| \| \| \| \| − 1.0 \| 0.85 \| 0.00172 \| 0.24844 \| 0.74984 \| \| − 1.5 \| 0.83 \| 0.000548 \| 0.246096 \| 0.753356 \| \| − 2.0 \| 0.82 \| 0.000174 \| 0.245348 \| 0.754478 \| \| − 2.3 \| 0.82 \| 0.000087 \| 0.245174 \| 0.754739 \| [tab:result-gc] lc\|ccc Cluster & N$^{\rm post-AGB}\_{\rm obs}$ & [M/H]$\_{\rm tracks}$ & N$^{\rm post-AGB}\_{\rm exp}$ NGC104 & 1 & − 1.00 & 0.08 – 0.65 NGC1851 & 1 & − 1.00 & 0.15 – 1.04 NGC5139 & 5 & − 1.50 & 1.31 – 13.58 NGC5272 & 1 & − 1.50 & 0.27 – 1.48 NGC5904 & 1 & − 1.50 & 0.38 – 2.11 NGC5986 & 1 & − 1.50 & 0.61 – 4.29 NGC6121 & 1 & − 1.00 & 0.19 – 1.32 NGC6205 & 1 & − 1.50 & 0.23 – 2.71 NGC6254 & 1 & − 1.50 & 0.26 – 1.44 NGC6397 & 1 & − 2.00 & 0.07 – 0.26 NGC6656 & 2 & − 1.50 & 0.59 – 6.65 NGC6712 & 2 & − 1.00 & 0.14 – 0.98 NGC6723 & 2 & − 1.00 & 0.12 – 0.83 NGC6752 & 0 & − 1.50 & 0.13 – 1.48 NGC6779 & 1 & − 2.00 & 0.29 – 1.13 NGC7078 & 2 & − 2.30 & 1.04 – 3.71 NGC7099 & 0 & − 2.30 & 0.17 – 0.64 [fig:alltg] The hot UV-bright phase contains stars in different evolutionary stages. The most luminous of these stars ((L/L⊙) $\gtrsim$ 3.1) are believed to be post-AGB stars, which go through a luminous UV-bright phase as they leave the AGB and move rapidly toward their final white dwarf state. Despite their short lifetimes ($\lesssim 10^{5}$yr), hot post-AGB stars ($T\_{\rm eff} >$ 7000K) can dominate the total UV flux of an old stellar population. UV-bright stars with luminosities 2.65 $\lesssim \log$ (L/L⊙)$\lesssim$ 3.1 are a mixture of low-mass post-AGB stars, which are stars that departed from the AGB before the beginning of the thermal pulses (post-early AGB stars), and post-EHB stars that managed to ignite the H-burning shell after the departure from the HB, but never reached the AGB. The post-early AGB population arises from hot HB stars with sufficient envelope mass to return to the AGB, but which peel off the AGB prior to the thermally pulsing phase. Less luminous hot UV-bright stars (1.8 $\lesssim \log$ (L/L⊙)$\lesssim$ 2.65) are mostly evolving from the EHB to the white dwarf domain (post-EHB stars), but some may also evolve from the hot end of the blue horizontal branch (BHB) to the AGB phase. The population of post-EHB stars is expected to be about 15–20% of the population of EHB stars. We now wish to compare the number of observed post-AGB stars with expectations from stellar evolution models. In the absence of a detailed study of the distribution of stars in the HB of each cluster it is not possible to perform detailed synthetic population simulations for each cluster. However, using the available information from the HST UV Legacy Survey of Galactic Globular Clusters we estimated the fractions of stars populating the red HB (RHB), BHB, and EHB, based on the HB morphology in the V, V − I colour-magnitude diagram (see Table[tab:par-gc] for results). Using these numbers, we estimated how many post-AGB stars should evolve from the RHB, BHB, and EHB in each cluster. The post-AGB evolutionary models computed for this work are an extension of the models presented by. The models were computed for values of [M/H] = − 1, − 1.5, − 2 and − 2.3 under the assumption of a scaled-solar mixture. The helium-metallicity relation was taken as in, that is, Y = 0.245 + 2 × Z. The initial masses of the models were chosen to correspond to ages between 11.5 to 12 Gyr. The higher or lower ages of some of the globular clusters (see Table[tab:par-gc]) are no problem because age does not influence the mass of the helium core at the time of the helium core flash, nor the following evolution. It might affect the mass of the hydrogen-rich envelope, but this mass is varied artifically in our models anyway to populate the whole horizontal branch. The corresponding initial parameters for the stellar evolution sequences are shown in Table[tab:evolini]. Different mass loss was applied to each model before the zero-age horizontal branch (ZAHB) in order to obtain different masses at the ZAHB and a complete coverage of the horizontal branch (from the RHB to the EHB) and post-HB evolution (see Table[tab:sequences] for details). Convective boundary mixing at all evolutionary stages and mass loss on the AGB were also adopted as in. Figure[fig:alltg] shows the atmospheric parameters derived in Sect.[sec:param] compared to the evolutionary tracks. The evolutionary stage derived from this comparison can be found in Table[tab:params]. The stars are colour-coded by their helium abundance because this helps to identify the evolutionary status of a star. Significantly sub-solar helium abundances due to diffusion are known for HB stars hotter than about 11000K. The diffusion patterns are erased once convection starts in the atmosphere as the star evolves towards cooler temperatures. A low helium abundance is therefore unlikely for stars that ascended the AGB (even if only partially). Fig.[fig:alltg] shows that helium-poor stars are found along post-(E)HB tracks and not along post-(e)AGB tracks. Helium-rich stars, on the other hand, are found only along the post-AGB tracks (except for the white dwarf merger). In order to compare the stellar evolution models with the observed number of stars, we adopted the evolutionary flux method (see for details). This method assumes that the late evolutionary stages are much shorter than the main-sequence evolution and that the number of “dying stars” can be equated to the number of stars leaving the main sequence. Under these assumptions, the number of stars Nk in a simple stellar population at each evolutionary stage k is given by $$N\_k= B\times L\_{\rm total}\times t\_k, \label{eq: evolflux}$$ where tk is the duration of the evolutionary stage k, $L\_{\rm total}$ is the total luminosity of the stellar population, and B stands for the specific evolutionary flux. For old stellar populations, like the globular clusters studied in this work, we can approximate B ≃ 2 × 10− 11 stars per year and per solar luminosity (see for details). In our case, we are interested in the time spent by the models in the region of the Hertzprung-Rusell (HR) diagram where we would observe them as hot UV-bright post-AGB stars. We define the region in the HR diagram corresponding to hot UV-bright post-AGB stars detected in optical colours as that defined by logL/L⊙ > 2.65 and $4.9<\log T\_{\rm eff}<3.845$ (grey zone in Fig.[fig:tracks]). Each computed track casts a timescale t corresponding to the time spent in that part of the HR diagram. With the parameters shown in Table[tab:par-gc] and with Eq.[eq: evolflux], we can compute for each cluster the expected numbers Nt predicted by each track t of a similar metallicity. It should be emphasized that Eq.[eq: evolflux] assumes that the whole cluster population evolved through that specific track k. Each post-HB track computed for the cluster metallicity gives a different expected number Nk (Eq.[eq: evolflux]), and the actual number of expected post-HB stars will depend on the frequency with which each specific track is followed in each cluster. Lacking a better characterization of the HB demographics, we can split the HB into the RHB, BHB, and EHB and then estimate the number of stars evolving from the RHB, BHB, or EHB using the relative populations of the RHB, BHB, and EHB in each cluster[4](#fn4) ($f\_{\rm RHB}$, $f\_{\rm BHB}$, and $f\_{\rm EHB}$ respectively, see Table[tab:par-gc]). Using all sequences evolving from the RHB, BHB, or EHB, we obtain a range of expected numbers for hot post-AGB stars evolving from each part of the HB ($N^{\rm RHB}$, $N^{\rm BHB}$, or $N^{\rm EHB}$, respectively). The actual range of expected UV-bright stars in the post-AGB region of the HR diagram (N) of a given cluster is then estimated as $$N= f\_{\rm EHB} \cdot N^{\rm EHB} + f\_{\rm BHB} \cdot N^{\rm BHB} + f\_{\rm RHB} \cdot N^{\rm RHB}. \label{eq: number}$$ [fig:tracks] The resulting ranges of expected numbers for each cluster are shown in Table[tab:result-gc]. We excluded NGC2808 from this investigation because we know that several luminous hot UV-bright stars are not included in our study. [fig:result-gc] Overall, the agreement between the number of post-AGB stars estimated (N$^{\rm post-AGB}\_{\rm exp}$, fourth column in Table[tab:result-gc]) and the observed number of post-AGB stars (N$^{\rm post-AGB}\_{\rm obs}$, second column in Table[tab:result-gc]) in each cluster is good (see also Fig.[fig:result-gc]). In 11 out of 17 cases ( ≈ 65%) the observed number of stars lies within the computed range, while in 3 other cases (NGC104, NGC6752, and NGC7099) the observed number is just an integer number next to the expected range. A more significant discrepancy is observed for NGC6712 and NGC6723, where the upper boundary of the predicted ranges is a factor ≈ 2 lower than the observed number. An even larger discrepancy is observed for NGC6397, for which the largest predicted number is about four times smaller than the observed one. In these three cases, the observed value is higher than the predicted ones, but a word of caution is in order. We have preselected for our sample those clusters that do harbour hot UV-bright stars. It is therefore possible that our clusters are biased to larger post-AGB numbers. Given the low number of the expected post-AGB stars and the fact that the observed number is necessarily an integer, it is difficult to point to this discrepancy as a discrepancy with the stellar evolution models. A full comparison of all clusters searched for hot UV-bright stars is required for firmer conclusions. Conclusions =========== Our results first confirm that spectroscopic observations of UV-bright stars are required to verify their evolutionary status, as is shown by the cases of NGC7099 − UVBS2 and NGC6121 − UVBS2. The need for metal-rich model spectra in the analysis of hot evolved stars has been reported elsewhere , and we can confirm that additional opacities are required to reproduce the Hei and Heii lines simultaneously and the UV spectra. This should be kept in mind when the contribution of hot post-AGB stars to the UV flux of evolved populations is estimated. We find general agreement for the number of observed hot post-AGB stars compared to predictions from evolutionary theory, although the numbers per cluster are low (at most two, except for ωCen, which has five). Some discrepancies are observed in the clusters NGC6397, NGC6712, and NGC6723, which show somewhat larger numbers than expected from the models. Because of the small-number statistics nature of the comparison, it is unclear, however, whether this is due to a problem with the models or a result of the way the sample was preselected. In order to improve our constraints on low-mass post-AGB stellar evolution models, the number of studied globular clusters needs to be increased. When the full set of 78 globular clusters with UV-imaging is studied, it will be possible to combine clusters with similar metallicities and HB morphologies to improve the number statistics in the comparisons. We are grateful to Nicole Reindl, who made her co-added spectrum of SDSSJ155610.40 + 254640.3 and its model spectrum available to us. We thank Simon Jeffery for the merger track and his explanations of the merger behaviour. We appreciate the anonymous referee’s comments that improved the readability and clarity of this paper substantially. M.M.M.B. is partially supported through ANPCyT grant PICT-2016-0053 and MinCyT-DAAD bilateral cooperation program through grant DA/16/07. This research has made use of NASA’s Astrophysics Data System Bibliographic Services and of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Flux calibration ================ GRIS\_1200B is a volume-phase holographic grism. Its response therefore depends on the position of the slit along the dispersion axis. The best solution for a flux calibration is to take the flux standard star at the same place on the detector as the science spectrum. This is done for data taken with the long slit by creating a 5 wide long slit with the slit blades of the MOS slitlets. For MOS observations with distributed slitlets, this would require taking several flux standard star observations, which is not feasible in service mode. To correct the response derived from standard stars taken at different positions than the science targets, we made use of the fact that the varying response is seen also in the spectral energy distribution of the screen flat fields taken for the MOS data (see and for a more detailed description). Therefore we took the ratio of the wavelength-calibrated master screen flat and normalized master flat and averaged it along the slit. Then we divided the extracted standard star spectra and the extracted science spectra by their flat-field spectra. This procedure will of course introduce the lamp spectrum into these spectra, but as long as the lamp spectrum does not change, this can be corrected for with the response curve. During this exercise we noted three potential problems: 1. The narrow-slit flat fields taken until June 22, 2012, show two emission lines at about 3944.2Å and 3961.8Å. The lines are not visible in the 5 flat fields taken for the standard stars. The smoothing of the master screen flat along the dispersion axis will smear out these lines to a radius of at least ± 10 pixels or ± 7Å (relative to the peak position). We therefore did not use these regions in our later analysis. 2. Between June 22 and July 22, 2012, the spectral energy distribution of the flat-field lamps changed. 3. Even between flat fields taken within 30 minutes of each other, the spectral energy distribution changed by some 20% (see Fig.[fig:specratio]). [fig:specratio] All extracted	arxiv_0000560
pixel values. A $\rm \sim4\sigma$ signal of $\rm 0.52\pm0.13\, mJy$ is obtained for this group, which is about one quarter of the value of the typical MAMBO detections. We ran the stacking with the non-detection group several times, omitting one source each stack, to see if this $\rm 4\sigma$ signal is due to a single source that is just below the detectability. The test shows stacking flux densities in the range of 0.47 to 0.56 mJy with rms values of ∼ 0.13 mJy. Thus it is unlikely that the average flux density is dominated by one object. No significant differences are found in the $\rm \langle f\_{250GHz}\rangle$ values when the radio-loud sources are excluded in each group. The average 1.4 GHz observed flux densities are $\rm 27\pm4$ to $\rm 37\pm4\, \mu Jy$ for the radio quiet sources in the three groups. We convert these measurements to the flux densities at rest frame 5 GHz assuming a power-law radio spectrum of $\rm f\_{\nu}\sim\nu^{-0.75}$ (Condon 1992). This gives an average radio loudness (i.e. average radio to optical emission ratio $\rm \langle R\rangle=\langle f\_{\nu,5GHz}\rangle/\langle f\_{\nu,4400}\rangle$) of $\rm \langle R\rangle\sim$0.7, which is consistent with the typical radio loudness values of $\rm 0.1<R<1$ found with local(Kellermann et al. 1989) and z ≤ 2 (Cirasuolo et al. 2003) radio quiet quasars. The derived q parameters (i.e. FIR to radio emission ratios) are all lower or at the edge of the range of $\rm q=2.34\pm0.7$ for typical star forming galaxies (Yun et al. 2001), but are consistent with the values of q < 2 found in the local Seyfert 1 galaxies (Ulvestad & Ho 2001). This result suggests that, as was found in the local universe and at z$\rm \sim$2 to 4, AGN power dominates the radio emission in a large fraction of these radio quiet quasars, i.e., the radio quiet AGNs are in fact not radio silent (Barvainis et al. 2005; Carilli et al. 2001; Petric et al. 2006). We plot the mean SED (normalized by the rest frame 1450$\rm \AA$ flux density) of the radio quiet sources in each group in Figure 2, together with the templates of low-z optical quasars from Elvis et al. (1994) and Richards et al. (2006). We also plot the average optical to near infrared emission based on the Spitzer photometry of 13 z ∼ 6 quasars from Jiang et al. (2006), who show that the ratio is consistent with the local templates. The FIR-to-1450$\rm \AA$ ratios of the three z ∼ 6 quasar groups have a range of ∼ 0.6 dex, and no obvious excess is seen between the value of the whole sample and the FIR end of the templates. However, due the lack of millimeter and submillimeter data, the available templates do not give a good constraint of the FIR-to-millimeter SED for the local optical quasars. Sanders et al. (1989) studied a sample of optically selected Palomar-Green quasars (PG quasars; Schmidt & Green 1983; Boroson & Green 1992), and their analysis suggest that the typical 100$\rm \,\mu$m to 1$\rm \,mm$ spectral index $\rm \alpha$ (($\rm f\_{\nu}\sim\nu^{\alpha}$)) is $\rm \geq2$. We extrapolate the FIR emission of the templates with this result (an example of $\rm \alpha=2$ is plotted in Figure 2). The extrapolation matchs the mean value of the z ∼ 6 MAMBO undetected quasars very well, while the average value of the MAMBO detection is ∼ 2.5 times higher. The SEDs of objects with 250 GHz detections ------------------------------------------- We plot the individual optical-to-radio spectral energy distribution (SEDs) for the ten sources that were detected by MAMBO in Figure 3. The FIR emission from the two strongest MAMBO detections, J1148+5251 and J0927+2001, was well measured with SHARC-II and SCUBA at shorter wavelengths, which show FIR emission bumps in the SEDs with dust temperatures of 55 K and 52 K, respectively (Robson et al. 2004; Beelen et al. 2006; Wang et al. 2008a). Similar excesses are also seen in the SEDs of another five sources: J0033-0125, J0840+5624, J1335+3533, J1425+3254, and J2054-0005, where the observed 250 GHz flux densities are $\rm \gtrsim0.5$ dex higher than those expected from the quasar templates fit to the observed optical emission. We model the FIR emission of these sources with a optically thin graybody assuming an emissivity index of $\rm \beta=1.6$ (Beelen et al. 2006). For J0927+2001 and J1148+5251, we directly adopt the dust temperatures fit in Wang et al. (2008a) and Beelen et al. (2006), respectively. For the other nine sources, we assume a dust temperature of 47K (Beelen et al. 2006; Wang et al. 2007). The model FIR SED was extrapolated to the radio band using the typical FIR to radio correlation defined by star forming galaxies, i.e. q=2.34 (Yun et al. 2001). A radio spectral index of -0.75 is adopted here (Condon 1992). Three out of the seven FIR excess sources have been detected at 1.4GHz with deep VLA observations, and the observed flux densities are all above the average for their 250 GHz flux and q=2.34, but the values are within the range in which star forming galaxies lie. Correlations between FIR emission and the AGN ============================================= To investigate the origin of the FIR emission, we compare the correlation between the FIR luminosities and the central AGN bolometric luminosities of these z ∼ 6 quasars to that of two samples of local quasars: a sample of forty-five local (z < 0.5) PG quasars from Hao et al. (2005), a sample of thirty-one z < 0.35 type I AGNs hosted in ULIRGs with IR luminosities $\rm L(8-1000\mu m)>10^{12}L\_{\odot}$ (IR quasars; Zheng et al. 2002; Hao et al. 2005). Hao et al. (2005) reported that the IR quasar sample showed stronger FIR emission and a shallower slope in the 60 μm-optical luminosity relationship when compared to the PG quasars (see also Hao et al. 2007). They attributed this result to the starburst-dominated FIR dust heating in the host galaxies of these IR quasars. We re-fit these FIR-AGN relationships using the FIR and AGN bolometric ($\rm L\_{bol}$) luminosities of the two samples. The bolometric luminosities are taken from Hao et al. (2005), which were derived from the extinction-corrected quasar optical emission (see also Zheng et al. 2002). The FIR luminosities of the IR quasars are taken from Zheng et al. (2002), while for the PG quasar sample we adopt the FIR luminosities and upper limits presented in Haas et al. (2003). All these data are corrected to our cosmology. We adopt the Expectation-Maximization method in the IRAF STSDAS package (Isobe et al. 1986) which can perform linear regression with censored data. The results are: $$\rm IR\,quasars:\,log L\_{FIR}=(0.35\pm0.08)log L\_{bol}+(7.9\pm1.0)$$ $$\rm PG\,quasars:\,log L\_{FIR}=(0.66\pm0.08)log L\_{bol}+(2.8\pm1.0)$$ Figure 4 compares the z ∼ 6 quasars to these two local quasar samples in the $\rm L\_{FIR}-L\_{bol}$ plane. All the millimeter detected sources, and the average of the non-detections are presented. For millimeter detections, $\rm L\_{FIR}$ was integrated using the modeled FIR SEDs described in Section 3.2 (see also Wang et al. 2007), and for non-detections we adopt the estimation of average FIR luminosity in Section 3. The AGN bolometric luminosities of these z ∼ 6 quasars are estimated based on optical and near-infrared observations. Jiang et al. (2006) presented $\rm L\_{bol}$ for the Spitzer observed sources. We adopted their results (corrected to the cosmology we adopted in this paper) and estimated $\rm L\_{bol}$ from the optical B-band luminosity for the other sources, $\rm L\_{bol}=10.4L\_{B}$ (Richards et al. 2006). The B-band luminosities are estimated with the rest frame 1450$\rm \AA$ magnitudes assuming a power-law spectrum of $\rm f\_{\nu}\sim\nu^{-0.5}$. We also convert $\rm L\_{bol}$ to black hole accretion rate $\rm \dot M$ using $$\rm L\_{bol}= \eta \dot M c^{2},$$ and an assumed efficiency η = 0.1. The derived parameters for the ten sources are listed in Table 4. Nine of the ten MAMBO-detected z ∼ 6 quasars stand close to the relationship derived from local IR quasars in the $\rm L\_{FIR}-L\_{bol}$ plot (Figure 4), while the other source J0818+0722 has a lower FIR luminosity, falling between the relationships of the IR and PG quasars. This may suggest that the strong millimeter detections at z ∼ 6 are the high-mass counterparts of the local IR selected quasars, and a similar starburst dust-heating mechanism dominates their FIR emission. However, these z ∼ 6 quasars are much more luminous in the optical than the two local quasars and the $\rm L\_{FIR}-L\_{bol}$ trends derived with the two local samples are getting mixed at high luminosity end. Thus extending the millimeter study to the optically fainter quasar population at z ∼ 6 will help to determine the $\rm L\_{FIR}-L\_{bol}$ correlation for the millimeter bright z ∼ 6 quasars and make a better comparison at low and high redshifts. On the other hand, the average value of the MAMBO non-detections is consistent with the trend defined by the local PG quasars. A link between millimeter detection and the strength of $\rm Ly\alpha$ emission =============================================================================== Omont et al. (1996b) studied a sample of z ∼ 4 optically bright quasars and found that the millimeter detected sources tend to have weaker UV emission lines compared to the average line strength of the millimeter non-detections. Bertoldi et al. (2003a) found a similar trend with MAMBO observations of five z ∼ 6 quasars. In this section, we investigate this effect with the $\rm Ly\alpha$ emission of full sample of z ∼ 6 quasars observed with MAMBO. The rest-frame $\rm Ly\alpha$ equivalent widths ($\rm EW\_{Ly\alpha}$) of five quasars in the SDSS deep strip were presented in Jiang et al. (2007a). The UV spectra of another 20 sources are available in the papers of Fan et al. (2000, 2001a, 2003, 2004, 2006a, 2008, in prep). We estimate $\rm EW\_{Ly\alpha}$ for 18 of them. The spectra have been corrected for Galactic extinction, adopting the extinction curve presented in Savage & Mathis (1979). We then fit the spectra using a method similar to that used in Jiang et al. (2007a), i.e. a power-law ($\rm f\_{\nu}\sim \nu^{-0.5}$) for the UV continuum and gaussians for the $\rm Ly\alpha$ and $\rm NV$ line emission. The sources J1044-0125 and J1048+4637 have broad absorption line features and their equivalent widths are therefore unavailable. We plot the histograms of $\rm EW\_{Ly\alpha}$ for MAMBO detections (solid line) and non-detections (dotted line) in Figure 5. Most of the MAMBO detections distribute in the region of $\rm log\,EW\_{Ly\alpha}(\AA)<1.5$ with a median value of 24 $\rm \AA$, while most non-detections have $\rm log\,EW\_{Ly\alpha}(\AA)$ $\rm >1.5$ with a median value of 68 $\rm \AA$ which is comparable to the average value of ∼ 70 $\rm \AA$ from the large optical quasar samples at lower redshifts (Schneider et al. 1991; Fan et al. 2001b). However, there is also one MAMBO non-detected source, J1621+5155, which shows very weak $\rm Ly\alpha$ emission with $\rm EW\_{Ly\alpha} < 5\AA$. The physical origin of this curious trend is not clear. According to the spectra study with samples of z ∼ 4 quasars presented in Omont et al. (1996b), similar weaker emission is also seen in other UV emission lines, such as the Si IV/O IV] and C IV lines. Though the UV line emission is weak in the MAMBO detected sources, there is no evidence of strong dust reddening in the UV continuum. We can speculate that intrinsically different physical conditions associated with the broad line emission cloud, or a special dust obscuration geometry that affects only the broad line region, may explain this effect. Further tests with other strong broad lines, such as the C IV, and Mg II lines in the UV, the Hα, and Hβ lines in the optical, and the Paschen lines in near-infrared, are required to give a better understand of this trend. In particular, the trend may disappear when tested with the near-infrared lines if it is due to dust absorption. Discussion ========== In this paper, we analyze the FIR properties of the current quasar sample at z ∼ 6. These quasars are mainly optically selected, and represent the luminous end of the quasar population at this early epoch. The X-ray to near-infrared SEDs of these sources are dominated by powerful AGN, just as in optical quasars at lower redshift (Jiang et al. 2006). We divide the sample into MAMBO 250 GHz detections of 10 sources and non-detections of 23 sources. Their FIR emission was compared to that of the low-z quasars, and with the results listed above we will further discuss the FIR dust heating and evolution stage of these z ∼ 6 quasars in this section. The average FIR emission of the 250 GHz non-detections ------------------------------------------------------ The average FIR emission of the MAMBO non-detections is recovered at $\rm 4\sigma$ by stacking the 250 GHz measurements of each source, and the mean SED is similar to the low-z quasar templates of Elvis et al. (1994) and Richards et al. (2006). This suggests that the emission from the outer part of AGN-heated dust torus may dominates the FIR SED in these sources, as was found in the low-z optically bright quasars (Sanders et al. 1989; Haas et al. 2003). However, we cannot rule out contributions from star formation in the quasar hosts. Further surveys of PAH and other star forming features will address the average contribution from star formation to FIR dust heating in these objects. The average FIR luminosity (42.5μm – 122.5μm, rest frame) derived from the stacking flux density is 1.2$\rm \times10^{12}\,L\_{\odot}$. If star formation is not the dominating dust heating source (i.e. contributes < 50%), the average star formation rate in the quasar hosts[4](#fn4) should be less than $\rm 200\,M\_{\odot}\,yr^{-1}$. The FIR emission of the strong 250 GHz detections ------------------------------------------------- Excess FIR emission is seen in seven of the MAMBO detected sources (see Figure 3). Their 250 GHz flux densities are much stronger ($\rm \gtrsim0.5$ dex) than the values expected from the quasar templates when normalized in the rest-frame UV. These sources show FIR-to-radio luminosity ratios within the range defined by typical star forming galaxies (Section 3.2, see also Carilli et al. 2004; Wang et al. 2007). The relation between FIR and AGN bolometric luminosity for these sources follows the trend defined by a sample of low-redshift IR luminous quasars which are hosted by ULIRGs (Section 4). The Spitzer near-infrared photometry by Jiang et al. (2006) detected the AGN-heated hot dust ( ∼ 1000 K) in some of these bright millimeter sources; the optical-to-near infrared SEDs of these sources match the low-z templates very well. The strong FIR emission bump in these sources indicates an additional warm dust component, with a dust temperature of 50 to 60 K. Brightness temperature arguments imply that the warm dust emission region must be extended over a few kpc (Wang et al. 2008a). Thus either a significant fraction of the dust heating is due to star formation in the host galaxy, or the AGN must heat dust over large scales. Intense star formation activity is suggested by other observations of these strong millimeter quasars. Molecular CO emission is detected in the three strongest z ∼ 6 millimeter sources: J1148+5251 (Bertoldi et al. 2003b; Walter et al. 2003, 2004), J0927+2001 (Carilli et al. 2007), and J0840+5624 (Wang et al. 2008b, in prep). These CO detections indicate the presence of $\rm \gtrsim10^{10}\,M\_{\odot}$ of molecular gas in the quasar hosts (Walter et al. 2003; Bertoldi et al. 2003b; Carilli et al. 2007), which can provide the fuel required for star formation rates of up to $\rm >10^{3}\,M\_{\odot}\,yr^{-1}$. Indeed, it is hard to see how the host galaxies can avoid massive star formation with so much molecular gas. The measurement of several CO transitions in J1148+5251 suggests a gas density of about $\rm 10^{4.5}\,cm^{-3}$ and a kinetic temperature of ∼ 100 K (Bertoldi et al. 2003b), which are similar to values in starburst environments found in other quasars and submillimeter galaxies at lower redshifts (Carilli et al. 2002; Solomon & Vanden Bout 2005; Riechers et al. 2006). In addition, recent high resolution observations of [C II] in J1148+5251 (Walter et al. 2008, in prep.; Carilli 2008) show an emission region extended over many kpc, i.e., comparable to that of the CO (3-2) line. Further studies also suggest connections between the CO and FIR emission. The FIR-to-CO luminosity ratios of J1148+5251 and J0927+2001 are about 500 and 650 $\rm L\_{\odot}$ $\rm (K\,km\,s^{-1}\,pc^{2})^{-1}$, respectively, which are consistent with the values found in other high-z CO emitting galaxies (Solomon & Vanden Bout 2005). In Figure 6, we indicate these two CO detected z ∼ 6 quasars (as stars) on the FIR-CO luminosity correlation plot discussed in Riechers et al. (2006). The two points fall a little above but within the observed scatters of the FIR-CO luminosity relation of the typical star forming galaxies at both low and high redshifts. All these facts suggest that massive star formation is an important source of the FIR dust heating. However, it is difficult to constrain what fraction of the FIR dust heating is contributed by star formation in these FIR luminous quasars. According to the FIR-to-CO luminosity correlation in Figure 6, at least $\rm 30-50\%$ of the FIR emission should originate from star formation, giving a corresponding star formation rate of $\rm \gtrsim 1000\,M\_{\odot}\,yr^{-1}$. The evolutionary stage of the z ∼ 6 quasars ------------------------------------------- Observations from optical to near-infrared of these z ∼ 6 quasars indicate that the central SMBHs have masses of $\rm \gtrsim10^{9}\,M\_{\odot}$ (Willott et al. 2003; Jiang et al. 2006). Based on the SMBH-bulge mass relationship (eg. Marconi & Hunt 2003), the final stellar bulge mass should be on order of $\rm 10^{12}\,M\_{\odot}$ for a mature system. The formation of such a massive stellar system may require a star formation rate of $\rm \gtrsim10^{3}\,M\_{\odot}\,yr^{-1}$ on time scales of a few hundred million years as was suggested by some numerical simulations (eg. Li et al. 2007a, b). Thus the average star formation rate of $\rm <200\, M\_{\odot}\,yr^{-1}$ found in the 250 GHz non-detected quasars can be either the case that the major bulge building has already been finished and the SMBH-bulge mass relationship seen at z=0 has already been established, or that the host galaxy is has not yet reached the massive star formation phase, and the growth of the stellar bulge is lagging behind that of the SMBH (see eg. Shields et al. 2006; Ho 2007). Deep and high-resolution imaging at near-IR wavelengths may provide a direct measurement of the stellar population in these z ∼ 6 quasar hosts. For the strong millimeter quasars at z ∼ 6, the FIR excess suggests a star formation rate of $\rm \gtrsim1000\,M\_{\odot}\,yr^{-1}$ from the host galaxies. It is likely that we are witnessing simultaneous building of the stellar bulge and growth of the SMBH in these objects. The case of J1148+5251 is the best documented: Willott et al. (2003) derived an estimate of $\rm 3\times10^9\,M\_{\odot}$ for the black hole mass, while Walter et al. (2004) estimated a dynamical mass of $\rm \sim5\times10^{10}\,M\_{\odot}$. The SMBH-bulge mass ratio is thus clearly more than one order of magnitude larger than the local SMBH-bulge relationship (eg. Marconi & Hunt 2003), as in several other high-z CO-detected QSOs (Shields et al. 2006; Ho 2007). However, the amount of gas detected in CO observations such as in J1148+5251, a few $\rm 10^{10}\,M\_{\odot}$, is far short of the value required to reach the bulge mass given by the local black hole-bulge relationship. To fully understand the evolutionary stage of these strong millimeter quasars at z ∼ 6, further observations are required. First, CO and [C II] searches and excitation studies should be extended to all the z ∼ 6 quasars with strong FIR excesses to determine the host galaxy ISM properties. Second, high resolution ($\rm \lesssim0.3''$) CO, [C II] and dust mapping should be made of all the CO detections. If the FIR-emitting dust and molecular gas are distributed similarly, the FIR emission is likely to be due to a starburst, while more compact FIR emission will argue for AGN dust heating. The resolved CO emission would also provide an estimate of the dynamical mass of the quasar hosts on kpc scales. This is a direct, and perhaps the only, way to test the black hole-bulge relationship at the highest redshifts. These observations can be done with sensitive submillimeter, millimeter and radio telescopes such as the PdBI, CARMA, and SMA, and the coming ALMA and EVLA. Conclusion ========== We study the SEDs of 33 quasars at z ∼ 6 from FIR to radio wavelengths. We conclude that, when averaged with the whole sample and the 250 GHz undetected sources, no significant difference is seen between the mean FIR-to-radio SEDs and the templates of low-redshift optically selected quasars. In particular, we extrapolate the FIR emission of the templates with the typical quasar FIR-to-millimeter spectrum (i.e. $\rm f\sim\nu^\alpha,\,\alpha\geq2$) from Sanders et al. (1989). This extrapolation is consistent with the average FIR emission of the 250 GHz undetected z ∼ 6 quasars very well (see Section 3.1 and Figure 2). This fact suggests that these sources have a similar AGN dominated dust heating mechanism. The average star formation rate from the host galaxies of the quasars undetected by MAMBO is estimated to be less than a few hundred $\rm M\_{\odot}\,yr^{-1}$. We detect a strong FIR excess in seven of the ten z ∼ 6 quasars detected in the millimeter band. These FIR luminous sources are likely to be the high-mass counterparts of local IR luminous quasars which are in transition between the starburst phase and the mature quasar phase. Star formation at rates of $\rm >10^{3}\,M\_{\odot}\,yr^{-1}$ may be the dominant mechanism to heat the dust that gives rise to the FIR emission. The millimeter detected sources also show weaker Lya emission compared to that of the non-detections, but the origin of this trend is not clear yet. Further observations of these FIR luminous quasars, especially spatially resolved line and dust studies, are required to further address questions of star formation and FIR dust heating in these extreme objects. The current sample of z ∼ 6 quasars is optically selected. There may be other quasar populations at z$\gtrsim$6 which are still buried in their starburst environment and are obscured in the optical. Sensitive IR and millimeter facilities, such as Spitzer, ALMA, and Herschel, may discover such objects and provide a more complete view of galaxy and SMBH evolution in the early universe. We acknowledge support from the Max-Planck Society and the Alexander von Humboldt Foundation through the Max-Planck-Forschungspreis 2005. Michael, A. Strauss acknowledges support of the National Science Foundation grant AST-0707266. X. Fan acknowledge support from a David and Lucile Packard Fellow in Science and Engineering. We thank James J. Condon for comments and suggestions. IRAM is funded by the Centre National de la Recherche Scientifique (France), the Max-Planck Gesellschaft (Germany), and the Instituto Geografico Nacional (Spain). The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated by Associated Universities, Inc. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. 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Y., Mao, S. et al. 2002, AJ, 124, 18 Zylka, R. 1998, MOPSI Users Manual, (IRAM: Grenoble) lcccccc Name & redshift &$\rm m\_{1450}$ & $\rm M\_{1450}$ & $\rm f\_{250GHz}$ & $\rm f\_{1.4GHz}$ & Referencea & & & & mJy & $\rm \mu$Jy & (1) & (2) & (3) & (4) & (5) & (6) & (7) J000552.34 − 000655.8 & 5.85&20.23 &-26.47 & 0.36 ± 0.48 & 40 ± 130& 1, \* ,2 J003311.40 − 012524.9 & 6.13&21.78 &-25.00 &1.13 ± 0.36 &-27 ± 19 & 3,3, \* J020332.39+001229.3 & 5.86&20.97 &-25.73 & 1.52 ± 0.67 & 195 ± 22 & 4, \* , \* J030331.40 − 001912.9 & 6.07&21.33 &-25.43 & 0.23 ± 0.51 & -85 ± 62 & 4, \* , \* J035349.76+010405.4 & 6.05&20.21 &-26.55 & 1.20 ± 0.46 &17 ± 19 & 4, \* , \* J081827.40+172251.8 & 6.00&19.34 &-27.40 & 1.19 ± 0.38 &123 ± 12 & 5, \* ,2 J084119.52+290504.5 & 5.96&19.61 &-27.12 & 1.00 ± 0.43 &43 ± 27 & 8, \* , \* J084229.23+121848.2 & 6.08&19.58 &-27.18 & 0.11 ± 0.55 &-8 ± 19 & 9, \* , \* J092721.82+200123.7 & 5.77&19.87 &-26.81 & 4.98 ± 0.75 &50 ± 11 & 5,2, \* J104433.04 − 012502.2 & 5.78&19.21 &-27.46 & 1.82 ± 0.43 & -15 ± 24 & 10, \* ,7 J142516.30+325409.0 & 5.85&20.62 &-26.08 & 2.27 ± 0.51 &20 ± 20 & 14, \* , \* J142738.59+331242.0 & 6.12&20.33 &-26.44 & 0.39 ± 0.66 &1730 ± 131 & 15, \* ,15 J160253.98+422824.9 & 6.07&19.86 &-26.90 & 1.41 ± 0.54 &60 ± 15 & 1, \* ,2 J162100.70+515544.8 & 5.71&19.89 &-26.77 & 0.30 ± 0.55 &-12 ± 21 & 9, \* , \* J163033.90+401209.6 & 6.05&20.64 &-26.12 & 0.80 ± 0.60 &14 ± 15 & 11,12, \* J164121.64+375520.5 & 6.04&21.30 &-25.45 & 0.08 ± 0.46 &-30 ± 32 & 3,3, \* J205406.42 − 000514.8 & 6.07&20.67 &-26.09 & 2.38 ± 0.53 &17 ± 23 & 4, \* , \* J231546.36 − 002357.5 & 6.12&21.31 &-25.46 & 0.28 ± 0.60 &31 ± 16 & 4, \* , \* J232908.28 − 030158.8 & 6.43&21.65 &-25.20 & 0.01 ± 0.50 &14 ± 22 & 3,3, \* Note – The detections are marked as boldface. aThe three references are for the optical, 250 GHz, and 1.4 GHz data, respectively. The asterisks denote new data reported in this paper. References for Table 1 and 2 – (1) Fan et al. 2004; (2) Wang et al. 2007; (3) Willott et al. 2007; (4) Jiang et al. 2008; (5) Fan et al. 2006a; (6) Fan et al. 2001a; (7) Petric et al. 2003; (8) Goto 2006; (9) Fan et al. 2008, in prep. (10) Fan et al. 2000; (11) Fan et al 2003; (12) Bertoldi et al. 2003a; (13) Carilli et al. 2004; (14) Cool et al. 2006; (15) McGreer et al. 2006. lcccccc Name & redshift &$\rm m\_{1450}$ & $\rm M\_{1450}$ & $\rm f\_{250GHz}$ & $\rm f\_{1.4GHz}$ & Referencea & & & & mJy & $\rm \mu$Jy & (1) & (2) & (3) & (4) & (5) & (6) & (7) J000239.39+255034.8 & 5.80&19.02 &-27.67 & 0.20 ± 0.88 & 89 ± 14 & 1,2,2 J083643.85+005453.3 & 5.81&18.81 &-27.88 &-0.39 ± 0.96 &1740 ± 40 & 6,7,7 J084035.09+562419.9 & 5.85&20.04 &-26.66 & 3.20 ± 0.64 &12 ± 9 & 5,2,2 J103027.10+052455.0 & 6.31&19.66 &-27.16 &-1.15 ± 1.13 & -3 ± 20 & 6,2,2 J104845.05+463718.3 & 6.20&19.25 &-27.55 & 3.00 ± 0.40 &7 ± 13 & 11,12,2 J113717.73+354956.9 & 6.01&19.63 &-27.12 & 0.10 ± 1.13 &9 ± 17 & 5,2,2 J114816.64+525150.2 & 6.42&19.03 &-27.82 & 5.00 ± 0.60 &55 ± 12 & 11,12,13 J125051.93+313021.9 & 6.13&19.64 &-27.14 & 0.07 ± 0.90 &37 ± 21 & 5,2,2 J130608.26+035626.3 & 6.02&19.55 &-27.19 &-1.05 ± 1.04 &14 ± 21 & 6,7,7 J133550.81+353315.8 & 5.95&19.89 &-26.84 & 2.34 ± 0.50 &35 ± 10 & 5,2,2 J141111.29+121737.4 & 5.93&19.97 &-26.75 & 1.00 ± 0.62 &61 ± 16 & 1,2,2 J143611.74+500706.9 & 5.83&20.16 &-26.54 &-0.21 ± 1.14 & 6 ± 16 & 5,2,2 J150941.78 − 174926.8 & 6.12&19.82 &-26.95 & 1.00 ± 0.46 &– & 3,3,- J162331.81+311200.5 & 6.25&20.13 &-26.67 & 0.17 ± 0.80 &24 ± 31 & 1,2,2 Note – The detections are marked as boldface. aThe three references are for the optical, 250 GHz, and 1.4 GHz data, respectively, and the corresponding literatures are listed at the end of Table 1. lccccccc Group & $\rm Number^{a}$ & $\rm <f\_{250GHz}>$ & $\rm <L\_{FIR}>$& $\rm Number^{b}$ & $\rm <f\_{1.4GHz}>$&$\rm <L\_{1.4GHz}>^{c}$ & q & & mJy & $\rm 10^{12}L\_{\odot}$ & & $\rm \mu$Jy &$\rm L\_{\odot}\,Hz^{-1}$ & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) Whole sample& 33 & 1.26 ± 0.10 & 2.9 ± 0.2 & 32 & 46 ± 3&0.031 ± 0.002 &1.41 ± 0.04 (radio quiet)& 30 & 1.29 ± 0.10 &3.0 ± 0.2 & 29 & 32 ± 3 & 0.022 ± 0.002 &1.57 ± 0.05 MAMBO detections& 10&2.73 ± 0.06& 6.4 ± 0.1 & 10 & 37 ± 4 &0.024 ± 0.003 & 1.84 ± 0.05 Non-detections&23& 0.52 ± 0.13& 1.2 ± 0.3 & 22 & 56 ± 4 &0.038 ± 0.003 & 0.94 ± 0.11 (radio quiet) &20& 0.51 ± 0.13& 1.2 ± 0.3 & 19 & 27 ± 4 &0.018 ± 0.003 & 1.24 ± 0.13 a Number of sources observed at 250 GHz. b Number of sources observed at 1.4 GHz. c A radio spectral index of -0.75 (Condon 1992) is adopted here to calculate the rest frame 1.4 GHz lunimosity. cccc Name & $\rm \frac{L\_{bol}}{10^{13}L\_{\odot}}$&$\rm \frac{\dot M}{M\_{\odot}\,yr^{-1}}$ & $\rm \frac{L\_{FIR}}{10^{12}L\_{\odot}}$ (1) & (2) & (3) & (4) J003311.40-012524.9 & 1.4&9.4 & 2.6 ± 0.8 J081827.40+172251.8 & 12.8&86.3 & 2.8 ± 0.9 J084035.09+562419.9 & 6.5&43.6 & 7.6 ± 1.5 J092721.82+200123.7 & 7.4&50.0 & 12.2 ± 2.8 J104433.04-012502.2 & 13.5&91.1 & 5.3 ± 0.8 J104845.05+463718.3 & 14.6&98.4 & 6.9 ± 0.9 J114816.64+525150.2 & 18.8&126.8 & 13.4 ± 3.4 J133550.81+353315.8 & 7.6&51.4 & 5.5 ± 1.2 J142516.30+325409.0 & 3.8&25.6 & 5.4 ± 1.2 J205406.42-000514.8 & 3.8&25.8 & 6.8 ± 1.1 Note – $\rm {\dot M}$ is the black hole accretion rate derived with equation (7) in Section 4. --- 1. derived from the absolute magnitude of $\rm M\_{1450}=-26.4$ presented in McGreen et al. (2006)[↩](#fnref1) 2. The $\rm 4400\AA$ and 5 GHz flux densities are derived from the 1450$\rm \AA$ magnitude and the observed 1.4 GHz flux density, assuming power law spectra indices of -0.5 in the optical and -0.75 in the radio, respectively.[↩](#fnref2) 3. A typical redshift value of z=6 is adopted here[↩](#fnref3) 4. The star formation rate is estimated with the empirical relationship from Kennicutt (1998), assuming a standard Salpeter initial mass function, i.e. $\rm SFR\sim 4.5L\_{IR}\,M\_{\odot}yr^{-1}$, where $\rm L\_{IR}$ is the infrared luminosity (8-1000 μm) in unit of $\rm 10^{44}\,erg\,s^{-1}$, and is $\rm\sim 1.5L\_{FIR}$ for warm dust emission.[↩](#fnref4) Thermal Emission from Warm Dust in the Most Distant Quasars =========================================================== We report new continuum observations of fourteen z ∼ 6 quasars at 250 GHz and fourteen quasars at 1.4 GHz. We summarize all recent millimeter and radio observations of the sample of the thirty-three quasars known with $\rm 5.71\le z\le 6.43$, and present a study of the rest frame far-infrared (FIR) properties of this sample. These quasars were observed with the Max Plank Millimeter Bolometer Array (MAMBO) at 250 GHz with mJy sensitivity, and $\rm 30\%$ of them were detected. We also recover the average 250 GHz flux density of the MAMBO undetected sources at $\rm 4\sigma$, by stacking the on-source measurements. The derived mean radio-to-UV spectral energy distributions (SEDs) of the full sample and the 250 GHz non-detections show no significant difference from that of lower-redshift optical quasars. Obvious FIR excesses are seen in the individual SEDs of the strong 250 GHz detections, with FIR-to-radio emission ratios consistent with that of typical star forming galaxies. Most 250 GHz-detected sources follow the $\rm L\_{FIR}$–$\rm L\_{bol}$ relationship derived from a sample of local IR luminous quasars ($\rm L\_{IR}>10^{12}L\_{\odot}$), while the average $\rm L\_{FIR}/L\_{bol}$ ratio of the non-detections is consistent with that of the optically-selected PG quasars. The MAMBO detections also tend to have weaker $\rm Ly\alpha$ emission than the non-detected sources. We discuss possible FIR dust heating sources, and critically assess the possibility of active star formation in the host galaxies of the z ∼ 6 quasars. The average star formation rate of the MAMBO non-detections is likely to be less than a few hundred $\rm M\_{\odot}\,yr^{-1}$, but in the strong detections, the host galaxy star formation is probably at a rate of $\rm \gtrsim10^{3}\,M\_{\odot}\,yr^{-1}$, which dominates the FIR dust heating. Introduction ============ More than thirty quasars at z ∼ 6 have been discovered (e.g., Fan et al. 2000, 2001a, 2003, 2004, 2006a; Jiang et al. 2007a; Willott et al. 2007). These objects are the earliest massive black hole accretion systems known (Jiang et al. 2006; 2007b), seen at an epoch close to the end of cosmic reionization (Fan et al. 2006b). They provide us with unique information on both the growth of supermassive black holes (SMBHs) and the formation of massive galaxies when the age of the universe was $\rm \lesssim1\,Gyr$. There are fundamental relationships between SMBH mass and bulge stellar mass/velocity dispersion in the local universe (e.g., Tremaine et al. 2002; Marconi & Hunt 2003; Hopkins et al. 2007), indicating that the formation of SMBHs and their spheroidal hosts are coupled. Active galactic nuclei (AGNs) buried in dusty starburst environments have been discovered in samples of low redshift ultraluminous infrared galaxies (ULIRGs), which are believed to be a transition phase between starburst mergers and typical optically bright AGNs (Sanders et al. 1988; Wu et al. 1998; Zheng et al. 2002). Studies of large samples of galaxies and AGNs also provide clues about the ’downsizing’ process in galaxy and SMBH formation, whereby the formation of the most massive systems occur at early epochs (Cowie et al. 1996; Heckman et al. 2004; Kauffmann & Heckman 2005). Examples of massive quasars hosted by interacting systems or bright submillimeter galaxies have been found at redshifts greater than 4, such as BR 1202-0725 at z=4.7 and BRI1335-0417 at z=4.4 (Omont et al. 1996a, 1996b; Benford et al. 1999; Beelen et al. 2006). These objects are all characterized by strong molecular CO line emission (Guilloteau et al. 1997, 1999; Carilli et al. 2002; Solomon & Vanden Bout 2005; Riechers et al. 2006) and FIR (Guilloteau et al. 1999) and radio (Momjian et al. 2005, 2007) emission originating from a starburst, with implied star formation rates of a few thousand $\rm M\_{\odot}\,yr^{-1}$. The results suggest that the galaxies are at an earlier evolution stage than are typical optically bright quasars in which the central AGN dominates the emission from X-ray to radio. These studies have been extended to z ∼ 6 quasar sample with sensitive submillimeter, millimeter, and radio telescopes (Priddey et al. 2003b; Bertoldi et al. 2003; Petric et al. 2003; Carilli et al. 2004; Wang et al. 2007). About 1/3 of these z ∼ 6 quasars were detected at millimeter wavelengths, at mJy sensitivity (Priddey et al. 2003b; Wang et al. 2007). The millimeter detections imply FIR luminosities of $\rm 10^{12}\sim10^{13}\,L\_{\odot}$ and dust masses of $\rm \gtrsim10^{8}\,M\_{\odot}$ in the quasar host galaxies (Bertoldi et al. 2003a, Beelen et al. 2006). Such huge dust masses require rapid metal and dust enrichment via a starburst in the early evolution of these galaxies (Bertoldi et al. 2003a; Walter et al. 2003; Venkatesan et al. 2006; Li et al. 2007b; Dwek et al. 2007). The heating sources of the FIR-emitting dust in the quasar systems at z ∼ 6 have become a key question. The reprocessed emission from star formation-heated dust can provide a direct estimate of the star formation rate, thus constraining the bulge building stage in these quasar hosts. Observations at submillimeter wavelengths imply that the FIR emission in the two sources with the strongest millimeter detections, SDSS J114816.64+525150.3 at z=6.42 (hereafter J1148+5251) and SDSS J092721.82+200123.7 at z=5.77 (hereafter J0927+2001), is from dust components with temperatures of 50 to 60 K (Beelen et al. 2006; Wang et al. 2008a). Large amounts of highly excited molecular CO were also detected in the host galaxies of these two sources (Bertoldi et al. 2003b; Walter et al. 2003; Carilli et al. 2007), as well as strong [C II] 158 μm ISM gas cooling line emission in J1148+5251 (Maiolino et al. 2005). These results suggest that star formation plays an important role in the heating of FIR-emitting warm dust. The implied star formation rate is $\rm \gtrsim10^{3}\,M\_{\odot}\,yr^{-1}$, which argues for active bulge building in these two z ∼ 6 quasars. We have a long standing program to study the dust and gas emission from the z ∼ 6 quasars (e.g., Wang et al. 2007, 2008a). In this paper, we report new observations and present an FIR emission study of the full sample of thirty-three z ∼ 6 quasars discovered to date, aiming (i) to give a general view of the FIR emission properties of the sample, and (ii) to further constrain the dust heating and star forming activity in the quasar hosts. We describe the full quasar sample, the new observations, and summarize the current millimeter and radio results in section 2. The analysis of the full sample is given in Sections 3, 4, and 5. We present a discussion of star formation in Section 6, and give a brief summary in Section 7. We adopt a $\rm \Lambda$-model cosmology with $\rm H\_{0}=71km\ s^{-1}\ Mpc^{-1}$, $\rm \Omega\_{M}=0.27$ and $\rm \Omega\_{\Lambda}=0.73$ throughout this paper (Spergel et al. 2007). Sample and observations ======================= The sample of z ∼ 6 quasars --------------------------- Thirty-three quasars have been discovered at z ∼ 6 ($\rm 5.71 \le z \le 6.43$, see Table 1 and 2). Twenty-two of these objects were selected from ∼ 8000 deg2 of the Sloan Digital Sky Survey (SDSS, York et al. 2000) imaging data (Fan et al. 2000, 2001a, 2003, 2004, 2006a, 2008, in prep.; Goto 2006). These objects typically have rest frame $\rm 1450\AA$ AB magnitudes $\rm m\_{1450}<20$. The SDSS imaging survey is using a dedicated 2.5m telescope and a large format CCD camera (Gunn et al. 2006) at the Apache Point Observatory in New Mexico to obtain images in five broad bands ($\rm u$, $\rm g$, $\rm r$, $\rm i$ and $\rm z$, centered at 3551, 4686, 6166, 7480 and 8932 $\rm \AA$, respectively; Fukugita et al. 1996) of high Galactic latitude sky in the Northern Galactic Cap. The imaging data are processed by the astrometric pipeline (Pier et al. 2003) and photometric pipeline (Lupton et al. 2001), resulting in astrometric calibration errors of $\rm < 0.1''$ rms per coordinate, and photometric calibration to better than 0.03 mag (Hogg et al. 2001, Smith et al. 2002, Ivezi' c et al. 2004, Tucker et al. 2006). These data have been made available to the public in a series of data releases (EDR: Stoughton et al. 2002; DR1 – DR6: Abazajian et al. 2003, 2004, 2005, Adelman-McCarthy et al. 2006, 2007, 2008). Five additional z ∼ 6 quasars were discovered in the SDSS Deep Equatorial Stripe (’Stripe 82’) by Jiang et al. (2007a). These five sources were selected from deep optical imaging of 260 deg2. These objects are about one magnitude fainter in the optical (i.e., $\rm 20<m\_{1450}<21.5$) than the primary SDSS sample. One source in this sample, J020332.39+001229.3 (hereafter J0203+0012), was also independently discovered in the UKIRT Infrared Deep Sky Survey (Venemans et al. 2007). Four new quasars at $\rm z>6$ have been published (Willott et al. 2007) from the Canada-France High-z Quasar Survey (CFHQS). These quasars were selected from imaging of 400 deg2; three of them have $\rm m\_{1450}>21$. The fourth source, CFHQS J1509-1749, is comparable in brightness to the faintest source in the primary SDSS sample, but the optical spectrum is much steeper than that of the typical quasars, indicating significant dust reddening. Another two z ∼ 6 quasars were discovered in the NOAO Deep Wide-Field Survey (Jannuzi et al. 2004) using optical, infrared, and radio data (McGreer et al. 2006; Cool et al. 2006). One of them, J1427+3312 at z=6.12, was selected by McGreer et al. (2006) from 4 deg2 by matching infrared and radio sources to optical counterparts. It is the brightest radio source in the z ∼ 6 quasar sample. The UV to optical spectrum of this object is likely to be reddened by dust, and the rest frame $\rm 1450\AA$ magnitude derived from near-infrared data is $\rm m\_{1450}=20.33$[1](#fn1). The other quasar was infrared selected from Spitzer mid-infrared data, and identified in the AGN and Galaxy Evolution Survey (Cool et al. 2006) with $\rm m\_{1450}=20.62$. New observations ---------------- Twenty-two sources in the z ∼ 6 quasar sample have published 250 GHz continuum observations with the Max-Planck Millimeter Bolometer Array (MAMBO, Kreysa et al. 1998) on the IRAM 30-m telescope (Bertoldi et al. 2003a; Petric et al. 2003; Wang et al. 2007; Willott et al. 2007; Priddey et al. 2008), and nineteen sources in the sample have published 1.4 GHz continuum observations with the VLA (Petric et al. 2003; Carilli et al. 2004; Wang et al. 2007; McGreer et al. 2006). We report our new MAMBO and VLA observations of totally nineteen z ∼ 6 quasars in this paper. These sources are unobserved or marginally detected in the previous observations, and we present new data for them at either 250 GHz or 1.4 GHz. We describe the new observations below and list the measurements for the nineteen quasars in Table 1. We also summarise the previous observations for other z ∼ 6 quasars in Table 2. We present new 250 GHz observations for fourteen z ∼ 6 quasars, including eleven sources that have never been previously observed. The new observations have been obtained with the MAMBO-II 117-element array in Winter 2006-2007, we adopted the same observing mode as was used in the previous observations (Wang et al. 2007), i.e., doing chopping photometry at 2 Hz with $\rm 32''$ throw in azimuth. We typically spent 1-3 hours on each source to achieve the sub-mJy rms noise level reached in previous observations. The data were reduced with the standard MOPSIC pipeline (Zylka 1998). Three sources, J1044-0125, J1425+3254, and J2054-0005, were detected from the new data at $\rm >4\sigma$ level. The MAMBO detection of J1044-0125 was recently reported in Priddey et al. (2008) from an independent observation with a 250 GHz flux density of $\rm 2.5\pm0.6$. We observed this source down to a lower rms noise level, and our result, $\rm 1.8\pm0.4\,mJy$, is a bit lower compared to the value in Priddey et al. (2008) but consistent within the error limits (see Table 1). By including the data from the new observations, we also confirmed the previous $\rm 3\sigma$ detection in the source J0818+1722 (Wang et al. 2007) with a lower rms of 0.4 mJy. New 1.4 GHz data were obtained with the VLA for fourteen quasars, including thirteen previously unobserved sources and a published marginally detected source, J0927+2001 ($\rm 3.2\sigma$, Wang et al. 2007). The new observations were carried out in 2006 and 2007 with the array in the A or B configurations, down to an rms noise level of $\rm \sim20\,\mu Jy$ for most of the sources. There is a ∼ 1 Jy source in the field of the quasar J0303-0019, leading to a high rms of $\rm 66\,\mu Jy$ in the image. All the data were reduced using AIPS. One source, J0203+0012, was detected by our new observation with a 1.4 GHz flux density of $\rm S\_{\nu}=195\pm22\,\mu Jy$, and the previous detection of J0927+2001 was confirmed at $\rm \gtrsim4\sigma$ level. There are fairly bright and extended radio sources in the 1.4 GHz field images of the CFHQS quasars J0033-0125 and J1641+3755 (Figure 1), though these two quasars are not detected. One double-lobe radio source is found about $\rm9'$ away from the optical position of the quasar J0033-0125, with integrated 1.4 GHz flux density of $\rm \sim23\,mJy$. This radio source was also detected in the NVSS survey (Condon et al. 1998) and FIRST survey (White et al. 1997), and there is no published deep optical data for this area yet. Five radio sources with integrated flux densities $\rm >10\,mJy$ are seen in the field of the source J1641+3755, including extended double radio sources, within 10$\rm '$ from the optical quasar position. According to the FIRST 1.4 GHz source counts (White et al. 1997), about 1.6 source with $\rm f\_{1.4GHz}>10\,mJy$ is expected within this amount of sky area. The brightest radio source in the field of J1641+3755 was identified as a radio-loud AGN with a spectroscopic redshift of z=0.162 (Best et al. 2005). Further optical imaging and spectrography will address the question of whether the bright radio sources in the fields of the two CFHQS quasars is due to foreground galaxy clusters. We notice that bright extended radio sources and overdensities were previously reported in the 1.4 GHz observations of the two SDSS z ∼ 6 quasars J1030+0524 and J1148+5251 (Petric et al. 2003; Carilli et al. 2004). Carilli et al. (2004) investigated the SDSS data of the J1148+5251 field, and however, there is no evidence for dense foreground clusters or gravitational lensing. Summary of the millimeter and radio results ------------------------------------------- In summary, all the thirty-three source in the z ∼ 6 quasar sample have been observed by MAMBO at 250 GHz. The observations have rms values in the range of $\rm 0.4-1.1\,mJy$ with a median value of about $\rm 0.6\,mJy$, and ten sources are detected (see Table 1 and 2). This yields a detection rate of $\rm 30\pm10\%$ which is consistent with the submillimeter and millimeter detection rates of optically bright quasars at redshifts 2 and 4 (Priddey et al. 2003a; Omont et al. 2001; 2003; Carilli et al. 2001). Thirty-two z ∼ 6 quasars have 1.4 GHz radio continuum observations by the VLA, with a median rms level of $\rm 20\,\mu Jy$, Ten out of the thirty-two sources have been detected, including four 250 GHz detections (see Table 1 and 2). Two of the radio detections, J0836+0054 and J1427+3312, have flux densities $\rm >1mJy$. These two sources, together with the new radio detection J0203+0012, have radio loudness[2](#fn2) $\rm R\equiv f\_{\nu,5GHz}/f\_{\nu,4400}\ge10$, and hence are radio loud according to the definition by Kellermann et al. (1989). Data at other wavelengths for the analysis in this paper were collected from the literature, including optical data from the original papers (Fan et al. 2000; 2001a; 2003; 2004; 2006a; 2008, in prep; Goto 2006; McGreer et al. 2006; Willott et al. 2007; Jiang et al. 2008), near-infrared data from Jiang et al. (2006), submillimeter data from Priddey et al. (2003b), Robson et al. (2004) and Beleen et al. (2006), and the millimeter continuum data from the PdBI CO observations (Carilli et al. 2007; Wang et al. 2008b, in prep.). The UV-to-radio SEDs of the z ∼ 6 quasars ========================================= Using the 250 GHz and 1.4 GHz continuum measurements, we derive the average FIR and radio emission of these z ∼ 6 quasars and compare it to the typical quasars at low redshift. The calculations are performed for three groups: (I) the whole sample of thirty-three sources, (II) the subsample of ten sources detected with MAMBO at $\rm \ge3\sigma$, and (III) the subsample of twenty-three sources not detected with MAMBO. Using the MAMBO 250GHz data, we calculate the weighted average 250GHz flux densities for each group, namely: $$\rm \left\langle f\_{250GHz}\right\rangle =\frac{\sum w\_{i}f\_{250GHz,i}}{\sum w\_{i}}$$ $$\rm w\_{i}=\frac{1}{\sigma^{2}\_{250GHz,i}}$$ $$\rm \left\langle \sigma\_{250GHz}\right\rangle=\left(\frac{\sum \left(w\_{i}\sigma\_{250GHz,i}\right)^2}{(\sum w\_{i})^2}\right)^{0.5}$$ where $\rm f\_{250GHz,i}$ is the MAMBO observed flux density for each source, and σ250GHz, i is the $\rm 1\sigma$ rms. For the calculation of $\rm \langle f\_{250GHz}\rangle$ of the whole sample and the subsample of 250 GHz non-detections, we weighed the data by $\rm 1/\sigma^{2}\_{250GHz,i}$ (equation (2)). However, equal weights were adopted (i.e. $\rm w\_{i}=1$) when calculate $\rm \langle f\_{250GHz}\rangle$ for the 250 GHz detections, as the observed rms values spread in a smaller range (0.36 ∼ 0.75 mJy) and we do not want to bias the average toward the weakest sources. The average flux density ratios between 250GHz and the rest frame 1450$\rm \AA$ are then calculated as $\rm \langle f\_{250GHz}\rangle/\langle f\_{1450}\rangle$, where the average 1450$\rm \AA$ flux density $\rm \langle f\_{1450}\rangle$ is derived from the AB magnitudes presented in the discovering papers (see Section 2.1). The average radio emission and 1.4GHz-to-1450$\rm \AA$ ratios for each group are calculated using the same method. The average FIR luminosities (42.5μm – 122.5μm, rest frame) for each group can be estimated[3](#fn3) as $\rm <L\_{FIR}>=2.34\times10^{12}(<f\_{250GHz}>/mJy)\,L\_{\odot}$, assuming a dust temperature of 47 K and an emissivity index $\rm \beta$ of 1.6 (Beelen et al. 2006). Based on this estimation, the mean FIR-to-radio ratios for each group can be calculated as (Helou et al. 1985): $$\rm q\equiv log\left(\frac{\langle L\_{FIR}\rangle}{3.75\cdot10^{12}\,L\_{\odot}}\right)-log\left(\frac{\langle L\_{1.4GHz}\rangle}{L\_{\odot}\cdot Hz^{-1}}\right)$$ The rest-frame 1.4 GHz radio luminosity densities ($\rm L\_{1.4GHz}$) are derived with the average radio flux densities, assuming a power-law radio spectrum of $\rm f\_{\nu}\sim\nu^{-0.75}$ (Condon 1992). The q values can be increased by ∼ 0.2 if we assume a flater radio spectrum of $\rm f\_{\nu}\sim\nu^{-0.5}$. The mean flux densities, luminosities, and q values are listed in Table 3. We exclude the three radio loud sources in the first and the third groups, and present the corresponding results in the second row of each group in Table 3. The ten MAMBO detected sources in the second group are all radio quiet (according to the definition in Kellermann et al. 1989). The mean SED ------------ The mean 250 GHz flux density is $\rm 1.26\pm0.10\, mJy$ for the whole sample, and $\rm 2.30\pm0.15\, mJy$ for the MAMBO detections. The mean 250 GHz flux density of MAMBO non-detections is recovered by averaging all the on-source pixel values. A $\rm \sim4\sigma$ signal of $\rm 0.52\pm0.13\, mJy$ is obtained for this group, which is about one quarter of the value of the typical MAMBO detections. We ran the stacking with the non-detection group several times, omitting one source each stack, to see if this $\rm 4\sigma$ signal is due to a single source that is just below the detectability. The test shows stacking flux densities in the range of 0.47 to 0.56 mJy with rms values of ∼ 0.13 mJy. Thus it is unlikely that the average flux density is dominated by one object. No significant differences are found in the $\rm \langle f\_{250GHz}\rangle$ values when the radio-loud sources are excluded in each group. The average 1.4 GHz observed flux densities are $\rm 27\pm4$ to $\rm 37\pm4\, \mu Jy$ for the radio quiet sources in the three groups. We convert these measurements to the flux densities at rest frame 5 GHz assuming a power-law radio spectrum of $\rm f\_{\nu}\sim\nu^{-0.75}$ (Condon 1992). This gives an average radio loudness (i.e. average radio to optical emission ratio $\rm \langle R\rangle=\langle f\_{\nu,5GHz}\rangle/\langle f\_{\nu,4400}\rangle$) of $\rm \langle R\rangle\sim$0.7, which is consistent with the typical radio loudness values of $\rm 0.1<R<1$ found with local(Kellermann et al. 1989) and z ≤ 2 (Cirasuolo et al. 2003) radio quiet quasars. The derived q parameters (i.e. FIR to radio emission ratios) are all lower or at the edge of the range of $\rm q=2.34\pm0.7$ for typical star forming galaxies (Yun et al. 2001), but are consistent with the values of q < 2 found in the local Seyfert 1 galaxies (Ulvestad & Ho 2001). This result suggests that, as was found in the local universe and at z$\rm \sim$2 to 4, AGN power dominates the radio emission in a large fraction of these radio quiet quasars, i.e., the radio quiet AGNs are in fact not radio silent (Barvainis et al. 2005; Carilli et al. 2001; Petric et al. 2006). We plot the mean SED (normalized by the rest frame 1450$\rm \AA$ flux density) of the radio quiet sources in each group in Figure 2, together with the templates of low-z optical quasars from Elvis et al. (1994) and Richards et al. (2006). We also plot the average optical to near infrared emission based on the Spitzer photometry of 13 z ∼ 6 quasars from Jiang et al. (2006), who show that the ratio is consistent with the local templates. The FIR-to-1450$\rm \AA$ ratios of the three z ∼ 6 quasar groups have a range of ∼ 0.6 dex, and no obvious excess is seen between the value of the whole sample and the FIR end of the templates. However, due the lack of millimeter and submillimeter data, the available templates do not give a good constraint of the FIR-to-millimeter SED for the local optical quasars. Sanders et al. (1989) studied a sample of optically selected Palomar-Green quasars (PG quasars; Schmidt & Green 1983; Boroson & Green 1992), and their analysis suggest that the typical 100$\rm \,\mu$m to 1$\rm \,mm$ spectral index $\rm \alpha$ (($\rm f\_{\nu}\sim\nu^{\alpha}$)) is $\rm \geq2$. We extrapolate the FIR emission of the templates with this result (an example of $\rm \alpha=2$ is plotted in Figure 2). The extrapolation matchs the mean value of the z ∼ 6 MAMBO undetected quasars very well, while the average value of the MAMBO detection is ∼ 2.5 times higher. The SEDs of objects with 250 GHz detections ------------------------------------------- We plot the individual optical-to-radio spectral energy distribution (SEDs) for the ten sources that were detected by MAMBO in Figure 3. The FIR emission from the two strongest MAMBO detections, J1148+5251 and J0927+2001, was well measured with SHARC-II and SCUBA at shorter wavelengths, which show FIR emission bumps in the SEDs with dust temperatures of 55 K and 52 K, respectively (Robson et al. 2004; Beelen et al. 2006; Wang et al. 2008a). Similar excesses are also seen in the SEDs of another five sources: J0033-0125, J0840+5624, J1335+3533, J1425+3254, and J2054-0005, where the observed 250 GHz flux densities are $\rm \gtrsim0.5$ dex higher than those expected from the quasar templates fit to the observed optical emission. We model the FIR emission of these sources with a optically thin graybody assuming an emissivity index of $\rm \beta=1.6$ (Beelen et al. 2006). For J0927+2001 and J1148+5251, we directly adopt the dust temperatures fit in Wang et al. (2008a) and Beelen et al. (2006), respectively. For the other nine sources, we assume a dust temperature of 47K (Beelen et al. 2006; Wang et al. 2007). The model FIR SED was extrapolated to the radio band using the typical FIR to radio correlation defined by star forming galaxies, i.e. q=2.34 (Yun et al. 2001). A radio spectral index of -0.75 is adopted here (Condon 1992). Three out of the seven FIR excess sources have been detected at 1.4GHz with deep VLA observations, and the observed flux densities are all above the average for their 250 GHz flux and q=2.34, but the values are within the range in which star forming galaxies lie. Correlations between FIR emission and the AGN ============================================= To investigate the origin of the FIR emission, we compare the correlation between the FIR luminosities and the central AGN bolometric luminosities of these z ∼ 6 quasars to that of two samples of local quasars: a sample of forty-five local (z < 0.5) PG quasars from Hao et al. (2005), a sample of thirty-one z < 0.35 type I AGNs hosted in ULIRGs with IR luminosities $\rm L(8-1000\mu m)>10^{12}L\_{\odot}$ (IR quasars; Zheng et al. 2002; Hao et al. 2005). Hao et al. (2005) reported that the IR quasar sample showed stronger FIR emission and a shallower slope in the 60 μm-optical luminosity relationship when compared to the PG quasars (see also Hao et al. 2007). They attributed this result to the starburst-dominated FIR dust heating in the host galaxies of these IR quasars. We re-fit these FIR-AGN relationships using the FIR and AGN bolometric ($\rm L\_{bol}$) luminosities of the two samples. The bolometric luminosities are taken from Hao et al. (2005), which were derived from the extinction-corrected quasar optical emission (see also Zheng et al. 2002). The FIR luminosities of the IR quasars are taken from Zheng et al. (2002), while for the PG quasar sample we adopt the FIR luminosities and upper limits presented in Haas et al. (2003). All these data are corrected to our cosmology. We adopt the Expectation-Maximization method in the IRAF STSDAS package (Isobe et al. 1986) which can perform linear regression with censored data. The results are: $$\rm IR\,quasars:\,log L\_{FIR}=(0.35\pm0.08)log L\_{bol}+(7.9\pm1.0)$$ $$\rm PG\,quasars:\,log L\_{FIR}=(0.66\pm0.08)log L\_{bol}+(2.8\pm1.0)$$ Figure 4 compares the z ∼ 6 quasars to these two local quasar samples in the $\rm L\_{FIR}-L\_{bol}$ plane. All the millimeter detected sources, and the average of the non-detections are presented. For millimeter detections, $\rm L\_{FIR}$ was integrated using the modeled FIR SEDs described in Section 3.2 (see also Wang et al. 2007), and for non-detections we adopt the estimation of average FIR luminosity in Section 3. The AGN bolometric luminosities of these z ∼ 6 quasars are estimated based on optical and near-infrared observations. Jiang et al. (2006) presented $\rm L\_{bol}$ for the Spitzer observed sources. We adopted their results (corrected to the cosmology we adopted in this paper) and estimated $\rm L\_{bol}$ from the optical B-band luminosity for the other sources, $\rm L\_{bol}=10.4L\_{B}$ (Richards et al. 2006). The B-band luminosities are estimated with the rest frame 1450$\rm \AA$ magnitudes assuming a power-law spectrum of $\rm f\_{\nu}\sim\nu^{-0.5}$. We also convert $\rm L\_{bol}$ to black hole accretion rate $\rm \dot M$ using $$\rm L\_{bol}= \eta \dot M c^{2},$$ and an assumed efficiency η = 0.1. The derived parameters for the ten sources are listed in Table 4. Nine of the ten MAMBO-detected z ∼ 6 quasars stand close to the relationship derived from local IR quasars in the $\rm L\_{FIR}-L\_{bol}$ plot (Figure 4), while the other source J0818+0722 has a lower FIR luminosity, falling between the relationships of the IR and PG quasars. This may suggest that the strong millimeter detections at z ∼ 6 are the high-mass counterparts of the local IR selected quasars, and a similar starburst dust-heating mechanism dominates their FIR emission. However, these z ∼ 6 quasars are much more luminous in the optical than the two local quasars and the $\rm L\_{FIR}-L\_{bol}$ trends derived with the two local samples are getting mixed at high luminosity end. Thus extending the millimeter study to the optically fainter quasar population at z ∼ 6 will help to determine the $\rm L\_{FIR}-L\_{bol}$ correlation for the millimeter bright z ∼ 6 quasars and make a better comparison at low and high redshifts. On the other hand, the average value of the MAMBO non-detections is consistent with the trend defined by the local PG quasars. A link between millimeter detection and the strength of $\rm Ly\alpha$ emission =============================================================================== Omont et al. (1996b) studied a sample of z ∼ 4 optically bright quasars and found that the millimeter detected sources tend to have weaker UV emission lines compared to the average line strength of the millimeter non-detections. Bertoldi et al. (2003a) found a similar trend with MAMBO observations of five z ∼ 6 quasars. In this section, we investigate this effect with the $\rm Ly\alpha$ emission of full sample of z ∼ 6 quasars observed with MAMBO. The rest-frame $\rm Ly\alpha$ equivalent widths ($\rm EW\_{Ly\alpha}$) of five quasars in the SDSS deep strip were presented in Jiang et al. (2007a). The UV spectra of another 20 sources are available in the papers of Fan et al. (2000, 2001a, 2003, 2004, 2006a, 2008, in prep). We estimate $\rm EW\_{Ly\alpha}$ for 18 of them. The spectra have been corrected for Galactic extinction, adopting the extinction curve presented in Savage & Mathis (1979). We then fit the spectra using a method similar to that used in Jiang et al. (2007a), i.e. a power-law ($\rm f\_{\nu}\sim \nu^{-0.5}$) for the UV continuum and gaussians for the $\rm Ly\alpha$ and $\rm NV$ line emission. The sources J1044-0125 and J1048+4637 have broad absorption line features and their equivalent widths are therefore unavailable. We plot the histograms of $\rm EW\_{Ly\alpha}$ for MAMBO detections (solid line) and non-detections (dotted line) in Figure 5. Most of the MAMBO detections distribute in the region of $\rm log\,EW\_{Ly\alpha}(\AA)<1.5$ with a median value of 24 $\rm \AA$, while most non-detections have $\rm log\,EW\_{Ly\alpha}(\AA)$ $\rm >1.5$ with a median value of 68 $\rm \AA$ which is comparable to the average value of ∼ 70 $\rm \AA$ from the large optical quasar samples at lower redshifts (Schneider et al. 1991; Fan et al. 2001b). However, there is also one MAMBO non-detected source, J1621+5155, which shows very weak $\rm Ly\alpha$ emission with $\rm EW\_{Ly\alpha} < 5\AA$. The physical origin of this curious trend is not clear. According to the spectra study with samples of z ∼ 4 quasars presented in Omont et al. (1996b), similar weaker emission is also seen in other UV emission lines, such as the Si IV/O IV] and C IV lines. Though the UV line emission is weak in the MAMBO detected sources, there is no evidence of strong dust reddening in the UV continuum. We can speculate that intrinsically different physical conditions associated with the broad line emission cloud, or a special dust obscuration geometry that affects only the broad line region, may explain this effect. Further tests with other strong broad lines, such as the C IV, and Mg II lines in the UV, the Hα, and Hβ lines in the optical, and the Paschen lines in near-infrared, are required to give a better understand of this trend. In particular, the trend may disappear when tested with the near-infrared lines if it is due to dust absorption. Discussion ========== In this paper, we analyze the FIR properties of the current quasar sample at z ∼ 6. These quasars are mainly optically selected, and represent the luminous end of the quasar population at this early epoch. The X-ray to near-infrared SEDs of these sources are dominated by powerful AGN, just as in optical quasars at lower redshift (Jiang et al. 2006). We divide the sample into MAMBO 250 GHz detections of 10 sources and non-detections of 23 sources. Their FIR emission was compared to that of the low-z quasars, and with the results listed above we will further discuss the FIR dust heating and evolution stage of these z ∼ 6 quasars in this section. The average FIR emission of the 250 GHz non-detections ------------------------------------------------------ The average FIR emission of the MAMBO non-detections is recovered at $\rm 4\sigma$ by stacking the 250 GHz measurements of each source, and the mean SED is similar to the low-z quasar templates of Elvis et al. (1994) and Richards et al. (2006). This suggests that the emission from the outer part of AGN-heated dust torus may dominates the FIR SED in these sources, as was found in the low-z optically bright quasars (Sanders et al. 1989; Haas et al. 2003). However, we cannot rule out contributions from star formation in the quasar hosts. Further surveys of PAH and other star forming features will address the average contribution from star formation to FIR dust heating in these objects. The average FIR luminosity (42.5μm – 122.5μm, rest frame) derived from the stacking flux density is 1.2$\rm \times10^{12}\,L\_{\odot}$. If star formation is not the dominating dust heating source (i.e. contributes < 50%), the average star formation rate in the quasar hosts[4](#fn4) should be less than $\rm 200\,M\_{\odot}\,yr^{-1}$. The FIR emission of the strong 250 GHz detections ------------------------------------------------- Excess FIR emission is seen in seven of the MAMBO detected sources (see Figure 3). Their 250 GHz flux densities are much stronger ($\rm \gtrsim0.5$ dex) than the values expected from the quasar templates when normalized in the rest-frame UV. These sources show FIR-to-radio luminosity ratios within the range defined by typical star forming galaxies (Section 3.2, see also Carilli et al. 2004; Wang et al. 2007). The relation between FIR and AGN bolometric luminosity for these sources follows the trend defined by a sample of low-redshift IR luminous quasars which are hosted by ULIRGs (Section 4). The Spitzer near-infrared photometry by Jiang et al. (2006) detected the AGN-heated hot dust ( ∼ 1000 K) in some of these bright millimeter sources; the optical-to-near infrared SEDs of these sources match the low-z templates very well. The strong FIR emission bump in these sources indicates an additional warm dust component, with a dust temperature of 50 to 60 K. Brightness temperature arguments imply that the warm dust emission region must be extended over a few kpc (Wang et al. 2008a). Thus either a significant fraction of the dust heating is due to star formation in the host galaxy, or the AGN must heat dust over large scales. Intense star formation activity is suggested by other observations of these strong millimeter quasars. Molecular CO emission is detected in the three strongest z ∼ 6 millimeter sources: J1148+5251 (Bertoldi et al. 2003b; Walter et al. 2003, 2004), J0927+2001 (Carilli et al. 2007), and J0840+5624 (Wang et al. 2008b, in prep). These CO detections indicate the presence of $\rm \gtrsim10^{10}\,M\_{\odot}$ of molecular gas in the quasar hosts (Walter et al. 2003; Bertoldi et al. 2003b; Carilli et al. 2007), which can provide the fuel required for star formation rates of up to $\rm >10^{3}\,M\_{\odot}\,yr^{-1}$. Indeed, it is hard to see how the host galaxies can avoid massive star formation with so much molecular gas. The measurement of several CO transitions in J1148+5251 suggests a gas density of about $\rm 10^{4.5}\,cm^{-3}$ and a kinetic temperature of ∼ 100 K (Bertoldi et al. 2003b), which are similar to values in starburst environments found in other quasars and submillimeter galaxies at lower redshifts (Carilli et al. 2002; Solomon & Vanden Bout 2005; Riechers et al. 2006). In addition, recent high resolution observations of [C II] in J1148+5251 (Walter et al. 2008, in prep.; Carilli 2008) show an emission region extended over many kpc, i.e., comparable to that of the CO (3-2) line. Further studies also suggest connections between the CO and FIR emission. The FIR-to-CO luminosity ratios of J1148+5251 and J0927+2001 are about 500 and 650 $\rm L\_{\odot}$ $\rm (K\,km\,s^{-1}\,pc^{2})^{-1}$, respectively, which are consistent with the values found in other high-z CO emitting galaxies (Solomon & Vanden Bout 2005). In Figure 6, we indicate these two CO detected z ∼ 6 quasars (as stars) on the FIR-CO luminosity correlation plot discussed in Riechers et al. (2006). The two points fall a little above but within the observed scatters of the FIR-CO luminosity relation of the typical star forming galaxies at both low and high redshifts. All these facts suggest that massive star formation is an important source of the FIR dust heating. However, it is difficult to constrain what fraction of the FIR dust heating is contributed by star formation in these FIR luminous quasars. According to the FIR-to-CO luminosity correlation in Figure 6, at least $\rm 30-50\%$ of the FIR emission should originate from star formation, giving a corresponding star formation rate of $\rm \gtrsim 1000\,M\_{\odot}\,yr^{-1}$. The evolutionary stage of the z ∼ 6 quasars ------------------------------------------- Observations from optical to near-infrared of these z ∼ 6 quasars indicate that the central SMBHs have masses of $\rm \gtrsim10^{9}\,M\_{\odot}$ (Willott et al. 2003; Jiang et al. 2006). Based on the SMBH-bulge mass relationship (eg. Marconi & Hunt 2003), the final stellar bulge mass should be on order of $\rm 10^{12}\,M\_{\odot}$ for a mature system. The formation of such a massive stellar system may require a star formation rate of $\rm \gtrsim10^{3}\,M\_{\odot}\,yr^{-1}$ on time scales of a few hundred million years as was suggested by some numerical simulations (eg. Li et al. 2007a, b). Thus the average star formation rate of $\rm <200\, M\_{\odot}\,yr^{-1}$ found in the 250 GHz non-detected quasars can be either the case that the major bulge building has already been finished and the SMBH-bulge mass relationship seen at z=0 has already been established, or that the host galaxy is has not yet reached the massive star formation phase, and the growth of the stellar bulge is lagging behind that of the SMBH (see eg. Shields et al. 2006; Ho 2007). Deep and high-resolution imaging at near-IR wavelengths may provide a direct measurement of the stellar population in these z ∼ 6 quasar hosts. For the strong millimeter quasars at z ∼ 6, the FIR excess suggests a star formation rate of $\rm \gtrsim1000\,M\_{\odot}\,yr^{-1}$ from the host galaxies. It is likely that we are witnessing simultaneous building of the stellar bulge and growth of the SMBH in these objects. The case of J1148+5251 is the best documented: Willott et al. (2003) derived an estimate of $\rm 3\times10^9\,M\_{\odot}$ for the black hole mass, while Walter et al. (2004) estimated a dynamical mass of $\rm \sim5\times10^{10}\,M\_{\odot}$. The SMBH-bulge mass ratio is thus clearly more than one order of magnitude larger than the local SMBH-bulge relationship (eg. Marconi & Hunt 2003), as in several other high-z CO-detected QSOs (Shields et al. 2006; Ho 2007). However, the amount of gas detected in CO observations such as in J1148+5251, a few $\rm 10^{10}\,M\_{\odot}$, is far short of the value required to reach the bulge mass given by the local black hole-bulge relationship. To fully understand the evolutionary stage of these strong millimeter quasars at z ∼ 6, further observations are required. First, CO and [C II] searches and excitation studies should be extended to all the z ∼ 6 quasars with strong FIR excesses to determine the host galaxy ISM properties. Second, high resolution ($\rm \lesssim0.3''$) CO, [C II] and dust mapping should be made of all the CO detections. If the FIR-emitting dust and molecular gas are distributed similarly, the FIR emission is likely to be due to a starburst, while more compact FIR emission will argue for AGN dust heating. The resolved CO emission would also provide an estimate of the dynamical mass of the quasar hosts on kpc scales. This is a direct, and perhaps the only, way to test the black hole-bulge relationship at the highest redshifts. These observations can be done with sensitive submillimeter, millimeter and radio telescopes such as the PdBI, CARMA, and SMA, and the coming ALMA and EVLA. Conclusion ========== We study the SEDs of 33 quasars at z ∼ 6 from FIR to radio wavelengths. We conclude that, when averaged with the whole sample and the 250 GHz undetected sources, no significant difference is seen between the mean FIR-to-radio SEDs and the templates of low-redshift optically selected quasars. In particular, we extrapolate the FIR emission of the templates with the typical quasar FIR-to-millimeter spectrum (i.e. $\rm f\sim\nu^\alpha,\,\alpha\geq2$) from Sanders et al. (1989). This extrapolation is consistent with the average FIR emission of the 250 GHz undetected z ∼ 6 quasars very well (see Section 3.1 and Figure 2). This fact suggests that these sources have a similar AGN dominated dust heating mechanism. The average star formation rate from the host galaxies of the quasars undetected by MAMBO is estimated to be less than a few hundred $\rm M\_{\odot}\,yr^{-1}$. We detect a strong FIR excess in seven of the ten z ∼ 6 quasars detected in the millimeter band. These FIR luminous sources are likely to be the high-mass counterparts of local IR luminous quasars which are in transition between the starburst phase and the mature quasar phase. Star formation at rates of $\rm >10^{3}\,M\_{\odot}\,yr^{-1}$ may be the dominant mechanism to heat the dust that gives rise to the FIR emission. The millimeter detected sources also show weaker Lya emission compared to that of the non-detections, but the origin of this trend is not clear yet. Further observations of these FIR luminous quasars, especially spatially resolved line and dust studies, are required to further address questions of star formation and FIR dust heating in these extreme objects. The current sample of z ∼ 6 quasars is optically selected. There may be other quasar populations at z$\gtrsim$6 which are still buried in their starburst environment and are obscured in the optical. Sensitive IR and millimeter facilities, such as Spitzer, ALMA, and Herschel, may discover such objects and provide a more complete view of galaxy and SMBH evolution in the early universe. We acknowledge support from the Max-Planck Society and the Alexander von Humboldt Foundation through the Max-Planck-Forschungspreis 2005. Michael, A. Strauss acknowledges support of the National Science Foundation grant AST-0707266. X. Fan acknowledge support from a David and Lucile Packard Fellow in Science and Engineering. We thank James J. Condon for comments and suggestions. IRAM is funded by the Centre National de la Recherche Scientifique (France), the Max-Planck Gesellschaft (Germany), and the Instituto Geografico Nacional (Spain). The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated by Associated Universities, Inc. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. 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Y., Mao, S. et al. 2002, AJ, 124, 18 Zylka, R. 1998, MOPSI Users Manual, (IRAM: Grenoble) lcccccc Name & redshift &$\rm m\_{1450}$ & $\rm M\_{1450}$ & $\rm f\_{250GHz}$ & $\rm f\_{1.4GHz}$ & Referencea & & & & mJy & $\rm \mu$Jy & (1) & (2) & (3) & (4) & (5) & (6) & (7) J000552.34 − 000655.8 & 5.85&20.23 &-26.47 & 0.36 ± 0.48 & 40 ± 130& 1, \* ,2 J003311.40 − 012524.9 & 6.13&21.78 &-25.00 &1.13 ± 0.36 &-27 ± 19 & 3,3, \* J020332.39+001229.3 & 5.86&20.97 &-25.73 & 1.52 ± 0.67 & 195 ± 22 & 4, \* , \* J030331.40 − 001912.9 & 6.07&21.33 &-25.43 & 0.23 ± 0.51 & -85 ± 62 & 4, \* , \* J035349.76+010405.4 & 6.05&20.21 &-26.55 & 1.20 ± 0.46 &17 ± 19 & 4, \* , \* J081827.40+172251.8 & 6.00&19.34 &-27.40 & 1.19 ± 0.38 &123 ± 12 & 5, \* ,2 J084119.52+290504.5 & 5.96&19.61 &-27.12 & 1.00 ± 0.43 &43 ± 27 & 8, \* , \* J084229.23+121848.2 & 6.08&19.58 &-27.18 & 0.11 ± 0.55 &-8 ± 19 & 9, \* , \* J092721.82+200123.7 & 5.77&19.87 &-26.81 & 4.98 ± 0.75 &50 ± 11 & 5,2, \* J104433.04 − 012502.2 & 5.78&19.21 &-27.46 & 1.82 ± 0.43 & -15 ± 24 & 10, \* ,7 J142516.30+325409.0 & 5.85&20.62 &-26.08 & 2.27 ± 0.51 &20 ± 20 & 14, \* , \* J142738.59+331242.0 & 6.12&20.33 &-26.44 & 0.39 ± 0.66 &1730 ± 131 & 15, \* ,15 J160253.98+422824.9 & 6.07&19.86 &-26.90 & 1.41 ± 0.54 &60 ± 15 & 1, \* ,2 J162100.70+515544.8 & 5.71&19.89 &-26.77 & 0.30 ± 0.55 &-12 ± 21 & 9, \* , \* J163033.90+401209.6 & 6.05&20.64 &-26.12 & 0.80 ± 0.60 &14 ± 15 & 11,12, \* J164121.64+375520.5 & 6.04&21.30 &-25.45 & 0.08 ± 0.46 &-30 ± 32 & 3,3, \* J205406.42 − 000514.8 & 6.07&20.67 &-26.09 & 2.38 ± 0.53 &17 ± 23 & 4, \* , \* J231546.36 − 002357.5 & 6.12&21.31 &-25.46 & 0.28 ± 0.60 &31 ± 16 & 4, \* , \* J232908.28 − 030158.8 & 6.43&21.65 &-25.20 & 0.01 ± 0.50 &14 ± 22 & 3,3, \* Note – The detections are marked as boldface. aThe three references are for the optical, 250 GHz, and 1.4 GHz data, respectively. The asterisks denote new data reported in this paper. References for Table 1 and 2 – (1) Fan et al. 2004; (2) Wang et al. 2007; (3) Willott et al. 2007; (4) Jiang et al. 2008; (5) Fan et al. 2006a; (6) Fan et al. 2001a; (7) Petric et al. 2003; (8) Goto 2006; (9) Fan et al. 2008, in prep. (10) Fan et al. 2000; (11) Fan et al 2003; (12) Bertoldi et al. 2003a; (13) Carilli et al. 2004; (14) Cool et al. 2006; (15) McGreer et al. 2006. lcccccc Name & redshift &$\rm m\_{1450}$ & $\rm M\_{1450}$ & $\rm f\_{250GHz}$ & $\rm f\_{1.4GHz}$ & Referencea & & & & mJy & $\rm \mu$Jy & (1) & (2) & (3) & (4) & (5) & (6) & (7) J000239.39+255034.8 & 5.80&19.02 &-27.67 & 0.20 ± 0.88 & 89 ± 14 & 1,2,2 J083643.85+005453.3 & 5.81&18.81 &-27.88 &-0.39 ± 0.96 &1740 ± 40 & 6,7,7 J084035.09+562419.9 & 5.85&20.04 &-26.66 & 3.20 ± 0.64 &12 ± 9 & 5,2,2 J103027.10+052455.0 & 6.31&19.66 &-27.16 &-1.15 ± 1.13 & -3 ± 20 & 6,2,2 J104845.05+463718.3 & 6.20&19.25 &-27.55 & 3.00 ± 0.40 &7 ± 13 & 11,12,2 J113717.73+354956.9 & 6.01&19.63 &-27.12 & 0.10 ± 1.13 &9 ± 17 & 5,2,2 J114816.64+525150.2 & 6.42&19.03 &-27.82 & 5.00 ± 0.60 &55 ± 12 & 11,12,13 J125051.93+313021.9 & 6.13&19.64 &-27.14 & 0.07 ± 0.90 &37 ± 21 & 5,2,2 J130608.26+035626.3 & 6.02&19.55 &-27.19 &-1.05 ± 1.04 &14 ± 21 & 6,7,7 J133550.81+353315.8 & 5.95&19.89 &-26.84 & 2.34 ± 0.50 &35 ± 10 & 5,2,2 J141111.29+121737.4 & 5.93&19.97 &-26.75 & 1.00 ± 0.62 &61 ± 16 & 1,2,2 J143611.74+500706.9 & 5.83&20.16 &-26.54 &-0.21 ± 1.14 & 6 ± 16 & 5,2,2 J150941.78 − 174926.8 & 6.12&19.82 &-26.95 & 1.00 ± 0.46 &– & 3,3,- J162331.81+311200.5 & 6.25&20.13 &-26.67 & 0.17 ± 0.80 &24 ± 31 & 1,2,2 Note – The detections are marked as boldface. aThe three references are for the optical, 250 GHz, and 1.4 GHz data, respectively, and the corresponding literatures are listed at the end of Table 1. lccccccc Group & $\rm Number^{a}$ & $\rm <f\_{250GHz}>$ & $\rm <L\_{FIR}>$& $\rm Number^{b}$ & $\rm <f\_{1.4GHz}>$&$\rm <L\_{1.4GHz}>^{c}$ & q & & mJy & $\rm 10^{12}L\_{\odot}$ & & $\rm \mu$Jy &$\rm L\_{\odot}\,Hz^{-1}$ & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) Whole sample& 33 & 1.26 ± 0.10 & 2.9 ± 0.2 & 32 & 46 ± 3&0.031 ± 0.002 &1.41 ± 0.04 (radio quiet)& 30 & 1.29 ± 0.10 &3.0 ± 0.2 & 29 & 32 ± 3 & 0.022 ± 0.002 &1.57 ± 0.05 MAMBO detections& 10&2.73 ± 0.06& 6.4 ± 0.1 & 10 & 37 ± 4 &0.024 ± 0.003 & 1.84 ± 0.05 Non-detections&23& 0.52 ± 0.13& 1.2 ± 0.3 & 22 & 56 ± 4 &0.038 ± 0.003 & 0.94 ± 0.11 (radio quiet) &20& 0.51 ± 0.13& 1.2 ± 0.3 & 19 & 27 ± 4 &0.018 ± 0.003 & 1.24 ± 0.13 a Number of sources observed at 250 GHz. b Number of sources observed at 1.4 GHz. c A radio spectral index of -0.75 (Condon 1992) is adopted here to calculate the rest frame 1.4 GHz lunimosity. cccc Name & $\rm \frac{L\_{bol}}{10^{13}L\_{\odot}}$&$\rm \frac{\dot M}{M\_{\odot}\,yr^{-1}}$ & $\rm \frac{L\_{FIR}}{10^{12}L\_{\odot}}$ (1) & (2) & (3) & (4) J003311.40-012524.9 & 1.4&9.4 & 2.6 ± 0.8 J081827.40+172251.8 & 12.8&86.3 & 2.8 ± 0.9 J084035.09+562419.9 & 6.5&43.6 & 7.6 ± 1.5 J092721.82+200123.7 & 7.4&50.0 & 12.2 ± 2.8 J104433.04-012502.2 & 13.5&91.1 & 5.3 ± 0.8 J104845.05+463718.3 & 14.6&98.4 & 6.9 ± 0.9 J114816.64+525150.2 & 18.8&126.8 & 13.4 ± 3.4 J133550.81+353315.8 & 7.6&51.4 & 5.5 ± 1.2 J142516.30+325409.0 & 3.8&25.6 & 5.4 ± 1.2 J205406.42-000514.8 & 3.8&25.8 & 6.8 ± 1.1 Note – $\rm {\dot M}$ is the black hole accretion rate derived with equation (7) in Section 4. --- 1. derived from the absolute magnitude of $\rm M\_{1450}=-26.4$ presented in McGreen et al. (2006)[↩](#fnref1) 2. The $\rm 4400\AA$ and 5 GHz flux densities are derived from the 1450$\rm \AA$ magnitude and the observed 1.4 GHz flux density, assuming power law spectra indices of -0.5 in the optical and -0.75 in the radio, respectively.[↩](#fnref2) 3. A typical redshift value of z=6 is adopted here[↩](#fnref3) 4. The star formation rate is estimated with the empirical relationship from Kennicutt (1998), assuming a standard Salpeter initial mass function, i.e. $\rm SFR\sim 4.5L\_{IR}\,M\_{\odot}yr^{-1}$, where $\rm L\_{IR}$ is the infrared luminosity (8-1000 μm) in unit of $\rm 10^{44}\,erg\,s^{-1}$, and is $\rm\sim 1.5L\_{FIR}$ for warm dust emission.[↩](#fnref4)	arxiv_0000579
Statistics of Satellite Galaxies Around Milky Way-Like Hosts ============================================================ We calculate the probability that a Milky-Way-like halo in the standard cosmological model has the observed number of Magellanic Clouds (MCs). The statistics of the number of MCs in the ΛCDM model are in good agreement with observations of a large sample of SDSS galaxies. Under the sub-halo abundance matching assumption of a relationship with small scatter between galaxy r-band luminosities and halo internal velocities ${v\_{\rm max}}$, we make detailed comparisons to similar measurements using SDSS DR7 data by. Models and observational data give very similar probabilities for having zero, one, and two MC-like satellites. In both cases, Milky Way-luminosity hosts have just a ∼ 10% chance of hosting two satellites similar to the Magellanic Clouds. In addition, we present a prediction for the probability for a host galaxy to have Nsats satellite galaxies as a function of the magnitudes of both the host and satellite. This probability and its scaling with host properties is significantly different from that of mass-selected objects because of scatter in the mass-luminosity relation and because of variations in the star formation efficiency with halo mass. Introduction ============ Understanding the satellite population of our own Milky Way galaxy is one of the outstanding problems in galaxy evolution studies today. Because these objects have weak gravitational potential wells, it is quite likely that different physical processes played a driving role in determining how such galaxies formed and evolved than for brighter, more massive objects. One manifestation of this is the well-known “missing satellites problem,” which refers to the fact that the number of satellite galaxies predicted from simulations appears to be much higher than the number actually observed around the Milky Way. One common way to address this problem has been to run simulations of individual objects using extremely high mass resolution. To date, roughly 10 of these ultra-high resolution N-body simulations have been run, which typically resolve individual Milky Way mass halos with upwards of 109 particles. The Magellanic Clouds (MCs) are a pair of well studied satellites of the Milky Way (MW) in the Southern Hemisphere with magnitudes MV = − 18.5 and -17.1 for the Large and Small Magellanic Clouds (LMC and SMC). While their large angular size makes detailed mass measurements somewhat difficult, observations constrain their maximum circular velocities to be around ${v\_{\rm max}}\sim 60$ km s− 1. Unlike the missing satellite problem, one common feature of these ultra-high resolution simulations is that they typically under-predict the number of massive satellites compared to the MCs in the MW. Indeed, most of these high resolution halos contain no objects of similar mass to either of the two MCs. While this should hardly be taken as a serious problem since the handful of high resolution simulated halos does not constitute a statistical sample, it is important to understand how typical or atypical the Milky Way is, and whether or not not there is an “extra satellites” problem at high satellite masses within the modern Lambda Cold Dark Matter (ΛCDM) paradigm. Because the MCs are relatively nearby and have been studied with high-precision instruments such as HST for some time, detailed 3-dimensional measurements of their velocities have been made. Some of these studies have indicated that the Magellanic Clouds may be on their first orbit around the Milky Way ; this may make such an extra satellite problem less worrisome if the presence of the Magellanic Clouds is a transient event. Regardless of the dynamical state of the Clouds, it is an important test of galaxy formation to understand the likelihood for MW-like systems to host massive satellite galaxies. Recent developments have afforded us the possibility to address this problem in more detail with both observations and theoretical models. From the observational side, wide area surveys such as the Sloan Digital Sky Survey have given us the ability to probe galaxy content for a large number of Milky Way-magnitude objects. The main theoretical effort has been in populating dark matter halos with galaxies using semi-analytic methods and hydrodynamic simulations. used a number of toy models to add galaxy properties to dark matter halos generated using a combination of Press-Schechter theory with semi-analytic models for tracking subhalo orbits. They found that it was very difficult to model objects as bright as Magellanic clouds without allowing for an extremely high star formation efficiency. This finding extended the underprediction of high-mass subhalos to an underprediction of luminous satellites, implying that the MW-MCs system is unusual. Similarly, explored a range of feedback models to add galaxies to some of the high resolution Aquarius halos. It was again difficult to readily reproduce halos with luminosities as bright as the MCs. Having looked in detail at a handful of simulated objects, however, this work indicates that there may be significant halo-to-halo scatter in the number of such massive objects. The idea of intrinsic scatter in the subhalo population was expanded on in. While this work did not concentrate specifically on MC-like subhalos, they did consider the range in number of subhalos with vmax, sub/vmax, host > 0.1. This work showed an extremely large variation (20-60) in the number of such massive subhalos a galaxy-sized halos would host. In contrast to the semi-analytic modeling just discussed, used hydrodynamic simulations to model the luminosity functions for the satellites around MW-like host halos. Their simulation identified 9 MW-like central galaxies and found that they, on average, have 1.6 satellites brighter than MV = − 16 and a third of them had satellites with luminosities comparable to the LMC. The recent Millennium-II and Bolshoi simulations have, for the first time, allowed us to probe cosmological volumes to understand the ΛCDM predictions for the satellite populations of MW-like halos. The properties of these simulations are summarized in Table 1. (hereafter BK10) quantified the likelihood for $10^{12} {\hbox{${\rm M}\_\odot$}}$ halos to host massive satellite galaxies in the Millennium-II simulation, finding that subhalos similar to the MCs are quite rare. In this paper, we expand on this work by making similar measurements for the Bolshoi simulation, which used WMAP7 cosmological parameters, and using an abundance matching technique to make detailed comparisons between the Bolshoi predictions and the measurements from. Our goal is to understand just how well ΛCDM reproduces the statistical properties of bright satellites. Note that this is the reverse question from the one that was asked in a companion paper,. That work assumed a satellite population and asked what the implications were for the properties of the host halo, including its mass. Here, we assume a host halo mass and ask about the implications for the subhalo population. There is no reason for both questions to give the same answer: while showed that a halo which hosts two MC-like satellites most likely has a mass near $1.2 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$, there is no reason to assume that a typical $1.2 \times 10^{12}{\hbox{${\rm M}\_\odot$}}$ halo will have the MCs as satellites. We begin by giving an overview of the Bolshoi simulation in §[sec:sims], and then investigate the properties of massive dark matter satellites around dark matter host halos in §[sec:halos], focusing on the mass ranges for hosts and satellites that are most relevant to the MW system. The analysis here is similar to that of BK10. In §[sec:luminosities], we assign galaxy luminosities to our suite of dark matter halos and extend the results for a sample with similar selection cuts as for observations. In this way, we are able to make detailed comparisons to the observational work of concerning the satellite population around MW-magnitude galaxies. Section [sec:comparisons] gives the results of this analysis — see especially Figure [fig:nsatsobssim]. Finally, in §[sec:generalproperties], we expand this study to include the satellite population of a more general distribution of hosts, and §[sec:conclusions] summarizes our conclusions. Throughout this paper, we adopt the convention h = 0.7 (the value that was used in the Bolshoi simulation) when reporting values from either simulations or observations. Simulations =========== We use the dark matter halos identified in the Bolshoi simulation. This simulation modeled a 250 ${h^{-1}\rm{Mpc}}$ comoving box using cosmological parameters similar to those derived by WMAP7 : Ωm = 0.27, ΩΛ = 0.73, σ8 = 0.82, n = 0.95, and h = 0.7. The simulation volume contains 20483 particles, each with a mass of $1.35 \times 10^8~{h^{-1}}{\hbox{${\rm M}\_\odot$}}$ and was run using the ART code. 180 snapshots from the simulation were saved and analyzed. One of the unique aspects of this simulation is the high level of spatial resolution employed, allowing objects to be resolved down to a physical scale of 1 ${h^{-1}\rm{kpc}}$. This gives us excellent ability to track halos as they merge with and are disrupted by larger objects, allowing us to track them even as they pass near the core of the host halo. A summary of these simulation parameters is presented in Table [table:simulations]. Because we discuss our work in the context of the BK10 results, we also present the same parameters for the Millennium II simulation, on which the BK10 results were based. Halos and subhalos were identified using the BDM algorithm. The algorithm identifies maxima in the density field and examines the neighboring region to identify bound particles. In this way, it treats both halos and subhalos identically. Subhalos are just identified as objects living within the virial radius of a larger objects. Because of the high level of mass and spatial resolution, BDM results in a halo catalog that is complete down to a maximum circular velocity ${v\_{\rm max}}= 50$ km s− 1, where ${v\_{\rm max}}= {\rm max}\left(\sqrt{{GM(<r)}/{r}}\right). $ This corresponds to a virial mass of roughly $10^{10}{h^{-1}}{\hbox{${\rm M}\_\odot$}}$. When BDM halos are identified, the ids of their 50 most bound particles are also stored to assist in producing merger trees. As discussed in Section [sec:sham], in order to assign galaxy luminosities to dark matter halos, we need to track the histories of dark matter substructures. This is done using merger trees created from the 180 snapshots of the Bolshoi simulation. The detailed algorithm for creating the merger trees is described in. Briefly, the algorithm works by first linking halos across time steps by tracking the 50 most bound particles of each halo. Some halos will not have any of their 50 most bound particles identified at a later timestep (e.g., very massive objects in which the 50 most bound particles change rapidly through stochastic processes), while some will have their particles distributed to multiple halos. The algorithm corrects for this by running a simple N-body calculation on the locations and masses of all halos in the simulation to predict where each halo should wind up at the next time step. Using this information, it is possible to link halos across multiple time steps with very high accuracy. \| \| Bolshoi \| MSII \| \| --- \| --- \| --- \| \| Box Size [Mpc] \| 357 \| 143 \| \| NP \| 20483 \| 21603 \| \| $M\_P [{\hbox{${\rm M}\_\odot$}}]$ \| 1.9 × 108 \| 9.8 × 106 \| \| h \| 0.7 \| 0.73 \| \| ΩM \| 0.27 \| 0.25 \| \| σ8 \| 0.82 \| 0.9 \| \| Force Resolution [kpc] \| 1.4 (proper) \| 1.4 (Plummer-equivalent softening) \| [table:simulations] Satellite Statistics for Dark Matter Subhalos ============================================= We begin by investigating the properties of dark matter satellites around dark matter hosts in the Bolshoi simulation, focusing on the mass range for hosts and satellites that is relevant to the MW system. In §[sec:massdep], we consider trends with host halo mass, and in §[sec:env] we investigate trends with halo environment. We then consider the distribution of the satellite number (§[sec:pois]). Finally, we extend this analysis to include a more observationally relevant selection based on galaxy luminosities in §[sec:luminosities]. Because the mass resolution of the Bolshoi simulation creates a halo catalog complete down to 50 km s− 1, roughly equivalent to the lower bound of the mass of the MCs, we begin by measuring the probability distribution for halos hosting Nsubs subhalos with ${v\_{\rm max}}$ larger than a given value. This measurement is made by identifying the virial radius of all distinct (non-subhalo) MW mass halos and counting the number of objects internal to their virial radii. We take the virial mass of the Milky Way to be $\log({\hbox{$M\_{\rm vir}$}}/{\hbox{${\rm M}\_\odot$}}) = 12.08 \pm 0.12$ (${\hbox{$M\_{\rm vir}$}}= 1.2 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$), the mass measured in, which is consistent with most results in the literature. This results in 36,000 MW analogs found in the Bolshoi volume. We present the resulting probability distribution of satellite counts as the colored points in the left panel of Figure [fig:vmaxcomp]. Here, the different colors represent different thresholds for satellite ${v\_{\rm max}}$: red, green, and blue represent all satellites with ${v\_{\rm max}}> $ 50, 60, and 70 km s− 1. Error bars were calculated using the bootstrap method and represent 95% confidence intervals. From Figure [fig:vmaxcomp], we can immediately see that the likelihood of hosting multiple satellites is somewhat low. For satellites with ${v\_{\rm max}}$ greater than 50 km s− 1, roughly 30% of simulated hosts contain one or more subhalos and just 8% have two or more. These numbers are in excellent agreement with the work of BK10, who performed a similar analysis on the Millennium II simulation, which was run using a different N-body code in a WMAP1 cosmology and with a very different subhalo identification algorithm. The probability of hosting multiple satellites drops precipitously with increasing satellite mass; e.g., the probability of a halo hosting two subhalos larger than ${v\_{\rm max}}= 70$km s− 1 is just 1%. For comparison, the black points in Figure [fig:vmaxcomp] represent measurements from, who measured the probability distribution for finding MC-luminosity satellites within 250 kpc around isolated MW-luminosity galaxies (the mean virial radius of our sample). We only plot these measurements out to Nsubs = 3 because, for a 250 kpc aperture, uncertainties from their background subtraction become extremely large for higher (lower probability) values of Nsubs. Here, the agreement for satellites larger than 50 km s− 1is striking, indicating that CDM can reproduce the statistics of MCs. We must, however, be careful about over-interpreting this result. In particular, their selection criteria are very different from our mass selection. We will return to this in §4.4 where we make a comparison directly to similarly selected samples. By comparison with the Bolshoi simulation, the existence of two satellites with ${v\_{\rm max}}$ larger than 50km s− 1 makes the MW almost a 2σ outlier. Understanding whether anything other than random chance is responsible for putting the MW in this slightly rare regime motivates further investigation into which other properties may correlate with satellite abundance. In particular, we investigate the correlation of the number of subhalos with host mass and environment. Mass dependence --------------- The correlation between subhalo abundance and host mass has been well studied. These studies have all found that, when scaled in units of μ = Msub/Mhost, larger halos contain more subhalos above a given μ than do smaller ones. This is the expected result from the hierarchical merging picture of CDM, in which larger objects grow through the merging of smaller objects and therefore form later. This later formation has a two-fold impact: it reduces the amount of time available to tidally strip a subhalo, and lowers the host concentration, further reducing the ability of a host to tidally strip its subhalos. Motivated by this, we present the abundance of LMC and SMC-mass subhalos for the 27,000 Bolshoi halos with ${\hbox{$M\_{\rm vir}$}}= 2.6\pm 0.9 \times 10^{12}{\hbox{${\rm M}\_\odot$}}$. The results can be seen in Figure [fig:vmaxcomp]. Here, the left panel shows the results for selecting hosts with ${\hbox{$M\_{\rm vir}$}}= 0.8-1.7\times 10^{12}{\hbox{${\rm M}\_\odot$}}$, while the right panel shows a mass range ${\hbox{$M\_{\rm vir}$}}= 1.7-3.4\times 10^{12}{\hbox{${\rm M}\_\odot$}}$. [t] [fig:vmaxcomp] For the more massive ${\hbox{$M\_{\rm vir}$}}\sim 2.6 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$ halos, the cumulative probability of hosting two or more 50 km s− 1 halos increases by a factor of 3 compared to the $1.2\times 10^{12} {\hbox{${\rm M}\_\odot$}}$ objects (from 8% to 26%), while jumping by a factor of 6 for 70km s− 1 subhalos (from 1% to nearly 6%). We emphasize that this is not a contradiction with the results of, which found that a halo with exactly two MC-like subhalos was likely to have mass $1.2\times 10^{12} {\hbox{${\rm M}\_\odot$}}$. While the fraction of more massive halos hosting two MC-like subhalos is larger, the steep slope of the mass function means that there are many more lower-mass halos. Additionally, aside from just considering ${v\_{\rm max}}$, selected halos with satellites like the MCs in terms of location and three-dimensional velocity, properties not considered in the present analysis. Again, these results are in excellent agreement with the work of BK10. While such an agreement is generally expected from pure collisionless N-body simulations, such agreement is not as trivial as it may appear. While the present work uses results from the Bolshoi simulation, run using the adaptive refinement ART code and the BDM halo finder, BK10 used the Millennium-II simulation run with the TreePM code Gadget-2 with substructures identified using the Subfind algorithm. Additionally, in both cases, we are concerned with subhalos close to the mass-limit for the simulations. Thus, the close agreement between these two works suggests that numerical effects in the simulations are well understood at this level. Environmental dependence ------------------------ Beyond simple host mass dependence, the size of the cosmological volume modeled by the Bolshoi simulation allows us to perform further detailed studies on properties impacting the satellite abundance distribution. Here, we quantify the environmental dependence of the subhalo population by splitting the sample on local density, defined by h,r = h,r - \|h \|h, where ρh, r is the Eulerian density of dark matter contained in halos larger than ${\hbox{$M\_{\rm vir}$}}= 2\times 10^{11}{\hbox{${\rm M}\_\odot$}}$ (roughly 1000 particles in Bolshoi) defined in a sphere of size r. In the present analysis, we take $r = 1.0 {h^{-1}}{\rm{Mpc}}$; we have performed similar analysis on a range of scales and find this to be the range where the galaxy halo distribution is the most sensitive to the presence of massive satellites. We compare the distribution P(Nsubs) for halos in the top and bottom quartile distributions to the mean relation in Figure [fig:nsatsdens]. There is a clear systematic trend: halos living in overdense regions (red points) are more likely to host massive subhalos than those in underdense regions (blue points). This is likely due to the fact that halos in denser environments live in more biased regions, where smaller density perturbations are amplified, making it easier for smaller halos to form and accrete onto intermediate mass hosts. This makes the deviations from the mean relation in Figure [fig:nsatsdens] a potentially observable manifestation of assembly bias, which posits that the properties of a given halo may be determined by properties other than mass, such as the halo environment. [fig:nsatsdens] Selecting for objects in a dense environment increases by almost 25% the probability of having Nsubs = 2 or more, from 9% to just over 11%; this effect becomes stronger for larger values of Nsubs. The Milky Way is somewhat overdense on this scale, having a massive companion, M31, as well as a number of surrounding dwarf galaxies. Recent estimates put the total mass of the local group at roughly $5\times 10^{12}{\hbox{${\rm M}\_\odot$}}$, solidly in a dense environment on the ∼ 1 Mpc scale. Even if we ignore the contribution to the local density from all neighboring objects except M31, the MW lies close to the top quartile of our measured local density distribution in Bolshoi on a ∼ 1 Mpc scale. We can begin to directly quantify the impact of such a massive neighbor on the subhalo population by identifying MW-mass halos which have M31-like neighbors with ${\hbox{$M\_{\rm vir}$}}= (1-3)\times 10^{12}{\hbox{${\rm M}\_\odot$}}$ at a distance of 700-900 kpc. These systems are represented by the green circles in Figure [fig:nsatsdens], which have a Nsubs = 2 probability enhanced by 2σ relative to the full sample. Thus, the local environment around the MW, which is dominated by the presence of M31, boosts the chance of seeing the Magellanic Clouds in the Milky Way. Probability distributions ------------------------- As an extension of the trends shown in Figure [fig:vmaxcomp], it is useful to generalize these results to arbitrary host and subhalo masses by fitting the probability distribution. A Poisson distribution seems to be the most natural assumption, and has been found previously in the literature. Deviations from a Poisson distribution can be quantified by the calculation of the second moment, α2, where 2 = N(N-1)N 2. For a Poisson distribution, α2 = 1. While studies have found some evidence for a deviation from Poisson, with α2 > 1, the distribution has been more fully quantified by BK10. This work found that the distribution could be much better modeled using a negative binomial distribution, P(N\|r,p) = (N+r) (r)(N+1) pr (1-p)N [eq:negativebinomial] where p = 1 1 + (22-1)N, r = 1 22 - 1, [eq:pr] although they note that a Poisson distribution is a reasonable fit when ⟨N⟩ is low, such as the case of the MCs. We plot fits to both a Poisson and a negative binomial distribution to our measurements as the dotted and dashed lines in Figure [fig:vmaxcomp]. We again see that, because ⟨N⟩ < 1, the Poisson distribution provides a reasonable fit, while the negative binomial distribution fit is excellent (at the expense of adding an additional parameter).[1](#fn1) Our results confirm those of BK10, in that the distribution becomes significantly more non-Poisson for larger values of ⟨N⟩. As a final comparison of the Bolshoi halos to previous work in the literature, we plot in Figure [fig:distributionfits] the trends of ⟨N⟩ (top panel, similar information to Figure 15 in ) and α2 (bottom panel) with mass ratio vacc, sub/vmax, host for a range of host masses. Here, vacc, sub is the maximum circular velocity, vmax, that the subhalo had at the epoch of accretion. As seen in the top panel, the value for ⟨N⟩ scales almost self-similarly, but with a slight increase in overall normalization as has been previously noted in the literature, indicating that more massive halos contain somewhat more subhalos, proportionally, than lower mass halos. [fig:distributionfits] More interesting is the trend in α2 with mass, shown in the lower panel of Figure [fig:distributionfits]. Here we see evidence that, for subhalos with $v\_{acc,sub}/v\_{max,host} {\lower.5ex\hbox{{$\; \buildrel > \over \sim \;$}}}0.25$, self-similarity breaks down much more significantly, where lower-mass hosts tend to have systematically higher values of α2, which implies increased scatter beyond Poisson statistics. There are a number of possible explanations for this trend; it may be due to the fact that lower mass objects exist in a wider range of environments than higher mass halos. As shown earlier in Figure [fig:nsatsdens], the immediate environment surrounding a halo can have a significant impact on its subhalo population. High-mass halos always form in regions of high bias, meaning that the environmental impact on a high mass object’s subhalo population is minimized. Low-mass objects are able to form in both biased and unbiased regions and in regions near to higher mass halos, creating a larger scatter in their subhalo populations. While not shown, we have made analogous plots to Figure [fig:nsatsdens] for hosts of higher mass and confirm that these objects typically display weaker environmental dependence in their subhalo populations. It is also interesting to note that previous studies, in particular and BK10, have not seen such a trend. There are a number of possible explanations for this, which include the improved $1 {h^{-1}\rm{kpc}}$ force resolution and $(250 {h^{-1}\rm{Mpc}})^3$ volume of the Bolshoi simulation, which allows for both better statistics and more accurate tracking of subhalos as they orbit around their hosts. The increased volume of Bolshoi is particularly important if the trend is driven by halo environment, since a larger volume is necessary to probe a full range of environments. Hints of this trend, however, can be noticed in Figure 9 of BK10. While not shown, nearly identical trends in ⟨N⟩ and α2 persist if we consider the mass ratio vmax, sub/vmax, host, using the maximum circular velocity of the subhalo at z = 0 instead of at the time of accretion. To summarize, in this section, we have shown that satellites comparable in size to the LMC and SMC are relatively rare in MW mass objects, occurring in roughly 9% of the potential hosts. Their abundance is dependent on a number of host properties, such as the host mass and environment. The latter can provide a boost at the 25% level. The proximity of M31 puts the Milky Way in a higher density environment, which increases the likelihood of an LMC/SMC pair to 12%. Finally, it was shown that for massive subhalos, the scatter in the expected number of subhalos systematically decreases with increasing host mass. This may be related to the environmental variations previously observed; lower mass halos are able to form in a wider range of environments than higher mass objects. In the next section, we turn to the task of comparing our simulation results with observations from the SDSS, which requires the adoption of an algorithm for adding magnitudes to simulated galaxies. Satellite Statistics for Galaxies ================================= We turn next to a detailed comparison of these results with observational measurements. The classical satellite galaxies around the MW have been well-studied, with the SMC and LMC being the only satellites brighter than MV = − 16. Indeed, the next brightest object, Sagittarius, with MV = − 13.1, is roughly 3 magnitudes dimmer than the SMC. Motivated by this, we search for the distribution of the number of satellites of MV ≤ − 16 in other galaxies, both simulated and real. In order to distinguish the results in this section from the previous sections, we adopt the formalism where Nsats refers to the number of satellite galaxies around a host, identified based on their luminosities. This contrasts with mass-selected dark matter subhalos, which we characterized in the previous sections using the nomenclature Nsubs. The Observational Sample ------------------------ Our recent companion paper studied the population of such LMC and SMC-luminosity objects in the SDSS. Here, MW analogs were identified in the SDSS main sample by locating objects with absolute magnitudes Mr = − 21.2 ± 0.2 with no brighter companions within a line-of-sight cylinder of radius 500 kpc and length 28 Mpc (corresponding to a redshift of ± 1000 km s− 1).[2](#fn2) The abundance of satellite galaxies was then measured by calculating the likelihood of having Nsats neighbors in the photometric sample with apparent magnitudes 2-4 dimmer in a fixed 150 kpc aperture. While we discuss details of their method below, our goal here is to compare our simulated results to their observational constraints. Modeling Galaxy Luminosities ---------------------------- Because internal kinematics are difficult to measure observationally for dwarfs outside the Local Group, the observations of L10 cannot be directly compared to the results from the previous section, which determined abundances based on the vmax of the satellites. Instead, we employ a SubHalo Abundance Matching (SHAM) algorithm to our simulation to map magnitudes onto dark matter halos. Here, luminosities are assigned by matching the number density of observed objects brighter than a given luminosity with the number density of simulated objects greater than a given ${v\_{\rm max}}$. For subhalos, ${v\_{\rm max}}$ at the time of accretion is used instead of the present ${v\_{\rm max}}(z=0)$. We also impose a log-normal scatter between halo mass and luminosity. In our base model, we assume a scatter of 0.16 dex, consistent with constraints on larger group and cluster mass objects. This algorithm has been shown to be extremely successful in matching several statistical properties of the galaxy distribution, including the luminosity dependence and redshift scaling of the two-point galaxy correlation function for galaxies brighter than Mr < − 19, the galaxy-mass correlation function, and galaxy three point statistics. have shown that a similar procedure can simultaneously match the luminosity function and the Tully-Fisher relation locally. Here, we match halos directly to the binned r-band luminosity function in the local Universe as measured by for galaxies with Mr < − 12.8. While there is some uncertainty in these measurements depending on the detailed surface brightness correction, the objects we are interested in here are bright enough that this effect is unimportant. This luminosity function also reproduces the more recent number densities measured in SDSS DR7 down to Mr ∼ − 17. The results of this procedure are shown in Figure [fig:mrvpeak]. Here, the red line shows the relation between vmax, acc and magnitude, while the blue points show ${v\_{\rm max}}$ at z = 0. The systematic movement to lower ${v\_{\rm max}}$ for many objects is due to tidal stripping of subhalos. The resulting simulated catalog has a distribution of galaxies that is complete down to Mr = − 15.3. Note that our pure-abundance matching measurements from Figure [fig:mrvpeak] gives most likely mass estimates for the LMC and SMC to be ${v\_{\rm max}}= 85\pm 6$ and 65 ± 6km s− 1, respectively. However, as shown in, ignoring the effects of baryons likely increases the maximum circular velocity by roughly 10% (see their Figure 5), bringing our SHAM mass estimates into excellent agreement with the observations Using luminosities assigned in this way, we now make measurements analogous to those of L10. We select for MW candidates in the same way as L10. First, we make a mock lightcone by taking the z = 0 SHAM catalog of the Bolshoi simulation, placing an observer at the center of the box, and adding redshift space distortions. We passively evolve galaxy magnitudes to the appropriate redshift using the SDSS constraint Mr(z) = Mr(z = 0.1) + Q(0.1 − 1), with Q = − 1.3. We then identify all isolated objects by selecting halos with Mr = − 21.2 ± 0.2 (the upper grey band in Figure [fig:mrvpeak] and excluding objects having a brighter neighbor in a cylinder of radius 500 kpc and length 28 Mpc. The resulting ${v\_{\rm max}}$ distribution is shown in Figure [fig:shamvmax]. This distribution is somewhat lower than the best current estimates for the maximum rotational velocity of the MW, ${\rm v}\_{rot} = 220$ km s− 1. This difference should not be surprising since ${v\_{\rm max}}$ includes only the dark matter, while ${\rm v}\_{rot}$ is the total rotational speed of an actual halo that includes the full impact of baryons. Further, have shown that, because the Milky Way sits near the peak of the star formation efficiency, baryons play a significant role in determining the circular velocity profile of MW mass halos, resulting in a value of ${v\_{\rm max}}$ that is underestimated in dark matter-only simulations. However, the virial mass of the galaxies that pass our MW selection peaks at $1.3 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$, in excellent agreement with the estimated MW mass from. We select for MC-analogs using a method similar to L10, identifying all satellites within the virial radius of our MW analogs that are 2-4 magnitudes dimmer (lower grey band in Figure [fig:mrvpeak]). The most likely ${v\_{\rm max}}$ values for our MC analogs are ∼ 80 and ∼ 70 km s− 1, in agreement with the 1-σ constraints from observations. [fig:mrvpeak] [fig:shamvmax] The Definitions of a Satellite ------------------------------ Before presenting detailed comparisons between the L10 observational data and our model, we note that the definition of a “satellite” differs between the two analyses. Because L10 used a photometric sample with hosts selected on luminosity and no spectroscopy for their satellites, the number of satellites was defined as the number of photometric pairs around a host. Specifically, they identified isolated MW-luminosity galaxies in the SDSS spectroscopic sample and counted the number of photometric pairs with Δmr = 2 − 4 within a fixed angular separation, nominally taken to be 150 kpc. They then applied a background subtraction to account for projection effects and correlated structures. While our SHAM catalog will let us select isolated host objects in an analogous way, because modeling the full distribution of galaxy apparrent magnitudes is beyond the scope of our model, we do not perform an identical calculation. Instead, we can select an appropriate definition of a satellite galaxy to mimic the observational effects. L10 presented two methods for background subtraction, which were used to calculate the number of satellites around a host galaxy. The first, which we refer to as isotropic subtraction, assumed that background objects had, on average, no dependence on position (implicitly ignoring the impact of correlated structures), to calculate an average background for their entire host population. Using this assumption, it is possible to calculate the average background to very high statistical accuracy. Their second method, which we refer to as annulus subtraction, measured the background object counts around each galaxy by considering the region just outside the projected region surrounding the galaxy. For completeness, we make comparisons to both methods. Because it removes the effects of correlated structures, comparison to the annulus subtraction method of L10 is straightforward. We define as a satellite galaxy any object within the appropriate spherical volume (taken to be 150 kpc in L10) that has an absolute magnitude that is 2-4 mag dimmer than its host. To compare to the isotropic subtraction method, which neglects the impact of correlated structures along the line of sight (see Section 3.2 of L10 for further discussion) requires additional modeling. We account for this by defining a satellite to be any object inside a cylinder of radius 150 kpc and length r0, where r0 is the correlation scale for the Milky Way host-satellite analog. This cross-correlation function between MW-luminous galaxies and objects 2-4 magnitudes dimmer is plotted in figure [fig:CrossCorrelation], and is well fit by a power law, (r) = (r/r0)- with r0 = 5.3 Mpc and γ = 1.7, with a similar correlation length to measurements for all galaxies of MW-like magnitudes inferred from. These authors measured r0 = 5.98 Mpc/h and γ = 1.9 for the autocorrelation function of galaxies with Mr = − 20.7 to − 21.7. We therefore consider a cylinder with radius 150 kpc and length 5.3 Mpc (the correlation length) to be the appropriate aperture for satellite selection to compare to the isotropic background subtraction of L10. Coincidentally, this correlation scale is very similar to the size of the expected redshift-space distortions. showed that a $\sim 10^{12} {\hbox{${\rm M}\_\odot$}}$ halo has a typical velocity dispersion of 125 km s− 1; at z = 0.08, where the bulk of our simulated halos live, this corresponds to a 3.7 Mpc spread due to the redshifts of the satellite population. A cylinder of this length thus represents the most accurate measurements one could make for the number of satellites around a host using spectroscopic data (without including some sort of additional background subtraction). [fig:CrossCorrelation] In Figure [fig:nsatscomp] we compare the P(Nsubs) distribution for four different satellite definitions: spherical and cylindrical apertures (which correspond to the L10 results using their annulus and isotropic background subtractions methods, respectively) as well as selecting satellites based on ${v\_{\rm max}}$ and abundance-matched magnitude. Here, the red circles consider subhalos within the virial radius of their host that have vmax = 50 − 80km s− 1. The black diamonds, blue triangles, and green boxes all use magnitude-selected satellites, counting objects 2-4 magnitudes dimmer than their host using the aforementioned SHAM model with cuts analogous to the observational criterion of L10. The red diamonds represent the “correct” number of satellites, using all objects inside the virial radius, The blue triangles use a cylindrical aperture, analogous to the isotropic background subtraction of L10, while the green boxes use a spherical aperture, equivalent to the annulus subtraction method of L10. All error bars were calculated using the jackknife method with 10 sub-regions. [fig:nsatscomp] A number of trends are readily apparent for the relations in Figure [fig:nsatscomp]. First, there is generally good agreement between the P(Nsubs) distribution for objects based on ${v\_{\rm max}}$ (red circles) with our SHAM magnitudes (black diamonds). This is to be expected if SHAM preserves the observed mass-luminosity relation in this regime, something that will only happen if an appropriate mass function is used. It is also interesting to note that our calculations for the cylindrical aperture (blue triangles) always lie above the other measurements for Nsubs ≥ 1. This shows that correlated structures are an important effect which is typically comparable to the signal from satellites that are formally inside the halo’s virial radius. Our measurements using a fixed 0.15 Mpc cylinder (green box) lie below other measurements for higher values of Nsats. This should be unsurprising, since the typical virial radius for our selected halos is roughly 250 kpc. It is interesting to note, however, that the cumulative probability for hosting (at least) one satellite differs by less than 5% when calculated using the virial radius instead of a radius of 150 kpc, while this difference increases rapidly for higher values of Nsats. All of these trends indicate the importance of careful background subtraction to eliminate this signal (such as the annulus subtraction of L10), or otherwise of comparing directly to simulations with similar selection criteria so as to properly interpret results. As concerns the length of our cylindrical aperture, we obtain almost identical results for any length beyond ∼ 1Mpc, beyond which the correlation function falls off substantially. Varying the cylinder length from 2 − 8 Mpc changes our results by roughly five percent, comparable to the size of our statistical errors. Thus, the comparison between satellites defined using our cylindrical aperture and the isotropic background subtraction technique of L10 should be robust. The numerical values for many of these probabilities — selection by rvir, as well as our cylindrical and fixed spherical apertures — are presented in Table [table:probabilities]. Note that, while we are only counting objects as satellites if they are 2-4 magnitudes dimmer than their hosts for the most accurate comparison with L10, these numbers are essentially identical to considering a threshold sample of objects 0-4 magnitudes dimmer than their (MW-like) hosts. Switching to this definition creates only percent-level changes, well within the statistical error bars. Comparisons Between Simulations and Observations ------------------------------------------------ Figure [fig:nsatsobssim] and Table [table:probabilities] show the main result of our analysis, making direct comparison with the results of L10. In the top panel, we compare the L10 P(Nsats) measurements using their isotropic background subtraction (black asterisks) with our SHAM results using subhalos inside a fixed cylinder with radius 150 kpc and length 5.3 Mpc (blue triangles; cylinder column in Table 2) that closely mimics their selection criteria. The bottom panel shows the comparison between the L10 annulus subtraction (black and gray asterisks) and our measurement of Nsats within a 150 kpc spherical aperture (green squares; Sphere column in Table 2). The black and gray asterisks in the lower panel represent the range of measurements based on uncertainty in selecting an optimal background subtraction. The difference between these sets of points thus gives an estimate for the size of the systematic errors in the measurements. Note that, after making their background subtraction, L10 also calculate a systematic adjustment due to catastrophic photo-z failures (see section 4.2 of L10). While we have included those corrections in Figure [fig:nsatsobssim], we have explicitly kept the lower error bar consistent with zero for any measurement that was consistent with zero at the 2-σ level before this systematic correction was included. For reference, Table [table:probabilities] also gives the P(Nsats) distribution for proper subhalos, i.e., objects within rvir that are 2-4 magnitudes dimmer than their host. The agreement between simulations and observations in Figure [fig:nsatsobssim] is generally very good. We take this to indicate a success for both the ΛCDM and SHAM models, particularly indicating, with excellent statistics, that the HOD predicted by Bolshoi for Milky Way-like objects at the massive end matches observations, and there is no evidence for either a missing or excess satellite problem for these objects. Considering our comparison between the cylindrical/isotropic sample, at no Nsats do the two results differ by more than ∼ 2.5σ, and in most situations agree to better than 1 σ. The comparison fares slightly worse for the spherical/annulus sample. Here, the simulations underpredicts the number of Nsats = 0 systems, while over-predicting the Nsats = 1 systems at the 3 σ level. While this could indicate a real disagreement of the simulation and galaxy formation model with the data at this mass scale, there are a number of other possibilities. First, it is possible that the annulus-subtraction method of L10 over-corrects the background subtraction. This has been tested in simulations, using their algorithm to estimate the number of objects within 150kpc around our halos centers using our cylindrical counts. The result, is an over preduction of the Nsats = 0 term by 6% and an underprediction of the Nsats = 1 term by 5%, larger than the estimated systematical uncertainties in Table 1 of L10 (the changes in the higher Nsats terms are of similar size to their estimated systematic uncertainty, and are therefore well characterized by the gray points in Figure [fig:nsatsobssim]). Such corrections would bring the these terms within 2- and 3- σ of our simulated estimates. It is also important to note that the predictions from simulations depend on the assumed scatter between mass and luminosity. This will be further discussed below. [fig:nsatsobssim] \| c \|\| c \| c c \| c c\| & rvira & Cylinderb & Isotropicc & Sphered & Annulusc P(0) & 55 ± 4% & 55 ± 5% & 66− 1.3+ 1.6 & 71 ± 3% & 81− 1.4+ 1.5 P(1) & 28 ± 3% & 28 ± 3% & 22− 1.9+ 1.7 & 23 ± 2% & 12− 1.6+ 1.8 P(2) & 11 ± 1.4% & 11 ± 2% & 8.9− 1.5+ 1.5 & 5.2 ± 1.6% & 3.5− 1.5+ 1.3 P(3) & 3.8 ± 0.7% & 3.8 ± 1.2% & 1.5− 0.7+ 1.0 & 1.0 ± 0.8% & 1.6− 1.6+ 0.9 P(4) & 1.55 ± 0.4% & 1.5 ± 1.1% & 0.90.5+ 0.7 & 0.2 ± 0.1% & 1.1− 1.1+ 0.6 ahalos contained within the virial radius of the host bhalos within a cylinder with radius 150 kpc and length 5.2 Mpc cL10 results counting objects within a projected 150 kpc and an isotropic background subtraction dhalos within a sphere of radius150 kpc eL10 results counting objects within a projected 150 kpc using an annulus background subtraction [table:probabilities] Looking at the SDSS/Bolshoi comparisons in more detail, we can further investigate the dependence of the P(Nsats) distribution on the aperture size, as done in L10. For this, we determine the probability of finding Nsats satellite galaxies in a spherical aperture around their host, where the radius is varied from 100 to 250 kpc using the annulus background subtraction. The results are shown in Figure [fig:nsatsrad]. Here, the red circles, orange squares, magenta triangles, and cyan inverse-triangles represent the probability of having 0, 1, 2, or 3 subhalos. Open symbols represent SDSS measurements, while filled symbols come from our Bolshoi+SHAM model, while the hatched lines represent the P(Nsats) distribution for satellites within the virial radius of their hosts. Again, we see good agreement between simulations and observations, although we note the persistence of higher values in the simulation for the Nsats = 1 measurement. However, the other values of Nsats are in much better agreement. [fig:nsatsrad] One of the appeals of the SHAM algorithm is that it has so few free parameters, in this particular case just the scatter (although there are additional implicit assumptions, e.g., about subhalo stripping and star formation after satellite accretion). Figure [fig:nsatsscatter] shows how the scatter impacts this result. Here, we present the probability distribution for an isolated host with Mr = − 21.2 to host Nsats satellites brighter than − 16.5 as we vary the scatter in the SHAM model from 0 to 0.3 dex using our cylindrical method for selecting satellites. While none of these models are ruled out by the data (L10 isotropic model), a number of systematic trends are present. In particular, there is a systematic increase in the probability of hosting 3 or more satellites, along with a decrease in the probability of hosting 1 satellite (by up to 5%). This trend is caused not by scatter in the mass–luminosity relation of the satellite galaxies, but in the mass–luminosity relation of the host objects. As scatter increases, higher and higher mass hosts scatter into our magnitude-selected sample, bringing with them their richer subhalo populations. As we shall discuss in the next section, this has the effect of creating a tail to the negative-binomial distribution that sets the probability for a host to have Nsats satellites. [fig:nsatsscatter] It is also worth noting that the dark blue points representing a scatter of σ = 0.3 dex look to be violating the trend of an increasing high-Nsats probability. This is not due to any special behavior in this regime, but more represents the limits of our simulation resolution. As scatter increases, objects with Mr − 5logh > − 16.5, which nominally have vacc = 90 km s− 1using a scatter-free SHAM, have infall masses as low as 65 km s− 1in our σ = 0.3 dex model. Because tidal stripping can reduce the ${v\_{\rm max}}$ of a subhalo by a factor close to 2, this puts such objects dangerously close to the Bolshoi completeness limit of 50 km s− 1. It should be noted, however, that the completeness of subhalos in the Bolshoi simulation is highly dependent on the host mass. Because of stripping of subhalos after accretion, we expect to be losing potentially 10-15scatter this high. It’s important to appreciate the high resolution necessary to fully model a galaxy population using SHAM. While a scatter of σ = 0.3 is likely ruled out for massive galaxies, if such scatter were appropriate the Bolshoi simulation would only be able to populate galaxies down to Mr ∼ − 17.5 while losing fewer than 10% of the objects. Finally, we should note that there is no compelling reason to assume that both massive and low-mass galaxies exhibit the same scatter in the mass-luminosity relation. We have investigated this by quantifying the variations in the P(Nsats) distribution where we keep the mass-luminosity scatter of the hosts fixed at 0.16 and vary the scatter of the subhalos from 0-0.3. While we see similar trends with, e.g., the Nsats = 1 term dropping with increased scatter, the effect is significantly weaker than shown in Figure [fig:nsatsscatter], where we allow the scatter in the host mass-luminosity relation to vary. While it is very possible that the scatter should be even higher than 0.3 for our satellite galaxies, the 50 km s− 1resolution limit of Bolshoi prevents us from exploring a wider range. Thus, over the ranger we are able to probe, the scatter in the host mass-luminosity relation, which sets the range of host masses sampled, has a larger impact that the scatter in the satellite mass-luminosity relation. Environmental Effects --------------------- We now revisit the discussion regarding the environmental impact on the subhalo population of §[sec:env], using observationally motivated environmental measures. Recall that in §[sec:env], we defined the local environment as the dark matter density in collapsed objects larger than $2 \times 10^{11} {\hbox{${\rm M}\_\odot$}}$. Here, we preform a similar analysis using luminosity as an environmental proxy. Specifically, using the isolated central galaxies from our SHAM catalog as selected above, we measure the total luminosity from galaxies brighter than Mr = − 20.8 (which corresponds to Mr − 5log(h) = − 20 in the usual SDSS units) within a sphere of 1 Mpc around our hosts, which we take to be our environment proxy. We then split the sample into our most and least luminous (i.e., dense) quartiles and re-measure the P(Nsats) distribution for satellites inside their host virial radii. The results are shown in Figure [fig:nsatsmagdens]. Here, the black points represent the distribution for all isolated hosts, while the red and blue represent those in high/low density environments. As before, we see a clear offset for galaxies in different environments. While the differences are only at the 1-2 sigma levels, high-density halos are systematically more likely to have massive subhalos than those in low density environments. In particular, the underdense halos are 14% more likely to host no massive subhalos than their dense counterparts. Similarly, halos in dense environments are twice as likely to host two or more subhalos than those in low density environments. This effect should therefore be an observable manifestation of assembly bias. [fig:nsatsmagdens] Statistics of Satellites in Luminosity-selected Hosts ===================================================== As noted previously, the inclusion of scatter in the halo ${v\_{\rm max}}$-galaxy luminosity SHAM creates a significant tail in the probability distribution for a host to have Nsats satellite galaxies. Consequently, the negative binomial distribution that was found to describe the $P(N\_{\rm subs})$ distribution for the subhalos in a mass-selected host does not produce a good fit to $P(N\_{\rm sats})$ when applied to luminosity-selected galaxies. Additionally, the change in star formation efficiency with host halo mass breaks the self-similarity that was observed for host halos. In this section, we quantify the P(Nsats) distribution for satellite galaxies around a luminosity-selected host. In Figure [fig:nsatswithfit], we show a series of probability distributions for a Mr = − 20.5 galaxy to host Nsats subhalos brighter than -19.8, -18.3 and -16.8 (equivalent to Mr − 5log(h) = − 19, − 17.5, − 16). This plot was generated using a scatter of σ = 0.16 dex. A low probability tail clearly extends to high Nsats. Such a tail was not present when considering ${v\_{\rm max}}$-selected hosts and subhalos in section [sec:pois]. This is emphasized by the dotted lines, which present the negative binomial distributions, equation [eq:negativebinomial], that characterized the P(Nsubs)distribution. While the negative binomial provided an excellent fit to the data for our mass-selected objects, it clearly fails here. While this tail may not appear to be a serious concern, since it is not important until P(Nsats) ∼ 1%, large surveys such as the SDSS contain more than 10,000 massive clusters, so even these small effects may be visible. [fig:nsatswithfit] [fig:fitparameters] As a better fit, we present a modified binomial distribution by adding a power law tail, P(N\|r,p,T,) = (N+r) (r)(N+1) pr (1-p)N + Te-N. [eq:fit] Here, the first term is just the negative binomial, with r and p functions of α2 and ⟨N⟩, as defined in Equation [eq:pr]. The final term models the high-Nsats tail that comes about as a result of the ${v\_{\rm max}}$-luminosity scatter, where T and τ control its slope and amplitude. The dashed lines in Figure [fig:nsatswithfit] show the best fits to Equation [eq:fit]. This clearly provides a significantly improved fit to the simulation measurements, albeit at the expense of adding two new fit parameters. Note, however, that the fit tends to have problems in the cross-over region and that the transition to the high-Nsats tail is somewhat more gradual than this model is able to reproduce. While we leave this for future investigations, it is worth noting that changing the scatter in our SHAM model has a direct impact on T and τ. High values of scatter will have a more pronounced tail, which will cause both the amplitude and slope of this tail to increase. For simplicity, for the rest of this work, we will limit ourselves to using a model with σ = 0.16 dex. While we could present our best-fitting parameters to the distributions in Figure [fig:nsatswithfit], a much more useful process is to model the P(Nsats) distribution as a function of both host magnitude and limiting satellite magnitude, as has been done for mass-selected subhalos. Unfortunately, this proves to be much more difficult for galaxies than for halos. Because the dark matter power spectrum is mostly featureless in the relevant regime, large halos are nearly self-similar versions of smaller halos. As an example of this, the top panel of Figure [fig:distributionfits] shows the mean number of satellites above a given mass cut for mass-selected host halos. This distribution exhibits a power-law behavior with nearly identical parameters regardless of the host mass. This is not the case for magnitude-selected objects. Figure [fig:fitparameters] presents the best-fit parameters for the satellite P(Nsats) distribution as modeled by Equation [eq:fit] around galaxies as a function of limiting subhalo luminosity (i.e., magnitude difference). The red, orange, yellow, dark green, green, and blue points represent hosts with Mr = − 20.3, − 20.8, − 21.3, − 21.8, − 22.3, and − 22.8. As can be clearly seen in the top left panel, which shows the trend of the mean number of satellite galaxies (⟨N⟩) with luminosity ratio, galaxies do not scale self-similarly. At a given luminosity ratio, brighter galaxies are significantly more likely to have a larger Nsats (expected number of satellites) than dimmer galaxies. This is a manifestation of the varying star formation efficiency with halo mass as seen in, e.g., Figure [fig:mrvpeak]. For masses above 1012, star formation efficiency becomes less efficient, so the rate of increase of Mr with vmax is shallow. At masses lower than 1012, star formation efficiency drops rapidly with decreasing mass, leading to a much steeper rate of change of Mr with respect to vmax. Thus, for a fixed bin in Mr, the corresponding logarithmic range in vmax will be much less for low-mass galaxies as compared to high-mass galaxies. As such, while the ⟨N⟩ values for mass-selected subhalos lie on a single line in Figure [fig:distributionfits], luminosity-selected halos do not scale in such a manner, resulting in the different lines in the top-left panel of Figure [fig:fitparameters]: brighter host galaxies have more satellites at a fixed luminosity ratio, Lsat/Lhost, than dimmer hosts. A number of the trends present in Figure [fig:fitparameters] can be readily understood in terms of simple models. As discussed above, the behavior of ⟨N⟩ is simply a manifestation of the changing star formation efficiency with halo mass. Similarly, the trends in τ can be understood though simple scaling relations. First, consider a fixed luminosity ratio, ∣Mr, host − Mr, lim∣. Since brighter hosts correspond to more massive halos, these objects live in a steeper part of the mass function, where a fixed scatter creates a broader selection of host masses, the larger of which are more likely to host bright satellites. Because of this, the exponential tail falls off more slowly, resulting in a lower τ. The trend of τ increasing with luminosity ratio for a fixed host mass. Guided by these observations, and noting that α2 and T don’t have strong trends with either host magnitude or limiting luminosity ratio, we propose a 7-parameter fit to generally describe the full P(N∣Mr, host, Mr, lim). We use our generalized negative-binomial plus power law as defined in Equation [eq:fit], and keep α2, T, and τ constant. However, we parameterize ⟨N⟩ as a linear function, the slope of which depends on Mr, host − Mr, lim and the offset of which depends on host magnitude, Mr, host. Our full fit for Equation [eq:fit] then becomes a function of the parameters N0.a, N0, b, N1, a, N1, b, α2, T0, and τ0 defined by: P(N\|Mr,host, Mr,lim) = (N+r) (r)(N+1) pr (1-p)N + T0e-0 N [eq:fullfit] where p &=& 1 1 + (22 - 1) N (Mr,host, Mr,lim) r &=& 1 22 - 1 [eq:parameters] N (Mr,host, Mr,lim)&=& -10N0,a + N0,b\(-Mr,host+5(h)) - & & (Mr,host-Mr,lim)10N1,a + N1,b\(-Mr,host+5(h)). We fit this functional form to the measured P(Nsats) distribution for all host halos down to Mr = − 18.5 and their satellites down to Mr = − 16 in Bolshoi stacked in bins of 0.5 mag. We present our best fitting parameters for Equation [eq:parameters] in table [table:parameters]. As a final validation of this result, Figure [fig:fitverification] shows the measured P(N) distributions for a range of Mr, host and Mr, lim, along with the fits from our model in Eq. [eq:fullfit]. Note that, because the slope and intercept of ⟨N⟩(Mr, host, Mr, lim) are linear functions in log space, the exact value of ⟨N⟩ is highly sensitive to the parameters N0, a, N0, b, N1, a, and N1, b. Because of this, it is necessary to constrain these parameters to 4 significant figures (which is easily done given the constraints from the Bolshoi simulation). While our nominal fit can be improved using a τ that is a power-law in Mr, host − Mr, lim with a normalization that depends on host mass, the actual fit does not significantly improve the reduced χ2, so we elect to hold it constant. \| parameter \| best-fit value \| \| --- \| --- \| \| N0, a \| -53.70 \| \| N0, b \| 40.11 \| \| N1, a \| -39.05 \| \| N1, b \| 29.25 \| \| α2 \| 1.3 \| \| T0 \| 1.3 × 10− 4 \| \| τ0 \| 0.023 \| [table:parameters] [fig:fitverification] Conclusions and Summary ======================= In this work, we have examined the likelihood for simulated MW-like dark matter halos to host substructures similar to the Magellanic Clouds. We have performed this analysis using both dark matter kinematic information and an abundance-matching technique which assigns galaxy luminosities to resolved dark matter halos and allows direct comparison with observations (e.g., L10). The main result of this work is that MW-like objects have a 5-11% chance to host two subhalos as large or as luminous as the SMC, in basic agreement with previous simulation results (e.g., BK10), as well as with observations. While we have extensively compared our results for the full probability distribution for a MW-luminous galaxy to host Nsatssatellite galaxies to the observational measurements from L10, finding good agreement. This results are also in qualitative agreement with previous results using smaller samples. Our results have a number of implications. First, this is a validation of the ΛCDM paradigm down to the level of $10^{10}{h^{-1}}{\hbox{${\rm M}\_\odot$}}$ objects. At this level, there is no indication that either the CDM paradigm or our standard picture of galaxy formation, including the relatively tight relationship between galaxy luminosities and halo masses, breaks down. In particular, it appears that the statistics of satellite galaxies at this mass in LCDM are in very good agreement with observations which sample a wide range of galaxy environments. One possible criticism of the model we have used for connecting galaxies to halos (SHAM) is that it may be so robust as to always reproduce the observed statistics, given its assumption of the correct global luminosity function. In order to evaluate this, it is important to keep track of the exact assumptions of the approach and how these relate to the particular statistics measured. First and foremost, SHAM assumes that ${v\_{\rm max}}$ of a halo or subhalo (at accretion into a larger system) is the only parameter that sets the luminosity of the galaxy it hosts. While this is robust to, e.g., a temporally-variable star formation efficiency with halo mass, it would differ from model in which there was environmental dependence to the galaxy formation recipe in addition to the inherent environmental dependence of the halo mass function. For example, if subhalos of a given mass evolve very differently than halos of the same mass in the field, our model would not produce good agreement. Additionally, if our simulated dark matter halo mass function was significantly off from reality (due to, for example, an incorrect σ8 or Ωm), the masses of the MCs as estimated from abundance matching would not provide a good match to the direct kinematic measurements, as we find here. Finally, this is a test of the SHAM treatment of subhalos in a mass regime lower than has been previously explored. We do see some manifestations of assembly bias with regard to the dark matter subhalo population inside MW-like galaxies: halos in higher density regions host more massive subhalos than those in lower density regions. This effect appears to be most pronounced for relatively local measurements of density, and suggests that the MW’s proximity to M31 boosts the likelihood for us to see a LMC/SMC pair by about 25%. The effect is strongest for densities measured on the scale of 1 Mpc. We note that such a boost seems to be present if we use either the total local mass or the total local luminosity as an environmental measure. This should make the effect an observable manifestation of assembly bias (though detecting it will require careful treatment of correlated structure). It is interesting that this result is in qualitative agreement with the work of, who saw a similar trend with environment but claimed it was most sensitive at the $\sim 5{h^{-1}}{\rm{Mpc}}$ scale. They note that we are in an underdense region on these scales, and cite this as a possible ingredient for solving the missing satellite problem. While our simulations do not allow us to determine the probabilities of finding objects smaller than the Magellanic Clouds, if our observed trend were extrapolated down to lower masses it would result in an opposite effect, where the presence of M31 could serve to exacerbate the missing satellite problem. Finally, we have generalized our measurements for the P(Nsats)distribution over a wide range of host and satellite luminosities. Our results show a significant deviation from the Poisson expectation for larger values of Nsats. While these deviations are typically present only in the low-likelihood tail of the distribution, the large volume of surveys like SDSS should allow tests of these predictions with observations. Deviations from Poisson statistics in the number of satellites may also have consequences for HOD modeling, which currently assume a Poisson distribution for the number of galaxies given the halo mass. RHW and MTB were supported by the National Science Foundation under grant NSF AST-0908883. RHW, PSB, and BFG received support from the U.S. Department of Energy under contract number DE-AC02-76SF00515. AK and JRP were supported by NASA ATP and NSF AST grants. The Bolshoi simulation was run on the NASA Advanced Supercomputing (NAS) Pleiades computer at NASA Ames Research Center. We thank Louie Strigari, Lulu Liu, and Anna Nierenberg for useful discussions. 59 natexlab#1#1 Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., Allam, S. S., Allende Prieto, C., An, D., Anderson, K. S. J., Anderson, S. F., Annis, J., Bahcall, N. A., & et al. 2009,, 182, 543 Battaglia, G., Helmi, A., Morrison, H., Harding, P., Olszewski, E. W., Mateo, M., Freeman, K. C., Norris, J., & Shectman, S. 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The value for the r-band absolute magnitudes was determined using the K-correction code of on a sample population of 500,000 SDSS galaxies.[↩](#fnref2) Statistics of Satellite Galaxies Around Milky Way-Like Hosts ============================================================ We calculate the probability that a Milky-Way-like halo in the standard cosmological model has the observed number of Magellanic Clouds (MCs). The statistics of the number of MCs in the ΛCDM model are in good agreement with observations of a large sample of SDSS galaxies. Under the sub-halo abundance matching assumption of a relationship with small scatter between galaxy r-band luminosities and halo internal velocities ${v\_{\rm max}}$, we make detailed comparisons to similar measurements using SDSS DR7 data by. Models and observational data give very similar probabilities for having zero, one, and two MC-like satellites. In both cases, Milky Way-luminosity hosts have just a ∼ 10% chance of hosting two satellites similar to the Magellanic Clouds. In addition, we present a prediction for the probability for a host galaxy to have Nsats satellite galaxies as a function of the magnitudes of both the host and satellite. This probability and its scaling with host properties is significantly different from that of mass-selected objects because of scatter in the mass-luminosity relation and because of variations in the star formation efficiency with halo mass. Introduction ============ Understanding the satellite population of our own Milky Way galaxy is one of the outstanding problems in galaxy evolution studies today. Because these objects have weak gravitational potential wells, it is quite likely that different physical processes played a driving role in determining how such galaxies formed and evolved than for brighter, more massive objects. One manifestation of this is the well-known “missing satellites problem,” which refers to the fact that the number of satellite galaxies predicted from simulations appears to be much higher than the number actually observed around the Milky Way. One common way to address this problem has been to run simulations of individual objects using extremely high mass resolution. To date, roughly 10 of these ultra-high resolution N-body simulations have been run, which typically resolve individual Milky Way mass halos with upwards of 109 particles. The Magellanic Clouds (MCs) are a pair of well studied satellites of the Milky Way (MW) in the Southern Hemisphere with magnitudes MV = − 18.5 and -17.1 for the Large and Small Magellanic Clouds (LMC and SMC). While their large angular size makes detailed mass measurements somewhat difficult, observations constrain their maximum circular velocities to be around ${v\_{\rm max}}\sim 60$ km s− 1. Unlike the missing satellite problem, one common feature of these ultra-high resolution simulations is that they typically under-predict the number of massive satellites compared to the MCs in the MW. Indeed, most of these high resolution halos contain no objects of similar mass to either of the two MCs. While this should hardly be taken as a serious problem since the handful of high resolution simulated halos does not constitute a statistical sample, it is important to understand how typical or atypical the Milky Way is, and whether or not not there is an “extra satellites” problem at high satellite masses within the modern Lambda Cold Dark Matter (ΛCDM) paradigm. Because the MCs are relatively nearby and have been studied with high-precision instruments such as HST for some time, detailed 3-dimensional measurements of their velocities have been made. Some of these studies have indicated that the Magellanic Clouds may be on their first orbit around the Milky Way ; this may make such an extra satellite problem less worrisome if the presence of the Magellanic Clouds is a transient event. Regardless of the dynamical state of the Clouds, it is an important test of galaxy formation to understand the likelihood for MW-like systems to host massive satellite galaxies. Recent developments have afforded us the possibility to address this problem in more detail with both observations and theoretical models. From the observational side, wide area surveys such as the Sloan Digital Sky Survey have given us the ability to probe galaxy content for a large number of Milky Way-magnitude objects. The main theoretical effort has been in populating dark matter halos with galaxies using semi-analytic methods and hydrodynamic simulations. used a number of toy models to add galaxy properties to dark matter halos generated using a combination of Press-Schechter theory with semi-analytic models for tracking subhalo orbits. They found that it was very difficult to model objects as bright as Magellanic clouds without allowing for an extremely high star formation efficiency. This finding extended the underprediction of high-mass subhalos to an underprediction of luminous satellites, implying that the MW-MCs system is unusual. Similarly, explored a range of feedback models to add galaxies to some of the high resolution Aquarius halos. It was again difficult to readily reproduce halos with luminosities as bright as the MCs. Having looked in detail at a handful of simulated objects, however, this work indicates that there may be significant halo-to-halo scatter in the number of such massive objects. The idea of intrinsic scatter in the subhalo population was expanded on in. While this work did not concentrate specifically on MC-like subhalos, they did consider the range in number of subhalos with vmax, sub/vmax, host > 0.1. This work showed an extremely large variation (20-60) in the number of such massive subhalos a galaxy-sized halos would host. In contrast to the semi-analytic modeling just discussed, used hydrodynamic simulations to model the luminosity functions for the satellites around MW-like host halos. Their simulation identified 9 MW-like central galaxies and found that they, on average, have 1.6 satellites brighter than MV = − 16 and a third of them had satellites with luminosities comparable to the LMC. The recent Millennium-II and Bolshoi simulations have, for the first time, allowed us to probe cosmological volumes to understand the ΛCDM predictions for the satellite populations of MW-like halos. The properties of these simulations are summarized in Table 1. (hereafter BK10) quantified the likelihood for $10^{12} {\hbox{${\rm M}\_\odot$}}$ halos to host massive satellite galaxies in the Millennium-II simulation, finding that subhalos similar to the MCs are quite rare. In this paper, we expand on this work by making similar measurements for the Bolshoi simulation, which used WMAP7 cosmological parameters, and using an abundance matching technique to make detailed comparisons between the Bolshoi predictions and the measurements from. Our goal is to understand just how well ΛCDM reproduces the statistical properties of bright satellites. Note that this is the reverse question from the one that was asked in a companion paper,. That work assumed a satellite population and asked what the implications were for the properties of the host halo, including its mass. Here, we assume a host halo mass and ask about the implications for the subhalo population. There is no reason for both questions to give the same answer: while showed that a halo which hosts two MC-like satellites most likely has a mass near $1.2 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$, there is no reason to assume that a typical $1.2 \times 10^{12}{\hbox{${\rm M}\_\odot$}}$ halo will have the MCs as satellites. We begin by giving an overview of the Bolshoi simulation in §[sec:sims], and then investigate the properties of massive dark matter satellites around dark matter host halos in §[sec:halos], focusing on the mass ranges for hosts and satellites that are most relevant to the MW system. The analysis here is similar to that of BK10. In §[sec:luminosities], we assign galaxy luminosities to our suite of dark matter halos and extend the results for a sample with similar selection cuts as for observations. In this way, we are able to make detailed comparisons to the observational work of concerning the satellite population around MW-magnitude galaxies. Section [sec:comparisons] gives the results of this analysis — see especially Figure [fig:nsatsobssim]. Finally, in §[sec:generalproperties], we expand this study to include the satellite population of a more general distribution of hosts, and §[sec:conclusions] summarizes our conclusions. Throughout this paper, we adopt the convention h = 0.7 (the value that was used in the Bolshoi simulation) when reporting values from either simulations or observations. Simulations =========== We use the dark matter halos identified in the Bolshoi simulation. This simulation modeled a 250 ${h^{-1}\rm{Mpc}}$ comoving box using cosmological parameters similar to those derived by WMAP7 : Ωm = 0.27, ΩΛ = 0.73, σ8 = 0.82, n = 0.95, and h = 0.7. The simulation volume contains 20483 particles, each with a mass of $1.35 \times 10^8~{h^{-1}}{\hbox{${\rm M}\_\odot$}}$ and was run using the ART code. 180 snapshots from the simulation were saved and analyzed. One of the unique aspects of this simulation is the high level of spatial resolution employed, allowing objects to be resolved down to a physical scale of 1 ${h^{-1}\rm{kpc}}$. This gives us excellent ability to track halos as they merge with and are disrupted by larger objects, allowing us to track them even as they pass near the core of the host halo. A summary of these simulation parameters is presented in Table [table:simulations]. Because we discuss our work in the context of the BK10 results, we also present the same parameters for the Millennium II simulation, on which the BK10 results were based. Halos and subhalos were identified using the BDM algorithm. The algorithm identifies maxima in the density field and examines the neighboring region to identify bound particles. In this way, it treats both halos and subhalos identically. Subhalos are just identified as objects living within the virial radius of a larger objects. Because of the high level of mass and spatial resolution, BDM results in a halo catalog that is complete down to a maximum circular velocity ${v\_{\rm max}}= 50$ km s− 1, where ${v\_{\rm max}}= {\rm max}\left(\sqrt{{GM(<r)}/{r}}\right). $ This corresponds to a virial mass of roughly $10^{10}{h^{-1}}{\hbox{${\rm M}\_\odot$}}$. When BDM halos are identified, the ids of their 50 most bound particles are also stored to assist in producing merger trees. As discussed in Section [sec:sham], in order to assign galaxy luminosities to dark matter halos, we need to track the histories of dark matter substructures. This is done using merger trees created from the 180 snapshots of the Bolshoi simulation. The detailed algorithm for creating the merger trees is described in. Briefly, the algorithm works by first linking halos across time steps by tracking the 50 most bound particles of each halo. Some halos will not have any of their 50 most bound particles identified at a later timestep (e.g., very massive objects in which the 50 most bound particles change rapidly through stochastic processes), while some will have their particles distributed to multiple halos. The algorithm corrects for this by running a simple N-body calculation on the locations and masses of all halos in the simulation to predict where each halo should wind up at the next time step. Using this information, it is possible to link halos across multiple time steps with very high accuracy. \| \| Bolshoi \| MSII \| \| --- \| --- \| --- \| \| Box Size [Mpc] \| 357 \| 143 \| \| NP \| 20483 \| 21603 \| \| $M\_P [{\hbox{${\rm M}\_\odot$}}]$ \| 1.9 × 108 \| 9.8 × 106 \| \| h \| 0.7 \| 0.73 \| \| ΩM \| 0.27 \| 0.25 \| \| σ8 \| 0.82 \| 0.9 \| \| Force Resolution [kpc] \| 1.4 (proper) \| 1.4 (Plummer-equivalent softening) \| [table:simulations] Satellite Statistics for Dark Matter Subhalos ============================================= We begin by investigating the properties of dark matter satellites around dark matter hosts in the Bolshoi simulation, focusing on the mass range for hosts and satellites that is relevant to the MW system. In §[sec:massdep], we consider trends with host halo mass, and in §[sec:env] we investigate trends with halo environment. We then consider the distribution of the satellite number (§[sec:pois]). Finally, we extend this analysis to include a more observationally relevant selection based on galaxy luminosities in §[sec:luminosities]. Because the mass resolution of the Bolshoi simulation creates a halo catalog complete down to 50 km s− 1, roughly equivalent to the lower bound of the mass of the MCs, we begin by measuring the probability distribution for halos hosting Nsubs subhalos with ${v\_{\rm max}}$ larger than a given value. This measurement is made by identifying the virial radius of all distinct (non-subhalo) MW mass halos and counting the number of objects internal to their virial radii. We take the virial mass of the Milky Way to be $\log({\hbox{$M\_{\rm vir}$}}/{\hbox{${\rm M}\_\odot$}}) = 12.08 \pm 0.12$ (${\hbox{$M\_{\rm vir}$}}= 1.2 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$), the mass measured in, which is consistent with most results in the literature. This results in 36,000 MW analogs found in the Bolshoi volume. We present the resulting probability distribution of satellite counts as the colored points in the left panel of Figure [fig:vmaxcomp]. Here, the different colors represent different thresholds for satellite ${v\_{\rm max}}$: red, green, and blue represent all satellites with ${v\_{\rm max}}> $ 50, 60, and 70 km s− 1. Error bars were calculated using the bootstrap method and represent 95% confidence intervals. From Figure [fig:vmaxcomp], we can immediately see that the likelihood of hosting multiple satellites is somewhat low. For satellites with ${v\_{\rm max}}$ greater than 50 km s− 1, roughly 30% of simulated hosts contain one or more subhalos and just 8% have two or more. These numbers are in excellent agreement with the work of BK10, who performed a similar analysis on the Millennium II simulation, which was run using a different N-body code in a WMAP1 cosmology and with a very different subhalo identification algorithm. The probability of hosting multiple satellites drops precipitously with increasing satellite mass; e.g., the probability of a halo hosting two subhalos larger than ${v\_{\rm max}}= 70$km s− 1 is just 1%. For comparison, the black points in Figure [fig:vmaxcomp] represent measurements from, who measured the probability distribution for finding MC-luminosity satellites within 250 kpc around isolated MW-luminosity galaxies (the mean virial radius of our sample). We only plot these measurements out to Nsubs = 3 because, for a 250 kpc aperture, uncertainties from their background subtraction become extremely large for higher (lower probability) values of Nsubs. Here, the agreement for satellites larger than 50 km s− 1is striking, indicating that CDM can reproduce the statistics of MCs. We must, however, be careful about over-interpreting this result. In particular, their selection criteria are very different from our mass selection. We will return to this in §4.4 where we make a comparison directly to similarly selected samples. By comparison with the Bolshoi simulation, the existence of two satellites with ${v\_{\rm max}}$ larger than 50km s− 1 makes the MW almost a 2σ outlier. Understanding whether anything other than random chance is responsible for putting the MW in this slightly rare regime motivates further investigation into which other properties may correlate with satellite abundance. In particular, we investigate the correlation of the number of subhalos with host mass and environment. Mass dependence --------------- The correlation between subhalo abundance and host mass has been well studied. These studies have all found that, when scaled in units of μ = Msub/Mhost, larger halos contain more subhalos above a given μ than do smaller ones. This is the expected result from the hierarchical merging picture of CDM, in which larger objects grow through the merging of smaller objects and therefore form later. This later formation has a two-fold impact: it reduces the amount of time available to tidally strip a subhalo, and lowers the host concentration, further reducing the ability of a host to tidally strip its subhalos. Motivated by this, we present the abundance of LMC and SMC-mass subhalos for the 27,000 Bolshoi halos with ${\hbox{$M\_{\rm vir}$}}= 2.6\pm 0.9 \times 10^{12}{\hbox{${\rm M}\_\odot$}}$. The results can be seen in Figure [fig:vmaxcomp]. Here, the left panel shows the results for selecting hosts with ${\hbox{$M\_{\rm vir}$}}= 0.8-1.7\times 10^{12}{\hbox{${\rm M}\_\odot$}}$, while the right panel shows a mass range ${\hbox{$M\_{\rm vir}$}}= 1.7-3.4\times 10^{12}{\hbox{${\rm M}\_\odot$}}$. [t] [fig:vmaxcomp] For the more massive ${\hbox{$M\_{\rm vir}$}}\sim 2.6 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$ halos, the cumulative probability of hosting two or more 50 km s− 1 halos increases by a factor of 3 compared to the $1.2\times 10^{12} {\hbox{${\rm M}\_\odot$}}$ objects (from 8% to 26%), while jumping by a factor of 6 for 70km s− 1 subhalos (from 1% to nearly 6%). We emphasize that this is not a contradiction with the results of, which found that a halo with exactly two MC-like subhalos was likely to have mass $1.2\times 10^{12} {\hbox{${\rm M}\_\odot$}}$. While the fraction of more massive halos hosting two MC-like subhalos is larger, the steep slope of the mass function means that there are many more lower-mass halos. Additionally, aside from just considering ${v\_{\rm max}}$, selected halos with satellites like the MCs in terms of location and three-dimensional velocity, properties not considered in the present analysis. Again, these results are in excellent agreement with the work of BK10. While such an agreement is generally expected from pure collisionless N-body simulations, such agreement is not as trivial as it may appear. While the present work uses results from the Bolshoi simulation, run using the adaptive refinement ART code and the BDM halo finder, BK10 used the Millennium-II simulation run with the TreePM code Gadget-2 with substructures identified using the Subfind algorithm. Additionally, in both cases, we are concerned with subhalos close to the mass-limit for the simulations. Thus, the close agreement between these two works suggests that numerical effects in the simulations are well understood at this level. Environmental dependence ------------------------ Beyond simple host mass dependence, the size of the cosmological volume modeled by the Bolshoi simulation allows us to perform further detailed studies on properties impacting the satellite abundance distribution. Here, we quantify the environmental dependence of the subhalo population by splitting the sample on local density, defined by h,r = h,r - \|h \|h, where ρh, r is the Eulerian density of dark matter contained in halos larger than ${\hbox{$M\_{\rm vir}$}}= 2\times 10^{11}{\hbox{${\rm M}\_\odot$}}$ (roughly 1000 particles in Bolshoi) defined in a sphere of size r. In the present analysis, we take $r = 1.0 {h^{-1}}{\rm{Mpc}}$; we have performed similar analysis on a range of scales and find this to be the range where the galaxy halo distribution is the most sensitive to the presence of massive satellites. We compare the distribution P(Nsubs) for halos in the top and bottom quartile distributions to the mean relation in Figure [fig:nsatsdens]. There is a clear systematic trend: halos living in overdense regions (red points) are more likely to host massive subhalos than those in underdense regions (blue points). This is likely due to the fact that halos in denser environments live in more biased regions, where smaller density perturbations are amplified, making it easier for smaller halos to form and accrete onto intermediate mass hosts. This makes the deviations from the mean relation in Figure [fig:nsatsdens] a potentially observable manifestation of assembly bias, which posits that the properties of a given halo may be determined by properties other than mass, such as the halo environment. [fig:nsatsdens] Selecting for objects in a dense environment increases by almost 25% the probability of having Nsubs = 2 or more, from 9% to just over 11%; this effect becomes stronger for larger values of Nsubs. The Milky Way is somewhat overdense on this scale, having a massive companion, M31, as well as a number of surrounding dwarf galaxies. Recent estimates put the total mass of the local group at roughly $5\times 10^{12}{\hbox{${\rm M}\_\odot$}}$, solidly in a dense environment on the ∼ 1 Mpc scale. Even if we ignore the contribution to the local density from all neighboring objects except M31, the MW lies close to the top quartile of our measured local density distribution in Bolshoi on a ∼ 1 Mpc scale. We can begin to directly quantify the impact of such a massive neighbor on the subhalo population by identifying MW-mass halos which have M31-like neighbors with ${\hbox{$M\_{\rm vir}$}}= (1-3)\times 10^{12}{\hbox{${\rm M}\_\odot$}}$ at a distance of 700-900 kpc. These systems are represented by the green circles in Figure [fig:nsatsdens], which have a Nsubs = 2 probability enhanced by 2σ relative to the full sample. Thus, the local environment around the MW, which is dominated by the presence of M31, boosts the chance of seeing the Magellanic Clouds in the Milky Way. Probability distributions ------------------------- As an extension of the trends shown in Figure [fig:vmaxcomp], it is useful to generalize these results to arbitrary host and subhalo masses by fitting the probability distribution. A Poisson distribution seems to be the most natural assumption, and has been found previously in the literature. Deviations from a Poisson distribution can be quantified by the calculation of the second moment, α2, where 2 = N(N-1)N 2. For a Poisson distribution, α2 = 1. While studies have found some evidence for a deviation from Poisson, with α2 > 1, the distribution has been more fully quantified by BK10. This work found that the distribution could be much better modeled using a negative binomial distribution, P(N\|r,p) = (N+r) (r)(N+1) pr (1-p)N [eq:negativebinomial] where p = 1 1 + (22-1)N, r = 1 22 - 1, [eq:pr] although they note that a Poisson distribution is a reasonable fit when ⟨N⟩ is low, such as the case of the MCs. We plot fits to both a Poisson and a negative binomial distribution to our measurements as the dotted and dashed lines in Figure [fig:vmaxcomp]. We again see that, because ⟨N⟩ < 1, the Poisson distribution provides a reasonable fit, while the negative binomial distribution fit is excellent (at the expense of adding an additional parameter).[1](#fn1) Our results confirm those of BK10, in that the distribution becomes significantly more non-Poisson for larger values of ⟨N⟩. As a final comparison of the Bolshoi halos to previous work in the literature, we plot in Figure [fig:distributionfits] the trends of ⟨N⟩ (top panel, similar information to Figure 15 in ) and α2 (bottom panel) with mass ratio vacc, sub/vmax, host for a range of host masses. Here, vacc, sub is the maximum circular velocity, vmax, that the subhalo had at the epoch of accretion. As seen in the top panel, the value for ⟨N⟩ scales almost self-similarly, but with a slight increase in overall normalization as has been previously noted in the literature, indicating that more massive halos contain somewhat more subhalos, proportionally, than lower mass halos. [fig:distributionfits] More interesting is the trend in α2 with mass, shown in the lower panel of Figure [fig:distributionfits]. Here we see evidence that, for subhalos with $v\_{acc,sub}/v\_{max,host} {\lower.5ex\hbox{{$\; \buildrel > \over \sim \;$}}}0.25$, self-similarity breaks down much more significantly, where lower-mass hosts tend to have systematically higher values of α2, which implies increased scatter beyond Poisson statistics. There are a number of possible explanations for this trend; it may be due to the fact that lower mass objects exist in a wider range of environments than higher mass halos. As shown earlier in Figure [fig:nsatsdens], the immediate environment surrounding a halo can have a significant impact on its subhalo population. High-mass halos always form in regions of high bias, meaning that the environmental impact on a high mass object’s subhalo population is minimized. Low-mass objects are able to form in both biased and unbiased regions and in regions near to higher mass halos, creating a larger scatter in their subhalo populations. While not shown, we have made analogous plots to Figure [fig:nsatsdens] for hosts of higher mass and confirm that these objects typically display weaker environmental dependence in their subhalo populations. It is also interesting to note that previous studies, in particular and BK10, have not seen such a trend. There are a number of possible explanations for this, which include the improved $1 {h^{-1}\rm{kpc}}$ force resolution and $(250 {h^{-1}\rm{Mpc}})^3$ volume of the Bolshoi simulation, which allows for both better statistics and more accurate tracking of subhalos as they orbit around their hosts. The increased volume of Bolshoi is particularly important if the trend is driven by halo environment, since a larger volume is necessary to probe a full range of environments. Hints of this trend, however, can be noticed in Figure 9 of BK10. While not shown, nearly identical trends in ⟨N⟩ and α2 persist if we consider the mass ratio vmax, sub/vmax, host, using the maximum circular velocity of the subhalo at z = 0 instead of at the time of accretion. To summarize, in this section, we have shown that satellites comparable in size to the LMC and SMC are relatively rare in MW mass objects, occurring in roughly 9% of the potential hosts. Their abundance is dependent on a number of host properties, such as the host mass and environment. The latter can provide a boost at the 25% level. The proximity of M31 puts the Milky Way in a higher density environment, which increases the likelihood of an LMC/SMC pair to 12%. Finally, it was shown that for massive subhalos, the scatter in the expected number of subhalos systematically decreases with increasing host mass. This may be related to the environmental variations previously observed; lower mass halos are able to form in a wider range of environments than higher mass objects. In the next section, we turn to the task of comparing our simulation results with observations from the SDSS, which requires the adoption of an algorithm for adding magnitudes to simulated galaxies. Satellite Statistics for Galaxies ================================= We turn next to a detailed comparison of these results with observational measurements. The classical satellite galaxies around the MW have been well-studied, with the SMC and LMC being the only satellites brighter than MV = − 16. Indeed, the next brightest object, Sagittarius, with MV = − 13.1, is roughly 3 magnitudes dimmer than the SMC. Motivated by this, we search for the distribution of the number of satellites of MV ≤ − 16 in other galaxies, both simulated and real. In order to distinguish the results in this section from the previous sections, we adopt the formalism where Nsats refers to the number of satellite galaxies around a host, identified based on their luminosities. This contrasts with mass-selected dark matter subhalos, which we characterized in the previous sections using the nomenclature Nsubs. The Observational Sample ------------------------ Our recent companion paper studied the population of such LMC and SMC-luminosity objects in the SDSS. Here, MW analogs were identified in the SDSS main sample by locating objects with absolute magnitudes Mr = − 21.2 ± 0.2 with no brighter companions within a line-of-sight cylinder of radius 500 kpc and length 28 Mpc (corresponding to a redshift of ± 1000 km s− 1).[2](#fn2) The abundance of satellite galaxies was then measured by calculating the likelihood of having Nsats neighbors in the photometric sample with apparent magnitudes 2-4 dimmer in a fixed 150 kpc aperture. While we discuss details of their method below, our goal here is to compare our simulated results to their observational constraints. Modeling Galaxy Luminosities ---------------------------- Because internal kinematics are difficult to measure observationally for dwarfs outside the Local Group, the observations of L10 cannot be directly compared to the results from the previous section, which determined abundances based on the vmax of the satellites. Instead, we employ a SubHalo Abundance Matching (SHAM) algorithm to our simulation to map magnitudes onto dark matter halos. Here, luminosities are assigned by matching the number density of observed objects brighter than a given luminosity with the number density of simulated objects greater than a given ${v\_{\rm max}}$. For subhalos, ${v\_{\rm max}}$ at the time of accretion is used instead of the present ${v\_{\rm max}}(z=0)$. We also impose a log-normal scatter between halo mass and luminosity. In our base model, we assume a scatter of 0.16 dex, consistent with constraints on larger group and cluster mass objects. This algorithm has been shown to be extremely successful in matching several statistical properties of the galaxy distribution, including the luminosity dependence and redshift scaling of the two-point galaxy correlation function for galaxies brighter than Mr < − 19, the galaxy-mass correlation function, and galaxy three point statistics. have shown that a similar procedure can simultaneously match the luminosity function and the Tully-Fisher relation locally. Here, we match halos directly to the binned r-band luminosity function in the local Universe as measured by for galaxies with Mr < − 12.8. While there is some uncertainty in these measurements depending on the detailed surface brightness correction, the objects we are interested in here are bright enough that this effect is unimportant. This luminosity function also reproduces the more recent number densities measured in SDSS DR7 down to Mr ∼ − 17. The results of this procedure are shown in Figure [fig:mrvpeak]. Here, the red line shows the relation between vmax, acc and magnitude, while the blue points show ${v\_{\rm max}}$ at z = 0. The systematic movement to lower ${v\_{\rm max}}$ for many objects is due to tidal stripping of subhalos. The resulting simulated catalog has a distribution of galaxies that is complete down to Mr = − 15.3. Note that our pure-abundance matching measurements from Figure [fig:mrvpeak] gives most likely mass estimates for the LMC and SMC to be ${v\_{\rm max}}= 85\pm 6$ and 65 ± 6km s− 1, respectively. However, as shown in, ignoring the effects of baryons likely increases the maximum circular velocity by roughly 10% (see their Figure 5), bringing our SHAM mass estimates into excellent agreement with the observations Using luminosities assigned in this way, we now make measurements analogous to those of L10. We select for MW candidates in the same way as L10. First, we make a mock lightcone by taking the z = 0 SHAM catalog of the Bolshoi simulation, placing an observer at the center of the box, and adding redshift space distortions. We passively evolve galaxy magnitudes to the appropriate redshift using the SDSS constraint Mr(z) = Mr(z = 0.1) + Q(0.1 − 1), with Q = − 1.3. We then identify all isolated objects by selecting halos with Mr = − 21.2 ± 0.2 (the upper grey band in Figure [fig:mrvpeak] and excluding objects having a brighter neighbor in a cylinder of radius 500 kpc and length 28 Mpc. The resulting ${v\_{\rm max}}$ distribution is shown in Figure [fig:shamvmax]. This distribution is somewhat lower than the best current estimates for the maximum rotational velocity of the MW, ${\rm v}\_{rot} = 220$ km s− 1. This difference should not be surprising since ${v\_{\rm max}}$ includes only the dark matter, while ${\rm v}\_{rot}$ is the total rotational speed of an actual halo that includes the full impact of baryons. Further, have shown that, because the Milky Way sits near the peak of the star formation efficiency, baryons play a significant role in determining the circular velocity profile of MW mass halos, resulting in a value of ${v\_{\rm max}}$ that is underestimated in dark matter-only simulations. However, the virial mass of the galaxies that pass our MW selection peaks at $1.3 \times 10^{12} {\hbox{${\rm M}\_\odot$}}$, in excellent agreement with the estimated MW mass from. We select for MC-analogs using a method similar to L10, identifying all satellites within the virial radius of our MW analogs that are 2-4 magnitudes dimmer (lower grey band in Figure [fig:mrvpeak]). The most likely ${v\_{\rm max}}$ values for our MC analogs are ∼ 80 and ∼ 70 km s− 1, in agreement with the 1-σ constraints from observations. [fig:mrvpeak] [fig:shamvmax] The Definitions of a Satellite ------------------------------ Before presenting detailed comparisons between the L10 observational data and our model, we note that the definition of a “satellite” differs between the two analyses. Because L10 used a photometric sample with hosts selected on luminosity and no spectroscopy for their satellites, the number of satellites was defined as the number of photometric pairs around a host. Specifically, they identified isolated MW-luminosity galaxies in the SDSS spectroscopic sample and counted the number of photometric pairs with Δmr = 2 − 4 within a fixed angular separation, nominally taken to be 150 kpc. They then applied a background subtraction to account for projection effects and correlated structures. While our SHAM catalog will let us select isolated host objects in an analogous way, because modeling the full distribution of galaxy apparrent magnitudes is beyond the scope of our model, we do not perform an identical calculation. Instead, we can select an appropriate definition of a satellite galaxy to mimic the observational effects. L10 presented two methods for background subtraction, which were used to calculate the number of satellites around a host galaxy. The first, which we refer to as isotropic subtraction, assumed that background objects had, on average, no dependence on position (implicitly ignoring the impact of correlated structures), to calculate an average background for their entire host population. Using this assumption, it is possible to calculate the average background to very high statistical accuracy. Their second method, which we refer to as annulus subtraction, measured the background object counts around each galaxy by considering the region just outside the projected region surrounding the galaxy. For completeness, we make comparisons to both methods. Because it removes the effects of correlated structures, comparison to the annulus subtraction method of L10 is straightforward. We define as a satellite galaxy any object within the appropriate spherical volume (taken to be 150 kpc in L10) that has an absolute magnitude that is 2-4 mag dimmer than its host. To compare to the isotropic subtraction method, which neglects the impact of correlated structures along the line of sight (see Section 3.2 of L10 for further discussion) requires additional modeling. We account for this by defining a satellite to be any object inside a cylinder of radius 150 kpc and length r0, where r0 is the correlation scale for the Milky Way host-satellite analog. This cross-correlation function between MW-luminous galaxies and objects 2-4 magnitudes dimmer is plotted in figure [fig:CrossCorrelation], and is well fit by a power law, (r) = (r/r0)- with r0 = 5.3 Mpc and γ = 1.7, with a similar correlation length to measurements for all galaxies of MW-like magnitudes inferred from. These authors measured r0 = 5.98 Mpc/h and γ = 1.9 for the autocorrelation function of galaxies with Mr = − 20.7 to − 21.7. We therefore consider a cylinder with radius 150 kpc and length 5.3 Mpc (the correlation length) to be the appropriate aperture for satellite selection to compare to the isotropic background subtraction of L10. Coincidentally, this correlation scale is very similar to the size of the expected redshift-space distortions. showed that a $\sim 10^{12} {\hbox{${\rm M}\_\odot$}}$ halo has a typical velocity dispersion of 125 km s− 1; at z = 0.08, where the bulk of our simulated halos live, this corresponds to a 3.7 Mpc spread due to the redshifts of the satellite population. A cylinder of this length thus represents the most accurate measurements one could make for the number of satellites around a host using spectroscopic data (without including some sort of additional background subtraction). [fig:CrossCorrelation] In Figure [fig:nsatscomp] we compare the P(Nsubs) distribution for four different satellite definitions: spherical and cylindrical apertures (which correspond to the L10 results using their annulus and isotropic background subtractions methods, respectively) as well as selecting satellites based on ${v\_{\rm max}}$ and abundance-matched magnitude. Here, the red circles consider subhalos within the virial radius of their host that have vmax = 50 − 80km s− 1. The black diamonds, blue triangles, and green boxes all use magnitude-selected satellites, counting objects 2-4 magnitudes dimmer than their host using the aforementioned SHAM model with cuts analogous to the observational criterion of L10. The red diamonds represent the “correct” number of satellites, using all objects inside the virial radius, The blue triangles use a cylindrical aperture, analogous to the isotropic background subtraction of L10, while the green boxes use a spherical aperture, equivalent to the annulus subtraction method of L10. All error bars were calculated using the jackknife method with 10 sub-regions. [fig:nsatscomp] A number of trends are readily apparent for the relations in Figure [fig:nsatscomp]. First, there is generally good agreement between the P(Nsubs) distribution for objects based on ${v\_{\rm max}}$ (red circles) with our SHAM magnitudes (black diamonds). This is to be expected if SHAM preserves the observed mass-luminosity relation in this regime, something that will only happen if an appropriate mass function is used. It is also interesting to note that our calculations for the cylindrical aperture (blue triangles) always lie above the other measurements for Nsubs ≥ 1. This shows that correlated structures are an important effect which is typically comparable to the signal from satellites that are formally inside the halo’s virial radius. Our measurements using a fixed 0.15 Mpc cylinder (green box) lie below other measurements for higher values of Nsats. This should be unsurprising, since the typical virial radius for our selected halos is roughly 250 kpc. It is interesting to note, however, that the cumulative probability for hosting (at least) one satellite differs by less than 5% when calculated using the virial radius instead of a radius of 150 kpc, while this difference increases rapidly for higher values of Nsats. All of these trends indicate the importance of careful background subtraction to eliminate this signal (such as the annulus subtraction of L10), or otherwise of comparing directly to simulations with similar selection criteria so as to properly interpret results. As concerns the length of our cylindrical aperture, we obtain almost identical results for any length beyond ∼ 1Mpc, beyond which the correlation function falls off substantially. Varying the cylinder length from 2 − 8 Mpc changes our results by roughly five percent, comparable to the size of our statistical errors. Thus, the comparison between satellites defined using our cylindrical aperture and the isotropic background subtraction technique of L10 should be robust. The numerical values for many of these probabilities — selection by rvir, as well as our cylindrical and fixed spherical apertures — are presented in Table [table:probabilities]. Note that, while we are only counting objects as satellites if they are 2-4 magnitudes dimmer than their hosts for the most accurate comparison with L10, these numbers are essentially identical to considering a threshold sample of objects 0-4 magnitudes dimmer than their (MW-like) hosts. Switching to this definition creates only percent-level changes, well within the statistical error bars. Comparisons Between Simulations and Observations ------------------------------------------------ Figure [fig:nsatsobssim] and Table [table:probabilities] show the main result of our analysis, making direct comparison with the results of L10. In the top panel, we compare the L10 P(Nsats) measurements using their isotropic background subtraction (black asterisks) with our SHAM results using subhalos inside a fixed cylinder with radius 150 kpc and length 5.3 Mpc (blue triangles; cylinder column in Table 2) that closely mimics their selection criteria. The bottom panel shows the comparison between the L10 annulus subtraction (black and gray asterisks) and our measurement of Nsats within a 150 kpc spherical aperture (green squares; Sphere column in Table 2). The black and gray asterisks in the lower panel represent the range of measurements based on uncertainty in selecting an optimal background subtraction. The difference between these sets of points thus gives an estimate for the size of the systematic errors in the measurements. Note that, after making their background subtraction, L10 also calculate a systematic adjustment due to catastrophic photo-z failures (see section 4.2 of L10). While we have included those corrections in Figure [fig:nsatsobssim], we have explicitly kept the lower error bar consistent with zero for any measurement that was consistent with zero at the 2-σ level before this systematic correction was included. For reference, Table [table:probabilities] also gives the P(Nsats) distribution for proper subhalos, i.e., objects within rvir that are 2-4 magnitudes dimmer than their host. The agreement between simulations and observations in Figure [fig:nsatsobssim] is generally very good. We take this to indicate a success for both the ΛCDM and SHAM models, particularly indicating, with excellent statistics, that the HOD predicted by Bolshoi for Milky Way-like objects at the massive end matches observations, and there is no evidence for either a missing or excess satellite problem for these objects. Considering our comparison between the cylindrical/isotropic sample, at no Nsats do the two results differ by more than ∼ 2.5σ, and in most situations agree to better than 1 σ. The comparison fares slightly worse for the spherical/annulus sample. Here, the simulations underpredicts the number of Nsats = 0 systems, while over-predicting the Nsats = 1 systems at the 3 σ level. While this could indicate a real disagreement of the simulation and galaxy formation model with the data at this mass scale, there are a number of other possibilities. First, it is possible that the annulus-subtraction method of L10 over-corrects the background subtraction. This has been tested in simulations, using their algorithm to estimate the number of objects within 150kpc around our halos centers using our cylindrical counts. The result, is an over preduction of the Nsats = 0 term by 6% and an underprediction of the Nsats = 1 term by 5%, larger than the estimated systematical uncertainties in Table 1 of L10 (the changes in the higher Nsats terms are of similar size to their estimated systematic uncertainty, and are therefore well characterized by the gray points in Figure [fig:nsatsobssim]). Such corrections would bring the these terms within 2- and 3- σ of our simulated estimates. It is also important to note that the predictions from simulations depend on the assumed scatter between mass and luminosity. This will be further discussed below. [fig:nsatsobssim] \| c \|\| c \| c c \| c c\| & rvira & Cylinderb & Isotropicc & Sphered & Annulusc P(0) & 55 ± 4% & 55 ± 5% & 66− 1.3+ 1.6 & 71 ± 3% & 81− 1.4+ 1.5 P(1) & 28 ± 3% & 28 ± 3% & 22− 1.9+ 1.7 & 23 ± 2% & 12− 1.6+ 1.8 P(2) & 11 ± 1.4% & 11 ± 2% & 8.9− 1.5+ 1.5 & 5.2 ± 1.6% & 3.5− 1.5+ 1.3 P(3) & 3.8 ± 0.7% & 3.8 ± 1.2% & 1.5− 0.7+ 1.0 & 1.0 ± 0.8% & 1.6− 1.6+ 0.9 P(4) & 1.55 ± 0.4% & 1.5 ± 1.1% & 0.90.5+ 0.7 & 0.2 ± 0.1% & 1.1− 1.1+ 0.6 ahalos contained within the virial radius of the host bhalos within a cylinder with radius 150 kpc and length 5.2 Mpc cL10 results counting objects within a projected 150 kpc and an isotropic background subtraction dhalos within a sphere of radius150 kpc eL10 results counting objects within a projected 150 kpc using an annulus background subtraction [table:probabilities] Looking at the SDSS/Bolshoi comparisons in more detail, we can further investigate the dependence of the P(Nsats) distribution on the aperture size, as done in L10. For this, we determine the probability of finding Nsats satellite galaxies in a spherical aperture around their host, where the radius is varied from 100 to 250 kpc using the annulus background subtraction. The results are shown in Figure [fig:nsatsrad]. Here, the red circles, orange squares, magenta triangles, and cyan inverse-triangles represent the probability of having 0, 1, 2, or 3 subhalos. Open symbols represent SDSS measurements, while filled symbols come from our Bolshoi+SHAM model, while the hatched lines represent the P(Nsats) distribution for satellites within the virial radius of their hosts. Again, we see good agreement between simulations and observations, although we note the persistence of higher values in the simulation for the Nsats = 1 measurement. However, the other values of Nsats are in much better agreement. [fig:nsatsrad] One of the appeals of the SHAM algorithm is that it has so few free parameters, in this particular case just the scatter (although there are additional implicit assumptions, e.g., about subhalo stripping and star formation after satellite accretion). Figure [fig:nsatsscatter] shows how the scatter impacts this result. Here, we present the probability distribution for an isolated host with Mr = − 21.2 to host Nsats satellites brighter than − 16.5 as we vary the scatter in the SHAM model from 0 to 0.3 dex using our cylindrical method for selecting satellites. While none of these models are ruled out by the data (L10 isotropic model), a number of systematic trends are present. In particular, there is a systematic increase in the probability of hosting 3 or more satellites, along with a decrease in the probability of hosting 1 satellite (by up to 5%). This trend is caused not by scatter in the mass–luminosity relation of the satellite galaxies, but in the mass–luminosity relation of the host objects. As scatter increases, higher and higher mass hosts scatter into our magnitude-selected sample, bringing with them their richer subhalo populations. As we shall discuss in the next section, this has the effect of creating a tail to the negative-binomial distribution that sets the probability for a host to have Nsats satellites. [fig:nsatsscatter] It is also worth noting that the dark blue points representing a scatter of σ = 0.3 dex look to be violating the trend of an increasing high-Nsats probability. This is not due to any special behavior in this regime, but more represents the limits of our simulation resolution. As scatter increases, objects with Mr − 5logh > − 16.5, which nominally have vacc = 90 km s− 1using a scatter-free SHAM, have infall masses as low as 65 km s− 1in our σ = 0.3 dex model. Because tidal stripping can reduce the ${v\_{\rm max}}$ of a subhalo by a factor close to 2, this puts such objects dangerously close to the Bolshoi completeness limit of 50 km s− 1. It should be noted, however, that the completeness of subhalos in the Bolshoi simulation is highly dependent on the host mass. Because of stripping of subhalos after accretion, we expect to be losing potentially 10-15scatter this high. It’s important to appreciate the high resolution necessary to fully model a galaxy population using SHAM. While a scatter of σ = 0.3 is likely ruled out for massive galaxies, if such scatter were appropriate the Bolshoi simulation would only be able to populate galaxies down to Mr ∼ − 17.5 while losing fewer than 10% of the objects. Finally, we should note that there is no compelling reason to assume that both massive and low-mass galaxies exhibit the same scatter in the mass-luminosity relation. We have investigated this by quantifying the variations in the P(Nsats) distribution where we keep the mass-luminosity scatter of the hosts fixed at 0.16 and vary the scatter of the subhalos from 0-0.3. While we see similar trends with, e.g., the Nsats = 1 term dropping with increased scatter, the effect is significantly weaker than shown in Figure [fig:nsatsscatter], where we allow the scatter in the host mass-luminosity relation to vary. While it is very possible that the scatter should be even higher than 0.3 for our satellite galaxies, the 50 km s− 1resolution limit of Bolshoi prevents us from exploring a wider range. Thus, over the ranger we are able to probe, the scatter in the host mass-luminosity relation, which sets the range of host masses sampled, has a larger impact that the scatter in the satellite mass-luminosity relation. Environmental Effects --------------------- We now revisit the discussion regarding the environmental impact on the subhalo population of §[sec:env], using observationally motivated environmental measures. Recall that in §[sec:env], we defined the local environment as the dark matter density in collapsed objects larger than $2 \times 10^{11} {\hbox{${\rm M}\_\odot$}}$. Here, we preform a similar analysis using luminosity as an environmental proxy. Specifically, using the isolated central galaxies from our SHAM catalog as selected above, we measure the total luminosity from galaxies brighter than Mr = − 20.8 (which corresponds to Mr − 5log(h) = − 20 in the usual SDSS units) within a sphere of 1 Mpc around our hosts, which we take to be our environment proxy. We then split the sample into our most and least luminous (i.e., dense) quartiles and re-measure the P(Nsats) distribution for satellites inside their host virial radii. The results are shown in Figure [fig:nsatsmagdens]. Here, the black points represent the distribution for all isolated hosts, while the red and blue represent those in high/low density environments. As before, we see a clear offset for galaxies in different environments. While the differences are only at the 1-2 sigma levels, high-density halos are systematically more likely to have massive subhalos than those in low density environments. In particular, the underdense halos are 14% more likely to host no massive subhalos than their dense counterparts. Similarly, halos in dense environments are twice as likely to host two or more subhalos than those in low density environments. This effect should therefore be an observable manifestation of assembly bias. [fig:nsatsmagdens] Statistics of Satellites in Luminosity-selected Hosts ===================================================== As noted previously, the inclusion of scatter in the halo ${v\_{\rm max}}$-galaxy luminosity SHAM creates a significant tail in the probability distribution for a host to have Nsats satellite galaxies. Consequently, the negative binomial distribution that was found to describe the $P(N\_{\rm subs})$ distribution for the subhalos in a mass-selected host does not produce a good fit to $P(N\_{\rm sats})$ when applied to luminosity-selected galaxies. Additionally, the change in star formation efficiency with host halo mass breaks the self-similarity that was observed for host halos. In this section, we quantify the P(Nsats) distribution for satellite galaxies around a luminosity-selected host. In Figure [fig:nsatswithfit], we show a series of probability distributions for a Mr = − 20.5 galaxy to host Nsats subhalos brighter than -19.8, -18.3 and -16.8 (equivalent to Mr − 5log(h) = − 19, − 17.5, − 16). This plot was generated using a scatter of σ = 0.16 dex. A low probability tail clearly extends to high Nsats. Such a tail was not present when considering ${v\_{\rm max}}$-selected hosts and subhalos in section [sec:pois]. This is emphasized by the dotted lines, which present the negative binomial distributions, equation [eq:negativebinomial], that characterized the P(Nsubs)distribution. While the negative binomial provided an excellent fit to the data for our mass-selected objects, it clearly fails here. While this tail may not appear to be a serious concern, since it is not important until P(Nsats) ∼ 1%, large surveys such as the SDSS contain more than 10,000 massive clusters, so even these small effects may be visible. [fig:nsatswithfit] [fig:fitparameters] As a better fit, we present a modified binomial distribution by adding a power law tail, P(N\|r,p,T,) = (N+r) (r)(N+1) pr (1-p)N + Te-N. [eq:fit] Here, the first term is just the negative binomial, with r and p functions of α2	arxiv_0000580
Intro-Stabilizing Byzantine Clock Synchronization in Heterogeneous IoT Networks [1](#fn1) ========================================================================================= For reaching dependable high-precision clock synchronization (CS) upon IoT networks, the distributed CS paradigm adopted in ultra-high reliable systems and the master-slave CS paradigm adopted in high-performance but unreliable systems are integrated. Meanwhile, traditional internal clock synchronization is also integrated with external time references to achieve efficient stabilization. Low network connectivity, low complexity, high precision, and high reliability are all considered. To tolerate permanent failures, the Byzantine CS is integrated with the common CS protocols. To tolerate transient failures, the self-stabilizing Byzantine CS is also extended upon open-world IoT networks. With these, the proposed intro-stabilizing Byzantine CS solution can establish and maintain synchronization with arbitrary initial states in the presence of permanent Byzantine faults. With the formal analysis and numerical simulations, it is shown that the best of the CS solutions provided for the ultra-high reliable systems and the high-performance unreliable systems can be well integrated upon IoT networks to derive dependable high-precision CS even across the traditional closed safety-boundary. clock synchronization, Internet of things, Byzantine fault, intro-stabilization Introduction ============ Clock synchronization (CS) is fundamental in designing traditional Distributed Real-Time Systems (DRTS) and today’s Real-Time Embedded Systems (RTES), Cyber-Physical Systems (CPS), Wireless Sensor Networks (WSN), Internet of Things (IoT), and many other distributed systems. In practice, by providing a sparse global time base for distributed applications, not only the communication efficiency can be improved with statically optimized Time Division Multiple Access (TDMA) schedules but the system design and verification can be greatly simplified in comparison with that of asynchronous real-time systems. Traditionally, as the required resources in providing the fault-tolerant CS service are often expensive, only a small number of real-world DRTS systems (such as the high-end safety-critical systems in avionics) can acquire some kind of reliable global time base. While most of the other DRTS systems can only be built upon some unreliable CS schemes or even be deployed under the traditional globally asynchronous architecture. With the rapid development of embedded computing and communication technologies, this situation changes drastically. On one aspect, the required communication and computation in implementing the fault-tolerant CS become more and more affordable, with the progress of low-end embedded Commercial-Off-The-Shelf (COTS) devices such as the Ethernet, embedded processers, and Field Programmable Gate Array (FPGA). On the other aspect, with the boost of everywhere computing and communicating, various modern DRTS systems are booming in accommodating ever-changing personal and social needs. As a result, being built upon multi-scales networks comprised of Wide Area Network (WAN), Local Area Network (LAN), and Personal Area Network (PAN) with different networking technologies including traditional Ethernet, Software Defined Network (SDN), Time-Sensitive Networking (TSN), Software-Defined Radio (SDR), and even Radio Frequency Identification (RFID), these modern DRTS systems exhibit great diversity and complexity. In this background, dependable CS would play a more and more critical role in seamlessly integrating the trustworthy services for the diversified DRTS applications. However, there is still a big gap between the dependability of the CS solutions provided in the emerging diversified DRTS and that provided in traditional DRTS. At one extreme, high-end DRTS (like trains and civil aircraft) often requires the Mean-Time-To-Failure (MTTF) to be significantly better than 109 hours. To satisfy this, distributed CS systems are often built upon small-scale communication networks with statically connected homogenous components. In this context, Byzantine-fault-tolerant CS (BFT-CS) solutions are provided with the assumption that a fraction of the distributed components can fail arbitrarily, or saying, under the full control of a malicious adversary. Further, self-stabilizing BFT-CS (SS-BFT-CS) solutions are also provided with tolerating both transient system-wide failures and an amount of permanent Byzantine component failures. At the other extreme, the CS schemes, such as Network Time Protocol (NTP) and Precision Time Protocol (PTP), referenced in the emerging IoT systems often inevitably run in large-scale open environment (such as the Internet) with dynamically connected members. In this context, the proposed CS solutions are seldom under the assumption of a non-cryptographic adversary or even just the common computationally limited attackers. Although some existing works deal with the attack-monitoring or reconfiguration problems upon open-world networks, the current results are far from being sufficient in concerning the possible far-reaching influence of future DRTS, especially the IoT systems. For example, with the ever-evolving communication technologies, today there are SDN, SDR, TSN, and various kinds of customized and non-standardized more intelligent switches and routers. As more and more core network functions are developed with programmable and flexible devices such as embedded processors and Field-Programmable Gate Array (FPGA), the failure modes of these devices are more and more unpredictable. However, malign faults are seldom considered in building practical IoT systems. For another example, some industrial safety-critical applications can be attacked by open-world hackers and result in loss of control (as the recent accident encountered by the Colonial Pipeline ). Especially in considering that there might be a great number of safety-critical applications to be built upon various IoT systems, the internal operations of these systems should be safe enough. In this respect, the dependability of CS solutions (such as the master-slave paradigm taken in PTP) proposed in the emerging IoT systems (might be with a massive number of sensors and actuators) is far below that of the SS-BFT-CS solutions taken in traditional DRTS. And this would expose the IoT systems to risks of uncovered common malfunctions, undetected attacks, or even undesired emergences in considering the so-called one-in-a-million events or just the unknown intelligent invaders with limited computational resources. Motivation ---------- To mitigate this gap, we aim to provide CS solutions with both high reliability and high performance upon the emerging IoT networks. Concretely, we would investigate the intro-stabilizing (IS) BFT-CS problem upon IoT networks where some kind of external clocks are expected to be utilized while such kind of external clocks is not always reliable. The so-called intro-stabilization is extended from the traditional concept self-stabilization to provide discreet use of the external resources like the open-world reference clocks. Meanwhile, the BFT-CS problem is investigated in sparsely connected low-degree IoT networks. Also, by leveraging the existing CS schemes like the PTP as low-layer primitives, we expect that the advantages of the original CS schemes, such as hardware-optimized time precision and computational efficiency, can be inherited in the overall CS systems. In presenting the IS-BFT-CS solution, we would also discuss the decoupled and easier error-detecting, correcting, fault-tolerant startup, and restartup procedures in the presence of the malicious adversary. With this, we expect that the reliability, efficiency, and synchronization qualities of the CS systems can be better integrated by complementing traditional BFT-CS solutions with widely available time references given in the open world. Main obstacles -------------- In considering the overall problem, firstly, as real-world IoT networks are often across the WAN, LAN, and PAN areas, the first-of-all question is how a dependable CS system can be deployed in such an all-scale network. From the traditional viewpoints, current assumptions about the open-world adversaries might be overoptimistic. For example, an unknown number of attackers arbitrarily distributed on the Internet may be very familiar with the provided CS algorithms. Meanwhile, they can often well-disguise themselves to attack the target systems. What is more, if these intelligent network neighbors can attack some system from somewhere of the open-world network for a while, it is no reason to think that they would not attack it intermittently from elsewhere of the network. In this situation, the attack-monitoring for the synchronization states and multi-source selection may be insufficient. Notice that such worst cases in the open world is much different from that in the closed world where all components are only exposed to physical permanent failures and unintentional system-wide transient failures within the strictly closed safety-boundary of the system. Secondly, in considering malign faults in practical IoT systems, as we can hardly restrict the kinds of hardware devices, networking schemes, or low-layer protocols in developing the CS systems, the failure modes of the synchronization nodes can hardly be restricted. In this context, it is safe to assume that these synchronization nodes can fail arbitrarily, for example, sending very different clock information and local states to different recipients. Meanwhile, with the fault-independence assumption of distributed systems, it is unlikely that more than a fixed number of synchronization nodes are faulty at the same time in a real-world IoT system, providing that this system is operated in a distributed and closed way. In this situation, it is often sufficient to assume that at most f nodes are faulty arbitrarily in the n-node distributed synchronization system while all the other n − f nodes are nonfaulty. With this, the core problem is to provide the desired distributed services with the nonfaulty nodes in the presence of up to f arbitrarily faulty nodes that are arbitrarily chosen and fully controlled by a malicious adversary. This is in line with the core abstraction of the classical Byzantine General Problem (BGP ). In the literature (and also in this paper), the arbitrary faults that happened in the distributed nodes are referred to as the Byzantine faults. Meanwhile, the distributed nodes being suffered from the Byzantine faults are referred to as the Byzantine nodes. Thus, in the IoT networks, to validate the assumption that there are at most f Byzantine nodes in the system, we do not allow the core BFT-CS algorithms to run across the WAN area. Meanwhile, as the terminal devices (such as the sensors and actuators) deployed in the PAN area of the IoT systems are often energy-constraint (like the passive RFID tags), they can hardly be utilized as synchronization servers physically. So, we only allow these low-power end devices to be passively synchronized, just like the thin clients and thick clients proposed in. We see that there are several existing synchronization protocols such as Flooding Time Synchronization Protocol (FTSP) and Timing-sync Protocol for Sensor Networks (TPSN) aiming for synchronizing the low-power end devices with the edge nodes, for example, the digital-twins, of the upper-layer networks. However, although these synchronization protocols pave the promising way for far-reaching observing, modeling, and controlling of the infinite physical world, they are mainly provided in the open-world wireless networks and take the master-slave paradigm, which cannot establish nor maintain the desired synchronization states of the system in the presence of the so-called Byzantine faults. So, in viewing the big picture, it is urgent to build a reliable synchronization between the so-called edge nodes. Thus, we confine the main problem in this paper as to synchronize the devices in the LAN area with sufficient dependability, precision, and accuracy, while also providing a minimized safe interface in optionally communicating with the upper-layer CS schemes and the lower-layer CS schemes. With this minimized safe interface, the CS of LAN can be better integrated with the existing CS schemes of WAN and PAN. Despite the whole problem, the confined LAN-layer CS problem is still nontrivial in IoT systems. From the traditional viewpoints, one main obstacle in implementing a BFT-CS solution upon a practical LAN network is the insufficient connectivity of real-world communication infrastructures. Namely, as is manifested in the classical Byzantine agreement (BA) problem, the network connectivity should be at least 2f + 1 in tolerating up-to f Byzantine faults. Alternatively, to mitigate this, practical high-end solutions also invest in designing specific hardware Byzantine filters. However, real-world LAN networks of IoT systems can hardly afford sufficiently high connectivity nor sufficiently designed Byzantine filters. Despite the limited network connectivity, there are also other obstacles. Firstly, the required computation, storage, and communication in executing the BFT-CS algorithms often grow fast with the increase of the system scale. As there can be a massive number of nodes being deployed in the IoT networks, high scalability of the BFT-CS solutions is desired. Besides, as real-world communication infrastructures of IoT are diversified in physical interfaces (such as wired, wireless, optical) and technical standards (such as legacy Ethernet, Gbit Ethernet, SDN, TSN), an additional obstacle is that not all devices in the heterogeneous network can be directly connected. Also, as the numbers of network interface controllers (NIC) in devices like the Ethernet switches are always bounded, only networks with bounded node-degrees can be provided. Last but not least, the precision and accuracy required in the CS might be far below the maximal possible delay experienced in the IoT networks, which means that the basic fault-tolerant CS solutions provided in bounded-delay message-passing networks cannot be directly employed in IoT networks. New possibilities ----------------- Nevertheless, there are also new possibilities. Firstly, with today’s modularized communication technologies, an embedded IoT device can be equipped with several NIC modules, such as the Wireless Fidelity (WIFI) module, the fast Ethernet module, and the Gbit Ethernet module, to perform diversified measuring, monitoring, and even modeling functions. In this background, these devices can often connect more than one kinds of communication infrastructures. As is shown in Fig. [fig:network], each computing device (for example, the leftmost blocks) is allowed to communicate with more than one kind of bridge devices (the colored blocks) in the typical real-world heterogeneous LAN network. Following our former work, such computing devices can be employed as multi-degree nodes in the LAN networks. ![image](network.png) [fig:network] Secondly, unlike the traditional high-reliable CS solutions deployed in the fly-by-wire applications, the required overall weight, volume, and power supplies of the CS systems in IoT applications can be largely relaxed. Also, the needed recovery time of the IoT systems can be largely relaxed in most real-world applications in comparison with that of avionics systems. Moreover, as there are often various available external time references (such as the NTP and GPS clocks) in common IoT systems, various strategies can be proposed to utilize these external time references. In this context, self-recovery is not required to be theoretically self-stabilizing but is expected to be more accessible, flexible, and still reliable. For example, it is promising to seek ways to utilize available external time references while avoiding the intelligent attackers to leverage this as a new way to sabotage the system. So, here the new problem is to efficiently synchronize the IoT networks with available external resources in the presence of various faults. Besides, as is investigated in, some easy fault-tolerant operations can also be performed on the side of the bridge devices (such as the customized Ethernet switches, SDN switches ) or at least be performed on some embedded server node (the rightmost blocks in Fig. [fig:network]) being connected to each kind of communication infrastructure. With this, each kind of communication infrastructure together with the server node connected to it can be viewed as an abstracted node (i.e., a single fault-containment region, FCR ) in the LAN area. By this trade-off, the original arbitrarily connected communication infrastructures can remain unchanged while the minimal network connectivity required in classical BFT-CS solutions can largely be supported to some extent in some kind of bounded-degree networks. Basic ideas and main contribution --------------------------------- In this paper, we provide an IS-BFT-CS solution upon IoT networks where the communication infrastructures are heterogeneous, and the computing devices and the bridge devices are all sparsely connected (with bounded node-degrees). Firstly, for the efficiency of networking, we only require that each kind of communication network be arbitrarily connected (which is also the minimum requirement in the original IoT networks), and there are more nonfaulty communication networks than faulty ones. With this, as it is unlikely that there are more than a half number of the communication networks being faulty at the same time, the reliability of the overall CS system can be enhanced. The basic idea is that, as we can deploy much more terminal nodes in the system than the available communication networks, the insufficient connectivity of the physical networks can be largely compensated by viewing the subnetworks as super nodes being inter-connected with a number of terminal nodes. With this shrinking operation, the abstracted network would gain sufficient connectivity at the expense of increased failure rates of the super nodes. Now by allowing almost half of the super nodes to fail arbitrarily, the networking problem and the fault-tolerance problem can be better balanced. Secondly, for the high-quality and efficient CS, we employ the original CS schemes as synchronization primitives to achieve high synchronization precision without changing the underlying realizations of the primitives. With this, the provided BFT CS algorithms can achieve synchronization precision in a similar order to the original CS schemes in the presence of Byzantine faults. The basic idea is that, although synchronization precision provided in SS-BFT-CS solutions is often restricted by the maximal message delays, this precision can be further improved in stabilized CS systems by utilizing high-precision CS protocols like PTP as underlying primitives. For this, as the stabilized CS system can provide well-separated semi-synchronous rounds, synchronous protocols such as the approximate agreement can be well simulated in a semi-synchronous manner with temporally well-separated remote clock readings. So, with the basic convergence property of the approximate agreement, the synchronization precision can be in the same order as the bounded errors of the remote clock readings and bounded clock drifts. Thirdly, for the efficiency of the IS-BFT-CS solution, the exact Byzantine agreement is avoided in establishing and maintaining the synchronization. Moreover, the required stabilization time only depends on the number of the communication networks (denoted as n1) and is independent of the number of the terminal devices (denoted as n0). Furthermore, once the system is stabilized, the complexity of computation, communication, and storage would be linear to max{n0, n1}. The basic idea is that, by constructing a closed safety boundary for the core CS system, the internal operations of the system within the closed safety boundary can be largely independent of the unknown open-world attacks. With this, we can safely utilize some open-world time resources in the presence of possible attacks from open-world intelligent adversaries, as long as the adversaries cannot know when the open-world time resources are utilized. Concretely, in the provided IS-BFT-CS solution, the open-world time resources are only utilized when the system is not stabilized. This kind of property of the CS system is not much investigated in the existing works but may improve the reliability of real-world CS systems without adding great investments. Paper layout ------------ In the rest of the paper, the related work is presented in Section [sec:Related], with emphasis on the integration of distributed BFT-CS provided for the ultra-reliable DRTS applications and the common master-slave CS provided for high-performance, high-precision, but unreliable applications. The system abstraction of the considered IoT networks is given in Section [sec:Model]. In Section [sec:NonSSBFTCS] and Section [sec:SSBFTCS], the basic non-stabilizing BFT-CS and the basic IS-BFT-CS algorithms are successively introduced. The worst-case analysis of these algorithms is presented in Section [sec:Analysis]. In Section [sec:Result], simulation results are also given in measuring the average performance of the IS-BFT-CS solution. Finally, the paper is concluded in Section [sec:Conclusion]. Related works ============= Classical problem and solutions ------------------------------- Dependable clock synchronization is a fundamental problem in building dependable DRTS applications. Traditionally, as the certification authorities in the aviation industry demand convincible proof in showing the MTTF of the certified system being better than 109 hours, significant efforts have been devoted to providing ultra-high reliable CS solutions. To this end, as it is impossible to exhibit the desired system dependability by testing more than 100000 years, distributed fault-tolerant methods are developed under the assumption that the MTTF of the independent hardware components might be with several orders of magnitude below (as can be experimentally observed) than that of the desired systems. Under such assumptions, real-world distributed fault-tolerant systems are built by deploying sufficiently redundant subsystems. Moreover, as one cannot easily show the behaviors of the faulty subsystems being under some restricted patterns, it is often necessary to assume that the faulty subsystems can fail arbitrarily, i.e., being Byzantine. In this context, classical BFT-CS algorithms are proposed in satisfying the dependability demanded in communities ranging from aviation, on-ground transportation, manufacturing industries, and other safety-critical realms. Besides the basic BFT, the CS algorithms running for the dependable DRTS applications are also required to be self-stabilizing in tolerating transient system-wide failures caused by uncovered transient disturbances such as some severe interference like lighting and other unforeseen environmental hazards. Namely, after the arbitrary transient disturbance, as long as a sufficient number of DRTS components are not physically damaged, synchronization should still be globally established between the undamaged components within the desired stabilization time. As all the variable values recorded in the RAM devices of the DRTS system can be arbitrarily altered during the transient disturbance, an SS-BFT-CS algorithm should work under all possible initial states of the system. In this context, several deterministic SS-BFT-CS algorithms with linear stabilization time have been proposed upon completely connected networks (CCN). Furthermore, to break the hard lower-bounds on the stabilization time and complexity of the message, probabilistic SS-BFT-CS solutions are also explored. From theory to reality ---------------------- However, most real-world industrial SS-BFT-CS solutions are not built upon pure SS-BFT-CS algorithms. For example, the Time-Triggered Architecture (TTA) takes a light-weight SS-BFT startup procedure where some kinds of hardware Byzantine filters, such as the central guardians in the Time-Triggered Protocol (TTP) or monitor-pairs in Time-Triggered Ethernet (TTEthernet), are employed. With this, the advantage is that the stabilization time and complexity of the CS algorithms can be reduced in accommodating the stringent requirement of avionics and automotive industries. However, the expense is that the hardware Byzantine filters should be implemented and verified very carefully in both the design and realization processes to show adequate assumption coverage. Except for some high-end safety-critical applications, most common DRTS applications cannot afford such a delicate implementation. Besides the SS-BFT startup problem, a more fundamental restriction in applying the classical BFT solutions in typical DRTS applications is the networking problem. As most of the efficient SS-BFT-CS solutions are built upon CCN, real-world systems should provide sufficient network connectivity in simulating the original SS-BFT-CS solutions. For this, the most straightforward networking scheme is to connect all the computing devices with a bus or a star topology. Obviously, the disadvantage of such a naive solution is that the bus or the central bridge device in the star topology forms a single point of failure, which goes far from the original intention of distributed fault-tolerance. A better networking scheme employs two stars or switches in eliminating the single point of failure. However, such a basic redundancy can only tolerate benign failures of the bridge devices. In the literature, there are also BFT solutions that tolerate Byzantine faults in both computing devices and bridge devices. But these BFT solutions are often based upon special localized broadcast devices and synchronous communication networks and do not aim for solving the SS-BFT-CS problem. In, an SS-BFT-CS solution that tolerates Byzantine faults in both computing devices and bridge devices is proposed with expected exponential stabilization time and relaxed synchronization precision. So an interesting question is how to safely reduce the stabilization time with available external time resources in the open-world networks. Lastly, in considering the synchronization precision, although classical BFT-CS solutions can provide some deterministic precision and accuracy under the assumption of bounded message delays and bounded clock drift rates, these original properties often need to be further optimized to support ultra-high synchronization requirements. For example, some prototype solution that integrates the time-triggered communication and the IEEE 1588 protocol exists in providing high synchronization precision for prototype TTEthernet, but without considering the BFT nor the self-stabilizing problem. Later in the standard TTEthernet, such high synchronization precision is supported with hardware-supported transparent clocks. However, restricted failure-mode of the Time-Triggered switches is required, which is then supposed to be supported with specially designed monitor-pairs (can be viewed as the hardware Byzantine filters ). Other high-precision CS solutions, such as the one provided in the White-Rabbit (WR) project, can even achieve sub-nanosecond precision by integrating both Synchronous Ethernet (SyncE) and PTP. But it is only provided in the master-slave paradigm without considering malign faults. In the extended PTP solutions, people also seek ways to enhance the reliability of PTP with redundant servers. But these solutions are not for the Byzantine fault tolerance problem nor the stabilization (self-stabilization or intro-stabilization) problem. As far as we know, there is no integration of SS-BFT-CS solution and IEEE 1588 upon sparsely connected network in DRTS applications without assuming some components generating benign faults only. The missing world for synchronizing IoT --------------------------------------- We can see that, for the CS problem, although the communication infrastructures of IoT are not better than that of traditional DRTS, they are not much worse, especially in the LAN area. But existing CS schemes proposed for IoT (such as PTP) are mainly derived from the server-client paradigm (including the master-slave one, the same below) proposed for the Internet and WSN, while seldom from the distributed paradigm proposed for traditional DRTS. However, the server-client CS schemes adopted on the Internet, such as the NTP and Simple NTP (SNTP), are not intentionally provided for real-time applications and can only provide best-effort services with coarse time precision. Meanwhile, the CS schemes provided for the WSN, such as the FTSP, TPSN and other wireless synchronization protocols, are mainly for large-scale dynamical networks consisting of tiny wireless devices with strictly restricted power-supply and physical communication radius. Besides, these CS schemes are provided mainly for real-time measurements but not for hard-real-time controls like the CPS applications. As a result, most of these CS schemes cannot tolerate Byzantine faults of some critical servers, masters, or other kinds of central nodes. This would gravely restrict the reliability of the emerging far-reaching large-scale IoT systems. For a simple example, some middle-layer NTP servers deployed in the CS systems may be attacked by some stealthy attackers (hard to detect) to send and relay inconsistent messages to all other nodes. However, the receivers cannot always distinguish the faulty messages from the correct ones without employing Byzantine fault-tolerance. Viewing the CS solutions provided for the Internet and the WSN as vivid instances of social world synchronization and physical world synchronization, respectively, we see a missing link between these two ultimate worlds in looking forward to the future dependable IoT applications. But unfortunately, this cannot be fixed by only adopting some other kind of server-client solutions, such as gPTP and ReversePTP. To mend this, just between the social world where the members are intellectually unrestricted and the physical world where the devices are physically restricted, there might be a better place where certainties can be built upon firm realistic foundations. Namely, in the words of the multi-layer networks, the internal CS (ICS) in the LAN should be as dependable as possible to minimize the influence of uncertainties raised from both the WAN and PAN sides. In this context, the main problem is to provide efficient high-reliable ICS upon the LAN networks of IoT while maintaining the advantages (high-precision, low-complexity, low-cost, etc.) of the original unreliable CS protocols (such as PTP or even the ultra-high-precision WR). Also, as external time is often available in the IoT systems, some kinds of external time references may be helpful. Further, providing that the ICS systems can be well designed, the remaining problem is integrating these systems with external CS (ECS). For this, integrations of ICS and ECS are provided in the literature. But up to now, with our limited knowledge, the SS-BFT (and IS-BFT) ICS solution upon heterogeneous IoT networks is still missing. System model and the main problem ================================= In this section, we give a basic model to characterize the discussed heterogeneous IoT network in handling the related CS problem. Generally, the whole IoT system N is constituted by three kinds of subsystems: the WAN systems, the LAN systems, and the PAN systems. For the confined CS problem, we first introduce the LAN system and then briefly introduce its interfaces to the other two kinds of systems. The LAN system -------------- As is presented in Fig. [fig:network], an LAN system (denoted as L) consists of n0 ≥ 6 terminal nodes (denoted as i ∈ V0 with V0 = {1, …, n0}) and a heterogeneous bridge network G. The heterogeneous bridge network G is comprised of n1 ≥ 3 disjoint (homogeneous) bridge subnetworks, denoted as Gs ∈ G for s ∈ S = {1, …, n1}. Each such bridge subnetwork Gs = (Bs, Es) consists of ∣Bs∣ connected bridge nodes, each denoted as bs, q ∈ Bs, and ∣Es∣ bidirectional communication channels. As G is heterogeneous, the bridge nodes bs1, q1 and bs2, q2 cannot be directly connected whenever s1 ≠ s2. The terminal nodes can be connected to the bridge nodes with bidirectional connections (denoted as E0) but with the node-degree of every terminal node being no more than d0. Also, the node-degree of every bridge node is no more than d1. Thus, the network topology of L is a bounded-degree undirected graph, denoted as H = (V0 ∪ B1 ∪ … ∪ Bn1, E0 ∪ E1 ∪ … ∪ En1). Generally, the bridge network G can also be wholly or partially homogeneous. Here we consider the worst cases. Practically, as the number of the communication infrastructures is often limited, we assume n1 is a fixed number equal to or greater than 3. For simplicity, we assume d0 = n1 and each i ∈ V0 is a synchronization server node being directly connected to the n1 bridge subnetworks. It is obvious that H can be extended with an O(logn0) diameter for any d1 ≥ 3. In providing backward compatibility, we assume that each bridge subnetwork Gs is directly connected to a network-manager node s ∈ S with a bidirectional communication channel (as the server nodes in Fig. [fig:network]). The terminal nodes V0 and the network-manager (manager for short) nodes S are all referred to as the computing nodes, as they can perform the required computation. The bridge nodes in a nonfaulty Gs can deliver the messages between the manager node s and the terminal nodes directly connected to Gs following the underlying CS protocol P and communication protocol C. When considering babbling-idiot failures of the terminal nodes, the bridge nodes are assumed to be able to perform some rate-constrained communication for the incoming messages from the terminal nodes. Concretely, P and C can be respectively interpreted as PTP (or even WR) and some rate-constrained Ethernet (such as IEEE AVB, AFDX, TTEthernet, OpenFlow, TSN ) or other customized protocols. In considering BFT of the terminal nodes, we assume up-to f0 nodes in V0 can fail arbitrarily since the real-time instant t = t0. For simplicity, the real-time t is assumed to be a universal physical time, such as the Newtonian time. And if not specified, the discussed time, instants, durations and time intervals are all referred to the real-time. For our purpose, we assume the system is in an arbitrary state at t0, and we only discuss the system since t0. With this, if a terminal node is not a Byzantine node, it is a nonfaulty node that always behaves according to P, C, and the provided upper-layer CS algorithms. Besides, as the failures of the communication channels between the computing nodes and the bridge nodes can be equivalent to the failures of the computing nodes, the communication channels between them are assumed reliable. In considering BFT of the bridge nodes and the manager nodes, as we allow that the bridges in each bridge subnetwork can be arbitrarily connected, each bridge subnetwork Gs together with the manager node s are deemed as a single FCR. Concretely, a bridge subnetwork Gs is nonfaulty during a time interval [t, tʹ] if and only if (iff) all bridge nodes and the internal communication channels in Gs are nonfaulty during [t, tʹ]. We say a bridge node b being nonfaulty during [t, tʹ] iff b correctly delivers the messages during [t, tʹ]. In supporting the bounded-delay model, to correctly deliver a message m in b, b is required to deliver m within a bounded message delay δ`q` in executing P and C. Practically, this bounded-delay requirement can be easily supported with rate-constrained Ethernet or even traditional Ethernet under low traffic loads. For CS, firstly, we assume that each nonfaulty computing node i is equipped with a hardware clock Hi. To approximately measure the time, each Hi can generate ticking events with a nominal frequency 1/TH, where TH is the nominal ticking cycle. As the accuracy of real-world clocks is imperfect, the actual ticking cycles of Hi are allowed to arbitrarily fluctuate within the range [(1 − ρ)TH, (1 + ρ)TH], where ρ ≥ 0 is the maximal drift-rate of the hardware clocks. At every instant t, the nonfaulty node i can read the hardware clock Hi as the number of the counted ticking events, denoted as Hi(t) and referred to as the hardware-time of i at t. In considering the stabilization problem, Hi(t0) is assumed to take arbitrary values in a finite set [[τmax]], where [[x]] = {0, 1, …, x − 1} is the set of the first x nonnegative integers. And since t0, Hi(t) would not be written outside the hardware clock and would monotonically increase with respect to t in counting the ticking events when Hi(t) < τmax − 1. When Hi(t) = τmax − 1, Hi would return to 0 in counting the next ticking event and then continue to count the following ticking events. As Hi is read-only, it can be used for realizing the timers with fixed timeouts. In performing clock adjustments in executing the CS algorithms, other kinds of clocks should be defined. For simplicity, the value of the local clock Ci at instant t can be defined as $C\_i(t)=(H\_i(t)+\textit{offset}^{C}\_i(t))\bmod \tau\_{max}$, where offsetiC(t) is the value of the local-offset variable offsetiC at t. In executing the CS algorithms, the local-time Ci(t) is allowed to be read (or saying Ci being used as input) at any t by the P protocol running in i. Also, Ci(t) is allowed to be written (or saying Ci being adjusted) at any t by the CS algorithms running in i. With this, the basic accuracy of Hi(t) can be shared in Ci(t) while the timers and the adjustments of the local clocks are decoupled. Sometimes, we also need one or more kinds of logical clocks for convenience. For example, by defining the logical clock of node i as $L\_i(t)=(C\_i(t)+\textit{offset}\_i(t))\bmod \tau\_{max}$, Li(t) is called the logical-time of i at t. Here, the difference of the logical-time and the local-time of i is represented as the logical-offset variable offseti in i. In this way, the basic accuracy and synchronization precision of Ci(t) can be shared in Li(t) while the unnecessary coupling between the P and the upper-layer CS algorithms can be avoided. It should be noted that the upper-layer CS algorithms are not completely decoupled with the underlying P protocol, as we allow the upper-layer CS algorithms to adjust Ci instead of Li (or equivalently, we allow the underlying P protocol to use Li instead of Ci as its input). But such coupling is made as small as possible and can be supported in real-world realizations such as the common embedded systems. Besides the L clocks, other kinds of clocks can also be defined upon the local-time Ci(t) or directly upon the hardware-time Hi(t). For example, we can define some alien clock of node i as $Y\_i(t)=(H\_i(t)+\textit{offset}^{Y}\_i(t))\bmod \tau\_{max}$ (can be specifically called the alien-time). In considering the stabilization problem, all the offset variables for the clocks can be arbitrary valued in [[τmax]] at t0. For convenience, as the hardware-times, local-times, logical-times, and alien-times are all circularly valued in [[τmax]], we define $\tau\_1\oplus \tau\_2=(\tau\_1+ \tau\_2)\bmod \tau\_{max}$ and $\tau\_1\ominus \tau\_2=(\tau\_1-\tau\_2)\bmod \tau\_{max}$. And to measure the difference of two such times τ1 and τ2, we define d̊(τ1, τ2) = min{τ1 ⊖ τ2, τ2 ⊖ τ1}. On the whole, by viewing each bridge subnetwork Gs together with the corresponding manager node s as an abstracted bridge node j ∈ V1 (V1 = {n0 + 1, …, n0 + n1}), H can be further simplified as a completely connected bipartite network (CCBN) G = (V, E) with V = V0 ∪ V1 and E making the complete bipartite topology Kn0, n1. An abstracted bridge node j ∈ V1 is nonfaulty iff Gs, s, and the communication channels between them are nonfaulty. The failures of the edges in E are equivalent to the failures of the nodes in V0. With this, we assume that up-to f0 = ⌊(n0 − 1)/5⌋ terminal nodes in V0 and f1 = ⌊(n1 − 1)/2⌋ abstracted bridge nodes in V1 can fail arbitrarily since t0. All faulty nodes in V0 and V1 are denoted as F0 and F1, respectively. The nonfaulty nodes are correspondingly denoted as U0 = V0 \ F0, U1 = V1 \ F1 and U = U0 ∪ U1. As the network diameter of each Gs can be bounded within O(logn0), the overall delay of a message from a node i ∈ U0 to a node j ∈ U1 (and vice versa) can be bounded within 2δ`p` + O(logn0)δ`q`, where δ`p` is an upper-bound of the processing delay for every message in every nonfaulty computing node. For convenience, we assume the maximal overall message delay between i and j is less than δ`d`. For discussing CS upon the abstracted CCBN G, the clocks of each s ∈ S are also used as the clocks of the corresponding node j ∈ V1. For convenience, we use `s`(j) = j − n0 to denote the corresponding manager node that is abstracted in j. Also, for every s ∈ S, we use `s`− 1(s) = s + n0 to denote the corresponding abstract node j ∈ V1. This is only for strictly differentiating j and s in avoiding possible confusion. No algorithm really needs to compute `s`(j) nor `s`− 1(s). Similarly, we also define `s`(Vʹ) = {`s`(j) ∣ j ∈ Vʹ} and `s`− 1(Sʹ) = {`s`− 1(s) ∣ s ∈ Sʹ} for every Vʹ ⊆ V1 and Sʹ ⊆ S, respectively. Upon existing works, the given assumptions can be practically supported with today’s COTS devices commonly used in IoT networks. Also, it is often easier to add more terminal nodes than to add more communication networks in the IoT networks. By allowing n0 > 5f0 and n1 > 2f1, the minimal realization of the IS-BFT-CS system only requires n1 = 3, which is easier to be supported in real-world systems than the minimal requirement of deterministic BA (DBA) upon CCN. The interfaces for the two sides -------------------------------- In the IoT system N, the LAN system L should connect to one or more lower-layer PAN systems for interconnecting the things. Moreover, L is often connected to one or more higher-layer WAN networks for interconnecting of more things, as is shown in Fig. [fig:network2]. ![image](network2.png) [fig:network2] For the lower-layer side of L, each terminal node i ∈ V0 in the network H can serve as a synchronization server for the connected PAN nodes which serve as synchronization clients. These PAN nodes can be low-power receivers, mobile stations, or even in-hand or wearable devices with dynamic accesses. Each terminal node i ∈ V0 can connect to more than one PAN network for scalability. In the overall synchronization system, the communication between the terminal nodes in V0 and the PAN nodes is unidirectional. Namely, each nonfaulty terminal node i ∈ U0 periodically broadcasts its current clock to the connected PAN nodes. Meanwhile, the messages from the PAN nodes are all ignored by U0 in the synchronization system. For the upper-layer side of L, firstly, each manager node s ∈ S in the network H can be configured as a synchronization client for the connected WAN nodes. These WAN nodes, denoted as Z with ∣Z∣ = n2, serve as time-abundant external synchronization stations. Namely, each node z ∈ Z can access at least one kind of external time (UTC, TAI, etc.) with well-configured timing devices (such as GPS receivers, PTP clients, or just NTP clients), providing that the node z is nonfaulty. For simplicity and without loss of generality, we assume the external time is represented as the universal physical time t. And z ∈ Z is nonfaulty during [t1, t2] iff every connected nonfaulty manager node s ∈ S always reads the reference clock of z (denoted as Rz(t)) with ∀t ∈ [t1, t2] : R̂z, s(t) ∈ [t − e0, t + e0], where e0 is the external time precision. In the overall CS system, each z ∈ Z can connect to more than one LAN network (like L) for scalability. Now at the side of L, each s ∈ S is typically connected to one node in Z. Each s can also connect to more than one node in Z to tolerate some permanent faults that happened in Z (such as shown in Fig. [fig:network2]). Obviously, if more than one-half of the nodes in Z is always nonfaulty, the BFT-CS problem is trivial by taking the majority from the timing information given by Z in every nonfaulty s ∈ S. In this case, we also say that the external time is available in s. However, as this timing information is from the open world, we cannot ensure that a sufficiently large number of nodes in Z would always withstand all intelligent attacks from the open world. So the external time is not always available in s. This differs from the transient failures that should be tolerated with self-stabilization in traditional DRTS. Namely, with the more realistic consideration of the open-world malignity, the intelligent attacks might be launched with an arbitrary frequency and deliberately designed intermittent periods. Here, to differentiate it from the traditional self-stabilization problem and the Byzantine General problem, we can view the open-world time references in the overall synchronization problem as some resources in some Dark Forest. Namely, the so-called Dark Forest might be a good (but sometimes being regarded as over-permissive) metaphor of the open-world resources (the forest) along with the unknown dangers (the darkness). We argue that this kind of problem is not well handled in the open world and it might also be over-optimistically neglected in the emerging large-scale IoT systems. In the context of the Dark Forest, the nodes in S should not always depend on the open-world timing information to update their clocks. Instead, at every instant t, each nonfaulty s ∈ S should select a subset Zs(t) ⊆ Z to decide its current time servers. And when Zs(t) = ∅, it indicates that s does not use any timing information given by Z at t. So, a pure ICS solution is provided if Zs(t) = ∅ always holds for every nonfaulty s ∈ S and every t, just as the traditional ICS solutions. And an external-time-based ICS solution is provided if Zs(t) = ∅ holds whenever the system is stabilized while Zs(t) can be nonempty when the system is not stabilized. In considering the dependability of the CS system in the context of the Dark Forest, the provided IS-BFT-CS solution is an external-time-based ICS solution. In this vein, the clocks Y`s`− 1(s) derived from R̂z, s(t) for all s ∈ S and z ∈ Z are called the alien clocks. When the external time is available in s, we also say Y`s`− 1(s) is available. The underlying protocols ------------------------ To the underlying CS protocol P, we assume that if two nonfaulty nodes i and j are connected by a nonfaulty bridge subnetwork Gs, j can synchronize i with P upon Gs and vice versa. Concretely, suppose that a point-to-point CS instance of P, denoted as Pj, i, runs between a server node j ∈ U and a client node i ∈ U since t0 and no other instance of P runs between i and j nor any adjustment of Cj happens. Then, by running Pj, i in the server node j and the client node i, i can remotely read the local clock Cj(t) as Ĉj, i(t). And if for all t ∈ [t0 + Δ0, + ∞) $$\begin{aligned} \label{eq:p\_precision}\mathring{d}(\hat{C}\_{j,i}(t)-C\_j(t))\leqslant \varepsilon\_{0}\end{aligned}$$ holds, we say P is with the synchronization precision ɛ0 and a stabilization time Δ0 (which includes the time for establishing the master/slave hierarchy and establishing the master-slave synchronization precision). Further, if for all δ ≤ Δ $$\begin{aligned} \label{eq:p\_accuracy}\|(\hat{C}\_{j,i}(t+\delta)\ominus\hat{C}\_{j,i}(t))-\delta\|\leqslant\varrho\_0\delta+\varepsilon\_{0}\end{aligned}$$ also holds, we say P is with the accuracy $\varrho\_0$ for ɛ0 and Δ. For Pj, i, we assume ɛ0 and Δ0 are all fixed numbers specified by the concrete realization of P. And as no adjustment of Cj happens, the accuracy $\varrho\_0$ of P can be no worse than ρ for ɛ0 and some Δ ≈ τmax (slightly less than τmax, the same below). In considering Byzantine faults, if j is faulty, Ĉj, i(t) would be an arbitrary value in [[τmax]] at any given t. Here, the nodes i and j can be arbitrary computing nodes that are directly connected to a bridge subnetwork. In considering adjustments of Cj, for simplicity, we assume that the P protocol updates the remote clocks with the instantaneous adjustments rather than the continuous adjustments. Namely, when j ∈ U and an adjustment of Cj(t) (shown as the solid curve in Fig. [fig:adjustments]) happens at t2, although there can be a period [t2, t3] during which Cj(t) might be measured in node i ∈ U as a value Ĉj, i(t) being arbitrarily distributed in the intersection of a vertical line and the two disjoint grey regions ABCD and AʹBʹCʹDʹ, this value cannot be inside the white region CDAʹBʹ at any given t ∈ [t2, t3]. Calling [t2, t3] as an updating span of Cj, for every such updating span, we require that the updating duration t3 − t2 is bounded by δ0, after which ([eq:pprecision]) and ([eq:paccuracy]) should hold until the beginning of the next updating span. This requirement can be supported in most real-world hardware PTP realizations. Also, we note that the realizations of P with continuous adjustments can also be accepted in the P-based CS algorithms provided in this paper. But to our aim, as the clock-updating time-bound δ0 should be as small as possible for reaching faster stabilization, instantaneous updating is preferred as it can often be much faster than the continuous one. Also, as we should consider the worst-case performance of the CS solution, software optimization of the synchronization precision would not much help. ![image](PTP.png) [fig:adjustments] Lastly, to the underlying communication protocol C, for every i, j ∈ U, `s`(j) can correctly communicate with i by sending messages to Gs and vice versa. In the abstracted CCBN, every node j ∈ U1 can send arbitrary message m to every i ∈ V0 at any instant t ≥ t0. For efficiency, j can also broadcast m to all nodes in V0. When i ∈ U0 receives such a message m, i can deduce the sender of m in V1 with the connected communication channels. Also, every j ∈ U1 can deduce the sender of m in V0 with the nonfaulty bridge network G`s`(j) and the fixed communication ports. The messages can be signature-free, just like the unauthenticated messages sent in standard Ethernet, but should be with bounded frequencies and bounded lengths. The synchronization problem --------------------------- Now assume n0 > 5f0, n1 > 2f1, and there are no more than f0 and f1 Byzantine nodes in respectively V0 and V1 since t0 (at which the system L can be with arbitrary initial system state). Then, the nodes in U should be synchronized with the desired synchronization precision ɛ1 and accuracy $\varrho\_{1}$ upon G since t1, where the actual stabilization time t1 − t0 is expected to be sufficiently small. Concretely, for the distributed CS, we say the X clocks (X can be C, L, or Y) of P are $(\varepsilon,\varrho,\Delta)$-synchronized during [t1, t2] iff $$\begin{aligned} \label{eq:s\_precision\_general}\mathring{d}(X\_i(t)-X\_j(t))\leqslant \varepsilon\\ \label{eq:s\_accuracy\_general}\|(X\_i(t')\ominus X\_i(t))-(t'-t)\|\leqslant \varrho\_{1} (t'-t) +\varepsilon\end{aligned}$$ hold for all i, j ∈ P and all tʹ, t ∈ [t1, t2] with 0 ≤ tʹ − t ≤ Δ. With this, it is required that the C clocks (and thus the L clocks) of U should be $(\varepsilon\_{1},\varrho\_{1},\Delta)$-synchronized during [t1, + ∞) with some Δ ≈ τmax. And when this happens, we say L is $(\varepsilon\_{1},\varrho\_{1})$-synchronized (and also stabilized) with the stabilization time Δ1 = t1 − t0. As the X clocks used in this paper are all with the same value range [[τmax]], Δ ≈ τmax can be a common parameter in all cases. So for simplicity we say the X clocks are $(\varepsilon,\varrho)$-synchronized when the X clocks of U are $(\varepsilon,\varrho,\Delta)$-synchronized. To avoid DBA, we do not always require the stabilization time being a deterministically fixed duration. Instead, a randomized stabilization time with an acceptable expectation Δ1 is also allowed. In the context of the IoT networks, as the alien clocks are often but not always available, we should seek some discreet ways to integrate the ICS system with the alien clocks. By assuming that the failures of the nodes in L are independent of those of the alien clocks, the new problem posed here is to construct some more efficient complementary system to integrate the closed-world resources with the open-world resources. The real-world scenario is that, with the minimized safe interface of L, the failures that happened in the ICS system can be largely assumed to be independent of that of the alien clocks. Meanwhile, as the external time sources are often maintained in good condition, and the external attacks can often be promptly detected and handled with attack-monitoring, the alien clocks can be available most of the time. So, when the ICS system experiences some transient system-wide failures (often caused by improper internal operations or some temporary device malfunctions), the probabilities of unavailable alien clocks are low. Thus, this kind of availability of the alien clocks can be leveraged to integrate traditional ICS and the open-world time resources more discreetly. Non-stabilizing BFT-CS algorithms upon G ========================================== In this section, we first provide some non-stabilizing BFT-CS algorithms built upon some particular initial system states. Then we will use some of these algorithms as building blocks for constructing the IS-BFT-CS solution in the following section. For simplicity, we will prefer the abstracted nodes V1 to the manager nodes S in describing the algorithms running in the abstracted bridge nodes, although the algorithms for V1 might actually run in the manager nodes in concrete realizations. BFT remote clock reading ------------------------ Firstly, to be compatible with the underlying protocol P, we give the definition of the initially δ-synchronized state. [defsynchronized] L is initially δ-synchronized upon G with P at t iff t is not in any updating span of Ci for all i ∈ U and $$\begin{aligned} \label{eq:def\_precision} t\geqslant t\_0+\Delta\_{0} \land \forall i,j\in U: \mathring{d}(C\_i(t),C\_j(t))\leqslant \delta\end{aligned}$$ Now suppose that the system L is initially δ`I`-synchronized (upon G with P, the same below) at t1. With this, to provide BFT-CS for the nonfaulty nodes in G, the most natural method is to run the P protocol for each pair of nodes j ∈ U1 and i ∈ U0 with j being the server and i being the client. Then, for every t ≥ t1, each node i ∈ U0 can remotely read the local clock Cj(t) of j ∈ U1 as Ĉj, i(t) in i at t with an error bounded by ɛ0. Now as the local clocks of the nodes in U are initially synchronized within δ`I`, every node i ∈ U0 knows $\mathring{d}(\hat{C}\_{j,i}(t),C\_i(t))\leqslant \hat\delta(t)$ with some bounded $\hat\delta(t)$ when j ∈ U1 and t ∈ [t1, t1 + kδ0] with k ≥ 1 being a bounded integer. Thus, by computing the actual difference of Ci(t) and Ĉj, i(t) as $\tau\_{j,i}(t)=\hat{C}\_{j,i}(t)\ominus(C\_i(t)\ominus\hat\delta(t))$, i knows the values τj, i(t) are within a bounded range for all remote nodes j ∈ U1. So, by taking the median of τj, i(t) for all j ∈ V1 in each node i, the returned values of the FTA (fault-tolerant averaging ) operations in all nodes i ∈ U0 would be in a bounded range. Following this simplest idea, denoting the underlying server-client P protocol running for the server j and client i as Pj, i (referred to as the forward P protocol), the basic BFT remote clock reading algorithm `BFT\_READ` is shown in Fig. [fig:algosimplest]. For simplicity, we assume that the algorithms are sequentially executed, in which a pending function (i.e., a function should but not yet be executed) in each node i ∈ U would not be executed during the ongoing execution (if it exists) of any function in i. If there are several pending functions in i, their execution orders can be arbitrarily scheduled as long as the overall maximal message delay is still bounded in δ`d`. [1]1em//#1 [1] *for every node i* ∈ U0: initialize at t** run Pj, i for each j ∈ V1; *readClock at t*** τ : = Ci(t); determine $\hat\delta(t)$ as δ; τj, i : = Ĉj, i(t) ⊖ (τ ⊖ δ); set τ as the median of τj, i for all j ∈ V1; offseti : = τ ⊖ δ *for every node j* ∈ U1: initialize at t** run Pj, i for each i ∈ V0; [fig:algosimplest] Note that the `BFT\_READ` algorithm does not require that the node i ∈ U0 must actually adjust its own clock with the `readClock` function. It depends on concrete applications. Sometimes, calling the `readClock` function in responding to some irregular local events in i would suffice. In other situations where the synchronized clocks are frequently referenced, the `readClock` function can also be called in i to periodically adjust the logical clock Li(t) in tracing the synchronized clock at any given t. As we allow n1 = 2f1 + 1, the median function is used to tolerant one Byzantine node in V1 without the convergence property. Obviously, the `BFT\_READ` algorithm along has several problems. Firstly, during each call of the `readClock` function, the bound $\hat\delta(t)$ is dynamically determined. Surely $\hat\delta(t)$ can also be always determined as a constant number. But as the local clocks of nodes in U1 would drift away from the initial synchronization precision δ`I` without further synchronization, the median taken for the circularly-valued remote clocks may not always be correct if $\hat\delta(t)$ is constant. Secondly, the median function can only ensure its outputs in nodes of U0 are within the range of the original inputs from U1. Now as the ranges of τj, i(t) for j ∈ U1 in each i would grow wider with the accumulated clock drifts in U1, the worst-case synchronization error δʹ(t) in U0 would grow larger accordingly. In overcoming this, the local clocks of nodes in U1 should also be periodically synchronized. The basic synchronizer ---------------------- To synchronize the local clocks of nodes in U1, here we want to simulate the synchronous approximate agreement upon the CCBN G with n0 > 3f0 and n1 > 2f1. Concretely, with the initial precision δ`I`, besides running the forward Pj, i protocols as clients, the nodes in U0 can also act as servers to reversely synchronize the nodes in U1 with the backward Pi, j protocols. The so-called backward Pi, j protocols are very like the ones proposed in ReversePTP. The main difference is that there are n1 nodes to be synchronized, not just the central node in ReversePTP. Despite this difference, both the ReversePTP instances and the common PTP instances can be employed in realizing the backward Pi, j protocols. Upon this, the basic BFT-CS algorithm (also called the basic synchronizer) `BFT\_SYNC` is shown in Fig. [fig:algosimplest2]. [1]1em//#1 [1] *for every node i* ∈ U0: initialize at t** run Pi, j and Pj, i for each j ∈ V1; offseti : = 0; reset timer τw; *at local-time k*τ0 + δ3** writeLogicalClock(V1, Ci(t), δ6);[code:syncread] set timer τw with δ4 ticks; *when timer τ*w is expired** Ci(t) : = Ci(t) ⊕ offseti; [code:syncoffsetupdate] offseti : = 0; *writeLogicalClock(R, τ, δ) at t*** τj, i : = min{Ĉj, i(t) ⊖ τ ⊕ δ, 2δ};[code:synccomputelogical] set τ as the median of {τj, i ∣ j ∈ V1}; offseti : = τ ⊖ δ *for every node j* ∈ U1: initialize at t** run Pj, i and Pi, j for each i ∈ V0; offsetj : = 0; reset timer τw; *at local-time k*τ0 + δ1** writeLogicalClock(V0, Cj(t), δ5);[code:backwardsyncread] set timer τw with δ2 ticks; *when timer τ*w is expired** Cj(t) : = Cj(t) ⊕ offsetj; [code:backwardsyncoffsetupdate] offsetj = 0; *writeLogicalClock(R, τ, δ) at t*** τi, j : = min{Ĉi, j(t) ⊖ τ ⊕ δ, 2δ};[code:backwardsynccomputelogical] τi, j : = 0; set τ1 and τ2 as the (f0 + 1)th smallest and largest τi, j;[code:backwardsyncfta] offsetj : = ((τ1 + τ2)/2) ⊖ δ [code:backwardsyncfta2] [fig:algosimplest2] During the initialization of the basic synchronizer, every nonfaulty node runs both the forward and backward P instances and resets its logical clocks and timers. Here we say a timer (such as the timer τw) is reset (denoted as τw = τmax) if it is closed and would not run again before the next scheduling of it. And we say a timer is set with δ if it is scheduled with a timeout δ after which the timer would be expired and reset. The timeout is counted with the ticks of the hardware clock in case it is affected by upper-layer clock adjustments. For clarity, all ticks referred to in this paper are the ticks of the hardware clocks. With this, for every i ∈ U0, at each local-time kτ0 + δ3 (for k ∈ [[τmax/τ0]] with $\tau\_{max}\bmod \tau\_0=0$), i reads the remote clocks and use the median of these readings as the logical clock of i. After another δ4 ticks, i adjusts its local clock with its logical clock. Similarly, for the backward synchronization, each node j ∈ U1 reads the remote clocks and uses the fault-tolerant averaging of these readings as its logical clock at each local-time kτ0 + δ1. Then, j uses it to adjust its local clock after another δ2 ticks. Note that in line [code:syncoffsetupdate] and line [code:backwardsyncoffsetupdate], we allow Ci(t) and Cj(t) to be adjusted by Li(t) and Lj(t), respectively. This is necessary as the underlying P protocol in the server nodes should use the adjusted clocks rather than the original freely-drifting ones to ensure the differences of the referenced clocks in all nonfaulty server nodes being always in a bounded range. But to avoid undesired asynchronous clock adjustments, firstly, the newly acquired clock values are not directly written to the local clocks. Instead, the new values are first written to the logical clocks (with lines [code:syncread] and [code:backwardsyncread]) and then written to the local clocks after some statically determined delays. This is for simulating the synchronous approximate agreement upon G with lines [code:backwardsyncfta] and [code:backwardsyncfta2]. And secondly, in lines [code:synccomputelogical] and [code:backwardsynccomputelogical]), the offsets of the logical clocks would always be within [0, 2δ]. With this, the adjustments of the local clocks would be no more than δ`I` even when the system is not initially δ`I`-synchronized. As the clock adjustments performed by the basic synchronizer are for maintaining some synchronized states of the system, these clock adjustments are called the basic adjustments. In Fig. [fig:clockdependency], the temporal dependencies of the referred clocks are described with the labeled arrows. The clocks Ci, Li and Ĉj, i (on the left side of Fig. [fig:clockdependency]) are of the node i ∈ U0. And the clocks Cj, Lj and Ĉi, j (on the right side of Fig. [fig:clockdependency]) are of the node j ∈ U1. For the forward synchronization, when Ci(t) = kτ0 + δ3 is satisfied, Li would be written in the `writeLogicalClock` function in i with the remote clock readings Ĉj, i from all j ∈ V1. Then, Ci would be written with Li after δ4 ticks in i. And then, for the backward synchronization, with the underlying Pi, j protocol, Ĉi, j can be updated with the adjusted Ci during the next δ0 time (here we assume that the actual delay can be arbitrarily distributed in [0, δ0]). So by properly setting δ1, Lj can be correctly written with the all updated Ĉi, j for all i ∈ U0. And by waiting for another δ2 ticks, Cj can be correctly written with Lj. ![image](clocks.png) [fig:clockdependency] So, the remaining problem is to determine the time parameters δ1, δ2, δ3, δ4, δ5, and δ6. Firstly, d̊(Ĉi, j(t), Cj(t)) ≤ δ5 and d̊(Ĉj, i(t), Ci(t)) ≤ δ6 should hold in executing line [code:syncread] and line [code:backwardsyncread] of the `BFT\_SYNC` algorithm with the initially δ`I`-synchronized state. Secondly, δ1, δ2, δ3, and δ4 should be determined to ensure the basic synchronization procedure simulating the desired synchronous approximate agreement, as is shown in Fig. [fig:basicprocess]. ![image](basicprocess.png) [fig:basicprocess] In Fig. [fig:basicprocess], the fastest and slowest nodes in U0 (U1) are denoted as i1 and i2 (j1 and j2), respectively. It should be noted that the actual slowest and fastest nodes can change over time. Here the case is just for describing the desired basic synchronization procedure. Arrows still represent the influences between the clocks. For example, the leftmost curved arrow (on the local-time of i1) represents that the local clock Ci is adjusted at δ3 + δ4 with the logical clock Li being written at δ3. And the straight arrows represent the clock distributions from the server-clocks to the client-clocks with the underlying P protocols. Here, as the local clocks of all nodes in U are initially synchronized within δ`I`, the synchronization phases (separated by the long dotted lines in Fig. [fig:basicprocess]) in the distributed nonfaulty nodes can be well-separated in real-time if the synchronization precision can be maintained within some fixed bounds. In Section [sec:Analysis], we would see that with properly configured time parameters, the desired synchronization precision can be maintained in L with the initially δ`I`-synchronized state. Note that the basic settings of the time parameters are for strictly separating the synchronization phases shown in Fig. [fig:basicprocess]. Actually, by setting one or both of the parameters δ4 and δ2 being 0, the synchronization procedure can also be realized in a rather wait-free manner. For simplicity, we take strictly separated synchronization phases in this paper. With this, a basic synchronization round of the initially δ`I`-synchronized L can be defined with any periodically appearing synchronization phase shown in Fig. [fig:basicprocess]. For instance, the time interval [t0ʹ, tʹ] can be viewed as a basic synchronization round of L. And for every i ∈ U, when Ci(t) ∈ [(k − 1)τ0, kτ0) with k ≥ 1, we say i is in its kth local basic synchronization round. The strong synchronizer ----------------------- Besides the basic synchronizer, an additional pulse synchronizer is provided with the `BFT\_PULSE\_SYNC` algorithm, as is shown in Fig. [fig:algopulse]. The basic synchronizer together with the pulse synchronizer are called the strong synchronizer. Here the readers might wonder why more than one synchronization algorithm is provided. Roughly speaking, with the strong synchronizer, we can provide some easier evidence that once such evidence is observed in a nonfaulty node j ∈ V1, j would know that the system would be stabilized in an expected way. Upon this, if all nodes in U1 observe such evidence for a sufficiently long time, the extra self-stabilizing procedure would not be performed. We would further explain this when we construct the stabilizer with this strong synchronizer. Here we first describe the `BFT\_PULSE\_SYNC` algorithm and its relationship to the `BFT\_SYNC` algorithm. For simplicity, we assume n0 > 5f0 for the `BFT\_PULSE\_SYNC` algorithm. [1]1em//#1 [1] *for every node i* ∈ U0: at local-time kk`pls`τ0** [code:pulsesyncpulsecon] send pulse-k to each j ∈ V1;[code:pulsesyncpulsesend0] always Pk = {j ∣ i receives pulse-k from j in the latest δ16 ticks}; [code:pulsesyncalwaysreceive0] [code:pulsesyncpreempitvecon0] k\* : = kʹ;[code:pulsesyncpreempitvestart0] set timer τw\* with δ17 ticks; *when timer τ*w\* is expired** τʹ : = k\*k`pls`τ0 + δ9; τj, iʹ = Ĉj, i(t) ⊖ τʹ; [code:pulsesyncinput0] set τ as the median of {τj, iʹ ∣ j ∈ V1}; Ci(t) : = τʹ ⊕ τ;[code:pulsesyncmedian] offseti = 0;[code:pulsesyncpreemptivedo0] set protecti as 1 until $C\_i\bmod \tau\_0=0$;[code:pulsesyncpreemptivestop0] *for every node j* ∈ U1: always** Pk = {i ∣ j receives pulse-k from i in the latest δ10 ticks}; [code:pulsesyncalwaysreceive] [code:pulsesyncpreempitvecon] k\* : = kʹ;[code:pulsesyncpreempitvestart] set timer τw\* with δ11 ticks; *when timer τ*w\* is expired** τʹ : = k\*k`pls`τ0 + δ8; τ : = Ĉi, j(t) ⊖ τʹ;[code:pulsesyncrawinput] τi, jʹ = τ; [code:pulsesyncinput] τi, jʹ : = 0; set τ1ʹ and τ2ʹ as the (f0 + 1)th smallest and largest τi, jʹ; Cj(t) : = τʹ ⊕ (τ1ʹ + τ2ʹ)/2; offsetj = 0;[code:pulsesyncpreemptivedo] set protectj as 1 until $C\_j\bmod \tau\_0=0$;[code:pulsesyncpreemptivestop] *at local-time k*k`pls`τ0 + δ12** send pulse-k\* to each i ∈ V0;[code:pulsesyncpulsesend] [fig:algopulse] As is shown in Fig. [fig:algopulse], firstly, by setting k`pls` > 1, the additional synchronization would be performed with a lower frequency than the basic synchronization. This is for well separating the additional pulse-like sparse synchronization events. Besides, with line [code:pulsesyncpulsecon] of the `BFT\_PULSE\_SYNC` algorithm (i would count the ticks since the beginning if i has not yet sent the first pulse), a node i ∈ U0 would not send any two pulses within δ15 ticks. This would provide some good properties for constructing the overall IS-BFT-CS solution. Here, when it runs in the desired way, this additional synchronization procedure adds some header rounds (or saying headers) into the original synchronization procedure, as is shown in Fig. [fig:headerbody]. In each such synchronization header (the yellow block in Fig. [fig:headerbody]), there should be at least n0 − 2f0 nodes in U0 sending their pulses (shown in Fig. [fig:headerbody] as bold arrows) in a short duration no more wider than δ10/(1 + ρ) − δ`d`. In this sense, these nodes in U0 are called a pulsing clique in U0, as all their pulses are within a sufficiently narrow duration. ![image](2stagesprocess.png) [fig:headerbody] Then, to perform the desired additional synchronization in the presence of such a pulsing clique, the lines from [code:pulsesyncalwaysreceive] to [code:pulsesyncpreemptivestop] of the `BFT\_PULSE\_SYNC` algorithm (denoted as the `B_1` block) should be executed with a higher priority than all lines of the `BFT\_SYNC` algorithm. Namely, when a node j ∈ U1 writes the logical clock Lj and adjusts the local clock Cj in executing the `B_1` block, any attempt to write Lj or Cj in the `BFT\_SYNC` algorithm would be preempted and canceled during a bounded time interval. This is for avoiding the undesired output of the FTA operation of the `BFT\_SYNC` algorithm to overwrite the desired output of the `B_1` block in the presence of a desired pulsing clique. Moreover, we use the a flag protectj ∈ {0, 1} (with the default value 0) in the algorithm for each node j ∈ U to indicate that if the clocks of j should be protected from being adjusted outside this algorithm. Concretely, the clocks of j can be adjusted outside the algorithm if and only if protectj is 0 at t in node j. Besides, the executions of the `B_1` block can also be preempted and canceled by themselves when two or more such executions are temporally overlapped. In other words, the latter execution of the `B_1` block always has the higher priority (with even canceling the cancelation of clock-writings implemented in the former executions). Thus, with a pulsing clique, the local clocks of all j ∈ U1 would be semi-synchronously adjusted with the line [code:pulsesyncpreemptivedo] of the `BFT\_PULSE\_SYNC` algorithm. And all these local clocks would at least be synchronized with the precision in the same order of δ10. Similarly, the nodes in U0 would also be synchronized with such a coarse precision in the presence of the desired pulsing clique. Then, although this precision could be coarser than the desired final synchronization precision, it is not a problem as the synchronization header is followed by the synchronization body in executing the `BFT\_SYNC` algorithm. Namely, in the synchronization body (shown in Fig. [fig:headerbody] as the green blocks following the leftmost yellow one), a (k`pls` − 1)-round synchronous approximate agreement is simulated (one such round is also shown in Fig. [fig:basicprocess] in [t0ʹ, tʹ]). With this, by the end of the synchronization body of the desired synchronization procedure, the local clocks (and also the logical clocks) of all nodes in U would be synchronized with the desired precision. In simulating the approximate agreement, the convergence rate can be further improved by employing the advanced FTA functions given in. Here the basic solution employs the basic FTA function (with convergence rate 1/2) for simplicity. Generally, this header-body synchronization procedure is called a two-stage synchronization procedure. To make this two-stage synchronization procedure work in the presence of a pulsing clique in U0, firstly, the first stage should deterministically bring all local clocks of the nodes in U1 into the expected coarser precision. This is implemented by the `BFT\_PULSE\_SYNC` algorithm by making all pulsing cliques in U0 being well-separated in real-time. Secondly, the second stage should deterministically simulate the synchronous approximate agreement. This is implemented by the `BFT\_SYNC` algorithm with an initially δ`I`-synchronized state at the end of the first stage. In Section [sec:Analysis], we would show that this procedure can be performed with properly configured time parameters. For simplicity, the header cycle (i.e., the nominal duration of a header) is also set as the basic cycle τ0 (i.e., the nominal duration of basic synchronization round). And a header can be viewed as a special kind of basic synchronization round. Basic IS-BFT-CS solution ======================== The `BFT\_SYNC` and `BFT\_PULSE\_SYNC` algorithms are not self-stabilizing, since either an initially δ`I`-synchronized state or a pulsing clique is required in executing these algorithms. For stabilization, the system should be synchronized in some desired time with all possible initial states. In this section, we provide a basic IS-BFT-CS solution. The problem of stabilization ---------------------------- As there might be no initially δ`I`-synchronized state at t0 nor any desired pulsing clique since t0, reliable synchronization cannot be established with only the strong but still non-stabilizing synchronizer. For stabilization, some kind of BFT stabilizers can be employed. The so-called BFT stabilizers, such as the ones proposed and utilized in, are able to convert non-stabilizing BFT protocols to the corresponding stabilizing ones. For example, the self-stabilizing DBA (SS-DBA) algorithm proposed in is used as a primitive in construction the deterministic SS-BFT-CS in. For some other examples, some resynchronization algorithms are used as BFT stabilizers in the SS-BFT-CS algorithms provided in. Obviously, if the manager nodes are fully connected, we can directly employ some existing SS-BFT-CS algorithms to construct the core synchronization system and then distribute the clocks of the manager nodes to the whole system. However, the existing SS-BFT-CS solutions have several disadvantages in the specific context of IoT networks. Firstly, building upon the classical bounded-delay assumption, all the existing SS-BFT-CS solutions are with synchronization precision no better than o(dʹ), where dʹ is the maximal message delay in the corresponding communication networks. In contrast, CS protocols such as PTP can often achieve better precision with several low-cost hardware and software optimizations. Secondly, most existing SS-BFT-CS solutions are constructed by periodically executing some kind of BA protocol, which generates additional complexity even when the system is stabilized. In contrast, CS protocols such as PTP require very sparse resources in maintaining the stabilized state of the system. Thirdly, although some randomized SS-BFT-CS algorithm does not rely on BA protocols, the expectation of the stabilization time is at least O(n), where n is the number of the synchronization nodes in the system. In contrast, CS protocols such as PTP trivially have a deterministic constant stabilization time. Fourthly, almost all existing SS-BFT-CS solutions require CCN in exchanging the synchronization messages. In contrast, the most common PTP protocols can run upon tree topologies without the message being exchanged between client nodes (with a pre-configured grandmaster). And lastly, migrating the SS-DBA-based BFT-stabilizer into CCBN is also not a trivial task and would generate many more messages, especially with n1 = 2f1 + 1. In contrast, some variants of PTP, such as ReversePTP, do not require exchanging any message between the clients in electing a new grandmaster. In mitigating these disadvantages, we provide a basic IS-BFT-CS solution upon CCBN with discreetly utilized external times. Generally, the overall framework of the IS-BFT-CS solutions is shown in Fig. [fig:framework]. With the developed strong synchronizer, the main problem here is to construct the BFT stabilizer. ![image](framework.png) [fig:framework] The BFT stabilizer should have the following properties. Firstly, the system should reach the stabilized state from an arbitrary initial state in the desired time. And secondly, for efficiency, once the system is stabilized, no nonfaulty node would detect the undesired system state. So no corrector would be called further. For this, the BFT stabilizer can be constructed in two steps. In the first step, we would construct some detector (or saying state-checker, monitor, etc.) to detect some undesired state of the system. In the second step, some corrector (or saying state-resetter, repairer) would be called to bring the state of the system into the desired one. As is required, no single point of failure is allowed. So the detector and corrector can only be implemented in a fault-tolerant way. Also, here we want to design the synchronizers, detectors, and correctors in a more decoupled way. One benefit of this would be that the basic building blocks would then be integrated with other extra supports such as external times and other resources more easily. And with the decoupled strong synchronizer and corrector, once the system is stabilized, the corrector would not be active before the happening of the next transient system-wide failure. With this, we expected that the synchronization precision and overall performance could be further improved. For this, the basic BFT stabilizer is built with the structure shown in Fig. [fig:stabilizer]. ![image](framework1.png) [fig:stabilizer] The basic detectors ------------------- To construct the BFT stabilizer, firstly, the `S\_DETECTOR` algorithm is provided in Fig. [fig:algosdetector] to act as the strong detector. For a concrete example, the set operation and the expired condition of the timer τd are implemented by the line [code:detectortimerset] and line [code:detectortimerexpired] of the `S\_DETECTOR` algorithm, respectively. Other timers (such as τw\* and τw) can be implemented similarly for fast stabilization. Here the timer τd is used as a watchdog timer to count the ticks passed since the last satisfaction of the condition in line [code:detectorsatis] of the `S\_DETECTOR` algorithm. So if τd is expired in j ∈ U1, j knows that the system is not stabilized, by which we say j is alerted. [1]1em//#1 [1] *for every node j* ∈ U1: always** Pk = {i ∣ j receives pulse-k from i in the latest δ13 ticks}; [code:detectorsatis] τd : = Hj(t) ⊕ (δ14 − 1); [code:detectortimerset] [code:detectortimerexpired] τd : = τmax; alertedj : = (τd = τmax); [fig:algosdetector] The detector is called strong (a slight abuse of the concept proposed in ) in that all possible undesired system states would be eventually detected in all nonfaulty nodes while some of the detected ones may not be actually the undesired cases. This kind of false alarm is largely inevitable in designing the detector in the presence of Byzantine nodes. But if the system is stabilized for a sufficiently long time, no false alarm would be generated. So it leaves for some kind of correctors to take appropriate actions in responding to the alarms (including the false-alarms) being generated in the strong detector. Similarly, other detectors can be designed to detect any other observable system states. For example, the clique detector `Q\_DETECTOR` shown in Fig. [fig:algowdetector] can tell if there is a possible pulsing clique in the current local basic synchronization round (the line [code:pulsesyncpreemptivestop0] and line [code:pulsesyncpreemptivestop] of the `BFT\_PULSE\_SYNC` algorithm can be realized similarly). The `S\_DETECTOR` and `Q\_DETECTOR` are called the basic detectors. [1]1em//#1 [1] *for every node j* ∈ U1: always** [code:detectorsatis2] pulsedj : = 1; pulsedj : = 0; [fig:algowdetector] The basic corrector ------------------- The basic corrector is constructed as the `H\_CORRECTOR` algorithm shown in Fig. [fig:algobasiccorrector] with the alien clocks (shown in the grey color in Fig. [fig:stabilizer] and given in Section [sec:Model]). Concretely, the `H\_CORRECTOR` algorithm running in every j ∈ U1 uses the alien clock Yj (in executing line [code:correctorcorrection] of the `H\_CORRECTOR` algorithm) as some kind of temporary synchronized clock to adjust Cj when the system is not stabilized. So, when the system is not stabilized, the alien clocks Yj for all j ∈ U1 are assumed to be at least coarsely synchronized. [1]1em//#1 [1] *for every node j* ∈ U1: at local-time (kk`pls` + 1)τ0** coinj : = random ({0, 1}); [code:correctorcointoss] [code:correctoreor] set protectj as 1; [code:correctorcorrectionbegin] Cj(t) : = Yj(t); [code:correctorcorrection] offsetj = 0;[code:correctorcorrectionend] set protectj as 0; [fig:algobasiccorrector] Generally, in the `H\_CORRECTOR` algorithm, not only the alien clocks but any kind of synchronized clocks can be employed to provide the reference clocks Yj for all j ∈ U1, providing that these clocks can be coarsely synchronized when L is not synchronized. However, as the stabilization time and message complexity of traditional SS-BFT-CS solutions are often prohibitively high, the alien clocks can be utilized here to reduce the stabilization time of the overall IS-BFT-CS system. Now providing that Yj are coarsely synchronized for all j ∈ U1 with the precision e1, the `H\_CORRECTOR` algorithm would mainly act as some kind of clock merger to merge Cj and Yj at some appropriate instants. Concretely, only if the node j observes the evidence of resynchronization (EoR), j would use its alien-time Yj(t) to overwrite its local-time Cj(t). To our aim, the EoR condition checked in line [code:correctoreor] (of the `H\_CORRECTOR` algorithm, the same below) can be configured in j as $$\begin{aligned} \label{eq:def\_eor} \textit{EoR}=(\textit{alerted}\_j\land (\neg\textit{pulsed}\_j\lor \textit{coin}\_j))\end{aligned}$$ to give chances for both fast and stable synchronization of Cj for all j ∈ U1. As ¬pulsedj implies alertedj when EoR is checked in node j, EoR can also be computed as ¬pulsedj ∨ (alertedj ∧ coinj). Roughly speaking, in running the intro-stabilizing BFT-CS algorithms with EoR, once any internal synchronizer (i.e., except the alien clocks) might work, the EoR condition is expected to be false, and thus the clock-merge operation is expected to be forbidden. When no such internal synchronizer works, the EoR condition is expected to be true, and thus the clock-merge operation is expected to be allowed. For this, besides the alertedj and pulsedj signals from the detectors, we also employ the coinj flag, which is the result of the coin tossed in executing line [code:correctorcointoss] when $C\_j(t)\bmod k\_\mathtt{pls}\tau\_0=\tau\_0$. This coinj flag is necessary as some node in U1 might observe pulsedj while some other nodes in U1 might observe ¬pulsedj during their overlapped header-body synchronization procedures. In this situation, there might be some nodes that want to be synchronized by the Y clocks while the others do not. To reconcile this, every node j ∈ U1 can toss an unbiased coin during every header-body synchronization procedure to decide if it would like to be synchronized by the Yj clock or not when alertedj ∧ pulsedj is observed. For better performance, some biased coins can also be employed. Here we take the unbiased coin for simplicity. Notice that we can also compute EoR as alertedj ∧ coinj to simplify the analysis. However, with this simplification, some non-worst-cases optimization is also sacrificed, as it is more likely that some nodes in U1 would observe ¬pulsedj when the system is not synchronized. Lastly, to be integrated with the strong synchronizer, the execution of the `H\_CORRECTOR` algorithm has a lower priority than that of the `BFT\_PULSE\_SYNC` algorithm. Namely, whenever the local clock Cj would be adjusted in executing the `BFT\_PULSE\_SYNC` algorithm, this adjustment would not be canceled by setting protectj as 1 in executing the `H\_CORRECTOR` algorithm. Meanwhile, the execution of the `H\_CORRECTOR` algorithm still has a higher priority than that of the `BFT\_SYNC` algorithm. It should be noted that as the protectj flag is set as 1 in executing the `BFT\_PULSE\_SYNC` only when $C\_j\bmod k\_\mathtt{pls}\tau\_0\leqslant \tau\_0$, the local state protectj = 1 set in executing the `H\_CORRECTOR` algorithm would not be changed by executing the `BFT\_PULSE\_SYNC` algorithm. Formal analysis =============== Now we show that by configuring the constant parameters referenced in the algorithms according to the constraints shown in Table [tab:parameterrelations] and Table [tab:relatedrelations] (some constraints are relaxed for simplicity), the provided algorithms make an IS-BFT-CS solution upon G. Some concrete configurations for the constant parameters are later given in Table [tab:parameters]. [htbp] [tab:parameterrelations] Nl &Constraints [re:delta1]&δ1 ≥ (θ1 + δ`d`)(1 + ρ) [re:delta2]&δ2 ≥ θ1(1 + ρ) [re:delta3]&δ3 ≥ θ3(1 + ρ) + δ1 [re:delta4]&δ4 ≥ θ1(1 + ρ) + 2ρθ4 [re:delta5]&δ5 ≥ δ`I` + 2ρθ2 + 2ɛ0 [re:delta6]&δ6 ≥ δ`I` + 2ρθ4 + 4ɛ0 [re:delta7]&δ7 ≥ ɛ1 + (δ`d` + δ11/(1 − ρ) + δ`p`)(1 + ρ) + ɛ0 [re:delta8]&δ8 = δ11(1 − ρ)/(1 + ρ) − ɛ0 [re:delta9]&δ9 = δ12 + δ17(1 − ρ)/(1 + ρ) − ɛ0 [re:delta10]&δ10 ≥ (σ7 + δ`d`)(1 + ρ) [re:delta11]&δ11 ≥ δ10 + δ`p` [re:delta12]&δ12 ≥ δ7 + δ8 + δ11 + 2δ`p` [re:delta13]&δ13 ≥ ɛ1 + δ`d`(1 + ρ) [re:delta14]&δ14 ≥ k`pls`(τ0 + 2δ`I`) + δ`p` [re:delta15]&δ15 = k`pls`(τ0 − 2δ`I`) − δ`p` ≥ (3σ1 + σ3)(1 + ρ) [re:delta16]&δ16 ≥ σ11 + δ`d`(1 + ρ) [re:delta17]&δ17 ≥ δ16 + δ`p` [re:tau0]&τ0 ≥ max{δ6 + (θ5 + δ0)(1 + ρ), δ12 + δ`I` + (σ12 + δ0)(1 + ρ)} [re:taumax]&$\tau\_{max}\geqslant 4\delta\_{14} \land \tau\_{max}\bmod(4k\_\mathtt{pls}\tau\_0)=0 $ [re:kpls]&k`pls` ≥ max{1 + ⌈logα((ɛ1/2 − ε`b`/(1 − α))/δ`I`)⌉, 3} [htbp] [tab:relatedrelations] Nl &Constraints [re:theta1]&θ1 = 2δ`I`/(1 − ρ) + δ`p` [re:theta2]&θ2 = θ1 + δ2/(1 − ρ) + δ`p` [re:theta3]&θ3 = θ2 + δ0 [re:theta4]&θ4 = (δ3 − δ1 + 2δ`I`)/(1 − ρ) + δ`p` [re:theta5]&θ5 = θ4 + δ4/(1 − ρ) + δ`p` [re:sigma1]&σ1 = δ10/(1 − ρ) + δ`d` [re:sigma2]&σ2 = δ15/(1 + ρ) − δ`d` − δ10/(1 − ρ) [re:sigma2g]&σ2 ≥ (k`pls` − 1)(τ0 + 2δ`I`)/(1 − ρ) [re:sigma3]&σ3 = 2δ`p` + δ11/(1 − ρ) [re:sigma4]&σ4 = 2δ`p` + δ17/(1 − ρ) [re:sigma5]&σ5 = σ7 + δ14/(1 − ρ) + δ`p` [re:sigma6]&σ6 = σ1 + σ2 + σ3 + (τ0 + δ1 + δ`I`)(1 + ρ) + δ`p` [re:sigma7]&σ7 = δ13/(1 − ρ) + δ`d` [re:sigma8]&σ8 = δ11/(1 + ρ) [re:sigma9]&σ9 = (σ3 − σ8 + δ`d`)(1 + ρ) [re:sigma10]&σ10 = δ12(1 + ρ) [re:sigma11]&σ11 = (δ7 + σ9 + 2ρσ10)(1 + ρ) + δ`p` [re:sigma12]&σ12 = σ3 + σ4 + σ10 + σ11 + δ0 [re:sigma13]&σ13 = 2δ`I`/(1 − ρ) + δ`p` + σ6 [re:sigma14]&σ14 = σ6 + (k`pls` − 1)T`max` [re:alpha]&α = (⌊(n0 − 2f0 − 1)/f0⌋ + 1)− 1 [re:epsilonb]&ε`b` = 11ɛ0 + ρ(3θ1 + 2θ5 + 4θ4 − 4θ3 + T`max`) [re:deltaI1]&δ`I` ≥ max{σ11 + 2ɛ0 + 2ρτ0, ɛ2 + 2ρk`pls`T`max`} [re:Tmin]&T`min` = (τ0 − δ6)/(1 + ρ) − δ`p` [re:Tmax]&T`max` = (τ0 + δ6)/(1 − ρ) + δ`p` [re:deltac]&Δ`C` = δ14/(1 − ρ) + δ`p` [re:varepsilon1varrho1]&ɛ1 > 2ε`b`/(1 − α), $\varrho\_{1}= \rho+\varepsilon\_{1}/T\_\mathtt{min}$ [re:Delta1]&Δ1 = Δ`C` + 3k`pls`τ0/η1 + σ14 Besides the constant parameters, each node i ∈ U also uses some local variables in running the algorithms. In the analysis, we use x(i) to denote the local variable x used in i ∈ U when it is needed to differentiate the different nodes. And the value of x (or x(i)) at t is denoted as x(t) (or x(i)(t)). For example, the value of offseti in running the `BFT\_SYNC` algorithm in node i ∈ U at t can be denoted as offseti(i)(t) (or simplified as offseti(t)). Also, we assume that each line of the algorithms is atomically executed. And when any line of the algorithms is being executed in i ∈ U at t, we assume that x(i)(t) takes the value after the execution of this line. As is shown in the algorithms, the timers can also be represented as local variables. In considering stabilization, all the local variables might have arbitrary values at t0. In the algorithms, as we always check the timers with value ranges as small as possible, all these timers can be locally stabilized within their scheduled ticks. And for all the other local variables, it is trivial to show that the history values recorded before t0 can be overwritten within the maximal scheduled ticks of the timers. So for convenience, we assume that all the local variables used in the algorithms are overwritten at least once since t0 at some instant t`C` > t0. With the provided algorithms, we have t`C` ∈ [t0, t0 + Δ`C`], where Δ`C` = δ14/(1 − ρ) + δ`p` is called the local recovery time. As all the local variables used in the algorithms can be recovered from all the possible incorrect values before t`C`, a node in U can be referred to as a correct node since t`C` (following and ). In the analysis, when the clocks are added or subtracted with small quantities (such as the timeouts, the reading errors, the message delays, the adjustment cycles), as τmax is assumed to be far greater than such quantities, the default (i.e., would be automatically performed in the computer) $\bmod$ operations can be ignored. Especially in representing the value range of the clocks, we would use the common operators + and − rather than ⊕ and ⊖ . For example, the value range [τ − δ, τ + δ] of a clock would actually be [τ − δ + τmax, τmax) ∪ [0, τ + δ] if τ − δ < 0 and would actually be [τ − δ, τmax) ∪ [0, τ + δ − τmax] if τ + δ ≥ τmax. But for simplicity, we would rather take the common representation [τ − δ, τ + δ]. Also, the default rounding operations on the discrete ticks are ignored in handling the multiplication and division operations (one can add an extra tick in each such operation to derive a sufficiently safe configuration). All these ignored operations (modular and rounding) can be trivially added when needed. Firstly, for the strong synchronizer, we give the definition of a synchronization point. [defwellsynchronized] t is a δ-synchronization point iff L is initially δ-synchronized at t, no pulse is being transmitted or processed in L at t, the timers τw, τw\* are all reset at t, and no line nor block of the `BFT\_SYNC` or `BFT\_PULSE\_SYNC` algorithm is being executed at t. For example, the vertical dotted lines t = t1, t = t3, t = t4, t = t6, and t = t7 in Fig. [fig:basicprocess] all correspond to some synchronization points. But t2 and t5 (in Fig. [fig:basicprocess]) are not synchronization points, since they are covered in some updating spans of the local clocks. Similarly, in Fig. [fig:headerbody], t2, t4, and t6 can be synchronization points while t1, t3, and t5 cannot be. Generally, with the synchronization points, the synchronization phases in the synchronized system can be well-separated. For analysis, here we further define some specific synchronization points between the separated synchronization phases. [defforerunner] t is a (δ, δʹ, k)-synchronization point iff t is a δ-synchronization point and ∃j ∈ U1 : Cj(t) = kτ0 + δʹ − δ. The basic synchronizer ---------------------- In this subsection, we assume the `BFT\_SYNC` algorithm runs alone (i.e., all the other algorithms are ignored here), and the system is in an initial δ`I`-synchronized state. With this, we show that the `BFT\_SYNC` algorithm can maintain the synchronized state of the system with the strictly separated synchronization phases shown in Fig. [fig:basicprocess]. Especially, we show that the synchronous approximate agreement can be simulated in CCBN with n0 > 3f0 and n1 > 2f1. The instants tx (for x = 1, 2, …, 6) referenced in Lemma [lemmasyncmaintain0] correspond to the ones shown in Fig. [fig:basicprocess]. As is mentioned, this is mainly for the ease of reading and can be further optimized for shorter synchronization cycles when it is needed. And for simplicity, we do not redefine the parameters given in Table [tab:parameterrelations] and Table [tab:relatedrelations] in the proofs. The readers can easily check the relations of the parameters used in the proofs with these tables. By this, we can also avoid the magic numbers and premature calculations being scattered in the proofs. [lemmasyncmaintain0] If there is a (δ, δ1, k)-synchronization point t0ʹ with some δ ≤ δ`I` and $k\in \mathbb Z^+$, then there is a (δʹ, δ1, k + 1)-synchronization point tʹ ∈ [t0ʹ + T`min`, t0ʹ + T`max`) with some δʹ ≤ αδ + ε`b` and L is (2δʹ, ρ)-synchronized during [t0ʹ, tʹ]. As t0ʹ is a (δ, δ1, k)-synchronization point, there is some j0 ∈ U1 satisfying Cj0(t0ʹ) = kτ0 + δ1 − δ and ∀j ∈ U : d̊(Cj0(t0ʹ), Cj(t0ʹ)) ≤ δ. So we have Cj(t0ʹ) ∈ [kτ0 + δ1 − 2δ, kτ0 + δ1] for all j ∈ U. As τw(j)(t0ʹ) = τmax and no line of the `BFT\_SYNC` algorithm is being executed at t0ʹ, the τw(j) timer would remain being closed and Lj and Cj would not be adjusted before some line (of the `BFT\_SYNC` algorithm, the same below) being executed in j since t0ʹ. As Ci(t0ʹ) ∈ [kτ0 + δ1 − 2δ, kτ0 + δ1] in every node i ∈ U0, Li and Ci would not be adjusted during [t0ʹ, t3) with t3 = t0ʹ + θ3 ≥ t0ʹ + θ2 + δ0 (see Table [tab:parameterrelations] and Table [tab:relatedrelations], the same below). During [t0ʹ, t3), as Cj(t0ʹ) ∈ [kτ0 + δ1 − 2δ, kτ0 + δ1], every node j ∈ U1 would read the remote clocks Ĉi, j(tjʹ) and write Lj in executing line [code:backwardsyncread] at some tjʹ ∈ [t0ʹ, t1) with t1 = t0ʹ + θ1. Denoting c̃i, j(t) = Ĉi, j(t) ⊖ (Cj(t) − δ5), as ∀i ∈ U0, ∀j ∈ U1 : d̊(Ĉi, j(t), Ci(t)) ≤ ɛ0 ∧ d̊(Ci(t), Cj(t)) ≤ δ0ʹ = δ + 2ρθ1 for all t ∈ [t0ʹ, t1], for every j ∈ U1 we have ∀i ∈ U0 : c̃i, j(t) ∈ [τj, τj + δ1ʹ] ∧ d̊(c̃i, j(t), c̃i, j(t1)) ≤ δ2ʹ with some τj ∈ [δ5 − δ1ʹ, δ5], δ1ʹ = δ0ʹ + 2ɛ0 < δ5, and δ2ʹ = 2ρθ1 + 2ɛ0 for all t ∈ [t0ʹ, t1]. So with the basic properties of the FTA function, as n0 > 3f0 and ∀j ∈ U1 : tjʹ ∈ [t0ʹ, t1), we have ∣offsetj(j))(tjʹ)∣ ≤ δ1ʹ and ∀j1, j2 ∈ U1 : d̊(Lj1(t1), Lj2(t1)) ≤ δ3ʹ with δ3ʹ = δ1ʹ/2 + 2δ2ʹ. Then, every node j ∈ U1 would execute line [code:backwardsyncoffsetupdate] during [t1, t2) with t2 = t0ʹ + θ2. So we have ∀j1, j2 ∈ U1 : d̊(Cj1(t2), Cj2(t2)) ≤ δ3ʹ + 2ρθ2 and ∀j ∈ U1 : τw(j)(t2) = τmax. Then, every node j ∈ U1 would not adjust Lj nor Cj and would not set τw(j) during [t2, t6) with t6 = t0ʹ + (τ0 − δ6)/(1 + ρ). So we have ∀j1, j2 ∈ U1 : d̊(Cj1(t), Cj2(t)) ≤ δ3ʹ + 2ρ(τ0 − δ6)/(1 + ρ) for all t ∈ [t2, t6). So as t3 − t2 ≥ δ0, we have ∀i ∈ U0, ∀j ∈ U1 : d̊(Ĉj, i(t), Cj(t)) ≤ ɛ0 for all t ∈ [t3, t6). And every node i ∈ U0 would read the remote clocks Ĉj, i and write Li with the median of the remote readings from V1 in executing line [code:syncread] at some tiʹ ∈ [t3, t4) with t4 = t0ʹ + θ4. Also, denoting c̃j, i(t) = Ĉj, i(t) ⊖ (Ci(t) − δ6), as ∀i ∈ U0, ∀j ∈ U1 : d̊(Ĉj, i(t), Cj(t)) ≤ ɛ0 ∧ d̊(Ci(t), Cj(t)) ≤ δ4ʹ = δ3ʹ + 2ρ(t4 − t1) for all t ∈ [t3, t4], for every i ∈ U0 we have ∀j ∈ U1 : c̃j, i(t) ∈ [τi, τi + δ5ʹ] ∧ d̊(c̃j, i(t), c̃j, i(t4)) ≤ δ6ʹ with some τi ∈ [δ6 − δ5ʹ, δ6], δ5ʹ = δ4ʹ + 2ɛ0 ≤ δ6, and δ6ʹ = 2ρ(t4 − t3) + 2ɛ0 for all t ∈ [t3, t4]. So with the basic properties of the median function, as n1 > 2f1 and ∀i ∈ U0 : tiʹ ∈ [t3, t4), we have ∀i, j ∈ U : d̊(Li(t4), Lj(t4)) ≤ δ7ʹ with δ7ʹ = δ5ʹ + 2δ6ʹ. Then, every node i ∈ U0 would execute line [code:syncoffsetupdate] during [t4, t5) with t5 = t0ʹ + θ5 ≤ t6 − δ0. So we have ∀i1, i2 ∈ U : d̊(Ci1(t5), Ci2(t5)) ≤ δ8ʹ with δ8ʹ = δ7ʹ + 2ρ(t5 − t4) and ∀i ∈ U : τw(j)(t5) = τmax. Then, as t6 − t5 ≥ δ0, we have ∀i ∈ U0, j ∈ U1 : d̊(Ĉi, j(t), Ci(t)) ≤ ɛ0 ∧ d̊(Ĉj, i(t), Cj(t)) ≤ ɛ0 for all t ∈ [t6, t7] with t7 ≥ t6 being the earliest instant satisfying Cj(t7) = (k + 1)τ0 + δ1 for some j ∈ U1. So there is a δʹ-synchronization point tʹ ∈ [t6, t7] satisfying ∃j0ʹ ∈ U1 : Cj0ʹ(tʹ) = (k + 1)τ0 + δ1 − δʹ. As the maximal difference of the logical clocks of the nodes in U is within 2δʹ during [t0ʹ, t7], in which every node in U adjusts its logical clock at most once with no more than 2δʹ ≤ δ6 clock-adjustment, L is (2δʹ, ρ)-synchronized during [t0ʹ, tʹ] with tʹ ∈ [t0ʹ + T`min`, t0ʹ + T`max`). [corollaryremainsync] If the premise of Lemma [lemmasyncmaintain0] holds, L would be (2δ(c), ρ(c))-synchronized since t + cT`max` with ρ(c) ≤ ρ + 2δ(c)/T`min` and δ(c) ≤ αcδ + ε`b`/(1 − α). Denote δ(0) = δ and δ(1) = δʹ for the parameters δ and δʹ used in Lemma [lemmasyncmaintain0], respectively. By applying Lemma [lemmasyncmaintain0], we have δ(1) = αδ(0) + ε`b` with t(1) ∈ [t + T`min`, t + T`max`). As the premise of Lemma [lemmasyncmaintain0] also holds for t(1), we have δ(2) = αδ(1) + ε`b` with t(2) ∈ [t(1) + T`min`, t(1) + T`max`). Iteratively, we have δ(c) = αcδ + ε`b`(1 − αc)/(1 − α) ≤ αcδ + ε`b`/(1 − α) with t(c) ∈ [t + cT`min`, t + cT`max`). As δ(0) ≤ δ`I`, we have δ(cʹ) ≤ δ(cʹ − 1) for all $c'\in \mathbb Z^+$. So the synchronization precision 2δ(c) and the accuracy ρ(c) ≤ ρ + 2δ(c)/T`min` can be maintained in L since t(c). Notice that in the provided algorithms, we use the basic FTA functions in simulating the basic approximate agreement, which achieves the basic convergence rate α0 = 1/2. For faster convergence, the FTA functions can be replaced as the advanced ones to achieve the convergence rate α = (⌊(n0 − 2f0 − 1)/f0⌋ + 1)− 1 (see for details). For example, with n0 > 5f0, we would get a better convergence rate α ≤ 1/4. And this is in line with our basic system settings. The strong synchronizer and strong detector ------------------------------------------- Now we show that with the strong synchronizer and the strong detector, if a node j ∈ U1 does not detect the undesired system state, i.e., alertedj(j)(t) = 0 holds for some t, then L can be deterministically synchronized in a finite time. In this subsection, we assume that only the `BFT\_SYNC`, `BFT\_PULSE\_SYNC`, and `S\_DETECTOR` algorithms run. Firstly, denoting the always guarded condition in line [code:pulsesyncpreempitvecon] of the `BFT\_PULSE\_SYNC` algorithm as `A_1` and the lines from [code:pulsesyncpreempitvestart] to [code:pulsesyncpreemptivestop] of the same algorithm in responding to the satisfaction of `A_1` as `B_1`, we give the definition of the semi-synchronous execution of the `B_1` block. [defsynchronous] The nodes in U1 perform a δ-synchronous execution of the `B_1` block during [t, tʹ] iff for every node j ∈ U1, there is an execution of the `B_1` block during [t, tʹ] such that for every line l ∈ `B_1`, l is not preempted or canceled and the execution instants of l are at most δ apart in all nodes of U1. Analogously, denoting the always guarded condition in line [code:pulsesyncpreempitvecon0] of the `BFT\_PULSE\_SYNC` algorithm as `A_0` and the lines from [code:pulsesyncpreempitvestart0] to [code:pulsesyncpreemptivestop0] as `B_0`, we can also define the semi-synchronous execution of the `B_0` block by taking place U1 as U0. Now we first show that the `A_1` condition would not be satisfied very frequently, and thus the `B_1` block would be eventually executed in the semi-synchronous way when the `A_1` condition is satisfied in all nodes of U1 during a sufficiently short period. [lemmasyncseparated] If the `A_1` condition is satisfied in nodes j1, j2 ∈ U1 (j1 and j2 can be the same or different nodes) at t1 and t2 with t2 > t1, then $t\_{2}-t\_{1}\notin(\sigma\_{1},\sigma\_{2}]$. Denote the sets Pkʹ satisfying the `A_1` condition at t1 and t2 as Pk1 and Pk2, respectively. As ∣Pkʹ∣ ≥ n0 − 2f0, there are at least n0 − 3f0 nonfaulty nodes in every such Pkʹ. As n0 > 5f0, there exists i0 ∈ U0 ∩ Pk1 ∩ Pk2 for otherwise it can only be 2(n0 − 3f0) ≤ n0 − f0. For every such i0, if its pulse is received in any j1 ∈ U1 at some tʺ, this pulse can only be sent at some tʹ ∈ [tʺ − δ`d`, tʺ] and thus can only be received in j2 ∈ U1 during [tʹ, tʹ + δ`d`]. So if the `A_1` condition is satisfied in j1 and j2 with receiving the same pulse of i0, then t2 − t1 ≤ σ1 holds. Otherwise, if two pulses are sent by any i ∈ U0 at t1ʹ and t2ʹ with t2ʹ > t1ʹ, with the condition in line [code:pulsesyncpulsecon], we have t2ʹ − t1ʹ ≥ ε with ε = δ15/(1 + ρ). So within any ε − δ`d` time, at most one pulse from i0 would be received in the nodes of U1. So if t2 − t1 > δ`d` + δ10/(1 − ρ), we have t2 − t1 > ε − δ`d` − δ10/(1 − ρ). [lemmasyncsynchronous] If the `A_1` condition is satisfied in every j ∈ U1 at some tjʹ ∈ [t0ʹ, t0ʹ	arxiv_0000594
Bowditch’s Q-conditions and Minsky’s primitive stability ======================================================== For the action of the outer automorphism group of the rank two free group on the corresponding variety of PSL(2, C) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky characters. One is introduced by Bowditch in 1998 (followed by Tan, Wong and Zhang in 2008) and the other by Minsky in 2013. We prove that these two domains are equal. We then show that they are contained in the set of characters having what we call the bounded intersection property. Introduction ============ Let Fn denote the free group on n ≥ 2 generators, and set G = PSL(2, C). The space of representations Hom(Fn, G) admits a natural action of Aut(Fn) × Aut(G) given by $$\begin{aligned} {5} {\mathsf{Aut}}({\mathsf{F}}\_n)&\times{\mathsf{Aut}}(G)&&:&\;{\mathrm{Hom}}({\mathsf{F}}\_n&,G)&&\longrightarrow&&\;{\mathrm{Hom}}({\mathsf{F}}\_n,G)\\ (&f,g)&&:&&\rho&&\longmapsto&&\;\;g\circ\rho\circ f^{-1}\end{aligned}$$ The G-character variety X(Fn) of Fn is then defined as the GIT quotient $$\begin{aligned} {\mathcal{X}}({\mathsf{F}}\_n)=\mathrm{Hom}({\mathsf{F}}\_n,G){\mathbin{/\mkern-6mu/}}\boldsymbol{\left(\right.}\{Id\}\times{\mathsf{Inn}}(G)\boldsymbol{\left.\right)}\end{aligned}$$ by the action of inner automorphisms of G. It then admits an action by the outer automorphism group Out(Fn) = Aut(Fn)/Inn(Fn), since the inner automorphism group Inn(Fn) acts trivially. Regarding the dynamical properties of the action of Out(Fn) on X(Fn), one of the classical results is that this action is properly discontinuous on the open subset S ⊂ X(Fn) consisting of Schottky characters. One may then ask if S is a maximal domain of discontinuity for the action. Recently, Minsky answered the question in the negative: an open subset PS, consisting of primitive stable characters, is strictly larger than S $${\mathcal{S}}\subsetneq{\mathcal{PS}}\subset{\mathcal{X}}({\mathsf{F}}\_n)$$ and admits a properly discontinuous action of Out(Fn). In particular, since S is known to be the interior of the closed subset consisting of characters coming from injective representations with discrete image, we see that PS contains characters coming from representations with non-discrete image. Thus Minsky’s result shows that the geometric decomposition (discrete vs. dense) of the characters is different from the dynamical decomposition (properly discontinuous vs. ergodic) in this case. For more results about dynamics on representation spaces we refer the reader to the survey papers of Goldman (for surfaces and locally homogeneous geometric structures on them), Lubotzky (for free groups and various target groups), and Canary (for word hyperbolic groups into semi-simple Lie groups with no compact factors). We will focus on the special case of n = 2 in the present paper, where another domain of discontinuity for the action of Out(F2) has been known to exist. Bowditch introduced a notion called the Q-conditions and investigated the type-preserving characters in X(F2) in an attempt to characterize the quasi-Fuchsian once-punctured torus groups. Generalizing Bowditch’s work, Tan, Wong and Zhang tested the Q-conditions for all characters in X(F2) and showed that the characters satisfying the Q-conditions form an open subset BQ ⊂ X(F2) and that Out(F2) acts on it properly discontinuously. As an interesting property of characters in BQ, Bowditch and Tan-Wong-Zhang showed that a variation of McShane’s identity holds for them. In the first part of the paper we study the relation between these two domains BQ and PS. Very roughly speaking, the primitive stability of ρ ∈ Hom(F2, PSL(2, C)) means that under the orbit map of the ρ-action on H3, the axes in F2 of primitive words are sent to uniform quasi-geodesics (Definitions [def:ps] and [def:altps]). On the other hand, the Q-conditions on ρ mean that for every primitive word, its cyclically reduced length and the translation distance of its ρ-action on H3 are comparable (Definition [def:bq] and Theorem [thm:bqpd]). It is not hard to see from the definitions that we have the inclusion PS ⊂ BQ ⊂ X(F2). A proof is given by Lupi in his thesis \Proposition 2.9, but we also provide a different proof in Proposition [prop:psbq] to make the paper self-contained. A number of authors questioned if these two conditions are actually equivalent. See \p.64, \§.7.1 and \§.4.2. Also compare \Problem 3.2. Our first theorem answers the question in the affirmative: [I] If a representation ρ* : F2 → PSL(2, C) satisfies the Q-conditions, then it is primitive stable. Consequently, we have the equality PS = BQ. We prove this in Section [sec:pf]. (a) Lupi showed the same equality in his thesis \Proposition 3.4 and Theorem 4.3 when the target group G* is restricted to PSL(2, R). His idea of proof, however, does not seem to directly generalize to the case of PSL(2, C). We discuss this issue in Section [sec:lupi]. (b) One may view Theorem [I] in analogy with a theorem of Delzant, Guichard, Labourie and Mozes that, for a word hyperbolic group acting on a metric space by isometries, an orbit map is a quasi-isometric embedding if and only if the action is displacing. In a sense, primitive stability (resp. the Q-conditions) is a weakening of the former (resp. the latter) property obtained by looking only at actions of primitive elements. We discuss this analogy in more detail in Section [sec:pspd]. Let us briefly present our ideas of proof. The definitions of primitive stability and the Q-conditions involve the conjugacy classes of primitive elements of F2. The so-called Christoffel words are representatives of such classes and have many desirable properties for our purposes. Without giving the precise definition (Definition [def:chris]) for now, let us exhibit a simple and motivational example in order to illustrate how to generate the Christoffel words and how the relevant geometry behaves asymptotically under the representations of Theorem [I]. at 30 280 at 108 154 at 194 162 at 222 214 at 74 206 at 152 110 at 300 134 at 146 279 at 12 128 at 156 8 at 494 100 at 460 160 at 556 46 at 690 60 at 394 60 at 560 240 at 790 120 at 950 90![A subgroup {\mathsf{W}}(p,q,r) of {\mathsf{Isom}}^+({\mathbb{H}}^2), a hyperbolic translation a=pq, and two operations L and R.](fig_torus1 "fig:") [fig:torus] See Figure [fig:torus]. The first picture shows a subgroup W(p, q, r) of Isom+(H2) generated by three orientation-preserving involutions p, q and r, whose fixed points are specified in the Klein projective model of H2. If the product pqr is of infinite order, the group W(p, q, r) is isomorphic to the free Coxeter group Z/2 \* Z/2 \* Z/2 of rank three and contains the free group F(a, b), where a = pq and b = qr, as an index two subgroup. Note that ab− 1a− 1b = pqr(qq)pqr = (pqr)2. The element pqr is an elliptic transformation in the picture; its fixed point and the fixed points of its three other conjugates form a quadrilateral. If pqr is parabolic or hyperbolic, then the quadrilateral is ideal or hyper-ideal, respectively. In all possible cases, however, the elements a = pq and b = qr are always hyperbolic translations and they identify the opposite edges of the quadrilateral. The quotient surface is a hyperbolic torus either with a cone-type singularity or with a cusp or with a funnel, depending on the nature of pqr. The second picture represents a hyperbolic translation a = pq as a directed edge. The third picture shows two operations L and R (left and right slides) which inductively generate other basis triples of involutions in W(p, q, r) as well as other directed edges starting with the initial ones. The Christoffel words are the hyperbolic translations represented by the directed edges so generated. They are primitive elements of F(a, b). (For a more detailed account, see Section [sec:chris] as well as Appendix [sec:a].) at 410 54 at 484 40 at 534 90 at 450 70 at 346 96 at 478 100 at 630 124 at 470 170 at 180 110 at 550 170 at 360 150![Idea of proof in the case of G={\mathsf{PSL}}(2,{\mathbb{R}})\cong{\mathsf{Isom}}^+({\mathbb{H}}^2). ](fig_torus2 "fig:") [fig:torus2] Figure [fig:torus2] exhibits an example of the iteration LRLR (read from right to left). Notice how quickly and uniformly the associated triangles degenerate with respect to the number of steps of LR-slides. In fact, if the initial triangle is acute then other triangles generated by the LR-slides are all obtuse. This asymptotic behavior of triangles immediately imply that the representation F(a, b) in Isom+(H2) satisfies the Q-conditions and, after a little more work, primitive stability as well. We remark that this process may be viewed as being reverse to the classical trace minimizing algorithm for checking discreteness of two-generator subgroups of PSL(2, R). See, for example, \Chapter 2 and the references therein. In the general case of representations in PSL(2, C), we exploit the fact (see Theorem [thm:coxeter]) that an irreducible representation of F(a, b) extends uniquely to the free Coxeter group W(p, q, r), thereby giving us a triple of complete geodesics in H3 which are axes of the involutions p, q* and r. For representations satisfying the Q-conditions (and hence irreducibility as well), we show that the triple of axes behaves analogously as in Figure [fig:torus2] under the LR-slides asymptotically. See Lemma [lem:angle] for a precise statement. Combined with the properties of Christoffel words, this enables us to show primitive stability. In the second part of the paper (Section [sec:bi]) we investigate a condition which we call the bounded intersection property. It is motivated by the work of Gilman and Keen, where they investigated palindromic elements of F2 in order to test the discreteness of irreducible representations ρ : F2 → PSL(2, C). Slightly strengthening their idea we consider the primitive elements that are palindromic in one of the three chosen bases (a, b), (b, c), (a, c) of F2, where abc = 1. The bounded intersection property is a certain geometric condition on the ρ-images of these palindromic primitive elements. See Definition [def:bi] for details. This property also defines a subset BI ⊂ X(F2), which we show in Proposition [prop:bi] to be invariant under the action of Out(F2). We do not know if BI is an open subset nor if Out(F2) acts properly discontinuously on it. Nevertheless, we prove in Section [sec:pf2] the following. [II] If ρ : F2 → PSL(2, C) satisfies the Q-conditions, then it has the bounded intersection property. That is, we have the inclusion BQ ⊂ BI. On the other hand, if ρ is discrete and faithful, and has the bounded intersection property, then it satisfies the Q-conditions. We were kindly informed by Caroline Series that she also proved Theorem [I] as well as a result similar to Theorem [II]. See. The rest of the paper is organized as follows. In Section [sec:2] we recall some classical results about automorphisms of F2. We introduce the Christoffel words in a basis of F2 and discuss their properties. In Section [sec:3] we mostly follow Fenchel and discuss some basic geometry of H3 with an emphasis on half-turns and Coxeter extensions. We also introduce the amplitude of a right-angled hexagon. In Section [sec:4] we recall the definition of primitive stability and that of the Q-conditions, and prove that a primitive stable representation satisfies the Q-conditions. In Section [sec:pf] we give a proof of Theorem [I]. In Section [sec:bi] we discuss palindromic elements of F2, introduce the bounded intersection property and prove Theorem [II]. In Appendix [sec:a] we discuss automorphisms of F2 in view of its Coxeter extension. #### Acknowledgments. The present work was initiated during our respective collaboration with Ser Peow Tan. We are deeply thankful to him for generously bridging us together and encouraging our collaboration. We would like to thank Misha Kapovich and Xin Nie for numerous helpful discussions, and Caroline Series for her interest and encouragement. We also thank the referee for many helpful suggestions that clarified our earlier argument. J. Lee would like to thank Inkang Kim for his support and encouragement; he was supported by the grant NRF-2014R1A2A2A01005574 and NRF-2017R1A2A2A05001002. B. Xu thanks Korea Institute for Advanced Study, where he finished the major part of the collaboration as a postdoc; he was supported by the grant DynGeo FNR INTER/ANR/15/11211745. The free group of rank two and Christoffel words ================================================ We recall classical facts about the (outer) automorphism group of the free group F2 of rank two. One often studies the group Out(F2) via its identification with the mapping class group of the one-holed torus. Since the representations of our interest are not (yet) known to be coming from a geometric structure on the surface, we do not discuss this identification here but rather proceed purely algebraically. The eventual goal is to introduce Christoffel bases, which are explicit representatives of conjugacy classes of bases of F2. Their nice properties, especially Lemma [lem:chris], will be essential in our proof of Theorem [I]. The automorphism group of ------------------------- The following results of Nielsen are all from his seminal paper. Let F2 denote the free group on two generators. A basis of F2 is an ordered pair of free generators. An element of F2 is called primitive if it is a member of a basis. The abelianization F2/[F2, F2] of F2 is isomorphic to the free abelian group Z2 of rank two. Bases and primitive elements of Z2 are defined analogously. To be specific, let us fix one basis (a, b) of F2 once and for all, and denote it by e = (a, b). In order to emphasize this we shall often write F2 = F(e) = F(a, b) interchangeably. The abelianization homomorphism $$\begin{aligned} \label{eqn:abel} \pi\_{\mathsf{e}}:{\mathsf{F}}\_2\to{\mathbb{Z}}^2\end{aligned}$$ is then given by πe(a) = (1, 0) and πe(b) = (0, 1). Nielsen showed the following. [Nielsen][thm:Nielsen] Let x and y be elements of F(a, b). 1. If the pair (x, y) is a basis of F(a, b) then the 2 × 2 matrix with columns πe(x) and πe(y) belongs to GL(2, Z), that is, has determinant ± 1. Conversely, any matrix in GL(2, Z) determines exactly one conjugacy class of a basis of F(a, b). 2. The pair (x, y) is a basis of F(a, b) if and only if the commutator xyx− 1y− 1 is conjugate either to the commutator aba− 1b− 1 or to its inverse bab− 1a− 1. Let us briefly discuss only the item (a) of the theorem. Note that a basis (x, y) of F2 corresponds to a unique automorphism f ∈ Aut(F2) defined by f(a) = x and f(b) = y. Nielsen found a set of generators of Aut(F2): they are the so-called elementary Nielsen transformations and correspond to the bases $$\begin{aligned} \label{eqn:elementary} (b,a),\; (a^{-1},b),\; (a,b^{-1}),\; (a,ab),\; (ab,b),\end{aligned}$$ respectively. Since an analogous result for Aut(Z2) ≅ GL(2, Z) is well-known, we see that the abelianization homomorphism πe induces a surjective homomorphism $${\mathsf{Aut}}({\mathsf{F}}\_2)\twoheadrightarrow{\mathsf{Aut}}({\mathbb{Z}}^2).$$ Nielsen also showed that the kernel of this map is exactly the group Inn(F2) of all inner automorphisms of F2. See also \Proposition I.4.5. Hence we obtain an isomorphism Out(F2) = Aut(F2)/Inn(F2) ≅ Aut(Z2) ≅ GL(2, Z). Therefore, the conjugacy classes of bases of F2 are in one-to-one correspondence with the bases of Z2. This implies, in turn, that the conjugacy classes of primitive elements of F2 are in one-to-one correspondence with the primitive elements of Z2. See also \Corollary 3.2. Unoriented primitive classes and the Farey triangulation -------------------------------------------------------- Both Bowditch’s Q-conditions and Minsky’s primitive stability are concerned with conjugacy classes of primitive elements of F2. As we shall see later in Section [sec:4], these conditions also make sense for a slightly bigger classes. To be more precise, we consider the equivalence relation ∼ on F2, where $$\begin{aligned} a'\sim a\quad\Longleftrightarrow\quad a'\text{ is conjugate either to }a\text{ or to }a^{-1}.\end{aligned}$$ Let us denote by Prim the set of all primitive elements of F2. For a primitive element x ∈ Prim we call its ∼ -equivalence class [x] ∈ Prim/∼ ⊂ F2/∼ the unoriented primitive class of x. In view of the above one-to-one correspondence between primitive conjugacy classes of F2 and primitive elements of Z2, we see that the unoriented primitive classes of F2 correspond to the projectivized primitive elements of Z2 $$\begin{aligned} \label{eqn:prim} {\mathsf{Prim}}/\_\sim\;\longleftrightarrow\;{\mathsf{P}}^1({\mathbb{Q}})={\mathbb{Q}}\cup\{\infty\}\end{aligned}$$ and the action of Aut(F2) on the unoriented primitive classes factors through ${\mathsf{PGL}}(2,{\mathbb{Z}})={\mathsf{GL}}(2,{\mathbb{Z}})/\{\pm{\textup{Id}}\}$: $${\mathsf{Aut}}({\mathsf{F}}\_2)\twoheadrightarrow{\mathsf{PGL}}(2,{\mathbb{Z}})\curvearrowright{\mathsf{P}}^1({\mathbb{Q}}).$$ Note that the kernel of this homomorphism is ${\mathsf{Inn}}^\mathfrak{e}({\mathsf{F}}\_2):={\mathsf{Inn}}({\mathsf{F}}\_2)\rtimes\langle\mathfrak{e}\rangle$, where e is an involutory automorphism of F2 = F(a, b) defined by e(a, b) = (a− 1, b− 1). [conv:slope] When we write an element p/q of Q for coprime integers p and q, we shall always assume q > 0. We formally write either 1/0 or ( − 1)/0 for the infinity ∞. at 304 155 at 0 160 at 0 140 at 156 292 at 150 16 at 246 262 at 240 42 at 68 262 at 68 42 at 270 234 at 44 234 at 276 76 at 38 76 at 210 280 at 102 280 at 206 28 at 102 28 at 550 155 at 420 170 at 420 134 at 482 240 at 476 70 at 545 232 at 540 76 at 420 232 at 418 76![(Left) The Farey triangulation {\mathcal{F}}. (Right) Its dual tree with rational numbers written inside the complementary regions.](fig_Farey "fig:") [fig:Farey] As the action of Aut(F2) on Prim/∼ factors through PGL(2, Z), we may visualize this action via the classical Farey triangulation. Let us take the upper half-plane of the complement P1(C) \ P1(R) as a conformal model of the hyperbolic plane H2. We denote $\overline{{\mathbb{H}}^2}={\mathbb{H}}^2\cup\partial\_\infty{\mathbb{H}}^2$, where the ideal boundary ∂∞H2 is identified with P1(R). The Farey triangulation F is a 2-dimensional simplicial complex realized as a subset $${\mathcal{F}}={\mathbb{H}}^2\cup{\mathsf{P}}^1({\mathbb{Q}})\subset\overline{{\mathbb{H}}^2}.$$ By definition, the set of 0-simplices are realized as P1(Q). We denote V = P1(Q) and call its elements the Farey vertices. The 1-simplices, called the Farey edges, are realized as complete geodesics connecting two Farey vertices p/q and r/s such that $$\det \begin{pmatrix}p&r\\q&s\end{pmatrix}=\pm1.$$ The 2-simplices, called the Farey triangles, are realized as ideal triangles bounded by three Farey edges. See Figure [fig:Farey]. For more about the Farey triangulation see, for example, \Chapter 1. Abusing notations we shall often identify the vertex set V = P1(Q) with the set Prim/∼ of unoriented primitive classes via V = Prim/∼. and say a Farey vertex [x] ∈ V. The conjugacy class of a basis (x, y) of F2 determines a directed* Farey edge connecting vertices from [x] to [y], which we simply denote by [x, y] instead of ([x], [y]). Note that three other conjugacy classes of bases (x, y− 1), (x− 1, y) and (x− 1, y− 1) also determine the same directed edge [x, y]. Compare Theorem [thm:Nielsen](a). By convention, however, whenever we write [x, y] it will be assumed that the specific representative basis (x, y) is chosen. The directed edge corresponding to the distinguished basis e = (a, b) will often be denoted by [e] = [a, b]; it runs from ∞ = 1/0 to 0 = 0/1. We identified the Farey triangulation F as a subset of the disk $\overline{{\mathbb{H}}^2}$. In most instances, however, we are interested only in its combinatorial structure and draw the pictures accordingly: for example, see Figure [fig:level]. [rem:topograph] The dual object of the Farey triangulation is the complete binary tree. If it is properly embedded in the plane, it is called the topograph (see and also ). The combinatorial structure of topograph is more convenient in Bowditch’s theory, where one investigates flows on the topograph. See Bowditch as well as and. Although we do not use the topograph directly, its tree structure underlies the inductive construction that we explain below using galleries and levels. Levels and Christoffel words ---------------------------- Recall that we fixed a basis e = (a, b) of F2 and that the distinguished directed Farey edge is denoted by [e] = [a, b]. This naturally leads us to consider the 2-fold reflectional symmetry of the Farey triangulation with respect to [e] among other symmetries in PGL(2, Z). From the embedding ${\mathcal{F}}\subset\overline{{\mathbb{H}}^2}$ of the Farey triangulation, we can talk about the sides of directed Farey edges. The set of Farey vertices lying on the left-hand (resp. right-hand) side of [x, y] will be denoted by V+[x, y] (resp. V−[x, y]). A single directed Farey edge [x, y] induces directions on other edges as follows. The circle ∂H2 is the union of two semi-circles connecting [x] and [y], both of which are endowed with the linear order running from [x] to [y]. We then direct other Farey edges according to this linear order; we say they are co-directed with [x, y]. For example, Figure [fig:level](Left) shows the directions induced by [e]. This will be used later for the inductive procedure in Definitions [def:chris] and [def:pal]. at 246 144 at 796 144 at 860 112 at 740 154 at 740 92 at 650 190 at 650 60 at 616 230![(Left) Farey edges with directions induced from the edge [{\mathsf{e}}]. (Right) Levels with respect to the edge [{\mathsf{e}}].](fig_Farey_levels "fig:") [fig:level] We now introduce the notion of levels relative to the distinguished edge [e]. A gallery is the union T1 ∪ T2 ∪ ⋯ ∪ Tk of Farey triangles such that the triangles Ti and Ti + 1 are adjacent along an edge for 1 ≤ i ≤ k − 1. The length of a gallery is the number of triangles it contains. (See Figure [fig:level](Right).) Let X denote either a vertex distinct from [a] or [b], or an edge distinct from [e], or a triangle. Then there exists a unique gallery of shortest length containing both X and [e]. The e-level Lve(X) of X is defined to be the length of this gallery. The edge [e] itself as well as its end points [a] and [b] are defined to be of e-level 0. There is no triangle of e-level 0. Most of the time the reference edge [e] is understood and we shall simply say levels instead of e-levels. While we defined the level combinatorially, it can also be defined using the geometry of H2 as in \Definition 3. A vertex of level k* > 0 is contained in a unique triangle of level k. A triangle of level k > 0 contains two edges of level k and one edge of level k − 1; if k > 1 its vertices are of level k, k − 1 and j (with j ≤ k − 2), respectively. A directed edge [x, y] of level k > 0 contains vertices of level k and j (with j ≤ k − 1), and is contained in two triangles of level k and of level k + 1, which will be denoted, respectively, by $$[w;x,y]\quad\textup{and}\quad[x,y;z],$$ where Lve[w] ≤ k − 1 and Lve[z] = k + 1. The union of these triangles will be denoted by [w; x, y; z] and called the quadrilateral determined by [x, y]. Following Bowditch \p.704, p.706 we use the notion of levels in order to define functions on V±[a, b] inductively. Let S* be a set and B : S × S → S a binary operation. Given an S-valued function f defined on the vertices [a] and [b] of level 0, we can inductively extend it to a map f : V+[a, b] → S as follows: a vertex [z] ∈ V+[a, b] of level k + 1 (k ≥ 0) is contained in a unique triangle [x, y; z] of level k + 1 such that [x, y] is co-directed with [a, b]. Then the levels of [x] and [y] are at most k. Assuming, inductively, the values f[x] and f[y] have already been defined, we define f[z] = B(f[x], f[y]). Similarly, we extend f to V−[a, b] starting with (possibly new) initial values of f on [a] and [b]. Note that the level function Lve itself can be defined in this way by setting S = Z and B(p, q) = max{p, q} + 1 starting with Lve[a] = Lve[b] = 0. More interesting examples are the Fibonacci function and the Christoffel function to be defined as follows. See Figure [fig:fibchris]. at 2 120 at 440 120 at 240 176 at 240 70 at 104 210 at 104 30 at 336 210 at 336 30 at 56 260 at 156 260 at 284 260 at 380 260 at 44 344 at 96 344 at 146 344 at 196 344 at 246 344 at 296 344 at 346 344 at 396 344 at 1126 124 at 494 140 at 494 94 at 836 166 at 844 84 at 630 210 at 980 210 at 630 40 at 986 40 at 556 260 at 726 256 at 886 256 at 1050 260 at 516 314 at 610 340 at 688 340 at 766 340 at 842 340 at 916 340 at 990 340 at 1090 310![(Left) The Fibonacci function. (Right) The Christoffel function.](fig_fibchris "fig:") [fig:fibchris] [def:chris] Let S be a set and B : S × S → S a binary operation. 1. Let S = N and B(p, q) = p + q. The e-Fibonacci function Fibe is defined inductively on V with initial values Fibe[a] = Fibe[b] = 1. 2. Let S = F(a, b) and B(w1, w2) = w1w2, the concatenation of w1 and w2. The e-Christoffel function Che is defined inductively on V+[a, b] with initial values Che[a] = a and Che[b] = b, and on V−[a, b] with initial values Che[a] = a− 1 and Che[b] = b. The values of Che are called e-Christoffel words (or Christoffel words in {a, b}). Strictly speaking, the value Che[a] is indeterminate and can be either a or a− 1 depending on the context. But this will not cause any confusion later. Before we delve into a few immediate properties of the Fibonacci and Christoffel functions, let us first adopt the following terminology: elements of the free semi-group generated by two symbols x and y are said to be positive in {x, y}. Let [w] ∈ V with Lve[w] > 0. Since Christoffel words are obtained by concatenation, it is clear that Che[w] is either positive in {a, b} (if [w] ∈ V+[a, b]) or positive in {a− 1, b} (if [w] ∈ V−[a, b]). This implies, in particular, that every e-Christoffel word is cyclically reduced, that is, its initial and terminal symbols are not inverse to each other. Observe also that if [w] ∈ V corresponds to p/q ∈ P1(Q) then πe(Che[w]) = (p, q) (see, and Convention [conv:slope]). One can use the continued fraction expansion of p/q to find the corresponding e-Christoffel word. (See Appendix [sec:cf] for an illustration of this reverse procedure.) Therefore, Che[w] is a cyclically reduced representative of its unoriented primitive class [w]. It is also apparent from the definitions that, for each vertex [w] ∈ V, we have $$\begin{aligned} \label{eqn:fib1} {\mathrm{Fib}}\_{\mathsf{e}}[w]=\|\,{\mathrm{Ch}}\_{\mathsf{e}}[w]\,\|\_{\mathsf{e}},\end{aligned}$$ where ∣ ⋅ ∣e denotes the word length with respect to the basis e = (a, b). See \Lemma 2.2.1. (As we shall see later in, the value Fibe[w] is also equal to the translation length ℓe(w) for [w] ∈ V.) We also observe that $$\begin{aligned} \label{eqn:fib2} {\mathrm{Fib}}\_{\mathsf{e}}[w]\ge {\mathrm{Lv}}\_{\mathsf{e}}[w]+1\end{aligned}$$ for all [w] ∈ V. Let us unravel the definition of e-Christoffel words a little further. Let [x, y; z] be a Farey triangle of level k* + 1 (k ≥ 0) appearing in the inductive procedure and assume the edge [x, y] of level k is co-directed with [e]. We then have Che[z] = Che[x]Che[y] by definition. If the pair (Che[x], Che[y]) is a basis of F2, then the pairs (Che[x], Che[z]) and (Che[z], Che[y]) are obtained by elementary Nielsen transformations (the last two in ) and thus are bases as well. Therefore, for any Farey edge [x, y] co-directed with [e], we see inductively that the pair (Che[x], Che[y]) is a basis. We call such a pair an e-Christoffel basis. [rem:monoid] The above inductive procedure basically amounts to considering the action of the free submonoid of Aut(F2) generated by two automorphisms corresponding to $\begin{psmallmatrix}1&1\\0&1\end{psmallmatrix}$ and $\begin{psmallmatrix}1&0\\1&1\end{psmallmatrix}$ in PGL(2, Z). For more details, see Appendix [sec:cf]. Suppose now k > 0 and let [w; x, y] be the other Farey triangle adjacent to [x, y]. It is of level k. Then there are two cases depending on whether Lv[x] < Lv[y] or Lv[x] > Lv[y]. In the former case we have Che[y] = Che[x]Che[w], and in the latter Che[x] = Che[w]Che[y]. To summarize, if (x, y) is an e-Christoffel basis with Lve[x, y] ≠ 0 then it determines the quadrilateral [x− 1y; x, y; xy]. In particular, Lve[x− 1y] < Lve[xy]. We extract this property and apply it to arbitrary bases. [def:acute] A basis f = (x, y) of F2 is said to be acute relative to the basis e = (a, b) if Lve[x− 1y] < Lve[xy]. For any basis (x, y) with Lve[x, y] ≠ 0, note that either (x, y) and (x− 1, y− 1) are acute relative to the basis e, or else (x− 1, y) and (x, y− 1) are acute relative to the basis e. Lastly, suppose [x, y] is a Farey edge co-directed with [e] and Lve[x, y] > 0. In order to simplify the notation let us define I[x, y] to be V+[x, y] if [x, y] lies on the left of [e], or to be V−[x, y] if [x, y] lies on the right of [e]. Then the following lemma is also immediate from the inductive construction of the Christoffel words. [lem:chris] Suppose [x, y] is a Farey edge co-directed with [e] and Lve[x, y] > 0. Then the e-Christoffel words in I[x, y] are positive words in {Che[x], Che[y]}. For example, the unoriented primitive class of the Christoffel word (a2b)2ab, which corresponds to 5/3 in {a, b}, lies in V+[a, ab]. Since (a2b)2ab = a(ab)a(ab)2, it corresponds to the positive slope 2/3 in {a, ab}. See Figure [fig:fibchris](Right). Although the Christoffel words are classical mathematical objects, the terminology was introduced rather recently in the 1990s. See and for more detailed accounts on the theory of Christoffel words. See also, and. In the literature of Kleinian groups and low-dimensional topology, the Christoffel words also appeared in, and (with correction ) as well as in (in a somewhat weaker form). Compare also. Level- partition of Farey vertices ---------------------------------- For each integer n ≥ 0 we associate a partition of the vertex set V as follows. Let V≤ n be the (finite) subset of V consisting of all vertices of level at most n. Under the embedding V = P1(Q) ⊂ P1(R), consider the complement of V≤ n in P1(R). It consists of 2n + 1 open intervals Ij (1 ≤ j ≤ 2n + 1). [def:partition] The level-n* partition* of V is defined to be the partition V = V≤ n ∪ I1 ∪ I2 ∪ ⋯I2n + 1, where Ij = Ij ∩ P1(Q) for 1 ≤ j ≤ 2n + 1. We still call each Ij an interval and cyclically order the indices as in Figure [fig:n-partition]. at 80 224 at 80 26 at 310 212 at 470 212 at 310 38 at 470 38 at 534 180 at 534 70 at 600 234 at 600 14 at 674 234 at 674 10 at 740 180 at 740 70![The level-n partitions of {\mathsf{V}} for n=0,1,2.](fig_n-partition "fig:") [fig:n-partition] Note that each interval Ij is either of the form V+[xj, yj] if 1 ≤ j ≤ 2n or of the form V−[xj, yj] if 2n + 1 ≤ j ≤ 2n + 1. As we did previously, we can simply write Ij = I[xj, yj] for all j. By Lemma [lem:chris] the vertices in Ij have representatives that are positive words in {Che[xj], Che[yj]}. This property will be essential in our proof of Theorem [I]. Representations of into ======================= We discuss some classical facts on irreducible actions of F2 on the hyperbolic space H3. Mostly following the book of Fenchel, we emphasize the use of half-turns and the Coxeter extension. We also introduce the notion of amplitude of a right-angled hexagon. Isometries of and half-turns ---------------------------- We recall some of the basic definitions and classical results on the orientation-preserving isometries of the 3-dimensional hyperbolic space (H3, dH). We mainly refer to the book by Fenchel. As a model of (H3, dH) we use the upper half-space model C × R+. The group Isom+(H3) of orientation preserving isometries of H3 can be identified, via the Poincaré extension, with the group ${\mathsf{PSL}}(2,{\mathbb{C}})={\mathsf{SL}}(2,{\mathbb{C}})/\{\pm{\textup{Id}}\}$ of projective automorphisms of P1(C). See \p.26 for more details. For a projective class X* ∈ PSL(2, C) we write ${\widetilde}{X}$ to denote one of its lifts in SL(2, C). Under the above identification, non-trivial isometries X ∈ PSL(2, C) of H3 are classified by the trace of their matrix representatives ${\widetilde}{X}\in{\mathsf{SL}}(2,{\mathbb{C}})$: X is elliptic if ${\mathsf{tr}}{\widetilde}{X}\in(-2,2)$, parabolic if ${\mathsf{tr}}{\widetilde}{X}\in\{-2,2\}$, and loxodromic if ${\mathsf{tr}}{\widetilde}{X}\notin[-2,2]$. See \p.46. If X* is elliptic or loxodromic, then it preserves a geodesic line AxisX in H3, called the axis of X. An elliptic isometry P ∈ PSL(2, C) is involutory, that is, of order two if and only if ${\mathsf{tr}}{\widetilde}{P}=0$ if and only if ${\widetilde}{P}^2=-{\textup{Id}}$; such an isometry will be called a half-turn. It is not hard to see that every orientation-preserving isometry X can be expressed X = PQ as a composition of two half-turns P and Q, although such a decomposition is never unique. See again \pp.46-47. Below we shall investigate this decomposition using matrices in SL(2, C). For our purpose we focus only on the case when the isometry X* is elliptic or loxodromic. The complex translation length of X λH(X) = ℓH(X) + iθH(X) ∈ A = C/2πiZ is the complex number such that its real part ℓH(X) ≥ 0 is the (real) translation length along the axis and its imaginary part θH(X) ∈ R/2πZ ≅ ( − π, π] is the rotation angle measured according to the right-hand rule. A half-turn P ∈ PSL(2, C) determines a geodesic AxisP in H3. The two lifts $\pm {\widetilde}{P}\in{\mathsf{SL}}(2,{\mathbb{C}})$ of P have zero trace. An important fact is that we can consistently associate each of these matrices with one of the orientations of the geodesic AxisP; when the oriented geodesic is determined by an ordered pair of its ideal end points (u, uʹ) ∈ P1(C) × P1(C), an explicit formula of the corresponding matrix is given in terms of u and uʹ in \p.64, (1)-(3). In order to emphasize this fact we shall deliberately abuse notation: a traceless matrix ${\widetilde}{P}\in{\mathsf{SL}}(2,{\mathbb{C}})$ will also denote the corresponding oriented* geodesic as well. Now let X = PQ be a decomposition of an elliptic or loxodromic isometry X into two half-turns P and Q. Then the axes of P and Q intersect the axis of X orthogonally. Take arbitrary lifts ${\widetilde}{P}, {\widetilde}{Q}\in{\mathsf{SL}}(2,{\mathbb{C}})$ and orient AxisX arbitrarily. Given such a triple $({\widetilde}{P},{\widetilde}{Q};{\mathsf{Axis}}\_X)$ of oriented geodesics, Fenchel \p.67 defines the width* from ${\widetilde}{Q}$ to ${\widetilde}{P}$ along AxisX $$\eta({\widetilde}{Q},{\widetilde}{P})=\eta({\widetilde}{Q},{\widetilde}{P};{\mathsf{Axis}}\_X):=\pm{\lambda}\_{\mathbb{H}}(H)\in{\mathbb{A}}={\mathbb{C}}/2\pi i{\mathbb{Z}}$$ as the signed complex translation length of H, where H is the isometry that has axis AxisX and maps ${\widetilde}{Q}$ to ${\widetilde}{P}$, and the sign is provided in accordance with the orientation of AxisX. (We shall always use the simpler notation $\eta({\widetilde}{Q},{\widetilde}{P})$ assuming implicitly that the orientation of AxisX is understood.) See Figure [fig:width]. at -40 180 at 130 110 at 172 196 at 140 66 at 106 0 at 220 10![The width \eta({\widetilde}{Q},{\widetilde}{P}).](fig_width "fig:") [fig:width] Let ${\widetilde}{H}\in{\mathsf{SL}}(2,{\mathbb{C}})$ be an arbitrary lift of H. Then Fenchel \p.68 shows that $$X=H^2=PQ, \quad{\widetilde}{H}^2=-{\widetilde}{P}{\widetilde}{Q}, \quad\frac{1}{2}{\mathsf{tr}}{\widetilde}{H}^2=\cosh\eta({\widetilde}{Q},{\widetilde}{P}).$$ Let us set ${\widetilde}{X}={\widetilde}{H}^2$ and assume that AxisX* is oriented naturally in the translation direction of X. Then $\eta({\widetilde}{Q},{\widetilde}{P})$ is one of the two half translation lengths of X, that is, ${\lambda}\_{\mathbb{H}}(X)/2=\eta({\widetilde}{Q},{\widetilde}{P})$ or $\eta({\widetilde}{Q},{\widetilde}{P})+\pi i$, and we have $$\begin{aligned} \label{eqn:cosh} \frac{1}{2}{\mathsf{tr}}{\widetilde}{X}=\cosh\eta({\widetilde}{Q},{\widetilde}{P})=\pm\cosh\frac{{\lambda}\_{\mathbb{H}}(X)}{2}.\end{aligned}$$ For a complex number η = ℓ + iθ (ℓ ≥ 0) we have coshη = coshℓcosθ + isinhℓsinθ. So we see that ∣coshη∣2 = cosh2ℓ − sin2θ = sinh2ℓ + cos2θ, in particular, sinhℓ ≤ ∣coshη∣ ≤ coshℓ. Thus from the above formula for ${\mathsf{tr}}{\widetilde}{X}$ we obtain $$\begin{aligned} \label{eqn:ltr} e^{\frac{1}{2}\ell\_{\mathbb{H}}(X)}-1 \le \|{\mathsf{tr}}{\widetilde}{X}\| \le e^{\frac{1}{2}\ell\_{\mathbb{H}}(X)}+1 < e^{\frac{1}{2}\ell\_{\mathbb{H}}(X)+1}\end{aligned}$$ For a number h > 0 its parallel angle α(h) ∈ (0, π/2) is defined by the formula $$\begin{aligned} \label{eqn:parallel\_angle} \sin{\alpha}(h) \cosh h=1.\end{aligned}$$ See Fenchel \p.92. A geometric meaning of α(h) is given in Figure [fig:parallelangle](Left). at 70 170 at 80 106 at 180 200 at 386 74 at 320 160 at 450 196 at 720 210 at 640 122 at 256 76 at 520 200![(Left) The parallel angle {\alpha}(h) of h. (Right) Proof of Lemma [lem:parallelangle].](fig_parallel_angle1 "fig:") [fig:parallelangle] Suppose a (totally geodesic) plane H intersects the axis of a loxodromic isometry Y* transversely. The following lemma says that if the (real) translation length of Y is large enough then, regardless of the rotation angle of Y, the plane H and its image Y(H) are ultra-parallel, that is, their closures in $\overline{{\mathbb{H}}^3}$ are disjoint. [lem:parallelangle] Let Y be a loxodromic isometry and H ⊂ H3 a plane intersecting AxisY with angle α ∈ (0, π/2]. If the angle α is greater than the parallel angle α(ℓH(Y)/2) of the half (real) translation length of Y, then the plane H and the image Y(H) are ultra-parallel. See Figure [fig:parallelangle](Right). We may express the isometry Y = PQ as a composition of two half-turns P and Q so that the axis ${\widetilde}{Q}$ is contained in the plane H. Let H0 denote the plane which contains the axis ${\widetilde}{P}$ and is orthogonal to AxisY. (Note that H0 is determined by the half translation length of Y independently of the rotation angle of Y.) Now consider another plane Hʹ which contains AxisY and is orthogonal to both H and H0. Then α is the angle between the geodesics H ∩ Hʹ and AxisY. By the definition of parallel angle, we conclude that the geodesics H ∩ Hʹ and H0 ∩ Hʹ are ultra-parallel in the plane Hʹ. Since both planes H and H0 are orthogonal to the plane Hʹ, we see that H and H0 are ultra-parallel as well. By symmetry under the half-turn P, we also conclude that P(H) = P(Q(H)) = Y(H) and P(H0) = H0 are ultra-parallel. Therefore, the two planes H and Y(H) are separated by H0 and are ultra-parallel. Irreducible representations and their Coxeter extensions -------------------------------------------------------- A homomorphism ρ : F(a, b) → PSL(2, C) represents an isometric action of F(a, b) on H3. Let (x, y) be another basis of F2. For simplicity we shall write X = ρ(x), Y = ρ(y), Z = ρ(xy)− 1. The trace of the commutator of X and Y is well-defined without ambiguity of sign for arbitrary lifts ${\widetilde}{X}$ and ${\widetilde}{Y}$. It is also independent of the choice of basis (x, y) by Theorem [thm:Nielsen](b), since trA = trA− 1 for A ∈ SL(2, C). We denote this number by κρ : = tr(XYX− 1Y− 1). By definition, the representation ρ is irreducible if its action on P1(C) has no fixed point. It is known that this is the case if and only if κρ ≠ 2. See, for example, \Proposition 2.3.1. Let W(p, q, r) = ⟨p, q, r* ∣ p2 = q2 = r2 = 1⟩ = Z/2 \* Z/2 \* Z/2 be the free (or universal) Coxeter group of rank three. The elements pq and qr freely generate a subgroup ⟨pq, qr⟩ of index two. (The reader may recall the discussion in the introduction and Figure [fig:torus].) Given a basis f = (x, y) of F2, let ψf : F(x, y) → W(p, q, r) denote the embedding defined by ψf(x) = pq and ψf(y) = qr. We say the group W(p, q, r), together with ψf, is the Coxeter extension of F2 associated with the basis f = (x, y). [rem:2fold] The lack of ``notational" 3-fold symmetry in the definition of ψf is intentional and this way we emphasize the 2-fold symmetry of the Farey triangulation. In Section [sec:pal], however, we will define ψf differently in order to respect a 3-fold symmetry. For irreducible representations of F2 into PSL(2, C) we have the following classical theorem. For a proof see, for example, \p.185 or \II, p.94. See also \Theorem B and Theorem 3.2.2 for a more comprehensive account. [Coxeter extension][thm:coxeter] Let f = (x, y) be a basis of F2. If ρ* : F(x, y) → PSL(2, C) is an irreducible representation, then there exists a unique representation ρf : W(p, q, r) → PSL(2, C) such that ρ = ρf ∘ ψf. We also call ρf the Coxeter extension of ρ associated with the basis f = (x, y). We simplify the notations by setting $$\begin{aligned} \label{eqn:pqr} P=\rho\_{\mathsf{f}}(p),\quad Q=\rho\_{\mathsf{f}}(q),\quad R=\rho\_{\mathsf{f}}(r).\end{aligned}$$ We then have X = PQ, Y = QR and Z = RP. Henceforth, we focus on the case of our most interest, namely, when X, Y and Z are all loxodromic. Then irreducibility of ρ implies that the half-turns P, Q and R are uniquely defined by the following properties. The axis AxisQ is the common perpendicular of AxisX and AxisY. Likewise, the axis AxisP (resp. AxisR) is the common perpendicular of AxisZ and AxisX (resp. AxisY). We denote by Hex(ρ, f) : = (AxisR, AxisY, AxisQ, AxisX, AxisP, AxisZ) the cyclically ordered sextuple of the geodesics, and call it the right-angled hexagon associated with ρ and f = (x, y). Right-angled hexagons and their amplitudes ------------------------------------------ We continue with the previous discussion, so assume ρ : F(x, y) → PSL(2, C) is irreducible and X, Y and Z are all loxodromic. Henceforth, we further assume that the geodesics AxisX, AxisY, and AxisZ are oriented naturally in accordance with the transformation directions of X, Y and Z. Given the right-angled hexagon Hex(ρ, f) we will be mostly interested in the angular part of the width η(AxisY, AxisX) between AxisX and AxisY. For the purpose of computation it will be convenient if we lift ρ to SL(2, C) and use the trace identities for matrices in SL(2, C). More precisely, we take arbitrary lifts ${\widetilde}{P}, {\widetilde}{Q}, {\widetilde}{R}\in{\mathsf{SL}}(2,{\mathbb{C}})$ of P, Q, R, respectively. This gives rise to a lift ${\widetilde{\rho}}:{\mathsf{F}}(x,y)\to{\mathsf{SL}}(2,{\mathbb{C}})$ of ρ defined by $$\begin{aligned} {\widetilde{\rho}}(x)={\widetilde}{X}=-{\widetilde}{P}{\widetilde}{Q},\quad{\widetilde{\rho}}(y)={\widetilde}{Y}=-{\widetilde}{Q}{\widetilde}{R}\end{aligned}$$ In this case we have ${\widetilde}{Z}={\widetilde}{Y}^{-1}{\widetilde}{X}^{-1}=(-{\widetilde}{R}{\widetilde}{Q})(-{\widetilde}{Q}{\widetilde}{P})=-{\widetilde}{R}{\widetilde}{P}$. Conversely, a lift ${\widetilde{\rho}}:{\mathsf{F}}(x,y)\to{\mathsf{SL}}(2,{\mathbb{C}})$ of ρ determines the lifts ${\widetilde}{P}, {\widetilde}{Q}, {\widetilde}{R}\in{\mathsf{SL}}(2,{\mathbb{C}})$ uniquely (up to reverting their signs simultaneously). The right-angled hexagon Hex(ρ, f) is now oriented (in the sense of Fenchel \p.79). The cyclically ordered sextuple of the oriented geodesics will be denoted by $${\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}}):=({\widetilde}{R},\; {\mathsf{Axis}}\_Y,\; {\widetilde}{Q},\; {\mathsf{Axis}}\_X,\; {\widetilde}{P},\; {\mathsf{Axis}}\_Z).$$ See Figure [fig:hexagon]. at 0 90 at 174 10 at 396 230 at 124 118 at 280 116 at 240 206![The right-angled hexagon {\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}}).](fig_hexagon "fig:") [fig:hexagon] For brevity let us set $$\eta\_X=\eta({\widetilde}{Q},{\widetilde}{P}),\quad \eta\_Y=\eta({\widetilde}{R},{\widetilde}{Q}),\quad \eta\_Z=\eta({\widetilde}{P},{\widetilde}{R}),\quad \eta\_Q=\eta({\mathsf{Axis}}\_Y,{\mathsf{Axis}}\_X).$$ In fact, the width η*Q is determined by ${\widetilde{\rho}}$ only up to sign, since we may change the signs of ${\widetilde}{P}, {\widetilde}{Q}, {\widetilde}{R}$ simultaneously and obtain the same lift ${\widetilde{\rho}}$. However, if we set $$\begin{aligned} \label{eqn:angle} \theta(\rho,{\mathsf{f}})=\|\mathrm{Im}\,\eta\_Q\|\in[0,\pi]\end{aligned}$$ this number is independent of the choice of the lift ${\widetilde{\rho}}$. We call it the angle associated to ρ and f. We shall use the following form of the law of Cosines for ${\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}})$: $$\begin{aligned} \cosh\eta\_Z =\cosh\eta\_X\cosh\eta\_Y+\sinh\eta\_X\sinh\eta\_Y\cosh\eta\_Q.\end{aligned}$$ See \p.83. Since we have from $$\begin{aligned} {\mathsf{tr}}{\widetilde}{X}&=-{\mathsf{tr}}{\widetilde}{P}{\widetilde}{Q}=2\cosh \eta\_X\\ {\mathsf{tr}}{\widetilde}{Y}&=-{\mathsf{tr}}{\widetilde}{Q}{\widetilde}{R}=2\cosh \eta\_Y\\ {\mathsf{tr}}{\widetilde}{Z}&=-{\mathsf{tr}}{\widetilde}{R}{\widetilde}{P}=2\cosh \eta\_Z\end{aligned}$$ the law of Cosines can be written as $$\begin{aligned} \label{eqn:cosine} \frac{2\,{\mathsf{tr}}{\widetilde}{Z}}{{\mathsf{tr}}{\widetilde}{X}{\mathsf{tr}}{\widetilde}{Y}}=\frac{\cosh\eta\_Z}{\cosh\eta\_X\cosh\eta\_Y} =1+\tanh\eta\_X\tanh\eta\_Y\cosh\eta\_Q,\end{aligned}$$ which we shall need later. Note that $$\begin{aligned} \label{eqn:tanh} \mathrm{Re}(\tanh\eta\_X)>0\textup{ if and only if }\mathrm{Re}(\eta\_X)>0,\end{aligned}$$ the latter of which is the case as we assumed X* is loxodromic and AxisX is oriented naturally. Lastly, to the oriented right-angled hexagon ${\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}})$, Fenchel \p.102 assigns a complex number, called its amplitude, defined by $$\begin{aligned} {\mathsf{am}}({\widetilde{\rho}},{\mathsf{f}})=-\frac{1}{2}{\mathsf{tr}}{\widetilde}{P}{\widetilde}{Q}{\widetilde}{R}.\end{aligned}$$ He then draws many formulas for the amplitude in geometric terms of the right-angled hexagon ${\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}})$. One of the formulas we shall need later is the following \p.103, (3): $$\begin{aligned} \label{eqn:amplitude} {\mathsf{am}}({\widetilde{\rho}},{\mathsf{f}})=-i\sinh\eta\_X\sinh\eta\_Y\sinh\eta\_Q.\end{aligned}$$ The amplitude is closely related to the commutator trace κρ as follows: $$\begin{aligned} \label{eqn:kappa} \begin{aligned} {\widetilde}{X}{\widetilde}{Y}^{-1}{\widetilde}{X}^{-1}{\widetilde}{Y}&={\widetilde}{P}{\widetilde}{Q}{\widetilde}{R}({\widetilde}{Q}{\widetilde}{Q}){\widetilde}{P}{\widetilde}{Q}{\widetilde}{R}=-({\widetilde}{P}{\widetilde}{Q}{\widetilde}{R})^2,\\ {\kappa}\_\rho={\mathsf{tr}}{\widetilde}{X}{\widetilde}{Y}^{-1}{\widetilde}{X}^{-1}{\widetilde}{Y}&=-{\mathsf{tr}}({\widetilde}{P}{\widetilde}{Q}{\widetilde}{R})^2=2-{\mathsf{tr}}^2{\widetilde}{P}{\widetilde}{Q}{\widetilde}{R}=2-4{\mathsf{am}}^2({\widetilde{\rho}},{\mathsf{f}}). \end{aligned}\end{aligned}$$ In particular, since κρ is a constant independent of the basis f = (x, y), so is the modulus $\|{\mathsf{am}}({\widetilde{\rho}},{\mathsf{f}})\|$. Primitive stability and the Q-conditions ======================================== We define two open conditions on isometric actions ρ : F2 → PSL(2, C) ≅ Isom+(H3) of F2 on (H3, dH): Minsky’s primitive stability (Section [sec:ps]) and Bowditch’s Q-conditions (Section [sec:bq]). In Section [sec:psbq] we show that the former implies the latter. The isometric action of on itself --------------------------------- Recall that we fixed a basis e = (a, b) of F2. As before, we write ∣w∣e = ∣w∣(a, b) to denote the word length of w ∈ F2 with respect to the basis e = (a, b). The word metric de on F2 is defined by de(u, v) = ∣u− 1v∣e for u, v ∈ F2. Then F2 acts freely and isometrically on the metric space (F2, de) by left multiplication. For an element w ∈ F2 its translation length ℓe(w) is defined by $$\begin{aligned} \label{eqn:le} \ell\_{\mathsf{e}}(w):=\inf\_{u\in{\mathsf{F}}\_2}d\_{\mathsf{e}}(u,wu) \left(=\inf\_{u\in{\mathsf{F}}\_2}\|u^{-1}wu\|\_{\mathsf{e}}=\|r\|\_{\mathsf{e}}=:\\|w\\|\_{\mathsf{e}}\right).\end{aligned}$$ Algebraically, this can be interpreted as the cyclically reduced length ∥w∥e of w, that is, the word length ∣r∣e of a cyclically reduced representative r of the conjugacy class of w. Thus the translation length ℓe(w) depends only on the ∼ -equivalence class of w. Note that if w ∈ F2 is not the identity then the action of w on (F2, de) is axial: there exists a (discrete) geodesic Z → (F2, de) which is invariant under the action of w and is translated non-trivially through the amount ℓe(w). We denote the image of this geodesic by Axise(w) (or simply Axis(w) when e is understood) and call it the axis of w. Note that if w = uru− 1 then Axis(w) = uAxis(r). If r is cyclically reduced, then Axis(r) contains the cyclic orbit {ri ∣ i ∈ Z} of the identity; in particular, it passes through the identity. Similarly, for any basis f = (x, y), we denote by Axisf(w) ⊂ (F2, df) the axis of a non-trivial element w with respect to f. [lem:axis] Let f = (x, y) be an e-Christoffel basis of F2 with Lve[x, y] > 0. If w is a positive word in {x, y}, then the axis Axisf(w) is of the form {gi ∣ i ∈ Z}, where g0 = 1 and gi− 1gi + 1 ∈ {x, y} for all i ∈ Z, and is contained in the axis Axis(w) ⊂ (F2, de). If w is positive in {x, y} then it is of the form w = s1s2…sn for some n > 0 where each si ∈ {x, y}. Since f = (x, y) is an e-Christoffel basis, x and y are e-Christoffel words and they are both positive in {a, b} or both positive in {a− 1, b}. (Recall the discussion after Definition [def:chris].) Therefore, in either case, w = s1s2…sn is linearly and cyclically reduced as a word in {a, b}. The lemma now follows. Minsky’s primitive stability ---------------------------- Minsky introduced the notation of primitive stability and studied its basic properties. Although the definition makes sense for the free group Fn of arbitrary rank n ≥ 2, we restrict to the case n = 2 for simplicity. Recall that we endowed F2 with a metric de, where e = (a, b) is the distinguished basis. Suppose a representation ρ : F2 → PSL(2, C) is given. Upon choosing a base point o ∈ H3 the orbit map τρ, o : F2 → H3 is defined by τρ, o(w) = ρ(w)(o). Obviously, it is ρ-equivariant: τρ, o(uw) = ρ(uw)(o) = ρ(u)(τρ, o(w)) for u, w ∈ F2. [def:ps] A representation ρ : F2 → PSL(2, C) is said to be primitive stable if there exist a point o ∈ H3 and constants M, c > 0 such that the τρ, o-images of the axes Axis(x) ⊂ (F2, de) of primitive elements x of F2 are (M, c)-quasi-geodesics in H3, that is, $$\frac{1}{M}d\_{\mathsf{e}}(u,v)-c\le d\_{\mathbb{H}}(\rho(u)(o),\rho(v)(o))\le M d\_{\mathsf{e}}(u,v)+c$$ for all u, v ∈ Axis(x). See \Definition 3.1. Note that the property is preserved under the PSL(2, C)-conjugacy action, so we can consider this definition in the character variety X(F2). As in the introduction we denote by PS ⊂ X(F2) the subset consisting of all PSL(2, C)-conjugacy classes of primitive stable representations. The definition of primitive stability can be simplified a little. First of all, from the equivariance of τ*ρ, o and the triangle inequality in H3, one can deduce that the constants (M, c) satisfying the upper inequality above exist for any isometric action ρ and for any choice of base point o ∈ H3. See \Lemma I.8.18 for example. Thus the lower inequality is the only essential requirement for primitive stability. Furthermore, the requirement for each primitive element is rather for each (unoriented) primitive conjugacy class. For if x* = uru− 1 then Axis(x) = uAxis(r) and hence τρ, o(Axis(x)) = ρ(u)τρ, o(Axis(r)) by the equivariance of τρ, o. So it suffices to test the inequality only for cyclically reduced representatives. To summarize, we can simplify the above definition as follows. Henceforth we shall always use this alternative definition. [Alternative][def:altps] A representation ρ : F2 → PSL(2, C) is said to be primitive stable if there exist a point o ∈ H3 and constants m, c > 0 satisfying the following condition: for each unoriented primitive class [x] ∈ V, there is a cyclically reduced representative x such that $$\begin{aligned} m\cdot d\_{\mathsf{e}}(u,v)-c\le d\_{\mathbb{H}}(\rho(u)(o),\rho(v)(o))\end{aligned}$$ for all u, v ∈ Axis(x). Bowditch’s Q-conditions ----------------------- Note that, for an element X ∈ PSL(2, C) and its two lifts $\pm{\widetilde}{X}\in{\mathsf{SL}}(2,{\mathbb{C}})$, we have ${\mathsf{tr}}(-{\widetilde}{X})=-{\mathsf{tr}}{\widetilde}{X}$ and ${\mathsf{tr}}{\widetilde}{X} = {\mathsf{tr}}{\widetilde}{X}^{-1}$. So, given a representation ρ : F2 → PSL(2, C), the complex modulus ∣tr(ρ(x))∣ is a well-defined function on the set V = Prim/∼ of unoriented primitive classes of F2. [def:bq] A representation ρ : F2 → PSL(2, C) is said to satisfy the Q-conditions if 1. ρ(x) is loxodromic for all [x] ∈ V, and 2. ∣trρ(x)∣ ≤ 2 for only finitely many [x] ∈ V. See \p.702 and \p.765. The conditions (i) and (ii) will be referred to as the first and second Q-condition, respectively. Since the trace function is invariant under conjugacy, this definition also makes sense in the character variety X(F2). We denote by BQ ⊂ X(F2) the subset consisting of all PSL(2, C)-conjugacy classes of representations satisfying the Q-conditions. One of the main results of Bowditch \Theorem 2 and Proposition 4.9 and Tan, Wong and Zhang \Theorem 3.3 is the following characterization of the second Q-condition. We rephrased their theorem using the equality ([eqn:fib1]). [thm:bqpd] Suppose a representation ρ : F2 → PSL(2, C) satisfies the first Q-condition. Then ρ satisfies the Q-conditions if and only if there exist uniform positive constants m = m(ρ) and c = c(ρ) such that log∣trρ(x)∣ ≥ m ⋅ ℓe(x) − c for every [x] ∈ V. As a consequence we see that, for a representation ρ satisfying the Q-conditions, the set { ± trρ(x) ∣ [x] ∈ V} is a discrete subset of C. In the course of proving the above theorem, the authors investigated flows on the topograph (see Remark [rem:topograph]) and showed the existence of a ``finite attracting subtree." See \Corollary 3.12 and Lemma 3.15 and \Lemmas 3.21 and 3.24. This fact can be reformulated in our terminology as follows: [prop:subtree] If a representation ρ : F2 → PSL(2, C) satisfies the Q-conditions, then there exists a uniform constant Nρ > 0 with the following property: if a Farey edge [x, y] has Lv[x, y] ≥ Nρ and determines the quadrilateral [w; x, y; z] then ∣trρ(w)∣ ≤ ∣trρ(z)∣. Consider the level-Nρ partition (see Definition [def:partition]). Very roughly speaking, then the proposition implies that, for a representation ρ satisfying the Q-conditions, if we start with a Farey triangle and run the process of minimizing the complex modulus of vertex traces then the process always ends up with only a finitely many Farey triangles that are of level ≤ Nρ. Compare with the PSL(2, R) case in the introduction: if the axes of two hyperbolic translations intersect, then the trace minimizing process always stops at an acute (or right-angled) triangle. Let (x, y) be an acute basis of F2 with respect to e = (a, b) so that the Farey edge [x, y] determines the quadrilateral [x− 1y; x, y; xy]. (See Definition [def:acute].) For an arbitrary lift ${\widetilde{\rho}}$ to SL(2, C) one can deduce the trace identity $${\mathsf{tr}}{\widetilde{\rho}}(x^{-1}y)+{\mathsf{tr}}{\widetilde{\rho}}(xy)={\mathsf{tr}}{\widetilde{\rho}}(x){\mathsf{tr}}{\widetilde{\rho}}(y)$$ from the Cayley-Hamilton theorem. (See, for example, \Theorem A and Section 2.2 for a proof.) Then the inequality in Proposition [prop:subtree] is easily seen to be equivalent to $$\begin{aligned} \label{eqn:re} \mathrm{Re}\left(\frac{{\mathsf{tr}}{\widetilde{\rho}}(xy)}{{\mathsf{tr}}{\widetilde{\rho}}(x){\mathsf{tr}}{\widetilde{\rho}}(y)}\right) \ge\frac{1}{2}.\end{aligned}$$ Primitive stability implies the Q-conditions -------------------------------------------- Although the converse to our Theorem [I] is well-known, we provide a proof for the sake of completeness. Our proof is essentially the same as the proof in \Proposition 4.0.5. [prop:psbq] If ρ : F2 → PSL(2, C) is primitive stable, then it satisfies the Q-conditions. Suppose ρ is primitive stable. By Definition [def:altps] (with v = 1 ∈ F2) there exist a point o ∈ H3 and constants m, c > 0 such that, for each unoriented primitive class [x] ∈ V, there is a cyclically reduced representative x such that $$\begin{aligned} m\|u\|\_{\mathsf{e}}-c\le d\_{\mathbb{H}}(o,\rho(u)(o))\end{aligned}$$ for all u ∈ Axis(x). In particular, the ρ-orbit {ρ(xn)(o) ∣ n ∈ Z} of the point o under the cyclic subgroup ⟨x⟩ generated by x is a quasi-geodesic. This is the case only if ρ(x) is loxodromic. Therefore, ρ satisfies the first Q-condition. Moreover, setting u = xn ∈ Axis(x) in the above inequality, we have $$\begin{aligned} m \|x^n\|\_{\mathsf{e}}-c \le d\_{\mathbb{H}}(o,\rho(x^n)(o))\end{aligned}$$ for any n ∈ N. Therefore, we see that $$\begin{aligned} \ell\_{{\mathbb{H}}}(\rho(x)) &\ge\lim\_{n\to\infty}\frac{1}{n}d\_{\mathbb{H}}(o,\rho(x^n)(o))\\ &\ge\lim\_{n\to\infty}\frac{1}{n}\left(m \|x^n\|\_{\mathsf{e}}-c\right)\\ &=m \|x\|\_{\mathsf{e}}\\ &=m \cdot\ell\_{\mathsf{e}}(x),\end{aligned}$$ where the first inequality (which is actually equality in the present case) follows from the triangle inequality and the fact that the limit on the right-hand side is independent of the point o. See, for example, \Section 10.6 or \Exercise II.6.6. Since $\|{\mathsf{tr}}\rho(x)\|\ge e^{\frac{1}{2}\ell\_{{\mathbb{H}}}(\rho(x))}-1$ by ([eqn:ltr]), we have $\|{\mathsf{tr}}\rho(x)\|\ge e^{\frac{m}{2}\ell\_{\mathsf{e}}(x)}-1$ for all classes [x] ∈ V. On the other hand, for [x] ∈ V we have ℓe(x) = ∣ Che[x] ∣e = Fibe[x] from and since Che[x] is cyclically reduced. Thus we can possibly have $e^{\frac{m}{2}\ell\_{\mathsf{e}}(x)}-1\le2$, that is, ${\mathrm{Fib}}\_{\mathsf{e}}[x]\le\frac{2}{m}\log3$ for only finitely many classes [x] ∈ V; recall Definition [def:chris](a) and Figure [fig:fibchris]. So we conclude that $$\|{\mathsf{tr}}\rho(x)\|\ge e^{\frac{m}{2}\ell\_{\mathsf{e}}(x)}-1>2$$ for all but finitely many classes [x] ∈ V, and thus ρ satisfies the second Q-condition. Since H3 is Gromov δ-hyperbolic, we may use the Morse lemma (stability of quasi-geodesics) to give an alternative proof of the above proposition as in \Proposition 2.9. See also \Proposition 2.1. The above proof is stronger in that it requires no assumption on the metric space on which F2 acts isometrically. The Q-conditions imply primitive stability ========================================== In this section we prove Theorem [I]. We first establish Lemma [lem:angle] as a consequence of Theorem [thm:bqpd] and Proposition [prop:subtree]. This lemma is crucial in that it enables us to consider the level-N partition of V (Definition [def:partition]) for some N > 0 and to focus only on one interval Ij thereof. This is done in Theorem [thm:main], which is in fact slightly more general than what is needed for the proof of Theorem [I], in the sense that, for such an interval Ij = I(x, y), it deals not only with the primitive classes [w] ∈ I(x, y) but with all positive words in x and y. The proof of Theorem [I] starts in Section [sec:pf1]. Lemma on the angle ------------------ Suppose ρ : F2 → PSL(2, C) satisfies the Q-conditions. It is known \Theorem 1.4 that ρ* is irreducible. Since ρ is irreducible and satisfies the first Q-condition, we can follow the discussion in Section [sec:amplitude]. In particular, the angle θ(ρ, f) is defined for any basis f = (x, y) of F2. See. The following lemma will be crucial in our proof of Theorem [I]. It says that, for an acute basis f, its angle θ(ρ, f) is bounded above by a quantity depending only on its level. The same statement also appears in \Proposition 4.12 with a different proof. Recall Definition [def:acute] for acute bases. [lem:angle] Suppose ρ* : F2 → PSL(2, C) satisfies the Q-conditions. Let f = (x, y) be a basis of F2 that is acute relative to the basis e = (a, b). Then for any ɛ ∈ (0, π) there exists an integer N > 0 such that if Lve[x, y] ≥ N then 0 ≤ θ(ρ, f) < ɛ. Let f = (x, y) be a basis with Lve[x, y] > 0. Without loss of generality, we may assume that Lve[x] < Lve[y]. (The other case Lve[x] > Lve[y] can be dealt with in a similar fashion switching the roles of x and y.) Then we have Lve[x, y] = Lve[y]. See Figure [fig:level](Right). Furthermore, by, and, we have the inequality ℓe(y) ≥ Lve[y] + 1 as well. We now proceed as in Section [sec:amplitude] and employ the same notations therein. We take a lift ${\widetilde{\rho}}$ to SL(2, C) and obtain the oriented right-angled hexagon ${\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}})$ by choosing a set of lifts of P, Q, R accordingly. Then, as we observed in and, the number $$\|{\mathsf{am}}({\widetilde{\rho}},{\mathsf{f}})\|=\|\sinh\eta\_X\sinh\eta\_Y\sinh\eta\_Q\|$$ is a constant which depends only on ρ. Below we shall adopt the following terminology: if a number a(f) is a function of a variable basis f, by uniform convergence (or divergence) of the number a(f) we mean that the convergence is controlled by a quantity depending only on the level Lve[f] of f. We first note that $$\sinh\eta\_X=0\quad\Longleftrightarrow\quad\eta\_X=0\textup{ or }i\pi\quad\Longleftrightarrow\quad{\mathsf{tr}}{\widetilde}{X}=2\cosh\eta\_X=\pm2.$$ Since ρ satisfies the Q-conditions, we know from Theorem [thm:bqpd] that the set $\{{\mathsf{tr}}{\widetilde}{X} \mid [x]\in{\mathsf{V}}\}$ does not accumulate at ± 2. Thus there exists a uniform bound ερ > 0 such that ∣sinhηX∣ ≥ ερ for all [x] ∈ V. On the other hand, by Theorem [thm:bqpd] again, we have $$\begin{aligned} \label{eqn:coshy} \log\|2\cosh\eta\_Y\|=\log\|{\mathsf{tr}}{\widetilde}{Y}\|\ge m \cdot\ell\_{\mathsf{e}}(y)-c\ge m({\mathrm{Lv}}\_{\mathsf{e}}[y]+1)-c\end{aligned}$$ for some uniform constants m, c > 0. So ∣sinhηY∣, as well as ∣coshηY∣, diverges uniformly to infinity as Lve[x, y] = Lve[y] tends to infinity. In conclusion, we have $$\|\sinh\eta\_Q\| =\frac{\|{\mathsf{am}}({\widetilde{\rho}},{\mathsf{f}})\|}{\|\sinh\eta\_X\sinh\eta\_Y\|} \le\frac{\|{\mathsf{am}}({\widetilde{\rho}},{\mathsf{f}})\|}{{\epsilon}\_\rho\|\sinh\eta\_Y\|} =\frac{\textup{Const.}}{\|\sinh\eta\_Y\|},$$ and thus sinhηQ must converge uniformly to 0 as Lve[x, y] → ∞. This means that the possible accumulation points of the set $$\{ \cosh\eta\_Q \mid {\mathsf{f}}\textup{ is a basis of }{\mathsf{F}}\_2 \}$$ are 1 and − 1. However, we claim that if we consider only the bases that are acute relative to e then − 1 cannot be an accumulation point of the corresponding subset. To see this, suppose on the contrary that there is a sequence of acute bases {fi = (xi, yi)}i such that coshηQi converges to − 1 as i → ∞. By passing to a subsequence we may further assume, without loss of generality, that Lve[xi] < Lve[yi] for all i. Now recall the equation : $$\frac{2\,{\mathsf{tr}}{\widetilde}{Z\_i}}{{\mathsf{tr}}{\widetilde}{X\_i}{\mathsf{tr}}{\widetilde}{Y\_i}} =1+\tanh\eta\_{X\_i}\tanh\eta\_{Y\_i}\cosh\eta\_{Q\_i},$$ where ${\widetilde}{Z\_i}=({\widetilde}{X\_i}{\widetilde}{Y\_i})^{-1}$. Note from that Re(tanhηXi) > 0, and from that tanhηYi converges to 1. Thus we have $\mathrm{Re}\left(\frac{{\mathsf{tr}}{\widetilde}{Z\_i}}{{\mathsf{tr}}{\widetilde}{X\_i}{\mathsf{tr}}{\widetilde}{Y\_i}}\right) <\frac{1}{2}$ for all sufficiently large i. But this contradicts Proposition [prop:subtree] and the inequality, since we would eventually have Lv[xi, yi] ≥ Nρ for large i. Therefore, we conclude that the set $$\{ \cosh\eta\_Q \mid {\mathsf{f}}\textup{ is an acute basis of }{\mathsf{F}}\_2\textup{ relative to }{\mathsf{e}}\}$$ has a unique accumulation point 1. This means that, for an acute basis f = (x, y), the width ηQ, as well as θ(ρ, f) = ∣Im ηQ∣, converges uniformly to 0 as Lve[x, y] → ∞. By a similar argument one can show that the widths $$\eta({\mathsf{Axis}}\_X,{\mathsf{Axis}}\_Z)\;\textup{ and }\;\eta({\mathsf{Axis}}\_Z,{\mathsf{Axis}}\_Y)$$ both converge uniformly to πi ∈ A = C/2πiZ as Lve[x, y] → ∞. See also \Proposition 4.12. In this sense we may say that, as Lve[f] → ∞, the hexagon Hex(ρ, f) asymptotically becomes a ``degenerate obtuse triangle" as in the case of PSL(2, R) ≅ Isom+(H2) illustrated in the introduction. Positive words in a high level Christoffel basis ------------------------------------------------ The following theorem is* our main theorem and contains the most general statement, from which Theorem [I] will follow almost immediately. [thm:main] Suppose ρ : F2 → PSL(2, C) satisfies the Q-conditions. Then there exists a number N > 0 with the following property: if f = (x, y) is an e-Christoffel basis of F2 with Lve[x, y] ≥ N then there exist a point of ∈ H3 and constants mf, cf > 0 such that, for every positive word w in {x, y}, the inequality $$\begin{aligned} m\_{\mathsf{f}}\cdot d\_{\mathsf{e}}(u,v)-c\_{\mathsf{f}}\le d\_{\mathbb{H}}(\rho(u)(o\_{\mathsf{f}}),\rho(v)(o\_{\mathsf{f}}))\end{aligned}$$ holds for all u, v ∈ Axis(w). Suppose ρ satisfies the Q-conditions. By Lemma [lem:angle] we can take N1 > 0 large enough that we have 0 ≤ θ(ρ, f) < π/4 for every acute basis f = (x, y) with Lve[x, y] ≥ N1. By Theorem [thm:bqpd], there are constants m, c > 0 such that log∣trρ(w)∣ ≥ m ⋅ ℓe(w) − c for all [w] ∈ V. We have from and that ℓe(w) = ∣ Che[w] ∣e = Fibe[w] ≥ Lve[w] + 1. Note also from that $\log\|{\mathsf{tr}}\rho(w)\|<\frac{1}{2}\ell\_{\mathbb{H}}(\rho(w))+1$. Thus we see that $$\frac{1}{2}\ell\_{{\mathbb{H}}}(\rho(w)) > \log\|{\mathsf{tr}}\rho(w)\| -1 \ge m\cdot\ell\_{\mathsf{e}}(w) -(c+1) \ge m ({\mathrm{Lv}}\_{\mathsf{e}}[w]+1) -(c+1)$$ for all [w] ∈ V. Take N2 > 0 big enough so that if Lve[w] ≥ N2 then the parallel angle of the half translation length $\frac{1}{2}\ell\_{{\mathbb{H}}}(\rho(w))$ is less than π/4. Set N = max{N1, N2}. Suppose f = (x, y) is an e-Christoffel basis and Lve[x, y] ≥ N. In particular, f is acute relative to e (see the discussion preceding Definition [def:acute]). Without loss of generality, we may further assume Lve[x] < Lve[y] so that Lve[x, y] = Lve[y]. Proceeding as in Section [sec:amplitude] we take a lift ${\widetilde{\rho}}$ to SL(2, C) and obtain the oriented right-angled hexagon ${\mathsf{Hex}}({\widetilde{\rho}},{\mathsf{f}})$ by choosing a set of lifts of P, Q, R accordingly. Take the (totally geodesic) plane H ⊂ H3 which contains the geodesic ${\widetilde}{Q}$ and is perpendicular to AxisX. See Figure [fig:hyperplane]. at 300 346 at 50 350 at 550 354 at 160 160 at 280 90 at 450 176 at 210 180 at 360 170 at 314 260 at 650 30 at 480 60 at 580 90 at 540 120![Taking the plane {\mathcal{H}}.](fig_hyperplane "fig:") [fig:hyperplane] Since the planes H and ρ(x)(H) are both orthogonal to AxisX, they are ultra-parallel. Since Lve[y] = Lve[x, y] ≥ N2, the parallel angle of $\frac{1}{2}\ell\_{{\mathbb{H}}}(\rho(y))$ is less than π/4. Since Lve[x, y] ≥ N1 and f is acute relative to e, we have 0 ≤ θ(ρ, f) < π/4. This means that the angle θʹ between H and AxisY is greater than π/4. By Lemma [lem:parallelangle] we see that the planes H and ρ(y)− 1(H) are also ultra-parallel. Everything combined, we can conclude that for any choice of g, h ∈ {x, y} the three planes $$\begin{aligned} \label{eqn:triple} \rho(g)^{-1}({\mathcal{H}}),\; {\mathcal{H}},\; \rho(h)({\mathcal{H}})\end{aligned}$$ are pairwise ultra-parallel and H separates the other two. Now we choose an arbitrary point o ∈ H as a base point and consider the associated orbit map τρ, o. Let w be a positive word in {x, y}. From Lemma [lem:axis] we know that the axis Axis(w) ⊂ (F2, de) properly contains the set Axisf(w) = {gi ∣ i ∈ Z}, where g0 = 1 and gi− 1gi + 1 ∈ {x, y} for all i ∈ Z. See Figure [fig:axis]. We claim that the points in the image τρ, o(Axisf(w)) in H3 belong to various translates of H which are pairwise ultra-parallel and well-ordered. In order to see this, note that any three consecutive points in Axisf(w) are of the form (gig− 1, gi, gih) for g, h ∈ {x, y}. Under the orbit map τρ, o, such a triple is mapped to (ρ(gi)ρ(g)− 1(o), ρ(gi)(o), ρ(gi)ρ(h)(o)). Since o ∈ H, we see that, up to isometry ρ(gi), these three points in this order belong to the three planes in ([eqn:triple]), respectively. The claim is proved. For an illustration of the claim, see Figure [fig:orbit], where the various translates of H are shown to be well-ordered in the case of the positive (Christoffel) word w = x2yxy. at 380 110 at 204 160 at 330 186 at 310 240 at 240 270 at 436 240 at 560 240 at 720 280 at 830 320![An example for the positive word w=x^2yxy.](fig_orbit "fig:") [fig:orbit] We set the following positive constants: $$\begin{aligned} d&=\min\{d\_{\mathbb{H}}({\mathcal{H}},\rho(x)({\mathcal{H}})),d\_{\mathbb{H}}({\mathcal{H}},\rho(y)({\mathcal{H}}))\},\\ D&=\max\{d\_{\mathbb{H}}(o,\rho(a)(o)),d\_{\mathbb{H}}(o,\rho(b)(o))\},\\ L&=\max\{\|x\|\_{\mathsf{e}},\|y\|\_{\mathsf{e}}\}.\end{aligned}$$ These constants do not depend on w, but they depend only on the representation ρ, the bases e = (a, b) and f = (x, y), and the chosen point o ∈ H. Let u, v ∈ Axis(w). Then there are indices j, k ∈ Z such that u and v lie in the subintervals [gj − 1, gj] and [gk, gk + 1] of Axis(w), respectively. Without loss of generality we may assume j − 1 ≤ k. Then u = gj − 1s and v = gkt where s and t are certain subwords either of x or of y. See Figure [fig:axis]. at 386 54 at 80 16 at 170 40 at 310 70 at 480 76 at 610 102 at 690 82 at 110 62 at 646 128![The discrete geodesic {\mathsf{Axis}}(w) in ({\mathsf{F}}_2,d_{\mathsf{e}}).](fig_axis "fig:") [fig:axis] From the triangle inequality, we have $$\begin{aligned} d\_{{\mathbb{H}}}&(\rho(u)(o),\rho(v)(o)) + 2 D L\\ &\ge d\_{{\mathbb{H}}}(\rho(g\_j)(o),\rho(u)(o)) + d\_{{\mathbb{H}}}(\rho(u)(o),\rho(v)(o)) + d\_{{\mathbb{H}}}(\rho(v)(o),\rho(g\_k)(o))\\ &\ge d\_{{\mathbb{H}}}(\rho(g\_j)(o),\rho(g\_k)(o)),\end{aligned}$$ and thus $$\begin{aligned} d\_{{\mathbb{H}}}(\rho(u)(o),\rho(v)(o)) &\ge d\_{{\mathbb{H}}}(\rho(g\_j)(o),\rho(g\_k)(o)) - 2 D L\\ &\ge d(k-j) - 2 D L\\ &\ge \frac{d}{L}d\_{\mathsf{e}}(u,v) - (2 d + 2 D L), \end{aligned}$$ since L(k − j + 2) ≥ de(gj − 1, gk + 1) ≥ de(u, v). The proof is complete if we set of = o, mf = d/L and cf = 2d + 2DL. Proof of Theorem [I] -------------------- [Proof of Theorem I] Suppose ρ satisfies the Q-conditions. In order to show ρ is primitive stable, choose any base point o ∈ H3 and consider the associated orbit map τρ, o. Take the number N > 0 guaranteed by Theorem [thm:main] and consider the level-N partition of V (Definition [def:partition]): V = V≤ N ∪ I1 ∪ I2 ∪ ⋯I2N + 1. Recall that each interval Ii is of the form Ii = I[xi, yi] and Lve[xi, yi] = N for 1 ≤ i ≤ 2N + 1. Fix an index i (1 ≤ i ≤ 2N + 1) and let [w] ∈ Ii = I[xi, yi]. Then the vertex [w] has a representative Che[w] which is an e-Christoffel word (Definition [def:chris]). In particular, Che[w] is cyclically reduced. Furthermore, the word Che[w] is a positive word in {Che[xi], Che[yi]} by Lemma [lem:chris], and the pair fi = (Che[xi], Che[yi]) is an e-Christoffel basis. Therefore, by Theorem [thm:main] again, there exist oi ∈ H3 and mi, ci > 0 such that mi ⋅ de(u, v) − ci ≤ dH(ρ(u)(oi), ρ(v)(oi)), for all u, v ∈ Axis(w). Let ri = dH(o, oi). Then, by the triangle inequality, we have $$\begin{aligned} d\_{\mathbb{H}}(\rho(u)(o),\rho(v)(o)) &\ge d\_{{\mathbb{H}}}(\rho(u)(o\_i),\rho(v)(o\_i)) -d\_{{\mathbb{H}}}(\rho(u)(o\_i),\rho(u)(o))-d\_{{\mathbb{H}}}(\rho(v)(o),\rho(v)(o\_i))\\ &\ge m\_i\cdot d\_{\mathsf{e}}(u,v)-(c\_i+2r\_i),\end{aligned}$$ for all u, v ∈ Axis(w). These constants mi, ci and ri depend only on the interval Ii. Let (m0, c0) be some positive constants for which the analogous inequalities of Definition [def:altps] hold for the finite number of vertices in V≤ N with respect to the base point o. By setting $$\begin{aligned} m&=\min\{m\_0,m\_i\mid 1\le i\le 2^{N+1}\}\\ c&=\max\{c\_0,c\_i+2r\_i\mid 1\le i\le 2^{N+1}\},\end{aligned}$$ we conclude that ρ satisfies Definition [def:altps] with constants m, c > 0 and base point o. On Lupi’s proof --------------- Lupi recently showed in his thesis \Proposition 3.4 and Theorem 4.3 that, for representations ρ* : F(a, b) → PSL(2, R), primitive stability is equivalent to the Q-conditions. In order to sketch his idea of proof, we first remark that, for PSL(2, R) representations, the only non-trivial case to analyze is the non-Schottky representations where the axes of hyperbolic transformations a and b intersect, namely, the case we discussed in the introduction. See Figure [fig:torus]. There we have a quadrilateral Q whose opposite edges are paired by a and b. Thus as a quotient of Q we obtain a hyperbolic torus with one cone-type singularity. The image of such representation is not discrete in general, unless the cone-angle is a rational multiple of π. Nevertheless, we may still think of Q as a sort of fundamental domain and investigate its translates under the action. In fact, a crucial observation made by Lupi is that, for a primitive element w with Axis(w) = {ui}i ∈ Z in F(a, b), the translates ρ(ui)(Q) of Q do not overlap and form an infinite staircase-like strip. See \Figure 3.3. From this one can easily see that the image of Axis(w) under an orbit map is a uniform quasi-geodesic. For PSL(2, C)-representations satisfying the Q-conditions, however, there seems to be no ``pseudo-fundamental domain" in H3 as simple as the quadrilateral Q* in H2. For example, consider a slight perturbation of the above example out of PSL(2, R) and its Coxeter extension W(p, q, r). We obtain four complete geodesics which are axes of pqr and of its three other conjugates as in Figure [fig:torus]. The four axes are in skew position so that, in general, no (totally geodesic) plane in H3 contains any two of them. So there is no direct analogue of the quadrilateral Q. We may try to form a domain using other types of surfaces than planes. For example, the three axes of pqr, p and qrp (see Figure [fig:torus] again) can be used to define a ruled surface, that is, the union of all complete geodesics which intersect the three axes simultaneously. For a slight perturbation of the above example, we may then form a ``twisted" quadrilateral prism bounded by such ruled surfaces. However, it seems rather hard to analyze such an object. For example, it is not immediate to know how long its combinatorial structure is preserved under perturbation and how far the distance is between two opposite surfaces of the quadrilateral. On primitive displacing actions ------------------------------- Let us explain why we may view Theorem [I] in analogy with a theorem of Delzant, Guichard, Labourie and Mozes. Let (X, dX) be a metric space and ρ : Γ → Isom(X) an isometric action of a group Γ. For an element w ∈ Γ its translation length ℓX(w) for the ρ-action is defined as ℓX(w) = infx ∈ XdX(x, ρ(w)(x)). Endow Γ with a word metric de for a generating set e, and consider the isometric action of Γ on itself by left multiplication. The translation length of w ∈ Γ for this action will be denoted by ℓe(w). The action ρ : Γ → Isom(X) is called displacing if there are positive constants m and c such that ℓX(w) ≥ m ⋅ ℓe(w) − c for all w ∈ Γ. Delzant et al showed that if Γ is word hyperbolic then the action ρ is displacing if and only if an orbit map of ρ is a quasi-isometric embedding. See \Lemma 2.0.1, Proposition 2.2.1, Corollary 4.0.6. Since the free group F2 is word hyperbolic, we may view our Theorem [I] in analogy with the above theorem. Namely, if we focus only on actions of primitive elements, the Q-conditions, by Theorem [thm:bqpd] and ([eqn:ltr]), can be thought of as a weakening of the displacing property, while primitive stability is obviously a weakening of quasi-isometric embedding. Primitive stability is originally defined by Minsky for free groups of arbitrary rank. The Q-conditions also make sense for such free groups, but Theorem [thm:bqpd] is only known to hold for F2. One may then ask if it holds for any Fn. Or, regardless, we rather consider the following definition: a representation ρ* : Fn → Isom(X) (n ≥ 2) is said to be primitive displacing if there are positive constants m and c such that ℓX(w) ≥ m ⋅ ℓe(w) − c for all primitive elements w ∈ Fn(e). Then it is natural to ask: For which metric space X, is it true that ρ : Fn → Isom(X) (n ≥ 2) is primitive stable if and only if it is primitive displacing? In fact, one direction, primitive stability implies primitive displacing, is true with the same proof as in Proposition [prop:psbq]. So the question is really about the other. Our Theorem [I] tells us that the equivalence is true for n = 2 and X = H3. The next case one may want to explore seems to be when n = 2 and X is a rank-1 symmetric space of non-compact type, e.g., H4 and CH2. In the absence of trigonometric rules, one may employ certain large-scale geometric arguments in order to draw a conclusion that is analogous to Lemma [lem:angle], assuming the primitive displacing property and using Theorem [thm:Nielsen](b). Bounded intersection property ============================= In this section we investigate representations ρ : F2 → PSL(2, C) satisfying a new condition which we call the bounded intersection property. It is described using primitive elements in F2 that are palindromic in a fixed basis. Our definition is motivated by the work of Gilman and Keen \Theorem 6.6, where they use palindromic elements of F2 to give a sufficient condition for a representation to be discrete. In Section [sec:pf2] we prove Theorem [II] that the bounded intersection property is implied by the Q-conditions. Tri-coloring ------------ For the forthcoming discussion we need the tri-coloring* of the Farey triangulation. We fix a basis e = (a, b) with abelianization πe : F2 → Z2 as in. Consider the homomorphism given by reduction modulo 2 $$\begin{aligned} {\mathsf{PGL}}(2,{\mathbb{Z}})\twoheadrightarrow{\mathsf{PGL}}(2,{\mathbb{Z}}/2)\cong{\mathsf{Sym}}\_3.\end{aligned}$$ Its kernel, the level two congruence subgroup PGL(2, Z/2)(2), is isomorphic to the ideal triangle reflection group T\(∞, ∞, ∞) ≅ Z/2 \ Z/2 \* Z/2 with fundamental domain any Farey triangle. Reduction modulo 2 $$\begin{aligned} {\mathsf{V}}={\mathsf{P}}^1({\mathbb{Q}})\twoheadrightarrow{\mathsf{P}}^1({\mathbb{Z}}/2)=\{0/1,\;1/1,\;1/0\}\end{aligned}$$ partitions the vertex set V into three PGL(2, Z/2)(2)-orbits ``colored" by 0/1, 1/1 and 1/0, respectively. Note that the three vertices of a Farey triangle are all colored differently. A Farey edge is then colored by the same color of its opposite vertex. See Figure [fig:pal1]. For example, the vertex [a] is colored by 1/0 (black), the vertex [b] by 0/1 (white), and the edge [a, b] by 1/1 (gray). If a vertex [w] ∈ V is colored by c ∈ {0/1, 1/1, 1/0} then we shall often say [w] is a c-vertex. Similarly for edges. For a more detailed discussion of tri-coloring, see \Section 2. Primitive palindromes --------------------- We shall be interested in primitive elements that are palindromic. A word w* = w(a, b) ∈ F2 is said to be palindromic in a basis e = (a, b) (or, simply, an e-palindrome) if it reads the same forward and backward, that is, $$\begin{aligned} \label{eqn:pal} w(a,b)^{-1}=w(a^{-1},b^{-1}).\end{aligned}$$ Gilman and Keen introduced an algorithm of finding a palindromic representative in each primitive conjugacy class whose cyclically reduced length is odd. This algorithm can be described by slightly modifying the inductive procedure for the Christoffel function (Definition [def:chris](b)) as follows. Co-direct all Farey edges with the directed edge [e] = [a, b] as in Section [sec:chris]. at 8 156 at 8 124 at 650 144 at 340 176 at 340 110 at 190 224 at 480 224 at 190 64 at 480 64 at 110 270 at 250 270 at 410 270 at 550 270 at 54 360 at 130 360 at 206 350 at 284 360 at 356 360 at 430 350 at 504 350 at 580 360 at 850 154 at 960 154 at 904 252 at 856 228 at 848 212 at 954 228 at 964 212 at 900 130 at 910 114 ![(Left) The {\mathsf{e}}-palindrome function. (Right) The {\widetilde}{{\mathsf{e}}}-palindromes. ](fig_pal1 "fig:") [fig:pal1] [def:pal] Let S = F(a, b) and define B : S × S → S by $$B(w\_1,w\_2)= \left\{ \begin{array}{ll} w\_2w\_1& \text{if $[w\_1,w\_2]$ and $[a,b]$ have the same color,}\\ w\_1w\_2& \text{otherwise.} \end{array} \right.$$ The e-palindrome function Pale is defined inductively on V+[a, b] with initial values Pale[a] = a and Pale[b] = b, and on V−[a, b] with initial values Pale[a] = a− 1 and Pale[b] = b. See Figure [fig:pal1](Left). Observe that Pale[w] is an e-palindrome if and only if [w] ∈ V has color 1/0 or 0/1 (black or white). On the other hand, Pale[w] is a product of two e-palindromes if and only if [w] has color 1/1 (gray). These are re-statements of \Theorem 2.1, since [w] has color 1/1 (gray) if and only if its cyclically reduced length ∥w∥e = Fibe[w] is even, that is, 1 + 1 (mod 2). We also note that, according to \Theorem 1.(1), Pale[w] and its inverse are the only e-palindromic representatives belonging to the unoriented conjugacy class [w] ∈ V. An e-palindromic basis (x, y) is a basis such that both x and y are e-palindromes. As with the e-Christoffel bases, it is apparent that if [x, y] is a Farey edge then the pair (Pale[x], Pale[y]) is a basis. If, in addition, [x, y] and [a, b] have the same color, then (Pale[x], Pale[y]) is an e-palindromic basis. Compare \Theorem 2 and \Theorem 5.5. [lem:pal1] Let f = (x, y) = (Pale[x], Pale[y]) be an e-palindromic basis (so that [x, y] and [a, b] have the same color). Then 1. the basis e = (a, b) is an f-palindromic basis; 2. the set of e-palindromes is the same as the set of f-palindromes. (a) Without loss of generality, we may assume that [x] and [y] are both in V+[a, b] and the edge [x, y] is co-directed with [a, b]. Let G denote the shortest gallery containing both [a, b] and [x, y]. Let {[gk] ∈ V ∣ 1 ≤ k ≤ n} be the set of all 1/1-vertices (gray vertices) in G. Then we can decompose G into a chain of simple galleries Gk (1 ≤ k ≤ n) G = G1 ∪ G2 ∪ ⋯ ∪ Gn, where each Gk is the star of [gk] in G, namely, the union of all Farey triangles in G containing [gk]. The intersection Gk ∩ Gk + 1 of two adjacent simple galleries is a gray edge [ak, bk], so that each Gk is the shortest gallery connecting two gray edges [ak − 1, bk − 1] and [ak, bk], where (a0, b0) = (a, b) and (an, bn) = (x, y). See Figure [fig:pal2](Bottom). at 66 342 at 150 342 at 260 342 at 340 342 at 66 206 at 200 206 at 340 206 at 380 280 at 492 342 at 626 342 at 770 342 at 804 280 at 492 206 at 570 206 at 680 206 at 770 206 at 54 166 at 54 30 at 224 166 at 224 30 at 374 166 at 374 30 at 524 166 at 524 30 at 674 166 at 674 30 at 850 166 at 850 30 ![(Top) Two simple galleries. (Bottom) A generic gallery connecting [a,b] and [x,y].](fig_pal2 "fig:") [fig:pal2] Note that the property of being palindromic is transitive in the following sense: if a word w = w(xʹ, yʹ) is an (xʹ, yʹ)-palindrome and if (xʹ, yʹ) is an (x, y)-palindromic basis, then w = w(x, y) is an (x, y)-palindrome. Thus, by induction, it is enough to prove the lemma when n = 1 and the gallery G is simple. There are two possibilities of a simple gallery: $$(x,y)=(a(ba)^i,a(ba)^{i+1})\quad\textup{or}\quad (x,y)=((ba)^{i+1}b,(ba)^ib)$$ for some integer i ≥ 0. See Figure [fig:pal2](Top). As the proofs are similar we shall consider only the first case. Then we have x− 1y = ba and $$\begin{aligned} a&=x(ba)^{-i}=x(y^{-1}x)^i,\\ b&=x^{-1}ya^{-1}=x^{-1}y(x^{-1}y)^ix^{-1}=(x^{-1}y)^{i+1}x^{-1}.\end{aligned}$$ Therefore, both a and b are f-palindromes. The proof is complete. (b) This follows from (a) and the transitive property of palindromes we noted above. Following \Section 2.2 we define a basic triple* as an ordered triple (x, y, z) ∈ Prim3 such that (x, y) is a basis and xyz = 1. A basis f = (x, y) extends uniquely to a basic triple (x, y, z), which we denote by ${\widetilde}{{\mathsf{f}}}$. It is clear that this correspondence is one-to-one between bases and basic triples. [def:fpal] Let ${\widetilde}{{\mathsf{f}}}=(x,y,z)$ be a basic triple. An element w ∈ F2 is said to be an ${\widetilde}{{\mathsf{f}}}$-palindrome if it is palindromic in either (x, y) or (y, z) or (z, x) (when written in these bases). Consider the basic triple ${\widetilde}{{\mathsf{e}}}=(a,b,c)$ associated to the distinguished basis e = (a, b). In addition to the (a, b)-palindrome function (Definition [def:pal]) we may consider the (b, c)-palindrome function and the (c, a)-palindrome function as well. See Figure [fig:pal1](Right), where the vertices [a], [b] and [c] are colored black, white and gray, respectively. We noted previously that Pal(a, b)[w] is an (a, b)-palindrome unless [w] is colored gray. Analogous statements are true of the functions Pal(b, c) and Pal(c, a). Thus, for example, a black vertex contains an (a, b)-palindrome and a (c, a)-palindrome. Therefore, we can conclude as follows. [lem:pal2] If a primitive element is not conjugate to a± 1, b± 1 or c± 1, then its conjugacy class contains at least one and at most two ${\widetilde}{{\mathsf{e}}}$-palindromes. One can actually show that the two ${\widetilde}{{	arxiv_0000605
lambda {\arrow{dd}{triple}}[swap]{\mu'\,\widehat{mT'\mu}} && \mu'\circ\lambda\lambda\circ T\mu {\arrow{dd}{triple}}{m\,(T\mu)} \\ \\ \mu'\circ\lambda\lambda\circ\mu T {\arrow{dr}{triple}}[swap]{m\,(\mu T)} && \lambda\circ\mu\circ T\mu {\arrow{dl}{triple}}{\lambda a} \\ & \lambda\circ\mu\circ\mu T \end{tikzcd}$$ which we call left unitality, right unitality, and associativity conditions, respectively, where + $\widehat{uT'\eta}$ denotes (λTʹ ηλ) ∘ (uTʹ λ), + $\widehat{Tu\lambda}$ denotes λTʹ (κ− 1 ∘ Tu) ∘ ληʹ− 1, + $\widehat{mT'\mu}$ denotes (λTʹ μλ) ∘ (mTʹ TTλ), and + $\widehat{Tm\lambda}$ denotes (λTʹ (κ− 1 ∘ Tm ∘ κκ)) ∘ (λμʹ− 1 TλTʹ TTλ). We can write the coherence conditions 2-dimensionally, as follows. The unitality conditions are the following two commutative prisms, $$\begin{tikzcd}[sep=large] & T' {\arrow{dr}{Rightarrow, ""{name=ID}}}[swap]{{\mathrm{id}}} {\arrow{r}{Rightarrow}}{\eta'T'} & \|[alias=TPTP]\| T'T' {\arrow{d}{Rightarrow}}{\mu'} \\ T {\arrow{ur}{Rightarrow}}{\lambda} {\arrow{dr}{Rightarrow}}[swap]{{\mathrm{id}}} && T' \\ & T {\arrow{ur}{Rightarrow}}[swap]{\lambda} {\arrow{\end}{triple, from=TPTP, to=ID, "\ell'", shorten <=0.3em, shorten >=0.2em}}{tikzcd} \qquad=\qquad \begin{tikzcd}[sep=large] & T' {\arrow{r}{Rightarrow}}{\eta'T'} & T'T' {\arrow{d}{Rightarrow}}{\mu'} \\ T {\arrow{r}{Rightarrow}}{\eta T} {\arrow{ur}{Rightarrow}}{\lambda} {\arrow{dr}{Rightarrow, ""{name=ID}}}[swap]{{\mathrm{id}}} {\arrow{urr}{phantom, bend left, ""{name=LET, below, pos=0.55}}} {\arrow{urr}{phantom, bend right, ""{name=ETLL, above, pos=0.45}}} & \|[alias=TT]\| TT {\arrow{ur}{Rightarrow}}{\lambda\lambda} {\arrow{d}{Rightarrow}}{\mu} {\arrow{r}{phantom, bend left=45, ""{name=LLM, below, pos=0.7}}} {\arrow{r}{phantom, bend right=45, ""{name=ML, above, pos=0.3}}} & T' \\ & T {\arrow{ur}{Rightarrow}}[swap]{\lambda} {\arrow{{\arrow{}}{}}{triple, from=TT, to=ID, "\ell", shorten <=0.3em, shorten >=0.2em}}[triple, from=LET, to=ETLL, shorten <= 1em, shorten >= 1em, "\widehat{uT'\eta}", swap] {\arrow{\end}{triple, from=LLM, to=ML, shorten <=0.5em, shorten >=0.5em, "m"}}{tikzcd}$$ where $\widehat{uT'\eta}$ denotes the following composite, $$\begin{tikzcd}[column sep=huge, row sep=large] T {\arrow[Rightarrow]{rr}{}}{\lambda} {\arrow[Rightarrow]{dd}{}}[swap]{\eta T} && T' {\arrow{dd}{Rightarrow, ""{name=ETAT}}}{\eta' T'} {\arrow{ddl}{Rightarrow}}[swap]{\eta T'} {\arrow{ddll}{triple, shorten <=7em, shorten >=4em}}[swap, pos=0.6]{\eta\_\lambda} \\ \\ TT {\arrow[Rightarrow]{r}{}}[swap]{T\lambda} & TT' {\arrow{{\arrow[Rightarrow]{}}{}}{triple, from=ETAT, "uT'", shorten <=2em, shorten >=2.5em}}{r}[swap]{\lambda T'} & T'T' \end{tikzcd}$$ and $$\begin{tikzcd}[sep=large] & T' {\arrow{d}{Rightarrow}}[swap, near end]{T'\eta'} {\arrow{dr}{Rightarrow, ""{name=ID, below}}}{{\mathrm{id}}} \\ T {\arrow{d}{Rightarrow}}[swap]{T\eta} {\arrow{ur}{Rightarrow}}{\lambda} {\arrow{r}{phantom, bend left, ""{name=LET, below, pos=0.7}}} {\arrow{r}{phantom, bend right, ""{name=ETLL, above, pos=0.4}}} & \|[alias=TPTP]\| T'T' {\arrow{r}{Rightarrow}}[swap, near start]{\mu'} & T' \\ \|[alias=TT]\| TT {\arrow{ur}{Rightarrow}}[swap]{\lambda\lambda} {\arrow{r}{Rightarrow}}[swap]{\mu} {\arrow{urr}{phantom, bend left=45, ""{name=LLM, below, pos=0.6}}} {\arrow{urr}{phantom, bend right=45, ""{name=ML, above, pos=0.4}}} & T {\arrow{ur}{Rightarrow}}[swap]{\lambda} {\arrow{{\arrow{}}{}}{triple, from=ID, to=TPTP, "r'", shorten <=0.3em, shorten >=0.2em, swap}}[triple, from=LET, to=ETLL, shorten <= 0.2em, shorten >= 0.2em, "\widehat{Tu\lambda}", swap, near end] {\arrow{\end}{triple, from=LLM, to=ML, shorten <=2em, shorten >=2em, "m"}}{tikzcd} \qquad=\qquad \begin{tikzcd}[sep=large] & T' {\arrow{dr}{Rightarrow, ""{name=IDP}}}{{\mathrm{id}}} \\ T {\arrow{d}{Rightarrow}}[swap]{\eta} {\arrow{ur}{Rightarrow}}{\lambda} {\arrow{dr}{Rightarrow, ""{name=ID, below}}}{{\mathrm{id}}} && T' \\ \|[alias=TT]\| TT {\arrow{r}{Rightarrow}}[swap]{\mu} & T {\arrow{ur}{Rightarrow}}[swap]{\lambda} {\arrow{\end}{triple, from=ID, to=TT, "r", shorten <=0.2em, shorten >=0.2em}}{tikzcd}$$ where $\widehat{Tu\lambda}$ denotes the following composite and the unfilled 3-cells are intended to be identities; $$\begin{tikzcd}[column sep=huge, row sep=large] T {\arrow[Rightarrow]{rr}{}}{\lambda} {\arrow[Rightarrow]{dd}{}}[swap]{T\eta} {\arrow{ddr}{Rightarrow, bend left=20, ""{name=TETAP, below}}}[near end]{T\eta'} {\arrow{ddr}{Rightarrow, bend right=20, ""{name=TCOMP, pos=0.4}}}[near end, swap, yshift=0.2ex, xshift=0.8ex]{T(\lambda\circ\eta)} && T' {\arrow[Rightarrow]{dd}{}}{T'\eta'} {\arrow{ddl}{triple, shorten <= 2em, shorten >= 2em}}{\lambda\_{\eta'}^{-1}} \\ \\ TT {\arrow[Rightarrow]{r}{}}[swap]{T\lambda} {\arrow{}{triple, from=TCOMP, swap, shorten <=1em, shorten >=1em, near start}}[inner sep=-0.5ex]{\kappa^{-1}} & TT' {\arrow[Rightarrow]{r}{}}[swap]{\lambda T'} & T'T' {\arrow{\end}{triple, from=TETAP, to=TCOMP, shorten <=0.5em, "Tu", pos=0.6}}{tikzcd}$$ associativity is the following commutative cube, $$\begin{tikzcd}[column sep=small] TTT {\arrow{rr}{Rightarrow}}{\lambda\lambda\lambda} {\arrow{dd}{Rightarrow}}[swap]{\mu T} && T'T'T' {\arrow{dd}{Rightarrow}}{\mu'T'} {\arrow{dr}{Rightarrow}}{T'\mu'} {\arrow{ddll}{triple, shorten <=2em, shorten >=2em}}{\widehat{mT'\mu}} \\ &&& T'T' {\arrow{dd}{Rightarrow}}{\mu'} {\arrow{dl}{triple, shorten <=0.5em, shorten >=0.5em}}{a'} \\ TT {\arrow{dr}{Rightarrow}}[swap]{\mu} {\arrow{rr}{Rightarrow}}{\lambda\lambda} && T'T' {\arrow{dr}{Rightarrow}}{\mu'} {\arrow{dl}{triple, shorten <=0.5em, shorten >=0.5em}}{m} \\ & T {\arrow{rr}{Rightarrow}}[swap]{\lambda} && T' \end{tikzcd} \qquad=\qquad \begin{tikzcd}[column sep=small] TTT {\arrow{dr}{Rightarrow}}[near end]{T\mu} {\arrow{rr}{Rightarrow}}{\lambda\lambda\lambda} {\arrow{dd}{Rightarrow}}[swap]{\mu T} && T'T'T' {\arrow{dr}{Rightarrow}}{T'\mu'} {\arrow{dl}{triple, shorten <=0.5em, shorten >=0.5em}}{\widehat{Tm\lambda}} \\ & TT {\arrow{dd}{Rightarrow}}{\mu} {\arrow{rr}{Rightarrow}}[swap]{\lambda\lambda} {\arrow{dl}{triple, shorten <=0.5em, shorten >=0.5em}}{a} && T'T' {\arrow{dd}{Rightarrow}}{\mu'} {\arrow{ddll}{triple, shorten <=2em, shorten >=2em}}{m} \\ TT {\arrow{dr}{Rightarrow}}[swap]{\mu} \\ & T {\arrow{rr}{Rightarrow}}[swap]{\lambda} && T' \end{tikzcd}$$ where $\widehat{mT'\mu}$ and $\widehat{Tm\lambda}$ denote the following compositions, respectively, $$\begin{tikzcd}[row sep=large] TTT {\arrow[Rightarrow]{dd}{}}[swap]{\mu T} {\arrow[Rightarrow]{r}{}}{TT\lambda} & TTT' {\arrow[Rightarrow]{r}{}}{T\lambda T'} {\arrow[Rightarrow]{dd}{}}{\mu T'} {\arrow{ddl}{triple, shorten <=2em, shorten >=2em}}{\mu\_\lambda} & TT'T' {\arrow[Rightarrow]{r}{}}{\lambda T'T'} & T'T'T' {\arrow[Rightarrow]{dd}{}}{\mu'T'} {\arrow{ddll}{triple, shorten <=3em, shorten >=3em}}{m T'} \\ \\ TT {\arrow[Rightarrow]{r}{}}[swap]{T\lambda} & TT' {\arrow[Rightarrow]{rr}{}}[swap]{\lambda T'} && T'T' \end{tikzcd}$$ and $$\begin{tikzcd}[row sep=large] TTT {\arrow[Rightarrow]{dd}{}}[swap]{T\mu} {\arrow[Rightarrow]{r}{}}{TT\lambda} {\arrow{ddrr}{Rightarrow, bend left=20, ""{name=TCUP, above, near end}, ""{name=TCDOWN, below}}}[pos=0.3, inner sep=0ex]{T(\mu'\circ\lambda T'\circ T\lambda)} {\arrow{ddrr}{Rightarrow, bend right=20, ""{name=TDUP, above}, ""{name=TDDOWN, below, near start}}}[swap, pos=0.7, inner sep=0ex]{T(T\lambda\circ\mu)} & TTT' {\arrow[Rightarrow]{r}{}}{T\lambda T'} & \|[alias=TR]\| TT'T' {\arrow[Rightarrow]{dd}{}}{T\mu'} {\arrow[Rightarrow]{r}{}}{\lambda T'T'} & T'T'T' {\arrow[Rightarrow]{dd}{}}{T'\mu'} {\arrow{ddl}{triple, shorten <=2em, shorten >=2em}}{\lambda\_{\mu'}^{-1}} \\ \\ \|[alias=BL]\| TT {\arrow[Rightarrow]{rr}{}}[swap]{T\lambda} && TT' {\arrow[Rightarrow]{r}{}}[swap]{\lambda T'} & T'T' {\arrow{{\arrow{}}{}}{triple, from=TR, to=TCUP, "\kappa\kappa", shorten <=0.5em, shorten >=0.5em}}[triple, from=TCDOWN, to=TDUP, "Tm", shorten <=0.3em, shorten >=0.3em] {\arrow{\end}{triple, from=TDDOWN, to=BL, "\kappa^{-1}", shorten <=0.5em, shorten >=0.5em}}{tikzcd}$$ Finally, 2-cells of pseudomonads are defined as follows. [monad2cell] Let K be a strict 2-category. Let (T, μ, η, ℓ, r, a) and (Tʹ, μʹ, ηʹ, ℓʹ, rʹ, aʹ) be pseudomonads on K. Let (λ, u, m) and (ξ, v, n) be (pseudo)morphisms T → Tʹ. A 2-cell of monads is a modification $$\begin{tikzcd} T {\arrow{rr}{Rightarrow, bend left, ""{name=L,below}}}{\lambda} {\arrow{rr}{Rightarrow, bend right, ""{name=LP,above}}}[swap]{\xi} && T' {\arrow{\end}{triple, from=L, to=LP, "t"}}{tikzcd}$$ such that the following diagrams commute, $$\begin{tikzcd}[row sep=small] & \lambda\circ\eta {\arrow{dd}{triple}}{t\,\eta} \\ \eta' {\arrow{ur}{triple}}{u} {\arrow{dr}{triple}}[swap]{v} \\ & \xi\circ \eta \end{tikzcd} \qquad \begin{tikzcd}[row sep=large] \mu'\circ (\lambda\lambda) {\arrow{r}{triple}}{m} {\arrow{d}{triple}}{\mu\,(tt)} & \lambda\circ\mu {\arrow{d}{triple}}{t\,\mu} \\ \mu'\circ (\xi\xi) {\arrow{r}{triple}}{n} & \xi\circ\mu \end{tikzcd}$$ which we call “unit” and “multiplication” conditions, respectively. As above, it may be helpful to write the conditions 2-dimensionally. The unit condition forms the following commutative “cone”, $$\begin{tikzcd}[row sep=large, column sep=small] & {\mathrm{id}}\_{{\mathsf{K}}} {\arrow{dr}{Rightarrow, ""{name=ETAP, pos=0.3}}}{\eta'} {\arrow{dl}{Rightarrow}}[swap]{\eta} & \\ \|[alias=T]\| T {\arrow{rr}{Rightarrow, bend left, ""{name=L,below}}}[pos=0.56]{\lambda} {\arrow{rr}{Rightarrow, bend right, ""{name=LP,above}}}[swap]{\xi} && T' {\arrow{{\arrow{}}{}}{triple, from=L, to=LP, "t"}}[triple, from=ETAP, to=T, "u", shorten <= 1.2em, shorten >= 2em, swap, near start, shift right=1] \end{tikzcd} \qquad=\qquad \begin{tikzcd}[row sep=large, column sep=small] & {\mathrm{id}}\_{{\mathsf{K}}} {\arrow{dr}{Rightarrow, ""{name=ETAP}}}{\eta'} {\arrow{dl}{Rightarrow}}[swap]{\eta} & \\ \|[alias=T]\| T {\arrow{rr}{Rightarrow, bend right}}[swap]{\xi} && T' {\arrow{\end}{triple, from=ETAP, to=T, "v", shorten <= 1.2em, shorten >= 1.5em, shift left}}{tikzcd}$$ and the multiplication condition forms the following commutative “cylinder”, $$\begin{tikzcd}[sep=large] TT {\arrow{r}{Rightarrow, bend left}}{\lambda\lambda} {\arrow{d}{Rightarrow}}[swap]{\mu} & \|[alias=TPTP]\| T'T' {\arrow{d}{Rightarrow}}{\mu'} \\ \|[alias=T]\| T {\arrow{r}{Rightarrow, bend left, ""{name=L,below}}}{\lambda} {\arrow{r}{Rightarrow, bend right, ""{name=LP,above}}}[swap]{\xi} & T' {\arrow{{\arrow{}}{}}{triple, from=TPTP, to=T, "m", shorten <= 1.5em, shorten >= 2em, swap, shift right=4, pos=0.4}}[triple, from=L, to=LP, "t"] \end{tikzcd} \qquad=\qquad \begin{tikzcd}[sep=large] TT {\arrow{r}{Rightarrow, bend left, ""{name=L,below, pos=0.54}}}{\lambda\lambda} {\arrow{r}{Rightarrow, bend right, ""{name=LP,above}}}[swap]{\xi\xi} {\arrow{d}{Rightarrow}}[swap]{\mu} & \|[alias=TPTP]\| T'T' {\arrow{d}{Rightarrow}}{\mu'} \\ \|[alias=T]\| T {\arrow{r}{Rightarrow, bend right}}[swap]{\xi} & T' {\arrow{{\arrow{}}{}}{triple, from=L, to=LP, "tt"}}[triple, from=TPTP, to=T, "n", shorten <= 2em, shorten >= 1.5em, shift left=4] \end{tikzcd}$$ In line with our usual convention for horizontal composition, tt denotes the following 3-cell. $$\begin{tikzcd}[sep=large] TT {\arrow{r}{Rightarrow, bend left, ""{name=TL, below}}}{T\lambda} {\arrow{r}{Rightarrow, bend right, ""{name=BL, above}}}[swap]{T\lambda} & TT' {\arrow{r}{Rightarrow, bend left, ""{name=TR, below}}}{\lambda T'} {\arrow{r}{Rightarrow, bend right, ""{name=BR, above}}}[swap]{\lambda T'} & T'T' {\arrow{{\arrow{}}{}}{triple, from=TL, to=BL, "Tt"}}[triple, from=TR, to=BR, "t T'"] \end{tikzcd}$$ Pseudoalgebras and their morphisms ---------------------------------- [defpseudoalgebra] Let (T, μ, η, ℓ, r, a) be a pseudomonad on K. A pseudoalgebra over the pseudomonad T consists of * an object A of K, together with * a morphism e : TA → A, and * invertible 2-cells $$\begin{tikzcd}[sep=large] A {\arrow{dr}{""{name=ID}}}[swap]{{\mathrm{id}}} {\arrow{r}{}}{\eta\_A} & TA {\arrow{{\arrow{}}{}}{Rightarrow, to=ID, "\iota", shorten >= 0.1em, shorten <= 0.3em}}{d}{e} \\ & A \end{tikzcd} \qquad\mbox{and}\qquad \begin{tikzcd}[sep=large] TTA {\arrow{d}{}}[swap]{\mu\_A} {\arrow{r}{}}{Te} & TA {\arrow{d}{}}{e} {\arrow{dl}{Rightarrow, shorten >= 1.2em, shorten <= 1.2em}}{\gamma}\\ TA {\arrow{r}{}}[swap]{e} & A \end{tikzcd}$$ such that * the following diagrams commute, $$\begin{tikzcd}[column sep=small] e \circ Te \circ T\eta\_A {\arrow{rr}{Rightarrow}}{\gamma \, (T\eta\_A)} {\arrow{dr}{Rightarrow,swap}}{e \, (\widehat{T\iota})} && e \circ\mu\_A \circ T\eta\_A \\ & e \circ {\mathrm{id}}\_{TA} {\arrow{ur}{Rightarrow}}[swap]{e \, r\_A} \end{tikzcd}$$ $$\begin{tikzcd}[column sep=tiny] & e\circ Te \circ TTe {\arrow{dl}{Rightarrow}}[swap]{\gamma\,(TTe)} {\arrow{dr}{Rightarrow}}{e\,(\widehat{T\gamma})}\\ e\circ\mu\_A\circ TTe {\arrow{dd}{Rightarrow}}[swap]{e\,\mu\_e} && e\circ Te\circ T\mu\_A {\arrow{dd}{Rightarrow}}{\gamma\,(T\mu\_A)} \\ \\ e\circ Te\circ\mu\_{TA} {\arrow{dr}{Rightarrow}}[swap]{\gamma\,\mu\_A} && e\circ\mu\_A\circ T\mu\_A {\arrow{dl}{Rightarrow}}{e\,a\_A} \\ & e\circ\mu\circ \mu\_{TA} \end{tikzcd}$$ which we call unit and multiplication conditions, respectively, where $\widehat{T\iota} = \upsilon^{-1}\circ T\iota\circ\kappa$, and $\widehat{T\gamma}=\kappa^{-1}\circ T\gamma\circ \kappa$. We call the pseudoalgebra normal if the 2-cell ι is the identity. We can write the coherence diagrams 2-dimensionally as well, as follows. Here is the unit condition, $$\begin{tikzcd}[column sep=small] TA {\arrow{rr}{}}{T\eta} {\arrow{dd}{}}[swap]{T\eta} {\arrow{ddrr}{}}[near end]{{\mathrm{id}}} && \|[alias=TTTUR]\| TTA {\arrow{dr}{}}{Te} {\arrow{dd}{}}{Te} \\ & \|[alias=ID]\| && TA {\arrow{dd}{}}{e} \\ \|[alias=TTTDL]\| TTA {\arrow{dr}{}}[swap]{\mu} {\arrow{rr}{}}[swap]{\mu} && TA {\arrow{dr}{}}{e} \\ & TA {\arrow{rr}{}}[swap]{e} && A {\arrow{{\arrow{}}{}}{Rightarrow, from=TTTUR, to=ID, "\widehat{T\iota}", shorten <= 0.5em, shorten >= 0.5em, swap}}[Rightarrow, from=ID, to=TTTDL, "r", shorten <= 0.5em, shorten >= 0.5em, swap] \end{tikzcd} \qquad=\qquad \begin{tikzcd}[column sep=small] TA {\arrow{rr}{}}{T\eta} {\arrow{dd}{}}[swap]{T\eta} {\arrow{dr}{}}[swap, near end]{T\eta} && TTA {\arrow{dr}{}}{Te} \\ & TTA {\arrow{rr}{}}{Te} {\arrow{dd}{}}{\mu} && \|[alias=TTUR]\| TA {\arrow{dd}{}}{e} \\ TTA {\arrow{dr}{}}[swap]{\mu} \\ & \|[alias=TTDL]\| TA {\arrow{rr}{}}[swap]{e} && A {\arrow{\end}{Rightarrow, from=TTUR, to=TTDL, "\gamma", shorten <= 2em, shorten >= 2em}}{tikzcd}$$ where the unfilled 2-cells are identities, and $\widehat{T\iota}$ denotes the following composite cell, $$\begin{tikzcd}[sep=large] & \|[alias=TTT]\| TTA {\arrow{ddr}{}}{Te} \\ \\ TA {\arrow{uur}{}}{T\eta} {\arrow{rr}{bend left=35, ""{name=COMP}}}[swap, pos=0.1, yshift=0.5ex]{T(e\circ \eta)} {\arrow{rr}{""{name=TID}}}[near start, swap]{T{\mathrm{id}}} {\arrow{rr}{bend right=35, ""{name=ID}}}[swap]{{\mathrm{id}}} && TA {\arrow{{\arrow{}}{}}{Rightarrow, from=TTT, to=COMP, "\kappa", shorten <= 0.5em, shorten >= 0em}}[Rightarrow, from=COMP, to=TID, "T\iota", shorten <= 0.5em, shorten >= 0em] {\arrow{\end}{Rightarrow, from=TID, to=ID, "\upsilon^{-1}", shorten <= 0.5em, shorten >= 0em}}{tikzcd}$$ and here is the multiplication condition, $$\begin{tikzcd}[column sep=small] TTTA {\arrow{rr}{}}{TTe} {\arrow{dd}{}}[swap]{\mu} && \|[alias=TTTUR]\| TTA {\arrow{dr}{}}{Te} {\arrow{dd}{}}{\mu} \\ &&& \|[alias=TTUR]\| TA {\arrow{dd}{}}{e} \\ \|[alias=TTTDL]\| TTA {\arrow{rr}{}}{Te} {\arrow{dr}{}}[swap]{\mu} && \|[alias=TT]\| TA {\arrow{dr}{}}{e} \\ & \|[alias=TTDL]\| TA {\arrow{rr}{}}[swap]{e} && A {\arrow{{\arrow{}}{}}{Rightarrow, from=TTUR, to=TT, "\gamma", shorten <= 0.5em, shorten >= 0.5em}}[Rightarrow, from=TT, to=TTDL, "\gamma", shorten <= 0.5em, shorten >= 0.5em] {\arrow{\end}{Rightarrow, from=TTTUR, to=TTTDL, "\mu\_e", shorten <= 2em, shorten >= 2em}}{tikzcd} \qquad=\qquad \begin{tikzcd}[column sep=small] TTTA {\arrow{rr}{}}{TTe} {\arrow{dd}{}}[swap]{\mu} {\arrow{dr}{}}[swap]{T\mu} && \|[alias=TTTUR]\| TTA {\arrow{dr}{}}{Te} \\ & \|[alias=TTT]\| TTA {\arrow{rr}{}}{Te} {\arrow{dd}{}}[swap]{\mu} && \|[alias=TTUR]\| TA {\arrow{dd}{}}{e} \\ \|[alias=TTTDL]\| TTA {\arrow{dr}{}}[swap]{\mu} \\ & \|[alias=TTDL]\| TA {\arrow{rr}{}}[swap]{e} && A {\arrow{{\arrow{}}{}}{Rightarrow, from=TTTUR, to=TTT, "\widehat{T\gamma}", shorten <= 0.5em, shorten >= 0.5em, swap}}[Rightarrow, from=TTT, to=TTTDL, "a", shorten <= 0.5em, shorten >= 0.5em, swap] {\arrow{\end}{Rightarrow, from=TTUR, to=TTDL, "\gamma", shorten <= 2em, shorten >= 2em}}{tikzcd}$$ where $\widehat{T\gamma}$ denotes the following composite cell. $$\begin{tikzcd}[sep=large] & \|[alias=TOP]\| TTA {\arrow{dr}{}}{Te} \\ TTTA {\arrow{ur}{}}{TTe} {\arrow{dr}{}}[swap]{T\mu} {\arrow{rr}{bend left=20, ""{name=COMPUP}}}[swap, pos=0.1, yshift=0.4ex]{T(e\circ Te)} {\arrow{rr}{bend right=20, ""{name=COMPDOWN}}}[pos=0.1, yshift=-0.6ex]{T(e\circ\mu)} && TA \\ & \|[alias=BOTTOM]\| TTA {\arrow{ur}{}}[swap]{Te} {\arrow{{\arrow{}}{}}{Rightarrow, from=TOP, to=COMPUP, "\kappa"}}[Rightarrow, from=COMPUP, to=COMPDOWN, "T\gamma", shorten <=0.5em] {\arrow{\end}{Rightarrow, from=COMPDOWN, to=BOTTOM, "\kappa^{-1}", shorten <=0.5em}}{tikzcd}$$ [defmorphalg] Let (A, eA, ιA, γA) and (B, eB, ιB, γB) be pseudoalgebras over the pseudomonad (T, μ, η, ℓ, r, a). A (strong) morphism of pseudoalgebras consists of * a morphism f : A → B of K, together with * an invertible 2-cell $$\begin{tikzcd} TA {\arrow{d}{}}[swap]{e\_A} {\arrow{r}{}}{Tf} & TB {\arrow{d}{}}{e\_B} {\arrow{dl}{Rightarrow, shorten <= 1em, shorten >= 1em}}{\phi} \\ A {\arrow{r}{}}[swap]{f} & B \end{tikzcd}$$ such that * the following diagrams commute, $$\begin{tikzcd} e\_B\circ\eta\_B\circ f {\arrow{d}{}}{\iota\_B} {\arrow{r}{}}{e\_B\,\eta\_f} & e\_B\circ Tf\circ\eta\_A {\arrow{d}{}}{\phi\,\eta\_A} \\ f & f\circ e\_A\circ\eta\_A {\arrow{l}{}}{f\,\iota\_A} \end{tikzcd} \qquad \begin{tikzcd}[row sep=small, column sep=-1em] & e\_B\circ Te\_B\circ TTf {\arrow{dr}{}}[near end]{e\_B\,(\widehat{T\phi})} {\arrow{dl}{}}[swap, near end]{\gamma\_B\,(TTf)} \\ e\_B\circ\mu\_B\circ TTf {\arrow{dd}{}}[swap]{e\_B\,\mu\_f} && e\_B\circ Tf\circ Te\_A {\arrow{dd}{}}{\phi\,(Te\_A)} \\ \\ e\_B\circ Tf \circ \mu\_A {\arrow{dr}{}}[swap]{\phi\,\mu\_A} && f\circ e\_A\circ Te\_A {\arrow{dl}{}}{f\,\gamma\_A} \\ & f\circ e\_A\circ\mu\_A \end{tikzcd}$$ which we call the unit and multiplication condition, respectively, where $\widehat{T\phi}=\kappa^{-1}\circ T\phi\circ \kappa$. As above, it may be helpful to draw the coherence conditions in a 2-dimensional way. The unit condition is the following commutative prism, $$\begin{tikzcd}[sep=large] & B {\arrow{r}{}}{\eta} {\arrow{dr}{""{name=ID, above}}}[swap]{{\mathrm{id}}} & TB {\arrow{d}{}}{e\_B} {\arrow{\\}{Rightarrow, to=ID, "\iota\_B", shorten <=0.3em, shorten >=0.3em}} A {\arrow{ur}{}}{f} {\arrow{dr}{}}[swap]{{\mathrm{id}}} && B \\ & A {\arrow{ur}{}}[swap]{f} \end{tikzcd} \qquad=\qquad \begin{tikzcd}[sep=large] & B {\arrow{r}{}}{\eta} {\arrow{d}{Rightarrow, shorten <=0.7em, shorten >=0.7em}}{\eta\_f} & TB {\arrow{d}{}}{e\_B} {\arrow{ddl}{Rightarrow, shorten <=3em, shorten >=3em}}{\phi} \\ A {\arrow{ur}{}}{f} {\arrow{r}{}}{\eta} {\arrow{dr}{""{name=ID, above}}}[swap]{{\mathrm{id}}} & TA {\arrow{ur}{}}{Tf} {\arrow{d}{}}{e\_A} {\arrow{&}{Rightarrow, to=ID, "\iota\_A", shorten <=0.3em, shorten >=0.3em}} B \\ & A {\arrow{ur}{}}[swap]{f} \end{tikzcd}$$ where again the unfilled 2-cell is an identity, and the multiplication condition is the following commutative cube, $$\begin{tikzcd}[column sep=small] TTA {\arrow{dd}{}}[swap]{\mu} {\arrow{rr}{}}{TTf} && TTB {\arrow{dd}{}}{\mu} {\arrow{dr}{}}{Te\_B} {\arrow{ddll}{Rightarrow, shorten <=2em, shorten >=2em}}{\mu\_f} \\ &&& TB {\arrow{dd}{}}{e\_B} {\arrow{dl}{Rightarrow, shorten <=0.5em, shorten >=0.5em}}{\gamma\_B} \\ TA {\arrow{rr}{}}{Tf} {\arrow{dr}{}}[swap]{e\_A} && TB {\arrow{dr}{}}{e\_B} {\arrow{dl}{Rightarrow, shorten <=0.5em, shorten >=0.5em}}{\phi} \\ & A {\arrow{rr}{}}[swap]{f} && B \end{tikzcd} \qquad=\qquad \begin{tikzcd}[column sep=small] TTA {\arrow{dr}{}}[swap]{Te\_A} {\arrow{dd}{}}[swap]{\mu} {\arrow{rr}{}}{TTf} && TTB {\arrow{dr}{}}{Te\_B} {\arrow{dl}{Rightarrow, shorten <=0.5em, shorten >=0.5em}}{\widehat{T\phi}} \\ & TA {\arrow{rr}{}}[swap]{Tf} {\arrow{dd}{}}{e\_A} {\arrow{dl}{Rightarrow, shorten <=0.5em, shorten >=0.5em}}{\gamma\_A} && TB {\arrow{dd}{}}{e\_B} {\arrow{ddll}{Rightarrow, shorten <=2em, shorten >=2em}}{\phi} \\ TA {\arrow{dr}{}}[swap]{e\_A} \\ & A {\arrow{rr}{}}[swap]{f} && B \end{tikzcd}$$ where $\widehat{T\phi}$ denotes the following composite cell. $$\begin{tikzcd}[sep=large] & \|[alias=TOP]\| TTB {\arrow{dr}{}}{Te\_B} \\ TTA {\arrow{ur}{}}{TTf} {\arrow{dr}{}}[swap]{Te\_A} {\arrow{rr}{bend left=20, ""{name=COMPUP}}}[swap, pos=0.1, yshift=0.4ex]{T(e\_B\circ Tf)} {\arrow{rr}{bend right=20, ""{name=COMPDOWN}}}[pos=0.1, yshift=-0.6ex]{T(f\circ e\_A)} && TB \\ & \|[alias=BOTTOM]\| TTA {\arrow{ur}{}}[swap]{Tf} {\arrow{{\arrow{}}{}}{Rightarrow, from=TOP, to=COMPUP, "\kappa"}}[Rightarrow, from=COMPUP, to=COMPDOWN, "T\phi", shorten <=0.5em] {\arrow{\end}{Rightarrow, from=COMPDOWN, to=BOTTOM, "\kappa^{-1}", shorten <=0.5em}}{tikzcd}$$ [alg2cell] Let (A, eA, ιA, γA) and (B, eB, ιB, γB) be pseudoalgebras over the pseudomonad (T, μ, η, ℓ, r, a). Let (f, ϕ) and (g, χ) be strong morphisms of pseudoalgebras. A 2-cell of pseudoalgebras is a 2-cell $$\begin{tikzcd}[sep=large] A {\arrow{r}{bend left, ""{name=F, below}}}{f} {\arrow{r}{bend right, ""{name=FP, above}}}[swap]{g} & B {\arrow{\end}{Rightarrow, from=F, to=FP, "\alpha"}}{tikzcd}$$ such that the following diagram commutes. $$\begin{tikzcd} e\_B\circ Tf {\arrow[Rightarrow]{r}{}}{\phi} {\arrow[Rightarrow]{d}{}}{e\_B\,T\alpha} & f\circ e\_A {\arrow[Rightarrow]{d}{}}{\alpha\,e\_A} \\ e\_B\circ Tg {\arrow[Rightarrow]{r}{}}{\chi} & g\circ e\_A \end{tikzcd}$$ If we draw the condition in a 2-dimensional way, we get the following commutative “cylinder”: $$\begin{tikzcd}[sep=large] TA {\arrow{r}{bend left}}{Tf} {\arrow{d}{}}[swap]{e\_A} & TB {\arrow{d}{}}{e\_B} {\arrow{dl}{Rightarrow, shorten <= 1em, shorten >= 2em, shift right=4, pos=0.4}}[swap]{\phi} \\ A {\arrow{r}{bend left, ""{name=F, below}}}{f} {\arrow{r}{bend right, ""{name=FP, above}}}[swap]{g} & B {\arrow{\end}{Rightarrow, from=F, to=FP, "\alpha"}}{tikzcd} \qquad=\qquad \begin{tikzcd}[sep=large] TA {\arrow{r}{bend left, ""{name=F, below, pos=0.52}}}{Tf} {\arrow{r}{bend right, ""{name=FP, above}}}[swap]{Tg} {\arrow{d}{}}[swap]{e\_A} & TB {\arrow{d}{}}{e\_B} {\arrow{{\arrow{}}{}}{Rightarrow, from=F, to=FP, "T\alpha"}}[Rightarrow, shorten <= 2em, shorten >= 1em, shift left=4, pos=0.55]{dl}{\chi} \\ A {\arrow{r}{bend right}}[swap]{g} & B \end{tikzcd}$$ It is immediate from the definitions to check that, given any object X of K, the object TX is canonically a pseudoalgebra, with structure morphism μX : TTX → TX and 2-cells ℓX : μX ∘ ηTX ⇒ idTX and aX : μX ∘ TμX ⇒ μX ∘ μTX. We call this a “free algebra”, analogously to the 1-dimensional case. Moreover, by naturality of μ, for any morphism f : X → Y of K, the morphism Tf : TX → TY gives a morphism of pseudoalgebras. Similarly, for every f, g : X → Y and α : f ⇒ g, the 2-cell Tα : Tf ⇒ Tg is automatically a 2-cell of algebras. Restriction of scalars for pseudomonads --------------------------------------- It is very well known that in the one-dimensional context, a morphism of monads induces a pullback functor between the algebras, sometimes named “restriction of scalars” after its instance in ring theory. A similar phenomenon occurs in two dimensions, as follows. We use this statement in, see there also for further context. [rescalthm] Let K be a (strict) 2-category. Let (T, μ, η, ℓ, r, a) and (Tʹ, μʹ, ηʹ, ℓʹ, rʹ, aʹ) be pseudomonads on K, and let (λ, u, m) be a pseudomorphism of monads from T to Tʹ. Each Tʹ-pseudoalgebra (A, eʹ, ιʹ, γʹ), defines canonically a T-pseudoalgebra structure on A with the following structure 1- and 2-cells. $$\label{drescal} e\coloneqq \begin{tikzcd} TA {\arrow{d}{}}{\lambda\_A} \\ T'A {\arrow{d}{}}{e'} \\ A \end{tikzcd} \qquad \iota\coloneqq \begin{tikzcd}[column sep=large] & \|[alias=TA]\| TA {\arrow{d}{}}{\lambda\_A} \\ A {\arrow{dr}{""{name=ID,above}}}[swap]{{\mathrm{id}}} {\arrow{r}{""{name=ETAP,above}}}[swap]{\eta'\_A} {\arrow{ur}{}}{\eta\_A} & \|[alias=TPA]\| T'A {\arrow{d}{}}{e'} \\ & A {\arrow{{\arrow{}}{}}{Rightarrow, from=TA, to=ETAP, "u^{-1}\_A", shorten <= 0.8em, shorten >= 0.5em, outer sep=-2pt}}[Rightarrow, from=TPA, to=ID, "\iota'", shorten <= 0.2em, shorten >= 0.2em] \end{tikzcd} \qquad \gamma\coloneqq \begin{tikzcd} TTA {\arrow{r}{}}{T\lambda\_A} {\arrow{dd}{}}[swap]{\mu\_A} & TT'A {\arrow{d}{}}[swap]{\lambda\_{T'A}} {\arrow{r}{}}{Te'} {\arrow{ddl}{Rightarrow, shorten <= 2em, shorten >= 2em, swap}}{m} & TA {\arrow{d}{}}{\lambda\_A} {\arrow{dl}{Rightarrow, shorten <= 0.5em, shorten >= 0.5em, swap}}{\lambda\_{e'}} \\ & T'T'A {\arrow{d}{}}[swap]{\mu'\_A} {\arrow{r}{}}{T'e'} & T'A {\arrow{d}{}}{e'} {\arrow{dl}{Rightarrow, shorten <= 0.5em, shorten >=0.5em}}{\gamma'} \\ TA {\arrow{r}{}}[swap]{\lambda\_A} & T'A {\arrow{r}{}}[swap]{e'} & A \end{tikzcd}$$ Moreover, this construction defines a 2-functor between the categories of pseudoalgebras λ\* : KTʹ → KT. In analogy with the 1-dimensional case, we call λ\* the restriction of scalar functor. The unit diagram for (A, e, ι, γ), obtained by plugging into the unit diagram of, can be decomposed in the following way, where the whiskerings have been suppressed for reasons of space. $$\begin{tikzpicture}[baseline= (a).base] \node[scale=0.55] (a) at (0,0){ \begin{tikzcd}[column sep=tiny, row sep=large] {e'\circ\lambda\_A\circ Te'\circ T\lambda\_A\circ T\eta\_A} &&[+10pt]& {e'\circ T'e'\circ(\lambda\lambda)\_A\circ T\lambda\_A} &&[-20pt]& {e'\circ \mu'\_A\circ (\lambda\lambda)\_A\circ T\lambda\_A} &&& {e'\circ\lambda\_A\circ\mu\_A\circ T\eta\_A} &&&\\ && {e'\circ\lambda\_A\circ Te'\circ T\eta'\_A} && {e'\circ T'e'\circ\lambda\_{T'A}\circ T\eta'\_A} && {e'\circ \mu'\_A\circ\lambda\_{T'A}\circ T\eta'\_A} \\ &&&&& {e'\circ T'e'\circ T'\eta'\_A\circ T\lambda\_A} & {e'\circ\mu'\_A\circ T\eta'\_A\circ\lambda\_A} \\ \\ &&&&&& {e'\circ\lambda\_A} \arrow[Rightarrow, "{\gamma'}", from=1-4, to=1-7] \arrow[Rightarrow, "{\lambda\_{e'}}", from=1-1, to=1-4] \arrow[Rightarrow, "{m\_A}", from=1-7, to=1-10] \arrow[Rightarrow, "{Tu^{-1}\_A\circ\kappa}"', from=1-1, to=2-3] \arrow[Rightarrow, "{Tu^{-1}\_A\circ\kappa}", from=1-4, to=2-5, near end] \arrow[Rightarrow, "{\lambda\_{e'}}", from=2-3, to=2-5] \arrow[Rightarrow, "{\lambda\_{\eta'}}", from=2-5, to=3-6, near end] \arrow[Rightarrow, "{\gamma'}", from=2-5, to=2-7] \arrow[Rightarrow, "{Tu^{-1}\_A\circ\kappa}", from=1-7, to=2-7] \arrow[Rightarrow, "{\lambda\_{\eta'}}", from=2-7, to=3-7] \arrow[Rightarrow, "{\gamma'}", from=3-6, to=3-7] \arrow[Rightarrow, "{r'\_A}"', from=5-7, to=3-7] \arrow[Rightarrow, "{\widehat{T'\iota'\_A}}", from=3-6, to=5-7] \arrow[Rightarrow, "{\widehat{T\iota'\_A}}"', from=2-3, to=5-7] \arrow[Rightarrow, "{r\_A}"', from=5-7, to=1-10] \end{tikzcd} }; \end{tikzpicture}$$ Now, * the region on the right commutes by the right unit condition of (λ, u, m), as in (recall that all the arrows of the diagram are invertible); * the triangle in the center bottom is the right unitality condition of the algebra (A, eʹ, ιʹ, γʹ), as in ; * the region on the bottom left commutes by pseudonaturality of λ; * finally, the remaining parallelograms commute by the interchange law. The multiplication diagram for (A, e, ι, γ), analogously obtained by plugging into the multiplication diagram of, can be decomposed as follows, where again the whiskerings have been suppressed, and the hat denotes the suitable application of the compositors. $$\begin{tikzpicture}[baseline= (a).base] \node[scale=.58] (a) at (0,0){ \begin{tikzcd}[column sep=-6em] &&& {e'\circ\lambda\_A\circ Te'\circ T\lambda\_A\circ TTe'\circ TT\lambda\_A} \\ && {e'\circ Te'\circ(\lambda\lambda)\_A\circ TTe'\circ TT\lambda\_A} && {e'\circ\lambda\_A\circ Te'\circ TTe'\circ T\lambda\_{T'A}\circ TT\lambda\_A} \\ & {e'\circ \mu'\_A\circ(\lambda\lambda)\_A\circ TTe'\circ TT\lambda\_A} && {e'\circ Te'\circ \lambda\_{T'A}\circ TTe'\circ T\lambda\_{T'A}\circ TT\lambda\_A} && {e'\circ\lambda\_A\circ Te'\circ T\mu'\_A\circ T\lambda\_{T'A}\circ TT\lambda\_A} \\ {e'\circ\lambda\_A\circ\mu\_A\circ TTe'\circ TT\lambda\_A} && {e'\circ \mu'\_A\circ \lambda\_{T'A}\circ TTe'\circ T\lambda\_{T'A}\circ TT\lambda\_A} &&&& {e'\circ\lambda\_A\circ Te'\circ T\lambda\_A\circ T\mu\_A} \\ &&& {e'\circ Te'\circ T'T'e\circ (\lambda\lambda\lambda)\_A} && {e'\circ T'e'\circ\lambda\_{T'A}\circ T\mu'\_A\circ T\lambda\_{T'A}\circ TT\lambda\_A} \\ && {e'\circ\mu'\_A\circ T'T'e\circ (\lambda\lambda\lambda)\_A} && {e'\circ T'e'\circ T'\mu'\_A\circ (\lambda\lambda\lambda)\_A} && {e'\circ T'e'\circ(\lambda\lambda)\_A\circ T\mu\_A} \\ &&&&& {e'\circ\mu'\_A\circ\lambda\_{T'A}\circ T\mu'\_A\circ T\lambda\_{T'A}\circ TT\lambda\_A} \\ {e'\circ\lambda\_A\circ Te'\circ\mu\_{T'A}\circ TT\lambda\_A} && {e'\circ T'e'\circ\mu'\_{T'A}\circ (\lambda\lambda\lambda)\_A} && {e'\circ \mu'\_A\circ T'\mu'\_A\circ (\lambda\lambda\lambda)\_A} && {e'\circ \mu'\_A\circ(\lambda\lambda)\_A\circ T\mu\_A} \\ & {e'\circ T'e'\circ(\lambda\lambda)\_A\circ\mu\_{TA}} && {e'\circ \mu'\_A\circ\mu'\_{T'A}\circ (\lambda\lambda\lambda)\_A} \\ {e'\circ\lambda\_A\circ Te'\circ T\lambda\_a\circ\mu\_{TA}} && {e'\circ \mu'\_A\circ(\lambda\lambda)\_A\circ\mu\_{TA}} &&&& {e'\circ \lambda\_A\circ\mu\_A\circ T\mu\_A} \\ & {e'\circ T'e'\circ(\lambda\lambda)\_A\circ\mu\_{TA}} && {} \\ && {e'\circ \mu'\_A\circ(\lambda\lambda)\_A\circ\mu\_{TA}} \\ &&& {e'\circ\lambda\_A\circ\mu\_A\circ\mu\_{TA}} \arrow[Rightarrow, "{\lambda\_{e'}}"', from=1-4, to=2-3, near end] \arrow[Rightarrow, "{\gamma'}"', from=2-3, to=3-2, near end] \arrow[Rightarrow, "{m\_A}"', from=3-2, to=4-1, near end] \arrow[Rightarrow, "{\widehat{T\lambda\_{e'}}}", from=1-4, to=2-5, near end] \arrow[Rightarrow, "{\widehat{T\gamma'}}", from=2-5, to=3-6, near end] \arrow[Rightarrow, "{\widehat{Tm\_A}}", from=3-6, to=4-7, near end] \arrow[Rightarrow, "{\widehat{T\lambda\_{e'}}}", from=2-3, to=3-4, near end] \arrow[Rightarrow, "{\lambda\_{e'}}"', from=2-5, to=3-4, near end] \arrow[Rightarrow, "{\widehat{T\lambda\_{e'}}}", from=3-2, to=4-3, near end] \arrow[Rightarrow, "{m\_A}"', from=12-3, to=13-4] \arrow[Rightarrow, "{\gamma'}"', from=11-2, to=12-3] \arrow[Rightarrow, "{\lambda\_{e'}}"', from=10-1, to=11-2] \arrow[Rightarrow, "{\widehat{Tm\_A}}", from=5-6, to=6-7, near end] \arrow[Rightarrow, "{\lambda\_{e'}}"', from=3-6, to=5-6] \arrow[Rightarrow, "{\gamma'}"', from=5-6, to=7-6] \arrow[Rightarrow, "{\lambda\_{\mu'}}", from=7-6, to=8-5] \arrow[Rightarrow, "{a'\_A}", from=8-5, to=9-4] \arrow[Rightarrow, "{\gamma'}", from=8-3, to=9-4, near end] \arrow[Rightarrow, "{\lambda\_{\mu'}}"', from=5-6, to=6-5, near end] \arrow[Rightarrow, "{\gamma'}"', from=6-5, to=8-5] \arrow[Rightarrow, "{\mu'\_{e'}}"', from=6-3, to=8-3] \arrow[Rightarrow, "{\gamma'}"', from=5-4, to=6-3, near end] \arrow[Rightarrow, "{\lambda\_{T'e'}}", from=3-4, to=5-4] \arrow[Rightarrow, "{\gamma'}"', from=3-4, to=4-3, near end] \arrow[Rightarrow, "{\lambda\_{T'e'}}"', from=4-3, to=6-3] \arrow[Rightarrow, "{\widehat{T'\gamma'}}", from=5-4, to=6-5, near end] \arrow[Rightarrow, "{\widehat{Tm\_A}}"', from=7-6, to=8-7] \arrow[Rightarrow, "{a\_A}", from=10-7, to=13-4] \arrow[Rightarrow, "{\gamma'}", from=6-7, to=8-7] \arrow[Rightarrow, "{\lambda\_{e'}}", from=4-7, to=6-7] \arrow[Rightarrow, "{m\_A}", from=8-7, to=10-7] \arrow[Rightarrow, "{m\_{T'A}}"', from=8-3, to=9-2, near end] \arrow[Rightarrow, "{\lambda\_{e'}}", from=8-1, to=9-2, near end] \arrow[Rightarrow, "{\mu\_\lambda}", from=9-2, to=11-2] \arrow[Rightarrow, "{m\_{T'A}}", from=9-4, to=10-3] \arrow[Rightarrow, "{\mu\_\lambda}", from=10-3, to=12-3] \arrow[Rightarrow, "{\gamma'}"', from=9-2, to=10-3] \arrow[Rightarrow, "{\mu\_\lambda}"', from=8-1, to=10-1] \arrow[Rightarrow, "{\mu\_{e'}}"', from=4-1, to=8-1] \end{tikzcd} }; \end{tikzpicture}$$ Now, * the bottom right region commutes by the associativity condition of (λ, u, m), as in ; * the top right hexagon commutes by pseudonaturality of λ; * the center hexagon commutes by the multiplication condition of the algebra (A, eʹ, ιʹ, γʹ), as in ; * the top left hexagon commutes by the modification property for m; * all the remaining parallelograms commute by the interchange law. Therefore, (A, e, ι, γ) is a T-pseudoalgebra. For functoriality, consider now Tʹ-pseudoalgebras (A, eʹA, ιʹA, γʹA) and (B, eʹB, ιʹB, γʹB), and a morphism of Tʹ-pseudoalgebras (f, ϕ) from A to B. The composite 2-cell $$\begin{tikzcd} TA {\arrow{r}{}}{Tf} {\arrow{d}{}}[swap]{\lambda\_A} & TB {\arrow{d}{}}{\lambda\_B} {\arrow{dl}{Rightarrow, "\lambda\_f", shorten <= 1em, shorten >= 1em, swap}} \\ T'A {\arrow{r}{}}{T'f} {\arrow{d}{}}[swap]{e'\_A} & T'B {\arrow{d}{}}{e'\_B} {\arrow{dl}{Rightarrow, "\phi", shorten <= 1em, shorten >= 1em}} \\ A {\arrow{r}{}}[swap]{f} & B \end{tikzcd}$$ makes f a morphism between the T-pseudoalgebra structures defined above. Indeed, the unit condition, obtained by plugging into, can be decomposed as follows, again omitting the whiskering. $$\begin{tikzcd}[column sep=small] {e'\_B\circ\lambda\_B\circ\eta\_B\circ f} && {e'\_B\circ \lambda\_B\circ Tf\circ\eta\_A} \\ {e'\_B\circ\eta'\_B\circ f} & {e'\_B\circ T'f\circ\eta'\_A} & {e'\_B\circ T'f\circ\lambda\_A\circ\eta\_A} \\ {f} & {f\circ e'\_A\circ\eta'\_A} & {f\circ e'\_A\circ\lambda\_A\circ\eta\_A} \arrow[Rightarrow, "{u^{-1}\_B}"', from=1-1, to=2-1] \arrow[Rightarrow, "{\iota'\_B}"', from=2-1, to=3-1] \arrow[Rightarrow, "{\iota'\_A}", from=3-2, to=3-1] \arrow[Rightarrow, "{u^{-1}\_A}", from=3-3, to=3-2] \arrow[Rightarrow, "{\lambda\_f}", from=1-3, to=2-3] \arrow[Rightarrow, "{\phi}", from=2-3, to=3-3] \arrow[Rightarrow, "{\eta\_f}", from=1-1, to=1-3] \arrow[Rightarrow, "{u^{-1}\_A}"', from=2-3, to=2-2] \arrow[Rightarrow, "{\phi}", from=2-2, to=3-2] \arrow[Rightarrow, "{\eta'\_f}", from=2-1, to=2-2] \end{tikzcd}$$ Now * the top region commutes by the modification property for u; * the bottom left rectangle commutes by the unit condition for (f, ϕ) as in ; * the bottom right rectangle commutes by the interchange law. Similarly, the multiplication condition can be decomposed as follows, where again the whiskerings are omitted. $$\begin{tikzpicture}[baseline= (a).base] \node[scale=.75] (a) at (0,0){ \begin{tikzcd}[column sep=-4em] &&& {e'\_B\circ\lambda\_B\circ Te'\_B\circ T\lambda\_B\circ TTf} \\ && {e'\_B\circ T'e'\_B\circ (\lambda\lambda)\_B\circ TTf} && {e'\_B\circ\lambda\_B\circ Te'\_B\circ TT'f\circ T\lambda\_A} \\ & {e'\_B\circ\mu'\_B\circ(\lambda\lambda)\_B\circ TTf} && {e'\_B\circ T'e'\_B\circ\lambda\_{T'B}\circ TT'f\circ T\lambda\_A} && {e'\_B\circ\lambda\_B\circ Tf\circ Te'\_A\circ T\lambda\_A} \\ {e'\_B\circ \lambda\_B\circ \mu\_B\circ TTf} && {e'\_B\circ\mu'\_B\circ\lambda\_{T'B}\circ TT'f\circ T\lambda\_A} &&& {} \\ &&& {e'\_B\circ T'e'\_B\circ T'T'f\circ(\lambda\lambda)\_A} && {e'\_B\circ T'f\circ\lambda\_A\circ Te'\_A\circ T\lambda\_A} \\ && {e'\_B\circ\mu'\_B\circ T'T'f\circ(\lambda\lambda)\_A} && {e'\_B\circ Tf\circ T'e'\_A\circ (\lambda\lambda)\_A} \\ &&&&& {f\circ e'\_A\circ\lambda\_A\circ Te'\_A\circ T\lambda\_A} \\ {e'\_B\circ\lambda\_B\circ Tf} && {e'\_B\circ T'f\circ \mu'\_A\circ(\lambda\lambda)\_A} && {f\circ e'\_A\circ T'e'\_A\circ(\lambda\lambda)\_A} \\ & {e'\_B\circ T'f\circ \lambda\_A\circ\mu\_A} && {f\circ e'\_A\circ\mu'\_A\circ(\lambda\lambda)\_A} \\ && {f\circ e'\_A\circ\lambda\_A\circ\mu\_A} \arrow[Rightarrow, "{\widehat{T\lambda\_f}}", from=1-4, to=2-5, near end] \arrow[Rightarrow, "{\widehat{T\phi}}", from=2-5, to=3-6, near end] \arrow[Rightarrow, "{\phi}", from=5-6, to=7-6] \arrow[Rightarrow, "{\lambda\_{e'\_A}}", from=7-6, to=8-5] \arrow[Rightarrow, "{\gamma'\_A}", from=8-5, to=9-4] \arrow[Rightarrow, "{\lambda\_f}", from=3-6, to=5-6] \arrow[Rightarrow, "{m\_A}", from=9-4, to=10-3] \arrow[Rightarrow, "{\lambda\_{e'\_B}}"', from=1-4, to=2-3, near end] \arrow[Rightarrow, "{\gamma'\_B}"', from=2-3, to=3-2, near end] \arrow[Rightarrow, "{m\_B}"', from=3-2, to=4-1, near end] \arrow[Rightarrow, "{\mu\_f}"', from=4-1, to=8-1] \arrow[Rightarrow, "{\lambda\_f}"', from=8-1, to=9-2] \arrow[Rightarrow, "{\phi}"', from=9-2, to=10-3] \arrow[Rightarrow, "{\widehat{T\lambda\_f}}", from=3-2, to=4-3, near end] \arrow[Rightarrow, "{\lambda\_f}"', from=4-3, to=6-3] \arrow[Rightarrow, "{\mu'\_f}"', from=6-3, to=8-3] \arrow[Rightarrow, "{m\_A}"', from=8-3, to=9-2, near end] \arrow[Rightarrow, "{\widehat{T\lambda\_f}}", from=2-3, to=3-4, near end] \arrow[Rightarrow, "{\gamma'\_B}"', from=3-4, to=4-3, near end] \arrow[Rightarrow, "{\lambda\_f}", from=3-4, to=5-4] \arrow[Rightarrow, "{\gamma'\_B}"', from=5-4, to=6-3, near end] \arrow[Rightarrow, "{\phi}", from=8-3, to=9-4, near end] \arrow[Rightarrow, "{\widehat{T'\phi}}", from=5-4, to=6-5, near end] \arrow[Rightarrow, "{\phi}", from=6-5, to=8-5] \arrow[Rightarrow, "{\lambda\_{e'\_B}}"', from=2-5, to=3-4, near end] \arrow[Rightarrow, "{\lambda\_{e'\_A}}"', from=5-6, to=6-5, near end] \end{tikzcd} }; \end{tikzpicture}$$ Now, * the region on the far left commutes by the modification property for m; * the hexagon on the bottom commutes by the multiplication condition for (f, ϕ) as in ; * the hexagon on the top right commutes by pseudonaturality of λ; * all the remaining parallelograms commute by the interchange law. This makes (f, ϕ λ) a pseudomorphism of T-pseudoalgebras. Finally, to prove 2-functoriality, let (f, ϕ) and (g, χ) be pseudomorphisms of Tʹ-pseudoalgebras A → B, and let α : f ⇒ g be a 2-cell of Tʹ-pseudoalgebras. We have that α is canonically also a 2-cell of T-pseudoalgebras, since the relevant diagram can be decomposed as follows, $$\begin{tikzcd} e'\_B\circ\lambda\_B\circ Tf {\arrow[Rightarrow]{d}{}}{T\alpha} {\arrow[Rightarrow]{r}{}}{\lambda\_f} & e'\_B\circ T'f\circ\lambda\_A {\arrow[Rightarrow]{r}{}}{\phi} {\arrow[Rightarrow]{d}{}}{T'\alpha} & f\circ e'\_A\circ\lambda\_A {\arrow[Rightarrow]{d}{}}{\alpha} \\ e'\_B\circ\lambda\_B\circ Tg {\arrow[Rightarrow]{r}{}}[swap]{\lambda\_g} & e'\_B\circ T'g\circ\lambda\_A {\arrow[Rightarrow]{r}{}}[swap]{\chi} & g\circ e'\_A\circ\lambda\_A \end{tikzcd}$$ again omitting the whiskerings, and now * the left square commutes by pseudonaturality of λ; * the right square commutes since α is a 2-cell of pseudoalgebras, as in. This action on pseudoalgebras, their morphisms and their 2-cells defines then a 2-functor from the 2-category of Tʹ-pseudoalgebras to the 2-category of T-pseudoalgebras (notice the direction). We encourage the readers more familiar with 2-dimensional diagrams to rewrite the proof using 2-cells and, for clarity, we suggest to dedicate one direction in each diagram to the transformation λ. --- 1. By “strict” here we mean “we give an isomorphism of hom-categories, not just an equivalence”. The colimit is still oplax, not strict.[↩](#fnref1) 2. These are known in the literature also as “cofinal”, “coinitial” and “final”, terms which may cause confusion. The “co” in “cofinal” (e.g. in “cofinal subnet”) does not denote duality, but rather, follows the Latin particle “cum” which means “with, together”. As such, we feel that “confinal” is both closer to the original etymology, and less prone to cause confusion. The term “confinal” (or in German, “konfinal”) was introduced by Hausdorff for the case of ordered sets, and it has been in use at least until.[↩](#fnref2) 3. Often a stronger statement is called “co-Yoneda lemma”, see.[↩](#fnref3) 4. This name comes from the theory of monoidal categories, where the diagram has the shape of a pentagon (in that case, the analogue of μ is strictly natural, and so the arrow corresponding to μμ is an identity).[↩](#fnref4) Kan extensions are partial colimits =================================== One way of interpreting a left Kan extension is as taking a kind of “partial colimit”, whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the “partial evaluations” sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the “restriction-of-scalars” construction of monads extends to the case of pseudoalgebras over pseudomonads, we define a morphism of monads between them, which we call “image”. This morphism allows us in particular to generalize the idea of “confinal functors”, i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass. Introduction ============ Kan extensions are a prominent tool of category theory, to the extent that, already in the preface to the first edition of, Mac Lane declared that “all concepts of category theory are Kan extensions”, a claim reinforced more recently in. However, they are also considered to be a notoriously slippery concept, especially by newcomers to the subject. One of the most powerful pictures that help understanding how they work may be the idea that Kan extensions, especially in their pointwise form, “replace parts of a diagram with their best approximations, either from the right or from the left”. In other words, Kan extensions can be seen as taking limits or colimits of “parts” of a diagram. The scope of this paper is making this intuition mathematically precise. We make use of the concept of partial evaluation, which was introduced in, and which is a way to formalize “partially computed operations” in terms of monads. The standard example is that “1 + 2 + 3 + 4” may be evaluated to “10”, but also partially evaluated to “3 + 7”, whereby parts of the given sum have been replaced by their sums. Just as monads on sets may be seen as encoding different algebraic structures and operations, here we consider pseudomonads on categories which encode the operation of taking colimits. We are in particular interested in two pseudomonads: the monad of diagrams and the monad of small presheaves (also known as the free (small) cocompletion monad). Both monads are known in the literature, but certainly not presented in sufficient detail as needed for our purposes. To make the paper more accessible, we therefore decided to spell out their definition in full detail, in and. The definitions of pseudomonads, pseudoalgebras, and their morphisms are also hard to find in the literature in sufficient detail. For this reason, to avoid any ambiguity, we have given a detailed account of them in. Readers who are familiar with these pseudomonads, and with the concepts of pseudomonads in general, may skip these sections, with the exception of, and, which contain new results. Here is what the novel content of this work consists of. First of all, we introduce the concept of “image presheaf”, which takes a diagram and forms a presheaf that can be considered the “free colimit” of the diagram. This induces a morphism of monads from the monad of diagrams to the monad of small presheaves, which in turn gives a “pullback” functor between the categories of algebras (we prove the 2-dimensional version of this statement in ). This morphism of monads is not injective in any sense. Indeed, it turns out that diagrams with isomorphic image presheaves have the same colimit, in a very strong sense, analogous to “differing by a confinal functor”. We indeed generalize the theory of confinal functors, and connect it to the theory of absolute colimits – both because we need that in order to prove the subsequent statements, and because it should be interesting for its own sake. We then turn to the central topic of this paper and study partial evaluations for both monads. We prove that partial evaluations for the monad of diagrams correspond to pointwise left Kan extensions along split opfibrations, by invoking the Grothendieck correspondence between split opfibrations and functors into Cat. For the monad of small presheaves, we show that partial evaluations correspond to pointwise left Kan extensions along arbitrary functors. This result may be summarized in the following way: given small presheaves P and Q on a locally small, small-cocomplete category, Q is a partial colimit of P if and only if they can be written as image presheaves of small diagrams D and Dʹ, in such a way that Dʹ is the left Kan extension of D along some functor. More concisely, Kan extensions are partial colimits, as claimed by the paper’s title. This result is analogous to, and was motivated by, an analogous result in measure theory involving probability monads, where partial evaluations (or “partial expectations”) correspond exactly to conditional expectations (). Indeed, one could say that “if coends are like integrals, then Kan extensions are like conditional expectations”. (See for more on this.) As usual, when one talks about free cocompletion, one has to be very careful with size issues. This is why some parts of this work, such as the proof of, appear to be rather technical. The payoff is that the main theorems of this work will hold for arbitrary (small) colimits in arbitrary (locally small) categories, beyond the trivial case of preorders. #### Outline. In we study the category of diagrams in a given category, and show that the construction gives a pseudomonad on the 2-category of locally small categories. While this construction seems to be known, its details don’t seem to have been spelled out previously. The content of, however, seems to be entirely new. We show that cocomplete categories, equipped with a choice of colimit for each diagrams, are pseudoalgebras over this pseudomonad, and that not all pseudoalgebras are of that form. In we define the concept of “image presheaf”, which can be seen as a “free colimit of a diagram”, or as a “colimit blueprint”. We show that “having the same image presheaf” is a strong and consistent generalization both of the theory of confinal functors (), and of the concept of absolute colimit (). As far as we know, this generalization is new. In we study small presheaves and show that they form a pseudomonad. Again, this is known, but here we spell out the construction in much greater detail than previous accounts have done. This enables us to establish the new result presented in : the image presheaf construction forms a morphism of pseudomonads from diagrams to presheaves. The principal new results of this paper appear in. and state that partial colimits for the monad of diagrams correspond to pointwise left Kan extensions of diagrams along split opfibrations. In we prove that partial colimits for the free cocompletion monad correspond to pointwise left Kan extensions of diagrams along arbitrary functors. Then, in, we compare this categorical result to the analogous measure-theoretic fact that partial expectations for probability monads correspond (in some cases) to conditional expectations of random variables. This is in line with the famous analogy between coends and integrals. Finally, in we recall the (known) definition of pseudomonads and pseudoalgebras, and of the categories they form, which we use in the rest of the paper. We also provide a 2-dimensional version of the “restriction of scalars” construction (), where a morphism of monads induces a functor between the categories of their algebras in opposite direction. As far as we know, this 2-dimensional version has not appeared in the literature previously.. #### Acknowledgements. The first author would like to thank Bartosz Milewski and David Jaz Myers for the insight on coends and weighted limits, Joachim Kock and Emily Riehl for enlightenment on some of the higher-dimensional aspects, and Tobias Fritz for further helpful insight. The first author would also like to thank Sean McKenna, as well as David Spivak and MIT as a whole, for all the support during the 2020 pandemic, and Nathanael Arkor for the development of the app Quiver, which proved to be very helpful in writing some of the diagrams in this document. The first author was affiliated to the Massachusetts Institute of Technology (MIT) for most of the time of writing. This research was partially funded by the Fields Institute (Canada) and by AFOSR grants FA9550-19-1-0113 and FA9550-17-1-0058 (U.S.A.). The second author acknowledges partial financial support by the Natural Science and Engineering Council of Canada under the Discovery Grants Program (grant no. 501260). #### Categorical setting, notation, and conventions. As it is to be expected when one talks about generic colimits, size issues are relevant. Here are our conventions. All the categories in this work (except CAT) are assumed locally small. We denote by Cat the 2-category of small categories, and by CAT the 2-category of possibly large, locally small categories. Note that CAT is itself larger than a large category (some authors call it a “huge” category). When we say “category”, without specifying the size, we will always implicitly refer to a possibly large, locally small category. Similarly, by “cocomplete category” we always mean a possibly large, locally small category which admits all small colimits. The monad of diagrams ===================== In this section we define the monad of diagrams. The first source for it that we are aware of is Guitart’s article, but without an explicit construction. We give in detail all the structure maps, and in we prove that cocomplete categories with a choice of colimits are pseudoalgebras (but not all pseudoalgebras are in this form). The notions of pseudomonad and pseudoalgebra that we use are given in detail in. Note that, differently from some of the literature, we use the following slightly generalized notion of morphism of diagrams (also used, for example, in Guitart’s original work ). Moreover, in order to avoid size issues, we require every diagram to be small. [defdiag] Let C be a locally small category. * We call a diagram in C a small category J together with a functor D : J → C. Throughout this work, all the diagrams will be implicitly assumed to be of this form (i.e. be small). * Given diagrams (J, D) and (J', Dʹ) in C, we call a morphism of diagrams a functor R : J → J' together with a natural transformation ρ : D → Dʹ ∘ R, i.e. a diagram in CAT as the following. $$\begin{tikzcd}[row sep=small] {{\mathsf{J}}} {\arrow{dr}{""{name=D,below}}}{D} {\arrow{dd}{}}[swap]{R} \\ & {{\mathsf{C}}} \\ {{\mathsf{J'}}} {\arrow{ur}{}}[swap]{D'} {\arrow{\end}{Rightarrow,from=D, "\rho", shorten <= 0.5em, shorten >= 1em, near start}}{tikzcd}$$ * Given diagrams (J, D) and (J', Dʹ) in D and morphisms of diagrams (R, ρ), (Rʹ, ρʹ) : (J, D) → (J', Dʹ), we call a 2-cell of diagrams a natural transformation α : R ⇒ Rʹ such that the following 2-cells are equal. $$\begin{tikzcd}[column sep=large] {{\mathsf{J}}} {\arrow{dr}{""{name=D,below, pos=0.7}}}{D} {\arrow{dd}{bend right, ""{name=RP, right}}}[swap]{R'} {\arrow{dd}{bend left, ""{name=R, left}}}[pos=0.4]{R} \\ & {{\mathsf{C}}} \\ {{\mathsf{J'}}} {\arrow{ur}{}}[swap]{D'} {\arrow{{\arrow{}}{}}{Rightarrow,from=D, "\rho", shorten <= 0.5em, shorten >= 1.5em, near start}}[Rightarrow,from=R,to=RP,"\alpha"] \end{tikzcd} \qquad=\qquad\begin{tikzcd}[column sep=large] {{\mathsf{J}}} {\arrow{dr}{""{name=D,below}}}{D} {\arrow{dd}{bend right}}[swap]{R'} \\ & {{\mathsf{C}}} \\ {{\mathsf{J'}}} {\arrow{ur}{}}[swap]{D'} {\arrow{\end}{Rightarrow,from=D, "\rho'", shorten <= 0.5em, shorten >= 1em, swap}}{tikzcd}$$ We denote by Diag(C) the 2-category of diagrams in C, their morphisms, and their 2-cells. (Sometimes we will still denote by Diag(C) the underlying 1-category.) Note that the definition of morphism of diagrams is slightly more general than just a natural transformation between parallel functors. This is still compatible with the traditional intuitive picture of “deforming a diagram into another one”, provided that one notices the following. In principle R may be not essentially surjective, so one should visualize the natural transformation ρ as “deforming” the figure drawn by D into a subfigure of the one drawn by Dʹ. Note moreover that: * For C locally small, Diag(C) is locally small too; * The forgetful functor Diag(C) → Cat given by the domain is a fibration (via precomposition), and it is an opfibration (via left Kan extensions) if and only if C is cocomplete (see for example ). In the rest of this section we show that ${{\mathsf{C}}}\rightsquigarrow{{\mathsf{Diag(C)}}}$ is part of a pseudomonad on CAT, and that cocomplete categories with a choice of colimit for each diagram are pseudoalgebras, with the structure map given by such chosen colimits. For the precise definitions of pseudomonads and pseudoalgebras, see. Functoriality ------------- We show that the assignment ${{\mathsf{C}}}\rightsquigarrow{{\mathsf{Diag(C)}}}$ is part of a 2-functor on CAT. First of all, let C and D be locally small categories, and let F : C → D be a functor. Consider now Diag(C) and Diag(D) as 1-categories. There is a (1-)functor F\* : Diag(C) → Diag(D) induced by F via postcomposition and whiskering, as follows. $$\begin{tikzcd} {{\mathsf{J}}} {\arrow{r}{}}{D} & {{\mathsf{C}}} \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} {{\mathsf{J}}} {\arrow{r}{}}{D} & {{\mathsf{C}}} {\arrow{r}{}}{F} & {{\mathsf{D}}} \end{tikzcd}$$ $$\begin{tikzcd}[row sep=small] {{\mathsf{J}}} {\arrow{dr}{""{name=D,below}}}{D} {\arrow{dd}{}}[swap]{R} \\ & {{\mathsf{C}}} \\ {{\mathsf{J'}}} {\arrow{ur}{}}[swap]{D'} {\arrow{\end}{Rightarrow,from=D, "\rho", shorten <= 0.5em, shorten >= 1em, near start}}{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] {{\mathsf{J}}} {\arrow{dr}{""{name=D,below}}}{D} {\arrow{dd}{}}[swap]{R} \\ & {{\mathsf{C}}} {\arrow{r}{}}{F} & {{\mathsf{D}}} \\ {{\mathsf{J'}}} {\arrow{ur}{}}[swap]{D'} {\arrow{\end}{Rightarrow,from=D, "\rho", shorten <= 0.5em, shorten >= 1em, near start}}{tikzcd}$$ Therefore, Diag is an endofunctor of CAT. The functor F\* : Diag(C) → Diag(D) extends to 2-cells giving a 2-functor, but we will not need this in order for Diag to be a pseudomonad CAT. On the other hand, we need to extend Diag to the 2-cells of CAT. So let C and D be locally small categories, let F, G : C → D be functors, and let α : F ⇒ G be a natural transformation. We have an induced natural transformation α\* : F\* ⇒ G\* induced via whiskering, as follows. To each diagram (J, D) in C, we assign the morphism of diagrams (idJ, αD) of D, i.e. $$\begin{tikzcd} {{\mathsf{J}}} {\arrow{r}{}}{D} & {{\mathsf{C}}} \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} {{\mathsf{J}}} {\arrow{r}{}}{D} & {{\mathsf{C}}} {\arrow{r}{bend left=35, ""{name=F,below}}}{F} {\arrow{r}{bend right=35, ""{name=G,above}}}[swap]{G} & {{\mathsf{D}}} {\arrow{\end}{Rightarrow, from=F, to=G, "\alpha"}}{tikzcd}$$ Naturality follows from naturality of α. This makes Diag a strict 2-functor CAT → CAT. Unit and multiplication ----------------------- ### The unit: one-object diagrams The unit of the monad is a map constructing “one-object diagrams”. In detail, let C be a locally small category. We construct the functor ηC : C → Diag(C) as follows. $$\begin{tikzcd} C \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} {{\mathsf{1}}} {\arrow{r}{}}{C} & {{\mathsf{C}}} \end{tikzcd}$$ $$\begin{tikzcd} C {\arrow{r}{}}{f} & C' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] {{\mathsf{1}}} {\arrow{dr}{""{name=D,below}}}{C} {\arrow{dd}{}}[swap]{{\mathrm{id}}} \\ & {{\mathsf{C}}} \\ {{\mathsf{1}}} {\arrow{ur}{}}[swap]{C'} {\arrow{\end}{Rightarrow,from=D, "f", shorten <= 0.5em, shorten >= 1em, near start}}{tikzcd}$$ For brevity, we will denote ηC simply by η. This is (strictly) natural in the category C: given a functor F : C → D, the following diagram commutes strictly. $$\begin{tikzcd} {{\mathsf{C}}} {\arrow{d}{}}{\eta} {\arrow{r}{}}{F} & {{\mathsf{D}}} {\arrow{d}{}}{\eta} \\ {{\mathsf{Diag(C)}}} {\arrow{r}{}}{F\_\} & {{\mathsf{Diag(D)}}} \end{tikzcd}$$ Indeed, both paths in the diagram give the following assignment, $$\begin{tikzcd} C \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} {{\mathsf{1}}} {\arrow{r}{}}{C} & {{\mathsf{C}}} {\arrow{r}{}}{F} & {{\mathsf{D}}} \end{tikzcd}$$ $$\begin{tikzcd} C {\arrow{r}{}}{f} & C' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] {{\mathsf{1}}} {\arrow{dr}{""{name=D,below}}}{C} {\arrow{dd}{}}[swap]{{\mathrm{id}}} \\ & {{\mathsf{C}}} {\arrow{r}{}}{F} & {{\mathsf{D}}} \\ {{\mathsf{1}}} {\arrow{ur}{}}[swap]{C'} {\arrow{\end}{Rightarrow,from=D, "f", shorten <= 0.5em, shorten >= 1em, near start}}{tikzcd}$$ using the fact that F(C) = F* ∘ C if we view C as a functor 1 → C, and that analogously Ff is given by whiskering f (seen as a natural transformation) with F. ### Diagrams of diagrams are lax cocones Let’s now turn to the multiplication. We first notice that an object of Diag(Diag(C)) is the same as a lax cocone in CAT with tip C, where the indexing category and all the categories appearing in the cone except C are required to be small. Let’s see how. Let J be a small category. A functor D : J → Diag(C) assigns to each object J of J a diagram in C, i.e. a small category D0J together with a functor D1J : D0J → C: $$\begin{tikzcd} J \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} D\_0J {\arrow{r}{}}{D\_1J} & {{\mathsf{C}}} \end{tikzcd}$$ and to each morphism j : J → Jʹ of J a morphism of diagrams, which amounts to a functor D0j : D0J → D0Jʹ together with a natural transformation D1j as below: $$\begin{tikzcd} J {\arrow{d}{}}[swap]{j} \\ J' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] D\_0J {\arrow{dr}{""{name=D,below}}}{D\_1J} {\arrow{dd}{}}[swap]{D\_0j} \\ & {{\mathsf{C}}} \\ D\_0J' {\arrow{ur}{}}[swap]{D\_1J'} {\arrow{\end}{Rightarrow,from=D, "D\_1j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ Moreover, since we want D to be a functor, we need it to preserve identities and composition, i.e. D0 needs to be a functor, and D1 needs to satisfy the conditions D1(idJ) = idD1J and D1(jʹ ∘ j) = D1jʹD0j ∘ D1J, which are exactly the conditions of lax naturality. In pictures, $$\begin{tikzcd} J {\arrow{d}{}}[swap]{{\mathrm{id}}} \\ J \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] D\_0J {\arrow{dr}{""{name=D,below}}}{D\_1J} {\arrow{dd}{}}[swap]{{\mathrm{id}}} \\ & {{\mathsf{C}}} \\ D\_0J {\arrow{ur}{}}[swap]{D\_1J} {\arrow{\end}{Rightarrow,from=D, "{\mathrm{id}}", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ $$\begin{tikzcd} J {\arrow{dd}{}}[swap]{j} \\ \\ J' {\arrow{dd}{}}[swap]{j'} \\ \\ J'' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[column sep=huge] D\_0J {\arrow{ddr}{""{name=D,below}}}{D\_1J} {\arrow{dd}{}}[swap]{D\_0j} \\ \\ D\_0J {\arrow{r}{""{name=MID}}}{D\_1J'} {\arrow{dd}{}}[swap]{D\_0j'} {\arrow{&}{Rightarrow,from=D, "D\_1j", shorten <= 0.2em, shorten >= 0.2em, swap, near start}} {{\mathsf{C}}} \\ \\ D\_0J'' {\arrow{uur}{}}[swap]{D\_1J''} {\arrow{\end}{Rightarrow,from=MID, "D\_1j'", shorten <= 1.5em, shorten >= 1.5em, swap, pos=0.38}}{tikzcd}$$ In other words, a functor D : J → Diag(C) consists of a functor D0 : J → Cat ⊆ CAT, together with a lax cocone D1 in CAT under D0 with tip C. A lax cocone is a lax natural transformation D1 : D0 ⇒ ΔC, where ΔC is the constant functor at C. ### The multiplication: the Grothendieck construction Given now D = (D0, D1) as above, take the Grothendieck construction $\groth D\_0$ of D0 : J → Cat ⊆ CAT, which we recall. * An object of $\groth D\_0$ consists of a pair (J, X) where J is an object of J and X is an object of the category D0J; * A morphism (J, X) → (Jʹ, Xʹ) of $\groth D\_0$ consists of a pair (j, f) where j : J → Jʹ is a morphism of J, and f : D0j(X) → Xʹ is a morphism of the category D0Jʹ. The short integral sign does not denote a coend here, it is standard for the Grothendieck construction (we use different sizes to avoid confusion, since both symbols are standard notation). Note that since J and all the D0J are small, $\groth D\_0$ is small too. Its set of objects is given by ∐J ∈ JD0J. Moreover: * For each object J of J, the inclusion maps $i\_J:D\_0J\to \groth D\_0$ defined by the coproduct above can be canonically made into functors via $$\begin{tikzcd} X {\arrow{d}{}}[swap]{f} \\ X' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} (J,X) {\arrow{d}{}}{({\mathrm{id}}\_J,f)} \\ (J,X') \end{tikzcd}$$ We will call the images of the iJ the fibers of $\groth D\_0$. * For each morphism j : J → Jʹ of J, there is a natural transformation $$\begin{tikzcd}[row sep=small] D\_0J {\arrow{dr}{""{name=D,below}}}[near end]{i\_J} {\arrow{dd}{}}[swap]{D\_0j} \\ & \groth D\_0 & & \\ D\_0J' {\arrow{ur}{}}[near end,swap]{i\_{J'}} {\arrow{\end}{Rightarrow,from=D, "i\_j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ whose component at each object X of D0J is given by $$\begin{tikzcd}[column sep=huge] (J,X) {\arrow{r}{}}{(j,{\mathrm{id}}\_{D\_0j(X)})} & (J', D\_0j(X)) \end{tikzcd}$$ * The iJ and ij assemble into a lax cocone $D\_0 \Rightarrow \Delta \groth D\_0$, i.e. the identity and composition conditions are satisfied. It is well known that $\groth D\_0$ is the oplax colimit of D0 in Cat, with the universal lax cocone given by the iJ. We now show that it is so also in CAT, and we also give a strict version of the universal property.[1](#fn1) [oplaxcolimit] Let D0 : J → Cat ⊆ CAT be a small diagram of small categories. Let D1 : D0 ⇒ C be a lax cocone over D0 in CAT, with tip C locally small (but not necessarily small). There is a unique functor $\groth D\_0\to {{\mathsf{C}}}$ such that * for all objects J of J, the following triangle commutes (strictly); $$\label{ijstrict} \begin{tikzcd} D\_0J {\arrow{d}{}}[swap]{i\_J} {\arrow{dr}{}}{D\_1J} \\ \groth D\_0 {\arrow{r}{}} & {{\mathsf{C}}} \end{tikzcd}$$ * for all morphisms j : J → Jʹ of J, the following 2-cells coincide. $$\label{ijlax} \begin{tikzcd}[column sep=large] D\_0J {\arrow{dr}{""{name=D,below}}}{D\_1J} {\arrow{dd}{}}[swap]{D\_0j} \\ & {{\mathsf{C}}} \\ D\_0J' {\arrow{ur}{}}[swap]{D\_1J'} {\arrow{\end}{Rightarrow,from=D, "D\_1j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd} \qquad=\qquad \begin{tikzcd} D\_0J {\arrow{drrr}{}}{D\_1J} {\arrow{dr}{""{name=D,below}}}[near end]{i\_J} {\arrow{dd}{}}[swap]{D\_0j} \\ & \groth D\_0 {\arrow{rr}{}} & & {{\mathsf{C}}} \\ D\_0J' {\arrow{ur}{}}[near end,swap]{i\_{J'}} {\arrow{urrr}{}}[swap]{D\_1J'} {\arrow{\end}{Rightarrow,from=D, "i\_j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ Denote this functor by μ(D0, D1), or more briefly by μ(D). It is a diagram in C. This will give the multiplication of the monad Diag. [Proof of ] Since we want the diagram [ijstrict] to commute strictly, the only possibility to define μ(D) on objects is as follows. For every object J of J, and for every object X of D0J, $$\mu(D) (J,X) \coloneqq D\_1J(X).$$ Just as well, for all the morphisms of $\groth D\_0$ in the fiber, i.e. in the form (idJ, f) for a morphism f : X → Xʹ of D0J, we are forced to define $$\mu(D) ({\mathrm{id}}\_J,f) \coloneqq D\_1J(f).$$ Moreover, since we want the condition to hold, for all morphisms j : J → Jʹ of J we have to require that μ(D) on the components of ij has to give the respective component of D1j. Explicitly, for each object X of D0J, $$\mu(D) (i\_j)\_X = \mu(D) (j, {\mathrm{id}}\_{D\_0j(X)}) \coloneqq (D\_1j)\_X.$$ The generic morphism (j, f) : (J, X) → (Jʹ, Xʹ), for j : J → Jʹ and f : D0j(X) → Xʹ can be decomposed as $$\begin{tikzcd}[column sep=huge] (J,X) {\arrow{r}{}}{(j,{\mathrm{id}}\_{D\_0j(X)})} & (J',D\_0j(X)) {\arrow{r}{}}{({\mathrm{id}}\_J,f)} & (J',X') \end{tikzcd}$$ and so we have determined the action of μ(D) for all morphisms of $\groth D\_0$. Functoriality of this assignment is routine. ### Functoriality and naturality of the multiplication We now have to show that the assignment (D0, D1) ↦ μ(D0, D1) is functorial, and that it is natural in the category C. To address functoriality, we need to look at morphisms in Diag(Diag(C)). Given diagrams D : J → Diag(C) and E : K → Diag(C), where J and K are small categories, a morphism of diagrams from D to E amounts to a functor F : J → K together with a natural transformation $$\begin{tikzcd}[row sep=small] {{\mathsf{J}}} {\arrow{dr}{""{name=D,below}}}{D} {\arrow{dd}{}}[swap]{F} \\ & {{\mathsf{Diag(C)}}} \\ {{\mathsf{K}}} {\arrow{ur}{}}[swap]{E} {\arrow{\end}{Rightarrow,from=D, "\phi", shorten <= 0.5em, shorten >= 1em, near start}}{tikzcd}$$ Explicitly, D consists of a functor D0 : J → Cat ⊆ CAT and a lax cocone D1 : D0 ⇒ C, and E has an analogous form. The natural transformation ϕ amounts to the following. For each object J of J, we have a morphism of diagrams $$\begin{tikzcd}[row sep=small] D\_0J {\arrow{dr}{""{name=D,below}}}{\phi\_0J} {\arrow{dd}{}}[swap]{D\_0J} \\ & {{\mathsf{C}}} \\ E\_0FJ {\arrow{ur}{}}[swap]{E\_1J} {\arrow{\end}{Rightarrow,from=D, "\phi\_1J", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ and for each morphism j : J → Jʹ of J, the following diagrams have to commute. First of all, this diagram of functors has to commute strictly. $$\begin{tikzcd}[sep=large] D\_0J {\arrow{d}{}}{D\_0j} {\arrow{r}{}}{\phi\_0J} & E\_0FJ {\arrow{d}{}}{E\_0Fj} \\ D\_0J' {\arrow{r}{}}{\phi\_0J'} & E\_0FJ' \end{tikzcd}$$ Moreover, the following composite 2-cells have to coincide, forming a commutative pyramid with 2-cells as lateral faces, whose square base is the commutative square just described above. $$\label{pyramid} \begin{tikzcd}[column sep=large] & {{\mathsf{C}}} \\ \\ \\ \phantom{E}\\ D\_0J {\arrow{dr}{}}[swap]{D\_0j} {\arrow{uuuur}{""{name=D1J}}}[near start]{D\_1J} & & \|[alias=E0FJP]\| E\_0FJ' {\arrow{uuuul}{}}[swap, near start]{E\_1FJ'} \\ & D\_0J {\arrow{{\arrow{}}{}}{Rightarrow, from=D1J, "D\_1j", shorten <= 2em, shorten >= 2em, swap}}[""{name=D1JP}]{uuuuu}[swap]{D\_1J'} {\arrow{ur}{}}[swap]{\phi\_0J'} {\arrow{\end}{Rightarrow, from=D1JP, to=E0FJP, "\phi\_1J'", shorten <= 2em, shorten >= 1.5em, swap, pos=0.6}}{tikzcd} \qquad=\qquad \begin{tikzcd}[column sep=large] & {{\mathsf{C}}} \\ \\ \\ & \|[alias=E0FJ]\| E\_0FJ {\arrow{uuu}{""{name=E1FJ}}}[swap, pos=0.4]{E\_1FJ} {\arrow{dr}{}}[swap]{E\_0Fj}\\ D\_0J {\arrow{ur}{}}[swap]{\phi\_0J} {\arrow{dr}{}}[swap]{D\_0j} {\arrow{uuuur}{""{name=D1J}}}[near start]{D\_1J} & & E\_0FJ' {\arrow{uuuul}{}}[swap, near start]{E\_1FJ'} {\arrow{\\}{Rightarrow, from=E1FJ, "E\_1Fj", shorten <= 2em, shorten >= 3em, pos=0.4}} & D\_0J {\arrow{ur}{}}[swap]{\phi\_0J'} {\arrow{\end}{Rightarrow, from=D1J, to=E0FJ, "\phi\_1J", shorten <= 0.5em, shorten >= 0em, swap, pos=0.7}}{tikzcd}$$ Form now the Grothendieck construction of E0. We can form a lax cocone over D0 with tip $\groth E\_0$ as follows. $$\begin{tikzcd} J \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd} D\_0J {\arrow{r}{}}{\phi\_0J} & E\_0FJ {\arrow{r}{}}{i\_{FJ}} & \groth E\_0 \end{tikzcd}$$ $$\begin{tikzcd} J {\arrow{d}{}}{j} \\ J' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] D\_0J {\arrow{r}{}}{\phi\_0J} {\arrow{dd}{}}[swap]{D\_0j} & E\_0FJ {\arrow{dr}{""{name=D,below}}}{i\_{FJ}} {\arrow{dd}{}}[swap]{E\_0Fj} \\ & & \groth E\_0 \\ D\_0J' {\arrow{r}{}}{\phi\_0J'} & E\_0FJ' {\arrow{ur}{}}[swap]{i\_{FJ'}} {\arrow{\end}{Rightarrow,from=D, "i\_{Fj}", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ Note that this is a lax cocone $D\_0\Rightarrow\groth E\_0$. By the universal property of the Grothendieck construction as an oplax colimit (), there is a unique functor $\groth D\_0\to \groth E\_0$ such that * for all objects J of J, the following square commutes; $$\label{squarestrict} \begin{tikzcd} D\_0J {\arrow{d}{}}[swap]{i\_J} {\arrow{r}{}}{\phi\_0J} & E\_0FJ {\arrow{d}{}}{i\_{FJ}} \\ \groth D\_0 {\arrow{r}{}} & \groth E\_0 \end{tikzcd}$$ * for all morphisms j : J → Jʹ of J, the following composite 2-cells coincide. $$\begin{tikzcd} D\_0J {\arrow{r}{}}{\phi\_0J} {\arrow{dd}{}}[swap]{D\_0j} & E\_0FJ {\arrow{dr}{""{name=D,below}}}{i\_{FJ}} {\arrow{dd}{}}[swap]{E\_0Fj} \\ & & \groth E\_0 \\ D\_0J' {\arrow{r}{}}{\phi\_0J'} & E\_0FJ' {\arrow{ur}{}}[swap]{i\_{FJ'}} {\arrow{\end}{Rightarrow,from=D, "i\_{Fj}", shorten <= 1em, shorten >= 2em, swap, near start}}{tikzcd} \qquad=\qquad \begin{tikzcd} D\_0J {\arrow{dd}{}}[swap]{D\_0j} {\arrow{dr}{""{name=D,below}}}[near end]{i\_J} {\arrow{r}{}}{\phi\_0J} & E\_0 FJ {\arrow{dr}{}}{i\_{FJ}}\\ & \groth D\_0 {\arrow{r}{}} & \groth E\_0\\ D\_0J' {\arrow{{\arrow{}}{}}{Rightarrow,from=D, "i\_j", shorten <= 1em, shorten >= 2em, swap, near start}}{ur}[near end,swap]{i\_{J'}} {\arrow{r}{}}[swap]{\phi\_0J'} & E\_0FJ' {\arrow{ur}{}}[swap]{i\_{FJ'}} \end{tikzcd}$$ Denote this functor by μ0(F, ϕ). This gives a triangle $$\begin{tikzcd}[row sep=small] \groth D\_0 {\arrow{dr}{""{name=D,below}}}{\mu(D)} {\arrow{dd}{}}[swap]{\mu\_0(F,\phi)} \\ & {{\mathsf{C}}} \\ \groth E\_0 {\arrow{ur}{}}[swap]{\mu(E)} \end{tikzcd}$$ which does not necessarily commute. In order to get a morphism of diagram μ(D) → μ(E) we need to fill the triangle above with a 2-cell which we form as follows. Consider the object (J, X) of $\groth D\_0$, where J is an object of J and X is an object of D0J. Note that, by the diagram [ijstrict], μ(D)(J, X) = D1J(X). Analogously, using the diagram [ijstrict] for E together with the diagram [squarestrict], μ(E)(μ0(J, X)) = μ(E)(J, ϕ0J(X)) = E1FJ(ϕ0J(X)). We now assign to the object (J, X) the morphism of C given by the component of ϕ1J at X, $$\begin{tikzcd} \mu(D)(J,X) = D\_1J(X) {\arrow{r}{}}{(\phi\_1J)\_X} & E\_1FJ(\phi\_0J(X)) = \mu(E)(\mu\_0(J,X)). \end{tikzcd}$$ Let’s now show that this assignment is natural. We will again test this first along the fibers, and then on the opcartesian morphisms of $\groth D\_0$. So let f : X → Y be a morphism of D0J. The following diagram commutes simply by naturality of ϕ1J. $$\begin{tikzcd} \mu(D)(J,X) {\arrow{d}{}}{\mu(D)({\mathrm{id}}\_J, f)} {\arrow[equal]{r}{}} & D\_1J(X) {\arrow{d}{}}{D\_1J(f)} {\arrow{r}{}}{(\phi\_1J)\_X} & E\_1FJ(\phi\_0J(X)) {\arrow{d}{}}{E\_1FJ(\phi\_0J(f))} {\arrow[equal]{r}{}} & \mu(E)(\mu\_0(J,X)) {\arrow{d}{}}{\mu(E)(\mu\_0({\mathrm{id}}\_J, f))} \\ \mu(D)(J,D\_0j(X)) {\arrow[equal]{r}{}} & D\_1J(X) {\arrow{r}{}}{(\phi\_1J)\_X} & E\_1FJ(\phi\_0J(X)) {\arrow[equal]{r}{}} & \mu(E)(\mu\_0(J,X)) \end{tikzcd}$$ Let now j : J → Jʹ be a morphism of J. We have to prove that the following diagram commutes. $$\begin{tikzcd}[column sep=huge] D\_1J(X) {\arrow{d}{}}{(D\_1j)\_X} {\arrow{r}{}}{(\phi\_1J)\_X} & E\_1FJ(\phi\_0J(X)) {\arrow{d}{}}{(E\_1Fj)\_{\phi\_0J(X)}} \\ D\_1J'(D\_0j(X)) {\arrow{r}{}}{(\phi\_1J')\_{D\_0j(X)}} & E\_1FJ'(\phi\_0J'(D\_0j(X))) \end{tikzcd}$$ This is however exactly, written out in components. Therefore we have a natural transformation, which we denote by μ1(F, ϕ), and we get a morphism of diagrams $$\begin{tikzcd}[row sep=small] \groth D\_0 {\arrow{dr}{""{name=D,below}}}{\mu(D)} {\arrow{dd}{}}[swap]{\mu\_0(F,\phi)} \\ & {{\mathsf{C}}} \\ \groth E\_0 {\arrow{ur}{}}[swap]{\mu(E)} {\arrow{\end}{Rightarrow,from=D, "\mu\_1", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ which makes μ functorial. (The identity and composition conditions follow by uniqueness.) The functor μ : Diag(Diag(C)) → Diag(C) is strictly natural in C. Let F : C → D be a functor. The induced functor $$\begin{tikzcd} {{\mathsf{Diag(Diag(C))}}} {\arrow{r}{}}{F\_{\\}} & {{\mathsf{Diag(Diag(D))}}} \end{tikzcd}$$ maps a lax cocone with tip C to the lax cocone with tip D obtained simply via postcomposition with F. $$\begin{tikzcd}[row sep=small] D\_0J {\arrow{dr}{""{name=D,below}}}{D\_1J} {\arrow{dd}{}}[swap]{D\_0j} \\ & {{\mathsf{C}}} \\ D\_0J' {\arrow{ur}{}}[swap]{D\_1J'} {\arrow{\end}{Rightarrow,from=D, "D\_1j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] D\_0J {\arrow{dr}{""{name=D,below}}}{D\_1J} {\arrow{dd}{}}[swap]{D\_0j} \\ & {{\mathsf{C}}} {\arrow{r}{}}{F} & {{\mathsf{D}}} \\ D\_0J' {\arrow{ur}{}}[swap]{D\_1J'} {\arrow{\end}{Rightarrow,from=D, "D\_1j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ In other words, F\* \(D0, D1) = (D0, F* ∘ D1). If we take the Grothendieck construction in both cases we get morphisms $\mu(D):\groth D\_0 \to {{\mathsf{C}}}$ and $\mu(F\_{\\}D):\groth D\_0\to {{\mathsf{D}}}$. By uniqueness (), necessarily μ(F\* \* D) = F ∘ μ(D), and therefore μ is strictly natural. Unitors and associators ----------------------- ### Left unitor Let D : J → C be a diagram. We can apply to it the unit ηDiag(C) : Diag(C) → Diag(Diag(C)) to form the diagram $$\begin{tikzcd} {{\mathsf{1}}} {\arrow{r}{}}{({{\mathsf{J}}}, D)} & {{\mathsf{Diag(C)}}} \end{tikzcd}$$ corresponding to the following (rather trivial) lax cocone in CAT with tip C. $$\begin{tikzcd} {{\mathsf{J}}} {\arrow{r}{}}{D} {\arrow{r}{}} & {{\mathsf{C}}} \end{tikzcd}$$ We view this as a lax cocone over the diagram J : 1 → CAT which maps the unique object of 1 to J. If we form the Grothendieck construction as prescribed by the multiplication of the monad, we get the category $\groth {{\mathsf{J}}}$ which is isomorphic to J, explicitly given as follows: * Objects are pairs ( • , X), where • is the unique object of 1 and X is an object of J; * A morphism ( • , X) → ( • , Y) is simply a morphism f : X → Y of J. The functor μ(η(D)) maps then ( • , X) to DX and f : ( • , X) → ( • , Y) to Df in C. The functor $\groth {{\mathsf{J}}}\to{{\mathsf{J}}}$ given by ( • , X) ↦ X is an isomorphism of categories. This defines an isomorphism of diagrams μC(ηDiag(C)(D)) → D, in the category Diag(C). Denote this isomorphism by ℓ. This is the map that we take as left unitor. The map ℓ induces a modification $\mu\circ(\eta\,{{\mathsf{Diag}}}) \Rrightarrow {\mathrm{id}}$. Let’s first show that ℓ is natural in the diagram D. If (R, ρ) : (J, D) → (K, E) is a morphism of diagrams, it’s easy to see that this diagram commutes strictly, $$\begin{tikzcd} \groth {{\mathsf{J}}} {\arrow{d}{}}{\tilde R} {\arrow{r}{}}{\ell} & {{\mathsf{J}}} {\arrow{d}{}}{R}\\ \groth {{\mathsf{K}}} {\arrow{r}{}}{\ell} & {{\mathsf{K}}} \end{tikzcd}$$ where $\tilde R$ is the functor mapping ( • , X) to ( • , RX), and acting similarly on morphisms. This commutative diagram induces an analogous commutative diagram in Diag(C), so that ℓ is a natural isomorphism of functors $$\begin{tikzcd}[column sep=small] {{\mathsf{Diag(C)}}} {\arrow{dr}{""{name=ID}}}[swap]{{\mathrm{id}}} {\arrow{r}{}}{\eta} & {{\mathsf{Diag(Diag(C))}}} {\arrow{{\arrow{}}{}}{Rightarrow, to=ID, "\ell", shorten >= 0.2em, shorten <= 0.2em}}{d}{\mu} \\ & {{\mathsf{Diag(C)}}} \end{tikzcd}$$ In order to show that ℓ is a modification, we have to show that ℓ is natural in the category C as well, in the sense that for each functor F : C → D the following composite 2-cells are equal, $$\begin{tikzcd}[column sep=tiny] & {{\mathsf{Diag(D)}}} {\arrow{r}{}}{\eta} & {{\mathsf{Diag(Diag(D))}}} {\arrow{d}{}}{\mu} \\ {{\mathsf{Diag(C)}}} {\arrow{r}{}}{\eta} {\arrow{ur}{}}{F\_\} {\arrow{dr}{""{name=ID2}}}[swap]{{\mathrm{id}}} & {{\mathsf{Diag(Diag(C))}}} {\arrow{ur}{}}{F\_{\\}} {\arrow{d}{}}{\mu} {\arrow{&}{Rightarrow, to=ID2, "\ell", shorten >= 0.2em, shorten <= 0.2em}} {{\mathsf{Diag(D)}}} \\ & {{\mathsf{Diag(C)}}} {\arrow{ur}{}}[swap]{F\_\} \end{tikzcd} \;=\quad\;\, \begin{tikzcd}[column sep=tiny] & {{\mathsf{Diag(D)}}} {\arrow{dr}{""{name=ID}}}[swap]{{\mathrm{id}}} {\arrow{r}{}}{\eta} & {{\mathsf{Diag(Diag(D))}}} {\arrow{{\arrow{}}{}}{Rightarrow, to=ID, "\ell", shorten >= 0.2em, shorten <= 0.2em}}{d}{\mu} \\ {{\mathsf{Diag(C)}}} {\arrow{ur}{}}{F\_\} {\arrow{dr}{""{name=ID2}}}[swap]{{\mathrm{id}}} & & {{\mathsf{Diag(D)}}} \\ & {{\mathsf{Diag(C)}}} {\arrow{ur}{}}[swap]{F\_\} \end{tikzcd}$$ where the squares without a 2-cell commute (by naturality). Explicitly, we have to check that given a diagram D : J → C, the following parallel functors coincide. $$\begin{tikzcd} \mu(\eta(F\_\D)) {\arrow{r}{shift left}}{F\_\\ell} {\arrow{r}{shift right}}[swap]{\ell} & F\_\* D \end{tikzcd}$$ Note however that F\* acts on the codomain of the diagram, by postcomposing with F, while ℓ acts on the domain of the diagram, mapping J to its isomorphic copy $\groth{{\mathsf{J}}}$. Therefore both arrows give the same morphism of diagrams, explicitly the following commutative diagram of CAT, $$\begin{tikzcd}[row sep=small] \groth {{\mathsf{J}}} {\arrow{dr}{""{name=D,below}}}{D} {\arrow{dd}{}}[swap]{\cong} \\ & {{\mathsf{C}}} {\arrow{r}{}}{F} & {{\mathsf{D}}} \\ {{\mathsf{J}}} {\arrow{ur}{}}[swap]{\tilde D} \end{tikzcd}$$ where $\tilde D(\bullet, X)=D(X)$, and the action on morphisms is similarly defined. Therefore ℓ is a modification. ### Right unitor Let D : J → C be a diagram. This time we apply to it the map η\* : Diag(C) → Diag(Diag(C)) given by the functor image of η under Diag. Explicitly, the result is the following lax cocone in CAT, with tip C, indexed by J via the constant functor Δ1 : J → CAT at 1. In pictures, $$\begin{tikzcd}[row sep=small] J {\arrow{dd}{}}[swap]{j} \\ \\ J' \end{tikzcd} \qquad\rightsquigarrow\qquad \begin{tikzcd}[row sep=small] {{\mathsf{1}}} {\arrow{dr}{""{name=D,below}}}{DJ} {\arrow{dd}{}} \\ & {{\mathsf{C}}} \\ {{\mathsf{1}}} {\arrow{ur}{}}[swap]{DJ'} {\arrow{\end}{Rightarrow,from=D, "j", shorten <= 0.5em, shorten >= 1em, swap, near start}}{tikzcd}$$ If we form the Grothendieck construction, this time we get the category $\groth {{\mathsf{\Delta 1}}}$ which is again isomorphic to J, explicitly given as follows: * Objects are pairs (X, • ), where X is an object of J and • is the unique object of 1; * A morphism (X, • ) → (Y, • ) is simply a morphism f : X → Y of J. The functor μ(η\(D)) maps (X, • ) to D*X and f : (X, • ) → (Y, • ) to Df in C. Analogously to the left unitor case, we have a functor $\groth \Delta{{\mathsf{1}}}\to{{\mathsf{J}}}$ given by (X, • ) ↦ X which induces an isomorphism of categories. This defines in turn an isomorphism of diagrams μ(η\(D)) → D, which we denote by r− 1 (and its inverse by r. See for the convention we are using). The map r* is the one that we take as right unitor. The map r induces a modification ${\mathrm{id}}\Rrightarrow\mu\circ\eta\_\* $. We omit the proof, since it is analogous to the case of ℓ. ### Associator In order to define the associator, we have to look at Diag(Diag(Diag(C))). So let D : J → Diag(Diag(C)) be a diagram which assigns to each object J of J a diagram of diagrams D1J : D0J → Diag(C), itself mapping the object K of D0J to the diagram (D1J)1 : (D1J)0K → C, which, in turn, maps an object L of (D1J)0K to the object (D1J)1(L) of C. We could depict the situation as follows. For brevity we omit the action on morphisms, which is similarly constructed. $$\begin{tikzcd}[row sep=0, column sep=tiny] {{{\mathsf{J}}}} && {{{\mathsf{Diag(Diag(C))}}}} \\ {J} & {\rightsquigarrow} & {\bigg\{ D\_0J } && {{{\mathsf{Diag(C)}}} \bigg\}} \\ && {K} & {\rightsquigarrow} & {\bigg\{ (D\_1J)\_0K} && {{{\mathsf{C}}} \bigg\}} \\ &&&& {L} & {\rightsquigarrow} & {((D\_1J)\_1K)(L)} \arrow["{D}", from=1-1, to=1-3] \arrow["{D\_1J}", from=2-3, to=2-5] \arrow["{(D\_1J)\_1K}", from=3-5, to=3-7] \end{tikzcd}$$ We can now take the Grothendieck construction at two different depths, which we can think of as “joining levels J and K” and “joining levels K and L”. The former way, which is μ(D) ∈ Diag(Diag(C)), gives the following diagram of diagrams (only two levels). $$\begin{tikzcd}[row sep=0, column sep=tiny] {\groth D\_0} && {{{\mathsf{Diag(C)}}}} \\ {(J\in{{\mathsf{J}}}, K\in D\_0J)} & {\rightsquigarrow} & {\bigg\{ (D\_1J)\_0K} && {{{\mathsf{C}}} \bigg\}} \\ && {L} & {\rightsquigarrow} & {((D\_1J)\_1K)(L)} \arrow["{\mu(D)}", from=1-1, to=1-3] \arrow["{(D\_1J)\_1K}", from=2-3, to=2-5] \end{tikzcd}$$ The latter way, which is μ\*D ∈ Diag(Diag(C)), gives instead the following diagram of diagrams, which is in general not isomorphic to the former. $$\begin{tikzcd}[row sep=0, column sep=tiny] {{{\mathsf{J}}}} && {{{\mathsf{Diag(C)}}}} \\ {J} & {\rightsquigarrow} & {\bigg\{ \groth (D\_1J)\_0} && {{{\mathsf{C}}} \bigg\}} \\ && {(K\in D\_0J, L\in (D\_1J)\_0K)} & {\rightsquigarrow} & {((D\_1J)\_1K)(L)} \arrow["{\mu\_\D}", from=1-1, to=1-3] \arrow["{\mu(D\_1J)}", from=2-3, to=2-5] \end{tikzcd}$$ If we apply once again the Grothendieck construction to the two, we do obtain isomorphic diagrams: $$\begin{tikzcd}[row sep=0, column sep=tiny] {\groth \mu(D)} && {{{\mathsf{C}}}} \\ {((J\in{{\mathsf{J}}}, K\in D\_0J), L\in (D\_1J)\_0K)} & {\rightsquigarrow} & {((D\_1J)\_1K)(L)} \arrow[from=1-1, to=1-3] \end{tikzcd}$$ and $$\begin{tikzcd}[row sep=0, column sep=tiny] {\groth \mu\_\D} && {{{\mathsf{C}}}} \\ {(J\in{{\mathsf{J}}}, (K\in D\_0J, L\in (D\_1J)\_0K))} & {\rightsquigarrow} & {((D\_1J)\_1K)(L)} \arrow[from=1-1, to=1-3] \end{tikzcd}$$ These are isomorphic as diagrams through the map $$\begin{tikzcd}[row sep=0, column sep=tiny] {\groth \mu\_\D} && {\groth \mu(D)} \\ {(J, (K, L))} & {\rightsquigarrow} & {((J, K), L)} \arrow[from=1-1, to=1-3] \end{tikzcd}$$ and so the following diagram commutes up to isomorphism. $$\begin{tikzcd}[sep=tiny] {{{\mathsf{Diag(Diag(Diag(C)))}}}} && {{{\mathsf{Diag(Diag(C))}}}} \\ & {\cong} \\ {{{\mathsf{Diag(Diag(C))}}}} && {{{\mathsf{Diag(C)}}}} \arrow["{\mu\_\}", from=1-1, to=1-3] \arrow["{\mu}"', from=1-1, to=3-1] \arrow["{\mu}", from=1-3, to=3-3] \arrow["{\mu}"', from=3-1, to=3-3] \end{tikzcd}$$ We call this isomorphism the associator, and denote it by a. Again, analogously as for the unitors, we have: The associators assemble to a modification $\mu\circ\mu\_\* \Rrightarrow \mu\circ\mu$. Higher coherence laws --------------------- The associator and unitors satisfy coherence conditions that are analogous to the ones of monoidal categories (see for the precise definition). We spell them out in detail for this case. ### Unit condition Instantiating the unit condition of in our case, we get the following statement, which reminds us of the unit condition of monoidal categories. * Consider a diagram of diagrams as follows. $$\begin{tikzcd}[row sep=0, column sep=tiny] {{{\mathsf{J}}}} && {{{\mathsf{Diag(C)}}}} \\ {J} & {\rightsquigarrow} & {\bigg\{ D\_0J} && {{{\mathsf{C}}} \bigg\}} \\ && {K} & {\rightsquigarrow} & {(D\_1J)(K)} \arrow["{D}", from=1-1, to=1-3] \arrow["{D\_1J}", from=2-3, to=2-5] \end{tikzcd}$$ Applying directly the Grothendieck construction we would have a diagram (J, K) ↦ (D1J)(K). However, we instead want to insert a “bullet” via the unitor, and this can be done in two ways. * We can either apply the (inverse of the) left unitor ℓ at depth K, and then take the Grothendieck construction, obtaining the following diagram. $$\begin{tikzcd}[row sep=0, column sep=tiny] {\groth (\mu\_\\eta\_\D)\_0} && {{{\mathsf{C}}}} \\ {(J,(\bullet, K))} & {\rightsquigarrow} & {(D\_1J)(K)} \arrow["{\mu(\mu\_\\eta\_\D)}", from=1-1, to=1-3] \end{tikzcd}$$ * Alternatively, we can apply the right unitor r at depth *J	arxiv_0000610
�0, z) = 9.15 I(x, 0, z) , S(x, 0, z) = N(x) − E(x, 0, z) − I(x, 0, z) − A(x, 0, z) . Finally, removed individuals are considered initially null everywhere in the network, R(x, 0, z) = 0.0, with all arcs assumed empty at the beginning of the simulation. To model the escalation of lockdown restrictions, starting from March 9, 2020, initial day of the northern Italy lockdown, the transmission rate βA is reduced by 15% (with a consequent update also of βI, 0 and therefore of βI) and coefficients kI = kA = 60, due to the public being increasingly aware of the epidemic risks. Furthermore, the percentage of commuting individuals is reduced by 60% in the whole network according to mobility data tracked through mobile phones and made available by Google. After the additional restrictions in place as of March 22, 2020, βA is ulteriorly reduced by 10% (again with a consequent rearrangement also of βI, 0 and βI) and coefficients k are increased to kI = kA = 70. [b!] [tab:convergencen1] l \| c c \| c c \| c c \| c c & & & & & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) 2 & 4.1539e-07 & & 1.3499e-08 & & 2.0021e-08 & & 1.2599e-11 & 4 & 6.1458e-08 & 2.7568 & 1.9974e-09 & 2.7567 & 1.7945e-09 & 3.4799 & 1.1296e-12 & 3.4794 6 & 2.5542e-09 & 7.8443 & 8.3016e-11 & 7.8443 & 8.4784e-11 & 7.5281 & 5.3366e-14 & 7.5282 8 & 1.2694e-10 & 10.4343 & 4.1258e-12 & 10.4343 & 4.2423e-12 & 10.4108 & 2.6703e-15 & 10.4108 10 & 6.7564e-12 & 13.1451 & 2.1959e-13 & 13.1451 & 6.8746e-14 & 18.4744 & 4.3271e-17 & 18.4744 [b!] [tab:convergencen3] l \| c c \| c c \| c c \| c c & & & & & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) 2 & 4.8912e-06 & & 1.0543e-06 & & 2.1399e-06 & & 4.8133e-08 & 4 & 3.8079e-07 & 3.6831 & 8.2083e-08 & 3.6830 & 7.5528e-08 & 4.8244 & 1.6996e-09 & 4.8237 6 & 1.7441e-08 & 7.6047 & 3.7596e-09 & 7.6047 & 4.2088e-09 & 7.1210 & 9.4709e-11 & 7.1211 8 & 9.1663e-10 & 10.2400 & 1.9759e-10 & 10.2400 & 2.4765e-10 & 9.8474 & 5.5728e-12 & 9.8474 10 & 4.3175e-11 & 13.6927 & 9.3069e-12 & 13.6927 & 1.4533e-11 & 12.7075 & 3.2703e-13 & 12.7075 0.49![Lodi (n_1)](convergence_II "fig:") [fig.convergencen1] 0.49![Bergamo (n_3)](convergence_AA "fig:") [fig.convergencen3] [fig.convergence] ### Convergence analysis As a first numerical test we analyze the numerical convergence of the stochastic Collocation method, discussed in details in Appendix [appendix:SCM]. We consider the above setting, but with a larger level of uncertainty, with μ1 = 10, to emphasize the convergence rates. The numerical results evaluated with an increased number of collocation points Mp are compared to a reference solution obtained using Mp = 50, in terms of expectation and variance of the state variables. The expected exponential convergence is shown in Tables [tab:convergencen1]–[tab:convergencen3] for chosen state variables, respectively for nodes n1 and n3 (taken as representative nodes), where L2 error norms and the related order of accuracy are presented. The result is highlighted by Fig. [fig.convergence], in which the spectral decay of the L∞ norm is observed in terms of both expected value and variance as the number of collocation points increases. 0.37![March 2, 2020](lombardia_results_t5 "fig:") [fig.lombardiaresultst5] 0.37![March 7, 2020](lombardia_results_t10 "fig:") [fig.lombardiaresultst10] 0.37![March 12, 2020](lombardia_results_t15 "fig:") [fig.lombardiaresultst15] 0.37![March 17, 2020](lombardia_results_t20 "fig:") [fig.lombardiaresultst20] 0.37![March 22, 2020](lombardia_results_t25 "fig:") [fig.lombardiaresultst25] 0.37![March 27, 2020](lombardia_results_t30 "fig:") [fig.lombardiaresultst30] [fig.lombardiaresults] Simulation of different scenarios of the spread of COVID-19 ----------------------------------------------------------- In the following, we consider several scenarios regarding the spread of the COVID-19 epidemic in the Lombardy region based on the data and parameters determined in the previous section. In particular, we will consider the baseline scenario, corresponding to the actual spread of the epidemic observed from the data, and two hypothetical scenarios corresponding to the absence of long-range mobility and the absence of restrictions. ### Baseline scenario Numerical results of the Lombardy network test, in its baseline configuration (presented in details at the beginning of Section [sect.LombardiaTest]), obtained using the sAP IMEX Runge-Kutta FV Collocation method with Mp = 6 points, are reported in Figs. [fig.lombardiaresults]–[fig.NetworkLombardiadata1]. In Fig. [fig.lombardiaresults] a first qualitative trend of the spatial and temporal spread of the epidemic is presented. Here it can be observed the heterogeneity of the diffusion of the virus, which has firstly mostly affected the city of Bergamo and only in a second moment prevailing in Milan. In Fig. [fig.NetworkLombardia] the expected evolution in time of the infected individuals, together with 95% confidence intervals, is shown for each node and for the whole Lombardy network, including exposed E, highly symptomatic subjects I and asymptomatic or weakly symptomatic people A. Each plot is also associated with the expected temporal evolution of the reproduction number R0(t), computed as described in Section [sect:R0]. One can see the capacity of the model to reproduce a very heterogeneous epidemic trend in the network analyzed, which is also reflected in the different ranges and patterns shown for the R0 of each province. It can also be observed the agreement between the evolution of the reproduction number and the epidemic spread. In particular, it is confirmed the decline of the daily number of infected as R0 reaches values below 1, as shown in the plots for Lodi, Bergamo and Cremona. On the other hand, the persistence of the virus in the complete network, and especially in Milan, is noticed until March 27, 2020 (last day of the simulation), where the reproduction number remains R0 > 1, confirming the consistency of the definition proposed for R0(t). 0.45![Lodi (n_1)](node1 "fig:") [fig.node1] 0.45![Milan (n_2)](node2 "fig:") [fig.node2] 0.45![Bergamo (n_3)](node3 "fig:") [fig.node3] 0.45![Brescia (n_4)](node4 "fig:") [fig.node4] 0.45![Cremona (n_5)](node5 "fig:") [fig.node5] 0.45![Lombardy (total network)](tot_network "fig:") [fig.totnetwork] [fig.NetworkLombardia] 0.45![Lodi (n_1)](node1_data "fig:") [fig.node1data] 0.45![Milan (n_2)](node2_data "fig:") [fig.node2data] 0.45![Bergamo (n_3)](node3_data "fig:") [fig.node3data] 0.45![Brescia (n_4)](node4_data "fig:") [fig.node4data] 0.45![Cremona (n_5)](node5_data "fig:") [fig.node5data] 0.45![Lombardy (total network)](tot_network_data "fig:") [fig.totnetworkdata] [fig.NetworkLombardiadata] 0.45![Lodi (n_1)](node1_data1 "fig:") [fig.node1data1] 0.45![Milan (n_2)](node2_data1 "fig:") [fig.node2data1] 0.45![Bergamo (n_3)](node3_data1 "fig:") [fig.node3data1] 0.45![Brescia (n_4)](node4_data1 "fig:") [fig.node4data1] 0.45![Cremona (n_5)](node5_data1 "fig:") [fig.node5data1] 0.45![Lombardy (total network)](tot_network_data1 "fig:") [fig.totnetworkdata1] [fig.NetworkLombardiadata1] As visible from Fig. [fig.NetworkLombardiadata], the lower bound of the confidence band of the cumulative amount in time of highly symptomatic individuals is comparable with the observed data of the Civil Protection Department of Italy. As expected, the mean value of the numerical result reports an higher amount of I, especially in Milan, the province most affected by the virus, due to the uncertainty of available data, which surely underestimate the real amount of infectious people, as discussed in Section [sect.LombardiaTest]. The comparison between the expected evolution in time of the cumulative amount of severe infectious with respect to the effective cumulative amount of total infectious people, including asymptomatic and mildly symptomatic individuals, is shown in Fig. [fig.NetworkLombardiadata1]. Here, it can be noticed how much of the spread of COVID-19 has actually been lost from the data of the first outbreak in Lombardy and the impact that the presence of asymptomatic or undetected subjects has had on the epidemic evolution. By the end of the simulation, indeed, E[A + RA]/E[I + A + R] = 0.92 in the network, indicating with RA the amount of recovered coming from the compartment A. This result is in line with WHO guidelines given during the first wave of COVID-19, stating that approximately 80% of the infected population was asymptomatic. It is here remarked, indeed, that the compartment A in the proposed model includes not only the asymptomatic people but also the mildly symptomatic, which would approximately be 12% of the total cases. In the results here presented, transmission coefficients αi, j(x), which define the behavior of commuters in the network, are imposed as deterministic and constant in time, meaning that the amount of individuals exiting from (and entering in) each city is the same in each instant of the simulation. Clearly, this assumption leads to a simplification of the description of the phenomenon of commuting, generally characterized by a peak in the early morning hours and a drastic drop in the night hours. What we represent is indeed the mean commuting trend during the day, which tends to assume a stationary solution in time. However, a more realistic behavior, characterized for example by periodic sinusoidal functions, leads to a slightly oscillatory trend in the reported curves of each compartment and also in the reproduction number R0. By selecting αi, j(x) values that define an average trend over the day, allows us to avoid this misleading representation, while maintaining the consistency of the results. ### No-mobility scenario To assess the effects of the mobility of commuters, a second scenario is investigated in which the whole population is not allowed to move out of the residence city. Numerical results of this test are reported in Fig. [fig.NetworkLombardianomobility] for two representative provinces of the network: Lodi and Milan. It can be observed the different evolution of the spread of the epidemic comparing trends presented in Fig. [fig.NetworkLombardianomobility] with the corresponding ones in Fig. [fig.NetworkLombardia] (for the first column of plots) and Fig. [fig.NetworkLombardiadata1] (for the second column of plots). As a consequence of the absence of commuting people in the network, the spread of COVID-19 in Milan results consistently damped. The province of Milan, in fact, is the one with the highest number of daily incomes of commuting workers and students, followed by Bergamo and Brescia, as shown in Table [tab:matrixnetworklombardia]. On the other hand, Lodi is the city with the highest amount of daily outcomes of commuters (30% of its total population). Thus, preventing the population from leaving the province shows a slight increase in local infections with respect to the baseline scenario. This result is not intended to suggest that a constraint on the mobility of people would be disadvantageous in fighting the spread of the epidemic. In fact, one must consider that the contagions shown in these Figures are due to a still present interaction of people at the provincial level, which has not been reduced in any way with respect to the previous scenario. Similar conclusions are drawn also in. These results are primarily intended to demonstrate how much the evolution of a pandemic can vary in response to changes in people’s mobility. 0.45![Lodi (n_1)](node1_nomobility "fig:") [fig.node1nomobility] 0.45![Lodi (n_1)](node1_data1_nomobility "fig:") [fig.node1data1nomobility] 0.45![Milan (n_2)](node2_nomobility "fig:") [fig.node2nomobility] 0.45![Milan (n_2)](node2_data1_nomobility "fig:") [fig.node2data1nomobility] [fig.NetworkLombardianomobility] 0.45![Milan (n_2)](node2_norestrictions "fig:") [fig.node2norestrictions] 0.45![Milan (n_2)](node2_data1_norestrictions "fig:") [fig.node2data1norestrictions] 0.45![Bergamo (n_3)](node3_norestrictions "fig:") [fig.node3norestrictions] 0.45![Bergamo (n_3)](node3_data1_norestrictions "fig:") [fig.node3data1norestrictions] 0.45![Lombardy (total network)](tot_network_norestrictions "fig:") [fig.totnetworknorestrictions] 0.45![Lombardy (total network)](tot_network_data1_norestrictions "fig:") [fig.totnetworkdata1norestrictions] [fig.NetworkLombardianorestrictions] ### No-restrictions scenario Finally, results in the event that no restrictions of any kind were applied by the government are presented in Fig. [fig.NetworkLombardianorestrictions] for the province of Milan, Bergamo and the complete Lombardy network. Comparing these results with the corresponding in Fig. [fig.NetworkLombardia] (for the first column of plots) and Fig. [fig.NetworkLombardiadata1] (for the second column of plots), it can be seen immediately the fundamental role that government restrictions have played in containing the spread of the disease (notice the different y-axis scale in the Figures). In fact, the cumulative number of infected people across the region in this scenario is almost 1.8 times higher than in the baseline scenario at the end of the simulation. This value indicates that the restrictions imposed during the first wave of COVID-19 in Italy, and the consequent increasing risk awareness in the population, have contributed to attenuate the spread of the virus by 43%, which is in agreement with the result presented in. Also the basic reproduction number R0 shows a totally different evolution in time, being far from reaching the value 1 at the end of the simulation. Indeed, on March 27, 2020 the virus still persists in its propagation, with mean value R0 = 1.50 when considering the whole network and, in particular, R0 = 1.90 in the province of Milan. Conclusions =========== In this paper, a stochastic transport model for the spread of an epidemic phenomenon described by a multi-population SEIAR compartmental dynamics on networks is presented. The starting point for the description of spatial motion and the interactions of the individuals has its roots in the kinetic theory of discrete velocity models. The model is structured on spatial networks, where nodes identify the locations of interest (cities in this case) and arcs represent the set of major mobility paths (roads and railways). Individuals are divided into commuters, who move on a suburban scale, and non-commuters, who act on an urban scale. In this way, we avoid unrealistic mass migration effects in which the entire population moves indiscriminately through the network. Thanks to the hyperbolicity of the resulting system, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic reaction-diffusion models proposed in literature, is removed. Nevertheless, we show that for small relaxation times and large characteristic speeds, in the diffusive regime, the proposed model recovers its parabolic nature. Since the derivation and the consequent definition of the basic reproduction number is well-established in the literature for ODE epidemic models, we resort on these results to introduce a derivation of R0 for spatial epidemic models in the case of no-inflow/outflow boundary conditions, validating the effectiveness of the resulting definition as an indicator of the viral growth rate. The model has been tested for the analysis of the emergence of COVID-19 in Italy, by simulating the propagation and evolution of the virus in the months of February and March 2020 in Lombardy. Indeed, it is in these early stages of the epidemic that uncertainty in data and transport dynamics played a key role. In order to study the effects of the uncertainties of the initial conditions and of the parameters involved in the model on the solution of the problem, a second-order stochastic asymptotic-preservative IMEX Runge-Kutta Finite Volume Collocation method has been used. It is shown that the proposed numerical scheme achieves spectral accuracy in the stochastic space by achieving an exponential convergence rate even in the case where uncertainty is present not only in the initial data but also in the nonlinear interaction terms. Numerical results of the baseline scenario were compared with observed data made available by the Italian Civil Protection Department, demonstrating that the proposed model is suitable to adapt to real settings and applications and showing its ability on capturing the heterogeneity underlying epidemics. Also alternative scenarios have been evaluated, considering both a total lockdown of extra-urban mobility and a total absence of restrictions at governmental level. The proposed methodology has the potential to be applied to epidemiological models structured in more compartments, such as those proposed in. In this context, it was decided to keep the compartmentalization as simple as possible given the lack of more specific observed data at the provincial level. As a matter of fact, for the single provinces during the first epidemic wave, the Italian Civil Protection Department has made available only the cumulative trend of the subjects tested positive to COVID-19. There is no distinction between individuals simply quarantined, hospitalized or even in intensive care. Similarly, no structured data are publicly available at the level of provinces for individuals who have died. Future developments foresee an extension of the model to include the age structure of the population, as this is an essential feature to correctly describe mobility flows (which mainly affect the younger part of the population) and consequently a more correct spread of viruses such as COVID-19. Another interesting aspect, especially for analysis related to recent developments of the COVID-19 pandemic, is certainly the extension of the model to take into account the effects of the viral load of individuals, by analyzing the effects of the so called super-spreaders, and the immunization of susceptible individuals through vaccination campaigns. Acknowledgements ================ This work was partially supported by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) PRIN 2017, project “Innovative numerical methods for evolutionary partial differential equations and applications”, code 2017KKJP4X. Derivation of the reproduction number ===================================== Recurring to the next-generation matrix (NGM) procedure, it is possible to compute the dominant eigenvalue of a positive linear operator, called next-generation operator, which permits to define the reproduction number when there are several compartments contributing to the spread of the infection. The generalization to space dependent models is non straightforward due to the nonlinear nature of the interactions. In the following we consider the situation where no-inflow/outflow boundary conditions are assumed. Given the spatial model defined in, let us consider the following subsystem of balance equations of the infectious compartments including both commuters and non-commuters $$\begin{aligned} \frac{\partial E\_T}{\partial t} + \frac{\partial J\_E}{\partial x} &= f\_I(S\_T, I\_T) +f\_A(S\_T,A\_T) -aE\_T \\ \frac{\partial I\_T}{\partial t} + \frac{\partial J\_I}{\partial x} &= a \sigma E\_T -\gamma\_I I\_T\\ \frac{\partial A\_T}{\partial t} + \frac{\partial J\_A}{\partial x} &= a(1- \sigma) E\_T -\gamma\_A A\_T\,. \end{aligned} \label{eq.EIA}$$ Considering, a deterministic framework, in which the epidemic parameters only depend on the variable x, thanks to our assumptions, integration over Ω leads to $$\begin{aligned} \frac{\partial}{\partial t}\int\_\Omega E\_T(x,t) dx &= \int\_\Omega f\_I(S\_T, I\_T) dx + \int\_\Omega f\_A(S\_T,A\_T) dx - \int\_\Omega a(x) E\_T(x,t) dx \\ \frac{\partial}{\partial t} \int\_\Omega I\_T(x,t) dx &= \int\_\Omega a(x) \sigma (x) E\_T(x,t) dx - \int\_\Omega \gamma\_I(x) I\_T(x,t) dx\\ \frac{\partial}{\partial t} \int\_\Omega A\_T(x,t) dx &= \int\_\Omega a(x) (1- \sigma (x)) E\_T(x,t) dx -\int\_\Omega \gamma\_A(x) A\_T(x,t) dx\,. \end{aligned} \label{eq.EIA\_int}$$ If we define the total densities $$\bar E = \int\_\Omega E\_T(x,t) dx \,, \qquad \bar I = \int\_\Omega I\_T(x,t) dx \,, \qquad \bar A = \int\_\Omega A\_T(x,t) dx$$ and the averaged epidemic operators, $$\bar f\_I(S\_T,I\_T) = \frac{\int\_\Omega f\_I(S\_T, I\_T) dx}{\int\_\Omega I\_T(x,t) dx} \,, \qquad \bar f\_A(S\_T,A\_T) = \frac{\int\_\Omega f\_A(S\_T, A\_T) dx}{\int\_\Omega A\_T(x,t) dx} \,,$$ $$\bar \gamma\_I = \frac{\int\_\Omega \gamma\_I(x) I\_T(x,t) dx}{\int\_\Omega I\_T(x,t) dx}\,, \qquad \bar \gamma\_A = \frac{\int\_\Omega \gamma\_A(x) A\_T(x,t) dx}{\int\_\Omega A\_T(x,t) dx}\,,$$ $$\bar a = \frac{\int\_\Omega a(x) E\_T(x,t) dx}{\int\_\Omega E\_T(x,t) dx}\,, \quad \hat a = \frac{\int\_\Omega a(x) \sigma(x) E\_T(x,t) dx}{\int\_\Omega E\_T(x,t) dx}\,, \quad \tilde a = \frac{\int\_\Omega a(x)(1- \sigma(x)) E\_T(x,t) dx}{\int\_\Omega E\_T(x,t) dx}\,,$$ system can be rewritten in the following ODE form $$\begin{aligned} \dot{\bar E} &= \bar f\_I(S\_T,I\_T) \bar I + \bar f\_A(S\_T,A\_T) \bar A - \bar a \bar E\\ \dot{\bar I} &= \hat a \bar E - \bar \gamma\_I \bar I\\ \dot{\bar A} &= \tilde a \bar E - \bar \gamma\_A \bar A \,. \end{aligned} \label{eq.EIA\_ode}$$ Following the NGM approach, the Jacobian matrix of the above obtained ODE system results $$\bf{J\_0} = \begin{pmatrix} \bar a &\bar f\_I(S\_T,I\_T) &\bar f\_A(S\_T,A\_T) \\ \hat a &-\bar \gamma\_I &0 \\ \tilde a &0 & -\bar \gamma\_A \end{pmatrix},$$ which can be decomposed into the following transmission matrix $\bf{T}$ and transition matrix $\bf{\Sigma}$, so that $\bf{J\_0} = \bf{T} + \bf{\Sigma}$ $$\bf{T} = \begin{pmatrix} 0 &\bar f\_I(S\_T,I\_T) &\bar f\_A(S\_T,A\_T) \\ 0 &0 &0 \\ 0 &0 &0 \end{pmatrix}, \qquad \bf{\Sigma} = \begin{pmatrix} \bar a &0 &0 \\ \hat a &-\bar \gamma\_I &0 \\ \tilde a &0 & -\bar \gamma\_A \end{pmatrix}.$$ The basic reproduction number R0 is the spectral radius of the next-generation operator, $\bf{K\_L} = -\bf T \bf \Sigma^{-1}$, and results composed by the sum of two components, deriving from the two compartments of the model contributing to the spread of the epidemic (I and A) $$R\_0 = \rho \left( \bf K\_L \right) = R\_0^I + R\_0^A \label{eq.R0\_1}$$ with $$R\_0^I = \frac{\bar f\_I(S\_T,I\_T) \, \hat a}{\bar\gamma\_I \, \bar a}, \quad R\_0^A= \frac{\bar f\_A(S\_T,A\_T) \, \tilde a}{\bar\gamma\_A \, \bar a} \,. \label{eq.R0\_IA}$$ and therefore, recalling the previous definitions $$\begin{aligned} R\_0 (t) &= \frac{\int\_\Omega f\_I(S\_T,I\_T) \,dx}{\int\_\Omega \gamma\_I(x) I\_T(x,t) \,dx} \cdot \frac{\int\_\Omega a(x)\sigma (x) E\_T(x,t) \,dx}{\int\_\Omega a(x) E\_T(x,t) \,dx} \\&+ \frac{\int\_\Omega f\_A(S\_T,A\_T) \,dx}{\int\_\Omega \gamma\_A(x) A\_T(x,t)\, dx} \cdot \frac{\int\_\Omega a(x)(1-\sigma(x)) E\_T(x,t) \,dx}{\int\_\Omega a(x) E\_T(x,t)\, dx} \,. \end{aligned} \label{eq:Rt}$$ Note that, the dependence of additional stochastic parameters will not modify the above reasoning leading to the same definition including dependence from the uncertainty variable $\boldsymbol{z} = (z\_1,\ldots,z\_{d})^T \in \mathbb{R}^{d}$. Stochastic Collocation Method ============================= Following a stochastic collocation approach, the solution of the stochastic problem can be computed employing a generalized Polynomial Chaos (gPC) expansion. Let us consider, for simplicity, a single source of uncertainty $z\in\RR$ and that the probability density function (PDF) of this random input, ρz : Γ → R+, is known. The approximated solution of the problem, ${\bf{Q}}^h(x,t,z)=({\bf{U}}\_c^h, {\bf{V}}^h,{\bf{U}}\_0^h)$, where ${\bf{U}}\_0^h$ is the state vector of non-commuters, referring to system, can be expressed as a finite series of orthonormal polynomials in terms of the stochastic parameter $$\label{eq:expansion} {\bf{Q}}^h(x,t,z)=\sum\_{j=1}^{M} \hat{\bf{Q}}\_j(x,t) \phi\_j(z),$$ where M is the number of terms of the truncated series and ϕj(z) are orthonormal polynomials, with respect to the measure $\rho\_z(z)\, {{\rm d}}z$. The expansion coefficients $\hat{\bf{Q}}\_j(x,t)$ is obtained as $$\label{eq:exp\_coeff\_int} \hat{\bf{Q}}\_j(x,t) = \int\_{\Gamma} {\bf{Q}} (x,t,z)\, \phi\_j(z)\, \rho\_z(z)\, {{\rm d}}z, \qquad j=1,\ldots,M.$$ Following the stochastic collocation method, the integrals for the expansion coefficients in Eq. are replaced by a suitable quadrature UMp characterized by the set {zn, wn}n = 1Mp, where zn is the n-th collocation point, wn is the corresponding weight and Mp represents the number of quadrature points. For instance, for a stochastic parameter with a uniform distribution the quadrature is defined by the Gauss-Legendre weights and nodes; while for a random variable associated to a Gaussian PDF, we will rely on a Gauss-Hermite quadrature, which reads $$\label{eq:exp\_coeff\_quad} \hat{\bf{Q}}\_j(x,t) \approx \mathcal{U}^{M\_p}\left[{\bf{Q}}^d(x,t,z)\, \phi\_j(z)\right] = \sum\_{n=1}^{M\_p} {\bf{Q}}^d(x,t,z\_n)\, \phi\_j(z\_n)\, w\_n, \qquad j=1,\ldots,M$$ where ${\bf{Q}}^d(x,t,z\_n)$, with n = 1, …, Mp, is the deterministic solution of the problem evaluated at the n-th collocation point. After the computation of the expansion coefficients, substituting eq. in eq. , an approximated solution is available. In particular, it is possible to continue with the post-processing step, evaluating the expectation of the variables of interest $$\label{eq:mean\_an} \mathbb{E}\left[{\bf{Q}}\right] =\int\_{\Gamma} {\bf{Q}}(x,t,z)\, \rho\_z(z)\, {{\rm d}}z,$$ which are approximated as $$\label{eq:mean\_apx} \mathbb{E}\left[{\bf{Q}}\right] \approx \mathbb{E}\left[{\bf{Q}}^h\right] =\int\_{\Gamma} {\bf{Q}}(x,t,z)\, \rho\_z(z)\, {{\rm d}}z \approx \sum\_{n=1}^{M\_p} {\bf{Q}}^d(x,t,z\_n)\, w\_n.$$ Similarly, other statistical quantities of interest can be computed, like the variance $$\label{eq:variance\_apx} \mathbb{V}\left[{\bf{Q}}\right] = \mathbb{E}\left[ \left( {\bf{Q}} - \mathbb{E}\left[{\bf{Q}}\right] \right)^2 \right] \approx \mathbb{E}\left[\left({\bf{Q}}^h \right)^2\right] - \mathbb{E}\left[{\bf{Q}}^h \right]^2.$$ When more than one stochastic parameter is considered in the problem, $\boldsymbol{z}\in \RR^{d}$, the joint PDF $\rho\_z\left(\boldsymbol{z}\right)$ of the random vector composed by the random parameters (assuming independence of the variables) is given by $$\rho\_z\left(\boldsymbol{z}\right) = \prod\_{k=1}^{d} \rho\_{z,k}\left(z\_k\right),$$ where ρz, k is the PDF of the k-th variable. The extension of the collocation method then follows in a similar way. AP-IMEX Runge-Kutta Finite Volume scheme ======================================== To compute the solution at each collocation point, an IMEX Runge-Kutta Finite Volume method for hyperbolic systems with multiscale relaxation is adopted. The IMEX discretization for the commuters’ dynamics, coupled with the equations for the non commuter population through identities, results with internal Runge-Kutta stages $$\begin{split} &{{\bf{U}}}\_c^{(k)} = {{\bf{U}}}\_c^n - \Delta t \sum\_{j=1}^{k} a\_{kj} \partial\_x {{\bf{V}}}^{(j)} + \Delta t \sum\_{j=1}^{k-1} \tilde{a}\_{kj} {{\bf{F}}}\_c\left({{\bf{U}}}\_T^{(j)},{{\bf{U}}}\_c^{(j)}\right) \\ &{{\bf{V}}}^{(k)} = {{\bf{V}}}^n - \Delta t \sum\_{j=1}^{k-1} \tilde{a}\_{kj} \left( \boldsymbol{\Lambda}^2 \partial\_x {{\bf{U}}}\_c^{(j)} - {{\bf{G}}}\left({{\bf{U}}}\_T^{(j)},{{\bf{V}}}^{(j)}\right)\right) + \Delta t \sum\_{j=1}^{k} a\_{kj} {{\bf{H}}}\left({{\bf{V}}}^{(j)}\right) \\ &{{\bf{U}}}\_0^{(k)} = {{\bf{U}}}\_0^n + \Delta t \sum\_{j=1}^{k-1} \tilde{a}\_{kj} {{\bf{F}}}\_0\left({{\bf{U}}}\_T^{(j)},{{\bf{U}}}\_0^{(j)}\right), \end{split} \label{eq.iterIMEX}$$ and final numerical solution $$\begin{split} & {{\bf{U}}}\_c^{n+1} = {{\bf{U}}}\_c^n - \Delta t \sum\_{k=1}^{s} b\_{k} \partial\_x {{\bf{V}}}^{(k)} + \Delta t \sum\_{k=1}^{s} \tilde{b}\_{k} {{\bf{F}}}\_c\left({{\bf{U}}}\_T^{(k)},{{\bf{U}}}\_c^{(k)}\right) \\ & {{\bf{V}}}^{n+1} = {{\bf{V}}}^n - \Delta t \sum\_{k=1}^{s} \tilde{b}\_{k} \left( \boldsymbol{\Lambda}^2 \partial\_x {{\bf{U}}}\_c^{(k)} - {{\bf{G}}}\left({{\bf{U}}}\_T^{(k)},{{\bf{V}}}^{(k)}\right) \right) + \Delta t \sum\_{k=1}^{s} b\_{k} {{\bf{H}}}\left({{\bf{V}}}^{(k)}\right) \\ & {{\bf{U}}}\_0^{n+1} = {{\bf{U}}}\_0^n + \Delta t \sum\_{k=1}^{s} \tilde{b}\_{k} {{\bf{F}}}\_0\left({{\bf{U}}}\_T^{(k)},{{\bf{U}}}\_0^{(k)}\right). \end{split} \label{eq.finalIMEX}$$ Here Δt = tn + 1 − tn is the time step size that follows the less restrictive between the standard hyperbolic Courant-Friedrichs-Levy condition, $\Delta t \leq {\mathsf{CFL}}\frac{\Delta x}{max\left\{\lambda\_i\right\}}$, and the parabolic stability restriction, $\Delta t \leq \nu \frac{\Delta x^2}{max\left\{D\_i\right\}}$, given by the diffusive components of the system, with CFL and ν suitable stability constants and $\Delta x = x\_{i+\frac{1}{2}} - x\_{i-\frac{1}{2}}$ space grid size. In this work, we fix CFL = 0.9 and $\nu = \frac{max\left\{D\_i\right\}}{2}$. The numerical scheme is characterized by two s × s matrices, $\tilde A = (\tilde a\_{kj})$, with $\tilde a\_{kj} = 0$ for j ≥ k, and A = (akj), with akj = 0 for j > k, and by the weights vectors $\tilde b = (\tilde b\_1,...,\tilde b\_s)^T$, b = (b1, ..., bs)T (with s identifying the number of the Runge-Kutta stages), which can be represented in the following Butcher notation c \| c $\tilde c$ & $\tilde A$ & $\tilde b^T$ c \| c c & A & bT where $\tilde c$ and c are the time coefficient vectors $$\tilde c\_k = \sum^{k-1}\_{j=1} \tilde a\_{kj}, \qquad \qquad c\_k = \sum^{k}\_{j=1} a\_{kj}.$$ The distribution of matrices $\tilde A$ and A permits to treat implicitly the stiff terms (hence those depending on the scaling parameters, τi and λi) and explicitly all the rest. Moreover, if a proper set of matrices $\tilde A, A$ and vectors $\tilde b, b$ are chosen, the AP property is satisfied, which means that the scheme maintains a consistent discretization of the asymptotic system in the diffusive (parabolic) regime (see ). For example, the second order GSA BPR(4,4,2) scheme proposed in satisfies the AP property and is defined by the following double Butcher tableau (explicit on the left and implicit on the right) $$\begin{tabular}{c \| c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 1/4 & 1/4 & 0 & 0 & 0 & 0 \\ 1/4 & 13/4 & -3 & 0 & 0 & 0 \\ 3/4 & 1/4 & 0 & 1/2 & 0 & 0 \\ 1 & 0 & 1/3 & 1/6 & 1/2 & 0 \\ \hline & 0 & 1/3 & 1/6 & 1/2 & 0 \\ \end{tabular} \hspace{1.0cm} \begin{tabular}{c \| c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 1/4 & 0 & 1/4 & 0 & 0 & 0 \\ 1/4 & 0 & 0 & 1/4 & 0 & 0 \\ 3/4 & 0 & 1/24 & 11/24 & 1/4 & 0 \\ 1 & 0 & 11/24 & 1/6 & 1/8 & 1/4 \\ \hline & 0 & 11/24 & 1/6 & 1/8 & 1/4 \\ \end{tabular} \label{eq:tables}$$ At each internal stage of the IMEX Runge-Kutta scheme, we apply a TVD Finite Volume discretization. To achieve second order accuracy also in space, while avoiding the occurrence of spurious oscillations, a classical minmod slope limiter has been adopted. Finally, let us remark that, given the non-intrusive nature of the stochastic collocation method which only requires the evaluation of the solutions of the corresponding deterministic problem at each collocation point, the AP property of the deterministic IMEX scheme is preserved, leading to a stochastic asymptotic-preserving scheme. --- 1. Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy ([email protected])[↩](#fnref1) 2. Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy ([email protected])[↩](#fnref2) Hyperbolic compartmental models for epidemic spread on networks with uncertain data: application to the emergence of Covid-19 in Italy ====================================================================================================================================== The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources, such as the amount of infectious individuals. In this paper we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest, such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interactions of a population partitioned, from an epidemiological point of view, on the basis of an extended compartmental structure and divided into commuters, moving on a suburban scale, and non-commuters, acting on an urban scale. Through a diffusive rescaling, the model allows us to recover classical diffusion equations related to commuting dynamics. The numerical solution of the resulting multiscale hyperbolic system with uncertainty is then tackled using a stochastic collocation approach in combination with a finite-volume IMEX method. The ability of the model to correctly describe the spatial heterogeneity underlying the spread of an epidemic in a realistic city network is confirmed with a study of the outbreak of COVID-19 in Italy and its spread in the Lombardy Region. Keywords: Kinetic transport equations, hyperbolic systems, epidemic models, network modelling, diffusion limit, uncertainty quantification, IMEX finite-volume methods, asymptotic-preserving scheme, COVID-19 Introduction ============ The advent of the COVID-19 pandemic, which still afflicts the entire world, has prompted many researchers in the mathematical field and beyond to propose epidemic models increasingly suitable for the study of the evolution of this specific coronavirus, to support the definition of the best strategies for the control of its spread. Most of the proposed models are rooted in the compartmental epidemiological modeling proposed by Kermack and McKendrick, based on systems of ordinary differential equations (ODE), which describe only the temporal evolution of the spread of the epidemic, neglecting the spatial component in favor of an assumption of homogeneity of population and territory. Recently, models have also been proposed in which we have moved away from the idea of a homogeneously mixed population to allow a better description in terms of social contact patterns, vulnerability and case fatality ratio, but still neglecting the spatial development of the epidemic spread. Generally, the concept of the average behavior of a population is sufficient to have a first reliable description of the development of an epidemic. However, the inclusion of the spatial component in epidemiological systems is crucial especially when there is a need to consider spatially heterogeneous interventions, as was and still is the case for the control of the spread of COVID-19. Some recent works have attempted to fill these gaps by proposing epidemiological models based on partial differential equations (PDE), in which the spatial structure is taken into account. In a two-dimensional (2D) space dependence is considered, obtaining models able to properly capture the complexity of the spatial transmission of the virus. On the other hand, meta-population network approaches have also been proposed, which result in systems of ODE that provide no information about the spatial characteristics of the epidemic spread along the pathways connecting the different populations. Another important aspect concerns the wide use of deterministic models, which, although more computationally efficient, are based on the assumption that initial conditions, boundary conditions and all the parameters involved are known. However, in practical applications, this assumption is rarely true. In the context of epidemic modeling, for example, initial conditions are certainly affected by uncertainty because data are limited and screening policy is always a matter of compromises. Also epidemic parameters, although normally estimated or calibrated, are candidates for being random variables. Therefore, when attempting to solve the problem of interest numerically, one must take into account these limitations, recurring to more consistent stochastic models. In this work, a hyperbolic transport model on networks with uncertain data capable to describe the spatial spread of an epidemic phenomenon defined by an extended compartmental dynamics is proposed. The model takes inspiration by the multiscale transport model recently introduced in by extending it to include more realistic compartmentalizations. In addition, the spatial structure of the system lays on the network modeling originally adopted in, in which nodes of the network identify locations of interests, depending on the spatial scale considered (villages, provinces, regions, nations), and arcs constitute the mobility paths connecting them (hence roads, railways, airline connections, etc.). Individuals moving on the network are subdivided into commuting and non-commuting, the latter acting only at the urban, nodal scale they belong to, not contributing in the transmission of the virus in the extra-urban domain. This particular feature prevents unrealistic mass migration effects where the entire population travels through the network. Furthermore, thanks to its hyperbolic structure the model avoids the unphysical feature of infinite propagation speeds typical of reaction-diffusion systems, still recovering the parabolic behaviour in the diffusive limit. The resulting model is solved numerically through an IMEX finite volume stochastic scheme that permits to preserve uniform accuracy with respect to the scaling parameters. More precisely, the numerical scheme combines a stochastic collocation method, a non-intrusive method which guarantees spectral convergence in the stochastic space and ease of implementation, avoiding the loss of important structural properties of the original system of governing equations, with an Implicit-Explicit (IMEX) Runge-Kutta finite-volume scheme that maintains the consistency in the asymptotic limit (i.e. the asymptotic-preserving (AP) property). The rest of the paper is organized as follows. In Section [sect:mathmodel] the mathematical model is introduced, first in its kinetic transport formulation and then deriving its macroscopic representation. Subsequently, the diffusion limit is formally computed and the details of the model extension to networks are presented. Moreover, a definition of the basic reproduction number of the epidemic is proposed, with the details of its derivation discussed in Appendix [appendix:R0]. In Section [sect:numres] we present an application to the first outbreak of COVID-19 in Italy and its spread in the Lombardy Region. First, to validate the proposed numerical approach a convergence study is performed. Next, the capacity of the proposed model to capture the spatial heterogeneous characteristics underlying COVID-19 epidemics is assessed through several simulations based on data reported by official sources. The details of the stochastic collocation approach are summarized in Appendix [appendix:SCM], whereas the IMEX finite-volume scheme is described in Appendix [appendix:IMEX]. Finally, conclusions and future perspectives are drawn in Section [sect:conclusion]. Mathematical model ================== A hyperbolic compartmental model for epidemic spread ---------------------------------------------------- The epidemiological starting point of our model is given by a compartmental structure with a SEIAR partitioning, in which the population is assumed to be divided in susceptible individuals S, exposed individuals in the latent period, which are not yet infectious E, infectious individuals manifesting severe symptoms I, infectious individuals with no/mild symptoms A and removed (deceased or healed and immune) individuals R. We assume to have a population with subjects having no prior immunity. The vital dynamics represented by births and deaths is neglected because of the time scale considered. This model differs from the classic SEIR for the presence of a subgroup of people who will never develop symptoms or very mild ones, which is an essential feature to take into account if one aims at analyzing the evolution of the COVID-19 pandemic. Indeed, individuals belonging to this group are very hard to be detected, deeply impacting the efficiency of the monitoring and making isolation, containment and tracing of individual cases very challenging. They tend to behave like simple susceptible, but in fact highly contributing in augmenting the spread of the virus. To account for the spatial movement of the population, the model has its roots within discrete velocity models in kinetic theory, and follows the approach in, where individuals of each compartment are subdivided in three classes, S± , 0, E± , 0, I± , 0, A± , 0, R± , 0, traveling in a one-dimensional space domain $\Omega\subseteq \RR$ with characteristic speeds + λi, − λi and 0 respectively, with i ∈ {S, E, I, A, R}. The total compartmental densities are defined as the sum of all the components of the subgroups $$\begin{split} S\_T=S\_++S\_-+S\_0,\qquad E\_T=E\_+&+E\_-+E\_0,\qquad I\_T=I\_++I\_-+I\_0,\\ A\_T=A\_++A\_-+A\_0,&\qquad R\_T=R\_++R\_-+R\_0\,. \label{eq:seiart} \end{split}$$ The discrete-velocity system of the SEIAR epidemic transport model for commuters, associated to relaxation times τi, i ∈ {S, E, I, A, R}, then reads $$\begin{aligned} \frac{\partial S\_{\pm}}{\partial t} \pm \lambda\_S \frac{\partial S\_{\pm}}{\partial x} &= -f\_I(S\_{\pm}, I\_T) -f\_A(S\_{\pm}, A\_T) + \frac{1}{2\tau\_S}\left(S\_\mp - S\_\pm\right)\\ \frac{\partial E\_{\pm}}{\partial t} \pm \lambda\_E \frac{\partial E\_{\pm}}{\partial x} &= f\_I(S\_{\pm}, I\_T) +f\_A(S\_{\pm},A\_T) -a E\_{\pm} + \frac{1}{2\tau\_E}\left(E\_\mp - E\_\pm\right)\\ \frac{\partial I\_{\pm}}{\partial t} \pm \lambda\_I \frac{\partial I\_{\pm}}{\partial x} &= a \sigma E\_{\pm} -\gamma\_I I\_{\pm} + \frac1{2\tau\_I}\left(I\_\mp - I\_\pm\right)\\ \frac{\partial A\_{\pm}}{\partial t} \pm \lambda\_A \frac{\partial A\_{\pm}}{\partial x} &= a(1-\sigma) E\_{\pm} -\gamma\_A A\_{\pm} + \frac{1}{2\tau\_A}\left(A\_\mp - A\_\pm\right)\\ \frac{\partial R\_{\pm}}{\partial t} \pm \lambda\_R \frac{\partial R\_{\pm}}{\partial x} &= \gamma\_I I\_{\pm} + \gamma\_A A\_{\pm} + \frac1{2\tau\_R}\left(R\_\mp - R\_\pm\right)\,. \end{aligned} \label{eq.SEIARkinetic}$$ This system is coupled with the following SEIAR model describing the evolution of a stationary population of non-commuters $$\begin{aligned} \frac{{{\rm d}}S\_{0}}{{{\rm d}}t} &= -f\_I(S\_{0}, I\_T) +f\_A(S\_{0}, A\_T) \\ \frac{{{\rm d}}E\_{0}}{{{\rm d}}t} &= f\_I(S\_{0}, I\_T) +f\_A(S\_{0}, A\_T) -a E\_0 \\ \frac{{{\rm d}}I\_{0}}{{{\rm d}}t} &= a \sigma E\_0 -\gamma\_I I\_{0}\\ \frac{{{\rm d}}A\_{0}}{{{\rm d}}t} &= a(1-\sigma) E\_0 - \gamma\_A A\_0 \\ \frac{{{\rm d}}R\_{0}}{{{\rm d}}t} &= \gamma\_I I\_0 + \gamma\_A A\_0\,. \end{aligned} \label{eq.SEIARkinetic\_noncommuters}$$ [fig.SEIAR] In the above system - all the epidemic densities S± , 0, E± , 0, I± , 0, A± , 0, R± , 0 depend on $(x,t,\boldsymbol{z})$, where (x, t) are the physical variables of space x ∈ Ω ⊆ R and time t > 0, while $\boldsymbol{z} = (z\_1,\ldots,z\_{d})^T \in \mathbb{R}^{d}$ is a random vector characterizing the possible sources of uncertainty due to independent parameters z1, …, zd. The same applies for the contact rate function f, defined with respect to the infectious compartments, I and A respectively, as $$f\_I(S,I)=\beta\_I \frac{S I^p}{1+k\_I I^p}, \qquad f\_A(S,A)=\beta\_A \frac{S A^p}{1+k\_A A^p},\qquad p\geq 1,$$ where $\beta\_I(x,t,\boldsymbol{z})$ and $\beta\_A(x,t,\boldsymbol{z})$ are the transmission rates, accounting for both number of contacts and probability of transmission, hence may vary based on the effects of government control actions, such as mandatory wearing of masks, shutdown of specific work/school activities, or full lockdowns ; $k\_I(x,t,\boldsymbol{z})$ and $k\_A(x,t,\boldsymbol{z})$ act as incidence damping coefficients based on the self-protective behavior of the individual that arises from awareness of the risk associated with the epidemic. Note that, the classic bilinear case corresponds to p = 1 and kI = kA = 0, even though it has been observed that an incidence rate that increases more than linearly with respect to the number of infectious can occur under certain circumstances. Parameters $\gamma\_I(x,t,\boldsymbol{z})$ and $\gamma\_A(x,t,\boldsymbol{z})$ are the recovery rates of highly symptomatic infected and of asymptomatic or mildly infected (inverse of the infectious periods), respectively, while $a(x,t,\boldsymbol{z})$ represents the inverse of the latency period and $\sigma(x,t,\boldsymbol{z})$ is the probability rate of developing severe symptoms. The flow chart of the SEIAR model considered in this work is also illustrated in Fig. [fig.SEIAR], where transition rates between compartments are clearly displayed. * The model allows to describe more realistically the typical dynamic of commuters, which regards only a small fraction of individuals, and to distinguish this from the epidemic process which, instead, involves the whole population, including non-commuters. The presence of a group of non-commuting population, indeed, permits to avoid that the whole population in a compartment moves indiscriminately in the full space originating an unrealistic mass migration effect. * The presence of uncertainty in the data included from the beginning in the modeling process could allow the compartmentalization of asymptomatic individuals to be eliminated by implicitly including them in the uncertainty about the number of infected individuals. This approach was proposed in. In our case, however, in order to highlight the link with similar models used in the literature, we chose to keep the asymptomatic compartment, which consequently will be the one affected by the highest level of uncertainty. Macroscopic formulation and diffusion limit ------------------------------------------- Introducing now the macroscopic variables Sc, Ec, Ic, Ac, Rc for the commuters, with Sc = S+ + S−, Ec = E+ + E−, Ic = I+ + I−, Ac = A+ + A−, Rc = R+ + R−, and defining the fluxes JS = λS(S+ − S−), JE = λE(E+ − E−), JI = λI(I+ − I−), JA = λA(A+ − A−), JR = λR(R+ − R−), a hyperbolic model underlying the macroscopic formulation of the spatial propagation of an epidemic, equivalent to the mesoscopic one, presented in system, is obtained: $$\begin{aligned} \frac{\partial S\_{c}}{\partial t} + \frac{\partial J\_S}{\partial x} &= -f\_I(S\_c, I\_T) -f\_A(S\_c, A\_T) \\ \frac{\partial E\_{c}}{\partial t} + \frac{\partial J\_E}{\partial x} &= f\_I(S\_c, I\_T) + f\_A(S\_c, A\_T) -a E\_{c} \\ \frac{\partial I\_{c}}{\partial t} + \frac{\partial J\_I}{\partial x} &= a \sigma E\_{c} - \gamma\_I I\_{c}\\ \frac{\partial A\_{c}}{\partial t} + \frac{\partial J\_A}{\partial x} &= a(1- \sigma) E\_{c} - \gamma\_A A\_{c}\\ \frac{\partial R\_{c}}{\partial t} + \frac{\partial J\_R}{\partial x} &= \gamma\_I I\_{c} + \gamma\_A A\_{c}\\ \frac{\partial J\_S}{\partial t} + \lambda\_S^2 \frac{\partial S\_c}{\partial x} &= -f\_I(J\_S, I\_T) -f\_A(J\_S, A\_T) - \frac1{\tau\_S} J\_S\\ \frac{\partial J\_E}{\partial t} + \lambda\_E^2 \frac{\partial E\_c}{\partial x} &= \frac{\lambda\_E}{\lambda\_S}\left(f\_I(J\_S, I\_T) + f\_A(J\_S, A\_T)\right) - a J\_E - \frac1{\tau\_E} J\_E\\ \frac{\partial J\_I}{\partial t} + \lambda\_I^2 \frac{\partial I\_c}{\partial x} &= \frac{\lambda\_I}{\lambda\_E} a \sigma J\_E - \gamma\_I J\_I-\frac1{\tau\_I} J\_I\\ \frac{\partial J\_A}{\partial t} + \lambda\_A^2 \frac{\partial A\_c}{\partial x} &= \frac{\lambda\_A}{\lambda\_E} a (1- \sigma) J\_E - \gamma\_A J\_A-\frac1{\tau\_A} J\_A\\ \frac{\partial J\_R}{\partial t} + \lambda\_R^2 \frac{\partial R\_c}{\partial x} &= \frac{\lambda\_R}{\lambda\_I} \gamma\_I J\_I + \frac{\lambda\_R}{\lambda\_A} \gamma\_A J\_A -\frac1{\tau\_R} J\_R \end{aligned} \label{eq.SEIARmacro}$$ Note that here the above system is coupled with the equations for the non commuter population through identities. Formally, system is a 1D system of stochastic balance laws which can be rewritten in compact form as: $$\begin{aligned} \partial\_t {{\bf{U}}}\_c(x,t,\boldsymbol{z}) + \partial\_x {{\bf{V}}}(x,t,\boldsymbol{z}) &= {{\bf{F}}}\_c({{\bf{U}}}\_T(x,t,\boldsymbol{z}),{{\bf{U}}}\_c(x,t,\boldsymbol{z})) \\ \partial\_t {{\bf{V}}}(x,t,\boldsymbol{z}) + \boldsymbol{\Lambda}^2(x,t)\, \partial\_x {{\bf{U}}}\_c (x,t,\boldsymbol{z}) &= {{\bf{G}}}({{\bf{U}}}\_T(x,t,\boldsymbol{z}),{{\bf{V}}}(x,t,\boldsymbol{z})) + {{\bf{H}}}({{\bf{V}}}(x,t,\boldsymbol{z})), \end{aligned} \label{systcompactform}$$ in which $${{\bf{U}}}\_T = \begin{pmatrix} S\_T \\ E\_T \\ I\_T \\ A\_T \\ R\_T \end{pmatrix}, \quad {{\bf{U}}}\_c = \begin{pmatrix} S\_c \\ E\_c \\ I\_c \\ A\_c \\ R\_c \end{pmatrix}, \quad {{\bf{V}}}= \begin{pmatrix} J\_S \\ J\_E \\ J\_I \\ J\_A \\ J\_R \end{pmatrix}, \quad \boldsymbol{\Lambda} = \begin{pmatrix} \lambda\_S &0 &0 &0 &0\\ 0 &\lambda\_E &0 &0 &0\\ 0 &0 &\lambda\_I &0 &0\\ 0 &0 &0 &\lambda\_A &0\\ 0 &0 &0 &0 &\lambda\_R \end{pmatrix},$$ $${{\bf{F}}}\_c({{\bf{U}}}\_T,{{\bf{U}}}\_c) = \begin{pmatrix} -f\_I(S\_c,I\_T) -f\_A(S\_c,A\_T) \\ f\_I(S\_c, I\_T) +f\_A(S\_c,A\_T) -aE\_c \\ a\sigma E\_c -\gamma\_I I\_c \\ a(1-\sigma) E\_c -\gamma\_A A\_c \\ \gamma\_I I\_c + \gamma\_A A\_c \end{pmatrix},$$ $${{\bf{G}}}({{\bf{U}}}\_T,{{\bf{V}}}) = \begin{pmatrix} -f\_I(J\_S, I\_T) -f\_A(J\_S, A\_T) \\ \frac{\lambda\_E}{\lambda\_S}\left(f\_I(J\_S, I\_T) + f\_A(J\_S, A\_T)\right) - a J\_E\\ \frac{\lambda\_I}{\lambda\_E} a\sigma J\_E - \gamma\_I J\_I \\ \frac{\lambda\_A}{\lambda\_E} a(1-\sigma) J\_E - \gamma\_A J\_A \\ \frac{\lambda\_R}{\lambda\_I} \gamma\_I J\_I + \frac{\lambda\_R}{\lambda\_A} \gamma\_A J\_A \end{pmatrix},\quad {{\bf{H}}}({{\bf{V}}}) = - \begin{pmatrix} {J\_S}/{\tau\_S} \\ {J\_E}/{\tau\_E}\\ {J\_I}/{\tau\_I} \\ {J\_A}/{\tau\_A} \\ {J\_R}/{\tau\_R} \end{pmatrix}.$$ It is easy to verify that system is symmetric hyperbolic in the sense of Friedrichs-Lax, with real finite characteristic velocities (eigenvalues) λi, i ∈ {S, E, I, A, R}, and a complete set of linearly independent eigenvectors. All the eigenvectors are associated with genuinely non-linear fields, defining shocks and rarefactions, and the Riemann invariants of the system correspond to the kinetic transport variables $$\begin{split} S^{\pm} = \frac{1}{2} \left( S\_c \pm \frac{J\_S}{\lambda\_S} \right), \quad E^{\pm} = \frac{1}{2} &\left( E\_c \pm \frac{J\_E}{\lambda\_E} \right), \quad I^{\pm} = \frac{1}{2} \left( I\_c \pm \frac{J\_I}{\lambda\_I} \right), \\ A^{\pm} = \frac{1}{2} \left( A\_c \pm \frac{J\_A}{\lambda\_A} \right) &,\quad R^{\pm} = \frac{1}{2} \left( R\_c \pm \frac{J\_R}{\lambda\_R} \right) \,. \end{split} \label{eq.RI}$$ ### Reproduction number The standard threshold of epidemic models is the well-known basic reproduction number R0, also called the basic reproduction ratio or the basic reproductive rate, which defines the expected number of secondary cases produced, in a completely susceptible population, by a typical infected individual during its entire period of infectiousness. For many deterministic infectious disease models, this number determines whether an infection can invade and persist in a new host population (R0 > 1) or cannot (R0 < 1). Its definition in the case of spatially dependent dynamics, as already noted in, is not straightforward particularly when considering its spatial dependence. In the following we will consider the following definition for the average value of the reproduction number on the domain Ω for t > 0: $$\begin{aligned} R\_0 (t) &= \frac{\int\_\Omega f\_I(S\_T,I\_T) \,dx}{\int\_\Omega \gamma\_I(x) I\_T(x,t) \,dx} \cdot \frac{\int\_\Omega a(x)\sigma (x) E\_T(x,t) \,dx}{\int\_\Omega a(x) E\_T(x,t) \,dx} \\&+ \frac{\int\_\Omega f\_A(S\_T,A\_T) \,dx}{\int\_\Omega \gamma\_A(x) A\_T(x,t)\, dx} \cdot \frac{\int\_\Omega a(x)(1-\sigma(x)) E\_T(x,t) \,dx}{\int\_\Omega a(x) E\_T(x,t)\, dx} \,. \end{aligned} \label{eq.R0\_2}$$ The derivation of the above expression for R0(t), computed following the next-generation matrix approach, is presented in detail in Appendix [appendix:R0]. In the two contributions of eq. it is possible to identify: * the transmission rates for compartment I and A: fI(ST, IT) and fA(ST, AT), respectively; * the mean time of staying in compartment I and A: γI− 1 and γA− 1, respectively; * the fraction of individuals passing from compartment E to I and A: aσ/a = σ and a(1 − σ)/a = 1 − σ, respectively. It is worth to underline that from definition it can be deduced that it is a combination of the growth of ET, IT and AT that determines the persistence of the epidemic, not solely the growth of ET in time, neither the growth of the simple sum ET + IT + AT. If no spatial dependence is assigned to variables and parameters, hence the ODE version of system is considered, and no social distancing effects are taken into account (i.e. kI = kA = 0) the reproduction number results in accordance with : $$R\_0 (t) = \sigma \frac{\beta\_I S\_T}{\gamma\_I} + (1-\sigma)\frac{\beta\_A S\_T}{\gamma\_A} \,.$$ ### Diffusion limit From a formal viewpoint, it can be shown that the proposed model recovers the parabolic behavior expected from standard space-dependent epidemic models in the diffusion limit. Introducing the diffusion coefficients Di = λi2τi, i ∈ {S, E, I, A, R} that characterize the diffusive transport mechanism of S, E, I, A, R respectively, and letting τi → 0, i ∈ {S, E, I, A, R}, while keeping the diffusion coefficients finite, from the last five equations of system we recover Fick’s law getting $$\begin{split} J\_S = - D\_S\frac{\partial S\_c}{\partial x}, \quad J\_E = - D\_E\frac{\partial E\_c}{\partial x}, \quad J\_I = - D\_I\frac{\partial I\_c}{\partial x}, \quad J\_A = - D\_A\frac{\partial A\_c}{\partial x}, \quad J\_R = - D\_R\frac{\partial R\_c}{\partial x}. \end{split}$$ Finally, inserting these results in the rest of the equations of system yields the following parabolic reaction-diffusion system for the commuters $$\begin{split} \frac{\partial S\_{c}}{\partial t} &= \frac{\partial}{\partial x}\left(D\_S \frac{\partial }{\partial x}S\_c \right) - f\_I(S\_{c}, I\_T) - f\_A(S\_{c}, A\_T)\\ \frac{\partial E\_{c}}{\partial t} &= \frac{\partial}{\partial x}\left(D\_E \frac{\partial }{\partial x}E\_c \right) +f\_I(S\_{c}, I\_T) +f\_A(S\_{c},A\_T) -aE\_{c} \\ \frac{\partial I\_{c}}{\partial t} &= \frac{\partial}{\partial x}\left(D\_I \frac{\partial }{\partial x}I\_c \right)+ a\sigma E\_{c} -\gamma\_I I\_{c}\\ \frac{\partial A\_{c}}{\partial t} &= \frac{\partial}{\partial x}\left(D\_A \frac{\partial }{\partial x}A\_c \right)+ a(1-\sigma) E\_{c} -\gamma\_A A\_{c}\\ \frac{\partial R\_{c}}{\partial t} &= \frac{\partial}{\partial x}\left(D\_R \frac{\partial }{\partial x}R\_c \right)+\gamma\_I I\_{c} +\gamma\_A A\_{c}\,. \end{split} \label{eq.SEIARdiffusive}$$ Therefore, the relaxation times can modify the nature of the behavior of the solution, which can result either hyperbolic or parabolic (when considering small relaxation times and large speeds). This feature of the model makes it particularly suitable for the description of the dynamics of human populations, which are characterized by movement at different spatial scales. It is therefore natural to assume τi = τi(x), i ∈ {S, E, I, A, R}, since in geographic areas densely populated we can assume a diffusive dynamics while in other areas or along the main arteries of communication a hyperbolic description will be more appropriate avoiding propagation of information at infinite speed. In the model description adopted in this Section, the relaxation times are assumed to be space dependent but independent of the state variables. More generally, it is possible to consider the relaxation times $$\tau\_i = \tilde\tau\_i/\kappa(i\_c,i\_T),\qquad i\in\{S,E,I,A,R\},$$ leading, as $\tilde \tau\_i \to 0$ and taking $\lambda\_i^2=\tilde D\_i/\tilde \tau\_i$, to nonlinear diffusion equations in the form where the diffusion coefficients $D\_i=\tilde D\_i \kappa(i\_c,i\_T)$, i ∈ {S, E, I, A, R}, depend on the state variables. A classical example is represented by the choice κ(ic, iT) = icα, i ∈ {S, E, I, A, R} which corresponds to a generalization of Carleman model. In the limit $\tilde\tau\_i\to 0$, for α = 0 we recover again the linear diffusion model whereas assuming α ∈ ( − 1, 0) we obtain a fast diffusion process considered by other authors in epidemiology. From a mathematical viewpoint we refer to for rigorous results concerning these kind of diffusion limits for generalized Carlemann models. Although interesting, here we do not explore further this direction. Extension to network modeling ----------------------------- The transport model here proposed can be extended to network approaches in the sense of those presented in. In the sequel we summarize the details of the network structure we adopted to characterize arcs and nodes with different sizes. Following, it is possible to structure a 1D network considering that the nodes of the network identify locations of interest such as municipalities, provinces or, in a wider scale, regions or nations, while the arcs, enclosing the 1D spatial dynamics, represent the paths linking each location to the others. In this way, the epidemic state of each node evolves in time influenced by the mobility of the commuting individuals, moving from the other locations included in the network, always considering a part of the population composed by non-commuting individuals which remain at the origin node. In order to prescribe the proper coupling between nodes and arcs, ensuring the conservation of total density (population) in the network and of fluxes at the interface, it is necessary to impose appropriate transmission conditions at each arc-node interface. ### Transmission conditions at nodes A network or a connected graph G = (N,A) is composed of a finite set of N nodes (or vertices) N and a finite set of A bidirectional arcs (or edges) A, such that an arc connects a pair of nodes. Let us parametrize the A arcs of the network as intervals ai = [0, Li], i = 1, …, A. Arcs are bidirectional, as the network is non-oriented, but an artificial orientation needs to be fixed in order to define a sign for the velocities and therefore the fluxes. For an incoming arc, Li is the abscissa of the node, whereas for an outgoing arc the same abscissa is 0. To define transmission conditions at a generic node n ∈ N having ai ∈ A, i = 1, …, Na, L incoming arcs and aj ∈ A, j = 1, …, Na, 0 outgoing arcs, we need to consider two kind of interfaces at the node: the interfaces neighboring incoming arcs (L, i) and the interface neighboring outgoing arcs (0, j). If variables are discontinuous across these interfaces, at time t + Δt, for each of them (1 + Na, L) new states originate at interfaces (L, i) and (1 + Na, 0) new states originate at interfaces (0, j). To compute them, we need to solve (2 + Na, L + Na, 0) Riemann problems, using the Riemann Invariants (kinetic variables) of the system, defined in Eqs. , and the principle of conservation of fluxes at interfaces. For each one of the compartments of commuting individuals of the SEIAR model here discussed, let us indicate, for ease of notation, with u the number of individuals of the compartment, with v the corresponding analytical flux, with λ its characteristic velocity, and with u± the Riemann Invariants. Introducing transmission coefficients, αi, j ∈ [0, 1], which represent the probability that an individual at a generic arc-node interface decides to move across that interface, from the j-th location to the i-th location, transmission conditions at the interfaces with an incoming arc (L, i) results for the arcs side $$\begin{split} &u^\\_{L,i} = u^+\_{L,i} + \sum\_{k=1}^{N\_{a,L}} \alpha\_{i,k} u\_{L,k}^+ + \alpha\_{i,n} u\_{n}^- \\ &v^\\_{L,i} = \lambda\_i \left(u^+\_{L,i} - \sum\_{k=1}^{N\_{a,L}} \alpha\_{i,k} u\_{L,k}^+ - \alpha\_{i,n} u\_{n}^- \right) \end{split} \label{sol.junction\_Li}$$ and for the node side (with the subscript n indicating the variable –or the location, when concerning transmission coefficients– of the node) $$\begin{split} &u^\\_{L,n} = u\_{n}^- + \sum\_{k=1}^{N\_{a,L}} \alpha\_{n,k} u\_{L,k}^+ + \alpha\_{n,n} u\_{n}^- \\ &v^\\_{L,n} = - \lambda\_n \left( u\_{n}^- - \sum\_{k=1}^{N\_{a,L}} \alpha\_{n,k} u\_{L,k}^+ - \alpha\_{n,n} u\_{n}^- \right). \end{split} \label{sol.junction\_Ln}$$ On the other hand, for the transmission conditions at the interfaces with an outgoing arc (0, j), we have for the arcs side $$\begin{split} &u^\\_{0,j} = u\_{0,j}^- + \sum\_{k=1}^{N\_{a,0}} \alpha\_{j,k} u\_{0,k}^- + \alpha\_{j,n} u\_{n}^+ \\ &v^\\_{0,j} = - \lambda\_j\left( u\_{0,j}^- - \sum\_{k=1}^{N\_{a,0}} \alpha\_{j,k} u\_{0,k}^- - \alpha\_{j,n} u\_{n}^+\right). \end{split} \label{sol.junction\_0j}$$ and for the node side $$\begin{split} &u^\\_{0,n} = u\_{n}^+ + \sum\_{k=1}^{N\_{a,0}} \alpha\_{n,k} u\_{0,k}^- + \alpha\_{n,n} u\_{n}^+ \\ &v^\\_{0,n} = \lambda\_n \left(u\_{n}^+ - \sum\_{k=1}^{N\_{a,0}} \alpha\_{n,k} u\_{0,k}^- - \alpha\_{n,n} u\_{n}^+ \right) \end{split} \label{sol.junction\_0n}$$ Notice that the condition differs when considering an incoming flow or an outgoing flow, due to the artificial orientation that has been set. Indeed, for each incoming arc, we need to use uL, i+ from the arc and un− from the node; while for each outgoing arc we consider u0, j− from the arc and un+ from the node. Furthermore, to guarantee the conservation of fluxes at the interface, ensuring that the global mass (population) of the system is conserved, the following must hold vL, n\* = ∑i = 1Na, LvL, i\, v0, n\ = ∑j = 1Na, 0v0, j\; which is fulfilled imposing at interfaces (L, i) λ*i = ∑k = 1Na, Lαk, iλk + αn, iλn, λn = ∑k = 1Na, Lαk, nλk + αn, nλn, and at interfaces (0, j) λn = ∑k = 1Na, 0αk, nλk + αn, nλn, λj = ∑k = 1Na, 0αk, jλk + αn, jλn. Nodes located at the inlet (outlet) end of the domain are without any incoming (outgoing) arcs. At these nodes, in order to ensure that there are no individuals entering or leaving the network (thus ensuring the preservation of the total population), we simply enforce the standard no-flux boundary condition, which consists of imposing at inlet nodes $$v^\\_{L,n} = 0, \qquad \qquad u^\\_{L,n} = u\_n - \frac{v\_{n}}{\lambda\_n}, \label{eq.zeroflux\_inlet}$$ and at outlet nodes $$v^\\_{0,n} = 0, \qquad \qquad u^\\_{0,n} = u\_n + \frac{v\_{n}}{\lambda\_n}. \label{eq.zeroflux\_outlet}$$ Numerical results ================= The multiscale transport SEIAR model - is solved using a second-order IMEX Runge-Kutta Finite Volume Collocation method (see Appendices [appendix:SCM] and [appendix:IMEX] for details). In particular, we will show that the numerical scheme is capable to reach spectral accuracy in the stochastic space, if the solution is sufficiently smooth in that space, and to preserve this accuracy in the diffusive (stiff) limit (i.e. stochastic AP property). An advantage related to the choice of the stochastic collocation method lies in its non-intrusive nature. This feature ensures ease of implementation, since the method requires only the evaluation of the solution of the corresponding deterministic problem, followed by a post-processing step. Thus, no major manipulation efforts of the deterministic computational code are required and the loss of important structural properties of the original problem is avoided. [fig.networklombardia] [b!] [tab:matrixnetworklombardia] c \| c c c c c \|\| c From/To &LO & MI &BG &BS &CR &Total commuters &– &56717 &– &– &13712 &70429 &&(80.53%) &&&(19.47%) &(30.97%) &55397 &– &74168 &26709 &21622 &177896 &(31.14%)&&(41.69%)&(15.01%)&(12.15%) &(5.45%) &– &76337 &– &78348 &12016 &166701 &&(45.79%)&&(47.00%)&(7.21%)& (15.04%) &– &26594 &70879 &– &16967 &114440 &&(23.24%)&(61.93%)&&(14.83%)&(9.12%) &13264 &23142 &12025 &17681 &– &66112 &(20.06%)&(35.00%)&(18.19%)&(26.75%)&&(18.58%) Application to the emergence of COVID-19 in Italy ------------------------------------------------- To analyze the effectiveness of the proposed approach in a realistic geographical and epidemic scenario, we design a numerical setting reproducing the evolution of the first outbreak of COVID-19 in the Lombardy Region of Italy, from February 27, 2020 to March 27, 2020, with respect to uncertainties underlying the initial conditions and chosen epidemic parameters. ### The Lombardy network A five-node network is considered, whose nodes represent the 5 main provinces interested by the epidemic outbreak in the first months of 2020: Lodi (n1), Milan (n2), Bergamo (n3), Brescia (n4) and Cremona (n5). The arcs aj connecting each node to the others identify the main set of routes and railways viable by commuters each day. A schematic representation of this network is shown in Fig. [fig.networklombardia]. Routes that connect cities that are not direct neighbors and that require crossing other provinces are to be considered as the sum of the two route sections. Thus, for instance, it can be seen from Fig. [fig.networklombardia] that the Milan–Brescia connection is actually identified by the sum of the Milan–Bergamo (a2) and Bergamo–Brescia (a3) arcs, indeed indicated as a2 + 3. Since the space unit of measurement adopted is 1 L = 102 km, the lengths of the single sections of arc result: La1 = 0.20, La2 = 0.30, La3 = 0.35, La4 = La5 = 0.40 . These arcs are discretized with a grid size Δxa = 0.01. The length of the spatial cell associated to each node is proportional to the dimension of the urbanized area of the corresponding province, hence: Δxn1 = 0.025, Δxn2 = 0.420, Δxn3 = 0.100, Δxn4 = 0.085, Δxn5 = 0.035. Transmission coefficients αi, j, as well as percentage of commuters belonging to each province, are imposed recurring to official national assessment of mobility flow. In particular, the matrix of commuters presented in Table [tab:matrixnetworklombardia] reflects mobility data provided by the Lombardy Region for the regional fluxes of year 2020, which is in agreement with the one derived from ISTAT data released in October, 2011, as also confirmed in. Notice that connections Lodi–Bergamo and Lodi–Brescia are not taken into account (as observable also from Fig. [fig.networklombardia]) because the amount of commuters along these routes is very low compared to the amount of individuals traveling the rest of the routes. [b!] [tab:ICnetworklombardia] \| City \| Total population (N) \| Infectious (i0) \| \| --- \| --- \| --- \| \| Lodi (n1) \| 227412 \| 159 \| \| Milan (n2) \| 3265327 \| 15 \| \| Bergamo (n3) \| 1108126 \| 72 \| \| Brescia (n4) \| 1255437 \| 10 \| \| Cremona (n5) \| 355908 \| 91 \| The characteristic speed associated to each arc is fixed to permit a full round trip in each origin-destination section within a day. Since the time unit of measurement adopted is 1 T = 1 day, for all the compartments we impose: λa1 = 0.4, λa2 = 0.6, λa3 = 0.7, λa4 = λa5 = 0.8, respectively for each arc. The characteristic speed of compartment I is fixed to zero in all the nodes of the network, λn = 0, while for the rest of the compartments, namely S, E, A, R, λn = 1 . In the arcs, the relaxation time is assigned so that the model recovers a hyperbolic regime, hence τr, a = 103. On the other hand, a parabolic setting is prescribed in the cities for commuters in order to correctly capture the diffusive behavior of the disease spread which typically occurs in highly urbanized zones, with τr, n = 10− 3. The reader is invited to refer to for further details on the sensitivity of the model to relaxation and speed parameters. ### Data fitting and uncertainty We simulate the spread of COVID-19 starting from February 27, 2020 until March 27, 2020, included. At the beginning of the pandemic, tracking of positive individuals cannot be considered reliable. Thus, the initial amount of infected people is the first information affected by uncertainty. To this aim, we introduce a single source of uncertainty z having uniform distribution, z ∼ U(0, 1) and the initial conditions for compartment I, at each node, are prescribed as I(x, 0, z) = i0(1 + μ1z) , with i0 density of infectious people on February 27, 2020, as given by data recorded by the Civil Protection Department of Italy, reported in Table [tab:ICnetworklombardia] for each node, together with the total inhabitants of each province given by ISTAT data of 2019. We chose to associate all infected persons detected to the I compartment. This choice is justified by the ongoing screening policy. Indeed, during the first wave of this pandemic in Italy, tests to assess the presence of SARS-CoV-2 virus were performed almost only on patients with consistent symptoms and fever. We select μ1 = 1, assuming no more than half of the actual highly symptomatic infected were detected at the very beginning of the outbreak of the pandemic. According to values used in, we fix γI = 1/14, γA = 2γI = 1/7, a = 1/3, considering these clinical parameters deterministic. We also assume the probability rate of developing severe symptoms σ = 1/12.5, as in. Since the first day of the simulation, the public was aware of the epidemic outbreak and recommendations such as washing hands often, not touching the face, avoiding handshakes and keeping distance had already been disseminated. Therefore, we initially set coefficients kI = kA = 30. The initial value of the transmission rate of asymptomatic/mildly infectious people is calibrated as the result of a least square problem, namely the L2 norm of the difference between the observed cumulative number of infected I(t) and the numerical evolution of the same compartment, through a simple deterministic SEIAR ODE model set up for the whole Lombardy Region, with the result βA = 0.545. Since after February 23, 2020, Codogno, city in the province of Lodi, is put under strict lockdown as “red zone”, a reduced contact rate is considered for the node of Lodi, with βA = 0.50. In the above fitting, the expected initial amount of exposed and asymptomatic/mildly symptomatic individuals is also estimated imposing an initial condition of one single exposed individual, obtaining e0 ≈ 10.23 i0 and a0 ≈ 9.15 i0. In particular this permits to have a rough estimate of the presumed day of outbreak of the epidemic which, in our case, would appear to have started on January 14, 2020. With the above setup, we obtain an initial expected value of the basic reproduction number in the whole network (as from definition given in Section [sect:R0]) R0 = 3.6, which is in agreement with estimations reported in. On the other hand, the transmission rate of compartment I, βI, is considered a random parameter, given that, at each node, the initial amount of severe symptomatic subjects I is affected by uncertainty, as previously discussed. Therefore we impose: βI(0, z) = βI, 0(1 + μ1μ2z) . Assuming that highly infectious subjects are mostly detected in the most optimistic scenario, being subsequently quarantined or hospitalized, the minimum value of the transmission rate of I is set βI, 0 = 0.03 βA, as in. Furthermore, μ2 = 0.06− 1, indicating that the more the number of infected in I increases relative to the observed value i0, the more the transmission rate of this compartment tends to that of compartment A, proportionally to the error in I. As a consequence, also initial conditions for compartments E, A and S are stochastic, depending on the initial amount of severe infectious at each location: E(x, 0, z) = 10.23 I(x, 0, z) , A(x, 0, z) = 9.15 I(x, 0, z) , S(x, 0, z) = N(x) − E(x, 0, z) − I(x, 0, z) − A(x, 0, z) . Finally, removed individuals are considered initially null everywhere in the network, R(x, 0, z) = 0.0, with all arcs assumed empty at the beginning of the simulation. To model the escalation of lockdown restrictions, starting from March 9, 2020, initial day of the northern Italy lockdown, the transmission rate βA is reduced by 15% (with a consequent update also of βI, 0 and therefore of βI) and coefficients kI = kA = 60, due to the public being increasingly aware of the epidemic risks. Furthermore, the percentage of commuting individuals is reduced by 60% in the whole network according to mobility data tracked through mobile phones and made available by Google. After the additional restrictions in place as of March 22, 2020, βA is ulteriorly reduced by 10% (again with a consequent rearrangement also of βI, 0 and βI) and coefficients k are increased to kI = kA = 70. [b!] [tab:convergencen1] l \| c c \| c c \| c c \| c c & & & & & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) 2 & 4.1539e-07 & & 1.3499e-08 & & 2.0021e-08 & & 1.2599e-11 & 4 & 6.1458e-08 & 2.7568 & 1.9974e-09 & 2.7567 & 1.7945e-09 & 3.4799 & 1.1296e-12 & 3.4794 6 & 2.5542e-09 & 7.8443 & 8.3016e-11 & 7.8443 & 8.4784e-11 & 7.5281 & 5.3366e-14 & 7.5282 8 & 1.2694e-10 & 10.4343 & 4.1258e-12 & 10.4343 & 4.2423e-12 & 10.4108 & 2.6703e-15 & 10.4108 10 & 6.7564e-12 & 13.1451 & 2.1959e-13 & 13.1451 & 6.8746e-14 & 18.4744 & 4.3271e-17 & 18.4744 [b!] [tab:convergencen3] l \| c c \| c c \| c c \| c c & & & & & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) & L2 & O(L2) 2 & 4.8912e-06 & & 1.0543e-06 & & 2.1399e-06 & & 4.8133e-08 & 4 & 3.8079e-07 & 3.6831 & 8.2083e-08 & 3.6830 & 7.5528e-08 & 4.8244 & 1.6996e-09 & 4.8237 6 & 1.7441e-08 & 7.6047 & 3.7596e-09 & 7.6047 & 4.2088e-09 & 7.1210 & 9.4709e-11 & 7.1211 8 & 9.1663e-10 & 10.2400 & 1.9759e-10 & 10.2400 & 2.4765e-10 & 9.8474 & 5.5728e-12 & 9.8474 10 & 4.3175e-11 & 13.6927 & 9.3069e-12 & 13.6927 & 1.4533e-11 & 12.7075 & 3.2703e-13 & 12.7075 0.49![Lodi (n_1)](convergence_II "fig:") [fig.convergencen1] 0.49![Bergamo (n_3)](convergence_AA "fig:") [fig.convergencen3] [fig.convergence] ### Convergence analysis As a first numerical test we analyze the numerical convergence of the stochastic Collocation method, discussed in details in Appendix [appendix:SCM]. We consider the above setting, but with a larger level of uncertainty, with μ1 = 10, to emphasize the convergence rates. The numerical results evaluated with an increased number of collocation points Mp are compared to a reference solution obtained using Mp = 50, in terms of expectation and variance of the state variables. The expected exponential convergence is shown in Tables [tab:convergencen1]–[tab:convergencen3] for chosen state variables, respectively for nodes n1 and n3 (taken as representative nodes), where L2 error norms and the related order of accuracy are presented. The result is highlighted by Fig. [fig.convergence], in which the spectral decay of the L∞ norm is observed in terms of both expected value and variance as the number of collocation points increases. 0.37![March 2, 2020](lombardia_results_t5 "fig:") [fig.lombardiaresultst5] 0.37![March 7, 2020](lombardia_results_t10 "fig:") [fig.lombardiaresultst10] 0.37![March 12, 2020](lombardia_results_t15 "fig:") [fig.lombardiaresultst15] 0.37![March 17, 2020](lombardia_results_t20 "fig:") [fig.lombardiaresultst20] 0.37![March 22, 2020](lombardia_results_t25 "fig:") [fig.lombardiaresultst25] 0.37![March 27, 2020](lombardia_results_t30 "fig:") [fig.lombardiaresultst30] [fig.lombardiaresults] Simulation of different scenarios of the spread of COVID-19 ----------------------------------------------------------- In the following, we consider several scenarios regarding the spread of the COVID-19 epidemic in the Lombardy region based on the data and parameters determined in the previous section. In particular, we will consider the baseline scenario, corresponding to the actual spread of the epidemic observed from the data, and two hypothetical scenarios corresponding to the absence of long-range mobility and the absence of restrictions. ### Baseline scenario Numerical results of the Lombardy network test, in its baseline configuration (presented in details at the beginning of Section [sect.LombardiaTest]), obtained using the sAP IMEX Runge-Kutta FV Collocation method with Mp = 6 points, are reported in Figs. [fig.lombardiaresults]–[fig.NetworkLombardiadata1]. In Fig. [fig.lombardiaresults] a first qualitative trend of the spatial and temporal spread of the epidemic is presented. Here it can be observed the heterogeneity of the diffusion of the virus, which has firstly mostly affected the city of Bergamo and only in a second moment prevailing in Milan. In Fig. [fig.NetworkLombardia] the expected evolution in time of the infected individuals, together with 95% confidence intervals, is shown for each node and for the whole Lombardy network, including exposed E, highly symptomatic subjects I and asymptomatic or weakly symptomatic people A. Each plot is also associated with the expected temporal evolution of the reproduction number R0(t), computed as described in Section [sect:R0]. One can see the capacity of the model to reproduce a very heterogeneous epidemic trend in the network analyzed, which is also reflected in the different ranges and patterns shown for the R0 of each province. It can also be observed the agreement between the evolution of the reproduction number and the epidemic spread. In particular, it is confirmed the decline of the daily number of infected as R0 reaches values below 1, as shown in the plots for Lodi, Bergamo and Cremona. On the other hand, the persistence of the virus in the complete network, and especially in Milan, is noticed until March 27, 2020 (last day of the simulation), where the reproduction number remains R0 > 1, confirming the consistency of the definition proposed for R0(t). 0.45![Lodi (n_1)](node1 "fig:") [fig.node1] 0.45![Milan (n_2)](node2 "fig:") [fig.node2] 0.45![Bergamo (n_3)](node3 "fig:") [fig.node3] 0.45![Brescia (n_4)](node4 "fig:") [fig.node4] 0.45![Cremona (n_5)](node5 "fig:") [fig.node5] 0.45![Lombardy (total network)](tot_network "fig:") [fig.totnetwork] [fig.NetworkLombardia] 0.45![Lodi (n_1)](node1_data "fig:") [fig.node1data] 0.45![Milan (n_2)](node2_data "fig:") [fig.node2data] 0.45![Bergamo (n_3)](node3_data "fig:") [fig.node3data] 0.45![Brescia (n_4)](node4_data "fig:") [fig.node4data] 0.45![Cremona (n_5)](node5_data "fig:") [fig.node5data] 0.45![Lombardy (total network)](tot_network_data "fig:") [fig.totnetworkdata] [fig.NetworkLombardiadata] 0.45![Lodi (n_1)](node1_data1 "fig:") [fig.node1data1] 0.45![Milan (n_2)](node2_data1 "fig:") [fig.node2data1] 0.45![Bergamo (n_3)](node3_data1 "fig:") [fig.node3data1] 0.45![Brescia (n_4)](node4_data1 "fig:") [fig.node4data1] 0.45![Cremona (n_5)](node5_data1 "fig:") [fig.node5data1] 0.45![Lombardy (total network)](tot_network_data1 "fig:") [fig.totnetworkdata1] [fig.NetworkLombardiadata1] As visible from Fig. [fig.NetworkLombardiadata], the lower bound of the confidence band of the cumulative amount in time of highly symptomatic individuals is comparable with the observed data of the Civil Protection Department of Italy. As expected, the mean value of the numerical result reports an higher amount of I, especially in Milan, the province most affected by the virus, due to the uncertainty of available data, which surely underestimate the real amount of infectious people, as discussed in Section [sect.LombardiaTest]. The comparison between the expected evolution in time of the cumulative amount of severe infectious with respect to the effective cumulative amount of total infectious people, including asymptomatic and mildly symptomatic individuals, is shown in Fig. [fig.NetworkLombardiadata1]. Here, it can be noticed how much of the spread of COVID-19 has actually been lost from the data of the first outbreak in Lombardy and the impact that the presence of asymptomatic or undetected subjects has had on the epidemic evolution. By the end of the simulation, indeed, E[A + RA]/E[I + A + R] = 0.92 in the network, indicating with RA the amount of recovered coming from the compartment A. This result is in line with WHO guidelines given during the first wave of COVID-19, stating that approximately 80% of the infected population was asymptomatic. It is here remarked, indeed, that the compartment A in the proposed model includes not only the asymptomatic people but also the mildly symptomatic, which would approximately be 12% of the total cases. In the results here presented, transmission coefficients αi, j(x), which define the behavior of commuters in the network, are imposed as deterministic and constant in time, meaning that the amount of individuals exiting from (and entering in) each city is the same in each instant of the simulation. Clearly, this assumption leads to a simplification of the description of the phenomenon of commuting, generally characterized by a peak in the early morning hours and a drastic drop in the night hours. What we represent is indeed the mean commuting trend during the day, which tends to assume a stationary solution in time. However, a more realistic behavior, characterized for example by periodic sinusoidal functions, leads to a slightly oscillatory trend in the reported curves of each compartment and also in the reproduction number R0. By selecting αi, j(x) values that define an average trend over the day, allows us to avoid this misleading representation, while maintaining the consistency of the results. ### No-mobility scenario To assess the effects of the mobility of commuters, a second scenario is investigated in which the whole population is not allowed to move out of the residence city. Numerical results of this test are reported in Fig. [fig.NetworkLombardianomobility] for two representative provinces of the network: Lodi and Milan. It can be observed the different evolution of the spread of the epidemic comparing trends presented in Fig. [fig.NetworkLombardianomobility] with the corresponding ones in Fig. [fig.NetworkLombardia] (for the first column of plots) and Fig. [fig.NetworkLombardiadata1] (for the second column of plots). As a consequence of the absence of commuting people in the network, the spread of COVID-19 in Milan results consistently damped. The province of Milan, in fact, is the one with the highest number of daily incomes of commuting workers and students, followed by Bergamo and Brescia, as shown in Table [tab:matrixnetworklombardia]. On the other hand, Lodi is the city with the highest amount of daily outcomes of commuters (30% of its total population). Thus, preventing the population from leaving the province shows a slight increase in local infections with respect to the baseline scenario. This result is not intended to suggest that a constraint on the mobility of people would be disadvantageous in fighting the spread of the epidemic. In fact, one must consider that the contagions shown in these Figures are due to a still present interaction of people at the provincial level, which has not been reduced in any way with respect to the previous scenario. Similar conclusions are drawn also in. These results are primarily intended to demonstrate how much the evolution of a pandemic can vary in response to changes in people’s mobility. 0.45![Lodi (n_1)](node1_nomobility "fig:") [fig.node1nomobility] 0.45![Lodi (n_1)](node1_data1_nomobility "fig:") [fig.node1data1nomobility] 0.45![Milan (n_2)](node2_nomobility "fig:") [fig.node2nomobility] 0.45![Milan (n_2)](node2_data1_nomobility "fig:") [fig.node2data1nomobility] [fig.NetworkLombardianomobility] 0.45![Milan (n_2)](node2_norestrictions "fig:") [fig.node2norestrictions] 0.45![Milan (n_2)](node2_data1_norestrictions "fig:") [fig.node2data1norestrictions] 0.45![Bergamo (n_3)](node3_norestrictions "fig:") [fig.node3norestrictions] 0.45![Bergamo (n_3)](node3_data1_norestrictions "fig:") [fig.node3data1norestrictions] 0.45![Lombardy (total network)](tot_network_norestrictions "fig:") [fig.totnetworknorestrictions] 0.45![Lombardy (total network)](tot_network_data1_norestrictions "fig:") [fig.totnetworkdata1norestrictions] [fig.NetworkLombardianorestrictions] ### No-restrictions scenario Finally, results in the event that no restrictions of any kind were applied by the government are presented in Fig. [fig.NetworkLombardianorestrictions] for the province of Milan, Bergamo and the complete Lombardy network. Comparing these results with the corresponding in Fig. [fig.NetworkLombardia] (for the first column of plots) and Fig. [fig.NetworkLombardiadata1] (for the second column of plots), it can be seen immediately the fundamental role that government restrictions have played in containing the spread of the disease (notice the different y-axis scale in the Figures). In fact, the cumulative number of infected people across the region in this scenario is almost 1.8 times higher than in the baseline scenario at the end of the simulation. This value indicates that the restrictions imposed during the first wave of COVID-19 in Italy, and the consequent increasing risk awareness in the population, have contributed to attenuate the spread of the virus by 43%, which is in agreement with the result presented in. Also the basic reproduction number R0 shows a totally different evolution in time, being far from reaching the value 1 at the end of the simulation. Indeed, on March 27, 2020 the virus still persists in its propagation, with mean value R0 = 1.50 when considering the whole network and, in particular, R0 = 1.90 in the province of Milan. Conclusions =========== In this paper, a stochastic transport model for the spread of an epidemic phenomenon described by a multi-population SEIAR compartmental dynamics on networks is presented. The starting point for the description of spatial motion and the interactions of the individuals has its roots in the kinetic theory of discrete velocity models. The model is structured on spatial networks, where nodes identify the locations of interest (cities in this case) and arcs represent the set of major mobility paths (roads and railways). Individuals are divided into commuters, who move on a suburban scale, and non-commuters, who act on an urban scale. In this way, we avoid unrealistic mass migration effects in which the entire population moves indiscriminately through the network. Thanks to the hyperbolicity of the resulting system, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic reaction-diffusion models proposed in literature, is removed. Nevertheless, we show that for small relaxation times and large characteristic speeds, in the diffusive regime, the proposed model recovers its parabolic nature. Since the derivation and the consequent definition of the basic reproduction number is well-established in the literature for ODE epidemic models, we resort on these results to introduce a derivation of R0 for spatial epidemic models in the case of no-inflow/outflow boundary conditions, validating the effectiveness of the resulting definition as an indicator of the viral growth rate. The model has been tested for the analysis of the emergence of COVID-19 in Italy, by simulating the propagation and evolution of the virus in the months of February and March 2020 in Lombardy. Indeed, it is in these early stages of the epidemic that uncertainty in data and transport dynamics played a key role. In order to study the effects of the uncertainties of the initial conditions and of the parameters involved in the model on the solution of the problem, a second-order stochastic asymptotic-preservative IMEX Runge-Kutta Finite Volume Collocation method has been used. It is shown that the proposed numerical scheme achieves spectral accuracy in the stochastic space by achieving an exponential convergence rate even in the case where uncertainty is present not only in the initial data but also in the nonlinear interaction terms. Numerical results of the baseline scenario were compared with observed data made available by the Italian Civil Protection Department, demonstrating that the proposed model is suitable to adapt to real settings and applications and showing its ability on capturing the heterogeneity underlying epidemics. Also alternative scenarios have been evaluated, considering both a total lockdown of extra-urban mobility and a total absence of restrictions at governmental level. The proposed methodology has the potential to be applied to epidemiological models structured in more compartments, such as those proposed in. In this context, it was decided to keep the compartmentalization as simple as possible given the lack of more specific observed data at the provincial level. As a matter of fact, for the single provinces during the first epidemic wave, the Italian Civil Protection Department has made available only the cumulative trend of the subjects tested positive to COVID-19. There is no distinction between individuals simply quarantined, hospitalized or even in intensive care. Similarly, no structured data are publicly available at the level of provinces for individuals who have died. Future developments foresee an extension of the model to include the age structure of the population, as this is an essential feature to correctly describe mobility flows (which mainly affect the younger part of the population) and consequently a more correct spread of viruses such as COVID-19. Another interesting aspect, especially for analysis related to recent developments of the COVID-19 pandemic, is certainly the extension of the model to take into account the effects of the viral load of individuals, by analyzing the effects of the so called super-spreaders, and the immunization of susceptible individuals through vaccination campaigns. Acknowledgements ================ This work was partially supported by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) PRIN 2017, project “Innovative numerical methods for evolutionary partial differential equations and applications”, code 2017KKJP4X. Derivation of the reproduction number ===================================== Recurring to the next-generation matrix (NGM) procedure, it is possible to compute the dominant eigenvalue of a positive linear operator, called next-generation operator, which permits to define the reproduction number when there are several compartments contributing to the spread of the infection. The generalization to space dependent models is non straightforward due to the nonlinear nature of the interactions. In the following we consider the situation where no-inflow/outflow boundary conditions are assumed. Given the spatial model defined in, let us consider the following subsystem of balance equations of the infectious compartments including both commuters and non-commuters $$\begin{aligned} \frac{\partial E\_T}{\partial t} + \frac{\partial J\_E}{\partial x} &= f\_I(S\_T, I\_T) +f\_A(S\_T,A\_T) -aE\_T \\ \frac{\partial I\_T}{\partial t} + \frac{\partial J\_I}{\partial x} &= a \sigma E\_T -\gamma\_I I\_T\\ \frac{\partial A\_T}{\partial t} + \frac{\partial J\_A}{\partial x} &= a(1- \sigma) E\_T -\gamma\_A A\_T\,. \end{aligned} \label{eq.EIA}$$ Considering, a deterministic framework, in which the epidemic parameters only depend on the variable x, thanks to our assumptions, integration over Ω leads to $$\begin{aligned} \frac{\partial}{\partial t}\int\_\Omega E\_T(x,t) dx &= \int\_\Omega f\_I(S\_T, I\_T) dx + \int\_\Omega f\_A(S\_T,A\_T) dx - \int\_\Omega a(x) E\_T(x,t) dx \\ \frac{\partial}{\partial t} \int\_\Omega I\_T(x,t) dx &= \int\_\Omega a(x) \sigma (x) E\_T(x,t) dx - \int\_\Omega \gamma\_I(x) I\_T(x,t) dx\\ \frac{\partial}{\partial t} \int\_\Omega A\_T(x,t) dx &= \int\_\Omega a(x) (1- \sigma (x)) E\_T(x,t) dx -\int\_\Omega \gamma\_A(x) A\_T(x,t) dx\,. \end{aligned} \label{eq.EIA\_int}$$ If we define the total densities $$\bar E = \int\_\Omega E\_T(x,t) dx \,, \qquad \bar I = \int\_\Omega I\_T(x,t) dx \,, \qquad \bar A = \int\_\Omega A\_T(x,t) dx$$ and the averaged epidemic operators, $$\bar f\_I(S\_T,I\_T) = \frac{\int\_\Omega f\_I(S\_T, I\_T) dx}{\int\_\Omega I\_T(x,t) dx} \,, \qquad \bar f\_A(S\_T,A\_T) = \frac{\int\_\Omega f\_A(S\_T, A\_T) dx}{\int\_\Omega A\_T(x,t) dx} \,,$$ $$\bar \gamma\_I = \frac{\int\_\Omega \gamma\_I(x) I\_T(x,t) dx}{\int\_\Omega I\_T(x,t) dx}\,, \qquad \bar \gamma\_A = \frac{\int\_\Omega \gamma\_A(x) A\_T(x,t) dx}{\int\_\Omega A\_T(x,t) dx}\,,$$ $$\bar a = \frac{\int\_\Omega a(x) E\_T(x,t) dx}{\int\_\Omega E\_T(x,t) dx}\,, \quad \hat a = \frac{\int\_\Omega a(x) \sigma(x) E\_T(x,t) dx}{\int\_\Omega E\_T(x,t) dx}\,, \quad \tilde a = \frac{\int\_\Omega a(x)(1- \sigma(x)) E\_T(x,t) dx}{\int\_\Omega E\_T(x,t) dx}\,,$$ system can be rewritten in the following ODE form $$\begin{aligned} \dot{\bar E} &= \bar f\_I(S\_T,I\_T) \bar I + \bar f\_A(S\_T,A\_T) \bar A - \bar a \bar E\\ \dot{\bar I} &= \hat a \bar E - \bar \gamma\_I \bar I\\ \dot{\bar A} &= \tilde a \bar E - \bar \gamma\_A \bar A \,. \end{aligned} \label{eq.EIA\_ode}$$ Following the NGM approach, the Jacobian matrix of the above obtained ODE system results $$\bf{J\_0} = \begin{pmatrix} \bar a &\bar f\_I(S\_T,I\_T) &\bar f\_A(S\_T,A\_T) \\ \hat a &-\bar \gamma\_I &0 \\ \tilde a &0 & -\bar \gamma\_A \end{pmatrix},$$ which can be decomposed into the following transmission matrix $\bf{T}$ and transition matrix $\bf{\Sigma}$, so that $\bf{J\_0} = \bf{T} + \bf{\Sigma}$ $$\bf{T} = \begin{pmatrix} 0 &\bar f\_I(S\_T,I\_T) &\bar f\_A(S\_T,A\_T) \\ 0 &0 &0 \\ 0 &0 &0 \end{pmatrix}, \qquad \bf{\Sigma} = \begin{pmatrix} \bar a &0 &0 \\ \hat a &-\bar \gamma\_I &0 \\ \tilde a &0 & -\bar \gamma\_A \end{pmatrix}.$$ The basic reproduction number R0 is the spectral radius of the next-generation operator, $\bf{K\_L} = -\bf T \bf \Sigma^{-1}$, and results composed by the sum of two components, deriving from the two compartments of the model contributing to the spread of the epidemic (I and A) $$R\_0 = \rho \left( \bf K\_L \right) = R\_0^I + R\_0^A \label{eq.R0\_1}$$ with $$R\_0^I = \frac{\bar f\_I(S\_T,I\_T) \, \hat a}{\bar\gamma\_I \, \bar a}, \quad R\_0^A= \frac{\bar f\_A(S\_T,A\_T) \, \tilde a}{\bar\gamma\_A \, \bar a} \,. \label{eq.R0\_IA}$$ and therefore, recalling the previous definitions $$\begin{aligned} R\_0 (t) &= \frac{\int\_\Omega f\_I(S\_T,I\_T) \,dx}{\int\_\Omega \gamma\_I(x) I\_T(x,t) \,dx} \cdot \frac{\int\_\Omega a(x)\sigma (x) E\_T(x,t) \,dx}{\int\_\Omega a(x) E\_T(x,t) \,dx} \\&+ \frac{\int\_\Omega f\_A(S\_T,A\_T) \,dx}{\int\_\Omega \gamma\_A(x) A\_T(x,t)\, dx} \cdot \frac{\int\_\Omega a(x)(1-\sigma(x)) E\_T(x,t) \,dx}{\int\_\Omega a(x) E\_T(x,t)\, dx} \,. \end{aligned} \label{eq:Rt}$$ Note that, the dependence of additional stochastic parameters will not modify the above reasoning leading to the same definition including dependence from the uncertainty variable $\boldsymbol{z} = (z\_1,\ldots,z\_{d})^T \in \mathbb{R}^{d}$. Stochastic Collocation Method ============================= Following a stochastic collocation approach, the solution of the stochastic problem can be computed employing a generalized Polynomial Chaos (gPC) expansion. Let us consider, for simplicity, a single source of uncertainty $z\in\RR$ and that the probability density function (PDF) of this random input, ρz : Γ → R+, is known. The approximated solution of the problem, ${\bf{Q}}^h(x,t,z)=({\bf{U}}\_c^h, {\bf{V}}^h,{\bf{U}}\_0^h)$, where ${\bf{U}}\_0^h$ is the state vector of non-commuters, referring to system, can be expressed as a finite series of orthonormal polynomials in terms of the stochastic parameter $$\label{eq:expansion} {\bf{Q}}^h(x,t,z)=\sum\_{j=1}^{M} \hat{\bf{Q}}\_j(x,t) \phi\_j(z),$$ where M is the number of terms of the truncated series and ϕj(z) are orthonormal polynomials, with respect to the measure $\rho\_z(z)\, {{\rm d}}z$. The expansion coefficients $\hat{\bf{Q}}\_j(x,t)$ is obtained as $$\label{eq:exp\_coeff\_int} \hat{\bf{Q}}\_j(x,t) = \int\_{\Gamma} {\bf{Q}} (x,t,z)\, \phi\_j(z)\, \rho\_z(z)\, {{\rm d}}z, \qquad j=1,\ldots,M.$$ Following the stochastic collocation method, the integrals for the expansion coefficients in Eq. are replaced by a suitable quadrature UMp characterized by the set {zn, wn}n = 1Mp, where zn is the n-th collocation point, wn is the corresponding weight and Mp represents the number of quadrature points. For instance, for a stochastic parameter with a uniform distribution the quadrature is defined by the Gauss-Legendre weights and nodes; while for a random variable associated to a Gaussian PDF, we will rely on a Gauss-Hermite quadrature, which reads $$\label{eq:exp\_coeff\_quad} \hat{\bf{Q}}\_j(x,t) \approx \mathcal{U}^{M\_p}\left[{\bf{Q}}^d(x,t,z)\, \phi\_j(z)\right] = \sum\_{n=1}^{M\_p} {\bf{Q}}^d(x,t,z\_n)\, \phi\_j(z\_n)\, w\_n, \qquad j=1,\ldots,M$$ where ${\bf{Q}}^d(x,t,z\_n)$, with n = 1, …, Mp, is the deterministic solution of the problem evaluated at the n-th collocation point. After the computation of the expansion coefficients, substituting eq. in eq. , an approximated solution is available. In particular, it is possible to continue with the post-processing step, evaluating the expectation of the variables of interest $$\label{eq:mean\_an} \mathbb{E}\left[{\bf{Q}}\right] =\int\_{\Gamma} {\bf{Q}}(x,t,z)\, \rho\_z(z)\, {{\rm d}}z,$$ which are approximated as $$\label{eq:mean\_apx} \mathbb{E}\left[{\bf{Q}}\right] \approx \mathbb{E}\left[{\bf{Q}}^h\right] =\int\_{\Gamma} {\bf{Q}}(x,t,z)\, \rho\_z(z)\, {{\rm d}}z \approx \sum\_{n=1}^{M\_p} {\bf{Q}}^d(x,t,z\_n)\, w\_n.$$ Similarly, other statistical quantities of interest can be computed, like the variance $$\label{eq:variance\_apx} \mathbb{V}\left[{\bf{Q}}\right] = \mathbb{E}\left[ \left( {\bf{Q}} - \mathbb{E}\left[{\bf{Q}}\right] \right)^2 \right] \approx \mathbb{E}\left[\left({\bf{Q}}^h \right)^2\right] - \mathbb{E}\left[{\bf{Q}}^h \right]^2.$$ When more than one stochastic parameter is considered in the problem, $\boldsymbol{z}\in \RR^{d}$, the joint PDF $\rho\_z\left(\boldsymbol{z}\right)$ of the random vector composed by the random parameters (assuming independence of the variables) is given by $$\rho\_z\left(\boldsymbol{z}\right) = \prod\_{k=1}^{d} \rho\_{z,k}\left(z\_k\right),$$ where ρz, k is the PDF of the k-th variable. The extension of the collocation method then follows in a similar way. AP-IMEX Runge-Kutta Finite Volume scheme ======================================== To compute the solution at each collocation point, an IMEX Runge-Kutta Finite Volume method for hyperbolic systems with multiscale relaxation is adopted. The IMEX discretization for the commuters’ dynamics, coupled with the equations for the non commuter population through identities, results with internal Runge-Kutta stages $$\begin{split} &{{\bf{U}}}\_c^{(k)} = {{\bf{U}}}\_c^n - \Delta t \sum\_{j=1}^{k} a\_{kj} \partial\_x {{\bf{V}}}^{(j)} + \Delta t \sum\_{j=1}^{k-1} \tilde{a}\_{kj} {{\bf{F}}}\_c\left({{\bf{U}}}\_T^{(j)},{{\bf{U}}}\_c^{(j)}\right) \\ &{{\bf{V}}}^{(k)} = {{\bf{V}}}^n - \Delta t \sum\_{j=1}^{k-1} \tilde{a}\_{kj} \left( \boldsymbol{\Lambda}^2 \partial\_x {{\bf{U}}}\_c^{(j)} - {{\bf{G}}}\left({{\bf{U}}}\_T^{(j)},{{\bf{V}}}^{(j)}\right)\right) + \Delta t \sum\_{j=1}^{k} a\_{kj} {{\bf{H}}}\left({{\bf{V}}}^{(j)}\right) \\ &{{\bf{U}}}\_0^{(k)} = {{\bf{U}}}\_0^n + \Delta t \sum\_{j=1}^{k-1} \tilde{a}\_{kj} {{\bf{F}}}\_0\left({{\bf{U}}}\_T^{(j)},{{\bf{U}}}\_0^{(j)}\right), \end{split} \label{eq.iterIMEX}$$ and final numerical solution $$\begin{split} & {{\bf{U}}}\_c^{n+1} = {{\bf{U}}}\_c^n - \Delta t \sum\_{k=1}^{s} b\_{k} \partial\_x {{\bf{V}}}^{(k)} + \Delta t \sum\_{k=1}^{s} \tilde{b}\_{k} {{\bf{F}}}\_c\left({{\bf{U}}}\_T^{(k)},{{\bf{U}}}\_c^{(k)}\right) \\ & {{\bf{V}}}^{n+1} = {{\bf{V}}}^n - \Delta t \sum\_{k=1}^{s} \tilde{b}\_{k} \left( \boldsymbol{\Lambda}^2 \partial\_x {{\bf{U}}}\_c^{(k)} - {{\bf{G}}}\left({{\bf{U}}}\_T^{(k)},{{\bf{V}}}^{(k)}\right) \right) + \Delta t \sum\_{k=1}^{s} b\_{k} {{\bf{H}}}\left({{\bf{V}}}^{(k)}\right) \\ & {{\bf{U}}}\_0^{n+1} = {{\bf{U}}}\_0^n + \Delta t \sum\_{k=1}^{s} \tilde{b}\_{k} {{\bf{F}}}\_0\left({{\bf{U}}}\_T^{(k)},{{\bf{U}}}\_0^{(k)}\right). \end{split} \label{eq.finalIMEX}$$ Here Δt = tn + 1 − tn is the time step size that follows the less restrictive between the standard hyperbolic Courant-Friedrichs-Levy condition, $\Delta t \leq {\mathsf{CFL}}\frac{\Delta x}{max\left\{\lambda\_i\right\}}$, and the parabolic stability restriction, $\Delta t \leq \nu \frac{\Delta x^2}{max\left\{D\_i\right\}}$, given by the diffusive components of the system, with CFL and ν suitable stability constants and $\Delta x = x\_{i+\frac{1}{2}} - x\_{i-\frac{1}{2}}$ space grid size. In this work, we fix CFL = 0.9 and $\nu = \frac{max\left\{D\_i\right\}}{2}$. The numerical scheme is characterized by two s × s matrices, $\tilde A = (\tilde a\_{kj})$, with $\tilde a\_{kj} = 0$ for j ≥ k, and A = (akj), with akj = 0 for j > k, and by the weights vectors $\tilde b = (\tilde b\_1,...,\tilde b\_s)^T$, b = (b1, ..., bs)T (with s identifying the number of the Runge-Kutta stages), which can be represented in the following Butcher notation c \| c $\tilde c$ & $\tilde A$ & $\tilde b^T$ c \| c c & A & bT where $\tilde c$ and c are the time coefficient vectors $$\tilde c\_k = \sum^{k-1}\_{j=1} \tilde a\_{kj}, \qquad \qquad c\_k = \sum^{k}\_{j=1} a\_{kj}.$$ The distribution of matrices $\tilde A$ and A permits to treat implicitly the stiff terms (hence those depending on the scaling parameters, τi and λi) and explicitly all the rest. Moreover, if a proper set of matrices $\tilde A, A$ and vectors $\tilde b, b$ are chosen, the AP property is satisfied, which means that the scheme maintains a consistent discretization of the asymptotic system in the diffusive (parabolic) regime (see ). For example, the second order GSA BPR(4,4,2) scheme proposed in satisfies the AP property and is defined by the following double Butcher tableau (explicit on the left and implicit on the right) $$\begin{tabular}{c \| c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 1/4 & 1/4 & 0 & 0 & 0 & 0 \\ 1/4 & 13/4 & -3 & 0 & 0 & 0 \\ 3/4 & 1/4 & 0 & 1/2 & 0 & 0 \\ 1 & 0 & 1/3 & 1/6 & 1/2 & 0 \\ \hline & 0 & 1/3 & 1/6 & 1/2 & 0 \\ \end{tabular} \hspace{1.0cm} \begin{tabular}{c \| c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 1/4 & 0 & 1/4 & 0 & 0 & 0 \\ 1/4 & 0 & 0 & 1/4 & 0 & 0 \\ 3/4 & 0 & 1/24 & 11/24 & 1/4 & 0 \\ 1 & 0 & 11/24 & 1/6 & 1/8 & 1/4 \\ \hline & 0 & 11/24 & 1/6 & 1/8 & 1/4 \\ \end{tabular} \label{eq:tables}$$ At each internal stage of the IMEX Runge-Kutta scheme, we apply a TVD Finite Volume discretization. To achieve second order accuracy also in space, while avoiding the occurrence of spurious oscillations, a classical minmod slope limiter has been adopted. Finally, let us remark that, given the non-intrusive nature of the stochastic collocation method which only requires the evaluation of the solutions of the corresponding deterministic problem at each collocation point, the AP property of the deterministic IMEX scheme is preserved, leading to a stochastic asymptotic-preserving scheme. --- 1. Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy ([email protected])[↩](#fnref1) 2. Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy ([email protected])[↩](#fnref2)	arxiv_0000611
Stars in the Open Cluster NGC 7142 =========================================== We present new discoveries of variable stars near the turnoff of the old open cluster NGC 7142. Contrary to previous studies, we detect eight contact or near contact eclipsing binaries (including three near the cluster turnoff), and most of these have good probability of being cluster members. We also identified one long-period variable that resides far to the red of the cluster giant branch, and four new detached eclipsing binaries. We have re-examined the question of distance and reddening for the cluster, and find that the distance is larger and reddening lower than in most previous studies. In turn this implies that NGC 7142 is probably slightly younger than M67, about 3 Gyr old. With an age of this size, NGC 7142 would be one of a small group of clusters with main sequence turnoff stars at the transition between convective and radiative cores. Introduction ============ Star clusters have long been testbeds for our understanding of the evolution of stars, the physics of the gas in their interiors, and the dynamics of the member stars. Variable stars can provide a great deal of additional information that allows us to tighten the screws on theoretical models attempting to predict cluster stellar characteristics from first principles. Pulsating stars can provide information about internal structure and stellar mass. Short-period semi-detached and contact binaries may contain encoded information about the dynamical history of the cluster. But perhaps most valuable of all are detached eclipsing binaries because they can provide high precision measurements of masses and radii for individual stars with a minimum of theoretical interpretation. Variability studies of NGC 7142 have previously been carried out by and. searched for low-amplitude variables on a single night of observations, finding eight stars with significant light variations. expected to find a number of short-period variable stars based on discoveries in clusters like NGC 188. V375 Cep was the only variable star they discovered in the cluster, although they lacked sufficient observations to determine a period. The variable is of interest because its photometry puts it very close to the turnoff of the cluster’s color-magnitude diagram, within the main sequence band. Masses for these stars would be valuable constraints on the cluster age, but in addition, one of the two stars could be significantly evolved from the zero-age main sequence. This means that the radius of the star could help refine the age of the cluster. [One other semi-regular variable, V582 Cep, was known to be in the field of the cluster, but was outside the field observed by us. In one of the earliest studies of the cluster, found that NGC 7142 was intermediate in age between NGC 7789 and M67, approximately 1.5 − 4 Gyr. found that it was intermediate in age between M67 and NGC 188, or approximately 4 − 5 Gyr. estimated an age of 4.9 Gyr by comparing synthetic color-magnitude diagrams (generated from model isochrones) with photometry. give an age of 4 ± 1 Gyr for NGC 7142 based on the magnitude difference between the subgiant branch and the red giant clump. Recently though, derived a much larger age of 6.9 ± 0.8 Gyr, using synthetic CMDs to match the turnoff color and red giant clump color and magnitude. Much of the disagreement may be the result of differential extinction in the cluster and the small numbers of evolved stars, making the reference points in the CMD difficult to identify. In order to improve the precision of the age determination for this old open cluster (and others), we have undertaken a program to identify and analyze age-sensitive eclipsing binary stars. This paper presents the results of our variable search, and a following study presents and analysis of the most promising candidates. Observational Material and Data Reduction ========================================= Photometry ---------- All of the photometry for this study was taken with the Mount Laguna Observatory 1 m telescope using a camera having a 2048 × 2048 pixel CCD and approximately a $13\farcm5\times13\farcm5$ field of view. Table [phottab] lists the nights during which images were taken for this article. 17 nights of data were taken in RC band for the purpose of identifying variable stars in the cluster, and subsequent nights were taken primarily to refine the ephemerides of the most important variables and to determine light curves in different filter bands. Differential photometry was undertaken using the image subtraction package ISIS. Our procedure is very similar to that of, so for most details, we refer the reader to that paper. Our image sets in B, V, RC, and IC filters were interpolated to a common image coordinate system, and then processed separately by filter. The output of ISIS is a difference flux measured on the subtracted images. To convert these difference fluxes into magnitudes, star fluxes were measured on the reference frame. ISIS’s algorithm is a modified aperture photometry routine that employs the reference image point-spread function (PSF) for weighting purposes. The reference PSF is first transformed to the seeing of the image under consideration. Only the portion within an aperture of radphot pixels will be used, but it is normalized to a larger aperture of rad\_aper pixels. The pixel values in the subtracted image are weighted by this transformed PSF. In this study, we used radphot = 4 pix and rad\_aper = 10 pix. We discovered a bug in the determination of uncertainties in the difference fluxes derived by ISIS. As part of the image subtraction process, the overall flux scaling from frame to frame is fit and corrected in ISIS. This scaling affects the size of the noise and therefore the flux uncertainties, but this was not done in the last version of ISIS. This had the unfortunate effect of depressing computed uncertainties on measurements with low fluxes. We fixed this error in our copy of the ISIS code (version 2.2), and all of the results here use this corrected version. ### Photometric Calibration We took calibration images under photometric conditions on the night of 25 October 2008. We observed the standard fields PG0231+051, SA 92, SA95, SA 98, and NGC 6940, and used standard values taken from. The standard fields were observed between 3 and 10 times per filter in BVI at airmasses that ranged from 1.034 to 1.998. All together this resulted in more than 4000 standard star observations per filter covering a color range $-0.5 \la (B-I) \la 5$. We derived aperture photometry from all frames using DAOPHOT, and made curve of growth corrections using the program DAOGROW. The observations were transformed to the standard system using the following equations: b = B + a0 + a1(B − I) + a2X v = V + b0 + b1(B − I) + b2X i = I + c0 + c1(B − I) + c2X where b, v, and i are instrumental magnitudes, B, V, and I are standard-system magnitudes, X is airmass, and ai, bi, and ci are coefficients determined from least-squares fits. Fig. [stetcomp] shows the residuals of the comparison of our standardized observations and the Stetson values. Clusters to be calibrated were observed with a range of exposures times on the same night. For NGC 7142, there were 9 observations in B (1 × 60 s, 5 × 120 s, and 3 × 300 s), 8 observations in V (3 × 60 s, 2 × 120 s, and 3 × 300 s), and 11 observations in I (4 × 10 s, 2 × 60 s, 2 × 120 s, and 3 × 300 s). Little attention had been paid to NGC 7142 in the past, with providing the most extensive set of photometry. Fig. [ctcomp] shows a comparison of our photometry with theirs. There are small zeropoint differences between the two studies, but more importantly, there is a clear trend with color. We believe our calibration is superior based on number of standards and improved technique. We have also compared to the more recent photometry by. The zeropoint differences are again small as shown in Fig. [janescomp], but there are also no obvious trends with color. The color-magnitude diagram for our field is shown in Fig. [cmd]. Note that much of the scatter probably comes from differential reddening resulting from the cluster’s proximity on the sky to the reflection nebula NGC 7129 (although the cluster is well behind the cloud). However, to help identify the cluster’s sparsely populated giant branch, we identified all the stars with radial velocities from spectra that have been presented in the literature. Spectra ------- Spectra for three variable star targets and three red clump star candidates were obtained at the Hobby-Eberly Telescope (HET) with the High Resolution Spectrograph (HRS, ) as part of normal queue scheduled observing. The observed stars are listed in Table [spectab]. HRS was configured to HRS\_30k\_central\_316g5936\_2as\_0sky\_IS0\_GC0\_2x3 or HRS\_30k\_central\_600g5822\_2as\_2sky\_IS0\_GC0\_2x3 to achieve R=30,000 spectra covering 4100Å to 7800Å or 4825Å to 6750Å respectively. Exposure times ranged from 600 seconds to 1200 seconds. The signal-to-noise was typically 25 per resolution element at 5800 Å. The spectra were reduced with IRAF ECHELLE scripts. The standard IRAF scripts for overscan removal, bias subtraction, flat fielding and scattered light removal were employed. For the HRS flat field we masked out the Li I, H I and Na D regions because the HET HRS flat field lamp suffered from faint emission lines. The spectra were combined into a single long spectrum for the blue and red chips. Radial velocities were determined from cross-correlation using the IRAF task fxcor using the solar spectra. The heliocentric correction was made using the IRAF task rvcorrect. The radial velocities are discussed in more detail in §[debs]. Variable Stars ============== We present the detected variable stars in Table [vartab]. We have started a new nomenclature for all detections (except the one known variable V375 Cep), ordered by V magnitude. identified 8 stars as “suspected” variable stars and an additional 12 as “potential” (less likely) variables. Even though all of their systems were in our field and all but five were faint enough to be within the dynamic range of our images, only two of the stars in their list were identified as variables in our observations. Contact and Near-Contact Binaries --------------------------------- Because our calibration observations were taken over two relatively short periods on one night, we determined corrections to the calibrated magnitudes for each of our close binary systems to give us values at maximum light. These corrections were as large as about 0.1 mag in a few cases, which is enough to affect judgements about cluster membership from CMD position. Short-period contact and near-contact binary star systems have only been found in open clusters older than about 600 Myr. For fainter systems, this comes about as a result of the long timescales for tidal interactions to produce orbital decay. At the bright end though, the evolution of one component of a binary can accelerate the process. NGC 7142 has 4 contact or near-contact systems near the cluster turnoff, comparable in number to systems fainter on the main sequence. Similar groupings of systems are seen in old open clusters like M67 and NGC 7789, supporting the idea that mass transfer (and potentially coalescence of the stars) can produce blue straggler stars. This does not appear to be universal among open clusters, however., for example, found that such systems are not concentrated at the turnoff, but can be found spread brightward (among blue stragglers) or faintward (along the main sequence). Before drawing conclusions, we can assess whether the contact binaries are likely to be members using the period-luminosity-color (PLC) relationship from MV = − 4.44logP + 3.02(B − V)0 + 0.12 and comparing the implied distance modulus to what is estimated from isochrones. The significant differential reddening in NGC 7142 will reduce the sharpness of this tool by introducing additional random scatter of approximately 0.15 mag, but in some cases we should be able to make strong statements. Based on our later discussion of the red giant clump (§[dist] and [aged]), we find that the mean reddening for the cluster appears to be E(B − V) ≈ 0.36, with a distance modulus (m − M)V = 13.18. V5 has an implied distance modulus about 2 mag smaller [(m − M)V = 11.1], and V13 has a value almost a magnitude larger (14.0). Two of the remaining systems are very consistent with cluster membership (V3: 13.1; V9: 13.0), while two others are possible members (V7: 13.5; V12: 13.4). Because NGC 7142 is similar in age to other clusters (such as M67) known to have contact binary members, these new detections are understandable, although the systems don’t seem to show any sign of being concentrated toward the cluster center. V3 (Fig. [lcwuma]): This is one of the systems that was “suspected” to be variable by that we confirm as variable (their star 279). Based on the PLC relation, this system may be a cluster blue straggler. There are slight but distinct differences between the depths of the two eclipses, and there are also signs of light curve variability in the R data (which covers the largest range of time) on a timescale of weeks. Strangely, these light curve variations seem to primarily affect the maxima and minima. V5 (Fig. [lcwuma]): This is a “potential” variable previously identified by that we confirm here (their star 170). The depths of the eclipses in different bands differ significantly, with eclipses getting deeper for bluer filter bands. In other close binaries, this might imply light curve variability, yet we don’t see strong evidence of this. V5 is interesting because it is quite red compared to the main sequence, but falls very near the base of the giant branch. However, the PLC relation implies that this is probably a foreground system. V6 and V7 (Figs. [lcv6] and [lcv7]): Both of these systems fall very close to the cluster turnoff, and their variability may be related to evolution-driven size changes if they turn out to be cluster members. The eclipses for V6 have very different depths ( ∼ 0.42 mag versus 0.19 mag in R) indicating that the stars are close, but probably not in physical contact. As a result, the two stars may not have exchanged significant amounts of mass and the mass of the primary star may be a future constraint on the cluster age. V7 is probably a W UMa-type variable with a low inclination. The light curve shows an amplitude of about 0.02 mag as well as hints that the photometric minimums have different depth. A period of about 0.695 d seems to be preferred, but cycle-to-cycle variability interferes with a definitive determination of the ephemeris. V9 (Fig. [lcwuma]): This system has fairly deep eclipses that are similar in depth. The combined photometry places it to the red of the main sequence but near the expected location of the equal-mass binary sequence. V12 (Fig. [lcwuma]): This object is toward the faint end of our sample, and its amplitude is small and possibly variable. The system does fall near the main sequence in the CMD though. It is difficult to find a period under those conditions, but a period of approximately 0.29 d appears to to the best job of phasing the observations. V13 (Fig. [lcwuma]): V13 appears to show a total eclipse, making a membership determination of interest. V13’s corrected system photometry places it close to the cluster main sequence, although this portion of the CMD is fairly heavily contaminated by the field star population. The distance modulus derived from the PLC relationship is almost a magnitude greater than estimates for the cluster, making it unlikely to be a member. The system shows clear variability from night to night. The most notable example occured during a 10 day break in observations in R when the system’s median value dropped by approximately 0.04 mag. During the course of the observations in R, the light curve maxima varied over a range of about 0.10 mag. V14 (Fig. [lcwuma]): This is the faintest of the near-contact systems, and it also showed significant changes in brightness and amplitude over the course of 2 months of observations in R. There appeared to be some correlation between eclipse depth and the brightness level at the quadratures. Detached Eclipsing Binaries --------------------------- Using spectroscopic data for three of the detached eclipsing systems (V1, V2, and V375 Cep), we simultaneously determined the center-of-mass radial velocity and the mass ratio by χ2 minimization of a conservation of momentum condition. For all measured radial velocity pairs, we minimized ∑(vA − vCoM + q(vB − vCoM))2/(σA2 + σB2) The results are shown in Table [spectab]. There has been relatively little radial velocity work done on NGC 7142 that can be used to judge the membership of the measured stars. measured radial velocities for 13 stars in the field of the cluster using relatively low-resolution spectra, finding a mean value of − 44 km s− 1 (12 km s− 1 standard deviation, and typical measurement errors of 10 − 15 km s− 1 ). measured 6 stars and found − 48.6 ± 1.1 km s− 1 , while found − 50.3 ± 0.3 km s− 1 from higher resolution spectra of 4 of the same stars. If NGC 7142 is still in virial equilibrium, the cluster velocity dispersion should be small ($\la 1$ km s− 1 ) given likely masses for the cluster. However, the number of stars with high precision radial velocities is still small and the distribution of field stars has not been characterized, so membership judgements using the radial velocities should still be made with care. V1 (Fig. [lcv1]): Primary and secondary eclipses were initially detected with comparable depths, making this very likely to be a double-lined spectroscopic binary with stars of comparable mass. We subsequently confirmed two detectable components in spectra taken with the HET. Although the system falls in the blue straggler portion of the color-magnitude diagram, there was a possibility that the components are main sequence stars because differential reddening may affect the position of the blend in the CMD. Our three radial velocities imply a mass ratio q = 0.96 ± 0.02 and a system velocity vr = − 17.0 ± 1.0 km s− 1 . The system velocity puts it far away from the likely cluster mean. Based on the smalll displacement of the secondary eclipse from phase 0.5, the binary also has at least a small eccentricity. Binaries with periods this short ( < 5 d) appear to be able to circularize on timescales much shorter than NGC 7142’s age, although on its own this is not conclusive because some known blue stragglers have short periods and significant eccentricity. But based on these indications, we conclude it is most likely not a member of the cluster. V2 (Fig. [v2]): This appears to be a rather extraordinary eclipsing binary system. It was first identified in a single deep ( ∼ 0.75 mag) eclipse in our observations from 2005. While characterizing the light curve of V375 Cep (see below) in 2008, we detected two additional eclipses (a primary and a secondary) separated by about 19.03 d. At that time, we noticed that the secondary eclipse ingress was significantly longer in duration than the primary eclipse, implying a significant eccentricity. However, the large separation between the first and second observed eclipses (1017.3 d) in combination with the eccentricity made it difficult to determine the period. Using the photometric constraints along with three subsequent radial velocity measurements, we were able to narrow down the possibilities by requiring i) no photometric observations at constant light overlapped with predicted eclipses, and ii) radial velocities should have a physically realistic distribution in phase (e.g., no sudden changes in direction implied when phased). Using these constraints, the system was observed on dates with eclipses predicted for the most likely periods until a period of 15.6505 days was finally confirmed. With this period, the secondary eclipse is centered near phase ϕ = 0.22, consistent with a large eccentricity. The spectra of the system to date confirm that it is double-lined and that the system velocity ( − 42.1 ± 0.6 km s− 1 ) falls near the mean cluster value, but different by about several km s− 1 . At this time, we have to regard V2 as a possible cluster member. In spite of the measured mass ratio (q = 0.99 ± 0.10), the two stars differ significantly in size, showing the effects of evolution. When the depths of the eclipses are considered, the implied inclination of the orbit is very close to $90\degr$. In the CMD, V2 falls in the blue straggler region, brighter and bluer than the turnoff. Both of the stars individually have magnitudes placing them near the turnoff, and one or both may still be bluer than the turnoff when their light is disentangled. At this time, however, we cannot rule out the possibility that they are on a line of sight having less than the average cluster reddening. The characteristics of the orbits in V2 don’t appear to distinguish between these possibilities: the period at which most binaries have circularized in M67 (which has similar age to NGC 7142) is approximately 12.1− 1.5+ 1.0 d. V375 Cep (Fig. [v375]): This system was the primary motivation for many of the observations we took, and so we have observations of primary and secondary eclipses in all photometric bands. The secondary eclipse is clearly detected, and we have shown that it is a double-lined spectroscopic binary. The spectroscopic data give vCoM = − 49.1 ± 1.7 km s− 1 and q = 0.69 ± 0.03, and a minimum χν2 = 0.98. vCoM is completely consistent with the cluster average (see the beginning of this section), making it a highly probable cluster member. We collected a small number of previous photometric observations from and to improve the accuracy of the ephemeris and test for the possibility of a nonlinear ephemeris using the 27 y baseline. For the densely observed light curves from our study, we used the method of to determine times of minima and the errors. Most of the observations by agree well with our phased light curve and confirm that they observed the system very near one of the primary minima. They had observations in and out of eclipse on the night of one eclipse, which allowed us to fit our BV light curves to their data and derive an approximate time of minimum. One additional observation in V on a different date also appears to have fallen near an eclipse minimum. If there are no errors in their Table 2, the separation between their two faintest measurements in V (26.83 d) would be consistent with a period of about 1.91647 d. However, the implied change in period over the 20 years since the observations seems unrealistically large. A linear ephemeris that satisfactorily matches our data and one of the eclipse observations would cause a disagreement of about 0.06 in phase for the other, which is slightly longer than the entire duration of an eclipse ingress. In addition, most of the eclipse measurements disagree significantly with a linear ephemeris based on our data. They appear to have caught three measurements during the egress of a primary eclipse on one night (HJD 2446650), although the steeper slope and larger depth of their eclipse of when compared to ours makes it hard to trust estimations of the time of minimum. We have therefore elected to assign asymmetric error bars to that point. Two faint measurements in B seem likely to have been taken during ingress and egress respectively, if the period has remained near 1.9 d. However, another observation in B later on one of those nights was not consistent with out-of-eclipse levels or with predictions based on our eclipse light curves. Unfortunately it appears that the early data on this variable is not of sufficient quality to test for nonlinearities in the ephemeris. V8 (Fig. [lcv8]): This is a binary with partial eclipses of quite small amplitude ( ∼ 0.03 − 0.04 mag). Eclipses were detected on some consecutive nights separated by about 1.0546 days, although we find the light curves phase together better when half this period is used. If the true period is indeed near 0.53 d, a secondary eclipse is not detected. This system falls near the expected position of the equal-mass binary sequence, so it has a reasonable probability of cluster membership. However, if it is a cluster member, it would need to be a triple system, as this would explain both the shallow eclipses (through the diluting effects of third light) and the position in the CMD brighter than the main sequence. A third star with brightness similar to the primary of the binary system would need to be on a wider orbit. If the system is not a cluster member, it could involve a star being eclipsed by a smaller faint object. The significant out-of-eclipse variation implies that the companion must be of stellar mass in order to significantly distort the primary. V11 (Fig. [lcv11]): Eclipses were detected in R at HJD 2453594.97, 2453598.83, 2453638.70, and 2453639.99, in V at 2454676.74, 2454678.02 (only ingress), and 2455081.93, and in B at 2454627.85. In two cases, eclipses of nearly equal depth ( ∼ 0.2 mag) were observed on consecutive nights, and in one additional case, eclipses were separated by about 3.85 d. These observations make the period very likely to be near 1.3 d. Phasing the observations to this period, it appears we may have detected a shallow secondary eclipse in I band only, centered at phase 0.5 with approximate depth 0.03 mag. This system is therefore likely to be a single-line spectroscopic binary with a fairly small mass ratio. Although we do not have radial velocity measurements for this system, the position of the system in the CMD implies that there is a good chance of cluster membership. The primary eclipses appear to have a short period of totality. We also see strong out-of-eclipse light curve variations. For example, observations in 2005 separated by a little more than a month show large differences in the out-of-eclipse phases. In addition, a later night of observations in R shows the system brightness increasing before an eclipse, contrary to other observations at similar phase. The variations may be due to strong spot activity driven by rapid rotation in a system with such a short orbital period. ### Other Variables V4 shows variability on multiple timescales and with different amplitude. We have averaged measurements for each night in Fig. [lcv4] to make these trends clearer. Because of the large numbers of observations, the errors in most of these averages are smaller than the sizes of the points. During individual nights of observation there are small but significant trends of about 0.01 mag, particularly noted in our B and V observations. On the other hand, our observations reveal longer timescale variations (tens of days) of greater amplitude (at least 0.6 mag in B, V, and R). Comparisons of observations of measurements in different filter bands should be regarded as approximate — the medians in each band generally do not correspond to the identical times in the oscillation cycle, and there is evidence that the variation amplitude varies from band to band. V4 appears to be an single star in our best seeing images. The CMD position of the star sets it apart — it is quite red even for a star on the giant branch, and even accounting for realistic amounts of differential reddening. On the night of our calibration observations (HJD 2454765), the star appears to have been in a bright phase: more than 0.6 mag above the median of our B measurements, and about 0.2 mag above the median in V. (Note that we generally did not take observations in multiple bands during one night, so the median measurements are probably not representative of corresponding points in the variability cycle of the star.) However, V4 resides in a similar position in the CMDs of and. Our calibration observations and the observations of these other photometric studies were taken close together in time, so that they should be fairly representative of the object’s instantaneous color. Because they place the star to the red of the cluster giant branch, the calibrated color cannot be representative of a static star if it is a cluster member as it would be redder than the Hayashi line. The nature of the variability and its CMD position are somewhat reminiscent of two variable members of the similarly old cluster M67. S1063 and S1113 sit below the subgiant branch of M67 and are both X-ray sources and spectroscopic binaries. S1113 shows fairly periodic variations, while S1063 does not. In both cases, the variations are of much smaller amplitude than we see in V4. We have not detected variability among stars below the subgiant branch in NGC 7142, and they would be impossible to identify based on CMD position alone, thanks to field star contamination. We also carefully examined the other object that resides near V4 in the CMD, but saw no sign of variability. Spectroscopic information would be valuable to verify the stellar nature of the star, to check cluster membership (both by measuring radial velocities and by checking the possibility that it is a background giant). V10 appears to be a quasiperiodic variable. On any given night the star shows a trend in brightness of up to about 0.1 mag from beginning to end. Fig. [lcv10] shows our observations in RC, but similar variation appears in other filters as well. Using the Lafler-Kinman method, we found the most likely periods of variability to be approximately 1.38 and 2.77 d, but we have not been able to find a period that phases all of the datasets satisfactorily. The object is well separated from other sources, so that contamination of the photometry is unlikely. Based on its CMD position redder than the equal-mass binary sequence, it is not likely to be a cluster member. However, its unusual variation may be worth further study. Discussion ========== Overall, little work has been done on NGC 7142 compared to other open clusters, but as will be seen below, there is reason to believe the cluster’s characteristics should be revised. Before addressing the age of the cluster with a traditional isochrone fit, we first review previous estimates of the metallicity, distance, and reddening. Metallicity ----------- In the last decade or so, spectroscopic measurements of NGC 7142’s metallicity have been moving more metal-rich. This trend of cluster metallicities moving more metal-rich with time is not unique to this cluster (see for NGC 6791, for example), but the fact that the cluster now seems to be slightly more metal-rich than the Sun is important for accurate age determination. The first spectroscopic measurement for NGC 7142 by ([Fe/H] = − 0.23 ± 0.13) as part of a larger survey of open clusters was later revised upward to − 0.10 ± 0.10 by based on a new calibration of Fe spectroscopic indices. computed an even higher value ( + 0.04 ± 0.06) by transforming the same data to a common metallicity scale based on DDO photometry. Later spectroscopic measurements ( + 0.08 ± 0.06, ; + 0.14 ± 0.01 from 4 giants, ) also indicated a super-solar metallicity, however. NGC 7142 also appears to be more metal rich than the well-studied cluster M67 ([Fe/H] = 0.00 ± 0.09, ; + 0.03 ± 0.07, ) in studies where the two clusters can be compared reliably. Distance and Reddening ---------------------- determined that the reddening for the cluster was E(B − V) = 0.41 using the UBV color-color diagram, and this has been the most commonly quoted value since. found evidence of differential reddening on the order of ΔE(B − V) = 0.1 based on the width of the observed main sequence. Even though there are clear signs of a cloud front stretching ENE to WSW (with the amount of gas appearing to decrease toward the SSE), did not find evidence of a systematic change in reddening across the face of the cluster. They interpreted this to mean that there is significant smaller scale variation. settled on E(B − V) = 0.43 for turnoff stars and 0.40 for giants, based on a survey of previous studies including the two above. derived E(B − V) = 0.32 ± 0.05 using a method depending largely on comparing photometry of the red giant clump and main sequence turnoff to those of synthetic CMDs generated from isochrones. The reddening derived from IRAS and COBE dust maps is significantly larger [E(B − V) = 0.51 ± 0.02] than any previous determination for the cluster, but this might be a better indication of the reddening of stars beyond NGC 7142. The dust map values do vary in a range covering 0.09 mag within $5\arcmin$ of the cluster center, lending some credence to the possibility of differential reddening for the cluster. There has not been much agreement among previous distance modulus determinations either. The first estimate by was (m − M)V ≃ 13.7. quoted a dereddened distance modulus (m − M)0 = 11.4 ± 0.9 when using E(B − V) = 0.41, corresponding to an uncorrected distance modulus of approximately (m − M)V = 12.67. derive (m − M)V = 12.95 from main sequence fitting, assuming E(B − V) = 0.43. derive (m − M)0 = 11.85 ± 0.05, or (m − M)V = 12.84. Given the disagreements, it is worth revisiting the issue here. Of the features in the CMD on NGC 7142, the red giant clump is probably the most clearly identifiable, and some stars within it have been identified spectroscopically as likely cluster members. We identified candidate clump stars in optical and 2MASS infrared CMDs (see Fig. [clumpcmd]), and the 6 stars are given in Table [rctab]. Obviously, the sample is small, and there may be a field star or first-ascent giant star present, given the amount of differential reddening across the cluster’s face. In fact, one star (CT 399) was rejected based on radial velocity measurement. We obtained single radial velocity measurements for three other red clump stars (listed in Table [spectab]), and one of them (JH 2222) is consistent with membership, while the other two have velocities that are separated from the cluster mean by ∼ 4 km s− 1 and whose memberships are still in question. However, by using the median magnitude and color of the sample in 2MASS photometry (as did in their study of open cluster clump stars), the problems related to membership determination should be minimized. The color of the clump is largely a function of age in the range of composition containing NGC 7142 [∂(J − KBB)0/∂logtage ≈ − 0.17 dex− 1] according to the open cluster data of. Based on their study, the clump is predicted to have a color (J − KBB)0 = 0.67 ± 0.03, where we have assumed an age similar to M67 (4 Gyr) with a conservative uncertainty of about 1.5 Gyr. For NGC 7142, we find a median observed color (J − KBB) = 0.80, for a reddening E(J − KBB) = 0.13 ± 0.03 or E(B − V) = 0.25 ± 0.06. Using the median magnitude for NGC 7142 stars (KBB = 10.42), and the median absolute value from for stars similar in age and metallicity to NGC 7142 (⟨MK⟩ = − 1.62 ± 0.06), we find (m − M)K = 12.04 ± 0.15. These values imply (m − M)0 = 11.96 or (m − M)V = 12.72 ± 0.19. The reader should note that the 6 most probable clump stars are separated into two groups in K magnitude and (J − K) color, and the medians split the difference here. In the optical CMD used by, there is a greater degree of consistency in the V magnitudes of the clump stars, which is somewhat odd considering the apparent differential reddening across the cluster. After comparing with the results of previous studies, the two results derived from the clump (this study and that of ) give the largest values for the distance and smallest values for the reddening (with the exception of the early estimates by ). Earlier estimates were probably affected by degeneracy between the effect of dust extinction and distance, particularly because other features (like the turnoff) are difficult to identify in the cluster’s CMDs. The larger distance implies that the turnoff is more luminous than previously thought and that the age will tend to be on the low side, as discussed below. Age --- Part of the motivation for this project involved finding detached eclipsing binary star systems that contained evolved stars. Precise measurements of the masses and radii for stars that have evolved off the main sequence can lead to precise age determinations for the cluster. A detailed analysis of the two detached eclipsing binary systems sitting at the cluster turnoff (V375 Cep and V2) and the implications for the cluster age will be presented in an upcoming paper. However, it is beneficial to have as reliable an age determination from isochrones and the CMD for comparison. With the current CMDs, it is not easy to tell whether NGC 7142 turnoff stars are high enough in mass to produce a significant “hook” feature associated with core convection on the main sequence. A detailed isochrone fit for a feature of that kind is complicated by the short evolution timescale for stars moving through the hook, and the difficulties are exacerbated by differential reddening across the face of the cluster, field star contamination, and confusion due to optical blends of stars or true binaries. There is a weak indication of a subgiant branch at V ≈ 15.8, sloping faintward from the massive end of the main sequence, but more work is needed to establish the cluster membership of these stars. As a first check, we compare with M67, a well-studied cluster with an age thought to be near that of NGC 7142. M67 has a low and well-determined reddening (E(B − V) = 0.041 ± 0.004; ), and a well-determined distance modulus [(m − M)V = 9.72 ± 0.05, (m − M)0 = 9.60 ± 0.03; ]. M67 also appears to be a transition object, with evidence of weak core convection for the most massive turnoff stars, and a sparsely populated subgiant branch. Because photometry is available in BVIC and JHKS, differences in median magnitudes of the clump of the clusters in different filters allow a good estimate of the difference in distance moduli. In the 2MASS filters where the effects of extinction are minimized, these differences converge to about 2.4 mag. With a small correction for the extinction, this implies (m − M)0, N7142 − (m − M)0, M67 = 2.3, and (m − M)0 = 11.9. We can then look for consistency when the color-magnitude diagrams of the clusters are overlaid in different filter combinations in order to check the earlier estimate of the reddening. This requires that the extinctions and reddenings in different filter combinations can be reliably related, which is a complicated subject. For example, the measured reddening should depend on the spectral type of the star being observed because the flux-weighted wavelength of detected photons (the “effective wavelength”) changes. The use of extinction relationships (including the use of RV = AV/E(B − V) values different from the canonical 3.1) did not produce consistent CMDs in optical bands. In Fig. [m67comp], we show the results of using the method of to compute reddenings and extinctions from E(V − I), which was itself estimated from the shifts between the red clump stars in the two clusters and assuming the clump stars are approximately K1-K2 III giants. Calculations were done using the York Extinction Solver[1](#fn1). As can be seen, the resulting CMDs show that the clumps, giant branch stars and main sequence lines largely agree in position. The expected extinctions and reddening for turnoff stars (near F5 V) only differ from the clump values by 0.02 (in the optical) at most. So we conclude that differential effects due to the differing spectral energy distributions of giants and dwarfs are a minor effect on the quality of the fit. Based on the comparison with M67, NGC 7142 appears to be a slightly younger cluster than M67. Even with some uncertainty in where to align the clumps of the two clusters, there is a group of NGC 7142 stars that reach up to about 0.75 mag brighter than the turnoff of M67. Using measured values for M67 and the results of the fit, our preferred values are E(B − V) = 0.32 ± 0.06, (m − M)0 = 11.9 ± 0.15, and (m − M)V = 12.96 ± 0.24. The color difference between the turnoff and the giant branch is similar for the two clusters, although M67 has a larger difference in magnitude between the clump and the turnoff, implying that NGC 7142 is younger. At the suggestion of the referee, we also compared NGC 7142 to the more metal-rich ([Fe/H] = + 0.43; ) open cluster NGC 6253. If the NGC 6253 photometry is shifted [ΔV = 1.1 mag, Δ(B − V) = 0.02] so that the red clumps are once again in approximate agreement, the main sequences also agree as shown in Fig. [n6253comp]. The brightness of subgiant star candidates in NGC 6253 indicate that that cluster is similar in age to or slightly younger than NGC 7142. Present indications are than NGC 6253 is around 3 Gyr old (see for a summary of determinations), although the large metallicity of the cluster complicates its analysis. In Fig. [isos1], we present some illustrative comparisons with model isochrones from two groups. Using distance and reddening derived from the M67 comparison above and [Fe/H] equal to the most recent determination by, we find adequate matches to the main sequence, but imperfect agreement with spectroscopically identified giants. Decreased [Fe/H] improves the agreement with the giant branch, but does not eliminate it for reasonable choices ([Fe/H] > − 0.1)., for example, describe in detail difficulties with the color-temperature transformations that must be used to place theoretical models in the observational plane. For the Hyades, which have [Fe/H] similar to NGC 7142, several transformation algorithms do a reasonable job of reproducing the main sequence. However, when it comes to the older and more metal-rich cluster NGC 6791, the authors found that the giant branch was too red when transformations were used (as in both sets of isochrones in Fig. [isos1]). Given the current uncertainties in the transformations and potential problems related to the composition of super-metal rich stars (such as non-solar abundance ratios and helium abundance), we will put off further discussion of the isochrones until full analyses of the eclipsing binary stars is completed. If the colors of main sequence stars are reliably modelled, the distance and reddening derived from the red giant clump imply an age near 3 Gyr for the cluster. Larger ages do not adequately model the bright end of the main sequence or potential subgiant branch stars (15.5 < V < 16 and 1 < B − V < 1.3). This is greatly different from the age of nearly 7 Gyr derived by, in spite of the fact that they derived a similar distance and reddening. It may be that their technique (which involved fitting sub-solar and solar-composition synthetic CMDs rather than super-solar ones) was affected by some of the same mismatches on the giant branch we pointed out above. If we tried to fit the color difference between the main sequence and red giant branch at the level of the turnoff with current isochrones, a much larger age would be obtained, but the red clump’s magnitude would be poorly matched. Conclusions =========== We have conducted an extensive variability search among stars in the field of the cluster NGC 7142. Among the most important discoveries were several detached eclipsing systems near the cluster turnoff (two of which have been verified as cluster members based on radial velocities), a faint eclipsing binary with shallow eclipses indicative of a small stellar companion, and a long period variable found far to the red of the cluster giant branch. The detached eclipsing binaries allow us to put NGC 7142 on a growing list of clusters with multiple double-lined binary stars that can provide masses and radii for further tests of stellar models. We have used the red giant clump in an attempt to derive a more reliable distance and mean reddening for the cluster. The values derived here fall at the low end of previous determinations for reddening, and at the high end for distance. Our preferred values [E(B − V) = 0.32 ± 0.06, E(V − I) = 0.46, (m − M)0 = 11.9 ± 0.15] come from a fit of the 2MASS CMD of M67 to that of NGC 7142. With this distance, models indicate that the cluster’s age is near 3 Gyr. Future work is needed to check the agreement of theoretical models with the CMD, however, as there are lingering difficulties with color-$T\_{\rm eff}$ transformations and with our understanding of the compositions of super-metal-rich stars. NGC 7142 has characteristics putting it in an interesting regime. It appears to be more metal rich than the Sun or the well-studied open cluster M67, similar in metallicity to the Hyades, and more metal-poor than the very metal-rich cluster NGC 6791. As such, it can help further test metal-rich stellar models. NGC 7142’s age also is interesting — it is similar in age to M67, making it a cluster with main sequence stars that are near the transition between massive stars with convective cores and low-mass stars with radiative cores. The main difficulties in deriving accurate cluster properties are the fairly heavy reddening and field star contamination of the field. With future membership surveys (radial velocity or proper motion) or with the use of Stromgren photometry, these difficulties can be overcome. This work has been funded through grant AST 09-08536 from the National Science Foundation to E.L.S. We would like to thank the Director of Mount Laguna Observatory (P. Etzel) for generous allocations of observing time. Infrastructure support for the observatory was generously provided by the National Science Foundation through the Program for Research and Education using Small Telescopes (PREST) under grant AST 05-19686. We would also like to thank A. Bostroem, C. Gabler, and J. B. Leep for assisting with the photometric calibration observations, and Mark Jeffries for taking some of the later time-series photometry. The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universitat Munchen, and Georg-August-Universitat Gottingen. 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B., Deng, L., & Lu, P. 2009,, 138, 680 R phased light curve for the probable contact binary V7. [lcv7] Light curves (magnitude minus the median value in each filter band) for the long-period variable V4 in B ( • ), V ( ○ ), RC ( × ), and IC ( \* ). Individual points are averages of measurements made during time intervals up to 0.05 d. Measurements in B and RC have been shifted to bring them into approximate agreement with V measurements on HJDs 2454419 and 2453653, respectively. [lcv4] Comparison of NGC 7142 photometry with likely single-star members of M67 ($\sq$; ). Other symbols have the same meaning as in Fig. [clumpcmd]. The photometry for M67 has been shifted in color to bring red clump stars into agreement in V − I, and the implied E(V − I) was used to compute reddenings and extinctions in other filters according to the method of. Reddening vectors are shown in each diagram. [m67comp] clcl\|clcl Aug. 10, 2005 & R & 3593.755 & 44 & Jul. 7, 2008 & Ic & 4655.698 & 158 Aug. 11, 2005 & R & 3594.687 & 39 & Jul. 8, 2008 & Ic & 4656.686 & 204 Aug. 13, 2005 & R & 3596.670 & 46 & Jul. 9, 2008 & Ic & 4657.684 & 96 Aug. 14, 2005 & R & 3597.659 & 19 & Jul. 27, 2008 & V & 4675.660 & 97 Aug. 15, 2005 & R & 3598.642 & 50 & Jul. 28, 2008 & V & 4676.655 & 103 Aug. 16, 2005 & R & 3599.666 & 42 & Jul. 29, 2008 & V & 4677.658 & 105 Aug. 17, 2005 & R & 3600.659 & 48 & Jul. 31, 2008 & V & 4678.653 & 107 Sep. 21, 2005 & R & 3635.749 & 40 & Aug. 18, 2008 & V & 4697.952 & 34 Sep. 22, 2005 & R & 3636.607 & 32 & Oct. 6, 2008 & VRK & 4746.674 & 11,10 Sep. 23, 2005 & R & 3637.613 & 68 & Oct. 25, 2008 & BVIC & 4765.686 & 9,8,11 Sep. 24, 2005 & R & 3638.612 & 30 & Sep. 6, 2009 & V & 5081.895 & 31 Sep. 25, 2005 & R & 3639.605 & 68 & Oct. 9, 2009 & V & 5114.592 & 41 Sep. 26, 2005 & R & 3640.704 & 28 & Oct. 16, 2009 & V & 5121.578 & 66 Sep. 27, 2005 & R & 3641.608 & 48 & Nov. 10, 2009 & V & 5146.575 & 52 Oct. 6, 2005 & R & 3650.607 & 38 & Nov. 29, 2009 & V & 5165.590 & 13 Oct. 7, 2005 & R & 3651.608 & 46 & May 19, 2010 & IC & 5336.815 & 37 Oct. 8, 2005 & R & 3652.597 & 33 & Jun. 7, 2010 & B & 5355.742 & 55 Oct. 9, 2005 & BVR & 3653.600 & 11,11,11 & Jun. 24, 2010 & V & 5372.709 & 68 Nov. 12, 2007 & BVRK & 4417.703 & 11,16,10 & Jul. 5, 2010 & R & 5383.674 & 76 Nov. 14, 2007 & BVRK & 4419.612 & 13,11,11 & Jul. 12, 2010 & IC & 5390.669 & 75 Jun. 8, 2008 & B & 4626.780 & 55 & Aug. 2, 2010 & VIC & 5411.665 & 12,61 Jun. 9, 2008 & B & 4627.796 & 50 & Aug. 3, 2010 & IC & 5412.660 & 11 Jun. 11, 2008 & B & 4629.764 & 57 & Aug. 14, 2010 & V & 5423.686 & 18 Jun. 12, 2008 & B & 4630.753 & 60 & Nov. 16, 2010 & VIC & 5517.570 & 70,5 Jun. 15, 2008 & B & 4633.746 & 60 & Jun. 23, 2011 & IC & 5736.721 & 48 Jun. 16, 2008 & B & 4634.747 & 57 & Jul. 12, 2011 & RIC & 5755.670 & 16,65 Jun. 17, 2008 & Ic & 4635.740 & 159 [phottab] lccr V1 & − 17.0 & 1.0 & 3 V2 & − 42.1 & 0.6 & 13 V375 Cep & − 49.1 & 1.7 & 21 JH2222& − 49.6 & 0.1 & 1 JH2288& − 44.0 & 0.1 & 1 JH1003& − 43.9 & 0.1 & 1 [spectab] rrcllllllll V1 & 395 & EA & 21:44:32.86 & +65:45:26.3 & 14.864 & 0.831 & 1.039 & 4.6691 & 2453639.93 & rv prob. nm V2 & 18 & EA & 21:44:29.62 & +65:48:43.9 & 15.310 & 0.797 & 1.010 & 15.6505 & 2453639.684 & deep eclipses; eccentric; rv mem V3 & 155 & EW & 21:45:15.16 & +65:49:24.3 & 15.38 & 0.72 & 0.93 & 0.580793 & 2453594.80 & straggler? V4 & 430 & LPV & 21:44:55.98 & +65:45:50.0 & 15.43 & 1.89 & 2.06 & & & VH Y V5 & 199? & EW & 21:45:30.03 & +65:46:42.3 & 15.80 & 1.18 & 1.38 & 0.3368235 & 2453596.782 & nearby faint star V6 & & EB & 21:44:13.21 & +65:45:01.3 & 15.71 & 0.91 & 1.15 & 0.441163 & 2453599.757 & V7 & 288 & EW & 21:45:10.77 & +65:44:41.3 & 15.95 & 0.89 & 1.12 & 0.69531? & & VH f & & EA & 21:44:54.74 & +65:43:55.3 & 16.115 & 0.877 & 1.080 & 1.909682 & 2454676.8071 & V375 Cep; rv mem V8 & & EA & 21:44:35.73 & +65:45:04.7 & 17.288 & 1.106 & 1.373 & 0.52730 & 2453596.985 & triple? V9 & & EW & 21:44:28.44 & +65:46:36.6 & 17.63 & 1.14 & 1.34 & 0.330233 & 2453593.918 & V10 & 356 & Irr & 21:45:40.73 & +65:43:40.0 & 17.63 & 1.25 & 1.53 & & & V11 & & EA & 21:44:17.92 & +65:50:04.9 & 17.906 & 1.055 & 1.304 & 1.28629 & 2453598.825 & V12 & & EW? & 21:44:55.46 & +65:52:59.0 & 18.330 & 1.157 & 1.340 & 0.2938 & 2453636.79 & V13 & & EW & 21:44:20.66 & +65:44:59.1 & 18.35 & 1.08 & 1.20 & 0.34207 & 2453598.89 & total eclipses, prob. nm V14 & & EW & 21:46:06.8 & +65:52:53.6 & & & & 0.29005 & 2453600.75 & not detected in calibrating obs. [vartab] rrcccccccl 421 & 1065 & 21:45:00.73 & +65:45:56.5 & 13.452 & 1.364 & 1.468 & 10.097 & 0.802 & I; rv, abund mem 170 & 1519 & 21:45:20.43 & +65:48:31.0 & 13.743 & 1.380 & 1.526 & 10.237 & 0.794 & O,rv mem? 203 & 1767 & 21:45:32.93 & +65:47:31.1 & 13.765 & 1.326 & 1.147 & 10.533 & 0.718 & rv, abund mem;nearby star affecting phot? & 2222 & 21:45:57.03 & +65:44:10.5 & 13.665 & 1.411 & 1.485 & 10.268 & 0.768 & rv mem,off CT field 93 & 1003 & 21:44:57.60 & +65:48:36.8 & 13.735 & 1.294 & 1.412 & 10.490 & 0.734 & rv mem? & 2288 & 21:46:00.93 & +65:49:06.8 & 13.887 & 1.360 & 1.476 & 10.526 & 0.736 & rv mem?,off CT field 399 & 591 & 21:44:36.70 & +65:43:19.1 & 13.448 & 1.375 & 1.531 & 9.998 & 0.845 & rv nm [rctab] lr\|lr\|lr ΔE(V − I) & 0.41 & Δτ1 & 0.335 & & ΔAB & 1.22 & Δ(B − V) & 0.28 & AV/E(B − V) & 3.33 ΔAV & 0.94 & & & AV/E(V − I) & 2.30 ΔAKs & 0.11 & ΔE(J − Ks) & 0.15 & AKs/E(J − Ks) & 0.76 [exttab] --- 1. http://www2.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/YorkExtinctionSolver/[↩](#fnref1) Variable Stars in the Open Cluster NGC 7142 =========================================== We present new discoveries of variable stars near the turnoff of the old open cluster NGC 7142. Contrary to previous studies, we detect eight contact or near contact eclipsing binaries (including three near the cluster turnoff), and most of these have good probability of being cluster members. We also identified one long-period variable that resides far to the red of the cluster giant branch, and four new detached eclipsing binaries. We have re-examined the question of distance and reddening for the cluster, and find that the distance is larger and reddening lower than in most previous studies. In turn this implies that NGC 7142 is probably slightly younger than M67, about 3 Gyr old. With an age of this size, NGC 7142 would be one of a small group of clusters with main sequence turnoff stars at the transition between convective and radiative cores. Introduction ============ Star clusters have long been testbeds for our understanding of the evolution of stars, the physics of the gas in their interiors, and the dynamics of the member stars. Variable stars can provide a great deal of additional information that allows us to tighten the screws on theoretical models attempting to predict cluster stellar characteristics from first principles. Pulsating stars can provide information about internal structure and stellar mass. Short-period semi-detached and contact binaries may contain encoded information about the dynamical history of the cluster. But perhaps most valuable of all are detached eclipsing binaries because they can provide high precision measurements of masses and radii for individual stars with a minimum of theoretical interpretation. Variability studies of NGC 7142 have previously been carried out by and. searched for low-amplitude variables on a single night of observations, finding eight stars with significant light variations. expected to find a number of short-period variable stars based on discoveries in clusters like NGC 188. V375 Cep was the only variable star they discovered in the cluster, although they lacked sufficient observations to determine a period. The variable is of interest because its photometry puts it very close to the turnoff of the cluster’s color-magnitude diagram, within the main sequence band. Masses for these stars would be valuable constraints on the cluster age, but in addition, one of the two stars could be significantly evolved from the zero-age main sequence. This means that the radius of the star could help refine the age of the cluster. [One other semi-regular variable, V582 Cep, was known to be in the field of the cluster, but was outside the field observed by us. In one of the earliest studies of the cluster, found that NGC 7142 was intermediate in age between NGC 7789 and M67, approximately 1.5 − 4 Gyr. found that it was intermediate in age between M67 and NGC 188, or approximately 4 − 5 Gyr. estimated an age of 4.9 Gyr by comparing synthetic color-magnitude diagrams (generated from model isochrones) with photometry. give an age of 4 ± 1 Gyr for NGC 7142 based on the magnitude difference between the subgiant branch and the red giant clump. Recently though, derived a much larger age of 6.9 ± 0.8 Gyr, using synthetic CMDs to match the turnoff color and red giant clump color and magnitude. Much of the disagreement may be the result of differential extinction in the cluster and the small numbers of evolved stars, making the reference points in the CMD difficult to identify. In order to improve the precision of the age determination for this old open cluster (and others), we have undertaken a program to identify and analyze age-sensitive eclipsing binary stars. This paper presents the results of our variable search, and a following study presents and analysis of the most promising candidates. Observational Material and Data Reduction ========================================= Photometry ---------- All of the photometry for this study was taken with the Mount Laguna Observatory 1 m telescope using a camera having a 2048 × 2048 pixel CCD and approximately a $13\farcm5\times13\farcm5$ field of view. Table [phottab] lists the nights during which images were taken for this article. 17 nights of data were taken in RC band for the purpose of identifying variable stars in the cluster, and subsequent nights were taken primarily to refine the ephemerides of the most important variables and to determine light curves in different filter bands. Differential photometry was undertaken using the image subtraction package ISIS. Our procedure is very similar to that of, so for most details, we refer the reader to that paper. Our image sets in B, V, RC, and IC filters were interpolated to a common image coordinate system, and then processed separately by filter. The output of ISIS is a difference flux measured on the subtracted images. To convert these difference fluxes into magnitudes, star fluxes were measured on the reference frame. ISIS’s algorithm is a modified aperture photometry routine that employs the reference image point-spread function (PSF) for weighting purposes. The reference PSF is first transformed to the seeing of the image under consideration. Only the portion within an aperture of radphot pixels will be used, but it is normalized to a larger aperture of rad\_aper pixels. The pixel values in the subtracted image are weighted by this transformed PSF. In this study, we used radphot = 4 pix and rad\_aper = 10 pix. We discovered a bug in the determination of uncertainties in the difference fluxes derived by ISIS. As part of the image subtraction process, the overall flux scaling from frame to frame is fit and corrected in ISIS. This scaling affects the size of the noise and therefore the flux uncertainties, but this was not done in the last version of ISIS. This had the unfortunate effect of depressing computed uncertainties on measurements with low fluxes. We fixed this error in our copy of the ISIS code (version 2.2), and all of the results here use this corrected version. ### Photometric Calibration We took calibration images under photometric conditions on the night of 25 October 2008. We observed the standard fields PG0231+051, SA 92, SA95, SA 98, and NGC 6940, and used standard values taken from. The standard fields were observed between 3 and 10 times per filter in BVI at airmasses that ranged from 1.034 to 1.998. All together this resulted in more than 4000 standard star observations per filter covering a color range $-0.5 \la (B-I) \la 5$. We derived aperture photometry from all frames using DAOPHOT, and made curve of growth corrections using the program DAOGROW. The observations were transformed to the standard system using the following equations: b = B + a0 + a1(B − I) + a2X v = V + b0 + b1(B − I) + b2X i = I + c0 + c1(B − I) + c2X where b, v, and i are instrumental magnitudes, B, V, and I are standard-system magnitudes, X is airmass, and ai, bi, and ci are coefficients determined from least-squares fits. Fig. [stetcomp] shows the residuals of the comparison of our standardized observations and the Stetson values. Clusters to be calibrated were observed with a range of exposures times on the same night. For NGC 7142, there were 9 observations in B (1 × 60 s, 5 × 120 s, and 3 × 300 s), 8 observations in V (3 × 60 s, 2 × 120 s, and 3 × 300 s), and 11 observations in I (4 × 10 s, 2 × 60 s, 2 × 120 s, and 3 × 300 s). Little attention had been paid to NGC 7142 in the past, with providing the most extensive set of photometry. Fig. [ctcomp] shows a comparison of our photometry with theirs. There are small zeropoint differences between the two studies, but more importantly, there is a clear trend with color. We believe our calibration is superior based on number of standards and improved technique. We have also compared to the more recent photometry by. The zeropoint differences are again small as shown in Fig. [janescomp], but there are also no obvious trends with color. The color-magnitude diagram for our field is shown in Fig. [cmd]. Note that much of the scatter probably comes from differential reddening resulting from the cluster’s proximity on the sky to the reflection nebula NGC 7129 (although the cluster is well behind the cloud). However, to help identify the cluster’s sparsely populated giant branch, we identified all the stars with radial velocities from spectra that have been presented in the literature. Spectra ------- Spectra for three variable star targets and three red clump star candidates were obtained at the Hobby-Eberly Telescope (HET) with the High Resolution Spectrograph (HRS, ) as part of normal queue scheduled observing. The observed stars are listed in Table [spectab]. HRS was configured to HRS\_30k\_central\_316g5936\_2as\_0sky\_IS0\_GC0\_2x3 or HRS\_30k\_central\_600g5822\_2as\_2sky\_IS0\_GC0\_2x3 to achieve R=30,000 spectra covering 4100Å to 7800Å or 4825Å to 6750Å respectively. Exposure times ranged from 600 seconds to 1200 seconds. The signal-to-noise was typically 25 per resolution element at 5800 Å. The spectra were reduced with IRAF ECHELLE scripts. The standard IRAF scripts for overscan removal, bias subtraction, flat fielding and scattered light removal were employed. For the HRS flat field we masked out the Li I, H I and Na D regions because the HET HRS flat field lamp suffered from faint emission lines. The spectra were combined into a single long spectrum for the blue and red chips. Radial velocities were determined from cross-correlation using the IRAF task fxcor using the solar spectra. The heliocentric correction was made using the IRAF task rvcorrect. The radial velocities are discussed in more detail in §[debs]. Variable Stars ============== We present the detected variable stars in Table [vartab]. We have started a new nomenclature for all detections (except the one known variable V375 Cep), ordered by V magnitude. identified 8 stars as “suspected” variable stars and an additional 12 as “potential” (less likely) variables. Even though all of their systems were in our field and all but five were faint enough to be within the dynamic range of our images, only two of the stars in their list were identified as variables in our observations. Contact and Near-Contact Binaries --------------------------------- Because our calibration observations were taken over two relatively short periods on one night, we determined corrections to the calibrated magnitudes for each of our close binary systems to give us values at maximum light. These corrections were as large as about 0.1 mag in a few cases, which is enough to affect judgements about cluster membership from CMD position. Short-period contact and near-contact binary star systems have only been found in open clusters older than about 600 Myr. For fainter systems, this comes about as a result of the long timescales for tidal interactions to produce orbital decay. At the bright end though, the evolution of one component of a binary can accelerate the process. NGC 7142 has 4 contact or near-contact systems near the cluster turnoff, comparable in number to systems fainter on the main sequence. Similar groupings of systems are seen in old open clusters like M67 and NGC 7789, supporting the idea that mass transfer (and potentially coalescence of the stars) can produce blue straggler stars. This does not appear to be universal among open clusters, however., for example, found that such systems are not concentrated at the turnoff, but can be found spread brightward (among blue stragglers) or faintward (along the main sequence). Before drawing conclusions, we can assess whether the contact binaries are likely to be members using the period-luminosity-color (PLC) relationship from MV = − 4.44logP + 3.02(B − V)0 + 0.12 and comparing the implied distance modulus to what is estimated from isochrones. The significant differential reddening in NGC 7142 will reduce the sharpness of this tool by introducing additional random scatter of approximately 0.15 mag, but in some cases we should be able to make strong statements. Based on our later discussion of the red giant clump (§[dist] and [aged]), we find that the mean reddening for the cluster appears to be E(B − V) ≈ 0.36, with a distance modulus (m − M)V = 13.18. V5 has an implied distance modulus about 2 mag smaller [(m − M)V = 11.1], and V13 has a value almost a magnitude larger (14.0). Two of the remaining systems are very consistent with cluster membership (V3: 13.1; V9: 13.0), while two others are possible members (V7: 13.5; V12: 13.4). Because NGC 7142 is similar in age to other clusters (such as M67) known to have contact binary members, these new detections are understandable, although the systems don’t seem to show any sign of being concentrated toward the cluster center. V3 (Fig. [lcwuma]): This is one of the systems that was “suspected” to be variable by that we confirm as variable (their star 279). Based on the PLC relation, this system may be a cluster blue straggler. There are slight but distinct differences between the depths of the two eclipses, and there are also signs of light curve variability in the R data (which covers the largest range of time) on a timescale of weeks. Strangely, these light curve variations seem to primarily affect the maxima and minima. V5 (Fig. [lcwuma]): This is a “potential” variable previously identified by that we confirm here (their star 170). The depths of the eclipses in different bands differ significantly, with eclipses getting deeper for bluer filter bands. In other close binaries, this might imply light curve variability, yet we don’t see strong evidence of this. V5 is interesting because it is quite red compared to the main sequence, but falls very near the base of the giant branch. However, the PLC relation implies that this is probably a foreground system. V6 and V7 (Figs. [lcv6] and [lcv7]): Both of these systems fall very close to the cluster turnoff, and their variability may be related to evolution-driven size changes if they turn out to be cluster members. The eclipses for V6 have very different depths ( ∼ 0.42 mag versus 0.19 mag in R) indicating that the stars are close, but probably not in physical contact. As a result, the two stars may not have exchanged significant amounts of mass and the mass of the primary star may be a future constraint on the cluster age. V7 is probably a W UMa-type variable with a low inclination. The light curve shows an amplitude of about 0.02 mag as well as hints that the photometric minimums have different depth. A period of about 0.695 d seems to be preferred, but cycle-to-cycle variability interferes with a definitive determination of the ephemeris. V9 (Fig. [lcwuma]): This system has fairly deep eclipses that are similar in depth. The combined photometry places it to the red of the main sequence but near the expected location of the equal-mass binary sequence. V12 (Fig. [lcwuma]): This object is toward the faint end of our sample, and its amplitude is small and possibly variable. The system does fall near the main sequence in the CMD though. It is difficult to find a period under those conditions, but a period of approximately 0.29 d appears to to the best job of phasing the observations. V13 (Fig. [lcwuma]): V13 appears to show a total eclipse, making a membership determination of interest. V13’s corrected system photometry places it close to the cluster main sequence, although this portion of the CMD is fairly heavily contaminated by the field star population. The distance modulus derived from the PLC relationship is almost a magnitude greater than estimates for the cluster, making it unlikely to be a member. The system shows clear variability from night to night. The most notable example occured during a 10 day break in observations in R when the system’s median value dropped by approximately 0.04 mag. During the course of the observations in R, the light curve maxima varied over a range of about 0.10 mag. V14 (Fig. [lcwuma]): This is the faintest of the near-contact systems, and it also showed significant changes in brightness and amplitude over the course of 2 months of observations in R. There appeared to be some correlation between eclipse depth and the brightness level at the quadratures. Detached Eclipsing Binaries --------------------------- Using spectroscopic data for three of the detached eclipsing systems (V1, V2, and V375 Cep), we simultaneously determined the center-of-mass radial velocity and the mass ratio by χ2 minimization of a conservation of momentum condition. For all measured radial velocity pairs, we minimized ∑(vA − vCoM + q(vB − vCoM))2/(σA2 + σB2) The results are shown in Table [spectab]. There has been relatively little radial velocity work done on NGC 7142 that can be used to judge the membership of the measured stars. measured radial velocities for 13 stars in the field of the cluster using relatively low-resolution spectra, finding a mean value of − 44 km s− 1 (12 km s− 1 standard deviation, and typical measurement errors of 10 − 15 km s− 1 ). measured 6 stars and found − 48.6 ± 1.1 km s− 1 , while found − 50.3 ± 0.3 km s− 1 from higher resolution spectra of 4 of the same stars. If NGC 7142 is still in virial equilibrium, the cluster velocity dispersion should be small ($\la 1$ km s− 1 ) given likely masses for the cluster. However, the number of stars with high precision radial velocities is still small and the distribution of field stars has not been characterized, so membership judgements using the radial velocities should still be made with care. V1 (Fig. [lcv1]): Primary and secondary eclipses were initially detected with comparable depths, making this very likely to be a double-lined spectroscopic binary with stars of comparable mass. We subsequently confirmed two detectable components in spectra taken with the HET. Although the system falls in the blue straggler portion of the color-magnitude diagram, there was a possibility that the components are main sequence stars because differential reddening may affect the position of the blend in the CMD. Our three radial velocities imply a mass ratio q = 0.96 ± 0.02 and a system velocity vr = − 17.0 ± 1.0 km s− 1 . The system velocity puts it far away from the likely cluster mean. Based on the smalll displacement of the secondary eclipse from phase 0.5, the binary also has at least a small eccentricity. Binaries with periods this short ( < 5 d) appear to be able to circularize on timescales much shorter than NGC 7142’s age, although on its own this is not conclusive because some known blue stragglers have short periods and significant eccentricity. But based on these indications, we conclude it is most likely not a member of the cluster. V2 (Fig. [v2]): This appears to be a rather extraordinary eclipsing binary system. It was first identified in a single deep ( ∼ 0.75 mag) eclipse in our observations from 2005. While characterizing the light curve of V375 Cep (see below) in 2008, we detected two additional eclipses (a primary and a secondary) separated by about 19.03 d. At that time, we noticed that the secondary eclipse ingress was significantly longer in duration than the primary eclipse, implying a significant eccentricity. However, the large separation between the first and second observed eclipses (1017.3 d) in combination with the eccentricity made it difficult to determine the period. Using the photometric constraints along with three subsequent radial velocity measurements, we were able to narrow down the possibilities by requiring i) no photometric observations at constant light overlapped with predicted eclipses, and ii) radial velocities should have a physically realistic distribution in phase (e.g., no sudden changes in direction implied when phased). Using these constraints, the system was observed on dates with eclipses predicted for the most likely periods until a period of 15.6505 days was finally confirmed. With this period, the secondary eclipse is centered near phase ϕ = 0.22, consistent with a large eccentricity. The spectra of the system to date confirm that it is double-lined and that the system velocity ( − 42.1 ± 0.6 km s− 1 ) falls near the mean cluster value, but different by about several km s− 1 . At this time, we have to regard V2 as a possible cluster member. In spite of the measured mass ratio (q = 0.99 ± 0.10), the two stars differ significantly in size, showing the effects of evolution. When the depths of the eclipses are considered, the implied inclination of the orbit is very close to $90\degr$. In the CMD, V2 falls in the blue straggler region, brighter and bluer than the turnoff. Both of the stars individually have magnitudes placing them near the turnoff, and one or both may still be bluer than the turnoff when their light is disentangled. At this time, however, we cannot rule out the possibility that they are on a line of sight having less than the average cluster reddening. The characteristics of the orbits in V2 don’t appear to distinguish between these possibilities: the period at which most binaries have circularized in M67 (which has similar age to NGC 7142) is approximately 12.1− 1.5+ 1.0 d. V375 Cep (Fig. [v375]): This system was the primary motivation for many of the observations we took, and so we have observations of primary and secondary eclipses in all photometric bands. The secondary eclipse is clearly detected, and we have shown that it is a double-lined spectroscopic binary. The spectroscopic data give vCoM = − 49.1 ± 1.7 km s− 1 and q = 0.69 ± 0.03, and a minimum χν2 = 0.98. vCoM is completely consistent with the cluster average (see the beginning of this section), making it a highly probable cluster member. We collected a small number of previous photometric observations from and to improve the accuracy of the ephemeris and test for the possibility of a nonlinear ephemeris using the 27 y baseline. For the densely observed light curves from our study, we used the method of to determine times of minima and the errors. Most of the observations by agree well with our phased light curve and confirm that they observed the system very near one of the primary minima. They had observations in and out of eclipse on the night of one eclipse, which allowed us to fit our BV light curves to their data and derive an approximate time of minimum. One additional observation in V on a different date also appears to have fallen near an eclipse minimum. If there are no errors in their Table 2, the separation between their two faintest measurements in V (26.83 d) would be consistent with a period of about 1.91647 d. However, the implied change in period over the 20 years since the observations seems unrealistically large. A linear ephemeris that satisfactorily matches our data and one of the eclipse observations would cause a disagreement of about 0.06 in phase for the other, which is slightly longer than the entire duration of an eclipse ingress. In addition, most of the eclipse measurements disagree significantly with a linear ephemeris based on our data. They appear to have caught three measurements during the egress of a primary eclipse on one night (HJD 2446650), although the steeper slope and larger depth of their eclipse of when compared to ours makes it hard to trust estimations of the time of minimum. We have therefore elected to assign asymmetric error bars to that point. Two faint measurements in B seem likely to have been taken during ingress and egress respectively, if the period has remained near 1.9 d. However, another observation in B later on one of those nights was not consistent with out-of-eclipse levels or with predictions based on our eclipse light curves. Unfortunately it appears that the early data on this variable is not of sufficient quality to test for nonlinearities in the ephemeris. V8 (Fig. [lcv8]): This is a binary with partial eclipses of quite small amplitude ( ∼ 0.03 − 0.04 mag). Eclipses were detected on some consecutive nights separated by about 1.0546 days, although we find the light curves phase together better when half this period is used. If the true period is indeed near 0.53 d, a secondary eclipse is not detected. This system falls near the expected position of the equal-mass binary sequence, so it has a reasonable probability of cluster membership. However, if it is a cluster member, it would need to be a triple system, as this would explain both the shallow eclipses (through the diluting effects of third light) and the position in the CMD brighter than the main sequence. A third star with brightness similar to the primary of the binary system would need to be on a wider orbit. If the system is not a cluster member, it could involve a star being eclipsed by a smaller faint object. The significant out-of-eclipse variation implies that the companion must be of stellar mass in order to significantly distort the primary. V11 (Fig. [lcv11]): Eclipses were detected in R at HJD 2453594.97, 2453598.83, 2453638.70, and 2453639.99, in V at 2454676.74, 2454678.02 (only ingress), and 2455081.93, and in B at 2454627.85. In two cases, eclipses of nearly equal depth ( ∼ 0.2 mag) were observed on consecutive nights, and in one additional case, eclipses were separated by about 3.85 d. These observations make the period very likely to be near 1.3 d. Phasing the observations to this period, it appears we may have detected a shallow secondary eclipse in I band only, centered at phase 0.5 with approximate depth 0.03 mag. This system is therefore likely to be a single-line spectroscopic binary with a fairly small mass ratio. Although we do not have radial velocity measurements for this system, the position of the system in the CMD implies that there is a good chance of cluster membership. The primary eclipses appear to have a short period of totality. We also see strong out-of-eclipse light curve variations. For example, observations in 2005 separated by a little more than a month show large differences in the out-of-eclipse phases. In addition, a later night of observations in R shows the system brightness increasing before an eclipse, contrary to other observations at similar phase. The variations may be due to strong spot activity driven by rapid rotation in a system with such a short orbital period. ### Other Variables V4 shows variability on multiple timescales and with different amplitude. We have averaged measurements for each night in Fig. [lcv4] to make these trends clearer. Because of the large numbers of observations, the errors in most of these averages are smaller than the sizes of the points. During individual nights of observation there are small but significant trends of about 0.01 mag, particularly noted in our B and V observations. On the other hand, our observations reveal longer timescale variations (tens of days) of greater amplitude (at least 0.6 mag in B, V, and R). Comparisons of observations of measurements in different filter bands should be regarded as approximate — the medians in each band generally do not correspond to the identical times in the oscillation cycle, and there is evidence that the variation amplitude varies from band to band. V4 appears to be an single star in our best seeing images. The CMD position of the star sets it apart — it is quite red even for a star on the giant branch, and even accounting for realistic amounts of differential reddening. On the night of our calibration observations (HJD 2454765), the star appears to have been in a bright phase: more than 0.6 mag above the median of our B measurements, and about 0.2 mag above the median in V. (Note that we generally did not take observations in multiple bands during one night, so the median measurements are probably not representative of corresponding points in the variability cycle of the star.) However, V4 resides in a similar position in the CMDs of and. Our calibration observations and the observations of these other photometric studies were taken close together in time, so that they should be fairly representative of the object’s instantaneous color. Because they place the star to the red of the cluster giant branch, the calibrated color cannot be representative of a static star if it is a cluster member as it would be redder than the Hayashi line. The nature of the variability and its CMD position are somewhat reminiscent of two variable members of the similarly old cluster M67. S1063 and S1113 sit below the subgiant branch of M67 and are both X-ray sources and spectroscopic binaries. S1113 shows fairly periodic variations, while S1063 does not. In both cases, the variations are of much smaller amplitude than we see in V4. We have not detected variability among stars below the subgiant branch in NGC 7142, and they would be impossible to identify based on CMD position alone, thanks to field star contamination. We also carefully examined the other object that resides near V4 in the CMD, but saw no sign of variability. Spectroscopic information would be valuable to verify the stellar nature of the star, to check cluster membership (both by measuring radial velocities and by checking the possibility that it is a background giant). V10 appears to be a quasiperiodic variable. On any given night the star shows a trend in brightness of up to about 0.1 mag from beginning to end. Fig. [lcv10] shows our observations in RC, but similar variation appears in other filters as well. Using the Lafler-Kinman method, we found the most likely periods of variability to be approximately 1.38 and 2.77 d, but we have not been able to find a period that phases all of the datasets satisfactorily. The object is well separated from other sources, so that contamination of the photometry is unlikely. Based on its CMD position redder than the equal-mass binary sequence, it is not likely to be a cluster member. However, its unusual variation may be worth further study. Discussion ========== Overall, little work has been done on NGC 7142 compared to other open clusters, but as will be seen below, there is reason to believe the cluster’s characteristics should be revised. Before addressing the age of the cluster with a traditional isochrone fit, we first review previous estimates of the metallicity, distance, and reddening. Metallicity ----------- In the last decade or so, spectroscopic measurements of NGC 7142’s metallicity have been moving more metal-rich. This trend of cluster metallicities moving more metal-rich with time is not unique to this cluster (see for NGC 6791, for example), but the fact that the cluster now seems to be slightly more metal-rich than the Sun is important for accurate age determination. The first spectroscopic measurement for NGC 7142 by ([Fe/H] = − 0.23 ± 0.13) as part of a larger survey of open clusters was later revised upward to − 0.10 ± 0.10 by based on a new calibration of Fe spectroscopic indices. computed an even higher value ( + 0.04 ± 0.06) by transforming the same data to a common metallicity scale based on DDO photometry. Later spectroscopic measurements ( + 0.08 ± 0.06, ; + 0.14 ± 0.01 from 4 giants, ) also indicated a super-solar metallicity, however. NGC 7142 also appears to be more metal rich than the well-studied cluster M67 ([Fe/H] = 0.00 ± 0.09, ; + 0.03 ± 0.07, ) in studies where the two clusters can be compared reliably. Distance and Reddening ---------------------- determined that the reddening for the cluster was E(B − V) = 0.41 using the UBV color-color diagram, and this has been the most commonly quoted value since. found evidence of differential reddening on the order of ΔE(B − V) = 0.1 based on the width of the observed main sequence. Even though there are clear signs of a cloud front stretching ENE to WSW (with the amount of gas appearing to decrease toward the SSE), did not find evidence of a systematic change in reddening across the face of the cluster. They interpreted this to mean that there is significant smaller scale variation. settled on E(B − V) = 0.43 for turnoff stars and 0.40 for giants, based on a survey of previous studies including the two above. derived E(B − V) = 0.32 ± 0.05 using a method depending largely on comparing photometry of the red giant clump and main sequence turnoff to those of synthetic CMDs generated from isochrones. The reddening derived from IRAS and COBE dust maps is significantly larger [E(B − V) = 0.51 ± 0.02] than any previous determination for the cluster, but this might be a better indication of the reddening of stars beyond NGC 7142. The dust map values do vary in a range covering 0.09 mag within $5\arcmin$ of the cluster center, lending some credence to the possibility of differential reddening for the cluster. There has not been much agreement among previous distance modulus determinations either. The first estimate by was (m − M)V ≃ 13.7. quoted a dereddened distance modulus (m − M)0 = 11.4 ± 0.9 when using E(B − V) = 0.41, corresponding to an uncorrected distance modulus of approximately (m − M)V = 12.67. derive (m − M)V = 12.95 from main sequence fitting, assuming E(B − V) = 0.43. derive (m − M)0 = 11.85 ± 0.05, or (m − M)V = 12.84. Given the disagreements, it is worth revisiting the issue here. Of the features in the CMD on NGC 7142, the red giant clump is probably the most clearly identifiable, and some stars within it have been identified spectroscopically as likely cluster members. We identified candidate clump stars in optical and 2MASS infrared CMDs (see Fig. [clumpcmd]), and the 6 stars are given in Table [rctab]. Obviously, the sample is small, and there may be a field star or first-ascent giant star present, given the amount of differential reddening across the cluster’s face. In fact, one star (CT 399) was rejected based on radial velocity measurement. We obtained single radial velocity measurements for three other red clump stars (listed in Table [spectab]), and one of them (JH 2222) is consistent with membership, while the other two have velocities that are separated from the cluster mean by ∼ 4 km s− 1 and whose memberships are still in question. However, by using the median magnitude and color of the sample in 2MASS photometry (as did in their study of open cluster clump stars), the problems related to membership determination should be minimized. The color of the clump is largely a function of age in the range of composition containing NGC 7142 [∂(J − KBB)0/∂logtage ≈ − 0.17 dex− 1] according to the open cluster data of. Based on their study, the clump is predicted to have a color (J − KBB)0 = 0.67 ± 0.03, where we have assumed an age similar to M67 (4 Gyr) with a conservative uncertainty of about 1.5 Gyr. For NGC 7142, we find a median observed color (J − KBB) = 0.80, for a reddening E(J − KBB) = 0.13 ± 0.03 or E(B − V) = 0.25 ± 0.06. Using the median magnitude for NGC 7142 stars (KBB = 10.42), and the median absolute value from for stars similar in age and metallicity to NGC 7142 (⟨MK⟩ = − 1.62 ± 0.06), we find (m − M)K = 12.04 ± 0.15. These values imply (m − M)0 = 11.96 or (m − M)V = 12.72 ± 0.19. The reader should note that the 6 most probable clump stars are separated into two groups in K magnitude and (J − K) color, and the medians split the difference here. In the optical CMD used by, there is a greater degree of consistency in the V magnitudes of the clump stars, which is somewhat odd considering the apparent differential reddening across the cluster. After comparing with the results of previous studies, the two results derived from the clump (this study and that of ) give the largest values for the distance and smallest values for the reddening (with the exception of the early estimates by ). Earlier estimates were probably affected by degeneracy between the effect of dust extinction and distance, particularly because other features (like the turnoff) are difficult to identify in the cluster’s CMDs. The larger distance implies that the turnoff is more luminous than previously thought and that the age will tend to be on the low side, as discussed below. Age --- Part of the motivation for this project involved finding detached eclipsing binary star systems that contained evolved stars. Precise measurements of the masses and radii for stars that have evolved off the main sequence can lead to precise age determinations for the cluster. A detailed analysis of the two detached eclipsing binary systems sitting at the cluster turnoff (V375 Cep and V2) and the implications for the cluster age will be presented in an upcoming paper. However, it is beneficial to have as reliable an age determination from isochrones and the CMD for comparison. With the current CMDs, it is not easy to tell whether NGC 7142 turnoff stars are high enough in mass to produce a significant “hook” feature associated with core convection on the main sequence. A detailed isochrone fit for a feature of that kind is complicated by the short evolution timescale for stars moving through the hook, and the difficulties are exacerbated by differential reddening across the face of the cluster, field star contamination, and confusion due to optical blends of stars or true binaries. There is a weak indication of a subgiant branch at V ≈ 15.8, sloping faintward from the massive end of the main sequence, but more work is needed to establish the cluster membership of these stars. As a first check, we compare with M67, a well-studied cluster with an age thought to be near that of NGC 7142. M67 has a low and well-determined reddening (E(B − V) = 0.041 ± 0.004; ), and a well-determined distance modulus [(m − M)V = 9.72 ± 0.05, (m − M)0 = 9.60 ± 0.03; ]. M67 also appears to be a transition object, with evidence of weak core convection for the most massive turnoff stars, and a sparsely populated subgiant branch. Because photometry is available in BVIC and JHKS, differences in median magnitudes of the clump of the clusters in different filters allow a good estimate of the difference in distance moduli. In the 2MASS filters where the effects of extinction are minimized, these differences converge to about 2.4 mag. With a small correction for the extinction, this implies (m − M)0, N7142 − (m − M)0, M67 = 2.3, and (m − M)0 = 11.9. We can then look for consistency when the color-magnitude diagrams of the clusters are overlaid in different filter combinations in order to check the earlier estimate of the reddening. This requires that the extinctions and reddenings in different filter combinations can be reliably related, which is a complicated subject. For example, the measured reddening should depend on the spectral type of the star being observed because the flux-weighted wavelength of detected photons (the “effective wavelength”) changes. The use of extinction relationships (including the use of RV = AV/E(B − V) values different from the canonical 3.1) did not produce consistent CMDs in optical bands. In Fig. [m67comp], we show the results of using the method of to compute reddenings and extinctions from E(V − I), which was itself estimated from the shifts between the red clump stars in the two clusters and assuming the clump stars are approximately K1-K2 III giants. Calculations were done using the York Extinction Solver[1](#fn1). As can be seen, the resulting CMDs show that the clumps, giant branch stars and main sequence lines largely agree in position. The expected extinctions and reddening for turnoff stars (near F5 V) only differ from the clump values by 0.02 (in the optical) at most. So we conclude that differential effects due to the differing spectral energy distributions of giants and dwarfs are a minor effect on the quality of the fit. Based on the comparison with M67, NGC 7142 appears to be a slightly younger cluster than M67. Even with some uncertainty in where to align the clumps of the two clusters, there is a group of NGC 7142 stars that reach up to about 0.75 mag brighter than the turnoff of M67. Using measured values for M67 and the results of the fit, our preferred values are E(B − V) = 0.32 ± 0.06, (m − M)0 = 11.9 ± 0.15, and (m − M)V = 12.96 ± 0.24. The color difference between the turnoff and the giant branch is similar for the two clusters, although M67 has a larger difference in magnitude between the clump and the turnoff, implying that NGC 7142 is younger. At the suggestion of the referee, we also compared NGC 7142 to the more metal-rich ([Fe/H] = + 0.43; ) open cluster NGC 6253. If the NGC 6253 photometry is shifted [ΔV = 1.1 mag, Δ(B − V) = 0.02] so that the red clumps are once again in approximate agreement, the main sequences also agree as shown in Fig. [n6253comp]. The brightness of subgiant star candidates in NGC 6253 indicate that that cluster is similar in age to or slightly younger than NGC 7142. Present indications are than NGC 6253 is around 3 Gyr old (see for a summary of determinations), although the large metallicity of the cluster complicates its analysis. In Fig. [isos1], we present some illustrative comparisons with model isochrones from two groups. Using distance and reddening derived from the M67 comparison above and [Fe/H] equal to the most recent determination by, we find adequate matches to the main sequence, but imperfect agreement with spectroscopically identified giants. Decreased [Fe/H] improves the agreement with the giant branch, but does not eliminate it for reasonable choices ([Fe/H] > − 0.1)., for example, describe in detail difficulties with the color-temperature transformations that must be used to place theoretical models in the observational plane. For the Hyades, which have [Fe/H] similar to NGC 7142, several transformation algorithms do a reasonable job of reproducing the main sequence. However, when it comes to the older and more metal-rich cluster NGC 6791, the authors found that the giant branch was too red when transformations were used (as in both sets of isochrones in Fig. [isos1]). Given the current uncertainties in the transformations and potential problems related to the composition of super-metal rich stars (such as non-solar abundance ratios and helium abundance), we will put off further discussion of the isochrones until full analyses of the eclipsing binary stars is completed. If the colors of main sequence stars are reliably modelled, the distance and reddening derived from the red giant clump imply an age near 3 Gyr for the cluster. Larger ages do not adequately model the bright end of the main sequence or potential subgiant branch stars (15.5 < V < 16 and 1 < B − V < 1.3). This is greatly different from the age of nearly 7 Gyr derived by, in spite of the fact that they derived a similar distance and reddening. It may be that their technique (which involved fitting sub-solar and solar-composition synthetic CMDs rather than super-solar ones) was affected by some of the same mismatches on the giant branch we pointed out above. If we tried to fit the color difference between the main sequence and red giant branch at the level of the turnoff with current isochrones, a much larger age would be obtained, but the red clump’s magnitude would be poorly matched. Conclusions =========== We have conducted an extensive variability search among stars in the field of the cluster NGC 7142. Among the most important discoveries were several detached eclipsing systems near the cluster turnoff (two of which have been verified as cluster members based on radial velocities), a faint eclipsing binary with shallow eclipses indicative of a small stellar companion, and a long period variable found far to the red of the cluster giant branch. The detached eclipsing binaries allow us to put NGC 7142 on a growing list of clusters with multiple double-lined binary stars that can provide masses and radii for further tests of stellar models. We have used the red giant clump in an attempt to derive a more reliable distance and mean reddening for the cluster. The values derived here fall at the low end of previous determinations for reddening, and at the high end for distance. Our preferred values [E(B − V) = 0.32 ± 0.06, E(V − I) = 0.46, (m − M)0 = 11.9 ± 0.15] come from a fit of the 2MASS CMD of M67 to that of NGC 7142. With this distance, models indicate that the cluster’s age is near 3 Gyr. Future work is needed to check the agreement of theoretical models with the CMD, however, as there are lingering difficulties with color-$T\_{\rm eff}$ transformations and with our understanding of the compositions of super-metal-rich stars. NGC 7142 has characteristics putting it in an interesting regime. It appears to be more metal rich than the Sun or the well-studied open cluster M67, similar in metallicity to the Hyades, and more metal-poor than the very metal-rich cluster NGC 6791. As such, it can help further test metal-rich stellar models. NGC 7142’s age also is interesting — it is similar in age to M67, making it a cluster with main sequence stars that are near the transition between massive stars with convective cores and low-mass stars with radiative cores. The main difficulties in deriving accurate cluster properties are the fairly heavy reddening and field star contamination of the field. With future membership surveys (radial velocity or proper motion) or with the use of Stromgren photometry, these difficulties can be overcome. This work has been funded through grant AST 09-08536 from the National Science Foundation to E.L.S. We would like to thank the Director of Mount Laguna Observatory (P. Etzel) for generous allocations of observing time. Infrastructure support for the observatory was generously provided by the National Science Foundation through the Program for Research and Education using Small Telescopes (PREST) under grant AST 05-19686. We would also like to thank A. Bostroem, C. Gabler, and J. B. Leep for assisting with the photometric calibration observations, and Mark Jeffries for taking some of the later time-series photometry. The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universitat Munchen, and Georg-August-Universitat Gottingen. 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M., Stiening, R., et al. 2006,, 131, 1163 Southworth, J., Maxted, P. F. L., & Smalley, B. 2004,, 349, 547 Southworth, J., & Clausen, J. V.2006,, 304, 199 Stetson, P. B. 2000,, 112, 925 Talamantes, A., Sandquist, E. L., Clem, J. L., Robb, R. M., Balam, D. D., & Shetrone, M. 2010,, 140, 1268 Taylor, B. J. 2007,, 133, 370 Thompson, I. B., Kaluzny, J., Pych, W., Burley, G., Krzeminski, W., Paczyński, B., Persson, S. E., & Preston, G. W. 2001,, 121, 3089 Tull, R. G. 1998,, 3355, 387 Twarog, B. A., Ashman, K. M., & Anthony-Twarog, B. J. 1997,, 114, 2556 Twarog, B. A., Anthony-Twarog, B. J., & De Lee, N. 2003,, 125, 1383 VandenBerg, D. A., Bergbusch, P. A., & Dowler, P. D. 2006,, 162, 375 VandenBerg, D. A., Casagrande, L., & Stetson, P. B. 2010,, 140, 1020 VandenBerg, D. A., & Clem, J. L. 2003,, 126, 778 van den Berg, M., Stassun, K. G., Verbunt, F., & Mathieu, R. D. 2002,, 382, 888 van den Bergh, S., & Heeringa, R. 1970,, 9, 209 Zhang, X. B., Deng, L., & Lu, P. 2009,, 138, 680 R phased light curve for the probable contact binary V7. [lcv7] Light curves (magnitude minus the median value in each filter band) for the long-period variable V4 in B ( • ), V ( ○ ), RC ( × ), and IC ( \* ). Individual points are averages of measurements made during time intervals up to 0.05 d. Measurements in B and RC have been shifted to bring them into approximate agreement with V measurements on HJDs 2454419 and 2453653, respectively. [lcv4] Comparison of NGC 7142 photometry with likely single-star members of M67 ($\sq$; ). Other symbols have the same meaning as in Fig. [clumpcmd]. The photometry for M67 has been shifted in color to bring red clump stars into agreement in V − I, and the implied E(V − I) was used to compute reddenings and extinctions in other filters according to the method of. Reddening vectors are shown in each diagram. [m67comp] clcl\|clcl Aug. 10, 2005 & R & 3593.755 & 44 & Jul. 7, 2008 & Ic & 4655.698 & 158 Aug. 11, 2005 & R & 3594.687 & 39 & Jul. 8, 2008 & Ic & 4656.686 & 204 Aug. 13, 2005 & R & 3596.670 & 46 & Jul. 9, 2008 & Ic & 4657.684 & 96 Aug. 14, 2005 & R & 3597.659 & 19 & Jul. 27, 2008 & V & 4675.660 & 97 Aug. 15, 2005 & R & 3598.642 & 50 & Jul. 28, 2008 & V & 4676.655 & 103 Aug. 16, 2005 & R & 3599.666 & 42 & Jul. 29, 2008 & V & 4677.658 & 105 Aug. 17, 2005 & R & 3600.659 & 48 & Jul. 31, 2008 & V & 4678.653 & 107 Sep. 21, 2005 & R & 3635.749 & 40 & Aug. 18, 2008 & V & 4697.952 & 34 Sep. 22, 2005 & R & 3636.607 & 32 & Oct. 6, 2008 & VRK & 4746.674 & 11,10 Sep. 23, 2005 & R & 3637.613 & 68 & Oct. 25, 2008 & BVIC & 4765.686 & 9,8,11 Sep. 24, 2005 & R & 3638.612 & 30 & Sep. 6, 2009 & V & 5081.895 & 31 Sep. 25, 2005 & R & 3639.605 & 68 & Oct. 9, 2009 & V & 5114.592 & 41 Sep. 26, 2005 & R & 3640.704 & 28 & Oct. 16, 2009 & V & 5121.578 & 66 Sep. 27, 2005 & R & 3641.608 & 48 & Nov. 10, 2009 & V & 5146.575 & 52 Oct. 6, 2005 & R & 3650.607 & 38 & Nov. 29, 2009 & V & 5165.590 & 13 Oct. 7, 2005 & R & 3651.608 & 46 & May 19, 2010 & IC & 5336.815 & 37 Oct. 8, 2005 & R & 3652.597 & 33 & Jun. 7, 2010 & B & 5355.742 & 55 Oct. 9, 2005 & BVR & 3653.600 & 11,11,11 & Jun. 24, 2010 & V & 5372.709 & 68 Nov. 12, 2007 & BVRK & 4417.703 & 11,16,10 & Jul. 5, 2010 & R & 5383.674 & 76 Nov. 14, 2007 & BVRK & 4419.612 & 13,11,11 & Jul. 12, 2010 & IC & 5390.669 & 75 Jun. 8, 2008 & B & 4626.780 & 55 & Aug. 2, 2010 & VIC & 5411.665 & 12,61 Jun. 9, 2008 & B & 4627.796 & 50 & Aug. 3, 2010 & IC & 5412.660 & 11 Jun. 11, 2008 & B & 4629.764 & 57 & Aug. 14, 2010 & V & 5423.686 & 18 Jun. 12, 2008 & B & 4630.753 & 60 & Nov. 16, 2010 & VIC & 5517.570 & 70,5 Jun. 15, 2008 & B & 4633.746 & 60 & Jun. 23, 2011 & IC & 5736.721 & 48 Jun. 16, 2008 & B & 4634.747 & 57 & Jul. 12, 2011 & RIC & 5755.670 & 16,65 Jun. 17, 2008 & Ic & 4635.740 & 159 [phottab] lccr V1 & − 17.0 & 1.0 & 3 V2 & − 42.1 & 0.6 & 13 V375 Cep & − 49.1 & 1.7 & 21 JH2222& − 49.6 & 0.1 & 1 JH2288& − 44.0 & 0.1 & 1 JH1003& − 43.9 & 0.1 & 1 [spectab] rrcllllllll V1 & 395 & EA & 21:44:32.86 & +65:45:26.3 & 14.864 & 0.831 & 1.039 & 4.6691 & 2453639.93 & rv prob. nm V2 & 18 & EA & 21:44:29.62 & +65:48:43.9 & 15.310 & 0.797 & 1.010 & 15.6505 & 2453639.684 & deep eclipses; eccentric; rv mem V3 & 155 & EW & 21:45:15.16 & +65:49:24.3 & 15.38 & 0.72 & 0.93 & 0.580793 & 2453594.80 & straggler? V4 & 430 & LPV & 21:44:55.98 & +65:45:50.0 & 15.43 & 1.89 & 2.06 & & & VH Y V5 & 199? & EW & 21:45:30.03 & +65:46:42.3 & 15.80 & 1.18 & 1.38 & 0.3368235 & 2453596.782 & nearby faint star V6 & & EB & 21:44:13.21 & +65:45:01.3 & 15.71 & 0.91 & 1.15 & 0.441163 & 2453599.757 & V7 & 288 & EW & 21:45:10.77 & +65:44:41.3 & 15.95 & 0.89 & 1.12 & 0.69531? & & VH f & & EA & 21:44:54.74 & +65:43:55.3 & 16.115 & 0.877 & 1.080 & 1.909682 & 2454676.8071 & V375 Cep; rv mem V8 & & EA & 21:44:35.73 & +65:45:04.7 & 17.288 & 1.106 & 1.373 & 0.52730 & 2453596.985 & triple? V9 & & EW & 21:44:28.44 & +65:46:36.6 & 17.63 & 1.14 & 1.34 & 0.330233 & 2453593.918 & V10 & 356 & Irr & 21:45:40.73 & +65:43:40.0 & 17.63 & 1.25 & 1.53 & & & V11 & & EA & 21:44:17.92 & +65:50:04.9 & 17.906 & 1.055 & 1.304 & 1.28629 & 2453598.825 & V12 & & EW? & 21:44:55.46 & +65:52:59.0 & 18.330 & 1.157 & 1.340 & 0.2938 & 2453636.79 & V13 & & EW & 21:44:20.66 & +65:44:59.1 & 18.35 & 1.08 & 1.20 & 0.34207 & 2453598.89 & total eclipses, prob. nm V14 & & EW & 21:46:06.8 & +65:52:53.6 & & & & 0.29005 & 2453600.75 & not detected in calibrating obs. [vartab] rrcccccccl 421 & 1065 & 21:45:00.73 & +65:45:56.5 & 13.452 & 1.364 & 1.468 & 10.097 & 0.802 & I; rv, abund mem 170 & 1519 & 21:45:20.43 & +65:48:31.0 & 13.743 & 1.380 & 1.526 & 10.237 & 0.794 & O,rv mem? 203 & 1767 & 21:45:32.93 & +65:47:31.1 & 13.765 & 1.326 & 1.147 & 10.533 & 0.718 & rv, abund mem;nearby star affecting phot? & 2222 & 21:45:57.03 & +65:44:10.5 & 13.665 & 1.411 & 1.485 & 10.268 & 0.768 & rv mem,off CT field 93 & 1003 & 21:44:57.60 & +65:48:36.8 & 13.735 & 1.294 & 1.412 & 10.490 & 0.734 & rv mem? & 2288 & 21:46:00.93 & +65:49:06.8 & 13.887 & 1.360 & 1.476 & 10.526 & 0.736 & rv mem?,off CT field 399 & 591 & 21:44:36.70 & +65:43:19.1 & 13.448 & 1.375 & 1.531 & 9.998 & 0.845 & rv nm [rctab] lr\|lr\|lr ΔE(V − I) & 0.41 & Δτ1 & 0.335 & & ΔAB & 1.22 & Δ(B − V) & 0.28 & AV/E(B − V) & 3.33 ΔAV & 0.94 & & & AV/E(V − I) & 2.30 ΔAKs & 0.11 & ΔE(J − Ks) & 0.15 & AKs/E(J − Ks) & 0.76 [exttab] --- 1. http://www2.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/YorkExtinctionSolver/[↩](#fnref1)	arxiv_0000615
Performance of Energy-Buffer Aided Incremental Relaying in Cooperative Networks =============================================================================== In this paper, we consider a two-hop cooperative network with a direct link based on an energy harvesting (EH) decode-and-forward relay. The energy-buffer equipped relay harvests energy from the ambience, and uses the harvest-store-use (HSU) architecture. Since it is known that using a discrete-state Markov chain to model the energy buffer is inaccurate even for moderate number of states, we use a discrete-time continuous-state space Markov chain instead. We derive the limiting distribution of energy for both incremental on-off policy (IOFP) and the incremental best-effort policy (IBEF), and use them to obtain expressions for outage probability and throughput. The corresponding expressions for non-incremental signalling follow as a special case. We show that stable buffers using IBEP harness a diversity of two as compared to those using IOFP, which attain a diversity of one. However, while buffers using IBEF are consequently more reliable than those with IOFP, their throughput performance is only marginally superior. Simulation results are presented to validate the derived analytical expressions. Introduction ============ Relays have been incorporated into several modern communication standards (e.g. LTE-A) due to their promise in increasing range and reliability of wireless networks. Relays are needed when the direct link is shadowed, when the source has power limitations, or when diversity gain needs to be harnessed. For this reason, cooperative relay networks have been extensively studied in recent years. Regenerative or decode-and-forward (DF) relays, as well as non-regenerative amplify-and-forward (AF) relays have been widely studied. Conventionally, relays are assumed to be equipped with a power source or a battery. Of late however, there has been considerable research interest in relays that are powered by energy harvesting (EH). In applications like mobile sensor networks and wireless body area networks, use of a nearby node to serve as a relay to communicate to a distant node is well motivated. In a 5G pico-cell communication framework, use of EH by nodes serving as relays is being explored. In these applications, the node serving as a relay is often battery powered, and use of its battery energy to provide relaying services is undesirable. This is because battery life-times are typically short, and replacement of batteries is often difficult (and in some applications, impossible). Use of EH to make the devices battery-less is therefore desirable. In other scenarios, the node serving as a relay may seek to supplement the harvested energy with as little battery energy as possible to ensure long battery lifetimes. Use of relays powered by EH is also motivated by green energy considerations - telecom is known to be a large consumer of energy. Analysis of performance of cooperative communication links with EH relays is clearly of considerable interest. EH is readily possible from natural sources like light, wind, vibration, and ambient radio frequency (RF) signals. Simultaneous wireless information and power transfer (SWIPT), which exploits the possibility of RF signals carrying both energy and information simultaneously, has attracted considerable research interest. However, while they enable communication with a battery-less relay, they are not energy efficient (green) due to path-loss in practical channels. The same is true of power beacons. In this paper, we focus on relays powered by ambient sources. Despite the apparent similarity between energy and data buffers,, there is one fundamental difference - while data buffers are essentially discrete, energy is a continuous variable. However, virtually all literature in this area is based on discrete-state energy buffers. It has been shown that for accurate representation, the number of states needs to be quite large (close to 200,), which make the approach difficult and cumbersome. Also, it is difficult to draw physical insights into performance of such systems with this approach. When the direct channel from source to destination is not shadowed, it is advantageous to combine the signals from source and relay optimally. In addition, use of incremental relaying to improve spectral efficiency is well motivated. While is based on harvest-use (HU) architecture, uses a discrete state space Markov chain to model the buffer (the relay harvests energy from source). Recently, it has been shown that a discrete-time continuous-state Markov chain model can be used to model the energy buffer accurately in point-to-point links. Performance depends on the policy used to operate the buffer. With the best-effort energy storage policy, the node transmits by drawing constant M amount of energy when sufficient energy exists, and drawing entire energy in the buffer otherwise. With the on-off storage policy, the node transmits using M amount of energy when it is available, and remains silent otherwise. Performance of single-hop wireless powered communication with both these models was analyzed, and derivation for the asymptotic distribution of energy in the buffer was presented in. In this paper, we analyze the performance of cooperative links with buffered relays using IBEP and IOFP, and derive expressions for asymptotic distributions of energy. Expressions for non-incremental signalling follow as a special case. To the best of our knowledge there has been no work on performance of cooperative links with energy buffers modeled using a discrete-time continuous-state Markov chain[2](#fn2). The significant contributions are as follows: 1. Unlike all prior work on EH relays with energy buffers that are based on discrete state Markov chain,, in this paper we analyze performance of cooperative links using a continuous state space Markov chain to model the energy buffer. Also, unlike most other works on EH, we do not ignore the direct path from source to destination, and optimally combine the direct and relayed signals at the destination. 2. For incremental signalling (IBEP as well as IOFP), we derive an expression for the limiting (steady-state) distribution of energy in the energy buffer, and establish conditions for its existence. 3. Expressions are derived for outage probabilities and throughput with both IBEP and IOFP. Expressions for non-incremental signalling follow as a special case. We compare performance with the HU architecture and direct (relay-less) transmission. 4. We establish that the IBEP attains a diversity order of two, while the IOFP attains a diversity of one, making links with the former policy more reliable. However, the throughput performance with IBEP is only marginally superior to that with IOFP. Notations: ---------- ${\cal CN}\sim \left(\mu,\sigma^{2}\right)$ denotes the complex Gaussian distribution with mean μ and variance σ2, ∼ denotes ``distributed as", and ∣x∣ denotes the absolute value of x. Pr{A} indicates probability of the event A. W( ⋅ ) is the principal branch of the Lambert-W function. min(a, b) denotes minimum of a, b. E{ ⋅ } denotes the expectation operator. FX(x) and fX(x) denote the cumulative distribution function (CDF) and the probability density function (PDF) of random variable X. System Model ============ The network (as depicted in Fig. [coop]) consists of source S, a DF relay R, and a destination D. While the source S is equipped with a power supply, the relay R is powered solely by harvested ambient energy. Quasi-static Rayleigh fading is assumed. Let dab denote the normalized distance between nodes with a ∈ [S, R], and b ∈ [R, D]. Denote by $h\_{\_{SR}}(i)\sim{\cal CN}(0,d\_{\_{SR}}^{-\alpha})$, $h\_{\_{RD}}(i)\sim{\cal CN}(0,d\_{\_{RD}}^{-\alpha})$, and $h\_{\_{SD}}(i)\sim{\cal CN}(0,d\_{\_{SD}}^{-\alpha})$ the channels in the ith signalling interval between S and R, R and D, and S and D respectively (where α is the path-loss exponent). [coop] Two types of EH architectures have been proposed in literature. With the HSU (Harvest-store-use) architecture, the relay is equipped with a rechargeable device (battery for example) that can store and release energy. Since storage devices cannot charge and discharge at the same time,, harvested energy is stored in a secondary energy buffer (SEB), which is typically a super-capacitor, and transferred to the primary energy buffer (PEB), which is typically a battery, at the end of the signalling interval in a negligible amount of time. On the other hand, with HU (Harvest-use) architecture, all the energy that is harvested is used in the same signalling interval (for example architectures based on super-capacitors). It is the HSU architecture that is of primary interest in this paper. For comparison purpose, we also consider direct (relay-less) transmission from source to destination, as well as the HU architecture. The reason for use of the feedback bit will become clear in what follows. For both HSU and HU architectures, the signalling interval is divided into two phases of equal duration. The overall signalling interval T is normalized to unity for convenience. In the first phase (phase I) of normalized duration 1/2, the source S transmits unit-energy information symbols xS(i) at rate R0 with power PS to the relay R and destination D. The received signals ySD(i) and ySR(i) at D and R in phase I of the $i\textsuperscript{th}$ signalling interval are given by: $$\begin{aligned} &y\_{\_{SD}}(i)=\sqrt{P\_{\_{S}}}h\_{\_{SD}}(i) x\_{\_{S}}(i) +n\_{\_{SD}}(i), &\text{Phase I}\\ &y\_{\_{SR}}(i)=\sqrt{P\_{\_{S}}}h\_{\_{SR}}(i) x\_{\_{S}}(i) +n\_{\_{SR}}(i), & \text{Phase I} \label{1-3}\end{aligned}$$ where $n\_{\_{SD}}(i),n\_{\_{SR}}(i) \sim {\cal CN}\left(0,\sigma^{2}\right)$ are additive noise samples. The instantaneous link signal to noise ratios (SNRs) in the $i\textsuperscript{th}$ signalling interval at D and R in the first phase are given by: $$\begin{aligned} &\gamma\_{\_{SD}}(i)=\displaystyle\frac{P\_{\_{S}}\|h\_{\_{SD}}(i)\|^{2}}{\sigma^{2}}, ~~\gamma\_{\_{SR}}(i)=\frac{P\_{\_{S}}\|h\_{\_{SR}}(i)\|^{2}}{\sigma^{2}}. \label{link\_snr}\end{aligned}$$ Let: $$\begin{aligned} \Gamma\_{th}^{\prime} &=& \left\{ \begin{array}{ll} \Gamma\_{th}=2^{2R\_{0}}-1 & \text{Incremental Signalling,}\\ \infty & \text{Non-incremental Signalling.} \end{array}\right.\end{aligned}$$ It will become apparent later that this definition allows us to analyze the performance of incremental and non-incremental relaying in a unified fashion. *Case γSD(i) ≥ Γt*h′*: If the received SNR at D is greater than (or equal to) the threshold Γt*h′ (signalling is successful in the first phase itself), D sends a positive acknowledgment (ACK), which is received by both R and S. On receipt of ACK, S directly transmits a new set of symbols to D in the second phase (phase II), making the overall SNR γD(i) = γSD(i). Note that an ACK can never be sent for non-incremental signalling. *Case γSD(i) < Γt*h′*: If received SNR at D is less than the threshold in the first phase, D sends a negative acknowledgment (NACK), which is received by both S and R. Clearly, a NACK is always sent for non-incremental signalling. In this case, the fixed DF R decodes x*S(i), re-encodes, and then transmits the information as unit-energy symbols xR(i) at rate R0 with power PR(i) to D using the energy harvested (hence the dependence of PR(i) on i). Clearly, the signal yRD(i) received by D in this phase is given by: $$\begin{aligned} &y\_{\_{RD}}(i)=\sqrt{P\_{\_{R}}(i)}h\_{\_{RD}}(i) x\_{\_{R}}(i) +n\_{\_{RD}}(i), & \text{Phase II}\end{aligned}$$ where $n\_{\_{RD}}(i) \sim {\cal CN}\left(0,\sigma^{2}\right).$ Clearly, γRD(i) = PR(i)∣hRD(i)∣2/σ2. The signals received from S and R are combined using maximal ratio combining (MRC) at D. The overall SNR γD(i) at D is given by γSD(i) + γRD(i). Limiting distribution of Energy with HSU Architecture ===================================================== In this section, we derive expressions for the limiting distributions of energy in the PEB with both IBEP and IOFP, assuming it to be of infinite-size. Since the energy harvested is typically small, this assumption is not limiting. Those with non-incremental signalling follow as special cases (with Γth′ = ∞), and will not be written explicitly. Limiting Distribution of IBEP ----------------------------- The incoming harvested energy X(i) is assumed to be an exponential random variable (as in,) with PDF fX(x). Denote by B(i) the energy level in the buffer in the ith signalling interval. Let M denote the energy drawn for every relay transmission. The PEB’s buffer update equation is then given by: $$\begin{aligned} B(i+1) & = & B(i)+X(i) \hspace{1.5 cm} \left( \gamma\_{\_{SD}}(i) \geq \Gamma^{\prime}\_{th}\right), \nonumber\\ B(i+1) & = & B(i)-M+X(i) \hspace{0.6 cm} (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\cap (B(i) \geq M), \nonumber\\ B(i+1) & = & X(i) \hspace{3 cm} (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\cap (B(i)<M). \label{BI2}\end{aligned}$$ Let W1, W2, W3 and W4 be defined as $W\_{1}=\dfrac {\sigma^2d\_{\_{RD}}^\alpha}{2M}$, $W\_{2}=\dfrac {\sigma^2d\_{\_{SD}}^\alpha}{P\_{\_{S}}}$, $W\_{3}=\dfrac {\sigma^2d\_{\_{RD}}^\alpha}{2}$ and $W\_{4}=\dfrac {\sigma^2d\_{\_{SR}}^\alpha}{P\_{\_{S}}}$. Let $$\begin{aligned} \phi\_{inc} = \frac{M\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right)}{\mathbb{E}{\{X(i)}\}}=M\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right), \label{phinc}\end{aligned}$$ where $\mathbb{E}{\{X(i)}\}=\displaystyle \frac{1}{\lambda\_{1}}$. Note that the equivalent constant ϕninc in the non-incremental case is given by ϕninc = Mλ1 (using Γth′ = ∞). The limiting distributions when ϕinc > 1 is presented in Theorem [TheoremLimitingPDF], and the case when ϕinc ≤ 1 is discussed in Theorem [TheoremLimitingPDF not exists]. [TheoremLimitingPDF] For an infinite-size PEB based on the IBEP as in, the limiting energy distribution exists when ϕinc = Mλ1(1 − exp( − W2Γth′)) > 1, and its limiting CDF G(x) satisfies the following integral equation: $$\begin{aligned} G(x)=\int\_{u=0}^{x}\left[ e^{-W\_{2}\Gamma^{\prime}\_{th}}f\_{X}\left( x-u\right)G\left(u\right)+\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)f\_{X}\left( u\right)G\left( x+M-u\right) \right] du, \label{limcdfintegr}\end{aligned}$$ whose solution is given by: $$\begin{aligned} G(x)=\left( 1-\exp\left( -Z\_{1}x\right) \right), \label{limcdf}\end{aligned}$$ where $$\qquad {Z\_{1}} = \displaystyle \frac{\mathcal{W}\left( -\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right) \lambda\_{1}M\exp\left( -\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) \lambda\_{1}M \right) \right) }{M} +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right) \lambda\_{1}.$$ Refer to Appendix [AppendixLimitingPDF]. [TheoremLimitingPDF not exists] For an infinite-size PEB based on the IBEP as in, the limiting energy distribution does not exist when ϕinc = Mλ1(1 − exp( − W2Γth′)) ≤ 1, and M amount of energy is almost always available for relay transmission in the PEB. Relay transmit power PR(i) equals 2M ($P\_{\_{R}}\left( i\right) =\dfrac{M}{\left( 1/2\right)}=2M$) almost surely. Refer to Appendix [AppendixLimitingPDF not exists]. Limiting Distribution of IOFP ----------------------------- Unlike the on-off policy described for point-to-point links described in, the relay R here becomes silent: 1) when it receives an ACK from the destination, or 2) when it does not have M amount of energy in its PEB but NACK is received from D. With IOFP, the energy buffer update equation of the PEB is given by: $$\begin{aligned} B(i+1) & = & B(i)+X(i) \hspace{1.9 cm} (\gamma\_{\_{SD}}(i) \geq \Gamma^{\prime}\_{th}), \nonumber\\ B(i+1) & = & B(i)-M+X(i) \hspace{1 cm} (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\cap (B(i)\geq M), \nonumber\\ B(i+1) & = &B(i)+X(i) \hspace{2.1 cm} (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\cap (B(i)<M). \label{BIonoff}\end{aligned}$$ The limiting distribution when ϕinc > 1 is presented in Theorem [TheoremLimitingPDF On-Off], and the case when ϕinc ≤ 1 is first discussed in Theorem [on-off not exists]. Let Ī be an indicator variable defined as: $\bar{I}=1 \hspace\{0.2 cm} \text{if} \hspace\{0.2 cm} B\left( i\right) \geq M $, and Ī = 0 otherwise. [on-off not exists] For the infinite-size PEB based on the IOFP as in, the limiting distribution does not exist when ϕinc ≤ 1. M amount of energy is almost always available for transmission in the PEB, and relay transmit power $P\_R\left( i\right) =\dfrac{M}{\left( 1/2\right)}=2M$ almost surely. Proof is along lines similar to that of Theorem [TheoremLimitingPDF not exists] with R having transmit energy ER(i) = MĪ, and is therefore omitted. [TheoremLimitingPDF On-Off] For an infinite-size PEB based on the IOFP as in, the limiting distribution of infinite-size PEB exists when ϕinc > 1, and its limiting CDF G(x) must satisfy following integral equation: $$\begin{aligned} G(x)= \begin{cases} e^{-W\_{2}\Gamma^{\prime}\_{th}}\displaystyle\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left( u\right)du +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) \left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\left( u\right)du \right. \\ \left.+\displaystyle \int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left(u\right)du \right] \hspace{2.9 cm} 0 \leq x < M \\ e^{-W\_{2}\Gamma^{\prime}\_{th}}\displaystyle\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left( u\right)du +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\left( u\right)du \right. \\ \left.+\displaystyle \int\_{u=0}^{M}F\_{X}\left( x-u\right)g\left(u\right)du \right] \hspace{2.9 cm} x \geq M \end{cases} \label{limcdfintsonoff}\end{aligned}$$ Its corresponding limiting PDF g(x) is given by: $$\begin{aligned} g(x)=\begin{cases} \dfrac{\left(1-e^{Qx}\right)}{M} \hspace{4.2 cm} 0 \leq x < M \\ \dfrac{-Qe^{Qx}}{M\left(Q+\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\right) } \hspace{2 cm} x \geq M, \end{cases} \label{limpdfsplifin}\end{aligned}$$ where Q is given by: $$Q = -\frac{\mathcal{W}\left( -\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right) \lambda\_{1}M\exp\left( -\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) \lambda\_{1}M \right) \right) }{M} -\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right) \lambda\_{1}.$$ For existence of the limiting distribution, Q should be negative (i.e. Q < 0). Refer to Appendix [AppendixLimitingPDF On-Off]. Outage and Throughput Analysis ============================== In this section, expressions are derived for outage and throughput with the HSU and HU architectures. HSU Architecture with IBEP -------------------------- With incremental relaying, the link is not in outage when γSD(i) ≥ Γth′. When γSD(i) < Γth′, the link is once again not in outage when both γSR(i) and γSD(i) + γRD(i) are greater than or equal to Γth. The outage probability PoutHSU for incremental relaying is given by : $$\begin{aligned} P\_{out}^{HSU}=\Pr\left\lbrace \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\}\Pr\{\min\left( \gamma\_{\_{SR}}(i),\gamma\_{\_{RD}}(i)+\gamma\_{\_{SD}}(i)\right) < \Gamma\_{th}\mid(\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\right\rbrace, \label{outin}\end{aligned}$$ where γSR(i), γSD(i) are given by and γRD(i) = PR(i)∣hRD(i)∣2/σ2. The throughput expression τHSU for incremental relaying with fixed DF protocol is given by: $$\begin{aligned} \tau^{HSU}= & \hspace{0.1 cm} R\_{0}\Pr\left\lbrace \gamma\_{\_{SD}}\left( i\right) \geq \Gamma^{\prime}\_{th}\right\rbrace \nonumber +\frac {R\_{0}}{2}\Pr\left\lbrace \gamma\_{\_{SD}}\left( i\right) <\Gamma^{\prime}\_{th}\right\rbrace {\nonumber}\\ &\times \Pr\left\lbrace \min\left( \gamma\_{\_{SR}}\left( i\right),\gamma\_{\_{RD}}\left( i\right) +\gamma\_{\_{SD}}\left( i\right)\right) \geq \Gamma\_{th}\mid \left( \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\right) \right\rbrace. \label{ta}\end{aligned}$$ In, the last term can be rewritten as: $$\begin{aligned} &\Pr\left\lbrace \gamma\_{\_{SD}}\left( i\right) <\Gamma^{\prime}\_{th}\right\rbrace \Pr\left\lbrace \min\left( \gamma\_{\_{SR}}\left( i\right),\gamma\_{\_{RD}}\left( i\right) +\gamma\_{\_{SD}}\left( i\right) \right) \geq \Gamma\_{th} \mid\left( \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\right) \right\rbrace {\nonumber}\\ &=\Pr\left\lbrace \min\left( \gamma\_{\_{SR}}\left( i\right),\gamma\_{\_{RD}}\left( i\right) +\gamma\_{\_{SD}}\left( i\right)\right) \geq \Gamma\_{th}, \gamma\_{\_{SD}}(i) < \Gamma^{\prime}\_{th} \right\rbrace {\nonumber}\\ &\overset{\ell}{=}\Pr\left\lbrace \gamma\_{\_{SD}}\left( i\right) <\Gamma^{\prime}\_{th}\right\rbrace-\Pr\left\lbrace \min\left( \gamma\_{\_{SR}}\left( i\right),\gamma\_{\_{RD}}\left( i\right) +\gamma\_{\_{SD}}\left( i\right)\right) < \Gamma\_{th}, \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th} \right\rbrace {\nonumber}\\ &=\Pr\left\lbrace \gamma\_{\_{SD}}\left( i\right) <\Gamma^{\prime}\_{th}\right\rbrace-\Pr\left\lbrace \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\right\rbrace \times {\nonumber}\\ & \hspace{2 cm }\Pr\left\lbrace \min(\gamma\_{\_{SR}}(i),\gamma\_{\_{RD}}(i)+\gamma\_{\_{SD}}(i))< \Gamma\_{th}\mid(\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\right\rbrace {\nonumber}\\ &\overset{n}{=}\Pr\left\lbrace \gamma\_{\_{SD}}\left( i\right) <\Gamma^{\prime}\_{th}\right\rbrace-P\_{out}^{HSU}=\Pr\left\lbrace \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\right\rbrace \bigg[1-\frac {P\_{out}^{HSU}}{\Pr\{\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\}}\bigg]. \label{outs} \end{aligned}$$ In the above, equality ℓ follows from the fact that for two events A and B, $\Pr\{A,B\}+\Pr\{\overline{A},B\}=\Pr\{B\}$, where $\overline{A}$ denotes the complement of A. Equation n follows from ([outin]). Using in gives: $$\begin{aligned} \tau^{HSU} =R\_{0}\Pr\{\gamma\_{\_{SD}}(i)\geq\Gamma^{\prime}\_{th}\}+\frac {R\_{0}}{2}\Pr\left\lbrace \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\right\rbrace \bigg[1-\frac {P\_{out}^{HSU}}{\Pr\{\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\}}\bigg],\label{throughput}\end{aligned}$$ where Pr{γSD(i) ≥ Γth′} = e− W2Γth′. In the following theorem, we present expressions for outage PoutHSU using the fact that PR(i) is given by: $$\begin{aligned} {P\_{\_{R}}(i)}= \begin{cases} \dfrac {\min\left( B(i),M\right) }{(1/2)}=2\min\left( B(i),M\right) \hspace{1cm} \hspace{1cm} \phi\_{inc} > 1.\\ \dfrac{M}{(1/2)}=2M \hspace{6 cm} \phi\_{inc} \leq 1 \end{cases} \label{reltrb}\end{aligned}$$ For an infinite-size PEB based on the IBEP, the outage probability PoutHSU with HSU architecture when ϕinc > 1 and ϕinc ≤ 1 are given by: $$\begin{aligned} P\_{out}^{HSU} =&\hspace{0.1 cm} (1-e^{-W\_{4}\Gamma\_{th}})\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) +e^{-W\_{4}\Gamma\_{th}} {\nonumber}\\ &\times\Bigg[e^{-Z\_{1}M}\left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right] }{(W\_{1}-W\_{2})}\right\rbrace {\nonumber}\\ &+(1-e^{-Z\_{1}M})\left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +I\_{HSU}\Bigg] \hspace{0.5cm} \hspace{0.5cm} \phi\_{inc} > 1 \label{hsuoutag1}\end{aligned}$$ where $$\begin{aligned} I\_{HSU}=W\_{2}Z\_{1}\int\_{x=0}^{M}\frac{xe^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}} -e^{-\frac{W\_{3}\Gamma\_{th}}{x}}] }{(xW\_{2}-W\_{3})}dx. \label{integ}\end{aligned}$$ $$\begin{aligned} P\_{out}^{HSU} =& \hspace{0.1 cm} (1-e^{-W\_{4}\Gamma\_{th}})\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) {\nonumber}\\ &+e^{-W\_{4}\Gamma\_{th}} \left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right] }{(W\_{1}-W\_{2})}\right\rbrace \hspace{0.8 cm} \phi\_{inc} \leq 1 \label{hsuoutage2}\end{aligned}$$ Refer to Appendix [outage HSU arch]. The integral IHSU has no closed-form expression, but is bounded, and can be evaluated numerically. The throughput τHSU is given by ([throughput]), where PoutHSU is given by and for ϕinc > 1 and ϕinc ≤ 1. As noted already, expressions for the non-incremental BEP (NIBEP) follow with Γth′ = ∞. HSU Architecture with IOFP -------------------------- With IOFP, the relay transmit power PR(i) is given by: $$\begin{aligned} P\_{\_{R}}\left( i\right)= \begin{cases} \dfrac{M\bar{I}}{\left( 1/2\right) }=2M\bar{I} \hspace{1.2 cm } \phi\_{inc} > 1\\ \dfrac{M}{\left( 1/2\right) }=2M \hspace{1.5 cm } \phi\_{inc} \leq 1. \end{cases} \hspace{1 cm} \label{onofftra}\end{aligned}$$ For an infinite-size buffer based on the IOFP, the outage probability PoutHSU when ϕinc > 1 is given by: $$\begin{aligned} P\_{out}^{HSU} =& \hspace{0.1 cm} \left( 1-e^{-W\_{4}\Gamma\_{th}}\right) \left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) +e^{-W\_{4}\Gamma\_{th}} {\nonumber}\\ &\times\Bigg[\frac{1}{\phi\_{inc}}\left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right] }{(W\_{1}-W\_{2})}\right\rbrace {\nonumber}\\ &+\left( 1-\frac{1}{\phi\_{inc}}\right) \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) \Bigg] \hspace{1 cm} \hspace{3.2 cm} \phi\_{inc} > 1. \label{hsuonoffoutag1}\end{aligned}$$ Using Theorem [on-off not exists], when ϕinc ≤ 1, the expression for outage probability is as in. Refer to Appendix [outage on-off]. The throughput τHSU is given by ([throughput]), where PoutHSU is given in and for ϕinc > 1 and ϕinc ≤ 1. As noted already, expressions for the non-incremental OFP (NIOFP) follow with Γth′ = ∞. HU Architecture with Incremental Relaying ----------------------------------------- In HU Incremental (HU Inc), there is no energy buffer at the relay node. In the first phase, the relay stores harvested energy in a super-capacitor. In the second phase, the relay transmits to the destination with all the harvested energy if NACK is received from destination. If ACK is received from destination, the relay remains silent (the harvested energy cannot be used in future time slots due to inability of the super-capacitor to store energy over long intervals). With the HU architecture the relay transmit power is given by:$ P\_{\_{R}}(i)=\frac {X(i)}{\left(1/2\right) }(1/2)=X(i)$. Substituting PR(i) into, the outage probability with HU architecture can be written as: $$\begin{aligned} P\_{out}^{HU} =& \hspace{0.1 cm}(1-e^{-W\_{4}\Gamma\_{th}})\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) +e^{-W\_{4}\Gamma\_{th}}\left[ \lambda\_{1}W\_{2}I\_{HU}+\left( 1-e^{-W\_{2}\Gamma\_{th}}\right)\right], \label{huouta}\end{aligned}$$ where $$I\_{HU}=\int\_{x=0}^{\infty}\frac{xe^{-\lambda\_{1}x}[e^{-W\_{2}\Gamma\_{th}}-e^{-\frac{2W\_{3}\Gamma\_{th}}{x}}]}{(xW\_{2}-2W\_{3})}dx.$$ Proof is omitted due to paucity of space. The integral IHU has no closed form expression, but is bounded, and can be evaluated numerically. The throughput τHU with HU architecture is given by ([throughput]) with PoutHU replacing PoutHSU: $$\begin{aligned} \tau^{HU} =& \hspace{0.1 cm} R\_{0}\Pr\left\lbrace \gamma\_{\_{SD}}(i)\geq\Gamma^{\prime}\_{th}\right\rbrace +\frac {R\_{0}}{2}\Pr\left\lbrace \gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\right\rbrace \left[ 1-\frac {P\_{out}^{HU}}{\Pr\{\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}\}}\right],\end{aligned}$$ where PoutHU is given by. Direct Transmission ------------------- In this scheme S directly transmits to D. The outage PoutDT and throughput τDT are given by: PoutDT = Pr(γSD(i) < Γth) = (1 − exp( − W2Γth)) and τDT = R0(1 − PoutDT) = R0exp( − W2Γth). Diversity Analysis for IBEP and IOFP ==================================== In this section we present the diversity order analysis for both IBEP and IOFP. For this reason, we use Γth′ = Γth throughout this section. IBEP ---- To achieve maximum throughput for stable buffers we need to use ϕinc greater than unity (but close to it). This implies that $M > \dfrac{\mathbb{E}{\{X(i)}\}}{\left( 1-e^{-W\_{2}\Gamma\_{th}}\right) }$. Clearly, when SNR $ \triangleq \dfrac{P\_{\_{S}}}{\sigma^2} \rightarrow \infty, W\_{2} \rightarrow 0,\hspace{0.2 cm} \text{and\ } W\_{4} \rightarrow 0.$ Note that when W2 → 0, M → ∞, which implies that $ W\_{1} \rightarrow 0 \hspace{0.2 cm} \text{and} \hspace{0.2 cm} Z\_{1} \rightarrow 0. $ Although Z1 and M are dependent on SNR, their product Z1M is independent of SNR (Z1M = W( − ϕincexp( − ϕinc)) + ϕinc is constant). This is a crucial observation, and will be used to establish the diversity order of IBEP starting with. We first note that the terms e− Z1M and (1 − e− Z1M) in are independent of SNR. We know that e− x ≈ 1 − x for small x, we get: $$\begin{aligned} e^{-W\_{2}\Gamma\_{th}} \approx \left(1-W\_{2}\Gamma\_{th} \right), \hspace{0.2 cm} e^{-W\_{4}\Gamma\_{th}} \approx \left(1-W\_{4}\Gamma\_{th} \right), \hspace{0.2 cm} e^{-W\_{1}\Gamma\_{th}} \approx \left(1-W\_{1}\Gamma\_{th} \right). \label{approximaatsnr}\end{aligned}$$ At high SNR, using, the second term present inside the curly brackets of can be approximated as follows: $$\begin{aligned} \dfrac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right] }{(W\_{1}-W\_{2})} \approx \dfrac{W\_{2}\left[ \left( 1-W\_{1}\Gamma\_{th}\right) -\left( 1-W\_{2}\Gamma\_{th}\right) \right] }{(W\_{1}-W\_{2})} \approx -W\_2\Gamma\_{th} \label{fistappr}\end{aligned}$$ The product W3Γth is independent of SNR. Furhter, since noise variance is small, the product W3Γth < 1 for any given target rate R0. Let $a\_{0}=W\_3 \Gamma\_{th}+\Delta, \text{where $> 0suchthat0$}$. Using this, the integral can be simplified as follows: $$\begin{aligned} I\_{HSU}&= W\_{2}\int\_{x=0}^{M}\frac{xZ\_{1}e^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}} -e^{-\frac{W\_{3}\Gamma\_{th}}{x}}] }{(xW\_{2}-W\_{3})}dx {\nonumber}\\ &=\underbrace{W\_{2}\int\_{x=0}^{a\_{0}}\frac{xZ\_{1}e^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}} -e^{-\frac{W\_{3}\Gamma\_{th}}{x}}] }{(xW\_{2}-W\_{3})}dx}\_{I\_{1}}+\underbrace{W\_{2}\int\_{x=a\_{0}}^{M}\frac{xZ\_{1}e^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}} -e^{-\frac{W\_{3}\Gamma\_{th}}{x}}] }{(xW\_{2}-W\_{3})}dx}\_{I\_{2}} {\nonumber}\\ & \overset{t}{\approx } W\_{2}\int\_{x=a\_{0}}^{M}\frac{xZ\_{1}e^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}} -e^{-\frac{W\_{3}\Gamma\_{th}}{x}}] }{(xW\_{2}-W\_{3})}dx {\nonumber}\\ & \approx W\_{2} \int\_{x=a\_{0}}^{M}\frac{xZ\_{1}e^{-Z\_{1}x}\left[ \left( 1-W\_{2}\Gamma\_{th}\right) -\left( 1-\frac{W\_{3}\Gamma\_{th}}{x}\right) \right] }{(xW\_{2}-W\_{3})}dx =- \Gamma\_{th}W\_{2} \int\_{x=a\_{0}}^{M} Z\_{1}e^{-Z\_{1}x} {\nonumber}\\ &= - \Gamma\_{th}W\_{2} \left[\left(1-e^{-Z\_{1}M} \right)- \left(1-e^{-Z\_{1}a\_{0}} \right) \right] \approx - \Gamma\_{th}W\_{2} \left(1-e^{-Z\_{1}M} \right). \label{approxima}\end{aligned}$$ In the above, approximation t results because I1 is insignificant when SNR is high (this is due to the fact that the upper limit in I1 is a0, which is small, and a constant independent of SNR). On the other hand, the upper limit in I2 is M, which increases with SNR. The last approximation in is due to the fact that at high SNR: Z1 → 0 and a0 < 1, making the term (1 − e− Z1a0) → 0. Using, and the outage probability expression given in can be approximated as follows: $$\begin{aligned} P\_{out}^{HSU} \approx &\hspace{0.2 cm} W\_{4}\Gamma\_{th}W\_{2}\Gamma\_{th} =K\_{0}\left( \frac{1}{\text{SNR}}\right) ^{2} \hspace{0.8cm} \phi\_{inc} > 1 \label{divbest}\end{aligned}$$ where K0 = dSDαdSRαΓth2. Clearly, the diversity order = $-\lim\limits\_{\text{SNR} \rightarrow \infty}\dfrac{\log P\_{out}^{HSU} }{\log \text {(SNR)} } =2 $. IOFP ---- Proceedings as with IBEP, the outage probability expression for the IOFP given in can be approximated at high SNR as follows: $$\begin{aligned} P\_{out}^{HSU} \approx & \hspace{0.2 cm} \frac{1}{\phi\_{inc}} \left( W\_{4}\Gamma\_{th}\right) \left( W\_{2}\Gamma\_{th}\right) + \left( 1-\frac{1}{\phi\_{inc}}\right) \left( W\_{2}\Gamma\_{th}\right) \hspace{0.4 cm} \phi\_{inc} > 1 \label{outa5}\end{aligned}$$ $$\begin{aligned} P\_{out}^{HSU} \approx & \hspace{0.2 cm} K\_{1}\left( \frac{1}{\text{SNR}}\right)^{2} +K\_{2} \left( \frac{1}{\text{SNR}}\right) \hspace{0.4 cm}\phi\_{inc} > 1 \label{outa6}\end{aligned}$$ where $K\_{1}=\dfrac{{\Gamma\_{th}}^{2}d\_{\_{SR}}^\alpha d\_{\_{SD}}^\alpha}{\phi\_{inc}}$, and $K\_{2}=\left( 1-\dfrac{1}{\phi\_{inc}}\right)\Gamma\_{th}d\_{\_{SD}}^\alpha$. Again can be approximated when SNR → ∞ as follows: $$\begin{aligned} P\_{out}^{HSU} \approx & \hspace{0.2 cm} K\_{2} \left( \frac{1}{\text{SNR}}\right) \hspace{0.4 cm}\phi\_{inc} > 1\end{aligned}$$ Clearly, the diversity order = $-\lim\limits\_{\text{SNR} \rightarrow \infty}\dfrac{\log P\_{out}^{HSU} }{\log \left(\text{SNR} \right) } =1 $. In the case of IOFP, the diversity order is limited to 1, due to the fact that the relay is silent in some signalling intervals when M amount of energy is not present in the buffer (note that M is large at high SNRs). As a result, the second term present in restricts the diversity order to 1. For the case of ϕinc ≤ 1, it is apparent from Theorems [TheoremLimitingPDF not exists] and [on-off not exists] that M amount of energy is almost always present in the buffer. Using it can be easily shown that diversity order is 2 for both the policies when the buffer is unstable. simulation and numerical results ================================ In this section we present simulations to validate the derived expressions and to obtain insights into performance. In all the simulations, S, R and D are assumed to be located in a two dimensional plane. S is located at (0,0) and D at (4,0). Further, path-loss exponent α = 3, and noise power σ2 = − 40 dB. Since performance plots are presented for both incremental and non-incremental relaying, we will find it convenient to use ϕ to represent either ϕinc or ϕninc. For the existence of a limiting distribution ϕ = ϕinc = ϕninc = 1.1 is chosen for all the figures except Fig. [outagevspinc]. .5 [limtinpdf] .5 [outavssnr] Fig. [limtinpdf] depicts the limiting distribution of energy with IOFP and IBEP for two different source SNRs (25 dB and 35 dB) assuming a fixed target rate R0 = 1.5 bits per channel use (bpcu). The analytical values are obtained using and derivative of. In this, and all figures that follow, the solid lines represent analytical plots, and the markers denote simulation values. It is evident from Fig. [limtinpdf] that when the source SNR increases, the tail of the limiting PDF increases. This is because at higher SNR, R needs to transmit in fewer time slots due to frequent ACK from D (R therefore accumulates more energy). Fig. [outavssnr] depicts the variation of outage probability with source SNR. If the transmit power of the source increases, the outage probability decreases for all three schemes (HSU, HU and DT) due to the strong direct link between S and D. As compared to DT and HU schemes, HSU yields better performance, clearly indicating that use of energy buffers is advantageous. One reason for this is that use of PEB with SEB in HSU allows R to charge and accumulate energy even when it is transmitting. Another reason is that at high source SNRs there is accumulation of energy at R due to frequent ACK from D. This is not possible with the HU scheme. From Fig. [outavssnr] it can also be observed that at low SNRs performance of incremental and non-incremental schemes with best-effort and on-off policies is quite similar. But when the SNR is higher, IBEF is clearly superior in performance to all the other schemes, and achieves full diversity of 2 even with stable buffers. IOFP achieves lower diversity since R is silent (because B(i) < M) in larger number of signalling intervals. .3 [thrptvstarge] .3 [thrptvsrtod] .3 [thrptvsmean] Fig. [thrptvstarge] depicts the variation of throughput with target rate R0. From Fig. [thrptvstarge] we observe that at very low target rates R0, HSU and HU yield practically the same throughput (which is quite intuitive). If the target rate R0 is increased, HSU performs better than HU. Further increase in R0 results in lower throughput performance. It can be seen that HSU schemes result in larger throughput as compared to HU and DT, and can support larger target rates R0 (due to use of the energy buffer). From Fig. [thrptvstarge] it can also be observed that for low-to-medium target rates, the performance of incremental schemes is superior to that of non-incremental schemes. For higher target rates, the performance is quite similar due to increase in frequency of NACK from D. Fig. [thrptvsrtod] depicts the variation of throughput with respect to R-D distance dRD keeping S-D distance dSD the same (dSD = dSR + dRD). From Fig. [thrptvsrtod] it can be observed that when the R is located away from D, throughput increases for both HSU and HU upto some point since the probability of successful decoding at R increases (path-loss dSRα decreases). However, as R moves closer to S, the increasing R-D distance causes loss in throughput as it reduces the probability of successful decoding at D (path-loss dRDα increases). Thus there exists an optimal R location for both HSU and HU as can be observed from Fig. [thrptvsrtod]. Note also that HSU results in higher throughputs as compared to HU when R is located further away from the D (this once again is due to the energy buffer). From Fig. [thrptvsrtod] it can also be observed that the performance of incremental schemes is always superior to that of the non-incremental schemes for a given R0. .5 [outagevspinc] .5 [thrptvssnr] Fig. [thrptvsmean] depicts the variation of throughput with respect to mean of the incoming EH process $\frac{1}{\lambda\_{1}}$ (in dB). Clearly, throughput increases with increase in $ \frac{1}{\lambda\_{1}}$ for HSU and HU. With HSU architecture, as the mean $(\frac{1}{\lambda\_{1}})$ increases, the energy stored at R increases. However, the energy drawn from the buffer (M) also simultaneously increases for both incremental and non-incremental schemes (since ϕ = ϕninc = ϕinc is a constant). For large $ \frac{1}{\lambda\_{1}}$ values, performance of HU is closer to that of HSU with on-off policy. From Fig. [thrptvsmean] it can also be observed that the incremental schemes result in performance that is superior to that of non-incremental schemes. Fig. [outagevspinc] shows that variation of outage vs. ϕinc for three different source SNRs (25, 30 and 35 dB). When ϕinc < 1, performance of IBEP and IOFP is the same (this is because M units of energy is available almost surely with both schemes). When ϕinc > 1, IBEP results in better performance (the gap is particularly pronounced at high SNRs). This is because M increases with increase in ϕinc, which causes IOFP to keep R slient in more signalling intervals due to paucity of energy. Note that the optimum ϕinc is close to 1. From Fig. [outagevspinc] it can also be observed that for high source SNR (35 dB), the performance of IOFP is very poor for ϕinc > 1 compared to that of IBEP. This is because a very high M is required at these SNRs, and the relay with IOFP has insufficient energy in some signalling intervals. With IBEP, a small amount of energy (B(i) < M) often sufices to ensure sufficient SNR at D. Fig. [thrptvssnr] depicts the variation of throughput vs. source SNR PS/σ2 assuming the optimum target rate R0 is used at each source SNR. For low SNRs (10 to 20 dB) performance of HSU incremental and HSU non-incremental schemes is quite similar due to frequent failure of the S-D direct link. But when the SNR increases further (beyond 20 dB) for the optimum target rate R0, there is increase in ACKs with incremental schemes, which enable them to yield higher throughput. From Fig. [thrptvssnr] it can also be observed that at very high SNR, the throughput gap between incremental and non-incremental is very high for optimum target rate R0 due to increase in frequency of ACKs from D with incremental signalling, which causes energy accumulation in the buffer (this energy is used for relaying in the signalling intervals in which NACK is received from D). At higher SNR, DT and HU incremental are also better than non-incremental transmission schemes. conclusion ========== In this paper, performance of energy buffer-aided incremental relaying is analyzed for fixed-rate transmission. A discrete-time continuous state-space Markov chain model is used for the energy buffer, and limiting distributions are derived for the stored energy with incremental best-effort and on-off policies. Expressions are derived for outage probabilities and throughput. Corresponding expressions for non-incremental schemes follow as special cases. It is shown that with IBEP, diversity of two is attained with stable buffers, while diversity of only one is attained with IOFP. Throughput with IBEP is marginally superior to that with IOFP. Proof of Theorem [TheoremLimitingPDF] ===================================== Using the total probability theorem, the CDF of B(i + 1) can be evaluated for energy storage model of as follows: $$\begin{aligned} &\Pr \{B\left( i+1\right) \leq x \} = \Pr \left\lbrace \left( B\left( i\right)+X\left( i\right) \right) \leq x \mid \gamma\_{\_{SD}}(i) \geq \Gamma^{\prime}\_{th}\right\rbrace \Pr \{\gamma\_{\_{SD}}(i) \geq \Gamma^{\prime}\_{th}\}{\nonumber}\\ &+ \Pr \left\lbrace (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}), (B(i)\geq M)\right\rbrace \Pr \left\lbrace \left( B\left( i\right)-M+X\left( i\right)\right) \leq x \mid (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}), (B(i)\geq M) \right\rbrace {\nonumber}\\ &+ \Pr \{(\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}), (B(i)<M)\} \Pr \{ X\left( i\right) \leq x \mid (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}), (B(i)<M) \}. \label{buffeq3}\end{aligned}$$ Let B(i) = U, X(i) = X and γSD(i) = γSD. Then can be written in terms of joint probabilities as: $$\begin{aligned} \Pr \{B\left( i+1\right) \leq x \}= & \Pr \{ U+X \leq x, (\gamma\_{\_{SD}}\geq \Gamma^{\prime}\_{th}) \}{\nonumber}\\ &+\Pr \{ U-M+X \leq x, (\gamma\_{\_{SD}}<\Gamma^{\prime}\_{th}), (U\geq M) \} {\nonumber}\\ & +\Pr \{ X \leq x, (\gamma\_{\_{SD}}<\Gamma^{\prime}\_{th}), (U<M) \}. \label{10} \end{aligned}$$ Let Gi(x) be the CDF of B(i). Then Gi + 1(x) is the CDF of B(i + 1) i.e. Pr{B(i + 1) ≤ x} = Gi + 1(x). After a long time (i → ∞), the buffer reaches its steady-state so that Gi(x) = Gi + 1(x) = G(x). In this state, can be written as follows: $$\begin{aligned} G(x)= &\int\_{y=\Gamma^{\prime}\_{th}}^{\infty}\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left( u\right)f\_{\gamma\_{\_{SD}}}\left( y\right) du dy {\nonumber}\\ &+ \int\_{y=0}^{\Gamma^{\prime}\_{th}}\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\left( u\right)f\_{\gamma\_{\_{SD}}}\left( y\right) du dy {\nonumber}\\ &+\int\_{y=0}^{\Gamma^{\prime}\_{th}}\int\_{u=0}^{M}F\_{X}\left( x\right)g\left( u\right)f\_{\gamma\_{\_{SD}}}\left( y\right) du dy, \label{11} \end{aligned}$$ where g(x) is the derivative of G(x). The PDFs of γSD and X are given by: fγSD(y) = W2exp( − W2y) and FX(x) = 1 − exp( − λ1x). Substituting fγSD(y) in we get: $$\begin{aligned} G(x)= \hspace{0.1 cm}&e^{-W\_{2}\Gamma^{\prime}\_{th}}\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left( u\right)du {\nonumber}\\ & +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\left( u\right)du + \int\_{u=0}^{M}F\_{X}\left( x\right)g\left(u\right)du \right]. \label{limcdfints} \end{aligned}$$ Consider the term ∫u = MM + xFX(x − u + M)g(u)du + ∫u = 0MFX(x)g(u)du. Integrating by parts and using G(0) = 0, the term can be simplified as follows: $$\begin{aligned} =\hspace{0.1 cm}& F\_{X}\left( x-u+M\right)G(u)\Bigm\|\_{u=M}^{M+x}+ \int\_{u=M}^{M+x}f\_{X}\left(x-u+M\right) G\left( u\right) du+F\_{X}\left( x\right) G\left( M\right) \\ = & -F\_{X}\left( x\right) G\left( M\right)+\int\_{u=M}^{M+x}f\_{X}\left(x-u+M\right) G\left( u\right) du+F\_{X}\left( x\right) G\left( M\right) \\ =& \int\_{u=M}^{M+x}f\_{X}\left(x-u+M\right) G\left( u\right) du. \end{aligned}$$ Similarly, $\displaystyle\int\_{u=0}^{x}F\_{X}\left( x-u\right) g\left( u\right)du\hspace{-0.02in}=\hspace{-0.02in}\int\_{u=0}^{x}G\left( u\right) f\_{X}\left( x-u\right) du$ so that can be simplified as: $$\begin{aligned} G(x)= \hspace{0.1 cm} & e^{-W\_{2}\Gamma^{\prime}\_{th}}\displaystyle\int\limits\_{u=0}^{x}f\_{X}\left( x-u\right)G\left( u\right)du +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\int\limits\_{u=M}^{M+x}f\_{X}\left( x-u+M\right)G\left( u\right)du. \label{13} \end{aligned}$$ Using v = x − u + M, the second integral in can be simplified as: ∫u = MM + xfX(x − u + M)G(u)du = ∫u = 0xfX(u)G(x + M − u)du. With this, can be simplified and rewritten as in. The integral equation can be solved by postulating a solution of the form,: $$\begin{aligned} G\left( x\right) =\sum\_{i}c\_{i}e^{ -Z\_{i}x}. \label{cdfl}\end{aligned}$$ Substituting fX(x) = λ1e− λ1x in and using we get: $$\begin{aligned} \sum\_{i}c\_{i}e^{-Z\_{i}x}=\hspace{0.1 cm}& e^{-W\_{2}\Gamma^{\prime}\_{th}}\lambda\_{1}e^{ -\lambda\_{1}x} \int\_{u=0}^{x}e^{ \lambda\_{1}u} \sum\_{i}c\_{i}e^{-Z\_{i}u}du {\nonumber}\\ &+\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}\int\_{u=0}^{x}e^{-\lambda\_{1}u} \sum\_{i}c\_{i}e^{ -Z\_{i}\left(x+M-u\right)}du.\end{aligned}$$ Exchanging the summation and integration, and simplifying, we get: $$\begin{aligned} \sum\_{i}c\_{i}e^{-Z\_{i}x}= &\sum\_{i}\left[ \frac {e^{-W\_{2}\Gamma^{\prime}\_{th}}\lambda\_{1}+\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}e^{ -Z\_{i}M} }{\left( \lambda\_{1}-Z\_{i}\right)} \right]c\_{i}e^{-Z\_{i}x} {\nonumber}\\ &-e^{-\lambda\_{1}x} \sum\_{i}\left[ \frac {e^{-W\_{2}\Gamma^{\prime}\_{th}}\lambda\_{1}+\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}e^{-Z\_{i}M} }{\left( \lambda\_{1}-Z\_{i}\right)} \right]c\_{i}. \end{aligned}$$ For both sides of the above equation to be equal, the following conditions need to be satisfied: $$\begin{aligned} \frac {e^{-W\_{2}\Gamma^{\prime}\_{th}}\lambda\_{1}+\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}e^{-Z\_{i}M} }{\left( \lambda\_{1}-Z\_{i}\right)}=1 \hspace{0.5cm} \forall i,\hspace{1 cm} \text{and} \hspace{1 cm} \sum\_{i}c\_{i}=0. \label{17}\end{aligned}$$ Clearly, has two roots, one default root is Z0 = 0 for i = 0, and other root Z1 for i = 1 can be seen to satisfy the following equation: $$\begin{aligned} e^{-W\_{2}\Gamma^{\prime}\_{th}}\lambda\_{1}+\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}e^{-Z\_{1}M}=\left( \lambda\_{1}-Z\_{1}\right),\\ \Rightarrow e^{ -Z\_{1}M} =\frac{-Z\_{1}M}{\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}M}+1. \label{othroot}\end{aligned}$$ With b = 1, $a=\dfrac{1}{\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}M}$, p = e, x = − Z1M, the above is of the form px = ax + b, whose solution is $ x=-\mathcal{W}\left( -a^{-1}p^{-\frac{b}{a}}\ln p\right)-\dfrac{b}{a} $, when p > 0 and p ≠ 1 (a ≠ 0), where W( ⋅ ) is Lambert-W function. This implies that: $$\begin{aligned} Z\_{1}=\frac{\mathcal{W}\left( -\frac{1}{a}\exp{\left( -\frac{1}{a}\right) }\right)}{M}+\frac{1}{aM}. \label{z1vas}\end{aligned}$$ For the limiting distribution to exist, Z1 should be positive (Z1 > 0) so that $$\begin{aligned} \hspace{1 cm} & \frac{\mathcal{W}\left( -\frac{1}{a}\exp{\left( -\frac{1}{a}\right) }\right)}{M}+\frac{1}{aM} >0 \label{22}\end{aligned}$$ Using the property of Lambert-W function $\mathcal{W}\left( -x\exp\left( -x\right)\right) =-x \hspace{0.1 cm}$ (0 < x ≤ 1), the condition for existence of limiting distribution can be seen to be: $$\begin{aligned} \frac{1}{a} > 1 &\Rightarrow \frac{1}{\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}M} <1 \Rightarrow M>\frac{1}{\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}} {\nonumber}\\ & \Rightarrow M > \frac {\mathbb{E}{\{X(i)}\}}{\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right) }.\end{aligned}$$ It is clear from ([phinc]) that the above implies that ϕinc > 1. From, we have: $$\begin{aligned} G\left( x\right) =\sum\_{i}c\_{i}e^{-Z\_{i}x}=c\_{0}e^{-Z\_{0}x} +c\_{1}e^{ -Z\_{1}x}=c\_{0}+c\_{1}e^{-Z\_{1}x}. \label{cf}\end{aligned}$$ Since G(x) is limiting CDF of energy stored in the PEB, G(0) = 0 and G(∞) = 1. The latter implies that c0 = 1 and the former shows that c0 + c1 = 0 (c1 = − 1). Substituting the values of c0, c1 in, we get G(x) = 1 − exp( − Z1x). After substituting for ‘a’ in we obtain the Z1 as in. Proof of Theorem [TheoremLimitingPDF not exists] ================================================ Let the S-D link outage indicator variable OSD(i) ∈ (0, 1) be defined as follows: $$\mathbb{O}\_{\_{SD}}\left( i\right)\triangleq\begin{cases} 1, & \text{if $\gamma\_{\_{SD}}\left( i\right) < \Gamma^{\prime}\_{th}$}\\ 0, & \text{if $\gamma\_{\_{SD}}\left( i\right) \geq \Gamma^{\prime}\_{th}$}. \end{cases} \label{indi}$$ Let ER(i) = min(B(i), M). With, the buffer update equations in can be rewritten as: $$\begin{aligned} B\left( i+1\right) - B\left( i\right) = X\left( i\right)-E\_{R}\left( i\right)\mathbb{O}\_{\_{SD}}\left( i\right). \label{buffequal}\end{aligned}$$ Ergodicity implies that $\lim\limits\_{N \rightarrow \infty}\dfrac{1}{N}\sum\_{i=1}^{N}\mathbb{O}\_{\_{SD}}\left( i\right)=\mathbb{E}\{\mathbb{O}\_{\_{SD}}\left( i\right)\}$ so that: $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}\mathbb{O}\_{\_{SD}}\left( i\right)= & \mathbb{E}\{\mathbb{O}\_{\_{SD}}\left( i\right)\}= 1\times\Pr\{\mathbb{O}\_{\_{SD}}\left( i\right)=1\}+0\times\Pr\{\mathbb{O}\_{\_{SD}}\left( i\right)=0\} {\nonumber}\\ =&\Pr\{\mathbb{O}\_{\_{SD}}\left( i\right)=1\}=\Pr\{\gamma\_{\_{SD}}\left( i\right) < \Gamma^{\prime}\_{th}\}=\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right). \label{33}\end{aligned}$$ From the law of conservation of flow, average arrival rate is always greater than or equal to average departure rate i.e. A ≥ D. A = D holds only when the buffer is non-absorbing. Using law of large numbers, the average arrival rate is given by: $$\begin{aligned} \mathcal{A}=\mathbb{E}\left\lbrace X\left( i\right) \right\rbrace =\lim\limits\_{N \rightarrow \infty}\dfrac{1}{N} \sum\_{i=1}^{N}X\left( i\right) \label{avgarrival} \end{aligned}$$ Similarly the average departure rate D is given by: $$\begin{aligned} \mathcal{D}=\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}E\_{R}\left( i\right)\mathbb{O}\_{\_{SD}}\left( i\right)=\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{B(i)\geq M}M\mathbb{O}\_{\_{SD}}\left( i\right) +\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{B(i)< M}B(i)\mathbb{O}\_{\_{SD}}\left( i\right)\end{aligned}$$ The relay transmit energy ER(i) is upper bounded by ER(i) ≤ M ⇒ $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}E\_{\_{R}}\left( i\right)\mathbb{O}\_{\_{SD}}\left( i\right) \leq \lim\limits\_{N \rightarrow \infty} \frac{1}{N}\sum\_{i=1}^{N}M\mathbb{O}\_{\_{SD}}\left( i\right) \label{upperbou}\end{aligned}$$ For ϕinc < 1 ⇒ it follows from that: $$\begin{aligned} \mathbb{E}\{X\left( i\right)\} > M\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\Rightarrow \lim\limits\_{N \rightarrow \infty} \frac{1}{N}\sum\_{i=1}^{N}X\left( i\right) > \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}M\mathbb{O}\_{\_{SD}}\left( i\right) \label{inequ} \end{aligned}$$ From and we get : $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}X\left( i\right) > \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}E\_{\_{R} }\left( i\right)\mathbb{O}\_{\_{SD}}\left( i\right) \label{buffabsor}\end{aligned}$$ From and we get: $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}X\left( i\right) > \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}E\_{\_{R} }\left( i\right)\mathbb{O}\_{\_{SD}}\left( i\right) \Rightarrow \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}B\left( i+1\right)>\lim\limits\_{N \rightarrow \infty} \frac{1}{N}\sum\_{i=1}^{N}B\left( i\right) \label{expe}\end{aligned}$$ Clearly, indicates that the buffer is in an absorbing state ( i.e. A > D), which makes it unstable. In this case the average buffer length increases with each signalling interval. After a certain number of signalling intervals, M amount of energy is present in the buffer almost surely i.e. Pr(B(i) ≥ M) → 1. For ϕinc = 1 ⇒ it follows from that: $$\begin{aligned} \mathbb{E}\{X\left( i\right)\} = M\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\Rightarrow \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}X\left( i\right) = \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}M \mathbb{O}\_{\_{SD}}(i) \label{inflow} \end{aligned}$$ From and we get : $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}X\left( i\right) \geq \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}E\_{\_{R}}\left( i\right)\mathbb{O}\_{\_{SD}}(i) \label{balcritical}\end{aligned}$$ In when inequality condition holds, the buffer is unstable i.e. A > D. Then from the buffer is in absorbing state, which is unstable. We can get almost surely M amount of energy from the buffer i.e. Pr(B(i) ≥ M) → 1. In if equality holds, then from, and we get: $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}M\mathbb{O}\_{\_{SD}}(i)=\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}X\left( i\right)=\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}E\_{\_{R}}\left( i\right)\mathbb{O}\_{\_{SD}}(i) \label{buffbala}\end{aligned}$$ Using ER(i) = min(B(i), M), gives: $$\begin{aligned} \lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{i=1}^{N}M\mathbb{O}\_{\_{SD}}(i)=\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{B(i) \geq M} \hspace{-0.3 cm} M\mathbb{O}\_{\_{SD}}(i)+\lim\limits\_{N \rightarrow \infty}\frac{1}{N}\sum\_{B(i) < M }\hspace{-0.3 cm}B(i)\mathbb{O}\_{\_{SD}}(i) \label{critically}\end{aligned}$$ Clearly, holds only when the buffer is at the edge of non-absorbing (but it is at the boundary of absorbing and non-absorbing state). This kind of buffers are termed critically balanced buffers. These buffers may be stable, sub-stable or unstable,. Usually the Markov chain sequences which are critically balanced are unstable. From it can be observed that the second term is zero, and that B(i) > M almost surely (ER(i) = M and Pr(B(i) ≥ M) → 1. Thus M amount of energy is almost always present in the buffer for ϕinc = 1. Proof of Theorem [TheoremLimitingPDF On-Off] ============================================ Using an approach similar to that used in Appendix [AppendixLimitingPDF] for IBEP, the CDF of B(i + 1) with IOFP can be obtained from energy storage model in as follows: $$\begin{aligned} \Pr \{B\left( i+1\right) \leq x \}= & \Pr \left\lbrace \left( B\left( i\right)+X\left( i\right) \right) \leq x, (\gamma\_{\_{SD}}(i)\geq \Gamma^{\prime}\_{th}) \right\rbrace {\nonumber}\\ &+\Pr \left\lbrace \left( B\left( i\right)-M+X\left( i\right) \right) \leq x, (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}), (B(i)\geq M) \right\rbrace {\nonumber}\\ & +\Pr \left\lbrace \left( B\left( i\right)+X\left( i\right)\right) \leq x, (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th}), \left( B(i)<M\right) \right\rbrace. \label{onoffsimp} \end{aligned}$$ With B(i) = U, X(i) = X and γSD(i) = γSD, can be written as: $$\begin{aligned} \Pr \{B\left( i+1\right) \leq x \}= & \Pr \{\left( U+X\right) \leq x, (\gamma\_{\_{SD}} \geq \Gamma^{\prime}\_{th}) \}{\nonumber}\\ &+\Pr \{\left( U-M+X\right) \leq x, (\gamma\_{\_{SD}}<\Gamma^{\prime}\_{th}), (U \geq M) \} {\nonumber}\\ & +\Pr \{\left( U+X \right) \leq x, (\gamma\_{\_{SD}}<\Gamma^{\prime}\_{th}), (U<M) \}. \label{onoffsolv} \end{aligned}$$ Let Gi(x) be the CDF of B(i), then Gi + 1(x) is the CDF of B(i + 1) i.e. Pr{B(i + 1) ≤ x} = Gi + 1(x). After a long time (i → ∞), the buffer reaches its steady-state so that Gi(x) = Gi + 1(x) = G(x). In this state, can be simplified as follows: $$\begin{aligned} G(x)= &\displaystyle\int\_{y=\Gamma^{\prime}\_{th}}^{\infty}\displaystyle\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left( u\right)f\_{\gamma\_{\_{SD}}}\left( y\right) du dy {\nonumber}\\ &+ \int\_{y=0}^{\Gamma^{\prime}\_{th}}\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\left( u\right)f\_{\gamma\_{\_{SD}}}\left( y\right) du dy {\nonumber}\\ &+\int\_{y=0}^{\Gamma^{\prime}\_{th}}\int\_{u=0}^{\min (x,M)}F\_{X}\left( x-u\right)g\left( u\right)f\_{\gamma\_{\_{SD}}}\left( y\right) du dy. \label{onoff11} \end{aligned}$$ In the upper limit of third term integral is min(x, M) since U < M and FX(0) = 0. Substituting fγSD(y) = W2exp( − W2y) in it can be simplified as in. Using fγSD(y), when x ≥ M, the function G(x) is: $$\begin{aligned} G(x)= & e^{-W\_{2}\Gamma^{\prime}\_{th}}\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\left( u\right)du +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\left( u\right)du \right. {\nonumber}\\ &\left.+ \int\_{u=0}^{M}F\_{X}\left( x-u\right)g\left(u\right)du \right] \hspace{2.9 cm} x \geq M \label{cdfonoff2} \end{aligned}$$ Let g(x) can be defined as: $$\begin{aligned} g\left( x\right) = \begin{cases} g\_{1}\left( x\right) \hspace{2.9 cm} 0 \leq x < M \\ g\_{2}\left( x\right) \hspace{2.9 cm} x \geq M \end{cases} \label{limpdfsplit} \end{aligned}$$ From and the function G(x) can be written as: $$\begin{aligned} G(x)= & \hspace{0.1 cm} e^{-W\_{2}\Gamma^{\prime}\_{th}}\int\_{u=0}^{M}F\_{X}\left( x-u\right)g\_{1}\left( u\right)du +e^{-W\_{2}\Gamma^{\prime}\_{th}}\int\_{u=M}^{x}F\_{X}\left( x-u\right)g\_{2}\left( u\right)du +{\nonumber}\\ &\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\_{2}\left( u\right)du + \int\_{u=0}^{M}F\_{X}\left( x-u\right)g\_{1}\left(u\right)du \right] \hspace{0.2 cm} x \geq M \label{cdfonoffpart2} \end{aligned}$$ After cancellation of some terms in and differentiating w.r.t. ‘x’ on both sides, we get: $$\begin{aligned} g\_{2}\left( x\right) = & \int\_{u=0}^{M}f\_{X}\left( x-u\right)g\_{1}\left( u\right)du +e^{-W\_{2}\Gamma^{\prime}\_{th}}\int\_{u=M}^{x}f\_{X}\left( x-u\right)g\_{2}\left( u\right)du +{\nonumber}\\ &\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\int\_{u=M}^{M+x}f\_{X}\left( x-u+M\right)g\_{2}\left( u\right)du \hspace{1.6 cm} x \geq M. \label{onoffintegr} \end{aligned}$$ We postulate that has a solution of the form g2(x) = keQx where k is a constant. Using g2(x) = keQx and fX(x) = λ1e− λ1x, can be simplified as follows: $$\begin{aligned} ke^{Qx} = & \hspace{0.1 cm} \lambda\_{1}e^{-\lambda\_{1}x} \int\_{u=0}^{M}e^{\lambda\_{1}u}g\_{1}\left( u\right)du +e^{-W\_{2}\Gamma^{\prime}\_{th}}\lambda\_{1}e^{-\lambda\_{1}x}\int\_{u=M}^{x}e^{\lambda\_{1}u}ke^{Qu}du +{\nonumber}\\ &\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\lambda\_{1}e^{-\lambda\_{1}x}\int\_{u=M}^{M+x}e^{\lambda\_{1}\left( u-M\right) }ke^{Qu}du \hspace{1.6 cm} x \geq M \label{onoffintsol} \end{aligned}$$ Simplifying we get: $$\begin{aligned} ke^{Qx} = & \hspace{0.1 cm} \lambda\_{1}e^{-\lambda\_{1}x}\left[ \int\_{u=0}^{M}e^{\lambda\_{1}u}g\_{1}\left( u\right)du -\frac{ke^{MQ}}{\left( Q+\lambda\_{1}\right) }\left[\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)+e^{M\lambda\_{1}}e^{-W\_{2}\Gamma^{\prime}\_{th}}\right] \right] {\nonumber}\\ &+\frac{k\lambda\_{1}e^{Qx}}{\left(Q+\lambda\_{1}\right)}\left[ e^{-W\_{2}\Gamma^{\prime}\_{th}}+e^{MQ}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) \right] \hspace{1.6 cm} x \geq M. \label{onoffequat} \end{aligned}$$ Both sides of are equal when: $$\begin{aligned} \int\_{u=0}^{M}e^{\lambda\_{1}u}g\_{1}\left( u\right)du=\frac{ke^{MQ}}{\left( Q+\lambda\_{1}\right) }\left[\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)+e^{M\lambda\_{1}}e^{-W\_{2}\Gamma^{\prime}\_{th}}\right], \end{aligned}$$ and $$\begin{aligned} \frac{Q+\lambda\_{1}}{\lambda\_{1}}= e^{-W\_{2}\Gamma^{\prime}\_{th}}+e^{MQ}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right). \label{onoffnonequ} \end{aligned}$$ Simplifying we get: $$\begin{aligned} \lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)e^{MQ}=Q+\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right). \label{onoffeqcond} \end{aligned}$$ One default solution of is Q = 0. However, this is not feasible solution for the limiting PDF (area under the PDF needs to be unity). The other solution of can be obtained as when ϕinc > 1. Similarly from and, when 0 ≤ x < M, the function G(x) is given by: $$\begin{aligned} G(x)= \hspace{0.1 cm}& e^{-W\_{2}\Gamma^{\prime}\_{th}}\int\_{u=0}^{x}F\_{X}\left( x-u\right)g\_{1}\left( u\right)du +\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right){\nonumber}\\ &\left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\_{2}\left( u\right)du + \int\_{u=0}^{x}F\_{X}\left( x-u\right)g\_{1}\left(u\right)du \right] \hspace{0.1 cm} 0 \leq x < M \label{cdfonoffpar} \end{aligned}$$ After cancellation of some terms in we get: $$\begin{aligned} G(x)= & \int\_{u=0}^{x}F\_{X}\left( x-u\right)g\_{1}\left( u\right)du +{\nonumber}\\ &\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left[\int\_{u=M}^{M+x}F\_{X}\left( x-u+M\right)g\_{2}\left( u\right)du \right] \hspace{0.6 cm} 0 \leq x < M. \label{onoffpdfsolv} \end{aligned}$$ Differentiating both sides of w.r.t. ‘x’ and using g2(x) = keQx and fX(x) = λ1e− λ1x, we get: $$\begin{aligned} g\_{1}(x)= & \hspace{0.1 cm} \lambda\_{1} \int\_{u=0}^{x}e^{-\lambda\_{1}\left( x-u\right) }g\_{1}\left( u\right)du +{\nonumber}\\ &\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left[\int\_{u=M}^{M+x}\lambda\_{1}e^{-\lambda\_{1}\left( x-u+M\right) }ke^{Qu}du \right] \hspace{0.6 cm} 0<x \leq M \label{cdfonoffpart} \end{aligned}$$ Simplifying the second integral in, we obtain: $$\begin{aligned} g\_{1}(x)=\lambda\_{1} \int\_{u=0}^{x}e^{-\lambda\_{1}\left( x-u\right) }g\_{1}\left( u\right)du+\frac{k\lambda\_{1}e^{QM}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)}{\left( Q+\lambda\_{1}\right) }\left[ e^{Qx}-e^{-\lambda\_{1}x}\right]. \label{onoffvolt} \end{aligned}$$ The above integral equation can be rewritten as follows: $$\begin{aligned} g\_{1}(x)=\lambda\_{1} \int\_{u=0}^{x}e^{-\lambda\_{1}\left( x-u\right) }g\_{1}\left( u\right)du+r\left( x\right), \label{onoffvolter} \end{aligned}$$ where $r(x)=\dfrac{k\lambda\_{1}e^{QM}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)}{\left(Q+\lambda\_{1}\right) }\left[ e^{Qx}-e^{-\lambda\_{1}x}\right]$. is a Volterra type integral equation of the second kind whose solution is given by : $$\begin{aligned} g\_{1}(x)=r\left( x\right)+\lambda\_{1}\int\_{t=0}^{x}r(t) dt. \label{valofrx} \end{aligned}$$ Substituting r(x) in we get: $$\begin{aligned} g\_{1}(x)=\dfrac{k\lambda\_{1}e^{QM}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)}{\left( Q+\lambda\_{1}\right) }\left[ e^{Qx}-e^{-\lambda\_{1}x}\right]+ \lambda\_{1}\int\_{t=0}^{x}\dfrac{k\lambda\_{1}e^{QM}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)}{\left(Q+\lambda\_{1}\right) }\left[ e^{Qt}-e^{-\lambda\_{1}t}\right]dt. \label{onoffint} \end{aligned}$$ Simplifying we get: $$\begin{aligned} g\_{1}(x)=\frac{k \lambda\_{1}e^{QM}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\left( e^{Qx}-1\right)}{Q}. \label{funcg1x} \end{aligned}$$ To find k in g2(x) and g1(x) we impose the unit area property of PDF g(x) i.e. $$\begin{aligned} \int\_{x=0}^{\infty}g\left( x\right) dx=\int\_{x=0}^{M}g\_{1}\left( x\right)dx +\int\_{x=M}^{\infty}g\_{2}\left( x\right) dx=1 \end{aligned}$$ $$\begin{aligned} \frac{k\lambda\_{1}e^{QM}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right) }{Q} \int\_{x=0}^{M}\left( e^{Qx}-1\right) dx +\int\_{x=M}^{\infty}ke^{Qx} dx=1. \label{onoffkval} \end{aligned}$$ Simplifying, the value of k is obtained as: $$\begin{aligned} k=\frac{-Q}{M\left(Q+\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\right)}. \end{aligned}$$ Substituting k value in and using we obtain: $$\begin{aligned} g\_{1}\left( x\right) =\frac{1}{M}\left( 1-e^{Qx}\right). \label{funcg1xfin} \end{aligned}$$ Similarly after substituting k value in g2(x) = keQx we get: $$\begin{aligned} g\_{2}\left( x\right) =\frac{-Qe^{Qx}}{M\left(Q+\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\right)}. \label{funcg2xfin} \end{aligned}$$ Substituting and in, we obtain. Outage analysis of HSU with IBEP ================================ Simplifying the outage probability expression, the final outage probability expression for incremental relaying with DF protocol is given by : $$\begin{aligned} P\_{out}= &\Pr\left\lbrace \gamma\_{\_{SR}}(i) < \Gamma\_{th}\right\rbrace \Pr\left\lbrace \gamma\_{\_{SD}}(i) < \Gamma^{\prime}\_{th}\right\rbrace {\nonumber}\\ &+\Pr\left\lbrace \gamma\_{\_{SR}}(i) \geq \Gamma\_{th}\right\rbrace \Pr\left\lbrace \gamma\_{\_{RD}}(i)+\gamma\_{\_{SD}}(i) < \Gamma\_{th} \right\rbrace. \label{outgen}\end{aligned}$$ Clearly, $\Pr\left\lbrace \gamma\_{\_{SR}}(i) < \Gamma\_{th}\right\rbrace=1-\exp\left( - W\_{4}\Gamma\_{th}\right), \hspace{0.1 cm} \Pr\left\lbrace \gamma\_{\_{SD}}(i) < \Gamma^{\prime}\_{th}\right\rbrace=1-\exp\left( - W\_{2}\Gamma^{\prime}\_{th}\right), \text{and} \\ \hspace{0.4 cm}\Pr\left\lbrace \gamma\_{\_{SR}}(i) \geq \Gamma\_{th}\right\rbrace=\exp\left( - W\_{4}\Gamma\_{th}\right).$ Further: $$\begin{aligned} \Pr\left\lbrace \gamma\_{\_{RD}}(i)+\gamma\_{\_{SD}}(i) < \Gamma\_{th} \right\rbrace=\Pr\left\lbrace \frac {P\_{\_{R}}(i)\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+\gamma\_{\_{SD}}(i) < \Gamma\_{th} \right\rbrace. \label{rel}\end{aligned}$$ Since PR(i) in depends on ϕinc, it will be convenient to study the cases when ϕinc > 1 and ϕinc ≤ 1 separately. $ \hspace{1 cm} \phi\_{inc} > 1 $ Substituting the relay transmit power PR(i) from into gives: $$\begin{aligned} \Pr\left\lbrace \frac {P\_{\_{R}}(i)\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+\gamma\_{\_{SD}}(i) < \Gamma\_{th} \right\rbrace =&\Pr\left\lbrace \frac {2\min\left( B\left( i\right),M\right)\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+\gamma\_{\_{SD}}(i) < \Gamma\_{th}\right\rbrace {\nonumber}\\ =&\Pr\left\lbrace \frac {2 B\left( i\right)\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+\gamma\_{\_{SD}}(i) < \Gamma\_{th},B\left( i\right)<M\right\rbrace {\nonumber}\\ +&\Pr\left\lbrace \frac {2 M\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+\gamma\_{\_{SD}}(i) < \Gamma\_{th},B\left( i\right)\geq M\right\rbrace \label{outpr}\end{aligned}$$ The first term in can be simplified using $ \Psi\_{\_{RD}}\left( i\right) \triangleq \dfrac{2\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}.$ Denote the PDFs of ΨRD(i) and γSD(i) by fΨRD(i)(y) = W3e− W3y and fγSD(i)(z) = W2e− W2z, and the limiting PDF of B(i) by g(x) = Z1e− Z1x. Then can be simplified as follows: $$\begin{aligned} &\Pr\left\lbrace B\left( i\right)\Psi\_{\_{RD}}\left( i\right) +\gamma\_{\_{SD}}(i) < \Gamma\_{th}, B\left( i\right)<M\right\rbrace {\nonumber}\\ &= \int\_{x=0}^{M} \int\_{y=0}^{\frac{\Gamma\_{th}-z}{x}}\int\_{z=0}^{\Gamma\_{th}}g\left( x\right) f\_{\Psi\_{\_{RD}}\left( i\right)}\left( y\right) f\_{\gamma\_{\_{SD}}\left( i\right)}\left( z\right) dx dy dz {\nonumber}\\ &=\left( 1-e^{-Z\_{1}M} \right) \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +Z\_{1}W\_{2}\int\_{x=0}^{M}\frac{xe^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}}-e^{-\frac {W\_{3}\Gamma\_{th}}{x}}]}{(xW\_{2}-W\_{3})}dx, \label{term1}\end{aligned}$$ where Z1 is given by. The second term in can be simplified using $ V\_{\_{RD}}\left( i\right) \triangleq \dfrac{2M\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}.$ Denote the PDF of VRD(i) by fVRD(i)(y) = W1e− W1y. Then the second term in can be simplified as follows: $$\begin{aligned} &\Pr\left\lbrace V\_{\_{RD}}\left( i\right) +\gamma\_{\_{SD}}(i) < \Gamma\_{th},B\left( i\right) \geq M\right\rbrace {\nonumber}\\ &= \int\_{x=M}^{\infty} \int\_{y=0}^{\Gamma\_{th}-z}\int\_{z=0}^{\Gamma\_{th}}g\left( x\right) f\_{V\_{\_{RD}}\left( i\right)}\left( y\right) f\_{\gamma\_{\_{SD}}\left( i\right)}\left( z\right) dx dy dz {\nonumber}\\ &=e^{-Z\_{1}M} \left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right]}{(W\_{1}-W\_{2})} \right\rbrace. \label{term2}\end{aligned}$$ Substituting, into then, we get: $$\begin{aligned} &\Pr\left\lbrace \gamma\_{\_{RD}}(i)+\gamma\_{\_{SD}}(i)\leq \Gamma\_{th}, \right\rbrace=e^{-Z\_{1}M} \left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right]}{(W\_{1}-W\_{2})} \right\rbrace {\nonumber}\\ &+\left( 1-e^{-Z\_{1}M} \right) \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +Z\_{1}W\_{2}\int\_{x=0}^{M}\frac{xe^{-Z\_{1}x}[e^{-W\_{2}\Gamma\_{th}}-e^{-\frac {W\_{3}\Gamma\_{th}}{x}}]}{(xW\_{2}-W\_{3})}dx. \label{ergc}\end{aligned}$$ Substituting in we obtain. $ \hspace{1 cm} \phi\_{inc} \leq 1 $ Substituting PR(i) from into and then, we get: $$\begin{aligned} &\Pr\left\lbrace \gamma\_{\_{RD}}(i)+\gamma\_{\_{SD}}(i) < \Gamma\_{th} \right\rbrace =\Pr\left\lbrace \frac {2 M\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+\gamma\_{\_{SD}}(i) < \Gamma\_{th}\right\rbrace{\nonumber}\\ &=\int\_{y=0}^{\Gamma\_{th}-z}\int\_{z=0}^{\Gamma\_{th}}f\_{V\_{\_{RD}}\left( i\right)}\left( y\right) f\_{\gamma\_{\_{SD}}\left( i\right)}\left(z\right) dy dz = \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right]}{(W\_{1}-W\_{2})} \label{term3}\end{aligned}$$ Substituting into we obtain. Outage analysis of HSU with IOFP ================================ [onoff outage infinite] When ϕinc > 1, substituting PR(i) from into we get: $$\begin{aligned} \Pr\left\lbrace \frac {P\_{\_{R}}(i)\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+ \gamma\_{\_{SD}}(i) < \Gamma\_{th} \right\rbrace =& \Pr\left\lbrace \frac {2 M\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+ \gamma\_{\_{SD}}(i) < \Gamma\_{th},B\left( i\right)\geq M\right\rbrace {\nonumber}\\ & +\Pr\left\lbrace \gamma\_{\_{SD}}(i)\leq \Gamma\_{th},B\left( i\right)<M\right\rbrace. \label{outpronoff}\end{aligned}$$ We first note that using g(x) from and, Pr{B(i) ≥ M} can be simplified as follows: $$\begin{aligned} \Pr\left\lbrace B\left( i\right)\geq M\right\rbrace=&\int\_{M}^{\infty}g\_{2}(x)dx=\int\_{M}^{\infty}\frac{-Qe^{Qx}}{M\left( Q+\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\right)}dx {\nonumber}\\ =&\frac{e^{QM}}{M\left( Q+\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)\right)}=\frac{e^{QM}}{M\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}}\right)e^{QM}} =\frac{1}{\phi\_{inc}}\end{aligned}$$ Let $ V\_{\_{RD}}\left( i\right) \triangleq \dfrac{2M\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}.$ The first term in can be simplified using independence of B(i), γSD(i) and ∣hRD(i)∣2 as follows: $$\begin{aligned} &\Pr\left\lbrace \frac {2 M\|h\_{\_{RD}}(i)\|^{2}}{\sigma^{2}}+ \gamma\_{\_{SD}}(i) < \Gamma\_{th},B\left( i\right) \geq M\right\rbrace = \Pr\left\lbrace V\_{\_{RD}}\left( i\right) + \gamma\_{\_{SD}}(i) < \Gamma\_{th}\right\rbrace \Pr\left\lbrace B\left( i\right) \geq M\right\rbrace{\nonumber}\\ &=\frac{1}{\phi\_{inc}} \left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right]}{(W\_{1}-W\_{2})} \right\rbrace. \label{onoffinfi1}\end{aligned}$$ The second term in can be simplified as follows: $$\begin{aligned} \Pr\left\lbrace \gamma\_{\_{SD}}(i) < \Gamma\_{th},B\left( i\right)<M\right\rbrace=&\Pr\left\lbrace \gamma\_{\_{SD}}(i) < \Gamma\_{th}\right\rbrace \Pr\left\lbrace B\left( i\right)<M\right\rbrace {\nonumber}\\ =& \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) \left( 1-\frac{1}{\phi\_{inc}}\right) \label{onoffinf2}\end{aligned}$$ Substituting and into, we get: $$\begin{aligned} \Pr\left\lbrace \gamma\_{\_{RD}}(i)+ \gamma\_{\_{SD}}(i)< \Gamma\_{th}, \right\rbrace= &\hspace{0.1 cm}\frac{1}{\phi\_{inc}} \left\lbrace \left( 1-e^{-W\_{2}\Gamma\_{th}}\right) +\frac{W\_{2}\left[ e^{-W\_{1}\Gamma\_{th}}-e^{-W\_{2}\Gamma\_{th}}\right]}{(W\_{1}-W\_{2})} \right\rbrace {\nonumber}\\ &+\left( 1-\frac{1}{\phi\_{inc}} \right) \left( 1-e^{-W\_{2}\Gamma\_{th}}\right). \label{onoffinfout}\end{aligned}$$ Substituting into, we obtain. --- 1. Dileep Bapatla and Shankar Prakriya are with the Department of Electrical Engineering, Indian Institute of Technology Delhi, (IIT Delhi) New Delhi 110016, India (e-mail: [email protected], [email protected]).[↩](#fnref1) 2. A conference version of this paper dealing only with IBEP was submitted to IEEE VTC 2019.[↩](#fnref2) Performance of Energy-Buffer Aided Incremental Relaying in Cooperative Networks =============================================================================== In this paper, we consider a two-hop cooperative network with a direct link based on an energy harvesting (EH) decode-and-forward relay. The energy-buffer equipped relay harvests energy from the ambience, and uses the harvest-store-use (HSU) architecture. Since it is known that using a discrete-state Markov chain to model the energy buffer is inaccurate even for moderate number of states, we use a discrete-time continuous-state space Markov chain instead. We derive the limiting distribution of energy for both incremental on-off policy (IOFP) and the incremental best-effort policy (IBEF), and use them to obtain expressions for outage probability and throughput. The corresponding expressions for non-incremental signalling follow as a special case. We show that stable buffers using IBEP harness a diversity of two as compared to those using IOFP, which attain a diversity of one. However, while buffers using IBEF are consequently more reliable than those with IOFP, their throughput performance is only marginally superior. Simulation results are presented to validate the derived analytical expressions. Introduction ============ Relays have been incorporated into several modern communication standards (e.g. LTE-A) due to their promise in increasing range and reliability of wireless networks. Relays are needed when the direct link is shadowed, when the source has power limitations, or when diversity gain needs to be harnessed. For this reason, cooperative relay networks have been extensively studied in recent years. Regenerative or decode-and-forward (DF) relays, as well as non-regenerative amplify-and-forward (AF) relays have been widely studied. Conventionally, relays are assumed to be equipped with a power source or a battery. Of late however, there has been considerable research interest in relays that are powered by energy harvesting (EH). In applications like mobile sensor networks and wireless body area networks, use of a nearby node to serve as a relay to communicate to a distant node is well motivated. In a 5G pico-cell communication framework, use of EH by nodes serving as relays is being explored. In these applications, the node serving as a relay is often battery powered, and use of its battery energy to provide relaying services is undesirable. This is because battery life-times are typically short, and replacement of batteries is often difficult (and in some applications, impossible). Use of EH to make the devices battery-less is therefore desirable. In other scenarios, the node serving as a relay may seek to supplement the harvested energy with as little battery energy as possible to ensure long battery lifetimes. Use of relays powered by EH is also motivated by green energy considerations - telecom is known to be a large consumer of energy. Analysis of performance of cooperative communication links with EH relays is clearly of considerable interest. EH is readily possible from natural sources like light, wind, vibration, and ambient radio frequency (RF) signals. Simultaneous wireless information and power transfer (SWIPT), which exploits the possibility of RF signals carrying both energy and information simultaneously, has attracted considerable research interest. However, while they enable communication with a battery-less relay, they are not energy efficient (green) due to path-loss in practical channels. The same is true of power beacons. In this paper, we focus on relays powered by ambient sources. Despite the apparent similarity between energy and data buffers,, there is one fundamental difference - while data buffers are essentially discrete, energy is a continuous variable. However, virtually all literature in this area is based on discrete-state energy buffers. It has been shown that for accurate representation, the number of states needs to be quite large (close to 200,), which make the approach difficult and cumbersome. Also, it is difficult to draw physical insights into performance of such systems with this approach. When the direct channel from source to destination is not shadowed, it is advantageous to combine the signals from source and relay optimally. In addition, use of incremental relaying to improve spectral efficiency is well motivated. While is based on harvest-use (HU) architecture, uses a discrete state space Markov chain to model the buffer (the relay harvests energy from source). Recently, it has been shown that a discrete-time continuous-state Markov chain model can be used to model the energy buffer accurately in point-to-point links. Performance depends on the policy used to operate the buffer. With the best-effort energy storage policy, the node transmits by drawing constant M amount of energy when sufficient energy exists, and drawing entire energy in the buffer otherwise. With the on-off storage policy, the node transmits using M amount of energy when it is available, and remains silent otherwise. Performance of single-hop wireless powered communication with both these models was analyzed, and derivation for the asymptotic distribution of energy in the buffer was presented in. In this paper, we analyze the performance of cooperative links with buffered relays using IBEP and IOFP, and derive expressions for asymptotic distributions of energy. Expressions for non-incremental signalling follow as a special case. To the best of our knowledge there has been no work on performance of cooperative links with energy buffers modeled using a discrete-time continuous-state Markov chain[2](#fn2). The significant contributions are as follows: 1. Unlike all prior work on EH relays with energy buffers that are based on discrete state Markov chain,, in this paper we analyze performance of cooperative links using a continuous state space Markov chain to model the energy buffer. Also, unlike most other works on EH, we do not ignore the direct path from source to destination, and optimally combine the direct and relayed signals at the destination. 2. For incremental signalling (IBEP as well as IOFP), we derive an expression for the limiting (steady-state) distribution of energy in the energy buffer, and establish conditions for its existence. 3. Expressions are derived for outage probabilities and throughput with both IBEP and IOFP. Expressions for non-incremental signalling follow as a special case. We compare performance with the HU architecture and direct (relay-less) transmission. 4. We establish that the IBEP attains a diversity order of two, while the IOFP attains a diversity of one, making links with the former policy more reliable. However, the throughput performance with IBEP is only marginally superior to that with IOFP. Notations: ---------- ${\cal CN}\sim \left(\mu,\sigma^{2}\right)$ denotes the complex Gaussian distribution with mean μ and variance σ2, ∼ denotes ``distributed as", and ∣x∣ denotes the absolute value of x. Pr{A} indicates probability of the event A. W( ⋅ ) is the principal branch of the Lambert-W function. min(a, b) denotes minimum of a, b. E{ ⋅ } denotes the expectation operator. FX(x) and fX(x) denote the cumulative distribution function (CDF) and the probability density function (PDF) of random variable X. System Model ============ The network (as depicted in Fig. [coop]) consists of source S, a DF relay R, and a destination D. While the source S is equipped with a power supply, the relay R is powered solely by harvested ambient energy. Quasi-static Rayleigh fading is assumed. Let dab denote the normalized distance between nodes with a ∈ [S, R], and b ∈ [R, D]. Denote by $h\_{\_{SR}}(i)\sim{\cal CN}(0,d\_{\_{SR}}^{-\alpha})$, $h\_{\_{RD}}(i)\sim{\cal CN}(0,d\_{\_{RD}}^{-\alpha})$, and $h\_{\_{SD}}(i)\sim{\cal CN}(0,d\_{\_{SD}}^{-\alpha})$ the channels in the ith signalling interval between S and R, R and D, and S and D respectively (where α is the path-loss exponent). [coop] Two types of EH architectures have been proposed in literature. With the HSU (Harvest-store-use) architecture, the relay is equipped with a rechargeable device (battery for example) that can store and release energy. Since storage devices cannot charge and discharge at the same time,, harvested energy is stored in a secondary energy buffer (SEB), which is typically a super-capacitor, and transferred to the primary energy buffer (PEB), which is typically a battery, at the end of the signalling interval in a negligible amount of time. On the other hand, with HU (Harvest-use) architecture, all the energy that is harvested is used in the same signalling interval (for example architectures based on super-capacitors). It is the HSU architecture that is of primary interest in this paper. For comparison purpose, we also consider direct (relay-less) transmission from source to destination, as well as the HU architecture. The reason for use of the feedback bit will become clear in what follows. For both HSU and HU architectures, the signalling interval is divided into two phases of equal duration. The overall signalling interval T is normalized to unity for convenience. In the first phase (phase I) of normalized duration 1/2, the source S transmits unit-energy information symbols xS(i) at rate R0 with power PS to the relay R and destination D. The received signals ySD(i) and ySR(i) at D and R in phase I of the $i\textsuperscript{th}$ signalling interval are given by: $$\begin{aligned} &y\_{\_{SD}}(i)=\sqrt{P\_{\_{S}}}h\_{\_{SD}}(i) x\_{\_{S}}(i) +n\_{\_{SD}}(i), &\text{Phase I}\\ &y\_{\_{SR}}(i)=\sqrt{P\_{\_{S}}}h\_{\_{SR}}(i) x\_{\_{S}}(i) +n\_{\_{SR}}(i), & \text{Phase I} \label{1-3}\end{aligned}$$ where $n\_{\_{SD}}(i),n\_{\_{SR}}(i) \sim {\cal CN}\left(0,\sigma^{2}\right)$ are additive noise samples. The instantaneous link signal to noise ratios (SNRs) in the $i\textsuperscript{th}$ signalling interval at D and R in the first phase are given by: $$\begin{aligned} &\gamma\_{\_{SD}}(i)=\displaystyle\frac{P\_{\_{S}}\|h\_{\_{SD}}(i)\|^{2}}{\sigma^{2}}, ~~\gamma\_{\_{SR}}(i)=\frac{P\_{\_{S}}\|h\_{\_{SR}}(i)\|^{2}}{\sigma^{2}}. \label{link\_snr}\end{aligned}$$ Let: $$\begin{aligned} \Gamma\_{th}^{\prime} &=& \left\{ \begin{array}{ll} \Gamma\_{th}=2^{2R\_{0}}-1 & \text{Incremental Signalling,}\\ \infty & \text{Non-incremental Signalling.} \end{array}\right.\end{aligned}$$ It will become apparent later that this definition allows us to analyze the performance of incremental and non-incremental relaying in a unified fashion. *Case γSD(i) ≥ Γt*h′*: If the received SNR at D is greater than (or equal to) the threshold Γt*h′ (signalling is successful in the first phase itself), D sends a positive acknowledgment (ACK), which is received by both R and S. On receipt of ACK, S directly transmits a new set of symbols to D in the second phase (phase II), making the overall SNR γD(i) = γSD(i). Note that an ACK can never be sent for non-incremental signalling. *Case γSD(i) < Γt*h′*: If received SNR at D is less than the threshold in the first phase, D sends a negative acknowledgment (NACK), which is received by both S and R. Clearly, a NACK is always sent for non-incremental signalling. In this case, the fixed DF R decodes x*S(i), re-encodes, and then transmits the information as unit-energy symbols xR(i) at rate R0 with power PR(i) to D using the energy harvested (hence the dependence of PR(i) on i). Clearly, the signal yRD(i) received by D in this phase is given by: $$\begin{aligned} &y\_{\_{RD}}(i)=\sqrt{P\_{\_{R}}(i)}h\_{\_{RD}}(i) x\_{\_{R}}(i) +n\_{\_{RD}}(i), & \text{Phase II}\end{aligned}$$ where $n\_{\_{RD}}(i) \sim {\cal CN}\left(0,\sigma^{2}\right).$ Clearly, γRD(i) = PR(i)∣hRD(i)∣2/σ2. The signals received from S and R are combined using maximal ratio combining (MRC) at D. The overall SNR γD(i) at D is given by γSD(i) + γRD(i). Limiting distribution of Energy with HSU Architecture ===================================================== In this section, we derive expressions for the limiting distributions of energy in the PEB with both IBEP and IOFP, assuming it to be of infinite-size. Since the energy harvested is typically small, this assumption is not limiting. Those with non-incremental signalling follow as special cases (with Γth′ = ∞), and will not be written explicitly. Limiting Distribution of IBEP ----------------------------- The incoming harvested energy X(i) is assumed to be an exponential random variable (as in,) with PDF fX(x). Denote by B(i) the energy level in the buffer in the ith signalling interval. Let M denote the energy drawn for every relay transmission. The PEB’s buffer update equation is then given by: $$\begin{aligned} B(i+1) & = & B(i)+X(i) \hspace{1.5 cm} \left( \gamma\_{\_{SD}}(i) \geq \Gamma^{\prime}\_{th}\right), \nonumber\\ B(i+1) & = & B(i)-M+X(i) \hspace{0.6 cm} (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\cap (B(i) \geq M), \nonumber\\ B(i+1) & = & X(i) \hspace{3 cm} (\gamma\_{\_{SD}}(i)<\Gamma^{\prime}\_{th})\cap (B(i)<M). \label{BI2}\end{aligned}$$ Let W1, W2, W3 and W4 be defined as $W\_{1}=\dfrac {\sigma^2d\_{\_{RD}}^\alpha}{2M}$, $W\_{2}=\dfrac {\sigma^2d\_{\_{SD}}^\alpha}{P\_{\_{S}}}$, $W\_{3}=\dfrac {\sigma^2d\_{\_{RD}}^\alpha}{2}$ and $W\_{4}=\dfrac {\sigma^2d\_{\_{SR}}^\alpha}{P\_{\_{S}}}$. Let $$\begin{aligned} \phi\_{inc} = \frac{M\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right)}{\mathbb{E}{\{X(i)}\}}=M\lambda\_{1}\left( 1-e^{-W\_{2}\Gamma^{\prime}\_{th}} \right), \label{phinc}\end{aligned}$$ where $\mathbb{E}{\{X(i)}\}=\displaystyle \frac{1}{\lambda\_{1}}$. Note that the equivalent constant ϕninc in the non-incremental case is given by ϕninc = Mλ1 (using Γth′ = ∞). The limiting distributions when ϕi*n	arxiv_0000616
Introduction ============ The Standard Model of particle physics provides viable parameterisation of most observable sub–atomic data to date. Nevertheless, it is clear that the Standard Model is merely an effective field theory description, albeit one that may be effective up to the Grand Unified Theory (GUT), or Planck, scales. The first evidence for the need to go beyond the Standard Model stems from the observation of non–vanishing neutrino masses, which requires the augmentation of the Standard Model spectrum with right–handed neutrinos, or the introduction of nonrenormalisable interactions, that are suppressed by some cutoff scale, at which the Standard Model ceases to provide viable parameterisation. Ultimately, the Standard Model is formulated using the framework of point quantum field theory, which is fundamentally incompatible with gravity. An early construction to accommodate non–trivial neutrino masses was the Pati–Salam model that sought to introduce a symmetry between quarks and leptons. This rightly celebrated insight was vindicated by experiments. It paved the road for the so–called Grand Unified Theories (GUTs) that aim to unify all the sub–atomic gauge interactions into a simple or semi–simple gauge group. Experimental constraints stemming from proton longevity and suppression of left–handed neutrino masses mandate that the unification scale is much higher than the electroweak scale, which is currently being actively probed by experiments. The GUT scale is in fact an order of magnitude or two below the Planck scale, the scale at which the gauge and gravitational interactions are of comparable strength. It is therefore sensible to construct GUT models in a framework that incorporates gravity into the fold. Building quantum field theory models at a scale which is one or two orders of magnitude below the Planck scale is fraught with uncertainties as quantum gravity effects may become dominant. String theory provides a framework that enables the construction of particle physics models within a perturbatively consistent theory of quantum gravity. String theory gives rise to a large number of vacuum solutions that may a priori be relevant for the particle physics data. The way forward is to explore the properties of classes of string vacua that share some common characteristics and to develop the tools to discern them from other classes. While the majority of models studied to date possess N = 1 spacetime supersymmetry, non–supersymmetric vacua that correspond to compactifications of the tachyon–free ten–dimensional SO(16) × SO(16) heterotic string have been of interest as well. It is well known that in addition to the tachyon–free vacua, string theory gives rise to tachyonic vacua in ten dimensions. Recently, it was argued that these ten–dimensional vacua may serve as good starting points for the construction of phenomenological models, provided that the tachyonic states are projected out from the physical spectrum in the four–dimensional models, and may offer some novel perspectives on some of the outstanding issues in string phenomenology. A three generation standard–like model in this class was presented in, and it was argued that all the moduli are projected out in the model and that it may therefore be stable. The free fermionic formulation of the heterotic string gave rise to a large space of phenomenological string vacua, which correspond to compactifications of the ten–dimensional heterotic string on Z2 × Z2 toroidal orbifolds. The initial constructions corresponded to isolated examples with SU(5) × U(1) (FSU5), SO(6) × SO(4) (PS), SU(3) × SU(2) × U(1)2 (SLM) and SU(3) × U(1) × SU(2)2 (LRS), unbroken subgroups of SO(10), with the GUT embedding of the electroweak hypercharge, producing the canonical prediction for the Weinberg angle sin2θW = 3/8 at $M\_{\rm String}$. Systematic methods to classify large numbers of free fermionic models were developed over the past two decades. In ref. the classification program was extended to tachyon–free models with unbroken SO(10) subgroup that may be regarded as compactifications of a tachyonic ten–dimensional vacuum. Numerous models with equal number of massless bosons and fermions were found and an analysis of the one loop vacuum amplitude was presented. Of particular interest was the observation of the misaligned supersymmetry phenomena, which guarantees the finiteness of the string amplitude in the absence of spacetime supersymmetry. In this paper we extend the free fermionic classification program to non–supersymmetric models with PS gauge symmetry. We develop the classification method both for compactifications of the SO(16) × SO(16) heterotic string as well as for compactifications of the tachyonic ten–dimensional vacuum. Preliminary development of the methodology in the first case was carried out in ref., but only with respect to simultaneous shifts of tori, whereas here we include independent shifts with respect to the six internal circles. In the fermionic language this entails enlarging the basis vectors that span the models to include shifts with respect to circles, rather than tori. We show that the –models cannot in fact give rise to viable models, due to the absence from the spectrum of the states required to break the PS gauge group to the Standard Model. We remark that this result is similar to the absence of viable supersymmetric models with SO(10) → SU(4) × SU(2) × U(1). A particular focus of our analysis here is the adaptation of the ‘fertility methodology’ to the class of non–supersymmetric PS models. These methods were applied in the case of the SLM and LRS models and is very efficient in extracting phenomenologically viable vacua. The new element of the fertility methodology in the non–supersymmetric models involves the extraction of tachyon–free models, which occur with low frequency in the total space of vacua. To contrast the cases of the –models with that of the S–models, we apply a parallel set of phenomenological criteria, despite the fact that they do not hold in the case of the –models due to the absence of Heavy Higgs scalar representations. In addition we analyse the partition function of the models. A particular interest is in models with equal number of massless bosons and fermions. We present exemplary three generation – and S–models that satisfy this criteria. Our paper is organised as follows: in Section [10D] we discuss the general structure of – vs S–models. In Section [StPS] we elaborate on the analysis of the massless spectrum in PS –models, whereas in Section [FertMeth] we discuss the fertility methodology in –models. The corresponding analysis in the S–models is discussed in Section [PSNormS]. In Section [PF] we discuss the analysis of the partition functions. Section [results] presents the results of the analysis and Section [conclude] concludes our paper. S vs S̃–Models ===================== We will explore four–dimensional non-supersymmetric Pati–Salam models via two distinct routes. Firstly, we will explore those PS models descending from the tachyonic ten–dimensional heterotic string which utilize the vector S̃. We will refer to these models as (PS) S̃-models. Secondly, we will explore models derived from the SO(16) × SO(16) heterotic string which contain the SUSY–generating basis vector S and thus we will refer to these as S–models. String models in the free fermionic formulation are defined in terms of boundary condition basis vectors and one–loop Generalised GSO (GGSO) phases. The SO(16) × SO(16) and E8 × E8 heterotic–models in ten dimensions are specified in terms of a common set of basis vectors v1=&=&{,1,…,6 \| 1,2,3, 1,…,5,1,…,8}, v2=z1&=&{1,…,5, 1,2,3 }, v3=z2&=&{1,…,8}, [tendbasisvectors] where we employed the common notation used in the free fermionic models. The spacetime supersymmetry generator arises from the combination S=+z1+z2 = {,1,…,6}. [tendsvector] The choice of GGSO phase ${C{z\_1\atopwithdelims[]z\_2}}=\pm 1$ selects between the E8 × E8 or SO(16) × SO(16) heterotic strings in ten dimensions. Eq. ([tendsvector]) implies that the breaking of spacetime supersymmetry in ten dimensions is correlated with the breaking pattern E8 × E8 → SO(16) × SO(16). The would–be tachyons in these models arise from the Neveu-Schwarz (NS) sector, by acting on the right–moving vacuum with a fermionic oscillator LaR, [untwistedtach] where a = 1, ⋯, 16. The GSO projection induced by the S–vector projects out these NS tachyons, producing tachyon–free models in both cases. In lower dimensions eq. ([tendsvector]) does not hold and the two breakings are not correlated. In lower dimensions the projection of the N = 1 supersymmetry generator from the S–sector is induced by an alternative phase. In this manner non–supersymmetric vacua are constructed. In the orbifold representation, these free fermion models correspond to compactifications of the ten–dimensional SO(16) × SO(16) heterotic string, where the detailed dictionary of ref., can be used to translate the free fermion data to the toroidal orbifold data. Tachyon producing sectors proliferate in the lower dimensional models, and can be projected out for special choices of the GGSO projection coefficients. As discussed in refs., the ten–dimensional tachyonic vacua in the free fermionic formulation are obtained by removing the S–vector from the construction, or by augmenting it with four periodic right–moving worldsheet fermions. These ten–dimensional configurations do contain the NS tachyons appearing in eq. ([untwistedtach]) or a subset of them, and are connected by interpolations or orbifolds along the lines of ref.. Similarly, the construction of the four–dimensional free fermion models that descend from the ten–dimensional tachyonic vacua amounts to removing the vector S from the set of basis vectors that are used to build the models. Alternatively, the S–vector can be augmented with a set of four right–moving periodic fermions. A convenient choice is given by = {1,2, 1,2, 3,4, 5,6 \|3,..., 6 }[newS] In this case there are no massless gravitinos, and the untwisted tachyonic states \|0L3,..., 6\|0R [stildetachstates] are invariant under the ${\tilde S}$–vector projection, and therefore are the NS tachyons that descend from the ten–dimensional vacuum. We denote the general map, which is induced by the exchange S, [stildemap] as the ${\tilde S}$–map. This map was used in the –based model of ref. and in the classification of the SO(10) GUT models in ref.. As for the S–based models that correspond to compactifications of the SO(16) × SO(16) heterotic string, the tachyonic states in the –models are projected out for discrete choices of the GGSO projection coefficients. Thus, these models should be taken on equal footing as the S–models. However, the characteristics of the spectra in the two cases are different. Our aim here is to explore these different characteristics. Pati–Salam S̃-Models ====================== Let us turn our attention to the classification set-up for the Pati–Salam S̃-models which descend from the 10D tachyonic heterotic string. We can build these models by first defining a set of 12 basis vectors that generate SO(10) GUT S̃-models which were used in the classification performed in $$\begin{aligned} \label{basisSt} \mathds{1}&=\{\psi^\mu,\ \chi^{1,\dots,6},y^{1,\dots,6}, w^{1,\dots,6}\ \| \ \overline{y}^{1,\dots,6},\overline{w}^{1,\dots,6}, \overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,8}\},\nonumber\\ \tilde{S}&=\{{\psi^\mu},\chi^{1,\dots,6} \ \| \ \overline{\phi}^{3,4,5,6}\},\nonumber\\ {e\_i}&=\{y^{i},w^{i}\; \| \; \overline{y}^{i},\overline{w}^{i}\}, \ \ \ i=1,...,6 \nonumber\\ {b\_1}&=\{\psi^\mu,\chi^{12},y^{34},y^{56}\; \| \; \overline{y}^{34}, \overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\\ {b\_2}&=\{\psi^\mu,\chi^{34},y^{12},y^{56}\; \| \; \overline{y}^{12}, \overline{y}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dots,5}\},\nonumber\\ {b\_3}&=\{\psi^\mu,\chi^{56},y^{12},y^{34}\; \| \; \overline{y}^{12}, \overline{y}^{34},\overline{\eta}^3,\overline{\psi}^{1,\dots,5}\},\nonumber\\ z\_1&=\{\overline{\phi}^{1,\dots,4}\},\nonumber \nonumber\end{aligned}$$ where we note that the {yi, wi ∣∣ ȳi, w̄i} are fermionised coordinates of the internal toroidal Γ6, 6 lattice such that i∂XLi = yiwi and the ei’s allow for all symmetric shifts of the six internal circles. Furthermore, we note the existence of a vector combination [z2] z2=+i=16 ei +k=13 bk+z1={\|5,6,7,8} in these models, which is typically its own basis vector in the supersymmetric classifications of. In order to extend this structure to only those models realising the SO(6) × SO(4) PS subgroup of SO(10) we supplement this basis with the vector: [alpha] ={\|4,5, \|1,2}. Having fixed our basis we can now specify a model by fixing a set of GGSO phases ${C{v\_i\atopwithdelims[]v\_j}}$, which for our Pati–Salam basis here involves 78 free phases with all others specified by modular invariance. Hence, the full space of models is of size 278 ∼ 3 × 1023 models. This is considerably larger than the space of supersymmetric Pati–Salam models classified in where supersymmetry constraints reduce the parameter space. With the basis and GGSO phases fixed we can then construct the modular invariant Hilbert space H of states $\ket{S\_\xi}$ for a model through the one–loop GGSO projections: $$\mathcal{H}=\bigoplus\_{\xi\in\Xi}\prod^{k}\_{i=1} \left\{ e^{i\pi v\_i\cdot F\_{\xi}}\ket{S\_\xi}=\delta\_{\xi} {C{\xi\atopwithdelims[]v\_i}}^\\ket{S\_\xi}\right\}$$ where the ξ* are sectors formed as a linear combination of the basis vectors which span an additive group Ξ, Fξ is the fermion number operator and δξ = 1, − 1 is the spin-statistics index. The sectors in the model can be characterised according to the left and right moving vacuum separately $$\begin{aligned} M\_L^2&=-\frac{1}{2}+\frac{\xi\_L \cdot\xi\_L}{8}+N\_L\\ M\_R^2 &=-1+\frac{\xi\_R \cdot\xi\_R}{8}+N\_R \nonumber\end{aligned}$$ where NL and NR are sums over left and right moving oscillators, respectively. Physical states must then additionally satisfy the Virasoro matching condition, ML2 = MR2. States not satisfying this condition are off–shell. It will be useful to distinguish between those features of our models depending on the presence of the PS breaking vector ([alpha]) and those features present at the level of SO(10) defined by the basis set ([basisSt]) and explored in detail in our recent work. This distinction will be crucial in how we define our methodology for efficiently scanning the space of PS S̃–models using the so–called ‘fertility’ methodology. Gauge Group and Enhancements ---------------------------- The untwisted sector gauge vector bosons for our PS S̃–models give rise to the gauge group SO(6)SO(4) U(1)1U(1)2U(1)3SO(4)4. The gauge group of a model may be enhanced by additional gauge bosons. The SO(10) enhancements arise from $\psi^\mu \{\bar{\lambda}^i\}\ket{z\_1},\psi^\mu \{\bar{\lambda}^i\}\ket{z\_2}$ and $\psi^\mu \ket{z\_1+z\_2}$ sectors, where λ̄i are right moving oscillators. These were discussed in ref. and so we will focus our analysis on enhancements at the PS level arising from: G= L {\|i}R, L {\|i}R L R, L R L R, L R where we have made use of the important vector combination: [xtilde] =b1+b2+b3={, 1,2,3,4,5,6 \| \|1,...,5,\|1,2,3} the properties of which are discussed in some detail in ref.. These enhancements are all ‘mixed’ enhancements in the sense that they affect both the observable and hidden gauge group factors. The presence of these gauge bosons is model–dependent since it depends on the survival of the sectors under the GGSO projections. It turns out that the survival of $\psi^\mu\ket{\alpha}\_L \otimes \{\bar{\psi}^{1,2,3},\bar{\eta}^{1,2,3}\}\ket{\alpha}\_R$ and $\psi^\mu\ket{z\_1+\alpha}\_L \otimes \{\bar{\psi}^{1,2,3},\bar{\eta}^{1,2,3}\}\ket{z\_1+\alpha}\_R $ is correlated exactly with the survival of the tachyonic sectors $\ket{\alpha}$ and $\ket{z\_1+\alpha}$. Hence, the absence of tachyons necessitates the projection of these mixed gauge enhancements. We note that this is similar to the result for mixed SO(10) enhancements $\psi^\mu\{\bar{\psi}^a\}\ket{z\_1},\psi^\mu\{\bar{\psi}^a\} \ket{z\_2},\ a=1,...,5 $, which must be projected for models free from $\ket{z\_1}$ and $\ket{z\_2}$ tachyonic sectors. However, the other right–moving oscillator cases for these enhancement sectors have to be checked carefully as they also can affect the observable gauge group. The analysis of the level–matched tachyonic sectors in our PS case is presented in the following section. Analysis of Tachyonic Sectors ----------------------------- Due to the absence of supersymmetry the analysis of whether on–shell tachyons arise in the spectrum of our models is of utmost importance. Since the details of how to project the tachyonic sectors within the SO(10) construction are shown in, we will focus on the tachyonic sectors at the PS level, i.e. those arising due to the inclusion of the α–vector. However, there is one amendment to the projection conditions in Section 4 of ref. required, which is that we must account for the fact that the α–vector can be used as a projector for many of the tachyonic sectors at the SO(10)–level. In particular, it can project any of the tachyonic sectors except for those involving z1. This is an important detail when we turn to the fertility analysis in Section [FertMeth]. Turning our attention to the tachyonic sectors introduced at the PS level, we note that on–shell tachyonic sectors will arise when [onshelltach] ML2=MR2<0, which corresponds to left and right products of ξL ⋅ ξL = 0, 1, 2, 3 and ξR ⋅ ξR = 0, 1, 2, 3, 4, 5, 6, 7. The presence of such tachyonic sectors in the physical spectrum indicates the instability of the string vacuum with respect to the background on which the theory is defined. We can deduce that there are 84 PS tachyonic sectors for us to consider once the SO(10) tachyons are projected. These sectors are displayed in Table [StTachSecs]. [!ht] \| Mass Level \| Sectors \| \| --- \| --- \| \| ( − 1/2, − 1/2) \| α, z1 + α \| \| ( − 3/8, − 3/8) \| ei + α, ei + z1 + α \| \| ( − 1/4, − 1/4) \| ei + ej + α, ei + ej + z1 + α \| \| ( − 1/8, − 1/8) \| ei + ej + ek + α, ei + ej + ek + z1 + α \| In Table [StTachConds] the conditions on the projection of these tachyonic sectors are delineated. We only show the cases when i = 1, j = 2 and k = 3 as the other combinations are similar. [!ht] \| Tachyonic Sector \| Survival conditions \| \| --- \| --- \| \| α \| ${C{\alpha\atopwithdelims[]e\_1}}={C{\alpha\atopwithdelims[]e\_2}}={C{\alpha\atopwithdelims[]e\_3}}={C{\alpha\atopwithdelims[]e\_4}}={C{\alpha\atopwithdelims[]e\_5}}={C{\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{\alpha\atopwithdelims[]z\_2}}={C{\alpha\atopwithdelims[]z\_1+\alpha+\tilde{x}}}=+1$ \| \| [0.75ex] z1 + α \| ${C{z\_1+\alpha\atopwithdelims[]e\_1}}={C{z\_1+\alpha\atopwithdelims[]e\_2}}={C{z\_1+\alpha\atopwithdelims[]e\_3}}={C{z\_1+\alpha\atopwithdelims[]e\_4}}={C{z\_1+\alpha\atopwithdelims[]e\_5}}={C{z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{z\_1+\alpha\atopwithdelims[]z\_2}}={C{z\_1+\alpha\atopwithdelims[]\alpha+\tilde{x}}}=+1$ \| \| e1 + α \| ${C{e\_1+\alpha\atopwithdelims[]e\_2}}={C{e\_1+\alpha\atopwithdelims[]e\_3}}={C{e\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+\alpha\atopwithdelims[]b\_1+z\_1+\alpha}}={C{e\_1+\alpha\atopwithdelims[]\tilde{x}+z\_1+\alpha}}={C{e\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + z1 + α \| ${C{e\_1+z\_1+\alpha\atopwithdelims[]e\_2}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_3}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+z\_1+\alpha\atopwithdelims[]b\_1+\alpha}}={C{e\_1+z\_1+\alpha\atopwithdelims[]\tilde{x}+\alpha}}={C{e\_1+z\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + α \| ${C{e\_1+e\_2+\alpha\atopwithdelims[]e\_3}}={C{e\_1+e\_2+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+\alpha\atopwithdelims[]b\_1+z\_1+\alpha}}={C{e\_1+e\_2+\alpha\atopwithdelims[]\tilde{x}+z\_1+\alpha}}={C{e\_1+e\_2+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + z1 + α \| ${C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_3}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]b\_1+\alpha}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]\tilde{x}+\alpha}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + e3 + α \| ${C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]\tilde{x}+z\_1+\alpha}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + e3 + z1 + α \| ${C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]\tilde{x}+\alpha}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| Massless Sectors ---------------- Now that we have a way to check that models are free of on–shell tachyons, we can turn our attention to the massless sectors and their representations. In Section 5 of a detailed analysis of the SO(10) massless sectors was presented. In this section we will only mention the sectors from the SO(10) level relevant for determining the observable phenomenology of a model. Otherwise we will focus on the new sectors to the PS models. ### Observable Massless Sectors The spinorial $\mathbf{16}/\overline{\mathbf{16}}$ representations of SO(10) arise from the 48 sectors (16 from each orbifold plane) $$\begin{aligned} \label{spins1} B\_{pqrs}^{(1)} &=& b\_1 + pe\_3 + qe\_4 + re\_5 + se\_6 \nonumber \\ &=& \{\psi^{\mu},\chi^{1,2},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \nonumber\\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{1},\bar{\psi}^{1,\ldots,5}\} \\ B\_{pqrs}^{(2)} &=& b\_2 + pe\_1 + qe\_2 + re\_5 + se\_6\nonumber \\ B\_{pqrs}^{(3)} &=& b\_3 + pe\_1 + qe\_2 + re\_3 + se\_4\nonumber\end{aligned}$$ where p, q, r, s = 0, 1 account for all combinations of shift vectors of the internal fermions {yi, wi ∣ ȳi, w̄i}. As in previous classifications, we can now write down generic algebraic equations to determine the number of 16 and $\overline{\mathbf{16}}$, N16 and $N\_{\overline{16}}$, as a function of the GGSO coefficients. To do this we first utilize the following projectors to determine which of the 48 spinorial sectors survive $$\begin{aligned} \label{SpinProjector}\nonumber P^1\_{pqrs} &=& \frac{1}{2^4}\prod\_{i=1,2}\left(1-{C{B^1\_{pqrs}\atopwithdelims[]e\_i}}^\* \right)\prod\_{a=1,2}\left(1-{C{B^1\_{pqrs}\atopwithdelims[]z\_a}}^\\right)\\ P^2\_{pqrs} &=& \frac{1}{2^4}\prod\_{i=3,4}\left(1-{C{ B^2\_{pqrs}\atopwithdelims[]e\_i}}^\\right)\prod\_{a=1,2}\left(1-{C{B^2\_{pqrs}\atopwithdelims[]z\_a}}^\\right)\\ \nonumber P^3\_{pqrs} &=& \frac{1}{2^4}\prod\_{i=5,6}\left(1-{C{B^3\_{pqrs}\atopwithdelims[]e\_i}}^\\right)\prod\_{a=1,2}\left(1-{C{B^3\_{pqrs}\atopwithdelims[]z\_a}}^\\right)\end{aligned}$$ where we recall that the vector $z\_2=\{\overline{\phi}^{5,6,7,8}\}$ is the combination defined in eq. ([z2]). Then we define the chirality phases $$\begin{aligned} X^1\_{pqrs} &=& -{C{B^1\_{pqrs}\atopwithdelims[]b\_2+(1-r)e\_5+(1-s)e\_6}}^\ \nonumber\\ X^2\_{pqrs} &=& -{C{B^2\_{pqrs}\atopwithdelims[]b\_1+(1-r)e\_5+(1-s)e\_6}}^\\\ X^3\_{pqrs} &=& -{C{B^3\_{pqrs}\atopwithdelims[]b\_1+(1-r)e\_3+(1-s)e\_4}}^\\nonumber\end{aligned}$$ to determine whether a sector will give rise to a 16 or a $\overline{\mathbf{16}}$. With these definitions we can write compact expressions for N16 and $N\_{\overline{16}}$ $$\begin{aligned} \label{16s} \begin{split} N\_{16} &= \frac{1}{2}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} P\_{pqrs}^A\left(1 + X^A\_{pqrs}\right) \\ N\_{\overline{16}} &= \frac{1}{2}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} P\_{pqrs}^A\left(1 - X^A\_{pqrs}\right). \end{split}\end{aligned}$$ In order to describe the phenomenological properties of the models under consideration, we need to also consider what happens to the components of the 16/ $\overline{\mathbf{16}}$’s as the SO(10) GUT is broken to the PS subgroup. Recall that the basis vector α ([alpha]) induces SO(10) gauge symmetry breaking. The spinorial 16/ $\overline{\mathbf{16}}$ representations of SO(10) decompose under the SO(6) × SO(4) ≡ SU(4) × SU(2) × SU(2) gauge group as: $$\begin{aligned} \label{sp\_deco} \mathbf{16}=&(\mathbf{4}, \mathbf{2}, \mathbf{1})+(\mathbf{\bar{4}}, \mathbf{1}, \mathbf{2})=n\_{4L}+n\_{\bar{4}R}\\ \mathbf{\overline{16}}=&(\mathbf{\bar{4}}, \mathbf{2}, \mathbf{1})+(\mathbf{4}, \mathbf{1}, \mathbf{2})=n\_{4R}+n\_{\bar{4}L} \nonumber\end{aligned}$$ The GGSO projection under α determines which of the two possible PS representations a particular 16 or $\mathbf{\overline{16}}$ will fall into. Furthermore, we can make the connection to the Standard Model by noting that after Higgsing the SU(2)R gauge symmetry, the 16 rep. decomposes as $$\begin{aligned} \label{sm\_deco} \mathbf{16} &= \ Q \left(\textbf{3}, \textbf{2}, \frac{1}{6}\right) + L\left(1, \textbf{2}, -\frac{1}{2}\right) \nonumber \\ &+ u^c\left(\overline{\textbf{3}}, \textbf{1}, -\frac{2}{3}\right) + d^c\left(\overline{\textbf{3}}, \textbf{1}, \frac{1}{3}\right) + e^c\left(\textbf{1}, \textbf{1},1\right) + \nu^c\left(\textbf{1}, \textbf{1}, 0\right)\end{aligned}$$ under the SM gauge group. A phenomenological issue arises due to the fact –maps makes the would-be scalar degrees of freedom S̃ + Bpqrsi massive. In the supersymmetric models, the 16 and $\mathbf{\overline{16}}$ include the scalar superpartners, in particular those of the $n\_{\bar{4}R}$ and n4R, which are used to break the intermediate PS gauge symmetry down to the Standard Model gauge group. Since there are no such states for our S̃–models and no suitable scalar exotic sectors or scalars from higher Kac–Moody level representations, the phenomenology of these models is seemingly incomplete. In the case of Standard–like S̃–models such as, however, the breaking of the additional U(1)Zʹ may be achieved by a scalar from the exotic sectors. Despite this gap in the analysis of these –models we continue with the analysis since our main aim here is to build up the tools for the classification for the – and S–models of Non-SUSY string models and to do a comparison of their key characteristics. The other key states for observable physics are obtained from the vectorial 10 representations of SO(10). We note that the vector x̃ defined in ([xtilde]) induces a map between the fermionic spinorial sectors Bpqrs1, 2, 3 and the bosonic vectorial sectors $$\begin{aligned} \label{vects} V\_{pqrs}^{(1)} &=& B\_{pqrs}^{(1)}+\tilde{x}\nonumber \\ &=&b\_2+b\_3 + pe\_3 + qe\_4 + re\_5 + se\_6 \nonumber \\ &=& \{\chi^{3,4,5,6},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \nonumber\\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{2,3}\} \\ V\_{pqrs}^{(2)} &=& B\_{pqrs}^{(2)}+\tilde{x}\nonumber \\ V\_{pqrs}^{(3)} &=& B\_{pqrs}^{(3)}+\tilde{x}\nonumber\end{aligned}$$ The observable states will arise from these sectors when there is a right moving oscillator of $\overline{\psi}^{a(\)}$, a* = 1, ..., 5. To determine the number of such observable vectorial sectors we use the projectors $$\begin{aligned} \label{VectProjector} \nonumber R^{(1)}\_{pqrs} &=& \frac{1}{2^4}\prod\_{i=1,2}\left(1 + {C{e\_i\atopwithdelims[]V^{(1)}\_{pqrs}}} \right) \prod\_{a=1,2}\left[1+ {C{z\_a\atopwithdelims[]V^{(1)}\_{pqrs}}}\right)\\ R^{(2)}\_{pqrs} &=& \frac{1}{2^4}\prod\_{i=3,4}\left(1+{C{e\_i\atopwithdelims[]V^{(2)}\_{pqrs}}} \right) \prod\_{a=1,2}\left(1 + {C{z\_a\atopwithdelims[]V^{(2)}\_{pqrs}}}\right)\\\end{aligned}$$ Then we can write the number of vectorial 10’s arising from these sectors as $$\begin{aligned} N\_{10}=\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1 }}R^A\_{pqrs}.\end{aligned}$$ A GGSO projection with the vector α can then be used to induce doublet-triplet splitting in these models and tell us whether a particular 10 produces a bidoublet or a triplet. In particular, the 10 representation is decomposed under SU(4) × SU(2)L × SU(2)R as: $$\begin{aligned} \textbf{10} = D(\mathbf{6},\mathbf{1},\mathbf{1})+h(\mathbf{1},\mathbf{2},\mathbf{2}) \label{vec\_deco}\end{aligned}$$ where the colored triplets are generated by the $\overline{\psi}^{1,2,3}\_{1/2}/\overline{\psi}^{\* 1,2,3}\_{1/2}$ and the bi–doublet is generated by $\overline{\psi}^{4,5}\_{1/2}/\overline{\psi}^{\4,5}\_{1/2}$ oscillators. We can write the number of bidoublets, n*h, and number of triplets, n6, algebraically as $$\begin{aligned} \label{DoubTrip} \begin{split} n\_{h} &= \frac{1}{2}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} R\_{pqrs}^A\left(1 - {C{V^A\_{pqrs}\atopwithdelims[]\alpha}} \right) \\ n\_6 &= \frac{1}{2}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} R\_{pqrs}^A\left(1 + {C{V^A\_{pqrs}\atopwithdelims[]\alpha}}\right). \end{split}\end{aligned}$$ ### Pati–Salam Exotics Having reviewed the key elements of the observable massless sectors, we can turn our attention to the new features of the massless spectrum due to the PS vector α. Such sectors are exotic in the sense they transform under both the hidden and observable gauge groups and carry fractional electric charge. For the purposes of our analysis we will only classify the fermionic exotic sectors and ensure that any potentially viable models are checked for the absence of chiral exotics. In a classification of supersymmetric models it would be sufficient to classify only the fermionic exotic sectors and then know that the equivalent scalar exotic sectors will be in a model with + S. However, in our non-supersymmetric models we have no control over how many scalar exotics will arise in a model when we check the fermionic exotic sectors. We note that the scalar exotics can always gain heavy mass since they are not chiral. First of all we note two fermionic vectorial exotic sectors [ExotVecs] = {\|i}, {\|i} . Then we have in total 196 fermionic spinorial exotics. Of these, 96 arise from sectors transforming as $(\mathbf{4},\mathbf{2},\mathbf{1}), (\mathbf{4},\mathbf{1},\mathbf{2}), (\overline{\mathbf{4}},\mathbf{2},\mathbf{1})$ or $(\overline{\mathbf{4}},\mathbf{1},\mathbf{2})$ where 4 and $\overline{\mathbf{4}}$ are spinorial and anti–spinorial representations of the observable SU(4), respectively, and the 2 are doublet representations of either the first or second hidden SO(4) factor. Explicitly, these sectors are $$\begin{aligned} \label{fermExot1} E\_{pqrs}^{(1)} &=& B^{(1)}\_{pqrs}+\alpha \nonumber \\ &=& \{\psi^{\mu},\chi^{1,2},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \nonumber\\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{1},\bar{\psi}^{1,2,3},\bar{\phi}^{1,2}\} \nonumber\\ E\_{pqrs}^{(2)} &=& B^{(2)}\_{pqrs}+\alpha \nonumber\\ E\_{pqrs}^{(3)} &=& B^{(3)}\_{pqrs}+\alpha\\ E\_{pqrs}^{(4)} &=& B^{(1)}\_{pqrs}+z\_1+\alpha \nonumber \\ &=& \{\psi^{\mu},\chi^{1,2},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \nonumber\\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{1},\bar{\psi}^{1,2,3},\bar{\phi}^{3,4}\} \nonumber\\ E\_{pqrs}^{(5)} &=& B^{(2)}\_{pqrs}+z\_1+\alpha \nonumber\\ E\_{pqrs}^{(6)} &=& B^{(3)}\_{pqrs}+z\_1+\alpha\nonumber\end{aligned}$$ A further 96 exotic sectors arise from $$\begin{aligned} \label{fermExot2} E\_{pqrs}^{(7)} &=& \tilde{S}+V^{(1)}\_{pqrs}+z\_1+\alpha \nonumber \\ &=& \{\psi^{\mu},\chi^{1,2},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \nonumber\\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{2,3},\bar{\psi}^{4,5},\bar{\phi}^{5,6}\}\nonumber\\ E\_{pqrs}^{(8)} &=& \tilde{S}+ V^{(2)}\_{pqrs}+z\_1+\alpha \nonumber\\ E\_{pqrs}^{(9)} &=& \tilde{S}+ V^{(3)}\_{pqrs}+z\_1+\alpha\\ E\_{pqrs}^{(10)} &=& \tilde{S}+ V^{(1)}\_{pqrs}+z\_1+z\_2+\alpha \nonumber \\ &=& \{\psi^{\mu},\chi^{1,2},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \nonumber\\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{2,3},\bar{\psi}^{4,5},\bar{\phi}^{7,8}\}\nonumber\\ E\_{pqrs}^{(11)} &=& \tilde{S}+ V^{(2)}\_{pqrs}+z\_1+z\_2+\alpha \nonumber\\ E\_{pqrs}^{(12)} &=& \tilde{S}+ V^{(3)}\_{pqrs}+z\_1+z\_2+\alpha\nonumber\end{aligned}$$ which give representations ((2, 1), (2, 1)), ((2, 1), (1, 2)), ((1, 2), (1, 2)) or ((1, 2), (2, 1)) under SU(2)L × SU(2)R × SO(4)3/4. Finally there are 4 additional fermionic spinorial exotics from the sectors: [rogueExots] +, +z2+, +, +z1+ which transform as a doublet under one of the observable SU(2) factors and doublets under three of the hidden SO(4) gauge factors. In regard to phenomenology, we are interested in ensuring that there are no chiral exotics in our models, which is why we look at the fermionic exotic sectors only. In order to check this we can classify the exotics in eq. ([fermExot1]) with the numbers $n\_4,n\_{\bar{4}}$ counting the number of $\mathbf{4}/\mathbf{\bar{4}}$’s under the SU(4) gauge factor and count the numbers n2L, n2R of (1, 2)/(2, 1)’s from equations ([fermExot2]) and ([rogueExots]) under the SU(2) × SU(2) gauge factors. Fertility Methodology in S̃–Models ==================================== The application of the fertility methodology was employed very successfully in the case of the supersymmetric standard–like models in and left-right symmetric models in. The methodology was particularly useful in these cases due to the scarcity of phenomenologically viable vacua, which was largely due to the abundance of exotic sectors in these models. The key notable feature of our S̃–models which make them ripe for a fertility methodology is the abundance of tachyonic models. It was noted in that the probability of tachyon–free SO(10) models was ∼ 5 × 10− 3, and we can similarly find that the probability for PS S̃–models to be tachyon–free is ∼ 2 × 10− 3, i.e. we have to filter out all but around 1 in 500 GGSO configurations. The fertility methodology involves splitting the full parameter space of, in this case, PS models into two components Π = ΠSO(10) × Πα and performing a classification in two steps. The first step is to take the SO(10) subspace ΠSO(10), which is the space of 266 independent GGSO phase configurations for the 12 basis vectors ([basisSt]), and checking for a set of phenomenological conditions which can solely be evaluated within this subspace. These conditions we call ‘fertility conditions’ and the SO(10) models satisfying them ‘fertile cores’. The fertility conditions we impose in our analysis of the –models are listed below. Once we have these SO(10) fertile cores, we perform the second step which is to evaluate them in the space Πα, which means iterating over all independent choices of the GGSO phases involving α around these cores. We then collect statistics for these PS models and find a much increased efficiency in finding models satisfying phenomenological constraints owing to this fertility methodology, compared with a random classification in the full space of PS models. S̃ Fertility Conditions ------------------------- The set of fertility conditions we will use to derive fertile SO(10) S̃ cores can be listed as follows: 1. Absence of the tachyonic sectors: [z1TachCondSt] z1, ei+z1, ei+ej+z1, ei+ej+ek+z1, ij k =1,2,3,4,5,6 using the set of projection conditions delineated in Section 4 of. 2. Constraints on SO(10) spinorial states $$\begin{aligned} \label{fertility1} n\_{4L}-n\_{\bar{4}L} = n\_{\bar{4}R} -n\_{4R}\geq 6 \,\ n\_{\bar{4}R}^F > 6.\end{aligned}$$ The first condition results in a high likelihood of having (at least) 3 fermion families and the second condition is used as a fermionic analogy to the condition for a Heavy Higgs, which would be $n\_{\bar{4}R}^B > 1$ if the –models had the scalar partners of B(1, 2, 3). 3. For the presence of a SM higgs, i.e. nh ≥ 1, and a D(6, 1, 1), i.e. n6 ≥ 1, we can impose $$\begin{aligned} \label{fertility2} N\_{10}\geq 2\end{aligned}$$ at the SO(10) level. 4. Presence of a top quark mass coupling (TQMC) which amounts to fixing the following GGSO coefficients $$\begin{aligned} \label{tqmcSO10} & &{C{b\_1\atopwithdelims[]e\_1}} = {C{b\_1\atopwithdelims[]e\_2}} = {C{b\_1\atopwithdelims[]z\_1}} = {C{b\_1\atopwithdelims[]z\_2}} = -1\,\nonumber\\ & &{C{b\_2\atopwithdelims[]e\_3}} = {C{b\_2\atopwithdelims[]e\_4}} = {C{b\_2\atopwithdelims[]z\_1}} = {C{b\_2\atopwithdelims[]z\_2}} = -1\,\\ & & {C{e\_5\atopwithdelims[]b\_1}}{C{e\_5\atopwithdelims[]b\_2}}={C{e\_6\atopwithdelims[]b\_1}}{C{e\_6\atopwithdelims[]b\_2}}=1\nonumber\\ & &{C{b\_1\atopwithdelims[]b\_2}} = -{C{e\_5\atopwithdelims[]b\_1}} {C{e\_6\atopwithdelims[]b\_1}}\nonumber,\end{aligned}$$ at the SO(10) level. Since z2 is a combination of basis vectors we can rewrite the conditions ${C{b\_1\atopwithdelims[]z\_2}} ={C{b\_2\atopwithdelims[]z\_2}} = -1$ in terms of the GGSO phases between basis vectors using the equations in ([tqmcSO10]) and ABK rules, choosing to fix the phases $$\begin{aligned} \begin{split}\label{z2TQMfixes} {C{b\_1\atopwithdelims[]b\_3}}&=-{C{b\_1\atopwithdelims[]e\_3}}{C{b\_1\atopwithdelims[]e\_4}}\\ {C{b\_2\atopwithdelims[]b\_3}}&=-{C{b\_2\atopwithdelims[]e\_1}}{C{b\_2\atopwithdelims[]e\_2}}. \end{split}\end{aligned}$$ Having listed the conditions 1)-4), we will now explain them in more detail. In regard to condition 1), we should note that these 42 tachyonic sectors ([z1TachCondSt]) involving z1 are the only SO(10) tachyonic sectors we can use as a fertility condition, since the others can be projected by an α GGSO projection as mentioned in Section [tsa]. For example, we could have an SO(10) core with a spinorial tachyon from the z2 sector which is in fact absent at the PS level due to ${C{z\_2\atopwithdelims[]\alpha}}=-1$. However, the z1 tachyonic sectors must be projected at the SO(10) level as a necessary condition on the absence of all tachyons at the PS level. In regard to the second part of condition 2), having a $n\_{4R}^B-n\_{\bar{4}R}^B$ present we typically call having a heavy Higgs, where we use the B superscript to make it clear that this is a bosonic (scalar) sector. However, as already mentioned, the –models have no such scalar components and so we cannot implement this condition. Instead we implement a condition on having the fermionic representations $n\_{\bar{4}R},n\_{4R}$ in the spectrum in order to draw a parallel with the Heavy Higgs condition which we can implement for the S–models analysed in the next section. We further must note that the condition n6 ≥ 1 in 3), relates to the so–called Missing Partner Mechanism of Pati–Salam models for which we also require the $n\_{4R}^{(B)},n\_{\bar{4}R}^{(B)}$. Again, since we have no Heavy Higgs for the –models we implement this n6 ≥ 1 condition purely to draw the parallel to the methodology we can apply for the S–models. In analogy with the Missing Partner Mechanism we will say that the requirement of n6 ≥ 1 is required for a ‘Heavy Triplet Constraint’. The conditions on the GGSO phases from 4) in equations ([tqmcSO10]) and ([z2TQMfixes]) for the existence of the TQMC is a necessary condition for the presence of a TQMC at the PS level and can be made into a sufficient condition by fixing the phases $$\begin{aligned} \label{tqmcPS} {C{b\_1\atopwithdelims[]\alpha}} = -{C{b\_2\atopwithdelims[]\alpha}} = -1\end{aligned}$$ at the PS level. The derivation of these results in a supersymmetric PS construction can be found in ref.. It is derived through choosing B00001 = b1 and B00002 = b2 to survive the GGSO projections ([SpinProjector]), along with a bidoublet Higgs from the sector V00003 = b1 + b2 surviving the projector ([DoubTrip]). Furthermore, using ([16s]) we can ensure b1 and b2 give rise to 16’s such that X00001 = X00002 = 1 through the condition ([tqmcSO10]). Then the GGSO phase conditions ([tqmcPS]) ensures that b1 generates a n4L (4, 2, 1), b2 generates a $n\_{\bar{4}R} \ (\mathbf{\bar{4}},\mathbf{1},\mathbf{2})$ and then b1 + b2 generates the h (1, 2, 2). We thus obtain the TQMC via the term: $$\begin{aligned} \label{tqmcSPot} n\_{4L}^{(1) F} n\_{\bar{4}R}^{(2) F} h^{(3)B}=Qu^c H^u+Q d^c H^d+L e^c H^d+L \nu^c H^u.\end{aligned}$$ Since we have guaranteed that a bidoublet Higgs arises from the sector b1 + b2 we can note this overlaps with condition 3) since we automatically have N10 ≥ 1. Having defined the fertility methodology and the details of the construction of PS S̃–models, in the next section we present the construction of the non-supersymmetric S–models which descend from the non–tachyonic SO(16) × SO(16) 10D string. The results of the classification of PS S̃–models and S–models are given in Section [results]. SO(16) × SO(16)–Derived 4D Models ========================================= We now turn to look at what we will refer to as the S–models, which are the non-supersymmetric class of models descending from the SO(16) × SO(16) heterotic string. We can employ the same basis as used for the classification of supersymmetric PS models in $$\begin{aligned} \label{basisS} \mathds{1}&=\{\psi^\mu,\ \chi^{1,\dots,6},y^{1,\dots,6}, w^{1,\dots,6}\ \| \ \overline{y}^{1,\dots,6},\overline{w}^{1,\dots,6}, \overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,8}\},\nonumber\\ S&=\{{\psi^\mu},\chi^{1,\dots,6} \},\nonumber\\ {e\_i}&=\{y^{i},w^{i}\; \| \; \overline{y}^{i},\overline{w}^{i}\}, \ \ \ i=1,...,6 \nonumber\\ {b\_1}&=\{\chi^{34},\chi^{56},y^{34},y^{56}\; \| \; \overline{y}^{34}, \overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\\ {b\_2}&=\{\chi^{12},\chi^{56},y^{12},y^{56}\; \| \; \overline{y}^{12}, \overline{y}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dots,5}\},\nonumber\\ z\_1&=\{\overline{\phi}^{1,\dots,4}\},\nonumber\\ z\_2&=\{\overline{\phi}^{5,\dots,8}\},\nonumber\\ \alpha&=\{\overline{\psi}^{4,5},\overline{\phi}^{1,2} \} \nonumber\end{aligned}$$ We will also make regular use of the combination x=1+S+i=16 ei +k=12 zk={\|1,...,5,\|1,2,3}. The non–SUSY models are those in which the gravitino is projected from the S massless sector. This means, in other words, that SUSY is broken by a GGSO phase in these S–models. Therefore, the total space of non–supersymmetric models is the total space of all PS models which is 213(13 − 1)/2 = 278 models[1](#fn1) minus the space of supersymmetric configurations. The necessary and sufficient condition for the presence of supersymmetry is the fixing of the 9 GGSO phases [SUSYphases] CSei=CSzn=CS=-1, with i = 1, ..., 6, m = 1, 2 and n = 1, 2. These phase choices ensure the gravitino is not projected. Therefore the total space of non–SUSY PS models has size 278 − 278 − 9 ∼ 3 × 1023. The untwisted sector gauge vector bosons for these models generate the gauge group SO(6)SO(4) U(1)1U(1)2U(1)3SO(4)2SO(8). which only differ from the S̃–models in that the gauge bosons generating the SO(8) hidden group factor $\psi^{\mu}\bar{\phi}^c\bar{\phi}^d\ket{NS}$, c, d = 5, 6, 7, 8, are unbroken unlike in the case of the S̃ projection. Additional gauge bosons arise from enhanced sectors which are exactly those listed in the supersymmetric case from Section 3.1 of. Analysis of Tachyonic Sectors ----------------------------- The tachyonic sectors for the S–models are identical to the S̃ case, however the projection/survival conditions differ. Having S in our basis ($\ref{basisS}$), means it can be used as a possible projector of all the tachyonic sectors depending on the GGSO phases involving S. Of course, in the SUSY case where eq. ([SUSYphases]) holds, all the tachyonic sectors are automatically projected by the S projection, which is the origin of SUSY models being tachyon–free in the free fermionic construction. Due to this special property of S, in terms of computational efficiency it makes sense to implement the S projection first on all the tachyonic sectors and then only apply the further GGSO projections to sectors surviving S. Concretely, for a generic tachyonic sector ξ, in which eq. ([onshelltach]) holds, we can first check CS= +1 -1 and if ξ survives this then we can move on to check the other GGSO projections for its survival/absence. The conditions for the survival of all SO(10) tachyonic sectors are detailed in Tables [NormSSO10] and [NormSSO10Vect] before the PS tachyonic sectors survival conditions are explicated in Table [NormSal]. [!ht] \|c\|l\| Tachyonic &Survival conditions Sector& z1&${C{z\_1\atopwithdelims[]e\_1}}={C{z\_1\atopwithdelims[]e\_2}}={C{z\_1\atopwithdelims[]e\_3}}={C{z\_1\atopwithdelims[]e\_4}}={C{z\_1\atopwithdelims[]e\_5}}={C{z\_1\atopwithdelims[]e\_6}}=+1$ &${C{z\_1\atopwithdelims[]S}}={C{z\_1\atopwithdelims[]z\_2}}={C{z\_1\atopwithdelims[]b\_1}}={C{z\_1\atopwithdelims[]b\_2}}={C{z\_1\atopwithdelims[]x}}=+1$ [0.75ex] e1 + z1 &${C{e\_1+z\_1\atopwithdelims[]e\_2}}={C{e\_1+z\_1\atopwithdelims[]e\_3}}={C{e\_1+z\_1\atopwithdelims[]e\_4}}={C{e\_1+z\_1\atopwithdelims[]e\_5}}={C{e\_1+z\_1\atopwithdelims[]e\_6}}=+1$ &${C{e\_1+z\_1\atopwithdelims[]S}}={C{e\_1+z\_1\atopwithdelims[]b\_1}}={C{e\_1+z\_1\atopwithdelims[]z\_2}}={C{e\_1+z\_1\atopwithdelims[]x}}=+1$ e1 + e2 + z1 &${C{e\_1+e\_2+z\_1\atopwithdelims[]e\_3}}={C{e\_1+e\_2+z\_1\atopwithdelims[]e\_4}}={C{e\_1+e\_2+z\_1\atopwithdelims[]e\_5}}={C{e\_1+e\_2+z\_1\atopwithdelims[]e\_6}}=+1$ &${C{e\_1+e\_2+z\_1\atopwithdelims[]S}}={C{e\_1+e\_2+z\_1\atopwithdelims[]b\_1}}={C{e\_1+e\_2+z\_1\atopwithdelims[]x}}={C{e\_1+e\_2+z\_1\atopwithdelims[]z\_2}}=+1$ e1 + e2 + e3 + z1 &${C{e\_1+e\_2+e\_3+z\_1\atopwithdelims[]e\_4}}={C{e\_1+e\_2+e\_3+z\_1\atopwithdelims[]e\_5}}={C{e\_1+e\_2+e\_3+z\_1\atopwithdelims[]e\_6}}=+1$ &${C{e\_1+e\_2+e\_3+z\_1\atopwithdelims[]S}}={C{e\_1+e\_2+e\_3+z\_1\atopwithdelims[]x}}={C{e\_1+e\_2+e\_3+z\_1\atopwithdelims[]z\_2}}=+1$ z2&${C{z\_2\atopwithdelims[]e\_1}}={C{z\_2\atopwithdelims[]e\_2}}={C{z\_2\atopwithdelims[]e\_3}}={C{z\_2\atopwithdelims[]e\_4}}={C{z\_2\atopwithdelims[]e\_5}}={C{z\_2\atopwithdelims[]e\_6}}=+1$ &${C{z\_2\atopwithdelims[]S}}={C{z\_2\atopwithdelims[]\alpha}}={C{z\_2\atopwithdelims[]z\_1}}={C{z\_2\atopwithdelims[]b\_1}}={C{z\_2\atopwithdelims[]b\_2}}={C{z\_2\atopwithdelims[]x}}=+1$ e1 + z2 &${C{e\_1+z\_2\atopwithdelims[]e\_2}}={C{e\_1+z\_2\atopwithdelims[]e\_3}}={C{e\_1+z\_2\atopwithdelims[]e\_4}}={C{e\_1+z\_2\atopwithdelims[]e\_5}}={C{e\_1+z\_2\atopwithdelims[]e\_6}}=+1$ &${C{e\_1+z\_2\atopwithdelims[]S}}={C{e\_1+z\_2\atopwithdelims[]\alpha}}={C{e\_1+z\_2\atopwithdelims[]b\_1}}={C{e\_1+z\_2\atopwithdelims[]x}}={C{e\_1+z\_2\atopwithdelims[]z\_1}}=+1$ e1 + e2 + z2 &${C{e\_1+e\_2+z\_2\atopwithdelims[]e\_3}}={C{e\_1+e\_2+z\_2\atopwithdelims[]e\_4}}={C{e\_1+e\_2+z\_2\atopwithdelims[]e\_5}}={C{e\_1+e\_2+z\_2\atopwithdelims[]e\_6}}=+1$ &${C{e\_1+e\_2+z\_2\atopwithdelims[]S}}={C{e\_1+e\_2+z\_2\atopwithdelims[]\alpha}}={C{e\_1+e\_2+z\_2\atopwithdelims[]b\_1}}={C{e\_1+e\_2+z\_2\atopwithdelims[]x}}={C{e\_1+e\_2+z\_2\atopwithdelims[]z\_1}}=+1$ e1 + e2 + e3 + z2 &${C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]e\_4}}={C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]e\_5}}={C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]e\_6}}=+1$ &${C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]S}}={C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]\alpha}}={C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]x}}={C{e\_1+e\_2+e\_3+z\_2\atopwithdelims[]z\_1}}=+1$ [!ht] \|c\|l\| Tachyonic &Survival conditions Sector& $\{ \bar{\lambda}\}\ket{e\_1} $ & $ S\_O=\Big\{{C{e\_1\atopwithdelims[]e\_2}},{C{e\_1\atopwithdelims[]e\_3}},{C{e\_1\atopwithdelims[]e\_4}},{C{e\_1\atopwithdelims[]e\_5}},{C{e\_1\atopwithdelims[]e\_6}},{C{e\_1\atopwithdelims[]z\_1}},{C{e\_1\atopwithdelims[]z\_2}},{C{e\_1\atopwithdelims[]\alpha}},{C{e\_1\atopwithdelims[]b\_1}},{C{e\_1\atopwithdelims[]x}}\Big\}$ λ̄ = ψ̄1, 2, 3/η̄1&$ \ {C{e\_1\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_1\atopwithdelims[]b\_1}}={C{e\_1\atopwithdelims[]x}}=-1$ λ̄ = ϕ̄1, 2&$\text{ \textbf{or} } \ {C{e\_1\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_1\atopwithdelims[]z\_1}}={C{e\_1\atopwithdelims[]\alpha}}=-1$ λ̄ = ȳ3, 4, 5, 6&$\text{ \textbf{or } } \ {C{e\_1\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_1\atopwithdelims[]b\_1}}={C{e\_1\atopwithdelims[]e\_{3,4,5,6}}}=-1$ λ̄ = ψ̄4, 5&$\text{ \textbf{or } } \ {C{e\_1\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=3 \ \text{s.t. } {C{e\_1\atopwithdelims[]b\_1}}={C{e\_1\atopwithdelims[]x}}={C{e\_1\atopwithdelims[]\alpha}}=-1$ λ̄ = else &$\text{ \textbf{or } }{C{e\_1\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=1$ [0.75ex] $\{\bar{\lambda}\} \ket{e\_{12}} $ &$S\_O=\Big\{{C{e\_{12}\atopwithdelims[]e\_3}},{C{e\_{12}\atopwithdelims[]e\_4}},{C{e\_{12}\atopwithdelims[]e\_5}},{C{e\_{12}\atopwithdelims[]e\_6}},{C{e\_{12}\atopwithdelims[]z\_1}},{C{e\_{12}\atopwithdelims[]z\_2}},{C{e\_{12}\atopwithdelims[]\alpha}},{C{e\_{12}\atopwithdelims[]b\_1}},{C{e\_{12}\atopwithdelims[]x}}\Big\}$ λ̄ = ψ̄1, 2, 3/η̄1&$\ {C{e\_{12}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_{12}\atopwithdelims[]b\_1}}={C{e\_{12}\atopwithdelims[]x}}=-1$ λ̄ = ϕ̄1, 2&$\text{ \textbf{or} } \ {C{e\_{12}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_{12}\atopwithdelims[]z\_1}}={C{e\_{12}\atopwithdelims[]\alpha}}=-1$ λ̄ = ȳ3, 4, 5, 6&$\text{ \textbf{or } } \ {C{e\_{12}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_{12}\atopwithdelims[]b\_1}}={C{e\_{12}\atopwithdelims[]e\_{3,4,5,6}}}=-1$ λ̄ = ψ̄4, 5&$\text{ \textbf{or } } \ {C{e\_{12}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=3 \ \text{s.t. } {C{e\_{12}\atopwithdelims[]b\_1}}={C{e\_{12}\atopwithdelims[]x}}={C{e\_{12}\atopwithdelims[]\alpha}}=-1$ λ̄ = else&$\text{ \textbf{or } }{C{e\_{12}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=1$ [0.75ex] $\{\bar{\lambda}\}\ket{e\_{123}} $ &$S\_O=\Big\{{C{e\_{123}\atopwithdelims[]e\_4}},{C{e\_{123}\atopwithdelims[]e\_5}},{C{e\_{123}\atopwithdelims[]e\_6}},{C{e\_{123}\atopwithdelims[]z\_1}},{C{e\_{123}\atopwithdelims[]z\_2}},{C{e\_{123}\atopwithdelims[]\alpha}},{C{e\_{123}\atopwithdelims[]x}}\Big\}$ λ̄ = ψ̄4, 5&$ \ {C{e\_{123}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_{123}\atopwithdelims[]\alpha}}={C{e\_{123}\atopwithdelims[]x}}=-1$ λ̄ = ϕ̄1, 2&$\text{ \textbf{or} } \ {C{e\_{123}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=2 \ \text{s.t. } {C{e\_{123}\atopwithdelims[]z\_1}}={C{e\_{123}\atopwithdelims[]\alpha}}=-1$ λ̄ = else&$\text{ \textbf{or} }{C{e\_{123}\atopwithdelims[]S}}=1 \ \text{ and } \ \#(x \in S\_O \| x=-1)=1$ [0.75ex] [!ht] \| Tachyonic sector \| Survival conditions \| \| --- \| --- \| \| α \| ${C{\alpha\atopwithdelims[]e\_1}}={C{\alpha\atopwithdelims[]e\_2}}={C{\alpha\atopwithdelims[]e\_3}}={C{\alpha\atopwithdelims[]e\_4}}={C{\alpha\atopwithdelims[]e\_5}}={C{\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{\alpha\atopwithdelims[]S}}={C{\alpha\atopwithdelims[]z\_2}}={C{\alpha\atopwithdelims[]b\_1+z\_1+\alpha}}={C{\alpha\atopwithdelims[]b\_2+z\_1+\alpha}}=+1$ \| \| [0.75ex] z1 + α \| ${C{z\_1+\alpha\atopwithdelims[]e\_1}}={C{z\_1+\alpha\atopwithdelims[]e\_2}}={C{z\_1+\alpha\atopwithdelims[]e\_3}}={C{z\_1+\alpha\atopwithdelims[]e\_4}}={C{z\_1+\alpha\atopwithdelims[]e\_5}}={C{z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{z\_1+\alpha\atopwithdelims[]S}}={C{z\_1+\alpha\atopwithdelims[]z\_2}}={C{z\_1+\alpha\atopwithdelims[]b\_1+\alpha}}={C{z\_1+\alpha\atopwithdelims[]b\_2+\alpha}}=+1$ \| \| e1 + α \| ${C{e\_1+\alpha\atopwithdelims[]e\_2}}={C{e\_1+\alpha\atopwithdelims[]e\_3}}={C{e\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+\alpha\atopwithdelims[]S}}={C{e\_1+\alpha\atopwithdelims[]b\_1+z\_1+\alpha}}={C{e\_1+\alpha\atopwithdelims[]\tilde{x}+z\_1+\alpha}}={C{e\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + z1 + α \| ${C{e\_1+z\_1+\alpha\atopwithdelims[]e\_2}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_3}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+z\_1+\alpha\atopwithdelims[]S}}={C{e\_1+z\_1+\alpha\atopwithdelims[]b\_1+\alpha}}={C{e\_1+z\_1+\alpha\atopwithdelims[]\tilde{x}+\alpha}}={C{e\_1+z\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + α \| ${C{e\_1+e\_2+\alpha\atopwithdelims[]e\_3}}={C{e\_1+e\_2+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+\alpha\atopwithdelims[]S}}={C{e\_1+e\_2+\alpha\atopwithdelims[]b\_1+z\_1+\alpha}}={C{e\_1+e\_2+\alpha\atopwithdelims[]\tilde{x}+z\_1+\alpha}}={C{e\_1+e\_2+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + z1 + α \| ${C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_3}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]S}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]b\_1+\alpha}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]\tilde{x}+\alpha}}={C{e\_1+e\_2+z\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + e3 + α \| ${C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]S}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]\tilde{x}+z\_1+\alpha}}={C{e\_1+e\_2+e\_3+\alpha\atopwithdelims[]z\_2}}=+1$ \| \| e1 + e2 + e3 + z1 + α \| ${C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]e\_4}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]e\_5}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]e\_6}}=+1$ \| \| \| ${C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]S}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]\tilde{x}+\alpha}}={C{e\_1+e\_2+e\_3+z\_1+\alpha\atopwithdelims[]z\_2}}=+1$ \| Massless Sectors ---------------- Since the basis is identical to that analysed in, the massless sectors are the same, and we will not repeat all the details here. The aspects of the analysis which deserve further exploration are those in which the absence of supersymmetry manifests itself. Of particular interest is how the breaking of SUSY by a GGSO phase in S–models causes differences to where it is broken explicitly in the S̃–models. The way of breaking SUSY in the S–models (by a GGSO phase) allows the possibility of spontaneous SUSY breaking of Scherk-Schwarz type, which cannot be the case for any –models. In particular, it is well known in orbifold compactifications that if a freely acting orbifold gives a mass contribution to a gravitino then the SUSY breaking is spontaneous and above the gravitino mass scale SUSY is effectively restored. In such cases, the orbifold model is an example of a stringy Scherk-Schwarz compactification. In order to see how this is implemented within our S–models, we could transform it into a orbifold model with freely acting action using the dictionary of. Such an analysis is done explicitly for several SO(10) models in ref. but doing so for the S–models under analysis is left for future work. As in the supersymmetric case, spinorial 16/$\overline{\mathbf{16}}$’s arise from the sectors $$\begin{aligned} \label{S16s} B\_{pqrs}^{(1)} &=& S + b\_1 + pe\_3 + qe\_4 + re\_5 + se\_6 \nonumber \\ &=& \{\psi^{\mu},\chi^{1,2},(1-p)y^3\bar{y}^3, pw^3\bar{w}^3,(1-q)y^4\bar{y}^4,qw^4\bar{w}^4, \\ & & ~~~ (1-r)y^5\bar{y}^5,rw^5\bar{w}^5,(1-s)y^6\bar{y}^6, sw^6\bar{w}^6,\bar{\eta}^{1},\bar{\psi}^{1,\ldots,5}\} \nonumber \\ B\_{pqrs}^{(2)} &=& S + b\_2 + pe\_1 + qe\_2 + re\_5 + se\_6\nonumber \\ B\_{pqrs}^{(3)} &=& S + b\_3 + pe\_1 + qe\_2 + re\_3 + se\_4\nonumber\end{aligned}$$ where p, q, r, s = 0, 1 and b3 = b1 + b2 + x and vectorial 10’s from the sectors $$\begin{aligned} V\_{pqrs}^{(I)} &=& B\_{pqrs}^{(I)}+x\,\ I=1,2,3 \label{ovsinovs}\end{aligned}$$ However, in the SUSY case we know that any states in the Hilbert space of a model will be accompanied by their superpartners, generated through the addition of the S vector. In the non–SUSY case under consideration here, due to the S projection, the would-be superpartners can either be projected or have mismatched charges. Since the same formulae can be found for the SUSY classifications we will write the equations for the key classification numbers related to the observable gauge factors without further explanation. From the spinorial $\mathbf{16}/\mathbf{\overline{16}}$ we get the PS numbers $$\begin{aligned} \label{SpinorialNumbers} n\_{4L} &=& \frac{1}{4}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} P\_{pqrs}^A\left(1 + X^A\_{pqrs}\right)\left(1 + {C{B^A\_{pqrs}\atopwithdelims[]\alpha}} \right)\\ n\_{4R} &=& \frac{1}{4}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} P\_{pqrs}^A\left(1 - X^A\_{pqrs}\right)\left(1 - {C{B^A\_{pqrs}\atopwithdelims[]\alpha}} \right) \\ n\_{\overline{4}L} &=& \frac{1}{4}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} P\_{pqrs}^A\left(1 - X^A\_{pqrs}\right)\left(1 + {C{B^A\_{pqrs}\atopwithdelims[]\alpha}} \right)\\ n\_{\overline{4}R} &=& \frac{1}{4}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1}} P\_{pqrs}^A\left(1 + X^A\_{pqrs}\right)\left(1 - {C{B^A\_{pqrs}\atopwithdelims[]\alpha}} \right).\end{aligned}$$ where XpqrsA determines the chirality $\mathbf{16}/\mathbf{\overline{16}}$ of a Bpqrs1, 2, 3 sector given by eq. ([S16s]) and PpqrsA are the projectors ([SpinProjector]) for Bpqrs1, 2, 3. Meanwhile, from the 10 we can write the bi–doublet and D(6, 1, 1) numbers as $$\begin{aligned} \label{bidoublet} n\_h &=& \frac{1}{2}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1 }}R^{(A)}\_{pqrs}\left(1 - {C{V^A\_{pqrs}\atopwithdelims[]\alpha}}\right) \\ n\_6 &=& \frac{1}{2}\sum\_{\substack{A=1,2,3 \\ p,q,r,s=0,1 }}R^{(A)}\_{pqrs}\left(1 + {C{V^A\_{pqrs}\atopwithdelims[]\alpha}}\right) \end{aligned}$$ where Rpqrs(A) are the vectorial projectors in this case. The other important aspects of these models that we will need for a phenomenological analysis are the absence of gauge enhancements affecting the observable gauge factors and the absence of chiral exotic sectors. The exotic sectors listed in Section 2 of the SUSY PS classification of are also present in our analysis. However, by analysing the fermionic exotics in our S–models we cannot say anything about the would-be superpartner scalar exotics. For example, in the SUSY case, if you find an absence of any fermionic exotic sectors then a model can be declared exophobic (free of all exotic sectors) since the scalar superpartners will also be absent. For our purposes we will not classify these scalar exotic sectors but only seek to ensure the absence of chiral exotics by inspecting the fermionic exotics. A further difference to our exotics analysis is that we have the exotic sectors [SExotics] {\|i}, {\|i} , , which in SUSY models are thought of as (gaugino) superpartners of mixed enhancement sectors generating additional gauge bosons. In the SUSY case, such sectors can be assured to be projected out whenever the enhancements are projected. Since the absence of observable enhancements is a fundamental phenomenological constraint to impose, once checked we can be sure that these sectors ([SExotics]) are projected too. For the non–SUSY case, we cannot be sure that the absence of enhancements to the observable gauge group will result in the projection of these exotic states ([SExotics]) so we must account for them when looking at the absence of a chiral exotic anomaly. Fertility Conditions -------------------- We will use a similar fertility methodology as in the S̃ case except in this case we can consistently enforce conditions on the presence of a Heavy Higgs to break the PS group and have the accompanying Missing Partner Mechanism. We can list the set of fertility conditions as follows: 1. Absence of the tachyonic sectors: [z1TachCondS] z1, ei+z1, ei+ej+z1, ei+ej+ek+z1 using the set of conditions delineated in Table $\ref{NormSSO10}$. 2. Constraints on SO(10) spinorial states related to the presence of complete fermion families and a Heavy Higgs $$\begin{aligned} \label{Sfertility1} n\_{4L}-n\_{\bar{4}L} = n\_{\bar{4}R} -n\_{4R}\geq 6 \,\ n\_{\bar{4}R}^{(B)} > 1.\end{aligned}$$ 3. For the presence of a SM higgs, i.e. nh ≥ 1, and a D(6, 1, 1), i.e. n6 ≥ 1, we can impose $$\begin{aligned} \label{Sfertility2} N\_{10}\geq 2\end{aligned}$$ at the SO(10) level. 4. Presence of a top quark mass coupling (TQMC) which amounts to fixing the following GGSO coefficients following the methodology of $$\begin{aligned} \label{Stqmc} {C{e\_i\atopwithdelims[]b\_1}}&=&-{C{e\_i\atopwithdelims[]S}}, \ \ \ i=1,2 \\ {C{e\_j\atopwithdelims[]b\_2}}&=&-{C{e\_j\atopwithdelims[]S}}, \ \ \ \ j=3,4 \\ {C{z\_k\atopwithdelims[]b\_1}}&=&-{C{z\_k\atopwithdelims[]S}}, \ \ \ {C{z\_k\atopwithdelims[]b\_2}}=-{C{z\_k\atopwithdelims[]S}}, \ \ \ \ k=1,2 \\ {C{b\_1\atopwithdelims[]e\_5}}&=&{C{b\_2\atopwithdelims[]e\_5}}, \ \ \ \ {C{b\_1\atopwithdelims[]e\_6}}={C{b\_2\atopwithdelims[]e\_6}} \end{aligned}$$ at the SO(10) level, and $$\begin{aligned} \label{StqmcPS} {C{b\_1\atopwithdelims[]\alpha}} =-{C{b\_2\atopwithdelims[]\alpha}} = {C{S\atopwithdelims[]\alpha}}\end{aligned}$$ at the PS level. A few comments on these conditions are in order. In condition 2) the equation $n\_{\bar{4}R}^{(B)} > 1$ is a necessary, but not sufficient, condition on the presence of a Heavy Higgs to break the PS gauge symmetry. We specify that these come from the bosonic sectors Bpqrs(1, 2, 3)B = S + Bpqrs(1, 2, 3)F. We note that any such sector, ξ, that survives the GGSO projections and has a GGSO phase ${C{\xi\atopwithdelims[]\alpha}}=1$ will give rise to the scalars with observable representations \|4,5 =: n\|4R(B)+n4R(B), where we employ the convenient notation $\binom{3}{even}=\{ \ket{\pm \pm \pm} \ \| \ \# (\ket{-}) \mod 2=0\}$. These $n\_{\bar{4}R}^{(B)}+n\_{4R}^{(B)}$ states are those we want to break the PS gauge group. Furthermore, with both of them we can allow for breaking along a D-flat direction, which enables hierarchical SUSY breaking via Scherk-Schwarz, although this would need to be evaluated carefully for a specific model. In condition 3) imposing n6 ≥ 1 is required for the implementation of the Missing Partner Mechanism of PS models. This occurs once the PS group is broken and the Heavy Higgs has acquired a VEV. With a (6, 1, 1) triplet/anti–triplet field we induce couplings to d/d̄ SM fields of the form ∼ ⟨n4R⟩dcD3 and ∼ ⟨n4R⟩d̄cD̄3. These form massive states at the GUT scale which protect proton decay from happening too quickly. Partition Function of Pati–Salam Models ======================================= The analysis of the partition function is particularly important in non–supersymmetric constructions, as it gives a complementary tool to analyse the structure of the theory. It also provides the necessary tools to count the total number of states at each mass level and hence check for the existence of on and off–shell tachyons in any specific model. Moreover, its integration over the fundamental domain corresponds to the cosmological constant. The partition function for free fermionic theories is given by the integral $$Z = \int\_\mathcal{F}\frac{d^2\tau}{\tau\_2^2}\, Z\_B \sum\_{\alpha,\beta} {C{\alpha\atopwithdelims[]\beta}} \prod\_{f} Z \sqbinom{\alpha(f)}{\beta(f)}, \label{ZInt}$$ where d2τ/τ22 is the modular invariant measure and ZB is the bosonic contribution arising from the worldsheet bosons. The sum gives the contributions from the worldsheet fermions and involves all combinations of basis vectors α, β from a given basis set, which in our case are the ones given in ([basisSt]) and ([basisS]). The integral is over the fundamental domain of the modular group F = {τ ∈ C ∣ ∣τ∣2 > 1 ∧ ∣τ1∣ < 1/2}, which ensures that only physically inequivalent geometries are counted. The above integral specifically represents the one–loop vacuum energy Λ of our theory. Note that this is the cosmological constant from the worldsheet point of view and hence is a dimensionless quantity. It is related to the spacetime cosmological constant, λ, by $\lambda = -\frac{1}{2}\mathcal{M}^4\Lambda$, where M = MString/2π. For simplicity, in the following we will refer to Λ as the cosmological constant. The best way to perform this integral is as presented in using the expansion of the η and θ functions in terms of the modular parameter τ, or more more precisely, in terms of q ≡ e2πiτ and q̄ ≡ e− 2πiτ̄. This leads to a series expansion of the one-loop partition function $$Z = \sum\_{n.m} a\_{mn} \int\_\mathcal{F} \frac{d^2\tau}{\tau\_2^3} \, q^m \bar{q}^n. \label{QPF}$$ The benefit of such an expansion to the partition function is that the amn represent the difference between bosonic and fermionic degrees of freedom at each mass level, i.e. amn = Nb − Nf. As expected, on–shell tachyonic states, i.e. states with m = n < 0, have an infinite contribution. On the other hand off–shell tachyonic states may contribute a finite value to the partition function. It is indeed known that such off–shell tachyonic states are necessarily present in the spectrum of non–SUSY theories and so all of our Pati–Salam models posses such states. It is also important to note that modular invariance constraints imposed on the basis and GGSO phases only allows states with m − n ∈ Z. In theories with spacetime supersymmetry, it is ensured that the bosonic and fermionic degrees of freedom are exactly matched at each mass level. That is, we necessarily have that amn = 0 for all m and n, which in turn causes the vanishing of the cosmological constant. For non–supersymmetric models like the ones introduced in Sections [StPS] and [PSNormS], this level-by-level cancellation is not ensured and so in general produce a non–zero value for Λ. This value, however, is specific to the fermionic point in the orbifold moduli space and hence moving away from this point can in principle change Λ as discussed in. An interesting possibility is to try to suppress the value of the cosmological constant so that we are guaranteed a small value. It has been shown in that such a mechanism may be possible for models which satisfy the constraint Nb0 = Nf0 or in the above language a00 = 0. These rely on a Scherk–Schwarz SUSY breaking in which the internal dimensions of the compactification space R are used to suppress the next–to leading order contributions to Λ in the large volume R → ∞ limit. This is because in this setup, the dependence of the cosmological constant on the compactification radius schematically becomes $$\Lambda \propto (N^0\_b - N^0\_f) \frac{1}{R^4} + \mathcal{O}(e^{-\alpha R^2}),$$ and thus in the large R limit, Λ is exponentially suppressed given that Nb0 = Nf0. Thus in our classification program, we also try to find models which fulfil this condition and so may be good candidates to further explore this idea. It is important to note, however, that since the above mechanism relies on a Scherk-Schwarz breaking, it can only be applied to (some) S–models. This is because in S̃–models, SUSY is broken explicitly. Results of Classification ========================= Using the methodology built-up in the previous sections, we can now turn towards analysing samples of PS S̃ and S–models with respect to standard phenomenological criteria. First of all, we perform a random classification in the space of PS S̃ and S–models. This works by generating random GGSO phase configurations and classifying according to the absence of tachyons and the classification numbers: $n\_{4L},n\_{\bar{4}L},n\_{4R},n\_{\bar{4}R}, n\_h, n\_6, n\_{4},n\_{\bar{4}},$ n2L, n2R which are common to S̃ and S–models and are defined in previous sections. For the S̃–models there are a further two classification numbers, nV2L, nV2R, relating to the vectorial exotics ([ExotVecs]). The results of the random classification for a sample of 109 GGSO configurations in both the S̃ and S–model cases are displayed in Table [RandomStStats] and [RandomSStats], respectively. [!htb] \|c\|l\|r\|c\|r\| &Constraints & & Probability & No Constraints & 109 & 1 (1)&+ Tachyon–free & 2038657 & 2.04 × 10− 3 (2)& + No Observable Enhancements & 2014917 & 2.01 × 10− 3 (3)&+ Complete Families & 572411 & 5.72 × 10− 4 (4)&+ No Chiral Exotics & 403989 & 4.04 × 10− 4 (5)&+ 3 Generations& 3074 & 3.07 × 10− 6 (6)&+ $n\_{\bar{4}R}^F,n\_{4R}^F$ Present& 346& 3.46 × 10− 7 (7)&+ SM Higgs& 314&3.14 × 10− 7 (8)&+ Heavy Triplet Constraint& 298&2.98 × 10− 7 (9)&+ TQMC & 289 & 2.89 × 10− 7 (10)&+ No Fermionic Exotics& 0 & 0 [!htb] \|c\|l\|r\|c\|r\| &Constraints & & Probability & No Constraints & 109 & 1 (1)&+ Tachyon–free & 10578258 & 1.06 × 10− 2 (2)& + No Observable Enhancements & 10246688 & 1.02 × 10− 2 (3)&+ Complete Families & 2730363 & 2.73 × 10− 3 (4)&+ No Chiral Exotics & 461666 & 4.62 × 10− 4 (5)&+ 3 Generations& 3103 & 3.10 × 10− 6 (6)&+ PS Higgs& 2684&2.68 × 10− 7 (7)&+ SM Higgs& 2263&2.26 × 10− 7 (8)&+ Partner Mechanism& 1934&1.93 × 10− 7 (9)&+ TQMC& 1878 & 1.88 × 10− 7 (10)&+ No Fermionic Exotics& 36& 3.6 × 10− 9 From these results we can first note a higher probability of ∼ 0.01 that an S–model is tachyon–free compared with an S̃–model which has probability ∼ 0.002. This is likely due to the power of GGSO phases involving S in projecting the tachyons. Another notable result is the absence of PS S̃–models without massless fermionic exotics in the final criteria (10), whereas we find vacua for the S–models where all fermionic exotic sectors are projected. As mentioned in previous sections, we are not classifying the scalar exotic sectors here and so we can not say whether there are exophobic models, which were found in the SUSY PS classification of. Due to the absence of SUSY it appears that finding such exophobic models would be much less likely due to having to check the bosonic exotic sectors independently to the fermionic ones. As explained in earlier sections, the PS –model analysis is unphysical in the sense that we implement the condition (6) for the presence of $n\_{\bar{4}R}^F,n\_{4R}^F$ and condition (8) for the presence of a D(6, 1, 1) in order to draw an analogy with the analysis of the S–models where a Heavy Higgs can be found from the bosonic component of the $\mathbf{16}/\mathbf{\overline{16}}$ and a Missing Partner Mechanism can be implemented requiring the D(6, 1, 1) state. The difference in condition (6) skew the numbers a bit since requiring a PS Higgs for the S–models is a weak condition, whereas the symbolic condition for a $n\_{\bar{4}R}^F,n\_{4R}^F$ in the –models is stronger. In order to find more models satisfying our phenomenological constraints we can now turn to utilizing the fertility methodology outlined in sections [StFertConds] and [SFertConds] for the S̃ and S cases, respectively. Explicitly, we do this by implementing the conditions 1) - 4) in sections [StFertConds] and [SFertConds] in the space of SO(10)–models, ΠSO(10), and collect 200,000 fertile S̃ SO(10) cores and 550,000 S cores. Once we have the cores, we look at the space Πα and, in particular, we iterate over all independent choices of α GGSO phases. When equations ([tqmcPS]) and ([StqmcPS]) are accounted for we are left with 10 independent such phases. However, a subtlety arises here due to how to avoid SUSY configurations in the S–models. We must allow for the possibility that some SO(10) cores are supersymmetric in the S–models because we can still get non-SUSY models in the case where SUSY is broken by α, i.e. ${C{S\atopwithdelims[]\alpha}}=+1$. The logical procedure to deal with this is to check if the condition ([SUSYphases]) holds for the SO(10) phases and then fix ${C{S\atopwithdelims[]\alpha}}=+1$ and iterate over the reduced Πα space spanned by 9 independent GGSO phases involving α. In our sample of S SO(10) cores we extracted 50,000 such SUSY cores and 500,000 non-SUSY cores The results of this fertility classification are displayed in Table [FertileStStats] and [FertileSStats]. [!htb] \|c\|l\|r\|c\|r\| &Constraints & & Probability & No Constraints & 204800000 & 1 (1)&+ Tachyon–free & 2417463 & 1.18 × 10− 2 (2)& + No Observable Enhancements & 2406298 & 1.17 × 10− 2 (3)&+ Complete Families & 623004 & 3.04 × 10− 3 (4)&+ No Chiral Exotics & 438280 & 2.14 × 10− 3 (5)&+ 3 Generations& 327463 & 1.60 × 10− 3 (6)&+ PS Higgs& 190766 &9.31 × 10− 4 (7)&+ SM Higgs& 190766 &9.31 × 10− 4 (8)&+ Partner Mechanism& 183753 & 8.97 × 10− 4 (9)&+ TQMC & 183753 & 8.97 × 10− 4 (10)&+ No Fermionic Exotics& 0 &0 [!htb] \|c\|l\|r\|c\|r\| &Constraints & & Probability & No Constraints & 537600000 & 1 (1)&+ Tachyon–free & 11770044 & 2.19 × 10− 2 (2)& + No Observable Enhancements & 11431950 & 2.12 × 10− 2 (3)&+ Complete Families & 3020242 & 5.62 × 10− 3 (4)&+ No Chiral Exotics & 723352 & 1.35 × 10− 3 (5)&+ 3 Generations& 488802 & 9.09 × 10− 4 (6)&+ PS Higgs& 444454 &8.27 × 10− 4 (7)&+ SM Higgs& 444454 &8.27 × 10− 4 (8)&+ Partner Mechanism& 384080 &7.14 × 10− 4 (9)&+ TQMC& 384080 &7.14 × 10− 4 (10)&+ No Fermionic Exotics& 16030 &2.98 × 10− 5 The first thing to note from these results is that owing to the fertility analysis the number of models meeting the phenomenological criteria has improved by around 3 orders of magnitude compared with the random classification. Furthermore, as expected, we see that all PS models derived in the fertility methodology will come with a SM Higgs and TQMC automatically. Taking a sample of models with complete families from the random classification in Table [RandomSStats] we can plot a graph displaying the frequency for the different number of generations. This is shown in Figure [NumGensGraphSt] and [NumGensGraphS]. [NumGensGraphSt]Frequency plot for the number of generations in a model from a sample of randomly generated models that satisfy criteria (1)-(3). [NumGensGraphS]Frequency plot for the number of generations in a model from a sample of 572411 randomly generated models that satisfy criteria (1)-(3). An aspect of the phenomenological data that’s worth displaying in more detail is the number of fermionic exotics. In particular, for models with complete families (satisfying criteria (1)-(3)) from the random classifications shown in Tables [RandomStStats] and [RandomSStats] we plot the total number of fermionic exotic states. The frequency plot for the S̃–models is displayed in Figure [NumExotsGraphSt] and S–models in Figure [NumExotsGraphS]. [NumExotsGraphSt]Frequency plot for the number of exotic states in a model from a sample of 572411 randomly generated models that satisfy criteria (1)-(3). [NumExotsGraphS]Frequency plot for the number of exotic states in a model from a sample of 1365922 randomly generated models that satisfy criteria (1)-(3). The notable feature of these exotics graphs is how they do not obey nice bell curve distributions. This is due to the vectorial exotics listed in eq. ([ExotVecs]) for S̃–models and ([SExotics]) for S–models. Without these, the graphs obeys a smooth bell curve of the taller bars. Another thing to note is that we do find S̃–models free of fermionic exotics in the sample, except none of them have more than two generations so don’t satisfy further phenomenological constraints. As already mentioned, it is reasonable to say that exophobic models in either class, S̃ or S, are much less likely to arise than in the SUSY case due to the scalar exotics needing to be checked independently. The final classification data we will display can be found in Appendix A, where the quantum numbers for models satisfying constraints (1)-(9) in Tables [RandomSStats] and [FertileSStats] are displayed. Results for Nb0 − Nf0 ----------------------------- As discussed in Section [PF], the constant term of the partition function, a00 = Nb0 − Nf0, is an important quantity. It quantifies the Bose–Fermi degeneracy at the massless level and so is of phenomenological significance. It also provides the leading order behaviour of the vacuum energy, and thus models with Nb0 − Nf0 can be of particular interest. Thus, in our classification program we have taken a close look at this value. Its distribution for a sample of 2 × 103 is shown in Figures [STDist] and [SDist] for S̃ and S–models respectively. We have included a random sample of non–tachyonic vacua along with a sample of models satisfying the criteria (1)-(9) in order to see what effect the imposition of certain phenomenological features have on this net Bose–Fermi degeneracy. The distributions show that for both sets of models it should be possible to find ones which satisfy the condition Nb0 = Nf0. Indeed, from the sample of 2 × 103 we have found 14 such models for both S̃ and S. [STDist] [SDist] Example Models with Nb0 = Nf0 ------------------------------------- Having found O(105) S̃ models satisfying all phenomenological constraints (1)-(9) we can do a subsequent search for such models that additionally satisfy the condition a00 = Nb0 − Nf0 = 0 which suppresses the leading order contribution to the cosmological constant of the model. The following GGSO phase configuration meets this condition $${C{v\_i\atopwithdelims[]v\_j}}= \begin{blockarray}{rrrrrrrrrrrrrr} &\mathds{1}& \tilde{S} & e\_1 & e\_2 & e\_3 & e\_4 & e\_5 & e\_6 & b\_1 & b\_2 & b\_3 & z\_1 & \alpha \\ \begin{block}{r(rrrrrrrrrrrrr)} \mathds{1}&-1& 1& 1& 1& -1& -1& -1& -1& 1& -1& 1& 1& -1\ \\ \tilde{S}& 1& -1& 1& -1& 1& -1& 1& 1& 1& -1& -1& 1& -1\ \\ e\_1& 1& 1& -1& -1& -1& 1& 1& 1& -1& -1& -1& -1& 1\ \\ e\_2& 1& -1& -1& -1& -1& 1& -1& -1& -1& -1& -1& -1& 1\ \\ e\_3& -1& 1& -1& -1& 1& 1& 1& 1& -1& -1& -1& -1& -1\ \\ e\_4& -1& -1& 1& 1& 1& 1& 1& -1& -1& -1& -1& -1& 1\ \\ e\_5& -1& 1& 1& -1& 1& 1& 1& -1& -1& -1& -1& 1& 1\ \\ e\_6& -1& 1& 1& -1& 1& -1& -1& 1& -1& -1& -1& -1& 1\ \\ b\_1&1& -1& -1& -1& -1& -1& -1& -1& 1& -1& -1& -1& 1\ \\ b\_2&-1& 1& -1& -1& -1& -1& -1& -1& -1& -1& -1& -1& -1\ \\ b\_3&1& 1& -1& -1& -1& -1& -1& -1& -1& -1& 1& -1& 1\ \\ z\_1&1& -1& -1& -1& -1& -1& 1& -1& -1& -1& -1& 1& 1\ \\ \alpha&-1& -1& 1& 1& -1& 1& 1& 1& -1& 1& -1& -1& -1\ \\ \end{block} \end{blockarray}$$ Furthermore, as desired, it enjoys a PS–breaking Higgs and 3 chiral generations with PS quantum numbers $n\_{4L}=3, n\_{\bar{4}L}=0, n\_{4R}=1$ and $n\_{\bar{4}R}=4$. Furthermore, the model has the necessary SM Higgs and D(6, 1, 1) for the unphysical ‘Heavy Triplet Constraint’, such that nh = 3 and n6 = 5. The model also has exotic quantum numbers $n\_{4}=n\_{\bar{4}}=1,n\_{2L}=n\_{2R}=6$ and nV2L = nV2R = 0 which allows for the generation of vector–like exotics at high mass scale so as not to have fractionally charged states at lower mass scales which violates experimental observation. The top quark mass coupling is guaranteed by the conditions ([Stqmc]) and ([StqmcPS]). We also note that the x̃–sector arises in the spectrum of this model. This is important since it is charged under the observable group and in this case generates four extra $n\_{\bar{4}R}$ and n4R charged under each U1, 2, 3 factor, leaving the number of generations still equal to 3. It was noted in that the x̃–sector corresponds to the sector producing the fermionic superpartners of the states from the x–sector, i.e. S + x, which enhance the SO(10) symmetry to E6. The x̃–sector therefore gives rise to the fermionic superpartners of the spacetime vector bosons from the x–sector, which do not arise in the construction of our S̃–models. We can also calculate the traces under the U(1)i = 1, 2, 3 associated with the right–moving currents η̄iη̄i, which are U(1)1=-24, U(1)2=-24 U(1)3=48. such that the combination U(1)1 + U(1)2 − U(1)3 is anomalous and we can choose U(1)1 − U(1)2 and U(1)1 + U(1)2 + U(1)3 as anomaly-free combinations. The partition function for this model is given by [PFStModel] Z = 2q0\|q-1 + 0q0\|q0 - 668q1/8\|q1/8 - 4224q1/4\|q1/4 + 32q3/8\|q-5/8 +, where we see that there are no on–shell tachyons and that we have equal number of bosons and fermions at the massless level as advertised. We can further note the off–shell model–independent term 2 q0q̄− 1 obtained from the so–called ‘proto–graviton’ resulting from the state $\psi^\mu \ket{0}\_L\otimes \ket{0}\_R$. This expression ([PFStModel]) integrates via ([QPF]) to give a cosmological constant Λ = − 166.42. Recall that, as described in Section [PF], this is the dimensionless worldsheet vacuum energy and hence has the opposite sign compared to the 4D spacetime cosmological constant. Turning our attention to the analysis of S–models, we can achieve Nb0 = Nf0 and meet the phenomenological constraints (1)-(9) with the following GGSO phase configuration $$\label{GGSOphasesS} {C{v\_i\atopwithdelims[]v\_j}}= \begin{blockarray}{rrrrrrrrrrrrrr} &\mathds{1}& S & e\_1 & e\_2 & e\_3 & e\_4 & e\_5 & e\_6 & b\_1 & b\_2 & z\_1 & z\_2 & \alpha \\ \begin{block}{r(rrrrrrrrrrrrr)} \mathds{1}&-1& -1& -1& 1& -1& 1& -1& -1& -1& 1& 1& 1& -1\ \\ S&-1& -1& -1& -1& -1& -1& -1& -1& -1& 1& 1& -1& -1\ \\ e\_1&-1& -1& 1& -1& 1& 1& 1& -1& 1& 1& 1& 1& 1\ \\ e\_2& 1& -1& -1& -1& 1& 1& 1& 1& 1& -1& 1& 1& -1\ \\ e\_3&-1& -1& 1& 1& 1& 1& 1& -1& -1& 1& -1& 1& 1\ \\ e\_4& 1& -1& 1& 1& 1& -1& 1& 1& -1& 1& 1& 1& -1\ \\ e\_5&-1& -1& 1& 1& 1& 1& 1& -1& -1& -1& 1& -1& 1\ \\ e\_6&-1& -1& -1& 1& -1& 1& -1& 1& 1& 1& -1& 1& -1\ \\ b\_1&-1& 1& 1& 1& -1& -1& -1& 1& -1& 1& -1& 1& -1\ \\ b\_2& 1& -1& 1& -1& 1& 1& -1& 1& 1& 1& -1& 1& 1\ \\ z\_1& 1& 1& 1& 1& -1& 1& 1& -1& -1& -1& 1& 1& 1\ \\ z\_2& 1& -1& 1& 1& 1& 1& -1& 1& 1& 1& 1& 1& -1\ \\ \alpha&-1& -1& 1& -1& 1& -1& 1& -1& 1& -1& -1& -1& -1\ \\ \end{block} \end{blockarray}$$ which enjoys a PS–breaking Higgs with $\#(n\_{4R}/n\_{\bar{4}R})=3$ and three chiral generations with quantum numbers $n\_{4L}=4, n\_{\bar{4}L}=1, n\_{4R}=1$ and $n\_{\bar{4}R}=4$. Furthermore the model has the necessary SM Higgs and D(6, 1, 1) for the Missing Partner Mechanism such that nh = 2, n6 = 2. The model also has exotic quantum numbers $n\_{4}=n\_{\bar{4}}=0 $ and n2L = n2R = 4, which ensures the absence of chiral exotic states. We note that exophobic vacua with Nb0 = Nf0 were not found in our analysis. The top quark mass coupling is guaranteed by the conditions ([Stqmc]) and ([StqmcPS]). The traces of U(1)1, 2, 3 for this model are given by: U(1)1=0, U(1)2=0, U(1)3=0 such that each of the U(1)1, 2, 3 are anomaly-free independently. For U(1)1 this cancellation occurs between the trace in the observable $\mathbf{16}/\overline{\mathbf{16}}$ and the trace from hidden sectors. For the U(1)2 and U(1)3 the cancellation happens in each type of sectors (observables, hidden, exotics) independently. Inspecting the GGSO phase matrix ([GGSOphasesS]) we see that supersymmetry is only broken by one phase ${C{S\atopwithdelims[]z\_1}}=1$. It is not too surprising that configurations close to supersymmetric ones are common origins of potentially viable models since they preserve some of the benefits from having supersymmetry. In particular, having most of the GGSO phases involving S equal to − 1 will help to ensure the absence of tachyons. The gravitino is of course projected but we note that the following states from the S sector: $\{\bar{\psi}^{1,2,3},\bar{\eta}^{1,2,3}\}\{\bar{\phi}^{3,4}\}\ket{S}$ and $\{\bar{\psi}^{4,5}\}\{\bar{\phi}^{1,2}\}\ket{S}$ are retained. The partition function for this S model is given by Z = 2q0\|q-1 + 0q0\|q0 + 16q1/8\|q1/8 - 192q1/4\|q1/4 +192q3/8\|q3/8 - 4q1/2\|q-1/2 +, resulting in a worldsheet vacuum energy Λ = − 62.66. We see that we indeed have Nb0 = Nf0, hence the lack of constant term in both models above. We also observe the necessary off–shell tachyon at q̄− 1 and the lack of physical tachyons. Conclusions =========== The free fermionic representation of the heterotic string in four dimensions gave rise to an abundance of three generation models with varying unbroken SO(10) subgroups and the canonical GUT embedding of the weak hypercharge. These models correspond to Z2 × Z2 orbifold of six–dimensional compactified tori at special points in the moduli space. The free fermionic formalism was used to develop a systematic classification of the Z2 × Z2 toroidal orbifolds, leading to numerous fundamental observations, among them: the construction of the first known string models that produce in the low energy effective field theory solely the spectrum of the Minimal Supersymmetric Standard Model ; the discovery of spinor–vector duality in the space of Z2 × Z2 orbifold compactifications ; the discovery of exophobic string models. In this paper we extended the systematic classification of free fermionic Z2 × Z2 orbifolds to non–supersymmetric Pati–Salam models. We pursued the construction of such models via two routes, based on the –models and S–models, where the first class descend from a tachyonic ten–dimensional vacuum, whereas the second correspond to compactifications of the ten–dimensional non–supersymmetric SO(16) × SO(16) heterotic string. A first task in the construction of non–supersymmetric models is to ensure that all the physical tachyonic states are projected out from the physical spectrum. Systematic classification rules were developed to analyse the tachyon producing sectors and to extract tachyon–free vacua in the two classes of models. tachyon–free models were found with probability 0.002 and 0.01 in and S–models, respectively. To facilitate the extraction of phenomenological vacua merits the development of the ‘fertility methodology’ that pre–selects SO(10) preserving GGSO configuration that are amenable for producing viable phenomenological models. We demonstrated that application of the ‘fertility methodology’ increases the frequency of models that satisfy key phenomenological criteria by three orders of magnitude. We note that whereas S–models contain the heavy Higgs scalar representations required to break the Pati–Salam gauge symmetry down to the Standard Model gauge	arxiv_0000618
) at the orange star for panel (c) and at the white and black stars for panels (d) and (e), respectively.](appc2.png)[figappc2]Dynamical phase diagram, continuum bands, and eigenvalues of the levitated nanoparticle array. (a,b) Dynamical phase diagrams with Ω = 3 and γ = 2 and Ω = 1 and γ = 2. respectively. Panel (b) is the same as Fig. [fig2](a). The blue, green, red, and gray-shaded regions indicate the underdamped, critical, overdamped, and dynamically unstable phase, respectively. The spectral singularity (SS) appears in the green-shaded region and on the green lines. (c) and (d) Continuum bands ω̃+ (magenta) and ω̃− (cyan), and eigenvalues ωl, +> (red) and ωl, −> (blue). (e) Continuum band ω̃+ (black), and eigenvalues ωl, +< (red). We set L = 8 in panels (c–e) and choose the values of (K, K̄) at the orange star for panel (c) and at the white and black stars for panels (d) and (e), respectively. In our work, we investigate the levitated nanoparticle array in the thermodynamic limit, L → ∞, and obtain the dynamical phase diagram and the continuum bands of the levitated nanoparticle array. We show the dynamical phase diagram with γ < 2Ω and γ > 2Ω in Figs. [figappc2](a) and [figappc2](b), respectively. We note that the critical phase extends over only a narrow region in Fig. [figappc2](a), while it clearly possesses a broad region in Fig. [figappc2](b). We here discuss how the finite-size effects can affect the band structures in the critical phase and the boundaries between the underdamped and dynamically unstable phases. To do so, we recall that the continuum bands of the infinite-size system is given by $$\tilde{\omega}\_\pm=\frac{i}{2}\gamma+\sqrt{\Omega^2+\frac{2}{m}\left(K-\sqrt{K^2-\bar{K}^2}\cos\theta\right)-\frac{\gamma^2}{4}}. \label{eqappc13}$$ First, we consider the parameters indicated by the orange star in Fig. [figappc2](a) and show the corresponding continuum bands ω̃± and the eigenvalues ωl, ±> in Fig. [figappc2](c). One can see that there is no degeneracy between ωl, +> and ωl, −> in a strict sense, simply because θl takes only the discrete values determined by Eq. ([eqappc9]). Nevertheless, we emphasize that the eigenvectors around the spectral singularity, where ω̃+ touches ω̃−, still exhibit the strong nonorthogonality which is a hallmark of the non-Hermitian degeneracy. This result is the same as in the case of the parameter indicated by the white star in Fig. [figappc2](b). Indeed, the eigenvalues in Fig. [figappc2](d) are similar to those in Fig. [figappc2](c). Next, we consider the parameters indicated by the white star in Fig. [figappc2](a) and show the corresponding continuum band ω̃+ and the eigenvalues ωl, +< in Fig. [figappc2](c). It is found that the finite-size system does not exhibit the dynamical instability because the imaginary parts of all the discrete eigenvalues become positive. In this sense, the finite-size effect can slightly modify the boundary between the dynamically unstable phase and the other phases. One can also infer from Fig. [figappc2](c) that a larger system size would be favorable to observe the dynamical instability. Nevertheless, we emphasize that the phase diagram of a finite-size system still remains qualitatively the same as in the result obtained in the thermodynamic limit. [secD]Periodic boundary condition ================================= ![[figappd]Dynamical phase diagram of the levitated nanoparticle array obtained from the conventional Bloch band theory. The blue and gray-shaded regions are the underdamped and dynamically unstable phases, respectively. The spectral singularity appears on the green line. We set the parameters to be \Omega=1 and \gamma=5.](appd.png)[figappd]Dynamical phase diagram of the levitated nanoparticle array obtained from the conventional Bloch band theory. The blue and gray-shaded regions are the underdamped and dynamically unstable phases, respectively. The spectral singularity appears on the green line. We set the parameters to be Ω = 1 and γ = 5. We consider the levitated nanoparticle array described by Eq. ([eqappa14]) with N = 1 and study the dynamical phase diagram under periodic boundary conditions to compare the result obtained in Fig. [fig2](a). We note that the characteristic equation of the system reads $$\begin{aligned} &&\frac{1}{m}\left[\left(K\_1-\bar{K}\_1\right)\beta+\left(K\_1+\bar{K}\_1\right)\beta^{-1}\right], \nonumber\\ &&+\left(\omega^2-i\gamma\omega-\Omega^2-\frac{2}{m}K\_1\right)=0. \label{eqappd1}\end{aligned}$$ After replacing β by eik for the real Bloch wavenumber k, we can obtain $$\omega\_\pm^\prime=\frac{i}{2}\gamma\pm\sqrt{\Omega^2-\frac{\gamma^2}{4}+\frac{2}{m}K\_1\left(1+\cos k\right)-\frac{2i}{m}\bar{K}\_1\sin k}, \label{eqappd2}$$ which reproduces the eigenvalues of the system with periodic boundary conditions. Let us suppose that γ > 2Ω for the sake of concreteness. The condition for the appearance of the spectral singularity can be then written as $$\Omega^2-\frac{\gamma^2}{4}+\frac{4}{m}K\_1=0, \label{eqappd3}$$ since it appears when k = 0. Remarkably, this indicates that the critical phase obtained in Fig. [fig2](a) disappears in the system with periodic boundary conditions. Said differently, the appearance of the spectral singularity requires the fine-tuning of the system parameters. Indeed, one can see from the dynamical phase diagram shown in Fig. [figappd] that only the underdamped and dynamically unstable phases appear in this case. Thus, modifying open boundaries to periodic boundaries drastically changes the behavior of the levitated nanoparticle array. [secE]Experimental feasibility ============================== We investigate the dynamical phase diagram of the levitated nanoparticle array with K/m, K̄/m ∈ [ − 10Ω2, 10Ω2] and γ = 5Ω in our work. We here discuss the experimental values to access the assumed parameter region in our setup. We note that the previous experiment realizing the coupled two levitated nanoparticles has achieved K/m, K̄/m ∈ [ − 0.1Ω2, 0.1Ω2] and γ ≃ 0.03Ω , where the power and wavelength of the trapping laser, and the gas pressure are set to be $P=400~{\rm mW}$, $\lambda=1064~{\rm nm}$, and $P\_{\rm gas}=1.5~{\rm mbar}$, respectively. Furthermore, the previous work has utilized silica nanoparticles, the radius, the polarizability, and the mass of which are set to be $r=105~{\rm nm}$, α = 3.48 × 10− 32 F ⋅ m2, and $m=1.07\times10^{-17}~{\rm kg}$, respectively. First, we explain a possible way to amplify the dipole-dipole couplings K/m, K̄/m compared to Ω2. To this end, we consider the ration between G/m and Ω2, which can be obtained from Eqs. ([eqappa8]) and ([eqappa20]) as follows: $$\frac{G}{m\Omega^2}=\frac{1}{\pi\varepsilon\_0}\left(\frac{k\_0^2w\_0^2}{2}-1\right)^2\alpha k\_0^3. \label{eqS43}$$ It is important that one can achieve K/m, K̄/m ∈ [ − 10Ω2, 10Ω2] by using the trapping laser with the half wavelength and making the radius of the particle 2.5 times larger than the one used in the previous experiment; we recall that the larger particle results in higher polarizability. Thus, we assume that the wavelength of the trapping laser is set to be $\lambda=532~{\rm nm}$, and the radius, the polarizability and the mass of the particles are set to be $r=260~{\rm nm}$, $\alpha=5.28\times10^{-31}~{\rm F\cdot m^2}$, and $m=1.62\times10^{-16}~{\rm kg}$, respectively, and the laser power is the same as in the previous experiment. We expect that our model captures qualitative features of the motion of the levitated nanoparticle even if its size is comparable to the wavelength of the trapping laser, because the dominant contribution to the forces which the particle feels still originates from the dipole moments in the particle. Meanwhile, to develop a quantitatively accurate theory, it should be necessary to modify the description to go beyond the dipolar approximation. Specifically, one should take into account the contributions from the higher-order moments (e.g., quadrupole moments). Second, we explain how to realize the friction coefficient γ = 5Ω. We note that, in the low-vacuum regime, the damping rate of a levitated nanoparticle is proportional to the square of the radius of the particle and the gas pressure. 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Phys. 82, 1767 ( 2010)](https://doi.org/10.1103/RevModPhys.82.1767) , @noop Classical electrodynamics ( Wiley, New York, 1999) , , , and , Optomechanics with levitated particles, @noop Rep. Prog. Phys. 83, 026401 ( 2020) Non-Hermitian physics of levitated nanoparticle array ===================================================== The ability to control levitated nanoparticles allows one to explore various fields of physics, including quantum optics, quantum metrology, and nonequilibrium physics. It has been recently demonstrated that the arrangement of two levitated nanoparticles naturally realizes the tunable nonreciprocal dipole-dipole interaction. Motivated by this development, we here propose and analyze an array of levitated nanoparticles as an ideal platform to study non-Hermitian physics in a highly controlled manner. We employ the non-Bloch band theory to determine the continuum bands of the proposed setup and investigate the non-Hermitian skin effect therein. In particular, we point out that the levitated nanoparticle array exhibits rich dynamical phases, including the dynamically unstable phase and the unconventional critical phase where the spectral singularity persists over a broad region of the controllable parameters. We also show that the long-range nature of the dipole-dipole interaction gives rise to the unique self-crossing point of the continuum band. [sec1]Introduction ================== A levitated nanoparticle is a laser trapped nanoscale dielectric particle smaller than wavelength of light . Recent experimental developments have allowed one to cool a levitated nanoparticle to ultracold temperatures and offered unique opportunities to study quantum mechanics of mesoscopic objects . Additionally, previous studies demonstrated the potential of a levitated nanoparticle to explore various fields of physics, such as nonequilibrium physics and quantum sensing . Remarkably, recent experimental studies have shown the possibility of realizing multi-particle setups . In particular, Ref. has reported a realization of an on-demand assembly of levitated nanoparticles, in which optical tweezers are used to trap and arrange the nanoparticles one by one. On another front, recent years have witnessed remarkable advances in our understandings of non-Hermitian systems, i.e., a class of nonequilibrium systems that can be effectively described by non-Hermitian operators . While non-Hermitian physics has been widely investigated in several fields of quantum science, such as ultracold atoms and photonics , its idea has also found numerous applications in classical systems realized in optics , mechanics , and electrical circuits . These previous studies uncovered rich non-Hermitian phenomena that have no counterparts to Hermitian systems. For instance, one-dimensional (1D) tight-binding systems with asymmetric hopping amplitudes exhibit the non-Hermitian skin effect , where the bulk eigenstates are localized at open boundaries, leading to the extreme boundary sensitivity of the eigenvalue. In this paper, we propose and analyze a 1D levitated nanoparticle array as an ideal platform to study previously unexplored regimes of non-Hermitian physics in a highly controlled manner. A prominent feature is that there exists the tunable nonreciprocal dipole-dipole interaction between particles, which is induced by the nonreciprocal interference originating from phase difference between the trapping lasers . The proposed system then realizes a 1D tight-binding model with arbitrarily tunable asymmetric hopping amplitudes that have possibly negative signs and long-range dependence. This high controllability allows one to explore the whole parameter region of non-Hermitian systems, thus opening the possibility to fully uncover the potential of non-Hermitian systems. The proposed setup should be contrasted to the previous non-Hermitian platforms where it remains challenging to realize long-range asymmetric and/or negative hopping amplitudes. To determine the continuum bands and the dynamical phase diagram of the levitated nanoparticle array, we invoke the non-Bloch band theory , a recently developed powerful tool to investigate models featuring the non-Hermitian skin effect. The non-Bloch band theory allows for calculating the asymptotic eigenvalues under open boundary conditions in the limit of a large system size. This makes contrast to the conventional Bloch band theory, where the band structure reproduces the eigenvalues under periodic boundary conditions. On the basis of this theoretical framework, we find that the levitated nanoparticle array exhibits rich dynamical phases, including the unconventional critical phase and the dynamically unstable phase. In the former, a remarkable feature is that the non-Hermitian degeneracy of the continuum bands known as the spectral singularity appears over a broad region of the parameters. The key ingredients of the latter are negative interparticle couplings, which were difficult to realize in the existing non-Hermitian platforms. Moreover, the proposed system can naturally realize the long-range hopping amplitudes originating from the dipole-dipole interaction. We show that this long-range nature leads to the unique self-crossing point of the continuum band, which corresponds to the singularity of the generalized Brillouin zone. [sec2]Levitated nanoparticle array ================================== ![[fig1]Schematic figure of the levitated nanoparticle array. The distance between the nearest-neighbor particles is d_0, and the mass of all the particles is m. All the trapping lasers have the power P and the wavelength \lambda. We set the phase of the nth trapping laser in the focal plane to be \phi+n\Delta\phi.](01.png)[fig1]Schematic figure of the levitated nanoparticle array. The distance between the nearest-neighbor particles is d0, and the mass of all the particles is m. All the trapping lasers have the power P and the wavelength λ. We set the phase of the nth trapping laser in the focal plane to be ϕ + nΔϕ. To be concrete, we consider a 1D array of the trapped levitated nanoparticles (Fig. [fig1]). The particles are equally spaced at the interval d0, and all the particles have the mass m. Let λ and P denote the wavelength and the power of all the trapping lasers, respectively. Furthermore, we assume that the motion of the particles along the plane perpendicular to the optical axis is frozen. We note that the distances between the particles are assumed to be much larger than a characteristic length scale in the collective behavior of particles . The interaction between the two particles arises due to the interference between the scattered electromagnetic field and the trapping laser. Since the scattered field acquires the phase kd0 during the propagation, the phase difference between the trapping lasers at the positions of the particles leads to the constructive and destructive interference depending on the propagation direction of the scattered field. It is this spatial asymmetry that renders the interparticle coupling nonreciprocal. We note that the effective open boundaries can be realized by arranging the tightly localized particles at the ends of the system (cf. Fig. [figappc1] in Appendix [secC]); the non-Hermitian skin effect then manifests itself as the large oscillation amplitudes close to the boundary regions. Due to the long-range nature of the dipole-dipole interaction, it is in general necessary to incorporate the couplings that reach up to Nth nearest neighbor particles. Altogether, the linearized equation of motion of the nth particle along the z axis is given by $$\begin{aligned} &&m\ddot{z}\_n+m\gamma\dot{z}\_n=-\left(m\Omega^2+2\sum\_{l=1}^NK\_l\right)z\_n \nonumber\\ &&+\sum\_{l=1}^N\left[\left(K\_l+\bar{K}\_l\right)z\_{n-l}+\left(K\_l-\bar{K}\_l\right)z\_{n+l}\right]. \label{eq1}\end{aligned}$$ Here, Ω is an intrinsic mechanical frequency of the particle proportional to $\sqrt{P}$, γ is a friction coefficient, and Kl and $\bar{K\_l}$ are the coupling strengths given by $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle K\_l=\frac{G}{lk\_0d\_0}\cos\left(lk\_0d\_0\right)\cos\left(l\Delta\phi\right), \vspace{3pt}\\ \displaystyle \bar{K}\_l=\frac{G}{lk\_0d\_0}\sin\left(lk\_0d\_0\right)\sin\left(l\Delta\phi\right), \end{array}\right. \label{eq2}\end{aligned}$$ where G has the dimension of a spring constant and is proportional to P, Δϕ is the optical phase difference between the nearest neighbor trapping lasers in the focal plane (Fig. [fig1]), and k0( = 2π/λ) is the wavenumber of the trapping laser. We explain the detailed derivation of Eq. ([eq1]) in Appendix [secA]. Importantly, the optical phase difference gives rise to the nonreciprocal couplings due to the nonzero K̄l. Thus, our setup is distinct from the array of levitated nanoparticles proposed in Ref. , which has investigated a non-Hermitian transport phenomenon with reciprocal couplings. Furthermore, one can infer from Eq. ([eq2]) that the couplings are long-range because the dipole-dipole interaction is proportional to the inverse of the distance between the particles. We note that the model can be mathematically mapped to a tight-binding model with gain and loss via a similarity transformation . The continuum bands of non-Hermitian tight-binding models can be obtained by invoking the non-Bloch band theory (cf. Appendix [secB]), which reproduces the eigenvalues under open boundary conditions. Specifically, the continuum bands are calculated from the generalized Brillouin zone spanned by β ≡ eik for the complex Bloch wavenumber k. We here apply the non-Bloch band theory to the levitated nanoparticle array; throughout this paper, we assume ∣KN∣ ≠ ∣K̄N∣. By substituting zn = ψneiωt to Eq. ([eq1]), the real-space eigenequation reads $$\begin{aligned} &&\frac{1}{m}\sum\_{l=1}^N\left[\left(K\_l-\bar{K}\_l\right)\psi\_{n+l}\left(K\_l+\bar{K}\_l\right)\psi\_{n-l}\right] \nonumber\\ &&+\left(\omega^2-i\gamma\omega-\Omega^2-\frac{2}{m}\sum\_{l=1}^NK\_l\right)\psi\_n=0. \label{eq3}\end{aligned}$$ Importantly, the ansatz of Eq. ([eq3]) can be taken as ψn = ∑j = 12N(βj)nϕ(j), where βj( = β) is the solution of the characteristic equation given by $$\begin{aligned} &&\frac{1}{m}\sum\_{l=1}^N\left[\left(K\_l-\bar{K}\_l\right)\beta^l+\left(K\_l+\bar{K}\_l\right)\beta^{-l}\right] \nonumber\\ &&+\left(\omega^2-i\gamma\omega-\Omega^2-\frac{2}{m}\sum\_{l=1}^NK\_l\right)=0. \label{eq5}\end{aligned}$$ We note that Eq. ([eq5]) is an algebraic equation for β of 2Nth degrees. The main result of the non-Bloch band theory is that the condition for the generalized Brillouin zone is obtained from the 2N solutions as follows: ∣βN∣ = ∣βN + 1∣ with ∣β1∣ ≤ ⋯ ≤ ∣β2N∣. The trajectories of βN and βN + 1 form the generalized Brillouin zone on the complex plane, which reveals the essential features of non-Hermitian systems (see, e.g., Refs. ). Then, we can calculate the continuum bands by combining Eq. ([eq5]) with the generalized Brillouin zone. [sec3]Dynamical phase diagram ============================= ![[fig2] Dynamical phase diagram and continuum bands of the levitated nanoparticle array. (a) Dynamical phase diagram exhibiting the underdamped, critical, overdamped, and dynamically unstable phases shown in the blue, green, red, and gray-shaded regions, respectively. The spectral singularity (SS) appears in the green-shaded region and on the green lines. \left\|K_1\right\|=\left\|\bar{K}_1\right\| is satisfied on the black dashed lines. We set the parameters to be \Omega=1 and \gamma=5. (b–g) Evolutions of the continuum bands along the arrows in panel (a). The magenta and cyan express \omega_+ and \omega_-, respectively. The numerical values in each panel specify \left(K_1,\bar{K}_1\right).](02.png)[fig2] Dynamical phase diagram and continuum bands of the levitated nanoparticle array. (a) Dynamical phase diagram exhibiting the underdamped, critical, overdamped, and dynamically unstable phases shown in the blue, green, red, and gray-shaded regions, respectively. The spectral singularity (SS) appears in the green-shaded region and on the green lines. ∣K1∣ = ∣K̄1∣ is satisfied on the black dashed lines. We set the parameters to be Ω = 1 and γ = 5. (b–g) Evolutions of the continuum bands along the arrows in panel (a). The magenta and cyan express ω+ and ω−, respectively. The numerical values in each panel specify (K1, K̄1). We start our analysis from the levitated nanoparticle array with the nearest-neighbor interaction, which corresponds to N = 1 in Eq. ([eq1]); in the following, we assume γ > 2Ω for the sake of concreteness. From Eq. ([eq6]), the generalized Brillouin zone can be given by the circle with the radius $r=\sqrt{\left\|\left(K\_1+\bar{K}\_1\right)/\left(K\_1-\bar{K}\_1\right)\right\|}$. The analytical form of the continuum bands reads $$\omega\_\pm=\frac{i}{2}\gamma\pm\sqrt{\Omega^2+\frac{2}{m}\left(K\_1-\sqrt{K\_1^2-\bar{K}\_1^2}\cos\theta\right)-\frac{\gamma^2}{4}}, \label{eq7}$$ where θ is a real number. Since each eigenmode contributes to the dynamics through the factor $e^{-{\rm Im}\left(\omega\_\pm\right)}e^{i{\rm Re}\left(\omega\_\pm\right)}$, we can show the dynamical phase diagram depending on K1/m and K̄1/m [Fig. [fig2](a)]. We here emphasize that the continuum bands discussed here can have direct experimental relevance. Indeed, the theoretical calculation of the eigenvalues in the coupled two levitated nanoparticles has explained well experimentally observed crossing/avoided crossing of the eigenspactra and the appearance of an exceptional point . Figures [fig2](b)–[fig2](d) and [fig2](e)–[fig2](g) plot the evolutions of the continuum bands along the black and white arrows indicated in Fig. [fig2](a), respectively. In the blue-shaded regions of Fig. [fig2](a), all the particles oscillate with the attenuation because ${\rm Re}\left(\omega\_\pm\right)\neq0$ and ${\rm Im}\left(\omega\_\pm\right)>0$ [Fig. [fig2](b)]. In contrast, in the red-shaded regions, their motion monotonically vanishes without oscillations because ${\rm Re}\left(\omega\_\pm\right)=0$ and ${\rm Im}\left(\omega\_\pm\right)>0$ [Fig. [fig2](d)]. For these reasons, we term the former (latter) the dynamical phase as the underdamped (overdamped) phase. Remarkably, we find the broad green-shaded region where the two branches coalesce at ${\rm Re}\left(\omega\_\pm\right)=0$ [Fig. [fig2](c)]; this degeneracy is called the spectral singularity. There, we find the crossover behavior where the overdamped behavior eventually sets in after the initial underdamped oscillations; we shall term this intermediate regime as the critical phase. One of its key characteristics is the presence of the particles near the boundaries which are strongly driven by the adjacent trapping lasers inducing the nonreciprocal couplings. It is worthwhile to mention that, with γ < 2Ω, the critical phase appears on the parameter region where the sign of K/m becomes negative [cf. Fig. [figappc2](a) in Appendix [secC]]. The transient phenomenon discussed here is supported by the non-Hermitian skin effect, since the critical phase disappears under periodic boundary conditions, as discussed in Appendix [secD]. As shown in Fig. [fig2](f), the spectral singularity also appears along the green vertical lines in Fig. [fig2](a). However, one would need the fine-tuning of the parameters in this case as indicated by Figs. [fig2](e)–[fig2](g), where the two continuum bands are recombined across the green line. Additionally, we also find the dynamically unstable phase as indicated by the gray-shaded region in Fig. [fig2](a). There, the driving forces from the adjacent trapping lasers give rise to the dynamical instability of the particles where the oscillation amplitudes diverge in the long-time limit, because negative hopping amplitudes cause the force that increasingly keeps away the particles from their equilibrium positions. We expect that nonlinear effects will eventually play a crucial role in this phase, since the amplification is eventually balanced by nonlinear suppression. The dynamically unstable phase discussed here is difficult to realize in previous non-Hermitian systems due to the lack of the ability to implement negative hopping amplitudes. It is worthwhile to mention that, in finite-size systems, the phase boundary between the overdamped and dynamically unstable phases can be slightly modified, as discussed in Appendix [secC]. [sec4]Nonreciprocal long-range interaction ========================================== ![[fig3]Continuum bands and generalized Brillouin zones of the levitated nanoparticle array at different \Delta\phi. (a–c) The continuum bands with the positive branch of the square root, and (d–f) the corresponding generalized Brillouin zones are shown. The red (blue) curves indicate the results for N=2 (N=1). In panels (d–f), the black dashed curve expresses the conventional Brillouin zone spanned by \beta\equiv e^{ik}~\left(k\in{\mathbb R}\right). The system parameters are set to be \lambda=1.064\times10^{-6}~{\rm m},d_0=10^{-5}~{\rm m},\Omega=10^5~{\rm s^{-1}}, and G/\left(mk_0d_0\right)=10^8~{\rm s^{-2}}.](03.png)[fig3]Continuum bands and generalized Brillouin zones of the levitated nanoparticle array at different Δϕ. (a–c) The continuum bands with the positive branch of the square root, and (d–f) the corresponding generalized Brillouin zones are shown. The red (blue) curves indicate the results for N = 2 (N = 1). In panels (d–f), the black dashed curve expresses the conventional Brillouin zone spanned by $\beta\equiv e^{ik}~\left(k\in{\mathbb R}\right)$. The system parameters are set to be $\lambda=1.064\times10^{-6}~{\rm m},d\_0=10^{-5}~{\rm m},\Omega=10^5~{\rm s^{-1}}$, and $G/\left(mk\_0d\_0\right)=10^8~{\rm s^{-2}}$. We next investigate how the long-range nature of the couplings can affect the continuum band and the corresponding generalized Brillouin zone; in the following, we neglect the friction for the sake of simplicity. We assume that the interaction reaches up to the next-nearest-neighbor particles, which corresponds to N = 2 in Eq. ([eq1]). In Fig. [fig3], we plot the continuum bands with the positive branch of the square root and the corresponding generalized Brillouin zone at different Δϕ. The black dashed curves in Figs. [fig3](d)–(f) indicate the conventional Brillouin zone formed by $\beta\equiv e^{ik}~\left(k\in\mathbb R\right)$. We note that the parameters considered in these calculations satisfy K1/m, K̄1/m, K2/m, K̄2/m ≪ Ω2 and have been experimentally realized in Ref. in the case of two particles. One can see from Figs. [fig3](d) and [fig3](f) that the generalized Brillouin zone with N = 2 forms a skewed closed curve with the cusps, at which it becomes indifferentiable, while the generalized Brillouin zone with N = 1 is merely a circle. Importantly, the cusps correspond to the self-crossing points of the continuum band [Figs. [fig3](a) and [fig3](c)] . Thus, the long-range nature of the nonreciprocal couplings can lead to these unconventional band structures. Meanwhile, at Δϕ = π/2, the generalized Brillouin zone becomes the unit circle independently of N [Fig. [fig3](e)], and there are no self-crossing points [Fig. [fig3](b)], where the non-Hermitian skin effect disappears. It is noteworthy that, in the case of Fig. [fig2](c), nonorthogonality of the eigenstates with the eigenvalues around the self-crossing points is stronger than that of the other eigenstates away from the self-crossing point. This is because the overlap of the left and right eigenvectors becomes minimum at the self-crossing point. Thereby, such eigenstates exhibit a striking response against perturbations , which indicates that the excitation modes around the self-crossing point could be utilized for a highly sensitive sensor. [sec5]Summary and Discussion ============================ In summary, we propose and analyze the levitated nanoparticle array as an ideal platform to study new realms of non-Hermitian physics in a highly controlled manner. We show that the system exhibits the unconventional critical phase, where the spectral singularity originating from the non-Hermitian skin effect persists over a broad parameter region. We also point out that the tunable dipole-dipole interaction allows for extremely nonreciprocal hopping amplitudes with possibly negative signs, which result in the dynamical instability. We finally reveal that the long-range nature of the couplings further enriches the non-Hermitian band structures, leading to the cusps of the generalized Brillouin zone and the self-crossing point of the continuum band. We expect that the levitated nanoparticle array will be an ideal platform to explore rich phenomena induced by nonreciprocal long-range hopping amplitudes thanks to its high tunability. For example, a nontrivial topological phase transition mediated by a topological semimetal phase with exceptional points has been proposed . Additionally, the number of branches from the self-crossing point of the continuum band increases as a coupling distance between two particles . Furthermore, it has been proposed that nonreciprocal long-range couplings give rise to bulk eigenstates which exhibit the crossover from a constant localization length to a system-size-dependent localization length . This previous work has also indicated that, since long-range couplings suppress the non-Hermitian skin effect, the scaling of entanglement entropy can change from an area law to a subextensive law. The continuum bands of the levitated nanoparticle array can be experimentally studied by measuring the power spectral density. It is thus feasible to directly observe the spectral singularity and the self-crossing points. Meanwhile, to access the strong coupling regime with $K/m\Omega^2={\rm O}\left(1\mathchar`-10\right)$ considered in Fig. [fig2], it is necessary to realize a stronger dipole-dipole interaction than the one already realized in Ref. . We expect that this should be made possible by increasing size of the particles and using lasers with a shorter wavelength. Also, we note that the friction coefficient can be controlled by changing the pressure of the surrounding gas, while thermal noise can be suppressed by keeping its temperature sufficiently low. We discuss in detail the experimental feasibility in Appendix [secE]. It is interesting to further explore various aspects and potentials of a levitated nanoparticle array, since this unique platform opens a new avenue of investigating sensing, nonlinearity, and nonequilibrium quantum physics. First, the critical phase can be potentially applicable for enhanced sensing because the strong nonorthogonality associated with the spectral singularity which one can observe in the array of a few particles (cf. see Appendix [secC]) is known to trigger the singular sensitivity to perturbation . Second, it is crucial to reveal the competition between nonreciprocity and nonlinearity , where a levitated nanoparticle array is expected to exhibit rich collective phenomena such as synchronization. Third, an extension of the present analysis to quantum regimes is an intriguing open problem. For instance, it merits further study to understand how the nonreciprocal couplings affect the entanglement between particles , how the dynamical instability of the levitated nanoparticle array can be utilized to squeeze the mechanical mode of a levitated nanoparticle , and whether or not the nonorthogonality is helpful for quantum metrology . K.Y. was supported by JSPS KAKENHI through Grant No. JP21J01409. Y.A. acknowledges support from the Japan Society for the Promotion of Science through Grant No. JP19K23424 and from JST FOREST Program (Grant No. JPMJFR222U, Japan). [secA]Equation of motion ======================== ![[figappa]Schematic figure of a single levitated nanoparticle, coupled two levitated nanoparticles, and an array of leviated nanoparticles. In panels (a), (b), and (c), the mass of the particle is m, and the trapping lasers have the power P and the wavelength \lambda. In panels (b) and (c), the distance between the particles is d_0. In panel (c), we set the phase of the nth trapping laser in the focal plane to be \phi+n\Delta\phi.](appa.png)[figappa]Schematic figure of a single levitated nanoparticle, coupled two levitated nanoparticles, and an array of leviated nanoparticles. In panels (a), (b), and (c), the mass of the particle is m, and the trapping lasers have the power P and the wavelength λ. In panels (b) and (c), the distance between the particles is d0. In panel (c), we set the phase of the nth trapping laser in the focal plane to be ϕ + nΔϕ. We derive the equation of motion of a single levitated nanoparticle, coupled two levitated nanoparticles, and an array of levitated nanoparticles as shown in Figs. [figappa](a), [figappa](b), and [figappa](c), respectively. We assume that the motion of the particles along the plane perpendicular to the optical axis is frozen. [secA-1]Single levitated nanoparticle ------------------------------------- We first focus on the motion of a single levitated nanoparticle along the z axis [Fig. [figappa](a)]. Let P and λ denote the power and the wavelength of the trapping laser, respectively. In this system, the subwavelength-sized particle with the refractive index n is surrounded by a medium with the refractive index n′, and it can be regarded as a point dipole with the electric dipole moment given by $${\bm p}\left({\bm r},t\right)=\alpha{\bm E}\left({\bm r},t\right). \label{eqappa1}$$ Here, ${\bm E}\left({\bm r},t\right)$ expresses the electric field of the trapping laser, and α is the polarizability of the particle. The latter can explicitly be written as $$\alpha=3\varepsilon\_0n^{\prime2}V\frac{n\_r^2-1}{n\_r^2+2}, \label{eqappa2}$$ where c is the speed of light in vacuum, V is the volume of the particle, and nr ≡ n/n′ is the relative refractive index of the particle. The electric field then acts on the point dipole, and the particle feels the gradient force given by $$\begin{aligned} {\bm F}\_{\rm grad}\left({\bm r}\right)&=&\left<\frac{1}{2}{\rm Re}\left[\left(\alpha{\bm E}^\ast\left({\bm r},t\right)\cdot{\bm\nabla}\right){\bm E}\left({\bm r},t\right)\right]\right>\_T \nonumber\\ &=&\frac{\alpha}{2c\varepsilon\_0n^\prime}{\bm\nabla}I\left({\bm r}\right), \label{eqappa3}\end{aligned}$$ where $\left<\cdots\right>\_T$ means time average, and $I\left({\bm r}\right)$ is the spatial profile of the intensity of the trapping laser. We note that the trapping laser is described by the Gaussian beam, of which the electric field is given by $$\begin{aligned} {\bm E}\left({\bm r},t\right)&=&{\bm E}\_0\frac{w\_0}{w\left(z\right)} \nonumber\\ &&\times\exp\left[-\frac{x^2+y^2}{\left\{w\left(z\right)\right\}^2}+ik\frac{x^2+y^2}{2R\left(z\right)}-i\zeta\left(z\right)\right] \nonumber\\ &&\times e^{i\left(k\_0z-\omega t\right)}, \label{eqappa4}\end{aligned}$$ where $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle w\left(z\right)=w\_0\sqrt{1+\left(\frac{z}{z\_0}\right)^2}, \vspace{5pt}\\ \displaystyle R\left(z\right)=z\left[1+\left(\frac{z\_0}{z}\right)^2\right], \vspace{5pt}\\ \displaystyle \zeta\left(z\right)=\tan^{-1}\left(\frac{z}{z\_0}\right). \end{array}\right. \label{eqappa5}\end{aligned}$$ Here, k0( = 2π/λ) is the wavenumber of the trapping laser, w0 is the beam waist, and z0 = πw02/λ is the Rayleigh length. The combination of Eqs. ([eqappa3]) and ([eqappa4]) leads to the z component of the gradient force given by $$F\_{{\rm grad},z}\left(z\right)=-\frac{P\alpha}{2\pi c\varepsilon\_0n^\prime w\_0^2z\_0^2}\frac{z}{\left[1+\left(z/z\_0\right)^2\right]^2}. \label{eqappa6}$$ In the vicinity of the focal plane of the trapping laser, the linearized equation of the motion of the levitated nanoparticle is obtained as $$m\ddot{z}+m\gamma\dot{z}=-m\Omega^2z, \label{eqappa7}$$ where $$\Omega^2=\frac{P\alpha}{2\pi c\varepsilon\_0n^\prime mw\_0^2z\_0^2}, \label{eqappa8}$$ and γ is a friction coefficient. We remark that the linearized motion of the particle exhibits the harmonic oscillation because the gradient force plays a role as the restoring force. [secA-2]Coupled two levitated nanoparticles ------------------------------------------- We next consider the equation of motion of the coupled two levitated nanoparticles [Fig. [figappa](b)]. We assume that the polarization direction of the particles points to the y axis. Then, in addition to the gradient force, both particles feel the interaction caused by the dipole radiation, which is called the optical binding force . First of all, we shall explain the mechanism of the optical binding force based on Ref. . Importantly, there exist two contributions to the optical binding force. The first contribution results from the combination of the electric field scattered from one particle to another particle and the electric dipole induced by the trapping laser. The second contribution originates from the acting of the trapping laser on the electric dipole induced by the scattered field. Specifically, for particle 1, the sum of these contributions leads to the form of the optical binding force as follows: $$\begin{aligned} &&{\bm F}\_{\rm bind}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2\right) \nonumber\\ &&=\left<\frac{1}{2}{\rm Re}\left[\left(\alpha{\bm E}^\ast\left({\bm r}\_1,t\right)\cdot{\bm\nabla}\_{{\bm r}\_1}\right){\bm E}\_{\rm sca}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2,t\right)\right]\right. \nonumber\\ &&\left.+\frac{1}{2}{\rm Re}\left[\left(\alpha{\bm E}\_{\rm sca}^{2\rightarrow1\ast}\left({\bm r}\_1,{\bm r}\_2,t\right)\cdot{\bm\nabla}\_{{\bm r}\_1}\right){\bm E}\left({\bm r}\_1,t\right)\right]\right>\_T. \nonumber\\ \label{eqappa9}\end{aligned}$$ Here, ${\bm E}\_{\rm sca}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2,t\right)$ is the electric field scattered from particle 2 to particle 1, and its form is written as $${\bm E}\_{\rm sca}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2,t\right)=G\left({\bm r}\_1-{\bm r}\_2\right)\alpha{\bm E}\left({\bm r}\_2,t\right), \label{eqappa10}$$ where $G\left({\bm r}\right)$ called the Green’s tensor is the electric field propagator between the two dipoles , and it is given by $$G\left({\bm r}\right)=\frac{e^{ik\_0r}}{4\pi\varepsilon\_0}\left[\frac{3{\bm r}\otimes{\bm r}-r^2}{r^5}\left(1-ik\_0r\right)+k\_0^2\frac{r^2-{\bm r}\otimes{\bm r}}{r^3}\right]. \label{eqappa11}$$ By using $$\begin{aligned} &&{\rm Re}\left[\left({\bm\nabla}\_{{\bm r}\_1}\times{\bm E}\left({\bm r}\_1,t\right)\right)\times{\bm E}\_{\rm sca}^{2\rightarrow1\ast}\left({\bm r}\_1,{\bm r}\_2,t\right)\right] \nonumber\\ &&-{\rm Re}\left[{\bm E}\_{\rm sca}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2,t\right)\times\left({\bm\nabla}\_{{\bm r}\_1}\times{\bm E}^\ast\left({\bm r}\_1,{\bm r}\_2,t\right)\right)\right] \nonumber\\ &&={\rm Re}\left[\left({\bm E}^\ast\left({\bm r}\_1,t\right)\cdot{\bm\nabla}\_{{\bm r}\_1}\right){\bm E}\_{\rm sca}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2,t\right)\right] \nonumber\\ &&+{\rm Re}\left[\left({\bm E}\_{\rm sca}^{2\rightarrow1\ast}\left({\bm r}\_1,{\bm r}\_2,t\right)\cdot{\bm\nabla}\_{{\bm r}\_1}\right){\bm E}\left({\bm r}\_1,t\right)\right] \nonumber\\ &&-{\rm Re}\left[{\bm\nabla}\_{{\bm r}\_1}\left({\bm E}^\ast\left({\bm r}\_1,t\right)\cdot{\bm E}\_{\rm sca}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2,t\right)\right)\right] \label{eqappa12}\end{aligned}$$ and ${\bm\nabla}\_{{\bm r}\_1}\times{\bm E}\left({\bm r}\_1,t\right)={\bm0}$, we can rewrite Eq. ([eqappa9]) to a brief form given by $$\begin{aligned} &&{\bm F}\_{\rm bind}^{2\rightarrow1}\left({\bm r}\_1,{\bm r}\_2\right) \nonumber\\ &&=\left<\frac{1}{2}{\bm\nabla}\_{{\bm r}\_1}{\rm Re}\left[\alpha{\bm E}^\ast\left({\bm r}\_1,t\right)G\left({\bm r}\_1-{\bm r}\_2\right)\alpha{\bm E}\left({\bm r}\_2,t\right)\right]\right>\_T. \nonumber\\ \label{eqappa13}\end{aligned}$$ In the following, we calculate the z component of the optical binding force. In the far-field regime with k0d0 ≫ 1, the dominant contribution from the Green’s tensor to the optical binding force is the term proportional to 1/r. Furthermore, in the vicinity of the focal plane, the z dependence of the electric field is approximated as $${\bm E}\left({\bm r}\_j,t\right)\approx{\bm E}\_0\exp\left[i\phi\_j+i\left(k\_0-\frac{1}{z\_0}\right)z\_j-i\omega t\right] \label{eqappa14}$$ for j = 1, 2, where ϕj expresses the optical phase at the focal plane. Substituting Eqs. ([eqappa11]) and ([eqappa14]) into Eq. ([eqappa13]), we can get the optical binding force along the z axis as follows: $$\begin{aligned} &&F\_{{\rm bind},z}^{2\rightarrow1}\left(z\_1,z\_2\right)\approx\frac{P\alpha^2k\_0^3\left(k\_0-1/z\_0\right)}{2\pi^2c\varepsilon\_0^2n^\prime w\_0^2} \nonumber\\ &&\times\sin\left[k\_0d\_0-\Delta\phi-\left(k\_0-\frac{1}{z\_0}\right)\left(z\_1-z\_2\right)\right], \nonumber\\ \label{eqappa15}\end{aligned}$$ where Δϕ ≡ ϕ1 − ϕ2. We can similarly obtain the optical binding force for particle 2 as follows: $$\begin{aligned} &&{\bm F}\_{\rm bind}^{1\rightarrow2}\left({\bm r}\_1,{\bm r}\_2\right) \nonumber\\ &&=\left<\frac{1}{2}{\bm\nabla}\_{{\bm r}\_2}{\rm Re}\left[\alpha{\bm E}^\ast\left({\bm r}\_2,t\right)G\left({\bm r}\_2-{\bm r}\_1\right)\alpha{\bm E}\left({\bm r}\_1,t\right)\right]\right>\_T. \nonumber\\ \label{eqappa16}\end{aligned}$$ Therefore, the explicit form of the optical binding force along the z axis is derived as $$\begin{aligned} &&F\_{{\rm bind},z}^{1\rightarrow2}\left(z\_1,z\_2\right)\approx\frac{P\alpha^2k\_0^3\left(k\_0-1/z\_0\right)}{2\pi^2c\varepsilon\_0^2n^\prime w\_0^2} \nonumber\\ &&\times\sin\left[k\_0d\_0+\Delta\phi+\left(k\_0-\frac{1}{z\_0}\right)\left(z\_1-z\_2\right)\right]. \nonumber\\ \label{eqappa17}\end{aligned}$$ Remarkably, the interaction between these two particles becomes nonreciprocal because $F\_{{\rm bind},z}^{2\rightarrow1}\left(z\_1,z\_2\right)\neq-F\_{{\rm bind},z}^{1\rightarrow2}\left(z\_2,z\_1\right)$. The key ingredient of this nonreciprocity is the interference between the trapping laser and the scattered field. Let Φj (j = 1, 2) denote the optical phase of the trapping laser at the position of the particle. The interference depends on the local phase difference ΔΦ ≡ Φ1 − Φ2 and the phase accumulation kd0 which the scattered field acquires during the propagation. Specifically, while the contribution of the interference is kd0 − ΔΦ within the propagation of the scattered field from particle 1 to particle 2, it becomes kd0 + ΔΦ in the opposite case. As a result, the interaction originating from the interference becomes spatially asymmetric. We can now obtain the linearized equation of motion of the coupled levitated nanoparticles as follows: $$\begin{aligned} \left\{ \begin{array}{l} m\ddot{z}\_1+m\gamma\dot{z}\_1=-\left(m\Omega^2+K+\bar{K}\right)z\_1+\left(K+\bar{K}\right)z\_2, \vspace{3pt}\\ m\ddot{z}\_2+m\gamma\dot{z}\_2=-\left(m\Omega^2+K-\bar{K}\right)z\_2+\left(K-\bar{K}\right)z\_1, \end{array}\right. \nonumber\\ \label{eqappa18}\end{aligned}$$ where Ω is the intrinsic mechanical frequency given by Eq. ([eqappa1]). The coupling constants are given by $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle K=\frac{G}{k\_0d\_0}\cos\left(k\_0d\_0\right)\cos\left(\Delta\phi\right), \vspace{3pt}\\ \displaystyle \bar{K}=\frac{G}{k\_0d\_0}\sin\left(k\_0d\_0\right)\sin\left(\Delta\phi\right), \end{array}\right. \label{eqappa19}\end{aligned}$$ and $$G=\frac{P\alpha^2k\_0^3\left(k\_0-1/z\_0\right)^2}{2\pi^2c\varepsilon\_0^2n^\prime w\_0^2}. \label{eqappa20}$$ [secA-3]Levitated nanoparticle array ------------------------------------ We finally study the arrangement of the multiple levitated nanoparticles at equal interval d0 [Fig. [figappa](c)]. In this system, the dipole-dipole interaction among the several particles arises from the multiple scattering of the trapping lasers. Nevertheless, the dominant contribution to the dynamics of the system comes from the interaction between two particles. Thus, we neglect higher-order scattering processes. This corresponds to approximating the optical binding force up to ${\cal O}\left(\left\|{\bm p}\right\|^2\right)$. We shall explain how one can derive the equation of motion of the levitated nanoparticle array. We set the optical phase of the nth trapping laser in the focal plane to be ϕ + nΔϕ. For the nth and n + lth particles, the phase difference between the trapping lasers is lΔϕ, and the distance between the particles is ld0. Hence, the interaction between these particles can be obtained by the same procedure as explained above. Due to the long-range nature of the dipole-dipole interaction, it is necessary to incorporate the couplings that reach up to Nth neighbor particles. Then, the equation of motion of the system is written as $$\begin{aligned} m\ddot{z}\_n+m\gamma\dot{z}\_n&=&-\left(m\Omega^2+2\sum\_{l=1}^NK\_l\right)z\_n \nonumber\\ &&+\sum\_{l=1}^N\left[\left(K\_l+\bar{K}\_l\right)z\_{n-l}+\left(K\_l-\bar{K}\_l\right)z\_{n+l}\right], \nonumber\\ \label{eqappa21}\end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle K\_l=\frac{G}{lk\_0d\_0}\cos\left(lk\_0d\_0\right)\cos\left(l\Delta\phi\right), \vspace{3pt}\\ \displaystyle \bar{K}\_l=\frac{G}{lk\_0d\_0}\sin\left(lk\_0d\_0\right)\sin\left(l\Delta\phi\right), \end{array}\right. \label{eqappa22}\end{aligned}$$ where the constant G is given in Eq. ([eqappa20]). [secB]Non-Bloch band theory =========================== We describe the physical meaning of the non-Bloch band theory for 1D non-Hermitian tight-binding models. First of all, we show that the condition for the generalized Brillouin zone can be interpreted as the condition that the “plane waves” form the standing wave, by using the specific model. We next prove that the recombination of the “plane waves” forming the standing wave occurs at the cusps of the generalized Brillouin zone. [secB-1]Condition for the generalized Brillouin zone ---------------------------------------------------- We consider the 1D non-Hermitian tight-binding model with asymmetric hopping amplitudes, the Hamiltonian of which reads $$\begin{aligned} H&=&\sum\_n\left(t\_{+,2}c\_{n+2}^\dag c\_n+t\_{+,1}c\_{n+1}^\dag c\_n\right. \nonumber\\ &&\left.+t\_{-,1}c\_n^\dag c\_{n+1}+t\_{-,2}c\_n^\dag c\_{n+2}\right), \label{eqappb1}\end{aligned}$$ where all the parameter are real numbers. The Schrödinger equation can be then written in the form of the real-space eigenequation given by t+ , 2ψn − 2 + t+ , 1ψn − 1 + t− , 1ψn + 1 + t− , 2ψn + 2 = Eψn, where ψn means the amplitude of the state at the site n. A general difference theory allows us to take the form of the linear combination ψn = ∑j(βj)nϕ(j), which corresponds to the plane-wave expansion, as the ansatz of Eq. ([eqappa2]), due to the spatial periodicity. Here, βj( = β) is the solution of the characteristic equation written as t+ , 2β− 2 + t+ , 1β− 1 + t− , 1β + t− , 2β2 = E, which is a quadratic equation for β. We note that it is necessary to combine the open boundary conditions obtained from Eqs. ([eqappb3]) and Eq. ([eqappb4]) to get the asymptotic set of the energy eigenvalues in the thermodynamic limit. Although the calculation is cumbersome for a large system size, the non-Bloch band theory allows us to avoid the procedure of calculating the continuum band . It has been shown that the values of β are restricted to lie on the closed curve so that the wavefunction satisfies the open boundary conditions. The closed curve, called the generalized Brillouin zone, is then formed by β = eik for the Bloch wavenumber k. It is important that, for the four solutions of Eq. ([eqappb4]), the condition for the generalized Brillouin zone reads ∣β2∣ = ∣β3∣ with ∣β1∣ ≤ ∣β2∣ ≤ ∣β3∣ ≤ ∣β4∣. We note that the trajectories of β2 and β3 satisfying Eq. ([eqappb5]) form the generalized Brillouin zone. Finally, the combination of Eqs. ([eqappb4]) and ([eqappb5]) gives the continuum band of the system. We exemplify the continuum band and the generalized Brillouin zone with the specific parameters as shown in Figs. [figappb](a) and [figappb](b). The Bloch wavenumber obtained by Eq. ([eqappb5]) becomes complex numbers, which indicates that the bulk eigenstates are localized at boundaries of the system due to the non-Hermitian skin effect. Meanwhile, the generalized Brillouin zone forms a unit circle, when the system becomes a Hermitian system: t+ , 1 = t− , 1 and t+ , 2 = t− , 2. This means that the Bloch wavenumber takes real values in Hermitian tight-binding systems, which is consistent with the result of the conventional Bloch band theory. It is remarkable that the condition for the generalized Brillouin zone can be interpreted in the viewpoint of the non-Hermitian skin effect. The physical meaning of Eq. ([eqappb5]) is that the localization lengths of the “plane waves” corresponding to β2 and β3 match each other, which leads to the formation of the standing wave by the interference of the “plane waves”. Furthermore, it is intriguing that the asymptotic energy eigenvalues of the open chain in the thermodynamic limit do not depend on any boundary conditions, since Eqs. ([eqappb4]) and ([eqappb5]) are independent of boundary conditions of the open chain. ![[figappb]Continuum band and trajectories of the solutions of the characteristic equation of the model ([eqappb1]). (a) Continuum band. (b) Generalized Brillouin zone (c) Sets of the solutions of the characteristic equation satisfying Eq. ([eqappb7]). We set the parameters to be t_{+,2}=0.9,t_{+,1}=0.3,t_{-,1}=0.7, and t_{-,2}=0.1.](appb.png)[figappb]Continuum band and trajectories of the solutions of the characteristic equation of the model ([eqappb1]). (a) Continuum band. (b) Generalized Brillouin zone (c) Sets of the solutions of the characteristic equation satisfying Eq. ([eqappb7]). We set the parameters to be t+ , 2 = 0.9, t+ , 1 = 0.3, t− , 1 = 0.7, and t− , 2 = 0.1. [secB-2]Cusps of the generalized Brillouin zone ----------------------------------------------- We nest explain the appearance mechanism of the cusp of the generalized Brillouin zone, at which it becomes indifferentiable. To this end, we investigate what happens if we impose ∣βi∣ = ∣βj∣ to the system, for some i and j among the four solutions of Eq. ([eqappb4]). We then show several sets of βi and βj satisfying Eq. ([eqappb7]) in Fig. [figappb](c). We note that the trajectory satisfying Eq. ([eqappb5]) among the sets is equivalent to the generalized Brillouin zone. Compared with Figs. [figappb](b) and (c), one can see that the cusps appear, when the three of the four solutions share the same absolute values. Namely, for ∣β1∣ < ∣β2∣ = ∣β3∣ < ∣β4∣, ∣β1∣ approaches ∣β2∣( = ∣β3∣) as we go around the generalized Brillouin zone, and the behavior of the solutions satisfying Eq. ([eqappb5]) eventually changes when ∣β1∣ = ∣β2∣ = ∣β3∣. Thus, the recombination of the “plane waves” forming the standing wave occurs at the cusps of the generalized Brillouin zone and the corresponding self-crossing points of the continuum band. [secC]Array of finite levitated nanoparticles ============================================= []![[figappc1]Schematic figure of the array of the L levitated nanoparticles. The distance between the nearest-neighbor particles is d_0, and the mass of all the particles is m. The trapping lasers have the power P and the wavelength \lambda. We set the phase of the nth trapping laser in the focal plane to be \phi+n\Delta\phi. At both boundaries of the system, we arrange the two levitated nanoparticles frozen in the motion in all the directions. The trapping lasers at the left and right boundaries have the optical phase \phi-\Delta\phi and \phi+L\Delta\phi in the focal plane, respectively.](appc1.png "fig:") We investigate the array of a finite number of levitated nanoparticles as shown in Fig. [figappc1]. We assume that this system includes only the interaction between the nearest-neighbor particles. Furthermore, at both boundaries of the system, we arrange the deeply trapped levitated nanoparticles, of which the motion in all the directions is frozen, and we set the optical phase of the trapping laser in the focal plane at the left and right boundaries to be ϕ − Δϕ and ϕ + (L + 1)Δϕ, respectively. In this case, these particles are coupled with the system so that they impose the fixed end boundary conditions on the system. [secC-1]Eigenvalue ------------------ We first show a way to calculate the eigenvalues of the system described by $$\begin{aligned} m\ddot{z}\_n+m\gamma\dot{z}\_n&=&-\left(m\Omega^2+2K\right)z\_n \nonumber\\ &&+\left(K+\bar{K}\right)z\_{n-l}+\left(K-\bar{K}\right)z\_{n+l} \nonumber\\ \label{eqappc1}\end{aligned}$$ with the fixed boundary condition given by z0 = zL + 1 = 0. In this equation, the coupling constants K and K̄ are given by Eq. ([eqappa22]). In the following, we suppose ∣K∣ ≠ ∣K̄∣. By assuming zn = ψneiωt, Eq. ([eqappc1]) is rewritten into $$\begin{aligned} &&\frac{1}{m}\left[\left(K-\bar{K}\right)\psi\_{n+l}\left(K+\bar{K}\right)\psi\_{n-l}\right] \nonumber\\ &&+\left(\omega^2-i\gamma\omega-\Omega^2-\frac{2}{m}K\right)\psi\_n=0. \label{eqappc2}\end{aligned}$$ From a general theory of a difference equation, we can take ψn = ∑j = 12(βj)nϕ(j) as an ansatz of Eq. ([eqappc2]). Here, βj( = β) is the solution of the characteristic equation given by $$\begin{aligned} &&\frac{1}{m}\left[\left(K-\bar{K}\right)\beta+\left(K+\bar{K}\right)\beta^{-1}\right] \nonumber\\ &&+\left(\omega^2-i\gamma\omega-\Omega^2-\frac{2}{m}K\right)=0. \label{eqappc4}\end{aligned}$$ We note that Eq. ([eqappc4]) is a quadratic equation for β. The boundary conditions, ψ0 = ψL + 1 = 0, tell us the condition that the combination coefficients ϕ(1) and ϕ(2) take nonzero values, and it is written as $$\left(\frac{\beta\_1}{\beta\_2}\right)^{L+1}=1. \label{eqappc5}$$ Then, we can obtain the explicit form of β1 and β2 from Eqs. ([eqappc4]) and ([eqappc5]). When (K + K̄)(K − K̄) > 0, the Vieta’s formula of Eq. ([eqappc4]) gives β1 = reiθl, β2 = re− iθl, where $$r\_+=\sqrt{\frac{K+\bar{K}}{K-\bar{K}}}, \label{eqappc7}$$ and $$\theta\_l=\frac{\pi l}{N+1}~\left(l=1,\dots,N\right). \label{eqappc8}$$ Hence, the eigenvalues of the system can be calculated as $$\omega\_{l,\pm}^>=\frac{i}{2}\gamma\pm\sqrt{\Omega^2+\frac{2}{m}\left(K-\sqrt{K^2-\bar{K}^2}\cos\theta\_l\right)-\frac{\gamma^2}{4}}. \label{eqappc9}$$ Similarly, when (K + K̄)(K − K̄) < 0, we obtain the form of β1 and β2 as follows: β1 = − ir′eiθl, β2 = − ir′e− iθl. Here, $$r\_-=\sqrt{\left\|\frac{K+\bar{K}}{K-\bar{K}}\right\|}, \label{eqappc11}$$ and θl is given by Eq. ([eqappc8]). The eigenvalue of the system in this case is written as $$\omega\_{l,\pm}^<=\frac{i}{2}\gamma\pm\sqrt{\Omega^2+\frac{2}{m}\left(K-i\sqrt{\left\|K^2-\bar{K}^2\right\|}\cos\theta\_l\right)-\frac{\gamma^2}{4}}. \label{eqappc12}$$ [secC-2]Finite-size effect -------------------------- ![[figappc2]Dynamical phase diagram, continuum bands, and eigenvalues of the levitated nanoparticle array. (a,b) Dynamical phase diagrams with \Omega=3 and \gamma=2 and \Omega=1 and \gamma=2. respectively. Panel (b) is the same as Fig. [fig2](a). The blue, green, red, and gray-shaded regions indicate the underdamped, critical, overdamped, and dynamically unstable phase, respectively. The spectral singularity (SS) appears in the green-shaded region and on the green lines. (c) and (d) Continuum bands \tilde{\omega}_+ (magenta) and \tilde{\omega}_- (cyan), and eigenvalues \omega_{l,+}^> (red) and \omega_{l,-}^> (blue). (e) Continuum band \tilde{\omega}_+ (black), and eigenvalues \omega_{l,+}^< (red). We set L=8 in panels (c–e) and choose the values of \left(K,\bar{K}\right) at the orange star for panel (c) and at the white and black stars for panels (d) and (e), respectively.](appc2.png)[figappc2]Dynamical phase diagram, continuum bands, and eigenvalues of the levitated nanoparticle array. (a,b) Dynamical phase diagrams with Ω = 3 and γ = 2 and Ω = 1 and γ = 2. respectively. Panel (b) is the same as Fig. [fig2](a). The blue, green, red, and gray-shaded regions indicate the underdamped, critical, overdamped, and dynamically unstable phase, respectively. The spectral singularity (SS) appears in the green-shaded region and on the green lines. (c) and (d) Continuum bands ω̃+ (magenta) and ω̃− (cyan), and eigenvalues ωl, +> (red) and ωl, −> (blue). (e) Continuum band ω̃+ (black), and eigenvalues ωl, +< (red). We set L = 8 in panels (c–e) and choose the values of (K, K̄) at the orange star for panel (c) and at the white and black stars for panels (d) and (e), respectively. In our work, we investigate the levitated nanoparticle array in the thermodynamic limit, L → ∞, and obtain the dynamical phase diagram and the continuum bands of the levitated nanoparticle array. We show the dynamical phase diagram with γ < 2Ω and γ > 2Ω in Figs. [figappc2](a) and [figappc2](b), respectively. We note that the critical phase extends over only a narrow region in Fig. [figappc2](a), while it clearly possesses a broad region in Fig. [figappc2](b). We here discuss how the finite-size effects can affect the band structures in the critical phase and the boundaries between the underdamped and dynamically unstable phases. To do so, we recall that the continuum bands of the infinite-size system is given by $$\tilde{\omega}\_\pm=\frac{i}{2}\gamma+\sqrt{\Omega^2+\frac{2}{m}\left(K-\sqrt{K^2-\bar{K}^2}\cos\theta\right)-\frac{\gamma^2}{4}}. \label{eqappc13}$$ First, we consider the parameters indicated by the orange star in Fig. [figappc2](a) and show the corresponding continuum bands ω̃± and the eigenvalues ωl, ±> in Fig. [figappc2](c). One can see that there is no degeneracy between ωl, +> and ωl, −> in a strict sense, simply because θl takes only the discrete values determined by Eq. ([eqappc9]). Nevertheless, we emphasize that the eigenvectors around the spectral singularity, where ω̃+ touches ω̃−, still exhibit the strong nonorthogonality which is a hallmark of the non-Hermitian degeneracy. This result is the same as in the case of the parameter indicated by the white star in Fig. [figappc2](b). Indeed, the eigenvalues in Fig. [figappc2](d) are similar to those in Fig. [figappc2](c). Next, we consider the parameters indicated by the white star in Fig. [figappc2](a) and show the corresponding continuum band ω̃+ and the eigenvalues ωl, +< in Fig. [figappc2](c). It is found that the finite-size system does not exhibit the dynamical instability because the imaginary parts of all the discrete eigenvalues become positive. In this sense, the finite-size effect can slightly modify the boundary between the dynamically unstable phase and the other phases. One can also infer from Fig. [figappc2](c) that a larger system size would be favorable to observe the dynamical instability. Nevertheless, we emphasize that the phase diagram of a finite-size system still remains qualitatively the same as in the result obtained in the thermodynamic limit. [secD]Periodic boundary condition ================================= ![[figappd]Dynamical phase diagram of the levitated nanoparticle array obtained from the conventional Bloch band theory. The blue and gray-shaded regions are the underdamped and dynamically unstable phases, respectively. The spectral singularity appears on the green line. We set the parameters to be \Omega=1 and \gamma=5.](appd.png)[figappd]Dynamical phase diagram of the levitated nanoparticle array obtained from the conventional Bloch band theory. The blue and gray-shaded regions are the underdamped and dynamically unstable phases, respectively. The spectral singularity appears on the green line. We set the parameters to be Ω = 1 and γ = 5. We consider the levitated nanoparticle array described by Eq. ([eqappa14]) with N = 1 and study the dynamical phase diagram under periodic boundary conditions to compare the result obtained in Fig. [fig2](a). We note that the characteristic equation of the system reads $$\begin{aligned} &&\frac{1}{m}\left[\left(K\_1-\bar{K}\_1\right)\beta+\left(K\_1+\bar{K}\_1\right)\beta^{-1}\right], \nonumber\\ &&+\left(\omega^2-i\gamma\omega-\Omega^2-\frac{2}{m}K\_1\right)=0. \label{eqappd1}\end{aligned}$$ After replacing β by eik for the real Bloch wavenumber k, we can obtain $$\omega\_\pm^\prime=\frac{i}{2}\gamma\pm\sqrt{\Omega^2-\frac{\gamma^2}{4}+\frac{2}{m}K\_1\left(1+\cos k\right)-\frac{2i}{m}\bar{K}\_1\sin k}, \label{eqappd2}$$ which reproduces the eigenvalues of the system with periodic boundary conditions. Let us suppose that γ > 2Ω for the sake of concreteness. The condition for the appearance of the spectral singularity can be then written as $$\Omega^2-\frac{\gamma^2}{4}+\frac{4}{m}K\_1=0, \label{eqappd3}$$ since it appears when k = 0. Remarkably, this indicates that the critical phase obtained in Fig. [fig2](a) disappears in the system with periodic boundary conditions. Said differently, the appearance of the spectral singularity requires the fine-tuning of the system parameters. Indeed, one can see from the dynamical phase diagram shown in Fig. [figappd] that only the underdamped and dynamically unstable phases appear in this case. Thus, modifying open boundaries to periodic boundaries drastically changes the behavior of the levitated nanoparticle array. [secE]Experimental feasibility ============================== We investigate the dynamical phase diagram of the levitated nanoparticle array with K/m, K̄/m ∈ [ − 10Ω2, 10Ω2] and γ = 5Ω in our work. We here discuss the experimental values to access the assumed parameter region in our setup. We note that the previous experiment realizing the coupled two levitated nanoparticles has achieved K/m, K̄/m ∈ [ − 0.1Ω2, 0.1Ω2] and γ ≃ 0.03Ω , where the power and wavelength of the trapping laser, and the gas pressure are set to be $P=400~{\rm mW}$, $\lambda=1064~{\rm nm}$, and $P\_{\rm gas}=1.5~{\rm mbar}$, respectively. Furthermore, the previous work has utilized silica nanoparticles, the radius, the polarizability, and the mass of which are set to be $r=105~{\rm nm}$, α = 3.48 × 10− 32 F ⋅ m2, and $m=1.07\times10^{-17}~{\rm kg}$, respectively. First, we explain a possible way to amplify the dipole-dipole couplings K/m, K̄/m compared to Ω2. To this end, we consider the ration between G/m and Ω2, which can be obtained from Eqs. ([eqappa8]) and ([eqappa20]) as follows: $$\frac{G}{m\Omega^2}=\frac{1}{\pi\varepsilon\_0}\left(\frac{k\_0^2w\_0^2}{2}-1\right)^2\alpha k\_0^3. \label{eqS43}$$ It is important that one can achieve K/m, K̄/m ∈ [ − 10Ω2, 10Ω2] by using the trapping laser with the half wavelength and making the radius of the particle 2.5 times larger than the one used in the previous experiment; we recall that the larger particle results in higher polarizability. Thus, we assume that the wavelength of the trapping laser is set to be $\lambda=532~{\rm nm}$, and the radius, the polarizability and the mass of the particles are set to be $r=260~{\rm nm}$, $\alpha=5.28\times10^{-31}~{\rm F\cdot m^2}$, and $m=1.62\times10^{-16}~{\rm kg}$, respectively, and the laser power is the same as in the previous experiment. We expect that our model captures qualitative features of the motion of the levitated nanoparticle even if its size is comparable to the wavelength of the trapping laser, because the dominant contribution to the forces which the particle feels still originates from the dipole moments in the particle. Meanwhile, to develop a quantitatively accurate theory, it should be necessary to modify the description to go beyond the dipolar approximation. Specifically, one should take into account the contributions from the higher-order moments (e.g., quadrupole moments). Second, we explain how to realize the friction coefficient γ = 5Ω. We note that, in the low-vacuum regime, the damping rate of a levitated nanoparticle is proportional to the square of the radius of the particle and the gas pressure. With the above parameter set, Ω approximately becomes twice larger than the value in the previous experiment, and it is possible to realize γ = 5Ω by setting the gas pressure to be $P\_{\rm gas}=80~{\rm mbar}$. Thus, we expect that the levitated nanoparticle array with the above experimental value set should allow one to realize the parameters assumed in our manuscript and to observe the predicted dynamical phases accordingly. 104ifxundefined [1] ifx#1 ifnum [1] #1firstoftwo secondoftwo ifx [1] #1firstoftwo secondoftwo ““#1””@noop [0]secondoftwosanitize@url [0]` 12`$12`&12`#12`12`\_12`%12@startlink[1]@endlink[0]@bib@innerbibempty , , , and , Observation of a single-beam gradient force optical trap for dielectric particles, @noop Opt. Lett. 11, 288 ( 1986) , , , and , Subkelvin parametric feedback cooling of a laser-trapped nanoparticle, [Phys. Rev. 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Herschel Observations of Dusty Debris Disks =========================================== We present results from two Herschel observing programs using the Photodetector Array Camera and Spectrometer. During three separate campaigns, we obtained Herschel data for 24 stars at 70, 100, and 160 μm. We chose stars that were already known or suspected to have circumstellar dust based on excess infrared emission previously measured with IRAS or Spitzer, and used Herschel to examine long-wavelength properties of the dust. Fifteen stars were found to be uncontaminated by background sources, and possess infrared emission most likely due to a circumstellar debris disk. We analyzed the properties of these debris disks to better understand the physical mechanisms responsible for dust production and removal. Seven targets were spatially resolved in the Herschel images. Based on fits to their spectral energy distributions, nine disks appear to have two temperature components. Of these nine, in three cases, the warmer dust component is likely the result of a transient process rather than a steady state collisional cascade. The dust belts at four stars are likely stirred by an unseen planet, and merit further investigation. Introduction ============ Debris disks are signposts of planetesimal formation and, as such, are crucial subjects for study when considering the evolution of a planetary system. Debris disks are typically identified via an excess infrared (IR) flux above the stellar photosphere. New debris disks have been discovered with five satellites, starting with the InfraRed Astronomical Satellite (IRAS) discovery of the first debris disk around Vega in 1984. In 1995, the Infrared Space Observatory (ISO; ) imaged the sky at wavelengths ranging from 2.5 to 240 μm, providing low-resolution long wavelength photometry. Subsequently, the Spitzer Space Telescope provided mid-IR spectroscopy with the Infrared Spectrograph (IRS; ), and photometry with the Multiband Imaging Photometer (MIPS; ), leading to the detection of over 100 new debris disks (e.g.,,, ). The Wide Field Survey Explorer (WISE; ) imaged the sky at 3.4, 4.6, 11, and 22μm, and was also most sensitive to warm and hot debris disks that peak in the mid-IR. In 2009, the Herschel Space Observatory began taking data, providing far-IR photometry with the Photodetector Array Camera and Spectrometer (PACS; ) and spectroscopy with the Spectral and Photometric Imaging Receiver (SPIRE; ) and the Heterodyne Instrument for the Far Infrared (HIFI; ). Where Spitzer was uniquely able to detect warm ( > 100 K) debris disks in the terrestrial planet zone, Herschel was sensitive to cooler disks ( < 100 K) at larger radial separations from their host stars. In this paper, we present Herschel observations of 24 stars initially identified with IRAS and/or Spitzer as definitely or possibly possessing a debris disk. One goal of our project was to search for cold dust components that would peak near the Herschel PACS wavelengths (70, 100, and 160 μm). In the case that there is no separate cold dust component present, Herschel photometry helps to better characterize the Rayleigh-Jeans tail of thermal emission from warm dust. Another goal was the identification of disks with double-belt debris systems; that is, systems containing an inner belt of warm or hot ( > 100K) dust and an outer belt of cold ( < 100 K) dust. Such systems would be a direct analog of our own Solar System, which hosts a Kuiper Belt that lies between 30-50 AU at ∼ 50 K and an Asteroid Belt at 3 AU and ∼ 175 K. A double-belt system may also be a signature of a planet (or planets) that lie in the gap between dust belts. Such systems have been discovered around HR 8799 (,, ) and HD 95086 (, ). Stellar Sample ============== The sample used in this work is an amalgamation of stellar samples from three different Herschel proposals (bzuckerman-OT1, jolofsson-OT1, and bzuckerman-OT2). In the first OT1 proposal (PI: B. Zuckerman), we observed A-F stars with known, very luminous mid-IR emission. Since mid-IR emission is known to originate from the terrestrial planet region, we wanted to use Herschel to search for accompanying cold dust in outer regions analogous to the Solar System’s Kuiper Belt. Four such stars were observed during this program. The second OT-1 proposal (PI: J. Olofsson) had very similar science goals, in that the authors were looking for cold dust components accompanying known warm debris disks. Six stars were observed for their program, of which we present two which fit into our initial selection criteria (luminous mid-IR emission). The objective of the OT-2 proposal was to follow up on a subsample of stars (from and ) that had been observed either with IRAS or Spitzer, and that had only one far-IR data point at 60 μm (IRAS) or 70 μm (Spitzer). Herschel observations were carried out to characterize the far-IR emission of the dust and, in the case of stars with apparent excess emission detected only at 60 μm with IRAS, to confirm or deny the existence of a dusty debris disk. IRAS is not only less sensitive than Herschel, but its large beam size made IRAS vulnerable to confusion by background sources (the Herschel PSF is 5.75′′ at 70 μm, while the average detector element size for IRAS at 60 μm was 5′ × 2′). Eighteen stars were observed during this OT-2 program. We collected information from the literature on binarity, distance from Earth, and stellar age. Literature data for target stars can be found in Table [table:stellar]. Observations ============ We observed 24 stars in total with PACS; simultaneous observations were obtained at 160 μm and either 70 μm or 100 μm. Although some stars were not detected at 160 μm, a 3σ upper limit at that wavelength helped to constrain cold dust temperatures. The observations were taken with a scan speed of 20$\arcsec$/sec. We used HIPE (Herschel Interactive Processing Environment, version 12.0; ) to reduce the data and produce the final maps. We used a pixel scale of 1$\arcsec$/pixel for 70 and 100 μm data and 2$\arcsec$/pixel for the 160 μm data. A high pass filter was applied to remove instrumental noise. To achieve the highest signal to noise in the resulting maps, we used a high pass filter radius of 30$\arcsec$ for 70μm data and 70$\arcsec$ for 100 and 160μm data. The larger radius filter was used for the longer wavelength data in order to be more aggressive in removing instrumental noise at those wavelengths while maintaining a high SNR. A mask was applied to avoid removing any flux within 15" of the target star. Aperture photometry was carried out with an aperture radius of 5$\arcsec$ for unresolved sources and 10$\arcsec$ for resolved sources, with a sky annulus extending from 20-40$\arcsec$. Aperture corrections were made to account for photospheric flux falling outside of the chosen aperture$\footnotemark{}$$\footnotetext{All corrections were taken from Herschel Release Note PICC-ME-TN-037, Table 15.}$. Errors on the photometric points are derived by placing apertures on empty regions of sky along the sky background annulus and measuring the r.m.s. of the background flux. Some targets presented flux at a significant offset from the stellar location. If the offset was larger than the pointing accuracy of Herschel ( ∼ 2$\arcsec$), we excluded it from further analysis. This was the case for two stars in our target list (HD 60234 and HD 203562). These two stars were taken from, and were 60μm excesses only. Observed fluxes are found in Table 2. We discuss potential sources of contamination in Section [contamination]. Herschel Non-Detections ----------------------- A non-detection with Herschel implies that any circumstellar dust would necessarily be < 35K, extremely tenuous (LIR/Lbol=τ < 10− 6), or both. The three stars mentioned in this section were not detected in any of the Herschel bands, and are not considered in the analysis in later sections. ### HD 70298 This star was reported by as a new debris disk candidate with a surprisingly substantial 60μm excess (τ=3.54E-04), given its old age ( > 3Gyr). The star was not observed by Spitzer, but a WISE excess was identified by at 22 μm (τ=4.35E-04). Close inspection of the WISE images shows that there is a nebulous IR source ∼ 15$\arcsec$ away from the target, which is inside the WISE contamination radius at 22 μm (2 × FWHM = 24$\arcsec$; ). We therefore consider the WISE and IRAS images to be contaminated. In the absence of any detectable flux at Herschel wavelengths, we report no IR excess around this star. ### HD 72660 This star was reported by as a new debris disk candidate based on an IRAS excess. The star was subsequently observed with Spitzer, and a mid-IR spectrum showed no IR excess consistent with the IRAS data point. WISE also saw no evidence of any warm excess in the near-IR. Even so, it remained possible that the IRAS photometry was catching the Wein tail of cold dust ( < 50 K), until our negative Herschel observations. ### HD 132950 This star was reported by as a new debris disk candidate with a large IR excess (τ=1.17E-03) and an old age ( ∼ 3 Gyr). A Spitzer IR spectrum shows no evidence of an IR excess consistent with the IRAS data point, and no warm excess was seen by WISE. In the absence of an IR excess seen with Herschel, we report no detectable dust around this star. Systems with No Detectable Dust ------------------------------- In addition to stars with no detectable flux at Herschel wavelengths, there is one system (HD 191692) for which Herschel detected the photosphere of the star, but no evidence of an IR excess was seen. This star was reported by to have a small (τ < 10− 5) IR excess seen at IRAS wavelengths, but unconfirmed by Spitzer. Their dust model fit to the IRAS photometry suggested the existence of dust at over 200 AU separation from the central star. reported the star as a binary with a separation of < 1 AU, meaning that the dust, if confirmed, would be circumbinary. report a large IR excess (τ > 10− 3) seen at WISE wavelengths. A close inspection of the WISE images shows a source ∼ 18$\arcsec$ to the NE of the target star, which is within the contamination radius of the WISE beam at 22μm. Thus we consider both the IRAS and the WISE images to be contaminated, and given that the Herschel images show no evidence of an IR excess, we report this star to have no detectable dust. Sources of Possible Contamination ================================= Of the remaining targets (those with detectable Herschel fluxes indicative of an IR excess), four (HD 8558, HD 13183, HD 80425, and HD 99945) have apparent cold dust disks with temperatures < 40K (see Section [SEDs]). To ensure that we are really seeing evidence of cold debris belts, we investigated several alternative sources of the apparent IR excess. Extragalactic Background ------------------------ presented Herschel observations of cold debris disks and investigated the possibility of confusion with a background galaxy. A typical galaxy below the confusion limit of Herschel ( ∼ 2.5mJy at 160 μm) would lie between z=0.94 and 1.2 and its IR emission would correspond to an apparent dust temperature which would peak between 20 and 29 K. Of the four stars with Herschel emission at 160μm, only two (HD 8558 and HD 80425) have F160 < 10 mJy and could be explained by confusion with a background galaxy at z ∼ 1. In addition, the position of a third star (HD 13183) is 2$\arcsec$ away from a galaxy detected with GALEX. Since the galaxy is within the confusion beam radius of PACS at 100μm (7.19$\arcsec$; ), we consider this target star to be contaminated, even though the source flux is 40mJy at 160 μm. These three stars are not included in Table [dustpars]. IR Cirrus --------- We must also investigate the possibility that excess IR emission is due to confusion with background cirrus, which is known to emit at ∼ 20 K. We checked each of our cold excess systems (HD 8558, HD 13183, HD 80425, and HD 99945) for evidence of cirrus in the Herschel and WISE images. HD 99945 is not explained by confusion with a background galaxy or nearby IR excess source. We also found no detectable cirrus in the Herschel or WISE images. This, combined with the fact that the 160 μm flux for this object is > 100mJy implies that the excess around the star is due to emission from a debris disk. HD 99945 is therefore included in Table 3. Of the four potential cold disks in our sample, only HD 99945 (the warmest of the four) survived to the final sample. Spectral Energy Distributions ============================= Spectral energy distributions (SEDs) were created using a fully automated photosphere-fitting technique that made use of the PHOENIX models. Stellar photospheres were fit using B, V, J, H, and K fluxes, while the mid- and far-IR photometry was used to fit the dust emission. We fit a simple blackbody to the IR photometry using Spitzer, WISE, and IRAS to supplement the Herschel data points whenever possible. For some systems, the long wavelength data (at 100 or 160 μm) falls below a simple blackbody curve. In these cases, we applied a modified blackbody described by: $$F\_{\nu}\propto\left(\frac{\nu\_{0}}{\nu}\right)^{\beta}B\_{\nu}(T\_{d})$$ where Td is the dust temperature. Uncertainties for Td are found using a Monte Carlo approach. For each SED, we generated 1000 simulated data sets by randomly drawing flux values from a single Gaussian distribution centered on the observed fluxes, where the width of the distribution is given by the observed uncertainty. The values and uncertainties for Td reported in Table 3 are the average and standard deviation of the ensemble of fits for each object. The resulting SEDs are found in Figure [fig:SEDs1]. Stellar Parameters ------------------ We used the best-fit model photospheres to determine the stellar temperature and radius. From there, we calculated a stellar luminosity using L\=4πR\2σTeff4. We assumed solar metallicity and log(g) for all 24 stars. Stellar masses were taken from the literature. Ages for stars in our sample were taken from moving group membership whenever possible. When moving group membership could not be determined, we used literature ages from stellar isochrones (for A-F-type stars). For all other stars, we relied on literature ages based on lithium abundance or X-ray activity. Stellar parameters are found in Table [table:stellar]. Dust Parameters --------------- For the 15 stars in our sample with dust detected by Herschel, we obtain a dust temperature and fractional IR luminosity (LIR/Lbol=τ) from a blackbody SED fit. Assuming blackbody dust grains, the orbital semi-major axis is: $$R\_{BB}=\frac{R\_{\}}{2}\left(\frac{T\_{\}}{T\_{BB}}\right)^{2}$$ For the purposes of this work, we define three dust temperature regions; hot dust ( > 200K), warm dust (100-200K), and cool dust ( < 100K). These temperature regions were chosen to correspond to the solar system’s zodiacal dust, asteroid belt, and Kuiper belt. Of the 15 systems in our sample that were detected by Herschel and were found to be uncontaminated, 9 stars show a cool dust component, 3 stars show a warm dust component, and 10 stars show a hot dust component (some stars show multiple dust components). It should be noted that the dust temperatures provided in Table [dustpars] are the best-fit values from SED fitting described in Section [SEDs]. In some cases, the best fit dust temperature is reported as being > 1000 K. We examined the variation in χ2 values for our SED fits as we varied the dust temperatures between 600-1200K, and find that the χ2 values drop significantly as one increases the dust temperature from 600 to 1000 K, but flatten out after 1000 K. Such model-suggested dust temperatures of $\gtrsim$1000 K are difficult to explain (especially since the hottest dust temperature around an extremely dusty star known to this point is the ∼ 800 K dust around V488 Per, reported by ) and we acknowledge that a one or even two ring model for these debris disks is very likely an over-simplification of the actual dust configuration. We calculated the blowout radius for a dust grain, ablow, defined as: $$a\_{blow}=\frac{3L\_{\}Q\_{PR}}{16\pi GM\_{\}c\rho}$$ where QPR is the radiation pressure coupling coefficient and ρ is the density of a typical dust particle. We assume QPR ∼ 1 for 2πa/λ > 1 and ρ ∼ 2.5 g cm− 3. Finally, we calculated the minimum mass of dust in the disk. This parameter is best measured using far-IR or sub-mm data, which probes the largest grains in the system. The large grains are where most of the mass of the dust is concentrated. However, we can calculate a minimum dust mass from Equation 4 from : $$M\_{d, min}\geq\frac{16}{3}\pi\tau\rho R\_{d}^{2}\langle a\rangle$$ where ⟨a⟩ = (5/3)ablow and Rd is the dust semi-major axis. Minimum dust mass values are calculated using the outermost dust belt parameters only, since the mass of the colder dust dominates the system in all cases. Resolved Disks ============== To determine whether a disk was resolved, we compared the radial profile of the star+disk to a reference PSF (see Figure [fig:profs]). To create the radial profile, we binned the pixels of each raw image by radius. The error bars in Figure [fig:profs] are the standard deviation of the fluxes within each radial bin. We consider each disk presented in Figure [fig:profs] to be resolved$\footnotemark{}$$\footnotetext{\scriptsize HD 121191 is marginally resolved, and further discussion can be found in Section \ref{HD121191}.}$. We performed a PSF subtraction and modeled the residual flux with a single narrow ring of dust (ΔRd=0.1Rd). The bright star α Cet was used as a reference PSF. The free parameters fit in our ring model were the semi-major axis, inclination, and position angle of the narrow ring. This model (convolved with the instrument PSF) was appropriate for all of our resolved systems, since residual maps showed no significant structure once the ring model was subtracted. Raw images, PSF-subtracted images, and residuals can be found in Figure [fig:mapsA]. Of the 15 stars in our sample, seven (HD 54341, HD 76543, HD 76582, HD 84870, HD 85672, HD 99945, and HD 121191) were resolved at either 70 or 100 μm. One system (HD 76582) was resolved at both 100 and 160 μm. Disk parameters determined by ring-fitting can be found in Table [resolvedtable]. Errors were determined by varying parameters until residuals increased by 1σ. Disk Radii ---------- The blackbody disk semi-major axes derived from SED fitting (RBB) often did not agree with those observed in the resolved images (Rimg). We define a semi-major axis ratio fR=Rimg/RBB. It is well-known that small blackbody particles tend to be super-thermal and thus blackbody SED-fitting will underestimate radial extent of the dust, often by a factor of 2-5. The steeper the size distribution of the dust grains, the more small grains are present in the disk, and the disk will appear much hotter than a blackbody. At first glance, then, it would seem that fR can probe the size distribution of the dust. If, however, the dust production mechanism is similar among all of our disks (see Sections 6-7), then it is reasonable to assume that the resulting size distribution would be similar as well. If that is true, the biggest factor affecting fR would be the spectral type or luminosity of the host star. If the host star is very luminous, it will remove the smallest grains from the system via radiation pressure; this will make the dust appear to act more like a blackbody. We compared our radius ratios to those of similar programs (,, ) in Figure [fig:radii]. We find that, in most cases, fR increases for lower-luminosity stars. One interesting feature to note in Figure [fig:radii] is that the value of fR (=Rimg/RBB) is well constrained for disks that are resolved in thermal emission, but there is much more scatter in the relationship between fR and Lbol for stars whose ``resolved" radii are derived from SED modeling of the Si emission (see Figure [fig:radii]) and/or scattered light. This may be because the scattered light images are most sensitive to the smallest grains, which can be found in a large, extended disk or halo, rather than in a narrow ring as described by a blackbody fit. It appears that the stars in our sample (red data points in Figure [fig:radii]) have a higher scatter in Lbol- fR space than do stars from. Notably, the biggest outliers (HD 121191 and HD 85672) are those stars for which we believe an unseen planet could be responsible for dust production (see Section [planetstirred]). Since the ratio fR depends heavily on the size of the grains themselves, it is possible that a planet-stirred disk has an inherently different grain size distribution than a self-stirred disk. Different grain size distribution could arise from a number of factors, including the collisional velocities of the < 100km bodies that produce the dust, and the composition of the dust itself. It is also possible that the Herschel images are sensitive to small grains further from the star (which would peak at Herschel wavelengths), while the average temperature of the grains in the system is actually higher than the temperature implied by the blackbody fit to the SED. Dust Production and Planet Formation ==================================== Having determined dust properties from the Herschel photometry, we turn our attention to the way in which the observed dust was produced. By the time a star is ∼ 10 Myr old, any primordial dust should have been completely depleted (used to form planetesimals, accreted onto the star, or blown out of the system by radiation pressure).$\footnotemark{}$ $\footnotetext{\scriptsize This idea has been challenged by recent discoveries of 30-40 Myr old disks with copious molecular gas present, suggesting that, for some stars, the protoplanetary stage may last longer than previously thought (e.g. \citealt{Kospal\_2013}, \citealt{Zuckerman\_2012}). Still, since there are so few stars that exhibit gas at 30-40Myr, we maintain that most primordial gas and dust should be gone by $~$10 Myr.}$ Thus, the dust we see in debris disks older than ∼ 10 Myr is likely second-generation, created in collisional processes. Furthermore, the lifetime of this dust against radiation pressure and drag forces is shorter than the lifetime of the star. This means that any dust seen in these systems must be replenished regularly. There are various mechanisms by which dust can be produced as a planetary system forms and evolves. First, the dust can be produced through steady-state collisions of small planetesimals. This occurs naturally as a final stage in the process of planet formation. Planet formation proceeds first through the growth of ∼ 1 km-sized planetesimals. Once the planetesimals grow large enough to start gravitationally focusing the primordial dust in their path, runaway growth can occur. Following this, a few large bodies accrete a large amount of dust, and as they grow, they become the dominant accreter in their orbital path - a process known as oligarchic growth. Once ∼ 1000 km-sized bodies have formed, they can begin to dynamically stir the remaining smaller ( < 100 km) planetesimals, driving them to collisional velocities high enough to be destructive. It is in these destructive collisions that the observed dust is produced. The resulting debris is collisionally ground down until the grains are small enough to be blown out of the system via radiation pressure. On their way out of the system, they can also collide with small planetestimals, creating an outward-moving collisional cascade. In the process described above, the stirring mechanism at work is the natural formation of 1000 km-sized bodies in the disk itself (hence, the disks are ``self-stirred", see Section [selfstirred]). However, one can also imagine a situation where small ( < 100km) bodies are stirred by a nearby planet (see Section [planetstirred]). In either case, one would expect moderate fractional IR luminosities (τ ∼ 10− 4) compared to those associated with protoplanetary disks (τ ∼ 10− 3). Alternatively, a belt of dust can be created through a giant impact between planetary embryos (e.g. ; see Section [giantimpacts]). In an even more extreme scenario, two fully formed rocky planets can undergo a catastrophic collision, resulting in large amounts of debris. Such a collision has been proposed to explain warm dust in orbit around BD +20 307 (, ). Such catastrophic collisions should produce a multitude of small grains. Similar collisions are also more likely to occur in the terrestrial planet formation zone, where disk surface density is high; such collisions result in production of warm or hot dust. A third possible source of hot dust in disk systems might be debris left behind by so-called ``star-grazing comets." This mechanism was suggested by who found that hot dust in a sample of Spitzer-observed systems had a characteristic temperature of 190K - near the sublimation temperature for icy bodies such as comets (see Section [comets]). Distinguishing Transient from Steady-State Events ------------------------------------------------- To distinguish dust created in a transient event from dust created in a steady-state process, one can examine the maximum fractional IR luminosity that can be attained through a steady-state process (τmax). This method was explored in. We examine the τmax method of. According to their Equation 20: $$\tau\_{max}=0.58 \times 10^{-9} R\_{d}^{7/3}\frac{\Delta R\_{d}}{R\_{d}}D^{1/2}Q^{5/6}e^{-5/3}M\_{\}^{-5/6}L\_{\}^{-1/2}t^{-1}$$ As in Equation 4, Rd is the semi-major axis of the dust, and here ΔRd is the radial width of the disk, D is the diameter of the largest planetesimal in the disk, Q is the specific incident energy required to destroy a particle, e is the eccentricity of the dust ring, and t is the stellar age. For our calculations, we use the outermost dust component (in cases where two components are needed to fit the observed IR flux). In all following calculations, if the dust was spatially resolved, we use the dust semi-major axis from the ring-fitting algorithm (Rimg, see Section 6.1). Otherwise, we use RBB from the SED fits. To model a steady-state collisional cascade, we follow the prescription of and assume that D=2000km, Q=200 J kg− 1, e=0.05, and ΔRd=0.1Rd. According to Wyatt et al., one should expect that observed dust was likely created in a transient event if τ/τmax > 1000. The τ/τmax > 1000 threshold is based on observations of known very luminous debris disks, while taking into consideration the assumptions that went into Equation 5. For the purposes of our analysis, we assume that debris disks with τ/τmax > 100 are likely to be transient in nature, while those with τ/τmax > 1000 cannot be explained by a steady-state collisional process alone (see Tables 3 and 5). We calculated τmax for the stars in our sample and found that three debris disks systems could not be explained by a steady-state process alone (all from OT1, and thus with the highest fractional IR luminosities and previously suspected to be the result of transient processes.); HD 15407, HD 113766, and BD +20 307. Since these are well-studied systems, we do not discuss their transient nature further in this paper. Furthermore, the disks at HD 23514 and HD124718 had 100 < τ/τmax < 1000, implying transient processes. Steady-State Collisions - Stirring Mechanisms --------------------------------------------- If the observed dust is produced in a steady-state collisional cascade, we would like to identify the trigger mechanism that starts the cascade. We consider collisions triggered by perturbations due to 1000 km-sized bodies within the disk itself (self-stirring), and collisions triggered by a nearby planet or distant stellar or substellar companion (planet-stirring). ### Self-Stirring In the case of self-stirring, 1000 km-sized bodies naturally form in the disk which dynamically stir the population of smaller ( < 100 km-sized) bodies to velocities high enough to cause destructive collisions. These initial collisions then trigger a collisional cascade which leads to the production of copious amounts of dust. Since the self-stirring mechanism requires the existence of 1000 km-sized bodies at approximately the same radial location of the dust, we can place limits on the parameters of a disk stirred by this mechanism. In Figure [fig:1000km], we examine the semi-major axis of 1000 km-sized bodies as a function of stellar age. Following the models of, we assume that 1000 km-sized planetesimals form at a time: $$t\_{1000}=145x\_{m}^{1.15}\left(\frac{R\_d}{80 AU}\right)^{3}\left(\frac{2M\_{\sun}}{M\_{\}}\right)^{3/2} [Myr]$$ As in, we vary two parameters in our models; the mass of the host star and a scaling factor (xm) related to the initial mass of the protoplanetary disk (where xm=1 corresponds to the minimum mass solar nebula). We plot several models for the star-disk system in Figure [fig:1000km], representing stars from 1.5-2.5M$\_{\sun}$ and a range of xm. For self-stirring to be responsible for the dust observed around the four labeled stars, the initial disk would need to be 30 times as massive as the minimum mass solar nebula or more. Since ∼ 30 × MMSN is approximately the protoplanetary disk mass of a high-mass star, we take this as an upper limit for our models. None of the stars in our sample have detected planets. ### Planet Stirring For disks that are unlikely to have formed 1000 km-sized bodies at the radial location of the observed dust, planet-stirring offers another mechanism for dust production (major planets could have formed near the dust at an earlier time). Since debris disks are often considered evidence of planet formation, one might expect a correlation between the existence of planets and detection of circumstellar dust. Unfortunately, with so few stars observed to have both a dust disk and at least one planet, this relationship has proved difficult to study. found little correlation between planet hosts and detection of an IR excess (the typical age of stars in their sample was ∼ 6 Gyr, whereas the stars in our sample are significantly younger). More recently, examined a large ( > 200 star) sample of Herschel-observed stars to look for correlations between the presence of a debris disk and: (1) the presence of low-mass planets, (2) the presence of high-mass planets, (3) metallicity, and (3) the presence of one or more stellar companions. Even with their large sample size, found no significant correlations between any of the aforementioned parameters. In our sample of 24 stars, none have detected planets (exoplanets.org; ). One can try to draw conclusions about the properties of a hypothetical unseen planet based on the observed dust properties. Unfortunately, it is difficult to resolve the degeneracy between planet mass, orbital eccentricity, and semi-major axis using simple models. This issue is treated in more detail in a theory paper motivated by the present results. It is possible that a belt of small planetesimals could be stirred up to collisional velocities by a distant stellar-mass companion (or brown dwarf) as opposed to a nearby planet. However, found that stars with debris disks are less likely to be found in binary systems, at least in a sample of FGK stars. This may be attributable to the fact that a companion will accelerate the evolution of the dust, making a debris disk detectable for a shorter period of time, earlier on in the evolution of the system. In any case, we do not consider companion stirring to be the cause of the disks in our sample. This possibility is also examined further in. Star-Grazing Comets ------------------- One suggested explanation for the existence of warm and hot dust disks is that so-called ``star-grazing comets" leave behind a cloud of debris. Should this be the case, one would expect most of this cometary debris to be at a radial separation from the star such that the dust is at a temperature of ∼ 150 K (the sublimation temperature for icy planetesimals). In Figure [fig:Thist], we compare the temperature of our disks to those of (Spitzer Catalog of Debris Disks). found that, in a sample of ∼ 70 stars, most of the warm dust components fell around ∼ 190 K.$\footnotemark{}$ $\footnotetext{\scriptsize The reason the dust temperature is higher than the sublimation temperature is that once the comets sublimate and leave behind grains, those grains appear to be hotter than the original comet because they are not emitting like blackbodies.}$ We do not find a similar trend in the Spitzer catalog taken as a whole. The disks in our Herschel sample are mostly cold ( < 100 K), peaking around 60 K. In fact, the Spitzer data from also shows a peak around 60K. A similar peak is found in the sample of, but no explanation was put forward. Perhaps this 60 K peak is simply an observational bias (a 60K dust belt would peak between 70-100 μm, close to both Spitzer and Herschel filter wavelengths). It may also be that disk detectability falls off at lower temperatures, producing a false ``peak" near 60K. also identified a peak at 60 K in a Spitzer sample of debris disks, but claim that the peak is not solely due to the dust temperature (see their Figure 5). They suggest that the cold dust temperatures follow a trend with spectral type; hotter, earlier-type stars have warmer outer debris belts. This implies that the peak at 60 K is not due to a temperature-dependent phenomenon such as sublimation. Regardless of the exact distribution of temperatures among our debris disks, we do not find a peak in dust temperature near the sublimation temperature for comets, and thus conclude that debris left by star-grazing comets does not contribute significantly to the dust seen in the present sample of stars or in the Spitzer Catalog of Debris Disks. Giant Impacts and Catastrophic Collisions ----------------------------------------- It is possible that the observed warm dust components are created during a giant impact between two planetary embryos in the terrestrial planet formation zone. It is generally believed that such impacts are an important part of our own Solar System’s history; a collision between the early Earth and a Mars-sized planetary embryo was likely responsible for the formation of our Moon. We also have evidence of large catastrophic collisions around stars other than our Sun; such is the case with BD+20 307 (, ). Even if the observed dust luminosity is smaller than τma*x (see Section [distinguishing]), the dust could still have been created in a catastrophic collision, but one in which the parent bodies did not entirely pulverize one another, and much of the mass of the parent body survived in large fragments. Two-Temperature Systems ======================= To determine whether the debris disks in our sample require a two-component fit, we examined the χ2 values for a single- and double- belt fit to the SEDs. In about half of the targets, we found that a double-belt fit resulted in a significantly lower χ2. In one case (HD 76543), we do not find any significant difference in the χ2 value for a single versus double belt fit, but decided to fit two belts in order to be consistent with previous studies in the literature. Nine stars in our sample showed evidence of a two-temperature component dust system based on the SED fits. While IRS spectra were not available in all cases, which limits our ability to properly judge whether two temperature components are truly needed, we based our SED fits on a χ2 analysis, with the understanding that our results would be improved by mid-IR spectroscopy. In most cases, we would interpret a two-temperature component fit as two spatially separated disks. Such systems can be explained in several ways; 1) the dust originated in a cold belt, and “leaked” into the warm belt via PR drag and/or scattering by large planetesimals, 2) there was once one extended disk, into which a gap was carved by intervening planets, 3) the dust may have originated in two belts independent of one another, analogous to the asteroid and Kuiper belts, or 4) there is only one belt of dust that presents as a multiple temperature component system due to a range of grain properties. Concerning possibility 4# above, a two temperature SED could be necessary even if the dust emission arises from a single narrow belt if the grains in the belt have multiple sizes. Since smaller grains radiate inefficiently at long wavelengths, they retain heat and appear to be at a hotter temperature than larger grains at the same radial location from the star. Four of the two temperature systems orbit A-type stars (HD 54341, HD 76543, HD 121191, and HD 131488). Since the blowout radius of grains around A-type stars is larger than that of solar-type stars$\footnotemark{}$$\footnotetext{\scriptsize An A-type star has a typical blowout grain size of a few microns, whereas a solar-type star has a blowout grain size of 1 $$m or smaller.}$, grains in a single disk around an A-type star would likely have a more limited distribution of grain sizes, and thus would not be able to mimic a spatially separated two temperature component dust belt. In addition, three stars (HD 15407, HD 76582, and HD 113766) are early or mid F-type, and are also likely to have a limited grain size distribution. We expect that these seven stars likely have two spatially separated belts. For the remaining two systems (HD 35650 and HD 23514), we calculated a temperature ratio (RT=Td, 2/Td, 1, see Table 3) to determine if the dust emission is coming from two separate belts. From Figure 4 of, we see that single-belt systems with two temperature components tend to have small (RT $\lesssim$ 5) values of RT. By contrast, spatially separated double-belt systems appear to have values of RT $\gtrsim$ 5; both HD 35650 (RT=11.9) and HD 23514 (RT=6.4) thus likely represents true spatially separated belts (see Figure [fig:doubles]). Comparison with Previous Studies ================================ We compared our results to previous studies in the literature. In particular, we compare literature results with our SED-fitting results only, since none of the stars that follow have ever previously been resolved. HD 15407 -------- This F5/K2 binary system has an age of 80 Myr, derived from high resolution measurements of lithium in the photospheres of both components, X-ray data, and UVW space velocities. Si emission features are seen in the IRS spectra. Melis et al. predicted (based on the IRS spectrum and the IRAS upper limits) that there would be no cold dust detected around HD 15407A. performed a detailed analysis of the emission features in the IRS spectra, and found that HD 15407A should have an extended belt (from 0.4-19.2 AU) with a grain size distribution proportional to n− 3.1. This is shallower than the typical -3.5 value assumed for most debris disks. Olofsson et al. calculate a dust mass of ∼ 7.7E-05 M$\_{\Earth}$. estimated that the dust should lie between 0.6 and 1.0 AU. We find a double-belt system around HD 15407 (Figure 1). Both dust components are hot (T ∼ 334K, 1022K) and lie within 1 AU of the central star. These results are consistent with the prediction of a warm, extended belt. studied the mid-IR emission features of HD 15407, and (based on the width of the disk), predicted that a cold dust belt should be found, that is responsible for feeding dust to the observed hot dust belt. The absence of a cold dust component in the Herschel observations implies that either there is not enough dust to be observable with Herschel, or the dust is unrealistically cold ( < 10K). The lack of a substantial dust belt would be consistent with a proposed model of transient dust. HD 23514 -------- HD 23514 is a Pleiades member with an IR excess detected by IRAS and Spitzer. Rhee et al. report a dust temperature of 750 K and an exceptionally high fractional IR luminosity of 2.0E-02. Notably, they also report the results of observations with the Michelle Spectrograph on Gemini North, that show an unusual emission feature peaking around 9 μm. This feature (attributed to SiO2) may indicate a recent major high-velocity collision involving a differentiated terrestrial planet. Rhee et al. found a lack of olivine and pyroxene in the spectra, which suggests that a steady-state collisional cascade involving an asteroid belt is likely not the cause of the observed dust. This model is only strengthened by the fact that we detected no cold dust component with Herschel, and is consistent with our calculations of τ/τmax ∼ 300. We fit two dust components to the observed emission at 168K and 1082K. report a substellar companion to HD23514 with a mass of ∼ 0.06M$\_{\sun}$ at a separation of ∼ 360 AU. notes that several stars with dominant warm dust components were later discovered to be members of wide binary systems. The connection between the wide binary and the presence of warm dust is not well understood at this time. HD 35650 -------- This star is a known AB Dor member, with age ∼ 100Myr. It was observed by Spitzer, and was found to have a 70 μm excess only. report a dust temperature of 60 K and τ ∼ 1.7E-04. We find that the cold ( ∼ 45 K) dust has τ=1.5E-04, consistent with previous results. We also find that the mid-IR flux could be fit with an additional hot dust component at 536 K (Figure 1). HD 35650 was observed by the Near-Infrared Coronagraphic Imager (NICI) with angular differential imaging (ADI), but was not found to have a substellar companion. HD 43989 -------- HD 43989 is a member of Tuc Hor, and as such has an age of ∼ 40 Myr. report Spitzer observations that show an excess at 24 μm and in the IRS spectrum, but only an upper limit at 70 μm. We find a dust temperature of 112 K and a τ of 7.4E-05. The star was observed for a planetary companion by the Multiple Mirror Telescope (MMT) with spectral differential imaging (SDI), but no such companion was found. HD 54341 -------- This A0 star was presented by. The system age is unreliable; the age based on isochrones disagrees with the age based on UVW space motions (while the isochrones suggest an age of 10 Myr, the UVWs suggest an older age). At the time, there was no MIPS data point, and the dust temperature and luminosity quoted in is unrealistic. It was also observed with adaptive optics at Lick Observatory and found not to be a member of a binary system. The system was observed by Spitzer, and fit a two-component dust belt, with dust temperatures of 246K and 60K, and fractional IR luminosities of 1.9E-05 and 1.7E-04, respectively. While the χ2 values for a single- and double- belt fit were similar, we chose to fit a double-belt to be consistent with Chen et al.’s results. We find that the two belts have temperatures of 168 K and 60 K. The discrepancy in the warm dust temperatures is likely due to the fact that we often did not weight the IRAS photometry highly, due to the fact that IRAS data points are typically inconsistent with other, more reliable photometry. HD 54341 is one of our resolved systems; we find that the resolved disk size ( ∼ 185 AU) is almost twice as big as the blackbody disk size ( ∼ 106 AU). HD 76543 -------- This A5 star was initially reported by to have an IRAS excess suggestive of a disk at 85 K with τ=1.04E-04. Based on its location in UVW space (well outside of the ``good box" of ) and its location on the HR diagram, this star is estimated to have an age of ∼ 400 Myr. The star was also observed with Spitzer. report a two-component disk with a cool belt at 81 K and a warm belt at 146 K. They find fractional IR luminosities of the two systems of 3E-05 and 1.7E-05 for the cool and warm belts, respectively. The addition of Herschel data helps to constrain the dust temperatures; we now find two belts of dust - one at 49 K and one at 105 K with fractional IR luminosities of 4.8E-05 and 3.6E-05, respectively. Any discrepancy between our dust temperatures and those of are likely due to the high number of free parameters needed to fit a double belt system; a different cool dust temperature would correspond to a different warm dust temperature. The dust temperature from does not take into account any Spitzer or Herschel data. HD 76582 -------- This F0 star was reported by to have an IRAS excess indicative of a disk at 85 K and τ=2.22E-04. This relatively high fractional IR luminosity is surprising, given its 300 Myr age. The star was also observed by Spitzer. These observations revealed a two-component disk, with an inner belt at 466 K (akin to a zodiacal dust cloud) and an outer belt at 78 K (akin to a Kuiper Belt). Our Herschel observations constrain the long wavelength behavior of the cold belt. We also obtain a two-component fit to our SEDs, with an inner belt at 132 K (possibly an asteroid belt analog) and an outer belt at 51 K (a close analog of the Kuiper Belt). The cold dust emission appears to peak at Herschel wavelengths; after a fit with a 51 K curve, dust at ∼ 450 K does not appear to be needed. Longer wavelength data will help to constrain the grain properties and dust mass. This star was targeted for sub-mm follow-up by the DUNES team. HD 84870 -------- HD 84870 was first reported by to host a 100 Myr old debris disk with an IRAS excess indicative of a dust belt at 85 K. Rhee et al. listed this A3 star as a binary, but describe the two stars in question as an optical pair (sep ∼ 30$\arcsec$) based on a study of their relative motions. report a slight Si emission at 10 μm (5σ) and 20 μm (12σ). They also report 70 μm photometry that suggests a double-belt architecture, with a hot inner belt at 553 K and a cool outer belt at 67 K. Our Herschel observations are slightly inconsistent with the Spitzer data point at 70 μm, and we do not find a need for a double-belt system. We fit a single belt at 54 K with τ=4.7E-04. HD 85672 -------- This A0 star was first reported as a debris disk by with a cool belt at 79 K and τ=4.82E-04. Our Herschel results (combined with new photometry from WISE) better constrain the temperature of the observed dust. We found that the long-wavelength photometry required a modified blackbody fit (λ0=107, β=1.1), resulting in a best-fit dust temperature of 77 K and τ=6.0E-04. HD 99945 -------- HD 99945 was initially reported by as a new debris disk discovered by IRAS. While our Herschel photometry at 100 μm is inconsistent with the IRAS data point at 60 μm, we are able to fit a belt of cold dust to the emission (T=38 K). Due to the low temperature, we examined this A2 star to determine if the emission might the due to contamination with a background galaxy or IR cirrus (see Sections 4.1 and 4.2). Due to the high far-IR flux and the fact that the emission is slightly too warm do be accounted for by a background galaxy at z ∼ 1, we believe that the excess emission indicates a belt of dust at 167 AU from the host star. HD 113766 --------- HD 113766 is a well-known, extremely bright debris disk with a hot dust component. The mid-IR photometry has been well-covered in the literature. Si emission features are seen in the IRS spectra. reported the Spitzer results, and found a single belt of dust at 330 K. observed this star with VLTI/MIDI and Herschel/PACS, and determined that, in order to fit the mid-IR data simultaneously with the far-IR photometry, two spatially separated belts were needed: a hot belt within 1 AU, made of small grains which cause emission features in the mid-IR, and an outer belt at 9-13 AU. Our results independently confirm this two-belt structure, and are consistent with the findings of regarding the transient nature of the dust. We report an inner belt at 0.2 AU and an outer belt at 2.3 AU. Our estimate for the semimajor axis of the cooler dust is smaller than that of, though Oloffson et al. used a modified blackbody fit to the dust, and attempted to fit their VLTI/MIDI data simultaneously. As discussed in, the binary companion to HD 113766A is likely too far from the disk to have any serious influence on the stirring of the disk material, though point out a possible relationship between distant companions and copious amounts of warm dust. HD 121191 and HD 131488 ----------------------- These A-type stars were OT-1 targets. report mid-IR imaging and spectroscopy from Gemini/T-ReCS. Most notably, they discovered that HD 121191 and HD 131488 both have unusual emission features that peak shortward of 8 μm. This emission feature does not appear to be due silica, olivines, or pyroxenes, all which show emission features peaking between 8-12 μm. Our Herschel observations confirm the two-belt structure of the debris disk around HD 121191, with a hot inner disk at 555 K and a warm disk at 118 K. There does not seem to be a cold dust component in this system. Melis et al. fit a two-belt debris disk to the SED of HD 131488, with one inner disk at 750 K and an outer disk at 100 K. Our Herschel observations confirm this double-belt nature of the debris disk, but with an inner belt at 570 K and an outer belt at 94 K. It should be noted that, while we believe HD 121191 is at least marginally resolved in the Herschel images, confirmation is needed to consider it to be a truly resolved disk since it appears as an extreme outlier in Figure [fig:radii]. HD 124718 --------- This G5 star was initially reported by to have an IRAS excess indicative of a dust belt at 85 K and with a large fractional IR luminosity (τ=2.11E-03), which was surprising given its old age ( > 500 Myr). WISE mid-IR photometry reveals an excess at 22 μm. Our Herschel observations are inconsistent with the IRAS data, and demonstrate that there is no accompanying cold dust belt. We fit a dust temperature of 179 K and τ=2.1E-04. This discrepancy is unsurprising, since the dust temperature and luminosity reported by Rhee et al. were based on a single data point in excess, before WISE results were available. BD+20 307 --------- The excess IR emission around this well-studied close G-type binary star system was first reported by primarily based on IRAS photometry at 12 and 25 μm (the 60 and 100 μm IRAS observations only returned upper limits). obtained optical spectra to better determine the system age. Their results suggest that BD+20 307 could, in fact, be several Gyr old. Thus, the massive quantity of hot dust is not a result of ongoing planet formation, but rather could be the result of a collision between two mature planets in the terrestrial planet zone. The absence of cold dust was also noted in. The Herschel SED confirms that there is no cold dust emitting at far-IR wavelengths, and helps to constrain the temperature of the dust (417 K) and to lend support to the planet-collision model. We did not attempt to fit the Spitzer IRS spectrum, since it is dominated by emission features. Thus, we adopt the τ value of 0.032 from, since our estimate of τ ( ∼ 0.01) does not include flux from the strong silicate emission feature. Conclusions =========== We observed 24 stars with the PACS camera on the Herschel Space Observatory. Two infrared sources detected with Herschel are offset from the target coordinates by > 2$\arcsec$. These are unlikely to be due to dusty debris belts. Three targets were not detected with Herschel, and are non-excess stars. One star was detected with Herschel, but shows no evidence of an IR excess. Two stars have low IR fluxes, and could possibly be explained by contamination by a background galaxy at z ∼ 1. One target star is clearly contaminated by a background galaxy. The remaining 15 stars were examined to determine dust properties and the possibility of ongoing planet formation. A summary of the results can be found in Table 5. * Nine stars (HD 15407, HD 23514, HD 35650, HD 54341, HD 76543, HD 76582, HD 113766, HD 121191, and HD 131488) appear to have dust components at two temperatures according to SED fitting. All appear to have spatially separated dust belts. * Three stars (HD 15407, HD 113766, and BD+20 307) have disks that cannot be explained by a steady-state collisional process alone. Two other stars (HD 23514 and HD 124718) are likely explained by a transient process. The other debris disks could be explained by steady-state collisional processes. * Dust belts at HD 54341, HD 84870, HD 85672, and HD 121191 are at large enough distances from their host stars that the dust could be stirred by a yet-unseen planet or binary companion. Stirring of these disks by putative 1000 km-sized bodies may be insufficient to explain the dust production in these systems. * The stars HD 54341, HD 76543, HD 76582, HD 84870, HD 85672, and HD 99945 are spatially resolved (or probably resolved as in the case of HD 121191), and a comparison between their blackbody radii (from SED-fitting) and resolved radii show that the latter are typically larger than the former, perhaps by up to a factor as large as 10 (HD 121191). * Six stars (HD 15407, HD 23514, HD113766, HD 121191, HD 124718, and BD+20 307) show no evidence of cold ( < 100 K) dust in the Herschel data. Acknowledgements ================ This research has made use of the Exoplanet Orbit Database, the Exoplanet Data Explorer at exoplanets.org, the SIMBAD database, and VIZIER search engine, operated by CDS in France. Partial support for this work was provided by a NASA grant to UCLA and by an NSF Graduate Research Fellowship to Laura Vican. We also thank the referee for their helpful and thorough feedback. cccccccccccc 15407&11696& F5 &55&1.4&1.6&6500&4.0&80& a & M10& Y 23514&& F6 &135&1.4&1.08&6300&1.6&100& b & R08 &Y 35650&25283& K6 &18&0.8&0.63&4300&0.1&70&b&Z11& 43989&30030& G0 &49&1.1&1.05&6100&1.3&30& b & Z11 & 54341&34276& A0 &93&2.4&1.83&9500&24.1&10&c&R07 & 76543&43970& A5 &49&1.9&1.83&8200&13.4&400&c&R07& 76582&44001& F0 &49&1.7&1.68&7700&8.8&300&c&R07& 84870&48164& A3 &90&1.6&1.63&7600&7.8&100&c&R07& 85672&48541& A0 &93&1.7&1.51&8000&8.3&30&c&R07& 99945&56253& A2 &60&1.8&1.79&7600&9.4&300&c&R07& 113766&& F3 &120&1.9&1.9&6100&4.4&15&b&C05& Y 121191&& A5 &130&1.8&1.83&7700&10.4&10&b&M13& N 124718&69682& G5 &61&1&0.96&5900&0.99& > 500 &a&S03& 131488&& A1 &150&2.2&2.06&8700&21.4&10&b&M13& N BD+20 307 &8920& G0 &92&1.9&1.26&6100&1.9&1000&a&Z08& Y 60234&36906& G0 &108&1.6&3.534&5900&13.3&600&d&R07& 70298&40938& F2 &71&&1.73&6500&4.7& > 3000&d&R07& 72660&42028& A1 &100&&2.24&9500&36.1&200&c&R07& 132950&73512& K2 &30&&0.73&4800&0.25&3000&...&R07& 203562&105570& A3 &110&2.7&3.734&8800&6.5&600&c&R07& 8558&6485& G6 &50&0.9&0.94&5800&0.9&30& b & Z11 & 13183&9892& G7 &50&0.9&0.95&5700&0.8&30& b & Z11 & 80425&45758& A5 &98&&2.544&7500&18.0&300&c&R07& 191692&99473& B9 &88&3.34&6.40&10500&2680&500&c&R07&Y ccccc 15407 & OT1bzuckerm1 & 70 & 55.7 & 4.5 && 100 & 24.4 & 3.8 && 160 & 52.6 & 14.9 23514 & OT1bzuckerm1 & 70 & 24.8 & 3.5 && 100 & 83.5 & 3.2 && 160 & 60.7 & 15.2 35650 & OT2bzuckerm2 &100 & 26.8 & 1.9 && 160 & 57.0 & 25.4 43989 & OT2bzuckerm2 & 70 & 10.5 & 3.0 && 160 & 49.4 & 11.4 54341 & OT2bzuckerm2 & 100 & 297 & 3.9 && 160 & 200 & 11 76543 & OT2bzuckerm2 & 100 & 302 & 6.5 && 160 & 268 & 52 76582 & OT2bzuckerm2 & 100 & 605 & 7.3 && 160 & 485 & 40.0 84870 & OT2bzuckerm2 & 100 & 287 & 3.8 && 160 & 246.0 & 56.0 85672 & OT2bzuckerm2 & 100 & 165 & 8.1 && 160 & 137 & 72 99945 & OT2bzuckerm2 & 100 & 159 & 4.0 && 160 & 232 & 66 113766 & OT1jolofsso1 &70 & 322 & 9.7 && 100 & 201 & 6.0 && 160 & 79 & 8.0 121191 & OT1bzuckerm1 & 70 & 246.6 & 4.7 && 160 & 37.8 & 25.2 124718 & OT2bzuckerm2 & 100 & 4.2 & 5.2 && 160 & 2.4 & 17.7 131488 & OT1bzuckerm1 & 100 & 336.6 & 5.8 && 160 & 191.3 & 24.0 BD+20 307 & OT1jolofsso1 & 70 & 47 & 3.0 & & 100 & 16 & 3.0 70298 & OT2bzuckerm2 & 100 & < 10.2& & & 160 & < 23.1& 72660 & OT2bzuckerm2 & 100 & < 13.2& & & 160 & < 38.7& 132950 & OT2bzuckerm2 & 100 & < 11.1& & & 160 & < 39.6& ccccccccccccc 15407&334 ± 39&1022 ± 79&1.41 ± 0.2&0.1 ± 0.02&2.10E-03 ± 1.57E-02&9.71E-03 ± 1.57E-02&3.3E-07&6360&6.56E-07&2.5 23514&168 ± 7&1082 ± 22&3.53 ± 0.2&0.1 ± 0.004&1.03E-03 ± 1.50E-03&1.87E-02 ± 1.50E-03&3.54E-06&290&2.02E-06&0.98 35650&45 ± 3&536 ± 19&13.4 ± 1.8&0.1 ± 0.007&1.48E-04 ± 8.54E-05&2.50E-03 ± 8.54E-05&6.71E-04&0.22&4.16E-06&0.11 43989&112 ± 13&&7.24 ± 2.3&&7.43E-05 ± 1.19E-05&&8.54E-05&0.87&6.12E-07&1.0 54341&60 ± 1&168 ± 25&106.7 ± 2.9&13.6 ± 4.1&2.61E-04 ± 8.88E-06&1.95E-05 ± 8.88E-06&1.64E-02&0.02&4.66E-04&8.7 76543&49 ± 7&105 ± 10&119.2 ± 20.2&25.9 ± 0.3&4.81E-05 ± 3.98E-06&3.64E-05 ± 3.98E-06&8.65E-04&0.06&1.07E-04&6.1 76582&51 ± 3&132 ± 33&89.1 ± 10.7&13.29 ± 5.5&1.71E-04 ± 1.19E-05&3.20E-05 ± 1.19E-05&7.92E-04&0.26&2.13E-04&4.5 84870&54 ± 2&&75.1 ± 5.4&&4.71E-04 ± 3.39E-05&&1.78E-03&0.26&4.17E-04&4.2 85672&77 ± 6&&37.9 ± 5.6&&5.99E-04 ± 1.03E-04&&1.11E-03&0.54&1.35E-04&4.2 99945&38 ± 3&&166.5 ± 27.7&&5.93E-05 ± 3.52E-06&&3.14E-03&0.02&2.58E-04&4.5 113766&267 ± 5&1209 ± 213&2.3 ± 0.11&0.2 ± 0.07&2.24E-02 ± 2.39E-03&1.05E-02 ± 2.39E-03&4.12E-06&5440&1.88E-05&2.0 121191&118 ± 2&555 ± 16&18.1 ± 0.5&0.8 ± 0.06&2.59E-03 ± 1.22E-04&2.09E-03 ± 1.22E-04&5.10E-04&5.1&1.34E-04&4.99 124718&179 ± 49&&2.4 ± 1.8&&2.14E-04 ± 3.05E-05&&4.16E-07&514&1.98E-07&0.9 131488&94 ± 1&570 ± 10&41.0 ± 0.8&1.1 ± 0.05&2.73E-03 ± 1.13E-04&2.78E-03 ± 1.13E-04&2.02E-03&1.4&7.22E-04&8.4 BD+20307&417 ± 14&&0.7 ± 0.05&&9.78E-03 ± 4.80E-03&&2.00E-08&490000&7.74E-07&0.9 ccccccc HD 54341 &100& 185 ± 18 & 2.0 ± 0.2 & 29 ± 20 & 72 ± 41&321.6 HD 76543 &100& 162 ± 11 & 3.3 ± 0.2 & 69 ± 9 & 86 ± 7&311.1 HD 76582 &100& 216 ± 6 & 4.4 ± 0.1 & 66 ± 3 & 103 ± 3&637.5 &160& 235 ± 33 & 4.8 ± 0.7 & 74 ± 15 & 103 ± 12&444.4 HD 84870 &100& 252 ± 16 & 2.8 ± 0.2 & 35 ± 10 & 147 ± 18&301.8 HD 85672 &100& 186 ± 28 & 2.0 ± 0.3 & 28 ± 36 &N/A &172.8 HD 99945 &100& 198 ± 18 & 3.3 ± 0.3 & 28 ± 16 & 92 ± 35&166.8 HD 121191 &70&195 ± 261 & 1.5 ± 0.2 & 40 ± 17 & 25 ± 27&295.6 c\|c\|c\|c\|c\|c\|c 15407&&Y&&&Y&Y 23514&&Y?&&&Y&Y 35650&&&&Y&Y& 43989&&&&Y&&N 54341&Y&&Y&&Y&N 76543&Y&&&Y&Y&N 76582&Y&&&Y&Y&N 84870&Y&&Y&&&N 85672&Y&&Y&&& 99945&Y&&&Y&& 113766&&Y&&&Y&Y 121191&Y&&Y&&Y&Y 124718&&Y?&&Y&& 131488&&&&Y&Y&Y BD+20307&&Y&&&&Y ![Spectral energy distributions (SEDs) of OT1 and OT2 stars. Green data points are B, V, J, H, and K flux densities from the Hipparcos and 2MASS catalogs. Dark blue data points are WISE data (3.4, 4.6, 11, and 22 \mum). Red data points are from IRAS. Cyan data points and blue spectra are from Spitzer. Finally, magenta data points are Herschel data points. Stellar temperature and radii were derived by fitting PHOENIX model stellar photospheres to photometric points. IR data is fit with one or two simple blackbodies for most targets, except in the case of HD 85672, which required a modified blackbody fit (red and blue dashed curves).](figure1 "fig:") [fig:SEDs1] ![Cont’d](figure2 "fig:") [fig:SEDs2] ![Cont’d](figure3 "fig:") [fig:SEDs3] ![Radial profiles were created to determine whether a disk was actually resolved. We compared the radial profile of the observed emission to a reference PSF (Alf Cet).](radialprofiles_hires "fig:") [fig:profs] ![Resolved disks were fit with a narrow ring of dust with four parameters: dust semi major axis, position angle, inclination, and brightness. ``PSF-subtracted" images were created by subtracting a stellar PSF which was created using a bright standard star (Alf Cet) and scaled to match the peak flux of the source. All images include the same square-root-stretch scale.](resolved1 "fig:") [fig:mapsA] ![Cont’d](resolved2 "fig:") [fig:mapsB] ![Left panel: Green points represent stars compiled by which were observed in scattered light. Orange points represent stars in whose ``resolved" radii are determined by emission feature - fitting in the SED. The solid black line represents a best-fit to the data from Booth et al. and Rodriguez et al. Right panel: Pink points represent stars compiled by which were resolved in thermal emission. Blue points represent stars from which were resolved by Herschel. Red points are stars in this work that are resolved at 70 or 100 \mum (see Section 6.1). The solid black line represents a best-fit to the data from Booth et al. and Rodriguez et al.. The large scatter in the red points about this best-fit line may be due to the fact that the stars in our sample with the largest discrepancies between R_{img} and R_{BB} are also the ones that we suspect may result from planet stirring.](figure4b "fig:") [fig:radii] ![Comparison of the fractional IR luminosity (\tau) of disks in our sample to the maximum dust luminosity achievable through steady-state collisions (\tau_{max}). If \tau/\tau_{max} >1000 (red dashed line), the dust was likely produced in a transient event such as a catastrophic collision. Stars meeting this criterion are labeled. We also consider that disks with \tau/\tau_{max} >100 (black dashed line) could have been created in a catastrophic collision.](taumax "fig:") [fig:taumax] ![Disk Radius vs. stellar age for stars in our sample. Curves represent predicted maximum radial separation of 1000km-sized bodies at a given stellar age (see Equation 4). ``x" is a scaling factor relative to the minimum mass solar nebula. Data points (circles) represent our resolved disks. Triangles represent unresolved targets for which we applied a scaling factor to the blackbody radius in order to obtain more realistic R_{disk} values. See section 7.1.1 for further discussion of this figure.](figure5b "fig:") [fig:1000km] ![Comparison of the characteristic temperatures of disks in our sample to those in the Spitzer Debris Disk Catalog. We find that our disks are colder, on average, than those in the Spitzer catalog. This makes sense, since Herschel is most sensitive to colder disks. Notably, we do not find a peak in the distribution of temperatures from the disks in the Spitzer catalog that corresponds to the sublimation temperature of icy planetesimals (\sim150 K). We do, however, note a peak around 60 K. The build-up of disks at 500 K in the single dust belt sample is due to the fact that Chen et al. assigned a dust temperature of 500 K for anything that appeared to be 500 K or hotter.](figure8 "fig:") [fig:Thist] ![The temperature ratio (R_{T}) between the inner and outer disks of double-belt systems can be used to distinguish spatially separated belts from single-belt systems with two grain populations. We conservatively assume that any systems with R_{T}>5 (dashed line) are truly spatially separated systems (see Section 8.1). Colored circles represent double-belt systems from our Herschel sample (OT1 and OT2). Blue triangles represent stars from with known (resolved) double-belt systems. Black crosses represent stars from with double-belt fits to the dust SEDs, but for which it is not known whether there are two truly separated belts present. The green square is HD 181327, which is known to host a single belt of dust with two different temperature grain populations.](figure10 "fig:") [fig:doubles] Herschel Observations of Dusty Debris Disks =========================================== We present results from two Herschel observing programs using the Photodetector Array Camera and Spectrometer. During three separate campaigns, we obtained Herschel data for 24 stars at 70, 100, and 160 μm. We chose stars that were already known or suspected to have circumstellar dust based on excess infrared emission previously measured with IRAS or Spitzer, and used Herschel to examine long-wavelength properties of the dust. Fifteen stars were found to be uncontaminated by background sources, and possess infrared emission most likely due to a circumstellar debris disk. We analyzed the properties of these debris disks to better understand the physical mechanisms responsible for dust production and removal. Seven targets were spatially resolved in the Herschel images. Based on fits to their spectral energy distributions, nine disks appear to have two temperature components. Of these nine, in three cases, the warmer dust component is likely the result of a transient process rather than a steady state collisional cascade. The dust belts at four stars are likely stirred by an unseen planet, and merit further investigation. Introduction ============ Debris disks are signposts of planetesimal formation and, as such, are crucial subjects for study when considering the evolution of a planetary system. Debris disks are typically identified via an excess infrared (IR) flux above the stellar photosphere. New debris disks have been discovered with five satellites, starting with the InfraRed Astronomical Satellite (IRAS) discovery of the first debris disk around Vega in 1984. In 1995, the Infrared Space Observatory (ISO; ) imaged the sky at wavelengths ranging from 2.5 to 240 μm, providing low-resolution long wavelength photometry. Subsequently, the Spitzer Space Telescope provided mid-IR spectroscopy with the Infrared Spectrograph (IRS; ), and photometry with the Multiband Imaging Photometer (MIPS; ), leading to the detection of over 100 new debris disks (e.g.,,, ). The Wide Field Survey Explorer (WISE; ) imaged the sky at 3.4, 4.6, 11, and 22μm, and was also most sensitive to warm and hot debris disks that peak in the mid-IR. In 2009, the Herschel Space Observatory began taking data, providing far-IR photometry with the Photodetector Array Camera and Spectrometer (PACS; ) and spectroscopy with the Spectral and Photometric Imaging Receiver (SPIRE; ) and the Heterodyne Instrument for the Far Infrared (HIFI; ). Where Spitzer was uniquely able to detect warm ( > 100 K) debris disks in the terrestrial planet zone, Herschel was sensitive to cooler disks ( < 100 K) at larger radial separations from their host stars. In this paper, we present Herschel observations of 24 stars initially identified with IRAS and/or Spitzer as definitely or possibly possessing a debris disk. One goal of our project was to search for cold dust components that would peak near the Herschel PACS wavelengths (70, 100, and 160 μm). In the case that there is no separate cold dust component present, Herschel photometry helps to better characterize the Rayleigh-Jeans tail of thermal emission from warm dust. Another goal was the identification of disks with double-belt debris systems; that is, systems containing an inner belt of warm or hot ( > 100K) dust and an outer belt of cold ( < 100 K) dust. Such systems would be a direct analog of our own Solar System, which hosts a Kuiper Belt that lies between 30-50 AU at ∼ 50 K and an Asteroid Belt at 3 AU and ∼ 175 K. A double-belt system may also be a signature of a planet (or planets) that lie in the gap between dust belts. Such systems have been discovered around HR 8799 (,, ) and HD 95086 (, ). Stellar Sample ============== The sample used in this work is an amalgamation of stellar samples from three different Herschel proposals (bzuckerman-OT1, jolofsson-OT1, and bzuckerman-OT2). In the first OT1 proposal (PI: B. Zuckerman), we observed A-F stars with known, very luminous mid-IR emission. Since mid-IR emission is known to originate from the terrestrial planet region, we wanted to use Herschel to search for accompanying cold dust in outer regions analogous to the Solar System’s Kuiper Belt. Four such stars were observed during this program. The second OT-1 proposal (PI: J. Olofsson) had very similar science goals, in that the authors were looking for cold dust components accompanying known warm debris disks. Six stars were observed for their program, of which we present two which fit into our initial selection criteria (luminous mid-IR emission). The objective of the OT-2 proposal was to follow up on a subsample of stars (from and ) that had been observed either with IRAS or Spitzer, and that had only one far-IR data point at 60 μm (IRAS) or 70 μm (Spitzer). Herschel observations were carried out to characterize the far-IR emission of the dust and, in the case of stars with apparent excess emission detected only at 60 μm with IRAS, to confirm or deny the existence of a dusty debris disk. IRAS is not only less sensitive than Herschel, but its large beam size made IRAS vulnerable to confusion by background sources (the Herschel PSF is 5.75′′ at 70 μm, while the average detector element size for IRAS at 60 μm was 5′ × 2′). Eighteen stars were observed during this OT-2 program. We collected information from the literature on binarity, distance from Earth, and stellar age. Literature data for target stars can be found in Table [table:stellar]. Observations ============ We observed 24 stars in total with PACS; simultaneous observations were obtained at 160 μm and either 70 μm or 100 μm. Although some stars were not detected at 160 μm, a 3σ upper limit at that wavelength helped to constrain cold dust temperatures. The observations were taken with a scan speed of 20$\arcsec$/sec. We used HIPE (Herschel Interactive Processing Environment, version 12.0; ) to reduce the data and produce the final maps. We used a pixel scale of 1$\arcsec$/pixel for 70 and 100 μm data and 2$\arcsec$/pixel for the 160 μm data. A high pass filter was applied to remove instrumental noise. To achieve the highest signal to noise in the resulting maps, we used a high pass filter radius of 30$\arcsec$ for 70μm data and 70$\arcsec$ for 100 and 160μm data. The larger radius filter was used for the longer wavelength data in order to be more aggressive in removing instrumental noise at those wavelengths while maintaining a high SNR. A mask was applied to avoid removing any flux within 15" of the target star. Aperture photometry was carried out with an aperture radius of 5$\arcsec$ for unresolved sources and 10$\arcsec$ for resolved sources, with a sky annulus extending from 20-40$\arcsec$. Aperture corrections were made to account for photospheric flux falling outside of the chosen aperture$\footnotemark{}$$\footnotetext{All corrections were taken from Herschel Release Note PICC-ME-TN-037, Table 15.}$. Errors on the photometric points are derived by placing apertures on empty regions of sky along the sky background annulus and measuring the r.m.s. of the background flux. Some targets presented flux at a significant offset from the stellar location. If the offset was larger than the pointing accuracy of Herschel ( ∼ 2$\arcsec$), we excluded it from further analysis. This was the case for two stars in our target list (HD 60234 and HD 203562). These two stars were taken from, and were 60μm excesses only. Observed fluxes are found in Table 2. We discuss potential sources of contamination in Section [contamination]. Herschel Non-Detections ----------------------- A non-detection with Herschel implies that any circumstellar dust would necessarily be < 35K, extremely tenuous (LIR/Lbol=τ < 10− 6), or both. The three stars mentioned in this section were not detected in any of the Herschel bands, and are not considered in the analysis in later sections. ### HD 70298 This star was reported by as a new debris disk candidate with a surprisingly substantial 60μm excess (τ=3.54E-04), given its old age ( > 3Gyr). The star was not observed by Spitzer, but a WISE excess was identified by at 22 μm (τ=4.35E-04). Close inspection of the WISE images shows that there is a nebulous IR source ∼ 15$\arcsec$ away from the target, which is inside the WISE contamination radius at 22 μm (2 × FWHM = 24$\arcsec$; ). We therefore consider the WISE and IRAS images to be contaminated. In the absence of any detectable flux at Herschel wavelengths, we report no IR excess around this star. ### HD 72660 This star was reported by as a new debris disk candidate based on an IRAS excess. The star was subsequently observed with Spitzer, and a mid-IR spectrum showed no IR excess consistent with the IRAS data point. WISE also saw no evidence of any warm excess in the near-IR. Even so, it remained possible that the IRAS photometry was catching the Wein tail of cold dust ( < 50 K), until our negative Herschel observations. ### HD 132950 This star was reported by as a new debris disk candidate with a large IR excess (τ=1.17E-03) and an old age ( ∼ 3 Gyr). A Spitzer IR spectrum shows no evidence of an IR excess consistent with the IRAS data point, and no warm excess was seen by WISE. In the absence of an IR excess seen with Herschel, we report no detectable dust around this star. Systems with No Detectable Dust ------------------------------- In addition to stars with no detectable flux at Herschel wavelengths, there is one system (HD 191692) for which Herschel detected the photosphere of the star, but no evidence of an IR excess was seen. This star was reported by to have a small (τ < 10− 5) IR excess seen at IRAS wavelengths, but unconfirmed by Spitzer. Their dust model fit to the IRAS photometry suggested the existence of dust at over 200 AU separation from the central star. reported the star as a binary with a separation of < 1 AU, meaning that the dust, if confirmed, would be circumbinary. report a large IR excess (τ > 10− 3) seen at WISE wavelengths. A close inspection of the WISE images shows a source ∼ 18$\arcsec$ to the NE of the target star, which is within the contamination radius of the WISE beam at 22μm. Thus we consider both the IRAS and the WISE images to be contaminated, and given that the Herschel images show no evidence of an IR excess, we report this star to have no detectable dust. Sources of Possible Contamination ================================= Of the remaining targets (those with detectable Herschel fluxes indicative of an IR excess), four (HD 8558, HD 13183, HD 80425, and HD 99945) have apparent cold dust disks with temperatures < 40K (see Section [SEDs]). To ensure that we are really seeing evidence of cold debris belts, we investigated several alternative sources of the apparent IR excess. Extragalactic Background ------------------------ presented Herschel observations of cold debris disks and investigated the possibility of confusion with a background galaxy. A typical galaxy below the confusion limit of Herschel ( ∼ 2.5mJy at 160 μm) would lie between z=0.94 and 1.2 and its IR emission would correspond to an apparent dust temperature which would peak between 20 and 29 K. Of the four stars with Herschel emission at 160μm, only two (HD 8558 and HD 80425) have F160 < 10 mJy and could be explained by confusion with a background galaxy at z ∼ 1. In addition, the position of a third star (HD 13183) is 2$\arcsec$ away from a galaxy detected with GALEX. Since the galaxy is within the confusion beam radius of PACS at 100μm (7.19$\arcsec$; ), we consider this target star to be contaminated, even though the source flux is 40mJy at 160 μm. These three stars are not included in Table [dustpars]. IR Cirrus --------- We must also investigate the possibility that excess IR emission is due to confusion with background cirrus, which is known to emit at ∼ 20 K. We checked each of our cold excess systems (HD 8558, HD 13183, HD 80425, and HD 99945) for evidence of cirrus in the Herschel and WISE images. HD 99945 is not explained by confusion with a background galaxy or nearby IR excess source. We also found no detectable cirrus in the Herschel or WISE images. This, combined with the fact that the 160 μm flux for this object is > 100mJy implies that the excess around the star is due to emission from a debris disk. HD 99945 is therefore included in Table 3. Of the four potential cold disks in our sample, only HD 99945 (the warmest of the four) survived to the final sample. Spectral Energy Distributions ============================= Spectral energy distributions (SEDs) were created using a fully automated photosphere-fitting technique that made use of the PHOENIX models. Stellar photospheres were fit using B, V, J, H, and K fluxes, while the mid- and far-IR photometry was used to fit the dust emission. We fit a simple blackbody to the IR photometry using Spitzer, WISE, and IRAS to supplement the Herschel data points whenever possible. For some systems, the long wavelength data (at 100 or 160 μm) falls below a simple blackbody curve. In these cases, we applied a modified blackbody described by: $$F\_{\nu}\propto\left(\frac{\nu\_{0}}{\nu}\right)^{\beta}B\_{\nu}(T\_{d})$$ where Td is the dust temperature. Uncertainties for Td are found using a Monte Carlo approach. For each SED, we generated 1000 simulated data sets by randomly drawing flux values from a single Gaussian distribution centered on the observed fluxes, where the width of the distribution is given by the observed uncertainty. The values and uncertainties for Td reported in Table 3 are the average and standard deviation of the ensemble of fits for each object. The resulting SEDs are found in Figure [fig:SEDs1]. Stellar Parameters ------------------ We used the best-fit model photospheres to determine the stellar temperature and radius. From there, we calculated a stellar luminosity using L\=4πR\2σTeff4. We assumed solar metallicity and log(g) for all 24 stars. Stellar masses were taken from the literature. Ages for stars in our sample were taken from moving group membership whenever possible. When moving group membership could not be determined, we used literature ages from stellar isochrones (for A-F-type stars). For all other stars, we relied on literature ages based on lithium abundance or X-ray activity. Stellar parameters are found in Table [table:stellar]. Dust Parameters --------------- For the 15 stars in our sample with dust detected by Herschel, we obtain a dust temperature and fractional IR luminosity (LIR/Lbol=τ) from a blackbody SED fit. Assuming blackbody dust grains, the orbital semi-major axis is: $$R\_{BB}=\frac{R\_{\}}{2}\left(\frac{T\_{\}}{T\_{BB}}\right)^{2}$$ For the purposes of this work, we define three dust temperature regions; hot dust ( > 200K), warm dust (100-200K), and cool dust ( < 100K). These temperature regions were chosen to correspond to the solar system’s zodiacal dust, asteroid belt, and Kuiper belt. Of the 15 systems in our sample that were detected by Herschel and were found to be uncontaminated, 9 stars show a cool dust component, 3 stars show a warm dust component, and 10 stars show a hot dust component (some stars show multiple dust components). It should be noted that the dust temperatures provided in Table [dustpars] are the best-fit values from SED fitting described in Section [SEDs]. In some cases, the best fit dust temperature is reported as being > 1000 K. We examined the variation in χ2 values for our SED fits as we varied the dust temperatures between 600-1200K, and find that the χ2 values drop significantly as one increases the dust temperature from 600 to 1000 K, but flatten out after 1000 K. Such model-suggested dust temperatures of $\gtrsim$1000 K are difficult to explain (especially since the hottest dust temperature around an extremely dusty star known to this point is the ∼ 800 K dust around V488 Per, reported by ) and we acknowledge that a one or even two ring model for these debris disks is very likely an over-simplification of the actual dust configuration. We calculated the blowout radius for a dust grain, ablow, defined as: $$a\_{blow}=\frac{3L\_{\}Q\_{PR}}{16\pi GM\_{\}c\rho}$$ where QPR is the radiation pressure coupling coefficient and ρ is the density of a typical dust particle. We assume QPR ∼ 1 for 2πa/λ > 1 and ρ ∼ 2.5 g cm− 3. Finally, we calculated the minimum mass of dust in the disk. This parameter is best measured using far-IR or sub-mm data, which probes the largest grains in the system. The large grains are where most of the mass of the dust is concentrated. However, we can calculate a minimum dust mass from Equation 4 from : $$M\_{d, min}\geq\frac{16}{3}\pi\tau\rho R\_{d}^{2}\langle a\rangle$$ where ⟨a⟩ = (5/3)ablow and Rd is the dust semi-major axis. Minimum dust mass values are calculated using the outermost dust belt parameters only, since the mass of the colder dust dominates the system in all cases. Resolved Disks ============== To determine whether a disk was resolved, we compared the radial profile of the star+disk to a reference PSF (see Figure [fig:profs]). To create the radial profile, we binned the pixels of each raw image by radius. The error bars in Figure [fig:profs] are the standard deviation of the fluxes within each radial bin. We consider each disk presented in Figure [fig:profs] to be resolved$\footnotemark{}$$\footnotetext{\scriptsize HD 121191 is marginally resolved, and further discussion can be found in Section \ref{HD121191}.}$. We performed a PSF subtraction and modeled the residual flux with a single narrow ring of dust (ΔRd=0.1Rd). The bright star α Cet was used as a reference PSF. The free parameters fit in our ring model were the semi-major axis, inclination, and position angle of the narrow ring. This model (convolved with the instrument PSF) was appropriate for all of our resolved systems, since residual maps showed no significant structure once the ring model was subtracted. Raw images, PSF-subtracted images, and residuals can be found in Figure [fig:mapsA]. Of the 15 stars in our sample, seven (HD 54341, HD 76543, HD 76582, HD 84870, HD 85672, HD 99945, and HD 121191) were resolved at either 70 or 100 μm. One system (HD 76582) was resolved at both 100 and 160 μm. Disk parameters determined by ring-fitting can be found in Table [resolvedtable]. Errors were determined by varying parameters until residuals increased by 1σ. Disk Radii ---------- The blackbody disk semi-major axes derived from SED fitting (RBB) often did not agree with those observed in the resolved images (Rimg). We define a semi-major axis ratio fR=Rimg/RBB. It is well-known that small blackbody particles tend to be super-thermal and thus blackbody SED-fitting will underestimate radial extent of the dust, often by a factor of 2-5. The steeper the size distribution of the dust grains, the more small grains are present in the disk, and the disk will appear much hotter than a blackbody. At first glance, then, it would seem that fR can probe the size distribution of the dust. If, however, the dust production mechanism is similar among all of our disks (see Sections 6-7), then it is reasonable to assume that the resulting size distribution would be similar as well. If that is true, the biggest factor affecting fR would be the spectral type or luminosity of the host star. If the host star is very luminous, it will remove the smallest grains from the system via radiation pressure; this will make the dust appear to act more like a blackbody. We compared our radius ratios to those of similar programs (,, ) in Figure [fig:radii]. We find that, in most cases, fR increases for lower-luminosity stars. One interesting feature to note in Figure [fig:radii] is that the value of fR (=Rimg/RBB) is well constrained for disks that are resolved in thermal emission, but there is much more scatter in the relationship between fR and Lbol for stars whose ``resolved" radii are derived from SED modeling of the Si emission (see Figure [fig:radii]) and/or scattered light. This may be because the scattered light images are most sensitive to the smallest grains, which can be found in a large, extended disk or halo, rather than in a narrow ring as described by a blackbody fit. It appears that the stars in our sample (red data points in Figure [fig:radii]) have a higher scatter in Lbol- fR space than do stars from. Notably, the biggest outliers (HD 121191 and HD 85672) are those stars for which we believe an unseen planet could be responsible for dust production (see Section [planetstirred]). Since the ratio fR depends heavily on the size of the grains themselves, it is possible that a planet-stirred disk has an inherently different grain size distribution than a self-stirred disk. Different grain size distribution could arise from a number of factors, including the collisional velocities of the < 100km bodies that produce the dust, and the composition of the dust itself. It is also possible that the Herschel images are sensitive to small grains further from the star (which would peak at Herschel wavelengths), while the average temperature of the grains in the system is actually higher than the temperature implied by the blackbody fit to the SED. Dust Production and Planet Formation ==================================== Having determined dust properties from the Herschel photometry, we turn our attention to the way in which the observed dust was produced. By the time a star is ∼ 10 Myr old, any primordial dust should have been completely depleted (used to form planetesimals, accreted onto the star, or blown out of the system by radiation pressure).$\footnotemark{}$ $\footnotetext{\scriptsize This idea has been challenged by recent discoveries of 30-40 Myr old disks with copious molecular gas present, suggesting that, for some stars, the protoplanetary stage may last longer than previously thought (e.g. \citealt{Kospal\_2013}, \citealt{Zuckerman\_2012}). Still, since there are so few stars that exhibit gas at 30-40Myr, we maintain that most primordial gas and dust should be gone by $~$10 Myr.}$ Thus, the dust we see in debris disks older than ∼ 10 Myr is likely second-generation, created in collisional processes. Furthermore, the lifetime of this dust against radiation pressure and drag forces is shorter than the lifetime of the star. This means that any dust seen in these systems must be replenished regularly. There are various mechanisms by which dust can be produced as a planetary system forms and evolves. First, the dust can be produced through steady-state collisions of small planetesimals. This occurs naturally as a final stage in the process of planet formation. Planet formation proceeds first through the growth of ∼ 1 km-sized planetesimals. Once the planetesimals grow large enough to start gravitationally focusing the primordial dust in their path, runaway growth can occur. Following this, a few large bodies accrete a large amount of dust, and as they grow, they become the dominant accreter in their orbital path - a process known as oligarchic growth. Once ∼ 1000 km-sized bodies have formed, they can begin to dynamically stir the remaining smaller ( < 100 km) planetesimals, driving them to collisional velocities high enough to be destructive. It is in these destructive collisions that the observed dust is produced. The resulting debris is collisionally ground down until the grains are small enough to be blown out of the system via radiation pressure. On their way out of the system, they can also collide with small planetestimals, creating an outward-moving collisional cascade. In the process described above, the stirring mechanism at work is the natural formation of 1000 km-sized bodies in the disk itself (hence, the disks are ``self-stirred", see Section [selfstirred]). However, one can also imagine a situation where small ( < 100km) bodies are stirred by a nearby planet (see Section [planetstirred]). In either case, one would expect moderate fractional IR luminosities (τ ∼ 10− 4) compared to those associated with protoplanetary disks (τ ∼ 10− 3). Alternatively, a belt of dust can be created through a giant impact between planetary embryos (e.g. ; see Section [giantimpacts]). In an even more extreme scenario, two fully formed rocky planets can undergo a catastrophic collision, resulting in large amounts of debris. Such a collision has been proposed to explain warm dust in orbit around BD +20 307 (, ). Such catastrophic collisions should produce a multitude of small grains. Similar collisions are also more likely to occur in the terrestrial planet formation zone, where disk surface density is high; such collisions result in production of warm or hot dust. A third possible source of hot dust in disk systems might be debris left behind by so-called ``star-grazing comets." This mechanism was suggested by who found that hot dust in a sample of Spitzer-observed systems had a characteristic temperature of 190K - near the sublimation temperature for icy bodies such as comets (see Section [comets]). Distinguishing Transient from Steady-State Events ------------------------------------------------- To distinguish dust created in a transient event from dust created in a steady-state process, one can examine the maximum fractional IR luminosity that can be attained through a steady-state process (τmax). This method was explored in. We examine the τmax method of. According to their Equation 20: $$\tau\_{max}=0.58 \times 10^{-9} R\_{d}^{7/3}\frac{\Delta R\_{d}}{R\_{d}}D^{1/2}Q^{5/6}e^{-5/3}M\_{\}^{-5/6}L\_{\}^{-1/2}t^{-1}$$ As in Equation 4, Rd is the semi-major axis of the dust, and here ΔRd is the radial width of the disk, D is the diameter of the largest planetesimal in the disk, Q is the specific incident energy required to destroy a particle, e is the eccentricity of the dust ring, and t is the stellar age. For our calculations, we use the outermost dust component (in cases where two components are needed to fit the observed IR flux). In all following calculations, if the dust was spatially resolved, we use the dust semi-major axis from the ring-fitting algorithm (Rimg, see Section 6.1). Otherwise, we use RBB from the SED fits. To model a steady-state collisional cascade, we follow the prescription of and assume that D=2000km, Q=200 J kg− 1, e=0.05, and ΔRd=0.1Rd. According to Wyatt et al., one should expect that observed dust was likely created in a transient event if τ/τmax > 1000. The τ/τmax > 1000 threshold is based on observations of known very luminous debris disks, while taking into consideration the assumptions that went into Equation 5. For the purposes of our analysis, we assume that debris disks with τ/τmax > 100 are likely to be transient in nature, while those with τ/τmax > 1000 cannot be explained by a steady-state collisional process alone (see Tables 3 and 5). We calculated τmax for the stars in our sample and found that three debris disks systems could not be explained by a steady-state process alone (all from OT1, and thus with the highest fractional IR luminosities and previously suspected to be the result of transient processes.); HD 15407, HD 113766, and BD +20 307. Since these are well-studied systems, we do not discuss their transient nature further in this paper. Furthermore, the disks at HD 23514 and HD124718 had 100 < τ/τmax < 1000, implying transient processes. Steady-State Collisions - Stirring Mechanisms --------------------------------------------- If the observed dust is produced in a steady-state collisional cascade, we would like to identify the trigger mechanism that starts the cascade. We consider collisions triggered by perturbations due to 1000 km-sized bodies within the disk itself (self-stirring), and collisions triggered by a nearby planet or distant stellar or substellar companion (planet-stirring). ### Self-Stirring In the case of self-stirring, 1000 km-sized bodies naturally form in the disk which dynamically stir the population of smaller ( < 100 km-sized) bodies to velocities high enough to cause destructive collisions. These initial collisions then trigger a collisional cascade which leads to the production of copious amounts of dust. Since the self-stirring mechanism requires the existence of 1000 km-sized bodies at approximately the same radial location of the dust, we can place limits on the parameters of a disk stirred by this mechanism. In Figure [fig:1000km], we examine the semi-major axis of 1000 km-sized bodies as a function of stellar age. Following the models of, we assume that 1000 km-sized planetesimals form at a time: $$t\_{1000}=145x\_{m}^{1.15}\left(\frac{R\_d}{80 AU}\right)^{3}\left(\frac{2M\_{\sun}}{M\_{\}}\right)^{3/2} [Myr]$$ As in, we vary two parameters in our models; the mass of the host star and a scaling factor (xm) related to the initial mass of the protoplanetary disk (where xm=1 corresponds to the minimum mass solar nebula). We plot several models for the star-disk system in Figure [fig:1000km], representing stars from 1.5-2.5M$\_{\sun}$ and a range of xm. For self-stirring to be responsible for the dust observed around the four labeled stars, the initial disk would need to be 30 times as massive as the minimum mass solar nebula or more. Since ∼ 30 × MMSN is approximately the protoplanetary disk mass of a high-mass star, we take this as an upper limit for our models. None of the stars in our sample have detected planets. ### Planet Stirring For disks that are unlikely to have formed 1000 km-sized bodies at the radial location of the observed dust, planet-stirring offers another mechanism for dust production (major planets could have formed near the dust at an earlier time). Since debris disks are often considered evidence of planet formation, one might expect a correlation between the existence of planets and detection of circumstellar dust. Unfortunately, with so few stars observed to have both a dust disk and at least one planet, this relationship has proved difficult to study. found little correlation between planet hosts and detection of an IR excess (the typical age of stars in their sample was ∼ 6 Gyr, whereas the stars in our sample are significantly younger). More recently, examined a large ( > 200 star) sample of Herschel-observed stars to look for correlations between the presence of a debris disk and: (1) the presence of low-mass planets, (2) the presence of high-mass planets, (3) metallicity, and (3) the presence of one or more stellar companions. Even with their large sample size, found no significant correlations between any of the aforementioned parameters. In our sample of 24 stars, none have detected planets (exoplanets.org; ). One can try to draw conclusions about the properties of a hypothetical unseen planet based on the observed dust properties. Unfortunately, it is difficult to resolve the degeneracy between planet mass, orbital eccentricity, and semi-major axis using simple models. This issue is treated in more detail in a theory paper motivated by the present results. It is possible that a belt of small planetesimals could be stirred up to collisional velocities by a distant stellar-mass companion (or brown dwarf) as opposed to a nearby planet. However, found that stars with debris disks are less likely to be found in binary systems, at least in a sample of FGK stars. This may be attributable to the fact that a companion will accelerate the evolution of the dust, making a debris disk detectable for a shorter period of time, earlier on in the evolution of the system. In any case, we do not consider companion stirring to be the cause of the disks in our sample. This possibility is also examined further in. Star-Grazing Comets ------------------- One suggested explanation for the existence of warm and hot dust disks is that so-called ``star-grazing comets" leave behind a cloud of debris. Should this be the case, one would expect most of this cometary debris to be at a radial separation from the star such that the dust is at a temperature of ∼ 150 K (the sublimation temperature for icy planetesimals). In Figure [fig:Thist], we compare the temperature of our disks to those of (Spitzer Catalog of Debris Disks). found that, in a sample of ∼ 70 stars, most of the warm dust components fell around ∼ 190 K.$\footnotemark{}$ $\footnotetext{\scriptsize The reason the dust temperature is higher than the sublimation temperature is that once the comets sublimate and leave behind grains, those grains appear to be hotter than the original comet because they are not emitting like blackbodies.}$ We do not find a similar trend in the Spitzer catalog taken as a whole. The disks in our Herschel sample are mostly cold ( < 100 K), peaking around 60 K. In fact, the Spitzer data from also shows a peak around 60K. A similar peak is found in the sample of, but no explanation was put forward. Perhaps this 60 K peak is simply an observational bias (a 60K dust belt would peak between 70-100 μm, close to both Spitzer and Herschel filter wavelengths). It may also be that disk detectability falls off at lower temperatures, producing a false ``peak" near 60K. also identified a peak at 60 K in a Spitzer sample of debris disks, but claim that the peak is not solely due to the dust temperature (see their Figure 5). They suggest that the cold dust temperatures follow a trend with spectral type; hotter, earlier-type stars have warmer outer debris belts. This implies that the peak at 60 K is not due to a temperature-dependent phenomenon such as sublimation. Regardless of the exact distribution of temperatures among our debris disks, we do not find a peak in dust temperature near the sublimation temperature for comets, and thus conclude that debris left by star-grazing comets does not contribute significantly to the dust seen in the present sample of stars or in the Spitzer Catalog of Debris Disks. Giant Impacts and Catastrophic Collisions ----------------------------------------- It is possible that the observed warm dust components are created during a giant impact between two planetary embryos in the terrestrial planet formation zone. It is generally believed that such impacts are an important part of our own Solar System’s history; a collision between the early Earth and a Mars-sized planetary embryo was likely responsible for the formation of our Moon. We also have evidence of large catastrophic collisions around stars other than our Sun; such is the case with BD+20 307 (, ). Even if the observed dust luminosity is smaller than τma*x (see Section [distinguishing]), the dust could still have been created in a catastrophic collision, but one in which the parent bodies did not entirely pulverize one another, and much of the mass of the parent body survived in large fragments. Two-Temperature Systems ======================= To determine whether the debris disks in our sample require a two-component fit, we examined the χ2 values for a single- and double- belt fit to the SEDs. In about half of the targets, we found that a double-belt fit resulted in a significantly lower χ2. In one case (HD 76543), we do not find any significant difference in the χ2 value for a single versus double belt fit, but decided to fit two belts in order to be consistent with previous studies in the literature. Nine stars in our sample showed evidence of a two-temperature component dust system based on the SED fits. While IRS spectra were not available in all cases, which limits our ability to properly judge whether two temperature components are truly needed, we based our SED fits on a χ2 analysis, with the understanding that our results would be improved by mid-IR spectroscopy. In most cases, we would interpret a two-temperature component fit as two spatially separated disks. Such systems can be explained in several ways; 1) the dust originated in a cold belt, and “leaked” into the warm belt via PR drag and/or scattering by large planetesimals, 2) there was once one extended disk, into which a gap was carved by intervening planets, 3) the dust may have originated in two belts independent of one another, analogous to the asteroid and Kuiper belts, or 4) there is only one belt of dust that presents as a multiple temperature component system due to a range of grain properties. Concerning possibility 4# above, a two temperature SED could be necessary even if the dust emission arises from a single narrow belt if the grains in the belt have multiple sizes. Since smaller grains radiate inefficiently at long wavelengths, they retain heat and appear to be at a hotter temperature than larger grains at the same radial location from the star. Four of the two temperature systems orbit A-type stars (HD 54341, HD 76543, HD 121191, and HD 131488). Since the blowout radius of grains around A-type stars is larger than that of solar-type stars$\footnotemark{}$$\footnotetext{\scriptsize An A-type star has a typical blowout grain size of a few microns, whereas a solar-type star has a blowout grain size of 1 $$m or smaller.}$, grains in a single disk around an A-type star would likely have a more limited distribution of grain sizes, and thus would not be able to mimic a spatially separated two temperature component dust belt. In addition, three stars (HD 15407, HD 76582, and HD 113766) are early or mid F-type, and are also likely to have a limited grain size distribution. We expect that these seven stars likely have two spatially separated belts. For the remaining two systems (HD 35650 and HD 23514), we calculated a temperature ratio (RT=Td, 2/Td, 1, see Table 3) to determine if the dust emission is coming from two separate belts. From Figure 4 of, we see that single-belt systems with two temperature components tend to have small (RT $\lesssim$ 5) values of RT. By contrast, spatially separated double-belt systems appear to have values of RT $\gtrsim$ 5; both HD 35650 (RT=11.9) and HD 23514 (RT=6.4) thus likely represents true spatially separated belts (see Figure [fig:doubles]). Comparison with Previous Studies ================================ We compared our results to previous studies in the literature. In particular, we compare literature results with our SED-fitting results only, since none of the stars that follow have ever previously been resolved. HD 15407 -------- This F5/K2 binary system has an age of 80 Myr, derived from high resolution measurements of lithium in the photospheres of both components, X-ray data, and UVW space velocities. Si emission features are seen in the IRS spectra. Melis et al. predicted (based on the IRS spectrum and the IRAS upper limits) that there would be no cold dust detected around HD 15407A. performed a detailed analysis of the emission features in the IRS spectra, and found that HD 15407A should have an extended belt (from 0.4-19.2 AU) with a grain size distribution proportional to n− 3.1. This is shallower than the typical -3.5 value assumed for most debris disks. Olofsson et al. calculate a dust mass of ∼ 7.7E-05 M$\_{\Earth}$. estimated that the dust should lie between 0.6 and 1.0 AU. We find a double-belt system around HD 15407 (Figure 1). Both dust components are hot (T ∼ 334K, 1022K) and lie within 1 AU of the central star. These results are consistent with the prediction of a warm, extended belt. studied the mid-IR emission features of HD 15407, and (based on the width of the disk), predicted that a cold dust belt should be found, that is responsible for feeding dust to the observed hot dust belt. The absence of a cold dust component in the Herschel observations implies that either there is not enough dust to be observable with Herschel, or the dust is unrealistically cold ( < 10K). The lack of a substantial dust belt would be consistent with a proposed model of transient dust. HD 23514 -------- HD 23514 is a Pleiades member with an IR excess detected by IRAS and Spitzer. Rhee et al. report a dust temperature of 750 K and an exceptionally high fractional IR luminosity of 2.0E-02. Notably, they also report the results of observations with the Michelle Spectrograph on Gemini North, that show an unusual emission feature peaking around 9 μm. This feature (attributed to SiO2) may indicate a recent major high-velocity collision involving a differentiated terrestrial planet. Rhee et al. found a lack of olivine and pyroxene in the spectra, which suggests that a steady-state collisional cascade involving an asteroid belt is likely not the cause of the observed dust. This model is only strengthened by the fact that we detected no cold dust component with Herschel, and is consistent with our calculations of τ/τmax ∼ 300. We fit two dust components to the observed emission at 168K and 1082K. report a substellar companion to HD23514 with a mass of ∼ 0.06M$\_{\sun}$ at a separation of ∼ 360 AU. notes that several stars with dominant warm dust components were later discovered to be members of wide binary systems. The connection between the wide binary and the presence of warm dust is not well understood at this time. HD 35650 -------- This star is a known AB Dor member, with age ∼ 100Myr. It was observed by Spitzer, and was found to have a 70 μm excess only. report a dust temperature of 60 K and τ ∼ 1.7E-04. We find that the cold ( ∼ 45 K) dust has τ=1.5E-04, consistent with previous results. We also find that the mid-IR flux could be fit with an additional hot dust component at 536 K (Figure 1). HD 35650 was observed by the Near-Infrared Coronagraphic Imager (NICI) with angular differential imaging (ADI), but was not found to have a substellar companion. HD 43989 -------- HD 43989 is a member of Tuc Hor, and as such has an age of ∼ 40 Myr. report Spitzer observations that show an excess at 24 μm and in the IRS spectrum, but only an upper limit at 70 μm. We find a dust temperature of 112 K and a τ of 7.4E-05. The star was observed for a planetary companion by the Multiple Mirror Telescope (MMT) with spectral differential imaging (SDI), but no such companion was found. HD 54341 -------- This A0 star was presented by. The system age is unreliable; the age based on isochrones disagrees with the age based on UVW space motions (while the isochrones suggest an age of 10 Myr, the UVWs suggest an older age). At the time, there was no MIPS data point, and the dust temperature and luminosity quoted in is unrealistic. It was also observed with adaptive optics at Lick Observatory and found not to be a member of a binary system. The system was observed by Spitzer, and fit a two-component dust belt, with dust temperatures of 246K and 60K, and fractional IR luminosities of 1.9E-05 and 1.7E-04, respectively. While the χ2 values for a single- and double- belt fit were similar, we chose to fit a double-belt to be consistent with Chen et al.’s results. We find that the two belts have temperatures of 168 K and 60 K. The discrepancy in the warm dust temperatures is likely due to the fact that we often did not weight the IRAS photometry highly, due to the fact that IRAS data points are typically inconsistent with other, more reliable photometry. HD 54341 is one of our resolved systems; we find that the resolved disk size ( ∼ 185 AU) is almost twice as big as the blackbody disk size ( ∼ 106 AU). HD 76543 -------- This A5 star was initially reported by to have an IRAS excess suggestive of a disk at 85 K with τ=1.04E-04. Based on its location in UVW space (well outside of the ``good box" of ) and its location on the HR diagram, this star is estimated to have an age of ∼ 400 Myr. The star was also observed with Spitzer. report a two-component disk with a cool belt at 81 K and a warm belt at 146 K. They find fractional IR luminosities of the two systems of 3E-05 and 1.7E-05 for the cool and warm belts, respectively. The addition of Herschel data helps to constrain the dust temperatures; we now find two belts of dust - one at 49 K and one at 105 K with fractional IR luminosities of 4.8E-05 and 3.6E-05, respectively. Any discrepancy between our dust temperatures and those of are likely due to the high number of free parameters needed to fit a double belt system; a different cool dust temperature would correspond to a different warm dust temperature. The dust temperature from does not take into account any Spitzer or Herschel data. HD 76582 -------- This F0 star was reported by to have an IRAS excess indicative of a disk at 85 K and τ=2.22E-04. This relatively high fractional IR luminosity is surprising, given its 300 Myr age. The star was also observed by Spitzer. These observations revealed a two-component disk, with an inner belt at 466 K (akin to a zodiacal dust cloud) and an outer belt at 78 K (akin to a Kuiper Belt). Our Herschel observations constrain the long wavelength behavior of the cold belt. We also obtain a two-component fit to our SEDs, with an inner belt at 132 K (possibly an asteroid belt analog) and an outer belt at 51 K (a close analog of the Kuiper Belt). The cold dust emission appears to peak at Herschel wavelengths; after a fit with a 51 K curve, dust at ∼ 450 K does not appear to be needed. Longer wavelength data will help to constrain the grain properties and dust mass. This star was targeted for sub-mm follow-up by the DUNES team. HD 84870 -------- HD 84870 was first reported by to host a 100 Myr old debris disk with an IRAS excess indicative of a dust belt at 85 K. Rhee et al. listed this A3 star as a binary, but describe the two stars in question as an optical pair (sep ∼ 30$\arcsec$) based on a study of their relative motions. report a slight Si emission at 10 μm (5σ) and 20 μm (12σ). They also report 70 μm photometry that suggests a double-belt architecture, with a hot inner belt at 553 K and a cool outer belt at 67 K. Our Herschel observations are slightly inconsistent with the Spitzer data point at 70 μm, and we do not find a need for a double-belt system. We fit a single belt at 54 K with τ=4.7E-04. HD 85672 -------- This A0 star was first reported as a debris disk by with a cool belt at 79 K and τ=4.82E-04. Our Herschel results (combined with new photometry from WISE) better constrain the temperature of the observed dust. We found that the long-wavelength photometry required a modified blackbody fit (λ0=107, β=1.1), resulting in a best-fit dust temperature of 77 K and τ=6.0E-04. HD 99945 -------- HD 99945 was initially reported by as a new debris disk discovered by IRAS. While our Herschel photometry at 100 μm is inconsistent with the IRAS data point at 60 μm, we are able to fit a belt of cold dust to the emission (T=38 K). Due to the low temperature, we examined this A2 star to determine if the emission might the due to contamination with a background galaxy or IR cirrus (see Sections 4.1 and 4.2). Due to the high far-IR flux and the fact that the emission is slightly too warm do be accounted for by a background galaxy at z ∼ 1, we believe that the excess emission indicates a belt of dust at 167 AU from the host star. HD 113766 --------- HD 113766 is a well-known, extremely bright debris disk with a hot dust component. The mid-IR photometry has been well-covered in the literature. Si emission features are seen in the IRS spectra. reported the Spitzer results, and found a single belt of dust at 330 K. observed this star with VLTI/MIDI and Herschel/PACS, and determined that, in order to fit the mid-IR data simultaneously with the far-IR photometry, two spatially separated belts were needed: a hot belt within 1 AU, made of small grains which cause emission features in the mid-IR, and an outer belt at 9-13 AU. Our results independently confirm this two-belt structure, and are consistent with the findings of regarding the transient nature of the dust. We report an inner belt at 0.2 AU and an outer belt at 2.3 AU. Our estimate for the semimajor axis of the cooler dust is smaller than that of, though Oloffson et al. used a modified blackbody fit to the dust, and attempted to fit their VLTI/MIDI data simultaneously. As discussed in, the binary companion to HD 113766A is likely too far from the disk to have any serious influence on the stirring of the disk material, though point out a possible relationship between distant companions and copious amounts of warm dust. HD 121191 and HD 131488 ----------------------- These A-type stars were OT-1 targets. report mid-IR imaging and spectroscopy from Gemini/T-ReCS. Most notably, they discovered that HD 121191 and HD 131488 both have unusual emission features that peak shortward of 8 μm. This emission feature does not appear to be due silica, olivines, or pyroxenes, all which show emission features peaking between 8-12 μm. Our Herschel observations confirm the two-belt structure of the debris disk around HD 121191, with a hot inner disk at 555 K and a warm disk at 118 K. There does not seem to be a cold dust component in this system. Melis et al. fit a two-belt debris disk to the SED of HD 131488, with one inner disk at 750 K and an outer disk at 100 K. Our Herschel observations confirm this double-belt nature of the debris disk, but with an inner belt at 570 K and an outer belt at 94 K. It should be noted that, while we believe HD 121191 is at least marginally resolved in the Herschel images, confirmation is needed to consider it to be a truly resolved disk since it appears as an extreme outlier in Figure [fig:radii]. HD 124718 --------- This G5 star was initially reported by to have an IRAS excess indicative of a dust belt at 85 K and with a large fractional IR luminosity (τ=2.11E-03), which was surprising given its old age ( > 500 Myr). WISE mid-IR photometry reveals an excess at 22 μm. Our Herschel observations are inconsistent with the IRAS data, and demonstrate that there is no accompanying cold dust belt. We fit a dust temperature of 179 K and τ=2.1E-04. This discrepancy is unsurprising, since the dust temperature and luminosity reported by Rhee et al. were based on a single data point in excess, before WISE results were available. BD+20 307 --------- The excess IR emission around this well-studied close G-type binary star system was first reported by primarily based on IRAS photometry at 12 and 25 μm (the 60 and 100 μm IRAS observations only returned upper limits). obtained optical spectra to better determine the system age. Their results suggest that BD+20 307 could, in fact, be several Gyr old. Thus, the massive quantity of hot dust is not a result of ongoing planet formation, but rather could be the result of a collision between two mature planets in the terrestrial planet zone. The absence of cold dust was also noted in. The Herschel SED confirms that there is no cold dust emitting at far-IR wavelengths, and helps to constrain the temperature of the dust (417 K) and to lend support to the planet-collision model. We did not attempt to fit the Spitzer IRS spectrum, since it is dominated by emission features. Thus, we adopt the τ value of 0.032 from, since our estimate of τ ( ∼ 0.01) does not include flux from the strong silicate emission feature. Conclusions =========== We observed 24 stars with the PACS camera on the Herschel Space Observatory. Two infrared sources detected with Herschel are offset from the target coordinates by > 2$\arcsec$. These are unlikely to be due to dusty debris belts. Three targets were not detected with Herschel, and are non-excess stars. One star was detected with Herschel, but shows no evidence of an IR excess. Two stars have low IR fluxes, and could possibly be explained by contamination by a background galaxy at z ∼ 1. One target star is clearly contaminated by a background galaxy. The remaining 15 stars were examined to determine dust properties and the possibility of ongoing planet formation. A summary of the results can be found in Table 5. * Nine stars (HD 15407, HD 23514, HD 35650, HD 54341, HD 76543, HD 76582, HD 113766, HD 121191, and HD 131488) appear to have dust components at two temperatures according to SED fitting. All appear to have spatially separated dust belts. * Three stars (HD 15407, HD 113766, and BD+20 307) have disks that cannot be explained by a steady-state collisional process alone. Two other stars (HD 23514 and HD 124718) are likely explained by a transient process. The other debris disks could be explained by steady-state collisional processes. * Dust belts at HD 54341, HD 84870, HD 85672, and HD 121191 are at large enough distances from their host stars that the dust could be stirred by a yet-unseen planet or binary companion. Stirring of these disks by putative 1000 km-sized bodies may be insufficient to explain the dust production in these systems. * The stars HD 54341, HD 76543, HD 76582, HD 84870, HD 85672, and HD 99945 are spatially resolved (or probably resolved as in the case of HD 121191), and a comparison between their blackbody radii (from SED-fitting) and resolved radii show that the latter are typically larger than the former, perhaps by up to a factor as large as 10 (HD 121191). * Six stars (HD 15407, HD 23514, HD113766, HD 121191, HD 124718, and BD+20 307) show no evidence of cold ( < 100 K) dust in the Herschel data. Acknowledgements ================ This research has made use of the Exoplanet Orbit Database, the Exoplanet Data Explorer at exoplanets.org, the SIMBAD database, and VIZIER search engine, operated by CDS in France. Partial support for this work was provided by a NASA grant to UCLA and by an NSF Graduate Research Fellowship to Laura Vican. We also thank the referee for their helpful and thorough feedback. cccccccccccc 15407&11696& F5 &55&1.4&1.6&6500&4.0&80& a & M10& Y 23514&& F6 &135&1.4&1.08&6300&1.6&100& b & R08 &Y 35650&25283& K6 &18&0.8&0.63&4300&0.1&70&b&Z11& 43989&30030& G0 &49&1.1&1.05&6100&1.3&30& b & Z11 & 54341&34276& A0 &93&2.4&1.83&9500&24.1&10&c&R07 & 76543&43970& A5 &49&1.9&1.83&8200&13.4&400&c&R07& 76582&44001& F0 &49&1.7&1.68&7700&8.8&300&c&R07& 84870&48164& A3 &90&1.6&1.63&7600&7.8&100&c&R07& 85672&48541& A0 &93&1.7&1.51&8000&8.3&30&c&R07& 99945&56253& A2 &60&1.8&1.79&7600&9.4&300&c&R07& 113766&& F3 &120&1.9&1.9&6100&4.4&15&b&C05& Y 121191&& A5 &130&1.8&1.83&7700&10.4&10&b&M13& N 124718&69682& G5 &61&1&0.96&5900&0.99& > 500 &a&S03& 131488&& A1 &150&2.2&2.06&8700&21.4&10&b&M13& N BD+20 307 &8920& G0 &92&1.9&1.26&6100&1.9&1000&a&Z08& Y 60234&36906& G0 &108&1.6&3.534&5900&13.3&600&d&R07& 70298&40938& F2 &71&&1.73&6500&4.7& > 3000&d&R07& 72660&42028& A1 &100&&2.24&9500&36.1&200&c&R07& 132950&73512& K2 &30&&0.73&4800&0.25&3000&...&R07& 203562&105570& A3 &110&2.7&3.734&8800&6.5&600&c&R07& 8558&6485& G6 &50&0.9&0.94&5800&0.9&30& b & Z11 & 13183&9892& G7 &50&0.9&0.95&5700&0.8&30& b & Z11 & 80425&45758& A5 &98&&2.544&7500&18.0&300&c&R07& 191692&99473& B9 &88&3.34&6.40&10500&2680&500&c&R07&Y ccccc 15407 & OT1bzuckerm1 & 70 & 55.7 & 4.5 && 100 & 24.4 & 3.8 && 160 & 52.6 & 14.9 23514 & OT1bzuckerm1 & 70 & 24.8 & 3.5 && 100 & 83.5 & 3.2 && 160 & 60.7 & 15.2 35650 & OT2bzuckerm2 &100 & 26.8 & 1.9 && 160 & 57.0 & 25.4 43989 & OT2bzuckerm2 & 70 & 10.5 & 3.0 && 160 & 49.4 & 11.4 54341 & OT2bzuckerm2 & 100 & 297 & 3.9 && 160 & 200 & 11 76543 & OT2bzuckerm2 & 100 & 302 & 6.5 && 160 & 268 & 52 76582 & OT2bzuckerm2 & 100 & 605 & 7.3 && 160 & 485 & 40.0 84870 & OT2bzuckerm2 & 100 & 287 & 3.8 && 160 & 246.0 & 56.0 85672 & OT2bzuckerm2 & 100 & 165 & 8.1 && 160 & 137 & 72 99945 & OT2bzuckerm2 & 100 & 159 & 4.0 && 160 & 232 & 66 113766 & OT1jolofsso1 &70 & 322 & 9.7 && 100 & 201 & 6.0 && 160 & 79 & 8.0 121191 & OT1bzuckerm1 & 70 & 246.6 & 4.7 && 160 & 37.8 & 25.2 124718 & OT2bzuckerm2 & 100 & 4.2 & 5.2 && 160 & 2.4 & 17.7 131488 & OT1bzuckerm1 & 100 & 336.6 & 5.8 && 160 & 191.3 & 24.0 BD+20 307 & OT1jolofsso1 & 70 & 47 & 3.0 & & 100 & 16 & 3.0 70298 & OT2bzuckerm2 & 100 & < 10.2& & & 160 & < 23.1& 72660 & OT2bzuckerm2 & 100 & < 13.2& & & 160 & < 38.7& 132950 & OT2bzuckerm2 & 100 & < 11.1& & & 160 & < 39.6& ccccccccccccc 15407&334 ± 39&1022 ± 79&1.41 ± 0.2&0.1 ± 0.02&2.10E-03 ± 1.57E-02&9.71E-03 ± 1.57E-02&3.3E-07&6360&6.56E-07&2.5 23514&168 ± 7&1082 ± 22&3.53 ± 0.2&0.1 ± 0.004&1.	arxiv_0000633
Dynamical friction for accelerated motion in a gaseous medium ============================================================= ### Dates: UNDER REVIEW [firstpage] Dynamical friction arises from the interaction of a perturber and the gravitational wake it excites in the ambient medium. This interaction is usually derived assuming that the perturber has a constant velocity. In realistic situations, motion is accelerated as for instance by dynamical friction itself. Here, we study the effect of acceleration on the dynamical friction force. We characterize the density enhancement associated with a constantly accelerating perturber with rectilinear motion in an infinite homogeneous gaseous medium and show that dynamical friction is not a local force and that its amplitude may depend on the perturber’s initial velocity. The force on an accelerating perturber is maximal between Mach 1 and Mach 2, where it is smaller than the corresponding uniform motion friction. In the limit where the perturber’s size is much smaller than the distance needed to change the Mach number by unity through acceleration, a subsonic perturber feels a force similar to uniform motion friction only if its past history does not include supersonic episodes. Once an accelerating perturber reaches large supersonic speeds, accelerated motion friction is marginally stronger than uniform motion friction. The force on a decelerating supersonic perturber is weaker than uniform motion friction as the velocity decreases to a few times the sound speed. Dynamical friction on a decelerating subsonic perturber with an initial Mach number larger than 2 is much larger than uniform motion friction and tends to a finite value as the velocity vanishes in contrast to uniform motion friction. hydrodynamics – ISM: general Introduction ============ Dynamical friction is the reaction force that a perturber feels from the gravitational wake it generates in the ambient medium. This fundamental interaction has therefore had a wide range of applications in the formation of astronomical structures. Applications to collisionless systems include mass segregation in star clusters (e.g. Kim and Morris 2003, Portegies et al 2004), galaxy distribution in hierarchical cluster formation (e.g. Frenk et al 1996, Bullock and Johnston 2005), the decay of satellites in galactic halos and galaxy mergers (Binney and Tremaine 2008 and references therein). Dynamical friction also occurs in collisional particle systems and was applied to the study of the growth of planetesimals (Hornung et al 1985, Stewart and Whetherill 1988), the eccentricity excitation of planetary embryo orbits (Ida 1990, Kokubo and Ida 1996, Namouni et al 1996), and the confinement of planetary rings (Goldreich and Tremaine 1982). For gaseous systems, applications include the orbital decay of compact stars orbiting supermassive black holes (Narayan 2000), and the heating of intercluster gas by decelerating supersonic galaxies (El-Zant et al. 2004, Faltenbacher et al 2005, Kim et al 2005). Following the pioneering work of Chandrasekhar (1943) on the gravitational drag experienced by a perturber in collisionless star systems, the derivations of dynamical friction assume that the perturber has a constant velocity in the ambient medium. The assumption of uniform motion is made for collisionless systems (Binney and Tremaine 2008), collisional particle systems (Hornung et al 1985, Stewart and Whetherill 1988, Ida 1990, Goldreich and Tremaine 1982) as well as gaseous media (Kim and Kim 2007, Kim et al 2008, Ostriker 1999). The resulting dynamical friction force which depends on the perturber’s velocity is subsequently used to model the evolution of the perturber’s motion whose velocity varies with time. Only the numerical simulations of the nonlinear fluid equations are free from this assumption (Sánchez-Salcedo and Brandenburg 1999, 2001). The neglect of the perturber’s acceleration in dynamical friction is justified by the small amplitude of the force that is proportional to the square of the pertuber’s mass. Neglecting acceleration is independent of the type of motion and is mainly motivated by the lack of an analytical framework where the perturber’s accelerated motion could be taken into account. To overcome this obstacle, we derive in Appendix A a new expression of dynamical friction that applies to a general type of motion and depends on the perturber’s motion and the boundary conditions but not the density perturbation. In particular, dynamical friction is shown to be non-local as it depends on the whole trajectory of the perturber. Although non-locality does not appear if the perturber’s velocity is constant, its origin is easy to identify: as the perturber’s wake propagates in the medium, a velocity change because of acceleration affects the density enhancement near the perturber and offsets the force from its local value as the far parts of the wake were generated by a different velocity. Therefore when acceleration is taken into account, the amplitude of dynamical friction may change with respect to the value derived assuming a constant velocity because in modelling the force on an accelerated perturber as that on a constant velocity perturber, one assumes that the gravitational wake produced by accelerated motion is similar to that of uniform motion. We shall see that this difference between the two wakes makes dynamical friction dependent on the initial velocity of the perturber. Another difference between the wakes generated by constant velocity motion and accelerated motion is the absence of a perturbed region with front-back symmetry with respect to a subsonic accelerating perturber. In effect a constant velocity perturber raises a gravitational wake in the ambient medium that is symmetric with respect to it and truncated by the sonic shockwave. This truncation defines two regions, one that lacks front-back symmetry located behind the perturber, and one with front-back symmetry around the perturber that becomes the largest region inside the wavefront for velocities smaller than half the sound speed. Dynamical friction arises from the former region as shown by Ostriker (1999). When acceleration is taken into account, front-back symmetry is lost everywhere. This would intuitively imply that dynamical friction on an accelerated perturber should be much larger than the value derived assuming uniform motion for subsonic perturbers. Curiously, we shall show that the force on an accelerating subsonic perturber initially at rest is similar to the uniform friction force. Lastly, accelerated motion friction would differ from uniform motion friction because the former never reaches steady state. In effect, dynamical friction on a constant subsonic velocity perturber is known to be time-independent (Ostriker 1999). It is the object of this paper to quantify how taking into account acceleration modifies dynamical friction on a perturber travelling in an infinite homogeneous gaseous medium. The perturber’s motion is assumed to have a constant acceleration, a choice that makes the analytical developments simpler. Physically such constant acceleration approximates the deceleration felt by the perturber because of dynamical friction as the force amplitude is small and the corresponding instantaneous velocity change is equally small. Constant acceleration also models the motion of a young star with an asymmetric jet system moving in interstellar gas (Hirth et al 1994, Hartigan and Morse 2007, Namouni 2005, 2007). In what follows, we characterize the density perturbation associated with accelerated motion in Section 2. We derive the force on an accelerated perturber in Section 3 using the general expression of dynamical friction (derived in Appendix A) that does not depend on the density enhancement caused by the perturber but only on the perturber’s motion and the medium’s boundary conditions. In particular we show that including acceleration makes dynamical friction dependent on the initial velocity of the perturber. Section 4 contains concluding remarks. In Appendix B, the new expression of dynamical friction derived in Appendix A is validated by applying it to uniform rectilinear motion and recovering the known results on dynamical friction. Density enhancement =================== Consider a point-like perturber of mass M on an accelerated rectilinear trajectory, of initial velocity V0 and acceleration A, that travels inside an inviscid isothermal gaseous medium of density ρ and sound speed c. Assuming the density perturbation associated with the perturber’s wake is small compared to the medium’s unperturbed density ρ0, the corresponding density enhancement $\varrho$ defined as $\rho=\rho\_0(1+\varrho)$ satisfies the standard forced sound wave equation (Appendix A, equation [wav2A]): $$\nabla^2\varrho-\frac{1}{c^2}\, \partial\_t^2\varrho=-\frac{{\cal H}(t)}{c^2}\nabla^2\phi\_p=-4\pi \frac{GM}{c^2}\, \delta^3[{\bm x}-{\bm \xi}(t)] {\cal H}(t), \label{waveeq}$$ where $\phi\_p= -GM/\|{\bm x}-{\bm \xi(t)}\|$ is the perturber’s gravitational potential, ${\cal H}(t)$ is the Heaviside function, G is the gravitational constant, ${\bm \xi}(t)=(At^2/2+V\_0t){\bm e}\_z$ is the perturber’s trajectory chosen along the z–direction and δ(x) is Dirac’s delta function. The solution of this equation is the Liénard-Wiechert potential which may be written in integral form using the retarded Green function as ([dens2A]): $$\begin{aligned} \varrho({\bm x},t) &=&\frac{GM}{c^2}\int\_{-\infty}^{+\infty} \frac{\delta \left[u-t+\|{\bm x}-{\bm \xi}(u)\|/c\right] {\cal H}(u)} {\|{\bm x}-{\bm \xi}(u)\|}\ {\rm d}u. \label{density}\end{aligned}$$ The acceleration A introduces two natural time and length scales given by: $$T=\frac{c}{A}, \ \ L=\frac{c^2}{A}. \label{scalings}$$ In these units, the perturber’s trajectory may be written as $\xi(t)=t^2/2+{\cal M}\_0t$ where ${\cal M}\_0$ is the initial Mach number. The Mach number is a function of time and reads: ${\cal M}=t+{\cal M}\_0$. The scaling time T is the duration to increase the Mach number by unity and the scaling length L is the corresponding travelled distance. Applied to the density enhancement these scalings yield: $$\varrho({\bm x},t)=\frac{GMA}{c^4}\int\_{-\infty}^{\infty} \frac{\delta \left(u-t+\left[R^2+\left(z-{\cal M}\_0u- u^{2}/2\right)^2\right]^{1/2}\right) {\cal H}(u)} {\left[R^2+\left(z-{\cal M}\_0u- u^{2}/2\right)^2\right]^{1/2}}\ {\rm d}u, \label{density2}$$ where R2 = x2 + y2 is the cylindrical radius; unless it is stated otherwise, all length and time variables and constants are normalized to ([scalings]). In order to derive the density profile we use the property of the δ-function α(u)δ[β(u)] = ∑iα(ui)δ(u − ui)/∣β′(ui)∣ where ui are the roots of β(u), and β′ is the derivative of β. The roots of the argument of the δ–function in equation ([density2]) satisfy the quartic equation: $$u^4/4+{\cal M}\_0u^3-(1+ z-{\cal M}\_0^2)\,u^2+2 (t-{\cal M}\_0z) u -t^2+r^2=0, \label{roots}$$ where r = (R2 + z2)1/2 is the spherical radius. This equation is the counterpart of the quadratic equation derived by Ostriker (1999) for uniform motion. The former may be reduced to the latter once time and length dimensions are restored. Equation ([roots]), however, cannot be solved analytically because its varying coefficients do not permit simple and unique expressions of the roots. Instead we determine the roots numerically and substitute them into the density expression obtained by applying the δ–function property cited above. The density expression reduces to: $$\varrho({\bm x},t)=\frac{GMA}{c^4}\sum\_{{\rm roots}\ u\_i {\rm\ of\ eq.\ (\ref{roots})} }\|t-u\_i-({\cal M}\_0+u\_i)(z-{\cal M}\_0 u\_i - u\_i^2/2)\|^{-1}.$$ Fig. (1) follows the level curves of $\varrho$ for perturbers initially at rest (${\cal M}\_0=0$) as they evolve from t = 0 to Mach 6. The front-back symmetry that characterizes the density enhancement of subsonic constant velocity motion in a uniform medium is absent for both subsonic and supersonic motion. The perturber reaches Mach 1 at z = 0.5 but remains inside the sonic wavefront of radius t as the latter has been propagating faster than the perturber’s velocity. Once the velocity is supersonic, a Mach cone is formed inside the sonic wavefront and a shallow density depression develops and trails the perturber (z ≤ − 1). Eventually as Mach 2 is reached the perturber catches up with the sonic shockwave and exits the sonic region to form an external Mach cone. Fig. (2) shows the level curves of $\varrho$ associated with decelerating perturbers with ${\cal M}\_0 \neq 0$ as a function time until the perturber’s velocity vanishes. The density enhancement of subsonic perturbers has a structure similar to that of uniform motion (Ostriker 1999) but lacks its front-back symmetry with respect to the perturber. We show in the following section that under certain circumstances, dynamical friction may be not significantly affected by the loss of front-back symmetry even as closed level curves around the perturber contribute to the force unlike those of uniform motion (see Introduction). The deceleration of supersonic perturbers modifies the structure of the density enhancement and reduces the extent of the Mach cone but does not erase it as the velocity vanishes (bottom row, right panel) indicating the presence of a finite friction force. Friction force ============== The reaction force exerted on a point-like perturber in a homogeneous gaseous medium by its gravitational wake is given by equation ([for3A]) and is written in dimensional variables as: $${\bm F}=\frac{{\cal H}(t)(GM)^2\rho\_0}{c^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r/c)], r\leq ct}r\sin\theta{\rm d}r\, {\rm d}\theta\, {\rm d}\varphi \ \frac{{\bm y}-{\bm \Delta}}{\|{\bm y}-{\bm \Delta}\|^3},\label{force}$$ where r, θ and φ are the spherical coordinates of the vector ${\bm y}$ defined as the relative position within the medium with respect to the perturber’s position at the retarded time t − r/c. It may be written as ${\bm y}={\bm x}-{\bm \xi}(t-r/c)$ where ${\bm x}$ is the position of fluid element in the medium. This definition of ${\bm y}$ influences the force integral through the boundary conditions of the medium denoted by ∂V. The phase ${\bm \Delta}={\bm {\bm\xi}}(t)-{\bm {\bm\xi}}(t-r/c)$, the upper radial cutoff at the retarded sonic shockwave r ≤ ct, and the boundary conditions are all that is necessary to compute the friction force. In particular, the absence of an explicit expression for the density enhancement is irrelevant for the derivation of the force. The derivation of expression ([force]) assumes that the perturber is assembled at t = 0 a fact that introduces the size of the retarded sonic shockwave as an upper cutoff. This assumption is not restrictive as it allows us to study the time-dependent evolution of dynamical friction as well approximate steady state behaviour (see section 3.2). It also describes turning on a numerical simulation of dynamical friction (Sánchez-Salcedo and Brandenburg 1999, 2001). For rectilinear motion along the z–direction in an infinite medium, the force is equally along ${\bm e}\_z$ and its expression can be reduced further by integrating over the longitude ϕ and the variable τ = cosθ as: $$F=\frac{2\pi{\cal H}(t)(GM)^2\rho\_0}{c^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r)], r\leq t}\ \frac{{\rm d}r} {\Delta^2}\ \left[\frac{ r-\tau\Delta}{\left\|1+(\Delta/r)^2-2(\Delta/r)\tau\right\|^{1/2}}\right]^{\tau\_{\rm max}}\_{\tau\_{\rm min}},\label{forcex}$$ where the scalings ([scalings]) have been applied; the perturber’s trajectory ${\bm \xi}(t)$, the time, t, and the position ${\bm x}$ and its spherical components are dimensionless hereafter. The expressions of $\tau\_{\rm min}$ and $\tau\_{\rm max}$ as a function of the perturber’s trajectory and the variable r are determined from the boundary conditions. For uniform rectilinear motion, ${\cal M}$ is constant and the phase $\Delta/r={\cal M} $. Calling r0 the gravitational potential’s small distance cutoff scaled to L ([scalings]), the integral that appears in the expression ([forcex]) may be written as (Appendix B, equation [ostriA]; Ostriker 1999): $$\begin{aligned} I\_0&=&{\cal M}^{-2}\left(\frac{1}{2}\left[1+{\cal M}-\|1-{\cal M}\|\right]\left[1+{\rm sign}(1-{\cal M})\right]-\log \left\|\frac{1+{\cal M}}{1-{\cal M}}\right\| -\left[1-{\rm sign} (1-{\cal M})\right] \ \log\left\|\frac{(1-{\cal M})t}{r\_0}\right\|\label{ostriker}\right).\end{aligned}$$ For accelerated motion, the phase $\Delta/r={\cal M}\_0+t-r/2$. In order to examine how acceleration modifies the dynamical friction force, we consider the three cases of perturbers initially at rest (${\cal M}\_0=0$), accelerating perturbers (${\cal M}\_0>0$) and decelerating perturbers (${\cal M}\_0<0$). Before deriving dynamical friction for accelerated motion, a choice of the small distance cutoff r0 must be made. This cutoff is usually taken as the perturber’s accretion radius or the system’s typical size. The former dimensional cutoff reads 2GM/V2 and is a function of the Mach number whereas the latter is a constant that does not depend on the flow’s variables. In this work, the perturber is treated as a compact object whose gravitational potential is truncated at small distances by the perturber’s size. To satisfy such a condition and be consistent with a point-like potential, the perturber’s size needs to be much smaller than the accretion radius 2GM/V2. This choice of cutoff is in agreement with the numerical simulation of the fluid equations in the context of Bondi-Hoyle-Lyttleton accretion (Edgar 2005). For larger size or extended objects, equation ([force]) no longer applies and dynamical friction may be obtained from the general expression derived for an arbitrary perturbing potential (Appendix A, equation [forc2A]). Perturbers initially at rest ---------------------------- In an infinite medium, the only relevant physical boundary condition is that of the small distance cutoff $\|{\bm x}-{\bm \xi} (t)\|\leq r\_0$.[2](#fn2) For a perturber initially at rest, ξ(t) = t2/2, Δ/r = t − r/2 and the Mach number is ${\cal M}=t$. The variables r and τ therefore need to satisfy the three conditions: (i) − 1 ≤ τ ≤ 1, (ii) τ ≤ [1 + (2t − r)2/4 − (r0/r)2]/(2t − r) ≡ f(r, t, r0), and (iii) r ≤ t. The last condition implies that Δ > 0. Examination of the three conditions shows $\tau\_{\rm min}=-1$, and that $\tau\_{\rm max}$ and the remaining bounds on r in the integral ([forcex]) depend on the value of t as the velocity increases. There are five time intervals that need to be considered: $$\begin{aligned} \mbox{I}&:& (1+2r\_0)^{1/2}-1\leq t\leq 1-(1-2r\_0)^{1/2}, \nonumber\\ \mbox{II}&:& 1-(1-2r\_0)^{1/2} \leq t \leq 1+(2r\_0)^{1/2}, \nonumber\\ \mbox{III}&:& 1+(2r\_0)^{1/2} \leq t \leq 1+(1-2r\_0)^{1/2}, \nonumber\\ \mbox{IV}&:& 1+(1-2r\_0)^{1/2}\leq t\leq 1+(1+2r\_0)^{1/2}, \nonumber\\ \mbox{V}&:& 1+(1+2r\_0)^{1/2} \leq t. \nonumber\end{aligned}$$ Case I is that of transitory motion where the density perturbation is in the perturber’s vicinity. Prior to t = (1 + 2r0)1/2 − 1, the force is zero as the three conditions above cannot be satisfied. The integration domain of r and $\tau\_{\rm max}$ are given as: $$\begin{aligned} r^f\_2\leq r\leq t&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\ \ \ \mbox{where} \ \ r\_2^{f}=1+t-[(1+t)^2-2r\_0]^{1/2}. \end{aligned}$$ Calling I the integral in equation ([forcex]) and using these boundaries for r, we find: $$\begin{aligned} I\_{\rm I}&=&\left[ \frac{t^2 r^3-2t^3r^2+(4t^2-4r\_0 t+ 2 r\_0^2)r - 2 r\_0^2 t }{2r\_0 t^2r(2t-r)} -\frac{2t-r\_0}{2t^3}\, \log\left\|\frac{r}{r-2t}\right\| \right] \_{r\_2^{f}}^{t}, \label{i0}\\ I\_{\rm I}&=&-\frac{r\_0(2+r^f\_2)-(2+r\_0+r^f\_2)t -r^f\_2t^2+t^3}{2r\_0t^2}-\frac{2t-r\_0}{2t^3}\, \log\left\|\frac{2t-r\_0}{tr^f\_2-r\_0}\right\|.\label{case1}\end{aligned}$$ Examination of the variation of $I\_{\rm I}$ as a function of time shows that it decays smoothly to zero as t = (1 + 2r0)1/2 − 1 is approached. The expression of the force in this time interval is the equivalent of that derived for uniform motion in Appendix B (equation [transient]) when $r\_0/\|1+{\cal M}\|\leq ct\leq r\_0/\|1-{\cal M}\|$ and the wake has not left the perturber’s vicinity. In that case, the force read $F\propto {\cal M}^{-2}\{ -\log[ct(1+{\cal M})/r\_0]+{\cal M}-{r\_0}/{2ct}+(1-{\cal M}^2)ct/2r\_0\}$; it helped explain the transient phase observed before saturation in the numerical simulation of the nonlinear fluid equations (Sánchez-Salcedo and Brandenburg 1999). Case II covers subsonic motion and the corresponding integration boundaries are: $$\begin{aligned} r^f\_2\leq r\leq r^c\_1&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0), \ \ \mbox{where}\ \ r\_1^{c}=t-1+[(t-1)^2+2r\_0]^{1/2}, \label{dom21}\\ r^c\_1\leq r\leq t&\mbox{with}& \tau\_{\rm max}=1.\label{dom22}\end{aligned}$$ The latter domain ([dom22]) does not contribute to the force. The corresponding integrand is proportional to ${\rm sign} (1-t+r/2)-1$ and the sign argument is positive for r > 2(t − 1). The expression of I for domain ([dom21]) may be obtained from equation ([i0]) by substituting r1c for t. The result is : $$\begin{aligned} I\_{\rm II}&=&\frac{2r\_0 t + (t-r\_0) ( r\_1^c+r\_2^f) -t^2 (r\_1^c-r\_2^f)}{2 r\_0 t^2}- \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{r\_1^ct+r\_0}{r\_2^ft-r\_0}\right\|. \label{case2}\end{aligned}$$ In the limit r0 ≪ 1 and t < 1, $I\_{\rm II}(r\_0\ll 1)=t^{-2}(2t -\log\|(1+t)/(1-t)\|)$. This is exactly the friction force acting on subsonic perturbers with uniform motion given by equation ([ostriker]). It is interesting that although the density enhancements of subsonic accelerated motion and subsonic uniform motion differ significantly, the corresponding friction forces are identical when the limit r0 ≪ 1 is taken for the accelerated motion friction. [3](#fn3) This result is the more surprising as the density enhancement associated with accelerated motion does not involve a region with front-back symmetry as explained in the Introduction leading one to guess wrongly that friction would be stronger for accelerated motion than it is for uniform motion. The explanation of this agreement comes from the friction force’s dependence on the phase Δ = ξ(t) − ξ(t − r/c) (dimensional variables). As the contribution of the force for subsonic motion comes mostly from the vicinity of the perturber’s retarded position ($\|{\bm y}\|\rightarrow 0$), we have r/c ≪ t and the phase becomes $\Delta=\dot\xi(t)r/c$. The latter is exactly the phase of uniform motion with a Mach number ${\cal M}=\dot \xi(t)/c$ regardless of the type of motion ξ(t) as long as, (1) r0 ≪ 1 which implies in dimensional variables that $\|\ddot\xi(t)\|\ll c^2/r\_0$, (2) the velocity remains subsonic and (3) the motion does not have a history with supersonic episodes (see section 3.3). Case III concerns motion between Mach 1 and Mach 2 where the perturber travels at supersonic speed but has not yet crossed the sonic shock wave. The integration domains are given as: $$\begin{aligned} r^f\_2\leq r\leq r^c\_2&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\ \ \mbox{where} \ \ r\_2^{c}=t-1-[(t-1)^2-2r\_0]^{1/2}, \label{dom31}\\ r^c\_2\leq r\leq r\_3^c&\mbox{with}& \tau\_{\rm max}=1,\ \ \mbox{where}\ \ \ r\_3^{c}=t-1+[(t-1)^2-2r\_0]^{1/2},\label{dom32}\\ r^c\_3\leq r\leq r^c\_1&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\label{dom33}\\ r^c\_1\leq r\leq t&\mbox{with}& \tau\_{\rm max}=1.\label{dom34}\end{aligned}$$ The integration radii satisfy r2f < r2c ≤ r3c ≤ 2(t − 1) ≤ r1c. This shows that domain ([dom34]) does not contribute a finite term to the force in the same way as domain ([dom22]). Domains ([dom31]) and ([dom33]) contribute terms similar to that of equation ([i0]) except that for the former t is replaced with r2c and for the latter t is replaced with r1c, and r2f with r3c. The second domain ([dom32]) contributes the term $I\_{(\ref{dom32})}$ given as: $$\begin{aligned} I\_{(\ref{dom32})}&=&\left[\frac{4}{(r-2t)t}-\frac{2}{t^2}\log\left\|\frac{r}{r-2t}\right\|\right]\_{r\_2^{c}}^{r\_3^{c}}, \label{i2}\end{aligned}$$ which when combined with the contributions of domains ([dom31]) and ([dom33]) yields: $$\begin{aligned} I\_{\rm III}&=&\frac{2r\_0 t -r\_0 (r\_1^{c}+r\_2^{f}-r\_2^{c}+r\_3^{c}) +t (r\_1^{c}+r\_2^{f}+r\_2^{c}-r\_3^{c})}{2 r\_0 t^2}-\frac{2t+r\_0}{2t^3}\,\log\left\|\frac{r\_3^{c}t-r\_0}{r\_2^{c}t-r\_0}\right\|\nonumber \\ && -\frac{r\_1^{c}-r\_2^{f}+r\_2^{c}-r\_3^{c}}{2 r\_0} - \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{r\_1^{c}t+r\_0}{r\_2^{f}t-r\_0}\right\|.\label{case3}\end{aligned}$$ Case IV defines a small time interval near Mach 2 where the perturber exits the sonic wavefront. The corresponding integration domains are: $$\begin{aligned} r^f\_2\leq r\leq r^c\_2&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\\ r^c\_2\leq r\leq r\_3^c&\mbox{with}& \tau\_{\rm max}=1,\\ r^c\_3\leq r\leq t&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0).\end{aligned}$$ The expression of I is obtained by summing up the terms associated with domains ([dom31]) and ([dom32]) of Case III, and substituting t for r1c in domain ([dom33]). The result is: $$\begin{aligned} I\_{\rm IV}&=&\frac{ r\_0(r\_2^c-r\_3^c-r\_2^f-2)+(2 + r\_0 + r\_2^c - r\_3^c + r\_2^f)t+( -r\_2^c + r\_3^c + r\_2^f)t^2-t^3}{2 r\_0 t^2}\nonumber \\ &&-\frac{2t+r\_0}{2t^3}\,\log\left\|\frac{r\_3^ct-r\_0}{r\_2^{c}t-r\_0}\right\| - \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{2t-r\_0}{r\_2^{f}t-r\_0}\right\|. \label{case4}\end{aligned}$$ Case V concerns supersonic motion that exited the sonic wavefront ${\cal M}>2$. The integration domains are given as: $$\begin{aligned} r^f\_2\leq r\leq r^c\_2&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\\ r^c\_2\leq r\leq t &\mbox{with}& \tau\_{\rm max}=1. \label{dom-v}\end{aligned}$$ The integral I is obtained as previously by combining the domains of Case IV and making the necessary boundary substitutions. The result is: $$I\_{\rm V}=\frac{ r\_0 (r\_2^{c}-r\_2^{f}-4) +(r\_2^{c}+r\_2^{f})t- (r\_2^{c}-r\_2^{f}) t^2}{2 r\_0 t^2}-\frac{2t+r\_0}{2t^3}\,\log\left\|\frac{2t+r\_0}{r\_2^{c}t-r\_0}\right\| - \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{2t-r\_0}{r\_2^{f}t-r\_0}\right\|. \label{case5}$$ For r0 ≪ 1 and t > 2, the scaled force reads: $$I\_{\rm V}(r\_0\ll 1)=-t^{-2} \left[2(1+\log 2) +\log\left\|\frac{t^2(t^2-1)}{r\_0^2}\right\|\right].\label{case5l}$$ This expression differs from that of equation ([ostriker]) through the first term in the bracket − 2(1 + log2)/t2 implying that the force on accelerated motion of a perturber initially at rest, is larger than that on uniform motion. Fig. (3) shows the friction force ([case1]), ([case2]),([case3]), ([case4]), ([case5]) as a function of time, or equivalently the Mach number, for the three values of the small distance cutoff r0 = 10− 2, 10− 4, and 10− 6. The force obtained by using the uniform motion expression, I0, ([ostriker]) with ${\cal M}=t$ is also shown for the same values of r0. For subsonic motion, the agreement between the two forces is best when r0 is smallest. The force I0 overestimates maximal drag by about 20%; it underestimates the drag on supersonic motion by the factor ∼ 2(1 + log2)/t2. Lastly, we remark that the derivation of dynamical friction in the five time intervals shows that there are no force singularities associated with accelerated motion such as that at Mach 1 for uniform motion ([ostriker]). Perturbers with positive acceleration ------------------------------------- To an accelerating perturber of trajectory, $\xi(t)={\cal M}\_0 t +t^2/2$ with ${\cal M}\_0>0$, corresponds the phase $\Delta/r={\cal M}-r/2$ where ${\cal M}=t+{\cal M}\_0$ is the Mach number. It could therefore seem sufficient to take the force expressions of perturbers initially at rest and substitute ${\cal M}$ for t. There is however a key difference between the two cases: the condition from the retarded sonic wavefront, r ≤ t, refers to time regardless of the type of motion. It should therefore be replaced with $r\leq {\cal M}-{\cal M}\_0$ that we refer to as condition (iv). This difference makes the force dependent on the initial velocity in contrast to the expressions derived assuming a constant velocity ([ostriker]). The physical reason for such a dependence was discussed in the Introduction: as the velocity varies, the far part of the wake that was launched in the medium corresponds to an earlier velocity. Combining the force from the far part of the wake to that near the perturber results in a force that has a memory of the earlier velocity. Inspection of the conditions (i,ii) of the previous section along with (iv) shows that the two cases ${\cal M}\_0>1$ and ${\cal M}\_0<1$ may be considered separately. In the following we do not derive the force expressions for the counterparts of Cases I and IV as their space extensions vanish when r0 tends to zero. For an accelerated perturber with an initial supersonic speed (${\cal M}\_0>1$), the boundary conditions (i, ii, iv) away from the transient phase (Case I) are identical to those of Case V ([dom-v]) except that in r2f and r2c, t is replaced with ${\cal M}$, and the outer integration domain is $r^c\_2\leq r\leq {\cal M}-{\cal M}\_0$. Provided that ${\cal M}>1+[(1-{\cal M}\_0)^2+2r\_0]^{1/2}$ that we call Case VI, the friction force is given as: $$\begin{aligned} I\_{\rm VI}&=&\frac{r\_0 {\cal M}\_0( 4 + r^c\_2 - r^f\_2)+[ {\cal M}\_0 (r^c\_2 +r^f\_2)+ r\_0 (r^c\_2 - r^f\_2-4)]{\cal M} +[r^c\_2+r^f\_2+{\cal M}\_0(r^f\_2-r^c\_2)]{\cal M}^2+(r^f\_2 -r^c\_2){\cal M}^3}{2 r\_0 {\cal M}^2({\cal M}+{\cal M}\_0)} \nonumber \\ &&-\frac{2{\cal M}+r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}+r\_0}{r\_2^{c}{\cal M}-r\_0}\right\| - \frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}-r\_0}{r\_2^{f}{\cal M}-r\_0}\right\| -\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\|.\label{case6}\end{aligned}$$ New terms depending the initial Mach number appear in the expression of the force. As ${\cal M}\_0$ is set to zero the force reduces to ([case5]) with $t={\cal M}$. This shows that including acceleration in the perturber’s motion makes the medium store the information about the perturber’s initial state which is in turn reflected in the reaction force felt by the perturber. Taking the limit r0 ≪ 1, the force becomes: $$\begin{aligned} I\_{\rm VI}(r\_0\ll 1)&=&-\frac{2({\cal M}- {\cal M}\_0)}{{\cal M}^2({\cal M}+{\cal M}\_0)}-\frac{2\log 2}{{\cal M}^2} -\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\| -\frac{1}{{\cal M}^2}\,\log\left\|\frac{{\cal M}^2({\cal M}^2-1)}{r\_0^2}\right\|.\label{case60}\end{aligned}$$ Three extra terms appear with respect to the uniform velocity expression ([ostriker]). As ${\cal M}\gg {\cal M}\_0$, the force tends to that of Case V ([case5l]) of perturbers initially at rest as the density perturbation of a larger Mach number dominates the initial density perturbation. For an accelerated perturber with an initial subsonic speed, the boundary conditions (i,ii,iv) show that the force expression is given as: $$\begin{aligned} 1-\left[(1-{\cal M}\_0)^2-2r\_0\right]^{1/2}\leq {\cal M}\leq1+(2r\_0)^{1/2}:&& I=I\_{\rm II},\label{case61}\\ 1+(2r\_0)^{1/2}\leq {\cal M}\leq 1+\left[(1-{\cal M}\_0)^2-2r\_0\right]^{1/2}:&& I=I\_{\rm III},\label{case62}\\ 1+\left[(1-{\cal M}\_0)^2+2r\_0\right]^{1/2}\leq {\cal M} :&& I=I\_{\rm VI},\label{case63}\end{aligned}$$ where in the expressions of $I\_{\rm II}$ and $I\_{\rm III}$, t is replaced with ${\cal M}$. We recover the general statement we made about the force on subsonic motion in Case II. In particular, the friction force on an initially subsonic perturber depends on the initial velocity as soon as the velocity is supersonic and becomes larger than 2c − V0. In Fig. (4) we show the friction force with r0 = 10− 4 for an initial subsonic velocity with ${\cal M}\_0=0.6$ using equations ([case61]), ([case62]) and ([case63]), and an initial supersonic velocity with ${\cal M}\_0=2$ using ([case6]). We remark that the dependence of dynamical friction on the initial velocity of the perturber is not an artifact of choosing to assemble the perturber at t = 0. This starting condition can be replaced with t = − T0 and T0 in turn can be made to tend to infinity thus approximating a steady state. Examination of the general force expression shows that T0 → ∞ is equivalent to the condition t = 0 where the limits ${\cal M} \gg 1$ and ${\cal M} \ll 1$ are taken for accelerating and decelerating perturbers respectively. For positively accelerating perturbers (Case IV), the force is independent of the initial Mach number as ${\cal M}\gg1$ ([case60]). In the next section, we show that for decelerating perturbers the dependence on ${\cal M}\_0$ remains as the velocity decays to zero. Decelerating perturbers ----------------------- For decelerating perturbers (${\cal M}\_0<0$), we seek the dynamical friction force from t = 0 until the velocity vanishes at $t=\|{\cal M}\_0\|$. The choice of a negative Mach number instead of a negative acceleration makes the derivations easier but results in a positive friction force (opposed to the direction of motion). The phase $\Delta/r={\cal M}-r/2$ is now negative and the boundary conditions that determine the integration domains of ([forcex]) are (i), (vi) and (v) $\tau\geq f(r,{\cal M},r\_0)$. The latter condition shows that $\tau\_{\rm max}=1$. Three cases need to be considered: $-1\leq {\cal M}\_0$, $-1-(1-2r\_0)^{1/2}\leq {\cal M}\_0 \leq -1$ and ${\cal M}\_0 \leq -1-(1-2r\_0)^{1/2}$. For subsonic motion ($-1\leq {\cal M}\_0$) and away from the transient regime (i.e. $-1+[(1+{\cal M}\_0)^2+2r\_0]^{1/2}\leq {\cal M}$), the integration domains are given as: $$\begin{aligned} r^c\_1\leq r\leq r^f\_2, &\mbox{with}&\tau\_{\rm min}=f(r,{\cal M},r\_0), \label{decesub1}\\ r^f\_2\leq r\leq {\cal M}-{\cal M}\_0, &\mbox{with}& \tau\_{\rm min}=-1.\label{decesub2} \end{aligned}$$ As for Case II (equations [dom21] and [dom22]), the latter domain ([decesub2]) does not contribute to the force. The corresponding integrand is proportional to $1-{\rm sign} (1+{\cal M}-r/2)$ and the sign argument is positive for $r<2({\cal M}+1)$. The contribution of domain ([decesub1]) is identical to that of Case II and $I=I\_{\rm II}$ ([case2]) with t replaced by ${\cal M}$. Dynamical friction on a decelerating subsonic perturber is therefore independent of its initial velocity. This confirms the general statement we made about subsonic motion in an infinite medium in Case II. For supersonic perturbers with an initial Mach number $-1-(1-2r\_0)^{1/2}\leq {\cal M}\_0 \leq -1$, we need to consider three cases depending on the value of the Mach number ${\cal M}$. The first which we call Case VII corresponds to $-1-[(1+{\cal M}\_0)^2-2r\_0]^{1/2}\leq {\cal M}\leq -1+(2r\_0)^{1/2}$ whose boundaries are: $$\begin{aligned} r^c\_1\leq r\leq r^f\_1, &\mbox{with}& \tau\_{\rm min}=f(r,{\cal M},r\_0),\ \ \mbox{where}\ \ r\_1^{f}={\cal M}+1+[({\cal M}+1)^2+2r\_0]^{1/2}, \label{decesup1}\\ r^f\_1\leq r\leq {\cal M}-{\cal M}\_0, &\mbox{with}& \tau\_{\rm min}=-1. \label{decesup2}\end{aligned}$$ Domain ([decesup1]) contributes a term similar to ([i0]) with r1c and r1f substituted for t and r2f respectively. Domain ([decesup2]) contributes a term similar to ([i2]) with r1f and ${\cal M}-{\cal M}\_0$ substituted for r3c and r2c respectively. The force is then given as: $$\begin{aligned} I\_{\rm VII}&=&\frac{{\cal M}\_0 r\_0 (r^f\_1- r^c\_1-4 ) + [{\cal M}\_0 (r^c\_1 + r^f\_1) + r\_0 (4 - r^c\_1 + r^f\_1)] {\cal M}+[r^c\_1 + r^f\_1 + {\cal M}\_0 (r^f\_1- r^c\_1)] {\cal M}^2+ ( r^f\_1-r^c\_1 ){\cal M}^3}{2r\_0{\cal M}^2({\cal M}+{\cal M}\_0)}+\nonumber\\ && +\frac{2{\cal M}+r\_0}{2 {\cal M}^3}\,\log\left\|\frac{2{\cal M}+r\_0}{r\_1^{f}{\cal M}+r\_0}\right\| +\frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}-r\_0}{r\_1^{c}{\cal M}+r\_0}\right\| +\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\| \label{case7}.\end{aligned}$$ The second case which we call Case VIII concerns $-1+(2r\_0)^{1/2}\leq {\cal M}\leq -1+[(1+{\cal M}\_0)^2-2r\_0]^{1/2}$. The corresponding integration domains are: $$\begin{aligned} r^c\_1\leq r\leq r^f\_2, &\mbox{with}& \tau\_{\rm min}=f(r,{\cal M},r\_0),\label{dom81}\\ r^f\_2\leq r\leq r^f\_3, &\mbox{with}& \tau\_{\rm min}=-1, \label{dom82}\\ r^f\_3\leq r\leq r^f\_1, &\mbox{with}& \tau\_{\rm min}=f(r,{\cal M},r\_0),\ \ \mbox{where}\ \ r\_3^{f}={\cal M}+1+[({\cal M}+1)^2-2r\_0]^{1/2},\label{dom83}\\ r^f\_1\leq r\leq {\cal M}-{\cal M}\_0, &\mbox{with}& \tau\_{\rm min}=-1.\label{dom84}\end{aligned}$$ The integration radii satisfy $r^c\_1< r\_2^{f}\leq r\_3^{f}\leq 2({\cal M}+1)\leq r\_1^{f}$. In contrast to Case III, this ordering shows that the second domain ([dom82]) does not contribute a finite term to the force. Domain ([dom81]) contributes a term equal to $I\_{\rm II}$, and domain ([dom83]) a term similar to ([i2]) where r3f and r1f are substituted for t and r2f respectively. The last domain ([dom84]) contributes a term similar to that of ([decesup2]) in Case VII. Summing up all terms, we find: $$\begin{aligned} I\_{\rm VIII}&=& \frac{{\cal M}\_0 r\_0 (r^f\_1 - r^f\_2 + r^f\_3- r^c\_1 -4)+ [{\cal M}\_0 (r^c\_1 + r^f\_1 + r^f\_2 - r^f\_3) + r\_0 (4 - r^c\_1 + r^f\_1 - r^f\_2 + r^f\_3)]{\cal M}}{2r\_0{\cal M}^2({\cal M}+{\cal M}\_0)}+\nonumber\\ && +\frac{[r^c\_1 + r^f\_1 + r^f\_2 - r^f\_3+ {\cal M}\_0 (r^f\_1 + r^f\_2 - r^f\_3 - r^c\_1) ]{\cal M}^2 +( r^f\_1 + r^f\_2 - r^f\_3-r^c\_1){\cal M}^3}{2r\_0{\cal M}^2({\cal M}+{\cal M}\_0)}+\nonumber\\ && +\frac{2{\cal M}+r\_0}{2 {\cal M}^3}\,\log\left\|\frac{2{\cal M}+r\_0}{r\_1^{f}{\cal M}+r\_0}\right\| + \frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}-r\_0}{r\_3^{f}{\cal M}-r\_0}\right\| +\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\|-\frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{r\_1^c{\cal M}+r\_0}{r\_2^f{\cal M}-r\_0}\right\|. \label{case8}\end{aligned}$$ The third case concerns $-1+[(1+{\cal M}\_0)^2+2r\_0]^{1/2}\leq {\cal M}\leq 0$ whose boundaries turn out to be those of subsonic motion ${\cal M}\_0>-1$. The corresponding force is therefore given by ([case2]) where t is replaced ${\cal M}$. Fig. (5, left panel) shows the force on a perturber with an initial velocity ${\cal M}\_0=-1.7$ using equations ([case7]), ([case8]) and ([case2]) alongside the uniform motion friction force ([ostriker]). In the supersonic regime the uniform motion force overestimates the drag by about 10% at its maximum. In the subsonic regime the force on accelerated motion departs significantly from the uniform velocity force before joining it at $\|{\cal M}\|\simeq 2+{\cal M}\_0= 0.3$ to assume the expression ([case2]). In particular, when the velocity vanishes so does the force. Lastly, for a supersonic perturber with an initial velocity ${\cal M}\_0 \leq -1-(1-2r\_0)^{1/2}$, we find that: $$\begin{aligned} -1-[(1+{\cal M}\_0)^2-2r\_0]^{1/2}\leq {\cal M}\leq -1+(2r\_0)^{1/2}:&& I=I\_{\rm VII},\label{case8x} \\ -1+(2r\_0)^{1/2}\leq {\cal M}\leq 0: && I=I\_{\rm VIII}.\label{case8x2}\end{aligned}$$ Fig. (5, right panel) shows the force on a perturber with an initial velocity ${\cal M}\_0=-4$ using equations ([case8x]) and ([case8x2]) alongside the uniform velocity friction force ([ostriker]). In the supersonic regime the uniform motion force overestimates the drag by about 20% at its maximum. In the subsonic regime the force on accelerated motion departs significantly from that on uniform motion and is finite at ${\cal M}=0$. Taking the limit where r0 is small, we find $I\_{\rm VII}(r\_0\ll 1)=-I\_{\rm VI}(r\_0\ll 1)$ and: $$\begin{aligned} I\_{\rm VIII}(r\_0\ll 1)&=& \frac{4(1+{\cal M}\_0+{\cal M})}{{\cal M}({\cal M}+{\cal M}\_0)} +\frac{1}{ {\cal M}^2}\,\log\left\|\frac{({\cal M}-{\cal M}\_0)^2(1-{\cal M})^2}{({\cal M}+{\cal M}\_0)^2(1+{\cal M})^3}\right\|.\end{aligned}$$ At ${\cal M}=0$, $I\_{\rm VIII}(r\_0\ll 1)=1-4/{\cal M}\_0^2$ in contrast to the uniform motion force which vanishes with the perturber’s velocity. The knowledge of the history of the perturber’s motion is therefore required to evaluate dynamical friction. Simply applying uniform motion dynamical friction force ([ostriker]) may yield incorrect results regarding the perturber’s evolution in the ambient medium as evidenced in Fig. (6) where we plot the relative error between the accelerated motion and uniform motion forces for decelerating supersonic perturbers. A similar conclusion was previously reached by Sánchez-Salcedo and Brandenburg (1999) who numerically tested the accuracy of the uniform motion friction formula ([ostriker]) using the full nonlinear fluid equations. Following the evolution of a decelerating initial perturber with ${\cal M}\_0=1.5$ up to ${\cal M}=0.6$, they found that equation ([ostriker]) underestimated the friction force as ${\cal M}$ became smaller than unity (see their Fig. 5) which corresponds to our Cases VII and VIII. The authors attributed this discrepancy to the ‘memory of the wake’. Concluding remarks ================== In this paper, we studied the influence of acceleration on dynamical friction in a gaseous medium. Our main conclusions are stated in the Abstract. This work also constitutes an illustration of the use of the general expression of dynamical friction derived in Appendix A. In particular, applying that formulation to a perturber with accelerated motion allowed us to illustrate how dynamical friction is not a local force and may depend the perturber’s initial state. The meaning of the initial state depends on the problem’s configuration. For instance, it may correspond to the time the perturber enters the gaseous medium or to the initial time of a numerical simulation of dynamical friction. We note that non-locality is not peculiar to rectilinear motion and should be true of the dynamical friction associated with rotational (circular or eccentric) motion. The main technical result is the set of expressions $I\_{\rm I}$ to $I\_{\rm VIII}$ that describe dynamical friction as a function of the initial and current velocities as the perturber is accelerated (positively or negatively) in the ambient medium. Our derivations have assumed a constant acceleration because the calculations are simpler to carry out in this case. Dynamical friction for constantly accelerating perturbers applies readily to the motion of young stars with asymmetric jet systems. We believe that the general features we found will hold for a general type of acceleration as long as the instant velocity change remains small. Including the deceleration generated by dynamical friction instead of a constant negative acceleration is likely to further reduce the friction force because the acceleration’s amplitude increases with time. The dependence of dynamical friction on the perturber’s size, r0, is most important for supersonic motion. In the force expressions, the dimensionless radius, r0, stands in for the ratio r0A/c2 of dimensional parameters. The reaction force therefore depends explicitly on acceleration even in the limit r0 ≪ c2/A as in Case VI. Dynamical friction on subsonic motion is independent of r0 when r0 ≪ c2/A. Using the force expression requires the knowledge of the perturber’s acceleration and the medium’s sound speed. In Fig. (3) to (5), the value r0 = 10− 4 was used so as to make accelerated motion friction closer to uniform motion friction (see Fig. 1). The assumption of constant velocity in the derivation of dynamical friction is also made for collisionless star systems. Although they have different magnitudes, the uniform motion friction forces in a gaseous medium and a collisionless star system behave similarly (as ∣F∣ ∝ V− 2 for supersonic motion and ∣F∣ ∝ V for subsonic motion; see Ostriker 1999, Sánchez-Salcedo and Brandenburg 2001). This similarity suggests that the conclusions we found are likely to hold for collisionless systems. Acknowledgments =============== The author thanks an anonymous referee and Scott Tremaine for useful comments. General expression of dynamical friction ======================================== In this appendix, we derive a general expression of dynamical friction on a perturber of gravitational potential ϕp with a general trajectory ${\bm \xi}(t)$ that travels inside an inviscid homogeneous gaseous medium. The perturber is set in motion at time t = 0. The medium’s response to the perturber’s presence is described by the standard fluid equations: $$\partial\_t\rho+{\bm \nabla} \cdot (\rho {\bm v})=0,\ \ \ \partial\_t {\bm v}+({\bm v}\cdot{\bm \nabla}) {\bm v}=-\frac{1}{\rho}{\bm \nabla} p-{\cal H}(t){\bm \nabla}{\phi\_p},$$ where ${\bm v}({\bm x},t)$ and $\rho({\bm x},t)$ are the perturbed velocity and density fields, p is the pressure, and ${\cal H}(t)$ is the Heaviside function. Prior to the perturber’s motion, the gaseous medium is assumed to be at rest $[i.e. {\bm v}({\bm x},t)=0]$, homogeneous of density ρ0, pressure p0 and isothermal with a sound speed c = (p0/ρ0)1/2. The linearization of medium’s equation in the vicinity of the rest state yields: $$\partial\_t\varrho+c{\bm \nabla} \cdot { \bm{\nu}}=0,\ \ \ \frac{1}{c}\partial\_t {\bm \nu}+{\bm \nabla} \varrho= -\frac{{\cal H}(t)}{c^2}{\bm \nabla} \phi\_p, \label{momentumeq}$$ where we used $\rho=\rho\_0(1+\varrho)$ and ${\bm v}=c {\bm \nu}$. The combined equations in turn yield the forced sound wave equation satisfied by the density perturbation $\varrho$ as: $$\nabla^2\varrho-\frac{1}{c^2}\, \partial\_t^2\varrho=-\frac{{\cal H}(t)}{c^2}\nabla^2\phi\_p \label{wavA}.$$ The solution of this equation is known in electromagnetism theory as the Liénard-Wiechert potential (Jackson 1998, Chapter 14) and represents the electric potential of a moving electric charge distribution. It is written in integral form using the retarded Green function as: $$\begin{aligned} \varrho({\bm x},t)&=&\frac{1}{4\pi c^2}\int\_{-\infty}^{+\infty}{\rm d}^3x^\prime {\rm d}u\ \frac{\delta \left[u-t+\|{\bm x}-{\bm x}^\prime\|/c\right] \ \nabla^2\phi\_p[{\bm x}^\prime - {\bm \xi}(u)]\ {\cal H}(u)} {\|{\bm x}-{\bm x}^\prime\|}. \label{densA}\end{aligned}$$ In reaction to the perturber’s excitation of the wake, the gaseous medium exerts a force on the perturber given as: $${\bm F}=\int\_{\partial V({\bm x})} \rho\_0\varrho {\cal H}(t) {\bm \nabla} \phi\_p {\rm d}^3x =\frac{{\cal H}(t)\rho\_0}{4\pi c^2} \int\_{\partial V({\bm x})}{\rm d}^3x{\rm d}^3x^\prime {\rm d}u\ \frac{\delta \left[u-t+\|{\bm x}-{\bm x}^\prime\|/c\right]\ \nabla^2\phi\_p[{\bm x}^\prime - {\bm \xi}(u)]\ {\bm \nabla} \phi\_p[{\bm x} - {\bm \xi}(t)]\ {\cal H}(u)} {\|{\bm x}-{\bm x}^\prime\|}\label{forc1A}$$ The symbol $\partial V({\bm x})$ denotes the boundaries of the spacial integration domain defined by the medium’s physical boundary as well as the lower distance cutoff in the perturber’s vicinity such as its size or its corresponding accretion radius. The boundaries $\partial V({\bm x})$ are used to truncate the perturber’s potential at small and large distances to avoid divergence. Applying the variable changes, ${\bm X}={\bm x}-{\bm \xi}(t)$ and ${\bm Y}={\bm x}-{\bm x}^\prime$ and integrating the force over u yields the general expression of dynamical friction as: $$\begin{aligned} {\bm F} &=&\frac{{\cal H}(t)G\rho\_0}{c^2} \int\_{\partial V[{\bm X}+{\bm \xi}(t)], \|{\bm Y}\|\leq ct}{\rm d}^3X\frac{{\rm d}^3Y}{\|{\bm Y}\|} \ \ \rho\_p({\bm X}-{\bm Y}+{\bm \Delta})\ {\bm \nabla} \phi\_p({\bm X}), \label{forc2A} \end{aligned}$$ where ρp is the perturber’s density distribution, ρp = ∇2ϕp/4πG, and ${\bm \Delta}={\bm \xi}(t)-{\bm \xi}(t-\|{\bm Y}\|/c)$. For a given perturbing potential, dynamical friction therefore depends only on the phase ${\bm \Delta}$ as well as the boundary conditions specified by ∂V and $\|{\bm Y}\|\leq ct$. The dependence of ${\bm \Delta}$ on the whole trajectory ${\bm \xi}(t)$ shows that dynamical friction is manifestly non-local. Deriving dynamical friction through equation ([forc2A]) is in contrast with the previous two-step methods of estimating dynamical friction for uniform motion by first calculating explicitly ([densA]) then applying an analytical or numerical calculation to the first integral in equation ([forc1A]) to derive the force (Ostriker 1999, Kim and Kim 2007, Kim et al 2008). As a result such methods obscure the true dependence of the force on the perturber’s motion and the boundary conditions. For a point-like perturber, $\phi\_p(\bm x)=-GM/\|{\bm x}\|$ and $\rho\_p ({\bm x}) =M \delta({\bm x})$, where M is the perturber’s mass. The forced sound wave equation, density enhancement and friction force are reduced to: $$\begin{aligned} &&\nabla^2\varrho-\frac{1}{c^2}\, \partial\_t^2\varrho=-\frac{1}{c^2}\nabla^2\phi=-4\pi \frac{GM}{c^2}\, \delta^3[{\bm x}-{\bm \xi}(t)] {\cal H}(t), \label{wav2A}\\ \varrho({\bm x},t) &=&\frac{GM}{c^2}\int\_{-\infty}^{+\infty} \frac{\delta \left[u-t+\|{\bm x}-{\bm \xi}(u)\|/c\right] {\cal H}(u)} {\|{\bm x}-{\bm \xi}(u)\|}\ {\rm d}u, \label{dens2A}\\ {\bm F}&=&\frac{{\cal H}(t)(GM)^2\rho\_0}{c^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r/c)], r\leq ct}r\sin\theta{\rm d}r\, {\rm d}\theta\, {\rm d}\varphi \ \frac{{\bm y}-{\bm \Delta}}{\|{\bm y}-{\bm \Delta}\|^3},\label{for3A} \end{aligned}$$ where ${\bm y}={\bm x}-{\bm \xi}(t-r/c)$. The condition on the time of perturbation (t ≥ 0) represented by the Heaviside function in equation ([wavA]) and the expression ([forc1A]) now implies that the spacial domain that enters the expression of the force is equal to the size of the wavefront, ct. However, the spacial domain refers to the relative position with respect to the perturber not at time t but at the retarded time t − r/c. This explains why the force ([for3A]) describes subsonic as well as supersonic motion that perturbs the medium outside the sonic wavefront of radius ct. In the force expression, the boundaries $\partial V[{\bm y}+{\bm \xi}(t-r/c)]$ are treated as truncation boundaries. In the next Appendix, the force expression is applied to a point-like perturber in uniform rectilinear motion to recover the corresponding known results. Application to uniform rectilinear motion ========================================= We validate the expressions ([forc2A]) and ([for3A]) by deriving dynamical friction for a perturber set in uniform rectilinear motion in an infinite homogeneous medium. The perturber’s trajectory is written as ${\bm \xi}(t)=Vt {\bm e}\_z$ where ${\bm e}\_z$ is the unit vector along z, the phase ${\bm \Delta}={\cal M} r {\bm e}\_z$ where ${\cal M}$ is the Mach number. The friction force is equally along ${\bm e}\_z$ and its expression is simplified as follows: $$\begin{aligned} {F}&=& \frac{{\cal H}(t)(GM)^2\rho\_0}{c^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r/c)], r\leq ct}r\sin\theta{\rm d}r\, {\rm d}\theta\, {\rm d}\varphi \ \frac{r\cos \theta-{\Delta}}{\|r^2+\Delta^2-2r\Delta\cos\theta\|^{3/2}},\\ F&=&\frac{2\pi {\cal H}(t)(GM)^2\rho\_0}{V^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r/c)], r\leq ct} \ \frac{{\rm d}r} {r} \left[\frac{1-{\cal M}\tau}{({1+{\cal M}^2-2 {\cal M}\tau})^{1/2}}\right]\_{\tau\_{\rm min}}^{\tau\_{\rm max}}, \label{forc3A}\end{aligned}$$ where in the last expression we performed the integrals over φ and τ = cosθ. The extremal values $ \tau\_{\rm min}$ and $\tau\_{\rm max}$ are determined from the boundary conditions. Calling r0 the perturber’s size, the force is truncated at small distances by $\|{\bm x}-{\bm\xi}(t)\|>r\_0$. Expressing these conditions in terms of the variables r and τ that appear in equation ([forc3A]) shows that we need to find the domains that satisfy: (i) − 1 ≤ τ ≤ 1, (ii) $r\_0\leq r(1+{\cal M}^2-2 {\cal M}\tau)^{1/2}$, and (iii) r ≤ ct. The first two conditions yield the following integration domains: $$\begin{aligned} \frac{r\_0}{\|1+{\cal M}\|} <r<\frac{r\_0}{\|1-{\cal M}\|} &\mbox{with}& \tau\_{\rm min}=-1,\ \ \ \tau\_{\rm max}=\frac{1+{\cal M}^2-(r\_0/r)^2}{2{\cal M}}, \label{dom1A}\\ \frac{r\_0}{\|1-{\cal M}\|} <r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &\mbox{with}& \tau\_{\rm min}=-1,\ \ \ \tau\_{\rm max}=1.\label{dom2A}\end{aligned}$$ The inner domain ([dom1A]) corresponds to the vicinity of the perturber’s retarded position. The radius $r\_0/\|1+{\cal M}\|$ defines a region around the perturber that does not contribute to dynamical friction as the first two conditions are not satisfied. The third condition depends on the distance ct travelled by the retarded wavefront. There are two force regimes: the local perturbation regime ($ct\leq r\_0/\|1-{\cal M}\|$) where the perturbation remains in the perturber’s vicinity and the distant perturbation regime ($r\_0/\|1-{\cal M}\|\leq ct$ ) where the perturbation has expanded beyond the perturber’s vicinity. Calling I the integral that appears in ([forc3A]), the integration for the distant perturbation regime extends over the two domains ([dom1A]) and ([dom2A]) with r ≤ ct and I is given as: $$\begin{aligned} I&=& \int^{\frac{r\_0}{\|1-{\cal M}\|}}\_{\frac{r\_0}{\|1+{\cal M}\|}} \left[\frac{1}{2r\_0}\left(1-{\cal M}^2+\frac{r\_0^2}{r^2}\right)-\frac{1}{r}\right]\ {\rm d}r-\left[1-{\rm sign} (1-{\cal M})\right]\int\_{\frac{r\_0}{\|1-{\cal M}\|}}^{ct}\frac{{\rm d}r}{r},\\I&=&-\log \left\|\frac{1+{\cal M}}{1-{\cal M}}\right\|+\frac{1}{2}\left[1+{\cal M}-\|1-{\cal M}\|\right]\left[1+{\rm sign}(1-{\cal M})\right] -\left[1-{\rm sign} (1-{\cal M})\right] \ \log\left\|\frac{(1-{\cal M})ct}{r\_0}\right\|. \label{ostriA}\end{aligned}$$ This expression agrees with those derived by Ostriker (1999) for subsonic and supersonic perturbers moving in an infinite gaseous medium using explicit expressions of the density perturbation. In particular, the force, $F\propto I/{\cal M}^2$, is time-independent and behaves as $-{\cal M}$ for small Mach numbers. We note the interesting result that only the vicinity of a subsonic perturber’s retarded position ([dom1A]) contributes to dynamical friction. We show in section 3 that this property implies that the force on subsonic perturbers with a general type of motion (not necessarily rectilinear or uniform) is similar to ([ostriA]) provided that the perturber’s motion history does not involve supersonic episodes and that the perturber’s acceleration $\|\ddot {\bm \xi}(t)\|$ is much smaller than c2/r0. For the local perturbation regime ($ct\leq r\_0/\|1-{\cal M}\|$), the force is zero before $t=r\_0/c\|1+{\cal M}\|$ and only the domain ([dom1A]) is relevant to the integral I which is written as: $$\begin{aligned} I&=& \int^{ct}\_{\frac{r\_0}{\|1+{\cal M}\|}} \left[\frac{1}{2r\_0}\left(1-{\cal M}^2+\frac{r\_0^2}{r^2}\right)-\frac{1}{r}\right]\ {\rm d}r= -\log\left[\frac{ct(1+{\cal M})}{r\_0}\right]+{\cal M}-\frac{r\_0}{2ct}+\frac{(1-{\cal M}^2)ct}{2r\_0}. \label{transient}\end{aligned}$$ In Fig. (7), we show the friction force as a function of t for subsonic and supersonic perturbers as t increases smoothly from 0 to a value in the distant perturbation regime. The time dependence of the force agrees with that obtained from the numerical simulation of the nonlinear fluid equations (Sánchez-Salcedo and Brandenburg 1999, Figure 2). However, the attempted fit of the force in the local perturbation regime from numerical simulations (Sánchez-Salcedo and Brandenburg 2001, Equation 22) does not reproduce the correct dependence of ([transient]) on ${\cal M}$, r0 and time. Binney J., Tremaine S., 2008, Galactic Dynamics, Princeton University Press, Princeton Bullock J. S., Johnston K. V., 2005, ApJ, 635, 931 Chandrasekhar S., 1943, ApJ, 97, 255 Edgar R. G., 2005, A&A, 434, 41 El-Zant A. A., Kim W.-T., Kamionkowski M., 2004, MNRAS, 354, 169 Faltenbacher A., Kravtsov A. V., Nagai D., Gottlöber S., 2005, MNRAS, 358, 139 Frenk C. S., Evrard A. E., White S. D. M., Summers F. J., 1996, ApJ, 472, 460 Goldreich P., Tremaine S., 1982, ARA&A, 20, 249 Hartigan P., Morse J., 2007, ApJ, 660, 426 Hirth G. A., Mundt R., Solf J., Ray T. P., 1994, ApJ, 427, L99 Hornung P., Pellat R., Barge P., 1985, Icar, 64, 295 Ida S., 1990, Icar, 88, 129 Jackson J. D., 1998, Classical Electrodynamics, John Wiley & Sons Inc, New York Kim W.-T., El-Zant A. A., Kamionkowski M., 2005, ApJ, 632, 157 Kim H., Kim W.-T., 2007, ApJ, 665, 432 Kim H., Kim W.-T., Sánchez-Salcedo F. J., 2008, ApJ, 679, L33 Kim S. S., Morris M., 2003, ApJ, 597, 312 Kokubo E., Ida S., 1996, Icar, 123, 180 Namouni F., 2005, AJ, 130, 280 Namouni F., 2007, ApJ, 659, 1505 Namouni F., Luciani J. F., Pellat R., 1996, A&A, 307, 972 Narayan R., 2000, ApJ, 536, 663 Ostriker E. C., 1999, ApJ, 513, 252 Portegies Zwart S. F., Baumgardt H., Hut P., Makino J., McMillan S. L. W., 2004, Natur, 428, 724 Sánchez-Salcedo F. J., Brandenburg A., 1999, ApJ, 522, L35 Sánchez-Salcedo F. J., Brandenburg A., 2001, MNRAS, 322, 67 Stewart G. R., Wetherill G. W., 1988, Icar, 74, 542 [f1] [f2] [f3] [f4] [f5] [f6] [f7] --- 1. E-mail: [email protected][↩](#fnref1) 2. The variables ${\bm x}$, ${\bm \xi}$ and r0 remain scaled to L ([scalings]).[↩](#fnref2) 3. We remind the reader that the uniform motion friction force on subsonic perturbers ([ostriker]) is independent of r0 and that no assumptions on r0 are made to derive its expression.[↩](#fnref3) Dynamical friction for accelerated motion in a gaseous medium ============================================================= ### Dates: UNDER REVIEW [firstpage] Dynamical friction arises from the interaction of a perturber and the gravitational wake it excites in the ambient medium. This interaction is usually derived assuming that the perturber has a constant velocity. In realistic situations, motion is accelerated as for instance by dynamical friction itself. Here, we study the effect of acceleration on the dynamical friction force. We characterize the density enhancement associated with a constantly accelerating perturber with rectilinear motion in an infinite homogeneous gaseous medium and show that dynamical friction is not a local force and that its amplitude may depend on the perturber’s initial velocity. The force on an accelerating perturber is maximal between Mach 1 and Mach 2, where it is smaller than the corresponding uniform motion friction. In the limit where the perturber’s size is much smaller than the distance needed to change the Mach number by unity through acceleration, a subsonic perturber feels a force similar to uniform motion friction only if its past history does not include supersonic episodes. Once an accelerating perturber reaches large supersonic speeds, accelerated motion friction is marginally stronger than uniform motion friction. The force on a decelerating supersonic perturber is weaker than uniform motion friction as the velocity decreases to a few times the sound speed. Dynamical friction on a decelerating subsonic perturber with an initial Mach number larger than 2 is much larger than uniform motion friction and tends to a finite value as the velocity vanishes in contrast to uniform motion friction. hydrodynamics – ISM: general Introduction ============ Dynamical friction is the reaction force that a perturber feels from the gravitational wake it generates in the ambient medium. This fundamental interaction has therefore had a wide range of applications in the formation of astronomical structures. Applications to collisionless systems include mass segregation in star clusters (e.g. Kim and Morris 2003, Portegies et al 2004), galaxy distribution in hierarchical cluster formation (e.g. Frenk et al 1996, Bullock and Johnston 2005), the decay of satellites in galactic halos and galaxy mergers (Binney and Tremaine 2008 and references therein). Dynamical friction also occurs in collisional particle systems and was applied to the study of the growth of planetesimals (Hornung et al 1985, Stewart and Whetherill 1988), the eccentricity excitation of planetary embryo orbits (Ida 1990, Kokubo and Ida 1996, Namouni et al 1996), and the confinement of planetary rings (Goldreich and Tremaine 1982). For gaseous systems, applications include the orbital decay of compact stars orbiting supermassive black holes (Narayan 2000), and the heating of intercluster gas by decelerating supersonic galaxies (El-Zant et al. 2004, Faltenbacher et al 2005, Kim et al 2005). Following the pioneering work of Chandrasekhar (1943) on the gravitational drag experienced by a perturber in collisionless star systems, the derivations of dynamical friction assume that the perturber has a constant velocity in the ambient medium. The assumption of uniform motion is made for collisionless systems (Binney and Tremaine 2008), collisional particle systems (Hornung et al 1985, Stewart and Whetherill 1988, Ida 1990, Goldreich and Tremaine 1982) as well as gaseous media (Kim and Kim 2007, Kim et al 2008, Ostriker 1999). The resulting dynamical friction force which depends on the perturber’s velocity is subsequently used to model the evolution of the perturber’s motion whose velocity varies with time. Only the numerical simulations of the nonlinear fluid equations are free from this assumption (Sánchez-Salcedo and Brandenburg 1999, 2001). The neglect of the perturber’s acceleration in dynamical friction is justified by the small amplitude of the force that is proportional to the square of the pertuber’s mass. Neglecting acceleration is independent of the type of motion and is mainly motivated by the lack of an analytical framework where the perturber’s accelerated motion could be taken into account. To overcome this obstacle, we derive in Appendix A a new expression of dynamical friction that applies to a general type of motion and depends on the perturber’s motion and the boundary conditions but not the density perturbation. In particular, dynamical friction is shown to be non-local as it depends on the whole trajectory of the perturber. Although non-locality does not appear if the perturber’s velocity is constant, its origin is easy to identify: as the perturber’s wake propagates in the medium, a velocity change because of acceleration affects the density enhancement near the perturber and offsets the force from its local value as the far parts of the wake were generated by a different velocity. Therefore when acceleration is taken into account, the amplitude of dynamical friction may change with respect to the value derived assuming a constant velocity because in modelling the force on an accelerated perturber as that on a constant velocity perturber, one assumes that the gravitational wake produced by accelerated motion is similar to that of uniform motion. We shall see that this difference between the two wakes makes dynamical friction dependent on the initial velocity of the perturber. Another difference between the wakes generated by constant velocity motion and accelerated motion is the absence of a perturbed region with front-back symmetry with respect to a subsonic accelerating perturber. In effect a constant velocity perturber raises a gravitational wake in the ambient medium that is symmetric with respect to it and truncated by the sonic shockwave. This truncation defines two regions, one that lacks front-back symmetry located behind the perturber, and one with front-back symmetry around the perturber that becomes the largest region inside the wavefront for velocities smaller than half the sound speed. Dynamical friction arises from the former region as shown by Ostriker (1999). When acceleration is taken into account, front-back symmetry is lost everywhere. This would intuitively imply that dynamical friction on an accelerated perturber should be much larger than the value derived assuming uniform motion for subsonic perturbers. Curiously, we shall show that the force on an accelerating subsonic perturber initially at rest is similar to the uniform friction force. Lastly, accelerated motion friction would differ from uniform motion friction because the former never reaches steady state. In effect, dynamical friction on a constant subsonic velocity perturber is known to be time-independent (Ostriker 1999). It is the object of this paper to quantify how taking into account acceleration modifies dynamical friction on a perturber travelling in an infinite homogeneous gaseous medium. The perturber’s motion is assumed to have a constant acceleration, a choice that makes the analytical developments simpler. Physically such constant acceleration approximates the deceleration felt by the perturber because of dynamical friction as the force amplitude is small and the corresponding instantaneous velocity change is equally small. Constant acceleration also models the motion of a young star with an asymmetric jet system moving in interstellar gas (Hirth et al 1994, Hartigan and Morse 2007, Namouni 2005, 2007). In what follows, we characterize the density perturbation associated with accelerated motion in Section 2. We derive the force on an accelerated perturber in Section 3 using the general expression of dynamical friction (derived in Appendix A) that does not depend on the density enhancement caused by the perturber but only on the perturber’s motion and the medium’s boundary conditions. In particular we show that including acceleration makes dynamical friction dependent on the initial velocity of the perturber. Section 4 contains concluding remarks. In Appendix B, the new expression of dynamical friction derived in Appendix A is validated by applying it to uniform rectilinear motion and recovering the known results on dynamical friction. Density enhancement =================== Consider a point-like perturber of mass M on an accelerated rectilinear trajectory, of initial velocity V0 and acceleration A, that travels inside an inviscid isothermal gaseous medium of density ρ and sound speed c. Assuming the density perturbation associated with the perturber’s wake is small compared to the medium’s unperturbed density ρ0, the corresponding density enhancement $\varrho$ defined as $\rho=\rho\_0(1+\varrho)$ satisfies the standard forced sound wave equation (Appendix A, equation [wav2A]): $$\nabla^2\varrho-\frac{1}{c^2}\, \partial\_t^2\varrho=-\frac{{\cal H}(t)}{c^2}\nabla^2\phi\_p=-4\pi \frac{GM}{c^2}\, \delta^3[{\bm x}-{\bm \xi}(t)] {\cal H}(t), \label{waveeq}$$ where $\phi\_p= -GM/\|{\bm x}-{\bm \xi(t)}\|$ is the perturber’s gravitational potential, ${\cal H}(t)$ is the Heaviside function, G is the gravitational constant, ${\bm \xi}(t)=(At^2/2+V\_0t){\bm e}\_z$ is the perturber’s trajectory chosen along the z–direction and δ(x) is Dirac’s delta function. The solution of this equation is the Liénard-Wiechert potential which may be written in integral form using the retarded Green function as ([dens2A]): $$\begin{aligned} \varrho({\bm x},t) &=&\frac{GM}{c^2}\int\_{-\infty}^{+\infty} \frac{\delta \left[u-t+\|{\bm x}-{\bm \xi}(u)\|/c\right] {\cal H}(u)} {\|{\bm x}-{\bm \xi}(u)\|}\ {\rm d}u. \label{density}\end{aligned}$$ The acceleration A introduces two natural time and length scales given by: $$T=\frac{c}{A}, \ \ L=\frac{c^2}{A}. \label{scalings}$$ In these units, the perturber’s trajectory may be written as $\xi(t)=t^2/2+{\cal M}\_0t$ where ${\cal M}\_0$ is the initial Mach number. The Mach number is a function of time and reads: ${\cal M}=t+{\cal M}\_0$. The scaling time T is the duration to increase the Mach number by unity and the scaling length L is the corresponding travelled distance. Applied to the density enhancement these scalings yield: $$\varrho({\bm x},t)=\frac{GMA}{c^4}\int\_{-\infty}^{\infty} \frac{\delta \left(u-t+\left[R^2+\left(z-{\cal M}\_0u- u^{2}/2\right)^2\right]^{1/2}\right) {\cal H}(u)} {\left[R^2+\left(z-{\cal M}\_0u- u^{2}/2\right)^2\right]^{1/2}}\ {\rm d}u, \label{density2}$$ where R2 = x2 + y2 is the cylindrical radius; unless it is stated otherwise, all length and time variables and constants are normalized to ([scalings]). In order to derive the density profile we use the property of the δ-function α(u)δ[β(u)] = ∑iα(ui)δ(u − ui)/∣β′(ui)∣ where ui are the roots of β(u), and β′ is the derivative of β. The roots of the argument of the δ–function in equation ([density2]) satisfy the quartic equation: $$u^4/4+{\cal M}\_0u^3-(1+ z-{\cal M}\_0^2)\,u^2+2 (t-{\cal M}\_0z) u -t^2+r^2=0, \label{roots}$$ where r = (R2 + z2)1/2 is the spherical radius. This equation is the counterpart of the quadratic equation derived by Ostriker (1999) for uniform motion. The former may be reduced to the latter once time and length dimensions are restored. Equation ([roots]), however, cannot be solved analytically because its varying coefficients do not permit simple and unique expressions of the roots. Instead we determine the roots numerically and substitute them into the density expression obtained by applying the δ–function property cited above. The density expression reduces to: $$\varrho({\bm x},t)=\frac{GMA}{c^4}\sum\_{{\rm roots}\ u\_i {\rm\ of\ eq.\ (\ref{roots})} }\|t-u\_i-({\cal M}\_0+u\_i)(z-{\cal M}\_0 u\_i - u\_i^2/2)\|^{-1}.$$ Fig. (1) follows the level curves of $\varrho$ for perturbers initially at rest (${\cal M}\_0=0$) as they evolve from t = 0 to Mach 6. The front-back symmetry that characterizes the density enhancement of subsonic constant velocity motion in a uniform medium is absent for both subsonic and supersonic motion. The perturber reaches Mach 1 at z = 0.5 but remains inside the sonic wavefront of radius t as the latter has been propagating faster than the perturber’s velocity. Once the velocity is supersonic, a Mach cone is formed inside the sonic wavefront and a shallow density depression develops and trails the perturber (z ≤ − 1). Eventually as Mach 2 is reached the perturber catches up with the sonic shockwave and exits the sonic region to form an external Mach cone. Fig. (2) shows the level curves of $\varrho$ associated with decelerating perturbers with ${\cal M}\_0 \neq 0$ as a function time until the perturber’s velocity vanishes. The density enhancement of subsonic perturbers has a structure similar to that of uniform motion (Ostriker 1999) but lacks its front-back symmetry with respect to the perturber. We show in the following section that under certain circumstances, dynamical friction may be not significantly affected by the loss of front-back symmetry even as closed level curves around the perturber contribute to the force unlike those of uniform motion (see Introduction). The deceleration of supersonic perturbers modifies the structure of the density enhancement and reduces the extent of the Mach cone but does not erase it as the velocity vanishes (bottom row, right panel) indicating the presence of a finite friction force. Friction force ============== The reaction force exerted on a point-like perturber in a homogeneous gaseous medium by its gravitational wake is given by equation ([for3A]) and is written in dimensional variables as: $${\bm F}=\frac{{\cal H}(t)(GM)^2\rho\_0}{c^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r/c)], r\leq ct}r\sin\theta{\rm d}r\, {\rm d}\theta\, {\rm d}\varphi \ \frac{{\bm y}-{\bm \Delta}}{\|{\bm y}-{\bm \Delta}\|^3},\label{force}$$ where r, θ and φ are the spherical coordinates of the vector ${\bm y}$ defined as the relative position within the medium with respect to the perturber’s position at the retarded time t − r/c. It may be written as ${\bm y}={\bm x}-{\bm \xi}(t-r/c)$ where ${\bm x}$ is the position of fluid element in the medium. This definition of ${\bm y}$ influences the force integral through the boundary conditions of the medium denoted by ∂V. The phase ${\bm \Delta}={\bm {\bm\xi}}(t)-{\bm {\bm\xi}}(t-r/c)$, the upper radial cutoff at the retarded sonic shockwave r ≤ ct, and the boundary conditions are all that is necessary to compute the friction force. In particular, the absence of an explicit expression for the density enhancement is irrelevant for the derivation of the force. The derivation of expression ([force]) assumes that the perturber is assembled at t = 0 a fact that introduces the size of the retarded sonic shockwave as an upper cutoff. This assumption is not restrictive as it allows us to study the time-dependent evolution of dynamical friction as well approximate steady state behaviour (see section 3.2). It also describes turning on a numerical simulation of dynamical friction (Sánchez-Salcedo and Brandenburg 1999, 2001). For rectilinear motion along the z–direction in an infinite medium, the force is equally along ${\bm e}\_z$ and its expression can be reduced further by integrating over the longitude ϕ and the variable τ = cosθ as: $$F=\frac{2\pi{\cal H}(t)(GM)^2\rho\_0}{c^2}\int\_{\partial V[{\bm y}+{\bm \xi}(t-r)], r\leq t}\ \frac{{\rm d}r} {\Delta^2}\ \left[\frac{ r-\tau\Delta}{\left\|1+(\Delta/r)^2-2(\Delta/r)\tau\right\|^{1/2}}\right]^{\tau\_{\rm max}}\_{\tau\_{\rm min}},\label{forcex}$$ where the scalings ([scalings]) have been applied; the perturber’s trajectory ${\bm \xi}(t)$, the time, t, and the position ${\bm x}$ and its spherical components are dimensionless hereafter. The expressions of $\tau\_{\rm min}$ and $\tau\_{\rm max}$ as a function of the perturber’s trajectory and the variable r are determined from the boundary conditions. For uniform rectilinear motion, ${\cal M}$ is constant and the phase $\Delta/r={\cal M} $. Calling r0 the gravitational potential’s small distance cutoff scaled to L ([scalings]), the integral that appears in the expression ([forcex]) may be written as (Appendix B, equation [ostriA]; Ostriker 1999): $$\begin{aligned} I\_0&=&{\cal M}^{-2}\left(\frac{1}{2}\left[1+{\cal M}-\|1-{\cal M}\|\right]\left[1+{\rm sign}(1-{\cal M})\right]-\log \left\|\frac{1+{\cal M}}{1-{\cal M}}\right\| -\left[1-{\rm sign} (1-{\cal M})\right] \ \log\left\|\frac{(1-{\cal M})t}{r\_0}\right\|\label{ostriker}\right).\end{aligned}$$ For accelerated motion, the phase $\Delta/r={\cal M}\_0+t-r/2$. In order to examine how acceleration modifies the dynamical friction force, we consider the three cases of perturbers initially at rest (${\cal M}\_0=0$), accelerating perturbers (${\cal M}\_0>0$) and decelerating perturbers (${\cal M}\_0<0$). Before deriving dynamical friction for accelerated motion, a choice of the small distance cutoff r0 must be made. This cutoff is usually taken as the perturber’s accretion radius or the system’s typical size. The former dimensional cutoff reads 2GM/V2 and is a function of the Mach number whereas the latter is a constant that does not depend on the flow’s variables. In this work, the perturber is treated as a compact object whose gravitational potential is truncated at small distances by the perturber’s size. To satisfy such a condition and be consistent with a point-like potential, the perturber’s size needs to be much smaller than the accretion radius 2GM/V2. This choice of cutoff is in agreement with the numerical simulation of the fluid equations in the context of Bondi-Hoyle-Lyttleton accretion (Edgar 2005). For larger size or extended objects, equation ([force]) no longer applies and dynamical friction may be obtained from the general expression derived for an arbitrary perturbing potential (Appendix A, equation [forc2A]). Perturbers initially at rest ---------------------------- In an infinite medium, the only relevant physical boundary condition is that of the small distance cutoff $\|{\bm x}-{\bm \xi} (t)\|\leq r\_0$.[2](#fn2) For a perturber initially at rest, ξ(t) = t2/2, Δ/r = t − r/2 and the Mach number is ${\cal M}=t$. The variables r and τ therefore need to satisfy the three conditions: (i) − 1 ≤ τ ≤ 1, (ii) τ ≤ [1 + (2t − r)2/4 − (r0/r)2]/(2t − r) ≡ f(r, t, r0), and (iii) r ≤ t. The last condition implies that Δ > 0. Examination of the three conditions shows $\tau\_{\rm min}=-1$, and that $\tau\_{\rm max}$ and the remaining bounds on r in the integral ([forcex]) depend on the value of t as the velocity increases. There are five time intervals that need to be considered: $$\begin{aligned} \mbox{I}&:& (1+2r\_0)^{1/2}-1\leq t\leq 1-(1-2r\_0)^{1/2}, \nonumber\\ \mbox{II}&:& 1-(1-2r\_0)^{1/2} \leq t \leq 1+(2r\_0)^{1/2}, \nonumber\\ \mbox{III}&:& 1+(2r\_0)^{1/2} \leq t \leq 1+(1-2r\_0)^{1/2}, \nonumber\\ \mbox{IV}&:& 1+(1-2r\_0)^{1/2}\leq t\leq 1+(1+2r\_0)^{1/2}, \nonumber\\ \mbox{V}&:& 1+(1+2r\_0)^{1/2} \leq t. \nonumber\end{aligned}$$ Case I is that of transitory motion where the density perturbation is in the perturber’s vicinity. Prior to t = (1 + 2r0)1/2 − 1, the force is zero as the three conditions above cannot be satisfied. The integration domain of r and $\tau\_{\rm max}$ are given as: $$\begin{aligned} r^f\_2\leq r\leq t&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\ \ \ \mbox{where} \ \ r\_2^{f}=1+t-[(1+t)^2-2r\_0]^{1/2}. \end{aligned}$$ Calling I the integral in equation ([forcex]) and using these boundaries for r, we find: $$\begin{aligned} I\_{\rm I}&=&\left[ \frac{t^2 r^3-2t^3r^2+(4t^2-4r\_0 t+ 2 r\_0^2)r - 2 r\_0^2 t }{2r\_0 t^2r(2t-r)} -\frac{2t-r\_0}{2t^3}\, \log\left\|\frac{r}{r-2t}\right\| \right] \_{r\_2^{f}}^{t}, \label{i0}\\ I\_{\rm I}&=&-\frac{r\_0(2+r^f\_2)-(2+r\_0+r^f\_2)t -r^f\_2t^2+t^3}{2r\_0t^2}-\frac{2t-r\_0}{2t^3}\, \log\left\|\frac{2t-r\_0}{tr^f\_2-r\_0}\right\|.\label{case1}\end{aligned}$$ Examination of the variation of $I\_{\rm I}$ as a function of time shows that it decays smoothly to zero as t = (1 + 2r0)1/2 − 1 is approached. The expression of the force in this time interval is the equivalent of that derived for uniform motion in Appendix B (equation [transient]) when $r\_0/\|1+{\cal M}\|\leq ct\leq r\_0/\|1-{\cal M}\|$ and the wake has not left the perturber’s vicinity. In that case, the force read $F\propto {\cal M}^{-2}\{ -\log[ct(1+{\cal M})/r\_0]+{\cal M}-{r\_0}/{2ct}+(1-{\cal M}^2)ct/2r\_0\}$; it helped explain the transient phase observed before saturation in the numerical simulation of the nonlinear fluid equations (Sánchez-Salcedo and Brandenburg 1999). Case II covers subsonic motion and the corresponding integration boundaries are: $$\begin{aligned} r^f\_2\leq r\leq r^c\_1&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0), \ \ \mbox{where}\ \ r\_1^{c}=t-1+[(t-1)^2+2r\_0]^{1/2}, \label{dom21}\\ r^c\_1\leq r\leq t&\mbox{with}& \tau\_{\rm max}=1.\label{dom22}\end{aligned}$$ The latter domain ([dom22]) does not contribute to the force. The corresponding integrand is proportional to ${\rm sign} (1-t+r/2)-1$ and the sign argument is positive for r > 2(t − 1). The expression of I for domain ([dom21]) may be obtained from equation ([i0]) by substituting r1c for t. The result is : $$\begin{aligned} I\_{\rm II}&=&\frac{2r\_0 t + (t-r\_0) ( r\_1^c+r\_2^f) -t^2 (r\_1^c-r\_2^f)}{2 r\_0 t^2}- \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{r\_1^ct+r\_0}{r\_2^ft-r\_0}\right\|. \label{case2}\end{aligned}$$ In the limit r0 ≪ 1 and t < 1, $I\_{\rm II}(r\_0\ll 1)=t^{-2}(2t -\log\|(1+t)/(1-t)\|)$. This is exactly the friction force acting on subsonic perturbers with uniform motion given by equation ([ostriker]). It is interesting that although the density enhancements of subsonic accelerated motion and subsonic uniform motion differ significantly, the corresponding friction forces are identical when the limit r0 ≪ 1 is taken for the accelerated motion friction. [3](#fn3) This result is the more surprising as the density enhancement associated with accelerated motion does not involve a region with front-back symmetry as explained in the Introduction leading one to guess wrongly that friction would be stronger for accelerated motion than it is for uniform motion. The explanation of this agreement comes from the friction force’s dependence on the phase Δ = ξ(t) − ξ(t − r/c) (dimensional variables). As the contribution of the force for subsonic motion comes mostly from the vicinity of the perturber’s retarded position ($\|{\bm y}\|\rightarrow 0$), we have r/c ≪ t and the phase becomes $\Delta=\dot\xi(t)r/c$. The latter is exactly the phase of uniform motion with a Mach number ${\cal M}=\dot \xi(t)/c$ regardless of the type of motion ξ(t) as long as, (1) r0 ≪ 1 which implies in dimensional variables that $\|\ddot\xi(t)\|\ll c^2/r\_0$, (2) the velocity remains subsonic and (3) the motion does not have a history with supersonic episodes (see section 3.3). Case III concerns motion between Mach 1 and Mach 2 where the perturber travels at supersonic speed but has not yet crossed the sonic shock wave. The integration domains are given as: $$\begin{aligned} r^f\_2\leq r\leq r^c\_2&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\ \ \mbox{where} \ \ r\_2^{c}=t-1-[(t-1)^2-2r\_0]^{1/2}, \label{dom31}\\ r^c\_2\leq r\leq r\_3^c&\mbox{with}& \tau\_{\rm max}=1,\ \ \mbox{where}\ \ \ r\_3^{c}=t-1+[(t-1)^2-2r\_0]^{1/2},\label{dom32}\\ r^c\_3\leq r\leq r^c\_1&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\label{dom33}\\ r^c\_1\leq r\leq t&\mbox{with}& \tau\_{\rm max}=1.\label{dom34}\end{aligned}$$ The integration radii satisfy r2f < r2c ≤ r3c ≤ 2(t − 1) ≤ r1c. This shows that domain ([dom34]) does not contribute a finite term to the force in the same way as domain ([dom22]). Domains ([dom31]) and ([dom33]) contribute terms similar to that of equation ([i0]) except that for the former t is replaced with r2c and for the latter t is replaced with r1c, and r2f with r3c. The second domain ([dom32]) contributes the term $I\_{(\ref{dom32})}$ given as: $$\begin{aligned} I\_{(\ref{dom32})}&=&\left[\frac{4}{(r-2t)t}-\frac{2}{t^2}\log\left\|\frac{r}{r-2t}\right\|\right]\_{r\_2^{c}}^{r\_3^{c}}, \label{i2}\end{aligned}$$ which when combined with the contributions of domains ([dom31]) and ([dom33]) yields: $$\begin{aligned} I\_{\rm III}&=&\frac{2r\_0 t -r\_0 (r\_1^{c}+r\_2^{f}-r\_2^{c}+r\_3^{c}) +t (r\_1^{c}+r\_2^{f}+r\_2^{c}-r\_3^{c})}{2 r\_0 t^2}-\frac{2t+r\_0}{2t^3}\,\log\left\|\frac{r\_3^{c}t-r\_0}{r\_2^{c}t-r\_0}\right\|\nonumber \\ && -\frac{r\_1^{c}-r\_2^{f}+r\_2^{c}-r\_3^{c}}{2 r\_0} - \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{r\_1^{c}t+r\_0}{r\_2^{f}t-r\_0}\right\|.\label{case3}\end{aligned}$$ Case IV defines a small time interval near Mach 2 where the perturber exits the sonic wavefront. The corresponding integration domains are: $$\begin{aligned} r^f\_2\leq r\leq r^c\_2&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\\ r^c\_2\leq r\leq r\_3^c&\mbox{with}& \tau\_{\rm max}=1,\\ r^c\_3\leq r\leq t&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0).\end{aligned}$$ The expression of I is obtained by summing up the terms associated with domains ([dom31]) and ([dom32]) of Case III, and substituting t for r1c in domain ([dom33]). The result is: $$\begin{aligned} I\_{\rm IV}&=&\frac{ r\_0(r\_2^c-r\_3^c-r\_2^f-2)+(2 + r\_0 + r\_2^c - r\_3^c + r\_2^f)t+( -r\_2^c + r\_3^c + r\_2^f)t^2-t^3}{2 r\_0 t^2}\nonumber \\ &&-\frac{2t+r\_0}{2t^3}\,\log\left\|\frac{r\_3^ct-r\_0}{r\_2^{c}t-r\_0}\right\| - \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{2t-r\_0}{r\_2^{f}t-r\_0}\right\|. \label{case4}\end{aligned}$$ Case V concerns supersonic motion that exited the sonic wavefront ${\cal M}>2$. The integration domains are given as: $$\begin{aligned} r^f\_2\leq r\leq r^c\_2&\mbox{with}& \tau\_{\rm max}=f(r,t,r\_0),\\ r^c\_2\leq r\leq t &\mbox{with}& \tau\_{\rm max}=1. \label{dom-v}\end{aligned}$$ The integral I is obtained as previously by combining the domains of Case IV and making the necessary boundary substitutions. The result is: $$I\_{\rm V}=\frac{ r\_0 (r\_2^{c}-r\_2^{f}-4) +(r\_2^{c}+r\_2^{f})t- (r\_2^{c}-r\_2^{f}) t^2}{2 r\_0 t^2}-\frac{2t+r\_0}{2t^3}\,\log\left\|\frac{2t+r\_0}{r\_2^{c}t-r\_0}\right\| - \frac{2t-r\_0}{2t^3}\,\log\left\|\frac{2t-r\_0}{r\_2^{f}t-r\_0}\right\|. \label{case5}$$ For r0 ≪ 1 and t > 2, the scaled force reads: $$I\_{\rm V}(r\_0\ll 1)=-t^{-2} \left[2(1+\log 2) +\log\left\|\frac{t^2(t^2-1)}{r\_0^2}\right\|\right].\label{case5l}$$ This expression differs from that of equation ([ostriker]) through the first term in the bracket − 2(1 + log2)/t2 implying that the force on accelerated motion of a perturber initially at rest, is larger than that on uniform motion. Fig. (3) shows the friction force ([case1]), ([case2]),([case3]), ([case4]), ([case5]) as a function of time, or equivalently the Mach number, for the three values of the small distance cutoff r0 = 10− 2, 10− 4, and 10− 6. The force obtained by using the uniform motion expression, I0, ([ostriker]) with ${\cal M}=t$ is also shown for the same values of r0. For subsonic motion, the agreement between the two forces is best when r0 is smallest. The force I0 overestimates maximal drag by about 20%; it underestimates the drag on supersonic motion by the factor ∼ 2(1 + log2)/t2. Lastly, we remark that the derivation of dynamical friction in the five time intervals shows that there are no force singularities associated with accelerated motion such as that at Mach 1 for uniform motion ([ostriker]). Perturbers with positive acceleration ------------------------------------- To an accelerating perturber of trajectory, $\xi(t)={\cal M}\_0 t +t^2/2$ with ${\cal M}\_0>0$, corresponds the phase $\Delta/r={\cal M}-r/2$ where ${\cal M}=t+{\cal M}\_0$ is the Mach number. It could therefore seem sufficient to take the force expressions of perturbers initially at rest and substitute ${\cal M}$ for t. There is however a key difference between the two cases: the condition from the retarded sonic wavefront, r ≤ t, refers to time regardless of the type of motion. It should therefore be replaced with $r\leq {\cal M}-{\cal M}\_0$ that we refer to as condition (iv). This difference makes the force dependent on the initial velocity in contrast to the expressions derived assuming a constant velocity ([ostriker]). The physical reason for such a dependence was discussed in the Introduction: as the velocity varies, the far part of the wake that was launched in the medium corresponds to an earlier velocity. Combining the force from the far part of the wake to that near the perturber results in a force that has a memory of the earlier velocity. Inspection of the conditions (i,ii) of the previous section along with (iv) shows that the two cases ${\cal M}\_0>1$ and ${\cal M}\_0<1$ may be considered separately. In the following we do not derive the force expressions for the counterparts of Cases I and IV as their space extensions vanish when r0 tends to zero. For an accelerated perturber with an initial supersonic speed (${\cal M}\_0>1$), the boundary conditions (i, ii, iv) away from the transient phase (Case I) are identical to those of Case V ([dom-v]) except that in r2f and r2c, t is replaced with ${\cal M}$, and the outer integration domain is $r^c\_2\leq r\leq {\cal M}-{\cal M}\_0$. Provided that ${\cal M}>1+[(1-{\cal M}\_0)^2+2r\_0]^{1/2}$ that we call Case VI, the friction force is given as: $$\begin{aligned} I\_{\rm VI}&=&\frac{r\_0 {\cal M}\_0( 4 + r^c\_2 - r^f\_2)+[ {\cal M}\_0 (r^c\_2 +r^f\_2)+ r\_0 (r^c\_2 - r^f\_2-4)]{\cal M} +[r^c\_2+r^f\_2+{\cal M}\_0(r^f\_2-r^c\_2)]{\cal M}^2+(r^f\_2 -r^c\_2){\cal M}^3}{2 r\_0 {\cal M}^2({\cal M}+{\cal M}\_0)} \nonumber \\ &&-\frac{2{\cal M}+r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}+r\_0}{r\_2^{c}{\cal M}-r\_0}\right\| - \frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}-r\_0}{r\_2^{f}{\cal M}-r\_0}\right\| -\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\|.\label{case6}\end{aligned}$$ New terms depending the initial Mach number appear in the expression of the force. As ${\cal M}\_0$ is set to zero the force reduces to ([case5]) with $t={\cal M}$. This shows that including acceleration in the perturber’s motion makes the medium store the information about the perturber’s initial state which is in turn reflected in the reaction force felt by the perturber. Taking the limit r0 ≪ 1, the force becomes: $$\begin{aligned} I\_{\rm VI}(r\_0\ll 1)&=&-\frac{2({\cal M}- {\cal M}\_0)}{{\cal M}^2({\cal M}+{\cal M}\_0)}-\frac{2\log 2}{{\cal M}^2} -\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\| -\frac{1}{{\cal M}^2}\,\log\left\|\frac{{\cal M}^2({\cal M}^2-1)}{r\_0^2}\right\|.\label{case60}\end{aligned}$$ Three extra terms appear with respect to the uniform velocity expression ([ostriker]). As ${\cal M}\gg {\cal M}\_0$, the force tends to that of Case V ([case5l]) of perturbers initially at rest as the density perturbation of a larger Mach number dominates the initial density perturbation. For an accelerated perturber with an initial subsonic speed, the boundary conditions (i,ii,iv) show that the force expression is given as: $$\begin{aligned} 1-\left[(1-{\cal M}\_0)^2-2r\_0\right]^{1/2}\leq {\cal M}\leq1+(2r\_0)^{1/2}:&& I=I\_{\rm II},\label{case61}\\ 1+(2r\_0)^{1/2}\leq {\cal M}\leq 1+\left[(1-{\cal M}\_0)^2-2r\_0\right]^{1/2}:&& I=I\_{\rm III},\label{case62}\\ 1+\left[(1-{\cal M}\_0)^2+2r\_0\right]^{1/2}\leq {\cal M} :&& I=I\_{\rm VI},\label{case63}\end{aligned}$$ where in the expressions of $I\_{\rm II}$ and $I\_{\rm III}$, t is replaced with ${\cal M}$. We recover the general statement we made about the force on subsonic motion in Case II. In particular, the friction force on an initially subsonic perturber depends on the initial velocity as soon as the velocity is supersonic and becomes larger than 2c − V0. In Fig. (4) we show the friction force with r0 = 10− 4 for an initial subsonic velocity with ${\cal M}\_0=0.6$ using equations ([case61]), ([case62]) and ([case63]), and an initial supersonic velocity with ${\cal M}\_0=2$ using ([case6]). We remark that the dependence of dynamical friction on the initial velocity of the perturber is not an artifact of choosing to assemble the perturber at t = 0. This starting condition can be replaced with t = − T0 and T0 in turn can be made to tend to infinity thus approximating a steady state. Examination of the general force expression shows that T0 → ∞ is equivalent to the condition t = 0 where the limits ${\cal M} \gg 1$ and ${\cal M} \ll 1$ are taken for accelerating and decelerating perturbers respectively. For positively accelerating perturbers (Case IV), the force is independent of the initial Mach number as ${\cal M}\gg1$ ([case60]). In the next section, we show that for decelerating perturbers the dependence on ${\cal M}\_0$ remains as the velocity decays to zero. Decelerating perturbers ----------------------- For decelerating perturbers (${\cal M}\_0<0$), we seek the dynamical friction force from t = 0 until the velocity vanishes at $t=\|{\cal M}\_0\|$. The choice of a negative Mach number instead of a negative acceleration makes the derivations easier but results in a positive friction force (opposed to the direction of motion). The phase $\Delta/r={\cal M}-r/2$ is now negative and the boundary conditions that determine the integration domains of ([forcex]) are (i), (vi) and (v) $\tau\geq f(r,{\cal M},r\_0)$. The latter condition shows that $\tau\_{\rm max}=1$. Three cases need to be considered: $-1\leq {\cal M}\_0$, $-1-(1-2r\_0)^{1/2}\leq {\cal M}\_0 \leq -1$ and ${\cal M}\_0 \leq -1-(1-2r\_0)^{1/2}$. For subsonic motion ($-1\leq {\cal M}\_0$) and away from the transient regime (i.e. $-1+[(1+{\cal M}\_0)^2+2r\_0]^{1/2}\leq {\cal M}$), the integration domains are given as: $$\begin{aligned} r^c\_1\leq r\leq r^f\_2, &\mbox{with}&\tau\_{\rm min}=f(r,{\cal M},r\_0), \label{decesub1}\\ r^f\_2\leq r\leq {\cal M}-{\cal M}\_0, &\mbox{with}& \tau\_{\rm min}=-1.\label{decesub2} \end{aligned}$$ As for Case II (equations [dom21] and [dom22]), the latter domain ([decesub2]) does not contribute to the force. The corresponding integrand is proportional to $1-{\rm sign} (1+{\cal M}-r/2)$ and the sign argument is positive for $r<2({\cal M}+1)$. The contribution of domain ([decesub1]) is identical to that of Case II and $I=I\_{\rm II}$ ([case2]) with t replaced by ${\cal M}$. Dynamical friction on a decelerating subsonic perturber is therefore independent of its initial velocity. This confirms the general statement we made about subsonic motion in an infinite medium in Case II. For supersonic perturbers with an initial Mach number $-1-(1-2r\_0)^{1/2}\leq {\cal M}\_0 \leq -1$, we need to consider three cases depending on the value of the Mach number ${\cal M}$. The first which we call Case VII corresponds to $-1-[(1+{\cal M}\_0)^2-2r\_0]^{1/2}\leq {\cal M}\leq -1+(2r\_0)^{1/2}$ whose boundaries are: $$\begin{aligned} r^c\_1\leq r\leq r^f\_1, &\mbox{with}& \tau\_{\rm min}=f(r,{\cal M},r\_0),\ \ \mbox{where}\ \ r\_1^{f}={\cal M}+1+[({\cal M}+1)^2+2r\_0]^{1/2}, \label{decesup1}\\ r^f\_1\leq r\leq {\cal M}-{\cal M}\_0, &\mbox{with}& \tau\_{\rm min}=-1. \label{decesup2}\end{aligned}$$ Domain ([decesup1]) contributes a term similar to ([i0]) with r1c and r1f substituted for t and r2f respectively. Domain ([decesup2]) contributes a term similar to ([i2]) with r1f and ${\cal M}-{\cal M}\_0$ substituted for r3c and r2c respectively. The force is then given as: $$\begin{aligned} I\_{\rm VII}&=&\frac{{\cal M}\_0 r\_0 (r^f\_1- r^c\_1-4 ) + [{\cal M}\_0 (r^c\_1 + r^f\_1) + r\_0 (4 - r^c\_1 + r^f\_1)] {\cal M}+[r^c\_1 + r^f\_1 + {\cal M}\_0 (r^f\_1- r^c\_1)] {\cal M}^2+ ( r^f\_1-r^c\_1 ){\cal M}^3}{2r\_0{\cal M}^2({\cal M}+{\cal M}\_0)}+\nonumber\\ && +\frac{2{\cal M}+r\_0}{2 {\cal M}^3}\,\log\left\|\frac{2{\cal M}+r\_0}{r\_1^{f}{\cal M}+r\_0}\right\| +\frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}-r\_0}{r\_1^{c}{\cal M}+r\_0}\right\| +\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\| \label{case7}.\end{aligned}$$ The second case which we call Case VIII concerns $-1+(2r\_0)^{1/2}\leq {\cal M}\leq -1+[(1+{\cal M}\_0)^2-2r\_0]^{1/2}$. The corresponding integration domains are: $$\begin{aligned} r^c\_1\leq r\leq r^f\_2, &\mbox{with}& \tau\_{\rm min}=f(r,{\cal M},r\_0),\label{dom81}\\ r^f\_2\leq r\leq r^f\_3, &\mbox{with}& \tau\_{\rm min}=-1, \label{dom82}\\ r^f\_3\leq r\leq r^f\_1, &\mbox{with}& \tau\_{\rm min}=f(r,{\cal M},r\_0),\ \ \mbox{where}\ \ r\_3^{f}={\cal M}+1+[({\cal M}+1)^2-2r\_0]^{1/2},\label{dom83}\\ r^f\_1\leq r\leq {\cal M}-{\cal M}\_0, &\mbox{with}& \tau\_{\rm min}=-1.\label{dom84}\end{aligned}$$ The integration radii satisfy $r^c\_1< r\_2^{f}\leq r\_3^{f}\leq 2({\cal M}+1)\leq r\_1^{f}$. In contrast to Case III, this ordering shows that the second domain ([dom82]) does not contribute a finite term to the force. Domain ([dom81]) contributes a term equal to $I\_{\rm II}$, and domain ([dom83]) a term similar to ([i2]) where r3f and r1f are substituted for t and r2f respectively. The last domain ([dom84]) contributes a term similar to that of ([decesup2]) in Case VII. Summing up all terms, we find: $$\begin{aligned} I\_{\rm VIII}&=& \frac{{\cal M}\_0 r\_0 (r^f\_1 - r^f\_2 + r^f\_3- r^c\_1 -4)+ [{\cal M}\_0 (r^c\_1 + r^f\_1 + r^f\_2 - r^f\_3) + r\_0 (4 - r^c\_1 + r^f\_1 - r^f\_2 + r^f\_3)]{\cal M}}{2r\_0{\cal M}^2({\cal M}+{\cal M}\_0)}+\nonumber\\ && +\frac{[r^c\_1 + r^f\_1 + r^f\_2 - r^f\_3+ {\cal M}\_0 (r^f\_1 + r^f\_2 - r^f\_3 - r^c\_1) ]{\cal M}^2 +( r^f\_1 + r^f\_2 - r^f\_3-r^c\_1){\cal M}^3}{2r\_0{\cal M}^2({\cal M}+{\cal M}\_0)}+\nonumber\\ && +\frac{2{\cal M}+r\_0}{2 {\cal M}^3}\,\log\left\|\frac{2{\cal M}+r\_0}{r\_1^{f}{\cal M}+r\_0}\right\| + \frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{2{\cal M}-r\_0}{r\_3^{f}{\cal M}-r\_0}\right\| +\frac{2}{{\cal M}^2}\log\left\|\frac{{\cal M}-{\cal M}\_0}{{\cal M}+{\cal M}\_0}\right\|-\frac{2{\cal M}-r\_0}{2{\cal M}^3}\,\log\left\|\frac{r\_1^c{\cal M}+r\_0}{r\_2^f{\cal M}-r\_0}\right\|. \label{case8}\end{aligned}$$ The third case concerns $-1+[(1+{\cal M}\_0)^2+2r\_0]^{1/2}\leq {\cal M}\leq 0$ whose boundaries turn out to be those of subsonic motion ${\cal M}\_0>-1$. The corresponding force is therefore given by ([case2]) where t is replaced ${\cal M}$. Fig. (5, left panel) shows the force on a perturber with an initial velocity ${\cal M}\_0=-1.7$ using equations ([case7]), ([case8]) and ([case2]) alongside the uniform motion friction force ([ostriker]). In the supersonic regime the uniform motion force overestimates the drag by about 10% at its maximum. In the subsonic regime the force on accelerated motion departs significantly from the uniform velocity force before joining it at $\|{\cal M}\|\simeq 2+{\cal M}\_0= 0.3$ to assume the expression ([case2]). In particular, when the velocity vanishes so does the force. Lastly, for a supersonic perturber with an initial velocity ${\cal M}\_0 \leq -1-(1-2r\_0)^{1/2}$, we find that: $$\begin{aligned} -1-[(1+{\cal M}\_0)^2-2r\_0]^{1/2}\leq {\cal M}\leq -1+(2r\_0)^{1/2}:&& I=I\_{\rm VII},\label{case8x} \\ -1+(2r\_0)^{1/2}\leq {\cal M}\leq 0: && I=I\_{\rm VIII}.\label{case8x2}\end{aligned}$$ Fig. (5, right panel) shows the force on a perturber with an initial velocity ${\cal M}\_0=-4$ using equations ([case8x]) and ([case8x2]) alongside the uniform velocity friction force ([ostriker]). In the supersonic regime the uniform motion force overestimates the drag by about 20% at its maximum. In the subsonic regime the force on accelerated motion departs significantly from that on uniform motion and is finite at ${\cal M}=0$. Taking the limit where r0 is small, we find $I\_{\rm VII}(r\_0\ll 1)=-I\_{\rm VI}(r\_0\ll 1)$ and: $$\begin{aligned} I\_{\rm VIII}(r\_0\ll 1)&=& \frac{4(1+{\cal M}\_0+{\cal M})}{{\cal M}({\cal M}+{\cal M}\_0)} +\frac{1}{ {\cal M}^2}\,\log\left\|\frac{({\cal M}-{\cal M}\_0)^2(1-{\cal M})^2}{({\cal M}+{\cal M}\_0)^2(1+{\cal M})^3}\right\|.\end{aligned}$$ At ${\cal M}=0$, $I\_{\rm VIII}(r\_0\ll 1)=1-4/{\cal M}\_0^2$ in contrast to the uniform motion force which vanishes with the perturber’s velocity. The knowledge of the history of the perturber’s motion is therefore required to evaluate dynamical friction. Simply applying uniform motion dynamical friction force ([ostriker]) may yield incorrect results regarding the perturber’s evolution in the ambient medium as evidenced in Fig. (6) where we plot the relative error between the accelerated motion and uniform motion forces for decelerating supersonic perturbers. A similar conclusion was previously reached by Sánchez-Salcedo and Brandenburg (1999) who numerically tested the accuracy of the uniform motion friction formula ([ostriker]) using the full nonlinear fluid equations. Following the evolution of a decelerating initial perturber with ${\cal M}\_0=1.5$ up to ${\cal M}=0.6$, they found that equation ([ostriker]) underestimated the friction force as ${\cal M}$ became smaller than unity (see their Fig. 5) which corresponds to our Cases VII and VIII. The authors attributed this discrepancy to the ‘memory of the wake’. Concluding remarks ================== In this paper, we studied the influence of acceleration on dynamical friction in a gaseous medium. Our main conclusions are stated in the Abstract. This work also constitutes an illustration of the use of the general expression of dynamical friction derived in Appendix A. In particular, applying that formulation to a perturber with accelerated motion allowed us to illustrate how dynamical friction is not a local force and may depend the perturber’s initial state. The meaning of the initial state depends on the problem’s configuration. For instance, it may correspond to the time the perturber enters the gaseous medium or to the initial time of a numerical simulation of dynamical friction. We note that non-locality is not peculiar to	arxiv_0000634
MULTILAYER TENSOR FACTORIZATION WITH APPLICATIONS TO RECOMMENDER SYSTEMS ======================================================================== , Recommender systems have been widely adopted by electronic commerce and entertainment industries for individualized prediction and recommendation, which benefit consumers and improve business intelligence. In this article, we propose an innovative method, namely the recommendation engine of multilayers (REM), for tensor recommender systems. The proposed method utilizes the structure of a tensor response to integrate information from multiple modes, and creates an additional layer of nested latent factors to accommodate between-subjects dependency. One major advantage is that the proposed method is able to address the “cold-start” issue in the absence of information from new customers, new products or new contexts. Specifically, it provides more effective recommendations through sub-group information. To achieve scalable computation, we develop a new algorithm for the proposed method, which incorporates a maximum block improvement strategy into the cyclic blockwise-coordinate-descent algorithm. In theory, we investigate both algorithmic properties for global and local convergence, along with the asymptotic consistency of estimated parameters. Finally, the proposed method is applied in simulations and IRI marketing data with 116 million observations of product sales. Numerical studies demonstrate that the proposed method outperforms existing competitors in the literature. [class=MSC] Introduction ============ Recommender systems have become very important in daily life due to high demand from the entertainment industry and business marketing which produce large amounts of data. In addition, applications of recommender systems have been greatly facilitated by the advancement of statistical and machine learning techniques. These applications include personalized marketing for internet users, merchandise recommendation for retail stores, and even individualized gene therapies. Each application involves collecting a wide variety of information, and successful exploitation of such rich information leads to more accurate recommendations. However, this also imposes unprecedented challenges to traditional methods due to the large size and complex structure of data. Therefore, more general and integrative recommender systems are urgently needed. The tensor, also called multidimensional array, is well-recognized as a powerful tool to represent complex and unstructured data. It is applied in many areas such as signal processing, neuroimaging, and psychometrics. In recommender systems, the tensor shows its flexibility to accommodate contextual information, and is also regarded as one of the most effective tools for developing context-aware recommender systems. Nevertheless, applying the tensor effectively to CARS remains a challenging problem. In matrix recommender systems, the singular value decomposition (SVD) method provides the best low-rank approximation, and is known to be arguably the most effective single procedure. This imposes a great challenge to generalize matrix decomposition to the tensor framework. One common approach to utilize the tensor structure is to apply latent factor models, where each user, item or context is assigned an individual latent factor to represent their characteristics quantitatively. However, several key issues have not been solved completely. The first is the “cold-start” problem, where available information is not sufficient to provide valid predictions for new users, items or contexts (in the rest of this paper, we use “subject” to denote a user, an item or a context in general). For instance, in latent factor modeling, a latent factor is not estimable if a subject is not available in the training set. The subject’s utilities can only be predicted through the average information from other subjects, which may lead to low prediction accuracy. Several solutions have been proposed for the traditional matrix recommendation techniques, for example, imputing pseudo ratings, supplementing artificial users and items, incorporating content-boosted information, or utilizing group information for new subjects. Nevertheless, the “cold-start” problem under CARS is quite challenging and has not been well-investigated. For example, viewers may have different movie-watching experiences with new friends or at a new theater, and stores’ sales volumes may vary under a new promotion strategy. Another critical issue involves solving the higher-order tensors (beyond the third-order) problem. Higher-order tensors are very useful, because we might be interested in integrating more than one contextual variable to fully utilize their subject-specific information. For example, when recommending a new destination to travelers, one has to consider several factors, e.g., the timing and the cost of the trip, and the crowdedness of the destination, in addition to travelers’ travel interests. While some existing methods provide a general methodology on high-order tensors, the implementation of higher-order tensors could be challenging. One obstacle is the high computational cost. Another obstacle is that higher-order tensors could result in higher missing rates due to fewer observations at each combination of contextual variables; and this could lead to non-convergence in computing for some existing methods. One possible solution is to choose only one special contextual variable in the tensor, and treat the rest of the contextual variables as linear covariates. However, this may lead to loss of information on subject-specific or group-wise interaction. In this paper, we propose a novel tensor factorization method, namely the recommendation engine of multilayers (REM). Specifically, we assume a tensor structure where each mode corresponds to user, item or a contextual variable, and each element of the tensor represents a utility, such as a rating or sales volume. The novelty of our method is that we add another layer which categorizes users, items and contexts into the same subgroups if they share similar characteristics. We quantify subgroup effects as random effects through nested-factors modeling, in addition to latent factors in the tensor factorization. Theoretically, we demonstrate the algorithmic properties of the proposed method, which converges to a stationary point from an arbitrary initial point, and local convergence to a local minimum with a linear convergence rate. The estimated parameter achieves asymptotic consistency under the L2-loss function and other more general circumstances. The proposed tensor factorization method has two significant advantages. First, it solves the “cold-start” problem effectively. For a new subject, even if its latent factors are not available, the nested factors from the corresponding subgroups can provide a group-specific estimate, which is more accurate than the average of the available observations. This finding is also supported by simulation studies where the proposed method is more effective than competing tensor factorization methods when the proportion of new subjects is high. Second, the proposed method is able to accommodate high-order tensors. The difficulty of applying high-order tensors is solved by the proposed nested factors which utilize group-wise information and hence are more robust to a higher missing rate. In addition, we propose a new algorithm that incorporates the maximum block improvement into the cyclic blockwise-coordinate-descent algorithm. This avoids the direct operation of large-scale tensors, and makes the estimation of high-order tensors feasible. Furthermore, a parallel computing strategy is implemented to calculate latent factors and nested factors for each subject and subgroup, respectively, which provides scalable and efficient computation. The rest of this paper is organized as follows. Section [Preparations] provides the background of the tensor factorization and the framework for the context-aware recommender systems. Section [A Multilayer Method] introduces the proposed method and algorithm. Theoretical properties are derived in Section [Theoretical Properties]. Section [Simulation Studies] presents simulation studies to validate the performance of the proposed method. In Section [realdata], we apply the proposed method to IRI marketing data. Section [Discussion] concludes with a discussion. Preparations ============ Notation and Tensor Background ------------------------------ In this subsection, we provide the background of the tensor, and introduce the notation for the proposed method. We define a d-th order tensor Y ∈ Rn1 × ⋯ × nd as a d-dimensional array, where each order is also called a mode. In the rest of this article, bold capital letters denote tensors, capital letters denote matrices, bold small letters denote vectors, and small letters denote scalars. That is: [CP] j=1r p1j p2j pdj, where ∘ represents the vector outer product, and $\mathbf p\_{\cdot j}^k$, j = 1, …, r, are nk-dimensional vectors corresponding to the k-th mode. Here r is called the rank of Y if the number of terms r is minimal. Equivalently, for each element of Y, we have: [CPelement] yi1i2id j=1r p1i1j p2i2j pdidj. For other properties of a tensor, see for an extensive review. Context-Aware Recommender Systems --------------------------------- In this subsection, we briefly review the tensor application to recommender systems, namely, context-aware recommender systems (CARS). See also for a comprehensive review of CARS. Let nk (k = 1, …, d) be the number of subjects for the k-th mode, that is, n1 is the number of users, n2 is the number of items, and n3, …, nd are the number of contexts for the (d − 2) contextual variables, respectively. For the CPD representation in ([CP]), let $P^k=(\mathbf p^k\_{\cdot 1},\ldots,\mathbf p^k\_{\cdot r})\_{n\_k \times r}$, where each row of P1 or P2 represents the r-dimensional latent factors for each user or item, and rows of P3, …, Pd are the latent factors for the contextual variables, respectively. In the rest of this article, we use $\mathbf p^k\_{i\_k}$ to represent the ik-th row of Pk, in contrast to $\mathbf p\_{\cdot j}^k$ being the j-th column of Pk. Each element of Y is defined as a utility, for example, a rating, a purchase or a sales volume, and is estimated via ([CPelement]) in CPD. Here the orthonormality for the latent factors is not required, and each utility yi1i2⋯id comprises user-, item- and context-specific information. In addition, the CPD requires estimation of only ∑k = 1dnkr parameters, which essentially performs a tensor dimension reduction procedure. In many applications, we may have a non-negative tensor $\mathbf Y$. Methods for non-negative CPD are proposed to address this issue, as such decompositions make results meaningful and interpretable. In recommender system problems, however, the direct interpretation of latent factors is less critical, as the relative scale or ranking is important for recommendation. Most importantly, improving prediction accuracy is the ultimate goal of a good recommender system for users. Therefore, non-negativity is not required here. Technically, we can always standardize $\mathbf Y$ prior to analysis. In addition, non-negative CPDs usually entail non-negative latent factors, which might restrict the parameter space to the non-negative orthant. In the proposed framework, an unbounded parameter space allows more flexibility for extensive search, which may lead to a more satisfactory convergence result. The most common loss function is the L2-loss, which is computationally efficient. Theoretical properties of other loss functions are also considered. Let Ω = {(i1, i2, …, id) : yi1i2⋯idis observed} be a set of indices corresponding to observed utilities, and ∣Ω∣ represent the sample size. Since the number of available observations for each subject might be smaller than the number of latent factors r, we adopt a penalty function using regularization. For example, when the L2-penalty is applied, we have: [CPcriterion] L(P1,…,Pd\|Y)=(i1,,id) (yi1id - i1id)2 +k=1dPkF2, where ŷi1⋯id = ∑j = 1rpi1j1⋯pidjd is the estimated utility provided by ([CPelement]), and ∥ ⋅ ∥F represents the Frobenius norm. Other regularization methods include, but are not limited to, the trend filtering penalty when d = 3, and the L0- and L1-penalty for sparse low-rank pursuit when d = 2. Algorithms for implementing the CPD include the cyclic coordinate descent algorithm, and the stochastic gradient descent method. Alternatively, propose a maximum block improvement algorithm that only updates the block with the maximum improvement in each iteration instead of updating each block cyclically. This strategy guarantees convergence to a stationary point, and ensures a fast convergence rate in many circumstances. A Multilayer Method =================== General Methodology ------------------- In this section, we develop the methodology for the proposed REM method. Specifically, we adopt the idea of nested factors from the design of experiments to capture between-subject dependency under the CPD framework ([CP]). We assume that subjects can be categorized into subgroups, where subjects within the same subgroup share similar characteristics and are dependent on each other. For subgrouping, we can incorporate prior information such as users’ demographic information, item categories and functionality, and contextual similarity. If this kind of information is not available, one can utilize the missing pattern of the tensor data, or the number of records from each user and on each item. As shown in, implicit information from the number of records may reflect subjects’ behavior that is not available elsewhere, and can help improve recommendation accuracy. In more general situations, clustering methods such as the k-means can be applied to determine the subgroups. See and on robust approaches to select the number of subgroups. Suppose the subgroup labels are given, then we formulate each utility as follows: [GSM] i1i2id = j=1r (p1i1j+q1i1j) (p2i2j+q2i2j) (pdidj+qdidj), where pikjk is the j-th latent factor for the ik-th subject from the k-th mode, and qikjk is the corresponding nested factor, j = 1, …, r, ik = 1, …, nk, and k = 1, …, d. We define the nk × r-dimensional matrix Qk similar to Pk as in Section [CARS22]. Notice that we have $\mathbf q^k\_{i\_k}=\mathbf q^k\_{i'\_k}$ if subjects ik and iʹk are from the same subgroup. We assume that the number of subgroups for the k-th mode is mk, which corresponds to mk unique values for qikjk. We use $\mathbf q\_{(u\_k)}^k$ occasionally to denote the nested factor associated with the subgroup uk, uk = 1, …, mk. Let P = ((P1)ʹ, …, (Pd)ʹ)ʹ and Q = ((Q1)ʹ, …, (Qd)ʹ)ʹ represent all parameters of interest. We define $L(P,Q\|\mathbf Y)=L(P^1,\ldots,P^d,Q^1,\ldots,Q^d\|\mathbf Y)$ as the overall criterion function: [pracloss] L(P,Q\|Y)= (i1,,id) (yi1id - i1id)2 +(PF2+QF2), where ŷi1⋯id is represented by ([GSM]) and λ is the penalization coefficient. Here we adopt the most commonly used L2-loss and L2-penalty for efficient computation, although other types of loss and penalty functions are also applicable. For example, one may consider the hinge loss or the ψ-loss for classification, and the absolute loss or Huber loss to achieve robust estimation. Meanwhile, if prior knowledge of the latent and nested factors is available, for example regarding sparsity or smoothness, then appropriate regularization methods can be applied. Parameter Training ------------------ In the following, we discuss how the model parameters are estimated. Mainly we are interested in finding a solution (P̂, Q̂) aiming at minimizing $L(P,Q\|\mathbf Y)$. Let Ωikk = {(i1, …, ik, …, id) : yi1⋯ik⋯idis observed} be the set of indices where the k-th mode index equals ik and the corresponding utilities are observed; Let I(uk)k be the set of subjects in the subgroup uk, uk = 1, …, mk. We assume that ∣I(uk)k∣ ≥ 2 for each uk. For each mode of the tensor, the partial derivatives of $L(\cdot\|\mathbf Y)$ have explicit forms with respect to the latent factors or the nested factors, which makes it feasible to apply the blockwise coordinate descent approach. That is, [estp] kik = pkik kik (yi1id - i1id)2 +pkik22, for ik = 1, …, nk, and [estq] k(uk) = qk(uk) ik kukkik (yi1id - i1id)2 +qk(uk)22, for uk = 1, …, mk, and k = 1, …, d. The estimation procedure of $\hat{\mathbf p}^k\_{i\_k}$ in ([estp]) is a ridge regression, and does not require knowing $\hat{\mathbf p}^k\_{i'\_k}$ for iʹk ≠ ik. Thus, parallel computation is applicable to calculate $\hat{\mathbf p}^k\_1,\ldots,\hat{\mathbf p}^k\_{n\_k}$ efficiently. This strategy is also applicable to obtaining the $\hat{\mathbf q}^k\_{(u\_k)}$’s in ([estq]). Therefore, the minimization of $L(P,Q\|\mathbf Y)$ can be done cyclically through estimating P and Q. Notice that Ω = ∪ ik = 1nkΩikk, and it is possible that Ωikk is empty for certain ik’s; that is, there is no observation on subject ik, as in the case of the “cold-start” problem. Algorithm --------- In contrast to matrix factorization,. First, the number of parameters and sample size for a tensor decomposition could be much greater than its matrix counterpart, which makes the computation of the MBI more intensive. Second, since the number of blocks increases significantly, the MBI algorithm may never update certain blocks due to small improvements along these directions, which leads to the estimated values corresponding to these blocks remaining the same as the initial values. To solve these problems, we propose a two-step algorithm, which estimates the latent-factors matrix P̂ and the nested-factors matrix Q̂ iteratively. Within the estimation of each matrix, we apply the MBI algorithm to find the optimal block direction with the largest improvement of estimations. Specifically, we propose the following algorithm aiming at minimizing ([pracloss]). Let (Ps, Qs) denote the estimated (P, Q) at the s-th iteration, then the improvement of estimations for updating the k-th mode is defined as [impI] Iks=1-, and [impJ] Jks=1-, where Pk \* and Qk \* are the attempted updates for the k-th mode, k = 1, …, d. $\overline{\mbox{\underline{\makebox[\textwidth]{\textbf{Algorithm 1:} A Two-Step Algorithm with Parallel Computing}}}}$ 1. (Initialization) Input all observed yi1⋯id’s, the rank r, the tuning parameter λ, initial value (P0, Q0) and a stopping criterion ɛ = 10− 4. 2. (Latent-factors update) At the s-th iteration (s ≥ 1), estimate Ps. 1. For each Pk, solve ([estp]) through parallel computing and obtain $P^{k\}=(\hat{\mathbf p}^k\_{1},\ldots,\hat{\mathbf p}^k\_{n\_k})'$. Calculate Isk* through ([impI]). 2. Assign Psk0 ← Pk0 \, if Isk0 = max{I*s1, …, Isd}. 3. (Nested-factors update) At the s-th iteration (s ≥ 1), estimate Qs. 1. For each Qk, solve ([estq]) through parallel computing and obtain $Q^{k\}=(\hat{\mathbf q}^k\_{1},\ldots,\hat{\mathbf q}^k\_{n\_k})'$. Calculate Jsk* through ([impJ]). 2. Assign Qsk0 ← Qk0 \, if Jsk0 = max{J*s1, …, Jsd}. 4. (Stopping Criterion) Stop if max{Is1, …, Isd, Js1, …, Jsd} < ɛ. Set (P̂1, …, P̂d, Q̂1, …, Q̂d) = (Ps1, …, Psd, Qs1, …, Qsd). Otherwise set s ← s + 1 and go to step 2. One advantage of the proposed algorithm is that it requires small memory storage. Note that at each iteration, only one subject’s information is required to estimate $\hat{\mathbf p}^k\_{i\_k}$ and one sub-group’s information is required to estimate $\hat{\mathbf q}^k\_{(u\_k)}$. Furthermore, the computational complexity of the proposed algorithm is no greater than 2dnitercridge, where niter is the number of iterations and cridge is the complexity of the ridge regression. In addition, since the MBI algorithm does not update blocks cyclically, it is able to discover and utilize “shortcuts” in optimization, which may significantly reduce the number of iterations. Implementation -------------- In this subsection, we address several implementation issues. In general, we split the data into a 50% training set, a 25% validation set and a 25% testing set, randomly. The tuning parameter λ is selected to minimize the root mean square error (RMSE) on the validation set, where the RMSE on a set Ω is defined as $\sqrt{\frac{1}{\|\Omega\|}\sum\_{\Omega}(y\_{i\_1\cdots i\_d}-\hat{y}\_{i\_1\cdots i\_d})^2}$. To improve prediction, we could specify λ differently for each row of P and Q. In our numerical study, we use λ uniformly for the latent factors in P, but use λ(uk)k = λ/∣I(uk)k∣ for each subgroup uk to penalize each parameter equally. Furthermore, we need to choose the number of latent factors r. In general, r is no smaller than the theoretical rank of the tensor in order to represent subjects’ characteristics sufficiently well. However, a large r may lead to intensive computation and possible non-convergence of algorithms. Moreover, it is important to determine which contextual variables should be integrated in the tensor. In Section [hotensor], we demonstrate that if the true tensor is of high-order, then assuming a low-order tensor structure may lead to a loss of information. That is, it is important to include key contextual variables in the tensor. On the other hand, applying a high-order tensor to formulate a low-order problem is unnecessary and may entail extra computational cost. In practice, we assume that the order of a tensor can be judged from prior knowledge. We acknowledge that determining the order of a tensor remains an open problem. Theoretical Properties ====================== This section develops theoretical properties for the proposed method. Our contributions are mainly on two aspects. One is on the convergence properties of the proposed algorithm, which converges to a stationary point from an arbitrary initial point, and local convergence to a local minimum (or a global minimum, if one exists) with linear convergence rate. The other contribution is on the statistical properties. We prove the asymptotic consistency of the estimated parameter under L2 loss and more general criterion functions. One well-known critical issue is the discrepancy between the algorithmic and statistical properties. In some existing works, the statistical framework may require the estimated parameter to be a global minimizer, which might not be existent or attainable. Our theoretical development, nevertheless, bridges this gap from two aspects. First and foremost, we relax the strict condition such that a global minimizer is no longer required to establish statistical properties, as long as the criterion function converges to its infimum asymptotically. Second, we provide technical solutions on finding possible global minima or satisfactory local minima, with additional computational cost. Identifiability --------------- Identifiability is critical for tensor representations. For recommender systems, although having identifiable latent factors does not improve prediction accuracy, it could still be important for algorithmic convergence which may lead to favorable statistical properties. Here we provide sufficient conditions to achieve identifiable latent factors prior to establishing theoretical properties. In the proposed framework, unidentifiability is attributed to four aspects. The first three aspects are elementary indeterminacies of scaling, permutation and addition, whereas the last one is the so called non-uniqueness of the CPD with more than one possible combination of rank-one tensors sum to Y after controlling for the three elementary indeterminacies. Let B = ((B1)ʹ, …, (Bd)ʹ)ʹ where Bk = Pk + Qk, k = 1, …, d. The scaling indeterminacy refers to non-uniqueness with respect to a scale change of each column vector of Bk. That is, for d diagonal scaling matrices Γk = diag(γ1k, …, γrk), k = 1, …, d, we have B̃ = ((B1Γ1)ʹ, …, (BdΓd)ʹ)ʹ such that ∏k = 1dγjk = 1 for j = 1, …, r. The permutation indeterminacy comes from an arbitrary r × r permutation matrix Π, such that $\tilde{B} =\big((B^1 \Pi)', \ldots,$ $(B^d \Pi)'\big)'$. In addition to scaling and permutation, the proposed tensor representation may suffer from the addition indeterminacy, that is, for arbitrary nk × r matrix Δk, P̃k = Pk + Δk and Q̃k = Qk − Δk. As a special case of d = 2, the elementary indeterminacy reduces to the non-singular transformation indeterminacy for a matrix. Specifically, for an r × r non-singular matrix Υ, B̃1 = B1Υ and B̃2 = B2(Υ− 1)ʹ. Nevertheless, this issue can always be solved by the singular value decomposition, which imposes orthonormality to column vectors. Therefore, we focus our attention on higher-order tensors with d ≥ 3, although some of the results continue to hold for a matrix. [invariant] Predicted values given by ([GSM]) are invariant with respect to scaling, permutation and addition indeterminacies. The proof is straightforward by applying the aforementioned definition of scaling, permutation and addition indeterminacies to ([GSM]), and is hence skipped. We introduce the concept of *k-rank, which is the Kruskal rank introduced in. Specifically, for a matrix A, the k-rank of A* is A={k:}. [identifiability] Suppose ∑k = 1dKBk ≥ 2r + (d − 1). Minimizers of L(P, Q∣Y) in P and Q are unique up to permutation almost surely. The above condition ∑k = 1dKBk ≥ 2r + (d − 1) is not strong. In numerical studies, factors in (Pk + Qk) are usually linearly independent, and hence we have KBk = r. Then this condition reduces to r ≥ 1 + 1/(d − 2) for d ≥ 3, which is achievable even for low-rank high-order tensors. As shown in Proposition [identifiability], the issue of scaling and addition indeterminacies is resolved almost surely through penalization imposed on L(P, Q∣Y). To treat the permutation indeterminacy, we rearrange r column vectors (p⋅ 1k + q⋅ 1k, p⋅ 2k + q⋅ 2k, …, p⋅ rk + q⋅ rk) for each mode such that ∑k = 1d∥p⋅ 1k + q⋅ 1k∥22 ≥ ∑k = 1d∥p⋅ 2k + q⋅ 2k∥22 ≥ ⋯ ≥ ∑k = 1d∥p⋅ rk + q⋅ rk∥22, which is analogous to imposing a descending order of eigenvalues as in matrix decomposition. The rearrangement of column vectors can be implemented during or after the proposed algorithm, since it does not affect the estimation procedure. We acknowledge that the above choice is arbitrary. One may carry out other rearrangements. For example, suggest imposing ordering based on the first element rather than the vector norm. Alternatively, one may utilize prior knowledge if available. In the rest of Section [Theoretical Properties], we assume that parameter (P, Q) is identifiable. Algorithmic Properties ---------------------- ### Convergence with an Arbitrary Initial Point This subsection investigates the convergence property of the proposed algorithm in terms of (P, Q) from an arbitrary initial point. We first establish the property in a compact parameter space and then generalize it to an unbounded open parameter space with extra conditions. Let D ⊂ R2r∑k = 1dnk be the parameter space of (P, Q). Then (P̃, Q̃) ∈ D is called a blockwise local minimizer of $L(\cdot\|\mathbf Y)$ if $$\tilde{P}^k=\mathop{{\arg\!\min}}\_{P^k}L(\tilde{P}^1,\ldots,\tilde{P}^{k-1},P^k,\tilde{P}^{k+1},\ldots,\tilde{P}^d,\tilde{Q}^1,\ldots,\tilde{Q}^d\|\mathbf Y),$$ and $$\tilde{Q}^k=\mathop{{\arg\!\min}}\_{Q^k}L(\tilde{P}^1,\ldots,\tilde{P}^d,\tilde{Q}^1,\ldots,\tilde{Q}^{k-1},Q^k,\tilde{Q}^{k+1},\ldots,\tilde{Q}^d\|\mathbf Y),$$ which is equivalent to a local minimizer along each block direction. In the rest of Section [Algorithmic Properties], we consider the criterion function $L(\cdot\|\mathbf Y)$ as defined in ([pracloss]). [stationary] Suppose D is compact, and the iterates obtained from Algorithm 1 have a cluster point (P̃, Q̃). Then (P̃, Q̃) is a blockwise local minimizer of the criterion function $L(\cdot\|\mathbf Y)$. In the following, we assume that D is open. We consider $L(\cdot\|\mathbf Y)$ at each of its block coordinates. Let $$L^k(P^k)=L(P^k\|\mathbf Y,P^{(-k)},Q) \mbox{ and } L^{k+d}(Q^k)=L(Q^k\|\mathbf Y,P,Q^{(-k)})$$ be $L(P,Q\|\mathbf Y)$ given (P( − k), Q) and (P, Q( − k)), respectively, where X( − k) = (X1, …, Xk − 1, Xk + 1, …, Xd) for X = P or Q, k = 1, …, d. Then Assumption [homo] implies that the improvement on (Pk)’s or (Qk)’s is not dominated by each other. [homo] Let {(Ps, Qs)}s ≥ 1 be a sequence of estimated parameters generated by Algorithm 1, where s represents the s-th iteration. Then O(maxk = 1, …, d∥∇Lk(Psk)∥F) ∼ O(maxk = 1, …, d∥∇Lk + d(Qsk)∥F). Furthermore, let H(Lk) be the Hessian matrix of Lk( ⋅ ), k = 1, …, 2d. We assume that ∥H(Lk)∥2 ≤ ζk, where the constant ζk > 0 is bounded above and may depend on all block coordinates of L( ⋅ ∣Y) except the k-th block. The following proposition leads to the convergence to a blockwise local minimizer. [gc] Suppose D is open and Assumption [homo] holds. Let ∥H(Lk)∥2 ≤ ζk for k = 1, …, 2d. Then the sequence {(Ps, Qs)}s ≥ 1 obtained from Algorithm 1 converges to a blockwise local minimizer of the criterion function $L(P,Q\|\mathbf Y)$. Notice that, since the parameter space is open, a blockwise local minimizer satisfies ∇L = 0, and hence is a special case of a stationary point. ### Local Convergence In this subsection, we provide the local convergence property of Algorithm 1. Specifically, we follow and show that Algorithm 1 converges to a local minimum at the linear rate, provided that an initial value is sufficiently close to the local minimizer. Moreover, the same property applies to a global minimum if it exists. Let (P̃, Q̃) be a local minimizer of L(P, Q∣Y), and H̃ = H(L(P̃, Q̃∣Y)) be the Hessian matrix at (P̃, Q̃). We define the energy norm based on H̃ as ∥(P, Q)∥E = ⟨vec(P, Q), H̃vec(P, Q)⟩1/2. [lc] Suppose D is open, and let (P̃, Q̃) ∈ D be a strict local minimizer of L(P, Q∣Y). For a small neighborhood V of (P̃, Q̃), suppose (Ps0, Qs0) ∈ V for some s0 ≥ 0. Then a sequence {(Ps, Qs)}s ≥ s0 ⊂ V obtained from Algorithm 1 exists, and converges to (P̃, Q̃) at least linearly in the energy norm. That is, there exists μ ∈ [0, 1), such that ∥(Ps + 1, Qs + 1) − (P̃, Q̃)∥E ≤ μ∥(Ps, Qs) − (P̃, Q̃)∥E. Suppose the tolerance error of Algorithm 1 is ε, and s0 = 0. Then the number of iterations has an upper bound: niter ∼ O({logε − log∥(P0, Q0) − (P̃, Q̃)∥E}/logμ). Several methods are available to obtain a good initial value. One suggestion is to increase the sample size ∣Ω∣. For an n1 × ⋯ × nd-dimensional tensor, the sample size can potentially reach ∣Ω∣ = n1⋯nd, while the number of parameters is r∑k = 1dnk. Since each step of the algorithm is a ridge regression, a larger sample size leads to more accurate estimation. However, sample size is not the only determinant. As discussed in and, there does not exist a best rank-r approximation for high-order tensors in general. Alternatively, one could employ the branch-and-bound technique, or utilize multiple random start points. These techniques could result in a satisfactory local minimum if computational capacity allows. Since the parameter space is open, a global minimum is also a local minimum if it exists. Therefore, Proposition [lc] applies if an initial value is in a small neighborhood of the global minimum. [lc2] Suppose D is open, and that a global minimizer (P̃, Q̃) of L(P, Q∣Y) exists. Let H̃ be positive definite. For a small neighborhood V of (P̃, Q̃), suppose (Ps0, Qs0) ∈ V for some s0 ≥ 0, then a sequence {(Ps, Qs)}s ≥ s0 ⊂ V obtained from Algorithm 1 exists, and converges at least linearly to (P̃, Q̃) in the energy norm. That is, there exists μ ∈ [0, 1), such that ∥(Ps + 1, Qs + 1) − (P̃, Q̃)∥E ≤ μ∥(Ps, Qs) − (P̃, Q̃)∥E. The proof is a straightforward exercise of Proposition [lc] given that ∇L = 0 and H̃ is positive. Nevertheless, the existence of a global minimum is not guaranteed. For instance, as demonstrated in and, a tensor might be approximated arbitrarily well by lower-rank tensors. also demonstrate that the best rank-r approximation problem could be ill-posed: for tensors with some rank r in certain tensor spaces, there is a strictly positive probability that a global minimum cannot be obtained. For the recommender system framework, this issue is further complicated since a large proportion of tensor entries are missing. In the proposed setting, however, the criterion function $L(\cdot\|\mathbf Y)$ is always bounded below by zero. Therefore, even if a global minimum does not exist, we can still minimize $L(\cdot\|\mathbf Y)$ such that it is sufficiently close to its infimum. Asymptotic Properties --------------------- In this subsection, we derive asymptotic properties for the proposed method. Specifically, we prove consistency of estimated parameters when the sample size goes to infinity. The result holds true under the L2-loss, or more general loss functions with additional smoothness conditions. As illustrated in the previous section, the global minimum of the criterion function may not exist, but the criterion function is bounded below by zero. We demonstrate that our asymptotic properties still hold even if the estimated parameter is not a global minimizer, as long as the criterion function converges to its infimum. ### Consistency under the L2-Loss In this section, we focus on the asymptotic properties of the predicted values instead of the latent and nested factors (P, Q), since prediction accuracy instead of parameter estimation is the primary concern. For this purpose, we re-define the parameter space. Suppose $\mathbf Y$ is a d-th order tensor with dimension n1 × ⋯ × nd. Let $\mathbf Y=\mathbf \Theta + \mathcal E$, where $\mathbf \Theta=\mbox{E}(\mathbf Y)$ is our primary interest, and $\mathcal E$ is an n1 × ⋯ × nd-dimensional random error. We assume that $\mathcal E$ has i.i.d elements with mean 0, variance σ2 and a finite moment generating function at an open interval containing 0. For an arbitrary element yi1i2⋯id of $\mathbf Y$, the L2-loss function is [l2loss] l(, yi1i2id)=(yi1i2id- i1i2id)2, where θi1i2⋯id = ∑j = 1r(pi1j1 + qi1j1)(pi2j2 + qi2j2)⋯(pidjd + qidjd) is the corresponding element in $\mathbf \Theta$ and is a function of P and Q. Notice that each $l(\mathbf \Theta, y\_{i\_1i\_2\cdots i\_d})$ relies on $\mathbf \Theta$ only through θi1i2⋯id. Since, in practice, the utilities are usually non-negative and finite, we assume that ∥(P, Q)∥∞ ≤ c0, where c0 is a positive constant. Let $J(\mathbf \Theta)$ be a non-negative penalty function. Then the overall criterion function is re-defined as [criterion] L(\|Y)=(i1,,id) l(, yi1i2id)+\|\|J(),, where λ∣Ω∣ is the penalization coefficient and S ⊆ Rn1 × ⋯ × nd is the parameter space for $\mathbf \Theta$. Let $\mathbf \Theta\_0$ be the unique true parameter. We assume that $\hat{{\mbox{\boldmath $\Theta$}}}\_{\|\Omega\|}$ is a sample estimator of $\mathbf \Theta\_0$ satisfying: [SIEVE] L(\|\|\|Y) L(\|Y) +\|\|, where lim∣Ω∣ → ∞τ∣Ω∣ = 0. Note that $\hat{{\mbox{\boldmath $\Theta$}}}\_{\|\Omega\|}$ is identifiable by Proposition [identifiability]. Condition ([SIEVE]) implies that $\hat{{\mbox{\boldmath $\Theta$}}}\_{\|\Omega\|}$ converges to a global minimizer of $L({\mbox{\boldmath $\Theta$}}\|\mathbf Y)$ as ∣Ω∣ → ∞. However, the verification of ([SIEVE]) could still be challenging in that L is non-convex in \boldmath $\Theta$ given missing values associated with Y. Nevertheless, for a finite sample, condition ([SIEVE]) does not impose any restrictions on the existence of a global minimum, and $\hat{{\mbox{\boldmath $\Theta$}}}\_{\|\Omega\|}$ is not required to be a global minimizer even if one exists. In practice, one could use a good initial point to sufficiently reduce the value of L, as discussed in Section 4.2.2, which would lead to more satisfactory numerical results. Let $l\_{\Delta}(\mathbf \Theta\|\cdot)=l(\mathbf \Theta, \cdot) - l(\mathbf \Theta\_0, \cdot)$ be the loss difference, and [K] K(0, )=i1=1n1id=1nd{l(, yi1i2id)} be the expected loss difference. Since $\mathbf \Theta\_0$ is the unique true parameter, we have $K(\mathbf \Theta\_0, \mathbf \Theta) \ge 0$ for all $\mathbf \Theta \in \mathcal{S}$ and K = 0 only if $\mathbf \Theta=\mathbf \Theta\_0$. We define the distance between $\mathbf \Theta$ and $\mathbf \Theta\_0$ as $\rho(\mathbf \Theta\_0, \mathbf \Theta)=K^{1/2}(\mathbf \Theta\_0, \mathbf \Theta)$, and let [V] V(0, )=i1=1n1id=1nd{l(, yi1i2id)}. Then, under the L2-loss, we have $K(\mathbf \Theta\_0, \mathbf \Theta)=\frac{1}{n\_1\cdots n\_d}\\|\mathbf \Theta\_0- \mathbf \Theta\\|^2$ and $V(\mathbf \Theta\_0, \mathbf \Theta)=\frac{4\sigma^2}{n\_1\cdots n\_d} \\|\mathbf \Theta\_0- \mathbf \Theta\\|^2$, where ∥ ⋅ ∥ stands for the Euclidean norm of the vectorized tensor. [l2convergence] Then we have: $$P(\rho(\hat{\mathbf \Theta}\_{\|\Omega\|},\mathbf \Theta\_0) \ge \eta\_{\|\Omega\|}) \leq 7\exp(-c\_1\|\Omega\| \eta\_{\|\Omega\|}^2),$$ where c1 ≥ 0 is a constant, η∣Ω∣ = max(ɛ∣Ω∣, λ∣Ω∣1/2), and $\varepsilon\_{\|\Omega\|} \sim \frac{1}{{\|\Omega\|}^{1/2}}$ is the best possible rate achieved when λ∣Ω∣ ∼ ɛ∣Ω∣2. Theorem 1 states that if the penalty term shrinks to zero at a rate no slower than the rate ɛ∣Ω∣2 as the sample size tends to infinity, then the proposed method can achieve the convergence rate of $\frac{1}{{\|\Omega\|}^{1/2}}$, the same rate as the maximum likelihood estimator. In addition, the development of the convergence rate is under the L2 distance, which, as a special case of the Kullback-Leiber information, is stronger than the commonly used Hellinger distance. ### Next we develop the estimation consistency under more general settings. For each element of $\mathbf Y$, we assume E(yi1⋯id) = ν(θi1⋯id), where ν( ⋅ ) is a mean function. For example, when $\mathbf Y$ is binary, we might adopt the logistic link $\nu(\theta)=\frac{\exp(\theta)}{1+\exp(\theta)}$, and when $\mathbf Y$ is ordinal, we have ν(θ) = exp(θ). We also assume Var(yi1⋯id) < ∞. In this general setting, l( ⋅ , ⋅ ) in ([criterion]) is not necessarily an L2-loss function. Let $K(\mathbf \Theta\_0, \mathbf \Theta)$ and $V(\mathbf \Theta\_0, \mathbf \Theta)$ be defined as in ([K]) and ([V]), respectively, and let $\rho(\mathbf \Theta\_0, \mathbf \Theta)=K^{1/2}(\mathbf \Theta\_0, \mathbf \Theta)$. Here K( ⋅ , ⋅ ) reduces to the Kullback-Leiber pseudo-distance if l( ⋅ , ⋅ ) corresponds to a log-likelihood. Let Wpα[a, b]n1 × ⋯ × nd be a Sobolev space with finite Lp-norm, where a and b are some constants and α is the parameter associated with the degree of smoothness of functions. [smoothness] For each yi1⋯id, suppose $$\|l(\mathbf \Theta\_0,y\_{i\_1 \cdots i\_d})-l(\mathbf \Theta,y\_{i\_1 \cdots i\_d})\| \le g(y\_{i\_1 \cdots i\_d}) \\|\mathbf \Theta\_0-\mathbf \Theta\\|,$$ where g( ⋅ ) satisfies $\emph{E}[\exp\{{t\_0 g(y\_{i\_1 \cdots i\_d})}\}] \le c\_2 < \infty$, for a constant c2 and some constants t0 around 0. In particular, there exists a constant c2ʹ > 0, such that E{g2(yi1, …, id)} ≤ c2ʹ for all yi1⋯id’s. [ball] Suppose there exist δ > 0 and β ∈ [0, 1), such that for a δ-ball centered at $\mathbf \Theta\_0$, we have $\rho(\mathbf \Theta\_0, \mathbf \Theta) \ge c\_3 \\|\mathbf \Theta\_0-\mathbf \Theta\\|^{\frac{1}{1+\beta}}$, where c3 ≥ 0 is a constant. Assumption [ball] indicates that in a neighborhood of the true parameter $\mathbf \Theta\_0$, the distance $\rho(\mathbf \Theta\_0,\cdot)$ is no smaller than the Euclidean distance up to a certain order. On the contrary, if $\rho(\mathbf \Theta\_0,\cdot)$ is dominated by the Euclidean distance for all neighborhoods of $\mathbf \Theta\_0$, then the convergence result under ρ( ⋅ , ⋅ ) can be shown similar to the proof of Theorem [l2convergence]. [generalconvergence] Let $\hat{\mathbf \Theta}\_{\|\Omega\|}$ be a sample estimator satisfying ([SIEVE]). Assume that lΔ ∈ Wpα[a, b]n1 × ⋯ × nd, where p > 2, and that Assumptions [smoothness] and [ball] hold. Then: $$P(\rho(\hat{\mathbf \Theta}\_{\|\Omega\|},\mathbf \Theta\_0) \ge \eta\_{\|\Omega\|}) \leq 7\exp(-c\_4\|\Omega\| \eta\_{\|\Omega\|}^2),$$ where c4 ≥ 0 is a constant, and η∣Ω∣ = max(ɛ∣Ω∣, λ∣Ω∣1/2) with $$\varepsilon\_{\|\Omega\|} \sim \begin{cases} \left(\frac{1}{{\|\Omega\|}^{1/2}}\right)^{\frac{2\omega}{2\omega+1}} & \emph{if } \omega>\frac{1}{2}\\ \left(\frac{1}{{\|\Omega\|}^{1/2}}\right)^{\omega} & \emph{if } \omega\leq\frac{1}{2} \end{cases}$$ being the best possible rate, which can be achieved when λ∣Ω∣ ∼ ɛ∣Ω∣2. Here ω = α/γ, and γ = ∑k = 1d(nk + mk)r is the total number of parameters. The assumption of p > 2 can be relaxed to p ≥ 2 if pω does not go to 0 as ∣Ω∣ → ∞. Notice that the convergence rate in Theorem [generalconvergence] becomes $\varepsilon\_{\|\Omega\|}\sim\frac{1}{{\|\Omega\|}^{1/2}}$ if ω = ∞, which is the convergence rate of the maximum likelihood estimator achieved in Theorem [l2convergence]. Simulation Studies ================== In this section, we perform simulations to compare the proposed method (REM) with five competing tensor factorization methods. Three methods are existing methods, namely, Bayesian probabilistic tensor factorization, the factorization machine, and the Gaussian process factorization machine.[1](#fn1) Since subgroup information is available, another naive but effective tensor factorization method is to conduct CPD for each subgroup separately, and combine the final result. This method is referred to as the groupwise Canonical Polyadic Decomposition (GCPD). The “Cold-Start” Problem ------------------------ The first simulation study is designed to compare the performance of each method under various severity levels of the “cold-start” problem. Specifically, we consider a third-order tensor with user, item and one contextual variable. We set the number of users n1 = 400, the number of items n2 = 1100 and the number of contexts n3 = 9. We assume that the users, items and contexts are from m1 = 10, m2 = 11 and m3 = 3 subgroups, respectively, and assume the number of latent factors r = 3. We generate each latent factor $\mathbf p\_{i\_k}^k \stackrel{iid}{\sim} N(0,\mathbf I\_r)$ for ik = 1, …, nk, and k = 1, 2, 3. To distinguish different subgroups, we set the nested factors as a simple ordered sequence, where $\mathbf q^1\_{(u\_1)}=(-5.5+{u\_1})\mathbf{1}\_r$, $\mathbf q^2\_{(u\_2)}=(-3.6+0.6{u\_2})\mathbf{1}\_r$, and $\mathbf q^3\_{(u\_3)}=(-4+2{u\_3})\mathbf{1}\_r$ for uk = 1, …, mk. Users, items and contexts are evenly assigned to each subgroup. For each simulation, we generate N = n1n2n3(1 − π0) entries out of the entire tensor, where π0 = 80%, 95% or 99% are the missing percentages. Furthermore, we use ϕcs to measure the severity of the “cold-start” problem, and ϕcs = 30%, 60% or 95% represents the proportion of the testing data whose utilities are about new items and are not available from the training set. Each utility is generated by yi1i2i3 = ∑j = 1r(pi1j1 + qi1j1)(pi2j2 + qi2j2)(pi3j3 + qi3j3)/3 + ɛ, where ɛ ∼ N(0, 1) is the random error. For all methods, we assume that r = 3 is known. For REM and GCPD, we also assume that the subgroup memberships are correctly specified, and the tuning parameter λ is pre-selected from grid points ranging from 1 to 11. Since the subgroup structure is assumed to be known, these two methods have more advantage and are expected to have better performance. For MF, we assume the same setting as the proposed method. For BPTF, we choose the number of Gibbs samples to be 50 and keep the remaining parameters by their default choices. For libFM, we use their default setting, and for GPFM, we select the radial basis function kernel with noise being equal to 10 and the standard deviation of latent variables being equal to 1. All methods are replicated by 200 simulation runs. Table [sim1] provides the performance of each method based on the root mean square error (RMSE) and the mean absolute error (MAE), where the MAE is defined as $\frac{1}{\|\Omega\|}\sum\_{\Omega}\|y\_{i\_1\cdots i\_d}-\hat{y}\_{i\_1\cdots i\_d}\|$. We observe that most methods perform worse when either the missing percentage or the severity of the “cold-start” problem increases. In contrast, the proposed method is relatively robust against these changes. Specifically, REM performs the best across all settings, especially in the worst setting when the missing percentage is 99% and the proportion of new items reaches 95%; that is, 95% of the items are new and are not available from the training set. In this scenario, the proposed method is at least 100% better than other methods in terms of both the RMSE and the MAE. High-Order Tensors ------------------ We design the second simulation study to evaluate the performance of the proposed method under fourth-order tensors. Specifically, we let the number of users and the number of items be the same, namely, 500 or 1000, and the number of contexts be 4 for the two contextual variables. The number of latent factors is r = 3, and each latent factor is generated from an i.i.d standard normal distribution. The nested factors for users and items are ( − 5.5 + u)1r for u = 1, …, 10, and the nested factors for the two contextual variables are − 0.25 ⋅ 1r and 0.25 ⋅ 1r corresponding to the two subgroups. Each utility is generated as yi1i2i3i4 = ∑j = 1r(pi1j1 + qi1j1)(pi2j2 + qi2j2)(pi3j3 + qi3j3)(pi4j4 + qi4j4)/4 + ɛ, where ɛ ∼ N(0, 1) is the random error. For each replication in the tensor, we assume that the missing percentage π0 = 95%, 97% or 99%,. Furthermore, we assume that 30% of the items are not available in the training set, that is, the “cold-start” severity level is fixed. The tuning parameter selection for each method is the same as in simulation study 1. All methods are replicated by 200 simulations. REM, GCPD, libFM and GPFM are able to utilize the fourth order of the tensor, while MF and BPTF use up to the second and the third order, respectively. Table [sim2] provides the comparisons of the proposed method and other methods under the fourth-order tensor setting. It is clear that REM has the overall best performance in terms of both RMSEs and MAEs. For the three correctly-specified competing methods, namely GCPD, libFM and GPFM, their performances are not significantly better than the misspecified methods, although they utilize all tensor information. In contrast, the proposed method provides much smaller RMSEs and MAEs in all settings. For example, when the missing percentage is 99% and the number of users and items are both equal to 1000, the existing methods perform mostly similar to that of the grand mean imputation, producing RMSEs slightly above 4 and MAEs around 2.6. That is, these methods do not utilize subject-specific information effectively. In contrast, the proposed method is able to improve on both the RMSE and the MAE by more than 100%. IRI Marketing Data ================== In this section, we apply the proposed method to IRI Marketing Data. There are 30 promotion strategies for various products to attract consumers. Each observation consists of a store ID, a product ID, a promotion strategy and the corresponding average sales volume. The 2447 stores are selected from 47 markets across the United States, where demographic information within two miles of each store is collected as well. The 161,114 products include all items sold from these stores during the 11-year period. These products can be classified into 31 categories, including beer, coffee, frozen pizza, paper towels, etc. The data have a 99% missing rate after being re-organized into a third-order tensor by store, product and promotion. The data are randomly split into a 50% training set, a 25% validation set and a 25% testing set. The random split is replicated 50 times. Sales volumes from each category of product are standardized before analysis to avoid large differences of sales volumes from different categories. We compare the proposed method with the existing methods listed in Section [cs problem]. In addition, we also compare it to the grand mean imputation where all missing elements in the tensor are imputed by the mean of the observed values. Tuning parameters for each method are selected from a wide range of grid points to minimize the RMSE on the validation set. For the proposed method, we classify stores, products and promotion strategies into subgroups based on their geographical locations (the first digit of the zipcode), product categories, and whether a price reduction is applied, respectively. Since the data are standardized, the grand mean imputation returns an RMSE close to 1, which can be regarded as a benchmark basis for comparison. Table [real data] indicates that the proposed REM has the best performance in terms of both RMSE and MAE. Specifically, REM improves the RMSE of the MF by the largest percentage, which demonstrates the great advantage of incorporating contextual information through tensor structure. In addition, the RMSE of the proposed method is less than that of BPTF and libFM, illustrating that REM has better prediction accuracy among the competing tensor factorization methods. GPFM does not converge due to the high missing rate and the large number of parameters involved. Meanwhile, BPTF has nearly the same MAE as the proposed method, but produces a larger RMSE, indicating that BPTF’s performance is possibly better than the proposed method for certain subjects, but has inferior performance for the rest of the subjects. Furthermore, the naive method GCPD has the second best performance. This might be explained by the following two reasons. One is that the GCPD utilizes additional subgroup information as does the REM, which is not applicable to the other competing methods. The other is that, unlike in the simulation studies, the size of the IRI data is large enough so that the number of observations within each subgroup is sufficient to make good estimations. Most of the numerical studies are implemented on Dell C8220 computing sleds equipped with two 10-core Intel Xeon E5-2670V2 processors and 64GB RAM. The running time for each method is provided in Table [real data]. Notice that BPTF’s method also requires large memory storage due to the high demand of the Gibbs samples. Reducing the number of Gibbs samples may lead to more efficient computation but less accurate predictions. Discussion ========== In this article, we propose a new tensor-based recommender system which makes recommendations through incorporating contextual information. A unique contribution of our method is that we achieve tensor completion through utilizing an additional layer of nested factors, in addition to applying the latent factors as in the commonly used Canonical Polyadic Decomposition. The nested factors incorporate subject-subject dependency, which are estimated based on subgrouping. This leads to a significant advantage in solving the “cold-start” problem effectively. That is, for a new subject, information from other subgroup members can be borrowed to make more accurate predictions even though the subject’s own information is not collected sufficiently. In addition, the proposed method is able to address high-order tensors which are beyond the third-order. Existing methods are not effective in handling the high-order tensor problem due to the high computational cost and the high missing rate. We propose a new algorithm that integrates maximum block improvement into the blockwise coordinate descent algorithm, which avoids operating high-order tensors directly and therefore achieves scalable computation. Moreover, the proposed nested factors borrow information from all members in the same subgroup and hence accommodate a high missing rate. The proposed method also shows excellent numerical performance and theoretical properties. In Section [realdata], the proposed method is applied to IRI marketing data which consists of 116 million observations. The proposed method improves prediction accuracy compared to existing methods with relatively small computational cost. In theory, we demonstrate the convergence properties of the proposed algorithm, which converges to a stationary point from an arbitrary initial point, and local convergence to a local minimum with linear convergence rate. The estimated parameter achieves asymptotic consistency under the L2-loss function and other more general circumstances. cccccc & & &ϕcs = 0.3&ϕcs = 0.6&ϕcs = 0.95 π0 = 80% &REM &RMSE &4.586 (0.562) &6.467 (0.823) &8.228 (1.052) & &MAE &2.210 (0.214) &3.606 (0.432) &5.292 (0.687) &MF &RMSE &10.307 (1.012)&10.279 (1.020)&10.373 (1.062) & &MAE &6.789 (0.658) &6.783 (0.673) &6.822 (0.700) &GCPD &RMSE &5.892 (0.754) &8.083 (0.851) &9.949 (0.974) & &MAE &2.781 (0.226) &4.467 (0.421) &6.389 (0.628) &BPTF &RMSE &5.793 (0.590) &8.142 (0.837) &10.192 (1.050) & &MAE &2.645 (0.222) &4.442 (0.438) &6.518 (0.685) &libFM &RMSE &10.361 (1.070)&10.373 (1.070)&10.355 (1.073) & &MAE &6.778 (0.707) &6.786 (0.713) &6.768 (0.718) &GPFM &RMSE &9.017 (1.391)&9.852 (1.894) &11.068 (1.406) & &MAE &5.215 (1.131) &5.624 (0.761) &6.992 (0.814) π0 = 95% &REM &RMSE &3.322 (0.510) &4.658 (0.773) &6.082 (1.055) & &MAE &1.760 (0.196) &2.702 (0.401) &3.943 (0.674) &MF &RMSE &10.768 (1.179)&10.728 (1.148)&10.656 (1.160) & &MAE &7.055 (0.761) &7.045 (0.751) &6.992 (0.761) &GCPD &RMSE &13.489 (2.059) &12.247 (1.632) &10.461 (1.153) & &MAE &6.213 (0.826) &6.386 (0.767) &6.638 (0.701) &BPTF &RMSE &5.847 (0.696) &8.173 (0.881) &10.197 (1.050) & &MAE &2.736 (0.411) &4.498 (0.507) &6.524 (0.683) &libFM &RMSE &10.384 (1.074)&10.395 (1.071)&10.359 (1.076) & &MAE &6.789 (0.717) &6.799 (0.716) &6.769 (0.719) &GPFM &RMSE &10.397 (1.072)&10.406 (1.070) &10.369 (1.078) & &MAE &6.812 (0.718) &6.822 (0.717) &6.792 (0.720) π0 = 99% &REM &RMSE &3.361 (1.032) &4.329 (1.379) &4.988 (1.582) & &MAE &1.865 (0.414) &2.588 (0.722) &3.289 (1.021) &MF &RMSE &12.647 (1.940)&12.612 (1.883)&12.379 (1.706) & &MAE &8.016 (1.071) &8.016 (1.050) &7.903 (0.981) &GCPD &RMSE &10.925 (1.191) &10.808 (1.124) &10.283 (1.058) & &MAE &6.748 (0.755) &6.808 (0.721) &6.685 (0.691) &BPTF &RMSE &8.246 (2.829) &9.243 (1.737) &10.268 (1.093) & &MAE &4.696 (1.917) &5.537 (1.256) &6.616 (0.723) &libFM &RMSE &10.387 (1.058) &10.389 (1.058)&10.342 (1.075) & &MAE &6.790 (0.707) &6.792 (0.713) &6.759 (0.717) &GPFM &RMSE &10.450 (1.068) &10.456 (1.072)&10.410 (1.084) & &MAE &6.874 (0.712) &6.878 (0.722) &6.845 (0.723) [sim1] ccccccc & &d̂ & &n1 = n2 = 500&n1 = n2 = 1000 π0 = 95% &REM &d̂ = 4 &RMSE &1.427 (0.351) &1.799 (0.570) & & &MAE &1.015 (0.138) &1.148 (0.202) &MF &d̂ = 2 &RMSE &4.040 (1.206) &4.099 (1.373) & & &MAE &2.582 (0.679) &2.601 (0.759) &GCPD&d̂ = 4&RMSE &4.691 (1.906) &4.565 (1.807) & & &MAE &2.599 (0.732) &2.395 (0.638) &BPTF &d̂ = 3&RMSE &3.944 (1.212) &3.896 (1.322) & & &MAE &2.472 (0.650) &2.454 (0.722) &libFM &d̂ = 4 &RMSE &4.057 (1.172) &4.136 (1.423) & & &MAE &2.578 (0.655) &2.585 (0.773) &GPFM&d̂ = 4&RMSE &3.958 (1.103) &3.052 (1.194) & & &MAE &2.533 (0.640) &1.916 (0.602) π0 = 97% &REM &d̂ = 4 &RMSE &1.512 (0.670) &1.689 (0.524) & & &MAE &1.050 (0.227) &1.108 (0.174) &MF &d̂ = 2 &RMSE &4.048 (1.210) &4.103 (1.375) & & &MAE &2.588 (0.682) &2.605 (0.761) &GCPD&d̂ = 4&RMSE &4.640 (1.792) &5.384 (1.876) & & &MAE &2.665 (0.795) &2.779 (0.699) &BPTF &d̂ = 3&RMSE &4.047 (1.264) &3.952 (1.345) & & &MAE &2.503 (0.662) &2.473 (0.726) &libFM &d̂ = 4 &RMSE &4.087 (1.214) &4.198 (1.371) & & &MAE &2.593 (0.677) &2.648 (0.768) &GPFM &d̂ = 4&RMSE &4.115 (1.215) &3.604 (1.128) & & &MAE &2.616 (0.682) &2.341 (0.634) π0 = 99% &REM &d̂ = 4 &RMSE &2.780 (1.915) &1.880 (1.540) & & &MAE &1.579 (0.742) &1.187 (0.433) &MF &d̂ = 2 &RMSE &4.108 (1.233) &4.124 (1.390) & & &MAE &2.625 (0.692) &2.621 (0.769) &GCPD&d̂ = 4&RMSE &4.168 (1.364) &4.453 (1.680) & & &MAE &2.567 (0.718) &2.611 (0.753) &BPTF&d̂ = 3&RMSE &4.399 (1.477) &4.174 (1.497) & & &MAE &2.620 (0.706) &2.540 (0.754) &libFM&d̂ = 4 &RMSE &4.124 (1.226) &4.164 (1.399) & & &MAE &2.613 (0.678) &2.624 (0.761) &GPFM&d̂ = 4&RMSE &4.121 (1.221) &4.248 (1.391) & & &MAE &2.619 (0.680) &2.678 (0.778) [sim2] cccc &RMSE &MAE &Comp. Time (hrs) REM &0.637 (0.009)&0.209 (0.001) &3.9 MF &0.969 (0.010)&0.371 (0.007) &0.7 GCPD &0.640 (0.010)&0.229 (0.001) &5.4 BPTF &0.782 (0.155)&0.209 (0.001) &8.4 libFM &0.705 (0.010)&0.236 (0.001) &0.5 GMI &1.000 (0.006)&0.392 (4.483 × 10− 5) &N/A ccc &RMSE &MAE MF &34.2% &43.7% GCPD &0.5% &8.7% BPTF &18.5% &0% libFM &9.6% &11.4% GMI &36.3% &46.7% [real data] Appendix ======== Recall that L(P, Q∣Y) = ∑(i1, ⋯, id) ∈ Ω(yi1⋯id − ŷi1⋯id)2 + λ∑k = 1d∑j = 1r(∥p⋅ jk∥22 + ∥q⋅ jk∥22), where ŷi1⋯id = ∑j = 1r(pi1j1 + qi1j1)(pi2j2 + qi2j2)⋯(pidjd + qidjd). Suppose there exist two minimizers of L( ⋅ , ⋅ ∣Y), namely (P, Q) and (P̃, Q̃). Under the assumption that ∑k = 1dKBk ≥ 2r + (d − 1), it follows from Theorem 3 of that (P, Q) and (P̃, Q̃) are identical with the exception of scaling, permutation and addition. Then based on Lemma [invariant], the ŷi1⋯id’s provided by (P, Q) and (P̃, Q̃) are identical. Therefore, L(P, Q∣Y) = L(P̃, Q̃∣Y) implies that [identi] k=1d j=1r(jk22+jk22)=k=1d j=1r(jk22+jk22). For scaling, suppose there exist some k1, k2 = 1, …, d, k1 ≠ k2, such that $\tilde{\mathbf{p}}\_{\cdot j}^{k\_1}=\alpha\_j \mathbf{p}\_{\cdot j}^{k\_1}$ and $\tilde{\mathbf{q}}\_{\cdot j}^{k\_1}=\alpha\_j \mathbf{q}\_{\cdot j}^{k\_1}$, whereas $\tilde{\mathbf{p}}\_{\cdot j}^{k\_2}=\frac{1}{\alpha\_j} \mathbf{p}\_{\cdot j}^{k\_2}$ and $\tilde{\mathbf{q}}\_{\cdot j}^{k\_2}=\frac{1}{\alpha\_j} \mathbf{q}\_{\cdot j}^{k\_2}$ for a constant αj > 0, j = 1, …, r. We have &&j=1r(jk122+jk122)+j=1r(jk222+jk222) &=& j=1rj2(jk122+jk122)+ j=1r(jk222+jk222). Then ([identi]) implies that αj = 1 almost surely, j = 1, …, r. For addition, similarly, suppose $\tilde{\mathbf{p}}\_{\cdot j}^k=\mathbf{p}\_{\cdot j}^k+{\mbox{\boldmath $\delta$}}\_j^k$ and $\tilde{\mathbf{q}}\_{\cdot j}^k=\mathbf{q}\_{\cdot j}^k-{\mbox{\boldmath $\delta$}}\_j^k$ for a constant vector \boldmath $\delta$jk, k = 1, …, d and j = 1, …, r. Then ([identi]) implies that &&k=1d j=1r(jk22+jk22) &=& k=1d j=1r(jk22+jk22+2jk22+2(jk)’jk-2(jk)’jk) &=& k=1d j=1r(jk22+jk22), which indicates that \boldmath $\delta$jk = 0 almost surely, k = 1, …, d and j = 1, …, r. This concludes that (P, Q) = (P̃, Q̃) almost surely, except for permutation. Recall that X( − k) = (X1, …, Xk − 1, Xk + 1, …, Xd) for X = P or Q, k = 1, …, d. For each Pk, let $$U^k(P^{(-k)},Q)= \mathop{{\arg\!\min}}\_{P^k}L(P^k\|\mathbf Y,P^{(-k)},Q).$$ Here Uk(P( − k), Q) is the unique minimizer of L( ⋅ ∣Y) along the direction of Pk, because each of its rows is a solution of the ridge regression ([estp]). Similarly, we define Vk(P, Q( − k)) as the unique minimizer of L( ⋅ ∣Y) along the direction of Qk. In this proof, let Ptk and Qtk denote the estimated Pk and Qk at the t-th attempted update instead of the t-th iteration, respectively. At each attempted update, only one block of (Pt1, …, Ptd, Qt1, …, Qtd) is updated. For example, if Ptk is updated, then Pt + 1k1 = Ptk1 and Qt + 1k2 = Qtk2 remain unchanged, for k1 ≠ k and k2 = 1, …, d. Recall that, at each iteration, the MBI algorithm retains only one update with the largest improvement in terms of the objective function value. In other words, only a subsequence of {(Pt1, …, Ptd, Qt1, …, Qtd)}t ≥ 1 is adopted by the algorithm and actually converges to the limit. That is, s (U1ts,…,Udts,V1ts,…,Vdts) = (1,…,d,1,…,d). Therefore, we shall focus our attention on the convergence property of this subsequence. We assume that (Pk)’s and (Qk)’s are updated iteratively, as described in Algorithm 1. Subsequently, similar to the proof of Theorem 3.1 of, we show that (P̃, Q̃) is a blockwise local minimizer. On the one hand, suppose that Pk is updated at the (ts + 1)-th iteration and Qkʹ is updated at the (ts + 2)-th iteration, where k is not necessarily equal to kʹ. Then, Pts + 1k = Uk(Pts( − k), Qts)and Qts + 2kʹ = Vkʹ(Pts + 1, Qts + 1( − kʹ)). Hence, &&L(P1ts,…,Pk-1ts,Uk((-k),),Pk+1ts,…,Pdts,Q1ts,…,Qdts\|Y) && L(P1ts,…,Pk-1ts,Uk(P(-k)ts,Qts),Pk+1ts,…,Pdts,Q1ts,…,Qdts\|Y) &=& L(P1ts+1,…,Pdts+1,Q1ts+1,…,Qdts+1\|Y) && L(P1ts+1,…,Pdts+1,Q1ts+1,…,Qk’-1ts+1,Qk’ts+2,Qk’+1ts+1,…,Qdts+1\|Y) &=& L(P1ts+2,…,Pdts+2,Q1ts+2,…,Qdts+2\|Y) && L(P1ts+1,…,Pdts+1,Q1ts+1,…,Qdts+1\|Y). As s → ∞, && L(1,…,k-1,Uk((-k),),k+1,…,d,1,…,d\|Y) &&L(1,…,d,1,…,d\|Y). On the other hand, by the definition of Uk(P̃( − k), Q̃), we have && L(1,…,k-1,Uk((-k),),k+1,…,d,1,…,d\|Y) &&L(1,…,d,1,…,d\|Y). Thus, the above inequality holds as an equality. In a similar way, we can show that the same equality holds for Vk(P̃, Q̃( − k)), k = 1, …, d. Therefore, by definition, (P̃1, …, P̃d, Q̃1, …, Q̃d) is a blockwise local minimizer. It can be seen that the estimation of latent factors and nested factors, as in ([estp]) and ([estq]), is similar to that of ridge regression. For the ik-th subject in the k-th mode, we define a hypothetical design matrix Zikk = (zik, ijk)∣Ωikk∣ × r, where i represents the i-th element of the sorted set {(i1, …, ik − 1, ik + 1, …, id) : (i1, …, id) ∈ Ω} and zik, ijk = ∏l ≠ k(piljl + qiljl). Thus, each Lk(Pk) can be represented by Lk(Pk) = ∑ik = 1nk{∥yik − Zikk(pikk + qikk)∥2 + λ∥pikk∥22}, k = 1, …, d, where yik is a vector of all yi1⋯id’s whose k-th index equals ik. Furthermore, we have the nkr-dimensional gradient $\nabla L^k= (\frac{\partial L^k}{\partial \mathbf{p}\_1^k},\ldots,\frac{\partial L^k}{\partial \mathbf{p}\_{n\_k}^k})$, where $$\frac{\partial L^k}{\partial \mathbf{p}\_{i\_k}^k}=-2\left\{\mathbf{y}\_{i\_k} - Z\_{i\_k}^k(\mathbf{p}\_{i\_k}^k+\mathbf{q}\_{i\_k}^k)\right\}' Z\_{i\_k}^k+2\lambda (\mathbf{p}\_{i\_k}^k)', \mbox{ }k=1,\ldots,d.$$ The representation of Lk + d(Qk), k = 1, …, d, can be deducted similarly. Next, we calculate the Hessian matrix of Lk( ⋅ ). The second partial derivatives are $$\frac{\partial^2 L^k}{\partial \mathbf{p}\_{i\_k}^k \partial \mathbf{p}\_{i\_l}^k} = \begin{cases} 2\left\{(Z\_{i\_k}^k)' Z\_{i\_k}^k+\lambda I\_r\right\} &\quad\mbox{if }k=l,\\ 0 &\quad\mbox{if } k \ne l, \\ \end{cases}$$ where Ir is an r-dimensional identity matrix. Therefore, the Hessian matrix of Lk( ⋅ ) is H(Lk) = 2{diag((Z1k)ʹZ1k, …, (Znkk)ʹZnkk) + λInkr}. It can be seen that H(Lk) is positive definite for λ > 0. Hence, each blockwise function Lk( ⋅ ) is strongly convex. For fixed (P1, …, Pd, Q1, …, Qd) and all X1, X2 ∈ Rnk × r satisfying Lk(X1), Lk(X2) ≤ Lk(Pk), this implies that [strongly convex] Lk(X1) Lk(X2) + Lk(X2), X1-X2 F + X1-X2F2, where k = 1, …, 2d, ξk is a constant, and ξk ≥ ɛ > 0 for a small constant ɛ. Notice that both Lk( ⋅ ) and ξk may depend on all block coordinates of $L(\cdot\|\mathbf Y)$ except for the k-th block. Meanwhile, since the Hessian matrix H(Lk) of Lk(X) is no longer a function of X, we have [Lip] Lk(X1) - Lk(X2)F &= H(Lk(X2)) (X1-X2)2 & k X1-X2F by Taylor expansion, where vec(X) represents the vectorized X. This holds true for fixed (P1, …, Pd, Q1, …, Qd) and all X1, X2 ∈ Rnk × r satisfying Lk(X1), Lk(X2) ≤ Lk(Pk). Now we verify the primary descent condition. Recall that {(Ps, Qs)}s ≥ 1 is a sequence of estimated parameters generated by Algorithm 1. Then similar to the proof of Theorem 2.3 in, we have: &&L(Ps,Qs\|)-L(Ps+1,Qs+1\|) &=& {L(Ps,Qs\|)-L(Ps+1,Qs\|)} + {L(Ps+1,Qs\|)-L(Ps+1,Qs+1\|)} &=& k=1,…,d { Lk(Pks\|Qks)-Lk(Pks+1\|Qks) } &&+ k=1,…,d { Lk(Qks\|Pks+1)-Lk(Qks+1\|Pks+1) } && k=1,…,d ( Pks-Pks+12F ) + k=1,…,d ( Qks-Qks+12F ), where Lk(Ps + 1k∣Qsk) is the same as Lk(Ps + 1k), but to emphasize that Qsk has not been updated. The inequality holds because of ([strongly convex]) with ∇Lk(Ps + 1k∣Qsk) = 0 and ∇Lk(Qs + 1k∣Ps + 1k) = 0. Furthermore, let ξs = mink = 1, …, 2dξsk. Then for any l1, l2 = 1, …, d, &&L(Ps,Qs\|)-L(Ps+1,Qs+1\|) && ( Pl1s-Pl1s+1F Pk1s-Pk1s+1F + Ql2s-Ql2s+1F Qk2s-Qk2s+1F ) && ( k=1,…,d Lk(Pks) F Pk1s-Pk1s+1F &&+ k=1,…,d Lk+d(Qks) F Qk2s-Qk2s+1F ), where k1, k2 ∈ {1, 2, …, d} are the two blocks that are actually updated in the (s + 1)-th iteration corresponding to Ps + 1k1 and Qs + 1k2, respectively, and the second inequality is resulted from ([Lip]). Then by the triangle inequality, (Ps,Qs) -(Ps+1,Qs+1) F &=& (Pk1s-Pk1s+1)+(Qk2s-Qk2s+1) F & & (Pk1s-Pk1s+1)F + (Qk2s-Qk2s+1) F. Assumption [homo] states that the improvement over Psk and Qsk is of the same order. This implies that there exists a sequence of constants {cs}s ≥ 1, such that mins ≥ 1cs ≥ ɛ > 0, and min{maxk = 1, …, d∥∇Lk(Psk)∥F, maxk = 1, …, d∥∇Lk + d(Qsk)∥F} ≥ csmaxk = 1, …, 2d∥∇Lk∥F. Therefore, by combining the two inequalities above, we have: &&L(Ps,Qs\|)-L(Ps+1,Qs+1\|) && k=1,…,2d LkF ( Pk1s-Pk1s+1F + Qk2s-Qk2s+1F ) && k=1,…,2d LkF (Ps,Qs) -(Ps+1,Qs+1) F. Finally, since ∥∇L∥F2 = ∑k = 12d∥∇Lk∥F2 ≤ 2dmaxk = 1, …, 2d∥∇Lk∥F2, we have the primary descent condition: L(Ps,Qs\|)-L(Ps+1,Qs+1\|) LF (Ps,Qs) -(Ps+1,Qs+1) F. Since cs and ξs are bounded below and ζs is bounded above, there exists a σ̄ > 0, such that $\frac{c\_s\xi\_s}{2 \sqrt{2d} \zeta\_s} \ge \bar{\sigma} >0$ for all s ≥ 1. By applying Proposition [identifiability] and the rearrangement method beneath Proposition [identifiability], the complementary descent condition is satisfied almost surely. Furthermore, since L(P, Q∣Y) is analytic in the neighborhood of (P̃, Q̃) by definition ([pracloss]), the cluster point (P̃, Q̃) satisfies the Lojasiewicz gradient inequality. That is, for all (P, Q)’s in some neighborhood of (P̃, Q̃), there exist constants c > 0 and κ ∈ (0, 1), such that ∥∇L(P, Q∣Y)∥F ≥ c∣L(P, Q∣Y) − L(P̃, Q̃∣Y)∣1 − κ. Then Theorem 3.2 of and Theorem 2.2 of hold. Therefore, the cluster point estimated from Algorithm 1 is a limit point. Then, following Lemma [stationary] and Section 2.3 of, since L( ⋅ ∣Y) is differentiable and the parameter space of L( ⋅ ∣Y) is open, the limit point of Algorithm 1 is a stationary point and a blockwise local minimizer. Without loss of generality, we assume that (P̃, Q̃) = 0 and L(P̃, Q̃∣Y) = 0. For a given (P, Q), we define the domain for the blockwise function Lk( ⋅ ) as Dk(P, Q) = {(P1, …, Pk − 1)} × Rnk × r × {(Pk + 1, …, Pd, Q1, …, Qd)}, and Dk + d(P, Q) = {(P1, …, Pd, Q1, …, Qk − 1)} × Rnk × r × {(Qk + 1, …, Qd)}, for k = 1, …, d. Then ∪ k = 1dDk(P, Q) and ∪ k = d + 12dDk(P, Q) represent the two search crosses for updating P and Q, respectively. Without loss of generality, we let the neighborhood V be a ball based on the energy norm and centered at (P̃, Q̃) = 0. Since (P̃, Q̃) is a strict local minimizer, we can set V small enough such that it is convex and {(Ps, Qs)}s ≥ s0 ⊂ V exists for an s0 ≥ 0. For the (s + 1)-th iteration of Algorithm 1, the search domain is defined as Ds + 1 = V ∩ ( ∪ k = 1dDk(Ps, Qs) ∪ k = d + 12dDk(Ps + 1, Qs)). Then by definition, $\displaystyle(P\_{s+1},Q\_{s+1}) =\mathop{\arg\min}\_{\mathcal{D}\_{s+1}} L(P,Q\|\mathbf{Y})$ exists and is in V. Let $\displaystyle{{\bf x}}\_{s+1} = \mathop{\arg\min}\_{\mathcal{D}\_{s+1}} \\|(P,Q)\\|\_E$ be a vector estimated from a block coordinate descent method that has the same block directions as Algorithm 1. It is known that $\{{{\bf x}}\_s\}\_{s \ge 1}$ converges linearly in the energy norm, e.g., under the almost-cyclic rule or the Gauss-Southwell rule. Then there exists μ1 ∈ [0, 1), such that for a large s, we have [rhs] xs+1E 1 (Ps,Qs)E. Meanwhile, if we re-arrange ${{\bf x}}\_{s+1}$ as in the format of (P, Q), then by the definition of (Ps + 1, Qs + 1), $$L(P\_{s+1},Q\_{s+1}\|\mathbf Y) \le L({{\bf x}}\_{s+1}\|\mathbf Y).$$ Therefore, by Lemma 3.2 of, for a small ɛ > 0, there exists δ > 0, such that V ⊂ B(δ), where B(δ) is a ball centered at 0 with energy-norm-based radius equal to δ. Then [lhs] (Ps+1,Qs+1)E (1+)xs+1E. Define μ = μ1(1 + ɛ). Let ɛ be small enough such that μ < 1. Then by ([rhs]) and ([lhs]): ∥(Ps + 1, Qs + 1)∥E ≤ μ∥(Ps, Qs)∥E. Proof of Theorem [l2convergence] ================================ Let $A(k\_1,k\_2)=\{\mathbf \Theta \in \mathcal{S}: k\_1 \leq \rho(\mathbf \Theta\_0, \mathbf \Theta) \leq 2k\_1, J(\mathbf \Theta) \leq k\_2\}$, and $\mathcal{F}(k\_1,k\_2)=\{l\_{\Delta}(\mathbf \Theta\|\cdot): \mathbf \Theta \in A(k\_1,k\_2)\}$. We first verify several conditions of Corollary 2 in. By definition, we have $\sup\_{A(k\_1,k\_2)} V(\mathbf\Theta\_0,\mathbf\Theta) \le c\_8 k\_1^2 =c\_8 k\_1^2\{1+(k\_1^2+k\_2)^{\beta\_1}\}$, and hence β1 = 0. In the rest of this section, all ci’s with i ∈ N are assumed to be non-negative constants. For a given yi1⋯id, we have $\|\theta\_{0,i\_1 \cdots i\_d}-\theta\_{i\_1 \cdots i\_d}\| \le \frac{1}{4\sigma^2} \mbox{Var}\{l(\mathbf\Theta,y\_{i\_1 \cdots i\_d})-l(\mathbf\Theta\_0,y\_{i\_1 \cdots i\_d})\}$ for ik = 1, …, nk, k = 1, …, d. Furthermore, $$\|l(\mathbf\Theta,y\_{i\_1 \cdots i\_d})-l(\mathbf\Theta\_0,y\_{i\_1 \cdots i\_d})\|=\|\theta\_{0,i\_1 \cdots i\_d}-\theta\_{i\_1 \cdots i\_d}\| \cdot \|2y\_{i\_1 \cdots i\_d}-\theta\_{0,i\_1 \cdots i\_d}-\theta\_{i\_1 \cdots i\_d}\|.$$ Define a new random variable w = ∣2yi1⋯id − θ0, i1⋯id − θi1⋯id∣, then we have E{exp(t0w)} < ∞ for t0 at an open interval containing 0. Now we verify that for a constant c9 > 0, we have $\sup\_{A(k\_1,k\_2)} \\|\mathbf\Theta\_0-\mathbf\Theta\\|\_{\sup} \le c\_9 (k\_1^2+k\_2)^{\beta\_2}$ for β2 ∈ [0, 1). Define $f\_0=f\_0(P,Q)=\mathbf\Theta-\mathbf\Theta\_0$. Recall that ∥(P, Q)∥∞ ≤ c0 and γ = ∑k = 1d(nk + mk)r is the total number of parameters. Since f0 is a quadratic function of elements of P and Q, we have f0 ∈ W2∞[ − c0, c0]γ where W2∞ is a Sobolev space, and $\\|f\_0\\|\_2=\rho(\mathbf\Theta\_0,\mathbf\Theta) \le c\_{10}$ for a constant c10 > 0. In addition, we have f0(α) = 0 for α = ∞. Therefore, based on Lemma 2 of, we get $$\\|f\_0\\|\_{\infty}=\\|\mathbf\Theta\_0-\mathbf\Theta\\|\_{\infty} \le 2c\_{10}.$$ The required conditions are fulfilled by defining c9 = 2c10 and β2 = 0. Next, we verify the assumption on the Hellinger metric entropy. Let N(ɛ, n) = {g1l, g1u, …, gnl, gnu} be a set of functions from the L2 space, where max1 ≤ i ≤ n∥giu − gil∥2 ≤ ɛ. Suppose for any function lΔ ∈ F(k1, k2), there exists i ∈ {1, …, n} such that gil ≤ lΔ ≤ giu almost surely. Then the Hellinger metric entropy is defined as H(ɛ, F) = log{n : minN(ɛ, n)}. Let $\omega=\frac{\alpha}{\gamma}=\infty$, then pω = ∞ > 1. Define ψ(k1, k2) = ∫L0U0H1/2(u, F)du/L0 where L0 = c12λ∣Ω∣(k12 + k2) and U0 = c13ɛ∣Ω∣(k12 + k2)(1 + max(β1, β2))/2. Based on Theorem 5.2 of, the Hellinger metric entropy is controlled by H(ɛ∣Ω∣, F) ≤ c11ɛ∣Ω∣− 0 = c11. Recall that β1 = β2 = 0. Then for fixed k1 and k2, we have $\psi(k\_1,k\_2)=\sqrt{c\_{11}} \frac{U\_0-L\_0}{L\_0} \sim \frac{\varepsilon\_{\|\Omega\|}-\lambda\_{\|\Omega\|}}{\lambda\_{\|\Omega\|}}$. Given that ψ ∼ ∣Ω∣1/2, the best possible rate is achieved at ɛ∣Ω∣ ∼ λ∣Ω∣1/2, that is, $$\varepsilon\_{\|\Omega\|} \sim \frac{1}{\|\Omega\|^{1/2}}.$$ The result in Theorem [l2convergence] then follows by applying Corollary 2 of. Proof of Theorem [generalconvergence] ===================================== Let A(k1, k2) and F(k1, k2) be defined as in the proof of Theorem [l2convergence]. We start by verifying conditions of Corollary 2 of. First, based on the definition of V( ⋅ , ⋅ ) and Assumption [smoothness], we have V(0,) &=&i1=1n1id=1nd{l(, yi1id)-l(0, yi1id)} && i1=1n1id=1nd {\|l(, yi1id)-l(0, yi1id)\|2} && i1=1n1id=1nd {g2(yi1 id)} 0-2 && c2’ 0-2. Therefore, we have $\sup\_{A(k\_1,k\_2)} V(\mathbf \Theta\_0,\mathbf \Theta) \le 4c\_2' k\_1^2 \{1+(k\_1^2+k\_2)^{\beta\_1}\}$ with β1 = 0. Furthermore, by Assumptions [smoothness] and [ball], we have l(\|yi1id)2 &=& [{\|l(, yi1id)-l(0, yi1id)\|2}]1/2 && c2’ 0- && c14 (0,)1+ for c14 = c2ʹ/c31 + β, a given element yi1, …, id in the tensor, and δ > 0 such that $\mathbf \Theta \in B\_{\delta}(\mathbf \Theta\_0)$. Then by Lemma 2 of, we have l(\|yi1id) && c15 l(\|yi1id)2(-p-1)/(-p-1+1/2) && c14c15(0,)22, where $\beta\_2=\frac{1+\beta}{2} \frac{\alpha-p^{-1}}{\alpha-p^{-1}+1/2}$. Since β ∈ [0, 1), we have β2 ∈ [0, 1). We now verify the condition on the Hellinger metric entropy. Recall that $\omega=\frac{\alpha}{\gamma}$. For any ω > 0, we have p > 2 implies $\frac{p}{1-p\omega} >2$. Therefore, based on Theorem 5.2 of, the Hellinger metric entropy is upper-bounded by H(ɛ∣Ω∣, F) ≤ c16ɛ∣Ω∣− 1/ω. Then we have: (k1,k2) &=& L0U0 u-du/L0 &~& C(k1,k2). The best possible rate is provided by setting ψ(k1, k2) ∼ ∣Ω∣1/2 and λ∣Ω∣ ∼ ɛ∣Ω∣2. Hence, \|\|--1-\|\|- ~\|\|1/2. That is, $$\varepsilon\_{\|\Omega\|} \sim \begin{cases} (\frac{1}{{\|\Omega\|}^{1/2}})^{\frac{2\omega}{2\omega+1}} & \mbox{if } \omega>\frac{1}{2}\\ (\frac{1}{{\|\Omega\|}^{1/2}})^{\omega} & \mbox{if } \omega\leq\frac{1}{2} \end{cases}.$$ Then the result follows by applying Corollary 2 of. --- 1. The codes can be obtained from <https://www.cs.cmu.edu/~lxiong/bptf/bptf.html>, <http://www.libfm.org/>, and <http://trungngv.github.io/gpfm/>, respectively.[↩](#fnref1) MULTILAYER TENSOR FACTORIZATION WITH APPLICATIONS TO RECOMMENDER SYSTEMS ======================================================================== , Recommender systems have been widely adopted by electronic commerce and entertainment industries for individualized prediction and recommendation, which benefit consumers and improve business intelligence. In this article, we propose an innovative method, namely the recommendation engine of multilayers (REM), for tensor recommender systems. The proposed method utilizes the structure of a tensor response to integrate information from multiple modes, and creates an additional layer of nested latent factors to accommodate between-subjects dependency. One major advantage is that the proposed method is able to address the “cold-start” issue in the absence of information from new customers, new products or new contexts. Specifically, it provides more effective recommendations through sub-group information. To achieve scalable computation, we develop a new algorithm for the proposed method, which incorporates a maximum block improvement strategy into the cyclic blockwise-coordinate-descent algorithm. In theory, we investigate both algorithmic properties for global and local convergence, along with the asymptotic consistency of estimated parameters. Finally, the proposed method is applied in simulations and IRI marketing data with 116 million observations of product sales. Numerical studies demonstrate that the proposed method outperforms existing competitors in the literature. [class=MSC] Introduction ============ Recommender systems have become very important in daily life due to high demand from the entertainment industry and business marketing which produce large amounts of data. In addition, applications of recommender systems have been greatly facilitated by the advancement of statistical and machine learning techniques. These applications include personalized marketing for internet users, merchandise recommendation for retail stores, and even individualized gene therapies. Each application involves collecting a wide variety of information, and successful exploitation of such rich information leads to more accurate recommendations. However, this also imposes unprecedented challenges to traditional methods due to the large size and complex structure of data. Therefore, more general and integrative recommender systems are urgently needed. The tensor, also called multidimensional array, is well-recognized as a powerful tool to represent complex and unstructured data. It is applied in many areas such as signal processing, neuroimaging, and psychometrics. In recommender systems, the tensor shows its flexibility to accommodate contextual information, and is also regarded as one of the most effective tools for developing context-aware recommender systems. Nevertheless, applying the tensor effectively to CARS remains a challenging problem. In matrix recommender systems, the singular value decomposition (SVD) method provides the best low-rank approximation, and is known to be arguably the most effective single procedure. This imposes a great challenge to generalize matrix decomposition to the tensor framework. One common approach to utilize the tensor structure is to apply latent factor models, where each user, item or context is assigned an individual latent factor to represent their characteristics quantitatively. However, several key issues have not been solved completely. The first is the “cold-start” problem, where available information is not sufficient to provide valid predictions for new users, items or contexts (in the rest of this paper, we use “subject” to denote a user, an item or a context in general). For instance, in latent factor modeling, a latent factor is not estimable if a subject is not available in the training set. The subject’s utilities can only be predicted through the average information from other subjects, which may lead to low prediction accuracy. Several solutions have been proposed for the traditional matrix recommendation techniques, for example, imputing pseudo ratings, supplementing artificial users and items, incorporating content-boosted information, or utilizing group information for new subjects. Nevertheless, the “cold-start” problem under CARS is quite challenging and has not been well-investigated. For example, viewers may have different movie-watching experiences with new friends or at a new theater, and stores’ sales volumes may vary under a new promotion strategy. Another critical issue involves solving the higher-order tensors (beyond the third-order) problem. Higher-order tensors are very useful, because we might be interested in integrating more than one contextual variable to fully utilize their subject-specific information. For example, when recommending a new destination to travelers, one has to consider several factors, e.g., the timing and the cost of the trip, and the crowdedness of the destination, in addition to travelers’ travel interests. While some existing methods provide a general methodology on high-order tensors, the implementation of higher-order tensors could be challenging. One obstacle is the high computational cost. Another obstacle is that higher-order tensors could result in higher missing rates due to fewer observations at each combination of contextual variables; and this could lead to non-convergence in computing for some existing methods. One possible solution is to choose only one special contextual variable in the tensor, and treat the rest of the contextual variables as linear covariates. However, this may lead to loss of information on subject-specific or group-wise interaction. In this paper, we propose a novel tensor factorization method, namely the recommendation engine of multilayers (REM). Specifically, we assume a tensor structure where each mode corresponds to user, item or a contextual variable, and each element of the tensor represents a utility, such as a rating or sales volume. The novelty of our method is that we add another layer which categorizes users, items and contexts into the same subgroups if they share similar characteristics. We quantify subgroup effects as random effects through nested-factors modeling, in addition to latent factors in the tensor factorization. Theoretically, we demonstrate the algorithmic properties of the proposed method, which converges to a stationary point from an arbitrary initial point, and local convergence to a local minimum with a linear convergence rate. The estimated parameter achieves asymptotic consistency under the L2-loss function and other more general circumstances. The proposed tensor factorization method has two significant advantages. First, it solves the “cold-start” problem effectively. For a new subject, even if its latent factors are not available, the nested factors from the corresponding subgroups can provide a group-specific estimate, which is more accurate than the average of the available observations. This finding is also supported by simulation studies where the proposed method is more effective than competing tensor factorization methods when the proportion of new subjects is high. Second, the proposed method is able to accommodate high-order tensors. The difficulty of applying high-order tensors is solved by the proposed nested factors which utilize group-wise information and hence are more robust to a higher missing rate. In addition, we propose a new algorithm that incorporates the maximum block improvement into the cyclic blockwise-coordinate-descent algorithm. This avoids the direct operation of large-scale tensors, and makes the estimation of high-order tensors feasible. Furthermore, a parallel computing strategy is implemented to calculate latent factors and nested factors for each subject and subgroup, respectively, which provides scalable and efficient computation. The rest of this paper is organized as follows. Section [Preparations] provides the background of the tensor factorization and the framework for the context-aware recommender systems. Section [A Multilayer Method] introduces the proposed method and algorithm. Theoretical properties are derived in Section [Theoretical Properties]. Section [Simulation Studies] presents simulation studies to validate the performance of the proposed method. In Section [realdata], we apply the proposed method to IRI marketing data. Section [Discussion] concludes with a discussion. Preparations ============ Notation and Tensor Background ------------------------------ In this subsection, we provide the background of the tensor, and introduce the notation for the proposed method. We define a d-th order tensor Y ∈ Rn1 × ⋯ × nd as a d-dimensional array, where each order is also called a mode. In the rest of this article, bold capital letters denote tensors, capital letters denote matrices, bold small letters denote vectors, and small letters denote scalars. That is: [CP] j=1r p1j p2j pdj, where ∘ represents the vector outer product, and $\mathbf p\_{\cdot j}^k$, j = 1, …, r, are nk-dimensional vectors corresponding to the k-th mode. Here r is called the rank of Y if the number of terms r is minimal. Equivalently, for each element of Y, we have: [CPelement] yi1i2id j=1r p1i1j p2i2j pdidj. For other properties of a tensor, see for an extensive review. Context-Aware Recommender Systems --------------------------------- In this subsection, we briefly review the tensor application to recommender systems, namely, context-aware recommender systems (CARS). See also for a comprehensive review of CARS. Let nk (k = 1, …, d) be the number of subjects for the k-th mode, that is, n1 is the number of users, n2 is the number of items, and n3, …, nd are the number of contexts for the (d − 2) contextual variables, respectively. For the CPD representation in ([CP]), let $P^k=(\mathbf p^k\_{\cdot 1},\ldots,\mathbf p^k\_{\cdot r})\_{n\_k \times r}$, where each row of P1 or P2 represents the r-dimensional latent factors for each user or item, and rows of P3, …, Pd are the latent factors for the contextual variables, respectively. In the rest of this article, we use $\mathbf p^k\_{i\_k}$ to represent the ik-th row of Pk, in contrast to $\mathbf p\_{\cdot j}^k$ being the j-th column of Pk. Each element of Y is defined as a utility, for example, a rating, a purchase or a sales volume, and is estimated via ([CPelement]) in CPD. Here the orthonormality for the latent factors is not required, and each utility yi1i2⋯id comprises user-, item- and context-specific information. In addition, the CPD requires estimation of only ∑k = 1dnkr parameters, which essentially performs a tensor dimension reduction procedure. In many applications, we may have a non-negative tensor $\mathbf Y$. Methods for non-negative CPD are proposed to address this issue, as such decompositions make results meaningful and interpretable. In recommender system problems, however, the direct interpretation of latent factors is less critical, as the relative scale or ranking is important for recommendation. Most importantly, improving prediction accuracy is the ultimate goal of a good recommender system for users. Therefore, non-negativity is not required here. Technically, we can always standardize $\mathbf Y$ prior to analysis. In addition, non-negative CPDs usually entail non-negative latent factors, which might restrict the parameter space to the non-negative orthant. In the proposed framework, an unbounded parameter space allows more flexibility for extensive search, which may lead to a more satisfactory convergence result. The most common loss function is the L2-loss, which is computationally efficient. Theoretical properties of other loss functions are also considered. Let Ω = {(i1, i2, …, id) : yi1i2⋯idis observed} be a set of indices corresponding to observed utilities, and ∣Ω∣ represent the sample size. Since the number of available observations for each subject might be smaller than the number of latent factors r, we adopt a penalty function using regularization. For example, when the L2-penalty is applied, we have: [CPcriterion] L(P1,…,Pd\|Y)=(i1,,id) (yi1id - i1id)2 +k=1dPkF2, where ŷi1⋯id = ∑j = 1rpi1j1⋯pidjd is the estimated utility provided by ([CPelement]), and ∥ ⋅ ∥F represents the Frobenius norm. Other regularization methods include, but are not limited to, the trend filtering penalty when d = 3, and the L0- and L1-penalty for sparse low-rank pursuit when d = 2. Algorithms for implementing the CPD include the cyclic coordinate descent algorithm, and the stochastic gradient descent method. Alternatively, propose a maximum block improvement algorithm that only updates the block with the maximum improvement in each iteration instead of updating each block cyclically. This strategy guarantees convergence to a stationary point, and ensures a fast convergence rate in many circumstances. A Multilayer Method =================== General Methodology ------------------- In this section, we develop the methodology for the proposed REM method. Specifically, we adopt the idea of nested factors from the design of experiments to capture between-subject dependency under the CPD framework ([CP]). We assume that subjects can be categorized into subgroups, where subjects within the same subgroup share similar characteristics and are dependent on each other. For subgrouping, we can incorporate prior information such as users’ demographic information, item categories and functionality, and contextual similarity. If this kind of information is not available, one can utilize the missing pattern of the tensor data, or the number of records from each user and on each item. As shown in, implicit information from the number of records may reflect subjects’ behavior that is not available elsewhere, and can help improve recommendation accuracy. In more general situations, clustering methods such as the k-means can be applied to determine the subgroups. See and on robust approaches to select the number of subgroups. Suppose the subgroup labels are given, then we formulate each utility as follows: [GSM] i1i2id = j=1r (p1i1j+q1i1j) (p2i2j+q2i2j) (pdidj+qdidj), where pikjk is the j-th latent factor for the ik-th subject from the k-th mode, and qikjk is the corresponding nested factor, j = 1, …, r, ik = 1, …, nk, and k = 1, …, d. We define the nk × r-dimensional matrix Qk similar to Pk as in Section [CARS22]. Notice that we have $\mathbf q^k\_{i\_k}=\mathbf q^k\_{i'\_k}$ if subjects ik and iʹk are from the same subgroup. We assume that the number of subgroups for the k-th mode is mk, which corresponds to mk unique values for qikjk. We use $\mathbf q\_{(u\_k)}^k$ occasionally to denote the nested factor associated with the subgroup uk, uk = 1, …, mk. Let P = ((P1)ʹ, …, (Pd)ʹ)ʹ and Q = ((Q1)ʹ, …, (Qd)ʹ)ʹ represent all parameters of interest. We define $L(P,Q\|\mathbf Y)=L(P^1,\ldots,P^d,Q^1,\ldots,Q^d\|\mathbf Y)$ as the overall criterion function: [pracloss] L(P,Q\|Y)= (i1,,id) (yi1id - i1id)2 +(PF2+QF2), where ŷi1⋯id is represented by ([GSM]) and λ is the penalization coefficient. Here we adopt the most commonly used L2-loss and L2-penalty for efficient computation, although other types of loss and penalty functions are also applicable. For example, one may consider the hinge loss or the ψ-loss for classification, and the absolute loss or Huber loss to achieve robust estimation. Meanwhile, if prior knowledge of the latent and nested factors is available, for example regarding sparsity or smoothness, then appropriate regularization methods can be applied. Parameter Training ------------------ In the following, we discuss how the model parameters are estimated. Mainly we are interested in finding a solution (P̂, Q̂) aiming at minimizing $L(P,Q\|\mathbf Y)$. Let Ωikk = {(i1, …, ik, …, id) : yi1⋯ik⋯idis observed} be the set of indices where the k-th mode index equals ik and the corresponding utilities are observed; Let I(uk)k be the set of subjects in the subgroup uk, uk = 1, …, mk. We assume that ∣I(uk)k∣ ≥ 2 for each uk. For each mode of the tensor, the partial derivatives of $L(\cdot\|\mathbf Y)$ have explicit forms with respect to the latent factors or the nested factors, which makes it feasible to apply the blockwise coordinate descent approach. That is, [estp] kik = pkik kik (yi1id - i1id)2 +pkik22, for ik = 1, …, nk, and [estq] k(uk) = qk(uk) ik kukkik (yi1id - i1id)2 +qk(uk)22, for uk = 1, …, mk, and k = 1, …, d. The estimation procedure of $\hat{\mathbf p}^k\_{i\_k}$ in ([estp]) is a ridge regression, and does not require knowing $\hat{\mathbf p}^k\_{i'\_k}$ for iʹk ≠ ik. Thus, parallel computation is applicable to calculate $\hat{\mathbf p}^k\_1,\ldots,\hat{\mathbf p}^k\_{n\_k}$ efficiently. This strategy is also applicable to obtaining the $\hat{\mathbf q}^k\_{(u\_k)}$’s in ([estq]). Therefore, the minimization of $L(P,Q\|\mathbf Y)$ can be done cyclically through estimating P and Q. Notice that Ω = ∪ ik = 1nkΩikk, and it is possible that Ωikk is empty for certain ik’s; that is, there is no observation on subject ik, as in the case of the “cold-start” problem. Algorithm --------- In contrast to matrix factorization,. First, the number of parameters and sample size for a tensor decomposition could be much greater than its matrix counterpart, which makes the computation of the MBI more intensive. Second, since the number of blocks increases significantly, the MBI algorithm may never update certain blocks due to small improvements along these directions, which leads to the estimated values corresponding to these blocks remaining the same as the initial values. To solve these problems, we propose a two-step algorithm, which estimates the latent-factors matrix P̂ and the nested-factors matrix Q̂ iteratively. Within the estimation of each matrix, we apply the MBI algorithm to find the optimal block direction with the largest improvement of estimations. Specifically, we propose the following algorithm aiming at minimizing ([pracloss]). Let (Ps, Qs) denote the estimated (P, Q) at the s-th iteration, then the improvement of estimations for updating the k-th mode is defined as [impI] Iks=1-, and [impJ] Jks=1-, where Pk \* and Qk \* are the attempted updates for the k-th mode, k = 1, …, d. $\overline{\mbox{\underline{\makebox[\textwidth]{\textbf{Algorithm 1:} A Two-Step Algorithm with Parallel Computing}}}}$ 1. (Initialization) Input all observed yi1⋯id’s, the rank r, the tuning parameter λ, initial value (P0, Q0) and a stopping criterion ɛ = 10− 4. 2. (Latent-factors update) At the s-th iteration (s ≥ 1), estimate Ps. 1. For each Pk, solve ([estp]) through parallel computing and obtain $P^{k\}=(\hat{\mathbf p}^k\_{1},\ldots,\hat{\mathbf p}^k\_{n\_k})'$. Calculate Isk* through ([impI]). 2. Assign Psk0 ← Pk0 \	arxiv_0000635
Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series ============================================================================================================= The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed exactly according to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. The present paper validates that the simplifying assumptions can in fact be safely made. For general triangular arrays of block maxima attracted to the Fréchet distribution, consistency and asymptotic normality is established for the maximum likelihood estimator of the parameters of the limiting Fréchet distribution. The results are specialized to the common setting of block maxima extracted from a strictly stationary time series. The case where the underlying random variables are independent and identically distributed is further worked out in detail. The results are illustrated by theoretical examples and Monte Carlo simulations. Introduction ============ For the analysis of extreme values, two fundamental approaches can be distinguished. First, the peaks-over-threshold method consists of extracting those values from the observation period which exceed a high threshold. To model such threshold excesses, asymptotic theory suggests the use of the Generalized Pareto distribution. Second, the block maxima method consists of dividing the observation period into a sequence of non-overlapping intervals and restricting attention to the largest observation in each time interval. Thanks to the extremal types theorem, the probability distribution of such block maxima is approximately Generalized Extreme-Value (GEV), popularized by. The block maxima method is particularly common in environmental applications, since appropriate choices of the block size yield a simple but effective way to deal with seasonal patterns. For both methods, honest theoretical justifications must take into account two distinct features. First, the postulated models for either threshold excesses or block maxima arise from asymptotic theory and are not necessarily accurate at sub-asymptotic thresholds or at finite block lengths. Second, if the underlying data exhibit serial dependence, then the same will likely be true for the extreme values extracted from those data. How to deal with both issues is well-understood for the peaks-over-threshold method. The model approximation can be justified under a second-order condition (see, e.g., for a vast variety of applications), while serial dependence is taken care of in or, among others. Excesses over large thresholds often occur in clusters, and such serial dependence usually has an impact on the asymptotic variances of estimators based on these threshold excesses. Surprisingly, perhaps, is that for the block maxima method, no comparable analysis has yet been done. With the exception of some recent articles, which we will discuss in the next paragraph, the commonly used assumption is that the block maxima constitute an independent random sample from a GEV distribution. The heuristic justification for assuming independence over time, even for block maxima extracted from time series data, is that for large block sizes, the occurrence times of the consecutive block maxima are likely to be well separated. A more accurate framework is that of a triangular array of block maxima extracted from a sequence of random variables, the block size growing with the sample size. While shows consistency of the maximum likelihood estimator for the parameters of the GEV distribution, show both consistency and asymptotic normality of the probability weighted moment estimators. In both papers, however, the random variables from which the block maxima are extracted are supposed to be independent and identically distributed. In many situations, this assumption is clearly violated. To the best of our knowledge, is the only reference treating both the approximation error and the time series character, providing large-sample theory of nonparametric estimators of extreme-value copulas based on samples of componentwise block maxima extracted out of multivariate stationary time series. The aim of the paper is to show the consistency and asymptotic normality of the maximum likelihood estimator for more general sampling schemes, including the common situation of extracting block maxima from an underlying stationary time series. For technical reasons explained below, we restrict attention to the heavy-tailed case. The block maxima paradigm then suggests to use the two-parametric Fréchet distribution as a model for a sample of block maxima extracted from that time series. The first (quite general) main result, Theorem [theo:asydis], is that for triangular arrays of random variables whose empirical measures, upon rescaling, converge in an appropriate sense to a Fréchet distribution, the maximum likelihood estimator for the Fréchet parameters based on those variables is consistent and asymptotically normal. The theorem can be applied to the common set-up discussed above of block maxima extracted from an underlying time series, and the second main result, Theorem [theo:asydis:stat], shows that, in this case, the asymptotic variance matrix is the inverse of the Fisher information of the Fréchet family: asymptotically, it is as if the data were an independent random sample from the Fréchet attractor. In this sense, our theorem confirms the soundness of the common simplifying assumption that block maxima can be treated as if they were serially independent. Interestingly enough, the result allows for time series of which the strong mixing coefficients are not summable, allowing for some long range dependence scenarios. Restricting attention to the heavy-tailed case is done because of the non-standard nature of the three-parameter GEV distribution. The issue is that the support of a GEV distribution depends on its parameters. Even for the maximum likelihood estimator based on an independent random sample from a GEV distribution, asymptotic normality has not yet been established. The article usually cited in this context is, although no formal result is stated therein. Even the differentiability in quadratic mean of the three-parameter GEV is still to be proven; only shows differentiability in quadratic mean for the one-parameter GEV family (shape parameter only) at the Gumbel distribution. We feel that solving all issues simultaneously (irregularity of the GEV model, finite block size approximation error and serial dependence) is a far too ample program for one paper. For that reason, we focus on the analytically simpler Fréchet family, while thoroughly treating the triangular nature of the array of block maxima and the issue of serial dependence within the underlying time series. In a companion paper, we consider the maximum likelihood estimator in the general GEV-model based on independent and identically distributed random variables sampled directly from the GEV distribution. The main focus of that paper is devoted to resolving the considerable technical issues arising from the dependence of the GEV support on its parameters. We will build up the theory in three stages. First, we consider general triangular arrays of observations that asymptotically follow a Fréchet distribution in Section [sec:tri]. Second, we apply the theory to the set-up of block maxima extracted from a strictly stationary time series in Section [sec:stat]. Third, we further specialize the results to the special case of block maxima formed from independent and identically distributed random variables in Section [sec:iid]. This section can hence be regarded as a continuation of by reinforcing consistency to asymptotic normality, albeit for the Fréchet domain of attraction only. We work out an example and present finite-sample results from a simulation study in Section [sec:ex]. The main proofs are deferred to Appendix [sec:proofs], while some auxiliary results concerning the Fréchet distribution are mentioned in Appendix [sec:aux]. The proofs of the less central results are postponed to a supplement. Triangular arrays of block maxima ================================= In this section, we summarize results concerning the maximum likelihood estimator for the parameters of the Fréchet distribution: given a sample of observations which are not all tied, the Fréchet likelihood admits a unique maximum (Subsection [subsec:tri:existsUnique]). If the observations are based on a triangular array which is approximately Fréchet distributed in the sense that certain functionals admit a weak law of large numbers or a central limit theorem, the maximum likelihood estimator is consistent or asymptotically normal, respectively (Subsections [subsec:tri:consistency] and [subsec:tri:asydis]). Proofs are given in Subsection [subsec:proofs:tri]. Existence and uniqueness ------------------------ Let Pθ denote the two-parameter Fréchet distribution on (0, ∞) with parameter θ = (α, σ) ∈ (0, ∞)2 = Θ, defined through its cumulative distribution function Gθ(x) = exp{ − (x/σ)− α}, x > 0. Its probability density function is equal to $$p\_\theta(x) = \frac{\alpha}{\sigma} \exp \{ - (x/\sigma)^{-\alpha} \} \, (x/\sigma)^{-\alpha-1}, \quad x>0,$$ with log-likelihood function ℓθ(x) = log(α/σ) − (x/σ)− α − (α + 1)log(x/σ), x > 0, and score functions $\dot \ell\_\theta=(\dot \ell\_{\theta,1}, \dot \ell\_{\theta,2})^T$, with $$\begin{aligned} \label{eq:score:alpha} \dot \ell\_{\theta,1}(x) &= \partial\_\alpha \ell\_\theta(x) = \alpha^{-1} + \left( (x/\sigma)^{-\alpha} - 1 \right) \log(x/\sigma), \\ \label{eq:score:sigma} \dot \ell\_{\theta,2}(x) &=\partial\_\sigma \ell\_\theta(x) = \left( 1 - (x/\sigma)^{-\alpha} \right) \alpha/\sigma.\end{aligned}$$ Let $\bm{x} = (x\_1, \ldots, x\_k) \in (0, \infty)^k$ be a sample vector to which the Fréchet distribution is to be fitted. Consider the log-likelihood function $$\label{eq:loglik} L( \theta \mid \bm{x} ) = \sum\_{i=1}^k \ell\_\theta( x\_i ).$$ Further, define $$\begin{aligned} \label{eq:Psik} \Psi\_k( \alpha \mid \bm{x} ) &= \frac{1}{\alpha} + \frac{{\frac{1}{k}\sum\_{i=1}^k}x\_i^{-\alpha} \log( x\_i )}{{\frac{1}{k}\sum\_{i=1}^k}x\_i^{-\alpha}} - {\frac{1}{k}\sum\_{i=1}^k}\log(x\_i), \\ \label{eq:sigmahat} {\sigma}( \alpha \mid \bm{x} ) &= \left( \frac{1}{k} \sum\_{i=1}^k x\_i^{-\alpha} \right)^{-1/\alpha}.\end{aligned}$$ [Existence and uniqueness] [lem:existsUnique] If the scalars x1, …, xk ∈ (0, ∞) are not all equal (k ≥ 2), then there exists a unique maximizer $$\hat{\theta}( \bm{x} ) = \bigl( \hat{\alpha}( \bm{x} ),\hat \sigma(\bm x) \bigr) = {\operatornamewithlimits{\arg\max}}\_{\theta \in \Theta} L( \theta \mid \bm{x} ).$$ We have $\hat \sigma(\bm x) = \sigma( \hat{\alpha}(\bm{x}) \mid \bm{x} )$ while $\hat{\alpha}( \bm{x} )$ is the unique zero of the strictly decreasing function $\alpha \mapsto \Psi\_k( \alpha \mid \bm{x})$: $$\label{eq:estimEq} \Psi\_k \bigl( \hat{\alpha}( \bm{x} ) \mid \bm{x} \bigr) = 0.$$ It is easily verified that the estimating equation for α is scale invariant: for any c ∈ (0, ∞), we have $\Psi\_k( \alpha \mid c\bm{x} ) = \Psi\_k( \alpha \mid \bm{x} )$. As a consequence, the maximum likelihood estimator for the shape parameter is scale invariant: $$\hat{\alpha}( c\bm{x} ) = \hat{\alpha}(\bm{x}).$$ Moreover, the estimator for σ is a scale parameter in the sense that $$\hat \sigma(c\bm x) = {\sigma}( \hat{\alpha}(c\bm{x}) \mid c\bm{x} ) = c \, {\sigma}( \hat{\alpha}(\bm{x}) \mid \bm{x} ) = c \, \hat\sigma(\bm x).$$ Until now, the maximum likelihood estimator is defined only in case not all xi values are identical. For definiteness, if x1 = … = xk, define $\hat{\alpha}( \bm{x} ) = \infty$ and $\hat{\sigma}(\bm x ) = \min(x\_1, \ldots, x\_k) = x\_1$. Consistency ----------- We derive a general condition under which the maximum likelihood estimator for the parameters of the Fréchet distribution is consistent. The central result, Theorem [theo:consistency] below, shows that, apart from a not-all-tied condition, the only thing that is required for consistency is a weak law of large numbers for the functions appearing in the estimating equation for the shape parameter. Suppose that for each positive integer n, we are given a random vector $\bm{X}\_n = (X\_{n,1}, \ldots, X\_{n,k\_n})$ taking values in (0, ∞)kn, where kn ≥ 2 is a positive integer sequence such that kn → ∞ as n → ∞. One may think of Xn, i as being (approximately) Fréchet distributed with shape parameter α0 > 0 and scale parameter σn > 0. This statement is made precise in Condition [cond:LLN] below. On the event that the kn variables Xn, i are not all equal, Lemma [lem:existsUnique] allows us to define $$\label{eq:MLE:shape} \hat{\alpha}\_n = \hat{\alpha}( \bm{X}\_n ),$$ the unique zero of the function $0 < \alpha \mapsto \Psi\_{k\_n}( \alpha \mid \bm{X}\_n )$. Further, as in, put $$\label{eq:MLE:scale} \hat{\sigma}\_n = {\sigma}( \hat{\alpha}\_n \mid \bm{X}\_n ) = \left( \frac{1}{k\_n} \sum\_{i=1}^{k\_n} X\_{n,i}^{-\hat{\alpha}\_n} \right)^{-1/\hat{\alpha}\_n}.$$ For definiteness, put α̂n = ∞ and σ̂n = Xn, 1 on the event {Xn, 1 = … = Xn, kn}. Subsequently, we will assume that this event is asymptotically negligible: limn → ∞Pr(Xn, 1 = … = Xn, kn) = 0. We refer to (α̂n, σ̂n) as the maximum likelihood estimator. The fundamental condition guaranteeing consistency of the maximum likelihood estimator concerns the asymptotic behavior of sample averages of f(Xn, i/σn) for certain functions f. For 0 < α− < α+ < ∞, consider the function class F1(α−, α+) = {x ↦ logx} ∪ {x ↦ x− α : α− < α < α+} ∪ {x ↦ x− αlogx : α− < α < α+}, all functions being from (0, ∞) into R. Let the arrow ‘${\rightsquigarrow}$’ denote weak convergence. [cond:LLN] There exist 0 < α− < α0 < α+ < ∞ and a positive sequence (σn)n ∈ N such that, for all f ∈ F1(α−, α+), $$\label{eq:LLN} \frac{1}{k\_n} \sum\_{i=1}^{k\_n} f( X\_{n,i}/\sigma\_n ) {\rightsquigarrow}\int\_0^\infty f(x) \, p\_{\alpha\_0, 1}(x) \, {\mathrm{d}}x, \qquad n \to \infty.$$ [Consistency] [theo:consistency] Let $\bm{X}\_n = (X\_{n,1}, \ldots, X\_{n,k\_n})$ be a sequence of random vectors in (0, ∞)kn, where kn → ∞. Assume that Equation and Condition [cond:LLN] hold. On the complement of the event {Xn, 1 = … = Xn, kn}, the random vector (α̂n, σ̂n) is the unique maximizer of the log-likelihood (α, σ) ↦ L(α, σ ∣ Xn, 1, …, Xn, kn). Moreover, the maximum likelihood estimator is consistent in the sense that $$( \hat{\alpha}\_n, \hat{\sigma}\_n / \sigma\_n ) {\rightsquigarrow}(\alpha\_0, 1), \qquad n \to \infty.$$ Asymptotic distribution ----------------------- We formulate a general condition under which the estimation error of the maximum likelihood estimator for the Fréchet parameter vector converges weakly. The central result is Theorem [theo:asydis] below. For 0 < α− < α+ < ∞, recall the function class F1(α−, α+) in and define another one: F2(α−, α+) = F1(α−, α+) ∪ {x ↦ x− α(logx)2 : α− < α < α+}. Furthermore, fix α0 > 0 and consider the following triple of real-valued functions on (0, ∞): H = {f1, f2, f3} = {x ↦ x− α0log(x), x ↦ x− α0, x ↦ logx}. The following condition strengthens Condition [cond:LLN]. [cond:CLT] There exist α0 ∈ (0, ∞) and a positive sequence (σn)n ∈ N such that the following two statements hold: 1. There exist 0 < α− < α0 < α+ < ∞ such that Equation holds for all f ∈ F2(α−, α+). 2. There exists a sequence 0 < vn → ∞ and a random vector $\bm Y=(Y\_1, Y\_2, Y\_3)^T$ such that, denoting $$\label{eq:Gn} {\mathbb{G}\_n}f = v\_n \left( \frac{1}{k\_n} \sum\_{i=1}^{k\_n} f( X\_{n,i} / \sigma\_n ) - \int\_0^\infty f(x) \, p\_{\alpha\_0,1}(x) \, {\mathrm{d}}x \right),$$ we have, for fj as in, $$\label{eq:Y} ({\mathbb{G}\_n}f\_1, {\mathbb{G}\_n}f\_2, \, {\mathbb{G}\_n}f\_3)^T {\rightsquigarrow}\bm{Y} , \qquad n \to \infty.$$ Let Γ be the Euler gamma function and let γ = − Γʹ(1) = 0.5772… be the Euler–Mascheroni constant. Recall Γʺ(2) = (1 − γ)2 + π2/6 − 1. Define the matrix $$\label{eq:M} M(\alpha\_0) = \frac{6}{\pi^2} \begin{pmatrix} \alpha\_0^2 & \alpha\_0(1-\gamma) & - \alpha\_0^2 \\ \gamma-1 & -(\Gamma''(2) +1) / \alpha\_0 & 1-\gamma \end{pmatrix}, \qquad \alpha\_0 \in (0, \infty).$$ [Asymptotic distribution] [theo:asydis] Let $\bm{X}\_n = (X\_{n,1}, \ldots, X\_{n,k\_n})$ be a sequence of random vectors in (0, ∞)kn, where kn → ∞. Assume that Equation and Condition [cond:CLT] hold. As n → ∞, the maximum likelihood estimator (α̂n, σ̂n) satisfies $$\label{eq:joint:delta} \begin{pmatrix} v\_n ( \hat{\alpha}\_n - \alpha\_0 ) \\ v\_n \, ( \hat{\sigma}\_n / \sigma\_n - 1 ) \end{pmatrix} = M(\alpha\_0) \begin{pmatrix} {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) \\ {\mathbb{G}\_n}x^{-\alpha\_0} \\ {\mathbb{G}\_n}\log(x) \end{pmatrix} + o\_p(1) {\rightsquigarrow}M(\alpha\_0) \bm{Y},$$ where $\bm{Y} = (Y\_1, Y\_2, Y\_3)^T$ and M(α0) are given in Equations and , respectively. For block maxima extracted from a strongly mixing stationary time series, Condition [cond:CLT] with $v\_n = \sqrt{k\_n}$, where kn denotes the number of blocks, will be derived from the Lindeberg central limit theorem. In that case, the distribution of $\bm{Y}$ is trivariate Gaussian with some mean vector $\mu\_{\bm Y}$ (possibly different from 0, see Theorem [theo:asydis:stat] below for details) and covariance matrix $$\begin{aligned} \label{eq:Sigma} \Sigma\_{\bm Y} &= \frac{1}{\alpha\_0^2} \begin{pmatrix} 1-4\gamma+\gamma^2+\pi^2/3 & \alpha\_0(\gamma - 2) & \pi^2/6-\gamma \\ \alpha\_0(\gamma - 2) & \alpha\_0^2 & -\alpha\_0 \\ \pi^2/6-\gamma &-\alpha\_0 & \pi^2/6 \end{pmatrix}.\end{aligned}$$ According to Lemma [lem:cov] below, the right-hand side in coincides with the covariance matrix of the random vector $\bigl( X^{-\alpha\_0} \log(X), \, X^{-\alpha\_0}, \, \log(X) \bigr)^T$, where X is Fréchet distributed with parameter (α0, 1). From Lemma [lem:fisher], recall the inverse of the Fisher information matrix of the Fréchet family at (α, σ) = (α0, 1): $$\label{eq:fisher} I\_{(\alpha\_0,1)}^{-1} = \frac{6}{\pi^2} \begin{pmatrix} \alpha\_0^2 & (\gamma-1) \\ (\gamma-1) & \alpha\_0^{-2} \{ (1-\gamma)^2 +\pi^2/6 \} \end{pmatrix}.$$ [add:asydis] If $\bm Y$ is normally distributed with covariance matrix $\Sigma\_{\bm Y}$ as in, then the limit $M(\alpha\_0) \bm{Y}$ in Theorem [theo:asydis] is also normally distributed and its covariance matrix is equal to the inverse of the Fisher information matrix of the Fréchet family, $M(\alpha\_0) \, \Sigma\_{\bm Y} \, M(\alpha\_0)^T = I\_{(\alpha\_0, 1)}^{-1}$. Block maxima extracted from a stationary time series ==================================================== Let (ξt)t ∈ Z be a strictly stationary time series, that is, for any k ∈ N and τ, t1, …, tk ∈ Z, the distribution of (ξt1 + τ, …, ξtk + τ) is the same as the distribution of (ξt1, …, ξtk). For positive integer i and r, consider the block maximum Mr, i = max(ξ(i − 1)r + 1, …, ξir). Abbreviate Mr, 1 = Mr. The classical block maxima method consists of choosing a sufficiently large block size r and fitting an extreme-value distribution to the sample of block maxima Mr, 1, …, Mr, k. The likelihood is constructed under the simplifying assumption that the block maxima are independent. The present section shows consistency and asymptotic normality of this method in an appropriate asymptotic framework. For the block maxima distribution to approach its extreme-value limit, the block sizes must increase to infinity. Moreover, consistency can only be achieved when the number of blocks grows to infinity too. Hence, we consider a positive integer sequence rn, to be thought of as a sequence of block sizes. The number of disjoint blocks of size rn that fit into a sample of size n is equal to kn = ⌊n/rn⌋, where ⌊x⌋ denotes the integer part of a real number x. Assume that both rn → ∞ and kn → ∞ as n → ∞. The theory will be based on an application of Theorem [theo:asydis] to the sample of left-truncated block maxima Xn, i = Mrn, i ∨ c (i = 1, …, kn), for some positive constant c specified below. The estimators α̂n and σ̂n are thus the ones in and, respectively. The reason for the left-truncation is that otherwise, some of the block maxima could be zero or negative. Asymptotically, such left-truncation does not matter, since all maxima will simultaneously diverge to infinity in probability (Condition [cond:small] below). In Section [sec:iid] below, we will specialize things further to the case where the random variables ξt are independent. In particular, we will simplify the list of conditions given in this section. The basic assumption is that the distribution of rescaled block maxima is asymptotically Fréchet. The sequence of scaling constants should possess a minimal degree of regularity. The assumption is satisfied in case the stationary distribution of the series is in the Fréchet domain of attraction and the series possesses a positive extremal index; see Remark [rem:extrIndex] below. [Domain of attraction] [cond:DA] The time series (ξt)t ∈ Z is strictly stationary and there exists a sequence (σn)n ∈ N of positive numbers with σn → ∞ and a positive number α0 such that $$\label{eq:DA:stat} M\_n / \sigma\_n {\rightsquigarrow}{\operatorname{\text{Fr\'echet}}}( \alpha\_0, 1 ), \qquad n \to \infty.$$ Moreover, σmn/σn → 1 for any integer sequence (mn)n ∈ N such that mn/n → 1 as n → ∞. The domain-of-attraction condition implies that, for every scalar c, we have Pr[Mn ≤ c] = Pr[Mn/σn ≤ c/σn] → 0 as n → ∞. In words, the block maxima become unboundedly large as the sample size grows to infinity. Still, out of a sample of kn block maxima, the smallest of the maxima might still be small, especially when the number of blocks is large, or, equivalently, the block sizes are not large enough. The following condition prevents this from happening. [All block maxima diverge] [cond:small] For every c ∈ (0, ∞), we have limn → ∞Pr[min(Mrn, 1, …, Mrn, kn) ≤ c] = 0. To control the serial dependence within the time series, we require that the Rosenblatt mixing coefficients decay sufficiently fast: for positive integer ℓ, put $$\alpha(\ell) = \sup \big\{ {\left\lvert{ \Pr(A \cap B) - \Pr(A) \Pr(B) }\right\rvert} : A \in \sigma( \xi\_t : t \le 0 ), B \in \sigma( \xi\_t : t \ge \ell ) \big\},$$ where σ( ⋅ ) denotes the σ-field generated by its argument. [α-Mixing with rate] [cond:alpha] We have limℓ → ∞α(ℓ) = 0. Moreover, there exists ω > 0 such that kn1 + ω α(rn) → 0, n → ∞. Condition [cond:alpha] can be interpreted as requiring the block sizes rn to be sufficiently large. For instance, if α(ℓ) = O(ℓ− a) for some a > 0, then is satisfied as soon as rn is of larger order than n(1 + ɛ)/(1 + a) for some 0 < ɛ < a; in that case, one may choose ω = ɛ. Note that the exponent a is allowed to be smaller than one, in which case the sequence of mixing coefficients is not summable. In order to be able to integrate to the limit, we require an asymptotic bound on certain moments of the block maxima; more precisely, on negative power moments in the left tail and on logarithmic moments in the right tail. [Moments] [cond:moment] There exists some ν > 2/ω with ω from Condition [cond:alpha] such that $$\begin{aligned} \limsup\_{n \to \infty} {\mathbb{E}}\bigl[ g\_{{\nu}, \alpha\_0}\bigl( (M\_n \vee 1)/\sigma\_n \bigr)\bigr] < \infty, \label{eq:mom1}\end{aligned}$$ where $g\_{{\nu}, \alpha\_0}(x) = \{ x^{-\alpha\_0} {\mathds{1}}(x\le e) + \log (x) {\mathds{1}}(x>e) \}^{2+{\nu}}$. An elementary argument shows that if Condition [cond:moment] holds, then Mn ∨ 1 in the limsup may be replaced by Mn ∨ c, for arbitrary c > 0. Moreover, note that the limiting Fréchet distribution satisfies ∫0∞xβ pα0, 1(x) dx < ∞ if and only if β is less than α0. In some scenarios, e.g., for the iid case considered in Section [sec:iid] or for the moving maximum process considered in Section [subsec:movmax], it can be shown that the following sufficient condition for Condition [cond:moment] is true: $$\begin{aligned} \label{eq:momall} \limsup\_{n \to \infty} {\mathbb{E}}\bigl[ \bigl\{ (M\_n \vee c)/\sigma\_n \bigr\}^\beta \bigr] < \infty\end{aligned}$$ for all c > 0 and all β ∈ ( − ∞, α0). In that case, Condition [cond:moment] is easily satisfied for any ν > 0. By Condition [cond:small] and Lemma [lem:ties], the probability that all block maxima Mrn, 1, …, Mrn, kn are larger than some positive constant c and that they are not all equal tends to unity. On this event, we can study the maximum likelihood estimators (α̂n, σ̂n) for the parameters of the Fréchet distribution based on the sample of block maxima. Fix c ∈ (0, ∞) and put Xn, i = Mrn, i ∨ c. Let Gn be the empirical process associated to Xn, 1/σrn, …, Xn, kn/σrn as in with $v\_n = \sqrt{k\_n}$. The empirical process is not necessarily centered, which is why we need a handle on its expectation. [Bias] [cond:bias] There exists c ∈ (0, ∞) such that for every function f in H defined in, the following limit exists: $$\label{eq:bias} \lim\_{n \to \infty} \sqrt{k\_n} \left( {\mathbb{E}}\bigl[ f \bigl( (M\_{r\_n} \vee c) / \sigma\_{r\_n} \bigr)\bigr] - \int\_0^\infty f(x) \, p\_{\alpha\_0, 1}(x) \, {\mathrm{d}}x \right) = B(f).$$ [theo:asydis:stat] Suppose that Conditions [cond:DA] up to [cond:bias] are satisfied and fix c as in Condition [cond:bias]. Then, with probability tending to one, there exists a unique maximizer (α̂n, σ̂n) of the Fréchet log-likelihood based on the block maxima Mrn, 1, …, Mrn, kn, and we have, as n → ∞, $$\begin{aligned} \begin{pmatrix} \sqrt{k\_n} \, ( \hat{\alpha}\_n - \alpha\_0 ) \\ \sqrt{k\_n} \, ( \hat{\sigma}\_n / \sigma\_{r\_n} - 1 ) \end{pmatrix} &= M(\alpha\_0) \begin{pmatrix} {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) \\ {\mathbb{G}\_n}x^{-\alpha\_0} \\ {\mathbb{G}\_n}\log(x) \end{pmatrix} + o\_p(1) {\rightsquigarrow}\mathcal{N}\_2 \bigl( M(\alpha\_0) \, B, \; I\_{\alpha\_0,1}^{-1} \bigr).\end{aligned}$$ Here, M(α0) and Iα0, 1− 1 are defined in Equations and, respectively, while B = (B(f1), B(f2), B(f3))T, where B(f) is the limit in and where f1, f2, f3 are defined in. The proof of Theorem [theo:asydis:stat] is given in Subsection [subsec:proofs:stat]. The conditions imposed in Theorem [theo:asydis:stat] are rather high-level. In the setting of a sequence of independent and identically distributed random variables, they can be brought down to analytical conditions on the tail of the stationary distribution function (Theorem [theo:ml]). Moreover, all conditions will be worked out in a moving maximum model in Section [subsec:movmax]. Still, we admit that for more common time series models, such as linear time series with heavy-tailed innovations or solutions to stochastic recurrence equations, checking the conditions in Theorem [theo:asydis:stat] may not be an easy matter. Especially the bias Condition [cond:bias], which requires quite detailed knowledge on the distribution of the sample maximum, may be hard to verify. Even in the i.i.d. case, where the distribution of the sample maximum is known explicitly, checking Condition [cond:bias] occupies more than three pages in the proof of Theorem [theo:ml] below. Interestingly, the asymptotic covariance matrix in Theorem [theo:asydis:stat] is unaffected by serial dependence and the asymptotic standard deviation of $\sqrt{k\_n} ( \hat{\alpha}\_n - \alpha\_0 )$ is always equal to $(\sqrt{6}/\pi) \times \alpha\_0 \approx 0.7797 \times \alpha\_0$. The reason for this invariance is that even for time series, maxima over large disjoint blocks are asymptotically independent because of the strong mixing condition. [Domain-of-attraction condition for positive extremal index] [rem:extrIndex] Let F be the cumulative distribution function of ξ1. Assume that there exist 0 < an → ∞ and α0 ∈ (0, ∞) such that limn → ∞Fn(anx) = exp( − x− α0), x ∈ (0, ∞). Moreover, assume that the sequence (ξt)t ∈ Z has extremal index ϑ ∈ (0, 1] : If un → ∞ is such that Fn(un) converges, then Pr(Mn ≤ un) = Fnϑ(un) + o(1), n → ∞. Note that we assume that ϑ > 0. Putting σn = ϑ1/α0an we obtain that Condition [cond:DA] is satisfied: for every x ∈ (0, ∞), we have $$\begin{aligned} \Pr( M\_n / \sigma\_n \le x ) &= F^{n\vartheta}( \sigma\_n x ) + o(1) \to \exp \bigl( - \vartheta (\vartheta^{1/\alpha\_0} x)^{-\alpha\_0} \bigr) = \exp ( - x^{-\alpha\_0} ), \qquad n \to \infty.\end{aligned}$$ Block maxima extracted from an iid sample ========================================= We specialize Theorem [theo:asydis:stat] to the case where the random variables ξ1, ξ2, … are independent and identically distributed with common distribution function F. In this setting, fitting extreme-value distributions to block maxima is also considered in (consistency of the maximum likelihood estimator in the GEV-family with γ > − 1) and (asymptotic normality of the probability weighted moment estimator in the GEV-family with γ < 1/2). Assume that F is in the maximum domain of attraction of the Fréchet distribution with shape parameter α0 ∈ (0, ∞): there exists a positive scalar sequence (an)n ∈ N such that, for every x ∈ (0, ∞), Fn(anx) → e− x− α0, n → ∞. Because of serial independence, the conditions in Theorem [theo:asydis:stat] can be simplified considerably. In addition, the mean vector of the asymptotic bivariate normal distribution of the maximum likelihood estimator can be made explicit. Required is a second-order reinforcement of in conjunction with a growth restriction on the number of blocks. Equation is equivalent to regular variation of − logF at infinity with index − α0 : we have F(x) < 1 for all x ∈ R and $$\label{eq:RV} \lim\_{u \to \infty} \frac{- \log F(ux)}{ -\log F(u)} = x^{-\alpha\_0}, \qquad x \in (0, \infty).$$ The scaling constants in may be chosen as any sequence (an)n ∈ N that satisfies limn → ∞n { − logF(an)} = 1. Being constructed from the asymptotic inverse of a regularly varying function with non-zero index, the sequence (an)n ∈ N is itself regularly varying at infinity with index 1/α0. The following condition reinforces and thus from regular variation to second-order regular variation. With − logF replaced by 1 − F, it appears for instance in in the context of the asymptotic distribution of the Hill estimator. For τ ∈ R, define hτ : (0, ∞) → R by $$\label{eq:htau} h\_\tau(x) = \int\_1^x y^{\tau - 1} \, {\mathrm{d}}y = \begin{cases} \dfrac{x^\tau - 1}{\tau}, & \text{if $\tau \neq 0$,} \\[1ex] \log(x), & \text{if $\tau = 0$.} \end{cases}$$ [Second-Order Condition] [cond:secor] There exists α0 ∈ (0, ∞), ρ ∈ ( − ∞, 0], and a real function A on (0, ∞) of constant, non-zero sign such that limu → ∞A(u) = 0 and such that, for all x ∈ (0, ∞), $$\label{eq:SV:2} \lim\_{u \to \infty} \frac{1}{A(u)} \left( \frac{-\log F(ux)}{-\log F(u)} - x^{-\alpha\_0} \right) = x^{-\alpha\_0} \, h\_\rho(x).$$ The function A can be regarded as capturing the speed of convergence in. The form of the limit function in may seem unnecessarily specific, but actually, it is not, as explained in Remark [rem:secor] below. Let ψ = Γʹ/Γ denote the digamma function and recall the Euler–Mascheroni constant γ = − Γʹ(1) = 0.5772…. To express the asymptotic bias of the maximum likelihood estimators, we will employ the functions b1 and b2 defined by $$\label{eq:b1} b\_1(x) = \begin{cases} (1+x) \, \Gamma ( x ) \{ \gamma + \psi(1+x) \}, & \text{if $x > 0$,} \\ \dfrac{\pi^2}{6}, & \text{if $x = 0$,} \end{cases}$$ and $$\label{eq:b2} b\_2(x) = \begin{cases} -\dfrac{\pi^2}{6x} + (1+x) \, \Gamma ( x ) \{ \Gamma''(2) + \gamma + (\gamma-1) \, \psi(1 + x)\}, & \text{if $x > 0$,} \\ 0, & \text{if $x = 0$}. \end{cases}$$ See Figure [fig:asybias] for the graphs of these two functions. For (α0, ρ) ∈ (0, ∞) × ( − ∞, 0], define the bias function $$\label{eq:bias:iid} B(\alpha\_0, \rho) = - \frac{6}{\pi^2} \begin{pmatrix} b\_1({\left\lvert{\rho}\right\rvert}/\alpha\_0) \\ b\_2({\left\lvert{\rho}\right\rvert}/\alpha\_0) / \alpha\_0^2 \end{pmatrix}.$$ The proof of the following theorem is given in Section [subsec:proofs:iid]. [theo:ml] Let ξ1, ξ2, … be independent random variables with common distribution function F satisfiying Condition [cond:secor]. Let the block sizes rn be such that rn → ∞ and kn = ⌊n/rn⌋ → ∞ as n → ∞ and assume that $$\label{eq:ka} \lim\_{n\to \infty} \sqrt{k\_n} \, A(a\_{r\_n}) = \lambda \in {\mathbb{R}}.$$ Then, with probability tending to one, there exists a unique maximizer (α̂n, σ̂n) of the Fréchet log-likelihood based on the block maxima Mrn, 1, …, Mrn, kn, and we have $$\label{eq:ml} \sqrt{k\_n} \begin{pmatrix} \hat{\alpha}\_n - \alpha\_0 \\ \hat{\sigma}\_n / a\_{r\_n} - 1 \end{pmatrix} {\rightsquigarrow}{\mathcal{N}}\_2 \left( \lambda \, B(\alpha\_0, \rho), \, I\_{(\alpha\_0,1)}^{-1} \right), \qquad n \to \infty,$$ where I(α0, 1)− 1 denotes the inverse of the Fisher information of the Fréchet family as in and with B(α0, ρ) as in. We conclude this section with a series of remarks on the second-order Condition [cond:secor] and its link to the block-size condition in and the mean vector of the limiting distribution in . [fig:asybias] [Second-order regular variation] [rem:secor] Let F satisfy. For x > 0 sufficiently large such that F(x) > 0, define L(x) by $$\begin{aligned} \label{eq:defL} - \log F(x) = x^{-\alpha\_0} \, L(x).\end{aligned}$$ In view of, the function L is slowly varying at infinity, that is, $$\lim\_{u \to \infty} \frac{L(ux)}{L(u)} = 1, \qquad x \in (0, \infty).$$ A second-order refinement of this would be that there exist A : (0, ∞) → (0, ∞) and h : (0, ∞) → R, the latter not identically zero, such that limu → ∞A(u) = 0 and $$\lim\_{u \to \infty} \frac{1}{A(u)} \left( \frac{L(ux)}{L(u)} - 1 \right) = h(x), \qquad x \in (0, \infty).$$ Writing g(u) = A(u) L(u), Theorem B.2.1 in (see also, Section 3.6) implies that there exists ρ ∈ R such that g and thus A = g/L are regularly varying at infinity with index ρ. Since A vanishes at infinity, necessarily ρ ≤ 0. Furthermore, there exists κ ∈ R \ {0} such that h(x) = κ hρ(x) for x ∈ (0, ∞), with hρ as in. Incorporating the constant κ into the function A, we can assume without loss of generality that κ = 1 and we arrive at Condition [cond:secor]. The function A then possibly takes values in ( − ∞, 0) rather than in (0, ∞). [Asymptotic mean squared error] According to and, the distribution of the estimation error α̂n − α0 is approximately equal to $$\mathcal{N} \left( -A(a\_{r\_n}) \, \frac{6}{\pi^2} \, b\_1( {\left\lvert{\rho}\right\rvert}/\alpha\_0 ), \; \frac{r\_n}{n} \, \frac{6}{\pi^2} \, \alpha\_0^2 \right).$$ The asymptotic mean squared error is therefore equal to $$\operatorname{AMSE}( \hat{\alpha}\_n ) = \operatorname{ABias}^2( \hat{\alpha}\_n ) + \operatorname{AVar}( \hat{\alpha}\_n ) = {\left\lvert{A(a\_{r\_n})}\right\rvert}^2 \, \frac{36}{\pi^4} \, b\_1( {\left\lvert{\rho}\right\rvert}/\alpha\_0 )^2 + \frac{r\_n}{n} \frac{6}{\pi^2} \alpha\_0^2.$$ The choice of the block size rn (or, equivalently, the number of blocks kn), thus involves a bias–variance trade-off; see Section [sec:ex]. Alternatively, if ρ and A(arn) could be estimated, then one could construct bias-reduced estimators, just as in the case of the Hill estimator (see, e.g.,, among others) or probability weighted moment estimators. [On the number of blocks] A version of is used in to prove asymptotic normality of probability weighted moment estimators. Equation also implies the following limit relation, which is imposed in and which we will be needing later on as well: $$\label{eq:klogk} \lim\_{n \to \infty} \frac{k\_n \log(k\_n)}{n} = 0.$$ Indeed, in view of Remark [rem:secor] and regular variation of (an)n ∈ N, the sequence (∣A(ar)∣)r ∈ N is regularly varying at infinity. Potter’s theorem then implies that there exists β > 0 such that r− β = o(∣A(ar)∣) as r → ∞. But then $\sqrt{k\_n} (r\_n)^{-\beta} = \sqrt{k\_n} \, o(\|A(a\_{r\_n})\|) = o(1)$ by and thus kn1/2 + β/nβ = o(1) as n → ∞. We obtain that kn1 + 1/(2β)/n = o(1), which implies . [No asymptotic bias] If λ = 0 in, then the limiting normal distribution in is centered and the maximum likelihood estimator is said to be asymptotically unbiased. If the index, ρ, of regular variation of the auxiliary function \|A\| is strictly negative (see Remark [rem:secor]), then a sufficient condition for λ = 0 to occur is that kn = O(nβ) for some β < \|ρ\|/(α0/2 + \|ρ\|). Examples and finite-sample results ================================== Verification of conditions in a moving maximum model ---------------------------------------------------- For many stationary time series models, the distribution of the sample maximum is a difficult object to work with. This is true even for linear time series models, since the maximum operator is non-linear. In such cases, checking the conditions of Section [sec:stat] may be a hard or even impossible task. An exception occurs for moving maximum models, where the sample maximum can be linked directly to maxima of the innovation sequence. Let (Zt)t ∈ Z be a sequence of independent and identically distributed random variables with common distribution function F in the domain of attraction of the Fréchet distribution with shape parameter α0 > 0, that is, such that is satisfied for some sequence an → ∞. Let p ∈ N, p ≥ 2, be fixed and let b1, …, bp be nonnegative constants, b1 ≠ 0 ≠ bp, such that ∑i = 1pbi = 1. We consider the moving maximum process ξt of order p, defined by ξt = max{b1Zt, b2Zt − 1, …, bpZt − p + 1}, t ∈ Z. A simple calculation (see also the proof of Lemma [lem:movmax] for the stationary distribution of ξt) shows that the extremal index of (ξt)t ∈ Z is equal to $$\theta= \left\{ \textstyle \sum\_{i=1}^p b\_i^{\alpha\_0} \right\}^{-1} {b\_{(p)}^{\alpha\_0}},$$ where b(p) = maxi = 1pbi. Let σn = b(p)an. The proof of the following lemma is given in Section [subsec:proofs:ex] in the supplementary material. [lem:movmax] The stationary time series (ξt)t ∈ Z satisfies Conditions [cond:DA], [cond:alpha] and [cond:moment]. If additionally is met, then Condition [cond:small] is satisfied as well. Finally, if F satisfies the Second-Order Condition [cond:secor], if is met and if kn = o(n2/3) as n → ∞, then Condition [cond:bias] is also satisfied, with B(f) denoting the same limit as in the iid case, that is, $B(f) = \bm \beta$ with $\bm \beta$ as in. As a consequence, Theorem [theo:asydis:stat] may be applied and the asymptotic bias of the maximum likelihood estimator is the same as specified in Theorem [theo:ml] for the case of independent and identically distributed random variables. Simulation results ------------------ We report on the results of a simulation study, highlighting some interesting features regarding the finite-sample performance of the maximum likelihood estimator. Attention is restricted to the estimation of the shape parameter, and particular emphasis is given to a comparison with the common Hill estimator, which is based on the competing peaks-over-threshold method. Its variance is of the order O(k− 1), where k is the number of upper order statistics taken into account for its calculation. The Hill estimator’s asymptotic variance is given by α02, which is larger than the asymptotic variance (6/π2) × α02 of the block maxima maximum likelihood estimator. Furthermore, numerical experiments (not shown) involving the probability weighted moment estimator showed a variance that was higher, in all cases considered, than the one of the maximum likelihood estimator. We consider three time series models for (ξt)t ∈ Z: independent and identically distributed random variables, the moving maximum process from Section [subsec:movmax], and the absolute values of a GARCH(1,1) time series. In the first two models, three choices are considered for the distribution function F of either the variables ξt in the first model and the innovations Zt in the second model: absolute values of a Cauchy-distribution, the standard Pareto distribution and the Fréchet(1,1) distribution itself. All three distribution functions are attracted to the Fréchet distribution with α0 = 1. For the moving maximum process, we fix p = 4 and bj = j/10 for j ∈ {1, 2, 3, 4}. The GARCH(1,1) model is based on standard normal innovations, that is, ξt = \|Zt\|, where Zt is the stationary solution of the equations $$\label{eq:GARCH} \left\{ \begin{array}{rcl} Z\_t &=& \varepsilon\_t \sigma\_t, \\ \sigma\_t^2 &=& \lambda\_0 + \lambda\_1 Z\_{t-1}^2 + \lambda\_2 \sigma\_{t-1}^2, \end{array} \right.$$ with ɛt, t ∈ Z, independent standard normal random variables. The parameter vector (λ0, λ1, λ2) is set to either (0.5, 0.367, 0.367) or (0.5, 0.08, 0.91). By, the stationary distribution associated to any of these two models is attracted to the Fréchet distribution with shape parameter being (approximately) equal to α0 = 5. We generate samples from all of the afore-mentioned models for a fixed sample size of n = 1 000. Based on N = 3 000 Monte Carlo repetitions, we obtain empirical estimates of the finite sample bias, variance and mean squared error (MSE) of the competing estimators. The results are summarized in Figure [fig:mse] for the iid and the moving maxima model, and in Figure [fig:mse2] for the GARCH-model. Additional details for the case of independent random sampling from the absolute value of a Cauchy distribution are provided in the Supplement, Section [sec:simulextra]. In general, (most of) the graphs nicely reproduce the bias-variance tradeoff, its characteristic form however varying from model to model. Consider the iid scenario: since the Hill estimator is essentially the maximum likelihood estimator in the Pareto family, it is to be expected that it outperforms the block maxima estimator. On the other hand, by max-stability of the Fréchet family, the block maxima estimator should outperform the Hill estimator for that family. These expectations are confirmed by the simulation results in the left column of Figure [fig:mse]. For the Cauchy distribution, it turns out that the block maxima maximum likelihood estimator shows a better performance. Now, consider the moving maxima time series scenarios (right column in Figure [fig:mse]). Compared to the iid case, we observe an increase in the mean squared error (note that the scale on the axis of ordinates is row-wise identical). The block maxima method clearly outperforms the Hill estimator, except for the Pareto model. The big increase in relative performance is perhaps not too surprising, as the data points from a moving maximum process are already (weighted) maxima, which principally favors the block maxima method with small block sizes. Finally, consider the GARCH models in Figure [fig:mse2]. While, as in line with the theoretical findings, the variance of the block maxima estimator is smaller than the one of the Hill estimator, the squared bias turns out to be substantially higher for a large range of values for k. The MSE-optimal point is smaller for the Hill estimator. [fig:mse] [fig:mse2] Proofs ====== Proofs for Section [sec:tri] ---------------------------- [Proof of Lemma [lem:existsUnique]] The proof extends the development in Section 2 of. First, fix α > 0 and consider the function $0 < \sigma \mapsto L(\alpha, \sigma \mid \bm{x})$. By Equation , its derivative is equal to $$\partial\_\sigma L(\alpha, \sigma \mid \bm{x}) = \sum\_{i=1}^k \partial\_\sigma \ell\_\theta( x\_i ) = (\alpha/\sigma) \left(k - \sigma^\alpha \sum\_{i=1}^k x\_i^{-\alpha}\right).$$ We find that $\partial\_\sigma L(\alpha, \sigma \mid \bm{x})$ is positive, zero, or negative according to whether σ is smaller than, equal to, or larger than ${\sigma}( \alpha \mid \bm{x} )$, respectively. In particular, for fixed α, the expression $L(\alpha, \sigma \mid \bm{x})$ is maximal at σ equal to ${\sigma}( \alpha \mid \bm{x} )$. Hence we need to find the maximum of the function $0 < \alpha \mapsto L( \alpha, {\sigma}(\alpha \mid \bm{x}) \mid \bm{x} )$. By, its derivative is given by $$\begin{aligned} \frac{d}{d \alpha} L( \alpha, {\sigma}(\alpha \mid \bm{x}) \mid \bm{x} ) = \sum\_{i=1}^k \partial\_\alpha \ell\_{\alpha, \sigma}(x\_i) \bigg\|\_{\sigma = {\sigma}(\alpha \mid \bm{x})} + \sum\_{i=1}^k \partial\_\sigma \ell\_{\alpha, \sigma}(x\_i) \biggl\|\_{\sigma = {\sigma}(\alpha \mid \bm{x})} \times \frac{d}{d\alpha} {\sigma}( \alpha \mid \bm{x} ).\end{aligned}$$ The second sum is equal to zero, by definition of ${\sigma}( \alpha \mid \bm{x} )$. We obtain $$\frac{d}{d \alpha} L( \alpha, {\sigma}(\alpha \mid \bm{x}) \mid \bm{x} ) = k \, \Psi\_k( \alpha \mid \bm{x} ),$$ with Ψk as in. This is the same expression as Eq. (2.3) in, with their xi replaced by our xi− 1. Differentiating once more with respect to α, we obtain that $$\begin{aligned} \label{eq:ddLalpha} \frac{d^2}{d \alpha^2} L( \alpha, {\sigma}(\alpha \mid \bm{x}) \mid \bm{x} ) = - \frac{k}{\alpha^2} - k \frac{\sum\_{i=1}^k x\_i^{-\alpha} (\log(x\_i))^2 \sum\_{i=1}^k x\_i^{-\alpha} - \left( \sum\_{i=1}^k x\_i^{-\alpha} \log(x\_i) \right)^2}{\left( \sum\_{i=1}^k x\_i^{-\alpha} \right)^2}.\end{aligned}$$ By the Cauchy–Schwartz inequality, the numerator of the big fraction is nonnegative, whence $$\frac{d^2}{d \alpha^2} L( \alpha, {\sigma}(\alpha \mid \bm{x}) \mid \bm{x} ) \le - \frac{k}{\alpha^2} < 0.$$ Hence, $\alpha \mapsto \Psi\_k(\alpha\mid \bm x)$ is strictly decreasing. For α → 0, this function diverges to ∞, whereas for α → ∞, it converges to log(min(x1, …, xk)) − k− 1∑i = 1klog(xi), which is less than zero given the assumptions on x1, …, xk. Hence, there exists a unique $\hat{\alpha}( \bm{x} ) \in (0, \infty)$ such that this function is zero. We conclude that the function $\theta \mapsto L(\theta \mid \bm{x})$ admits a unique maximum at $\hat{\theta}( \bm{x} )$. Fix α0 ∈ (0, ∞). Let P denote the Fréchet distribution with parameter θ0 = (α0, 1), with support X = (0, ∞). The tentative limit of the functions $\alpha \mapsto \Psi\_k( \alpha \mid \bm{x} )$ is the function $$\Psi( \alpha ) = \frac{1}{\alpha} + \frac{\int\_0^\infty x^{-\alpha} \log(x) \, {\mathrm{d}}P(x)}{\int\_0^\infty x^{-\alpha} \, {\mathrm{d}}P(x)} - \int\_0^\infty \log(x)\, {\mathrm{d}}P(x).$$ Let Γ be the gamma function and let ψ = Γʹ/Γ be the digamma function. [lem:Psi] Fix α0 ∈ (0, ∞). We have $$\label{eq:Psi} \Psi(\alpha) = \frac{1}{\alpha\_0} \bigl( \psi(1) - \psi(\alpha/\alpha\_0) \bigr), \qquad \alpha \in (0, \infty).$$ As a consequence, Ψ : (0, ∞) → R is a decreasing bijection with a unique zero at α = α0. [Proof of Lemma [lem:Psi]] By Lemma [lem:moments], $$\begin{aligned} \Psi(\alpha) &= \frac{1}{\alpha} + \frac{(-\alpha\_0^{-1}) \Gamma'(1 + \alpha/\alpha\_0)}{\Gamma(1 + \alpha/\alpha\_0)} - (-\alpha\_0^{-1}) \Gamma'(1) = \frac{1}{\alpha\_0} \left( (\alpha/\alpha\_0)^{-1} - \psi(1 + \alpha/\alpha\_0) + \psi(1) \right).\end{aligned}$$ The digamma function satisfies the recurrence relation $\psi(x+1) = \psi(x) + \frac{1}{x}$. Equation follows. The final statement follows from the fact that the digamma function ψ : (0, ∞) → R is an increasing bijection. [Proof of Theorem [theo:consistency]] By Lemma [lem:existsUnique], we only have to show the claimed convergence. Define a random function Ψn on (0, ∞) by $$\label{eq:Psin} \Psi\_n( \alpha ) = \Psi\_{k\_n}( \alpha \mid \bm{X}\_n ) = \Psi\_{k\_n}( \alpha \mid \bm{X}\_n / \sigma\_n ),$$ with Ψk( ⋅ ∣ ⋅ ) as in. Recall Ψ in. The hypotheses imply that, for each α ∈ (α−, α+), $$\Psi\_n( \alpha ) {\rightsquigarrow}\Psi( \alpha ), \qquad n \to \infty.$$ By Lemma [lem:Psi], the limit Ψ(α) is positive, zero, or negative according to whether α is less than, equal to, or greater than α0. Moreover, the function Ψn is decreasing and Ψn(α̂n) = 0; see the proof of Lemma [lem:existsUnique]. Let δ > 0 be such that α− < α0 − δ < α0 + δ < α+. Since $\Psi\_n( \alpha\_0 - \delta ) {\rightsquigarrow}\Psi( \alpha\_0 - \delta ) > 0$ as n → ∞, we find $$\begin{aligned} \Pr[ \hat{\alpha}\_n \le \alpha\_0 - \delta ] \le \Pr[ \Psi\_n( \alpha\_0 - \delta ) \le 0 ] \to 0, \qquad n \to \infty.\end{aligned}$$ Similarly, Pr[α̂n ≥ α0 + δ] → 0 as n → ∞. We can choose δ > 0 arbitrarily small, thereby concluding that $\hat{\alpha}\_n {\rightsquigarrow}\alpha\_0$ as n → ∞. Second, Condition [cond:LLN] also implies that, for each α ∈ (α−, α+) and as n → ∞, $$\begin{aligned} \frac{1}{\sigma\_n} \left( \frac{1}{k\_n} \sum\_{i=1}^{k\_n} X\_{n,i}^{-\alpha} \right)^{-1/\alpha} &= \left( \frac{1}{k\_n} \sum\_{i=1}^{k\_n} (X\_{n,i}/\sigma\_n)^{-\alpha} \right)^{-1/\alpha} \\ &{\rightsquigarrow}\left( \int\_0^\infty x^{-\alpha} \, p\_{\alpha\_0,1}(x) \, {\mathrm{d}}x \right)^{-1/\alpha} = \bigl( \Gamma(1 + \alpha/\alpha\_0) \bigr)^{-1/\alpha},\end{aligned}$$ where we used Lemma [lem:moments] for the last identity. Both the left-hand and right-hand sides are continuous, nonincreasing functions of α. Since $\hat{\alpha}\_n {\rightsquigarrow}\alpha\_0$ as n → ∞ and since the right-hand side evaluates to unity at α = α0, a standard argument then yields $$\frac{\hat \sigma\_n}{\sigma\_n}=\frac{1}{\sigma\_n} \left( \frac{1}{k\_n} \sum\_{i=1}^{k\_n} X\_{n,i}^{-\hat{\alpha}\_n} \right)^{-1/\hat{\alpha}\_n} {\rightsquigarrow}1, \qquad n \to \infty. \qedhere$$ The proof of Theorem [theo:asydis] is decomposed into a sequence of lemmas. Recall Ψn and Ψ in and, respectively, and define $\dot{\Psi}\_n( \alpha ) = ({\mathrm{d}}/ {\mathrm{d}}\alpha) \Psi\_n( \alpha )$ and $\dot{\Psi}(\alpha) = ({\mathrm{d}}/ {\mathrm{d}}\alpha) \Psi(\alpha)$. By, $$\label{eq:dPsin} \dot{\Psi}\_n(\alpha) = - \frac{1}{\alpha^2} - \frac{{\mathbb{P}\_n}[x^{-\alpha} (\log x)^2] \, {\mathbb{P}\_n}x^{-\alpha} - ({\mathbb{P}\_n}x^{-\alpha} \log x)^2 }{({\mathbb{P}\_n}x^{-\alpha})^2},$$ where Pn denotes the empirical distribution of the points (Xn, i/σn)i = 1kn and where $${\mathbb{P}\_n}f = \frac{1}{k\_n} \sum\_{i=1}^{k\_n} f(X\_{n,i} / \sigma\_n).$$ The asymptotic distribution of $v\_n( \hat \alpha\_n - \alpha\_0)$ can be derived from the asymptotic behavior of $\dot \Psi\_n$ and vnΨn, which is the subject of the next two lemmas, respectively. [Slope] [lem:dPsin] Let $\bm{X}\_n = (X\_{n,1}, \ldots, X\_{n,k\_n})$ be a sequence of random vectors in (0, ∞)kn, where kn → ∞. Suppose that Equation and Condition [cond:CLT](i) are satisfied. If α̃n is a random sequence in (0, ∞) such that $\tilde{\alpha}\_n {\rightsquigarrow}\alpha\_0$ as n → ∞, then $$\dot{\Psi}\_n( \tilde{\alpha}\_n ) {\rightsquigarrow}\dot{\Psi}( \alpha\_0 ) = - \frac{\pi^2}{6 \alpha\_0^2}, \qquad n \to \infty.$$ For α ∈ (0, ∞) and m ∈ {0, 1, 2}, define fm, α(x) = x− α(logx)m, x ∈ (0, ∞), with (logx)0 = 1 for all x ∈ (0, ∞). Suppose that we could show that, for m ∈ {0, 1, 2} and some ɛ > 0, $$\label{eq:uniformly} \sup\_{\alpha : {\left\lvert{\alpha - \alpha\_0}\right\rvert} \le {\varepsilon}} {\left\lvert{ {\mathbb{P}\_n}f\_{m,\alpha} - \int\_0^\infty f\_{m,\alpha}(x) \, p\_{\alpha\_0}(x) \, {\mathrm{d}}x }\right\rvert} {\rightsquigarrow}0, \qquad n \to \infty.$$ Then from weak convergence of α̃n to α0, Slutsky’s lemma (, Lemma 2.8) and Lemma [lem:moments] below, it would follow that $$\dot{\Psi}\_n( \tilde{\alpha}\_n ) {\rightsquigarrow}- \frac{1}{\alpha\_0^2} - \frac{\alpha\_0^{-2} \Gamma''(2) \, \Gamma(2) - (\alpha\_0^{-1} \Gamma'(2))^2}{(\Gamma(2))^2}, \qquad n \to \infty.$$ Since Γ(2) = 1, Γʹ(2) = 1 − γ and Γʺ(2) = (1 − γ)2 + π2/6 − 1, the conclusion would follow. It remains to show. We consider the three cases m ∈ {0, 1, 2} separately. Let ɛ > 0 be small enough such that α− < α0 − ɛ < α0 + ɛ < α+. First, let m = 0. The maps α ↦ (Pnf0, α)1/α and α ↦ (∫0∞f0, α pα0, 1)1/α are monotone by Lyapounov’s inequality [i.e., ∥f∥r ≤ ∥f∥s for 0 < r < s, where ∥f∥r = (∫X∣f∣r dμ)1/r denotes the Lr-norm of some real-valued function f on a measurable space (X, μ)], and the second one is also continuous by Lemma [lem:moments]. Pointwise convergence of monotone functions to a monotone, continuous limit implies locally uniform convergence. This property easily extends to weak convergence, provided the limit is nonrandom. We obtain $$\sup\_{\alpha : {\left\lvert{\alpha - \alpha\_0}\right\rvert} \le {\varepsilon}} {\left\lvert{ ({\mathbb{P}\_n}f\_{0,\alpha})^{1/\alpha} - \left(\int\_0^\infty f\_{0,\alpha}(x) \, p\_{\alpha\_0}(x) \, {\mathrm{d}}x\right)^{1/\alpha} }\right\rvert} {\rightsquigarrow}0, \qquad n \to \infty.$$ Uniform continuity of the map (y, α) ↦ yα on compact subsets of (0, ∞)2 then yields for m = 0. Second, let m = 1. The maps α ↦ Pnf1, α and α ↦ ∫0∞f1, α pα0, 1 are continuous and nonincreasing (their derivatives are nonpositive). Pointwise weak convergence at each α ∈ (α−, α+) then yields for m = 1. Finally, let m = 2. With probability tending to one, not all variables Xn, i are equal to σn, and thus Pn(logx)2 > 0. On the latter event, we have $${\mathbb{P}\_n}x^{-\alpha} (\log x)^2 = {\mathbb{P}\_n}(\log x)^2 \, \left\{ \left( \frac{{\mathbb{P}\_n}x^{-\alpha} (\log x)^2}{{\mathbb{P}\_n}(\log x)^2} \right)^{1/\alpha} \right\}^\alpha.$$ By Lyapounov’s inequality, the expression in curly braces is nondecreasing in α. For each α ∈ (α−, α+), it converges weakly to {Γʺ(1 + α/α0)/Γʺ(1)}1/α, which is nondecreasing and continuous in α; see Lemma [lem:moments]. It follows that $$\sup\_{\alpha : {\left\lvert{\alpha - \alpha\_0}\right\rvert} \le {\varepsilon}} {\left\lvert{ \left( \frac{{\mathbb{P}\_n}x^{-\alpha} (\log x)^2}{{\mathbb{P}\_n}(\log x)^2} \right)^{1/\alpha} - \left( \frac{\Gamma''(1 + \alpha/\alpha\_0)}{\Gamma''(1)} \right)^{1/\alpha} }\right\rvert} {\rightsquigarrow}0, \qquad n \to \infty.$$ Equation for m = 2 follows. [lem:Psinweak2] Assume Condition [cond:CLT]. Then, as n → ∞, $$\begin{aligned} \label{eq:Psinweak2} v\_n \, \Psi\_n( \alpha\_0 ) = {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) + \frac{1-\gamma}{\alpha\_0} \, {\mathbb{G}\_n}x^{-\alpha\_0} - {\mathbb{G}\_n}\log(x) + o\_p(1).\end{aligned}$$ The expression on the right converges weakly to $Y \equiv Y\_1 + \frac{1-\gamma}{\alpha\_0} Y\_2 - Y\_3$. Recall that $$\Psi\_n( \alpha\_0 ) = \Psi\_{k\_n}( \alpha\_0 \mid \bm{X}\_n / \sigma\_n ) = \frac{1}{\alpha\_0} + \frac{{\mathbb{P}\_n}x^{-\alpha\_0} \log(x)}{{\mathbb{P}\_n}x^{-\alpha\_0}} - {\mathbb{P}\_n}\log(x).$$ Define ϕ : R × (0, ∞) × R → R by $$\phi(y\_1, y\_2, y\_3) = \frac{1}{\alpha\_0} + \frac{y\_1}{y\_2} - y\_3.$$ The previous two displays allow us to write $$\Psi\_n( \alpha\_0 ) = \phi \bigl( {\mathbb{P}\_n}x^{-\alpha\_0} \log(x), \, {\mathbb{P}\_n}x^{-\alpha\_0}, \, {\mathbb{P}\_n}\log(x) \bigr).$$ Recall Lemma [lem:moments] and put $$\begin{aligned} \bm{y}\_0 &= \bigl( - \alpha\_0^{-1} \Gamma'(2), \Gamma(2), - \alpha\_0^{-1} \Gamma'(1) \bigr) = \bigl( \alpha\_0^{-1} (\gamma - 1), \, 1, \, \alpha\_0^{-1} \gamma \bigr).\end{aligned}$$ As already noted in the proof of Lemma [lem:Psi], we have $ \phi ( \bm{y}\_0 ) = \alpha\_0^{-1} + \alpha\_0^{-1} (\gamma-1) - \alpha\_0^{-1} \gamma = 0. $ As a consequence, $$v\_n \, \Psi\_n( \alpha\_0 ) = v\_n \bigl\{ \phi \bigl( {\mathbb{P}\_n}x^{-\alpha\_0} \log(x), \, {\mathbb{P}\_n}x^{-\alpha\_0}, \, {\mathbb{P}\_n}\log(x) \bigr) - \phi ( \bm{y}\_0 ) \bigr\}.$$ In view of Condition [cond:CLT] and the delta method, as n → ∞, $$v\_n \, \Psi\_n( \alpha\_0 ) = \dot{\phi}\_1( \bm{y}\_0 ) \, {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) + \dot{\phi}\_2( \bm{y}\_0 ) \, {\mathbb{G}\_n}x^{-\alpha\_0} + \dot{\phi}\_3( \bm{y}\_0 ) \, {\mathbb{G}\_n}\log(x) + o\_p(1),$$ where $\dot{\phi}\_j$ denotes the first-order partial derivative of ϕ with respect to yj for j ∈ {1, 2, 3}. Elementary calculations yield $$\begin{aligned} \dot{\phi}\_1( \bm{y}\_0 ) &= 1, & \dot{\phi}\_2( \bm{y}\_0 ) &= \alpha\_0^{-1}(1 - \gamma), & \dot{\phi}\_3( \bm{y}\_0 ) &= -1.\end{aligned}$$ The conclusion follows by Slutsky’s lemma. [Asymptotic expansion for the shape parameter] [prop:shape] Assume that the conditions of Theorem [theo:asydis] hold. Then, with Y as defined in Lemma [lem:Psinweak2], $$\label{eq:shape:delta} v\_n \, \bigl( \hat{\alpha}\_n - \alpha\_0 \bigr) = \frac{6 \alpha\_0^2}{\pi^2} \, v\_n \, \Psi\_n( \alpha\_0 ) + o\_p(1) {\rightsquigarrow}\frac{6 \alpha\_0^2}{\pi^2} \, Y, \qquad n \to \infty.$$ Recall that, with probability tending to one, α̂n is the unique zero of the random function α ↦ Ψn(α). Recall that $\dot{\Psi}\_n$ in is the derivative of Ψn. With probability tending to one, we have, by virtue of the mean-value theorem, $$\begin{aligned} 0 = \Psi\_n( \hat{\alpha}\_n ) &= \bigl( \Psi\_n( \hat{\alpha}\_n ) - \Psi\_n( \alpha\_0 ) \bigr) + \Psi\_n( \alpha\_0 ) = ( \hat{\alpha}\_n - \alpha\_0 ) \, \dot{\Psi}\_n( \tilde{\alpha}\_n ) + \Psi\_n( \alpha\_0 );\end{aligned}$$ here α̃n is a convex combination of α̂n and α0. Since $\dot{\Psi}\_n( \alpha ) \le -1/\alpha^2 < 0$ (argument as in the proof of Lemma [lem:existsUnique]), we can write $$v\_n \, \bigl( \hat{\alpha}\_n - \alpha\_0 \bigr) = - \frac{1}{\dot{\Psi}\_n( \tilde{\alpha}\_n )} \, v\_n \, \Psi\_n( \alpha\_0 ).$$ By weak consistency of α̂n, we have $\tilde{\alpha}\_n {\rightsquigarrow}\alpha\_0$ as n → ∞. Lemma [lem:dPsin] then gives $\dot{\Psi}\_n( \tilde{\alpha}\_n ) {\rightsquigarrow}- \pi^2 / (6 \alpha\_0^2)$ as n → ∞. Apply Lemma [lem:Psinweak2] and Slutsky’s lemma to conclude. [Proof of Theorem [theo:asydis] and Addendum [add:asydis]] Combining Equations and yields $$v\_n \, ( \hat{\alpha}\_n - \alpha\_0 ) = \frac{6 \alpha\_0^2}{\pi^2} \left( {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) + \frac{1-\gamma}{\alpha\_0} \, {\mathbb{G}\_n}x^{-\alpha\_0} - {\mathbb{G}\_n}\log(x) \right) + o\_p(1)$$ as n → ∞. This yields the first row in. By definition of σ̂n, we have (σ̂n/σn)− α̂n = Pnx− α̂n. Consider the decomposition $$\label{eq:scale:decomp} v\_n \, \bigl( ( \hat{\sigma}\_n / \sigma\_n )^{-\hat{\alpha}\_n} - 1 \bigr) = v\_n \, \bigl( {\mathbb{P}\_n}x^{-\hat{\alpha}\_n} - {\mathbb{P}\_n}x^{-\alpha\_0} \bigr) + v\_n \, \bigl( {\mathbb{P}\_n}x^{-\alpha\_0} - 1 \bigr).$$ By the mean value theorem, there exists a convex combination, α̃n, of α̂n and α0 such that Pnx− α̂n − Pnx− α0 = − (α̂n − α0) Pnx− α̃nlog(x). By the argument for the case m = 1 in the proof of Lemma [lem:dPsin], we have $${\mathbb{P}\_n}x^{-\tilde{\alpha}\_n} \log(x) {\rightsquigarrow}- \frac{1}{\alpha\_0} \Gamma'(2) = - \frac{1-\gamma}{\alpha\_0}, \qquad n \to \infty.$$ By Proposition [prop:shape] and Lemma [lem:Psinweak2], it follows that, as n → ∞, $$\begin{aligned} v\_n \, \bigl( {\mathbb{P}\_n}x^{-\hat{\alpha}\_n} - {\mathbb{P}\_n}x^{-\alpha\_0} \bigr) &= v\_n \, ( \hat{\alpha}\_n - \alpha\_0 ) \, \frac{1-\gamma}{\alpha\_0} + o\_p(1) \\ &= \frac{6 \alpha\_0 \, (1-\gamma)}{\pi^2} \, v\_n \, \Psi\_n( \alpha\_0 ) + o\_p(1) \\ &= \frac{6 \alpha\_0 \, (1-\gamma)}{\pi^2} \, \left( {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) + \frac{1-\gamma}{\alpha\_0} \, {\mathbb{G}\_n}x^{-\alpha\_0} - {\mathbb{G}\_n}\log(x) \right) + o\_p(1).\end{aligned}$$ This expression in combination with yields, as n → ∞, $$\begin{gathered} \label{eq:scale:aux} v\_n \, \bigl( ( \hat{\sigma}\_n / \sigma\_n )^{-\hat{\alpha}\_n} - 1 \bigr) \\ = \frac{6 \alpha\_0 \, (1-\gamma)}{\pi^2} \, \left( {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) + \frac{1-\gamma}{\alpha\_0} \, {\mathbb{G}\_n}x^{-\alpha\_0} - {\mathbb{G}\_n}\log(x) \right) + {\mathbb{G}\_n}x^{-\alpha\_0} + o\_p(1).\end{gathered}$$ Write Zn = (σ̂n/σn)− α̂n, which converges weakly to 1 as n → ∞. By the mean value theorem, $$\begin{aligned} v\_n \, ( \hat{\sigma}\_n / \sigma\_n - 1 ) &= v\_n \, (Z\_n^{-1/\hat{\alpha}\_n} - 1) = v\_n \, (Z\_n - 1) \, (-1/\hat{\alpha}\_n) \, \tilde{Z}\_n^{-1/\hat{\alpha}\_n - 1},\end{aligned}$$ where Z̃n is a random convex combination of Zn and 1. But then $\tilde{Z}\_n {\rightsquigarrow}1$ as n → ∞, whence, by consistency of α̂n and Slutsky’s lemma, vn (σ̂n/σn − 1) = ( − 1/α0) vn ((σ̂n/σn)− α̂n − 1) + op(1), n → ∞. Combinining this with, we find $$\begin{gathered} v\_n \, ( \hat{\sigma}\_n / \sigma\_n - 1 ) = - \frac{6 (1-\gamma)}{\pi^2} \, \left( {\mathbb{G}\_n}x^{-\alpha\_0} \log(x) + \frac{1-\gamma}{\alpha\_0} \, {\mathbb{G}\_n}x^{-\alpha\_0} - {\mathbb{G}\_n}\log(x) \right) \\ - \alpha\_0^{-1} {\mathbb{G}\_n}x^{-\alpha\_0} + o\_p(1)\end{gathered}$$ as n → ∞. This is the second row in. The proof of Addendum [add:asydis] follows from a tedious but straightforward calculation. Proofs for Section [sec:stat] ----------------------------- [Block maxima rarely show ties] [lem:ties] Under Conditions [cond:DA] and [cond:alpha], for every c ∈ (0, ∞), we have Pr[Mrn, 1 ∨ c = Mrn, 3 ∨ c] → 0 as n → ∞. [Proof of Lemma [lem:ties]] By the domain-of-attraction condition combined with the strong mixing property, the sequence of random vectors ((Mrn, 1 ∨ c)/σrn, (Mrn, 3 ∨ c)/σrn) converges weakly to the product of two independent ${\operatorname{\text{Fr\'echet}}}(\alpha\_0, 1)$ random variables. Apply the Portmanteau lemma – the set {(x, y) ∈ R2 : x = y} is closed and has zero probability in the limit. [Moments of block maxima converge] [lem:moment] Under Conditions [cond:DA] and [cond:moment], we have, for every c ∈ (0, ∞), $$\lim\_{n \to \infty} {\mathbb{E}}[ f \bigl( ( M\_n \vee c) / \sigma\_n \bigr) ] = \int\_0^\infty f(x) \, p\_{\alpha\_0,1}(x) \, {\mathrm{d}}x$$ for every measurable function f : (0, ∞) → R which is continuous almost everywhere and for which there exist 0 < η < ν such that ${\left\lvert{ f(x) }\right\rvert} \le g\_{\eta, \alpha\_0}(x) = \{ x^{-\alpha\_0} {\mathds{1}}(x\le e) +\log (x) {\mathds{1}}(x>e)\}^{2+\eta}$. [Proof of Lemma [lem:moment]] An elementary argument shows that we may replace Mn ∨ 1 by Mn ∨ c in. Since c/σn → 0 as n → ∞, the sequence (Mn ∨ c)/σn converges weakly to the ${\operatorname{\text{Fr\'echet}}}(\alpha\_0, 1)$ distribution in view of Condition [cond:DA]. The result follows from Example 2.21 in. In order to separate maxima over consecutive blocks by a time lag of at least ℓ, we clip off the final ℓ − 1 variables within each block: Mr, i[ℓ] = max{ξt : (i − 1)r + 1 ≤ t ≤ ir − ℓ + 1}. Clearly, Mr, i ≥ Mr, i[ℓ]. The probability that the maximum over a block of size r is attained by any of the final ℓ − 1 variables should be small; see Lemma [lem:clipping:2] below. [Short blocks are small] [lem:clipping] Assume Condition [cond:DA]. If ℓn = o(rn) and if α(ℓn) = o(ℓn/rn) as n → ∞, then for all ɛ > 0, Pr[Mℓn ≥ ɛσrn] = O(ℓn/rn), n → ∞. [Proof of Lemma [lem:clipping]] Let Fr be the cumulative distribution function of Mr. By, for every u > 0, Pr[Frn(Mℓn) ≥ u] = O(ℓn/rn), n → ∞. Fix ɛ > 0. By assumption, limn → ∞Frn(ɛσrn) = exp( − ɛ− α0). For sufficiently large n, we have $$\begin{aligned} \Pr[ M\_{\ell\_n} \ge {\varepsilon}\sigma\_n ] &\le \Pr[ F\_{r\_n}( M\_{\ell\_n} ) \ge F\_{r\_n} ({\varepsilon}\sigma\_n ) ] \le \Pr[ F\_{r\_n}( M\_{\ell\_n} ) \ge \exp( - {\varepsilon}^{-\alpha\_0} ) / 2 ].\end{aligned}$$ Set u = exp( − ɛ− α0)/2 in to arrive at. [Clipping doesn’t hurt] [lem:clipping:2] Assume Condition [cond:DA]. If ℓn = o(rn) and if α(ℓn) = o(ℓn/rn) as n → ∞, then Pr[Mrn > Mrn − ℓn] → 0, n → ∞. [Proof of Lemma [lem:clipping:2]] Recall Lemma [lem:clipping]. For every ɛ > 0 we have, by stationarity, Pr[Mrn > Mrn − ℓn] ≤ Pr[Mrn − ℓn ≤ ɛσrn] + Pr[Mℓn > ɛσrn]. Since σrn − ℓn/σrn → 1 as a consequence of Condition [cond:DA] and the fact that ℓn = o(rn) as n → ∞, the first term converges to exp( − ɛ− α0) as n → ∞, whereas the second one converges to 0 by Lemma [lem:clipping]. Since ɛ > 0 was arbitrary, Equation follows. [Proof of Theorem [theo:asydis:stat]] We apply Theorem [theo:asydis] and Addendum [add:asydis] to the array Xn, i = Mrn, i ∨ c and $v\_n = \sqrt{k\_n}$, where c ∈ (0, ∞) is arbitrary and i ∈ {1, …, kn}. By Condition [cond:small], we have limn → ∞Pr[∀i = 1, …, kn : Xn, i = Mrn, i] = 1. The not-all-tied property has been established in Lemma [lem:ties]. We need to check Condition [cond:CLT], and in particular that the distribution of the random vector $\bm{Y}$ in is $\mathcal{N}\_3( B, \Sigma\_{\bm{Y}} )$ with B as in the statement of Theorem [theo:asydis:stat] and $\Sigma\_{\bm{Y}}$ as in. Essentially, the proof employs the Bernstein big-block-small-block method in combination with the Lindeberg central limit theorem. Let $\ell\_n= \max\{ s\_n, {\lfloor r\_n \sqrt{ \alpha(s\_n)} \rfloor} \}$, where $s\_n= {\lfloor \sqrt{r\_n} \rfloor}$. Clearly, $$\begin{aligned} \label{eq:mixing:ellr} \ell\_n \to \infty, \quad \ell\_n=o(r\_n) \quad \text{and} \quad \alpha(\ell\_n) = o(\ell\_n/r\_n), \text{ as } n \to \infty. \end{aligned}$$ Consider the truncated and rescaled block maxima $$\begin{aligned} Z\_{r,i} &= (M\_{r\_n,i} \vee c) / \sigma\_r, \qquad Z\_{r,i}^{[\ell\_n]} = (M\_{r\_n,i}^{[\ell\_n]} \vee c) / \sigma\_r,\end{aligned}$$ with Mr, i[ℓn] as in. Consider the following empirical and population probability measures: $$\begin{aligned} {\mathbb{P}\_n}f &= \frac{1}{k\_n} \sum\_{i=1}^{k\_n} f(Z\_{r\_n,i}), & P\_n f &= {\mathbb{E}}[f(Z\_{r\_n,i})], \\ {\mathbb{P}\_n}^{[\ell\_n]} f &= \frac{1}{k\_n} \sum\_{i=1}^{k\_n} f(Z\_{r\_n,i}^{[\ell\_n]}), & P\_n^{[\ell\_n]} f &= {\mathbb{E}}[f(Z\_{r\_n,i}^{[\ell\_n]})].\end{aligned}$$ Abbreviate the tentative limit distribution by $P = {\operatorname{\text{Fr\'echet}}}(\alpha\_0, 1)$. We will also need the following empirical processes: $$\begin{aligned} {\mathbb{G}\_n}&= \sqrt{k\_n} ( {\mathbb{P}\_n}- P ) && \text{(uncentered)}, \\ \tilde{{\mathbb{G}}}\_n &= \sqrt{k\_n} ( {\mathbb{P}\_n}- P\_n ) && \text{(centered)}, \\ \tilde{{\mathbb{G}}}\_n^{[\ell\_n]} &= \sqrt{k\_n} ( {\mathbb{P}\_n}^{[\ell\_n]} - P\_n^{[\ell\_n]} ) && \text{(centered)}.\end{aligned}$$ Finally, the bias arising from the finite block size is quantified by the operator $$B\_n = \sqrt{k\_n} (P\_n - P).$$ Proof of Condition [cond:CLT](i). Choose η ∈ (2/ω, ν) and 0 < α− < α0 < α+. Additional constraints on α+ will be imposed below, while the values of η and α− do not matter. Recall the function class F2(α−, α+) in . For every f ∈ F2(α−, α+), we just need to show that Pnf = Pf + op(1), n → ∞. The domain-of-attraction property (Condition [cond:DA]) and the asymptotic moment bound (Condition [cond:moment]) yield E[Pnf] = Pnf → Pf, n → ∞, by uniform integrability, see Lemma [lem:moment] (note that ∣f∣ is bounded by a multiple of g0, α0 if α+ is chosen suitably small: α+ < 2α0 must be satisfied). Further, $${\mathbb{P}\_n}f - P\_n f = \frac{1}{\sqrt{k\_n}} \tilde{{\mathbb{G}}}\_n f.$$ Below, see, we will show that $$\label{eq:tildeGnf} \tilde{{\mathbb{G}}}\_n f = \tilde{{\mathbb{G}}}\_n^{[\ell\_n]} f + o\_p(1) = O\_p(1) + o\_p(1) = O\_p(1), \qquad n \to \infty.$$ It follows that, as required, Pnf = (Pnf − Pnf) + Pnf = op(1) + Pf + o(1) = Pf + op(1), n → ∞. Proof of Condition [cond:CLT](ii). We can decompose the empirical process Gn in a stochastic term and a bias term: $$\begin{aligned} {\mathbb{G}\_n}&= \sqrt{k\_n} ( {\mathbb{P}\_n}- P\_n ) + \sqrt{k\_n} ( P\_n - P ) = \tilde{{\mathbb{G}}}\_n + B\_n.\end{aligned}$$ For f ∈ H = {f1, f2, f3}, the bias term Bnf converges to B(f) thanks to Condition [cond:bias]. It remains to treat the stochastic term $\tilde{{\mathbb{G}}}\_n f$, for all f ∈ F2(α−, α+) [in view of the proof of item (i); see above]. We will show that the finite-dimensional distributions of $\tilde{{\mathbb{G}}}\_n$ converge to those of a P-Brownian bridge, G, i.e., a zero-mean, Gaussian stochastic process with covariance function given by $$\operatorname{cov}( {\mathbb{G}}f, {\mathbb{G}}g ) = P \bigl( (f - Pf) (g - Pg) \bigr) = \operatorname{cov}\_P \bigl( f(X), g(X) \bigr), \qquad f, g \in \mathcal{F}\_2 ( \alpha\_-, \alpha\_+ ).$$ Decompose the stochastic term in two parts: $$\label{eq:Delta} \tilde{{\mathbb{G}}}\_n = \tilde{{\mathbb{G}}}\_n^{[\ell\_n]} + \Delta\_n.$$ We will show that Δn converges to zero in probability and that the finite-dimensional distributions of $\tilde{{\mathbb{G}}}\_n^{\scriptscriptstyle [\ell\_n]}$ converge to those of G. First, we treat the main term, $\tilde{{\mathbb{G}}}\_n^{\scriptscriptstyle [\ell\_n]}$. By the Cramér–Wold device, it suffices to show that $\tilde{{\mathbb{G}}}\_n^{\scriptscriptstyle [\ell\_n]} g {\rightsquigarrow}{\mathbb{G}}g$ as n → ∞, where g is an arbitrary linear combination of functions f ∈ F2(α−, α+). Define $$\phi\_{ni}(t) = \exp\big[ -{\mathrm{i}}t k\_n^{-1/2} \{ g(Z\_{r\_n,i}^{[\ell\_n]}) - P\_n^{[\ell\_n]} g \} \big],$$ with i the imaginary unit. Note that the characteristic function of $\tilde{{\mathbb{G}}}\_n^{\scriptscriptstyle [\ell\_n]} g$ can be written as t ↦ E[∏i = 1knϕni(t)]. Successively applying Lemma 3.9 in, we obtain that ∣E[∏i = 1knϕni(t)] − ∏i = 1knE[ϕni(t)]∣ ≤ 2πknmaxi = 1knα(σ{ϕni(t)}, σ{∏j = i + 1knϕnj(t)}), where α(A1, A2) denotes the alpha-mixing coefficient between the sigma-fields A1 and A2. Since the maxima $Z\_{r,i}^{\scriptscriptstyle[\ell\_n]}$ over different blocks are based on observations that are at least ℓn observations apart, the expression on the right-hand side of the last display is of the order O(knα(ℓn)), which converges to 0 as a consequence of Equation . We can conclude that the weak limit of $\tilde{{\mathbb{G}}}\_n^{\scriptscriptstyle [\ell\_n]} g$ is the same as the one of $$\tilde {{\mathbb{H}}}\_n^{ [\ell\_n]} g = \sqrt{k\_n}\left\{ \frac{1}{k\_n} \sum\_{i=1}^{k\_n} g(\bar Z\_{r\_n,i}^{[\ell\_n]}) - P\_n^{[\ell\_n]}g \right\},$$ where $\bar Z\_{r\_n,i}^{\scriptscriptstyle[\ell\_n]}$ are independent over i ∈ N and have the same distribution as $Z\_{r\_n,i}^{\scriptscriptstyle[\ell\_n]}$. By the classical central limit theorem for row wise independent triangular arrays, the weak limit of $\tilde {{\mathbb{H}}}\_n^{\scriptscriptstyle [\ell]} g$ is Gg: first, its variance $$\operatorname{Var}(\tilde {{\mathbb{H}}}\_n^{ [\ell\_n]} g) = P\_n^{[\ell\_n]}g^2 - (P\_n^{[\ell\_n]} g)^2$$ converges to Var(Gg) by Lemma [lem:moment]. Note that the square of any linear combination g of functions f ∈ F2(α−, α+) can be bounded by a multiple of gη, α0, after possibly decreasing the value of α+ > α0. Second, the Lyapunov Condition is satisfied: for all δ > 0, $$\begin{aligned} \frac{1}{k\_n^{1+\delta/2}} \sum\_{i=1}^{k\_n} {\mathbb{E}}\big[\| g(\bar Z\_{r\_n,i}^{[\ell\_n]}) - P\_n^{[\ell\_n]}g \|^{2+\delta} \big] \end{aligned}$$ converges to 0 as n → ∞ again as a consequence of Lemma [lem:moment], as ∣g∣2 + δ can also be bounded by a multiple of gη, α0 if δ ∈ (0, η) and α+ > α0 are chosen sufficiently small. Now, consider the remainder term Δn in. Since $\tilde{{\mathbb{G}}}\_n f$ and $\tilde{{\mathbb{G}}}\_n^{[\ell\_n]} f$ are centered, so is Δnf, and $$\begin{aligned} {\mathbb{E}}[ (\Delta\_n f)^2 ] &= \operatorname{var}( \Delta\_n f ) = \frac{1}{k\_n} \operatorname{var}\left( \sum\_{i=1}^{k\_n} \Delta\_{r\_n,i}^{[\ell\_n]} f \right),\end{aligned}$$ where Δr, i[ℓn]f = f(Zr, i) − f(Zr, i[ℓn]). By stationarity and the Cauchy–Schwartz inequality, $$\begin{aligned} \nonumber {\mathbb{E}}[ (\Delta\_n f)^2 ] &= \operatorname{var}\left( \Delta\_{r\_n,1}^{[\ell\_n]} f \right) + \frac{2}{k\_n} \sum\_{h=1}^{k\_n-1} (k\_n - h) \, \operatorname{cov}\left( \Delta\_{r\_n,1}^{[\ell\_n]} f, \Delta\_{r\_n,1+h}^{[\ell\_n]} f \right) \\ \label{eq:MSEdecomp} &\le 3 \operatorname{var}\left( \Delta\_{r\_n,1}^{[\ell\_n]} f \right) + 2 \sum\_{h=2}^{k\_n-1} {\left\lvert{ \operatorname{cov}\left( \Delta\_{r\_n,1}^{[\ell\_n]} f, \Delta\_{r\_n,1+h}^{[\ell\_n]} f \right) }\right\rvert}.\end{aligned}$$ Please note that we left the term h = 1 out of the sum; whence the factor three in front of the variance term. Since ℓn = o(rn) as n → ∞ by Condition [cond:alpha], we have σrn − ℓn + 1/σrn → 1 as n → ∞ by Condition [cond:DA]. The asymptotic moment bound in Condition [cond:moment] then ensures that we may choose δ ∈ (2/ω, ν) and α+ > α0 such that, for every f ∈ F2(α−, α+), we have, by Lemma [lem:moment], limsupn → ∞E[\|Δrn, 1[ℓn]f\|2 + δ] < ∞. On the event that Mrn, 1 = Mrn − ℓn + 1, we have Δrn, 1[ℓn]f = 0. The mixing rate in together with Lemma [lem:clipping:2] then imply Δrn, 1[ℓn]f = op(1), n → ∞. Lyapounov’s inequality and the asymptotic moment bound then ensure that limn → ∞E[\|Δrn, 1[ℓn]f\|2 + δ] = 0, f ∈ F2(α−, α+). Recall Lemma 3.11 in : for random variables ξ and η and for numbers p, q ∈ [1, ∞] such that 1/p + 1/q < 1, \|cov(ξ, η)\| ≤ 10 ∥ξ∥p ∥η∥q {α(σ(ξ), σ(η))}1 − 1/p − 1/q, where α(A1, A2) denotes the strong mixing coefficient between two σ-fields A1 and A2. Use inequality with p = q = 2 + δ to bound the covariance terms in : E[(Δnf)2] ≤ 3 ∥Δrn, 1[ℓn]f∥22 + 20 kn ∥Δrn, 1[ℓn]f∥2 + δ2 {α(rn)}δ/(2 + δ). In view of and Condition [cond:alpha], the right-hand side converges to zero since ω < 2/δ. Proof of Theorem [theo:ml] -------------------------- [Proof of Theorem [theo:ml]] We apply Theorem [theo:asydis:stat]. To this end, we verify its conditions. Proof of Condition [cond:DA]. The second-order regular variation condition implies the first-order one in, which is in turn equivalent to weak convergence of partial maxima as in. Condition [cond:DA] follows with scaling sequence σn = an. The latter sequence is regularly varying with index 1/α0, which implies that limn → ∞amn/an = 1 whenever limn → ∞mn/n = 1. Proof of Condition [cond:small]. For any real c we have, since logF(c) < 0 and since log(kn) = o(rn) by, $$\begin{aligned} \Pr[ \min( M\_{r\_n,1}, \ldots, M\_{r\_n,k\_n} ) \le c ] \le k\_n \, F^{r\_n}( c ) = \exp \{ \log(k\_n) + r\_n \log F(c) \} \to 0, \qquad n \to \infty.\end{aligned}$$ Proof of Condition [cond:alpha]. Trivial, since α(ℓ) = 0 for integer ℓ ≥ 1. Proof of Condition [cond:moment]. This follows from Lemma [lem:blockmoments] in the supplementary material (which in turn is a variant of Proposition 2.1(i) in ), where we prove that the sufficient Condition is satisfied. Proof of Condition [cond:bias]. Recall Remark [rem:secor] and therein the functions L and g(u) = A(u)L(u). We begin by collecting some non-asymptotic bounds on the function L. Fix δ ∈ (0, α0). Potter’s theorem (, Theorem 1.5.6) implies that there exists some constant xʹ(δ) > 0 such that, for all u ≥ xʹ(δ) and x ≥ xʹ(δ)/u, $$\begin{aligned} \label{eq:L1b} \frac{L(u)}{L(ux)} \le (1+\delta)\, \max( x^{-\delta}, x^{\delta}). \end{aligned}$$ As a consequence of Theorem B.2.18 in, accredited to, there exists some further constant xʺ(δ) > 0 such that, for all u ≥ xʺ(δ) and x ≥ xʺ(δ)/u, $$\begin{aligned} \label{eq:L2b} \left\| \frac{L(ux) - L(u)}{g(u)} \right\| \le \ c(\delta) \, \max( x^{\rho - \delta}, x^{\rho+\delta}),\end{aligned}$$ for some constant c(δ) > 0. Define x(δ) = max{xʹ(δ), xʺ(δ), 1}. We are going to show Condition [cond:bias] for c = x(δ) and σrn = arn. For i = 1, …, kn, define Xn, i = Mrn, i ∨ x(δ). Let Pn denote the common distribution of the rescaled, truncated block maxima Xn, i/arn and let P denote the Fréchet(α0, 1) distribution. Write $B\_n = \sqrt{k\_n} (P\_n - P)$ and define the three-by-one vector $\bm{\beta}$ by $$\label{eq:beta} \bm{\beta} = \frac{\lambda }{{\left\lvert{\rho}\right\rvert} \alpha\_0} \begin{pmatrix} 2 - \gamma - \Gamma(2+\tfrac{{\left\lvert{\rho}\right\rvert}}{\alpha\_0}) - \Gamma'(2+\tfrac{{\left\lvert{\rho}\right\rvert}}{\alpha\_0}) \\[1ex] \alpha\_0 \Gamma(2+\tfrac{{\left\lvert{\rho}\right\rvert}}{\alpha\_0}) - \alpha\_0 \\[1ex] 1 - \Gamma(1+\tfrac{{\left\lvert{\rho}\right\rvert}}{\alpha\_0}) \end{pmatrix}$$ if ρ < 0 and by $$\bm{\beta} = \frac{\lambda }{\alpha\_0^2} \begin{pmatrix} \gamma - (1-\gamma)^2 - \pi^2/6 \\ \alpha\_0(1 - \gamma) \\ \gamma \end{pmatrix}$$ if ρ = 0. We will show that $$\label{eq:bias:Y:iid} \lim\_{n \to \infty} \bigl( B\_n x^{-\alpha\_0} \log x, \, B\_n x^{-\alpha\_0}, \, B\_n \log x \bigr)^T = \bm{\beta}.$$ Elementary calculations yield that $M(\alpha\_0) \, \bm{\beta} = \lambda \, B( \alpha\_0, \rho)$ as required in. Equation can be shown coordinatewise. We begin by some generalities. For any f ∈ H as in, we can write, for arbitrary x, x0 ∈ (0, ∞), $$f(x) = \begin{cases} f(x\_0) - \int\_x^{x\_0} f'(y) \, dy, & \text{if $0 < x \le x\_0$}, \\[1ex] f(x\_0) + \int\_{x\_0}^x f'(y) \, dy, & \text{if $x\_0 < x < \infty$}. \end{cases}$$ By Fubini’s theorem, with Gn and G denoting the cdf-s of Pn and P, respectively, $$\begin{aligned} P f &= \int\_{(0, x\_0]} f(x) \, dP(x) + \int\_{(x\_0, \infty)} f(x) \, dP(x) \\ &= f(x\_0) - \int\_{x \in (0, x\_0]} \int\_{y=x}^{x\_0} f'(y) \, dy \, dP(x) + \int\_{x \in (x\_0, \infty)} \int\_{y=x\_0}^x f'(y) \, dy \, dP(x) \\ &= f(x\_0) - \int\_{y=0}^{x\_0} \int\_{x \in (0, y]} \, dP(x) \, f'(y) \, dy+ \int\_{y=x\_0}^\infty \int\_{x \in (y, \infty)} \, dP(x) \, f'(y) \, dy \\ &= f(x\_0) - \int\_0^{x\_0} G(y) \, f'(y) \, dy + \int\_{x\_0}^\infty \left\{ 1 - G(y) \right \} \, f'(y) \, dy,\end{aligned}$$ and the same formula holds with P and G replaced by Pn and Gn, respectively. We find that $$B\_n f = \sqrt {k\_n}(P\_n-P) f = - \int\_0^\infty \sqrt{k\_n} \left\{ G\_n(y) - G(y) \right \} \, f'(y) \, dy.$$ Note that $$\begin{aligned} G(y) &= \exp ( - y^{-\alpha\_0} ) \, {{\mathds{1}}}\_{(0, \infty)}(y), \qquad G\_n(y) = F^{r\_n}(a\_{r\_n} y) \, {{\mathds{1}}}\_{[x(\delta)/a\_{r\_n}, \infty)}(y),\end{aligned}$$ From the definition of L in, we can write, for y ≥ x(δ)/arn, $$G\_n(y) = \exp \left( - y^{-\alpha\_0} r\_n \{ - \log F(a\_{r\_n}) \} \, \frac{L(a\_{r\_n}y)}{L(a\_{r\_n})} \right).$$ For the sake of brevity, we will only carry out the subsequent parts of the proof in the case where F is ultimately continuous, so that rn { − logF(arn)} = 1 for all sufficiently large n. In that case, Bnf = Jn1(f) + Jn2(f) where $$\begin{aligned} J\_{n1}(f) &= \sqrt{k\_n} \int\_0^{x(\delta)/a\_{r\_n}} \exp(-y^{-\alpha\_0})f'(y)\, dy, \\ J\_{n2}(f) &= - \sqrt{k\_n} \int\_{x(\delta)/a\_{r\_n}} ^\infty \left[ \exp\left( -y^{-\alpha\_0} \frac{L(a\_{r\_n}y)}{L(a\_{r\_n})} \right) - \exp(-y^{-\alpha\_0}) \right] f'(y)\, dy, \end{aligned}$$ Let us first show that Jn1(f) converges to 0 for any f ∈ H. For that purpose, note that any f ∈ H satisfies ∣fʹ(x)∣ ≤ Kx− α0 − ɛ − 1 for any ɛ < 1 and for some constant K = K(ɛ) > 0. As a consequence, by, for sufficiently large n, $$\max\_{ f\in {\mathcal{H}}} \|J\_{n1}(f) \| \le \{ \lambda+o(1) \} \frac{K}{A(a\_{r\_n})} \int\_0^{x(\delta) /a\_{r\_n}} \exp(-y^{-\alpha\_0}) y^{-\alpha\_0 - {\varepsilon}- 1} \, dy.$$ Since A(x) is bounded from below by a multiple of xρ − ɛ for sufficiently large x (by Remark [rem:secor] and Potter’s theorem), the expression on the right-hand side of the last display can be easily seen to converge to 0 for n → ∞. For the treatment of Jn2, note that $$\begin{aligned} J(f,\rho) &\equiv \int\_0^\infty h\_\rho(y) \exp\left( -y^{-\alpha\_0} \right) y^{-\alpha\_0}f'(y) \, dy \\ &= \begin{cases} \int\_0^\infty h\_\rho(y) \exp\left( -y^{-\alpha\_0} \right) y^{-2\alpha\_0-1} (1-\alpha\_0\log y) \, dy &, f(y)=y^{-\alpha\_0} \log y\\ \int\_0^\infty h\_\rho(y) \exp\left( -y^{-\alpha\_0} \right) (-\alpha\_0y^{-2\alpha\_0-1}) \, dy &, f(y)=y^{-\alpha\_0} \\ \int\_0^\infty h\_\rho(y) \exp\left( -y^{-\alpha\_0} \right) y^{-\alpha\_0-1} \, dy &, f(y)= \log y \end{cases} \\ &= \begin{cases} {\mathbb{E}}[h\_\rho(Y) Y^{-\alpha\_0}(\alpha\_0^{-1}- \log Y) ] &, f(y)=y^{-\alpha\_0} \log y\\ - {\mathbb{E}}[h\_\rho(Y) Y^{-\alpha\_0}] &, f(y)=y^{-\alpha\_0} \\ \alpha\_0^{-1} {\mathbb{E}}[h\_\rho(Y) ] &, f(y)= \log y, \end{cases}\end{aligned}$$ where Y denotes a Fréchet(α0, 1) random variable. By Lemma [lem:moments] this implies $$\begin{aligned} J(x^{-\alpha\_0} \log x, \rho) &= \frac{1}{\rho\alpha\_0} \left\{ \	arxiv_0000638
Impact of Frequentist and Bayesian Methods on Survey Sampling Practice: A Selective Appraisal ============================================================================================= According to Hansen, Madow and Tepping [J. Amer. Statist. Assoc. 78 (1983) 776–793], “Probability sampling designs and randomization inference are widely accepted as the standard approach in sample surveys.” In this article, reasons are advanced for the wide use of this design-based approach, particularly by federal agencies and other survey organizations conducting complex large scale surveys on topics related to public policy. Impact of Bayesian methods in survey sampling is also discussed in two different directions: nonparametric calibrated Bayesian inferences from large samples and hierarchical Bayes methods for small area estimation based on parametric models. . Introduction ============ Sample surveys have long been conducted to obtain reliable estimates of finite population descriptive parameters, such as totals, means, ratios and quantiles, and associated standard errors and normal theory intervals with large enough sample sizes. Probability-sampling designs and randomization (repeated sampling) inference, also called the design-based approach, played a dominant role, especially in the production of official statistics, ever since the publication of the landmark paper by Neyman () which laid the theoretical foundations of the design-based approach. Neyman’s approach was almost universally accepted by practicing survey statisticians and it also inspired various important theoretical contributions, mostly motivated by practical and efficiency considerations. In this paper I will first provide some highlights of the design-based approach, for handling sampling errors, to demonstrate its significant impact on survey sampling practice, especially on the production of official statistics (Sections [2] and [3.1]). Model-dependent approaches (Section [3.2]) that lead to conditional inferences more relevant and appealing than repeated sampling inferences have also been advanced (Brewer, ; Royall, ). Unfortunately, for large samples such approaches may perform very poorly under model misspecifications; even small model deviations can cause serious problems (, ). On the other hand, model-dependent approaches can play a vital role in small area (domain) estimation, where the area-specific sample sizes are very small or even zero and make the design-based area-specific direct estimation either very unreliable or not feasible. Demand for reliable small area statistics has greatly increased in recent years and to meet this growing demand, federal statistical agencies and other survey organizations are currently paying considerable attention to producing small area statistics using models and methods that can “borrow strength” across areas. Hierarchical Bayes (HB) model-dependent methods are particularly attractive in small area estimation because of their ability to handle complex modeling and provide “exact” inferences on desired parameters (Section [5]). I will highlight some HB developments in small area estimation that seem to have a significant impact on survey practice. I will also discuss the role of nonparametric Bayesian methods for inferences, based on large area-specific sample sizes, especially those providing Bayesian inferences that can be also justified under the design-based framework (Section [4.2]). Models are needed, regardless of the approach used, to handle nonsampling errors that include measurement errors, coverage errors and missing data due to nonresponse. In the design-based approach, a combined design and modeling approach is used to minimize the reliance on models, in contrast to fully model-dependent approaches (Section [3.4]). For simplicity, I will focus on descriptive parameters, but survey data are also increasingly used for analytical purposes, in particular, to study relationships and making inferences on model parameters under assumed super-population models. For example, social and health scientists are interested in fitting linear and logistic regression models to survey data and then making inferences on the model parameters taking account of the survey design features (Section [3.3]). Design-Based Approach: Early Landmark Contributions =================================================== In this section I will highlight some early landmark contributions to the design-based approach that had major impact on survey practice. Prior to Neyman (), sampling was implemented either by “balanced” sampling through purposive selection or by probability sampling with equal inclusion probabilities. Such a method was called the “representative method.” Bowley () studied stratified sampling with proportional sample size allocation, leading to a representative sample with equal probabilities. Neyman () broke through this restrictive setup by relaxing the condition of equal probabilities and introducing the ideas of efficiency and optimal sample size allocation in his theory of stratified random sampling. He also demonstrated that balanced purposive sampling may perform poorly if the underlying model assumptions are violated. Neyman proposed normal theory confidence intervals for large samples such that the frequency of errors in the confidence statements based on all possible stratified random samples that could be drawn does not exceed the limit prescribed in advance “whatever the unknown properties of the finite population.” He broadened the definition of representative method by calling any method of sampling that satisfies the above frequency statement as representative. It is interesting to note that Neyman advocated distribution-free design-based inferences for survey sampling in contrast to his own fundamental work on parametric inference, including the Neyman–Pearson theory of hypothesis testing and confidence intervals. The possibility of developing efficient probability sampling designs by minimizing total cost subject to a specified precision of an estimator or maximizing precision for a given cost, taking account of operational considerations, and making distribution-free inferences (point estimation, variance estimation and large sample confidence intervals) through the design-based approach were soon recognized.This, in turn, led to a significant increase in the number and type of surveys taken by probability sampling and covering large populations. In the early stages, the primary focus was on sampling errors. I now list a few important post-Neyman theoretical developments in the design-based approach. As early as 1937, Mahalanobis used multistage sampling designs for crop surveys in India. His classic 1944 paper (Mahalanobis, ) presents a rigorous theoretical setup and a generalized approach to the efficient design of sample surveys of different crops in Bengal, India, with emphasis on variance and cost functions. Mahalanobis considered a geographical region of finite area and defined a field consisting of “a finite number, say, N0, of basic cells arranged in a definite space or geographic order together with a single value (or a set of values in the multivariate case) of z for each basic cell,” where z is the variable of interest (say, crop yield). Under this setup, he studied four different probability sampling designs for selecting a sample of cells (called quads): unitary unrestricted, unitary configurational, zonal unrestricted and zonal configurational. In modern terminology, the four designs correspond to simple random sampling, stratified random sampling, single-stage cluster sampling and single-stage stratified cluster sampling, respectively. He developed realistic cost functions depending on particular situations. He also extended the theoretical setup to subsampling of clusters (which he named as two-stage sampling). We refer the reader to Murthy () for a detailed account of the 1944 paper and other contributions of Mahalanobis to sample surveys. Hall () provides a scholarly historical account of the pioneering contributions of Mahalanobis to the early development of survey sampling in India. Mahalanobis was instrumental in establishing the National Sample Survey of India and the famous Indian Statistical Institute. Survey statisticians at the U.S. Census Bureau, under the leadership of Morris Hansen, made fundamental contributions to survey sampling theory and practice during the period 1940–1970, and many of those methods are still widely used in practice. This period is regarded as the golden era of the Census Bureau. Hansen and Hurwitz () developed the basic theory of stratified two-stage cluster sampling with one cluster or primary sampling unit (PSU) within each stratum drawn with probability proportional to a size measure (PPS) and then subsampled at a rate that ensures self-weighting (equal overall probabilities of selection). This method provides approximately equal interviewer work loads which are desirable in terms of field operations. It can also lead to significant variance reduction by controlling the variability arising from unequal PSU sizes without actually stratifying by size and thus allowing stratification on other variables to further reduce the variance. The Hansen–Hurwitz method, with some modifications, has been widely used for designing large-scale socio-economic, health and agricultural surveys throughout the world. Many large-scale surveys are repeated over time, such as the monthly Current Population Survey (CPS), and rotation sampling with partial replacement of ultimate units (e.g., households) is used to reduce response burden. Hansen et al. () developed simple but efficient composite estimators under rotation sampling in the context of stratified multistage sampling. Rotation sampling and composite estimation are widely used in large-scale surveys. Prior to the 1950s, the primary focus was on estimating totals, means and ratios for the whole population and large planned subpopulations such as US states or provinces in Canada. Woodruff () developed a unified design-based approach for constructing confidence intervals on quantiles using only the estimated distribution function and the associated standard error. This ingenious method is to general probability sampling designs and performs well in terms of coverage probabilities in many cases. Woodruff intervals can also be used to obtain standard errors of estimated quantiles (Rao and Wu, ; Francisco and Fuller, ). Because of those features, the Woodruff method had a significant impact on survey practice. However, the method should not be treated as a black box for constructing confidence intervals on quantiles, because it can perform poorly in some practical situations. For example, it performed very poorly under stratified random sampling when the population is stratified by a concomitant variable x highly correlated with the variable of interest y (, ). The failure of the Woodruff method in this case stems from the fact that the standard error of the estimated distribution function at the quantile will be too small due to zero contributions to the standard error from most strata. () showed that the bootstrap method for stratified random sampling performs better than the Woodruff method in this case, but in other situations the Woodruff method is better. Attention was also given to inferences for unplanned subpopulations (also called domains) such as age–sex groups within a state. Hartley () and Durbin () developed simple, unified theories for domain estimation applicable to general designs, requiring only existing formulae for population totals and means. After the consolidation of basic design-based sampling theory, Hansen et al. () and others paid attention to measurement errors in surveys. They developed basic theories under additive measurement error models with minimal model assumptions on the observed responses treated as random variables. Total variance of an estimator is decomposed into sampling variance, simple response variance and correlated response variance (CRV) due to interviewers. The CRV was shown to dominate the total variance when the number of interviewers is small, and the 1950 U.S. Census interviewer variance validation study showed that this component is indeed large for small areas. Partly for this reason, self-enumeration by mail was first introduced in the 1960 U.S. Census to reduce the CRV component. Earlier, Mahalanobis () developed the method of interpenetrating subsamples (called replicated sampling by Deming, ) and used it extensively in large-scale surveys in India for assessing both sampling and interviewer errors. By assigning the subsamples at random to interviewers, the total variance can be estimated and interviewer differences assessed. It should be clear from the above brief description of early developments that much of the basic sampling theory was developed by official statisticians or those closely associated with official statistics. Theory was driven by the need to solve real problems and often theory was not challenging enough to attract academic researchers to survey sampling. As a result, university researchers paid little attention to survey sampling in those days with few exceptions (e.g., Iowa State University under the leadership of Cochran, Jessen and Hartley). Some Recent Design-Based and Non-Bayesian Developments ====================================================== Model-Assisted Approach ----------------------- We first give a brief account of the model-assisted approach that uses a working model to find efficient estimators. However, the associated inferences are design-based. Consider a finite population U consisting of N elements labeled 1, …, N with associated values y1, …, yN of a variable of interest y. Under a probability sampling design, the inclusion probabilities π1, …, πN are all strictly positive and a basic estimator of the total Y = ∑i ∈ Uyi is of the form Ŷ = ∑i ∈ sdiyi, where s denotes a sample and di = πi− 1 are the so-called design weights (Horvitz and Thompson, ; Narain, ). For example, in the Neyman stratified random sampling design, the design weights are equal to the inverse of the sampling fractions within strata and vary across strata, while in the Hansen et al. two-stage cluster sampling design, the design weights are all equal. Design unbiasedness of estimators is not insisted upon (contrary to statements in some papers on inferential issues of sampling theory) because it “often results in much larger MSE than necessary” (, ). Instead, design consistency is deemed necessary for large samples. Strategies (design and estimation) that appeared reasonable are entertained (accounting for costs) and relative properties are carefully studied by analytical and/or empirical methods, mainly through the comparison of mean squared error (MSE) or anticipated MSE under plausible population models on the variables yi treated as random variables. This is essentially the basis of the repeated sampling (or designbased) approach. In recent years, a model-assisted repeated sampling approach has received the attention of survey practitioners. In this approach, a working population model is used to find efficient design-consistent estimators. For example, suppose the working model is a linear regression model of the form yi = xʹiβ + ɛi; i = 1, …, N, with model errors ɛi assumed to be uncorrelated with mean zero and variance proportional to a known constant qi, where xi is a vector of auxiliary variables with known population total X. Under model ([eq1]), the best linear unbiased estimator (BLUE) of the model parameter β, based on the census values {(yi, xi); i ∈ U}, is given by the “census” regression coefficient $$B = \biggl(\sum\_{i \in U} x\_{i} x'\_{i}/q\_{i}\biggr)^{ - 1}\biggl(\sum\_{i \in U} x\_{i} y\_{i}/q\_{i}\biggr).$$ A predictor of yi under the working model is then given by ŷi = xʹiB̂ for i = 1, …, N where B̂ is the design-weighted estimator of B: $$\hat{B} = \biggl(\sum\_{i \in s} d\_{i} x\_{i}x'\_{i}/q\_{i}\biggr)^{ - 1}\biggl(\sum\_{i \in s} d\_{i} x\_{i}y\_{i}/q\_{i}\biggr).$$ By writing the total as Y = ∑i ∈ Uŷi + ∑i ∈ Uei where ei = yi − ŷi denotes the prediction error, a design-based estimator of Y is given by ∑i ∈ Uŷi + ∑i ∈ sdiei. We can express this estimator as a generalized regression (GREG) estimator Ŷgr = Ŷ + B̂ʹ(X − X̂), where X̂ = ∑i ∈ sdixi (Särndal, Swensson and Wretman, ). The GREG estimator ([e2]) is design-consistent regardless of the validity of the working model (Robinson and Sarndal, ) under certain regularity conditions provided X is precisely correct. If the working model provides a good fit to the data, then the residuals ei should be less variable than the response values yi and the GREG estimator is likely to be significantly more efficient than the basic design-weighted estimator Ŷ. The estimator ([e2]) may also be expressed as a weighted sum ∑i ∈ swiyi, where wi = digi with $$\label{e3} g\_{i} = 1 + (X - \hat{X})'\biggl(\sum\_{i \in s} d\_{i} x\_{i}x'\_{i}/q\_{i}\biggr)^{ - 1}x\_{i}q\_{i}^{ - 1}.$$ The adjustment factors gi, popularly known as the g-weights, ensure the calibration property ∑i ∈ swixi = X so that the GREG estimator when applied to the sample values xi agrees with the known total X. This property is attractive to the user when the vector X contains user-specified totals. The assumption of a working linear regression model ([eq1]) can be relaxed by adopting more flexible working models. For example, Breidt, Claeskens and Opsomer () proposed a nonparametric model-assisted approach based on a penalized spline (P-spline) regression working model and showed that the resulting estimators are design-consistent and more efficient than the usual GREG estimators based on linear regression working models when the latter are incorrectly specified. Also, the P-spline model-assisted estimators were shown to be approximately as efficient as the GREG estimators when the linear regression working model is correctly specified. The P-spline approach can be easily implemented using existing estimation packages for GREG because the underlying model is closely related to a linear regression model. It offers a wider scope for the model-assisted approach because it makes minimal assumptions on the regression of y on x without assuming a specific parametric form. Under the model-assisted approach, design-consistent variance estimators are obtained either by a Taylor linearization method or by a resampling method (when applicable), provided the probability sampling design ensures strictly positive joint inclusion probabilities πij, i ≠ j. Using the estimator and associated standard error, asymptotically valid normal theory intervals are obtained regardless of the validity of the working model. Most large-scale surveys are multipurpose and observe multiple variables of interest, and the same working model may not hold for all the variables of interest. In that case, a model-assisted approach may lead to possibly different gi and hence different calibration weights wi associated with the variables, that is, the calibration weights are of the form wij associated with the variable j and sample unit i. However, survey users prefer to use a commonweight wi for all variables of interest. This is often accomplished by minimizing a suitable distance measure between di and wi for i ∈ s subject to userspecified calibration constraints, say, ∑i ∈ swizi = Z, without appealing to any working model, where Z is the vector of known totals associated with the user-specified variables z. For example, a chi-squared distance measure leads to common calibrationweights wi of the form digi where gi is given by ([e3]) with xi replaced by zi (Deville and Sarndal, ). Thus, calibration estimation in this case corresponds to using model-assisted estimation based on a linear regression model ([eq1]) with zi as the vector of predictor variables. Calibration estimation has attracted the attention of users due to its ability to produce common calibration weights and accommodate an arbitrary number of user-specified calibration (or benchmark) constraints, for example, calibration to the marginal counts of several post-stratification variables. Several national statistical agencies have developed software designed to compute calibration weights: GES (Statistics Canada), LIN WEIGHT (Statistics Netherlands), CALMAR (INSEE, France) and CLAN97 (Statistics Sweden). Sarndal () says, “Calibration has established itself as an important methodological instrument in large-scale production of statistics.” Brakel and Bethlehem () noted that the use of common calibration weights for estimation in multipurpose surveys makes the calibration method “very attractive to produce timely official releases in a regular production environment.” Unfortunately, the model-free calibration approach can lead to erroneous inferences for some of the response variables, even in fairly large samples if the underlying working linear regression model uses an incorrect or incomplete set of auxiliary variables, unlike the model-assisted approach that uses a working model obtained after some model checking. For example, suppose that the underlying model is a quadratic regression model of y on x and the distribution of x is highly skewed. Also, suppose that the user-specified calibration constraints are the known population size N and the known population total X. In this case, the calibration estimator of the total Y under simple random sampling is the familiar simple linear regression estimator with x as the predictor variable. On the other hand, a model-assisted estimator under the quadratic regression working model is given by a multiple linear regression estimator with x1 = x and x2 = x2 as the predictor variables, assuming the total of x2 is also known. () demonstrated that the coverage performance of the normal theory interval associated with the calibration estimator is poor even in fairly large samples, unlike the coverage performance of the normal theory intervals associated with the model-assisted estimator. The coverage performance depends on the skewness of the residuals from the fitted model and in the case of calibration estimation the skewness of residuals after fitting simple linear regression remains large, whereas the skewness of residuals after fitting quadratic regression is small even if y and x are highly skewed. This simple example demonstrates that the population structure does matter in design-based inferences and that it should be taken into account through a model-assisted approach based on suitable working models. But the model-assisted approach has the practical limitation that the weight wi may vary across variables in surveys with multiple variables of interest, unlike in the calibration approach. Also, for complex working models, such as the P-spline, all the population values of the predictor variables should be known in order to implement the model-assisted approach. The model-assisted approach is essentially design-based, unlike the model-dependent approach (Section [3.2]) that can provide conditional inferences referring to the particular sample, s, of units selected. Such conditional inferences may be more relevant and appealing than the unconditional repeated sampling inferences used in the design-based approach. Model-Dependent Approach ------------------------ The frequentist model-dependent approach to inference assumes that the population structure obeys a specified population model and that the same model holds for the sample, that is, no sample selection bias with respect to the assumed population model. Sampling design features are often incorporated into the model to reduce or eliminate the sample selection bias (see Section [3.3] for some difficulties to implement this in practice). Typically, distributional assumptions are avoided by focusing on point estimation, variance estimation and associated normal theory confidence intervals, as in the case of design-based inferences. As a result, models used specify only the mean function and the variance function of the variable of interest, y.We refer the reader to Valliant, Dorfman and Royall () for an excellent account of the model-dependent approach. As noted in Section [1], model-dependent strategies may perform poorly in large samples when the population model is not correctly specified; even small deviations from the assumed model that are not easily detectable through routine model checking can cause serious problems. In the Hansen, Madow and Tepping () example of an incorrectly specified population model, the best linear unbiased prediction (BLUP) estimator of the mean is not design consistent under their stratified simple random sampling design with near optimal sample allocation (commonly used to handle highly skewed populations such as business populations). As a result, model-dependent confidence intervals exhibited poor performance: for n = 100, coverage was around 70% compared to nominal level of 95%, while coverage for model-assisted intervals was 94.4%. To get around this difficulty, Little () proposed restricting attention to models that hold for the sample and for which the BLUP estimator is design consistent. For example, in the () example, the BLUP of the mean under a model with means differing across strata is identical to the traditional stratified mean which is design consistent. But it seems not possible even to find a suitable model under which the widely used combined ratio estimator of the mean under the stratified random sampling is the BLUP estimator. The combined ratio estimator is a model-assisted estimator under a ratio working model with a common slope. It allows a large number of strata with few sample units from each stratum and yet remains design consistent, unlike the separate ratio estimator which is the BLUP estimator under a ratio model with separate slopes across strata: E(yhi∣xhi) = βhxhi, V(yhi∣xhi) = σ2xhi, where yhi and xhi denote the values of the variable of interest y and an auxiliary variable x for the unit i in stratum h. Moreover, the BLUP estimator under this model requires the knowledge of the strata population means X̄h, whereas the combined ratio estimator requires only the overall population mean X̄. It is also not clear how one proceeds to formulate suitable models for general sampling designs that lead to design-consistent BLUP estimators. Further, the main focus has been on point estimation and it is not clear how one should proceed with variance estimation and setting up confidence intervals that have repeated sampling validity. In this context, Pfeffermann () says, “I presume that these are supposed to be computed under the corrected model as well. Are we guaranteed that they are sufficiently accurate under the model? Do we need to robustify them separately?” Little (), in his rejoinder to Pfeffermann’s comment, says that he advocates using some replication method for variance estimation and then appealing to normal approximation for confidence intervals. Clearly, further work is needed to address the above issues. Note that if parametric assumptions are made, such as normality of model errors, then it is possible to make exact Bayesian inferences by introducing suitable priors on the model parameters (Section [4]). Some recent work on the model-dependent approach focused on avoiding misspecification of the mean function E(y∣x) = m(x) by using P-spline models. Zheng and Little (, ) studied single stage PPS sampling using a P-spline model, based on the size measure x used in PPS sampling, to represent the regression function m(x), and a specified function of the size measure as the variance function. In a simulation study, they compared the performance of the usual linear GREG estimator and the P-spline model-based estimators, not necessarily design-consistent, and showed that the P-spline model-based estimators are generally more efficient than the GREG or the NHT estimator in terms of design MSE even for large samples. However, the simulation study did not consider model-assisted estimators corresponding to their P-spline model. The simulations also showed that the design-bias for their P-spline estimators is minor, even though the estimators are not design-consistent, and hence the authors conclude that “design consistency may not be of paramount importance.” On the other hand, in the () study, their model-assisted P-spline estimator is sometimes much better, and never worse, than the corresponding P-spline model-based estimator under stratified random sampling. The latter estimator is not design-consistent under the P-spline model considered by (). As noted above, a main advantage of the frequentist model-dependent approach is that it leads to inferences conditional on the selected sample of units, s, unlike the unconditional design-based approach. However, it is possible to develop a conditional model-assisted approach that allows us to restrict the reference set of samples to a “relevant” subset of all possible samples specified by the design. Conditionally valid inferences for large samples can then be obtained. Rao () and Casady and Valliant () developed an “optimal” linear regression estimator that is asymptotically valid under the conditional setup. We refer the reader to Kalton () for compelling arguments for favoring design-based approaches (possibly model-assisted and/or conditional) to handle sampling errors. Smith () named the traditional repeated sampling inference as “procedural inference” and argued that procedural inference is the correct approach for surveys in the public domain. Analysis of Complex Survey Data ------------------------------- Data collected from large-scale socio-economic,health and other surveys are being extensively used for analysis purposes, such as inferences on the regression parameters of linear and logistic regression population models. Ignoring the survey design features and using standard methods can lead to erroneous on model parameters because of sample selection bias caused by informative sampling. It is tempting to expand the models by including among the predictors all the design variables that define the selection process at the various levels and then ignore the design and apply standard methods to the expanded model. The main difficulties with this approach, advocated by some leading researchers, are the following, among others (Pfeffermann and Sverchkov, ): (1) Not all design variables may be known or accessible to the analyst. (2) Too many design variables can lead to difficulties in making inferences from the expanded models. (3) The expanded model may no longer be of scientific interest to the analyst. The design-based approach can provide asymptotically valid repeated sampling inferences without changing the analyst model. A unified approach based on survey-weighted estimating equations leads to design-consistent estimators of the “census” or finite population parameters, which in turn estimate the associated model parameters. Further, using resampling methods for variance estimation, such as the jackknife and the bootstrap for survey data, asymptotically valid design-based inferences on the census parameters can be implemented. The same methods may also be applicable for inference on the model parameters, in many cases of large-scale surveys. In the other cases, it is necessary to estimate the model variance of the census parameters from the sample. The estimate of the total variance is then given by the sum of this estimate and the resampling variance estimate. In practice, the data file would contain for each sampled unit the variables of interest and predictor variables, final weights after adjustment for unit nonresponse and the corresponding replication weights, for example, bootstrap weights. The analyst can use software that handles survey weights (such as SAS) to obtain point estimates from the final weights and the corresponding point estimates for each bootstrap replicate using bootstrap weights. The variability of the bootstrap point estimates provides asymptotically valid standard errors for designs commonly used in large-scale surveys. Details of the methods are not provided due to space limitations, but the reader is referred to Rao (, Section 6) for a succinct account of analysis of survey data using resampling methods for variance estimation and normal theory confidence intervals. Design-based approach using resampling methods is extensively used in practice and software is also available (e.g., WesVar, Stata). The design-based approach has also been applied to make inferences on the regression parameters and the variance parameters of multilevel models from data obtained from multistage sampling designs corresponding to the levels of the model. For example, in an education study of students, schools (first-stage units or clusters) may be selected with probabilities proportional to school size and students (second-stage units) within selected schools by stratified random sampling. Again, ignoring the sampling design and using traditional methods for multilevel models that ignore the design can lead to erroneous inferences in the presence of sample selection bias. In the design-based approach, estimation of variance parameters of the model is more difficult than that of regression parameters and the necessary information for estimating variance parameters is often not provided in public-use data files which typically report only the final weight for each sample unit. Widely used design-based methods have been proposed in the literature (e.g., Pfeffermann et al.,, and Rabe-Hesketh and Skrondal, ) to handle variance parameters that require the weights within sampled clusters in addition to the weights associated with the clusters. Some of those methods can be implemented using the Stata program gllamm. Unfortunately, the resulting estimators of variance parameters may not be design-model consistent when the sample sizes within clusters are small, even for two-level linear models. Korn and Graubard () demonstrated the bias problem and proposed a different method for simple two-level or three-level models involving only a common mean as the fixed effect. This method first obtains the census parameters and then estimates those parameters. It worked well in empirical studies even for small within-cluster sample sizes. Rao, Verret and Hidiroglou () proposed a weighted estimating equations (WEE) approach for general two-level linear models that uses within-cluster joint inclusion probabilities, similar to Korn and Graubard (). The WEE method leads to design-model consistent estimators of variance parameters even for smallwithin-cluster sample sizes, provided the number of sample clusters is large. It performed well in empirical studies compared to the other methods proposed in the literature. () also proposed a unified approach based on a weighted log-composite likelihood that can handle generalized linear multilevel models and small within-cluster sample sizes. This method is currently under investigation. A drawback of the design-based approach to the analysis of survey data is that it may lead to loss in efficiency when the final weights vary considerably across the sampling units. Alternative approaches that can reduce the variability of the weights and thus lead to more efficient estimators have also been proposed (e.g., Pfeffermann and Sverchkov, ; Fuller,, Chapter 6). We refer the reader to Pfeffermann () and Rao et al. () for overviews on the role of sampling weights in the analysis of survey data. Nonsampling Errors ------------------ Survey practitioners have to rely on models, regardless of the approach used, to handle nonsampling errors that include measurement errors, coverage errors and missing data due to unit nonresponse and item nonresponse. In the design-based approach, a combined design and modeling approach is used to minimize the reliance on models, in contrast to fully model-dependent approaches that will have similar difficulties noted in the previous subsections. As mentioned in Section [1], Hansen et al. () studied measurement errors under minimal model assumptions on the observed responses treated as random variables, and their discovery that the correlated response variance due to interviewers dominates the total variance when the number of interviewers is small led to the adoption of self enumeration by mail in the 1960 U.S. Census. Inference in the presence of missing survey data, particularly item nonresponse, has attracted a lot of ; see Little and Rubin () for an excellent account of missing data methods. To handle item nonresponse, imputation of missing data is often used because of its practical advantages. In the design-based approach, traditional weighted estimators of a total or a mean are computed from the completed data set, leading to an imputed estimator. Often imputed values are generated from an imputation model assumed to hold for the respondents under a missing at random (MAR) response mechanism. Under this setup, the imputed estimator is unbiased or asymptotically unbiased under the combined design and model set up. Reiter, Raghunathan and Kinney () demonstrated the importance of incorporating sampling design features into the imputation model in order to make the model hold for the sample and then for the respondents under the assumed MAR response mechanism. An alternative approach avoids imputation models but assumes a model for the response mechanism. For example, a popular method consists of forming imputation classes (according to the values of estimated response probabilities under a specified response model) and assuming that the missing values are missing completely at random (MCAR) within classes. The missing values are then imputed by selecting donors at random from the observed values within classes. It may be also possible to develop imputation methods that make an imputed estimator doubly robust in the sense that it is valid either under an assumed imputation model or under an assumed response mechanism (e.g., see Haziza and Rao, ). Doubly robust estimation has attracted considerable attention in the nonsurvey literature (see, e.g.,, ). Variance estimation under imputation for missing survey data has attracted a lot of attention because treating the imputed values as if observed and then applying standard variance formulae can often lead to serious underestimation because the additional variability due to estimating the missing values is not taken into account. Methods that can lead to asymptotically valid variance estimators under single imputation for missing data have been proposed under the above setups. We refer the reader to Kim and Rao () for a unified approach to variance estimation under the imputation model approach, and to Haziza () for an excellent overview of imputation for survey data and associated methods for inference. Rubin () proposed multiple imputation to account for the underestimation when applying standard formulae treating the imputed values as if observed. Under this approach, M ( ≥ 2) imputed values are generated for a missing item, leading to M completed data sets. Rubin recommends the use of traditional design-based estimators and variance estimators, computed from each of the completed data sets, although multiple imputation ideas are based on a Bayesian perspective: “We restrict attention to standard scientific surveys and standard complete data statistics” (Rubin,, page 113). Multiple imputation estimator ŶMI of a total Y is taken as the average of the M estimators ŶI1, …, ŶIM, and its estimator of variance is given by vMI = v̄M + (1 + M− 1)bM, where v̄M is the average of the M naïve variance estimators vI1, …, vIM and bM = (M − 1)− 1∑m = 1M(ŶIm − ŶMI)2. Rubin gives design-based conditions for “proper imputation” that ensure the repeated sampling validity of the estimator ŶMI and the associated variance estimator vMI under a posited response mechanism. Unfortunately, there are some difficulties in developing imputation methods satisfying Rubin’s conditions for “proper imputation” with complex survey data (see, e.g., Kim et al., ). Nevertheless, multiple imputation shows how Bayesian ideas can be integrated to some extent with the traditional design-based approach that is widely used in practice. Bayesian Approaches =================== This section provides an account of both parametric and nonparametric Bayesian (and pseudo-Bayesian) approaches to inference from survey data, focusing on descriptive finite population parameters. Parametric Bayesian Approach ---------------------------- As noted in Section [3.2], the frequentist model-dependent approach mostly avoided distributional assumptions by specifying only the mean function and the variance function of the variable of interest. Under a specified distribution on the assumed model, Bayesian inferences can be easily implemented, provided the model holds for the sample. Royall and Pfeffermann () studied Bayesian inference on the population mean assuming normality and flat (diffuse) priors on the parameters of a linear regression model. Their focus was on the posterior mean and the posterior variance and, hence, results were similar to those of Royall () without the normality assumption and priors on model parameters. However, exact credible intervals on the mean and other parameters of interest can be obtained conditional on the observed data, using a parametric Bayesian setup. It can be implemented even under complex modeling, using powerful Monte Carlo Markov chain (MCMC) methods to simulate samples from the posterior distributions of interest. Scott and Smith () obtained the posterior mean and the posterior variance of the population mean under linear models with random effects, normality and diffuse priors on the model parameters. Their posterior mean is also the BLUP estimator without the normality assumption when the variance parameters of the model are known. In the frequentist approach, estimates of variance parameters are substituted in the BLUP to get the empirical BLUP (EBLUP) estimator which is different (but close to) to the posterior mean. However, the Bayesian approach also provides the posterior variance which is typically different from the estimated mean squared prediction error (MSPE) of the EBLUP estimator; several different methods of estimating MSPE have been proposed in the context of small area estimation (Rao,, Chapter 7). A simulation study by Bellhouse and Rao () showed that any gain in efficiency of the posterior mean (or the BLUP) over traditional design-based estimators is likely to be small in practice. However, by regarding the clusters as small areas of interest, the Scott–Smith approach provides models linking the small areas and the resulting estimators of small area means can lead to significant efficiency gains over direct area-specific estimators. Random cluster effect models are now extensively used to construct efficient small area estimators by “borrowing strength” across small areas using auxiliary information (Section [5]). It may be noted that the empirical Bayes (EB) approach to inference from random cluster effect models is similar to EBLUP, but it can handle general parametric random cluster effect models and does not require the linearity assumption used in the BLUP method. The EB approach is essentially frequentist, unlike the Bayesian approach that requires the specification of priors on the model parameters. It may be more appropriate to name “empirical Bayes” as “empirical best” without changing the abbreviation EB (Jiang and Lahiri, ). Sedransk () studied regression models with random slopes β1, …, βL ∼ i.i.d.N(ν, σβ2), using a prior distribution on ν specified as N(β0, σ02). He then followed the Scott and Smith () approach and obtained posterior mean and posterior variance of the finite population total. He applied the method to data on banks from the U.S. Federal Reserve Board to estimate a current monetary total making use of extensive historical data to specify the values of β0, σ02 and thus arrive at an informative prior which in turn leads to more efficient posterior inferences compared to those based on a noninformative prior, provided the informative prior is correctly specified. Malec and Sedransk () extended Scott–Smith results to three-stage sampling. () applied the Bayesian approach to obtain order-restricted estimators of the age composition of a population of Atlantic cod, using MCMC methods. Sedransk () lists possible uses of parametric Bayesian methods for sample surveys, including the above application to estimation from establishment surveys, “optimal” sample allocation and small area estimation from data pooled from independent surveys (see Section [5]). () report an interesting application of the Bayesian approach to make inferences from multilevel models under informative sampling. In this case, the multilevel sample model induced by informative sampling is more complicated than the corresponding population model and, as a result, frequentist methods are difficult to implement. On the other hand, the authors show that the Bayesian approach, using noninformative priors on the model parameters indexing the sample model and applying MCMC methods, is efficient and convenient for handling such complex sample models, although computer intensive. This application is an example where the Bayesian approach offers computational advantage over the corresponding frequentist approach. Nonparametric Bayesian Approaches --------------------------------- For multipurpose large-scale surveys, parametric Bayesian methods based on distributional assumptions have limited value because of the difficulties in validating the parametric assumptions. It may be more appealing to use a nonparametric Bayesian approach, but this requires the specification of a nonparametric likelihood function based on the full sample data {(i, yi), i ∈ s} and a prior distribution on the parametric vector (y1, …, yN). The likelihood function based on the full sample data, however, is noninformative in the sense that all possible unobserved values of the parameter vector have the same likelihood function (Godambe, ). One way out of this difficulty is to take a Bayesian route by assuming an informative (exchangeable) prior on the N-dimensional parameter vector and combine it with the noninformative likelihood (Ericson, ; Binder, ) to get an informative posterior, but inferences do not depend on the sample design; Ericson argued that an exchangeable prior assumption may be reasonable under simple random sampling. Ericson () focused on the posterior mean and the posterior variance of the population mean Ȳ which approximately agree, under prior vagueness, with the usual formulae under the design-based approach. In the case of stratified sampling with known strata differences, priors within strata are assumed to be exchangeable. Meeden and Vardeman () used a Polya posterior (PP) over the unobserved, assuming that “unseen are like the seen” (equivalent to exchangeability). In this case, the posterior “does not arise from a single prior distribution” (Meeden, ) and, hence, it is called a pseudo-posterior. It is also similar to the Bayesian bootstrap (Rubin, ; Lo, ). The Polya posterior is a flexible tool and methods based on PP have reasonable design-based properties under simple random sampling. PP approach permits Bayesian interval estimation for the mean and any other parameters of interest through simulation of many finite populations from PP. The general interval estimation feature of the PP approach is attractive. Meeden () extended the PP approach to utilize auxiliary population information (x1, …, xN) by making a strong prior assumption that the ratios ri = yi/xi are exchangeable and obtained point and interval estimators for the population median. Empirical results under simple random sampling are given to show that the resulting Bayesian intervals perform well in terms of design-based coverage. () developed a constrained Polya posterior to generate simulated populations that are consistent with the known population mean of an auxiliary variable x, using MCMC methods. This approach permits the use of known population auxiliary information and leads to more efficient Bayesian inferences. Nelson and Meeden () adapted PP to incorporate prior knowledge that the population median belongs to some interval. Meeden () studied two-stage cluster sampling (balanced case) and his two-stage PP-based results for the posterior mean and the posterior variance are very close to standard design-based results, but it is not clear how readily Meeden’s approach extends to the unbalanced case with unequal cluster sizes. Although the PP approach is attractive and seems to provide calibrated Bayesian inferences at least for some simple sampling designs, it is unlikely to be used in the production of official statistics because of the underlying assumption that “unseen are like the seen” and each case needs to be studied carefully to develop suitable PP. Also, it is not clear how this method can handle complex designs, such as stratified multistage sampling designs, or even single stage unequal probability sampling without replacement with nonnegligible sampling fractions, and provide design-calibrated Bayesian inferences., the PP approach may be useful for some specialized surveys and when inferences are desired on a variety of finite population parameters associated with the variable of interest y or prior knowledge on the parameters is available, as in the case of Nelson and Meeden (). An alternative approach is to start with an informative likelihood based on reduced data. For example, under simple random sampling, it may be reasonable to suppress the labels i from the full data {(i, yi), i ∈ s} and use the likelihood based on the reduced data {yi, i ∈ s}; see Hartley and Rao () and Royall (). On the other hand, for stratified random sampling, labels within strata are suppressed but strata labels are retained because of known strata differences. Hartley and Rao () proposed a “scale-load” approach for inference on the mean Ȳ. Under this approach, the y-values are assumed to belong to a finite set of possible values {h1, …, hD} for some finite D (unspecified). Then Nt is the scale load of ht and the population mean is expressed in terms of the scale loads as $\bar{Y} =\break N^{ - 1}\sum\_{t = 1}^{D} N\_{t} h\_{t}$. Reduced sample data under simple random sampling without replacement is represented by the sample scale loads nt, t = 1, …, D, and the resulting likelihood function is the hyper-geometric likelihood L(N1, …, ND). If the sampling fraction is negligible, then the likelihood is simply the multinomial likelihood which is the same as the empirical likelihood (EL) of Owen (). In the case of stratified random sampling, the likelihood function is the product of hyper-geometric likelihoods corresponding to the different strata h = 1, …, L. Hartley and Rao focused primarily on design-based inferences, but also briefly studied Bayesian inference under simple random sampling using a compound-multinomial prior on the scale loads $N\_{1},\ldots,\break N\_{D}$. Hoadley () obtained the compound-multinomial prior, denoted CMtn(Nd; νd, d = 1, …, D), by first assuming that the finite population $\{ y\_{i},i \!=\! 1,\ldots,\break N\}$ is a random sample from an infinite population with unknown probabilities $p\_{d} = P(y\_{i} = h\_{d}),d =\break 1,\ldots,D$, and then using a Dirichlet prior with parameters νd ( > 0) on the probabilities pd, d = 1, …, D. The posterior distribution of Nd − nd, d = 1, …, D, given the data nd, d = 1, …, D, is the compound multinomial CMtn(Nd − nd; νd + nd, d = 1, …, D). Using this posterior distribution, Hartley and Rao () obtained the posterior mean and the posterior variance of the population mean Ȳ. Under a diffuse prior with the vd close to zero, the results are identical to those of Ericson (). However, there are fundamental differences in the two approaches in the sense that under exchangeability the conditional distribution of the sample scale loads nd, given the population scale loads Nd, is equal to the hyper-geometric likelihood of Hartley–Rao for any sampling design, whereas in the Hartley–Rao approach this conditional distribution and the resulting posterior of the Nd are derived under simple random sampling, and hence depend on the sampling design. Rao and Ghangurde () studied Bayesian optimal sample allocation, by minimizing the expected posterior variance of the mean, for stratified simple random sampling and some other cases including two-phase sampling to handle the nonresponse problem. Attention was given to data-based priors obtained by combining diffuse priors with likelihoods based on pilot samples. Aitkin () used the scale-load framework and obtained Bayesian intervals on the population mean under simple random sampling, by using a compound-multinomial prior with vd = 0 on the observed scale loads nd > 0 and then simulating a large number of samples from the resulting posterior distribution. This approach is similar to the simulation approach used by Meeden and Vardeman (), but the posterior intervals depend on the design via the likelihood function. As in the case of Meeden and Vardman, the simulation method can be applied to other parameters of interest. Also, the simulation method readily extends to stratified simple random sampling. The scale-load approach is promising but somewhat limited in applicability, in the sense that the scale-load likelihoods cannot be obtained easily for complex sampling designs. To handle complex sampling designs, Rao and Wu () used a pseudo-EL approach, proposed by Wu and Rao (), to obtain “calibrated” pseudo-Bayesian intervals in the sense that the intervals have asymptotically correct designbased coverage probabilities. The pseudo-EL approach uses the survey weights and the design effect (via the effective sample size n\) in defining the profile pseudo-EL function for the mean θ* = Ȳ. Let d̃i(s) = di/∑j ∈ sdj be the normalized weights and n\* = n/(deff), where deff is the ratio of the estimated variance of the weighted mean ∑i ∈ sd̃i(s)yi to the estimate of the variance under simple random sampling. Then the profile pseudo empirical log-likelihood function for θ is given by lPEL(θ) = n\∑i* ∈ sd̃i(s)log{p̂i(θ)}, where the p̂i(θ) maximize ∑i ∈ sd̃i(s)logpi subject to pi > 0, ∑i ∈ spi = 1 and ∑i ∈ spiyi = θ. We refer the reader to Rao and Wu () for an overview of EL methods used for inference from survey data. By combining the profile pseudo-EL function for the population mean from ([e4]) with a flat prior on the mean, one can get pseudo-Bayesian intervals that have asymptotically correct design-based coverage probabilities. Also, it may be easier to specify informative priors on the mean if historical information on the mean is available. The proposed approach can incorporate known auxiliary population information in the construction of pseudo-Bayesian intervals using the basic design weights or using weights already calibrated by the known auxiliary information. The latter is more appealing because, in practice, data files report the calibrated weights. One limitation of the Rao–Wu method for complex designs is that the pseudo-EL depends on the design effects which may not be readily available. Lazar () proposed the Bayesian profile EL approach for the case of independent and identically distributed (i.i.d.) observations. It should be noted that even in the i.i.d. case a “matching” prior on the mean that provides higher order coverage accuracy for the intervals does not exist when using the nonparametric Bayesian profile-EL (Fang and Mukerjee, ). Therefore, the main advantage of the approach is to get “exact” pseudo-Bayesian intervals that are also calibrated in the sense of first order coverage accuracy in the design-based framework. Small Area Estimation: HB Approach ================================== Methods for small area (or domain) estimation have received much attention in recent years due to growing demand for reliable small area statistics. Traditional area-specific direct estimation methods (either design-based or model based or Bayesian) are not suitable in the small area context because of small (or even zero) area-specific sample sizes. As a result, it is necessary to use indirect estimation methods that borrow information across related areas through linking models based on survey data and auxiliary information, such as recent census data and current administrative records. Advocates of design-based methods indeed acknowledge the need for models in small area estimation. For example, () remark, “If the assumed model accurately represents the state of nature, useful inferences can be based on quite small samples at least for certain models.” Linking models based on linear mixed models and generalized linear mixed models with random small area effects are currently used extensively, in conjunction with empirical best linear unbiased prediction (EBLUP), parametric empirical Bayes (EB) and hierarchical Bayes (HB) methods for estimation of small area means and other small area parameters of interest. A detailed treatment of small area estimation methods is given in Rao (). We focus here on HB methods to highlight the significant impact of Bayesian methods on small area estimation. In the HB approach, model parameters are treated as random variables and assigned a prior distribution. Typically, noninformative priors are used, but one must make sure that the resulting posteriors are proper because some priors on the variance parameters can lead to improper posteriors (see Rao,, Section 10.2.4, for a discussion on the choice of priors). The posterior distribution of a small area parameter of interest is then obtained from the prior and the likelihood function generated from the data and the assumed model. Typically, closed-form expressions for desired posterior distributions do not exist, but powerful MCMC methods are now available for simulating samples from the desired posterior distribution and then computing the desired posterior summaries. Rao (, Chapter 10) gives a detailed account of the HB methods in the small area context; see also the review paper by Datta (), Section 3. A significant advantage of the HB approach is that it is straightforward and the inferences are “exact,” unlike in the EB approach. Moreover, it can handle complex small area models using MCMC methods. Availability of powerful MCMC methods and software, such as WinBUGS, also makes HB attractive to the user. Extensive HB model diagnostic tools are also available, but some of the default HB model-checking measures that are widely used may not be necessarily good for detecting model deviations. For example, the commonly used posterior predictive p-value (PPP) for checking goodness of fit may not be powerful enough to detect nonnormality of random effects (Sinharay and Stern, ) because this measure makes “double use” of data in the sense of first generating values from the predictive posterior distribution and then calculating the p-value. Bayarri and Castellanos () say, “Double use of the data can result in an extreme conservatism of the resulting p-values.” Alternative measures, such as the partial PPP and the conditional PPP (Bayarri and Berger, ), attempt to avoid double use of data, but those measures are more difficult to implement than the PPP, especially for the small area models. Browne and Draper () suggested the use of prior-free, frequentist methods in the model exploration phase and then the HB for inference based on the selected models using possibly diffuse priors on the model parameters. However, many Bayesians may not agree with this suggestion because of the orientation of frequentist tests of goodness of fit to rejecting null hypotheses, as noted by a referee. To illustrate the HB approach for small area estimation, we focus on a basic area-level model with two components, a sampling model and a linking model, requiring only area-specific (direct) designbased estimators ȳiw of small area means Ȳi and associated area-level covariates zi (i = 1, …, m). The linking model is of the form g(Ȳi) = zʹiβ + vi, where the random effects vi ∼ i.i.d.N(0, A) and g( ⋅ ) is a specified link function. The sampling model assumes that g(ȳiw) = g(Ȳi) + ẽi, where the sampling errors ẽi∣Ȳi are assumed to be independent N(0, Di) with known sampling variances Di. The assumptions of zero mean sampling errors and known sampling variances may be both quite restrictive in practice. The first difficulty may be circumvented by using the sampling model ȳiw = Ȳi + ei, where the sampling errors ei are assumed to be independent normal with zero means, which simply says that the direct estimators are design unbiased or nearly design unbiased, as in the case of a GREG estimator, for large overall sample size. The second assumption of known sampling variances is more problematic and the usual practice to get around this problem is to model the estimated sampling variances (using generalized variance functions) and then treat the resulting smoothed estimates as the true variances Di. Bell () studied the sensitivity of small area inferences to errors in the specification of the true variances. The original model, called the Fay–Herriot (FH) model, is a matched model in the sense that the sampling model matches the linking model and the combined model is simply a special case of a linear mixed model. On the other hand, the alternative sampling model is not necessarily matched to the linking motirdel and in this case the two models are “mismatched.” For simplicity, we focus on the matched case, but the HB approach readily extends to the more complex case of mismatched models and also to models that allow the sampling variance to depend on the area mean Ȳi (You and Rao, ). Attractive features of area level models are that the sampling design is taken into account through the direct estimators ȳiw and that the direct estimators and the associated area level covariates are more readily available to the users than the corresponding unit level sample data. For example, the U.S. Small Area Income and Poverty Estimation (SAIPE) Program used the FH model to estimate county level poverty counts of school-age children by employing direct estimates for sampled counties from the Current Population Survey and associated county level auxiliary information from tax records, food stamps programs and other administrative sources (see Rao,, Chapter 7, for ). Bayesians have used the area level models extensively through the HB approach, in spite of the limitations mentioned above, because of their practical advantages (Rao,, Chapter 10). In the HB approach, a flat prior on the model parameters β and A is often specified as f(β, A) ∝ f(A) and f(A) ∝ 1, and the resulting posterior summaries (means, variances and credible intervals) for the means Ȳi are obtained. Bell () studied matched models in the context of estimating the proportion of school-age children in poverty at the state level in the US, using the survey proportions ȳiw = piw based on the Current Population Survey (CPS) data for 1989–1993 and area level covariates zi related to Ȳi = Pi. Bell found that the maximum likelihood (ML) and restricted ML (REML) estimates of A turned out to be zero for the first four years (1989–1992) and the resulting EB estimates of state poverty rates attached zero weight to the direct estimate piw regardless of the CPS state sample sizes ni (number of households). This problem with EB based on ML or REML can be circumvented by using the HB approach. Bell used the above flat prior and obtained the posterior mean which always attached nonzero weight to the direct estimate. Further, the posterior variance is well behaved (smallest for California with the largest ni), unlike the estimated mean squared prediction error (MSPE) of the EB estimator. It is possible, however, to develop EB methods that always lead to nonzero estimates of A. Morris () proposed to multiply the residual likelihood function of A by the factor A and maximize this adjusted likelihood function. The resulting estimator of A is always positive and gets around the difficulty with REML. Li and Lahiri () used an adjusted profile likelihood function which also leads to positive estimates of A. They also established asymptotic consistency of the estimator and obtained a nearly unbiased estimator of the mean squared prediction error (MSPE) of the associated EB estimator of Ȳi. () studied frequentist properties of HB by deriving a moment-matching prior on A, in the sense that the resulting posterior variance is nearly unbiased for the MSPE of the HB estimator of the small area mean. The moment-matching prior is given by f(A) ∝ (A + Di)2∑l = 1m(A + Dl)− 2. This prior depends collectively on the sampling variances Dl for all the areas as well as on the area-specific sampling varianceDi. Note that the matching prior is designed for inference on area i and, hence, its dependence on Di should not be problematic. The matching prior ([e5]) reduces to the flat prior f(A) ∝ 1 in the special case of equal sampling variances Di = D. However, in the application considered by Bell (), maxDi/minDi is as large as 20. Ganesh and Lahiri () derived a single matching prior such that a weighted posterior variance over the areas tracks the corresponding weighted MSPE for specified weights. By letting the weights be one for area i and zero for the remaining areas, the resulting prior is identical to ([e5]). Datta () has shown that the previous moment-matching priors also ensure matching property for interval estimation in the sense that the coverage probability of the credible interval tracks the corresponding coverage probability of the normal interval based on the EB estimator and its estimated MSPE. Further work on matching priors in the context of small area estimation would be useful. Mismatched models are often more realistic for practical applications, as they allow flexibility in formulating the linking model. A recent application of HB under mismatched models is to the estimation of adult literacy levels for all states and counties in the US, using data from the National Assessment of Adult Literacy and literacy-related auxiliary data (Mohadjer et al., ). Bizier et al. () used mismatched models and the HB approach to produce estimates of disability rates for health regions and selected municipalities in Canada. A variety of applications of HB under complex modeling have been reported in the literature (see Rao,, Chapter 10, for work prior to 2003). Nandram and Choi () and () studied extensions of HB to handle nonignorable nonresponse and applied the methods to data from the National Health and Nutrition Examination Survey (NHANES III) to produce small area estimates. Raghunathan et al. () applied the HB approach to combine data from two independent surveys [Behavioral Risk Factor Surveillance System (BRFSS) and the National Health Interview Survey (NHIS)] for the years 1997–2000 to produce yearly prevalence estimates at the county level for six outcomes. BRFSS is a large telephone survey covering almost all US counties, but the nonresponse rates are high and also nontelephone households are not covered. On the other hand, NHIS is a smaller personal interview survey with lower nonresponse rates and covers nontelephone households. In this application, direct survey weighted county estimates of proportions from the two surveys were transformed using the inverse sine transformation and the sampling variances were taken as (4ñd)− 1, where ñd denotes the effective sample size for a particular domain d (calculated as the actual domain sample size nd divided by the estimated design effect which is the ratio of the estimated variance under the given design to the binomial estimated variance). The resulting sampling model was then combined with a suitable linking model to obtain county estimates of the prevalence rates, using diffuse proper priors on the model parameters and MCMC. This application attempts to account for possible noncoverage bias and obtain efficient county estimates by combining data from two independent surveys. It may be noted that the model used here is an extension of the basic FH area level model and the application demonstrates how design-based and Bayesian approaches can be fruitfully integrated in small area estimation. HB methods studied in the literature have been largely parametric, based on specified distributions for the data. However, Meeden () extended his noninformative Bayesian approach, based on the Polya posterior (PP), to small area estimation in some simple cases. Extension of this approach to handle complex models is not likely to be easy in practice. Concluding Remarks ================== I have provided an appraisal of the role of Bayesian and frequentist methods in sample surveys. My opinion is that for domains (subpopulations) with sufficiently large samples, a traditional design-based frequentist approach that makes effective use of auxiliary information, through calibration or assistance from working models, will remain as the preferred approach in the large-scale production of official statistics from complex surveys. Nonsampling errors can be handled using a combined design and model approach with minimal use of models. But the designbased approach, using survey weights, is not a panacea even for large samples and yet “many people ask too much of the weights” (Lohr, ), prompting statements like, “Survey weighting is a mess” (Gelman, ). As Lohr () noted, survey weighting is not a mess as long as the weighting is not stretched to a limit as in the case of a very large number of post-stratified cells leading to very small or even zero cell sample sizes, thus making weighting at the cell level unstable or even not feasible (Gelman, ). Alternative weighting methods can be used in those situations to get around this problem. For example, by calibrating to the marginal counts of the post-stratification variables instead of the cell counts leads to a calibration estimator with stable weights which should perform well for estimating population totals or means. Also, the resulting weights do not depend on the response values, thus ensuring internal consistency, unlike the hierarchical regression method proposed by Gelman () based on models involving random effects. Recent work on nonparametric Bayesian methods that can be used for both Bayesian and design-based inferences looks promising, at least for some specialized surveys. For small area estimation, the hierarchical Bayes (HB) approach offers a lot of promise because of its ability to handle complex small area models and provide “exact” inferences. However, the choice of noninformative priors that can provide frequentist validity is not likely to be easy in practice when complex modeling is involved. Also, caution needs to be exercised in the routine use of popular HB model-checking methods. Acknowledgments =============== This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. My thanks are due to an associate editor and two referees for constructive comments and suggestions. 107 [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] (). [auto:STB\|2010-11-18\|09:18:59] ().. In., [mr] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59],, ().. [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ().,. [mr], ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] ().. [auto:STB\|2010-11-18\|09:18:59], ()... [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ().. [auto:STB\|2010-11-18\|09:18:59] ().. 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[auto:STB\|2010-11-18\|09:18:59] ().. In.. [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59], ()... [auto:STB\|2010-11-18\|09:18:59], ()... [mr] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ().. In. ().,. [mr], ()... [mr],,, ()... [mr] ()... [mr],,,,, ()... [auto:STB\|2010-11-18\|09:18:59] ().. [mr] ()..,. [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ()... [mr] ().. In. [mr] (). [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59], ()... [auto:STB\|2010-11-18\|09:18:59], (). [auto:STB\|2010-11-18\|09:18:59],, ()... [auto:STB\|2010-11-18\|09:18:59], ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ()... [mr] ()... [mr] ()..,. [auto:STB\|2010-11-18\|09:18:59] ()... [mr], ()..,. [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [mr], ()..,. [mr] ()... [mr] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] ()... [auto:STB\|2010-11-18\|09:18:59] ()... Impact of Frequentist and Bayesian Methods on Survey Sampling Practice: A Selective Appraisal ============================================================================================= According to Hansen, Madow and Tepping [J. Amer. Statist. Assoc. 78 (1983) 776–793], “Probability sampling designs and randomization inference are widely accepted as the standard approach in sample surveys.” In this article, reasons are advanced for the wide use of this design-based approach, particularly by federal agencies and other survey organizations conducting complex large scale surveys on topics related to public policy. Impact of Bayesian methods in survey sampling is also discussed in two different directions: nonparametric calibrated Bayesian inferences from large samples and hierarchical Bayes methods for small area estimation based on parametric models. . Introduction ============ Sample surveys have long been conducted to obtain reliable estimates of finite population descriptive parameters, such as totals, means, ratios and quantiles, and associated standard errors and normal theory intervals with large enough sample sizes. Probability-sampling designs and randomization (repeated sampling) inference, also called the design-based approach, played a dominant role, especially in the production of official statistics, ever since the publication of the landmark paper by Neyman () which laid the theoretical foundations of the design-based approach. Neyman’s approach was almost universally accepted by practicing survey statisticians and it also inspired various important theoretical contributions, mostly motivated by practical and efficiency considerations. In this paper I will first provide some highlights of the design-based approach, for handling sampling errors, to demonstrate its significant impact on survey sampling practice, especially on the production of official statistics (Sections [2] and [3.1]). Model-dependent approaches (Section [3.2]) that lead to conditional inferences more relevant and appealing than repeated sampling inferences have also been advanced (Brewer, ; Royall, ). Unfortunately, for large samples such approaches may perform very poorly under model misspecifications; even small model deviations can cause serious problems (, ). On the other hand, model-dependent approaches can play a vital role in small area (domain) estimation, where the area-specific sample sizes are very small or even zero and make the design-based area-specific direct estimation either very unreliable or not feasible. Demand for reliable small area statistics has greatly increased in recent years and to meet this growing demand, federal statistical agencies and other survey organizations are currently paying considerable attention to producing small area statistics using models and methods that can “borrow strength” across areas. Hierarchical Bayes (HB) model-dependent methods are particularly attractive in small area estimation because of their ability to handle complex modeling and provide “exact” inferences on desired parameters (Section [5]). I will highlight some HB developments in small area estimation that seem to have a significant impact on survey practice. I will also discuss the role of nonparametric Bayesian methods for inferences, based on large area-specific sample sizes, especially those providing Bayesian inferences that can be also justified under the design-based framework (Section [4.2]). Models are needed, regardless of the approach used, to handle nonsampling errors that include measurement errors, coverage errors and missing data due to nonresponse. In the design-based approach, a combined design and modeling approach is used to minimize the reliance on models, in contrast to fully model-dependent approaches (Section [3.4]). For simplicity, I will focus on descriptive parameters, but survey data are also increasingly used for analytical purposes, in particular, to study relationships and making inferences on model parameters under assumed super-population models. For example, social and health scientists are interested in fitting linear and logistic regression models to survey data and then making inferences on the model parameters taking account of the survey design features (Section [3.3]). Design-Based Approach: Early Landmark Contributions =================================================== In this section I will highlight some early landmark contributions to the design-based approach that had major impact on survey practice. Prior to Neyman (), sampling was implemented either by “balanced” sampling through purposive selection or by probability sampling with equal inclusion probabilities. Such a method was called the “representative method.” Bowley () studied stratified sampling with proportional sample size allocation, leading to a representative sample with equal probabilities. Neyman () broke through this restrictive setup by relaxing the condition of equal probabilities and introducing the ideas of efficiency and optimal sample size allocation in his theory of stratified random sampling. He also demonstrated that balanced purposive sampling may perform poorly if the underlying model assumptions are violated. Neyman proposed normal theory confidence intervals for large samples such that the frequency of errors in the confidence statements based on all possible stratified random samples that could be drawn does not exceed the limit prescribed in advance “whatever the unknown properties of the finite population.” He broadened the definition of representative method by calling any method of sampling that satisfies the above frequency statement as representative. It is interesting to note that Neyman advocated distribution-free design-based inferences for survey sampling in contrast to his own fundamental work on parametric inference, including the Neyman–Pearson theory of hypothesis testing and confidence intervals. The possibility of developing efficient probability sampling designs by minimizing total cost subject to a specified precision of an estimator or maximizing precision for a given cost, taking account of operational considerations, and making distribution-free inferences (point estimation, variance estimation and large sample confidence intervals) through the design-based approach were soon recognized.This, in turn, led to a significant increase in the number and type of surveys taken by probability sampling and covering large populations. In the early stages, the primary focus was on sampling errors. I now list a few important post-Neyman theoretical developments in the design-based approach. As early as 1937, Mahalanobis used multistage sampling designs for crop surveys in India. His classic 1944 paper (Mahalanobis, ) presents a rigorous theoretical setup and a generalized approach to the efficient design of sample surveys of different crops in Bengal, India, with emphasis on variance and cost functions. Mahalanobis considered a geographical region of finite area and defined a field consisting of “a finite number, say, N0, of basic cells arranged in a definite space or geographic order together with a single value (or a set of values in the multivariate case) of z for each basic cell,” where z is the variable of interest (say, crop yield). Under this setup, he studied four different probability sampling designs for selecting a sample of cells (called quads): unitary unrestricted, unitary configurational, zonal unrestricted and zonal configurational. In modern terminology, the four designs correspond to simple random sampling, stratified random sampling, single-stage cluster sampling and single-stage stratified cluster sampling, respectively. He developed realistic cost functions depending on particular situations. He also extended the theoretical setup to subsampling of clusters (which he named as two-stage sampling). We refer the reader to Murthy () for a detailed account of the 1944 paper and other contributions of Mahalanobis to sample surveys. Hall () provides a scholarly historical account of the pioneering contributions of Mahalanobis to the early development of survey sampling in India. Mahalanobis was instrumental in establishing the National Sample Survey of India and the famous Indian Statistical Institute. Survey statisticians at the U.S. Census Bureau, under the leadership of Morris Hansen, made fundamental contributions to survey sampling theory and practice during the period 1940–1970, and many of those methods are still widely used in practice. This period is regarded as the golden era of the Census Bureau. Hansen and Hurwitz () developed the basic theory of stratified two-stage cluster sampling with one cluster or primary sampling unit (PSU) within each stratum drawn with probability proportional to a size measure (PPS) and then subsampled at a rate that ensures self-weighting (equal overall probabilities of selection). This method provides approximately equal interviewer work loads which are desirable in terms of field operations. It can also lead to significant variance reduction by controlling the variability arising from unequal PSU sizes without actually stratifying by size and thus allowing stratification on other variables to further reduce the variance. The Hansen–Hurwitz method, with some modifications, has been widely used for designing large-scale socio-economic, health and agricultural surveys throughout the world. Many large-scale surveys are repeated over time, such as the monthly Current Population Survey (CPS), and rotation sampling with partial replacement of ultimate units (e.g., households) is used to reduce response burden. Hansen et al. () developed simple but efficient composite estimators under rotation sampling in the context of stratified multistage sampling. Rotation sampling and composite estimation are widely used in large-scale surveys. Prior to the 1950s, the primary focus was on estimating totals, means and ratios for the whole population and large planned subpopulations such as US states or provinces in Canada. Woodruff () developed a unified design-based approach for constructing confidence intervals on quantiles using only the estimated distribution function and the associated standard error. This ingenious method is to general probability sampling designs and performs well in terms of coverage probabilities in many cases. Woodruff intervals can also be used to obtain standard errors of estimated quantiles (Rao and Wu, ; Francisco and Fuller, ). Because of those features, the Woodruff method had a significant impact on survey practice. However, the method should not be treated as a black box for constructing confidence intervals on quantiles, because it can perform poorly in some practical situations. For example, it performed very poorly under stratified random sampling when the population is stratified by a concomitant variable x highly correlated with the variable of interest y (, ). The failure of the Woodruff method in this case stems from the fact that the standard error of the estimated distribution function at the quantile will be too small due to zero contributions to the standard error from most strata. () showed that the bootstrap method for stratified random sampling performs better than the Woodruff method in this case, but in other situations the Woodruff method is better. Attention was also given to inferences for unplanned subpopulations (also called domains) such as age–sex groups within a state. Hartley () and Durbin () developed simple, unified theories for domain estimation applicable to general designs, requiring only existing formulae for population totals and means. After the consolidation of basic design-based sampling theory, Hansen et al. () and others paid attention to measurement errors in surveys. They developed basic theories under additive measurement error models with minimal model assumptions on the observed responses treated as random variables. Total variance of an estimator is decomposed into sampling variance, simple response variance and correlated response variance (CRV) due to interviewers. The CRV was shown to dominate the total variance when the number of interviewers is small, and the 1950 U.S. Census interviewer variance validation study showed that this component is indeed large for small areas. Partly for this reason, self-enumeration by mail was first introduced in the 1960 U.S. Census to reduce the CRV component. Earlier, Mahalanobis () developed the method of interpenetrating subsamples (called replicated sampling by Deming, ) and used it extensively in large-scale surveys in India for assessing both sampling and interviewer errors. By assigning the subsamples at random to interviewers, the total variance can be estimated and interviewer differences assessed. It should be clear from the above brief description of early developments that much of the basic sampling theory was developed by official statisticians or those closely associated with official statistics. Theory was driven by the need to solve real problems and often theory was not challenging enough to attract academic researchers to survey sampling. As a result, university researchers paid little attention to survey sampling in those days with few exceptions (e.g., Iowa State University under the leadership of Cochran, Jessen and Hartley). Some Recent Design-Based and Non-Bayesian Developments ====================================================== Model-Assisted Approach ----------------------- We first give a brief account of the model-assisted approach that uses a working model to find efficient estimators. However, the associated inferences are design-based. Consider a finite population U consisting of N elements labeled 1, …, N with associated values y1, …, yN of a variable of interest y. Under a probability sampling design, the inclusion probabilities π1, …, πN are all strictly positive and a basic estimator of the total Y = ∑i ∈ Uyi is of the form Ŷ = ∑i ∈ sdiyi, where s denotes a sample and di = πi− 1 are the so-called design weights (Horvitz and Thompson, ; Narain, ). For example, in the Neyman stratified random sampling design, the design weights are equal to the inverse of the sampling fractions within strata and vary across strata, while in the Hansen et al. two-stage cluster sampling design, the design weights are all equal. Design unbiasedness of estimators is not insisted upon (contrary to statements in some papers on inferential issues of sampling theory) because it “often results in much larger MSE than necessary” (, ). Instead, design consistency is deemed necessary for large samples. Strategies (design and estimation) that appeared reasonable are entertained (accounting for costs) and relative properties are carefully studied by analytical and/or empirical methods, mainly through the comparison of mean squared error (MSE) or anticipated MSE under plausible population models on the variables yi treated as random variables. This is essentially the basis of the repeated sampling (or designbased) approach. In recent years, a model-assisted repeated sampling approach has received the attention of survey practitioners. In this approach, a working population model is used to find efficient design-consistent estimators. For example, suppose the working model is a linear regression model of the form yi = xʹiβ + ɛi; i = 1, …, N, with model errors ɛi assumed to be uncorrelated with mean zero and variance proportional to a known constant qi, where xi is a vector of auxiliary variables with known population total X. Under model ([eq1]), the best linear unbiased estimator (BLUE) of the model parameter β, based on the census values {(yi, xi); i ∈ U}, is given by the “census” regression coefficient $$B = \biggl(\sum\_{i \in U} x\_{i} x'\_{i}/q\_{i}\biggr)^{ - 1}\biggl(\sum\_{i \in U} x\_{i} y\_{i}/q\_{i}\biggr).$$ A predictor of yi under the working model is then given by ŷi = xʹiB̂ for i = 1, …, N where B̂ is the design-weighted estimator of B: $$\hat{B} = \biggl(\sum\_{i \in s} d\_{i} x\_{i}x'\_{i}/q\_{i}\biggr)^{ - 1}\biggl(\sum\_{i \in s} d\_{i} x\_{i}y\_{i}/q\_{i}\biggr).$$ By writing the total as Y = ∑i ∈ Uŷi + ∑i ∈ Uei where ei = yi − ŷi denotes the prediction error, a design-based estimator of Y is given by ∑i ∈ Uŷi + ∑i ∈ sdiei. We can express this estimator as a generalized regression (GREG) estimator Ŷgr = Ŷ + B̂ʹ(X − X̂), where X̂ = ∑i ∈ sdixi (Särndal, Swensson and Wretman, ). The GREG estimator ([e2]) is design-consistent regardless of the validity of the working model (Robinson and Sarndal, ) under certain regularity conditions provided X is precisely correct. If the working model provides a good fit to the data, then the residuals ei should be less variable than the response values yi and the GREG estimator is likely to be significantly more efficient than the basic design-weighted estimator Ŷ. The estimator ([e2]) may also be expressed as a weighted sum ∑i ∈ swiyi, where wi = digi with $$\label{e3} g\_{i} = 1 + (X - \hat{X})'\biggl(\sum\_{i \in s} d\_{i} x\_{i}x'\_{i}/q\_{i}\biggr)^{ - 1}x\_{i}q\_{i}^{ - 1}.$$ The adjustment factors gi, popularly known as the g-weights, ensure the calibration property ∑i ∈ swixi = X so that the GREG estimator when applied to the sample values xi agrees with the known total X. This property is attractive to the user when the vector X contains user-specified totals. The assumption of a working linear regression model ([eq1]) can be relaxed by adopting more flexible working models. For example, Breidt, Claeskens and Opsomer () proposed a nonparametric model-assisted approach based on a penalized spline (P-spline) regression working model and showed that the resulting estimators are design-consistent and more efficient than the usual GREG estimators based on linear regression working models when the latter are incorrectly specified. Also, the P-spline model-assisted estimators were shown to be approximately as efficient as the GREG estimators when the linear regression working model is correctly specified. The P-spline approach can be easily implemented using existing estimation packages for GREG because the underlying model is closely related to a linear regression model. It offers a wider scope for the model-assisted approach because it makes minimal assumptions on the regression of y on x without assuming a specific parametric form. Under the model-assisted approach, design-consistent variance estimators are obtained either by a Taylor linearization method or by a resampling method (when applicable), provided the probability sampling design ensures strictly positive joint inclusion probabilities πij, i ≠ j. Using the estimator and associated standard error, asymptotically valid normal theory intervals are obtained regardless of the validity of the working model. Most large-scale surveys are multipurpose and observe multiple variables of interest, and the same working model may not hold for all the variables of interest. In that case, a model-assisted approach may lead to possibly different gi and hence different calibration weights wi associated with the variables, that is, the calibration weights are of the form wij associated with the variable j and sample unit i. However, survey users prefer to use a commonweight wi for all variables of interest. This is often accomplished by minimizing a suitable distance measure between di and wi for i ∈ s subject to userspecified calibration constraints, say, ∑i ∈ swizi = Z, without appealing to any working model, where Z is the vector of known totals associated with the user-specified variables z. For example, a chi-squared distance measure leads to common calibrationweights wi of the form digi where gi is given by ([e3]) with xi replaced by zi (Deville and Sarndal, ). Thus, calibration estimation in this case corresponds to using model-assisted estimation based on a linear regression model ([eq1]) with zi as the vector of predictor variables. Calibration estimation has attracted the attention of users due to its ability to produce common calibration weights and accommodate an arbitrary number of user-specified calibration (or benchmark) constraints, for example, calibration to the marginal counts of several post-stratification variables. Several national statistical agencies have developed software designed to compute calibration weights: GES (Statistics Canada), LIN WEIGHT (Statistics Netherlands), CALMAR (INSEE, France) and CLAN97 (Statistics Sweden). Sarndal () says, “Calibration has established itself as an important methodological instrument in large-scale production of statistics.” Brakel and Bethlehem () noted that the use of common calibration weights for estimation in multipurpose surveys makes the calibration method “very attractive to produce timely official releases in a regular production environment.” Unfortunately, the model-free calibration approach can lead to erroneous inferences for some of the response variables, even in fairly large samples if the underlying working linear regression model uses an incorrect or incomplete set of auxiliary variables, unlike the model-assisted approach that uses a working model obtained after some model checking. For example, suppose that the underlying model is a quadratic regression model of y on x and the distribution of x is highly skewed. Also, suppose that the user-specified calibration constraints are the known population size N and the known population total X. In this case, the calibration estimator of the total Y under simple random sampling is the familiar simple linear regression estimator with x as the predictor variable. On the other hand, a model-assisted estimator under the quadratic regression working model is given by a multiple linear regression estimator with x1 = x and x2 = x2 as the predictor variables, assuming the total of x2 is also known. () demonstrated that the coverage performance of the normal theory interval associated with the calibration estimator is poor even in fairly large samples, unlike the coverage performance of the normal theory intervals associated with the model-assisted estimator. The coverage performance depends on the skewness of the residuals from the fitted model and in the case of calibration estimation the skewness of residuals after fitting simple linear regression remains large, whereas the skewness of residuals after fitting quadratic regression is small even if y and x are highly skewed. This simple example demonstrates that the population structure does matter in design-based inferences and that it should be taken into account through a model-assisted approach based on suitable working models. But the model-assisted approach has the practical limitation that the weight wi may vary across variables in surveys with multiple variables of interest, unlike in the calibration approach. Also, for complex working models, such as the P-spline, all the population values of the predictor variables should be known in order to implement the model-assisted approach. The model-assisted approach is essentially design-based, unlike the model-dependent approach (Section [3.2]) that can provide conditional inferences referring to the particular sample, s, of units selected. Such conditional inferences may be more relevant and appealing than the unconditional repeated sampling inferences used in the design-based approach. Model-Dependent Approach ------------------------ The frequentist model-dependent approach to inference assumes that the population structure obeys a specified population model and that the same model holds for the sample, that is, no sample selection bias with respect to the assumed population model. Sampling design features are often incorporated into the model to reduce or eliminate the sample selection bias (see Section [3.3] for some difficulties to implement this in practice). Typically, distributional assumptions are avoided by focusing on point estimation, variance estimation and associated normal theory confidence intervals, as in the case of design-based inferences. As a result, models used specify only the mean function and the variance function of the variable of interest, y.We refer the reader to Valliant, Dorfman and Royall () for an excellent account of the model-dependent approach. As noted in Section [1], model-dependent strategies may perform poorly in large samples when the population model is not correctly specified; even small deviations from the assumed model that are not easily detectable through routine model checking can cause serious problems. In the Hansen, Madow and Tepping () example of an incorrectly specified population model, the best linear unbiased prediction (BLUP) estimator of the mean is not design consistent under their stratified simple random sampling design with near optimal sample allocation (commonly used to handle highly skewed populations such as business populations). As a result, model-dependent confidence intervals exhibited poor performance: for n = 100, coverage was around 70% compared to nominal level of 95%, while coverage for model-assisted intervals was 94.4%. To get around this difficulty, Little () proposed restricting attention to models that hold for the sample and for which the BLUP estimator is design consistent. For example, in the () example, the BLUP of the mean under a model with means differing across strata is identical to the traditional stratified mean which is design consistent. But it seems not possible even to find a suitable model under which the widely used combined ratio estimator of the mean under the stratified random sampling is the BLUP estimator. The combined ratio estimator is a model-assisted estimator under a ratio working model with a common slope. It allows a large number of strata with few sample units from each stratum and yet remains design consistent, unlike the separate ratio estimator which is the BLUP estimator under a ratio model with separate slopes across strata: E(yhi∣xhi) = βhxhi, V(yhi∣xhi) = σ2xhi, where yhi and xhi denote the values of the variable of interest y and an auxiliary variable x for the unit i in stratum h. Moreover, the BLUP estimator under this model requires the knowledge of the strata population means X̄h, whereas the combined ratio estimator requires only the overall population mean X̄. It is also not clear how one proceeds to formulate suitable models for general sampling designs that lead to design-consistent BLUP estimators. Further, the main focus has been on point estimation and it is not clear how one should proceed with variance estimation and setting up confidence intervals that have repeated sampling validity. In this context, Pfeffermann () says, “I presume that these are supposed to be computed under the corrected model as well. Are we guaranteed that they are sufficiently accurate under the model? Do we need to robustify them separately?” Little (), in his rejoinder to Pfeffermann’s comment, says that he advocates using some replication method for variance estimation and then appealing to normal approximation for confidence intervals. Clearly, further work is needed to address the above issues. Note that if parametric assumptions are made, such as normality of model errors, then it is possible to make exact Bayesian inferences by introducing suitable priors on the model parameters (Section [4]). Some recent work on the model-dependent approach focused on avoiding misspecification of the mean function E(y∣x) = m(x) by using P-spline models. Zheng and Little (, ) studied single stage PPS sampling using a P-spline model, based on the size measure x used in PPS sampling, to represent the regression function m(x), and a specified function of the size measure as the variance function. In a simulation study, they compared the performance of the usual linear GREG estimator and the P-spline model-based estimators, not necessarily design-consistent, and showed that the P-spline model-based estimators are generally more efficient than the GREG or the NHT estimator in terms of design MSE even for large samples. However, the simulation study did not consider model-assisted estimators corresponding to their P-spline model. The simulations also showed that the design-bias for their P-spline estimators is minor, even though the estimators are not design-consistent, and hence the authors conclude that “design consistency may not be of paramount importance.” On the other hand, in the () study, their model-assisted P-spline estimator is sometimes much better, and never worse, than the corresponding P-spline model-based estimator under stratified random sampling. The latter estimator is not design-consistent under the P-spline model considered by (). As noted above, a main advantage of the frequentist model-dependent approach is that it leads to inferences conditional on the selected sample of units, s, unlike the unconditional design-based approach. However, it is possible to develop a conditional model-assisted approach that allows us to restrict the reference set of samples to a “relevant” subset of all possible samples specified by the design. Conditionally valid inferences for large samples can then be obtained. Rao () and Casady and Valliant () developed an “optimal” linear regression estimator that is asymptotically valid under the conditional setup. We refer the reader to Kalton () for compelling arguments for favoring design-based approaches (possibly model-assisted and/or conditional) to handle sampling errors. Smith () named the traditional repeated sampling inference as “procedural inference” and argued that procedural inference is the correct approach for surveys in the public domain. Analysis of Complex Survey Data ------------------------------- Data collected from large-scale socio-economic,health and other surveys are being extensively used for analysis purposes, such as inferences on the regression parameters of linear and logistic regression population models. Ignoring the survey design features and using standard methods can lead to erroneous on model parameters because of sample selection bias caused by informative sampling. It is tempting to expand the models by including among the predictors all the design variables that define the selection process at the various levels and then ignore the design and apply standard methods to the expanded model. The main difficulties with this approach, advocated by some leading researchers, are the following, among others (Pfeffermann and Sverchkov, ): (1) Not all design variables may be known or accessible to the analyst. (2) Too many design variables can lead to difficulties in making inferences from the expanded models. (3) The expanded model may no longer be of scientific interest to the analyst. The design-based approach can provide asymptotically valid repeated sampling inferences without changing the analyst model. A unified approach based on survey-weighted estimating equations leads to design-consistent estimators of the “census” or finite population parameters, which in turn estimate the associated model parameters. Further, using resampling methods for variance estimation, such as the jackknife and the bootstrap for survey data, asymptotically valid design-based inferences on the census parameters can be implemented. The same methods may also be applicable for inference on the model parameters, in many cases of large-scale surveys. In the other cases, it is necessary to estimate the model variance of the census parameters from the sample. The estimate of the total variance is then given by the sum of this estimate and the resampling variance estimate. In practice, the data file would contain for each sampled unit the variables of interest and predictor variables, final weights after adjustment for unit nonresponse and the corresponding replication weights, for example, bootstrap weights. The analyst can use software that handles survey weights (such as SAS) to obtain point estimates from the final weights and the corresponding point estimates for each bootstrap replicate using bootstrap weights. The variability of the bootstrap point estimates provides asymptotically valid standard errors for designs commonly used in large-scale surveys. Details of the methods are not provided due to space limitations, but the reader is referred to Rao (, Section 6) for a succinct account of analysis of survey data using resampling methods for variance estimation and normal theory confidence intervals. Design-based approach using resampling methods is extensively used in practice and software is also available (e.g., WesVar, Stata). The design-based approach has also been applied to make inferences on the regression parameters and the variance parameters of multilevel models from data obtained from multistage sampling designs corresponding to the levels of the model. For example, in an education study of students, schools (first-stage units or clusters) may be selected with probabilities proportional to school size and students (second-stage units) within selected schools by stratified random sampling. Again, ignoring the sampling design and using traditional methods for multilevel models that ignore the design can lead to erroneous inferences in the presence of sample selection bias. In the design-based approach, estimation of variance parameters of the model is more difficult than that of regression parameters and the necessary information for estimating variance parameters is often not provided in public-use data files which typically report only the final weight for each sample unit. Widely used design-based methods have been proposed in the literature (e.g., Pfeffermann et al.,, and Rabe-Hesketh and Skrondal, ) to handle variance parameters that require the weights within sampled clusters in addition to the weights associated with the clusters. Some of those methods can be implemented using the Stata program gllamm. Unfortunately, the resulting estimators of variance parameters may not be design-model consistent when the sample sizes within clusters are small, even for two-level linear models. Korn and Graubard () demonstrated the bias problem and proposed a different method for simple two-level or three-level models involving only a common mean as the fixed effect. This method first obtains the census parameters and then estimates those parameters. It worked well in empirical studies even for small within-cluster sample sizes. Rao, Verret and Hidiroglou () proposed a weighted estimating equations (WEE) approach for general two-level linear models that uses within-cluster joint inclusion probabilities, similar to Korn and Graubard (). The WEE method leads to design-model consistent estimators of variance parameters even for smallwithin-cluster sample sizes, provided the number of sample clusters is large. It performed well in empirical studies compared to the other methods proposed in the literature. () also proposed a unified approach based on a weighted log-composite likelihood that can handle generalized linear multilevel models and small within-cluster sample sizes. This method is currently under investigation. A drawback of the design-based approach to the analysis of survey data is that it may lead to loss in efficiency when the final weights vary considerably across the sampling units. Alternative approaches that can reduce the variability of the weights and thus lead to more efficient estimators have also been proposed (e.g., Pfeffermann and Sverchkov, ; Fuller,, Chapter 6). We refer the reader to Pfeffermann () and Rao et al. () for overviews on the role of sampling weights in the analysis of survey data. Nonsampling Errors ------------------ Survey practitioners have to rely on models, regardless of the approach used, to handle nonsampling errors that include measurement errors, coverage errors and missing data due to unit nonresponse and item nonresponse. In the design-based approach, a combined design and modeling approach is used to minimize the reliance on models, in contrast to fully model-dependent approaches that will have similar difficulties noted in the previous subsections. As mentioned in Section [1], Hansen et al. () studied measurement errors under minimal model assumptions on the observed responses treated as random variables, and their discovery that the correlated response variance due to interviewers dominates the total variance when the number of interviewers is small led to the adoption of self enumeration by mail in the 1960 U.S. Census. Inference in the presence of missing survey data, particularly item nonresponse, has attracted a lot of ; see Little and Rubin () for an excellent account of missing data methods. To handle item nonresponse, imputation of missing data is often used because of its practical advantages. In the design-based approach, traditional weighted estimators of a total or a mean are computed from the completed data set, leading to an imputed estimator. Often imputed values are generated from an imputation model assumed to hold for the respondents under a missing at random (MAR) response mechanism. Under this setup, the imputed estimator is unbiased or asymptotically unbiased under the combined design and model set up. Reiter, Raghunathan and Kinney () demonstrated the importance of incorporating sampling design features into the imputation model in order to make the model hold for the sample and then for the respondents under the assumed MAR response mechanism. An alternative approach avoids imputation models but assumes a model for the response mechanism. For example, a popular method consists of forming imputation classes (according to the values of estimated response probabilities under a specified response model) and assuming that the missing values are missing completely at random (MCAR) within classes. The missing values are then imputed by selecting donors at random from the observed values within classes. It may be also possible to develop imputation methods that make an imputed estimator doubly robust in the sense that it is valid either under an assumed imputation model or under an assumed response mechanism (e.g., see Haziza and Rao, ). Doubly robust estimation has attracted considerable attention in the nonsurvey literature (see, e.g.,, ). Variance estimation under imputation for missing survey data has attracted a lot of attention because treating the imputed values as if observed and then applying standard variance formulae can often lead to serious underestimation because the additional variability due to estimating the missing values is not taken into account. Methods that can lead to asymptotically valid variance estimators under single imputation for missing data have been proposed under the above setups. We refer the reader to Kim and Rao () for a unified approach to variance estimation under the imputation model approach, and to Haziza () for an excellent overview of imputation for survey data and associated methods for inference. Rubin () proposed multiple imputation to account for the underestimation when applying standard formulae treating the imputed values as if observed. Under this approach, M ( ≥ 2) imputed values are generated for a missing item, leading to M completed data sets. Rubin recommends the use of traditional design-based estimators and variance estimators, computed from each of the completed data sets, although multiple imputation ideas are based on a Bayesian perspective: “We restrict attention to standard scientific surveys and standard complete data statistics” (Rubin,, page 113). Multiple imputation estimator ŶMI of a total Y is taken as the average of the M estimators ŶI1, …, ŶIM, and its estimator of variance is given by vMI = v̄M + (1 + M− 1)bM, where v̄M is the average of the M naïve variance estimators vI1, …, vIM and bM = (M − 1)− 1∑m = 1M(ŶIm − ŶMI)2. Rubin gives design-based conditions for “proper imputation” that ensure the repeated sampling validity of the estimator ŶMI and the associated variance estimator vMI under a posited response mechanism. Unfortunately, there are some difficulties in developing imputation methods satisfying Rubin’s conditions for “proper imputation” with complex survey data (see, e.g., Kim et al., ). Nevertheless, multiple imputation shows how Bayesian ideas can be integrated to some extent with the traditional design-based approach that is widely used in practice. Bayesian Approaches =================== This section provides an account of both parametric and nonparametric Bayesian (and pseudo-Bayesian) approaches to inference from survey data, focusing on descriptive finite population parameters. Parametric Bayesian Approach ---------------------------- As noted in Section [3.2], the frequentist model-dependent approach mostly avoided distributional assumptions by specifying only the mean function and the variance function of the variable of interest. Under a specified distribution on the assumed model, Bayesian inferences can be easily implemented, provided the model holds for the sample. Royall and Pfeffermann () studied Bayesian inference on the population mean assuming normality and flat (diffuse) priors on the parameters of a linear regression model. Their focus was on the posterior mean and the posterior variance and, hence, results were similar to those of Royall () without the normality assumption and priors on model parameters. However, exact credible intervals on the mean and other parameters of interest can be obtained conditional on the observed data, using a parametric Bayesian setup. It can be implemented even under complex modeling, using powerful Monte Carlo Markov chain (MCMC) methods to simulate samples from the posterior distributions of interest. Scott and Smith () obtained the posterior mean and the posterior variance of the population mean under linear models with random effects, normality and diffuse priors on the model parameters. Their posterior mean is also the BLUP estimator without the normality assumption when the variance parameters of the model are known. In the frequentist approach, estimates of variance parameters are substituted in the BLUP to get the empirical BLUP (EBLUP) estimator which is different (but close to) to the posterior mean. However, the Bayesian approach also provides the posterior variance which is typically different from the estimated mean squared prediction error (MSPE) of the EBLUP estimator; several different methods of estimating MSPE have been proposed in the context of small area estimation (Rao,, Chapter 7). A simulation study by Bellhouse and Rao () showed that any gain in efficiency of the posterior mean (or the BLUP) over traditional design-based estimators is likely to be small in practice. However, by regarding the clusters as small areas of interest, the Scott–Smith approach provides models linking the small areas and the resulting estimators of small area means can lead to significant efficiency gains over direct area-specific estimators. Random cluster effect models are now extensively used to construct efficient small area estimators by “borrowing strength” across small areas using auxiliary information (Section [5]). It may be noted that the empirical Bayes (EB) approach to inference from random cluster effect models is similar to EBLUP, but it can handle general parametric random cluster effect models and does not require the linearity assumption used in the BLUP method. The EB approach is essentially frequentist, unlike the Bayesian approach that requires the specification of priors on the model parameters. It may be more appropriate to name “empirical Bayes” as “empirical best” without changing the abbreviation EB (Jiang and Lahiri, ). Sedransk () studied regression models with random slopes β1, …, βL ∼ i.i.d.N(ν, σβ2), using a prior distribution on ν specified as N(β0, σ02). He then followed the Scott and Smith () approach and obtained posterior mean and posterior variance of the finite population total. He applied the method to data on banks from the U.S. Federal Reserve Board to estimate a current monetary total making use of extensive historical data to specify the values of β0, σ02 and thus arrive at an informative prior which in turn leads to more efficient posterior inferences compared to those based on a noninformative prior, provided the informative prior is correctly specified. Malec and Sedransk () extended Scott–Smith results to three-stage sampling. () applied the Bayesian approach to obtain order-restricted estimators of the age composition of a population of Atlantic cod, using MCMC methods. Sedransk () lists possible uses of parametric Bayesian methods for sample surveys, including the above application to estimation from establishment surveys, “optimal” sample allocation and small area estimation from data pooled from independent surveys (see Section [5]). () report an interesting application of the Bayesian approach to make inferences from multilevel models under informative sampling. In this case, the multilevel sample model induced by informative sampling is more complicated than the corresponding population model and, as a result, frequentist methods are difficult to implement. On the other hand, the authors show that the Bayesian approach, using noninformative priors on the model parameters indexing the sample model and applying MCMC methods, is efficient and convenient for handling such complex sample models, although computer intensive. This application is an example where the Bayesian approach offers computational advantage over the corresponding frequentist approach. Nonparametric Bayesian Approaches --------------------------------- For multipurpose large-scale surveys, parametric Bayesian methods based on distributional assumptions have limited value because of the difficulties in validating the parametric assumptions. It may be more appealing to use a nonparametric Bayesian approach, but this requires the specification of a nonparametric likelihood function based on the full sample data {(i, yi), i ∈ s} and a prior distribution on the parametric vector (y1, …, yN). The likelihood function based on the full sample data, however, is noninformative in the sense that all possible unobserved values of the parameter vector have the same likelihood function (Godambe, ). One way out of this difficulty is to take a Bayesian route by assuming an informative (exchangeable) prior on the N-dimensional parameter vector and combine it with the noninformative likelihood (Ericson, ; Binder, ) to get an informative posterior, but inferences do not depend on the sample design; Ericson argued that an exchangeable prior assumption may be reasonable under simple random sampling. Ericson () focused on the posterior mean and the posterior variance of the population mean Ȳ which approximately agree, under prior vagueness, with the usual formulae under the design-based approach. In the case of stratified sampling with known strata differences, priors within strata are assumed to be exchangeable. Meeden and Vardeman () used a Polya posterior (PP) over the unobserved, assuming that “unseen are like the seen” (equivalent to exchangeability). In this case, the posterior “does not arise from a single prior distribution” (Meeden, ) and, hence, it is called a pseudo-posterior. It is also similar to the Bayesian bootstrap (Rubin, ; Lo, ). The Polya posterior is a flexible tool and methods based on PP have reasonable design-based properties under simple random sampling. PP approach permits Bayesian interval estimation for the mean and any other parameters of interest through simulation of many finite populations from PP. The general interval estimation feature of the PP approach is attractive. Meeden () extended the PP approach to utilize auxiliary population information (x1, …, xN) by making a strong prior assumption that the ratios ri = yi/xi are exchangeable and obtained point and interval estimators for the population median. Empirical results under simple random sampling are given to show that the resulting Bayesian intervals perform well in terms of design-based coverage. () developed a constrained Polya posterior to generate simulated populations that are consistent with the known population mean of an auxiliary variable x, using MCMC methods. This approach permits the use of known population auxiliary information and leads to more efficient Bayesian inferences. Nelson and Meeden () adapted PP to incorporate prior knowledge that the population median belongs to some interval. Meeden () studied two-stage cluster sampling (balanced case) and his two-stage PP-based results for the posterior mean and the posterior variance are very close to standard design-based results, but it is not clear how readily Meeden’s approach extends to the unbalanced case with unequal cluster sizes. Although the PP approach is attractive and seems to provide calibrated Bayesian inferences at least for some simple sampling designs, it is unlikely to be used in the production of official statistics because of the underlying assumption that “unseen are like the seen” and each case needs to be studied carefully to develop suitable PP. Also, it is not clear how this method can handle complex designs, such as stratified multistage sampling designs, or even single stage unequal probability sampling without replacement with nonnegligible sampling fractions, and provide design-calibrated Bayesian inferences., the PP approach may be useful for some specialized surveys and when inferences are desired on a variety of finite population parameters associated with the variable of interest y or prior knowledge on the parameters is available, as in the case of Nelson and Meeden (). An alternative approach is to start with an informative likelihood based on reduced data. For example, under simple random sampling, it may be reasonable to suppress the labels i from the full data {(i, yi), i ∈ s} and use the likelihood based on the reduced data {yi, i ∈ s}; see Hartley and Rao () and Royall (). On the other hand, for stratified random sampling, labels within strata are suppressed but strata labels are retained because of known strata differences. Hartley and Rao () proposed a “scale-load” approach for inference on the mean Ȳ. Under this approach, the y-values are assumed to belong to a finite set of possible values {h1, …, hD} for some finite D (unspecified). Then Nt is the scale load of ht and the population mean is expressed in terms of the scale loads as $\bar{Y} =\break N^{ - 1}\sum\_{t = 1}^{D} N\_{t} h\_{t}$. Reduced sample data under simple random sampling without replacement is represented by the sample scale loads nt, t = 1, …, D, and the resulting likelihood function is the hyper-geometric likelihood L(N1, …, ND). If the sampling fraction is negligible, then the likelihood is simply the multinomial likelihood which is the same as the empirical likelihood (EL) of Owen (). In the case of stratified random sampling, the likelihood function is the product of hyper-geometric likelihoods corresponding to the different strata h = 1, …, L. Hartley and Rao focused primarily on design-based inferences, but also briefly studied Bayesian inference under simple random sampling using a compound-multinomial prior on the scale loads $N\_{1},\ldots,\break N\_{D}$. Hoadley () obtained the compound-multinomial prior, denoted CMtn(Nd; νd, d = 1, …, D), by first assuming that the finite population $\{ y\_{i},i \!=\! 1,\ldots,\break N\}$ is a random sample from an infinite population with unknown probabilities $p\_{d} = P(y\_{i} = h\_{d}),d =\break 1,\ldots,D$, and then using a Dirichlet prior with parameters νd ( > 0) on the probabilities pd, d = 1, …, D. The posterior distribution of Nd − nd, d = 1, …, D, given the data nd, d = 1, …, D, is the compound multinomial CMtn(Nd − nd; νd + nd, d = 1, …, D). Using this posterior distribution, Hartley and Rao () obtained the posterior mean and the posterior variance of the population mean Ȳ. Under a diffuse prior with the vd close to zero, the results are identical to those of Ericson (). However, there are fundamental differences in the two approaches in the sense that under exchangeability the conditional distribution of the sample scale loads nd, given the population scale loads Nd, is equal to the hyper-geometric likelihood of Hartley–Rao for any sampling design, whereas in the Hartley–Rao approach this conditional distribution and the resulting posterior of the Nd are derived under simple random sampling, and hence depend on the sampling design. Rao and Ghangurde () studied Bayesian optimal sample allocation, by minimizing the expected posterior variance of the mean, for stratified simple random sampling and some other cases including two-phase sampling to handle the nonresponse problem. Attention was given to data-based priors obtained by combining diffuse priors with likelihoods based on pilot samples. Aitkin () used the scale-load framework and obtained Bayesian intervals on the population mean under simple random sampling, by using a compound-multinomial prior with vd = 0 on the observed scale loads nd > 0 and then simulating a large number of samples from the resulting posterior distribution. This approach is similar to the simulation approach used by Meeden and Vardeman (), but the posterior intervals depend on the design via the likelihood function. As in the case of Meeden and Vardman, the simulation method can be applied to other parameters of interest. Also, the simulation method readily extends to stratified simple random sampling. The scale-load approach is promising but somewhat limited in applicability, in the sense that the scale-load likelihoods cannot be obtained easily for complex sampling designs. To handle complex sampling designs, Rao and Wu () used a pseudo-EL approach, proposed by Wu and Rao (), to obtain “calibrated” pseudo-Bayesian intervals in the sense that the intervals have asymptotically correct designbased coverage probabilities. The pseudo-EL approach uses the survey weights and the design effect (via the effective sample size n\) in defining the profile pseudo-EL function for the mean θ* = Ȳ. Let d̃i(s) = di/∑j ∈ sdj be the normalized weights and n\* = n/(deff), where deff is the ratio of the estimated variance of the weighted mean ∑i ∈ sd̃i(s)yi to the estimate of the variance under simple random sampling. Then the profile pseudo empirical log-likelihood function for θ is given by lPEL(θ) = n\∑i* ∈ sd̃i(s)log{p̂i(θ)}, where the p̂i(θ) maximize ∑i ∈ sd̃i(s)logpi subject to pi > 0, ∑i ∈ spi = 1 and ∑i ∈ spiyi = θ. We refer the reader to Rao and Wu () for an overview of EL methods used for inference from survey data. By combining the profile pseudo-EL function for the population mean from ([e4]) with a flat prior on the mean, one can get pseudo-Bayesian intervals that have asymptotically correct design-based coverage probabilities. Also, it may be easier to specify informative priors on the mean if historical information on the mean is available. The proposed approach can incorporate known auxiliary population information in the construction of pseudo-Bayesian intervals using the basic design weights or using weights already calibrated by the known auxiliary information. The latter is more appealing because, in practice, data files report the calibrated weights. One limitation of the Rao–Wu method for complex designs is that the pseudo-EL depends on the design effects which may not be readily available. Lazar () proposed the Bayesian profile EL approach for the case of independent and identically distributed (i.i.d.) observations. It should be noted that even in the i.i.d. case a “matching” prior on the mean that provides higher order coverage accuracy for the intervals does not exist when using the nonparametric Bayesian profile-EL (Fang and Mukerjee, ). Therefore, the main advantage of the approach is to get “exact” pseudo-Bayesian intervals that are also calibrated in the sense of first order coverage accuracy in the design-based framework. Small Area Estimation: HB Approach ================================== Methods for small area (or domain) estimation have received much attention in recent years due to growing demand for reliable small area statistics. Traditional area-specific direct estimation methods (either design-based or model based or Bayesian) are not suitable in the small area context because of small (or even zero) area-specific sample sizes. As a result, it is necessary to use indirect estimation methods that borrow information across related areas through linking models based on survey data and auxiliary information, such as recent census data and current administrative records. Advocates of design-based methods indeed acknowledge the need for models in small area estimation. For example, () remark, “If the assumed model accurately represents the state of nature, useful inferences can be based on quite small samples at least for certain models.” Linking models based on linear mixed models and generalized linear mixed models with random small area effects are currently used extensively, in conjunction with empirical best linear unbiased prediction (EBLUP), parametric empirical Bayes (EB) and hierarchical Bayes (HB) methods for estimation of small area means and other small area parameters of interest. A detailed treatment of small area estimation methods is given in Rao (). We focus here on HB methods to highlight the significant impact of Bayesian methods on small area estimation. In the HB approach, model parameters are treated as random variables and assigned a prior distribution. Typically, noninformative priors are used, but one must make sure that the resulting posteriors are proper because some priors on the variance parameters can lead to improper posteriors (see Rao,, Section 10.2.4, for a discussion on the choice of priors). The posterior distribution of a small area parameter of interest is then obtained from the prior and the likelihood function generated from the data and the assumed model. Typically, closed-form expressions for desired posterior distributions do not exist, but powerful MCMC methods are now available for simulating samples from the desired posterior distribution and then computing the desired posterior summaries. Rao (, Chapter 10) gives a detailed account of the HB methods in the small area context; see also the review paper by Datta (), Section 3. A significant advantage of the HB approach is that it is straightforward and the inferences are “exact,” unlike in the EB approach. Moreover, it can handle complex small area models using MCMC methods. Availability of powerful MCMC methods and software, such as WinBUGS, also makes HB attractive to the user. Extensive HB model diagnostic tools are also available, but some of the default HB model-checking measures that are widely used may not be necessarily good for detecting model deviations. For example, the commonly used posterior predictive p-value (PPP) for checking goodness of fit may not be powerful enough to detect nonnormality of random effects (Sinharay and Stern, ) because this measure makes “double use” of data in the sense of first generating values from the predictive posterior distribution and then calculating the p-value. Bayarri and Castellanos () say, “Double use of the data can result in an extreme conservatism of the resulting p-values.” Alternative measures, such as the partial PPP and the conditional PPP (Bayarri and Berger, ), attempt to avoid double use of data, but those measures are more difficult to implement than the PPP, especially for the small area models. Browne and Draper () suggested the use of prior-free, frequentist methods in the model exploration phase and then the HB for inference based on the selected models using possibly diffuse priors on the model parameters. However, many Bayesians may not agree with this suggestion because of the orientation of frequentist tests of goodness of fit to rejecting null hypotheses, as noted by a referee. To illustrate the HB approach for small area estimation, we focus on a basic area-level model with two components, a sampling model and a linking model, requiring only area-specific (direct) designbased estimators ȳiw of small area means Ȳi and associated area-level covariates zi (i = 1, …, m). The linking model is of the form g(Ȳi) = zʹiβ + vi, where the random effects vi ∼ i.i.d.N(0, A) and g( ⋅ ) is a specified link function. The sampling model assumes that g(ȳiw) = g(Ȳi) + ẽi, where the sampling errors ẽi∣Ȳi are assumed to be independent N(0, Di) with known sampling variances Di. The assumptions of zero mean sampling errors and known sampling variances may be both quite restrictive in practice. The first difficulty may be circumvented by using the sampling model ȳiw = Ȳi + ei, where the sampling errors ei are assumed to be independent normal with zero means, which simply says that the direct estimators are design unbiased or nearly design unbiased, as in the case of a GREG estimator, for large overall sample size. The second assumption of known sampling variances is more problematic and the usual practice to get around this problem is to model the estimated sampling variances (using generalized variance functions) and then treat the resulting smoothed estimates as the true variances Di. Bell () studied the sensitivity of small area inferences to errors in the specification of the true variances. The original model, called the Fay–Herriot (FH) model, is a matched model in the sense that the sampling model matches the linking model and the combined model is simply a special case of a linear mixed model. On the other hand, the alternative sampling model is not necessarily matched to the linking motirdel and in this case the two models are “mismatched.” For simplicity, we focus on the matched case, but the HB approach readily extends to the more complex case of mismatched models and also to models that allow the sampling variance to depend on the area mean Ȳi (You and Rao, ). Attractive features of area level models are that the sampling design is taken into account through the direct estimators ȳiw and that the direct estimators and the associated area level covariates are more readily available to the users than the corresponding unit level sample data. For example, the U.S. Small Area Income and Poverty Estimation (SAIPE) Program used the FH model to estimate county level poverty counts of school-age children by employing direct estimates for sampled counties from the Current Population Survey and associated county level auxiliary information from tax records, food stamps programs and other administrative sources (see Rao,, Chapter 7, for ). Bayesians have used the area level models extensively through the HB approach, in spite of the limitations mentioned above, because of their practical advantages (Rao,, Chapter 10). In the HB approach, a flat prior on the model parameters β and A is often specified as f(β, A) ∝ f(A) and f(A) ∝ 1, and the resulting posterior summaries (means, variances and credible intervals) for the means Ȳi are obtained. Bell () studied matched models in the context of estimating the proportion of school-age children in poverty at the state level in the US, using the survey proportions ȳiw = piw based on the Current Population Survey (CPS) data for 1989–1993 and area level covariates zi related to Ȳi = Pi. Bell found that the maximum likelihood (ML) and restricted ML (REML) estimates of A turned out to be zero for the first four years (1989–1992) and the resulting EB estimates of state poverty rates attached zero weight to the direct estimate piw regardless of the CPS state sample sizes ni (number of households). This problem with EB based on ML or REML can be circumvented by using the HB approach. Bell used the above flat prior and obtained the posterior mean which always attached nonzero weight to the direct estimate. Further, the posterior variance is well behaved (smallest for California with the largest ni), unlike the estimated mean squared prediction error (MSPE) of the EB estimator. It is possible, however, to develop EB methods that always lead to nonzero estimates of A. Morris () proposed to multiply the residual likelihood function of A by the factor A and maximize this adjusted likelihood function. The resulting estimator of A is always positive and gets around the difficulty with REML. Li and Lahiri () used an adjusted profile likelihood function which also leads to positive estimates of A. They also established asymptotic consistency of the estimator and obtained a nearly unbiased estimator of the mean squared prediction error (MSPE) of the associated EB estimator of Ȳi. () studied frequentist properties of HB by deriving a moment-matching prior on A, in the sense that the resulting posterior variance is nearly unbiased for the MSPE of the HB estimator of the small area mean. The moment-matching prior is given by f(A) ∝ (A + Di)2∑l = 1m(A + Dl)− 2. This prior depends collectively on the sampling variances Dl for all the areas as well as on the area-specific sampling varianceDi. Note that the matching prior is designed for inference on area i and, hence, its dependence on Di should not be problematic. The matching prior ([e5]) reduces to the flat prior f(A) ∝ 1 in the special case of equal sampling variances Di = D. However, in the application considered by Bell (), maxDi/minDi is as large as 20. Ganesh and Lahiri () derived a single matching prior such that a weighted posterior variance over the areas tracks the corresponding weighted MSPE for specified weights. By letting the weights be one for area i and zero for the remaining areas, the resulting prior is identical to ([e5]). Datta () has shown that the previous moment-matching priors also ensure matching property for interval estimation in the sense that the coverage probability of the credible interval tracks the corresponding coverage probability of the normal interval based on the EB estimator and its estimated MSPE. Further work on matching priors in the context of small area estimation would be useful. Mismatched models are often more realistic for practical applications, as they allow flexibility in formulating the linking model. A recent application of HB under mismatched models is to the estimation of adult literacy levels for all states and counties in the US, using data from the National Assessment of Adult Literacy and literacy-related auxiliary data (Mohadjer et al., ). Bizier et al. () used mismatched models and the HB approach to produce estimates of disability rates for health regions and selected municipalities in Canada. A variety of applications of HB under complex modeling have been reported in the literature (see Rao,, Chapter 10, for work prior to 2003). Nandram and Choi () and () studied extensions of HB to handle nonignorable nonresponse and applied the methods to data from the National Health and Nutrition Examination Survey (NHANES III) to produce small area estimates. Raghunathan et al. () applied the HB approach to combine data from two independent surveys [Behavioral Risk Factor Surveillance System (BRFSS) and the National Health Interview Survey (NHIS)] for the years 1997–2000 to produce yearly prevalence estimates at the county level for six outcomes. BRFSS is a large telephone survey covering almost all US counties, but the nonresponse rates are high and also nontelephone households are not covered. On the other hand, NHIS is a smaller personal interview survey with lower nonresponse rates and covers nontelephone households. In this application, direct survey weighted county estimates of proportions from the two surveys were transformed using the inverse sine transformation and the sampling variances were taken as (4ñd)− 1, where ñd denotes the effective sample size for a particular domain d (calculated as the actual domain sample size nd divided by the estimated design effect which is the ratio of the estimated variance under the given design to the binomial estimated variance). The resulting sampling model was then combined with a suitable linking model to obtain county estimates of the prevalence rates, using diffuse proper priors on the model parameters and MCMC. This application attempts to account for possible noncoverage bias and obtain efficient county estimates by combining data from two independent surveys. It may be noted that the model used here is an extension of the basic FH area level model and the application demonstrates how design-based and Bayesian approaches can be fruitfully integrated in small area estimation. HB methods studied in the literature have been largely parametric, based on specified distributions for the data. However, Meeden () extended his noninformative Bayesian approach, based on the Polya posterior (PP), to small area estimation in some simple cases. Extension of this approach to handle complex models is not likely to be easy in practice. Concluding Remarks ================== I have provided an appraisal of the role of Bayesian and frequentist methods in sample surveys. My opinion is that for domains (subpopulations) with sufficiently large samples, a traditional design-based frequentist approach that makes effective use of auxiliary information, through calibration or assistance from working models, will remain as the preferred approach in the large-scale production of official statistics from complex surveys. Nonsampling errors can be handled using a combined design and model approach with minimal use of models. But the designbased approach, using survey weights, is not a panacea even for large samples and yet “many people ask too much of the weights” (Lohr, ), prompting statements like, “Survey weighting is a mess” (Gelman, ). As Lohr () noted, survey weighting is not a mess as long as the weighting is not stretched to a limit as in the case of a very large number of post-stratified cells leading to very small or even zero cell sample sizes, thus making weighting at the cell level unstable or even not feasible (Gelman, ). Alternative weighting methods can be used in those situations to get around this problem. For example, by calibrating to the marginal counts of the post-stratification variables instead of the cell counts leads to a calibration estimator with stable weights which should perform well for estimating population totals or means. Also, the resulting weights do not depend on the response values, thus ensuring internal consistency, unlike the hierarchical regression method proposed by Gelman () based on models involving random effects. Recent work on nonparametric Bayesian methods that can be used for both Bayesian and design-based inferences looks promising, at least for some specialized surveys. For small area estimation, the hierarchical Bayes (HB) approach offers a lot of promise because of its ability to handle complex small area models and provide “exact” inferences. However, the choice of noninformative priors that can provide frequentist validity is not likely to be easy in practice when complex modeling is involved. Also, caution needs to be exercised in the routine use of popular HB model-checking methods. Acknowledgments =============== This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. My thanks are due to an associate editor and two referees for constructive comments and suggestions. 107 [auto:STB\|2010-11-18\|09:18:59] ()... [mr] ()... [mr] ()... [auto:STB\|2010-11-18\|09:18:59] (). [auto:STB\|2010-11-18\|09:18:59] ().. In., [m	arxiv_0000644
Are local ULIRGs powered by AGN? The sub-kpc view of the 220GHz continuum. PUMA II ================================================================================== Introduction ============ Ultraluminous infrared galaxies (ULIRGs; $L\_{\rm IR}>10^{12}$) are among the most luminous objects in the local Universe. The majority of local ULIRGs are major gas-rich mergers at different evolutionary stages: from interacting systems with two nuclei separated by few kpc to more advanced mergers with a single nucleus (see and references therein). A classic evolutionary scenario suggests that merging ULIRGs evolve into a quasar that quenches the star-formation (SF) and, after that, the merger remnant becomes an intermediate-mass elliptical galaxy (e.g., ). However, recent observations and simulations indicate that mergers do not always quench the SF and also that disks can regrow in mergers remnants (e.g., ). This suggests that ULIRGs can have more varied evolutionary paths than that suggested by the classic scenario. Local ULIRGs might also be scaled down versions of the dusty star-forming mergers detected at z > 2 (e.g., ). Therefore, local ULIRGs are excellent targets for detailed studies of the physical processes that shape the, possibly diverse, evolutionary paths of merging gas-rich galaxies, which were important in the high-z universe. One key property of local ULIRGs are their extremely compact ( < 1kpc; e.g., ) and dust obscured nuclei ($A\_{\rm v}$>1000mag in some cases based on their nuclear molecular gas column densities; e.g., ). Because of this extreme obscuration most of the radiation produced in their nuclei, either by an active galactic nucleus (AGN) or by SF, is absorbed by dust and re-emitted in the infrared (IR) spectral range. For this reason, it is not straightforward to determine the dominant power source (AGN vs. SF) of local ULIRGs. Mid-IR studies, which are less affected by extinction than optical and near-IR works, suggest that ULIRGs are mostly powered by SF (e.g., ), although the AGN contribution increases with increasing luminosities (e.g., ). However, for $A\_{\rm v}$>1000mag, the mid-IR extinction is still very high, $A\_{\rm 15\mu m}$>50mag (e.g., ), and a large part of the emission could be completely obscured even in the mid-IR which. This could prevent an accurate determination of the AGN and SF contributions to the $L\_{\rm IR}$ of local ULIRGs using mid-IR observations. Alternatively, it is possible to investigate what powers local ULIRGs by measuring the size of the region that emits the bulk of the $L\_{\rm IR}$ as well as the molecular gas content (i.e., the fuel for SF) of this region. These quantities are needed to determine the nuclear IR luminosity and gas surface densities. Finding IR luminosity densities well above the limit of a maximal starburst (e.g., ) can be used to infer the presence of an obscured AGN and to estimate its luminosity (see e.g., ). In this paper, we analyze high-resolution ( ∼ 400pc) ALMA CO(2–1) and ∼ 220GHz continuum observations of sample of 23 local ULIRGs. We measure the size of the ∼ 220GHz ( ∼ 1400) continuum, which we link with the bulk of the $L\_{\rm IR}$ in these sources, and we estimate the nuclear cold molecular gas content from the CO(2–1) emission. We use these results to calculate their nuclear luminosity and molecular gas densities. These ALMA observations of a representative sample of local ULIRGs are part of the “Physics of ULIRGs with MUSE and ALMA” (PUMA) project. The main goals of this project are: a) to establish the impact of massive outflows in the evolution of ULIRGs (negative and positive feedback); and b) to determine what drives this feedback (AGN vs. SF) during the entire merging process (from early stages to advanced mergers). To do so, we combine sub-kpc resolution adaptive optics-assisted VLT/MUSE optical integral field spectroscopy and CO(2–1) ALMA data to trace the multi-phase structure of the massive outflows as well as to investigate basic properties of the ULIRGs like their main power source. The first MUSE results on the spatially resolved stellar kinematics and the ionized outflow phase were presented by while the detailed analysis of the Arp 220 MUSE data was presented in. Likewise, presented the first ALMA results on the spatially resolved cold molecular outflows detected in three of these local ULIRGs. The paper is organized as follows. We briefly describe the PUMA sample in Sect. [s:sample]. The ALMA observations and data reduction are presented in Sect. [s:data]. In Sect. [s:dataanalysis], we derive the spatial properties of the ∼ 220GHz continuum and the CO(2–1) emission, and fit the IR and radio spectral energy distributions (SEDs) of the ULIRGs. Sect. [s:discussion] investigates the origin of the high luminosity and molecular gas surface densities in the ULIRG nuclei, the relation between the ∼ 220GHz continuum size with the mid-IR excess emission, the 9.7 silicate absorption, and the broad-band IRAS colors. We also estimate the radiation pressure in these nuclei. The main conclusions are summarized in Sect. [s:conclusions]. Throughout this article we assume the following cosmology: $H\_{\rm 0} = 70$kms− 1Mpc− 1, $\Omega\_{\rm m}=0.3$, and $\Omega\_{\rm \Lambda}=0.7$. [tp] [tbl:sample] lcccccccccc IRAS name & Nucleus & R.A. & Dec. & v$\_{\rm CO}$ & z & $d\_{\rm L}$ & Scale & $\log L\_{\rm IR}$ & Class. & Morph. & & (ICRS) & (ICRS) & (kms− 1) & & (Mpc) & (kpcarcsec− 1) & (L⊙) 00091 − 0738 & & & & & 0.1181 & 550 & 2.13 & 12.34 & HII & I & S & 00 11 43.272 & − 07 22 07.35 & 31637 & & & & & ⋯ & & N & 00 11 43.302 & − 07 22 06.18 & 31686 & & & & & ⋯ & 00188 − 0856 & - & 00 21 26.513 & − 08 39 25.99 & 34136 & 0.1285 & 602 & 2.29 & 12.42 & Sy2 & M 00509+1225 & - & 00 53 34.934 & + 12 41 35.94 & 17265 & 0.0611 & 273 & 1.18 & 11.87 & Sy1 & M 01572+0009 & - & 01 59 50.251 & + 00 23 40.88 & 42077 & 0.1633 & 782 & 2.80 & 12.65 & Sy1 & M F05189 − 2524 & - & 05 21 01.400 & − 25 21 45.30 & 12285 & 0.0427 & 188 & 0.84 & 12.10 & Sy2 & M 07251 − 0248 & & & & & 0.0878 & 400 & 1.64 & 12.45 & & I & W & 07 27 37.532 & − 02 54 54.38 & 24201 & & & & & HII & & E & 07 27 37.613 & − 02 54 54.25 & 24193 & & & & & HII & 09022 − 3615 & - & 09 04 12.706 & − 36 27 01.93 & 16856 & 0.0596 & 266 & 1.15 & 12.33 & HII & M F10190 + 1322 & & & & & 0.0763 & 345 & 1.45 & 12.04 & & I & W & 10 21 42.493 & + 13 06 53.83 & 21336 & & & & & HII & & E & 10 21 42.754 & + 13 06 55.61 & 21167 & & & & & HII & 11095 − 0238 & & & & & 0.1064 & 491 & 1.95 & 12.33 & & I & SW & 11 12 03.359 & − 02 54 23.29 & 28806 & & & & & LINER & & NE & 11 12 03.383 & − 02 54 22.94 & 28863 & & & & & LINER & F12072 − 0444 & & & & & 0.1284 & 601 & 2.29 & 12.48 & & I & S & 12 09 45.13 & − 05 01 14.6† & & & & & & Sy2 & & N & 12 09 45.13 & − 05 01 13.5† & & & & & & Sy2 & F12112 + 0305 & & & & & 0.0730 & 329 & 1.39 & 12.32 & LINER & I & SW & 12 13 45.940 & + 02 48 39.12 & 20448 & & & & & ⋯ & & NE & 12 13 46.057 & + 02 48 41.55 & 20322 & & & & & ⋯ & 13120 − 5453 & - & 13 15 06.323 & − 55 09 22.82 & 9046 & 0.0311 & 136 & 0.62 & 12.27 & Sy2 & M F13451 + 1232 & & & & & 0.1217 & 568 & 2.19 & 12.31 & & I & W & 13 47 33.36 & + 12 17 24.2† & & & & & & Sy2 & & E & 13 47 33.50 & + 12 17 23.8† & & & & & & LINER & F14348 − 1447 & & & & & 0.0826 & 375 & 1.55 & 12.41 & & I & SW & 14 37 38.281 & − 15 00 24.23 & 22911 & & & & & LINER & & NE & 14 37 38.397 & − 15 00 21.27 & 22809 & & & & & LINER & F14378 − 3651 & - & 14 40 59.013 & − 37 04 31.93 & 19113 & 0.0681 & 306 & 1.30 & 12.15 & Sy2 & M F15327+2340 & & & & & 0.0181 & 78 & 0.37 & 12.19 & & M & W & 15 34 57.224 & + 23 30 11.44 & 5290 & & & & & LINER & & E & 15 34 57.293 & + 23 30 11.29 & 5380 & & & & & LINER & 16090 − 0139 & - & 16 11 40.419 & − 01 47 06.35 & 35352 & 0.1337 & 629 & 2.37 & 12.62 & HII & M 16155 + 0146 & & & & & 0.1330 & 625 & 2.36 & 12.24 & & I & NW & 16 18 09.364 & + 01 39 21.75 & 35188 & & & & & Sy2 & & SE & 16 18 09.54 & + 01 39 19.7† & ⋯ & & & & & ⋯ & 17208 − 0014 & - & 17 23 21.957 & − 00 17 00.88 & 12304 & 0.0428 & 189 & 0.84 & 12.43 & LINER & M F19297 − 0406 & & & & & 0.0856 & 390 & 1.61 & 12.45 & & I & S & 19 32 22.30 & − 04 00 01.8† & 23685 & & & & & HII & & N & 19 32 22.309 & − 04 00 01.03 & 23589 & & & & & HII & 19542 + 1110 & - & 19 56 35.785 & + 11 19 05.03 & 17629 & 0.0625 & 280 & 1.20 & 12.09 & LINER & M 20087 − 0308 & - & 20 11 23.866 & − 02 59 50.72 & 28600 & 0.1055 & 487 & 1.93 & 12.47 & LINER & M 20100 − 4156 & & & & & 0.1298 & 609 & 2.31 & 12.66 & & I & NW & 20 13 29.48 & − 41 47 32.6† & 34428 & & & & & HII & & SE & 20 13 29.556 & − 41 47 35.21 & 34453 & & & & & LINER & 20414 − 1651 & - & 20 44 18.159 & − 16 40 16.82 & 23962 & 0.0869 & 396 & 1.63 & 12.24 & HII & M F22491 − 1808 & & & & & 0.0776 & 352 & 1.47 & 12.23 & & I & W & 22 51 49.24 & − 17 52 23.7† & ⋯ & & & & & HII & & E & 22 51 49.349 & − 17 52 24.13 & 21600 & & & & & HII & [t] [tbl:almaobs] lcccccccc IRAS name & Synthesized beam & Beam FWHM & Sensitivity & Obs freq. & ALMA Band & ALMA Project ID & (arcsec × arcsec,∘) & (pc) & (μJybeam− 1) & (GHz) & & 00091 − 0738 & 0.31x0.23, − 83 & 570 & 45 & 194.2 & 5 & 2018.1.00699.S 00188 − 0856 & 0.13x0.12, − 46 & 290 & 18 & 192.4 & 5 & 2018.1.00699.S 00509 + 1225 & 0.31x0.28, 24 & 340 & 24 & 232.6 & 6 & 2018.1.00699.S 01572 + 0009 & 0.16x0.13, 69 & 410 & 33 & 188.2 & 5 & 2018.1.00699.S F05189 − 2524 & 0.52x0.42, − 70 & 390 & 34 & 236.7 & 6 & 2018.1.00699.S 07251 − 0248 & 0.27x0.24, − 50 & 420 & 23 & 228.3 & 6 & 2018.1.00699.S 09022 − 3615 & 0.30x0.27, − 87 & 330 & 24 & 232.9 & 6 & 2018.1.00699.S F10190 + 1322 & 0.30x0.27, 5 & 410 & 23 & 229.4 & 6 & 2018.1.00699.S 11095 − 0238 & 0.31x0.24, − 86 & 540 & 29 & 196.2 & 5 & 2018.1.00699.S F12112 + 0305 & 0.30x0.26, − 76 & 390 & 21 & 231.1 & 6 & 2016.1.00170.S 13120 − 5453 & 0.65x0.65, 0 & 400 & 150 & 239.6 & 6 & 2016.1.00777.S F14348 − 1447 & 0.29x0.25, 89 & 420 & 22 & 229.1 & 6 & 2016.1.00170.S F14378 − 3651 & 0.36x0.24, 84 & 390 & 41 & 231.1 & 6 & 2018.1.00699.S F15327 + 2340 & 1.27x0.81, 40 & 370 & 290 & 226.9 & 6 & 2015.1.00113.S 16090 − 0139 & 0.20x0.16, − 85 & 420 & 37 & 193.3 & 5 & 2018.1.00699.S 16155 + 0146 & 0.26x0.14, − 72 & 440 & 50 & 193.4 & 5 & 2018.1.00699.S 17208 − 0014 & 0.47x0.47, 0 & 400 & 240 & 237.8 & 6 & 2018.1.00486.S F19297 − 0406 & 0.27x0.26, 79 & 420 & 18 & 228.5 & 6 & 2018.1.00699.S 19542 + 1110 & 0.35x0.30, − 59 & 390 & 26 & 232.3 & 6 & 2018.1.00699.S 20087 − 0308 & 0.31x0.25, − 70 & 540 & 17 & 196.3 & 5 & 2018.1.00699.S 20100 − 4156 & 0.18x0.12, 58 & 350 & 25 & 193.9 & 5 & 2018.1.00699.S 20414 − 1651 & 0.18x0.14, − 54 & 260 & 27 & 228.4 & 6 & 2018.1.00699.S F22491 − 1808 & 0.39x0.29, − 89 & 500 & 26 & 229.6 & 6 & 2015.1.00263.S Sample of local ULIRGs ====================== The PUMA sample is a volume-limited (z < 0.165; d < 800Mpc) representative sample of 25 local ULIRGs (38 individual nuclei). These objects were selected to examine the most relevant parameters for the feedback processes: (1) the main power source (AGN vs. SF); (2) the interaction stage (from interacting pairs to advanced mergers); and (3) the IR luminosity. The parent sample is the 1Jy ULIRG sample extended to southern objects by. Our sample is limited to objects with Dec. between − 65∘ and + 25∘ which is appropriate for ALMA. We selected 12 interacting systems (nuclear separation > 1kpc) and 13 mergers with nuclear separations < 1kpc. Half of the objects in each interaction stage category were selected to be dominated by AGN based on mid-IR spectroscopy. The selected objects uniformly cover the ULIRG luminosity range between 1012.0 and 1012.7L⊙. See Table [tbl:sample] and for more details. So far, we have obtained ALMA CO(2–1) and ∼ 220GHz continuum observations for 92% of the systems in the sample (23 systems with 34 individual nuclei). The CO(2–1) emission is detected in 33 nuclei and the continuum in 29 (see Sect. [s:dataanalysis]). In addition to our VLT/MUSE-AO optical integral field spectroscopy, the majority of the targets have extensive ancillary multi-wavelength data which include mid- and far-IR (e.g., ), radio (e.g., ), and X-ray (e.g., ) observations. Observations and data reduction =============================== ALMA observations ----------------- We obtained ALMA 12-m array CO(2–1) 230.538GHz and continuum observations for 23 out of the 25 PUMA ULIRGs. ALMA observations for the remaining two ULIRGs have been scheduled but are not available at the time of writing. These observations were mainly conducted as part of our programs 2015.1.00263.S, 2016.1.00170.S, and 2018.1.00699.S (PI: M. Pereira-Santaella). For 13120 − 5453 and F15327+2340 (Arp 220), we used archive data from programs 2016.1.00777.S (PI: K. Sliwa) and 2015.1.00113.S (PI: N. Scoville), respectively. In addition, we complemented this dataset with higher angular resolution data for 17208 − 0014 from program 2018.1.00486.S (PI: M. Pereira-Santaella). Observations of three of the ULIRGs in our sample (F12112 + 0305, F14348 − 1447, and F22491 − 1808) have been already presented in, but we include them here for completeness. We aimed to have a similar spatial resolution of ∼ 400pc in all the systems, so the synthesized beam full-width half-maximum (FWHM) varies between 012 and 1 depending on the distance of each target. We used a single 12-m array configuration with baselines set to achieve the required angular resolution. The maximum recoverable scale is about 10 times the beam FWHM (i.e., 4kpc). Depending on the redshift, the CO(2–1) transition lies in the ALMA Band 5 or Band 6. Details on the observations are listed in Table [tbl:almaobs]. We defined four 1.875GHz bandwidth spectral windows with 2 to 8MHz (3–10kms− 1) channels, depending on the targeted spectral feature. One spectral window was centered at the sky frequency of 12CO(2–1) 230.538GHz. The remaining spectral windows were centered at the frequency of nearby transitions (e.g., CS(5–4), H30α, SiO(5–4)) when possible or at a “line-free” spectral range. We used the ALMA reduction software CASA (v5.6.1; ) to calibrate the data using the standard pipeline. The absolute flux accuracy of Band 5 and 6 data is ∼ 10% (ALMA Technical Handbook). For the CO(2–1) spectral window, we subtracted a constant continuum level estimated from the line emission free channels in the uv plane. The data were cleaned using the tclean CASA task and the Brigss weighting with robustness parameters between − 0.5 and 2.0 to match the required ∼ 400pc spatial resolution. For two systems (13120 − 5453 and 17208 − 0014), the largest synthesized beam provides a spatial resolution better than 400pc, so we used the imsmooth task to convolve the cubes with a Gaussian and obtained the desired spatial resolution. For F15327+2340 (Arp 220), we only used the compact configuration data which provide a ∼ 370pc spatial resolution comparable to that of the other ULIRGs in our sample. The channel width of the final cubes is ∼ 10kms− 1 and the pixel sizes are about a sixth of the beam FWHM (i.e., between 20 and 120mas). In addition to the line data cubes, we produced continuum images using spectral windows where no emission or absorption lines were present. The continuum sensitivities range from 18 to 290μJybeam− 1 with more sensitive data for the more distant objects (see Table [tbl:almaobs]). In this paper, we primarily focus on the analysis of the continuum and CO(2–1) maps. In a future paper (Lamperti et al. in prep.), we will present the detailed analysis of the line data. [!ht] ALMA continuum observation (first panel), best-fit model (second panel), residual emission after subtracting the continuum model (third panel), and the integrated CO(2–1) emission (moment 0) from Lamperti et al. in prep. (fourth panel) for 00188 − 0856 as an example. The two contour levels in the first panel indicate the 3σ and the 0.5 × peak emission levels. In the second panel, the individual components of the best-fit model are presented as a black circle (point source model) and as a dashed ellipse (deconvolved Gaussian model). The black crosses in the first, third, and fourth panels mark the fitted location of the continuum peak. The red hatched ellipses represent the beam FWHM. The units are μJybeam− 1 for the continuum panels and Jykms− 1beam− 1 for the CO(2–1) panel. The continuum model fits for the whole sample are shown in Fig. [fig:apxalmamodels].[fig:almamodels] Ancillary Spitzer data ------------------------ To complete the spectral energy distribution (SED) of the ULIRGs (Sect. [ss:sedmodel]), we used mid-IR Spitzer data. In particular we used the 5.2–38 low-resolution (R ∼ 60 − 130) spectra from the Infrared Spectrograph (IRS; ) and the 70 and 160 images from the Multiband Imaging Photometer (MIPS; ). We downloaded the calibrated IRS spectra for all the systems in our sample from the Cornell Atlas of Spitzer/Infrared Spectrograph Sources (CASSIS; ) and measured the flux at 34, which is approximately at the middle point between the 24 and 70 photometric points in log scale and avoids the noisier long-wavelength edge of the IRS spectrum. We also downloaded the calibrated MIPS images for 5 systems from the Spitzer Heritage Archive[1](#fn1). The ULIRG systems appear as point-sources at the MIPS angular resolution (18and 40 at 70 and 160, respectively). For the 70 image, we used a 35 radius aperture and a 39–65 background annulus and then multiplied the flux by 1.24 to account for the aperture correction factor (see Table 4.14 of the MIPS Instrument Handbook). For the 160 images, we subtracted a global background emission level and used a 60 radius aperture. We applied a 1.40 aperture correction factor which is appropriate for sources with temperatures between 30 and 150K (see Table 4.15 of the MIPS Instrument Handbook). The measured Spitzer IRS and MIPS fluxes are listed in Appendix[apx:spitzerfluxes]. Distribution of the ∼ 220GHz continuum half-light radius ($r\_{\rm cont}$; see Table [tbl:almacontmodel]). Upper limits are indicated with arrows. The red (black) histogram bars and arrows correspond to galaxies classified as AGN (non-AGN) from optical spectroscopy. The sizes of the systems whose ALMA flux might have high ( > 40%) non-thermal synchrotron contributions are marked in blue (see Sect. [ss:discalmacomp]).[fig:histosize] Data analysis ============= ALMA continuum model. Size and flux ----------------------------------- We modeled the ALMA 220–250GHz continuum images to determine the flux, size, and position of the detected emitting regions. In general, these regions are compact (FWHMs similar to the beam size) and their morphological structure is barely resolved. Therefore, we used simple models consisting of a point-source, a Gaussian, point-source + Gaussian, or 2 Gaussians. These models were convolved with the beam and compared with the observations to determine a χ2 value. Then, we minimized the χ2 by varying the fluxes, sizes, and positions of the model components. We tried these 4 models for each nucleus and selected that with the lowest reduced χ2. The best-fit models reproduce quite well the observed emission. The median (mean) reduced χ2 is 1.1 (1.4), the maximum is 3.1, and no significant structures are seen in the residual images (see Fig. [fig:almamodels]). This figure also shows the best-fit model whose parameters are listed in Table [tbl:almacontmodel]. The best-fit positions are presented in Table [tbl:sample]. Based on these parameters, we computed the half-light radius of the ∼ 220GHz continuum, $r\_{\rm cont}$, which is defined as the radius of the region that contains 50% of the observed flux. For 11 nuclei whose model includes a point-source, the half-light radius is not well defined because the point source contributes > 50% to the total flux. Therefore, to estimate the size upper limit in these cases, we performed a series of simulations. First, we subtracted, when present, the extended Gaussian component of the model. Then, we used circular Gaussian models with fixed FWHM from 2 to 6pixels[2](#fn2), that were convolved with the beam, and obtained the χ2 variation as function of the model FWHM. Finally, we estimated the 3σ FWHM upper limit as the FWHM at which the χ2 increases by 9.0 [3](#fn3) with respect to the minimum χ2. The FWHM upper limits are also included in Table [tbl:almacontmodel]. Fig. [fig:histosize] shows the distribution of the half-light radius and the upper limits. The measured $r\_{\rm cont}$ range from < 50pc to 350pc with a median value of 80–100pc. The $r\_{\rm cont}$ of AGN and non-AGN objects, based on optical spectroscopy, are similar. Mid-IR observations already indicate that ULIRGs are very compact ( < 1kpc; ). Our higher angular resolution ALMA continuum data suggest that they are even more compact. ### Higher resolution observations The CO(2–1) and 230GHz continuum emission of F15327+2340 (Arp 220) have been observed by ALMA at much higher resolution (8pc) than the data used in this paper. However, these data are still unpublished. Instead, we can compare with the 20–40pc 2.7mm ( ∼ 110GHz) continuum observations presented by and. These authors measure deconvolved Gaussian FWHM of 160–180mas for the East nucleus and 74–114mas for the West nucleus. From the low resolution data, we obtained 2 × $r\_{\rm cont}$=400mas and < 260mas for the East and West nuclei, respectively. Therefore, we recover a reliable upper limit size for the bright and compact source in the West nucleus. For the East nucleus, we derive a size 2 times larger. However, it is possible that part of extended emission detected in the low-resolution data is filtered out, or too faint, in the ten times higher resolution published observations. In addition, 17208 − 0014 was observed as part of another program which aimed to obtain ∼ 100pc spatial resolution CO(2–1) and continuum data. These observations will be analyzed in detail in a future paper. However, here we use the high resolution continuum data (120pc vs. 400pc), to test whether the source size derived from the low-resolution data ($r\_{\rm cont}<75$pc for this object) is consistent with the size measured in the high-resolution data. Fig. [fig:comphilo] compares the low- and the high-resolution maps. The difference between the continuum fluxes in both images measured using a 1 radius aperture is < 9% (42.7 ± 0.2 vs. 46.3 ± 0.3mJy). This indicates that the higher resolution data do not miss significant low surface brightness emission. This figure also shows that the continuum emission peak measured in the original low-resolution data (Table [tbl:sample]) appears slightly shifted (30mas or 26pc) in the high-resolution image. This is possibly because now we start to spatially resolve the inner structure of the nucleus and multiple smaller regions appear. We applied the same model fitting procedure described in Sect. [ss:almamodel] to the higher resolution image. The original model consisted of a point source plus a Gaussian (see Table [tbl:almacontmodel] and Fig. [fig:apxalmamodels]). For the high-resolution data, we used 2 Gaussians since the emission core is resolved. This “core” Gaussian has a flux of 17.6 ± 1.1mJy and a circularized FWHM of 74 ± 3mas (62pc; $r\_{\rm cont}=31$pc). It contains about 40% of the 247GHz continuum emission from 17208 − 0014, so the $r\_{\rm cont}<$75pc upper limit we estimated from the low-resolution data (Table [tbl:almacontmodel]) seems to be consistent with what is observed at higher angular resolution. The result for these two objects supports that our method to estimate the $r\_{\rm cont}$ can produce realistic values, even below the beam size. Comparison between the 400pc resolution 247GHz continuum image analyzed in Sect. [ss:almamodel] (left panel) and the higher resolution (120pc) data (rigth panel) available for 17208 − 0014. The black cross is the position of the center measured on the 400pc image.The contours are as in Fig. [fig:almamodels]. The hatched red ellipses correspond to the beam FWHM of each image. The color scales are in mJybeam− 1.[fig:comphilo] Growth curve of the CO(2–1) moment 0 map for 00188 − 0856 as an example. The circles correspond to the observed flux within a circular aperture of radius r. The red line is the best fit model (Sect. [ss:nuclearmol]). The deconvolved best fit profile is shown in blue and its effective radius, $r\_{\rm CO}=$FWHM/2, is indicated by the vertical blue dotted line. The dashsed green line marks the ∼ 220GHz continuum radius, $r\_{\rm cont}$ for comparison. The CO(2–1) model fits for the whole sample are shown in Fig. [fig:apxcomodels].[fig:comodels] Half light radius of the 220GHz continuum $r\_{\rm cont}$ vs. 0.5 × FWHM of the CO(2–1) emission $r\_{\rm CO}$. Red (black) symbols mark systems classified as AGN (non-AGN) based on optical spectroscopy. Filled symbols correspond to nuclei in interacting systems and empty symbols to nuclei in mergers (see Table [tbl:sample]). Blue encircled symbols are galaxies with excess non-thermal emission whose continuum size estimates might be inaccurate (Sect. [ss:discalmacomp]). The green line is best linear fit excluding the nuclei with $r\_{\rm cont}$ upper limits. The dotted line indicates the 1:1 relation.[fig:sizecontco] [t] [tbl:almaco21] lcccccccc IRAS name & Nucleus & $r\_{\rm CO}$ & $S\_{\rm CO}$ & $\log M\_{\rm H\_2}$ & $\log \Sigma\_{\rm H\_2}$ & & (pc) & (Jykms− 1) & (M⊙) & (M⊙pc− 2) 00091 − 0738 & S & 217 ± 9 & 11.6 ± 0.1 & 9.22 ± 0.04 & 3.74 ± 0.05 & N & 421 ± 9 & 8.00 ± 0.41 & 9.05 ± 0.05 & 3.01 ± 0.06 00188 − 0856 & - & 321 ± 5 & 15.9 ± 0.2 & 9.43 ± 0.04 & 3.61 ± 0.05 00509 + 1225 & - & 281 ± 9 & 22.7 ± 0.4 & 8.92 ± 0.04 & 3.23 ± 0.06 01572 + 0009 & - & 285 ± 6 & 5.12 ± 0.07 & 9.15 ± 0.04 & 3.44 ± 0.04 F05189 − 2524 & - & 264 ± 8 & 71.3 ± 0.9 & 9.11 ± 0.04 & 3.46 ± 0.06 07251 − 0248 & W & 343 ± 13 & 12.5 ± 1.7 & 8.99 ± 0.07 & 3.12 ± 0.09 & E & 255 ± 13 & 31.7 ± 0.4 & 9.39 ± 0.04 & 3.78 ± 0.06 09022 − 3615 & - & 397 ± 7 & 153 ± 2 & 9.73 ± 0.04 & 3.74 ± 0.04 F10190 + 1322 & W & 470 ± 9 & 17.0 ± 0.8 & 9.00 ± 0.05 & 2.85 ± 0.05 & E & 418 ± 9 & 52.6 ± 0.8 & 9.49 ± 0.04 & 3.44 ± 0.05 11095 − 0238 & SW & 480 ± 11 & 20.1 ± 0.8 & 9.36 ± 0.05 & 3.20 ± 0.05 & NE & 365 ± 11 & 22.5 ± 0.5 & 9.41 ± 0.04 & 3.49 ± 0.05 F12112 + 0305 & SW & 262 ± 6 & 17.2 ± 0.5 & 8.96 ± 0.05 & 3.33 ± 0.05 & NE & 325 ± 6 & 72.2 ± 0.9 & 9.58 ± 0.04 & 3.76 ± 0.05 13120 − 5453 & - & 462 ± 5 & 472 ± 5 & 9.65 ± 0.04 & 3.52 ± 0.05 F14348 − 1447 & SW & 384 ± 9 & 53.9 ± 0.8 & 9.57 ± 0.04 & 3.60 ± 0.04 & NE & 350 ± 9 & 33.3 ± 1.3 & 9.36 ± 0.05 & 3.47 ± 0.05 F14378 − 3651 & - & 447 ± 7 & 40.9 ± 0.6 & 9.28 ± 0.04 & 3.18 ± 0.05 F15327 + 2340† & - & 370 ± 4 & 1360 ± 7 & 9.63 ± 0.04 & 3.69 ± 0.05 16090 − 0139 & - & 374 ± 8 & 43.9 ± 0.6 & 9.91 ± 0.04 & 3.96 ± 0.06 16155 + 0146 & NW & 64 ± 12 & 2.24 ± 0.03 & 8.61 ± 0.04 & 4.19 ± 0.16 & SE & ⋯ & < 0.04 & < 6.81 & ⋯ 17208 − 0014 & - & 364 ± 9 & 338 ± 6 & 9.79 ± 0.04 & 3.86 ± 0.05 F19297 − 0406 & S & ⋯ & 1.2 ± 0.3 & 7.93 ± 0.11 & ⋯ & N & 455 ± 8 & 81.7 ± 1.3 & 9.78 ± 0.04 & 3.66 ± 0.05 19542 + 1110 & - & 231 ± 4 & 35.4 ± 0.2 & 9.14 ± 0.04 & 3.61 ± 0.04 20087 − 0308 & - & 413 ± 4 & 59.7 ± 0.6 & 9.83 ± 0.04 & 3.80 ± 0.05 20100 − 4156 & NW & 364 ± 8 & 0.939 ± 0.042 & 8.21 ± 0.05 & 2.29 ± 0.05 & SE & 210 ± 4 & 21.1 ± 0.1 & 9.56 ± 0.04 & 4.12 ± 0.05 20414 − 1651 & - & 262 ± 5 & 32.2 ± 0.5 & 9.39 ± 0.04 & 3.75 ± 0.06 F22491 − 1808 & W & ⋯ & 0.41 ± 0.02 & 7.38 ± 0.02 & ⋯ & E & 280 ± 5 & 45.4 ± 0.3 & 9.44 ± 0.04 & 3.74 ± 0.05 Nuclear molecular gas --------------------- Figs. [fig:almamodels] and [fig:apxalmamodels] show that the CO(2–1) emission is more extended than the continuum and also that it has a more complex morphology. As a consequence, the simple set of models used to fit the continuum (Sect. [ss:almamodel]) does not reproduce the CO(2–1) emission properly. Therefore, we considered a different approach to determine the size and flux of the nuclear CO(2–1) emission. We used the CO(2–1) moment 0 maps (Lamperti et al. in prep.) to extract the flux in concentric apertures centered at the continuum peak. This produces an azimuthally averaged growth curve for the CO(2–1) emission (see Fig. [fig:comodels]). To fit this curve, we simulated a 2D circular Gaussian model, which was convolved with the beam, and we compared the model growth curve with the observed one. From this fit, we obtained the deconvolved circularized FWHM of the CO(2–1) emission and the nuclear flux. Then, we used a ULIRG-like $\alpha\_{\rm CO}$ conversion factor (0.78M⊙${\rm (K\,km\,s^{-1}\,pc^{-2})^{-1}}$) and a CO 2–1 to 1–0 ratio $r\_{\rm 21}=0.91$ to estimate the molecular gas mass. We limited this growth curve to the central ∼ 1.5kpc, so the most extended emission of the systems is not included in the measured flux. Nevertheless, we are mostly interested in the CO(2–1) sizes and fluxes of the nuclear regions detected in the continuum and these are well covered by the apertures used ( ∼ 700pc aperture radius vs. $r\_{\rm cont}<$350pc). The nuclear CO(2–1) deconvolved sizes, $r\_{\rm CO}$, fluxes, molecular gas masses, and molecular gas surface densities, $\Sigma\_{\rm H\_2}$, are presented in Table[tbl:almaco21]. The CO(2–1) radius $r\_{\rm CO}$ ranges from 60 to 500pc (median 320pc). Fig. [fig:sizecontco] shows that $r\_{\rm CO}$ is larger then the continuum size $r\_{\rm cont}$. The median $r\_{\rm CO}$/$r\_{\rm cont}$ ratio is 2.5 ± 1.1. As for the continuum size, we do not find significant differences between the $r\_{\rm CO}$ of AGN and non-AGN nuclei. If we exclude the nuclei with upper limits for $r\_{\rm cont}$, there is a good correlation between the CO and continuum sizes (Spearman’s rank correlation coefficient $r\_{\rm s}=0.72$, probability of no correlation p = 7 × 10− 4). The best linear fit is $r\_{\rm CO}=(0.70\pm 15)\times r\_{\rm cont}$ + (240 ± 30)pc. Spectral energy distribution fit -------------------------------- SED fit for 00188 − 0856 as an example. The data points correspond to the radio (green diamonds), ALMA ∼ 220GHz from this paper (red circle), and IR (remaining points) observations. The solid green line is the 5–38 Spitzer/IRS spectrum. The IR observations are color coded as follows: Spitzer/IRS synthetic photometry at 34 (green circle); Spitzer/MIPS (yellow circles); and Herschel/SPIRE (green triangles). The solid red line is the best gray-body fit and the dashed blue line represents the best power-law fit to the non-thermal radio emission. Only the encircled symbols have been used for the fits (i.e., the ALMA point is excluded from the SED fit). The dot-dashed green line represents the expected maximum free-free emission assuming that all the $L\_{\rm IR}$ is produced by SF (see Sect. [ss:sedmodel]). The SED fits for the whole sample are shown in Fig. [fig:apxsedmodels].[fig:sedmodels] [tbl:bibfluxes] lcccccccc IRAS name & IR & Radio 00091 − 0738 & M90 & H15, H21 00188 − 0856 & PS21, P16 & H15, H21 00509 + 1225 & P15 & H15, B89 01572 + 0009 & P15, P16 & H15, B89 F05189 − 2524 & P16, C17 & C90, C91 07251 − 0248 & C17 & C98 09022 − 3615 & P16, C17 & C98 F10190 + 1322 & M90, PS21 & H15 11095 − 0238 & P16, PS21 & H15 F12072 − 0444 & M90, P16 & H15 F12112 + 0305 & C17 & C90, C91 13120 − 5453 & P16, C17 & W94, M07 F13451 + 1232 & PS21 & S98 F14348 − 1447 & P16, C17 & C90, C91 F14378 − 3651 & P16, C17 & C98, M03 F15327 + 2340 & M90, C17 & C91, C98, BM15 16090 − 0139 & M90, K01, P16 & C98 16155 + 0146 & M90 & H15 17208 − 0014 & P16, C17 & B06 F19297 − 0406 & P16, C17 & C98 19542 + 1110 & C17 & C98, L11 20087 − 0308 & M90, P16 & W98, C98, M17 20100 − 4156 & PS21, P16 & C96 20414 − 1651 & M90, P16 & C98, N03 F22491 − 1808 & P16, C17 & C90, C91, H21 ### Infrared SED We fitted the IR spectral energy distribution of these ULIRGs to determine the expected dust emission at the ALMA frequency (220–250GHz ≡ 1400–1200). We used published IR photometry from Herschel, ISO, and IRAS (see Table [tbl:bibfluxes]), as well as Spitzer IRS and MIPS data (see Sect. [ss:dataspitzer] and Appendix[apx:spitzerfluxes]). For observation at similar wavelengths, we gave preference to the data with the highest angular resolution. We fitted the far-IR SED using a single-temperature gray-body (e.g., equations 1 and 2 of ). We assumed a fixed β = 1.8, but allowed the optical depth to vary. This model reproduces well the observed far-IR SED between 30 and 500 for most objects. At shorter wavelengths, some of these ULIRGs have excess mid-IR emission which has been associated with warmer dust due to an AGN (e.g., ). Since we are interested in the longer wavelength emission to compare with the ALMA observation, we only used in the fit the photometric points between 34 and 500 to avoid any bias due to this mid-IR excess. We note that we excluded the ALMA continuum flux from the SED fit. For four systems (00509+1225, 01572+0009, F05189-2524, and F13451+1232), we started the fit at 70 because the excess mid-IR emission was clear even at the 34 photometric point. Also, for F13451+1232 (which hosts the radio source 4C + 12.50), we excluded the 500 flux because it has a noticeable contribution from non-thermal emission. We show the best-fit models in Figs. [fig:sedmodels] and [fig:apxsedmodels] and the model parameters are presented in Table [tbl:sedresult]. To compute the total IR luminosity between 6 and 1500 (rest-frame), we first integrated the gray body emission. Then we subtracted the gray body model to the IRS spectrum and obtained the 6–20 mid-IR excess, $\Delta L\_{\rm 6-20\mu m}$. The total $L\_{\rm IR}$ is the addition of the gray body emission and the mid-IR excess. The total $L\_{\rm IR}$ and the $\Delta L\_{\rm 6-20\mu m}$/$L\_{\rm IR}$ ratio are listed in Tables [tbl:sample] and [tbl:sedresult], respectively. ### Non-thermal synchrotron emission At the frequency of the ALMA observations, it is possible to have a significant contribution from non-thermal synchrotron emission. To estimate this contribution, we used published radio observations of our sample of ULIRGs with frequencies between 1 and 40GHz (Table [tbl:bibfluxes]). All the systems have been observed at least at 1.4GHz, except 13120 − 5453 which only has 4.85 and 0.843GHz data. For objects with more than one radio observation (16 systems), we fitted a power law to the radio data. We obtained spectral indexes between − 0.3 and − 1.0 with a mean index of − 0.62 (see Table [tbl:sedresult]), which are similar to the spectral indexes found in a sample of 31 local ULIRGs by. For the remaining objects with just one radio observation at 1.4GHz (9 systems), we assumed the mean spectral index between 1.4 and 22.5GHz (α22.51.4 = − 0.671) measured in local ULIRGs. We find that the non-thermal emission contributes between 4 and 55% (median 20%) of the ALMA continuum flux (Table [tbl:sedresult]). In this fit, we ignored that the free-free emission (see below) can also affect (flatten) the spectral index (e.g., ). Therefore, if the free-free emission is strong compared to the synchrotron, our “non-thermal” contribution estimate would be closer to the combined free-free + synchrotron total emission. For 20087 − 0308, the predicted non-thermal flux is 2.8 times higher than the observed ALMA flux. However, this source shows radio fluxes between 1.4 and 4.85GHz that are not fully compatible with a power-law (Fig. [fig:apxsedmodels]). This suggests that either the power-law model is not adequate for this source, that it presents variable radio emission, or that some of the radio fluxes are not reliable. ### Free-free thermal emission The ALMA continuum measurements can also include a contribution from thermal free-free emission. The free-free emission is related to the ionizing photon rate from young stars and can trace the SFR (e.g., ). We used the relation between the SFR and the free-free radio emission to estimate its contribution at the ALMA frequency (equation 11 from ). The SFR of the ULIRGs was derived from the total IR luminosity using the calibration. The free-free contributions are presented in Table [tbl:sedresult]. However, we note that this free-free emission estimate is an upper limit because we ignored the potential AGN contribution to the $L\_{\rm IR}$, and also because, in the dusty nuclear regions of ULIRGs, a fraction of the ionizing photons can be absorbed by dust grains instead of ionizing H atoms and, therefore, reduce the actual free-free emission (e.g., ). Actually, with these assumptions, six systems (five classified as AGN and one as LINER) have predicted free-free upper limits above 90–100% of the observed ALMA flux. This indicates that a large part of their $L\_{\rm IR}$ likely comes from an AGN and it cannot be directly translated into SFR and subsequently into free-free emission. [t] [tbl:sedresult] lccccccccc IRAS name & $T\_{\rm dust}\slash$K & $\tau\_{\rm 350\mu m}$ & $\log \Delta L\_{\rm 6-20\mu m}\slash L\_{\rm IR}$ & $\alpha\_{\rm non-thermal}$ & $F\_{\rm dust}\slash F\_{\rm ALMA}$ & $F\_{\rm free-free}\slash F\_{\rm ALMA}$ & $F\_{\rm non-thermal}\slash F\_{\rm ALMA}$ 00091 − 0738 & 78 & 0.40 & − 1.45 ± 0.19 & − 0.33 ± 0.05 & 0.22 ± 0.04 & < 0.24 & 0.20 ± 0.03 00188 − 0856 & 71 & 0.71 ± 0.09 & − 1.23 ± 0.10 & − 0.79 ± 0.05 & 1.35 ± 0.14 & < 0.67 & 0.21 ± 0.05 00509 + 1225 & 57 & 0.66 ± 0.17 & − 0.34 ± 0.01 & − 0.68 ± 0.05 & 4.39 ± 0.56 & < 1.77 & 0.16 ± 0.03 01572 + 0009 & 77 & 0.42 ± 0.17 & − 0.73 ± 0.11 & − 1.02 ± 0.05 & 0.90 ± 0.08 & < 1.16 & 0.15 ± 0.04 F05189 − 2524 & 66 & 0.28 ± 0.17 & − 0.72 ± 0.10 & − 0.54 ± 0.10 & 1.31 ± 0.10 & < 0.99 & 0.28 ± 0.12 07251 − 0248 & 75 & 0.50 ± 0.08 & − 1.54 ± 0.25 & − 0.671 & 0.61 ± 0.06 & < 0.27 & 0.04 09022 − 3615 & 65 & 0.30 ± 0.04 & − 1.10 ± 0.06 & − 0.671 & 1.39 ± 0.10 & < 0.70 & 0.40 F10190 + 1322 & 61 & 0.40 & − 1.33 ± 0.06 & − 0.671 & 1.49 ± 0.18 & < 0.46 & 0.17 11095 − 0238 & 85 & 0.27 ± 0.05 & − 1.26 ± 0.20 & − 0.671 & 0.53 ± 0.04 & < 1.06 & 0.55 F12072 − 0444 & 82 & 0.36 ± 0.06 & − 0.93 ± 0.08 & − 0.671 & ⋯ & ⋯ & ⋯ F12112 + 0305 & 63 & 0.36 ± 0.05 & − 1.63 ± 0.16 & − 0.51 ± 0.07 & 1.14 ± 0.09 & < 0.41 & 0.23 ± 0.07 13120 − 5453 & 58 & 0.37 ± 0.05 & − 1.45 ± 0.07 & − 0.63 ± 0.05 & 1.82 ± 0.15 & < 0.50 & 0.38 ± 0.10 F13451 + 1232 & 70 & 0.64 ± 0.24 & − 0.66 ± 0.05 & − 0.54 ± 0.03 & ⋯ & ⋯ & ⋯ F14348 − 1447 & 64 & 0.35 ± 0.04 & − 1.62 ± 0.14 & − 0.75 ± 0.06 & 1.62 ± 0.12 & < 0.65 & 0.17 ± 0.06 F14378 − 3651 & 66 & 0.32 ± 0.05 & − 1.48 ± 0.18 & − 1.00 ± 0.25 & 1.95 ± 0.14 & < 0.91 & 0.08 ± 0.10 F15327 + 2340 & 63 & 0.57 ± 0.10 & − 1.86 ± 0.24 & − 0.53 ± 0.05 & 0.73 ± 0.06 & < 0.26 & 0.13 ± 0.02 16090 − 0139 & 69 & 0.39 ± 0.05 & − 1.31 ± 0.08 & − 0.671 & 0.70 ± 0.05 & < 0.54 & 0.25 16155 + 0146 & 95 & 0.40 & − 0.86 ± 0.13 & − 0.671 & 0.56 ± 0.05 & < 1.20 & 0.49 17208 − 0014 & 62 & 0.38 ± 0.06 & − 1.73 ± 0.17 & − 0.47 ± 0.09 & 0.89 ± 0.08 & < 0.28 & 0.20 ± 0.06 F19297 − 0406 & 67 & 0.48 ± 0.07 & − 1.50 ± 0.12 & − 0.671 & 1.46 ± 0.11 & < 0.49 & 0.16 19542 + 1110 & 69 & 0.35 ± 0.05 & − 1.40 ± 0.16 & − 0.57 ± 0.04 & 1.47 ± 0.09 & < 0.72 & 0.32 ± 0.04 20087 − 0308 & 61 & 0.46 ± 0.08 & − 1.45 ± 0.09 & − 0.42 ± 0.08 & 0.82 ± 0.06 & < 0.37 & 2.79 ± 1.24 20100 − 4156 & 80 & 0.48 ± 0.09 & − 1.49 ± 0.22 & − 0.671 & 0.56 ± 0.05 & < 0.52 & 0.15 20414 − 1651 & 66 & 0.54 ± 0.08 & − 1.68 ± 0.21 & − 0.70 ± 0.06 & 1.32 ± 0.12 & < 0.38 & 0.15 ± 0.04 F22491 − 1808 & 75 & 0.33 ± 0.04 & − 1.66 ± 0.26 & − 0.39 ± 0.08 & 0.65 ± 0.04 & < 0.43 & 0.18 ± 0.05 Mean & 70 ± 9 & 0.43 ± 0.13 & − 1.30 ± 0.08 & ⋯ & 1.21 ± 0.83 & < 0.77 & 0.34 ± 0.55 Median & 67 ± 7 & 0.38 ± 0.12 & − 1.45 ± 0.07 & ⋯ & 1.14 ± 0.62 & < 0.65 & 0.20 ± 0.08 Discussion ========== ALMA continuum as tracer of the IR luminosity --------------------------------------------- We aim to determine the physical size and luminosity surface density of the regions that emit the bulk of the IR luminosity in local ULIRGs. Far-infrared telescopes, that detect the peak of the IR emission, lack the angular resolution to spatially resolve it, although it is possible to infer the size of the far-IR emission through indirect methods like the modeling of far-IR OH absorptions (e.g., ). In this section, we investigate if the ∼ 220GHz ALMA continuum, which provides much higher angular resolutions, can be used as a proxy of the IR emission to obtain direct estimates of the IR emitting region sizes. However, using the ∼ 220GHz continuum to trace the size of ULIRGs is not straightforward. At this frequency, the continuum includes emission from dust, which is connected to the IR luminosity, but it may also include contributions from free-free and synchrotron emissions, which might not be directly related to the IR luminosity. This is important because, even if the IR luminosity of local ULIRGs is thought to be dominated by SF, the AGN contribution increases with increasing $L\_{\rm IR}$, and the synchrotron AGN emission could affect the ∼ 220GHz source sizes. In addition, ALMA is an interferometer, so part of the emission might be filtered out. In this section, we study the impact of these effects on the measured source sizes. ### Filtered out flux In general, due to the limited coverage of the uv plane, interferometric observations filter out extended large scale emission. Therefore, it might be possible that extended continuum emission from our ULIRGs is missing in the measured fluxes (Sect. [ss:almamodel]). To evaluate this possibility we compare the maximum recoverable scale of our observations (about 4kpc; Sect. [ss:dataalma]) with the sizes of the detected sources. The radii range from < 60 to 300pc (Sect. [ss:almamodel] and Table [tbl:almacontmodel]), which are ≪ 4kpc. If > 4kpc structures were actually present in these ULIRGs, we would expect to detect, in addition to these very compact sources, intermediate size structures ( ∼ 2kpc FHWM) which are not seen in the continuum images. This suggests that the 220GHz continuum of ULIRGs is intrinsically compact and that we can recover most of the continuum emission with these data. It also possible that we do not detect extended low-surface brightness continuum emission due to the observations sensitivity. To quantify its possible impact, we assume an emitting area with a 4 × 4kpc2 size (the typical Hα effective radius of ULIRGs is < 2kpc; ). From the continuum sensitivity, Table [tbl:almaobs], we estimated extended emission 3σ upper limits which are on average 8% of the measured fluxes and up to 20–75% for the 4 faint nuclei with $f\_{\rm 220\,GHz}<0.7$mJy (F10190+1322 W; 11095 − 0238 SW, F12112+0305 SW, and 16155+0146 NW; see Table [tbl:almacontmodel]). Therefore, if low-surface brightness emission is present, its contribution would be small for the great majority of the nuclei (at least for 25 out of 29). ### Dust, free-free, and synchrotron contributions At the frequencies of the ALMA observations (190–250GHz), in addition to the Rayleigh-Jeans tail of the IR dust emission, a contribution from thermal free-free, and non-thermal synchrotron emission is possible. In particular, we explore whether the free-free or synchrotron emissions could bias the measured sizes toward more compact sizes in the case of a starburst nucleus. #### Dust We first estimate the dust contribution. Based on the IR SED modeling (Sect. [ss:sedmodel]), we found that the ALMA flux densities are just slightly lower than the extrapolation of the IR gray-body fit (median ratio of 1.14; Table [tbl:sedresult]). We note that we did not use the ALMA flux in the gray-body fit. Thus, a possible interpretation for the good agreement between the data and the model prediction is that the ALMA flux comes from the long-wavelength tail of the dust gray-body emission. If this is the case, we could use the high-resolution ALMA data to determine the size of the IR emitting regions. This good agreement between the ALMA continuum flux and the IR SED extrapolation was also found by using 260GHz observations of local ULIRGs at comparable spatial resolutions. #### Free-Free The ALMA emission can include free-free emission as well. The free-free emission is produced by ionized hydrogen usually associated with star-forming regions (see Sect. [ss:irsed]). We estimated an upper limit for the free-free emission assuming that all the $L\_{\rm IR}$ is produced by SF (i.e., ignoring the possible AGN contribution). This assumption also implies that dust does not absorb any ionizing photon (Sect. [ss:irsed]). The upper limit for the free-free contribution has a median value of < 65% (Table [tbl:sedresult]). But even if the free-free emission dominates the ALMA flux, it should not affect the region size estimates since, for this free-free upper limit estimation, both IR and free-free emissions should be co-spatial as they have a common star-formation origin. Moreover, taking into account the AGN contribution to the $L\_{\rm IR}$ and the effect of absorption of UV photons by dust would reduce this upper limit and, therefore, the possible impact of the free-free emission on the source size measurements. #### Synchrotron Some contribution from synchrotron emission is possible too. If this synchrotron emission is produced by supernovae (i.e., related to star-forming regions), the ALMA regions sizes should not be affected since the IR emission and the supernovae (SNe) should have similar spatial distributions. Alternatively, AGN can produce strong synchrotron emission. In our sample, only F13451+1232 (4C + 12.50) has excess radio emission with respect to the radio-IR relation (see ). Actually, the F13451+1232 ∼ 220GHz emission is dominated by synchrotron radiation (see Fig. [fig:apxsedmodels]) and, therefore, cannot be directly used as a proxy of the IR emitting region. For the remaining objects, we estimated a median non-thermal contribution, which includes both SNe and AGN emission, of 20% and up to 40–60% in four objects: one of the starbursts (09022 − 3615), one LINER (11095 − 0238), and 2 out of the 7 systems classified as AGN in the optical (13120 − 5453, and 16155 + 0146). The latter suggests that the optical detection of an AGN does not imply that the ∼ 220GHz emission is always dominated by synchrotron AGN emission in local ULIRGs. However, how the AGN synchrotron emission affects the ALMA source sizes is unclear. AGN jets producing synchrotron emission have sizes ranging from pc to few kpc in radio-quiet AGN. For instance, the synchrotron radio jet emission from the AGN ULIRG 01572+0009 (PG 0157+001) has a ∼ 7kpc diameter, although according to our SED modeling, the synchrotron contribution at 220GHz is small, about 0.15, and should not affect the estimated size in this object. For the 5 systems with a high non-thermal contribution, we find that their sizes and luminosity and molecular gas surface densities do not differ from those of the rest of the sample (see Fig. [fig:histosize]). ### Summary It seems likely that these ∼ 220GHz continuum ALMA observations trace the IR emitting region for the majority of local ULIRGs and that the filtered out flux due to the interferometric observations is small. The good agreement between the IR gray-body extrapolation and the ALMA fluxes supports this. Free-free and SNe synchrotron emissions could contribute to the ALMA flux, but since they have a star-formation origin they should not affect the size estimates for a starburst ULIRG. Synchrotron emission from AGN could bias the size measurements, although we do not find significant differences in size between the 5 systems with high synchrotron emission and the rest of the sample. Therefore, in the following sections we assume that the size of the ∼ 220GHz continuum is equivalent to the size of the region which emits the bulk of the IR luminosity in these ULIRGs. [t] Cold molecular gas surface density ($\Sigma\_{\rm H\_2}$) vs. IR luminosity surface density ($\Sigma\_{L\_{\rm IR}}$). Galaxy symbols are as in Fig. [fig:sizecontco]. The dotted black lines indicate the maximum luminosity from an instantaneous starburst using 100% (ε = 1) or 20% (ε = 0.2) of the available cold molecular gas (see Sect. [ss:thickSB]). The orange solid line is the Eddington luminosity limit. For points above this line, the radiation pressure is stronger than gravity (Sect. [ss:Eddlimit]). The dashed blue lines indicate the 1 and 10Myr depletion times for reference assuming that the IR luminosity is produced by SF. The column density ($N\_{\rm H}$) axis is calculated from $\Sigma\_{\rm H\_2}$ assuming a uniform mass distribution (i.e., $N\_{\rm H} = 2 \times \Sigma\_{\rm H\_2}\slash m({\rm H\_2})$ where $m({\rm H\_2})$ is the H2 molecular weight).[fig:lirdensity] Extreme nuclear IR luminosity densities --------------------------------------- Using the half-light radius, $r\_{\rm cont}$, of the ALMA continuum and half of the IR luminosity (Sects. [ss:almamodel] and [ss:sedmodel]), we calculated the luminosity surface density, $\Sigma\_{L\_{\rm IR}}$, in the nuclear regions of the ULIRGs. For systems with 2 nuclei, we estimated their IR luminosity fraction using their relative ALMA continuum fluxes. In 70% of the interacting systems, the luminosity is completely dominated ( > 90% of the total luminosity) by one of the nuclei. These fractions and the resulting surface densities are listed in Table [tbl:lirdensity]. We find $\log \Sigma\_{L\_{\rm IR}}\slash (L\_\odot$kpc− 2) between 11.5 and 14.3 with a median value of 13.2. If this IR luminosity is produced by SF, it corresponds to $\Sigma\_{\rm SFR}=2500$M⊙yr− 1kpc− 2 using the SFR calibration. These values are much higher (1–2 orders of magnitude) than the densities found in local starburst LIRGs, even when they are observed at higher angular resolutions of ∼ 100pc (e.g., ). Similarly, z ∼ 3–6 sub-mm galaxies have lower surface densities, 150–1300M⊙yr− 1kpc− 2 when observed at ∼ kpc resolutions (e.g., ). At higher resolutions, ∼ 200pc, these sub-mm galaxies have $\Sigma\_{\rm SFR}$ between 100 and 3000M⊙yr− 1kpc− 2 which are still lower than the majority of the local ULIRGs (e.g., ). The observed $\Sigma\_{L\_{\rm IR}}$ range is comparable to that measured in local ULIRGs using radio and IR observations at similar angular resolution. Using 33GHz radio data, found a median surface density of 1012.8L⊙kpc− 2 in a sample of 22 local interacting/merging systems with $\log L\_{\rm IR}\slash L\_\odot>11.6$. Similarly, estimated nuclear $\Sigma\_{L\_{\rm IR}}>$1012.8L⊙kpc− 2 for 10 local ULIRGs based on the modeling of the far-IR OH absorptions. In addition, mid-IR ground-based studies of ULIRGs derive maximum luminosity densities between 1012.1 and 1014.6L⊙kpc− 2. For the western nucleus of F15327+2340 (Arp 220 W), using higher resolution data ( < 50pc) a luminosity density about 1014.3 − 15.5L⊙kpc− 2 has been estimated. Therefore, our results are compatible with previous findings and confirm that the luminosity density in the nucleus of local ULIRGs is much higher than in other local and high-z starbursts when measured at 100–1000pc scales. [t] [tbl:lirdensity] lcccccccc IRAS name & Nucleus & Frac. & $\log \Sigma\_{L\_{\rm IR}}$ & $t\_{\rm dep}$ & Above Eddington & Optical AGN? & CON & HCN14 & & & (L⊙kpc− 2) & (Myr) & limit? 00091 − 0738 & S & 0.97 & > 13.67 & < 0.78 & Y & N & ⋯ & Y & N & < 0.03 & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ 00188 − 0856 & - & 1.00 & 13.15 ± 0.08 & 2.0 ± 0.1 & Y & Y & ⋯ & N 00509 + 1225 & - & 1.00 & 13.30 ± 0.09 & 0.65 ± 0.09 & Y & Y & ⋯ & N 01572 + 0009 & - & 1.00 & 14.30 ± 0.18 & 0.10 ± 0.01 & Y & Y& ⋯ & N F05189 − 2524 & - & 1.00 & > 13.70 & < 0.4 & Y & Y & ⋯ & Y 07251 − 0248 & W & 0.09 & 12.74 ± 0.07 & 1.6 ± 0.3 & Y & N & ⋯ & Y & E & 0.91 & > 13.78 & < 0.65 & Y & N & ⋯ 09022 − 3615 & - & 1.00 & 12.70 ± 0.24 & 7.3 ± 0.7 & N & N & N & N F10190 + 1322 & W & 0.12 & 11.53 ± 0.11 & 14.3 ± 1.8 & N & N & ⋯ & N & E & 0.88 & 12.50 ± 0.08 & 6.0 ± 0.7 & N & N & ⋯ 11095 − 0238& SW & 0.32 & > 12.88 & < 1.3 & Y & N & ⋯ & Y & NE & 0.68 & > 13.22 & < 1.2 & Y & N & ⋯ F12112 + 0305 & SW & 0.09 & 12.27 ± 0.12 & 7.5 ± 0.8 & N & N & ⋯ & N & NE & 0.91 & > 14.00 & < 0.38 & Y & N &? 13120 − 5453 & - & 1.00 & 12.59 ± 0.23 & 5.8 ± 0.7 & N & Y & N & N F14348 − 1447 & SW & 0.63 & > 13.83 & < 0.39 & Y & N & N & N & NE & 0.37 & 13.19 ± 0.08 & 1.3 ± 0.2 & Y & N & N F14378 − 3651 & - & 1.00 & 12.29 ± 0.07 & 5.2 ± 0.6 & N & Y &? & N F15327 + 2340 & W & 0.71 & > 13.89 & 0.42 ± 0.05 & Y & N & Y & Y & E & 0.29 & 13.07 ± 0.07 & 2.7 ± 0.3 & N & N &? & 16090 − 0139 & - & 1.00 & 12.94 ± 0.07 & 7.0 ± 0.9 & N & N& ⋯ & Y 16155 + 0146 & NW & 0.80 & > 13.26 & < 5.8 &? & Y& ⋯ & N & SE & < 0.2 & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ 17208 − 0014 & - & 1.00 & > 13.88 & < 0.65 & Y & N & Y & Y F19297 − 0406 & S & < 0.01 & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & Y & N & 0.99 & 13.13 ± 0.07 & 2.3 ± 0.2 & N & N & ⋯ 19542 + 1110 & - & 1.00 & 13.22 ± 0.07 & 1.7 ± 0.2 & Y & N & ⋯ & N 20087 − 0308 & - & 1.00 & 13.32 ± 0.06 & 2.0 ± 0.2 & Y & N & ⋯ & N 20100 − 4156 & NW & < 0.02 & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & Y & SE & 0.98 & 14.04 ± 0.07 & 0.78 ± 0.09 & Y & N & ⋯ 20414 − 1651 & - & 1.00 & 13.80 ± 0.11 & 0.60 ± 0.08 & Y & N & ⋯ & N F22491 − 1808 & W & < 0.02 & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & Y & E & 0.98 & > 13.69 & < 0.74 & Y & N & Y To investigate the origin of this high $\Sigma\_{L\_{\rm IR}}$ values, we discuss two alternatives: an optically thick starburst and the presence of an obscured AGN. ### Optically thick starburst In Fig. [fig:lirdensity], we plot the $\Sigma\_{L\_{\rm IR}}$ vs. $\Sigma\_{\rm H\_2}$ (Sect. [ss:nuclearmol]) relation for our sample of local ULIRGs. Based on theoretical models, a $\Sigma\_{L\_{\rm IR}}$ of ∼ 1013L⊙kpc− 2 has been suggested as the maximum $\Sigma\_{L\_{\rm IR}}$ of warm (T < 200K) optically thick starbursts. The maximum $\Sigma\_{L\_{\rm IR}}$ for these warm starbursts is similar to the median value found in our sample. Hot (T > 200K) optically thick starbursts could have $\Sigma\_{L\_{\rm IR}}\sim10^{15}$L⊙kpc− 2 when $\Sigma\_{\rm H\_2}>10^6$pc− 2. However, we measure $\Sigma\_{\rm H\_2}\ll10^6$pc− 2, so these local ULIRGs might be more similar to the warm optically thick starburst models. Although the $\Sigma\_{L\_{\rm IR}}$ are similar to the maximum for a warm optically thick starburst, when combined with the $\Sigma\_{\rm H\_2}$, the resulting depletion times, $\Sigma\_{\rm H\_2}$/$\Sigma\_{\rm SFR}$, would be shorter ( < 1–15Myr; see Table [tbl:lirdensity]) than those measured in other LIRG starbursts ( > 30–100Myr; ). We also estimated the maximum $\Sigma\_{L\_{\rm IR}}$ that a starburst can produce for a given $\Sigma\_{\rm H\_2}$. Using starburst99, we find that the maximum luminosity produced by a solar metallicity instantaneous burst, assuming a initial mass function (IMF), is ∼ 1060L⊙M⊙− 1 at an age of ∼ 2.2Myr (see also ). In Fig. [fig:lirdensity], we show this limit (ε = 1 dotted black line). This limit assumes that all the molecular gas is instantaneously transformed into stars (i.e., 100% efficient SF). In reality, stellar feedback dissipates the molecular clouds before a 100% efficiency is achieved, so the maximum luminosity from a starburst would be lower than this limit. Based on magneto-hydrodynamic simulations, the maximum efficiency per free-fall time is about 20% (e.g., ) which is shown in Fig. [fig:lirdensity] too (ε = 0.2). This figure shows that 70% of the nuclei are above the 100% efficiency limit (ε = 1) and all of them are above this 20% efficiency limit. In addition, if the nuclear luminosity is produced by a compact and intense starburst, a continuous supply of molecular gas to the nucleus would be required to sustain the ongoing SFR level. Otherwise, it would not be possible to achieve the observed $\Sigma\_{L\_{\rm IR}}$ which implies depletion times < 1Myr in 30% of the sample and < 15Myr in all of them. However, the high radiation pressure in the nucleus, which is compatible with being above the Eddington limit for 67% of the nuclei (see Sect. [ss:Eddlimit]), could prevent these massive gas inflows. ### Or obscured AGN? We find that 70% of the ULIRGs have $\Sigma\_{L\_{\rm IR}}$/$\Sigma\_{\rm H\_2}$ ratios above the limit for a ε = 1 efficient starburst. Also, 65% have $\Sigma\_{L\_{\rm IR}}$ above the theoretical value of an optically thick warm starburst ( ∼ 1013L⊙kpc− 2). These fractions are similar for systems optically classified as AGN and starbursts (see Fig. [fig:lirdensity]). These results suggest that what produces the bulk of the IR luminosity in these local ULIRGs is not a standard starburst. Alternatively, an AGN could dominate the $L\_{\rm IR}$ of these objects. This possibility has also been suggested because of the high $\Sigma\_{L\_{\rm IR}}$ in the ULIRG nuclei derived from mid-IR data (e.g., ) or from the mm continuum in Arp 220 (e.g., ). From $\Sigma\_{\rm H\_2}$, we estimate that the nuclear H column densities are moderate, between 1023 and 1024cm− 2 (see Fig. [fig:lirdensity]). These values are lower than the Compton thick limit ( ∼ 2 × 1024cm− 2), so we would expect the AGN X-ray emission not to be completely absorbed. observed a sample of local U/LIRGs with Chandra at 0.5–7keV. Twelve out of our 25 systems are part of their sample. They found AGN evidence in 6 out of the 12, but they estimated low AGN contributions to the $L\_{\rm IR}$ (3–20%). However, it is possible that the actual $N\_{\rm H}$ that obscures these AGN is actually higher and could absorb the 0.5–7keV X-ray emission. Our $N\_{\rm H}$ estimates are based on molecular gas observations at ∼ 400pc resolution, but the obscuring molecular torus could be smaller, as observed in local Seyfert galaxies (median diameter of 40pc; ), so our $N\_{\rm H}$ values might be underestimated. To minimize the effects of the obscuring column density, we also considered the Swift-BAT 105-Month 14–195keV survey. This higher energy X-ray band is less affected by $N\_{\rm H}$ than the Chandra 0.5–7keV range. The 5σ sensitivity of the survey is 8.4 × 10− 12ergs− 1cm− 2. Only one source in our sample, F05189 − 2524, is detected at 14–195keV. For the rest of the targets, the Swift-BAT survey implies $L\_{\rm 14-195\,keV} <$1043.3 − 44.8ergs− 1, depending on the distance. Assuming that the bolometric AGN luminosity is $L\_{\rm AGN}\sim12\times L\_{\rm 14-195\,keV}$, the 5σ upper limits would correspond to $L\_{\rm AGN}<$1044.3 − 45.9ergs− 1 = 1010.7 − 12.3L⊙. This would result in an AGN contribution $L\_{\rm AGN}$/$L\_{\rm IR}<$0.45 for all but one of these ULIRGs and a median upper limit of < 0.25. More sensitive NuSTAR > 10keV observations were presented by. Their sample contains four of our ULIRGs, three of them already classified as Sy in the optical. One is undetected (F14378 − 3651), in F15327+2340 (Arp 220) no AGN evidence is found, although a very deeply buried AGN is still possible, and F05189 − 2524 and 13120 − 5453 are Compton-thin and -thick AGN, respectively. ULIRGs are known to be hard X-ray underluminous and AGN in mergers are also heavily obscured. The combination of these two factors could explain why the AGN in these sources, if present, remain mostly undetected in X-ray observations. As discussed before, the high $\Sigma\_{L\_{\rm IR}}$/$\Sigma\_{\rm H\_2}$ nuclear ratios cannot be easily explained by a starburst, even if an optically thick one is considered. These ALMA data would be consistent with an AGN dominating the IR luminosity, but it is not possible to confirm that an AGN is present in the nuclei of the majority of these ULIRGs. The non-detection of these possible AGN in ultra-hard X-ray observations could indicate extremely high obscuring column densities. Higher angular resolution ALMA data could spatially resolve the obscuring material and establish its actual column density as in nearby Seyfert galaxies (e.g., ). [t] Logarithm of the excess mid-IR emission vs. size of the 220GHz continuum (left panel) and size of the CO(2–1) emission (right panel). Galaxy symbols are as in Fig. [fig:sizecontco]. The solid orange line is the best linear fit to the non-AGN (black) points. In the left panel, the red box (a) marks a region of this diagram solely occupied by optically detected AGN. The blue box (b) indicates the location of very compact ULIRG nuclei (mostly unresolved by our data) with negligible excess mid-IR emission. The green box shows the location of more extended nuclei with higher mid-IR excess.[fig:emirr50] ### Systematic uncertainties There are several assumptions which could bias the $\Sigma\_{L\_{\rm IR}}$ and $\Sigma\_{\rm H\_2}$ values presented in Fig. [fig:lirdensity]. For instance, the $r\_{\rm CO}$ size used to estimate the nuclear $\Sigma\_{\rm H\_2}$ is larger than the $r\_{\rm cont}$ used for $\Sigma\_{L\_{\rm IR}}$. This is because the CO(2–1) emission is more extended than the ∼ 220GHz continuum (=2.5 ± 1.1; see Sect. [ss:nuclearmol]) and it is not possible to exactly determine the amount of CO(2–1) within the $r\_{\rm cont}$ region since, in most cases, both $r\_{\rm cont}$ and $r\_{\rm CO}$ are smaller than the beam size. Therefore, $\Sigma\_{\rm H\_2}$ are averaged over larger regions than $\Sigma\_{\rm L\_{\rm IR}}$ and the true nuclear $\Sigma\_{\rm H\_2}$ could be higher. Consequently, the real star-formation efficiency (depletion time) could be lower (longer). Actually,, by modeling the far-IR OH absorptions, estimated nuclear $\Sigma\_{\rm H\_2}$ between 103.8 and 104.7pc− 2 for 10 local ULIRGs. These values are ∼ 5 times higher than our average $\Sigma\_{\rm H\_2}$ derived from CO(2–1) using ∼ 400pc resolution data. In addition, 8 of our targets are part of a survey to detect vibrationally excited HCN emission to identify compact obscured nuclei (CONs). The HCN-vib (3–2) 267.199GHz line is detected in 5 of these systems (63 ± 18%; see Table [tbl:lirdensity] and ). Similarly, the mid-IR HCN-vib 14 absorption, which populates the levels originating the HCN-vib emission, is detected in 10 (6 interacting systems and 4 mergers) out of the 23 systems (43 ± 10% globally, 60 ± 15% of the interacting systems and 31 ± 13% of the advanced mergers; Table [tbl:lirdensity]). The presence of these HCN-vib spectral features suggests the presence of extreme nuclear column densities too. Higher resolution CO(2–1) data will help us to establish the cold molecular gas density more accurately. The already available high resolution CO(2–1) data for 17208 − 0014 (120pc resolution; see Sect. [ss:almamodelhires]), suggest a factor of 2 higher nuclear $\Sigma\_{\rm H\_2}$ (Pereira-Santaella in prep.). However, for this galaxy, the nuclear $\Sigma\_{L\_{\rm IR}}$ also increases by a factor of ∼ 5 using these data, so the resulting $\Sigma\_{\rm SFR}$/$\Sigma\_{\rm H\_2}$ ratio would be even higher and reinforce the need for an obscured AGN in this object. Another uncertainty related to the $\Sigma\_{\rm H\_2}$ is the $\alpha\_{\rm CO}$ conversion factor. We assumed a ULIRG-like factor, which is relatively low compared to the conversion factor used for normal galaxies (see e.g., ). High nuclear column densities can produce self-absorbed CO(2–1) line profiles as seen in the compact nuclei of some local LIRGs (e.g.,,, ). If the nuclear CO(2–1) emission of these ULIRGs is self-absorbed, the assumed ULIRG-like $\alpha\_{\rm CO}$ conversion factor could result in underestimated $\Sigma\_{\rm H\_2}$ values. For this paper, we opt to use the standard ULIRG-like $\alpha\_{\rm CO}$, as it is typically done in local ULIRG studies, but we will investigate the presence of self-absorbed CO(2–1) profiles in these targets in a future paper. The depletion times depend on the assumed $\alpha\_{\rm CO}$ conversion factor. If the actual $\alpha\_{\rm CO}$ of these objects is similar to the Milky Way factor (i.e., 5 times higher; ), the molecular gas mass will be 5 times higher, and the depletion times 5 times longer. However, the median $t\_{\rm dep}$ would still be very short ( < 6Myr). Finally, the maximum luminosity for a starburst (dotted lines in Fig. [fig:lirdensity]) is calculated using a IMF. If the IMF in ULIRGs is top-heavy as suggested by some works (see e.g., ), the maximum starburst luminosity per unit of molecular gas could be higher. For example, the maximum luminosity for an IMF truncated at 30would produce ∼ 10 times more luminosity per unit of molecular gas than the standard IMF. For an ε = 0.2 efficiency, this top-heavy IMF could explain the $\Sigma\_{L\_{\rm IR}}$/$\Sigma\_{\rm H\_2}$ ratios observed in about half of the nuclei, although these starbursts would have radiation pressures above the Eddington limit and they would not be stable on short timescales of few Myr (see Sect. [ss:Eddlimit]). Same as left panel of Fig. [fig:emirr50], but galaxies with more than 50% AGN contribution based on mid-IR observations are indicated in red.[fig:emirr50mir] Distribution of the 220GHz continuum radius $r\_{\rm cont}$ (left) and the CO(2–1) emission radius $r\_{\rm CO}$ (right) for the nuclei in interacting ULIRGs. The upper limits for the 220GHz continuum are included in the first bin and indicated by the green shaded area.[fig:histointersizes] [t] Relation between the 9.7 silicate absorption and the 220GHz continuum (left) and the CO(2-1) (middle) sizes. The right panel shows the 220GHz continuum size vs. IRAS 12/25 color relation. Symbols as in Fig. [fig:sizecontco]. For the IRAS 12/25 color relation (right), the best linear fit to the non-AGN objects (black symbols) is $\log f\_{\rm 12}$/$f\_{\rm 25} = (-1.29\pm0.04) + (1.6\pm0.3)\times 10^{-3}\times r\_{\rm cont}\slash {\rm pc}$. [fig:irasr50] Continuum and CO(2–1) sizes vs. mid-infrared excess --------------------------------------------------- In the left panel of Fig. [fig:emirr50], we show the relation between the excess mid-IR emission (Sect. [ss:irsed]) and the 220GHz continuum size, $r\_{\rm cont}$. We note that the excess mid-IR emission is estimated from the system integrated Spitzer/IRS spectra. Therefore, in interacting systems where a nuclei dominates the total $L\_{\rm IR}$, the mid-IR excess of the secondary nucleus cannot be directly estimated from this integrated spectrum. Thus, we excluded the 8 nuclei in interacting systems with IR luminosity fractions < 20% (see Table [tbl:lirdensity]). In this diagram, we can classify the ULIRG nuclei in two main groups: compact objects ($r\_{\rm cont}<$120pc) with high mid-IR excess (logarithm > − 0.9; red box); and objects following a linear relationship between the two properties with decreasing excess mid-IR emission for decreasing $r\_{\rm cont}$ (blue line). The first group exclusively contains objects classified as AGN in the optical. We note that the 2 interacting systems in our sample with no ALMA data, F12072 − 0444 and F13451 + 1232, are optical AGN with $\log \Delta L\_{\rm 6-20\mu m}$/$L\_{\rm IR}>-0.9$ and could lie in this region of the diagram as well. The location of these AGN in this diagram (red box) is what is expected if warm dust (which emits in the mid-IR) from a compact torus produces a significant part of their total $L\_{\rm IR}$. Actually, the mid-IR $f\_{15\mu m}\slash f\_{30\mu m}$ color, which behaves similar to the mid-IR excess, have been used to estimate the AGN luminosity contribution in ULIRGs (e.g., ). For the remaining nuclei there is a correlation ($r\_{\rm s}=0.87$ and p = 1 × 10− 3) between the continuum size and the mid-IR excess with increasing sizes for increasing mid-IR excess emission. The size of these sources ranges from < 70pc to ∼ 300pc and the best linear fit is $\log \Delta L\_{\rm 6-20\mu m}$/$L\_{\rm IR} = (-1.8\pm0.1) + (2.2\pm0.6)\times 10^{-3}\times r\_{\rm cont}\slash {\rm pc}$. In Fig. [fig:emirr50] (left panel), we highlighted (blue box) sources with $r\_{\rm cont}<120$pc. This box contains 46% of the sample (12 nuclei out of 26). They have sizes comparable to the AGN (red box), although most of them only have an upper limit $r\_{\rm cont}$, so their real size could be lower than that of the AGN. Opposite to most AGN, which have excess mid-IR emission between 15 and 70% of the $L\_{\rm IR}$, these objects have negligible mid-IR excess ( < 3%). A majority of the nuclei in interacting systems (60%) lie in this blue box while it only includes 30% of the mergers. Actually, mergers following this relation have larger radii (median 160pc) than the interacting systems (median 90pc). This suggests that during the early phases of the interaction (nuclear separation > 1kpc), most of the activity (AGN and/or SF) occurs in compact regions where most of the mid-IR emission is absorbed by dust and then re-emitted in the far-IR. Then, in more advanced merger stages (nuclear separation < 1kpc), the activity appears on more extended regions (green box in Fig. [fig:emirr50]), unless an optically detected AGN is present (red box sources). In Fig. 9, we indicate the systems with AGN contributions > 50% based on mid-IR observations. The majority of the optical AGN are also classified as AGN by these mid-IR diagnostics (5 out of 7) and only 3 non-Sy are identified as mid-IR AGN. In addition, this figure shows that there is a good correlation between the mid-IR AGN classification and the mid-IR excess. This is because these mid-IR diagnostics are based on the detection of warm dust which is closely related to the mid-IR excess definition used here. Therefore, if the mid-IR emission of the AGN in these ULIRGs is absorbed (see also Sect. [ss:contsiliras]), these mid-IR diagnostics could fail to detect them. The right panel of Fig. [fig:emirr50] shows the excess mid-IR emission as function of the molecular gas emission size $r\_{\rm CO}$. We do not find a significant correlation between the molecular gas size and the mid-IR excess ($r\_{\rm s}=0.34$ and p = 0.12). The molecular gas emission size is similar in the interacting systems and advanced mergers (median ∼ 350pc). On the contrary, hydrodynamic simulations predict that strong torques during the first passage can pile up the gas in ∼ kpc-scale starbursts and later, during the final-coalescence, in more compact sub-kpc starbursts (e.g., ). We find that, for the interacting systems, the molecular gas and the continuum size distributions are very different (see Fig. [fig:histointersizes]). The radii of the molecular gas emission is uniformly distributed over the whole observed range (mostly between 200 and 500pc; right panel of Fig. [fig:histointersizes]) while the continuum size distribution peaks at compact radii ( < 100pc; left panel of Fig. [fig:histointersizes]). This suggests that, even if there is a global correlation between the continuum and the CO emission sizes for the whole sample of ULIRGs (see Sect. [ss:nuclearmol] and Fig. [fig:histosize]), for the interacting systems, the two sizes seem to be decoupled. An obscured AGN, which do not need large amounts of molecular gas to produce high luminosities, could explain why these interacting nuclei have very compact continuum sources but a more extended molecular gas distribution. Continuum size vs. silicate absorption and IR colors ---------------------------------------------------- In this section, we explore possible relations between the 220GHz continuum size and other IR tracers. As in Sect. [ss:mirexcess], we excluded from this analysis nuclei with contributions < 20% to the total IR luminosity of the systems. We first consider the 9.7 silicate absorption. We use the 9.7 silicate strengths from the Infrared Database of Extragalactic Observables from Spitzer (IDEOS; Table [tbl:spitzerfluxes];, Spoon et al. in prep.). Deep silicate absorptions have been associated with an evolutionary phase in which the obscuring molecular cocoon created during the early interaction phases have not been yet shed (e.g., ). Therefore, we could expect a relation between the silicate absorption and the size of the cold molecular or dust continuum emissions. However, we find no significant correlation between them (see left and middle panels of Fig. [fig:irasr50]). To explain this absence of correlation, we argue that to measure deep 9.7 silicate absorptions, some mid-IR radiation must escape the nuclear molecular cocoon. As shown in Sect. [ss:mirexcess], the fraction of the total IR emission that is emitted in the mid-IR is greatly reduced in the most compact nuclei (i.e., the mid-IR emission is absorbed and then re-emitted in the far-IR). Therefore, the 9.7 silicate absorption is not necessarily extreme in these compact sources since their nuclear mid-IR emission is possibly absorbed and what is observed in the mid-IR is likely produced by less obscured external regions. The presence of several CONs in our sample (see ), is also consistent with the mid-IR emission of the hot nucleus being obscured. Likewise, ground-based mid-IR spectroscopy of local AGN ULIRGs showed that their silicate absorptions are produced by dust not directly associated with the AGN. This scenario is also supported by radiative transfer models of compact nuclei where the mid-IR emission is completely absorbed for objects with very high column densities ($N\_{\rm H\_2}>$1025cm− 2; see fig. 2 of ). We also studied possible correlations between the broad-band IRAS colors and the size of the continuum and cold molecular emissions. Most of the ULIRGs are undetected by IRAS at 12, for this reason we computed synthetic fluxes for the four IRAS bands (12, 25, 60, and 100) form the Spitzer/IRS spectrum and the IR SED model (Sect. [ss:sedmodel]). We tried the 6 IRAS color combinations. Only the $f\_{\rm 12}$/$f\_{\rm 25}$ color shows a significant correlation with the continuum size (Fig. [fig:irasr50] right). The $f\_{\rm 12}$/$f\_{\rm 25}$ ratio is actually related to the mid-IR excess emission. The 12 flux would trace the mid-IR emission while at 25, the emission is dominated by the IR gray-body for most of the non-AGN ULIRGs. Therefore, this relation would be equivalent to that presented in Sect. [ss:mirexcess] with the mid-IR excess emission. Nuclear radiation pressure. Eddington limit ------------------------------------------- We find very high $\Sigma\_{L\_{\rm IR}}$ in the nuclei of the ULIRGs, so it is important to determine if the radiation pressure can overcome the gravity attraction in these objects. To do so, we calculated the Eddington limit in the $\Sigma\_{\rm H\_2}$-$\Sigma\_{L\_{\rm IR}}$ plane by fitting the model results reported by. These models assume spherical symmetry and accurately determine the force due to radiation pressure once the equilibrium $T\_{\rm dust}$ profile across the source is calculated. The models assume a density profile ∼ r− 1, such that $\Sigma\_{\rm H\_2}$ does not depend on the source radius. Our fitting, shown in Fig. [fig:lirdensity] with an orange line, is approximately valid for $\Sigma\_{\rm H\_2}\lesssim3\times10^4$M⊙pc− 2, and assumes an intermediate molecular gas fraction with respect to the total mass, $f\_{\rm g} = 0.3$, and a gas-to-dust ratio of $f\_{\rm gd}=100$ by mass. In the optically thick limit, the Eddington luminosity is proportional to $f\_{\rm g}^{-1/2}\times f\_{\rm gd}$. Fig. [fig:lirdensity] shows that half of the nuclei, 19 out of the 29, are above the estimated Eddington limit (see also Table [tbl:lirdensity]). This suggests that the radiation pressure can be stronger than gravity in these nuclei and, therefore, massive gas outflows are expected. This result is consistent with the detection of massive molecular outflows in local ULIRGs and also supports that radiation pressure plays a relevant role as a potential launching mechanism of the outflows in local ULIRGs. Conclusions =========== We have analyzed new high-resolution ALMA ∼ 220GHz and CO(2–1) observations of a representative sample of 23 local ULIRGs (34 individual nuclei) as part of the Physics of ULIRGs with MUSE and ALMA (PUMA) project (see also ). The main results of this work are the following: 1. We modeled the ∼ 220GHz (190–250GHz) continuum emission of these ULIRGs. We find that the median deconvolved half light radius ($r\_{\rm cont}$) is 80–100pc and about 40% (11/29 with continuum detection) of the nuclei are not resolved by these data. From the IR and radio SED modeling, we obtain that the ALMA ∼ 220GHz continuum fluxes are in good agreement, within a factor of 2 (median ratio of 1.1 ± 0.7), with the extrapolation of the dust far-IR gray body emission. This suggests that the ∼ 220GHz continuum traces the regions emitting the bulk of the IR luminosity in these objects. We estimate that the contributions from synchrotron ( ∼ 20%) and free-free emission ( < 65%) to the ALMA flux are not likely to bias the measured sizes. Using the ∼ 220GHz continuum size, we calculate IR luminosity densities, $\Sigma\_{L\_{\rm IR}}$, in the range 1011.5 − 1014.3L⊙kpc− 2 (median 1013.2L⊙kpc− 2), which is equivalent to $\Sigma\_{\rm SFR}=2500$M⊙yr− 1kpc− 2. This is similar to the range derived from previous radio and ground-based mid-IR observations and 1–2 orders of magnitude brighter than local and high-z starbursts. 2. Similarly, we measure deconvolved CO(2–1) emission sizes, $r\_{\rm CO}$, between 60 and 700pc. These are on average 2.5 ± 1.1 times larger than the ∼ 220GHz continuum size. We find no differences between systems optically classified as AGN or starburst or between interacting systems and advanced mergers. Using a ULIRG-like $\alpha\_{\rm CO}$ conversion factor, we find nuclear molecular gas surface densities, $\Sigma\_{\rm H\_2}$, in the range 102.9 − 104.2M⊙pc− 2. 3. If the $L\_{\rm IR}$ is produced by SF, the $\Sigma\_{L\_{\rm IR}}$/$\Sigma\_{\rm H\_2}$ ratios imply extremely short molecular gas depletion times ( < 1–15Myr). In addition, 70% of the nuclei would have SF efficiencies above the maximum for a starburst (ε = 1) and all of them would have ε > 0.2, which is the maximum efficiency per free-fall time predicted by simulations. These findings suggests that the bulk of the IR luminosity of these ULIRGs does not originate in a nuclear starburst. An obscured AGN with $L\_{\rm AGN}$/$L\_{\rm IR}> 0.5$ would be an alternative energy source. 4. For the compact nuclei ($r\_{\rm cont}<$120pc) in interacting system with low mid-IR excess emission, the $r\_{\rm cont}$ is not correlated with the cold molecular gas emission, $r\_{\rm CO}$, which varies between 200 and 500pc. This could support the presence of a deeply embedded AGN which, opposed to star-formation, would not require large amounts of cold molecular gas to produce the observed high IR luminosities. 5. The presence of compact and extremely embedded nuclei is supported by the detection of the HCN-vib 14 absorption in 43 ± 10% of the sample. The detection rate is higher, 60 ± 15%, in interacting systems than in advanced mergers, 31 ± 13%. 6. The ULIRG nuclei can be classified in two groups in the 220GHz continuum size, $r\_{\rm cont}$, vs. mid-IR excess emission, $\Delta L\_{\rm 6-20\mu m}$, diagram. $\Delta L\_{\rm 6-20\mu m}$ is the 6–20μm emission excess after subtracting the far-IR gray-body contribution in this wavelength range. These two groups are: (a) compact ($r\_{\rm cont}<$120pc) nuclei with high mid-IR excess emission (log$\Delta L\_{\rm 6-20\mu m}$/$L\_{\rm IR}$> − 0.9) which are optically classified AGN; and (b) objects that follow a relation with decreasing $r\_{\rm cont}$ for decreasing mid-IR excess emission. A majority of the nuclei in interacting systems (60%) that follow this relation have $r\_{\rm cont}<$120pc, which are at the lower end of the $r\_{\rm cont}$-mid-IR excess relation, while only 30% of the mergers have these compact continuum emission. Mergers following this relation have larger sizes on average. This suggest that in the early stages of the interaction (nuclear separation > 1kpc) most of the activity occurs in very compact regions while, in more advanced merger stages, the activity is more extended unless and optically detected AGN is present. 7. We find no correlation between the 9.7 silicate absorption and the ∼ 220GHz continuum or CO(2–1) sizes. The relatively faint mid-IR emission of the most compact nuclei could prevent the presence of deep silicate absorptions in their mid-IR spectra and this could hinder the use of this absorption feature to find some obscured nuclei. 8. We find that 67% (19/29) of the nuclei have nuclear radiation pressures above the estimated Eddington limit. This is consistent with the presence of massive molecular outflows in ULIRGs and supports that radiation pressure can have a relevant role in the outflow launching process. We thank the referee for the useful comments and suggestions. We are grateful to A. Hernán-Caballero and H. Spoon for providing measurements from the IDEOS database. MPS and IL acknowledge support from the Comunidad de Madrid through the Atracción de Talento Investigador Grant 2018-T1/TIC-11035 and PID2019-105423GA-I00 (MCIU/AEI/FEDER,UE). AA-H and SG-B acknowledge support through grant PGC2018-094671-B-I00 (MCIU/AEI/FEDER,UE). MP is supported by the Programa Atracción de Talento de la Comunidad de Madrid via grant 2018-T2/TIC-11715. AL acknowledges the support from Comunidad de Madrid through the Atracción de Talento Investigador Grant 2017-T1/TIC-5213. SA, LC, MP, and AL acknowledge support from the Spanish Ministerio de Economía y Competitividad through grants ESP2017-83197-P and PID2019-106280GB-I00. This work was done under project No. MDM-2017-0737 Unidad de Excelencia “María de Maeztu”- Centro de Astrobiología (INTA-CSIC). D. Rigopoulou acknowledges support from STFC through grant ST/S000488/1. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.00113.S, ADS/JAO.ALMA#2015.1.00263.S, ADS/JAO.ALMA#2016.1.00170.S, ADS/JAO.ALMA#2016.1.00777.S, ADS/JAO.ALMA#2018.1.00486.S, and ADS/JAO.ALMA#2018.1.00699.S. 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K., Cao, C., Lu, N., et al. 2015,, 799, 11 Mid-IR Spitzer observations of the ULIRGs =========================================== [h] [tbl:spitzerfluxes] lcccccccc IRAS name & $S\_{\rm 9.7\mu m}$ & IRS 34 & MIPS 70& MIPS 160 00091 − 0738 & − 3.20 ± 0.12 & 1.07 & ⋯ & ⋯ 00188 − 0856 & − 2.72 ± 0.08 & 0.80 & 2.65 & 1.91 00509 + 1225 & 0.30 ± 0.01 & 1.34 & ⋯ & ⋯ 01572 + 0009 & 0.19 ± 0.02 & 1.17 & ⋯ & ⋯ F05189 − 2524 & − 0.32 ± 0.01 & 7.30 & ⋯ & ⋯ 07251 − 0248 & − 2.77 ± 0.06 & 2.62 & ⋯ & ⋯ 09022 − 3615 & − 1.04 ± 0.01 & 3.53 & ⋯ & ⋯ F10190 + 1322 & − 1.09 ± 0.02 & 0.85 & 3.55 & ⋯ 11095 − 0238 & − 3.43 ± 0.07 & 1.54 & 2.95 & 1.28 F12072 − 0444 & − 1.42 ± 0.02 & 1.20 & ⋯ & ⋯ F12112 + 0305 & − 1.35 ± 0.03 & 2.06 & ⋯ & ⋯ 13120 − 5453 & − 1.18 ± 0.01 & 10.1 & ⋯ & ⋯ F13451 + 1232 & − 0.32 ± 0.02 & 1.03 & 2.04 & 1.41 F14348 − 1447 & − 1.89 ± 0.03 & 1.98 & ⋯ & ⋯ F14378 − 3651 & − 1.44 ± 0.05 & 1.85 & ⋯ & ⋯ F15327 + 2340 & − 2.74 ± 0.01 & 34.4 & ⋯ & ⋯ 16090 − 0139 & − 2.65 ± 0.06 & 1.07 & ⋯ & ⋯ 16155 + 0146 & − 2.41 ± 0.07 & 0.75 & ⋯ & ⋯ 17208 − 0014 & − 1.56 ± 0.01 & 8.67 & ⋯ & ⋯ F19297 − 0406 & − 1.49 ± 0.04 & 2.12 & ⋯ & ⋯ 19542 + 1110 & − 0.97 ± 0.02 & 2.19 & ⋯ & ⋯ 20087 − 0308 & − 1.82 ± 0.04 & 0.99 & ⋯ & ⋯ 20100 − 4156 & − 2.79 ± 0.06 & 1.81 & 4.56 & 2.33 20414 − 1651 & − 1.62 ± 0.08 & 1.25 & �	arxiv_0000656
\boldmath $e$1, \boldmath $e$2, \boldmath $e$3. Now the modes with a wave vector ${\bf k}$ may be classified as follows: 1. Transverse modes (T) are divergence-free vector fields consisting of a superposition of sinusoidal modulations of \boldmath $e$2 and \boldmath $e$3. 2. Longitudinal modes (L) are irrotational vector fields consisting of a superposition of sinusoidal modulations of \boldmath $e$2 and \boldmath $e$3. 3. Momentum modes (P) are vector fields consisting of sinusoidal modulations of \boldmath $e$1. Due to the random orientations of the particle velocities, a P mode is not easily recognized as fundamental mode in a simulation. However, it may be numerically transformed into an easily recognizable periodic shape, as will be shown in Fig. [P10] below. The subspaces spanned by these modes are denoted by $T ({\bf n})$, $L ({\bf n})$, and $P ({\bf n})$, respectively, where the wave vector for a periodic box with sides Lx, Ly is ${\bf k} = (2 \pi n\_x / L\_x, 2 \pi n\_y/ L\_y)$, and ${\bf n} \equiv (n\_x, n\_y)$. nx and ny are integers. For the same ${\bf k}$, the values of the Lyapunov exponents for the longitudinal and momentum modes coincide. These modes actually belong to a combined subspace $LP({\bf n}) \equiv L({\bf n}) \oplus P({\bf n})$. The dimensions of the subspaces $T ({\bf n})$ and $LP({\bf n})$ determine the multiplicity of the exponents associated with the T and LP modes. For a periodic boundary, say in x direction, the multiplicity is 2 for the T, and 4 for the LP modes. For details we refer to Ref. . The transverse modes are stationary in space and time, but the L and P modes are not. In the following, we only consider the case of periodic boundaries. Any tangent vector from an LP subspace observed in a simulation is a combination of a pure L and a pure P mode. The dynamics of such an LP pair may be represented as a rotation in a two-dimensional space with a constant angular frequency proportional to the wave number of the mode, $$\omega\_{\bf n} = v k\_{\bf n}, \nonumber$$ where the L mode is continuously transformed into the P mode and vice versa. Since the P mode is modulated with the velocity field of the particles, the experimentally observed mode pattern becomes periodically blurred when its character is predominantly P. Since during each half of a period, during which all spacial sine and cosine functions change sign, the mode is offset in the direction of the wave vector ${\bf k}$, which gives it the appearance of a traveling wave with an averaged phase velocity v. If reflecting boundaries are used, standing waves are obtained. The phase velocity v is about one third of the sound velocity, but otherwise seems to be unrelated to it. The basis vectors spanning the subspaces of the T and LP modes may be reconstructed from the experimental data with a least square method. As an example, we consider the L(1,0) mode of a system consisting of 198 hard disks at a density 0.7 in a rectangular periodic box with an aspect ratio 2/11. The LP(1,0) subspace includes the tangent vectors for 388 ≤ ℓ ≤ 391 with four identical Lyapunov exponents. The mutually orthogonal basis vectors spanning the corresponding four-dimensional subspace are viewed as vector fields in position space and are given by $$L(1,0): \left( \begin{array}{l} 1\\ 0 \end{array} \right) \cos\left(\frac{2 \pi q\_x}{ L\_x }\right), \;\; \left( \begin{array}{l} 1\\ 0 \end{array} \right) \sin\left(\frac{2 \pi q\_x}{ L\_x }\right), \nonumber$$ $$P(1,0): \left( \begin{array}{l} p\_x\\ p\_y \end{array} \right) \cos\left(\frac{2 \pi q\_x}{ L\_x }\right), \;\; \left( \begin{array}{l} p\_x\\ p\_y \end{array} \right) \sin\left(\frac{2 \pi q\_x}{ L\_x }\right), \nonumber$$ where, for simplicity, we have omitted the normalization. If the Gram-Schmidt vectors are used for the reconstruction, the components corresponding to the cosine are shown for L(1,0) in Fig. [L10], for P(1,0) in Fig. [P10]. In the latter case, the mode structure is visible due to the division by px and py as indicated. This also explains the larger scattering of the points. The components proportional to the sine are analogous, but are not shown. A related analysis may be carried out for the covariant vectors. Recently, Morriss and coworkers considered in detail the tangent-space dynamics of the zero modes and of the mode-forming perturbations over a time τ, during which many collisions occur. In addition, they enforced the mutual orthogonality of the subspaces of the modes and of their conjugate pairs (for which the Lyapunov exponents only differ by the sign), with the central manifold by invoking an (inverse) Gram-Schmidt procedure, which starts with the zero modes and works outward towards modes with larger exponents. With this procedure they were able to construct the modes in the limit of large τ and to obtain (approximate) values for the Lyapunov exponents for the first few Lyapunov modes (counted away from the null space). The agreement with simulation results is of the order of a few percent for the T modes, and slightly worse for the LP modes. [L10] [P10] For the modes with a wave length large compared to the inter-particle spacing, a continuum limit for the perturbation vectors leads to a set of partial differential equations for the perturbation fields, whose solutions give analytical expressions for the modes in accordance with the boundary conditions applied. This procedure works for the stationary transverse modes, as well as for the time-dependent LP modes. For the latter, the partial differential equations for the continuous perturbation fields assume the form of a wave equation with traveling wave solutions. For quasi-one-dimensional systems (with narrow boxes such that particles may not pass each other), the phase velocity is found to depend on the particle density via the mean particle separation and the mean time between collisions (of a particle). For fully two-dimensional models of hard disks, the wave velocity becomes proportional to the collision frequency of a particle, and inversely proportional to the mean squared distance of a particle to all its neighbors in the first coordination shell. This is an interesting result, since it provides a long-looked-for connection between a mode property – the wave velocity – with other density-dependent microscopic quantities of a fluid. Rototranslation =============== Rough hard disks ---------------- Up to now we were concerned with simple fluids allowing only translational particle dynamics. As a next step toward more realistic models, we consider fluids, for which the particles may also rotate and exchange energy between translational and rotational degrees of freedom. In three dimensions, the first model of that kind, rough hard spheres, was already suggested by Bryan in 1894. Arguably, it constitutes the simplest model of a molecular fluid. It was later treated by Pidduck and, in particular, by Chapman and Cowling, who derived the collision map in phase space for two colliding spheres with maximum possible roughness. The latter requires that the relative surface velocity at the point of contact of the collision partners is reversed, leaving their combined (translational plus rotational) energy, and their combined linear and angular momenta invariant. The thermodynamics and the dynamical properties of that models were extensively studied by O’Dell and Berne. They also considered generalizations to partial roughness in-between smooth and maximally-rough spheres. Here we consider the simplest two-dimensional version of that model, maximally-rough hard disks in a planar box with periodic boundaries. Explicit expressions for the collision maps in phase space and in tangent space may be obtained from our previous work. The coupling between translation and rotation is controlled by a single dimensionless parameter $$\kappa = \frac{4I}{ m \sigma^2},$$ where I is the principal moment of inertia for a rotation around an axis through the center, and m and σ denote the mass and diameter of a disk, respectvely. κ may take values between zero and one: 0, if all the mass is concentrated in the center, 1/2 for a uniform mass distribution, and 1, if all the mass is located on the perimeter of the disk. The rotation requires to add angular momentum Ji to the other independent variables ${\bf q}\_i, {\bf p}\_i$ of a particle i. The Ji show up in the collision map, and their perturbations δJi affect the tangent space dynamics. If one is only interested in the global Lyapunov exponents, this is all what is needed, since it constitutes an autonomous system. We refer to it as the J-version of the rough-disk model. Its phase-space dimension D = 5N. Since D may be an odd number, it is obvious that one cannot ask questions about the symplectic nature of the model. For this it is necessary to include also the disk orientations Θi conjugate to the angular momenta in the list of independent variables. The angles do not show up in the collision map of the reference trajectory, but their perturbations, δΘi, affect the collision map in tangent space. Now the phase space has 6N dimensions. We refer to this representation as the JΘ-version. Before entering the discussion of the Lyapunov spectra for large systems, let us clarify the number of vanishing exponents first. [zeroexponents] As an illuminating case, we plot in Fig. [zeroexponents] some spectra for the JΘ-version of a rough disk system with only 4 particles and 24 exponents. For positive κ, when translation and rotation are intimately coupled, the number of vanishing exponents is determined by the intrinsic symmetries of space and time-translation invariance and by the number of particles. For periodic boundaries in x and y directions (full red dots in Fig. [zeroexponents]), space-translation invariance contributes four, time-translation invariance another two vanishing exponents. Furthermore, the invariance of the collision map with respect to an arbitrary rotation of a particle adds another N zero exponents, four in our example. Altogether, this gives 10 vanishing exponents, as is demonstrated in Fig. [zeroexponents] by the red points. For reflecting boundaries in x and y directions (small blue dots in Fig. [zeroexponents]) there is no space-translation invariance, and the total number of vanishing exponents is limited to six. If κ vanishes, the rotational energy also vanishes, and all variables connected with rotation cease to be chaotic and contribute another 2N zeroes to the spectrum. Together with the intrinsic-symmetry contributions, this amounts to 2N + 6 = 14 vanishing exponents for periodic boundaries (open purple circles in Fig. [zeroexponents]), respective 2N + 2 = 10 for reflecting boundaries (not shown). Note that we have plotted the full spectra including the positive and negative branches in this figure. For the J-version, and for an odd number of particles, the discussion is slightly more involved for which we refer to Ref. . Next we turn to the discussion of large systems, for which deviations from equipartition are negligible. We use reduced units for which m, σ and the kinetic translational (and rotational) temperatures are unity. Lyapunov exponents are given in units of $\sqrt{K/Nm\sigma^2}.$ The system we first consider in the following consists of N = 88 rough hard disks with a density ρ = 0.7 in a periodic box with an aspect ratio A = 2/11. For comparison, spectra for 400-particle systems are given in Ref. . Since we are here primarily interested in the global exponents, the simulations are carried out with the J-version of the model. [spectrumrough] [centerrough] In Fig. [spectrumrough] we show the positive branches of the Lyapunov spectra for a few selected values of κ. Although the exponents are only defined for integer ℓ, smooth lines are drawn for clarity, and a reduced index is used on the abscissa. There are 5N = 440 exponents, of which the first half constitute the positive branch shown in the figure. An enlargement of the small exponents near the center of the spectrum is provided in Fig. [centerrough] for selected values of κ. There we have also included the negative branches in such a way to emphasize conjugate pairing. For κ = 0, which corresponds to freely rotating smooth disks without roughness, the Lyapunov spectrum is identical to that of pure smooth disks without any rotation, if N = 88 vanishing exponents are added to the spectrum. The simulation box is long enough (due to the small aspect ratio) for Lyapunov modes to develop along the x direction. This may be verified by the step structure of the spectrum due to the small degenerate exponents. If κ is increased, the Lyapunov modes quickly disappear, and hardly any trace of them is left for κ = 0.1. For small positive κ the spectrum is decomposed into a rotation-dominated regime for 2N < ℓ < 3N and a translation-dominated regime for ℓ ≤ 2N and ℓ ≥ 3N. It is, perhaps, surprising that, with the exception of the maximum exponent λ1, already a small admixture of rotational motion significantly reduces the translation-dominated exponents, whereas the rotation-dominated exponents only gradually increase. To study this further, we plot in the Figs. [maxkappa] and [KSkappa] the isothermal κ-dependence of the maximum exponent, λ1, and of the Kolmogorov-Sinai entropy per particle, hKS/N, respectively, for various fluid densities as indicated by the labels. These data are for a system consisting of 400 rough disks. λ1 and, hence, dynamical chaos tends to decrease with κ for low densities, and always increases for large densities. At the same time, the KS-entropy always decreases with κ. This means that mixing becomes less effective and the time for phase-space mixing goes up, the more the rotational degrees of freedom interfere with the translational dynamics. [maxkappa] Before leaving the rough hard-disk case, a slightly disturbing observation for that system should be mentioned. In Fig. [symplectic] we have demonstrated that the local GS Lyapunov exponents of the smooth hard-disk model display symplectic symmetry. Unexpectedly, for the rough hard-disks this is not the case. This is shown in Fig. [roughsymplectic]. where the maximum and minimum local exponents do not show the expected symmetry, and neither do the other conjugate pairs which are not included in the figure. If the collision map for two colliding particles in tangent space is written in matrix form, $\delta {\bf \Gamma}' = {\cal M} \delta {\bf \Gamma},$ where $\delta {\bf \Gamma}$ and $\delta {\bf \Gamma}' $ are the perturbation vectors immediately before and after a collision, respectively, the matrix ${\cal M}$ does not obey the symplectic condition ${\cal S}^{\dagger} {\cal M} {\cal S } = {\cal M}.$ Here, ${\cal S}$ is the anti-symmetric symplectic matrix, and † means transposition. For a more extensive discussion we refer to Refs. . [KSkappa] [roughsymplectic] Hard dumbbells -------------- A slightly more realistic model for linear molecules are “smooth” hard dumbbells, whose geometry is shown in Fig. [dumbbellgeometry]. As with hard disks, the dynamics is characterized by hard encounters with impulsive forces perpendicular to the surface. Between collisions, the particles translate and rotate freely, which makes the simulation rather efficient. In the following we restrict to the planar version of this model, for which the ”molecule” consists of two fused disks as shown in Fig. [dumbbellgeometry]. The state space is spanned by the center-of-mass (CM) coordinates ${\bf q}\_i$, the translational momenta ${\bf p}\_i$, the orientation angles Θi, and the angular momenta Ji, i = 1, 2, …, N, and has 6N dimensions. The Lyapunov spectra for this model were first computed by Milanović et al.. Related work for rigid diatomic molecules interacting with soft repulsive site-site potentials was carried out by Borzsák et al. and Kum et al.. ![ Geometry of a hard dumbbell diatomic. All quantities for this model are given in terms of reduced quantities, for which the disk diameter \sigma, the molecular mass m, and the total kinetic energy per molecule, K/N, are equal to unity. The molecular anisotropy is defined by d/\sigma.](figure_16.jpg "fig:") [dumbbellgeometry] Here we restrict the discussion to homogeneous dumbbells, for which the molecular mass m is uniformly distributed over the union of the two disks. The moment of inertia for rotation around the center of mass becomes $$I(d \le \sigma) = \frac{m \sigma^2}{4} \frac{ 3 d w + [d^2 + (\sigma^2/2)] [\pi + 2 \arctan(d/w)]}{ 2 d w + \sigma^2 [\pi + 2 \arctan(d/w)]}, \nonumber$$ and $$I(d > \sigma) = \frac{m}{4}\left[\frac{\sigma^2}{2}+d^2\right], \nonumber$$ where $w = \sqrt{\sigma^2 - d^2}$ is the molecular waist. I(d) monotonously increases with d. For d → 0 it converges to that of a single disk with mass m and diameter σ, and with a homogeneous mass density. Results for other mass distributions may be found in Ref. . d plays a similar role as the coupling parameter κ for the rough-disk model. The results are summarized in Fig. [surface], where the Lyapunov spectra, positive branches only, for a system of 64 dumbbells in a periodic square box at an intermediate gas density of 0.5 are shown for various molecular anisotropies d. The system is too small for Lyapunov modes to be clearly visible (although a single step due to a transverse mode may be observed for d > 0.01 even in this case). Most conspicuous, however, is the widening gap in the spectra between the indices l = 2N − 3 = 125 and l = 126 for d < dc ≈ 0.063. It separates the translation-dominated exponents (1 ≤ l ≤ 125) from the rotation-dominated exponents (126 ≤ l ≤ 189). The gap disappears for d > dc. ![ Anisotropy dependence of the Lyapunov spectra for a system of 64 planar hard dumbbells in a square box with periodic boundaries. d is the molecular anisotropy (in reduced units). Only the positive branches of the spectra with 3N = 192 exponents are shown. The number density is 0.5. The Lyapunov index is denoted by l. (From Ref. ). ](figure_17.jpg "fig:") [surface] It was observed in Ref. that for d < dc the perturbation vectors associated with the translation- and rotation-dominated exponents predominantly point into the subspace belonging to center-of-mass translation and molecular rotation, respectively, and very rarely rotate into a direction belonging to the other subspace. For anisotropies d > dc, however, one finds that the offset vectors for all exponents spend comparable times in both subspaces, which is taken as an indication of strong translation - rotation coupling. dc is found to increase with the density. The anisotropy dependence of some selected Lyapunov exponents is shown in Fig. [aniso]. The horizontal lines for l = 1 and l = 2N − 3 = 125 indicate the values of the maximum and smallest positive exponents for a system of 64 hard disks with the same density, to which the respective exponents of the dumbbell system converge with d → 0. The smooth hard-disk data were taken from Ref. . [aniso] It was shown in the Refs. that two “phase transitions” exist for the hard dumbbells with an anisotropy of d = 0.25, a first at a number density n1 = 0.75 from a fluid to a rotationally-disordered solid, and a second, at n2 ≈ 0.775, from the orientationally-disordered solid to a crystal with long-range orientational order. Both transitions were observed by computing orientational correlation functions. The first transition at n1 makes itself felt by large fluctuations of the Lyapunov exponents during the simulation and a slow convergence, but does not show up in the density dependence of the Lyapunov exponents. The second transition at n2 does give rise to steps in the density-dependence curves of both λ1 and of λ189 (which is the smallest positive exponent). These steps are even more noticeable when viewed as a function of the collision frequency instead of the density, and the step is significantly larger for λ189 than for λ1. This indicates that the collective long-wavelength perturbations are much stronger affected by the locking of dumbbell orientations than the large exponents, which – due to localization – measure only localized dynamical events. Here, dynamical systems theory offers a new road to an understanding of collective phenomena and phase transitions, which has not been fully exploited yet. [low] Unlike the rough hard disks, the hard dumbbell systems generate well-developed Lyapunov modes for their small exponents. Instructive examples for N = 400 homogeneous hard dumbbells with an anisotropy d = 0.25 are given in the Refs. . The classification of the modes in terms of transversal (T) and longitudinal-momentum (LP) modes is completely analogous to that for hard disks, although some modifications due to the presence of rotational degrees of freedom have not yet been worked out in all detail. In Fig. [low], we show, as an example, the smallest positive exponents of a system of N = 150 dumbbells with a density n = 0.5 in an elongated (along the x axis) periodic box with an aspect ratio A = 1/32. [trans446] Only modes with wave vectors parallel to x develop, which facilitates the analysis. The spectrum displays the typical step structure due to the mode degeneracies with multiplicity two, for the T modes, and four, for the LP modes. For this system, typical instantaneous mode patterns are shown in the Figs. [trans446] and [trans443], the former for a T mode with l = 446, the latter for an LP mode with l = 443. In both cases, the wavelength is equal to the box length in x direction. Please note that the system appears strongly contracted along the x axis. [trans443] Stationary particle systems out of equilibrium ============================================== One of the most interesting – and historically first – studies of Lyapunov spectra for many-particle systems is for stationary non-equilibrium states. If a system is driven away from equilibrium by an external or thermal force, the irreversibly generated heat needs to be removed and transferred to a heat bath. Otherwise the system would heat up and would never reach a steady state. Such a scheme may assume the following equations of motion, $$\begin{aligned} \dot{{\bf q}}\_i &=& {\bf p\_i}/m\_i + {\cal{Q}}(\{{\bf q}\},\{{\bf p}\}) X(t) \nonumber\\ \dot{{\bf p}}\_i &=& - \partial \Phi/\partial {\bf q}\_i + {\cal{P}}(\{{\bf q}\},\{{\bf p}\}) X(t) - s\_i\zeta {\bf p}\_i, \label{thermo}\\ \nonumber\end{aligned}$$ where ${\cal{Q}} X$ and ${\cal{P}} X$ represent the driving, and $-\zeta {\bf p}\_i$ the thermostat or heat bath acting on a particle i selected with the switch si ∈ {0, 1}. ζ fluctuates and may be positive or negative, whether kinetic energy has to be extracted from – or added to – the system. Averaged over a long trajectory, ⟨ζ⟩ needs to be positive. $\zeta(\{{\bf q}\},\{{\bf p}\})$ may be a Lagrange multiplier, which minimizes the thermostatting force according to Gauss’ principle of least constraint and keeps either the kinetic energy or the total internal energy a constant of the motion. Or it may be a single independent variable in an extended phase space such as for the Nosé-Hoover thermostat. There are excellent monographs describing these schemes. They are rather flexible and allow for any number of heat baths. Although it is not possible to construct such thermostats in the laboratory, they allow very efficient non equilibrium molecular dynamics (NEMD) simulations and are believed to provide an accurate description of the nonlinear transport involved. Simple considerations lead to the following thermodynamic relations for the stationary non-equilibrium state generated in this way after initial transients have decayed: $$\begin{aligned} &&\left\langle \frac{ d \ln \delta V^{(D)} }{dt } \right\rangle = \left\langle \frac{\partial}{\partial {\bf \Gamma}} \cdot \dot{{\bf \Gamma}} \right\rangle \nonumber = \sum\_{\ell =1}^{D} \lambda\_{\ell} = - d \sum\_{i=1}^N s\_i \left\langle \zeta \right\rangle \nonumber \\ &=& -\frac{1}{k\_B T} \left\langle \frac{dQ}{dt} \right\rangle = - \frac{1}{k\_B} \left\langle \frac{dS\_{irr}}{dt} \right\rangle < 0. \label{ne} \\ \nonumber\end{aligned}$$ Here, d is the dimension of the physical space, and d∑isi becomes the number of thermostatted degrees of freedom. D is the phase space dimension, and dQ/dt is the rate with which heat is transferred to the bath and gives rise to a positive rate of entropy production, ⟨dSirr/dt⟩. T is the kinetic temperature enforced by the thermostat. These relations show that an infinitesimal phase volume shrinks with an averaged rate, which is determined by the rate of heat transfer to the bath, which, in turn, is regulated by the thermostat variable ζ. As a consequence, there exists a multifractal attractor in phase space, to which all trajectories converge for t → ∞, and on which the natural measure resides. For Axiom A flows or even non-uniform hyperbolic systems, these states have been identified with the celebrated Sinai-Ruelle-Bowen (SRB) states. This result is quite general for dynamically thermostatted systems and has been numerically confirmed for many transport processes in many-body systems. The computation of the Lyapunov spectrum is essential for the understanding of this mechanism and, still, is the only practical way to compute the information dimension of the underlying attractor. Such a system is special in the sense that its stationary measure is singular and resides on a fractal attractor with a vanishing phase space volume. The ensuing Gibbs entropy is diverging to minus infinity, which explains the paradoxical situation that heat is continuously transferred to the bath giving rise to a positive rate of irreversible entropy production $\dot{S}\_{irr}$. The fact that there is a preferred direction for the heat flow is in accordance with the Second Law of thermodynamics, which these systems clearly obey. The divergence of the Gibbs entropy also indicates that the thermodynamic entropy, namely the derivative of the internal energy with respect to the entropy, is not defined for such stationary non-equilibrium states. ![(Color online) One-dimensional Frenkel-Kontorova conductivity model described in the main text. Three Poincaré maps for the planes defining the first octant are shown. The system is in a stationary non-equilibrium state. Reprinted from Ref. ](figure_22.jpg "fig:") [frenkel] We demonstrate such a fractal attractor for a one-dimensional Frenkel-Kontorova conductivity model of a charged particle in a sinusoidal potential, which is subjected to a constant applied field X and a Nosé-Hoover thermostat. The equations of motion are $$\dot{q} = p; \;\;\; \dot{p} = -\sin(q) + X - \zeta p; \;\;\; \dot{\zeta} = p^2 -1. \nonumber$$ The phase space is three-dimensional. In Fig. [frenkel] three Poincaré maps for mutually orthogonal planes defining the first octant of the phase space are shown. The singularity spectrum for this model has been computed in Ref. , and the multifractal nature of the stationary phase-space measure has been established. The sum of all Lyapunov exponents becomes negative for stationary non-equilibrium states. To demonstrate this effect, we show in Fig. [sllod] [sllod] the Lyapunov spectra of a system of 36 particles subjected to planar shear flow for various shear rates $\dot{\varepsilon}$ as indicated by the labels. Homogeneous SLLOD mechanics in combination with a Gaussian energy control has been used, which keeps the internal energy of the fluid a constant of the motion. The figure emphasizes the conjugate pairing symmetry in the non equilibrium stationary state: Since all particles are homogeneously affected by the same thermostat, the pair sum of conjugate exponents becomes negative and is the same for all pairs. With the Kaplan-Yorke formula, the information dimension is found to be smaller than the dimension of the phase space, a clear indication of a multi-fractal phase-space distribution. Final remarks ============= Here we have described some of our studies of simple fluids, the results so far obtained, and the new questions they raise. Most of our simulations are for planar systems due to the numerical effort required for the computation of Lyapunov spectra, but the results are expected to carry over to three-dimensional systems. What have these studies taught us about fluids? For equilibrium systems, the results about the density dependence of the maximum Lyapunov exponents and of the mixing time in Sec. [hardwca] clearly provide deeper and quantitative insight into the foundation and limitations of more familiar models for dense fluids. Still, the behavior of the exponents near phase transitions has not been satisfactorily exploited yet and should be the topic for further research. It is interesting to note that time-dependent local Lyapunov exponents have been used to identify atypical trajectories with very high or very low chaoticity. Such trajectories, although rare and difficult to locate in phase space, may be of great physical significance. For example, in phase traditions, or in chemical reactions, they may lead over unstable saddle points connecting a quasi-stable state, or chemical species, with another. Tailleur and Kurchan invented a very promising algorithm, Lyapunov weighted dynamics, for that purpose, in which a swarm of phase points evolves according to the natural dynamics. Periodically, trajectories are killed, or cloned, with a rate determined by their local exponents Λ1. As a consequence, trajectories are weighted according to their chaoticity. A slightly modified version of this algorithm, Lyapunov weighted path sampling, has been suggested by Geiger and Dellago, and has been successfully applied to the isomerizing dynamics of a double-well dimer in a solvent. Since the turn of the century, Lyapunov modes have raised a lot of interest. But it has been surprisingly difficult to connect them to other hydrodynamic properties. Most recently, however, some progress has been made. But this problem will still occupy researchers for some time. Another interesting and promising field is the study of systems with qualitatively different degrees of freedom, such as translation and rotation. Our studies of the Lyapunov instability of rough disks and of linear dumbbell molecules has opened another road to investigate translation-rotation coupling. An extension of these studies to vibrating molecules and to more complex systems with internal rotation, such as rotational isomer dynamics in butane, are promising topics for future research. Arguably, the most important impact Lyapunov spectra have achieved is in non-equilibrium statistical mechanics. In combination with dynamical thermostats, it is possible to generate stationary states far from equilibrium and to link the rate of entropy production with the time-averaged friction coefficients and the logarithmic rate of phase-volume contraction. See Eq.([ne]). The existence of a multi-fractal phase-space distribution with an information dimension smaller than the phase space dimension provides a geometrical interpretation of the Second Law of thermodynamics. The importance of this result is reflected in the voluminous literature on this subject. There are many other applications of local Lyapunov exponents and tangent vectors in meteorology, in the geological science, and in other fields. For more details we refer to Ref. , which is a recent collection of review articles and applications. Acknowledgments =============== We thank Prof. Christoph Dellago and Prof. Willliam G. Hoover for many illuminating discussions. We also thank Drs. Ljubo Milanović, Jacobus van Meel, Robin Hirschl and Christina Forster, whose enthusiasm and dedication provided the basis for the present discussion. 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To a large extent, the royal road to gain an understanding of fluids or solids has been statistical mechanics. Based on interaction potentials obtained from experiments and quantum mechanical simulations, sophisticated perturbation theories are capable of providing a quantitative description of the static structural properties of such fluids on the atomistic scale. Computer simulations have provided guidance and invaluable insight to unravel the intricate local structure and even the non-equilibrium dynamics of “simple” liquids including hydrodynamic flows and shock waves. At present, this program is being extended to ever more complex molecular fluids and/or to systems confined to particular geometries such as interfaces and pores, and to fluids out of thermal equilibrium. We have raised a related question, “What is liquid?” in 1998 in a similar context. A fluid differs from a solid by the mobility of its particles, and this ability to flow is a collective phenomenon. The flow spreads rapidly in phase space, which constitutes the fundamental instability characteristic of a gas or liquid. About thirty years ago, dynamical systems theory provided new tools for the characterization of the chaotic evolution in a multi-dimensional phase space, which were readily applied to liquids shortly after. The main idea is to follow not only the evolution of a state in phase space but, simultaneously, the evolution of various tiny perturbations applied to that state at an initial time and to measure the growth or decay of such perturbations. The study of the Lyapunov stability, or instability, of a trajectory with respect to small perturbations always present in real physical systems is hoped to provide new and alternative insight into the theoretical foundation and limitation of dynamical models for dense gases and liquids, of phase transitions involving rare events, and of the working of the Second Law of thermodynamics for stationary non-equilibrium systems. It is the purpose of this review to assess how far this hope has become true and how much our understanding of fluids has gained from the study of the Lyapunov instability in phase space. The structure of a simple fluid is essentially determined by the steep (e.g. ∝ r− 12) repulsive part of the pair potential, which can be well approximated by a discontinuous hard core. In perturbation theories, this hard potential may be taken as a reference potential with the long-range attractive potential ( ∝ − r− 6) acting as a perturbation. Hard disk fluids in two dimensions, and hard sphere systems in three, are easy to simulate and are paradigms for simple fluids. The first molecular dynamics simulations for hard sphere systems were carried out by Alder and Wainwright in 1957, the first computation of Lyapunov spectra for such systems by Dellago, Posch and Hoover in 1996. There exist numerous extensions of the hard-sphere model to include rotational degrees of freedom and various molecular geometries. Arguably the simplest, which still preserves spherical symmetry, are spheres with a rough surface, so-called ”rough hard spheres”. In another approach, fused dumbbell diatomics are used to model linear molecules. Both models are used to study the energy exchange between translational and rotational degrees of freedom and, hence, rotation-translation coupling for molecules with moderate anisotropy. Other approaches more suitable for larger molecular anisotropies involve spherocylinders, ellipsoids and even inflexible hard lines. To our knowledge, only for the first two schemes, namely for rough hard disks and for planar hard dumbbells, extensive studies of the Lyapunov spectra have been carried out. We shall come back to this work below. There are numerous papers for repulsive soft-potential systems and for Lennard-Jones systems from various authors, in which an analysis of the Lyapunov instability has been carried out. We shall make reference to some of this work in the following sections. The paper is organized as follows. In the next section we introduce global and local (time dependent) Lyapunov exponents and review some of their properties. Of particular interest are the symmetry properties of the local exponents and of the corresponding perturbation vectors in tangent space. Both the familiar orthonormal Gram-Schmidt vectors and the covariant Lyapunov vectors are considered. In Sec. [hardwca] we study planar hard disk systems over a wide range of densities, and compare them to analogous fluids interacting with a smooth repulsive potential. There we also demonstrate that the largest (absolute) Lyapunov exponents are generated by perturbations, which are strongly localized in physical space: only a small cluster of particles contributes at any instant of time. This localization persists in the thermodynamic limit. Sec. [hydro] is devoted to another property of the perturbations associated with the small exponents, the so-called Lyapunov modes. In a certain sense, the Lyapunov modes are analogous to the Goldstone modes of fluctuating hydrodynamics (such as the familiar sound and heat modes). However, it is surprisingly difficult to establish a connection between these two different viewpoints. In Sec. [roto] particle systems with translational and rotational dynamics are considered. Two simple planar model fluids are compared, namely gases of rough hard disks and of hard-dumbbell molecules. Close similarities, but also surprising differences, are found. Most surprising is the fast disappearance of the Lyapunov modes and the breakdown of the symplectic symmetry of the local Gram-Schmidt exponents for the rough hard disk systems, if the moment of inertia is increased from zero. In Sec. [nonequilibrium] we summarize some of the results for stationary systems far from thermal equilibrium, for which the Lyapunov spectra have been valuable guides for our understanding. For dynamically thermostatted systems in stationary non-equilibrium states, they provide a direct link with the Second Law of thermodynamics due to the presence of a multifractal phase-space distribution. We conclude with a few short remarks in Sec. [outlook]. Lyapunov exponents and perturbation vectors =========================================== Let ${\bf \Gamma}(t)= \{{\bf p},{\bf q}\}$ denote the instantaneous state of a dynamical particle system with a phase space M of dimension D. Here, ${\bf p}$ and ${\bf q}$ stand for the momenta and positions of all the particles. The motion equations are usually written as a system of first-order differential equations, $$\dot{\bf \Gamma} = {\bf F}({\bf \Gamma}), \label{motion}$$ where ${\bf F}$ is a (generally nonlinear) vector-valued function of dimension D. The formal solution of this equation is written as ${\bf \Gamma}(t) = \phi^t ({\bf \Gamma}(0))$, where the map $\phi^t: {\bf \Gamma} \to {\bf \Gamma}$ defines the flow in M, which maps a phase point ${\bf \Gamma}(0)$ at time zero to a point ${\bf \Gamma}(t)$ at time t. Next, we consider an arbitrary infinitesimal perturbation vector $\delta {\bf \Gamma}(t)$, which depends on the position ${\bf \Gamma}(t)$ in phase space and, hence, implicitly on time. It evolves according to the linearized equations of motion, $$\dot{\delta {\bf \Gamma}} = {\cal J}({\bf \Gamma}) \delta {\bf \Gamma}. \label{linearized}$$ This equation may be formally solved according to $$\delta{\bf \Gamma}(t) = D\phi^t \vert\_{{\bf \Gamma}(0)}\; \delta{\bf \Gamma}(0), \label{evolution}$$ where Dϕt defines the flow in tangent space and is represented by a real but generally non-symmetric D × D matrix. The dynamical (or Jacobian) matrix, $${\cal J}({\bf \Gamma}) \equiv \frac{ \partial {\bf F}}{ \partial {\bf \Gamma}}, \label{jacobian}$$ determines, whether the perturbation vector $ \delta {\bf \Gamma}(t) $ has the tendency to grow or shrink at a particular point ${\bf \Gamma}(t)$ in phase space the system happens to be at time t. Accordingly, the matrix element $$\delta {\bf \Gamma}^{\dagger}({\bf \Gamma}) {\cal J({\bf \Gamma})} \delta {\bf \Gamma}({\bf \Gamma}) \label{lyageneral}$$ turns out to be positive or negative, respectively. Here, † means transposition. If, in addition, the perturbation is normalized, $\|\|\delta {\bf \Gamma}\|\| = 1$, and points into particular directions in tangent space to be specified below, this matrix element turns out to be a local rate for the growth or decay of $\|\| \delta {\bf \Gamma}(t) \|\|$ and will be referred to as a local Lyapunov exponent $\Lambda({\bf \Gamma})$ at the phase point ${\bf \Gamma}$. Covariant Lyapunov vectors -------------------------- In 1968 Oseledec published his celebrated multiplicative ergodic theorem, in which he proved that under very general assumptions (which apply to molecular fluids) the tangent space decomposes into subspaces E(j) with dimension mj, $$TM({\bf \Gamma}) = E^{(1)}({\bf \Gamma}) \oplus E^{(2)}({\bf \Gamma}) \oplus \cdots \oplus E^{(L)}({\bf \Gamma}), \label{split}$$ for almost all ${\bf \Gamma} \in M$ (with respect to the Lesbegue measure), such that ∑j = 1Lmj = D. These subspaces evolve according to $$D \phi^t \vert\_{{\bf \Gamma}(0)} \; E^{(j)}\left({\bf \Gamma}(0)\right) = E^{(j)}\left( {\bf \Gamma}(t)\right), \label{defcov}$$ and are said to be covariant, which means that they co-move – and in particular co-rotate – with the flow in tangent space. In general they are not orthogonal to each other. If $ {{\mbox{\boldmath $v$}}}\left({\bf \Gamma}(0)\right) \in E^{(j)}\left({\bf \Gamma}(0)\right)$ is a vector in the subspace $E^{(j)}\left({\bf \Gamma}(0)\right)$, it evolves according to $$D \phi^t \vert\_{{\bf \Gamma}(0)} \; {\mbox{\boldmath $v$}}\left({\bf \Gamma}(0)\right) = {\mbox{\boldmath $v$}}\left( {\bf \Gamma}(t)\right) \in E^{(j)}\left({\bf \Gamma}(t)\right). \label{defcovv}$$ The numbers $$^{(\pm)}\lambda^{(j)} = \lim\_{t \rightarrow \pm \infty} \dfrac{1}{\vert t \vert} \, \ln \frac{ \big\\|{{\mbox{\boldmath $v$}}} \left({\bf \Gamma}(t)\right) \big\\|} { \big\\|{{\mbox{\boldmath $v$}}} \left({\bf \Gamma}(0)\right) \big\\|} \label{covlambda}$$ for j ∈ {1, 2, …, L} exist and are referred to as the (global) Lyapunov exponents. Here, the upper index ( + ) or ( − ) indicates, whether the trajectory is being followed forward or backward in time, respectively. If a subspace dimension mj is larger than one, then the respective exponent is called degenerate with multiplicity mj. If all exponents are repeated according to their multiplicity, there are D exponents altogether, which are commonly ordered according to size, $$\begin{aligned} ^{(+)}\lambda\_1& \ge & \cdots \ge ^{(+)}\lambda\_D, \label{e1}\\ ^{(-)}\lambda\_1& \ge & \cdots \ge ^{(-)}\lambda\_D, \label{e2} \\ \nonumber\end{aligned}$$ where the subscripts are referred to as Lyapunov index. The vetors \boldmath $v$ℓ generating λℓ according to $$^{(\pm)}\lambda\_{\ell} = \lim\_{t \rightarrow \pm \infty} \dfrac{1}{\vert t \vert} \, \ln \frac{ \big\\|{{\mbox{\boldmath $v$}}}^{\ell} \left({\bf \Gamma}(t)\right) \big\\|} { \big\\|{{\mbox{\boldmath $v$}}}^{\ell} \left({\bf \Gamma}(0)\right) \big\\|}; \;\;\; \ell = 1,2,\dots,D, \label{covlambda1}$$ are called covariant Lyapunov vectors. This notation, which treats the tangent space dynamics in terms of a set of vectors, is more convenient for algorithmic purposes, although the basic theoretical objects are the covariant subspaces E(j); j = 1, 2, …, L. Degenerate Lyapuov exponents appear, if there exist intrinsic continuous symmetries (such as invariance of the Lagrangian with respect to time and/or space translation, giving rise to energy and/or momentum conservation, respectively). For particle systems such symmetries almost always exist. Some consequences will be discussed below. The global exponents for systems evolving forward or backward in time are related according to ( + )λℓ = − ( − )λD + 1 − ℓ; ℓ ∈ {1, 2, …, D}. The subspaces E(j) and, hence, the covariant Lyapunov vectors \boldmath $v$ℓ generally are not pairwise orthogonal. If in Eq. ([covlambda]) the time evolution is not followed from zero to infinity but only over a finite time interval τ > 0, so-called covariant finite-time dependent Lyapunov exponents are obtained, $$^{(\pm)}\bar{\Lambda}\_{\ell}^{\tau,\mbox{cov}} = \dfrac{1}{ \tau} \, \ln \, \big\\| \, D\phi^{ \pm \tau}\vert\_{\textrm{\smallskip{{{\mbox{\boldmath $\Gamma$}}}}(0)}} \, \; \, {{\mbox{\boldmath $v$}}}^{\ell} \left({\bf \Gamma}(0)\right) \, \big\\|. \label{covftlambda}$$ In the limit τ → 0 so-called local Lyapunov exponents are generated, $$\begin{aligned} ^{(\pm)}\Lambda\_{\ell}^{\mbox{cov}}\left({\bf \Gamma}\right) & = & \lim\_{t \rightarrow \pm 0} \dfrac{1}{\vert t \vert} \, \ln \, \big\\| \, \ D\phi^t\vert\_{\textrm{\smallskip{{{\mbox{\boldmath $\Gamma$}}}}}} \, \; \, {{\mbox{\boldmath $v$}}}^{\ell} \left({\bf \Gamma}\right) \, \big\\| \nonumber \\ & = & \left[ ^{(\pm)}{{\mbox{\boldmath $v$}}}^{\ell}({\bf \Gamma}) \right]^{\dagger} {\cal J({\bf \Gamma})} \;\; ^{(\pm)}{{\mbox{\boldmath $v$}}}^{\ell}({\bf \Gamma}) \label{covllambda}\end{aligned}$$ where $\vert\vert{{\mbox{\boldmath $v$}}}^{\ell}({\bf \Gamma})\|\| = 1$ is required. The second equality has a structure as in Eq. ([lyageneral]), applied to the covariant vectors, and indicates that the local exponents are point functions, which only depend on ${\bf \Gamma}$. The finite-time-dependent exponents may be viewed as time averages of local exponents over a stretch of trajectory during the finite time τ, the global exponents as time averages over an infinitely-long trajectory. For the latter to become dynamical invariants, one requires ergodicity, which we will always assume in the following for lack of other evidence. Orthogonal Lyapunov vectors --------------------------- Another definition of the Lyapunov exponents, also pioneered by Oseledec, is via the real and symmetrical matrices $$\lim\_{t \to \pm \infty} \left[ D \phi^{t}\vert\_{{\bf \Gamma}(0)}^{\dagger} D \phi^{t}\vert\_{{\bf \Gamma}(0)} \right]^{\frac{1}{2\|t\|}}, \label{Oseledec1}$$ which exist with probability one (both forward and backward in time). Their (real) eigenvalues involve the global Lyapunov exponents, exp(( ± )λ1) ≥ ⋯ ≥ exp(( ± )λD), where, as before, degenerate eigenvalues are repeated according to their multiplicities. For non-degenerate and degenerate eigenvalues the > and = signs apply, respectively. The corresponding eigenspaces, U±(j); j = 1, …, L, are pairwise orthogonal and provide two additional decompositions of the tangent space at almost every point in phase space, one forward ( + ) and one backward ( − ) in time: $$TM({\bf \Gamma}) = U\_{\pm}^{(1)}({\bf \Gamma}) \oplus U\_{\pm}^{(2)}({\bf \Gamma}) \oplus \cdots \oplus U\_{\pm}^{(L)}({\bf \Gamma}). \label{splitalt}$$ In each case, the dimensions mj of the Oseledec subspaces U±(j); j = 1, …, L, have a sum equal to the phase-space dimension D: ∑j = 1Lmj = D. These subspaces are not covariant. The classical algorithm for the computation of (global) Lyapunov exponents carefully keeps track of all d-dimensional infinitesimal volume elements δV(d) which (almost always) evolve according to δV(d)(t) ≈ δV(d)(0)exp(∑ℓ = 1dλℓt). This is algorithmically achieved with the help of a set of perturbation vectors $^{(\pm)}{{\mbox{\boldmath $g$}}}\_{\ell}({\bf \Gamma}); \;\; \ell = 1,\dots,D$, which are either periodically re-orthonormalized with a Gram-Schmidt (GS) procedure (equivalent to a QR-decomposition), or are continuously constrained to stay orthonormal with a triangular matrix of Lagrange multipliers. For historical reasons we refer to them as GS-vectors. They have been shown to be the spanning vectors for the Oseledec subspaces U±(j). Accordingly, they are orthonormal but not covariant. Analogous to Eqs. ([covftlambda]) and ([covllambda]), one may define finite-time-dependent GS Lyapunov exponents $ ^{(\pm)}\bar{\Lambda}\_{\ell}^{\tau,\mbox{GS}}$ and local GS exponents $ ^{(\pm)}\Lambda\_{\ell}^{\mbox{GS}}\left({\bf \Gamma}\right)$, where the latter are again point functions, $$^{(\pm)}\Lambda\_{\ell}^{\mbox{GS}}\left({\bf \Gamma}\right) = [ ^{(\pm)}{{\mbox{\boldmath $g$}}}^{\ell}({\bf \Gamma}) ]^{\dagger}\; {\cal J({\bf \Gamma})} \;\;^{(\pm)}{{\mbox{\boldmath $g$}}}^{\ell}({\bf \Gamma}).$$ As before, the finite-time and global GS exponents are time averages of the respective local exponents over a finite respective infinite trajectory. From all GS vectors only $^{(\pm)}{{\mbox{\boldmath $g$}}}\_{1}({\bf \Gamma})$, which is associated with the maximum global exponent, evolves freely without constraints which might affect its orientation in tangent space. Therefore it agrees with $^{(\pm)}{{\mbox{\boldmath $v$}}}\_{1}({\bf \Gamma})$ and is also covariant. Also the corresponding local exponents agree for ℓ = 1, but generally not for other ℓ. However, the local covariant exponents may be computed from the local GS exponents, and vice versa, if the angles between them are known. As an illustration of the relation between covariant Lyapunov vectors and orthonormal GS vectors, we consider a simple two-dimensional example, the Hénon map, $$\begin{aligned} x\_{n+1} &=& a - x\_n^2 + b \, y\_n \nonumber \enspace, \\ y\_{n+1} &=& x\_n \enspace, \nonumber\end{aligned}$$ with a = 1.4 and b = 0.3. [henonmap] The light-blue line in Fig. [henonmap] represents the Hénon attractor, which is known to coincide with its unstable manifold. The black line is a finite-time approximation of its stable manifold. Let O denote a point on the attractor. The GS vectors at this point are indicated by \boldmath $g$1(O) and \boldmath $g$2(O), where the former is also covariant and identical to \boldmath $v$1(O) (parallel to the unstable manifold). As required, the covariant vector \boldmath $v$2(O) is tangent to the stable manifold (which was determined by a different method). For the computation of the covariant vectors at O, it is necessary to follow and store the reference trajectory and the GS vectors sufficiently far into the future up to a point F in our example. In Fig. [henonmap] the GS vectors at this point are denoted by \boldmath $g$1(F) and \boldmath $g$2(F). Applying an algorithm by Ginelli et al., to which we shall return below, a backward iteration to the original phase point O yields the covariant vectors, \boldmath $v$2(O) in particular. Symmetry properties of global and local exponents ------------------------------------------------- For ergodic systems the global exponents are averages of the local exponents over the natural measure in phase space and, according to the multiplicative ergodic theorem for chaotic systems, they do not depend on the metric and the norm one applies to the tangent vectors. Also the choice of the coordinate system (Cartesian or polar, for example) does not matter. For practical reasons the Euclidian (or L2) norm will be used throughout in the following. It also does not matter, whether covariant or Gram-Schmidt vectors are used for the computation. The global exponents are truly dynamical invariants. This is not the case for the local exponents. They depend on the norm and on the coordinate system. And they generally depend on the set of basis vectors, covariant or GS, as was mentioned before. Furthermore, some symmetry properties of the equations of motion are reflected quite differently by the two local representations. * Local Gram-Schmidt exponents: During the construction of the GS vectors, the changes of the differential volume elements δV(d) following Eq. ([volume]) are used to compute the local exponents. If the total phase volume is conserved as is the case for time-independent Hamiltonian systems, the following sum rule holds for almost all ${\bf \Gamma}$: $$\sum\_{\ell = 1}^D \Lambda\_{\ell}^{\mbox{GS}}({\bf \Gamma}) = 0. \label{sum\_rule}$$ In this symplectic case we can even say more. For each positive local GS exponent there exists a negative local GS exponent such that their pair sum vanishes : $$^{(\pm)}\Lambda\_{\ell}^{\mbox{GS}}({\bf \Gamma}) = -^{(\pm)}\Lambda\_{D + 1 -\ell }^{\mbox{GS}}({\bf \Gamma}),\;\; \ell = 1,\dots,D. \label{gsfwd}$$ As indicated, such a symplectic local pairing symmetry is found both forward and backward in time. Non-symplectic systems do not display that symmetry. On the other hand, the re-orthonormalization process tampers with the orientation and rotation of the GS vectors and destroys all consequences of the time-reversal invariance property of the original motion equations. * Local covariant exponents: During their construction, only the norm of the covariant perturbation vectors needs to be periodically adjusted for practical reasons, but the angles between them remain unchanged. This process effectively destroys all information concerning the d-dimensional volume elements. Thus, no symmetries analogous to Eq. ([gsfwd]) exist. Instead, the re-normalized covariant vectors faithfully preserve the time-reversal symmetry of the equations of motion, which is reflected by $$^{(\mp)}\Lambda\_{\ell}^{\mbox{cov}}({\bf \Gamma}) = -^{(\pm)}\Lambda\_{D+1 -\ell}^{\mbox{cov}}({\bf \Gamma}), \; \; \ell = 1,\dots,D, \label{local\_symmetry}$$ regardless, whether the system is symplectic or not. This means that an expanding co-moving direction is converted into a collapsing co-moving direction by an application of the time-reversal operation. The set of all global Lyapunov exponents, ordered according to size, is referred to as the Lyapunov spectrum. For stationary Hamiltonian systems in thermal equilibrium the (global) conjugate pairing rule holds, $$^{(\pm)}\lambda\_{\ell} = -^{(\pm)}\lambda\_{D + 1 -\ell }, \label{cpr} \nonumber$$ which is a consequence of Eq. ([gsfwd]) and of the fact that the global exponents are the phase-space averages of these quantities. In such a case only the first half of the spectrum containing the positive exponents needs to be computed. In the following we shall refer to this half as the positive branch of the spectrum. The subspaces spanned by the covariant (or GS) vectors associated with the strictly positive/negative global exponents are known as the unstable/stable manifolds. Both are covariant and are linked by the time-reversal transformation, which converts one into the other. Numerical considerations ------------------------ The computation of the Gram-Schmidt exponents is commonly carried out with the algorithm of Benettin et al. and Shimada et al.. Based on this classical approach, a reasonably efficient algorithm for the computation of covariant exponents has been recently developed by Ginelli et al.. Some computational details for this method may also be found in Refs. . An alternative approach is due to Wolfe and Samelson, which was subsequently applied to Hamiltonian systems with many degrees of freedom. The considerations of the previous section are for time-continuous systems based on the differential equations ([motion]). They may be readily extended to maps such as systems of hard spheres, for which a pre-collision state of two colliding particles is instantaneously mapped to an after-collision state. Between collisions the particles move smoothly. With the linearized collision map the time evolution of the perturbation vectors in tangent space may be constructed. In numerical experiments of stationary systems, the initial orientation of the perturbation vectors is arbitrary. There exists a transient time, during which the perturbation vectors converge to their proper orientation on the attractor. All symmetry relations mentioned above refer to this well-converged state and exclude transient conditions. For the computation of the full set of exponents, the reference trajectory and D perturbation vectors (each of dimension D) have to be followed in time, which requires D(D + 1) equations to be integrated. Present computer technology limits the number of particles to about 104 for time-continuous systems, and to 105 for hard-body systems.. In all applications below appropriate reduced units will be used. To ease the notation, we shall in the following omit to indicate the forward-direction in time by ( + ), if there is no ambiguity. Two-dimensional hard-disk and WCA fluids in equilibrium ======================================================= [comparison] We are now in the position to apply this formalism to various models for simple fluids. First we consider two moderately dense planar gases, namely a system of elastic smooth hard disks with a pair potential $$\phi\_{HD} = \left\{\begin{array}{ll} \infty, &\qquad r \leq \sigma, \\ 0,&\qquad r>\sigma, \end{array}\right. \nonumber$$ and a two-dimensional Weeks-Chandler-Anderson (WCA) gas interacting with a repulsive soft potential $$\phi\_{WCA}=\left\{\begin{array}{ll}4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}- \left(\frac{\sigma}{r}\right)^6\right]+\epsilon, &\qquad r \leq 2^{1/6}\sigma, \\ 0,&\qquad r>2^{1/6}\sigma. \end{array}\right. \nonumber$$ In Fig. [comparison] the positive branches of their (global) Lyapunov spectra are compared. Both gases consist of N = 400 particles in a square box with side length L and periodic boundaries, and have the same density, ρ = N/L2 = 0.4, and temperature, T = 1. Although, as expected, for low densities the Lyapunov spectra are rather insensitive to the interaction potential, the comparison of Fig. [comparison] for moderately dense gases already reveals a rather strong sensitivity. In particular, the maximum exponent λ1 is much lower for the WCA fluid, which means that deterministic chaos is significantly reduced in systems with smooth interaction potentials. This difference becomes even more pronounced for larger densities, as will be shown below. ![Localization of the perturbation vector {{\mbox{\boldmath $g$}}}^1 in physical space (the red square) for a planar gas of 102,400 soft repulsive disks. The quantity \mu^{1}, measuring the contribution of individual particles to the perturbation vector associated with the maximum GS exponent, is plotted in the vertical direction at the position of the particles. ](figure_3.jpg "fig:") [localsurface] The maximum (minimum) Lyapunov exponent denotes the rate of fastest perturbation growth (decay) and is expected to be dominated by the fastest dynamical events such as a locally enhanced collision frequency. To prove this expectation one may ask how individual particles contribute to the growth of the perturbations determined by \boldmath $v$ℓ or \boldmath $g$ℓ. Writing the normalized perturbation vectors in terms of the position and momentum perturbations of all particles, ${{\mbox{\boldmath $v$}}}^{\ell} = \{\delta {\bf q}\_i^{(\ell)}, \delta {\bf p}\_i^{(\ell)}; i=1,\dots, N \}$, the quantity $$\mu\_i^{(\ell)} \equiv \left( \delta {\bf q}\_i^{(\ell)}\right)^2 + \left( \delta {\bf p}\_i^{(\ell)}\right)^2 \label{gamma}$$ is positive, bounded, and obeys the sum rule ∑i = 1Nμi(ℓ) = 1 for any ℓ. It may be interpreted as a measure for the activity of particle i contributing to the perturbation in question. Equivalent relations apply for \boldmath $g$ℓ. If μi(1) is plotted - vertical to the simulation plane - at the position of particle i, surfaces such as in Fig. [localsurface] are obtained. They are strongly peaked in a small domain of the physical space indicating strong localization of \boldmath $v$1 ≡ \boldmath $g$1. It means that only a small fraction of all particles contributes to the perturbation growth at any instant of time. This property is very robust and persists in the thermodynamic limit in the sense that the fraction of particles, which contribute significantly to the formation of \boldmath $v$1 varies as a negative power of N. For example, Fig. [localsurface] is obtained for a system of 102,400 (!) smooth disks interacting with a pair potential similar to the WCA potential. This dynamical localization of the perturbation vectors associated with the large Lyapunov exponents is a very general phenomenon and has been also observed for one-dimensional models of space-time chaos and Hamiltonian lattices. [W] For ℓ > 1, the localization for \boldmath $v$ℓ differs from that of \boldmath $g$ℓ. This may be seen by using a localization measure due to Taniguchi and Morriss, $$W = \frac{1}{N} \exp\langle S \rangle; \;\;\;\; S({\bf \Gamma}(t)) = -\sum\_{i=1}^N \mu\_i^{(\ell)} \ln \mu\_i^{(\ell)}, \nonumber$$ which is bounded according to 1/N ≤ W ≤ 1. The lower and upper bounds correspond to complete localization and delocalization, respectively. S is the Shannon entropy for the ‘measure’ defined in Eq. ([gamma]), and ⟨…⟩ denotes a time average. The localization spectra for the two sets of perturbation vectors are shown in Fig. [W] for a rather dense system of hard disks in two dimensions. One may infer from the figure that localization is much stronger for the covariant vectors (red line) whose orientations in tangent space are only determined by the tangent flow and are not constrained by periodic re-orthogonalization steps as is the case for the Gram-Schmidt vectors (blue line). One further observes that the localization of the covariant vectors becomes less and less pronounced the more the regime of coherent Lyapunov modes, located in the center of the spectrum, is approached. Next we turn our attention to the maximum Lyapunov exponent λ1 and to the Kolmogorov-Sinai entropy. Both quantities are indicators of dynamical chaos. The KS-entropy is the rate with which information about an initial state is generated, if a finite-precision measurement at some later time is retraced backward along the stable manifold. According to Pesin’s theorem it is equal to the sum of the positive Lyapunov exponents, hKS = ∑λℓ > 0λℓ. It also determines the characteristic time for phase space mixing according to τmix = 1/hKS. In Fig. [dens] we compare the isothermal density dependence of the maximum Lyapunov exponent λ1 (top panel), of the smallest positive exponent λ2N − 3 (middle panel), and of the Kolmogorov-Sinai (KS) entropy per particle hKS/N (bottom panel) for planar WCA and HD fluids. Both systems contain N = 375 particles in a periodic box with aspect ratio 0.6 and with a temperature T = 1. As expected, these quantities for the two fluids agree at small densities, but differ significantly for large ρ. [dens] For hard disks, van Zon and van Beijeren and de Wijn used kinetic theory to obtain expressions for λ1 and hKS/N to leading orders of ρ. They agree very well with computer simulations of Dellago et al.. The regime of larger densities, however, is only qualitatively understood. λ1HD and hKSHD/N increase monotonically due to the increase of the collision frequency ν. The (first-order) fluid-solid transition shows up as a step between the freezing point of the fluid ( ρfHD = 0.88 ) and the melting point of the solid (ρmHD = 0.91 ) (not shown in the figure). These steps disappear, if instead of the density the collision frequency is plotted on the abscissa. λ1HD and hKSHD/N diverge at the close-packed density due to the divergence of ν. For the WCA fluid, both the maximum exponent and the Kolmogorov-Sinai entropy have a maximum as a function of density, and become very small when the density of the freezing point is approached, which happens for ρfWCA = 0.89. This behavior is not too surprising in view of the fact that a harmonic solid is not even chaotic and λ1 vanishes. The maximum of λ1WCA occurs at a density of about 0.9ρfWCA and confirms earlier results for Lennard-Jones fluids. At this density chaos is most pronounced possibly due to the onset of cooperative dynamics characteristic of phase transitions. At about the same density, mixing becomes most effective as is demonstrated by the maximum of the KS-entropy. The latter is related to the mixing time in phase space according to τmix = 1/hKS. It is interesting to compare the Lyapunov time τλ = 1/λ1, which is a measure for the system to “forget” its past, with the (independently-measured) time between collisions of a particle, τc = 1/ν. In Fig. [tau] such a comparison is shown for a three-dimensional system of 108 hard spheres in a cubic box with periodic boundary conditions. Also included is the behavior of τKS ≡ N/hKS. For small densities, we have τλ ≪ τc, and subsequent collisions are uncorrelated. This provides the basis for the validity of lowest-order kinetic theory (disregarding correlated collisions). For densities ρ > 0.4 the Lyapunov time is progressively larger than the collision time and higher-order corrections such as ring collisions become important. Also the lines for τλ and τKS cross even before, at a density 0.1. This is a consequence of the shape change of the Lyapunov spectrum with density : the small positive exponents grow faster with ρ than the large exponents. It also follows that neither the Lyapunov time nor the mixing time determine the correlation decay of - say - the particle velocities. For lower densities, for which ring collisions are not important, the decay of the velocity autocorrelation function z(t) is strictly dominated by the collision time. In Ref. we also demonstrate that for hard-particle systems the time τλ does not provide an upper bound for the time for which correlation functions may be reliably computed. [tau] For later reference, we show in Fig. [symplectic] the time-dependence of the local exponents for ℓ = 1 and ℓ = D = 4N = 16 of a system consisting of four smooth hard disks. The symplectic symmetry as given by Eq. ([gsfwd]) is well obeyed. [symplectic] Lyapunov modes ============== Already the first simulations of the Lyapunov spectra for hard disks revealed an interesting new step-like structure of the small positive and negative exponents in the center of the spectrum, which is due to degenerate exponents. A similar structure was also found for hard dumbbell systems, to which we come back in Sec. [dumbb]. The explanation for this behavior lies in the fact that the perturbation vectors for these exponents are characterized by coherent sinusoidal patterns spread out over the whole physical space (the simulation cell). We have referred to these collective patterns as Lyapunov modes. The modes are interpreted as a consequence of a spontaneous breaking of continuous symmetries and, hence, are intimately connected with the zero modes spanning the central manifold. They appear as sinusoidal modulations of the zero modes in space – with wave number k ≠ 0 – due to the spontaneous breaking of the continuous symmetries. For k → 0, the modes reduce to a linear superposition of the zero modes, and their Lyapunov exponent vanishes. The experimentally observed wave vectors depend on the size of the system and the nature of the boundaries (reflecting or periodic). Our discovery of the Lyapunov modes triggered a number of theoretical approaches. Eckmann and Gat were the first to provide theoretical arguments for the existence of the Lyapunov modes in transversal-invariant systems. Their model did not have a dynamics in phase space but only an evolution matrix in tangent space, which was modeled as a product of independent random matrices. In another approach, McNamara and Mareschal isolated the six hydrodynamic fields related to the invariants of the binary particle collisions and the vanishing exponents, and used a generalized Enskog theory to derive hydrodynamic evolution equations for these fields. Their solutions are the Lyapunov modes. In a more detailed extension of this work restricted to small densities, a generalized Boltzmann equation is used for the derivation. de Wijn and van Beijeren pointed out the analogy to the Goldstone mechanism of constructing (infinitesimal) excitations of the zero modes and the Goldstone modes of fluctuating hydrodynamics. They used this analogy to derive the modes within the framework of kinetic theory. With a few exceptions, a rather good agreement with the simulation results was achieved, at least for low densities. Finally, Taniguchi, Dettmann, and Morriss approached the problem from the point of view of periodic orbit theory and master equations. The modes were observed for hard-ball systems in one, two, and three dimensions, for planar hard dumbbells, and also for one- and two-dimensional soft particles. A formal classification for smooth hard-disk systems has been given by Eckmann et al.. If the 4N components of a tangent vector are arranged according to $$\delta {\bf \Gamma}\,=\,\left( \delta q\_x,\delta q\_y ; \delta p\_x,\delta p\_y \right), \nonumber$$ the six orthonormal zero modes, which span the central manifold and are the generators of the continuous symmetry transformations, are given by $$\begin{aligned} {\mbox{\boldmath $e$}}\_1 &=& \dfrac{1}{\sqrt{2 K}\,} \, ({p\_x},{p\_y};\,0,0),\nonumber \\ {\mbox{\boldmath $e$}}\_2 &=& \dfrac{1}{\sqrt{N}\,} \, (1,0 ;\ 0,0),\nonumber \\ {\mbox{\boldmath $e$}}\_3 &=& \dfrac{1}{\sqrt{N}\,} \, (0,1 ;\,0,0), \nonumber \\ {\mbox{\boldmath $e$}}\_4 &=& \dfrac{1}{\sqrt{2 K}\,} \, (0,0 ;\,{p\_x},{p\_y} ),\nonumber \\ {\mbox{\boldmath $e$}}\_5 &=& \dfrac{1}{\sqrt{N}\,} \, (0,0 ;\,1,0),\nonumber \\ {\mbox{\boldmath $e$}}\_6 &=& \dfrac{1}{\sqrt{N}\,} \, (0,0 ;\,0,1).\nonumber \\ \nonumber \end{aligned}$$ \boldmath $e$1 corresponds to a shift of the time origin, \boldmath $e$4 to a change of energy, \boldmath $e$2 and \boldmath $e$3 to a uniform translation in the x and y directions, respectively, and \boldmath $e$5 and \boldmath $e$6 to a shift of the total momentum in the x and y directions, respectively. The six vanishing Lyapunov exponents associated with these modes have indices 2N − 2 ≤ i ≤ 2N + 3 in the center of the spectrum. Since, for small-enough k, the perturbation components of a particle obey $\delta {\bf p} = C \delta {\bf q} $ for the unstable perturbations (λ > 0), and $\delta {\bf q} = - C \delta {\bf p} $ for the stable perturbations (λ < 0), where C > 0 is a number, one may restrict the classification to the $\delta {\bf q}$ part of the modes and, hence, to the basis vectors \boldmath $e$1, \boldmath $e$2, \boldmath $e$3. Now the modes with a wave vector ${\bf k}$ may be classified as follows: 1. Transverse modes (T) are divergence-free vector fields consisting of a superposition of sinusoidal modulations of \boldmath $e$2 and \boldmath $e$3. 2. Longitudinal modes (L) are irrotational vector fields consisting of a superposition of sinusoidal modulations of \boldmath $e$2 and \boldmath $e$3. 3. Momentum modes (P) are vector fields consisting of sinusoidal modulations of \boldmath $e$1. Due to the random orientations of the particle velocities, a P mode is not easily recognized as fundamental mode in a simulation. However, it may be numerically transformed into an easily recognizable periodic shape, as will be shown in Fig. [P10] below. The subspaces spanned by these modes are denoted by $T ({\bf n})$, $L ({\bf n})$, and $P ({\bf n})$, respectively, where the wave vector for a periodic box with sides Lx, Ly is ${\bf k} = (2 \pi n\_x / L\_x, 2 \pi n\_y/ L\_y)$, and ${\bf n} \equiv (n\_x, n\_y)$. nx and ny are integers. For the same ${\bf k}$, the values of the Lyapunov exponents for the longitudinal and momentum modes coincide. These modes actually belong to a combined subspace $LP({\bf n}) \equiv L({\bf n}) \oplus P({\bf n})$. The dimensions of the subspaces $T ({\bf n})$ and $LP({\bf n})$ determine the multiplicity of the exponents associated with the T and LP modes. For a periodic boundary, say in x direction, the multiplicity is 2 for the T, and 4 for the LP modes. For details we refer to Ref. . The transverse modes are stationary in space and time, but the L and P modes are not. In the following, we only consider the case of periodic boundaries. Any tangent vector from an LP subspace observed in a simulation is a combination of a pure L and a pure P mode. The dynamics of such an LP pair may be represented as a rotation in a two-dimensional space with a constant angular frequency proportional to the wave number of the mode, $$\omega\_{\bf n} = v k\_{\bf n}, \nonumber$$ where the L mode is continuously transformed into the P mode and vice versa. Since the P mode is modulated with the velocity field of the particles, the experimentally observed mode pattern becomes periodically blurred when its character is predominantly P. Since during each half of a period, during which all spacial sine and cosine functions change sign, the mode is offset in the direction of the wave vector ${\bf k}$, which gives it the appearance of a traveling wave with an averaged phase velocity v. If reflecting boundaries are used, standing waves are obtained. The phase velocity v is about one third of the sound velocity, but otherwise seems to be unrelated to it. The basis vectors spanning the subspaces of the T and LP modes may be reconstructed from the experimental data with a least square method. As an example, we consider the L(1,0) mode of a system consisting of 198 hard disks at a density 0.7 in a rectangular periodic box with an aspect ratio 2/11. The LP(1,0) subspace includes the tangent vectors for 388 ≤ ℓ ≤ 391 with four identical Lyapunov exponents. The mutually orthogonal basis vectors spanning the corresponding four-dimensional subspace are viewed as vector fields in position space and are given by $$L(1,0): \left( \begin{array}{l} 1\\ 0 \end{array} \right) \cos\left(\frac{2 \pi q\_x}{ L\_x }\right), \;\; \left( \begin{array}{l} 1\\ 0 \end{array} \right) \sin\left(\frac{2 \pi q\_x}{ L\_x }\right), \nonumber$$ $$P(1,0): \left( \begin{array}{l} p\_x\\ p\_y \end{array} \right) \cos\left(\frac{2 \pi q\_x}{ L\_x }\right), \;\; \left( \begin{array}{l} p\_x\\ p\_y \end{array} \right) \sin\left(\frac{2 \pi q\_x}{ L\_x }\right), \nonumber$$ where, for simplicity, we have omitted the normalization. If the Gram-Schmidt vectors are used for the reconstruction, the components corresponding to the cosine are shown for L(1,0) in Fig. [L10], for P(1,0) in Fig. [P10]. In the latter case, the mode structure is visible due to the division by px and py as indicated. This also explains the larger scattering of the points. The components proportional to the sine are analogous, but are not shown. A related analysis may be carried out for the covariant vectors. Recently, Morriss and coworkers considered in detail the tangent-space dynamics of the zero modes and of the mode-forming perturbations over a time τ, during which many collisions occur. In addition, they enforced the mutual orthogonality of the subspaces of the modes and of their conjugate pairs (for which the Lyapunov exponents only differ by the sign), with the central manifold by invoking an (inverse) Gram-Schmidt procedure, which starts with the zero modes and works outward towards modes with larger exponents. With this procedure they were able to construct the modes in the limit of large τ and to obtain (approximate) values for the Lyapunov exponents for the first few Lyapunov modes (counted away from the null space). The agreement with simulation results is of the order of a few percent for the T modes, and slightly worse for the LP modes. [L10] [P10] For the modes with a wave length large compared to the inter-particle spacing, a continuum limit for the perturbation vectors leads to a set of partial differential equations for the perturbation fields, whose solutions give analytical expressions for the modes in accordance with the boundary conditions applied. This procedure works for the stationary transverse modes, as well as for the time-dependent LP modes. For the latter, the partial differential equations for the continuous perturbation fields assume the form of a wave equation with traveling wave solutions. For quasi-one-dimensional systems (with narrow boxes such that particles may not pass each other), the phase velocity is found to depend on the particle density via the mean particle separation and the mean time between collisions (of a particle). For fully two-dimensional models of hard disks, the wave velocity becomes proportional to the collision frequency of a particle, and inversely proportional to the mean squared distance of a particle to all its neighbors in the first coordination shell. This is an interesting result, since it provides a long-looked-for connection between a mode property – the wave velocity – with other density-dependent microscopic quantities of a fluid. Rototranslation =============== Rough hard disks ---------------- Up to now we were concerned with simple fluids allowing only translational particle dynamics. As a next step toward more realistic models, we consider fluids, for which the particles may also rotate and exchange energy between translational and rotational degrees of freedom. In three dimensions, the first model of that kind, rough hard spheres, was already suggested by Bryan in 1894. Arguably, it constitutes the simplest model of a molecular fluid. It was later treated by Pidduck and, in particular, by Chapman and Cowling, who derived the collision map in phase space for two colliding spheres with maximum possible roughness. The latter requires that the relative surface velocity at the point of contact of the collision partners is reversed, leaving their combined (translational plus rotational) energy, and their combined linear and angular momenta invariant. The thermodynamics and the dynamical properties of that models were extensively studied by O’Dell and Berne. They also considered generalizations to partial roughness in-between smooth and maximally-rough spheres. Here we consider the simplest two-dimensional version of that model, maximally-rough hard disks in a planar box with periodic boundaries. Explicit expressions for the collision maps in phase space and in tangent space may be obtained from our previous work. The coupling between translation and rotation is controlled by a single dimensionless parameter $$\kappa = \frac{4I}{ m \sigma^2},$$ where I is the principal moment of inertia for a rotation around an axis through the center, and m and σ denote the mass and diameter of a disk, respectvely. κ may take values between zero and one: 0, if all the mass is concentrated in the center, 1/2 for a uniform mass distribution, and 1, if all the mass is located on the perimeter of the disk. The rotation requires to add angular momentum Ji to the other independent variables ${\bf q}\_i, {\bf p}\_i$ of a particle i. The Ji show up in the collision map, and their perturbations δJi affect the tangent space dynamics. If one is only interested in the global Lyapunov exponents, this is all what is needed, since it constitutes an autonomous system. We refer to it as the J-version of the rough-disk model. Its phase-space dimension D = 5N. Since D may be an odd number, it is obvious that one cannot ask questions about the symplectic nature of the model. For this it is necessary to include also the disk orientations Θi conjugate to the angular momenta in the list of independent variables. The angles do not show up in the collision map of the reference trajectory, but their perturbations, δΘi, affect the collision map in tangent space. Now the phase space has 6N dimensions. We refer to this representation as the JΘ-version. Before entering the discussion of the Lyapunov spectra for large systems, let us clarify the number of vanishing exponents first. [zeroexponents] As an illuminating case, we plot in Fig. [zeroexponents] some spectra for the JΘ-version of a rough disk system with only 4 particles and 24 exponents. For positive κ, when translation and rotation are intimately coupled, the number of vanishing exponents is determined by the intrinsic symmetries of space and time-translation invariance and by the number of particles. For periodic boundaries in x and y directions (full red dots in Fig. [zeroexponents]), space-translation invariance contributes four, time-translation invariance another two vanishing exponents. Furthermore, the invariance of the collision map with respect to an arbitrary rotation of a particle adds another N zero exponents, four in our example. Altogether, this gives 10 vanishing exponents, as is demonstrated in Fig. [zeroexponents] by the red points. For reflecting boundaries in x and y directions (small blue dots in Fig. [zeroexponents]) there is no space-translation invariance, and the total number of vanishing exponents is limited to six. If κ vanishes, the rotational energy also vanishes, and all variables connected with rotation cease to be chaotic and contribute another 2N zeroes to the spectrum. Together with the intrinsic-symmetry contributions, this amounts to 2N + 6 = 14 vanishing exponents for periodic boundaries (open purple circles in Fig. [zeroexponents]), respective 2N + 2 = 10 for reflecting boundaries (not shown). Note that we have plotted the full spectra including the positive and negative branches in this figure. For the J-version, and for an odd number of particles, the discussion is slightly more involved for which we refer to Ref. . Next we turn to the discussion of large systems, for which deviations from equipartition are negligible. We use reduced units for which m, σ and the kinetic translational (and rotational) temperatures are unity. Lyapunov exponents are given in units of $\sqrt{K/Nm\sigma^2}.$ The system we first consider in the following consists of N = 88 rough hard disks with a density ρ = 0.7 in a periodic box with an aspect ratio A = 2/11. For comparison, spectra for 400-particle systems are given in Ref. . Since we are here primarily interested in the global exponents, the simulations are carried out with the J-version of the model. [spectrumrough] [centerrough] In Fig. [spectrumrough] we show the positive branches of the Lyapunov spectra for a few selected values of κ. Although the exponents are only defined for integer ℓ, smooth lines are drawn for clarity, and a reduced index is used on the abscissa. There are 5N = 440 exponents, of which the first half constitute the positive branch shown in the figure. An enlargement of the small exponents near the center of the spectrum is provided in Fig. [centerrough] for selected values of κ. There we have also included the negative branches in such a way to emphasize conjugate pairing. For κ = 0, which corresponds to freely rotating smooth disks without roughness, the Lyapunov spectrum is identical to that of pure smooth disks without any rotation, if N = 88 vanishing exponents are added to the spectrum. The simulation box is long enough (due to the small aspect ratio) for Lyapunov modes to develop along the x direction. This may be verified by the step structure of the spectrum due to the small degenerate exponents. If κ is increased, the Lyapunov modes quickly disappear, and hardly any trace of them is left for κ = 0.1. For small positive κ the spectrum is decomposed into a rotation-dominated regime for 2N < ℓ < 3N and a translation-dominated regime for ℓ ≤ 2N and ℓ ≥ 3N. It is, perhaps, surprising that, with the exception of the maximum exponent λ1, already a small admixture of rotational motion significantly reduces the translation-dominated exponents, whereas the rotation-dominated exponents only gradually increase. To study this further, we plot in the Figs. [maxkappa] and [KSkappa] the isothermal κ-dependence of the maximum exponent, λ1, and of the Kolmogorov-Sinai entropy per particle, hKS/N, respectively, for various fluid densities as indicated by the labels. These data are for a system consisting of 400 rough disks. λ1 and, hence, dynamical chaos tends to decrease with κ for low densities, and always increases for large densities. At the same time, the KS-entropy always decreases with κ. This means that mixing becomes less effective and the time for phase-space mixing goes up, the more the rotational degrees of freedom interfere with the translational dynamics. [maxkappa] Before leaving the rough hard-disk case, a slightly disturbing observation for that system should be mentioned. In Fig. [symplectic] we have demonstrated that the local GS Lyapunov exponents of the smooth hard-disk model display symplectic symmetry. Unexpectedly, for the rough hard-disks this is not the case. This is shown in Fig. [roughsymplectic]. where the maximum and minimum local exponents do not show the expected symmetry, and neither do the other conjugate pairs which are not included in the figure. If the collision map for two colliding particles in tangent space is written in matrix form, $\delta {\bf \Gamma}' = {\cal M} \delta {\bf \Gamma},$ where $\delta {\bf \Gamma}$ and $\delta {\bf \Gamma}' $ are the perturbation vectors immediately before and after a collision, respectively, the matrix ${\cal M}$ does not obey the symplectic condition ${\cal S}^{\dagger} {\cal M} {\cal S } = {\cal M}.$ Here, ${\cal S}$ is the anti-symmetric symplectic matrix, and † means transposition. For a more extensive discussion we refer to Refs. . [KSkappa] [roughsymplectic] Hard dumbbells -------------- A slightly more realistic model for linear molecules are “smooth” hard dumbbells, whose geometry is shown in Fig. [dumbbellgeometry]. As with hard disks, the dynamics is characterized by hard encounters with impulsive forces perpendicular to the surface. Between collisions, the particles translate and rotate freely, which makes the simulation rather efficient. In the following we restrict to the planar version of this model, for which the ”molecule” consists of two fused disks as shown in Fig. [dumbbellgeometry]. The state space is spanned by the center-of-mass (CM) coordinates ${\bf q}\_i$, the translational momenta ${\bf p}\_i$, the orientation angles Θi, and the angular momenta Ji, i = 1, 2, …, N, and has 6N dimensions. The Lyapunov spectra for this model were first computed by Milanović et al.. Related work for rigid diatomic molecules interacting with soft repulsive site-site potentials was carried out by Borzsák et al. and Kum et al.. ![ Geometry of a hard dumbbell diatomic. All quantities for this model are given in terms of reduced quantities, for which the disk diameter \sigma, the molecular mass m, and the total kinetic energy per molecule, K/N, are equal to unity. The molecular anisotropy is defined by d/\sigma.](figure_16.jpg "fig:") [dumbbellgeometry] Here we restrict the discussion to homogeneous dumbbells, for which the molecular mass m is uniformly distributed over the union of the two disks. The moment of inertia for rotation around the center of mass becomes $$I(d \le \sigma) = \frac{m \sigma^2}{4} \frac{ 3 d w + [d^2 + (\sigma^2/2)] [\pi + 2 \arctan(d/w)]}{ 2 d w + \sigma^2 [\pi + 2 \arctan(d/w)]}, \nonumber$$ and $$I(d > \sigma) = \frac{m}{4}\left[\frac{\sigma^2}{2}+d^2\right], \nonumber$$ where $w = \sqrt{\sigma^2 - d^2}$ is the molecular waist. I(d) monotonously increases with d. For d → 0 it converges to that of a single disk with mass m and diameter σ, and with a homogeneous mass density. Results for other mass distributions may be found in Ref. . d plays a similar role as the coupling parameter κ for the rough-disk model. The results are summarized in Fig. [surface], where the Lyapunov spectra, positive branches only, for a system of 64 dumbbells in a periodic square box at an intermediate gas density of 0.5 are shown for various molecular anisotropies d. The system is too small for Lyapunov modes to be clearly visible (although a single step due to a transverse mode may be observed for d > 0.01 even in this case). Most conspicuous, however, is the widening gap in the spectra between the indices l = 2N − 3 = 125 and l = 126 for d < dc ≈ 0.063. It separates the translation-dominated exponents (1 ≤ l ≤ 125) from the rotation-dominated exponents (126 ≤ l ≤ 189). The gap disappears for d > dc. ![ Anisotropy dependence of the Lyapunov spectra for a system of 64 planar hard dumbbells in a square box with periodic boundaries. d is the molecular anisotropy (in reduced units). Only the positive branches of the spectra with 3N = 192 exponents are shown. The number density is 0.5. The Lyapunov index is denoted by l. (From Ref. ). ](figure_17.jpg "fig:") [surface] It was observed in Ref. that for d < dc the perturbation vectors associated with the translation- and rotation-dominated exponents predominantly point into the subspace belonging to center-of-mass translation and molecular rotation, respectively, and very rarely rotate into a direction belonging to the other subspace. For anisotropies d > dc, however, one finds that the offset vectors for all exponents spend comparable times in both subspaces, which is taken as an indication of strong translation - rotation coupling. dc is found to increase with the density. The anisotropy dependence of some selected Lyapunov exponents is shown in Fig. [aniso]. The horizontal lines for l = 1 and l = 2N − 3 = 125 indicate the values of the maximum and smallest positive exponents for a system of 64 hard disks with the same density, to which the respective exponents of the dumbbell system converge with d → 0. The smooth hard-disk data were taken from Ref. . [aniso] It was shown in the Refs. that two “phase transitions” exist for the hard dumbbells with an anisotropy of d = 0.25, a first at a number density n1 = 0.75 from a fluid to a rotationally-disordered solid, and a second, at n2 ≈ 0.775, from the orientationally-disordered solid to a crystal with long-range orientational order. Both transitions were observed by computing orientational correlation functions. The first transition at n1 makes itself felt by large fluctuations of the Lyapunov exponents during the simulation and a slow convergence, but does not show up in the density dependence of the Lyapunov exponents. The second transition at n2 does give rise to steps in the density-dependence curves of both λ1 and of λ189 (which is the smallest positive exponent). These steps are even more noticeable when viewed as a function of the collision frequency instead of the density, and the step is significantly larger for λ189 than for λ1. This indicates that the collective long-wavelength perturbations are much stronger affected by the locking of dumbbell orientations than the large exponents, which – due to localization – measure only localized dynamical events. Here, dynamical systems theory offers a new road to an understanding of collective phenomena and phase transitions, which has not been fully exploited yet. [low] Unlike the rough hard disks, the hard dumbbell systems generate well-developed Lyapunov modes for their small exponents. Instructive examples for N = 400 homogeneous hard dumbbells with an anisotropy d = 0.25 are given in the Refs. . The classification of the modes in terms of transversal (T) and longitudinal-momentum (LP) modes is completely analogous to that for hard disks, although some modifications due to the presence of rotational degrees of freedom have not yet been worked out in all detail. In Fig. [low], we show, as an example, the smallest positive exponents of a system of N = 150 dumbbells with a density n = 0.5 in an elongated (along the x axis) periodic box with an aspect ratio A = 1/32. [trans446] Only modes with wave vectors parallel to x develop, which facilitates the analysis. The spectrum displays the typical step structure due to the mode degeneracies with multiplicity two, for the T modes, and four, for the LP modes. For this system, typical instantaneous mode patterns are shown in the Figs. [trans446] and [trans443], the former for a T mode with l = 446, the latter for an LP mode with l = 443. In both cases, the wavelength is equal to the box length in x direction. Please note that the system appears strongly contracted along the x axis. [trans443] Stationary particle systems out of equilibrium ============================================== One of the most interesting – and historically first – studies of Lyapunov spectra for many-particle systems is for stationary non-equilibrium states. If a system is driven away from equilibrium by an external or thermal force, the irreversibly generated heat needs to be removed and transferred to a heat bath. Otherwise the system would heat up and would never reach a steady state. Such a scheme may assume the following equations of motion, $$\begin{aligned} \dot{{\bf q}}\_i &=& {\bf p\_i}/m\_i + {\cal{Q}}(\{{\bf q}\},\{{\bf p}\}) X(t) \nonumber\\ \dot{{\bf p}}\_i &=& - \partial \Phi/\partial {\bf q}\_i + {\cal{P}}(\{{\bf q}\},\{{\bf p}\}) X(t) - s\_i\zeta {\bf p}\_i, \label{thermo}\\ \nonumber\end{aligned}$$ where ${\cal{Q}} X$ and ${\cal{P}} X$ represent the driving, and $-\zeta {\bf p}\_i$ the thermostat or heat bath acting on a particle i selected with the switch si ∈ {0, 1}. ζ fluctuates and may be positive or negative, whether kinetic energy has to be extracted from – or added to – the system. Averaged over a long trajectory, ⟨ζ⟩ needs to be positive. $\zeta(\{{\bf q}\},\{{\bf p}\})$ may be a Lagrange multiplier, which minimizes the thermostatting force according to Gauss’ principle of least constraint and keeps either the kinetic energy or the total internal energy a constant of the motion. Or it may be a single independent variable in an extended phase space such as for the Nosé-Hoover thermostat. There are excellent monographs describing these schemes. They are rather flexible and allow for any number of heat baths. Although it is not possible to construct such thermostats in the laboratory, they allow very efficient non equilibrium molecular dynamics (NEMD) simulations and are believed to provide an accurate description of the nonlinear transport involved. Simple considerations lead to the following thermodynamic relations for the stationary non-equilibrium state generated in this way after initial transients have decayed: $$\begin{aligned} &&\left\langle \frac{ d \ln \delta V^{(D)} }{dt } \right\rangle = \left\langle \frac{\partial}{\partial {\bf \Gamma}} \cdot \dot{{\bf \Gamma}} \right\rangle \nonumber = \sum\_{\ell =1}^{D} \lambda\_{\ell} = - d \sum\_{i=1}^N s\_i \left\langle \zeta \right\rangle \nonumber \\ &=& -\frac{1}{k\_B T} \left\langle \frac{dQ}{dt} \right\rangle = - \frac{1}{k\_B} \left\langle \frac{dS\_{irr}}{dt} \right\rangle < 0. \label{ne} \\ \nonumber\end{aligned}$$ Here, d is the dimension of the physical space, and d∑isi becomes the number of thermostatted degrees of freedom. D is the phase space dimension, and dQ/dt is the rate with which heat is transferred to the bath and gives rise to a positive rate of entropy production, ⟨dSirr/dt⟩. T is the kinetic temperature enforced by the thermostat. These relations show that an infinitesimal phase volume shrinks with an averaged rate, which is determined by the rate of heat transfer to the bath, which, in turn, is regulated by the thermostat variable ζ. As a consequence, there exists a multifractal attractor in phase space, to which all trajectories converge for t → ∞, and on which the natural measure resides. For Axiom A flows or even non-uniform hyperbolic systems, these states have been identified with the celebrated Sinai-Ruelle-Bowen (SRB) states. This result is quite general for dynamically thermostatted systems and has been numerically confirmed for many transport processes in many-body systems. The computation of the Lyapunov spectrum is essential for the understanding of this mechanism and, still, is the only practical way to compute the information dimension of the underlying attractor. Such a system is special in the sense that its stationary measure is singular and resides on a fractal attractor with a vanishing phase space volume. The ensuing Gibbs entropy is diverging to minus infinity, which explains the paradoxical situation that heat is continuously transferred to the bath giving rise to a positive rate of irreversible entropy production $\dot{S}\_{irr}$. The fact that there is a preferred direction for the heat flow is in accordance with the Second Law of thermodynamics, which these systems clearly obey. The divergence of the Gibbs entropy also indicates that the thermodynamic entropy, namely the derivative of the internal energy with respect to the entropy, is not defined for such stationary non-equilibrium states. ![(Color online) One-dimensional Frenkel-Kontorova conductivity model described in the main text. Three Poincaré maps for the planes defining the first octant are shown. The system is in a stationary non-equilibrium state. Reprinted from Ref. ](figure_22.jpg "fig:") [frenkel] We demonstrate such a fractal attractor for a one-dimensional Frenkel-Kontorova conductivity model of a charged particle in a sinusoidal potential, which is subjected to a constant applied field X and a Nosé-Hoover thermostat. The equations of motion are $$\dot{q} = p; \;\;\; \dot{p} = -\sin(q) + X - \zeta p; \;\;\; \dot{\zeta} = p^2 -1. \nonumber$$ The phase space is three-dimensional. In Fig. [frenkel] three Poincaré maps for mutually orthogonal planes defining the first octant of the phase space are shown. The singularity spectrum for this model has been computed in Ref. , and the multifractal nature of the stationary phase-space measure has been established. The sum of all Lyapunov exponents becomes negative for stationary non-equilibrium states. To demonstrate this effect, we show in Fig. [sllod] [sllod] the Lyapunov spectra of a system of 36 particles subjected to planar shear flow for various shear rates $\dot{\varepsilon}$ as indicated by the labels. Homogeneous SLLOD mechanics in combination with a Gaussian energy control has been used, which keeps the internal energy of the fluid a constant of the motion. The figure emphasizes the conjugate pairing symmetry in the non equilibrium stationary state: Since all particles are homogeneously affected by the same thermostat, the pair sum of conjugate exponents becomes negative and is the same for all pairs. With the Kaplan-Yorke formula, the information dimension is found to be smaller than the dimension of the phase space, a clear indication of a multi-fractal phase-space distribution. Final remarks ============= Here we have described some of our studies of simple fluids, the results so far obtained, and the new questions they raise. Most of our simulations are for planar systems due to the numerical effort required for the computation of Lyapunov spectra, but the results are expected to carry over to three-dimensional systems. What have these studies taught us about fluids? For equilibrium systems, the results about the density dependence of the maximum Lyapunov exponents and of the mixing time in Sec. [hardwca] clearly provide deeper and quantitative insight into the foundation and limitations of more familiar models for dense fluids. Still, the behavior of the exponents near phase transitions has not been satisfactorily exploited yet and should be the topic for further research. It is interesting to note that time-dependent local Lyapunov exponents have been used to identify atypical trajectories with very high or very low chaoticity. Such trajectories, although rare and difficult to locate in phase space, may be of great physical significance. For example, in phase traditions, or in chemical reactions, they may lead over unstable saddle points connecting a quasi-stable state, or chemical species, with another. Tailleur and Kurchan invented a very promising algorithm, Lyapunov weighted dynamics, for that purpose, in which a swarm of phase points evolves according to the natural dynamics. Periodically, trajectories are killed, or cloned, with a rate determined by their local exponents Λ1. As a consequence, trajectories are weighted according to their chaoticity. A slightly modified version of this algorithm, Lyapunov weighted path sampling, has been suggested by Geiger and Dellago, and has been successfully applied to the isomerizing dynamics of a double-well dimer in a solvent. Since the turn of the century, Lyapunov modes have raised a lot of interest. But it has been surprisingly difficult to connect them to other hydrodynamic properties. Most recently, however, some progress has been made. But this problem will still occupy researchers for some time. Another interesting and promising field is the study of systems with qualitatively different degrees of freedom, such as translation and rotation. Our studies of the Lyapunov instability of rough disks and of linear dumbbell molecules has opened another road to investigate translation-rotation coupling. An extension of these studies to vibrating molecules and to more complex systems with internal rotation, such as rotational isomer dynamics in butane, are promising topics for future research. Arguably, the most important impact Lyapunov spectra have achieved is in non-equilibrium statistical mechanics. In combination with dynamical thermostats, it is possible to generate stationary states far from equilibrium and to link the rate of entropy production with the time-averaged friction coefficients and the logarithmic rate of phase-volume contraction. See Eq.([ne]). The existence of a multi-fractal phase-space distribution with an information dimension smaller than the phase space dimension provides a geometrical interpretation of the Second Law of thermodynamics. The importance of this result is reflected in the voluminous literature on this subject. There are many other applications of local Lyapunov exponents and tangent vectors in meteorology, in the geological science, and in other fields. For more details we refer to Ref. , which is a recent collection of review articles and applications. Acknowledgments =============== We thank Prof. Christoph Dellago and Prof. Willliam G. Hoover for many illuminating discussions. We also thank Drs. Ljubo Milanović, Jacobus van Meel, Robin Hirschl and Christina Forster, whose enthusiasm and dedication provided the basis for the present discussion. 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Phys. 66, 2821 (1977). C.S. Pangali and B.J. Berne, J. Chem. Phys. 67, 4561 (1977). J.A. Montgomery, Jr. and B.J. Berne, J. Chem. Phys. 67, 4580 (1977). J. van Meel and H.A. Posch, Phys. Rev. E 80, 016206 (2009). A. Bellemans, J. Orban, and D. Van Belle, Molec. Phys. 39, 781 (1980). Lj. Milanović, Dynamical instability of two-dimensional molecular fluids: hard dumbbells, Ph.D. thesis, University of Vienna (2001). I. Borzsák, H.A. Posch, and A. Baranyai, Phys. Rev. E 53, 3694 (1996). O. Kum, Y.H. Shin, and E.K. Lee, Phys. Rev. E 58, 7243 (1998). B.L. Holian, W.G. Hoover and H.A. Posch, Phys. Rev. Lett. 59, 10 (1987). D.J. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids, 2nd edition, Cambridge University Press, Cambridge (2008). Wm.G. Hoover and C.G. Hoover, Time Reversibility, Computer Simulation, Algorithms, Chaos, 2nd edition, World Scientific Publishing Company, Singapore (2012). H.A. Posch and Ch. Forster, ” Lyapunov Instability if Fluids”, in Collective Dynamics of Nonlinear and Disordered Systems, G. Radons, W. Just, and P. Häussler eds, Springer Verlag, Berlin (2005). J. Tailleur and J. Kurchan, Nature Phys. 3, 203 (2007). P. Geiger and C. Dellago, Chem. Phys. 375, 309 (2010). H.A. Posch and S. Toxvaerd, unpublished. M. Cencini and F. Ginelli eds., Special issue on Lyapunov analysis: from dynamical systems theory to applications, J. Phys. A, 46 (2013).	arxiv_0000661
Software for enumerative and analytic combinatorics =================================================== ### 2013 We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, Pólya theory, combinatorial species, and asymptotics. We describe the capabilities that the packages offer as well as some of the algorithms used, and provide links to original documentation. Most of the packages are freely downloadable from the web. Note: In this document, to refer to webpages we place URL links in footnotes, and for all other types of referent we use standard endnote references. Introduction ============ In an opinion article[1](#fn1) posted to his website in 1999, Doron Zeilberger challenged mathematicians to rethink the role of computers in mathematics. “Everything that we can prove today will soon be provable, faster and better, by computers,” he says, then gives the following advice: > > The real work of us mathematicians, from now until, roughly, fifty years from now, when computers won’t need us anymore, is to make the transition from human-centric math to machine-centric math as smooth and efficient as possible. …We could be much more useful than we are now, if, instead of proving yet another theorem, we would start teaching the computer everything we know, so that it would have a headstart. …Once you learned to PROGRAM (rather than just use) Maple (or, if you insist Mathematica, etc.), you should immediately get to the business of transcribing your math-knowledge into Maple. > > > If futurist Ray Kurzweil’s predictions for artificial intelligence progress[2](#fn2) are to be believed, Zeilberger’s suggestions will turn out to be sound. (However, utilitarians would urge us to consider the risks such technology would present.[3](#fn3) [4](#fn4)) What is certain even today is that those who use mathematics, given the rise of computers, can ask themselves if they are failing to capitalize on a productive division of labor between man and machine. is no longer necessary to spend three years of Sundays factoring the Mersenne number 267 − 1, like F. N. Cole did in the 1900s, and neither is it necessary to use error-prone pen and paper methods to perform an ever-growing set of mathematical procedures. To illustrate this statement, this document examines symbolic computation (a.k.a. computer algebra) software packages for enumerative and algebraic combinatorics. We start, in Section [sec:Enumerativeandanalyticcombinatorics], with an overview of the fields of enumerative and analytic combinatorics. Then we go into more detail on the scope of the document in Section [sec:notes]. In Sections [sec:basiccombinatorialobjects]–[sec:asymptotics] we cover packages relating to basic combinatorial objects, symbolic combinatorics, Pólya theory, combinatorial species, and asymptotics. Finally, we offer concluding observations and remarks in Section [sec:conc]. Enumerative and analytic combinatorics ====================================== Welcome to the fields of enumerative and analytic combinatorics, a.k.a. combinatorial and asymptotic enumeration! In order to completely cover what mathematicians think of when they think of these fields (and to avoid saying “enumerative” or “combinatorial”), we break up our discussion into two parts which we call counting, the more mathematical side, and enumeration, the more algorithmic side. Counting -------- Counting, the oldest mathematical subject, is enumeration in the mathematical sense: the study of the cardinalities of finite sets. The most basic principle of counting is the addition rule : [Addition rule] If S is a set and A1, A2, …, An is a partition of S, then ∣S∣ = ∣A1∣ + ∣A2∣ + ⋯ + ∣An∣. Other similar basic ways of counting may be familiar from introductory probability, and indeed, many concepts overlap between counting and discrete probability.[5](#fn5) Elements of the body of work on counting can be roughly categorized based on three criteria: whether they deal with exact or asymptotic results, whether they speak in terms of generating functions or their coefficients, and whether they make use of bijections or manipulations. ### Exact vs. asymptotic counting Boldface symbols refer to (possibly terminating) 1-based sequences, i.e. $ \boldsymbol{a} = (a\_1, a\_2, \dots) $. Generally, counting problems involve a triple $ (S, \boldsymbol{p}, N) $ comprising a countable set S of objects, a sequence $ \boldsymbol{p} = (p\_1, p\_2, \dots) $ of functions pi : S → Z≥ 0, and a set N of sequences $ \boldsymbol{n} $ for which $ \boldsymbol{p}^{-1}(\boldsymbol{n}) = \{ s \in S : p\_1(s) = n\_1, p\_2(s) = n\_2, \dots \} \subseteq S$ is finite. The problem is to answer the question “For $\boldsymbol{n} \in N$, how many objects does the set $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $ contain?” The definition of an exact answer was given by Herb Wilf: a polynomial-time algorithm that computes the number. [exa:signature] If π ∈ Sn is a permutation, then let cj(π) be the number of cycles of π with size j. The signature of π is $ \boldsymbol{c}(\pi) = (c\_1(\pi), c\_2(\pi), \dots) $. Let S be the set of all permutations, and for s ∈ S, let $ \boldsymbol{p}(s) $ be the signature of s. Let n ≥ 1 and let $ \boldsymbol{n} = ($[[](http://en.wikipedia.org/wiki/Iverson_bracket)j = n])j ≥ 0. Then $ \|\boldsymbol{p}^{-1}(\boldsymbol{n})\| = (n-1)! $. The expression (n − 1)! immediately suggests a polynomial-time algorithm to compute $ \|\boldsymbol{p}^{-1}(\boldsymbol{n})\| $. Exact answers can be classified by ansatz, meaning the form of the sequence $ (\|\boldsymbol{p}^{-1}(\boldsymbol{n})\|)\_{ \boldsymbol{n} \in N} $, which is generally determined by the “simplest” reccurence relation it satisfies. Exact answers are not the end of the story, however, partly because of applications to the analysis of algorithms. In the single-parameter case, i.e. $ \boldsymbol{p} = (p) $ and N = ((0), (1), (2), …), an asymptotic answer is a relation between f : n ↦ ∣p− 1(n)∣ and “simple” functions, which holds as n → ∞. Often the “simple” functions come from the logarithmico-exponential class L of Hardy, and the relation is f(n) ∼ g(n) as n → ∞, where g ∈ L. A more substantial relation is a full asymptotic series : [def:asymptoticseries] Given a sequence of functions $ \boldsymbol{g} $ with gk + 1(n) = o(gk(n)) as n → ∞ for all k, and real numbers c1, c2, …, the statement $$f(n) \sim \boldsymbol{c}\cdot \boldsymbol{g}(n) = c\_1 g\_1(n) + c\_2 g\_2(n) + c\_3 g\_3(n) + \cdots$$ is called an asymptotic series for f, and it means $$\begin{aligned} f(n) &= O(g\_1(n)) \\ f(n) &= c\_1 g\_1(n) + O(g\_2(n)) \\ f(n) &= c\_1 g\_1(n) + c\_2 g\_2(n) + O(g\_3(n)) \\ f(n) &= c\_1 g\_1(n) + c\_2 g\_2(n) + c\_3 g\_3(n) + O(g\_4(n))\\ &\vdots\end{aligned}$$ (as n → ∞.) An asymptotic answer in the multiple-parameter case is more complicated; it generally involves (possibly just some moments of) a continuous approximation to a discrete probability distribution as one parameter approaches infinity, or an asymptotic relation which holds as one or more parameters approach infinity at various rates. ### Generating functions vs. their coefficients Given a triple $ (S, \boldsymbol{p}, N) $, let f : N → Z≥ 0 be defined $ f( \boldsymbol{n} ) = \| \boldsymbol{p}^{-1} ( \boldsymbol{n} )\| $. A (type u) generating function of S with $ \boldsymbol{z} = (z\_1, z\_2, \dots) $ marking $ \boldsymbol{p} $ is the element of the ring $ \mathbb{Q}[[\boldsymbol{z}]] $ $$F( \boldsymbol{z} ) = \sum\_{\boldsymbol{n} \succeq \boldsymbol{0}} f(\boldsymbol{n}) u(\boldsymbol{n}) \boldsymbol{z}^{\boldsymbol{n}},$$ where $ \boldsymbol{z}^{\boldsymbol{n}} = z\_1^{n\_1} z\_2^{n\_2} \cdots $. We call $ F( \boldsymbol{z} ) $ an ordinary generating function iff $ u(\boldsymbol{n}) = 1 $ for all $ \boldsymbol{n} \succeq \boldsymbol{0} $, and we call it an exponential generating function iff $ u(\boldsymbol{n}) = (n\_1!)^{-1} $ for all $ \boldsymbol{n} \succeq \boldsymbol{0} $. Let $ F( \boldsymbol{z}) $ be the ordinary generating function for words of length n on the alphabet [1..k], with zj marking the number of occurrences of j, 1 ≤ j ≤ k. We have $$F(\boldsymbol{z}) = (z\_1 + z\_2 + \cdots + z\_k)^n.$$ To define convergence of sequences of generating functions, the norm on formal power series used in this document is defined for $ f( \boldsymbol{z} ) \neq 0 $ as $${\lVertf(\boldsymbol{z})\rVert} = 2^{-k}, \text{ where } k= \max \left\{ j \in \mathbb{Z}\_{\geq 0} : \forall i \in [0..j], [z^i]f(z,z, \dots) = 0 \right\},$$ and ∥0∥ = 0. It can be efficient to initially make statements about $ F( \boldsymbol{z} ) $ instead of working directly with f, for a variety of reasons [6](#fn6), and extracting $ f(\boldsymbol{n}) $ exactly from a suffiently simple representation of $ F( \boldsymbol{z} ) $ can be done in polynomial time. In addition, there are very widely applicable theorems for obtaining the asymptotics of f from $ F(\boldsymbol{z}) $, which involve using the power series $ F(\boldsymbol{z}) $ to define a complex function analytic at the origin. Since generating functions are heavily used in both the exact and asymptotic worlds, Wilf says in his book, “To omit the analytical (i.e. asymptotic) parts of [counting with generating functions] …is like listening to a stereo broadcast of, say, Beethoven’s Ninth Symphony, using only the left audio channel. The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous.” ### Bijective combinatorics vs. manipulatorics To prove two sets have equal cardinality, often one of two methods is used. First, if one has algebraic expressions for the cardinality of each set, one may perform algebraic manipulations upon them until they are syntactically equivalent. Second, one may give an explicit bijection between them. The following example demonstrates each method on the same problem: A partition is called odd if all its parts are odd and it is called distinct if all its parts are distinct. Let f(n) and g(n) be the number of odd and distinct partitions of size n respectively, and let us define the ordinary generating functions F(z) = ∑n ≥ 0f(n)zn and G(z) = ∑n ≥ 0g(n)zn. Since $$\begin{aligned} G(z) &= \prod\_{n \geq 0 } (1 + z^n) \\ &= \prod\_{n \geq 0} \frac{1-z^{2n}}{1-z^n} \\ &= \frac{\prod\_{n \geq 0} (1-z^{2n}) }{ \prod\_{ n \geq 0 } (1-z^{2n}) \prod\_{n \geq 0} (1-z^{2n + 1})} \\ &= \prod\_{n \geq 0} \frac{1}{1-z^{2n+1}} \\ &= F(z),\end{aligned}$$ we have [zn]G(z) = [zn]F(z), and thus f(n) = g(n), for all n ≥ 0. There is also a bijective proof of this fact, due to Glaisher, which we sketch. The function from distinct partitions to odd partitions is defined as follows: Given a distinct partition, write each of its parts as 2rs, where s is odd, and replace each by 2r copies of s. This function is invertible, with inverse computable as follows: Take an odd part a which occurs m times, and write m in base 2, i.e. m = (sk⋯s1)2, then replace the m copies of a by the k parts 2s1a, …, 2ska. Arguably, manipulations are not combinatorics, hence the name “manipulatorics”. Indeed, here the borders between algebra, analysis, combinatorics, the analysis of algorithms, and other fields are blurry. Usually, bijections are used to give exact answers, but bijections can also be used in the context of asymptotics, as described in, for example. Standard textbooks for the field of counting include. Enumeration ----------- Enumeration is the field of computer science dealing with algorithms that generate the elements of finite sets. Various types of enumeration problems can be proposed for a given triple $ (S, \boldsymbol{p}, N) $; the most common ones are ranking and unranking, random generation, exhaustive listing, and iteration, which we define in that order below. We call an ordering of $ \boldsymbol{p}^{-1}(\boldsymbol{n}) \subseteq S $ a bijection between $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $ and the integers $ [1..\|\boldsymbol{p}^{-1}(\boldsymbol{n})\| ] $. A ranking algorithm computes this bijection and an unranking algorithm computes its inverse. In random generation, discrete distributions (always uniform distributions in this document) are specified on the sets $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $, and an algorithm is required to take as input $ \boldsymbol{n} \in N $ and return a random variate drawn from $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $ according to the distribution. Unranking algorithms can be used for random generation, since an integer can be generated at random and then unranked to give a random object. An exhaustive listing algorithm takes as input $ \boldsymbol{n} \in N $ and returns a list of all elements in $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $. Generally, as with random generation, if the $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $ are well defined, a brute force algorithm is trivial, and the problem lies in designing an efficient algorithm. Iteration is the problem of, given $ \boldsymbol{n} \in N$ and an object $s \in \boldsymbol{p}^{-1}(\boldsymbol{n}) $, generating the next object sʹ in a certain ordering of $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $. It is related to the problem of exhaustive listing since any iteration algorithm immediately leads to an exhaustive listing algorithm, and more importantly, finding a particular ordering of $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $ often leads to the most efficient exhaustive listing algorithms. If one employs an iteration algorithm repeatedly, in an exhaustive listing algorithm, one aims for an ordering that takes constant amortized time for each iteration. A Gray code is an ordering in which sucessive objects differ in some prespecified small way and thus is perfect for exhaustive listing through iteration. A few notes on the following packages ===================================== In this document we focus on general-purpose software of wide interest to mathematicians, mathematics students, and perhaps those outside the field. Many packages have been created for solving particular counting and enumeration problems, such as Lara Pudwell’s enumeration schemes packages[7](#fn7), Donald Knuth’s OBDD[8](#fn8) which enumerates perfect matchings of bipartite graphs, and many of Zeilberger’s numerous Maple packages[9](#fn9); such packages are outside the scope of this document. There are some relatively general-purpose packages which did not make it into the document but should to be mentioned for completeness, though: The Mathematica package Omega implements MacMahon’s Partition Analysis[10](#fn10), and there is a Maple analog written by Zeilberger called LinDiophantus[11](#fn11); an algorithm by Guo-Niu Han able to cover more general expressions than Omega was implemented in Maple[12](#fn12) and Mathematica[13](#fn13) packages; the package RLangGFun translates from rational generating functions to regular languages[14](#fn14), and the regexpcount package from the INRIA Algorithms Group translates in the other direction[15](#fn15); and Zeilberger has written a number of packages related to the umbral transfer matrix method, an infinite-matrix generalization of the transfer matrix method.[16](#fn16) This document excludes some software related to the intersection of algebraic and enumerative combinatorics, and all software related to power series summation and manipulation. General-purpose software is currently skewed towards manipulations, away from bijections. Indeed, one would not expect complicated bijections such as the proofs of Theorem 1 in (balanced trees and rooted triangulations) or Theorem 4.34 in (indecomposable 1342-avoiding n-permutations and β(0, 1)-trees on n vertices) to be obtainable by symbolic methods any time soon. However, in this document, we include as many algorithms with a bijective flavor as possible. (Ultimately all computations may be considered manipulations, but we refer to a qualitative difference in mathematical content.) Finally, we note that plenty of basic algorithms from combinatorics, as well as advanced algorithms from the symbolic computation literature, have not been implemented in a published package. (Actually, whether or not an algorithm has been “implemented in a published package” has a fuzzy value. For example, some have been implemented, published, but are now gone, while others have been published and are available, but are written in obscure languages that are unfamiliar or difficult to obtain compilers for. The sentence is true even in the loosest sense, however.) Basic combinatorial objects =========================== Mathematical background ----------------------- Algorithmically counting and enumerating the basic objects of combinatorics like graphs, trees, set partitions, integer partitions, integer compositions, subsets, and permutations is implemented in various packages and discussed in various books (and many papers). As an example, describes over 30 permutation generation algorithms published as of 1977. A complete comparison of the enumeration algorithms implemented in the packages in this document could be its own project. For the packages mentioned in this document that enumerate basic objects, we do not give full details on the algorithms used. For more information, see the packages’ documentation. However, in the rest of this subsection we provide some examples of two general concepts, first mentioned in Section [sec:enumeration], which guide the discovery of enumeration algorithms: orderings and Gray codes. ### Orderings Many combinatorial objects can be ordered lexicographically. Lexicographic order, a.k.a. lex order, applies when the objects can be represented as words over an ordered alphabet.[17](#fn17) If w = w1w2⋯wn and v = v1v2⋯vn are words then, in lexicographic order, then w < v iff w1 < v1 or there is some k ∈ [1..n − 1] such that wj = vj for 1 ≤ j ≤ k and wk + 1 < vk + 1. Permutations are a clear example of a case where this order applies, and iterating through permutations in lexicographic order is an easy exercise, see for a solution. Co-lexicographic order, a.k.a. co-lex order, is related: If w and wʹ are words, w ≤ wʹ in co-lexicographic order iff rev(w) ≤ rev(wʹ) in lexicographic order (where rev reverses words). Another order, cool-lex order, applies to binary words containing exactly k copies of 1, which we can think of as k-subsets. Generating the next binary word in cool-lex order is done as follows: Find the shortest prefix ending in 010 or 011, or the entire word if no such prefix exists. Then cyclically shift it one position to the right. Since the shifted portion of the string consists of at most four contiguous runs of 0’s and 1’s, each succesive binary word can be generated by transposing only one or two pairs of bits. Thus cool-lex order for k-subsets is a Gray code. ### Gray codes Unrestricted subsets have a Gray code that is very easy to understand, called the standard reflected Gray code, in which, as above, we represent subsets as binary words. Say we want to construct a Gray code Gn of subsets of a size-n set, and suppose we already have a Gray code Gn − 1 of subsets of the last n − 1 elements of the set. Concatenate Gn − 1 with a reversed copy of Gn − 1 with the first element of the set added to each subset. Then all subsets differ by one from their neighbors, including the center, where the subsets are identical except for the first element. Combinat (Maple) ---------------- Author: Maplesoft Website: <http://www.maplesoft.com/support/help/Maple/view.aspx?path=combinat> The combinat package, which is distributed with Maple, has routines for counting, listing, randomly generating, and ranking and unranking basic combinatorial objects such as permutations, k-subsets, unrestricted subsets, integer partitions, set partitions, and integer compositions.[18](#fn18) Like Combinatorica and unlike Sage and the Combinatorial Object Server, combinat does not offer a wide range of restrictions that can be placed on the objects. As mentioned in Section [sec:combstruct], most of the functionality of the combinat package is also covered by Combstruct. Most types of objects can only be enumerated in a single ordering, but unrestricted subsets (in binary word form) can be listed in Gray code order with the `graycode` function. We can use `graycode` to print all subsets of a size-3 set: > printf(cat($\hspace{4pt}\grave{ }\hspace{2pt}$ %.3d$\hspace{4pt}\grave{ }\hspace{2pt}$$8), op(map(convert, graycode(3), binary))) `000 001 011 010 110 111 101 100` We note that outside the combinat package, Maple includes support for random graph generation, which is comparable to, for example, Mathematica’s. For more information on Maple, see the Appendix. Combinatorial Object Server --------------------------- Author: Frank Ruskey Last modified: May 2011 Website: <http://theory.cs.uvic.ca/cos.html> The Combinatorial Object Server (COS) is a website that runs on the University of Victoria’s domain. It has a web interface for easily specifying a set of basic combinatorial objects and viewing an exhaustive listing of all objects in the set (see Figure [fig:cos] on page ). Objects available include permutations, derangements, involutions, k-subsets, unrestricted subsets, set partitions, trees, necklaces, and unlabeled graphs. On each type of object, there is a set of restrictions that can be placed. Integer partitions, for example, can be restricted by largest part, and whether the parts must be odd, distinct, or odd and distinct. There is also a wide variety of output formats for the objects. Permutations, for example, can be printed in one line notation, cycle notation, permutation matrix form, standard Young tableau form and more. The order of output can sometimes be specified, too. Combinations, for example, can be shown in Gray code, lexicographic, co-lexicographic, cool-lex, transposition, or adjacent transposition orders. Combinatorica: basic combinatorial objects ------------------------------------------ Authors: Sriram Pemmaraju and Steven Skiena Download: <http://www.cs.uiowa.edu/~sriram/Combinatorica/NewCombinatorica.m> Last modified: 2006 Website: <http://www.cs.sunysb.edu/~skiena/combinatorica/> Combinatorica is a Mathematica package for discrete mathematics. In development since 1990, it includes over 450 functions in the areas of Pólya theory, permutations and algebraic combinatorics, basic combinatorial objects, graph algorithms, and graph plotting. A book was written by the package authors, which is the definitive source of information on Combinatorica. The Combinatorica package has been included with releases of Mathematica since Mathematica version 4.2, although some of Combinatorica’s functionality has recently been redone and built into the Mathematica kernel in Mathematica 8. For information on Mathematica as a programming language, see the Appendix. Combinatorica has support for counting and enumeration with permutations, k-subsets, unrestricted subsets, integer partitions, integer compositions, set partitions, Young tableaus and graphs. For each type of object, Combinatorica generally offers rules for counting, iteration and listing in one or two orderings, and random generation. Combinatorica does not provide as many ways to specify restrictions on the objects as COS or Sage. The function `GrayCodeSubsets` exhaustively lists all subsets of a set in standard reflected Gray code order: `In[1]:= GrayCodeSubsets[{1, 2, 3, 4}]` Out[1]:= {{},{4},{3,4},{3},{2,3},{2,3,4},{2,4},{2},{1,2},{1,2,4}, `{1,2,3,4},{1,2,3},{1,3},{1,3,4},{1,4},{1}}` One may wonder if such a Gray code is unique, and one can find this out by first noticing that Gray codes for the subsets of a size-n set are in bijection with Hamiltonian paths in the n-dimensional hypercube. Combinatorica includes a database of common graphs, including `Hypercube[n]`, and also has the `HamiltonianCycle` rule which replaces `HamiltonianCycle[graph, All]` with a list of all Hamiltonian cycles in `graph`. So to find out if the Gray code order above is unique, one can find the length of the list of Hamiltonian paths in the 4-dimensional hypercube: `In[2]:= Length[HamiltonianCycle[Hypercube[4], All]]` `Out[2]:= 2688` It is definitely not! The number of Hamiltonian cycles in an n-dimensional hypercube is not known, even asymptotically. Sage: basic combinatorial objects --------------------------------- Download: <http://www.sagemath.org/download.html> Online access: <http://www.sagenb.org/> Website: <http://sagemath.org/doc/reference/combinat/index.html> Sage is a free, open-source computer algebra system (CAS) first released in 2005. Sage integrates many specialized open-source symbolic and numeric packages, such as Maxima, GAP, SciPy, and NumPy, written in various languages and allows them all to be called from a unified Python interface. In addition, it has native support for a wide and quickly expanding range of mathematical fields, including combinatorics. This section covers Sage’s capabilities for counting and enumerating basic combinatorial objects; see Section [sec:sagespecies] for Sage’s combinatorial species capabilities. Sage uses object-oriented programming to implement a category-theoretic hierarchy of categories and objects.[19](#fn19) [20](#fn20) Sage’s support for combinatorial objects, which is part of a migration of the MuPAD-Combinat project[21](#fn21), which has reached end-of-life, to Sage, is based on the category called `EnumeratedSets`[22](#fn22). Classes of basic combinatorial structures (such as k-subsets, unrestricted subsets, signed and unsigned integer compositions, necklaces, integer partitions, permutations, ordered and unordered set partitions, words and subwords) all belong to the category `EnumeratedSets` which implies that sets of objects from those categories can be constructed which inherit at least the following methods: 1. `cardinality()` - the cardinality of the set, 2. `list()` - a list of all elements, 3. `unrank(n)` - the `n`th object in an ordering, 4. `rank(e)` - the rank of the object `e`, 5. `first()` - the first object in the ordering, 6. `next(e)` - the next object after `e` in an ordering, 7. `random_element()` - an object chosen at random according to the uniform distribution. Of course, each class of combinatorial object built in to the system may also implement many more methods. Many classes allow restrictions to be specified, but only the default ordering is available. The `Partitions()` static method is called to construct an object representing a set of integer partitions specified by its arguments.[23](#fn23) For example, `Partitions(4)` returns all integer partitions of 4, while `Partitions(4, max_part=2)` returns all partitions of 4 with maximum part size 2: `Partitions(4, max_part=2).cardinality()` `3` `sage: Partitions(4, max_part=2).list()` `[[2, 2], [2, 1, 1], [1, 1, 1, 1]]` `sage: Partitions(4, max_part=2).random_element()` `[2,2]` Sage also provides several implementations of counting functions, separate from the `EnumeratedSets` category.[24](#fn24) These include the partition-theoretic counting functions for number of set partitions, and ordered and unordered integer partitions, and the set-theoretic counting functions for number of subsets, arrangements, derangements and permutations of a multiset. Symbolic combinatorics ====================== Mathematical background ----------------------- Let S be a set of objects, with a parameter $\boldsymbol{p}=(p) $. We define a new set S< 2 > = S × S, with parameter $ \boldsymbol{p}^{<2>} = (p^{<2>}),$ where p< 2 >((s1, s2)) = p(s1) + p(s2) for all s1, s2 ∈ S. Let f(n) = ∣p− 1(n)∣, f< 2 >(n) = ∣(p< 2 >)− 1(n)∣. Then if F(z) is the ordinary generating function F(z) = ∑n ≥ 0f(n)zn, and F< 2 >(z) is the ordinary generating function F< 2 >(z) = ∑n ≥ 0f< 2 >(n)zn, we have, simply, F< 2 >(z) = F(z)2. It turns out that many other correspondences exist between the structure of a set of objects and its generating function. This document includes sections for two frameworks that develop this idea: the theory of combinatorial species which is the focus of Section [sec:species], and symbolic combinatorics, which is described below. The central concept of symbolic combinatorics[25](#fn25) is the combinatorial class. A combinatorial class is a countable set on which a parameter called size is defined, such that the number of elements of any given size n ≥ 0 is finite. If A is a combinatorial class, the size of an element α ∈ A is denoted ∣α∣. We denote the set of elements of size n in A by An, and denote its cardinality by an = ∣An∣. The counting sequence of a combinatorial class A is the sequence (an)n ≥ 0. There are two types of combinatorial class, unlabeled and labeled. ### Unlabeled classes The word class in this section refers to an unlabeled combinatorial class, which can be thought of as a set of objects made up of nodes without unique labels (think graphs). This will become rigorous as we proceed. The (ordinary) generating function of a class A with z marking size is the formal power series A(z) = ∑n ≥ 0anzn. A (k-ary) combinatorial construction Φ is a function that maps combinatorial classes B< 1 >, B< 2 >, …, B< k > to a new class A = Φ(B< 1 >, B< 2 >, …, B< k >). The combinatorial construction Φ is admissible iff the counting sequence of A only depends on the counting sequences of the arguments B< 1 >, B< 2 >, …, B< k >. If a construction Φ is admissible, there exists a corresponding operator Ψ on generating functions such that if A = Φ(B< 1 >, B< 2 >, …, B< k >), then A(z) = Ψ(B< 1 >(z), B< 2 >(z), …, B< k >(z)). The basic admissible constructions for unlabeled classes are called sum, product, sequence, powerset, multiset, and cycle. The definitions and corresponding generating function operators for all of these can be found in ; here we only describe the first three. 1. The sum of two classes A and B is written A + B and is formed by the discriminated union[26](#fn26) of A and B, with size inherited from the summands. The generating function of A + B is A(z) + B(z). 2. The product of two classes is written A × B and is formed by the cartesian product of A and B, with size defined additively. This is the construction used, where we saw that the generating function for A × B is A(z)B(z). 3. Finally, the sequence construction $ \textsc{Seq} $ is defined on classes with no elements of size 0. For a class A, the value $ \textsc{Seq}( \mathcal{A}) $ is the set of all finite sequences of elements in A, with size defined additively. The generating function for $ \textsc{Seq}( \mathcal{A}) $ is $$1 + A(z) + A(z)^2 + \cdots = \frac{1}{1-A(z)}.$$ The basic combinatorial constructions can be modified (restricted) in a number of ways. For example, we can fix k and define a construction $ \textsc{Seq}\_{ \geq k} $ that constructs sequences of length at least k, or for another example we could construct products containing an element of even size from one class and an element of odd size from another, etc. Let E be the class with a single element ε of size 0, called the neutral object, and let Z be the class with a single element ζ of size 1, called an atom. A specification for an r-tuple of classes is a collection of r equations $$\begin{aligned} \mathcal{A}^{<1>} &= \Phi\_1(\mathcal{A}^{<1>}, \dots, \mathcal{A}^{<r>}) \\ \mathcal{A}^{<2>} &= \Phi\_2(\mathcal{A}^{<1>}, \dots, \mathcal{A}^{<r>}) \\ &\cdots \\ \mathcal{A}^{<r>} &= \Phi\_r(\mathcal{A}^{<1>}, \dots, \mathcal{A}^{<r>}),\end{aligned}$$ where each Φi(⋯) represents an expression built from the A’s using the (possibly restricted) basic admissible constructions, as well as the classes E and Z. Let T be the class of nonempty unlabeled plane trees, with the size of a tree being the number of nodes. Then T satisfies the specification $$\mathcal{T} = \mathcal{Z} \times \textsc{Seq}(\mathcal{T}),$$ since an object in T is a single root with a sequence of subtrees. This specification implies T(z) = z(1 − T(z))− 1, and thus $ T(z) = \frac{1}{2} \left(1-\sqrt{1-4 z}\right) $. ### Labeled classes An object in a labeled class is labeled, meaning, if it has size n, each of its n indivisible components is labeled with a unique integer from the set [1..n]. A rigorous way to define such classes begins with a different definition for the elementary classes: Again, let E be the class with one neutral object ε of size 0, but now let Z be the class containing one labeled element ζ of size 1, a labeled atom. Then, labeled classes can be defined by specifications as above if we define some useful constructions on labeled classes. There is indeed a set of admissible constructions for labeled classes that is analogous to that for the unlabeled case. The sum of two classes is defined the same as for unlabeled classes, but for the others, we first need to define the product of two labeled objects: Given two labeled objects, α and β, the labeled product α ⋆ β is the set of all pairs (αʹ, βʹ) where αʹ and βʹ are relabeled versions of α and β such that order is perserved in the relabelings and each number in [1..∣α∣ + ∣β∣] appears as a label in either αʹ or βʹ. This concept leads to the definition of the product, sequence, set and cycle constructions, the first of which we define here; the rest can be found in. The product of labeled classes A and B is the set A ⋆ B = ⋃α ∈ A, β ∈ B(α ⋆ β), with size defined additively. As with the unlabeled case, the usefulness of defining a labeled class in terms of a specification comes from the fact that labeled constructions correspond to relatively “simple” operators on generating functions — exponenential generating functions in the labeled case. The (exponential) generating fuction of a labeled class A with z marking size is the formal power series $$A(z) = \sum\_{n \geq 0} a\_n \frac{z^n}{n!}.$$ For example, the generating function for A ⋆ B is A(z)B(z). We define a labeled binary tree as a labeled tree in which every internal node has two children. Let B be the labeled class representing such trees. Since an object in B is either a node with no children or a node with two children, we have B = Z + Z ⋆ B ⋆ B, which implies that B(z) = z + zB(z)2, and thus $B(z) = (1-\sqrt{1-4 z^2})/2 z $. ### Multiple parameters Unlabeled and labeled classes can be augmented with parameters other than size. For example, if there is one more parameter φ, we can redefine the original elementary classes so that E = {ε}, where ∣ε∣ = φ(ε) = 0, and Z = {ζ}, where ∣ζ∣ = 1, φ(ζ) = 0, and define a new φ- class P = {π}, where ∣π∣ = 0, φ(π) = 1. Size is the only parameter with respect to which a structure is “labeled” or “unlabeled” (and the only parameter which may be marked by a variable with a factorial below it in the generating function for the class), so those words can still be used unambiguously to refer to a class with more than one parameter. All constructions for labeled and unlabeled classes discussed so far have defined size additively, e.g. the size of an object is the sum of the sizes of its components. All of these constructions can be defined on multi-parameter classes with non-size parameters defined additively, just like size. The generating function equations they correspond to are the same as the single-parameter ones, except the generating functions may be multivariate (with a different variable marking each parameter). Let A be a single-parameter unlabeled combinatorial class. One can define a new class, B, consisting of sequences of elements of A, with size and an additional parameter φ such that φ(β) is the number of elements of A in β, for all β ∈ B. Then $ \mathcal{B} = \textsc{Seq}(\mathcal{P} \times \mathcal{A} ) $, and, if u marks φ, A(z, u) = (1 − uB(z))− 1. For more details, and for information on constructions where additional parameters are not defined additively, see. Given a labeled or unlabeled class A with additional parameter φ, one can define a random variable Xn to be an object chosen at random from An according to the uniform distribution. If u marks φ in the bivariate generating function A(z, u), then we have the syntactically simple relation $$\mathbf{E}[\varphi(X\_n)] = \frac{[z^n] A^{(0,1)}(z,u)\|\_{u=1}}{[z^n] A(z,1)}.$$ Higher factorial moments are obtained similarly. Techniques for obtaining limiting distributions from multivariate generating functions also exist: see. However, note that these methods for obtaining probabilistic facts apply for any multivariate generating function, whether or not it was obtained with symbolic combinatorics. Combstruct ---------- Authors: INRIA Algorithms Group[27](#fn27) and Maplesoft Website: <http://www.maplesoft.com/support/help/Maple/view.aspx?path=combstruct> Combstruct is a Maple package originally developed by the INRIA Algorithms Group which is now distributed with the most recently released version of Maple, Maple 16. Its functionality has changed over time, but today, it includes the capabilities of ALAS from LUO (Section [sec:alas]) along with the ability to enumerate, both randomly and exhaustively, the objects of a given size from a specified combinatorial class. Combstruct also extends ALAS by supporting translation from multiple-parameter specifications to mulivariate generating functions. Combstruct can also count, randomly and exhaustively enumerate, and iterate through a small set of built-in, predefined structures. In fact, Combstruct provides most of the functionality of Maple’s combinat[28](#fn28) package for working with basic combinatorial structures, but with syntax that is unified with that for working with classes of objects defined by the user by combinatorial specifications. We elaborate on these areas of functionality in the rest of this section. Combstruct allows the user to create labeled or unlabeled single-parameter combinatorial specifications, and augment the specifications with additional parameters separately. To specify a class, the elementary classes E and Z can be used, as well as the constructions sum, product, set, powerset, sequence, cycle and substitution. The constructions set, powerset, sequence and cycle can be restricted by an inequality or equality relation on the number of components allowed in the objects. It is possible, of course, to define a great many types of basic combinatorial object, as well as more complicated objects, with such specifications. The `gfeqns` command returns the system of equations over generating functions corresponding to a well-defined single-parameter specification (see Definition [defin:welldefined]), and `gfsolve` attempts to return explicit expressions for the generating functions. The `gfseries` command returns the initial values of the counting sequences of the classes. We create a specification `bintreespec` for labeled binary trees, where `B` is the class of labeled binary trees and `Z` is the atomic class: `>bintreespec := {B=Union(Z, Prod(Z, B, B)), Z=Atom }:` (The `Union` construction is the same as sum.) Then we use `gfsolve` to get the generation functions: `>gfsolve(bintreespec, labeled, z)` `B(z) = -(1/2)(-1+sqrt(1-4z^2))/z, Z(z) = z` Combstruct’s `draw` command takes a single-parameter specification, the name of a class A, and an integer n ≥ 0 and returns a object chosen at random from the set An according to the uniform distribution. For information on the algorithms used, see the documentation. To add additional parameters to a single-parameter specification, Combstruct allows the user to use an attribute grammar (see the documentation and for more information on attribute grammars; also, note that it is also possible to augment single-parameter specifications in Combstruct using a more limited method based on defining and using new atomic classes as described in the ). The `agfeqns` command returns the system of equations over multivariate generating functions corresponding to a given specification and attribute grammar. The `agfseries` command returns the initial values of the multidimensional sequences. The `agfmomentsolve` command takes an integer k ≥ 0 and a set of equations over n multivariate generating functions $A^{<i>}(z, \boldsymbol{u}), 1 \leq i \leq n $ and attempts to return explicit expressions for $ D\_{\boldsymbol{u}}^k A^{<i>}(z, \boldsymbol{u})\|\_{\boldsymbol{u} = \boldsymbol{1}}, 1 \leq i \leq n$. A number of in-depth examples of Combstruct in action are available at the INRIA Algorithms Group’s website[29](#fn29). Encyclopedia of Combinatorial Structures ---------------------------------------- Authors: Frédéric Chyzak, Alexis Darrasse, and Stéphanie Petit Download: <http://algo.inria.fr/libraries/#down> Last modified: July 2000 — original version[30](#fn30), 2011 — online version[31](#fn31) The Encyclopedia of Combinatorial Structures started out as a Maple package written by Stéphanie Petit as part of the INRIA Algorithm Group’s algolib and in 2009 a web interface for it was created by Alexis Darrasse and Frédéric Chyzak at <http://algo.inria.fr/encyclopedia/intro.html>. The Encyclopedia is a database of counting sequences of specifiable combinatorial structures. For each sequence in the database, the following fields, if available, are either computed or stored: 1. Name 2. Combinatorial specification, in syntax 3. Initial values, obtained with Combstruct’s `count` 4. Generating function, obtained with Combstruct’s gfsolve 5. A linear reccurrence relation, if applicable, obtained with gfun’s `holexprtodiffeq` and `diffeqtorec` 6. A closed-form expression, obtained with either Maple’s `rsolve` or gfun’s `ratpolytocoeff` 7. Dominant asymptotic term as n → ∞ computed by ’s `equivalent` 8. Description of combinatorial structure 9. References, such as entry in Sloane’s Encyclopedia of Integer Sequences[32](#fn32) (Note that gfun is the name of a Maple package developed by the INRIA Algorithms Group, for more information, see [33](#fn33).) It is possible to search the database by initial values of the sequence, keywords, generating function, or closed form of the sequence. Lambda-Upsilon-Omega (LUO): symbolic combinatorics -------------------------------------------------- Authors: Bruno Salvy and Paul Zimmermann Download: <http://www.loria.fr/~zimmerma/software/luoV2.1.tar.gz> Last modified: May 1995[34](#fn34) Website: <http://algo.inria.fr/libraries/libraries.html#luo> LUO is a software project started in the late 1980s designed to automatically analyze algorithms. It is no longer heavily used; most of its functionality is available in and the `equivalent` command in.[35](#fn35) In this document, we focus on the subset of its capabilities related to enumerative and analytic combinatorics, omitting a discussion of its capabilities for algorithm analyis. is made up of two modules: the algebraic analyzer (ALAS) and the analytic analyzer (ANANAS). ALAS takes as input a combinatorial specification, either labeled or unlabeled, and outputs a system of equations over generating functions for the classes in the specification. Then an intermediate process attempts to solve the equations explicitly; if successful, it passes the solutions to ANANAS. ANANAS then employs a routine (which became `equivalent` in gdev) on the expressions which returns asymptotic expressions for the coefficients. ALAS is described in further detail below, and ANANAS is described in Section [sec:ananas] Official documentation for LUO comes in the form of a main article discussing the algorithms used, but not the code, a cookbook containing a summary of and a selection of examples of algorithms being analyzed, and some notes on the code and usage of ALAS, including a reference for writing programs in the syntax that the system can analyze, and examples of ALAS in action. As mentioned, ALAS, written by Paul Zimmermann, is a tool for translating specifications to systems of equations over generating functions. That is, the user supplies a list of r equations $$\begin{aligned} \mathcal{A}^{<1>} &= \Phi\_1(\mathcal{A}^{<1>}, \dots, \mathcal{A}^{<r>}) \\ \mathcal{A}^{<2>} &= \Phi\_2(\mathcal{A}^{<1>}, \dots, \mathcal{A}^{<r>}) \\ &\cdots \\ \mathcal{A}^{<r>} &= \Phi\_r(\mathcal{A}^{<1>}, \dots, \mathcal{A}^{<r>}),\end{aligned}$$ where each Φi(⋯) represents an expression built from the A’s using the basic admissible constructions, as well as the classes E and Z. Some restricted constructions are also allowed. The valuation of a class A< i > is the minimum size of an object in A< i > built according to the specification. [defin:welldefined] A combinatorial specification is well defined iff it satisfies the two properties 1. each class has finite valuation, and 2. for each class A< i > and n ≥ 0, the number of objects of size n in A< i > built according to the specification is finite. Since there is nothing preventing the user from supplying a non-well defined specification, it would be nice if the well-definedness of a specification could be checked programmaticly and indeed it can and ALAS does this. See for details. Once ALAS verifies that the specification is well-defined, it proceeds to generate the corresponding equations over generating functions using simple replacement rules. These generating function equations can then be used to (among other things) compute initial values of the counting sequences. The following theorem is proved in : The number of arithmetic operations necessary for computing all the counting sequences associated with all classes A< i > up to size n is O(rn2). Pólya theory ============ Mathematical background ----------------------- Pólya theory is the study of counting symmetric objects, which makes use of group theory and generating functions. It was originally developed by John Redfield, then refounded by George Pólya, who used it to count chemical compounds, among other objects, and after whom the central theorem (Theorem [thm:polya] on page ) is named. An action of a group H on a set X is a homomorphism θ : H → S(X) from H to the symmetric group on X. In this section, the groups we work with are assumed to be the images of group actions on finite sets, i.e. permutation groups. Let G be such a group on a set X. Define an equivalence relation ≡ on X by x ≡ yiff ∃g ∈ G : g(x) = y. The set of of X under ≡ is written X/G. Elements of X/G are called the orbits of X under G, and the orbit of an element x ∈ X is the equivalence class represented by x and is denoted by G(x). The stabilizer Gx of an element x ∈ X is Gx = {g ∈ G : g(x) = x}. Our first lemmas bring counting into the picture: [Orbit-stabilizer] [lem:orbit] If G is a permutation group on X then for all x ∈ X, ∣G(x)∣ = ∣G∣/∣Gx∣. For any x ∈ X, consider the mapping f : G → X defined g ↦ g(x). Then there is a bijection between the image of f, which is G(x), and the set of left cosets of Gx, which has cardinality ∣G∣/∣Gx∣, given by h(x) ↦ hGx for all h(x) ∈ G(x). [Burnside] [lem:burnside] The number of orbits of a permutation group G on a set X is $$\| X / G \| = \frac{1}{\|G\|} \sum\_{g \in G} \operatorname{fix}(g),$$ where fix(g) is the number of fixed points of g. $$\begin{aligned} \sum\_{g \in G} \operatorname{fix}(g) &= \sum\_{g \in G} \sum\_{x \in X} [g(x) = x] \\ &= \sum\_{x \in X} \sum \_{g \in G} [g(x) = x] \\ &= \sum\_{x \in X} \| G\_x \|\\ &= \sum\_{x \in X} \frac{\|G\|}{\|G(x)\|} \\ &= \|G\| \sum\_{x \in X} \frac{1}{\|G(x)\|} \\ &= \|G\| \sum\_{A \in X/G} \sum\_{x \in A} \frac{1}{\|A\|} \\ &= \|G\| \sum\_{A \in X/G} \|A\| \frac{1}{\|A\|} \\ &= \|G\| \|X / G\| \qedhere\end{aligned}$$ A graph automorphism of a graph G = (V, E) is a bijection π : V → V such that {(π(v1), π(v2))∣(v1, v2) ∈ E} = E. The set of all automorphisms of G forms a group, aut(G). Say we are given a labeled graph G, and we would like to find the number of relabelings (bijections on the vertex set) that yield distinct graphs. Let X be set of all relabelings, and let θ be the induced action of aut(G) on X, i.e. if π ∈ X and g ∈ aut(G), then θ(g)(π) = (g(π1), g(π2), …). Then the number of such relabelings, which is the number of isomorphic graphs on the same vertex set, is ∣X/θ(aut(G))∣. (The packages nauty[36](#fn36), saucy[37](#fn37) and bliss[38](#fn38), as well as some of the packages in this section, can compute the automorphism group of a graph. The code that nauty uses is also included in GRAPE.[39](#fn39)) We now begin working up to the central theorem of Pólya theory: a generalization of Lemma [lem:burnside] for counting weighted colorings of an object. Let g ∈ G be a permutation, and let $ \boldsymbol{c}(g) $ be the of g. Then the cycle index of g is the formal monomial $$z(g; \boldsymbol{s} ) = \boldsymbol{s}^{\boldsymbol{c}(g) }.$$ The cycle index of the whole permutation group G is the terminating formal power series $$Z(G; \boldsymbol{s}) = \frac{1}{\|G\|} \sum\_{g \in G} z(g; \boldsymbol{s}).$$ Let Φ = {ϕ1, ϕ2, …} be a countable set of “colors”, each of which has a non-negative weight w(ϕi), such that w− 1(n) is finite for all n ≥ 0. We define the color-counting generating function a(t) = ∑n ≥ 0antn, where an = ∣w− 1(n)∣. Let F = ΦX be the set of functions from X → Φ; we call these functions colorings. The total weight of a coloring f ∈ F is ∑x ∈ Xw(f(x)). Define an action θ of G on F by g ↦ (f ↦ (f ∘ g− 1)), and let the coloring-counting generating function be b(t) = ∑n ≥ 0bntn, where bn is the number of orbits of θ(G) on F with total weight n. [lem:polya] Say ∣X∣ = m. Then the generating function for functions from X to Φ fixed by a permutation θ(g) with total weight marked by t is z(g; a(t), a(t2), …, a(tm)). A function is fixed by θ(g) iff it is constant on the cycles of g. So a function fixed by θ(g) is specified by, for each i, a mapping between each cycle of size i and Φ. For each cycle size i, these mappings have generating function a(ti)ci(g) since there are ci(g) cycles of size i and each color of weight k contributes a function of weight ki that is constant on the i-cycle. The overall generating function is thus $$a(t)^{c\_1(g)} a(t^2)^{c\_2(g)} \cdots a(t^m) = z(g; a(t), a(t^2), \dots, a(t^m) ). \qedhere$$ [Pólya enumeration theorem, single variable version] [page:polyatheorem] [thm:polya] Say ∣X∣ = m, then b(t) = Z(G; a(t), a(t2), …, a(tm)). Immediate from Lemmas [lem:burnside] and [lem:polya]. A necklace of size n is a cycle of colored beads that can be flipped over or rotated and still be considered the same object. In this example, we count black and white necklaces of size 4 with a given number of black beads. We can use as our set X = {1, 2, 3, 4}, and use as our permutation group on X the dihedral group D4 with 8 elements. We can let weight be the number of black beads, so that a(t) = 1 + t. Since D4 has cycle index $$Z(\mathcal{D}\_4; s\_1, s\_2, s\_3, s\_4) = \frac{s\_1^4+2 s\_1^2 s\_2+3 s\_2^2+2 s\_4}{8},$$ we have b(t) = Z(D4; 1 + t, 1 + t2, 1 + t3, 1 + t4) = 1 + t + 2t2 + t3 + t4. From this generating function, we can also read off that there are 6 necklaces of size 4 with (at most) two colors of bead, as shown in Figure [fig:necklaces], taken from MathWorld[40](#fn40). ![The six necklaces with four beads and two colors. [fig:necklaces]](necklaces)The six necklaces with four beads and two colors. [fig:necklaces] [exa:graphs] Suppose we would like to know the number of unlabeled graphs with 3 vertices and k edges, 0 ≤ k ≤ 3, up to isomorphism. Pólya’s enumeration theorem can be used if we represent unlabeled graphs with 3 vertices as 2-colorings of the set E of all $ \binom{3}{2} $ possible edges, where the color black represents an edge and the color white represents no edge. Total weight of a coloring equals the number of black edges, so our color-counting generating function is a(t) = 1 + t. The group of permutations on E that we need to quotient out is isomorphic to S3, the symmetric group on the three vertices. The group S3 has cycle index $$Z( \mathcal{S}\_3; s\_1, s\_2, s\_3) = \frac{s\_1^3 + 3 s\_1 s\_2 + 2 s\_3}{3!},$$ thus b(t) = Z(S3; 1 + t, 1 + t2, 1 + t3) = 1 + t + t2 + t3, and we see that there is exactly one graph, up to isomorphism, with 3 vertices and k edges, 0 ≤ k ≤ 3. Each of these is shown in Figure [fig:graphs]. ![The four unlabeled graphs with 3 vertices. [fig:graphs]](graphs)The four unlabeled graphs with 3 vertices. [fig:graphs] For more information on Pólya theory, a more general version of Theorem [thm:polya], and many more examples, see [41](#fn41). A note on Pólya theory packages ------------------------------- Our goal in this section is to describe packages designed for Pólya theory-type counting, rather than general computational group theory. Capable software for the latter includes GAP, Magma, Maple, Mathematica, and Sage. In the words of the authors of Combinatorica, “Our aim in introducting permutation groups into Combinatorica is primarily for solving combinatorial enumeration problems. We make no attempt to efficiently represent permutation groups or to solve many of the standard computational problems in group theory.” COCO ---- Download: <http://www.win.tue.nl/~aeb/ftpdocs/math/coco/coco-1.2a.tar.gz> COCO is a package for doing computations with permutation groups which was designed to investigate coherent configurations, which are a certain type of edge–colored complete graphs. COCO includes routines for, among other things, finding the automorphism group of an edge–colored complete graph and, given a base set and a permutation group on that set, computing the induced permutation group on a set of combinatorial structures over the base set. For more information, see its documentation and. Combinatorica: Pólya theory --------------------------- Authors: Sriram Pemmaraju and Steven Skiena Download: <http://www.cs.uiowa.edu/~sriram/Combinatorica/NewCombinatorica.m> Last modified: 2006 Website: <http://www.cs.sunysb.edu/~skiena/combinatorica/> Combinatorica has Pólya-theoretic functionality that integrates with the rest of its capabilities, such as basic combinatorial objects, a description of which, along with an overview of the Combinatorica package can be found in Section [sec:combinatoricabasicobjects]. Combinatorica has built-in rules representing the symmetric, cyclic, dihedral and alternating groups, as well as the ability to create groups from other groups or simply from a set of permutations. Combinatorica can compute the automorphism group of a graph. The rules `Orbits` and `OrbitRepresentatives` take a set and a permutation group, and optionally how the group acts on the set (the default action being the identity map), and return the set of orbits and representatives of those orbits, respectively. The `OrbitRepresentatives` rule can be used to list all distinct necklaces of size 4 with 2 colors. To do so, we evaluate the rule with the group `DihedralGroup[4]`, which is D4, and the set of words of length 4 over the letters R and B: `In[1]:= OrbitRepresentatives[DihedralGroup[4], Strings[{R, B}, 4]]` `Out[1]:= {{R,R,R,R},{B,B,B,B},{B,B,B,R},{B,B,R,R},{B,R,B,R},{B,R,R,R}}` The rule `CycleStructure` gives the cycle index of a single permutation, and `CycleIndex` gives the cycle index of a permutation group. Combinatorica comes with special rules for cycle index of symmetric, alternating, cyclic and dihedral groups which work much faster than `CycleIndex`. For Pólya’s enumeration theorem, we have the rule `OrbitInventory`, which takes the cycle index $ G( Z; \boldsymbol{s} ) $ of a group G and a list (w1, w2, …, wm) of expressions and returns G(Z; ∑i = 1mwi, ∑i = 1mwi2, …). Letting G = D4 and setting w1 = 1, w2 = t, we can obtain the generating function for 2-colored necklaces of size 4: `In[2]:= dihedralGroupCycleIndex = DihedralGroupIndex[4, x];` `colorEnumerator = {1, t};` `In[3]:= OrbitInventory[dihedralGroupCycleIndex, x, colorEnumerator]` `Out[3]:= 1 + t + 2 t^2 + t^3 + t^4` It turns out that Combinatorica has a built-in rule for this type of result: `In[4]:= NecklacePolynomial[4, colorEnumerator, Dihedral]` `Out[4]:= 1 + t + 2 t^2 + t^3 + t^4` As our last example of Combinatorica in action, we count the number of unlabeled graphs with 3 vertices. To apply Pólya’s theorem, we define the set of all 2-subsets of [1..3], representing all $ \binom{3}{2} $ edges: `In[5]:= set = KSubsets[Range[3], 2]` `Out[5]:= {{1,2},{1,3},{2,3}}` The group of permutations to quotient out is the set containing each permutation of `set` obtainable by applying a permutation g ∈ S3 to each 2-set in `set`. Combinatorica has the rule `KSubsetGroup` to create such a group: `In[6] := group = KSubsetGroup[SymmetricGroup[3], set];` We stated that this group is isomorphic to S3. We can prove this with the Combinatorica rule `MultiplicationTable`, which takes a set and an operation and gives the group multiplication table of the group they form: `In[7] := MultiplicationTable[SymmetricGroup[3], Permute] // TableForm` `Out[7]//Tableform=` $$\begin{array}{llllll} {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{6}}} \\ {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{4}}} \\ {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{5}}} \\ {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{2}}} \\ {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{3}}} \\ {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{1}}} \end{array}$$ `In[8] := MultiplicationTable[group, Permute] // TableForm` `Out[8]//Tableform=` $$\begin{array}{llllll} {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{6}}} \\ {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{4}}} \\ {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{5}}} \\ {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{2}}} \\ {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{1}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{3}}} \\ {{\fontshape{n}\texttt{6}}} & {{\fontshape{n}\texttt{5}}} & {{\fontshape{n}\texttt{4}}} & {{\fontshape{n}\texttt{3}}} & {{\fontshape{n}\texttt{2}}} & {{\fontshape{n}\texttt{1}}} \end{array}$$ In order to apply Pólya’s enumeration theorem, we use `CycleIndex` to compute the cycle index of `group`: `In[9] := cycleIndex = CycleIndex[group, x]` `Out[9] := x[1]^3/6 + x[1] x[2]/2 + x[3]/3` Now `OrbitInventory` can reproduce our result: `In[10] := OrbitInventory[cycleIndex,x,{1,t}]` `Out[10] := 1 + t + t^2 + t^3` We did not actually have to do all this to find the total number, since Combinatorica includes the rules `NumberOfGraphs` and `ListGraphs` to count and list all nonisomorphic graphs with a given number of vertices. `In[11] := NumberOfGraphs[3]` `Out[11] := 4` GraphEnumeration ---------------- Author: Doron Zeilberger Download: <http://www.math.rutgers.edu/~zeilberg/tokhniot/GraphEnumeration> Last Modified: July 20, 2010 Website: <http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/GE.html> This Maple package implements the Pólya-theoretic methods of to solve various counting problems of the following type: How many nonisomorphic graphs are there with n vertices, m edges, and some property P? The types of graphs that GraphEnumeration counts include unlabeled connected graphs according to the number of edges, unlabeled regular k-hyper-graphs according to edges, unlabeled rooted trees, unlabeled trees, all (connected or general) unlabeled simple graphs with a given degree sequence, and all (connected or general) unlabeled multi-graphs with a given degree sequence. PermGroup --------- Author: Thomas Bayer Last modified: June 21, 2004 Website: <http://www.risc.jku.at/research/combinat/software/PermGroup/> PermGroup is a Mathematica package for permutation groups, group actions and counting. Documentation is scant, but like Combinatorica, PermGroup has built-in rules for the most commonly used permutation groups, methods for creating custom groups, and ways to use them to solve counting problems involving symmetry. PermGroup implements the Schreier-Sims algorithm[42](#fn42) for computing group orders. The examples below illustrate some of PermGroup’s functionality, for more details consult the documentation and code. To obtain the generating function for 2-colored necklaces of length 4 with t marking the number of black beads, we can use the `DihedralGroupCIPoly` rule which returns the cycle index for the dihedral group. `In[1] := DihedralGroupCIPoly[4, x]` `Out[1] := 1/4 (x[1]^2 x[2] + x[2]^2) + 1/8 (x[1]^4 + x[2]^2 + 2 x[4])` Now we can replace xn with 1 + tn: In[2] := % /. {x[n\_] -> 1 + t^n} `Out[2] := 1/4 ((1 + t)^2 (1 + t^2) + (1 + t^2)^2) + 1/8 ((1 + t)^4 +` `(1 + t^2)^2 + 2 (1 + t^4))` Finally, we expand the polynomial and get something familar: In[3] := Expand@% `Out[3] := 1 + t + 2 t^2 + t^3 + t^4` To obtain the generating function for unlabeled graphs with 3 vertices, we first define the set of all 2-subsets of [1..3], representing all $ \binom{3}{2} $ edges: `In[4] := graphs3 = PermGroup\hspace{4pt}\grave{ }\hspace{2pt}PairSet[3]` `Out[4] := {{1,2},{1,3},{2,3}}` We let `s3` represent the symmetric group S3 using the `SymmGroup` rule: `In[5] := s3 = PermGroup\hspace{4pt}\grave{ }\hspace{2pt}Generate[PermGroup\hspace{4pt}\grave{ }\hspace{2pt}SymmGroup[3]];` Again, the group we really need is the induced group of `s3` on `graphs3`. PermGroup can give this to us with the `TransformGroup` rule: `In[6] := g3 = TransformGroup[s3, graphs3, PairSetAction];` Now we could compute the cycle index polynomial of `g3` and substitute in 1 + tn as above, but PermGroup offers the `PolyaEnumeration` rule to save some work. The expression `PolyaEnumeration[group, x, list]` evaluates to the cycle index of `group` in `x[1], x[2], …` with `x[i]` replaced with `Total[list^i]`. `In[7] := Expand[PolyaEnumeration[g3, x, {1, t}]]` `Out[7] := 1 + t + t^2 + t^3` Combinatorial species ===================== Mathematical background ----------------------- The theory of combinatorial species, “the most fruitful unifying concept in enumerative combinatorics of this quarter-century” according to Zeilberger,[43](#fn43) is a theoretical framework in the language of which a great diversity of families of combinatorial objects and their generating functions can be described. While the theory of species and symbolic combinatorics (the subject of Section [sec:symboliccombinatorics]) have a lot in common, one of the important differences is that in the theory of species, labeled and unlabeled objects are explicitly connected through Pólya theory. Martin Rubey, co-author of the Aldor-Combinat package described in the next section, said in 2008: > > Combstruct and MuPAD-Combinat really had “usability” as primary goal. As one consequence, they do not implement species, but rather “combinatorial classes”, that is, collections of objects with a size function. > > > The main drawback of that method is that you cannot treat labelled and unlabelled objects uniformly, there is no such concept as an isomorphism type, which, in my opinion, is the main strength of (ordinary) species.[44](#fn44) > > > Connections between symbolic combinatorics and Pólya theory have been made [45](#fn45), but they are not as central to the theory as they are in species. The article is “self-contained and can be used as a dictionary between the theory of species and the symbolic method of Flajolet and Sedgewick”. We begin with the definition of a species: A species (of structures) F is a pair of functions, 1. the first of which maps each finite set U to a finite set F[U], and 2. the second of which maps each bijection π : U → V to a function F[π] : F[U] → F[V]. The functions F[π] must satisfy the following properties: 1. for each pair of bijections π : U → V and τ : V → W, F[τ ∘ π] = F[τ] ∘ F[π], 2. and if IdU : U → U is the identity function, F[IdU] = IdF[U]. We note that if H is a group of permutations on U, then the second function of F is a group homomorphism from H to S(F[U]), i.e. an action of H on F[U]. We denote the image of this action by F[U; H]. An element s ∈ F[U] is called an F-structure on U, or alternatively a structure of species F* on U. The function F[π] is called the transport* of F-structures along π. If s ∈ F[U], we use the notation π ⋅ s to denote F[π](s). Note that ∣U∣ = ∣V∣ implies ∣F[U]∣ = ∣F[V]∣. Whereas in symbolic combinatorics we had the classes E and Z, in combinatorial species, the most elementary species are the empty set species E, where $$E[U] = \begin {cases} \{\emptyset\} & \text{if } \|U\|=0, \\ \emptyset & \text{otherwise}, \end {cases}$$ and the singleton species S, where $$S[U] = \begin {cases} \{u\} & \text{if } U=\{u\}, \\ \emptyset & \text{otherwise}. \end {cases}$$ Consider two F-structures s1 ∈ F[U] and s2 ∈ F[V]. A bijection π : U → V is called an isomorphism of s1 to s2 iff s2 = π ⋅ s1. If there is such a bijection, we write s1 ∼ s2 and we say these two structures have the same isomorphism type. The isomorphism types of a species are isomorphism classes and thus equivalence classes. An F-structure s ∈ F[U] on a set U can be referred to as a labeled structure, whereas an isomorphism type of F-structures can be referred to unlabeled structure. Attentive readers will be detecting a whiff of Pólya theory by now, which will become stronger as we define the generating functions associated with a species. The exponential generating function of a species F is the formal power series $$F(z) = \sum\_{n \geq 0} f(n) \frac{z^n}{n!},$$ where f(n) = ∣F[[1..n]]∣. [defin:t] Denote by T(Fn) the F[[1..n]]/F[[1..n]; Sn] of unlabeled F-structures on [1..n]. Then the isomorphism type generating function of a species F is the formal power series F̃(z) = ∑n ≥ 0f̃(n)zn, where f̃(n) = ∣T(Fn)∣. As in Lemma [lem:burnside], if π is a permutation, let fix(π) be the number of fixed points of π. Let Sn denote the group of permutations of [1..n]. Then the cycle index generating function of a species F is the formal power series $$Z\_F(\boldsymbol{z}) = \sum\_{n \geq 0}\frac{1}{n!} \sum\_{\pi \in S\_n} \operatorname{fix}(F[\pi]) \boldsymbol{z}^{\boldsymbol{c}(\pi)}.$$ For any species F we have 1. F(z) = ZF(z, 0, 0, …) 2. F̃(z) = ZF(z, z2, z3, …). We prove each part separately. 1. We have $$\begin{aligned} Z\_F(z,0,0,\dots) &= \sum\_{n \geq 0}\frac{1}{n!} \sum\_{\pi \in S\_n} \operatorname{fix}(F[\pi]) z^{c\_1(\pi)} 0^{c\_2(\pi)} 0^{c\_3(\pi)} \cdots \\ &= \sum\_{n \geq 0} \frac{1}{n!} \operatorname{fix}(F[\operatorname{Id}\_{[1..n]}]) z^n \\ &= \sum\_{n \geq 0} f(n) \frac{z^n}{n!}.\end{aligned}$$ 2. Now, $$\begin{aligned} Z\_F(z, z^2, z^3, \dots ) &= \sum\_{n \geq 0}\frac{1}{n!} \sum\_{\pi \in S\_n} \operatorname{fix}(F[\pi]) z^{c\_1(\pi)} z^{2 c\_2(\pi)} z^{3 c\_3(\pi)} \cdots \\ &= \sum\_{n \geq 0}\frac{1}{n!} \sum\_{\pi \in S\_n} \operatorname{fix}(F[\pi]) z^n\end{aligned}$$ Let G = F[[1..n], Sn] be the image of the action of Sn on F[[1..n]] given by the second function of F. Then we have $$\begin{aligned} \sum\_{n \geq 0}\frac{1}{n!} \sum\_{\pi \in S\_n} \operatorname{fix}(F[\pi]) z^n &= \sum\_{n \geq 0}\frac{1}{n!} \frac{n!}{\|G\|} \sum\_{\operatorname{fix}(F[\pi]) \in G} \operatorname{fix}(F[\pi]) z^n \\ &= \sum\_{n \geq 0}\frac{1}{\|G\|} \sum\_{\tau \in G} \operatorname{fix}(\tau) z^n \\ &= \sum\_{n \geq 0} \tilde{f}(n) z^n,\end{aligned}$$ where the last equality follows from Lemma [lem:burnside]. [defin:speciesequality] Let F and G be two species. An isomorphism of F to G is a family of bijections bU : F[U] → G[U] which satisfies the following condition: for any bijection π : U → V between two finite sets and any F-structure s ∈ F[U], we must have π ⋅ bU(s) = bV(π ⋅ s). The two species are then said to be isomorphic and we write F = G. As with combinatorial classes from symbolic combinatorics, there is a set of operations on species which have corresponding operations on their generating functions (all three types). These operations include addition, multiplication, composition, differentiation, pointing, cartesian product, and functorial composition, just as in symbolic combinatorics. For more information on these, and all other parts of the theory of species, see the 457-page book devoted to the subject. The species analogs of combinatorial specifications are systems of equations, which can be created using the operators just mentioned and the notion of equality from Definition [defin:speciesequality]. ### Weighted species The species equivalent of multiple parameters on combinatorial classes is weighted species, which we briefly discuss. A weighting function w : F[U] → R[t] maps objects in F[U] to monomials in t over a ring R ⊆ C, and an R[t]-weighted set is a pair (S, w), where w : S → R[t] is a weighting function on S. For such a pair (S, w), let ∣S∣w = ∑s ∈ Sw(s). Let R ⊆ C be a ring. An R[t]-weighted species F is a pair of functions, 1. the first of which maps each finite set U to an R[t]-weighted set (F[U], wU) such that ∣F[U]∣wU converges, and 2. the second of which maps each bijection π : U → V to a function F[π] : (F[U], wU) → (F[V], wV), that preserves weights. The functions F[π] must satisfy the following properties: 1. for each pair of bijections π : U → V and τ : V → W, F[τ ∘ π] = F[τ] ∘ F[π], 2. and if IdU : U → U is the identity function, F[IdU] = IdF[U]. Let F be a weighted species, with weight functions wn : F[[1..n]] → R[t], for n ≥ 0. Then the exponential generating function of F is $$F\_w(z) = \sum\_{n \geq 0} \|F[[1..n]]\|\_{w\_n} \frac{z^n}{n!};$$ the isomorphsim type generating function of F is F̃w(z) = ∑n ≥ 0∣T(Fn)∣wzn, where T(Fn) is defined as in Definition [defin:t]; and the cycle index generating function of F is $$Z\_{F\_w}( \boldsymbol{z} ) = \sum\_{n \geq 0} \frac{1}{n!} \left( \sum\_{ \pi \in \mathcal{S}\_n} \|\operatorname{Fix}(F[\pi])\|\_{w\_n} \boldsymbol{z}^{\boldsymbol{c}(\pi)} \right),$$ where, if π ∈ Sn, Fix(F[π]) is the set of F-structures on [1..n] fixed by F[π]. Operations on weighted species can be defined similarly to the unweighted case. For more information, see. Let L≥ 1 be the species of linear orders resricted to sets of at least one element; let S be the singleton species with weight 1; and let W be the singleton species with weight q. Then the species T of ordered trees with number of internal nodes marked by q satisfies T = S + W ⋅ (L≥ 1 ∘ T) where + is addition (the species analog of sum), ⋅ is multiplication (the analog of product), and ∘ is composition (the analog of substitution). Two notable packages for species, Darwin, and Devmol have been described in the literature, but no longer exist. Aldor-Combinat -------------- Authors: Ralf Hemmecke and Martin Rubey Website: <https://portal.risc.jku.at/Members/hemmecke/aldor/combinat> A project which started in 2006, Aldor-Combinat is an unfinished package primarily for working with species, which is based on MuPAD-Combinat’s implementation of symbolic combinatorics. It is written in the Aldor language to be used with the Axiom[46](#fn46) computer algebra system. The only included documentation contains many of the implementation details of the package, but it is designed to be read by Aldor developers.[47](#fn47) Like Sage uses for basic combinatorial objects, Aldor-Combinat uses a category-theoretic model to organize its functionality. All species in Aldor-Combinat are objects in the `CombinatorialSpecies` category and they must implement methods returning exponential generating series, isomorphism type generating series, cycle index generating series, a listing of structures, and a listing of isomorphism types. Aldor-Combinat has the following species predefined: empty set, set, singleton, linear order, cycle, permutation, and set partition (whose isomorphism types are integer partitions). We can compute the initial coefficients of the exponential generating function of the singleton species: `L == Integer;` `E == EmptySetSpecies L;` `gse: OrdinaryGeneratingSeries := generatingSeries $ E;` `import from Integer;` `le: List Integer := [coefficient(gse, n) for n in 0..3];` This assigns `[1, 0, 0, 0]` to `le`. We can compute something similar for the singleton species: `M == Integer;` `S == SingletonSpecies M;` `gss: ExponentialGeneratingSeries := generatingSeries $ S;` `import from Integer;` `ls: List Integer := [coefficient(gss, n) for n in 0..3];` This assigns `[0, 1, 0, 0]` to `ls`. Let F be a species. Define, for each n ≥ 0, the species F* restricted to n* by $$F\_n[U] = \begin{cases} F[U] & \text{if } \|U\|=n, \\ \emptyset & \text{otherwise}, \end{cases}$$ for all finite sets U. Aldor-Combinat allows species to be restricted with `RestrictedSpecies`, which is useful for defining other species. Species may be defined implicitly or explicitly with the following operations: addition, multiplication, composition, and functorial composition. Sage: combinatorial species --------------------------- Download: <http://www.sagemath.org/download.html> Website: <http://sagemath.org/doc/reference/combinat/species.html> This section covers Sage’s species functionality. For information on Sage’s basic combinatorial objects functionality and an overview Sage as a CAS, see Section [sec:alas]. Sage’s capabilities for working with combinatorial species, which began development around 2008, are based on those in Aldor-Combinat (covered in the previous section). A project roadmap describes future plans for the project.[48](#fn48) Sage offers built-in classes representing the cycle, partition, permutation, linear-order, set, and subset species. Also, the `CharacteristicSpecies(n)` method returns the characteristic species on `n`, which yields one object on sets of size `n`, and no objects on any other set (i.e. the restriction to `n` of the set species X, defined X[U] = U). We assign an empty set species to `E`: `sage: E = species.EmptySetSpecies()` We list all structures of `X` on the empty set and the set {1, 2}: `sage: E.structures([]).list()` `[{}]` `sage: E.structures([1,2]).list()` `[]` We find the first four coefficients of the exponential generating function of `X`: `sage: E.generating_series().coefficients(4)` `[1, 0, 0, 0]` Since the empty set species is isomorphic to the characteristic species on 0, we get identical output using `CharacteristicSpecies(0)`: `sage: C0 = species.CharacteristicSpecies(0)` `sage: C0.structures([]).list()` `[{}]` `sage: C0.structures([1,2]).list()` `[]` `sage: C0.generating_series().coefficients(4)` `[1, 0, 0, 0]` The characteristic species on 1 is isomorphic to the singleton species: `sage: C1 = species.CharacteristicSpecies(1)` `sage: C1.structures([1]).list()` `[1]` `sage: C1.structures([1,2]).list()` `[]` `sage: C1.generating_series().coefficients(4)` `[0, 1, 0, 0]` The operations on species of addition, multiplication, composition, and functorial composition are supported. Species can be defined explicitly or implicitly (`define`). When species are created, a weight can be specified, as well as a restriction on the size of the set. Each species object `B` implements a number of useful methods: `B.structures(set)` is the set of `B`-structures on the set `set`, `B.isotypes(set)` is the set of equivalence classes of `B` structures (isomorphism types) on the set `set`, `cycle_index_series()` is the cycle index series, `generating_series()` is the exponential series, and `isotype_generating_series()` is the isomorphism type series. We wish to count unlabeled ordered trees by total number of nodes and number of internal nodes. To achieve this, we begin by assigning a weight of 1 to the leaves and q to internal nodes, which are each singleton species: `sage: q = QQ[’q’].gen()` `sage: leaf = species.SingletonSpecies()` `sage: internal_node = species.SingletonSpecies(weight=q)` Now we define a species `T` representing the trees, defined as `leaf + internal_nodeL(T)`, where `L` is the species of linear orders restricted to sets of size 1 or greater: `sage: L = species.LinearOrderSpecies(min=1)` `sage: T = species.CombinatorialSpecies()` `sage: T.define(leaf + internal_nodeL(T))` All that remains, since the trees are unlabeled, is to compute the coefficients of the isomorphism type generating function: `sage: T.isotype_generating_series().coefficients(6)` `[0, 1, q, q^2 + q, q^3 + 3q^2 + q, q^4 + 6q^3 + 6q^2 + q]` Further examples may be found in a Sage species demo by Mike Hansen, one of the developers.[49](#fn49) Asymptotics =========== Mathematical background ----------------------- ### Asymptotic scales and series Let P* be an open set and let L be a limit point of P. Let SP be a set of functions from P to C. Then the set SP is an asymptotic scale at L iff for every pair of different functions ϕ1, ϕ2 ∈ SP, the limit of ϕ1(x)/ϕ2(x) as x → L is either 0 or + ∞. [exa:polys] If P = Z≥ 0 ∪ {∞}, L = ∞, then an asymptotic scale is SP = {n ↦ 1, n ↦ n, n ↦ n2, …}. The following definition, which comes from (as does the previous definition), generalizes Definition [def:asymptoticseries] for terminating series: Let SP be an asymptotic scale at L and let f be a complex valued function on P. Then f is said to admit a terminating asymptotic series at L iff there exists a sequence (a1, …, an) of complex numbers and a sequence (ϕ1, …, ϕn) of elements of SP such that $$\forall { i \in \{1,\dots, n \}, \phi \in S\_P}: \qquad \phi\_i(x) = o(\phi(x)) \implies f(x) - \sum\_{j=1}^{i} a\_j \phi\_j(x) = o(\phi(x)) \text{ as } x \rightarrow L.$$ With P, L, and SL defined as in Example [exa:polys], let $f\_k(n) = [z^n] \frac{z^k}{(1-z)^k}$ be the number of integer compositions of n with k parts. Then a terminating asymptotic series for fk(n) at L consists of the complex numbers $ \left( \frac{1}{(k-1)!}, -\frac{k(k-1)}{2(k-1)!} \right) $ and functions (n ↦ nk − 1, n ↦ nk − 2) since $$f\_k(n) = \frac{n^{k-1}}{(k-1)!} - \frac{k(k-1) n^{k-2}}{2 (k-1)!} + o(n^{k-2}).$$ Various CASes offer support for generating and working with asymptotic series expansions, including Maple, Mathematica, Maxima, and MATLAB’s Symbolic Math Toolbox (the current product containing the code of MuPAD)[50](#fn50); in this section we include a couple of implementations of special interest. ### Singularity analysis Singularity analysis is a method due to Flajolet and Odlyzko which shows how to compute asymptotic expressions for the coefficients of some classes of generating functions. In order to apply the theorem, we forget about formal power series and treat generating functions as elements of Cω(C; 0) instead of Q[[z]]. In its simplest form, the method can be expressed as the following theorem: [Singularity analysis] Fix the range of Arg to be [ − π, π). Let F ∈ Cω(C; 0) be a function analytic in a domain $$D = \left\{z : \|z\| \leq s\_1, \|\operatorname{Arg}(z-s)\| > \frac{\pi}{2} - \eta \right\},$$ where s, s1 > s, and η are three positive real numbers. Assume that, with σ(u) = uαlogβu and $ \alpha \notin \{0, -1, -2, \dots \} $, we have $$F(z) \sim \sigma \left( \frac{1}{1-z/s} \right) \qquad \text{ as } z \rightarrow s \text{ in } D.$$ Then, as n → ∞, the Maclaurin coefficients of F satisfy $$[z^n]F(z) \sim s^{-n} \frac{\sigma(n)}{n \Gamma(\alpha)}.$$ Generalizations are discussed at length in. ### Saddle-point asymptotics The saddle-point method is another way of computing asympotic expressions for the coefficients of generating functions. In, W.K. Hayman defined a set of functions called the H-admissible functions for which the following theorem holds: If F is a function defined around the origin by F(z) = ∑n ≥ 0f(n)zn and F is H-admissible, then $$f(n) \sim \frac{F(r(n))}{r(n)^n \sqrt{2 \pi b(r(n))}}, \text{ as } n \rightarrow \infty$$ where r(n) is the smallest positive root of r(n)Fʹ(r(n))/F(r(n)) = n and b(r) = rDr(rFʹ(r)). Useful information on which functions are H-admissible is given by the following theorem (also from ): The following are three properties of H-admissible functions. 1. Let α and β1 be positive real numbers, and let β2 and β3 be real numbers. Then F defined by $$F(z) = \exp \left( \beta\_1 (1-z)^{-\alpha} \left( \frac{1}{z} \log \frac{1}{1-z} \right)^{\beta\_2} \left( \frac{2}{z} \log \left(\frac{1}{z} \log\frac{1}{1-z} \right) \right)^{\beta\_3} \right)$$ is H-admissible. 2. If F and G are H-admissible, and P is a polynomial function with real coefficients and positive leading coefficient, then exp(F), F + G, F + P, P(F), and P ⋅ F are H-admissible. 3. If P is a polynomial function such that P(z) cannot be written as P(Q(zk)) for any polynomial function Q and k > 1, then exp(P) is H-admissible. The saddle-point method are discussed in further detail in. ### Power series coefficients and the radius of convergence [from ] Given a sequence (a0, a1, a2, …) of rational numbers and r ∈ [0, ∞], we write an ⋈ rn iff limsupn ≥ 0∣an∣1/n = r, where 1/∞ = 0, 1/0 = ∞. Given a sequence (a0, a1, a2, …) of rational numbers and r ∈ [0, ∞], we have an ⋈ riff an ∼ rnc(n), where limsup∣c(n)∣1/n = 1. In other words fn ⋈ r iff 1/r is the radius of convergence of ∑n ≥ 0anzn. Given a function F with Maclaurin series expansion F(z) = ∑n ≥ 0f(n)zn having radius of convergence r, with a finite set of singularities at distance r from the origin, the dominant directions of F is the set Θ where θ ∈ Θ ⇔ θ ∈ [ − π, π)and Fis singular at reiθ. Gdev ---- Author: Bruny Salvy Download: <http://algo.inria.fr/libraries/#down> Webpage: <http://algo.inria.fr/libraries/#gdev> Last modified: March 2003[51](#fn51) Gdev is a package for computing asymptotic expressions for functions and generating function coefficients, which, except for the `equivalent` function (described below), has been superseeded by the MultiSeries package (see Section [sec:multiseries]).[52](#fn52). For more information about the capabilities of gdev see an article written by Bruno Salvy. Many packages provide routines for computing Taylor expansions, but the capabilities of gdev’s function `gdev` go well beyond that particular asymptotic scale. The function `gdev` takes a function given by an explicit expression and, automatically choosing the asymptotic scale to use, finds an asymptotic expansion around a point, which ends with a big O term. The function `equivalent`, originally in from LUO, extracts an asymptotic expression for the coefficients of an explictly given generating function using singularity analysis or the saddle-point method. See the next section for more information on the underlying algorithm. Lambda-Upsilon-Omega (LUO): asymptotics --------------------------------------- Authors: Bruno Salvy and Paul Zimmermann Download: <http://www.loria.fr/~zimmerma/software/luoV2.1.tar.gz> Last modified: May 1995 Website: <http://algo.inria.fr/libraries/libraries.html#luo> This section covers ANANAS, the component of LUO for extracting asymptotic information from generating functions. To find out about LUO’s other component, ALAS, and how ANANAS fits in to the LUO system, see Section [sec:alas]. Once ALAS produces a system of equations for A< 1 >(z), …, A< r >(z), an attempt to solve it is made. If an explicit expression is obtained, it is passed to ANANAS, written by Bruno Salvy. Not all explicit expressions can be handled by ANANAS, however — in particular, not all expressions produced by ALAS. The set E of functions is the set of generating functions corresponding to labeled classes defined by well-defined explicit specifications. That is, the functions containing the monomial functions z ↦ 1 and z ↦ z, and closed under the operations { + , × , Q, L, E}, where $$Q(F) = \frac{1}{1-F}, \qquad L(F) = \log\frac{1}{1-F}, \qquad E(F) = \exp(F).$$ ANANAS can handle some functions not in E, but for the rest of this section, we only consider ANANAS’s behavior on functions in E, since the documentation focuses on this case. The set E is subdivided into three disjoint sets $$\mathscr{E} = \mathscr{E}\_{AL} \hspace{3pt} \dot{\cup} \hspace{3pt} \mathscr{E}\_{\text{entire}} \hspace{3pt} \dot\cup \hspace{3pt} \mathscr{E}\_{\text{other}},$$ where Eentire is the set of entire functions and EAL is the set of functions with algebraic-logarithmic (AL) growth, meaning that F ∈ EAL iff F has radius of convergence 0 < r < ∞ and there exist α ∈ R, k ∈ Z≥ 0 such that, $$F(z) \sim \frac{1}{(1 -z/r)^\alpha} \log^k \frac{1}{1 -z/{r}} \qquad \text{ as } z \rightarrow {r^-}.$$ ANANAS determines which subset of E a function is a member of and proceeds accordingly. For the details of the steps involved in this choice, see. ANANAS is able to handle some functions from Eentire and Eother using the saddle-point method, but the coverage is not as complete as for EAL, where singularity analysis is very effective. We briefly describe the algorithm used for the class EAL; for information on what ANANAS does with Eentire and Eother, see. Given a function in EAL, ANANAS performs the algorithm Equivalent: F is an expression corresponding to a function in EAL 3.1 Obtain an algebraic-logarithmic asymptotic expansion of F around r 3.2 Apply singularity analysis 4.1 Apply a singularity-analysis style result for functions with more than one dominant singularity Step 1 turns out to be simple; it takes the form of the Radius algorithm: F is an expression corresponding to a function in EAL For more details on the rest of Equivalent, see. MultiSeries ----------- Authors: Bruno Salvy and Maplesoft Download: <http://algo.inria.fr/libraries/#down> Website: <http://www.maplesoft.com/support/help/Maple/view.aspx?path=MultiSeries> Originally developed by Bruno Salvy at the INRIA Algorithms Group and now built into Maple releases, the MultiSeries Maple package provides functionality for computing asymptotic series expansions that superseeds that provided in gdev, the subject of Section [sec:gdev]. MultiSeries contains the function `multiseries`, the sucessor to gdev’s `gdev`, and four special cases of `multiseries` that cover the same functionality of core Maple functions: `asympt`, `series`, `limit`, and `taylor`. According to the Maple documentation, “the simplest use of the package is by overriding the standard `asympt`, `series`, `limit` using `with`. The corresponding MultiSeries functions are often more powerful than the default ones, but require more computational time.” (Presumably `taylor` can be used in the same way.) The `multiseries` function can take as arguments an expression `expr`, the limit point `a`, the variable `x` in the expression tending to `a`, and the truncation order `n`. It returns, like `gdev`, an asymptotic expansion of `expr` as `x` tends to `a` ending in a big O term, in terms of an asymptotic scale automatically chosen. It is also possible to specify some extra options such as choosing the path of approach, and there is support for choosing the asymptotic scale used.[53](#fn53) Conclusions =========== A number of kinds of conclusions can be made from this document. First, recall that we with the claim that “[it is no longer] necessary to use error-prone pen and paper methods to perform an ever-growing set of mathematical procedures”. Indeed, the packages in this document cover a lot of ground, and much of the progress is recent, but there is massive potential for more work both with mathematical algorithms that have been implemented and those that have not. Note that there are few packages available for any one CAS/programming language. The average mathematician cannot be expected to be familiar with more than one system, if any. There is therefore a need for projects to be ported to systems without those capabilities, preferably the big three CASs: Maple, Mathematica and Sage. Next, future package-writers can note that not all of the packages in this document come with sufficient documentation for all uses. To be maximally helpful, two forms of documentation should be available: one for mere users, and one for developers or other people interested in the implementation details. Users want to know what the package can do and how to do those things. Developers wishing to port or extend the software want to know the algorithms and code used. Lastly, as casual observations, we can say that learning how to use a package can be a good way to learn a mathematical concept or increase one’s understanding, and also that these packages are for the most part not huge collaborative efforts — often there is only one author, sometimes two. Writing symbolic packages is not trivial, since it requires skills from multiple disciplines, namely mathematics, computer science, and software engineering. However, it is clear that the future of the field is bright, and the global project to replace pen and paper, and ultimately more and more of the problem solving, theorem proving and question answering now done only by human mathematicians will continue to progress and advance significantly in the years to come. Acknowledgements ================ The author thanks Daniel Panario and Brett Stevens for their help with this document. Appendix ======== Maple 16 -------- Maple is a modern commercial CAS whose development began in 1980 by the Symbolic Computation Group at the University of Waterloo, and, since 1988, has been developed and sold by Waterloo Maple Inc., a.k.a. Maplesoft. The most recent version as of time of writing is Maple 16, released in 2012. Mathematica 8 ------------- Mathematica, another modern commercial CAS, is developed by Wolfram Research. It began as a project called Symbolic Manipulation Program created by Stephen Wolfram around 1979 at Caltech, and then after Wolfram Research was founded in 1987, was renamed Mathematica. Version 1.0 of Mathematica was released in 1988; the most recent release was Mathematica 8, in 2010. Mathematica is a Turing complete term rewriting system, which means that all Mathematica programs compute by manipulating expressions (terms). When an expression is evaluated by Mathematica, it searches through a sequence of stored pattern-replacement rules (in which form Mathematica’s mathematical routines exist) looking for a match between a pattern and a subexpression of the expression. When a rule’s pattern matches a subexpression, that subexpression is replaced according to the rule, and the new expression is evaluated. For more on term rewriting systems, see, and for more on Mathematica programming, see its documentation and. --- 1. <http://www.math.rutgers.edu/~zeilberg/Opinion36.html>[↩](#fnref1) 2. <http://en.wikipedia.org/wiki/Predictions_made_by_Ray_Kurzweil>[↩](#fnref2) 3. <http://singinst.org/research/publications>[↩](#fnref3) 4. <http://www.existential-risk.org/>[↩](#fnref4) 5. <http://planetmath.org/encyclopedia/Combinatorics.html>[↩](#fnref5) 6. <http://web.mit.edu/~qchu/Public/TopicsInGF.pdf>[↩](#fnref6) 7. <http://faculty.valpo.edu/lpudwell/maple.html>[↩](#fnref7) 8. <http://www-cs-staff.stanford.edu/~knuth/programs.html>[↩](#fnref8) 9. <http://www.math.rutgers.edu/~zeilberg/programs.html>[↩](#fnref9) 10. <http://www.risc.jku.at/research/combinat/software/Omega/index.php>[↩](#fnref10) 11. <http://www.math.rutgers.edu/~zeilberg/tokhniot/LinDiophantus>[↩](#fnref11) 12. <http://www-irma.u-strasbg.fr/~guoniu/software/omega.html>[↩](#fnref12) 13. <http://www.risc.jku.at/research/combinat/software/GenOmega/index.php>[↩](#fnref13) 14. <http://www.risc.jku.at/research/combinat/software/RLangGFun/index.php>[↩](#fnref14) 15. <http://algo.inria.fr/libraries/libraries.html#regexpcount>[↩](#fnref15) 16. <http://www.math.rutgers.edu/~zeilberg/programs.html>[↩](#fnref16) 17. <http://planetmath.org/encyclopedia/DictionaryOrder.html>[↩](#fnref17) 18. <http://www.maplesoft.com/support/help/Maple/view.aspx?path=combinat>[↩](#fnref18) 19. <http://www.sagemath.org/doc/reference/sage/categories/category.html>[↩](#fnref19) 20. <http://www.sagemath.org/doc/reference/sage/categories/primer.html>[↩](#fnref20) 21. <http://mupad-combinat.sourceforge.net/>[↩](#fnref21) 22. <http://www.sagemath.org/doc/reference/sage/categories/enumerated_sets.html>[↩](#fnref22) 23. <http://sagemath.org/doc/reference/sage/combinat/partition.html>[↩](#fnref23) 24. <http://sagemath.org/doc/reference/sage/combinat/combinat.html>[↩](#fnref24) 25. <http://en.wikipedia.org/wiki/Symbolic_combinatorics>[↩](#fnref25) 26. <http://en.wikipedia.org/wiki/Disjoint_union>[↩](#fnref26) 27. <http://algo.inria.fr/index.html>[↩](#fnref27) 28. <http://www.maplesoft.com/support/help/Maple/view.aspx?path=combinat>[↩](#fnref28) 29. <http://algo.inria.fr/libraries/autocomb/>[↩](#fnref29) 30. <http://algo.inria.fr/salvy/index.html>[↩](#fnref30) 31. <http://algo.inria.fr/encyclopedia/intro.html>[↩](#fnref31) 32. <http://www.research.att.com/~njas/sequences/>[↩](#fnref32) 33. <http://algo.inria.fr/libraries/#gfun>[↩](#fnref33) 34. <http://algo.inria.fr/salvy/index.html>[↩](#fnref34) 35. <http://algo.inria.fr/salvy/index.html>[↩](#fnref35) 36. <http://cs.anu.edu.au/~bdm/nauty/>[↩](#fnref36) 37. <http://vlsicad.eecs.umich.edu/BK/SAUCY/>[↩](#fnref37) 38. <http://www.tcs.hut.fi/Software/bliss/>[↩](#fnref38) 39. <http://www.gap-system.org/Packages/grape.html>[↩](#fnref39) 40. <http://mathworld.wolfram.com/Necklace.html>[↩](#fnref40) 41. <http://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem>[↩](#fnref41) 42. <http://en.wikipedia.org/wiki/Schreier-Sims_algorithm>[↩](#fnref42) 43. <http://www.math.rutgers.edu/~zeilberg/khaver.html>[↩](#fnref43) 44. <http://www.mail-archive.com/[email protected]/msg00503.html>[↩](#fnref44) 45. <http://www.mathematik.uni-stuttgart.de/~riedelmo/papers/collier.pdf>[↩](#fnref45) 46. <http://axiom-developer.org/>[↩](#fnref46) 47. <http://www.risc.jku.at/people/hemmecke/AldorCombinat/combinat.html>[↩](#fnref47) 48. <http://trac.sagemath.org/sage_trac/ticket/10662>[↩](#fnref48) 49. <http://sage.math.washington.edu/home/mhansen/CombinatorialSpeciesDemo.html>[↩](#fnref49) 50. <http://www.mathworks.com/help/toolbox/mupad/stdlib/asympt.html>[↩](#fnref50) 51. <http://algo.inria.fr/salvy/index.html>[↩](#fnref51) 52. <http://algo.inria.fr/salvy/index.html>[↩](#fnref52) 53. <http://www.maplesoft.com/support/help/Maple/view.aspx?path=MultiSeries%2fmultiseries>[↩](#fnref53) Software for enumerative and analytic combinatorics =================================================== ### 2013 We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, Pólya theory, combinatorial species, and asymptotics. We describe the capabilities that the packages offer as well as some of the algorithms used, and provide links to original documentation. Most of the packages are freely downloadable from the web. Note: In this document, to refer to webpages we place URL links in footnotes, and for all other types of referent we use standard endnote references. Introduction ============ In an opinion article[1](#fn1) posted to his website in 1999, Doron Zeilberger challenged mathematicians to rethink the role of computers in mathematics. “Everything that we can prove today will soon be provable, faster and better, by computers,” he says, then gives the following advice: > > The real work of us mathematicians, from now until, roughly, fifty years from now, when computers won’t need us anymore, is to make the transition from human-centric math to machine-centric math as smooth and efficient as possible. …We could be much more useful than we are now, if, instead of proving yet another theorem, we would start teaching the computer everything we know, so that it would have a headstart. …Once you learned to PROGRAM (rather than just use) Maple (or, if you insist Mathematica, etc.), you should immediately get to the business of transcribing your math-knowledge into Maple. > > > If futurist Ray Kurzweil’s predictions for artificial intelligence progress[2](#fn2) are to be believed, Zeilberger’s suggestions will turn out to be sound. (However, utilitarians would urge us to consider the risks such technology would present.[3](#fn3) [4](#fn4)) What is certain even today is that those who use mathematics, given the rise of computers, can ask themselves if they are failing to capitalize on a productive division of labor between man and machine. is no longer necessary to spend three years of Sundays factoring the Mersenne number 267 − 1, like F. N. Cole did in the 1900s, and neither is it necessary to use error-prone pen and paper methods to perform an ever-growing set of mathematical procedures. To illustrate this statement, this document examines symbolic computation (a.k.a. computer algebra) software packages for enumerative and algebraic combinatorics. We start, in Section [sec:Enumerativeandanalyticcombinatorics], with an overview of the fields of enumerative and analytic combinatorics. Then we go into more detail on the scope of the document in Section [sec:notes]. In Sections [sec:basiccombinatorialobjects]–[sec:asymptotics] we cover packages relating to basic combinatorial objects, symbolic combinatorics, Pólya theory, combinatorial species, and asymptotics. Finally, we offer concluding observations and remarks in Section [sec:conc]. Enumerative and analytic combinatorics ====================================== Welcome to the fields of enumerative and analytic combinatorics, a.k.a. combinatorial and asymptotic enumeration! In order to completely cover what mathematicians think of when they think of these fields (and to avoid saying “enumerative” or “combinatorial”), we break up our discussion into two parts which we call counting, the more mathematical side, and enumeration, the more algorithmic side. Counting -------- Counting, the oldest mathematical subject, is enumeration in the mathematical sense: the study of the cardinalities of finite sets. The most basic principle of counting is the addition rule : [Addition rule] If S is a set and A1, A2, …, An is a partition of S, then ∣S∣ = ∣A1∣ + ∣A2∣ + ⋯ + ∣An∣. Other similar basic ways of counting may be familiar from introductory probability, and indeed, many concepts overlap between counting and discrete probability.[5](#fn5) Elements of the body of work on counting can be roughly categorized based on three criteria: whether they deal with exact or asymptotic results, whether they speak in terms of generating functions or their coefficients, and whether they make use of bijections or manipulations. ### Exact vs. asymptotic counting Boldface symbols refer to (possibly terminating) 1-based sequences, i.e. $ \boldsymbol{a} = (a\_1, a\_2, \dots) $. Generally, counting problems involve a triple $ (S, \boldsymbol{p}, N) $ comprising a countable set S of objects, a sequence $ \boldsymbol{p} = (p\_1, p\_2, \dots) $ of functions pi : S → Z≥ 0, and a set N of sequences $ \boldsymbol{n} $ for which $ \boldsymbol{p}^{-1}(\boldsymbol{n}) = \{ s \in S : p\_1(s) = n\_1, p\_2(s) = n\_2, \dots \} \subseteq S$ is finite. The problem is to answer the question “For $\boldsymbol{n} \in N$, how many objects does the set $ \boldsymbol{p}^{-1}(\boldsymbol{n}) $ contain?” The definition of an exact answer was given by Herb Wilf: a polynomial-time algorithm that computes the number. [exa:signature] If π ∈ Sn is a permutation, then let cj(π) be the number of cycles of π with size j. The signature of π is $ \boldsymbol{c}(\pi) = (c\_1(\pi), c\_2(\pi), \dots) $. Let S be the set of all permutations, and for s ∈ S, let $ \boldsymbol{p}(s) $ be the signature of s. Let n ≥ 1 and let $ \boldsymbol{n} = ($[[](http://en.wikipedia.org/wiki/Iverson_bracket)j = n])j ≥ 0. Then $ \|\boldsymbol{p}^{-1}(\boldsymbol{n})\| = (n-1)! $. The expression (n − 1)! immediately suggests a polynomial-time algorithm to compute $ \|\boldsymbol{p}^{-1}(\boldsymbol{n})\| $. Exact answers can be classified by ansatz, meaning the form of the sequence $ (\|\boldsymbol{p}^{-1}(\boldsymbol{n})\|)\_{ \boldsymbol{n} \in N} $, which is generally determined by the “simplest” reccurence relation it satisfies. Exact answers are not the end of the story, however, partly because of applications to the analysis of algorithms. In the single-parameter case, i.e. $ \boldsymbol{p} = (p) $ and N = ((0), (1), (2), …), an asymptotic answer is a relation between f : n ↦ ∣p− 1(n)∣ and “simple” functions, which holds as n → ∞. Often the “simple” functions come from the logarithmico-exponential class L of Hardy, and the relation is f(n) ∼ g(n) as n → ∞, where g ∈ L. A more substantial relation is a full asymptotic series : [def:asymptoticseries] Given a sequence of functions $ \boldsymbol{g} $ with gk + 1(n) = o(gk(n)) as n → ∞ for all k, and real numbers c1, c2, …, the statement $$f(n) \sim \boldsymbol{c}\cdot \boldsymbol{g}(n) = c\_1 g\_1(n) + c\_2 g\_2(n) + c\_3 g\_3(n) + \cdots$$ is called an asymptotic series for f, and it means $$\begin{aligned} f(n) &= O(g\_1(n)) \\ f(n) &= c\_1 g\_1(n) + O(g\_2(n)) \\ f(n) &= c\_1 g\_1(n) + c\_2 g\_2(n) + O(g\_3(n)) \\ f(n) &= c\_1 g\_1(n) + c\_2 g\_2(n) + c\_3 g\_3(n) + O(g\_4(n))\\ &\vdots\end{aligned}$$ (as n → ∞.) An asymptotic answer in the multiple-parameter case is more complicated; it generally involves (possibly just some moments of) a continuous approximation to a discrete probability distribution as one parameter approaches infinity, or an asymptotic relation which holds as one or more parameters approach infinity at various rates. ### Generating functions vs. their coefficients Given a triple $ (S, \boldsymbol{p}, N) $, let f : N → Z≥ 0 be defined $ f( \boldsymbol{n} ) = \| \boldsymbol{p}^{-1} ( \boldsymbol{n} )\| $. A (type u) generating function of S with $ \boldsymbol{z} = (z\_1, z\_2, \dots) $ marking $ \boldsymbol{p} $ is the element of the ring $ \mathbb{Q}[[\boldsymbol{z}]] $ $$F( \boldsymbol{z} ) = \sum\_{\boldsymbol{n} \succeq \boldsymbol{0}} f(\boldsymbol{n}) u(\boldsymbol{n}) \boldsymbol{z}^{\boldsymbol{n}},$$ where $ \boldsymbol{z}^{\boldsymbol{n}} = z\_1^{n\_1} z\_2^{n\_2} \cdots $. We call $ F( \boldsymbol{z} ) $ an ordinary generating function iff $ u(\boldsymbol{n}) = 1 $ for all $ \boldsymbol{n} \succeq \boldsymbol{0} $, and we call it an exponential generating function iff $ u(\boldsymbol{n}) = (n\_1!)^{-1} $ for all $ \boldsymbol{n} \succeq \boldsymbol{0} $. Let $ F( \boldsymbol{z}) $ be the ordinary generating function for words of length n on the alphabet [1..k], with zj marking the number of occurrences of j, 1 ≤ j ≤ k. We have $$F(\boldsymbol{z}) = (z\_1 + z\_2 + \cdots + z\_k)^n.$$ To define convergence of sequences of generating functions, the norm on formal power series used in this document is defined for $ f( \boldsymbol{z} ) \neq 0 $ as $${\lVertf(\boldsymbol{z})\rVert} = 2^{-k}, \text{ where } k= \max \left\{ j \in \mathbb{Z}\_{\geq 0} : \forall i \in [0..j], [z^i]f(z,z, \dots) = 0 \right\},$$ and ∥0∥ = 0. It can be efficient to initially make statements about $ F	arxiv_0000662
was obtained from the hydrostatic atmospheres returned by SPINOR. The B value of 350 G in the filament is comparable (within a factor of 2) to the field strengths found in AR filaments by in the chromosphere sampled by the He I 10830 $\AA$ triplet. The azimuthal orientation of the magnetic field within the filament is almost invariant across the whole filament. Also, the orientation is not aligned with the sunspot’s canopy, but rather almost parallel to the axis of the PIL, indicating sheared magnetic fields. This excludes the possibility that the field we assign to the filament could merely be a low-lying part of the sunspot’s canopy. [imgcanopy] [slice] Fig. [slice] displays a vertical cut through a typical MFC and qualitatively illustrates many properties analysed in more detail in the following sections. The MFC is composed of nearly vertical kG magnetic fields that decrease with height, whilst the MFC expands. The apparent asymmetric expansion of the feature at − 288X arises from the merging of the feature’s canopy with the canopy of a nearby MFC. Both the temperature and the microturbulence are enhanced at mid-photospheric layers within the MFC, but even more so at the interface between it and the surrounding quiet Sun, where strong downflows are also present. The feature lies between two granules, which can be identified in the temperature image at log(τ) = 0. The pixel-to-pixel variations seen in Fig. [slice] are sizeable, but statistically the results are quite robust in that they apply to most plage MFCs. Magnetic field strength ----------------------- Figures [imgBIncl]a-c indicate that the MFCs in plage regions are composed of magnetic fields on the order of kG in the lower and middle photosphere. This is confirmed by the histograms of magnetic field strength in Fig. [genB], which are restricted to pixels selected using the magnetic field thresholds defined in Sect. 4. Besides histograms of B of $\emph{core}$ MFC fields at each optical depth, the histogram of the $\emph{canopy}$ pixels at log(τ) = − 2.3 is plotted as well. Histograms of the magnetic field strength for the $\emph{canopy}$ at log(τ) = 0 and − 0.9 have been omitted as at these heights the atmosphere is similar to the quiet Sun or contains other, weaker fields that are analysed in Sect. 4.9. According to Fig. [genB] the magnetic field strength at log(τ) = − 0.9 has an average value of 1520 G. At this height the two Fe I absorption lines show the greatest response to all the fitted parameters, making the results from this node the most robust and comparable to results obtained from Milne-Eddington (ME) inversions of this line pair. As expected, the average magnetic field strength in $\emph{core}$ pixels decreases with decreasing optical depth, so whilst at log(τ) = 0 the average field strength is 1660 G, at log(τ) = − 2.3 the average field strength drops to 1180 G. [genB] Fig. [genB] also reveals that the widths of the histograms using $\emph{core}$ pixels decreases with height. At log(τ) = 0 the FWHM of the histogram is 800 G that then subsequently decreases to 580 G at log(τ) = − 0.9 and to 400 G log(τ) = − 2.3, which is half the value measured at log(τ) = 0. The comparatively broad distribution at log(τ) = 0 appears to be intrinsic to the MFCs. Large MFCs display a magnetic field gradient across the feature, beginning at one kG at its border and rising to over two kG within the space of $\approx0\farcs5$. Smaller MFCs, too, often decompose into several smaller features when higher magnetic field thresholds are used. However, it cannot be completely ruled out, that the field strengths of the smallest MFCs are partially underestimated due to the finite resolution of SOT/SP. At greater heights neighbouring MFCs merge to create a more homogenous magnetic field with a smaller lateral gradient in B and appear to loose some of their underlying complexity. Nonetheless, differences in the distance between neighbouring MFCs still can lead to an inhomogeneous magnetic field strength above the merging height of the field, which itself strongly depends on this distance. The distribution of the $\emph{canopy}$ pixels indicated by the $\emph{dashed}$ line in Fig. [genB] reveals that B in the $\emph{canopy}$ is generally much weaker than in the $\emph{core}$ pixels. The distribution also has no obvious cut-off save for the arbitrary 300 G threshold, suggesting that the MFCs keep expanding with height in directions in which they are not hindered by neighbouring magnetic features. Magnetic field gradient ----------------------- The change in the peak magnetic field strength in each of the coloured histograms in Fig. [genB] (see Sect. 4.1) indicates that the magnetic field strength of the $\emph{core}$ pixels decreases with height. This was investigated further, first, by correlating the relative decrease, d, of the magnetic field for each $\emph{core}$ pixel using $$d = \frac{B(\log(\tau)=-2.3)}{B(\log(\tau)=0)},$$ against B(log(τ) = 0), which is displayed in Fig. [ftexpanB]. The dashed line in Fig. [ftexpanB] shows the d predicted by a thin flux-tube model with B=2000 G at log(τ) = 0. The thin flux-tube model is identical to the plage model described by. [ftexpanB] Fig. [ftexpanB] reveals that only magnetic fields above about 2000 G at log(τ) = 0 have a d close to the plage thin flux-tube model. The greatest limiting factor regarding the calculation of d appears to suffer from a lack of magnetic flux conservation between the log(τ) = 0 & − 2.3 nodes. The flux of isolated magnetic features at different log(τ) was calculated by drawing a box around it, which was large enough to easily encompass the entire MFC at all heights. Whilst the flux between the lower two nodes agreed within 5%, the log(τ) = − 2.3 node consistently contained 20% more flux than the log(τ) = 0 node. This flux discrepancy between the nodes remained unchanged when the flux from a box containing several merged MFCs was used. Without this flux discrepancy, a $\emph{core}$ pixel with 2500 G at log(τ) = 0 would have a d value of 0.44 instead of 0.55, which would bring it close to the predicted value of the plage flux-tube model. Between the lower two log(τ) nodes, where the flux is roughly conserved, the decrease in the magnetic field strength with height of such a pixel closely follows the thin flux-tube approximation. The $\emph{core}$ pixels with weakest B are found at the edges of their respective MFC and thus may already be partially part of the canopy, due to the limited spatial resolution. This would reduce the vertical field strength gradient, thus increasing d in particular for $\emph{core}$ pixels close to one kG. The small opposite polarity patches described in Sect. 4.9 allow the possibility of two opposite polarity magnetic fields to exist within a weak $\emph{core}$ pixel at the boundary of the MFC in the lower nodes. The Stokes V signals from those two fields would at least partially cancel each other, leading to a reduction in the retrieved B value (and apparent magnetic flux) in the lower two log(τ) nodes and hence to an increase in d. Another contribution to the mismatch in Fig. [ftexpanB] is that some of the smallest $\emph{core}$ pixel patches are flux tubes which are not fully resolved by Hinode, in particular in the lower two layers. The expansion of such an unresolved flux tube would then take place primarily within the pixel, leading to a nearly unchanging B in all three log(τ) nodes, i.e. a d close to unity. The inversion also returns a geometric scale for each pixel. However, the inversion process only prescribes hydrostatic equilibrium within each pixel, but does not impose horizontal pressure balance across pixels. Therefore, each pixel has an individual geometric height scale,that can be off-set with respect to other pixels. This makes the comparison of gradients in (for example) B between pixels with very different atmospheres difficult. However, the $\emph{core}$ pixels found at the very centre of a MFC, with B > 2000 G, have very similar atmospheres, allowing the estimation of a common gradient in B. The gradient, ΔB, in the magnetic field of these $\emph{core}$ pixels is − 2.6 ± 0.5 G/km between log(τ) = 0 and log(τ) = − 2.3. As expected from Fig. [ftexpanB], this ΔB is smaller than the gradient given by the thin flux tube model, which takes a value of ΔB = − 3.9 G/km over the same interval in log(τ), but if magnetic flux conservation is imposed then the gradient of the inversion agrees with the thin-tube approximation. Expansion of magnetic features with height ------------------------------------------ Fig. [imgBIncl] and, in particular, Fig. [imgcanopy] demonstrate that the MFCs in the FOV expand with height. Furthermore, inclination and azimuth, reveal that MFCs generally expand in all directions and are not subjected to extreme foreshortening effects or deformations, (see Fig. [genAziIncl] discussed in detail in Sect. 4.6), except due to other nearby MFCs (see Sect. 4.7). This raises the question of how close this expansion is to that of a model thin flux-tube. Several methods were tested to find a robust measure of the change in size of a magnetic feature with height. The most obvious method, the conservation of magnetic flux with height, was found to be unreliable to estimate the expansion of the magnetic features (see Sect. 4.2). The expansion of the MFCs was, therefore, estimated in the following way. At log(τ) = 0 the size of a magnetic feature was arbitrarily defined by the number of pixels that harboured a magnetic field of at least 900 G. Thresholds above and below this value ( ± 200 G) did not significantly alter the results. Then all the magnetic field values in the log(τ) = 0 image were normalized by the maximum field strength in the feature and the ratio, rt, was calculated using $r\_t=\frac{900 G}{B\_{max}(\tau=1)}$. Each subsequent log(τ) layer above log(τ) = 0 was in turn normalized by its own Bmax(τ) value. The expansion of a magnetic feature could then be tracked by the total number of pixels at a given log(τ) layer where $\frac{B\_(\tau)}{B\_{max}(\tau)}>r\_t$. This method assumes that the MFCs follow a self-similar structure with height. The thin-tube approximation as well as some other models follow this principle. Rather than tracking the expansion of a feature using only the three log(τ) nodes returned by the inversion, the change of the magnetic field with height was tracked using a finer grid of log(τ) layers, with a log(τ) increment of 0.1. This finer log(τ) grid was created using the same spline interpolation between the three nodes as was used during the fitting by the inversion procedure (see Fig. [slice]). The 300 G threshold used to select $\emph{canopy}$ pixels in other parts of this investigation was not imposed here in order to avoid an artificial obstruction of the expansion. Finally, the relative expansion of a feature was calculated using $$\frac{R(\tau)}{R\_{0}} = \sqrt{\frac{A(\tau)}{A\_{0}}},$$ where R and A are the radius and area, respectively, of the flux tube at optical depth τ, R0 and A0 at log(τ) = 0. Five isolated magnetic features were selected from within the field of view. The number of selected features is small since most magnetic features have merged with other features at log(τ) = − 2.3, as demonstrated in Figs. [imgBIncl] & [imgcanopy]. The R(τ)/R0 of the five selected features are represented in Fig. [ftexpanR] by the $\emph{dotted}$ curves, while [ftexpanR] the $\emph{solid}$ line shows the relative radius of the 0th order thin flux-tube plage model of. All the $\emph{dotted}$ lines in Fig. [ftexpanR] follow the expansion predicted by the plage model reasonably well. Interestingly, the thin flux-tube model for the solar network, dashed line in Fig. [ftexpanR], did not fit the expansion of the observed MFCs as well as the plage model ($\emph{solid}$ line in Fig. [ftexpanR]). The reduced relative expansion of the selected features above log(τ) = − 2 when compared to the model may be an indication of the merging of features limiting the expansion at those heights. Another possibility is that a zeroth order model is not sufficient to describe the expansion of the selected features, especially in higher layers, since the lateral variation of the magnetic field within the tube is no longer negligible in higher order flux tube models. Given that the majority of MFCs merge with nearby MFCs, it follows that the majority of MFCs are expected to depart from the expansion displayed by isolated MFCs. Velocities ---------- [imgVel] Figure [imgVel]a-c displays the LOS velocities retrieved by the inversion at the three log(τ) nodes. The small FOV for this figure, representing a typical plage region, was chosen to better highlight the striking differences between velocities in field-free and kG regions and the unusual velocities recorded at the interface between these two regions. The $\emph{core}$ pixels in the images are enclosed by the $\emph{thin}$ black contour lines. Outside the areas harbouring $\emph{core}$ pixels, the typical granular convection pattern can be seen at log(τ) = 0 & − 0.9. The LOS velocities in the top node outline only traces of the stronger granules and display some similarities with chromospheric observations, albeit with smaller velocity amplitudes. The $\emph{dotted}$ contour lines outline the $\emph{canopy}$ fields. The plasma within the majority of $\emph{core}$ pixels is nearly at rest in all log(τ) nodes, as Figs. [imgVel]a-c qualitatively indicate. Histograms of the LOS velocities of these pixels displayed in Fig. [genVel] support this assertion. The mean and median velocities of these $\emph{core}$ pixels are 0.8 km/s and 0.6 km/s, respectively, at log(τ) = 0. They drop to 0.2 km/s and 0.2 km/s at log(τ) = − 0.9, and to 0.1 km/s and 0.0 km/s at log(τ) = − 2.3. [genVel] The $\emph{red}$ histogram in Fig. [genVel], corresponding to the log(τ) = 0 layer, also contains a significant fraction of pixels with downflows larger than 1 km/s. The tail of faster downflows is mainly responsible for the larger than average velocity at log(τ) = 0. The individual MFCs, one of which is displayed in Fig. [imgVelzoom], were inspected further to determine the location and nature of these fast downflows at logτ = 0. Fig. [imgVelzoom]a reveals the $\emph{core}$ pixels of a MFC, enclosed by the black contour line, to be surrounded by a ring of strong downflows. Other MFCs show similar downflow rings at log(τ) = 0 and can often be identified based on such a ring alone. Downflows that exceed 1 km/s are never found within a MFC, but occasionally a $\emph{core}$ pixel located at the edge of a MFC can coincide with a strong downflow, giving rise to the tail of strong downflows seen in the $\emph{red}$ histogram in Fig. [genVel]. Furthermore, the velocities within the rings are not of uniform magnitude. Often some portions of a ring show very fast downflows, up to 10 km/s, whilst others can harbour flows of barely 1 km/s. The portions featuring the fastest downflows do not have a positional preference with respect to its MFC, of which Fig. [imgVelzoom] is one example, which precludes that the magnitude of the downflow is affected by the viewing geometry. The downflow ring seen at log(τ) = 0 in Fig. [imgVelzoom]a is still visible at log(τ) = − 0.9, in Fig. [imgVelzoom]b. However, the pixels with the fastest downflows in the ring seem to be located further away from the $\emph{core}$ pixels in this layer, when compared to Fig. [imgVelzoom]a. It appears that the downflow ring shifts outwards as the magnetic field of the MFC expands with height (see Sect. 4.3). Also, the ring appears to be wider at this height. At log(τ) = − 2.3 the ring can no longer be identified. A quantitative picture of the LOS velocities within these rings was gained by analyzing the pixels, which directly adjoin the $\emph{core}$ pixels. Only the lower two layers were analysed and are displayed in Fig. [Velring], since the rings are no longer present at log(τ) = − 2.3. LOS velocities of up to 10 km/s were found within these rings at log(τ) = 0 and the mean and median values for the corresponding histogram are 2.44 km/s and 2.16 km/s, respectively. For comparison, the fastest downflow in the quiet Sun region, see the black box in Fig. [cont], is only 6 km/s. At log(τ) = − 0.9 the rings no longer contain downflow velocities faster than those found in the quiet Sun at the same logτ layer, but the histogram of the velocities in the ring still has a mean LOS velocity of 0.84 km/s and a median of 0.77 km/s. We further investigated whether the fast downflow velocities at log(τ) = 0 in these rings attain supersonic values. Since the SPINOR code calculates a full atmosphere, including density and pressure, for each pixel, we were able to directly calculate a local sound speed for each pixel over its entire log(τ) range, using the same approach as. The adiabatic index, also needed to calculate the sound speed, was acquired from look-up tables produced by the MURaM MHD simulation code. Supersonic velocities were found in pixels with downflows exceeding 8 km/s at log(τ) = 0 and the fastest downflows, reaching 10 km/s, have a Mach number of 1.25. Higher log(τ) layers did not show any supersonic velocities in any of the pixels. Since the highest speed found in the quiet Sun reach up to 6 km/s, no supersonic velocities were consequently found in the quiet Sun. Furthermore, we determined that 2.5% of a MFC’s downflow ring contain supersonic velocities. Pixels containing supersonic velocities are coloured black in Fig. [imgVel]. Whilst the downflow rings seen at log(τ) = 0 & − 0.9 are generally not traceable at log(τ) = − 2.3, many MFCs additionally have downflows in the form of a plume-like feature, which can be traced through all three log(τ) layers. An example of such a plume-like feature is displayed in Fig. [imgVelplume]. The plume lies just outside the $\emph{core}$ pixels at log(τ) = 0 (Fig. [imgVelplume]a), but is considerably further away, at the boundary of the canopy at log(τ) = − 2.3 (Fig. [imgVelplume]c), and appears to trace the expansion of the MFC. It also increases in size with height. At all heights the LOS velocities of the feature are high when compared to their immediate surroundings, but only at log(τ) = 0 are the velocities in the feature higher than can be found in the quiet Sun. As with the downflow rings the velocities progressively increase with depth. [imgVelzoom] [Velring] [imgVelplume] Temperature ----------- The temperatures at each of the three log(τ) nodes is displayed in Fig. [imgT] for the same FOV as in Fig. [imgVel]. Fig. [imgT]a corresponds to the temperature at log(τ) = 0 and exhibits the familiar granulation pattern. The positions of $\emph{core}$ pixels are revealed by the black contour line in the images and show that they (the $\emph{core}$ pixels) are predominantly found within the comparatively cool intergranular lanes. Figures [imgT]b & [imgT]c display the temperature at log(τ) = − 0.9 and − 2.3, respectively. Both images indicate the high temperatures within MFCs when compared to the quiet Sun at these heights. Furthermore, the temperature map at log(τ) = − 2.3 not only reveals the comparatively hot MFCs but also a reversed granulation pattern. [imgT] The higher temperatures within MFCs at log(τ) = − 0.9 and − 2.3, compared to the quiet Sun, are illustrated more quantitatively by the histograms in Fig. [T]. At both those layers the average temperature is around 300 K higher within $\emph{core}$ pixels, with average temperatures of 5690 K and 5070 K at log(τ) = − 0.9 and − 2.3, respectively, than in quiet Sun pixels, where the average temperatures at the same log(τ) heights are 5290 K and 4780 K, respectively. Fig. [T]c further demonstrates that the average temperature in $\emph{canopy}$ pixels at log(τ) = − 2.3 is only slightly lower than the temperatures of $\emph{core}$ pixels at the same height, with a mean of 5000 K. Only at log(τ) = 0 is the average quiet Sun temperature higher, at 6410 K, than in the $\emph{core}$ pixels, which have a mean temperature of 6270 K. Quiet Sun pixels harbouring downflows ($\emph{dotted red}$ histogram in Fig. [T]a) have a slightly lower mean temperature at 6240 K than MFCs, which are also located predominantly in downflowing regions. The MFC temperature histograms at log(τ) = 0 have been artificially curtailed by the 5800 K threshold imposed at the beginning of the investigation to remove pores. However, the bulk of plage pixels is well above this threshold. Quiet Sun pixels in upflowing regions at log(τ) = 0 ($\emph{dotted blue}$ histogram in Fig. [T]a) have a mean temperature of 6590 K, well above the values of MFCs. 0.5 0.5 0.5 [T] The temperature gradient within $\emph{core}$ pixels was studied further, first, by taking the ratio of the log(τ) = 0 and − 2.3 temperatures. A histogram of these ratios is seen in the left panel of Fig. [expanT]. The average ratio for $\emph{core}$ pixels is 0.81 ± 0.02, which demonstrates that the majority of $\emph{core}$ pixels have a very similar temperature stratification. The ratios were then compared to the temperature ratio obtained from a plage flux tube model derived by, which is shown by the $\emph{dotted}$ line in the left panel in Fig. [expanT]. The temperature ratio of the model, which has a ratio of 0.79, agrees reasonably well with the inversion results. The thin flux-tube network model, has a ratio of 0.7 between the same log(τ) heights and lies outside the histogram. The temperature stratification of MFCs studied in this investigation significantly deviate from the network model’s prediction, which is not surprising given that we are studying strong plage. The temperature stratification of various models is depicted in the right panel of Fig. [expanT]. The $\emph{dotted}$ and $\emph{dot-dashed}$ curves represent the plage and network flux-tube models, while the $\emph{solid}$ line in the same panel shows the typical temperature stratification obtained from $\emph{core}$ pixels by the inversion. Whilst quantitatively the model and inversion results agree quite well, in particular at the three log(τ) nodes, qualitatively there is an important difference. The temperature stratification of the model has a notable bend between log(τ) = − 1 and − 1.5, which is entirely absent from the inversion result. However, such a bend can only be reproduced by the inversion by employing at least four nodes, which would in turn introduce too many free parameters, when inverting only 2 spectral lines. 0.5 0.5 [expanT] Whilst on average the temperature stratification of a MFC follows that a thin flux tube model, a closer inspection of the MFC’s cross-sections reveals that the temperature within a MFC is not uniformly distributed at all. The highest temperatures appear to be preferentially located at the edges of the MFCs. At log(τ) = 0, Fig. [imgT]a, the temperature gradients across a MFC are strongest, whilst higher layers display an ever more uniform distribution of temperatures. Nevertheless, even at log(τ) = − 2.3, Fig. [imgT]c, some areas within the MFCs have an enhanced temperature when compared to their immediate surroundings. These enhanced temperature areas can usually be traced through all three layers and become smaller in size in deeper layers, e.g. at − 242.5X and − 235Y or at − 238.5X and − 240Y. The $\emph{white}$ areas in Fig. [imgT]a indicate the pixels containing supersonic velocities (see Sect. 4.4). Many examples can be found in Fig. [imgT] of a match between the location of supersonic velocities and a nearby (1-2 pixels away, but always within the MFC) local temperature enhancement. However, there is no one-to-one relationship between the two. At numerous locations throughout the plage region, e.g. at − 238X and − 237.5Y in Fig. [imgT], there is a clear local enhanced of the temperature across all three log(τ) layers, but no nearby supersonic velocity. Furthermore, no linear relationship seems to exist between the magnitude of the temperature enhancement (in any layer) and the magnitude of the nearby supersonic downflow. Inclination & Azimuth --------------------- The inclinations of the magnetic field plotted in Figs. [imgBIncl] are inclinations in the observer’s frame of reference. Due to the small heliocentric angle (⟨θ⟩ = 13o) a qualitative picture of the inclinations of plage magnetic features can still be gained from those figures. However, a conversion of these inclinations to local solar coordinates is necessary to find the inclination of the magnetic fields with respect to the solar surface. The conversion of the inclinations and azimuths retrieved by the inversion to local solar coordinates is not straightforward. Whilst the inclination of the magnetic field is uniquely defined the azimuth has an inherent 180∘ ambiguity. Therefore, when converting to local solar coordinates one is forced to choose between one of two possible solutions for the magnetic field vector. Many codes, requiring various amounts of manual input, have been developed to solve the 180∘ ambiguity. The reader is directed to and for overviews. An additional challenge facing these codes is that they generally use the output of an ME inversion as an input. The output of the ME inversion does not contain any information on the change of the magnetic field vector over the formation height of the inverted absorption line. The SPINOR code, however, provides this information, which in turn allows the canopies of magnetic features to be identified as is shown in Fig. [imgcanopy]. The azimuths retrieved by the inversion were spatially very smooth in all the nodes, indicating that the azimuth was well defined in the regions containing magnetic fields. The $\emph{canopy}$ pixels shown in Fig. [imgcanopy] form continuous rings around the various $\emph{cores}$. By assuming that the magnetic field in each $\emph{canopy}$ pixel originates from the largest patch of $\emph{core}$ pixels in its immediate vicinity, the direction of the magnetic field vector of each $\emph{canopy}$ pixel can be determined unambiguously as long as the polarity of the corresponding $\emph{core}$ patch is known. The polarity of a patch of $\emph{core}$ pixels can be determined easily from their Stokes V spectra. In essence, the magnetic field vector of a $\emph{canopy}$ pixel points towards a patch of $\emph{core}$ pixels if it has a negative polarity and, in turn, away from it if the patch of $\emph{core}$ pixels has a positive polarity. With the help of this process we were able to resolve the azimuth ambiguity of the canopy pixels without having to make any further assumptions on the properties of the magnetic field. This process was repeated for each $\emph{canopy}$ pixel individually. Since the canopies of the various plage features are greatest at log(τ) = − 2.3, the magnetic field vector of the $\emph{canopy}$ pixels was determined at this log(τ) height. Once the magnetic field vectors of the $\emph{canopy}$ pixels were determined, the vectors of the $\emph{core}$ could be obtained using an acute-angle method. This method works by performing dot products, using the two possible vectors in an undetermined pixel, with those surrounding pixels whose vector was already determined. The dot products corresponding to each of the two possible solutions were then summed. The vector associated with the smallest sum was subsequently selected as the correct vector for that pixel. The pixels that were surrounded by the largest number of already determined field vectors were given preference. Once the magnetic field vectors of both the $\emph{canopy}$ and $\emph{core}$ pixels at log(τ) = − 2.3 were known, the vectors at log(τ) = 0 & − 0.9 could also be determined. The now known vector at log(τ) = − 2.3 in each pixels was used to perform dot products with the two possible vectors in the next lower log(τ) layer. The vector with the smaller dot product was subsequently chosen as the correct magnetic field vector. The 180∘ ambiguity of the magnetic field vector at log(τ) = 0 & − 0.9 was removed for only those pixels where B>700 G in either layer. The resolution of all the vectors is entirely automatic and only the definition of the $\emph{core}$ and $\emph{canopy}$ pixels for the initial input is manual, but followed the definition given in Sect. 3. Also, no smoothing of the azimuths is performed at any point. The converted inclinations and azimuths in local solar coordinates after the resolution of the 180∘ ambiguity are plotted in Fig. [genAziIncl]. [genAziIncl] Figures [genAziIncl]a-c show the resolved azimuths, Φ, at all three nodes. Several ’azimuth centres’ can be readily identified, in particular at log(τ) = − 2.3. In combination with Figures [genAziIncl]d-f it can be seen that most of the ’azimuth centres’ have vertical fields in their $\emph{cores}$ that become more horizontal towards the edges of a MFC. ’Azimuth centres’ tend to be either relatively isolated features or large ones. Most of the MFCs tend to be elongated with the field expanding roughly perpendicular away from the long axis of the structure over most of its length and directed radially away at the ends. [genIncl] A more quantitive picture of the general inclinations of the $\emph{core}$ pixels can be obtained through their histograms, depicted in Fig. [genIncl]. The distributions of the inclinations have their peak between 10∘ and 15∘ for all log(τ) nodes. The mean inclination for each log(τ) layer is 22∘, 18∘ and 23∘ with decreasing optical depth. The median values have a similar progression with optical depth, taking values of 19∘, 17∘ and 21∘ respectively. Fig. [genIncl] also reveals that the $\emph{canopy}$ pixels are significantly more horizontal, with a peak at 25∘ and a very extended tail reaching up to 90∘. The mean inclination for the $\emph{canopy}$ fields is 39∘ with a median of 36∘, which demonstrates quantitatively the more horizontal nature of the $\emph{canopy}$ when compared to the $\emph{core}$ fields. The largest inclinations are found at the edges of the canopies as expected for an expanding flux tube or flux sheet. [genAzi] The histograms of Φ of $\emph{core}$ pixels are depicted in Fig. [genAzi]. None of the four distributions are homogeneous and show a consistent under-representation of the W and partly the N directions. These two directions are, however, expected to be under-represented due to the viewing geometry, as the region was located in the SE at the time of the observation. The azimuth distributions from MFCs found at the northern edge of the field of view show a more homogeneous distribution, as expected. Fig. [genAzi] also shows that the peak of each azimuth distribution is shifted with respected to other distributions, suggesting that the direction of the magnetic field vector of individual pixels in MFCs appears to rotate with height. Several tests were carried to determine the nature of this rotation, after which a solar origin as well as an inversion based error seem unlikely. Several instrumental effects such as cross-talks or differences in the spectral dispersion between the Stokes parameters were found to be capable in causing the observed rotation. However, a more in depth investigation regarding this matter is required. Effect of the sunspot --------------------- The majority of the MFCs show no obvious and conclusive influence of the nearby sunspot’s magnetic field, which stretches well beyond the spot’s visible boundary in the form of a low-lying magnetic canopy, as can be seen in Fig. [imgcanopy]. Similar extended sunspot canopies were already found in earlier studies. A few MFCs in the FOV, however, are noticeably influenced by the sunspot. The most striking of these features is located at − 217X, − 125Y in Figs. [imgBIncl] & [imgcanopy], where an extensive loop system can be identified. These loops have horizontal fields and connect several pores with positive polarity to the negative polarity sunspot. The magnetic field strengths found in this loop system can reach values as high as 1000 G at log(τ) = − 2.3 in a few places and Fig. [imgcanopy] indicates that they are suspended above relatively field-free gas, since they are identified as $\emph{canopy}$ pixels. MFCs close to the sunspot also posses highly deformed canopies, which are elongated towards the spot if the MFC has the opposite polarity of the spot. Such MFCs can be seen at − 220X, − 170Y and histograms of their inclinations and azimuths are presented in Fig. [nrspot]. 0.5 0.5 [nrspot] The field in the $\emph{core}$ pixels of these MFCs is more inclined than on average; compare with Fig. [genIncl]. The mean inclinations of the field at the three layers from log(τ) = 0 to log(τ) = − 2.3 are 31∘, 32∘ and 35∘, respectively. These average inclinations are about 10∘ larger than for MFCs found further away from the spot. In particular the inclinations of the $\emph{canopy}$ pixels in Fig. [nrspot] reveal the effect of the sunspot upon these magnetic fields even more strongly. The mean inclination of the canopy fields is 56∘ and the median value is 57∘, which is more than 15∘ larger than the average in Fig. [genIncl]. The azimuth distributions in Fig. [nrspot] clearly display the influence of the sunspot as all the distributions both from the $\emph{core}$ and $\emph{canopy}$ pixels show a clear preferred orientation towards the spot. It is likely that the canopies of these MFCs interact with the sunspot’s canopy, even if the fields of both magnetic structures do not appear to be directly connected over the log(τ) range considered in this investigation. From the examples given in this section the following picture emerges. Those magnetic features that are in the immediate vicinity of the sunspot, i.e. if there are no further kG magnetic fields in between them and the spot, show a clear deformation of their canopy and have inclination and azimuth distributions that either predominantly point towards or away from the spot, depending on the polarity. The spot’s influence on the orientation of magnetic features could be observed up to $20"$ away from the sunspot’s outer penumbral boundary, provided there were no other magnetic features in between. Any magnetic feature situated behind this ’first row’ of kG features is effectively shielded from the spot in the photosphere and then behaves closer to an isolated magnetic feature, affected only by its nearest neighbours. Weak opposite polarity fields next to MFCs ------------------------------------------ [imgIncl] Figures [imgIncl]a-c show the LOS inclinations of the magnetic field over a small FOV (same as Figs. [imgVel], [imgT] & [imgmicro]). In this blow-up the fields in the MFCs ($\emph{black}$ contour lines) display mainly vertical orientations, shown in $\emph{blue}$, but a closer inspection of the images corresponding to the log(τ) = 0 & − 0.9 layers exhibit that many MFCs are adjoint by magnetic patches with an inclination opposite to the MFC, as can be seen from their $\emph{red}$ colour, e.g. at − 237.5X and − 238.5Y. Further examination of Fig. [imgIncl] reveals that these small opposite polarity patches are hidden beneath the canopy of the nearest MFC. As a result the small $\emph{red}$ patches are absent in Fig. [imgIncl]c. The vast majority of these small opposite polarity patches have B<100 G at log(τ) = 0 & − 0.9, whereas the canopies above them have B>300 G. This means that the Stokes spectra of these pixels are dominated by the canopy fields. Only the deconvolved Stokes profiles returned by the inversion, as displayed in Fig. [spectra], show a small polarity reversal in the wings of the Stokes V profile. [spectra] 0.5 0.5 [reversal] Nonetheless a number of pixels in these small opposite polarity patches have field strengths in excess of 100 G at log(τ) = 0. These pixels were analysed further by producing histograms of their field strengths and LOS velocities, which are depicted in Fig. [reversal]a & b, respectively. At log(τ) = 0 the mean and median of the histogram are only 280 G and 190 G, respectively. These values drop to 160 G and 70 G at log(τ) = − 0.9 before rising to the considerably higher strengths of the $\emph{canopy}$ fields at log(τ) = − 2.3. The histograms in Fig. [reversal]b, corresponding to the lower two log(τ) layers, demonstrate that these weak opposite polarity fields are predominantly located in the strong downflows surrounding the MFCs. The mean and median velocities of these pixels at log(τ) = 0 are 1.6 km/s and 1.3 km/s, respectively. At log(τ) = − 0.9 the mean and median velocities are 1.3 km/s and 1.2 km/s. The velocities at log(τ) = − 2.3 are, as expected, the same on average to those seen in the $\emph{dashed}$ histogram in Fig. [genVel], since at this height the LOS velocity inside the MFC is sampled and, hence, mostly weak flows are seen. The flux stored in these opposite polarity patches is tyically below 1% of the flux of their parent MFCs. Microturbulence --------------- The inclusion of micro- and macro-turbulent velocities is common when fitting photospheric absorption lines and is an indication of unresolved fine structure. Quiet Sun Hinode SOT/SP observations containing only weak magnetic fields have been fitted without the inclusion of a microturbulence term. In areas with kG magnetic fields, however, an excess of turbulent velocities was found by. The total rms value for turbulent velocities has a range of 1.0 km/s to 3.5 km/s in areas containing kG magnetic fields, depending on the spectral line and on the line strength. found, using lines in the infrared, that rms turbulent velocities between 3.0 km/s and 3.5 km/s were necessary to fit the observed profile shapes. The dependence on spectral line strength suggests a height dependent turbulent velocity. Therefore, we carried out the inversions with a height dependent microturbulence, while forcing the macroturbulence to zero. This approach turned out to give satisfactory fits to the line profiles. Restricting ourselves to microturbulence alone is also in line with the improved spatial resolution and stable observing conditions of the Hinode satellite. Any remaining non-thermal broadening in the spectral line profiles is assumed to be caused by unresolved velocities within the resolution element. A mixture of up- and downflows with a large correlation length along the LOS is less likely at high resolution, than in the low resolution data analysed in the earlier investigations. [imgmicro] Figure [imgmicro] displays the microturbulent velocities, ξmic, retrieved by the inversion at the three log(τ) heights. In all three panels areas with an increased ξmic are found to coincide with areas of strong magnetic fields. The quietest areas are found to have the lowest turbulent velocities, supporting the finding of that turbulent velocities are low in the quiet Sun. In particular, Fig. [imgmicro]b reveals that although $\emph{core}$ pixels display enhanced microturbulence, the largest turbulent velocities are preferentially situated at the edges of a magnetic feature, forming a narrow halo or ring around MFCs. A closer inspection of Fig. [imgmicro]c reveals that the same is also true at log(τ) = − 2.3, although the halo is less well marked at this height. At log(τ) = 0 the halo of large microturbulence around the MFCs is more extended, although large microturbulent velocities are found everywhere. In this inversion the upper limit for the microturbulence was set at 5 km/s. If a higher upper limit was set then microturbulent velocities of up to 9 km/s were retrieved in the halo by the inversion at log(τ) = 0. 0.5 0.5 0.5 [micro] [Microring] The distribution of ξmic, found in $\emph{core}$ pixels, are given by the $\emph{solid}$ histograms in Fig. [micro]. These distributions can be compared to $\emph{dotted}$ histograms depicting ξmic obtained from quiet Sun pixels. The differences in ξmic between the quiet Sun and MFCs, suggested in Fig. [imgmicro], are confirmed by Fig. [micro]. At log(τ) = 0 ξmic in the quiet Sun is on average slightly larger, with a mean ξmic of 3.1 km/s, than in MFCs, which have an average velocity of 2.8 km/s. In higher layers the micro- turbulence in MFCs is significantly larger than in the quiet Sun. Within the MFCs the average ξmic are 1.9 km/s and 1.3 km/s at log(τ) = − 0.9 and log(τ) = − 2.3, whilst in the quiet Sun the average ξmic values are 0.8 km/s and 0.4 km/s in the two layers, respectively. Note that the canopy pixels require a similar microturbulence as the $\emph{core}$ pixels at the same optical depth. Fig. [micro] also reveals that in the upper two layers there are many pixels in the quiet Sun which require no microturbulent broadening at all. The microturbulent velocities found in pixels, which are immediately adjacent to $\emph{core}$ pixels are displayed in Fig. [Microring]. The pixels used in this figure are identical to the ones used in Fig. [Velring], and are, therefore, the pixels featuring the highest LOS downflow velocities. Again, only the microturbulence in the lower two log(τ) layers are shown in Fig. [Microring] (due to the overlying canopy at log(τ) = − 2.3). The average and median microturbulent velocities are 4.4 km/s and 5.0 km/s at log(τ) = 0 and 2.2 km/s and 2.3 km/s at log(τ) = − 0.9. These values are higher than those in the quiet Sun and also than those found within the $\emph{core}$ pixels, demonstrating that the highest microturbulence is located at the edges of MFCs. Both the average and median values of ξmic at log(τ) = 0 in the surrounding, downflowing pixels are likely to be lower limits due to the upper bound of 5 km/s set on the microturbulence in the inversion. The microturbulence retrieved by the inversion could, but does not necessarily imply the existence of turbulence or unresolved convective processes taking place within the magnetic features. It may point to unresolved waves in the MFC (e.g. surface waves could account for the higher ξmic near the boundaries of the MFCs), or it may be due to unresolved horizontal velocity gradients strongest at the boundaries. Another possibility may be, at least in part, a signal of magnetic reconnection between the opposite polarity fields found in this study (see Sect. 4.9). Although it cannot be completely ruled out that inaccuracies in the damping constants of Fe I 6301.5 $\AA$ and 6302.5 $\AA$ may contribute to the deduced ξmic, such inaccuracies are unable to explain the excess in ξmic in the magnetic features, since the spectral lines are significantly weakened there, so that damping becomes less important. Discussion ========== In the preceding sections we sought to ascertain some of the characteristic properties of the magnetic field in solar plage. Stokes I, Q, U and V profiles of AR 10953 observed by the SOT/SP aboard Hinode were analysed using the SPINOR inversion code, extended by, to perform a spatially coupled 2D inversion, which removes the influence of the instrument’s PSF in parallel with inverting the data. The inversion was able to retrieve and reproduce many of the previously deduced characteristic properties of magnetic flux concentrations (MFCs) in plage regions. The typical magnetic field strength of MFCs was found to be in the kG range and took on an average value of 1520 G at log(τ) = − 0.9, where the 6302 $\AA$ line pair is most sensitive to the magnetic field. Similar photospheric magnetic field values were found previously by. These values are, however, somewhat larger than those found using the Fe I 5250.2/5247.1 $\AA$ line pair, which is likely related to differences in the heights of formation of the two line pairs. Since no magnetic filling factor is introduced the obtained B values are lower limits to the true field strengths. The magnetic field is found to drop more slowly with height when compared to a thin-tube (zeroth order) model. This lack of agreement stems from a lack of flux conservation with height, by as much as 20%, that primarily affects the upper log(τ) node in the inversion. If this increase in magnetic flux is compensated then the stratification of the strongest magnetic fields with strengths higher than 2000 G, found at the centres of MFCs, do agree with the thin flux-tube model. When only the lower two log(τ) layers are considered the model and the inversion also agree well. For magnetic fields < 2000 G at log(τ) = 0 other effects start to play a role. The expansion of five isolated MFCs was compared to the ideal expansion of two 0th order thin flux tube models. The expansion of the selected MFCs and the models agreed well, supporting previous results reported by. At the highest inverted layers the field did expand somewhat less rapidly than in the model, however, probably due to the interactions (merging) with other magnetic features. The inversion demonstrates that the majority of MFCs in the strong plage investigated here merge with neighbouring MFCs already in the middle photosphere. Above the merging height B drops more slowly or not at all and the field also does not expand very much. The fact that a zeroth order flux tube is too simple to describe a MFC and expands more rapidly than more realistic models that include curvature forces may also contribute. The LOS velocities obtained from the inversion show that the bulk of the magnetized gas in MFCs is essentially at rest and shows only weak downflows typically on the order of 200 m/s at log(τ) = − 0.9, which agrees with the results of. Some of the $\emph{core}$ and $\emph{canopy}$ pixels show upflows or downflows of up to 1 km/s, which were also observed by. They are in most cases found within the MFCs and could be the result of oscillations or other transient events occurring within MFCs. The MFCs are each surrounded by a ring of strong downflows, which can be readily observed at the log(τ) = 0 and − 0.9 nodes in Figs. [imgVel] & [imgVelzoom]. These downflow rings were also seen to shift outwards in higher log(τ) layers as the MFC expands. A downflow ring has an average velocity of 2.4 km/s at log(τ) = 0, but parts of the ring achieve supersonic velocities up to 10 km/s, corresponding to a Mach number of 1.25, in the same log(τ) layer. On average 2.5% of a MFC’s ring contains downflows reaching supersonic velocities. Some of these downflows appear to overlap with the magnetic field, which could be the signature of entrainment, or may be a result of insufficient spatial resolution to cleanly resolve the magnetic boundary of MFCs. It is unlikely that they indicate convective collapse, since the Stokes V profiles of these pixels do not have the characteristic third lobe reported by. Downflows at the edges of magnetic features were previously found by and. They are also compatible with the downflows inferred from the modelling of Stokes V asymmetries. These downflows are strongest at log(τ) = 0, gradually weaken higher up in the atmosphere and have completely disappeared at log(τ) = − 2.3, suggesting that they are associated with the granular convection pattern. However, downflows exceeding 6 km/s as well as supersonic velocities are only found at the boundaries of MFCs and never in the quiet Sun, indicating that the granular downflows are strengthened by the presence of the MFC. We may be seeing a similar effect as at the edges of light-bridge granules, which display extremely strong downflows, probably because of a combination of radiative losses into the magnetic feature, which leads to faster flows, and because the Wilson depression allows us to see deeper layers with faster flows. The downflows around MFCs also appear qualitatively similar to the downflows observed around pores by and and, therefore, might show a high temporal variation as well. Several MFCs also possess a plume-like downflow features characterised by localised strong downflows traceable through all log(τ) layers. These plumes become larger in size and smaller in magnitude with height. They also trace the expansion of their MFC by shifting outwards with height. The origin of these plumes might be tied to a particular active granule in the vicinity, or may be caused by plasma pouring down from the chromosphere. A reconnection event in the chromosphere might also give rise to such a structure, but the rapid increase of LOS velocities with depth appears to be incompatible with such a scenario. Located within the downflow rings surrounding the MFCs, we found small magnetic patches bearing the opposite polarity to the main MFC to which they adjoin. These opposite polarity patches are only visible in the lower two log(τ) nodes, as shown in Fig. [imgIncl], supporting the notion that they are intimately connected to the downflows in which they are immersed. MHD simulations carried out by and predict such patches in the vicinity of strong magnetic field concentrations,. Indirect evidence was also found in a network patch by, although the exact spatial proximity to the MFC could not be established in that publication due to the low spatial resolution. However, the reversal of the Stokes V amplitude in the wings of the 630 nm line pair is only seen in the deconvolved profiles produced by the inversion, where the Stokes V spectrum displays three lobes. An additional complication is the masking of these magnetic fields by the canopy of the main MFC. The canopy has a typical field strength of 300 G or higher at locations overlying such weak opposite polarity fields. The canopy thus produces a strong signal in Stokes Q, U and V, whereas the field strengths of the opposite polarity patches at log(τ) = 0 are typically well below $\emph{canopy}$ values. This also prevents the selection of these patches via a typical amplitude threshold in the Stokes profile, so that we cannot completely rule out that even those opposite polarity patches with B>100 G are an artifact of the inversion. The opposite polarities typically carry less than 1% of the flux present in their parent MFCs. Further observations performed with a higher spatial resolution are necessary to ascertain the existence of these small opposite polarity patches. Such observations would be particularly useful in the 1.56 μm lines, due to their large Zeeman sensitivity and low formation height. The 1.56 μm data analysed by, although of low spatial resolution, provide some support for our results. The close coexistence of such opposite polarities might give rise to current sheets and could lead to reconnection events. We have introduced a novel method for the resolution of the 180∘ azimuth ambiguity applicable to largely unipolar regions. It makes use of the basic thin flux-tube structure of MFCs and assumes that the divergence of the magnetic field tends towards zero. With this method we find that the $\emph{core}$ pixels of MFCs have typical inclinations, relative to local solar coordinates, between 10∘ and 15∘ in all three log(τ) nodes of the inversion, which agrees well with earlier inclination results found by and. The distribution of the azimuths shows a preference for the eastern direction, which can be attributed to a LOS effect. MFCs located closest to the disc centre in the field of view showed the most homogeneous azimuth distribution, which supports the conclusion of that in general MFCs have no preferred orientation. MFCs close to the sunspot did, however, display azimuth distributions that were either predominantly directed towards or away from the spot depending on their polarity. The canopies of these MFCs were also irregular and elongated. This demonstrates that a nearby sunspot has a direct impact on the properties of MFCs even in the middle photosphere. The inversion allowed us to clearly differentiate between magnetic fields which form magnetic canopies from those which form the core or root of a MFC. A separate canopy could be identified for all MFCs as well as for the pores and sunspot. The canopies were found to harbour weaker, more horizontal magnetic fields, with inclinations as high as Γ = 80∘. The typical LOS velocities in the canopies are identical to the LOS velocities found at the core of a MFC. Many magnetic canopies were found to lie above essentially field free regions. The canopies found here agree with the model proposed by for the production of the Stokes V area asymmetry and the observational results presented by and. All magnetic features were observed to expand with height and many isolated MFCs merged to form comparatively large expanses of magnetic field at log(τ) = − 2.3. We therefore expect that at least some of the magnetic features display a similarity to the wine-glass model described by. The average temperature stratification within MFCs most closely followed the empirical plage flux-tube model of. Temperatures were hotter by 300 K within the MFCs when compared to the quiet Sun in the upper two log(τ) layers. However, a closer inspection of MFCs revealed that the temperature is not homogeneous in any log(τ) across a MFC. The highest temperatures in a MFC are typically positioned at its edges and concentrated into isolated points at log(τ) = 0. Often supersonic downflows were located in the vicinity of these temperature enhancements. This raises the possibility that the two might be connected, since the supersonic flows likely create a shockfront below the log(τ) = 0 layer. However, there were also cases of temperature enhancements without a nearby supersonic velocity. It is likely that a clear picture of the effect of these supersonic downflows upon an MFC can only be determined together with an analysis of a time series of data. The inversion procedure included a depth-dependent microturbulent velocity, with the macroturbulence set to zero. Particularly high values of the microturbulence, partly in excess of 5 km/s at log(τ) = 0, were found at the edges of the magnetic features. The typical microturbulent velocities in MFCs obtained by the inversion has an average value of 2.8 km/s at log(τ) = 0 and becomes weaker with height. Such values are on a par with the broadenings presented by and. This suggests the presence of strong unresolved velocities around the MFCs. Also, although its amplitude decreases rapidly with height, ξmic is at least a factor of 2 larger in the MFC than in the quiet Sun in the upper two log(τ) layers. These large microturbulence values may be telling us that the strong downflows at the edges of the MFCs are associated with vigorous turbulent motions, or it may be a signature of waves travelling along the MFC. Finally, the microturbulence returned by the inversion code may be signalling strong horizontal velocity gradients, e.g. across the boundary of MFCs. A possible geometry for such gradients is a strong downflow outside and a weak one inside. Conclusion ========== In this investigation we have been able to confirm many previously obtained properties associated with magnetic fields, which form solar plage. At the same time, we have also uncovered new features associated with kG magnetic fields. KiloGauss fields in plage are strong, vertical (10∘ − 15∘) magnetic fields on the order of 1.5 kG, which expand with height similar to a thin flux-tube forming extensive canopies as low as the upper photosphere. These fields are further characterised by being on average 300 K hotter than their surroundings in the middle and upper photosphere, in agreement with empirical flux-tube models, and contain weak, on average 200 m/s, plasma flows within them. This investigation has also discovered several new properties of these fields. Each magnetic flux concentration is surrounded by a ring containing strong downflows, on average 2.4 km/s in the lower photosphere. A typical ring shifts outward with height as the field expands and parts of it can attain supersonic velocities in the lower photosphere. Co-spatial with these rings we found enhanced microturbulent velocities decreasing with height as well as magnetic patches situated beneath the canopy with a polarity opposite to the main plage-forming fields. The plasma temperature of the MFCs was not uniform across their cross sections, displaying instead temperature enhancements at their edges. Magnetic field concentrations located close to larger features such as pores or a sunspot displayed an asymmetric canopy and more inclined magnetic fields. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). This work has been partially supported by the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea. D.B. acknowledges a PhD fellowship of the International Max Planck Research School on Physical Processes in the Solar System and Beyond (IMPRS). Properties of solar plage from a spatially coupled inversion of Hinode SP data ============================================================================== The properties of magnetic fields forming an extended plage region in AR 10953 were investigated. Stokes spectra of the Fe I line pair at 6302 $\AA$ recorded by the spectropolarimeter aboard the Hinode satellite were inverted using the SPINOR code. The code performed a 2D spatially coupled inversion on the Stokes spectra, allowing the retrieval of gradients in optical depth within the atmosphere of each pixel, whilst accounting for the effects of the instrument’s PSF. Consequently, no magnetic filling factor was needed. The inversion results reveal that plage is composed of magnetic flux concentrations (MFCs) with typical field strengths of 1520 G at log(τ) = − 0.9 and inclinations of 10∘ − 15∘. The MFCs expand by forming magnetic canopies composed of weaker and more inclined magnetic fields. The expansion and average temperature stratification of isolated MFCs can be approximated well with an empirical plage thin flux tube model. The highest temperatures of MFCs are located at their edges in all log(τ) layers. Whilst the plasma inside MFCs is nearly at rest, each is surrounded by a ring of downflows of on average 2.4 km/s at log(τ) = 0 and peak velocities of up to 10 km/s, which are supersonic. The downflow ring of an MFC weakens and shifts outwards with height, tracing the MFC’s expansion. Such downflow rings often harbour magnetic patches of opposite polarity to that of the main MFC with typical field strengths below 300 G at log(τ) = 0. These opposite polarity patches are situated beneath the canopy of their main MFC. We found evidence of a strong broadening of the Stokes profiles in MFCs and particularly in the downflow rings surrounding MFCs (expressed by a microturbulence in the inversion). This indicates the presence of strong unresolved velocities. Larger magnetic structures such as sunspots cause the field of nearby MFCs to be more inclined. Introduction ============ In a typical active region on the solar disc three types of features can be identified most easily at visible wavelengths: sunspots, pores and plage. Whilst sunspots and pores are defined by their characteristic darkening of the continuum intensity, plage appears brighter than the surrounding quiet Sun mainly in spectral lines, or as faculae near the solar limb in the continuum. It has been known since the work of that sunspots and pores harbour magnetic fields with strengths of the order of kG. showed that plage, too, is associated with magnetic fields, but it was only realised much later that it is also predominantly composed of kG magnetic features. The kG magnetic fields, or magnetic flux concentrations (MFCs), in plage are often considered to take the form of small flux tubes or sheets, and considerable effort has gone into the details that determine their structure and dynamics. The convective collapse mechanism is thought to concentrate the field on kG values, whereby the plasma inside the tube is evacuated and the magnetic field is concentrated. This mechanism may not always lead to kG fields, however. The diameter of an individual kG flux tube is expected to be a few 100 km or less, although a lower limit for kG fields may exist. In the internetwork quiet Sun, diameters typically do not exceed 100 km, necessitating an instrument with an angular resolution of $0\farcs15$ or better to fully resolve an individual flux tube. Owning to the comparatively small lateral size of these flux tubes, they are commonly treated using a thin flux tube model, where the lateral variation in the atmospheric parameters inside the tube is smaller than the pressure scale height. The 3D radiative MHD simulations by give rise to magnetic concentrations with properties that are close to those of the (2nd order) thin-tube approximation. More complex flux-tube models have also been proposed (see and references therein). Despite their small size, many of the general properties of flux tubes residing in plage have nonetheless been determined by observations. This has been achieved by analyzing the polarisation of the light that is produced by the Zeeman effect in areas containing magnetic field. Thus, and for example, used the deep photospheric infrared Fe I 1.56 μm line to find magnetic field strengths of 1400 − 1700 G directly from the splitting of this strongly Zeeman sensitive line. Field strengths of around 1400 G were also obtained by and when using lines in the visible, such as the 6302 $\AA$ line pair, whilst values of only 1000 − 1100 G were found by with the 5250 $\AA$ lines, which are formed somewhat higher in the photosphere. Finally, the Mg I 12.3 μm lines, used by, returned values as low as 200 − 500 G in plage regions, which are fully consistent with the kG fields observed in the aforementioned lines due to the even greater formation height of the Mg I 12.3 μm lines, their different response to unresolved magnetic fields and the merging of neighbouring flux tubes. The expansion with height of MFCs has also been investigated. used SOT/SP images recorded at increasing μ-values and examined the change in the Stokes V signal of MFCs in the quiet Sun network. The variations of the Stokes V signal across a MFC was, with the help of MHD simulations, found to be compatible with a thin flux tube approximation. analysed the Stokes V area asymmetry across a large network patch recorded with IMaX aboard $\emph{\sc Sunrise}$ on the solar disc centre and found that the internal structure of the large network patch was likely to be more complex than that of a simple thin flux tube. A similar conclusion concerning the internal structure of plages was reached by. also examined the change of the Stokes V area asymmetry of a network patch situated at disc centre using SOT/SP and showed that it was surrounded by a magnetic canopy. analysed thin flux tubes and sheets produced by MHD simulations and concluded that a 2nd order flux tube approximation is necessary to accurately describe the structure of the magnetic features. showed that the relative expansion of sunspot canopies is close to that of a thin flux tube, which could imply that the relative expansion of all flux tubes is similar. The inclination with respect to the solar surface of MFCs in plage was found to be predominantly vertical, with typical inclinations of 10∘, which can be attributed to the magnetic buoyancy of the flux tubes, although MFCs with highly inclined magnetic fields were also found. The azimuthal orientation of MFCs was shown to have no preferred direction and to form so called ’azimuth centres’, although did find a preferred E − W orientation. The potential existence of mass motions inside MFCs has been fueled by the observation of significant asymmetries in the Stokes Q, U, and particularly V profiles, in both amplitude and area. However, showed that within a MFC no stationary mass motions stronger than 300 m/s are present. This result was confirmed by using Milne-Eddington (ME) inversion results based on data of the Advanced Stokes Polarimeter (ASP). Stokes profiles in plage display a marked asymmetry between the areas of their blue and red lobes. This asymmetry is thought to result from the interplay between the magnetic element and the convecting plasma in which it is immersed. Following this scenario showed that low resolution Stokes I and V profiles of the Mg I b2 line can be fitted with a combination of atmospheres representing a magnetic flux tube expanding with height, containing no significant flows, which is surrounded by strong downflows of up to 5 km/s representing the field-free convecting plasma around it. This scenario is generally supported by magneto-hydrodynamic (MHD) simulations performed by e.g., although some of the simulated magnetic features do display internal downflows. More recently, observations performed at higher spatial resolution using the Swedish Solar Telescope have further confirmed the basic picture. found, by placing a slit across a plage-like feature, downflows in the range of 1 − 3 km/s at the edges of the feature. Similarly, observed, using SOT/SP, that pores, too, are surrounded by strong downflows in the photosphere. The relationship between the magnetic field strength and continuum intensity of plage was studied extensively by using ME inversions of SOT/SP data and a clear dependence of the continuum intensity on the magnetic field strength was found. Furthermore, the granular convection in plage areas has an abnormal appearance. and concluded that the high spatial density of the kG magnetic fields causes a suppression of the convection process. As illustrated by the above papers, which are only a small sample of the rich literature on this topic, there has been significant progress in our knowledge of plage in the last two decades. Nonetheless, no comprehensive study of plage properties using inversions has been published since the work of, which was based on $1\arcsec$ resolution data from the Advanced Stokes Polarimeter (ASP). In the following sections we aim to both test and expand upon our knowledge of the typical characteristics associated with plage using the results provided by the recently developed and powerful spatially coupled inversion method applied to Hinode SOT/SP observations. We concentrate here on the strong-field magnetic elements and do not discuss the horizontal weak-field features also found in active region plage areas. Data ==== The data set used in this investigation was recorded by the spectropolarimeter, which forms part of the solar optical telescope (SOT/SP) aboard the Hinode satellite. The observation was performed on the 30th of April 2007, UT 18:35:18 - 19:39:53, using the normal observation mode, hence, a total exposure time of 4.8 s per slit position and an angular resolution of $0\farcs3$ was achieved. All four Stokes parameters, I, Q, U and V, were recorded at each slit position with a noise level of 1 × 10− 3Ic. The field of view contains a fully developed sunspot of the active region (AR) 10953 with an extended plage forming region trailing it. During the observation the spot was located in the southern hemisphere towards the east limb, − 190X, − 200Y, at μ = 0.97 (μ = cos(∣θ∣), where θ is the heliocentric angle). A normalized continuum image of the investigated region used in the inversion is shown in Fig. [cont]. The data were reduced using the standard sp\_prep routine from the solar software package. Inversion Method ================ The region of the SOT/SP scan containing most of the plage was inverted using the SPINOR code, which uses response functions in order to perform a least-squares fitting of the Stokes spectra. It is based upon the STOPRO routines described by. The SPINOR code was extended by to perform spatially coupled inversions using the point-spread-function (PSF) of SOT/SP. Such spatially coupled inversions have already been successfully applied to Hinode SOT/SP data of sunspots by. We employ the same PSF used by these authors, which is based on the work of. The size of the inverted area, corresponding to that shown in Fig. [cont], is the largest that can currently be inverted in a single run by the employed code due to computer memory limitations. The inversion code allows the recovery of thermal, magnetic and velocity gradients with optical depth, among others, which reveal themselves by the strengths, shapes and asymmetries present in the SOT/SP Stokes profiles. The stratification of each atmospheric parameter with optical depth is calculated using a spline interpolation through preset log(τ) nodes, where the code can modify a pixel’s atmosphere. The resultant full atmosphere is then used to solve the radiative transfer equation and the emergent synthetic spectra are fitted iteratively by a Levenberg-Marquardt algorithm that minimizes the χ2 merit function. The choice of the log(τ) nodes is important for achieving a credible atmospheric stratification. Three nodes were chosen. A larger number of log(τ) nodes produced more complex atmospheres at the expense of the uniqueness of the solution, while fewer log(τ) nodes failed to fit the asymmetries in the observed spectra. The chosen nodes corresponded to optical depths at log(τ) = 0, − 0.9 and − 2.3 based on calculated contribution functions of the 630 nm line pair. The contribution functions were obtained from an empirical atmosphere simulating a plage pixel, i.e. containing magnetic field of 2000 G at log(τ) = 0 and satisfying the thin-tube approximation at all heights. We also carried out inversions with nodes at slightly different log(τ) values, to see if a better combination was available, but did not find one for plage. At each of the three chosen nodes the temperature, T, magnetic field strength, B, inclination relative to the line-of-sight (LOS), γ, azimuth, ψ, line-of-sight velocity, v, and micro turbulence, ξmic, were fitted, leading to 18 free parameters in total. We stress that no macro turbulent broadening was allowed and a fixed μ-value of μ = 0.97 was assumed during the inversion. The influence of straylight from neighbouring pixels is taken into account by the PSF and the simultaneous coupled inversion of all pixels. Consequently, no magnetic filling factor was introduced in the inversion. A common problem affecting this inversion process is the possibility that the fitting algorithm finds a solution that corresponds to a local χ2 minimum. This is particularly so if the initial guess atmosphere for a pixel is far from the global minimum. In an effort to ensure that the solution for each pixel of the inversion corresponds to the global minimum, the inversion process was performed a total of four times with each inversion performing 12 iterations. Save for the initial inversion each successive inversion used the smoothed results of the previous inversion as an initial input, thereby ensuring that the initial guess for each pixel is closer to the global minimum (under the assumption that the inversion does reach the global minimum by itself for the majority of pixels, but runs into danger of falling into a local minimum for a minority). After the fourth inversion process the mean χ2 value of all the pixels could not be decreased any further, e.g. by inverting the scan a successive time. [cont] Results ======= In this section we describe the various results obtained from the inversion. First we give a general overview of the output of the inversion followed by subsections dealing with more specific points. Figure [cont] provides an overview of the continuum intensity, Figs. [imgBIncl]a-c of the magnetic field strength returned by the inversion, and Figs. [imgBIncl]d-f of the inclination, γ, of the magnetic field vector. All these figures [imgBIncl] display the entire field of view (FOV) to which the inversion code was applied. Fig. [cont] reveals that part of the sunspot’s penumbra as well as pores of various sizes are contained in the FOV and had to be excluded from the analysis. The sunspot’s penumbra and the largest pores in the image were cut out by excluding the lower right hand side of the FOV from the analysis. However, many of the pores in the figure are only a few pixels in size, illustrated by the contour lines in Figs. [cont] & [imgBIncl]a-c and are often entirely embedded within a larger magnetic feature. These small pores were removed from the analysis using a temperature threshold of T < 5800 K at log(τ) = 0. Both higher and lower temperature thresholds, ± 150 K, were tested with insignificant effect upon the following results. The 5800 K threshold was finally chosen since it corresponds to the lowest temperature found in the quiet Sun, (see $\emph{black}$ box in Fig. [cont]). An (alternative) intensity threshold to remove the pores yielded statistically similar results. The area within the $\emph{black}$ box serves as a quiet Sun reference throughout the following sections, since it is almost devoid of kG magnetic fields, although hG fields are present. The LOS velocities were corrected by subtracting an offset of 140 m/s, which was obtained by assuming that the pores are at rest on average. Figures [imgBIncl]a-c reveal that all MFCs expand significantly with height, suggesting that many pixels harbour magnetic fields only in higher layers of the atmosphere, indicating the presence of magnetic canopies. Therefore, all MFC pixels in the inversion result were subsequently divided into two populations: $\emph{core}$ pixels and $\emph{canopy}$ pixels. The $\emph{core}$ pixels were defined by a positive magnetic field gradient with optical depth, i.e. a magnetic field strength decreasing with height, and an absolute magnetic field strength, B, > 1000 G at log(τ) = 0. Higher thresholds at log(τ) = 0 merely reduced the number of selected pixels but did not provide results that differ qualitatively from those presented here. The $\emph{canopy}$ pixels were defined by a negative magnetic field gradient with optical depth and an absolute magnetic field strength, B, above 300 G at log(τ) = − 2.3. A threshold < 300 G at log(τ) = − 2.3 caused the selection of a large number of pixels that were not directly connected to MFCs forming plage regions. These ’extra’ pixels were predominantly associated with the sunspot penumbra and canopy, a filament and with weak horizontal magnetic fields found on top of granules in the few quiet Sun areas in Fig. [cont]. This last group is likely related to the weak horizontal fields found in plage by. All selected pixels have a Stokes Q, U or V amplitude of at least 5σ, where σ = 1 × 10− 3Ic. The location of $\emph{core}$ and $\emph{canopy}$ pixels using these thresholds is illustrated in Fig. [imgcanopy]. The figure shows that the $\emph{canopy}$ pixels associated with MFCs surround the $\emph{core}$ pixels. The sunspot’s canopy forms an extended ring around it and photospheric loops between the sunspot and adjacent, opposite polarity pores with field strengths of up to 1000 G at the loop apex can be seen. Such a loop structure is located at approximately − 215X, − 200Y. The sunspot’s canopy extends particularly far, as elongated finger-like structures at − 210X, − 160Y. These fingers are presumably very low lying loops connecting the spot to MFCs. The clear division between these various magnetic structures is only possible by applying the inversion code in 2D coupled mode, since the inversion requires no secondary atmosphere and/or a filling factor and the remaining single atmosphere is given the freedom to differentiate between $\emph{core}$ and $\emph{canopy}$ fields. Most of the MFCs in Figs. [imgBIncl]d-f have the same, positive, polarity (shown in blue). Only in the lower right hand corner of the field-of-view (FOV) can MFCs of negative polarity be found (shown in red in Figs. [imgBIncl]d-f). The dominant polarity of the MFCs is opposite to that of the sunspot. Between the MFCs of opposite polarities a polarity inversion line (PIL) can be seen, stretching from approximately − 212X, − 245Y to − 227X, − 210Y in Figs. [cont] & [imgBIncl]. Hα images, not shown here, indicate the presence of a filament along this PIL. The photospheric part of this filament is visible in Fig. [imgBIncl] at log(τ) = − 2.3 in the form of predominantly horizontal magnetic fields (Fig. [imgBIncl]f) of around 350 G. The atmosphere below the PIL is almost free of magnetic field (Fig. [imgBIncl]a). The location of the filament is clearly seen as the elongated canopy-like structure following the PIL in Fig. [imgcanopy]. For a more detailed analysis of this filament the reader is referred to. Here we can add to their findings that, although the filament’s magnetic field reaches down into the photosphere, it is largely restricted to layers more than roughly 200 km above the solar surface. This geometrical height was obtained from the hydrostatic atmospheres returned by SPINOR. The B value of 350 G in the filament is comparable (within a factor of 2) to the field strengths found in AR filaments by in the chromosphere sampled by the He I 10830 $\AA$ triplet. The azimuthal orientation of the magnetic field within the filament is almost invariant across the whole filament. Also, the orientation is not aligned with the sunspot’s canopy, but rather almost parallel to the axis of the PIL, indicating sheared magnetic fields. This excludes the possibility that the field we assign to the filament could merely be a low-lying part of the sunspot’s canopy. [imgcanopy] [slice] Fig. [slice] displays a vertical cut through a typical MFC and qualitatively illustrates many properties analysed in more detail in the following sections. The MFC is composed of nearly vertical kG magnetic fields that decrease with height, whilst the MFC expands. The apparent asymmetric expansion of the feature at − 288X arises from the merging of the feature’s canopy with the canopy of a nearby MFC. Both the temperature and the microturbulence are enhanced at mid-photospheric layers within the MFC, but even more so at the interface between it and the surrounding quiet Sun, where strong downflows are also present. The feature lies between two granules, which can be identified in the temperature image at log(τ) = 0. The pixel-to-pixel variations seen in Fig. [slice] are sizeable, but statistically the results are quite robust in that they apply to most plage MFCs. Magnetic field strength ----------------------- Figures [imgBIncl]a-c indicate that the MFCs in plage regions are composed of magnetic fields on the order of kG in the lower and middle photosphere. This is confirmed by the histograms of magnetic field strength in Fig. [genB], which are restricted to pixels selected using the magnetic field thresholds defined in Sect. 4. Besides histograms of B of $\emph{core}$ MFC fields at each optical depth, the histogram of the $\emph{canopy}$ pixels at log(τ) = − 2.3 is plotted as well. Histograms of the magnetic field strength for the $\emph{canopy}$ at log(τ) = 0 and − 0.9 have been omitted as at these heights the atmosphere is similar to the quiet Sun or contains other, weaker fields that are analysed in Sect. 4.9. According to Fig. [genB] the magnetic field strength at log(τ) = − 0.9 has an average value of 1520 G. At this height the two Fe I absorption lines show the greatest response to all the fitted parameters, making the results from this node the most robust and comparable to results obtained from Milne-Eddington (ME) inversions of this line pair. As expected, the average magnetic field strength in $\emph{core}$ pixels decreases with decreasing optical depth, so whilst at log(τ) = 0 the average field strength is 1660 G, at log(τ) = − 2.3 the average field strength drops to 1180 G. [genB] Fig. [genB] also reveals that the widths of the histograms using $\emph{core}$ pixels decreases with height. At log(τ) = 0 the FWHM of the histogram is 800 G that then subsequently decreases to 580 G at log(τ) = − 0.9 and to 400 G log(τ) = − 2.3, which is half the value measured at log(τ) = 0. The comparatively broad distribution at log(τ) = 0 appears to be intrinsic to the MFCs. Large MFCs display a magnetic field gradient across the feature, beginning at one kG at its border and rising to over two kG within the space of $\approx0\farcs5$. Smaller MFCs, too, often decompose into several smaller features when higher magnetic field thresholds are used. However, it cannot be completely ruled out, that the field strengths of the smallest MFCs are partially underestimated due to the finite resolution of SOT/SP. At greater heights neighbouring MFCs merge to create a more homogenous magnetic field with a smaller lateral gradient in B and appear to loose some of their underlying complexity. Nonetheless, differences in the distance between neighbouring MFCs still can lead to an inhomogeneous magnetic field strength above the merging height of the field, which itself strongly depends on this distance. The distribution of the $\emph{canopy}$ pixels indicated by the $\emph{dashed}$ line in Fig. [genB] reveals that B in the $\emph{canopy}$ is generally much weaker than in the $\emph{core}$ pixels. The distribution also has no obvious cut-off save for the arbitrary 300 G threshold, suggesting that the MFCs keep expanding with height in directions in which they are not hindered by neighbouring magnetic features. Magnetic field gradient ----------------------- The change in the peak magnetic field strength in each of the coloured histograms in Fig. [genB] (see Sect. 4.1) indicates that the magnetic field strength of the $\emph{core}$ pixels decreases with height. This was investigated further, first, by correlating the relative decrease, d, of the magnetic field for each $\emph{core}$ pixel using $$d = \frac{B(\log(\tau)=-2.3)}{B(\log(\tau)=0)},$$ against B(log(τ) = 0), which is displayed in Fig. [ftexpanB]. The dashed line in Fig. [ftexpanB] shows the d predicted by a thin flux-tube model with B=2000 G at log(τ) = 0. The thin flux-tube model is identical to the plage model described by. [ftexpanB] Fig. [ftexpanB] reveals that only magnetic fields above about 2000 G at log(τ) = 0 have a d close to the plage thin flux-tube model. The greatest limiting factor regarding the calculation of d appears to suffer from a lack of magnetic flux conservation between the log(τ) = 0 & − 2.3 nodes. The flux of isolated magnetic features at different log(τ) was calculated by drawing a box around it, which was large enough to easily encompass the entire MFC at all heights. Whilst the flux between the lower two nodes agreed within 5%, the log(τ) = − 2.3 node consistently contained 20% more flux than the log(τ) = 0 node. This flux discrepancy between the nodes remained unchanged when the flux from a box containing several merged MFCs was used. Without this flux discrepancy, a $\emph{core}$ pixel with 2500 G at log(τ) = 0 would have a d value of 0.44 instead of 0.55, which would bring it close to the predicted value of the plage flux-tube model. Between the lower two log(τ) nodes, where the flux is roughly conserved, the decrease in the magnetic field strength with height of such a pixel closely follows the thin flux-tube approximation. The $\emph{core}$ pixels with weakest B are found at the edges of their respective MFC and thus may already be partially part of the canopy, due to the limited spatial resolution. This would reduce the vertical field strength gradient, thus increasing d in particular for $\emph{core}$ pixels close to one kG. The small opposite polarity patches described in Sect. 4.9 allow the possibility of two opposite polarity magnetic fields to exist within a weak $\emph{core}$ pixel at the boundary of the MFC in the lower nodes. The Stokes V signals from those two fields would at least partially cancel each other, leading to a reduction in the retrieved B value (and apparent magnetic flux) in the lower two log(τ) nodes and hence to an increase in d. Another contribution to the mismatch in Fig. [ftexpanB] is that some of the smallest $\emph{core}$ pixel patches are flux tubes which are not fully resolved by Hinode, in particular in the lower two layers. The expansion of such an unresolved flux tube would then take place primarily within the pixel, leading to a nearly unchanging B in all three log(τ) nodes, i.e. a d close to unity. The inversion also returns a geometric scale for each pixel. However, the inversion process only prescribes hydrostatic equilibrium within each pixel, but does not impose horizontal pressure balance across pixels. Therefore, each pixel has an individual geometric height scale,that can be off-set with respect to other pixels. This makes the comparison of gradients in (for example) B between pixels with very different atmospheres difficult. However, the $\emph{core}$ pixels found at the very centre of a MFC, with B > 2000 G, have very similar atmospheres, allowing the estimation of a common gradient in B. The gradient, ΔB, in the magnetic field of these $\emph{core}$ pixels is − 2.6 ± 0.5 G/km between log(τ) = 0 and log(τ) = − 2.3. As expected from Fig. [ftexpanB], this ΔB is smaller than the gradient given by the thin flux tube model, which takes a value of ΔB = − 3.9 G/km over the same interval in log(τ), but if magnetic flux conservation is imposed then the gradient of the inversion agrees with the thin-tube approximation. Expansion of magnetic features with height ------------------------------------------ Fig. [imgBIncl] and, in particular, Fig. [imgcanopy] demonstrate that the MFCs in the FOV expand with height. Furthermore, inclination and azimuth, reveal that MFCs generally expand in all directions and are not subjected to extreme foreshortening effects or deformations, (see Fig. [genAziIncl] discussed in detail in Sect. 4.6), except due to other nearby MFCs (see Sect. 4.7). This raises the question of how close this expansion is to that of a model thin flux-tube. Several methods were tested to find a robust measure of the change in size of a magnetic feature with height. The most obvious method, the conservation of magnetic flux with height, was found to be unreliable to estimate the expansion of the magnetic features (see Sect. 4.2). The expansion of the MFCs was, therefore, estimated in the following way. At log(τ) = 0 the size of a magnetic feature was arbitrarily defined by the number of pixels that harboured a magnetic field of at least 900 G. Thresholds above and below this value ( ± 200 G) did not significantly alter the results. Then all the magnetic field values in the log(τ) = 0 image were normalized by the maximum field strength in the feature and the ratio, rt, was calculated using $r\_t=\frac{900 G}{B\_{max}(\tau=1)}$. Each subsequent log(τ) layer above log(τ) = 0 was in turn normalized by its own Bmax(τ) value. The expansion of a magnetic feature could then be tracked by the total number of pixels at a given log(τ) layer where $\frac{B\_(\tau)}{B\_{max}(\tau)}>r\_t$. This method assumes that the MFCs follow a self-similar structure with height. The thin-tube approximation as well as some other models follow this principle. Rather than tracking the expansion of a feature using only the three log(τ) nodes returned by the inversion, the change of the magnetic field with height was tracked using a finer grid of log(τ) layers, with a log(τ) increment of 0.1. This finer log(τ) grid was created using the same spline interpolation between the three nodes as was used during the fitting by the inversion procedure (see Fig. [slice]). The 300 G threshold used to select $\emph{canopy}$ pixels in other parts of this investigation was not imposed here in order to avoid an artificial obstruction of the expansion. Finally, the relative expansion of a feature was calculated using $$\frac{R(\tau)}{R\_{0}} = \sqrt{\frac{A(\tau)}{A\_{0}}},$$ where R and A are the radius and area, respectively, of the flux tube at optical depth τ, R0 and A0 at log(τ) = 0. Five isolated magnetic features were selected from within the field of view. The number of selected features is small since most magnetic features have merged with other features at log(τ) = − 2.3, as demonstrated in Figs. [imgBIncl] & [imgcanopy]. The R(τ)/R0 of the five selected features are represented in Fig. [ftexpanR] by the $\emph{dotted}$ curves, while [ftexpanR] the $\emph{solid}$ line shows the relative radius of the 0th order thin flux-tube plage model of. All the $\emph{dotted}$ lines in Fig. [ftexpanR] follow the expansion predicted by the plage model reasonably well. Interestingly, the thin flux-tube model for the solar network, dashed line in Fig. [ftexpanR], did not fit the expansion of the observed MFCs as well as the plage model ($\emph{solid}$ line in Fig. [ftexpanR]). The reduced relative expansion of the selected features above log(τ) = − 2 when compared to the model may be an indication of the merging of features limiting the expansion at those heights. Another possibility is that a zeroth order model is not sufficient to describe the expansion of the selected features, especially in higher layers, since the lateral variation of the magnetic field within the tube is no longer negligible in higher order flux tube models. Given that the majority of MFCs merge with nearby MFCs, it follows that the majority of MFCs are expected to depart from the expansion displayed by isolated MFCs. Velocities ---------- [imgVel] Figure [imgVel]a-c displays the LOS velocities retrieved by the inversion at the three log(τ) nodes. The small FOV for this figure, representing a typical plage region, was chosen to better highlight the striking differences between velocities in field-free and kG regions and the unusual velocities recorded at the interface between these two regions. The $\emph{core}$ pixels in the images are enclosed by the $\emph{thin}$ black contour lines. Outside the areas harbouring $\emph{core}$ pixels, the typical granular convection pattern can be seen at log(τ) = 0 & − 0.9. The LOS velocities in the top node outline only traces of the stronger granules and display some similarities with chromospheric observations, albeit with smaller velocity amplitudes. The $\emph{dotted}$ contour lines outline the $\emph{canopy}$ fields. The plasma within the majority of $\emph{core}$ pixels is nearly at rest in all log(τ) nodes, as Figs. [imgVel]a-c qualitatively indicate. Histograms of the LOS velocities of these pixels displayed in Fig. [genVel] support this assertion. The mean and median velocities of these $\emph{core}$ pixels are 0.8 km/s and 0.6 km/s, respectively, at log(τ) = 0. They drop to 0.2 km/s and 0.2 km/s at log(τ) = − 0.9, and to 0.1 km/s and 0.0 km/s at log(τ) = − 2.3. [genVel] The $\emph{red}$ histogram in Fig. [genVel], corresponding to the log(τ) = 0 layer, also contains a significant fraction of pixels with downflows larger than 1 km/s. The tail of faster downflows is mainly responsible for the larger than average velocity at log(τ) = 0. The individual MFCs, one of which is displayed in Fig. [imgVelzoom], were inspected further to determine the location and nature of these fast downflows at logτ = 0. Fig. [imgVelzoom]a reveals the $\emph{core}$ pixels of a MFC, enclosed by the black contour line, to be surrounded by a ring of strong downflows. Other MFCs show similar downflow rings at log(τ) = 0 and can often be identified based on such a ring alone. Downflows that exceed 1 km/s are never found within a MFC, but occasionally a $\emph{core}$ pixel located at the edge of a MFC can coincide with a strong downflow, giving rise to the tail of strong downflows seen in the $\emph{red}$ histogram in Fig. [genVel]. Furthermore, the velocities within the rings are not of uniform magnitude. Often some portions of a ring show very fast downflows, up to 10 km/s, whilst others can harbour flows of barely 1 km/s. The portions featuring the fastest downflows do not have a positional preference with respect to its MFC, of which Fig. [imgVelzoom] is one example, which precludes that the magnitude of the downflow is affected by the viewing geometry. The downflow ring seen at log(τ) = 0 in Fig. [imgVelzoom]a is still visible at log(τ) = − 0.9, in Fig. [imgVelzoom]b. However, the pixels with the fastest downflows in the ring seem to be located further away from the $\emph{core}$ pixels in this layer, when compared to Fig. [imgVelzoom]a. It appears that the downflow ring shifts outwards as the magnetic field of the MFC expands with height (see Sect. 4.3). Also, the ring appears to be wider at this height. At log(τ) = − 2.3 the ring can no longer be identified. A quantitative picture of the LOS velocities within these rings was gained by analyzing the pixels, which directly adjoin the $\emph{core}$ pixels. Only the lower two layers were analysed and are displayed in Fig. [Velring], since the rings are no longer present at log(τ) = − 2.3. LOS velocities of up to 10 km/s were found within these rings at log(τ) = 0 and the mean and median values for the corresponding histogram are 2.44 km/s and 2.16 km/s, respectively. For comparison, the fastest downflow in the quiet Sun region, see the black box in Fig. [cont], is only 6 km/s. At log(τ) = − 0.9 the rings no longer contain downflow velocities faster than those found in the quiet Sun at the same logτ layer, but the histogram of the velocities in the ring still has a mean LOS velocity of 0.84 km/s and a median of 0.77 km/s. We further investigated whether the fast downflow velocities at log(τ) = 0 in these rings attain supersonic values. Since the SPINOR code calculates a full atmosphere, including density and pressure, for each pixel, we were able to directly calculate a local sound speed for each pixel over its entire log(τ) range, using the same approach as. The adiabatic index, also needed to calculate the sound speed, was acquired from look-up tables produced by the MURaM MHD simulation code. Supersonic velocities were found in pixels with downflows exceeding 8 km/s at log(τ) = 0 and the fastest downflows, reaching 10 km/s, have a Mach number of 1.25. Higher log(τ) layers did not show any supersonic velocities in any of the pixels. Since the highest speed found in the quiet Sun reach up to 6 km/s, no supersonic velocities were consequently found in the quiet Sun. Furthermore, we determined that 2.5% of a MFC’s downflow ring contain supersonic velocities. Pixels containing supersonic velocities are coloured black in Fig. [imgVel]. Whilst the downflow rings seen at log(τ) = 0 & − 0.9 are generally not traceable at log(τ) = − 2.3, many MFCs additionally have downflows in the form of a plume-like feature, which can be traced through all three log(τ) layers. An example of such a plume-like feature is displayed in Fig. [imgVelplume]. The plume lies just outside the $\emph{core}$ pixels at log(τ) = 0 (Fig. [imgVelplume]a), but is considerably further away, at the boundary of the canopy at log(τ) = − 2.3 (Fig. [imgVelplume]c), and appears to trace the expansion of the MFC. It also increases in size with height. At all heights the LOS velocities of the feature are high when compared to their immediate surroundings, but only at log(τ) = 0 are the velocities in the feature higher than can be found in the quiet Sun. As with the downflow rings the velocities progressively increase with depth. [imgVelzoom] [Velring] [imgVelplume] Temperature ----------- The temperatures at each of the three log(τ) nodes is displayed in Fig. [imgT] for the same FOV as in Fig. [imgVel]. Fig. [imgT]a corresponds to the temperature at log(τ) = 0 and exhibits the familiar granulation pattern. The positions of $\emph{core}$ pixels are revealed by the black contour line in the images and show that they (the $\emph{core}$ pixels) are predominantly found within the comparatively cool intergranular lanes. Figures [imgT]b & [imgT]c display the temperature at log(τ) = − 0.9 and − 2.3, respectively. Both images indicate the high temperatures within MFCs when compared to the quiet Sun at these heights. Furthermore, the temperature map at log(τ) = − 2.3 not only reveals the comparatively hot MFCs but also a reversed granulation pattern. [imgT] The higher temperatures within MFCs at log(τ) = − 0.9 and − 2.3, compared to the quiet Sun, are illustrated more quantitatively by the histograms in Fig. [T]. At both those layers the average temperature is around 300 K higher within $\emph{core}$ pixels, with average temperatures of 5690 K and 5070 K at log(τ) = − 0.9 and − 2.3, respectively, than in quiet Sun pixels, where the average temperatures at the same log(τ) heights are 5290 K and 4780 K, respectively. Fig. [T]c further demonstrates that the average temperature in $\emph{canopy}$ pixels at log(τ) = − 2.3 is only slightly lower than the temperatures of $\emph{core}$ pixels at the same height, with a mean of 5000 K. Only at log(τ) = 0 is the average quiet Sun temperature higher, at 6410 K, than in the $\emph{core}$ pixels, which have a mean temperature of 6270 K. Quiet Sun pixels harbouring downflows ($\emph{dotted red}$ histogram in Fig. [T]a) have a slightly lower mean temperature at 6240 K than MFCs, which are also located predominantly in downflowing regions. The MFC temperature histograms at log(τ) = 0 have been artificially curtailed by the 5800 K threshold imposed at the beginning of the investigation to remove pores. However, the bulk of plage pixels is well above this threshold. Quiet Sun pixels in upflowing regions at log(τ) = 0 ($\emph{dotted blue}$ histogram in Fig. [T]a) have a mean temperature of 6590 K, well above the values of MFCs. 0.5 0.5 0.5 [T] The temperature gradient within $\emph{core}$ pixels was studied further, first, by taking the ratio of the log(τ) = 0 and − 2.3 temperatures. A histogram of these ratios is seen in the left panel of Fig. [expanT]. The average ratio for $\emph{core}$ pixels is 0.81 ± 0.02, which demonstrates that the majority of $\emph{core}$ pixels have a very similar temperature stratification. The ratios were then compared to the temperature ratio obtained from a plage flux tube model derived by, which is shown by the $\emph{dotted}$ line in the left panel in Fig. [expanT]. The temperature ratio of the model, which has a ratio of 0.79, agrees reasonably well with the inversion results. The thin flux-tube network model, has a ratio of 0.7 between the same log(τ) heights and lies outside the histogram. The temperature stratification of MFCs studied in this investigation significantly deviate from the network model’s prediction, which is not surprising given that we are studying strong plage. The temperature stratification of various models is depicted in the right panel of Fig. [expanT]. The $\emph{dotted}$ and $\emph{dot-dashed}$ curves represent the plage and network flux-tube models, while the $\emph{solid}$ line in the same panel shows the typical temperature stratification obtained from $\emph{core}$ pixels by the inversion. Whilst quantitatively the model and inversion results agree quite well, in particular at the three log(τ) nodes, qualitatively there is an important difference. The temperature stratification of the model has a notable bend between log(τ) = − 1 and − 1.5, which is entirely absent from the inversion result. However, such a bend can only be reproduced by the inversion by employing at least four nodes, which would in turn introduce too many free parameters, when inverting only 2 spectral lines. 0.5 0.5 [expanT] Whilst on average the temperature stratification of a MFC follows that a thin flux tube model, a closer inspection of the MFC’s cross-sections reveals that the temperature within a MFC is not uniformly distributed at all. The highest temperatures appear to be preferentially located at the edges of the MFCs. At log(τ) = 0, Fig. [imgT]a, the temperature gradients across a MFC are strongest, whilst higher layers display an ever more uniform distribution of temperatures. Nevertheless, even at log(τ) = − 2.3, Fig. [imgT]c, some areas within the MFCs have an enhanced temperature when compared to their immediate surroundings. These enhanced temperature areas can usually be traced through all three layers and become smaller in size in deeper layers, e.g. at − 242.5X and − 235Y or at − 238.5X and − 240Y. The $\emph{white}$ areas in Fig. [imgT]a indicate the pixels containing supersonic velocities (see Sect. 4.4). Many examples can be found in Fig. [imgT] of a match between the location of supersonic velocities and a nearby (1-2 pixels away, but always within the MFC) local temperature enhancement. However, there is no one-to-one relationship between the two. At numerous locations throughout the plage region, e.g. at − 238X and − 237.5Y in Fig. [imgT], there is a clear local enhanced of the temperature across all three log(τ) layers, but no nearby supersonic velocity. Furthermore, no linear relationship seems to exist between the magnitude of the temperature enhancement (in any layer) and the magnitude of the nearby supersonic downflow. Inclination & Azimuth --------------------- The inclinations of the magnetic field plotted in Figs. [imgBIncl] are inclinations in the observer’s frame of reference. Due to the small heliocentric angle (⟨θ⟩ = 13o) a qualitative picture of the inclinations of plage magnetic features can still be gained from those figures. However, a conversion of these inclinations to local solar coordinates is necessary to find the inclination of the magnetic fields with respect to the solar surface. The conversion of the inclinations and azimuths retrieved by the inversion to local solar coordinates is not straightforward. Whilst the inclination of the magnetic field is uniquely defined the azimuth has an inherent 180∘ ambiguity. Therefore, when converting to local solar coordinates one is forced to choose between one of two possible solutions for the magnetic field vector. Many codes, requiring various amounts of manual input, have been developed to solve the 180∘ ambiguity. The reader is directed to and for overviews. An additional challenge facing these codes is that they generally use the output of an ME inversion as an input. The output of the ME inversion does not contain any information on the change of the magnetic field vector over the formation height of the inverted absorption line. The SPINOR code, however, provides this information, which in turn allows the canopies of magnetic features to be identified as is shown in Fig. [imgcanopy]. The azimuths retrieved by the inversion were spatially very smooth in all the nodes, indicating that the azimuth was well defined in the regions containing magnetic fields. The $\emph{canopy}$ pixels shown in Fig. [imgcanopy] form continuous rings around the various $\emph{cores}$. By assuming that the magnetic field in each $\emph{canopy}$ pixel originates from the largest patch of $\emph{core}$ pixels in its immediate vicinity, the direction of the magnetic field vector of each $\emph{canopy}$ pixel can be determined unambiguously as long as the polarity of the corresponding $\emph{core}$ patch is known. The polarity of a patch of $\emph{core}$ pixels can be determined easily from their Stokes V spectra. In essence, the magnetic field vector of a $\emph{canopy}$ pixel points towards a patch of $\emph{core}$ pixels if it has a negative polarity and, in turn, away from it if the patch of $\emph{core}$ pixels has a positive polarity. With the help of this process we were able to resolve the azimuth ambiguity of the canopy pixels without having to make any further assumptions on the properties of the magnetic field. This process was repeated for each $\emph{canopy}$ pixel individually. Since the canopies of the various plage features are greatest at log(τ) = − 2.3, the magnetic field vector of the $\emph{canopy}$ pixels was determined at this log(τ) height. Once the magnetic field vectors of the $\emph{canopy}$ pixels were determined, the vectors of the $\emph{core}$ could be obtained using an acute-angle method. This method works by performing dot products, using the two possible vectors in an undetermined pixel, with those surrounding pixels whose vector was already determined. The dot products corresponding to each of the two possible solutions were then summed. The vector associated with the smallest sum was subsequently selected as the correct vector for that pixel. The pixels that were surrounded by the largest number of already determined field vectors were given preference. Once the magnetic field vectors of both the $\emph{canopy}$ and $\emph{core}$ pixels at log(τ) = − 2.3 were known, the vectors at log(τ) = 0 & − 0.9 could also be determined. The now known vector at log(τ) = − 2.3 in each pixels was used to perform dot products with the two possible vectors in the next lower log(τ) layer. The vector with the smaller dot product was subsequently chosen as the correct magnetic field vector. The 180∘ ambiguity of the magnetic field vector at log(τ) = 0 & − 0.9 was removed for only those pixels where B>700 G in either layer. The resolution of all the vectors is entirely automatic and only the definition of the $\emph{core}$ and $\emph{canopy}$ pixels for the initial input is manual, but followed the definition given in Sect. 3. Also, no smoothing of the azimuths is performed at any point. The converted inclinations and azimuths in local solar coordinates after the resolution of the 180∘ ambiguity are plotted in Fig. [genAziIncl]. [genAziIncl] Figures [genAziIncl]a-c show the resolved azimuths, Φ, at all three nodes. Several ’azimuth centres’ can be readily identified, in particular at log(τ) = − 2.3. In combination with Figures [genAziIncl]d-f it can be seen that most of the ’azimuth centres’ have vertical fields in their $\emph{cores}$ that become more horizontal towards the edges of a MFC. ’Azimuth centres’ tend to be either relatively isolated features or large ones. Most of the MFCs tend to be elongated with the field expanding roughly perpendicular away from the long axis of the structure over most of its length and directed radially away at the ends. [genIncl] A more quantitive picture of the general inclinations of the $\emph{core}$ pixels can be obtained through their histograms, depicted in Fig. [genIncl]. The distributions of the inclinations have their peak between 10∘ and 15∘ for all log(τ) nodes. The mean inclination for each log(τ) layer is 22∘, 18∘ and 23∘ with decreasing optical depth. The median values have a similar progression with optical depth, taking values of 19∘, 17∘ and 21∘ respectively. Fig. [genIncl] also reveals that the $\emph{canopy}$ pixels are significantly more horizontal, with a peak at 25∘ and a very extended tail reaching up to 90∘. The mean inclination for the $\emph{canopy}$ fields is 39∘ with a median of 36∘, which demonstrates quantitatively the more horizontal nature of the $\emph{canopy}$ when compared to the $\emph{core}$ fields. The largest inclinations are found at the edges of the canopies as expected for an expanding flux tube or flux sheet. [genAzi] The histograms of Φ of $\emph{core}$ pixels are depicted in Fig. [genAzi]. None of the four distributions are homogeneous and show a consistent under-representation of the W and partly the N directions. These two directions are, however, expected to be under-represented due to the viewing geometry, as the region was located in the SE at the time of the observation. The azimuth distributions from MFCs found at the northern edge of the field of view show a more homogeneous distribution, as expected. Fig. [genAzi] also shows that the peak of each azimuth distribution is shifted with respected to other distributions, suggesting that the direction of the magnetic field vector of individual pixels in MFCs appears to rotate with height. Several tests were carried to determine the nature of this rotation, after which a solar origin as well as an inversion based error seem unlikely. Several instrumental effects such as cross-talks or differences in the spectral dispersion between the Stokes parameters were found to be capable in causing the observed rotation. However, a more in depth investigation regarding this matter is required. Effect of the sunspot --------------------- The majority of the MFCs show no obvious and conclusive influence of the nearby sunspot’s magnetic field, which stretches well beyond the spot’s visible boundary in the form of a low-lying magnetic canopy, as can be seen in Fig. [imgcanopy]. Similar extended sunspot canopies were already found in earlier studies. A few MFCs in the FOV, however, are noticeably influenced by the sunspot. The most striking of these features is located at − 217X, − 125Y in Figs. [imgBIncl] & [imgcanopy], where an extensive loop system can be identified. These loops have horizontal fields and connect several pores with positive polarity to the negative polarity sunspot. The magnetic field strengths found in this loop system can reach values as high as 1000 G at log(τ) = − 2.3 in a few places and Fig. [imgcanopy] indicates that they are suspended above relatively field-free gas, since they are identified as $\emph{canopy}$ pixels. MFCs close to the sunspot also posses highly deformed canopies, which are elongated towards the spot if the MFC has the opposite polarity of the spot. Such MFCs can be seen at − 220X, − 170Y and histograms of their inclinations and azimuths are presented in Fig. [nrspot]. 0.5 0.5 [nrspot] The field in the $\emph{core}$ pixels of these MFCs is more inclined than on average; compare with Fig. [genIncl]. The mean inclinations of the field at the three layers from log(τ) = 0 to log(τ) = − 2.3 are 31∘, 32∘ and 35∘, respectively. These average inclinations are about 10∘ larger than for MFCs found further away from the spot. In particular the inclinations of the $\emph{canopy}$ pixels in Fig. [nrspot] reveal the effect of the sunspot upon these magnetic fields even more strongly. The mean inclination of the canopy fields is 56∘ and the median value is 57∘, which is more than 15∘ larger than the average in Fig. [genIncl]. The azimuth distributions in Fig. [nrspot] clearly display the influence of the sunspot as all the distributions both from the $\emph{core}$ and $\emph{canopy}$ pixels show a clear preferred orientation towards the spot. It is likely that the canopies of these MFCs interact with the sunspot’s canopy, even if the fields of both magnetic structures do not appear to be directly connected over the log(τ) range considered in this investigation. From the examples given in this section the following picture emerges. Those magnetic features that are in the immediate vicinity of the sunspot, i.e. if there are no further kG magnetic fields in between them and the spot, show a clear deformation of their canopy and have inclination and azimuth distributions that either predominantly point towards or away from the spot, depending on the polarity. The spot’s influence on the orientation of magnetic features could be observed up to $20"$ away from the sunspot’s outer penumbral boundary, provided there were no other magnetic features in between. Any magnetic feature situated behind this ’first row’ of kG features is effectively shielded from the spot in the photosphere and then behaves closer to an isolated magnetic feature, affected only by its nearest neighbours. Weak opposite polarity fields next to MFCs ------------------------------------------ [imgIncl] Figures [imgIncl]a-c show the LOS inclinations of the magnetic field over a small FOV (same as Figs. [imgVel], [imgT] & [imgmicro]). In this blow-up the fields in the MFCs ($\emph{black}$ contour lines) display mainly vertical orientations, shown in $\emph{blue}$, but a closer inspection of the images corresponding to the log(τ) = 0 & − 0.9 layers exhibit that many MFCs are adjoint by magnetic patches with an inclination opposite to the MFC, as can be seen from their $\emph{red}$ colour, e.g. at − 237.5X and − 238.5Y. Further examination of Fig. [imgIncl] reveals that these small opposite polarity patches are hidden beneath the canopy of the nearest MFC. As a result the small $\emph{red}$ patches are absent in Fig. [imgIncl]c. The vast majority of these small opposite polarity patches have B<100 G at log(τ) = 0 & − 0.9, whereas the canopies above them have B>300 G. This means that the Stokes spectra of these pixels are dominated by the canopy fields. Only the deconvolved Stokes profiles returned by the inversion, as displayed in Fig. [spectra], show a small polarity reversal in the wings of the Stokes V profile. [spectra] 0.5 0.5 [reversal] Nonetheless a number of pixels in these small opposite polarity patches have field strengths in excess of 100 G at log(τ) = 0. These pixels were analysed further by producing histograms of their field strengths and LOS velocities, which are depicted in Fig. [reversal]a & b, respectively. At log(τ) = 0 the mean and median of the histogram are only 280 G and 190 G, respectively. These values drop to 160 G and 70 G at log(τ) = − 0.9 before rising to the considerably higher strengths of the $\emph{canopy}$ fields at log(τ) = − 2.3. The histograms in Fig. [reversal]b, corresponding to the lower two log(τ) layers, demonstrate that these weak opposite polarity fields are predominantly located in the strong downflows surrounding the MFCs. The mean and median velocities of these pixels at log(τ) = 0 are 1.6 km/s and 1.3 km/s, respectively. At log(τ) = − 0.9 the mean and median velocities are 1.3 km/s and 1.2 km/s. The velocities at log(τ) = − 2.3 are, as expected, the same on average to those seen in the $\emph{dashed}$ histogram in Fig. [genVel], since at this height the LOS velocity inside the MFC is sampled and, hence, mostly weak flows are seen. The flux stored in these opposite polarity patches is tyically below 1% of the flux of their parent MFCs. Microturbulence --------------- The inclusion of micro- and macro-turbulent velocities is common when fitting photospheric absorption lines and is an indication of unresolved fine structure. Quiet Sun Hinode SOT/SP observations containing only weak magnetic fields have been fitted without the inclusion of a microturbulence term. In areas with kG magnetic fields, however, an excess of turbulent velocities was found by. The total rms value for turbulent velocities has a range of 1.0 km/s to 3.5 km/s in areas containing kG magnetic fields, depending on the spectral line and on the line strength. found, using lines in the infrared, that rms turbulent velocities between 3.0 km/s and 3.5 km/s were necessary to fit the observed profile shapes. The dependence on spectral line strength suggests a height dependent turbulent velocity. Therefore, we carried out the inversions with a height dependent microturbulence, while forcing the macroturbulence to zero. This approach turned out to give satisfactory fits to the line profiles. Restricting ourselves to microturbulence alone is also in line with the improved spatial resolution and stable observing conditions of the Hinode satellite. Any remaining non-thermal broadening in the spectral line profiles is assumed to be caused by unresolved velocities within the resolution element. A mixture of up- and downflows with a large correlation length along the LOS is less likely at high resolution, than in the low resolution data analysed in the earlier investigations. [imgmicro] Figure [imgmicro] displays the microturbulent velocities, ξmic, retrieved by the inversion at the three log(τ) heights. In all three panels areas with an increased ξmic are found to coincide with areas of strong magnetic fields. The quietest areas are found to have the lowest turbulent velocities, supporting the finding of that turbulent velocities are low in the quiet Sun. In particular, Fig. [imgmicro]b reveals that although $\emph{core}$ pixels display enhanced microturbulence, the largest turbulent velocities are preferentially situated at the edges of a magnetic feature, forming a narrow halo or ring around MFCs. A closer inspection of Fig. [imgmicro]c reveals that the same is also true at log(τ) = − 2.3, although the halo is less well marked at this height. At log(τ) = 0 the halo of large microturbulence around the MFCs is more extended, although large microturbulent velocities are found everywhere. In this inversion the upper limit for the microturbulence was set at 5 km/s. If a higher upper limit was set then microturbulent velocities of up to 9 km/s were retrieved in the halo by the inversion at log(τ) = 0. 0.5 0.5 0.5 [micro] [Microring] The distribution of ξmic, found in $\emph{core}$ pixels, are given by the $\emph{solid}$ histograms in Fig. [micro]. These distributions can be compared to $\emph{dotted}$ histograms depicting ξmic obtained from quiet Sun pixels. The differences in ξmic between the quiet Sun and MFCs, suggested in Fig. [imgmicro], are confirmed by Fig. [micro]. At log(τ) = 0 ξmic in the quiet Sun is on average slightly larger, with a mean ξmic of 3.1 km/s, than in MFCs, which have an average velocity of 2.8 km/s. In higher layers the micro- turbulence in MFCs is significantly larger than in the quiet Sun. Within the MFCs the average ξmic are 1.9 km/s and 1.3 km/s at log(τ) = − 0.9 and log(τ) = − 2.3, whilst in the quiet Sun the average ξmic values are 0.8 km/s and 0.4 km/s in the two layers, respectively. Note that the canopy pixels require a similar microturbulence as the $\emph{core}$ pixels at the same optical depth. Fig. [micro] also reveals that in the upper two layers there are many pixels in the quiet Sun which require no microturbulent broadening at all. The microturbulent velocities found in pixels, which are immediately adjacent to $\emph{core}$ pixels are displayed in Fig. [Microring]. The pixels used in this figure are identical to the ones used in Fig. [Velring], and are, therefore, the pixels featuring the highest LOS downflow velocities. Again, only the microturbulence in the lower two log(τ) layers are shown in Fig. [Microring] (due to the overlying canopy at log(τ) = − 2.3). The average and median microturbulent velocities are 4.4 km/s and 5.0 km/s at log(τ) = 0 and 2.2 km/s and 2.3 km/s at log(τ) = − 0.9. These values are higher than those in the quiet Sun and also than those found within the $\emph{core}$ pixels, demonstrating that the highest microturbulence is located at the edges of MFCs. Both the average and median values of ξmic at log(τ) = 0 in the surrounding, downflowing pixels are likely to be lower limits due to the upper bound of 5 km/s set on the microturbulence in the inversion. The microturbulence retrieved by the inversion could, but does not necessarily imply the existence of turbulence or unresolved convective processes taking place within the magnetic features. It may point to unresolved waves in the MFC (e.g. surface waves could account for the higher ξmic near the boundaries of the MFCs), or it may be due to unresolved horizontal velocity gradients strongest at the boundaries. Another possibility may be, at least in part, a signal of magnetic reconnection between the opposite polarity fields found in this study (see Sect. 4.9). Although it cannot be completely ruled out that inaccuracies in the damping constants of Fe I 6301.5 $\AA$ and 6302.5 $\AA$ may contribute to the deduced ξmic, such inaccuracies are unable to explain the excess in ξmic in the magnetic features, since the spectral lines are significantly weakened there, so that damping becomes less important. Discussion ========== In the preceding sections we sought to ascertain some of the characteristic properties of the magnetic field in solar plage. Stokes I, Q, U and V profiles of AR 10953 observed by the SOT/SP aboard Hinode were analysed using the SPINOR inversion code, extended by, to perform a spatially coupled 2D inversion, which removes the influence of the instrument’s PSF in parallel with inverting the data. The inversion was able to retrieve and reproduce many of the previously deduced characteristic properties of magnetic flux concentrations (MFCs) in plage regions. The typical magnetic field strength of MFCs was found to be in the kG range and took on an average value of 1520 G at log(τ) = − 0.9, where the 6302 $\AA$ line pair is most sensitive to the magnetic field. Similar photospheric magnetic field values were found previously by. These values are, however, somewhat larger than those found using the Fe I 5250.2/5247.1 $\AA$ line pair, which is likely related to differences in the heights of formation of the two line pairs. Since no magnetic filling factor is introduced the obtained B values are lower limits to the true field strengths. The magnetic field is found to drop more slowly with height when compared to a thin-tube (zeroth order) model. This lack of agreement stems from a lack of flux conservation with height, by as much as 20%, that primarily affects the upper log(τ) node in the inversion. If this increase in magnetic flux is compensated then the stratification of the strongest magnetic fields with strengths higher than 2000 G, found at the centres of MFCs, do agree with the thin flux-tube model. When only the lower two log(τ) layers are considered the model and the inversion also agree well. For magnetic fields < 2000 G at log(τ) = 0 other effects start to play a role. The expansion of five isolated MFCs was compared to the ideal expansion of two 0th order thin flux tube models. The expansion of the selected MFCs and the models agreed well, supporting previous results reported by. At the highest inverted layers the field did expand somewhat less rapidly than in the model, however, probably due to the interactions (merging) with other magnetic features. The inversion demonstrates that the majority of MFCs in the strong plage investigated here merge with neighbouring MFCs already in the middle photosphere. Above the merging height B drops more slowly or not at all and the field also does not expand very much. The fact that a zeroth order flux tube is too simple to describe a MFC and expands more rapidly than more realistic models that include curvature forces may also contribute. The LOS velocities obtained from the inversion show that the bulk of the magnetized gas in MFCs is essentially at rest and shows only weak downflows typically on the order of 200 m/s at log(τ) = − 0.9, which agrees with the results of. Some of the $\emph{core}$ and $\emph{canopy}$ pixels show upflows or downflows of up to 1 km/s, which were also observed by. They are in most cases found within the MFCs and could be the result of oscillations or other transient events occurring within MFCs. The MFCs are each surrounded by a ring of strong downflows, which can be readily observed at the log(τ) = 0 and − 0.9 nodes in Figs. [imgVel] & [imgVelzoom]. These downflow rings were also seen to shift outwards in higher log(τ) layers as the MFC expands. A downflow ring has an average velocity of 2.4 km/s at log(τ) = 0, but parts of the ring achieve supersonic velocities up to 10 km/s, corresponding to a Mach number of 1.25, in the same log(τ) layer. On average 2.5% of a MFC’s ring contains downflows reaching supersonic velocities. Some of these downflows appear to overlap with the magnetic field, which could be the signature of entrainment, or may be a result of insufficient spatial resolution to cleanly resolve the magnetic boundary of MFCs. It is unlikely that they indicate convective collapse, since the Stokes V profiles of these pixels do not have the characteristic third lobe reported by. Downflows at the edges of magnetic features were previously found by and. They are also compatible with the downflows inferred from the modelling of Stokes V asymmetries. These downflows are strongest at log(τ) = 0, gradually weaken higher up in the atmosphere and have completely disappeared at log(τ) = − 2.3, suggesting that they are associated with the granular convection pattern. However, downflows exceeding 6 km/s as well as supersonic velocities are only found at the boundaries of MFCs and never in the quiet Sun, indicating that the granular downflows are strengthened by the presence of the MFC. We may be seeing a similar effect as at the edges of light-bridge granules, which display extremely strong downflows, probably because of a combination of radiative losses into the magnetic feature, which leads to faster flows, and because the Wilson depression allows us to see deeper layers with faster flows. The downflows around MFCs also appear qualitatively similar to the downflows observed around pores by and and, therefore, might show a high temporal variation as well. Several MFCs also possess a plume-like downflow features characterised by localised strong downflows traceable through all log(τ) layers. These plumes become larger in size and smaller in magnitude with height. They also trace the expansion of their MFC by shifting outwards with height. The origin of these plumes might be tied to a particular active granule in the vicinity, or may be caused by plasma pouring down from the chromosphere. A reconnection event in the chromosphere might also give rise to such a structure, but the rapid increase of LOS velocities with depth appears to be incompatible with such a scenario. Located within the downflow rings surrounding the MFCs, we found small magnetic patches bearing the opposite polarity to the main MFC to which they adjoin. These opposite polarity patches are only visible in the lower two log(τ) nodes, as shown in Fig. [imgIncl], supporting the notion that they are intimately connected to the downflows in which they are immersed. MHD simulations carried out by and predict such patches in the vicinity of strong magnetic field concentrations,. Indirect evidence was also found in a network patch by, although the exact spatial proximity to the MFC could not be established in that publication due to the low spatial resolution. However, the reversal of the Stokes V amplitude in the wings of the 630 nm line pair is only seen in the deconvolved profiles produced by the inversion, where the Stokes V spectrum displays three lobes. An additional complication is the masking of these magnetic fields by the canopy of the main MFC. The canopy has a typical field strength of 300 G or higher at locations overlying such weak opposite polarity fields. The canopy thus produces a strong signal in Stokes Q, U and V, whereas the field strengths of the opposite polarity patches at log(τ) = 0 are typically well below $\emph{canopy}$ values. This also prevents the selection of these patches via a typical amplitude threshold in the Stokes profile, so that we cannot completely rule out that even those opposite polarity patches with B>100 G are an artifact of the inversion. The opposite polarities typically carry less than 1% of the flux present in their parent MFCs. Further observations performed with a higher spatial resolution are necessary to ascertain the existence of these small opposite polarity patches. Such observations would be particularly useful in the 1.56 μm lines, due to their large Zeeman sensitivity and low formation height. The 1.56 μm data analysed by, although of low spatial resolution, provide some support for our results. The close coexistence of such opposite polarities might give rise to current sheets and could lead to reconnection events. We have introduced a novel method for the resolution of the 180∘ azimuth ambiguity applicable to largely unipolar regions. It makes use of the basic thin flux-tube structure of MFCs and assumes that the divergence of the magnetic field tends towards zero. With this method we find that the $\emph{core}$ pixels of MFCs have typical inclinations, relative to local solar coordinates, between 10∘ and 15∘ in all three log(τ) nodes of the inversion, which agrees well with earlier inclination results found by and. The distribution of the azimuths shows a preference for the eastern direction, which can be attributed to a LOS effect. MFCs located closest to the disc centre in the field of view showed the most homogeneous azimuth distribution, which supports the conclusion of that in general MFCs have no preferred orientation. MFCs close to the sunspot did, however, display azimuth distributions that were either predominantly directed towards or away from the spot depending on their polarity. The canopies of these MFCs were also irregular and elongated. This demonstrates that a nearby sunspot has a direct impact on the properties of MFCs even in the middle photosphere. The inversion allowed us to clearly differentiate between magnetic fields which form magnetic canopies from those which form the core or root of a MFC. A separate canopy could be identified for all MFCs as well as for the pores and sunspot. The canopies were found to harbour weaker, more horizontal magnetic fields, with inclinations as high as Γ = 80∘. The typical LOS velocities in the canopies are identical to the LOS velocities found at the core of a MFC. Many magnetic canopies were found to lie above essentially field free regions. The canopies found here agree with the model proposed by for the production of the Stokes V area asymmetry and the observational results presented by and. All magnetic features were observed to expand with height and many isolated MFCs merged to form comparatively large expanses of magnetic field at log(τ) = − 2.3. We therefore expect that at least some of the magnetic features display a similarity to the wine-glass model described by. The average temperature stratification within MFCs most closely followed the empirical plage flux-tube model of. Temperatures were hotter by 300 K within the MFCs when compared to the quiet Sun in the upper two log(τ) layers. However, a closer inspection of MFCs revealed that the temperature is not homogeneous in any log(τ) across a MFC. The highest temperatures in a MFC are typically positioned at its edges and concentrated into isolated points at log(τ) = 0. Often supersonic downflows were located in the vicinity of these temperature enhancements. This raises the possibility that the two might be connected, since the supersonic flows likely create a shockfront below the log(τ) = 0 layer. However, there were also cases of temperature enhancements without a nearby supersonic velocity. It is likely that a clear picture of the effect of these supersonic downflows upon an MFC can only be determined together with an analysis of a time series of data. The inversion procedure included a depth-dependent microturbulent velocity, with the macroturbulence set to zero. Particularly high values of the microturbulence, partly in excess of 5 km/s at log(τ) = 0, were found at the edges of the magnetic features. The typical microturbulent velocities in MFCs obtained by the inversion has an average value of 2.8 km/s at log(τ) = 0 and becomes weaker with height. Such values are on a par with the broadenings presented by and. This suggests the presence of strong unresolved velocities around the MFCs. Also, although its amplitude decreases rapidly with height, ξmic is at least a factor of 2 larger in the MFC than in the quiet Sun in the upper two log(τ) layers. These large microturbulence values may be telling us that the strong downflows at the edges of the MFCs are associated with vigorous turbulent motions, or it may be a signature of waves travelling along the MFC. Finally, the microturbulence returned by the inversion code may be signalling strong horizontal velocity gradients, e.g. across the boundary of MFCs. A possible geometry for such gradients is a strong downflow outside and a weak one inside. Conclusion ========== In this investigation we have been able to confirm many previously obtained properties associated with magnetic fields, which form solar plage. At the same time, we have also uncovered new features associated with kG magnetic fields. KiloGauss fields in plage are strong, vertical (10∘ − 15∘) magnetic fields on the order of 1.5 kG, which expand with height similar to a thin flux-tube forming extensive canopies as low as the upper photosphere. These fields are further characterised by being on average 300 K hotter than their surroundings in the middle and upper photosphere, in agreement with empirical flux-tube models, and contain weak, on average 200 m/s, plasma flows within them. This investigation has also discovered several new properties of these fields. Each magnetic flux concentration is surrounded by a ring containing strong downflows, on average 2.4 km/s in the lower photosphere. A typical ring shifts outward with height as the field expands and parts of it can attain supersonic velocities in the lower photosphere. Co-spatial with these rings we found enhanced microturbulent velocities decreasing with height as well as magnetic patches situated beneath the canopy with a polarity opposite to the main plage-forming fields. The plasma temperature of the MFCs was not uniform across their cross sections, displaying instead temperature enhancements at their edges. Magnetic field concentrations located close to larger features such as pores or a sunspot displayed an asymmetric canopy and more inclined magnetic fields. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). This work has been partially supported by the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea. D.B. acknowledges a PhD fellowship of the International Max Planck Research School on Physical Processes in the Solar System and Beyond (IMPRS).	arxiv_0000669
Numerical signatures of ultra-local criticality in a one dimensional Kondo lattice model ======================================================================================== Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling JK at fixed doping x. At large positive JK, we confirm the expected conventional Luttinger liquid phase with $2k\_F=\frac{1+x}{2}$ (in units of 2π), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the JK ≤ 0 side, our simulation finds the existence of a fractional Luttinger liquid (LL\) phase with $2k\_F=\frac{x}{2}$, accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL\) phase in higher dimensions. The LL\* phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive JK. Then we mainly focus on the ‘critical regime’ between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of JK, the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035J) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z = + ∞. The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference. Introduction ============ The study of quantum phase transition between a small Fermi surface phase and a large Fermi surface phase is a central topic in modern quantum condensed matter physics and may be closely related to the strange metals observed in heavy Fermion systems and in hole-doped high Tc cuprates. The standard Landau-Ginzburg theory involves the onset of a symmetry-breaking order and its fluctuation. However, a number of experiments in heavy Fermion systems do not appear to be consistent with the simple spin-density-wave (SDW) approach. It was suggested that the transition in heavy fermion systems may be characterized by a jump in Fermi surface volume resulting from Kondo breakdown, rather than fluctuations in symmetry-breaking orders. There have been many attempts to formulate a framework of an exotic transition following different approaches, such as extended dynamical mean field theory (EDMFT), fractionalization and slave boson theory, ancilla qubit theory. However, a well-established theoretical description of such a Kondo breakdown transition is still elusive. In this paper, we take a microscopic approach to avoid uncontrolled approximations usually existing in low-energy effective field theory methods. Specifically, we will numerically simulate a one-dimensional Kondo lattice model using density matrix renormalization group (DMRG). DMRG has been demonstrated to be an unbiased method with excellent performance in one dimension (1D). Therefore, the numerical results should be reliable. The only question is whether there is anything interesting in a 1D model. We will show that the answer is yes and we find a critical point or phase which seems to support local criticality behaviour. We note that there already exist a few numerical studies of the Kondo lattice model in one dimension, but to our best knowledge, there is no detailed study of how a Kondo breakdown phase at negative JK evolves to the Luttinger liquid in the large positive JK at a generic filling. The model we study consists of a t-J model of itinerant electron and a Heisenberg model of spin 1/2 chain. They couple to each other through a Kondo coupling JK. At a density x for the itinerant electron, we vary JK to study the phase diagram. In the JK ≤ 0 side, the ground state has one charge mode and two spin modes (C1S2), where the localized spin 1/2 moments provide an additional gapless mode with momentum Q = π. The itinerant electron forms a Luttinger liquid with $2k\_F^\=\frac{x}{2}$ (in units of 2π). The phase is an analogue of the fractional Fermi liquid (FL\) phase in higher dimension and we call it fractional Luttinger liquid (LL\). In the large positive J*K we find the expected Luttinger liquid (LL) phase with $2k\_F=\frac{1+x}{2}$ (in units of 2π), which is an analogue of the heavy Fermi liquid (HFL) phase in the higher dimensional Kondo lattice model. Therefore, we have the same problem of small to large Fermi surface evolution as in higher dimensions. Complexity arises in one dimension because the LL\* phase is unstable at small positive JK and transitions to a Luther-Emery liquid (LE) phase with a spin gap and only one gapless charge mode. The LE phase is best described as a descendant of the LL\* phase. It is similar to a superconductor phase in a higher dimension and above the energy scale of the spin gap it smoothly connects to the LL\* phase. We note, that in the heavy Fermion experiments, the transitions between the small and large Fermi surface metals are typically covered by a superconductor dome. Thus, the situation in 1D is similar to higher dimension and we will try to understand the nature of the evolution from the LE phase to the LL phase upon increasing JK. The hope is that there may also be a ‘strange metal’ critical point or a phase in between. As the LE phase descends from the LL\* phase and we are not aware of any way to construct it from the LL phase, we do not expect any obvious continuous transition between the LE and LL phases. Indeed, we find that there is either a first-order transition or an intermediate region in between. We will focus on the latter case and provide evidence of local criticality behaviour beyond the familiar Luttinger liquid or conformal field theory (CFT) descriptions. At one point (or a narrow region) of JK, we find that the spin gap is almost vanishing, while there is still a finite correlation length in equal time spin-spin correlation function in real space. Meanwhile, the dynamical spin structure factor S(ω, q) ∼ ImχS(ω, q) obtained from the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a range of the momentum space above an energy cutoff (around 0.035J, J is the Heisenberg spin coupling) imposed by the numerical accuracy itself. Such behaviour resembles what is called local criticality. We note, that in the literature sometimes local criticality is also used for the case where only the self-energy is momentum independent, while there is still a significant spatial correlation. In this weaker case, the dynamical exponent is still finite. The behaviour in our model is closer to a stronger definition with an infinite dynamical exponent. Therefore, we follow Ref. and call it ultra-local criticality to be distinguished from the weaker definition. The discovery of ultra-local critical spin fluctuations above a small energy scale is quite remarkable, as this phenomenon is not generally believed to be possible in a reasonable model with translation invariance and a finite-dimensional Hilbert space at each site. The existence of ultra-local criticality also has significant implications for our understanding of the strange metal. For example, it may be a loophole of the anomaly approach of non-Fermi liquid and it is known that ultra-local critical spin fluctuations with a constant spectral function over frequency can lead to a marginal Fermi liquid and linear T resistivity. On the experimental side, similar local critical behaviours have been discovered in neutron scattering measurements of some heavy Fermion materials. One may worry that the experimental results arise from disorder effects. Our numerical observation of similar local critical behaviours in a clean model strongly suggests that such a phenomenon may likely be intrinsic and does not need disorders. On the theoretical side, similar behaviour has been discussed in holographic theory from the gravity side and dubbed as ‘semi-local quantum fluid’. However, we are not aware of a well-established theory of ultra-local criticality for a local and translation invariant quantum lattice model directly. We hope our numerical confirmation of the existence of ultra-local criticality will stimulate theoretical efforts in this direction. Lastly, we propose to simulate the Kondo lattice model in a bilayer optical lattice with a potential difference, which hopefully will provide more information at finite temperatures and higher dimensions. Layer selective Mott localization and Kondo lattice model in bilayer optical lattice ==================================================================================== [h]1 [fig:kondolattice] Here we first propose to simulate a Kondo lattice model in bilayer optical lattice, as has been experimentally realized in Ref. . One new requirement now is that we need to add a potential difference Δ between the two layers. The system is described by a bilayer Hubbard model: $$\begin{aligned} H&=\Delta\sum\_i n\_{i;1}-t\sum\_{a=1,2}\sum\_{\sigma=\uparrow,\downarrow}\sum\_{\langle ij \rangle}( c^\dagger\_{i;a \sigma}c\_{j;a \sigma}+h.c.)\notag \\ &-t\_{12}\sum\_{a=1,2}\sum\_{\sigma=\uparrow,\downarrow}\sum\_{\langle ij \rangle}(c^\dagger\_{i;1\sigma}c\_{j;2\sigma}+c^\dagger\_{i;2\sigma}c\_{j;1\sigma}+h.c.)\notag \\ &-t\_{\perp}\sum\_{a,\sigma}\sum\_{i}(c^\dagger\_{i;1\sigma}c\_{i;2\sigma}+h.c.)-\mu \sum\_{a=1,2} \sum\_i n\_{i;a}\notag \\ &+\frac{U}{2}\sum\_{a}\sum\_i n\_{i;a}(n\_{i;a}-1)+U'\sum\_i n\_{i;1}n\_{i;2}, \label{eq:bilayer\_Hubbard\_model}\end{aligned}$$ where ni; a = ∑σci; aσ†ci; aσ is the density at site i for layer a = 1, 2. ni = ni; 1 + ni; 2 is the total density at site i. We also define the average density $n=\frac{1}{N\_s}\sum\_i n\_i$, where Ns is the total number of sites in the system. Here a = 1, 2 labels the two layers and t⊥ is the inter-layer vertical tunnelling. A non-zero Δ > 0 is caused by a displacement field or a potential difference between the two layers. We will stay in the limit U > > t and U > > Uʹ. We assume t⊥, t < Δ < U − Uʹ. At density n = 1, we have a Mott insulator with one particle at the layer 2. Then at density n = 1 + x with x ∈ (0, 1), the doped additional particle enters the layer 1 to reduce the on-site Hubbard U. In this case the layer 2 is always Mott localized and provides a spin 1/2 moment. The itinerant electron in the layer 1 is described by a t − J model which then couples to the local moment of the layer 2 through a Kondo coupling. At low energy we can deal with an effective Kondo lattice model: $$\begin{aligned} H&=-t P\sum\_{<i,j>,\sigma}(c^\dag\_{i, \sigma}c\_{j, \sigma}+h.c)P+J\_c \sum\_{\langle ij \rangle}\vec{S^e\_i}\cdot \vec{S^e\_j}\notag \\ &~~~+(V-\frac{1}{4}J\_c )\sum\_{\langle ij \rangle} n\_i n\_j+ J \sum\_{<i,j>}\vec{S\_i}\cdot \vec{S}\_j\notag \\ &~~~+J\_K\sum\_{i}\vec{S^e\_i}\cdot \vec{S}\_i+J\_{cs} \sum\_{\langle ij \rangle}\vec{S^e\_i}\cdot \vec{S}\_j+\vec{S}\_i\cdot \vec{S^e\_j}, \label{eqn:H}\end{aligned}$$ where P is the projection operator to forbid double occupancy in the familiar t-J model. $\vec{S^e\_i}=\frac{1}{2}\sum\_{\sigma \sigma'=\uparrow,\downarrow} c^\dagger\_{i;\sigma}\vec \sigma\_{\sigma \sigma'} c\_{i;\sigma'}$ is the spin operator of the itinerant electron. The first two lines describe a t-J model in the first layer and a Heisenberg spin 1/2 model in the second layer. We will call these two layers C layer and S layer respectively in what follows. The third line includes the inter-layer Kondo coupling. We have $J\_c=J=\frac{4 t^2}{U}$, $J\_{cs}=2\frac{t\_{12}^2}{U-U'-\Delta}+2\frac{t\_{12}^2}{U+U'+\Delta}$ and $J\_K=2\frac{t\_{\perp}^2}{U-U'-\Delta}+2\frac{t\_{\perp}^2}{U-U'+\Delta}$. Jc is the super-exchange term within the t-J layer. V is a repulsive interaction which we will set to zero later. J is the Heisenberg term in the spin layer. JK is the on-site Kondo coupling. Jcs is the nearest neighbour Kondo coupling arising from off-diagonal interlayer hopping. We will later see that Jcs does not change the physics qualitatively. Fig. [fig:kondolattice]. shows the geometry and the corresponding couplings pictorially. In the rest of the paper, we fix t = 1, J = Jc = 0.5 and study how the system evolves as we change Kondo coupling JK. We will simulate the model with both finite and infinite DMRG. The bond dimension varies from 500 to 8000 depending on parameters. The typical truncation error is at order 10− 8 or even smaller. Phase diagram ============= [h]1 [fig:phasediagram] We start by providing an illustrated phase diagram of the model in Fig. [fig:phasediagram]. Previous calculations have found a dominant ferromagnetic phase in the conventional Kondo lattice model with J = 0. Here we use J = 0.5t to get rid of the FM order. Then the phase diagram is dramatically different from that of the conventional Kondo lattice model with J = 0. At JK = 0, we can start from the layer decoupled phase. We know the itinerant electron in the C layer just forms a spinful Luttinger liquid, while the spin moments in the S layer form a gapless phase with one spin mode. We can dub this phase C2S1 because it has two charge modes and one spin mode. The itinerant electrons in the C layer form a Fermi surface with $2k\_F^\=\frac{x}{2} \times 2\pi$, which is different from the required value of the Luttinger theorem by 1/2 of 2π. This feature is similar to the fractional Fermi liquid (FL\) phase discussed in higher dimensions. Therefore, we dub this phase a fractional Luttinger liquid (LL\). The LL\ phase is stable in the negative JK regime. However, it is unstable to a spin-gapped Luther Emery (LE) liquid phase with a finite positive JK. In the large positive JK, we recover the expective Luttinger liquid (LL) as an analogue of the heavy Fermi liquid in higher dimensions. The LL phase has a Fermi surface with $2k\_F=\frac{1+x}{2} \times 2\pi$, satisfying the Yamanaka-Oshikawa theorem. Note that the central charge for the LL\, LE, and LL phases are c* = 3, 1, 2 respectively and they can be labelled as C1S2, C1S0, C1S1. [h]1 [fig:combined68113] [tbp]![Results for the intermediate region I at x=\frac{7}{31}, J_{cs}=0.5 J, V=0. (a) \Delta_S L for a few system sizes obtained from finite DMRG with bond dimension m=2000. L is the system size and \Delta_S is the spin gap. \Delta_S L is proportional to the inverse of the uniform spin susceptibility. The two dashed lines are at J_K=2.05 and J_K=2.7, which mark the phase boundaries of the intermediate region I. (b) Central charge fit from infinite DRMG with unit cell L=31. The central charge is c=1,3,2 for the three phases when increasing J_K. Here the bond dimension m varies from 500 to 2000 and we fit the central charge with the formula S=\frac{c}{6} \log \xi, where \xi is the correlation length obtained from the transfer matrix technique. (c) Spin-spin correlation function in momentum space. Here we use the total spin operator \vec{S^t}=\vec S+\vec{S^e}. (d) Density-density correlation function in momentum space. In (c)(d) the black dashed vertical line labels q=2k_F^=\frac{x}{2}\times 2\pi. The red dashed line labels q=4k_F=x \times 2\pi. The blue dashed line labels q=2k_F=\frac{1+x}{2}\times 2\pi. In (c)(d) the lines of different J_K are shifted, so the absolute value of the y-axis is meaningless. ](intermediate_results.png "fig:") [fig:phase1] Although the LL\ phase is unstable to the spin-gapped LE phase, one can view the LE phase as a descendant of the LL\* phase. Above the energy scale of the spin gap, we can still think of this phase as a LL\* phase with a small Fermi surface. Therefore we can ask how the small Fermi surface changes to the large Fermi surface in the large JK regime. In the regime of intermediate filling x ∈ (0.33, 0.43) the transition appeared to be of the first order, labelled as the red line in Fig. [fig:phasediagram]. As evidence of the first order transition, the spin gap Δs jumps to zero discontinuously and other physical quantities such as $V=2\langle \vec{S}\_i \cdot \vec{S^e\_i} \rangle$ also experience a jump, as shown in Figure [fig:combined68113]. The central charge changes from c = 1 in the LE phase to c = 2 in the LL phase directly at the transition. Intermediate region I --------------------- At small doping $x<\frac{1}{3}$ the LE phase evolves to the LL phase through an intermediate region I. Region I has a central charge c = 3 and a finite spin susceptibility, in agreement with a conformal field theory (CFT) description with both gapless charge and spin modes. We list results for intermediate region I at $x=\frac{7}{31}$, Jcs = 0.5J in Fig. [fig:phase1]. In Fig. [fig:phase1](a) we plot ΔSL from finite DMRG, where ΔS is the spin gap and L is the system size. ΔS is obtained from E(Szt = 1) − E(Szt = 0), where E(Szt = m) is the ground state energy of the sector of the total spin Sz = m sector. Note that the total Szt = Sz + Sze component is conserved in our calculation, so we can target a state at each Szt = m. It is known that the inverse of the uniform spin susceptibility χS− 1 ∝ ΔSL. When JK < 2.05, we can see that ΔSL increases with the system size L, indicating a finite spin gap in agreement with the LE phase. But when JK > 2.05, ΔSL is constant with system size, indicating a finite uniform spin susceptibility. This is expected from the scaling $\Delta\_S \sim \frac{1}{L}$ of a conformal field theory (CFT) description. Inside the intermediate region I, we find that the central charge is c = 3 from the infinite DMRG result in Fig. [fig:phase1](b). This central charge is larger than both the LE phase (c = 1) on the left and the LL phase (c = 2) on the right. One natural interpretation is that there are two Fermi surfaces per spin component in the intermediate region I, leading to two charge and two spin modes. Then one of the four modes gets gapped, giving c = 3. In the appendix, we will argue that a simple mean-field theory is able to explain the existence of several Fermi surfaces and show how a flat band scenario is able to explain a finite coherence length in the gapless system. To support the above picture, we indeed find that the peak of the spin-spin correlation function ⟨S⃗(q) ⋅ S⃗(q)⟩ is still at $2k\_F=\frac{1+x}{2}$ (in units of 2π) in the intermediate region (see Fig. [fig:phase1](c)), while the peak of density-density correlation functions ⟨N(q)N( − q)⟩ shifts from 2kF to 4kF gradually in the intermediate phase I, as shown in Fig. [fig:phase1](d). A gradually changing momentum is a signature of a split Fermi surface. The phase may be labelled as C1S2 or C2S1. We conjecture that it is C2S1 and there is only one spin mode, given that the peak of the spin-spin correlation function seems to be pinned at 2kF. But more analysis is needed to fully understand how a spin mode gets gapped starting from four modes. Except for the unusually odd central charge, the phase is otherwise consistent with a CFT. It easily converges in our numerical calculation with expected CFT behaviour. Dip of inverse charge compressibility and Luttinger parameter ------------------------------------------------------------- Overall at a generic filling, we find dips in both the inverse charge compressibility κc− 1 and the Luttinger parameter Kc in the intermediate regime, shown in Fig. [fig:luttinger]. They are extracted using the formulas κc− 1 = ∂μ/∂n = L(E(N + 2) + E(N − 2) − 2E(N))/4 and ⟨N(q)N( − q)⟩ = Kcq/2π at small q. We find $K\_c<\frac{1}{3}$ quite generically, indicating strong repulsive interaction. Given that $\kappa\_c=\frac{\pi K\_c}{\upsilon\_c}$ with υc as the charge velocity in a Luttinger liquid, a dip of both κc− 1 and Kc means that the velocity υc goes down even faster than Kc. This also means that the Drude weight Dc ∝ Kcυc gets much smaller in the intermediate region. All of these properties suggest that the intermediate region has a large repulsive interaction and slow charge velocity. It is a region where the charge compressibility tends to become large, while the Drude weight tends to vanish. ![a) Charge Luttinger parameter for different dopings x. The inset shows density-density correlations at small q at commensurate doping x=37/113 \approx 0.33 to demonstrate the quadratic behaviour. b) Inverse compressibility for different dopings x. The inset shows inverse compressibility for a commensurate doping x=37/113 \approx 0.33(blue line) and incommensurate x=35/113\approx 0.31(red line). The maximum bond dimension m=1000(finite DMRG). ](luttinger.png "fig:") [fig:luttinger] Charge density wave at commensurate filling ------------------------------------------- One consequence of the small Luttinger parameter Kc is that the ground states at commensurate filling such as $x=\frac{1}{3},\frac{1}{2}$ are charge density wave(CDW) insulators. That is because the umclapp terms become relevant for a small Luttinger parameter. To identify the insulating nature, we computed the inverse charge compressibility κc− 1 In the insulating phase we have κc− 1 = LΔc/2 which means that inverse compressibility diverges when L → ∞. As shown in the inset of Fig. [fig:luttinger](b), at commensurate filling κc− 1 is significantly larger than the corresponding one at incommensurate filling nearby. Moreover, in the insulator phase we expect that ⟨N(q)N( − q)⟩ ∼ q2 at small q. This is indeed the case as shown in the inset of Fig. [fig:luttinger](a). Unconventional criticality around the region I ============================================== After we have a general understanding of the global phase diagram in the (JK, x) parameter space, we now zoom in on the ‘critical region’ to understand the evolution from the LE phase to the LL phase. As shown in the phase diagram, we never find a direct continuous transition between the LE phase and the LL phase. Instead, we find either a first-order transition or another intermediate phase. The intermediate phase I appears to be well described by a CFT with c = 3. Below we are interested in how the spin gap closes when approaching this intermediate phase I, starting from the Luther-Emery liquid phase at small JK. Surprisingly we find that the transition between the Luther-Emery liquid phase and the c = 3 intermediate phase (in Region I of fig. [fig:phasediagram]) is not described by a usual conformal field theory(CFT). First, when approaching the critical point between the LE and the intermediate region I around JK = 2.05, ΔsL vanishes and the uniform spin susceptibility diverges, shown in Fig. [fig:phase1](a). This is already unexpected from a usual critical point described by CFT, where we should still expect $\Delta\_S \propto \frac{1}{L}$ and a finite ΔSL. Besides, when approaching the critical point from the spin-gapped phase, the correlation length remains finite, shown in Fig. [fig:corrlenQCP1]. At JK = 2.05, we still have a very small correlation length ( ξS ≈ 2) in spin channel corresponding to ξS− 1 ≈ 0.46. Then across the critical point, ξS− 1 jumps to 0. This is a clear signature that the dynamical exponent z at the critical point must be larger than 1, because otherwise in a relativistic critical theory we should expect the inverse spin correlation length ξS− 1 ∝ ΔS and also vanishes from the spin gapped side. ![Spin-Spin correlation functions when approaching the critical point J_K=2.05 from the spin-gapped Luther Emery liquid phase. At J_K=2.05, we still have a finite correlation length \xi_S^{-1}\approx 0.46. Then \xi^{-1}_S jumps to 0 into the intermediate phase with c=3. The jump of the correlation length from finite to infinite around J_K^c\approx 2.05 is in contrast to the continuous vanishing of \Delta_S L in Fig. [fig:phase1](a), indicating a critical point with dynamical exponent z>1, and probably z=+\infty. The parameters are the same as in Fig. [fig:phase1] with system size L=124 in finite DMRG. ](corr_len_qcp_I.png "fig:") [fig:corrlenQCP1] In summary, when approaching the critical point JKc ≈ 2.05 from the LE phase, we find that the spin gap ΔS goes to zero continuously, indicating a divergent correlation length ξt in the time direction. But the correlation length ξS in the real space stays finite (around 2 even at JK → JKc − ε, with ε an infinitesimal number). Then if we use the conventional scaling ξt ∼ ξSz, we reach a striking conclusion that z = + ∞. At small JK the spin excitation has a dispersion ω2 = c2(δq)2 + Δ2 with δq = q − π. Initially Δ increases with JK, but then Δ decreases when JK is close to the critical point JKc ≈ 2.05. Note in this ansatz the inverse of the real space spin correlation length is ξS− 1 = Δ/c. So the only way that ξS− 1 can stay finite is that the velocity c also vanishes along with the gap Δ. So we are in an unusual situation: the gap is closing while the dispersion of the excitation also becomes flat. We leave it to the future to develop an analytical theory of this kind of exotic criticality. Evidence of ultra-local criticality around region II ==================================================== The intermediate region I in Fig. [fig:phasediagram] seems to be well described by a CFT. In contrast, the intermediate region II (see Fig. [fig:phasediagram].) is much more exotic. In the following, we provide numerical evidence for ultra-local criticality around region II with dynamical exponent z = + ∞. We will also show evidence of gapless spin fluctuations in a range of momentum space in the dynamical spin structure factor. Inside region II, there is a small sub-region coloured red in Fig. [fig:phasediagram]. This small regime has a re-entrance of a spin gap. In the other places of region II, there is no spin gap. Instead, the inverse spin susceptibility even vanishes. We will show that it has a weak ferromagnetic moment and/or spin glass behaviour. We will discuss these two cases separately. But both of them have signatures of ultra-local criticality at a critical point (or region) of JK. ultra-local criticality between LE and LE2 phase ------------------------------------------------ ![Numerical results at J_{cs}=0 with x=\frac{7}{11}. (a) Spin gap \Delta_S from finite DMRG with system size L=99,110,121. The bond dimension is m=2000. One can see that there is a re-entrance of spin gapped phase when J_K>1.42, which we dub as LE_2. The dashed line is at J_K=1.42. (b) The inverse spin correlation length \xi_S^{-1} obtained from infinite DMRG for bond dimension m up to 3000. We use a unit cell size L=22. The dashed line is at J_K=1.42. \xi_S is obtained from the transfer matrix technique in the charge sector (Q,S_z^t)=(0,1). The value at m=\infty is extrapolated with the formula \xi^{-1}_S(m)=\xi^{-1}_S(m=\infty)+a \frac{1}{m}+b\frac{1}{m^2}. (c) Central charge from infinite DMRG. We use the relationship S=\frac{c}{6}\log \xi, where \xi is the correlation length in the charge sector (Q,S_z)=(0,0), which serves as an effective length scale of the infinite DMRG. We then get c(m)=6\frac{\partial S}{\partial \log \xi} where the derivative is calculated with the values from two nearby bond dimensions. For example, c(m=3000) is calculated from m=2500,3000. The dashed vertical line is at J_K=1.42. The two dashed horizontal lines label c=1,2. (d) The growth of the entanglement entropy S and the correlation length \xi with the bond dimension m. Here \xi_N is from the sector (Q,S_z)=(0,0) and should correspond to the density operator. We can see that S\sim \log m as in a typical critical phase, while \xi_N saturates to around 21. We also plot the correlation length \xi_S in the sector (Q,S_z)=(0,1) and \xi_C in the sector (Q,S_z)=(1,\frac{1}{2}). One can see that the single electron correlation length \xi_C is around 3 as in finite DMRG. For the spin correlation length \xi_S, it reaches \xi_S\approx 12.5 for m=6000, only slightly larger than the value from finite DMRG (see Fig. [fig:correlationJcs0] below). ](case1_ultra_local.png "fig:") [fig:case1results] We first look at the subregion inside region II coloured red in Fig. [fig:phasediagram]. We list results in Fig. [fig:case1results] at $x=\frac{7}{11}$ for Jcs = 0. We can see two spin-gapped phases with central charge c = 1. In between them, the central charge seems to approach 2 (see Fig. [fig:case1results](b)). There is one point of JK = 1.42 which has entanglement entropy growing faster than logξ in infinite DMRG so one can not extract a reasonable central charge (see Fig. [fig:case1results](c)). As shown in Fig. [fig:case1results](d), the entanglement entropy S scales as S ∼ logm with the bond dimension m, as in a usual CFT. However, the correlation length ξN (obtained in the sector (Q, Szt) = (0, 0)) has a tendency of saturation with logm. This indicates deviation from CFT behaviour at JK = 1.42. We also find that the spin correlation length ξS in real space is finite at JK = 1.42, shown in Fig. [fig:case1results](b). In Fig. [fig:correlationJcs0] we fit the correlation length also from finite DMRG at JK = 1.42. We confirm that the spin correlation length ξS ≈ 10. We also confirm that ξS actually becomes shorter with a larger bond dimension (see the Appendix). The single electron Green function has an even shorter correlation length of around 3 (see Fig. [fig:correlationJcs0](c)). The correlation length in the density channel ξN also appears to be finite with ξN ≈ 20 from Fig. [fig:correlationJcs0](d). This is also consistent with the infinite DMRG result in Fig. [fig:case1results](d). In summary from both finite and infinite DMRG, we find that the correlation lengths in single electron, spin and density channels are all finite, which are around 3, 10 and 20 respectively. ![(a)(b)(c) Correlation function and fitted correlation lengths in finite DMRG with system size L=132,154,176 for J_K=1.42 at filling x=\frac{7}{11}. The bond dimension is m=6000. (d) Evolution of the density-density correlation function with the bond dimension at L=176. Here we fix x_0=L/4. One can see that the correlation length \xi_N becomes shorter with increasing bond dimension, suggesting a finite correlation length also in the density channel. ](correlation_Jcs0_x=7_11.png "fig:") [fig:correlationJcs0] Here we will mainly focus on the spin channel. A finite correlation length ξS ≈ 10 is in contradiction with a vanishing spin gap (see Fig. [fig:case1results](a)) at JK = 1.42 if we assume an usual relativistic scaling ΔS ∝ ξS− 1 with dynamical exponent z = 1. This suggests a dynamical exponent z > 1 in the spin-spin correlation. To further check the dynamical exponent, we plot the imaginary part of the dynamical spin susceptibility Imχ+ −(ω, q) in Fig. [fig:spinspectroscopy], which is proportional to dynamical spin structure factor. The results are obtained from the TEBD algorithm (see the appendix for details). We apply the operator S−(L/2) to the ground state and then evolve the system under e− iHt to obtain ⟨S†(x, t)S−(L/2, 0)⟩. Imχs(ω, q) is then calculated by Fourier transformation. The total evolve time is T = 100 with a step δt = 0.15 for each TEBD step. The maximal bond dimension is set to be m = 500 in the calculation. In Fig. [fig:spinspectroscopy](a)(b) we get the expected spectroscopy results for the LL and LL\* phase. One can see the gapless mode at $2k\_F=\frac{1+x}{2}\times 2\pi$ for the LL phase and the gapless mode at $2k\_F^\=\frac{x}{2}\times 2\pi$ for the LL\ phase at JK = 0. For the LL\* the dominant spectral weight is actually at the momentum Q = π from the local spin moments. Around the gapless momentum, the dispersion is linear in agreement with a dynamical exponent z = 1. In contrast, at JK = 1.42, there are gapless spin fluctuations in a range of momentum instead of just one single momentum point. From the line cut at several momenta (see Fig. [fig:spinspectroscopy](d)), we can see that the spectral weight grows when decreasing ω for a range of momentum until it reaches a cutoff energy scale (around 0.035J) below which our calculation can not resolve. We note that the TEBD calculation is not quantitatively accurate, but qualitatively these results suggest that there is no dispersion within our numerical resolution. [tbp]![Dynamical spin structure factor \text{Im} \chi_{+-}(\omega,q) at J_{cs}=0 and x=\frac{7}{11}. Here we use a system size of L=110 in finite DMRG. (a) \text{Im} \chi_{+-}(\omega,q) for the LL phase at J_K=2. There is a gapless mode at q=0 and 2k_F=\frac{1+x}{2}\times 2\pi. (b) LL* phase at J_K=0. Note that the colour bar is significantly larger than other plots due to the large contribution from \mathbf Q=\pi from the local spin moments. There is also a gapless mode at q=2k_F^=\frac{x}{2}\times 2\pi corresponding to a small Fermi surface, but its spectral weight is smaller than that at q=\pi. (c) The unusual ultra-local critical behaviour at J_K=1.42. (d) Line cuts along several momenta q (in units of \frac{2\pi}{a}) at J_K=1.42. The vertical dashed line is at 0.035 J. Below 0.035 J the spectral weight vanishes as proportional to \omega, but this is due to numerical accuracy with a finite time evolution. In (a)(b)(c), the vertical red dashed line is at 2k_F=\frac{1+x}{2} \times 2\pi, while the blue dashed line is at 2k_F^=\frac{x}{2}\times 2\pi. In the TEBD calculation, the total time is T=100 with a step \delta t=0.15 and the maximal bond dimension is m=500. We use \eta=0.035 for the damping term. ](spin_spectroscopy_Jcs0_x=7_11.png "fig:") [fig:spinspectroscopy] ![(a)Spin gap \Delta_S from different system sizes at filling x=\frac{7}{11}, J_K=1.42 and J_{cs}=0. \Delta_S is obtained as E(S_z=1)-E(S_z=0). The bond dimension m ranges from 1000 to 6000. For L=176, the bond dimension is up to 8000. The dashed horizontal line indicates a gap of 2\times 10^{-4}. (b) Inverse charge compressibility \kappa_c^{-1} with J_K obtained from finite dmrg with bond dimension m=2000. \kappa_c=\frac{\partial n}{\partial \mu}. For finite size L with N number of electron, we use the formula: \kappa_c^{-1}=\frac{L}{4}(E(N+2)+E(N-2)-2E(N)). At J_K=1.42, \kappa^{-1}_c remains finite, indicating that this is still a compressible phase. Actually, the compressibility is largest around J_K=1.42.](spin_gap_ikappa_Jcs0_x_7_11.png "fig:") [fig:spingapikappa] As the dynamical spin structure factor ImχS(ω, q) is not accurate at the low energy limit, it is not clear whether the local criticality can survive down to zero energy limit or not. To understand the property at the zero energy limit, we need to rely on the ground state calculation. From the ground state calculation in finite DMRG (shown in Fig. [fig:correlationJcs0]) we already know that the spin correlation length is finite with ξS ≈ 10 at JK = 1.42. For a regular phase, we must also have a finite spin gap ΔS ∝ ξS− 1. We scale the spin gap at JK = 1.42 with bond dimension up to m = 6000 (m = 8000 for L = 176) in Fig. [fig:spingapikappa](a). The conclusion is that there is an almost zero spin gap ΔS ≈ 2 × 10− 4. We conjecture that the gapless modes in a finite momentum region found in ImχS(ω, q) can survive down to this scale ΔS ≈ 2 × 10− 4. Note that ΔS may still become truly zero if JK is fine-tuned to a critical point JKc ≈ 1.42. Because the calculation is quite time-consuming, it is impossible for us to do a dense sampling around JK = 1.42. Therefore it is still an open question whether the minimal spin gap is truly zero or not. However, even if there is a gap $\Delta\_S \lesssim 2\times 10^{-4}$ at true JKc, the ultra-local criticality behaviour still applies for the temperature scale above it. Given that almost any experimental measurement is likely performed well above this energy scale, we may conclude that ultra-local criticality exists for practical purposes. Lastly, we comment on the density correlations. The inverse charge compressibility κc− 1 in Fig. [fig:spingapikappa](b) shows a dip around JK = 1.42, indicating that this point is still a compressible phase with zero charge gap. Meanwhile in Fig. [fig:case1results](d) and Fig. [fig:correlationJcs0](d), we find that the correlation length in the density channel ξN is also finite. It is then possible that there is also ultra-local critical behaviour in the density channel. An intermediate weak ferromagnetic phase and ultra-local criticality -------------------------------------------------------------------- In the previous subsection, we find two spin-gapped domes for doping x around 0.61 − 0.63. Away from this narrow doping regime, we do not find another LE2 phase. Instead, there is a very narrow but finite intermediate region which hosts a very weak ferromagnetic (FM) moment and also ultra-local criticality around the phase boundary. ![(a)\Delta_S L with J_K for J_{cs}=0.5 J, x=\frac{4}{7} from finite DMRG for a few system sizes. \Delta_S=E(S_z=1)-E(S_z=0) is the spin gap and \Delta_S L is proportional to the inverse of the uniform static spin susceptibility. Here bond dimension is m=2000. (b) \Delta_S L at x=\frac{4}{7} for J_{cs}=0. Here bond dimension is m=2000. (c) \Delta_S L at x=\frac{21}{31} and J_{cs}=0.5 J. (d) \Delta_S(S_z=2) L at x=\frac{21}{31} and J_{cs}=0.5 J. \Delta_S(S_z=2)=E(S_z=2)-E(S_z=0). In (c)(d) the two dashed vertical lines label J_K=1.095 and J_K=1.15.](weak_FM_finite_DMRG.png "fig:") [fig:weakFM] In Fig. [fig:weakFM] we show that the inverse spin susceptibility χS− 1 ∝ ΔSL goes to basically zero in an intermediate region of JK at $x=\frac{4}{7}$ for both Jcs = 0 and Jcs = 0.5J. It also happens at a larger filling $x=\frac{21}{31}$ (see Fig. [fig:weakFM](c)(d)), suggesting that this is a quite generic phenomenon. In the following we focus on the parameter $x=\frac{21}{31}$ and Jcs = 0.5J. A vanishing ΔSL in the intermediate region indicates a divergence of the uniform spin susceptibility $\chi\_s=\frac{\partial S\_z}{\partial h}$ where h is the Zeeman field. It usually signatures an FM phase. However, here we find that the FM moment is very small and only at the order of 1% (1/L). For an FM phase, we expect that ΔS(Sz) = 0 for Sz < ML, where M is the ferromagnetic moment per site. In Fig. [fig:weakFM](d) we find that the gap of two spin flips becomes finite in this intermediate region except in a much smaller interval. Especially at the two boundaries JK = 1.095 and JK = 1.15 there is a finite ΔS(Sz = 2), indicating that $M<\frac{2}{L}$ if there is an FM order. ![Spin-spin correlation functions from finite DMRG results at J_{cs}=0.5 J, x=\frac{21}{31}. (a) J_K=1.09; (b) J_K=1.1; (c) J_K=1.12; (d) J_K=1.14. ](weak_FM_correlation.png "fig:") [fig:weakFMcorrelation] In Fig. [fig:weakFMcorrelation] we provide spin-spin correlation functions from JK = 1.09 to JK = 1.14 for Jcs = 0.5J and $x=\frac{21}{31}$. They are obtained from finite DMRG, so boundary effects may matter here. One can see that the correlation function saturates to a small but finite value, which should be identified as M2, where M is the FM moment. We can see that around JK = 1.12 − 1.14, the FM magnetic moment M is at order 10− 2. At JK = 1.09, it is much smaller and at order M ∼ 10− 4, which even decreases with the system size L. We note that a small FM moment of M ∼ 10− 2 is roughly polarizing one spin in the entire system with L ∼ 100. Hence it is even not clear whether this weak FM moment survives to the thermodynamic limit. We tend to conjecture that the weak FM moment is only a secondary effect in this region. Despite the weak FM moment, we still discover ultra-local criticality behaviour in the dynamical spin structure factor at the two boundaries of this intermediate phase. In Fig. [fig:spectroscopyweakFM1] we plot the imaginary dynamical spin susceptibility at the right boundary JK = 1.15 and the left boundary JK = 1.1 of the weak FM phase at Jcs = 0.5J and $x=\frac{21}{31}$. At JK = 1.15, one can see gapless spin modes in a range of momentum around q = 0. At JK = 1.1, the gapless spin modes are mainly concentrated at q = π. From the line cuts at fixed momentum in Fig. [fig:spectroscopyweakFM1](d) we can see that Imχ(ω, q) in a range of q around q = π have constant spectral weight at intermediate energy and then grows at lower energy at JK = 1.1. Within our numerical resolution, we can not see obvious dispersion, which suggests z = ∞ at least above the energy scale corresponding to our energy resolution (around 0.035J). Spin fluctuations around q = π should be mainly from the localized spin moments, suggesting that they are not fully Kondo screened at this parameter. ![Dynamical spin susceptibility \chi_{+-}(\omega,q) at the two boundaries of the weak FM phase at J_{cs}=0.5 J and x=\frac{21}{31}. The system size is L=62 in this calculation. (a) \text{Im} \chi_{+-}(\omega,q) at J_K=1.15. (b) Line cuts of \text{Im}\chi_{+-}(\omega,q) at several q at J_K=1.15. (c) (d) are at J_K=1.1. The calculation is done from the TEBD algorithm with total evolution time T=200 with a step \delta t=0.1. The bond dimension is m=2000. We include a damping term e^{- \eta t} with \eta=0.025 when performing the Fourier transformation along the time direction. The dashed vertical line is at \omega=0.035J. the momentum q is in units of 2\pi/a, where a is the lattice constant.](spectroscopy_weak_FM1.png "fig:") [fig:spectroscopyweakFM1] Inside the weak FM phase, we already know that there is a very small weak FM moment M ∼ 1%. However, we find the real part of the dynamical spin susceptibility is still dominated by q = π instead of q = 0 as can be seen in Fig. [fig:spectroscopyweakFM2]. At JK = 1.08, the system is still in a spin-gapped phase, one can see that the imaginary part of the dynamical spin susceptibility has spectral weights mainly around q = π with the spin gap already very small. Correspondingly, the real part of the dynamical spin susceptibility is largest around q = π. The real susceptibility at q = 0 is very weak here. We can also approach the weak FM phase from large JK. At JK = 1.15 (see Fig. [fig:spectroscopyweakFM2](d)) we find the real part of the dynamical spin susceptibility is large around q = π and around $q=2k\_F=\frac{1+x}{2}$. The susceptibility at q = 0 is again quite small here. Then when we decrease JK to JK = 1.14, there is some feature in Reχ+ −(ω, q) around q = 0, but the region with largest susceptibility is still around q = π (see Fig. [fig:spectroscopyweakFM2](c)). All of these results suggest that the weak FM moment is likely only a secondary effect, not the main property of this region. ![(a) \text{Re} \chi_{+-}(\omega,q) at J_K=1.08. (b)\text{Im} \chi_{+-}(\omega,q) at J_K=1.08. (c)\text{Re} \chi_{+-}(\omega,q) at J_K=1.14. (d)\text{Re} \chi_{+-}(\omega,q) at J_K=1.15. The parameters are the same as in Fig. [fig:spectroscopyweakFM1]. ](spectroscopy_weak_FM2.png "fig:") [fig:spectroscopyweakFM2] Next, we try to offer a possible explanation for the weak FM order. In the appendix, we will show that the dynamical spin structure factors inside the weak FM phase (such as JK = 1.12 and JK = 1.14) have gapless spin fluctuations in a region around q = 0. Such many gapless fluctuations may couple to the boundary of the system and order at certain small momentum including q = 0. Our interpretation is that the spin modes get dispersion-less in a region around q = 0 and thus is very easy to be stuck in a profile with zero momentum or a small momentum. In real experiments with even weak disorder, we conjecture the system will develop a spin glass order. However, we note that the weak FM or spin glass order has only a very small moment and we should still expect ultra-local critical behaviour above a very small energy scale. Discussion ========== Here we discuss the implications of the unusual behaviour we found at intermediate JK (phase boundary between LE and region I (or region II) of Fig. [fig:phasediagram]). One common property of the intermediate narrow region is that the spin velocity υs apparently becomes small and even vanishes. So the question is how to understand a vanishing spin velocity or spatial correlations. The usual way of dealing with a Kondo lattice model is through Abrikosov fermion theory: $$\vec S\_i=\frac{1}{2} f^\dagger\_{i;\sigma} \vec \sigma\_{\sigma \sigma'} f\_{i;\sigma'}$$ Then a Kondo screened phase at large JK is captured by a simple mean-field ansatz: HM = − b∑ici; σ†fi; σ where b ≠ 0 describes a Kondo-screened phase. If we consider a model with J = 0 for the localized spin and also ignore the Ruderman–Kittel–Kasuya–Yosida(RKKY) interaction, then the f band is perfectly flat. In this case, one expects $b \sim e^{- A \frac{t}{J\_K}}$ and one can get a very small velocity in the JK → 0 limit. This is the usual heavy Fermion picture. However, the model considered in this paper is different. We have a quite sizable J = 0.5t between the localized spin moments. Therefore, in the above theory, there is a sizable velocity υf for the f band itself and one should not expect a vanishing velocity in the mean field picture. A simple way to describe the Kondo breakdown transition is to let b vanish at a critical JKc starting from the large JK phase. However, in this picture, one does not expect υf (or the bandwidth of the f band) to vanish at the Kondo breakdown transition. Therefore, we still expect dispersion in spin fluctuations. This is in contrast to our discovery where we find dispersion-less gapless spin fluctuations in a range of momentum space within our energy resolution. For example, in Fig. [fig:spectroscopyweakFM1](c) we can see a gapless spin fluctuation continuum in a range of (ω, q) space around q = π. In 1D we indeed expect that the localized spin moments contribute a gapless mode at q = π in the Kondo breakdown phase, but it should have a strong dispersion proportional to J (for example, see Fig. [fig:spinspectroscopy](b) for the result at JK = 0). It is clear that the dispersion (or spatial correlation) of the localized spin moments also gets suppressed when approaching the critical regime. This is beyond the usual mean-field theory where only the hybridization cσ†fσ vanishes, while the bandwidth of the f band is still proportional to J. The lesson we learned is that in the `critical regime` between small JK and large Jk, the spatial correlation of local spin moments can get suppressed and then they fluctuate locally in real space. This is a property not captured by any theory we are aware of and thus offers a theoretical challenge. One can of course argue that this property may be special to this one-dimensional model and is irrelevant to higher dimensions. However, we note that similar features were observed in some neutron scattering experiments of higher dimensional heavy fermion material. This suggests that the ‘ultra-local critical’ phenomenon may be universal and relevant also for the higher dimensions. Conclusion ========== In summary, we present the DMRG results of a one-dimensional Kondo lattice model. Through varying the Kondo coupling JK, we studied how the Kondo breakdown phase evolves to the Kondo-screened Luttinger liquid phase. Around the intermediate regime, we discover signatures of ultra-local criticality with dynamical exponent z = + ∞ in the spin fluctuations. Similar phenomena have been reported in neutron scattering experiments of certain heavy fermion materials. Momentum-independent density fluctuations have also been observed at optical doping of hole-doped cuprates. However, the inevitable existence of disorder in real materials complicates the interpretations of these experimental results. Our numerical observations in a simple translation invariant model suggest that ultra-local criticality can arise intrinsically without the disorder. The fact that it shows up even in a 1D model may suggest that the phenomenon is quite universal around small to large Fermi surface transition and may be dimension independent. Theoretically, it was proposed that local criticality may be key to the solution of the mysterious strange metal phase. We plan to study the transport properties of this model in the near future to test this idea. Given the simplicity of a one-dimensional model, we hope future work on the current model will lead to progress on a better understanding of ultra-local criticality and strange metal. Acknowledgement =============== We thank Subir Sachdev for the discussions and comments on the manuscript. YHZ thanks Collin Broholm, T. Senthil and Mingru Yang for the discussions. AN thanks Pavel A. Volkov, Brenden Roberts and Alexei M. Tsvelik for the discussions. YHZ was supported by the National Science Foundation under Grant No. DMR-2237031. AN was supported by the U.S. National Science Foundation grant No. DMR-2002850. The DMRG simulations were performed using the TeNPy Library(version 0.10.0). Part of the numerical simulation was carried out at the Advanced Research Computing at Hopkins (ARCH) core facility (rockfish.jhu.edu), which is supported by the National Science Foundation (NSF) grant number OAC 1920103. Spin correlation length at the critical point near the intermediate region I ============================================================================ In this Appendix, we elaborate more on the transition between the LE phase and the intermediate region I phase. [h]0.45 a) [h]0.45 b) [fig:coherence] Fig. [fig:coherence](a) shows the spin gap in the LE phase, gradually vanishing as we approach the transition point. Fig. [fig:coherence](b) shows the inverse coherence length, extracted from the spin-spin correlation function. Right at the transition point JKc ≈ 2.0 we again observe the signature of the ultra criticality: the spin gap is almost zero, while the inverse coherence length is finite ξs− 1 ≈ 0.4. Assuming that transition between LE and I phase is of BKT type, the action in the massive phase is $$S=\frac{1}{2\pi K c}\frac{1}{\beta \Omega}\sum\_{k,n}\left( \omega\_n^2+c^2k^2+\Delta^2\right)\phi^\\_{k,n}\phi\_{k,n}.$$ Based on this action, the most general relation between the coherence length and the gap is ξ*s− 1 = Δ/c. The natural way to obtain finite coherence length at zero spin gap is to have c → 0. Note that c is different from vs, shown in Fig. [fig:phase1](a) of the main text. While vs is the property of the gapless state, c measures the dispersion of the excitation in the gapped phase. Therefore, c = 0 implies having a band of dispersionless excitations at energy Δ above the ground state. More detailed studies of this transition would be the purpose of our future work. ![(a)(b) Spin-Spin correlation function with bond dimension at system size L=154 and L=176. Here we set the initial value x_0=L/4. One can see that the correlation length \xi_S becomes shorter when increasing the bond dimension for L=176. ](convergence_Jcs_0_x_7_11.png "fig:") [fig:convergenceJcs0spinspin] Convergence of spin correlation length at $J\_K=1.42, J\_{cs}=0, x=\frac{7}{11}$ ================================================================================ ![(a)(b) Total spin \langle S^z_t(x) \rangle for system size L=154,176 at J_K=1.42, J_{cs}=0 and x=\frac{7}{11}.](convergence_Sz_Jcs0_x_7_11.png "fig:") [fig:convergenceJcs0Sz] We focus on Jcs = 0, JK = 1.42 at $x=\frac{7}{11}$. In the main text, we show that the spin correlation length is finite while the spin gap is very small. Here we provide more evidence that the spin correlation length is indeed finite. In Fig. [fig:convergenceJcs0spinspin] we show the spin-spin correlation function with bond dimension. At L = 176, we find that the correlation becomes more short-ranged when increasing the bond dimension, in agreement with a finite correlation length at the infinite bond dimension limit. ![Re-entrance of spin gapped phase at a different filling x=\frac{19}{31}.](spin_gap_x_19_31_Jcs0.png "fig:") [fig:anotherfilling] We also plot ⟨Stz(x)⟩ in Fig. [fig:convergenceJcs0Sz]. For a model with SU(2) spin rotation symmetry, we should expect ⟨Stz(x)⟩ = 0 in the ground state. At the intermediate regime such as JK = 1.42, ⟨Stz(x)⟩ vanishes quite slowly with the bond dimension. We need to use the bond dimension m = 8000 for L = 176. Lastly in fig. [fig:anotherfilling] we show that the same re-entrance of spin-gapped phase also exists at a different but close filling $x=\frac{19}{31}$ with Jcs = 0. TEBD calculation of the dynamical spin susceptibility ===================================================== We want to calculate the dynamical spin susceptibility: $$\chi\_{ij}(t-t')=i \langle [\vec S\_i(t),\vec S\_j(t')] \theta(t-t')$$ We can decompose χij to be: χij(t − tʹ) = χijzz(t − tʹ) + χijyy(t − tʹ) + χijzz(t − tʹ) We define: Cab; ij(t) = ⟨Sia(t)Sjb(0)⟩ − ⟨Sia⟩⟨Sjb⟩ We will mainly focus on calculation χij+ −(t − tʹ). We have the following identity: $$\chi\_{xx;ij}(t)+\chi\_{yy;ij}(t)=i\theta(t)(\frac{1}{2}(\chi\_{+-;ij}(t)+\chi\_{-+;ij}(t)))$$ where χ+ − ; ij(t) = iθ(t)(C+ − ; ij(t) − C+ − ; ji( − t)) χ− + ; ij(t) = iθ(t)(C− + ; ij(t) − C− + ; ji( − t)) ![\text{Im}\chi_{+-}(\omega,q). (a) J_K=3, J_{cs}=0.5 J and x=\frac{21}{93} with L=93. This is a Luttinger phase. The red and blue dashed lines are at q=2k_F=\frac{1+x}{2}\times 2\pi and q=2k_F^=\frac{x}{2}\times 2k_F respectively. (b) J_K=1, J_{cs}=0, x=\frac{7}{11} with L=110. This is in the spin-gapped Luther-Emery phase. In both calculations, we use bond dimension m=500, total time T=100 and a step \delta t=0.15. We use \eta=0.025 in performing the Fourier transformation along the time direction. ](spectroscopy_easy.png "fig:") [fig:spectroscopyeasy] We have the equation: C+ − ; i*j\(t) = C+ − ; j*i( − t) Therefore χ+ − ; ij(ω) = i∫0∞(C+ − ; ij(t) − C+ − ; ij\(t))eiω*t χ− + ; ij(ω) = i∫0∞(C− + ; ij(t) − C− + ; ij\(t))eiω*t In practice our evolution is limited to a finite time T, we use the following formula: χ− + ; ij(ω) = i∫0T(C− + ; ij(t) − C− + ; ij\(t))eiωte− η*t where η is a damping term. ![Change of \text{Im}\chi_{+-}(\omega,q) with the bond dimension. Here J_{cs}=0.5, J_K=1.1, L=62 and x=\frac{42}{62}. (a)(c) bond dimension m=1000. (b)(d) bond dimension m=2000. ](spectroscopy_bond_dimension.png "fig:") [fig:spectroscopyweakbonddimension] For our model, the above two quantities are the same. Therefore we focus on χ+ −. By doing Fourier transformation also in real space, we get: χ+ −(ω, q) = ∑iχ+ − ; i, j = L/2(ω)cos(q(i − L/2)) Here we use cos(qx) instead of eiqx by assuming the inversion symmetry respect to x = L/2. This can remove the artificial inversion breaking from numerical inaccuracy. In our calculation, we use the TEBD algorithm with T = 200 (assuming the hopping t = 1) with a step δt = 0.1. The largest bond dimension is m = 500 − 2000. When doing the Fourier transformation, we typically use η = 0.035. Note that without η we will find oscillations in χ(ω) because of the finite time T. To benchmark the calculation, we first try two points deep inside the LL and the spin-gapped LE phase, shown in Fig. [fig:spectroscopyeasy]. Here we only use bond dimension m = 500, but one can see the results are already quite reasonable. In (a) we find gapless mode at q = 2kF and q = 0, typical behaviour of a Luttinger liquid phase. There are also features at higher harmonics. In (b) we find a clear spin gap in the LE phase. These results demonstrate the validity of the TEBD calculation. To check that the TEBD results converge with the bond dimension even in the intermediate regime with ultra-local criticality, we plot Imχ+ −(ω, q) at the left boundary of the weak FM phase at $J\_{cs}=0.5, J\_K=1.1, x=\frac{42}{62}$ in Fig. [fig:spectroscopyweakbonddimension]. We can see that the results are qualitatively the same for bond dimensions m = 1000 and m = 2000. Both show dispersionless gapless spin fluctuations around q = π. The weak FM phase ================= In this section we add more discussions on the weak FM phase in the region II. More results at $J\_{cs}=0.5 J, x=\frac{21}{31}$ ------------------------------------------------ ![Dynamical spin susceptibility \chi_{+-}(\omega,q) at J_K=1.14 and J_K=1.12. (a)(c) \text{Re} \chi_{+-}(\omega,q). (b)(d) \text{Im}\chi_{+-}(\omega,q). One can see gapless spectral weights in a region around q=0. ](spectroscopy_weak_FM3.png "fig:") [fig:spectroscopyweakFM3] ![Line cut at fixed q of \text{Im} \chi_{+-}(\omega,q) inside the weak FM phase at J_K=1.12 and J_K=1.14. ](weak_FM_spectral_weight_appendix.png "fig:") [fig:weakFMspectralweightappendix] Here in Fig. [fig:spectroscopyweakFM3] and Fig. [fig:weakFMspectralweightappendix] we show more data to supplement the discussions in the main text on the weak FM regime for the parameter Jcs = 0.5J, V = 0 at the filling $x=\frac{21}{31}$. These results are again obtained from the TEBD calculation with system size L = 62. From the Imχ+ −(ω, q) at Jk = 1.12, 1.14 inside the weak FM phase, we can see gapless spin fluctuations in a region around q = 0. Another filling --------------- ![Infinite DMRG results at J_{cs}=0.5 J, x=\frac{4}{7}. We use unit cell size 22. (a) Central charge obtained from c=6 \frac{\partial S}{\partial \xi}. (b) The inverse of the spin correlation length \xi^{-1}_S. \xi_S is obtained from the transfer matrix technique in the sector (Q,S_z)=(0,1). (c) Scaling of the entanglement entropy and correlation lengths with bond dimension m at J_K=1.3. \xi_N, \xi_S, \xi_C correspond to density, spin and single electron operators, obtained from the sector (Q,S_z)=(0,0), (0,1), (1,\frac{1}{2}) respectively. (d) Scaling of the entanglement entropy and correlation lengths with the bond dimension m at J_K=1.31.](weak_FM_idmrg.png "fig:") [fig:weakFMidmrg] ![Log-x plot of the spin spin correlation function \langle S^\dagger(x) S^-(0)\rangle from infintie DMRG at x=\frac{4}{7}, J_{cs}=0.5J. (a) J_K=1.3. (b) J_K=1.31. ](weak_FM_spin_spin.png "fig:") [fig:weakFMspinspin] We also provide the results from a different filling $x=\frac{4}{7}$ for the weak FM region. From infinite DMRG in Fig. [fig:weakFMidmrg], we can see that there is a c = 2 region (with spin gap) in JK ∈ [1.295, 1.305] and a c = 3 region in JK ∈ [1.32, 1.38]. Compared to the finite DMRG in the main text (Fig. [fig:weakFM](a)), ΔSL approaches zero in the whole c = 2 region and the left part of the c = 3 region. Between JK = 1.31 and JK = 1.32, we again see that the entanglement entropy S grows with logm while the correlation length ξN saturates. In the region JK ∈ [1.295, 1.305], there is a quite short correlation length in the spin channel, as shown in Fig. [fig:weakFMspinspin](a). This is again at odds with the vanishing ΔSL from our finite DMRG calculation in Fig. [fig:weakFM](a). In this case, the charge correlation length is infinite and we have a regular central charge c = 2, presumably from two charge modes. However, the spin modes are not simply gapped given that ΔSL goes to zero even faster than any power of 1/L. We conjecture that in the spin channel there is still ultra-local criticality. When JK > 1.31, we find that ∣⟨S†(x)S−(0)⟩ seems to saturate to a finite value in the large x limit, indicating a very small FM moment. TEBD calculation of the single electron spectral density ======================================================== In this appendix, we demonstrate the existence of the small and large Fermi momenta by calculating the electron spectral function. The spectral function is defined as Ak(ω) = − ImGkR(ω) and GkR(t) = ⟨{ck(t), ck†(0)}⟩. The calculations were done for the Kondo-Heisenberg model to ensure that the Luther-Emery phase at JK → 0 coincides with the free electrons phase. $$H=-t \sum\_{<i,j>,\sigma}(c^\dag\_{i, \sigma}c\_{j, \sigma}+h.c)+J\_K\sum\_{i}\vec{S^e\_i} \vec{S}\_i+J\sum\_{i} \vec{S}\_i \vec{S}\_j$$ We studied the model at x = 0.25 and observed the same phases as in Figure [fig:phasediagram], with phase I being in the region JK ≈ (2.5, 2.8). Fig. [fig:TEBDCC] a) shows the small Kondo coupling regime with the spectral function simply matching the free electron band with small Fermi momentum 2kF = x/2. The spin gap should also be present but it is too small to be distinguishable. At larger JK the layer of spins starts to interfere and the band dispersion is modified, see Fig. [fig:TEBDCC] b). A similar band reconstruction happens if we implement a mean-field theory. Fig. [fig:TEBDCC] c) shows the dispersion in phase I which has a c = 3 central charge and a split in Fermi momentum. We do not clearly see two Fermi momenta, but the bands are strongly reconstructed and the accuracy is not enough to make a definite conclusion. Finally, at large Kondo coupling the system is in LL phase with a large Fermi momentum 2kF = (1 + x)/2, see Fig. [fig:TEBDCC] d). ![TEBD calculation of electron spectral function for Kondo-Heisenberg model. Figures a)-d) correspond to the Kondo coupling J_K=1,2.3,2.6,4 correspondingly, while other parameters are J=0.5,t=1. The length of the chain was L=80 and the number of electrons n_e=20, while the bond dimension m=500 with the time evolution t=12 and additional linear prediction interpolation.](TEBD_CC.png "fig:") [fig:TEBDCC] To obtain better frequency resolution we used a linear prediction algorithm, see. We extrapolated to times 3 times larger than the initial computation. The energies also need to be shifted by the chemical potential. We computed it separately by using the formula μc = (E(ne + 2) − E(ne − 2))/4. Mean-field theory ================= In this appendix, we show how some of the observations in the main part of the paper can be explained using simple mean-field analysis. As in the previous appendix, we start with the simpler Kondo-Heisenberg Hamiltonian $$H=-t \sum\_{<i,j>,\sigma}c^\dag\_{i, \sigma}c\_{j, \sigma}+\sum\_{<i,j>} \left(J \vec{S\_i} \vec{S}\_j+J\_{cs} \vec{S^e\_i} \vec{S}\_j \right)+J\_K\sum\_{i}\vec{S^e\_i} \vec{S}\_i.$$ We fractionalize spin S⃗ = 1/2fα†σαβfβ and use Pauli identities to obtain $$H=-2t \cos(k a) c^\dag\_k c\_k-\frac{J}{2}f^\dag\_{i \alpha}f\_{j \alpha}f^\dag\_{j \beta}f\_{i \beta}-\frac{J\_{cs}}{2}f^\dag\_{i \alpha}c\_{j \alpha}c^\dag\_{j \beta}f\_{i \beta}-\frac{J\_K}{2}f^\dag\_{i \alpha}c\_{i \alpha}c^\dag\_{i \beta}f\_{i \beta}.$$ After taking a large M limit (where M is a number of spin indices), we arrive at the following saddle-point equations: $$\begin{split} & P=-J\_K \langle f^\dag\_i c\_i\rangle=-\frac{J\_K}{V} \sum\_k\langle f^\dag\_k c\_k\rangle, \\ & Q=-J \langle f^\dag\_i f\_{i+1}\rangle=-\frac{J}{V} \sum\_k \cos(ka)\langle f^\dag\_k f\_k\rangle, \\ & P R=-J\_{cs} \langle f^\dag\_i c\_{i+1}\rangle=-\frac{J\_{cs}}{V} \sum\_k \cos(ka)\langle f^\dag\_k c\_k\rangle, \\ &\frac{1}{2}=\frac{1}{V}\sum\_k \langle f^\dag\_k f\_k\rangle. \\ \end{split} \label{eqn:mean\_field}$$ The corresponding free Hamiltonian is H = ( − 2tcos(ka) − μ)ck†ck + (2Qcos(ka) + λ)fk†fk + P(1 + 2Rcos(ka))ck†fk + P(1 + 2Rcos(ka))fk†ck. We take x = 2ρc = 0.7, J = 0.5,Jcs = 0.25 and t = 1, which is close to the parameters in the paper, and study the phase diagram as a function of JK. This requires solving Eq. [eqn:meanfield] self-consistently. At small JK < 1.4 there is only a trivial solution for the mean-field P = 0 which corresponds to an LL\* phase with a small Fermi surface kF = πx/2. For the intermediate JK ∈ [1.4, 2] there is a nontrivial solution which corresponds to a nontrivial hybridized Fermi-surfaces, see Fig. [fig:meanfieldx07](a). There is another solution with two Fermi surfaces, see Fig.[fig:meanfieldx07](b), which proves to be unstable after a comparison of Free energies. Finally, at large JK > 2, there is a nontrivial solution with a large Fermi surface kF = π(1 + x)/2, which correspond to an LL phase, see Fig.[fig:meanfieldx07](c). [h]0.32 (a) Two FS solution at JK = 1.8. [h]0.32 (b) Unstable solution at JK = 2.7. [h]0.32 (c) One large FS solution at JK = 3. [fig:meanfieldx07] The point of the analysis above is to show that a simple mean-field model is able to capture certain properties of the phase diagram, such as a transition from an LL\* to an LL phase through the intermediate phase with two FS. Though certain features, such as the spin gap, are missing from the above analysis, the rough boundaries of the phases match our DMRG predictions in Fig. [fig:phasediagram]. Now we address another question, stated in Section [sec:criticality] of the paper, namely the existence of the phase with zero gap and finite coherence length. To evaluate the coherence length, we computed the density-density response of the Hamiltonian in Eq. [eq:Hmf]. Equal time density-density correlation in Fourier space is: ⟨N(q)N( − q)⟩ = ∑k, ωn, qnGc(k + q, ωn + qn)Gc(k, ωn) For the Kondo-Heisenberg model, the Green function Gc is $$G\_c=\frac{1}{i \omega\_n-E\_+}\frac{E\_+-e\_f}{E\_+-E\_-}+\frac{1}{i \omega\_n-E\_-}\frac{E\_--e\_f}{E\_--E\_+}=\frac{u\_+^2}{i \omega\_n-E\_+}+\frac{u\_-^2}{i \omega\_n-E\_-},$$ where E+ and E− are the eigenvalues of the Hamiltonian H. The full density-density correlation functions are $$\begin{split} \langle N(q) N(-q)\rangle=&\sum\_{k}(n\_F(E\_{+q})-n\_F(E\_+)n\_B(E\_{+q}-E\_+)u\_+^2 u\_{+q}^2+(n\_F(E\_{-q})-n\_F(E\_-)n\_B(E\_{-q}-E\_-)u\_-^2 u\_{-q}^2+\\ &+(n\_F(E\_{+q})-n\_F(E\_-)n\_B(E\_{+q}-E\_-)u\_-^2 u\_{+q}^2+(n\_F(E\_{-q})-n\_F(E\_+)n\_B(E\_{-q}-E\_+)u\_+^2 u\_{-q}^2, \end{split}$$ where each term correspond to scattering from the band Ei to the band Ej, i, j = + , − . The leading contribution will be given by the scattering in the lower band E− → E−, since this band hosts zero-energy excitations. We computed the density-density response for the simple mean-field model in Fig. [fig:meanfielddd](a) at two temperatures. At small temperatures, we observed a typical power law decay with an infinite coherence length, see Fig. [fig:meanfielddd](b). However, at finite temperatures comparable to the dispersion of the f electron, the density-density correlation demonstrated an exponential decay, with finite coherence length and zero spin gap. Though our DMRG calculations are done at zero temperature, there is always a finite error in energy calculation δEs. Therefore, we can successfully explain the existence of a finite coherence length and zero spin gap by assuming that the dispersion of the f electron is even smaller Q ≪ δEs. δEs is very small in our calculation, and the f fermion band should be almost flat, which is quite a surprise for a finite J. [h]0.32 (a) Hybridized bands. [h]0.32 (b) Small temperature T = 0.001. [h]0.32 (c) Finite temperature T = 0.1. [fig:meanfielddd] Numerical signatures of ultra-local criticality in a one dimensional Kondo lattice model ======================================================================================== Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling JK at fixed doping x. At large positive JK, we confirm the expected conventional Luttinger liquid phase with $2k\_F=\frac{1+x}{2}$ (in units of 2π), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the JK ≤ 0 side, our simulation finds the existence of a fractional Luttinger liquid (LL\) phase with $2k\_F=\frac{x}{2}$, accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL\) phase in higher dimensions. The LL\* phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive JK. Then we mainly focus on the ‘critical regime’ between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of JK, the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035J) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z = + ∞. The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference. Introduction ============ The study of quantum phase transition between a small Fermi surface phase and a large Fermi surface phase is a central topic in modern quantum condensed matter physics and may be closely related to the strange metals observed in heavy Fermion systems and in hole-doped high Tc cuprates. The standard Landau-Ginzburg theory involves the onset of a symmetry-breaking order and its fluctuation. However, a number of experiments in heavy Fermion systems do not appear to be consistent with the simple spin-density-wave (SDW) approach. It was suggested that the transition in heavy fermion systems may be characterized by a jump in Fermi surface volume resulting from Kondo breakdown, rather than fluctuations in symmetry-breaking orders. There have been many attempts to formulate a framework of an exotic transition following different approaches, such as extended dynamical mean field theory (EDMFT), fractionalization and slave boson theory, ancilla qubit theory. However, a well-established theoretical description of such a Kondo breakdown transition is still elusive. In this paper, we take a microscopic approach to avoid uncontrolled approximations usually existing in low-energy effective field theory methods. Specifically, we will numerically simulate a one-dimensional Kondo lattice model using density matrix renormalization group (DMRG). DMRG has been demonstrated to be an unbiased method with excellent performance in one dimension (1D). Therefore, the numerical results should be reliable. The only question is whether there is anything interesting in a 1D model. We will show that the answer is yes and we find a critical point or phase which seems to support local criticality behaviour. We note that there already exist a few numerical studies of the Kondo lattice model in one dimension, but to our best knowledge, there is no detailed study of how a Kondo breakdown phase at negative JK evolves to the Luttinger liquid in the large positive JK at a generic filling. The model we study consists of a t-J model of itinerant electron and a Heisenberg model of spin 1/2 chain. They couple to each other through a Kondo coupling JK. At a density x for the itinerant electron, we vary JK to study the phase diagram. In the JK ≤ 0 side, the ground state has one charge mode and two spin modes (C1S2), where the localized spin 1/2 moments provide an additional gapless mode with momentum Q = π. The itinerant electron forms a Luttinger liquid with $2k\_F^\=\frac{x}{2}$ (in units of 2π). The phase is an analogue of the fractional Fermi liquid (FL\) phase in higher dimension and we call it fractional Luttinger liquid (LL\). In the large positive J*K we find the expected Luttinger liquid (LL) phase with $2k\_F=\frac{1+x}{2}$ (in units of 2π), which is an analogue of the heavy Fermi liquid (HFL) phase in the higher dimensional Kondo lattice model. Therefore, we have the same problem of small to large Fermi surface evolution as in higher dimensions. Complexity arises in one dimension because the LL\* phase is unstable at small positive JK and transitions to a Luther-Emery liquid (LE) phase with a spin gap and only one gapless charge mode. The LE phase is best described as a descendant of the LL\* phase. It is similar to a superconductor phase in a higher dimension and above the energy scale of the spin gap it smoothly connects to the LL\* phase. We note, that in the heavy Fermion experiments, the transitions between the small and large Fermi surface metals are typically covered by a superconductor dome. Thus, the situation in 1D is similar to higher dimension and we will try to understand the nature of the evolution from the LE phase to the LL phase upon increasing JK. The hope is that there may also be a ‘strange metal’ critical point or a phase in between. As the LE phase descends from the LL\* phase and we are not aware of any way to construct it from the LL phase, we do not expect any obvious continuous transition between the LE and LL phases. Indeed, we find that there is either a first-order transition or an intermediate region in between. We will focus on the latter case and provide evidence of local criticality behaviour beyond the familiar Luttinger liquid or conformal field theory (CFT) descriptions. At one point (or a narrow region) of JK, we find that the spin gap is almost vanishing, while there is still a finite correlation length in equal time spin-spin correlation function in real space. Meanwhile, the dynamical spin structure factor S(ω, q) ∼ ImχS(ω, q) obtained from the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a range of the momentum space above an energy cutoff (around 0.035J, J is the Heisenberg spin coupling) imposed by the numerical accuracy itself. Such behaviour resembles what is called local criticality. We note, that in the literature sometimes local criticality is also used for the case where only the self-energy is momentum independent, while there is still a significant spatial correlation. In this weaker case, the dynamical exponent is still finite. The behaviour in our model is closer to a stronger definition with an infinite dynamical exponent. Therefore, we follow Ref. and call it ultra-local criticality to be distinguished from the weaker definition. The discovery of ultra-local critical spin fluctuations above a small energy scale is quite remarkable, as this phenomenon is not generally believed to be possible in a reasonable model with translation invariance and a finite-dimensional Hilbert space at each site. The existence of ultra-local criticality also has significant implications for our understanding of the strange metal. For example, it may be a loophole of the anomaly approach of non-Fermi liquid and it is known that ultra-local critical spin fluctuations with a constant spectral function over frequency can lead to a marginal Fermi liquid and linear T resistivity. On the experimental side, similar local critical behaviours have been discovered in neutron scattering measurements of some heavy Fermion materials. One may worry that the experimental results arise from disorder effects. Our numerical observation of similar local critical behaviours in a clean model strongly suggests that such a phenomenon may likely be intrinsic and does not need disorders. On the theoretical side, similar behaviour has been discussed in holographic theory from the gravity side and dubbed as ‘semi-local quantum fluid’. However, we are not aware of a well-established theory of ultra-local criticality for a local and translation invariant quantum lattice model directly. We hope our numerical confirmation of the existence of ultra-local criticality will stimulate theoretical efforts in this direction. Lastly, we propose to simulate the Kondo lattice model in a bilayer optical lattice with a potential difference, which hopefully will provide more information at finite temperatures and higher dimensions. Layer selective Mott localization and Kondo lattice model in bilayer optical lattice ==================================================================================== [h]1 [fig:kondolattice] Here we first propose to simulate a Kondo lattice model in bilayer optical lattice, as has been experimentally realized in Ref. . One new requirement now is that we need to add a potential difference Δ between the two layers. The system is described by a bilayer Hubbard model: $$\begin{aligned} H&=\Delta\sum\_i n\_{i;1}-t\sum\_{a=1,2}\sum\_{\sigma=\uparrow,\downarrow}\sum\_{\langle ij \rangle}( c^\dagger\_{i;a \sigma}c\_{j;a \sigma}+h.c.)\notag \\ &-t\_{12}\sum\_{a=1,2}\sum\_{\sigma=\uparrow,\downarrow}\sum\_{\langle ij \rangle}(c^\dagger\_{i;1\sigma}c\_{j;2\sigma}+c^\dagger\_{i;2\sigma}c\_{j;1\sigma}+h.c.)\notag \\ &-t\_{\perp}\sum\_{a,\sigma}\sum\_{i}(c^\dagger\_{i;1\sigma}c\_{i;2\sigma}+h.c.)-\mu \sum\_{a=1,2} \sum\_i n\_{i;a}\notag \\ &+\frac{U}{2}\sum\_{a}\sum\_i n\_{i;a}(n\_{i;a}-1)+U'\sum\_i n\_{i;1}n\_{i;2}, \label{eq:bilayer\_Hubbard\_model}\end{aligned}$$ where ni; a = ∑σci; aσ†ci; aσ is the density at site i for layer a = 1, 2. ni = ni; 1 + ni; 2 is the total density at site i. We also define the average density $n=\frac{1}{N\_s}\sum\_i n\_i$, where Ns is the total number of sites in the system. Here a = 1, 2 labels the two layers and t⊥ is the inter-layer vertical tunnelling. A non-zero Δ > 0 is caused by a displacement field or a potential difference between the two layers. We will stay in the limit U > > t and U > > Uʹ. We assume t⊥, t < Δ < U − Uʹ. At density n = 1, we have a Mott insulator with one particle at the layer 2. Then at density n = 1 + x with x ∈ (0, 1), the doped additional particle enters the layer 1 to reduce the on-site Hubbard U. In this case the layer 2 is always Mott localized and provides a spin 1/2 moment. The itinerant electron in the layer 1 is described by a t − J model which then couples to the local moment of the layer 2 through a Kondo coupling. At low energy we can deal with an effective Kondo lattice model: $$\begin{aligned} H&=-t P\sum\_{<i,j>,\sigma}(c^\dag\_{i, \sigma}c\_{j, \sigma}+h.c)P+J\_c \sum\_{\langle ij \rangle}\vec{S^e\_i}\cdot \vec{S^e\_j}\notag \\ &~~~+(V-\frac{1}{4}J\_c )\sum\_{\langle ij \rangle} n\_i n\_j+ J \sum\_{<i,j>}\vec{S\_i}\cdot \vec{S}\_j\notag \\ &~~~+J\_K\sum\_{i}\vec{S^e\_i}\cdot \vec{S}\_i+J\_{cs} \sum\_{\langle ij \rangle}\vec{S^e\_i}\cdot \vec{S}\_j+\vec{S}\_i\cdot \vec{S^e\_j}, \label{eqn:H}\end{aligned}$$ where P is the projection operator to forbid double occupancy in the familiar t-J model. $\vec{S^e\_i}=\frac{1}{2}\sum\_{\sigma \sigma'=\uparrow,\downarrow} c^\dagger\_{i;\sigma}\vec \sigma\_{\sigma \sigma'} c\_{i;\sigma'}$ is the spin operator of the itinerant electron. The first two lines describe a t-J model in the first layer and a Heisenberg spin 1/2 model in the second layer. We will call these two layers C layer and S layer respectively in what follows. The third line includes the inter-layer Kondo coupling. We have $J\_c=J=\frac{4 t^2}{U}$, $J\_{cs}=2\frac{t\_{12}^2}{U-U'-\Delta}+2\frac{t\_{12}^2}{U+U'+\Delta}$ and $J\_K=2\frac{t\_{\perp}^2}{U-U'-\Delta}+2\frac{t\_{\perp}^2}{U-U'+\Delta}$. Jc is the super-exchange term within the t-J layer. V is a repulsive interaction which we will set to zero later. J is the Heisenberg term in the spin layer. JK is the on-site Kondo coupling. Jcs is the nearest neighbour Kondo coupling arising from off-diagonal interlayer hopping. We will later see that Jcs does not change the physics qualitatively. Fig. [fig:kondolattice]. shows the geometry and the corresponding couplings pictorially. In the rest of the paper, we fix t = 1, J = Jc = 0.5 and study how the system evolves as we change Kondo coupling JK. We will simulate the model with both finite and infinite DMRG. The bond dimension varies from 500 to 8000 depending on parameters. The typical truncation error is at order 10− 8 or even smaller. Phase diagram ============= [h]1 [fig:phasediagram] We start by providing an illustrated phase diagram of the model in Fig. [fig:phasediagram]. Previous calculations have found a dominant ferromagnetic phase in the conventional Kondo lattice model with J = 0. Here we use J = 0.5t to get rid of the FM order. Then the phase diagram is dramatically different from that of the conventional Kondo lattice model with J = 0. At JK = 0, we can start from the layer decoupled phase. We know the itinerant electron in the C layer just forms a spinful Luttinger liquid, while the spin moments in the S layer form a gapless phase with one spin mode. We can dub this phase C2S1 because it has two charge modes and one spin mode. The itinerant electrons in the C layer form a Fermi surface with $2k\_F^\=\frac{x}{2} \times 2\pi$, which is different from the required value of the Luttinger theorem by 1/2 of 2π. This feature is similar to the fractional Fermi liquid (FL\) phase discussed in higher dimensions. Therefore, we dub this phase a fractional Luttinger liquid (LL\). The LL\ phase is stable in the negative JK regime. However, it is unstable to a spin-gapped Luther Emery (LE) liquid phase with a finite positive JK. In the large positive JK, we recover the expective Luttinger liquid (LL) as an analogue of the heavy Fermi liquid in higher dimensions. The LL phase has a Fermi surface with $2k\_F=\frac{1+x}{2} \times 2\pi$, satisfying the Yamanaka-Oshikawa theorem. Note that the central charge for the LL\, LE, and LL phases are c* = 3, 1, 2 respectively and they can be labelled as C1S2, C1S0, C1S1. [h]1 [fig:combined68113] [tbp]![Results for the intermediate region I at x=\frac{7}{31}, J_{cs}=0.5 J, V=0. (a) \Delta_S L for a few system sizes obtained from finite DMRG with bond dimension m=2000. L is the system size and \Delta_S is the spin gap. \Delta_S L is proportional to the inverse of the uniform spin susceptibility. The two dashed lines are at J_K=2.05 and J_K=2.7, which mark the phase boundaries of the intermediate region I. (b) Central charge fit from infinite DRMG with unit cell L=31. The central charge is c=1,3,2 for the three phases when increasing J_K. Here the bond dimension m varies from 500 to 2000 and we fit the central charge with the formula S=\frac{c}{6} \log \xi, where \xi is the correlation length obtained from the transfer matrix technique. (c) Spin-spin correlation function in momentum space. Here we use the total spin operator \vec{S^t}=\vec S+\vec{S^e}. (d) Density-density correlation function in momentum space. In (c)(d) the black dashed vertical line labels q=2k_F^=\frac{x}{2}\times 2\pi. The red dashed line labels q=4k_F=x \times 2\pi. The blue dashed line labels q=2k_F=\frac{1+x}{2}\times 2\pi. In (c)(d) the lines of different J_K are shifted, so the absolute value of the y-axis is meaningless. ](intermediate_results.png "fig:") [fig:phase1] Although the LL\ phase is unstable to the spin-gapped LE phase, one can view the LE phase as a descendant of the LL\* phase. Above the energy scale of the spin gap, we can still think of this phase as a LL\* phase with a small Fermi surface. Therefore we can ask how the small Fermi surface changes to the large Fermi surface in the large JK regime. In the regime of intermediate filling x ∈ (0.33, 0.43) the transition appeared to be of the first order, labelled as the red line in Fig. [fig:phasediagram]. As evidence of the first order transition, the spin gap Δs jumps to zero discontinuously and other physical quantities such as $V=2\langle \vec{S}\_i \cdot \vec{S^e\_i} \rangle$ also experience a jump, as shown in Figure [fig:combined68113]. The central charge changes from c = 1 in the LE phase to c = 2 in the LL phase directly at the transition. Intermediate region I --------------------- At small doping $x<\frac{1}{3}$ the LE phase evolves to the LL phase through an intermediate region I. Region I has a central charge c = 3 and a finite spin susceptibility, in agreement with a conformal field theory (CFT) description with both gapless charge and spin modes. We list results for intermediate region I at $x=\frac{7}{31}$, Jcs = 0.5J in Fig. [fig:phase1]. In Fig. [fig:phase1](a) we plot ΔSL from finite DMRG, where ΔS is the spin gap and L is the system size. ΔS is obtained from E(Szt = 1) − E(Szt = 0), where E(Szt = m) is the ground state energy of the sector of the total spin Sz = m sector. Note that the total Szt = Sz + Sze component is conserved in our calculation, so we can target a state at each Szt = m. It is known that the inverse of the uniform spin susceptibility χS− 1 ∝ ΔSL. When JK < 2.05, we can see that ΔSL increases with the system size L, indicating a finite spin gap in agreement with the LE phase. But when JK > 2.05, ΔSL is constant with system size, indicating a finite uniform spin susceptibility. This is expected from the scaling $\Delta\_S \sim \frac{1}{L}$ of a conformal field theory (CFT) description. Inside the intermediate region I, we find that the central charge is c = 3 from the infinite DMRG result in Fig. [fig:phase1](b). This central charge is larger than both the LE phase (c = 1) on the left and the LL phase (c = 2) on the right. One natural interpretation is that there are two Fermi surfaces per spin component in the intermediate region I, leading to two charge and two spin modes. Then one of the four modes gets gapped, giving c = 3. In the appendix, we will argue that a simple mean-field theory is able to explain the existence of several Fermi surfaces and show how a flat band scenario is able to explain a finite coherence length in the gapless system. To support the above picture, we indeed find that the peak of the spin-spin correlation function ⟨S⃗(q) ⋅ S⃗(q)⟩ is still at $2k\_F=\frac{1+x}{2}$ (in units of 2π) in the intermediate region (see Fig. [fig:phase1](c)), while the peak of density-density correlation functions ⟨N(q)N( − q)⟩ shifts from 2kF to 4kF gradually in the intermediate phase I, as shown in Fig. [fig:phase1](d). A gradually changing momentum is a signature of a split Fermi surface. The phase may be labelled as C1S2 or C2S1. We conjecture that it is C2S1 and there is only one spin mode, given that the peak of the spin-spin correlation function seems to be pinned at 2kF. But more analysis is needed to fully understand how a spin mode gets gapped starting from four modes. Except for the unusually odd central charge, the phase is otherwise consistent with a CFT. It easily converges in our numerical calculation with expected CFT behaviour. Dip of inverse charge compressibility and Luttinger parameter ------------------------------------------------------------- Overall at a generic filling, we find dips in both the inverse charge compressibility κc− 1 and the Luttinger parameter Kc in the intermediate regime, shown in Fig. [fig:luttinger]. They are extracted using the formulas κc− 1 = ∂μ/∂n = L(E(N + 2) + E(N − 2) − 2E(N))/4 and ⟨N(q)N( − q)⟩ = Kcq/2π at small q. We find $K\_c<\frac{1}{3}$ quite generically, indicating strong repulsive interaction. Given that $\kappa\_c=\frac{\pi K\_c}{\upsilon\_c}$ with υc as the charge velocity in a Luttinger liquid, a dip of both κc− 1 and Kc means that the velocity υc goes down even faster than Kc. This also means that the Drude weight Dc ∝ Kcυc gets much smaller in the intermediate region. All of these properties suggest that the intermediate region has a large repulsive interaction and slow charge velocity. It is a region where the charge compressibility tends to become large, while the Drude weight tends to vanish. ![a) Charge Luttinger parameter for different dopings x. The inset shows density-density correlations at small q at commensurate doping x=37/113 \approx 0.33 to demonstrate the quadratic behaviour. b) Inverse compressibility for different dopings x. The inset shows inverse compressibility for a commensurate doping x=37/113 \approx 0.33(blue line) and incommensurate x=35/113\approx 0.31(red line). The maximum bond dimension m=1000(finite DMRG). ](luttinger.png "fig:") [fig:luttinger] Charge density wave at commensurate filling ------------------------------------------- One consequence of the small Luttinger parameter Kc is that the ground states at commensurate filling such as $x=\frac{1}{3},\frac{1}{2}$ are charge density wave(CDW) insulators. That is because the umclapp terms become relevant for a small Luttinger parameter. To identify the insulating nature, we computed the inverse charge compressibility κc− 1 In the insulating phase we have κc− 1 = LΔc/2 which means that inverse compressibility diverges when L → ∞. As shown in the inset of Fig. [fig:luttinger](b), at commensurate filling κc− 1 is significantly larger than the corresponding one at incommensurate filling nearby. Moreover, in the insulator phase we expect that ⟨N(q)N( − q)⟩ ∼ q2 at small q. This is indeed the case as shown in the inset of Fig. [fig:luttinger](a). Unconventional criticality around the region I ============================================== After we have a general understanding of the global phase diagram in the (JK, x) parameter space, we now zoom in on the ‘critical region’ to understand the evolution from the LE phase to the LL phase. As shown in the phase diagram, we never find a direct continuous transition between the LE phase and the LL phase. Instead, we find either a first-order transition or another intermediate phase. The intermediate phase I appears to be well described by a CFT with c = 3. Below we are interested in how the spin gap closes when approaching this intermediate phase I, starting from the Luther-Emery liquid phase at small JK. Surprisingly we find that the transition between the Luther-Emery liquid phase and the c = 3 intermediate phase (in Region I of fig. [fig:phasediagram]) is not described by a usual conformal field theory(CFT). First, when approaching the critical point between the LE and the intermediate region I around JK = 2.05, ΔsL vanishes and the uniform spin susceptibility diverges, shown in Fig. [fig:phase1](a). This is already unexpected from a usual critical point described by CFT, where we should still expect $\Delta\_S \propto \frac{1}{L}$ and a finite ΔSL. Besides, when approaching the critical point from the spin-gapped phase, the correlation length remains finite, shown in Fig. [fig:corrlenQCP1]. At JK = 2.05, we still have a very small correlation length ( ξS ≈ 2) in spin channel corresponding to ξS− 1 ≈ 0.46. Then across the critical point, ξS− 1 jumps to 0. This is a clear signature that the dynamical exponent z at the critical point must be larger than 1, because otherwise in a relativistic critical theory we should expect the inverse spin correlation length ξS− 1 ∝ ΔS and also vanishes from the spin gapped side. ![Spin-Spin correlation functions when approaching the critical point J_K=2.05 from the spin-gapped Luther Emery liquid phase. At J_K=2.05, we still have a finite correlation length \xi_S^{-1}\approx 0.46. Then \xi^{-1}_S jumps to 0 into the intermediate phase with c=3. The jump of the correlation length from finite to infinite around J_K^c\approx 2.05 is in contrast to the continuous vanishing of \Delta_S L in Fig. [fig:phase1](a), indicating a critical point with dynamical exponent z>1, and probably z=+\infty. The parameters are the same as in Fig. [fig:phase1] with system size L=124 in finite DMRG. ](corr_len_qcp_I.png "fig:") [fig:corrlenQCP1] In summary, when approaching the critical point JKc ≈ 2.05 from the LE phase, we find that the spin gap ΔS goes to zero continuously, indicating a divergent correlation length ξt in the time direction. But the correlation length ξS in the real space stays finite (around 2 even at JK → JKc − ε, with ε an infinitesimal number). Then if we use the conventional scaling ξt ∼ ξSz, we reach a striking conclusion that z = + ∞. At small JK the spin excitation has a dispersion ω2 = c2(δq)2 + Δ2 with δq = q − π. Initially Δ increases with JK, but then Δ decreases when JK is close to the critical point JKc ≈ 2.05. Note in this ansatz the inverse of the real space spin correlation length is ξS− 1 = Δ/c. So the only way that ξS− 1 can stay finite is that the velocity c also vanishes along with the gap Δ. So we are in an unusual situation: the gap is closing while the dispersion of the excitation also becomes flat. We leave it to the future to develop an analytical theory of this kind of exotic criticality. Evidence of ultra-local criticality around region II ==================================================== The intermediate region I in Fig. [fig:phasediagram] seems to be well described by a CFT. In contrast, the intermediate region II (see Fig. [fig:phasediagram].) is much more exotic. In the following, we provide numerical evidence for ultra-local criticality around region II with dynamical exponent z = + ∞. We will also show evidence of gapless spin fluctuations in a range of momentum space in the dynamical spin structure factor. Inside region II, there is a small sub-region coloured red in Fig. [fig:phasediagram]. This small regime has a re-entrance of a spin gap. In the other places of region II, there is no spin gap. Instead, the inverse spin susceptibility even vanishes. We will show that it has a weak ferromagnetic moment and/or spin glass behaviour. We will discuss these two cases separately. But both of them have signatures of ultra-local criticality at a critical point (or region) of JK. ultra-local criticality between LE and LE2 phase ------------------------------------------------ ![Numerical results at J_{cs}=0 with x=\frac{7}{11}. (a) Spin gap \Delta_S from finite DMRG with system size L=99,110,121. The bond dimension is m=2000. One can see that there is a re-entrance of spin gapped phase when J_K>1.42, which we dub as LE_2. The dashed line is at J_K=1.42. (b) The inverse spin correlation length \xi_S^{-1} obtained from infinite DMRG for bond dimension m up to 3000. We use a unit cell size L=22. The dashed line is at J_K=1.42. \xi_S is obtained from the transfer matrix technique in the charge sector (Q,S_z^t)=(0,1). The value at m=\infty is extrapolated with the formula \xi^{-1}_S(m)=\xi^{-1}_S(m=\infty)+a \frac{1}{m}+b\frac{1}{m^2}. (c) Central charge from infinite DMRG. We use the relationship S=\frac{c}{6}\log \xi, where \xi is the correlation length in the charge sector (Q,S_z)=(0,0), which serves as an effective length scale of the infinite DMRG. We then get c(m)=6\frac{\partial S}{\partial \log \xi} where the derivative is calculated with the values from two nearby bond dimensions. For example, c(m=3000) is calculated from m=2500,3000. The dashed vertical line is at J_K=1.42. The two dashed horizontal lines label c=1,2. (d) The growth of the entanglement entropy S and the correlation length \xi with the bond dimension m. Here \xi_N is from the sector (Q,S_z)=(0,0) and should correspond to the density operator. We can see that S\sim \log m as in a typical critical phase, while \xi_N saturates to around 21. We also plot the correlation length \xi_S in the sector (Q,S_z)=(0,1) and \xi_C in the sector (Q,S_z)=(1,\frac{1}{2}). One can see that the single electron correlation length \xi_C is around 3 as in finite DMRG. For the spin correlation length \xi_S, it reaches \xi_S\approx 12.5 for m=6000, only slightly larger than the value from finite DMRG (see Fig. [fig:correlationJcs0] below). ](case1_ultra_local.png "fig:") [fig:case1results] We first look at the subregion inside region II coloured red in Fig. [fig:phasediagram]. We list results in Fig. [fig:case1results] at $x=\frac{7}{11}$ for Jcs = 0. We can see two spin-gapped phases with central charge c = 1. In between them, the central charge seems to approach 2 (see Fig. [fig:case1results](b)). There is one point of JK = 1.42 which has entanglement entropy growing faster than logξ in infinite DMRG so one can not extract a reasonable central charge (see Fig. [fig:case1results](c)). As shown in Fig. [fig:case1results](d), the entanglement entropy S scales as S ∼ logm with the bond dimension m, as in a usual CFT. However, the correlation length ξN (obtained in the sector (Q, Szt) = (0, 0)) has a tendency of saturation with logm. This indicates deviation from CFT behaviour at JK = 1.42. We also find that the spin correlation length ξS in real space is finite at JK = 1.42, shown in Fig. [fig:case1results](b). In Fig. [fig:correlationJcs0] we fit the correlation length also from finite DMRG at JK = 1.42. We confirm that the spin correlation length ξS ≈ 10. We also confirm that ξS actually becomes shorter with a larger bond dimension (see the Appendix). The single electron Green function has an even shorter correlation length of around 3 (see Fig. [fig:correlationJcs0](c)). The correlation length in the density channel ξN also appears to be finite with ξN ≈ 20 from Fig. [fig:correlationJcs0](d). This is also consistent with the infinite DMRG result in Fig. [fig:case1results](d). In summary from both finite and infinite DMRG, we find that the correlation lengths in single electron, spin and density channels are all finite, which are around 3, 10 and 20 respectively. ![(a)(b)(c) Correlation function and fitted correlation lengths in finite DMRG with system size L=132,154,176 for J_K=1.42 at filling x=\frac{7}{11}. The bond dimension is m=6000. (d) Evolution of the density-density correlation function with the bond dimension at L=176. Here we fix x_0=L/4. One can see that the correlation length \xi_N becomes shorter with increasing bond dimension, suggesting a finite correlation length also in the density channel. ](correlation_Jcs0_x=7_11.png "fig:") [fig:correlationJcs0] Here we will mainly focus on the spin channel. A finite correlation length ξS ≈ 10 is in contradiction with a vanishing spin gap (see Fig. [fig:case1results](a)) at JK = 1.42 if we assume an usual relativistic scaling ΔS ∝ ξS− 1 with dynamical exponent z = 1. This suggests a dynamical exponent z > 1 in the spin-spin correlation. To further check the dynamical exponent, we plot the imaginary part of the dynamical spin susceptibility Imχ+ −(ω, q) in Fig. [fig:spinspectroscopy], which is proportional to dynamical spin structure factor. The results are obtained from the TEBD algorithm (see the appendix for details). We apply the operator S−(L/2) to the ground state and then evolve the system under e− iHt to obtain ⟨S†(x, t)S−(L/2, 0)⟩. Imχs(ω, q) is then calculated by Fourier transformation. The total evolve time is T = 100 with a step δt = 0.15 for each TEBD step. The maximal bond dimension is set to be m = 500 in the calculation. In Fig. [fig:spinspectroscopy](a)(b) we get the expected spectroscopy results for the LL and LL\* phase. One can see the gapless mode at $2k\_F=\frac{1+x}{2}\times 2\pi$ for the LL phase and the gapless mode at $2k\_F^\=\frac{x}{2}\times 2\pi$ for the LL\ phase at JK = 0. For the LL\* the dominant spectral weight is actually at the momentum Q = π from the local spin moments. Around the gapless momentum, the dispersion is linear in agreement with a dynamical exponent z = 1. In contrast, at JK = 1.42, there are gapless spin fluctuations in a range of momentum instead of just one single momentum point. From the line cut at several momenta (see Fig. [fig:spinspectroscopy](d)), we can see that the spectral weight grows when decreasing ω for a range of momentum until it reaches a cutoff energy scale (around 0.035J) below which our calculation can not resolve. We note that the TEBD calculation is not quantitatively accurate, but qualitatively these results suggest that there is no dispersion within our numerical resolution. [tbp]![Dynamical spin structure factor \text{Im} \chi_{+-}(\omega,q) at J_{cs}=0 and x=\frac{7}{11}. Here we use a system size of L=110 in finite DMRG. (a) \text{Im} \chi_{+-}(\omega,q) for the LL phase at J_K=2. There is a gapless mode at q=0 and 2k_F=\frac{1+x}{2}\times 2\pi. (b) LL* phase at J_K=0. Note that the colour bar is significantly larger than other plots due to the large contribution from \mathbf Q=\pi from the local spin moments. There is also a gapless mode at q=2k_F^=\frac{x}{2}\times 2\pi corresponding to a small Fermi surface, but its spectral weight is smaller than that at q=\pi. (c) The unusual ultra-local critical behaviour at J_K=1.42. (d) Line cuts along several momenta q (in units of \frac{2\pi}{a}) at J_K=1.42. The vertical dashed line is at 0.035 J. Below 0.035 J the spectral weight vanishes as proportional to \omega, but this is due to numerical accuracy with a finite time evolution. In (a)(b)(c), the vertical red dashed line is at 2k_F=\frac{1+x}{2} \times 2\pi, while the blue dashed line is at 2k_F^=\frac{x}{2}\times 2\pi. In the TEBD calculation, the total time is T=100 with a step \delta t=0.15 and the maximal bond dimension is m=500. We use \eta=0.035 for the damping term. ](spin_spectroscopy_Jcs0_x=7_11.png "fig:") [fig:spinspectroscopy] ![(a)Spin gap \Delta_S from different system sizes at filling x=\frac{7}{11}, J_K=1.42 and J_{cs}=0. \Delta_S is obtained as E(S_z=1)-E(S_z=0). The bond dimension m ranges from 1000 to 6000. For L=176, the bond dimension is up to 8000. The dashed horizontal line indicates a gap of 2\times 10^{-4}. (b) Inverse charge compressibility \kappa_c^{-1} with J_K obtained from finite dmrg with bond dimension m=2000. \kappa_c=\frac{\partial n}{\partial \mu}. For finite size L with N number of electron, we use the formula: \kappa_c^{-1}=\frac{L}{4}(E(N+2)+E(N-2)-2E(N)). At J_K=1.42, \kappa^{-1}_c remains finite, indicating that this is still a compressible phase. Actually, the compressibility is largest around J_K=1.42.](spin_gap_ikappa_Jcs0_x_7_11.png "fig:") [fig:spingapikappa] As the dynamical spin structure factor ImχS(ω, q) is not accurate at the low energy limit, it is not clear whether the local criticality can survive down to zero energy limit or not. To understand the property at the zero energy limit, we need to rely on the ground state calculation. From the ground state calculation in finite DMRG (shown in Fig. [fig:correlationJcs0]) we already know that the spin correlation length is finite with ξS ≈ 10 at JK = 1.42. For a regular phase, we must also have a finite spin gap ΔS ∝ ξS− 1. We scale the spin gap at JK = 1.42 with bond dimension up to m = 6000 (m = 8000 for L = 176) in Fig. [fig:spingapikappa](a). The conclusion is that there is an almost zero spin gap ΔS ≈ 2 × 10− 4. We conjecture that the gapless modes in a finite momentum region found in ImχS(ω, q) can survive down to this scale ΔS ≈ 2 × 10− 4. Note that ΔS may still become truly zero if JK is fine-tuned to a critical point JKc ≈ 1.42. Because the calculation is quite time-consuming, it is impossible for us to do a dense sampling around JK = 1.42. Therefore it is still an open question whether the minimal spin gap is truly zero or not. However, even if there is a gap $\Delta\_S \lesssim 2\times 10^{-4}$ at true JKc, the ultra-local criticality behaviour still applies for the temperature scale above it. Given that almost any experimental measurement is likely performed well above this energy scale, we may conclude that ultra-local criticality exists for practical purposes. Lastly, we comment on the density correlations. The inverse charge compressibility κc− 1 in Fig. [fig:spingapikappa](b) shows a dip around JK = 1.42, indicating that this point is still a compressible phase with zero charge gap. Meanwhile in Fig. [fig:case1results](d) and Fig. [fig:correlationJcs0](d), we find that the correlation length in the density channel ξN is also finite. It is then possible that there is also ultra-local critical behaviour in the density channel. An intermediate weak ferromagnetic phase and ultra-local criticality -------------------------------------------------------------------- In the previous subsection, we find two spin-gapped domes for doping x around 0.61 − 0.63. Away from this narrow doping regime, we do not find another LE2 phase. Instead, there is a very narrow but finite intermediate region which hosts a very weak ferromagnetic (FM) moment and also ultra-local criticality around the phase boundary. ![(a)\Delta_S L with J_K for J_{cs}=0.5 J, x=\frac{4}{7} from finite DMRG for a few system sizes. \Delta_S=E(S_z=1)-E(S_z=0) is the spin gap and \Delta_S L is proportional to the inverse of the uniform static spin susceptibility. Here bond dimension is m=2000. (b) \Delta_S L at x=\frac{4}{7} for J_{cs}=0. Here bond dimension is m=2000. (c) \Delta_S L at x=\frac{21}{31} and J_{cs}=0.5 J. (d) \Delta_S(S_z=2) L at x=\frac{21}{31} and J_{cs}=0.5 J. \Delta_S(S_z=2)=E(S_z=2)-E(S_z=0). In (c)(d) the two dashed vertical lines label J_K=1.095 and J_K=1.15.](weak_FM_finite_DMRG.png "fig:") [fig:weakFM] In Fig. [fig:weakFM] we show that the inverse spin susceptibility χS− 1 ∝ ΔSL goes to basically zero in an intermediate region of JK at $x=\frac{4}{7}$ for both Jcs = 0 and Jcs = 0.5J. It also happens at a larger filling $x=\frac{21}{31}$ (see Fig. [fig:weakFM](c)(d)), suggesting that this is a quite generic phenomenon. In the following we focus on the parameter $x=\frac{21}{31}$ and Jcs = 0.5J. A vanishing ΔSL in the intermediate region indicates a divergence of the uniform spin susceptibility $\chi\_s=\frac{\partial S\_z}{\partial h}$ where h is the Zeeman field. It usually signatures an FM phase. However, here we find that the FM moment is very small and only at the order of 1% (1/L). For an FM phase, we expect that ΔS(Sz) = 0 for Sz < ML, where M is the ferromagnetic moment per site. In Fig. [fig:weakFM](d) we find that the gap of two spin flips becomes finite in this intermediate region except in a much smaller interval. Especially at the two boundaries JK = 1.095 and JK = 1.15 there is a finite ΔS(Sz = 2), indicating that $M<\frac{2}{L}$ if there is an FM order. ![Spin-spin correlation functions from finite DMRG results at J_{cs}=0.5 J, x=\frac{21}{31}. (a) J_K=1.09; (b) J_K=1.1; (c) J_K=1.12; (d) J_K=1.14. ](weak_FM_correlation.png "fig:") [fig:weakFMcorrelation] In Fig. [fig:weakFMcorrelation	arxiv_0000680
Critical cluster volumes in hierarchical percolation ======================================================== We consider long-range Bernoulli bond percolation on the d-dimensional hierarchical lattice in which each pair of points x and y are connected by an edge with probability 1 − exp( − β∥x − y∥− d − α), where 0 < α < d is fixed and β ≥ 0 is a parameter. We study the volume of clusters in this model at its critical point β = βc, proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up-to-constants estimates on the tail of the volume of the cluster of the origin, denoted K, at criticality, namely $$\P\_{\beta\_c}(\|K\|\geq n) \asymp \begin{cases} n^{-(d-\alpha)/(d+\alpha)} & d < 3\alpha\\ n^{-1/2}(\log n)^{1/4} & d=3\alpha \\ n^{-1/2} & d>3\alpha. \end{cases}$$ In particular, we compute the critical exponent δ to be (d + α)/(d − α) when d is below the upper-critical dimension dc = 3α and establish the precise order of polylogarithmic corrections to scaling at the upper-critical dimension itself. Interestingly, we find that these polylogarithmic corrections are not those predicted to hold for nearest-neighbour percolation on $\Z^6$ by Essam, Gaunt, and Guttmann (J. Phys. A 1978). Our work also lays the foundations for the study of the scaling limit of the model: In the high-dimensional case d ≥ 3α we prove that the sized-biased distribution of the volume of the cluster of the origin inside a box converges under suitable normalization to a chi-squared random variable, while in the low-dimensional case d < 3α we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in ℓp \ {0} if and only if p > 2d/(d + α). Introduction ============ A central goal of mathematical physics and statistical mechanics is to understand critical phenomena: the intricate, fractal-like behaviour exhibited by many systems at and near points of phase transition. Mathematically, such critical phenomena are often described by power laws, and the computation of the critical exponents governing these power laws is one of the core projects guiding the field. A particularly fascinating aspect of critical behaviour is its dependence on the dimension of the lattice on which the model is defined: Typically, each given model of interest, such as Bernoulli percolation or the Ising model, has an upper-critical dimension dc (which is 6 for percolation and 4 for the Ising model) such that in dimensions d > dc the model has mean-field critical behaviour, meaning roughly that it has the same critical behaviour on $\Z^d$ as in ‘geometrically trivial’ settings such as the complete graph or the binary tree. In contrast, for d < dc the critical behaviour of the model should be significantly influenced by the finite-dimensional nature of the lattice, with critical exponents that are distinct from their mean-field values. At the upper-critical dimension itself it is expected that mean-field behaviour almost holds, so that exponents take their mean-field values but quantities of interest scale in a way that differs from their mean-field scaling by polylogarithmic factors. On the other hand, once the dimension is fixed it is expected that all relevant large-scale critical behaviours are universal, meaning in particular that nearest-neighbour Bernoulli bond percolation on any two Euclidean lattices of the same dimension should have the same critical exponents. See e.g. for a general overview in the context of percolation and for background on the high-dimensional theory. Although this story is uncontroversial at a heuristic level, its rigorous verification has been extremely difficult. In the specific case of Bernoulli percolation we now have a fairly good understanding in two dimensions and high dimensions, while there seems to be a complete lack of tools to adequately address the problem either in intermediate dimensions 2 < d < 6 or at the upper-critical dimension d = 6. Even at a heuristic level, there are no exact values conjectured for critical exponents in intermediate dimensions nor any reason to expect that closed-form expressions for these exponents should exist. In this paper we study critical percolation on the hierarchical lattice (defined in ), a discrete analogue of d-dimensional p-adic space Qpd which exhibits similar phenomena to critical percolation on $\Z^d$ but is significantly easier to study due to the very large amount of symmetry it enjoys. We focus on the distribution of critical cluster volumes, establishing a precise description of these distributions in both finite and infinite volume and in all three regimes d < dc, d = dc, and d > dc. In particular, we compute the critical exponent δ which governs the power law decay of the tail of the volume of the cluster of the origin via $\P\_{\beta\_c}(\|K\|\geq n) \approx n^{-1/\delta}$ as well as the precise polylogarithmic corrections to this power law decay at the upper-critical dimension. We believe that our results concerning the volume tail at and below the upper-critical dimension are the first of their kind for Bernoulli percolation in any context outside the mean-field[1](#fn1) or planar settings. Interestingly, we find that the logarithmic corrections to scaling at the upper-critical dimension are not the same as those predicted to hold for nearest-neighbour percolation on $\Z^6$, in contrast to what is known to occur in other models such as weakly self-avoiding walk and the φ4 model. (See for details.) Our work also lays the groundwork for the study of the scaling limit of the model as discussed in detail in. All our results build upon our analysis of the critical two-point function in our earlier work, the behaviour of which is much simpler than that of the cluster volume tails and is not sensitive to the difference between the low-dimensional and high-dimensional regimes. The study of hierarchical models of statistical mechanics goes back fifty years to the work of Dyson and Baker, who independently introduced hierarchical interactions as simplifications of long-range Euclidean interactions in the context of the Ising model. Since then, hierarchical and p-adic models have attracted a great deal of interest throughout mathematical and theoretical physics, with Dyson’s paper having over 1000 citations. They are particularly popular in the context of rigorous renormalization group[2](#fn2) analyses of critical phenomena, a topic we discuss in more detail in. We refer the reader to for comprehensive overviews of the use of hierarchical and p-adic models in physics, to for detailed overviews of the rigorous renormalization group analysis of hierarchical spin systems, and to Tao’s blog post for a broad informal discussion of the use of hierarchical models in other parts of mathematics. Closest to the topic of the present paper, several significant works have studied hierarchical models at their upper-critical dimension, establishing asymptotic Gaussianity for both Ising and φ4 and computing logarithmic corrections to scaling for weakly self-avoiding walk and the φ4 model. These hierarchical works also played important roles guiding subsequent work establishing analogous results in the Euclidean case as surveyed in. All these works are, however, centred in a crucial way around spin systems, with weakly self-avoiding walk having been analyzed at its upper-critical dimension only via an equivalent supersymmetric spin system. Since percolation is not known to have any exact spin system representations, it does not fit into this framework and requires a new suite of tools to be developed for its study. Besides the need to move beyond the setting of spin systems, there are several further important technical differences between our work and the previous literature on critical phenomena in hierarchical models. Indeed, most significant previous work on critical behaviour for hierarchical spin systems has required the model under consideration to be a “small perturbation of a Gaussian free field” in some appropriate sense. For example, the computations of the logarithmic corrections to scaling for the φ4 model and weakly self-avoiding walk at the upper-critical dimension as summarized in require the relevant parameters describing the quartic perturbation to the Gaussian measure or the energetic cost of self-intersections to be small as appropriate, and do not apply to the Ising model (which can be thought of as a strong-coupling limit of the φ4 model) or to strictly self-avoiding walk. Indeed, in an exception that proves the rule, Hara, Hattori, and Watanabe proved that the hierarchical Ising model is asymptotically Gaussian at the upper-critical dimension using a computer-assisted proof in which the renormalization group map is iterated 70 times numerically and the output is shown to satisfy an appropriate perturbative criterion for asymptotic Gaussianity (which in turn built on the work of Bleher and Sinai and Newman ). Similar restrictions apply to the study of renormalization group fixed points below the upper-critical dimension as described in, where the analysis is carried out under the assumption that the dimension is very close to the critical dimension. In contrast, our analysis of hierarchical percolation is completely non-perturbative, and does not require any conditions on the parameters used to define the model. Moreover, we believe that our paper is the first to give a a reasonably uniform and complete treatment of all three cases d < dc, d = dc, and d > dc for a specific statistical mechanics model. As such, we are optimistic that some of the new techniques we develop can also be used to make new advances for spin systems, particularly for the Ising and Potts models via their random-cluster model representations., which establishes a kind of “marginal triviality” theorem for hierarchical percolation at the upper critical dimension, may be particularly interesting from this perspective; it is inspired in part by the recent breakthrough result of Aizenman and Duminil-Copin on marginal triviality for the Ising model on $\Z^4$, although the details of the proof are very different and significantly simpler. Before moving on, let us stress again that a key motivation behind the study of hierarchical models is that they provide insight into the behaviour of Euclidean models. Indeed, significant advances on the understanding of the critical two-point function for long-range percolation on $\Z^d$, which is believed to have the same critical exponents as hierarchical percolation in certain regimes as discussed in detail in, have very recently been made by the author and by Bäumler and Berger, with both papers making progress primarily by finding ways to implement parts of the hierarchical analysis of in the Euclidean setting. As such, we are optimistic that the methods we develop here will lead to new results about long-range percolation on $\Z^d$ and perhaps in the more distant future to new results about nearest-neighbour percolation also. The model --------- Two graphical representations of the hierarchical lattice H22, which can be identified with H41 by a bijection that transforms distances by a power. On the left, the vertices of $\bbH^1\_4$ are represented by the leaves of the tree, and the distance between two distinct leaves is 4 to the power of the height of their most recent common ancestor. On the right, the distance between two points is equal to the side-length of the smallest distinguished dyadic box containing both points. For each d ≥ 1 and L ≥ 2, the hierarchical lattice $\mathbbm{H}^d\_L$ is defined to be the countable abelian group $\bigoplus\_{i=1}^\infty \mathbb{T}^d\_L = \{x =(x\_1,x\_2,\ldots) \in (\mathbb{T}^d\_L)^\N : x\_i =0$ for all but finitely many i ≥ 0}, where $\mathbb{T}^d\_L=(\Z/L\Z)^d$ is the discrete torus of side length L, equipped with the group-invariant ultrametric $$\\|y-x\\| := \begin{cases} 0 & x=y\\ L^{h(x,y)} & x \neq y \end{cases} \qquad \text{ where }h(x,y)=\max\{i \geq 1: x\_i \neq y\_i\}.$$ We refer to ultrametric balls of radius $L^n \mathbbm{1}(n>0)$ in $\bbH^d\_L$ as **n-blocks*, and write Λn* for the n-block containing the origin. When n ≥ 1, each n-block Λ contains Ld (n − 1)-blocks which we call the children of Λ. As a metric space, the hierarchical lattice can also be defined recursively by taking Λ0 = {0} and, for each n ≥ 0, taking Λn + 1 to be the disjoint union of Ld copies of Λn with distances ∥x − y∥ = Ln + 1 for every pair x, y ∈ Λn + 1 belonging to separate copies of Λn. When L = p is prime one can think of $\bbH^1\_p$ as a discrete analogue of d-dimensional p-adic space Qp just as Z is a discrete analogue of R. See e.g. for background on this perspective in the context of statistical mechanics. We say that a kernel $J: \bbH^d\_L \times \bbH^d\_L \to [0,\infty)$ is translation-invariant if J(x, y) = J(0, y − x) for every $x,y\in \bbH^d\_L$, that J is symmetric if J(x, y) = J(y, x) for every $x,y\in \bbH^d\_L$, and that J is integrable if $\sum\_{y\in \bbH^d\_L} J(x,y)<\infty$ for every $x\in \bbH^d\_L$. We say that a translation-invariant kernel J is radially symmetric if it is invariant under all isometries of $\bbH^d\_L$, or equivalently if J(x) can be expressed as a function of ∥x∥. Given a symmetric, integrable kernel $J:\bbH^d\_L \times \bbH^d\_L \to [0,\infty)$ and β ≥ 0, long-range percolation on $\bbH^d\_L$ is defined to be the random graph with vertex set $\bbH^d\_L$ in which each pair {x, y} is included as an edge of the graph independently at random with inclusion probability 1 − e− βJ(x, y). Edges that are included in this random graph are also referred to as open. (Note that the hierarchical lattices $\bbH^d\_L$ and $\bbH^1\_{L^d}$ are related by a bijection that transforms distances by a dth power, so that long-range percolation on $\bbH^d\_L$ with exponent α is equivalent to long-range percolation on $\bbH^1\_{L^d}$ with exponent α/d.) We write $\P\_\beta=\P\_{\beta,J}$ and $\E\_\beta = \E\_{\beta,J}$ for probabilities and expectations taken with respect to the law of the resulting random graph. (For most of the paper we will fix β = βc and drop it from notation.) The integrability of J implies that this graph is locally finite (i.e., has finite vertex degrees) almost surely. The connected components of the resulting random graph are known as clusters and the critical probability βc = βc(d, L, J) is defined by $$\beta\_c = \inf\bigl\{\beta \geq 0: \text{ there exists an infinite cluster with positive probability}\bigr\},$$ which is always positive when J is translation-invariant and integrable. For translation-invariant, symmetric kernels J satisfying J(x, y) ∼ A∥x − y∥− d − α as x − y → ∞ for some α > 0 and A > 0, the critical parameter βc is finite if and only if 0 < α < d, and in this case the phase transition is continuous in the sense that there are no infinite clusters at criticality. This continuity theorem was made quantitative in our recent series of papers (see also ), where we showed in particular that for radially-symmetric kernels of this form with 0 < α < d the critical connection probabilities always satisfy $$\label{eq:two\_point\_intro} \P\_{\beta\_c}(x \leftrightarrow y) \asymp \\|x-y\\|^{-d+\alpha},$$ where {x ↔ y} denotes the event that x and y are connected by an open path; the point-to-point connection probability $\P\_{\beta\_c}(x \leftrightarrow y)$ is often referred to as the two-point function. This was used to prove that the model has mean-field critical behaviour when d > 3α and does not have mean-field critical behaviour when d < 3α, so that dc = 3α may be regarded as the upper-critical dimension of the model. Note however that both the two-point function estimate and its proof are completely unaffected by the distinction between the high-dimensional (d > 3α) and low-dimensional (d < 3α) regimes and that there are no polylogarithmic corrections to the two-point function at the upper-critical dimension d = 3α. As mentioned above, the goal of this paper is to understand more refined properties of the model at criticality, all of which will exhibit different behaviours in the three cases d < 3α, d = 3α, and d > 3α. For notational convenience and clarity of exposition, we will work throughout the paper with the specific choice of translation-invariant kernel $$J(x,y) = J(y-x) = \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|y-x\\|^{-d-\alpha} = \sum\_{n = h(y-x)}^\infty L^{-(d+\alpha)n}.$$ Of course one could equivalently consider the kernel J(x, y) = ∥y − x∥− d − α (or any other constant rescaling thereof), since multiplying the kernel by a constant is equivalent to a change of the parameter β. We expect our analysis to extend to other translation-invariant kernels satisfying J(x, y) ∼ A∥y − x∥− d − α for some constant A (i.e., that our results are universal), but do not pursue this here. Our main results concern the distribution of the volume of critical clusters both in infinite volume and inside a block, with the finite-volume results being used in the proof of the infinite-volume results. In order to ensure as much symmetry as possible, we will work with a slightly different notion of ‘the cluster inside a block’ than used in, which we now introduce. For each n ≥ 0 and each n-block Λ, we take ωΛ to be a percolation configuration on Λ in which each potential edge is included independently at random with inclusion probability 1 − exp( − βL− (d + α)n), and take ωΛ and ωΛʹ to be independent for any two distinct blocks Λ and Λʹ. Note that the union ⋃ΛωΛ is distributed as Bernoulli-β bond percolation on the hierarchical lattice with the kernel $J(x,y)=J(y-x)= \sum\_{n \geq h(y-x)} L^{-(d+\alpha)n} = \frac{L^{d+\alpha}}{L^{d+\alpha}-1}\\|x-y\\|^{-d-\alpha}$ as defined above. For each block Λ, we also define ηΛ = ⋃{ωΛʹ : Λʹ ⊆ Λ} = ωΛ ∪ ⋃{ηΛʹ : Λʹa child of $\Lambda$}, write ηn = ηΛn, and write Kn for the cluster of the origin in ηn. Be careful to note that this notation is not consistent with that used in, where Kn denoted the cluster of the origin in the restriction of ω to Λn; the cluster Kn as we define it is always contained in the cluster Kn as defined in. Asymptotic notation. We now briefly introduce our conventions concerning asymptotic notation that will be used throughout the paper. We write ≍ , ≼ , and ≽ for equalities and inequalities holding to within positive multiplicative constants depending on the parameters d, L, and α and, if relevant, on the index of the moment being estimated, but not on any other parameters (such as the scale on which the model is being studied). The emphasized clause of the previous statement means that if we write e.g. $\E\_{\beta\_c} \|K\_n\|^p \asymp L^{(2p-1)\alpha n}$ then the implicit constants may depend on p. Although this is not standard, it significantly lightens the notation throughout the paper and will hopefully lead to very little confusion. Landau’s asymptotic notation is used similarly, so that e.g. if f is a non-negative function then “f(n) = O(n) for every n ≥ 1” and “f(n) ≼ n for every n ≥ 1” both mean that there exists a positive constant C such that f(n) ≤ Cn for every n ≥ 1. We also write f(n) = o(g(n)) to mean that f(n)/g(n) → 0 as n → ∞ and write f(n) ∼ g(n) to mean that f(n)/g(n) → 1 as n → ∞. Results ------- We now state our main theorems, which we describe separately in the three cases d < dc, d > dc, and d = dc. We recall that the critical exponents δ and η, if they exist, are defined by the relations $$\begin{aligned} \P\_{\beta\_c}(\|K\|\geq n) &= n^{-1/\delta\pm o(1)} & \text{ as $n\to\infty$ and}\\ \P\_{\beta\_c}(x \leftrightarrow y) &= \\|x-y\\|^{-d+2-\eta \pm o(1)} & \text{ as $x-y\to\infty$;}\end{aligned}$$ The results of imply in particular that η is well-defined an equal to 2 − α for every 0 < α < d. Low dimensions. We first describe our results in the low-dimensional case d < dc = 3α, where we obtain precise up-to-constants estimates on both the moments of the cluster of the origin inside a block and on the tail of the cluster of the origin. Let us stress again that, in accordance with the conventions on asymptotic notation used throughout the paper, the implicit constants appearing here may depend on the choice of index p. [thm:volumelowdim] Let d ≥ 1 and L ≥ 2, let d/3 < α < d, and consider critical percolation on the hierarchical lattice $\bbH^d\_L$ with kernel $J(x,y)= \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$. For each integer p ≥ 1 the estimates $$\E\_{\beta\_c} \|K\_n\|^p \asymp \left(L^{\alpha+\frac{d+\alpha}{2}(p-1)}\right)^n \qquad \text{ and } \qquad \P\_{\beta\_c} (\|K\|\geq k) \asymp k^{-\frac{d-\alpha}{d+\alpha}}$$ hold for all integers n, k ≥ 1. In particular, the critical exponent δ is well-defined and equal to (d + α)/(d − α). We believe that this is the first time the exponent δ has been computed for a Bernoulli percolation model that is neither mean-field nor planar. Previously, we showed in that the exponent δ satisfies δ ≥ (d + α)/(d − α) if it is well-defined, but the proof did not establish a pointwise lower bound on $\P\_{\beta\_c} (\|K\|\geq k)$. In the other direction, it was shown in (see also ) that δ satisfies the (non-sharp) upper bound δ ≤ 2d/(d − α) whenever it is well-defined; this remains the best known estimate for long-range percolation on $\Z^d$. Regarding the moments of ∣Kn∣, the asymptotics of the first moment were established in and the methods of that paper together with the universal tightness theorem (which is reviewed in detail in ) easily imply that the claimed upper bound on the pth moment holds for each p ≥ 1, while the lower bounds are new. At a technical level, the most important intermediate results going into the proof of are that the maximum cluster size in an n-block is typically of order $L^{\frac{d+\alpha}{2}n}$ () and that clusters significantly smaller than this characteristic size do not contribute significantly to the mean of ∣Kn∣ (). The first of these intermediate results complements, which implies that $L^{\frac{d+\alpha}{2}n}$ is always an upper bound on the maximum cluster size in an n-block for every 0 < α < d. The fact that this upper bound is sharp for d < 3α but not for d ≥ 3α can be thought of as the primary driver for the distinction between the high-dimensional and low-dimensional regimes; this perspective is developed at length in. Given these two intermediate results it is rather easy to conclude the desired bounds on moments, while computing the tail of the volume still requires a novel and non-trivial argument that is given in. The equality δ = (d + α)/(d − α) follows heuristically from the equality η = 2 − α established in together with the scaling and hyperscaling relations, which are believed to always relate η and δ via (2 − η)(δ + 1) = d(δ − 1) for percolation below the upper-critical dimension; see for background. While the scaling and hyperscaling relations have been established unconditionally for planar percolation models, they are known for nearest-neighbour models only conditionally under appropriate hyperscaling postulates. These postulates amount roughly to the assertion that there are O(1) macroscopic clusters on each scale whose geometry determines most the interesting feature of the model. While we do not explicitly frame our proofs in terms of hyperscaling, the arguments of can be thought of as establishing and applying an appropriate hyperscaling postulate () for low-dimensional hierarchical percolation. High dimensions. We next describe our results in the high-dimensional case d > dc = 3α. In this case, the results of already establish that the triangle condition holds at criticality and hence that the model has mean-field critical behaviour with $\P\_{\beta\_c}(\|K\|\geq k) \asymp k^{-1/2}$. Nevertheless, our methods still yield significant new content in this case, and in particular establish precise asymptotic estimates on moments of all orders for the size of the cluster of the origin inside a block. For each n ≥ − 1 we write n!! for the double factorial n!! : = ∏k = 0⌊n/2⌋ − 1(n − 2k), i.e., the product of all positive integers less than n that have the same parity as n, with the convention that 0!! = ( − 1)!! = 1. [thm:highdimmomentsmain] Let d ≥ 1 and L ≥ 2, let 0 < α < d/3, and consider critical percolation on the hierarchical lattice $\bbH^d\_L$ with kernel $J(x,y)= \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$. There exists a constant A = A(d, α, L) such that $$\E\_{\beta\_c}\|K\_n\|^p \sim (2p-3)!! A^{p-1} \left(\frac{L^\alpha-1}{L^\alpha \beta\_c}\right) L^{(2p-1)\alpha n}$$ as n → ∞ for each integer p ≥ 1. The rate of convergence in this asymptotic formula depends on the choice of p. The relevance of the double-factorial term (2p − 3)!! appearing here for the scaling limit of the model is discussed in. The same double-factorial term also appears in the critical dimension as discussed below, where the precise determination of leading constants is an important step in the determination of the order of polylogarithmic corrections. [remark:freevsperiodic] Roughly speaking, our results in the high-dimensional case show that the ‘typical large clusters’ in an n-block have size of order L2αn and that there are order L(d − 3α)n such clusters. While the largest cluster in a block is presumably larger than this characteristic size by a factor of order log#{large clusters} ≍ n due to entropic fluctuations[3](#fn3), it is the large number of characteristic-size clusters, not the largest cluster, that drive most interesting behaviours of the model. This is consistent with what happens in critical high-dimensional percolation on a box [ − r, r]d in $\Z^d$ with free boundary conditions, where there are order rd − 6 ‘typical large clusters’ of characteristic size r4. It is not the same behaviour observed in the critical Erds-Rényi graph or high-dimensional torus, where there are O(1) large clusters of size (volume)2/3. In light of this disparity, the critical high-dimensional hierarchical model should be compared not with the critical Erds-Rényi graph G(N, 1/N), but rather with the Erds-Rényi graph G(N, p) with p = (1 − N− α/d)N− 1, which is significantly below the scaling window p = (1 ± O(N− 1/3))N− 1 when d > 3α. See for a discussion of how to define ‘periodic boundary conditions’ for hierarchical percolation, which should lead to the largest cluster in an n-block having size $L^{\frac{2}{3}dn}$ in the high-dimensional case. The critical dimension. We now describe our results in the upper-critical dimension d = dc = 3α. These are the most technically challenging results of the paper, with the proofs drawing heavily on the techniques developed in both the low- and high-dimensional cases. Our main results in this case compute precise asymptotics on the moments of ∣Kn∣, which are then applied to prove up-to-constant estimates on the tail of the volume. Besides the results of, which show that there is no logarithmic correction to scaling for the two-point function in hierarchical percolation at the upper-critical dimension, we believe this is the first time logarithmic corrections at the upper-critical dimension have been rigorously determined for any Bernoulli percolation model that does not continue to exhibit exact mean-field behaviour at the upper-critical dimension (as the model considered in does). [thm:criticaldimmoments] Let d ≥ 1 and L ≥ 2, let α = d/3, and consider critical percolation on the hierarchical lattice $\bbH^d\_L$ with kernel $J(x,y)= \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$. There exists a positive constant A = A(d, L) given explicitly by $$A=\sqrt{ \frac{L^{\alpha}-1} {\beta\_c(5L^{4\alpha}-2 L^{\alpha}-3)}}$$ such that $$\E\_{\beta\_c} \|K\_n\|^p \sim (2p-3)!! A^{p-1}\left(\frac{L^\alpha-1}{L^\alpha \beta\_c}\right) n^{-\frac{p-1}{2}}L^{(2p-1)\frac{d}{3}n}$$ as n → ∞ for each p ≥ 1. [thm:criticaldimvolumetailmain] Let d ≥ 1 and L ≥ 2, let α = d/3, and consider critical percolation on the hierarchical lattice $\bbH^d\_L$ with kernel $J(x,y)= \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$. The tail of the critical cluster volume admits the estimate $$\P\_{\beta\_c} (\|K\|\geq k) \asymp (\log k)^{1/4} k^{-1/2}$$ for every k ≥ 2. [remark:largeclusterscriticaldimension] The algebraic factor $n^{-\frac{p-1}{2}}$ appearing in should be thought of as a polylogarithmic correction since we are working on an exponential scale. Intuitively, our results show that at scale n the behaviour of the model is driven by a collection of Θ(n) ‘typical large clusters’ of size $\Theta(n^{-1/2} L^{\frac{2}{3}dn})$; although entropic fluctuations should presumably push the largest cluster to be larger than this characteristic size by a factor of order log#{large clusters} ≍ logn, this largest cluster should not play an important role in determining other quantities of interest. We expect the constant A appearing in to be universal in the sense that the same constant would arise for any radially symmetric J satisfying $J(x,y)\sim \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|y-x\\|^{-d-\alpha}$ as y − x → ∞, albeit with a value of βc that is sensitive to the precise choice of J. On the other hand, the analogous constant A appearing in the high-dimensional case d > 3α is not expected to be universal since it arises as an infinite product (see ) whose value is determined primarily by the small-scale behaviour of the model. [remark:wronglogs] The logarithmic correction to scaling established in is not the same as predicted to hold for nearest-neighbour percolation on $\Z^6$ by Essam, Gaunt, and Guttmann, namely $$\P\_{\beta\_c} (\|K\|\geq k) \asymp (\log k)^{2/7} k^{-1/2}.$$ In fact, a tension between our results and these predictions was already present in the two-point function results of : There is no logarithmic correction to scaling in the hierarchical model, while the predictions of together with standard heuristic scaling theory arguments lead to the predicted scaling $$\P\_{\beta\_c}(x\leftrightarrow y) \asymp (\log \\|x-y\\|)^{1/21} \\|x-y\\|^{-d+2}$$ for the critical two-point function for nearest-neighbour percolation on $\Z^6$. This disparity between the hierarchical and nearest-neighbour models for percolation is in stark contrast to weakly self-avoiding walk and the φ4 model, where the logarithmic corrections to scaling at the upper-critical dimension are the same for the hierarchical and nearest-neighbour models as surveyed in. The predictions of are consistent with those obtained in several related works in the physics literature (some of which use completely different methods) and are very likely to be correct. We believe that the right way to think about the disparity is as follows: Hierarchical models can essentially never have logarithmic corrections to their two-point functions, so that one should expect the logarithmic corrections for other quantities to be the same for the Euclidean and hierarchical models only if the Euclidean models do not have any logarithmic corrections to their two-point functions either. The fact that there are such corrections for percolation and no such corrections for φ4 should be thought of as a special feature of φ4 rather than a pathological feature of percolation. In the physics literature, percolation at the upper-critical dimension is studied either by 1) applying a renormalization group analysis to the φ3 model, which is believed to belong to the same universality class as percolation but not to satisfy any exact equivalences at the discrete level, or 2) applying a renormalization group analysis to the q-state Potts model for an integer q ≥ 2 before taking q → 1 in the exponent formulae obtained at the end of the calculation. We stress that our proof is not a rigorous implementation of either of these heuristic approaches, and does not rely on any isomorphism theorem relating percolation to a spin system. We conjecture that the same logarithmic corrections computed here also appear in long-range percolation on $\Z^d$ with d < 6 and α = d/3. Indeed, the related conjecture that there are no logarithmic corrections to the critical two-point function has already been established for d = 1, 2 in. Since these logarithmic corrections do not coincide with those predicted to hold for nearest-neighbour percolation on $\Z^6$, this suggests that the ɛ-expansions derived in e.g. cannot be applied as-is to long-range models, and it may be interesting to revisit the numerical results of in light of this. A glimpse of the scaling limit ------------------------------ We now discuss the consequences of our work for the scaling limit of the model. First, in the high-dimensional case d ≥ 3α, easily yield the following corollary regarding the scaling limit of the size-biased law of the cluster volume inside a block. [Chi-squared limit law for high-dimensional size-biased cluster volumes] [cor:ChiSquaredmain] Let d ≥ 1 and L ≥ 2, let 0 < α ≤ d/3, and consider critical percolation on the hierarchical lattice $\bbH^d\_L$ with kernel $J(x,y)= \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$. For each n ≥ 0, let Qn be the probability measure on [0, ∞) defined by size-biasing the law of ∣Kn∣ under $\P\_{\beta\_c}$ and rescaling the resulting size-biased random variable by its mean $\hat \E\_{\beta\_c}\|K\_n\|=\E\_{\beta\_c}\|K\_n\|^2/\E\_{\beta\_c}\|K\_n\|$, so that $$\int\_0^\infty F(x) \dif \mathbb{Q}\_n (x) = \frac{1}{\E\_{\beta\_c}\|K\_n\|}\E\_{\beta\_c} \left[\|K\_n\|\cdot F\!\left(\frac{\E\_{\beta\_c}\|K\_n\|^{\phantom{2}}}{\E\_{\beta\_c}\|K\_n\|^2}\|K\_n\|\right)\right]$$ for each Borel measurable function F : [0, ∞) → [0, ∞). Then Qn converges as n → ∞ to the law of a chi-squared distribution with one degree of freedom, that is, the law of the square of a standard normal random variable. [Proof of given ] It follows from that if 0 < α ≤ d/3 then $$\label{eq:moment\_asymptotics\_p\_vs\_2/1} \E\_{\beta\_c}\|K\_n\|^{p} \sim (2p-3)!! \frac{(\E\_{\beta\_c}\|K\_n\|^{2})^{p-1}}{(\E\_{\beta\_c}\|K\_n\|)^{p-2}}$$ as n → ∞ for each fixed p ≥ 2. (Note in particular that this estimate holds in both the d > 3α and d = 3α cases despite the differing asymptotics of $\E\_{\beta\_c}\|K\_n\|^{2}$.) It follows from and the definition of $\bbQ\_n$ that the moments of $\bbQ\_n$ satisfy $$\int x^p \dif \bbQ\_n(x) = \left(\frac{\E\_{\beta\_c}\|K\_n\|^2}{\E\_{\beta\_c}\|K\_n\|}\right)^{-p} \frac{1}{\E\_{\beta\_c}\|K\_n\|} \E\_{\beta\_c} \|K\_n\|^{p+1} \sim (2p-1)!!$$ as n → ∞ for each fixed p ≥ 1. This implies that $(\bbQ\_n)\_{n\geq 1}$ is tight and, by dominated convergence, that any subsequential distributional limit of $\bbQ\_n$ has pth moment (2p − 1)!! for every p ≥ 1. The claim follows since, by Carleman’s criterion, the chi-squared distribution with one degree of freedom is the unique distribution on [0, ∞) having pth moment (2p − 1)!! for every p ≥ 1. We conjecture that the size-biased cluster Kn converges as a metric measure space to a continuum random tree of chi-squared volume under appropriate rescaling (where the appropriate scaling factors will include polylogarithmic terms at the upper-critical dimension). The fact that we expect to see trees in the scaling limit, in contrast to the scaling limit of critical Erds-Rényi graphs, is related to the discussion in. Note that the analogous problem for high-dimensional percolation (i.e., convergence of large critical clusters to the CRT) remains open despite significant partial progress ; see for a detailed discussion. The same chi-squared limiting distribution also appears in slightly subcritical branching processes. Indeed, if Z is the total progeny of a Poisson(1 − ɛ) branching process and $\hat \P\_{1-{\varepsilon}}$ is the size-biased law of Z then, since the unbiased law of Z follows the Borel distribution, we can express the probability mass function of $\hat \P$ exactly as $$\hat\P\_{1-{\varepsilon}}(Z=n) = {\varepsilon}\frac{e^{-(1-{\varepsilon})n}(1-{\varepsilon})^{n-1} n^{n-1}}{(n-1)!}.$$ A simple calculation using Stirling’s formula then yields that $$\hat \P\_{1-{\varepsilon}}(Z=n) \sim \frac{{\varepsilon}^2}{\sqrt{2\pi \lambda}} e^{-\frac{1}{2}\lambda} \qquad \text{ as ${\varepsilon}\downarrow 0$ and $n\to \infty$ with $n \sim \lambda {\varepsilon}^{-2}$},$$ so that if we divide Z by its mean under $\hat \P$ and take the limit as ɛ ↓ 0 we obtain a chi-squared distribution with one degree of freedom exactly as in. We now turn to the low-dimensional case d < 3α. In this setting there is no explicit candidate for what the scaling limit of the model ought to be, and the identification of such a limit appears to be a very difficult problem. Nevertheless, our results provide an important first step towards the study of such a scaling limit by establishing tightness of the appropriately normalized list of cluster sizes in a block. We write ℓ↓p for the subspace of ℓp formed by sequences that are non-negative and (weakly) decreasing. [cor:ell2tightness] Let d ≥ 1 and L ≥ 2, let d/3 < α < d, and consider critical percolation on the hierarchical lattice $\bbH^d\_L$ with kernel $J(x,y)= \frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$. For each n, i ≥ 0 let ∣Kn, i∣ be the size of the ith largest cluster in ηBn, setting ∣Kn, i∣ = 0 if there are fewer than i clusters. Then the family of sequence-valued random variables $$\Bigl\{ L^{-\frac{d+\alpha}{2}n}\left(\|K\_{n,1}\|,\,\|K\_{n,2}\|,\,\|K\_{n,3}\|,\,\ldots \right) : n \geq 0\Bigr\}$$ is tight in ℓ↓p \ {0} if and only if p > 2d/(d + α), and in particular is tight in ℓ↓2 \ {0}. We highlight the ℓ↓2 tightness provided by this corollary since, by a theorem of Aldous, this is precisely what is needed for the “renormalization group map” to extend continuously to the set of subsequential limits of the model, a property that might plausibly play a central role in the future study of the model. This perspective is discussed in detail in. In we slightly strengthen the conclusion of to show that any subsequential weak limit of the normalized ordered sequence of cluster sizes $L^{-\frac{d+\alpha}{2}n}(\|K\_{n,1}\|$, ∣Kn, 2∣, ∣Kn, 3∣, …) is supported on sequences all of whose entries are positive. together with the afforementioned theorem of Aldous imply in particular that, for each fixed m ≥ 1, the conditional distribution of the sizes of the m largest clusters in Λn given the entire percolation configuration in each child of Λn is approximately determined by the sizes of the largest M clusters in each of these children of Λn when M is a large constant and n is large. This gives further weight to the intuitive statement, which is related to the validity of the hyperscaling relations, that when d < 3α all interesting features of the model are driven by the O(1) large clusters that have volume of order $L^{\frac{d+\alpha}{2}n}$, with clusters of volume $o(L^{\frac{d+\alpha}{2}n})$ being negligible for most purposes. This is in stark contrast to the cases d > 3α and d = 3α as discussed in. Organization and proof overview ------------------------------- We now briefly overview the rest of the paper. 1. In we introduce our reframing of hierarchical percolation as an infinite recursive system of multiplicative coalescents, which describe the evolution of the cluster sizes in hierarchical percolation as we continuously increase the weight of scale-n edges from 0 up to their final values of L− (d + α)n. In particular, we discuss how the multiplicative coalescent can be seen as an infinite system of ODEs governing the evolution of sums of powers of cluster sizes. This perspective, with hierarchical percolation thought of as an infinite-dimensional dynamical system, will be used throughout the rest of the paper. 2. In we review the universal tightness theorem of and prove an ℓp generalization of this theorem, which applies to percolation on arbitrary weighted graphs. We summarise the consequences of these theorems for hierarchical percolation in. 3. In we introduce the hydrodynamic condition[4](#fn4), which is defined to hold when the typical size of the largest cluster in the box Λn, denoted Mn satisfies $M\_n =o(L^{\frac{d+\alpha}{2}n})$, a strict improvement of the upper bound $M\_n=O(L^{\frac{d+\alpha}{2}n})$ proven in ; over the course of the paper, we eventually show that this condition holds if and only if d ≥ 3α, and can be seen as the chief driver of the distinction between the low-dimensional and high-dimensional regimes. In this section, we first show that the hydrodynamic condition is a simple consequence of the results of and the tree-graph inequalities of Aizenman and Newman in the high-dimensional case d > 3α. We then show that the hydrodynamic condition can be used to significantly simplify the infinite system of ODEs described in, allowing for a precise analysis of the asymptotics of the moments $\E\|K\_n\|^p$ over the course of this section including a complete proof of. Since the resulting asymptotics are not consistent with the results of in the low-dimensional case d < 3α, we also deduce that the hydrodynamic condition does not hold in this case. 4. In we prove our results concerning the low-dimensional case d < 3α. First, in we sharpen the failure of the hydrodynamic condition into a pointwise lower bound $M\_n \succeq L^{\frac{d+\alpha}{2}n}$ by a careful quantitative treatment of the arguments used to study the assymptotics of the second moment in. The main idea is to show that if this bound fails to hold on a large enough number of consecutive scales, we can approximately simplify the relevant ODEs, as we can in the high-dimensional case, and deduce estimates that are known to be false in the low-dimensional case. Then, in, we prove our results on the volume tail and its with the aid of an important supporting technical result on the ‘negligibility of mesoscopic clusters’, which is morally related to hyperscaling hypotheses such as those used in and is proven via analysis of differential inequalities for certain truncated moments. 5. In we prove our results concerning the upper-critical dimension d = 3α. This section draws heavily on the tools developed to study both the low-dimensional and high-dimensional cases. First, in we prove that the hydrodynamic condition holds in the critical dimension. This can be thought of as a ‘marginal triviality’ result à la. Its proof uses both the techniques introduced in and a new differential inequality for the susceptibility originating in and derived from the OSSS inequality to obtain a contradiction under the assumption that the hydrodynamic condition does not hold. In we prove, which establishes precise asymptotics on the moments of ∣Kn∣ at the upper-critical dimension. This is done by establishing more precise approximations of the infinite system of ODEs introduced in than those used in, including precise asymptotics on the second-order terms in these approximations, from which these asymptotics can be extracted. Finally, we use these moment estimates to deduce the voluime tail estimates of using an argument very similar to that used in the low-dimensional case. 6. In, we conclude the paper by discussing several open problems and directions for future research. The multiplicative coalescent as an infinite system of ODEs =========================================================== Throughout the paper we will work with an equivalent description of hierarchical percolation in terms of an infinite recursive system of multiplicative coalescents. The multiplicative coalescent is a partition-valued Markov process that was introduced by Aldous as a means to study the Erds-Rényi random graph and has since been the subject of several significant works including. For our purposes the encoding in terms of the multiplicative coalescent amounts mostly to a change of notation, albeit one that we find very useful, and we will not need to engage much with the previous literature on the topic. This literature may however be very useful in the future study of hierarchical percolation as we discuss in. Let Ω be a finite set and let X0 be a partition of Ω. The multiplicative coalescent (Xt)t ≥ 0 is a continuous-time Markov process in which each two blocks A and B of Xt of the partition merge at rate ∣A∣ ⋅ ∣B∣. In other words, the multiplicative coalescent is the unique continuous-time Markov chain on the set of partitions of Ω satisfying $$\begin{gathered} \P\_{X\_0}({X}\_{t+{\varepsilon}}=P'\mid {X}\_t=P) \\= \begin{cases} {\varepsilon}\|A\|\cdot\|B\| + o({\varepsilon}) &\text{ if $P'=P\cup\{A \cup B\} \setminus \{A,B\}$ for some distinct $A,B \in P$}\\ 1-{\varepsilon}\sum\_{\substack{A,B \in P\\ \text{ distinct}}} \|A\|\cdot \|B\| + o({\varepsilon}) &\text{ if $P'=P$}\\ o({\varepsilon}) &\text{ otherwise}. \end{cases}\end{gathered}$$ Equivalently, the multiplicative coalescent is the unique stochastically continuous process on the set $\cP(\Omega)$ of partitions of Ω such that if $F:\cP(\Omega)\to \R$ is a function then $\E\_{X\_0}F(X\_t)$ depends smoothly on t ∈ [0, ∞) with $\E\_{X\_0}(F(X\_0))=F(X\_0)$ and $$\begin{aligned} \label{eq:verygeneralODE} \frac{d}{dt} \E\_{X\_0} \left[F(X\_t)\right] = \frac{1}{2}\E\_{X\_0}\left[ \sum\_{\substack{A,B \in X\_t\\\text{distinct}}} \|A\| \|B\| \Bigl( F\bigl(X\_t\cup\{A \cup B\} \setminus \{A,B\}\bigr) -F(X\_t) \Bigr)\right]\end{aligned}$$ for every t ≥ 0. The factor of 1/2 appearing here accounts for the fact that each unordered pair of distinct sets appears in the sum twice. When started with the trivial partition X0 = {{x} : x ∈ Ω}, the multiplicative coalescent is equivalent to the component partition of the Erds-Rënyi random graph: We can couple the multiplicative coalescent (Xt)t ≥ 0 with the monotone coupling of Bernoulli percolation on the complete graph (ωp)p ∈ [0, 1] so that $$X\_t = \{\text{clusters of $\omega\_{1-e^{-t}}$}\}$$ for every t ≥ 0. Similarly, if we initialize $X\_0=\{\text{clusters of $H$}\}$ for some subgraph of the complete graph over Ω, then (Xt)t ∈ [0, 1] is distributed as $(\{\text{clusters of $1-e-t H$}\})\_{t\geq 0}$. While the component structure of the Erds-Rënyi random graph is described by a single multiplicative coalescent, the component structure of hierarchical percolation can be described by an infinite recursive system of multiplicative coalescents. Fix β ≥ 0, let $\bbH^d\_L$ be the hierarchical lattice, and let α > 0. For each n ≥ 0 let $\cB\_n$ be the collection of n-blocks in $\bbH^d\_L$ and define tn = βL− (d + α)n. We will define a family of processes $$\mathfrak{X}=\bigl((X\_{\Lambda,t})\_{t=0}^{t\_n} : \Lambda \in \cB\_n \text{ for some $n\geq 0$}\bigr),$$ so that for each n ≥ 0 and each block $\Lambda \in \cB\_n$, the process (XΛ, t)t = 0tn takes values in the set of partitions of Λ. We define this family of processes recursively, with the initial state of the process associated to a block determined by the final states of the processes associated to the children of that block. When n = 0, the blocks of $\cB\_0$ are singletons and we have that XΛ, t = {Λ} for every $\Lambda \in \cB\_0$ and 0 ≤ t ≤ t0. Recursively, conditional on the sigma-algebra generated by the processes $((X\_{\Lambda,t})\_{t=0}^{t\_k} : \Lambda \in \cB\_n \text{ for some $0k n$})$, we take the processes $((X\_{\Lambda,t})\_{t=0}^{t\_{n+1}} : \Lambda \in \cB\_{n+1})$ to be conditionally independent and take (XΛ, t)t = 0tn + 1 to be a multiplicative coalescent on the block Λ initialized with $$X\_{\Lambda,0}=\bigcup\bigl\{ X\_{\Lambda',t\_n}: \Lambda' \text{ a child of $\Lambda$}\bigr\}.$$ This definition ensures that we can couple X with hierarchical percolation as defined in so that XΛ, tn = {clusters of $\eta\_\Lambda$} for every n ≥ 0 and every block $\Lambda\in \cB\_n$. We stress that the times tn = βL− (d + α)n all depend on the parameter β, which we will almost always take to be βc. We write Xn, t = XΛn, t where Λn is the n-block containing the origin. [remark:intermediatetpercolation] Given an n-block Λn and 0 ≤ t ≤ tn, the partition XΛ, t is distributed as the partition into clusters associated to percolation on the weighted graph with vertex set Λ and edge weights $$J\_{n,t}(x,y):= \frac{t}{t\_n} L^{-(d+\alpha )n} + \sum\_{m=h(x,y)}^{n-1}L^{-(d+\alpha )m}.$$ This observation allows us to apply results concerning percolation on general weighted graphs (such as the universal tightness theorem, the Harris-FKG inequality, and the BK inequality) to the intermediate configurations XΛ, t with 0 ≤ t ≤ tn. An infinite system of ODEs -------------------------- Given a partition P of a finite set Ω, we define the ℓp norm of P by $$\\|P\\|\_p = \begin{cases} \sup\_{A\in P}\|A\| & p = \infty\\ \left(\sum\_{A\in P}\|A\|^p\right)^{1/p} & 1\leq p <\infty, \end{cases}$$ so that ∥P∥p is the ℓp norm of the vector of component sizes of P. The p-norms of the partitions arising in the infinite recursive system of multiplicative coalescents X defined above are naturally related to the moments of clusters in hierarchical percolation by the formula $$\label{eq:moments\_to\_norms} \E \|K\_n\|^p = L^{-dn}\sum\_{x\in \Lambda\_n} \E \|K\_n(x)\|^p = L^{-dn} \E \left[\sum\_{C \in \sC\_n} \|C\|^{p+1} \right] = L^{-dn} \E\\|X\_{n,t\_n}\\|\_{p+1}^{p+1},$$ where we recall that Kn(x) is the cluster of x in ηΛn and $\sC\_n$ is the partition of Λn into the clusters of ηΛn. As such, most quantities of interest in the hierarchical percolation model can be computed in terms of the expectations $\E\\|X\_{n,t}\\|\_{p}^{p}$, which we will spend much of the paper estimating. In light of the relation, we will sometimes refer to $\E\\|X\_{n,t}\\|\_{p}^{p}$ as the *(p* − 1)th moment** of Xn, t; we will always take p to be an integer in such an expression unless specified otherwise. We will make extensive use of the following formula. [lem:ODE1] Let (Xt)t ≥ 0 be the multiplicative coalescent on some finite set initialized at some partition X0. Then $$\begin{aligned} \frac{d}{dt} \E \\|X\_t\\|\_p^p = \E \left[\frac{1}{2}\sum\_{k=1}^{p-1} \binom{p}{k} \\|X\_t\\|\_{k+1}^{k+1}\\|X\_t\\|\_{p-k+1}^{p-k+1}-(2^{p-1}-1)\\|X\_t\\|\_{p+2}^{p+2}\right]\end{aligned}$$ for every t ≥ 0 and every integer p ≥ 2. [Proof of ] It follows from that $$\begin{aligned} 2\frac{d}{dt} \E \\|X\_t\\|\_p^p &= \E\left[ \sum\_{\substack{A,B \in X\_t\\\text{distinct}}} \|A\| \|B\| \Bigl( (\|A\|+\|B\|)^p-\|A\|^p-\|B\|^p \Bigr)\right]\\ &= \E\left[ \sum\_{\substack{A,B \in X\_t\\\text{distinct}}} \sum\_{k=1}^{p-1} \binom{p}{k} \|A\|^{k+1}\|B\|^{p-k+1}\right].\end{aligned}$$ To conclude, we write the sum over distinct elements of Xt as the difference of the sum over all pairs of elements of Xt and the sum over the diagonal to obtain that $$\begin{aligned} 2\frac{d}{dt} \E \\|X\_t\\|\_p^p &= \E\left[ \sum\_{A,B\in X\_t} \sum\_{k=1}^{p-1} \binom{p}{k} \|A\|^{k+1}\|B\|^{p-k+1}\right]- \E\left[ \sum\_{A\in X\_t} \sum\_{k=1}^{p-1} \binom{p}{k} \|A\|^{p+2}\right]\\ &=\sum\_{k=1}^{p-1} \binom{p}{k}\E\left[\\|X\_t\\|\_{k+1}^{k+1}\\|X\_t\\|\_{p-k+1}^{p-k+1} \right] - (2^{p}-2)\E\left[\\|X\_t\\|\_{p+2}^{p+2}\right]\end{aligned}$$ as claimed. More generally, the derivative of any expression of the form $\E \prod\_{i=1}^k \\|X\_t\\|\_{p\_i}^{p\_i}$ can be expressed in terms of other expectations of the same form. We define a multi-index to be a finitely supported function p : {2, 3, 4…} → {0, 1, 2, …}, define the degree of a multi-index p to be deg(p) = ∑p ≥ 2p ⋅ p(p), and define ∥X∥pp = ∏p ≥ 2∥X∥pp ⋅ p(p). We refer to expectations of the form $\E \\|X\\|\_{\mathbf{p}}^{\mathbf{p}}$ as multimoments. An elaboration on the calculation used to prove shows that there exist integer coefficients A(p, q) indexed by pairs of multi-indices such that $$\begin{aligned} \label{eq:ODE\_general} \frac{d}{dt} \E\\|X\_t\\|\_{{\mathbf{p}}}^{{\mathbf{p}}} = \frac{1}{2}\sum\_{\deg(\mathbf{q})=\deg(\mathbf{p})+2} A({\mathbf{p}},\mathbf{q})\E \\|X\_t\\|\_{\mathbf{q}}^{\mathbf{q}}\end{aligned}$$ for every multi-index p; we do not write down the (rather complicated) general formula for A(p, q) since it will not be needed in the remainder of the paper. It can be shown that multimoments of the form $\E \\|X\_t\\|\_{{\mathbf{p}}}^{{\mathbf{p}}}$ completely characterise the distribution of the component sizes in the multiplicative coalescent Xt, so that there is a sense in which the model is equivalent to the infinite system of linear ODEs. Similarly, the infinite recursive system of multiplicative coalescents representing percolation on the hierarchical lattice is completely described by the system of equations $$\begin{aligned} \label{eq:ODE\_general\_hierarchical} \frac{d}{dt} \E\\|X\_{n,t}\\|\_{{\mathbf{p}}}^{{\mathbf{p}}} = \frac{1}{2}\sum\_{\deg(\mathbf{q})=\deg(\mathbf{p})+2} A({\mathbf{p}},\mathbf{q})\E \\|X\_{n,t}\\|\_{\mathbf{q}}^{\mathbf{q}}\end{aligned}$$ for each multi-index p, n ≥ 0 and 0 ≤ t ≤ tn, and $$\label{eq:discrete\_step\_general} \E\\|X\_{n+1,0}\\|\_{{\mathbf{p}}}^{{\mathbf{p}}} = \sum\_{\substack{{\mathbf{p}}\_1,\ldots,{\mathbf{p}}\_{L^d}\\\sum\_{i=1}^{L^d} {\mathbf{p}}\_i = {\mathbf{p}}}} \left[ \prod\_{p\geq 2} \binom{{\mathbf{p}}(p)}{{\mathbf{p}}\_1(p),\ldots,{\mathbf{p}}\_{L^d}(p)}\right]\prod\_{i=1}^{L^d} \E\\|X\_{n,t\_n}\\|\_{{\mathbf{p}}\_i}^{{\mathbf{p}}\_i}$$ for each multi-index p and n ≥ 0, where the second equation holds since Xn + 1, 0 is distributed as the disjoint union of Ld copies of Xn, tn. (We do not include a full derivation of this equation since we will not make use of the general case.) When considering moments rather than more general multimoments, the relation admits the simpler expression $$\label{eq:discrete\_step\_simple} \E \\|X\_{n+1,0}\\|\_{p}^{p} = L^d \cdot \E \\|X\_{n,t\_n}\\|\_{p}^{p},$$ which again follows immediately from the fact that Xn + 1, 0 is distributed as the disjoint union of Ld copies of Xn, tn. Of course, the perspective discussed above is not helpful if the relevant system of equations is intractably difficult to study, as it seems likely to be in general. In we introduce the hydrodynamic condition, which we eventually show holds for hierarchical percolation if and only if d ≥ 3α. We prove that if the hydrodynamic condition holds then $\E \\|X\_{n,t}\\|\_{{\mathbf{p}}}^{{\mathbf{p}}} \sim \prod\_{p\geq 2} (\E \\|X\_{n,t}\\|\_{p}^{p})^{{\mathbf{p}}(p)}$ for every multi-index p and that $\E\\|X\_{n,t}\\|\_{p+2}^{p+2} = o(\E\\|X\_{n,t}\\|\_{k+1}^{k+1}\\|X\_{n,t}\\|\_{p-k+1}^{p-k+1})$ as n → ∞ for each 1 ≤ k ≤ p − 1, so that for large n the system is approximately governed by the exactly solvable system of ODEs $$\begin{aligned} \label{eq:approximate\_ODE} \frac{d}{dt} \E \\|X\_{n,t}\\|\_p^p \sim \frac{1}{2}\sum\_{k=1}^{p-1} \binom{p}{k}\E\left[ \\|X\_{n,t}\\|\_{k+1}^{k+1}\right]\E\left[\\|X\_{n,t}\\|\_{p-k+1}^{p-k+1}\right].\end{aligned}$$ When d > 3α it is easily justified from the results of that this approximation holds with errors exponentially small in n, allowing for a fairly straightforward analysis of this case. In the critical case d = 3α the errors are merely polynomially small in n and a much more refined analysis is necessary; even showing that the errors go to zero is a highly non-trivial matter that is dealt with in. Moreover, to compute the logarithmic corrections to scaling for d = 3α, the first-order approximation is not sufficient and one must instead expand the relevant ODEs to second order (see ). On the other hand, when d < 3α, we will see that all terms appearing in are of the same order, and that it is not the case that $\E \\|X\_{n,t}\\|\_{{\mathbf{p}}}^{{\mathbf{p}}} \sim \prod\_{p\geq 2} (\E \\|X\_{n,t}\\|\_{p}^{p})^{{\mathbf{p}}(p)}$. This suggests that the system ([eq:ODEgeneralhierarchical]-[eq:discretestepgeneral]) is much more difficult to study in this case than when d ≥ 3α, perhaps intractably so. Surprisingly, the same ODE perspective is nevertheless still useful in this regime: In we prove our sharp lower bounds on the maximum cluster size by assuming for contradiction[5](#fn5) that such a bound does not hold, showing that this allows us to approximately simplify the relevant ODEs as in, and then showing that the outputs of this analysis are inconsistent with the results of when d < 3α. The maximal cluster size and the ℓp universal tightness theorem ================================================================= In this section we briefly review the universal tightness theorem of and prove a generalization of this theorem to ℓp norms other than the ℓ∞ norm. Roughly speaking, the universal tightness theorem states that on any weighted graph, the largest cluster in Bernoulli percolation is of the same order as its median with high probability, with exponential upper tail bounds that hold uniformly over the choice of graph. Since its introduction in the universal tightness theorem has already found several further applications in various other contexts and we hope the ℓp version of the universal tightness theorem we prove here will be similarly applicable in the future. We begin by stating the ℓ∞ universal tightness theorem of. Since the theorems discussed here apply uniformly to all weighted graphs, we will temporarily leave our main setting of hierarchical percolation, discussing the applications of these results to the hierarchical setting in. Let G = (V, E, J) be a countable weighted graph and let ω be Bernoulli-β bond percolation on G or some β ≥ 0, in which each edge e of G is open with probability 1 − e− βJ(e). We write ${\mathbf P}\_\beta={\mathbf P}\_\beta^G$ and ${\mathbf E}\_\beta={\mathbf E}\_\beta^G$ for probabilities and expectations taken with respect to the law of ω. Let Λ ⊆ V be finite and non-empty (often we will take G to be finite and take Λ = V) and consider the random partition of Λ given by $\sC=\sC(\omega,\Lambda)=\{C \cap \Lambda : C$ a cluster of ω}, where we stress that the clusters of ω are computed with respect to the entire graph G and may involve connections that go outside of the distinguished set Λ. We consider the random variable $$\|K\_\mathrm{max}(\Lambda)\|= \\|\sC\\|\_\infty=\max\{\|K\_v \cap \Lambda\| : v\in V\}=\max\{\|K\_v \cap \Lambda\| : v\in \Lambda\}$$ and for each β ≥ 0 define the typical value $M\_\beta(\Lambda):=\min \{n \geq 0 : {\mathbf P}\_\beta(\|K\_\mathrm{max}(\Lambda)\| \geq n)\leq e^{-1} \}$. The universal tightness theorem states that ∣Kmax(Λ)∣ is of the same order as its typical value with high probability, where all relevant constants are universal over all weighted graphs. Moreover, the distribution of the volume of a specific cluster exhibits ‘exponential damping’ in a universal way above the typical value of the maximum cluster size. [Universal tightness of the maximum cluster size] [thm:universaltightness] Let G = (V, E, J) be a countable weighted graph and let Λ ⊆ V be finite and non-empty. Then the inequalities $$\begin{aligned} {\mathbf P}\_\beta\Bigl(\|K\_\mathrm{max}(\Lambda)\| \geq \alpha M\_\beta(\Lambda)\Bigr) &\leq \exp\left(-\frac{1}{9}\alpha \right) \label{eq:BigClusterUnrooted} \\ \text{and} \qquad {\mathbf P}\_\beta\Bigl(\|K\_\mathrm{max}(\Lambda)\| < {\varepsilon}M\_\beta(\Lambda) \Bigr) &\leq 27 {\varepsilon}\qquad \phantom{\text{and}} \qquad \label{eq:SmallMaximum}\end{aligned}$$ hold for every β ≥ 0, α ≥ 1, and 0 < ɛ ≤ 1. Moreover, the inequality $$\label{eq:BigClusterRooted} {\mathbf P}\_\beta\Bigl(\|K\_u \cap \Lambda\| \geq \alpha M\_\beta(\Lambda)\Bigr) \leq e \cdot {\mathbf P}\_\beta\Bigl(\|K\_u \cap \Lambda\| \geq M\_\beta(\Lambda)\Bigr) \exp\left(-\frac{1}{9}\alpha \right)$$ holds for every β ≥ 0, α ≥ 1, and u ∈ V. An elementary consequence of this theorem is that for each p ≥ 1 there exist universal positive constants cp and Cp such that $$\label{eq:max\_moment} c\_p M\_\beta(\Lambda) \leq {\mathbf E}\_\beta \left[\|K\_\mathrm{max}(\Lambda)\|^p\right]^{1/p} \leq C\_p M\_\beta(\Lambda)$$ for very weighted graph G = (V, E, J), every finite non-empty set Λ ⊆ V, and every β ≥ 0. Similarly, one also has that there exist universal constants Cp, k such that $$\label{eq:rooted\_cluster\_moment\_max} {\mathbf E}\_\beta \|K\_v \cap \Lambda\|^{p+k} \leq C\_{p,k} M\_\beta(\Lambda)^k {\mathbf E}\_\beta \|K\_v \cap \Lambda\|^{p}$$ for very weighted graph G = (V, E, J), every finite non-empty set Λ ⊆ V, every β ≥ 0, and every v ∈ V. (If desired one may easily compute explicit values of these constants from.) We now state our new ℓp generalization of the universal tightness theorem. Since $\|K\_\mathrm{max}(\Lambda)\|=\\|\sC\\|\_\infty$, the p = ∞ case of this theorem follows immediately from together with. [Universal tightness of ℓp norms] [thm:lpuniversaltightness] There exist universal positive constants c and A such that the following holds. Let G = (V, E, J) be a countable weighted graph, let Λ ⊆ V be finite and non-empty. Let ω be Bernoulli bond percolation on G and consider the random partition of Λ given by $\sC=\{C \cap \Lambda : C$ a cluster of ω}. Then the inequalities $$\begin{aligned} {\mathbf P}\_\beta\Bigl(\\|\sC\\|\_p \geq \alpha {\mathbf E}\_\beta\\|\sC\\|\_p\Bigr) \leq A \exp\left(-c\alpha \right) \qquad \text{and} \qquad {\mathbf P}\_\beta\Bigl(\\|\sC\\|\_p < {\varepsilon}{\mathbf E}\_\beta\\|\sC\\|\_p \Bigr) \leq A {\varepsilon}\end{aligned}$$ hold for every 1 ≤ p ≤ ∞, β ≥ 0, α ≥ 1, and 0 < ɛ ≤ 1. Note that the constants c and C appearing here do not depend on the choice of index 1 ≤ p ≤ ∞. The proof will rely on the same combinatorial lemma used to prove. [ , Lemma 2.4] [lem:divideandconquer] Let G = (V, E) be a connected, locally finite graph, let k ≥ 1, and let A be a finite subset of V such that ∣A∣ ≥ 3k. Then there exists m ≥ 3k − 1 + 1 and a collection {Ei : 1 ≤ i ≤ m} of disjoint, non-empty subsets of E such that the following hold: 1. For each 1 ≤ i ≤ m, the subgraph of G spanned by Ei is connected. 2. Every vertex in V is incident to some edge in ⋃i = 1mEi. 3. The set Vi of vertices incident to an edge of Ei satisfies $$3^{-k} \leq \frac{\|A \cap V\_i\|}{\|A\|} < 3^{-k+1}$$ for each 1 ≤ i ≤ m. implies the following deterministic fact from which we will deduce. [cor:Lpdivideandconquer] Let G = (V, E) be a locally finite graph, let A be a finite subset of V, and let $\sC(G,A)=\{C \cap A : C$ a connected component of G}. If 1 ≤ p < ∞ and k ≥ 1 is an integer such that $\\|\sC(G,A)\\|\_p \geq 3^k$ then there exists m ≥ 3k − 1 and a collection of edge-disjoint subgraphs H1, …, Hm of G such that $\\|\sC(H\_i,A)\\|\_p \geq 3^{-k} \\|\sC(G,A)\\|\_p$ for every 1 ≤ i ≤ m. [Proof of ] Let $\sC\_{\geq k}$ be the set of connected components C of G satisfying ∣V(C) ∩ A∣ ≥ 3k and let $\sC\_{<k}$ be the set of connected components C of G satisfying ∣V(C) ∩ A∣ < 3k. Applying, we can decompose each component $C\in \sC\_{\geq k}$ into mC ≥ 3k − 1 + 1 edge-disjoint connected subgraphs H1(C), …, HmC(C) each of whose vertex sets satisfy ∣V(Hi(C)) ∩ A∣ ≥ 3− k∣V(C) ∩ A∣. For each 1 ≤ i ≤ 3k − 1 + 1, let Hi be the subgraph of G with vertex set V and with edge set equal to the union of the edge sets of the graphs $\{H\_i(C): C\in \sC\_{\geq k}\}$, so that H1, …, H3k − 1 + 1 are edge-disjoint. Since for each 1 ≤ i ≤ 3k − 1 + 1 the collection of graphs $\{H\_i(C): C\in \sC\_{\geq k}\}$ all have disjoint vertex sets, we can bound $$\begin{aligned} \\|\sC(H\_i,A)\\|\_p^p &\geq \sum\_{C \in \sC\_{\geq k}} \|V(H\_i(C)) \cap A\|\_p^p + \|A \cap \bigcup\_{C\in \sC\_{<k}} V(C)\| \\ &\geq 3^{-pk} \sum\_{C \in \sC\_{\geq k}} \|V(C) \cap A\|\_p^p + 3^{-pk} \sum\_{C \in \sC\_{< k}} \|V(C) \cap A\|\_p^p = 3^{-pk} \\|\sC(G,A)\\|\_p^p\end{aligned}$$ for each 1 ≤ i ≤ 3k − 1 + 1 as claimed, where the bound on the first term follows by choice of H and the bound on the second term follows since ∣V(C) ∩ A∣pp ≤ 3(p − 1)k∣V(C) ∩ A∣ ≤ 3pk∣V(C) ∩ A∣ for every $C \in \sC\_{<k}$. [lem:lpuniversaltightnessgeneral] Let G = (V, E, J) be a countable weighted graph, let Λ ⊆ V be finite and non-empty, let ω be Bernoulli-β bond percolation on G, and let $\sC=\{C \cap \Lambda : C$ a cluster of ω} be the partition of Λ into clusters. Then the inequality $$\begin{aligned} {\mathbf P}\_\beta\bigl(\\|\sC\\|\_p \geq 3^k \lambda \bigr) &\leq {\mathbf P}\_\beta\bigl(\\|\sC\\|\_p \geq \lambda \bigr)^{3^{k-1}+1}\end{aligned}$$ holds for every 1 ≤ p ≤ ∞, β ≥ 0, λ ≥ 1, and k ≥ 0. The proof will rely on the BK inequality and the attendant notion of disjoint witnesses, which we now recall. Given (not necessarily distinct) increasing subsets A1, …, Ak of {0, 1}E, the disjoint occurrence A1 ∘ ⋯ ∘ Ak is defined to be the set of ω ∈ {0, 1}E such that there exist disjoint sets W1, …, Wk ⊆ {e : ω(e) = 1} such that $$(\omega'(e)=1 \text{ for every $e\in W\_i$}) \Rightarrow (\omega' \in A\_i) \qquad \text{for every $\omega' \in \{0,1\}^E$ and $1 \leq i \leq k$.}$$ The sets W1, …, Wk are referred to as disjoint witnesses for A1, …, Ak. The BK inequality (see also ) states that if G = (V, E, J) is a finite weighted graph and A1, …, Ak ⊆ {0, 1}E are increasing events then $${\mathbf P}\_\beta(A\_1 \circ \cdots \circ A\_k) \leq \prod\_{i=1}^k {\mathbf P}\_\beta(A\_i)$$ for every β ≥ 0. [Proof of ] It suffices to consider the case 1 ≤ p < ∞, the case p = ∞ having already been handled by. Let G = (V, E, J) be a finite weighted graph, let Λ ⊆ V, and let 1 ≤ p < ∞. Suppose that the event $\{\\|\sC\\|\geq 3^k \lambda\}$ holds for some λ ≥ 1 and k ≥ 1. Applying to the open subgraph ω yields that there exists m ≥ 3k − 1 + 1 and m edge-disjoint subgraphs H1, …, Hm of ω such that $\\|\sC(H\_i,\Lambda)\\|\_p \geq \lambda$ for every 1 ≤ i ≤ m, where $\sC(H\_i,\Lambda)=\{C\cap \Lambda: $ C a connected component of Hi}. Thus, if Ei denotes the edge set of Hi then the sets E1, …, Em are all disjoint witnesses for the event $\\|\sC\\|\_p\geq \lambda$, so that $$\label{eq:BK\_inclusion1} \{ \\|\sC\\|\_p\geq 3^k \lambda \} \subseteq \underbrace{\{\\|\sC\\|\_p\geq \lambda\} \circ \cdots \circ \{\\|\sC\\|\_p\geq \lambda\}}\_{\text{$3^{k-1}+1$ copies}}$$ for every λ ≥ 1 and k ≥ 1. Taking probabilities on both sides and applying the BK inequality completes the proof when G is finite. The infinite case follows straightforwardly by taking limits over exhaustions and we omit the details. [Proof of ] Let Mp = Mp(β) be the 1/e quartile of the distribution of $\\|\sC\\|\_p$: $$M\_p = \inf\left\{ x \geq 0: {\mathbf P}\_\beta(\\|\sC\\|\_p \geq x) \leq e^{-1}\right\},$$ so that ${\mathbf P}\_\beta(\\|\sC\\|\_p \geq M\_p) \geq e^{-1}$ and ${\mathbf P}\_\beta(\\|\sC\\|\_p \geq 2M\_p) < e^{-1}$. We have by that $${\mathbf P}\_\beta(\\|\sC\\|\_p \geq 2\cdot 3^k M\_p ) \leq \exp\left[ -3^{k-1}-1\right]$$ and $${\mathbf P}\_\beta(\\|\sC\\|\_p \geq 3^{-k} M\_p) \geq {\mathbf P}\_\beta(\\|\sC\\|\_p \geq M\_p)^{1/(3^{k-1}+1)} \geq e^{-1/(3^{k-1}+1)}.$$ Taking k = ⌊log3α/2⌋ or k = ⌈log31/ɛ⌉ as appropriate, it follows that there exist universal positive constants c1 and C1 such that $${\mathbf P}\_\beta(\\|\sC\\|\_p \geq \alpha M\_p) \leq C\_1 e^{-c\_1\alpha} \qquad \text{ and } \qquad {\mathbf P}\_\beta(\\|\sC\\|\_p \leq {\varepsilon}M\_p) \leq C\_1 {\varepsilon}$$ for every α ≥ 1 and ɛ > 0. Integrating these estimates yields moreover that there exist universal positive constants c2 and C2 such that $c\_2 M\_p \leq {\mathbf E}\_\beta \\|\sC\\|\_p \leq C\_2M\_p$, so that $$\begin{aligned} {\mathbf P}\_\beta(\\|\sC\\|\_p \geq \alpha {\mathbf E}\_\beta \\|\sC\\|\_p) \leq {\mathbf P}\_\beta(\\|\sC\\|\_p \geq \alpha c\_2 M\_p) \leq C\_1 e^{-c\_1 c\_2 \alpha}\end{aligned}$$ and $$\begin{aligned} {\mathbf P}\_\beta(\\|\sC\\|\_p \leq {\varepsilon}{\mathbf E}\_\beta \\|\sC\\|\_p ) \leq {\mathbf P}\_\beta(\\|\sC\\|\_p \leq {\varepsilon}C\_2 M\_p) \leq C\_1 C\_2 {\varepsilon}.\end{aligned}$$ as claimed. In most examples, the cluster of the origin has heavy-tailed behaviour at criticality. If we think of the origin as being taken uniformly at random over all vertices, can be interpreted as saying that most the randomness in this heavy tail comes from the location of the origin, with the entire ensemble of cluster sizes in any finite region always having light-tailed behaviour in an appropriate sense. Consequences for hierarchical percolation ----------------------------------------- We now return to our primary setting, in which d, L, and 0 < α < d are fixed and we consider the infinite system of multiplicative coalescents encoding critical hierarchical percolation as defined in. We make note of the following elementary consequences of in this setting, which applies to the intermediate configurations Xn, t by. For each n ≥ 0 we define $$M\_n = \min \left\{m \geq 0:\P(\\|X\_{n,t\_n}\\|\_\infty \geq m) \leq e^{-1}\right\}$$ to be the typical vlaue of the largest cluster size in ηn, noting that Mn ≥ 2 for every n ≥ 0 with equality when n = 0. [Sums of powers of cluster sizes are well-behaved][cor:universaltightnessX] 1. For each sequence of positive integers p = (p1, …, pk) there exists a constant Cp such that $$\prod\_{i=1}^k \E\left[ \\|X\_{n,t}\\|\_{p\_i}^{p\_i}\right] \leq \E\left[\prod\_{i=1}^k \\|X\_{n,t}\\|\_{p\_i}^{p\_i}\right] \leq C\_{\mathbf{p}}\prod\_{i=1}^k \E\left[ \\|X\_{n,t}\\|\_{p\_i}^{p\_i}\right] \label{eq:taking\_out\_the\_product}$$ for every n ≥ 0 and 0 ≤ t ≤ tn. 2. For each pair of positive integers p1, p2 there exists a constant Cp1, p2 such that $$\label{eq:pulling\_out\_max} \E\left[ \\|X\_{n,t}\\|\_{p\_1+p\_2}^{p\_1+p\_2}\right] \leq \E\left[ \\|X\_{n,t}\\|\_{p\_1}^{p\_1} \\|X\_{n,t}\\|\_\infty^{p\_2}\right] \leq C\_{p\_1,p\_2} M\_{n}^{p\_2} \E\left[ \\|X\_{n,t}\\|\_{p\_1}^{p\_1}\right]$$ holds for every pair of positive integers p1, p2 ≥ 1, n ≥ 0, and 0 ≤ t ≤ tn. 3. For each positive integer p there exists a positive constant cp such that $$\label{eq:moment\_max\_lower} \E\left[ \\|X\_{n,t}\\|\_{p}^{p}\right] \geq \E\left[ \\|X\_{n,t}\\|\_{\infty}^{p}\right] \geq c\_p M\_{n-1}^p$$ holds for every positive integer p ≥ 1, n ≥ 0, and 0 ≤ t ≤ tn. [Proof of ] implies that there exists a universal constant C such that $$\E\left[\\|X\_{n,t}\\|\_{p}^{kp}\right] \leq C^k k! \E\left[\\|X\_{n,t}\\|\_{p}^{p}\right]^k$$ for every p, k ≥ 1, n ≥ 0 and 0 < t < tn. The inequalities and follow immediately from this estimate and Hölder’s inequality (noting that $\E\\|X\_{n,t}\\|\_\infty \leq \E\\|X\_{n,t\_n}\\|\_\infty \preceq M\_n$), while follows directly from the definitions since $\E\\|X\_{n,t}\\|\_\infty \geq \E\\|X\_{n-1,t\_{n-1}}\\|\_\infty \succeq M\_{n-1}$. For our purposes, will mean that we never have to worry about issues of non-uniform integrability when analyzing the asymptotics of ∥Xn, t∥pp, saving us from various technical issues that might arise otherwise. In particular, the inequality will be used extensively when studying the consequences of the hydrodynamic condition in the next section. --- Notation: For the rest of the body of the paper,, we will work exclusively with hierarchical percolation on $\bbH^d\_L$ with kernel $J(x,y)=\frac{L^{d+\alpha}}{L^{d+\alpha}-1} \\|x-y\\|^{-d-\alpha}$ and the equivalent multiplicative coalescent process X = ((XΛ, t)t = 0tn : n ≥ 0, Λ and n-block), regarding the parameters d, α, and L as fixed and working only at the critical parameter β = βc. As such, we will not continue to write these hypotheses in the statements of our theorems and lemmas. Moreover, when applying asymptotic notation to quantities depending on both the scale n and the time parameter 0 ≤ t ≤ tn we will always take all estimates to be uniform in the choice of 0 ≤ t* ≤ tn given n. On the other hand, we will continue to not* require estimates to be uniform in the choice of the index p when estimating moments. Thus, for example, ``$\E \\|X\_{n,t}\\|\_p^p \sim f\_p(n,t)$ for each integer p ≥ 1" means that for every integer p ≥ 1 and every ɛ > 0 there exists N = N(ɛ, p, d, L, α) such that $\|\E \\|X\_{n,t}\\|\_p^p/f\_p(n,t)-1\| \leq {\varepsilon}$ for every n ≥ N and 0 ≤ t ≤ tn. The hydrodynamic condition and its consequences =============================================== In this section we introduce the hydrodynamic condition, prove that this condition holds in the high-dimensional case d > 3α, and establish the main consequences of the condition. We will ultimately see that all the distinctions between the low-dimensional and high-dimensional regimes can be explained in terms of whether or not the hydrodynamic condition holds. Before proceeding, let us formally state the main results of in the manner that is most useful to us going forward. For each n ≥ 0 we let Λn denote the n-block containing the origin and recall that Mn denotes the typical size of the largest cluster in ηn as defined in. Recall that we are now always taking β = βc and suppressing the choice of β from our notation. [thm:paper1restatement] $M\_n \preceq L^{\frac{d+\alpha}{2}n}$ and $\E\|K\_n\| \asymp L^{\alpha n}$ for every n ≥ 0. [Proof of ] This is essentially proven in, although a little care is needed to deduce our statement from the ones given there since our definition of ‘clusters inside a block’ (and in particular of Kn) differs from the ones given there as explained at the end of : In our notation, the results of concern the clusters of the restriction of ω to Λn while we will always work with the clusters of the configuration ηn. Since ηn is contained in the restriction of ω to Λn, the upper bounds of are implied by the upper bounds of. While the lower bound $\E\|K\_n\| \preceq L^{\alpha n}$ does not a priori follow from the lower bound of since the inclusion goes in the wrong direction, one can easily verify that the proof (which is very short) goes through straightforwardly for our modified definition of ∣Kn∣. While the bound $M\_n \preceq L^{\frac{d+\alpha}{2}n}$ holds for all 0 < α < d, it is not sharp when d ≥ 3α. This is easily established from the results of under the stronger assumption that d > 3α: [lem:treegraphM] Mn ≼ nL2αn for every n ≥ 1. [Proof of ] This bound is a consequence of the tree-graph inequality method of Aizenman and Newman and was mentioned in. Indeed, it follows from the tree-graph inequalities (see ) that it is very hard for ∣Kn∣ to be much larger than the square of its first moment in the sense that $$\P(\|K\_n\| \geq m) \leq \frac{\sqrt{2}\E \|K\_n\|}{m}\exp\left[-\frac{m}{4(\E \|K\_n\|)^2}\right]$$ for every n ≥ 1, and m ≥ 1. (This bound holds for percolation on any transitive weighted graph, with any value of β ≥ 0.) It follows by a union bound that $$\P(\|K\_n^\mathrm{max}\| \geq m) \leq \sum\_{x\in \Lambda\_n} \P(\|K\_n(x)\| \geq m) \leq \frac{\sqrt{2} L^{dn} \E\|K\_n\|}{m}\exp\left[-\frac{m}{4(\E\_\beta \|K\_n\|)^2}\right]$$ for every n ≥ 1, and m ≥ 1. Since $\E \|K\_n\| \asymp L^{\alpha n}$, we deduce that there exist positive constants A1 and a such that $$\P(\|K\_n^\mathrm{max}\| \geq m) \leq \frac{A\_1 L^{(d+\alpha)n}}{m}\exp\left[-a L^{-2\alpha n} m\right]$$ for every n ≥ 1 and m ≥ 1. The claim follows since the right hand side is smaller than 1/e when m = ⌈A2nL2αn⌉ for a suitably large constant A2. Note that if Λ is an n-block, Λ1 and Λ2 are children of Λ, and C1 and C2 are clusters in ηΛ1 and ηΛ2 respectively, then there is an edge of ωB connecting C1 and C2 with probability of order min{1, L− (d + α)∣C1∣∣C2∣}. This leads the hierarchical percolation model to have very different asymptotic behaviours in the two cases $M\_n \asymp L^{\frac{d+\alpha}{2}n}$ and $M\_n \ll L^{\frac{d+\alpha}{2}n}$: In the first case there exist pairs of clusters in each scale that have a good probability to be connected by an edge when passing to the next scale, while in the second case no such clusters exist with high probability. We shall see moreover that the evolution of the recursive system of multiplicative coalescents X defined in is approximately deterministic under the condition $M\_n \ll L^{\frac{d+\alpha}{2}n}$. This property is very important to our analysis and therefore merits a memorable name: We say that the hydrodynamic condition holds if $M\_n=o(L^{\frac{d+\alpha}{2}n})$ as n → ∞. One of the most important technical results of the paper is as follows. The hydrodynamic condition holds if and only if d ≥ 3α. Note that the d > 3α case of the theorem follows immediately from since $L^{2\alpha}<L^{\frac{d+\alpha}{2}}$ when d > 3α. The critical case d = 3α is significantly more difficult to prove and is established in. In we prove that if d < 3α then $M\_n \succeq L^{\frac{d+\alpha}{2}n}$ for every n ≥ 0, which is strictly stronger than the negation of the hydrodynamic condition. The word ``hydrodynamic" is used here by analogy with the theory of hydrodynamic limits, in which the trajectories of Markov processes converge to those of deterministic dynamical systems. The fact that the process $\fX$ is approximately deterministic under the hydrodynamic condition is hinted at by the following proposition; we will see a much more wide-ranging generalisation of this phenomenon in. A precise asymptotic expression for the variance of ∥Xn, t∥22 will later be proven in. [prop:hydrodynamicvariance] If the hydrodynamic condition holds then $$\sqrt{{{\mathrm{Var}}}(\\|X\_{n,t}\\|\_2^2)} = o\left(\E\\|X\_{n,t}\\|\_2^2\right)$$ as n → ∞. The proof of this proposition will apply the following two inequalities, which we will use extensively throughout the paper. [lem:variance] If F, G : [0, ∞) → [0, ∞) are increasing then $$\begin{gathered} 0\leq \E \left[\sum\_{A\in X\_{n,t}} \|A\|F(\|A\|) \sum\_{B\in X\_{n,t}}\|B\|G(\|B\|)\right] - \E \left[\sum\_{A\in X\_{n,t}} \|A\|F(\|A\|)\right]\E\left[\sum\_{B\in X\_{n,t}}\|B\|G(\|B\|)\right]\\ \leq \E \left[\sum\_{A\in X\_{n,t}} \|A\|^2F(\|A\|)G(\|A\|)\right]\end{gathered}$$ for every n ≥ 0 and 0 ≤ t ≤ tn. In particular, the inequalities $$0\leq \E \left[\\|X\_{n,t}\\|\_p^p\\|X\_{n,t}\\|\_q^q\right] - \E \left[\\|X\_{n,t}\\|\_p^p\right]\E\left[\\|X\_{n,t}\\|\_q^q\right] \leq \E\\|X\_{n,t}\\|\_{p+q}^{p+q}$$ hold for every n ≥ 0, 0 ≤ t ≤ tn, and p ≥ 1. [lem:sumoverdistinctpairs] If F, G : [0, ∞) → [0, ∞) are increasing then $$\E\left[\sum\_{\substack{A,B \in X\_{n,t}\\\text{\emph{distinct}}}} \|A\|\|B\|F(\|A\|)G(\|B\|) \right] \leq \E\left[\sum\_{A\in X\_{n,t}}\|A\|F(\|A\|)\right]\E\left[\sum\_{B\in X\_{n,t}}\|B\|G(\|B\|)\right]$$ for every n ≥ 0 and 0 ≤ t ≤ tn. We will first prove, which will be applied in the proof of. [Proof of ] Fix n ≥ 0 and 0 ≤ t ≤ tn, and recall by that we can think of Xn, t as the partition into clusters of Bernoulli percolation on an appropriately defined weighted graph G with vertex set Λn. Letting K(x) = Kn, t(x) denote the cluster of x in this model for each x ∈ Λn, we can write $$\begin{aligned} \E\left[\sum\_{\substack{A,B \in X\_{n,t}\\\text{distinct}}} \|A\|F(\|A\|)\|B\|G(\|B\|)\right] &= \sum\_{x,y \in \Lambda\_n} \E \left[ F(\|K(x)\|)G(\|K(y)\|) \mathbbm{1}(x\nleftrightarrow y) \right],\end{aligned}$$ where ``$x\nleftrightarrow y$" means that x is not connected to y in this percolation model. Consider one such choice of x, y ∈ Λn contributing to this sum. If x is not connected to y then the conditional distribution of K(y) given K(x) is equal to the distribution of the cluster of y in percolation on the subgraph of G induced by the complement of K(x), which is stochastically dominated by the unconditioned distribution of the cluster of y. As such, we have that $$\E\left[G(\|K(y)\|)\mid K(x)\right] \leq \E[G(\|K(y)\|)] \qquad \text{ a.s.\ on the event that $x \nleftrightarrow y$}$$ and hence that $$\begin{aligned} \E \left[ F(\|K(x)\|)G(\|K(y)\|) \mathbbm{1}(x \nleftrightarrow y) \right] &= \E \left[ F(\|K(x)\|) \mathbbm{1}(x \nleftrightarrow y) \E\left[G(\|K(y)\|)\mid K(x)\right]\right]\\ &\leq \E \left[ F(\|K(x)\|) \mathbbm{1}(x \nleftrightarrow y) \right] \E\left[G(\|K(y)\|)\right]\\ &\leq \E \left[ F(\|K(x)\|) \right] \E\left[G(\|K(y)\|)\right],\end{aligned}$$ where the final inequality follows by Harris-FKG since F is increasing and $\{x\nleftrightarrow y\}$ is decreasing. Summing this estimate yields that $$\begin{aligned} \E\left[\sum\_{\substack{A,B \in X\_{n,t}\\\text{distinct}}} \|A\|F(\|A\|)\|B\|G(\|B\|)\right] &\leq \sum\_{x,y \in \Lambda\_n} \E \left[ F(\|K(x)\|) \right] \E\left[G(\|K(y)\|)\right]\\ &= \E\left[\sum\_{A\in X\_{n,t}}\|A\|F(\|A\|)\right]\E\left[\sum\_{B\in X\_{n,t}}\|B\|G(\|B\|)\right]\end{aligned}$$ as claimed. [Proof of ] The lower bound follows from Harris-FKG since ∑A ∈ Xn, t∣A∣F(∣A∣) is an increasing function of Xn, t when F is increasing. For the upper bound, we can expand $$\begin{gathered} \E \left[\sum\_{A\in X\_{n,t}} \|A\|F(\|A\|) \sum\_{B\in X\_{n,t}}\|B\|G(\|B\|)\right] = \E \left[\sum\_{\substack{A,B \in X\_{n,t}\\\text{distinct}}} \|A\|F(\|A\|)\|B\|G(\|B\|)\right] \\+ \E \left[\sum\_{A\in X\_{n,t}} \|A\|^2F(\|A\|)G(\|A\|)\right]\end{gathered}$$ and apply to bound the first term on the right hand side. [Proof of ] For the claim concerning the variance of ∥Xn, t∥22, we apply and of to obtain that $${{\mathrm{Var}}}(\\|X\_{n,t}\\|\_2^2) = \E\\|X\_{n,t}\\|\_2^{4} - \E\left[\\|X\_{n,t}\\|\_2^{2}\right]^2 \leq \E\\|X\_{n,t}\\|\_{4}^{4} \preceq M\_n^2 \E\\|X\_{n,t}\\|\_{2}^{2}.$$ As such, the ratio of the variance to the mean squared is O(L− (d + α)nMn2), which is o(1) under the hydrodynamic condition. In the remainder of this section we study the asymptotics of the moments $\E\\|X\_{n,t}\\|\_p^p$ under the hydrodynamic condition. While this is obviously important in the cases d > 3α and d = 3α, the techniques we develop will also be important in the low-dimensional case d < 3α, where they are used in particular to establish that the hydrodynamic condition does not hold. The mean -------- In this section we study the asymptotics of the expected sum of squares $\E \\|X\_{n,t}\\|\_2^2$. Since $\E\\|X\_{n,t\_n}\\|\_2^2 = L^{dn} \E\|K\_n\|$ and $L^d \E\\|X\_{n-1,t\_{n-1}}\\|\_2^2 = \E\\|X\_{n,0}\\|\_2^2 \leq \E \\|X\_{n,t}\\|\_2^2 \leq \E\\|X\_{n,t\_n}\\|\_2^2$ for each n ≥ 1, we have by that $$\label{eq:first\_moment\_restatement\_4.1} \E \\|X\_{n,t}\\|\_2^2 \asymp L^{(d+\alpha)n}$$ for every n ≥ 0 and 0 ≤ t ≤ tn. We will now argue that one can establish much more precise estimates on $\E \\|X\_{n,t}\\|\_2^2$ under the hydrodynamic condition. To begin, note that yields the differential equation $$\begin{aligned} \frac{d}{dt} \E \\|X\_{n,t}\\|\_2^2 = \E \left[ \\|X\_{n,t}\\|\_{2}^{4} - \\|X\_{n,t}\\|\_{4}^{4}\right] = \left(1-\cE\_{2,n,t}\right)\E\left[\\|X\_{n,t}\\|\_2^2\right]^2 \label{eq:sum\_of\_squares\_E\_diff\_eq}\end{aligned}$$ where we define $$\cE\_{2,n,t} := \frac{\E[\\|X\_{n,t}\\|\_2^2]^2+\E[\\|X\_{n,t}\\|\_4^4] - \E[\\|X\_{n,t}\\|\_2^4]}{\E[\\|X\_{n,t}\\|\_2^2]^2} = \frac{\E[\\|X\_{n,t}\\|\_4^4] - {{\mathrm{Var}}}(\\|X\_{n,t}\\|\_2^2)}{\E[\\|X\_{n,t}\\|\_2^2]^2}$$ To make use of this, we first prove the following elementary bounds on the error term $\cE\_{2,n,t}$. [lem:E21] The error term $\cE\_{2,n,t}$ satisfies $$0\leq \cE\_{2,n,t} \leq \frac{\E[\\|X\_{n,t}\\|\_4^4]}{\E[\\|X\_{n,t}\\|\_2^2]^2}.$$ [Proof of ] The upper bound follows from Jensen’s inequality, while the lower bound follows from the p = 2 case of. [cor:E22] There exists a universal constant C such that $\cE\_{2,n,t} \leq C L^{-(d+\alpha)n}M\_n^2$. In particular, if the hydrodynamic condition holds then $\cE\_{2,n,t} = o(1)$ as n → ∞. (Recall that estimates of this form are always taken to be uniform in the choice of 0 ≤ t ≤ tn, so that the statement given here means that $\sup\_{0\leq t \leq t\_n}\cE\_{2,n,t} = o(1)$ as n → ∞.) [Proof of ] This follows immediately from together with of and. We next show that yields an exact formula for $\E\\|X\_{n,t}\\|\_2^2 $ in terms of the errors $\cE\_{2,n,t}$. [lem:sumofsquaresexactexpression] The equality $$L^{-(d+\alpha)n}\E\\|X\_{n,t}\\|\_2^2 = \frac{1}{\beta\_c}\left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n} - \frac{1}{t\_n}\int\_t^{t\_n} \cE\_{2,n,s} \dif s - \sum\_{m=1}^\infty \frac{L^{-\alpha m}}{t\_{n+m}}\int\_0^{t\_{n+m}}\cE\_{2,n+m,s}\dif s \right)^{-1}$$ holds for every n ≥ 0 and 0 ≤ t ≤ tn. Together with, this lemma has the following immediate corollary. [lem:sumofsquaresexacthydrodynamicasymptotic] If the hydrodynamic condition holds then $$\E\\|X\_{n,t}\\|\_2^2 \sim \frac{1}{\beta\_c}\left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n}\right)^{-1} L^{(d+\alpha)n} = \left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n}\right)^{-1} t\_n^{-1}$$ for all 0 ≤ t ≤ tn as n → ∞. When d > 3α, implies that the errors in this approximation are exponentially small in n (equivalently, polynomially small in the side-length of the block), while we will see that they are merely polynomially small in n (equivalently, polylogarithmically small in the side-length of the block) in the critical case d = 3α (a precise asymptotic estimate on the second order term is given in ). Since $\cE\_{2,n,t}$ is non-negative by, we also deduce that the lower bound of always holds exactly, whether or not the hydrodynamic condition is satisfied. [lem:sumofsquareslowerbound] The lower bound $$\E\\|X\_{n,t}\\|\_2^2 \geq \frac{1}{\beta\_c}\left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n}\right)^{-1} L^{(d+\alpha)n} = \left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n}\right)^{-1} t\_n^{-1}$$ holds for all 0 ≤ t ≤ tn as n → ∞. We now prove. [Proof of ] We begin by proving the t = 0 case of the equality. The differential equation can be rewritten $$\label{eq:one\_over\_sum\_of\_squares\_diff\_eq}\frac{d}{dt}\frac{1}{\E\\|X\_{n,t}\\|\_2^2} = -1 + \cE\_{2,n,t},$$ and since $\E\\|X\_{n+1,0}\\|\_2^2 = L^d \E\\|X\_{n,t\_n}\\|\_2^2$ it follows that $$\begin{aligned} \frac{L^{(d+\alpha)(n+1)}}{\E\\|X\_{n+1,0}\\|\_2^2} &= \frac{L^{(d+\alpha)n+\alpha}}{ \E\\|X\_{n,t\_n}\\|\_2^2} \\&= \frac{L^{(d+\alpha)n+\alpha}}{\E\\|X\_{n,0}\\|\_2^2} - L^{(d+\alpha)n+\alpha}t\_n + L^{(d+\alpha)n+\alpha}\int\_0^{t\_n} \cE\_{2,n,s} \dif s \\&= \frac{L^{(d+\alpha)n+\alpha}}{\E\\|X\_{n,0}\\|\_2^2} - \beta\_c L^\alpha\left(1- \frac{1}{t\_n}\int\_0^{t\_n} \cE\_{2,n,s} \dif s \right)\end{aligned}$$ for every n ≥ 0. Rearranging, we obtain that $$\frac{L^{(d+\alpha)n}}{\E\\|X\_{n,0}\\|\_2^2} = \beta\_c \left(1- \frac{1}{t\_n}\int\_0^{t\_n} \cE\_{2,n,s} \dif s \right) + \frac{1}{L^\alpha} \cdot \frac{L^{(d+\alpha)(n+1)}}{\E\\|X\_{n+1,0}\\|\_2^2}$$ and hence inductively that $$\frac{L^{(d+\alpha)n}}{\E\\|X\_{n,0}\\|\_2^2} = \beta\_c \sum\_{m=0}^k L^{-\alpha m} \left(1- \frac{1}{t\_{n+m}}\int\_0^{t\_{n+m}} \cE\_{2,n+m,s} \dif s \right) + \frac{1}{L^{\alpha(k+1)}} \cdot \frac{L^{(d+\alpha)(n+k+1)}}{\E\\|X\_{n+k+1,0}\\|\_2^2}$$ for every n, k ≥ 0. Since $L^{(d+\alpha)n}(\E\\|X\_{n,0}\\|\_2^2)^{-1} $ is bounded away from zero by, we can take the limit as k → ∞ to obtain that $$\frac{L^{(d+\alpha)n}}{\E\\|X\_{n,0}\\|\_2^2} = \beta\_c \sum\_{m=0}^\infty L^{-\alpha m} \left(1- \frac{1}{t\_{n+m}}\int\_0^{t\_{n+m}} \cE\_{2,n+m,s} \dif s \right),$$ which is equivalent to the t = 0 case of the claim. For other values of t, we simply integrate to obtain that $$\begin{aligned} \frac{L^{(d+\alpha)n}}{\E\\|X\_{n,t}\\|\_2^2} &= \frac{L^{(d+\alpha)n}}{\E\\|X\_{n,0}\\|\_2^2} - L^{(d+\alpha)n} t + L^{(d+\alpha) n} \int\_0^t \cE\_{2,n,t} \dif s \\&= \beta\_c \frac{t\_n-t}{t\_n} - \frac{\beta\_c}{t\_n} \int\_t^{t\_n} \cE\_{2,n,t} \dif s + \beta\_c \sum\_{m=1}^\infty L^{-\alpha m} \left(1- \frac{1}{t\_{n+m}}\int\_0^{t\_{n+m}} \cE\_{2,n+m,s} \dif s \right),\end{aligned}$$ which is equivalent to the claim. The second moment ----------------- In this subsection we build upon our analysis of $\E\\|X\_{n,t}\\|\_2^2$ in the previous section to prove asymptotic estimates on $\E\\|X\_{n,t}\\|\_3^3$, which we will then apply to study higher powers in the next subsection. While the results of this section are not strictly needed in the study of the d < 3α case, the same ideas used in the proof will appear again there as part of a more complicated situation, so that if the reader is primarily interested in the low-dimensional case they are still strongly encouraged to read this proof. [prop:hydrodynamicsumofcubes] If the hydrodynamic condition holds then $$\E\\|X\_{n,t}\\|\_3^3 = L^{(d+3\alpha)n+o(n)} \qquad \text{ and } \qquad \frac{\E\\|X\_{n,t}\\|\_3^3}{\E\\|X\_{n,0}\\|\_3^3} \sim \left(1-\frac{t}{t\_n} \frac{L^\alpha-1}{L^\alpha}\right)^{-3}$$ as n → ∞. If moreover d > 3α then there exists a constant A such that $$\E\\|X\_{n,t}\\|\_3^3 \sim A \left(1-\frac{t}{t\_n} \frac{L^\alpha-1}{L^\alpha}\right)^{-3} L^{(d+\alpha)n}$$ as n → ∞. Before proving this proposition, let us note the following immediate corollary, the conclusion of which will be strengthened in. [cor:lowdimensionsnothydrodynamic] The hydrodynamic condition does not hold when d < 3α. [Proof of ] It follows from the estimate of together with the estimates of as stated in that $$\E\\|X\_{n,t\_n}\\|\_3^3 \preceq M\_n \E\\|X\_{n,t\_n}\\|\_2^2 \preceq L^{\frac{3}{2}(d+\alpha)n}$$ for every n ≥ 0. When d < 3α this bound is not consistent with the asymptotic estimate $\E\\|X\_{n,t}\\|\_3^3 = L^{(d+3\alpha)n+o(n)}$, and it follows from that the hydrodynamic condition does not hold in this case. We now turn to the proof of. [Proof of ] The p = 3 case of admits the simplified expression $$\begin{aligned} \frac{d}{dt} \E \\|X\_{n,t}\\|\_3^3 &= 3\E \left[ \\|X\_{n,t}\\|\_{2}^{2}\\|X\_{n,t}\\|\_{3}^{3}-\\|X\_{n,t}\\|\_{5}^{5}\right].\end{aligned}$$ We rewrite this equation as $$\label{eq:diff\_eq\_E3} \frac{d}{dt} \log \E \\|X\_{n,t}\\|\_3^3 = 3(1-\cE\_{3,n,t})\E \\|X\_{n,t}\\|\_{2}^{2} \\= 3(1-\cE\_{3,n,t})(1+\cH\_{n,t}) \left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n}\right)^{-1} t\_n^{-1}$$ where we define $$\cE\_{3,n,t} := \frac{\E \\|X\_{n,t}\\|\_{5}^{5} + \E \\|X\_{n,t}\\|\_{2}^{2} \E \\|X\_{n,t}\\|\_{3}^{3} - \E[ \\|X\_{n,t}\\|\_{2}^{2} \\|X\_{n,t}\\|\_{3}^{3}]}{\E \\|X\_{n,t}\\|\_{2}^{2} \E \\|X\_{n,t}\\|\_{3}^{3}}$$ and $$\cH\_{n,t} := \left(\frac{L^\alpha}{L^\alpha-1}-\frac{t}{t\_n}\right) t\_n \E\\|X\_{n,t}\\|\_2^2 -1.$$ (Although it is not particularly important at this moment, we note that imply that the errors $\cE\_{3,n,t}$ and $\cH\_{n,t}$ are both non-negative.),, and imply that $\cH\_{n,t}=o(1)$ under the hydrodynamic condition and that $\cH\_{n,t}=O(n^2 L^{-(d-3\alpha)n})$ is exponentially small in n when d > 3α. Moreover, we have analogously to that $$0 \leq \cE\_{3,n,t} \leq \frac{\E \\|X\_{n,t}\\|\_{5}^{5}}{\E \\|X\_{n,t}\\|\_{2}^{2} \E \\|X\_{n,t}\\|\_{3}^{3}},$$ where the upper bound follows from Jensen’s inequality and the lower bound follows by the same reasoning as. Applying the estimate of it follows that $$\label{eq:E3\_upper} \cE\_{3,n,t} \preceq L^{-(d+\alpha)n}M\_n^2$$ and hence that if the hydrodynamic condition is satisfied then $\cE\_{3,n,t}=o(1)$ as n → ∞. If additionally d > 3α then it follows from that $\cE\_{3,n,t} = O(n^2 L^{-(d-3\alpha)n})$ is exponentially small in n. Integrating yields that if the hydrodynamic condition holds then $$\begin{aligned} \E \\|X\_{n,t}\\|\_3^3 &= \exp\left[ \frac{3}{t\_n} \int\_0^t (1-\cE\_{3,n,s})(1+\cH\_{n,s}) \left(\frac{L^\alpha}{L^\alpha-1}-\frac{s}{t\_n}\right)^{-1} \dif s \right] \E\\|X\_{n,0}\\|\_3^3 \nonumber\\ &= \exp\left[ \frac{3}{t\_n} \int\_0^t \left(\frac{L^\alpha}{L^\alpha-1}-\frac{s}{t\_n}\right)^{-1} \dif s \pm O\left(L^{-(d+\alpha)n}M\_n^2\right) \right] \E\\|X\_{n,0}\\|\_3^3 \nonumber\\ & \sim \exp\left[ \frac{3}{t\_n} \int\_0^t \left(\frac{L^\alpha}{L^\alpha-1}-\frac{s}{t\_n}\right)^{-1} \dif s \right] \E\\|X\_{n,0}\\|\_3^3 \nonumber\\ &= \exp\left[ -3 \log \left(1-\frac{t}{t\_n} \frac{L^\alpha-1}{L^\alpha}\right) \right] \E\\|X\_{n,0}\\|\_3^3\label{eq:X3\_asymptotics\_proof}\end{aligned}$$ as required. Taking t = tn it follows in particular that $$\label{eq:X3\_asymptotics\_proof2} \E \\|X\_{n+1,0}\\|\_3^3 = L^d \E\\|X\_{n,t\_n}\\|\_3^3 \sim L^{d+3\alpha} \E \\|X\_{n,0}\\|\_3^3,$$ which implies the claim that $\E\\|X\_{n,t}\\|\_3^3 = L^{(d+3\alpha)n+o(n)}$ as n → ∞. Now suppose that d > 3α. In this case, the error in the approximation is exponentially small	arxiv_0000690
iders and target drones. Air-defence systems & (a) All land-based surface-to-air missile systems, and (b) all anti-aircraft guns with a calibre of more than 40 mm or with multiple barrels with a combined caliber of at least 70 mm. This includes self-propelled systems on armoured or unarmoured chassis. Anti-submarine warfare weapons & Rocket launchers, multiple rocket launchers and mortars for use against submarines, with a calibre equal to or above 100 mm. Armoured vehicles & All vehicles with integral armour protection, including all types of tank, tank destroyer, armoured car, armoured personnel carrier, armoured support vehicle and infantry fighting vehicle. Vehicles with very light armour protection (such as trucks with an integral but lightly armoured cabin) are excluded. Artillery & Naval, fixed, self-propelled and towed guns, howitzers, multiple rocket launchers and mortars, with a calibre equal to or above 100 mm. Engines & (a) Engines for military aircraft, for example, combat-capable aircraft, larger military transport and support aircraft, including large helicopters; (b) Engines for combat ships -,fast attack craft, corvettes, frigates, destroyers, cruisers, aircraft carriers and submarines; (c) Engines for most armoured vehicles - generally engines of more than 200 horsepower output.\* Missiles & (a) All powered, guided missiles and torpedoes, and (b) all unpowered but guided bombs and shells. This includes man-portable air defence systems and portable guided anti-tank missiles. Unguided rockets, free-fall aerial munitions, anti-submarine rockets and target drones are excluded. Sensors & (a) All land-, aircraft- and ship-based active (radar) and passive (e.g. electro-optical) surveillance systems with a range of at least 25 kilometres, with the exception of navigation and weather radars, (b) all fire-control radars, with the exception of range-only radars, and (c) anti-submarine warfare and anti-ship sonar systems for ships and helicopters.\* Ships & (a) All ships with a standard tonnage of 100 tonnes or more, and (b) all ships armed with artillery of 100-mm calibre or more, torpedoes or guided missiles, and (c) all ships below 100 tonnes where the maximum speed (in kmh) multiplied with the full tonnage equals 3500 or more. Exceptions are most survey ships, tugs and some transport ships Other & (a) All turrets for armoured vehicles fitted with a gun of at least 12.7 mm calibre or with guided anti-tank missiles, (b) all turrets for ships fitted with a gun of at least 57-mm calibre, and (c) all turrets for ships fitted with multiple guns with a combined calibre of at least 57 mm, and (d) air refueling systems as used on tanker aircraft.\* [coverage] [ht] \| Country \| Code \| Included \| Country \| Code \| Included \| Country \| Code \| Included \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Afghanistan \| AFG \| 1950 - 2016 \| German Dem. Rep. \| GDR \| 1950 - 1991 \| Pakistan \| PAK \| 1950 - 2016 \| \| Albania \| ALB \| 1950 - 2016 \| Germany \| DEU \| 1950 - 2016 \| Panama \| PAN \| 1950 - 2016 \| \| Algeria \| DZA \| 1962 - 2016 \| Ghana \| GHA \| 1957 - 2016 \| Papua New Guin. \| PNG \| 1975 - 2016 \| \| Angola \| AGO \| 1975 - 2016 \| Greece \| GRC \| 1950 - 2016 \| Paraguay \| PRY \| 1950 - 2016 \| \| Argentina \| ARG \| 1950 - 2016 \| Guatemala \| GTM \| 1950 - 2016 \| Peru \| PER \| 1950 - 2016 \| \| Armenia \| ARM \| 1991 - 2016 \| Guinea \| GIN \| 1958 - 2016 \| Philippines \| PHL \| 1950 - 2016 \| \| Australia \| AUS \| 1950 - 2016 \| Guinea-Bissau \| GNB \| 1973 - 2016 \| Poland \| POL \| 1950 - 2016 \| \| Austria \| AUT \| 1950 - 2016 \| Guyana \| GUY \| 1966 - 2016 \| Portugal \| PRT \| 1950 - 2016 \| \| Azerbaijan \| AZE \| 1991 - 2016 \| Haiti \| HTI \| 1950 - 2016 \| Qatar \| QAT \| 1971 - 2016 \| \| Bahrain \| BHR \| 1971 - 2016 \| Honduras \| HND \| 1950 - 2016 \| Romania \| ROM \| 1950 - 2016 \| \| Bangladesh \| BGD \| 1971 - 2016 \| Hungary \| HUN \| 1950 - 2016 \| Russia \| RUS \| 1992 - 2016 \| \| Belarus \| BLR \| 1991 - 2016 \| India \| IND \| 1950 - 2016 \| Rwanda \| RWA \| 1962 - 2016 \| \| Belgium \| BEL \| 1950 - 2016 \| Indonesia \| IDN \| 1950 - 2016 \| Saudi Arabia \| SAU \| 1950 - 2016 \| \| Benin \| BEN \| 1961 - 2016 \| Iran \| IRN \| 1950 - 2016 \| Senegal \| SEN \| 1960 - 2016 \| \| Bhutan \| BTN \| 1950 - 2016 \| Iraq \| IRQ \| 1950 - 2016 \| Serbia \| SRB \| 1992 - 2016 \| \| Bolivia \| BOL \| 1950 - 2016 \| Ireland \| IRL \| 1950 - 2016 \| Sierra Leone \| SLE \| 1961 - 2016 \| \| Bosnia Herzegov. \| BIH \| 1992 - 2016 \| Israel \| ISR \| 1950 - 2016 \| Singapore \| SGP \| 1965 - 2016 \| \| Botswana \| BWA \| 1966 - 2016 \| Italy \| ITA \| 1950 - 2016 \| Slovakia \| SVK \| 1993 - 2016 \| \| Brazil \| BRA \| 1950 - 2016 \| Jamaica \| JAM \| 1962 - 2016 \| Slovenia \| SVN \| 1991 - 2016 \| \| Bulgaria \| BGR \| 1950 - 2016 \| Japan \| JPN \| 1950 - 2016 \| Solomon Islands \| SLB \| 1978 - 2016 \| \| Burkina Faso \| BFA \| 1960 - 2016 \| Jordan \| JOR \| 1950 - 2016 \| Somalia \| SOM \| 1960 - 2016 \| \| Burundi \| BDI \| 1962 - 2016 \| Kazakhstan \| KAZ \| 1991 - 2016 \| South Africa \| ZAF \| 1950 - 2016 \| \| Cambodia \| KHM \| 1953 - 2016 \| Kenya \| KEN \| 1963 - 2016 \| Soviet Union \| SUN \| 1950 - 1991 \| \| Cameroon \| CMR \| 1960 - 2016 \| North Korea \| PRK \| 1950 - 2016 \| Spain \| ESP \| 1950 - 2016 \| \| Canada \| CAN \| 1950 - 2016 \| South Korea \| KOR \| 1950 - 2016 \| Sri Lanka \| LKA \| 1950 - 2016 \| \| Cape Verde \| CPV \| 1975 - 2016 \| Kuwait \| KWT \| 1961 - 2016 \| Sudan \| SDN \| 1956 - 2016 \| \| Central Afr. Rep. \| CAF \| 1960 - 2016 \| Kyrgyzstan \| KGZ \| 1991 - 2016 \| Suriname \| SUR \| 1975 - 2016 \| \| Chad \| TCD \| 1960 - 2016 \| Laos \| LAO \| 1950 - 2016 \| Swaziland \| SWZ \| 1968 - 2016 \| \| Chile \| CHL \| 1950 - 2016 \| Latvia \| LVA \| 1991 - 2016 \| Sweden \| SWE \| 1950 - 2016 \| \| China \| CHN \| 1950 - 2016 \| Lebanon \| LBN \| 1950 - 2016 \| Switzerland \| CHE \| 1950 - 2016 \| \| Colombia \| COL \| 1950 - 2016 \| Lesotho \| LSO \| 1966 - 2016 \| Syria \| SYR \| 1950 - 2016 \| \| Comoros \| COM \| 1975 - 2016 \| Liberia \| LBR \| 1950 - 2016 \| Taiwan \| TWN \| 1950 - 2016 \| \| DR Congo \| ZAR \| 1960 - 2016 \| Libya \| LBY \| 1951 - 2016 \| Tajikistan \| TJK \| 1991 - 2016 \| \| Congo \| COG \| 1960 - 2016 \| Lithuania \| LTU \| 1990 - 2016 \| Tanzania \| TZA \| 1961 - 2016 \| \| Costa Rica \| CRI \| 1950 - 2016 \| Luxembourg \| LUX \| 1950 - 2016 \| Thailand \| THA \| 1950 - 2016 \| \| Cote dIvoire \| CIV \| 1960 - 2016 \| Macedonia \| MKD \| 1991 - 2016 \| Timor-Leste \| TMP \| 2002 - 2016 \| \| Croatia \| HRV \| 1991 - 2016 \| Madagascar \| MDG \| 1960 - 2016 \| Togo \| TGO \| 1960 - 2016 \| \| Cuba \| CUB \| 1950 - 2016 \| Malawi \| MWI \| 1964 - 2016 \| Trinidad Tobago \| TTO \| 1962 - 2016 \| \| Cyprus \| CYP \| 1960 - 2016 \| Malaysia \| MYS \| 1957 - 2016 \| Tunisia \| TUN \| 1956 - 2016 \| \| Czech Republic \| CZR \| 1993 - 2016 \| Mali \| MLI \| 1960 - 2016 \| Turkey \| TUR \| 1950 - 2016 \| \| Czechoslovakia \| CZE \| 1950 - 1991 \| Mauritania \| MRT \| 1960 - 2016 \| Turkmenistan \| TKM \| 1991 - 2016 \| \| Denmark \| DNK \| 1950 - 2016 \| Mauritius \| MUS \| 1968 - 2016 \| Uganda \| UGA \| 1962 - 2016 \| \| Djibouti \| DJI \| 1977 - 2016 \| Mexico \| MEX \| 1950 - 2016 \| Ukraine \| UKR \| 1991 - 2016 \| \| Dominican Rep. \| DOM \| 1950 - 2016 \| Moldova \| MDA \| 1991 - 2016 \| Un. Arab Emirates \| ARE \| 1971 - 2016 \| \| Ecuador \| ECU \| 1950 - 2016 \| Mongolia \| MNG \| 1950 - 2016 \| United Kingdom \| GBR \| 1950 - 2016 \| \| Egypt \| EGY \| 1950 - 2016 \| Morocco \| MAR \| 1956 - 2016 \| United States \| USA \| 1950 - 2016 \| \| El Salvador \| SLV \| 1950 - 2016 \| Mozambique \| MOZ \| 1975 - 2016 \| Uruguay \| URY \| 1950 - 2016 \| \| Equatorial Guin. \| GNQ \| 1968 - 2016 \| Myanmar \| MMR \| 1950 - 2016 \| Uzbekistan \| UZB \| 1991 - 2016 \| \| Eritrea \| ERI \| 1993 - 2016 \| Namibia \| NAM \| 1990 - 2016 \| Venezuela \| VEN \| 1950 - 2016 \| \| Estonia \| EST \| 1991 - 2016 \| Nepal \| NPL \| 1950 - 2016 \| Vietnam \| VNM \| 1976 - 2016 \| \| Ethiopia \| ETH \| 1950 - 2016 \| Netherlands \| NLD \| 1950 - 2016 \| South Vietnam \| SVM \| 1950 - 1975 \| \| Fiji \| FJI \| 1970 - 2016 \| New Zealand \| NZL \| 1950 - 2016 \| Yemen \| YEM \| 1991 - 2016 \| \| Finland \| FIN \| 1950 - 2016 \| Nicaragua \| NIC \| 1950 - 2016 \| North Yemen \| NYE \| 1950 - 1991 \| \| France \| FRA \| 1950 - 2016 \| Niger \| NER \| 1960 - 2016 \| South Yemen \| SYE \| 1950 - 1991 \| \| Gabon \| GAB \| 1960 - 2016 \| Nigeria \| NGA \| 1960 - 2016 \| Yugoslavia \| YUG \| 1950 - 1992 \| \| Gambia \| GMB \| 1965 - 2016 \| Norway \| NOR \| 1950 - 2016 \| Zambia \| ZMB \| 1964 - 2016 \| \| Georgia \| GEO \| 1991 - 2016 \| Oman \| OMN \| 1950 - 2016 \| Zimbabwe \| ZWE \| 1950 - 2016 \| [countinc] Details on the estimation procedure ----------------------------------- The recent implementation of Generalised Additive Models (GAM) in the `R` package `mgcv` allows for smooth varying coefficients as proposed by. These models can be represented in GAMs by multiplying the smooths by a covariate (in the given application the smooths of time are multiplied by the covariates. See for more details. The functions for the smooths are based on P-Splines as proposed by, giving low rank smoothers using a B-spline basis using a simple difference penalty applied to the parameters. For the smooth time-varying coefficients on the fixed effects a maximum number of 65 knots is used, combined with a second-order P-spline basis (quadratic splines) and a first-order difference penalty on the coefficients. The non-linear random smooths are estimated similar to those proposed by. As a basic idea, one views the individual smooths as splines with random coefficients, i.e. each country has a random effect, that is in fact a function of time that is approximated by regression splines. The parameters of the splines are assumed to be normally distributed with mean zero and the same variance for all curves, which translates into having the same smoothness parameter for all curves. This concept is implemented efficiently in the GAM structure of the `mgcv` package by using the nesting of the smooth within the respective actor. In order to avoid overfitting and keeping computation tractable, a first-order penalty with nine knots is employed. The smoothness selection is done for all smooths by the restricted maximum likelihood criterion (REML). As the data set is rather big with more than 1.3 million observations in the formation model, the fitting procedure of the model is computationally expensive and was virtually impossible with standard implementations in `R` before the introduction of the `bam()` function in the `mgcv` package in 2016 that needs less memory and is much faster than other comparable packages. The estimation routine employs techniques as proposed in. Those methods use discretization of covariate values and iterative updating schemes that require only subblocks of the model matrix to the computed at once which allows for the application of parallelization tools. For all computations we also used the statistical programming language `R` (). Important packages used for visualization of networks and computation of network statistics are the `statnet` suite of network analysis packages () as well as the package `igraph` (). For the Tables the `stargazer` package from was employed. For the model evaluation and visualization we used the `PRROC` package of. Details on the PCA of the time-varying smooth random effects ------------------------------------------------------------ For the analysis of the smooth random effects we are following the discretization approach of. As noted in Section [heterogen] we assume the random effects ϕi(t) (ϕi+(t) and ϕi−(t) in the formation and the persistence model, respectively) to be realizations of a stochastic process Φ = {ϕ(t), t ∈ τ}, for i = 1, ..., N individual countries and τ = [1951, 2016]. In order to summarize the information provided by these functions we are searching for a weight function β(t) that gives us the principal component scores ϕi = ∫τβ(t)ϕi(t)dt. In order to do so, the weight function ξ1(t) among all possible functions β(t) must be found that maximizes N− 1∑i = 1N(∫τβ(t)ϕi(t)dt)2 subject to the constraint ∫τξ12(t)dt = 1. From our model we get N individual estimated functions ϕ̂i(t) for all observations (countries) and can discretize the functions ϕ̂i(t) on a grid. We use T = 100 equidistant points {t1, ..., t100} on the interval τ of length ∣τ∣ = T. This gives a discretized (N × T) time series matrix $\hat{\Phi}$ with N country specific observations in the rows and the estimated functions, evaluated at the discrete time points, in the columns: $$\hat{\Phi}=\begin{pmatrix} \hat{\phi}\_1(t\_1) & \cdots & \hat{\phi}\_1(t\_{100})\\ \vdots & \ddots & \vdots \\ \hat{\phi}\_N(t\_1) & \cdots& \hat{\phi}\_N(t\_{100}) \end{pmatrix}$$ Therefore, in fact we are searching for a solution for the discrete approximation of ∫τβ(t)ϕ̂i(t)dt ≈ (T/T)∑j = 1Tβ(tj)ϕ̂i(tj) = ∑j = 1Tβ̃(tj)ϕ̂i(tj) such that the solution ξ̃1 that maximizes the mean square satisfies ∣∣ξ̃1∣∣2 = 1. This is now a standard problem, with the solution ξ̃1 being found by the eigenvector that corresponds to the largest eigenvalue of the covariance matrix of $\hat{\Phi}$. Out-of-sample-predictions for simulated networks ------------------------------------------------ As a standard principle in network analysis, a model should be able the reflect global network characteristics. We evaluate six of them for our out-of-sample forecasts. The first three characteristics are related to the number of actors that are actively engaged in the arms trade. The statistic Size is defined as the count of predicted edges in each year. This measure helps to evaluate the ability of the model to predict amount of realized arms trade in each year. As it is also of interest to measure how dense the predicted arms trade network is, we include Density, relating the size of the network to the number of edges that could have potentially realized. We define the Order of the network as number of actors that are engaged in either exporting or importing arms. The results will provide an impression whether the model has the ability not only to classify the right amount of edges (as in Size), but also their nesting within the countries. As we have emphasized the importance of local network statistics we evaluate whether the local network statistics are able to generate the corresponding global statistics. Therefore, we include the Mean Indegree (being the same as Mean Outdegree), as well as the share of Reciprocity. In order to evaluate the accuracy of our predictions with respect to triangular relationships we furthermore include the measure Transitivity, that divides the number of triangles by the number of connected triples in the graph. In this statistic, the direction of the edges is ignored. The analysis of this measure gives an impression how well the two chosen transitivity measures capture the overall clustering in the network. The results are presented in Figure [figsim1]. In each of the six panels we see the respective network statistics plotted against time. The solid red line gives the network statistics, evaluated at the real MCW network. The boxplots show the network statistics, evaluated for each year for the 1.000 simulated networks. Comparison of realized and simulated network topologies. The boxplots give statistics from the simulated networks. The solid line gives the statistics for the real networks. Number of edges (Size), number of active countries (Order), number of realized transfers relative to possible transfers (Density), average indegree (Indegree), share of reciprocated transfers and ratio of undirected triangles relative to connected triples (Transitivity). [figsim1] Supplementary Material ====================== Different threshold values -------------------------- ### Distribution of TIVs In Figure [figdenstiy] we present a kernel density estimate (KDE) of the pooled TIVs for the whole time period. The distribution of the TIVs is highly skewed and has a long tail. Therefore, we give a logarithmic representation. In order to give an impression of the left tail, Table [quantiles] provides the lower quantiles of the distribution. From this it can be seen that roughly 20% of all observations are below a threshold of 3. [figdenstiy] [!htbp] ccccc 0% & 4.75% & 9.5% & 14.25% & 19% 0.020 & 0.700 & 1.332 & 2.200 & 3 [quantiles] In order to demonstrate the effect of different binarization thresholds on the estimated coefficients we pursue the following strategy. As a baseline we use the “original effects” from the paper with a threshold of zero and plot them in solid black together with two standard error confidence bounds in dark grey. Additionally, we include the estimated coefficients with thresholds incrementing from zero to three in steps of 0.5 as dashed black lines. (i) By comparison of the solid line with the dashed lines it can be seen how strong the point-estimates vary with different thresholds. (ii) If the dashed lines are within the confidence intervals in dark grey they can be said to be statistically indistinguishable from the original estimates. (iii) Furthermore, we show how the confidence bounds of the new estimates exceed the ones of the original estimation, displayed in light grey. These areas represent the highest upper limit and the lowest lower limit that exceeds the original confidence bounds. Hence, if only the dark grey confidence bound is visible, then the effects of all estimates are the same, if the light grey confidence bound is above and/or below the dark grey, this means that the bounds of the estimates with higher thresholds are wider. ### Fixed effects with different thresholds [figcomp1] The degree-related statistics are shown in the top four panels of Figures [figcomp1]. We find no significant changes of the results as the dashed lines stay in almost all cases within the original confidence bounds. An exception is the senders’ outdegree in the formation model but even with the highest threshold this effect does not become significant. On the contrary, partly the results get even more clear. For example the outdegree effect for the receiver in the formation and dissolution model (second row) becomes even more negative in tendency with increasing thresholds. For the reciprocity effect in the third row we find that the coefficients stay almost the same for all different thresholds of binarization. The same applies for the transitivity effect in the formation model (left panel in the fourth row). Here the upper confidence bound even indicates a potentially higher effect. For the transitivity in the persistence model (right panel in the fourth row) we find that the effect becomes insignificant in the beginning if we set roughly 20% of the lowest observations to zero. Otherwise the effect stays significant and very close to the point estimates of the original estimation. For the Shared Suppliers Effect (the two panels at the bottom) the results are very similar to the transitivity effects, i.e. we find an potentially stronger effect in the formation model (left panel at the bottom) and an insignificant effect for the first years in the persistence model (right panel) with the highest binarization threshold. Given that the network statistics are constructed from the network and therefore directly and potentially strongly affected by different thresholds, the robustness of the effects is reassuring. Only if we replace almost one fifth of the existing edges by zeros the effects of the hypderdyadic statistics in the persistence model start to become partly insignificant. However, the affected statistics are in line with our theoretical expectation that the network effects matter mostly for the formation. From this result it might not come as a surprise that the effects of the non-network related covariates are even more robust because their construction is not affected by the thinning of the network. We can confirm with Figure [figcomp2] that shows virtually no noteworthy changes of the effects. [figcomp2] Different time windows ---------------------- In the paper we assume that the STERGM process applies to two consecutive years. As a robustness check, we define the periods t and t + 1 such that they contain multiple years. If we take the years 2013, 2014, 2015 and 2016 as an example for time windows of length two, we set Yijt − 1, t = 1 if country i exports arms to j in 2013 or 2014 and Yijt, t − 1 = 0 if country i has not exported to j neither in 2015 nor in 2016. We also extend this concept such that we combine three years into one period. For the non-network related covariates we are using the time averages for the respective time windows for continuous variables (e.g. if a period contains two years, the average of the GDP in these two years is taken) and we set binary variables to one if the respective feature was present in all year (e.g. the indicator for a formal alliance is one if the alliance was present in all two or three years). The corresponding estimates can be seen in Figures [figwindow1] and [figwindow2]. (i) The Figures are constructed such that the baseline is given by the “original effects” from the paper. These effects are plotted in solid black together with two standard error confidence bounds in dark grey. (ii) The coefficients with two or three years within one time period are given in dotted (two years) and dotted-dashed (three years). (iii) The area where the original confidence bounds are exceeded is given in light grey. In Figure [figwindow1] it can be seen that the dotted and dashed-dotted line rarely is outside of the dark grey confidence bound in the panels of rows one to three. One really noteworthy exception applies to the transitivity effect in the formation model. Here the estimates with periods containing three years behave somewhat more time-constant than the original estimates. This is, however, a natural result because with the broader time windows major changes in the data as the collapse of the Soviet Union become more smooth. All in all, the panels clearly show that the coefficients are very robust and do not change fundamentally. This impression is confirmed by Figure [figwindow2] where again no notable exceptions can be found influence the interpretation in terms of variation with time or significance. This is again in line with our theoretical expectations since the coefficient for the military expenditures of the receiver is almost not affected by different time windows. [figwindow1] [figwindow2] Model without random effects ---------------------------- In the main article we mentioned that the inclusion of the random effects leads to a vanishing global effect of the senders outdegree. I.e once controlled for the sender-specific random effect the coefficient on the outdegree statistic is insignificant. Here, we show that once we exclude random effects from the models, all results are very robust with the exception of the coefficient on the outdegree. This can be seen in Figures [fignoraneff1] and [fignoraneff2] with coefficients that are very comparable to the ones from the main paper. The main exception is given by the senders outdegree (top panels in Figure [fignoraneff1]). Here the coefficients are now in both models positive and significant. Especially in the formation effect the coefficient is very high. This shows that there is indeed a global effect of the senders outdegree that, however, vanishes if one controls for country-specific heterogeneity. [fignoraneff1] [fignoraneff2] Comparison of different methods ------------------------------- ### Theoretical discussion Besides the STERGM, there are many other models suitable for dynamic networks. In the following, we provide an overview of alternative approaches with a discussion of their suitability for the given dataset. Latent Space Models: A potential alternative is given by the family of latent space models (e.g. ; ). However, it is hard to model arms trade appropriately within this model class. The need of estimating time-varying coefficients results in yearly separate estimations. The resulting yearly latent space representations are not very stable and do not work well with isolates, i.e. countries without transfers in a given year. If one goes for a panel approach with this model class (see the package `amen` by ) one must accept that the coefficients stay the same for the whole time-period and, even more problematic, that the positions within one latent space are sufficient to capture all network dependencies for the time-period 1950-2016, which is clearly a heroic assumption. On top of that, the latent space approach does not allow for the evaluation of complex network statistics we are interested in. Stochastic-Actor Oriented Models: Stochastic-Actor Oriented Model (SAOM, see e.g. ) are built for modelling dynamic networks and have the virtue of allowing for the estimation of dyadic and hyper-dyadic network effects. However, the model is tailored for social networks and some of its’ assumptions are very problematic for modelling the arms trade network. Most importantly, it assumes actor-homogeneity, an assumption that is clearly violated for the dataset. Additionally, the SAOM fundamentally builds on the idea that the networks observed represent snapshots of a continuous underlying process of edge formation and persistence, i.e. we would need to assume that between t and t + 1 multiple changes could have been realized in the network. This is not an acceptable assumption because if a transfer between i and j was recorded in t and t + 1, it is not meaningful to assume that there exists an in-between state where the transfer has the change to disappear and re-emerged multiple times. Otherwise, if no transfer is recorded in t and t + 1 we have no reason to believe that there was trade in between. In the basic description of the SAOM () the authors write: “A foundational assumption of the models discussed in this paper is that the network ties are not brief events, but can be regarded as states with a tendency to endure over time. Many relations commonly studied in network analysis naturally satisfy this requirement of gradual change, such as friendship, trust, and cooperation.” Apparently it is hard to argue that recording whether there was a transfer between two countries in a given year can be viewed as an enduring state. Exponential Random Graph Models: Our model is in fact motivated by recent advances within the exponential random graph model (ERGM) family, i.e. the TERGM (,, ) and the STERGM (). However, our model differs from the standard cross-sectional ERGM (but also from the conventional TERGM and STERGM) because it does not allow for simultaneous network dependencies. In general, a static cross-sectional ERGM seems to be an implausible choice for the modelling of a dynamic network with strong actor heterogeneity. A dynamic formation and the TERGM, however, can be constructed in a very similar manner as the STERGM. Those models are natural candidate model for comparison. ### Candidate models In the following we present several alternative candidate models by increasing level of complexity. Autoregressive ERGM (Model 1): The most simplistic stochastic model is an autoregressive model that assumes time-dependence of all individual dyads such that $$\label{logit1} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=\theta\_0+\theta\_1y\_{ij}^{t-1,t}.$$ This temporal dependence structure can be interpreted as a cross sectional ERGM with the lagged response as a dyadic exogenous covariate or as a TERGM with a dyadic stability term (see e.g. ). It is motivated by the idea that the probability of a transfer in t might change if there was a transfer in t − 1. TERGM with covariates (Model 2): We can change model ([logit1]) by including all the network effects and covariates as specified in Section 3.2 of the main paper: $$\label{logit2} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})$$ In this formulation, we include the autoregressive component only indirectly, the lagged network statistics give some information about the network embedding of a transfer but not whether there was a preceding transfer. TERGM with covariates and random effects (Model 3): With the inclusion of smooth time-varying random effects for the sender and the receiver we have $$\begin{split} \label{logit3} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=&\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})\\&+\phi\_{i,sender}(t)+\phi\_{j,receiver}(t). \end{split}$$ The main difference to the STERGM is now that we do not model the processes of formation and persistence separately but within one model and that we do not include the information on the lagged response here, that is implicitly included in the STERGM mechanics. TERGM with covariates and dyadic stability (Model 4): The inclusion of the lagged response to the TERGM with covariates gives $$\label{logit4} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})+\theta\_1y\_{ij}^{t-1,t}.$$ TERGM with network effects, dyadic stability and random effects (Model 5): And as a last step we allow for sender- and receiver-specific random effects in the TERGM with network statistics and lagged response: $$\label{logit5} \begin{split} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}= x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=&\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})+\theta\_1y\_{ij}^{t-1,t}\\&+\phi\_{i,sender}(t)+\phi\_{j,receiver}(t). \end{split}$$ STERGM without random effects (Model 6): Together with our main model from the paper (Model 7), the STERGM with random effects we additionally include a STERGM without random effects. The formal representations are given in the main paper in equations (4) and (5). [t!] \| Number \| Model Name \| lagged edge \| covariates \| rand. eff. \| PR % \| ROC % \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| 1 \| Autoreg. ERGM \| yes \| no \| no \| 0 \| 0 \| \| 2 \| TERGM \| no \| yes \| no \| 0 \| 0 \| \| 3 \| TERGM \| no \| yes \| yes \| 0 \| 1.54% \| \| 4 \| TERGM with dyadic stability \| yes \| yes \| no \| 4.62% \| 6.15% \| \| 5 \| TERGM with dyadic stability \| yes \| yes \| yes \| 18.46% \| 44.62% \| \| 6 \| STERGM \| implicit \| yes \| no \| 24.62% \| 6.15% \| \| 7 \| STERGM \| implicit \| yes \| yes \| 52.31% \| 41.54% \| [models] ### Comparison of out-of-sample predictions We evaluate the proposed models in the following way. In a first step, we fit coefficients based on the information of t − 1 to the response in t. In the second step we use the fitted models in order to predict the edges in t + 1. As a result we obtain probabilistic out-of-sample predictions. The evaluation is done with area under the curve (AUC) measures for the Receiver-Operating-Characteristic (ROC) curve and the Precision-Recall (PR) curve. All models are fitted using the package `mgcv` (; Version 1.8-24) and evaluated with the package `PRROC` (; Version 1.3). Table [models] gives and overview of the predictive performance of all models included. In the two rightmost columns we present the share of years where the respective model has the highest out-of-sample predictive power. It can be seen that evaluated by the PR, the STERGM with random effects clearly represents the superior model with the highest predictive performance. Judged by the ROC the performance of the TERGM and the STERGM (both with random effects and as a model class) are very similar. We give a detailed description of the results below. In order to give a clear impression of how the out-of-sample fits are related we provide multiple plots where the baseline model, the STERGM with covariates and random effects, is always indicated in solid red. [figroc1] In Figure [figroc1] we compare the AUC values of the autoregressive ERGM (Model 1), the TERGM with covariates (Model 2) and with random effects (Model 3) with the baseline STERGM model (Model 7). The Precision Recall AUC values are shown on the left hand side of Figure [figroc1] and provide a clear message since all selected candidate models have AUC values clearly below the STERGM used in the paper (Model 7). Looking at the AUC values from the ROC measure (right panel) shows that the two TERGM models (Models 2 and 3) come partly close to the baseline model in the first years of the observational period, while the simplistic autoregressive ERGM has considerable lower AUC values. However as a general picture the STERGM with random effects is clearly the superior model. Including the lagged response as explanatory variable makes the TERGM models (Models 4 and 5) pretty similar to our baseline model (Model 7). [figroc2] Naturally, this is reflected in Figure [figroc2]. However, the STERGM model is more flexible because it allows for different coefficients for the processes of formation and persistence. This fact leads to superior predictions of the STERGM model (Model 7) in the left panel of Figure [figroc2]. The red line provides the upper boundary in most time points, but there are some instances where the autoregressive TERGMs (Models 4 and 5) provide the better predictions when evaluated with the PR curve. If the AUC measure for the ROC curve is compared, we see again that the predictions of the TERGM partly outperform the STERGM. Nevertheless, this is mostly the case in the beginning of the observational period and it seems like the superiority of the STERGM increases with the more recent periods. In Figure [figroc3] the predictive performance of the STERGM with (Model 7) and without (Model 6) random effects is compared. Although the AUC values are very close to each other, the model that includes the random effects clearly provides better out-of-sample predictions. On the right hand side of Figure [figroc3] the contrast becomes visible more clearly and in almost all years our baseline model provides the better predictions. All in all we conclude the following. (i) The STERGM gives, among all other candidate models the best predictions when judged by Precision Recall, being the more important measure when predicting rare vents. (ii) Furthermore, the STERGM has a much richer interpretation than the TERGM and (iii) the random effects provide an substantial benefit to the inferential part of the model. We therefore conclude that the choice of the STERGM with random effects seems to be very appropriate regarding both, the predictive performance as well as the ability to gain new insights. [figroc3] --- 1. Department of Statistics, Ludwig-Maximilians-Universität München, 80539 Munich, Germany, [email protected][↩](#fnref1) 2. Department of Political Science, Ludwig-Maximilians-Universität München, 80538 Munich, Germany, [email protected][↩](#fnref2) 3. Department of Statistics, Ludwig-Maximilians-Universität München, 80539 Munich, Germany, [email protected][↩](#fnref3) 4. The authors gratefully acknowledge funding provided by the german research foundation (DFG) for the project International Trade of Arms: A Network Approach.[↩](#fnref4) A Dynamic Separable Network Model with Actor Heterogeneity: An Application to Global Weapons Transfers ========================================================================================================== #1 In this paper we propose to extend the separable temporal exponential random graph model (STERGM) to account for time-varying network- and actor-specific effects. Our application case is the network of international major conventional weapons transfers, based on data from the Stockholm International Peace Research Institute (SIPRI). The application is particularly suitable since it allows to distinguish the potentially differing driving forces for creating new trade relationships and for the endurance of existing ones. In accordance with political economy models we expect security- and network-related covariates to be most important for the formation of transfers, whereas repeated transfers should prevalently be determined by the receivers’ market size and military spending. Our proposed modelling approach corroborates the hypothesis and quantifies the corresponding effects. Additionally, we subject the time-varying heterogeneity effects to a functional principal component analysis. This serves as exploratory tool and allows to identify countries that stand out by exceptional increases or decreases of their tendency to import and export weapons. Arms Transfers, Functional Principal Component Analysis, Generalized Additive Mixed Model, Security and Defence Network, Varying Coefficient Model Introduction ============ In this paper we present a data-driven extension of the separable temporal exponential random graph model (STERGM, ) applied appropriately to a highly relevant case: The international weapons exchange. The STERGM allows to differentiate between the formation, i.e. new arms trades, and the persistence of existing edges, i.e. continued arms transfers. To introduce into the field, we first sketch and motivate network analysis for (arms) trade data. We then put the model in a broader context of statistical network models, supplemented by a description and discussion of international arms trade. Trade networks Statistical network analysis provides a good framework to conceptualize international trade systems. highlight the enormous interdependencies of economic transactions and propose a network approach for capturing the systemic complexity. Gravity models, as standard approach in econometrics for modelling trade data (), are usually focussed on dyadic relations. Hence, the models exclude highly important hyper-dyadic dependencies, and especially indirect relations. showed that gravity models of international trade are, therefore, necessarily incomplete. In particular, they demonstrated that analysing the determinants of link creation is highly important as the binary network carries information that goes beyond the classical gravity model representation. demonstrated that trade networks are commodity-specific, i.e. their topologies are quite different across commodities - leading us to conclude that there is also a need to consider arms transfers separately. This is theoretically challenging since arms transfers constitute a very special trade relationship. The transferred products and services can potentially lead to deadly quarrels between or within states, or they may contribute to stabilization and deterrence. The delivery is not always a purely economic exchange but may also serve the support of aligned countries or groups. In sum, the exchange of weapons is a politically sensible and security-related, but also an economically beneficial relationship. For this reason, we make use of flexible statistical models for network data that allow us to investigate the special incentives in the international arms trade network. Statistical network models Statistical models that are suitable for temporal networks have been developed just in the recent years, and different techniques have been proposed. were the first to extend the static exponential random graph model (ERGM, ; ) to discrete-time Markov chain models, see also. or also consider network dynamics on a discrete time scale. They propose the temporal exponential random graph model (TERGM) which makes use of a Markov structure conditioning on previous network statistics as covariates in the model. A related approach is presented by, discussing assumptions that allow for circumventing the often computationally intractable fitting process of dynamic network models by applying logistic regression models. expand the model using Bayesian methods which allows the parameters in the dynamic network model to change with time. A general perspective on dynamic networks is provided by. It also includes models for continuous time, such as stochastic actor-oriented models (SAOM, see ) or dynamic stochastic block models (SBM, see for instance ). A recent novel modelling strategy for networks observed at discrete time points has been proposed by. They do not model the state of the network itself but rather focus on network changes which either occur because of the formation of new edges or because of the (non-)persistence of existing ones. Assuming independence between the two processes, conditional on the previous network, leads to the so called separable TERGM. The separation is motivated by the fact that the two processes under study are highly likely to be driven by different mechanisms and factors. The authors argue that the inclusion of a stability term (being mathematically equivalent to the inclusion of the lagged edge values as explanatory variable) in a TERGM could lead to ambiguous conclusions because it is not clear whether a positive stability parameter means that non-existing ties remain non-existent (no formation) or whether existent ties remain existent (persistence). For many real world dynamic networks the process change with time and therefore the assumption of stationarity seems to be inappropriate. This is especially the case for network data that span a long time period and potentially subject to structural breaks. Under such conditions it appears necessary to allow the model parameters to change with time. We take up this idea and extend the STERGM by allowing for time-varying coefficients. More specifically, we propose to rely on so called generalized additive models (GAM). This model class has been proposed by and extended fundamentally by to allow for smooth, semi-parametric modelling of time-varying parameters in a generalized regression framework (see also ). Furthermore, the assumption of node homogeneity must be regarded as questionable. We therefore allow for heterogeneity in the model (see for a discussion on node heterogeneity). Accordingly, we follow the p2-model developed by and enrich the STERGM with functional time-varying random effects () which leads to smooth node-specific effects. We propose to investigate the fitted functional heterogeneity effects with techniques from functional data analysis (FDA), see for instance. This allows to identify countries (nodes) that have fundamentally changed their role in the arms-trading network over the observation period. Global weapons transfers [usecase] At present, there are only a few empirical binary network analyses of the international arms trade. pioneered in analysing topological features of the binary arms trade network. Their descriptive network analysis is supplemented by an empirical investigation using a binarized gravity model without considering network dependencies. In this article we build on the recently published paper by that uses a TERGM. However, our approach extends the TERGM in many aspects. Most importantly, we treat dynamic dependencies in a fundamentally different way. In, the authors found that previous arms trading has a highly determining impact on the occurrence of subsequent transfers due to the enormous inertia. This finding implies that the information whether trade happened in the preceding time period(s) has a considerable impact on the probability to trade again, leading to the same ambiguities as mentioned in the stability term example by. In order to disentangle the driving network formation forces due to pure inertia, we propose to incorporate this distinction directly in the model. More precisely, the STERGM allows us to investigate whether the mechanisms that result in transfers being formed without immediate predecessor differ from those that lead to consecutive transfers. This is also of practical importance because governments carefully reflect the decision whether to authorize arms transfers based on economic and security considerations. Furthermore, they continuously reconsider this decision whether to maintain such trade relations or whether to dissolve because the importer potentially jeopardises strategic interests or violates once shared normative standards (see for the general model and for as well as for normative considerations). We expect several necessary conditions to hold for the formation of transfers: the receiving country must be considered at least marginally trustworthy and politically and economically reliable. Hence, passing a threshold of trustworthiness is required for formation, i.e building new trades. The special role of trustworthiness in arms transfers stems from the fact that security concerns play an important role when governments decide whether to license the delivery. We expect network statistics, as well as regime dissimilarity and formal alliances to play a prominent role in the formation stage to raise a relationship above the minimum threshold level of reservation. Follow-up trades and their repetition should then be rather dominated by economic considerations like the size of a receiver economy and by the size of the military expenditures (see ). While differentiation between formation and repetition, respectively, legitimates the use of the STERGM per se, our extensions of the model towards time-varying coefficients are important and in our view inevitable because the observational time covers more than 65 years. Hence, the introduction of smooth dynamic effects is needed to build a realistic model. Given the dynamic evolution of the network, the historical developments and the presence of at least one system-wide structural break with the collapse of the Soviet Union, we expect that the generative mechanisms change over time and differ with respect to the included variables if we compare the pre- and post cold war time period (see also and ). Finally, we argue that not all network activities and trades can be explained by observables and, thus, unobserved heterogeneity remains. We expect primarily actor-specific heterogeneity which is accentuated by systematic historical accounts (; ). This highlights the self-reinforcing tendencies of technological advantages of highly developed countries which results in strong heterogeneity of the countries’ abilities to export (and import). Therefore, the inclusion of actor-specific random effects seems necessary and we expect strong heterogeneity among the countries with respect to imports and exports. We proceed as follows. Section [descrip] presents the data provided by the Stockholm International Peace Research Institute (SIPRI). Section [model] introduces the statistical models used to analyse the data. Section [results] provides the results and their interpretation. Section [conc] concludes the paper. Data description and preprocessing ================================== [degreedist] Data on the international trade of major conventional weapons (MCW) are provided by the Stockholm International Peace Research Institute (see ). They include for example aircrafts, armoured vehicles and ships (see Table [coverage] in the Appendix [descrannex] for an overview of the types of arms). The countries included and their three-digit country codes are given in Table [countinc] of Appendix [descrannex]. Note that we have excluded all non-state organizations like the Khmer Rouge or the Lebanon Palestinian Rebels from the dataset as well as countries with no reliable covariate information available. Figure [Fig-network-1] in the Appendix [descrannex] shows binary networks for the years 2015 and 2016 and Figure [sumstat] in the Appendix [descrannex] provides a collection of summary statistics for the networks. We focus on the binary occurrence of trade thereby disregarding the exact transfer volumes and follow and in setting the edge value to one if there is a trade flow greater zero between two countries and zero else. Additionally, we re-estimated our model with different thresholds and found that the results are quite robust, for details see the Supplementary Material. [fig:d1] The analysis of the degree distributions is of vital interest in statistical network analysis () and gives important insights into the basic properties of the network under study. With more than 65 networks to analyse, we compute the period-average degree distribution and provide information on the minimal and maximal value of the realized degree distribution. This is represented in a log-log version in Figure [degreedist] for both, outdegree and indegree. The plot shows the enormous heterogeneity in the networks. Most of the countries have no exports at all with a time-average share of 78% of countries exhibiting outdegree zero, while the outdegree distribution has a long tail, indicating that there are a few countries, having a very high outdegree. The highest observed outdegree in a year is 66 and is observed for the United States. Other countries with exceptional high outdegree for almost the whole time period are Russia (Soviet Union), France, Germany, United Kingdom, China, Italy and Canada. In the right plot, the indegree distribution can be seen. Here the pattern is different. The highest value observed in a year is 16 and corresponds to Saudi Arabia. In contrast to the outdegree distribution, the countries with a high indegree are changing with time. In the beginning of the observational period the countries with the highest indegree were Germany, Indonesia, Italy, Turkey and Australia, but in more recent times these are the United Arab Emirates, Saudi Arabia, Singapore, Thailand and Oman. In Figure [fig:d1] we provide a graphical representation of the change and stability patterns in the network. On the left hand side we present the share of observations (vertical axis) against the number of subsequent transfers (i.e. repeated transfers) on the horizontal axis. Out of roughly 19, 000 recorded trading instances only 33% do not have at least one consecutive transfer in the follow-up year of a trade. Looking on the right hand side of Figure [fig:d1] we visualize the share of observations (vertical axis) that has at least as much subsequent transfers as indicated by the horizontal axis. It can be seen that roughly the same share of observations (35%) lasts at least five periods and almost 10% of all dyadic relations last more than 20 consecutive years without any interruption. Therefore, a differentiated approach to the explanation of formation and persistence could be fruitful in this application case. Model ===== Dynamic formation and Persistence model --------------------------------------- In this section we formalize our network model. Let Yt be the network at time point t, which consists of a set of actors, labelled as At and a set of directed edges, represented through the index set Et = {(i, j) : i, j ∈ At}. Note that this is a slight misuse of index notation since Yijt does not necessarily refer to the (i, j)-th element if we consider Yt as adjacency matrix. This is because the actor set At is allowed to change with time, so that i and j are not running indices from 1 to nt, where nt is the number of elements in At. Instead indices i and j represent the i-th and j-th country, respectively. We define Yijt = 1 if country i exports weapons to country j and since self-loops are meaningless, elements Yiit are not defined. We aim to model the network in t based on the previous year network in t − 1. To do so we have to take into account that the actor sets At − 1 and At may differ. In particular we have to consider the case of newly formed countries. New countries of interest are those that are present in t but do not provide information about their network embedding in the previous period. For exports this is not a concern as it is almost never the case that a new country starts sending arms immediately after entering the network. Notable exceptions are Russia, the Czech Republic and Slovakia. However, these countries have clear defined predecessor states (the Soviet Union and Czechoslovakia) which can be used in order to gain information about the position of these countries in the precedent network. Regarding the imports, there is a share of countries that start receiving arms immediately with entering the network. Notwithstanding, those transactions represent a share of less then 0.3% of the observed trade flows. Therefore, we regard this cases as negligible and include in the model only countries where information on the current and previous time period is available. We formalize this approach by defining Yt, t − 1 as the subgraph of Yt with actor set Bt, t − 1 = At ∩ At − 1 containing $n\_{t,t-1} \coloneqq \|B^{t,t-1}\|$ elements. Accordingly, Yt − 1, t represents the subgraph of Yt − 1 with actor set Bt, t − 1. Note that both subgraphs share the same set of actors and Yt − 1 = Yt − 1, t if At − 1 and At coincide. From a modelling perspective, we follow and assume that the network in t can be modelled given preceding networks, using a first-order Markov structure to describe transition dynamics for those actors included in the set Bt, t − 1. Furthermore, we want to identify the driving forces of a transfer in t if there was a preceding transfer in t − 1 in the persistence model while the formation model considers the process of forming a trade relationship without a preceding transfer, i.e. biannual data. The notion of formation and persistence can be amended by using broader time windows. We demonstrate the robustness of our results with respect to broader time windows in the Supplementary Material. Let Y+ = Yt, t − 1 ∪ Yt − 1, t represent the formation network, that consists of edges that are either present in t or in t − 1. For the persistence network, we define Y− = Yt, t − 1 ∩ Yt − 1, t, being the network that consists of edges that are present in t and in t − 1. Based on the actor set Bt, t − 1 and given the formation and persistence network as well as the network in t − 1 the network in t is uniquely defined by Yt, t − 1 = Y+ ∖ (Yt − 1, t ∖ Y−) = Y− ∪ (Y+ ∖ Yt − 1, t). Note that both, Y+ as well as Y− depend on time t as well, which we omitted in the notation for ease of readability. We assume that for each discrete time step, the processes of formation and persistence are separable. That is, the process that drives the formation of edges does not interact with the process of the persistence of the edges conditional on the previous network. Formally this is given by the conditional independence of Y+ and Y−: $$\begin{split} &P(Y^{t,t-1}=y^{t,t-1}\|Y^{t-1,t}=y^{t-1,t};\theta)=\\& P(Y^{+}=y^{+}\|Y^{t-1,t}=y^{t-1,t};\theta^+)P(Y^{-}=y^{-}\|Y^{t-1,t}=y^{t-1,t};\theta^-), \end{split}$$ where the lower case letters denote the realizations of the random networks and θ = (θ+, θ−) gives the parameters of the model. We will also include non-network related covariates in our analysis, but we suppress this here in the notation for simplicity. Note that it is not possible to use the lagged response as predictor, as by construction Yijt − 1, t = 1 ⇒ Yij+ = 1 and Yijt − 1, t = 0 ⇒ Yij− = 0. That is, an edge that existed in t − 1 cannot be formed newly and an edge that was not existent in t − 1 cannot be dissolved. It follows that the formation model exclusively focuses on the binary variables Yij+ with (i, j) ∈ E+ = {(i, j) : i, j ∈ Bt, t − 1, Yijt − 1, t = 0}. This assures that in t − 1 no edge between actors i and j was present and both actors are observable at both time points. Equivalently, the model for Y− consists of observations Yij− with (i, j) ∈ E− = {(i, j) : i, j ∈ Bt, t − 1, Yijt − 1, t = 1}, assuring that only edges that could potentially persist enter the model. The time-dependence of E+ and E− is omitted for ease of readability. If we use an ERGM for the transition, this would yield the following probability model for the formation $$P(Y^+=y^+\|Y^{t-1,t}=y^{t-1,t};\theta^+)=\frac{\exp\{\theta^+g(y^{+},y^{t-1,t})\}}{\sum\_{\tilde{y}^+ \in \mathcal{Y}^+(y^{t-1,t})}\exp\{\theta^+g(\tilde{y}^+,y^{t-1,t})\}}.$$ The sum in the denominator is over all possible formation networks from the set of potential edges that can form given the network yt − 1, t. The inner product θ+g(y+, yt − 1), relates a vector of statistics g( ⋅ ) to the parameter vector θ+. The analogous model is assumed for the persistence of edges and not explicitly given here for the interest of space. We will subsequently work with a simplified model which is computationally much more tractable. We assume that the formation or persistence of an edge at time point t does solely depend on the past state but not on the current state of the network. This is achieved by restricting the statistics such, that they decompose to g(y+, yt − 1, t) = ∑(i, j) ∈ E+yij+g̃ij(yt − 1, t) for some statistics g̃( ⋅ ). This assumption is extensively discussed by and can be well justified by the notion that the lagged network accounts for the major share of the dependency among the edges in the current network. It also allows for intuitive interpretations as can be seen as follows. Let Y− ij+ represent the formation network Y+, excluding the entry Yij+. Then, for (i, j) ∈ E+ the following logistic model holds $$\label{logit} \begin{split} \log\bigg{\{} \frac{P(Y^+\_{ij}=1\|Y^+\_{-ij}=y^+\_{-ij},Y^{t-1,t}=y^{t-1,t};\theta^+)}{P(Y^+\_{ij}=0\|Y^+\_{-ij}=y^+\_{-ij},Y^{t-1,t}=y^{t-1,t};\theta^+)} \bigg{\}} &=\log\bigg{\{} \frac{P(Y^+\_{ij}=1\|Y^{t-1,t}=y^{t-1,t};\theta^+)}{P(Y^+\_{ij}=0\|Y^{t-1,t}=y^{t-1,t};\theta^+)}\bigg{\}}\\&=\theta^+\tilde{g}\_{ij}(y^{t-1,t}). \end{split}$$ Note that model ([logit]) describes network dynamics, but the model itself is static. Hence we model dynamics but do not allow for dynamics in the model itself. This is a very implausible restriction which we give up by allowing the model parameters to change with time t, that is we replace the parameter θ+ by θ+(t), representing a smooth function in time. In other words, we allow the parameters in the model to smoothly interact with time. This leads to a time-varying coefficient model in the style as proposed by. The focus of interest is therefore not only on the formation and persistence of edges (trade flows) but also on how these effects change in the 67 years long observation period. Network statistics and explanatory variables -------------------------------------------- From a statistical point of view, network statistics are required in order to capture network dependencies. However, as social network literature has shown, network statistics usually are not just statistical controls but convey substantial meaning (see e.g. ). In the given context, they can be motivated by political, strategic and economic arguments that refer to real-world processes (see ). Note, that we norm all network statistics (with the exception of Reciprocity) to be within a percentage range between 0 and 100, this is necessary in order to make the statistics independent from the varying network size and allows to compare them. Outdegree: The outdegree of a node is a standard statistic in network models. Formally, the outdegree of actor i at time point t − 1 is defined as $$outdeg\_{t-1,i}=\frac{100}{ n\_{t,t-1}-1}\sum\_{k \in B^{t,t-1}} y^{t-1,t}\_{ik}.$$ The arms trade network exhibits a highly oligopolistic structure with a few high-intensity traders, hence a positive coefficient for the outdegree of the sender (sender.outdegt − 1, i) is plausible. However, we incorporate country-specific random effects in the model and it is therefore not clear whether the senders’ outdegree as a global measure is still of relevance once controlled for the random country heterogeneity. Only few advanced countries within NATO export and import at the same time. They have a highly differentiated portfolio, rendering specialization economically reasonable and strategically non-hazardous. In order to better represent this world-wide asymmetry we include the outdegree of the importer (receiver.outdegt − 1, j). This should not be captured by the random effects and we expect a clear negative effect, indicating that strong exporters seldom match with strong importers. Reciprocity: This statistic is intended to detect whether there is a general tendency of arms transfers to be mutual. The statistic measures whether the potential receiver was a sender in the dyadic relationship in the previous period: recipt − 1, ij = yjit − 1, t. Reciprocation is an essential mechanism in human relations in general, and in trade more specifically. Similar as noted above, in the context of arms transfers, especially highly developed countries exhibit this feature. Since this group of countries is rather small, and specialization-induced transfers between developed countries do not lead to continuous inflows we expect this mechanism to be rather visible at the formation stage, whereas it should not be a dominant feature for permanent repetition. Transitivity: Hyperdyadic trade relationships are an effective mechanism for pooling risks in buyer-seller networks () and for the emergence of generalized trust which is especially important in exchanging security goods. As a measure for higher-order dependencies we include transitivity, defined as $$trans\_{t-1,ij}=\frac{100}{n\_{t,t-1}-2}\sum\_{k \in B^{t,t-1}, k \neq i,j}y^{t-1,t}\_{ik}y^{t-1,t}\_{kj}.$$ This statistic essentially counts the directed two-paths from i to j in t − 1 and can be interpreted as a direct application of the Friend of a Friend logic from social networks to arms trade. Clearly, this kind of network embeddedness of weapons transfer deals is important for establishing for new ones but is also likely to be relevant for the continuation of already existing ones. Shared Suppliers: We also include a statistic that we call shared-suppliers in this context. This statistic counts the shared number of actors that export to a given pair of countries: $$sup\_{t-1,ij}=\frac{100}{n\_{t,t-1}-2}\sum\_{k \in B^{t,t-1}, k \neq i,j}y^{t-1,t}\_{ki}y^{t-1,t}\_{kj}.$$ This statistic allows to investigate whether two countries that share multiple suppliers have the tendency to engage in trade with each other. Such a pattern is likely to be induced by a general hierarchy in the network (see ). While the first tier consists of strong exporters, the second tier is populated by countries with the ability to produce and export that are nevertheless mainly supplied by the big exporters. Countries with many shared partners are likely to engage in trade with each other but on the other hand they are typically dependent on imports from the first tier. Therefore, relationships among those countries are rather of a sporadic nature and unlikely to endure. Consequently, we expect a positive coefficient in the formation model and a negative one in the persistence model. Naturally, the network of international arms trade is not exclusively driven by endogenous network processes but also influenced by variables from the realms of politics and economics. We lag all exogenous covariates by one year, first in order to be consistent with the idea that the determination of the network in t is based on the preceding time period and second, to account for the time lag between the ordering and the delivery of MCW. Formal Alliance: We regard dyadic formal alliances (including defence agreements and non-aggression pacts) as an important security related criteria that plays a central role for the formation during the cold war period. Therefore, the binary variable allianceij is included in the model, being one if countries i and j had a formal alliance in the previous period. Given the restriction that the data is available only until 2012 () we extrapolate the data, thereby assuming that the formal alliances did not change between 2012 and 2015. Regime Dissimilarity: Another important security related variable that potentially acts on the formation of arms transfers is given by the differences in political regimes between two potential trading partners. Hence we include the so called polity IV score, ranging from the spectrum − 10 (hereditary monarchy) to + 10 (consolidated democracy). This data can be downloaded as annual cross-national time-series until 2015, see for the data and as a basic reference. In our model we operationalise the distance between political regimes by using the absolute differences between the scores: poldiffij = ∣polityi − polityj∣. GDP: Following the standard gravity model, we include market sizes and distance in our model. The standard measure for market size is the gross domestic product (GDP, in millions). We include the GDP in logarithmic form for the sender (gdpi) as well as the receiver (gdpj). The GDP data are taken from and merged from the year 2010 on with recent real GDP data from the World Bank real GDP dataset (). Clearly, the market size and economic reliability of the exporter is a prerequisite for forming and maintaining arms exports. Distance: For gravity models applied to trade in commercial goods, there exists mounting empirical evidence that distance is a relevant factor for determining trade relations (). We do not expect that trade costs and geographical distance impede arms trade because arms transfers establish world-wide alignments of exporters pursuing global strategic interests. Nevertheless, we include the logarithmic distance between capital cities in kilometres () in order to fulfil the gravity model specification. Military Expenditures: We propose to include military expenditures of the sending and receiving country. This measure can be used as representing the size of the defence industrial base of the exporter, and the spending power and the intensity of the threat perceptions of the importing country. Accordingly, military expenditure is added separately for the exporter and the importer in logarithmic form (milexi, milexj). With regard to the distinction between formation and persistence, our expectation is related to the hypothesis that countries with high military expenditures are attractive customers for repeated importing. We therefore expect a positive and high coefficient for the military expenditures of the importer in the persistence model. The data are available from in the national material capabilities data set with as the basic reference on the data. Modelling heterogeneity ----------------------- The proposed network model assumes homogeneity, meaning that all differences between nodes in the network are fully described by the gravity model, enriched by security related criteria and network statistics as proposed above. However, the arms transfer network exhibits a rather small number of countries that are high-intensity exporters and a large number of countries that are restricted to imports. Furthermore, there are some countries that change their relative position in the trade network during the course of time. This mirrors a substantial amount of dynamic heterogeneity which need to be taken into account. This dynamic heterogeneity is accommodated by the inclusion of latent country effects, capturing the unobserved heterogeneity. We to follow the idea of and model country specific random curves which are fitted with penalized splines. This can be written in a mixed model representation such that the smooth country-specific effects are constructed using a B-spline basis with (a-priori) normally distributed spline coefficients. We follow the modelling strategy of and assume that the model includes two time-dependent random coefficients ϕi, sender+(t) and ϕj, receiver+(t). The effects are assumed to be a realization of a stochastic process with continuous and integrable functions. For each sender and receiver in both models the country-specific curves are given by ϕi(t) = B(t)ai where B(t) = (B1(t), ..., BQ(t)) is a B-spline basis covering the time range of observations and ai = (ai1, ..., aiQ) is the coefficient vector. We impose the prior distribution ai ∼ N(0, σa2DQ), i.i.d.for i = 1, ..., n where DQ is the inverse of a difference based penalty matrix which guarantees smoothness of the fitted curves ϕi(t) (see e.g. , for details on smoothing with B-splines). Note that for time windows where a country did not exist, the corresponding B-spline does take value zero, so that no heterogeneity effect is present. Complete model and estimation ----------------------------- Putting all the above elements together, the specification of the formation model of equation ([logit]) is given by $$\begin{split} \theta^+(t)\tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t}\_{ij})=&\theta\_0^+ sender.outdeg\_{t-1,i}\theta\_1^+(t)+ receiver.outdeg\_{t-1,j}\theta\_2^+(t)\\&+ recip\_{t-1,ij}\theta\_3^+(t) + trans\_{t-1,ij} \theta\_4^+(t)+sup\_{t-1,ij} \theta\_5^+(t)\\ &+ distance\_{t-1,ij } \theta\_5^+(t)+alliance\_{t-1,ij} \theta\_6^+(t)+poldiff\_{t-1,ij} \theta\_7^+(t)\\ &+ gdp\_{t-1,i } \theta\_8^+(t)+gdp\_{t-1,j} \theta\_9^+(t)+milex\_{t-1,i}\theta\_{10}^+(t)+milex\_{t-1,j} \theta\_{11}^+(t)\\ &+ \phi^+\_{i,sender}(t)+\phi^+\_{j,receiver}(t). \end{split}$$ Analogously we get the persistence model. Estimation is carried out with spline smoothing. That is, we replace the coefficients by θk(t) = B(t)uk, where uk is penalized through uk ∼ N(0, σ2D). Like above, the penalty matrix is appropriately chosen (see e.g. ) and B(t) is a B-spline basis. Hence, smooth functions and smooth random heterogeneity can be estimated in a coherent framework (see ). The entire model can be integrated in the flexible generalized additive model (GAM) framework provided by (see also ) which is implemented in the `mgcv` package (version 1.8-28) by. The identification of the smooth components and the intercept term is ensured by a “sum-to-zero” constraint (). For further details see the Appendix [estimdetail]. Results ======= Time-varying fixed effects -------------------------- The results of the time-varying effects are grouped into network-related covariates (presented in Figure [resnet]) and political and economic covariates (presented in Figure [respolecon]). The left columns give the coefficients for the formation model and the right columns for the persistence model, respectively. In the case of the network statistics, a schematic representation of the corresponding network effects is added on the right hand side. The values for the coefficients are presented as solid lines with shaded regions, indicating two standard error bounds. The zero-line is indicated as dashed line and the estimates for time-constant coefficients are given by the dotted horizontal line. Note that the coefficients at a given time point can be interpreted just as the coefficients in a simple logit model. Additionally, for the same coefficient (or coefficients with the same norming) in the formation and persistence model, the effect size can be compared directly. .84 .16 [fig:sub2] [resnet] (see Figure [resnet]) : The senders’ outdegree has a coefficient that is almost time-constant and close to zero for both models. This stands in contrast to the findings of, where a strong effect is present. Hence, once controlled for country-specific heterogeneity (especially the sender-specific country effect), no population-level outdegree effect for the exporter is present (we show in the Supplementary Material that the effect is indeed present when country-specific heterogeneity is excluded). However, the inclusion of country-specific sender and receiver effects does not affect the effect of the receivers’ outdegree and the coefficient is consistently negative, and slightly increasing over time in the formation model. For the persistence model, we find a less pronounced but significant negative effect. We interpret this as clear evidence that countries with a high outdegree are comparatively less frequently importing, and importers usually have relatively less frequent export relations. According to our experience this specification captures the trade asymmetries of the oligopolistic market better than just specifying the indegrees of the receiver. Reciprocity: Controlling for the distinguished asymmetrical nature of the weapons transfers, we identify a positive and significant impact of reciprocity in the formation model. Reciprocity in repeated transfers is only a relevant feature after the breakdown of the bipolar block structure. We conclude that the asymmetric structure is more present in persistent trade relations with importing countries that are typically dependent on big exporters. Transitivity: Looking at three-node statistics it can be seen that the variable transitivity has a positive impact on the formation and persistence. In the formation model, the effect is insignificant in the first years. This may be influenced by the clear hegemony of the United States and the Soviet Union, respectively, immediately after World War II which did not require a shared control over the recipient country, because the donor was powerful enough to secure the terms of a deal. In the 1980s middle power countries became technologically more advanced and especially in the West, they joined the US in delivering to other countries. The pronounced change between 1990 and 2010 can be explained by the break up of the two hostile blocs and the interruption of long-standing arm-trading partnerships leading to a fundamental reorganization until 2010 when the effect came back to the level of 1990. Although these arguments are also valid for the persistence model, we see that transitivity is less relevant for ongoing, repeated transfers (the time constant effect in the formation model has twice the size as the one in the persistence model). This impression is also strengthened by the fact that the coefficient is not subject to changes over time. Shared Suppliers: The coefficients related to the shared suppliers corroborate our expectation that many shared suppliers lead to the formation of transfers (positive and significant coefficient for the whole time period in the formation model). This indeed mirrors the phenomenon described above: there is a hierarchy of producing countries in the world. Receiver countries i and j should become acquainted with these technologies and should have similar levels of production capacities. This allows them to exchange arms. Also, the act of receiving both from the same supplier means that this country places trust to both receivers - such that this facilitates trust giving one to another. On the other hand, in the persistence model, the effect is indeed significantly negative and virtually zero from the 1975 on, showing that repetitive trading is not promoted by many shared suppliers. [respolecon] (see Figure [respolecon]) : The impact of a bilateral formal alliances on the formation of a transfer is positive and significant for both, the formation and with a more modest effect for the persistence, corroborating our expectation that formal alliances are most relevant for the formation, i.e. by passing the required threshold of starting weapons transfers. The required threshold of trustfulness to start seems to decline over time for the initiation. Hence, while formal alliances play a central role for arms trading after the second world war, the formation of arms trades is less and less influenced by the existence of a formal alliance by the sending and receiving state. However, given there exists an alliance, the impact (despite being smaller) continues to be relevant for repeated transfers. This is an important insight as we show for the first time that formalized alliance actually breed a dense web of arms transfers. Regime Dissimilarity: For the formation model, the coefficient on the absolute difference of the polity scores is all along negative, significant and shows some time variation. With the decay of the eastern bloc, the resistance to send new arms to dissimilar regimes increases until 2000. After that, the absolute effect of different polity scores declines again, coming back to the long-term constant effect. Interestingly, we find that regime dissimilarity is irrelevant in the persistence model, showing that given a relationship is started, repetition does no more require regimes to exhibit shared governance values. GDP: As expected, the coefficients on the logarithmic GDP for sender and receiver are positive and constant for both models. However, the effect for the senders’ GDP is much stronger in the formation model, showing that indeed mostly economically strong countries are able to open new markets for arms exports. Together, the coefficients support the “gravity hypothesis”, i.e. greater economic power and market sizes of the sender as well as the receiver increases the probability of forming and maintaining trade relations. However, given a transfer relation is started, this effect becomes smaller for repetition. Distance: In accordance with previous insights (), the results on the logarithmic distance contradicts the standard gravity model and distance proves to be insignificant in both models. Military Expenditures: For the military expenditures of the sender, we find very comparable and declining effects that become insignificant from 1990 on in both models. This indicates that with the end of the cold war the dominance of exporting countries with high military budgets has decreased. For the receivers’ military expenditures in the formation model, the effect is positive and turns significant with time. This clearly illustrates that the military expenditures of the receiver are not as important in the Cold War period where super powers often granted military assistance. Only with end of the 1980s there begins a marketization of the weapons transfers with suppliers demanding money for delivery. Given there is a preceding exchange, we find a very strong effect for the military expenditures of the receiver for the full observational period, indicating that the availability of huge military expenditures is a key for understanding the continuous yearly inflow of weapons. Overall, the results confirm our initial hypothesis. Judged by the size of the coefficients and their significance we find that the network statistics (reciprocity, transitivity, shared suppliers) and security related covariates (formal alliance, regime dissimilarity) prove to be highly influential in the formation model. On the other hand, we find weaker (or insignificant) network effects in the persistence model combined with a high dominance of the GDP and especially the military expenditures of the receiving country. This is not to say that we regard for example the positive effect of transitivity or alliances in the persistence model as irrelevant for repeated trading since the special nature of arms trading clearly demands trust for the formation and the persistence of transfers but the effects nevertheless show that the two processes are guided by different mechanisms that attach different priorities to security-related and economic variables. Time-varying smooth random effects ---------------------------------- ### Functional component analysis We now pay attention to the actor-specific heterogeneity. In Figure [smooth1], the country-specific effects for the sender, as well as the receiver countries are visualized for the formation model on the left and the persistence model on the right. Note that in these plot we have truncated the curves for the years where countries are not existent. [smooth1] At a first sight, interpretation of these plots looks clumsy. We therefore retrieve information by employing a functional principal component analysis to the multivariate time series of random effects seen in Figure [smooth1] (see also and the Appendix [detailsmooth]). The results are shown in Figure [prin1] for the formation model and in Figure [prin2] for the persistence model. On the left hand side the scores of the first two principal components are plotted, where the latter are visualized on the right hand side. The share of variance explained by the respective component is provided in the brackets. The basic idea of the approach is to show the effect of the principal components as perturbations from the mean random effects curves. By adding (the “+” line) or subtracting (the “-” line) a multiple of the principal component curve we get the visualized perturbation from the mean. The first principal component is close to be constant and represents the share of variance induced by different overall levels of the random effect curves. The dynamic of the random effects is captured by the second principal component, delivering a tendency for an upward movement if positive and downward if negative. Hence, looking on the horizontal axes, we see countries that build up their arm trade links over the years as exporters (importers) on the right hand side while countries that are reluctant to building up export (import) links are plotted on the left hand side. Looking on the vertical axes, we see countries that decrease their role as exporter (importer) over the time on the bottom, and vice versa countries that increase the number of export (import) links over time on the top. All these effects are conditional on the remaining covariate effects discussed before. Hence, these random effects capture the remaining heterogeneity not included in the remaining model. [prin1] [prin2] ### Results of the functional component analysis Because of the great amount of information condensed in Figures [prin1] and [prin2] we restrict our interpretation to a few global patterns and selected countries that take either very special positions in the arms trade network (high or low values for component 1) or exhibit variation over time (high or low values for component 2). Overall regarding the different levels of the random effects, it can already be seen in Figure [smooth1], that the heterogeneity is much more pronounced in the formation model in comparison to the persistence model. Furthermore, in the formation model, the countries differ more strongly in their ability to export in comparison to their ability to import while this contrast is not present in the persistence model. A global pattern regarding the dynamics of the sender effect becomes visible since the top left in Figure [prin1] looks like a lying mushroom. That is, countries that started on a low level (i.e. negative component 1) show, with the exception of Japan (JPN) and Turkey (TUR), not very much upward or downward variability (i.e. low level for component 2). In contrast, countries that have a random effect above zero move more strongly up or down with time. This means that the export dynamics are mainly driven by countries with relatively high sender effects. Figures [prin1] and [prin2] show very well that fundamental changes of the system are driven by the end of the cold war. This can be seen exemplary regarding the position of the Soviet Union (SUN) and Czechoslovakia (CZE) in the top left in Figures [prin1] and [prin2] (both with a high level for component 1 and a low level for component 2). This mirrors that these countries left the system shortly after the collapse of the eastern bloc. However, this turning point affected not only exporters but also importers and consequently the representation of the receiver effects of the formation model at the bottom left of Figure [prin1] is populated with (former) socialist countries such as Cuba (CUB), Ukraine (UKR), North Korea (PRK), Yugoslavia (YUG) and Moldova (MDA). Additionally, we find a prominent position for Romania (ROM), being a country that has a high level (high value for component 1) but decreased its’ tendency to be a receiver in persistent trade relations (low value for component 2) in Figure [prin2]. However, while some of the countries of the eastern block ceased to exist or strongly reduced their exports or imports we also find a contrary pattern. Countries like Ukraine (UKR) and Bulgaria (BGR) have managed to increase their sender effect in the formation as well as in the persistence model with time (high value for component 1 and component 2 in the top left of Figures [prin1] and [prin2]). This indicates that some left overs of the collapsed Soviet Union defence industries sold off their stocks and rushed into the global market of military products. Besides the massive shift initiated by the end of the cold war, we see that some dominant exporting countries, especially Great Britain (GBR), France (FRA) and Egypt (EGY), lost importance over time. This countries can be found in the fourth quadrant of the top left panels in Figures [prin1] and [prin2], meaning their high sender effects decreased strongly with time. This might seem surprising since France and Great Britain are still among the countries with the highest exported volumes. However, France and Great Britain have left their dominance over former colonies leading to a loss of control over many potential importers. The general pattern also carries over to their receiver effects. Looking at the scores of Great Britain (GBR) and France (FRA) at the bottom left of Figure [prin2] we see a strong decrease of their receiver effects in the persistence model. Apart from global patterns, some countries exhibit exceptional scores that can be traced back to country-specific circumstances. We find that Japan (JPN) stands out among the countries with the lowest proclivity to import (see the low scores for components 1 and 2 at the bottom left of Figure [prin1]). Even more pronounced is the astonishing low tendency to export, mirrored by Japan’s sender effect in the persistence models (Figure [prin2], top left) and the strongly declining sender effect in the formation model (Figure [prin1], top left). This stands in contrast to the fact that Japan is among the wealthiest countries with a highly developed export industry and is clearly due to the highly restrictive arms export principles introduced in 1967, and tightened in 1976. This ban on exports was only lifted in 2014 (; ). Another, very notable case is Israel (ISR), being somehow the opposite pole in comparison to Japan (JPN). The sender effects on the top left of Figures [prin1] and [prin2] show that Israel (ISR) has an outstanding tendency to establish and maintain arms exports. On the other hand, Israel (ISR) takes a very polar position in the bottom left of Figure [prin1] as a consequence of a strongly decreased (i.e. low level for component 2) receiver effect in the formation model. These results reflect the country’s path of developing highly internationally competitive weapons systems and its’ rise to be one of the most important exporters. This stands in contrast to countries like Mexico (MEX), being the country with the least tendency to form new trade exports (top left in Figure [prin1]). It appears that this country is not able to be a relevant player in the market despite being among the worlds’ largest economies. We consider these special paths as induced by cumulative advantages and learning over time in the one case (Israel), whereas in the case of Mexico (MEX) we observe the stickiness and path inertia of a country having not been able to make its defence products sold externally. There remain many other interesting cases. For example the rise of South Africa (ZAF) as an exporter in the formation model (top left in Figure [prin1]), mirroring the history of the country, being initially dependent on imports and now among the major exporters of MCW. We also find that Ireland (IRL) strongly increased its tendency to be a persistent importer after its entry to the European Union (bottom left in Figure [prin2]) while Germany (DEU) and Canada (CAN) strongly increased their roles as persistent exporters (top left in Figure [prin2]). Model evaluation ---------------- [figcombined] The evaluation of the out-of-sample predictive power is based on the following steps. We first fit the formation model as well as the persistence model, based on the information in t − 1, to the data in t and use the estimated coefficients for the prediction of new formation or persistence of existing ties in t + 1. As the predictions are probabilistic by their nature, we weight the recall (true positive rate) against the false positive rate for varying threshold levels, yielding the ROC curve and the AUC for each year of prediction. Because arms transfers can be regarded as rare events we also compute the PR curve and the corresponding AUC. The results are plotted in Figure [figcombined] with the AUC values that correspond to the PR curves on the left and the one corresponding to the ROC curves on the right. The first row gives the evaluation of the formation model and the second row shows the persistence model. While the AUC values in the formation model are very high when evaluated at the ROC curves they are much lower with the PR curves. This is a consequence of being right quite frequently if a zero is predicted, while its is hard to forecast the actual transfers in the next period in case of the formation model. Interestingly, the opposite holds for the persistence model. In the combined version at the bottom of Figure [figcombined] the AUC values derived from the PR curve show that the model does quite well. Additionally, we evaluate how well global network structures like the mean outdegree, the share of reciprocity and observed transitivity can be mirrored by the predictions using a simulation-based approach (see ). To do so, we fit the models for the transition between t − 1 and t and simulate from the formation model an the persistence model 1, 000 times based on the information in t. Then, based on equation ([construct]), the predicted network for t + 1 is constructed. From this, we evaluate global network characteristics and compare them to the actual characteristics from the true MCW trade network in t + 1. The corresponding Figure [figsim1] is given in the Appendix [ospsim]. The results are reassuring and the simulated networks mirror the real network properties in an acceptable way. Clearly the proposed model is not the only suitable network model. Alternatively, it is possible to analyse the data with a STERGM without random effects and with various variants of the ERGM or the TERGM with and without random effect. We discuss this extensively in the Supplementary Material and show that the out-of-sample predictive power of our model is superior to all other fitted candidate models. Conclusion ========== In this paper we employ a dynamic separable network model as introduced by and add techniques proposed by and. This enables us to study the process of formation and persistence separately as well as the inclusion of time-varying coefficients and smooth time-varying random effects that are further analysed by methods from functional data analysis as described in. Applied to the discretized MCW networks from 1950 to 2016 we find that the mechanisms leading to formation and persistence differ fundamentally. Most importantly, the formation is driven by network effects and security related variables, while the persistence of transfers is dominated the military expenditures of the receiving country. A careful analysis of the random effects exhibits a high variation among the countries as well as along the time dimension. By using functional principal component analysis we decompose the functional time series of smooth random effects in order find countries that have increased or decreased their relative importance in the network. The evaluation of the fit confirms that the chosen model is able to give good out-of-sample predictions. Appendix ======== Descriptives ------------ [Fig-network-1] In Figure [Fig-network-1], the binary network is shown for the years 2015 and 2016. Table [coverage] gives the categories of arms that are included in the analysis. All types with explanations are taken from. The 171 countries that are included in our analysis can be found in Table [countinc], together with the three-digit country codes that are used to abbreviate countries in the paper. In addition to that, the time periods, for which we coded the countries as existent are included. Note that the SIPRI data set contains more than 171 arm trading entities but we excluded non-states and countries with no (reliable) covariates available. In the covariate GDP some missings are present in the data. No time series of covariates for the selected countries is completely missing (those countries are excluded from the analysis) and the major share of them is complete but there are series with some missing values. This is sometimes the case in the year 1990 and/or 1991 where the former socialist countries splitted up or had some transition time. In other cases values at the beginning or at the end of the series are missing. We have decided on three general rules to fill the gaps: First, if a value for a certain country is missing in t but there are values available in t − 1 and t + 1, the mean of those is used. If the values are missing at the end of the observational period, the last value observed is taken. In case of missing values in the beginning, the first value observed is taken. The series on military expenditures are imputed similarly, using linear interpolation by employing the `R` package `imputeTS` by. The number of countries included each year in the network is provided in the upper left panel of Figure [sumstat]. It can be seen that the network is growing almost each year until 1992, with two big leaps that show the effects of the decolonization, beginning in 1960 and the end of the Soviet Union after 1991. Typical descriptive statistics for the analysis of networks are Density, Reciprocity and Transitivity, all shown in Figure [sumstat]. The Density is defined as the number of edges divided by the number of possible edges. Reciprocity is defined as the share of trade flows being reciprocal. Transitivity is defined as the ratio of triangles and connected triples in the graph. [sumstat] [t!] l p11cm Type & Explanation Aircraft & All fixed-wing aircraft and helicopters, including unmanned aircraft with a minimum loaded weight of 20 kg. Exceptions are microlight aircraft, powered and unpowered gliders and target drones. Air-defence systems & (a) All land-based surface-to-air missile systems, and (b) all anti-aircraft guns with a calibre of more than 40 mm or with multiple barrels with a combined caliber of at least 70 mm. This includes self-propelled systems on armoured or unarmoured chassis. Anti-submarine warfare weapons & Rocket launchers, multiple rocket launchers and mortars for use against submarines, with a calibre equal to or above 100 mm. Armoured vehicles & All vehicles with integral armour protection, including all types of tank, tank destroyer, armoured car, armoured personnel carrier, armoured support vehicle and infantry fighting vehicle. Vehicles with very light armour protection (such as trucks with an integral but lightly armoured cabin) are excluded. Artillery & Naval, fixed, self-propelled and towed guns, howitzers, multiple rocket launchers and mortars, with a calibre equal to or above 100 mm. Engines & (a) Engines for military aircraft, for example, combat-capable aircraft, larger military transport and support aircraft, including large helicopters; (b) Engines for combat ships -,fast attack craft, corvettes, frigates, destroyers, cruisers, aircraft carriers and submarines; (c) Engines for most armoured vehicles - generally engines of more than 200 horsepower output.\* Missiles & (a) All powered, guided missiles and torpedoes, and (b) all unpowered but guided bombs and shells. This includes man-portable air defence systems and portable guided anti-tank missiles. Unguided rockets, free-fall aerial munitions, anti-submarine rockets and target drones are excluded. Sensors & (a) All land-, aircraft- and ship-based active (radar) and passive (e.g. electro-optical) surveillance systems with a range of at least 25 kilometres, with the exception of navigation and weather radars, (b) all fire-control radars, with the exception of range-only radars, and (c) anti-submarine warfare and anti-ship sonar systems for ships and helicopters.\* Ships & (a) All ships with a standard tonnage of 100 tonnes or more, and (b) all ships armed with artillery of 100-mm calibre or more, torpedoes or guided missiles, and (c) all ships below 100 tonnes where the maximum speed (in kmh) multiplied with the full tonnage equals 3500 or more. Exceptions are most survey ships, tugs and some transport ships Other & (a) All turrets for armoured vehicles fitted with a gun of at least 12.7 mm calibre or with guided anti-tank missiles, (b) all turrets for ships fitted with a gun of at least 57-mm calibre, and (c) all turrets for ships fitted with multiple guns with a combined calibre of at least 57 mm, and (d) air refueling systems as used on tanker aircraft.\* [coverage] [ht] \| Country \| Code \| Included \| Country \| Code \| Included \| Country \| Code \| Included \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Afghanistan \| AFG \| 1950 - 2016 \| German Dem. Rep. \| GDR \| 1950 - 1991 \| Pakistan \| PAK \| 1950 - 2016 \| \| Albania \| ALB \| 1950 - 2016 \| Germany \| DEU \| 1950 - 2016 \| Panama \| PAN \| 1950 - 2016 \| \| Algeria \| DZA \| 1962 - 2016 \| Ghana \| GHA \| 1957 - 2016 \| Papua New Guin. \| PNG \| 1975 - 2016 \| \| Angola \| AGO \| 1975 - 2016 \| Greece \| GRC \| 1950 - 2016 \| Paraguay \| PRY \| 1950 - 2016 \| \| Argentina \| ARG \| 1950 - 2016 \| Guatemala \| GTM \| 1950 - 2016 \| Peru \| PER \| 1950 - 2016 \| \| Armenia \| ARM \| 1991 - 2016 \| Guinea \| GIN \| 1958 - 2016 \| Philippines \| PHL \| 1950 - 2016 \| \| Australia \| AUS \| 1950 - 2016 \| Guinea-Bissau \| GNB \| 1973 - 2016 \| Poland \| POL \| 1950 - 2016 \| \| Austria \| AUT \| 1950 - 2016 \| Guyana \| GUY \| 1966 - 2016 \| Portugal \| PRT \| 1950 - 2016 \| \| Azerbaijan \| AZE \| 1991 - 2016 \| Haiti \| HTI \| 1950 - 2016 \| Qatar \| QAT \| 1971 - 2016 \| \| Bahrain \| BHR \| 1971 - 2016 \| Honduras \| HND \| 1950 - 2016 \| Romania \| ROM \| 1950 - 2016 \| \| Bangladesh \| BGD \| 1971 - 2016 \| Hungary \| HUN \| 1950 - 2016 \| Russia \| RUS \| 1992 - 2016 \| \| Belarus \| BLR \| 1991 - 2016 \| India \| IND \| 1950 - 2016 \| Rwanda \| RWA \| 1962 - 2016 \| \| Belgium \| BEL \| 1950 - 2016 \| Indonesia \| IDN \| 1950 - 2016 \| Saudi Arabia \| SAU \| 1950 - 2016 \| \| Benin \| BEN \| 1961 - 2016 \| Iran \| IRN \| 1950 - 2016 \| Senegal \| SEN \| 1960 - 2016 \| \| Bhutan \| BTN \| 1950 - 2016 \| Iraq \| IRQ \| 1950 - 2016 \| Serbia \| SRB \| 1992 - 2016 \| \| Bolivia \| BOL \| 1950 - 2016 \| Ireland \| IRL \| 1950 - 2016 \| Sierra Leone \| SLE \| 1961 - 2016 \| \| Bosnia Herzegov. \| BIH \| 1992 - 2016 \| Israel \| ISR \| 1950 - 2016 \| Singapore \| SGP \| 1965 - 2016 \| \| Botswana \| BWA \| 1966 - 2016 \| Italy \| ITA \| 1950 - 2016 \| Slovakia \| SVK \| 1993 - 2016 \| \| Brazil \| BRA \| 1950 - 2016 \| Jamaica \| JAM \| 1962 - 2016 \| Slovenia \| SVN \| 1991 - 2016 \| \| Bulgaria \| BGR \| 1950 - 2016 \| Japan \| JPN \| 1950 - 2016 \| Solomon Islands \| SLB \| 1978 - 2016 \| \| Burkina Faso \| BFA \| 1960 - 2016 \| Jordan \| JOR \| 1950 - 2016 \| Somalia \| SOM \| 1960 - 2016 \| \| Burundi \| BDI \| 1962 - 2016 \| Kazakhstan \| KAZ \| 1991 - 2016 \| South Africa \| ZAF \| 1950 - 2016 \| \| Cambodia \| KHM \| 1953 - 2016 \| Kenya \| KEN \| 1963 - 2016 \| Soviet Union \| SUN \| 1950 - 1991 \| \| Cameroon \| CMR \| 1960 - 2016 \| North Korea \| PRK \| 1950 - 2016 \| Spain \| ESP \| 1950 - 2016 \| \| Canada \| CAN \| 1950 - 2016 \| South Korea \| KOR \| 1950 - 2016 \| Sri Lanka \| LKA \| 1950 - 2016 \| \| Cape Verde \| CPV \| 1975 - 2016 \| Kuwait \| KWT \| 1961 - 2016 \| Sudan \| SDN \| 1956 - 2016 \| \| Central Afr. Rep. \| CAF \| 1960 - 2016 \| Kyrgyzstan \| KGZ \| 1991 - 2016 \| Suriname \| SUR \| 1975 - 2016 \| \| Chad \| TCD \| 1960 - 2016 \| Laos \| LAO \| 1950 - 2016 \| Swaziland \| SWZ \| 1968 - 2016 \| \| Chile \| CHL \| 1950 - 2016 \| Latvia \| LVA \| 1991 - 2016 \| Sweden \| SWE \| 1950 - 2016 \| \| China \| CHN \| 1950 - 2016 \| Lebanon \| LBN \| 1950 - 2016 \| Switzerland \| CHE \| 1950 - 2016 \| \| Colombia \| COL \| 1950 - 2016 \| Lesotho \| LSO \| 1966 - 2016 \| Syria \| SYR \| 1950 - 2016 \| \| Comoros \| COM \| 1975 - 2016 \| Liberia \| LBR \| 1950 - 2016 \| Taiwan \| TWN \| 1950 - 2016 \| \| DR Congo \| ZAR \| 1960 - 2016 \| Libya \| LBY \| 1951 - 2016 \| Tajikistan \| TJK \| 1991 - 2016 \| \| Congo \| COG \| 1960 - 2016 \| Lithuania \| LTU \| 1990 - 2016 \| Tanzania \| TZA \| 1961 - 2016 \| \| Costa Rica \| CRI \| 1950 - 2016 \| Luxembourg \| LUX \| 1950 - 2016 \| Thailand \| THA \| 1950 - 2016 \| \| Cote dIvoire \| CIV \| 1960 - 2016 \| Macedonia \| MKD \| 1991 - 2016 \| Timor-Leste \| TMP \| 2002 - 2016 \| \| Croatia \| HRV \| 1991 - 2016 \| Madagascar \| MDG \| 1960 - 2016 \| Togo \| TGO \| 1960 - 2016 \| \| Cuba \| CUB \| 1950 - 2016 \| Malawi \| MWI \| 1964 - 2016 \| Trinidad Tobago \| TTO \| 1962 - 2016 \| \| Cyprus \| CYP \| 1960 - 2016 \| Malaysia \| MYS \| 1957 - 2016 \| Tunisia \| TUN \| 1956 - 2016 \| \| Czech Republic \| CZR \| 1993 - 2016 \| Mali \| MLI \| 1960 - 2016 \| Turkey \| TUR \| 1950 - 2016 \| \| Czechoslovakia \| CZE \| 1950 - 1991 \| Mauritania \| MRT \| 1960 - 2016 \| Turkmenistan \| TKM \| 1991 - 2016 \| \| Denmark \| DNK \| 1950 - 2016 \| Mauritius \| MUS \| 1968 - 2016 \| Uganda \| UGA \| 1962 - 2016 \| \| Djibouti \| DJI \| 1977 - 2016 \| Mexico \| MEX \| 1950 - 2016 \| Ukraine \| UKR \| 1991 - 2016 \| \| Dominican Rep. \| DOM \| 1950 - 2016 \| Moldova \| MDA \| 1991 - 2016 \| Un. Arab Emirates \| ARE \| 1971 - 2016 \| \| Ecuador \| ECU \| 1950 - 2016 \| Mongolia \| MNG \| 1950 - 2016 \| United Kingdom \| GBR \| 1950 - 2016 \| \| Egypt \| EGY \| 1950 - 2016 \| Morocco \| MAR \| 1956 - 2016 \| United States \| USA \| 1950 - 2016 \| \| El Salvador \| SLV \| 1950 - 2016 \| Mozambique \| MOZ \| 1975 - 2016 \| Uruguay \| URY \| 1950 - 2016 \| \| Equatorial Guin. \| GNQ \| 1968 - 2016 \| Myanmar \| MMR \| 1950 - 2016 \| Uzbekistan \| UZB \| 1991 - 2016 \| \| Eritrea \| ERI \| 1993 - 2016 \| Namibia \| NAM \| 1990 - 2016 \| Venezuela \| VEN \| 1950 - 2016 \| \| Estonia \| EST \| 1991 - 2016 \| Nepal \| NPL \| 1950 - 2016 \| Vietnam \| VNM \| 1976 - 2016 \| \| Ethiopia \| ETH \| 1950 - 2016 \| Netherlands \| NLD \| 1950 - 2016 \| South Vietnam \| SVM \| 1950 - 1975 \| \| Fiji \| FJI \| 1970 - 2016 \| New Zealand \| NZL \| 1950 - 2016 \| Yemen \| YEM \| 1991 - 2016 \| \| Finland \| FIN \| 1950 - 2016 \| Nicaragua \| NIC \| 1950 - 2016 \| North Yemen \| NYE \| 1950 - 1991 \| \| France \| FRA \| 1950 - 2016 \| Niger \| NER \| 1960 - 2016 \| South Yemen \| SYE \| 1950 - 1991 \| \| Gabon \| GAB \| 1960 - 2016 \| Nigeria \| NGA \| 1960 - 2016 \| Yugoslavia \| YUG \| 1950 - 1992 \| \| Gambia \| GMB \| 1965 - 2016 \| Norway \| NOR \| 1950 - 2016 \| Zambia \| ZMB \| 1964 - 2016 \| \| Georgia \| GEO \| 1991 - 2016 \| Oman \| OMN \| 1950 - 2016 \| Zimbabwe \| ZWE \| 1950 - 2016 \| [countinc] Details on the estimation procedure ----------------------------------- The recent implementation of Generalised Additive Models (GAM) in the `R` package `mgcv` allows for smooth varying coefficients as proposed by. These models can be represented in GAMs by multiplying the smooths by a covariate (in the given application the smooths of time are multiplied by the covariates. See for more details. The functions for the smooths are based on P-Splines as proposed by, giving low rank smoothers using a B-spline basis using a simple difference penalty applied to the parameters. For the smooth time-varying coefficients on the fixed effects a maximum number of 65 knots is used, combined with a second-order P-spline basis (quadratic splines) and a first-order difference penalty on the coefficients. The non-linear random smooths are estimated similar to those proposed by. As a basic idea, one views the individual smooths as splines with random coefficients, i.e. each country has a random effect, that is in fact a function of time that is approximated by regression splines. The parameters of the splines are assumed to be normally distributed with mean zero and the same variance for all curves, which translates into having the same smoothness parameter for all curves. This concept is implemented efficiently in the GAM structure of the `mgcv` package by using the nesting of the smooth within the respective actor. In order to avoid overfitting and keeping computation tractable, a first-order penalty with nine knots is employed. The smoothness selection is done for all smooths by the restricted maximum likelihood criterion (REML). As the data set is rather big with more than 1.3 million observations in the formation model, the fitting procedure of the model is computationally expensive and was virtually impossible with standard implementations in `R` before the introduction of the `bam()` function in the `mgcv` package in 2016 that needs less memory and is much faster than other comparable packages. The estimation routine employs techniques as proposed in. Those methods use discretization of covariate values and iterative updating schemes that require only subblocks of the model matrix to the computed at once which allows for the application of parallelization tools. For all computations we also used the statistical programming language `R` (). Important packages used for visualization of networks and computation of network statistics are the `statnet` suite of network analysis packages () as well as the package `igraph` (). For the Tables the `stargazer` package from was employed. For the model evaluation and visualization we used the `PRROC` package of. Details on the PCA of the time-varying smooth random effects ------------------------------------------------------------ For the analysis of the smooth random effects we are following the discretization approach of. As noted in Section [heterogen] we assume the random effects ϕi(t) (ϕi+(t) and ϕi−(t) in the formation and the persistence model, respectively) to be realizations of a stochastic process Φ = {ϕ(t), t ∈ τ}, for i = 1, ..., N individual countries and τ = [1951, 2016]. In order to summarize the information provided by these functions we are searching for a weight function β(t) that gives us the principal component scores ϕi = ∫τβ(t)ϕi(t)dt. In order to do so, the weight function ξ1(t) among all possible functions β(t) must be found that maximizes N− 1∑i = 1N(∫τβ(t)ϕi(t)dt)2 subject to the constraint ∫τξ12(t)dt = 1. From our model we get N individual estimated functions ϕ̂i(t) for all observations (countries) and can discretize the functions ϕ̂i(t) on a grid. We use T = 100 equidistant points {t1, ..., t100} on the interval τ of length ∣τ∣ = T. This gives a discretized (N × T) time series matrix $\hat{\Phi}$ with N country specific observations in the rows and the estimated functions, evaluated at the discrete time points, in the columns: $$\hat{\Phi}=\begin{pmatrix} \hat{\phi}\_1(t\_1) & \cdots & \hat{\phi}\_1(t\_{100})\\ \vdots & \ddots & \vdots \\ \hat{\phi}\_N(t\_1) & \cdots& \hat{\phi}\_N(t\_{100}) \end{pmatrix}$$ Therefore, in fact we are searching for a solution for the discrete approximation of ∫τβ(t)ϕ̂i(t)dt ≈ (T/T)∑j = 1Tβ(tj)ϕ̂i(tj) = ∑j = 1Tβ̃(tj)ϕ̂i(tj) such that the solution ξ̃1 that maximizes the mean square satisfies ∣∣ξ̃1∣∣2 = 1. This is now a standard problem, with the solution ξ̃1 being found by the eigenvector that corresponds to the largest eigenvalue of the covariance matrix of $\hat{\Phi}$. Out-of-sample-predictions for simulated networks ------------------------------------------------ As a standard principle in network analysis, a model should be able the reflect global network characteristics. We evaluate six of them for our out-of-sample forecasts. The first three characteristics are related to the number of actors that are actively engaged in the arms trade. The statistic Size is defined as the count of predicted edges in each year. This measure helps to evaluate the ability of the model to predict amount of realized arms trade in each year. As it is also of interest to measure how dense the predicted arms trade network is, we include Density, relating the size of the network to the number of edges that could have potentially realized. We define the Order of the network as number of actors that are engaged in either exporting or importing arms. The results will provide an impression whether the model has the ability not only to classify the right amount of edges (as in Size), but also their nesting within the countries. As we have emphasized the importance of local network statistics we evaluate whether the local network statistics are able to generate the corresponding global statistics. Therefore, we include the Mean Indegree (being the same as Mean Outdegree), as well as the share of Reciprocity. In order to evaluate the accuracy of our predictions with respect to triangular relationships we furthermore include the measure Transitivity, that divides the number of triangles by the number of connected triples in the graph. In this statistic, the direction of the edges is ignored. The analysis of this measure gives an impression how well the two chosen transitivity measures capture the overall clustering in the network. The results are presented in Figure [figsim1]. In each of the six panels we see the respective network statistics plotted against time. The solid red line gives the network statistics, evaluated at the real MCW network. The boxplots show the network statistics, evaluated for each year for the 1.000 simulated networks. Comparison of realized and simulated network topologies. The boxplots give statistics from the simulated networks. The solid line gives the statistics for the real networks. Number of edges (Size), number of active countries (Order), number of realized transfers relative to possible transfers (Density), average indegree (Indegree), share of reciprocated transfers and ratio of undirected triangles relative to connected triples (Transitivity). [figsim1] Supplementary Material ====================== Different threshold values -------------------------- ### Distribution of TIVs In Figure [figdenstiy] we present a kernel density estimate (KDE) of the pooled TIVs for the whole time period. The distribution of the TIVs is highly skewed and has a long tail. Therefore, we give a logarithmic representation. In order to give an impression of the left tail, Table [quantiles] provides the lower quantiles of the distribution. From this it can be seen that roughly 20% of all observations are below a threshold of 3. [figdenstiy] [!htbp] ccccc 0% & 4.75% & 9.5% & 14.25% & 19% 0.020 & 0.700 & 1.332 & 2.200 & 3 [quantiles] In order to demonstrate the effect of different binarization thresholds on the estimated coefficients we pursue the following strategy. As a baseline we use the “original effects” from the paper with a threshold of zero and plot them in solid black together with two standard error confidence bounds in dark grey. Additionally, we include the estimated coefficients with thresholds incrementing from zero to three in steps of 0.5 as dashed black lines. (i) By comparison of the solid line with the dashed lines it can be seen how strong the point-estimates vary with different thresholds. (ii) If the dashed lines are within the confidence intervals in dark grey they can be said to be statistically indistinguishable from the original estimates. (iii) Furthermore, we show how the confidence bounds of the new estimates exceed the ones of the original estimation, displayed in light grey. These areas represent the highest upper limit and the lowest lower limit that exceeds the original confidence bounds. Hence, if only the dark grey confidence bound is visible, then the effects of all estimates are the same, if the light grey confidence bound is above and/or below the dark grey, this means that the bounds of the estimates with higher thresholds are wider. ### Fixed effects with different thresholds [figcomp1] The degree-related statistics are shown in the top four panels of Figures [figcomp1]. We find no significant changes of the results as the dashed lines stay in almost all cases within the original confidence bounds. An exception is the senders’ outdegree in the formation model but even with the highest threshold this effect does not become significant. On the contrary, partly the results get even more clear. For example the outdegree effect for the receiver in the formation and dissolution model (second row) becomes even more negative in tendency with increasing thresholds. For the reciprocity effect in the third row we find that the coefficients stay almost the same for all different thresholds of binarization. The same applies for the transitivity effect in the formation model (left panel in the fourth row). Here the upper confidence bound even indicates a potentially higher effect. For the transitivity in the persistence model (right panel in the fourth row) we find that the effect becomes insignificant in the beginning if we set roughly 20% of the lowest observations to zero. Otherwise the effect stays significant and very close to the point estimates of the original estimation. For the Shared Suppliers Effect (the two panels at the bottom) the results are very similar to the transitivity effects, i.e. we find an potentially stronger effect in the formation model (left panel at the bottom) and an insignificant effect for the first years in the persistence model (right panel) with the highest binarization threshold. Given that the network statistics are constructed from the network and therefore directly and potentially strongly affected by different thresholds, the robustness of the effects is reassuring. Only if we replace almost one fifth of the existing edges by zeros the effects of the hypderdyadic statistics in the persistence model start to become partly insignificant. However, the affected statistics are in line with our theoretical expectation that the network effects matter mostly for the formation. From this result it might not come as a surprise that the effects of the non-network related covariates are even more robust because their construction is not affected by the thinning of the network. We can confirm with Figure [figcomp2] that shows virtually no noteworthy changes of the effects. [figcomp2] Different time windows ---------------------- In the paper we assume that the STERGM process applies to two consecutive years. As a robustness check, we define the periods t and t + 1 such that they contain multiple years. If we take the years 2013, 2014, 2015 and 2016 as an example for time windows of length two, we set Yijt − 1, t = 1 if country i exports arms to j in 2013 or 2014 and Yijt, t − 1 = 0 if country i has not exported to j neither in 2015 nor in 2016. We also extend this concept such that we combine three years into one period. For the non-network related covariates we are using the time averages for the respective time windows for continuous variables (e.g. if a period contains two years, the average of the GDP in these two years is taken) and we set binary variables to one if the respective feature was present in all year (e.g. the indicator for a formal alliance is one if the alliance was present in all two or three years). The corresponding estimates can be seen in Figures [figwindow1] and [figwindow2]. (i) The Figures are constructed such that the baseline is given by the “original effects” from the paper. These effects are plotted in solid black together with two standard error confidence bounds in dark grey. (ii) The coefficients with two or three years within one time period are given in dotted (two years) and dotted-dashed (three years). (iii) The area where the original confidence bounds are exceeded is given in light grey. In Figure [figwindow1] it can be seen that the dotted and dashed-dotted line rarely is outside of the dark grey confidence bound in the panels of rows one to three. One really noteworthy exception applies to the transitivity effect in the formation model. Here the estimates with periods containing three years behave somewhat more time-constant than the original estimates. This is, however, a natural result because with the broader time windows major changes in the data as the collapse of the Soviet Union become more smooth. All in all, the panels clearly show that the coefficients are very robust and do not change fundamentally. This impression is confirmed by Figure [figwindow2] where again no notable exceptions can be found influence the interpretation in terms of variation with time or significance. This is again in line with our theoretical expectations since the coefficient for the military expenditures of the receiver is almost not affected by different time windows. [figwindow1] [figwindow2] Model without random effects ---------------------------- In the main article we mentioned that the inclusion of the random effects leads to a vanishing global effect of the senders outdegree. I.e once controlled for the sender-specific random effect the coefficient on the outdegree statistic is insignificant. Here, we show that once we exclude random effects from the models, all results are very robust with the exception of the coefficient on the outdegree. This can be seen in Figures [fignoraneff1] and [fignoraneff2] with coefficients that are very comparable to the ones from the main paper. The main exception is given by the senders outdegree (top panels in Figure [fignoraneff1]). Here the coefficients are now in both models positive and significant. Especially in the formation effect the coefficient is very high. This shows that there is indeed a global effect of the senders outdegree that, however, vanishes if one controls for country-specific heterogeneity. [fignoraneff1] [fignoraneff2] Comparison of different methods ------------------------------- ### Theoretical discussion Besides the STERGM, there are many other models suitable for dynamic networks. In the following, we provide an overview of alternative approaches with a discussion of their suitability for the given dataset. Latent Space Models: A potential alternative is given by the family of latent space models (e.g. ; ). However, it is hard to model arms trade appropriately within this model class. The need of estimating time-varying coefficients results in yearly separate estimations. The resulting yearly latent space representations are not very stable and do not work well with isolates, i.e. countries without transfers in a given year. If one goes for a panel approach with this model class (see the package `amen` by ) one must accept that the coefficients stay the same for the whole time-period and, even more problematic, that the positions within one latent space are sufficient to capture all network dependencies for the time-period 1950-2016, which is clearly a heroic assumption. On top of that, the latent space approach does not allow for the evaluation of complex network statistics we are interested in. Stochastic-Actor Oriented Models: Stochastic-Actor Oriented Model (SAOM, see e.g. ) are built for modelling dynamic networks and have the virtue of allowing for the estimation of dyadic and hyper-dyadic network effects. However, the model is tailored for social networks and some of its’ assumptions are very problematic for modelling the arms trade network. Most importantly, it assumes actor-homogeneity, an assumption that is clearly violated for the dataset. Additionally, the SAOM fundamentally builds on the idea that the networks observed represent snapshots of a continuous underlying process of edge formation and persistence, i.e. we would need to assume that between t and t + 1 multiple changes could have been realized in the network. This is not an acceptable assumption because if a transfer between i and j was recorded in t and t + 1, it is not meaningful to assume that there exists an in-between state where the transfer has the change to disappear and re-emerged multiple times. Otherwise, if no transfer is recorded in t and t + 1 we have no reason to believe that there was trade in between. In the basic description of the SAOM () the authors write: “A foundational assumption of the models discussed in this paper is that the network ties are not brief events, but can be regarded as states with a tendency to endure over time. Many relations commonly studied in network analysis naturally satisfy this requirement of gradual change, such as friendship, trust, and cooperation.” Apparently it is hard to argue that recording whether there was a transfer between two countries in a given year can be viewed as an enduring state. Exponential Random Graph Models: Our model is in fact motivated by recent advances within the exponential random graph model (ERGM) family, i.e. the TERGM (,, ) and the STERGM (). However, our model differs from the standard cross-sectional ERGM (but also from the conventional TERGM and STERGM) because it does not allow for simultaneous network dependencies. In general, a static cross-sectional ERGM seems to be an implausible choice for the modelling of a dynamic network with strong actor heterogeneity. A dynamic formation and the TERGM, however, can be constructed in a very similar manner as the STERGM. Those models are natural candidate model for comparison. ### Candidate models In the following we present several alternative candidate models by increasing level of complexity. Autoregressive ERGM (Model 1): The most simplistic stochastic model is an autoregressive model that assumes time-dependence of all individual dyads such that $$\label{logit1} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=\theta\_0+\theta\_1y\_{ij}^{t-1,t}.$$ This temporal dependence structure can be interpreted as a cross sectional ERGM with the lagged response as a dyadic exogenous covariate or as a TERGM with a dyadic stability term (see e.g. ). It is motivated by the idea that the probability of a transfer in t might change if there was a transfer in t − 1. TERGM with covariates (Model 2): We can change model ([logit1]) by including all the network effects and covariates as specified in Section 3.2 of the main paper: $$\label{logit2} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})$$ In this formulation, we include the autoregressive component only indirectly, the lagged network statistics give some information about the network embedding of a transfer but not whether there was a preceding transfer. TERGM with covariates and random effects (Model 3): With the inclusion of smooth time-varying random effects for the sender and the receiver we have $$\begin{split} \label{logit3} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=&\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})\\&+\phi\_{i,sender}(t)+\phi\_{j,receiver}(t). \end{split}$$ The main difference to the STERGM is now that we do not model the processes of formation and persistence separately but within one model and that we do not include the information on the lagged response here, that is implicitly included in the STERGM mechanics. TERGM with covariates and dyadic stability (Model 4): The inclusion of the lagged response to the TERGM with covariates gives $$\label{logit4} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})+\theta\_1y\_{ij}^{t-1,t}.$$ TERGM with network effects, dyadic stability and random effects (Model 5): And as a last step we allow for sender- and receiver-specific random effects in the TERGM with network statistics and lagged response: $$\label{logit5} \begin{split} \log\bigg{\{} \frac{P(Y^{t,t-1}\_{ij}=1\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}= x^{t-1,t};\theta)}{P(Y^{t,t-1}\_{ij}=0\|Y^{t-1,t}=y^{t-1,t},X^{t-1,t}=x^{t-1,t};\theta)}\bigg{\}}=&\theta \tilde{g}\_{ij}(y^{t-1,t},x^{t-1,t})+\theta\_1y\_{ij}^{t-1,t}\\&+\phi\_{i,sender}(t)+\phi\_{j,receiver}(t). \end{split}$$ STERGM without random effects (Model 6): Together with our main model from the paper (Model 7), the STERGM with random effects we additionally include a STERGM without random effects. The formal representations are given in the main paper in equations (4) and (5). [t!] \| Number \| Model Name \| lagged edge \| covariates \| rand. eff. \| PR % \| ROC % \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| 1 \| Autoreg. ERGM \| yes \| no \| no \| 0 \| 0 \| \| 2 \| TERGM \| no \| yes \| no \| 0 \| 0 \| \| 3 \| TERGM \| no \| yes \| yes \| 0 \| 1.54% \| \| 4 \| TERGM with dyadic stability \| yes \| yes \| no \| 4.62% \| 6.15% \| \| 5 \| TERGM with dyadic stability \| yes \| yes \| yes \| 18.46% \| 44.62% \| \| 6 \| STERGM \| implicit \| yes \| no \| 24.62% \| 6.15% \| \| 7 \| STERGM \| implicit \| yes \| yes \| 52.31% \| 41.54% \| [models] ### Comparison of out-of-sample predictions We evaluate the proposed models in the following way. In a first step, we fit coefficients based on the information of t − 1 to the response in t. In the second step we use the fitted models in order to predict the edges in t + 1. As a result we obtain probabilistic out-of-sample predictions. The evaluation is done with area under the curve (AUC) measures for the Receiver-Operating-Characteristic (ROC) curve and the Precision-Recall (PR) curve. All models are fitted using the package `mgcv` (; Version 1.8-24) and evaluated with the package `PRROC` (; Version 1.3). Table [models] gives and overview of the predictive performance of all models included. In the two rightmost columns we present the share of years where the respective model has the highest out-of-sample predictive power. It can be seen that evaluated by the PR, the STERGM with random effects clearly represents the superior model with the highest predictive performance. Judged by the ROC the performance of the TERGM and the STERGM (both with random effects and as a model class) are very similar. We give a detailed description of the results below. In order to give a clear impression of how the out-of-sample fits are related we provide multiple plots where the baseline model, the STERGM with covariates and random effects, is always indicated in solid red. [figroc1] In Figure [figroc1] we compare the AUC values of the autoregressive ERGM (Model 1), the TERGM with covariates (Model 2) and with random effects (Model 3) with the baseline STERGM model (Model 7). The Precision Recall AUC values are shown on the left hand side of Figure [figroc1] and provide a clear message since all selected candidate models have AUC values clearly below the STERGM used in the paper (Model 7). Looking at the AUC values from the ROC measure (right panel) shows that the two TERGM models (Models 2 and 3) come partly close to the baseline model in the first years of the observational period, while the simplistic autoregressive ERGM has considerable lower AUC values. However as a general picture the STERGM with random effects is clearly the superior model. Including the lagged response as explanatory variable makes the TERGM models (Models 4 and 5) pretty similar to our baseline model (Model 7). [figroc2] Naturally, this is reflected in Figure [figroc2]. However, the STERGM model is more flexible because it allows for different coefficients for the processes of formation and persistence. This fact leads to superior predictions of the STERGM model (Model 7) in the left panel of Figure [figroc2]. The red line provides the upper boundary in most time points, but there are some instances where the autoregressive TERGMs (Models 4 and 5) provide the better predictions when evaluated with the PR curve. If the AUC measure for the ROC curve is compared, we see again that the predictions of the TERGM partly outperform the STERGM. Nevertheless, this is mostly the case in the beginning of the observational period and it seems like the superiority of the STERGM increases with the more recent periods. In Figure [figroc3] the predictive performance of the STERGM with (Model 7) and without (Model 6) random effects is compared. Although the AUC values are very close to each other, the model that includes the random effects clearly provides better out-of-sample predictions. On the right hand side of Figure [figroc3] the contrast becomes visible more clearly and in almost all years our baseline model provides the better predictions. All in all we conclude the following. (i) The STERGM gives, among all other candidate models the best predictions when judged by Precision Recall, being the more important measure when predicting rare vents. (ii) Furthermore, the STERGM has a much richer interpretation than the TERGM and (iii) the random effects provide an substantial benefit to the inferential part of the model. We therefore conclude that the choice of the STERGM with random effects seems to be very appropriate regarding both, the predictive performance as well as the ability to gain new insights. [figroc3] --- 1. Department of Statistics, Ludwig-Maximilians-Universität München, 80539 Munich, Germany, [email protected][↩](#fnref1) 2. Department of Political Science, Ludwig-Maximilians-Universität München, 80538 Munich, Germany, [email protected][↩](#fnref2) 3. Department of Statistics, Ludwig-Maximilians-Universität München, 80539 Munich, Germany, [email protected][↩](#fnref3) 4. The authors gratefully acknowledge funding provided by the german research foundation (DFG) for the project International Trade of Arms: A Network Approach.[↩](#fnref4)	arxiv_0000692
6]). As typical, starting from the unperturbed profile, a shell of denser gas creates, in this case at a radius of ≃ 800 pc, particularly evident as a dip in the otherwise monotonically decreasing profile (see the -2 Myr red line in the top left panel). The shell falls towards the center, progressively cooling, so that the soft X-ray emission weighted temperature decreases (Fig. [f2], bottom panels). A first AGN burst is produced before the shell reaches the center, and the resulting outward moving shock empties and heats the gas in the inner regions of the galaxy, on a time scale of ≈ 1 Myr. The subsequent snapshot in Fig. [f6], at +6 Myr after the first burst, shows the high central temperature typical of the outburst phase: $\tp(R)\sim $few keV within a radius of ∼ 50 − 100 pc. When the shock enters the radiative snow-plow phase, the associated secondary dense shell interacts with the first one still falling, and a series of direct and reflected shock waves are produced, accompanied by accretions events and star formation. The profiles between -2 Myr and +18 Myr (not shown) are very irregular, and consisting of a series of dips propagating outwards, with the temperature in the central region quickly and alternately increasing and decreasing. As already noticed for Fig. [f2], both hotter and colder regions are continuously created, the hottest ones within ∼ 1 kpc (mainly due to shocks), and the coldest ones due to the cooling gas in the shells. Each sub-burst is stronger than the previous one; finally, a last shock from the center concludes the phenomenon, halting star formation and leaving behind a very hot nucleus (the +66 Myr red line), while emptying the rest of the galaxy [see also the Σ(R) profile at +66 Myr in Fig. [f11]]. Then, during the following ≃ 100 Myr, the gas resumes the temperature typical of quiescent times (the +202 Myr green line). The inner very hot phase lasts for $\lsim 0.1$ Gyr. During the bursts, cosmic rays are shock-accelerated, and the inner regions look similar to a gigantic supernova remnant. Only thermal X-rays are considered here, while those due to synchrotron emission from accelerated particles due to shocks were not computed (see Jiang et al. 2010). The profiles in the right panels of Fig. [f6] show the binned aperture temperature profiles, for the same times as for the left panels. The binning smears the small-scale features, but the major distinctive characteristics, as the temperature dip when the first shell forms, and the large temperature drop outside the center, at +18 Myr and +66 Myr, are still well detectable. ### Comparison with observed profiles Negative radial gradients, as shown by the model temperature during quiescence (Fig. [f5]), are common among ellipticals, as revealed most recently by Chandra observations (e.g., Kim & Fabbiano 2003, Humphrey et al. 2006, Sansom et al. 2006, Fukazawa et al. 2006, Diehl & Statler 2008b, Nagino & Matsushita 2009). In a large collection of temperature profiles (Diehl & Statler 2008b, Athey 2007), those cases with negative gradients show a temperature that roughly halves from the innermost bin (that in general extends out to a radius of 0.5-a few kpc from the center) to the outer galactic region; this is roughly the same behavior shown by the models. Observed temperatures range between 0.5 and 1 keV at the innermost radial bin, where the model temperature is 0.8-1 keV (Fig. [f6], right panels). Interestingly, the galaxies with a negative gradient reside in the field or in less dense environments, as the outer Virgo regions (Matsushita 2001, Fukazawa et al. 2006, Diehl & Statler 2008b), and all have[5](#fn5) $\lx<$few × 1040 erg s− 1, both characteristics that match those of the models. More complex temperature profiles are also common, and could correspond to phases of AGN activity. For example, the so-called “hybrid” profiles show a central negative gradient, until the temperature reaches a minimum, and an outer positive gradient; for example in NGC1316, NGC4552, NGC7618 the temperature has a minimum of ∼ 0.4-0.5 keV at ∼ few kpc, and rises both going towards the center (of ∼ 0.2-0.3 keV), and going outwards (Diehl & Statler 2008b). Also in the models, after each major burst, when the last shock is moving outwards and fading, leaving a hot, rapidly cooling core, there is a drop in the temperature profile at a radius of ∼ 1 kpc or more (Fig. [f6], bottom panels, black and red profiles). However, in the models the temperature reaches > 1 keV at the center, larger than the observed values (the comparison also depends on the binning used for the profiles, though). Interestingly, preliminary results from 2D simulations (Novak et al. 2010), show lower central temperatures than in Fig. [f6], during outbursts, due to the fragmentation of the cold shell, that causes a lower gas compression while falling to the center. Other observed profiles keep roughly isothermal, or are roughly flat out to $\sim\re$ and then increase outward (e.g., Humphrey et al. 2006, Diehl & Statler 2008b, Nagino & Matsushita 2009). Such a positive outer gradient is shown by galaxies in high-density environments, suggesting the influence of circumgalactic hot gas; also, these galaxies have typically a larger hot gas content than the models. Environmental confining effects, currently not included in the models, are expected to increase the gas luminosity, and produce outer postitive gradients (see, e.g., Sarazin & White 1987, Vedder et al. 1988). It has been proposed that galaxies could behave according to a scenario where weak radio AGN distribute their heat locally and host negative inner temperature gradients, whereas more luminous radio AGNs heat the gas more globally through a jet or rising bubbles, and produce a flat profile, or a positive gradient (Diehl & Statler 2008b). Our models during quiescence could correspond to the weak AGN phase; also, after an outburst, the temperature profile can be flattish (excluding the innermost bin) out to a few kpc, and show a positive gradient outer of this radius (see the red line of Fig. [f6], bottom right panel). However, without a confining environment, an exploratory investigation conducted by us shows that the kinetic heating of a bright radio AGN could cause a major degassing, rather than just a reversal of the temperature gradient from negative to flat or positive. Likely, a gas rich environment providing confinement for the galactic coronae is a necessary condition to observe a bright (extended) radio source, and the confinement in turn also helps produce a positive temperature gradient in the outer galactic regions. Both aspects (the jet and a dense environment) will be implemented in the models in the future. In conclusion, the addition of a jet to the simulations could have positive effects, if it heats the gas outside $\sim\re$, and reduces the accretion rate and then $\lbh$ during the stationary hot accretion phase (Sect. [nuc]); it should not increase the temperature at the center, though, since this is already on the upper end of those observed even during quiescence. Brightness profiles ------------------- Figure [f9] shows the evolution of the X-ray surface brightness profile of the gas Σ(R) for, at the same quiescent times of the temperature profiles in Fig. [f5], for the soft (0.3-2 keV) and hard (2-8 keV) bands. In the hard band, Σ(R) is always comparable to or lower than a profile representing the unresolved stellar emission due to low mass X-ray binaries, calculated for a deep ($\gsim 200$ ksec) Chandra pointing of a galaxy of the same optical luminosity as and distant $\lsim 20$ Mpc. Thus, hard emission during quiescent times would be difficult to distinguish from the contribution of unresolved binaries, even with Chandra. In both bands Σ(R) becomes flatter in shape with time increasing, since the emission level decreases mostly in the central galactic regions (within $\re$). This important effect is produced by the nuclear outbursts, that remove gas from the center. In analogy with the discussion of the temperature profiles, Fig. [f11] shows Σ(R) just before, during, and after the last major nuclear burst at 7.498 Gyr, with times counted from the first accretion event. At -2 Myr the formation of the off-center cold shell produces the characteristic feature of a sharp decrease of Σ in the hard band, and a bright rim in the soft band. The subsequent curves show the shock moving outwards after the major burst, and the presence of very hot gas at the center, revealed by the central peak in the hard band. The disturbances in the Σ profiles due to the shells launched by the repeated sub-bursts are much more visible in the soft than in the hard band, as particularly apparent at +18 Myr. At this time, note the remarkable presence of a hot center surrounded by a denser and colder shell, producing a sharp peak in Σ(R), at R = 600 − 700 pc, and a sharp dip in $\ta(R)$ in Fig. [f6]. The final two times (+66 Myr and +202 Myr) show the result of the degassing caused by the passage of the shock waves: the gas density is low, and subsonic perturbations remain at a radius of $\gsim 10$ kpc. A different representation of the profiles during the burst phases is given by Fig. [f12], where $\Sigma\_{\rm UM}(R)$ resulting from the unsharp masking procedure (Eq. [conv]) is shown. Fluctuations in brightness that can be clearly distinguished are that due to the cold shell (time -2 Myr), and, after the burst, the negative off-center region in $\Sigma\_{\rm UM}(R)$ delimited by a sharp positive peak (times +6, +18, +66 Myr); the latter resembles what is commonly called a hot gas “cavity”. Note that, when implemented in 2D simulations, the same input physics adopted for the present class of models gives conical hot gas outflows from the nucleus, during outbursts: hot gas is ejected along the symmetry axis, so that elongated “cavities” (i.e., regions with a gas density decrement) are created (Novak et al. 2010). In addition, in their study of a model very similar to, Jiang et al. (2010) found after a burst taking place at 6.5 Gyr, a cavity filled with radio emitting particles of ∼ 4.4 kpc in size, detectable during the first ∼ 10 Myr after the burst. Therefore, we expect that cavities in the hot gas, such as those seen as off-center minima in the profiles of Figs. [f11] and [f12] at times from 6 to 66 Myr after the burst, should be fairly bright in the radio and would have, in X-rays, two essentially hollow conical lobes. They should even show non-thermal X-ray emission of the type seen in the Crab nebula (Jiang et al. 2010). Finally, taking at face value the results of the present analysis and the estimates on the radio detectability of Jiang et al. (2010), hot gas cavities seem more long-lasting than their radio detectability. Coming to the observability of the predicted features, one major property of the models is the decrease of Σ(R) in the central galactic regions, produced by the nuclear outbursts, with respect to models without feedback. Interestingly, bright ellipticals imaged with Chandra (e.g., Loewenstein et al. 2001) show a brightness profile that is quite flat within the central ∼ 1 kpc, a feature impossible to reproduce with pure inflow models, while it resembles the profile of the “pre-burst” phase (black lines in Fig. [f11], left panels), or at the end of a burst (+202 Myr). To better illustrate this point, Fig. [f10] shows the different shape of Σ(R) for and for a model with the same $\lb$, σ and $\re$, but without AGN feedback, and the sole SNIa’s heating included. The two Σ(R) profiles considered refer to the present quiescent epoch, yet the difference in steepness is clear. Figure [f10] also shows the observed Σ(R) for an elliptical at the periphery of the Virgo cluster, with $\lx$ close to that of ; the agreement between model and observation is very good. Another major prediction is given by the disturbances in the profile during an outburst; these keep above the level of the unresolved stellar emission, for a deep observation of a nearby galaxy with Chandra, as shown by Fig. [f11]. The central spike in Σ(R), during the high-temperature and high-density phase at the center, is confined within ∼ 100 pc, and then likely to remain a central unresolved feature even in galaxies observed at the high angular resolution of Chandra. Disturbances as shells and ripples farther out in the galaxy last $\lsim 0.2$ Gyrs, and are more likely to be observed. Given the typical duration of these features, and the presence of 3–4 major outbursts during the whole evolution, statistically they should be present, and possibly revealed by current X-ray observations, in ≃ 5 − 10% of the galaxies with $\lb$ similar to the model galaxy. In fact, many nearby galaxies show a disturbed appearance, as most recently revealed by studies based on Chandra data (e.g., Finoguenov & Jones 2001, Forman et al. 2005, Soria et al. 2006, Diehl & Statler 2007, Nulsen et al. 2009, Baldi et al. 2009, Dunn et al. 2010). The ISM morphology of a large set of galaxies (54) shows a level of disturbance correlated with the radio luminosity derived by the NRAO VLA Sky Survey (thus including both pointlike and extended radio emission; Diehl & Statler 2008a). Also, many of the best studied gas-rich galaxies show decrements in the X-ray surface brightness map, identified as cavities formed when AGN jets inflate radio lobes and displace surrounding gas; in many cases the cavities are filled with radio plasma, and surrounded by armlike features, sometimes classified as shocks. The hot gas disturbances have then been generally attributed to jet activity. As discussed above, we expect that if we took two cones from our solutions, they would give “lobes”. These would produce shocks at the edges and the lobes would be filled with radio emitting particles. There are also a few galaxies without currently evident extended radio emission, but with signs of an outflow and hot central gas, as NGC4552 (Machacek et al. 2006). This galaxy shows a weak core radio source unresolved by the VLBA, and in the (unsharp masked) X-ray image two conspicuous ringlike features at 1.3 kpc from the galaxy center, surrounding two cavities; these features have been found consistent with shocked gas driven outward by recent nuclear activity, as in a bipolar nuclear outflow. The gas temperature in the central ∼ 100 pc of the galaxy is 1 ± 0.2 keV, hotter than elsewhere in the galaxy, suggesting that we may be directly observing the reheating of the galaxy ISM by the outburst (Machacek et al. 2006). These characteristics resemble those predicted by the models for the temperature and the surface brighntess during the afterburst phase. Discussion and conclusions ========================== The hot gas properties of massive ellipticals, with regard to luminosity and temperature, and their spatial distributions, allow us to derive insights into the hot gas evolutionary status, and its link with the host galaxy. Since the cooling times are short compared to the galaxy age, it is now commonly accepted that repeated cooling catastrophes have occurred in the past, accompanied by central starbursts and AGN outbursts. The interest of a better understanding of this phenomenon is obvious, as it is directly linked to galaxy formation and evolution, and to the growth of the central MBH. Unfortunately, a complete theoretical picture is still missing, so that modeling coupled with a close comparison with observations is crucial in this field. In fact, these feedback events must leave signatures on the X-ray properties of the galaxies; indeed, the observed temperature and brightness profiles often cannot be fit easily with smooth profiles, or cooling flow profiles (e.g., Sarazin 2011, Statler 2011). In this work we have calculated the observational properties in the X-ray band of two galaxy models, representative of a class of detailed simulations of physically based models for the investigation of feedback modulated accretion in isolated galaxies. The feedback physics includes the combined effects of radiation and AGN winds (Paper III). The observational properties derived provide good matches to what observed in general for the local universe, and account for a few otherwise puzzling observed properties; on the other hand, they also evidence the need for changes or additions to the input physics. The main results of the present investigation are the following: 1) After an evolution of ≃ 10 Gyr, the models are typically in a permanent quiescent phase. The bolometric nuclear emission is very sub-Eddington (l ≃ 10− 4), within the range observed for l, though the most frequently observed values (l ≈ 10− 5 or less) are somewhat lower. Unfortunately, uncertainties in the bolometric correction to apply to observed nuclear luminosities, appropriate for a spectral energy distribution at low emission levels, do not allow to make a stringent comparison between modelled and observed values. However, also the nuclear X-ray emission $L\_{\rm BH,X}$ of the models, estimated as $L\_{\rm BH,X} \lsim 0.2\lbh$ as should be appropriate for low luminosity nuclei, and $L\_{\rm BH,X}/\ledd$, tend to lie on the upper end of what is observed. Thus in real galaxies an additional mechanism may be required to reduce further the mass available for accretion; this could be provided by the mechanical feedback of a (nuclear) jet, and/or by a thermally driven wind from a RIAF. Alternatively, the switch from disc to RIAF behavior (that is $l\propto \dot m^2$) should occur at a larger l than assumed here (e.g., at l ≃ 0.1 rather than l ≃ 0.01). Simulations of ram-pressure stripping effects, instead, showed that a reduction in the accretion rate is not attained, because in the quiescent hot accretion regime, the accreting mass comes mainly from the stellar mass losses within the central ∼ 100 parsecs of the galaxy, that are not affected by stripping (Shin et al. 2010b). 2) The X-ray luminosity of the ISM oscillates in phase with the nuclear luminosity, though with much broader peaks; at the present epoch, $\lx$ lies at the lower end of the large observed range for galaxies of $\lb$ similar to that of the models. The degassing/heating in the models may then be too efficient, or a larger/more concentrated gravitating mass, or a confining external medium, are needed. However, when the gas luminosities during the whole evolution are considered, the observed $\lx$ range is better reproduced. This is even more true if also an additional model of larger σ, and same $\lb$, is included. Part of the observed large variation in $\lx$ for galaxies of a given $\lb$ could then be explained by nuclear activity. 3) The average ISM temperature is within the observed large range for the model σ; when estimated within $\re$, the model temperatures reproduce better the upper half of those observed. Modifications to the models by the addition of a jet or an external medium, as suggested in the previous points, should then not increase the average temperature within $\re$. 4) During quiescence, the profiles of the gas temperature and brightness resemble those observed for many local galaxies. Especially remarkable is the lack of the steep brightness profile shape typical of inflowing models, due to the frequent removal of gas from the galactic central regions, and to the heating provided by mechanical feedback (that is always present, even during quiescent phases). The models show negative temperature gradients, that are common for isolated galaxies; the addition of a jet or a confining agent should change the temperature profiles into a flat or outwardly increasing profile, as also frequently observed for galaxies in rich environments. 5) During outbursts, disturbances are predicted in the temperature and brightness profiles; the ISM resumes the smooth appearance of steady and low-luminosity hot accretion on the MBH on a time-scale of $\lsim 200$ Myrs. The most conspicuous variations with respect to smooth profiles are within current detection capabilities, and could correspond to (part of) the widespread disturbances observed in galaxies of the local Universe. In particular, shocked hot gas should be seen at the galactic center (within ∼ 100 pc), possibly not resolved; this would be a certain sign of prior AGN activity. These hot bubbles could be revealed by emission of cosmic-rays, in a structure similar to a gigantic supernova remnant. Preliminary results from 2D runs show bipolar nuclear outflows, that should be seen as conical cavities extending from the galactic center, and may be called jet-like features. 6) The duty-cycle of nuclear activity is of the order of a few × (10− 3 − 10− 2), depending on the assumed mechanical feedback efficiency; in general, a burst cycle lasts for ≈ 107 yrs. These duty-cycle values are broadly consistent with the fraction of active galaxies measured in observational works, though reported values for the local universe are somewhat lower, for the MBH mass of the models. In order to make a more consistent comparison with observations, the dataset of models should be increased, and the duty cycle computed only for the last 2–3 Gyrs; this will reduce the duty cycle, as the models are characterized by a declining nuclear activity. 7) The duty-cycle of perturbances in the ISM is of the order of 5-10%, from their average number and duration. This duty-cycle likely increases with galaxy mass, because an outburst has a greater impact in less massive (and less gas-rich) systems, which then are “on” for a shorter time (Ciotti & Ostriker 2011). The ISM duty-cycle, and its trend with galaxy mass, compare reasonably well with preliminary estimates obtained from a large sample of hot gas coronae in elliptical galaxies observed with Chandra (Nulsen et al. 2009): the fraction of galaxies with X-ray cavities in the hot gas is $\lsim 10$% when $\lx<10^{41}$ erg s− 1 (as for the models), and reaches ∼ 25% in the most luminous ones. The presence of cavities has been attributed to the action of jets inflating radio lobes and displacing the surrounding gas. Cavities can also be created in the scenario presented here, as hinted for by 2D simulations, due to bipolar nuclear outflows. 7) Two diagnostic planes have been constructed. In the first one, the nuclear luminosity $\lbh$ and the ISM luminosity $\lx$ are followed during the whole model evolution. The points representative of the models populate a wedge region, that should then be occupied when observing a large set of galaxies. Another plane shows the evolution of $\lx$ versus the average gas temperature $\ta$; here the most populated region is that of a large $\lx$ variation (factor of ∼ 10) for $\ta$ keeping between 0.4 and 0.6 keV. Clearly, a larger set of models is to be explored, in order to better establish the final gas properties (as gas content, nuclear and ISM duty-cycle, etc.) to be compared with those of a statistically large sample. 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M., Konidaris, N., Koo, D., Davis, M. 2006, ApJ 648, 281 Yu, Q., Tremaine, S. 2002, MNRAS 335, 965 [modref] \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Model \| $\epswM$ \| $<\epsw>$ \| $<\eps\_{\rm EM}>$ \| $\log \Delta\mbh$ \| logΔM\* \| $\log \Delta M\_{\rm w}$ \| $\log \mgas$ \| $\log \lbhefopt/\ledd$ \| \| (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (8) \| (9) \| \| \| 10− 3 \| 2.0 10− 5 \| 0.105 \| 8.74 \| 9.74 \| 10.27 \| 9.68 \| -5.13 \| \| \| 3 × 10− 4 \| 1.2 × 10− 5 \| 0.133 \| 9.05 \| 10.22 \| 10.31 \| 9.34 \| -5.43 \| [tab2] \| \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Model \| $\mbh$ \| log$\lbh $ \| l \| $L\_{\rm BH,X} $ \| $L\_{\rm BH,X}/L\_{Edd}$ \| \| duty cycle \| \| $\log L\_{\rm X} $ \| \| \| (M⊙) \| (erg s− 1) \| (10− 4) \| (erg s− 1) \| \| Bol \| Opt \| UV \| (erg s− 1) \| \| (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (7) \| (7) \| (8) \| \| \| 8.4 × 108 \| 43.39 \| 2.0 \| $\lsim 5\times 10^{42}$ \| $\lsim 2\times 10^{-5}$ \| 6.3 × 10− 3 \| 3.2 × 10− 3 \| 3.0 × 10− 3 \| 40.1 \| \| \| 1.4 × 109 \| 43.38 \| 1.0 \| $\lsim 5\times 10^{42}$ \| $\lsim 2\times 10^{-5}$ \| 4.8 × 10− 2 \| 1.8 × 10− 2 \| 8.6 × 10− 3 \| 39.6 \| -0.8truecm [f1] -1truecm -3truecm [f2] 2truecm [his] -1 truecm -9.2truecm [f3] -1 truecm -9.2truecm [uff] -5truecm -1truecm -3.5truecm [hisT] -6.truecm -1. truecm -3.5truecm [f5] -4.7truecm -0.9truecm -3.5truecm -4.8truecm -0.9truecm -3.6truecm [f6] -0.3truecm [f9] -1.5truecm -14.2truecm 8truecm [f11] -1.7truecm -14.2truecm 7truecm [f12] -6truecm -1truecm [f10] --- 1. In the code the gas is allowed to cool down to a minimum temperature of 104 K.[↩](#fnref1) 2. A similar fraction ( ∼ 52%) of the sample of red sequence galaxies of the SDSS (r < 17.77, median redshift z = 0.1) have detectable line emission (Yan et al. 2006).[↩](#fnref2) 3. The model evolution of course also changes, because of the smaller direct radiative feedback, and the smaller indirect mechanical feedback, that in the models is a function of $\lbh$. The bursts are more extended in time, with a larger accreted MBH mass, which reduces slightly also the final l ratio.[↩](#fnref3) 4. Alternatively, the duty-cycle can be obtained as a luminosity weighted average over a chosen time interval, with very similar results.[↩](#fnref4) 5. Except for NGC6482, the remnant of a fossil group.[↩](#fnref5) X-ray properties expected from AGN feedback in elliptical galaxies ================================================================== The evolution of the interstellar medium (ISM) of elliptical galaxies experiencing feedback from accretion onto a central supermassive black hole has been studied recently with high-resolution 1D hydrodynamical simulations; these included cooling, heating and radiative pressure effects on the gas, specific for an average AGN spectral energy distribution, a RIAF-like radiative efficiency, mechanical energy, mass and momentum input from AGN winds, and the effects of starbursts associated with accretion. Here we focus on the observational properties of the models in the soft and hard X-ray bands, specifically on 1) the nuclear X-ray luminosity; 2) the global X-ray luminosity and temperature of the hot ISM; 3) its temperature and X-ray brightness profiles, during quiescence, and before, during and after an outburst. After an evolution of ∼ 10 Gyr, the bolometric nuclear emission is very sub-Eddington (l ∼ 10− 4), and within the range observed, though larger than the most frequently observed values. The nuclear bursts last for ≈ 107 yrs, and the duty-cycle of nuclear activity is a few × (10− 3 − 10− 2), when calculated over the last 6 Gyr. The ISM thermal luminosity $\lx$ oscillates in phase with the nuclear one, but drawing much broader peaks; a comparison with observed $\lx$ values, for galaxies of optical luminosity similar to that of the models, shows that this behavior helps reproduce statistically the observed large $\lx$ variation. At the present epoch, the largest observed $\lx$ could be reproduced only by adding an external confining medium. The average gas temperature is within the observed range; when limited to within $\re$, its values lie on the upper half of those observed. In quiescence, the temperature profile has a negative gradient; thanks to past outbursts, the brightness profile lacks the steep shape typical of inflowing models. After outbursts, disturbances are predicted in the temperature and brightness profiles (as analyzed also with the unsharp masking). Most significantly, during major accretion episodes, a hot bubble from shocked hot gas is inflated at the galaxy center (within ≈ 100 pc); the bubble would be conical in shape, in real galaxies, and show radio emission. Its detection in the X-rays is within current capabilities, though it would likely remain unresolved. The ISM resumes its smooth appearance on a time-scale of ≈ 200 Myr; the duty-cycle of perturbances in the ISM is of the order of 5-10%. The present analysis reveals an agreement of the models with the observations, but also evidences that additional input physics is important in the ISM-black hole coevolution, to fully account for the properties of real galaxies. The main insertions to the models would be a confining external medium and a jet. The jet will reduce further the mass available for accretion (and then l), and may help, together with an external pressure, to produce flat or positive temperature gradient profiles (that are common among galaxies in high density environments). Alternatively to the jet, a reduction of l can be obtained if the switch from high to low radiative efficiency takes place at a larger l ( ≃ 0.1) than routinely assumed ( ≃ 0.01). Introduction ============ Supermassive black holes (MBHs) at the centers of bulges and elliptical galaxies play an important role in the processes of galaxy formation and evolution (e.g., Cattaneo et al. 2009), as testified by remarkable correlations between host galaxy properties and the MBH masses (e.g., Magorrian et al. 1998, Ferrarese & Merritt 2000, Gebhardt et al. 2000, Graham et al. 2001) and as supported by many theoretical studies (e.g., Silk & Rees 1998, Haiman, Ciotti & Ostriker 2004; Merloni et al. 2004, Sazonov et al. 2005, Di Matteo, Springel & Hernquist 2005, Hopkins et al. 2006, Somerville et al. 2008, Kormendy et al. 2009). An important aspect of the coevolution process is the radiative and mechanical feedback by the accreting MBH onto the galactic interstellar mediun (ISM) that is continuously replenished by normal stellar mass losses, at a rate of the order of ≈ 1M⊙ yr− 1 in a medium-mass galaxy. In absence of feedback from a central MBH (and stripping from the intracluster medium in case of satellite galaxies), this ISM would develop a flow directed towards the galactic center, accreting $\gsim 1 M\_{\odot}$ yr− 1 in a process similar to a “cooling flow”. Instead, in low mass galaxies, type Ia supernova (SNIa) heating is able to sustain a low-luminosity, global galactic wind (e.g., Ciotti et al. 1991, David et al. 1991, Pellegrini & Ciotti 1998), and the central MBH is in a state of permanent, highly sub-Eddington hot accretion (Ciotti & Ostriker 2011). Therefore, in medium to high mass galaxies, feedback is required by the following empirical arguments: 1) the large amount of gas lost by the passively evolving stellar population during the galaxies’ lifetime is not observed (e.g., Peterson & Fabian 2006), and just $\lsim $1% of the mass made available by stars is contained in the masses of present epoch MBHs; 2) bright AGNs, as would be expected given the predicted mass accretion rate, are not commonly seen in the spheroids of the local Universe (e.g., Fabian & Canizares 1988, Pellegrini 2005). Thus AGN feedback is required just on the basis of a mass balance argument. Since over a large part of the galaxies extent, and for a large fraction of their lifetime, the ISM cooling time tcool is much lower than the galaxy age, one early quasar phase cannot be the solution to the cooling flow problem. The solution requires either steady heating, or heating with bursts on a timescale Δt ≈ tcool. Unfortunately, on a purely theoretical ground, how much radiative and mechanical energy and momentum output from the MBH can effectively interact with the surrounding ISM, and what are the resultant MBH masses, is difficult to establish. Recently, the interaction of the MBH output with the inflowing gas has been studied with high-resolution 1D hydrodynamical simulations in a series of papers (Ciotti & Ostriker 2007; Ciotti, Ostriker & Proga 2009, 2010 hereafter Papers I and III; Shin, Ostriker & Ciotti 2010a,b; Ostriker et al. 2010; Jiang et al. 2010), that are currently being extended to 2D treatments (Novak, Ostriker & Ciotti 2010). These simulations implement a physically based detailed treatment of the radiative energy and momentum input from the MBH into the ISM, consistent both with observed average AGN spectra and theoretical calculations of radiation transport; they also include starformation, and a modelling of the mechanical energy and momentum feedback from AGN winds. The combined effects of radiative and mechanical feedback produce recurrent AGN burst phases accompanied by starformation, spaced apart by longer phases of relative quiescence. A cycle repeats with the galaxy seen alternately as an AGN/starburst for a small fraction of the time, and as a “normal” elliptical for much longer intervals. Accretion fueled feedback thus proves effective in suppressing long lasting cooling flows and in maintaining MBH masses within the range observed today, since the gas is mostly lost in outflows or consumed in starbursts. Remarkably, while star formation is suppressed when the AGN is in the low-luminosity state, it is enhanced by the strong AGN outbursts, consistent with observations (Schawinski et al. 2009). Note finally that a major role in producing global degassing, and in regulating the flow evolution, is also played by the SNIa’s heating. The previous papers, with the exception of Pellegrini, Ciotti & Ostriker (2009), were mainly dedicated to the study of the accretion physics and the feedback effects. Here instead we focus on the appearance that the models would have if observed in the X-ray band, both in quiescence and during an outburst of activity. We concentrate on two models (named and ) extracted from the suite of cases presented in Paper III (Table 1 therein), characterized by a mechanical feedback efficiency dependent on the Eddington scaled accretion luminosity. and were considered particularly successful, since their input parameters agree with previous theoretical studies or observations (as, e.g., for the AGN wind opening angle, and the peak value of mechanical efficiency, that are in accord with those estimated from 2D and 3D numerical simulations; Proga, Stone & Kallman 2000, Proga & Kallman 2004, Benson & Babul 2009), and, at the same time, their final properties (as mass fraction of a younger stellar population, MBH mass, etc.) are in reasonable accord with observations. The only difference in the input physics of and is the maximum value of the mechanical efficiency. Since in the simulations the treatment of feedback is physically based, not tuned to reproduce observations, any agreement or discrepancy of the resulting model properties with X-ray observations is relevant to improving our understanding of the MBH-ISM coevolution, putting further constraints on the input ingredients, and possibly telling us what additional physics may be important in the problem. However, we stress that the models describe an isolated galaxy, where ram-pressure stripping (in case of satellite galaxies) and intracluster medium pressure confinement (in case of group or cluster central galaxies) are not taken into account (see Shin et al. 2010b). The main observational signatures investigated here include the nuclear and gaseous emissions, and the ISM temperature and brightness profiles in the quiescent phases, and before, during, after a burst. Particular attention is paid to the appearance and detectability of central hot bubbles, with diameters of ∼ a hundred parsecs, that are produced by the models during outbursts, and to various kinds of disturbances in the hot ISM. The analysis is performed in the soft and hard X-ray bands, also after unsharp-masking. These predictions are relevant for their observational consequences, since the high angular resolution of the Chandra satellite has allowed us to obtain the best definition ever for the hot gas properties of the galaxies of the local universe, by separating the contributions of stellar sources and hot gas, and the emission coming from different spatial regions within galaxies (e.g., Fabbiano 2011). In particular, in several elliptical galaxies various kinds of hot gas disturbances have been detected, likely resultant from nuclear activity (e.g., Finoguenov & Jones 2001, Jones et al. 2002, Forman et al. 2005, Machacek et al. 2006, O’Sullivan et al. 2007, Million et al. 2010). At the same time, nuclear emission values have been detected down to very low luminosities, comparable to those of X-ray binaries (e.g., Loewenstein et al. 2001, Gallo et al. 2010, Pellegrini 2010). The paper is organized as follows. Section 2 summarizes the main evolutionary phases of the representative models (and ) considered; Section 3 describes how the observational properties of the models are derived; Section 4 presents a comparison of the nuclear luminosities with existing observations; Section 5 discusses the evolution of the X-ray luminosity and emission-weighted temperature of the ISM; Section 6 presents the projected temperature and surface brightness profiles at representative times during quiescence and a nuclear outburst. Finally, in Section 7 we summarize and discuss the main results. Two representative models: main features ======================================== The basic ideas behind the present class of models for feedback modulated accretion flows have been introduced in Ciotti & Ostriker (1997, 2001), and Ostriker & Ciotti (2005), and developed in detail in the papers listed in the Introduction; a comprehensive recent discussion is given in Ciotti & Ostriker (2011). Here we focus on models and from Paper III (where the description of the numerical code and the input physics is given). The initial parameter values and the main final properties of the models are given in Table [modref]. The two models refer to an isolated elliptical galaxy placed on the Fundamental Plane, with a projected central stellar velocity dispersion σ = 260 km s− 1, a total B-band luminosity $\lb=5\times 10^{10}L\_{\rm B\odot}$, and an effective radius $\re=6.9$ kpc. The stellar density profile is described by a Jaffe (1983) law, and the dark halo profile is such that the total (stellar+dark) mass density profile scales as ρ ∝ r− 2 at large radii; all relevant dynamical properties used in the code are discussed in Ciotti, Morganti & de Zeeuw (2009). The dark-to-visible mass ratio is one within $\re$, and the resulting stellar mass-to-light ratio is $M\_\/\lb=5.8$. Finally, a standard SNIa’s rate declining with time t* as t− 1.1 is assumed. The initial MBH mass is set to 10− 3 the initial stellar mass M\, i.e., it is $2.9\times 10^8\Msun$. The simulations begin at a galaxy age of ∼ 2 Gyr (that is a redshift z* ∼ 2, the exact value depending on the epoch of elliptical galaxy formation, usually put at $z\gsim 2$). The efficiency for producing radiation (Soltan 1982, Yu & Tremaine 2002) of material accreting on the MBH at the rate $\mdot$ is $$\epsilon =0.2\times {A \dot m \over 1+A\dot m }, \label{effic}$$ where $\dot m=\mdot/\dot M\_{\rm Edd}$ is the Eddington-scaled accretion rate. Thus, for A = 100, one has ε ∼ 0.2 at large mass accretion rates $\dot m \gg 0.01$, and ε declining as for radiatively inefficient accretion flows (RIAFs, Narayan & Yi 1994), as $\epsilon \sim 20 \dot m$, for $\dot m\lsim 0.01$ (see also Sect. [nuc] for additional comments on this choice). The mechanical feedback implemented is that of the Broad Line Region winds (leading to outflow velocities of ≃ 104 km/s, similar to what observed, e.g. Chartas et al. 2003, Crenshaw et al. 2003), and reproduces the main features of the numerical modeling by Proga (2003). In particular, the mechanical efficiency scales with the Eddington ratio $l=\lbh/\ledd$ (where $\lbh =\epsilon\mdot c^2$ is the instantaneous bolometric accretion luminosity), reaching a maximum value $\epswM$ of 3 × 10− 4 (for ), and 10− 3 (for ), when $l\gsim 2$; also, the aperture solid angle of the conical nuclear wind increases at increasing l. Note that the values of the mechanical efficiency in cols. (2) and (3) of Tab. [modref] are to be contrasted with the generally higher fixed efficiency of 5 × 10− 3 commonly adopted in the literature (e.g., Hopkins et al. 2005, Di Matteo et al. 2005, Johansson et al. 2009). The mechanical output of a nuclear jet is also computed, but not added to the hydrodynamical equations, and it will be inserted in a future work. The evolution of the gas flows is obtained integrating the time-dependent (1D) Eulerian equations of hydrodynamics, with a logarithmically spaced and staggered radial grid, extending from 2.5 pc from the central MBH to 250 kpc. It is most important that the resolution is high enough that the inner boundary is within the Bondi radius (Bondi 1952); if this is not ensured, the accretion rate will be calculated incorrectly. Thus “Bondi accretion” is not assumed; the correct, time-dependent accretion rate is computed from the hydrodynamical equations. The code derives self-consistenly the source and sink terms of mass, momentum and energy associated with the evolving stellar population (stellar mass losses, SNIa events), the temporal steepening of the stellar velocity dispersion within the sphere of influence of the MBH as a consequence of its growth, the star formation during nuclear starbursts, and finally accretion and MBH feedback. Needless to say, the code conserves mass, energy, and momentum (e.g., Ostriker et al. 2010). Gas heating and cooling are calculated for a photoionized plasma in equilibrium with an average quasar spectral energy distribution (as detailed by Sazonov et al. 2005), and the resulting radiation pressure and absorption/emission are computed and distributed over the ISM from numerical integration of the radiative transport equation. The effects of radiation pressure on dust due to the starburst luminosity in the optical, UV and Infrared are also considered. A circumnuclear accretion disk is modeled at the level of “sub-grid” physics, and a set of differential equations describing the mass flow on the disk, its star formation rate, mass ejection and finally MBH accretion are solved at each time-step. The resulting evolution of $\lbh$ for and is shown in Fig. [f1]. These models have fairly standard values of the input parameters, and their general properties in the X-ray band, are typical of the class of 1D models investigated in Paper III. At the beginning, the galaxy is replenished by gas produced by the mass return from the evolving stars. Soon AGN outbursts develop, due to accretion of this gas, accompanied by star formation, and followed by degassing and a precipitous drop of the nuclear accretion rate. The outbursts are separated by long time intervals during which the galaxy is replenished again by gas from the stellar mass losses. The behavior of the gas during an outburst is almost independent of the specific burst episode considered. The outburst precursor is the off-center growth of a thin shell of dense gas (at a radius of ∼ 0.5 − 1 kpc) that progressively cools below X-ray emitting temperatures and falls towards the center; compression of the enclosed gas follows and a central burst is triggered, even before the cold shell reaches the center[1](#fn1). In less than a million years, a radiative shock originates from the center and quickly (in ≈ 106 yrs) produces an outward moving shell that collides with the original shell falling in. Reflected shock waves carry fresh material for accretion, and produce new sub-bursts. This leads to the rich temporal structure of each outburst event, especially visible in model (bottom panel, Fig. [f1]). Eventually, the cold material left after starformation (Ciotti & Ostriker 2007) is accreted in a final burst, a major shock leaves behind a very hot and dense center, and causes a substantial galaxy degassing. In general, while radiative effects mainly work on the kpc scale, mechanical feedback from the AGN winds is more concentrated and affects the ISM on the ∼ 100 pc scale (see Fig. 11 in Paper III). During the next few tens of million years, the gas cools, resumes its subsonic velocity, the density starts increasing again due to stellar mass losses, and the cycle repeats. At late epochs, the gas flows finally set in a state of steady, hot and low-luminosity accretion. Observational properties of the models ====================================== The observational model properties considered in Paper III were the nuclear bolometric, optical and UV luminosities, and the X-ray luminosity of the ISM within $10\re$. The latter was evaluated fiducially just by cutting the bolometric gas emission below a threshold temperature of 5 × 106 K. In the following, we calculate the X-ray luminosity of the nucleus ($L\_{\rm BH,X}$, Sect. [nuc]), and, with a more detailed and realistic procedure, the total luminosity and emission weighted temperature of the hot gas ($\lx$ and $<\tx >$, Sect. [global]); we also calculate the temperature and the surface brightness profiles during quiescent times, and during an outburst (Sect. [proj]). We briefly describe below how the observational gas properties are calculated from the numerical outputs for the gas density and temperature. The X-ray emission of the different model components is calculated over the energy range of 0.3-8 keV (the Chandra sensitivity band), and also in two separate bands, 0.3-2 keV (“soft”) and 2-8 keV (“hard”), typically used in studies of the nuclear and gaseous properties. In practice, at any given time, the volume gas luminosity is calculated from the gas density and temperature distribution on the numerical grid as $$\lx=4\pi \int\_0^{\infty}\Emis (r) r^2 dr, \label{lumeq}$$ where the emissivity is given by $\Emis(r) = n\_{\rm e}(r) n\_{\rm H}(r) \Lambda [T(r), Z]$, $n\_{\rm e}$ and $n\_{\rm H}$ are the number densities of electrons and hydrogen, and Λ(T, Z) is the cooling function. The cooling function is calculated over the two energy intervals by means of the radiative emission code APEC, valid for hot plasmas at the collisional ionization equilibrium (Smith et al. 2001), as available in the XSPEC package for the analysis of the X-ray data. For simplicity we choose constant abundance at the solar value, and the solar abundance ratios of Grevesse & Sauval (1998), which is consistent with observed gas metallicities (e.g., Loewenstein & Davis 2010). In order to speed-up the analysis, we derived with APEC the values of Λ, for each energy band, for a large set of temperatures in the range 0.1-16 keV; then we obtained two very accurate non-linear fits of the tabulated values (with maximum deviations < 1%, see Ciotti & Pellegrini 2008). These fits were used to compute the integral in ([lumeq]), and in every other integration where the emissivity is needed. For example the emission weighted temperature for the whole galactic volume was computed as $$<\tx> = {4\pi\over\lx}\int\_0^{\infty}\Emis (r) T(r) r^2 dr. \label{teq}$$ The surface brightness profile Σ(R), the emission weighted projected temperature profile $\tp(R)$, and the associated emission weighted aperture temperature profile $\ta (R) $, were obtained by numerical integration of the simulation outputs as $$\Sigma (R)=2\int\_R^{\infty}{\Emis(r) r\over\sqrt{r^2-R^2}}dr, \label{sig}$$ $$\tp(R) = {2\over \Sigma (R)} \int\_R^{\infty}{T(r)\Emis(r) r\over\sqrt{r^2-R^2}}dr, \label{tprof}$$ $$\ta (R) ={\int\_0^R \tp (R')\Sigma(R')R' dR' \over\int\_0^R \Sigma(R') R' dR' }. \label{ta}$$ The accuracy of the integrations above is verified by checking that the surface integral of Σ(R) over the whole grid recovers the same luminosity calculated via equation ([lumeq]), and that $\ta(\infty)=<\tx>$ within few percent (Ciotti & Pellegrini 2008). The surface integral of Σ(R) is also used to compute the gas emission within the optical effective radius $R\_{\rm e}$, and in equation ([ta]) to compute the average temperature within the optical effective radius. In order to highlight local departures from the mean ISM brightness profile, and to evidence major brightness features that could be revealed by observations, “fluctuation” profiles have been also created. These have been constructed with a technique similar to the so-called unsharp masking, frequently adopted in observational analysis (e.g., Fabian et al. 2003). In practice, the brightness profiles Σ(R) have been convolved with a 2D Gaussian of dispersion σ: $${\rm PSF}={e^{-{R^2\over 2\sigma^2}}\over 2\pi\sigma^2}, \label{psf}$$ so that the resulting surface brightness profile can be written as $$\Sigma\_{\rm obs}(R)=\int\_0^{\infty} {\rm I}\_0\left({RR'\over\sigma^2}\right) \,\, {e^{-{R'^2+R^2\over2\sigma^2}} \over \sigma^2} \,\, \Sigma (R')R'dR', \label{conv}$$ where I0 is the zeroth-order modified Bessel function of first kind. In the analysis of the simulations, the integral above is solved numerically, after a careful choice of σ. As expected, a too large σ produces an almost featureless profile, while a too small σ reproduces the unprocessed profile. After some attempts, it turned out that, in order to highlight local features, the optimal choice is that of a σ equal, at each gridpoint, to the sum of the lengths of the immediately preceding and subsequent grid intervals. The “unsharp-masked” profile is then defined in a natural way as $$\Sigma\_{\rm UM}(R)\equiv {\Sigma (R)\over \Sigma\_{\rm obs}(R)}-1. \label{conv}$$ Nuclear luminosities ==================== Figure [f1] shows the time evolution of the nuclear bolometric accretion luminosity $\lbh$ for and (whose input parameters differ only for the maximum value of the mechanical efficiency $\epsilon\_{\rm w}^{\rm M}$, that is respectively 10− 3 and 3 × 10− 4, Tab. [modref]). Strong intermittencies at an earlier epoch, with $\lbh$ reaching the Eddington value, become rarer and rarer with time, as the mass return rate from the stellar population declines, until a smooth, hot, and very sub-Eddington accretion phase establishes. The different mechanical efficiency is responsible for the sharp bursts in model, and the more time-extended and structured bursts in model (Paper III). Towards the present epoch, at a galaxy age of 12 Gyr, the mass accretion rate on the MBH for both models is $\mdot\approx 0.02\, \Msun$/yr, that translates into an Eddington scaled accretion rate $\dot m \simeq 1.7\times 10^{-3}$ and $\dot m \simeq 1.2\times 10^{-3}$ respectively for and. The value of $\dot m$ of is a bit smaller than for, because of its larger final $\mbh$ (Tab. [tab2]), a consequence of the weaker mechanical feedback. At the present time, and during the interburst periods, accretion has then entered the RIAF regime, and the radiative efficiency is ε ≃ 0.02; the nuclear bolometric luminosity is $\lbh=2.4\times 10^{43}$ erg s− 1 for both models, and the corresponding Eddington ratios are l ≃ 2 × 10− 4 () and l ≃ 10− 4 (), see Tab. [tab2]. These results agree with the observation that in the local universe massive MBHs are mostly radiatively quiescent, and the fraction of them at luminosities approaching their Eddington limit is negligible (e.g., Ho 2008). For example, in the sample of nuclei of the Palomar Spectroscopic Survey of northern galaxies, a nearly complete sample, magnitude limited at $B\_{\rm T} \leq 12.5$ mag, ∼ 50% of ellipticals show detectable emission line nuclei[2](#fn2), but mostly of low level ($L\_{\rm H\alpha}<10^{40}$ erg/s; Ho 2008). For this sample, the nuclear bolometric luminosity $(L\_{\rm bol,nuc})$ was derived from the observed nuclear 2-10 keV emission, using the correction $L\_{\rm bol,nuc}/L\_{\rm X,nuc}=15.8$ (Ho 2009). It was found that elliptical galaxies span a large range of $L\_{\rm bol,nuc}$, from 1038 to 1043 erg s− 1, with a median value of $L\_{\rm bol,nuc}\simeq 1.7\times 10^{40}$ erg s− 1 and a mean value of 4.6 × 1041 erg s− 1; the Eddington scaled luminosity l has a median value of l = 1.2 × 10− 6 (mean l = 1.2 × 10− 5), with l < 10− 3 for elliptical galaxies, and extending down to l = 10− 8. The representative models tend therefore to lie on the upper end of the observed distributions of $L\_{\rm bol,nuc}$ and l, a result that probably remains true regardless of the uncertainty in the bolometric correction from the 2-10 keV band. We will return on this point (already noticed in Papers I and III) in the Conclusions. Another way of comparing the model $\lbh$ with observed values, is to estimate $L\_{\rm BH,X}$ of the models, and compare it directly with observed $L\_{\rm X,nuc}$ values. Accurate $L\_{\rm X,nuc}$ measurements in a large number have been derived recently for elliptical galaxies of the local universe, based on Chandra data (e.g., Gallo et al. 2010, Pellegrini 2010); in the 0.3–10 keV band, $L\_{\rm X,nuc}$ ranges from $\gsim 10^{38}$ erg s− 1 to ∼ 1042 erg s− 1, with most of $L\_{\rm X,nuc}/\ledd$ as low as 10− 6–10− 8. In particular, for the final MBH masses of the models (Tab. [tab2]), it is observed that $10^{38}\lsim L\_{\rm X,nuc}$(erg s$^{-1})\lsim 10^{42}$, and $10^{-9}\lsim L\_{\rm X,nuc}/\ledd \lsim 10^{-4}$, both from the sample of the Hubble Virgo Cluster Survey (Gallo et al. 2010), and that of 112 early type galaxies within ∼ 60 Mpc (Pellegrini 2010). The 0.3-10 keV nuclear luminosity $L\_{\rm BH,X}$ of the models at the present epoch can be derived adopting a correction factor appropriate for the spectral energy distribution of a radiatively inefficient accretion flow, i.e., $L\_{\rm BH,X} \lsim 0.2\lbh$ for low luminosity AGNs (Mahadevan 1997). This gives $L\_{\rm BH,X} \lsim 5\times 10^{42}$ erg s− 1, and $L\_{\rm BH,X}/\ledd\lsim 2\times 10^{-5}$. Also in the X-ray band, then, the model nuclear luminosity tends to be larger than what typically observed; $L\_{\rm BH,X}/\ledd$ is within the observed range, though lying in its upper end. All this may suggest that in real galaxies an additional mechanism is reducing further the mass available for accretion, as could be provided by a nuclear jet, and/or a thermally driven wind from a RIAF (Blandford & Begelman 1999). In the latter case, only a fraction of the gas within the accretion radius actually reaches the MBH; the binding energy released by the accreting gas is transported radially outward to drive away the remainder in the form of a wind. Alternatively, it may be that the quadratic dependence of l on $\dot m$ (Sect. [repmod]) sets in at a higher value of $\dot m$ than adopted in Eq. [effic], for which it starts at $\dot m\sim 10^{-2}$. Within the uncertainties on the observational and theoretical input, we might have chosen the constant A = 10 rather than 100; in this way, the quadratic dependence would have set in at $l \lsim 0.1$ rather than $l\lsim 0.01$. This would have reduced the late time nuclear luminosity by a factor of ≃ 9, as confirmed by a supplementary run[3](#fn3) of models and with A = 10. Finally, another interesting - albeit more difficult - comparison with observational results can be done using the duty-cycle. The latter can be calculated as the fraction of the total time spent by the AGN in the “on” state, defined by a luminosity greater than $\ledd/30$ in a given band, over some temporal baseline (Paper III)[4](#fn4). So doing, the duty-cycle turns out to be a small number (Tab. [tab2]); for example, for a temporal baseline ending at the present epoch (i.e., at 13 Gyr in Fig. [f1]), the duty-cycle of model is zero when starting from 9 Gyr (at a redshift z ≈ 0.45, for a flat universe with H0 = 71 km s− 1 Mpc− 1, ΩM = 0.27, ΩΛ = 0.73), as no burst occurs after ≃ 7.5 Gyr; when starting from 6 Gyr (z ≈ 1), it is ≃ 6.3 × 10− 3, 3.2 × 10− 3, and 3.0 × 10− 3 respectively in the bolometric, optical (absorbed), and UV (absorbed) bands. For model, in the same bands, the duty-cycle is respectively ≃ 4.9 × 10− 2, 1.7 × 10− 2, 7.1 × 10− 3 (when starting at 9 Gyr), and ≃ 4.8 × 10− 2, 1.8 × 10− 2, 8.6 × 10− 3 (when starting at 6 Gyr). The duty-cycle decreases from the bolometric, to the optical, to the UV bands, due to the different values of the opacity in these bands. These duty-cycle values are broadly consistent with the fraction of active galaxies measured in observational works. For example, Greene & Ho (2007) estimated the (mass dependent) number of active galaxies, using broad-line luminosities from SDSS DR4, for galaxies with z < 0.352 (age of the universe $\gsim 10$ Gyr); statistically speaking, the fraction of active systems can be interpreted as a duty cycle for MBHs in a given mass range. Greene & Ho report duty-cycle values of the order of 4 × 10− 3 for 107 M⊙ MBHs, declining at increasing mass. Similar duty-cycle values of ∼ 2 × 10− 3, decreasing at increasing MBH mass, are reported by Heckman et al. (2004). The duty-cyles of the models tend to be larger than those observed; however, the comparison is limited by the small number of models considered, and the fact that the only way to compute duty-cycles different from zero is to extend the temporal baseline back in the past. A more consistent comparison with observations can be made with an increased dataset of models, and computing the duty cycle for the last 2–3 Gyrs. Clearly this procedure will reduce the duty cycle, as the models are characterized by a declining nuclear activity. Again, the computed duty cycle would have been reduced significantly also if we had raised the threshold for RIAF-like behavior of the radiative efficiency to l = 0.1. Luminosity and Temperature of the Gas ===================================== We describe here the time evolution of the global thermal luminosity and temperature of the ISM. Luminosity evolution -------------------- The top panels of Fig. [f2] show the time evolution of the total gas emission $\lx$ for models (left) and (right). Red lines refer to the soft (0.3-2 keV) band, and blue lines to the hard (2-8 keV) band; for reference, the scaled-down $\lbh$ (black dotted line) is also shown. The luminosity evolution of the gas for the two models is qualitatively similar, with peaks in $\lx$ coinciding with those in $\lbh$. The sudden increase of the gas emission during outbursts is due to the increase in temperature and density in the central galactic regions ( ≃ 102 − 103 pc), caused by radiative gas heating (Compton and photoionization), and by compression due to direct and reflected shock waves, produced by mechanical and radiative feedback, that are associated with the AGN and the starburst. For short times, most of the luminosity in the peaks of $\lx$ originates from a very small region at the galactic center ( ≈ 100 pc), thus it is observationally indistinguishable from the luminosity of the nucleus (see also Sect. [brilx]). The hard emission oscillates in phase with the soft one, and presents the same overall behavior, but keeps at a level almost 2 orders of magnitude lower. Hard emission during the outburst, as shown in Fig. [f2], would be detectable with Chandra, if centrally concentrated (see also Sect. [brilx] below). However, hard emission during quiescent times would be difficult to distinguish from the contribution of unresolved binaries, even with Chandra, if extended (e.g., Pellegrini et al. 2007, Trinchieri et al. 2008). A comparison of the peaks in $\lx$ and $\lbh$ reveals differences and analogies. While $\lbh$ shows sharp and sudden spikes at the outbursts (increasing by 2 or more orders of magnitude in ≈ 106 − 7 yr), and is almost constant between them, $\lx$ increases slowly but steadily between outbursts, when the galaxy is replenished by the stellar mass losses. The peaks in $\lx$ become broader with increasing time, but not weaker; for example, the increase of $\lx$ during the last major outburst of is the largest one, with the largest amount of gas heated and then removed from the galaxy in its entire life. Instead, when the burst ends, $\lx$ has the same abrupt decrease as $\lbh$, due to the density drop following the final (and usually strongest) sub-burst in each major accretion event. Another similarity is that both $\lbh$ and $\lx$ show sharper and “cleaner” bursts in than in : more radiatively dominated feedback bursts (as in ) are richer in temporal substructure, because it takes longer for the cold shells to be destroyed, thus more star formation and MBH accretion occurs (Paper III). The quiescent values of $\lx$ during the past few Gyr remain at the same level of ∼ 1040 erg s− 1 for model, and decrease by a small factor of $\lsim 2$ to reach a present epoch $\lx=2\times 10^{39}$ erg s− 1 for. Previous compilations of observed $\lx$ values (Fabbiano et al. 1992, O’Sullivan et al. 2001, Diehl & Statler 2007, Mulchaey & Jeltema 2010) for early type galaxies of the local universe, residing in all kinds of environments (from the field to groups to clusters as Virgo and Fornax), show a range of $\lx$ from 1040 erg s− 1 up to 1043 erg s− 1, at a B-band optical luminosity of $\lb =5\times 10^{10}L\_{\rm B,\odot}$ as the model galaxy. $\lx$ values larger than a few × 1041 erg s− 1 belong to galaxies at the center of groups, or clusters, or subclusters, for which an important contribution from the intragroup or intracluster medium, or a confining action, is likely (e.g., Renzini et al. 1993, Brighenti & Mathews 1998, Brown & Bregman 2000, Helsdon et al. 2001). However, the final $\lx$ of and is on the lower end of the range of observed values; this indicates that degassing is important in the models, and for many real cases it must be impeded. In the simulations this could be obtained for example by adding the external pressure from an outer medium (e.g., Vedder et al. 1988), and it will be considered in future works. ### $\lx-\lb$ and $\lx-\lbh$ Real galaxies show a wide range of $\lx$ values, and the observed $\lx$ variation has remained a subject of debate for years (e.g., Fabbiano et al. 1989, Ciotti et al. 1991, White & Sarazin 1991; Pellegrini 2011). Thus, we check here whether the $\lx$ variation in the models during their evolution can (partly) account for the large observed one. For this check we considered the range of hot gas emission for the largest sample of early type galaxies of the local universe (O’Sullivan et al. 2001), after having excluded AGN-dominated cases, and central dominant cluster or group galaxies. Only galaxies with optical luminosites within a range close to that of the model galaxy have been considered (i.e., from logLB = 10.5 to logLB = 10.8, for which there are 43 galaxies). The discrete stellar sources contribution estimated by O’Sullivan et al. (2001) has been removed. The distribution of the observed X-ray emission values so obtained is then compared with that built for the soft X-ray emission of the models, during the epoch from 2 to 12 Gyrs (Fig. [his]). Each emission value enters the histogram with the fraction of the chosen epoch during which it is present; the hypothesis underlying this comparison is that statistically an observed galaxy can be catched in anyone of the different phases shown in the past 2–12 Gyr by the representative models. Figure [his] shows that $\lx$ of the models keeps within the observed range, but it does not exceed significantly ∼ 1041 erg s− 1, while a fraction of galaxies populates this region. Model has a ∼ constant LX in the past few Gyrs, so that it populates mostly a couple of bins; model (that experiences more outbursts) reaches a wider coverage of the observed LX range, but its LX distribution extends more to lower LX values than to larger ones, due to a larger overall degassing. Larger $\lx$ for the models can be obtained when considering that real galaxies of similar $\lb$ can have different values of the central stellar velocity dispersion and effective radius. These differences determine a variety of flow evolution, and then of $\lx$ (Ciotti & Ostriker 2011). Following this idea, Fig. [his] (bottom left) shows the histogram of one possible variation to model, that with σ = 280 km s− 1; this indeed reaches larger $\lx$ values. Finally, the bottom right panel shows the average of the histograms of the three models, and shows how this average reproduces the observed histogram reasonably well. Overall, this analysis shows that the large observed variation in $\lx$ at similar optical $\lb$ has another contributing factor, that is nuclear activity, in addition to those already put forward. Another useful diagram for an observational comparison is given by the relationship between $\lbh$ and $\lx$; this is shown in Fig. [f3], considering all the available temporal outputs, for model. The analogous figure for model is very similar, with the only difference being the broader temporal extension of the bursts. One can recognize the interburst times, when the galaxy is replenished by stellar mass losses, as periods with $\lbh$ almost constant, and $\lx$ increasing (which produces almost vertical lines). During outbursts, the soft $\lx$ and $\lbh$ abruptly increase and then decrease, which produces loops running clockwise on the right of each vertical line; these loops are occupied for a very short time. As time proceeds, the vertical lines (the interburst periods) shift towards lower $\lbh$. The final quiescent period is described by a line (the most crowded with numbers) where $\lbh$ and $\lx$ both decrease, though $\lbh$ decreases faster than $\lx$. The more the outbursts are confined at earlier epochs, the more the final weak correlation between $\lx$ and $\lbh$ extends with time; instead, the more the outbursts extend towards the present epoch, the more a trend is expected for $\lx$ to increase with $\lbh$ with a large scatter around it (as produced by the vertical lines, where galaxies reside most of the time, and by the loops). For a sample of early type galaxies of the local universe, the relationship between $\lx$ and the nuclear emission $L\_{\rm X,nuc}$ indeed shows a weak trend, dominated by a large scatter (Pellegrini 2010); in the present framework, such an observation could be explained with many galaxies being still in the phases made of the vertical rise followed by the loop. The comparison cannot be pushed farther, though, since the $\lbh$ values – when converted to an X-ray band – are typically larger than the observed $L\_{\rm X,nuc}$ (Sect. [nuc]). Temperature evolution and $\lx-\ta$ ----------------------------------- Another important global property of the ISM typically observed is the emission weighted aperture temperature $\ta$ (eq. [ta]), here calculated within the optical $\re$, and within $10\re$. The time evolution of these temperature diagnostics is shown in the bottom panels of Fig. [f2], in parallel with the luminosity evolution; red and blue lines again refer to the 0.3–2 keV and 2–8 keV bands, respectively. Note that $\ta(10\re)$ in the two bands is indistinguishable from the corresponding global emission weighted temperature $<\tx>$ (that is calculated but not shown in Fig. [f2]), as expected given that the density profile is steeply decreasing outward (Paper III); thus $\ta(10\re)$ is a good proxy for $<\tx>$. Also, temperatures computed for the whole 0.3–8 keV band (not shown) are always very close to those weighted with the 0.3–2 keV emission, except for those short burst times during which there is a very hot gas component. As for $\lx$, also for $\ta$ the temperature peaks of are significantly more structured in time than those of. Three main characteristic features of $\ta$, common to this class of models, can be pointed out. The first is the complex behavior of $\ta$ during outbursts, with variations going in opposite directions for the two bands. This is due to the coexistence of hot (the central bubble) and cold (the radiative shells) ISM phases during the bursts, as revealed by the temporal evolution of the radial profiles of the gas density and temperature (Paper III). The sharp and high peaks in the hard band (blue) correspond to the onset of very hot regions at the center, while the decrements in the soft band (red) are due to a dense cold shell created immediately before the major burst, and to cold gas accumulated by the passage of radiative shock waves produced by the outburst (see also Sect. [tempx]). A second, observationally relevant feature is that in each band $\ta(\re)$ is higher than $\ta(10\re)$, which is especially evident in the interburst periods (Fig. [f2]). For example, in the quiescent phases, both models show a similar $\ta(10\re)\simeq 0.4-0.5$ keV in the soft band, while $\ta(\re)\sim 0.7$ keV. This is explained by the radial temperature distribution decreasing outward (Sect. [tempx]). Finally, for model Fig. [uff] shows the relationship between $\lx$ and $\ta(10\re)$, both calculated for the 0.3–8 keV band; such a diagram is often produced in observational works, for galaxies of all σ (e.g., Pellegrini 2011). During the long interburst epochs, $\lx$ increases with little variation of $\ta$. In the short burst episodes, $\lx$ and $\ta$ reach a maximum, and then follow a clockwise loop on the right, reach back the original $\ta$ value, move left of it, and follow a counterclockwise smaller loop. Then the cycle repeats with $\lx$ increasing and $\ta$ almost constant. During the last few Gyr, $\lx$ remains at ∼ 1040 erg s− 1 and $\ta \sim 0.5 $ keV. We now compare these results with observations. In the largest sample of global, soft X-ray emission weighted temperatures, derived from ROSAT observations, $<\tx>$ is in the range ∼ 0.4 − 0.8 keV, for galaxies with σ ≃ 260 km s− 1, as for the model galaxy (O’Sullivan et al. 2003); $\ta(10\re)$ of the models during quiescence, in the soft band, is within this range, though on its lower end. However various factors tend to bias-high these observed temperatures, as incomplete subtraction of the hard stellar emission due to binaries, or hard AGN emission; in addition, many of the sample galaxies reside in high density environments, with possible contamination from the hotter group/cluster medium, and a temperature profile that is commonly rising outwards (e.g., Diehl & Statler 2008, Nagino & Matsushita 2009), a behavior opposite to that of the models, that refer to isolated galaxies (Sect. [tempx]). Chandra observations, with large sensitivity and much higher angular resolution ( ∼ 1′′), allowed for an accurate subtraction of all the AGN and stellar sources contributions from the total emission, giving measurements of the gas temperature of unprecedented accuracy (e.g., Boroson et al. 2011). For example, Boroson et al. derived global, 0.3–8 keV emission weighted temperatures, for a few galaxies with σ ∼ 260 km s− 1, ranging from 0.3 to 0.6 keV; $\ta(10\re)$ of the models agrees very well with this result, falling in the middle of the observed range. Coming to temperature estimates for more central regions, Athey (2007) derived 0.35–8 keV emission weighted temperatures within $\re$, for 53 galaxies with Chandra observations. For a selection of 20 galaxies with $\lb$ similar to that of the model galaxy, from log $\lb (L\_{B,\odot})=10.5$ to 10.8, the average temperature is 0.61 ± 0.03 keV (calculated weighting each measurement with its uncertainty). Figure [hisT] shows a full comparison between the distribution of these temperatures and $\ta(\re)$, calculated for the 0.3–8 keV band, during the evolution of and. The model $\ta(\re)$ tends to be concentrated in the upper half of the observed distribution; this result, if confirmed with a larger set of simulated galaxies, indicates that heating in the central galactic region of the models may be too efficient. In conclusion, the global temperatures of the models fall in the middle of the range of values recently observed, while the model $\ta(\re)$ reproduces easily the larger observed values and less easily the lower ones. Projected quantities: temperature and brightness profiles ========================================================= We consider here the radial profiles of the ISM temperature and surface brightness, as they would be revealed for the models by observations. For simplicity we restrict the discussion to model, whose sharp bursts allow for an easier presentation; the results are substantially similar for. The profiles are constructed using Eqs. [sig]–[conv], both during the quiescent phases, that occupy most of the ISM evolution, and during an outburst. The recurrent burst phases are temporally limited, but represent a central aspect of the models, thus they are devoted special attention. Statistically, the feedback features should be present, and possibly revealed by current X-ray observations, in ≃ 5 − 10% of the isolated galaxies with $\lb$ similar to that of the models (Sect. [brilx] below). In the following we present snapshots of the projected profiles during quiescence, at an age of 3, 6.5, 9 and 12 Gyr, and centered on the last outburst at 7.5 Gyr. Temperature profiles -------------------- The emission weighted projected temperature profiles $\tp(R)$ during quiescent interburst times are smooth, with the temperature monotonically decreasing for increasing radius (Fig. [f5]). From an age of 6 Gyr onwards, the temperature keeps at ≃ 1 keV at a radius of ≃ 100 pc, and at ∼ 0.4 keV at a radius of ≃ 30 kpc. Figure [f5] (right panel) shows the corresponding aperture temperature profiles $\ta (R)$ calculated in bins (i.e., in Eq. [ta] the numerator and denominator are evaluated over radial bins), chosen to reproduce the typical binning used for Chandra observations of nearby galaxies (Humphrey et al. 2006, Diehl & Statler 2007). Temperature profiles with negative gradients had already been found for gas inflows in steep galactic potentials without feedback, due to compressional heating (e.g., Pellegrini & Ciotti 1998). As typical of models without a central MBH, in that case the temperature also had a central drop. In the present computations, instead, the combined effects of the gravitational field of the MBH, and of the high injection temperature of the stellar mass losses (a consequence of the stellar velocity dispersion that is enhanced by the presence of the MBH, within its sphere of influence), keep the gas temperature increasing approaching the center, even outside the burst episodes (see also Pellegrini 2011). A temperature profile that keeps smoothly decreasing towards the center for a long time, as expected in classical cooling flows, is never shown by the model runs. With time increasing, the value of the central temperature does not increase significantly after 6 Gyr, since it is mainly determined by the gravitational effects of the MBH, whose mass remains approximately constant. The external $\tp(R)$ values instead steadily increase with time, since they are determined by the SNIa’s heating; the latter has a secular trend that produces a time-increasing specific heating of the mass return rate (i.e., $L\_{\rm SNIa}/\dot M\_\\propto t^{0.2}$, where $L\_{\rm SNIa}$ is the energy injected per unit time by SNIa’s supernova explosions, and $\dot M\_\$ is the stellar mass loss injected per unit time; e.g., see Pellegrini 2011). We now focus on the evolution during the last outburst (Fig. [f6]). As typical, starting from the unperturbed profile, a shell of denser gas creates, in this case at a radius of ≃ 800 pc, particularly evident as a dip in the otherwise monotonically decreasing profile (see the -2 Myr red line in the top left panel). The shell falls towards the center, progressively cooling, so that the soft X-ray emission weighted temperature decreases (Fig. [f2], bottom panels). A first AGN burst is produced before the shell reaches the center, and the resulting outward moving shock empties and heats the gas in the inner regions of the galaxy, on a time scale of ≈ 1 Myr. The subsequent snapshot in Fig. [f6], at +6 Myr after the first burst, shows the high central temperature typical of the outburst phase: $\tp(R)\sim $few keV within a radius of ∼ 50 − 100 pc. When the shock enters the radiative snow-plow phase, the associated secondary dense shell interacts with the first one still falling, and a series of direct and reflected shock waves are produced, accompanied by accretions events and star formation. The profiles between -2 Myr and +18 Myr (not shown) are very irregular, and consisting of a series of dips propagating outwards, with the temperature in the central region quickly and alternately increasing and decreasing. As already noticed for Fig. [f2], both hotter and colder regions are continuously created, the hottest ones within ∼ 1 kpc (mainly due to shocks), and the coldest ones due to the cooling gas in the shells. Each sub-burst is stronger than the previous one; finally, a last shock from the center concludes the phenomenon, halting star formation and leaving behind a very hot nucleus (the +66 Myr red line), while emptying the rest of the galaxy [see also the Σ(R) profile at +66 Myr in Fig. [f11]]. Then, during the following ≃ 100 Myr, the gas resumes the temperature typical of quiescent times (the +202 Myr green line). The inner very hot phase lasts for $\lsim 0.1$ Gyr. During the bursts, cosmic rays are shock-accelerated, and the inner regions look similar to a gigantic supernova remnant. Only thermal X-rays are considered here, while those due to synchrotron emission from accelerated particles due to shocks were not computed (see Jiang et al. 2010). The profiles in the right panels of Fig. [f6] show the binned aperture temperature profiles, for the same times as for the left panels. The binning smears the small-scale features, but the major distinctive characteristics, as the temperature dip when the first shell forms, and the large temperature drop outside the center, at +18 Myr and +66 Myr, are still well detectable. ### Comparison with observed profiles Negative radial gradients, as shown by the model temperature during quiescence (Fig. [f5]), are common among ellipticals, as revealed most recently by Chandra observations (e.g., Kim & Fabbiano 2003, Humphrey et al. 2006, Sansom et al. 2006, Fukazawa et al. 2006, Diehl & Statler 2008b, Nagino & Matsushita 2009). In a large collection of temperature profiles (Diehl & Statler 2008b, Athey 2007), those cases with negative gradients show a temperature that roughly halves from the innermost bin (that in general extends out to a radius of 0.5-a few kpc from the center) to the outer galactic region; this is roughly the same behavior shown by the models. Observed temperatures range between 0.5 and 1 keV at the innermost radial bin, where the model temperature is 0.8-1 keV (Fig. [f6], right panels). Interestingly, the galaxies with a negative gradient reside in the field or in less dense environments, as the outer Virgo regions (Matsushita 2001, Fukazawa et al. 2006, Diehl & Statler 2008b), and all have[5](#fn5) $\lx<$few × 1040 erg s− 1, both characteristics that match those of the models. More complex temperature profiles are also common, and could correspond to phases of AGN activity. For example, the so-called “hybrid” profiles show a central negative gradient, until the temperature reaches a minimum, and an outer positive gradient; for example in NGC1316, NGC4552, NGC7618 the temperature has a minimum of ∼ 0.4-0.5 keV at ∼ few kpc, and rises both going towards the center (of ∼ 0.2-0.3 keV), and going outwards (Diehl & Statler 2008b). Also in the models, after each major burst, when the last shock is moving outwards and fading, leaving a hot, rapidly cooling core, there is a drop in the temperature profile at a radius of ∼ 1 kpc or more (Fig. [f6], bottom panels, black and red profiles). However, in the models the temperature reaches > 1 keV at the center, larger than the observed values (the comparison also depends on the binning used for the profiles, though). Interestingly, preliminary results from 2D simulations (Novak et al. 2010), show lower central temperatures than in Fig. [f6], during outbursts, due to the fragmentation of the cold shell, that causes a lower gas compression while falling to the center. Other observed profiles keep roughly isothermal, or are roughly flat out to $\sim\re$ and then increase outward (e.g., Humphrey et al. 2006, Diehl & Statler 2008b, Nagino & Matsushita 2009). Such a positive outer gradient is shown by galaxies in high-density environments, suggesting the influence of circumgalactic hot gas; also, these galaxies have typically a larger hot gas content than the models. Environmental confining effects, currently not included in the models, are expected to increase the gas luminosity, and produce outer postitive gradients (see, e.g., Sarazin & White 1987, Vedder et al. 1988). It has been proposed that galaxies could behave according to a scenario where weak radio AGN distribute their heat locally and host negative inner temperature gradients, whereas more luminous radio AGNs heat the gas more globally through a jet or rising bubbles, and produce a flat profile, or a positive gradient (Diehl & Statler 2008b). Our models during quiescence could correspond to the weak AGN phase; also, after an outburst, the temperature profile can be flattish (excluding the innermost bin) out to a few kpc, and show a positive gradient outer of this radius (see the red line of Fig. [f6], bottom right panel). However, without a confining environment, an exploratory investigation conducted by us shows that the kinetic heating of a bright radio AGN could cause a major degassing, rather than just a reversal of the temperature gradient from negative to flat or positive. Likely, a gas rich environment providing confinement for the galactic coronae is a necessary condition to observe a bright (extended) radio source, and the confinement in turn also helps produce a positive temperature gradient in the outer galactic regions. Both aspects (the jet and a dense environment) will be implemented in the models in the future. In conclusion, the addition of a jet to the simulations could have positive effects, if it heats the gas outside $\sim\re$, and reduces the accretion rate and then $\lbh$ during the stationary hot accretion phase (Sect. [nuc]); it should not increase the temperature at the center, though, since this is already on the upper end of those observed even during quiescence. Brightness profiles ------------------- Figure [f9] shows the evolution of the X-ray surface brightness profile of the gas Σ(R) for, at the same quiescent times of the temperature profiles in Fig. [f5], for the soft (0.3-2 keV) and hard (2-8 keV) bands. In the hard band, Σ(R) is always comparable to or lower than a profile representing the unresolved stellar emission due to low mass X-ray binaries, calculated for a deep ($\gsim 200$ ksec) Chandra pointing of a galaxy of the same optical luminosity as and distant $\lsim 20$ Mpc. Thus, hard emission during quiescent times would be difficult to distinguish from the contribution of unresolved binaries, even with Chandra. In both bands Σ(R) becomes flatter in shape with time increasing, since the emission level decreases mostly in the central galactic regions (within $\re$). This important effect is produced by the nuclear outbursts, that remove gas from the center. In analogy with the discussion of the temperature profiles, Fig. [f11] shows Σ(R) just before, during, and after the last major nuclear burst at 7.498 Gyr, with times counted from the first accretion event. At -2 Myr the formation of the off-center cold shell produces the characteristic feature of a sharp decrease of Σ in the hard band, and a bright rim in the soft band. The subsequent curves show the shock moving outwards after the major burst, and the presence of very hot gas at the center, revealed by the central peak in the hard band. The disturbances in the Σ profiles due to the shells launched by the repeated sub-bursts are much more visible in the soft than in the hard band, as particularly apparent at +18 Myr. At this time, note the remarkable presence of a hot center surrounded by a denser and colder shell, producing a sharp peak in Σ(R), at R = 600 − 700 pc, and a sharp dip in $\ta(R)$ in Fig. [f6]. The final two times (+66 Myr and +202 Myr) show the result of the degassing caused by the passage of the shock waves: the gas density is low, and subsonic perturbations remain at a radius of $\gsim 10$ kpc. A different representation of the profiles during the burst phases is given by Fig. [f12], where $\Sigma\_{\rm UM}(R)$ resulting from the unsharp masking procedure (Eq. [conv]) is shown. Fluctuations in brightness that can be clearly distinguished are that due to the cold shell (time -2 Myr), and, after the burst, the negative off-center region in $\Sigma\_{\rm UM}(R)$ delimited by a sharp positive peak (times +6, +18, +66 Myr); the latter resembles what is commonly called a hot gas “cavity”. Note that, when implemented in 2D simulations, the same input physics adopted for the present class of models gives conical hot gas outflows from the nucleus, during outbursts: hot gas is ejected along the symmetry axis, so that elongated “cavities” (i.e., regions with a gas density decrement) are created (Novak et al. 2010). In addition, in their study of a model very similar to, Jiang et al. (2010) found after a burst taking place at 6.5 Gyr, a cavity filled with radio emitting particles of ∼ 4.4 kpc in size, detectable during the first ∼ 10 Myr after the burst. Therefore, we expect that cavities in the hot gas, such as those seen as off-center minima in the profiles of Figs. [f11] and [f12] at times from 6 to 66 Myr after the burst, should be fairly bright in the radio and would have, in X-rays, two essentially hollow conical lobes. They should even show non-thermal X-ray emission of the type seen in the Crab nebula (Jiang et al. 2010). Finally, taking at face value the results of the present analysis and the estimates on the radio detectability of Jiang et al. (2010), hot gas cavities seem more long-lasting than their radio detectability. Coming to the observability of the predicted features, one major property of the models is the decrease of Σ(R) in the central galactic regions, produced by the nuclear outbursts, with respect to models without feedback. Interestingly, bright ellipticals imaged with Chandra (e.g., Loewenstein et al. 2001) show a brightness profile that is quite flat within the central ∼ 1 kpc, a feature impossible to reproduce with pure inflow models, while it resembles the profile of the “pre-burst” phase (black lines in Fig. [f11], left panels), or at the end of a burst (+202 Myr). To better illustrate this point, Fig. [f10] shows the different shape of Σ(R) for and for a model with the same $\lb$, σ and $\re$, but without AGN feedback, and the sole SNIa’s heating included. The two Σ(R) profiles considered refer to the present quiescent epoch, yet the difference in steepness is clear. Figure [f10] also shows the observed Σ(R) for an elliptical at the periphery of the Virgo cluster, with $\lx$ close to that of ; the agreement between model and observation is very good. Another major prediction is given by the disturbances in the profile during an outburst; these keep above the level of the unresolved stellar emission, for a deep observation of a nearby galaxy with Chandra, as shown by Fig. [f11]. The central spike in Σ(R), during the high-temperature and high-density phase at the center, is confined within ∼ 100 pc, and then likely to remain a central unresolved feature even in galaxies observed at the high angular resolution of Chandra. Disturbances as shells and ripples farther out in the galaxy last $\lsim 0.2$ Gyrs, and are more likely to be observed. Given the typical duration of these features, and the presence of 3–4 major outbursts during the whole evolution, statistically they should be present, and possibly revealed by current X-ray observations, in ≃ 5 − 10% of the galaxies with $\lb$ similar to the model galaxy. In fact, many nearby galaxies show a disturbed appearance, as most recently revealed by studies based on Chandra data (e.g., Finoguenov & Jones 2001, Forman et al. 2005, Soria et al. 2006, Diehl & Statler 2007, Nulsen et al. 2009, Baldi et al. 2009, Dunn et al. 2010). The ISM morphology of a large set of galaxies (54) shows a level of disturbance correlated with the radio luminosity derived by the NRAO VLA Sky Survey (thus including both pointlike and extended radio emission; Diehl & Statler 2008a). Also, many of the best studied gas-rich galaxies show decrements in the X-ray surface brightness map, identified as cavities formed when AGN jets inflate radio lobes and displace surrounding gas; in many cases the cavities are filled with radio plasma, and surrounded by armlike features, sometimes classified as shocks. The hot gas disturbances have then been generally attributed to jet activity. As discussed above, we expect that if we took two cones from our solutions, they would give “lobes”. These would produce shocks at the edges and the lobes would be filled with radio emitting particles. There are also a few galaxies without currently evident extended radio emission, but with signs of an outflow and hot central gas, as NGC4552 (Machacek et al. 2006). This galaxy shows a weak core radio source unresolved by the VLBA, and in the (unsharp masked) X-ray image two conspicuous ringlike features at 1.3 kpc from the galaxy center, surrounding two cavities; these features have been found consistent with shocked gas driven outward by recent nuclear activity, as in a bipolar nuclear outflow. The gas temperature in the central ∼ 100 pc of the galaxy is 1 ± 0.2 keV, hotter than elsewhere in the galaxy, suggesting that we may be directly observing the reheating of the galaxy ISM by the outburst (Machacek et al. 2006). These characteristics resemble those predicted by the models for the temperature and the surface brighntess during the afterburst phase. Discussion and conclusions ========================== The hot gas properties of massive ellipticals, with regard to luminosity and temperature, and their spatial distributions, allow us to derive insights into the hot gas evolutionary status, and its link with the host galaxy. Since the cooling times are short compared to the galaxy age, it is now commonly accepted that repeated cooling catastrophes have occurred in the past, accompanied by central starbursts and AGN outbursts. The interest of a better understanding of this phenomenon is obvious, as it is directly linked to galaxy formation and evolution, and to the growth of the central MBH. Unfortunately, a complete theoretical picture is still missing, so that modeling coupled with a close comparison with observations is crucial in this field. In fact, these feedback events must leave signatures on the X-ray properties of the galaxies; indeed, the observed temperature and brightness profiles often cannot be fit easily with smooth profiles, or cooling flow profiles (e.g., Sarazin 2011, Statler 2011). In this work we have calculated the observational properties in the X-ray band of two galaxy models, representative of a class of detailed simulations of physically based models for the investigation of feedback modulated accretion in isolated galaxies. The feedback physics includes the combined effects of radiation and AGN winds (Paper III). The observational properties derived provide good matches to what observed in general for the local universe, and account for a few otherwise puzzling observed properties; on the other hand, they also evidence the need for changes or additions to the input physics. The main results of the present investigation are the following: 1) After an evolution of ≃ 10 Gyr, the models are typically in a permanent quiescent phase. The bolometric nuclear emission is very sub-Eddington (l ≃ 10− 4), within the range observed for l, though the most frequently observed values (l ≈ 10− 5 or less) are somewhat lower. Unfortunately, uncertainties in the bolometric correction to apply to observed nuclear luminosities, appropriate for a spectral energy distribution at low emission levels, do not allow to make a stringent comparison between modelled and observed values. However, also the nuclear X-ray emission $L\_{\rm BH,X}$ of the models, estimated as $L\_{\rm BH,X} \lsim 0.2\lbh$ as should be appropriate for low luminosity nuclei, and $L\_{\rm BH,X}/\ledd$, tend to lie on the upper end of what is observed. Thus in real galaxies an additional mechanism may be required to reduce further the mass available for accretion; this could be provided by the mechanical feedback of a (nuclear) jet, and/or by a thermally driven wind from a RIAF. Alternatively, the switch from disc to RIAF behavior (that is $l\propto \dot m^2$) should occur at a larger l than assumed here (e.g., at l ≃ 0.1 rather than l ≃ 0.01). Simulations of ram-pressure stripping effects, instead, showed that a reduction in the accretion rate is not attained, because in the quiescent hot accretion regime, the accreting mass comes mainly from the stellar mass losses within the central ∼ 100 parsecs of the galaxy, that are not affected by stripping (Shin et al. 2010b). 2) The X-ray luminosity of the ISM oscillates in phase with the nuclear luminosity, though with much broader peaks; at the present epoch, $\lx$ lies at the lower end of the large observed range for galaxies of $\lb$ similar to that of the models. The degassing/heating in the models may then be too efficient, or a larger/more concentrated gravitating mass, or a confining external medium, are needed. However, when the gas luminosities during the whole evolution are considered, the observed $\lx$ range is better reproduced. This is even more true if also an additional model of larger σ, and same $\lb$, is included. Part of the observed large variation in $\lx$ for galaxies of a given $\lb$ could then be explained by nuclear activity. 3) The average ISM temperature is within the observed large range for the model σ; when estimated within $\re$, the model temperatures reproduce better the upper half of those observed. Modifications to the models by the addition of a jet or an external medium, as suggested in the previous points, should then not increase the average temperature within $\re$. 4) During quiescence, the profiles of the gas temperature and brightness resemble those observed for many local galaxies. Especially remarkable is the lack of the steep brightness profile shape typical of inflowing models, due to the frequent removal of gas from the galactic central regions, and to the heating provided by mechanical feedback (that is always present, even during quiescent phases). The models show negative temperature gradients, that are common for isolated galaxies; the addition of a jet or a confining agent should change the temperature profiles into a flat or outwardly increasing profile, as also frequently observed for galaxies in rich environments. 5) During outbursts, disturbances are predicted in the temperature and brightness profiles; the ISM resumes the smooth appearance of steady and low-luminosity hot accretion on the MBH on a time-scale of $\lsim 200$ Myrs. The most conspicuous variations with respect to smooth profiles are within current detection capabilities, and could correspond to (part of) the widespread disturbances observed in galaxies of the local Universe. In particular, shocked hot gas should be seen at the galactic center (within ∼ 100 pc), possibly not resolved; this would be a certain sign of prior AGN activity. These hot bubbles could be revealed by emission of cosmic-rays, in a structure similar to a gigantic supernova remnant. Preliminary results from 2D runs show bipolar nuclear outflows, that should be seen as conical cavities extending from the galactic center, and may be called jet-like features. 6) The duty-cycle of nuclear activity is of the order of a few × (10− 3 − 10− 2), depending on the assumed mechanical feedback efficiency; in general, a burst cycle lasts for ≈ 107 yrs. These duty-cycle values are broadly consistent with the fraction of active galaxies measured in observational works, though reported values for the local universe are somewhat lower, for the MBH mass of the models. In order to make a more consistent comparison with observations, the dataset of models should be increased, and the duty cycle computed only for the last 2–3 Gyrs; this will reduce the duty cycle, as the models are characterized by a declining nuclear activity. 7) The duty-cycle of perturbances in the ISM is of the order of 5-10%, from their average number and duration. This duty-cycle likely increases with galaxy mass, because an outburst has a greater impact in less massive (and less gas-rich) systems, which then are “on” for a shorter time (Ciotti & Ostriker 2011). The ISM duty-cycle, and its trend with galaxy mass, compare reasonably well with preliminary estimates obtained from a large sample of hot gas coronae in elliptical galaxies observed with Chandra (Nulsen et al. 2009): the fraction of galaxies with X-ray cavities in the hot gas is $\lsim 10$% when $\lx<10^{41}$ erg s− 1 (as for the models), and reaches ∼ 25% in the most luminous ones. The presence of cavities has been attributed to the action of jets inflating radio lobes and displacing the surrounding gas. Cavities can also be created in the scenario presented here, as hinted for by 2D simulations, due to bipolar nuclear outflows. 7) Two diagnostic planes have been constructed. In the first one, the nuclear luminosity $\lbh$ and the ISM luminosity $\lx$ are followed during the whole model evolution. The points representative of the models populate a wedge region, that should then be occupied when observing a large set of galaxies. Another plane shows the evolution of $\lx$ versus the average gas temperature $\ta$; here the most populated region is that of a large $\lx$ variation (factor of ∼ 10) for $\ta$ keeping between 0.4 and 0.6 keV. Clearly, a larger set of models is to be explored, in order to better establish the final gas properties (as gas content, nuclear and ISM duty-cycle, etc.) to be compared with those of a statistically large sample. 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M., Konidaris, N., Koo, D., Davis, M. 2006, ApJ 648, 281 Yu, Q., Tremaine, S. 2002, MNRAS 335, 965 [modref] \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Model \| $\epswM$ \| $<\epsw>$ \| $<\eps\_{\rm EM}>$ \| $\log \Delta\mbh$ \| logΔM\* \| $\log \Delta M\_{\rm w}$ \| $\log \mgas$ \| $\log \lbhefopt/\ledd$ \| \| (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (8) \| (9) \| \| \| 10− 3 \| 2.0 10− 5 \| 0.105 \| 8.74 \| 9.74 \| 10.27 \| 9.68 \| -5.13 \| \| \| 3 × 10− 4 \| 1.2 × 10− 5 \| 0.133 \| 9.05 \| 10.22 \| 10.31 \| 9.34 \| -5.43 \| [tab2] \| \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Model \| $\mbh$ \| log$\lbh $ \| l \| $L\_{\rm BH,X} $ \| $L\_{\rm BH,X}/L\_{Edd}$ \| \| duty cycle \| \| $\log L\_{\rm X} $ \| \| \| (M⊙) \| (erg s− 1) \| (10− 4) \| (erg s− 1) \| \| Bol \| Opt \| UV \| (erg s− 1) \| \| (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (7) \| (7) \| (8) \| \| \| 8.4 × 108 \| 43.39 \| 2.0 \| $\lsim 5\times 10^{42}$ \| $\lsim 2\times 10^{-5}$ \| 6.3 × 10− 3 \| 3.2 × 10− 3 \| 3.0 × 10− 3 \| 40.1 \| \| \| 1.4 × 109 \| 43.38 \| 1.0 \| $\lsim 5\times 10^{42}$ \| $\lsim 2\times 10^{-5}$ \| 4.8 × 10− 2 \| 1.8 × 10− 2 \| 8.6 × 10− 3 \| 39.6 \| -0.8truecm [f1] -1truecm -3truecm [f2] 2truecm [his] -1 truecm -9.2truecm [f3] -1 truecm -9.2truecm [uff] -5truecm -1truecm -3.5truecm [hisT] -6.truecm -1. truecm -3.5truecm [f5] -4.7truecm -0.9truecm -3.5truecm -4.8truecm -0.9truecm -3.6truecm [f6] -0.3truecm [f9] -1.5truecm -14.2truecm 8truecm [f11] -1.7truecm -14.2truecm 7truecm [f12] -6truecm -1truecm [f10] --- 1. In the code the gas is allowed to cool down to a minimum temperature of 104 K.[↩](#fnref1) 2. A similar fraction ( ∼ 52%) of the sample of red sequence galaxies of the SDSS (r < 17.77, median redshift z = 0.1) have detectable line emission (Yan et al. 2006).[↩](#fnref2) 3. The model evolution of course also changes, because of the smaller direct radiative feedback, and the smaller indirect mechanical feedback, that in the models is a function of $\lbh$. The bursts are more extended in time, with a larger accreted MBH mass, which reduces slightly also the final l ratio.[↩](#fnref3) 4. Alternatively, the duty-cycle can be obtained as a luminosity weighted average over a chosen time interval, with very similar results.[↩](#fnref4) 5. Except for NGC6482, the remnant of a fossil group.[↩](#fnref5)	arxiv_0000696
Singular measure traveling waves in an epidemiological model with continuous phenotypes ======================================================================================= We consider the reaction-diffusion equation ut = uxx + μ(∫ΩM(y, z)u(t, x, z)dz − u) + u(a(y) − ∫ΩK(y, z)u(t, x, z)dz), where u = u(t, x, y) stands for the density of a theoretical population with a spatial ($x\in\mathbb R$) and phenotypic ($y\in\Omega\subset \mathbb R^n$) structure, M(y, z) is a mutation kernel acting on the phenotypic space, a(y) is a fitness function and K(y, z) is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation, and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $ c^\:=2\sqrt{-\lambda\_1} $, where λ1 is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation. Introduction ============ In this work we consider the reaction-diffusion equation: u*t = uxx + μ(M ⋆ u − u) + u(a(y) − K ⋆ u), where t > 0, $x\in\mathbb R$, y ∈ Ω for a bounded domain $\Omega \subset \mathbb R^n$, u = u(t, x, y), μ > 0 is a positive constant, a = a(y) is a continuous function, M = M(y, z) and K = K(y, z) are integration kernels, and the ⋆ operation is defined by. After discussing the existence of stationary states for, we construct measure-valued traveling waves and show the existence of a singularity for a subclass of parameters. Equation describes an asexual population living on a linear space, represented by the variable x. Several genotypes exist in the population, yielding a continuum of phenotypes, represented by the y variable. We denote $ \Omega\subset \mathbb R^n $ the set of all reachable phenotypes. Our basic assumption is that the fitness (or intrinsic growth rate) of each individual is a function a(y) of its phenotype. We also assume the existence of an underlying mutation process, by which an individual of phenotype z ∈ Ω may give birth to an individual of phenotype y ∈ Ω, with probability M(y, z). Such mutations are expected to occur at rate μ > 0. Finally, the individuals are in competition for e.g. a finite resource, and we denote K(y, z) the cost on the fitness of y caused by the presence of z. In the context of epidemiology, u(t, x, y) can be thought as a density of hosts at point x, infected with a pathogen of trait y. Equation is particularly relevant in this context, since evidences suggest that pathogens (like e.g. viruses ) can be subject to rapid evolution, which may then occur at the same time scale as the propagation of the epidemic. Moreover, equation can easily be derived from a host-pathogen microscopic model in which we neglect the influence of the pathogen on the hosts’ motility. The study of asymptotic propagation in biological models can be traced back to the seminal works of Fisher and Kolmogorov, Petrovsky, and Piskunov, who investigated simultaneously the equation: ut = uxx + u(1 − u), where u = u(t, x) stands for the density of a spatially structured theoretical population. They have shown, in particular, that for any compactly supported initial condition, the solution u(t, x) invades the whole space with constant speed c = 2 (such a result is often called spreading); and that there exists a particular solution to, which consists of a fixed profile shifting along the axis at speed c, $ u(t,x)=\tilde u(x-ct) $, and connecting the unstable state 0 near + ∞ to the stable state 1 near − ∞ (such a particular solution is called traveling wave). Since then, these results have been generalized to a variety of related models: see e.g., and the references therein. In the last decades, there has been an increasing interest in propagation models that take into account a multiplicity of different species. The main problems in the field include the replacement of a species by competitive interaction (see e.g. ), predation, adaptation to climate change, or cooperation. This last class of cooperative reaction-diffusion system has lead to particularly strong results, since its properties are somewhat comparable to those of scalar equations. In a recent work, the authors investigated the existence of traveling waves in the spatially homogeneous epidemiological model: $$\label{eq:Gri-Rao} \begin{system}{rcl} w\_t&=&w\_{xx}+w(1-(w+m)) + \mu(m-w) \\ m\_t&=&m\_{xx}+rm\left(1-\frac{w+m}{K}\right)+\mu(w-m), \end{system}$$ where w and m stand for a density of hosts infected by a wild type and mutant pathogen, respectively. Though this system is not globally cooperative, the authors managed to prove the existence and to compute the minimal speed of traveling waves as a function of the principal eigenvalue λ of the associated principal eigenvalue problem: $$\left(\begin{matrix} 1-\mu & \mu \\ \mu & r-\mu\end{matrix}\right){\left(\begin{matrix}w \\ m \end{matrix}\right)} + \lambda{\left(\begin{matrix}w \\ m \end{matrix}\right)}=0,$$ via the formula $ c=2\sqrt{-\lambda} $. Intuitively, the spatial dynamics is then guided by the linearized system far away from the front (such a traveling wave is sometimes called a pulled front ). Since then, these results have been extended to a more general class of systems in. Equation can be seen as the continuous limit of system with a large number of equations. Since we aim at computing the propagation speed for this equation, we turn to the associated principal eigenvalue problem: μ(M ⋆ u − u) + u(a(y) + λ) = 0. This problem has been investigated in and, where the author shows an unexpected concentration phenomenon occurring for very natural fitness functions: if $$\label{eq:intro-cond-conc} \frac{1}{\sup\_{z\in\Omega} a(z) - a(y)}\in L^1(\Omega),$$ and μ is small enough, there exists no continuous eigenfunction associated to, but rather singular measure eigenvectors with a singularity concentrated on the maximum of fitness $ \Omega\_0:=\{y\in\overline\Omega\,\|\, a(y)=\sup\_{z\in\Omega}a(z) \} $. According to, this phenomenon happens when a(y) is sufficiently steep near its global maximum, and is highly dependant on the Euclidean dimension of Ω. For instance, if n = 1, a concentration may appear at the optimum y = 0 for the particular fitness function $ a(y)=1-\sqrt{\|y\|} $, when a(y) = 1 − ∣y∣ always yields continuous eigenfunctions; if n = 2, a(y) = 1 − ∣y∣ may induce concentration, but a(y) = 1 − ∣y∣2 cannot. In dimension n = 3 or higher, smooth fitness functions such as a(y) = 1 − ∣y∣2 may induce concentration. A similar phenomenon, and in particular the critical mutation rate under which concentration appears for a sufficiently steep fitness function, has been discussed by Waxman and Peck. The nonlocal competition term − K ⋆ u(y) in is quite standard in models involving competition between different phenotypes. Many models focus on the case where the competition is simply the integral of the distribution — this corresponds to K(y, z) = 1. As an example, the nonlocal Fisher-KPP equation ut − Δu = μu(1 − Φ \* u), where Φ(y) is usually in $L^1(\mathbb R^n) $ with possibly additional restrictions, has attracted a lot of attention in the past. Nonlocal competition also appears in numerous other studies in population genetics and population dynamics. In general, the qualitative behavior of traveling waves, and the long-time behavior of the solutions to the parabolic equation, are still difficult to handle. Recent advances have been made towards a better understanding of the asymptotic location of the front for the solutions to the parabolic equations, see for the nonlocal Fisher-KPP equation; for the cane toads equation. In the case of the nonlocal Fisher-KPP equation, the existence of traveling waves has been established and the associated minimal speed characterized in. The convergence towards a stationary state on the back of the wave, has been shown in for small μ or when the Fourier transform of the competition kernel is positive (in which cases one can prove the stability of the constant steady state u ≡ 1); and more recently, in a perturbative case, the convergence in long time has been shown for solutions to the parabolic equation. In the general case, the convergence towards a stationary solution on the back of the wave is far from being clear. The situation is similar in the case of many other models involving nonlocal competition. As an indication that spreading happens, in the present paper we construct traveling waves for equation which travel at the expected spreading speed. One of the main difficulties we encountered studying equation is the lack of a regularizing effect in the mutation operator M ⋆ u. This phenomenon is confirmed by the existence of traveling waves having a nontrivial singular part — in particular, there is no hope for asymptotic regularity. This lack of regularity also makes it more difficult to apply some of the techniques commonly used in the study of reaction-diffusion equations (in particular, taking the limit of a subsequence of large shifts of a solution). Finally, the non-compactness of the time-1 map prevents an application of the spreading results of Weinberger. One other challenging issue is the absence of a comparison principle for equation, because of the nonlocal competition term. As in many other studies involving a nonlocal competition, this prevents a precise study of the long-time behavior of the solutions to the Cauchy problem and the behavior at the back of the waves (see also the above paragraph). To show that the traveling waves stay away from 0 on the back, we introduce a secondary problem, constructed by increasing self-competition in equation, which satisfies a comparison principle and serves as a sub-solution factory. To overcome the lack of regularity, we approximate the solutions of by a vanishing viscosity method. We choose the zero Neumann boundary conditions for the approximating problem because they behave well with respect to the integration across the domain. Finally, we introduce a weak notion of traveling waves which admit singularities. As we will see below, there is little hope to obtain more regularity in general, since there exist traveling waves for equation which present an actual singularity. As far as we know, the present work constitutes the first construction of a measure-valued traveling wave in a reaction-diffusion equation. Main results and comments ========================= Function spaces and basic notions --------------------------------- Throughout this document we use a number of function spaces that we make precise here to avoid any confusion. Whenever X is a subset of a Euclidean space, we will denote C(X), Cb(X), C0(X), Cc(X) the space of continuous functions, bounded continuous functions, continuous functions vanishing at ∞ and continuous functions with compact support over X, respectively. Notice that if X is compact, then those four function spaces coincide. Whenever $ X \subset\mathbb R^d $ is a Borel set, we define M1(X) as the set of all Borel-regular measures over X. Let us recall that M1(X) is the topological dual of C0(X), by Riesz’s representation theorem. In our context, M1(X) coincides with the set of Borel measures that are inner and outer regular. We will thus call Radon measure an element of M1(X). When p ∈ M1(X), we say that the equality p = 0 holds in the sense of measures if ∀ψ ∈ Cc(X), ∫Xψ(x)p(dx) = 0. We now define the notion of transition kernel (see ), which is crucial for our notion of traveling wave: [Transition kernel][def:TK] We say that $ u\in M^1(\mathbb R\times X) $ has a transition kernel if there exists a function k(x, dy) such that 1. for any Borel set A ⊂ X, k( ⋅ , A) is a measurable function, and 2. for almost every $x\in\mathbb R $ (with respect to the Lebesgue measure on $ \mathbb R $), k(x, ⋅ ) ∈ M1(X) and u(dx, dy) = k(x, dy)dx in the sense of measures, i.e. for any $ \varphi\in C\_c(\mathbb R\times X)$, the following equality holds $$\int\_{\mathbb R\times X} \varphi(x,y)u(dx,dy)=\int\_\mathbb R\int\_X\varphi(x,y)k(x,dy)dx.$$ For simplicity, if the measure u has a transition kernel, we will often say that u is a transition kernel and use directly the notation u(dx, dy) = u(x, dy)dx. We denote f ⋆ g the function: $$\label{eq:def\_star} f\star g (y):=\int\_{\overline\Omega}f(y,z)g(dz)$$ whenever $ f: \overline\Omega^2\to \mathbb R $ and g is a measure on $ \overline\Omega$. If g is continuous or L1(Ω) we use the convention g(dz) : = g(z)dz in the above formula. Remark that the operation ⋆ is not the standard convolution, though both notions share many properties. Finally, for y ∈ ∂Ω we will call ν(y) or simply ν the outward normal unit vector of Ω. Main results ------------ Our main result is the existence of a measure traveling wave, possibly singular, for equation. Before stating the result, let us give our assumptions, as well as subsidiary results. [Minimal assumptions][hyp:gen] 1. $ \Omega\subset\mathbb R^n $ is a bounded connected open set with C3 boundary. For simplicity we assume 0 ∈ Ω. 2. M = M(y, z) is a Cα positive function $ \overline\Omega\times\overline\Omega\to\mathbb R $ satisfying $$\forall z\in\overline\Omega, \int\_{\overline\Omega} M(y,z) d y = 1.$$ In particular, 0 < m0 ≤ M(y, z) ≤ m∞ < + ∞ for any $ (y,z)\in\overline\Omega\times\overline\Omega$. 3. K = K(y, z) is a Cα positive function $ \overline\Omega\times\overline\Omega\to\mathbb R $. In particular, we have 0 < k0 ≤ K(y, z) ≤ k∞ < + ∞ for any $(y, z)\in\overline\Omega\times\overline\Omega$. 4. $ a =a(y)\in C^{\alpha }(\overline\Omega) $ is a non-constant function with $\sup\_{y\in\overline\Omega} a(y)>0$. We assume that a(0) = supa. In particular, − ∞ < infa < supa < + ∞ holds. 5. We let $\Omega\_0:=\left\{y\in\overline\Omega\,\|\, a(y)=a(0)=\sup\_{z\in\overline\Omega} a(z)\right\}$ be the set of maximal value for a and assume Ω0 ⊂ ⊂ Ω. 6. $ 0<\mu<\sup a-\underset{z\in\partial\Omega}\sup a^+(z) $. We are particularly interested in a more restrictive set of assumptions, under which we hope to see a concentration phenomenon in : [Concentration hypothesis][hyp:dega] In addition to Assumption [hyp:gen], we suppose $$y\mapsto \frac{1}{\sup\_{z\in\Omega} a(z)-a(y)}\in L^1(\Omega).$$ Let us introduce the principal eigenvalue problem that guides our analysis: [Principal eigenvalue][def:vpp] We call principal eigenvalue associated with the real number: $$\label{eq:vpp} \lambda\_1:=\sup\{\lambda\,\|\, \exists \varphi\in C(\overline\Omega), \varphi>0 \text{ s.t. }\mu(M\star \varphi - \varphi) +(a(y)+\lambda)\varphi \leq 0\}.$$ Clearly, λ1 is well-defined and we have λ1 ≤ − (supa − μ) by evaluating at y = 0. Though we call λ1 the principal eigenvalue, we stress that λ1 is not always associated with a usual eigenfunction. In particular, Coville, in his work, gives conditions on the coefficients of under which there exists no associated eigenfunction. We will recall and extend these results in section [ssec:eigen]. [On the principal eigenvalue][prop:vpp] Under Assumption [hyp:gen], there exists a unique $ \lambda\in\mathbb R $ such that the equation μ(M ⋆ φ − φ) + (a(y) + λ)φ = 0 has a nonnegative nontrivial solution in the sense of measures, and λ = λ1. Moreover, under Assumption [hyp:dega], there exists μ0 > 0 such that if μ < μ0, we have λ1 = − (supa − μ) and, in this case, there exists a nonnegative measure φ solution to with a non-trivial singular part concentrated in Ω0. The most part of Proposition [prop:vpp] comes from the work of Coville. Our contribution to the result is the uniqueness of the real number λ such that there exists a nonnegative nontrivial measure solution to. We use this uniqueness result several times in the paper, in particular, in many of the arguments involving a vanishing viscosity; for instance in the proofs of Theorem [thm:eigen] and Theorem [thm:survival]. As well-known in KPP situations, we expect the sign of λ1 to dictate the long-time persistence of solutions to equation. In particular, when λ1 > 0, we expect that any nonnegative solution to the Cauchy problem starting from a positive bounded initial condition goes to 0 as t → ∞. Indeed, in this case there exists a positive continuous function ψ > 0 such that $$\mu(M\star \psi - \psi) +\left(a+\frac{\lambda\_1}{2}\right)\psi \leq 0.$$ One can check that $ Ce^{-\frac{\lambda\_1}{4}t}\psi(y) $ and u(t, x, y) are respectively a super- and subsolution of the equation ut = uxx + μ(M ⋆ u − u) + a(y)u. with ordered initial data (for C large enough). The result is then a consequence of the comparison principle satisfied by the (linear) above equation. In the λ1 = 0 case, we expect extinction as in the λ1 > 0 case. This is generally the case for scalar reaction-diffusion equations, as well as in the case of some systems (see in particular ). However, the usual strategy, which consists in establishing a contradiction by studying the least multiple of the principal eigenfunction which lies above the ω-limit set of a solution to, seems difficult to apply here. Indeed we lack three of the main ingredients for this argument: a Harnack inequality, compactness, and a L∞ bound on the orbit which would allow us to place a multiple of the principal eigenvector above the ω-limit set. Thus, in the present paper, we leave this particular point open. Note however that, in the case where M is symmetric (M(y, z) = M(z, y)), an argument similar to the one employed in may lead to an actual proof, by working directly on the parabolic problem. In the present paper we focus on the λ1 < 0 case, in which we expect survival of the population. To confirm this scenario, we first prove the existence of a nonnegative nontrivial stationary state for equation. [Survival of the population][thm:survival] Let Assumption [hyp:gen] hold and assume further λ1 < 0. Then, there exists a nonnegative nontrivial stationary state for equation, i.e. a nonnegative nontrivial measure $ p\in M^1(\overline\Omega) $ which satisfies μ(M ⋆ p − p) + p(a(y) − K ⋆ p) = 0 in the sense of measures. Under the hypothesis for concentration (Assumption [hyp:dega]) and in the special case where the competition kernel K(y, z) is independent of the trait y, Bonnefon, Coville and Legendre have shown that the solution to has a singularity concentrated in Ω0 when μ is small. A key argument was a separation of variables method, allowed by the assumption K(y, z) = K(z). Here we show that the concentration phenomenon occurs under a more general hypothesis on K, namely that the trait y ∈ Ω0 suffers less from the competition than any other trait. Since Ω0 also maximizes the basic reproductive ratio a(y), it seems natural to expect concentration in Ω0 in this case. [Nonlinear concentration][hyp:dom] In addition to Assumption [hyp:dega], we suppose that $$\forall (y,z)\in\overline\Omega\times\overline\Omega,\quad K(0, z)\leq K(y,z).$$ [Concentration on dominant trait][thm:concentration] Let Assumption [hyp:dom] hold, and assume λ1 < 0. Then, there exists μ0 > 0 such that, for any μ < μ0, the measure p, constructed in Theorem [thm:survival], has a singular part concentrated in Ω0. To better characterize the spatial dynamics of solutions to, we are going to construct traveling waves for. [Traveling wave][def:TW] A traveling wave for equation is a couple (c, u) where $ c\in\mathbb R $ and u is a locally finite transition kernel (see Definition [def:TK]) defined on $ \mathbb R\times\overline\Omega $. We require that (c, u) satisfies: − cux − uxx = μ(M ⋆ u − u) + u(a − K ⋆ u) in the sense of distributions, and that the measure u satisfies the limit conditions: $$\begin{aligned} \label{eq:limit-TW-left} \underset{\bar x\to+\infty}{\liminf}\int\_{\mathbb R\times\overline\Omega} \psi(x+\bar x,y)u(dx, dy)>0, \\ \label{eq:limit-TW-right} \underset{\bar x\to-\infty}{\limsup}\int\_{\mathbb R\times\overline\Omega} \psi(x+\bar x,y)u(dx, dy)=0 \end{aligned}$$ for any positive test function $ \psi\in C\_c(\mathbb R\times\overline\Omega)$. Condition differs from the usual behavior of traveling waves as defined, for instance, in, in which the convergence to a stationary state is required. Because of the nonlocal competition, indeed, it is very difficult to prove that a solution to equation converges to a stationary state when t → ∞. Imposing a weak condition like on the back of the wave is the usual way to go around this issue. One can refer for instance to, where a similar condition is imposed on the back of traveling waves. We are now in the position to state our main result, which concerns the existence of a traveling wave for. [Existence of a traveling wave][thm:TW] Under Assumption [hyp:gen] and if λ1 < 0, there exists a traveling wave (c, u) for with $c=c^\:=2\sqrt{-\lambda\_1} $. As it is the case in many nonlocal problem, the uniqueness and stability of the traveling waves are unknown. In this paper, we focus on the construction of a traveling wave for c* = c\. Altough this is expected, we leave the construction of traveling waves for c* > c\* for future work, as well as a proof of the non-existence of traveling waves for c < c\.In the general case, it seems very involved to determine whether u* has a singular part or not. Nevertheless, there are some particular cases where singular traveling waves do exist. [Traveling waves with a singular part] In the special case where K is independent from y (K(y, z) = K(z)), a separation of variables argument — see for a related argument— allows us to construct traveling waves that actually have a singular part in $ \overline\Omega $. From Proposition [prop:vpp], under Assumption [hyp:dega], there is μ0 > 0 such that, for any μ < μ0, there exists a measure eigenvector $ \varphi\in M^1(\overline\Omega) $ with a singular part concentrated in Ω0. We choose such a φ with normalization $ \int\_{\overline\Omega}K(z)\varphi(dz) = 1 $. If moreover λ1 < 0, then there exists a positive front ρ, connecting − λ1 to 0, for the Fisher-KPP equation − ρxx − cρx = ρ( − λ1 − ρ) for any $ c\geq 2\sqrt{-\lambda\_1} $. If we define u(x, dy) : = ρ(x)φ(dy), we see that u matches the definition of a traveling wave. Hence for any $ x\in\mathbb R $, u(x, ⋅ ) possesses a singular part concentrated in Ω0. The organization of the paper is as follows. In Section [sec:eigen] we study related eigenvalue problems for which concentration may occur. Section [sec:stat] is devoted to the construction of stationary states through a bifurcation method. Last, we construct a (possibly singular) measure traveling wave in Section [sec:TW]. On the principal eigenvalue problem =================================== In this section, we prove Proposition [prop:vpp], which allows an approximation by an elliptic Neumann eigenvalue problem in Theorem [thm:eigen] of crucial importance for the construction of steady states in Section [sec:stat]. The principal eigenvalue of nonlocal operators ---------------------------------------------- Under Assumption [hyp:gen], Coville et al. have extensively studied the principal eigenvalue problem associated with. We summarize and extend the results in. Our contribution is to show the uniqueness of the principal eigenvalue as a solution to in the sense of measures. [On the principal eigenproblem ][thm:eigenpair] 1. [item:eigenpair-unique] Let Assumption [hyp:gen] be satisfied. Then, there exists a unique $ \lambda\in\mathbb R $ such that admits a nonnegative nontrivial Radon measure solution, and λ = λ1. 2. [item:eigenpair-dega] Let Assumption [hyp:dega] hold, and let − γ1 be the principal eigenvalue[1](#fn1)of the operator $$\mathcal M[\psi]:=\int\_\Omega \mu M(y,z)\frac{\psi(z)}{\sup a-a(z)}dz,$$ acting on ψ ∈ Cb(Ω). Then the following holds: 1. [item:eigenpair-C0] γ1 > 1 if, and only if, λ1 < − (supa − μ). In this case, any solution to in the sense of measures is a pointwise solution. 2. [item:eigenpair-L1] γ1 = 1 if, and only if, λ1 = − (supa − μ) and there exists a nonnegative nontrivial function φ ∈ L1(Ω) solution to almost everywhere. In this case, φ is unique (up to multiplication by a positive constant). 3. [item:eigenpair-M1] γ1 < 1 if, and only if, λ1 = − (supa − μ) and there exists a nonnegative singular measure $ \varphi\in M^1(\overline\Omega) $ solution to. In this case, any nonnegative nontrivial solution to has a singularity concentrated in Ω0. The existence of a measure-valued solution to has been shown in. Here we focus on the uniqueness of λ. We first prove the uniqueness of λ when the complement of Assumption [hyp:dega] holds, by showing that any eigenvector is in fact a continuous eigenfunction. Then, we show that uniqueness holds under Assumption [hyp:dega]. Finally we prove the trichotomy in item [item:eigenpair-dega]. Step 1: Let the complement of Assumption [hyp:dega] hold, i.e. $ \frac{1}{\sup a-a(y)}\not\in L^1(\Omega)$. Let $ \varphi\in M^1(\overline\Omega) $ be a nonnegative nontrivial Radon measure solution to. Then by the Lebesgue-Radon-Nikodym Theorem, there exists a nonnegative φac ∈ L1(Ω) and a nonnegative measure $ \varphi\_s\in M^1(\overline\Omega) $, which is singular with respect to the Lebesgue measure on Ω, such that: φ = φacdy + φs. Equation is then equivalent to the following system: $$\label{eqlem:twomeasures} \begin{system}{lr} \mu M\star \varphi + (a(y)-\mu+\lambda)\varphi\_{ac}=0 &\quad a.e. (dy) \\ a(y)-\mu+\lambda = 0 &\quad a.e.(\varphi\_s). \end{system}$$ This readily shows that (a(y) − μ + λ)φac = − μM ⋆ φ is a continuous negative function and in particular λ ≤ − (supa − μ). We distinguish two cases: Case 1: Assume first that λ < − (supa − μ). Then the second line of implies $ \mathrm{supp\,}\varphi\_s=\varnothing $, i.e. φs ≡ 0. In this case we have $\varphi\_{ac}(y)=\frac{\mu M\star \varphi\_{ac}(y)}{-\lambda - (a(y)-\mu) }$, which is a positive continuous function since the kernel M(y, z) is itself continuous. A classical comparison argument (such as the one presented below on Step 2 case 1) then shows λ = λ1. Case 2: Assume λ = − (supa − μ). Then $$\varphi\_{ac}(y)=\frac{\mu M\star \varphi}{\sup a-a(y)},$$ and since μ(M ⋆ φ)(y) ≥ μm0∫Ωφ(dz) > 0, this implies φac ∉ L1(Ω), which contradicts the definition of φac. We have thus shown the uniqueness of the real number λ such that there exists a solution (λ, φ) to. Step 2: Let Assumption [hyp:dega] hold. We first establish that γ1 is well-defined, then resume the proof. The operator $ \mathcal M $ defined above is compact by virtue of the Arzelà-Ascoli Theorem. Since for any ψ ≥ 0, $ \psi\not\equiv 0 $, we have $$\begin{aligned} \forall y\in\overline\Omega, \quad\mathcal M[\psi](y)&=\int\_\Omega\mu M(y,z)\frac{\psi(z)}{\sup a-a(z)}dz \\ &\geq \mu m\_0\int\_\Omega\frac{\psi(z)}{\sup a-a(z)}dz>0, \end{aligned}$$ $ \mathcal M $ satisfies the hypotheses of the Krein-Rutman Theorem, which ensures that the real number γ1, defined by $\mathcal M[\Psi]=\gamma\_1\Psi$ for a positive Ψ ∈ Cb(Ω), is well-defined and positive. Let us resume the proof. Let (λ, φ) be a solution to in the sense of measures. Then, as above, by Lebesgue-Radon-Nikodym Theorem, there exists a nonnegative φac ∈ L1(Ω) and a nonnegative measure φs ⊥ dy, such that φ = φacdy + φs. In this context, equation is equivalent to system, and in particular we have λ ≤ − (supa − μ). We subdivide the rest of the proof in two cases. 1. Let us first assume λ < − (supa − μ). Then it follows from equation that φs ≡ 0. Moreover, $ \varphi\_{ac}(y)=\frac{\mu (M\star \varphi)(y)}{-\lambda - (a(y)-\mu)} $ is then a positive bounded continuous function and satisfies: μ(M ⋆ φac − φac) + (a(y) + λ)φac = 0 in the classical sense. Let us show that λ = λ1. Let $ ({\overline\lambda}, {\overline\varphi})\in \mathbb R\times {C(\Omega)} $ be a supersolution to, i.e. $ \overline\varphi>0 $ and $$\mu M\star {\overline\varphi} + {\overline\varphi}(a(y)-\mu+{\overline\lambda}) \leq 0.$$ Then $ {\overline\varphi}(0){(-a(0)+\mu-{\overline\lambda})} \geq \mu M\star {\overline\varphi}>0 $ and thus $ {\overline\lambda} <-(a(0)-\mu)=-(\sup a-\mu) $. Moreover $\overline\varphi(y)\geq \frac{\mu M\star\overline\varphi(y)}{\mu-a(y)-\overline\lambda}\geq\frac{\mu m\_0 \int\varphi}{-(\inf a + \overline\lambda-\mu)}>0$ and thus $\overline\varphi$ is uniformly bounded from below. In particular, $\alpha:=\sup\{\zeta>0\,\|\, \forall y\in\overline\Omega, \zeta\varphi\_{ac}(y)\leq {\overline\varphi}(y)\}$ is well-defined and positive. By definition of α we have $ \alpha\varphi\_{ac}(y)\leq {\overline\varphi}(y) $ for any $ y\in\overline\Omega $, and there exists a converging sequence $ \Omega\ni y\_n\to y\in\overline\Omega $ such that $ \alpha\varphi\_{ac}(y\_n)-{\overline\varphi}(y\_n)\to 0 $. Up to further extraction φac(yn) converges to a positive limit that we denote φac(y). We have then $$\begin{aligned} 0&\geq\mu\int\_{\overline\Omega}M(y\_n, z)\big({\overline\varphi}(z)-\alpha\varphi\_{ac}(z)\big)dz \\ &\quad + \big({\overline\varphi}(y\_n)-\alpha\varphi\_{ac}(y\_n)\big)(a(y\_n)-\mu) +{\overline\lambda}\overline\varphi(y\_n) - \lambda\alpha\varphi\_{ac}(y\_n)\\ &\geq 0 + \big({\overline\varphi}(y\_n)-\alpha\varphi\_{ac}(y\_n)\big)(a(y\_n)-\mu) +{\overline\lambda}\overline\varphi(y\_n) - \lambda\alpha\varphi\_{ac}(y\_n) \\ &= ({\overline\lambda} - \lambda)\alpha\varphi\_{ac}(y)+o\_{n\to\infty}(1). \end{aligned}$$ Taking the limit n → ∞, we have shown $ {\overline\lambda}\leq \lambda $. Hence, $$\lambda \geq \sup\{{\overline\lambda}\,\|\, \exists \psi\in C(\Omega), \psi>0 \text{ s.t. } \mu(M\star\psi - \psi) + \psi(a(y)+{\overline\lambda})\leq 0\} = \lambda\_1.$$ The reverse inequality λ ≤ λ1 is clear since φac is a supersolution to. Thus λ = λ1. In this case, we notice that $$\mathcal M[(\sup a-a(y))\varphi\_{ac}] = \mu M\star \varphi\_{ac}>(\sup a-a(y))\varphi\_{ac}.$$ Hence, by a classical comparison argument, γ1 > 1. 2. Let us assume now λ = − (supa − μ). We define the auxiliary function Ψ(y) : = φac(y)(supa − a(y)) = μ(M ⋆ φ). Then Ψ is a nontrivial positive bounded continuous function which satisfies: $$\mathcal M[\Psi]-\Psi=\mu(M\star \varphi\_{ac}-\varphi\_{ac})+(a(y)+\lambda)\varphi\_{ac}=-\mu M\star \varphi\_s\leq 0.$$ Thus, by a classical comparison argument, γ1 ≤ 1. We claim that λ1 = λ. As above, φac is a supersolution to, and thus λ ≤ λ1. Assume by contradiction that λ1 < λ. By the existence property, there exists a continuous function φ1 > 0 associated with λ1. Since λ1 < λ = − (supa − μ), point 1 above then applies to (λ1, φ1) and we have γ1 > 1. This is a contradiction. Hence λ = λ1. Step 3: We show [item:eigenpair-C0], [item:eigenpair-L1], and [item:eigenpair-M1]. Assume λ1 < − (supa − μ). Then, γ1 > 1, and the fact that any measure eigenvector is a continuous eigenfunction has been shown in Step 2. Assume λ1 = − (supa − μ) and φs ≡ 0. Let Ψ(y) : = (supa − a(y))φac(y). Then, by a straightforward computation, Ψ satisfies Ψ(y) = μM ⋆ φ(y), which shows that Ψ is bounded and continuous. We remark that: $$\mathcal M[\Psi]-\Psi=\mu M\star \varphi\_{ac}-(\sup a -a )\varphi\_{ac}=-\mu M\star \varphi\_s=0.$$ By the Krein-Rutman Theorem, we have γ1 = 1 and φ ≡ φac is unique up to multiplication by a scalar. Assume that λ1 = − (supa − μ) and $ \varphi\_s\not\equiv 0 $. Let Ψ(y) : = (supa − a(y))φac(y), then $$\mathcal M[\Psi]-\Psi=-\mu M\star \varphi\_s<0$$ and thus γ1 < 1. Notice that in this case, the second line in equation implies by definition $ \varphi\_s\left(\{y\in\overline\Omega\,\|\, a(y)\neq \sup a\}\right) = 0 $, hence supp φs ⊂ Ω0. Since we have investigated all the possibilities (recall λ ≤ − (supa − μ)), the equivalence holds in each case. This finishes the proof of Theorem [thm:eigenpair]. The critical mutation rate -------------------------- In this subsection we investigate further the linear eigenvalue problem, with λ = λ1 as compelled by Theorem [thm:eigenpair], under Assumption [hyp:dega]. We introduce the notion of critical mutation rate, which distinguishes between the existence of a bounded continuous eigenfunction for equation and the existence of a singular measure. [Critical mutation rate][thm:smallmu] Let Assumption [hyp:dega] hold. Then, there exists μ0 = μ0(Ω, M, supa − a) such that for any 0 < μ < μ0, problem has only singular measures solutions with a singularity concentrated in Ω0 (in which case λ1 = − (supa − μ) from Theorem [thm:eigenpair]), whereas for μ > μ0 equation has only bounded continuous eigenfunctions. Finally, $ \mu\_0 = \frac{1}{\gamma\_1^1} $ where − γ11 is the principal eigenvalue of the operator $$\mathcal M^1[\psi]=\int\_\Omega M(y,z)\frac{\psi(z)}{\sup a-a(z)}dz,$$ acting on bounded continuous functions. Let us define, for ψ ∈ Cb(Ω), $\mathcal M^\mu[\psi]=\mu\int\_\Omega M(y,z)\frac{\psi(z)}{\sup a-a(z)}dz$. Then by the Krein-Rutman Theorem there exists a unique principal eigenpair ( − γ1μ, Φμ) satisfying γ1μ > 0, Φμ(y) > 0, supΦμ = 1 and $ \mathcal M^\mu[\Phi^\mu]=\gamma\_1^\mu \Phi^\mu$. Since $ \mathcal M^\mu = \mu\mathcal M^1 $, we deduce from the uniqueness of ( − γ1μ, Φμ) that the equalities γ1μ = μγ11 and Φμ = Φ1 hold for any μ > 0. The result then follows from the trichotomy in Theorem [thm:eigenpair] We can now summarize our findings and prove Proposition [prop:vpp]. [Proof of Proposition [prop:vpp]] The first part, under Assumption [hyp:gen], follows from Proposition [thm:eigenpair], while the second part, under Assumption [hyp:dega], follows from Theorem [thm:smallmu]. We prove below that μ0 is linked to the steepness of the fitness function a near its maximum. This property will be used in the proof of Theorem [thm:concentration]. [Monotony of μ0][cor:compmucrit] Let Assumption [hyp:dega] hold and b be a continuous function on $ \overline\Omega $, satisfying $$\forall y\in\overline\Omega,\quad \sup a-a(y)\leq \sup b-b(y).$$ Then we have μ0(Ω, M, supa − a) ≤ μ0(Ω, M, supb − b), where μ0 is defined in Theorem [thm:smallmu]. It follows from our assumptions that, for $ y\in\overline\Omega$: $$\label{eqcor:compmucrit} 0<\frac{1}{{\sup b}-b(y)}\leq \frac{1}{{\sup a}-a(y)}.$$ In particular $ y\mapsto\frac{1}{{\sup b}-b(y)}\in L^1(\Omega) $. Thus Theorem [thm:smallmu] can be applied with both a and b. We claim that γ1b ≤ γ1a, where γ1b, γ1a denote the first eigenvalue of the operator $ \mathcal M\_b[\psi]= \int\_\Omega M(y,z)\frac{\psi(z)}{\sup b-b(z)}dz $ and $ \mathcal M\_a[\psi]=\int\_\Omega M(y,z)\frac{\psi(z)}{\sup a-a(z)}dz $ acting on the function ψ ∈ Cb(Ω), respectively. Indeed, let φa ∈ Cb(Ω), φa > 0 satisfy $ \int\_\Omega M(y,z) \frac{\varphi^a(z)}{{\sup a}-a(z)}dz=\gamma\_1^a\varphi^a(y) $ and φb ∈ Cb(Ω), φb > 0 respectively satisfy $ \int\_\Omega M(y,z) \frac{\varphi^b(z)}{{\sup b}-b(z)}dz=\gamma\_1^b\varphi^b(y)$. Up to multiplication by a positive constant, we assume without loss of generality that φb ≤ φa and that there exists $ y\in\overline\Omega $ satisfying φb(y) = φa(y) = 1. At this point, we have $$\gamma\_1^b = \int\_\Omega M(y,z) \frac{\varphi^b(z)}{{\sup b}-b(z)}dz \leq \int\_\Omega M(y,z) \frac{\varphi^a(z)}{{\sup a}-a(z)}dz=\gamma\_1^a.$$ We conclude that $$\mu\_0(\Omega, M, \sup a-a)=\frac{1}{\gamma\_1^a}\leq \frac{1}{\gamma\_1^b} = \mu\_0(\Omega, M, \sup b-b)$$ which finishes the proof of Corollary [cor:compmucrit]. Approximation by a degenerating elliptic eigenvalue problem ----------------------------------------------------------- Here we show that the previously introduced principal eigenvalue can be approximated by an elliptic Neumann eigenvalue. [Approximating λ1 by vanishing viscosity][thm:eigen] Let Assumption [hyp:gen] hold, and (λ1ɛ, φɛ(y) > 0) be the solution to the principal eigenproblem: $$\label{eq:eigeneps} \begin{system}{lr} -\varepsilon\Delta \varphi^\varepsilon -\mu(M\star \varphi^\varepsilon - \varphi^\varepsilon) = a(y)\varphi^\varepsilon + \lambda\_1^\varepsilon\varphi^\varepsilon & \text{ in } \Omega \\ \frac{\partial\varphi^\varepsilon}{\partial \nu}=0 & \text{ on } \partial \Omega, \end{system}$$ with ∫Ωφɛ(z)dz = 1, where ν is the unit normal vector. Then limɛ → 0λ1ɛ = λ1, where λ1 is the principal eigenvalue defined by. We divide the proof into three steps. Step 1: We show that λ1ɛ is bounded when ɛ → 0. Integrating equation by parts, we have 0 = ∫Ω(λ1ɛ + a(y))φɛdy. In particular, the function a(y) + λ1ɛ takes both nonnegative and nonpositive values. Hence, we have − supa ≤ λ1ɛ ≤ − infa, and (λ1ɛ)ɛ > 0 is bounded. Step 2: We identify the limit of converging subsequences. Let λ1ɛn be a converging sequence and λ10 : = limλ1ɛn. Then φɛn satisfies, for any $ \psi\in C^2(\overline\Omega) $, $$\int\_\Omega -{\varepsilon\_n} \varphi^{\varepsilon\_n}\Delta \psi dy - \varepsilon\_n \int\_{\partial\Omega}\varphi^{\varepsilon\_n}\frac{\partial\psi}{\partial\nu}dS - \int \mu(M\star \varphi^{\varepsilon\_n}-\varphi^{\varepsilon\_n})\psi-a(y)\varphi^{\varepsilon\_n}\psi =\lambda\_1^{\varepsilon\_n}\int\varphi^{\varepsilon\_n}\psi.$$ Let $$\label{eq:F0} F\_0:=\left\{ \psi\in C^2(\overline\Omega)\,\|\,\forall y\in\partial\Omega, \frac{\partial\psi}{\partial\nu}(y)=0\right\}$$ denote the space of functions in C2(Ω) with zero boundary flux as in Lemma [lem:zerofluxdense] item [item:zeroflux-Omega]. For ψ ∈ F0, this equation becomes: ∫Ω − ɛnφɛnΔψdy − ∫μ(M ⋆ φɛn − φɛn)ψ − a(y)φɛnψ = λ1ɛn∫φɛnψ. Since ∫Ωφɛn(y)dy = 1 and $ \overline\Omega $ is compact and by Prokhorov’s Theorem, the sequence (φɛn) is precompact for the weak topology in $ M^1(\overline\Omega) $, and there exists a weakly convergent subsequence φɛnʹ, which converges to a nonnegative Radon measure φ. Since $ 1\in C\_c(\overline\Omega)$, we have $\lim \int\_{\overline\Omega}\varphi^{\varepsilon\_n'}=\int\_{\overline\Omega}\varphi (dy) = 1$. Hence φ is non-trivial. Moreover, we have μ∫Ω∫ΩM(y, z)dφ(z)ψ(y)dy + ∫Ω(a(y) − μ)ψ(y)dφ(y) + λ10∫Ωψ(y)dφ(y) = 0 for any test function ψ ∈ F0. Since F0 is densely embedded in $ C\_b(\overline\Omega) $ by Lemma [lem:zerofluxdense], holds for any $\psi\in C\_b(\overline\Omega) $. Applying Proposition [prop:vpp], we have then λ10 = λ1. Step 3: Conclusion. We have shown that for any sequence ɛn → 0, there exists a subsequence ɛʹn → 0 such that λ1ɛʹn → λ1. Thus λ1ɛ → λ1 when ɛ → 0. Stationary states in trait ========================== This section deals with stationary states for. In particular, we prove Theorem [thm:survival] and Theorem [thm:concentration] via a bifurcation argument. Regularized solutions --------------------- We investigate the existence of positive solutions p = p(y) to the following problem $$\label{eq:evolstat-beta} \begin{system}{ll} -\varepsilon \Delta p - \mu(M\star p-p)=p(a(y)-K\star p-\beta p) & \text{ in } \Omega \\ \frac{\partial p}{\partial\nu}=0 & \text{ on } \partial\Omega, \end{system}$$ for any β ≥ 0. We prove the existence of positive solutions for when λ1ɛ < 0. We plan to let ɛ → 0 with β = 0 in Section [sec:evolstat], in order to prove the existence of stationary solutions to. The reason why we include a weight β ≥ 0 on the competition term in equation is that solutions to the latter will be used as subsolutions in the construction of traveling waves in Section [sec:TW]. Throughout this subsection we denote (λ1ɛ, φɛ) the eigenpair of the regularized problem, solving. Notice that (λ1ɛ, φɛ) is independent from β. Our main result is the following: [Regularized steady states][thm:ex-stat-eps] Let Assumption [hyp:gen] hold, ɛ > 0, (λ1ɛ, φɛ) be defined by, and β ≥ 0. 1. [item:exstat-extinction] Assume λ1ɛ > 0. Then 0 is the only nonnegative solution to. 2. [item:exstat-survival] Assume λ1ɛ < 0. Then there exists a positive solution to for any β ≥ 0. Item [item:exstat-extinction] is rather trivial and we will discuss it later in the proof of Theorem [thm:ex-stat-eps]. The actual construction in the case λ1ɛ < 0 is more involved. Our method is inspired by the similar situation in. We start by establishing a priori estimates on the solutions p to. [A priori estimates on p][lem:apriorip] Let Assumption [hyp:gen] hold, let ɛ > 0, β ≥ 0 and p be a nonnegative nontrivial solution to. Then: 1. [item:apriorip-positive] p is positive. 2. [item:apriorip-boundinfty] If β = 0, there exists a positive constant C = C(Ω, ɛ, μ, ‖a‖L∞, m∞, k0, k∞) such that ‖p‖L∞ ≤ C. If β > 0 then we have $ \sup p\leq \frac{\sup a}{\beta} $. Point [item:apriorip-positive] follows from the strong maximum principle. We turn our attention to point [item:apriorip-boundinfty]. Assume first β > 0. Let $ y\in\overline\Omega$ such that p(y) = supz ∈ Ωp(z) and assume by contradiction that $ p(y)>\frac{\sup a}{\beta} $. If y ∈ Ω, then we have 0 ≤ − ɛΔyp(y) − μ(M ⋆ p − p) = p(a(y) − K ⋆ p − βp) < 0 which is a contradiction. If y ∈ ∂Ω, then μ(M ⋆ p − p) ≤ 0 and a − K ⋆ p − βp ≤ 0 in a neighbourhood of y, and thus − ɛΔp − (a(y) − K ⋆ p − βp)p ≤ 0 in a neighbourhood of y. It follows from Hopf’s Lemma that $ \frac{\partial p}{\partial\nu}(y)>0 $, which contradicts the Neumann boundary conditions satisfied by p. Hence $ \sup p\leq \frac{\sup a}{\beta} $. We turn our attention to the case β = 0, which is more involved. We divide the proof in four steps. Step 1: We establish a bound on ∫Ωp(y)dy. Integrating over Ω, we have $$\int\_\Omega a(y)p(y)\mathrm dy - \int\_\Omega\int\_\Omega p(y)K(y,z)p(z)\mathrm dy\mathrm dz =\beta\int\_\Omega p^2(y)dy\geq 0.$$ Thus ∫a(y)p(y)dy ≥ k0(∫Ωp(y)dy)2 and $$\label{eq:apriorip-mass} \int\_\Omega p(y)dy\leq \frac{ \sup a}{k\_0}.$$ Step 2: We reduce the problem to a boundary estimate. By a direct application of the local maximum principle, for any ball BR(y) ⊂ Ω, there exists a constant C = C(R, ɛ, ‖a‖L∞, k0, k∞, μ, m∞) > 0 such that $ \underset{B\_{R/2}(y)}{\sup}\,p\leq C $. This shows an interior bound for any point at distance R from ∂Ω. To show that this estimate does not degenerate near the boundary, we use a coronation argument. Let d(y, ∂Ω) : = infz ∈ ∂Ω∣y − z∣, and ΩR : = {y ∈ Ω ∣ d(y, ∂Ω) < R} for any R > 0. As noted in, the function y ↦ d(y, ∂Ω) has C3 regularity on a tubular neighbourhood of ∂Ω. In particular, ∂ΩR ∖ ∂Ω is C3 for R small enough, since ∇d ≠ 0 in this neighbourhood. Moreover, by the comparison principle in narrow domains, the maximum principle holds for the operator − ɛΔv − (a(y) − μ)v in ΩR provided ∣ΩR∣ is small enough, meaning that if v satisfies − ɛΔv − (a(y) − μ)v ≥ 0 in ΩR and v ≥ 0 on ∂ΩR, then v ≥ 0. In particular, we choose R small enough for this property to hold. At this point, p ≤ C in Ω ∖ ΩR and comparison holds in ΩR. Step 3: We construct a supersolution. Notice that, in contrast with where Dirichlet boundary conditions are used, we need an additional argument to deal with the Neumann boundary conditions. Since the comparison principle holds in the narrow domain ΩR, the Fredholm alternative implies that, for any δ ∈ (0, 1], there exists a unique (classical) solution to the system: $$\begin{system}{ll} -\varepsilon\Delta v^\delta-(a(y)-\mu)v^\delta = \mu m\_\infty \frac{\sup a}{k\_0} & \text{ in } \Omega\_R \\ v^\delta=C & \text{ on }\partial \Omega\_R{\backslash}\partial\Omega \\ \delta v^\delta + (1-\delta)\frac{\partial v^\delta}{\partial\nu}=\delta & \text{ on }~\partial\Omega. \end{system}$$ As a result of the classical Schauder interior and boundary estimates, the mapping δ ↦ vδ is continuous from (0, 1] to Cb(ΩR). Moreover, vδ is positive for δ ∈ (0, 1] by virtue of the maximum principle. Next, still by a direct application of the Schauder estimates, (vδ)0 < δ ≤ 1 is precompact and there exists a sequence δn → 0 and v ∈ C2 such that vδn → v in $ C^2\_{loc}(\Omega\_R)\cap C^1(\overline\Omega\_R) $. Then v ≥ 0 satisfies: $$\begin{system}{ll} -\varepsilon\Delta v-(a(y)-\mu)v = \mu m\_\infty \frac{\sup a}{k\_0} & \text{ in } \Omega\_R \\ v=C & \text{ on }\partial \Omega\_R{\backslash}\partial\Omega \\ \frac{\partial v}{\partial\nu}=0 & \text{ on }\partial\Omega. \end{system}$$ By a direct application of the strong maximum principle and Hopf’s Lemma, we have v > 0 on $\overline\Omega\_R $. Step 4: We show that p ≤ v on ΩR. Let p be a solution to and select α : = inf{ζ > 0 ∣ ζv ≥ pin ΩR}. Assume by contradiction that α > 1. Then there exists $ y\_0\in\overline\Omega\_R $ such that the equality p(y0) = αv(y0) holds, and αv − p ≥ 0. In particular y0 is a zero minimum for the function αv − p. Because of the boundary conditions satisfied by p and v, y0 cannot be in ∂ΩR. y0 is then an interior local minimum to αv − p and thus $$\begin{aligned} 0&\geq -\varepsilon\Delta (\alpha v-p)(y\_0) = (a(y\_0)-\mu)(\alpha v-p)(y\_0)+\alpha \mu m\_\infty\frac{ \sup a}{k\_0} \\ &\quad- \mu (M\star p)(y\_0) +p(y\_0)(K\star p)(y\_0) \\ &> \alpha \mu m\_\infty\frac{ \sup a}{k\_0}-\mu (M\star p)(y\_0)\geq 0, \end{aligned}$$ using estimate, which is a contradiction. Thus α ≤ 1. This shows that p ≤ v. Since v is a bounded function, we have our uniform bound for p in ΩR. In Ω ∖ ΩR, we have p ≤ C. This ends the proof of Lemma [lem:apriorip]. In order to proceed to the proof of Theorem [thm:ex-stat-eps], we yet need an additional technical remark. [Fréchet differentiability at 0][lem:Frechet] Let Assumption [hyp:gen] hold, β ≥ 0 and $$\begin{array}{rccl} G:&C\_b(\Omega)&\to&C\_b(\Omega) \\ &p(y) & \mapsto & p(y) (K\star p)(y) + \beta p^2(y), \end{array}$$ then G is Fréchet differentiable at p = 0 and its derivative is DG(p) = 0. This comes from the remark $$\begin{aligned} \left\|\int\_\Omega K(y,z)p(z)dzp(y)+p^2(y)\right\|&\leq \int\_\Omega K(y,z)\|p(z)\|dz \|p(y)\| + \beta p^2(y)\\ & \leq k\_\infty\|\Omega\|\Vert p\Vert\_{C\_b(\Omega)}^2 + \beta \Vert p\Vert\_{C\_b(\Omega)}^2 \qedhere \end{aligned}$$ [Proof of Theorem [thm:ex-stat-eps]] Step 1: We prove item [item:exstat-extinction]. We assume λ1ɛ > 0. We recall that (λ1ɛ, φɛ) is the solution to with the normalization ∫Ωφɛ(y)dy = 1. Let p > 0 be a nonnegative solution to in Ω. Since p is bounded and φɛ is positive on $ \overline\Omega $, the quantity α : = inf{ζ > 0 ∣ ζφɛ > p} is well-defined and finite. Then, there exists $ y\_0\in\overline\Omega $ such that p(y0) = αφɛ(y0). Remark that y0 is a minimum to the nonnegative function αφɛ − p. If y0 ∈ ∂Ω, then Hopf’s Lemma implies $ \frac{\partial (\alpha\varphi^\varepsilon - p)}{\partial\nu}(y\_0)<0 $, which contradicts the Neumann boundary conditions satisfied by p and φɛ. Thus y0 ∈ Ω. Evaluating equation, we have: $$\begin{aligned} 0&\geq-\varepsilon \Delta(\alpha\varphi^\varepsilon - p)(y\_0)=\mu\big(M\star (\alpha\varphi^\varepsilon-p) - (\alpha\varphi^\varepsilon-p)\big)\\ &\quad +a(y\_0)\big(\alpha\varphi^\varepsilon(y\_0)-p(y\_0)\big)+p(y\_0)(K\star p)(y\_0)+\beta p^2(y\_0) + \lambda\_1^\varepsilon \alpha\varphi^\varepsilon \\ &\geq p(y\_0)(K\star p)(y\_0)+\beta p^2(y\_0) + \lambda\_1^\varepsilon \alpha\varphi^\varepsilon >0 \end{aligned}$$ which is a contradiction. Step 2 : We prove item [item:exstat-survival]. We assume λ1ɛ < 0. We argue as in : if the nonlinearity is negligible near 0 and we can prove local boundedness of the solutions in L∞, then we can prove existence through a bifurcation argument. This requires a topological result stated in Appendix [appendix:topo]. More precisely, for $ \alpha\in\mathbb R $ and p ∈ Cb(Ω), we let $ F(\alpha, p)=\tilde p $ where $ \tilde p $ is the unique solution to: $$\begin{system}{ll} -\varepsilon\Delta \tilde p + (\sup a-a(y))\tilde p - \mu (M\star \tilde p-\tilde p) = \alpha p - G(p) & \text{ in } \Omega \\ \frac{\partial \tilde p}{\partial \nu} = 0 & \text{ on } \partial \Omega \end{system}$$ where G is as in Lemma [lem:Frechet]. Notice that supa − a(y) ≥ 0, so comparison applies and the operator F is well-defined due to the Fredholm alternative. In particular, for each $ \alpha\in\mathbb R $, F(α, ⋅ ) is Fréchet differentiable near 0 and its derivative is the linear operator αT, where $ Tp=\tilde p $ and $ \tilde p $ is defined by: $$\begin{system}{ll} -\varepsilon\Delta \tilde p + (\sup a-a(y))\tilde p - \mu (M\star \tilde p - \tilde p) = p & \text{ in } \Omega \\ \frac{\partial \tilde p}{\partial \nu} = 0 & \text{ on } \partial \Omega. \end{system}$$ Let C : = {p ∈ Cb(Ω) ∣ p ≥ 0}. By a classical comparison argument, T maps the cone C ∖ {0} into Int C = {p ∈ Cb(Ω) ∣ p > 0}. By virtue of the Krein-Rutman Theorem, T has a first[2](#fn2) eigenvalue λ(T) (satisfying Tψ = λ(T)ψ for a ψ > 0) and we have the formula $ \lambda(T) = \frac{1}{\lambda\_1^\varepsilon + \sup a}$. We now check one by one the hypotheses of Theorem [thm:orientedbifurcation]: 1. Clearly we have F(α, 0) = 0 for any $ \alpha\in\mathbb R$. 2. It follows from Lemma [lem:Frechet] that G is Fréchet differentiable near 0 with derivative 0. As a consequence, F(α, ⋅ ) is Fréchet differentiable near 0 with derivative αT. 3. T satisfies the hypotheses of the Krein-Rutman Theorem. 4. It follows from Lemma [lem:apriorip] that the solutions to F(α, p) = p are locally uniformly bounded in α. 5. Since any nontrivial nonnegative fixed point p is positive, there is no nontrivial fixed point in the boundary of C. Thus, applying Theorem [thm:orientedbifurcation], there exists a branch of solutions $ \mathcal C $ connecting $ \alpha=\frac{1}{\lambda(T)} $ to either α → + ∞ or α → − ∞. By the uniqueness in the Krein-Rutman Theorem, if λα denotes the principal eigenvalue associated with F(α, p) = p, we have $ \lambda^\alpha=\lambda\_1^\varepsilon+\sup a-\alpha=\frac{1}{\lambda(T)}-\alpha$. In particular, for α < − supa − λ1ɛ, we deduce from Step 1 that there cannot exist a solution to F(α, p) = p in C. Thus $ \mathcal C $ connects $\frac{1}{\lambda(T)} $ to + ∞. In particular, there exists a positive solution for $ \alpha = \sup a = \frac{1}{\lambda(T)}-\lambda\_1^\varepsilon>\frac{1}{\lambda(T)} $, which solves. This ends the proof of Theorem [thm:ex-stat-eps]. We now prove a lower estimate for solutions to, which is crucial for the construction of traveling waves, but will not be used in the meantime. We stress that in the lemma below, the constant ρβ is independent from ɛ. [pɛ, β does not vanish][lem:beta-lowerbound] Let Assumption [hyp:gen] be satisfied, let β > 0 and λ1 < 0. Let finally pɛ, β be a solution to. Then, there exists constants ɛ0 = ɛ0(Ω, μ, M, a) > 0 and ρβ = ρβ(Ω, M, a, β) > 0 such that if ɛ ≤ ɛ0, then infΩpɛ, β ≥ ρβ. This proof is inspired by the one of. #### Step 1: Setting of an approximating eigenvalue problem. Here we introduce an approximating eigenvalue problem, that will be used to estimate from below the solutions to. Let δ > 0, ɛ > 0, aδ(y) : = min(a(y), supa − δ) and (λδ, ɛ, φδ, ɛ) be the principal eigenpair solving the problem $$\label{eq:beta-lowerbound2} \begin{system}{ll} \varepsilon\Delta\varphi^{\delta, \varepsilon} + \mu (M\star \varphi^{\delta, \varepsilon}-\varphi^{\delta, \varepsilon}) + (a^\delta(y)+\lambda^{\delta, \varepsilon})\varphi^{\delta, \varepsilon}=0 & \text{ in } \Omega \\ \frac{\partial \varphi^{\delta, \varepsilon}}{\partial \nu}=0 & \text{ on } \partial\Omega, \end{system}$$ with ∫Ωφδ, ɛ(y)dy = 1. It follows from Theorem [thm:eigen] that λδ, ɛ converges to the principal eigenvalue λδ, 0 of the operator ψ ↦ μ(M ⋆ ψ − ψ) + aδ(y)ψ when ɛ → 0. λδ, 0, in turn, converges to λ1 when δ → 0 by Lipschitz continuity. Thus we may approximate λ1 by λδ, ɛ for δ > 0 and ɛ > 0 small enough. Since $ y\mapsto \frac{1}{\sup a^\delta - a^\delta(y)}\not\in L^1(\Omega) $, it follows from (which can be adapted in our context; see ) that there exists a continuous eigenfunction associated with λδ, 0. In this case shows the strict upper bound λδ, 0 < − supaδ + μ = − supa + δ + μ. In what follows we fix the real number δ > 0 small enough so that the inequality $\delta < \frac{1}{2}\min\left(\mu,\sup a-\inf a, \sup a-\underset{\partial\Omega}{\sup}~ a^+-\mu\right) $ holds, together with $ \lambda^{\delta, 0} \leq \frac{3\lambda\_1}{4} $. We define η : = − λδ, 0 − supa + δ + μ > 0. Since λδ, ɛ → λδ, 0 as ɛ → 0, we fix ɛ0 > 0 such that for any 0 < ɛ < ɛ0, $ \|\lambda^{\delta, \varepsilon} - \lambda^{\delta, 0}\|\leq \frac{-\lambda\_1}{4} $, and $ \lambda^{\delta, \varepsilon}\leq \lambda^{\delta, 0} + \frac{\eta}{2}$. Finally, integrating equation, we have 0 = ∫Ω(aδ(y) + λδ, ɛ)φδ, ɛ(y)dy, thus the function aδ(y) + λδ, ɛ takes nonpositive and nonnegative values. This shows infa = infaδ ≤ − λδ, ɛ ≤ supaδ = supa − δ. Step 2: Estimates from above and from below of φδ, ɛ. Let us establish some upper and lower bounds for φδ, ɛ. Since φδ, ɛ is continuous on $ \overline\Omega$, there exists $ y\_0\in\overline\Omega $ such that $ \varphi^{\delta, \varepsilon}(y\_0)=\inf\_{z\in\overline\Omega}\varphi^{\delta, \varepsilon}(z) $. If y0 ∈ ∂Ω, then it follows from Hopf’s Lemma that $ \frac{\partial\varphi^{\delta, \varepsilon}}{\partial\nu}(y\_0) < 0 $, which contradicts the Neumann boundary conditions satisfied by φδ, ɛ (recall that a(y0) + λδ, ɛ < 0 for y0 ∈ ∂Ω). We conclude that y0 ∈ Ω. Thus we can evaluate equation : $$\begin{aligned} 0\geq -\varepsilon\Delta\varphi^{\delta, \varepsilon}(y\_0) &= \mu\left(M\star \varphi^{\delta, \varepsilon}-\varphi^{\delta, \varepsilon}\right)+\big(a^\delta(y\_0)+\lambda^{\delta, \varepsilon}\big)\varphi^{\delta, \varepsilon}, \\ (\sup a - \inf a+\mu)\varphi^{\delta, \varepsilon}(y\_0)&\geq \big(-\lambda^{\delta, \varepsilon} - a^\delta(y\_0)+\mu\big)\varphi^{\delta, \varepsilon}(y\_0)\geq \mu m\_0\int\_\Omega \varphi^{\delta, \varepsilon}, \\ \min\_{z\in\Omega}\varphi^{\delta, \varepsilon}(z)&\geq \frac{\mu m\_0}{\sup a-\inf a+\mu}. \end{aligned}$$ Similarly, there exists y0 ∈ Ω such that φδ, ɛ(y0) = maxz ∈ Ωφδ, ɛ(z). Evaluating equation, we get (recalling $ a^\delta(y\_0)-\mu + \lambda^{\delta, \varepsilon} \leq -\frac{\eta}{2}<0 $): $$\begin{aligned} 0\leq-\varepsilon \Delta \varphi^{\delta, \varepsilon} &= \mu M\star \varphi^{\delta, \varepsilon} + \big(a^\delta(y\_0)-\mu+\lambda^{\delta, \varepsilon}\big)\varphi^{\delta, \varepsilon}(y\_0), \\ \frac{\eta}{2}\varphi^{\delta, \varepsilon}&\leq \mu m\_\infty\int\_\Omega \varphi^{\delta, \varepsilon}=\mu m\_\infty, \\ \max\_{z\in\Omega}\varphi^{\delta, \varepsilon}(z)&\leq 2\frac{\mu m\_\infty}{\eta}. \end{aligned}$$ Hence, for 0 < ɛ ≤ ɛ0 and y0 ∈ Ω, we have shown that $$\frac{\mu m\_0}{\sup a-\inf a+\mu}\leq \varphi^{\delta, \varepsilon}(y\_0)\leq 2\frac{\mu m\_\infty}{\eta}.$$ Step 3: Lower estimate for pɛ, β. We are now in a position to derive a lower bound for pɛ, β. Since pɛ, β > 0 in $ \overline\Omega $, we can define α : = sup{ζ > 0 ∣ ∀y ∈ Ω, ζφδ, ɛ(y) ≤ pɛ, β(y)}. Assume by contradiction that $ \alpha< \alpha\_0:=\min\left(\frac{m\_0\eta}{2 k\_\infty m\_\infty}, \frac{-\lambda^{\delta, \varepsilon}\eta}{2\beta\mu m\_\infty+\eta k\_\infty}\right) $. By definition of α there exists $ y\_0\in\overline\Omega$ such that αφδ, ɛ(y0) = p(y0). Assume y0 ∈ ∂Ω, then it follows from Hopf’s Lemma that $ \frac{\partial(p^{\varepsilon, \beta}-\alpha \varphi^{\delta, \varepsilon})}{\partial\nu}(y\_0)<0 $, which contradicts the Neumann boundary conditions satisfied by pɛ, β and φδ, ɛ. Thus y0 ∈ Ω. We have: $$\begin{aligned} 0&\geq -\varepsilon\Delta(p^{\varepsilon, \beta}-\alpha\varphi^{\delta, \varepsilon})(y\_0)=\mu \left(M\star (p^{\varepsilon, \beta}-\alpha\varphi^{\delta, \varepsilon}) - (p^{\varepsilon, \beta}-\varphi^{\delta, \varepsilon})\right) \\ &\quad + p^{\varepsilon, \beta}\big(a(y\_0)-K\star p^{\varepsilon, \beta}-\beta p^{\varepsilon, \beta}\big)-\big(\lambda^{\delta, \varepsilon}+a^\delta(y\_0)\big)\alpha\varphi^{\delta, \varepsilon} \\ &=\int\_\Omega \big(\mu M(y\_0,z)-\alpha \varphi^{\delta, \varepsilon}(y\_0)K(y\_0, z)\big)\big(p^{\varepsilon, \beta}(z)-\alpha\varphi^{\delta, \varepsilon}(z)\big)dz\\ &\quad-\alpha\varphi^{\delta, \varepsilon}(y\_0)\int\_\Omega K(y\_0,z)\big(\alpha\varphi^{\delta, \varepsilon}(z)\big)dz\\ &\quad+\alpha\varphi^{\delta, \varepsilon}\big(a(y\_0)-a^\delta(y\_0)\big)-\beta \left(p^{\varepsilon,\beta}\right)^2 -\lambda^{\delta, \varepsilon}\alpha\varphi^{\delta, \varepsilon}(y\_0). \end{aligned}$$ By definition, $ \mu M(y\_0,z)-\alpha \varphi^{\delta, \varepsilon}(y\_0)K(y\_0,z)\geq \mu m\_0-k\_\infty\frac{m\_0\eta}{2 k\_\infty m\_\infty}\frac{2\mu m\_\infty}{\eta}= 0$ for any z ∈ Ω, and thus, recalling a(y0) ≥ aδ(y0), 0 ≥ − αλδ, ɛφδ, ɛ(y0) − α2φδ, ɛ(y0)(βφδ, ɛ + ∫ΩK(y0, z)φδ, ɛ(z)dz), $$\left(2\beta\frac{\mu m\_\infty}{\eta} + k\_\infty\right)\alpha\geq \alpha\left(\beta \varphi^{\delta, \varepsilon}(y\_0) + \int\_\Omega K(y\_0,z)\varphi^{\delta, \varepsilon}(z)dz\right)\geq -\lambda^{\delta, \varepsilon},$$ which is a contradiction since $ \alpha < \alpha\_0= \min\left(\frac{m\_0\eta}{2 k\_\infty m\_\infty}, \frac{-\lambda^{\delta, \varepsilon}\eta}{2\beta\mu m\_\infty+\eta k\_\infty}\right)$. We conclude that α ≥ α0 and thus (recalling $ \lambda^{\delta, \varepsilon}\leq \frac{\lambda\_1}{2}$) $$\begin{aligned} \min\_{y\in\Omega}p^{\varepsilon, \beta}(y)&\geq \alpha\_0 \min\_{y\in\Omega}\varphi^{\delta, \varepsilon}(y) \\ &\geq\min\left(\frac{m\_0\eta}{2 k\_\infty m\_\infty}, \frac{(-\lambda\_1)\eta}{4\beta\mu m\_\infty+2\eta k\_\infty}\right)\frac{\mu m\_0}{\sup a-\inf a+\mu}>0. \end{aligned}$$ Since this lower bound is independent from ɛ, this ends the proof of Lemma [lem:beta-lowerbound]. Construction of a stationary solution at ɛ = 0 ------------------------------------------------ In this section we assume λ1 < 0. Then, Theorem [thm:ex-stat-eps] guarantees the existence of a positive solution to for ɛ small enough, since λ1ɛ → λ1 as ɛ → 0 (recall Theorem [thm:eigen]). In this context, we expect the solution constructed in Theorem [thm:ex-stat-eps] with β = 0 to converge weakly to a (possibly singular) Radon measure, solution to. Here we prove this result, and complete the proof of Theorem [thm:survival]. In particular, in this section we assume β = 0. Before we can prove Theorem [thm:survival], we need a series of estimates on the previously constructed solutions pɛ : = pɛ, 0. [Estimates on the mass][lem:lowupintpeps] Let Assumption [hyp:gen] hold, let ɛ > 0, λ1ɛ < 0, and pɛ be a solution to equation with β = 0. Then $$\label{eq:lowupinteps} \frac{-\lambda\_1^\varepsilon}{k\_\infty}\leq \int\_\Omega p^\varepsilon(y)dy\leq \frac{\sup a}{k\_0}.$$ The upper bound in equation has been established in Lemma [lem:apriorip]. We turn our attention to the lower estimate. We assume by contradiction that λ1ɛ + k∞∫Ωpɛ(y)dy < 0. Let (λ1ɛ, φɛ > 0) be the solution to the eigenproblem, normalized with ∫φɛ = 1. Then, we define the real number α : = sup{ζ > 0 ∣ ∀y ∈ Ω, ζφɛ ≤ pɛ} > 0, which is then well-defined since pɛ > 0 and φɛ is bounded. By definition of α we have αφɛ ≤ pɛ and there exists a point $ y\_0\in\overline\Omega $ such that pɛ(y0) = αφɛ(y0). If y0 ∈ ∂Ω, since y0 is a maximum point for the function αφɛ − pɛ, then it follows from Hopf’s Lemma that $ \frac{\partial \alpha \varphi^\varepsilon-p^\varepsilon}{\partial \nu}(y\_0)>0 $, which contradicts the Neumann boundary conditions satisfied by pɛ and φɛ. Thus y0 ∈ Ω and we compute $$\begin{aligned} 0&\leq -\mu \big(M\star (\alpha\varphi^\varepsilon-p^\varepsilon)-(\alpha\varphi^\varepsilon-p^\varepsilon)\big)-\varepsilon\Delta(\alpha\varphi^\varepsilon-p^\varepsilon) \\ &=\lambda^\varepsilon\_1\alpha\varphi^\varepsilon+(K\star p^\varepsilon)p^\varepsilon+a(y\_0)(\alpha\varphi^\varepsilon-p^\varepsilon)= \big(\lambda\_1^\varepsilon + (K\star p^\varepsilon)(y\_0)\big)p^\varepsilon(y\_0), \end{aligned}$$ which is a contradiction since λ1 + (K ⋆ p)(y0) ≤ λ1 + k∞∫Ωpɛ(y)dy < 0. This finishes the proof of Lemma [lem:lowupintpeps]. [Proof of Theorem [thm:survival]] It follows from Lemma [lem:lowupintpeps] that the family (pɛ)0 < ɛ ≤ 1 of solutions to with ɛ > 0 and β = 0 is uniformly bounded in $ M^1(\overline\Omega) $. Hence, applying Prokhorov’s Theorem, (pɛ)0 < ɛ < 1 is precompact for the weak topology in $ M^1(\overline\Omega)$, and there exists a sequence pɛn (with ɛn → 0) and a measure p such that $ p^{\varepsilon\_n}\rightharpoonup p $ in the sense of measures. In particular, taking ψ = 1, we recover the estimate of Lemma [lem:lowupintpeps]: $ 0<\frac{-\lambda\_1}{k\_\infty}\leq \int\_{\overline\Omega}p(dy)\leq \frac{\sup a}{k\_0}$. Hence p is non-trivial. Let us show that p is indeed a solution to. Multiplying equation by ψ ∈ F0, where F0 is the set of functions with zero boundary flux as defined in, and integrating by parts, we get $$\begin{aligned} \label{eqthm:survival} -\varepsilon\_n\int\_\Omega p^{\varepsilon\_n}\Delta \psi dy &= \int\_\Omega \mu(M\star p^{\varepsilon\_n}-p^{\varepsilon\_n})\psi+a(y) p^{\varepsilon\_n}\psi dy \\ &\quad{-\int\_{\Omega}(K\star p^{\varepsilon\_n})(y) \psi(y)p^{\varepsilon\_n}(y) dy.}\nonumber \end{aligned}$$ Since $ \Delta \psi\in C\_b(\overline\Omega) $ and ∫Ωpɛn is bounded uniformly in n, the left-hand side of goes to 0 when n → ∞. Moreover since $ \psi(y)(a(y)-\mu)\in C(\overline\Omega) $, then by definition the convergence $\int\_{\Omega} \psi(y)(a(y)-\mu)p^{\varepsilon\_n}(y)dy\to\_{n\to\infty}\int\_{\overline\Omega}\psi(y)(a(y)-\mu)p(dy)$ holds. We turn our attention to the term ∫ΩM ⋆ pɛn(y)ψ(y)dy. We notice that ∫Ω∫ΩM(y, z)pɛn(z)dzψ(y)dy = ∫Ωpɛn(z)∫ΩM(y, z)ψ(y)dy = ∫Ωpɛn(z)M̌ ⋆ ψ(z)dz, where $\check M(y,z)=M(z, y)$. Since $\check M\star\psi(z) $ is a valid test function, we have indeed ∫ΩM ⋆ pɛn(y)ψ(y)dy → ∫ΩM ⋆ p(y)ψ(y)dy. We turn to the convergence of the nonlinearity ∫Ω(K ⋆ pɛn)(y)ψ(y)pɛn(y)dy. Since the sequence pɛn appears twice in this term, the above argument cannot be used directly. Therefore, we first show a stronger convergence for K ⋆ pɛn, namely that it converges uniformly to a continuous limit. We notice that $$\begin{aligned} \left\|(K\star p^{\varepsilon\_n})(y)-(K\star p^{\varepsilon\_n})(y')\right\| =& \left\|\int\_\Omega \big(K(y,z)-K(y', z)\big)p^{\varepsilon\_n}(z)dz\right\|\\ \leq & 2\Vert K(y, \cdot)-K(y', \cdot)\Vert\_{C\_b(\overline\Omega)}\frac{\sup a}{k\_0}. \end{aligned}$$ Thus, the modulus of continuity of K ⋆ pɛn is uniformly bounded. Up to the extraction of a subsequence, K ⋆ pɛn converges in $ C\_b(\overline\Omega) $ to a limit which we identify as K ⋆ p (by using another test function and the weak convergence $p^{\varepsilon\_n}\rightharpoonup p$). Along this subsequence, we have then limn → ∞∫Ω(K ⋆ pɛn)(y)ψ(y)pɛn(y)dy = ∫Ω(K ⋆ p)ψ(y)p(dy). We have shown that equation is satisfied for any ψ ∈ F0. Applying Lemma [lem:zerofluxdense], F0 is densely embedded in $ C\_b(\overline\Omega) $. Equation is thus satisfied for any $ \psi\in C\_b(\overline\Omega) $. This ends the proof of Theorem [thm:survival]. Proof of Theorem [thm:concentration] ------------------------------------ Under Assumption [hyp:dom] and if μ is small enough according to Theorem [thm:smallmu], we can actually prove that the measure solution to is concentrated in Ω0 (Theorem [thm:concentration]). To do so, we write a solution p to as an eigenvector for a problem similar to, and make use of Theorem [thm:eigenpair]. [Proof of Theorem [thm:concentration]] Let $ p\in M^1(\overline\Omega) $, $ p\not\equiv 0 $ be a nonnegative solution to. Define b(y) : = a(y) − (K ⋆ p)(y). Then b is a continuous function and we have $$\forall y\in\overline\Omega, b(y)=a(y)-\int\_{\overline\Omega} K(y,z)p(dz)\leq a(0)-\int\_{\overline\Omega}K(0, z)p(dz) = b(0),$$ as a result of Assumption [hyp:dom]. This shows that supb = b(0). Next we compute: $$b(0)-b(y) = a(0)-a(y) + \int\_{\overline\Omega}\big(K(y,z)-K(0,z)\big) p(dz) \geq a(0)-a(y).$$ Thus, b satisfies Assumption [hyp:dega]. We remark that p solves μ(M ⋆ p − p) + b(y)p = 0 in the sense of measures. Thus p is a solution to with a replaced by b. Applying Corollary [cor:compmucrit], since μ < μ0(Ω, M, supa − a), then μ < μ0(Ω, M, supb − b) and thus the only solutions to are singular measures which singular part is concentrated in {y ∈ Ω ∣ b(y) = supb}. Let us show that {y ∣ b(y) = supb} ⊂ Ω0. Let $ y\in\overline\Omega $ such that y ∉ Ω0. Then a(y) < a(0) and b(y) = a(y) − (K ⋆ p)(y) < a(0) − (K ⋆ p)(y) ≤ a(0) − (K ⋆ p)(0) = supb, which shows that y ∉ {y ∣ b(y) = supb}. This ends the proof of Theorem [thm:concentration]. Construction of traveling waves =============================== In this section, we prove our main result Theorem [thm:TW]. To construct the desired measure traveling wave, we first consider a regularized problem in a box − l ≤ x ≤ l, y ∈ Ω. Construction of a solution in a box ----------------------------------- Here we aim at constructing solutions (c, u = u(x, y)) to $$\label{pb:box-beta} \begin{system}{ll} -\varepsilon\Delta\_yu-u\_{xx}-cu\_x= \mu(M\star u-u) \hspace{2cm} \\ \hfill+ u(a(y)-K\star u-\beta u) & \text{ in } (-l, l)\times\Omega\\ \nabla\_y u(x,y)\cdot\nu=0 & \text{ on } (-l,l)\times\partial\Omega \\ u(l,y)=0 & \text{ in } \Omega \\ u(-l,y)=p(y) & \text{ in } \Omega, \end{system}$$ for β ≥ 0 and p solving. Notice that any solution to with β > 0 is a subsolution to with β = 0. In particular, we will use some solutions to with β > 0 to get lower estimates on solutions to with β = 0. In contrast with, we use a global continuation theorem (in the proof of Theorem [thm:solbox] below) instead of a topological degree to construct solutions to the local problem. Though both arguments have the same topological basis, we believe that this is an improvement of the usual method, since it spares the need to explicitly compute the topological degree associated with. Let us also introduce the following quantity, which is the minimal speed for traveling waves (as we will show later): $$\label{def:minimal-speed} c^\\_\varepsilon:=2\sqrt{-\lambda\_1^\varepsilon}.$$ Our result is the following: [Existence of solutions in the box][thm:solbox] Let Assumption [hyp:gen] hold, ɛ* > 0 be such that λ1ɛ < 0, and β ≥ 0. Then, there exists a nonnegative solution to. Moreover, let $ l\_0:=\frac{\pi}{\sqrt{-\lambda\_1^\varepsilon}}>0 $, $ \tau\_0:=\frac{-\lambda\_1^\varepsilon}{2}>0 $. Then, for any 0 < τ < τ0, there exists $ \bar l(\tau)\geq l\_0 + 1 $ such that if $l>\bar l(\tau) $, there exists a nonnegative solution (c, u) to with 0 < c ≤ cɛ\, which also satisfies the normalization condition $$\label{eq:norm} \sup\_{(x, y)\in(-l\_0, l\_0)\times \Omega}\left(\int\_\Omega K(y,z)u(x,z)\mathrm dz+\beta u(x,y)\right) = \tau.$$ Before we prove Theorem [thm:solbox], we need to establish some a priori* estimates on the solutions to. For technical reasons, we actually study the solutions to $$\label{pb:box-beta-pos} \begin{system}{ll} -\varepsilon\Delta\_yu-u\_{xx}-cu\_x =\sigma \big( \mu (M\star u-u) \hspace{1.5cm} \\ \hfill+ u\chi\_{u\geq 0}(a(y)-K\star u-\beta u)\big) & \text{ in } (-l,l)\times\Omega \\ \nabla\_y u(x,y)\cdot\nu=0 & \text{ on } (-l,l)\times\partial\Omega \\ u(l,y)=0 & \text{ in } \Omega \\ u(-l,y)=p(y) & \text{ in } \Omega, \end{system}$$ where χu ≥ 0 = 0if u ≤ 0, χu ≥ 0 = 1if u > 0, and σ ∈ (0, 1]. We introduce the positive-part cutoff involving χ in Problem in order to ensure that the nontrivial solutions to this problem are positive. [A priori estimates on the solutions to ][lem:aprioribox] Let Assumption [hyp:gen] hold, ɛ > 0 such that λ1ɛ < 0, β ≥ 0, and ∣c∣ ≤ cɛ\. We define $ l\_0:=\frac{\pi}{\sqrt{-\lambda\_1^\varepsilon} }$. Let u* be a solution to, then 1. [item:aprioribox-regular] $ u \in C^2\_{loc}\big((-l, l)\times\Omega\big)\cap C^1\_{loc}\big((-l,l)\times\overline\Omega\big)\cap C\_b\big([-l,l]\times\overline\Omega\big) $. 2. [item:aprioribox-positive] u is positive in $ (-l, l)\times \overline\Omega $. 3. [item:aprioribox-boundint] For any x ∈ [ − l, l], we have $ \int\_\Omega u(x, y)dy\leq \frac{\sup a}{k\_0} $. 4. [item:aprioribox-boundinfty] There exists a positive constant Cɛ, independent from c, l and σ, such that we have ‖u‖Cb(( − l, l) × Ω) ≤ Cɛ. If β > 0, then we have the estimate $ \Vert u\Vert\_{C\_b((-l, l)\times\Omega)}\leq \frac{\sup a}{\beta} $. 5. [item:aprioribox-minspeed] If σ = 1, c = 0, and l > l0, then $$\sup\_{(x, y)\in(-l\_0, l\_0)\times \Omega}\left(\int\_\Omega K(y,z)u(x,z)\mathrm dz+\beta u(x,y)\right) > \frac{-\lambda\_1^\varepsilon}{2}.$$ Remark that for this result to hold, u needs only be defined on ( − l0, l0) × Ω. 6. [item:aprioribox-maxspeed] If σ = 1 and c = cɛ\, then there exists a constant A* (independent from l) and $ \lambda:=\frac{c\_\varepsilon^\}{2}>0 $ such that ∀(x, y) ∈ ( − l, l) × Ω, u* ≤ Ae− λ(x + l). In particular for any $ l\geq \bar l(\tau):= \frac{1}{\lambda}\ln\left(\frac{\tau}{2A(k\_\infty\int\_\Omega\varphi^\varepsilon + \beta\sup\_\Omega\varphi^\varepsilon)}\right)-l\_0 $ and $ 0<\tau\leq \tau\_0= \frac{-\lambda\_1^\varepsilon}{2} $, we have $$\sup\_{(x, y)\in(-l\_0, l\_0)\times \Omega}\left(\int\_\Omega K(y,z)u(x,z)\mathrm dz+\beta u(x,y)\right) < \tau.$$ Item [item:aprioribox-regular] holds by a direct application of, and item [item:aprioribox-positive] by a classical comparison argument. Let us resume to the remaining items. Item [item:aprioribox-boundint]: By the estimate in Lemma [lem:lowupintpeps], we have $ \int\_\Omega p(y)dy\leq\frac{\sup a}{k\_0} $. Assume that the function x ↦ ∫Ωu(x, y)dy has a maximal value at x0 ∈ ( − l, l), then integrating over Ω we have $$\begin{aligned} 0&\leq -\frac{d^2 \int\_\Omega u(x\_0,y)dy}{dx^2}-c\frac{d \int\_\Omega u(x\_0,y)dy}{dx}\\ &=\sigma \int\_\Omega a(y)u(x\_0,y) - (K\star u)(x\_0,y)u(x\_0, y)dy, \end{aligned}$$ and thus: $$\begin{aligned} k\_0\left(\int\_\Omega u(x\_0,y)dy\right)^2&\leq \int\_\Omega\int\_{\Omega} K(y,z)u(x\_0,z)u(x\_0,y)dydz \\ &= \int\_\Omega a(y)u(x\_0,y)dy\leq \sup a\int\_\Omega u(x\_0,y)dy. \end{aligned}$$ This shows item [item:aprioribox-boundint]. Item [item:aprioribox-boundinfty]: Assume first β > 0 and let u(x0, y0) = supu at $ (x\_0,y\_0)\in[-l,l]\times \overline\Omega $. Assume by contradiction that $ u(x\_0,y\_0)>\frac{\sup a}{\beta} $. If x0 = − l, since p satisfies the upper bound $ \sup p\leq \frac{\sup a}{\beta} $ by the estimate in Lemma [lem:apriorip] item [item:apriorip-boundinfty], we have a contradiction. If x0 = + l, since u(x0, y0) = 0, we have a contradiction. Assume x0 ∈ ( − l, l). If y0 ∈ ∂Ω, then it follows from Hopf’s Lemma that $ \frac{\partial u}{\partial\nu}(x\_0,y\_0)>0 $, which is a contradiction. Thus y0 ∈ Ω. Now, testing at (x0, y0), we have $$\begin{aligned} 0&\leq -\varepsilon \Delta\_y u(x\_0,y\_0) -u\_{xx}(x\_0,y\_0)-cu\_x(x\_0,y\_0)-\sigma \mu\big((M\star u)-u\big) (x\_0,y\_0)\\ &=\sigma u(x\_0,y\_0)\big(a(y\_0)-(K\star u)(x\_0,y\_0)-\beta u(x\_0,y\_0)\big)<0 \end{aligned}$$ which is a contradiction. Thus $ u\leq \frac{\sup a}{\beta} $. We turn our attention to the case β = 0. In this case, we construct a supersolution as in Lemma [lem:apriorip]. Recalling that u satisfies Dirichlet boundary conditions at x = ± l, the local maximum principle up to the boundary shows the existence of C = C(Ω, R, ɛ, ‖a‖L∞, k0, k∞, μ, cɛ\) such that for any x* ∈ [ − l, l], y ∈ Ω with d(y, ∂Ω) ≥ R, we have the estimate $\underset{B\_{R/2}(x,y)}{\sup}\,u\leq C$. To show that this estimate does not degenerate near the boundary, we use the same kind of supersolution as in Lemma [lem:apriorip]. Let ΩR : = {y ∈ Ω ∣ d(y, ∂Ω) < R} for any R > 0. We select R small enough so that ΩR has a C3 boundary and the comparison principle holds in the narrow domain ΩR. Let us stress that since σ ∈ (0, 1), R can be chosen uniformly in σ. This allows us to construct a positive solution to $$\begin{system}{ll} -\varepsilon\Delta v-\sigma (a(y)-\mu)v = \mu m\_0 \frac{m\_\infty \sup a}{k\_0} & \text{ in }\Omega \\ v=C & \text{ on }\partial \Omega\_R{\backslash}\partial\Omega \\ \frac{\partial v}{\partial\nu}=0 & \text{ on }~\partial\Omega, \end{system}$$ which is bounded uniformly in σ, as we did in the proof of Lemma [lem:apriorip]. Now we select α : = inf{ζ > 0 ∣ ∀x ∈ ( − l, l), ∀y ∈ Ω, ζv(y) ≥ u(x, y)}. Assume by contradiction that α > 1. Then there exists $ (x\_0, y\_0)\in [-l,l]\times\overline\Omega $ such that u(x0, y0) = αv(y0). If x0 = l then u = 0, which is a contradiction. If x0 = − l, since u( − l, y0) = p(y0) solves, we argue as in Lemma [lem:apriorip] and get a contradiction. We are left to investigate the case x0 ∈ ( − l, l). If y0 ∈ ∂Ω, since (x0, y0) is a minimum to the function αv − u, then by Hopf’s Lemma we have $ \frac{\partial (\alpha v-u)}{\partial\nu}(x\_0,y\_0)<0 $ which is a contradiction. Thus y0 ∉ ∂Ω. Since α > 1 and u ≤ C on ∂ΩR ∖ ∂Ω, then y0 ∈ ΩR. Now (x0, y0) is a local minimum to αv − u and thus $$\begin{aligned} 0&\geq -\varepsilon\Delta (\alpha v-u)(x\_0,y\_0) = \sigma(a(y\_0)-\mu)(\alpha v-u)(x\_0,y\_0)+\alpha \mu m\_0 \frac{m\_\infty \sup a}{k\_0}\\ &\quad - \sigma \mu (M\star u)(x\_0,y\_0) +u(x\_0,y\_0)(K\star u)(x\_0,y\_0) \\ &> \alpha \mu m\_0 \frac{m\_\infty \sup a}{k\_0}-\sigma\mu (M\star u)(x\_0,y\_0)\geq 0 \end{aligned}$$ which is a contradiction. Thus α ≤ 1. This shows that u(x, y) ≤ v(y) in ( − l, l) × ΩR. Since v is bounded uniformly in σ, we have our uniform bound for u in [ − l, l] × ΩR. In the rest of the domain ( − l, l) × Ω ∖ ΩR, we have u ≤ C. This proves item [item:aprioribox-boundinfty], with Cɛ : = max(supy ∈ ΩRv(y), C). Item [item:aprioribox-minspeed]: This proof is similar to the one in. Assume by contradiction that $\sup\_{(x, y)\in(-l\_0, l\_0)\times \Omega}\left(\int\_\Omega K(y,z)u(x,z)\mathrm dz+\beta u(x,y)\right) \leq \tau\_0$. Then u satisfies: − uxx − ɛΔyu − μ(M ⋆ u − u) − a(y)u ≥ − τ0u in ( − l0, l0). Define $ \psi(x,y):=\cos\left(\frac{\pi}{2l\_0}x\right)\varphi^{\varepsilon}(y) $, where φɛ is the principal eigenfunction solution to satisfying supy ∈ Ωφɛ = 1. Since u is positive in $ [-l\_0, l\_0]\times \overline\Omega $, we can define the real number α : = sup{ζ > 0 ∣ ∀(x, y) ∈ ( − l0, l0) × Ω, ζψ(x, y) ≤ u(x, y)}, and we have α > 0. Then, there exists $ (x\_0,y\_0)\in [-l\_0, l\_0]\times \overline\Omega $ such that αψ(x0, y0) = u(x0, y0). Because of the boundary conditions satisfied by u and ψ, (x0, y0) has to be in ( − l0, l0) × Ω. Since (x0, y0) is the minimum of u − αψ, we have $$\begin{aligned} 0&\geq -\varepsilon \Delta\_y(u-\alpha\psi)(x\_0,y\_0) -(u-\alpha\psi)\_{xx}(x\_0,y\_0) \\ &\quad -\mu\big(M\star(u-\alpha\psi)-(u-\alpha\psi)\big)(x\_0,y\_0) -a(y\_0)(u-\alpha\psi)(x\_0,y\_0) \\ &\geq-\tau\_0 u(x\_0,y\_0) +\alpha\left(- \lambda\_1^\varepsilon-\left(\frac{\pi}{2l\_0}\right)^2\right)\psi(x\_0,y\_0) \\ &=\left(\frac{-3\lambda\_1^\varepsilon}{4}-\tau\_0\right)u(x\_0,y\_0)>0, \end{aligned}$$ since $ -\tau\_0= \frac{\lambda\_1^\varepsilon}{2} $. This is a contradiction. This proves item [item:aprioribox-minspeed]. Item [item:aprioribox-maxspeed]: Let $ \psi(x, y):=e^{-\frac{c^\\_\varepsilon}{2} x} \varphi^\varepsilon(y) $ with (λ1ɛ, φ*ɛ) solution to. Then ψ satisfies: − cɛ\ψx* − ψxx − ɛΔyψ − μ(M ⋆ ψ − ψ) = a(y)ψ. Since ψ > 0 on $ [-l, l]\times \overline\Omega $, there exists ζ > 0 such that ζψ ≥ u on ( − l, l) × Ω. Let us select α : = inf{ζ > 0 ∣ ∀(x, y) ∈ ( − l0, l0) × Ω, ζψ(x, y) ≥ u(x, y)}. By definition of α we have αψ ≥ u and there exists $ (x\_0,y\_0)\in [-l, l]\times \overline\Omega $ such that αψ(x0, y0) = u(x0, y0). Because of the boundary conditions satisfied by u and ψ, (x0, y0) has to be in [ − l, l) × Ω. If x0 ∈ ( − l, l), we have: $$\begin{aligned} 0&\leq -\varepsilon \Delta\_y(u-\alpha\psi)(x\_0,y\_0) -(u-\alpha\psi)\_{xx}(x\_0,y\_0) \\ &\quad -\mu\big(M\star(u-\alpha\psi)-(u-\alpha\psi)\big)(x\_0,y\_0) -a(y\_0)(u-\alpha\psi)(x\_0,y\_0) \\ &<0 \end{aligned}$$ which is a contradiction. We conclude that x0 = − l0 and thus $ \alpha\leq \frac{\sup\_\Omega p}{\inf\_\Omega \varphi^\varepsilon}e^{-\frac{c\_\varepsilon^\}{2} l} $. By definition of α, we can then write: $$u(x,y)\leq \alpha e^{-\frac{c\_\varepsilon^\}{2} x}\varphi^\varepsilon(y)\leq \frac{\sup\_\Omega p}{\inf\_\Omega \varphi^\varepsilon}e^{-\frac{c^\\_\varepsilon}{2}(x+l)}\varphi^\varepsilon(y)$$ which concludes the proof of item [item:aprioribox-maxspeed]. We are now in the position to prove Theorem [thm:solbox], by using the global continuation principle. [Proof of Theorem [thm:solbox]] For $ c\in\mathbb R $ and $ u\in C\_b\big((-l, l)\times\Omega\big)$. We define $ F(c,u)=\tilde u $ where $ \tilde u $ solves: $$\label{eqthm:solbox-1} \begin{system}{ll} -\tilde u\_{xx}-c\tilde u\_x-\varepsilon\Delta\_y\tilde u = \mu M\star u \hspace{2cm} \\ \hfill+u\chi\_{u\geq 0}(a(y)-\mu-K\star u-\beta u) & \text{ in } (-l,l)\times\Omega \\ \frac{\partial\tilde u}{\partial\nu}(x,y)=0 & \text{ on } (-l,l)\times\partial \Omega \\ \tilde u(l,y)=0 &\text{ in } \Omega \\ \tilde u(-l,y)=p(y).&\text{ in } \Omega. \end{system}$$ It follows from that for any u* the function $ \tilde u $ is well-defined and belongs to the space $ C\_b([-l, l]\times \overline\Omega)\cap W^{2, p}\_{loc}([-l, l]\times \overline\Omega{\backslash}\{-l,l\}\times \partial\Omega) $ for any p > 0. Step 1: Let us briefly show that F is in fact a compact operator. Since the right-hand side of the first equation in is bounded, it is easily seen that the function $ (x, y)\mapsto \big(1+\gamma (x+l)^\alpha\big) p(y) $ is a local supersolution to equation near x = l for 0 ≤ α < α0, γ ≥ γ0 > 0, where α0 and γ0 depend only on ‖u‖Cb(( − l, l) × Ω), a bound for c and the data and coefficients of the problem. Similarly, $ (1-\gamma(x+l)^\alpha\big) p(y) $ is a local subsolution, provided α is chosen small enough. Thus the inequality $\big(1-\gamma (x+l)^\alpha\big) p(y) \leq \tilde u(x,y) \leq \big(1+\gamma (x+l)^\alpha\big) p(y) $ holds for α > 0 small enough. In particular, the function $ x\in [-l, 0]\mapsto \tilde u(x,y) $ is uniformly in Cα for $ y\in\overline\Omega $. It then follows from (and the classical interior Sobolev embeddings) that $ \tilde u\in C^\alpha([-l, 0]\times\overline\Omega) $. Regularity near x = l can be shown the same way. Thus $ \tilde u\in C^\alpha([-l, l]\times\overline\Omega) $ where α depends only on a bound for ‖u‖Cb(( − l, l) × Ω) and c, and the data and coefficients of problem. In particular, F maps bounded sets of $ \mathbb R\times C\_b((-l, l)\times \Omega) $ into relatively compact sets in Cb(( − l, l) × Ω). Step 2: We aim at applying the Leray-Schauder fixed-point theorem to F(0, ⋅ ). We remark that the solutions to u = σF(0, u) for σ ∈ (0, 1] are in fact the solutions to. In particular, Lemma [lem:aprioribox] gives us a positive constant C > 0 such that any solution to satisfies the inequality ‖u‖Cb( − l, l) × Ω ≤ C. Let G : = {u ∈ Cb(( − l, l) × Ω) ∣ ‖u‖Cb(( − l, l) × Ω) ≤ 2C}, then 1. G is a bounded open subset of the Banach space Cb(( − l, l) × Ω), 2. 0 ∈ G, 3. F(0, ⋅ ) : G → Cb(( − l, l) × Ω) is a compact mapping, and 4. applying Lemma [lem:aprioribox], there is no solution to u = σF(0, u) with u ∈ ∂G and σ ∈ (0, 1]. Thus the Leray-Schauder fixed-point Theorem applies and we have ind(F(0, ⋅ ), G) = 1, where ind is the Leray-Schauder fixed-point index. Step 3: Let us now check that the hypotheses of the global continuation principle are satisfied. We have: 1. F is a compact mapping from $ (0, c\_\varepsilon^\)\times \overline G $ into C*b(( − l, l) × Ω), 2. applying Lemma [lem:aprioribox], there is no solution to u = F(c, u) with u ∈ ∂G and c ∈ [0, cɛ\], and 3. ind	arxiv_0000698
The Physics of the Intergalactic Medium[1](#fn1) ================================================ Intergalactic space is filled with a pervasive medium of ionized gas, the Intergalactic Medium (IGM). A residual neutral fraction is detected in the spectra of Quasi-Stellar Objects at both low and high redshifts, revealing a highly fluctuating medium with temperatures characteristic of photoionized gas. The statistics of the fluctuations are well-reproduced by numerical gravity-hydrodynamics simulations within the context of standard cosmological structure formation scenarios. As such, the study of the IGM offers an opportunity to probe the nature of the primordial density fluctuations on scales unavailable to other methods. The simulations also suggest the IGM is the dominant reservoir of baryons produced by the Big Bang, and so the principal source of the matter from which galaxies formed. The detection of metal systems within the IGM shows that it was enriched by evolved stars early in its history, demonstrating an intimate connection between galaxy formation and the IGM. The author presents a comprehensive review of the current understanding of the structure and physical properties of the IGM and its relation to galaxies, concluding with comments on prospects for furthering the study of the IGM using future ground-based facilities and space-based experiments. INTRODUCTION ============ According to the Big Bang theory, the primordial hydrogen and helium created in the Universe first materialized in the form of an extremely hot ionized gas. By the time the Universe was three hundred thousand years old, adiabatic expansion cooling brought the temperature of the primordial plasma down until the hydrogen and helium recombined. The radiation last scattered at this time appears today as the Cosmic Microwave Background (CMB). The search for the IGM began well before the discovery of the CMB with an attempt by to detect the hyperfine 21cm absorption signature from hydrogen along the line of sight to the extragalactic radio galaxy Cygnus A. Although no detection was made, combining the optical depth with the measured temperature of the CMB discovered in 1965 would have been sufficient to exclude the possibility that the Universe was closed by baryons, with an upper limit on the baryon density of only 20% of the closure density required for an Einstein-deSitter (flat) universe. [2](#fn2) Nearly coincident with the discovery of the CMB, however, a considerably improved measurement of the density of intergalactic neutral hydrogen was made. Soon after the discovery of the first Quasi-Stellar Object (QSO), reported a small decrement in a QSO spectrum shortward of its emission line. Attributing the decrement to the resonance scattering of radiation from the QSO by intergalactic neutral hydrogen, Gunn and Peterson demonstrated that the cosmic mass density of neutral hydrogen was exceedingly smaller than the Einstein-deSitter density. In fact, it was far smaller than the spatially averaged hydrogen of all the stars in the Universe. If the Big Bang theory was correct, it meant either that galaxy formation was an extraordinarily efficient process, sweeping up all but a tiny residue of the primordial hydrogen, or that the gas was reionized. These two themes, the detection of intergalactic gas through the 21cm signature in the radio or through Lyman resonance scattering lines in the optical or ultraviolet (UV), continue to dominate studies of the Intergalactic Medium (IGM). To date, almost all that is known about the structure of the IGM has been derived from optical and UV data. This situation is expected to change dramatically in the near future with the development of low frequency radio arrays like the LOw Frequency ARray (LOFAR)[3](#fn3), the Murchison Widefield Array (MWA)[4](#fn4), the Primeval Structure Telescope (PaST/21CMA) [5](#fn5), the Precision Array to Probe the Epoch of Reionization (PAPER) [6](#fn6), and a possible Square Kilometre Array (SKA). [7](#fn7) A primary science driver of these instruments is the direct imaging of the IGM prior to the completion of the Epoch of Reionization. Most of this review focuses on the current understanding of the state of the IGM as determined from optical and UV measurements. The observations have relied almost exclusively on the spectra of QSOs, although IGM absorption features have also been detected in the spectra of Gamma-Ray Bursts (GRBs), and indeed played a key role in establishing the extragalactic character of some bursts. Almost immediately after Gunn and Peterson publicized their finding, it was recognized that individual absorption features should appear from neutral hydrogen concentrated into cosmological structures. Absorption features had in fact been detected in higher resolution QSO spectra, but these were identified with intervening ionized metal absorption systems, as was expected if galaxies had hot halos of ionized gas: the lines of sight to background QSOs were expected to pass through such hot halos and intercept any ionized gas clouds within them. The features, however, were uncomfortably common, hinting at a class of unknown structures not associated with galaxies. The resonance line features continued to prove elusive until 1971, when recognized several absorption features shortward of the emission line of a QSO as lines[8](#fn8) arising in a population of discrete absorption systems also showing metal features. The lines form a plethora of distinct absorption features in the spectra of high redshift QSOs; they are collectively known today as the forest. The properties of the forest came under increasing scrutiny, with the first systematic survey conducted several years later by, convincingly demonstrating through the homogeneity of the observed properties of the absorbers their intergalactic origin, as opposed to clouds ejected by the QSOs observed. Although limited by the resolution of the spectrograph, the measured velocity widths of the features corresponded to gas temperatures of a few to several 104 K, characteristic photoionization temperatures for gas of a primordial composition, for which there is no significant cooling by metals. The number of features per comoving length was shown to increase with redshift, demonstrating that the systems were evolving. The past decade has witnessed a dramatic improvement in precision studies of the forest with the advent of 8–10 m class telescopes, particularly the Keck 10-m and the 8.2m Very Large Telescope (VLT). For the first time, the individual absorption features in the forest were spectroscopically resolved over their full range. Velocity widths of $\sim25\kms$ are typical. The neutral hydrogen column densities of the absorbers range from roughly $10^{12}-10^{22}\,{\rm cm^{-2}}$. The highest column density systems, the Damped Absorbers (DLAs), are of particular interest for galaxy formation, as they are suspected of containing the neutral gas that formed the bulk of the stars in present day galaxies. As the number per length of absorption systems increases along a line of sight with increasing redshift, so does the mean flux decrement in a background QSO spectrum due to scattering. The QSO spectra measured as part of the Sloan Digital Sky Survey (SDSS) [9](#fn9) show a rapid rise in the flux decrement at $z\gta5.5$, suggesting that the epoch of reionization may lie not far beyond z ≃ 6. Many of the hydrogen features also show absorption lines from metals, including carbon, silicon, nitrogen, oxygen, magnesium, iron and others. The abundances of the metals, however, are at most about 10% of solar at low redshifts, and as low as 1% at high redshifts, indicating that the absorption systems are comprised largely of primordial material. The primordial nature of the gas received further important confirmation in 1994 with the discovery by of intergalactic helium using the Hubble Space Telescope (HST) through the detection of absorption. Because the baryons in the IGM are detected only though their absorption signatures, the physical structures that give rise to the features must be modeled. Early models characterized the systems as discrete clouds of gas, with most of the focus on either clouds pressure-confined by a hot medium, or gravitationally confined in a dark matter halo. At the time it was believed that the absorption systems accounted for only a few percent of the baryons produced in the Big Bang, much like galaxies, their visible counterparts. A paradigm shift in the models occurred in the mid 1990s. Measurements of coincident absorption features along neighboring lines of sight suggested sizes of tens to hundreds of kiloparsecs for the absorbers, much larger than expected for clouds confined by pressure or dark matter halos. A radical transformation in the understanding of the nature and structure of the IGM was initiated by numerical simulations of cosmological structure formation. Today essentially all of the baryons produced in the Big Bang are believed to have undergone the same gravitational instability process initiated by primordial density fluctuations that was responsible for the formation of galaxies. The computation of the structure of the IGM has been converted into an initial value problem similar to that of the CMB fluctuations. Fluctuations in the CMB are solved for by following the gravitational instability growth of a postulated spectrum of primordial dark matter density fluctuations. The growth of structure in the IGM is now treated as the nonlinear extension of these computations. The result is a network of filamentary structures, the so-called “cosmic web”. The forest is believed to be a signature of the cosmic web. Early simulations broadly reproduced the statistics of the forest spectacularly well. An immediate conclusion was that at $z\gta1.5$, some 90% of the baryons produced in the Big Bang are contained within the IGM, with only 10% in galaxies, galaxy clusters or possibly locked up in an early generation of compact stars. Soon after the discovery of intervening absorption features, it was recognized that they provided potentially powerful tests of fundamental properties of the Universe. The split in the fine structure lines of the metals was used to set constraints on the variability of the fine structure constant. A bunching of absorption features near z ≃ 1.9 (now known to be fortuitous), gave rise to the (re)introduction of a cosmological constant to account for the numerous features as multiple images due to lensing. The expected primordial composition of the IGM offered the potential of placing constraints on the photon-to-baryon ratio of the Universe through measurements of the intergalactic deuterium abundance. More recently, the success of the models has inspired attempts to exploit the forest as a new means to constrain cosmological structure formation models and obtain stringent constraints on the cosmological parameters. The description of the IGM by the simulations, however, is far from complete. There remain many unsolved problems. The current simulations do not reproduce all the observed properties of the IGM. The absorption lines are predicted to be substantially narrower than measured. This likely stems from the principal outstanding missing piece of physics, the reionization of the IGM. Not only must hydrogen be ionized, but helium as well. The ionization heats the gas through the photoelectric effect. Detailed radiative transfer computations are required to recover the temperatures, for which there is still limited success. The sources of the reionization and the epochs of reionization, both of hydrogen and helium, are still not firmly established. The origin of the metal absorption systems in the diffuse IGM is still unknown, although it is widely expected they were deposited by winds from galaxies, possibly driven by intense episodes of star formation. As such, the metal absorption lines in principle offer an important means of studying the history of cosmic star formation. Most fundamentally, the relation of the IGM to the galaxies that form from it is still mostly unknown, but offers perhaps the most exciting prospects for new research directions. The purpose of this review is to describe the progress made in the understanding of the origin of structure within the IGM, with a view to presenting the underlying physics that determines the structure. An understanding of the physics is necessary for future progress. The past decade has revealed the IGM to be a complex dynamical arena involving interactions between the IGM gas, galaxies and QSOs. It is becoming increasingly apparent that the separation of these systems into distinct and isolated entities is an artificial construct. Galaxies and QSOs originated from the IGM, and their radiation and outflows impacted upon it. Any complete understanding of the origin of these systems requires a unified treatment. In this way, the IGM resembles the interstellar medium of disk galaxies in which the gaseous component is intimately linked to the stars and their evolution and impact. Interpreting the increasingly refined observations requires detailed modeling, which relies on large-scale numerical computations involving gas, radiative processes, and gravity. The physics involved is intermediate in complexity between that of the CMB and galaxy formation, rendering the IGM a bridge between these extreme scales of cosmological structure formation. Unraveling the processes that led to the formation and structure of the IGM may thus serve as a crucial step in the solution of the much more involved problem of galaxy formation. [fig:citations] A search of the literature for papers on the IGM since the Gunn & Peterson (1965) measurement produces close to 6000 references.[10](#fn10) While this review does not have the space to describe the observational methods used to measure the IGM, it should be recognized that progress in the understanding of the properties of the IGM is indebted to advances in observational techniques. This is well illustrated by a plot of the number of refereed publications in the field against time. Periods of relatively steady output are punctuated by four leaps. The first occurs at the end of the 1960s and beginning of the 1970s with the introduction of image tube spectrographs coupled with integrating television systems for photon counting, which greatly facilitated the taking of spectra. The next occurs in the mid-1970s with the development of x-ray astronomy following the launch of the UHURU satellite in 1970 and the recognition that galaxy clusters contain an extended and pervasive medium of hot, radiating gas. Another sharp rise occurs in the early 1990s following the launch of the Hubble Space Telescope in 1990, the installation of EMMI and its echelle spectroscope on the New Technology Telescope (NTT), and the delivery of the HIRES spectrometer to the Keck telescope. The fourth occurred in 2000 with the introduction of the UV and Visible Echelle Spectrograph (UVES) to the Very Large Telescope (VLT), the launch of the Far Ultraviolet Spectroscopic Explorer (FUSE), and the beginning of operations of the Sloan Digital Sky Survey. The latter in particular triggered a surge of activity in reionization studies following the discovery of high redshift QSOs (z > 6) with spectra indicating a rapid rise in the effective optical depth of the IGM to photons, hinting that the Epoch of Reionization was being approached. This is indicated by a rise in IGM reionization papers in Fig. [fig:citations], a trend which continues today, fueled in part by the growing interest in the influence of reionization on the CMB fluctuations measured by the Wilkinson Microwave Anisotropy Probe (WMAP). The next major leap may well come from the anticipated radio measurements. The rapidly rising tide of IGM studies has brought along with it several reviews. This is fortunate, as it is impossible in a single review to cover all areas thoroughly today. Early reviews of the QSO absorption line literature, now largely historical, were provided by and. and provide reviews of the forest that are still largely up-to-date in the sense that most of the topics currently engaging the community are treated, with developments since mostly improvements in accuracy and in the details of the numerical models. Reviews of the low redshift IGM are provided by and. have reviewed the current understanding of Damped Absorbers, an absorber class of special concern as it represents the closest link to the gaseous component forming present day galaxies. A new series of reviews followed the recent explosion of activity on the reionization of the IGM in anticipation of the detection of the Epoch of Reionization (EoR) through its high redshift 21cm signature. An early review in this area is by. Since then, current observational and theoretical aspects of reionization have been exhaustively covered by,, and. Rather than repeating the wide range of IGM phenomenology already covered by previous reviewers, I concentrate here on the physics underlying the structure of the IGM. One aim is to describe the physical underpinnings of current numerical simulations as required for future simulations to further progress. As observations are crucial for constraining any models, I begin by giving a broad overview of the current observational situation. The next section describes the physics of ionization equilibrium, followed by a discussion of the UV metagalactic background that maintains the ionization of the IGM. A brief review of early models of the forest absorbers is then presented, followed by a discussion of current numerical models. The reionization of the IGM is then summarized along with means for its detection. This is followed by a discussion of the connection between galaxies and the IGM before concluding with observational and theoretical prospects for the future. Unless stated otherwise, the cosmological parameters Ωm = 0.3 and Ωv = 0.7 are assumed for, respectively, the ratios of the present day matter density and the present day vacuum energy density to the current Einstein-deSitter density $\rho\_{\rm crit}(0)=3H\_0^2/8\pi G$, a Hubble Constant of $H\_0=100h\,{\rm km\,s^{-1}\,Mpc^{-1}}$ with h = 0.7 (${\rm Mpc} = {\rm Megaparsec}\simeq 3.0856\times10^{22}\,{\rm m}$), and a baryon density of Ωbh2 = 0.022. These values are consistent within the errors with the current best estimates for a flat Universe based on CMB measurements using Year-5 WMAP data of Ωm = 0.279 ± 0.014, Ωv = 0.721 ± 0.015, h = 0.701 ± 0.013, and Ωbh2 = 0.02265 ± 0.00059, or intergalactic D/H measurements, giving Ωbh2 = 0.021 ± 0.001. Observations ============ Resonance absorption lines -------------------------- The IGM is detected through the absorption features it produces in the spectrum of a background bright source of light (typically a QSO). The production of the absorption features is governed by the equation of radiative transfer through the IGM, conventionally expressed in terms of the specific intensity of a background radiation source. The specific intensity $I\_\nu({\bf r},t,\nhat)$ is defined as the rate at which energy crosses a unit area per unit solid angle per unit time as carried by photons of energy $h\_{\rm P}\nu$ traveling in the direction $\nhat$ relative to some fiducial direction. The equation of radiative transfer for $I\_\nu({\bf r},t,\nhat)$ in a medium with attenuation coefficient $\alpha\_\nu({\bf r},t,\nhat)$ is $$\begin{aligned} \frac{1}{c}\frac{\partial I\_\nu({\bf r},t,\nhat)}{\partial t}+\nhat\cdot {\bf \nabla}I\_\nu({\bf r},t,\nhat) = \nonumber\\ -\alpha\_\nu({\bf r},t,\nhat) I\_\nu({\bf r},t,\nhat) + j\_\nu({\bf r},t,\nhat). \label{eq:tdRT}\end{aligned}$$ Because of its use below, a radiative source term has been included through the emission coefficient $j\_\nu({\bf r},t,\nhat)$, which describes the local specific luminosity per solid angle per unit volume emitted by sources. Normally the random orientation of atoms will ensure jν is isotropic, but not always, as for instance if the source includes a scattering term, so that a dependence on $\nhat$ is included to account for the possibility of anisotropic sources. A central source like a QSO or galaxy will in fact generally radiate anisotropically. In general, the attenuation of the radiation field is due both to the absorption of photons and their scattering out of the beam. The attenuation coefficient is then $$\alpha\_\nu({\bf r},t,\nhat) = \rho({\bf r},t)\kappa\_\nu({\bf r},t,\nhat) + n({\bf r},t)\sigma\_\nu({\bf r},t,\nhat), \label{eq:atten}$$ where $\rho({\bf r},t)$ is the mass density of the medium, $\kappa\_\nu({\bf r},t,\nhat)$ is the opacity, which expresses the absorption cross section per unit mass, $n({\bf r},t)=\rho({\bf r},t)/\bar m$ is the number density of scattering particles of mean mass $\bar m$, and $\sigma\_\nu({\bf r},t,\nhat)$ is the scattering cross section. In general, the scattering could arise from more than one type of particle, in which case nσν would be replaced by a sum over particle species i of density ni and cross section σν(i). In a static medium, $\alpha\_\nu({\bf r},t,\nhat)$ is generally isotropic, but not always. As an example, the alignment of atoms in a strong magnetic field would absorb anisotropically. In a moving medium, an anisotropic contribution to the attenuation will be produced by the dependence of Doppler scattering on the relative motion of the radiation and the fluid. The value of Iν at any given time t and position s along the direction $\nhat$ will be given by any incoming intensity $I\_\nu^{\rm inc}$ at position s0 at time $t\_{\rm ret}=t-(s-s\_0)/c$, attenuated by intervening material at positions s′ at the retarded times $t^\prime\_{\rm ret}=t-(s-s^\prime)/c$, along with contributions from sources at positions s′′ that emitted at the retarded times $t^{\prime\prime}\_{\rm ret}=t-(s-s^{\prime\prime})/c$, followed by attenuation. The formal solution to Eq. ([eq:tdRT]) is then $$\begin{aligned} I\_\nu(s,t) = \left(I\_\nu^{\rm inc}\right)\_{s\_0,t\_{\rm ret}} \exp\left[-\int\_{s\_0}^s\,ds^\prime\, (\alpha\_\nu)\_{s^\prime,t\_{\rm ret}^\prime} \right]\nonumber\\ +\int\_{s\_0}^s\,ds^{\prime\prime}\, \left(j\_\nu\right)\_{s^{\prime\prime},t\_{\rm ret}^{\prime\prime}} \exp\left[-\int\_{s^{\prime\prime}}^s\,ds^\prime\, (\alpha\_\nu)\_{s^\prime,t\_{\rm ret}^\prime} \right] \label{eq:tdRT-sol}\end{aligned}$$ as may be verified by direct substitution. In a uniformly expanding (or contracting) medium described by the scale factor a(t), Eq. ([eq:tdRT]) must be modified to $$\begin{aligned} \frac{1}{c}\frac{\partial I\_\nu({\bf r},t,\nhat)}{\partial t}+ \frac{1}{c}\frac{\dot a}{a}\left[3I\_\nu({\bf r},t,\nhat)-\nu\frac{\partial I\_\nu({\bf r},t,\nhat)}{\partial\nu}\right]\nonumber\\ + \nhat\cdot {\bf \nabla}I\_\nu({\bf r},t,\nhat) = -\alpha\_\nu({\bf r},t,\nhat) I\_\nu({\bf r},t,\nhat) + j\_\nu({\bf r},t,\nhat), \label{eq:tdRT-exp}\end{aligned}$$ where the frequency redshifts according to ν ∝ a(t)− 1. The solution becomes $$\begin{aligned} I\_\nu(s,t) = \left(I\_{\nu\_0}^{\rm inc}\right)\_{s\_0,t\_0} \left[\frac{a(t\_0)}{a(t)}\right]^3\exp\left[-\int\_{s\_0}^s \,ds^\prime\, (\alpha\_{\nu^\prime})\_{s^\prime,t^\prime} \right]\nonumber\\ +\int\_{s\_0}^s\,ds^{\prime\prime}\, \left(j\_{\nu^{\prime\prime}}\right)\_{s^{\prime\prime},t^{\prime\prime}} \left[\frac{a(t^{\prime\prime})}{a(t)}\right]^3 \exp\left[-\int\_{s^{\prime\prime}}^s\,ds^\prime\, (\alpha\_{\nu^\prime})\_{s^\prime,t^\prime} \right], \label{eq:tdRT-exp-sol}\end{aligned}$$ where ν0 = νa(t)/a(t0), ν′ = νa(t)/a(t′), and ν′′ = νa(t)/a(t′′). In the case of no intermediate sources, jν = 0 and the received intensity depends only on how the incident intensity is modified by the intervening gas. This is the situation for a single background source such as a QSO. The Lyman absorption features arise from the scattering of resonance line photons received from the background QSO through a medium of scatterers with number density $n({\bf r},t)$. The frequency dependent scattering cross section for a scatterer at rest is given by the Lorentz profile[11](#fn11) $$\sigma\_\nu=\left(\frac{\pi e^2}{m\_e c}\right)\left[\frac{1}{4\pi\epsilon\_0} \right]f\_{lu} \frac{(\Gamma\_{ul}/4\pi^2)}{(\nu-\nu\_{lu})^2+(\Gamma\_{ul}/4\pi)^2}, \label{eq:Lorentz}$$ where νlu is the resonance line frequency, corresponding to the wavelength λlu, of the transition between an upper level broadened by radiation damping to a sharp lower level (the ground state), Γul = (glflu/gu)[1/4πε0]8π2e2/(mecλlu2) is the damping width of the upper level, where gl and gu are the respective statistical weights of the lower and upper levels, flu is the upward oscillator strength, e is the electric charge of an electron, and me the electron mass.[12](#fn12) The cross section is normalized according to $$\sigma\equiv\int\_{-\infty}^\infty\, d\nu\, \sigma\_\nu = \left(\frac{\pi e^2}{m\_e c}\right)\left[\frac{1}{4\pi\epsilon\_0}\right]f\_{lu}. \label{eq:sigmanu-norm}$$ For λlu in m, Γul = 6.670 × 10− 5(glflu/gu)λlu− 2. For the Lyman series, $$g\_1f\_{1n}=\frac{2^9n^5(n-1)^{2n-4}}{3(n+1)^{2n+4}}, \label{eq:gfLy}$$ where n is the quantum number of the upper state. For the transition, gl = 2, gu = 6 and flu = 0.4162⋯. For hydrogen Lyα, λlu = 1215.67Å and $\Gamma\_{ul}=6.262\times10^8\,{\rm s}^{-1}$. For Lyα, λlu is smaller by a factor of 4, and Γul larger by a factor of 16. In general, the atoms will not be at rest. At the very least they will generally undergo thermal motions described by a Maxwellian velocity distribution corresponding to their temperature T. They may additionally take part in an ordered flow of velocity ${\bf v}$, but this may be accounted for by shifting the line center frequency ν to ν(1 − v∣∣/c), to first order accuracy in v/c, where v∣∣ is the line-of-sight velocity. In addition, there may be a disordered turbulent, or so-called microturbulent, component in some collapsed or shocked regions. Ignoring non-thermal velocities, the profile including thermal motions is found by convolving Eq. ([eq:Lorentz]) with a Maxwellian. The result, which neglects additional frequency dependences far from line center, is the Voigt function $$H(a,x)=\frac{a}{\pi}\int\_{-\infty}^\infty\,dy \frac{e^{-y^2}}{(x-y)^2+a^2}, \label{eq:phiV}$$ where a = Γul/(4πΔνD) is the ratio of the damping width to the Doppler width, and x = (ν − νlu)/ΔνD is the frequency shift from line center in units of the Doppler width ΔνD = νlub/c with Doppler parameter $$b=\left(\frac{2k\_{\rm B}T}{m\_a}\right)^{1/2}, \label{eq:bparam}$$ where ma is the mass of the scattering particle and $k\_{\rm B}$ is Boltzmann’s constant. A kinematic component such as microturbulence is often accounted for by adding in quadrature the characteristic kinematic velocity to the thermal component of the Doppler parameter $$b= (b\_{\rm th}^2 + b\_{\rm kin}^2)^{1/2}, \label{eq:bbparam}$$ where $b\_{\rm th}$ is the thermal contribution from Eq. ([eq:bparam]). An expansion approximation for the Voigt function in a is provided by. Fast methods for arbitrary a are described by, and. Various evaluation methods are compared by. The resonance line scattering cross section is $$\sigma\_\nu = \left(\frac{\pi e^2}{m\_e c}\right) \left[\frac{1}{4\pi\epsilon\_0}\right]f\_{lu}\varphi\_V(a,\nu)= \sigma\varphi\_V(a,\nu), \label{eq:sigmaV}$$ where $$\varphi\_V(a,\nu)=\frac{1}{\Delta\nu\_D}\phi\_V(a,x) =\frac{1}{\pi^{1/2}\Delta\nu\_D}H(a,x) \label{eq:Vprof}$$ is the Voigt profile, normalized to ∫0∞ dν φV(a, ν) = 1. For hydrogen, a ≃ 0.0472T− 1/2. For Lyα, a is larger by a factor of 8. The intensity of the background QSO is attenuated by the factor e− τν, where Eq. ([eq:tdRT-exp-sol]) gives τν = ∫s0sds′ n(s′, t′)σν′. For a homogeneous and isotropic background Universe expanding with scale factor a(t), radiation emitted by the source at time t0 and rest frame frequency ν0 > νlu, where νlu is the resonance line frequency of a (rest frame) Lyman transition, will be scattered by the medium at epoch t′ given by ν′ = ν0a(t0)/a(t′) = νlu. Thus all of the received QSO intensity will be attenuated at the time t at all observed frequencies ν in the range νlu > ν > νlua(t0)/a(t), or wavelengths λ in the range λlu < λ < λlua(t)/a(t0), where λlu is the (rest frame) wavelength of the particular Lyman transition, producing a trough in the spectrum of the QSO. This is the Gunn-Peterson effect. Intervening metal absorption, assuming solar abundances distributed uniformly throughout intergalactic space, would produce similar troughs. The corresponding optical depth is given by noting that ds′ = cdt′ = cH− 1(t′)da(t′)/a(t′), where $H(t^\prime)= \dot a(t^\prime)/a(t^\prime)$ is the Hubble parameter at time t′. Changing the integration variable to a′ = a(t′) and setting z = νlu/ν − 1, it follows from Eq. ([eq:taunu]) that, taking ⟨n⟩ to denote the mean number density of particles in the lower state, $$\begin{aligned} \tau\_\nu&=&c\int\,\frac{da^\prime}{a^\prime} H(a^\prime)^{-1}\sigma\_{\nu/a^\prime-\nu\_{lu}} \langle n(a^\prime)\rangle\nonumber\\ &=&\frac{g\_u}{g\_l}\frac{1}{8\pi}\frac{\lambda\_{lu}^3\Gamma\_{ul} \langle n(z)\rangle}{H(z)}\nonumber\\ &=&1.191\times10^4\frac{(1+z)^3}{[\Omega\_m(1+z)^3+\Omega\_K(1+z)^2 +\Omega\_v]^{1/2}}\nonumber\\ &&\times\left(\frac{f\_{lu}\lambda}{506.0\AA}\right) \left(\frac{g\_u/g\_l}{3}\right) \left(\frac{\langle n\rangle}{\langle n\_{\rm H}\rangle}\right), \label{eq:tauGP}\end{aligned}$$ after noting from Eq. ([eq:sigmanu-norm]) that σ = (gu/gl)Γulλlu2/8π. The Hubble parameter H(z) at redshift z is related to the value of the Hubble constant today H0 by H(z) = H0E(z), where E(z) = [Ωm(1 + z)3 + ΩK(1 + z)2 + Ωv]1/2. Here, Ωm is the mass density of the Universe expressed as a fraction of the Einstein-de Sitter critical density, and ΩK and Ωv are similar fractions arising from the curvature and vacuum energy contributions, respectively. They satisfy the identity Ωm + ΩK + Ωv = 1. In the last line of Eq. ([eq:tauGP]), the result was scaled to the present day mean cosmic hydrogen density $n\_{\rm H}(0)=\rho\_{\rm H}(0)/m\_{\rm H}=(1-Y)\Omega\_b \rho\_{\rm crit}(0)/m\_{\rm H}\simeq0.186\,{\rm m^{-3}}$, where $m\_{\rm H}$ is the mass of a hydrogen atom and assuming a baryon density of Ωbh2 = 0.022 (see § [subsubsec:deuterium] below) and cosmic helium mass fraction Y ≃ 0.2477 ± 0.0029, and to a Hubble Constant of $H\_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$ (§ [subsubsec:evolution] below). Eq. ([eq:tauGP]) is the inverse of the Sobolev parameter for a homogeneously and isotropically expanding medium, and was first derived in the context of cosmological hydrogen photon scattering by. [fig:LyaF] Observations show that in fact the IGM is not homogeneous, but that the baryons have coalesced, creating a fluctuating density field n(s, t). The discreteness of the absorption lines measured in the spectra of high redshift QSOs suggests they originate in distinct localized regions. The resulting collection of absorption features, shown in Fig. [fig:LyaF], is known as the forest. Denoting the centers of the absorbing regions by the positions si, the optical depth becomes τν = ∑iτν(i), where τν(i) = ∫si − Δs/2si + Δs/2ds′ n(s′, ti)σν′. Here, the integral has been localized to a region of width Δs around si, and ν′ = νa(t)/a(ti), where ti corresponds to the epoch of coordinate si in an expanding universe. The absorption features thus probe discrete spatial structures in the IGM. Expressing the optical depth as a sum of discrete absorption systems is, however, an approximation, as the intergalactic gas forms an evolving spatial continuum. Gas distributed over a wide spatial range, and even at different epochs, can therefore affect the same observed frequency ν. However, for $z\lta3.5$, observations show the discrete absorber approximation provides a reasonably good description of the hydrogen forest. Eq. ([eq:taunu-i]) may be further simplified to τν = π1/2τ0⟨ϕV(a, x)⟩, where $$\tau\_0 = \frac{N\sigma\lambda}{\pi^{1/2}b} = \frac{\sqrt{\pi}e^2} {m\_e c}\left[\frac{1}{4\pi\epsilon\_0}\right]\frac{N}{b}\lambda f\_{lu} \label{eq:tau0}$$ is the optical depth at line centre. Here, N is the column density ∫ ds′ n(s′, t′), and ⟨ϕV(a, x)⟩ is averaged over the line of sight, weighted by the density. This may differ from ϕV(a, x) when the temperature or large-scale macroscopic velocity field varies along the line of sight. Tabulations of λflu and related atomic data for resonance lines for a variety of elements are available in the literature. For hydrogen, $$\tau^{\rm HI}\_0\simeq0.38\left(\frac{N\_{\rm HI}}{10^{13}\,{\rm cm}^{-2}}\right) \left(\frac{b}{20\kms}\right)^{-1}. \label{eq:tau0-HI}$$ [fig:Ftau] At redshifts z > 3.5, the blending of absorption lines makes it increasingly difficult to attribute the absorption to distinguishable systems. By z > 5.5, the individual lines have essentially all merged, forming in effect a Gunn-Peterson trough (though still not corresponding to a completely neutral IGM). Absorption features at intermediate redshifts not easily deblended into subcomponents may be characterized by an equivalent width wλ. This is defined to be the width, expressed in units of wavelength, a square-well absorption feature with zero flux at its bottom must have to match the integrated area of the detected feature under the continuum: $$w\_\lambda = \int\_0^\infty\,d\lambda \left(1 - e^{-\tau\_\nu}\right) = \frac{\lambda^2}{c}\int\_0^\infty\,d\nu \left(1 - e^{-\tau\_\nu}\right). \label{eq:w}$$ For a Voigt profile, the equivalent width is related to the column density through the line-center optical depth according to $$\frac{w\_\lambda}{\lambda}=2\frac{b}{c}F(a,\tau\_0), \label{eq:wFtau0}$$ where F(a, τ0) = ∫0∞ dx {1 − exp[ − π1/2τ0ϕV(a, x)]}. The relationship between the equivalent width and the column density is known as the “curve of growth.” For τ0 < 104, the absorption profiles are well-approximated as Doppler in shape, corresponding to ϕV(a, x) ≃ π− 1/2e− x2, and F(a, τ0) may be denoted more simply by F(τ0). The function F(τ0) may be expressed as the power series $$F(\tau\_0)=\frac{\sqrt\pi}{2}\sum\_{n=1}^\infty\,(-1)^{n-1} \frac{\tau\_0^n}{n!\sqrt{n}}. \label{eq:Ftau0Dlin}$$ For very optically thin lines (τ0 < < 1), $F(\tau\_0)\simeq(\sqrt{\pi}/2)\tau\_0$, and the equivalent width is related linearly to the column density, $$\frac{w\_\lambda}{\lambda}=\frac{\pi e^2}{m\_e c^2} \left[\frac{1}{4\pi\epsilon\_0}\right]N\lambda f\_{lu} \simeq8.85\times10^{-13}N\lambda f\_{lu}, \label{eq:wlin}$$ where Nλ has the dimension ${\rm cm}^{-1}$. Such features are said to lie on the “linear” part of the curve of growth. For 6 < τ0 < 300, better than 1% accuracy is provided by the asymptotic series $$F(\tau\_0)\sim\left(\ln\tau\_0\right)^{1/2}\left[1+ \frac{0.2886}{\ln\tau\_0}-\frac{0.1335}{(\ln\tau\_0)^2}+\cdots\right]. \label{eq:Ftau0Dlog}$$ For at a temperature T = 104 K, F(a, τ0) deviates from the logarithmic approximation Eq. ([eq:Ftau0Dlog]) by less than 10% for 2 < τ0 < 5000. Such features are said to lie on the “logarithmic” part of the curve of growth. They are also referred to as “saturated” lines, since an increase in the column density has only a small change on the shape of the absorption feature. The measurement of accurate column densities is notoriously difficult for these features, necessitating the search for higher order resonance lines on a more linear part of the curve of growth if accurate column densities are desired. On the other hand, the equivalent width is nearly directly proportional to the Doppler width, allowing an accurate determination of the Doppler parameter. For very large values of τ0, the Lorentzian radiation damping wings of the Voigt profile dominate the line profile, which in this limit is well approximated by ϕV(a, x) ≃ a/(πx2). In this case, the equivalent width is given approximately by $$\frac{w\_\lambda}{\lambda}\simeq\frac{2}{c}\left(\lambda^2N\sigma \frac{\Gamma\_{ul}}{4\pi}\right)^{1/2} = 2\pi^{1/4}\frac{b}{c}(a\tau\_0)^{1/2}. \label{eq:Ftau0sqrt}$$ Because of the square-root dependence on column density, this limit is referred to as the “square-root” part of the curve of growth. While the independence of the equivalent width on the Doppler width of the feature leaves b poorly determined, the stronger dependence on the column density permits a more accurate determination of the column density from the equivalent width, or from line-profile fitting, than is possible for systems on the logarithmic part. The square-root approximation Eq. ([eq:Ftau0sqrt]), however, is accurate only for very large values of τ0. It is more than 25% too low for τ0 < 104. Better than 1% accuracy is achieved for τ0 > 2 × 105. Absorption line properties -------------------------- ccccccccc & && $\frac{dN^d}{dz}=N\_0(1+z)^\gamma$ Absorber class&$N\_{\rm HI}$ (cm− 2)&$b^a\, (\rm km\,s^{-1})$ &$n^b\,({\rm m^{-3}})$&$T^b\,({\rm K})$&Size (kpc) &[M/H]c&N0&γ Lyα forest &$\lta10^{17}$ & 15–60 & 0.01 − 1000 &5000 − 50000 &15–1000(?) & -3.5 – -2 &6.1&2.47 LLS &1017 − 1019 & ∼ 15 & ∼ 103 − 104 & ∼ 30000 &–& -3 – -2 &0.3&1.50 Super LLS &1019 − 2 × 1020 & ∼ 15 & ∼ 104 & ∼ 10000 &–& -1 – +0.6 &0.03&1.50 DLA & > 2 × 1020 & ∼ 15 & ∼ 107; ∼ 104 & ∼ 100; ∼ 10000& ∼ 10 − 20(?)& -1.5 – -0.8 & ∼ 0.03& ∼ 1.5 [tab:absprops] \| \| \| --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| ### Physical properties of absorption line systems The absorption features comprising the forest are broadly classified into three main types: forest systems, Damped Absorbers (DLAs), and Lyman Limit Systems (LLSs). The classification is based primarily on the physical origin of the features. The forest systems, by far the most common, are generally well-fit by Doppler line profiles. The much rarer DLAs have sufficiently high hydrogen column densities that they show the radiation damping wings of the Lorentz profile, and require the Voigt line profile for accurate fitting. The intermediate LLSs have a sufficiently large column density to absorb photons with energies above the photoelectric edge, or Lyman limit. The classifications are not strictly exclusive. Damped absorbers of course produce Lyman Limit Systems. Lower column density Lyman Limit Systems will produce a forest feature. For convenience, the features, however, are treated as distinct, corresponding typically to the column density ranges shown in Table [tab:absprops]. Recently, a subclass of super Lyman Limit Systems (sLLSs) has been introduced, corresponding to systems with $10^{19}<N\_{\rm HI}<2\times10^{20}\, {\rm cm^{-2}}$.[13](#fn13) These systems are convenient for statistical studies since their column densities are more easily determined than their lower column density counterparts because the damping wings begin to appear. They are sometimes also referred to as sub-damped absorbers, but there is an important physical distinction between them and the general class of DLAs: the hydrogen in the DLAs is essentially neutral, while the sLLSs are sufficiently penetrated by the UV metagalactic background as to be partially ionized. It is therefore reasonable to distinguish the DLAs as an entirely unique class of absorber. [fig:dNdNHI] [fig:bdist] A great number of surveys of QSO intervening absorption systems have been carried out, building up a large inventory of their statistical properties. Systematic surveys of the forest were conducted by,,,,,,, and. Surveys of Lyman Limit Systems were conducted by, and. Surveys of Damped Absorbers were conducted by,,,,,,,,,,,, and. A radio survey of intervening absorption systems was carried out by. Surveys for galaxies associated with intervening absorption systems were conducted for galaxies associated with diffuse absorption systems at z < 1 by,,, and, with Damped Absorbers by,, and, and with metal absorption systems at z < 1 by,, and, and at z > 1 by. The neutral hydrogen column densities of the absorbers are measured to range from roughly $10^{12}-10^{22}\,{\rm cm^{-2}}$. Lower column density systems may exist, but are difficult to detect. An upper cut-off at $3-5\times10^{21}\,{\rm cm^{-2}}$ is suggested by. It was early recognized by that the distribution function of the column densities is a near perfect power law, $dN/dN\_{\rm HI}\propto N\_{\rm HI}^{-\beta}$, with β = 1.5 − 1.7. A recent determination for absorption systems in the redshift range 0.5 < z < 1.9 is shown in Fig. [fig:dNdNHI]. Although there may be small deviations from a perfect power law, the nearness to a single power law over such an enormous dynamic range strongly suggests a single formation mechanism. The measured Doppler velocities b range over about $10<b<100\,{\rm km\,s^{-1}}$, with the vast majority concentrated between $15-60\kms$. A typical distribution is shown in Fig. [fig:bdist], using the line list for Q0000–26 from. Temperatures may in principle be inferred from Eq. ([eq:bparam]), but doing so is hampered by two difficulties: 1. the systems may be broadened by a kinematic component and 2. the absorption features may be blends of more than a single system. Evidence for kinematic broadening is found when metal features are also detected (see below). In general, there is no unique fit to an absorption feature, particularly in the presence of blending: several statistically acceptable fits are possible, and these will change as the signal-to-noise ratio or spectral resolution changes. Indeed, Eq. ([eq:taunu]) shows that each absorption feature itself may be regarded as the blending of an infinity of smaller features. It is only because of clumpiness of the IGM that the features may be localized (Eq. [[eq:taunu-i]]), yet internal structure is still visible when multiple narrower metal features are detected in a single system. rllllll QSO name & $z\, {\rm range}$ & no. lines & ⟨b⟩ ($\kms$) & γ & δ & $p\_{\rm KS}$ Q0000–26 & 3.42 < z < 3.98 & 334 & 32.8 & − 7.15 & 2.13 & 0.36 Q0014 + 813 & 2.70 < z < 3.20 & 262 & 33.9 & − 8.86 & 2.58 & 0.99 Q0302–003 & 2.62 < z < 3.11 & 266 & 33.9 & − 8.75 & 2.56 & 0.97 Q0636 + 680 & 2.54 < z < 3.03 & 312 & 29.3 & − 7.24 & 2.23 & 0.30 Q0956 + 122 & 2.62 < z < 3.11 & 256 & 31.6 & − 8.05 & 2.42 & 0.53 HS 1946 + 7658 & 2.42 < z < 2.98 & 399 & 32.5 & − 7.18 & 2.16 & 0.50 [tab:bfit] Broadening is also expected from the line finding and fitting procedure. The systematic influence of procedures used to locate and fit the absorption lines on the resulting distributions of Doppler parameters has received limited attention. The potential usefulness of the Doppler parameter distribution for extracting physical information about the IGM, such as its temperature distribution, merits further study of the effect of fitting algorithms on the derived distribution. Artificially large b-values, for example, have been reproduced through Monte Carlo simulated spectra with much narrower gaussian distributions for the b-values. The Monte Carlo models assume Poisson distributed line centers. Allowing for clustering of the line centers (see below) may lead to even broader distributions. The measured distributions may be fit by a lognormal function. As an illustration, the best fitting lognormal function $f(\xi)=\exp(-0.5\xi^2)/\sqrt{2\pi}$, where ξ = − 7.15 + 2.13ln(b), provides a statistically acceptable description of the b distribution of Q0000–26 for systems lying between the restframe and wavelengths of the QSO, excluding a small region possibly influenced by the proximity effect (see § [subsubsec: proximity]). The fit distribution is shown in Fig. [fig:bdist]. The KS test yields a probability for obtaining as large a difference as found between the predicted and measured cumulative distributions of $p\_{\rm KS}=0.361$. A lognormal distribution will result when the error on a quantity is proportional to the magnitude of the quantity. For example, consider an absorption system with an intrinsic Doppler parameter b. If the iterations of the nonlinear fitting procedure of the remaining features in the spectrum each perturb the first feature by an amount Δb ∝ b, then Δlnb will change by a randomly distributed amount. Fitting several features will then result in a sum of random changes to Δlnb. By the central limit theorem, a normal distribution for lnb will result. A comparison between the measured b-values and their errors from shows a positive correlation. The Spearman rank-order correlation coefficient is rs = 0.31, with p( > rs) = 9.3 × 10− 9. A similar result is found for HS 1946 + 7658, for which a correlation between the Doppler parameters and their errors is found with rs = 0.28 and p( > rs) = 7.4 × 10− 9. A lognormal distribution again provides an acceptable fit to the Doppler parameter distribution. The results for this and several other distributions measured in high resolution Keck spectra are shown in Table [tab:bfit]. In all cases, a lognormal distribution provides a good fit. Although not conclusive, these results suggest that the measured Doppler parameters may in part be induced by a lognormal-generating stochastic process. It is noted that the process need not arise entirely from the line-fitting, but could also originate from the physical processes that gave rise to the structures that produce discrete absorption features. Uncertainty in the origin of the Doppler parameter distribution leaves the physical interpretation of the Doppler parameters with some ambiguity. Although their relation to the gas temperature is not straightforward, the Doppler parameters may usually be used legitimately to set upper limits on the gas temperature. (Even an upper limit is not always guaranteed, since noise affecting a weak line may leave it too narrow, or even produce an artificial narrow line.) For pure Doppler broadening, the range $20-60\kms$ corresponds to temperatures of 2.4 × 104 − 3.8 × 104 K, the range expected from photoionization at the moderate overdensities expected for the absorbers. Cooler temperatures are possible, however, particularly if the gas has been expanding sufficiently fast for adiabatic cooling to be appreciable. The element of indeterminacy in the measurements of the line parameters has made it difficult to search for evolution in the distribution of Doppler parameters. Evolution is especially interesting in the context of late reionization, which may have occurred at z ≃ 3 − 3.5. Evolution of the Doppler parameter distribution, however, could also result if aborption systems with different physical characteristics dominate the detected and fit absorption lines at different epochs. Numerical simulations suggest the systems that give rise to a given column density range shift to structures of different gas densities, and therefore different gas temperatures, and different sizes as the Universe expands and as the intensity of the photoionizing UV background evolves. The difficulty of interpreting any change in the Doppler parameter distribution in terms of sudden heating is exacerbated by an increase in line blending with increasing redshift. Using an analysis based on the lower cut-off in the Doppler parameter distribution, report evolution in the inferred IGM gas temperature over the range 2.0 < z < 4.5, peaking at z ≃ 3, consistent with the onset of reionization at this redshift. Similar results are obtained from a separate analysis by. also suggest evolution in the gas temperature based on an increase in the lower cut-off of the Doppler parameter distribution from z = 3.8 to z = 3.3. But the values for the cut-off they derive are discrepant with those of, which casts uncertainty on either interpretation. show there is considerable sample variance in the cut-off at the same redshift, and attribute the discrepancy with to this variance. On the other hand, find no significant evolution in the b-distribution over the redshift range 1.5 < z < 3.6 using the same sample as. This result is confirmed by the nearly identical parameters obtained for the best fit lognormal distributions between the highest and lowest redshift samples in Table [tab:bfit]. do argue for the presence of a second broader population at $z\lta0.4$. find no evidence for evolution in the Doppler parameters over the redshift range 0 < z < 2. Damped Absorbers show a more complex physical structure. Measurements of the 21cm hyperfine absorption signature produce a range of spin temperatures, from values as low as TS ≃ 200 K to lower limits of $T\_S\gta9000$ K, with the lower values occurring only at $z\lta3$. provide a tabulation from the literature. The presence of high ionization metal species show there is a second warmer component present as well, with T ≃ 104 K (see below). suggest a two-component interstellar medium consisting of a warm neutral medium (WNM) at T ≃ 8000 K in pressure equilibrium with a cold neutral medium (CNM) at T ≃ 100 K. A summary of the absorption line properties, as well as of the physical characteristics discussed below, is provided in Table [tab:absprops]. ### Evolution in the number density of the forest [fig:dNdzlow] [fig:dNdzhigh] The number of absorption systems per unit redshift increases with redshift. Some evolution is expected as a result of the expansion of the Universe. For a proper number density na(z) of absorption systems at redshift z, with proper cross section σa(z), the expected number of absorbers per unit proper length is dN/dlp = na(z)σa(z). The proper line element is related to redshift according to dlp/dz = c/[H(z)(1 + z)], where H(z) = H0E(z) is the Hubble parameter (see § [subsec: absorption-lines] above). For a flat Universe (ΩK = 0) and standard cosmological parameters, E(z) ≃ 0.55(1 + z)3/2[1 + 2.3/(1 + z)3]1/2. The evolution in the number density is then given by $$\frac{dN}{dz} \simeq (2100\,{\rm Mpc})n\_{a, {\rm c}}(z)\sigma\_a(z)(1+z)^{1/2} \biggl[1+\frac{2.3}{(1+z)^3}\biggr]^{-1/2}, \label{eq:dNdz}$$ where $n\_{a, {\rm c}}(z)=n\_a(z)(1+z)^{-3}$ is the comoving number density of systems. For a constant comoving number density and fixed proper cross-section, only moderate evolution is expected, dN/dz ∝ (1 + z)1/2. At low redshifts, this gives a reasonable description of the evolution. Using Hubble Space Telescope observations, find for all system with equivalent widths above 0.24Å, dN/dz = (32.7 ± 4.2)(1 + z)0.26 ± 0.22 for z < 1.5. Subsequent higher spectral resolution surveys using HST have extended the statistics to systems with equivalent widths below 0.1Å at z < 0.1. The large survey of obtains dN/dz = 129− 5+ 6 down to 0.030Å at z < 0.4 with ⟨z⟩ ≃ 0.14. For z > 1.5, however, the number density evolves considerably faster than the constant comoving rate. obtain dN/dz = 6.1(1 + z)2.47 ± 0.18 for systems with column densities in the range $13.64<\log\_{10} N\_{\rm HI} <17$. This corresponds to considerable evolution in the product $n\_{a, {\rm c}}(z)\sigma\_a(z)$. A smooth transition is found over the redshift range 0.5 < z < 1.9 by, with a dependence of the evolution rate on column density. High column density systems corresponding to Lyman Limit Systems evolve somewhat more slowly than lower column density systems. find for systems with τL > 1, dN/dz = 0.25+ 0.17− 0.10(1 + z)1.50 ± 0.39 over the redshift range 0.32 < z < 4.11. Based on a larger, higher redshift sample, find somewhat more rapid evolution, with dN/dz = 0.07+ 0.13− 0.04(1 + z)2.45− 0.65+ 0.75 over the redshift range 2.4 < z < 5. Damped Absorbers show evolution comparable to that of the Lyman Limit Systems, with dN/dz increasing by about a factor of 2 from z = 2.5 to z = 4. As will be discussed in § [sec:simuls] below, numerical simulations show that the structure of the IGM evolves, although in the comoving frame the structure is remarkably stable. Because dN/dz is fit for a fixed column density range, the diminishing gas density towards decreasing redshifts will result in fewer systems satisfying the column density threshold, so that dN/dz will decrease towards decreasing redshift. Evolution in the ionizing background will also affect the number of absorption systems lying above the column density threshold, and this is a major factor in the evolution of dN/dz. The slowdown at z < 1.5 in the evolution is in fact attributed predominantly to a reduction in the intensity of the ionizing background: as the ionizing rate decreases, fewer systems will slip below the column density threshold than under pure density evolution. As a result, the decrease in dN/dz towards lower z slows down. The difference in the rate of evolution between low column density and high column density systems found by (compare Figs. [fig:dNdzlow] and [fig:dNdzhigh]), shows that structural evolution in the IGM must also play a role. ### Characteristic sizes and spatial correlations Multiple QSOs in proximity to each other on the sky provide an invaluable means for estimating the physical sizes of the regions in the IGM that give rise to the absorption features. Especially useful have been multiply imaged lensed quasars. Using high resolution spectra of a pair of images of a lensed QSO with a separation of 3 arcsecs, detected a large number of absorbers coincident between the two lines of sight. All the clearly detected coincident systems show comparable equivalent widths between the lines of sight, suggesting the same physical structures are being probed. These authors set 2σ lower limits of about 70 kpc for weak absorption systems ($\log\_{10} N\_{\rm HI}<14$), and of about 15 kpc for strong systems ($\log\_{10} N\_{\rm HI}>15$), assuming the clouds are spherical. For disk-like systems, the lower limits are about twice as large. Other efforts resulted in similar lower limits. Multiple lines of sight with wider separations of about 1 arcminute suggest still larger sizes of 0.3–1 Mpc, and possibly as large as 40 Mpc (comoving). report a likely size of about 10–30 kpc for a single Damped Absorber, comparable to the lower limit obtained by direct imaging measurements of a separate system by. A difficulty with interpreting the larger Megaparsec-scale sizes for the forest systems is that the equivalent widths of systems with common redshifts along the neighboring lines of sight do not always match well over these scales. An alternative interpretation of the coinciding redshifts is that they indicate underlying spatial correlations between the absorption systems rather than their spatial coherence. Measurements of the auto-correlation function between absorption systems along single lines of sight suggest correlations on scales of a few hundred $\kms$ scales. An anti-correlation at separations of $500-1000\kms$ has been disputed by on the basis of the limited lines of sight analysed, but the issue appears not fully resolved. In the presence of correlations, the error on the correlation function depends on the 3 and 4-pt functions, which are not well-measured. A more recent analysis at z < 2 detects only weak positive correlations on the few hundred $\kms$ scale, and no clear anti-correlation. Transverse positive spatial correlations over comparable scales at ⟨z⟩ = 2.14 were obtained by and at 0.4 < z < 0.9 by. Coherence in absorption lines along lines of sight separated by ∼ 2h− 1 Mpc at 2.6 < z < 3.8 has been detected by. A characteristic size of 70 kpc for systems with $N\_{\rm HI}\simeq10^{14}\,{\rm cm^{-2}}$ suggests a large fraction of the baryons are contained within the forest. Taking the size to correspond to a characteristic line-of-sight thickness, the corresponding number density is $n\_{\rm HI}\simeq5\times10^{-4}\,{\rm m^{-3}}$. Ionization equilibrium in a UV background producing an ionization rate of $\Gamma\_{\rm HI}\simeq\Gamma\_{\rm HI, -12}10^{-12}$ atoms per second (§ [subsubsec: sources] below), gives a neutral fraction of $x\_{\rm HI}\simeq (1.2n\_{\rm HI}\alpha\_A/\Gamma\_{\rm HI})^{1/2} \simeq1.5\times10^{-5}N^{1/2}\_{\rm HI, 14}T\_4^{-0.37}\Gamma\_{\HI, -12}^{-1/2}$, where $\alpha\_A\simeq4\times10^{-19}\,{\rm m^3\,s^{-1}}T\_4^{-0.75}$ is the radiative recombination rate for gas at a temperature T4 = T/104 K and $N\_{\rm HI, 14}=N\_{\rm HI}/10^{14}\,{\rm cm^{-2}}$. The ratio of the baryonic mass density of the forest to the critical Einstein-deSitter density $\rho\_{\rm crit}(z)=\rho\_{\rm crit}(0)H(z)^2/ H\_0^2$ is $$\begin{aligned} \Omega\_{\rm Ly\alpha} &=& \frac{1.4m\_{\rm H}}{\rho\_{\rm crit}} \int dN\_{\rm HI}\frac{\partial^2{\cal N}}{\partial N\_{\rm HI}\partial z} \frac{N\_{\rm HI}}{x\_{\rm HI}}\biggl(\frac{dl\_p}{dz}\biggr)^{-1}\nonumber\\ &=& 1.4m\_{\rm H}\frac{8\pi G}{3cH(z)}(1+z) \int dN\_{\rm HI}\frac{\partial^2{\cal N}}{\partial N\_{\rm HI}\partial z} \frac{N\_{\rm HI}}{x\_{\rm HI}}\nonumber\\ &\simeq&3.0\times10^{-5}N\_0h^{-1}\Omega\_m^{-1/2}T\_4^{0.37} \Gamma^{1/2}\_{\rm HI, -12}\nonumber\\ &\times&(1+z)^{\gamma-1/2} \ln\biggl(\frac{N\_{\rm HI, max}}{N\_{\rm HI, min}}\biggr)\nonumber\\ &\simeq&0.06T^{0.37}\_4\Gamma^{1/2}\_{\rm HI, -12}, \label{eq:OmLya}\end{aligned}$$ for dN/dz = N0(1 + z)γ with N0 = 6.1 and γ = 2.47, and $dN/dN\_{\rm HI}\propto N\_{\rm HI}^{-1.5}$ with $N\_{\rm HI, min}=10^{13.64}\,{\rm cm^{-2}}$ and $N\_{\rm HI, max}=10^{17}\,{\rm cm^{-2}}$, evaluated at z = 3 with Ωm = 0.3 and h = 0.7. This is comparable to estimates for the baryon density of the Universe (see § [subsubsec:deuterium] below). The result is not conclusive, however. Since the size estimates are in the transverse direction, a much smaller value for $\Omega\_{\rm Ly\alpha}$ is possible if the structures are flattened, down by the aspect ratio of the thickness to breadth of the systems. ### Deuterium absorption systems High column density absorption systems offer the opportunity to measure the primordial deuterium abundance ${\rm D/H}$, a sensitive indicator of the cosmic baryon density, or equivalently the nucleon to photon ratio η, according to Big Bang nucleosynthesis. To date, a relatively limited number of accurate determinations of η from the deuterium abundance have been possible. A line of sight must meet several criteria to be useful for obtaining a clean measurement : the hydrogen column density must be sufficiently large that deuterium is detectable; the velocity width of the system must be sufficiently narrow so as not to encroach on the $82\kms$ offset deuterium feature; the structure of the system must not be so complicated that a weak interloping feature could be mistaken for deuterium; and the background QSO must be sufficiently bright to obtain a high signal-to-noise ratio measurement. These combined criteria whittle down the number of acceptable QSO lines-of-sight to about 1% of those available at z = 3. Despite the difficulty, several accurate determinations have been made. The current best estimate for ${\rm D/H}$ is $\log\_{10} {\rm D/H} = -4.55\pm0.04$. Using standard Big Bang nucleosynthesis models, this translates to a baryon density parameter of Ωbh2 = 0.0213 ± 0.0013 ± 0.0004, where the 1σ errors refer to the error on the ${\rm D/H}$ measurement and the uncertainties in the nuclear reaction rates, respectively. This corresponds to a nucleon to photon ratio of η = (5.8 ± 0.3) × 10− 10. This compares favorably to the determination from the 5-year WMAP measurements of the CMB fluctuations, combined with Baryonic Acoustic Oscillations and Type Ia supernovae constraints, of η ≃ (6.2 ± 0.2) × 10− 10. ### Helium absorption systems The UV spectral capability of the Hubble Space Telescope made it possible for the first time to measure intergalactic λ304. Using the spectrum of Q0302 − 003 at $z\_{\rm em}\simeq3.286$ observed by the Faint Object Camera, found $\tau\_{\rm HeII}>1.7$ (90% confidence) at z = 3.2, revised to a 95% confidence interval of $1.5 < \tau\_{\rm HeII}<3.0$ by and to $\tau\_{\rm HeII} > 4.8$ by after more detailed modeling of the absorption trough. Because the same sources which ionize the should also ionize, virtually all the intergalactic helium will be in the form of either or, depending on whether or not there were sources hard enough to fully ionize helium. Remarkably, using the spectrum of HS 1700 + 6416 ($z\_{\rm em}\simeq2.7$) observed by the Hopkins Ultraviolet Telescope (HUT), measured the more moderate optical depth of $\tau\_{\rm HeII}\simeq1.00\pm0.07$ over the somewhat lower redshift range z = 2.2 − 2.6. This led to the speculation that was ionized around z ≃ 3. Subsequent measurements of the spectrum of HE 2347 − 4342 ($z\_{\rm em}\simeq2.885$) have shown the optical depth to be very patchy at $z\lta3$, as if a second reionization epoch were being observed, that of. Observations of PKS 1935 − 692 ($z\_{\rm em}\simeq3.18$) using HST and of HE 2347 − 4342 using the Far Ultraviolet Spectroscopic Explorer have detected a forest, permitting more detailed assessments of the fluctuations in the to column densities. Measurements of the forest also provide an opportunity to probe the underlying IGM velocity field. Using combined FUSE and VLT data, were able to identify isolated absorption systems in underdense regions with common and features. A comparison between the fit Doppler parameters gives the ratio $b\_{\rm HeII}/b\_{\rm HI}=0.95\pm0.12$. According to Eq. ([eq:bbparam]), this indicates a velocity field in underdense regions with kinematic motions that dominate the thermal. Subsequent measurements continue to confirm the patchy nature of the optical depth, including HST observations of HE 2347 − 4342, SDSS J2346 − 0016 ($z\_{\rm em}\simeq3.50$), and Q1157 + 3143 ($z\_{\rm em}\simeq3.01$), and a FUSE observation of HS 1700 + 6416. As some of these references point out, and as will be discussed in § [subsubsec:He-constraints] below, the interpretation in terms of reionization is not unambiguous: the patchiness may also arise from fluctuations in the radiation field. ### Metal absorption systems The earliest identifications of intervening absorption systems were for elements heavier than helium, so called metals. Common are carbon, nitrogen, silicon and iron, but to date the list extends much further, including oxygen, magnesium, neon, and sulfur. The metals provide invaluable insight into the structure of the IGM in several different ways. The widths of metal absorption features allow direct estimates of the temperature of the IGM and its small-scale velocity structure. The narrow widths of features were used by to demonstrate the absorbers had the characteristic temperatures of photoionized gas rather than collisionally ionized. The metal systems probe the impact star formation has had on the IGM, as the metals were most likely transported into the IGM through galactic winds, or introduced in situ by local small-scale regions forming first generation, Population III, stars. In the nearby Universe, they indicate the presence of shocked gas, as may accompany the formation of galaxy clusters. As such, the metals in principle document the history of cosmic structure and star formation. Ratios of the metal column densities may be used to constrain the spectral shape of the metagalactic ionizing UV background, which in turn puts constraints on the possible sources of the background and their relative contributions. Lastly, metal absorption lines offer a unique opportunity to search for variability in the constants of nature. Tables of the resonance lines most easily detectable in intervening absorption systems are provided by and. Searches for intervening metal absorbers have been facilitated by the exploitation of atomic line doublets for some of the stronger species. The most common doublets used are λλ2796, 2803 Å and λλ1548, 1551 Å. These are among the strongest lines detected in the forest. They also have the important advantage of wavelengths redward of Lyα, placing them outside the forest and avoiding this potential source of confusion. Searches for absorption systems are by far the most common because of their established connection to galaxies. absorber surveys were carried out by,,,,,,,,,,,, and. Surveys of absorption systems were conducted by and. Absorption features by λλ1032, 1038Å are more difficult to identify since they lie within the forest, and so are not easily distinguished from the forest. Searches have been successful, however, and have moreover discovered a second IGM environment. While at high redshifts the systems probe the component of the IGM containing, at low redshifts they are associated with a warm-hot intergalactic medium (WHIM), that is most likely collisionally ionized rather than photoionized. A search for absorption systems as an indicator of a WHIM at high redshift was conducted by. Efforts are currently underway to detect an intergalactic x-ray absorption line forest using the XMM-Newton and Chandra x-ray satellites. Detections of low redshift have been reported by and. Attempts to infer the origin of the metals, whether distributed by galactic outflows or introduced locally by Population III stars, have led to various efforts to estimate the range in ambient metallicities and to determine their spatial distribution. Updated solar abundances useful for modeling the abundances of the metal absorption systems are provided by,, and. Early attempts to tease out weak metal features were based either on a spectral shift-and-stack approach to generate a high signal-to-noise ratio composite line or through a pixel-by-pixel statistical analysis of the spectra. A comparison of these two techniques is provided by. An alternative is to perform long integrations of bright QSOs to obtain high resolution, high signal-to-noise ratio spectra. With signal-to-noise ratios of 200–300 per spectral resolution element, detected metal lines in 70% of the forest systems. Absorption systems with column densities as low as $N\_{\rm HI}>10^{14}\,{\rm cm}^{-2}$ show features, while systems with $N\_{\rm HI}>4\times10^{13}\,{\rm cm}^{-2}$ show features. Although the inferred metallicities are model dependent (fixed in part by the assumed spectrum and intensity of the ionizing background), the measurements suggest metallicities as low as 3 × 10− 4 that of the Sun. No evidence for a metallicity floor could be discovered. The metallicity distribution inferred is consistent with a lognormal distribution with a mean of 0.006 solar. The mechanism and degree of metal mixing in the diffuse IGM are unclear. While the mixing of metals in stellar interiors by convection and diffusion may produce a homogeneous composition, preserved in supervovae ejecta and winds, it is far from clear that the mixing of the ejecta will result in uniform metallicity. The mixing process is still poorly understood in the Interstellar Medium, and much more so in the IGM. Just as stirring a drop of milk in a cup of coffee will result in the mixing and intermingling of the two fluids, so may the stirring produced by dynamical instabilities such as the Kelvin-Helmholtz and Rayleigh-Taylor mix metal enriched stellar ejecta with the surrounding primordial gas. Insufficient mixing, however, rather than a uniform mixture, will instead result in striations or patches of high metallicity gas entrained within the primordial gas of essentially zero metallicity. find evidence for just such inadequate mixing for diffuse absorption systems (with a median upper limit of 13.3 on $\log\_{10} N\_{\rm HI}$), at z ≃ 2.3. By combining measurements with upper limits on,,,, and, they place robust constraints on the metallicities and physical properties of the absorption systems, with a median lower limit on the metallicities of ${\rm [C/H]}>-0.42$ and a median upper limit on the absorber sizes of $R\lta1.5$ kpc. They obtain typical median lower and upper limits on the hydrogen densities of $n\_{\rm H}>100\,{\rm m^{-3}}$ and $n\_{\rm H}<1000\,{\rm m^{-3}}$, a range that suggests a cloud size of about 100 pc. The clouds, however, would likely be transient, either quickly dispersing if freely expanding or shorn apart by dynamical instabilities if moving relative to a confining medium, on a timescale of about 107 yrs. arrive at a picture in which new clouds are continuously created through dynamical or thermal instabilities. In this picture, nearly all the metals in the IGM could be processed through just such a cloud phase. The dispersed clouds would retain their coherence as small patches, but of too low column density to be detectable. The resulting IGM metallicity would persist as patchy, not mixed. [fig:civhi] Matching the metal absorption lines to the corresponding features frequently reveals multicomponent substructure obscured by the width of the feature. An example from QSO 1422 + 231 is shown in Fig. [fig:civhi]. The systems show evidence for spatial clustering on scales of up to $\sim300\kms$. This is consistent with a 2σ lower limit of ∼ 30 kpc for the clustering of and systems obtained by based on a statistical analysis of coincident metal systems in neighboring lines of sight to a lensed QSO. An analysis of a triply imaged quasar at z = 3.9 by shows that high ionization systems like and absorbers retain coherence over scales of ∼ 100 kpc. By contrast, low ionization systems like and absorbers show coherence only over much smaller scales. Assuming a spherical geometry, they are estimated to have sizes of 1.5–4.4 kpc (95% c.i.). Using similar measurements, find systems at $1.5\lta z\lta3$ are coherent over scales of about 300 pc, with column densities along neighboring lines of sight differing by 50% for separations of about 1 kpc. Detectable levels of extend over scales of several kiloparsecs. Damped Absorbers show a wide range of metals with both low and high ionization states, including boron, carbon, nitrogen, oxygen, magnesium, aluminum, silicon, phosphorus, sulfur, chlorine, argon, titanium, chromium, manganese, iron, cobalt, nickel, copper, zinc, germanium, arsenic and krypton. The metallicities evolve from about 0.03 solar at z ≃ 4 to 0.15 solar at z ≃ 1, with few above 0.3 solar. No DLA is found with a metallicity smaller than 0.0025 solar, distinguishing them as a population from the forest. By contrast, super Lyman Limit Systems are found to have metallicities ranging from 0.1 solar to super-solar abundances with as much as 5 times solar in zinc. As both these classes of systems are likely related to galaxies, further discussion of them is deferred to § [sec:abs-gal]. The presence of the metal lines in absorbers of all column densities in principle offers an opportunity to obtain a nearly model-free measurement of the gas temperature. Allowing for kinetic broadening of the Doppler parameters following Eq. ([eq:bbparam]), the measurements of the Doppler parameters from features corresponding to two or more elements permits the thermal and kinematic broadening components to be separated. (In principle, instrumental broadening could be incorporated into the kinematic term.) Given total Doppler parameter measurements bi and bj for two species i and j of masses mi and mj, the temperature may be solved for as $$T=\frac{m\_i m\_j}{2k\_{\rm B}}\frac{b\_i^2-b\_j^2}{m\_j-m\_i}. \label{eq:Tfromb}$$ Applying this method to and features detected in three high redshift QSOs, and assuming that the and features trace the same systems with the same temperature and velocity structure, obtain a typical temperature of T ≈ 38000 K and typical kinetic motions of $b\_{\rm kin}\approx6\kms$ for a select subsample of especially well-fit features. The large column densities of the systems (typically exceeding $10^{12}\,{\rm cm^{-2}}$, M. Rauch, personal communication), suggest large column densities of $N\_{\rm HI}>10^{15}\,{\rm cm^{-2}}$. A few other systems selected for quiescent velocity fields as part of a program of deuterium abundance measurements yield somewhat cooler temperatures. The Doppler parameters of and measured by in a system with $\log\_{10}N\_{\rm HI}=16.7$ yields a temperature of T ≈ 30000 − 32000 K. For a Lyman Limit System with $\log\_{10}N\_{\rm HI}=17.1$ at z = 0.7, the Doppler parameter measured for compared with that of gives T ≈ 31000 K and $b\_{\rm kin}\approx13\kms$, assuming the hydrogen and metals sample the same velocity field, which is not an obvious assumption to make. For a super Lyman Limit System with $\log\_{10} N\_{\rm HI}=19.42$, find from a simultaneous fit to, and T = 1.15 ± 0.02 × 104 K and $b\_{\rm kin}=2.56\pm0.12\kms$. In all cases, the temperatures obtained are consistent with photoionization heated gas. Although direct gas density measurements are not possible, models generally suggest that higher column density systems correspond to higher densities, especially at these high values. The temperatures support the expected trend of decreasing temperature with increasing density for sufficiently dense absorption systems. The small kinetic contributions to the line broadening are in contrast to what is found for absorption systems, which appear to be dominated by kinetic broadening. The features, however, were chosen for their clear isolation in the spectrum, and so may probe physically underdense regions, in contrast to the higher column density systems above. An attempt to estimate temperatures from a larger sample of and systems has not been able to produce secure values, but nonetheless shows that only ∼ 45% of the lines are dominated by kinematic broadening. Intergalactic metal absorption systems may be used to probe the time variability of fundamental physical quantities over cosmological timescales. The excitation of ionic fine-structure levels by the CMB allows metal lines in the forest to be exploited for measuring the evolution of the CMB temperature. Because of the possible influence of other excitation mechanisms, the measurements are strictly speaking upper limits. Existing excitation temperature measurements are T = 7.4 ± 0.8 K at z = 1.776, T = 7.9 ± 1.0 K at z = 1.9731, T = 15.7 ± 3.5 K at z = 2.34, and T = 12.1− 3.2+ 1.7 K (95% c.i.) at z = 3.025. These values are all consistent with a linearly evolving CMB temperature of T(z) = T0(1 + z), expected for a homogeneous and isotropic expanding universe, with T0 = 2.725 K. Detected resonance line multiplets have been used to place constraints on the time variation of fundamental constants. used the fine structure wavelength splittings of the lines near λ1260Å and λ1527Å and the lines near λ1394Å to show that the fine structure constant (α = [1/4πε0]e2/ℏc) varies by less than 5% between z = 0 and z = 1.95. Tentative evidence for an evolving fine structure constant between 1 < z < 1.6 of Δα/α ≃ ( − 1.9 ± 0.5) × 10− 5 was reported by using measurements of and transitions. The value was later revised to Δα/α = ( − 5.7 ± 1.1) × 10− 6 over the redshift range 0.2 < z < 4.2, suggestive of a positive detection of a variation. There is by no means concensus on the detection. and report Δα/α = ( − 0.6 ± 0.6) × 10− 6 over the redshift range 0.4 < z < 2.3. Their analysis method has been contested by and, and defended by. In an independent effort using a modified technique designed to eliminate some systematics, obtain Δα/α = (5.4 ± 2.5) × 10− 6 between z = 0 and z = 1.7. Including the hyperfine splitting of the hydrogen 21cm line permits constraints on the variation of x ≡ α2gpμ, where μ is the electron to proton mass ratio and gp is the proton gyromagnetic ratio. The current best limits are, for a linear fit over the redshift range 0.2 < z < 2.4, $\dot x/x = (-1.43\pm1.27)\times10^{-15}\,{\rm yr}^{-1}$. Combining 21cm data and OH 18cm data, obtain a limit on the effective combination F ≡ gp(α2/μ)1.57 of $\Delta F/F=(0.44\pm0.36^{\rm stat}\pm1.0^{\rm syst})\times10^{-5}$ between z = 0 and z ≈ 0.7. Direct flux statistics ---------------------- While the absorption line parameters relate most directly to the physical characteristics of the absorption systems, measuring similar parameters in numerical simulations can be very computationally demanding, particularly when building up sufficient statistics to make such a comparison meaningful. More straightforward comparisons may be made using the flux statistics directly. The statistical measures in most use are the mean transmitted flux, the transmitted flux per pixel probability distribution, the flux power spectrum and its Fourier transform, the flux autocorrelation function. An additional analysis based on wavelets permits a quick means of gauging the distribution of widths of the absorption features. ### Mean transmitted flux By far the simplest statistic is the mean transmitted flux through the IGM. introduced two measurements, DA, the mean flux of a background source removed by scattering between the and transitions in the restframe of the QSO (but avoiding emission line wings), and DB, the mean flux removed between and the Lyman edge. These definitions are straightforwardly generalized to the fraction of flux removed between each successive pair of Lyman transitions. Removing the effect on higher orders by lower orders, however, is not straightforward but may be accomplished in a statistical sense. The measurement DA may be evaluated as DA = ⟨1 − fi/ci⟩, where fi is the measured flux in pixel i, ci is the corresponding predicted continuum of the background object, normally a QSO, and the average is carried out over a given wavelength (or corresponding redshift) interval between and in the restframe of the QSO. Considerable care must be taken in measuring the contribution at a given redshift from alone. Systematics that would bias the result if not corrected for include: 1. a low fraction near the QSO due to its own ionizing intensity (the “proximity effect”), 2. contamination by metal absorption systems, 3. the large stochastic influence of Lyman Limit Systems and Damped Absorbers, the latter of which produce a divergent mean, and 4. evolution of the absorption in the region measured, including over individual pixels of the spectrum. A detailed discussion is provided by. For a spectrum that may be decomposed into individual absorption features with equivalent widths $w({\bf x})$ depending on some set of parameters ${\bf x}$, such as column density or Doppler parameter, the mean of the transmitted flux ${\cal T\_\nu}\equiv\exp(-\tau\_\nu)$ is given by $$\langle{\cal T\_\nu}\rangle=\langle \exp(-\tau\_\nu)\rangle = \exp(-\tau\_l), \label{eq:transmission-lines}$$ with the effective optical depth τl due to the lines given by $$\begin{aligned} \tau\_l &=& \int\, d{\bf x} w({\bf x}) \frac{d{\cal N}({\bf x})}{d\lambda}\nonumber\\ &=& \frac{1+z}{\lambda\_0}\int\, dw \frac{\partial^2{\cal N}}{\partial w\, \partial z}w, \label{eq:taul}\end{aligned}$$ where $d{\cal N}({\bf x})/d\lambda$ is the number of systems per unit rest wavelength per unit volume $d{\bf x}$ in parameter space, here represented in terms of the number per (proper) equivalent width per unit redshift $\partial^2{\cal N}/{\partial w\,\partial z}$. The variance $\langle {\cal T}^2\rangle - \langle {\cal T}\rangle^2$ of the transmission depends on the details of the line profiles. Expressions for idealized examples are provided by and. It is instructive to consider a comparison between Eq. ([eq:taul]) and the Gunn-Peterson optical depth ([eq:tauGP]). Using Eq. ([eq:dNdz]) for the number density evolution given by $dN/dz=\int\,dw {\partial^2{\cal N}}/{\partial w\, \partial z}$ and Eq. ([eq:wlin]) for systems on the linear part of the curve-of-growth, τl may be recast in the form of the Gunn-Peterson optical depth $$\tau\_l=\frac{3}{8\pi}\frac{\lambda\_{lu}^3\Gamma\_{ul} \langle n(z)\rangle}{H(z)}, \label{eq:taul-GP}$$ for a mean distributed scattering gas density $\langle n(z)\rangle= 8\pi f\_{lu} n\_a(z)\sigma\_a(z){\bar N}(z) = 8\pi f\_{lu}Q\_a(z){\bar n}(z)$, where ${\bar N}={\bar n}L$ is the mean column density of the absorption systems with mean atomic density $\bar n$ and line of sight thickness L, and Qa = naσaL is the porosity of the absorbers, equivalent to the spatial filling factor of the absorption systems for Qa < 1. For a column density distribution varying like $dN/dN\_{\rm HI}\propto N\_{\rm HI}^{-1.5}$, the effective absorber optical depth (excluding DLAs) is dominated by saturated systems, with F(τ0) ≈ 1 − 2. In this case, Eq. ([eq:wFtau0]) gives w/λ0 = 2(b/c)F(τ0) ≈ 3b/c. If the absorber velocity widths scale like the Hubble expansion across them, b ≈ fHL, Eq. ([eq:taul]) gives instead of Eq. ([eq:taul-GP]), τl ≃ 2Qa(z)f(z)F(τ0) ≃ 3Qa(z)f(z). In practice the effective optical depth will scale somewhat between the linear and saturated line limits. It will not in general scale like the Gunn-Peterson optical depth Eq. ([eq:tauGP]), as is often asserted in the literature. For dN/dz ∝ (1 + z)γ, Eq. ([eq:taul]) instead gives the dependence τl ∝ (1 + z)1 + γ. This is corroborated by estimates of the mean transmission due to scattering from all components, whether or not they may be modeled as individual Voigt profile absorption systems. More generally, the mean transmission may be expressed as $$\langle{\cal T\_\nu}\rangle \equiv \exp(-\tau\_{\rm eff}). \label{eq:transmission}$$ From an analysis of scattering by the IGM in the redshift range 2.5 < z < 4.3 using moderate resolution spectra, obtained $\tau\_{\rm eff}=A(1+z)^{1+\gamma}$ with A = 0.0175 − 0.0056γ ± 0.0002 and γ = 2.46 ± 0.37. The evolution exponent γ agrees closely with the finding of for dN/dz above. A similar result was obtained by using high resolution spectra: A = 0.0144 − 0.00471γ and γ = 2.43 ± 0.17 (as inferred by ). give the slight variation based on an expanded data set of A = 0.0032 ± 0.0009 and γ = 2.37 ± 0.20. Because the implied probability distribution for $\langle {\cal T}\rangle$ is skewed, the statistical expectation value for the mean transmission must be computed from these expressions for $\tau\_{\rm eff}$ using Monte Carlo integration. A good fit to the evolution of $\tau\_{\rm eff}$ based on the data compiled by, extended to lower redshifts using the data of, is given by $$\begin{aligned} \tau\_{\rm eff} &=& 0.0164(1+z)^{1.1};\quad 0 < z < 1.2\\ &=& 0.00211(1+z)^{3.7};\quad 1.2 < z < 4\nonumber\\ &=& 0.00058(1+z)^{4.5};\quad 4 < z < 5.5\nonumber \label{eq:taul\_ev}\end{aligned}$$ . The evolution is found to vary from relatively slowly at low redshifts, as expected for discrete saturated features, to the more rapid evolution of the Gunn-Peterson effect at high redshifts as the porosity of the absorption systems approaches unity and increasingly diffuse gas dominates the effective optical depth. A dip in the evolution of $\tau\_{\rm eff}$ at z ≃ 3.2 was reported by based on a sample of 1061 moderate-resolution QSO spectra measured by the SDSS. The feature was interpreted as evidence for sudden heating due to the onset of reionization. This interpretation is not supported by recent simulations. The feature was not confirmed by using an alternative analysis method applied to 3035 SDSS QSO spectra. Using a sample of 86 high-resolution high signal-to-noise ratio spectra, however, find a feature similar to that reported by. The existence of the feature appears still not to be entirely secure, but tentatively suggests that a physical change in the state of the IGM occurred at z ≃ 3.2, possibly reflecting a change in the ionization rate, perhaps owing to the growing influence of QSOs (§ [subsubsec: sources]). ### Statistics based on pixel fluxes More detailed transmission information may be extracted from the use of the full spectrum, including the probability distribution of the transmission per pixel, the flux power spectrum, the autocorrelation function of the transmission, and a wavelet decomposition. Because these statistical measures are best interpreted in conjunction with numerical simulations, they are discussed in more detail in § [sec:simuls]. Here they are only defined. The pixel transmission distribution (PTD) and its variants, like the transmission flux probability distribution function (TFPDF), simply quantify the frequency of occurrence of a given flux value in each pixel of the spectrum, normalized by the continuum, often expressed in frequency bins. It was introduced by to search for underlying intergalactic absorption in excess of the amount accounted for by discrete absorption systems. Several tabulations are available in the literature: ,,, and. Second order statistics include the flux autocorrelation function and the flux power spectrum, in analogy to similar statistical descriptions of the clustering of galaxies. The flux transmission autocorrelation function was introduced by to investigate properties of the forest. For a velocity separation v, it is defined by $$\xi\_{\rm F}(v) = \langle f(v^\prime)f(v^\prime-v)\rangle/\langle f\rangle^2-1, \label{eq:xiF}$$ where f(v) denotes the pixel flux at velocity v, and ⟨f⟩ is the measured mean flux at the relevant redshift.[14](#fn14) Several estimates of $\xi\_{\rm F}$ are provided in the literature:,,,,,, and. The results generally show significant correlations over scales of a few Megaparsecs (comoving). Similar to the flux autocorrelation function is the flux cross-correlation function along neighboring lines of sight to QSOs near each other on the sky. Cross-correlations of similar strength to the autocorrelations are found, extending over similar spatial scales. The flux power spectrum has received considerable attention as a means of constraining cosmological parameters. It is a measure of the variance in the amplitude of the Fourier transform coefficients of the transmitted flux. Defining δf = f − ⟨f⟩ and its Fourier transform $${\hat{\delta f}}(k)=\frac{1}{\Delta v}\int\_{-\Delta v/2}^{\Delta v/2}\,dv \delta f(v)e^{ikv}, \label{eq:FTflux}$$ over the velocity interval Δv, the flux power spectrum is $$P\_{\rm F}(k)=(\Delta v)\langle\vert{\hat{\delta f}}\vert^2 \rangle/\langle f\rangle^2. \label{eq:PFk}$$ The dimensionless form $kP\_{\rm F}(k)/\pi$ is often more convenient. It forms a Fourier transform pair with the flux autocorrelation function through $$\frac{kP\_{\rm F}(k)}{\pi} = \frac{k}{\pi}\int\_{-\Delta v/2}^{\Delta v/2}\,dv\xi\_F(v)e^{ikv}. \label{D2FxiF}$$ The reader should be aware that several different conventions for $P\_{\rm F}(k)$ exist in the literature, some without a mean flux normalization, some differing by a factor of 2π, and some referring to an effective 3D flux power spectrum extracted from the 1D flux power spectrum in a manner analogous to the relation between the 1D and 3D mass power spectra. Many tabulations of $P\_{\rm F}(k)$ are available: ,,, and. A substantial effort was made to measure $P\_{\rm F}(k)$ from the spectra of high redshift quasars discovered by the Sloan Digital Sky Survey. A shortcoming of the Fourier transform is that it loses spatial information. In the case of a spectrum showing the forest, the positions of individual absorption features are lost. A wavelet transform is analogous to a Fourier transform, but retains local information, giving in effect a description of the power on different scales as a function of position. A wavelet is a square-integrable function, ψ(x), defined in real space such that ψjk ≡ 2j/2ψ(2jx − k) (where j and k are integers) forms an orthonormal basis for the set of square-integrable functions. It satisfies ∫− ∞∞dx ψ(x) = 0, and is normally concentrated near x = k2− j. The wavelet coefficients are defined by wjk = ∫ dxf(x)ψjk(x); the original function may be recovered through f(x) = ∑j, kwjkψjk(x). The wavelet transform effects a localized multiscale rendition of the data. Although not commonly used, wavelets are in principle a powerful tool for analysing the forest, producing a statistical description similar to the distribution of b-values. The discrete wavelet transform, like the fast Fourier transform, may be computed extremely rapidly, permitting a large number of spectra to be analyzed much more quickly than by performing a full absorption line analysis. This is an advantage over full Voigt profile line fitting for assessing the match of simulations to observations. Ionization Equilibrium ====================== Ionization ---------- ### Hydrogen and helium The absorption signatures of the IGM are produced by atoms and ions with bound electrons, either through resonance scattering or photoelectric absorption. The amount of scattering or absorption depends on the ionization structure of the IGM. In this section, the photoionization of hydrogen and helium is discussed. In hot environments like galaxy clusters, collisional ionization can dominate. Collisional ionization equilibrium may play a particularly important role in the physics of the WHIM and is treated more generally in the section below on metal ionization. Photons with energies exceeding the ionization potential of a bound electron in a hydrogen atom will ionize the neutral hydrogen atoms in the IGM at the rate per neutral atom $$\Gamma\_{\rm HI} = c\int\_{\nu\_{\rm T}}^\infty\,d\nu \frac{u\_\nu}{h\_{\rm P}\nu}a\_\nu^{\rm HI}, \label{eq:GHI}$$ where $a\_\nu^{\rm HI}$ is the hydrogen photoelectric cross section, $\nu\_{\rm T}$ is the threshold frequency required to ionize hydrogen (the Lyman limit), and uν is the specific energy density of the ambient radiation field. The specific energy density is related to the specific intensity of the radiation field $I\_\nu({\bf r},t,{\hat{\bf n}})$ by uν = 4πJν/c, where $J\_\nu({\bf r},t)=(1/4\pi)\oint\,d{\bf\Omega} I\_\nu({\bf r},t,{\hat{\bf n}})$ is the angle-averaged specific intensity. Free electrons will be radiatively captured by the protons at the rate per proton $n\_e\alpha\_{\rm HII}(T)$, where $\alpha\_{\rm HII}$, often referred to as the Case A radiative recombination coefficent αA(T), is the total rate coefficient for radiative capture summed over recombinations to all energy levels. Expressed in terms of the ionization fractions $x\_{\rm HI}=n\_{\rm HI}/ n\_{\rm H}$ and $x\_{\rm HII}=n\_{\rm HII}/ n\_{\rm H}$, where $n\_{\rm H}= n\_{\rm HI} + n\_{\rm HII}$ is the total hydrogen number density, the ionization rate equations become $$\begin{aligned} \frac{dx\_{\rm HI}}{dt} &=& -x\_{\rm HI}\Gamma\_{\rm HI} +x\_{\rm HII}n\_e\alpha\_A(T), \nonumber\\ \frac{dx\_{\rm HII}}{dt} &=& -\frac{dx\_{\rm HI}}{dt}. \label{eq:phionizeH}\end{aligned}$$ The time-derivatives are lagrangian, so that Eqs. ([eq:phionizeH]) are valid in the presence of velocity flows. In equilibrium, the neutral fraction will be $$x\_{\rm HI}^{\rm eq}=\frac{n\_e\alpha\_A(T)}{\Gamma\_{\rm HI} + n\_e\alpha\_A(T)}. \label{eq:phionizeHI-eq}$$ For a pure hydrogen gas, $n\_e=n\_{\rm HII}$, and the rate equation for $x\_{\rm HI}$ becomes $$\frac{dx\_{\rm HI}}{dt}-t\_{\rm rec}^{-1}x\_{\rm HI}^2 +(\Gamma\_{\rm HI} + 2t\_{\rm rec}^{-1})x\_{\rm HI}-t\_{\rm rec}^{-1}=0, \label{eq:phionize-xiHI}$$ where $t\_{\rm rec}=1/(n\_{\rm H}\alpha\_A)$ is the total radiative recombination time for the gas. The equilibrium neutral fraction may be expressed as $$x\_{\rm HI}^{\rm eq}=\left(\frac{1}{2}\phi + 1\right) -\left[\left(\frac{1}{2}\phi + 1\right)^2-1\right]^{1/2}, \label{eq:phionize-xiHI-eq}$$ where $\phi=\Gamma\_{\rm HI}t\_{\rm rec}$ denotes the number of photoionizations per neutral atom over a recombination time. It is instructive to consider the case of constant ϕ. Eq. ([eq:phionize-xiHI]) then has the general solution $$x\_{\rm HI} = x\_{\rm HI}^{\rm eq} - \frac{\eta}{\left(1 - \delta\_0^{-1}\eta\right)\exp(\eta\tau) - 1}, \label{eq:phionizeH-sol}$$ where $\eta = \phi + 2 - x\_{\rm HI}^{\rm eq}$, $\tau=t/t\_{\rm rec}$, and $\delta\_0 = x\_{\rm HI}(0) - x\_{\rm HI}^{\rm eq}(0)$ is the initial deviation of the neutral hydrogen fraction from its equilibrium value. For a fast photoionization rate (ϕ ≫ 1), $x\_{\rm HI}^{\rm eq}\simeq \phi^{-1}\ll 1$ and $x\_{\rm HI}\simeq x\_{\rm HI}^{\rm eq} + \delta\_0 \exp(-\Gamma\_{\rm HI}t)$. An initial deviation δ0 from equilibrium decays exponentially. The neutral fraction approaches its equilibrium value on the photoionization rate timescale $\Gamma\_{\rm HI}^{-1}$, regardless of whether equilibrium is established through net photoionizations or net recombinations. Ionization equilibrium will be maintained provided any change to the gas (such as source variability or the expansion of the gas) occurs on a time scale long compared with the photoionization time scale. By contrast, for a slow photoionization rate ϕ ≪ 1, $x\_{\rm HI}^{\rm eq}\simeq 1$ and the neutral fraction approaches equilibrium on the recombination timescale, $x\_{\rm HI}\simeq 1 + \delta\_0\exp(-t/t\_{\rm rec})$ for \|δ0\| ≪ 1 (δ0 < 0). The photoionization equations for helium are analogous to Eqs. ([eq:phionizeH]). Denoting the respective,, and fractions by $x\_{\rm HeI}$, $x\_{\rm HeII}$ and $x\_{\rm HeIII}$, they are given by $$\begin{aligned} \frac{dx\_\HeIs}{dt} &=& -x\_\HeIs \Gamma\_\HeIs + x\_\HeIIs n\_e\alpha\_\HeIIs, \nonumber \\ \frac{dx\_\HeIIs}{dt}&=& -\frac{dx\_\HeIs}{dt} - \frac{dx\_\HeIIIs}{dt}, \nonumber \\ \frac{dx\_\HeIIIs}{dt}&=& x\_\HeIIs \Gamma\_\HeIIs - x\_\HeIIIs n\_e\alpha\_\HeIIIs, \label{eq:phionizeHe}\end{aligned}$$ where $\alpha\_\HeIIs$ and $\alpha\_\HeIIIs$ are the total recombination rates to all levels of and, respectively, and $\Gamma\_\HeIs$ and $\Gamma\_\HeIIs$ are the respective photoionization rates. The total electron density is $n\_e = n\_{\rm HII} + n\_{\rm HeII} + 2n\_{\rm HeIII}$. rl $\alpha\_\HIIs$ & $2.065{\times 10^{-17}} T^{-{{\frac{1}{2}}}} \left(6.414 - \frac{1}{2}\ln\, T + 8.68{\times 10^{-3}} T^{{\frac{1}{3}}}\right)$ $\alpha\_\HeIIs$ & $3.294{\times 10^{-17}} \left\{\left(\frac{T}{15.54}\right)^{{\frac{1}{2}}}\left[1+\left(\frac{T}{15.54}\right)^{{\frac{1}{2}}}\right]^{0.309} \left[ 1+\left(\frac{T}{3.676{\times 10^{7}}}\right)^{{\frac{1}{2}}}\right]^{1.691} \right\}^{-1} +1.9{\times 10^{-9}} \left( 1 + 0.3 e^{\frac{-9.4{\times 10^{4}}}{T}} \right) e^{\frac{-4.7{\times 10^{5}}}{T}} T^{-\frac{3}{2}}$ $\alpha\_\HeIIIs$ & $8.260{\times 10^{-17}} T^{-{{\frac{1}{2}}}} \left( 7.107 - \frac{1}{2}\ln\,T + 5.47{\times 10^{-3}} T^{{\frac{1}{3}}}\right)$ $\beta\_\HIIs$ & $2.851{\times 10^{-40}} T^{{{\frac{1}{2}}}} \left(5.914 -\frac{1}{2}\ln\, T + 0.01184 T^{{{\frac{1}{3}}}} \right)$ $\beta\_\HeIIs $ & $1.55{\times 10^{-39}} T^{0.3647} +1.24{\times 10^{-26}} \left( 1 + 0.3 e^{\frac{-9.4{\times 10^{4}}}{T}} \right) e^{\frac{-4.7{\times 10^{5}}}{T}} T^{-\frac{3}{2}}$ $\beta\_\HeIIIs$ & $1.140{\times 10^{-39}} T^{{\frac{1}{2}}}\left( 6.607 - \frac{1}{2}\ln\, T + 7.459{\times 10^{-3}} T^{{\frac{1}{3}}}\right)$ [table.RecombCoef] The Case A radiative recombination coefficient for hydrogenic ions is provided by, from which the coefficients for $\alpha\_\HIIs(T)$ and $\alpha\_\HeIIIs(T)$ may be derived. (In general, the radiative recombination coefficient αZ(T) for a hydrogenic ion of atomic number Z is related to that of hydrogen by $\alpha\_Z(T)=Z\alpha\_\HIIs(T/Z^2)$.) For elements He and beyond, the recombination coefficient is the sum of two rates. One is the usual radiative capture term, in which e− + Xn + 1 → Xn + γ for a species X with n electrons. The other arises from dielectronic recombination, in which a fast incoming electron excites an ion, which becomes trapped in a weakly bound autoionizing state upon capturing the electron, then decays by the emission of a resonance line photon: $e^{-}+X^{n+1}\rightarrow {X^n}^\* \rightarrow X^n+h\_{\rm P}\nu\_{ul}$. The probability for emission of the photon is suppressed by the factor $\exp(-h\_{\rm P}\nu\_{ul}/k\_{\rm B}T)$; usually excitation is followed by the reverse reaction. A form for the radiative recombination rate coefficient contribution to $\alpha\_\HeIIs(T)$ for → is provided by, while a form for the dielectronic term is provided by. The recombination coefficients are listed in Table [table.RecombCoef]. Recombination to may also occur through charge transfer, ${\rm He}^+ + {\rm H}^0 \rightarrow {\rm He}^0 + {\rm H}^+ + \Delta E$, where ΔE is the energy defect. Where hydrogen is highly ionized, this is a negligible effect compared with radiative recombinations. For partially ionized hydrogen, however, as for example within a Damped Absorber, or the IGM prior to its complete reionization, the effect may play an important role. The rate coefficient is provided by. The reverse process, charge transfer ionization, is suppressed by a Boltzmann factor and so is normally negligible except at high temperatures. rllll & $a\_T [\m^2]$ & νT [Hz] & β & $\s$ $a\_\HIs$ & 6.30 × 10− 22 & 3.282 × 1015 & 1.34 & 2.99 $a\_\HeIs$ & 7.83 × 10− 22 & 5.933 × 1015 & 1.66 & 2.05 $a\_\HeIIs$ & 1.58 × 10− 22 & 1.313 × 1016 & 1.34 & 2.99 [table.CrossSection] The exact photoionization cross section for hydrogenic atoms with atomic number Z in the ground state is $$a\_Z(\nu)=\frac{A\_0}{Z^2}\left(\frac{\nu\_T}{\nu}\right)^4 \frac{e^{4-(4{\rm tan^{-1}}\epsilon)/\epsilon}}{1-e^{-2\pi/\epsilon}}, \label{eq:sigmaHZ}$$ for $h\_{\rm P}\nu>h\_{\rm P}\nu\_T=13.60Z^2$eV, the threshold energy for ionization, and where $A\_0 = (2^9 \pi / 3 e^4) \alpha \pi a\_0^2= 6.30 \times 10^{-22}\,{\rm m}^2$, where a0 = [4πε0]ℏ2/mee2 is the Bohr radius, α = [1/4πε0](e2/ℏc) is the fine structure constant, and ε = (ν/νT − 1)1/2. For most practical applications, however, an approximate form is preferred. The ionization cross sections are well-approximated by the form given by, $$\label{eq.CrossSection} a(\nu) = a\_T \left[ \beta \left(\frac{\nu}{\nu\_T}\right)^{-s} + (1-\beta)\left(\frac{\nu}{\nu\_T}\right)^{-s - 1}\right].$$ The parameters for hydrogen and helium are listed in Table [table.CrossSection]. For results requiring very high accuracy, more precise fits to the photoionization cross sections are provided by, including results for an exhaustive list of metals. ### Metals The large number of ionization states of metals results in more complex processes involved in their photoionization. These include autoionization and Auger electron production in addition to dielectronic recombination. In an autoionization, an excited ion with one or two electrons lying in the photoionization continuum is produced following a photoionization event in which a deep shell electron is ejected. The excited ion spontaneously ionizes through a radiationless transition and the ejection of multiple Auger electrons. Although there has been considerable progress in accurate estimates of all the relevant rate coefficients, many remain poorly determined either experimentally or theoretically, requiring interpolation (and sometimes extrapolation) from coefficients known for related species. These uncertainties can affect the predicted metal abundances in the absorption systems. The effect of uncertainty in the dielectronic recombination rates on commonly measured metal features is discussed by. Some of the methods of approximation are described in the literature for the rate coefficients (see below). Sophisticated computer codes have been developed to solve for the equilibrium ionization states. A widely used publicly-distributed code is CLOUDY, which solves the radiative transfer problem for plane-parallel and spherical geometries. The photoionization of intervening absorption systems has been described by, whose formulation is followed here. Denote by xij the fraction of a particular element i in the $j^{\rm th}$ ionization state, so that j runs from 1 (neutral) to Zi + 1 (fully ionized) for an element of atomic number Zi. Then ∑j = 1Zi + 1xij = 1. The general rate equation for xij is $$\begin{aligned} \frac{dx\_{ij}}{dt} &=& -x\_{ij}\biggl[n\_e(c\_{ij}+a\_{ij}+r\_{ij-1}+ d\_{ij-1}) + p\_{ij}\nonumber\\ &&+n\_{\rm HII}c\_{ij}^{\rm H}+n\_{\rm HeII}c\_{ij}^{\rm He} +\sum\_{k=j+2}^{Z\_i+1}\,q\_{ijk}\nonumber\\ &&\phantom{{=}-x\_{ij}\biggl[}+n\_{\rm HI}r\_{i,j-1}^{\rm H}+ n\_{\rm HeI}r\_{i,j-1}^{\rm He}\biggr]\nonumber\\ &&+x\_{ij+1}\biggl[n\_e(r\_{ij}+d\_{ij})+n\_{\rm HI}r\_{ij}^{\rm H}+ n\_{\rm HeI}r\_{ij}^{\rm He}\biggr]\nonumber\\ &&+x\_{ij-1}\biggl[n\_e(c\_{ij-1}+a\_{ij-1})+p\_{ij-1}\nonumber\\ &&\phantom{{=}+x\_{ij-1}\biggl[}+n\_{\rm HII}c\_{ij-1}^{\rm H} +n\_{\rm HeII}c\_{ij-1}^{\rm He}\biggr]\nonumber\\ &&+\sum\_{l=1}^{j-2}x\_{il}q\_{ilj}, \label{eq:ionizemetal}\end{aligned}$$ where cij is the collisional ionization rate, aij the autoionization rate, pij the photoionization rate without post-ionization Auger-electron ejection, rij the radiative recombination rate, dij the dielectronic recombination rate, $c\_{ij}^{\rm H}$ the rate of charge transfer ionization with $\HII$, $c\_{ij}^{\rm He}$ the rate of charge transfer ionization with $\HeII$, $r\_{ij}^{\rm H}$ the rate of charge transfer recombination with $\HI$, and $r\_{ij}^{\rm He}$ the rate of charge transfer recombination with $\HeI$. Photoionization from deep shells followed by Auger-electron ejection, resulting in a change from an initial ionization state j to a final state k ≥ j + 2, occurs at the rate qijk. The convention that rij corresponds to recombination to ionization state j and cij corresponds to collisional ionization of state j is used. Because of the low metallicity of the IGM, the electron density may normally be well-approximated by the electrons lost only by hydrogen and helium. Using the above notation for the various species and taking i = 1 for hydrogen and i = 2 for helium, Eqs. ([eq:phionizeH]) and ([eq:phionizeHe]) are modified, allowing for collisional ionization and the charge transfer reactions, to $$\begin{aligned} \frac{dx\_{11}}{dt} &=& -x\_{11}\biggl(p\_{11}+n\_ec\_{11} +\sum\_{i\ne1}\sum\_{j=2}^{Z\_i+1}r\_{ij-1}^{\rm H}n\_{ij}\biggr)\nonumber\\ &&+ x\_{12}\biggl(n\_er\_{11} + \sum\_{i\ne1}\sum\_{j=1}^{Z\_i} c\_{ij}^{\rm H}n\_{ij}\biggr),\nonumber\\ \frac{dx\_{12}}{dt} &=& -\frac{dx\_{11}}{dt},\nonumber\\ \frac{dx\_{21}}{dt} &=& -x\_{21} \biggl[p\_{21} + n\_e(c\_{21} + a\_{21}) +q\_{213}\nonumber\\ &&\phantom{{=}-x\_{21}}+\sum\_{i\ne2}\sum\_{j=2}^{Z\_i+1}n\_{ij}r\_{ij-1}^{\rm He}\biggr]\nonumber\\ && + x\_{22}\biggl[n\_e(r\_{21} + d\_{21}) + \sum\_{i\ne2}\sum\_{j=1}^{Z\_i}n\_{ij}c\_{ij}^{\rm He}\biggr],\nonumber\\ \frac{dx\_{22}}{dt} &=& -\frac{dx\_{21}}{dt} - \frac{dx\_{23}}{dt},\nonumber\\ \frac{dx\_{23}}{dt} &=& -x\_{23} \biggl[n\_e(r\_{22} + d\_{22}) +n\_{11}r\_{22}^{\rm H}\biggr] + x\_{21}q\_{213}\nonumber\\ &&+x\_{22}\biggl[n\_e(c\_{22} + a\_{22}) + p\_{22} +n\_{12}c\_{22}^{\rm H}\biggr], \label{eq:ionizeHHe}\end{aligned}$$ where the first sum of the double sums is over all elements except the excluded one indicated. Noting that the photoionization rates are proportional to the energy density of the radiation field, it follows from the structure of Eqs. ([eq:ionizemetal]) and ([eq:ionizeHHe]) that the equilibrium fractions are determined by the ratio $U=n\_\gamma/n\_{\rm H}$ for a given spectral shape of the radiation field with photon number density nγ, and for a given set of elemental abundances. The ratio U is know as the ionization parameter. Larger values for U generally produce higher ionization stages in the metals, although these also depend on the spectral shape of the radiation field. Comparisons with measured ion column densities may then be used to constrain the natures of the radiation field and the absorption systems. Fitting formulas to collisional ionization, radiative recombination and dielectronic recombination rate coefficients are provided by. Improved values for many of these rates have since then become available, and have been referenced above. Power law fits to collisional and radiative recombination rates involving hydrogen and helium are provided by. Fitting formulas to more recent rates for the elements hydrogen through nickel are provided by, along with equilibrium ionization fractions for metals in a hot environment, for which collisional ionization will normally dominate photoionization. A comprehensive compilation of the literature is available at D. Verner’s Atomic Data for Astrophysics website, [15](#fn15) as described in. Since then, improved estimates have been obtained for some rates, including laboratory measured values of dielectronic recombination rates ( and references therein). A comprehensive review of the atomic physics relevant to x-ray lines, as will arise from a warm-hot IGM, is provided by. Thermal equilibrium ------------------- In the presence of heat transfer at the rate (G − L)/n per particle, where G and L are the thermal gain and loss functions per volume and n is the particle density, the second law of thermodynamics requires the entropy per particle s to change at the rate $$\frac{ds}{dt}=\frac{1}{nT}(G-L), \label{eq:dsdt}$$ where T is the temperature of the system. For an ideal gas, the entropy per particle is $s=(\gamma-1)^{-1}k\_{\rm B}\ln(p/\rho^\gamma) + s\_0$, where p is the gas pressure, ρ is the mass density, γ is the ratio of specific heats at constant pressure to constant volume (γ = 5/3 for a monatomic gas), $k\_{\rm B}$ is Boltzmann’s constant, and s0 is an arbitrary additive constant. For numerical computations, it is often useful to express the energy equation in its entropic form Eq. ([eq:dsdt]), but through the entropy parameter $$\label{eq.Entropy} S\_E \equiv \frac{p}{\rho^\gamma}. \label{eq:S}$$ It follows from Eq. ([eq:dsdt]) that $$\frac{dS\_E}{dt} = (\gamma-1)\rho^{-\gamma}(G-L).$$ The gas temperature is related to SE by $$T = \frac{\bar m}{k\_{\rm B}} S\_E \rho^{\gamma-1} \label{eq:TS}$$ where $\bar m$ is the mean mass per particle. Photoionization will provide heat through the excess energy absorbed by an electron above the threshold energy $h\_{\rm P}\nu\_T$ required to ionize a given atom. For a single species i of density ni, the photoionization heating rate is $$G\_i = n\_i c\int\_{\nu^i\_T}^{\infty}\frac{d\nu}{\nu}u\_\nu \sigma\_i(\nu) (\nu-\nu^i\_T), \label{eq:PhotoionizationHeating}$$ where $h\_{\rm P}\nu^i\_T$ is the ionization potential of species i. The total heating rate from all species is $$G = G\_\HIs + G\_\HeIs + G\_\HeIIs. \label{eq:netHeating}$$ The contributions due to the photoionization of the metals is normally negligible because of their low abundances. Cooling is provided by recombinations, collisional excitation of the excited levels in neutral hydrogen, and inverse Compton scattering off CMB photons. At high temperatures, free-free losses and collisionally excited line radiation from atoms and ions other than neutral hydrogen may become important contributions as well. As for heating, all species contribute to cooling, giving the total cooling rate $$L = L\_\HIIs + L\_\HeIIs + L\_\HeIIIs + L\_{eH} + L\_C + L\_{\rm line} + L\_{ff}.$$ For a single species, recombinations radiate the electron energy $\sim k\_{\rm B}T$ as photons at the rate Li = neniβi(T). Recombination cooling coefficients βi(T) are provided by for recombinations to and. An expression for the radiative contribution to $\beta\_\HeIIs$ for recombinations to is provided by. The dielectronic contribution may be approximated as $\beta^{\rm diel}\_\HeIIs = 3 \ry \alpha^{\rm diel}\_\HeIIs$ using the second part of the expression for $\alpha\_\HeIIs$ in Table [table.RecombCoef], which expresses the dielectronic component. The total recombination cooling coefficients are provided in Table [table.RecombCoef]. The cooling rate due to the collisional excitation of by electrons is given to an accuracy of 3% over the temperature range 4000 < T < 12000 K by the expression : $$L\_{eH} = 7.3{\times 10^{-32}} \J \m^3 \s^{-1} n\_e n\_\HIs e^{-118400 / T}.$$ For temperatures relevant to the IGM, cooling losses due to collisional ionization of hydrogen and collisional excitation and ionization losses from helium are negligible. Similarly, for the low metallicities encountered in the IGM, radiative losses from metals are normally negligible, although they may possibly be important in the WHIM. Cooling is also produced by the Compton scattering of CMB photons off the electrons, which is especially important at high redshifts. The cooling rate is $$L\_C = 4 n\_e c\frac{\sigma\_T}{m\_e c^2} a T^4\_{\rm CMB}(z) k\_{\rm B}\left[T - T\_{\rm CMB}(z)\right],$$ where $\sigma\_T=(8\pi/3)(e^2/[4\pi\epsilon\_0]m\_ec^2)^2= 6.65\times10^{-29}\,{\rm m^2}$ is the Thomson cross section, a is the radiation density constant, me is the electron mass, c is the speed of light, and $T\_{\rm CMB}$ is the temperature of the cosmic microwave background. In the WHIM, cooling by thermal bremsstrahlung radiation, or free-free emission, due to the acceleration of a charge in the Coulomb field of another charge may be significant. For an ion of number density ni and charge Zi, the cooling rate is $$L\_{ff} = 1.43\times10^{-40} \J \m^3 \s^{-1} T^{1/2}n\_e n\_i Z\_i^2{\bar g}\_B, \label{eq:tbremss}$$ where ${\bar g}\_B$ is a frequency average of the velocity-averaged Gaunt factor, and ranges between 1.1 and 1.5. provide the fitting formula ${\bar g}\_B=0.79464 + 0.1243\log\_{10}(T/Z\_i^2)$ for T/Zi2 < 3.2 × 105 K and ${\bar g}\_B=2.13164 - 0.1240\log\_{10}(T/Z\_i^2)$ for T/Zi2 > 3.2 × 105 K. Cooling rates including collisionally excited metallic line losses and their individual emissivities are provided by for solar abundances. Non-solar values may be interpolated with the zero-metallicity limit for which atomic cooling is determined only by the hydrogen and helium losses. Photoelectric heating by x-rays is automatically included in Eq. ([eq:PhotoionizationHeating]). X-rays, however, may introduce two other sources of energy exchange: gains through Compton heating and losses through the effect of secondary electrons. Sources of high energy radiation, such as a hard x-ray background, may inject additional energy into the IGM through Compton heating. For an IGM of electron density ne and temperature T, an x-ray background with energy density uνX will heat the gas through Compton scatterings at the rate $$G\_C=n\_ec\frac{\sigma\_T}{m\_ec^2}\int\_0^\infty\,d\nu u\_\nu^X(h\_{\rm P}\nu-	arxiv_0000703
3] indicates a general lack of electrons with E > 10 TeV. One possibility of addressing this issue is simply reduced cooling by lowering the magnetic field. Yet even in Figures [fig:SED2] and [fig:ratio2] with a low 6 μG field the Region 12 model is far steeper than the data due to excessive cooling. In addition, Figures [fig:SED3] and [fig:ratio3] demonstrate that higher fields are needed in the inner zones, which only serves to hasten synchrotron burn-off. Particle re-acceleration as particles traverse the nebula might explain the VHE flux ∼ 80 pc from the pulsar, perhaps due to reverse shock interactions. Such acceleration is highly speculative, however, and beyond the scope of this paper. A more standard method of maintaining the high energy particle population far from the pulsar, which we explore below, is rapid energy-dependent diffusion with mean free path λ ∝ Ee. For this model we select seven variables: velocity radial profile index α, magnetic profile index β, initial spin period P0, braking index n, electron power-law index p, electron exponential cutoff Ecut, and mean free path λ. Figure [fig:SED4] shows the SED of the best fit model, scaled to display the aggregate gamma-ray data. This aggregate data is discussed in Section 5. Figure [fig:ratio4] displays the flux and index ratios for this model. The overall fit to the twelve spectral regions is far superior to previous models, particularly in the outer nebula. The largest departure of the model from the data is the X-ray spectral index of Region 3, though we remind the reader that the very hard Region 3 X-ray index reported in Table 4 may reflect residual background contamination. Table 6 lists the fit parameters, while Table 7 lists derived parameters from the fit (injection time, final magnetic field, and χ2). As expected from the scaling relations of section 3.1, the particle residence time of Zone 1 is well under 1000 years. The braking index n and initial spin period P0 yield an age of 40 ± 9 kyr and a mean expansion velocity of $2000 \, \rm{km \, s^{-1}}$. The mean free path of $\lambda = 1.8 (E\_e/100 \, \rm{TeV})$ parsec predicts an escape timescale of $\tau\_{esc} \approx 90 \, (R / 10 \, \rm{pc})^2 (E\_e/100 \, \rm{TeV})^{-1} \, \rm{year}$. This model does have some difficulty matching the H.E.S.S. flux and slope in Region 1, which is due to the rapid diffusive loss of high energy particles. Most likely, the simple diffusive model employed vastly oversimplifies the complex nature of this source, and diffusion coefficients vary in both space and time. Nevertheless, the χ2/d.o.f. for Model 4 indicates that the general fit is quite good. As a check of our magnetic field model, we also allow the normalization at the termination shock to vary from its adopted value of $400 \, \mu \rm{G}$. The best fit model with this extra fit parameter yields a termination shock field of $320 \, \mu \rm{G}$ with no improvement of the fit statistic (χ2/d.o.f. = 118/69, versus χ2/d.o.f. = 119/70 for the field fixed. Fitting the magnetic field normalization therefore does not yield a markedly better fit, and the termination shock value of $400 \, \mu \rm{G}$ appears quite reasonable. One further check of the magnetic field model is achieved by fitting a model with diffusion over uniform magnetic field (simply replacing the magnetic field radial index of Model 4 with a constant magnetic field value). This model cannot adequately match the data, yielding a best fit of χ2/d.o.f. = 429/70, and rules out uniform magnetic field as a viable option. The adopted IR photon density of ≈ 1 eV$\rm \, cm^{-3}$ could easily vary by a factor of ∼ 2, though varying the photon density does not qualitatively change the results presented here. A higher (lower) IR photon density means fewer (more) high energy electron are required to match the HESS data, implying a slightly softer (harder) electron spectral index or lower (higher) high energy cutoff. [tpb!] 1.0cZZZZZZZc Model ${\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & α (v ∝ rα) & B/β (B ∝ rβ) & P0 (ms) & $n \; (\dot{\Omega} \propto \Omega^n) $ & p & Ecut (TeV) & $\lambda \; (\frac{E\_e}{100 \,{\rm TeV}}) \, {\rm pc}$ & χ2/d.o.f. $1 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − & $2.9 \pm 0.3 \tablenotemark{a}$ & $15\tablenotemark{b}$ & $3\tablenotemark{b}$ & 2.42 ± 0.06 & 3800 ± 800 & − & 575/74 $2 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − 0.36 ± 0.03 & $5.6\pm0.4\tablenotemark{a}$ & 12 ± 3 & 2.5 ± 0.1 & 2.30 ± 0.01 & 130 ± 10 & − & 236/71 $3 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − 0.60 ± 0.08 & − 0.74 ± 0.02 & 14 ± 7 & 2.2 ± 0.1 & 2.31 ± 0.02 & 190 ± 30 & − & 217/71 $4 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − 0.51 ± 0.09 & − 0.69 ± 0.01 & 13 ± 7 & 1.9 ± 0.2 & 2.24 ± 0.02 & 230 ± 90 & 1.8 ± 0.3 & 119/70 [htpb!] c@ c@ c@ c@ c@ c Zone & R (pc) & $\delta t\_i \,(\rm{year})\tablenotemark{a}$ & $B \, (\mu \rm{G})\tablenotemark{b}$ & χ2 & # data points 1 & 7 & 600 & 12 & 23 & 12 2 & 14 & 1200 & 6.6 & 13 & 11 3 & 21 & 1500 & 4.8 & 6.6 & 9 4 & 28 & 1900 & 3.8 & 4.6 & 5 5 & 35 & 2200 & 3.2 & 8.0 & 5 6 & 42 & 2500 & 2.8 & 10 & 5 7 & 49 & 2900 & 2.5 & 8.9 & 5 8 & 56 & 3300 & 2.2 & 11 & 5 9 & 63 & 3800 & 2.1 & 7.4 & 5 10 & 70 & 4400 & 1.9 & 6.4 & 5 11 & 77 & 5500 & 1.8 & 16 & 5 12 & 84 & 9900 & 1.7 & 4.4 & 5 ### Relativistic Maxwellian Plus Power-Law Injection Spectrum The SED model described in the previous section takes into account only X-ray and VHE H.E.S.S. data, though ideally the model should be consistent with Fermi LAT data as well. Note that while we do not fit to the LAT points in Figure [fig:SED4], the rough agreement between these data and the integrated (total) model flux in the LAT band is encouraging. However, the Fermi LAT photon index of Γ = 1.4 ± 0.2 predicts an electron index of p = 2Γ − 1 = 1.8 ± 0.4. This is somewhat harder than the electron index of p = 2.2 utilized in Figures [fig:SED4] and [fig:ratio4], and an injection spectrum with fewer low energy electrons might plausibly improve the fit to the LAT data. One option is adopting the relativistic Maxwellian plus power-law tail electron spectrum proposed by, and described in modeling terms by. Fitting with this model requires one additional parameter (the temperature of the Maxwellian), bringing the total up to eight. The best fit model with this injection spectrum yields a fit somewhat worse than the power-law case (χ2/d.o.f. = 123/69, versus 122/70 for the power-law spectrum). In addition, an extremely short initial spin period of P0 ∼ 2 ms is required, and the low kT = 0.2 TeV results in a severe over-prediction of GeV flux and a far worse fit to the LAT data than the power-law case. found an adequate fit to the aggregate data with a one-zone model, yet the inclusion of multiple zones seems to wash out any advantage of a relativistic Maxwellian electron spectrum. Discussion ========== As an initial check of the validity of the scalings applied to both the Suzaku and H.E.S.S. data, we refit Model 4 with the raw data. Using the original H.E.S.S. data points and X-ray data taken from regions coinciding with the HESS quarter annuli, the best fit parameters are within 1σ of the values listed in Table 6 for Model 4, except for the magnetic field index. The different H.E.S.S. flux ratios between interior zones results in a magnetic field index that differs from the Model 4 value by +0.02, or 2σ. Despite the similarity in fit parameters, however, the fit statistic increases dramatically, from χ2/d.o.f. = 119/70 for Model 4, to χ2/d.o.f. = 169/70. Nearly all of this increase in χ2 comes from the inner two regions. The H.E.S.S. excess map (Figure 1b) reveals that there appears to be significantly more flux $6\arcmin-12\arcmin$ from the pulsar than within $6\arcmin$ of the pulsar. Yet due to the small original H.E.S.S. extraction region for the $6\arcmin-12\arcmin$ region (which is 25% smaller than the $0-6\arcmin$ region), the reported H.E.S.S. fluxes are identical for the inner two regions. The use of identical fluxes for the inner two zones therefore results in a worse fit for the model, which assumes radial variations in the flow speed and a greater number of particles farther from the pulsar. Nevertheless, the commonality of the fit parameters with the raw and scaled data provides some support for the scaled approach taken throughout this paper. One further check of the X-ray scaling is to fit the scaled H.E.S.S. data and raw, unscaled X-ray data. With this slightly lower X-ray flux the best fit parameters are well within errors for Model 4, and give a nearly identical χ2. Of the four models scenarios described above, the final model which incorporates diffusion provides by the far the best fit to the data. The relativistic Maxwellian injection spectrum yields a somewhat worse fit statistic to the simple power-law case, and though the power-law injection spectrum also better matches the Fermi LAT data, definitive discrimination between lepton injection spectra is not possible at this stage. Yet we can convincingly argue that the pulsar wind must inject significant numbers of electrons up to energies of at least 100 TeV, the low energy power-law index must be p ∼ 2 in order to match the Fermi LAT data, that a high percentage of the pulsar spin-down energy must go into leptons, and that some mechanism (we invoke diffusion) maintains high energy particles far from the pulsar. In the following discussion we consider the simple power-law spectrum injection spectrum (Model 4) shown in Figures [fig:SED4] and [fig:ratio4]. The sum of fluxes from the twelve spatial zones has been scaled up some 25% in Figure [fig:SED4] due to the fact that our spectral extraction regions do not capture the entirety of the counts within the aggregate nebula. This scaled flux ideally should match the Fermi LAT and H.E.S.S. aggregate data points (which both represent larger spatial regions), which appears to be the case for Figure [fig:SED4]. Computing the fit statistic for the aggregate H.E.S.S. and LAT data gives χ2 = 27 for the 16 gamma-ray data points (assuming only statistical errors on the data points, and ignoring the LAT upper limit). Given the very small statistical errors on the H.E.S.S. data points the agreement between gamma-ray data and the summed model curves is remarkable. For comparison, the best fit model utilizing the relativistic Maxwellian injection spectrum nets a χ2 = 50 for the 16 gamma-ray data points. The ‘missing’ ≈ 25% of flux implies a missing population of particles contributing to the IC flux of the nebula. Such particles have likely diffused to the north and east of the pulsar exterior to our extraction regions. A tweak of the initial pulsar spin-down from 13 ms to 11 ms (well within errors) provides ≈ 40% greater spin-down energy, more than enough to enough to raise the model bulk nebular flux to the observed level. The models proposed above utilize η = 0.8, and we find an adequate fit to the X-ray and H.E.S.S. data for P0 = 13 ms. Neglecting the ‘missing’ energy component, this connotes a total of 1.1 × 1050 erg of pulsar spin-down energy. Disregarding losses, we therefore expect of order 1049 erg in magnetic field energy. While the magnetic field evolution and profile adopted in our model does not take into account the total energy budget of the pulsar, a reasonable model should nevertheless contain ∼ 1049 erg of magnetic energy for the choice of η = 0.8. Computing the magnetic field energy from the final magnetic field profile $B(r,T) = 400 (r/0.03 \, \rm{pc})^{-0.69} \, \mu \rm{G}$ yields an energy of 4.4 × 1048 erg, which is quite reasonable given that the magnetic field likely experiences some losses due to expansion. Even though the vast majority of the pulsar energy initially goes into particles, summing the energy content of the final electron spectrum reveals that electron cooling losses leave only 6.9 × 1048 erg remaining in the form of leptons. We can also compute the value of σ from the assumed termination shock radius of 0.03 pc and $400 \, \mu \rm{G}$ field strength. Adopting a flow speed at the termination shock of c/3 and a spherical geometry, one achieves an estimate of $7\times 10^{35} \, \rm{erg \, s^{-1}}$ of magnetic energy flux at the termination shock. For the pulsar spindown value of $\dot{E} = 2.8\times10^{36} \, \rm{erg \, s^{-1}}$ this yields σ ≈ 0.3. This value is consistent with the adopted value of η = 0.8, which corresponds to a value of σ = (1 − η)/η = 0.25. One further consistency check for our model comprises comparing the available spin-down power to the total energy present in the nebula plus cooling and escape losses. Indeed, summing the leptonic energy, magnetic energy, cooling losses, and escape losses yields 1.1 × 1050 erg, precisely the energy available from the pulsar. The energy lost due to adiabatic cooling goes into accelerating a shell of swept-up material, and we can use estimates of this swept-up mass as another check of the validity of the SED model. For the adopted SNR density profile of a constant density inner core surrounded by a ρ ∝ r− 9 outer envelope the transition between the two density regimes propagates out at a constant velocity vt, such that : $$\begin{aligned} v\_t \approx 6500 \left( \frac{E\_{SN}}{3 \times 10^{51} \, \rm{erg}} \right) ^{1/2} \left( \frac{M\_{ej}}{8 M\_{\odot}} \right)^{-1/2} \rm{km \, s^{-1}.}\end{aligned}$$ This is much faster than the Model 4 outer PWN expansion of $2000 \, \rm{km \, s^{-1}}$, so we can treat the PWN as always expanding into the constant density ejecta core, with density : $$\begin{aligned} \rho\_{ej}(t) & \approx & 3.7 \times 10^{-29} \left( \frac{E\_{SN}}{3 \times 10^{51} \, \rm{erg}} \right) ^{-3/2} \left( \frac{M\_{ej}}{8 M\_{\odot}} \right)^{5/2} \nonumber \\ & & \times \left( \frac{t}{20 \, \rm{kyr}} \right) ^{-3} \, \, \rm{g \, cm^{-3}.}\end{aligned}$$ The amount of mass swept up by the PWN over its lifetime can be approximated as: $$\begin{aligned} M\_{sw} = \int\_{t\_i}^T \rho\_{ej}(t) \, \zeta 4 \pi r^2 dr,\end{aligned}$$ with ti the initial timestep, r = voutert and ζ the filling factor of the PWN, equal to 0.31 for the adopted morphology. Integrating Equation 25 over the 40 kyr lifetime of the pulsar provides a very rough estimate of 1M⊙ of swept up mass. Accelerating 1 M⊙ to $2000 \, \rm{km \, s^{-1}}$ (the mean expansion velocity for a pulsar age of 40 kyr) requires some 4 × 1049 erg. For the parameters selected in Model 4, the SED model posits a total adiabatic loss of 3.4 × 1049 erg, quite close to the value quoted above. Recall that in the SED model adiabatic losses depend only on the assumed PWN size evolution (Equation 17), and do not take into account any SNR properties. Therefore the fact that the two adiabatic loss estimates are not glaringly different provides some support to the soundness of the SED model. The computed pressure within the entire nebula can be used as a further sanity check on our model by comparing to the expected PWN pressure after 40 kyr. Given the degree of particle cooling, the model’s final particle pressure ($P\_{pwn} = \frac{E\_p}{3 V\_{pwn}} = 1.0 \times 10^{-13} \, \rm{erg \, cm^{-3}}$) is less than the magnetic pressure ($\frac{B^2}{8 \pi} = 1.9 \times 10^{-13} \, \rm{erg \, cm^{-3}}$). Within a freely expanding SNR the PWN pressure scales as : $$\begin{aligned} P\_{pwn} \approx 3/25 \, \rho\_{ej}(t) v\_{pwn}^2, \end{aligned}$$ with ρej(t) given by Equation 24, and vpwn the PWN expansion velocity. After 40 kyr this predicts a mean nebular pressure of $P\_{pwn} \approx 2 \times 10^{-13} \, \rm{erg \, cm^{-3}}$, in good agreement with the computed values. The adopted termination shock radius of 0.03 pc predicts a pressure of $\approx 1 \times 10^{-9} \, \rm{erg \, cm^{-3}}$ at the termination shock (Equation 19), significantly higher than the mean pressure of $\approx 3 \times 10^{-13} \, \rm{erg \, cm^{-3}}$. This high pressure at the shock rapidly dissipates away from the pulsar, however, with the bulk of the nebula nearly isobaric at late times; each of the 10 outer zones has a pressure within ≈ 50% of the mean nebular pressure, as evidenced in the pressure and energy plot of Figure [fig:pen]. This plot illustrates that the magnetic pressure exceeds particle pressure for the inner regions, and that the total particle energy exceeds the bulk magnetic field energy in the nebula. The total magnetic field pressure scales as Ppwn, B ∝ t− 3.0, while the total particle pressure evolves as Ppwn, p ∝ t− 2.9. This is precisely the Ppwn ∝ t− 3 evolution predicted by Equations 24 and 26 for a freely expanding PWN. In addition to the PWN pressure and energy we can also compute a variety of quantities which vary over the pulsar lifetime. We calculate the mean magnetic field evolution as $\overline{B} \propto t^{-1.6}$, which is quite similar to the B ∝ t− 1.7 evolution computed by in their single-zone model during the free expansion phase. The various luminosities in the nebula are plotted in Figure [fig:lum], revealing that at late times the total luminosity of the nebula exceeds the pulsar spin-down power. This figure also demonstrates that after a few thousand years adiabatic losses overtake synchrotron radiation to provide the dominant cooling mechanism, though particles of energy Ee > 10 TeV primarily lose energy from synchrotron and IC cooling since for both the cooling timescale τ ∝ Ee− 1. Indeed, at late time the IC cooling actually exceeds the synchrotron cooling due to the low mean magnetic field of $\approx 2 \, \mu \rm{G}$. We construct radial flux profiles by summing the model fluxes in the Suzaku and H.E.S.S. wavebands within each region, which can be compared with the data points plotted in Figure 2. We are also free to compute the profile in any arbitrary energy range, and so in Figure [fig:modradial] we plot the projected surface brightness in the Fermi LAT 1-100 GeV range. Since the LAT energy range lies in the uncooled regime, the flux primarily tracks the energy content of electrons. As a result, the LAT flux peaks toward the outer realm of the nebula where the majority of electrons were injected when the pulsar was much younger with a far higher $\dot{E}$, and this gives that a rather uniform counts map in the LAT energy band. Comparison of point source and extended Gaussian morphologies in the 10 − 100 GeV range by yielded a significance of ≈ 8.5σ for a source extension of 0.56∘ ± 0.07∘ centered 0.5∘ southwest of the pulsar. Similar results are achieved with a uniform disk morphology, though the extension increases to 0.67∘ ± 0.02∘. Fitting the source to a spatial template provided by the H.E.S.S. excess map (E > 200 GeV) decreases the test statistic by 42 ≈ 6.5σ, implying that the LAT emission shows a distinctly different morphology from the H.E.S.S. emission. Indeed, the LAT source extension of ∼ 0.6∘ centered 0.5∘ southwest of the pulsar is remarkably consistent with Figure [fig:modradial]. Conclusions =========== Analysis of recently public Suzaku X-ray data covering the majority of the very bright VHE source HESS J1825 − 137 indicate that X-rays extend only ≈ 1/4 as far as VHE gamma-rays. The variable extent of the nebula in different wavebands, as well as the observed VHE spectral softening away from the pulsar, is naturally explained by electron cooling losses. Time-dependent 3-D modeling of the twelve data regions allows one to constrain the electron injection properties, spin-down behavior of the pulsar, nebular velocity profile, and magnetic field profile. A choice of 80% of the pulsar spin-down going into electrons is consistent with the data, though due to particle cooling at the current epoch magnetic field energy is ≈ 1/2 × the particle energy in the nebula. This cooling also results in a higher magnetic than particle pressure, with a nearly isobaric particle pressure profile in the outer ten zones and the total pressure falling as Ppwn ∝ t− 3. For the best fit model (Model 4) an initial spin period of ≈ 13 ms provides adequate pulsar spin-down energy to power the nebula, and combined with the braking index of n = 1.9 this predicts an age of 40 kyr and an expansion velocity of $2000 \, \rm{km \, s^{-1}}$. The interior velocity profile is found to scale as v(r) ∼ r− 0.5, while the magnetic field profile scales as $B(r,t) \propto r^{-0.7} \, \dot{E}(t)^{1/2}$, with the field falling from $\approx 400 \, \mu \rm{G}$ at the termination shock to $\approx 2 \, \mu \rm{G}$ in the far reaches of the nebula and the mean field scaling as $\overline{B} \propto t^{-1.6}$. The Fermi LAT photon index of 1.4 connotes an electron index near the canonical 2.0. However, fitting to the X-ray/TeV and the higher energy portion of the electron spectrum, we find a somewhat softer power law index of p ≈ 2.2. A relativistic Maxwellian electron spectrum provides a somewhat worse fit to the X-ray and H.E.S.S. data compared to a simple power-law injection, and the power-law case requires fewer fit parameters and better matches the Fermi LAT data. Regardless of injection spectrum, the existence of 10 TeV photons in the outer reaches of the nebula requires substantial particle diffusion; adopting a diffusion timescale of $\tau\_{esc} \approx 90 \, (R / 10 \, \rm{pc})^2 (E\_e/100 \, \rm{TeV})^{-1} \, \rm{year}$ allows an adequate fit to the data. This is in contrast to the common assumption of a purely toroidal PWN magnetic field which could contain particles extremely efficiently. The model predicts a rather uniform Fermi LAT surface brightness out to ∼ 1∘ from the pulsar, in good agreement with the recently discovered LAT source centered 0.5∘ southwest of PSR J1826 − 1334, with extension 0.6 ± 0.1∘. The multi-zone time-dependent SED model fitting approach applied in the paper extends simple one-zone models to allow investigation of spatially dependent properties such as flow speed, magnetic field, photon index, and flux levels. The growing number of sources with spatially resolved X-ray and VHE measurements (e.g. Vela-X, HESS J1303-631) are therefore prime targets for such multi-zone modeling. Acknowledgments: We thank Stefan Funk and Marianne Lemoine-Goumard for helpful discussions on PWN modeling and Fermi LAT data. We also would like to thank the referee for very thorough and helpful comments. This work was supported in part by the U.S. Department of Energy contract to SLAC no. DE-AC02-76SF00515 and by NASA grant NNX10AP65G. Abdo, A. A. et al., 2010c, ApJ, 713, 146 Ackermann, M., et al. 2011,, 726, 35 Aharonian, F., et al. 2006,, 460, 365 Blondin, J. M., Chevalier, R. A., & Frierson, D. M. 2001,, 563, 806 Blumenthal, G. R., & Gould, R. J. 1970, Reviews of Modern Physics, 42, 237 Bogovalov, S. V., & Khangoulyan, D. V. 2002, Astronomy Letters, 28, 373 Bucciantini, N., Amato, E., Bandiera, R., Blondin, J. 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The multi-wavelength data places significant constraints on electron injection, transport, and cooling within the nebula. The large size and high nebular energy budget imply a relatively rapid initial pulsar spin period of 13 ± 7 ms and an age of 40 ± 9 kyr. The relative fluxes of each VHE zone can be explained by advective particle transport with a radially decreasing velocity profile with v(r) ∝ r− 0.5. The evolution of the cooling break requires an evolving magnetic field which also decreases radially from the pulsar, $B(r,t) \propto r^{-0.7} \, \dot{E}(t)^{1/2}$. Detection of 10 TeV flux ∼ 80 pc from the pulsar requires rapid diffusion of high energy particles with $\tau\_{esc} \approx 90 \, (R / 10 \, \rm{pc})^2 (E\_e/100 \, \rm{TeV})^{-1} \, \rm{year}$, contrary to the common assumption of toroidal magnetic fields with strong magnetic confinement. The model predicts a rather uniform Fermi LAT surface brightness out to ∼ 1∘ from the pulsar, in good agreement with the recently discovered LAT source centered 0.5∘ southwest of PSR J1826 − 1334 with extension 0.6 ± 0.1∘. Introduction ============ Pulsar wind nebulae (PWNe) confine magnetized pulsar winds via a termination shock, whereupon particles are re-accelerated to extremely high energies and emit radiation across the electromagnetic spectrum. Many PWNe are spatially resolved in the radio, X-ray, and even very high energy (VHE) wavebands, and thereby provide an excellent laboratory to study not only pulsar winds and dynamics, but also shock processes, the ambient medium, magnetic field evolution, and particle transport. An archetypal PWN system is HESS J1825 − 137 associated with the energetic pulsar PSRJ1826 − 1334, which possesses an energy-dependent TeV morphology thought to be caused by relic electrons from earlier epochs. This system is also a prime example of offset VHE PWNe, with the H.E.S.S. flux centered 0.3∘ south of the pulsar and VHE emission extending 1.2∘ to the south. The TeV emission softens from a photon index of Γ = 1.8 near the pulsar to Γ = 2.5 in the outer reaches of the nebula. PSR J1826 − 1334 was detected in the Jodrell Bank 20 cm radio survey, with period 101 ms, spin-down power $\dot{E} = 2.8 \times 10^{36} \, \rm{erg \, s^{-1}}$, and characteristic age τc = 21.4 kyr. The dispersion measure distance of the pulsar is ∼ 3.9kpc. X-ray observations of the PWN with XMM-Newton showed a compact core with a hard photon index (Γ = 1.6− 0.2+ 0.1) of size 30ʺ embedded in a larger diffuse structure of extension ∼ 5ʹ extending to the south of the pulsar with a softer photon index of Γ ≈ 2.3. Chandra observations by revealed the compact core to be composed of a bright inner component $\approx 7\arcsec \times 3 \arcsec$ surrounded by an outer component elongated in the east-west direction. More recent X-ray observations with Suzaku revealed diffuse X-ray emission of Γ ≈ 2.0 extending 15ʹ south of the pulsar with little sign of spectral softening. HESS J1825 − 137 has not been detected in the radio. Using archival VLA observations of PSR J1826 − 1334 measured a proper motion of $440 \, d\_4 \, {\rm km \, s^{-1}}$ (with distance d = 4d4 kpc) at a position angle ≈ 100deg east of north. This places the pulsar birthsite some $ 8 \arcmin = 9 \, d\_4 \, {\rm pc}$ to the west for an age of 20 kyr (see Figure 1), and rules out pulsar proper motion as the cause of the large southerly extension of HESS J1825 − 127. advocated that the offset is instead likely the result of an asymmetric reverse shock returning from the north, as proposed by to explain the similar structure observed in Vela-X. Indeed, found a compelling spatial correlation between a molecular cloud discovered in CO studies and the VHE source. The distance to this cloud is measured as 4 kpc, and with a total mass of ≈ 4 × 105 M⊙ naturally provides the ISM inhomogeneity to the north required for the reverse shock scenario. The fact that emission extends roughly perpendicular to the pulsar motion suggests that the reverse shock velocity must be significantly greater than the pulsar proper motion. The Fermi Large Area Telescope (LAT) recently detected HESS J1825 − 137 using 20 months of survey data. The gamma-ray emission detected by the LAT boasts a similar size to the H.E.S.S. source, with extension 0.56∘ ± 0.07∘ for an assumed Gaussian model. The 1 − 100 GeV LAT spectrum of this source is well described by a power-law with a spectral index of 1.4 ± 0.2. Single-zone spectral energy distribution (SED) modeling of the X-ray and gamma-ray data showed that ≈ 5 × 1049 erg injected in the form of electrons evolved over 25 kyr could reproduce the broadband aggregate spectrum, and that the hard LAT spectrum was consistent with both a simple power-law electron injection spectrum, and with a relativistic Maxwellian with a power-law tail. In this paper we describe SED modeling of the X-ray and gamma-ray data, with particular emphasis on the spatial variations in spectra. In Section 2 we overview the X-ray and VHE data, Section 3 details the SED model, Section 4 describes results of SED fitting, and Section 5 discusses the conclusions that can be drawn from model fitting. Data Analysis ============= Suzaku Data ----------- Diffuse Suzaku X-ray emission surrounding PSR J1826 − 1334 was analyzed by. The $18 \arcmin$ Suzaku XIS field of view is broad enough to encompass the bright core of HESS J1825 − 137, and reported emission nearly to the edge of all four XIS chips. Concurrently, a dedicated background pointing $35 \arcmin$ northeast of the initial exposure and outside the H.E.S.S. emission region was used for spectral backgrounds. More recently, a greater portion of the H.E.S.S. excess was covered by a grid of three Suzaku pointings (unpublished at the time of submission) to the South and West of the initial pointing (see Table 1). A further Suzaku exposure centered on PSR J1826 − 1334 (also unpublished) expanded the data coverage of the PWN core. These five images therefore cover $\approx 40 \arcmin \times 40\arcmin$ of the brightest portion of HESS J1825 − 137, and are listed in Table 1. A number of point sources are observed in the Suzaku mosaic, five of which (A-E) were noted by. These sources are listed in Table 2, as well as displayed in Figure 1. Sources E and N have no cataloged counterpart. [b!] c@ c@ c@ c@ c@ c@ No. & Obs ID & Date & Pointing & Location & Time (ks) 0 & 501045010 & 2006-10-19 & 276.90, -13.29 & BG & 52.1 1 & 501044010 & 2006-10-17 & 276.51, -13.71 & H.E.S.S. core & 50.3 2 &503028010 & 2008-10-15 & 276.50, -14.01 & S of obs #1 & 57.2 3 & 503029010 & 2008-10-17 & 276.20, -13.72 & W of obs #1 & 57.2 4 & 503030010 & 2008-10-19 & 276.19, -13.99 & SW of obs #1 & 57.2 5 & 503086010& 2009-03-19 & 276.57, -13.59 & PSR & 52.1 [b!] c@ c@ \|c@ c Source & Name/Location & Source & Name/Location A& 2XMM J182557.9-134755 & H & 2XMM J182539.9-133130 B & 2XMM J182617.1-134111 & I & 2RXP J182535.2-135049 C & 2XMM J182629.5-133648 & J & 2RXP J182458.3-133747 D & 2XMM J182620.9-134426 & K & 2RXP J182509.9-134621 E & 18:26:12 -13:48:33 & L & 2RXP J182601.4-140423 F & 2RXP J182640.4-133911 & M & 2RXP J182436.5-140429 G& 2RXP J182642.6-133625 & N & 18:24:14 -13:58:15 We utilize the standard pipeline screened events, and analyze the XIS chips with XSelect version 2.4. The four most recent pointings occurred after the loss of the XIS2 chip, so these pointings include the two remaining front side illuminated chips (XIS0, XIS3) as well as the back side illuminated chip (XIS1). All observations operate the XIS in normal clocking full window mode, with the data split between two editing modes: 3 × 3 and 5 × 5. An exposure corrected mosaic of the five Suzaku exposures is shown in Figure 1. Diffuse emission clearly extends $\sim 18 \arcmin$ south of the pulsar position, with very little emission beyond this, or to the north of the pulsar. [tbp!] H.E.S.S. Data ------------- The twelve H.E.S.S. spectral extraction regions extend in $6 \arcmin$ chunks 1.2∘ from the pulsar ($0.1^{\circ} = 7.0 \, d\_4 \, \rm{pc}$), covering a 90∘ sector from approximately 190∘ − 280∘, measured counterclockwise from North (, Figure 4). While these regions capture the majority of the VHE emission, inspection of the H.E.S.S. excess map indicates that a sizable fraction of the emission lies to the southeast of the pulsar and outside of the extraction regions. An accurate value of the ratio of X-ray to gamma-ray flux is important for modeling purposes (see Section 3.1). Therefore, to account for the missing VHE flux, and to allow better comparison with the Suzaku X-ray data (which extends beyond the 90∘ sector) and LAT GeV gamma-ray data (which is drawn from a very large extraction region), we extend the quarter annuli an extra 45∘ such that they now cover 145∘ − 280∘. Barring a full reanalysis of the H.E.S.S. data (which is not possible given the propriety nature of H.E.S.S. data) the best we can do is to scale the amplitude of each slice’s spectrum by the ratio of total image counts in the original versus extended region. As a check of this method, we compute the image counts ratios between slices. For the original (unscaled) H.E.S.S. regions, the ratio of image counts between any two regions lies within the error bounds of the published H.E.S.S. flux ratios between those two regions. This lends some credence to the assumption that the flux within a region scales directly with the number of counts in the H.E.S.S. excess map. The scaling factors are listed in Table 3 (along with X-ray scaling factors, described in Section 2.3). These expanded regions capture 81% of the counts within the 0.8∘ radius region used to extract the aggregate H.E.S.S. spectrum. The VHE flux data points of the SED plots (Figures [fig:SED1], [fig:SED2], [fig:SED3], [fig:SED4]) are therefore taken to be the rescaled points from Figure 4 of, and summing the rescaled flux data points of these regions yields data points ≈ 25% below the aggregate H.E.S.S. data. This is quite close to the expected 19% difference expected from the counts ratio in the H.E.S.S. excess map. Some of the missing flux is the northeast extension of the nebula, but some is likely contributed by adjacent regions to the north. estimate the systematic error on the flux scale to be 20%, though for SED fitting we utilize statistical errors only. Figure 1 shows the H.E.S.S. excess map and spectral extraction regions. [b!] c@ c@ c@ c@ c Region & X-Ray Scaling & H.E.S.S. Scaling & Γ & Scaled Flux $0 \arcmin -6 \arcmin {\rule{0pt}{2.6ex}}$ & 1.13 & 1.0 & 1.83 ± 0.09 & 2.9 ± 0.3 $6 \arcmin -12 \arcmin$ & 1.16 & 1.5 & 1.96 ± 0.08 & 4.4 ± 0.4 $12 \arcmin -18 \arcmin$ & 1.27 & 1.4 & 2.20 ± 0.06 & 6.0 ± 0.4 $18 \arcmin -24 \arcmin$ & & 1.4 & 2.25 ± 0.06 & 7.4 ± 0.4 $24 \arcmin -30 \arcmin$ & & 1.4 & 2.32 ± 0.07 & 8.7 ± 0.5 $30 \arcmin -36 \arcmin$ & & 1.4 & 2.37 ± 0.06 & 9.7 ± 0.5 $36 \arcmin -42 \arcmin$ & & 1.3 & 2.36 ± 0.08 & 7.4 ± 0.5 $42 \arcmin -48 \arcmin$ & & 1.4 & 2.41 ± 0.09 & 7.6 ± 0.6 $48 \arcmin -54 \arcmin$ & & 1.2 & 2.42 ± 0.09 & 6.1 ± 0.5 $54 \arcmin -60 \arcmin$ & & 1.2 & 2.59 ± 0.13 & 6.2 ± 0.6 $60 \arcmin -66 \arcmin$ & & 1.0 & 2.43 ± 0.09 & 5.9 ± 0.5 $66 \arcmin -72 \arcmin {\rule[-1.2ex]{0pt}{0pt}}$ & & 1.0 & 2.45 ± 0.35 & 2.9 ± 0.7 X-ray Spectrum -------------- Motivated by SED modeling, we extract Suzaku X-ray spectra from $6 \arcmin$ wedges to mimic the regions of. Spectral fitting is accomplished with Xspec version 12.6. For background subtraction we follow the same procedure as (a more in-depth discussion can be found in this paper) and define the same region as these authors: a $9 \arcmin$ circle on the background exposure, excising $1.8 \arcmin$ radius circles around the four point sources noted by these authors. The non-uniformity of the background is taken into account internally within Xspec. Spectra are extracted with XSelect, while ARF and RMF response files are generated with the xisrmfgen and xissimarfgen functions, which internally take into account the different responses of each chip. The ASCA FTOOL addascaspec is utilized to combine the spectra and responses of the XIS front side illuminated chips. The XIS1 back side illuminated chip possesses a markedly different response function from the front side illuminated chips, and so is analyzed in parallel rather than added to the other chips. Initially, we extract and fit the spectrum of the $1.5 \arcmin$ circle surrounding PSR J1826 − 1334 titled Region A by. Fitting an absorbed power-law model we achieve a fit well within errors of the parameters reported by these authors (Γ = 1.7 ± 0.1, unabsorbed 1 − 8 keV flux of $1.2 \pm 0.2 \times 10^{-12} \, \rm{erg \, cm^{-2} \, s^{-1}}$). We subsequently define regions mimicking the inner $6 \arcmin$ circle surrounded by $6 \arcmin$ thick partial annuli shown in Figure 1. From these regions we excise $1.8 \arcmin$ radius circles around the 5 point sources defined by. For the innermost $6 \arcmin$ circle we use both the original dataset (501044010), as well as the most recent dataset (503086010) and jointly fit the absorption and power-law index. Since the original dataset does not encompass the entirety of the $6 \arcmin$ circle we allow the power-law normalization to vary between the two datasets, though we quote the flux from the entire circle. The $6 \arcmin - 12 \arcmin$ region follows the same procedure, using both datasets 501044010 and 503086010. For overlapping regions (exposure 1 and exposure 5) the joint spectral fits to both exposures are completely consistent (albeit with smaller errors) in both index and total flux (when taking into account the differing areas of the regions) with fits to the individual exposures. All fits are to an absorbed power-law from 1 − 8 keV with quoted single parameter 68% (1σ) errors. We fit the $6 \arcmin - 12 \arcmin$ region with NH fixed at the value determined in the innermost region, and find values completely consistent (albeit with smaller errors) with fitting with NH free. Though NH may vary over the nebula, we see no evidence of this, and IR maps show no appreciable gradient over the nebula, so for simplicity we adopt a uniform column density. The pulsar will contribute flux to the innermost region, though with a 1 − 8 keV flux of $\sim 2 \times 10^{-14} \, \rm{erg \, cm^{-2} \, s^{-1}}$, the pulsar presents only a minor perturbation to the PWN flux. The $12 \arcmin - 18 \arcmin$ wedge stretches across three different pointings and suffers from a much lower surface brightness than the inner two regions, greatly complicating flux extraction from this region. Due to the limited statistics, we fix the absorption to the best fit value from the innermost region. Extracting flux from the portion of this region which lies within the original pointing (501044010) yields a ∼ 5σ detection. No significant emission further south and west of the pulsar is seen in any of the three exposures. In fact, we can use the apparent lack of flux in the southwest exposure furthest from the pulsar (503030010) to estimate the uncertainty in the background. Selecting a background region from this chip alters flux estimates by ≈ 1% for Region 1 ($0 \arcmin - 6 \arcmin$), ≈ 3% for Region 2, and ≈ 26% for Region 3. An additional source of error stems from the uncertainty in the ARF calculation for complex extended sources, which leads to uncertainty in the source flux, which we estimate at ≈ 10%. Note that background subtraction errors dominate for Zone 3; in fact, the relatively hard spectrum fit here may reflect residual background contamination. In the SED plots we show regrouped X-ray data with the ARF and background systematic errors summed in quadrature with 1σ statistical errors. To be perfectly consistent, for modeling purposes we should utilize identical regions for both X-ray and VHE gamma-ray data. The excised point sources and slight portion of the H.E.S.S. spectral extraction regions which extends beyond the Suzaku exposures therefore require a small adjustment to the X-ray flux level. The raw X-ray data is therefore rescaled by the fraction of missing area, with the scaling factors shown in Table 3. Table 4 and the SED plots therefore display the rescaled fluxes (with errors adjusted accordingly) in order to remain fully consistent with the H.E.S.S. data. [tb!] c@ c@ c@ c@ c Region & NH & Γ & Unabs. Flux & χ2/d.o.f. $0 \arcmin -6 \arcmin {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & 1.26 ± 0.06 & 1.93 ± 0.05 & 4.90− 0.32+ 0.34 & 196/249 $6 \arcmin -12 \arcmin {\rule{0pt}{2.6ex}}$ & 1.12− 0.10+ 0.11 & 2.03 ± 0.09 & 3.05− 0.34+ 0.39 & 144/173 $6 \arcmin -12 \arcmin$ & 1.3 & 2.17 ± 0.05 & 3.18 ± 0.15 & 146/174 $12 \arcmin -18 \arcmin {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & 1.3 & 1.78 ± 0.18 & 1.04− 0.19+ 0.22 & 44/59 [pwnspec] The X-ray flux falls off rapidly with radial distance from the pulsar, likely the result of both lower magnetic fields as well as cooling losses, as discussed below. Figure 2 shows a radial profile of fluxes in the X-ray and TeV energy bands, clearly indicating the much greater extent of the nebula in the H.E.S.S. regime. For this figure, we fix NH at $1.3\times 10^{22} \, \rm{cm^{-2}}$ and extract X-ray spectra from $3\arcmin$ width regions for greater detail than the $6\arcmin$ regions shown in Table 4. SED Modeling ============ Motivation ---------- applied a one-zone SED model to the new Fermi LAT data of HESS J1825 − 137 and found an adequate fit to the aggregate gamma-ray and X-ray data. One-zone models can help constrain certain global nebula properties, such as the total energy injected in the form of electrons, the electron spectrum in the uncooled regime, and the mean magnetic field. Yet such models by definition assume a completely homogeneous nebula, contrary to the differing X-ray and VHE sizes and spectral steepening observed in both X-rays and gamma-rays. To further investigate these phenomena we implement a spatially resolved model of HESS J1825 − 137. We begin by investigating how simple scaling relations inform model implementation. Starting assumptions include: the source of all particles and magnetic field in the nebula is PSR J1826 − 1334, ambient photon fields are static, and the nebula is 4 kpc distant. Let us first consider the electrons present in the innermost zone (Zone 1) adjacent the pulsar. In a transverse magnetic field these particles synchrotron radiate photons at mean energy: $$\begin{aligned} E\_{\gamma} \approx 2.2 (E\_e/100 \, {\rm TeV})^2 (B/10 \, \mu {\rm G}) \, \, {\rm keV.}\end{aligned}$$ We scale with a low field of $10 \, \mu {\rm G}$ since, as we shall see later, even in the innermost zone the mean field is only $\sim 10 \, \mu {\rm G}$ for the best fit model. Such particles cool rapidly, with a lifetime ($\tau \equiv E/\dot{E}$) of: $$\begin{aligned} \tau\_{sync} \approx 820 \, (E\_e/100 \, {\rm TeV})^{-1} (B/10 \, \mu {\rm G})^{-2} \, \, {\rm year.}\end{aligned}$$ Therefore the cooling timescale for electrons radiating synchrotron photons at mean energy Eγ is: $$\begin{aligned} \tau\_{sync} \approx 1200 \, (E\_{\gamma}/1 \, {\rm keV})^{-1/2} (B/10 \, \mu {\rm G})^{-3/2} \, \, {\rm year.}\end{aligned}$$ The flat (Γ ≈ 2) X-ray spectrum extending to 10 keV of Region 1 indicates that the electrons responsible for this emission are largely uncooled, implying $\tau \lesssim \tau\_{sync} \approx 380 \, (B/10 \, \mu {\rm G})^{-3/2} \, \, {\rm year}$. The inner region therefore must be dominated by particles recently injected by the pulsar. In the uncooled regime the synchrotron spectral flux scales with the magnetic field and number of electrons as: F(E)sync ∝ ne × B3/2. Over timescales short compared to the 21 kyr spin-down age of the pulsar the number of electrons ne in Zone 1 is roughly proportional to the time period over which electrons have been injected δt, so F(E)sync ∝ δt × B3/2, a relation we will return to momentarily. Electrons inverse Compton (IC) upscatter CMB photons to a mean energy of: $$\begin{aligned} E\_{\gamma} \approx 0.32 \, (E\_e/10 \, {\rm TeV})^{2} \, \, {\rm TeV.}\end{aligned}$$ For young electrons IC cooling is of little consequence, since 10 TeV electrons scattering off the CMB cool in $\tau\_{IC} \approx 140 \, (E\_e/10 \, {\rm TeV})^{-1} \, \, {\rm kyr}$. We scale to 10 TeV rather than 100 TeV since at 100 TeV the Thomson regime is no longer realized and Klein-Nishina corrections must be incorporated. For example, while the above scaling relation for inverse Compton cooling holds for electrons with Ee ≤ 10 TeV, 100 TeV electrons cool slowly over 36 kyr, rather than the 14 kyr predicted by this relation. We therefore consider only synchrotron cooling in the following discussion. The lowest energy H.E.S.S. point at Eγ ≈ 0.3 TeV requires electrons of energy Ee ≈ 10 TeV, which synchrotron cool slowly at a rate of $\tau\_{sync} \approx 8200 \, (B/10 \, \mu {\rm G})^{-2} \, \, {\rm year}$. Therefore, electrons responsible for the low energy H.E.S.S. points cool quite slowly compared to those responsible for the X-ray flux. The electrons responsible for the lowest energy H.E.S.S. points (and therefore all the Fermi LAT flux points as well) are radiatively uncooled over time spans of a few thousand years, and provide a good indication of just how many electrons are present in the inner nebula. This is due to the fact that for Ee ≤ 10 TeV the Thomson limit holds and IC flux scales linearly with the number of electrons (ne) present: F(E)IC ∝ ne. The exact number of electrons at 10 TeV will depend on the electron injection spectrum parameters as well as the injection time δt, and very weakly on the magnetic field due to feeble synchrotron cooling, so F(E)IC ∝ δt. The X-ray data breaks the degeneracy in magnetic field since F(E)sync ∝ δt × B3/2, so between X-ray and gamma-ray data the electron injection spectrum, age, and magnetic field are very well constrained. For modeling the data, this means that once an injection spectrum is selected, the gamma-ray flux determines the injection time, while the ratio of X-ray to gamma-ray flux determines the magnetic field. Pulsar Spin-down ---------------- As pulsars spin down, they dissipate rotational kinetic energy according to: $$\begin{aligned} \dot{E} = I \Omega \dot{\Omega},\end{aligned}$$ with Ω the angular frequency and I the neutron star’s moment of inertia, assumed to be $10^{45} \, \rm{g \, cm^2}$. This energy goes into a magnetized particle wind such that for a braking index of n: $$\begin{aligned} \dot{\Omega} \propto \Omega^n.\end{aligned}$$ For spin-down via magnetic dipole radiation n = 3, though n has only been confidently measured for five pulsars, in each case falling between 2 < n < 3 ( and references therein). Integrating Equation 6 yields the age of the system : $$\begin{aligned} T = \frac{P}{(n-1) \mid \dot{P} \mid} \left( 1- \left( \frac{P\_0}{P} \right)^{(n-1)} \right),\end{aligned}$$ where P0 is the initial spin period, and $\dot{P}$ the period derivative. For P0 ≪ P and n = 3 this equation reduces to the characteristic age of the pulsar $\tau\_c \equiv P/2\dot{P}$. The spin-down luminosity of the pulsar evolves as : $$\begin{aligned} \dot{E}(t) = \dot{E\_0} \left( 1 + \frac{t}{\tau\_{0}} \right) ^{-\frac{(n+1)}{(n-1)}},\end{aligned}$$ with the initial spin-down timescale defined as: $$\begin{aligned} \tau\_{0} \equiv \frac{P\_0}{(n-1) \mid \dot{P\_0} \mid} \,.\end{aligned}$$ Given that the current P, $\dot{P}$, and $\dot{E}$ are known, once an initial period P0 and braking index n are selected the age T and spin-down history of the system are determined according to the equations above. The energy flux in the magnetized wind $\dot{E}$ is split between electromagnetic $\dot{E\_B}$ and particle energy $\dot{E\_{e}}$, with: $$\begin{aligned} \dot{E}\_{e} & = & \eta \dot{E} \nonumber \\ \dot{E}\_{B} & = & (1 - \eta) \dot{E}.\end{aligned}$$ Their ratio is commonly referred to as the wind magnetization parameter: $$\begin{aligned} \sigma \equiv \dot{E}\_B / \dot{E}\_{e} = (1-\eta)/\eta,\end{aligned}$$ with σ typically thought to be ≪ 1 past the termination shock. The particle energy is presumed to go into an electron/positron plasma with an exponentially cutoff power-law spectrum: $$\begin{aligned} \frac{dN}{dE} \propto E^{-p} e^{-E/E\_{cut}} \; \; \; \; \; E > E\_{min},\end{aligned}$$ with Emin the minimum particle energy. We initially select Emin as 100 GeV, which is well within the realm of minimum particle energies considered by in magnetohydrodynamical modeling of the Crab Nebula. The normalization of this power-law varies with time following Equation 8, though we treat the index, minimum energy, and cutoff energy as static. We begin by modeling pure power-law injection, although we later consider an injection spectrum composed of a relativistic Maxwellian with a non-thermal tail, as proposed by. Spatial Evolution ----------------- Pulsar winds flow out at relativistic speeds until the ram pressure of the wind is balanced by the internal pressure of the PWN at the termination shock. High-resolution X-ray images can help illuminate the location of this termination shock, and Chandra images of the pulsar reveal that the brightest inner PWN component stretches some $7 \arcsec \times 3 \arcsec$ ($0.14 \times 0.06 \, \rm{pc^2}$ at 4 kpc), with the short axis roughly oriented toward the pulsar birthsite. The overall extent of the equatorial flow is generally 2 − 3 times larger than the termination shock radius, and we accordingly adopt a value of rts = 0.03 pc for the radius of the termination shock. Past the termination shock the pulsar wind expands radially at a much reduced velocity. In the model of of the rapid rotation of the pulsar wraps up magnetic field lines, resulting in a primarily toroidal field at and beyond the termination shock. The field direction is therefore approximately perpendicular to the wind flow velocity. Since particles effectively cannot cross toroidal magnetic field lines, the bulk flow of particles predicts an ordered layering of particle ages with increasing distance from the pulsar, of age: $$\begin{aligned} t(r) = \int\_0^r v(r')^{-1} \, dr'.\end{aligned}$$ with v(r) the velocity profile. On larger scales, we consider the evolution of the parent supernova remnant (SNR) of PSR J1826 − 1334. Early in the pulsar lifetime the SNR expands freely into the ambient medium, typically at a velocity of $\sim 10,000 \, \rm{km \, s^{-1}}$. The large ∼ 80 pc size of the VHE PWN and lack of any observed SNR shell led to propose a very large SNR of radius ≈ 120 pc resulting from an energetic supernova explosion (ESN = 3 × 1051 erg) of age ≈ 40 kyr, and a very low ambient medium density of $n\_0 \approx 0.001 \, \rm{cm^{-3}}$. Early on, a spherically symmetric PWN expands rapidly into unshocked supernova ejecta with nearly constant velocity : $$\begin{aligned} v\_{pwn} & \approx & 3100 \left( \frac{\dot{E\_0}} {10^{39} \, \rm{erg \, s^{-1}}} \right)^{1/5} \left( \frac{E\_{SN}} {3 \times 10^{51} \, \rm{erg}} \right)^{3/10} \nonumber \\ & & \times \left( \frac{M\_{ej}} {8 M\_{\odot}} \right)^{-1/2} \left( \frac{t} {1 \, \rm{kyr}} \right)^{1/5} \; \rm{km \, s^{-1}.}\end{aligned}$$ As the SNR expands and sweeps up more mass, it eventually reaches a point where the swept up ISM material is approximately equal in mass to the mass of the supernova ejecta. At this point the SNR enters the Sedov-Taylor phase as the swept-up material begins to dominate the SNR dynamics. Adopting a uniform density ISM, for the standard core-collapse supernovae assumption of a constant density inner core surrounded by a ρ ∝ r− 9 outer envelope the transition to the Sedov-Taylor phase occurs at : $$\begin{aligned} t\_{S-T} & \approx & 7100 \, \left( \frac{E\_{SN}}{3 \times 10^{51} \, \rm{erg}} \right) ^{-1/2} \left( \frac{M\_{ej}}{8 M\_{\odot}} \right)^{5/6} \nonumber \\ & & \times \left( \frac{n\_0}{0.001 \, \rm{cm^{-3}}} \right) ^{-1/3} \, \rm{year.}\end{aligned}$$ In this phase a reverse shock is driven deep into the SNR interior, and for a SNR expanding into a homogenous medium the reverse shock will impact the PWN, crushing it and leading to a series of reverberations which can last many thousands of years, as demonstrated compellingly in. Calculations by yielded an upper limit on the reverse shock collision with the PWN at trs < 5 tS − T. A supernova explosion into an evacuated cavity of $n\_0 = 0.001 \, \rm{cm^{-3}}$ therefore yields a maximum PWN-reverse shock collision time of ≈ 35 kyr (nearly double the characteristic age of the pulsar), implying that the PWN could to this day be rapidly expanding to the southwest into unshocked SNR ejecta. Of course, reflection off of the molecular cloud discovered by could result in a reverse shock collision on a timescale reduced by an order of magnitude from this estimate. Axisymmetric 2-D simulations by showed that SNR expansion into an inhomogeneous medium leads to a PWN offset from the pulsar position and mixing of the thermal gas of the supernova ejecta and the relativistic gas of the PWN. Rayleigh-Taylor instabilities aid in the mixing and disruption of the nebula as bubbles of swept up material penetrate the PWN. The simulations of showed a cometary nebula ∼ 20 kyr after the reverse shock interaction, and nearly complete disruption of the nebula after ∼ 50 kyr. These simulations do not take into account the magnetic field that is mixed in with the relativistic particles, and which can dampen instabilities along magnetic field lines. Indeed, recent simulations by suggest that the growth of instabilities is highly suppressed for high σ flows, and significant structure may exist even some ∼ 20 kyr after reverse shock interaction. This suggests that the ordered layering of particle ages predicted by Equation 13 may remain valid in the case of an anisotropic reverse shock. If one invokes PWN expansion within a freely expanding SNR to account for the ∼ 80 pc size of the PWN, a mean velocity of $3100 {\rm \, km \, s^{-1}}$ over 25 kyr is required. While this velocity is quite high for a PWN, it is permitted by Equation 14 for a very low density ambient medium. The $17 \arcmin \, (20 d\_4 \, \rm{pc})$ displacement of the H.E.S.S. centroid south of PSR J1826 − 1334 cannot be explained by pure expansion, however, and requires some mechanism to achieve this offset. One possibility for the southerly displacement is the existence of a powerful jet rapidly transporting particles far from the pulsar. X-ray images show no indication of any such feature, however. In addition, PWN jets are typically aligned with the pulsar spin axis, and a southern jet would be roughly perpendicular to the inferred pulsar proper motion, also typically aligned with the pulsar spin axis. A more palatable alternative is that the southerly offset from the pulsar is due to crushing by the reverse shock. In this picture the reverse shock quickly reflects off of the molecular cloud to the north, and then impacts the PWN in a few thousand years. The mechanism responsible for the PWN displacement, be it direct interaction with the reverse shock or a resultant anisotropy from the shock, is evidently still at work since the young X-ray emitting electrons are clearly concentrated south of the pulsar. For a reverse shock returning from the North, particles initially north of the pulsar will be displaced south and mixed throughout the nebula. The fact that the H.E.S.S. spectral index of HESS J1825 − 137 increases monotonically from the pulsar hints that such mixing is likely far from complete. Precisely how the reverse shock crushing and displacement alters any ordered layering of the PWN requires detailed 2-D or 3-D magnetohydrodynamical simulations and is beyond the scope of this paper. We ignore such effects, and hence tacitly assume that for HESS J1825 − 137 reverse shock interactions serve primarily to displace rather than compress and mix the nebula. Morphology ---------- The shape of the nebula as viewed from earth resembles a circular sector. While the 3-dimensional morphology of HESS J1825 − 137 is uncertain, if the morphology results from a shock returning from the north, this implies nebular axial symmetry about the normal to the shock front. One simple geometry which matches the Suzaku and H.E.S.S. morphology is a spherical wedge of opening angle 135∘ (solid angle 3.88 steradians) with symmetry axis along the H.E.S.S. symmetry axis 17∘ counterclockwise from North. The molecular cloud which presumably gives rise to the assymetric reverse shock is located at the same 4 kpc distance as the pulsar, so we assume that the shock normal is perpendicular to the line of sight. For modeling purposes we split this wedge into twelve 3-D zones which mimic the spectral extraction regions. Given the axisymmetry, the twelve spatial zones form a series of nested bowls, or partial spherical wedges, of thickness 7.0d4 pc. Similar to the spectral extraction regions, we label Zone 1 adjacent to the pulsar (extending 0 − 7 pc), and Zone 12 the outermost spherical wedge (77 − 84 pc from the pulsar). Henceforth we refer to Regions as the 2-D projections on the sky of the 3-D spatial Zones. Figure 3 demonstrates the adopted morphology, with the symmetry axis oriented vertically for clarity. The final volume of each bubble is determined by the spherical wedge morphology discussed above. These volumes are listed in Table 5. Another, more digestible, way to address the size of each bubble is to consider the effective radius of each bubble, defined as the radius of a sphere with volume equivalent to the final volume Req = (3V/4π)1/3. These effective radii listed radially outward are: 4.7 pc, 9.0 pc, 13 pc, 16 pc, 19 pc, 21 pc, 24 pc, 26 pc, 28 pc, 31 pc, 33 pc, 35 pc. When projected into the sky, the counts from each bowl will overlap into multiple spectral extraction regions. In order to determining the degree of overlap we perform a Monte Carlo integration by projecting the 3-D spatial zones onto the plane of the sky. For example, consider Zone 8 extending $49 \, \rm{pc} - 56 \, \rm{pc}$ from the pulsar. We populate this zone with a large number of random points, then project each point onto the appropriate plane defined by the viewing angle. We then overlay the spectral extraction regions onto this projection and determine the fraction of counts which lie within each bin. Assuming each bin is spatially uniform, the fraction of counts within each region is equivalent to the fraction of flux. For all zones save the innermost, significant flux from each spatial zone is found in the interior spectral extraction regions, as demonstrated by Figure 4. Table 5 shows the fraction of flux from each zone which falls within each spectral region. [tb!] 1.0lZZZZZZZZZZZZc & Reg 1 & Reg 2 & Reg 3 & Reg 4 & Reg 5 & Reg 6 & Reg 7 & Reg 8 & Reg 9 & Reg 10 & Reg 11 & Reg 12 & V (pc3) Zone 1 ${\rule{0pt}{2.6ex}}$ & 1.00 & & & & & & & & & & & & 440 Zone 2 & 0.14 & 0.86 & & & & & & & & & & & 3100 Zone 3 & 0.01 & 0.30 & 0.69 & & & & & & & & & & 4800 Zone 4 & & 0.07 & 0.34 & 0.59 & & & & & & & & & 16000 Zone 5 & & 0.01 & 0.13 & 0.34 & 0.52 & & & & & & & & 27000 Zone 6 & & & 0.05 & 0.15 & 0.32 & 0.47 & & & & & & & 40000 Zone 7 & & & 0.02 & 0.08 & 0.16 & 0.30 & 0.44 & & & & & & 56000 Zone 8 & & & & 0.04 & 0.09 & 0.16 & 0.30 & 0.41 & & & & & 75000 Zone 9 & & & & 0.02 & 0.06 & 0.10 & 0.15 & 0.28 & 0.39 & & & & 96000 Zone 10 & & & & 0.01 & 0.04 & 0.07 & 0.10 & 0.15 & 0.27 & 0.37 & & & 120000 Zone 11 & & & & & 0.02 & 0.05 & 0.07 & 0.10 & 0.15 & 0.26 & 0.35 & & 150000 Zone 12 & & & & & 0.01 & 0.03 & 0.05 & 0.07 & 0.10 & 0.15 & 0.25 & 0.34 & 180000 Spatial Model ------------- Motivated by the morphology adopted in the previous section, we treat the PWN as a series of twelve expanding bubbles with the radial extent of each bubble accounted for by a preferential southerly expansion. As per the discussion in Section 3.3, this southerly expansion is assumed to result from an asymmetric reverse shock returning from the north. Particles are injected into each bubble for a period of time δti, after which the bubble is assumed to detach from the pulsar and direct injection of particles is discontinued. Given the distinct possibility that in the southerly direction SNR is still freely expanding (leading to nearly free expansion for the PWN), for simplicity we assume constant velocity expansion of the outer boundary of the nebula $v\_{outer} = 84 \, \rm{pc} / T$, with T the pulsar age determined by Equation 7. Despite the adoption of a constant outer expansion velocity, particles will still decelerate over time as they traverse the nebula and encounter supernova ejecta. In the interior of the nebula we therefore consider a radial velocity profile $$\begin{aligned} v(r,t) = v\_{outer} \, \left( \frac{r}{R\_{outer}} \right) ^{\alpha} \, \left( \frac{t}{T} \right) ^{-\alpha},\end{aligned}$$ with α a fit parameter, and the time dependence determined by the requirement that vouter remain constant. Integrating Equation 16 yields the position of various bubbles with time. The final extent of each bubble (e.g. Zone 1 covers 0 − 7 pc from the pulsar) determines the required injection time for each bubble via Equation 13. Once injection ends, the bulk evolution of each bubble is primarily adiabatic, as IC cooling is relatively weak and the magnetic field is reduced to the point that synchrotron losses are severe only for the highest energy particles. For relativistic particles with adiabatic index 4/3 the particle pressure is Ppwn = Ee/(3Vpwn), and for purely adiabatic expansion PpwnVpwn4/3 = constant. For all time steps particle adiabatic losses are therefore computed via $$\begin{aligned} E\_e \propto V\_{pwn}^{-1/3}.\end{aligned}$$ The sound speed in the relativistic gas of the PWN ($c/\sqrt{3}$) is far greater than the expansion velocity of the PWN, implying that pressure differences will rapidly equilibrate and that the PWN is nearly isobaric. The energy lost due to adiabatic expansion goes into accelerating supernova ejecta at the PWN-SNR interface, and so we consider adiabatic losses to be shared equally among all particles present in the nebula, with the fraction of energy lost by each particle a function only of the volume evolution of the aggregate nebula. Magnetic Field Model -------------------- In principle, one can use the injected magnetic field energy (Equation 10) and spatial evolution of the PWN zones to compute the magnetic field ($E\_B = \frac{B^2}{8 \pi} V$, see ). Due to the uncertainty in how the magnetic field is distributed and transported, and the possibility of field amplification in the post-shock flow, we adopt a somewhat simpler magnetic field evolution. A static and uniform magnetic field presents the simplest possible magnetic field structure, yet the nebular magnetic field is expected to decrease from the termination shock to the far reaches of the nebula, as well as decrease with time as the pulsar spins down. Near the pulsar one common simple formulation is a dipolar magnetic field (B ∝ r− 3) out to the pulsar light cylinder, with a toroidal field (B ∝ r− 1) out to the termination shock. The surface magnetic field is estimated from the period and period derivative as $$\begin{aligned} B\_s = 3.2 \times 10^{19} \, (P \dot{P})^{1/2} \, \rm{G}.\end{aligned}$$ For PSR J1826 − 1334 this gives a current value of $B\_s = 2.8 \times 10^{12} \, \rm{G}$ for a surface radius rs = 10 km. The light cylinder is currently located at $r\_{lc} = c/\Omega = 4.8 \times 10^8 \, \rm{cm}$. At the wind termination shock the field is amplified by a factor of ∼ 3, which gives $B\_{ts} \approx 400 \, \mu \rm{G}$ for rts = 0.03 pc, and the assumed geometry of B ∝ r− 3 out to the light cylinder and B ∝ r− 1 from the light cylinder to the termination shock. The termination shock occurs at radius: $$\begin{aligned} r\_{ts} = \left( \frac{\dot{E} \epsilon}{4 \pi c P\_{pwn}} \right) ^{\frac{1}{2}},\end{aligned}$$ where ε is the filling factor of the pulsar wind. For an isotropic wind ε = 1, though more common is a scaling with co-latitude θ of $\rm{sin}^2 \theta$, yielding ε = 3/2. The key feature here is that the termination shock radius scales as $r\_{ts} \propto (\dot{E}/P\_{pwn})^{1/2}$. One can estimate the PWN pressure as Ppwn ∝ ρejv2 (Equation 26), with ρej ∝ t− 3 (Equation 24). The assumed constant velocity expansion therefore gives Ppwn ∝ t− 3, which is identical to the Ppwn ∝ t− 3.0 scaling computed by the model prior to reverse shock interactions. The termination shock radius accordingly scales roughly as $r\_{ts} \propto \dot{E}(t)^{1/2} t^{3/2}$, and for a braking index near n = 2 (where $\dot{E}(t) \propto t^{-3}$) the termination shock remains approximately static. Furthermore, since our SED model focuses on emission from the entire nebula (which dwarfs rts), the precise value of rts has little effect on the resultant SED. For simplicity, we therefore assume a constant value of rts = 0.03 pc. Adopting a dipolar followed by toroidal magnetic field geometry out to the termination shock, $B\_{ts} = 3 B\_{lc} \, r\_{lc} \, r\_{ts}^{-1} = 3 B\_s \, r\_s^3 \, r\_{lc}^{-2} \, r\_{ts}^{-1} \propto P^{-3/2} \, \dot{P}^{1/2} \propto \dot{E}(t)^{1/2}$. Therefore $B\_{ts} \propto \dot{E}(t)^{1/2}$ was much higher at earlier times when the pulsar spin-down was greatest. Past the termination shock the field continues to decrease, though likely at a slower rate than B ∝ 1/r, since this would put the field at $\ll 1 \, \mu \rm{G}$ in the outermost reaches of the nebula. Instead, we adopt a behavior of: $$\begin{aligned} B(r,t) = 400 \, \left( \frac{r}{0.03 \, \rm{pc}} \right)^{\beta} \left( \frac{\dot{E}(t)}{2.8\times 10^{36}} \right)^{1/2} \, \mu \rm{G}\end{aligned}$$ past the termination shock, where β is a fit parameter and the $400 \, \mu \rm{G}$ coefficient is the approximate value at the termination shock. We should emphasize that this magnetic field model does not conserve magnetic flux, since we assume that some magnetic field may be generated in the post-shock flow. The mean magnetic field within the boundaries of each zone is determined via a volume integration of Equation 20, and this is the value used at each time step since each zone is treated as homogeneous in the SED code. Diffusion --------- A certain fraction of the injected particles will be lost from the nebula due to diffusive escape. For each zone we compute the diffusive flux between zones by solving: $$\begin{aligned} J\_i(E\_e,t) = -D \, \frac{\partial n\_i(E\_e,t)}{\partial R} \, \, \, \rm{cm^{-2} \, s^{-1},}\end{aligned}$$ with Ji(Ee, t) the diffusive flux in particles per $\rm{cm^2} \, s^{-1}$ for zone i, ni(Ee, t), the particle density, and D the diffusion coefficient. The concentration gradient ∂ni(Ee, t)/∂R is evaluated as the concentration difference between adjacent zones divided by the distance between zone midpoints. Solving Equation 21 for each zone at every time step gives the number of particles diffusing between zones. Hence, time permitting particles typically diffuse sequentially from inner zones to outer zones, stepping from Zone 1 to Zone 12. The PWN-SNR interface will likely trap the majority of high-energy particles within the PWN. In addition, the surrounding SNR is expected to contain significant numbers of high energy particles. We therefore expect that particles will diffuse rather slowly out of the PWN and into the SNR, and so we arbitrarily set the density in the surrounding SNR nSNR(Ee, t) = 0.8 × n12(Ee, t), which allows particles to slowly escape the PWN from Zone 12. The final SED depends very weakly on nSNR(Ee, t), however, given the high degree of cooling and large final size of the nebula. Toroidal magnetic field lines tend to contain particles very effectively, and even with relatively rapid Bohm diffusion $\tau\_{esc} \approx 40,000 \, (R / 10 \, \rm{pc})^2 (E\_e/100 \, \rm{TeV})^{-1} \, (B/10 \, \mu \rm{G}) \, \rm{year}$. In the Bohm limit the cross-field mean free path λ equals the particle gyroradius, which is exceedingly small: $\lambda = 0.01 (E\_e/100 \, \rm{TeV}) \, (B/10 \, \mu \rm{G})^{-1}$ parsec. If the field line structures contain radial components, however, particles diffuse far more rapidly. Turbulence and mixing caused by the passage of the reverse shock might provide the necessary disruption to the magnetic field structure to achieve radial field components. Even in young PWNe where significant reverse shock interaction has likely not yet occurred, however, the simple model of purely toroidal magnetic field has been challenged by comparing X-ray observations of PWNe with the predictions of the model. Chandra observations of 3C 58 allowed comparison of the radial X-ray spectral index variation with the predictions of. For any reasonable values of σ these authors found the observed photon index increased much slower in 3C 58 than expected from the model of. A similar deviation is observed for G21.5 − 0.9, implying that the magnetic field structure deviates from a purely toroidal structure. The characteristic time scale for particle diffusion is determined by both the size of the nebula as well as diffusion coefficient D, such that τesc ≈ R2/6D. Alternately, the mean free path of a particle λ ≡ 3D/c. The electron diffusion coefficient is frequently parameterized in power-law form D(Ee) = D0(Ee/Ee, 0)δ. For Bohm diffusion δ = 1, while energy independent diffusion of course corresponds to δ = 0. To account for diffusion in a non-toroidal magnetic field we select a Bohm-type diffusion with δ = 1, such that mean free path and escape time scale as: $$\begin{aligned} \lambda & = & a \, E\_e \nonumber \\ \tau\_{esc} & = & R^2 (2 a c E\_e)^{-1}\end{aligned}$$ with a left as a free parameter in the model such that the mean free path is allowed to vary from the Bohm value to account for complex nebular field structure. For simplicity we ignore any perturbation the magnetic field might have on the escape timescale, since clearly particles must primarily move along rather than across magnetic field lines. Photon Fields ------------- Common practice in PWN SED modeling has been to adjust the dust IR energy density to maximize the agreement between model and data. We instead fix the photon densities at published values, reducing the number of model fit parameters. Adopting three primary photon fields (CMBR, far IR (dust), and starlight), we employ the interstellar radiation mapcube within the GALPROP suite to estimate the photon fields at the Galactic radius of PSR J1826 − 1334. A distance of 4 kpc in the direction of PSR J1826 − 1334 corresponds to a Galactic radius of 4.7 kpc; at this radius, dust IR photons typically peak at T ≈ 32 K with a density of ≈ 1 eV$\rm \, cm^{-3}$, while stellar photons peak at T ≈ 2500 K with a density of ≈ 4 eV$\rm \, cm^{-3}$. At these densities IR and CMB photons contribute approximately equally to TeV inverse Compton emission, while stellar photons are inconsequential. Synchrotron self-Compton emission is negligible for the magnetic fields present in HESS J1825 − 137. SED Model --------- A number of recent papers have applied various models to investigate the broadband emission from PWNe (,,,,, )., in particular, constructed a detailed 1-D dynamical model of PWNe magnetic field, pressure, size, energetics and spectral evolution inside a host SNR. These authors demonstrated their model by fixing parameters to mimic the Crab Nebula. While this model certainly helps shed light on compact PWNe such as the Crab, the markedly different sizes of X-ray and TeV emission regions in HESS J1825 − 137 (along with most other middle aged Vela-like PWN) pose a challenge to 1-D models. We therefore treat the nebula as a series of twelve expanding bubbles in order to match the twelve H.E.S.S. spatial regions. Each of these bubbles is treated as homogeneous, and filled with particles (assumed to be electrons) as well as magnetic field. We compute SEDs from evolving the electron populations over various lifetimes in a series of time steps, extending the single-zone approach taken in,,. The twelve-zone modeling approach permits investigation of the twelve H.E.S.S. spectral extraction regions, yet the number of spatial zones need not be limited to the number of data regions and in theory one could utilize any number of zones and then project those zones onto the sky for SED construction. The spatial grid size is completely arbitrary, and hence the final SED plot should be independent of grid size. A finer grid of zones vastly increases computation time, however, and provides little new information given the limited number of data regions. Nevertheless, as a consistency check we double the number of zones (to 24) and plot the resultant SED. As expected, the model curves change imperceptibly in all but the innermost region. The slight alteration in Region 1 is due to the steep variation in the magnetic field near the pulsar. This lack of dependence of the SED on the spatial zone size provides a check of the SED model. Model Implementation -------------------- Incorporating diffusion and projection effects between zones necessitates that all zones evolve in parallel, rather than independently. Evolution begins with Zone 12 at t = ti, 12 = 0, with escaping particles simply lost from the nebula. Once injection ends for Zone 12, at ti, 11, Zone 11 begins to evolve, with its escaping particles injected into Zone 12. Once the injection period for Zone 11 ends, Zone 10 begins the injection phase, with Zones 11 and 12 in the cooling phases. This process is repeated until the current age of the pulsar is reached, with all twelve zones evolving simultaneously. Figure 5 shows a schematic of the evolution of the nebula for three spatial zones. To compute the PWN SED, injected electrons are evolved through both the injection and cooling phases of each bubble’s lifetime, with the age of each phase computed via Equation 13. Each zone shares the same injection spectrum, modulated by the spin down power. Cooling is computed from synchrotron, inverse Compton, and adiabatic losses (Equation 17). Evolution occurs in a series of time steps, such that during the injection phase at each time step we perform the following procedures: 1. Calculate the size of the zone by integrating Equation 16 2. Compute the mean magnetic field by integrating over the volume of the zone via Equation 20 3. Inject pulsar wind particles into the zone according to Equation 10 4. Inject particles diffusing from adjacent zones according to Equation 21 (if including diffusion) 5. At each particle energy E, compute δE from synchrotron, inverse Compton, and adiabatic losses 6. Calculate the subsequent particle spectrum at time t + δt: E(t + δt) = E − δE 7. Remove escaping particles according to Equation 21 (if including diffusion) Once injection ends, cooling is computed for the remainder of the evolution, following the same steps above (save for step 3). At each time step this process is repeated for all zones with ti > t. When evolution completes, the synchrotron, and IC fluxes are computed from the final electron spectrum and magnetic field for each zone. Finally, the flux within each region is tabulated using the coefficients of Table 5. Model fitting is achieved by minimizing the χ2 between model and data using the downhill simplex method described in. Data consists of both X-ray and H.E.S.S. spectral flux points (77 in total): 12 in Region 1, 11 in Region 2, 9 in Region 3, and 5 data points for all exterior regions. For each ensemble of N variable parameters we evolve the system over the pulsar lifetime and calculate χ2 between model curves and flux data points. The simplex routine subsequently varies the parameters of interest to minimize the fit statistic. We estimate parameter errors by computing χ2 for a sampling of points near the best fit values and using these points to fit the N − dimensional ellipsoid describing the surface of Δχ2 = 2.71. Under the assumption of Gaussian errors, the projected size of this Δχ2 = 2.71 ellipsoid onto each parameter axis defines the 90% multi-parameter (projected) error. Results ======= Uniform Expansion, Constant Magnetic Field (Model 1) ---------------------------------------------------- We begin by considering a very simple model of constant magnetic field ( ∼ 3 μG), uniform expansion ($v(r,t)= \rm{constant}$), and ignore diffusive effects. Uniform expansion implies that each zone undergoes expansion for $1/12^{\rm{th}}$ of the pulsar lifetime. Injection begins at pulsar birth (t = 0) for Zone 12, while Zone 1 begins evolution at T − T/12, where T is the current age of the pulsar. The large size of the PWN implies a relatively high age, and the braking index and P0 selected in Table 6 give a total age of 21 kyr. This corresponds to a mean expansion velocity of $3900 \, \rm{km \, s^{-1}}$. We initially adopt a minimum particle energy Emin = 100 GeV, and electron injection fraction η = 0.8, which is a reasonable value for the expected particle dominated wind. The model gives a poor match to the data, apparent in the SED plot of Figure [fig:SED1]. Applying the full fitting machinery to five variables (magnetic field B, initial spin period P0, braking index n, electron power-law index p, electron cutoff Ecut) yields an extremely low initial spin period < 1 ms. Figure [fig:SED1] therefore fixes the braking index (n = 3) and initial spin period (P0 = 15 ms) to reasonable values; the other three variables are fit and 90% multi-parameter error estimates are computed, as shown in Table 6. The fit is still quite poor, and the fit value of Ecut = 3200 TeV is implausibly large. Inspection of Figure [fig:SED1] indicates that the normalization of each individual zone is highly skewed. The innermost zone has far too much VHE flux, while Zone 12 at the nebular frontier has far too little VHE flux. Velocity Profile, Constant Magnetic Field (Model 2) --------------------------------------------------- Though each region occupies the same $6\arcmin$ width, the normalization issues of Figure [fig:SED1] intimate that the injection time of the zones must vary with distance from the pulsar, and hence the flow speed in the nebula must not be constant. We therefore adopt a velocity profile in the nebula of v(r, t) = rα C(t) (Equation 16). The best fit with this model and six variables is shown in Table 7. The radial velocity profile results in a lengthening of the injection phase with increasing radial distance from the pulsar, and the values n and P0 yield an age of 28 kyr. This model is shown in Figure [fig:SED2], is far superior to Figure [fig:SED1], and reproduces the VHE steepening trend moderately well for the inner six regions. Yet the outer model curves are all far steeper than the VHE data, implying an excess of synchrotron cooling for the ∼ 40 TeV particles required to upscatter CMB photons to the ∼ 5 TeV highest energy H.E.S.S. data point (Equation 4). As a result, in the outermost regions (presumably corresponding to the oldest electrons), electrons of energy ∼ 40 TeV have almost completely burned off. In the innermost region the X-ray flux is significantly under-produced, suggesting the field must be $> 5 \, \mu \rm{G}$ near the pulsar. Evidently a non-uniform magnetic field profile is required, with higher field in the interior to match X-ray fluxes and lower field in the exterior in order to reduce synchrotron cooling for the oldest electrons. In order to better visualize the results of the SED fitting, we also show the ratios between the model zone fluxes and spectral indices and the measured data values (Figure [fig:ratio2]). The model flux is computed by integrating the flux in the desired energy band (either 1 − 8 keV or 0.25 − 10 TeV) and model indices are computed by fitting to a power-law in the appropriate band. The model fluxes and indices are compared to the data from Tables 3 and 4. Velocity Profile, Magnetic Field Profile (Model 3) -------------------------------------------------- In hopes of improving the fit of Figure [fig:SED2], we invoke a radial magnetic field profile of the form $B(r,t) \propto r^{\beta} \, \dot{E}(t)^{1/2}$ (Equation 20), such that as particles traverse the nebula they experience an ever decreasing magnetic field. Similar to section 4.2, we select six fit parameters, though instead of fitting the final magnetic field value we fit the magnetic profile index β. The best fit model yields an age of 34 kyr, and Table 6 reveals an improved fit statistic over the constant magnetic field model. The non-constant magnetic field does a much better job of matching the innermost X-ray data (see Figures [fig:SED3] and [fig:ratio3] ), though we still have difficulty matching the VHE indices in the outer (purple) zones. Velocity Profile, Magnetic Field Profile, Diffusion (Model 4) ------------------------------------------------------------- The severe under-prediction of VHE flux and over-prediction of VHE index in the outer two zones of Figures [fig:SED3] and [fig:ratio3] indicates a general lack of electrons with E > 10 TeV. One possibility of addressing this issue is simply reduced cooling by lowering the magnetic field. Yet even in Figures [fig:SED2] and [fig:ratio2] with a low 6 μG field the Region 12 model is far steeper than the data due to excessive cooling. In addition, Figures [fig:SED3] and [fig:ratio3] demonstrate that higher fields are needed in the inner zones, which only serves to hasten synchrotron burn-off. Particle re-acceleration as particles traverse the nebula might explain the VHE flux ∼ 80 pc from the pulsar, perhaps due to reverse shock interactions. Such acceleration is highly speculative, however, and beyond the scope of this paper. A more standard method of maintaining the high energy particle population far from the pulsar, which we explore below, is rapid energy-dependent diffusion with mean free path λ ∝ Ee. For this model we select seven variables: velocity radial profile index α, magnetic profile index β, initial spin period P0, braking index n, electron power-law index p, electron exponential cutoff Ecut, and mean free path λ. Figure [fig:SED4] shows the SED of the best fit model, scaled to display the aggregate gamma-ray data. This aggregate data is discussed in Section 5. Figure [fig:ratio4] displays the flux and index ratios for this model. The overall fit to the twelve spectral regions is far superior to previous models, particularly in the outer nebula. The largest departure of the model from the data is the X-ray spectral index of Region 3, though we remind the reader that the very hard Region 3 X-ray index reported in Table 4 may reflect residual background contamination. Table 6 lists the fit parameters, while Table 7 lists derived parameters from the fit (injection time, final magnetic field, and χ2). As expected from the scaling relations of section 3.1, the particle residence time of Zone 1 is well under 1000 years. The braking index n and initial spin period P0 yield an age of 40 ± 9 kyr and a mean expansion velocity of $2000 \, \rm{km \, s^{-1}}$. The mean free path of $\lambda = 1.8 (E\_e/100 \, \rm{TeV})$ parsec predicts an escape timescale of $\tau\_{esc} \approx 90 \, (R / 10 \, \rm{pc})^2 (E\_e/100 \, \rm{TeV})^{-1} \, \rm{year}$. This model does have some difficulty matching the H.E.S.S. flux and slope in Region 1, which is due to the rapid diffusive loss of high energy particles. Most likely, the simple diffusive model employed vastly oversimplifies the complex nature of this source, and diffusion coefficients vary in both space and time. Nevertheless, the χ2/d.o.f. for Model 4 indicates that the general fit is quite good. As a check of our magnetic field model, we also allow the normalization at the termination shock to vary from its adopted value of $400 \, \mu \rm{G}$. The best fit model with this extra fit parameter yields a termination shock field of $320 \, \mu \rm{G}$ with no improvement of the fit statistic (χ2/d.o.f. = 118/69, versus χ2/d.o.f. = 119/70 for the field fixed. Fitting the magnetic field normalization therefore does not yield a markedly better fit, and the termination shock value of $400 \, \mu \rm{G}$ appears quite reasonable. One further check of the magnetic field model is achieved by fitting a model with diffusion over uniform magnetic field (simply replacing the magnetic field radial index of Model 4 with a constant magnetic field value). This model cannot adequately match the data, yielding a best fit of χ2/d.o.f. = 429/70, and rules out uniform magnetic field as a viable option. The adopted IR photon density of ≈ 1 eV$\rm \, cm^{-3}$ could easily vary by a factor of ∼ 2, though varying the photon density does not qualitatively change the results presented here. A higher (lower) IR photon density means fewer (more) high energy electron are required to match the HESS data, implying a slightly softer (harder) electron spectral index or lower (higher) high energy cutoff. [tpb!] 1.0cZZZZZZZc Model ${\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & α (v ∝ rα) & B/β (B ∝ rβ) & P0 (ms) & $n \; (\dot{\Omega} \propto \Omega^n) $ & p & Ecut (TeV) & $\lambda \; (\frac{E\_e}{100 \,{\rm TeV}}) \, {\rm pc}$ & χ2/d.o.f. $1 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − & $2.9 \pm 0.3 \tablenotemark{a}$ & $15\tablenotemark{b}$ & $3\tablenotemark{b}$ & 2.42 ± 0.06 & 3800 ± 800 & − & 575/74 $2 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − 0.36 ± 0.03 & $5.6\pm0.4\tablenotemark{a}$ & 12 ± 3 & 2.5 ± 0.1 & 2.30 ± 0.01 & 130 ± 10 & − & 236/71 $3 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − 0.60 ± 0.08 & − 0.74 ± 0.02 & 14 ± 7 & 2.2 ± 0.1 & 2.31 ± 0.02 & 190 ± 30 & − & 217/71 $4 {\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}$ & − 0.51 ± 0.09 & − 0.69 ± 0.01 & 13 ± 7 & 1.9 ± 0.2 & 2.24 ± 0.02 & 230 ± 90 & 1.8 ± 0.3 & 119/70 [htpb!] c@ c@ c@ c@ c@ c Zone & R (pc) & $\delta t\_i \,(\rm{year})\tablenotemark{a}$ & $B \, (\mu \rm{G})\tablenotemark{b}$ & χ2 & # data points 1 & 7 & 600 & 12 & 23 & 12 2 & 14 & 1200 & 6.6 & 13 & 11 3 & 21 & 1500 & 4.8 & 6.6 & 9 4 & 28 & 1900 & 3.8 & 4.6 & 5 5 & 35 & 2200 & 3.2 & 8.0 & 5 6 & 42 & 2500 & 2.8 & 10 & 5 7 & 49 & 2900 & 2.5 & 8.9 & 5 8 & 56 & 3300 & 2.2 & 11 & 5 9 & 63 & 3800 & 2.1 & 7.4 & 5 10 & 70 & 4400 & 1.9 & 6.4 & 5 11 & 77 & 5500 & 1.8 & 16 & 5 12 & 84 & 9900 & 1.7 & 4.4 & 5 ### Relativistic Maxwellian Plus Power-Law Injection Spectrum The SED model described in the previous section takes into account only X-ray and VHE H.E.S.S. data, though ideally the model should be consistent with Fermi LAT data as well. Note that while we do not fit to the LAT points in Figure [fig:SED4], the rough agreement between these data and the integrated (total) model flux in the LAT band is encouraging. However, the Fermi LAT photon index of Γ = 1.4 ± 0.2 predicts an electron index of p = 2Γ − 1 = 1.8 ± 0.4. This is somewhat harder than the electron index of p = 2.2 utilized in Figures [fig:SED4] and [fig:ratio4], and an injection spectrum with fewer low energy electrons might plausibly improve the fit to the LAT data. One option is adopting the relativistic Maxwellian plus power-law tail electron spectrum proposed by, and described in modeling terms by. Fitting with this model requires one additional parameter (the temperature of the Maxwellian), bringing the total up to eight. The best fit model with this injection spectrum yields a fit somewhat worse than the power-law case (χ2/d.o.f. = 123/69, versus 122/70 for the power-law spectrum). In addition, an extremely short initial spin period of P0 ∼ 2 ms is required, and the low kT = 0.2 TeV results in a severe over-prediction of GeV flux and a far worse fit to the LAT data than the power-law case. found an adequate fit to the aggregate data with a one-zone model, yet the inclusion of multiple zones seems to wash out any advantage of a relativistic Maxwellian electron spectrum. Discussion ========== As an initial check of the validity of the scalings applied to both the Suzaku and H.E.S.S. data, we refit Model 4 with the raw data. Using the original H.E.S.S. data points and X-ray data taken from regions coinciding with the HESS quarter annuli, the best fit parameters are within 1σ of the values listed in Table 6 for Model 4, except for the magnetic field index. The different H.E.S.S. flux ratios between interior zones results in a magnetic field index that differs from the Model 4 value by +0.02, or 2σ. Despite the similarity in fit parameters, however, the fit statistic increases dramatically, from χ2/d.o.f. = 119/70 for Model 4, to χ2/d.o.f. = 169/70. Nearly all of this increase in χ2 comes from the inner two regions. The H.E.S.S. excess map (Figure 1b) reveals that there appears to be significantly more flux $6\arcmin-12\arcmin$ from the pulsar than within $6\arcmin$ of the pulsar. Yet due to the small original H.E.S.S. extraction region for the $6\arcmin-12\arcmin$ region (which is 25% smaller than the $0-6\arcmin$ region), the reported H.E.S.S. fluxes are identical for the inner two regions. The use of identical fluxes for the inner two zones therefore results in a worse fit for the model, which assumes radial variations in the flow speed and a greater number of particles farther from the pulsar. Nevertheless, the commonality of the fit parameters with the raw and scaled data provides some support for the scaled approach taken throughout this paper. One further check of the X-ray scaling is to fit the scaled H.E.S.S. data and raw, unscaled X-ray data. With this slightly lower X-ray flux the best fit parameters are well within errors for Model 4, and give a nearly identical χ2. Of the four models scenarios described above, the final model which incorporates diffusion provides by the far the best fit to the data. The relativistic Maxwellian injection spectrum yields a somewhat worse fit statistic to the simple power-law case, and though the power-law injection spectrum also better matches the Fermi LAT data, definitive discrimination between lepton injection spectra is not possible at this stage. Yet we can convincingly argue that the pulsar wind must inject significant numbers of electrons up to energies of at least 100 TeV, the low energy power-law index must be p ∼ 2 in order to match the Fermi LAT data, that a high percentage of the pulsar spin-down energy must go into leptons, and that some mechanism (we invoke diffusion) maintains high energy particles far from the pulsar. In the following discussion we consider the simple power-law spectrum injection spectrum (Model 4) shown in Figures [fig:SED4] and [fig:ratio4]. The sum of fluxes from the twelve spatial zones has been scaled up some 25% in Figure [fig:SED4] due to the fact that our spectral extraction regions do not capture the entirety of the counts within the aggregate nebula. This scaled flux ideally should match the Fermi LAT and H.E.S.S. aggregate data points (which both represent larger spatial regions), which appears to be the case for Figure [fig:SED4]. Computing the fit statistic for the aggregate H.E.S.S. and LAT data gives χ2 = 27 for the 16 gamma-ray data points (assuming only statistical errors on the data points, and ignoring the LAT upper limit). Given the very small statistical errors on the H.E.S.S. data points the agreement between gamma-ray data and the summed model curves is remarkable. For comparison, the best fit model utilizing the relativistic Maxwellian injection spectrum nets a χ2 = 50 for the 16 gamma-ray data points. The ‘missing’ ≈ 25% of flux implies a missing population of particles contributing to the IC flux of the nebula. Such particles have likely diffused to the north and east of the pulsar exterior to our extraction regions. A tweak of the initial pulsar spin-down from 13 ms to 11 ms (well within errors) provides ≈ 40% greater spin-down energy, more than enough to enough to raise the model bulk nebular flux to the observed level. The models proposed above utilize η = 0.8, and we find an adequate fit to the X-ray and H.E.S.S. data for P0 = 13 ms. Neglecting the ‘missing’ energy component, this connotes a total of 1.1 × 1050 erg of pulsar spin-down energy. Disregarding losses, we therefore expect of order 1049 erg in magnetic field energy. While the magnetic field evolution and profile adopted in our model does not take into account the total energy budget of the pulsar, a reasonable model should nevertheless contain ∼ 1049 erg of magnetic energy for the choice of η = 0.8. Computing the magnetic field energy from the final magnetic field profile $B(r,T) = 400 (r/0.03 \, \rm{pc})^{-0.69} \, \mu \rm{G}$ yields an energy of 4.4 × 1048 erg, which is quite reasonable given that the magnetic field likely experiences some losses due to expansion. Even though the vast majority of the pulsar energy initially goes into particles, summing the energy content of the final electron spectrum reveals that electron cooling losses leave only 6.9 × 1048 erg remaining in the form of leptons. We can also compute the value of σ from the assumed termination shock radius of 0.03 pc and $400 \, \mu \rm{G}$ field strength. Adopting a flow speed at the termination shock of c/3 and a spherical geometry, one achieves an estimate of $7\times 10^{35} \, \rm{erg \, s^{-1}}$ of magnetic energy flux at the termination shock. For the pulsar spindown value of $\dot{E} = 2.8\times10^{36} \, \rm{erg \, s^{-1}}$ this yields σ ≈ 0.3. This value is consistent with the adopted value of η = 0.8, which corresponds to a value of σ = (1 − η)/η = 0.25. One further consistency check for our model comprises comparing the available spin-down power to the total energy present in the nebula plus cooling and escape losses. Indeed, summing the leptonic energy, magnetic energy, cooling losses, and escape losses yields 1.1 × 1050 erg, precisely the energy available from the pulsar. The energy lost due to adiabatic cooling goes into accelerating a shell of swept-up material, and we can use estimates of this swept-up mass as another check of the validity of the SED model. For the adopted SNR density profile of a constant density inner core surrounded by a ρ ∝ r− 9 outer envelope the transition between the two density regimes propagates out at a constant velocity vt, such that : $$\begin{aligned} v\_t \approx 6500 \left( \frac{E\_{SN}}{3 \times 10^{51} \, \rm{erg}} \right) ^{1/2} \left( \frac{M\_{ej}}{8 M\_{\odot}} \right)^{-1/2} \rm{km \, s^{-1}.}\end{aligned}$$ This is much faster than the Model 4 outer PWN expansion of $2000 \, \rm{km \, s^{-1}}$, so we can treat the PWN as always expanding into the constant density ejecta core, with density : $$\begin{aligned} \rho\_{ej}(t) & \approx & 3.7 \times 10^{-29} \left( \frac{E\_{SN}}{3 \times 10^{51} \, \rm{erg}} \right) ^{-3/2} \left( \frac{M\_{ej}}{8 M\_{\odot}} \right)^{5/2} \nonumber \\ & & \times \left( \frac{t}{20 \, \rm{kyr}} \right) ^{-3} \, \, \rm{g \, cm^{-3}.}\end{aligned}$$ The amount of mass swept up by the PWN over its lifetime can be approximated as: $$\begin{aligned} M\_{sw} = \int\_{t\_i}^T \rho\_{ej}(t) \, \zeta 4 \pi r^2 dr,\end{aligned}$$ with ti the initial timestep, r = voutert and ζ the filling factor of the PWN, equal to 0.31 for the adopted morphology. Integrating Equation 25 over the 40 kyr lifetime of the pulsar provides a very rough estimate of 1M⊙ of swept up mass. Accelerating 1 M⊙ to $2000 \, \rm{km \, s^{-1}}$ (the mean expansion velocity for a pulsar age of 40 kyr) requires some 4 × 1049 erg. For the parameters selected in Model 4, the SED model posits a total adiabatic loss of 3.4 × 1049 erg, quite close to the value quoted above. Recall that in the SED model adiabatic losses depend only on the assumed PWN size evolution (Equation 17), and do not take into account any SNR properties. Therefore the fact that the two adiabatic loss estimates are not glaringly different provides some support to the soundness of the SED model. The computed pressure within the entire nebula can be used as a further sanity check on our model by comparing to the expected PWN pressure after 40 kyr. Given the degree of particle cooling, the model’s final particle pressure ($P\_{pwn} = \frac{E\_p}{3 V\_{pwn}} = 1.0 \times 10^{-13} \, \rm{erg \, cm^{-3}}$) is less than the magnetic pressure ($\frac{B^2}{8 \pi} = 1.9 \times 10^{-13} \, \rm{erg \, cm^{-3}}$). Within a freely expanding SNR the PWN pressure scales as : $$\begin{aligned} P\_{pwn} \approx 3/25 \, \rho\_{ej}(t) v\_{pwn}^2, \end{aligned}$$ with ρej(t) given by Equation 24, and vpwn the PWN expansion velocity. After 40 kyr this predicts a mean nebular pressure of $P\_{pwn} \approx 2 \times 10^{-13} \, \rm{erg \, cm^{-3}}$, in good agreement with the computed values. The adopted termination shock radius of 0.03 pc predicts a pressure of $\approx 1 \times 10^{-9} \, \rm{erg \, cm^{-3}}$ at the termination shock (Equation 19), significantly higher than the mean pressure of $\approx 3 \times 10^{-13} \, \rm{erg \, cm^{-3}}$. This high pressure at the shock rapidly dissipates away from the pulsar, however, with the bulk of the nebula nearly isobaric at late times; each of the 10 outer zones has a pressure within ≈ 50% of the mean nebular pressure, as evidenced in the pressure and energy plot of Figure [fig:pen]. This plot illustrates that the magnetic pressure exceeds particle pressure for the inner regions, and that the total particle energy exceeds the bulk magnetic field energy in the nebula. The total magnetic field pressure scales as Ppwn, B ∝ t− 3.0, while the total particle pressure evolves as Ppwn, p ∝ t− 2.9. This is precisely the Ppwn ∝ t− 3 evolution predicted by Equations 24 and 26 for a freely expanding PWN. In addition to the PWN pressure and energy we can also compute a variety of quantities which vary over the pulsar lifetime. We calculate the mean magnetic field evolution as $\overline{B} \propto t^{-1.6}$, which is quite similar to the B ∝ t− 1.7 evolution computed by in their single-zone model during the free expansion phase. The various luminosities in the nebula are plotted in Figure [fig:lum], revealing that at late times the total luminosity of the nebula exceeds the pulsar spin-down power. This figure also demonstrates that after a few thousand years adiabatic losses overtake synchrotron radiation to provide the dominant cooling mechanism, though particles of energy Ee > 10 TeV primarily lose energy from synchrotron and IC cooling since for both the cooling timescale τ ∝ Ee− 1. Indeed, at late time the IC cooling actually exceeds the synchrotron cooling due to the low mean magnetic field of $\approx 2 \, \mu \rm{G}$. We construct radial flux profiles by summing the model fluxes in the Suzaku and H.E.S.S. wavebands within each region, which can be compared with the data points plotted in Figure 2. We are also free to compute the profile in any arbitrary energy range, and so in Figure [fig:modradial] we plot the projected surface brightness in the Fermi LAT 1-100 GeV range. Since the LAT energy range lies in the uncooled regime, the flux primarily tracks the energy content of electrons. As a result, the LAT flux peaks toward the outer realm of the nebula where the majority of electrons were injected when the pulsar was much younger with a far higher $\dot{E}$, and this gives that a rather uniform counts map in the LAT energy band. Comparison of point source and extended Gaussian morphologies in the 10 − 100 GeV range by yielded a significance of ≈ 8.5σ for a source extension of 0.56∘ ± 0.07∘ centered 0.5∘ southwest of the pulsar. Similar results are achieved with a uniform disk morphology, though the extension increases to 0.67∘ ± 0.02∘. Fitting the source to a spatial template provided by the H.E.S.S. excess map (E > 200 GeV) decreases the test statistic by 42 ≈ 6.5σ, implying that the LAT emission shows a distinctly different morphology from the H.E.S.S. emission. Indeed, the LAT source extension of ∼ 0.6∘ centered 0.5∘ southwest of the pulsar is remarkably consistent with Figure [fig:modradial]. Conclusions =========== Analysis of recently public Suzaku X-ray data covering the majority of the very bright VHE source HESS J1825 − 137 indicate that X-rays extend only ≈ 1/4 as far as VHE gamma-rays. The variable extent of the nebula in different wavebands, as well as the observed VHE spectral softening away from the pulsar, is naturally explained by electron cooling losses. Time-dependent 3-D modeling of the twelve data regions allows one to constrain the electron injection properties, spin-down behavior of the pulsar, nebular velocity profile, and magnetic field profile. A choice of 80% of the pulsar spin-down going into electrons is consistent with the data, though due to particle cooling at the current epoch magnetic field energy is ≈ 1/2 × the particle energy in the nebula. This cooling also results in a higher magnetic than particle pressure, with a nearly isobaric particle pressure profile in the outer ten zones and the total pressure falling as Ppwn ∝ t− 3. For the best fit model (Model 4) an initial spin period of ≈ 13 ms provides adequate pulsar spin-down energy to power the nebula, and combined with the braking index of n = 1.9 this predicts an age of 40 kyr and an expansion velocity of $2000 \, \rm{km \, s^{-1}}$. The interior velocity profile is found to scale as v(r) ∼ r− 0.5, while the magnetic field profile scales as $B(r,t) \propto r^{-0.7} \, \dot{E}(t)^{1/2}$, with the field falling from $\approx 400 \, \mu \rm{G}$ at the termination shock to $\approx 2 \, \mu \rm{G}$ in the far reaches of the nebula and the mean field scaling as $\overline{B} \propto t^{-1.6}$. The Fermi LAT photon index of 1.4 connotes an electron index near the canonical 2.0. However, fitting to the X-ray/TeV and the higher energy portion of the electron spectrum, we find a somewhat softer power law index of p ≈ 2.2. A relativistic Maxwellian electron spectrum provides a somewhat worse fit to the X-ray and H.E.S.S. data compared to a simple power-law injection, and the power-law case requires fewer fit parameters and better matches the Fermi LAT data. Regardless of injection spectrum, the existence of 10 TeV photons in the outer reaches of the nebula requires substantial particle diffusion; adopting a diffusion timescale of $\tau\_{esc} \approx 90 \, (R / 10 \, \rm{pc})^2 (E\_e/100 \, \rm{TeV})^{-1} \, \rm{year}$ allows an adequate fit to the data. This is in contrast to the common assumption of a purely toroidal PWN magnetic field which could contain particles extremely efficiently. The model predicts a rather uniform Fermi LAT surface brightness out to ∼ 1∘ from the pulsar, in good agreement with the recently discovered LAT source centered 0.5∘ southwest of PSR J1826 − 1334, with extension 0.6 ± 0.1∘. The multi-zone time-dependent SED model fitting approach applied in the paper extends simple one-zone models to allow investigation of spatially dependent properties such as flow speed, magnetic field, photon index, and flux levels. The growing number of sources with spatially resolved X-ray and VHE measurements (e.g. Vela-X, HESS J1303-631) are therefore prime targets for such multi-zone modeling. Acknowledgments: We thank Stefan Funk and Marianne Lemoine-Goumard for helpful discussions on PWN modeling and Fermi LAT data. We also would like to thank the referee for very thorough and helpful comments. This work was supported in part by the U.S. Department of Energy contract to SLAC no. DE-AC02-76SF00515 and by NASA grant NNX10AP65G. Abdo, A. A. et al., 2010c, ApJ, 713, 146 Ackermann, M., et al. 2011,, 726, 35 Aharonian, F., et al. 2006,, 460, 365 Blondin, J. M., Chevalier, R. A., & Frierson, D. M. 2001,, 563, 806 Blumenthal, G. R., & Gould, R. J. 1970, Reviews of Modern Physics, 42, 237 Bogovalov, S. V., & Khangoulyan, D. V. 2002, Astronomy Letters, 28, 373 Bucciantini, N., Amato, E., Bandiera, R., Blondin, J. 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The Equilibrium Existence Duality:Equilibrium with Indivisibilities & Income Effects ==================================================================================== ### 17th June 2020 We show that, with indivisible goods, the existence of competitive equilibrium fundamentally depends on agents’ substitution effects, not their income effects. Our Equilibrium Existence Duality allows us to transport results on the existence of competitive equilibrium from settings with transferable utility to settings with income effects. One consequence is that net substitutability—which is a strictly weaker condition than gross substitutability—is sufficient for the existence of competitive equilibrium. We also extend the “demand types” classification of valuations to settings with income effects and give necessary and sufficient conditions for a pattern of substitution effects to guarantee the existence of competitive equilibrium. JEL Codes: C62, D11, D44 Introduction ============ This paper shows that, when goods are indivisible and there are income effects, the existence of competitive equilibrium fundamentally depends on agents’ substitution effects—i.e., the effects of compensated price changes on agents’ demands. We provide general existence results that do not depend on income effects. In contrast to the case of divisible goods, competitive equilibrium does not generally exist in settings with indivisible goods. Moreover, most previous results about when equilibrium does exist with indivisible goods assume that utility is transferable—ruling out income effects but allowing tractable characterizations of (Pareto-)efficient allocations and aggregate demand that can be exploited to analyze competitive equilibrium.[1](#fn1) But understanding the role of income effects is important for economies with indivisible goods, as these goods may comprise large fractions of agents’ budgets. Furthermore, in the presence of income effects, the distribution of wealth among agents affects both Pareto efficiency and aggregate demand, making it necessary to develop new methods to analyze competitive equilibrium with indivisible goods. The cornerstone of our analysis is an application of the relationship between Marshallian and Hicksian demand. As in classical demand theory, Hicksian demand is defined by fixing a utility level and minimizing the expenditure of obtaining it. We combine Hicksian demands to construct a family of “Hicksian economies” in which prices vary, but agents’ utilities—rather than their endowments—are held constant. Our key result, which we call the Equilibrium Existence Duality, states that competitive equilibria exist for all endowment allocations if and only if competitive equilibria exist in the Hicksian economies for all utility levels. Preferences in each Hicksian economy reflect agents’ substitution effects. Therefore, by the Equilibrium Existence Duality, the existence of competitive equilibrium fundamentally depends on substitution effects. Moreover, as fixing a utility level precludes income effects, agents’ preferences are quasilinear in each Hicksian economy. Hence, the Equilibrium Existence Duality allows us to transport (and so generalize) any necessary or sufficient condition for equilibrium existence from settings with transferable utility to settings with income effects.[2](#fn2) In particular, our most general existence result gives a necessary and sufficient condition for a pattern of agents’ substitution effects to guarantee the existence of competitive equilibrium in the presence of income effects. Consider, for example, the case of substitutable goods in which each agent demands at most one unit of each good. With transferable utility, substitutability is sufficient for the existence of competitive equilibrium and defines a maximal domain for existence. With income effects, showed that competitive equilibrium exists under gross substitutability. The Equilibrium Existence Duality tells us that, with income effects, competitive equilibrium in fact exists under net substitutability and that net substitutability defines a maximal domain for existence. Moreover, we show that gross substitutability implies net substitutability; the reverse direction is not true in the presence of income effects. An implication of our results is that it is unfortunate that, and much of the subsequent literature, used the term “gross substitutes” to refer to a condition on quasilinear preferences. Indeed, gross and net substitutability are equivalent without income effects, and our work shows that it is net substitutability, not gross substitutability, that is critical to the existence of competitive equilibrium with substitutes.[3](#fn3) To appreciate the distinction between gross and net substitutability, suppose that Martine owns a house and is thinking about selling her house and buying one of two different other houses: a spartan one and a luxurious one. If the price of her own house increases, she may wish to buy the luxurious house instead of the spartan one—exposing a gross complementarity between her existing house and the spartan one. However, Martine regards the houses as net substitutes: the complementarity emerges entirely due an income effect. Competitive equilibrium is therefore guaranteed to exist in economies with Martine if all other agents see the goods as net substitutes, despite the presence of gross complementarities. Our most general equilibrium existence theorem characterizes the combinations of substitution effects that guarantee the existence of competitive equilibrium. It is based on classification of valuations into “demand types.” A demand type is defined by the set of vectors that summarize the possible ways in which demand can change in response to a small generic price change. For example, the set of all substitutes valuations forms a demand type, as does the set of all complements valuations, etc. Applying ’s taxonomy to changes in Hicksian demands, we see that their definition easily extends to general utility functions, capturing agents’ substitution effects. Examples of demand types in our setting with income effects, therefore, include the set of all net substitutes preferences, the set of all net complements preferences, etc. The Equilibrium Existence Duality then makes it straightforward that the Unimodularity Theorem[4](#fn4)—which encompasses many standard results on the existence of competitive equilibrium as special cases[5](#fn5)—is unaffected by income effects. Therefore, as with the case of substitutes, conditions on complementarities and substitutabilities that guarantee the existence of competitive equilibrium in settings with transferable utility translate to conditions on net complementarities and substitutabilities that guarantee the existence of competitive equilibrium in settings with income effects. In particular, there are patterns of net complementarities that are compatible with the existence of competitive equilibrium. Our results may have significant implications for the design of auctions that seek competitive equilibrium outcomes, and in which bidders face financing constraints. For example, they suggest that versions of the Product-Mix Auction, used by the Bank of England since the Global Financial Crisis, may work well in this context. Several other papers have considered the existence of competitive equilibrium in the presence of indivisibilities and income effects.,, and showed the existence of competitive equilibrium in a housing market economy in which agents have unit demand and endowments. Building on those results,,, and analyzed settings with multiple goods, but restricted attention to separable preferences. By contrast, our results—even for the case of substitutes—allow for interactions between the demand for different goods. We also clarify the role of net substitutability for the existence of competitive equilibrium. In a different direction, proved a version of the sufficiency direction of the Unimodularity Theorem for settings with income effects. also defined domains of preferences using an optimization problem that turns out to be equivalent to the expenditure minimization problem. However, they did not note the connection to the expenditure minimization problem or Hicksian demand, and, as a result, did not interpret their sufficient conditions in terms of substitution effects or establish the role of substitution effects in determining the existence of equilibrium. We proceed as follows. Section [sec:setting] describes our setting—an exchange economy with indivisible goods and money. Section [sec:EED] develops the Equilibrium Existence Duality. Since the existing literature has focused mostly on the case in which indivisible goods are substitutes, we consider that case in Section [sec:subst]. Section [sec:demTypes] develops demand types for settings with income effects and states our Unimodularity Theorem with Income Effects. Section [sec:auctions] remarks on implications for auction design, and Section [sec:conclusion] is a conclusion. Appendix [app:EEDproof] proves the Equilibrium Existence Duality. Appendix [app:grossToNet] proves the connection between gross and net. Appendices [app:dualDemPrefs] and [app:maxDomain] adapt the proofs of results from the literature to our setting. The Setting =========== We work with a model of exchange economies with indivisibilities—adapted to allow for income effects. There is a finite set J of agents, a finite set I of indivisible goods, and a divisible numéraire that we call “money.” We allow goods to be undesirable, i.e., to be ``bads." We fix a total endowment $\tot \in \mathbb{Z}^I$ of goods in the economy.[6](#fn6) Preferences and Marshallian Demand ---------------------------------- Each agent j ∈ J has a finite set $\Feas{j} \subseteq \Z^I$ of feasible bundles of indivisible goods and a lower bound $\feas{j} \ge -\infty$ on her consumption of money. As bundles that specify negative consumption of some goods can be feasible, our setting implicitly allows for production.[7](#fn7) The principal cases of $\feas{j}$ are $\feas{j} = -\infty$, in which case all levels of consumption of money are feasible, and $\feas{j} = 0$, in which case the consumption of money must be positive. Hence, the set of feasible consumption bundles for agent j is $\Feans{j} = (\feas{j},\infty) \times \Feas{j}$. Given a bundle $\bunn \in \Feans{j},$ we let $\numer$ denote the amount of money in $\bunn$ and $\bun$ denote the bundle of goods specified by $\bunn,$ so $\bunn = (\defbunn)$. The utility levels of agent j lie in the range $\feasUtil{j}$, where $-\infty \le \minu{j} < \maxu{j} \le \infty.$ Furthermore, each agent j has a utility function $\utilFn{j}: \Feans{j} \to \feasUtil{j}$ that we assume to be continuous and strictly increasing in $\numer$, and to satisfy $$\label{eq:ulimits} \lim\_{\numer \to (\feas{j})^+} \util{j}{\defbunn} = \minu{j} \quad \text{and} \quad \lim\_{\numer \to \infty} \util{j}{\defbunn} = \maxu{j}$$ for all $\bun \in \Feas{j}.$ Condition ([eq:ulimits]) requires that some consumption of money above the minimum level $\feas{j}$ be essential to agent j.[8](#fn8) We let $\pnumer = 1$. Given an endowment $\bunndow = (\defbunndow) \in \Feans{j}$ of a feasible consumption bundle and a price vector $\p \in \mathbb{R}^I,$ agent j’s Marshallian demand for goods is $$\dM{j}{\p}{\bunndow} = \left\{\bun^\* \lgiv \bunn^\* \in \argmax\_{\bunn \in X^j \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}\lgivend\right\}.$$ As usual, Marshallian demand is given by the set of bundles of goods that maximize an agent’s utility, subject to a budget constraint, given a price vector and an endowment. An income effect is a change in an agent’s Marshallian demand induced by a change in her money endowment, holding prices fixed.[9](#fn9) Our setup is flexible enough to capture a wide range of preferences with and without income effects, as the following two examples illustrate. [Quasilinear Utility] [eg:quasilin] Given a valuation $\valFn{j}: \Feas{j} \to \mathbb{R},$ letting $\feas{j} = \minu{j}= -\infty$ and $\maxu{j} =\infty,$ one obtains a quasilinear utility function given by $$\util{j}{\defbunn} = \numer + \val{j}{\bun}.$$ When agents utility functions are quasilinear, they do not experience income effects. When all agents have quasilinear utility functions, we say that utility is transferable. [Quasilogarithmic Utility] [eg:quasilog] Given a function $\quasivalFn{j}: \Feas{j} \to (-\infty,0)$, which we call a quasivaluation,[10](#fn10) and letting $\minu{j}= -\infty$, $\maxu{j} =\infty,$ and $\feas{j} = 0,$ there is a quasilogarithmic utility function given by $$\util{j}{\bunn} = \log \numer - \log(- \quasival{j}{\bun}).$$ Unlike with quasilinear utility functions, agents with quasilogarithmic utility functions exhibit income effects. Hicksian Demand,, and the -------------------------- The concept of Hicksian demand from consumer theory plays a key role in our analysis. Given a utility level $\ub \in \feasUtil{j}$ and a price vector $\p,$ agent j’s Hicksian demand for goods is $$\label{eq:costMin} \dH{j}{\p}{\ub} = \left\{\bun^\* \lgiv \bunn^\* \in \argmin\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn\lgivend\right\}.$$ As in the standard case with divisible goods, Hicksian demand is given by the set of bundles of goods that minimize the expenditure of obtaining a utility level given a price vector. A substitution effect is a change in an agent’s Hicksian demand induced by a change in prices, holding her utility level fixed. As in classical demand theory, Marshallian and Hicksian demand are related by the duality between the utility maximization and expenditure minimization problems. Specifically, a bundle of goods is expenditure-minimizing if and only if it is utility-maximizing.[11](#fn11) [Relationship between Marshallian and Hicksian Demand] [fac:dualDem] Let $\p$ be a price vector. 1. For all endowments $\bunndow$, we have that $\dM{j}{\p}{\bunndow} = \dH{j}{\p}{\ub},$ where $$\ub = \max\_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ 2. For all utility levels $\ub$ and endowments $\bunndow$ with $$\pall \cdot \bunndow = \min\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn,$$ we have that $\dH{j}{\p}{\ub} = \dM{j}{\p}{\bunndow}.$ If an agent has a quasilinear utility function, then, as she experiences no income effects, her Marshallian and Hicksian demands coincide and do not depend on endowments or utility levels. Under quasilinearity, we therefore refer to both Marshallian and Hicksian demand simply as demand, which we denote by $\dQL{j}{\p}$. Formally, if j has quasilinear utility with valuation $\valFn{j},$ defining $\dQL{j}{\p}$ as the solution to the quasilinear maximization problem $$\label{eqn:quasilin} \dQL{j}{\p} = \argmax\_{\bun \in \Feas{j}} \{\val{j}{\bun} - \p \cdot \bun\},$$ we have that $\dM{j}{\p}{\bunndow} = \dQL{j}{\p}$ for all endowments $\bunndow$ and that $\dH{j}{\p}{\ub} = \dQL{j}{\p}$ for all utility levels $\ub$. We next show that the interpretation of the expenditure minimization problem as a quasilinear maximization problem persists in the presence of income effects. Specifically, we can rewrite the expenditure minimization problem of Equation ([eq:costMin]) as a quasilinear optimization problem by using the constraint to solve for $\numer$ as a function of $\bun$. Formally, for a bundle $\bun \in \Feas{j}$ of goods and a utility level $\ub \in \feasUtil{j},$ we let $\cf{j}{\bun}{\ub} = \util{j}{\cdot,\bun}^{-1}(\ub)$ denote the level of consumption of money (or savings) needed to obtain utility level $\ub$ given $\bun.$[12](#fn12) By construction, we have that $$\dH{j}{\p}{\ub} = \argmin\_{\bun \in \Feas{j}} \left\{\cf{j}{\bun}{\ub} + \p \cdot \bun\right\}.$$ It follows that agent j’s expenditure minimization problem at utility level $\ub$ can be written as a quasilinear maximization problem for the valuation $-\cf{j}{\cdot}{\ub}$, which we therefore call the Hicksian valuation. The of agent j at utility level u is $\valHDef{j} = -\cf{j}{\cdot}{\ub}$. Note that $\cf{j}{\cdot}{\ub}$ is continuous and strictly increasing in $\ub,$ and hence $\valHDef{j}$ is continuous and strictly decreasing in $\ub$. The following lemma formally states that agent j’s Hicksian demand at utility level $\ub$ is the demand correspondence of an agent with valuation $\valHDef{j}$. [lem:dHvalH] For all price vectors $\p$ and utility levels $\ub$, we have that $$\dH{j}{\p}{\ub} = \argmin\_{\bun \in \Feas{j}} \left\{\cf{j}{\bun}{\ub} + \p \cdot \bun\right\} = \argmax\_{\bun \in \Feas{j}} \left\{\valH{j}{\bun}{\ub} - \p \cdot \bun\right\}.$$ As $\utilFn{j}(\bunn)$ is strictly increasing in $\numer,$ we have that $$\dH{j}{\p}{\ub} = \left\{\bun^\* \lgiv \bunn^\* \in \argmin\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} = \ub} \pall \cdot \bunn\lgivend\right\}.$$ Applying the substitution $\numer = \cf{j}{\bun}{\ub}=-\valH{j}{\bun}{\ub}$ to remove the constraint from the minimization problem yields the lemma. It follows from Lemma [lem:dHvalH] that an agent’s at a utility level gives rise to a quasilinear utility function that reflects the agent’s substitution effects at that utility level. Lemma [lem:dHvalH] also yields a relationship between the family of and income effects. Indeed, by Fact [fac:dualDem], an agent’s income effects correspond to changes in her Hicksian demand induced by changes in her utility level, holding prices fixed. By Lemma [lem:dHvalH], these changes in Hicksian demand reflect the changes in the that are induced by the changes in utility levels. Hence, the at each utility level determine an agent’s substitution effects, while the variation of the with the utility level captures her income effects. To illustrate how an agent’s family of reflects her income effects, we consider the cases of quasilinear and quasilogarithmic utility. [Example [eg:quasilin] continued] With quasilinear utility, the at utility level $\ub$ is $\valH{j}{\bun}{\ub} = \val{j}{\bun} - \ub.$ Changes in $\ub$ do not affect the relative values of bundles under $\valH{j}{\cdot}{\ub}$, so changes in the utility level do not affect Hicksian demand. Indeed, there are no income effects. By construction, a utility function $\util{j}{\bunn}$ is quasilinear in $\numer$ if and only if $\cf{j}{\bun}{\ub}$ is quasilinear in $\ub$—or, equivalently, $\valH{j}{\bun}{\ub}$ is quasilinear in $-\ub$. In general, it follows from Fact [fac:dualDem] and Lemma [lem:dHvalH] that agent j’s preferences exhibit income effects if and only if $\cf{j}{\bun}{\ub}$—or, equivalently, $\valH{j}{\bun}{\ub}$—is not additively separable between $\bun$ and $\ub$. [Example [eg:quasilog] continued] [eg:quasilogDualVal] With quasilogarithmic utility, the at utility level $\ub$ is $\valH{j}{\bun}{\ub} = e^{\ub} \quasival{j}{\bun}.$ In this case, each is a positive linear transformation of $\quasivalFn{j}$. Income effects are reflected by the fact that $\valH{j}{\bun}{\ub}$ is not additively separable between $\bun$ and $\ub$. We use Lemma [lem:dHvalH] to convert preferences with income effects into families of valuations. It turns out that each continuously decreasing family of valuations is the family of of a utility function, so a utility function can be represented equivalently by a family of. [Duality for Preferences] [fac:dualPrefs] Let $F: \Feas{j} \times \feasUtil{j} \to (-\infty,-\feas{j})$ be a function. There exists a utility function $\utilFn{j}: \Feans{j} \to \feasUtil{j}$ whose at each utility level $\ub$ is $F(\cdot,\ub)$ if and only if for each $\bun \in \Feas{j},$ the function $F(\bun,\cdot)$ is continuous, strictly decreasing, and satisfies[13](#fn13)[14](#fn14) $$\label{eq:vlimits} \lim\_{\ub \to (\minu{j})^+} F(\bun,\ub) = -\feas{j} \quad \text{and} \quad \lim\_{\ub \to (\maxu{j})^-} F(\bun,\ub) = -\infty.$$ Finally, we combine the families of to form a family of, in each of which utility is transferable and agents choose consumption bundles to minimize the expenditure of obtaining given utility levels. The for a profile of utility levels $(\ubj)\_{j \in J}$ is the transferable utility economy in which agent j’s valuation is $\valH{j}{\cdot}{\ubj}$. The family of consists of the ``duals" of the original economy in which income effects have been removed and are given by substitution effects. Like the construction of, the construction of the allows us to convert economies with income effects to families of economies with transferable utility and is a key step of our analysis. The Equilibrium Existence Duality ================================= We now turn to the analysis of in exchange economies. consists of an endowment $\bunndowj \in \Feans{j}$ for each agent j such that $\sum\_{j \in J} \bundowj = \tot,$ where $\tot$ is the total endowment. Given, a specifies a price vector such that markets for goods clear when agents maximize utility. By Walras’s Law, it follows that the market for money clears as well. Given $(\bunndowj)\_{j \in J}$, a consists of a price vector $\p$ and a bundle $\bunj \in \dM{j}{\p}{\bunndowj}$ for each agent such that $\sum\_{j \in J} \bunj = \tot$. In transferable utility economies, a consists of a price vector $\p$ and a bundle $\bunj \in \dQL{j}{\p}$ for each agent such that $\sum\_{j \in J} \bunj = \tot$. In this case, the does not affect because endowments do not affect (Marshallian) demand. We therefore omit the when considering in transferable utility economies in which exists—i.e., $\tot \in \sum\_{j \in J} \Feas{j}$. On the other hand, the total endowment $\tot$ affects even when utility is transferable. Recall that utility is transferable in the. Furthermore, by Lemma [lem:dHvalH], a in the for a profile $(\ubj)\_{j \in J}$ of utility levels consists of a price vector $\p$ and a bundle $\bunj \in \dH{j}{\p}{\ubj}$ for each agent such that $\sum\_{j \in J} \bunj = \tot$. Thus, agents act as if they minimize expenditure in in the.[15](#fn15) Building on Fact [fac:dualDem] and Lemma [lem:dHvalH], our Equilibrium Existence Duality connects the equilibrium existence problems in the original economy (which can feature income effects) and the (in which utility is transferable). Specifically, we show that always exists in the original economy if and only if it always exists in the. Here, we hold agents’ preferences and the total endowment (of goods) fixed but allow the to vary. [Equilibrium Existence Duality] [thm:existDualExchange] Suppose that the total endowment and the sets of feasible bundles are such that exists. exist for all if and only if exist in the for all profiles of utility levels. By Lemma [lem:dHvalH], agents’ substitution effects determine their preferences in each. Therefore, Theorem [thm:existDualExchange] tells us that any condition that ensures the existence of can be written as a condition on substitution effects alone. That is, substitution effects fundamentally determine whether exists. Both directions of Theorem [thm:existDualExchange] also have novel implications for the analysis of in economies with indivisibilities. As demands in the are given by Hicksian demand in the original economy (Lemma [lem:dHvalH]), the ``if“ direction of Theorem [thm:existDualExchange] implies that every condition on demand $\dQLFn{j}$ that guarantees the existence of in settings with transferable utility translates into a condition on Hicksian demand $\dHFn{j}$ that guarantees the existence of in settings with income effects. In Sections [sec:subst] and [sec:demTypes], we use the ``if” direction of Theorem [thm:existDualExchange] to obtain new domains for the existence of with income effects from previous results on the existence of in settings with transferable utility. Conversely, the ``only if" direction of Theorem [thm:existDualExchange] shows that if a condition on demand defines a maximal domain for the existence of in settings with transferable utility, then the translated condition on Hicksian demand defines a maximal domain for the existence of in settings with income effects. In Sections [sec:subst] and [sec:demTypes], we also use this implication to derive new maximal domain results for settings with income effects. To prove the ``only if" direction of Theorem [thm:existDualExchange], we exploit a version of the Second Fundamental Theorem of Welfare Economics for settings with indivisibilities. To understand connection to the existence problem for the, note that the existence of in the is equivalent to the conclusion of the Second Welfare Theorem—i.e., that each Pareto-efficient allocation can be supported in an equilibrium with endowment transfers—as the following lemma shows.[16](#fn16) [lem:dualEconSWT] Suppose that the total endowment and the sets of feasible bundles are such that exists. exist in the for all profiles of utility levels if and only if, for each Pareto-efficient allocation $(\bunnj)\_{j \in J}$ with $\sum\_{j \in J} \bunj = \tot$, there exists a price vector $\p$ such that $\bunnj \in \dM{j}{\p}{\bunnj}$ for all agents j. We prove Lemma [lem:dualEconSWT] in Appendix [app:EEDproof]. Intuitively, as utility is transferable in the, variation in utility levels between plays that same role as endowment transfers in the Second Welfare Theorem. It is well-known that the conclusion of the Second Welfare Theorem holds whenever exist for all .[17](#fn17) It follows that always exists in the whenever it always exists in the original economy, which is the ``only if" direction of Theorem [thm:existDualExchange]. We use a different argument to prove the ``if" direction. Our strategy is to show that there exists a profile of utility levels and a in the corresponding in which all agents’ expenditures equal their budgets in the original economy. To do so, we apply a topological fixed-point argument that is similar in spirit to standard proofs of the existence of. Specifically, we consider an auctioneer who, for a given profile of candidate equilibrium utility levels, evaluates agents’ expenditures over all in the and adjusts candidate equilibrium utility levels upwards (resp. downwards) for agents who under- (resp. over-) spend their budgets.[18](#fn18) The existence of in the ensures that the process is nonempty-valued, and the transferability of utility in the ensures that the process is convex-valued. Kakutani’s Fixed Point Theorem implies the existence of a fixed-point utility profile. By construction, there exists a in the corresponding at which agents’ expenditures equal the values of their endowments. By Lemma [lem:dHvalH], agents must be maximizing utility given their endowments at this equilibrium, and hence once obtains a in the original economy. The details of the argument are in Appendix [app:EEDproof]. Examples -------- We next illustrate the power of Theorem [thm:existDualExchange] using the two examples. Our first example is a “housing market” in which agents have unit-demand preferences, may be endowed with a house, and can experience arbitrary income effects. We can use Theorem [thm:existDualExchange] to reduce the existence problem to the assignment game of —reproving a result originally due to. [A Housing Market—] [eg:house] For each agent j, let $\Feas{j} \subseteq \{\zero\} \cup \{\e{i} \mid i \in I\}$ be nonempty. In this case, in, utility is transferable and agents have unit demand for the goods. As the does not affect when utility is transferable, the results of imply that exist in the for all profiles of utility levels (provided that exists). Hence, Theorem [thm:existDualExchange] implies that exist for all endowment allocations—even in the presence of income effects. In the second example, we revisit the quasilogarithmic utility functions from Example [eg:quasilog]. We provide sufficient conditions on agents’ quasivaluations for to exist. These conditions are related to, but not in general implied by, the conditions developed in Sections [sec:subst] and [sec:demTypes]. [Existence of with Quasilogarithmic Preferences] [eg:quasilogExist] For each agent j, let $\quasivalFn{j}: \Feas{j} \to (-\infty,0)$ be a quasivaluation. Let agent j’s utility function be quasilogarithmic for the quasivaluation $\quasivalFn{j}$, as in Example [eg:quasilog]. In this case, agent j’s at each utility level is a positive linear transformation of $\quasivalFn{j}$ (Example [eg:quasilogDualVal]). Hence, by Theorem [thm:existDualExchange], exist for all as long as exists when utility is transferable and each agent j’s valuation is an (agent-dependent) positive linear transformation of $\quasivalFn{j}$—e.g., if the quasivaluations $\quasivalFn{j}$ are all valuations, or all valuations of a unimodular demand type. Additionally, in the case in which one unit of each good is available in total (i.e., $\totComp{i} = 1$ for all goods i), showed that exists when utility is transferable and all agents have sign-consistent tree valuations. Hence, if one unit of each good is available in total, then Theorem [thm:existDualExchange] implies that exist with quasilogarithmic utility for all if all agents’ quasivaluations are sign-consistent tree valuations. In the remainder of the paper, we use Theorem [thm:existDualExchange] to develop novel conditions on preferences that ensure the existence of. The Case of Substitutes ======================= In this section, we apply the Equilibrium Existence Duality (Theorem [thm:existDualExchange]) to prove a new result regarding the existence of with substitutable indivisible goods and income effects: we show that a form of net is sufficient for, and in fact defines a maximal domain for, the existence of. We begin by reviewing previous results on the existence of under (gross). We then derive our existence theorem for and relate it to the previous results. In this section, we focus on the case in which each agent demands at most one unit of each good. Formally, we say that an agent j demands at most one unit of each good if $\Feas{j} \subseteq \{0,1\}^I$. We extend to the case in which agents can demand multiple units of some goods in Section [sec:ssub]. and the Existence of --------------------- We recall a notion of for preferences over indivisible goods from, which extends the condition from classical demand theory. It requires that uncompensated increases in the price of a good weakly raise demand for all other goods. With quasilinear utility, the modifier “gross” can be dropped—as in classical demand theory (see also Footnote 1 in ). [] [def:gsub] Suppose that agent j demands at most one unit of each good. 1. [part:gsub] A utility function $\utilFn{j}$ is a utility function at endowment $\bundow \in \Feas{j}$ of goods if for all money endowments $\numerdow > \feas{j}$, price vectors $\p$, and λ > 0, whenever $\dM{j}{\p}{\bunndow} = \{\bun\}$ and $\dM{j}{\p + \lambda \e{i}}{\bunndow} = \{\bunpr\},$ we have that $\bunprComp{k} \ge \bunComp{k}$ for all goods $k \not= i$.[19](#fn19) 2. A valuation is a valuation for which the corresponding quasilinear utility function is a utility function.[20](#fn20) Technically, Definition [def:gsub] imposes a condition on the locus of prices at which Marshallian demand is single-valued—following,,, and.[21](#fn21) It is well-known that when utility is transferable, exists under. [fac:subExist] Suppose that utility is transferable and that exists. If each agent demands at most one unit of each good and has a valuation, then exists.[22](#fn22) Moreover, the class of valuations forms a maximal domain for the existence of in transferable utility economies. Specifically, if an agent has a non- valuation, then may not exist when the other agents have valuations. Technically, we require that one unit of each good be present among agents’ endowments (i.e., that $\totComp{i} = 1$ for all goods i) as complementarities between goods that are not present are irrelevant for the existence of. [fac:subMaxDomain] Suppose that $\totComp{i} = 1$ for all goods i. If ∣J∣ ≥ 2, agent j demands at most one unit of each good, and $\valFn{j}$ is not a valuation, then there exist sets $\Feas{k} \subseteq \{0,1\}^I$ of feasible bundles and valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ for agents $k \not= j$, for which there exists but no.[23](#fn23) While Fact [fac:subMaxDomain] shows that there is no domain strictly containing the domain of valuations for which the existence of can be guaranteed in transferable utility economies, it does not rule out the existence of other domains for which the existence of can be guaranteed. For example,,, and gave examples of domains other than for which the existence of is guaranteed. Generalizing Fact [fac:subExist] to settings with income effects, showed that exists for $(\bunndowj)\_{j \in J}$ if each agent j’s utility function is a utility function at her endowment $\bundowj$ of goods.[24](#fn24) However, did not offer a maximal domain result for. In the next section, we show that does not actually drive existence of with substitutable indivisible goods. and the Existence of --------------------- In light of Theorem [thm:existDualExchange] and Fact [fac:subExist], exists if agents’ Hicksian demands satisfy an appropriate condition—i.e., if preferences satisfy a net analogue of. We build on Definition [def:gsub] to define a concept of for settings with indivisibilities. is a version of the condition from classical consumer theory. It requires that compensated increases in the price of a good (i.e., price increases that are offset by compensating transfers) weakly raise demand for all other goods. [] [def:nsub] Suppose that agent j demands at most one unit of each good. A utility function $\utilFn{j}$ is a utility function if for all utility levels $\ub$, price vectors $\p$, and λ > 0, whenever $\dH{j}{\p}{\ub} = \{\bun\}$ and $\dH{j}{\p + \lambda \e{i}}{\ub} = \{\bunpr\}$, we have that $\bunprComp{k} \ge \bunComp{k}$ for all goods $k \not= i$. For quasilinear utility functions, coincides with (gross). More generally, can be expressed as a condition on. [rem:netSubDual] By Lemma [lem:dHvalH], if an agent demands at most one unit of each good, then she has a utility function if and only if her at all utility levels are valuations. We can apply Fact [fac:dualPrefs] and Remark [rem:netSubDual] to construct large classes of preferences with income effects from families of valuations. There are several rich families of valuations, including endowed assignment valuations and matroid-based valuations. This leads to a large class of quasilogarithmic utility functions. [Example [eg:quasilog] continued] [eg:quasilogSubs] A quasilogarithmic utility function $\utilFn{j}$ is a utility function if and only if the quasivaluation $\quasivalFn{j}$ is a valuation.[25](#fn25) More generally, in light of Fact [fac:dualPrefs] and Remark [rem:netSubDual], each family of valuations leads to a class of utility functions with income effects consisting of the utility functions whose all belong to the family. These classes are defined by conditions on substitution effects and do not restrict income effects. By contrast, places substantial restrictions on the form of income effects.[26](#fn26) To understand the difference between gross and, we compare the conditions in a setting in which agents have unit demand for goods. [Example [eg:house] continued] [eg:houseGrossNet] Consider an agent, Martine, who owns a house i1 and is considering selling it to purchase (at most) one of houses i2 and i3. If Martine experiences income effects, then her choice between i2 and i3 generally depends on the price she is able to procure for her house i1. For example, if i3 is a more luxurious house than i2, then Martine may only demand i3 if the value of her endowment is sufficiently large—i.e., if the price of her house i1 is sufficiently high. As a result, when Martine is endowed with i1, she does not generally have preferences: increases in the price of i1 can lower Martine’s demand for i2. That is, Martine can regard i2 as a gross complement for i1. In contrast, Martine has preferences—no compensated increase in the price of i1 could make Martine stop demanding i2—a condition that holds generally in the housing market economy.[27](#fn27) Note also that, unlike, generally depends on endowments: if Martine were not endowed a house, she would have preferences. While Example [eg:houseGrossNet] shows that does not imply, it turns out that implies. [prop:ssubst] If agent j demands at most one unit of each good and there exists an endowment $\bundow$ of goods at which $\utilFn{j}$ is a utility function, then $\utilFn{j}$ is utility function. Proposition [prop:ssubst] and Example [eg:houseGrossNet] show that (at any one endowment of goods) implies but places additional restrictions on income effects. Nevertheless, the restrictions on substitution effects alone, entailed by, are sufficient for the existence of. [thm:netSubExist] If all agents demand at most one unit of each good and have utility functions, then exist for all. Theorem [thm:netSubExist] is an immediate consequence of the Equilibrium Existence Duality and the existence of in transferable utility economies under. Remark [rem:netSubDual] implies that the agents’ at all utility levels are valuations. Hence, Fact [fac:subExist] implies that exist in the for all profiles of utility levels if exists. The theorem follows by the “if” direction of Theorem [thm:existDualExchange]. As implies (Proposition [prop:ssubst]), the existence of under is a special case of Theorem [thm:netSubExist]. But Theorem [thm:netSubExist] is more general: as Example [eg:houseGrossNet] shows, allows for forms of gross complementarities between goods, in addition to. The following example illustrates how the distinction between and relates to the existence of when agents caan demand multiple goods. [ versus and the Existence of ] [eg:grossVersusNetNumerical] There are two goods and the total endowment is $\tot = (1,1)$. There are two agents, which we call j and k, and j’s feasible set of consumption bundles of goods is $\Feas{j} = \{0,1\}^2$. We consider the price vectors $\p = (2,2)$ and $\ppr = (4,2)$ and consider two examples in which agent j’s Marshallian demand changes from (1, 1) to (0, 0) as prices change from $\p$ to $\ppr$—a. But the consequences for the existence of are different across the two cases. In Case [eg:numericalComp], the reflects a net complementarity for j, and may not exist if k sees goods as net substitutes. In Case [eg:numericalLog], the reflects only an income effect for j, as in Example [eg:houseGrossNet], so is guaranteed to exist if k sees goods as net substitutes. 1. [eg:numericalComp] Suppose that j has a quasilinear utility function with valuation given by $$\val{j}{\bun} = \begin{cases} 0 & \text{if } \bun = (0,0),(0,1),(1,0)\\ 5 & \text{if } \bun = (1,1). \end{cases}$$ Here, $\valFn{j}$ is not a valuation because $\dQL{j}{\p} = \{(1,1)\}$ while $\dQL{j}{\ppr} = \{(0,0)\}$: i.e., increasing the price of the first good can lower j’s demand for the second good. If $\Feas{k} = \{(0,0),(0,1),(1,0)\}$ and agent k has a quasilinear utility function with a valuation given by $$\label{eq:kUnitDem} \val{k}{\bun} = \begin{cases} 0 & \text{if } \bun = (0,0)\\ 4 & \text{if } \bun = (1,0)\\ 3 & \text{if } \bun = (0,1), \end{cases}$$ then no exists.[28](#fn28) 2. [eg:numericalLog] Suppose instead that $\utilFn{j}$ is quasilogarithmic (as defined in Example [eg:quasilog]) with quasivaluation given by $$\quasival{j}{\bun} = \begin{cases} -11 & \text{if } \bun = (0,0)\\ -7 & \text{if } \bun = (0,1)\\ -4 & \text{if } \bun = (1,0)\\ -1 & \text{if } \bun = (1,1). \end{cases}$$ At the endowment $\bundowj = (0,1)$ of goods, $\utilFn{j}$ is not a utility function as, letting $\numerdowj = 3,$ we have that $\dM{j}{\p}{\bunndowj} = \{(1,1)\}$ while $\dM{j}{\ppr}{\bunndowj} = \{(0,0)\}$.[29](#fn29) That is, increasing the price of the first good can lower j’s Marshallian demand for the second good. By contrast, as $\quasivalFn{j}$ is a valuation, Example [eg:quasilogSubs] implies that $\utilFn{j}$ is a utility function: the is entirely due to an income effect. For example, at the utility level $$\ub = \max\_{\bunn \in \Feans{j} \mid \pprall \cdot \bunn \le \pprall \cdot \bunndowj} \util{j}{\bunn} = \log \frac{5}{11},$$ we have that $\dH{j}{\p}{u} = \{(1,0)\}$ and that $\dH{j}{\ppr}{\ub} = \{(0,0)\}$,[30](#fn30) so the decrease in the Marshallian demand for the second good as prices change from $\p$ to $\ppr$ at the endowment $\bunndowj$ reflects an income effect. By Theorem [thm:netSubExist], exists whenever k has a utility function. For example, if k has a quasilinear utility function with a valuation given by Equation ([eq:kUnitDem]), then for the defined by $\bundowj = (0,1)$, $\bundowag{k} = (1,0)$, and $\numerdowj = \numerdowag{k} = 3,$ the price vector (3, 2) and the allocation of goods defined by $\bunj = (1,0)$ and $\bunag{k} = (0,1)$ comprise a.[31](#fn31) In Case [eg:numericalLog], agent j has preferences—leading to the guaranteed existence of when agent k has preferences. By contrast, in Case [eg:numericalComp], agent j does not have preferences—and may not exist when k has preferences. In general, net substitutability forms a maximal domain for the existence of. Specifically, if an agent does not have preferences, then may not exist when the other agents have quasilinear preferences. [prop:netSubstMaxDomain] Suppose that $\totComp{i} = 1$ for all goods i. If ∣J∣ ≥ 2, agent j demands at most one unit of each good, and $\utilFn{j}$ is not a utility function, then there exist sets $\Feas{k} \subseteq \{0,1\}^I$ of feasible bundles and valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ for agents $k \not= j$, and for which no exists. Proposition [prop:netSubstMaxDomain] is an immediate consequence of the Equilibrium Existence Duality and the fact that defines a maximal domain for the existence of with transferable utility. By Remark [rem:netSubDual], there exists a utility level $\ub$ at which agent j’s $\valHDef{j}$ is not a valuation. Fact [fac:subMaxDomain] implies that there exist feasible sets $\Feas{k} \subseteq \{0,1\}^I$ and valuations $\valFn{k}$ for agents $k \not= j$, for which exists but no would exist with transferable utility if agent j’s valuation were $\valHDef{j}$. With those sets $\Feas{k}$ of feasible bundles and valuations $\valFn{k}$ for agents $k \not= j,$ the “only if” direction of Theorem [thm:existDualExchange] implies that there exists for which no exists. Proposition [prop:netSubstMaxDomain] entails that any domain of preferences that contains all quasilinear preferences and guarantees the existence of must lie within the domain of preferences. Therefore, Proposition [prop:netSubstMaxDomain] and Theorem [thm:netSubExist] suggest that is the most general way to incorporate income effects into a condition to ensure the existence of. By contrast, the relationship between the nonexistence of and failures of depends on why fails. can fail due to substitution effects that reflect net complementarities, as in Example [eg:grossVersusNetNumerical][eg:numericalComp], or due to income effects, as in Example [eg:grossVersusNetNumerical][eg:numericalLog]. If the failure of reflects a net complementarity, then Proposition [prop:netSubstMaxDomain] tells us that may not exist if the other agents have quasilinear preferences, as in Example [eg:grossVersusNetNumerical][eg:numericalComp]. On the other hand, the failure of is only due to income effects, then Theorem [thm:netSubExist] tells us that exists if the other agents have preferences (e.g., quasilinear preferences), as in Example [eg:grossVersusNetNumerical][eg:numericalLog]. Demand Types and the Unimodularity Theorem ========================================== In this section, we characterize exactly what conditions on patterns of substitution effects guarantee the existence of. Specifically, we consider classification of valuations into “demand types” based on sets of vectors that summarize the possible ways in which demand can change in response to a small generic price change. We first review the definition of demand types from. We then extend the concept of demand types to settings with income effects, and develop a version of the Unimodularity Theorem that allows for income effects and characterizes which demand types guarantee the existence of (see also ). A special case of the Unimodularity Theorem with Income Effects extends Theorem [thm:netSubExist] to settings in which agents can demand multiple units of some goods. Demand Types and the Unimodularity Theorem with Transferable Utility -------------------------------------------------------------------- We first review the concept of demand types for quasilinear settings, as developed by. An integer vector is primitive if the greatest common divisor of its components is 1. By focusing on the directions of demand changes, we can restrict to primitive demand change vectors. A is a set D ⊆ ZI of primitive such that if $\dvec \in\mathcal{D}$ then $- \dvec \in\mathcal{D}$. [Demand Types for Valuations] [def:demTypeTU] Let $\valFn{j}$ be a valuation. 1. A bundle $\bun$ is uniquely demanded by agent j** if there exists a price vector $\p$ such that $\dQL{j}{\p} = \{\bun\}.$ 2. A pair $\{\bun,\bunpr\}$ of uniquely demanded bundles are adjacently demanded by agent j** if there exists a price vector $\p$ such that $\dQL{j}{\p}$ contains $\bun$ and $\bunpr$ but no other bundle that is uniquely demanded by agent j. 3. If D is a, then $\valFn{j}$ is of demand type D if for all pairs $\{\bun,\bunpr\}$ that are adjacently demanded by agent j, the difference $\bunpr - \bun$ is a multiple of an element of D.[32](#fn32) [fig:demTypeDef] For intuition, suppose that a small price change causes a change in demand. Then, generically, demand changes between adjacently demanded bundles. Thus, the demand type vectors represent the possible directions of changes in demand in response to small generic price changes (see Proposition 3.3 in for a formal statement). To illustrate Definition [def:demTypeTU], we consider an example. [eg:demTypeDef] Suppose that there are two goods and let $$\Feas{j} = \{0,1,2,3\}^2 \ssm \{(2,3),(3,2),(3,3)\}.$$ Consider the valuation defined by $\val{j}{\bun} = \bunComp{1} + \bunComp{2}$. As Figure [fig:demTypeDef] illustrates, the uniquely demanded bundles are (0, 0), (0, 3), (1, 3), (3, 0), and (3, 1). When $1 = \pComp{1} < \pComp{2},$ agent j’s demand is $\dQL{j}{\p} = \{(0,0),(1,0),(2,0),(3,0)\}$. Hence, as the bundles (1, 0) and (2, 0) are not uniquely demanded, the bundles (0, 0) and (3, 0) are adjacently demanded. As a result, for $\valFn{j}$ to be of demand type D, the set D must contain the vector (1, 0), which is the primitive proportional to the demand change (3, 0) − (0, 0) = (3, 0). Similarly, the bundles (0, 0) and (0, 3) are adjacently demanded, and any D such that $\valFn{j}$ is of demand type D must contain the vector (0, 1). When $\pComp{1} < \pComp{2} = 1,$ demand is $\dQL{j}{\p} = \{(3,0),(3,1)\}$. Hence, the bundles (3, 0) and (3, 1) are adjacently demanded. Similarly, the bundles (0, 3) and (1, 3) are adjacently demanded. These facts respectively imply, again, that (0, 1) and (1, 0) are in any D such that Vj is of demand type D. Last, when p1 = p2 < 1, agent j’s demand is $\dQL{j}{\p} = \{(1,3),(2,2),(3,1)\}$. Hence, as the bundle (2, 2) is not uniquely demanded, the bundles (1, 3) and (3, 1) are adjacently demanded. As a result, for $\valFn{j}$ to be of demand type D, the set D must contain the vector (1, − 1), which is the primitive proportional to the demand change (3, 1) − (1, 3) = (2, − 2). By contrast, the bundles (0, 0) and (3, 1) are not adjacently demanded: the only price vector at which agent j demands them both is $\p = (1,1),$ but $\dQL{j}{1,1}$ also contains the uniquely demanded bundles (0, 3), (1, 3), and (3, 0). Similarly, the bundles (0, 0) and (1, 3) are not adjacently demanded. Hence, $$\mathcal{D} = \pm \left\{\begin{bmatrix}1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1\end{bmatrix}, \begin{bmatrix} 1 \\ -1\end{bmatrix}\right\}$$ is the minimal D such that $\valFn{j}$ is of demand type D. Consider any valuation of the same demand type D as in Example [eg:demTypeDef], and a change in price from $\p$ to $\ppr=\p+\lambda \e{1}$ for some λ > 0. For generic choices of $\p$ and λ, the demand at any price on the straight line from pI to pIʹ either is unique, or demonstrates the adjacency of two bundles uniquely demanded at prices on this line. The change in demand between such bundles must therefore be a multiple of an element of D (by Definition [def:demTypeTU]). Moreover, since only the price of good 1 is changing and that price is increasing, the law of demand entails that demand for good 1 must strictly decrease upon any change in demand.[33](#fn33) Thus, the change in demand between the two consecutive uniquely demanded bundles must be a positive multiple of either ( − 1, 0) or ( − 1, 1). Therefore, demand for good 2 must (weakly) increase, reflecting between the goods. This two-good example is a special case of an important class of demand types. [The Strong Substitutes Demand Type] [eg:ssubDemType] The strong substitutes consists of all vectors in $\Z^I$ with at most one + 1 component, at most one − 1 component, and no other nonzero components. As illustrated in Example [eg:demTypeDef], this captures one-to-one substitution between goods through demand type vectors with one component of 1 and one component of − 1. Furthermore, if an agent k demands at most one unit of each good, then $\valFn{k}$ is a valuation if and only if it is of the strong substitutes demand type (see Theorems 2.1 and 2.4 in ). In settings in which agents can demand multiple units of each good, a form of concavity is needed to ensure the existence of. A valuation is concave if, under that valuation, each bundle of goods that is a convex combination of feasible bundles of goods is demanded at some price vector. For the formal definition, we let $\conv(T)$ denote the convex hull of a set T ⊆ RI. [Concavity][def:concave] A valuation $\valFn{j}$ is concave if for each bundle $\bun \in \conv(\Feas{j}) \cap \mathbb{Z}^n,$ there exists a price vector $\p$ such that $\bun \in \dQL{j}{\p}$. In Section [sec:subOld], we discussed that guarantees the existence of in transferable utility economies when agents demand at most one unit of each good. Generalizing that result, identified a necessary and sufficient condition for the concave valuations of a demand type to form a domain for the guaranteed existence of. [Unimodularity] A set of vectors in $\Z^I$ is unimodular if every linearly independent subset can be extended to be a basis for $\R^I$, of, such that any square matrix whose columns are these vectors has determinant ± 1. For example, the in Example [eg:demTypeDef] is unimodular, while the $$\label{eq:subsComp} \pm \left\{\begin{bmatrix} 1 \\ -1\end{bmatrix}, \begin{bmatrix}1 \\ 1\end{bmatrix}\right\}$$ is not unimodular, because $$\left\|\begin{matrix} 1 & 1\\ -1 & 1 \end{matrix}\right\| = 2.$$ The in ([eq:subsComp]) represents that the two goods can be substitutable or complementary for agents—a possibility that can cause to fail to exist, as in Example [eg:grossVersusNetNumerical][eg:numericalComp]. showed that the unimodularity of a is precisely the condition for the corresponding demand type to guarantee the existence of. [Unimodularity Theorem with Transferable Utility] [fac:unimod] Let D be a. exist for all finite sets J of agents with concave valuations of demand type D and for all total endowments for which exist if and only if D is unimodular.[34](#fn34) used conditions on the ranges of agents’ demand correspondences to describe classes of concave valuations, which correspond to the concave valuations of unimodular demand types;[35](#fn35) they formulated a version of the ``if" direction of Fact [fac:unimod] with those conditions.[36](#fn36) As showed, the strong substitutes is unimodular. Therefore, in light of Example [eg:ssubDemType], the existence of in transferable utility economies in which agents demand at most one unit of each good and have substitutes valuations (Fact [fac:subExist]) is a special case of Fact [fac:unimod]. Moreover, Fact [fac:unimod] is strictly more general: as showed, there are unimodular for which the existence of cannot be deduced from the corresponding result for strong substitutes by applying a change of basis to the space of bundles of goods.[37](#fn37) To illustrate the additional generality, we discuss an example of such a demand type.[38](#fn38) [eg:5D] There are five goods. Consider the $$\mathcal{D} = \pm\left\{\begin{bmatrix}1 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 0 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ -1 \\ 1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ -1 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \\ -1 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ 0 \\ 0 \\ 1 \\ -1\end{bmatrix}, \begin{bmatrix}-1 \\ 1 \\ 0 \\ 0 \\ 1\end{bmatrix}\right\}.$$ Intuitively, this allows for independent changes in the demand for each good (through the first five vectors), as well as for substitution from a good to the bundle consisting of its two neighbors if the goods are arranged in a circle (through the last five vectors). This is unimodular, and cannot be obtained from the strong substitutes by a change of basis of the space of integer bundles of goods (see, e.g., Section 19.4 of ). Moreover, the demand types defined by maximal, unimodular turn out to define maximal domains for the existence of in settings with transferable utility. Here, we say that a unimodular is maximal if it is not strictly contained in another unimodular. [fac:unimodMaxDomain] Let D be a maximal unimodular. If ∣J∣ ≥ 2 and $\valFn{j}$ is non-concave or not of demand type D, then there exist sets $\Feas{k}$ of feasible bundles and concave valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ of demand type D for agents $k \not= j$, as well as a total endowment, for which there exists but no.[39](#fn39) While Fact [fac:unimod] shows that there exist valuations in each non-unimodular demand type for which does not exist, Fact [fac:unimodMaxDomain] shows that for every valuation outside a maximal unimodular demand type, there exist concave valuations within the demand type that lead to non-existence. Hence, the necessity direction of Fact [fac:unimod], together with Fact [fac:unimodMaxDomain], provide complementary perspectives on the way in which can fail to exist outside the context of unimodular demand types. Demand Types and the Unimodularity Theorem with Income Effects -------------------------------------------------------------- We now use Fact [fac:dualPrefs] to extend the demand types framework to settings with income effects. [Demand Types with Income Effects] [def:demTypeIncEff] An agent’s preferences are of demand type D if her at all utility levels are of demand type D. Lemma [lem:dHvalH] leads to an economic interpretation of Definition [def:demTypeIncEff]: a utility function is of demand type D if D summarizes the possible ways in which Hicksian demand can change in response to a small generic price change. In particular, Definition [def:demTypeIncEff] extends the concept of demand types to settings with income effects by placing conditions on substitution effects. Indeed, Definition [def:demTypeIncEff] considers only the properties of at each utility level (which, by Lemma [lem:dHvalH], reflect substitution effects), and not how an agent’s vary with her utility level (which, by Fact [fac:dualDem] and Lemma [lem:dHvalH], reflects income effects). translated their conditions on the ranges of agents’ demand correspondences from quasilinear settings to settings with income effects by using Fact [fac:dualPrefs] in an analogous manner (see Assumption 3ʹ in ). However, the economic interpretation in terms of substitution effects that Lemma [lem:dHvalH] leads to was not clear from formulation. As with the case of transferable utility, a concavity condition is needed to ensure the existence of. With income effects, the relevant condition is a version of the quasiconcavity condition from classical demand theory for settings with indivisible goods. We define quasiconcavity based on concavity and duality.[40](#fn40) [Quasiconcavity] [def:quasiConc] An agent’s utility function is quasiconcave if her at all utility levels are concave. As with the case of transferable utility, unimodularity is a necessary and sufficient condition for the existence of to be guaranteed for all quasiconcave preferences of a demand type when income effects are present. [Unimodularity Theorem with Income Effects] [thm:unimod] Let D be a. exist for all finite sets J of agents with quasiconcave utility functions of demand type D, for all total endowments, and for all if and only if D is unimodular. The “only if” direction of Theorem [thm:unimod] is a special case of the Unimodularity Theorem with Transferable Utility (Fact [fac:unimod]). The “if” direction of Theorem [thm:unimod] is an immediate consequence of the Equilibrium Existence Duality and Fact [fac:unimod]. [Proof of the ``if" direction of Theorem [thm:unimod]] Consider a finite set J of agents with quasiconcave preferences of demand type D and a total endowment for which exists. By definition, the agents’ at all utility levels are concave and of demand type D. Hence, exist in the for all profiles of utility levels by the “if” direction of Fact [fac:unimod]. By the “if” direction of Theorem [thm:existDualExchange], must therefore exist in the original economy for all. proved a version of the “if” direction of Theorem [thm:unimod] under the assumptions that utility functions are monotone in goods, that consumption of goods is nonnegative, and that the total endowment is strictly positive (see Theorems 2 and 4 in ).[41](#fn41) Note that they formulated their result in terms of Fact [fac:dualPrefs] and a condition on the ranges of demand correspondences (see their Assumption 3ʹ) instead of in terms of unimodular demand types. approach was to show the existence of in a convexified economy and that, under unimodularity, in the convexified economy give rise to in the original economy. In contrast, our approach of using the Equilibrium Existence Duality illuminates the role of substitution effects in ensuring the existence of. Moreover, it yields a maximal domain result for unimodular demand types with income effects. [prop:netUnimodMaxDomain] Let D be a maximal unimodular. If ∣J∣ ≥ 2 and $\utilFn{j}$ is not quasiconcave or not of demand type D, then there exist sets $\Feas{k}$ of feasible bundles and concave valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ of demand type D for agents $k \not= j$, as well as a total endowment and, for which no exists. Proposition [prop:netUnimodMaxDomain] is an immediate consequence of the Equilibrium Existence Duality and the maximal domain result for unimodular demand types under the transferability of utility. By definition, there exists a utility level $\ub$ at which agent j’s $\valHDef{j}$ is non-concave or not of demand type D. In either case, Fact [fac:unimodMaxDomain] implies that there exist sets $\Feas{k}$ of feasible bundles and concave valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ of demand type D for agents $k \not= j$, and a total endowment for which exists but no would exist with transferable utility if agent j’s valuation were $\valHDef{j}$. With those sets $\Feas{k}$ of feasible bundles and valuations $\valFn{k}$ for agents $k \not= j$ and that total endowment, the ``only if" direction of Theorem [thm:existDualExchange] implies that there exists for which no exists. Intuitively, Proposition [prop:netUnimodMaxDomain] and Theorem [thm:unimod] suggest that Definition [def:demTypeIncEff] is the most general way to incorporate income effects into unimodular demand types from the quasilinear setting and ensure the existence of. Indeed, Proposition [prop:netUnimodMaxDomain] entails that any domain of preferences that contains all concave quasilinear preferences of a maximal, unimodular demand type and guarantees the existence of must lie within the corresponding demand type constructed in Definition [def:demTypeIncEff]. The Strong Substitutes Demand Type and Net with Multiple Units --------------------------------------------------------------- We now use the case of Theorem [thm:unimod] for the strong substitutes demand type to extend Theorem [thm:netSubExist] to settings in which agents can demand multiple units of some goods. In such settings, if utility is transferable, the condition needed to ensure the existence of is —the condition requiring that agents see units of goods as . As and showed, there is a close relationship between strong (net) and the strong substitutes demand type.[42](#fn42) [] 1. A valuation is a valuation if it corresponds to a valuation when each unit of each good is regarded as a separate good. 2. A utility function is a utility function if it corresponds to a utility function when each unit of each good is regarded as a separate good. [fac:ssubDemTypeConc] A valuation (resp. utility function) is a strong (net) valuation (resp. utility function) if and only if it is concave (resp. quasiconcave) and of the strong substitutes demand type.[43](#fn43)[44](#fn44) As the strong substitutes is unimodular, the existence of under is therefore a special case of the Unimodularity Theorem with Income Effects. [cor:snsubExist] If all agents have utility functions, then exist for all. Corollary [cor:snsubExist] can also be proven directly using the Equilibrium Existence Duality and the existence of under in transferable utility economies. Theorem [thm:netSubExist] is the special case of Corollary [cor:snsubExist] for settings in which agents demand at most one unit of each good. As there are unimodular unrelated to the strong substitutes (such as the one in Example [eg:5D]), Theorem [thm:unimod] is strictly more general than Corollary [cor:snsubExist] (and hence Theorem [thm:netSubExist]). In particular, Theorem [thm:unimod] also illustrates that certain patterns of net complementarities can also be compatible with the existence of. As the strong substitutes is maximal as a unimodular (see, e.g., Example 9 in ), Proposition [prop:netUnimodMaxDomain] yields a maximal domain result for. [cor:snsubMax] If ∣J∣ ≥ 2 and $\utilFn{j}$ is not a utility function, then there exist valuations $\valFn{k}$ for agents $k \not= j$, as well as a total endowment and, for which no exists. Auction Design ============== Our work has several implications for auction design. First, our perspective of analyzing preferences by using the expenditure/minimization problem may yield new approaches for extending auction bidding languages to allow for income effects. Second, our equilibrium existence results suggest that some auctions with competitive equilibrium pricing may work well for indivisible goods even in the presence of financing constraints. One set of examples are Product-Mix Auctions, such as the one implemented by the Bank of England[45](#fn45)—these implement competitive equilibrium allocations assuming that the submitted sealed bids represent bidders’ actual preferences, since truth-telling is a reasonable approximation in these auctions when there are sufficiently many bidders. However, while we have shown that gross complementarities do not lead to the nonexistence of competitive equilibrium, they do create problems for dynamic auctions. When agents see goods as gross substitutes, iteratively increasing the prices of over-demanded goods leads to a competitive equilibrium. In contrast, when there are gross complementarities between goods, increases in the price of an over-demanded good can lead to other goods being under-demanded due to an income effect. So, even though competitive equilibrium always exists when agents see goods as (strong) net substitutes, it may not be possible to find a competitive equilibrium using a monotone, dynamic auction. In particular, simple “activity rules” that require bidders to bid on a smaller total number of units of goods as prices increase may result in inefficient outcomes. So, the Product-Mix Auction approach of finding competitive equilibrium based on a single round of sealed bids seems especially useful in the presence of income effects. Conclusion ========== The Equilibrium Existence Duality is a useful tool for analyzing economies with indivisible goods. It is based on the relationship between Marshallian and Hicksian demands, and on an interpretation of Hicksian demand in terms of a quasilinear maximization problem. The Equilibrium Existence Duality shows that competitive equilibrium exists (for all endowment allocations) if and only if competitive equilibrium exists in each of a family of Hicksian economies. An application is that it is net substitutability, not gross substitutability, that is relevant to the existence of equilibrium. And extending the demand types classification of valuations allows us to state a Unimodularity Theorem with Income Effects that gives conditions on the patterns of substitution effects that guarantee the existence of competitive equilibrium. In short, with income effects, just as without them, existence does not depend on agents seeing goods as substitutes; rather, substitution effects are fundamental to the existence of competitive equilibrium. Our results point to a number of potential directions for future work. First, it would be interesting to investigate applications of the Equilibrium Existence Duality to other results on the existence of equilibrium with transferable utility—such as those of,, and. Second, our results could be used to further develop auction designs that find competitive equilibrium outcomes given the submitted bids, such as Product-Mix Auction. More broadly, our approach may lead to new results about the properties of economies with indivisibilities and income effects. @fact=@theorem @lemma=@theorem @proposition=@theorem @corollary=@theorem @definition=@theorem @claim=@theorem @remark=@theorem @example=@theorem Proof of Theorem [thm:existDualExchange] and Lemma [lem:dualEconSWT] ==================================================================== We prove the following result, which combines Theorem [thm:existDualExchange] and Lemma [lem:dualEconSWT]. [thm:existDualExchangeSWT] Suppose that the total endowment and the sets of feasible bundles are such that exists. The following are equivalent. 1. [cond:marshall] exist for all. 2. [cond:SWT] For each Pareto-efficient allocation $(\bunnj)\_{j \in J}$ with $\sum\_{j \in J} \bunj = \tot$, there exists a price vector $\p$ such that $\bunnj \in \dM{j}{\p}{\bunnj}$ for all agents j. 3. [cond:hicks] exist in the for all profiles of utility levels. The remainder of this appendix is devoted to the proof of Theorem [thm:existDualExchangeSWT]. Proof of the [cond:marshall][cond:SWT] Implication in Theorem [thm:existDualExchangeSWT] ---------------------------------------------------------------------------------------- The proof of this implication is essentially identical to the proof of Theorem 3 in. Consider a Pareto-efficient allocation $(\bunnj)\_{j \in J}$ with $\sum\_{j \in J} \bunnj = \tot.$ Let agent j’s endowment be $\bunndowj = \bunnj$. By Statement [cond:marshall] in the theorem, there exists a, say consisting of the price vector $\p$ and the allocation $(\hbunj)\_{j \in J}$ of goods. By the definition of, we have that $\hbunj \in \dM{j}{\p}{\bunj}$ for all agents j. In particular, letting $\hnumerj = \numerj - \p \cdot (\hbunj - \bunj)$ for each agent j, we have that $\sum\_{j \in J} \hbunnj = \sum\_{j \in J} \bunnj$ and that $\util{j}{\hbunnj} \ge \util{j}{\bunnj}$ for all agents j. As the allocation $(\bunnj)\_{j \in J}$ is Pareto-efficient, we must have that $\util{j}{\hbunnj} = \util{j}{\bunnj}$ for all agents j. It follows that $\bunj \in \dM{j}{\p}{\bunnj}$ for all agents j—as desired. Proof of the [cond:SWT][cond:hicks] Implication in Theorem [thm:existDualExchangeSWT] ------------------------------------------------------------------------------------- Let $(\ubj)\_{j \in J}$ be a profile of utility levels. Consider any allocation $(\bunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ of goods with $\sum\_{j \in J} \bunj = \tot$ that minimizes $$\sum\_{j \in J} \cf{j}{\bunj}{\ubj}$$ over all allocations $(\hbunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ of goods with $\sum\_{j \in J} \hbunj = \tot.$ Such an allocation exists because each set $\Feas{j}$ is finite and exists. For each agent j, let $\numerj = \cf{j}{\bunj}{\ubj}$—so $\util{j}{\bunnj} = \ubj$. [cl:dualPE] The allocation $(\bunnj)\_{j \in J}$ is Pareto-efficient. Consider any allocation $(\hbunnj)\_{j \in J} \in \bigtimes\_{j \in J} \Feans{j}$ with $\sum\_{j \in J} \hbunj = \tot$, and $\util{j}{\hbunnj} \ge \util{j}{\bunnj} = \ubj$ for all agents j with strict inequality for some j = j1. As $\cf{j}{\hbunj}{\cdot}$ is strictly increasing for each agent j, we must have that $$\hnumerj = \cf{j}{\hbunj}{\util{j}{\hbunnj}} \ge \cf{j}{\hbunj}{\ubj}$$ for all agents j with strict inequality for j = j1. Hence, we must have that $$\sum\_{j \in J} \hnumerj > \sum\_{j \in J} \cf{j}{\hbunj}{\ubj} \ge \sum\_{j \in J} \cf{j}{\bunj}{\ubj} = \sum\_{j \in J} \numerj,$$ where the second inequality follows from the definition of $(\bunj)\_{j \in J}$, so the allocation $(\bunnj)\_{j \in J}$ cannot be Pareto-dominated. By Claim [cl:dualPE] and Statement [cond:SWT] in the theorem, there exists a price vector $\p$ such that $\bunj \in \dM{j}{\p}{\bunnj}$ for all agents j. Fact [fac:dualDem] implies that $\bunj \in \dH{j}{\p}{\ubj}$ for all agents j. By Lemma [lem:dHvalH], it follows that the price vector $\p$ and the allocation $\left(\bunj\right)\_{j \in J}$ of goods comprise a in the for the profile $(\ubj)\_{j \in J}$ of utility levels. Proof of the [cond:hicks][cond:marshall] Implication in Theorem [thm:existDualExchangeSWT] ------------------------------------------------------------------------------------------ Let $(\bunndowj)\_{j \in J}$ be an endowment allocation. For each agent j, we define a utility level $\umin{j} = \util{j}{\bunndowj}$ and let $$\begin{aligned} K^j &= \numerdowj - \min\_{\bun \in \Feas{j}} \cf{j}{\bun}{\umin{j}},\end{aligned}$$ which is non-negative by construction. Furthermore, let K = 1 + ∑j ∈ JKj and let $$\umax{j} = \max\_{\bun \in \Feas{j}} \util{j}{\numerdowj + K, \bun}.$$ Given a profile $\ubvec = (\ubj)\_{j \in J}$ of utility levels, let $$\payoffs{\ubvec} = \left\{\left(\begin{array}{l} \cf{j}{\bunj}{\ubj} - \numerdowj\\ + \p \cdot (\bunj - \bundowj) \end{array}\right)\_{j \in J} \lgiv \begin{array}{l} \left(\p,(\bunj)\_{j \in J}\right) \text{ is a competitive}\\ \text{equilibrium in the \dualecon}\\ \text{for the profile } (\ubj)\_{j \in J} \text{ of utility levels} \end{array}\lgivend \right\}$$ denote the set of profiles of net expenditures over all in the for the profile $(\ubj)\_{j \in J}$ of utility levels. As discussed in Section [sec:EED], the strategy of the proof is to solve for a profile $\ubvec = (\ubj)\_{j \in J}$ of utility levels such that $\zero \in \payoffs{\ubvec}$. We first show that the correspondence $\payoffFn: \bigtimes\_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values. We then apply a topological fixed point argument to show that there exists a profile $\ubvec = (\ubj)\_{j \in J} \in \bigtimes\_{j \in J} [\umin{j},\umax{j}]$ of utility levels such that $\zero \in \payoffs{\ubvec}$. We conclude the proof by constructing a for the endowment allocation $(\bunndowj)\_{j \in J}$ in the original economy from a in the for the profile $(\ubj)\_{j \in J}$ of utility levels. ### Proof of the Regularity Conditions for $\payoffFn$. We begin by proving that the correspondence $\payoffFn: \bigtimes\_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values. We actually give explicit bounds for the range of $\payoffFn$. Let $$\overline{M} = \max\_{j \in J} \left\{\cf{j}{\bundowj}{\umax{j}} - \numerdowj\right\}$$ and let $$\underline{M} = \sum\_{j \in J} \left(\min\_{\bun \in \Feas{j}} \left\{\cf{j}{\bun}{\umin{j}}\right\} - \numerdowj\right) - (\|J\|-1) \overline{M}.$$ [cl:regT] The correspondence $\payoffFn: \bigtimes\_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values and range contained in $[\underline{M},\overline{M}]^J$. The proof of Claim [cl:regT] uses the following technical description of $\payoffFn$. [cl:hicksEqHelp] Let $\ubvec = (\ubj)\_{j \in J} \in \bigtimes\_{j \in J} [\umin{j},\umax{j}]$ be a profile of utility levels and let $(\bunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ be an allocation of goods with $\sum\_{j \in J} \bunj = \tot$. If $(\bunj)\_{j \in J}$ minimizes $$\sum\_{j \in J} \cf{j}{\hbunj}{\ubj}$$ over all allocations $(\hbunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ of goods with $\sum\_{j \in J} \hbunj = \tot$, then we have that $$\payoffs{\ubvec} = \left\{\rgivend \left(\cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj)\right)\_{j \in J} \rgiv \p \in \mathcal{P}\right\},$$ where $$\mathcal{P} = \left\{\p \lgiv \cf{j}{\bunj}{\ubj} + \p \cdot \bunj \le \cf{j}{\bunpr}{\ubj} + \p \cdot \bunpr \text{ for all } j \in J \text{ and } \bunpr \in \Feas{j} \lgivend\right\}.$$ By construction, we have that $$\mathcal{P} = \left\{\p \lgiv \begin{array}{l} \left(\p,(\bunnj)\_{j \in J}\right) \text{ is a \ce\ in the}\\ \text{ \dualecon\ for the profile } (\ubj)\_{j \in J} \text{ of utility levels} \end{array}\lgivend \right\}.$$ A standard lemma regarding in transferable utility economies shows that in the for the profile $(\ubj)\_{j \in J}$ of utility levels, if $\left(\p,(\hbunnj)\_{j \in J}\right)$ is a, then so is $\left(\p,(\bunj)\_{j \in J}\right)$.[46](#fn46) In this case, we have that $$\cf{j}{\bunj}{\ubj} + \p \cdot \bunj = \cf{j}{\hbunj}{\ubj} + \p \cdot \hbunj,$$ and hence that $$\cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj) = \cf{j}{\hbunj}{\ubj} - \numerdowj + \p \cdot (\hbunj - \bundowj),$$ for all agents j. The claim follows. [Proof of Claim [cl:regT]] It suffices to show that $\payoffFn$ has convex values, range contained in $[\underline{M},\overline{M}]^J$, and a closed graph. We first show that $\payoffs{\ubvec}$ is convex for all $\ubvec \in \bigtimes\_{j \in J} [\umin{j},\umax{j}]$. We use the notation of Claim [cl:hicksEqHelp] to prove this assertion. Note that P is the set of solutions to a set of linear inequalities, and is hence convex. Claim [cl:hicksEqHelp] implies that $\payoffs{\ubvec}$ is the set of values of a linear function on P—so it follows that $\payoffs{\ubvec}$ is convex as well. We next show that $\payoffs{\ubvec} \subseteq [\underline{M},\overline{M}]^J$ holds for all $\ubvec \in \bigtimes\_{j \in J} [\umin{j},\umax{j}].$ We again use the notation of Claim [cl:hicksEqHelp]. Let $\ubvec \in \bigtimes\_{j \in J} [\umin{j},\umax{j}]$ and $\payoffvec \in \payoffs{\ubvec}$ be arbitrary. By Claim [cl:hicksEqHelp], there exists $\p \in \mathcal{P}$ such that $$\payoff{j} = \cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj)$$ for all agents j. Note that for all agents j, we must have that $$\payoff{j} \le \cf{j}{\bundowj}{\ubj} - \numerdowj \le \cf{j}{\bundowj}{\umax{j}} - \numerdowj \le \overline{M},$$ where the first inequality holds due to the definition of P, the second inequality holds because $\cf{j}{\bundowj}{\cdot}$ is strictly increasing, and the third inequality holds due to the definition of $\overline{M}$. Furthermore, as $\sum\_{j \in J} \bunj = \tot = \sum\_{j \in J} \bundowj$, we have that $$\begin{aligned} \sum\_{j \in J} \payoff{j} &= \sum\_{j \in J} (\cf{j}{\bunj}{\ubj} - \numerdowj).\end{aligned}$$ It follows that $$\begin{aligned} \payoff{j} &= \sum\_{k \in J} (\cf{k}{\bunag{k}}{\ubag{k}} - \numerdowag{k}) - \sum\_{k \in J \ssm \{j\}} \payoff{k}\\ &\ge \sum\_{k \in J} (\cf{k}{\bunag{k}}{\umin{k}} - \numerdowag{k}) - \sum\_{k \in J \ssm \{j\}} \payoff{k}\\ &\ge \sum\_{k \in J} (\cf{k}{\bunag{k}}{\umin{k}} - \numerdowag{k}) - (\|J\|-1) \overline{M}\\ &\ge \underline{M}\end{aligned}$$ for all agents j, where the first inequality holds because $\cf{k}{\bunag{k}}{\cdot}$ is increasing for each agent k, the second inequality holds because $\payoff{k} \le \overline{M}$ for all agents k, and the third inequality holds due to the definition of $\underline{M}$. Last, we show that $\payoffFn$ has a closed graph. Our argument uses the following version of Farkas’s Lemma. [Page 200 of [47](#fn47)] [fac:farkas] Let L1, L2 be disjoint, finite sets and, for each ℓ ∈ L1 ∪ L2, let $\v^\ell \in \R^I$ be a vector and let αℓ be a scalar. There exist scalars λℓ for ℓ ∈ L1 ∪ L2 with λℓ ≥ 0 for ℓ ∈ L2 such that $$\sum\_{\ell \in L\_1 \cup L\_2} \lambda\_\ell \v^\ell = \zero \quad \text{and} \sum\_{\ell \in L\_1 \cup L\_2} \lambda\_\ell \alpha\_\ell < 0$$ if and only if there does not exist a vector $\p \in \R^I$ such $\v^\ell \cdot \p \le \alpha\_\ell$ for all ℓ ∈ L1 ∪ L2 with equality for all ℓ ∈ L1. Consider a sequence ${\ubvec\_{(1)}},{\ubvec\_{(2)}},\ldots \in \bigtimes\_{j \in J} [\umin{j},\umax{j}]$ of profiles of utility levels. For each m, let ${\payoffvec\_{(m)}} \in \payoffs{{\ubvec\_{(m)}}}$. Suppose that ${\ubvec\_{(m)}} \to \ubvec$ and ${\payoffvec\_{(m)}} \to \payoffvec$ as m → ∞. We need to show that $\payoffvec \in \payoffs{\ubvec}$. As each set $\Feas{j}$ is finite and exists, by passing to a subsequence we can assume that there exists an allocation $(\bunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ of goods with $\sum\_{j \in J} \bunj = \tot$ that, for each m, minimizes $$\sum\_{j \in J} \cf{j}{\hbunj}{{\ubj\_{(m)}}}$$ over all allocations $(\hbunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ of goods with $\sum\_{j \in J} \hbunj = \tot.$ By the continuity of $\cf{j}{\hbunj}{\ub}$ in $\ub$ for each agent j, the allocation $(\bunj)\_{j \in J}$ minimizes $$\sum\_{j \in J} \cf{j}{\hbunj}{\ubj}$$ over all allocations $(\hbunj)\_{j \in J} \in \bigtimes\_{j \in J} \Feas{j}$ of goods with $\sum\_{j \in J} \hbunj = \tot.$ Suppose for sake of deriving a contradiction that $\payoffvec \notin \payoffs{\ubvec}$. Let L1 = J and let $L\_2 = \bigcup\_{j \in J} \{j\} \times \Feas{j}.$ Define vectors $\v^\ell \in \mathbb{R}^I$ for ℓ ∈ L1 ∪ L2 by $$\v^\ell = \begin{cases} \bunj - \bundowj & \text{ for } \ell = j \in L\_1\\ \bunj - \bunpr & \text{ for } \ell = (j,\bunpr) \in L\_2\\ \end{cases}$$ and scalars αℓ for ℓ ∈ L1 ∪ L2 by $$\alpha\_\ell = \begin{cases} \cf{j}{\bunj}{\ubj} - \numerdowj - \payoff{j} & \text{ for } \ell = j \in L\_1\\ \cf{j}{\bunpr}{\ubj} - \cf{j}{\bunj}{\ubj} & \text{ for } \ell = (j,\bunpr) \in L\_2. \end{cases}$$ By Claim [cl:hicksEqHelp], there does not exist a price vector $\p$ such that $\v^\ell \cdot \p \le \alpha\_\ell$ for all ℓ ∈ L1 ∪ L2 with equality for all ℓ ∈ L1. The “if” direction of Fact [fac:farkas] therefore guarantees that there exist scalars λℓ for ℓ ∈ L1 ∪ L2 with λℓ ≥ 0 for all ℓ ∈ L2 such that $$\sum\_{\ell \in L\_1 \cup L\_2} \lambda\_\ell \v^\ell = \zero \quad \text{and} \quad \sum\_{\ell \in L\_1 \cup L\_2} \lambda\_\ell \alpha\_\ell < 0.$$ By the definition of the scalars αℓ, we have that $$\sum\_{j \in J} \lambda\_j \left(\cf{j}{\bunj}{\ubj} - \numerdowj - \payoff{j}\right) + \sum\_{j \in J} \sum\_{\bunpr \in \Feas{j}} \lambda\_{j,\bunpr} \left(\cf{j}{\bunpr}{\ubj} - \cf{j}{\bunj}{\ubj}\right) < 0.$$ Due the continuity of $\cf{j}{\hbunj}{\ub}$ in $\ub$ for each agent j and because ${\ubvec\_{(m)}} \to \ubvec$ and ${\payoffvec\_{(m)}} \to \payoffvec$ as m → ∞, there must exist m such that $$\sum\_{j \in J} \lambda\_j \left(\cf{j}{\bunj}{{\ubj\_{(m)}}} - \numerdowj - {\payoff{j}\_{(m)}}\right) + \sum\_{j \in J} \sum\_{\bunpr \in \Feas{j}} \lambda\_{j,\bunpr} \left(\cf{j}{\bunpr}{{\ubj\_{(m)}}} - \cf{j}{\bunj}{{\ubj\_{(m)}}}\right) < 0.$$ Defining scalars αʹℓ for ℓ ∈ L1 ∪ L2 by $$\begin{aligned} \alpha'\_\ell &= \begin{cases} \cf{j}{\bunj}{{\ubj\_{(m)}}} - \numerdowj - {\payoff{j}\_{(m)}} & \text{ for } \ell = j \in L\_1\\ \cf{j}{\bunpr}{{\ubj\_{(m)}}} - \cf{j}{\bunj}{{\ubj\_{(m)}}} & \text{ for } \ell = (j,\bunpr) \in L\_2, \end{cases}\end{aligned}$$ we have that $$\sum\_{\ell \in L\_1 \cup L\_2} \lambda\_\ell \v^\ell = \zero \quad \text{and that} \quad \sum\_{\ell \in L\_1 \cup L\_2} \lambda\_\ell \alpha'\_\ell < 0.$$ The “only if” implication of Fact [fac:farkas] therefore guarantees that there does not exist a price vector $\p$ such that $\v^\ell \cdot \p \le \alpha'\_\ell$ for all ℓ ∈ L1 ∪ L2 with equality for all ℓ ∈ L1. By Claim [cl:hicksEqHelp], it follows that ${\payoffvec\_{(m)}} \notin \payoffs{{\ubvec\_{(m)}}}$—a contradiction. Hence, we can conclude that $\payoffvec \in \payoffs{\ubvec}$—as desired. ### Completion of the Proof of the [cond:marshall][cond:SWT] Implication in Theorem [thm:existDualExchangeSWT]. We first solve for a profile $\ubvec = (\ubj)\_{j \in J}$ of utility levels such that $\zero \in \payoffs{\ubvec}.$ [cl:fpEq] Under Statement [cond:hicks] in Theorem [thm:existDualExchangeSWT], there exists a profile $\ubvec = (\ubj)\_{j \in J}$ of utility levels such that $\zero \in \payoffs{\ubvec}$. To prove Claim [cl:fpEq], we apply a topological fixed point argument. Consider the compact, convex set $$Z = [\underline{M},\overline{M}]^J \times \bigtimes\_{j \in J} [\umin{j},\umax{j}].$$ As $\payoffs{\ubvec} \subseteq [\underline{M},\overline{M}]^J$ for all $\ubvec \in \bigtimes\_{j \in J} [\umin{j},\umax{j}],$ we can define a correspondence $\Phi: Z \toto Z$ by $$\Phi(\payoffvec,\ubvec) = \payoffs{\ubvec} \times \argmin\_{\hubvec \in \bigtimes\_{j \in J} [\umin{j},\umax{j}]} \left\{\sum\_{j \in J} \payoff{j} \hubj\right\}.$$ Claim [cl:regT] guarantees that $T: \bigtimes\_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values. Statement [cond:hicks] in Theorem [thm:existDualExchangeSWT] ensures that the correspondence T has non-empty values. Because $\bigtimes\_{j \in J} [\umin{j},\umax{j}]$ is compact and convex, it follows that the correspondence Φ is upper hemicontinuous and has non-empty, compact, convex values as well. Hence, Kakutani’s Fixed Point Theorem guarantees that Φ has a fixed point $(\payoffvec,\ubvec)$. By construction, we have that $\payoffvec \in \payoffs{\ubvec}$ and that $$\label{eq:fpUtil} \ubj \in \argmin\_{\hubj \in [\umin{j},\umax{j}]} \payoff{j} \hubj$$ for all agents j. It suffices to prove that $\payoffvec = \zero$. Let $\left(\p,(\bunnj)\_{j \in J}\right)$ be a in the for the profile $(\ubj)\_{j \in J}$ of utility levels with $$\label{eq:fpPayoff} \cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj) = \payoff{j}$$ for all agents j. As $\ubj \ge \umin{j}$ and $\cf{j}{\bunj}{\cdot}$ is increasing for each agent j, it follows from Equation ([eq:fpPayoff]) and the definition of Kj that $$\begin{aligned} \payoff{j} &= \cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj) \nonumber\\ &\ge \cf{j}{\bunj}{\umin{j}} - \numerdowj + \p \cdot (\bunj - \bundowj) \nonumber\\ &\ge \p \cdot (\bunj-\bundowj) - K^j \label{eq:tlb}\end{aligned}$$ for all agents j. Next, we claim that $\payoff{j} \le 0$ for all agents j. If $\payoff{j} > 0,$ then Equation ([eq:fpUtil]) would imply that $\ubj = \umin{j}$. But as $\payoffvec \in \payoffs{\ubvec},$ it would follow that $$\payoff{j} \le \cf{j}{\bundowj}{\umin{j}} - \numerdowj + \p \cdot (\bundowj - \bundowj) = \cf{j}{\bundowj}{\umin{j}} - \numerdowj = 0,$$ where the last equality holds due to the definitions of $\cfFn{j}$ and $\umin{j},$ so we must have that $\payoff{j} \le 0$ for all agents j. As $(\bunj)\_{j \in J}$ is the allocation of goods in a, we have that $\sum\_{j \in J} \bunj = \tot = \sum\_{j \in J} \bundowj$ and hence that $$\sum\_{j \in J} \p \cdot (\bunj - \bundowj) = 0 \ge \sum\_{j \in J} \payoff{j},$$ where the inequality holds because $\payoff{j} \le 0$ for all agents j. It follows that for all agents j, we have that $$\payoff{j} - \p \cdot (\bunj - \bundowj) \le \sum\_{k \in J \ssm \{j\}} (\p \cdot (\bunag{k} - \bundowag{k}) - \payoff{k}) \le \sum\_{k \in J \ssm \{j\}} K^k \le \sum\_{k \in J} K^k < K,$$ where the second inequality follows from Equation ([eq:tlb]), the third inequality holds because Kj ≥ 0, and the fourth inequality holds due to the definition of K. Hence, by Equation ([eq:fpPayoff]), we have that $$\cf{j}{\bunj}{\ubj} = \numerdowj + \payoff{j} - \p \cdot (\bunj - \bundowj) < \numerdowj + K$$ for all agents j. Since utility is strictly increasing in the consumption of money, it follows that $$\ubj = \util{j}{\cf{j}{\bunj}{\ubj},\bunj} < \util{j}{\numerdowj + K,\bunj} \le \umax{j},$$ where the equality holds due to the definition of $\cfFn{j}$ and the second inequality holds due to the definition of $\umax{j}$. Equation ([eq:fpUtil]) then implies that $\payoff{j} \ge 0$ for all agents j, so we must have that $\payoff{j} = 0$ for all agents j. By Claim [cl:fpEq], there exists a profile $\ubvec = (\ubj)\_{j \in J}$ of utility levels and a $(\p,(\bunj)\_{j \in j})$ in the corresponding with $$\label{eq:tFpFinal} \numerdowj = \cf{j}{\bunj}{\ubj} + \p \cdot (\bunj - \bundowj)$$ for all agents j. Lemma [lem:dHvalH] implies that $\bunj \in \dH{j}{\p}{\ubj}$ for all agents j, and we have that $\util{j}{\numerdowj - \p \cdot (\bunj - \bundowj),\bunj} = \ubj$ for all agents j by Equation ([eq:tFpFinal]) and the definition of $\cfFn{j}$. It follows from Fact [fac:dualDem] that $\bunj \in \dM{j}{\p}{\bunndowj}$ for all agents j, so the price vector $\p$ and the allocation $(\bunj)\_{j \in J}$ of goods comprise a in the original economy for the endowment allocation $(\bunndowj)\_{j \in J}$. Proof of Proposition [prop:ssubst] ================================== We actually prove a stronger statement. [cl:marshallNetWeak] Suppose that agent j demands at most one unit of each good and let $\bundow \in \Feas{j}$. A utility function $\utilFn{j}$ is a utility function if for all money endowments $\numerdow > \feas{j},$ price vectors $\p,$ and 0 < μ < λ, whenever 1. [cond:start] $\dM{j}{\p}{\bunndow} = \{\bun\}$, 2. [cond:end] $\dM{j}{\p + \lambda \e{i}}{\bunndow} = \{\bunpr\}$, 3. [cond:middle] $\{\bun,\bunpr\} \subseteq \dM{j}{\p + \mu \e{i}}{\bunndow}$, and 4. [cond:ineq] $\bunprComp{i} < \bunComp{i}$, we have that $\bunprComp{k} \ge \bunComp{k}$ for all goods $k \not= i.$ To complete the proof of the proposition from Claim [cl:marshallNetWeak], we work in the setting of Claim [cl:marshallNetWeak]. Note that, for the endowment $\bundow$ of goods, $\utilFn{j}$ is a utility function when $\bunprComp{k} \ge \bunComp{k}$ holds for all goods $k \not= i$ under Conditions [cond:start] and [cond:end]. This property clearly implies that $\bunprComp{k} \ge \bunComp{k}$ holds for all goods $k \not= i$ under Conditions [cond:start], [cond:end], [cond:middle], and [cond:ineq], and hence that $\utilFn{j}$ is utility function by Claim [cl:marshallNetWeak]. The proposition therefore follows from Claim [cl:marshallNetWeak]. It remains to prove Claim [cl:marshallNetWeak]. In the argument, we use the following characterization of valuations. [Theorems 2.1 and 2.4 in ; Theorems 3.9 and 4.10(iii) in ] [fac:subsDemCplx] Suppose that agent j demands at most one unit of each good. A valuation $\valFn{j}$ is a valuation if and only if for all price vectors $\p$ with $\|\dQL{j}{\p}\| = 2,$ writing $\dQL{j}{\p} = \{\bun,\bunpr\},$ the difference $\bunpr - \bun$ is a vector with at most one positive component and at most one negative component. [Proof of Claim [cl:marshallNetWeak]] We prove the contrapositive. Suppose that $\utilFn{j}$ is not a utility function. We show that there exists a money endowment $\numerdow,$ a price vector $\p,$ price increments 0 < μ < λ, and goods $i \not= k$ such that Conditions [cond:start], [cond:end], [cond:middle], and [cond:ineq] from the statement hold but $\bunprComp{k} < \bunComp{k}.$ By Remark [rem:netSubDual], there exists a utility level $\ub$ such that $\valHDef{j}$ is not a valuation. Hence, by Lemma [lem:dHvalH] and the ``if" direction of Fact [fac:subsDemCplx] for $\valFn{j} = \valHDef{j}$, there exists a price vector $\hp$ such that $\|\dH{j}{\hp}{\ub}\| = 2$, and writing $\dH{j}{\hp}{\ub} = \{\bun,\bunpr\},$ the difference $\bunpr - \bun$ has at least two positive components or at least two negative components. Without loss of generality, we can assume that the difference $\bunpr - \bun$ has at least two negative components. Suppose that $\bunprComp{i} < \bunComp{i}$ (so Condition [cond:ineq] holds) and that $\bunprComp{k} < \bunComp{k},$ where i, k ∈ I are distinct goods. Define a money endowment $\numerdow$ by $$\numerdow = \cf{j}{\bun}{\ub} + \hp \cdot (\bun - \bundow) = \cf{j}{\bunpr}{\ub} + \hp \cdot (\bunpr - \bundow);$$ Fact [fac:dualDem] implies that $\dM{j}{\hp}{\bunndow} = \{\bun,\bunpr\}$. Let μ be such that $$\dM{j}{\hp - \mu \e{i}}{\bunndow}, \dM{j}{\hp + \mu \e{i}}{\bunndow} \subseteq \{\bun,\bunpr\};$$ such a μ exists due to the upper hemicontinuity of $\dMFn{j}$. Let $\p = \hp - \mu \e{i},$ let λ = 2μ, and let $\ppr = \p + \lambda \e{i} = \hp + \mu \e{i}$. By construction, we have that $\{\bun,\bunpr\} \subseteq \dM{j}{\p + \mu \e{i}}{\bunndow} = \dM{j}{\hp}{\bunndow}$, so Condition [cond:middle] holds. It remains to show that $\dM{j}{\p}{\bunndow} = \{\bun\}$ and that $\dM{j}{\ppr}{\bunndow} = \{\bunpr\}$. As j demands at most one unit of each good, we must have that $\bunComp{i} = 1$ and that $\bunprComp{i} = 0.$ We divide into cases based on the value of $\bundowComp{i}$ to show that $$\label{eq:subsIneqs} \begin{aligned} \util{j}{\numerdow - \p \cdot (\bun - \bundow),\bun} &> \util{j}{\numerdow - \p \cdot (\bunpr - \bundow),\bunpr}\\ \util{j}{\numerdow - \ppr \cdot (\bunpr - \bundow),\bunpr} &> \util{j}{\numerdow - \ppr \cdot (\bun - \bundow),\bun}. \end{aligned}$$ $\bundowComp{i} = 0$. In this case, we have that $$\begin{aligned} \util{j}{\numerdow - \p \cdot (\bun - \bundow),\bun} &> \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &=\util{j}{\numerdow - \p \cdot (\bunpr - \bundow),\bunpr},\end{aligned}$$ where the inequality holds because $\pComp{i} < \hpComp{i}$ and $\bunComp{i} > \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunprComp{i} = \bundowComp{i}.$ Similarly, we have that $$\begin{aligned} \util{j}{\numerdow - \ppr \cdot (\bun - \bundow),\bun} &< \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &= \util{j}{\numerdow - \ppr \cdot (\bunpr - \bundow),\bunpr},\end{aligned}$$ where the inequality holds because $\pprComp{i} > \hpComp{i}$ and $\bunComp{i} > \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunprComp{i} = \bundowComp{i}.$ $\bundowComp{i} = 1$. In this case, we have that $$\begin{aligned} \util{j}{\numerdow - \p \cdot (\bunpr - \bundow),\bunpr} &< \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &= \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \p \cdot (\bun - \bundow),\bun}\end{aligned}$$ where the inequality holds because $\pComp{i} < \hpComp{i}$ and $\bunprComp{i} < \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunComp{i} = \bundowComp{i}.$ Similarly, we have that $$\begin{aligned} \util{j}{\numerdow - \ppr \cdot (\bunpr - \bundow),\bunpr} &> \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &= \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \ppr \cdot (\bun - \bundow),\bun},\end{aligned}$$ where the inequality holds because $\pprComp{i} > \hpComp{i}$ and $\bunprComp{i} < \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunComp{i} = \bundowComp{i}.$ As $\bundow \in \Feas{j} \subseteq \{0,1\}^I,$ the cases exhaust all possibilities. Hence, we have proven that Equation ([eq:subsIneqs]) must hold. As $\dM{j}{\p}{\bunndow},\dM{j}{\ppr}{\bunndow} \subseteq \{\bun,\bunpr\},$ we must have that $\dM{j}{\p}{\bunndow} = \{\bun\}$ and that $\dM{j}{\ppr}{\bunndow} = \{\bunpr\}$—so Conditions [cond:start] and [cond:end] hold, as desired. FOR ONLINE PUBLICATION Proofs of Facts [fac:dualDem] and [fac:dualPrefs] ================================================= Proof of Fact [fac:dualDem] --------------------------- We begin by proving two technical claims. [cl:dualDemHelpMtoH] Let $\bunndow \in \Feans{j}$ be an endowment and let $\ub$ be a utility level. If $$\label{eq:marshallForProofOfDual1} \ub = \max\_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn},$$ then we have that $$\pall \cdot \bunndow = \min\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn$$ and that $\dM{j}{\p}{\bunndow} \subseteq \dH{j}{\p}{\ub}$. Letting $\bunpr \in \dM{j}{\p}{\bunndow}$ be arbitrary and $\numerpr = \numerdow - \p \cdot (\bunpr - \bundow),$ we have that $\util{j}{\bunnpr} = u$ and that $\pall \cdot \bunnpr \le \pall \cdot \bunndow$ by construction. It follows that $$\pall \cdot \bunndow \ge \min\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn.$$ Suppose for the sake of deriving a contradiction that there exists $\bunndpr \in \Feans{j}$ with $\pall \cdot \bunndpr < \pall \cdot \bunndow$ and $\util{j}{\bunndpr} \ge \ub.$ Then, we have that $\numerdpr < \numerdow + \p \cdot (\bundpr - \bundow)$; write $\numertpr = \numerdow + \p \cdot (\bundpr - \bundow),$ so $\numertpr > \numerdpr.$ Since $\utilFn{j}$ is strictly increasing in consumption of money, it follows that $\util{j}{\numertpr,\bundpr} > u$—contradicting Equation ([eq:marshallForProofOfDual1]) as $\numertpr + \p \cdot \bundpr = \pall \cdot \bunndow.$ Hence, we can conclude that $$\pall \cdot \bunndow = \min\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn.$$ Since $\util{j}{\bunnpr} = \ub$ and $\pall \cdot \bunnpr = \pall \cdot \bunndow,$ it follows that $\bunpr \in \dH{j}{\p}{\ub}$. Since $\bunpr \in \dM{j}{\p}{\bunndow}$ was arbitrary, we can conclude that $\dM{j}{\p}{\bunndow} \subseteq \dH{j}{\p}{\ub}$. [cl:dualDemHelpHtoM] Let $\bunndow \in \Feans{j}$ be an endowment and let $\ub$ be a utility level. If $$\label{eq:hicksForProofOfDual1} \pall \cdot \bunndow = \min\_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn,$$ then we have that $$\ub = \max\_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}$$ and that $\dH{j}{\p}{\ub} \subseteq \dM{j}{\p}{\bunndow}$. Let $\bunpr \in \dH{j}{\p}{\ub}$ be arbitrary and $\numerpr = \cf{j}{\bunpr}{\ub}$. We have that $\util{j}{\bunnpr} \ge \ub$ and that $\pall \cdot \bunnpr = \pall \cdot \bunndow$ by construction. It follows that $$\ub \le \max\_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ We next show that $$\ub = \max\_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ Suppose for sake of deriving a contradiction that there exists $\bunndpr \in \Feans{j}$ with $\pall \cdot \bunndpr \le \pall \cdot \bunndow$ and $\util{j}{\bunndpr} > \ub.$ By definition of $\cfFn{j}$, we know that $\numerdpr > \cf{j}{\bundpr}{\ub}$. Letting $\numertpr = \cf{j}{\bundpr}{\ub}$, we have that $\numertpr + \p \cdot \bundpr < \pall \cdot \bunndow$, which contradicts Equation ([eq:hicksForProofOfDual1]) as $\util{j}{\numertpr,\bundpr} = \ub.$ Hence, we can conclude that $$\ub = \max\_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ Since $\util{j}{\bunpr} = u$ and $\pall \cdot \bunpr = \pall \cdot \bunndow,$ it follows that $\bunpr \in \dM{j}{\p}{\bunndow}$. Since $\bunpr \in \dH{j}{\p}{\ub}$ was arbitrary, we can conclude that $\dH{j}{\p}{\ub} \subseteq \dM{j}{\p}{\bunndow}$. Let $\bunndow \in \Feans{j}$ be an endowment and let $\ub$ be a utility level. By Claims [cl:dualDemHelpMtoH] and [cl:dualDemHelpHtoM], Conditions ([eq:marshallForProofOfDual1]) and ([eq:hicksForProofOfDual1]), are equivalent, and under these equivalent conditions, we have that $\dM{j}{\p}{\bunndow} \subseteq \dH{j}{\p}{\ub}$ and that $\dH{j}{\p}{\ub} \subseteq \dM{j}{\p}{\bunndow}$. Hence, we must have that $\dM{j}{\p}{\bunndow} = \dH{j}{\p}{\ub}$ under the equivalent Conditions ([eq:marshallForProofOfDual1]) and ([eq:hicksForProofOfDual1])—as desired. Proof of Fact [fac:dualPrefs] ----------------------------- We prove the “if” and “only if” directions separately. ### Proof of the ``If" Direction. We define a utility function $\utilFn{j}$ implicitly by $$\util{j}{\bunn} = F(\bun,\cdot)^{-1}(-\numer),$$ which is well-defined, continuous, and strictly increasing in $\numer$ by the Inverse Function Theorem because $F(\bun,\cdot)$ is continuous, strictly decreasing, and satisfies Condition ([eq:vlimits]). Condition ([eq:ulimits]) holds because F is defined over the entirety of $\Feas{j} \times \feasUtil{j}.$ ### Proof of the ``Only If" Direction. We define $F:\Feas{j} \times (\minu{j},\maxu{j}) \to (-\infty,-\feas{j})$ implicitly by $$F(\bun,\ub) = -\util{j}{\cdot,\bun}^{-1}(\numer),$$ which is well-defined, continuous, and strictly decreasing in $\ub$ by the Inverse Function Theorem because $\util{j}{\cdot,\bun}$ is continuous, strictly increasing, and satisfies Condition ([eq:ulimits]). Condition ([eq:vlimits]) holds because $\utilFn{j}$ is defined over the entirety of $\Feans{j}$. Proofs of the Maximal Domain and Necessity Results for Settings with Transferable Utility ========================================================================================= In this appendix, we supply proofs of Facts [fac:subMaxDomain] and [fac:unimodMaxDomain], as well as the ``only if" direction of Fact [fac:unimod]. Utility is transferable throughout this appendix. We use the concept of a pseudo-equilibrium price vector. [] Suppose that utility is transferable. A pseudo-equilibrium price vector is a price vector $\p$ such that $$\tot\in \conv\left(\sum\_{j \in J} D^j(\p)\right).$$ There is a connection between pseudo-equilibrium price vectors,, and the existence problem. [Theorem 18 in ; Lemma 2.19 in ] [fac:pseudoEquil] If utility is transferable and the total endowment is such that a competitive equilibrium exists, then, for each pseudo-equilibrium price vector $\p$, there exists an allocation $(\bunj)\_{j \in J}$ such that $\p$ and $(\bunj)\_{j \in J}$ comprise a. The nonexistence of may therefore be demonstrated by using the contrapositive of Fact [fac:pseudoEquil]. Our arguments use valuations that are linear on their domain. That is, let $\mathbf{t}\_I\in\R^I$, let $X^j\_I\subseteq\Z^n$ be finite, and let $V^j=V^{j,\mathbf{t}\_I}\mathbf:X^j\_I\rightarrow\R$ be given by $V^{j,\mathbf{t}\_I}(\mathbf{x}\_I)\coloneq\mathbf{t}\_I\cdot\mathbf{x}\_I$ for all xI ∈ XIj. Recalling Equation ([eqn:quasilin]) for demand sets in the quasilinear case, we observe that, for each $\mathbf{s}\_I\in\R^I$, we have that $$\label{eqn:faces} D^j(\mathbf{t}\_I-\mathbf{s}\_I)=\argmax\_{\mathbf{x}\_I\in X^j\_I}\left(\mathbf{t}\_I\cdot\mathbf{x}\_I-(\mathbf{t}\_I-\mathbf{s}\_I)\cdot\mathbf{x}\_I\right)=\argmax\_{\mathbf{x}\_I\in X^j\_I}\mathbf{s}\_I\cdot\mathbf{x}\_I.$$ [lem:linConv] If $\conv(X^j\_I)\cap\Z^I=X^j\_I$, then Vj, tI is concave for all $\mathbf{t}\_I\in\R^I$. Observe by Equation ([eqn:faces]) that $D^j(\mathbf{t}\_I)=\argmax\_{\mathbf{x}\_I\in X^j\_I}\mathbf{0}\cdot\mathbf{x}\_I=X^j\_I$. So, if $\mathbf{x}\_I\in\conv(X^j\_I)\cap\Z^I=X^j\_I$ then xI ∈ Dj(tI). By Definition [def:concave], we know Vj, tI is concave. We will also make use of an alternative characterization of concavity. [Lemma 2.11 in ] [fac:conc] A valuation $\valFn{j}$ is concave if and only if $\conv\left(\dQL{j}{\p}\right) \cap \Z^I = \dQL{j}{\p}$ for all price vectors $\p$. Additional Facts regarding Unimodularity and Demand Types --------------------------------------------------------- The following results are especially useful in the proof of the “only if” direction of Fact [fac:unimod], and the proof of Fact [fac:unimodMaxDomain]. We seek to construct pseudo-equilibrium price vectors (the total endowment is in the convex hull of aggregate demand) that are not price vectors (the total endowment is not demanded on aggregate). Failure of unimodularity allows such constructions because of the following property. [See, e.g., Fact 4.9 in ] [fac:unimodSet] A demand type vector set D is unimodular if and only if there is no linearly independent subset {d1, …, dr} of D such that there exists $\mathbf{z}=\sum\_{\ell=1}^r \alpha\_\ell \dvec^\ell\in\Z^I$ with αℓ ∈ (0, 1) for ℓ = 1, …, r. To see the connection to Fact [fac:pseudoEquil] and to existence of competitive equilibrium, suppose that {d1, …, dr} and z are as in Fact [fac:unimodSet]. If $\tot=\mathbf{z}$ and if $D^j\_M(\p,\bunndowj)=\{\mathbf{0},\mathbf{d}^j\}$ for j = 1, …, r, then $\p$ is a pseudo-equilibrium price vector but there is no competitive equilibrium at $\p$. generalized Fact [fac:subsDemCplx] to the general case of transferable utility. [Proposition 2.20 in ] [fac:demTypeCplx] Let $\valFn{j}$ be a valuation of demand type D. For any price $\ppr$, if $\conv(\dQL{j}{\ppr})$ has an edge E, then the difference between the extreme points of E is proportional to a demand type vector, and there exists a price $\p$ such that $\conv(\dQL{j}{\p})=E$. Moreover if $\dvec$ is in the minimal D, such that $\valFn{j}$ is of demand type D, then there exists a price vector $\p$ such that $\conv(\dQL{j}{\p})$ is a line segment, the difference between whose endpoints is proportional to $\dvec.$ We also demonstrate now the following useful corollary of Fact [fac:demTypeCplx]. [cor:edges] Let Vj = Vj, tI for some $\mathbf{t}\_I\in\R^I$, and let D be the minimal demand type vector set such that Vj is of demand type D. Then D consists of the primitive integer vectors in the directions of the edges of the polytope $\conv(X^j\_I)$. Observe that Dj(tI) = XIj and so, by Fact [fac:demTypeCplx], each edge of $\conv(X^j\_I)$ is proportional to a vector in D. Conversely, if d ∈ D then, by Fact [fac:demTypeCplx], there exists a price $\p$ such that $\conv(D^j(\p))$ is a line segment, the difference between whose endpoints is proportional to d. But writing $\mathbf{s}\_I=\mathbf{t}\_I-\p$, we see from Equation ([eqn:faces]) that $D^j(\p)=\argmax\_{x\_I\in X^j\_I} \mathbf{s}\_I\cdot\mathbf{x}\_I$ which tells us (cf. e.g. , Section 2.4) that E is an edge of $\conv(X^j\_I)$. Our proofs of Facts [fac:subMaxDomain] and [fac:unimodMaxDomain], and the “only if” direction of Fact [fac:unimod], now follow the same structure. Within each argument, we address a demand type which is not unimodular. Observe by Fact [fac:unimodSet] that when unimodularity fails for a set of vectors D, then there exist polytopes, with integer vertices and whose edge directions are in D, that contain a non-vertex integer vector, z. We use Corollary [cor:edges] construct valuations of the appropriate demand type such that, at some price $\p$, the convex hull of the aggregate demand set is a polytope with these properties; and such that there exists a feasible endowment allocation is the total endowment is the non-vertex integer vector z. Thus $\p$ is a pseudo-equilibrium price. Moreover, we design our individual valuations so that $\p$ is not a competitive equilibrium. The contrapositive of Fact [fac:pseudoEquil] can then be applied to show the non-existence of competitive equilibrium. Proof of Fact [fac:subMaxDomain] -------------------------------- By Fact [fac:subsDemCplx], there exists a price vector $\p$ such that $\dQL{j}{\p} = \{\bunpr,\bunpr+\norm\}$, where $\norm$ has at least two positive components or at least two negative components. Identify I with {1, …, ∣I∣} and without loss of generality assume that g1, g2 < 0. Because agent j demands at most one unit of each good, we know that $\bunpr,\bunpr+\norm\in\{0,1\}^{\|I\|}$ and so $\norm\in \{-1,0,1\}^{\|I\|}$. We conclude both that g1 = g2 = − 1 and that xʹ1 = xʹ2 = 1. Let $k \in J \ssm \{j\}$ be arbitrary. For agents $j' \in J \ssm \{j,k\},$ let $\Feas{j'} = \{\zero\}$, let $\valFn{j'}$ be arbitrary, and let $\bundowag{j'} = 0$. Let $X^k\_I\coloneq\{\rgivend\mathbf{x}\_I\in\{0,1\}^{\|I\|}\rgiv x\_1+x\_2\leq 1\}$ and let $\mathbf{t}\_I\coloneq\p-\mathbf{e}^1-\mathbf{e}^2$. Let $V^k = V^{k,{\bf t}\_I},$ which is a valuation by Example	arxiv_0000713
Two Extraordinary Substellar Binaries at the T/Y Transition andthe Y-Band Fluxes of the Coolest Brown Dwarfs ============================================================================================================== Using Keck laser guide star adaptive optics imaging, we have found that the T9 dwarf WISE J1217+1626 and T8 dwarf WISE J1711+3500 are exceptional binaries, with unusually wide separations ( ≈ 0.8, 8–15 AU), large near-IR flux ratios ( ≈ 2–3 mags), and small mass ratios ( ≈ 0.5) compared to previously known field ultracool binaries. Keck/NIRSPEC H-band spectra give a spectral type of Y0 for WISE J1217+1626B, and photometric estimates suggest T9.5 for WISE J1711+3500B. The WISE J1217+1626AB system is very similar to the T9+Y0 binary ; these two systems are the coldest known substellar multiples, having secondary components of ≈ 400 K and being planetary-mass binaries if their ages are $\lesssim$1 Gyr. Both WISE J1217+1626B and have strikingly blue Y − J colors compared to previously known T dwarfs, including their T9 primaries. Combining all available data, we find that Y − J color drops precipitously between the very latest T dwarfs and the Y dwarfs. The fact that this is seen in (coeval, mono-metallicity) binaries demonstrates that the color drop arises from a change in temperature, not surface gravity or metallicity variations among the field population. Thus, the T/Y transition established by near-IR spectra coincides with a significant change in the ≈ 1 fluxes of ultracool photospheres. One explanation is the depletion of potassium, whose broad absorption wings dominate the far-red optical spectra of T dwarfs. This large color change suggests that far-red data may be valuable for classifying objects of $\lesssim$500 K. Introduction ============ Binaries have played a central role in the study of the coolest dwarfs over the last two decades. Each advance that has led to creation of a later spectral type has benefitted from discovery of companions to higher mass objects. The first L dwarf, GD 165B, was found as a companion to a white dwarf. At the time, its unusual optical spectrum compared to late-M dwarfs was puzzling, but with the subsequent discovery of more L dwarfs, it became apparent that GD 165B was the prototype of a new spectral type. The first T dwarf, Gl 229B, was found around an M dwarf, and its uniqueness was evident from the strong and in its near-IR spectrum. Just last year, two low-mass companions were found with temperatures of only ≈ 300–400 K, one being a tight (3 AU) companion to the T9.5 dwarf and the other being a very wide (2500 AU) companion to the white dwarf GJ 3483. Spectroscopy of these objects has not been possible yet, due to the small angular separation (0.11) of and the lack of a near-IR detection for GJ 3483B (a.k.a. WD 0806 − 661B). However, their exceptionally faint absolute magnitudes showed these two objects were novel compared to all previously known T dwarfs. Their temperatures, as inferred from their absolute magnitudes using evolutionary models, are far cooler than the previous record-holder, the ≈ 520 K T10 dwarf, and place the two companions in terra incognita where theoretical models predict the onset of new photospheric signatures including absorption in the near-IR, the disappearance of neutral alkali lines, and the formation of water clouds. More recently, have identified five free-floating objects with sufficiently distinct near-IR spectra to propose classification as Y dwarfs, based on the likely presence of absorption in the blue wing of their H-band continuua. While reliable parallaxes are not yet available for these objects and thus their absolute magnitudes are uncertain, model atmosphere fits indicate temperatures of ≈ 300–500 K. Their novel near-IR appearance may be a precursor to even more significant changes along the remaining temperature gap down to Jupiter (124 K; ). As part of our ongoing effort to study low-temperature objects using substellar binaries, we present here the discovery of two extraordinarily well-resolved binaries that shed light on the transition from the latest T dwarfs to the Y dwarfs. The primaries, WISEPC J121756.91+162640.2 and WISEPA J171104.60+350036.8 (hereinafter WISE J1217+1626 and WISE J1711+3500), were found from a mid-IR search of the solar neighborhood by, who asssigned integrated-light near-IR spectral types of T9 and T8, respectively. We also present improved near-IR imaging of. Our characterization of these three binaries includes resolved photometry in the Y band, which spans the ≈ 1.0 transmission window of the Earth’s atmosphere. A broadband filter in this window is a new capability for adaptive optics imaging at Keck, just installed in September 2011. One of the prime motivating factors for its procurement, as a collaboration between one of us (M. Liu) and Keck Observatory, is to explore the ≈ 1 fluxes of the coolest substellar objects, which can be diagnostic of surface gravity, metallicity, and photospheric chemistry. Observations [sec:observations] =============================== Keck-II/NIRC2 Adaptive Optics Imaging ------------------------------------- We imaged WISE J1217+1626, WISE J1711+3500, and over 3 nights in January and April 2012 using the laser guide star adaptive optics (LGS AO) system of the 10-meter Keck II Telescope on Mauna Kea, Hawaii. For WISE J1217+1626, we used the facility IR camera NIRC2 with its wide field-of-view optics ($40.8\arcsec\times40.8\arcsec$), and for WISE J1711+3500 and we used the narrow field-of-view optics ($10.2\arcsec\times10.2\arcsec$). Conditions were photometric, with 0.8 seeing in January and 0.4–0.6 seeing in April. The LGS provided the wavefront reference source for AO correction, with the tip-tilt motion measured simultaneously using R = 15.5, 14.4, and 15.5 mag stars from the USNO-B1.0 catalog located 38, 43, and 55away from WISE J1217+1626, WISE J1711+3500, and, respectively. The LGS brightness, as measured by the flux incident on the AO wavefront sensor, was equivalent to a V ≈ 9.5 − 10.0 mag star. The LGS was pointed at the center of the NIRC2 field-of-view for all observations. We obtained images with all or a subset of the broad-band Y (1.02 ), J (1.25 ), H (1.64 ), and K (2.20 ) filters of the Mauna Kea Observatories (MKO) photometric system. We also obtained data with the medium-band CH4s filter (central wavelength of 1.592 , 0.126 wide), which is positioned around the H-band peak in the spectra of T dwarfs. Finally, for WISE J1217+1626, we obtained data with NIRC2’s custom “z” filter (central wavelength of 1.031 , 0.048 wide), which covers the cleanest portion of the Y-band atmospheric window. To avoid confusion with optical filters using the same letter but spanning different wavelengths, we refer to this as the z1.1 filter. For each filter, we obtained a set of dithered images, offsetting the telescope by a few arcseconds between every pair of images. The images were analyzed in the same fashion as our previous work. The raw images were reduced using standard methods for flat-fielding and sky-subtraction. The binary’s flux ratios and relative astrometry were derived by fitting an analytic model of the point spread function as the sum of three elliptical gaussians. For the very well-resolved WISE binaries, we fitted all the individual images and adopted the averages of the results as the final measurements and the standard deviations as the errors. (The two WISE binaries are so wide that simple aperture photometry gave essentially the same results, as the contamination of the secondary by the primary is negligible.) For, we fitted the stacked mosaic to maximize the S/N of the faint, close secondary and derived uncertainties from Monte Carlo simulations, as we did in. For NIRC2’s wide camera, we adopted a pixel scale of 39.884 ± 0.039 mas/pixel and an orientation for the detector’s + y axis of 0.16 ± 0.09. For the narrow camera, we adopted a pixel scale of 9.963 ± 0.005 mas/pixel and an orientation for the detector’s + y axis of 0.13 ± 0.07. The relative astrometry was corrected for (the very small) instrumental optical distortion based on a solution by B. Cameron (priv. comm.). Table [table:keck] presents our final measurements, and Figure [fig:images] shows our reduced images. For all three binaries, the astrometry from the different filters shows excellent agreement. For WISE J1217+1626, our measurements have values of 0.89 and 3.46 (5 degrees of freedom) for the separation and PA, respectively, when compared to the hypothesis of a constant value for each quantity. For WISE J1711+3500, the values are 0.2 and 3.10 (4 degrees of freedom) for separation and PA, respectively, and for they are 2.3 and 0.17 (2 degrees of freedom). Also, our J and CH4s flux ratios are consistent with those obtained in 2010 by, with the new J-band data being a significant improvement. Given that the uncertainties in the resolved measurements are likely dominated by systematic errors in deblending the two components of this tight binary, we forgo averaging the 2010 and 2012 measurements and simply choose for each filter the measurement with the smaller error in our analysis (Section 3). Gemini/NIRI Photometry of WISE J1217+1626 ----------------------------------------- We obtained seeing-limited Y- and K-band integrated-light photometry of WISE J1217+1626AB using the facility near-IR camera NIRI on the Gemini-North 8.1-m telescope on Mauna Kea, Hawaii. These observations complement the J- and H-band photometry published in. Our NIRI observations were obtained as queue observing program GN-2012A-DD-2 on 01 April 2012 UT. The seeing was 0.7 FWHM, and the binary was marginally resolved. We obtained five 60-s exposures at Y-band and eighteen 30-s exposures at K-band, taken using a dither pattern with 10 offsets. Data were reduced in a standard fashion and flatfielded with calibration lamps on the telescope. For photometric calibration, we observed the UKIRT faint standard FS 21, using K-band magnitudes from and Y-band magnitudes from the UKIRT online catalog.[1](#fn1) We measured YNIRI = 18.55 ± 0.03 mag and K = 18.80 ± 0.04 mag for the integrated-light system. As discussed in the Appendix, we apply a small shift to transform the NIRI Y-band photometry to the MKO system, giving a final value of YMKO = 18.38 ± 0.04 mag. Keck-II/NIRSPEC Spectroscopy of WISE J1217+1626 ----------------------------------------------- We obtained resolved seeing-limited spectroscopy of WISE J1217+1626AB with the facilty near-IR spectrograph NIRSPEC on the Keck-II Telescope on 12 April 2012 UT. Seeing conditions were excellent, with 0.5–0.7 FWHM in the optical as measured from the DIMM at the nearby Canada-France-Hawaii Telescope, suitable to resolve the two components of the binary. Very light cirrus was present in the sky. We used the low spectral resolution mode with the 0.38 × 42 slit and the NIRSPEC-5 blocking filter, which spans the H-band. The achieved spectral resolution (λ/δλ) was 2300 ± 200 based on measurements of bright isolated sky emission lines. The slit was oriented along the binary PA which did not match the parallactic angle. However, the binary was observed at airmass 1.0–1.1 so systematic errors due to differential chromatic refraction were negligible. We obtained 12 exposures of WISE J1217+1626AB, each with a 300 s integration time, nodding the telescope in an ABBA fashion. Immediately afterwards, we observed the A0V star HD 109055 for telluric calibration and then obtained calibration observations of an internal flat-field source, argon arc lamps, and a dark frame. During the spectroscopic observations of both the science target and A0V calibrator, the NIRSPEC image quality was afflicted by an unusual guiding program, causing the formation of double-peaked PSFs in the spatial direction on the detector. The PSF’s secondary peak was typically about 2/3 the flux of the main peak and in the direction opposite the binary’s B component. Basic image reduction was performed with the REDSPEC reduction package.[2](#fn2) This included bad pixel removal, spatial rectification of the images, the creation of a wavelength solution, pairwise image subtraction, and flat fielding. Residual sky line emission in the pair-subtracted images was removed by subtracting a constant value at each wavelength as measured from a spatial region 10 pixels in width on either side of the target spectrum. Because of the close separation of the binary pair and the unusual double-peaked PSF, we extracted 1-d spectra from the 2-d rectified images using custom IDL routines. We found that modeling the PSF as the sum of two gaussians provided an excellent match to the data. The 2-gaussian PSF of the two binary components differed only in total flux and position, as we fixed the FWHM and the relative amplitude of the 2 gaussians to be the same for each binary component. To determine the PSF characteristics of each image, we fit this 2-gaussian PSF model to the collapsed 1.55-1.60 spatial profile of the individual reduced images using the nonlinear, least-squares curve fitting package MPFIT written in IDL. The fitting results were used to fix all parameters of the PSF except the flux ratio of the binary. This model was then fit to the spatial profile as a function of wavelength, while median-averaging the spectrum in wavelength bins to boost the S/N. We found this model to be an excellent fit to the data, as the subtraction of a 2-d image based on the model fits from the original data produced essentially a pure noise image, without any coherent residuals. The resulting individual spectra for WISE J1217+1626 A and B were computed at each wavelength bin by summing the flux of their respective 2-gaussian PSFs. To construct the final spectra, we combined the individual extracted spectra by first scaling each one to the median-averaged spectrum from 1.55–1.65 . Then at each wavelength we computed the weighted average, with the weights inversely proportional to the FWHM of the PSF model. The purpose of the weighting is to favor data obtained in better seeing conditions. We adopted the standard error of the individual spectra as the measurement uncertainties for the final spectrum. The spectra of the A0V calibrator star were extracted by summing flux in a single fixed-sized aperture and then combined. This was used to correct the spectra of WISE J1217+1626A and B for telluric absorption using the xtellcor\_general routine in the Spextool reduction package for IRTF. To test the sensitivity of the results to the fitting procedure, we experimented with wavelength bin sizes of 5, 10, 15, and 20 pixels (0.0014, 0.0028, 0.0042, 0.0055 ). We also extracted the spectra by allowing the FWHM of each gaussian in the 2-gaussian PSF model vary. The results were all consistent within the spectral measurement uncertainties. We use the 10-pixel binned data for our final analysis, as the resulting sampling gives a resolution (R ≈ 200) compared to other published near-IR spectra of late-T and Y dwarfs. Figure [fig:spectra] shows the reduced spectra of the two components. The peak S/N are about 60 and 15 for the primary and secondary, respectively. As a cross-check on our reduction, we also compare the sum of our spectra with the integrated-light spectrum from and find excellent agreement, as shown in the inset of Figure [fig:spectra]. We also compute an H-band flux ratio from our NIRSPEC spectra of 2.25 ± 0.08 mag, in accord with our Keck LGS imaging (2.20 ± 0.04 mag). IRTF/SpeX Photometry and Spectroscopy of WISE J1711+3500[sec:spex] ------------------------------------------------------------------ To improve upon the modest S/N data for WISE J1711+3500 presented in, we obtained near-IR photometry and low-resolution (R ≈ 100) spectroscopy of WISE J1711+3500. Such data are important for our analysis, since the Keck LGS imaging provides only flux ratios for the two components. Computing the resolved magnitudes requires multi-band integrated-light photometry, which can be synthesized from a near-IR spectrum and photometry in a single band. We used the facilty instrument SpeX at NASA’s Infrared Telescope Facility located on Mauna Kea, Hawaii on 2012 April 20 UT. For imaging, we obtained 9 dithered exposures of WISE J1711+3500 and the UKIRT faint standard FS 27 using the MKO J-band filter. Conditions were photometric with good seeing, 0.6–0.7FWHM in J-band as measured from the reduced images. Data were reduced in a standard fashion, and we used aperture photometry to determine an apparent magnitude of JMKO = 17.59 ± 0.03 mag, with the uncertainty determined from the quadrature sum of the standard error of the fluxes from the individual images of the science target and photometric calibrator. reported J2MASS = 17.89 ± 0.13 mag, which is consistent with our measurement given the ≈ 0.3 mag offset expected for late-T dwarfs between the MKO and 2MASS filter systems. Our near-IR spectrum (described below) finds J2MASS − JMKO = 0.26 mag, consistent with the 0.30 ± 0.13 mag difference between our photometry and that of Kirkpatrick et al. For spectroscopy, we used Spex in prism mode with the 0.8 slit, obtaining 0.8–2.5 spectra in a single order. WISE J1711+3500 was nodded along the slit in an ABBA pattern with individual exposure times of 180 sec for a total exposure time of 48 min and observed over an airmass range of 1.04–1.06. We observed the A0V star GAT 7 contemporaneously for telluric calibration. All spectra were reduced using version 3.4 of the SpeXtool software package. The S/N per pixel in the final reduced spectrum is about 35, 50, 25, and 8 in the YJHK peaks, respectively. Figure [fig:spectra1711] shows that our spectrum agrees well with that of. We use our spectrum and J-band photometry to synthesize the integrated-light magnitudes in the YHK bands on the MKO system (Table [table:wisetwo]). Results [sec:results] ===================== Evidence for Companionship -------------------------- With only one epoch of imaging, it is not possible to determine if the companions seen in the Keck LGS images are co-moving with the our two WISE targets. But the resolved photometry of the systems (and spectra in the case of WISE J1217+1626AB) provides overwhelming circumstantial evidence that the binaries are true physical associations. Specifically, the nearly identical flux ratios in the CH4s and H-band images indicates that the secondary components also have strong methane absorption, with basically no flux redward of the 1.6 flux decrement seen in T dwarfs. The possibility of two unassociated T dwarfs having such a small angular separation is diminishingly small. identified 54 T dwarfs in an area of 700 degs2 to a depth of J = 19.0 mag. Nine of these objects are spectral type T7 or later, where the H-band methane absorption is saturated in medium-band filter photometry (Figure 4 of ). Assuming a uniform space density of late-T dwarfs, this means a surface density of 0.20 late-T dwarfs per degs to a depth of J = 21.0 mag. Within the field of view of the NIRC2 wide and narrow cameras, this corresponds to probabilities of 2.6 × 10− 5 and 1.6 × 10− 6 for an unassociated T dwarf to be next to WISE J1217+1626 and WISE J1711+3500, respectively. This is highly unlikely. Note that this calculation is an a posteriori one based on a discovery of a companion to a individual target. In fact, we need to consider the total number of objects that we have imaged with sensitivity to late-T dwarf companions. Our Keck LGS program to date has targeted about 100 field T dwarfs using the NIRC2 narrow camera. Thus the probability of an unassociated late-T companion being contiguous to any single object in our survey is 1.6 × 10− 4, respectively. However, note that both these “single-object” and “total-survey” calculations overstate the odds of a chance association, since they do not consider that the photometric distance of a background late-T dwarf would not be compatible with those of the science targets. We conclude that both WISE systems are physical binaries. We searched for common proper motion companions within 10$\arcmin$ of WISE J1217+1626, using the proper motion reported in Kirkpatrick et al. (2011). We found no comoving objects in the Hipparcos, Tycho, LSPM-N, or NLTT catalogs. The reported proper motion of WISE J1711+3500 is consistent with zero, obviating a search for comoving companions. Photometric Distances [sec:distance] ------------------------------------ We can estimate photometric distances to our two binaries using the late-T primary components, and then as a result we also know the distances to the T/Y secondary components. To compute a photometric distance for WISE J1217+1626AB, we compare the apparent magnitudes of component A to the six T8.5 and T9 dwarfs with parallaxes summarized in and using the spectral types for the latest T dwarfs. We include the T8.5 dwarfs (1) to compensate for the small sample of only two T9 dwarfs with parallaxes and MKO near-IR photometry (UGPS J0722 − 0540 and CFBDSIR J1458+1013A, which differ from each other by ≈ 0.9 mag in absolute magnitudes) and (2) to encompass the current uncertainties in the spectral typing of the very coolest brown dwarfs. A weighted average of the absolute magnitudes gives M(Y, J, H, K) = 18.74 ± 0.38, 17.88 ± 0.35, 18.21 ± 0.34, 18.42 ± 0.39 mag, where the uncertainty here is the RMS scatter of the six objects. We use the J- and H-band data for the photometric distance, as the Y and K bands are more influenced by variations in metallicity and surface gravity. Both bands give an identical distance modulus of (m − M) = 0.10 ± 0.35 mag (10.5 ± 1.7 pc) for WISE J1217+1626A. In a similar fashion, for WISE J1711+3500AB we use the weighted average of J- and H-band absolute magnitudes for T8 dwarfs, choosing the three objects that are “normal” (not young or low metallicity). These give distance moduli of 1.35 ± 0.35 and 1.41 ± 0.32 mag, respectively. We take the average and adopt the slightly larger RMS for a final distance modulus of 1.38 ± 0.35 mag (19 ± 3 pc) for WISE J1711+3500A. As expected, our photometric distances are larger than the 6.7 pc and 17.0 pc estimates for WISE J1217+1626 and WISE J1711+3500, respectively, given by which were based on integrated-light W2-band photometry assuming a single object. Note that if the large flux ratios we observe in the near-IR ( ≈ 2–3 mag) were also representative of the mid-IR flux ratios, we would expect better agreement between our distances and those of Kirkpatrick et al. In other words, the secondary components in these systems must be relatively brighter in the mid-IR and thus contribute a larger portion of the W2-band flux, as expected based on the very red mid-IR colors of the coolest brown dwarfs. Spectral Types [sec:spectra] ---------------------------- ### WISE J1217+1626AB Our NIRSPEC spectra of both components of WISE J1217+1626AB exhibit the deep and absorption in the H-band that is characteristic of the coolest brown dwarfs. Figure [fig:typing] compares our H-band spectra with the T9 and Y0 spectroscopic standards proposed by. Component A shows excellent agreement with their T9 standard. The spectral type for this component is identical to the integrated-light type, as expected given the large near-IR flux ratios. Component B shows good match with the Y0 standard WISE J1738+2732. In particular, the hallmark of the Y spectral type proposed by is the enhanced absorption at 1.53–1.58 seen in the low-resolution spectra of the coolest WISE discoveries that is distinct from the latest T dwarfs. They attribute this feature to, though a definitive identification is still pending. As seen in Figure [fig:typing], WISE J1217+1626B shows the same enhancd absorption on the blue side as the Y0 standard, and thus we classify it as Y0. The red side of the H-band continuum for WISE J1217+1626B does not appear to be as sharply truncated as for the Y0 standard. However, the other Y0 dwarf from the with reasonable S/N, WISE J1405+5534, also has a similar redward extent, which motivated a classification of “Y0 (pec?)” in their discovery paper. Given the modest S/N of our spectrum, we defer a decision on whether WISE J1217+1626B is spectroscopically peculiar. Followup resolved spectroscopy with broader wavelength coverage will help refine the spectral typing, especially as the width of the J-band continuum appears to decrease with later-type objects. ### WISE J1711+3500AB In the absence of resolved spectroscopy for WISE J1711+3500AB, we consider the flux ratios and estimated absolute magnitudes of the components to estimate their spectral types. The large near-IR flux ratios ( ≈ 2.8 mag) suggests strongly that the T8 integrated-light spectrum is dominated by the primary. This is corroborated with simple numerical experiments combining spectra of late-T and Y0 dwarfs using the observed H-band flux ratio as a constraint. The > 10 × fainter secondary has negligible impact on the near-IR spectrum. Thus we safely classify WISE J1711+3500A as T8. For the secondary, we use its near-IR absolute magnitude as estimated from the photometric distance to the primary. This is obviously an uncertain process, though preliminary parallaxes from suggest that the near-IR absolute magnitudes drop off quickly at the T/Y transition, which lessens the impact of uncertainties in the photometric distance. WISE J1711+3500B has M(H) = 19.58 ± 0.36 mag, which is intermediate between the two T9 and two Y0 dwarfs with parallaxes in Figure 12 of. (Their figure has plotted as the sole T9.5, but as discussed in the next subsection this object is more likely to be typed as Y0.) In terms of its Y − J color and estimated color-magnitude diagram position, WISE J1711+3500B also appears to be intermediate between the T9 and Y0 dwarfs (Section [sec:phot]). Thus we estimate a spectral type of T9.5. ### At the time was discovered to be a binary, the near-IR absolute magnitudes of its secondary component were fainter than the coolest known dwarf, which was classified as T10 at the time. This motivated an initial spectral type estimate of > T10 for the secondary by, with the primary type assumed to be identical to the integrated-light type of T9.5. While the secondary type was unknown given the absence of resolved spectroscopy, the monotonic behavior of near-IR absolute magnitudes among the T dwarfs demonstrated the object was exceptionally low temperature. Since then, there have been several late-T and Y dwarfs identified from the WISE dataset, which has warranted a reclassification of the integrated-light spectrum to T9 by. Given this new state of knowledge, we reexamine here the resolved spectral types.. Based on the latest H-band absolute magnitudes from, estimated types of T8.5 and T9.5 for the two components of. However, closer examination of the data indicates these types are too early. As is the case with WISE J1711+3500A, the large near-IR flux ratios ( ≈ 2 mag) of indicate the T9 integrated-light spectrum is dominated by the primary. This is corroborated by the aforementioned experiments of summing spectra and by the model atmosphere fitting in, which found a blended-light spectrum had negligible impact on the derived physical parameters. As for the secondary, it has M(H) = 19.99 ± 0.23 mag, in good agreement with the two Y0 dwarfs in with parallaxes. This absolute magnitude also agrees well with that estimated for the (spectroscopically typed) Y0 dwarf WISE J1217+1626B, based on our photometric distance for WISE J1217+1626A. Thus we estimate the near-IR spectral types for the two components of are T9 and Y0. Resolved Photometry[sec:phot] ----------------------------- Tables [table:wiseone], [table:wisetwo], and [table:cfbds] summarize the resolved photometry for our binaries, derived from the Keck LGS imaging and the available YJHK integrated-light photometry. Figure [fig:colorcolor] shows the YJH color-color plot comparing our binary sample to the known field T dwarfs. For the field sample, we use the data for 80 T dwarfs compiled by, one late-T dwarf (WISE J1617+1807) from,[3](#fn3) three recent late-T discoveries from the UKIDSS survey, and one Y0 dwarf (WISE J1405+5534) from.[4](#fn4) For the objects originally classified as T8.5–T10 in the Leggett et al. compilation, we use the slightly earlier spectral types of T8–T9 proposed by. We also queried the UKIDSS[5](#fn5) Large Area Survey Data Release 9 (DR9, Table [table:ukidss]) in order to add photometry for five late-T dwarfs from Kirkpatrick et al. and for the T8 dwarf PSO J043.5+02 (a.k.a. WISE J0254+0223, found independently by and ). Finally, we computed YJH synthetic photometry for the six T9–T9.5 and two Y0 dwarfs (WISE J0410+1502 [Y0] and WISE J1541 − 2250 [Y0.5]) in and with sufficently blue coverage in their spectra to include the Y-band. While WISE J1217+1626A, WISE J1711+3500A, and WISE J1711+3500B reside in the loci of the coolest known objects, the unusual nature of WISE J1217+1626B is apparent in the YJH color-color diagram, being distinguished by its very blue Y − J color compared to all previously known T dwarfs. Only the three field Y dwarfs in our sample are comparable. This blueness is also emphasized in the near-IR color-magnitude diagram (Figure [fig:cmd]), where component B is fainter and significantly bluer in Y − J than any T dwarf, based on the photometric distance for the binary. The resolved Y − J colors for show a similarly pronounced blue difference between the primary and secondary component. This is seen most clearly in the relative flux ratios, Δ(Y − J) = 0.43 ± 0.02 mag for WISE J1217+1626AB and 0.47 ± 0.15 mag for. (The integrated-light Y-band photometry for from UKIDSS is only S/N = 5 so comparing the resolved colors of the two components is more noisy than comparing their flux ratios, which depend only on the Keck LGS data.) To further highlight this phenomenon, Figure [fig:yj] shows the Y − J colors of the coolest brown dwarfs as a function of spectral type, based on published photometry and our synthesized photometry (regardless of whether the objects have parallaxes or not). and have noted that the Y − J colors of the latest T dwarfs (T8–T9) are bluer than the earlier-type objects, and found that the Y-band peak of two Y0 dwarfs are relatively brighter compared to the T9 standard. Figure [fig:yj] shows that in fact the blueward trend in Y − J crosses a precipice at the T/Y transition, declining dramatically ( ≈ 0.5 mag) over only half a spectral subclass, from T9.5 to Y0. This is discussed further in Section [sec:discussion]. Physical Properties from Evolutionary Models [sec:properties] ------------------------------------------------------------- We use evolutionary models from and to derive physical properties for the two binaries, assuming ages of 1 and 5 Gyr. As inputs, we use both the J-band absolute magnitude and the estimated. The uncertainties in these quantities are propagated into the calculations in a Monte Carlo fashion. We use J-band as it represents the peak flux of the near-IR spectral energy distribution. When using the J-band absolute magnitudes for our calculations, we are implicitly using the bolometric corrections from the underlying model atmospheres. To compute, we adopt a J-band bolometric correction of 1.75 ± 0.27 mag for WISE J1217+1626A (T9), WISE J1217+1626B (Y0), and WISE J1711+3500B( ≈ T9.5), based on the coolest T dwarfs as described in. This is expected to somewhat underestimate the, given the increasingly large fraction of the luminosity emitted in the mid-IR at colder temperatures. However, given the lack of actual bolometric measurements for objects as cool as WISE J1217+1626B, adopting a fixed value is also conservative. For the hotter WISE J1711+3500A (T8), we adopt J-band bolometric corrections from the four T7.5–T8.5 dwarfs with parallaxes: Gl 570D (T7.5; BCJ = 2.60 ± 0.13 mag), 2MASS J1217 − 0311 (T7.5, 2.64 ± 0.13 mag), and 2MASS J0415 − 0935 (T8; 2.54 ± 0.13 mag) from and Wolf 940B (T8.5, 2.09 ± 0.12 mag) from. The unweighted average and RMS are 2.47 ± 0.26 mag. Tables [table:wiseone] and [table:wisetwo] present the results derived from the evolutionary models. Altogether, the components of WISE J1217+1626AB are remarkably similar to those of, with near-IR absolute magnitudes that differ by only ≈ 0.2 mag; thus, the inferred physical properties of these two systems are very similar. Note that the results from the models should be somewhat preferred over the Lyon/COND models of. The predicted near-IR locus of Burrows et al. is a better match to the location of (Figure 7 of ), though the two sets of models basically give consistent results for the physical properties. Overall, the inferred temperature of WISE J1217+1626B is exceptionally low, 350–470 K depending on the choice of theoretical model and system age. Its inferred mass is 5–20 , overlapping the old gas-giant planets found by radial velocity and transit surveys and the young ones found by direct imaging. As expected based on its brighter estimated absolute magnitudes, we find WISE J1711+3500B is somewhat hotter (420–540 K) and more massive (9–26 ) than WISE J1217+1626A. Both components of WISE J1217+1626AB would reside in the planetary-mass regime and have retained their initial deuterium abundance for ages of $\lesssim$1 Gyr. In fact, the two components of WISE J1217+1626AB may possess different deuterium abundances, as their estimated masses could straddle the deuterium-burning limit (11–14 ; ). For WISE J1711+3500AB, the primary likely lies above the D-burning limit, and the secondary would lie below it for ages of $\lesssim$1 Gyr. Measuring the differential deuterium abundance in these binaries would help constrain their ages, analogous to the binary lithium test proposed by — see for a discussion of the very similar case of. Orbital Periods [sec:orbit] --------------------------- To estimate the orbital periods of the two new WISEbinaries, we adopt the statistical conversion factor between projected separation and true semi-major axis from. They offer several choices, based on the underlying eccentricity distribution and degree of completeness to finding binaries by imaging (“discovery bias”). We adopt the eccentricity distribution from their compilation of ultracool visual binaries with high quality orbits and assume no discovery bias, appropriate for such well-resolved systems. This gives a multiplicative correction factor of 1.15− 0.31+ 0.81 (68.3% confidence limits) for converting the projected separation into semi-major axis (compared to 1.10− 0.35+ 0.92 for a uniform eccentricity distribution with no discovery bias). For ages of 1–5 Gyr, the resulting orbital period estimates range from 120–210 yr and 260–430 yr for the two binaries, albeit with significant uncertainties (Tables [table:wiseone] and [table:wisetwo]). Dynamical masses can be realized from orbit monitoring covering $\gtrsim$30% of the period. Thus, several decades of monitoring will be needed to determine the visual orbits. Orbital motion of is clearly detected in our new data, taken two years after the discovery epoch in. Comparing the data from the filters with the best astrometry in the two epochs, we measure a change in separation of 16 ± 3 mas and in position angle of $8.5\pm1.6{\mbox{$^{\circ}$}}$. Such motion is not surprising, given the ≈ 30 yr orbital period estimated at the time of discovery. Discussion [sec:discussion] =========================== Theoretical models predict the near-IR spectra of ≈ 300–500 K objects to be sensitive to new physical processes. have identified changes in near-IR low-resolution spectra of Y dwarfs that they ascribe to. This molecule may also be present in high-resolution near-IR spectra of . Another anticipated signature is the removal of the very broad wings of the 0.77 doublet that dominate the far-red optical continuum. Down to ≈ 500 K, find that the far-red colors, as tracked by the izYJ filters, continue to be very red with only a hint of saturation. At somewhat cooler temperatures, below 400 K the Cs, K, Li and Rb neutral alkalis are expected to be depleted as they turn into chloride gases and solids, or other halide or hydroxide gases, and Na condenses as Na2S. Given the strength of the absorption wings, depletion of this element should be manifested by a change in the broad-band colors. Figures [fig:colorcolor], [fig:cmd], and [fig:yj] show that the Y − J colors of WISE J1217+1626B and the other two Y dwarfs are significantly bluer than all known T dwarfs, consistent with the signature of such depletion. Indeed, the sharp change in Y − J is reminscent of similar behavior seen in near-IR JHK colors at the L/T transition due the onset of photospheric methane absorption, though the change in Y − J occurs over a small range of spectral subclasses. Based on the photometric distance to its primary, the location of WISE J1217+1626B on the color-magnitude diagram also suggests a temperture low enough for potassium depletion. Figure [fig:cmd] compares the available photometry with updated version of the models from D. Saumon (priv. comm.). These updated models include the latest collision-induced absorption and opacities. However, the models are calculated in chemical equilibrium, which overestimates the strengths of the absorption in the near-IR (which is significant) and the mid-IR assuming non-equilibrium conditions are prevalent, as expected. The cloud-free model sequences show (Y − J) decreasing with decreasing temperature through the T dwarf sequence. The J-band absolute magnitudes of the Y0 dwarfs are comparable to the ≈ 400 K models, though the observed Y − J colors are somewhat bluer than the models. The predictions from have much redder Y − J colors than the color-magnitude data. Note that the Saumon & Marley models include the formation of iron, silicate and corundum clouds (which are important for L dwarfs and the L/T transition) as well as the depletion of the neutral alkalis. However, they do not include any opacity due to sulphide or chloride condensates, such as Na2S and KCl which are formed as Na and K are depleted. While such condensates have been historically ignored in ultracool atmosphere modeling, have found the resulting clouds, especially those of Na2S, significantly affect the near-IR spectra of mid/late-T dwarfs and provide a better match to the JHK data. As shown in Figure [fig:cmd], current cloud-free models agree reasonably well in {Y − J, M(J)}. But the mismatch for the latest-type objects may indicate that other factors in addition to K → KCl depletion play a role in the ≈ 1 fluxes, including non-equilibrium abundances and the opacity of sulfate condensates. Conclusions =========== WISE J1217+1626AB and WISE J1711+3500AB are remarkably well-resolved binaries compared to previously known T dwarf binaries (Figure [fig:binaries]), with only the T1+T6 binary being comparably wide in angular separation (0.73 at discovery; ). The projected physical separations of 8 and 15 AU for the WISE binaries are also remarkable, with only the T2.5+T4.0 binary SIMP J1619275+031350AB being comparable (15 AU separation; ) among T dwarfs. Finally, their mass ratios of ≈ 0.5 make them very rare among field ultracool binaries, where nearly equal-mass systems are most prevalent (e.g., see compilation in ). The fact that these two systems are so unusual in their separation and mass ratios compared to nearly all previously known substellar binaries opens the question of whether these initial discoveries among the late-T dwarfs is a harbinger of a change in binary properties at such cool temperatures. An answer awaits a larger survey of such targets at comparable sensitivity. Overall, WISE J1217+1626AB appears to be essentially a clone of the coolest known binary to date,, except for the difference in the projected separation. The blue Y − J color of its secondary component is unmatched by any previously known T dwarf, including the primary component. Synthetic photometry of field Y0–Y0.5 dwarfs also gives very blue colors, and also shows a much bluer Y − J color than its primary. Examining the complete ensemble of available Y − J color data, we find a sharp drop to the blue between the T9.5 and Y0 dwarfs. We suggest this may be the signature of potassium depletion, which removes a significant opacity source at far-red optical wavelengths and is expected for objects of $\lesssim$400 K. It is a fortuitous coincidence that the change in near-IR spectra that demarcates the T/Y transition (currently suspected to be due to ) also coincides with this change in photospheric chemistry. However, in contrast to the relatively subtle changes in the near-IR spectra between T9 and Y0, the Y − J color drops by ≈ 0.5 mag. This suggests that far-red spectra might be more discerning than near-IR spectra for classification in this temperature regime, as discussed by and. Future followup will be essential for reaping the full scientific value of these new binaries. Near-IR low resolution spectra will readily improve the spectral types. Water clouds are also expected to condense below ≈ 400 K, further motivating detailed spectroscopic characterization to search for their signature. The wide separation of these two binaries means that resolved near-IR and optical spectra can be obtained in good seeing conditions, without the need for AO. The binary nature of these systems also provides a unique avenue for analysis. Among the Y dwarfs, model atmosphere fitting indicates a significant range in temperatures (350–500 K), surface gravities ( > 1 dex range), and masses (3–30 ) that is in contrast to the modest differences in their near-IR spectra. It is unclear whether such dispersion represents intrinsic variations in the physical properties of Y dwarfs, failings in the model atmospheres, or both. For binaries, the two components have a common (albeit unknown) metallicity and very similar surface gravity (Tables [table:wiseone] and [table:wisetwo]). Thus resolved spectra will cleanly probe the effect of decreasing temperature, without the uncertainties arising from second parameter variations (age/gravity or metallicity) present in the free-floating population. In a similar vein, spectrophotometric monitoring of the two components may shed light on the nature of the clouds at the T/Y transition. Finally, the photometric distances of 10–20 pc place WISE J1217+1626AB and WISE J1711+3500AB easily within reach of near-IR parallax measurements, and we have begun such monitoring using the Canada-France-Hawaii Telescope. A direct distance will place the four components in context with other late-T and Y dwarfs, through both the absolute magnitude-spectral type relation and color-magnitude diagrams. Given the unusually small mass ratios of ≈ 0.5 for these systems, placing the components on the H-R diagram will also uniquely test the joint accuracy of current evolutionary and atmospheric models, using the constraint that models must indicate the two components of each binary are coeval. We thank Scott Dahm, Terry Stickel, Jim Lyke, and the Keck Observatory staff for assistance with LGS AO observing; the Gemini Observatory staff for obtaining the NIRI photometry through queue observing; and Kimberly Aller for assistance with the IRTF/SpeX observing. We thank Andrew Stephens for providing the Gemini/NIRI cold filter profile; James R. A. Davenport for distributing his IDL implementation of the “cubehelix” color scheme; Caroline Morley, Jonathan Fortney and collaborators for sharing results in advance of publication; and Didier Saumon and Isabelle Baraffe for providing expanded sets of evolutionary models. This publication makes use of data products from the Wide-Field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Our research has employed the 2MASS data products; NASA’s Astrophysical Data System; the SIMBAD database operated at CDS, Strasbourg, France; and the Spex Prism Spectral Libraries maintained by Adam Burgasser at http://pono.ucsd.edu/ ∼ adam/browndwarfs/spexprism. This research was supported by NSF grant AST09-09222 awarded to MCL. Support for this work to TJD was provided by NASA through Hubble Fellowship grant awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. Finally, the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Facilities: Y-Band Filter Offsets [sec:yband] =================================== The near-IR spectra of T dwarfs are highly structured and include strong absorption bands due to. These bands are proximate to the strong telluric absorption features in the Earth’s atmosphere. The telluric features have served to define the commonly used near-IR photometric bandpasses, though the exact wavelength ranges of the filters vary. have computed photometric shifts between different near-IR filter systems in the ubiquitous JHK bands. The shifts can be quite large ( ≈ 0.4 mag) depending on the particular filters being considered. The Y-band in the ≈ 1 atmospheric window is also subject to such effects. The measurements in this paper involve Y-band photometry from three instruments: Keck/NIRC2 (resolved photometry of three binaries), Gemini/NIRI (integrated-light photometry of WISE J1217+1626AB), and UKIRT/WFCAM (integrated-light photometry of late-T dwarfs from the literature). The three filters are very similiar, but they are distinct (Figure [fig:yband]). The differences are especially significant for the late-T and Y dwarfs as the peak of their continuum in this bandpass falls near the red edge of the filter, so for a given object the Y-band magnitudes will be brightest for the UKIRT/WFCAM filter and faintest for the Keck/NIRC2 filter. To assess the offsets between filters, we synthesize colors in the various Y-band filters using spectra of T8–Y0 dwarfs with sufficient blue wavelength coverage, primarily from and. We compute a Gemini-UKIRT shift of YNIRI − YWFCAM = 0.17 ± 0.03 mag, where the uncertainty is the RMS of the synthesized colors. We apply this small offset to our Gemini/NIRI photometry for a final value of YMKO = 18.38 ± 0.04 mag for WISE J1217+1626AB(Table [table:wiseone]). We assume here that UKIRT/WFCAM serves as the de facto definition for the Y band of the MKO photometric system, given that the UKIDSS project carried out with this instrument has produced the largest set of Y-band photometry by far. Note that applying this shift does not affect one of our main results, namely the very blue (Y − J) color of WISE J1217+1626B compared to previously known objects, since the secondary is unusually blue relative to both the field objects and its T9 primary even if we apply no shift at all. We also compute a Keck-UKIRT shift of YNIRC2 − YWFCAM = 0.27 ± 0.03 mag. Figure [fig:yband] indicates that for Y0 dwarfs, the WFCAM flux should be systematically different than for the Keck or Gemini fluxes given the stronger water absorption cutoff in the red edge of the bandpass for later-type objects, but no convincing offset was detected between the two Y dwarfs (WISE J0410+1502 and WISE J1541 − 2250) and the T8–T9.5 dwarfs in our compilation, especially given the low S/N spectra of the latest-type objects. Therefore, we assume the color offset has no spectral type dependence, and hence the flux ratios between the binary components in the YNIRC2 and YWFCAM filters are assumed to be the same. If there is such a dependence, Figure [fig:yband] indicates that shifting from Keck/NIRC2 to UKIRT/WFCAM would make WISE J1217+1626B and even bluer relative to their primaries. Finally, we synthesize the offsets between these Y-band filters over the entire T spectral class by adding the low-resolution spectra from the SpeX Prism Library. Figure [fig:yoffset] shows the results. Robust linear fitting to the synthetic photometry gives the following relations: $$Y\_{NIRI} - Y\_{WFCAM} = 0.090 + 0.010 \times (Tsubclass) \qquad\sigma = 0.015~{\rm mag}$$ $$Y\_{NIRC2} - Y\_{WFCAM} = 0.133 + 0.017 \times (Tsubclass) \qquad\sigma = 0.019~{\rm mag}$$ $$Y\_{NIRC2} - Y\_{NIRI} = 0.042 + 0.007 \times (Tsubclass) \qquad\sigma = 0.006~{\rm mag}$$ where Tsubclass is 0 for T0, 1 for T1, 10 for Y0, etc. and σ gives the RMS about the fitted relation. The fit is valid for T0 to Y0.5. 73 natexlab#1#1 Artigau, É., et al. 2011,, 739, 48 Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. 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L., et al. 2006,, 118, 297 5ex 5ex -1in 2ex -1in -1in 2ex -1in -2ex -1in -2ex -3ex Integrated-light spectral type versus J-band flux ratio (a proxy for mass and temperature ratio) for known mid/late-T dwarf binaries, based on the compilation of and more recent discoveries by and. (For WISE J0458+63AB and, we use the revised spectral types proposed by of T8.5 and T9, respectively.) The plotting symbol sizes and colors represent the projected separation of the binaries. The circles represent ground-based photometry, either on the 2MASS or MKO system, and the squares represent objects with HST/NICMOS F110W photometry. Only CFBDSIR 1458+10AB has comparably cold components to WISE J1217+1626AB and WISE J1711+3500AB, but it has a much tighter projected separation (0.11).[fig:binaries] -2ex 1ex lcccccccc 2012-Jan-29 & YNIRC2 & 6 × 180.0 & 1.03 & 130 ± 5 & 0.042 ± 0.003 & 758.2 ± 1.4 (1.2) & 14.50 ± 0.13 (0.10) & 1.666 ± 0.011 & z1.1 & 6 × 300.0 & 1.12 & 144 ± 9 & 0.030 ± 0.006 & 759.2 ± 3.3 (3.2) & 14.23 ± 0.26 (0.25) & 1.69 ± 0.05 & J & 7 × 60.0 & 1.01 & 115 ± 6 & 0.069 ± 0.007 & 757.1 ± 1.6 (1.5) & 14.32 ± 0.10 (0.03) & 2.10 ± 0.02 & CH4s & 7 × 60.0 & 1.00 & 114 ± 6 & 0.087 ± 0.012 & 758.5 ± 1.9 (1.7) & 14.43 ± 0.22 (0.20) & 2.15 ± 0.04 & H & 7 × 60.0 & 1.01 & 119 ± 10 & 0.082 ± 0.017 & 757.3 ± 2.5 (2.4) & 14.37 ± 0.29 (0.27) & 2.20 ± 0.04 & K & 6 × 180.0 & 1.06 & 109 ± 3 & 0.17 ± 0.04 & 760.7 ± 6.2 (6.2) & 14.27 ± 0.26 (0.24) & 2.16 ± 0.13 2012-Apr-12 & YNIRC2 & 8 × 180.0 & 1.05 & 74 ± 7 & 0.025 ± 0.006 & 777.2 ± 8.1 (8.0) & 328.74 ± 0.20 (0.19) & 2.71 ± 0.11 & J & 7 × 180.0 & 1.04 & 61 ± 6 & 0.059 ± 0.004 & 779.6 ± 1.1 (1.0) & 328.40 ± 0.08 (0.07) & 2.83 ± 0.06 & CH4s & 8 × 120.0 & 1.04 & 55 ± 2 & 0.137 ± 0.018 & 779.8 ± 2.7 (2.6) & 328.51 ± 0.19 (0.19) & 2.63 ± 0.11 & H & 8 × 120.0 & 1.04 & 56 ± 5 & 0.125 ± 0.006 & 780.0 ± 2.0 (1.9) & 328.43 ± 0.23 (0.23) & 2.83 ± 0.09 2012-Apr-13 & K & 5 × 180.0 & 1.04 & 60.0 ± 1.5 & 0.387 ± 0.107 & 780.7 ± 2.5 (2.4) & 328.49 ± 0.09 (0.09) & 3.08 ± 0.16 2012-Apr-13 & YNIRC2 & 5 × 180.0 & 1.07 & 65 ± 6 & 0.057 ± 0.040 & 125.6 ± 1.8 (1.8) & 317.4 ± 1.3(1.3) & 1.55 ± 0.14 & J & 3 × 180.0 & 1.03 & 73 ± 2 & 0.061 ± 0.008 & 127.2 ± 1.4 (1.4) & 318.1 ± 1.1(1.1) & 2.02 ± 0.07 & CH4s & 7 × 180.0 & 1.12 & 70 ± 6 & 0.178 ± 0.059 & 123.3 ± 2.2 (2.2) & 317.8 ± 0.8(0.8) & 1.86 ± 0.13 lcc -0.5in Photometric distance (pc) & Projected separation (AU) & YMKO (mag) & JMKO (mag) & HMKO (mag) & KMKO (mag) & Near-IR spectral type & T9 ± 0.5 & Y0 ± 0.5 YMKO (mag) & 18.59 ± 0.04 & 20.26 ± 0.04 JMKO (mag) & 17.98 ± 0.02 & 20.08 ± 0.03 HMKO (mag) & 18.31 ± 0.05 & 20.51 ± 0.06 KMKO (mag) & 18.94 ± 0.04 & 21.10 ± 0.12 (Y − J)MKO (mag) & 0.62 ± 0.04 & 0.18 ± 0.05 (J − H)MKO (mag) & − 0.34 ± 0.05 & − 0.44 ± 0.07 (H − K)MKO (mag) & − 0.62 ± 0.07 & − 0.58 ± 0.14 (J − K)MKO (mag) & − 0.96 ± 0.05 & − 1.02 ± 0.12 M(YMKO) (mag) & 18.49 ± 0.35 & 20.16 ± 0.35 M(JMKO) (mag) & 17.88 ± 0.35 & 19.98 ± 0.35 M(HMKO) (mag) & 18.21 ± 0.35 & 20.41 ± 0.36 M(KMKO) (mag) & 18.84 ± 0.35 & 21.00 ± 0.37 log(Lbol/L⊙) [from J band] & − 5.95 ± 0.18 & − 6.79 ± 0.18 log(Lbol, B/Lbol, A) [from J band] & Mass (MJup) & 11.5 ± 1.1, 29 ± 3 & 7.4 ± 0.5, 18.4 ± 1.0 q ( ≡ MB/MA) & Teff(K) & 490 ± 30, 530 ± 30 & 381 ± 13, 402 ± 11 log(g) (cgs) & 4.39 ± 0.03, 4.95 ± 0.05 & 4.18 ± 0.03, 4.68 ± 0.03 Orbital period (yr) & Mass (MJup) & 14.4 ± 1.8, 35 ± 3 & 8.3 ± 0.9, 20 ± 2 q ( ≡ MB/MA) & Teff(K) & 610 ± 40, 660 ± 40 & 430 ± 30, 470 ± 30 log(g) (cgs) & 4.54 ± 0.07, 5.07 ± 0.05 & 4.27 ± 0.05, 4.77 ± 0.05 Orbital period (yr) & Mass (MJup) & 13 ± 3, 33 ± 5 & 5.5 ± 1.2, 13 ± 3 q ( ≡ MB/MA) & Teff(K) & 580 ± 70, 630 ± 70 & 350 ± 40, 370 ± 50 log(g) (cgs) & 4.47 ± 0.10, 5.04 ± 0.09 & 4.07 ± 0.10, 4.54 ± 0.11 Orbital period (yr) & lcc Photometric distance (pc) & Projected separation (AU) & YMKO (mag) & JMKO (mag) & HMKO (mag) & KMKO (mag) & Near-IR spectral type & T8 ± 0.5 & (T9.5 ± 0.5) YMKO (mag) & 18.60 ± 0.03 & 21.31 ± 0.11 JMKO (mag) & 17.67 ± 0.03 & 20.50 ± 0.06 HMKO (mag) & 18.13 ± 0.03 & 20.96 ± 0.09 KMKO (mag) & 18.30 ± 0.03 & 21.38 ± 0.15 (Y − J)MKO (mag) & 0.93 ± 0.04 & 0.81 ± 0.12 (J − H)MKO (mag) & − 0.46 ± 0.04 & − 0.46 ± 0.11 (H − K)MKO (mag) & − 0.17 ± 0.04 & − 0.42 ± 0.18 (J − K)MKO (mag) & − 0.63 ± 0.04 & − 0.88 ± 0.17 M(YMKO) (mag) & 17.22 ± 0.35 & 19.93 ± 0.37 M(JMKO) (mag) & 16.29 ± 0.35 & 19.12 ± 0.36 M(HMKO) (mag) & 16.75 ± 0.35 & 19.58 ± 0.36 M(KMKO) (mag) & 16.92 ± 0.35 & 20.00 ± 0.38 log(Lbol/L⊙) [from J band] & − 5.60 ± 0.17 & − 6.45 ± 0.18 log(Lbol, B/Lbol, A) [from J band] & Mass (MJup) & 19 ± 3, 44 ± 4 & 8.7 ± 0.8, 22 ± 2 q ( ≡ MB/MA) & Teff(K) & 675 ± 50, 680 ± 40 & 420 ± 20, 440 ± 30 log(g) (cgs) & 4.65 ± 0.07, 5.18 ± 0.05 & 4.26 ± 0.03, 4.76 ± 0.05 Orbital period (yr) & Mass (MJup) & 23 ± 2, 48 ± 3 & 10.7 ± 0.9, 26 ± 2 q ( ≡ MB/MA) & Teff(K) & 810 ± 50, 870 ± 60 & 500 ± 30, 540 ± 30 log(g) (cgs) & 4.78 ± 0.05, 5.27 ± 0.04 & 4.39 ± 0.04, 4.90 ± 0.05 Orbital period (yr) & Mass (MJup) & 19 ± 3, 44 ± 6 & 8.1 ± 1.6, 20 ± 4 q ( ≡ MB/MA) & Teff(K) & 730 ± 80, 800 ± 90 & 430 ± 50, 460 ± 50 log(g) (cgs) & 4.70 ± 0.09, 5.21 ± 0.08 & 4.26 ± 0.10, 4.76 ± 0.10 Orbital period (yr) & lcc Distance (pc) & Projected separation (AU) & YMKO (mag) & JMKO (mag) & HMKO (mag) & KMKO (mag) & Near-IR spectral type & T9 ± 0.5 & (Y0 ± 0.5) YMKO (mag) & 20.81 ± 0.21 & 22.36 ± 0.24 JMKO (mag) & 19.83 ± 0.02 & 21.85 ± 0.06 HMKO (mag) & 20.18 ± 0.10 & 22.51 ± 0.16 KMKO (mag) & 20.63 ± 0.24 & 22.83 ± 0.30 (Y − J)MKO (mag) & 0.99 ± 0.21 & 0.52 ± 0.25 (J − H)MKO (mag) & − 0.35 ± 0.10 & − 0.66 ± 0.17 (H − K)MKO (mag) & − 0.45 ± 0.26 & − 0.32 ± 0.34 (J − K)MKO (mag) & − 0.81 ± 0.24 & − 0.99 ± 0.30 M(YMKO) (mag) & 18.29 ± 0.27 & 19.84 ± 0.29 M(JMKO) (mag) & 17.31 ± 0.17 & 19.33 ± 0.18 M(HMKO) (mag) & 17.66 ± 0.20 & 19.99 ± 0.23 M(KMKO) (mag) & 18.11 ± 0.30 & 20.31 ± 0.34 log(Lbol/L⊙) [from J band] & − 5.72 ± 0.13 & − 6.53 ± 0.13 log(Lbol, B/Lbol, A) [from J band] & lcccccc WISE J0049+0441 & 1 & L9 & 16.903 ± 0.012 & 15.767 ± 0.007 & 14.801 ± 0.006 & 14.131 ± 0.005 WISE J0254+0223 & 2,3 & T8 & 16.999 ± 0.014 & 15.916 ± 0.008 & 16.29 ± 0.02 & 16.73 ± 0.05 WISE J0750+2725 & 1 & T8.5 & 19.75 ± 0.09 & 18.73 ± 0.05 & 19.00 ± 0.06 & … WISE J0929+0409 & 1 & T6.5 & 18.00 ± 0.02 & 16.868 ± 0.014 & 17.37 ± 0.07 & 17.40 ± 0.09 WISE J1311+0122 & 1 & T9 & 19.89 ± 0.10 & 18.97 ± 0.08 & …& … WISE J2226+0440 & 1 & T8 & 18.04 ± 0.03 & 16.899 ± 0.019 & 17.45 ± 0.07 & 17.24 ± 0.09 WISE J2344+1034 & 1 & T9 & 19.88 ± 0.12 & 18.84 ± 0.09 & 19.24 ± 0.29 & … --- 1. http://www.jach.hawaii.edu/UKIRT/astronomy/calib/phot\_cal/fs\_ZY\_MKO\_wfcam.dat[↩](#fnref1) 2. http://www2.keck.hawaii.edu/inst/nirspec/redspec[↩](#fnref2) 3. The T9 dwarf WISE J1614+1739 in Kirkpatrick et al. also has YJH photometry but the resulting colors are unusual, with Y − J = 0.35 ± 0.12 and J − H = 0.61 ± 0.22 mag, which place it far from the color locus of all other T dwarfs. (The object would lie off the boundaries of Figure [fig:colorcolor] to the upper left.). However, the near-IR spectrum appears similar to other late-T dwarfs, and synthetic photometry of the spectrum gives Y − J = 0.77 mag and J − H = − 0.26 mag. Thus the photometry seems erroneous, so we exclude it.[↩](#fnref3) 4. The Morley et al. Y-band photometry was obtained with Gemini/NIRI, so we apply the shift computed in the Appendix to transform it to the MKO system.[↩](#fnref4) 5. UKIDSS (UKIRT Infrared Deep Sky Survey) is described in. UKIDSS uses the UKIRT Wide Field Camera (WFCAM; ), and a photometric system described in. The pipeline processing and science archive are described in Dye et al. (2006) and.[↩](#fnref5) Two Extraordinary Substellar Binaries at the T/Y Transition andthe Y-Band Fluxes of the Coolest Brown Dwarfs ============================================================================================================== Using Keck laser guide star adaptive optics imaging, we have found that the T9 dwarf WISE J1217+1626 and T8 dwarf WISE J1711+3500 are exceptional binaries, with unusually wide separations ( ≈ 0.8, 8–15 AU), large near-IR flux ratios ( ≈ 2–3 mags), and small mass ratios ( ≈ 0.5) compared to previously known field ultracool binaries. Keck/NIRSPEC H-band spectra give a spectral type of Y0 for WISE J1217+1626B, and photometric estimates suggest T9.5 for WISE J1711+3500B. The WISE J1217+1626AB system is very similar to the T9+Y0 binary ; these two systems are the coldest known substellar multiples, having secondary components of ≈ 400 K and being planetary-mass binaries if their ages are $\lesssim$1 Gyr. Both WISE J1217+1626B and have strikingly blue Y − J colors compared to previously known T dwarfs, including their T9 primaries. Combining all available data, we find that Y − J color drops precipitously between the very latest T dwarfs and the Y dwarfs. The fact that this is seen in (coeval, mono-metallicity) binaries demonstrates that the color drop arises from a change in temperature, not surface gravity or metallicity variations among the field population. Thus, the T/Y transition established by near-IR spectra coincides with a significant change in the ≈ 1 fluxes of ultracool photospheres. One explanation is the depletion of potassium, whose broad absorption wings dominate the far-red optical spectra of T dwarfs. This large color change suggests that far-red data may be valuable for classifying objects of $\lesssim$500 K. Introduction ============ Binaries have played a central role in the study of the coolest dwarfs over the last two decades. Each advance that has led to creation of a later spectral type has benefitted from discovery of companions to higher mass objects. The first L dwarf, GD 165B, was found as a companion to a white dwarf. At the time, its unusual optical spectrum compared to late-M dwarfs was puzzling, but with the subsequent discovery of more L dwarfs, it became apparent that GD 165B was the prototype of a new spectral type. The first T dwarf, Gl 229B, was found around an M dwarf, and its uniqueness was evident from the strong and in its near-IR spectrum. Just last year, two low-mass companions were found with temperatures of only ≈ 300–400 K, one being a tight (3 AU) companion to the T9.5 dwarf and the other being a very wide (2500 AU) companion to the white dwarf GJ 3483. Spectroscopy of these objects has not been possible yet, due to the small angular separation (0.11) of and the lack of a near-IR detection for GJ 3483B (a.k.a. WD 0806 − 661B). However, their exceptionally faint absolute magnitudes showed these two objects were novel compared to all previously known T dwarfs. Their temperatures, as inferred from their absolute magnitudes using evolutionary models, are far cooler than the previous record-holder, the ≈ 520 K T10 dwarf, and place the two companions in terra incognita where theoretical models predict the onset of new photospheric signatures including absorption in the near-IR, the disappearance of neutral alkali lines, and the formation of water clouds. More recently, have identified five free-floating objects with sufficiently distinct near-IR spectra to propose classification as Y dwarfs, based on the likely presence of absorption in the blue wing of their H-band continuua. While reliable parallaxes are not yet available for these objects and thus their absolute magnitudes are uncertain, model atmosphere fits indicate temperatures of ≈ 300–500 K. Their novel near-IR appearance may be a precursor to even more significant changes along the remaining temperature gap down to Jupiter (124 K; ). As part of our ongoing effort to study low-temperature objects using substellar binaries, we present here the discovery of two extraordinarily well-resolved binaries that shed light on the transition from the latest T dwarfs to the Y dwarfs. The primaries, WISEPC J121756.91+162640.2 and WISEPA J171104.60+350036.8 (hereinafter WISE J1217+1626 and WISE J1711+3500), were found from a mid-IR search of the solar neighborhood by, who asssigned integrated-light near-IR spectral types of T9 and T8, respectively. We also present improved near-IR imaging of. Our characterization of these three binaries includes resolved photometry in the Y band, which spans the ≈ 1.0 transmission window of the Earth’s atmosphere. A broadband filter in this window is a new capability for adaptive optics imaging at Keck, just installed in September 2011. One of the prime motivating factors for its procurement, as a collaboration between one of us (M. Liu) and Keck Observatory, is to explore the ≈ 1 fluxes of the coolest substellar objects, which can be diagnostic of surface gravity, metallicity, and photospheric chemistry. Observations [sec:observations] =============================== Keck-II/NIRC2 Adaptive Optics Imaging ------------------------------------- We imaged WISE J1217+1626, WISE J1711+3500, and over 3 nights in January and April 2012 using the laser guide star adaptive optics (LGS AO) system of the 10-meter Keck II Telescope on Mauna Kea, Hawaii. For WISE J1217+1626, we used the facility IR camera NIRC2 with its wide field-of-view optics ($40.8\arcsec\times40.8\arcsec$), and for WISE J1711+3500 and we used the narrow field-of-view optics ($10.2\arcsec\times10.2\arcsec$). Conditions were photometric, with 0.8 seeing in January and 0.4–0.6 seeing in April. The LGS provided the wavefront reference source for AO correction, with the tip-tilt motion measured simultaneously using R = 15.5, 14.4, and 15.5 mag stars from the USNO-B1.0 catalog located 38, 43, and 55away from WISE J1217+1626, WISE J1711+3500, and, respectively. The LGS brightness, as measured by the flux incident on the AO wavefront sensor, was equivalent to a V ≈ 9.5 − 10.0 mag star. The LGS was pointed at the center of the NIRC2 field-of-view for all observations. We obtained images with all or a subset of the broad-band Y (1.02 ), J (1.25 ), H (1.64 ), and K (2.20 ) filters of the Mauna Kea Observatories (MKO) photometric system. We also obtained data with the medium-band CH4s filter (central wavelength of 1.592 , 0.126 wide), which is positioned around the H-band peak in the spectra of T dwarfs. Finally, for WISE J1217+1626, we obtained data with NIRC2’s custom “z” filter (central wavelength of 1.031 , 0.048 wide), which covers the cleanest portion of the Y-band atmospheric window. To avoid confusion with optical filters using the same letter but spanning different wavelengths, we refer to this as the z1.1 filter. For each filter, we obtained a set of dithered images, offsetting the telescope by a few arcseconds between every pair of images. The images were analyzed in the same fashion as our previous work. The raw images were reduced using standard methods for flat-fielding and sky-subtraction. The binary’s flux ratios and relative astrometry were derived by fitting an analytic model of the point spread function as the sum of three elliptical gaussians. For the very well-resolved WISE binaries, we fitted all the individual images and adopted the averages of the results as the final measurements and the standard deviations as the errors. (The two WISE binaries are so wide that simple aperture photometry gave essentially the same results, as the contamination of the secondary by the primary is negligible.) For, we fitted the stacked mosaic to maximize the S/N of the faint, close secondary and derived uncertainties from Monte Carlo simulations, as we did in. For NIRC2’s wide camera, we adopted a pixel scale of 39.884 ± 0.039 mas/pixel and an orientation for the detector’s + y axis of 0.16 ± 0.09. For the narrow camera, we adopted a pixel scale of 9.963 ± 0.005 mas/pixel and an orientation for the detector’s + y axis of 0.13 ± 0.07. The relative astrometry was corrected for (the very small) instrumental optical distortion based on a solution by B. Cameron (priv. comm.). Table [table:keck] presents our final measurements, and Figure [fig:images] shows our reduced images. For all three binaries, the astrometry from the different filters shows excellent agreement. For WISE J1217+1626, our measurements have values of 0.89 and 3.46 (5 degrees of freedom) for the separation and PA, respectively, when compared to the hypothesis of a constant value for each quantity. For WISE J1711+3500, the values are 0.2 and 3.10 (4 degrees of freedom) for separation and PA, respectively, and for they are 2.3 and 0.17 (2 degrees of freedom). Also, our J and CH4s flux ratios are consistent with those obtained in 2010 by, with the new J-band data being a significant improvement. Given that the uncertainties in the resolved measurements are likely dominated by systematic errors in deblending the two components of this tight binary, we forgo averaging the 2010 and 2012 measurements and simply choose for each filter the measurement with the smaller error in our analysis (Section 3). Gemini/NIRI Photometry of WISE J1217+1626 ----------------------------------------- We obtained seeing-limited Y- and K-band integrated-light photometry of WISE J1217+1626AB using the facility near-IR camera NIRI on the Gemini-North 8.1-m telescope on Mauna Kea, Hawaii. These observations complement the J- and H-band photometry published in. Our NIRI observations were obtained as queue observing program GN-2012A-DD-2 on 01 April 2012 UT. The seeing was 0.7 FWHM, and the binary was marginally resolved. We obtained five 60-s exposures at Y-band and eighteen 30-s exposures at K-band, taken using a dither pattern with 10 offsets. Data were reduced in a standard fashion and flatfielded with calibration lamps on the telescope. For photometric calibration, we observed the UKIRT faint standard FS 21, using K-band magnitudes from and Y-band magnitudes from the UKIRT online catalog.[1](#fn1) We measured YNIRI = 18.55 ± 0.03 mag and K = 18.80 ± 0.04 mag for the integrated-light system. As discussed in the Appendix, we apply a small shift to transform the NIRI Y-band photometry to the MKO system, giving a final value of YMKO = 18.38 ± 0.04 mag. Keck-II/NIRSPEC Spectroscopy of WISE J1217+1626 ----------------------------------------------- We obtained resolved seeing-limited spectroscopy of WISE J1217+1626AB with the facilty near-IR spectrograph NIRSPEC on the Keck-II Telescope on 12 April 2012 UT. Seeing conditions were excellent, with 0.5–0.7 FWHM in the optical as measured from the DIMM at the nearby Canada-France-Hawaii Telescope, suitable to resolve the two components of the binary. Very light cirrus was present in the sky. We used the low spectral resolution mode with the 0.38 × 42 slit and the NIRSPEC-5 blocking filter, which spans the H-band. The achieved spectral resolution (λ/δλ) was 2300 ± 200 based on measurements of bright isolated sky emission lines. The slit was oriented along the binary PA which did not match the parallactic angle. However, the binary was observed at airmass 1.0–1.1 so systematic errors due to differential chromatic refraction were negligible. We obtained 12 exposures of WISE J1217+1626AB, each with a 300 s integration time, nodding the telescope in an ABBA fashion. Immediately afterwards, we observed the A0V star HD 109055 for telluric calibration and then obtained calibration observations of an internal flat-field source, argon arc lamps, and a dark frame. During the spectroscopic observations of both the science target and A0V calibrator, the NIRSPEC image quality was afflicted by an unusual guiding program, causing the formation of double-peaked PSFs in the spatial direction on the detector. The PSF’s secondary peak was typically about 2/3 the flux of the main peak and in the direction opposite the binary’s B component. Basic image reduction was performed with the REDSPEC reduction package.[2](#fn2) This included bad pixel removal, spatial rectification of the images, the creation of a wavelength solution, pairwise image subtraction, and flat fielding. Residual sky line emission in the pair-subtracted images was removed by subtracting a constant value at each wavelength as measured from a spatial region 10 pixels in width on either side of the target spectrum. Because of the close separation of the binary pair and the unusual double-peaked PSF, we extracted 1-d spectra from the 2-d rectified images using custom IDL routines. We found that modeling the PSF as the sum of two gaussians provided an excellent match to the data. The 2-gaussian PSF of the two binary components differed only in total flux and position, as we fixed the FWHM and the relative amplitude of the 2 gaussians to be the same for each binary component. To determine the PSF characteristics of each image, we fit this 2-gaussian PSF model to the collapsed 1.55-1.60 spatial profile of the individual reduced images using the nonlinear, least-squares curve fitting package MPFIT written in IDL. The fitting results were used to fix all parameters of the PSF except the flux ratio of the binary. This model was then fit to the spatial profile as a function of wavelength, while median-averaging the spectrum in wavelength bins to boost the S/N. We found this model to be an excellent fit to the data, as the subtraction of a 2-d image based on the model fits from the original data produced essentially a pure noise image, without any coherent residuals. The resulting individual spectra for WISE J1217+1626 A and B were computed at each wavelength bin by summing the flux of their respective 2-gaussian PSFs. To construct the final spectra, we combined the individual extracted spectra by first scaling each one to the median-averaged spectrum from 1.55–1.65 . Then at each wavelength we computed the weighted average, with the weights inversely proportional to the FWHM of the PSF model. The purpose of the weighting is to favor data obtained in better seeing conditions. We adopted the standard error of the individual spectra as the measurement uncertainties for the final spectrum. The spectra of the A0V calibrator star were extracted by summing flux in a single fixed-sized aperture and then combined. This was used to correct the spectra of WISE J1217+1626A and B for telluric absorption using the xtellcor\_general routine in the Spextool reduction package for IRTF. To test the sensitivity of the results to the fitting procedure, we experimented with wavelength bin sizes of 5, 10, 15, and 20 pixels (0.0014, 0.0028, 0.0042, 0.0055 ). We also extracted the spectra by allowing the FWHM of each gaussian in the 2-gaussian PSF model vary. The results were all consistent within the spectral measurement uncertainties. We use the 10-pixel binned data for our final analysis, as the resulting sampling gives a resolution (R ≈ 200) compared to other published near-IR spectra of late-T and Y dwarfs. Figure [fig:spectra] shows the reduced spectra of the two components. The peak S/N are about 60 and 15 for the primary and secondary, respectively. As a cross-check on our reduction, we also compare the sum of our spectra with the integrated-light spectrum from and find excellent agreement, as shown in the inset of Figure [fig:spectra]. We also compute an H-band flux ratio from our NIRSPEC spectra of 2.25 ± 0.08 mag, in accord with our Keck LGS imaging (2.20 ± 0.04 mag). IRTF/SpeX Photometry and Spectroscopy of WISE J1711+3500[sec:spex] ------------------------------------------------------------------ To improve upon the modest S/N data for WISE J1711+3500 presented in, we obtained near-IR photometry and low-resolution (R ≈ 100) spectroscopy of WISE J1711+3500. Such data are important for our analysis, since the Keck LGS imaging provides only flux ratios for the two components. Computing the resolved magnitudes requires multi-band integrated-light photometry, which can be synthesized from a near-IR spectrum and photometry in a single band. We used the facilty instrument SpeX at NASA’s Infrared Telescope Facility located on Mauna Kea, Hawaii on 2012 April 20 UT. For imaging, we obtained 9 dithered exposures of WISE J1711+3500 and the UKIRT faint standard FS 27 using the MKO J-band filter. Conditions were photometric with good seeing, 0.6–0.7FWHM in J-band as measured from the reduced images. Data were reduced in a standard fashion, and we used aperture photometry to determine an apparent magnitude of JMKO = 17.59 ± 0.03 mag, with the uncertainty determined from the quadrature sum of the standard error of the fluxes from the individual images of the science target and photometric calibrator. reported J2MASS = 17.89 ± 0.13 mag, which is consistent with our measurement given the ≈ 0.3 mag offset expected for late-T dwarfs between the MKO and 2MASS filter systems. Our near-IR spectrum (described below) finds J2MASS − JMKO = 0.26 mag, consistent with the 0.30 ± 0.13 mag difference between our photometry and that of Kirkpatrick et al. For spectroscopy, we used Spex in prism mode with the 0.8 slit, obtaining 0.8–2.5 spectra in a single order. WISE J1711+3500 was nodded along the slit in an ABBA pattern with individual exposure times of 180 sec for a total exposure time of 48 min and observed over an airmass range of 1.04–1.06. We observed the A0V star GAT 7 contemporaneously for telluric calibration. All spectra were reduced using version 3.4 of the SpeXtool software package. The S/N per pixel in the final reduced spectrum is about 35, 50, 25, and 8 in the YJHK peaks, respectively. Figure [fig:spectra1711] shows that our spectrum agrees well with that of. We use our spectrum and J-band photometry to synthesize the integrated-light magnitudes in the YHK bands on the MKO system (Table [table:wisetwo]). Results [sec:results] ===================== Evidence for Companionship -------------------------- With only one epoch of imaging, it is not possible to determine if the companions seen in the Keck LGS images are co-moving with the our two WISE targets. But the resolved photometry of the systems (and spectra in the case of WISE J1217+1626AB) provides overwhelming circumstantial evidence that the binaries are true physical associations. Specifically, the nearly identical flux ratios in the CH4s and H-band images indicates that the secondary components also have strong methane absorption, with basically no flux redward of the 1.6 flux decrement seen in T dwarfs. The possibility of two unassociated T dwarfs having such a small angular separation is diminishingly small. identified 54 T dwarfs in an area of 700 degs2 to a depth of J = 19.0 mag. Nine of these objects are spectral type T7 or later, where the H-band methane absorption is saturated in medium-band filter photometry (Figure 4 of ). Assuming a uniform space density of late-T dwarfs, this means a surface density of 0.20 late-T dwarfs per degs to a depth of J = 21.0 mag. Within the field of view of the NIRC2 wide and narrow cameras, this corresponds to probabilities of 2.6 × 10− 5 and 1.6 × 10− 6 for an unassociated T dwarf to be next to WISE J1217+1626 and WISE J1711+3500, respectively. This is highly unlikely. Note that this calculation is an a posteriori one based on a discovery of a companion to a individual target. In fact, we need to consider the total number of objects that we have imaged with sensitivity to late-T dwarf companions. Our Keck LGS program to date has targeted about 100 field T dwarfs using the NIRC2 narrow camera. Thus the probability of an unassociated late-T companion being contiguous to any single object in our survey is 1.6 × 10− 4, respectively. However, note that both these “single-object” and “total-survey” calculations overstate the odds of a chance association, since they do not consider that the photometric distance of a background late-T dwarf would not be compatible with those of the science targets. We conclude that both WISE systems are physical binaries. We searched for common proper motion companions within 10$\arcmin$ of WISE J1217+1626, using the proper motion reported in Kirkpatrick et al. (2011). We found no comoving objects in the Hipparcos, Tycho, LSPM-N, or NLTT catalogs. The reported proper motion of WISE J1711+3500 is consistent with zero, obviating a search for comoving companions. Photometric Distances [sec:distance] ------------------------------------ We can estimate photometric distances to our two binaries using the late-T primary components, and then as a result we also know the distances to the T/Y secondary components. To compute a photometric distance for WISE J1217+1626AB, we compare the apparent magnitudes of component A to the six T8.5 and T9 dwarfs with parallaxes summarized in and using the spectral types for the latest T dwarfs. We include the T8.5 dwarfs (1) to compensate for the small sample of only two T9 dwarfs with parallaxes and MKO near-IR photometry (UGPS J0722 − 0540 and CFBDSIR J1458+1013A, which differ from each other by ≈ 0.9 mag in absolute magnitudes) and (2) to encompass the current uncertainties in the spectral typing of the very coolest brown dwarfs. A weighted average of the absolute magnitudes gives M(Y, J, H, K) = 18.74 ± 0.38, 17.88 ± 0.35, 18.21 ± 0.34, 18.42 ± 0.39 mag, where the uncertainty here is the RMS scatter of the six objects. We use the J- and H-band data for the photometric distance, as the Y and K bands are more influenced by variations in metallicity and surface gravity. Both bands give an identical distance modulus of (m − M) = 0.10 ± 0.35 mag (10.5 ± 1.7 pc) for WISE J1217+1626A. In a similar fashion, for WISE J1711+3500AB we use the weighted average of J- and H-band absolute magnitudes for T8 dwarfs, choosing the three objects that are “normal” (not young or low metallicity). These give distance moduli of 1.35 ± 0.35 and 1.41 ± 0.32 mag, respectively. We take the average and adopt the slightly larger RMS for a final distance modulus of 1.38 ± 0.35 mag (19 ± 3 pc) for WISE J1711+3500A. As expected, our photometric distances are larger than the 6.7 pc and 17.0 pc estimates for WISE J1217+1626 and WISE J1711+3500, respectively, given by which were based on integrated-light W2-band photometry assuming a single object. Note that if the large flux ratios we observe in the near-IR ( ≈ 2–3 mag) were also representative of the mid-IR flux ratios, we would expect better agreement between our distances and those of Kirkpatrick et al. In other words, the secondary components in these systems must be relatively brighter in the mid-IR and thus contribute a larger portion of the W2-band flux, as expected based on the very red mid-IR colors of the coolest brown dwarfs. Spectral Types [sec:spectra] ---------------------------- ### WISE J1217+1626AB Our NIRSPEC spectra of both components of WISE J1217+1626AB exhibit the deep and absorption in the H-band that is characteristic of the coolest brown dwarfs. Figure [fig:typing] compares our H-band spectra with the T9 and Y0 spectroscopic standards proposed by. Component A shows excellent agreement with their T9 standard. The spectral type for this component is identical to the integrated-light type, as expected given the large near-IR flux ratios. Component B shows good match with the Y0 standard WISE J1738+2732. In particular, the hallmark of the Y spectral type proposed by is the enhanced absorption at 1.53–1.58 seen in the low-resolution spectra of the coolest WISE discoveries that is distinct from the latest T dwarfs. They attribute this feature to, though a definitive identification is still pending. As seen in Figure [fig:typing], WISE J1217+1626B shows the same enhancd absorption on the blue side as the Y0 standard, and thus we classify it as Y0. The red side of the H-band continuum for WISE J1217+1626B does not appear to be as sharply truncated as for the Y0 standard. However, the other Y0 dwarf from the with reasonable S/N, WISE J1405+5534, also has a similar redward extent, which motivated a classification of “Y0 (pec?)” in their discovery paper. Given the modest S/N of our spectrum, we defer a decision on whether WISE J1217+1626B is spectroscopically peculiar. Followup resolved spectroscopy with broader wavelength coverage will help refine the spectral typing, especially as the width of the J-band continuum appears to decrease with later-type objects. ### WISE J1711+3500AB In the absence of resolved spectroscopy for WISE J1711+3500AB, we consider the flux ratios and estimated absolute magnitudes of the components to estimate their spectral types. The large near-IR flux ratios ( ≈ 2.8 mag) suggests strongly that the T8 integrated-light spectrum is dominated by the primary. This is corroborated with simple numerical experiments combining spectra of late-T and Y0 dwarfs using the observed H-band flux ratio as a constraint. The > 10 × fainter secondary has negligible impact on the near-IR spectrum. Thus we safely classify WISE J1711+3500A as T8. For the secondary, we use its near-IR absolute magnitude as estimated from the photometric distance to the primary. This is obviously an uncertain process, though preliminary parallaxes from suggest that the near-IR absolute magnitudes drop off quickly at the T/Y transition, which lessens the impact of uncertainties in the photometric distance. WISE J1711+3500B has M(H) = 19.58 ± 0.36 mag, which is intermediate between the two T9 and two Y0 dwarfs with parallaxes in Figure 12 of. (Their figure has plotted as the sole T9.5, but as discussed in the next subsection this object is more likely to be typed as Y0.) In terms of its Y − J color and estimated color-magnitude diagram position, WISE J1711+3500B also appears to be intermediate between the T9 and Y0 dwarfs (Section [sec:phot]). Thus we estimate a spectral type of T9.5. ### At the time was discovered to be a binary, the near-IR absolute magnitudes of its secondary component were fainter than the coolest known dwarf, which was classified as T10 at the time. This motivated an initial spectral type estimate of > T10 for the secondary by, with the primary type assumed to be identical to the integrated-light type of T9.5. While the secondary type was unknown given the absence of resolved spectroscopy, the monotonic behavior of near-IR absolute magnitudes among the T dwarfs demonstrated the object was exceptionally low temperature. Since then, there have been several late-T and Y dwarfs identified from the WISE dataset, which has warranted a reclassification of the integrated-light spectrum to T9 by. Given this new state of knowledge, we reexamine here the resolved spectral types.. Based on the latest H-band absolute magnitudes from, estimated types of T8.5 and T9.5 for the two components of. However, closer examination of the data indicates these types are too early. As is the case with WISE J1711+3500A, the large near-IR flux ratios ( ≈ 2 mag) of indicate the T9 integrated-light spectrum is dominated by the primary. This is corroborated by the aforementioned experiments of summing spectra and by the model atmosphere fitting in, which found a blended-light spectrum had negligible impact on the derived physical parameters. As for the secondary, it has M(H) = 19.99 ± 0.23 mag, in good agreement with the two Y0 dwarfs in with parallaxes. This absolute magnitude also agrees well with that estimated for the (spectroscopically typed) Y0 dwarf WISE J1217+1626B, based on our photometric distance for WISE J1217+1626A. Thus we estimate the near-IR spectral types for the two components of are T9 and Y0. Resolved Photometry[sec:phot] ----------------------------- Tables [table:wiseone], [table:wisetwo], and [table:cfbds] summarize the resolved photometry for our binaries, derived from the Keck LGS imaging and the available YJHK integrated-light photometry. Figure [fig:colorcolor] shows the YJH color-color plot comparing our binary sample to the known field T dwarfs. For the field sample, we use the data for 80 T dwarfs compiled by, one late-T dwarf (WISE J1617+1807) from,[3](#fn3) three recent late-T discoveries from the UKIDSS survey, and one Y0 dwarf (WISE J1405+5534) from.[4](#fn4) For the objects originally classified as T8.5–T10 in the Leggett et al. compilation, we use the slightly earlier spectral types of T8–T9 proposed by. We also queried the UKIDSS[5](#fn5) Large Area Survey Data Release 9 (DR9, Table [table:ukidss]) in order to add photometry for five late-T dwarfs from Kirkpatrick et al. and for the T8 dwarf PSO J043.5+02 (a.k.a. WISE J0254+0223, found independently by and ). Finally, we computed YJH synthetic photometry for the six T9–T9.5 and two Y0 dwarfs (WISE J0410+1502 [Y0] and WISE J1541 − 2250 [Y0.5]) in and with sufficently blue coverage in their spectra to include the Y-band. While WISE J1217+1626A, WISE J1711+3500A, and WISE J1711+3500B reside in the loci of the coolest known objects, the unusual nature of WISE J1217+1626B is apparent in the YJH color-color diagram, being distinguished by its very blue Y − J color compared to all previously known T dwarfs. Only the three field Y dwarfs in our sample are comparable. This blueness is also emphasized in the near-IR color-magnitude diagram (Figure [fig:cmd]), where component B is fainter and significantly bluer in Y − J than any T dwarf, based on the photometric distance for the binary. The resolved Y − J colors for show a similarly pronounced blue difference between the primary and secondary component. This is seen most clearly in the relative flux ratios, Δ(Y − J) = 0.43 ± 0.02 mag for WISE J1217+1626AB and 0.47 ± 0.15 mag for. (The integrated-light Y-band photometry for from UKIDSS is only S/N = 5 so comparing the resolved colors of the two components is more noisy than comparing their flux ratios, which depend only on the Keck LGS data.) To further highlight this phenomenon, Figure [fig:yj] shows the Y − J colors of the coolest brown dwarfs as a function of spectral type, based on published photometry and our synthesized photometry (regardless of whether the objects have parallaxes or not). and have noted that the Y − J colors of the latest T dwarfs (T8–T9) are bluer than the earlier-type objects, and found that the Y-band peak of two Y0 dwarfs are relatively brighter compared to the T9 standard. Figure [fig:yj] shows that in fact the blueward trend in Y − J crosses a precipice at the T/Y transition, declining dramatically ( ≈ 0.5 mag) over only half a spectral subclass, from T9.5 to Y0. This is discussed further in Section [sec:discussion]. Physical Properties from Evolutionary Models [sec:properties] ------------------------------------------------------------- We use evolutionary models from and to derive physical properties for the two binaries, assuming ages of 1 and 5 Gyr. As inputs, we use both the J-band absolute magnitude and the estimated. The uncertainties in these quantities are propagated into the calculations in a Monte Carlo fashion. We use J-band as it represents the peak flux of the near-IR spectral energy distribution. When using the J-band absolute magnitudes for our calculations, we are implicitly using the bolometric corrections from the underlying model atmospheres. To compute, we adopt a J-band bolometric correction of 1.75 ± 0.27 mag for WISE J1217+1626A (T9), WISE J1217+1626B (Y0), and WISE J1711+3500B( ≈ T9.5), based on the coolest T dwarfs as described in. This is expected to somewhat underestimate the, given the increasingly large fraction of the luminosity emitted in the mid-IR at colder temperatures. However, given the lack of actual bolometric measurements for objects as cool as WISE J1217+1626B, adopting a fixed value is also conservative. For the hotter WISE J1711+3500A (T8), we adopt J-band bolometric corrections from the four T7.5–T8.5 dwarfs with parallaxes: Gl 570D (T7.5; BCJ = 2.60 ± 0.13 mag), 2MASS J1217 − 0311 (T7.5, 2.64 ± 0.13 mag), and 2MASS J0415 − 0935 (T8; 2.54 ± 0.13 mag) from and Wolf 940B (T8.5, 2.09 ± 0.12 mag) from. The unweighted average and RMS are 2.47 ± 0.26 mag. Tables [table:wiseone] and [table:wisetwo] present the results derived from the evolutionary models. Altogether, the components of WISE J1217+1626AB are remarkably similar to those of, with near-IR absolute magnitudes that differ by only ≈ 0.2 mag; thus, the inferred physical properties of these two systems are very similar. Note that the results from the models should be somewhat preferred over the Lyon/COND models of. The predicted near-IR locus of Burrows et al. is a better match to the location of (Figure 7 of ), though the two sets of models basically give consistent results for the physical properties. Overall, the inferred temperature of WISE J1217+1626B is exceptionally low, 350–470 K depending on the choice of theoretical model and system age. Its inferred mass is 5–20 , overlapping the old gas-giant planets found by radial velocity and transit surveys and the young ones found by direct imaging. As expected based on its brighter estimated absolute magnitudes, we find WISE J1711+3500B is somewhat hotter (420–540 K) and more massive (9–26 ) than WISE J1217+1626A. Both components of WISE J1217+1626AB would reside in the planetary-mass regime and have retained their initial deuterium abundance for ages of $\lesssim$1 Gyr. In fact, the two components of WISE J1217+1626AB may possess different deuterium abundances, as their estimated masses could straddle the deuterium-burning limit (11–14 ; ). For WISE J1711+3500AB, the primary likely lies above the D-burning limit, and the secondary would lie below it for ages of $\lesssim$1 Gyr. Measuring the differential deuterium abundance in these binaries would help constrain their ages, analogous to the binary lithium test proposed by — see for a discussion of the very similar case of. Orbital Periods [sec:orbit] --------------------------- To estimate the orbital periods of the two new WISEbinaries, we adopt the statistical conversion factor between projected separation and true semi-major axis from. They offer several choices, based on the underlying eccentricity distribution and degree of completeness to finding binaries by imaging (“discovery bias”). We adopt the eccentricity distribution from their compilation of ultracool visual binaries with high quality orbits and assume no discovery bias, appropriate for such well-resolved systems. This gives a multiplicative correction factor of 1.15− 0.31+ 0.81 (68.3% confidence limits) for converting the projected separation into semi-major axis (compared to 1.10− 0.35+ 0.92 for a uniform eccentricity distribution with no discovery bias). For ages of 1–5 Gyr, the resulting orbital period estimates range from 120–210 yr and 260–430 yr for the two binaries, albeit with significant uncertainties (Tables [table:wiseone] and [table:wisetwo]). Dynamical masses can be realized from orbit monitoring covering $\gtrsim$30% of the period. Thus, several decades of monitoring will be needed to determine the visual orbits. Orbital motion of is clearly detected in our new data, taken two years after the discovery epoch in. Comparing the data from the filters with the best astrometry in the two epochs, we measure a change in separation of 16 ± 3 mas and in position angle of $8.5\pm1.6{\mbox{$^{\circ}$}}$. Such motion is not surprising, given the ≈ 30 yr orbital period estimated at the time of discovery. Discussion [sec:discussion] =========================== Theoretical models predict the near-IR spectra of ≈ 300–500 K objects to be sensitive to new physical processes. have identified changes in near-IR low-resolution spectra of Y dwarfs that they ascribe to. This molecule may also be present in high-resolution near-IR spectra of . Another anticipated signature is the removal of the very broad wings of the 0.77 doublet that dominate the far-red optical continuum. Down to ≈ 500 K, find that the far-red colors, as tracked by the izYJ filters, continue to be very red with only a hint of saturation. At somewhat cooler temperatures, below 400 K the Cs, K, Li and Rb neutral alkalis are expected to be depleted as they turn into chloride gases and solids, or other halide or hydroxide gases, and Na condenses as Na2S. Given the strength of the absorption wings, depletion of this element should be manifested by a change in the broad-band colors. Figures [fig:colorcolor], [fig:cmd], and [fig:yj] show that the Y − J colors of WISE J1217+1626B and the other two Y dwarfs are significantly bluer than all known T dwarfs, consistent with the signature of such depletion. Indeed, the sharp change in Y − J is reminscent of similar behavior seen in near-IR JHK colors at the L/T transition due the onset of photospheric methane absorption, though the change in Y − J occurs over a small range of spectral subclasses. Based on the photometric distance to its primary, the location of WISE J1217+1626B on the color-magnitude diagram also suggests a temperture low enough for potassium depletion. Figure [fig:cmd] compares the available photometry with updated version of the models from D. Saumon (priv. comm.). These updated models include the latest collision-induced absorption and opacities. However, the models are calculated in chemical equilibrium, which overestimates the strengths of the absorption in the near-IR (which is significant) and the mid-IR assuming non-equilibrium conditions are prevalent, as expected. The cloud-free model sequences show (Y − J) decreasing with decreasing temperature through the T dwarf sequence. The J-band absolute magnitudes of the Y0 dwarfs are comparable to the ≈ 400 K models, though the observed Y − J colors are somewhat bluer than the models. The predictions from have much redder Y − J colors than the color-magnitude data. Note that the Saumon & Marley models include the formation of iron, silicate and corundum clouds (which are important for L dwarfs and the L/T transition) as well as the depletion of the neutral alkalis. However, they do not include any opacity due to sulphide or chloride condensates, such as Na2S and KCl which are formed as Na and K are depleted. While such condensates have been historically ignored in ultracool atmosphere modeling, have found the resulting clouds, especially those of Na2S, significantly affect the near-IR spectra of mid/late-T dwarfs and provide a better match to the JHK data. As shown in Figure [fig:cmd], current cloud-free models agree reasonably well in {Y − J, M(J)}. But the mismatch for the latest-type objects may indicate that other factors in addition to K → KCl depletion play a role in the ≈ 1 fluxes, including non-equilibrium abundances and the opacity of sulfate condensates. Conclusions =========== WISE J1217+1626AB and WISE J1711+3500AB are remarkably well-resolved binaries compared to previously known T dwarf binaries (Figure [fig:binaries]), with only the T1+T6 binary being comparably wide in angular separation (0.73 at discovery; ). The projected physical separations of 8 and 15 AU for the WISE binaries are also remarkable, with only the T2.5+T4.0 binary SIMP J1619275+031350AB being comparable (15 AU separation; ) among T dwarfs. Finally, their mass ratios of ≈ 0.5 make them very rare among field ultracool binaries, where nearly equal-mass systems are most prevalent (e.g., see compilation in ). The fact that these two systems are so unusual in their separation and mass ratios compared to nearly all previously known substellar binaries opens the question of whether these initial discoveries among the late-T dwarfs is a harbinger of a change in binary properties at such cool temperatures. An answer awaits a larger survey of such targets at comparable sensitivity. Overall, WISE J1217+1626AB appears to be essentially a clone of the coolest known binary to date,, except for the difference in the projected separation. The blue Y − J color of its secondary component is unmatched by any previously known T dwarf, including the primary component. Synthetic photometry of field Y0–Y0.5 dwarfs also gives very blue colors, and also shows a much bluer Y − J color than its primary. Examining the complete ensemble of available Y − J color data, we find a sharp drop to the blue between the T9.5 and Y0 dwarfs. We suggest this may be the signature of potassium depletion, which removes a significant opacity source at far-red optical wavelengths and is expected for objects of $\lesssim$400 K. It is a fortuitous coincidence that the change in near-IR spectra that demarcates the T/Y transition (currently suspected to be due to ) also coincides with this change in photospheric chemistry. However, in contrast to the relatively subtle changes in the near-IR spectra between T9 and Y0, the Y − J color drops by ≈ 0.5 mag. This suggests that far-red spectra might be more discerning than near-IR spectra for classification in this temperature regime, as discussed by and. Future followup will be essential for reaping the full scientific value of these new binaries. Near-IR low resolution spectra will readily improve the spectral types. Water clouds are also expected to condense below ≈ 400 K, further motivating detailed spectroscopic characterization to search for their signature. The wide separation of these two binaries means that resolved near-IR and optical spectra can be obtained in good seeing conditions, without the need for AO. The binary nature of these systems also provides a unique avenue for analysis. Among the Y dwarfs, model atmosphere fitting indicates a significant range in temperatures (350–500 K), surface gravities ( > 1 dex range), and masses (3–30 ) that is in contrast to the modest differences in their near-IR spectra. It is unclear whether such dispersion represents intrinsic variations in the physical properties of Y dwarfs, failings in the model atmospheres, or both. For binaries, the two components have a common (albeit unknown) metallicity and very similar surface gravity (Tables [table:wiseone] and [table:wisetwo]). Thus resolved spectra will cleanly probe the effect of decreasing temperature, without the uncertainties arising from second parameter variations (age/gravity or metallicity) present in the free-floating population. In a similar vein, spectrophotometric monitoring of the two components may shed light on the nature of the clouds at the T/Y transition. Finally, the photometric distances of 10–20 pc place WISE J1217+1626AB and WISE J1711+3500AB easily within reach of near-IR parallax measurements, and we have begun such monitoring using the Canada-France-Hawaii Telescope. A direct distance will place the four components in context with other late-T and Y dwarfs, through both the absolute magnitude-spectral type relation and color-magnitude diagrams. Given the unusually small mass ratios of ≈ 0.5 for these systems, placing the components on the H-R diagram will also uniquely test the joint accuracy of current evolutionary and atmospheric models, using the constraint that models must indicate the two components of each binary are coeval. We thank Scott Dahm, Terry Stickel, Jim Lyke, and the Keck Observatory staff for assistance with LGS AO observing; the Gemini Observatory staff for obtaining the NIRI photometry through queue observing; and Kimberly Aller for assistance with the IRTF/SpeX observing. We thank Andrew Stephens for providing the Gemini/NIRI cold filter profile; James R. A. Davenport for distributing his IDL implementation of the “cubehelix” color scheme; Caroline Morley, Jonathan Fortney and collaborators for sharing results in advance of publication; and Didier Saumon and Isabelle Baraffe for providing expanded sets of evolutionary models. This publication makes use of data products from the Wide-Field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Our research has employed the 2MASS data products; NASA’s Astrophysical Data System; the SIMBAD database operated at CDS, Strasbourg, France; and the Spex Prism Spectral Libraries maintained by Adam Burgasser at http://pono.ucsd.edu/ ∼ adam/browndwarfs/spexprism. This research was supported by NSF grant AST09-09222 awarded to MCL. Support for this work to TJD was provided by NASA through Hubble Fellowship grant awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract	arxiv_0000721
On the Worst-Case Complexity of TimSort ======================================= TimSortis an intriguing sorting algorithm designed in 2002 for Python, whose worst-case complexity was announced, but not proved until our recent preprint. In fact, there are two slightly different versions of TimSortthat are currently implemented in Python and in Java respectively. We propose a pedagogical and insightful proof that the Python version runs in time O(nlogn). The approach we use in the analysis also applies to the Java version, although not without very involved technical details. As a byproduct of our study, we uncover a bug in the Java implementation that can cause the sorting method to fail during the execution. We also give a proof that Python’s TimSortrunning time is in O(n + nH), where H is the entropy of the distribution of runs (i.e. maximal monotonic sequences), which is quite a natural parameter here and part of the explanation for the good behavior of TimSorton partially sorted inputs. Finally, we evaluate precisely the worst-case running time of Python’s TimSort, and prove that it is equal to 1.5nH + O(n). Introduction ============ TimSortis a sorting algorithm designed in 2002 by Tim Peters , for use in the Pythonprogramming language. It was thereafter implemented in other well-known programming languages such as Java. The algorithm includes many implementation optimizations, a few heuristics and some refined tuning, but its high-level principle is rather simple: The sequence S to be sorted is first decomposed greedily into monotonic runs (i.e. nonincreasing or nondecreasing subsequences of S as depicted on Figure [fig:runs]), which are then merged pairwise according to some specific rules. $ S=(~\underbrace{12,10,7,5}\_{\text{first run}}, ~\underbrace{7,10,14,25,36}\_{\text{second run}}, ~\underbrace{3,5,11,14,15,21,22}\_{\text{third run}}, ~\underbrace{20,15,10,8,5,1}\_{\text{fourth run}}~) $ The idea of starting with a decomposition into runs is not new, and already appears in Knuth’s NaturalMergeSort , where increasing runs are sorted using the same mechanism as in MergeSort. Other merging strategies combined with decomposition into runs appear in the literature, such as the MinimalSortof (see also for other considerations on the same topic). All of them have nice properties: they run in O(nlogn) and even O(n + nlogρ), where ρ is the number of runs, which is optimal in the model of sorting by comparisons , using the classical counting argument for lower bounds. And yet, among all these merge-based algorithms, TimSortwas favored in several very popular programming languages, which suggests that it performs quite well in practice. TimSortrunning time was implicitly assumed to be O(nlogn), but our unpublished preprint contains, to our knowledge, the first proof of it. This was more than ten years after TimSortstarted being used instead of QuickSortin several major programming languages. The growing popularity of this algorithm invites for a careful theoretical investigation. In the present paper, we make a thorough analysis which provides a better understanding of the inherent qualities of the merging strategy of TimSort. Indeed, it reveals that, even without its refined heuristics,[1](#fn1) this is an effective sorting algorithm, computing and merging runs on the fly, using only local properties to make its decisions. We first propose in Section [sec:analysis1] a new pedagogical and self-contained exposition that TimSortruns in time O(n + nlogn), which we want both clear and insightful. In fact, we prove a stronger statement: on an input consisting of ρ runs of respective lengths r1, …, rρ, we establish that TimSortruns in O(n + nH) ⊆ O(n + nlogρ) ⊆ O(n + nlogn), where H = H(r1/n, …, rρ/n) and H(p1, …, pρ) = − ∑i = 1ρpilog2(pi) is the binary Shannon entropy. We then refine this approach, in Section [sec:analysis2], to derive precise bounds on the worst-case running time of TimSort, and we prove that it is equal to 1.5nH + O(n). This answers positively a conjecture of . Of course, the first result follows from the second, but since we believe that each one is interesting on its own, we devote one section to each of them. To introduce our last contribution, we need to look into the evolution of the algorithm: there are actually not one, but two main versions of TimSort. The first version of the algorithm contained a flaw, which was spotted in : while the input was correctly sorted, the algorithm did not behave as announced (because of a broken invariant). This was discovered by De Gouw and his co-authors while trying to prove formally the correctness of TimSort. They proposed a simple way to patch the algorithm, which was quickly adopted in Python, leading to what we consider to be the real TimSort. This is the one we analyze in Sections [sec:analysis1] and [sec:analysis2]. On the contrary, Java developers chose to stick with the first version of TimSort, and adjusted some tuning values (which depend on the broken invariant; this is explained in Sections [presentation] and [sec:java]) to prevent the bug exposed by . Motivated by its use in Java, we explain in Section [sec:java] how, at the expense of very complicated technical details, the elegant proofs of the Python version can be twisted to prove the same results for this older version. While working on this analysis, we discovered yet another error in the correction made in Java. Thus, we compute yet another patch, even if we strongly agree that the algorithm proposed and formally proved in (the one currently implemented in Python) is a better option. TimSort core algorithm ====================== [t] $\operatorname{\texttt{runs}}{\ensuremath{\leftarrow}}$ a run decomposition of S $\operatorname{{\mathcal{R}}}{\ensuremath{\leftarrow}}$ an empty stack ()$\operatorname{\texttt{runs}}\neq \emptyset$[algline:beginloop] remove a run r from $\operatorname{\texttt{runs}}$ and push r onto $\operatorname{{\mathcal{R}}}$ merge\_collapse($\operatorname{{\mathcal{R}}}$)[algline:endloop] ()$\operatorname{\texttt{height}}(\operatorname{{\mathcal{R}}}) \neq 1$ merge\_force\_collapse($\operatorname{{\mathcal{R}}}$) The idea of TimSortis to design a merge sort that can exploit the possible “non randomness” of the data, without having to detect it beforehand and without damaging the performances on random-looking data. This follows the ideas of adaptive sorting (see for a survey on taking presortedness into account when designing and analyzing sorting algorithms). The first feature of TimSortis to work on the natural decomposition of the input sequence into maximal runs. In order to get larger subsequences, TimSortallows both nondecreasing and decreasing runs, unlike most merge sort algorithms. Then, the merging strategy of TimSort (Algorithm [alg:TimSortMainLoop]) is quite simple yet very efficient. The runs are considered in the order given by the run decomposition and successively pushed onto a stack. If some conditions on the size of the topmost runs of the stack are not satisfied after a new run has been pushed, this can trigger a series of merges between pairs of runs at the top or right under. And at the end, when all the runs in the initial decomposition have been pushed, the last operation is to merge the remaining runs two by two, starting at the top of the stack, to get a sorted sequence. The conditions on the stack and the merging rules are implemented in the subroutine called merge\_collapse detailed in Algorithm [alg:mergecollapse]. This is what we consider to be TimSortcore mechanism and this is the main focus of our analysis. [t] Another strength of TimSortis the use of many effective heuristics to save time, such as ensuring that the initial runs are not to small thanks to an insertion sort or using a special technique called “galloping” to optimize the merges. However, this does not interfere with our analysis and we will not discuss this matter any further. Let us have a closer look at Algorithm [alg:mergecollapse] which is a pseudo-code transcription of the merge\_collapse procedure found in the latest version of Python (3.6.5). To illustrate its mechanism, an example of execution of the main loop of TimSort(lines [algline:beginloop]-[algline:endloop] of Algorithm [alg:TimSortMainLoop]) is given in Figure [fig:ts-python-exec]. As stated in its note , Tim Peter’s idea was that: > > “The thrust of these rules when they trigger merging is to balance the run lengths as closely as possible, while keeping a low bound on the number of runs we have to remember.” > > > To achieve this, the merging conditions of merge\_collapse are designed to ensure that the following invariant[2](#fn2) is true at the end of the procedure: $$\begin{aligned} {\mathsf{r}}\_{i+2}&>&{\mathsf{r}}\_{i+1}+{\mathsf{r}}\_{i}, \label{eq:inv1}\\ {\mathsf{r}}\_{i+1}&>&{\mathsf{r}}\_{i}.\label{eq:inv2}\end{aligned}$$ This means that the runs lengths ri on the stack grow at least as fast as the Fibonacci numbers and, therefore, that the height of the stack stays logarithmic (see Lemma [lm:h-is-small], section [sec:analysis1]). Note that the bound on the height of the stack is not enough to justify the O(nlogn) running time of TimSort. Indeed, without the smart strategy used to merge the runs “on the fly”, it is easy to build an example using a stack containing at most two runs and that gives a Θ(n2) complexity: just assume that all runs have size two, push them one by one onto a stack and perform a merge each time there are two runs in the stack. We are now ready to proceed with the analysis of TimSortcomplexity. As mentioned earlier, Algorithm [alg:mergecollapse] does not correspond to the first implementation of TimSortin Python, nor to the current one in Java, but to the latest Python version. The original version will be discussed in details later, in Section [sec:java]. ccccccccccccccc [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 24 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 18 24 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 50 18 24 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 50 42 ; at (A.north) #2; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 92 ; at (A.north) #3; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 6 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 4 6 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 8 4 6 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 8 10 20 28 92 ; at (A.north) #2; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 18 20 28 92 ; at (A.north) #5; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 38 28 92 ; at (A.north) #4; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 66 92 ; at (A.north) #3; & [every node/.style=align=center,text width=1em] (A) [matrix of nodes,nodes=draw] 1 66 92 ; at (A.north) #1; (-3.8,0.2) – (-2.3,0.2); (2.0,0.2) – (5.3,0.2); (-3.05,-0.1) node; (3.65,-0.1) node; ; TimSort runs in =============== At the first release of TimSort , a time complexity of O(nlogn) was announced with no element of proof given. It seemed to remain unproved until our recent preprint , where we provide a confirmation of this fact, using a proof which is not difficult but a bit tedious. This result was refined later in , where the authors provide lower and upper bounds, including explicit multiplicative constants, for different merge sort algorithms. Our main concern is to provide an insightful proof of the complexity of TimSort, in order to highlight how well designed is the strategy used to choose the order in which the merges are performed. The present section is more detailed than the following ones as we want it to be self-contained once TimSorthas been translated into Algorithm [alg:TS translated] (see below). [t] $\operatorname{\texttt{runs}}{\ensuremath{\leftarrow}}$ the run decomposition of S $\operatorname{{\mathcal{R}}}{\ensuremath{\leftarrow}}$ an empty stack ()$\operatorname{\texttt{runs}}\neq \emptyset$ remove a run r from $\operatorname{\texttt{runs}}$ and push r onto $\operatorname{{\mathcal{R}}}$ As our analysis is about to demonstrate, in terms of worst-case complexity, the good performances of TimSortdo not rely on the way merges are performed. Thus we choose to ignore their many optimizations and consider that merging two runs of lengths r and rʹ requires both r + rʹ element moves and r + rʹ element comparisons. Therefore, to quantify the running time of TimSort, we only take into account the number of comparisons performed. In particular, aiming at computing precise bounds on the running time of TimSort, we follow and define the merge cost for merging two runs of lengths r and rʹ as r + rʹ, i.e., the length of the resulting run. Henceforth, we will identify the time spent for merging two runs with the merge cost of this merge. [thm:complexity n + n H] Let C be the class of arrays of length n, whose run decompositions consist of ρ monotonic runs of respective lengths r1, …, rρ. Let H(p1, …, pρ) = − ∑i = 1ρpilog2(pi) be the binary Shannon entropy, and let H = H(r1/n, …, rρ/n). The running time of TimSorton arrays in C is O(n + nH). From this result, we easily deduce the following complexity bound on TimSort, which is less precise but more simple. [thm:complexity n log n] The running time of TimSorton arrays of length n that consist of ρ monotonic runs is O(n + nlogρ), and therefore O(nlogn). The function f : x ↦ − xln(x) is concave on the interval R> 0 of positive real numbers, since its second derivative is fʺ(x) = − 1/x. Hence, when p1, …, pρ are positive real numbers that sum up to one, we have $H(p\_1,\ldots,p\_\rho) = {\textstyle\sum\_{i=1}^\rho f(p\_i)/\ln(2)} \leqslant \rho f(1/\rho)/\ln(2) = \log\_2(\rho)$. In particular, this means that H ≤ log2(ρ), and therefore that TimSortruns in time O(n + nlogρ). Since ρ ≤ n, it further follows that O(n + nlogρ) ⊆ O(n + nlogn) = O(nlogn), which completes the proof. Before proving Theorem [thm:complexity n + n H], we first show that it is optimal up to a multiplicative constant, by recalling the following variant of a result from . [proposition:optimality] For every algorithm comparing only pairs of elements, there exists an array in the class C whose sorting requires at least nH − 3n element comparisons. In the comparison model, at least log2(∣C∣) element comparisons are required for sorting all arrays in C. Hence, we prove below that log2(∣C∣) ≥ nH − 3n. Let π = (π1, …, πρ) be a partition of the set {1, …, n} into ρ subsets of respective sizes r1, …, rρ; we say that π is nice if maxπi > minπi + 1 for all i ≤ ρ − 1. Let us denote by P the set of partitions π of {1, …, n} such that ∣πi∣ = ri for all i ≤ ρ, and by N the set of nice partitions. Let us transform every partition π ∈ P into a nice partition as follows. First, by construction of the run decomposition of an array, we know that r1, …, rρ − 1 ≥ 2, and therefore that minπi < maxπi for all i ≤ ρ − 1. Then, for all i ≤ ρ − 1, if maxπi < minπi + 1, we exchange the partitions to which belong maxπi and minπi + 1, i.e., we move maxπi from the set πi to πi + 1, and minπi + 1 from πi + 1 to πi. Let π\* be the partition obtained after these exchanges have been performed. Observe that π\* is nice, and that at most 2ρ − 1 partitions π ∈ P can be transformed into π\. This proves that 2ρ* − 1∣N∣ ≥ ∣P∣. Let us further identify every nice partition π\* with an array in C, which starts with the elements of π1\* (listed in increasing order), then of π2\, …, π*ρ\. We thereby define an injective map from N to C, which proves that ∣C∣ ≥ ∣N∣. Finally, variants of the Stirling formula indicate that $(k/e)^k \leqslant k! \leqslant e \sqrt{k} (k/e)^k$ for all k* ≥ 1. This proves that $$\begin{aligned} \log\_2(\|{\mathcal{C}}\|) & \geqslant \log\_2(\|{\mathcal{C}}\|) \geqslant (1 - \rho) + \log\_2(\|\mathcal{P}\|) \\ & \geqslant (1 - \rho) + n \log\_2(n) - \rho \log\_2(e) - {\textstyle\sum\_{i=1}^\rho} (r\_i+1/2) \log\_2(r\_i) \\ & \geqslant n {\mathcal{H}}+ (1 - \rho - \rho \log\_2(e)) - 1/2 {\textstyle\sum\_{i=1}^\rho} \log\_2(r\_i).\end{aligned}$$ By concavity of the function x ↦ log2(x), it follows that $\textstyle\sum\_{i=1}^\rho \log\_2(r\_i) \leqslant \rho \log\_2(n/\rho)$. One checks easily that the function x ↦ xlog2(n/x) takes its maximum value at x = n/e, and since n ≥ ρ, we conclude that log2(∣C∣) ≥ nH − (1 + log2(e) + log2(e)/e)n ≥ nH − 3n. We focus now on proving Theorem [thm:complexity n + n H]. The first step consists in rewriting Algorithm [alg:TimSortMainLoop] and Algorithm [alg:mergecollapse] in a form that is easier to deal with. This is done in Algorithm [alg:TS translated]. For any input, Algorithms [alg:TimSortMainLoop] and [alg:TS translated] perform the same comparisons. The only difference is that Algorithm [alg:mergecollapse] was changed into the `while` loop of lines 5 to 10 in Algorithm [alg:TS translated]. Observing the different cases, it is straightforward to verify that merges involving the same runs take place in the same order in both algorithms. Indeed, if r3 < r1, then r3 ≤ r1 + r2, and therefore line 5 is triggered in Algorithm [alg:mergecollapse], so that both algorithms merge the 2nd and 3rd runs. On the contrary, if r3 ≥ r1, then both algorithms merge the 1st and 2nd runs if and only if r2 ≤ r1 or r3 ≤ r1 + r2 (or r4 ≤ r2 + r3). [rem:main-loop] Proving Theorem [thm:complexity n log n] only requires analyzing the main loop of the algorithm (lines 3 to 10). Indeed, computing the run decomposition (line 1) can be done on the fly, by a greedy algorithm, in time linear in n, and the final loop (line 11) might be performed in the main loop by adding a fictitious run of length n + 1 at the end of the decomposition. In the sequel, for the sake of readability, we also omit checking that h is large enough to trigger the cases #2 to #5. Once again, such omissions are benign, since adding fictitious runs of respective lengths 8n, 4n, 2n and n (in this order) at the beginning of the decomposition would ensure that h ≥ 4 during the whole loop. We sketch now the main steps of our proof, i.e., the amortized analysis of the main loop. A first step is to establish the invariant and , ensuring an exponential growth of the run lengths within the stack. Elements of the input array are easily identified by their starting position in the array, so we consider them as well-defined and distinct entities (even if they have the same value). The height of an element in the stack of runs is the number of runs that are below it in the stack: the elements belonging to the run Ri in the stack S = (R1, …, Rh) have height h − i, and we recall that the length of the run Ri is denoted by ri. [lm:invariant I] At any step during the main loop of TimSort, we have ri + ri + 1 < ri + 2 for all i ∈ {3, …, h − 2}. We proceed by induction. The proof consists in verifying that, if the invariant holds at some point, then it still holds when an update of the stack occurs in one of the five situations labeled #1 to #5 in the algorithm. This can be done by a straightforward case analysis. We denote by $\overline{{\mathcal{S}}}=(\overline{R}\_1,\ldots, \overline{R}\_{\overline{h}})$ the new state of the stack after the update: If Case #1 just occurred, a new run $\overline{R}\_1$ was pushed. This implies that none of the conditions of Cases #2 to #5 hold in S, otherwise merges would have continued. In particular, we have r2 + r3 < r4. As $\overline{r}\_i=r\_{i-1}$ for all i ≥ 2, and since the invariant holds for S, it also holds for $\overline{{\mathcal{S}}}$. If one of the Cases #2 to #5 just occurred, $\overline{r}\_i=r\_{i+1}$ for all i ≥ 3. Since the invariant holds for S, it must also hold for $\overline{{\mathcal{S}}}$. [cor:invariant I’] During the main loop of TimSort, whenever a run is about to be pushed onto the stack, we have ri ≤ 2(i + 1 − j)/2rj for all integers i ≤ j ≤ h. Since a run is about to be pushed, none of the conditions of Cases #2 to #5 hold in the stack S. Hence, we have r1 < r2, r1 + r2 < r3 and r2 + r3 < r4, and Lemma [lm:invariant I] further proves that ri + ri + 1 < ri + 2 for all i ∈ {3, …, h − 2}. In particular, for all i ≤ h − 2, we have ri < ri + 1, and thus 2ri ≤ ri + ri + 1 ≤ ri + 2. It follows immediately that ri ≤ 2− kri + 2k ≤ 2− kri + 2k + 1 for all integers k ≥ 0, which is exactly the statement of Corollary [cor:invariant I’]. Corollary [cor:invariant I’] will be crucial in proving that the main loop of TimSortcan be performed for a merge cost O(n + nH). However, we do not prove this upper bound directly. Instead, we need to distinguish several situations that may occur within the main loop. Consider the sequence of Cases #1 to #5 triggered during the execution of the main loop of TimSort. It can be seen as a word on the alphabet {#1, …, #5} that starts with #1, which completely encodes the execution of the algorithm. We split this word at every #1, so that each piece corresponds to an iteration of the main loop. Those pieces are in turn split into two parts, at the first occurrence of a symbol #3, #4 or #5. The first half is called a starting sequence and is made of a #1 followed by the maximal number of #2’s. The second half is called an ending sequence, it starts with #3, #4 or #5 (or is empty) and it contains no occurrence of #1 (see Figure [fig:sequence] for an example). $ \underbrace{\#1\;\#2\;\#2\;\#2}\_{\text{starting seq.}} ~~\underbrace{\#3\;\#2\;\#5\;\#2\;\#4\;\#2}\_{\text{ending seq.}} ~~~\underbrace{\#1\;\#2\;\#2\;\#2\;\#2\;\#2}\_{\text{starting seq.}} ~~\underbrace{\#5\;\#2\;\#3\;\#3\;\#4\;\#2}\_{\text{ending seq.}} $ We bound the merge cost of starting sequences first, and will deal with ending sequences afterwards. [lm:starting] The cost of all merges performed during the starting sequences is O(n). More precisely, for a stack S = (R1, …, Rh), we prove that a starting sequence beginning with a push of a run R of size r onto S uses at most γr comparisons in total, where γ is the real constant 2∑j ≥ 1j/2j/2. After the push, the stack is $\overline{{\mathcal{S}}} = (R,R\_1,\ldots,R\_h)$ and, if the starting sequence contains k ≥ 1 letters, i.e. k − 1 occurrences of #2, then this sequence amounts to merging the runs R1, R2, …, Rk. Since no merge is performed if k = 1, we assume below that k ≥ 2. More precisely, the total cost of these merges is $$C = (k-1)r\_1+(k-1)r\_2 + (k-2)r\_3+\ldots + r\_k \leqslant {\textstyle\sum\_{i=1}^k} (k+1-i)r\_i.$$ The last occurrence of Case #2 ensures that r > rk, hence applying Corollary [cor:invariant I’] to the stack S = (R1, …, Rh) shows that r ≥ rk ≥ 2(k − 1 − i)/2ri for all i = 1, …, k. It follows that $$C / r \leqslant {\textstyle\sum\_{i=1}^k} (k+1-i)2^{(i+1-k)/2} = 2 {\textstyle\sum\_{j=1}^k} j 2^{-j/2} < \gamma.$$ This concludes the proof, since each run is the beginning of exactly one starting sequence, and the sum of their lengths is n. Now, we must take care of run merges that take place during ending sequences. The cost of merging two runs will be taken care of by making run elements pay tokens: whenever two runs of lengths r and rʹ are merged, r + rʹ tokens are paid (not necessarily by the elements of those runs that are merged). In order to do so, and to simplify the presentation, we also distinguish two kinds of tokens, the $\diamondsuit$-tokens and the $\heartsuit$-tokens, which can both be used to pay for comparisons. Two $\diamondsuit$-tokens and one $\heartsuit$-token are credited to an element when its run is pushed onto the stack or when its height later decreases because of a merge that took place during an ending sequence: in the latter case, all the elements of R1 are credited when R1 and R2 are merged, and all the elements of R1 and R2 are credited when R2 and R3 are merged. Tokens are spent to pay for comparisons, depending on the case triggered: * Case #2: every element of R1 and R2 pays 1 $\diamondsuit$. This is enough to cover the cost of merging R2 and R3, because r1 > r3 in this case, and therefore r2 + r1 ≥ r2 + r3. * Case #3: every element of R1 pays 2 $\diamondsuit$. In this case r1 ≥ r2, and the cost is r1 + r2 ≤ 2r1. * Cases #4 and #5: every element of R1 pays 1 $\diamondsuit$and every element of R2 pays 1 $\heartsuit$. The cost r1 + r2 is exactly the number of tokens spent. [lm:balance] The balances of $\diamondsuit$-tokens and $\heartsuit$-tokens of each element remain non-negative throughout the main loop of TimSort. In all four cases #2 to #5, because the height of the elements of R1 and possibly the height of those of R2 decrease, the number of credited $\diamondsuit$-tokens after the merge is at least the number of $\diamondsuit$-tokens spent. The $\heartsuit$-tokens are spent in Cases #4 and #5 only: every element of R2 pays one $\heartsuit$-token, and then belongs to the topmost run $\overline{R}\_1$ of the new stack $\overline{{\mathcal{S}}}=(\overline{R}\_1,\ldots, \overline{R}\_{h-1})$ obtained after merging R1 and R2. Since $\overline{R}\_{i} = R\_{i+1}$ for i ≥ 2, the condition of Case #4 implies that $\overline{r}\_1\geqslant \overline{r}\_2$ and the condition of Case #5 implies that $\overline{r}\_1+\overline{r}\_2\geqslant \overline{r}\_3$: in both cases, the next modification of the stack $\overline{{\mathcal{S}}}$ is another merge, which belongs to the same ending sequence. This merge decreases the height of $\overline{R}\_1$, and therefore decreases the height of the elements of R2, who will regain one $\heartsuit$-token without losing any, since the topmost run of the stack never pays with $\heartsuit$-tokens. This proves that, whenever an element pay one $\heartsuit$-token, the next modification is another merge during which it regains its $\heartsuit$-token. This concludes the proof by direct induction. Finally, consider some element belonging to a run R. Let S be the stack just before pushing the run R, and let $\overline{S} = (\overline{R}\_1,\ldots,\overline{R}\_h)$ be the stack just after the starting sequence of the run R (i.e., the starting sequence initiated when R is pushed onto S) is over. Every element of R will be given at most 2h $\diamondsuit$-tokens and h $\heartsuit$-tokens during the main loop of the algorithm. [lm:h-is-small] The height of the stack when the starting sequence of the run R is over satisfies the inequality h ≤ 4 + 2log2(n/r). Since none of the runs $\overline{R}\_3,\ldots,\overline{R}\_h$ has been merged during the starting sequence of R, applying Corollary [cor:invariant I’] to the stack S proves that $\overline{r}\_3 \leqslant 2^{2-h/2} \overline{r}\_h \leqslant 2^{2-h/2} n$. The run R has not yet been merged either, which means that $r = \overline{r}\_1$. Moreover, at the end of this starting sequence, the conditions of case #2 do not hold anymore, which means that $\overline{r}\_1 \leqslant \overline{r}\_3$. It follows that $r = \overline{r}\_1 \leqslant \overline{r}\_3 \leqslant 2^{2-h/2} n$, which entails the desired inequality. Collecting all the above results is enough to prove Theorem [thm:complexity n + n H]. First, as mentioned in Remark [rem:main-loop], computing the run decomposition can be done in linear time. Then, we proved that the starting sequences of the main loop have a merge cost O(n), and that the ending sequences have a merge cost O(∑i = 1ρ(1 + log(n/ri))ri) = O(n + nH). Finally, the additional merges of line 11 may be taken care of by Remark [rem:main-loop]. This concludes the proof of the theorem. Refined analysis and precise worst-case complexity ================================================== The analysis performed in Section [sec:analysis1] proves that TimSortsorts arrays in time O(n + nH). Looking more closely at the constants hidden in the O notation, we may in fact prove that the cost of merges performed during an execution of TimSortis never greater than 6nH + O(n). However, the lower bound provided by Proposition [proposition:optimality] only proves that the cost of these merges must be at least nH + O(n). In addition, there exist sorting algorithms whose merge cost is exactly nH + O(n). Hence, TimSortis optimal only up to a multiplicative constant. We focus now on finding the least real constant κ such that the merge cost of TimSortis at most κnH + O(n), thereby proving a conjecture of . [thm:complexity 1.5 n + n H] The merge cost of TimSorton arrays in C is at most κnH + O(n), where κ = 3/2. Furthermore, κ = 3/2 is the least real constant with this property. The rest of this Section is devoted to proving Theorem [thm:complexity 1.5 n + n H]. The theorem can be divided into two statements: one that states that TimSortis asymptotically optimal up to a multiplicative constant of κ = 3/2, and one that states that κ is optimal. The latter statement was proved in . Here, we borrow their proof for the sake of completeness. [pro:kappa-optimal-BuKno18] There exist arrays of length n on which the merge cost of TimSortis at least 3/2nlog2(n) + O(n). The dynamics of TimSortwhen sorting an array involves only the lengths of the monotonic runs in which the array is split, not the actual array values. Hence, we identify every array with the sequence of its run lengths. Therefore, every sequence of run lengths ⟨r1, …, rρ⟩ such that r1, …, rρ − 1 ≥ 2, rρ ≥ 1 and r1 + … + rρ = n represents at least one possible array of length n. We define inductively a sequence of run lengths R(n) as follows: $${\mathcal{R}}(n) = \begin{cases}\langle n \rangle & \text{if } 1 \leqslant n \leqslant 6, \\ {\mathcal{R}}(k) \cdot {\mathcal{R}}(k-2) \cdot \langle 2\rangle & \text{if } n = 2k \text{ for some } k \geqslant 4, \\ {\mathcal{R}}(k) \cdot {\mathcal{R}}(k-1) \cdot \langle 2\rangle & \text{if } n = 2k+1 \text{ for some } k \geqslant 3, \end{cases}$$ where the concanetation of two sequences s and t is denoted by s ⋅ t. Then, let us apply the main loop of TimSort on an array whose associated monotonic runs have lengths r = ⟨r1, …, rρ⟩, starting with an empty stack. We denote the associated merge cost by c(r) and, if $\overline{{\mathcal{S}}} = (\overline{R}\_1,\ldots, \overline{R}\_{\overline{h}})$ is the stack obtained after the main loop has been applied, we denote by s(r) the sequence $\langle \overline{r}\_1,\ldots,\overline{r}\_{\overline{h}}\rangle$. An immediate induction shows that, if r1 ≥ r2 + … + rρ + 1, then c(r) = c(⟨r2, …, rρ⟩) and s(r) = ⟨r1⟩ ⋅ s(⟨r2, …, rρ⟩). Similarly, if r1 ≥ r2 + … + rρ + 1 and r2 ≥ r3 + … + rρ + 1, then c(r) = c(⟨r3, …, rρ⟩) and s(r) = ⟨r1, r2⟩ ⋅ s(⟨r3, …, rρ⟩). Consequently, and by another induction on n, it holds that s(R(n)) = ⟨n⟩ and that $$c({\mathcal{R}}(n)) = \begin{cases} 0 & \text{if } 1 \leqslant n \leqslant 6, \\ c({\mathcal{R}}(k)) + c({\mathcal{R}}(k-2)) + 3k & \text{if } n = 2k \text{ for some } k \geqslant 4, \\ c({\mathcal{R}}(k)) + c({\mathcal{R}}(k-1)) + 3k+2 & \text{if } n = 2k+1 \text{ for some } k \geqslant 3. \end{cases}$$ Let ux = c(R(⌊x⌋)) and vx = (ux − 4 − 15/2)/x − 3log2(x)/2. An immediate induction shows that c(R(n)) ≥ c(R(n + 1)) for all integers n ≥ 0, which means that x ↦ ux is non-decreasing. Then, we have un = un/2 + u(n − 3)/2 + ⌈3n/2⌉ for all integers n ≥ 6, and therefore ux ≥ 2ux/2 − 2 + 3(x − 1)/2 for all real numbers x ≥ 6. Consequently, for x ≥ 11, it holds that xvx = ux − 4 − 3xlog2(x)/2 − 15/2 ≥ 2ux/2 − 4 + 3(x − 5)/2 − 3xlog2(x)/2 − 15/2 = xvx/2. This proves that vx ≥ vx/2, from which it follows that vx ≥ inf{vt : 11/2 ≤ t < 11}. Since vt = − 15/(2t) − 3log2(t)/2 ≥ − 15/11 − 3log2(11)/2 ≥ − 7 for all t ∈ [11/2, 11), we conclude that vx ≥ − 7 for all x ≥ 11, and thus that c(R(n)) = un ≥ (n + 4)vn + 4 + 3(n + 4)log2(n + 4)/2 ≥ 3nlog2(n)/2 − 7(n + 4), thereby proving Proposition [pro:kappa-optimal-BuKno18]. It remains to prove the first statement of Theorem [thm:complexity 1.5 n + n H]. Our initial step towards this statement consists in refining Lemma [lm:invariant I]. This is the essence of Lemmas [lm:invariant II] to [lm:invariant IV]. [lm:invariant II] At any step during the main loop of TimSort, if h ≥ 4, we have r2 < r4 and r3 < r4. We proceed by induction. The proof consists in verifying that, if the invariant holds at some point, then it still holds when an update of the stack occurs in one of the five situations labeled #1 to #5 in the algorithm. This can be done by a straightforward case analysis. We denote by S = (R1, …, Rh) the stack just before the update, and by $\overline{{\mathcal{S}}}=(\overline{R}\_1,\ldots, \overline{R}\_{\overline{h}})$ the new state of the stack after the update: If Case #1 just occurred, a new run $\overline{R}\_1$ was pushed. This implies that the conditions of Cases #2 and #4 did not hold in S, otherwise merges would have continued. In particular, we have $\overline{r}\_2=r\_1<r\_3=\overline{r}\_4$ and $\overline{r}\_3=r\_2<r\_1+r\_2<r\_3=\overline{r}\_4$. If one of the Cases #2 to #5 just occurred, it holds that $\overline{r}\_2 \leqslant r\_2+r\_3$, that $\overline{r}\_3=r\_4$ and that $\overline{r}\_4=r\_5$. Since Lemma [lm:invariant I] proves that r3 + r4 < r5, it follows that $\overline{r}\_2 \leqslant r\_2+r\_3 < r\_3+r\_4 < r\_5 = \overline{r}\_4$ and that $\overline{r}\_3 = r\_4<r\_3+r\_4<r\_5 = \overline{r}\_4$. [lm:invariant III] At any step during the main loop of TimSort, and for all i ∈ {3, …, h}, it holds that r2 + … + ri − 1 < ϕ ri. Like for Lemmas [lm:invariant I] and [lm:invariant II], we proceed by induction and verify that, if the invariant holds at some point, then it still holds when an update of the stack occurs in one of the five situations labeled #1 to #5 in the algorithm. Let us denote by S = (R1, …, Rh) the stack just before the update, and by $\overline{{\mathcal{S}}}=(\overline{R}\_1,\ldots, \overline{R}\_{\overline{h}})$ the new state of the stack after the update: If Case #1 just occurred, then we proceed by induction on i ≥ 3. First, for i = 3, since the conditions for Cases #3 and #4 do not hold in S, we know that $\overline{r}\_2 = r\_1 < r\_2 = \overline{r}\_3$ and that $\overline{r}\_2 + \overline{r}\_3 = r\_1+r\_2 < r\_3 = \overline{r}\_4$. Then, for i ≥ 5, Lemma [lm:invariant I] states that ri − 2 + ri − 1 < ri, and therefore 1. if $\overline{r}\_{i-1} \leqslant \phi^{-1} \, \overline{r}\_i$, then $\overline{r}\_2+\ldots+\overline{r}\_{i-1} < (\phi + 1) \overline{r}\_{i-1} = \phi^2 \overline{r}\_{i-1} \leqslant \phi \overline{r}\_i$, and 2. if $\overline{r}\_{i-1} \geqslant \phi^{-1} \, \overline{r}\_i$, then $\overline{r}\_{i-2} \leqslant (1-\phi^{-1}) \ \overline{r}\_i = \phi^{-2} \, \overline{r}\_i$, and thus $\overline{r}\_2+\ldots+\overline{r}\_{i-1} < (\phi+1) \, \overline{r}\_{i-2} + \overline{r}\_{i-1} \leqslant \phi \, \overline{r}\_{i-2} + \overline{r}\_i \leqslant (\phi^{-1} + 1) \overline{r}\_i = \phi \, \overline{r}\_i$. Hence, in that case, it holds that $\overline{r}\_2+\ldots+\overline{r}\_{i-1} < \phi \, \overline{r}\_i$ for all i ∈ {3, …, h}. If one of the Cases #2 to #5 just occurred, it holds that $\overline{r}\_2 \leqslant r\_2+r\_3$ and that $\overline{r}\_j = r\_{j+1}$ for all j ≥ 3. It follows that $\overline{r}\_2+\ldots+\overline{r}\_{i-1} \leqslant r\_2+\ldots+r\_i < \phi \, r\_{i+1} = \overline{r}\_i$. We could also have derived directly Lemma [lm:invariant II] from Lemma [lm:invariant III], by noting that ϕ2 r2 = (ϕ + 1)r2 < ϕ r2 + ϕ r3 < ϕ2 r4. [lm:invariant IV] After every merge that occurred during an ending sequence, we have r1 < ϕ2r2. Once again, we proceed by induction. We denote by S = (R1, …, Rh) the stack just before an update occurs, and by $\overline{{\mathcal{S}}}=(\overline{R}\_1,\ldots, \overline{R}\_{\overline{h}})$ the new state of the stack after after the update: If Case #2 just occurred, then this update is not the first one within the ending sequence, hence $\overline{r}\_1 = r\_1 < \phi^2 \, r\_2 < \phi^2 (r\_2 + r\_3) = \phi^2 \, \overline{r}\_2$. If one of the Cases #2 to #5 just occurred, then r1 ≤ r3 and Lemma [lm:invariant III] proves that r2 < ϕ r3, which proves that $\overline{r}\_1 = r\_1 + r\_2 < (\phi+1) r\_3 = \phi^2 \, \overline{r}\_2$. [lm:invariant V] After every merge triggered by Case #2, we have r2 < ϕ2r1. We denote by S = (R1, …, Rh) the stack just before an update triggered by Case #2 occurs, and by $\overline{{\mathcal{S}}}=(\overline{R}\_1,\ldots, \overline{R}\_{\overline{h}})$ the new state of the stack after after the update. It must hold that r1 > r3 and Lemma [lm:invariant III] proves that r2 < ϕ r3. It follows that $\overline{r}\_2 = r\_2 + r\_3 < (\phi+1) r\_3 = \phi^2 \, r\_3 < \phi^2 \, r\_1 = \phi^2 \, \overline{r}\_1$. Our second step towards proving the first statement of Theorem [thm:complexity 1.5 n + n H] consists in identifying which sequences of merges an ending sequence may be made of. More precisely, in the proof of Lemma [lm:balance], we proved that every merge triggered by a case #4 or #5 must be followed by another merge, i.e., it cannot be the final merge of an ending sequence. We present now a variant of this result, which involves distinguishing between merges triggered by a case #2 and those triggered by a case #3, #4 or #5. Hence, we denote by ${$\#$\textsc{X}\xspace}$ every #3, #4 or #5. [lm:(X2)\X\] No ending sequence contains two conscutive #2’s, nor does it contain a subsequence of the form #X#X#2. Every ending sequence starts with an update #X, where #Xis equal to #3, #4 or #5. Hence, it suffices to prove that no ending sequence contains a subsequence t of the form #X#X#2 or #X#2#2. Indeed, for the sake of contradiction, assume that it does, and let S = (R1, …, Rh) be the stack just before t starts. We distinguish two cases, depending on the value of t: If t is the sequence ${$\#$\textsc{X}\xspace}\;{$\#$\textsc{X}\xspace}\;\#2$, it must hold that r1 + r2 < r4 and that r1 + r2 + r3 ≥ r5, as illustrated in Figure [fig:XX2] (top). Since Lemma [lm:invariant I] proves that r3 + r4 < r5, it follows that r1 + r2 + r3 ≥ r5 > r3 + r4 > r1 + r2 + r3, which is impossible. If t is the sequence ${$\#$\textsc{X}\xspace}\;\#2\;\#2$, it must hold that r1 < r3 and that r1 + r2 ≥ r5, as illustrated in Figure [fig:XX2] (bottom). Since Lemmas [lm:invariant I] and [lm:invariant II] prove that r3 + r4 < r5 and that r2 < r4, it comes that r1 + r2 ≥ r5 > r3 + r4 > r1 + r2, which is also impossible. [scale=0.45] iin 0,...,3 (9\i,0) – (9\i+2,0) – (9\i+2,5) – (9\i,5) – cycle; in 1,...,5 at (9\i+1,5--0.05) $r\_\j$; iin 0,...,2 in 1,..., (9\i,) – (9\i+2,); (9\3,2) – (9\3+2,2); (2.1,2.5) – (8.9,2.5); (11.1,2.5) – (17.9,2.5); (20.1,2.5) – (26.9,2.5); at (5.5,2.5-0.05) merge #X; at (14.5,2.5+0.05) r1 + r2 < r4; at (14.5,2.5-0.05) merge #X; at (23.5,2.5+0.05) r1 + r2 + r3 ≥ r5; at (23.5,2.5-0.05) merge #2; at (10,3.5) +; at (19,3.5) +; at (19,2.5) +; at (28,3.5) +; at (28,2.5) +; at (28,0.5) +; [scale=0.45] iin 0,...,3 (9\i,0) – (9\i+2,0) – (9\i+2,5) – (9\i,5) – cycle; in 1,...,5 at (9\i+1,5--0.05) $r\_\j$; iin 0,...,1 in 1,..., (9\i,) – (9\i+2,); (9\2,1) – (9\2+2,1); (9\2,3) – (9\2+2,3); (9\3,3) – (9\3+2,3); (2.1,2.5) – (9\1-0.1,2.5); (9\1+2.1,2.5) – (9\2-0.1,2.5); (9\2+2.1,2.5) – (9\3-0.1,2.5); at (9\0.5+1,2.5+0.05) r1 ≤ r3; at (9\0.5+1,2.5-0.05) merge #X; at (9\1+9\0.5+1,2.5-0.05) merge #2; at (9\2+9\0.5+1,2.5+0.05) r1 + r2 ≥ r5; at (9\2+9\0.5+1,2.5-0.05) merge #2; at (9\1+1,3.5) +; at (9\2+1,3.5) +; at (9\2+1,1.5) +; at (9\3+1,3.5) +; at (9\3+1,1.5) +; at (9\3+1,0.5) +; [fig:XX2] Our third step consists in modifying the cost allocation we had chosen in Section [sec:analysis1], which is not sufficient to prove Theorem [thm:complexity 1.5 n + n H]. Instead, we associate to every run R* its potential, which depends only on the length r of the run, and is defined as pot(r) = 3rlog2(r)/2. We also call potential of a set of runs the sum of the potentials of the runs it is formed of, and potential variation of a (sequence of) merges the increase in potential caused by these merge(s). We shall prove that the potential variation of every ending sequence dominates its merge cost, up to a small error term. In order to do this, let us study more precisely individual merges. Below, we respectively denote by Δpot(m) and c(m) the potential variation and the merge cost of a merge m. Then, we say that m is a balanced merge if c(m) ≤ Δpot(m). In the next Lemmas, we prove that most merges are balanced or can be grouped into sequences of merges that are balanced overall. [lm:delta-cost I] Let m be a merge between two runs R and Rʹ. If ϕ− 2 r ≤ rʹ ≤ ϕ2 r, then m is balanced. Let x = r/(r + rʹ): we have Φ < x < 1 − Φ, where Φ = 1/(1 + ϕ2). Then, observe that Δ(m) = 3(r + rʹ)H(x)/2, where H(x) = − xlog2(x) − (1 − x)log2(x) is the binary Shannon entropy of a Bernoulli law of parameter x. Moreover, the function z ↦ H(z) = H(1 − z) is increasing on [0, 1/2]. It follows that H(x) ≥ H(Φ) ≈ 0.85 > 2/3, and therefore that Δ(m) > r + rʹ = c(m). [lm:delta-cost II] Let m be a merge that belongs to some ending sequence. If m is a merge #2, then m is balanced and, if m is followed by another merge mʹ, then mʹ is also balanced. Lemma [lm:(X2)\X\] ensures that m was preceded by another merge m⋆, which must be a merge #X. Denoting by S = (R1, …, Rh) the stack just before the merge m⋆ occurs, the update m consists in merging the runs R3 and R4. Then, it comes that r1 ≤ r3 and that r1 + r2 > r4, while Lemma [lm:invariant II] and [lm:invariant III] respectively prove that r3 < r4 and that r2 < ϕ r3. Hence, we both have r3 < r4 and r4 < r1 + r2 < (1 + ϕ)r3 = ϕ2 r3, and Lemma [lm:delta-cost II] proves that m is balanced. Then, if m is followed by another merge mʹ, Lemma [lm:(X2)\X\] proves that mʹ is also a merge #X, between runs of respective lengths r1 + r2 and r3 + r4. Note that r1 ≤ r3 and that r1 + r2 > r4. Since Lemma [lm:invariant II] proves that r2 < r4 and that r3 < r4, it follows that 2(r1 + r2) > 2r4 > r3 + r4 > r1 + r2 and, using the fact that 2 < 1 + ϕ = ϕ2, Lemma [lm:delta-cost II] therefore proves that m is balanced. [lm:delta-cost III] Let m be a merge #Xbetween two runs R1 and R2 such that r1 < ϕ− 2 r2. Then, m is followed by another merge mʹ, and c(m) + c(mʹ) ≤ Δpot(m) + Δpot(mʹ). Let m⋆ be the update the immediately precedes m. Let also S⋆ = (R1⋆, …, Rh⋆⋆), S = (R1, …, Rh) and Sʹ = (Rʹ1, …, Rʹhʹ) be the respective states of the stack just before m⋆ occurs, just before m occurs and just after m occurs. Since r1 < ϕ− 2 r2, Lemma [lm:invariant V] proves that m⋆ is either an update #1 or a merge #X. In both cases, it follows that r2 < r3 and that r2 + r3 < r4. Indeed, if m⋆ is an update #1, then we must have r2 = r1⋆ < r2⋆ = r3 and r2 + r3 = r1⋆ + r2⋆ < r3⋆ = r4, and if mʹ is a merge #X, then Lemmas [lm:invariant I] and [lm:invariant II] respectively prove that r2 + r3 = r3⋆ + r4⋆ < r5⋆ = r4 and that r2 = r3⋆ < r4⋆ = r3. Then, since m is a merge #X, we also know that r1 ≤ r3. Since r1 < ϕ− 2 r2 and r2 + r3 < r4, this means that r1 + r2 ≥ r3. It follows that rʹ2 = r3 ≤ r1 + r2 = rʹ1 and that rʹ1 = r1 + r2 ≤ r2 + r3 < r4 = rʹ3. Consequently, the merge m must be followed by a merge mʹ, which is triggered by case #3. Finally, let x = r1/(r1 + r2) and y = (r1 + r2)/(r1 + r2 + r3). It comes that c(m) + c(mʹ) = (r1 + r2 + r3)(1 + y) and that Δpot(m) + Δpot(mʹ) = 3(r1 + r2 + r3)(yH(x) + H(y)) /2, where we recall that H is the binary Shannon entropy function, with H(t) = − tlog2(t) − (1 − t)log2(t). The above inequalities about r1, r2 and r3 prove that 0 ≤ 2 − 1/y ≤ x ≤ 1/(1 + ϕ2). Since H is increasing on the interval [0, 1/2], and since 1 + ϕ2 ≥ 2, it follows that Δpot(m) + Δpot(mʹ) ≥ 3(r1 + r2 + r3)(yH(2 − 1/y) + H(y)) /2. Hence, let F(y) = 3(yH(2 − 1/y) + H(y)) /2 − (1 + y). We shall prove that F(y) ≥ 0 for all y ≥ 0 such that 0 ≤ 2 − 1/y ≤ 1/(1 + ϕ2), i.e., such that 1/2 ≤ y ≤ (1 + ϕ2)/(1 + 2ϕ2). To that mean, observe that Fʺ(y) = 3/ ((1 − y)(1 − 2y)ln(2)) < 0 for all y ∈ (1/2, 1). Thus, F is concave on (1/2, 1). Since F(1/2) = 0 and F(3/4) = 1/2, it follows that F(y) ≥ 0 for all y ∈ [1/2, 3/4]. Checking that (1 + ϕ2)/(1 + 2ϕ2) < 3/4 completes the proof. [lm:delta-cost IV] Let m be the first merge of the ending sequence associated with a run R. Let R1 and R2 be the runs that m merges together. If r1 > ϕ2 r2, it holds that c(m) ≤ Δpot(m) + r. By definition of m, we have R = R1, and thus r = r1 ≥ r2. Hence, it follows that Δpot(m) = rlog((r + r2)/r) + r2log((r + r2)/r2) ≥ r2log((r + r2)/r2) ≥ r2 = c(m) − r. [pro:delta-cost] Let s be the ending sequence associated with a run R, and let Δpot(s) and c(s) be its potential variation and its merge cost. It holds that c(s) ≤ Δpot(s) + r. Let us group the merges of s as follows: 1. if m is an unbalanced merge #Xbetween two runs R1 and R2 such that r1 < r2, then m is followed by another merge mʹ, and we group m and mʹ together;[delta-cost:case:1] 2. otherwise, and if m has not been grouped with its predecessor, it forms its own group. In case , Lemma [lm:delta-cost IV] ensures that mʹ itself cannot be grouped with another merge. This means that our grouping is unambiguous. Then, let g be such a group, with potential variation Δpot(g) and merge cost c(g). Lemmas [lm:delta-cost I] to [lm:delta-cost IV] prove that c(g) ≤ Δpot(g) + r if g is formed of the first merge of s only, and that c(g) ≤ Δpot(g) in all other cases. Proposition [pro:delta-cost] follows. Collecting all the above results is enough to prove Theorem [thm:complexity 1.5 n + n H]. First, like in Section [sec:analysis1], computing the run decomposition and merging runs in starting sequences has a cost O(n), and the final merges of line 11 may be taken care of by Remark [rem:main-loop]. Second, by Proposition [pro:delta-cost], ending sequences have a merge cost dominated by Δpot + n, where Δpot is the total variation of potential during the algorithm. Observing that Δpot = − 3/2∑i = 1ρrilog2(ri/n) = − 3nH/2 concludes the proof of the theorem. About the Java version of TimSort ================================= cccccccccccc [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 24 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 18 24 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 50 18 24 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 50 42 ; at (A.north) #2; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 92 ; at (A.north) #3; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 6 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 4 6 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 8 4 6 20 28 92 ; at (A.north) #1; & [every node/.style=align=center,text width=1em,inner sep=2.5pt,row 2 column 1/.style=text=red,row 3 column 1/.style=text=red,row 4 column 1/.style=text=red] (A) [matrix of nodes,nodes=draw] 8 10 20 28 92 ; at (A.north) #2; & [every node/.style=align=center,text width=1em,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 1 8 10 20 28 92 ; at (A.north) #1; [fig:invariantbug] Algorithm [alg:mergecollapse] (and therefore Algorithm [alg:TS translated]) does not correspond to the original TimSort. Before release 3.4.4 of Python, the second part of the condition (in blue) in the test at line [algline:newcond] of merge\_collapse (and therefore merge case #5 of Algorithm [alg:TS translated]) was missing. This version of the algorithm worked fine, meaning that it did actually sort arrays, but the invariant given by Equation did not hold. Figure [fig:invariantbug] illustrates the difference caused by the missing condition when running Algorithm [alg:TS translated] on the same input as in Figure [fig:ts-python-exec]. This was discovered by de Gouw et al. when trying to prove the correctness of the Java implementation of TimSort(which is the same as in the earlier versions of Python). And since the Java version of the algorithm uses the (wrong) invariant to compute the maximum size of the stack used to store the runs, the authors were able to build a sequence of runs that causes the Java implementation of TimSortto crash. They proposed two solutions to fix TimSort: reestablish the invariant, which led to the current Python version, or keep the original algorithm and compute correct bounds for the stack size, which is the solution that was chosen in Java 9 (note that this is the second time these values had to be changed). To do the latter, the developers used the claim in that the invariant cannot be violated for two consecutive runs on the stack, which turns out to be false,[3](#fn3) as illustrated in Figure [fig:invstillbroken]. Thus, it is still possible to cause the Java implementation to fail: it uses a stack of runs of size at most 49 and we were able to compute an example requiring a stack of size 50 (see <http://igm.univ-mlv.fr/~pivoteau/Timsort/Test.java>), causing an error at runtime in Java’s sorting method. ccccccccccccc [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 25 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 16 25 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 8 16 25 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 7 8 16 25 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 26 7 8 16 25 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 26 15 16 25 83 109 ; at (A.north) #2; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 26 31 25 83 109 ; at (A.north) #2; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 26 56 83 109 ; at (A.north) #2; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 2 26 56 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt] (A) [matrix of nodes,nodes=draw] 27 2 26 56 83 109 ; at (A.north) #1; & [every node/.style=align=center,text width=15pt,inner sep=2.5pt,row 2 column 1/.style=text=red,row 3 column 1/.style=text=red,row 4 column 1/.style=text=red,row 5 column 1/.style=text=red] (A) [matrix of nodes,nodes=draw] 27 28 56 83 109 ; at (A.north) #2; [fig:invstillbroken] Even if the bug we highlighted in Java’s TimSortis very unlikely to happen, this should be corrected. And, as advocated by de Gouw et al. and Tim Peters himself,[4](#fn4) we strongly believe that the best solution would be to correct the algorithm as in the current version of Python, in order to keep it clean and simple. However, since this is the implementation of Java’s sort for the moment, there are two questions we would like to tackle: Does the complexity analysis holds without the missing condition? And, can we compute an actual bound for the stack size? We first address the complexity question. It turns out that the missing invariant was a key ingredient for having a simple and elegant proof. [pro:domination] At any time during the main loop of Java’s TimSort, if the stack of runs is (R1, …, Rh) then we have r3 < r4 < … < rh and, for all i ≥ 3, we have $(2 + \sqrt{7}) r\_i \geqslant r\_2 + \ldots + r\_{i-1}$. [Proof ideas] The proof of Proposition [pro:domination] is much more technical and difficult than insightful, and therefore we just summarize its main steps. As in previous sections, this proof relies on several inductive arguments, using both inductions on the number of merges performed, on the stack size and on the run sizes. The inequalities r3 < r4 < … < rh come at once, hence we focus on the second part of Proposition [pro:domination]. Since separating starting and ending sequences was useful in Section [sec:analysis2], we first introduce the notion of stable stacks: a stack S is stable if, when operating on the stack S = (R1, …, Rh), Case #1 is triggered (i.e. Java’s TimSortis about to perform a run push operation). We also call obstruction indices the integers i ≥ 3 such that ri ≤ ri − 1 + ri − 2: although they do not exist in Python’s TimSort, they may exist, and even be consecutive, in Java’s TimSort. We prove that, if i − k, i − k + 1, …, i are obstruction indices, then the stack sizes ri − k − 2, …, ri grow “at linear speed”. For instance, in the last stack of Figure [fig:invstillbroken], obstruction indices are 4 and 5, and we have r2 = 28, r3 = r2 + 28, r4 = r3 + 27 and r5 = r4 + 26. Finally, we study so-called expansion functions, i.e. functions f : [0, 1] ↦ R such that, for every stable stack S = (R1, …, Rh), we have r2 + … + rh − 1 ≤ rhf(rh − 1/rh). We exhibit an explicit function f such that $f(x) \leqslant 2+\sqrt{7}$ for all x ∈ [0, 1], and we prove by induction on rh that f is an expansion function, from which we deduce Proposition [pro:domination]. Once Proposition [pro:domination] is proved, we easily recover the following variant of Lemmas [lm:invariant I] and [lm:h-is-small]. [lem:height-J-Java] At any time during the main loop of Java’s TimSort, if the stack is (R1, …, Rh) then we have $r\_2 / (2 + \sqrt{7}) \leqslant r\_3 < r\_4 < \ldots < r\_h$ and, for all i ≥ j ≥ 3, we have ri ≥ δi − j − 4rj, where $\delta = \left( 5/(2+\sqrt{7}) \right)^{1/5} > 1$. Furthermore, at any time during an ending sequence, including just before it starts and just after it ends, we have $r\_1 \leqslant (2 + \sqrt{7}) r\_3$. The inequalities $r\_2 / (2 + \sqrt{7}) \leqslant r\_3 < r\_4 < \ldots < r\_h$ are just a (weaker) restatement of Proposition [pro:domination]. Then, for j ≥ 3, we have $(2+\sqrt{7}) r\_{j+5} \geqslant r\_j+\ldots+r\_{j+4} \geqslant 5 r\_j$, i.e. rj + 5 ≥ δ5rj, from which one gets that ri ≥ δi − j − 4rj. Finally, we prove by induction that $r\_1 \leqslant (2 + \sqrt{7}) r\_3$ during ending sequences. First, when the ending sequence starts, $r\_1 < r\_3 \leqslant (2+\sqrt{7}) r\_3$. Before any merge during this sequence, if the stack is S = (R1, …Rh), then we denote by $\overline{{\mathcal{S}}} = (\overline{R}\_1,\ldots,\overline{R}\_{h-1})$ the stack after the merge. If the invariant holds before the merge, in Case #2, we have $\overline{r}\_1 = r\_1 \leqslant (2+\sqrt{7}) r\_3 \leqslant (2+\sqrt{7}) r\_4 = (2+\sqrt{7}) \overline{r}\_3$; and using Proposition [pro:domination] in Cases #3 and #4, we have $\overline{r}\_1 = r\_1 + r\_2$ and r1 ≤ r3, hence $\overline{r}\_1 = r\_1 + r\_2 \leqslant r\_2 + r\_3 \leqslant (2+\sqrt{7}) r\_4 = (2+\sqrt{7}) \overline{r}\_3$, concluding the proof. We can then recover a proof of complexity for the Java version of TimSort, by following the same proof as in Sections [sec:analysis1] and [sec:analysis2], but using Lemma [lem:height-J-Java] instead of Lemmas [lm:invariant I] and [lm:h-is-small]. [thm:complexity n log rho - java] The complexity of Java’s TimSorton inputs of size n with ρ runs is O(n + nlogρ). Another question is that of the stack size requirements of Java’s TimSort, i.e. computing hmax. A first result is the following immediate corollary of Lemma [lem:height-J-Java]. [cor:bad-size] On an input of size n, Java’s TimSortwill create a stack of runs of maximal size hmax ≤ 7 + logδ(n), where $\delta = \left( 5/(2+\sqrt{7}) \right)^{1/5}$. At any time during the main loop of Java’s TimSorton an input of size n, if the stack is (R1, …, Rh) and h ≥ 3, it follows from Lemma [lem:height-J-Java] that n ≥ rh ≥ δh − 7r3 ≥ δh − 7. Unfortunately, for integers smaller than 231, Corollary [cor:bad-size] only proves that the stack size will never exceed 347. However, in the comments of Java’s implementation of TimSort,[5](#fn5) there is a remark that keeping a short stack is of some importance, for practical reasons, and that the value chosen in Python – 85 – is “too expensive”. Thus, in the following, we go to the extent of computing the optimal bound. It turns out that this bound cannot exceed 86 for such integers. This bound could possibly be refined slightly, but definitely not to the point of competing with the bound that would be obtained if the invariant of Equation were correct. Once more, this suggests that implementing the new version of TimSortin Java would be a good idea, as the maximum stack height is smaller in this case. [thm:good-size] On an input of size n, Java’s TimSortwill create a stack of runs of maximal size hmax ≤ 3 + logΔ(n), where $\Delta = (1+\sqrt{7})^{1/5}$. Furthermore, if we replace Δ by any real number Δʹ > Δ, the inequality fails for all large enough n. [Proof ideas] The first part of Theorem [thm:good-size] is proved as follows. Ideally, we would like to show that ri + j ≥ Δjri for all i ≥ 3 and some fixed integer j. However, these inequalities do not hold for all i. Yet, we prove that they hold if i + 2 and i + j + 2 are not obstruction indices, and i + j + 1 is an obstruction index, and it follows quickly that rh ≥ Δh − 3. The optimality of Δ is much more difficult to prove. It turns out that the constants $2+\sqrt{7}$, $(1+\sqrt{7})^{1/5}$, and the expansion function referred to in the proof of Proposition [pro:domination] were constructed as least fixed points of non-decreasing operators, although this construction needed not be explicit for using these constants and function. Hence, we prove that Δ is optimal by inductively constructing sequences of run sizes that show that limsup{log(rh)/h} ≥ Δ; much care is required for proving that our constructions are indeed feasible. Conclusion ========== At first, when we learned that Java’s QuickSort had been replaced by a variant of MergeSort, we thought that this new algorithm – TimSort – should be really fast and efficient in practice, and that we should look into its average complexity to confirm this from a theoretical point of view. Then, we realized that its worst-case complexity had not been formally established yet and we first focused on giving a proof that it runs in O(nlogn), which we wrote in a preprint . In the present article, we simplify this preliminary work and provide a short, simple and self-contained proof of TimSort’s complexity, which sheds some light on the behavior of the algorithm. Based on this description, we were also able to answer positively a natural question, which was left open so far: does TimSortruns in O(n + nlogρ), where ρ is the number of runs? We hope our theoretical work highlights that TimSortis actually a very good sorting algorithm. Even if all its fine-tuned heuristics are removed, the dynamics of its merges, induced by a small number of local rules, results in a very efficient global behavior, particularly well suited for almost sorted inputs. Besides, we want to stress the need for a thorough algorithm analysis, in order to prevent errors and misunderstandings. As obvious as it may sound, the three consecutive mistakes on the stack height in Java illustrate perfectly how the best ideas can be spoiled by the lack of a proper complexity analysis. Finally, following , we would like to emphasize that there seems to be no reason not to use the recent version of TimSort, which is efficient in practice, formally certified and whose optimal complexity is easy to understand. Appendix ======== Proofs ------ We provide below complete proofs of the results mentioned in Section [sec:java]. In what follows, we will often refer to so-called stable stacks: we say that a stack S = (R1, …, Rh) is stable if r1 + r2 < r3 and r1 < r2, i.e. if the next operation that will be performed by TimSortis a push operation (Case #1). ### Proving Proposition [pro:domination] Aiming to prove Proposition [pro:domination], and keeping in mind that studying stable stacks may be easier than studying all stacks, a first step is to introduce the following quantities. [def:alphabeta] Let n be a positive integer. We denote by αn (resp., βn), the smallest real number m such that, in every stack (resp., stable stack) S = (R1, …, Rh) obtained during an execution of TimSort, and for every integer i ∈ {1, …, h} such that ri = n, we have r2 + … + ri − 1 ≤ m × ri; if no such real number exists, we simply set αn = + ∞ (resp., βn = + ∞). By construction, αn ≥ βn for all n ≥ 1. The following lemma proves that αn ≤ βn. [lem:inv-K] At any time during the main loop of TimSort, if the stack is (R1, …, Rh), then we have (a) ri < ri + 1 for all i ∈ {3, 4, …, h − 1} and (b) r2 + … + ri − 1 ≤ βnri for all n ≥ 1 and i ≤ h such that ri = n. Assume that (a) and (b) do not always hold, and consider the first moment where some of them do not hold. When the main loop starts, both (a) and (b) are true. Hence, from a stack S = (R1, …, Rh), on which (a) and (b) hold, we carried either a push step (Case #1) or a merging step (Cases #2 to #4), thereby obtaining the new stack $\overline{{\mathcal{S}}} = (\overline{R}\_1,\ldots,\overline{R}\_{\overline{h}})$. We consider separately these two cases: * After a push step, we have $\overline{h} = h+1$, r1 + r2 < r3 (otherwise, we would have performed a merging step instead of a push step) and $\overline{r}\_i = r\_{i-1}$ for all i ≥ 2. It follows that $\overline{r}\_3 = r\_2 < r\_1 + r\_2 < r\_3 = \overline{r}\_4$, and that $\overline{r}\_i = r\_{i-1} < r\_i = \overline{r}\_{i+1}$ for all i ≥ 4. This proves that $\overline{{\mathcal{S}}}$ satisfies (a). In addition, the value of $\overline{r}\_1$ has no impact on whether $\overline{{\mathcal{S}}}$ satisfies (b). Hence, we may assume without loss of generality that $\overline{r}\_1 < \min\{\overline{r}\_2,\overline{r}\_3-\overline{r}\_2\}$ (up to doubling the size of every run ever pushed onto the stack so far and setting $\overline{r}\_1 = 1$), thereby making $\overline{{\mathcal{S}}}$ stable. This proves that $\overline{{\mathcal{S}}}$ satisfies (b). * After a merging step, we have $\overline{h} = h-1$, $\overline{r}\_2 \leqslant r\_2 + r\_3$ and $\overline{r}\_i = r\_{i+1}$ for all i ≥ 3. Hence, $\overline{r}\_i = r\_{i+1} < r\_{i+2} = \overline{r}\_{i+1}$ for all i ≥ 3, and $\overline{{\mathcal{S}}}$ satisfies (a). Furthermore, we have $0 \leqslant \beta\_{\overline{r}\_2} \overline{r}\_2$, and $\overline{r}\_2 + \overline{r}\_3 + \ldots + \overline{r}\_i \leqslant r\_2 + r\_3 + \ldots + r\_{i+1} \leqslant \beta\_n r\_{i+2} = \beta\_n \overline{r}\_{i+1}$ whenever i ≥ 1 and $\overline{r}\_{i+1} = r\_{i+2} = n$. This proves that $\overline{{\mathcal{S}}}$ also satisfies (b). Hence, in both cases, (a) and (b) also hold in $\overline{{\mathcal{S}}}$, which contradicts our assumption and completes the proof. For all integers n ≥ 1, we have αn = βn. It remains to prove that αn ≤ α∞ for all n ≥ 1, where $\alpha\_\infty = 2 + \sqrt{7}$. This is the object of the next results. What makes Java’s TimSortmuch harder to study than Python’s TimSortis the fact that, during the execution of Java’s TimSortalgorithm, we may have stacks S = (R1, …, Rh) on which the invariant : ri > ri − 1 + ri − 2 fails for many integers i ≥ 3, possibly consecutive. In Section [sec:java], such integers were called obstruction indices of the stack S. Hence, we focus on sequences of consecutive obstruction indices. [lemma:hard-b] Let S = (R1, …, Rh) be a stable stack obtained during the main loop of Java’s TimSort. Assume that i − k, i + 1 − k, …, i are consecutive obstruction indices of S, and that αn ≤ α∞ for all n ≤ ri − 1. Then, $$r\_{i-k-2} \leqslant \frac{\alpha\_\infty + 1 - k}{\alpha\_\infty + 2} r\_{i-1}.$$ Let T be the number of merge or push operations performed between the start of the main loop and the creation of the stack S. For all k ∈ {0, …, T} and all j ≥ 1, we denote by Sk the stack after k operations have been performed. We also denote by Pj, k the jth bottom-most run of the stack Sk, and by pj, k the size of Pj, k; we set Pj, k = ∅ and pj, k = 0 if Sk has fewer than j runs. Finally, for all j ≤ h, we set tj = min{k ≥ 0 ∣ ∀ℓ ∈ {k, …, T}, pj, ℓ = pj, T}. First, observe that tj < tj + 2 for all j ≤ h − 2, because a run can be pushed or merged only in top or 2nd-to-top position. Second, if tj ≥ tj + 1 for some j ≤ h − 1, then the runs Pj, tj, Pj + 1, tj are the two top runs of Stj. Since none of the runs P1, …, Pj + 1 is modified afterwards, it follows, if j ≥ 2, that pj + 1 + pj = pj + 1, tj + pj, tj < pj − 1, tj = pj − 1, and therefore that h + 2 − j is not an obstruction index. Conversely, let m0 = h + 3 − i. We just proved that tm0 − 2 < tm0 and also that tm0 − 1 < tm0 < … < tm0 + k. Besides, for all m ∈ {m0, …, m0 + k}, we prove that the tmth operation was a merge operation of type #2. Indeed, if not, then the run Pm, tm would be the topmost run of Stm; since the runs Pm − 1 and Pm − 2 were not modified after that, we would have pm + pm − 1 < pm − 2, contradicting the fact that h + 3 − m is an obstruction index. In particular, it follows that pm + 1, tm = pm + 2, tm − 1 ≥ pm, tm − 1 and that pm = pm, tm ≤ pm − 1, tm − pm + 1, tm = pm − 1 − pm + 1, tm. Moreover, for m = m0, observe that pm = pm, tm = pm, tm − 1 + pm + 1, tm − 1. Applying Lemma [lem:inv-K] on the stacks ST and Stm − 1, we know that pm, tm − 1 ≤ pm ≤ pm − 2 − 1 = ri − 1 and that pp + 1, tm − 1 ≤ apm, tm − 1pm, tm − 1 ≤ α∞pm, tm − 1, which proves that pm ≤ (α∞ + 1)pm, tm − 1 ≤ (α∞ + 1)pm + 1, tm, i.e., pm0 ≤ (α∞ + 1)pm0 + 1, tm0. Henceforth, we set κ = pm0 + 1, tm0. In addition, for all m ∈ {m0 + 1, …, m0 + k}, observe that the sequence (pm + 1, k)tm ≤ k ≤ T is non-decreasing. Indeed, when tm ≤ k, and therefore ti ≤ k for all i ≤ m, the run pm + 1, k can only be modified by being merged with another run, thereby increasing in size. This proves that pm + 2, tm + 1 ≥ pm + 1, tm + 1 − 1 ≥ pm + 1, tm. Hence, an immediate induction shows that pm + 1, tm ≥ pm0 + 1, tm0 = κ for all m ∈ {m0, …, m0 + k}, and it follows that pm ≤ pm − 1 − κ. Overall, this implies that ri − k − 2 = pm0 + k ≤ pm0 − kκ. Note that pm0 ≤ min{(α∞ + 1)κ, pm0 − 1 − pm0 + 1, tm0} = min{(α∞ + 1)κ, pm0 − 1 − κ}. It follows that ri − k − 2 ≤ min{(α∞ + 1)κ, pm0 − 1 − κ} − kκ ≤ min{(α∞ + 1 − k)κ, ri − 1 − (k + 1)κ}, whence (α∞ + 2)ri − k − 2 ≤ (k + 1)(α∞ + 1 − k)κ + (α∞ + 1 − k)(ri − 1 − (k + 1)κ) = (α∞ + 1 − k)ri − 1. Lemma [lemma:hard-b] paves the way towards a proof by induction that αn ≤ α∞. Indeed, a first, immediate consequence of Lemma [lemma:hard-b], is that, provided that αn ≤ α∞ for all n ≤ ri − 1, then the top-most part (R1, …, Ri) may not contain more than α∞ + 2 (and therefore no more than 6) consecutive obstruction indices. This suggests that the sequence r1, …, ri should grow “fast enough”, which might then be used to prove that αri ≤ α∞. We present below this inductive proof, which relies on the following objects. [def:expansion] We call expansion function the function f : [0, 1] → R≥ 0 defined by $$f : x \to \begin{cases} (1+\alpha\_\infty) x & \text{if $0 \leqslant x \leqslant 1/2$} \\ x + \alpha\_\infty(1-x) & \text{if $1/2 \leqslant x \leqslant \alpha\_\infty/(2\alpha\_\infty-1)$} \\ \alpha\_\infty x & \text{if $\alpha\_\infty/(2\alpha\_\infty-1) \leqslant x \leqslant 1$.}\end{cases}$$ In the following, we denote by θ the real number α∞/(2α∞ − 1). Let us first prove two technical results about the expansion function. [lem:f] We have α∞x ≤ f(x) for all x ∈ [0, 1], f(x) ≤ f(1/2) for all x ∈ [0, θ], f(x) ≤ f(1) for all x ∈ [0, 1], x + α∞(1 − x) ≤ f(x) for all x ∈ [1/2, 1] and x + α∞(1 − x) ≤ f(1/2) for all x ∈ [1/2, 1]. Since f is piecewise linear, it is enough to check the above inequalities when x is equal to 0, 1/2, θ or 1, which is immediate. [lem:f2] For all real numbers x, y ∈ [0, 1] such that x(y + 1) ≤ 1, we have x(1 + f(y)) ≤ min{f(1/2), f(x)}. We treat three cases separately, relying in each case on Lemma [lem:f]: * if 0 ≤ x ≤ 1/2, then x(1 + f(y)) ≤ x(1 + f(1)) = (1 + α∞)x = f(x) ≤ f(1/2); * if 1/2 < x ≤ 1 and f(1/2) < f(y), then θ ≤ y ≤ 1, hence x(1 + f(y)) = x + α∞xy ≤ x + α∞(1 − x) ≤ min{f(x), f(1/2)}; * if 0 ≤ f(y) ≤ f(1/2), and since α∞ ≥ 1, we have x(1 + f(y)) ≤ x(1 + f(1/2)) = x(3 + α∞)/2 ≤ x(1 + α∞) ≤ f(x); if, furthermore, y ≤ 1/2, then $$\begin{aligned} x (1+f(y)) & \leqslant (1+(1+\alpha\_\infty) y) / (1+y) = (1+\alpha\_\infty) - \alpha\_\infty / (1+y) \\ & \leqslant (1+\alpha\_\infty) - 2\alpha\_\infty/3 = (3+\alpha\_\infty)/3,\end{aligned}$$ and if 1/2 ≤ y, then x(1 + f(y)) ≤ (1 + f(1/2))/(1 + y) ≤ 2(1 + f(1/2))/3 = (3 + α∞)/3; since α∞ ≥ 3, it follows that x(1 + f(y)) ≤ (3 + α∞)/3 ≤ (1 + α∞)/2 = f(1/2) in both cases. Using Lemma [lemma:hard-b] and the above results about the expansion function, we finally get the following result, of which Proposition [pro:domination] is an immediate consequence. [lem:4b] Let S = (R1, …, Rh) be a stable stack obtained during the main loop of Java’s TimSort. For all integers i ≥ 2, we have r1 + r2 + … + ri − 1 ≤ rif(ri − 1/ri), where f is the expansion function. In particular, we have αn = βn ≤ α∞ for all integers n ≥ 1. Lemma [lem:f] proves that 2x ≤ α∞x ≤ yf(x/y) whenever 0 < x ≤ y, and therefore the statement of Lemma [lem:4b] is immediate when i ≤ 3. Hence, we prove Lemma [lem:4b] for i ≥ 4, and proceed by induction on ri = n, thereby assuming that αn − 1 exists. Let x = ri − 1/ri and y = ri − 2/ri − 1. By Lemma [lem:inv-K], and since the stack S is stable, we know that ri − 2 < ri − 1 < ri, and therefore that x < 1 and y < 1. If i is not an obstruction index, then we have ri − 2 + ri − 1 ≤ ri, i.e., x(1 + y) ≤ 1 and, using Lemma [lem:f2], it follows that r1 + … + ri − 1 = (r1 + … + ri − 2) + ri − 1 ≤ f(y)ri − 1 + ri − 1 = x(1 + f(y))ri ≤ f(x)ri. On the contrary, if i is an obstruction index, let k be the smallest positive integer such that i − k is not an obstruction index. Since the stack S is stable, we have r1 + r2 < r3, which means that 3 is not an obstruction index, and therefore i − k ≥ 3. Let u = ri − k − 1/ri − k and v = ri − k − 2/ri − k − 1. By construction, we have ri − k − 2 + ri − k − 1 ≤ ri − k, i.e., u(1 + v) ≤ 1. Using Lemma [lemma:hard-b], and since ri − k − 1 < ri and αri − 1 ≤ f(1) = α∞ by induction hypothesis, we have $$\begin{aligned} r\_1+\ldots+r\_{i-1} & = (r\_1 + \ldots + r\_{i-k-2}) + r\_{i-k-1} + \ldots + r\_{i-1} \leqslant r\_{i-k-1} f(v) + r\_{i-k-1} + \ldots + r\_{i-1} \\ & \leqslant r\_{i-k} u (1 + f(v)) + r\_{i-k} + \ldots + r\_{i-1} \leqslant r\_{i-k} f(1/2) + r\_{i-k} + \ldots + r\_{i-1} \\ & \leqslant \frac{1}{\alpha\_\infty+2} \left((\alpha\_\infty + 3-k) f(1/2) + \sum\_{j=1}^k (\alpha\_\infty+3-j) \right) r\_{i-1} \\ & \leqslant \frac{1}{2(\alpha\_\infty+2)} \left(\alpha\_\infty^2 + (4+k)\alpha\_\infty - k^2 + 4k + 3\right)r\_{i-1}.\end{aligned}$$ The function g : t → α∞2 + (4 + t)α∞ − t2 + 4t + 3 takes its maximal value, on the real line, at t = (α∞ + 4)/2 ∈ (4, 5). Consequently, for all integers k, and since α∞ ≤ 5, we have g(k) ≤ max{g(4), g(5)} = α∞2 + max{8α∞ + 3, 9α∞ − 2} = α∞2 + 8α∞ + 3. Then, observe that $2(\alpha\_\infty+2)\alpha\_\infty = 30 + 12 \sqrt{7} = \alpha\_\infty^2 + 8 \alpha\_\infty + 3$. It follows that $$r\_1+\ldots+r\_{i-1} \leqslant \frac{\alpha\_\infty^2 + 8 \alpha\_\infty + 3}{2(\alpha\_\infty+2)} r\_{i-1} = \alpha\_\infty x r\_i \leqslant f(x) r\_i.$$ Hence, regardless of whether i is an obstruction index or not, we have r1 + … + ri − 1 ≤ f(x)ri ≤ f(1)ri = α∞ri, which completes the proof. ### Proving the first part of Theorem [thm:good-size] We prove below the inequality of Theorem [thm:good-size]; proving that that the constant Δ used in Theorem [thm:good-size] is optimal will be the done in the next section. In order to carry out this proof, we need to consider some integers of considerable interest. Let S = (R1, …, Rh) be a stable stack of runs. We say that an integer i ≥ 1 is a growth index if i + 2 is not an obstruction index, and that i is a strong growth index if i is a growth index and if, in addition, i + 1 is an obstruction index. Note that h an h − 1 are necessarily growth indices, since h + 1 and h + 2 are too large to be obstruction indices. Our aim is now to prove inequalities of the form ri + j ≥ Δjri, where 3 ≤ i ≤ i + j ≤ h. However, such inequalities do not hold in general, hence we restrict the scope of the integers i and i + j, which is the subject of the two following results. [lem:non-obstruction-growth] Let i and j be positive integers such that i + 2 ≤ j ≤ h. If no obstruction index k exists such that i + 2 ≤ k ≤ j, then 2Δj − i − 2ri ≤ rj. For all n ≥ 0, let Fn denote the nth Fibonacci number, defined by F0 = 0, F1 = 1 and Fn + 2 = Fn + Fn + 1 or, alternatively, by $F\_n = (\phi^n - (-\phi)^{-n})/\sqrt{5}$, where $\phi = (1+\sqrt{5})/2$ is the Golden ratio. Observe now that Fj − i + 1ri ≤ Fj − i − 1ri + Fj − iri + 1 ≤ Fj − i − 2ri + 1 + Fj − i − 1ri + 2 ≤ … ≤ F0rj − 1 + F1rj = rj. Moreover, for all n ≥ 3, we have Fn = 2Fn/F3 = 2ϕn − 3(1 − ( − 1)nϕ− 2n)/(1 − ϕ− 6) ≥ 2ϕn − 3. Since Δ < ϕ, it follows that 2Δj − i − 2ri ≤ Fj − i + 1ri ≤ rj. [lem:next-growth] Let i and j be positive integers such that 1 ≤ i ≤ j ≤ h. If i is a growth index and j is a strong growth index, then Δj − iri ≤ rj. Without loss of generality, let us assume that i < j and that there is no strong growth index k such that i < k < j. Indeed, if such an index k exists, then a simple induction completes the proof of Lemma [lem:next-growth]. Let ℓ be the largest integer such that ℓ ≤ j and ℓ is not an obstruction index. Lemmas [lemma:hard-b] and [lem:4b] prove that (α∞ + 2)rℓ ≤ (α∞ + 2 + ℓ − j)rj and that (α∞ + 2)rℓ − 1 ≤ (α∞ + 1 + ℓ − j)rj. The latter inequality proves that j − ℓ ≤ ⌊α∞ + 1⌋ = 5. By construction, we have i + 2 ≤ ℓ, and no integer k such that i + 2 ≤ k ≤ ℓ is an obstruction index. Hence, Lemma [lem:non-obstruction-growth] proves that 2(α∞ + 2)Δℓ − i − 2ri ≤ (α∞ + 2)rℓ ≤ (α∞ + 2 + ℓ − j)rj. Moreover, simple numerical computations, for j − ℓ ∈ {0, …, 5}, prove that Δj − ℓ + 2 ≤ 2(α∞ + 2)/(α∞ + 2 + ℓ − j), with equality when j − ℓ = 3. It follows that Δj − iri = Δj − ℓ + 2Δℓ − i − 2ri ≤ rj. Finally, the inequality of Theorem [thm:good-size] is an immediate consequence of the following result. Let S = (R1, …, Rh) be a stack obtained during the main loop of Java’s TimSort. We have rh ≥ Δh − 3. Let us first assume that S is stable. Then, r1 ≥ 1, and 1 is a growth index. If there is no obstruction index, then Lemma [lem:non-obstruction-growth] proves that rh ≥ 2Δh − 3r1 ≥ Δh − 2. Otherwise, let ℓ be the largest obstruction index. Then, ℓ − 1 is a strong growth index, and Lemma [lem:next-growth] proves that rℓ − 1 ≥ Δℓ − 2r1 ≥ Δℓ − 2. If ℓ = h, then rh ≥ rℓ − 1 ≥ Δh − 2, and if ℓ ≤ h − 1, then Lemma [lem:non-obstruction-growth] also proves that rh ≥ 2Δh − ℓ − 1rℓ − 1 ≥ Δh − ℓΔℓ − 2 = Δh − 2. Finally, if S is not stable, the result is immediate for h ≤ 3, hence we assume that h ≥ 4. The stack S was obtained by pushing a run onto a stable stack Sʹ of size at least h − 1, then merging runs from Sʹ into the runs R1 and R2. It follows that Rh was already the largest run of Sʹ, and therefore that Rh ≥ Δh − 3. ### Proving the second part of Theorem [thm:good-size] We finally focus on proving that the constant Δ of Theorem [thm:good-size] is optimal. The most important step towards this result consists in proving that α∞ = limn → ∞αn, with the real numbers αn introduced in Definition [def:alphabeta] and $\alpha\_\infty = 2 + \sqrt{7}$. Since it is already proved, in Lemma [lem:4b], that αn ≤ α∞ for all n ≥ 1, it remains to prove that α∞ ≤ liminfn → ∞αn. We obtain this inequality by constructing explicitly, for k large enough, a stable sequence of runs (R1, …, Rh) such that rh = k and r2 + … + rh − 1 ≈ α∞k. Hence, we focus on constructing sequences of runs. In addition, let us consider the stacks of runs created by the main loop of Java’s TimSorton a sequence of runs P1, …, Pn. We say below that the sequence P1, …, Pk produces a stack of runs S = (R1, …, Rh) if the stack S is obtained after each of the runs P1, …, Pn has been pushed; observe that the sequence P1, …, Pn produces exactly one stable stack. We also say that a stack of runs is producible if it is produced by some sequence of runs. Finally, in what follows, we are only concerned with run sizes. Hence, we abusively identify runs with their sizes. For instance, in Figure [fig:invstillbroken], the sequence (109, 83, 25, 16, 8, 7, 26, 2, 27) produces the stacks (27, 2, 26, 56, 83, 109) and (27, 28, 56, 83, 109); only the latter stack is stable. We review now several results that will provide us with convenient and powerful ways of constructing producible stacks.	arxiv_0000725
.7)X10 (80.7,10.7)X01 (90.7,10.7)X11 (80.7,20.7)X02 (90.7,20.7)X12 (113,14) ⇒ (120,0)(150,0) (120,0)(120,40) (130,0)(130,40) (120,10)(130,10) (120,20)(130,20) (120,30)(130,30) (120,40)(150,40) (150,0)(150,40) (130,30)(1,0)20 (140,0)(0,1)40 (130,20)(1,0)20 (130,10)(1,0)20 [fig:APhalfbc] Here we use a sweeping procedure. Let $v\_h\in V\_{h0}^{\rm MC}$. First, by Lemma [lem:ght], we have that vh∣K11 = α11 ⋅ φX11∣K11 + γ11 ⋅ φ0, K11 with some constants α11 and γ11. Therefore, vh = vh1, 1 + α11 ⋅ φX11∣K11 + γ11 ⋅ φ0, K11 with $v\_h^{1,1}\in V\_{h0}^{\rm MC}$ and vh1, 1 vanishing on K11. Second, vh1, 1 = vh1, 2 + α12 ⋅ φX12 + γ12φ0, K12 with $v\_h^{1,2}\in V\_{h0}^{\rm MC}$ and vh1, 2 vanishing on K11 and K12. Furthermore, repeat this process on all the cells of the first column, and we obtain that vh = vh1 + ∑j = 1n − 1α1jφX1j + ∑j = 1nγ1jφ0, K1j, where $v\_h^1\in V\_{h0}^{\rm MC}$ and vh1 vanishes on the whole column G1. Finally, we repeat the process from G1 to Gn, and obtain that vh = ∑i = 1m − 1∑j = 1n − 1αijφXij + ∑i = 1m∑j = 1nγijφ0, Kij. Hence the result. Let Gh be a m × n rectangular subdivision of Ω. Then, we have $$V\_h^{\rm MC}=V^{\rm BL}\_h+ ({\rm{span}}\{\varphi\_{0,K}\}\_{K\in\mathcal{G}\_h}\oplus {\rm{span}}\{\varphi\_{i}^{x}\}\_{i=1}^m\oplus {\rm{span}}\{\varphi\_{j}^{y}\}\_{j=1}^n).$$ It is obvious that $V\_h^{\rm MC} \supset V^{\rm BL}\_h+ (\text{span}\{\varphi\_{0,K}\}\_{K\in\mathcal{G}\_h}\oplus \text{span}\{\varphi\_{i}^x\}\_{i=1}^m\oplus \text{span}\{\varphi\_{j}^y\}\_{j=1}^n)$. Here we have noted that, if φ1 + φ2 + φ3 = 0 with $\varphi\_1\in {\rm{span}}\{\varphi\_{0,K}\}\_{K\in\mathcal{G}\_h}$, φ2 ∈ span{φix}i = 1m, and φ3 ∈ span{φjy}j = 1n, then φ1 = φ2 = φ3 = 0. We only have to show the other direction. Let $v\_h\in V\_h^{\rm MC}$. First, by, there exists unique constants α00, α01, α10, α11, κ1, and σ1, such that vh∣K11 = α00φX00∣K11 + α01φX01∣K11 + α10φX10∣K11 + α11φX11∣K11 + κ1φ1x∣K11 + σ1φ1y∣K11. Thus, we have vh = vh1, 1 + α00φX00 + α01φX01 + α10φX10 + α11φX11 + κ1φ1x + σ1φ1y with $v\_h^{1,1}\in V\_h^{\rm MC}$ and vh1, 1∣K11 = 0. Second, by Lemma [lem:ght], we have vh1, 1∣K12 = α02φX02∣K12 + α12φX12∣K12 + γ12φ0, K12 + σ2φ2y∣K12. Therefore, we obtain vh1, 1 = vh1, 2 + α02φX02 + α12φX12 + γ12φ0, K12 + σ2φ2y with $v\_h^{1,2}\in V\_h^{\rm MC}$ and vh1, 2∣K11 ∪ K12 = 0. (150,43)(-1,-1) (0,0)(1,0)30 (0,10)(1,0)30 (0,30)(1,0)30 (0,20)(1,0)30 (0,40)(1,0)30 (0,0)(0,1)40 (10,0)(0,1)40 (30,0)(0,1)40 (20,0)(0,1)40 (-1,-1) • (9,-1) • (-1,9) • (9,9) • (0.7,0.7)X00 (10.7,0.7)X10 (0.7,10.7)X01 (10.7,10.7)X11 (4.5,4.5)K11 (33,14) ⇒ (40,0)(50,0) (40,0)(40,10) (50,10)(50,0) (40,10)(50,10) (50,0)(1,0)20 (50,10)(1,0)20 (40,20)(1,0)30 (40,30)(1,0)30 (40,40)(1,0)30 (40,10)(0,1)30 (50,10)(0,1)30 (60,0)(0,1)40 (70,0)(0,1)40 (44,4.5)K11 (44,14.5)K12 (73,14) ⇒ (80,0)(80,20) (90,0)(90,20) (80,0)(90,0) (80,10)(90,10) (80,20)(90,20) (90,0)(1,0)20 (90,10)(1,0)20 (90,20)(1,0)20 (80,30)(1,0)30 (80,40)(1,0)30 (80,20)(0,1)20 (90,20)(0,1)20 (100,0)(0,1)40 (110,0)(0,1)40 (83,4.5)K11 (93,4.5)K21 (113,14) ⇒ (120,0)(130,0) (120,0)(120,40) (130,0)(130,40) (120,10)(130,10) (120,20)(130,20) (120,30)(130,30) (120,40)(130,40) (130,30)(1,0)20 (140,0)(0,1)40 (130,20)(1,0)20 (150,0)(0,1)40 (130,0)(1,0)20 (130,10)(1,0)20 (130,40)(1,0)20 (139,18.59)Gh Furthermore, repeat this process on the column G1, and we obtain $$\begin{aligned} \label{eq:A12} v\_h=v\_h^1+\sum\_{j=0}^n(\alpha\_0^j\varphi\_{X\_0^j}+\alpha\_1^j\varphi\_{X\_1^j})+(\sum\_{j=1}^n \gamma\_1^j\varphi\_{0,K\_1^j}) - \varphi\_{0,K\_1^1}+\sum\_{j=1}^n \sigma\_j\varphi\_j^y+\kappa\_1\varphi\_1^x,\end{aligned}$$ where $v\_{h}^1\in V\_h^{\rm MC}$ and vh1∣G1 = 0. (150,45)(-1,-1) (0,0)(1,0)40 (0,10)(1,0)40 (0,30)(1,0)40 (0,20)(1,0)40 (0,40)(1,0)40 (0,0)(0,1)40 (10,0)(0,1)40 (30,0)(0,1)40 (20,0)(0,1)40 (40,0)(0,1)40 (0.5,0.5)X00 (10.51,0.5)X10 (0.51,10.51)X01 (10.51,10.51)X11 (4.5,4.5)K11 (43,19) ⇒ (50,0)(60,0) (50,0)(50,40) (60,0)(60,40) (50,10)(60,10) (50,20)(60,20) (50,30)(60,30) (50,40)(60,40) (60,30)(1,0)30 (70,0)(0,1)40 (60,20)(1,0)30 (80,0)(0,1)40 (90,0)(0,1)40 (60,0)(1,0)30 (60,10)(1,0)30 (60,40)(1,0)30 (69,18.59)Gh (93,19) ⇒ (100,0)(140,0) (100,0)(100,40) (110,0)(110,40) (100,10)(140,10) (100,20)(110,20) (100,30)(110,30) (100,40)(110,40) (120,0)(120,10) (130,0)(130,10) (140,0)(140,10) (110,30)(1,0)30 (110,20)(1,0)30 (120,10)(0,1)30 (130,10)(0,1)30 (140,10)(0,1)30 (110,40)(1,0)30 (119,18.59)Gh Similarly, repeat this process on the row G1, and we have $$\begin{aligned} \label{eq:A13} v\_h^1=\tilde{v}\_h^{1,1}+\sum\_{i=2}^m(\alpha\_i^0\varphi\_{X\_i^0}+\alpha\_i^1\varphi\_{X\_i^1})+\sum\_{i=2}^m \gamma\_i^1\varphi\_{0,K\_i^1}+\sum\_{i=2}^m \kappa\_i\varphi\_i^x,\end{aligned}$$ with $\tilde{v}\_h^{1,1}\in V\_{h}^{\rm MC}$, and vh1, 1∣G1 ∪ G1 = 0. Finally, by the same technique as used in the proof of Theorem [thm:hombcMC], we can prove that $$\begin{aligned} \label{eq:A14} \tilde{v}\_h^{1,1}=\sum\_{\substack{2\leqslant i\leqslant m\\ 2\leqslant j\leqslant n}}\alpha\_i^j\varphi\_{X\_i^j}+\sum\_{\substack{2\leqslant i\leqslant m\\ 2\leqslant j\leqslant n}}\gamma\_i^j\varphi\_{0,K\_i^j}.\end{aligned}$$ A combination of,, and leads to $$v\_h=\sum\_{\substack{0\leqslant i\leqslant m \\ 0\leqslant j\leqslant n}}\alpha\_i^j\varphi\_{X\_i^j}+(\sum\_{\substack{1\leqslant i\leqslant m \\ 1\leqslant j\leqslant n}}\gamma\_i^j\varphi\_{0,K\_i^j}) - \gamma\_1^1\varphi\_{0,K\_1^1} +\sum\_{i=1}^m\kappa\_i\varphi\_i^x+\sum\_{j=1}^n\sigma\_j\varphi\_j^y.$$ Hence the result. From the above two theorems, it holds that ${\rm dim} (V\_{h0}^{\rm MC} )= 2mn-m-n+1$ and ${\rm dim} (V\_{h}^{\rm MC} )= 2mn+2m+2n$. [comparisionMC] Define $V^{(2)}\_h:=\Big\{v\_h\in H^1(\Omega): v\_h\|\_K \in P\_2(K), \forall K \in\mathcal{G}\_h\Big\}$, and Vh0(2) : = Vh(2) ∩ H01(Ω). That is, Vh(2) and Vh0(2) are conforming P2 element spaces. Then, 1. $V^{(2)}\_h=V^{\rm BL}\_h+{\rm span}\{\varphi\_i^x,\varphi^y\_j\}$, and $\ V^{(2)}\_{h0}=V^{\rm BL}\_{h0};$ 2. $V\_h^{\rm MC}=V^{(2)}\_h+{\rm span}\{\varphi\_{0,K}\}\_{K\in\mathcal{G}\_h}$, and $ V\_{h0}^{\rm MC}=V^{(2)}\_{h0}+{\rm span}\{\varphi\_{0,K}\}\_{K\in\mathcal{G}\_{h}}.$ [pro:MCWIL] Let $V\_h^{\rm W}$ be the Wilson element space and $V\_{h0}^{\rm W}$ be its homogeneous subspace. Then 1. $\nabla\_hV\_h^{\rm MC}\subset \nabla\_hV\_h^{\rm W}$, and $ \nabla\_hV\_{h0}^{\rm MC}\subset\nabla\_hV\_{h0}^{\rm W};$ 2. $V\_h^{\rm MC}\setminus V\_h^{\rm W}\neq \varnothing$, $V\_h^{\rm W}\setminus V\_h^{\rm MC}\neq \varnothing$, $V\_{h0}^{\rm MC}\setminus V\_{h0}^{\rm W}\neq \varnothing$, and $V\_{h0}^{\rm W}\setminus V\_{h0}^{\rm MC}\neq \varnothing$. Construction of basis functions for the reduced rectangular Morley (RRM) element space ====================================================================================== Let Ω ∈ R2 be rectangle domain divided by m × n rectangles. Define the reduced rectangular Morley element spaces as $$\begin{aligned} \begin{split} V\_{h}^{\rm{R}} := \Big\{w\in L^{2}(\Omega) : w\|\_{K} \in P\_{2}(K), & \ w\_{h}(a)\text{ is continuous at } a\in \mathcal{N}\_{h}^{i}, \text{and}\\ & \fint\_{e}\partial\_{n\_{e}} w\_{h} {\,d}s \text{ is continuous across } e \in \mathcal{E}\_{h}^{i}\Big\}; \end{split}\end{aligned}$$ $$\begin{aligned} V\_{hs}^{\rm{R}} :=\Big\{w\_{h}\in V\_{h}^{\rm{R}} : w\_{h}(a)\text{ vanishes at } a\in \mathcal{N}\_{h}^{b} \Big\}.\end{aligned}$$ In this section, we will present an available set of basis functions of $V\_{hs}^{\rm{R}}$. Compatibility conditions ------------------------ By a pure linear algebra argument, the following description holds for P2(K). () Let K be a rectangle with vertices ai and edges Γl (i, l = 1 : 4). Denote its length and width by L and H, respectively; see Figure [fig:subdivision] (Left). Then, given αi, βi ∈ R, there exists a uniquely p ∈ P2(K) satisfying $$p(a\_{i}) = \alpha\_{i}, \fint\_{\Gamma\_{1}}\partial\_{x}p=\beta\_{1}, \fint\_{\Gamma\_{2}}\partial\_{y}p=\beta\_{2}, \fint\_{\Gamma\_{3}}\partial\_{x}p=\beta\_{3}, \fint\_{\Gamma\_{4}}\partial\_{y}p=\beta\_{4}.$$ if and only if the following compatibility conditions are satisfied on K, $$\begin{aligned} \frac{\alpha\_{3}-\alpha\_{4}}{L} + \frac{\alpha\_{2}-\alpha\_{1}}{L} = \beta\_{1}+\beta\_{3};\label{eq:c1}\\ \frac{\alpha\_{3}-\alpha\_{2}}{H} + \frac{\alpha\_{4}-\alpha\_{1}}{H} = \beta\_{2}+\beta\_{4}.\label{eq:c2}\end{aligned}$$ Recall the definitions of Gh, Gi, and Gj in Appendix A. Also, the vertices are labeled by Xij, the midpoints on any edge by Yij and Zij, and the cells by Kij; see Figure [fig:subdivision] (Right). Next we present some local or global functions in $V\_{hs}^{\rm{R}}$ by giving their value on Xij ∈ Ehi and derivative on the midpoint of e ∈ Eh. (140,90)(-60,-5)(-60,0)(1,0)35 (-60,30)(1,0)35 (-60,0)(0,1)30 (-25,0)(0,1)30 (-60.8,-0.8) • (-60.8,29.2) • (-25.8,-0.8) • (-25.8,29.2) • (-59,1)a1 (-59,31)a4 (-24,1)a2 (-24,31)a3 (-59,14)Γ1 (-24,14)Γ3 (-41,1)Γ2 (-41,31)Γ4 (-42,13)K (0,0)(1,0)80 (0,20)(1,0)80 (0,40)(1,0)80 (0,60)(1,0)80 (0,80)(1,0)80 (0,0)(0,1)80 (20,0)(0,1)80 (40,0)(0,1)80 (60,0)(0,1)80 (80,0)(0,1)80 (-7.5,10)H1 (-7.5,30)H2 (-7.5,50)⋯ (-7.5,70)Hn (10,83)L1 (30,83)L2 (50,83)⋯ (70,83)Lm (19,8.5) • (9,19) • (19,19) • (-1,8.5) • (9,-1) • (19,39) • (39,39) • (39,19) • (22,10)Z21 (2,10)Z11 (12,21)Y12 (12,1)Y11 (21,21)X11 (21,41)X12 (41,21)X21 (41,41)X22 (83,9)G1 (83,29)G2 (83,49)⋯ (83,69)Gn (9,-6)G1 (29,-6)G2 (49,-6)⋯ (69,-6)Gm (9,9)K11 (29,9)K21 (9,29)K12 (29,29)K22 [fig:subdivision] Patterns of supports of basis functions in $V\_{hs}^{\rm{R}}$ ------------------------------------------------------------- Associated with Gh, we present some patterns, i.e., the support sets of basis functions in the RRM element space. To begin with, we introduce some notations. An interior edge, denoted by $e\_{{\rm bot},i}$ (2 ≤ i ≤ m − 1), is called a bottom interior edge if its two endpoints are interior points, and its lower opposite edge is on the bottom of Gh. A top interior edge $e\_{{\rm top},i}$, a left interior edge $e\_{{\rm lef},j}$, and a right interior edge $e\_{{\rm rig},j}$ are defined in a similar way (2 ≤ j ≤ n − 1). In the following lemmas, we always denote, by ω, a generic patch with boundaries Γl (l = 1 : 4), anticlockwise. [lem:basis1] Let $e\_{{\rm bot},i}$ (2 ≤ i ≤ m − 1) be a bottom interior edge with endpoints Xi − 11 and Xi1. Let ω be a 3 × 2 cells patch as shown in Figure [fig:APfunction11], left. Define $$V\_{{\rm bot},i}^{\rm R} :=\Big \{v\_{h}\in V\_{hs}^{\rm{R}}(\omega) : \fint\_{e} \partial\_{n\_{e}}v\_{h} {\,d}s =0, \ \forall e \subset \Gamma\_{l}, \ l = 1,3,4 \Big \}.$$ Likewise, we can define $V\_{{\rm top},i}^{\rm R}$ (see Figure [fig:APfunction11], right), $V\_{{\rm lef},j}^{\rm R}$, and $V\_{{\rm rig},j}^{\rm R}$ (see Figure [fig:APfunction12], left and right), where 2 ≤ i ≤ m − 1 and 2 ≤ j ≤ n − 1. Then we have ${\rm dim}(V\_{{\rm bot},i}^{\rm R})= {\rm dim}(V\_{{\rm top},i}^{\rm R}) = {\rm dim}(V\_{{\rm lef},j}^{\rm R}) ={\rm dim}(V\_{{\rm rig},j}^{\rm R}) = 1$. First, we consider $V\_{{\rm bot},i}^{\rm R}$. Let the geometric features of ω be represented as in Figure [fig:APfunction11], left. Given $\varphi \in V\_{{\rm bot},i}^{\rm R}$, denote by xij : = φ(Xij), yij : = ∂yφ(Yij), and zij : = ∂xφ(Zij). Apply conditions and on every element, we have, row by row, $$\begin{aligned} & \frac{x\_{i-1}^1}{L\_{i-1}} = z\_{i}^1,\ \frac{x\_{i-1}^1}{H\_{1}} = y\_{i-1}^1+y\_{i-1}^2,\ \frac{x\_{i}^1-x\_{i-1}^1}{L\_{i}} = z\_{i}^1+z\_{i+1}^1,\ \frac{x\_{i}^1+x\_{i-1}^1}{H\_{1}} = y\_{i}^1+y\_{i}^2, \\ & -\frac{x\_{i}^1}{L\_{i+1}} = z\_{i+1}^1,\ \frac{x\_{i}^1}{H\_{1}} = y\_{i+1}^1+y\_{i+1}^2,\ \frac{x\_{i-1}^1}{L\_{i-1}} = z\_{i}^2,\ -\frac{x\_{i-1}^1}{H\_{2}} = y\_{i-1}^2, \\ & \frac{x\_{i}^1-x\_{i-1}^1}{L\_{i}} = z\_{i}^2+z\_{i+1}^2,\ -\frac{x\_{i-1}^1+x\_{i}^1}{H\_{2}} = y\_{i}^2,\ -\frac{x\_{i}^1}{L\_{i+1}} = z\_{i+1}^2,\ -\frac{x\_{i}^1}{H\_{2}} = y\_{i+1}^2.\end{aligned}$$ Rewrite the system after adjusting the order, $$\begin{aligned} \frac{1}{L\_{i-1}}x\_{i-1}^1 - z\_{i}^1 =0,\label{eq:1}\end{aligned}$$ $$\begin{aligned} \frac{1}{H\_{1}}x\_{i-1}^1 - y\_{i-1}^1 - y\_{i-1}^2 =0,\end{aligned}$$ $$\begin{aligned} -\frac{1}{L\_{i}}x\_{i-1}^1 + \frac{1}{L\_{i}}x\_{i}^1 - z\_{i}^1 - z\_{i+1}^1 =0,\label{eq:3}\end{aligned}$$ $$\begin{aligned} \frac{1}{H\_{1}}x\_{i-1}^1 + \frac{1}{H\_{1}}x\_{i}^1 - y\_{i}^1 - y\_{i}^2 =0,\end{aligned}$$ $$\begin{aligned} - \frac{1}{L\_{i+1}}x\_{i}^1 - z\_{i+1}^1 =0,\label{eq:5}\end{aligned}$$ $$\begin{aligned} \frac{1}{H\_{1}}x\_{i}^1 - y\_{i+1}^1 - y\_{i+1}^2 =0,\end{aligned}$$ $$\begin{aligned} \frac{1}{L\_{i-1}}x\_{i-1}^1 - z\_{i}^2 = 0,\label{eq:7}\end{aligned}$$ $$\begin{aligned} -\frac{1}{H\_{2}}x\_{i-1}^1 - y\_{i-1}^2 = 0,\end{aligned}$$ $$\begin{aligned} -\frac{1}{L\_{i}}x\_{i-1}^1 + \frac{1}{L\_{i}}x\_{i}^1 - z\_{i}^2 -z\_{i+1}^2= 0,\label{eq:9}\end{aligned}$$ $$\begin{aligned} -\frac{1}{H\_{2}}x\_{i-1}^1 -\frac{1}{H\_{2}}x\_{i}^1- y\_{i}^2= 0,\end{aligned}$$ $$\begin{aligned} -\frac{1}{L\_{i+1}}x\_{i}^1 - z\_{i+1}^2 = 0,\label{eq:11}\end{aligned}$$ $$\begin{aligned} -\frac{1}{H\_{2}}x\_{i}^1 - y\_{i+1}^2 = 0.\end{aligned}$$ It is straightforward to verify that $\eqref{eq:3} - \eqref{eq:1} -\eqref{eq:5} -\big[\eqref{eq:9} - \eqref{eq:7} -\eqref{eq:11}\big ] =0.$ The system admits a one-dimension solution space. Thus we obtain ${\rm dim} (V\_{{\rm bot},i}^{\rm R}) =1$. Likewise, we have ${\rm dim} (V\_{{\rm top},i}^{\rm R}) = {\rm dim} (V\_{{\rm lef},j}^{\rm R} )= {\rm dim} (V\_{{\rm rig},j}^{\rm R}) =1 \ (\forall \ 2\leqslant i \leqslant m-1, 2\leqslant j \leqslant n-1$). (150,50)(-90,-5)(-85,40)(-25,40) (-85,0)(-85,40) (-25,0)(-25,40) (-85,0)(1,0)60 (-85,20)(1,0)60 (-65,0)(0,1)40 (-45,0)(0,1)40 (-77,-5)Li − 1 (-58,-5)Li (-38,-5)Li + 1 (-92,9)H1 (-92,29)H2 (-77,-1) • (-77,19) • (-76,1.5)Yi − 11 (-76,21.5)Yi − 12 (-56,-1) • (-56,19) • (-55,1.5)Yi1 (-55,21.5)Yi2 (-55,15)$e\_{{\rm bot},i}$ (-35,-1) • (-35,19) • (-33.5,1.5)Yi + 11 (-33.5,21.5)Yi + 12 (-66,9) • (-66,19) • (-66,29) • (-64,9)Zi1 (-64,29)Zi2 (-46,9) • (-46,19) • (-46,29) • (-44,9)Zi + 11 (-44,29)Zi + 12 (-64,21.5)Xi − 11 (-44,21.5)Xi1 (0,0)(60,0) (0,0)(0,40) (60,0)(60,40) (0,20)(1,0)60 (0,40)(1,0)60 (20,0)(0,1)40 (40,0)(0,1)40 (8,-5)Li − 1 (27,-5)Li (46,-5)Li + 1 (-8,9)Hn − 1 (-7,29)Hn (8,19) • (8,39) • (9,21.5)Yi − 1n (9,41.5)Yi − 1n + 1 (29,19) • (29,39) • (31.5,21.5)Yin (30,41.5)Yin + 1 (30,15)$e\_{{\rm top},i}$ (50,19) • (50,39) • (52,21.5)Yi + 1n (51.5,41.5)Yi + 1n + 1 (19,9) • (19,19) • (19,29) • (21,9)Zin − 1 (21,29)Zin (39,9) • (39,19) • (39,29) • (41,9)Zi + 1n − 1 (41,29)Zi + 1n (20,21.5)Xi − 1n − 1 (40.5,21.5)Xin − 1 Illustration of the column patterns and row patterns of basis functions [fig:APfunction11] (135,70)(-80,-5)(-75,0)(-25,0) (-75,60)(-25,60) (-25,0)(-25,60) (-75,0)(0,1)60 (-50,0)(0,1)60 (-75,20)(1,0)50 (-75,40)(1,0)50 (-51,19) • (-51,39) • (-49,21.5)X1j − 1 (-49,41.5)X1j (-76,9) • (-76,29) • (-76,49) • (-74,9)Z1j − 1 (-74,29)Z1j (-74,49)Z1j + 1 (-51,9) • (-51,29) • (-51,49) • (-49,9)Z2j − 1 (-49,29)Z2j (-59,29)$e\_{{\rm lef},j}$ (-49,49)Z2j + 1 (-63.5,19) • (-38.5,19) • (-62.5,21.5)Y1j (-37.5,21.5)Y2j (-63.5,39) • (-38.5,39) • (-62.5,41.5)Y1j + 1 (-37.5,41.5)Y2j + 1 (-62.5,-5)L1 (-37.5,-5)L2 (-85,9)Hj − 1 (-85,29)Hj (-85,49)Hj + 1 (0,0)(50,0) (0,60)(50,60) (0,0)(0,60) (50,0)(0,1)60 (25,0)(0,1)60 (0,20)(1,0)50 (0,40)(1,0)50 (24,19) • (24,39) • (26,22)Xm − 1j − 1 (26,42)Xm − 1j (24,9) • (24,29) • (24,49) • (26,9)Zmj − 1 (26,29)Zmj (16,29)$e\_{{\rm rig},j}$ (26,49)Zmj + 1 (49,9) • (49,29) • (49,49) • (51,9)Zm + 1j − 1 (51,29)Zm + 1j (51,49)Zm + 1j + 1 (11.5,19) • (37.5,19) • (10.5,22)Ym − 1j (36.5,22)Ymj (12.5,39) • (37.5,39) • (10.5,42.5)Ym − 1j + 1 (36.5,42.5)Ymj + 1 (11.5,-5)Lm − 1 (36.5,-5)Lm (-8.5,9)Hj − 1 (-8.5,29)Hj (-8.5,49)Hj + 1 [fig:APfunction12] With similar procedures in Lemma [lem:basis2], the following two lemmas can be obtained. [lem:basis2] Let Xm − 1n − 1 be an interior node in the northeast corner. Let ω be a 2 × 2 patch as shown in Figure [fig:APfunction3and4], left. Define $V\_{X}^{\rm R} :=\Big\{v\_{h}\in V\_{hs}^{\rm{R}}(\omega): \fint\_{e} \partial\_{n\_{e}}v\_{h} {\,d}s =0,\ \forall e \subset \Gamma\_{l}\ (l = 1,2)\Big\}$. Then ${\rm dim}(V\_{X}^{\rm R}) =1$. ()[lem:basis3] Let ω be a 3 × 3 patch as shown in Figure [fig:APfunction3and4], right. Define $V\_{K\_{i}^{j}}^{\rm R} :=\Big \{v\_{h}\in V\_{hs}^{\rm{R}}(\omega): \fint\_{e} \partial\_{n\_{e}}v\_{h} {\,d}s =0, \ \forall e \subset \Gamma\_{i} \ (1 \leqslant l \leqslant 4 )\Big \}$, where 2 ≤ i ≤ m − 1 and 2 ≤ j ≤ n − 1. Then ${\rm dim} (V\_{K\_{i}^{j}}^{\rm R} )=1$. (160,85)(-80,-5)(-75,0)(-75,50) (-75,0)(-25,0) (-75,50)(1,0)50 (-25,0)(0,1)50 (-75,25)(1,0)50 (-50,0)(0,1)50 (-51,24) • (-49.9,27)Xm − 1n − 1 (-49,14.5)Zmn − 1 (-49,39.5)Zmn (-63.5,24) • (-63.5,49) • (-38.5,24) • (-38.5,49) • (-62.5,26.5)Ym − 1n (-62.5,51.5)Ym − 1n + 1 (-36.5,26.5)Ymn (-37.5,51.5)Ymn + 1 (-51,11.5) • (-51,36.5) • (-26,11.5) • (-26,36.5) • (-24,14.5)Zm + 1n − 1 (-24,39.5)Zm + 1n (0,0)(0,75) (0,0)(75,0) (0,75)(75,75) (75,0)(75,75) (0,25)(1,0)75 (0,50)(1,0)75 (25,0)(0,1)75 (50,0)(0,1)75 (24,24) • (24,49) • (49,24) • (49,49) • (26,26.5)Xi − 1j − 1 (26,51.5)Xi − 1j (51,26.5)Xij − 1 (51,51.5)Xij (36.5,37.5)Kij (12.5,24) • (37.5,24) • (62.5,24) • (14,26.5)Yi − 1j (38.5,26.5)Yij (63.5,26.5)Yi + 1j (12.5,49) • (37.5,49) • (62.5,49) • (14,51.5)Yi − 1j + 1 (38.5,51.5)Yij + 1 (63.5,51.5)Yi + 1j + 1 (24,12.5) • (24,37.5) • (24,62.5) • (26,12.5)Zij − 1 (26,37.5)Zij (26,62.5)Zij + 1 (51,12.5)Zi + 1j − 1 (51,37.5)Zi + 1j (51,62.5)Zi + 1j + 1 (-63,-5)Lm − 1 (-38,-5)Lm (12,-5)Li − 1 (37,-5)Li (62,-5)Li + 1 (-80,12)Hn − 1 (-80,37)Hn (-5,12)Hj − 1 (-5,37)Hj (-5,62)Hj + 1 [fig:APfunction3and4] [lem:1times3] Let ω be a 3 × 1 or 1 × 3 patch in Figure [fig:AP1times3], left and right, respectively. Define $$V\_{\omega} := \Big\{v\_{h}\in V\_{hs}^{\rm R}(\omega): \fint\_{e}\partial\_{n\_{e}}v\_{h} {\,d}s =0, \ \forall e \subset \Gamma\_{l} \ ( l = 1,2,3)\Big\}.$$ Then, ${\rm dim}(V\_{\omega}) =1$ and vh(X1) = 0 implies vh ≡ 0 on ω. (110,65)(-85,-5) (-80,20)(1,0)60 (-60,0)(0,1)20 (-40,0)(0,1)20 (-80,0)(-20,0) (-80,0)(-80,20) (-20,0)(-20,20) (-61,18.5) • (-41,18.5) • (-61,22)X1 (-41,22)X2 (-71,18.5) • (-51,18.5) • (-31,18.5) • (-71,22)Y1 (-51,22)Y2 (-31,22)Y3 (-61,9) • (-41,9) • (-66,9)Z1 (-46,9)Z2 (-51,-5)Γ2 (-85,9)Γ1 (-19,9)Γ3 (-51,28)Γ4 (20,0)(0,1)60 (0,20)(1,0)20 (0,40)(1,0)20 (0,0)(20,0) (0,60)(20,60) (0,0)(0,60) (19,18.5) • (19,38.5) • (19,59) • (19,9) • (19,29) • (19,48.5) • (9,18.5) • (9,38.5) • [fig:AP1times3] [lem:basis4] If we denote ω = Gi the union of elements in the i-th column with boundaries ⋃l = 14Γl (see Figure [fig:APfunction4], left), and define $$V\_{{\rm col},i}^{\rm R} := \Big\{v\_{h} \in V\_{hs}^{\rm{R}}(\omega): \fint\_{e} \partial\_{n\_{e}}v\_{h} {\,d}s =0, \ \forall e \subset \Gamma\_{l} \ (l = 1,3) \Big\},$$ then ${\rm dim}(V\_{{\rm col},i}^{\rm R}) =1$. If ω = Gj (see Figure [fig:APfunction4], right), and define $$V\_{{\rm row},j}^{\rm R} :=\Big\{v\_{h} \in V\_{hs}^{\rm{R}}(\omega): \fint\_{e} \partial\_{n\_{e}}v\_{h} {\,d}s =0,\ \forall e \subset \Gamma\_{l} \ (l = 2,4) \Big\},$$ then ${\rm dim}(V\_{{\rm row},j}^{\rm R}) =1$. Obviously, $V\_{{\rm col},i}^{\rm R} \ne \varnothing$. Assume that $v\_{h} \in V\_{{\rm col},i}^{\rm R}$. We denote yij : = vh(Yij), where 1 ≤ j ≤ n + 1; see Figure [fig:APfunction4] (Left). Apply and to vh on each element in Gi, and we obtain yij = − yij + 1 (1 ≤ j ≤ n). It proves that ${\rm dim}(V\_{{\rm col},i}^{\rm R}) =1$. Similarly, assume that $w\_{h} \in V\_{{\rm col},i}^{\rm R}$ and denote zij : = wh(Zij) (1 ≤ i ≤ m + 1). We have zij = − zi + 1j (1 ≤ i ≤ m); see Figure [fig:APfunction4] (Right). Thus we obtain ${\rm dim}(V\_{{\rm row},j}^{\rm R}) =1$. (90,50)(10,-5) (20,0)(1,0)10 (20,10)(1,0)10 (20,20)(1,0)10 (20,30)(1,0)10 (20,40)(1,0)10 (20,0)(20,40) (30,0)(30,40) (23.5,-4)Li (31.5,4)H1 (31.5,14)H2 (31.5,25)... (31.5,34)Hn (24,1.5)Yi1 (23,12)Yi2 (24,22)Yi3 (23,42)Yin + 1 (24,-0.5) • (24,9.5) • (24,19.5) • (24,29.5) • (24,39.5) • (50,10)(90,10) (50,20)(90,20) (50,10)(0,1)10 (60,10)(0,1)10 (70,10)(0,1)10 (80,10)(0,1)10 (90,10)(0,1)10 (46,14)Z1j (56,14)Z2j (66,14)Z3j (84,14)Zm + 1j (49.5,14) • (59.5,14) • (69.5,14) • (79.5,14) • (89.5,14) • (53.5,6)L1 (63.5,6)L2 (73.5,6)... (83.5,6)Lm (91.5,14)Hj [fig:APfunction4] Here and throughout this paper, we do not distinct between $V\_{{\rm bot},i}^{\rm R},$ $V\_{{\rm top},i}^{\rm R},$ $V\_{{\rm lef},j}^{\rm R},$ $V\_{{\rm rig},j}^{\rm R},$ $V\_{X}^{\rm R},$$V\_{K\_{i}^{j}}^{\rm R},$ $V\_{{\rm col},i}^{\rm R},$ $V\_{{\rm row},j}^{\rm R}$ and their respective extension onto the whole domain by zero. Each of them is a subspace in $V\_{hs}^{\rm{R}}$. Structure of the RRM element space ---------------------------------- [thm:APbasisofRRM] Let Gh be a m × n rectangular subdivision of Ω. Define $V\_{btlr}^{\rm R} : = \oplus\_{i,j} \Big(V\_{{\rm bot},i}^{\rm R} \oplus V\_{{\rm top},i}^{\rm R}\oplus V\_{{\rm lef},j}^{\rm R} \oplus V\_{{\rm rig},j}^{\rm R} \Big)$, $V\_{K}^{\rm R} := \oplus\_{i,j} V\_{K\_{i}^{j}}^{\rm R} $, and $V\_{{\rm glob}}^{\rm R} := V\_{{\rm col},m-1}^{\rm R} \oplus V\_{{\rm col},m}^{\rm R} \oplus V\_{{\rm row},n-1}^{\rm R}\oplus V\_{{\rm row},n}^{\rm R}$, where 2 ≤ i ≤ m − 1, and 2 ≤ j ≤ n − 1. Then we have $V\_{hs}^{\rm{R}} = V\_{btlr}^{\rm R} \oplus V\_{{\rm glob}}^{\rm R} \oplus V\_{K}^{\rm R} \oplus V\_{X}^{\rm R}$. Here we utilize the sweeping procedure again, and divide the proof process into four steps. Given $v \in V\_{hs}^{\rm{R}}$. We begin with the boundaries of Gh; see Figure [fig:APElim12]. Recall the definition of $e\_{{\rm bot},i}$, $e\_{{\rm top},i}$, $e\_{{\rm lef},j}$, and $e\_{{\rm rig},j}$. Associated with $e\_{{\rm bot},2}$, $V\_{{\rm bot},2}^{\rm R}$ is a subspace of V defined in Lemma [lem:basis1]. There exists a unique function $\varphi\_{b,1}\in V\_{{\rm bot},2}^{\rm R}$, such that φb, 1(Y11) = v(Y11). Then, v1 = v − φb, 1 with v1 ≡ 0 on the bottom edge of K11. Repeat the procedure for Y21,..., Ym − 21, and we obtain vm − 2 = v − ∑i = 2m − 1φb, i with vm − 2 vanishing on the first m − 2 edges on the bottom boundary of Gh. Similarly, we obtain functions φt, i, φl, j, and φr, j. Therefore, w1 = v − ∑i = 2m − 1φb, i − ∑i = 2m − 1φt, i − ∑j = 2n − 1φl, j − ∑j = 2n − 1φr, j with w1 vanishing on the dotted edges; see Figure [fig:APElim12] (Middle). There exists uniquely $\varphi\_{{\rm col},i}$ in $V\_{{\rm col},i}^{\rm R}$ and $\varphi\_{{\rm row},j}$ in $V\_{{\rm row},j}^{\rm R}$, which satisfy $\varphi\_{{\rm col},i}(Y\_{i}^{1}) = w\_{1}(Y\_{i}^{1})$, and $\varphi\_{{\rm row},j}(Z\_{1}^{j}) = w\_{1}(Z\_{1}^{j})$, respectively. Then $w\_{2} = w\_{1}-\sum\_{i = m-1}^{m} \varphi\_{{\rm col},i} - \sum\_{j = n-1}^{n} \varphi\_{{\rm row},j}$ with w2 vanishing on the dotted edges; see Figure [fig:APElim12] (Right). Consider elements in the first column G1; see Figure [fig:AP1times3]. Note that K22 is the only interior element whose 3 × 3 patch contains K11. There exists uniquely $\varphi\_{1}^{1}\in V\_{K\_{2}^{2}}^{\rm R}$, such that φ11∣K11 = w2∣K11. Therefore, w2 − φ11 vanishes on K11. Next, there exists a unique $\varphi\_{1}^{2} \in V\_{K\_{2}^{3}}^{\rm R} $, such that w2 − φ11 − φ12 vanishes on on K11 ∪ K12. Repeating the procedure for G1, we obtain w2 − ∑j = 1n − 2φ1j, which vanishes on ∪ j = 1n − 2K1j. Notice that ∪ j = n − 2nK1j is a 1 × 3 patch ω stated in Lemma [lem:1times3]; see Figure [fig:APElim34] (Left). Since w2 − ∑j = 1n − 2φ1j vanishes on X1n − 2, it vanishes on ω and further G1. By repeating the procedure along the column Gi (i = 2, …, m − 2), we derive w3 : = w2 − ∑i = 1m − 2∑j = 1n − 2φij, which vanishes on ⋃i = 1m − 2Gi. Especially, w3(Xm − 2j) = 0 (1 ≤ j ≤ n − 2). Since ∪ i = m − 2mKi1 forms a 1 × 3 patch ω (see Figure [fig:APElim34], middle) and w3(Xm − 21) = 0, thus it vanishes on ω. Find other 1 × 3 or 3 × 1 patch ω, which satisfies the conditions in Lemma [lem:1times3], in columns Gm − 2, Gm − 1, and Gm. Therefore, we derive that w3 vanishes on Gh except for a 2 × 2 patch in the northeast corner; see Figure [fig:APElim34] (Right). (160,50)(-10,-5) (0,0)(1,0)40 (0,10)(1,0)40 (0,20)(1,0)40 (0,30)(1,0)40 (0,40)(1,0)40 (0,0)(0,1)40 (10,0)(0,1)40 (20,0)(0,1)40 (30,0)(0,1)40 (40,0)(0,1)40 (-7,4)G1 (-7,14)... (-8,24)Gn − 1 (-7,34)Gn (4,-5)G1 (14,-5)... (22,-5)Gm − 1 (33,-5)Gm (42,19) ⇒ (50,0)(70,0) (50,0)(50,20) (50,40)(70,40) (90,0)(90,20) (70,0)(1,0)20 (70,40)(1,0)20 (50,20)(0,1)20 (90,20)(0,1)20 (50,10)(1,0)40 (50,20)(1,0)40 (50,30)(1,0)40 (60,0)(0,1)40 (70,0)(0,1)40 (80,0)(0,1)40 (92,19) ⇒ (100,0)(140,0) (100,0)(100,40) (100,40)(120,40) (140,0)(140,20) (100,10)(1,0)40 (100,20)(1,0)40 (100,30)(1,0)40 (110,0)(0,1)40 (120,0)(0,1)40 (130,0)(0,1)40 (120,40)(1,0)20 (140,20)(0,1)20 [fig:APElim12] (160,50)(-10,-5) (0,0)(0,40) (0,0)(40,0) (0,10)(10,10) (10,0)(10,10) (40,0)(40,20) (0,40)(20,40) (10,10)(1,0)30 (0,20)(1,0)40 (0,30)(1,0)40 (10,10)(0,1)30 (20,0)(0,1)40 (30,0)(0,1)40 (20,40)(1,0)20 (40,20)(0,1)20 (9,19) • (4,25)ω (-7,4)G1 (-7,14)... (-8,24)Gn − 1 (-7,34)Gn (4,-5)G1 (14,-5)... (22,-5)Gm − 1 (33,-5)Gm (42,19) ⇒ (50,0)(70,0) (50,0)(50,40) (50,40)(70,40) (90,0)(90,20) (60,0)(60,40) (50,10)(60,10) (50,20)(60,20) (50,30)(60,30) (70,0)(90,0) (70,40)(1,0)20 (90,20)(0,1)20 (60,10)(1,0)30 (60,20)(1,0)30 (60,30)(1,0)30 (70,0)(0,1)40 (80,0)(0,1)40 (69,9) • (69,19) • (69,29) • (74,4)ω (92,19) ⇒ (100,0)(140,0) (100,10)(140,10) (100,20)(140,20) (100,30)(120,30) (110,0)(110,40) (120,0)(120,40) (130,0)(130,20) (100,0)(100,40) (100,40)(120,40) (140,0)(140,20) (120,30)(1,0)20 (130,20)(0,1)20 (120,40)(1,0)20 (140,20)(0,1)20 (124,4)ω (124,14)ω (114,24)ω [fig:APElim34] From Lemma [lem:basis2], there exists a unique function $\varphi\_{m-1}^{n-1}\in V\_{X}^{\rm R}$ and w3 − φm − 1n − 1 = 0. This way, we have verified that $v = (\sum\_{i=2}^{m-1}\varphi\_{b,i} + \sum\_{i=2}^{m-1}\varphi\_{t,i} + \sum\_{j=2}^{n-1}\varphi\_{l,j} + \sum\_{j=2}^{n-1}\varphi\_{r,j}) + (\sum\_{i = m-1}^{m} \varphi\_{{\rm col},i} + \sum\_{j = n-1}^{n} \varphi\_{{\rm row},j}) + \sum\_{i=1}^{m-2}\sum\_{j=1}^{n-2}\varphi^j\_{i} + \varphi\_{m-1}^{n-1}$, and this representation by these mn + 1 functions is unique. From Theorem [thm:APbasisofRRM], it holds that ${\rm dim} V\_{hs}^{\rm{R}} = mn+1$. [pro:relationof3] It holds for $V\_{h}^{\rm{W}}$, $V\_{h0}^{\rm{W}}$, $V\_{h}^{\rm{M}}$, $V\_{hs}^{\rm{M}}$, $V\_{h}^{\rm{R}}$, and $V\_{hs}^{\rm{R}}$ that $$\begin{aligned} V\_{h}^{\rm{R}} & = V\_{h}^{\rm{M}} \cap V\_{h}^{\rm{W}};\\ V\_{hs}^{\rm{R}} & = V\_{hs}^{\rm{M}} \cap V\_{h0}^{\rm{W}}.\end{aligned}$$ --- 1. Zeng and C.-S. Zhang are partially supported by National Key Research and Development Program 2016YFB0201304, China. C.-S. Zhang is also supported by the Key Research Program of Frontier Sciences of CAS. S. Zhang is partially supported by National Natural Science Foundation, 11471026 and 11871465, China[↩](#fnref1) Optimal quadratic element on rectangular grids for H1 problems ================================================================ [1](#fn1) In this paper, a piecewise quadratic finite element method on rectangular grids for H1 problems is presented. The proposed method can be viewed as a reduced rectangular Morley (RRM) element. For the source problem, the convergence rate of this scheme is proved to be O(h2) in the energy norm on uniform grids over a convex domain. A lower bound of the L2-norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings. Introduction ============ The design and capacity analysis of the discretization schemes for the source problem (say, the boundary value problem) and the eigenvalue problem are key issues in numerical analysis and, in general, of approximation theory. When the approximation of functions in Sobolev spaces is performed using piecewise polynomials defined on a domain partition, lower-degree polynomials are often preferred in order to achieve a simpler interior structure. A finite element scheme with polynomials of the total degree no more than k, denoted by Pk, is called optimal if it achieves O(hk + 1 − m) accuracy in the energy norm for Hm elliptic problems. In this paper, we present an optimal quadratic element scheme for the H1 problems, including the source problem and the eigenvalue problem, on rectangular grids, and present its error analysis. The study of optimal finite element schemes has been attracting wide interests. For the case wherein the grid comprises simplexes, there are already some systematic results. It is known that the Lagrange finite elements of arbitrary degree on domains of arbitrary dimension are optimal conforming elements for second-order elliptic problems. At the same time, a systematic family of minimal-degree nonconforming finite elements is proposed by, where m-th degree polynomials work for 2m-th order elliptic problems in Rn for any n ≥ m. Known as the Wang-Xu or Morley-Wang-Xu family, these elements are constructed based on the perfect matching between the dimension of m-th degree polynomials and the dimension of (n − k)-subsimplexes with 1 ≤ k ≤ m. The generalisation to the cases where n < m is attracting increasing research interest (see, e.g., ). These spaces can be naturally used for both the source problem and the eigenvalue problem. On the other hand, to clarify the capacity of the schemes clearly, some kinds of extremal analysis have also been conducted, including, e.g., lower bounds of the error estimates and guaranteed bounds of the computed eigenvalues. We refer to, e.g., for a general analysis of the lower bounds of the discretization error for piecewise polynomials, and for specific analysis with certain finite element schemes. We refer to, e.g., for the computed guaranteed bounds of certain eigenvalue problems. The extremal analysis is naturally used on or ready to be generalized to optimal schemes. When the grid comprises shapes other than simplexes, the design of optimal schemes becomes more complicated. We would like to recall that, Qk (rather than Pk) polynomials are used for 2k-th order problems on Rn rectangular grids by, which form minimal conforming element spaces. For biharmonic equation, some low-degree rectangular elements have been designed, including the rectangular Morley element and incomplete P3 element. Very recently, a space, consisting of exactly piecewise quadratic polynomials, is constructed and shown convergent for the biharmonic equation on general quadrilateral grids, which forms a convergent scheme of the minimal degree. Also, there have been several rectangle elements for H1 problems in the literature. In, an enriched quadratic nonconforming element on rectangles is introduced, and second-order error is shown, which is generalized to higher orders by. Another second-order quadratic element is given by, where the spline technique is used, but the shape function space on a cell is not exactly P2. The Wilson element is the first quadratic quadrilateral nonconforming element. Despite its superior performance in practice, as shown in, its global asymptotic convergence rate is the same as that of the bilinear element, due to low internal continuity. Generally, this deficiency can be compensated by equipping the piecewise quadratic polynomial with second order moment-continuity across the internal edges. In this way, the moment-continuous (MC) element space is defined. However, it is proved in Appendix [sec:appA], that the MC element space possesses essentially the same accuracy as that of the bilinear element space, and thus it fails to reach second-order convergence rate. To our best knowledge, it remains open whether an optimal scheme can be constructed with degrees higher than minimal even on rectangular grids and for H1 problems. In this paper, we study the optimal finite element construction for the H1 problems, and present a finite element space comprising piecewise quadratic polynomials on uniform rectangular grids that can provide O(h2) convergence in energy norm for the source and eigenvalue problems. The computed eigenvalues are lower bounds of the exact ones, which can be proved theoretically and verified numerically. Only rectangular grids are taken into consideration herein, but if a quadrilateral grid is only a sufficiently small perturbation of a uniform one, then an optimal convergence rate could be expected on it. Moreover, the finite element functions cannot be described with free rein cell by cell. Similar to the elements described in and in many spline-type methods, the number of continuity restrictions of the finite element function is greater than the dimension of the local polynomial space. We believe this difficulty is not abnormal for low-degree schemes. In general, these cells can share interfaces with more neighbour cells, and more continuity restrictions will strengthen the requirement for higher-degree polynomials, generally higher than the order of the underlying Sobolev space. Thus, constructing consistent finite elements in the formulation of Ciarlet’s triple is difficult with m-th degree polynomials for Hm problems even on rectangular grids. Here we utilize some non-standard technical approaches to overcome the difficulty for both implementation and especially theoretical analysis. The main difficulty is that the local interpolation is too difficult, if ever possible, to be established, which plays a fundamental role in the approximation error analysis for the source problem and the guaranteed bounds analysis for the eigenvalue problem. Notice that the space constructed herein can be viewed as a reduced rectangular Morley element space. Similar to the approach in but with technical modifications, we can determine that the finite element functions are discrete stream functions of the discrete divergence-free functions constructed in a study ; using this exact relation, we can perform the approximation estimation indirectly. Also, the discretization of the eigenvalue problem can be viewed as an inner approximation of the rectangular Morley element scheme, and this helps avoid the direct dependence on an interpolation. This newly-designed routine method of theoretical analysis can be potentially used to find out other optimal schemes. Finally, we remark that, two examples, namely the rectangular Morley (RM) element and the reduced rectangular Morley (RRM) element, are reported in this paper that when used for the eigenvalue problem, errors of the eigenvalues and eigenfunctions are of the same order. This unusual performance is due to the fact that no nontrivial conforming finite element subspace can be found contained in these two spaces. The rest of the paper is organized as follows. In Section [sec:pre], some preliminaries are given and some related low-degree rectangle elements are reviewed. In Section [sec:rmrev], the rectangular Morley element is revisited. In Section [sec:rrmscheme], a reduced rectangular Morley element scheme is presented for both source problem and eigenvalue problem. In Section [sec:analysisrrm], the convergence analysis and lower bound properties are shown for the RRM element scheme. In Section [sec:concdisc], some concluding remarks and discussions are given. In contrast to a general implementation approach in Section [sec:analysisrrm], concise sets of basis functions of the MC element and the RRM element are presented in the appendix. Preliminaries ============= Notations --------- Throughout this paper, we use Ω for a simply-connected polygonal domain in R2. We use ∇, $\curl$, $\dv$, and ∇2 for the gradient operator, curl operator, divergence operator, and Hessian operator, respectively. As usual, we use H2(Ω), H02(Ω), H1(Ω), H01(Ω), and L2(Ω) for certain Sobolev spaces. Specifically, we denote $\displaystyle L^2\_0(\Omega):=\Big\{w\in L^2(\Omega):\int\_\Omega w dx=0\Big\}$, $\undertilde{H}{}^1\_0(\Omega):=\big(H^1\_0(\Omega)\big)^2$, and $\uH{}^1\_{\bf n}(\Omega):=\Big\{v \in \big(H^1(\Omega)\big)^{2}:v\cdot {\bf n}\big\|\_{\partial \Omega}=0\Big\}$. Denote, by $\uH{}^{-1}(\Omega)$ and H− 1(Ω), the dual spaces of $\uH{}^1\_0(\Omega)$ and H01(Ω), respectively. We use $``\undertilde{~}"$ for vector valued quantities in the present paper, and $\uv{}^1$ and $\uv{}^2$ for the two components of the function $\uv$. We utilize the subscript $``\cdot\_h"$ to indicate the dependence on grids. Particularly, an operator with the subscript ’$``\cdot\_h"$ implies the operation is done cell by cell. Finally, $\lesssim$, $\gtrsim$, and $\cequiv$ respectively denote ≤ , ≥ , and = up to a generic positive constant, which might depend on the shape-regularity of subdivisions, but not on the mesh-size h. Let Gh be in a regular family of quadrilateral grids of domain Ω. Let Nh be the set of all vertices, Nh = Nhi ∪ Nhb, with Nhi and Nhb comprising the interior vertices and the boundary vertices, respectively. Similarly, let Eh = Ehi⋃Ehb be the set of all the edges, with Ehi and Ehb comprising the interior edges and the boundary edges, respectively. For an edge e, ne is a unit vector normal to e and $\boldsymbol{\tau}\_e$ is a unit tangential vector of e such that $\mathbf{n}\_e\times \boldsymbol{\tau}\_e>0$. On the edge e, we use ⟦ ⋅ ⟧e for the jump across e. If e ⊂ ∂Ω, then ⟦ ⋅ ⟧e is the evaluation on e. The subscript ⋅ e can be dropped when there is no ambiguity brought in. Some rectangular finite element spaces -------------------------------------- Suppose that K ⊂ R2 represents a rectangle with sides parallel to the two axis respectively. Let (xK, yK) be the barycenter of K. Let hx, K, hy, K be the length of K in the x and y directions, respectively. Let ai and ei denote an vertex and an edge of K, respectively. Let $h := \max\limits\_{K \in \mathcal{G}\_{h}}\max\{h\_{x,K},h\_{y,K} \}$ be the mesh size of Gh. When Gh is uniform, we denote hx : = hx, K and hy : = hy, K. Let Pl(K) denote the space of polynomials on K of total degree no bigger than l. Let Ql(K) denote the space of polynomials of degree no bigger than l in each variable. Similarly, we define spaces Pl(e) and Ql(e) on an edge e. ### The Q1 element The Q1 element is defined by (K, PKBL, DKBL) with the following properties: * K is a rectangle; * PKBL = Q1(K); * for any v ∈ H1(K), $D\_{K}^{\text{BL}} =\big\{v(a\_{i})\big\}\_{i=1:4}$. Define the Q1 element space as $$V\_{h}^{\text{BL}} :=\Big\{w\_{h}\in H^{1}(\Omega) : w\_{h}\|\_{K} \in Q\_{1}(K),\ \forall K\in \mathcal{G}\_{h} \Big\}.$$ Associated with H01(Ω), we define $ V\_{h0}^{\text{BL}} :=\Big\{w\_{h}\in V\_{h}^{\text{BL}} : w\_{h}(a) = 0, \ \forall a\in \mathcal{N}\_{h}^{b}\Big\}. $ ### The Park–Sheen (PS) element The PS element is a piecewise linear nonconforming finite element space for H1 problems. It is defined as $$V\_{h}^{\rm{PS}} :=\Big\{w\_{h}\in L^{2}(\Omega) : w\_{h}\|\_{K} \in P\_{1}(K),\ \forall K\in \mathcal{G}\_{h}, \mbox{ and }\fint\_{e} {\llbracket {w\_{h}} \rrbracket} {\,d}s =0, \ \forall e \in \mathcal{E}\_{h}^{i} \Big\}.$$ Associated with H01(Ω), we define $ V\_{h0}^{\rm{PS}} :=\Big\{w\_{h} \in V\_{h}^{\rm{PS}} : \fint\_{e} w\_{h} {\,d}s =0, \ \forall e\in \mathcal{E}\_{h}^{b}\Big\}. $ ### The rotated Q1 ($Q\_{1}^{\rm{rot}}$) element The $Q\_{1}^{\rm{rot}}$ element is defined by $(K,P\_{K}^{\rm{rQ}},D\_{K}^{\rm{rQ}})$ with the following properties: * K is a rectangle; * $P\_{K}^{\rm{rQ}} = P\_{1}(K) + \text{span}\{x^{2}-y^{2}\}$; * for any v ∈ H1(K), $D\_{K}^{\rm{rQ}} = \Big\{\fint\_{e\_{i}}v {\,d}s \Big\}\_{i=1:4}$. Define the $Q\_{1}^{\rm{rot}}$ element space as $$V\_{h}^{\text{rQ}} := \Big\{w\_{h}\in L^{2}(\Omega) : w\_{h}\|\_{K} \in P\_{K}^{\rm rQ},\ \forall K \in \mathcal{G}\_{h}, \mbox{ and } \fint\_{e} {\llbracket {w\_{h}} \rrbracket}=0 {\,d}s, \ \forall e \in \mathcal{E}\_{h}^{i} \Big\}.$$ Associated with H01(Ω), we define $ V\_{h0}^{\text{rQ}} :=\Big\{w\_{h} \in V\_{h}^{\text{rQ}} : \fint\_{e} w\_{h} {\,d}s =0, \ \forall e\in \mathcal{E}\_{h}^{b}\Big\}. $ ### The Lin–Tobiska–Zhou (LTZ) element The LTZ element() is defined by $(K,P\_{K}^{\rm{LTZ}},D\_{K}^{\rm{LTZ}})$ with the following properties: * K is a rectangle; * $P\_{K}^{\rm{LTZ}} = P\_{1}(K) + \text{span}\{x^{2}, y^{2}\}$; * for any v ∈ H1(K), $D\_{K}^{\rm{LTZ}} = \Big\{\fint\_{K} v {\,d}x d y, \ \fint\_{e\_{1}}v {\,d}s, \ \ldots,\ \fint\_{e\_{4}}v {\,d}s \Big\}$. Define the LTZ element space as $$V\_{h}^{\rm{LTZ}} :=\Big\{w\_{h}\in L^{2}(\Omega) : w\_{h}\|\_{K} \in P\_{K}^{\rm{LTZ}}, \ \forall K\in \mathcal{G}\_{h}, \mbox{ and } \fint\_{e} {\llbracket {w\_{h}} \rrbracket} {\,d}s =0, \ \forall e \in \mathcal{E}\_{h}^{i} \Big\}.$$ Associated with H01(Ω), we define $ V\_{h0}^{\rm{LTZ}} := \Big\{w\_{h} \in V\_{h}^{\rm{LTZ}} : \fint\_{e}w\_{h} {\,d}s =0, \ \forall e\in \mathcal{E}\_{h}^{b} \Big\}, $ and associated with $\uH{}\_\mathbf{n}^1(\Omega)$, we define $ \uV{}\_{h{\bf n}}^{\rm LTZ}:= \Big\{\uv{}\_h\in \big(V\_h^{\rm LTZ}\big)^2:\int\_e\uv{}\_h\cdot\mathbf{n}=0, \ \forall e\in\mathcal{E}\_h^b \Big\}.$ ### The Wilson element The Wilson element is defined by $(K,P\_{K}^{\rm{W}},D\_{K}^{\rm{W}})$ with with the following properties: * K is a rectangle; * $P\_{K}^{\rm{W}} = P\_{2}(K)$; * for any v ∈ H2(K), $D\_{K}^{\rm{W}} = \Big\{ v(a\_{1}),\ \ldots, v(a\_{4}),\ \fint\_{K}\partial\_{xx}v {\,d}x d y, \ \fint\_{K}\partial\_{yy}v {\,d}x d y \Big\}$. Define the Wilson element space as $$V\_{h}^{\rm{W}} :=\Big\{w\_{h}\in L^{2}(\Omega) : w\_{h}\|\_{K} \in P\_{K}^{\rm{W}},\ \forall K\in \mathcal{G}\_{h}, \mbox{ and } w\_{h} \mbox{ is continuous at any } a\in \mathcal{N}\_{h}^{i}\Big\}.$$ Associated with H01(Ω), we define $ V\_{h0}^{\rm{W}} :=\Big\{w\_{h}\in V\_{h}^{\rm{W}} : w\_{h}(a)=0,\ \forall a\in \mathcal{N}\_{h}^{b}\Big\}. $ ### The moment-continuous (MC) element Associated with Gh, the MC element space is defined as $$V\_h^{\rm MC}:=\Big\{w\_h\in L^2(\Omega):w\_h\|\_K\in P\_2(K), \mbox{ and } w\_h \mbox{ is\ moment-continuous on }\mathcal{G}\_h\Big\}.$$ Associated with H01(Ω), we define $$V\_{h0}^{\rm MC}:=\Big\{w\_h\in V\_h^{\rm MC}: w\_h \mbox{ is\ moment-homogeneous on } \mathcal{G}\_h\Big\}.$$ A piecewise quadratic polynomial function w is moment-continuous of second-order, if ∫e⟦w⟧ev ds = 0, ∀v ∈ P1(e), e ∈ Ehi. Moreover, w is moment-homogeneous of second-order, if ∫ewv ds = 0, ∀v ∈ P1(e), e ∈ Ehb. A piecewise quadratic function vh belongs to $V\_{h0}^{\rm MC}$ if and only if vh is continuous at the second-order Gauss points of any e ∈ Ehi and vanishes on the Gauss points of any e ∈ Ehb. [thm:hombc] If Gh is a m × n rectangular subdivision of Ω, then ${\rm dim} (V\_{h0}^{\rm MC} )= 2mn-m-n+1$. [thm:nobc] If Gh be a m × n rectangular subdivision of Ω, then ${\rm dim} (V\_{h}^{\rm MC} )= 2mn+2m+2n$. Detailed proof of Theorems [thm:hombc] and [thm:nobc] are put in Appendix [sec:appA], and available sets of basis functions of $V\_{h0}^{\rm MC}$ and $V\_{h}^{\rm MC}$ are also presented there. Some technical lemmas --------------------- In addition to these spaces above, we denote $$\begin{aligned} & \mathcal{L}^0\_{h}:=\Big\{q\in L^2(\Omega):q\|\_K\in P\_0(K)\Big\}, \quad \mathcal{L}^0\_{h0}:=\mathcal{L}^0\_{h}\cap L^2\_0(\Omega), \\ & \uV{}^{\rm BL}\_{h\bf n}:=\Big\{\uv\in (V^{\rm BL}\_h)^2:\uv\cdot\mathbf{n}\|\_{\partial \Omega}=0\Big\}, \quad \uV{}^{\rm rQ}\_{h\bf n}:=\Big\{\uv\in (V^{\rm rQ}\_h)^2:\int\_e\uv\cdot\mathbf{n}=0,\ \forall\,e\in\mathcal{E}\_h^b\Big\}, \\ & \uV{}^{\rm PS}\_{h\bf n}:=(V^{\rm PS}\_h)^{2}\cap \uV{}^{\rm rQ}\_{h\bf n}, \quad \uV{}\_{h\bf n}^{\rm LTZ}:= \Big\{\uv{}\in (V\_h^{\rm LTZ})^2:\int\_e\uv{}\cdot\mathbf{n}=0,\ e\in\mathcal{E}\_h^b \Big\}.\end{aligned}$$ Let $\Pi^{\rm rQ}\_h$ and $\undertilde{\Pi}{}^{\rm rQ}\_h$ be the nodal interpolation associated with $V^{\rm rQ}\_h$ and $(V{}^{\rm rQ}\_{h})^2$, respectively. (,)[lem:converRQ] For the $Q\_{1}^{\rm{rot}}$ element, we have * $\big\|\Pi^{\rm rQ}\_h v \big\|\_{1,h} \lesssim \|v\|\_{1,h}$, ∀v ∈ H01(Ω) ∩ H2(Ω); * $\big\\| \Pi^{\rm rQ}\_h v - v \big\\|\_{0,\Omega} + h\big\|\Pi^{\rm rQ}\_h v - v\big\|\_{1,h} \lesssim h^{2}\|v\|\_{2,\Omega}$, ∀v ∈ H01(Ω) ∩ H2(Ω). () [lem:relationBLPS]The following relationships hold. $$\Pi^{\rm rQ}\_h V^{\rm BL}\_{h}=V^{\rm PS}\_h \ \mbox{ and }\ \undertilde{\Pi}{}^{\rm rQ}\_h\uV{}^{\rm BL}\_{h\bf n}=\uV{}^{\rm PS}\_{h\bf n}.$$ H1 elliptic problems and nonconforming finite element approximation --------------------------------------------------------------------- In this paper, we consider the following model problems: • Source problem: with f ∈ Q : = L2(Ω), ρ ∈ L∞(Ω), and ρ ≥ c0 > 0, $$\label{eq:Poisson} \left\{ \begin{split} -\Delta u &= \rho f & \text{in} &\; \Omega, \\ u & = 0 & \text{on} & \; \partial\Omega. \end{split} \right.$$ Its weak form is given by: Find u ∈ V : = H01(Ω) satisfying a(u, v) = b(f, v), ∀v ∈ V, where $ a(u,v) = \int\_{\Omega} \nabla u \cdot \nabla v {\,d}\emph{xdy}$ and $b(f,v) = \int\_{\Omega} \rho f v {\,d}\emph{xdy}.$ • Eigenvalue problem: with ρ ∈ L∞(Ω) and ρ ≥ c0 > 0, $$\label{eq:eigen} \left\{ \begin{split} -\Delta u &= \lambda \rho u & \text{in} &\; \Omega, \\ u & = 0 & \text{on} & \; \partial\Omega. \end{split} \right.$$ Its weak form is given by: Find (λ, u) ∈ R × V with ∥u∥0, ρ = 1, such that a(u, v) = λb(u, v), ∀v ∈ V, where $\\|v\\|\_{0,\rho} := b(v,v)^{\frac{1}{2}}$ defines a norm over V equivalent to the usually L2 norm. From, the eigenvalue problem has a sequence of eigenvalues $$\begin{aligned} 0 < \lambda\_{1} \leqslant \lambda\_{2} \leqslant \cdots \leqslant \lambda\_{k} \leqslant \cdots, \text{ satisfying } \lim\_{k\to \infty} \lambda\_{k} = \infty, \end{aligned}$$ and corresponding eigenfunctions $$\begin{aligned} u\_{1}, \ u\_{2},\ \cdots, \ u\_{k},\cdots, \text{ satisfying } b(u\_{i},u\_{j})=\delta\_{ij}.\end{aligned}$$ For a certain eigenvalue λj of, we define $$M(\lambda\_{j}) = 􏰛\{w \in V : w \text{ is an eigenfunction of }\eqref{eq:varEigen} \text{ corresponding to }\lambda\_{j}\}.$$ Given an discrete space Vh defined on Gh, the discretization schemes are • for the source problem: Find uh ∈ Vh, such that $$\begin{aligned} \label{eq:dvarP} a\_{h}(u\_{h},v\_{h}) = b(f,v\_{h}), \quad \forall v\_{h}\in V\_{h},\end{aligned}$$ • for the eigenvalue problem: Find (λh, uh) ∈ R × Vh with ∣∣uh∣∣0, ρ = 1, such that $$\begin{aligned} \label{eq:dEigen} a\_{h}(u\_{h},v\_{h}) = \lambda\_{h}b(u\_{h},v\_{h}), \quad \forall v\_{h}\in V\_{h}.\end{aligned}$$ Let dimVh = N. The discrete eigenvalue problem has a sequence of eigenvalues 0 < λ1, h ≤ λ2, h ≤ ⋯ ≤ λN, h, and corresponding eigenfunctions u1, h, u2, h, ⋯, uN, h, satisfying b(ui, h, uj, h) = δij. () [lem:EGregularity] Suppose that Ω is a rectangular region and ρ is smoothing enough. If (λj, uj) is an eigen-pair of, then uj ∈ C5, α(Ω). The rectangular Morley (RM) element revisited ============================================= The RM element space -------------------- The RM element is defined by $(K,P\_{K}^{\rm{M}},D\_{K}^{\rm{M}})$ with the following properties: * K is a rectangle; * $P\_{K}^{\rm{M}} = P\_{2}(K) + \text{span}\{x^{3},y^{3}\}$; * for any v ∈ H2(K), $D\_{K}^{\rm{M}} =\big\{ v(a\_{i}), \fint\_{e\_{i}}\partial\_{n\_{e\_{i}}}v {\,d}s \big\}\_{i=1:4}$. Define the RM element space as $$\begin{split} V\_{h}^{\rm{M}} := \Big\{w\_{h}\in L^{2}(\Omega) : w\_{h}\|\_{K} \in P\_{K}^{\rm{M}}, \ & w\_{h}(a)\mbox{ is continuous at any } a\in \mathcal{N}\_{h}^{i}, \\ & \mbox{and} \fint\_{e}\partial\_{n\_{e}} w\_{h} {\,d}s \mbox{ is continuous across any } e \in \mathcal{E}\_{h}^{i} \Big\}. \end{split}$$ Associated with H01(Ω), we define $V\_{hs}^{\rm{M}} :=\Big\{w\_{h}\in V\_{h}^{\rm{M}} : w\_{h}(a)=0, \forall a\in \mathcal{N}\_{h}^{b} \Big\}.$ ()[lem:consisRM] Denote Eh(w, vh) : = ah(w, vh) + (Δw, vh) with w ∈ Vand vh ∈ Vh. The following estimates hold. * For any shape-regular rectangular grid, it holds for any $v\_{h} \in V\_{hs}^{\rm{M}}$ that $$\begin{aligned} \| E\_{h}(v, v\_{h}) \| \lesssim \sum\_{K\in \mathcal{G}\_{h}} h\_{K}^{2} \|v\|\_{2,K} \|v\_{h}\|\_{2,K}\lesssim h \|v\|\_{2,\Omega} \|v\_{h}\|\_{1,h}, \quad \forall v\in H^{2}(\Omega)\cap H\_{0}^{1}(\Omega).\end{aligned}$$ * For any uniform rectangular grid, it holds for any $v\_{h} \in V\_{hs}^{\rm{M}}$ that $$\begin{aligned} \| E\_{h}(v, v\_{h}) \| \lesssim h^{k-1} \|v\|\_{k,\Omega} \|v\_{h}\|\_{1,h}, \quad \forall v\in H^{k}(\Omega)\cap H\_{0}^{1}(\Omega), \quad k = 2,3.\end{aligned}$$ For the RM element, there is a refined property of the interpolation operator $\Pi\_{h}^{\rm{M}}: V \mapsto V\_{hs}^{\rm{M}}$. () [lem:propertyPiRM] Assume that Gh is uniform. For any w ∈ H01(Ω) ∩ H3(Ω) with $\big\\| \frac{\partial^{2}w}{\partial x\partial y } \big\\|\_{0,\rho} \ne 0$, if h is small enough, then $$\begin{aligned} \label{eq:propertyPiRM} a\_{h}(w-\Pi\_{h}^{\rm{M}}w,\Pi\_{h}^{\rm{M}}w) \geqslant \alpha h^{2},\end{aligned}$$ where α > 0 is a constant independent of h. [cor:expansion2] Under the conditions in Lemma [lem:propertyPiRM], there exists α1 > 0, such that $$\begin{aligned} \label{eq:property1PiRM} a\_{h}(w-\Pi\_{h}^{\rm{M}}w,w)\geqslant \alpha\_{1} h^{2}.\end{aligned}$$ It follows from Lemma [lem:propertyPiRM] and $\big\|w-\Pi\_{h}^{\rm{M}}w\big\|\_{1,h} \lesssim h^{2}$ immediately. Hence we obtain an interesting and intuitive conclusion: $$a\_{h}(u-\Pi\_{h}^{\rm M}u,u) \cequiv \big\|u-\Pi\_{h}^{\rm M}u\big\|\_{1,h} \big\|u\big\|\_{1,h}, \text{ when } h \text{ is small enough}.$$ By standard argument, we can prove the exact sequence which reads $$\begin{array}{ccccccccc} 0 & \longrightarrow & V\_{hs}^{\rm{M}} & \xrightarrow{\curl\_h} & \uV{}\_{h\bf n}^{\rm LTZ} & \xrightarrow{\dv\_h} & \mathcal{L}\_h^{1,-1} & \longrightarrow & 0, \end{array}$$ where Lh1, − 1 = {q ∈ L02(Ω) : q∣K ∈ P1(K)}. The RM element scheme for the H1 eigenvalue problem ----------------------------------------------------- ### Expanded representation of the difference between energy of states Simple calculations yield $$\label{eq:xpyh} \begin{split} a(\upsilon\_1,\upsilon\_1) + a\_{h}(\upsilon\_2,\upsilon\_2) &= 2a\_{h}(\upsilon\_1,\upsilon\_2) + a\_{h}(\upsilon\_1-\upsilon\_2,\upsilon\_1-\upsilon\_2)\\ & = 2a\_{h}(\upsilon\_3,\upsilon\_2) + 2a\_{h}(\upsilon\_1-\upsilon\_3,\upsilon\_3)\\ & \quad + 2a\_{h}(\upsilon\_1-\upsilon\_3,\upsilon\_2-\upsilon\_3) + a\_{h}(\upsilon\_1-\upsilon\_2,\upsilon\_1-\upsilon\_2), \end{split}$$ $${\label{eq:yhmx}} \begin{split} a\_{h}(\upsilon\_2,\upsilon\_2) - a(\upsilon\_1,\upsilon\_1) & = \left[2a\_{h}(\upsilon\_3,\upsilon\_2) - 2a(\upsilon\_1,\upsilon\_1)\right] + 2a\_{h}(\upsilon\_1-\upsilon\_3,\upsilon\_3)\\ & \quad + 2a\_{h}(\upsilon\_1-\upsilon\_3,\upsilon\_2-\upsilon\_3) + a\_{h}(\upsilon\_1-\upsilon\_2,\upsilon\_1-\upsilon\_2), \end{split}$$ $$\label{eq:xmyh} \begin{split} a(\upsilon\_1,\upsilon\_1) - a\_{h}(\upsilon\_2,\upsilon\_2) & = 2a\_{h}(\upsilon\_3-\upsilon\_2,\upsilon\_2) + 2a\_{h}(\upsilon\_1-\upsilon\_3,\upsilon\_3)\\ & \quad + 2a\_{h}(\upsilon\_1-\upsilon\_3,\upsilon\_2-\upsilon\_3) + a\_{h}(\upsilon\_1-\upsilon\_2,\upsilon\_1-\upsilon\_2). \end{split}$$ Let u be the solution of the source problem or the eigenvalue problem and uh be its approximation. Let υ1 = u, υ2 = uh and υ3 = Πhu, where Πh : V ↦ Vh0 is an interpolation operator. We use to obtain an expansion of b( − f, u − uh) and to obtain an expansion of λ − λh. • For the source problem: Let u and uh be the solutions of and, respectively. It holds that b( − f, u − uh) = ah(uh, uh) − a(u, u). From [2ah(Πhu, uh) − 2a(u, u)] = 2b(f, Πhu − u), the formula becomes $$\label{eq:uhmu} \begin{split} a\_{h}(u\_{h},u\_{h}) - a(u,u) & = 2b(f,\Pi\_{h}u - u) + 2a\_{h}(u-\Pi\_{h}u,\Pi\_{h}u)\\ & \quad + 2a\_{h}(u-\Pi\_{h}u,u\_{h}-\Pi\_{h}u) + a\_{h}(u-u\_{h},u-u\_{h}), \end{split}$$ Analyze the items on the right-hand-side of. Suppose that u ∈ H3(Ω) ∩ H01(Ω). With the second term 2ah(u − Πhu, Πhu) not considered, the rest items in are of high order than ∣u − uh∣1, h. Therefore, 2ah(u − Πhu, Πhu) becomes the dominant factor to determine whether b( − f, u − uh) is of higher order than ∣u − uh∣1, h. • For the eigenvalue problem: Let u and uh be the solutions of and, which satisfy b(u, u) = b(uh, uh) = 1. It holds that λ − λh = a(u, u) − ah(uh, uh). Notice that 2b(uh, u − uh) = 2b(uh, u) − b(uh, uh) − b(u, u) = − b(u − uh, u − uh). Thus we can obtain $$\begin{split} 2a\_{h}(\Pi\_{h}u-u\_{h},u\_{h}) & = 2\lambda\_{h}b(u\_{h}, \Pi\_{h}u-u\_{h}) = 2\lambda\_{h}b(u\_{h}, \Pi\_{h}u-u) + 2\lambda\_{h}b(u\_{h}, u- u\_{h})\\ & = 2\lambda\_{h}b(u\_{h}, \Pi\_{h}u-u) - \lambda\_{h}b(u-u\_{h}, u- u\_{h}). \end{split}$$ Based on these above, becomes $$\label{eq:umuh} \begin{split} a(u,u) - a\_{h}(u\_{h},u\_{h}) & = 2\lambda\_{h}b(u\_{h}, \Pi\_{h}u-u) - \lambda\_{h}b(u-u\_{h}, u- u\_{h}) + 2a\_{h}(u-\Pi\_{h}u,\Pi\_{h}u)\\ & \quad + 2a\_{h}(u-\Pi\_{h}u,u\_{h}-\Pi\_{h}u) + a\_{h}(u-u\_{h},u-u\_{h}). \end{split}$$ Analyze the items on the right-hand-side of. Similarly, 2ah(u − Πhu, Πhu) is also the dominant factor to determine whether λ − λh is of higher order than ∣u − uh∣1, h. ### Analysis of the RM element for the eigenvalue problem. Based on the error estimates of the rectangular Morley element scheme for the source problem (see ), the following estimates for the eigenvalue problem follows by standard argument. [thm:EerrorRM] Let λj be the j-th eigenvalue of, and $(\lambda\_{j,h}^{\rm{M}},u\_{j,h}^{\rm{M}}) \in \mathbb{R} \times V\_{hs}^{\rm{M}}$ be the j-th eigen-pair of with $\\|u\_{j,h}^{\rm{M}}\\|\_{0,\rho} =1$. It holds that * if M(λj) ⊂ H01(Ω) ∩ H2(Ω), then there exists uj ∈ M(λj) with ∥uj∥0, ρ = 1, such that $$\begin{aligned} \big\|\lambda\_{j}-\lambda\_{j,h}^{\rm{M}}\big\| \lesssim h^{2}, \quad \big\\|u\_{j}-u\_{j,h}^{\rm{M}}\big\\|\_{0,\rho}\lesssim h^{2}, \mbox{ and } \big\|u\_{j}-u\_{j,h}^{\rm{M}}\big\|\_{1,h}\lesssim h ;\end{aligned}$$ * if the mesh is uniform and M(λj) ⊂ H01(Ω) ∩ H3(Ω), then there exists uj ∈ M(λj) with ∥uj∥0, ρ = 1, such that $ \big\|u\_{j}-u\_{j,h}^{\rm{M}}\big\|\_{1,h} \lesssim h^{2}. $ Moreover, we obtain the lower-bound property of eigenvalue approximations by the RM element. [thm:lowerM] Let λj and $ \lambda\_{j,h}^{\rm{M}} $ be an exact eigenvalue and its approximation by the RM element. Suppose that uj ∈ H01(Ω) ∩ H3(Ω) and the mesh is uniform. When h is small enough, we have $$\begin{aligned} \label{eq:lowerM} \lambda\_{j} - \lambda\_{j,h}^{\rm{M}} \geqslant C\_{\rm M}h^{2}. \end{aligned}$$ where $C\_{\rm M}$ is a positive constant independent of h. We have the basic expansion by, which generalizes the identity introduced by, $$\label{eq:eigenExpanM} \begin{split} \lambda\_{j} - \lambda\_{j,h}^{\rm{M}} =& \big\|u\_{j}-u\_{j,h}^{\rm{M}} \big\|^{2}\_{1,h} - \lambda\_{j,h}^{\rm{M}}\big\\|u\_{j}-u\_{j,h}^{\rm{M}}\big\\|^{2}\_{0,\rho} - 2\lambda\_{j,h}^{\rm{M}}b(u\_{j} - \Pi\_{h}^{\rm{M}}u\_{j},u\_{j,h}^{\rm{M}}) \\ &+ 2a\_{h}(u\_{j} - \Pi\_{h}^{\rm{M}}u\_{j},\Pi\_{h}^{\rm{M}}u\_{j}) + 2a\_{h}(u\_{j} - \Pi\_{h}^{\rm{M}}u\_{j},u\_{j,h}^{\rm{M}}-\Pi\_{h}^{\rm{M}}u\_{j}). \end{split}$$ From Theorem [thm:EerrorRM], the first two terms can be bounded as $$\big\\|u\_{j}-u\_{j,h}^{\rm{M}}\big\\|\_{0,\rho}^{2} \lesssim \big\|u\_{j}-u\_{j,h}^{\rm{M}}\big\|\_{1,h}^{2} \lesssim h^{4}.$$ From a standard interpolation theory in, the assumption $\big\\|u\_{j,h}^{\rm{M}}\big\\|\_{0,\rho} = 1$, and Theorem [thm:EerrorRM] (b), the third and last terms have the estimates below $$\begin{aligned} b(u\_{j}-\Pi\_{h}^{\rm{M}}u\_{j},u\_{j,h}^{\rm{M}}) & \lesssim \big\\|u\_{j}-\Pi\_{h}^{\rm{M}}u\_{j}\big\\|\_{0,\rho} \big\\|u\_{j,h}^{\rm{M}}\big\\|\_{0,\rho} \lesssim h^{3}, \\ a\_{h}(u\_{j}-\Pi\_{h}^{\rm{M}}u\_{j}, u\_{j,h}^{\rm{M}}-\Pi\_{h}^{\rm{M}}u\_{j}) & \lesssim \big\|u\_{j}-\Pi\_{h}^{\rm{M}}u\_{j}\big\|\_{1,h}\big\|u\_{j,h}^{\rm{M}} - \Pi\_{h}^{\rm{M}}u\_{j}\big\|\_{1,h} \lesssim h^{4}. \end{aligned}$$ When h is small enough, it follows from Lemma [lem:propertyPiRM] that $$a\_{h}(u\_{j}-\Pi\_{h}^{\rm{M}}u\_{j},\Pi\_{h}^{\rm{M}}u\_{j})\geqslant \alpha h^{2}.$$ Thus, $a\_{h}(u\_{j}-\Pi\_{h}^{\rm{M}}u\_{j},\Pi\_{h}^{\rm{M}}u\_{j})$ becomes the dominant term on the right-hand-side of. Hence the result. Reduced rectangular Morley element space for H1 problems ========================================================== Reduced rectangular Morley element space ---------------------------------------- We introduce an reduced rectangular Morley (RRM) element space by $$\begin{split} V\_{h}^{\rm{R}} := \Big\{w\_{h}\in L^{2}(\Omega) : w\_{h}\|\_{K} \in P\_{2}(K), & \; w\_{h}(a)\mbox{ is continuous at any } a\in \mathcal{N}\_{h}^{i}, \\ & \mbox{and }\fint\_{e}\partial\_{n\_{e}} w\_{h} {\,d}s \mbox{ is continuous across any } e \in \mathcal{E}\_{h}^{i}\Big\}, \end{split}$$ and, associated with H01(Ω), define $$V\_{hs}^{\rm{R}} :=\big\{w\_{h}\in V\_{h}^{\rm{R}} : w\_{h}(a)=0,\forall a\in \mathcal{N}\_{h}^{b} \big\}.$$ [thm:basisofRRM] If Gh is a m × n rectangular subdivision of Ω, then ${\rm dim} (V\_{hs}^{\rm {R}} )= mn+1$. Detailed proof of Theorems [thm:basisofRRM] and an available set of basis functions of $V\_{hs}^{\rm {R}}$ are put in Appendix [sec:appB]. For any function vh in the RRM element space, the number of continuity restrictions across internal edges is greater than ${\rm dim}\big(P\_{2}(K)\big)$, which makes it a nontrivial task to find out a set of basis functions of $V\_{hs}^{\rm {R}}$, and it is not even easy to tell if the space contains non-zero functions. Actually, the proof of Theorem [thm:basisofRRM] in Appendix [sec:appB] ensures that the RRM element space is non-zero. From the analysis therein, the supports of the basis functions in $V\_{hs}^{\rm {R}}$ are not completely local, making it complicated to construct an interpolation operator from V to $V\_{hs}^{\rm {R}}$, which, however, plays a fundamental role in the approximation error analysis. Since a non-convex domain which can be covered by a rectangular subdivision can be considered as a combination of several rectangular regions, a nontrivial RRM element space can still be expected on it. Approximation property of the RRM element space ----------------------------------------------- The main result of this subsection is the theorem below. [thm:approRRM] Given u ∈ H01(Ω) ∩ H3(Ω), we have $$\inf\_{v\_h\in V\_{hs}^{\rm R}}\|u-v\_h\|\_{1,h}\lesssim h^{\alpha}\|u\|\_{1+\alpha,\Omega},\ \ \alpha=1,2.$$ We postpone the proof of Theorem [thm:approRRM] after some technical lemmas. Let $\uf\in \big(\uH{}^1\_{\bf n}(\Omega)\big)'$. We firstly consider the regularity of the Stokes problem: Find $(\uu,p)\in \uH{}^1\_{\bf n}(\Omega)\times L^2\_0(\Omega)$, such that $$\label{eq:stokes} \left\{ \begin{array}{lll} (\nabla\uu,\nabla \uv)+(p,\dv\uv)&=(\uf,\uv), &\forall\,\uv\in \uH{}^1\_{\bf n}(\Omega), \\ (q,\dv\uu)&=0,&\forall\,q\in L^2\_0(\Omega). \end{array} \right.$$ [lem:regstokes] Let Ω be a rectangle. If $\uf\in \uL^2(\Omega)$, then $(\uu,p)\in \uH^2(\Omega)\times H^1(\Omega)$. As $\dv\uu=0$, there exists a unique φ ∈ H2(Ω) ∩ H01(Ω), such that $\curl\varphi=\uu$. Moreover, φ solves the biharmonic equation: $$\begin{aligned} (\nabla\curl\varphi,\nabla\curl\psi)=(\uf,\curl\psi),\quad\forall\,\psi\in H^2(\Omega)\cap H^1\_0(\Omega).\end{aligned}$$ By the regularity theory of the biharmonic equation (see ), we have φ ∈ H3(Ω), and $\\|\varphi\\|\_{3,\Omega} \lesssim \sup\limits\_{\psi\in H^1\_0(\Omega)\setminus\{0\}} {{{ \renewcommand{\arraystretch}{1.375} \begingroup\displaystyle \rule[0pt]{0pt}{11pt}( \uf,\curl \psi)\endgroup \over\displaystyle\rule[-3pt]{0pt}{11pt}\\|\psi\\|\_{1,\Omega} }} }\lesssim \\|\uf\\|\_{0,\Omega}$. Furthermore, $\nabla p=\uf+\Delta\uu$, and $\\|p\\|\_{1,\Omega}\lesssim \\|\uf+\Delta\uu\\|\_{0,\Omega}\lesssim \\|\uf\\|\_{0,\Omega}$. The proof is completed. A related finite element problem is to find $(\uu{}\_h,q\_h)\in \uV{}^{\rm BL}\_{h\bf n}\times\mathcal{L}^0\_{h0}$, such that $$\label{eq:stokesq1p0} \left\{ \begin{array}{ll} (\nabla\uu{}\_h,\nabla\uv{}\_h)+(p\_h,\dv\uv{}\_h)&=(\uf,\uv{}\_h) \\ (q\_h,\dv\uu{}\_h)&=0. \end{array} \right.$$ To ensure the convergence of the finite element scheme in Theorem [thm:stokes], we need the following hypothesis: #### Hypothesis RT A rectangular grid Gh is called to satisfy the hypothesis Hypothesis RT if and only if it is generated by refining a grid G4h twice. [thm:stokes] Let Gh be a grid that satisfies Hypothesis RT. Let $(\uu,p)$ and $(\uu{}\_h,p\_h)$ be the solutions of and, respectively. If $(\uu,p)\in \uH^2(\Omega)\times H^1(\Omega)$, then $$\begin{aligned} \|\uu-\uu{}\_h\|\_{1,\Omega}\lesssim h(\|u\|\_{2,\Omega}+\|p\|\_{1,\Omega}),\end{aligned}$$ and further $$\begin{aligned} \\|\uu-\uu{}\_h\\|\_{0,\Omega}\lesssim h^2(\|u\|\_{2,\Omega}+\|p\|\_{1,\Omega}).\end{aligned}$$ Based on Lemma [lem:regstokes], the proof of Theorem [thm:stokes] is just a duplication of the proofs of, and we omit the details here. [thm:approdf] Let Gh be a grid that satisfies Hypothesis RT. Given $\uw\in \uH{}^1\_{\bf n}(\Omega)\cap \uH^2(\Omega)$ satisfying $\dv\uw=0$. It holds that $$\begin{aligned} \inf\_{\uw{}\_h\in{\undertilde{V}}{}^{\rm PS}\_{h\bf n},\ \dv\_h\uw{}\_h=0}h\|\uw-\uw{}\_h\|\_{1,\Omega}+\\|\uw-\uw{}\_h\\|\_{0,\Omega}\lesssim h^2\|\uw\|\_{2,\Omega}.\end{aligned}$$ Let $\uu$ be the exact velocity of . Denote p0 ≡ 0. It can be verified directly that the pair $(\uu,p\_{0}\equiv0)\in \uH{}^1\_{\bf n}(\Omega)\times L^2\_0(\Omega)$ solves the equation $$\left\{ \begin{array}{ll} (\nabla\uu,\nabla\uv)+(p\_{0},\dv\uv)&=(-\Delta \uu,\uv) \\ (q,\dv\uu)&=0, \end{array} \right.$$ Let $(\uy{}\_h,p\_{0h})\in \uV{}^{\rm BL}\_{h\bf n}\times\mathcal{L}^0\_{h0}$ solve $$\left\{ \begin{array}{ll} (\nabla\uy{}\_h,\nabla\uv{}\_h)+(p\_{0h},\dv\uv{}\_h)&=(-\Delta \uu,\uv{}\_h) \\ (q\_h,\dv\uy{}\_h)&=0, \end{array} \right.$$ then $ h\|\uw-\uy{}\_h\|\_{1,\Omega}+\\|\uw-\uy{}\_h\\|\_{0,\Omega}\leqslant Ch^2\|\uu\|\_{2,\Omega}. $ Set $\uw{}\_h:=\undertilde{\Pi}{}^{\rm rQ}\_{h}\uy{}\_h$, then, from Lemma [lem:relationBLPS], $\uw{}\_h\in \uV{}^{\rm PS}\_{h\bf n}$. Moreover, it is easy to verify that $$(\dv\_h\uw{}\_h,q\_h)=(\dv\uy{}\_h,q\_h)=0,\quad \forall\,q\_h\in \mathcal{L}^0\_{h0},$$ namely $\dv\_h\uw{}\_h=0$. Furthermore, $$\|\uw-\uw{}\_h\|\_{1,h}\leqslant \big\|\uw-\undertilde{\Pi}{}^{\rm rQ}\_{h}\uw\big\|\_{1,\Omega} + \big\|\undertilde{\Pi}{}^{\rm rQ}\_{h}(\uw-\uy{}\_h)\big\|\_{1,h} \lesssim h \|\uw\|\_{2,\Omega},$$ and $$\\|\uw-\uw{}\_h\\|\_{0,\Omega}=\big\\|\uw-\uy{}\_h+\big(\undertilde{\rm Id}-\undertilde{\Pi}{}^{\rm rQ}\_{h}\big)(\uy{}\_h-\uw)+\big(\undertilde{\rm Id}-\undertilde{\Pi}{}^{\rm rQ}\_{h}\big)\uw\big\\|\_{0,\Omega}\lesssim h^2\|\uw\|\_{2,\Omega}.$$ Hence the result. [lem:strdf] For the $\curl$ of space $V\_{hs}^{\rm {R}}$, it can be depicted as a special subspace of the vector Park–Sheen element space, i.e., $$\curl\_h V\_{hs}^{\rm {R}}=\Big\{\uz{}\_h\in \uV{}^{\rm PS}\_{h\bf n}:\dv\_h\uz{}\_h=0\Big\}.$$ Firstly, by standard argument, we can prove the exact sequence which reads $$\begin{array}{ccccccccc} 0 & \longrightarrow & V\_{hs}^{\rm{M}} & \xrightarrow{\curl\_h} & \uV{}\_{h\bf n}^{\rm LTZ} & \xrightarrow{\dv\_h} & \mathcal{L}\_h^{1,-1} & \longrightarrow & 0, \end{array}$$ where Lh1, − 1 = {q ∈ L02(Ω) : q∣K ∈ P1(K)}. This way, given $\uv{}\_h\in\uV{}^{\rm PS}\_{h\mathbf n}\subset \uV{}\_{h\bf n}^{\rm LTZ}$ with $\dv\_h\uv{}\_h=0$, there exists $w\_h\in V\_{hs}^{\rm{M}}$, such that $\curl\_hw\_h=\uv{}\_h$. Since $\uv{}\_h$ is piecewise linear polynomial, wh is piecewise quadratic, namely $w\_h\in V\_{hs}^{\rm {R}}$. On the other hand, it is evident that $\curl\_hV\_{hs}^{\rm {R}}\subset\big\{\uz{}\_h\in \uV{}^{\rm PS}\_{h\bf n}:\dv\_h\uz{}\_h=0\big\}$. Hence the result. #### Proof of Theorem [thm:approRRM] By Theorem [thm:approdf] and Lemma [lem:strdf], $$\begin{gathered} \quad\inf\_{v\_h\in V\_{hs}^{\rm{R}}}\|u-v\_h\|\_{1,h}=\inf\_{v\_h\in V\_{hs}^{\rm{R}}}\\|\curl u-\curl v\_h\\|\_{0,\Omega} \\ =\inf\_{\uw{}\_h\in\undertilde{V}{}^{\rm PS}\_{h\bf n},\ \dv\_h\uw{}\_h=0}\\|\curl u-\uw{}\_h\\|\_{0,h}\lesssim h^\alpha\|\curl u\|\_{\alpha,\Omega}\lesssim h^{\alpha}\|u\|\_{1+\alpha,\Omega},\ \ \alpha=1,2.\quad\end{gathered}$$ The proof is completed. Convergence analysis of the RRM element schemes =============================================== Optimal convergence for the source problem ------------------------------------------ For the RM element space $V\_{hs}^{\rm{M}}$ and the RRM element space $V\_{hs}^{\rm{R}}$, the discrete source problems are given as: Find $u\_{h}^{\rm{M}} \in V\_{hs}^{\rm{M}}$, such that $$\begin{aligned} \label{eq:sourceRM} a\_{h}(u\_{h}^{\rm{M}},v\_{h}) = b(f,v\_{h}), \quad \forall v\_{h}\in V\_{hs}^{\rm{M}};\end{aligned}$$ Find $u\_{h}^{\rm{R}} \in V\_{hs}^{\rm{R}}$, such that $$\begin{aligned} \label{eq:sourceRRM} a\_{h}(u\_{h}^{\rm{R}},v\_{h}) = b(f,v\_{h}), \quad \forall v\_{h}\in V\_{hs}^{\rm{R}}.\end{aligned}$$ It is obvious that $V\_{hs}^{\rm{R}} = V\_{hs}^{\rm{M}} \cap V\_{h0}^{\rm{W}}$, and we infer that the RRM element is a quadratic nonconforming element on rectangles with a second-order convergence rate in the energy norm. We will verify this assertion strictly in this section. [thm:errorRRM] Let Gh be a grid that satisfies Hypothesis RT. Let u and $u\_{h}^{\rm{R}}$ be the solutions of and, respectively. It holds that * if u ∈ H01(Ω) ∩ H2(Ω), then $\big\|u-u\_{h}^{\rm{R}}\big\|\_{1,h} \lesssim h \|u\|\_{2,\Omega} \; \text{and} \; \big\\|u-u\_{h}^{\rm{R}}\big\\|\_{0,\rho} \lesssim h^{2} \|u\|\_{2,\Omega} ;$ * if u ∈ H01(Ω) ∩ H3(Ω) and the mesh is uniform, then $\big\|u-u\_{h}^{\rm{R}}\big\|\_{1,h} \lesssim h^{2}\|u\|\_{3,\Omega}.$ (a) By the Strang lemma, we have $$\begin{aligned} \label{eq:strangRRM} \big\|u-u\_{h}^{\rm R}\big\|\_{1,h} \lesssim \inf\_{v\_{h}\in V\_{hs}^{\rm{R}}} \|u - v\_{h}\|\_{1,h}+ \sup\_{w\_{h}\in V\_{hs}^{\rm{R}},w\_{h} \ne 0} \frac{E\_{h}(u,w\_{h})}{\|w\_{h}\|\_{1,h}}.\end{aligned}$$ For the first term in the right hand side of, we have from Theorem [thm:approRRM] that $$\begin{aligned} \label{eq:approRRM} \inf\_{v\_{h}\in V\_{hs}^{\rm{R}}} \|u - v\_{h}\|\_{1,h} \lesssim h\|u\|\_{2,\Omega}.\end{aligned}$$ For the second term, we have from Lemma [lem:consisRM] and $V\_{hs}^{\rm{R}} \subset V\_{hs}^{\rm{M}}$ that $$\begin{aligned} \label{eq:consisRRM} \|E\_{h}(u,w\_{h})\| \lesssim \sum\_{K\in \mathcal{G}\_{h}} h\_{K}^{2} \|u\|\_{2,K} \|w\_{h}\|\_{2,K}\lesssim h \|u\|\_{2,\Omega}\|w\_{h}\|\_{1,h}, \quad \forall w\_{h} \in V\_{h}^{\rm{R}}.\end{aligned}$$ Submit and into, and we obtain $ \big\|u-u\_{h}^{\rm R}\big\|\_{1,h}\lesssim h \|u\|\_{2,\Omega}, \mbox{ where } u \in H\_{0}^{1}(\Omega) \cap H^{2}(\Omega). $ Given g ∈ Q, let ϕg ∈ V and $\phi\_{gh} \in V\_{hs}^{\rm{R}}$ be the solutions of the two problems below, respectively, $$\begin{aligned} a(v,\phi\_{g}) = b(g,v),\quad \forall v \in V; \quad a\_{h}(v\_{h},\phi\_{gh}) = b(g,v\_{h}), \quad \forall v\_{h} \in V\_{hs}^{\rm{R}}.\end{aligned}$$ By the Nitsche-Lascaux-Lesaint lemma (see e.g., ), it holds that $$\begin{aligned} \label{eq:NitscheRRM} \big\\|u-u\_{h}^{\rm R}\big\\|\_{0,\rho} \lesssim \big\|u-u\_{h}^{\rm R}\big\|\_{1,h} \sup\_{g\in Q, g\ne 0} \	arxiv_0000727
�= [1 + Q2/Λπ, ρ2]− 1. Apart from the ππγ and ρπγ form factors, the model is parameter free, as the coupling constants at the vertices (such as gρπγ) are well determined by precise studies and analyses in the resonance region. The model gives a good description of the W- and t-dependences of then available π+ and π− photoproduction data, including the spin asymmetries, and of the earlier electroproduction data. [h] The VGL predictions have been compared to our measured cross sections and the ones taken at DESY in Ref. . For the discussion in this paper, the data for and are reproduced in Fig. [fig:xsecvgl], together with the results of the model calculations. The VGL cross sections were evaluated at the same $\overline W$ and $\overline Q^2$ values as the data, resulting in the discontinuities shown. The values of shown are determined by the fitting of the VGL model to the measured -values at the five values of t at each, resulting in values between 0.37 and 0.51. The value of is more poorly known. Calculations with both =0.600 and 1.500 are shown, where the upper value is taken from the application of the VGL model to kaon electroproduction. The model gives an overall good description of our data and those of, but the description of the t-dependence of the data is worse at =0.60 and 0.70. The poorer description of the data by the VGL model at lower and W may be due to contributions from resonances, which are not included explicitly in the Regge model. This is supported by the fact that the discrepancy in the t-dependence of the data is strongest at the lowest value, at higher the resonance form factor supposedly reducing such contributions. The values of are severely underestimated, especially at larger, even when taking a hard ρπγ form factor. Since the data at the real-photon point are well described, this suggests that another mechanism, whose contribution increases with, is at play. Recently the VGL model was extended by including a hard scattering between the virtual photon and a quark, the latter hadronizing in combination with the spectator diquark into a pion plus residual nucleon. With plausible assumptions, a good description of was obtained, with no influence on. Those results support the idea that the discrepancy in the magnitude of, which increases with, and the discrepancy in the slope of with − t, which decreases with, are not directly related. Strategies for dealing with the latter discrepancy when extracting the pion form factor are discussed in Sec. [sec:fpivgl]. We also considered a modification to the VGL Regge model published by J.M. Laget in 2004. Laget introduces a t-dependent factor into the pion form factor which is related to the pion saturating Regge trajectory, approaching -1 as t → − ∞. The effect of this modification is to boost by 40% for the largest − t spanned by our data (=2.703, − t=0.365 ), and converging with the unmodified calculation at small − t. The effect on is under 1% for the largest − t covered by our data, and is negligible at − tmin. [h] Another recent development is the effective Lagrangian model of Faessler, Gutsche, Lyubovitskij and Obukhovsky (FGLO, Ref. ). This is a modified Born Term Model, in which an effective Lagrangian is used to describe nucleon, pion, ρ and photon degrees of freedom. The (combined) effect of s- and u-channel contributions, which interferes with the pion t-pole, is modeled using a constituent quark model. The authors show that the ρ t-pole contribution is very important in the description of the magnitude of. When comparing vector and tensor representations of the ρ contribution, the latter was found to give better results. Unlike the VGL model, the cross section depends here also on the ρ exchange, because of the interference of the π and tensor ρ exchange contributions. The model contains a few free parameters, such as the renormalization constant of the Kroll-Ruderman contact term used to model the s(u)-channel, and t-dependent strong meson-nucleon vertices, which are parameterized in monopole form, as are the electromagnetic form factors. The corresponding parameters were adjusted so as to give overall good agreement with our and data. As in case of the VGL model, a detailed comparison of the FGLO model results to the measured data is given in Ref. , while the results for and are also shown in Fig. [fig:xsecobu]. The values of used were determined by the fitting of the model to the t-bins at each, while keeping the other parameters fixed at the values assigned by the authors. In some cases, this results in different values than shown in Ref.. However, it should be kept in mind that the FGLO model cross sections also depend on other parameters, which have been adjusted by the authors of the model to give good agreement to our and data. To the best of our knowledge, the =0.7 data of Ref. were not taken into account when these parameters were determined. Generally, the agreement of the FGLO model with the data is rather good except for the =0.60 [Fpi-1] and 0.70 measurements. There is a serious discrepancy in the Q2- and W-dependence of the data. For around 0.7, the model agrees fairly well with the data at W = 1.95 GeV, but it over-predicts the =0.70, W = 2.19 GeV data by a large factor. On the other hand, for =1.60, the W = 1.95 GeV data are under-predicted by about a factor of two, while those at W = 2.22 GeV are reproduced, and the W = 2.22 GeV data for =2.45 are under-predicted again by 20-60%. This indicates some problem in the description of the Q2, W-dependences of the ρ exchange used to describe. Because of the ρ − π interference, the problems with the description of also affect the calculation. This makes it hard to estimate how reliable the values of would be if extracted from the data using this model. $\bf F\_{\pi}$ Results [sec:fpivgl] =================================== As already discussed, the separated cross sections versus t over some range of and W are the actual observables measured by the experiment, and the extraction of the pion form factor from these data is inherently model dependent. Ideally, one would like to have a variety of reliable electroproduction models to choose from, so that the model dependence of the extracted values can be better understood. Since the VGL Regge model is able, without fitted parameters, to provide a good description of both π+ and π− photoproduction data, and of electroproduction data over a range in W, t, and, it is our opinion that at the moment only this model has shown itself to be sufficiently reliable to enable its use to extract pion form factor values from the data. Therefore, we will use this model to determine our values. Clearly, the values determined are strictly within the context of the VGL Regge model, and other values may result if other, better models become available in the future. $\bf W\approx 2.2$ GeV Data --------------------------- \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| Q2 \| W \| \| Fπ \| \| () \| (GeV) \| () \| \| \| 0.60 \| 1.95 \| 0.458 ± 0.031− 0.068+ 0.255 \| 0.433 ± 0.017− 0.036+ 0.137 \| \| 0.75 \| 1.95 \| 0.388 ± 0.038− 0.053+ 0.135 \| 0.341 ± 0.022− 0.031+ 0.078 \| \| 1.00 \| 1.95 \| 0.454 ± 0.034− 0.040+ 0.075 \| 0.312 ± 0.016− 0.019+ 0.035 \| \| 1.60 \| 1.95 \| 0.485 ± 0.038− 0.027+ 0.035 \| 0.233 ± 0.014− 0.010+ 0.013 \| \| 0.35 \| 2.10 \| 0.601 ± 0.060 \| 0.632 ± 0.023 \| \| 0.70 \| 2.19 \| 0.627 ± 0.058− 0.085+ 0.096 \| 0.473 ± 0.023− 0.034+ 0.038 \| \| 1.60 \| 2.22 \| 0.513 ± 0.033− 0.022+ 0.052 \| 0.243 ± 0.012− 0.008+ 0.019 \| \| 2.45 \| 2.22 \| 0.491 ± 0.035− 0.024+ 0.045 \| 0.167 ± 0.010− 0.007+ 0.013 \| As shown in Fig. [fig:xsecvgl], the VGL model does a good job of describing the t-dependence of the cross sections at W ≈ 2.2 GeV, Q2 = 0.35, 1.60 and 2.45. In these cases, the extraction of the pion form factor from the data is straightforward: the value of in the model is varied until the agreement of the model with the data is optimized. The mean $\overline Q^2$ and $\overline W$ values of the data for each t-bin are used when evaluating the model. is then calculated from Eqn. [eqn:monopole], using the best-fit and the nominal values. These are listed in the last two lines of Table [tab:fpi]. The experimental statistical and systematic uncertainties were propagated to the uncertainties as follows. The statistical and t, ε-uncorrelated systematic uncertainties[1](#fn1) were applied to the data prior to the fitting of the VGL model to the data. This yields the best-fit value and its associated fitting uncertainty. The effects of the t-correlated, ε-uncorrelated, and the t, ε-correlated systematic uncertainties on the fit were determined by investigating the variation in values allowed by fitting to the lowest − t bin only. Of these, the ε-uncorrelated, t-correlated systematic uncertainty is amplified by 1/Δε in the L-T separation, while the t, ε-correlated uncertainty is not. The resulting uncertainties are added in quadrature to the fitting error, yielding the first uncertainty listed in Table [tab:fpi]. This value is also propagated to according to the monopole parameterization, yielding the first uncertainty listed. In order to check if the extracted value of depends on the t-range used, the VGL model (i.e., the value of ) was fitted separately to each point from Fpi-2 and DESY, and the corresponding values of determined. In order to remove the natural variation of with the $\overline Q^2$ of each bin, the nominal values were used in the monopole equation. A plot of the obtained versus t is shown in Fig. [fig:fpi2check]. Also indicated as the shaded band is the value with the uncertainty that is obtained if one fits to all of the t-bins simultaneously. Except perhaps at =0.70, the data show no residual t-dependence beyond the statistical fluctuation. $\bf W=1.95$ GeV Data --------------------- As already shown in Sec. [sec:vgl], the VGL model does not fully describe the t-dependence of our data at W = 1.95 GeV. The difficulty, as far as the extraction is concerned, is that there is no theoretical guidance for the assumed interfering background not included in the VGL model, even if one assumes that it is due to resonances. Virtually nothing is known about the L/T character of resonances at W = 1.95 GeV, let alone how they may influence through their interference with the π-pole amplitude. Given this lack of theoretical guidance, we are forced to make some assumptions in extracting from these data. Our guiding principle is to minimize these assumptions to the greatest extent possible. The form factor extraction method that we have adopted for these data relies on the single assumption that the contribution of the background is smallest at the kinematic endpoint tmin. Our best estimate of for the W = 1.95 GeV data is determined in the following manner. Using the value of as a free parameter, the VGL model was fitted to each t-bin separately, yielding $\Lambda^2\_\pi(\overline{Q^2},\overline{W},t)$ values as shown in Fig. [fig:fpi1lpi]. The values of tend to decrease as − t increases, presumably because of an interfering background not included in the VGL model. Since the pole cross section containing increases strongly with decreasing − t, we assume that the effect of this background will be smallest at the lowest value of ∣t∣ allowed by the experimental kinematics, ∣tmin∣. Thus, an extrapolation of to this physical limit is used to obtain our best estimate of. The value of at tmin is obtained by a linear fit to the data in Fig. [fig:fpi1lpi]. The resulting and values for the Fpi-1 data are listed in Table [tab:fpi]. The first uncertainty listed includes both the experimental and the linear fit extrapolation uncertainties. Since Fig. [fig:fpi2check] suggests also a dependence (at larger − t) between the VGL calculation and the =0.70 data of Ref., this extraction method was also applied to those data. The result obtained when extrapolating to tmin is listed in Table [tab:fpi]. The value of () is 11(20)% larger than if the VGL model was simply fit to all data points. Applying the same procedure to our W = 2.22 GeV data, it was found that the resulting values of () would be 1(2)% larger, which is statistically insignificant, confirming that the t-dependence of those data is well described by the VGL model. Model Uncertainty Estimate -------------------------- The fact that we used an additional assumption for the cases where the VGL model does not completely describe the t-dependence of the data causes an additional uncertainty in the extracted value, which we term ‘model uncertainty’. This model uncertainty, which is within the context of the VGL model, should be distinguished from the general model uncertainty discussed in section [sec:eepimodels], which would result when using different models. In order to make a quantitative estimate of this additional uncertainty, the spread in extracted values of (and thus ) was investigated by assuming specific forms of the interfering background missing in the VGL model. An effective upper limit for is obtained by assuming that the background yields a constant, negative, contribution to. For each value of, this background and the value of were fit together to the data, assuming that the background is constant with t. The fitted contribution of the background was found to drop strongly with increasing. A second possibility is to assume, besides the VGL amplitude, a t-independent interfering background amplitude, fitting for every the magnitude and phase of the latter, together with the value of. Although the fitting uncertainties are very large, the results suggest an interfering amplitude whose magnitude decreases monotonically with increasing. In this case, the interference between the background amplitude and the VGL amplitude, which depends on their relative phase, does not necessarily result in a net negative cross section contribution to. The estimated model uncertainty is determined from the spread of the values and their uncertainties at each, obtained with these two choices of background. To keep the number of degrees of freedom the same in both cases, the background was fixed to the value giving the best χ2, and and its uncertainty were then determined in a one-parameter fit of the VGL model plus background to the data. Since the statistical uncertainties of the data are already taken into account in the first given uncertainty in Table [tab:fpi], the contribution of the statistical uncertainties of the data were quadratically removed from the uncertainties given by the fit. The model uncertainties at each are then taken as the range plus corrected fitting uncertainty given by these two methods, relative to the value of determined from the extrapolation to tmin. This procedure was applied to all data except those of Ref., yielding the model uncertainties listed as the second (asymmetric) uncertainty in Table [tab:fpi]. No model uncertainty was calculated for the =0.35 data from DESY because the t-range spanned by those data (only 0.03 ) was too small for this procedure to be reliably applied. For the W = 1.95 GeV data, the model uncertainty in the extracted value drops from about 20% at =0.60 to about 5% at 1.60. To be consistent, the same procedure was applied to the W = 2.22 GeV data, which yielded model uncertainties of about 5% at both =1.60 and 2.45. These rapidly dropping uncertainties with increasing reflect the smaller discrepancy of the VGL calculation with the t-dependence of the data at larger values of and W. These findings are at least compatible with the idea that resonance contributions, which presumably have a form factor that drops rapidly with, are responsible. They also suggest that our extraction methods are robust, when the background contribution is small, as appears to be the case at the higher value of W. Discussion and Comparison with Empirical Fits [sec:monopole] ------------------------------------------------------------ The form factors extracted from the Fpi-1 and Fpi-2 data with the use of the VGL model are shown in Fig. [fig:q2fpimono], along with the reanalyzed =0.70 data of Ref., the elastic scattering measurements of Ref., and the =0.35 data of Ref.. The Cornell data of Refs. are not included because, as discussed in section [sec:early], they have large unknown systematic uncertainties. The excellent agreement between the =1.6 form factor values obtained from our W = 1.95, 2.22 GeV data, despite their significantly different tmin and W values, indicates that the model uncertainties from the use of the VGL model seem to be under control, at least in this -range. Also shown is a more recently obtained value at =2.15 , which was also extracted with the use of the VGL model. The solid line shown in Fig. [fig:q2fpimono] is the monopole fit obtained by Amendolia et al. from their elastic scattering data. This curve is given by $$F\_{mono}=\frac{1}{1+\frac{r\_{mono}^2Q^2}{6\hbar^2c^2}},$$ where rmono2 = 0.431 fm2 is their best-fit squared pion charge radius. Fig. [fig:q2fpimono] indicates a systematic departure of the data from the monopole curve above Q2 ≈ 1.5. This departure may have implications for theoretical approaches that assume the validity of the monopole parameterization over a wide range of. To illustrate the departure from the monopole curve, as well as to provide an empirical fit that describes the data over the measured range, we also show in Fig [fig:q2fpimono] a fit which includes a small dipole component, Ffit = 85%Fmono + 15%Fdip, where $$F\_{dip}=\frac{1}{\bigl(1+\frac{r\_{dip}^2Q^2}{12\hbar^2c^2}\bigr)^2}$$ , and rdip2 = 0.411 fm2. This dipole parameterization has nearly the same χ2 for the elastic scattering data as the monopole curve shown, but it drops much more rapidly with. The combined monopole plus dipole fit is consistent with our intermediate data, while maintaining the quality of fit to the elastic scattering data. Since a monopole parameterization does not converge to the pQCD asymptotic limit (Eqn. [eqn:asymp]), it is expected to fail at some point. Similarly, we should expect this empirical monopole+dipole parameterization to show its limitations when additional high data become available. Comparison with Model Calculations [sec:models] =============================================== The pion form factor can be calculated relatively easily in a large number of theoretical approaches which help advance of our knowledge of hadronic structure. In this sense, plays a role similar to that of the positronium atom in QED. Here, we compare our extracted values to a variety of calculations, selected to provide a representative sample of the approaches used. Perturbative QCD ---------------- The most firmly grounded approach for the calculation of is that of pQCD. The large behavior of the pion form factor has already been given in Eqn. [eqn:pqcd]. By making use of model-independent dimensional arguments, the infinitely-large behavior of the pion’s quark wave function (distribution amplitude, or DA) is identified as ϕπ(x, Q2 → ∞) → 6fπx(1 − x) whose normalization is fixed from the π+ → μ+νμ decay constant. Eqn. [eqn:asymp] follows from this expression. Neither of these equations is expected to describe the pion form factor in the kinematic regime of our data, and so much effort has been expended to extend the calculation of to experimentally accessible. In this case, the pion DA, ϕπ(x, $Q^2$), must be determined at finite. Additional effects, such as quark transverse momentum and Sudakov suppression (essentially a suppression of large quark-quark separation configurations in elastic scattering processes) must be taken into account. A number of authors have performed leading-twist next-to-leading order (NLO) analyses of at finite. The hard contributions to expand as a leading order part of order αs and an NLO part of order αs2. Bakulev, Passek-Kumericki, Schroers and Stefanis have investigated the dependence of the form of the DA on the form factors, using data from a variety of experiments. These were the πγγ transition form factor data from CLEO and CELLO, as well as our data. Their results are insensitive to the shape of the DA near x = 1/2, while its behavior at x = 0, 1 is decisive. The resulting hard contribution to the pion form factor is only slightly larger than that calculated with the asymptotic DA in all considered schemes. The result of their study, shown as Fπhard in Fig. [fig:pqcd], is far below our data. The drop at low is due to their choice of infrared renormalization, which is not necessarily shared by other calculations. To bring the calculation into agreement with the experimental data, a soft component must also be added. The treatment of the soft contribution to the pion DA is model-dependent. The authors estimate this soft contribution using a local quark-hadron duality model. This soft estimate, along with the sum of the hard and soft contributions, are also shown in Fig. [fig:pqcd]. The interplay at intermediate between the hard and soft components can be non-trivial, as demonstrated by Braun, Khodjamirian, and Maul, using a light-cone sum rule approach. Their results support a pion DA that is close to the asymptotic expression, but they find that strong cancellations between soft terms and hard terms of higher twist lead to the paradoxical conclusion that the nonperturbative effects in the pion form factor can be small, and the soft contributions large, simultaneously. Because of complications such as these, different theoretical viewpoints on whether the higher-twist mechanisms dominate until very large momentum transfer, or not, remain. Lattice QCD ----------- Unlike QCD-based models, in which confinement must be explicitly added, Lattice QCD allows calculation from first principles. However, while lattice QCD is based on the QCD Lagrangian, it involves a number of approximations. Errors are introduced because space and time are crudely discretized on the lattice. This error is controlled by the use of improved lattice QCD actions. To allow a more rapidly converging action, and hence reduce CPU usage, the pion mass used is significantly larger than the physical pion mass. Chiral extrapolation errors are introduced when the lattice results, determined with large pion mass, are extrapolated to physical values. Finally, quenching errors are introduced when disconnected quark loops are neglected. The first lattice simulations of were done in the 1980’s. These pioneering works were primarily a proof of principle of the lattice technique, and were restricted to < 1 . These results are consistent with the low experimental data, within the large statistical uncertainties of these pioneering calculations. Spurred by advances in CPU power and lattice techniques, as well as the availability of new experimental data, a number of groups have returned to the calculation of on the lattice. Of these, we compare our data to the recent unquenched simulations of Brommel, et al.. They performed simulations for a wide range of pion masses and lattice spacings, so that both the chiral and continuum limits could be studied. However, the lowest pion mass used in the simulations was 400 MeV, so the chiral extrapolation is significant. The authors fitted the -dependence of each lattice configuration with a monopole form for the pion form factor and determined the corresponding monopole mass. They then extrapolated these masses to the one corresponding to the physical pion mass to obtain a chiral monopole mass value of 0.727 ± 0.016 GeV. The (monopole) form factor calculated with that mass (including its uncertainty) is indicated by the shaded band in Fig. [fig:latdisp], cut off at the highest point of the lattice simulation. This result begins to trend away from the Q2 > 1.5 experimental data. It remains to be seen how these results would be affected by our Sec. [sec:monopole] comments on the applicability of the monopole parameterization in this range. Dispersion Relation with QCD Constraint --------------------------------------- Dispersion relations are based on constraints posed by causality and analyticity, and relate the timelike and spacelike domains of the pion form factor on the complex plane. In principle the technique is exact, but our incomplete knowledge of the scattering amplitudes over the whole complex plane, and in particular the incomplete understanding of the contribution of all of the poles in the timelike region, creates uncertainties. Authors address these uncertainties by imposing additional constraints, such as the role of higher timelike resonances like the ρ‴, or chiral perturbation constraints near the spacelike threshold, or that must approach its expected asymptotic value at very large . We compare the data to the dispersion relation analysis of B.V. Geshkenbein in Fig. [fig:latdisp]. The displayed uncertainty band is obtained by assuming different distributions of zeroes in the complex s-plane. This results in a band that grows with, with the ‘no zeroes’ curve lying nearly at the lower end of the band. Our highest data lie above the ‘no zeroes’ curve, but below the ‘improved maximum ’ limit. QCD Sum Rules ------------- The QCD sum rule approach is designed to interpolate between the perturbative and non-perturbative sectors using dispersion relation methods in combination with the operator-product expansion. While the practical implementation of this approach cannot claim to be rigorously derived from QCD, its intuitive value is that it provides a bridge between the low- and high-energy properties of QCD. A number of authors have applied this technique with good success to the pion form factor. In the calculation of Radyushkin, QCD sum rules were used to give a local quark-hadron duality estimate of the soft wave function $$F^{soft}\_{\pi}(Q^2)=1-\frac{1+6 s\_0/Q^2}{(1+4s\_0/Q^2)^{3/2}}, \label{eqn:qsr\_soft}$$ where the duality interval, s0, which within the QCD sum rule approach is determined by the magnitude of the quark and gluon condensates, was taken as 4π2fπ2 ≈ 0.7. This soft calculation, shown in Fig. [fig:qsrdse], under-estimates the data by about 25%. For the hard contribution, a simple model based on the interpolation between the behavior near Q2 = 0 (related by the Ward identity to the O(αs) term of the 2-point correlator) and the asymptotic behavior was used $$F^{hard}\_{\pi}(Q^2)=\frac{\alpha\_s}{\pi}\frac{1}{(1+Q^2/2s\_0)}.$$ The sum, Fπsoft + Fπhard, is in excellent agreement with the data. More recently, Braguta, Lucha and Melikhov have replaced the simple ansatz leading to Eqn. [eqn:qsrsoft] with an expression including explicit corrections up to O(αs). Since the higher-order corrections needed to apply these results with authority to the intermediate region are beyond the capacity of their two-loop calculation, there is a model dependence in their numerical result, which is reflected in the two different curves for s0 = 0.65 and $s\_0=\frac{4\pi^2f\_{\pi}^2}{1+\alpha\_s(Q^2)/\pi}$ shown in Fig. [fig:qsrdse]. Bethe-Salpeter Equation ----------------------- The Bethe-Salpeter equation (BSE) is the conventional formalism for the treatment of relativistic bound states. In this formalism, a meson is described by a covariant wavefunction, which depends on the four momenta of its constituent quarks. Although formally correct, complications arise as the interplay between different configurations, such as q-q̄ and q-q̄-g, are implicitly buried in the potential and scattering amplitudes used in analyzing hadronic processes, and as a result, these potentials and scattering amplitudes are nearly intractable. The light-front Bethe-Salpeter model is a means to handle this problem by breaking the BSE into separate hard and soft components. A variety of models incorporating a confining potential which dominates at low, and a QCD-based interaction which dominates at high, are given in Refs.. Another approach is to use the Dyson-Schwinger equation (DSE) to obtain dressed quark propagators which may be used in the solution of the BSE. The Dyson-Schwinger approach to nonperturbative QCD has many advantages. It is consistent with quark and gluon confinement, it automatically generates dynamical chiral symmetry breaking, and the solution is Poincare invariant. In the work of Maris, Tandy, and Roberts, the meson Bethe-Salpeter amplitudes and quark-photon vertex are obtained as solutions of the homogeneous and inhomogeneous BSE, and the dressed quark propagators are obtained from the quark DSE. The model parameters are fixed by requiring fπ and mπ to be in good agreement with the data and then rπ and are predicted with no further adjustment of parameters. Their calculation is shown in Fig. [fig:qsrdse]. It is in excellent agreement with our data up to =1.60. To extend the validity of the model to higher, a more complete description that takes meson loop corrections into account self-consistently is required. Local Quark-Hadron Duality -------------------------- Quark-hadron duality relations link the hadronic structure information contained in exclusive form factors and inclusive structure functions by making strong assumptions of locality. While local quark-hadron duality is an expected consequence of QCD at asymptotically large momenta, it is not at all clear how well it could work at finite . And if it does, it may be due to accidental cancellations of higher twist effects. Nevertheless, it is worthwhile to compare predictions based on quark-hadron duality with the measured data, especially since duality is expected to work better at higher, in contrast to many other approaches. The approximate relationship between the pion elastic form factor and the pion structure function F2π = νW2π was found by Moffat and Snell, [$F\_\pi$($Q^2$)]2 ≈ ∫1ωmaxF2π(ω)dω, where ω = 1/x, and the upper limit of integration is chosen to select the elastic contribution to the inclusive structure function. In applying this formula use is made of the Drell-Yan-West relation, which is based on a field-theoretic parton model that predates QCD. It predicts that if the asymptotic behavior of a form factor is (1/$Q^2$)n, the corresponding structure function should behave as (1 − x)2n − 1 as x → 1. This leads to the prediction F2π(x → 1) ∼ (1 − x). The existence of Drell-Yan F2π data allows a quantitative test of Eqn. [eqn:moffat] using only phenomenological input. Calculations based on the leading-order analysis of F2π data by Ref., and the next-to-leading order analysis of Ref., are shown in Fig. [fig:ducqm]. In both cases, the magnitude of the prediction is dependent on the value chosen for the inelastic cutoff ωmax (and corresponding Wmax) in Eqn. [eqn:moffat]. Local duality is expected not to work at lower. This is reflected in the poor description of the Q2 < 1 form factor data. However, above Q2 > 2, the next-to-leading order analysis is consistent with our data. Constituent-Quark Model ----------------------- There are many calculations using a variety of constituent-quark models. The differences in approach typically involve differences in the treatment of the quark wave functions, or the inclusion of relativistic effects. Fig. [fig:ducqm] shows the result of calculations by Cardarelli et al. and by Hwang. Both are relativistic quark models on the light front. Ref. uses the effective qq̄ Hamlitonian of, which contains a one-gluon-exchange term and a linear confining term, and which describes a large set of meson spectroscopic data. Use of this interaction results in large high-momentum components, and is strongly overpredicted (upper dot-dashed curve in Fig. [fig:ducqm]). This can be cured in a way that is consistent with the notion of a constituent quark, by assuming a form factor for the latter. Taking a monopole form for the latter and adjusting the mass parameter so that the measured pion charge radius is reproduced, results in the lower dot-dashed curve shown. The model of Ref. allows a consistent and fully relativistic treatment of quark spins and center-of-mass motion to be carried out. A power-law wave function is used, whose parameters are determined from experimental data on the charged pion decay constant, the neutral pion two-photon decay width, and the charged pion electromagnetic radius. The charge and transition form factors of the charged pion and the branching ratios of all observed decay modes of the neutral pion are then predicted. The calculation is in very good agreement with our data. Li and Riska asked if the empirical data exclude the presence of a significant sea-quark configuration in the charged pion. They performed a constituent quark model calculation which was extended to include explicit sea-quark components in the pion wave function. They found that these sea-quark contributions grew with increasing, because they allowed the momentum transfer to be shared by a greater number of constituents, and so were less-suppressed at high than configurations which involved only a q̄q pair. Although their analysis was model-dependent, they found that our data allowed an approximate 20 ± 20% sea-quark component, with the data point at =2.45 providing the greatest constraint. Holographic QCD --------------- A recent theoretical development is the AdS/CFT correspondence between weakly-coupled string states defined on a 5-dimensional anti-de Sitter space-time ($\rm AdS\_5$) and a strongly-coupled conformal field theory (CFT) in physical space-time. The goal of holographic QCD models is to find a weakly-coupled theory for which the dual strongly-coupled theory is as close to QCD as possible, and so allow analytic solutions of hadronic properties in the non-perturbative regime to be performed. In these models, confinement is simulated by imposing boundary conditions on the extra fifth dimension z. In the “hard-wall” variant, confinement is modeled by a hard cutoff at a finite value z = z0 = 1/ΛQCD. This has the advantage of simplicity but produces the unphysical Regge trajectory Mn2 ∼ n2. The “soft-wall” variant replaces the hard-wall boundary with an oscillator-type potential, and produces the more phenomenologically realistic Regge behavior Mn2 ∼ n. Several authors have applied holographic models to the pion form factor. Complications arise when one introduces spontaneous and explicit chiral symmetry breaking effects into the soft-wall holographic QCD model. Refs. take different approaches to this problem. Grigoryan and Radyushkin consider only the hard-wall variant, and then estimate a soft-wall correction from their previous vector meson study. They conclude that a full analytic calculation would likely follow the data only in the Q2 < 1 region, while overshooting it above Q2 ∼ 2. The calculations by Kwee and Lebed are numerical. Both the hard-wall and the soft-wall calculations predict charge radii that are too small, especially for the soft-wall case (see Fig. [fig:adscft]). By allowing the parameters of the soft-wall model (originally fixed by mρ, mπ, and fπ) to vary, they find it is possible to describe at either high or low, but not both. Issues in ongoing discussions on the AdS/CFT approach include the applicability of this model to the larger region where partonic degrees of freedom become appreciable, and the treatment of chiral symmetry breaking. Summary and Outlook =================== Values for the charged pion form factor, Fπ(Q2), have been extracted for =0.60-2.45 from the longitudinal cross sections σL(t) for the 1H(e, e′π+)n reaction recently measured at JLab. values were also extracted from older experimental data acquired at DESY. The Cornell data are not included in this analysis because these were not obtained in a true L/T-separation, but instead by subtracting a certain assumption for. In addition, the higher data have excessively large values of − tmin. The form factor extraction requires the use of a model incorporating both the π+ production mechanism as well as the effect of the nucleon. Several approaches to extract from the data, including the Chew-Low extrapolation method, various types of Born Term Models, and newer models utilizing Regge trajectories and effective Lagrangians, were reviewed. By using specially generated test data, it was found that extrapolating to the pole at t = mπ2, as is done in the Chew-Low method, cannot be used in practice, because there is no way to determine the order of the polynomial to use for the extrapolation, and because even small uncertainties in the measured cross sections lead to a large uncertainty in. From the models available for determining from the measured values of, the VGL Regge model was chosen, since it contains no ad hoc parameters, and its validity has been well established over a wide kinematic range in t and W for both electroproduction and photoproduction data. The VGL model gives a rather good description of both the t and the W dependence of the JLab data at values of > 1, but especially at Q2 = 0.60 the fall-off of the data with − t is steeper than that of the model. In the cases where the VGL model described well the t-dependence of the data, the value of was determined by fitting the model to the data. Otherwise, the value of was determined by extrapolating the fit of the model to t = tmin. An additional ‘model uncertainty’ has been estimated by using two different assumptions for an interfering background that could be responsible for this discrepancy between the data and VGL model. The fact that the discrepancy, and hence the model uncertainty, is very small at higher values of and W suggests that effects from nucleon resonances play a role in the data at lower and W. It is stressed that the cross sections are the actual observables measured by the experiment, and that the extracted values of are inherently dependent on the model used to extract them. The development of additional models for the 1H(e, e′π+)n reaction would allow further exploration of the model dependence of the extraction of from the same cross section data. On the experimental front, proposed measurements after the completion of the JLab upgrade are expected to better establish the validity of any used model by investigating, for example, the W-dependence of the results. The results for, extracted from our data and from the DESY data with the use of the VGL model, are presented together with their experimental and model uncertainties. Above Q2 ≈ 1.5, these data are systematically below the monopole parameterization based on the empirical pion charge radius. 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However, the extraction of from data requires a model of the 1H(e, e′π+)n reaction, and thus is inherently model dependent. Therefore, a detailed description of the extraction of the charged pion form factor from electroproduction data obtained recently at Jefferson Lab is presented, with particular focus given to the dominant uncertainties in this procedure. Results for are presented for =0.60-2.45. Above Q2 = 1.5, the values are systematically below the monopole parameterization that describes the low data used to determine the pion charge radius. The pion form factor can be calculated in a wide variety of theoretical approaches, and the experimental results are compared to a number of calculations. This comparison is helpful in understanding the role of soft versus hard contributions to hadronic structure in the intermediate regime. Introduction ============ There is much interest in trying to understand the structure of hadrons, both mesons and baryons, in terms of their constituents, the quarks and gluons. However, this structure is too complicated to be calculated rigorously in Quantum Chromodynamics (QCD) because perturbative QCD (pQCD) methods are not applicable in the confinement regime. Chiral Perturbation Theory can give valuable insights, but it is limited to small values of the photon virtuality. Hence, in the intermediate regime one has to resort to models like the constituent quark model or methods employing Light-Cone (LC) dynamics or the Bethe-Salpeter (plus Dyson-Schwinger) equation, or to other approaches such as the use of dispersion relations or (QCD or LC) sum rules. Transitions and (transition) form factors are crucial elements for gauging the ideas underlying these QCD-based models. For example, the constituent quark model gives a fairly good description of the meson and baryon spectrum and some transitions, but quark effective form factors are typically required when describing hadronic form factors in the experimentally accessible region. In this framework, the study of hadronic form factors can thus be viewed as a study of the transition from constituent to current quark degrees of freedom. As exemplified by the many calculations of it, the electric form factor of the pion,, is one of the best observables for the investigation of the transition of QCD effective degrees of freedom in the soft regime, governed by all kinds of quark-gluon correlations at low, to the perturbative (including next-to-leading order and transverse corrections) regime at higher. In contrast to the nucleon, the asymptotic normalization of the pion wave function is known from pion decay. The hard part of the π+ form factor can be calculated within the framework of pQCD as the sum of logarithms and powers of $$F\_{\pi}(Q^2)=\frac{4\pi C\_F\alpha\_s(Q^2)}{Q^2} \Biggl\|\Sigma^{\infty}\_{n=0}a\_n \Biggl(log(\frac{Q^2}{\Lambda^2})\Biggr)^{-\gamma\_n}\Biggr\|^2 [1+O(\alpha\_s(Q^2),m/Q^2)], \label{eqn:pqcd}$$ which in the Q2 → ∞ limit becomes $$F\_{\pi}(Q^2) \overrightarrow{\_{Q^2 \rightarrow \infty}} \frac{16 \pi \alpha\_s(Q^2) f\_{\pi}^2}{Q^2}, \label{eqn:asymp}$$ where fπ = 93 MeV is the pion decay constant. Because the pion’s q̄q valence structure is relatively simple, the transition from “soft” (non-perturbative) to “hard” (perturbative) QCD is expected to occur at significantly lower values of for than for the nucleon form factors. Some estimates suggest that pQCD contributions to the pion form factor are already significant at Q2 ≥ 5. On the other hand, a recent analysis indicates that non-perturbative contributions dominate the pion form factor up to relatively large values of, giving more than half of the pion form factor up to =20. Thus, there is an ongoing theoretical debate on the interplay of these hard and soft components at intermediate, and high quality experimental data are needed to help guide this discussion. In this work, we concentrate exclusively on the spacelike region of the pion form factor. For recent measurements in the timelike region see Ref.. At low values of, where it is governed by the charge radius of the pion, has been determined up to =0.253 from the scattering of high-energy pions by atomic electrons. For the determination of the pion form factor at higher values of, one has to use high-energy electroproduction of pions on a nucleon, i.e., employ the 1H(e, e′π+)n reaction. For selected kinematic conditions, the longitudinal cross section is very sensitive to the pion form factor. In this way, data for have been obtained for values of up to 10 at Cornell . However, those data suffer from relatively large statistical and systematic uncertainties. More precise data were obtained at the Deutsches Elektronen-Synchrotron (DESY). With the availability of high-intensity electron beams, combined with accurate magnetic spectrometers at the Thomas Jefferson National Accelerator Facility (JLab), it has been possible to determine L/T separated cross sections with high precision. The measurement of these cross sections in the regime of =0.60-1.60 [Experiment Fpi-1 ] and =1.60-2.45 [Experiment Fpi-2 ] are described in detail in the preceding paper. In this paper, it is discussed how to determine from measured longitudinal cross sections, the values determined from the JLab and DESY data are presented, and the results of various theoretical calculations are compared with the experimental data. Since the pion in the proton is virtual (off its mass-shell), the extraction of from the measured electroproduction cross sections requires some model or procedure. In the next section, the methods that have been used to determine from the data are discussed. Section [sec:fpivgl] presents the adopted extraction method and the values of thus determined, including a full discussion of the uncertainties resulting from the experimental data and those from the adopted extraction procedure. Various model calculations of are discussed and compared to the data in section [sec:models]. In the final section, some conclusions are drawn and an outlook for the future is given. Methods of determining the pion charge form factor from data [sec:eepimodels] ============================================================================= The measurement of the pion form factor is challenging. As stated in the introduction, at low can be measured in a model-independent manner via the elastic scattering of π+ from atomic electrons, such as has been done up to =0.253 at Fermilab and at the CERN SPS. It is not possible to access significantly higher values of with this technique because of limitations in the energy of the pion beam together with the unfavorable momentum transfer. Therefore, at higher values of must be determined from pion electroproduction on the proton. The dependence on enters the cross section via the t-channel process, in which the incident electron scatters from a virtual pion, bringing it on-shell. This process dominates near the pion-pole at t = mπ2, where t is the Mandelstam variable. The physical region for t in pion electroproduction is negative, so measurements should be performed at the smallest attainable values of − t. To minimize background contributions, it is also necessary to separate out the longitudinal cross section, via a Rosenbluth L/T(/LT/TT) separation. The minimum physical value of − t, − tmin, is non-zero and increases with increasing and decreasing value of the invariant mass, W, of the produced pion-nucleon system. Carlson and Milana have estimated an approximate upper limit for the value of − tmin of the data appropriate for the extraction of the pion form factor by studying the competing non-pole QCD processes, which may complicate the extraction of at higher. They found that the background ratio MpQCD/Mpole rises dramatically once − tmin > 0.20. Their concern stemmed from the large value of − t in some of the Cornell results, which have − tmin > 0.4 . Therefore, reliable measurements should be performed at smaller − t and thus higher W (for a fixed ). The results presented in this paper respect this − tmin < 0.20 upper limit. It is yet to be determined if reliable measurements can be made in the future at larger − t. The value of Fπ(Q2) can then be determined from the data by trying to extrapolate the measured longitudinal cross sections at small values of − t to the pole at t = mπ2 = 0.02, or by comparing the measured longitudinal cross section at small values of − t to the best available model for the 1H(e, e′π+)n reaction, adjusting the value of in the latter. The presence of the nucleon and its structure complicates the theoretical model used, and so an unavoidable implication of this method is that the extracted pion form factor values are model dependent. The differential cross sections versus t over some range of and W are the actual observables measured by the experiment. It is important to note that in all cases the use of a model to extract is justified only if the model correctly predicts the t-dependence and magnitude of the data as well as the dependence on the invariant mass W of the photon-nucleon system. Chew-Low Extrapolation Method ----------------------------- Frazer originally proposed that be extracted from via a kinematic extrapolation to the pion-pole, and that this be done in an analytical manner using the so called Chew-Low extrapolation. The used Born formula is not gauge invariant, but in principle should give, nonetheless, when extrapolating to the pole. The last serious attempt to extract the space-like pion form factor from electroproduction data via the Chew-Low method was by Devenish and Lyth in 1972. Most of the data used were unseparated cross sections. The extrapolation failed to produce a reliable result, because different polynomial fits that were equally likely in the physical region gave divergent values of the form factor when extrapolated to the pion-pole at t = mπ2. Since then, the quality of the π+ electroproduction data-set has improved immensely, and separated longitudinal cross sections can now be used, avoiding the complications stemming from the other parts of the cross section. Therefore, it has been suggested to us that it may be appropriate to revisit the Chew-Low extrapolation method. However, before trying this method on the new data, it should be tested to see how reliably one can extrapolate to the pole. We start with high precision ‘pseudodata’ generated as a function of − t with the VGL Regge model. This model gives a fair to good description of a wide body of pion photo- and electroproduction data (see section [sec:vgl]). The kinematic conditions for the test are Q2 = 1.594 and W = 2.213 GeV, similar to our Fpi-2 data. The input value of the pion form factor in the model was Fπ = 0.244. The model cross sections were then used in a Chew-Low type extrapolation, with the challenge being to see if the Chew-Low extrapolation is able to reproduce (within fitting uncertainties) the input -value. The basis of the Chew-Low method is the Born-term model (BTM) formula for the pion-pole contribution to. We use the BTM of Actor, Korner and Bender, where pion-pole contribution to is given by $$N \frac{d\sigma\_L}{dt} = 4 \hbar c\ (e g\_{\pi NN})^2\ \frac{-t\ }{(t-m\_{\pi}^2)^2}\ Q^2 F\_{\pi}^2(Q^2), \label{eqn:chewlow}$$ where e2/(4πℏc) = 1/137 and N, which depends on the flux factor used in the definition of $\frac{d\sigma\_L}{dt}$, is given in our case by $$N=32\pi (W^2-m\_p^2)\sqrt{(W^2-m\_p^2)^2+Q^4+2Q^2(W^2+m\_p^2)}$$ . A monopole parameterization of the gπNN form factor is typically used to determine its value at t-values away from the pion-pole $$g\_{\pi NN}(t)=g\_{\pi NN}(m\_{\pi}^2) \Bigl(\frac{\Lambda\_{\pi}^2-m\_{\pi}^2}{\Lambda\_{\pi}^2-t}\Bigr),$$ where gπNN(mπ2) is the experimental value of 13.4. This is also the value used in the VGL calculations. We use the Λπ = 0.80 GeV result from the QCD Sum Rules calculation by T. Meissner, but because of the extrapolation to the pole the final result does not depend significantly upon the value chosen. (Color online) Linear (dotted), quadratic (dashed) and cubic (solid line) extrapolations of F2 to the pole as computed from Eqn.. The boxes are a VGL Regge model calculation for at fixed W = 2.213 GeV and Q2 = 1.594, calculated with =0.244. The lower limit of the box range is the kinematic endpoint of these, W values, while the upper limit is given by the t-range of our experiment. The input value in the model is indicated by the bullet placed at the pion-pole. [fig:chewlow] For the Chew-Low extrapolation, one plots the value of $$F^2=\frac{N}{4\hbar c\ (e g\_{\pi NN})^2} \frac{(t-m\_{\pi}^2)^2}{-Q^2 m\_{\pi}^2}\frac{d\sigma\_L}{dt} \label{eqn:chewlow2}$$ versus − t, which for a pure pole cross section gives a straight line passing through the origin, with value Fπ(Q2) at the pole (t = mπ2). Other contributions to the cross section, which have to be present, because the pole contribution alone is not gauge invariant, will change this behavior, but since they do not contain the $\frac{1}{(t-m\_{\pi}^2)^2}$ factor, they will not influence the value of F2 at the pole. However, it is not a priori given that the behavior as function of − t is linear, quadratic, or of higher order, thus introducing a ‘model’ (extrapolation) uncertainty. Values of F2 for the generated pseudodata, together with linear, quadratic and cubic extrapolations to the pole are shown in Fig. [fig:chewlow]. Also shown is the input form factor value in the VGL model, plotted at the pion-pole. Quadratic and higher-order extrapolations are almost indistinguishable and give a very good description of the (pseudo)data, but miss the input value of. This was true for all cases that were investigated, from Q2 = 0.60 to 2.45, the deviation from the input -value being 6-15%, depending on the case and the order of the extrapolation polynomial. Overall, there was no consistent trend for the order of polynomial which was best able to reproduce the input form factor value. This study indicates that even if is very well known over a range of physically-accessible t, the Chew Low extrapolation yields inconsistent results. The extrapolated result depends greatly upon the choice of quadratic cubic, or higher-order function, which all give a very good description of the data in the physical region. This indicates that the t-dependence of data in the physical region is insufficient to uniquely constrain the extrapolation through the unphysical region to the pole, even if the data have small relative uncertainties. Furthermore, even though modern data such as the JLab data are much more precise than those previously available, they still comprise 4-6 t-bins only, each with statistical and systematic uncertainties of 5-10%. Therefore, any polynomial extrapolation of such data to the pole will be more unreliable than the pseudodata test case shown here. Therefore, the Chew-Low extrapolation technique cannot be used to reliably determine the pion form factor from a realistic data set. Early Extractions of $\bf F\_{\pi}$ [sec:early] ----------------------------------------------- Brown et al. at CEA were the first to embrace the use of theoretical input to determine from their data. They used the model of Berends, which includes the dominant isovector Born term, with corrections for t values away from the pole by means of fixed-t dispersion relations. This model was also used by Bebek et al. for the analysis of the first two sets of Cornell data . The model gave a fair description of the data, but systematically underpredicted the LT term of the cross section and the t-dependence of the data. Until then, data were obtained at one (larger) value of the photon polarization parameter ε only. In the third Cornell experiment , data were taken at low values of ε, so that in combination with the earlier data an L/T separation could be performed at -values of 1.19, 2.00 and 3.32 . The value of was found to be substantially larger than predicted by Berends, especially at larger. The values obtained for had such large error bars that they were not used to determine. Instead, use was made of the observation that within the experimental error bars the -dependence of the forward transverse cross section was satisfactorily reproduced by the -dependence of the total virtual-photoproduction cross section. Therefore, () was parameterized with the overall scale as a free parameter, and the parameterized values then used to subtract from the measured unseparated cross sections to obtain. These data at the lowest value of − t were used to determine, assuming that is given there by the t-channel one pion-exchange Born term. This was done for all data obtained at CEA and Cornell. The uncertainties in thus obtained and presented in Ref. are statistical ones only, and do not include the contribution from the uncertainty in the value of used in the subtraction. Especially at the larger values of, these are considered to be substantial, as can be seen from Fig. 4 of Ref. . The DESY experiments produced high-quality separated cross sections at Q2 = 0.35, W = 2.10 GeV and Q2 = 0.70, W = 2.19 GeV. Both of these experiments used the generalized Born Term Model of Gutbrod and Kramer to determine. This BTM incorporates t, s, and u-channel diagrams for the γv + p → π+ + n reaction, giving a fair description of the magnitude of the measured unseparated cross sections, but failing to describe and. However, Gutbrod and Kramer found that when treating the magnitude of the nucleon form factor GEp(Q2) as a free parameter, a much better description of the then available data was obtained. In addition, they included a factor et/M2 in order to improve the description of the t dependence of the data. The justification given is that the nucleon is far off its mass-shell, whereas the pion is near to its pole. This generalized BTM gave a good overall description of the DESY data. However, at Q2 = 0.70, nucleon form factors about 50% above their on-mass-shell values were needed. The size of the modification needed at Q2 = 0.35 is not given. Newer Models [sec:vgl] ---------------------- More recently, two new models for the 1H(e, e′π+)n reaction have become available. In Refs. , Vanderhaeghen, Guidal and Laget (VGL) have presented a Regge model for pion production in which the pole-like propagators of Born term models are replaced with Regge propagators, i.e., the interaction is effectively described by the exchange of a family of particles with the same quantum numbers instead of a single particle. If the same vertices and coupling constants are used, the Regge model and the BTM calculations agree at the pole of the exchanged particle, but away from the pole the Regge model provides a superior description of the available data. For forward pion production, the dominant exchanges are the π and ρ trajectories. These determine the t-dependence of the cross section without the use of a gπNN(t) factor. At low values of − t, as covered by this work, is completely determined by the π trajectory, while is also sensitive to the ρ exchange contribution. Since the t-channel π diagram is by itself not gauge invariant, the s-channel (for π+ production) or u-channel (for π− production) nucleon exchange diagram was also Reggeized, to ensure gauge invariance of their sum. The VGL model was first applied to pion photoproduction and later extended to electroproduction , with monopole forms for the ππγ and ρπγ form factors: Fπ, ρ(Q2) = [1 + Q2/Λπ, ρ2]− 1. Apart from the ππγ and ρπγ form factors, the model is parameter free, as the coupling constants at the vertices (such as gρπγ) are well determined by precise studies and analyses in the resonance region. The model gives a good description of the W- and t-dependences of then available π+ and π− photoproduction data, including the spin asymmetries, and of the earlier electroproduction data. [h] The VGL predictions have been compared to our measured cross sections and the ones taken at DESY in Ref. . For the discussion in this paper, the data for and are reproduced in Fig. [fig:xsecvgl], together with the results of the model calculations. The VGL cross sections were evaluated at the same $\overline W$ and $\overline Q^2$ values as the data, resulting in the discontinuities shown. The values of shown are determined by the fitting of the VGL model to the measured -values at the five values of t at each, resulting in values between 0.37 and 0.51. The value of is more poorly known. Calculations with both =0.600 and 1.500 are shown, where the upper value is taken from the application of the VGL model to kaon electroproduction. The model gives an overall good description of our data and those of, but the description of the t-dependence of the data is worse at =0.60 and 0.70. The poorer description of the data by the VGL model at lower and W may be due to contributions from resonances, which are not included explicitly in the Regge model. This is supported by the fact that the discrepancy in the t-dependence of the data is strongest at the lowest value, at higher the resonance form factor supposedly reducing such contributions. The values of are severely underestimated, especially at larger, even when taking a hard ρπγ form factor. Since the data at the real-photon point are well described, this suggests that another mechanism, whose contribution increases with, is at play. Recently the VGL model was extended by including a hard scattering between the virtual photon and a quark, the latter hadronizing in combination with the spectator diquark into a pion plus residual nucleon. With plausible assumptions, a good description of was obtained, with no influence on. Those results support the idea that the discrepancy in the magnitude of, which increases with, and the discrepancy in the slope of with − t, which decreases with, are not directly related. Strategies for dealing with the latter discrepancy when extracting the pion form factor are discussed in Sec. [sec:fpivgl]. We also considered a modification to the VGL Regge model published by J.M. Laget in 2004. Laget introduces a t-dependent factor into the pion form factor which is related to the pion saturating Regge trajectory, approaching -1 as t → − ∞. The effect of this modification is to boost by 40% for the largest − t spanned by our data (=2.703, − t=0.365 ), and converging with the unmodified calculation at small − t. The effect on is under 1% for the largest − t covered by our data, and is negligible at − tmin. [h] Another recent development is the effective Lagrangian model of Faessler, Gutsche, Lyubovitskij and Obukhovsky (FGLO, Ref. ). This is a modified Born Term Model, in which an effective Lagrangian is used to describe nucleon, pion, ρ and photon degrees of freedom. The (combined) effect of s- and u-channel contributions, which interferes with the pion t-pole, is modeled using a constituent quark model. The authors show that the ρ t-pole contribution is very important in the description of the magnitude of. When comparing vector and tensor representations of the ρ contribution, the latter was found to give better results. Unlike the VGL model, the cross section depends here also on the ρ exchange, because of the interference of the π and tensor ρ exchange contributions. The model contains a few free parameters, such as the renormalization constant of the Kroll-Ruderman contact term used to model the s(u)-channel, and t-dependent strong meson-nucleon vertices, which are parameterized in monopole form, as are the electromagnetic form factors. The corresponding parameters were adjusted so as to give overall good agreement with our and data. As in case of the VGL model, a detailed comparison of the FGLO model results to the measured data is given in Ref. , while the results for and are also shown in Fig. [fig:xsecobu]. The values of used were determined by the fitting of the model to the t-bins at each, while keeping the other parameters fixed at the values assigned by the authors. In some cases, this results in different values than shown in Ref.. However, it should be kept in mind that the FGLO model cross sections also depend on other parameters, which have been adjusted by the authors of the model to give good agreement to our and data. To the best of our knowledge, the =0.7 data of Ref. were not taken into account when these parameters were determined. Generally, the agreement of the FGLO model with the data is rather good except for the =0.60 [Fpi-1] and 0.70 measurements. There is a serious discrepancy in the Q2- and W-dependence of the data. For around 0.7, the model agrees fairly well with the data at W = 1.95 GeV, but it over-predicts the =0.70, W = 2.19 GeV data by a large factor. On the other hand, for =1.60, the W = 1.95 GeV data are under-predicted by about a factor of two, while those at W = 2.22 GeV are reproduced, and the W = 2.22 GeV data for =2.45 are under-predicted again by 20-60%. This indicates some problem in the description of the Q2, W-dependences of the ρ exchange used to describe. Because of the ρ − π interference, the problems with the description of also affect the calculation. This makes it hard to estimate how reliable the values of would be if extracted from the data using this model. $\bf F\_{\pi}$ Results [sec:fpivgl] =================================== As already discussed, the separated cross sections versus t over some range of and W are the actual observables measured by the experiment, and the extraction of the pion form factor from these data is inherently model dependent. Ideally, one would like to have a variety of reliable electroproduction models to choose from, so that the model dependence of the extracted values can be better understood. Since the VGL Regge model is able, without fitted parameters, to provide a good description of both π+ and π− photoproduction data, and of electroproduction data over a range in W, t, and, it is our opinion that at the moment only this model has shown itself to be sufficiently reliable to enable its use to extract pion form factor values from the data. Therefore, we will use this model to determine our values. Clearly, the values determined are strictly within the context of the VGL Regge model, and other values may result if other, better models become available in the future. $\bf W\approx 2.2$ GeV Data --------------------------- \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| Q2 \| W \| \| Fπ \| \| () \| (GeV) \| () \| \| \| 0.60 \| 1.95 \| 0.458 ± 0.031− 0.068+ 0.255 \| 0.433 ± 0.017− 0.036+ 0.137 \| \| 0.75 \| 1.95 \| 0.388 ± 0.038− 0.053+ 0.135 \| 0.341 ± 0.022− 0.031+ 0.078 \| \| 1.00 \| 1.95 \| 0.454 ± 0.034− 0.040+ 0.075 \| 0.312 ± 0.016− 0.019+ 0.035 \| \| 1.60 \| 1.95 \| 0.485 ± 0.038− 0.027+ 0.035 \| 0.233 ± 0.014− 0.010+ 0.013 \| \| 0.35 \| 2.10 \| 0.601 ± 0.060 \| 0.632 ± 0.023 \| \| 0.70 \| 2.19 \| 0.627 ± 0.058− 0.085+ 0.096 \| 0.473 ± 0.023− 0.034+ 0.038 \| \| 1.60 \| 2.22 \| 0.513 ± 0.033− 0.022+ 0.052 \| 0.243 ± 0.012− 0.008+ 0.019 \| \| 2.45 \| 2.22 \| 0.491 ± 0.035− 0.024+ 0.045 \| 0.167 ± 0.010− 0.007+ 0.013 \| As shown in Fig. [fig:xsecvgl], the VGL model does a good job of describing the t-dependence of the cross sections at W ≈ 2.2 GeV, Q2 = 0.35, 1.60 and 2.45. In these cases, the extraction of the pion form factor from the data is straightforward: the value of in the model is varied until the agreement of the model with the data is optimized. The mean $\overline Q^2$ and $\overline W$ values of the data for each t-bin are used when evaluating the model. is then calculated from Eqn. [eqn:monopole], using the best-fit and the nominal values. These are listed in the last two lines of Table [tab:fpi]. The experimental statistical and systematic uncertainties were propagated to the uncertainties as follows. The statistical and t, ε-uncorrelated systematic uncertainties[1](#fn1) were applied to the data prior to the fitting of the VGL model to the data. This yields the best-fit value and its associated fitting uncertainty. The effects of the t-correlated, ε-uncorrelated, and the t, ε-correlated systematic uncertainties on the fit were determined by investigating the variation in values allowed by fitting to the lowest − t bin only. Of these, the ε-uncorrelated, t-correlated systematic uncertainty is amplified by 1/Δε in the L-T separation, while the t, ε-correlated uncertainty is not. The resulting uncertainties are added in quadrature to the fitting error, yielding the first uncertainty listed in Table [tab:fpi]. This value is also propagated to according to the monopole parameterization, yielding the first uncertainty listed. In order to check if the extracted value of depends on the t-range used, the VGL model (i.e., the value of ) was fitted separately to each point from Fpi-2 and DESY, and the corresponding values of determined. In order to remove the natural variation of with the $\overline Q^2$ of each bin, the nominal values were used in the monopole equation. A plot of the obtained versus t is shown in Fig. [fig:fpi2check]. Also indicated as the shaded band is the value with the uncertainty that is obtained if one fits to all of the t-bins simultaneously. Except perhaps at =0.70, the data show no residual t-dependence beyond the statistical fluctuation. $\bf W=1.95$ GeV Data --------------------- As already shown in Sec. [sec:vgl], the VGL model does not fully describe the t-dependence of our data at W = 1.95 GeV. The difficulty, as far as the extraction is concerned, is that there is no theoretical guidance for the assumed interfering background not included in the VGL model, even if one assumes that it is due to resonances. Virtually nothing is known about the L/T character of resonances at W = 1.95 GeV, let alone how they may influence through their interference with the π-pole amplitude. Given this lack of theoretical guidance, we are forced to make some assumptions in extracting from these data. Our guiding principle is to minimize these assumptions to the greatest extent possible. The form factor extraction method that we have adopted for these data relies on the single assumption that the contribution of the background is smallest at the kinematic endpoint tmin. Our best estimate of for the W = 1.95 GeV data is determined in the following manner. Using the value of as a free parameter, the VGL model was fitted to each t-bin separately, yielding $\Lambda^2\_\pi(\overline{Q^2},\overline{W},t)$ values as shown in Fig. [fig:fpi1lpi]. The values of tend to decrease as − t increases, presumably because of an interfering background not included in the VGL model. Since the pole cross section containing increases strongly with decreasing − t, we assume that the effect of this background will be smallest at the lowest value of ∣t∣ allowed by the experimental kinematics, ∣tmin∣. Thus, an extrapolation of to this physical limit is used to obtain our best estimate of. The value of at tmin is obtained by a linear fit to the data in Fig. [fig:fpi1lpi]. The resulting and values for the Fpi-1 data are listed in Table [tab:fpi]. The first uncertainty listed includes both the experimental and the linear fit extrapolation uncertainties. Since Fig. [fig:fpi2check] suggests also a dependence (at larger − t) between the VGL calculation and the =0.70 data of Ref., this extraction method was also applied to those data. The result obtained when extrapolating to tmin is listed in Table [tab:fpi]. The value of () is 11(20)% larger than if the VGL model was simply fit to all data points. Applying the same procedure to our W = 2.22 GeV data, it was found that the resulting values of () would be 1(2)% larger, which is statistically insignificant, confirming that the t-dependence of those data is well described by the VGL model. Model Uncertainty Estimate -------------------------- The fact that we used an additional assumption for the cases where the VGL model does not completely describe the t-dependence of the data causes an additional uncertainty in the extracted value, which we term ‘model uncertainty’. This model uncertainty, which is within the context of the VGL model, should be distinguished from the general model uncertainty discussed in section [sec:eepimodels], which would result when using different models. In order to make a quantitative estimate of this additional uncertainty, the spread in extracted values of (and thus ) was investigated by assuming specific forms of the interfering background missing in the VGL model. An effective upper limit for is obtained by assuming that the background yields a constant, negative, contribution to. For each value of, this background and the value of were fit together to the data, assuming that the background is constant with t. The fitted contribution of the background was found to drop strongly with increasing. A second possibility is to assume, besides the VGL amplitude, a t-independent interfering background amplitude, fitting for every the magnitude and phase of the latter, together with the value of. Although the fitting uncertainties are very large, the results suggest an interfering amplitude whose magnitude decreases monotonically with increasing. In this case, the interference between the background amplitude and the VGL amplitude, which depends on their relative phase, does not necessarily result in a net negative cross section contribution to. The estimated model uncertainty is determined from the spread of the values and their uncertainties at each, obtained with these two choices of background. To keep the number of degrees of freedom the same in both cases, the background was fixed to the value giving the best χ2, and and its uncertainty were then determined in a one-parameter fit of the VGL model plus background to the data. Since the statistical uncertainties of the data are already taken into account in the first given uncertainty in Table [tab:fpi], the contribution of the statistical uncertainties of the data were quadratically removed from the uncertainties given by the fit. The model uncertainties at each are then taken as the range plus corrected fitting uncertainty given by these two methods, relative to the value of determined from the extrapolation to tmin. This procedure was applied to all data except those of Ref., yielding the model uncertainties listed as the second (asymmetric) uncertainty in Table [tab:fpi]. No model uncertainty was calculated for the =0.35 data from DESY because the t-range spanned by those data (only 0.03 ) was too small for this procedure to be reliably applied. For the W = 1.95 GeV data, the model uncertainty in the extracted value drops from about 20% at =0.60 to about 5% at 1.60. To be consistent, the same procedure was applied to the W = 2.22 GeV data, which yielded model uncertainties of about 5% at both =1.60 and 2.45. These rapidly dropping uncertainties with increasing reflect the smaller discrepancy of the VGL calculation with the t-dependence of the data at larger values of and W. These findings are at least compatible with the idea that resonance contributions, which presumably have a form factor that drops rapidly with, are responsible. They also suggest that our extraction methods are robust, when the background contribution is small, as appears to be the case at the higher value of W. Discussion and Comparison with Empirical Fits [sec:monopole] ------------------------------------------------------------ The form factors extracted from the Fpi-1 and Fpi-2 data with the use of the VGL model are shown in Fig. [fig:q2fpimono], along with the reanalyzed =0.70 data of Ref., the elastic scattering measurements of Ref., and the =0.35 data of Ref.. The Cornell data of Refs. are not included because, as discussed in section [sec:early], they have large unknown systematic uncertainties. The excellent agreement between the =1.6 form factor values obtained from our W = 1.95, 2.22 GeV data, despite their significantly different tmin and W values, indicates that the model uncertainties from the use of the VGL model seem to be under control, at least in this -range. Also shown is a more recently obtained value at =2.15 , which was also extracted with the use of the VGL model. The solid line shown in Fig. [fig:q2fpimono] is the monopole fit obtained by Amendolia et al. from their elastic scattering data. This curve is given by $$F\_{mono}=\frac{1}{1+\frac{r\_{mono}^2Q^2}{6\hbar^2c^2}},$$ where rmono2 = 0.431 fm2 is their best-fit squared pion charge radius. Fig. [fig:q2fpimono] indicates a systematic departure of the data from the monopole curve above Q2 ≈ 1.5. This departure may have implications for theoretical approaches that assume the validity of the monopole parameterization over a wide range of. To illustrate the departure from the monopole curve, as well as to provide an empirical fit that describes the data over the measured range, we also show in Fig [fig:q2fpimono] a fit which includes a small dipole component, Ffit = 85%Fmono + 15%Fdip, where $$F\_{dip}=\frac{1}{\bigl(1+\frac{r\_{dip}^2Q^2}{12\hbar^2c^2}\bigr)^2}$$ , and rdip2 = 0.411 fm2. This dipole parameterization has nearly the same χ2 for the elastic scattering data as the monopole curve shown, but it drops much more rapidly with. The combined monopole plus dipole fit is consistent with our intermediate data, while maintaining the quality of fit to the elastic scattering data. Since a monopole parameterization does not converge to the pQCD asymptotic limit (Eqn. [eqn:asymp]), it is expected to fail at some point. Similarly, we should expect this empirical monopole+dipole parameterization to show its limitations when additional high data become available. Comparison with Model Calculations [sec:models] =============================================== The pion form factor can be calculated relatively easily in a large number of theoretical approaches which help advance of our knowledge of hadronic structure. In this sense, plays a role similar to that of the positronium atom in QED. Here, we compare our extracted values to a variety of calculations, selected to provide a representative sample of the approaches used. Perturbative QCD ---------------- The most firmly grounded approach for the calculation of is that of pQCD. The large behavior of the pion form factor has already been given in Eqn. [eqn:pqcd]. By making use of model-independent dimensional arguments, the infinitely-large behavior of the pion’s quark wave function (distribution amplitude, or DA) is identified as ϕπ(x, Q2 → ∞) → 6fπx(1 − x) whose normalization is fixed from the π+ → μ+νμ decay constant. Eqn. [eqn:asymp] follows from this expression. Neither of these equations is expected to describe the pion form factor in the kinematic regime of our data, and so much effort has been expended to extend the calculation of to experimentally accessible. In this case, the pion DA, ϕπ(x, $Q^2$), must be determined at finite. Additional effects, such as quark transverse momentum and Sudakov suppression (essentially a suppression of large quark-quark separation configurations in elastic scattering processes) must be taken into account. A number of authors have performed leading-twist next-to-leading order (NLO) analyses of at finite. The hard contributions to expand as a leading order part of order αs and an NLO part of order αs2. Bakulev, Passek-Kumericki, Schroers and Stefanis have investigated the dependence of the form of the DA on the form factors, using data from a variety of experiments. These were the πγγ transition form factor data from CLEO and CELLO, as well as our data. Their results are insensitive to the shape of the DA near x = 1/2, while its behavior at x = 0, 1 is decisive. The resulting hard contribution to the pion form factor is only slightly larger than that calculated with the asymptotic DA in all considered schemes. The result of their study, shown as Fπhard in Fig. [fig:pqcd], is far below our data. The drop at low is due to their choice of infrared renormalization, which is not necessarily shared by other calculations. To bring the calculation into agreement with the experimental data, a soft component must also be added. The treatment of the soft contribution to the pion DA is model-dependent. The authors estimate this soft contribution using a local quark-hadron duality model. This soft estimate, along with the sum of the hard and soft contributions, are also shown in Fig. [fig:pqcd]. The interplay at intermediate between the hard and soft components can be non-trivial, as demonstrated by Braun, Khodjamirian, and Maul, using a light-cone sum rule approach. Their results support a pion DA that is close to the asymptotic expression, but they find that strong cancellations between soft terms and hard terms of higher twist lead to the paradoxical conclusion that the nonperturbative effects in the pion form factor can be small, and the soft contributions large, simultaneously. Because of complications such as these, different theoretical viewpoints on whether the higher-twist mechanisms dominate until very large momentum transfer, or not, remain. Lattice QCD ----------- Unlike QCD-based models, in which confinement must be explicitly added, Lattice QCD allows calculation from first principles. However, while lattice QCD is based on the QCD Lagrangian, it involves a number of approximations. Errors are introduced because space and time are crudely discretized on the lattice. This error is controlled by the use of improved lattice QCD actions. To allow a more rapidly converging action, and hence reduce CPU usage, the pion mass used is significantly larger than the physical pion mass. Chiral extrapolation errors are introduced when the lattice results, determined with large pion mass, are extrapolated to physical values. Finally, quenching errors are introduced when disconnected quark loops are neglected. The first lattice simulations of were done in the 1980’s. These pioneering works were primarily a proof of principle of the lattice technique, and were restricted to < 1 . These results are consistent with the low experimental data, within the large statistical uncertainties of these pioneering calculations. Spurred by advances in CPU power and lattice techniques, as well as the availability of new experimental data, a number of groups have returned to the calculation of on the lattice. Of these, we compare our data to the recent unquenched simulations of Brommel, et al.. They performed simulations for a wide range of pion masses and lattice spacings, so that both the chiral and continuum limits could be studied. However, the lowest pion mass used in the simulations was 400 MeV, so the chiral extrapolation is significant. The authors fitted the -dependence of each lattice configuration with a monopole form for the pion form factor and determined the corresponding monopole mass. They then extrapolated these masses to the one corresponding to the physical pion mass to obtain a chiral monopole mass value of 0.727 ± 0.016 GeV. The (monopole) form factor calculated with that mass (including its uncertainty) is indicated by the shaded band in Fig. [fig:latdisp], cut off at the highest point of the lattice simulation. This result begins to trend away from the Q2 > 1.5 experimental data. It remains to be seen how these results would be affected by our Sec. [sec:monopole] comments on the applicability of the monopole parameterization in this range. Dispersion Relation with QCD Constraint --------------------------------------- Dispersion relations are based on constraints posed by causality and analyticity, and relate the timelike and spacelike domains of the pion form factor on the complex plane. In principle the technique is exact, but our incomplete knowledge of the scattering amplitudes over the whole complex plane, and in particular the incomplete understanding of the contribution of all of the poles in the timelike region, creates uncertainties. Authors address these uncertainties by imposing additional constraints, such as the role of higher timelike resonances like the ρ‴, or chiral perturbation constraints near the spacelike threshold, or that must approach its expected asymptotic value at very large . We compare the data to the dispersion relation analysis of B.V. Geshkenbein in Fig. [fig:latdisp]. The displayed uncertainty band is obtained by assuming different distributions of zeroes in the complex s-plane. This results in a band that grows with, with the ‘no zeroes’ curve lying nearly at the lower end of the band. Our highest data lie above the ‘no zeroes’ curve, but below the ‘improved maximum ’ limit. QCD Sum Rules ------------- The QCD sum rule approach is designed to interpolate between the perturbative and non-perturbative sectors using dispersion relation methods in combination with the operator-product expansion. While the practical implementation of this approach cannot claim to be rigorously derived from QCD, its intuitive value is that it provides a bridge between the low- and high-energy properties of QCD. A number of authors have applied this technique with good success to the pion form factor. In the calculation of Radyushkin, QCD sum rules were used to give a local quark-hadron duality estimate of the soft wave function $$F^{soft}\_{\pi}(Q^2)=1-\frac{1+6 s\_0/Q^2}{(1+4s\_0/Q^2)^{3/2}}, \label{eqn:qsr\_soft}$$ where the duality interval, s0, which within the QCD sum rule approach is determined by the magnitude of the quark and gluon condensates, was taken as 4π2fπ2 ≈ 0.7. This soft calculation, shown in Fig. [fig:qsrdse], under-estimates the data by about 25%. For the hard contribution, a simple model based on the interpolation between the behavior near Q2 = 0 (related by the Ward identity to the O(αs) term of the 2-point correlator) and the asymptotic behavior was used $$F^{hard}\_{\pi}(Q^2)=\frac{\alpha\_s}{\pi}\frac{1}{(1+Q^2/2s\_0)}.$$ The sum, Fπsoft + Fπhard, is in excellent agreement with the data. More recently, Braguta, Lucha and Melikhov have replaced the simple ansatz leading to Eqn. [eqn:qsrsoft] with an expression including explicit corrections up to O(αs). Since the higher-order corrections needed to apply these results with authority to the intermediate region are beyond the capacity of their two-loop calculation, there is a model dependence in their numerical result, which is reflected in the two different curves for s0 = 0.65 and $s\_0=\frac{4\pi^2f\_{\pi}^2}{1+\alpha\_s(Q^2)/\pi}$ shown in Fig. [fig:qsrdse]. Bethe-Salpeter Equation ----------------------- The Bethe-Salpeter equation (BSE) is the conventional formalism for the treatment of relativistic bound states. In this formalism, a meson is described by a covariant wavefunction, which depends on the four momenta of its constituent quarks. Although formally correct, complications arise as the interplay between different configurations, such as q-q̄ and q-q̄-g, are implicitly buried in the potential and scattering amplitudes used in analyzing hadronic processes, and as a result, these potentials and scattering amplitudes are nearly intractable. The light-front Bethe-Salpeter model is a means to handle this problem by breaking the BSE into separate hard and soft components. A variety of models incorporating a confining potential which dominates at low, and a QCD-based interaction which dominates at high, are given in Refs.. Another approach is to use the Dyson-Schwinger equation (DSE) to obtain dressed quark propagators which may be used in the solution of the BSE. The Dyson-Schwinger approach to nonperturbative QCD has many advantages. It is consistent with quark and gluon confinement, it automatically generates dynamical chiral symmetry breaking, and the solution is Poincare invariant. In the work of Maris, Tandy, and Roberts, the meson Bethe-Salpeter amplitudes and quark-photon vertex are obtained as solutions of the homogeneous and inhomogeneous BSE, and the dressed quark propagators are obtained from the quark DSE. The model parameters are fixed by requiring fπ and mπ to be in good agreement with the data and then rπ and are predicted with no further adjustment of parameters. Their calculation is shown in Fig. [fig:qsrdse]. It is in excellent agreement with our data up to =1.60. To extend the validity of the model to higher, a more complete description that takes meson loop corrections into account self-consistently is required. Local Quark-Hadron Duality -------------------------- Quark-hadron duality relations link the hadronic structure information contained in exclusive form factors and inclusive structure functions by making strong assumptions of locality. While local quark-hadron duality is an expected consequence of QCD at asymptotically large momenta, it is not at all clear how well it could work at finite . And if it does, it may be due to accidental cancellations of higher twist effects. Nevertheless, it is worthwhile to compare predictions based on quark-hadron duality with the measured data, especially since duality is expected to work better at higher, in contrast to many other approaches. The approximate relationship between the pion elastic form factor and the pion structure function F2π = νW2π was found by Moffat and Snell, [$F\_\pi$($Q^2$)]2 ≈ ∫1ωmaxF2π(ω)dω, where ω = 1/x, and the upper limit of integration is chosen to select the elastic contribution to the inclusive structure function. In applying this formula use is made of the Drell-Yan-West relation, which is based on a field-theoretic parton model that predates QCD. It predicts that if the asymptotic behavior of a form factor is (1/$Q^2$)n, the corresponding structure function should behave as (1 − x)2n − 1 as x → 1. This leads to the prediction F2π(x → 1) ∼ (1 − x). The existence of Drell-Yan F2π data allows a quantitative test of Eqn. [eqn:moffat] using only phenomenological input. Calculations based on the leading-order analysis of F2π data by Ref., and the next-to-leading order analysis of Ref., are shown in Fig. [fig:ducqm]. In both cases, the magnitude of the prediction is dependent on the value chosen for the inelastic cutoff ωmax (and corresponding Wmax) in Eqn. [eqn:moffat]. Local duality is expected not to work at lower. This is reflected in the poor description of the Q2 < 1 form factor data. However, above Q2 > 2, the next-to-leading order analysis is consistent with our data. Constituent-Quark Model ----------------------- There are many calculations using a variety of constituent-quark models. The differences in approach typically involve differences in the treatment of the quark wave functions, or the inclusion of relativistic effects. Fig. [fig:ducqm] shows the result of calculations by Cardarelli et al. and by Hwang. Both are relativistic quark models on the light front. Ref. uses the effective qq̄ Hamlitonian of, which contains a one-gluon-exchange term and a linear confining term, and which describes a large set of meson spectroscopic data. Use of this interaction results in large high-momentum components, and is strongly overpredicted (upper dot-dashed curve in Fig. [fig:ducqm]). This can be cured in a way that is consistent with the notion of a constituent quark, by assuming a form factor for the latter. Taking a monopole form for the latter and adjusting the mass parameter so that the measured pion charge radius is reproduced, results in the lower dot-dashed curve shown. The model of Ref. allows a consistent and fully relativistic treatment of quark spins and center-of-mass motion to be carried out. A power-law wave function is used, whose parameters are determined from experimental data on the charged pion decay constant, the neutral pion two-photon decay width, and the charged pion electromagnetic radius. The charge and transition form factors of the charged pion and the branching ratios of all observed decay modes of the neutral pion are then predicted. The calculation is in very good agreement with our data. Li and Riska asked if the empirical data exclude the presence of a significant sea-quark configuration in the charged pion. They performed a constituent quark model calculation which was extended to include explicit sea-quark components in the pion wave function. They found that these sea-quark contributions grew with increasing, because they allowed the momentum transfer to be shared by a greater number of constituents, and so were less-suppressed at high than configurations which involved only a q̄q pair. Although their analysis was model-dependent, they found that our data allowed an approximate 20 ± 20% sea-quark component, with the data point at =2.45 providing the greatest constraint. Holographic QCD --------------- A recent theoretical development is the AdS/CFT correspondence between weakly-coupled string states defined on a 5-dimensional anti-de Sitter space-time ($\rm AdS\_5$) and a strongly-coupled conformal field theory (CFT) in physical space-time. The goal of holographic QCD models is to find a weakly-coupled theory for which the dual strongly-coupled theory is as close to QCD as possible, and so allow analytic solutions of hadronic properties in the non-perturbative regime to be performed. In these models, confinement is simulated by imposing boundary conditions on the extra fifth dimension z. In the “hard-wall” variant, confinement is modeled by a hard cutoff at a finite value z = z0 = 1/ΛQCD. This has the advantage of simplicity but produces the unphysical Regge trajectory Mn2 ∼ n2. The “soft-wall” variant replaces the hard-wall boundary with an oscillator-type potential, and produces the more phenomenologically realistic Regge behavior Mn2 ∼ n. Several authors have applied holographic models to the pion form factor. Complications arise when one introduces spontaneous and explicit chiral symmetry breaking effects into the soft-wall holographic QCD model. Refs. take different approaches to this problem. Grigoryan and Radyushkin consider only the hard-wall variant, and then estimate a soft-wall correction from their previous vector meson study. They conclude that a full analytic calculation would likely follow the data only in the Q2 < 1 region, while overshooting it above Q2 ∼ 2. The calculations by Kwee and Lebed are numerical. Both the hard-wall and the soft-wall calculations predict charge radii that are too small, especially for the soft-wall case (see Fig. [fig:adscft]). By allowing the parameters of the soft-wall model (originally fixed by mρ, mπ, and fπ) to vary, they find it is possible to describe at either high or low, but not both. Issues in ongoing discussions on the AdS/CFT approach include the applicability of this model to the larger region where partonic degrees of freedom become appreciable, and the treatment of chiral symmetry breaking. Summary and Outlook =================== Values for the charged pion form factor, Fπ(Q2), have been extracted for =0.60-2.45 from the longitudinal cross sections σL(t) for the 1H(e, e′π+)n reaction recently measured at JLab. values were also extracted from older experimental data acquired at DESY. The Cornell data are not included in this analysis because these were not obtained in a true L/T-separation, but instead by subtracting a certain assumption for. In addition, the higher data have excessively large values of − tmin. The form factor extraction requires the use of a model incorporating both the π+ production mechanism as well as the effect of the nucleon. Several approaches to extract from the data, including the Chew-Low extrapolation method, various types of Born Term Models, and newer models utilizing Regge trajectories and effective Lagrangians, were reviewed. By using specially generated test data, it was found that extrapolating to the pole at t = mπ2, as is done in the Chew-Low method, cannot be used in practice, because there is no way to determine the order of the polynomial to use for the extrapolation, and because even small uncertainties in the measured cross sections lead to a large uncertainty in. From the models available for determining from the measured values of, the VGL Regge model was chosen, since it contains no ad hoc parameters, and its validity has been well established over a wide kinematic range in t and W for both electroproduction and photoproduction data. The VGL model gives a rather good description of both the t and the W dependence of the JLab data at values of > 1, but especially at Q2 = 0.60 the fall-off of the data with − t is steeper than that of the model. In the cases where the VGL model described well the t-dependence of the data, the value of was determined by fitting the model to the data. Otherwise, the value of was determined by extrapolating the fit of the model to t = tmin. An additional ‘model uncertainty’ has been estimated by using two different assumptions for an interfering background that could be responsible for this discrepancy between the data and VGL model. The fact that the discrepancy, and hence the model uncertainty, is very small at higher values of and W suggests that effects from nucleon resonances play a role in the data at lower and W. It is stressed that the cross sections are the actual observables measured by the experiment, and that the extracted values of are inherently dependent on the model used to extract them. The development of additional models for the 1H(e, e′π+)n reaction would allow further exploration of the model dependence of the extraction of from the same cross section data. On the experimental front, proposed measurements after the completion of the JLab upgrade are expected to better establish the validity of any used model by investigating, for example, the W-dependence of the results. The results for, extracted from our data and from the DESY data with the use of the VGL model, are presented together with their experimental and model uncertainties. Above Q2 ≈ 1.5, these data are systematically below the monopole parameterization based on the empirical pion charge radius. The data are also compared to a selection of calculations, including those based on pQCD, Lattice QCD, Dispersion Relations, QCD Sum Rules, Bethe-Salpeter Equation, Local Quark-Hadron Duality, Constituent-Quark Model, and Holographic QCD. There has been tremendous progress in the theory of hadronic structure physics in the past decade, as evident by the many new approaches under development. However at present, the intermediate regime remains a significant challenge. Several different approaches concur that up to at least =2.5, the data are far above the estimated ‘hard’ (perturbative) contribution, and that ‘soft’ (non-perturbative) contributions likely dominate in this region. Data expected to be taken after the completion of the JLab upgrade, up to at least Q2 = 6.0, are expected to indicate whether the higher-twist mechanisms dominate until very large momentum transfer, or not. Acknowledgments =============== The authors thank Drs. 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WT \| $\bf{0.80}$ \| 0.09 \| 1.00 \| 0.53 \| 0.53 \| -0.06 \| 0.64 \| \| TKWT \| 0.27 \| 0.38 \| 0.53 \| 1.00 \| -0.11 \| -0.09 \| -0.09 \| \| SPSM \| $\bf{0.74}$ \| 0.02 \| 0.53 \| -0.11 \| 1.00 \| -0.5 \| $\bf{0.83}$ \| \| KPS \| 0.022 \| -0.22 \| -0.06 \| -0.09 \| -0.5 \| 1.00 \| 0.014 \| \| KPSM \| $\bf{0.93}$ \| -0.07 \| 0.64 \| -0.09 \| $\bf{0.83}$ \| 0.01 \| 1.00 \| [CorrPheno] Figures from Sec. 6.2.1 ----------------------- ![image](Scatterplotlin1999.jpeg) [fig:mesh1] ![image](Scatterplotlin2001.jpeg) [fig:mesh1] Table for Sec. 6.4.1 -------------------- [h] [h]0.45 ccccccc Size& d̂VC & $\widehat{ERM}\_{1}$ & $\widehat{ERM}\_{2}$ & AIC & BIC & CV 1& 40& 4& 12& -1055& -1043& 0.00352 2& 40& 4& 12& -1053& -1037& 0.00352 3& 40& 4& 12& -1051& -1032& 0.00353 4& 39& 4& 11& -1050& -1026& 0.00353 5& 34& 4& 10& -1048& -1020& 0.04558 6& 34& 4& 10& -1046& -1015& 0.04206 7& 34& 4& 10& -1044& -1009& 0.04195 8& 36& 4& 11& -1042& -1003& 0.04272 9& 37& 4& 11& -1040& -997& 0.04294 10& 36& 4& 11& -1038& -991& 0.04350 11& 36& 4& 11& -1043& -992& 0.07936 12& 37& 4& 11& -1042& -987& 0.08302 13& 36& 4& 11& -1040& -981& 0.08296 14& 37& 4& 11& -1041& -979& 0.06465 15& 35& 4& 11& -1040& -973& 0.06295 16& 37& 4& 11& -1038& -968& 0.06338 17& 37& 4& 11& -1036& -962& 0.06401 18& 35& 4& 11& -1035& -956& 0.06202 19& 36& 4& 11& -1033& -951& 0.06527 20& 36& 4& 11& -1031& -945& 0.06451 21& 34& 4& 10& -1029& -939& 0.06535 22& 35& 4& 11& -1028& -933& 0.06692 23& 37& 4& 11& -1026& -928& 0.06749 24& 35& 4& 11& -1024& -922& 0.06673 25& 37& 4& 11& -1023& -917& 0.06469 26& 37& 4& 11& -1023& -913& 0.06055 27& 37& 4& 11& -1021& -907& 0.06135 28& 37& 4& 11& -1021& -903& 0.05328 29& 37& 4& 11& -1020& -899& 0.03878 30& 36& 4& 11& -1019& -893& 0.04111 31& 34& 4& 10& -1018& -888& 0.04418 32& 35& 4& 10& -1042& -908& 0.01195 33& 35& 4& 10& -1042& -908& 0.01195************ [LinSNP101] [h]0.45 \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Size \| ĥ \| $\widehat{ERM}\_{1}$ \| $\widehat{ERM}\_{2}$ \| AIC \| BIC \| CV \| \| 11 \| 36 \| 4 \| 11 \| -1043 \| -992 \| 0.07936 \| \| 12 \| 37 \| 4 \| 11 \| -1042 \| -987 \| 0.08302 \| \| 13 \| 36 \| 4 \| 11 \| -1040 \| -981 \| 0.08296 \| \| 14 \| 37 \| 4 \| 11 \| -1041 \| -979 \| 0.06465 \| \| 15 \| 35 \| 4 \| 11 \| -1040 \| -973 \| 0.06295 \| \| 16 \| 37 \| 4 \| 11 \| -1038 \| -968 \| 0.06338 \| \| 17 \| 37 \| 4 \| 11 \| -1036 \| -962 \| 0.06401 \| \| 18 \| 35 \| 4 \| 11 \| -1035 \| -956 \| 0.06202 \| \| 19 \| 36 \| 4 \| 11 \| -1033 \| -951 \| 0.06527 \| \| 20 \| 36 \| 4 \| 11 \| -1031 \| -945 \| 0.06451 \| [LinSNP201] [h]0.45 \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Size \| ĥ \| $\widehat{ERM}\_{1}$ \| $\widehat{ERM}\_{2}$ \| AIC \| BIC \| CV \| \| 21 \| 34 \| 4 \| 10 \| -1029 \| -939 \| 0.065345476 \| \| 22 \| 35 \| 4 \| 11 \| -1028 \| -933 \| 0.066919227 \| \| 23 \| 37 \| 4 \| 11 \| -1026 \| -928 \| 0.067491940 \| \| 24 \| 35 \| 4 \| 11 \| -1024 \| -922 \| 0.066729472 \| \| 25 \| 37 \| 4 \| 11 \| -1023 \| -917 \| 0.064685020 \| \| 26 \| 37 \| 4 \| 11 \| -1023 \| -913 \| 0.060548876 \| \| 27 \| 37 \| 4 \| 11 \| -1021 \| -907 \| 0.061349332 \| \| 28 \| 37 \| 4 \| 11 \| -1021 \| -903 \| 0.053281068 \| \| 29 \| 37 \| 4 \| 11 \| -1020 \| -899 \| 0.038782094 \| \| 30 \| 36 \| 4 \| 11 \| -1019 \| -893 \| 0.041109422 \| \| 31 \| 34 \| 4 \| 10 \| -1018 \| -888 \| 0.044183504 \| \| 32 \| 35 \| 4 \| 10 \| -1042 \| -908 \| 0.011947595 \| \| 33 \| 35 \| 4 \| 10 \| -1042 \| -908 \| 0.011947595 \| [Linsnp301] [SNPLin01] Table for Sec. 6.4.2 -------------------- [h] [h]0.45 ccccccc Size& d̂VC & $\widehat{ERM}\_{1}$ & $\widehat{ERM}\_{2}$ & AIC & BIC & CV 1& 33& 8& 16& -9142& -9160& 0.001801809 2& 33& 8& 16& -9136& -9159& 0.001802400 3& 33& 8& 16& -9129& -9159& 0.001802509 4& 33& 8& 16& -9122& -9157& 0.001804594 5& 33& 8& 16& -9114& -9156& 0.001806903 6& 32& 7& 16& -9109& -9156& 0.001806764 7& 33& 8& 16& -9101& -9154& 0.001807267 8& 33& 8& 16& -9095& -9153& 0.001811197 9& 33& 8& 16& -9087& -9151& 0.001814232 10& 32& 7& 16& -9081& -9152& 0.001815581 11& 32& 7& 16& -9073& -9150& 0.001820324 12& 32& 7& 16& -9067& -9149& 0.001820803 13& 32& 7& 16& -9061& -9149& 0.001822245 14& 31& 7& 15& -9053& -9147& 0.001822267 15& 31& 7& 15& -9045& -9145& 0.001823113 16& 31& 7& 15& -9042& -9148& 0.001822037 17& 43& 8& 18& -9035& -9146& 0.001827415 18& 43& 8& 18& -9027& -9144& 0.001828095 19& 43& 8& 18& -9019& -9142& 0.001828721 20& 31& 7& 15& -9012& -9141& 0.001828733 21& 32& 7& 16& -9004& -9139& 0.001835135 22& 32& 7& 16& -8996& -9137& 0.001838472 23& 32& 7& 16& -8989& -9136& 0.001838386 24& 31& 7& 15& -8982& -9134& 0.001846458 25& 43& 8& 18& -8978& -9137& 0.001844896 26& 42& 8& 18& -8974& -9138& 0.001843160 27& 42& 8& 18& -8966& -9137& 0.001843012 28& 41& 8& 17& -8959& -9135& 0.001843302 29& 41& 8& 17& -8951& -9133& 0.001847333 30& 41& 8& 17& -8944& -9132& 0.001848343 31& 40& 8& 17& -8944& -9138& 0.001847684 32& 43& 8& 18& -8939& -9139& 0.001850055************ [Gene1] [h]0.45 \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Size \| ĥ \| $\widehat{ERM}\_{1}$ \| $\widehat{ERM}\_{2}$ \| AIC \| BIC \| CV \| \| 11 \| 32 \| 7 \| 16 \| -9073 \| -9150 \| 0.001820324 \| \| 12 \| 32 \| 7 \| 16 \| -9067 \| -9149 \| 0.001820803 \| \| 13 \| 32 \| 7 \| 16 \| -9061 \| -9149 \| 0.001822245 \| \| 14 \| 31 \| 7 \| 15 \| -9053 \| -9147 \| 0.001822267 \| \| 15 \| 31 \| 7 \| 15 \| -9045 \| -9145 \| 0.001823113 \| \| 16 \| 31 \| 7 \| 15 \| -9042 \| -9148 \| 0.001822037 \| \| 17 \| 43 \| 8 \| 18 \| -9035 \| -9146 \| 0.001827415 \| \| 18 \| 43 \| 8 \| 18 \| -9027 \| -9144 \| 0.001828095 \| \| 19 \| 43 \| 8 \| 18 \| -9019 \| -9142 \| 0.001828721 \| \| 20 \| 31 \| 7 \| 15 \| -9012 \| -9141 \| 0.001828733 \| [Gene2] [h]0.45 ccccccc Size& ĥ & $\widehat{ERM}\_{1}$ & $\widehat{ERM}\_{2}$ & AIC & BIC 21& 32& 7& 16& -9004& -9139& 0.001835135 22& 32& 7& 16& -8996& -9137& 0.001838472 23& 32& 7& 16& -8989& -9136& 0.001838386 24& 31& 7& 15& -8982& -9134& 0.001846458 25& 43& 8& 18& -8978& -9137& 0.001844896 26& 42& 8& 18& -8974& -9138& 0.001843160 27& 42& 8& 18& -8966& -9137& 0.001843012 28& 41& 8& 17& -8959& -9135& 0.001843302 29& 41& 8& 17& -8951& -9133& 0.001847333 30& 41& 8& 17& -8944& -9132& 0.001848343 31& 40& 8& 17& -8944& -9138& 0.001847684 32& 43& 8& 18& -8939& -9139& 0.001850055 [Gene3] [Gene] Table for Sec 6.5.1 ------------------- [h] [h]0.45 cccccccc Size& d̂VC & $\widehat{ERM}\_{1}$ & $\widehat{ERM}\_{2}$ & AIC & BIC & CV 1& 39& 4& 11& -1056& -1040& 0.003523772 2& 39& 4& 11& -1054& -1034& 0.003523852 3& 39& 4& 11& -1052& -1028& 0.003528050 4& 39& 4& 11& -1051& -1023& 0.003531868 5& 35& 4& 11& -1049& -1017& 0.045582343 6& 38& 4& 11& -1047& -1012& 0.042056585 7& 34& 4& 10& -1045& -1006& 0.041950879 8& 36& 4& 11& -1044& -1000& 0.042716568 9& 36& 4& 11& -1042& -995& 0.042944324 10& 35& 4& 11& -1040& -989& 0.043503630 11& 36& 4& 11& -1043& -988& 0.079361885 12& 35& 4& 11& -1042& -983& 0.083019021 13& 35& 4& 11& -1040& -977& 0.082959969 14& 37& 4& 11& -1041& -975& 0.064653613 15& 35& 4& 11& -1040& -969& 0.062946594 16& 36& 4& 11& -1039& -964& 0.063382297 17& 36& 4& 11& -1037& -958& 0.064013291 18& 37& 4& 11& -1035& -953& 0.062024613 19& 34& 4& 10& -1034& -947& 0.065271013 20& 36& 4& 11& -1032& -942& 0.064513608 21& 35& 4& 11& -1030& -936& 0.065345476 22& 36& 4& 11& -1028& -930& 0.066919227 23& 36& 4& 11& -1027& -924& 0.067491940 24& 37& 4& 11& -1025& -919& 0.066729472 25& 35& 4& 11& -1024& -914& 0.064685020 26& 35& 4& 11& -1024& -910& 0.060548876 27& 36& 4& 11& -1022& -904& 0.061349332 28& 34& 4& 10& -1022& -900& 0.053281068 29& 34& 4& 10& -1022& -896& 0.038782094 30& 38& 4& 11& -1020& -891& 0.041109422 31& 34& 4& 10& -1019& -885& 0.044183504 32& 35& 4& 10& -1040& -903& 0.011947595 33& 35& 4& 10& -1040& -903& 0.011947595************ [phenosnpdesiGene12001] [h]0.45 \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Size \| ĥ \| $\widehat{ERM}\_{1}$ \| $\widehat{ERM}\_{2}$ \| AIC \| BIC \| \| 11 \| 37 \| 4 \| 11 \| -1043 \| -988 \| \| 12 \| 37 \| 4 \| 11 \| -1042 \| -983 \| \| 13 \| 38 \| 4 \| 11 \| -1040 \| -977 \| \| 14 \| 35 \| 4 \| 11 \| -1041 \| -975 \| \| 15 \| 35 \| 4 \| 11 \| -1040 \| -969 \| \| 16 \| 35 \| 4 \| 11 \| -1039 \| -964 \| \| 17 \| 36 \| 4 \| 11 \| -1037 \| -958 \| \| 18 \| 34 \| 4 \| 10 \| -1035 \| -953 \| \| 19 \| 36 \| 4 \| 11 \| -1034 \| -947 \| \| 20 \| 35 \| 4 \| 11 \| -1032 \| -942 \| [phenosnpdesiGene22001] [h]0.45 \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Size \| ĥ \| $\widehat{ERM}\_{1}$ \| $\widehat{ERM}\_{2}$ \| AIC \| BIC \| \| 21 \| 36 \| 4 \| 11 \| -1030 \| -936 \| \| 22 \| 36 \| 4 \| 11 \| -1028 \| -930 \| \| 23 \| 36 \| 4 \| 11 \| -1027 \| -924 \| \| 24 \| 34 \| 4 \| 10 \| -1025 \| -919 \| \| 25 \| 37 \| 4 \| 11 \| -1024 \| -914 \| \| 26 \| 35 \| 4 \| 11 \| -1024 \| -910 \| \| 27 \| 36 \| 4 \| 11 \| -1022 \| -904 \| \| 28 \| 36 \| 4 \| 11 \| -1022 \| -900 \| \| 29 \| 35 \| 4 \| 11 \| -1022 \| -896 \| \| 30 \| 36 \| 4 \| 11 \| -1020 \| -891 \| \| 31 \| 37 \| 4 \| 11 \| -1019 \| -885 \| \| 32 \| $\bf{36}$ \| 4 \| 11 \| -1040 \| -903 \| [phenosnpdesiGene32001] [phenosnpdesiGene01] Table for Sec. 6.5.2 -------------------- --- 1. Tour de France Data was collected by Bertrand Clarke. More information can be found at http://www.letour.fr/.[↩](#fnref1) 2. There actually is an 8th covariate. Sex is a nominal variable with 3 categories: Male, Female and Infant. We did not include it in our analysis because we wanted to limit attention to continuous variables in this example.[↩](#fnref2) Model Selection via the VC Dimension ======================================== We derive an objective function that can be optimized to give an estimator of the Vapnik-Chervonenkis dimension for model selection in regression problems. We verify our estimator is consistent. Then, we verify it performs well compared to seven other model selection techniques. We do this for a variety of types of data sets. Vapnik-Chervonenkis dimension, Model Selection, Bayesian information criterion, Sparsity methods, empirical risk minimization, multi-type data. Complexity and Model Selection ============================== Model selection is often the first problem that must be addressed when analyzing data. In -closed problems, see, the analyst posits a list of models and assumes one of them is true. In such cases, model selection is any procedure that uses data to identify one of the models on the model list. There is a vast literature on model selection in this context including information based methods such as the Aikaikie Information Criterion (AIC), the Bayes information criterion (BIC), residual based methods such as Mallows Cp or branch and bound, and code length methods such as the two-stage coding proposed by. We also have computational search methods such as simulated annealing and genetic algorithms. In addition, cross-validation (CV) is often used with non-parametric methods such as recursive partitioning, neural networks (aka deep learning) and kernel methods. A less well developed approach to model selection is via complexity as assessed by the Vapnik-Chervonenkis (VC) dimension, here denoted by dVC. Its earliest usage seems to be in. A translation into English was published as. The VC dimension was initially called an index of a collection of sets with respect to a sample and was developed to provide sufficient conditions for the uniform convergence of empirical distributions. It was extended to provide a sense of dimension for function spaces, particularly for functions that represented classifiers. After two decades of development this formed the foundation for the field of Statistical Learning Theory. Amongst others, Statistical Learning Theory has the key concepts of empirical risk minimization (ERM) and structural risk minimization (SRM). The idea behind ERM is to find an action e.g., an indicator function, that minimizes the empirical risk, see, p. 8. The point of SRM is to consider the solutions to the ERM problem that have the smallest complexity, for instance as measured by the VC dimension, see, p. 10. Seeking a tradeoff between error and complexity has long been a guiding principle in statistics; see who helped develop the ‘$R^2\_{\sf adj}$’ often used in linear regression. Although, the VC dimension goes back to 1968, it wasn’t until that a method for estimating dVC was proposed in the classification context. Specifically, given a collection C of classifiers, tried to estimate the VC dimension of C by deriving an objective function based on the expected value of the maximum difference between two empirical evaluations of a single loss function, here denoted by Δ. The two empirical values come from dividing a given data set into a first and second part. The objective function proposed by depends on dVC, the sample size n, and several constants that had to be determined. Using their objective function, they derived an estimator d̂VC for dVC given a class C of classifiers. This algorithm treated possible sample sizes as design points n1, n2, ⋯, nL and requires one level of bootstrapping. Despite the remarkable contribution of, the objective function was over-complex and the algorithm did not give a tight enough bound on Δ. Later, Vapnik and his collaborators suggested a fix to tighten the bound on Δ. We do not use this here; it is unclear if this ‘fix’ will work in classification, let alone regression. Choosing the design points is a nontrivial source of variability in the estimate of dVC. So, proposed an algorithm, based on extensive simulations, to generate optimal values of n1, n2, ⋯, nL, given L. They argued that non-uniform values of the nl’s gave better results than the uniform nl’s used in. More recently, in a pioneering paper that deserves more recognition that it has received, established the consistency of the estimator d̂VC for dVC in the classification context. The main reason the estimator for dVC did not become more widely used despite the result in is, we suggest, that it was too unstable because the objective function did not bound Δ tightly enough. In addition, the form of the objective function in is more complicated and less well-motivated than our result Theorem [Theo23and4]. The reason is that the derivation in uses conditional probabilities, one of which goes to zero quite quickly (with n). So, it contributes negligibly to the upper bound. Our derivation ignores the conditioning and bounds a form of Δ that is typically larger than that used in. Optimizing a factor in our bound also contributes to generating a tighter bound on the error. Our consistency proof is a simplification of the proof of the main result. Accordingly, we obtain a slower rate of consistency, but the probability of correct model selection still goes to one. As a generality, consistency results can be regarded as special cases of probably approximately correct (PAC) bounds, i.e., an estimator converges to its true value in the correct probability on the underlying measure space. Thus, loosely speaking, we show that P(∥dVC − d̂VC∥ > δ) < ε as n → ∞ For any pre-assigned δ, ε > 0. More general PAC bounds replace the fixed δ and ε with δ = δn and ε = εn. Usually, PAC bounds are used to express stronger results, see for instance who obtained a PAC bound of the form P(D(Q̂∥Q) > 1/n) < ε where Q̂ is an estimator of the distribution Q and D is the relative entropy. In some cases of parameter estimation with independent data X1 : n = (X1, …, Xn) one can obtain stronger results such as Pθ(W(N(θ, ε)∣X1 : n)c > e− γ1n) < e− γ2n for large n, where θ is the true value of a parameter, W( ⋅ ∣X1 : n) is the posterior, N(θ, ε) is an ε-neighborhood of θ, and γ1, γ2 > 0. Such results are easier to obtain for discrete parameters than for continuous parameters. This may explain why are able to get an exponentially small form for ε for the estimator in a classification context. We have established simple consistency because this is enough for our purposes (regression) and we were unable to extend the more sophisticated proof in. Our overall strategy is to derive an objective function for estimating dVC in the regression setting that provides, we think, a tighter bound on a modified form of Δ. To convert from classification to regression, we discretize the loss used for regression into m intervals (the case m = 1 would then apply to classification). To get a tighter bound, we change the form of Δ from what used and we optimize over the leading factor in our upper bound. To use our estimator, we use an extra layer of bootstrapping so the quantity we empirically optimize represents the quantity we derive theoretically more accurately. The extra layer of bootstrapping stabilizes our estimator of dVC and appears to reduce its dependency on the nl’s. If the models are nested in order of increasing VC dimension, it is straightforward to choose the model with VC dimension closest to our estimate d̂VC. Otherwise, we can convert a non-nested problem to the nested case by ordering the inclusion of the covariates using a shrinkage method such as the ‘smoothly clipped absolute deviation’ (SCAD, ), or correlation (see ), and use our d̂VC as before. Even when we force a model list to be nested, our model selection method performs well compared to a range of competitors including ’s original method, two forms of empirical risk minimization (denoted $\widehat{ERM}\_{1}$ and $\widehat{ERM}\_{2}$), AIC, BIC, CV (10-fold), SCAD, and adaptive LASSO (ALASSO, ). Our general findings indicate that in realistic settings, model selection via estimated VC dimension, when properly done, is fully competitive with existing methods and, unlike them, rarely gives abberant results. This manuscript is structured as follows. In Sec. [chap:bounds] we present the main theory justifying our estimator. In Subsec. [sec:Ext:Vap:Bounds] we discretize bounded loss functions so that upper bounds for the distinct regions of the expected supremal difference of empirical losses can be derived and in Subsec. [estimator] we define our estimator of the VC dimension and give an algorithm for how to compute it. In Sec. [ProofOfConsistency] use ’s consistency theorem to motivate our consistency theorem for d̂VC. In Sec. [chap:Numerical:Studies] we compare our method for model selection to AIC, BIC, CV, $\widehat{ERM}\_{1}$, and $\widehat{ERM}\_{2}$. In this context, we suggest criteria to guide the selection of design points. Our comparisons also include simplifying non-nested model lists by using correlation, SCAD, and ALASSO. In Sec. [GenreConcl] we discuss our overall findings in the context of model selection. Deriving an optimality criterion for estimating VC dimension ============================================================ We are going to bound a form of Δ for use in regression. This is the bound we will use to derive an estimator of the VC dimension. In Sec. [sec:Ext:Vap:Bounds], we present our alterantive version of the bounds and in Sec. [estimator], we present an estimator of dVC. Extension of the bounds to regression -------------------------------------- Let Z = (X, Y) be a random variable with outcomes z = (x, y) assuming values in $\cal{X} \times \cal{Y}$. The first entry, X = x, is regarded as an explanatory variable leading to Y = y. Let D be a set of 2n independent and identical (IID) copies of Z and write D = D1 ∪ D2 where D1 = {Z1, …, Zn} is the first half and D2 = {Zn + 1, …, Z2n} is the second half. Writing Zi = (Xi, Yi) for i = 1…2n, let Q(Zi, α) = L(Yi, f(Xi, α)), for a bounded real valued loss function L and α ∈ Λ. We assume that Λ is a connected compact set in a finite dimensional real space and that the interior of Λ is open and connected. Also, we assume the continuous functions f( ⋅ ∣ α) are parametrized by α continuously and one-to-one. Thus, in our examples, Λ will be the parameter space for a class of regression functions f( ⋅ ∣ α). For convenience we assume L, and hence Q, are also continuous. Now, for ℓ = 1, 2 let αℓ ∈ Λ and write $$Q\_{1}\left(Z\_{i}\right) = Q\_{1}\left(Z\_{i},\alpha\_{1}\right)= L\left(Y\_{i},f\left(X\_{i},\alpha\_{1}\right)\right) \quad \hbox{and}\quad Q\_{2}\left(Z\_{i}\right)=Q\_{2}\left(Z\_{i},\alpha\_{2}\right)= L\left(Y\_{i},f\left(X\_{i},\alpha\_{2}\right)\right)$$ where, in the first expression, it is assumed that Zi ∈ D1 and in the second expression it is assumed that Zi ∈ D2. Also, assume 0 ≤ Q( ⋅ , α) ≤ B for all α ∈ Λ and some B ∈ R. Denote the discretization of Qℓ using m disjoint intervals (with union [0, B)) given by $$\label{E11} Q\_{\ell}(Z) = Q^{m}\_{\ell j}\left(z\_{i},\alpha\_{\ell}\right)=\sum\_{j = 0}^{m-1}\frac{(2j+1)B}{2m}\mathbb{I}\left[Q\_{\ell}(Z)\in I\_{j}^{m}\right],$$ where ℓ = 1, 2, zi ∈ Dℓ and j = 0, 1, ⋯, m − 1. Note that the numbers $\frac{(2j+1)B}{2m}$ are the midpoints of the uniform left-closed, right-open partition of [0, B) into m sub-intervals, here denoted Ijm. In, I[ ⋅ ] is an indicator function taking value 1 when its argument is true and taking value 0 when its argument is false. Now, let $$\label{nu1and2} \nu\_{1}\left(D^{1}\right)= \frac{1}{n}\sum\_{i=1}^{n}Q\_1\left(Z\_{i}\right) \quad \hbox{and} \quad \nu\_{2}\left(D^{2}\right)= \frac{1}{n}\sum\_{i=1}^{n}Q\_2\left(Z\_{n+i}\right)$$ be the empirical risk using the first and second half of the sample, respectively. Observe that the empirical counts of the data points whose losses land in Ijm are $$N^m\_{1j} = \sum\_{i=1}^{n}\mathbb{I}\left[Q\_{1}\left(Z\_{n+i}\right)\in I\_j^m\right] \quad \hbox{and} \quad N\_{2j}^{m} = \sum\_{i=1}^{n}\mathbb{I}\left[Q\_{2}\left(Z\_{i}\right)\in I\_j^m\right].$$ This means we are evaluating the α1 model on the second half of the data and the α2 model on the first half of the data. So, we have expressions for the empirical losses of the discretized loss function: $$\label{nu1and2discretized} \nu\_{1}^{m}\left(D^2\right) = \frac{1}{n}\sum\_{j = 0}^{m - 1}N\_{2,j}^{m}\frac{(2j+1)B}{2m} \quad \hbox{and} \quad \nu\_{2}^{m}\left(D^1\right) = \frac{1}{n}\sum\_{j = 0}^{m - 1}N\_{1,j}^{m}\frac{(2j+1)B}{2m}.$$ We begin by controlling Δ, the expected supremal difference between two evaluations of a bounded loss function, to be formally defined in. Let ε > 0 and let $$\label{Intervalnu1and2} \nu\_{1,j}^m\left(Z\_{n+1:2n}\right) = \frac{1}{n}N\_{2,j}^{m}\frac{(2j+1)B}{2m} \quad \hbox{and} \quad \nu\_{2,j}^m\left(Z\_{1:n}\right) = \frac{1}{n}N\_{1,j}^{m}\frac{(2j+1)B}{2m}.$$ We define the events Aε = ⋃m = 0m − 1Aε, m, where Aε, m = {Z1 : 2n∣supα1, α2 ∈ Λ[ν1m(Zn + 1 : 2n) − ν2m(Z1 : n)] ≥ ε}. Since Aε is defined on the entire range of our loss function, and we want to partition the range into m disjoint intervals, write $$\begin{aligned} A\_{\epsilon,m}\subseteq \left\{Z\_{1:2n}~\|~ \exists~ j : \sup\_{\alpha\_{1},~\alpha\_{2}\in ~ \Lambda}\left(\nu\_{1j}^{m}\left(Z\_{n+1:2n}\right)-\nu\_{2j}^{m}\left(Z\_{1:n},\alpha\_{2},m\right)\right)\ge\frac{\epsilon}{m}\right\} \subseteq \bigcup\_{j=0}^{m-1} A\_{\epsilon,m,j},\end{aligned}$$ where $ A\_{\epsilon,m,j} = \left\{Z\_{1:2n}~\|~ \sup\_{\alpha\_{1},~\alpha\_{2}~\in\Lambda}\left(\nu\_{1j}^{m}\left(Z\_{n+1:2n}\right)- \nu\_{2j}^{m}\left(Z\_{1:n}\right)\right)\ge\frac{\epsilon}{m}\right\}$. Since Λ is compact, Λ × Λ is compact, and the supremum over Λ × Λ in Aε, m, j will be achieved at some (α1jm, α2jm) = argsupα1, α2 ∈ Λ(ν1jm(zn + 1 : 2n) − ν2jm(z1 : n)) ∈ Λ × Λ where zn + 1 : 2n and z1 : n are outcomes of Zn + 1 : 2n and Z1 : n respectively. Next, fix any value j ∈ {0, 1, …, m − 1}. For any fixed z2n, and any given α1 and α2, form the vector of length 2n of the form $$\begin{aligned} && \left(Q\_{1j}^m\left(Z\_{n+1:2n}, \alpha\_{1}\right), Q\_{2j}^m\left(Z\_{1:n}, \alpha\_{2}\right)\right) \nonumber \\ \nonumber &=& \left(Q^m\_{1j}\left(Z\_{n+1}, \alpha\_{1}\right), \ldots, Q^m\_{1j}\left(Z\_{n+1}, \alpha\_{1}\right), Q^m\_{2j}\left(Z\_{1}, \alpha\_{2}\right), \ldots, Q^m\_{2j}\left(Z\_{n}, \alpha\_{2}\right)\right).\end{aligned}$$ For any α1, α1′, α2, and α2′ ∈ Λ, define $$\begin{aligned} && \left(\alpha\_{1}, \alpha\_2\right) \sim \left(\alpha\_1^\prime, \alpha^\prime\_{2}\right) \Longleftrightarrow \nonumber \\ && \left(Q\_{1j}^m\left(Z\_{n+1:2n}, \alpha\_{1}\right), Q\_{2j}^m\left(Z\_{1:n}, \alpha\_{2}\right)\right) = \left(Q\_{1j}^m\left(Z\_{n+1:2n}, \alpha^\prime\_{1}\right), Q\_{2j}^m\left(Z\_{1:n}, \alpha^\prime\_{2}\right)\right). \nonumber\end{aligned}$$ So, for any fixed Z1 : 2n = z1 : 2n it is seen that ∼ is an equivalence relation on Λ × Λ and therefore partitions Λ × Λ into disjoint equivalence classes. Denote the number of these classes by NjΛ = NjΛ(z1 : 2n) = NjΛ(z1, z2, …, z2n). For fixed j, let Kj(z1 : 2n) be the number of equivalence classes in Λ × Λ and for a given α\* = (α1, α2) ∈ Λ × Λ write [α\] to be the equivalence class that contains it. Now, for k* = 1…, Kj let αjk\* = (α1jk\, α2j*k\) = argsupα1j, α2j* ∈ (Λ × Λ)k(ν1jm(Zn + 1 : 2n) − ν2jm(Z1 : n)), where (Λ × Λ)k is the k-th equivalence class. Now, ⋃k = 1Kj[αjk\] = Λ × Λ and [αjk\] ∩ [αjk′\] = ϕ* unless k = k′. Clearly, Kj = NjΛ(z1 : 2n). To make use of the above partitioning of Aε, consider mapping the underlying measure space space ${\cal{Z}}\_{1:2n}$ onto itself using (2n)! distinct permutations T = Ti. Then, if f is integrable with respect to the distribution of Zi, its Riemann-Stieltjes integral satisfies $$\int\_{{\cal{Z}}^{2n}}^{}f\left(Z\_{1:2n}\right)\emph{dF}\left(Z\_{1:2n}\right) = \int\_{{\cal{Z}}^{2n}}^{}f\left(T\_{i}Z\_{1:2n}\right)\emph{dF}\left(Z\_{1:2n}\right),$$ and this gives $$\label{permid} \int\_{{\cal{Z}}^{2n}}^{}f\left(Z\_{1:2n}\right)\emph{dF}\left(Z\_{1:2n}\right) = \int\_{{\cal{Z}}^{2n}}^{}\frac{\sum\_{i=1}^{(2n)!}f\left(T\_{i}Z\_{1:2n}\right)}{(2n)!}\emph{dF}\left(Z\_{1:2n}\right).$$ Our first main result is structured the same way as in, however, the differences are in the details. For instance, our equivalence class is defined on Λ × Λ, we use a cross-validation form of the error, we discretized the loss function, and our result leads to Theorem [Theo23and4] that only has one term, whereas the corresponding result in has three terms. [NewProp2] Let $\epsilon \ge 0,\quad m \in \mathcal{N}, \hbox{and},\quad d\_{VC} = VC\left( \left\{Q\left(\cdot,\alpha\right): \alpha \in \Lambda\right\}\right)$. If dVC is finite, then $$\label{NewUPBO} P\left(A\_{\epsilon,m}\right) \leq 2m\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\exp\left\{-\frac{n\epsilon^2}{m^{2}}\right\}.$$ The technique used to prove is similar to the proof of Theorem 4.1 in giving bounds for the uniform convergence of the empirical risk. The hypotheses of Theorem 4.1 in require only the existence of the key quantities e.g the annealed entropy, and the growth function. Our only extra condition is that dVC is finite. Let Δjm(TiZ1 : 2n, αj\) = ν1j*m(TiZn + 1 : 2n, α1j\) − ν2j*m(TiZ1 : n, α2j\), where $$\begin{aligned} \alpha\_{j}^{\} = \left(\alpha\_{1j}^\, \alpha\_{2j}^\\right)= \arg\underset{\left(\alpha\_{1j}, \alpha\_{2j}\right) \in \Lambda \times \Lambda} \max \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha\_{j}\right) \in \Lambda \times \Lambda. \nonumber\end{aligned}$$ Using some manipulations, we have $$\begin{aligned} \label{UpAep} P\left(A\_{\epsilon,m}\right) &\leq& P\left(\bigcup\_{j=0}^{m-1}A\_{\epsilon,m,j}\right)\leq \sum\_{j=0}^{m-1}P\left(A\_{\epsilon,m,j}\right)\nonumber\\\nonumber\\ &=& \sum\_{j=0}^{m-1}P\left(\left\{Z\_{1:2n}:\sup\_{\alpha\_{1},~\alpha\_{2}\in\Lambda}\left(\nu\_{1j}^{m}(Z\_{n+1:2n},\alpha\_{1})-\nu\_{2j}^{m}(Z\_{1n},\alpha\_{2})\right)\geq\frac{\epsilon}{m}\right\}\right). \nonumber\end{aligned}$$ Continuing the equality gives that the RHS equals $$\begin{aligned} \sum\_{j=0}^{m-1}P\left(\left\{Z\_{1:2n}:\left(\nu\_{1j}^{m}(T\_{i}Z\_{n+1:2n},\alpha\_{1j}^{\})-\nu\_{2j}^{m}(T\_{i}Z\_{1:n},\alpha\_{2j}^{\})\right)\geq\frac{\epsilon}{m}\right\}\right) \nonumber\end{aligned}$$ $$\begin{aligned} &=& \sum\_{j=0}^{m-1}P\left(\left\{Z\_{1:2n}: \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha\_{j}^{\}\right)\geq\frac{\epsilon}{m}\right\}\right)\nonumber\\ &=& \frac{1}{(2n)!}\sum\_{j=0}^{m-1}\sum\_{i=1}^{(2n)!}P\left(\left\{Z\_{1:2n}: \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha\_{j}^{\}\right)\geq\frac{\epsilon}{m}\right\}\right) \nonumber \\ &=& \frac{1}{(2n)!}\sum\_{j=0}^{m-1}\sum\_{i=1}^{(2n)!}\int\_{}^{}\emph{I}\_{\left\{Z\_{1:2n}: \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha\_{j}^{\}\right)\geq\frac{\epsilon}{m}\right\}}\left(z^{2n}\right)\emph{dP}(Z\_{1:2n}). \label{permprobbdd}\end{aligned}$$ Using the properties of the equivalence relation ∼ , for each fixed j* and Z1 : 2n we have the inequality $$\begin{aligned} \label{Indfnbdd} \emph{I}\_{\left\{Z\_{1:2n}:\Delta^m\_j\left(T\_{i}Z\_{1:2n},\alpha^{\}\_{j}\right)\geq\epsilon\right\}}(Z\_{1:2n}) &\leq& \emph{I}\_{\left\{Z\_{1:2n}: \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha^{\}\_{j1},m\right)\ge \frac{\epsilon}{m}\right\}}\left(z\_{1:2n}\right) \nonumber \\ &+& \dots + \emph{I}\_{\left\{Z\_{2n}:\Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha^{\}\_{jN^{\Lambda}\_{j}(z\_{1:2n})}\right)\ge\frac{\epsilon}{m}\right\}}\left(z\_{1:2n}\right) \nonumber\\ &=& \sum\_{k=1}^{N^{\Lambda}\_{j}(Z\_{1:2n})}\emph{I}\_{\left\{Z\_{1:2n}: \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha^{\}\_{jk}\right)\ge\frac{\epsilon}{m}\right\}}\left(Z\_{1:2n}\right)\end{aligned}$$ where $$\begin{aligned} A\_{\epsilon,m,j,k} &=& \left\{Z\_{1:2n}: \Delta\_{j}^{m}\left(T\_{i}Z\_{1:2n},\alpha^{\}\_{kj}\right)\ge\frac{\epsilon}{m}\right\} \\ &=& \left\{Z\_{1:2n}: \sup\_{\left(\alpha\_{1j}, \alpha\_{2j}\right)\in\left(\Lambda \times \Lambda\right)\_{k}}\left(\nu\_{1j}^{m}(T\_{i}Z\_{n+1:2n},\alpha\_{1j})-\nu\_{2j}^{m}(T\_{i}Z\_{1:n},\alpha\_{2j})\right)\ge\frac{\epsilon}{m}\right\}\end{aligned}$$ and for fixed j* and each k, [αjk\] = (Λ × Λ)k. Now, using, is bounded by $$\begin{aligned} P\left(A\_{\epsilon,m}\right) &\leq& \frac{1}{(2n)!}\sum\_{j=0}^{m-1}\sum\_{i=1}^{(2n)!}\int\_{}^{}\sum\_{k=1}^{N\_{j}^{\Lambda}(Z\_{1:2n})}\emph{I}\_{\left\{Z\_{1:2n}:\Delta\_{j}^{m}(T\_{i}Z\_{1:2n},\alpha^{\}\_{kj})\ge\frac{\epsilon}{m}\right\}}\left(Z\_{1:2n}\right) \emph{dP}\left(Z\_{1:2n}\right) \nonumber\\ &=& \int\_{}^{}\sum\_{j=0}^{m-1}\sum\_{k=1}^{N\_{j}^{\Lambda}(Z\_{1:2n})}\left[\frac{1}{(2n)!}\sum\_{i=1}^{(2n)!}\emph{I}\_{\left\{Z\_{1:2n}:\Delta\_{j}^{\}(T\_{i}Z\_{1:2n},\alpha\_{kj}^{\})\geq \frac{\epsilon}{m}\right\}}\left(Z\_{1:2n}\right)\right]\emph{dP}\left(Z\_{1:2n}\right) \nonumber. \label{overallprobbdd}\end{aligned}$$ The expression in square brackets is the fraction of the number of the (2n)! permutations Ti of Z1 : 2n for which Aε, m, j, k is closed under Ti for any fixed equivalence class (Λ × Λ)k. Following Sec. 4.13, it equals $$\label{E77} \Gamma\_{j} = \sum\_{k}^{}\frac{\binom{N^{m}\_{j}}{k} \binom{2n-N^{m}\_{j}}{N^{m}\_{j}-k}}{\binom{2n}{n}} \nonumber$$ where the summation is over k’s in the set $$\left\{k: \left\|\frac{k}{n}-\frac{N^m\_j -k}{n}\right\|\ge\frac{\epsilon}{m}\right\} \cap \{ k : \max(0, N\_j^m -n) \leq k \leq \min(N\_j^m, n) \}$$ where Njm = N1jm + N2jm. Here, Γj is the probability of choosing exactly k sample data points whose losses fall in interval Ij. From Sec. 4.13 in, we have $\Gamma\_{j} \le 2\exp\left(-\frac{n\epsilon^{2}}{m^2}\right)$. So, using this in gives that P(Aε, m) is upper bounded by $$\begin{aligned} && \int\_{}^{}\sum\_{j=0}^{m-1}\sum\_{k=1}^{N\_{j}^{\Lambda}(Z\_{1:2n})}2\exp\left(-\frac{n\epsilon^{2}}{m^2}\right)\emph{dP}(Z\_{1:2n}) = 2\exp\left(-\frac{n\epsilon^{2}}{m^2}\right)\int\_{}^{}\sum\_{j=0}^{m-1}\sum\_{k=1}^{N\_{j}^{\Lambda}(z^{2n})}\emph{dP}(Z\_{1:2n}) \nonumber \\ &=& 2\exp\left(-\frac{n\epsilon^{2}}{m^2}\right)\sum\_{j=0}^{m-1}\int\_{}^{}\sum\_{k=1}^{N\_{j}^{\Lambda}(Z\_{1:2n})}\emph{dP}(Z\_{1:2n}) = 2\exp\left(-\frac{n\epsilon^{2}}{m^2}\right)\sum\_{j=0}^{m-1}\int\_{}^{}N\_{j}^{\Lambda}(Z\_{1:2n})\emph{dP}(Z\_{1:2n}) \nonumber \\ &=& 2\exp\left(-\frac{n\epsilon^{2}}{m^2}\right)\sum\_{j=0}^{m-1}E\left(N\_{j}^{\Lambda}(Z\_{1:2n})\right). \label{probbddd}\end{aligned}$$ Recall from that the annealed entropy is defined as HannΛ(2n) = lnENΛ(z1, …, z2n), and the growth function is defined as GΛ(2n) = lnsupz1, z2, …, z2nNΛ(z1, …, z2n). These two quantities satisfy HannΛ(2n) ≤ GΛ(2n). Now, Theorem 4.3 from p.145 gives $$H\_{ann}(2n) = \ln\left(E\left(N\_{j}^{\Lambda}(z\_{1:2n})\right)\right)\leq G(2n)\leq d\_{VC}\ln\left(\frac{2ne}{d\_{VC}}\right) \Rightarrow E\left(N\_{j}^{\Lambda}(z^{2n})\right)\leq \left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}.$$ Using this m times in gives the Theorem. We can use Theorem [NewProp2] to give an upper bound on the unknown true risk via the following propositions. Let R(αk) be the true unknown risk at αk and Remp(αk) be the empirical risk at αk. [subProp1] For any η ∈ (0, 1), with probability at least 1 − η, the inequality $$\begin{aligned} \label{EquivProp1} R\left(\alpha\_{k}\right) \leq R\_{emp}\left(\alpha\_{k}\right) + m\sqrt{\frac{1}{n}\log\left(\left(\frac{2m}{\eta}\right)\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\right)} \end{aligned}$$ holds simultaneously for all functions Q(z, αk), k = 1, 2, ⋯, K. This inequality follows from the additive Chernoff bound and suggests that the best model will be the one that minimizes the RHS of inequality. The use of in model selection as a form of risk minimization because as dVC increases the second term on the right increases. This limits the size of dVC; we denoted this technique by ERM1 since a penalized empirical risk is being minimized. (It would also be accurate to call this structural risk minimization but it is more standard amongst people who work in model selection in both computer science and statistics to regard it as a form of ERM.) To obtain inequality, we equate the RHS of Proposition [NewProp2] to a positive number 0 ≤ η ≤ 1. Thus: $$\eta = 2m\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\exp\left(-\frac{n\epsilon^{2}}{m^{2}}\right).$$ Solving for ε gives $$\begin{aligned} \label{solution} \epsilon &=& m\sqrt{\frac{1}{n}\log\left(\left(\frac{2m}{\eta}\right)\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\right)}.\end{aligned}$$ Proposition [subProp1] can be obtained from the additive Chernoff bounds, expression 4.4 in as follows $$\begin{aligned} \label{proofsubProp1} R\left(\alpha\_{k}\right) &\leq& R\_{emp}\left(\alpha\_{k}\right) + \epsilon.\end{aligned}$$ Using in inequality, completes the proof. Parallel to Prop. [subProp1], we have the following for the multiplicative case. [subProp2] For any η ∈ (0, 1), with probability 1 − η, the inequality $$\label{EquivalentProp2} R\left(\alpha\_{k}\right)\le R\_{emp}\left(\alpha\_{k}\right) + \frac{m^{2}}{2n}\log\left(\frac{2m}{\eta}\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\right)\left(1+\sqrt{1+\frac{4nR\_{emp}\left(\alpha\_{k}\right)}{m^{2}\log\left(\frac{2m}{\eta}\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\right)}}\right)$$ holds simultaneously for all K functions in the set Q(z, αk), k = 1, 2, …, K. This follows from the multiplicative Chernoff bound and suggests that the best model will be the one that minimizes the right hand side (RHS) of. Analogous to we refer to the use of in model selection as ERM2. Let ε, η > 0. Then, inequality (4.18) in gives, with probability at least 1 − η, that $$\begin{aligned} \frac{R\left(\alpha\_{k}\right)-R\_{emp}\left(\alpha\_{k}\right)}{\sqrt{R\left(\alpha\_{k}\right)}} \leq \epsilon. \end{aligned}$$ Routine algebraic manipulations and completing the square give $$\begin{aligned} \left(R\left(\alpha\_{k}\right)-0.5\left(\epsilon^{2}+2R\_{emp}\left(\alpha\_{k}\right)\right)\right)^{2}-0.25\left(\epsilon^{2}+2R\_{emp}\left(\alpha\_{k}\right)\right)^{2} \leq -R^{2}\_{emp}\left(\alpha\_{k}\right). \end{aligned}$$ Taking the square root on both sides and re-arranging gives $$\begin{aligned} R\left(\alpha\_{k}\right) \leq R\_{emp}\left(\alpha\_{k}\right) + 0.5\epsilon^{2}\left(1 + \sqrt{1 + \frac{4R\_{emp}\left(\alpha\_{k}\right)}{\epsilon^{2}}}\right) \end{aligned}$$ Using in the last inequality completes the proof of the Proposition. The proof of Propositions [subProp1] and [subProp2] are easy and can be found in. Formally, let Δm = E(supα1, α2 ∈ Λ∣ν1m(Zn + 1 : 2n, α1) − ν2m(Z1 : n, α2)∣) and Δ = E(supα1, α2 ∈ Λ∣ν1(Zn + 1 : 2n, α1) − ν2(Z1 : n, α2)∣). Obviously, Δm ≈ Δ provided that $m, n, \hbox{and}~ d\_{VC} \rightarrow \infty$ at appropriate rates and the argument of Δm satisfies appropriate uniform integrability conditions. In fact, we do not use Δm → Δ. For our purpose, the following bounds are sufficient. They are important to our methodology because they bound the expected maximum difference between two values of the empirical losses by an expression that can be used to estimate the VC dimension. [Theo23and4] 1. If dVC < ∞, we have $$\Delta\_{m} \leq m\sqrt{\frac{1}{n}\ln\left(2m^{3}\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\right)}\\ + \frac{1}{m\sqrt{n\ln\left(2m^3\left(\frac{2ne}{d\_{VC}}\right)^{d\_{VC}}\right)}}$$ 2. If dVC ≤ ∞, and $$D\_{p}\left(\alpha\right) = \int\_{0}^{\infty}\sqrt[p]{P\left\{Q\left(z,\alpha\right)\geq c\right\}}dc \leq \infty$$ where 1 < p ≤ 2 is some fixed parameter, we have $$\Delta \leq \frac{D\_{p}(\alpha^{\})2^{2.5+\frac{1}{p}}\sqrt{d\_{VC}\ln\left(\frac{ne}{d\_{VC}}\right)}}{n^{1-\frac{1}{p}}} + \frac{16D\_{p}(\alpha^{\})2^{2.5+\frac{1}{p}}}{n^{1-\frac{1}{p}}\sqrt{d\_{VC}\ln\left(\frac{ne}{d\_{VC}}\right)}}. \label{Hannbd}$$ 3. Assume that dVC → ∞, $\frac{n}{d\_{VC}} \rightarrow \infty$, m → ∞, ln(m) = o(n), $$D\_{p}\left(\alpha\right) = \int\_{0}^{\infty}\sqrt[p]{P\left\{Q(z,\alpha)\ge c\right\}} dc\leq \infty$$ where p = 2. Then we have that $$\label{objfn} \Delta \leq \min\left(1,8D\_{p}(\alpha^{\})\right)\sqrt{\frac{d\_{VC}}{n}\ln\left(\frac{2ne}{d\_{VC}}\right)}.$$ The proof of Theorem [Theo23and4] can be found in in Appendices A1–A3. It rests on using the integral of probabilities identity and the bound in Theorem [NewProp2]. An Estimator of the VC Dimension -------------------------------- The upper bound from Theorem [Theo23and4] can be written as $$\label{UPPTHEO6} \Phi\_{d\_{VC}}(n)=\min\left(1,8D\_{p}(\alpha^{\})\right)\sqrt{\frac{d\_{VC}}{n}\log\left(\frac{2ne}{d\_{VC}}\right)}.$$ This expression is meaningfully different from the form derived in and studied in. Moreover, although min(1, 8Dp(α\)) does not affect the optimization, it might not be the best constant for the inequality in. So, we replace it with an arbitrary constant c* over which we optimize to make our upper bound as tight as possible. In our computations, we let c vary from 0.01 to 100 in steps of size 0.01. However, we have observed in practice that the best value of ĉ is usually between 1 and 8. The technique that we use to estimate d̂VC is also different from that in. Indeed, our Algorithm [Algo1] below accurately encapsulates the way the LHS of is formed unlike the algorithm in. In particular, we use two bootstrapping procedures, one as a proxy for calculating expectations and the second as a proxy for calculating a maximum. Moreover, we split the dataset into two subsets. Using the first dataset, we fit model I and using the second we fit model II. To explain how we find our estimate of the RHS of from Theorem [Theo23and4], we start by replacing the sample size n in with a specified value of design point, so that the only unknown is dVC. Thus, formally, we replace by $$\Phi\_{d\_{VC}}^{\}(n\_{l}) = \widehat{c}\sqrt{\frac{d\_{VC}}{n\_{l}}\log\left(\frac{2n\_{l}e}{d\_{VC}}\right)},$$ where ĉ* is the optimal data driven constant. If we knew the left hand side (LHS) of, even computationally, we could use it to estimate dVC. However, in general we don’t know the LHS of. Instead, we generate one observation of the form ξ(nl) = ΦdVC\(n*l) + ε(nl) for each design point nl by bootstrapping and denoted the realized values by ξ̂(nl). In, we assume ε(nl) has a mean zero, but an otherwise unknown, distribution. We can therefore obtain a list of values of ξ̂(nl) for the elements of NL. In effect, we are assuming that ΦdVC\(n*l) provides a tight bound on Δ, and hence Δm as suggested by Theorem [Theo23and4]. Our algorithm is as follows. Algorithm #1: Inputs: A collection of regression models G = {gβ : β ∈ β}, a data set, two integers b1 and b2 for the number of bootstrap samples, Integer m for the number of disjoint intervals use to discretize the losses, A set of design points NL = {n1, n2, ⋯, nl}. 1 For each l = 1, 2, ⋯, L do; 2 Take a bootstrap sample of size 2nl (with replacement) from our dataset; 3 Randomly divide the bootstrap data into two groups $~G\_{1}~ \hbox{and}~ G\_{2}$ of size nl each; 4 Fit two models one for G1 and one for G2; 5 Compute the square error of each model using the covariate and the response from the other model, thus: For instance $$SE\_{1} = \left(predict(Model\_{1}, x\_{2})-y\_{2}\right)^2,~ \hbox{and}~ SE\_{2} = \left(predict(Model\_{2}, x\_{1})-y\_{1}\right)^2$$ where (x1, y1) ranges over G1 and (x2, y2) ranges over G2. So, there are nl values of SE1 and nl values of SE2. 6 Discretize the loss function i.e., put each $SE\_{1} ~ \hbox{and} ~ SE\_{2}$ in one of the m disjoint intervals; 7 Estimate ν1jm(Znl + 1 : 2nl) and ν2jm(Z1 : nl) using the SE1’s and SE2’s respectively in the intervals j = 0, 1, …m − 1; 8 Compute the differences ∣ν1jm(znl + 1 : 2nl) − ν2jm(z1 : nl)∣ for j = 0, 1, …, m − 1; 9 Repeat Steps 1 − 8 b1 times, take the mean interval-wise and sum it across all intervals so we have: rb1(nl) = ∑j = 0m − 1mean∣ν1jm(zn1 + 1 : 2nl) − ν2jm(z1 : nl)∣; 10 Repeat Steps 1 − 9 b2 times to rb1, i(nl) for i = 1, …, b2 and form $$\hat{\xi}(n\_{l})= \frac{1}{b\_{2}}\sum\_{i=1}^{b\_{2}}r\_{b\_1,i}(n\_{l}).$$ It is seen that Step 9 uses a mean even though the definition of Δm and Δ (see and ) has a supremum inside the expectation. This is intentional because using a supremum within each interval gave a worse estimator. We suggest that summing the mean over the intervals performs well because it is not too far from the supermum and is more stable. Note that this algorithm is parallelizable because different nl can be sent to different nodes to speed the process of estimating ξ̂( ⋅ ) for all nl. After obtaining ξ̂(nl) for each value of nl, we estimate dVC by minimizing the squared distance between ξ̂(nl) and ΦdVC\(n*l). Our objective function is $$\begin{aligned} f\_{n\_{l}}(d\_{VC}) = \sum\_{l=1}^{L}\left(\hat{\xi}(n\_{l}) - \hat{c}\sqrt{\frac{d\_{VC}}{n\_{l}}\log\left(\frac{2n\_{l}e}{d\_{VC}}\right)}\right)^{2}, \label{objective}\end{aligned}$$ where L is the number of design points. Optimizing usually only leads to numerical solutions and in our work below, we set b1 = b2 = W for convenience. Proof of Consistency ==================== In this section, we provide a proof of consistency for the estimator d̂VC for dVC that we presented in Subsec. [estimator]. In many respects, the structure of this proof should be credited to. Our contribution is to adapt to our stable estimator for the regression context. We begin with some notation and definitions. Let Φ = {ϕdVC, c : H × I → R} where dVC ∈ H = (1, M] for some large M ∈ N, c ∈ I = [a, b] ⊂ R, and $$\begin{aligned} \label{phidvc} \phi\_{d\_{vc},c}\left(n\_{l}\right) = c\sqrt{\frac{d\_{VC}}{n\_{l}}\log\left(\frac{2n\_{l}e}{d\_{VC}}\right)}\end{aligned}$$ as derived in Subsec. [estimator] (see expression ). In expression, we assume L values n1, …, nL have been pre-specified. Fix a value of c and let ϕc be the corresponding elements of Φ. The proof holds for each fixed c and if we optimize over c to obtain ĉ as explained in Subsec. [estimator], the convergence of d̂VC to the true value dVC will only be faster. Without loss of generality, we assume that because Φ is compact we can choose R > supdVC∥ϕdVC∥L where the norm ∥ ⋅ ∥L is derived from the inner product $$\langle f, g \rangle = \frac{1}{L}\sum\_{l=1}^{L}f\left(n\_{l}\right)g\left(n\_{l}\right)$$ for real valued functions of a real variable. Thus ϕdVC = (ϕdVC(n1), …, ϕdVC(nL)) (where the subscript c on the ϕdVC, c(nL)ʹs in expression have been dropped for ease of notation). Now we can consider the class $$\begin{aligned} \label{Rbound} \Phi\_{c}(R) = \left\{\phi \in \Phi\_{c} : {\left\lVert\phi - \phi\_{d\_{VC}}\right\rVert}\_{Q} < R\right\},\end{aligned}$$ where ϕdVC is the element of Φc corresponding to the correct value of dVC. For a given nl, we have $$\begin{aligned} \hat{\xi}(n\_{l})= \frac{1}{b\_{2}}\sum\_{i=1}^{b\_{2}}r\_{b\_1,i}(n\_{l})\end{aligned}$$ where rb1, i(nl) is the i bootstrapped value of the integrand of Δm for each nl, i = 1, …, W and l = 1, …, L. In vector form, write ξ̂ = (ξ̂(n1), …, ξ̂(nL)). Using, each ξ̂(nl) can be represented as $$\begin{aligned} \hat{\xi}\left(n\_{l}\right) = \phi\_{d\_{VC}}\left(n\_{l}\right) + \epsilon\left(n\_{l}\right).\end{aligned}$$ We have the following result. [Lemma1consis] Let t ≥ 0 and let $\epsilon n\_{l} = \frac{1}{W}$, where εinl’s are independent with mean zero and satisfy $\epsilon\_{i} n\_{l} \in \left(\frac{jB}{m}, \frac{(j+1)B}{m}\right]$. Then at any design point nl, we have $$\label{chernoffLemma} E\left(\exp\left(t\epsilon\left(n\_{l}\right)\right)\right) \leq \exp\left(\frac{t^{2}B^{2}}{8Wm^{2}}\right).$$ Fix j = 0, 1, 2, ⋯, m − 1. Now, for any t ≥ 0 and i ≥ 0, and $\epsilon\_{i}\left(n\_{l}\right) \in \left(\frac{jB}{m}, \frac{(j+1)B}{m}\right]$, using Hoeffding’s Lemma we have $$E\left(\exp\left(t\epsilon\_{i}\left(n\_{l}\right)\right)\right)\leq \exp\left(\frac{t^{2}B^{2}}{8m^{2}}\right).$$ Thus, we have $$\begin{aligned} \label{consisLemma1} E\left(\exp\left(t\epsilon n\_{l}\right)\right) &=& E\left(\exp\left(\frac{t}{W}\sum\_{i=1}^{W}\epsilon\_{i} n\_{l}\right)\right) \nonumber \\ &=& E\left(\Pi\_{i=1}^{W}\exp\left(\frac{t}{W}\epsilon\_{i} n\_{l}\right)\right) \nonumber \\ &=& \Pi\_{i=1}^{W}E\left(\exp\left(\frac{t}{W}\epsilon\_{i}n\_{l}\right)\right) \nonumber \\ &\leq & \Pi\_{i=1}^{W}\exp\left(\frac{t^{2}B^{2}}{8W^{2}m^{2}}\right) \nonumber \\ &=& \exp\left(\frac{t^{2}B^{2}}{8Wm^{2}}\right) \end{aligned}$$ [Prop1Consistency] Suppose that {εl ≡ ε(nl), l = 1, 2…L} is a set of random variables satisfying Lemma [Lemma1consis]. Then for any γ ∈ Rk and ρ ≥ 0, we have $$\label{Prop1ConEq1} P\left(\mid \sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l} \mid\geq \rho \right) \leq \exp\left(-\frac{2Wm^{2}\rho^{2}}{B^{2}\sum\_{l=1}^{L}\gamma\_{l}^{2}}\right)$$ If X is a non-negative random variable and a > 0. Let Φ be a monotonically increasing non-negative function for the non-negative reals, then using Markov’s inequality, we have $$P\left(\|X\| > a \right) \leq \frac{E\left(\Phi(\left\|X\right\|)\right)}{\Phi(a)}.$$ For any t > 0, we have $$\begin{aligned} P\left(\mid \sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l} \mid\geq \rho \right) &=& P\left(\left\|t\sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l}\right\| \geq t\rho \right) \nonumber \\ &\leq& \frac{E\left(\exp\left(\left\|\sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l}\right\|\right)\right)}{\exp(t\rho)} \nonumber \\ &\leq& \frac{E\left(\exp\left(t\sum\_{l=1}^{k}\left\|\epsilon\_{l}\gamma\_{l}\right\|\right)\right)}{\exp(t\rho)} \nonumber \\ &=& \frac{E\left(\Pi\_{l=1}^{k}\exp\left(t\|\epsilon\_{l}\|\|\gamma\_{l}\|\right)\right)}{\exp(t\rho)} \nonumber \\ &=& \frac{\Pi\_{l=1}^{k}E\left(\exp\left(\|t\gamma\_{l}\|\|\epsilon\_{l}\|\right)\right)}{\exp(t\rho)} \nonumber \\ &\leq& \frac{\Pi\_{l=1}^{k}\exp\left(\frac{t^{2}B^{2}\gamma\_{l}^{2}}{8Wm^{2}}\right)}{\exp(t\rho)} \nonumber \\ &=& \exp\left(\frac{t^{2}B^{2}}{8Wm^{2}}\sum\_{l=1}^{k}\gamma\_{l}^{2} - t\rho\right) \label{eqProp1conc22} \end{aligned}$$ The RHS of [eqProp1conc22] attains its minimum for $$\label{consiT2} t = \frac{4Wm^{2}\rho}{B^{2}\sum\_{l=1}^{k}\gamma\_{l}^{2}}$$ $$\begin{aligned} P\left(\mid \sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l} \mid\geq \rho \right) &\leq& 2 P\left(\sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l} \geq \rho \right) \nonumber \\ &=& 2P\left(\exp\left(t\sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l}\right) \geq \exp(t\rho)\right) \nonumber \\ &\leq& \frac{2E\left(\exp\left(t\sum\_{l=1}^{k}\epsilon\_{l}\gamma\_{l}\right)\right)}{\exp(t\rho)} \nonumber \\ &=& \frac{2E\left(\Pi\_{l=1}^{k}\exp\left(t\epsilon\_{l}\gamma\_{l}\right)\right)}{\exp(t\rho)} \nonumber \\ &=& \frac{2\Pi\_{l=1}^{k}E\left(\exp\left(t\epsilon\_{l}\gamma\_{l}\right)\right)}{\exp(t\rho)} \nonumber \\ &\leq& \frac{2\Pi\_{l=1}^{k}\exp\left(\frac{t^{2}B^{2}\gamma\_{l}^{2}}{8Wm^{2}}\right)}{\exp(t\rho)} \nonumber \\ &=& 2\exp\left(\frac{t^{2}B^{2}}{8Wm^{2}}\sum\_{l=1}^{k}\gamma\_{l}^{2} - t\rho\right) \label{eqProp1conc} \end{aligned}$$ Since [eqProp1conc] is true for all t, the RHS of [eqProp1conc] attains its minimum for $$\label{consiT} t = \frac{8Wm^{2}\rho}{2B^{2}\sum\_{l=1}^{k}\gamma\_{l}^{2}}$$ Replacing t by its value in [eqProp1conc] gives the statement of the proposition. Let Q = QL be the empirical norm corresponding to the empirical L2 inner product defined by $$\langle f,g\rangle\_{Q} = \frac{1}{L}\sum\_{l=1}^{L}f\left(n\_{l}\right)g\left(n\_{l}\right).$$ [DefEntropy] Φ = {ϕdVC, c : I × H → R} where I ∈ [a, b] ⊂ R+, dVC ∈ (1, M] ⊂ N. For the purpose of our proof, initially we fix c so ϕ = ϕ(dVC). For s = 0, 1, 2, ⋯, let {ϕjs}j = 1Ns be the minimal 2− sR-covering set of (2− sR, Φ, ∥ ⋅ ∥ Q) for given s ∈ I and R > 0. Now, Ns = N(2− sR, Φ, Q) is the smallest number of balls of radius 2− sR that cover Φ. For each dVC, there exists a ϕjs ∈ {ϕ1s, ϕ2s, ⋯, ϕNss} such that ∥ ϕdVC − ϕjs ∥ Q ≤ 2− sR. Because Φ is compact we choose R > 0 so that supdVC ∥ ϕdVC ∥ Q ≤ R. Now consider the class Φ(R) = {ϕ ∈ Φ : ∥ ϕdVC − ϕdVC ∥ Q ≤ R}, i.e. the elements of Φ within R of ϕdVC. obviously j = j(dVC) and as derived in Theorem [Theo23and4], let $$\label{phibound} \phi\_{d\_{VC}}\left(n\right) = c\sqrt{\frac{d\_{VC}}{n}\log\left(\frac{2ne}{d\_{VC}}\right)}.$$ Let ϕjs and ϕjs − 1 be indexed so that the jth element of ϕ1s, ϕ2s, ⋯, ϕNss and the jth of ϕ1s − 1, ϕ2s − 1, ⋯, ϕNs − 1s − 1 are as close as possible. Using chaining, we have ϕjS = ∑s = 1S(ϕjs − ϕjs − 1). By the triangle inequality, we have $$\begin{aligned} \label{trienquality} \parallel\phi\_{j}^{s}-\phi\_{j}^{s-1}\parallel &\leq& \parallel\phi\_{j}^{s} - \phi\_{d\_{VC}} \parallel\_{Q\_{n}} + \parallel\phi\_{d\_{VC}} - \phi\_{j}^{s-1}\parallel\_{Q\_{n}}\nonumber\\ &\leq& 2^{-s}R + 2^{-s+1}R \nonumber\\ &=& 3 \cdot 2^{-s}R\end{aligned}$$ [TheoConsis] Given the conclusion of Proposition [Prop1Consistency] suppose that the true value dVC ∈ (0, M]. Then, $$\begin{aligned} \label{concistheo} P\left(\parallel\phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\parallel\_{L} \geq \delta\right) \leq 2\sum\_{s=1}^{S}\exp\left(-\frac{Wm^{2}L\delta^{4}}{18B^{2}2^{-2s}R^{2}}\right).\end{aligned}$$ [TheoConsis] Suppose the true dVC ∈ (0, M] and that ∀i = 1, …, W, ∀l = 1, …, L, rb1, i(nl) ∼ N(ϕdVC(nl), σ2) and independent, E(ε(nl)) = 0, Var(ε(nl)) = σ2. Then, on Φc(R), as n → ∞, m → ∞ and W = W(n) → ∞ at suitable rates we have that $$\begin{aligned} \label{deuxprime} P\left({\left\lVert\phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\right\rVert}\_{L} \geq \delta\right) = {\cal{O}}\left(\frac{1}{W}\right). \end{aligned}$$ Remark: Thus, pre-asymptotically, we get $$\begin{aligned} P\left({\left\lVert\phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\right\rVert}> \delta\right) \leq \frac{2^{2}\sigma^{2}}{\delta W}\sum\_{s=1}^{S}2^{s} \nonumber\end{aligned}$$ In fact, the rb1, i(nl)’s are only approximately independent N(ϕdVC(nl), σ2). However, as n increases they become closer and closer to being independent N(ϕdVC(nl), σ2), assuming ϕdVC(nl) is a tight enough upper bound, as n, m → ∞ at appropriate rates. Also, it is seen that if L = L(n) is increasing then ∥ ⋅ ∥L averages the evaluations of of more and more components of, say, ϕd̂VC. In the limit, this can be exihibited as an integral, i.e. as a quadratic norm. So, ∥ ⋅ ∥L can be regarded as an approximation of a L2-space norm that strengthens as a norm (or inner product) as n → ∞. In Theorem [TheoConsis], if we controlled the distance between ∥ ⋅ ∥L and its limit, we would get a stronger mode for consistency. By definition of ϕdVC, we have $$\begin{aligned} \label{eqmerlin} \sum\_{l = 1}^{L}\left(\hat{\xi}\left(n\_{l}\right) - c\sqrt{\frac{\hat{d}\_{VC}}{n\_{l}}\log\left(\frac{2n\_{l}e}{\hat{d}\_{VC}}\right)}\right)^2\leq \sum\_{l = 1}^{L}\left(\hat{\xi}\left(n\_{l}\right) - c\sqrt{\frac{d\_{VC}}{n\_{l}}\log\left(\frac{2n\_{l}e}{\hat{d}\_{VC}}\right)}\right)^{2}\end{aligned}$$ or more compactly ∥ξ̂ − ϕd̂VC∥L2 ≤ ∥ξ̂ − ϕdVC∥L2. Expanding both sides of gives $$\begin{aligned} \sum\_{l =1}^{L}\left(\phi^{2}\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}^{2}\left(n\_{l}\right) \right) \leq 2\sum\_{l=1}^{L}\hat{\xi}\left(n\_{l}\right)\left(\phi\_{\hat{d}\_{VC}} -\phi\_{d\_{VC}}\left(n\_{l}\right)\right) \nonumber\end{aligned}$$ and hence $$\begin{aligned} {\left\lVert\phi\_{\hat{d}\_{VC}}\right\rVert}^{2}\_{L} - {\left\lVert\phi\_{d\_{VC}}\right\rVert}^{2}\_{L} &\leq& \frac{2}{L}\sum\_{l=1}^{L}\left(\phi\_{d\_{VC}}\left(n\_{l}\right) + \epsilon\left(n\_{l}\right)\right)\left(\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right)\right)\nonumber \\ &=& \frac{2}{L}\sum\_{l=1}^{L}\left(\phi\_{d\_{VC}}\left(n\_{l}\right)\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) -\phi^{2}\_{d\_{VC}}\left(n\_{l}\right) \right. \nonumber \\ && \quad \quad \quad + \left. \epsilon\left(n\_{l}\right)\left(\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right)\right)\right) \nonumber\end{aligned}$$ Rearranging gives $$\begin{aligned} {\left\lVert\phi\_{\hat{d}\_{VC}}\right\rVert}\_{L}^{2} - 2\langle\epsilon, \phi\_{\hat{d}\_{VC}}\rangle + {\left\lVert\phi\_{{d}\_{VC}}\right\rVert}^{2}\_{L} \leq 2\langle \epsilon, \phi\_{d\_{VC}} - \phi\_{\hat{d}\_{VC}}\rangle\_{Q}\nonumber\end{aligned}$$ where ε = (ε(n1), …, ε(nL)), i.e. $$\begin{aligned} {\left\lVert\phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\right\rVert}^{2}\_{L} \leq 2\langle \epsilon, \phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\rangle\_{Q}.\end{aligned}$$ It is seen that the LHS is the main quantity we want to control. We have $$\begin{aligned} \label{three} P\left({\left\lVert\phi\_{\hat{d}\_{VC}}-\phi\_{d\_{VC}}\right\rVert}\_{L} > \delta \right) &\leq& P\left(\langle \epsilon, \phi\_{d\_{VC}} - \phi\_{\hat{d}\_{VC}}\rangle \ge \frac{\delta^2}{2}\right) \nonumber \\ &\leq& P\left({\left\lVert\epsilon\right\rVert}\_{L}{\left\lVert\phi\_{d\_{VC}} - \phi\_{\hat{d}\_{VC}}\right\rVert}\_L >\frac{\delta^2}{2}\right) \nonumber \\ &\leq &\frac{2R^2}{\delta^2} E{\left\lVert\epsilon\right\rVert}^{2}\_{L}\end{aligned}$$ using the Cauchy-Schwarz inequality, the bound in, and Markov’s inequality. By construction, we have that $$\begin{aligned} \label{five} E{\left\lVert\epsilon\right\rVert}^{2}\_{L} &=& \frac{1}{L}\sum\_{l=1}^{L}E\left(\epsilon^{2}\left(n\_l\right)\right)\nonumber\\ &=& \frac{1}{L}\sum\_{l=1}^{L}E\left[\left(\frac{1}{W}\sum\_{i=1}^{W}r\_{b\_{1},i}(n\_l)\right) - \phi(n\_{l})\right]^2\nonumber\\ &=& \frac{1}{L}\sum\_{l=1}^{L}E\left[\frac{1}{W}\sum\_{i=1}^{W}\left(r\_{b\_{1},i}(n\_l)-\phi(n\_l)\right)\right]^2\nonumber\\ &=&\frac{1}{LW^2}\sum\_{l=1}^{L}\sum\_{i=1}^{W}E\left(r\_{b\_{1},1}(n\_l) - \phi(n\_l)\right)^2\nonumber\\ &=&\frac{1}{LW^2}\sum\_{l=1}^{L}\sum\_{i=1}^{W}Var\left(r\_{b\_{1},1}(n\_l)\right)\nonumber\\ &=& \frac{\sigma^2}{W}\end{aligned}$$ Using in gives $$\begin{aligned} P\left({\left\lVert\phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\right\rVert}\_L\ge \delta\right) \leq \frac{2R^2 \sigma^2}{\delta^2 W} \end{aligned}$$ in which the upper bound decreases as n increases because W(n) is increasing, thereby giving. ∥ξ̂(nl) − ϕd̂VC∥2 ≤ ∥ξ̂(nl) − ϕdVC∥. Expanding both sides of, we have $$\begin{aligned} \\|\hat{\xi}\left(n\_{l}\right)\\|^{2} -2\langle\hat{\xi}\left(n\_{l}\right), \phi\_{\hat{d}\_{VC}}\rangle +\\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2} &\leq& \\|\hat{\xi}\left(n\_{l}\right)\\|^{2} -2\langle\hat{\xi}\left(n\_{l}\right), \phi\_{d\_{VC}}\rangle +\\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} \nonumber \end{aligned}$$ so, $$\begin{aligned} \label{inequality} \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2} - \\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} &\leq& 2\langle\hat{\xi}\left(n\_{l}\right), \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right)\rangle.\end{aligned}$$ Using, becomes $$\label{eq1} \begin{split} \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2} - \\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} & \leq 2\langle\phi\_{d\_{VC}}\left(n\_{l}\right) + \epsilon\_{l}, \nonumber \\ &\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right) \rangle \nonumber \\ & = 2\langle\phi\_{d\_{VC}}\left(n\_{l}\right),\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\rangle \nonumber \\ &-2\\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2}\nonumber \\ & + 2\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \rangle \nonumber\\ \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2} -2\langle\phi\_{d\_{VC}}\left(n\_{l}\right), \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\rangle + \\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} & = 2\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \rangle \nonumber \\ \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} &\leq 2\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \rangle \end{split}$$ $$\begin{aligned} \label{lasteq} \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2} - \\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} &\leq& 2\langle\phi\_{d\_{VC}}\left(n\_{l}\right) + \epsilon\_{l}, \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right) \rangle \nonumber\\ &=& 2\langle\phi\_{d\_{VC}}\left(n\_{l}\right),\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\rangle -2\\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2}\nonumber \\ &+& 2\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \rangle \nonumber\\ \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2} -2\langle\phi\_{d\_{VC}}\left(n\_{l}\right), \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\rangle + \\|\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} &=& 2\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \rangle \nonumber \\ \\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2} &\leq& 2\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \rangle\end{aligned}$$ Let $R\left(\frac{2^{s+1}\delta}{2}\right) = \left\{\phi\_{d\_{VC}}: \frac{2^{s}\delta}{2}\leq \\|\phi\_{d\_{VC}}\left(n\_{l}\right)-\phi\_{d\_{VC}}\left(n\_{l}\right)\\|\leq\frac{2^{s+1}\delta}{2}\right\}$, and assume $\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\in R\left(\frac{2^{s+1}\delta}{2}\right)$. Let $\phi\_{\hat{d}\_{VC}}^{s} = \phi\_{\hat{d}\_{VC}}\in R\left(\frac{2^{s+1}\delta}{2}\right)$ and define ϕd̂VC1 = ϕdVC where the ϕd̂VCs’s are canonical representations of a Ns(2− sR) covering of the compact set Φ(R). Using, and Proposition [Prop1Consistency] we have $$\begin{aligned} \label{mereq} P\left[\\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) -\phi\_{d\_{VC}}\left(n\_{l}\right)\\|> \delta\right] &=& P\left[\\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) -\phi\_{d\_{VC}}\left(n\_{l}\right)\\|^{2}> \delta^{2}\right] \nonumber\\ &\leq & P\left[\langle\epsilon\_{l}, \phi\_{d\_{VC}}\left(n\_{l}\right) -\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\rangle > \frac{1}{2}\delta^{2}\right] \nonumber\\ &\leq& P\left[\left\|\frac{1}{L}\sum\_{l=1}^{L}\epsilon\_{l}\left(\phi\_{d\_{VC}}\left(n\_{l}\right) -\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)\right)\right\| > \frac{1}{2}\delta^{2}\right] \nonumber\\ &\leq& P\left[\left\|\frac{1}{L}\sum\_{l=1}^{L}\sum\_{s=1}^{S}\epsilon\_{l}\left(\phi\_{d\_{VC}}\left(n\_{l}\right) - \phi\_{\hat{d}\_{VC}}\left(n\_{l}\right) \right)\right\| > \frac{1}{2}\delta^{2}\right]. \end{aligned}$$ Using the identity ϕdVC(nl) − ϕd̂VC(nl) = ∑s = 1Sϕd̂VCs(nl) − ϕd̂VCs − 1(nl), based on the rings $R\left(\frac{2^{s+1}\delta}{2}\right)$ the RHS of is upper bounded by $$\begin{aligned} \sum\_{s=1}^{S} P\left[\left\|\sum\_{l=1}^{L}\epsilon\_{l}\left(\phi^{s}\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi^{s-1}\_{\hat{d}\_{VC}}\left(n\_{l}\right) \right)\right\| > \frac{L}{2}\delta^{2}\right] &=& 2\sum\_{s=1}^{S}\exp\left(\frac{-2Wm^{2}L^{2}\delta^{4}}{4B^{2}\sum\_{l=1}^{L}\left(\phi^{s}\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi^{s-1}\_{\hat{d}\_{VC}}\left(n\_{l}\right)\right)^{2}}\right) \nonumber\\ &=& 2\sum\_{s=1}^{S}\exp\left(-\frac{2Wm^{2}L^{2}\delta^{4}}{4B^{2}L\\|\phi^{s}\_{\hat{d}\_{VC}}\left(n\_{l}\right) - \phi^{s-1}\_{\hat{d}\_{VC}}\left(n\_{l}\right)\\|^{2}}\right)\nonumber\\ &\leq& 2\sum\_{s=1}^{S}\exp\left(-\frac{Wm^{2}L\delta^{4}}{18B^{2}2^{-2s}R^{2}}\right)\end{aligned}$$ Note this proof allows S to increase provided the rate of increase is slow enough with n i.e., there exist sequences so that we can set S = S(n) and still retain. A notable difference between and the corresponding theorem in is that our simplified result effectively only gives $$\begin{aligned} P\left({\left\lVert\phi\_{\hat{d}\_{VC}} - \phi\_{d\_{VC}}\right\rVert}\_{L}\ge \delta\right) = \mathcal{O}\left(\frac{1}{W}\right)\end{aligned}$$ rather than O(e− γW) for some γ > 0, a much faster rate. We conjecture that the more sophisticated techniques used in could be adapted to our setting and thereby give an exponentially fast rate of convergence of d̂VC to dVC in probability. However, as yet, we have not been able to show this. Also, although it is suppressed in the notation, our result implicitly requires m → ∞ to justify the use of ϕdVC. As a consequence, we can show that our d̂VC is consistent. Suppose that ϕdVC( ⋅ ) is Lipschitz i.e. ∀n, ∃κ = κ(n) so that κ(n)∣dVC − dVCʹ∣ ≤ ∣ϕdVC(nl) − ϕdVCʹ(nl)∣, where κ(n) is bounded on compact sets. Since the form of ϕdVC(nl) is known from, it is clear that the uniform Lipschitz condition we have assumed actually holds at least for appropriately chosen compact sets. We also observed that for c ∈ C there exists a neighborhood B(c, εl) on which is true. Cover C × H by balls of the form B(c, εl) × {dVC} finitely many will be enough since C × H is compact. Using Theorem [TheoConsis], we can show that our d̂VC is consistent. Suppose that ϕdVC( ⋅ ) is Lipschitz i.e. ∀nl, ∃κ = κ(nl) so that κ(n)∣dVC − dVCʹ∣ ≤ ∣ϕdVC(nl) − ϕdVCʹ(nl)∣, where κ(n) is bounded on compact sets. Since the form of ϕdVC(nl) is known from, it is clear that the uniform Lipschitz condition we have assumed actually holds at least for appropriately chosen compact sets. We also observe that for c ∈ I there exists a neighborhood B(c, εl), η > 0, on which is true. Cover I × H by sets of the form B(c, η) × {dVC}; finitely many will be enough since I × H is compact. [Theo7] Given that the assumptions of Theorem [TheoConsis] hold and that ϕdVC( ⋅ ) is Lipschitz, we have, as n → ∞, that $$\label{consistencyresult} P\left(\left\|\hat{d\_{VC}} - d\_{VC}\right\|\geq \delta\right) \leq \frac{2 R^2 \sigma^2}{\delta^2\kappa W}\sum\_{s=1}^{S}2^s = {\cal{O}}\left(\frac{1}{W}\right),~ as ~ W\_n \rightarrow \infty,$$ where $\kappa = \sqrt{\frac{1}{L}\sum\_{l=1}^{L}\kappa\left(n\_{l}\right)}$ is the overall Lipshitz constant. Since all L of the ϕdVC(nl)’s are at least locally Lipschitz, their local Lipschitz inequlities can be summarized by an inequality of the form κ(nl)∣dVC − dVCʹ∣ ≤ ∣ϕdVC(nl) − ϕdVCʹ(nl)∣. So, using we have $$\label{blip} \begin{split} \left\|d\_{VC}-d\_{VC}'\right\|\sqrt{\frac{1}{L}\sum\_{l=1}^{L}\kappa\left(n\_{l}\right)} & \leq \sqrt{\frac{1}{L}\sum\_{l=1}^{L}\left(\phi\_{d\_{VC}}\left(n\_{l}\right)-\phi\_{d\_{VC}^{'}}\left(n\_{l}\right)\right)^2} \\ & =\\|\phi\_{d\_{VC}}\left(n\_{l}\right)-\phi\_{d\_{VC}^{'}}\left(n\_{l}\right)\\|\_L, \end{split}$$ where dVC is the true value and dVCʹ is any other value in H, and any extra constant from the local Lipschitz factors are assumed to have been absorbed into the κ(nl)’s as needed. Let $\kappa = \sqrt{\frac{1}{L}\sum\_{l=1}^{L}\kappa\left(n\_{l}\right)}$. Using Theorem [TheoConsis], and we have $$\label{ccprofend} \begin{split} P\left(\left\|\hat{d}\_{VC}-d\_{VC}\right\|\geq \delta\right) & \leq P\left[\\|\phi\_{\hat{d}\_{VC}}\left(n\_{l}\right)-\phi\_{d\_{VC}}\left(n\_{l}\right)\\|\_{L}\geq \delta \kappa\right] \\ & \leq \frac{2R^2\sigma^2}{\kappa \delta^2 W}\sum\_{s=1}^{S}2^{s}, \end{split}$$ where the last upper bound decreases as W = Wn → ∞ as n → ∞, giving. Simulation Studies ================== For any model, we can evaluate the LHS of from Theorem [Theo23and4] by Algorithm #1 in Sec. [Algo1]. Then, we can use nonlinear regression in to find d̂VC. So, it is seen that d̂VC is a function of the conjectured model. In principle, for any given model class, the VC dimension can be found, so our method can be applied. Since our goal is to estimate the true VC dimension, when a conjectured model P( ⋅ ∣ β) is linear and correct, we expect VC(P( ⋅ ∣ β)) ≅ d̂VC. By the same logic, if P( ⋅ ∣ β) is far from the true model, we expect VC(P( ⋅ ∣ β)) ≫ d̂VC or VC(P( ⋅ ∣ β)) ≪ d̂VC. This suggests we estimate dVC by seeking d̂VC = argmink∣VC(Pk( ⋅ ∣ β)) − d̂VC, k∣ ≤ t, where {Pk( ⋅ ∣ β)∣k = 1, 2, ⋯, K} is some set of models and d̂VC, k is calculated using model k, t is a positive and usually small number that such that t ≤ 2. In the case of linear model, with q = 1, 2, ⋯, Q explanatory variables, we get d̂VC = argminq∣q − d̂VC, q∣ ≤ t where d̂VC, q is the estimated VC dimension for model of size q. Note that can identify a good model even when consistency fails. The reason is that only requires a minimum at the VC dimension not convergence to the true VC dimension which may be any model under consideration. Here, we use a variation on by choosing the smallest local minimum of ∣q − d̂VC, q∣, effectively setting t = 0. In practice, this does not always give a unique model and this may not be desireable. On the other hand, choosing t > 0 can give a collection of models, which may be desireable in some settings. Our simulations are based on linear models, since for these we know the VC dimension equals the number of parameters in the model, see. To establish notation, we write the regression function as a linear combination of the covariates Xj, j = 0, 1, ⋯, p, y = f(x, β) = β0 + β1x1 + β2x2 + ⋯ + βpxp = ∑j = 0pβjxj. Given a dataset, {(xi, yi), i = 1, 2, …, n}, the matrix representation is Y = Xβ + ε where Y is the n × 1 vector of response values, X is the n × (1 + p) matrix with rows (1, x1, 1, x1, 2, …, x1, p), β = (β0, β1, …, βp)T is the vector of model parameters, and ε = (ε1, ε2, …, εn)T is a n × 1 mean zero Gaussian random vector. Now, the least squares estimator β̂ is given by β̂ = (XʹX)− 1XʹY. Our simulated data is analogous. We write Y = β0x0 + β1x1 + β2x2 + ⋯ + βpxp + ε ~N(0,=0.4)x0 = 1, j~N(= 5, = 3), xj ~N(=5, x = 2), in which all β’s, x’s and ε’s are independently generated. We center and scale all our variables, including the response. Initially, we use a nested sequence of model lists. If our covariates were highly correlated, before applying our method we could de-correlate them by sphering, i.e. transforming the covariates using their covariance matrix so they become approximately uncorrelated with variance one, see p. 144. Analysis of Synthetic Data -------------------------- In Subsec. [PropoWorks], we begin by presenting simulation results to verify our estimator for VC dimension is consistent for the VC dimension of the true model. Of course, since our results are only simulations, we do not always get perfect consistency; sometimes our d̂VC is off by one. (As we note later, this can often be corrected if the sample size is larger.) In Subsec. [Dependency], we will look at simulations where results do not initially appear to be consistent with the theory. However, we show that for larger values of p, larger values of n are needed. Also, as p increases, we must choose nl’s that are properly spread out over [0, n]. and these nl’s seem to matter less as n → ∞. We suggest this is necessary because is only an upper bound that conjecture tightens as n increases. ### Some first examples In this subsection, we implement simulations for model sizes $p = 15, 30, 40, 50 ~ \hbox{and}~ 60$ and we present the results for all these cases for six model selection techniques AIC, BIC, CV $\widehat{ERM}\_{1}$, $\widehat{ERM}\_2$ and VC dimension (VCD). For p = 15, 30, we use a sample size of n = 400. The design points are NL = {50, 100, 150, 200, 250, 300, 400}; m = 10; and the number of bootstrap samples is W = 50. For $p = 40, 50 ~\hbox{and}~60$, we use a sample size of n = 600. The design points are NL = (75, 150, 225, 300, 375, 450, 525, 600); m = 10; and the number of bootstrap samples is W = 50. For these cases, we fit two sets of models; the first set uses a subset of our covariates to estimate the VC dimension, and in the second set, we added some ‘decoys’ (their β’s in the generation of the responses are zeros). Outputs of these simulations are given in Figs. [erm15]-[erm60], in which $\widehat{ERM}\_{1}$, $\widehat{ERM}\_{2}$, VCD, AIC, BIC, and CV, are used to identify a good model. ![Upper: Values of \widehat{ERM}_{1}, \widehat{ERM}_{2}, and VC dimension. Lower: Values of AIC, BIC, and CV for p=15, \sigma_{\epsilon}=0.4, \sigma_{\beta}=3, \sigma_{x} = 2](P15N400NL50to400by50 "fig:") [erm15] ![Upper: Values of \widehat{ERM}_{1}, \widehat{ERM}_{2}, and VC dimension (VCD). Lower: AIC, BIC, and CV values for p=30, \sigma_{\epsilon}=0.4, \sigma_{\beta}=3, \sigma_{x} = 2](P30N400NL50to400by50 "fig:") [erm30] ![Upper: Values of \widehat{ERM}_{1}, \widehat{ERM}_{2} and VC dimension (VCD). Lower: Values of AIC, BIC, and CV, for p=40, \sigma_{\epsilon} = 0.4, \sigma_{\beta}=3, \sigma_{x} = 2](P40N600NL75to600by75 "fig:") [erm40] ![Upper: Values of \widehat{ERM}_{1}, \widehat{ERM}_{2}, and VC dimension (VCD). Lower: Values of AIC, BIC, and CV for p=50, \sigma_{\epsilon} = 0.4, \sigma_{\beta}=3, \sigma_{x} = 2](P50N600NL75to600by75 "fig:") [erm50] ![Upper: Values of \widehat{ERM}_{1}, \widehat{ERM}_{2}, and VC dimension (VCD). Lower: Values of AIC, BIC, and CV for p=60, \sigma_{\epsilon} = 0.4, \sigma_{\beta}=3, \sigma_{x} = 2](P60N600NL75to600By75 "fig:") [erm60] By examining Figs [erm15] to [erm60], we see that, for each given true model of pre-specified size, we fitted a list of nested models. When the size of the conjectured model is strictly less than that of the true model, the estimated VC dimension equals the minimum value of the design points, and the values of the AIC, BIC, CV, $\widehat{ERM}\_{1}$, $\widehat{ERM}\_{2}$ are high. These latter values typically decrease as the conjectured models become similar to the true model. For this range of model sizes, when the conjectured model exactly matches the true model, the estimated VC dimension (d̂VC) is closest to the true value. Except for Fig. [erm60], the biggest discrepancy (of size 2) occurs for p = 50; by contrast, for every other case the difference between the true value and the estimated VC dimension is at most one. Correspondingly, when the size of the conjectured of the true model is above the size of the true model, all six methods increase. The difference between our method (VCD) and $\widehat{ERM}\_1$, $\widehat{ERM}\_2$, AIC, BIC and CV is that the VCD values are much higher than the other values and tend to increase. Thus, they more decisively indicate which model should be taken as true. In Figs. [erm15]-[erm50], the qualitative behaviour of $\widehat{ERM}\_{1}$, $\widehat{ERM}\_{2}$ or BIC is the same. In fact, when the conjectured model is a subset of the true model, we see a consistent and substantial decrease of $\widehat{ERM}\_{1}$, $\widehat{ERM}\_{2}$ or BIC, and a sudden drop in these statistics when the conjectured model perfectly matches the true model. This sudden drop can be regarded as an indicator of the true model. After this point, $\widehat{ERM}\_{1}$ usually flatlines. However, $\widehat{ERM}\_{2}$ can still often has smallest value occur at the sudden drop because, $\widehat{ERM}\_{2}$ often increases (albeit slowly) from its minimum as p increases. We also see that BIC has good power of discrimination since its smallest values occur at the true model. As we add decoys, the values of BIC are bigger than those of the true model, although sometimes not by much. The smallest discrepancy between the size dVC of the model and d̂VC usually occurs at the true model. This indicates that d̂VC is consistent. In addition, even though the VCD values generally increase as the size of the conjectured model exceeds the size of the true model, in some cases, past a certain value dVC, the VCD value may flatline as well. The problem with flatlining or even decreasing past a certain value of dVC occurs mostly when n is not large enough relative to p. In Figure [erm60], the behaviour of all six model selection methods is qualitatively the same, however, d̂VC is far from the true value. We suggest that this discrepancy occurs because the sample size is too small compared to p and the choice of the design points is poor. Indeed, we suggest that in practice the effect of poorly chosen of design points is exacerbated when n is not large enough. For instance, in Fig. [ERM60700], we see that n = 700 as opposed to n = 600, and very similar design points, the VCD performs better, rising when the conjectured models are too large. For the present, we note that Figure [erm60] gives us estimates of the true model using $\widehat{ERM}\_{1}$ and $\widehat{ERM}\_{2}$ when the sample size is N = 600, and NL takes on values from 75 to 600 in steps of size 75. We observe that $\	arxiv_0000730
Online Influence Maximization under Independent Cascade Model with Semi-Bandit Feedback ======================================================================================= We study the online influence maximization problem in social networks under the independent cascade model. Specifically, we aim to learn the set of “best influencers” in a social network online while repeatedly interacting with it. We address the challenges of (i) combinatorial action space, since the number of feasible influencer sets grows exponentially with the maximum number of influencers, and (ii) limited feedback, since only the influenced portion of the network is observed. Under a stochastic semi-bandit feedback, we propose and analyze ${\tt{IMLinUCB}}$, a computationally efficient UCB-based algorithm. Our bounds on the cumulative regret are polynomial in all quantities of interest, achieve near-optimal dependence on the number of interactions and reflect the topology of the network and the activation probabilities of its edges, thereby giving insights on the problem complexity. To the best of our knowledge, these are the first such results. Our experiments show that in several representative graph topologies, the regret of ${\tt{IMLinUCB}}$ scales as suggested by our upper bounds. ${\tt{IMLinUCB}}$ permits linear generalization and thus is both statistically and computationally suitable for large-scale problems. Our experiments also show that ${\tt{IMLinUCB}}$ with linear generalization can lead to low regret in real-world online influence maximization. Introduction ============ Social networks are increasingly important as media for spreading information, ideas, and influence. Computational advertising studies models of information propagation or diffusion in such networks . Viral marketing aims to use this information propagation to spread awareness about a specific product. More precisely, agents (marketers) aim to select a fixed number of influencers (called seeds or source nodes) and provide them with free products or discounts. They expect that these users will influence their neighbours and, transitively, other users in the social network to adopt the product. This will thus result in information propagating across the network as more users adopt or become aware of the product. The marketer has a budget on the number of free products and must choose seeds in order to maximize the influence spread, which is the expected number of users that become aware of the product. This problem is referred to as influence maximization (IM) . For IM, the social network is modeled as a directed graph with the nodes representing users, and the edges representing relations (e.g., friendships on Facebook, following on Twitter) between them. Each directed edge (i, j) is associated with an activation probability ${\overline}{w}(i, j)$ that models the strength of influence that user i has on user j. We say a node j is a downstream neighbor of node i if there is a directed edge (i, j) from i to j. The IM problem has been studied under a number of diffusion models . The best known and studied are the models in , and in particular the independent cascade (IC) model. In this work, we assume that the diffusion follows the IC model and describe it next. After the agent chooses a set of source nodes S, the independent cascade model defines a diffusion (influence) process: At the beginning, all nodes in S are activated (influenced); subsequently, every activated node i can activate its downstream neighbor j with probability ${\overline}{w}(i, j)$ once, independently of the history of the process. This process runs until no activations are possible. In the IM problem, the goal of the agent is to maximize the expected number of the influenced nodes subject to a cardinality constraint on S. Finding the best set S is an NP-hard problem, but under common diffusion models including IC, it can be efficiently approximated to within a factor of 1 − 1/e. In many social networks, however, the activation probabilities are unknown. One possibility is to learn these from past propagation data . However in practice, such data are hard to obtain and the large number of parameters makes this learning challenging. This motivates the learning framework of IM bandits, where the agent needs to learn to choose a good set of source nodes while repeatedly interacting with the network. Depending on the feedback to the agent, the IM bandits can have (1) full-bandit feedback, where only the number of influenced nodes is observed; (2) node semi-bandit feedback, where the identity of influenced nodes is observed; or (3) edge semi-bandit feedback, where the identity of influenced edges (edges going out from influenced nodes) is observed. In this paper, we give results for the edge semi-bandit feedback model, where we observe for each influenced node, the downstream neighbors that this node influences. Such feedback is feasible to obtain in most online social networks. These networks track activities of users, for instance, when a user retweets a tweet of another user. They can thus trace the propagation (of the tweet) through the network, thereby obtaining edge semi-bandit feedback. The IM bandits problem combines two main challenges. First, the number of actions (possible sets) S grows exponentially with the cardinality constraint on S. Second, the agent can only observe the influenced portion of the network as feedback. Although IM bandits have been studied in the past (see Section [sec:related work] for an overview and comparison), there are a number of open challenges . One challenge is to identify reasonable complexity metrics that depend on both the topology and activation probabilities of the network and characterize the information-theoretic complexity of the IM bandits problem. Another challenge is to develop learning algorithms such that (i) their performance scales gracefully with these metrics and (ii) are computationally efficient and can be applied to large social networks with millions of users. In this paper, we address these two challenges under the IC model with access to edge semi-bandit feedback. We refer to our model as an independent cascade semi-bandit (ICSB). We make four main contributions. First, we propose ${\tt{IMLinUCB}}$, a UCB-like algorithm for ICSBs that permits linear generalization and is suitable for large-scale problems. Second, we define a new complexity metric, referred to as maximum observed relevance for ICSB, which depends on the topology of the network and is a non-decreasing function of activation probabilities. The maximum observed relevance C\* can also be upper bounded based on the network topology or the size of the network in the worst case. However, in real-world social networks, due to the relatively low activation probabilities , C\* attains much smaller values as compared to the worst case upper bounds. Third, we bound the cumulative regret of ${\tt{IMLinUCB}}$. Our regret bounds are polynomial in all quantities of interest and have near-optimal dependence on the number of interactions. They reflect the structure and activation probabilities of the network through C\* and do not depend on inherently large quantities, such as the reciprocal of the minimum probability of being influenced (unlike ) and the cardinality of the action set. Finally, we evaluate ${\tt{IMLinUCB}}$ on several problems. Our empirical results on simple representative topologies show that the regret of ${\tt{IMLinUCB}}$ scales as suggested by our topology-dependent regret bounds. We also show that ${\tt{IMLinUCB}}$ with linear generalization can lead to low regret in real-world online influence maximization problems. Influence Maximization under Independence Cascade Model ======================================================= In this section, we define notation and give the formal problem statement for the IM problem under the IC model. Consider a directed graph G = (V, E) with a set V = {1, 2, …, L} of L = ∣V∣ nodes, a set E = {1, 2, …, ∣E∣} of directed edges, and an arbitrary binary weight function ${{\bf w}}:{\mathcal{E}}\rightarrow \{ 0,1 \}$. We say that a node v2 ∈ V is reachable from a node v1 ∈ V under ${{\bf w}}$ if there is a directed path[1](#fn1) p = (e1, e2, …, el) from v1 to v2 in G satisfying ${{\bf w}}(e\_i)=1$ for all i = 1, 2, …, l, where ei is the i-th edge in p. For a given source node set S ⊆ V and ${{\bf w}}$, we say that node v ∈ V is influenced if v is reachable from at least one source node in S under ${{\bf w}}$; and denote the number of influenced nodes in G by $f({\mathcal{S}}, {{\bf w}})$. By definition, the nodes in S are always influenced. The influence maximization (IM) problem is characterized by a triple $\left({\mathcal{G}}, K, {\overline}{w} \right)$, where G is a given directed graph, K ≤ L is the cardinality of source nodes, and ${\overline}{w}: {\mathcal{E}}\rightarrow [0,1]$ is a probability weight function mapping each edge e ∈ E to a real number ${\overline}{w}(e) \in [0,1]$. The agent needs to choose a set of K source nodes S ⊆ V based on $\left({\mathcal{G}}, K, {\overline}{w} \right)$. Then a random binary weight function ${{\bf w}}$, which encodes the diffusion process under the IC model, is obtained by independently sampling a Bernoulli random variable ${{\bf w}}(e) \sim \mathrm{Bern}\left({\overline}{w}(e) \right)$ for each edge e ∈ E. The agent’s objective is to maximize the expected number of the influenced nodes: $\max\_{{\mathcal{S}}:\, \|{\mathcal{S}}\|=K} f({\mathcal{S}}, {\overline}{w})$, where $f({\mathcal{S}}, {\overline}{w}) \stackrel{\Delta}{=}{\mathbb{E}}\_{{{\bf w}}} \left[f({\mathcal{S}}, {{\bf w}}) \right]$ is the expected number of influenced nodes when the source node set is S and ${{\bf w}}$ is sampled according to ${\overline}{w}$.[2](#fn2) It is well-known that the (offline) IM problem is NP-hard, but can be approximately solved by approximation/randomized algorithms under the IC model. In this paper, we refer to such algorithms as oracles to distinguish them from the machine learning algorithms discussed in following sections. Let Sopt be the optimal solution of this problem, and ${\mathcal{S}}^\={{\tt{ORACLE}}}({\mathcal{G}}, K, {\overline}{w})$ be the (possibly random) solution of an oracle ${{\tt{ORACLE}}}$. For any α, γ* ∈ [0, 1], we say that ${{\tt{ORACLE}}}$ is an (α, γ)-approximation oracle for a given (G, K) if for any ${\overline}{w}$, $f({\mathcal{S}}^\, {\overline}{w}) \geq \gamma f({{\mathcal{S}}^{\mathrm{opt}}}, {\overline}{w})$ with probability at least α. Notice that this further implies that ${\mathbb{E}}\left[f({\mathcal{S}}^\, {\overline}{w}) \right] \geq \alpha \gamma f({{\mathcal{S}}^{\mathrm{opt}}}, {\overline}{w})$. We say an oracle is exact if α = γ = 1. Influence Maximization Semi-Bandit ================================== In this section, we first describe the IM semi-bandit problem. Next, we state the linear generalization assumption and describe ${\tt{IMLinUCB}}$, our `UCB`-based semi-bandit algorithm. Protocol -------- The independent cascade semi-bandit (ICSB) problem is also characterized by a triple $\left({\mathcal{G}}, K, {\overline}{w} \right)$, but ${\overline}{w}$ is unknown to the agent. The agent interacts with the independent cascade semi-bandit for n rounds. At each round t = 1, 2, …, n, the agent first chooses a source node set St ⊆ V with cardinality K based on its prior information and past observations. Influence then diffuses from the nodes in St according to the IC model. Similarly to the previous section, this can be interpreted as the environment generating a binary weight function ${{\bf w}}\_t$ by independently sampling ${{\bf w}}\_t(e) \sim \mathrm{Bern} \left( {\overline}{w}(e) \right)$ for each e ∈ E. At round t, the agent receives the reward $f({\mathcal{S}}\_t, {{\bf w}}\_t)$, that is equal to the number of nodes influenced at that round. The agent also receives edge semi-bandit feedback from the diffusion process. Specifically, for any edge e = (u1, u2) ∈ E, the agent observes the realization of ${{\bf w}}\_t(e)$ if and only if the start node u1 of the directed edge e is influenced in the realization ${{\bf w}}\_t$. The agent’s objective is to maximize the expected cumulative reward over the n steps. Linear generalization --------------------- Since the number of edges in real-world social networks tends to be in millions or even billions, we need to exploit some generalization model across activation probabilities to develop efficient and deployable learning algorithms. In particular, we assume that there exists a linear-generalization model for the probability weight function ${\overline}{w}$. That is, each edge e ∈ E is associated with a known feature vector xe ∈ ℜd (here d is the dimension of the feature vector) and that there is an unknown coefficient vector θ\* ∈ ℜd such that for all e ∈ E, ${\overline}{w}(e)$ is ``well approximated" by xe\scriptscriptstyle Tθ\. Formally, we assume that $\rho \stackrel{\Delta}{=}\max\_{e \in {\mathcal{E}}} \| {\overline}{w}(e) -x\_{e}{^\mathsf{\scriptscriptstyle T}}\theta^\ \|$ is small. In Section [sec:FB-experiment], we see that such a linear generalization leads to efficient learning in real-world networks. Note that all vectors in this paper are column vectors. Similar to the existing approaches for linear bandits , we exploit the linear generalization to develop a learning algorithm for ICSB. Without loss of generality, we assume that ∥xe∥2 ≤ 1 for all e ∈ E. Moreover, we use ${{\bf X}}\in \Re^{\|{\mathcal{E}}\| \times d}$ to denote the feature matrix, i.e., the row of ${{\bf X}}$ associated with edge e is xe\scriptscriptstyle T. Note that if a learning agent does not know how to construct good features, it can always choose the naïve feature matrix ${{\bf X}}={{\bf I}}\in \Re^{\|{\mathcal{E}}\| \times \|{\mathcal{E}}\|}$ and have no generalization model across edges. We refer to the special case ${{\bf X}}={{\bf I}}\in \Re^{\|{\mathcal{E}}\| \times \|{\mathcal{E}}\|}$ as the tabular case. ${\tt{IMLinUCB}}$ algorithm --------------------------- In this section, we propose Influence Maximization Linear UCB (${\tt{IMLinUCB}}$), detailed in Algorithm [alg:imb]. Notice that ${\tt{IMLinUCB}}$ represents its past observations as a positive-definite matrix (Gram matrix) ${{\bf M}}\_t \in \Re^{d \times d}$ and a vector Bt ∈ ℜd. Specifically, let ${{\bf X}}\_t$ be a matrix whose rows are the feature vectors of all observed edges in t steps and Yt be a binary column vector encoding the realizations of all observed edges in t steps. Then ${{\bf M}}\_t={{\bf I}}+\sigma^{-2} {{\bf X}}\_t{^\mathsf{\scriptscriptstyle T}}{{\bf X}}\_t$ and $B\_t={{\bf X}}\_t{^\mathsf{\scriptscriptstyle T}}Y\_t$. At each round t, ${\tt{IMLinUCB}}$ operates in three steps: First, it computes an upper confidence bound Ut(e) for each edge e ∈ E. Note that Proj[0, 1]( ⋅ ) projects a real number into interval [0, 1] to ensure that Ut ∈ [0, 1]∣E∣. Second, it chooses a set of source nodes based on the given ${{\tt{ORACLE}}}$ and Ut, which is also a probability-weight function. Finally, it receives the edge semi-bandit feedback and uses it to update ${{\bf M}}\_t$ and Bt. It is worth emphasizing that ${\tt{IMLinUCB}}$ is computationally efficient as long as ${{\tt{ORACLE}}}$ is computationally efficient. Specifically, at each round t, the computational complexities of both Step 1 and 3 of ${\tt{IMLinUCB}}$ are O(∣E∣d2).[3](#fn3) [t] [alg:imb] Input: graph G, source node set cardinality K, oracle ${{\tt{ORACLE}}}$, feature vector xe’s, and algorithm parameters σ, c > 0, Initialization: B0 ← 0 ∈ ℜd, ${{\bf M}}\_0 \leftarrow \mathbf{I} \in \Re^{d \times d}$ 1. set ${\overline}{\theta}\_{t-1} \leftarrow \sigma^{-2} {{\bf M}}\_{t-1}^{-1} B\_{t-1}$ and the UCBs as $ U\_t(e) \leftarrow \mathrm{Proj}\_{[0,1]} \left( x\_{e}{^\mathsf{\scriptscriptstyle T}}{\overline}{\theta}\_{t-1} + c \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \right) $ for all e ∈ E 2. choose ${\mathcal{S}}\_t \in {{\tt{ORACLE}}}({\mathcal{G}}, K, U\_t)$, and observe the edge-level semi-bandit feedback 3. update statistics: (a) initialize ${{\bf M}}\_t \leftarrow {{\bf M}}\_{t-1}$ and Bt ← Bt − 1 (b) for all observed edges e ∈ E, update ${{\bf M}}\_t \leftarrow {{\bf M}}\_t + \sigma^{-2} x\_{e} x\_{e}{^\mathsf{\scriptscriptstyle T}}$ and $B\_t \leftarrow B\_t + x\_{e} {{\bf w}}\_t(e)$ It is worth pointing out that in the tabular case, ${\tt{IMLinUCB}}$ reduces to ${{\tt{CUCB}}}$, in the sense that the confidence radii in ${\tt{IMLinUCB}}$ are the same as those in ${{\tt{CUCB}}}$, up to logarithmic factors. That is, ${{\tt{CUCB}}}$ can be viewed as a special case of ${\tt{IMLinUCB}}$ with ${{\bf X}}={{\bf I}}$. Performance metrics ------------------- Recall that the agent’s objective is to maximize the expected cumulative reward, which is equivalent to minimizing the expected cumulative regret. The cumulative regret is the loss in reward (accumulated over rounds) because of the lack of knowledge of the activation probabilities. Observe that in each round t, ${\tt{IMLinUCB}}$ needs to use an approximation/randomized algorithm ${{\tt{ORACLE}}}$ for solving the offline IM problem. Naturally, this can lead to O(n) cumulative regret, since at each round there is a non-diminishing regret due to the approximation/randomized nature of ${{\tt{ORACLE}}}$. To analyze the performance of ${\tt{IMLinUCB}}$ in such cases, we define a more appropriate performance metric, the scaled cumulative regret, as $R^{\eta}(n)=\sum\_{t=1}^n {\mathbb{E}}\left[ \textstyle R\_t^{\eta} \right]$, where n is the number of steps, η > 0 is the scale, and $R\_t^{\eta} = f({{\mathcal{S}}^{\mathrm{opt}}}, {{\bf w}}\_t) -\frac{1}{\eta} f({\mathcal{S}}\_t, {{\bf w}}\_t)$ is the η-scaled realized regret Rtη at round t. When η = 1, Rη(n) reduces to the standard expected cumulative regret R(n). Analysis ======== In this section, we give a regret bound for ${\tt{IMLinUCB}}$ for the case when ${\overline}{w}(e) = x\_{e}{^\mathsf{\scriptscriptstyle T}}\theta^\$ for all e* ∈ E, i.e., the linear generalization is perfect. Our main contribution is a regret bound that scales with a new complexity metric, maximum observed relevance, which depends on both the topology of G and the probability weight function ${\overline}{w}$, and is defined in Section [sec:complexity]. We highlight this as most known results for this problem are worst case, and some of them do not depend on probability weight function at all. Maximum observed relevance -------------------------- We start by defining some terminology. For given directed graph G = (V, E) and source node set S ⊆ V, we say an edge e ∈ E is relevant to a node v ∈ V \ S under S if there exists a path p from a source node s ∈ S to v such that (1) e ∈ p and (2) p does not contain another source node other than s. Notice that with a given S, whether or not a node v ∈ V \ S is influenced only depends on the binary weights ${{\bf w}}$ on its relevant edges. For any edge e ∈ E, we define NS, e as the number of nodes in V \ S it is relevant to, and define PS, e as the conditional probability that e is observed given S, $$\begin{aligned} \label{eqn:N} N\_{{\mathcal{S}}, e} \stackrel{\Delta}{=} \textstyle \sum\_{v \in {\mathcal{V}}\setminus {\mathcal{S}}} \mathbf{1} \left \{ \text{$e$ is relevant to $v$ under ${\mathcal{S}}$} \right \} \quad \text{and} \quad P\_{{\mathcal{S}}, e} \stackrel{\Delta}{=} {\mathbb{P}}\left( \text{$e$ is observed}\ \middle \| \, {\mathcal{S}}\right).\end{aligned}$$ Notice that NS, e only depends on the topology of G, while PS, e depends on both the topology of G and the probability weight ${\overline}{w}$. The maximum observed relevance C\* is defined as the maximum (over S) 2-norm of NS, e’s weighted by PS, e’s, $$\begin{aligned} \label{eqn:complexity} {C\_\}\stackrel{\Delta}{=} \textstyle \max\_{{\mathcal{S}}:\, \|{\mathcal{S}}\|=K}\sqrt{ \sum\_{e \in {\mathcal{E}}} N^2\_{{\mathcal{S}}, e} P\_{{\mathcal{S}}, e} }.\end{aligned}$$ As is detailed in the proof of Lemma [lemma:worst] in Appendix [sec:proofmain], C\ arises in the step where Cauchy-Schwarz inequality is applied. Note that C\* also depends on both the topology of G and the probability weight ${\overline}{w}$. However, C\* can be bounded from above only based on the topology of G or the size of the problem, i.e., L = ∣V∣ and ∣E∣. Specifically, by defining ${C\_{{\mathcal{G}}}}\stackrel{\Delta}{=} \textstyle \max\_{{\mathcal{S}}:\, {\left\|{\mathcal{S}}\right\|} = K} \sqrt{\sum\_{e \in {\mathcal{E}}} N\_{{\mathcal{S}}, e}^2} $, we have $${C\_\}\leq {C\_{{\mathcal{G}}}}= \textstyle \max\_{{\mathcal{S}}:\, {\left\|{\mathcal{S}}\right\|} = K} \sqrt{\sum\_{e \in {\mathcal{E}}} N\_{{\mathcal{S}}, e}^2} \leq (L - K) \sqrt{\|{\mathcal{E}}\|} ={\mathcal{O}}\left( L \sqrt{\|{\mathcal{E}}\|} \right) = {\mathcal{O}}\left(L^2 \right),$$ where CG is the maximum/worst-case (over ${\overline}{w}$) C\ for the directed graph G, and the maximum is obtained by setting ${\overline}{w}(e)=1$ for all e ∈ E. Since CG is worst-case, it might be very far away from C\* if the activation probabilities are small. Indeed, this is what we expect in typical real-world situations. Notice also that if $\max\_{e \in {\mathcal{E}}} {\overline}{w}(e) \rightarrow 0$, then PS, e → 0 for all $e \notin {\mathcal{E}}({\mathcal{S}})$ and PS, e = 1 for all e ∈ E(S), where E(S) is the set of edges with start node in S, hence we have $ {C\_\}\rightarrow C\_{{\mathcal{G}}}^0 \stackrel{\Delta}{=} \textstyle \max\_{{\mathcal{S}}:\, \|{\mathcal{S}}\|=K}\sqrt{ \sum\_{e \in {\mathcal{E}}({\mathcal{S}})} N^2\_{{\mathcal{S}}, e} }$. In particular, if K* is small, CG0 is much less than CG in many topologies. For example, in a complete graph with K = 1, CG = Θ(L2) while $C\_{{\mathcal{G}}}^0= \Theta(L^{\frac{3}{2}})$. Finally, it is worth pointing out that there exist situations $({\mathcal{G}}, {\overline}{w})$ such that C\* = Θ(L2). One such example is when G is a complete graph with L nodes and ${\overline}{w}(e) = L/(L+1)$ for all edges e in this graph. [fig:graphs] To give more intuition, in the rest of this subsection, we illustrate how CG, the worst-case C\, varies with four graph topologies in Figure [fig:graphs]: bar, star, ray, and grid, as well as two other topologies: general tree and complete graph. We fix the node set V = {1, 2, …, L} for all graphs. The bar graph (Figure [fig:graphs]a) is a graph where nodes i* and i + 1 are connected when i is odd. The star graph (Figure [fig:graphs]b) is a graph where node 1 is central and all remaining nodes i ∈ V \ {1} are connected to it. The distance between any two of these nodes is 2. The ray graph (Figure [fig:graphs]c) is a star graph with $k = \left\lceil\sqrt{L - 1}\right\rceil$ arms, where node 1 is central and each arm contains either ⌈(L − 1)/k⌉ or ⌊(L − 1)/k⌋ nodes connected in a line. The distance between any two nodes in this graph is ${\mathcal{O}}(\sqrt{L})$. The grid graph (Figure [fig:graphs]d) is a classical non-tree graph with O(L) edges. To see how CG varies with the graph topology, we start with the simplified case when K = ∣S∣ = 1. In the bar graph (Figure [fig:graphs]a), only one edge is relevant to a node v ∈ V \ S and all the other edges are not relevant to any nodes. Therefore, CG ≤ 1. In the star graph (Figure [fig:graphs]b), for any s, at most one edge is relevant to at most L − 1 nodes and the remaining edges are relevant to at most one node. In this case, ${C\_{{\mathcal{G}}}}\leq \sqrt{L^2 + L} = {\mathcal{O}}(L)$. In the ray graph (Figure [fig:graphs]c), for any s, at most ${\mathcal{O}}(\sqrt{L})$ edges are relevant to L − 1 nodes and the remaining edges are relevant to at most ${\mathcal{O}}(\sqrt{L})$ nodes. In this case, ${C\_{{\mathcal{G}}}}= {\mathcal{O}}(\sqrt{L^\frac{1}{2} L^2 + L L}) = {\mathcal{O}}(L^\frac{5}{4})$. Finally, recall that for all graphs we can bound CG by ${\mathcal{O}}(L \sqrt{\|{\mathcal{E}}\|})$, regardless of K. Hence, for the grid graph (Figure [fig:graphs]d) and general tree graph, ${C\_{{\mathcal{G}}}}={\mathcal{O}}(L^{\frac{3}{2}})$ since ∣E∣ = O(L); for the complete graph CG = O(L2) since ∣E∣ = O(L2). Clearly, CG varies widely with the topology of the graph. The second column of Table [table:topology] summarizes how CG varies with the above-mentioned graph topologies for general K = ∣S∣. Regret guarantees ----------------- Consider C\* defined in Section [sec:complexity] and recall the worst-case upper bound ${C\_\}\leq (L -K) \sqrt{\|{\mathcal{E}}\|}$, we have the following regret guarantees for ${\tt{IMLinUCB}}$. [thm:main] Assume that (1) ${\overline}{w}(e) = x\_{e}{^\mathsf{\scriptscriptstyle T}}\theta^\$ for all e ∈ E and (2) ${{\tt{ORACLE}}}$ is an (α, γ)-approximation algorithm. Let D be a known upper bound on ∥θ\∥2, if we apply ${\tt{IMLinUCB}}$ with σ* = 1 and $$\label{eqn:c\_lower} c = \sqrt{d \log \left ( 1+ \frac{n \|{\mathcal{E}}\|}{d}\right) + 2 \log \left ( n(L+1-K) \right)} + D,$$ then we have $$\begin{aligned} R^{\alpha \gamma}(n) \leq & \, \frac{2c {C\_\}}{\alpha \gamma} \sqrt{ d n \|{\mathcal{E}}\| \log\_2 \left( 1+ \frac{n \|{\mathcal{E}}\|}{d}\right)} + 1 = \, {\widetilde}{{\mathcal{O}}} \left( d {C\_\}\sqrt{\|{\mathcal{E}}\| n} /(\alpha \gamma) \right) \label{bound:general} \\ \leq & \, {\widetilde}{{\mathcal{O}}} \left( d (L-K) \|{\mathcal{E}}\| \sqrt{n} /(\alpha \gamma) \right). \label{bound:size}\end{aligned}$$ Moreover, if the feature matrix ${{\bf X}}={{\bf I}}\in \Re^{\|{\mathcal{E}}\| \times \|{\mathcal{E}}\|}$ (i.e., the tabular case), we have $$\begin{aligned} R^{\alpha \gamma}(n) \leq & \, \frac{2c {C\_\}}{\alpha \gamma} \sqrt{ n \|{\mathcal{E}}\| \log\_2 \left( 1+ n \right)} + 1 = \, {\widetilde}{{\mathcal{O}}} \left( \|{\mathcal{E}}\| {C\_\}\sqrt{n} /(\alpha \gamma) \right) \label{bound:tabular} \\ \leq & \, {\widetilde}{{\mathcal{O}}} \left( (L-K) \|{\mathcal{E}}\|^{\frac{3}{2}} \sqrt{n} /(\alpha \gamma) \right). \label{bound:tabular\_size} \end{aligned}$$ Please refer to Appendix [sec:proofmain] for the proof of Theorem [thm:main], that we outline in Section [sec:proofsketch]. We now briefly comment on the regret bounds in Theorem [thm:main]. Topology-dependent bounds: Since C\* is topology-dependent, the regret bounds in Equations [bound:general] and [bound:tabular] are also topology-dependent. Table [table:topology] summarizes the regret bounds for each topology[4](#fn4) discussed in Section [sec:complexity]. Since the regret bounds in Table [table:topology] are the worst-case regret bounds for a given topology, more general topologies have larger regret bounds. For instance, the regret bounds for tree are larger than their counterparts for star and ray, since star and ray are special trees. The grid and tree can also be viewed as special complete graphs by setting ${\overline}{w}(e)=0$ for some e ∈ E, hence complete graph has larger regret bounds. Again, in practice we expect C\* to be far smaller due to activation probabilities. [t] \|c\|c\|c\|c\| topology & CG (worst-case C\) & Rαγ(n) for general ${{\bf X}}$ & Rαγ(n) for ${{\bf X}}={{\bf I}}$ bar graph & ${\mathcal{O}}(\sqrt{K})$ & ${\widetilde}{{\mathcal{O}}} \left( d K \sqrt{n} / (\alpha \gamma) \right)$ & ${\widetilde}{{\mathcal{O}}} \left( L \sqrt{K n} / (\alpha \gamma) \right)$ star graph &${\mathcal{O}}( L \sqrt{K})$ & $ {\widetilde}{{\mathcal{O}}} \left( d L^{\frac{3}{2}} \sqrt{Kn} / (\alpha \gamma) \right)$ & $ {\widetilde}{{\mathcal{O}}} \left( L^{2} \sqrt{Kn} / (\alpha \gamma) \right)$ ray graph & ${\mathcal{O}}( L^{\frac{5}{4}} \sqrt{K})$ & ${\widetilde}{{\mathcal{O}}} \left( d L^{\frac{7}{4}} \sqrt{Kn} / (\alpha \gamma) \right)$ & $ {\widetilde}{{\mathcal{O}}} \left( L^{\frac{9}{4}} \sqrt{Kn} / (\alpha \gamma) \right)$ tree graph &${\mathcal{O}}(L^{\frac{3}{2}})$ & ${\widetilde}{{\mathcal{O}}} \left( d L^{2} \sqrt{n} / (\alpha \gamma) \right)$ & ${\widetilde}{{\mathcal{O}}} \left( L^{\frac{5}{2}} \sqrt{n} / (\alpha \gamma) \right)$ grid graph & ${\mathcal{O}}(L^{\frac{3}{2}})$ & ${\widetilde}{{\mathcal{O}}} \left( d L^{2} \sqrt{n} / (\alpha \gamma) \right)$ & ${\widetilde}{{\mathcal{O}}} \left( L^{\frac{5}{2}} \sqrt{n} / (\alpha \gamma) \right)$ complete graph &O(L2) &${\widetilde}{{\mathcal{O}}} \left( d L^3 \sqrt{n} / (\alpha \gamma) \right) $ & ${\widetilde}{{\mathcal{O}}} \left( L^4 \sqrt{n} / (\alpha \gamma) \right) $ [table:topology] Tighter bounds in tabular case and under exact oracle:* Notice that for the tabular case with feature matrix ${{\bf X}}={{\bf I}}$ and d = ∣E∣, ${\widetilde}{{\mathcal{O}}} ( \sqrt{\|{\mathcal{E}}\|} )$ tighter regret bounds are obtained in Equations [bound:tabular] and [bound:tabularsize]. Also notice that the ${\widetilde}{{\mathcal{O}}}(1/(\alpha \gamma))$ factor is due to the fact that ${{\tt{ORACLE}}}$ is an (α, γ)-approximation oracle. If ${{\tt{ORACLE}}}$ solves the IM problem exactly (i.e., α = γ = 1), then Rαγ(n) = R(n). Tightness of our regret bounds: First, note that our regret bound in the bar case with K = 1 matches the regret bound of the classic ${\tt LinUCB}$ algorithm. Specifically, with perfect linear generalization, this case is equivalent to a linear bandit problem with L arms and feature dimension d. From Table [table:topology], our regret bound in this case is ${\widetilde}{{\mathcal{O}}} \left( d \sqrt{ n} \right)$, which matches the known regret bound of ${\tt LinUCB}$ that can be obtained by the technique of . Second, we briefly discuss the tightness of the regret bound in Equation [bound:size] for a general graph with L nodes and ∣E∣ edges. Note that the ${\widetilde}{{\mathcal{O}}}(\sqrt{n})$-dependence on time is near-optimal, and the ${\widetilde}{{\mathcal{O}}}(d)$-dependence on feature dimension is standard in linear bandits, since ${\widetilde}{{\mathcal{O}}}(\sqrt{d})$ results are only known for impractical algorithms. The ${\widetilde}{{\mathcal{O}}}(L-K)$ factor is due to the fact that the reward in this problem is from K to L, rather than from 0 to 1. To explain the ${\widetilde}{{\mathcal{O}}}(\|{\mathcal{E}}\|)$ factor in this bound, notice that one ${\widetilde}{{\mathcal{O}}}(\sqrt{\|{\mathcal{E}}\|})$ factor is due to the fact that at most ${\widetilde}{{\mathcal{O}}}(\|{\mathcal{E}}\|)$ edges might be observed at each round (see Theorem [theorem:graph]), and is intrinsic to the problem similarly to combinatorial semi-bandits ; another ${\widetilde}{{\mathcal{O}}}(\sqrt{\|{\mathcal{E}}\|})$ factor is due to linear generalization (see Lemma [lemma:worst]) and might be removed by better analysis. We conjecture that our $ {\widetilde}{{\mathcal{O}}} \left( d (L-K) \|{\mathcal{E}}\| \sqrt{n} /(\alpha \gamma) \right)$ regret bound in this case is at most ${\widetilde}{{\mathcal{O}}} (\sqrt{\|{\mathcal{E}}\| d} )$ away from being tight. Proof sketch ------------ We now outline the proof of Theorem [thm:main]. For each round t ≤ n, we define the favorable event $\xi\_{t-1}= \{ \|x\_{e}{^\mathsf{\scriptscriptstyle T}}({\overline}{\theta}\_{\tau-1} - \theta^\)\| \leq c \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{\tau-1}^{-1} x\_{e}}, \, \forall e \in {\mathcal{E}}, \, \forall \tau \leq t \}$, and the unfavorable event ${\overline}{\xi}\_{t-1}$ as the complement of ξ*t − 1. If we decompose E[Rtαγ], the (αγ)-scaled expected regret at round t, over events ξt − 1 and ${\overline}{\xi}\_{t-1}$, and bound Rtαγ on event ${\overline}{\xi}\_{t-1}$ using the naïve bound Rtαγ ≤ L − K, then, $$\begin{aligned} {\mathbb{E}}[R^{\alpha \gamma}\_t] \leq \, {\mathbb{P}}\left ( \xi\_{t-1} \right) {\mathbb{E}}\left[ R^{\alpha \gamma}\_t \middle \| \xi\_{t-1} \right]+ \, {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K]. \nonumber\end{aligned}$$ By choosing c as specified by Equation [eqn:clower], we have ${\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K] < 1/n$ (see Lemma [lemma:concentration] in the appendix). On the other hand, notice that by definition of ξt − 1, ${\overline}{w}(e) \leq U\_t (e)$, ∀e ∈ E under event ξt − 1. Using the monotonicity of f in the probability weight, and the fact that ${{\tt{ORACLE}}}$ is an (α, γ)-approximation algorithm, we have $$\begin{aligned} {\mathbb{E}}\left[ R^{\alpha \gamma}\_t \middle \| \xi\_{t-1} \right] \leq {\mathbb{E}}\left[ f({\mathcal{S}}\_t, U\_t)- f({\mathcal{S}}\_t, {\overline}{w}) \middle \| \xi\_{t-1} \right] /(\alpha \gamma). \nonumber\end{aligned}$$ The next observation is that, from the linearity of expectation, the gap $f({\mathcal{S}}\_t, U\_t)- f({\mathcal{S}}\_t, {\overline}{w})$ decomposes over nodes v ∈ V \ St. Specifically, for any source node set S ⊆ V, any probability weight function w : E → [0, 1], and any node v ∈ V, we define f(S, w, v) as the probability that node v is influenced if the source node set is S and the probability weight is w. Hence, we have $$f({\mathcal{S}}\_t, U\_t)- f({\mathcal{S}}\_t, {\overline}{w}) = \textstyle \sum\_{v \in {\mathcal{V}}\setminus {\mathcal{S}}\_t} \left[ f({\mathcal{S}}\_t, U\_t, v)- f({\mathcal{S}}\_t, {\overline}{w}, v)\right] .$$ In the appendix, we show that under any weight function, the diffusion process from the source node set St to the target node v can be modeled as a Markov chain. Hence, weight function Ut and ${\overline}{w}$ give us two Markov chains with the same state space but different transition probabilities. $f(S\_t, U\_t, v) - f(S\_t, {\overline}{w}, v)$ can be recursively bounded based on the state diagram of the Markov chain under weight function ${\overline}{w}$. With some algebra, Theorem [theorem:graph] in Appendix [sec:proofmain] bounds $f({\mathcal{S}}\_t, U\_t, v) - f({\mathcal{S}}\_t, {\overline}{w}, v)$ by the edge-level gap $ U\_t (e) -{\overline}{w} (e)$ on the observed relevant edges for node v, $$\begin{aligned} \label{eqn:outline\_1} f({\mathcal{S}}\_t, U\_t, v) - f({\mathcal{S}}\_t, {\overline}{w}, v) \leq \textstyle \sum\_{e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}} {\mathbb{E}}\left[ \mathbf{1} \left \{ O\_{t}(e) \right \} \left[ U\_t (e) -{\overline}{w} (e) \right] \middle \| {\mathcal{H}}\_{t-1}, {\mathcal{S}}\_t \right],\end{aligned}$$ for any t, any ``history" (past observations) Ht − 1 and St such that ξt − 1 holds, and any v ∈ V \ St, where ESt, v is the set of edges relevant to v and Ot(e) is the event that edge e is observed at round t. Based on Equation [eqn:outline1], we can prove Theorem [thm:main] using the standard linear-bandit techniques (see Appendix [sec:proofmain]). Experiments =========== In this section, we present a synthetic experiment in order to empirically validate our upper bounds on the regret. Next, we evaluate our algorithm on a real-world Facebook subgraph. Stars and rays -------------- [b]0.6 [fig:growth] [b]0.35 [fig:FB-regret] [fig:experiments] In the first experiment, we evaluate ${\tt{IMLinUCB}}$ on undirected stars and rays (Figure [fig:graphs]) and validate that the regret grows with the number of nodes L and the maximum observed relevance C\* as shown in Table [table:topology]. We focus on the tabular case (${{\bf X}}={{\bf I}}$) with K = ∣S∣ = 1, where the IM problem can be solved exactly. We vary the number of nodes L; and edge weight ${\overline}{w}(e)=\omega$, which is the same for all edges e. We run ${\tt{IMLinUCB}}$ for n = 104 steps and verify that it converges to the optimal solution in each experiment. We report the n-step regret of ${\tt{IMLinUCB}}$ for 8 ≤ L ≤ 32 in Figure [fig:growth]. Recall that from Table [table:topology], $R(n) = {\widetilde}{{\mathcal{O}}}(L^2)$ for star and $R(n) = {\widetilde}{{\mathcal{O}}}(L^\frac{9}{4})$ for ray. We numerically estimate the growth of regret in L, the exponent of L, in the log-log space of L and regret. In particular, since log(f(L)) = plog(L) + log(c) for any f(L) = cLp and c > 0, both p and log(c) can be estimated by linear regression in the new space. For star graphs with ω = 0.8 and ω = 0.7, our estimated growth are respectively O(L2.040) and O(L2.056), which are close to the expected ${\widetilde}{{\mathcal{O}}}(L^2)$. For ray graphs with ω = 0.8 and ω = 0.7, our estimated growth are respectively O(L2.488) and O(L2.467), which are again close to the expected ${\widetilde}{{\mathcal{O}}}(L^\frac{9}{4})$. This shows that maximum observed relevance C\* proposed in Section [sec:complexity] is a reasonable complexity metric for these two topologies. Subgraph of Facebook network ---------------------------- In the second experiment, we demonstrate the potential performance gain of ${\tt{IMLinUCB}}$ in real-world influence maximization semi-bandit problems by exploiting linear generalization across edges. Specifically, we compare ${\tt{IMLinUCB}}$ with ${{\tt{CUCB}}}$ in a subgraph of Facebook network from . The subgraph has L = \|V\| = 327 nodes and \|E\| = 5038 directed edges. Since the true probability weight function ${\overline}{w}$ is not available, we independently sample ${\overline}{w}(e)$’s from the uniform distribution U(0, 0.1) and treat them as ground-truth. Note that this range of probabilities is guided by empirical evidence in . We set n = 5000 and K = 10 in this experiment. For ${\tt{IMLinUCB}}$, we choose d = 10 and generate edge feature xe’s as follows: we first use ${\tt node2vec}$ algorithm to generate a node feature in ℜd for each node v ∈ V; then for each edge e, we generate xe as the element-wise product of node features of the two nodes connected to e. Note that the linear generalization in this experiment is imperfect in the sense that $\min\_{\theta \in \Re^d} \max\_{e \in {\mathcal{E}}} \|{\overline}{w}(e) - x\_e^T \theta\| >0$. For both ${{\tt{CUCB}}}$ and ${\tt{IMLinUCB}}$, we choose ${{\tt{ORACLE}}}$ as the state-of-the-art offline IM algorithm proposed in . To compute the cumulative regret, we compare against a fixed seed set S\* obtained by using the true ${\overline}{w}$ as input to the oracle proposed in . We average the empirical cumulative regret over 10 independent runs, and plot the results in Figure [fig:FB-regret]. The experimental results show that compared with ${{\tt{CUCB}}}$, ${\tt{IMLinUCB}}$ can significantly reduce the cumulative regret by exploiting linear generalization across ${\overline}{w}(e)$’s. Related Work ============ There exist prior results on IM semi-bandits . First, Lei et al. gave algorithms for the same feedback model as ours. The algorithms are not analyzed and cannot solve large-scale problems because they estimate each edge weight independently. Second, our setting is a special case of stochastic combinatorial semi-bandit with a submodular reward function and stochastically observed edges . Their work is the closest related work. Their gap-dependent and gap-free bounds are both problematic because they depend on the reciprocal of the minimum observation probability p\* of an edge: Consider a line graph with ∣E∣ edges where all edge weights are 0.5. Then 1/p\* is 2∣E∣ − 1. On the other hand, our derived regret bounds in Theorem [thm:main] are polynomial in all quantities of interest. A very recent result of Wang and Chen removes the 1/p\* factor in for the tabular case and presents a worst-case bound of ${\widetilde}{{\mathcal{O}}}(L \|{\mathcal{E}}\| \sqrt{n})$, which in the tabular complete graph case improves over our result by ${\widetilde}{{\mathcal{O}}}(L)$. On the other hand, their analysis does not give structural guarantees that we provide with maximum observed relevance C\* obtaining potentially much better results for the case in hand and giving insights for the complexity of IM bandits. Moreover, both Chen et al. and Wang and Chen do not consider generalization models across edges or nodes, and therefore their proposed algorithms are unlikely to be practical for real-world social networks. In contrast, our proposed algorithm scales to large problems by exploiting linear generalization across edges. IM bandits for different influence models and settings: There exist a number of extensions and related results for IM bandits. We only mention the most related ones (see for a recent survey). Vaswani et al. proposed a learning algorithm for a different and more challenging feedback model, where the learning agent observes influenced nodes but not the edges, but they do not give any guarantees. Carpentier and Valko give a minimax optimal algorithm for IM bandits but only consider a local model of influence with a single source and a cascade of influences never happens. In related networked bandits , the learner chooses a node and its reward is the sum of the rewards of the chosen node and its neighborhood. The problem gets more challenging when we allow the influence probabilities to change, when we allow the seed set to be chosen adaptively, or when we consider a continuous model. Furthermore, Sigla et al. treats the IM setting with an additional observability constraints, where we face a restriction on which nodes we can choose at each round. This setting is also related to the volatile multi-armed bandits where the set of possible arms changes. Vaswani et al. proposed a diffusion-independent algorithm for IM semi-bandits with a wide range of diffusion models, based on the maximum-reachability approximation. Despite its wide applicability, the maximum reachability approximation introduces an additional approximation factor to the scaled regret bounds. As they have discussed, this approximation factor can be large in some cases. Lagrée et al. treat a persistent extension of IM bandits when some nodes become persistent over the rounds and no longer yield rewards. This work is also a generalization and extension of recent work on cascading bandits, since cascading bandits can be viewed as variants of online influence maximization problems with special topologies (chains). #### Acknowledgements The research presented was supported by French Ministry of Higher Education and Research, Nord-Pas-de-Calais Regional Council, Inria and Univertät Potsdam associated-team north-european project Allocate, and French National Research Agency projects ExTra-Learn (n.ANR-14-CE24-0010-01) and BoB (n.ANR-16-CE23-0003). We would also like to thank Dr.Wei Chen and Mr.Qinshi Wang for pointing out a mistake in an earlier version of this paper. [1]begintheorem#1endtheorem Proof of Theorem [thm:main] =========================== In the appendix, we prove a slightly stronger version of Theorem [thm:main], which also uses another complexity metric E\* defined as follows: Assume that the graph G = (V, E) includes m disconnected subgraphs G1 = (V1, E1), G2 = (V2, E2), …, Gm = (Vm, Em), which are in the descending order based on the number of nodes ∣Ei∣’s. We define E\* as the number of edges in the first min{m, K} subgraphs: $$\begin{aligned} \label{eqn:mre\_def} {E\_\}= \sum\_{i=1}^{\min \{m, K \}} \|{\mathcal{E}}\_i\|.\end{aligned}$$ Note that by definition, E\ ≤ ∣E∣. Based on E\, we have the following slightly stronger version of Theorem [thm:main]. [thm:stronger] Assume that (1) ${\overline}{w}(e) = x\_{e}{^\mathsf{\scriptscriptstyle T}}\theta^\$ for all e ∈ E and (2) ${{\tt{ORACLE}}}$ is an (α, γ)-approximation algorithm. Let D be a known upper bound on ∥θ\∥2. If we apply ${\tt{IMLinUCB}}$ with σ* = 1 and $$\label{eqn:stronger:c\_lower} c \geq \sqrt{d \log \left ( 1+ \frac{n {E\_\}}{d}\right) + 2 \log \left ( n(L+1-K) \right)} + D,$$ then we have $$\begin{aligned} \label{bound:stronger:general} R^{\alpha \gamma}(n) \leq \, \frac{2c {C\_\}}{\alpha \gamma} \sqrt{ d n {E\_\}\log\_2 \left( 1+ \frac{n {E\_\}}{d}\right)} + 1 = \, {\widetilde}{{\mathcal{O}}} \left( d {C\_\}\sqrt{{E\_\}n}/(\alpha \gamma) \right).\end{aligned}$$ Moreover, if the feature matrix is of the form X = I ∈ ℜ∣E∣ × ∣E∣ (i.e., the tabular case), we have $$\begin{aligned} \label{bound:stronger:tabular} R^{\alpha \gamma}(n) \leq \, \frac{2c {C\_\}}{\alpha \gamma} \sqrt{ n \|{\mathcal{E}}\| \log\_2 \left( 1+ n \right)} + 1 = \, {\widetilde}{{\mathcal{O}}} \left( \|{\mathcal{E}}\| {C\_\}\sqrt{n} /(\alpha \gamma) \right).\end{aligned}$$ Since E\* ≤ ∣E∣, Theorem [thm:stronger] implies Theorem [thm:main]. We prove Theorem [thm:stronger] in the remainder of this section. We now define some notation to simplify the exposition throughout this section. [def:fv] For any source node set S ⊆ V, any probability weight function w : E → [0, 1], and any node v ∈ V, we define f(S, w, v) as the probability that node v is influenced if the source node set is S and the probability weight function is w. Notice that by definition, f(S, w) = ∑v ∈ Vf(S, w, v) always holds. Moreover, if v ∈ S, then f(S, w, v) = 1 for any w by the definition of the influence model. [def:Ot] For any round t and any directed edge e ∈ E, we define event Ot(e) = {edge $e$ is observed at round $t$}. Note that by definition, an directed edge e is observed if and only if its start node is influenced and observed does not necessarily mean that the edge is active. Proof of Theorem [thm:stronger] ------------------------------- Let Ht be the history (σ-algebra) of past observations and actions by the end of round t. By the definition of Rtαγ, we have $$\begin{aligned} {\mathbb{E}}\left[ R\_t^{\alpha \gamma} \middle \| {\mathcal{H}}\_{t-1} \right]=& f({{\mathcal{S}}^{\mathrm{opt}}}, {\overline}{w})- \frac{1}{\alpha \gamma} {\mathbb{E}}\left[ f( {\mathcal{S}}\_t, {\overline}{w}) \middle \| \, {\mathcal{H}}\_{t-1} \right],\end{aligned}$$ where the expectation is over the possible randomness of St, since ${{\tt{ORACLE}}}$ might be a randomized algorithm. Notice that the randomness coming from the edge activation is already taken care of in the definition of f. For any t ≤ n, we define event ξt − 1 as $$\begin{aligned} \label{eqn:F\_t} \xi\_{t-1}= \left \{ \|x\_{e}{^\mathsf{\scriptscriptstyle T}}({\overline}{\theta}\_{\tau-1} - \theta^\)\| \leq c \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{\tau-1}^{-1} x\_{e}}, \, \forall e \in {\mathcal{E}}, \, \forall \tau \leq t \right \},\end{aligned}$$ and ${\overline}{\xi}\_{t-1}$ as the complement of ξ*t − 1. Notice that ξt − 1 is Ht − 1-measurable. Hence we have $${\mathbb{E}}[R^{\alpha \gamma}\_t] \leq {\mathbb{P}}\left ( \xi\_{t-1} \right) {\mathbb{E}}\left[ f({{\mathcal{S}}^{\mathrm{opt}}}, {\overline}{w})- f({\mathcal{S}}\_t, {\overline}{w})/(\alpha \gamma) \middle \| \xi\_{t-1} \right] + {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K].$$ Notice that under event ξt − 1, ${\overline}{w}(e) \leq U\_t (e)$, ∀e ∈ E, for all t ≤ n, thus we have $$f({{\mathcal{S}}^{\mathrm{opt}}}, {\overline}{w}) \leq f({{\mathcal{S}}^{\mathrm{opt}}}, U\_t ) \leq \max\_{{\mathcal{S}}: \, \|{\mathcal{S}}\|=K} f({\mathcal{S}}, U\_t ) \leq \frac{1}{\alpha \gamma} {\mathbb{E}}\left[ f( {\mathcal{S}}\_t, U\_t ) \middle \| \, {\mathcal{H}}\_{t-1} \right],$$ where the first inequality follows from the monotonicity of f in the probability weight, and the last inequality follows from the fact that ${{\tt{ORACLE}}}$ is an (α, γ)-approximation algorithm. Thus, we have $$\begin{aligned} {\mathbb{E}}[R^{\alpha \gamma}\_t] \leq \frac{{\mathbb{P}}\left ( \xi\_{t-1} \right)}{\alpha \gamma} {\mathbb{E}}\left[ f({\mathcal{S}}\_t, U\_t)- f({\mathcal{S}}\_t, {\overline}{w}) \middle \| \xi\_{t-1} \right] + {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K].\end{aligned}$$ Notice that based on Definition [def:fv], we have $$f({\mathcal{S}}\_t, U\_t)- f({\mathcal{S}}\_t, {\overline}{w}) = \sum\_{v \in {\mathcal{V}}\setminus {\mathcal{S}}\_t} \left[ f({\mathcal{S}}\_t, U\_t, v)- f({\mathcal{S}}\_t, {\overline}{w}, v)\right].$$ Recall that for a given graph G = (V, E) and a given source node set S ⊆ V, we say an edge e ∈ E and a node v ∈ V \ S are relevant if there exists a path p from a source node s ∈ S to v such that (1) e ∈ p and (2) p does not contain another source node other than s. We use ES, v ⊆ E to denote the set of edges relevant to node v under the source node set S, and use VS, v ⊆ V to denote the set of nodes connected to at least one edge in ES, v. Notice that ${\mathcal{G}}\_{{\mathcal{S}}, v} \stackrel{\Delta}{=} \left( {\mathcal{V}}\_{{\mathcal{S}}, v},{\mathcal{E}}\_{{\mathcal{S}}, v} \right)$ is a subgraph of G, and we refer to it as the relevant subgraph of node v under the source node set S. Based on the notion of relevant subgraph, we have the following theorem, which bounds $f({\mathcal{S}}\_t, U\_t, v) - f({\mathcal{S}}\_t, {\overline}{w}, v)$ by edge-level gaps $ U\_t (e) -{\overline}{w} (e)$ on the observed edges in the relevant subgraph GSt, v for node v; [theorem:graph] For any t, any history Ht − 1 and St such that ξt − 1 holds, and any v ∈ V \ St, we have $$f({\mathcal{S}}\_t, U\_t, v) - f({\mathcal{S}}\_t, {\overline}{w}, v) \leq \sum\_{e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}} {\mathbb{E}}\left[ \mathbf{1} \left \{ O\_{t}(e) \right \} \left[ U\_t (e) -{\overline}{w} (e) \right] \middle \| {\mathcal{H}}\_{t-1}, {\mathcal{S}}\_t \right],$$ where ESt, v is the edge set of the relevant subgraph GSt, v. Please refer to Section [sec:prooflemma1] for the proof of Theorem [theorem:graph]. Notice that under favorable event ξt − 1, we have $U\_t (e) - {\overline}{w}(e) \leq 2c \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}}$ for all e ∈ E. Therefore, we have $$\begin{aligned} {\mathbb{E}}[R^{\alpha \gamma}\_t] \leq & \, \frac{2c}{\alpha \gamma} {\mathbb{P}}\left ( \xi\_{t-1} \right) {\mathbb{E}}\left[ \sum\_{v \in {\mathcal{V}}\setminus {\mathcal{S}}\_t} \sum\_{e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}} \mathbf{1} \{ O\_{t}(e) \} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \middle \| \xi\_{t-1} \right] + {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K] \nonumber \\ \leq & \, \frac{2c}{\alpha \gamma} {\mathbb{E}}\left[ \sum\_{v \in {\mathcal{V}}\setminus {\mathcal{S}}\_t} \sum\_{e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}} \mathbf{1} \{ O\_{t}(e)\} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \right] + {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K] \nonumber \\ =& \, \frac{2c}{\alpha \gamma} {\mathbb{E}}\left[ \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e)\} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \sum\_{v \in {\mathcal{V}}\setminus {\mathcal{S}}\_t} \mathbf{1} \left \{ e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v} \right \} \right] + {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K] \nonumber \\ = & \, \frac{2c}{\alpha \gamma} {\mathbb{E}}\left[ \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N\_{{\mathcal{S}}\_t, e} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \right] + {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) [L-K],\end{aligned}$$ where NSt, e = ∑v ∈ V \ S1{e ∈ ESt, v} is defined in Equation [eqn:N]. Thus we have $$\begin{aligned} \label{eq:regret\_after\_ubc} R^{\alpha \gamma}(n) \leq \frac{2c}{\alpha \gamma} {\mathbb{E}}\left[ \sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N\_{{\mathcal{S}}\_t, e} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \right] + [L-K]\sum\_{t=1}^n {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right).\end{aligned}$$ In the following lemma, we give a worst-case bound on $\sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N\_{{\mathcal{S}}\_t, e} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}}$. [lemma:worst] For any round t = 1, 2, …, n, we have $$\sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N\_{{\mathcal{S}}\_t, e} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \leq \sqrt{\left(\sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N^2\_{{\mathcal{S}}\_t, e} \right) \frac{d {E\_\}\log \left( 1+ \frac{n {E\_\}}{d \sigma^2}\right)}{\log \left(1 + \frac{1}{\sigma^2} \right)}}\cdot$$ Moreover, if X = I ∈ ℜ∣E∣ × ∣E∣, then we have $$\sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N\_{{\mathcal{S}}\_t, e} \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}} \leq \sqrt{ \left(\sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N^2\_{{\mathcal{S}}\_t, e} \right) \frac{\|{\mathcal{E}}\| \log \left( 1+ \frac{n}{\sigma^2}\right)}{\log \left(1 + \frac{1}{\sigma^2} \right)}}\cdot$$ Please refer to Section [sec:prooflemma2] for the proof of Lemma [lemma:worst]. Finally, notice that for any t, $${\mathbb{E}}\left[ \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N^2\_{{\mathcal{S}}\_t, e} \middle \| {\mathcal{S}}\_t \right] = \sum\_{e \in {\mathcal{E}}} N^2\_{{\mathcal{S}}\_t, e} {\mathbb{E}}\left[ \mathbf{1} \{ O\_{t}(e) \} \middle \| {\mathcal{S}}\_t \right] = \sum\_{e \in {\mathcal{E}}} N^2\_{{\mathcal{S}}\_t, e} P\_{{\mathcal{S}}\_t, e} \leq {C\_\}^2,$$ thus taking the expectation over the possibly randomized oracle and Jensen’s inequality, we get $$\begin{aligned} {\mathbb{E}}\left[ \sqrt{ \sum\_{t=1}^n \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N^2\_{{\mathcal{S}}\_t, e} }\right] \leq \, \sqrt{ \sum\_{t=1}^n {\mathbb{E}}\left[ \sum\_{e \in {\mathcal{E}}} \mathbf{1} \{ O\_{t}(e) \} N^2\_{{\mathcal{S}}\_t, e} \right] } \leq \, \sqrt{ \sum\_{t=1}^n {C\_\}^2 } = {C\_\}\sqrt{n}.\end{aligned}$$ Combining the above with Lemma [lemma:worst] and, we obtain $$\begin{aligned} R^{\alpha \gamma}(n) \leq \frac{2c {C\_\}}{\alpha \gamma} \sqrt{ \frac{d n {E\_\}\log \left( 1+ \frac{n {E\_\}}{d \sigma^2}\right)}{\log \left(1 + \frac{1}{\sigma^2} \right)}} + [L-K]\sum\_{t=1}^n {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right).\end{aligned}$$ For the special case when X = I, we have $$\begin{aligned} R^{\alpha \gamma}(n) \leq \frac{2c {C\_\}}{\alpha \gamma} \sqrt{ \frac{ n \|{\mathcal{E}}\| \log \left( 1+ \frac{n}{\sigma^2}\right)}{\log \left(1 + \frac{1}{\sigma^2} \right)}} + [L-K]\sum\_{t=1}^n {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right).\end{aligned}$$ Finally, we need to bound the failure probability of upper confidence bound being wrong $\sum\_{t=1}^n {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right)$. We prove the following bound on ${\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right)$: [lemma:concentration] For any t* = 1, 2, …, n, any σ > 0, any δ ∈ (0, 1), and any $$c \geq \frac{1}{\sigma} \sqrt{d \log \left ( 1+ \frac{n {E\_\}}{d \sigma^2}\right) + 2 \log \left (\frac{1}{\delta} \right)} + \\|\theta^\ \\|\_2,$$ we have ${\mathbb{P}}\left( {\overline}{\xi}\_{t-1} \right) \leq \delta$. Please refer to Section [sec:prooflemma3] for the proof of Lemma [lemma:concentration]. From Lemma [lemma:concentration], for a known upper bound D on ∥θ\∥2, if we choose σ* = 1 and $c \geq \sqrt{d \log \left ( 1+ \frac{n {E\_\}}{d}\right) + 2 \log \left ( n(L+1-K) \right)} + D $, which corresponds to $\delta=\frac{1}{n(L+1-K)}$ in Lemma [lemma:concentration], then we have $$[L-K]\sum\_{t=1}^n {\mathbb{P}}\left ( {\overline}{\xi}\_{t-1} \right) <1.$$ This concludes the proof of Theorem [thm:stronger]. Proof of Theorem [theorem:graph] -------------------------------- Recall that we use GSt, v* = (VSt, v, ESt, v) to denote the relevant subgraph of node v under the source node set St. Since Theorem [theorem:graph] focuses on the influence from St to v, and by definition all the paths from St to v are in GSt, v, thus, it is sufficient to restrict to GSt, v and ignore other parts of G in this analysis. We start by defining some useful notations. Influence Probability with Removed Nodes: Recall that for any weight function w : E → [0, 1], any source node set S ⊂ V and any target node v ∈ V, f(S, w, v) is the probability that S will influence v under weight w (see Definition [def:fv]). We now define a similar notation for the influence probability with removed nodes. Specifically, for any disjoint node set V1, V2 ⊆ VSt, v ⊆ V, we define h(V1, V2, w) as follows: * First, we remove nodes V2, as well as all edges connected to/from V2, from GSt, v, and obtain a new graph Gʹ. * h(V1, V2, w) is the probability that V1 will influence the target node v in graph Gʹ under the weight (activation probability) w(e) for all e ∈ Gʹ. Obviously, a mathematically equivalent way to define h(V1, V2, w) is to define it as the probability that V1 will influence v in GSt, v under a new weight ${\widetilde}{w}$, defined as $${\widetilde}{w}(e)=\left \{ \begin{array}{ll} 0 & \text{if $e$ is from or to a node in ${\mathcal{V}}\_2$} \\ w(e) & \text{otherwise} \end{array} \right.$$ Note that by definition, f(St, w, v) = h(St, ∅, w). Also note that h(V1, V2, w) implicitly depends on v, but we omit v in this notation to simplify the exposition. Edge Set E(V1, V2): For any two disjoint node sets V1, V2 ⊆ VSt, v, we define the edge set E(V1, V2) as $${\mathcal{E}}({\mathcal{V}}\_1, {\mathcal{V}}\_2)=\left \{ e=(u\_1, u\_2): \, e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}, \, u\_1 \in {\mathcal{V}}\_1, \text{ and } u\_2 \notin {\mathcal{V}}\_2 \right \}.$$ That is, E(V1, V2) is the set of edges in GSt, v from V1 to VSt, v \ V2. Diffusion Process: Note that under any edge activation realization ${{\bf w}}(e)$, e ∈ ESt, v, on the relevant subgraph GSt, v, we define a finite-length sequence of disjoint node sets ${\mathcal{S}}^0, {\mathcal{S}}^1, \ldots, {\mathcal{S}}^{{\widetilde}{\tau}}$ as $$\begin{aligned} {\mathcal{S}}^0 \stackrel{\Delta}{=} & {\mathcal{S}}\_t \nonumber \\ {\mathcal{S}}^{\tau+1} \stackrel{\Delta}{=} & \left \{u\_2 \in {\mathcal{V}}\_{{\mathcal{S}}\_t, v}: \, u\_2 \notin \cup\_{\tau'=0}^{\tau} {\mathcal{S}}^{\tau'} \text{ and } \exists e=(u\_1, u\_2) \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}\text{ s.t. } u\_1 \in {\mathcal{S}}^{\tau} \text{ and }{{\bf w}}(e)=1 \right \}, \end{aligned}$$ $\forall \tau=0, \ldots, {\widetilde}{\tau}-1$. That is, under the realization ${{\bf w}}(e)$, e ∈ ESt, v, Sτ + 1 is the set of nodes directly activated by Sτ. Specifically, any node u2 ∈ Sτ + 1 satisfies $u\_2 \notin \bigcup\_{\tau'=0}^{\tau} {\mathcal{S}}^{\tau'}$ (i.e. it was not activated before), and there exists an activated edge e from Sτ to u2 (i.e. it is activated by some node in Sτ). We define ${\mathcal{S}}^{{\widetilde}{\tau}}$ as the first node set in the sequence s.t. either ${\mathcal{S}}^{{\widetilde}{\tau}}=\emptyset$ or $v \in {\mathcal{S}}^{{\widetilde}{\tau}}$, and assume this sequence terminates at ${\mathcal{S}}^{{\widetilde}{\tau}}$. Note that by definition, ${\widetilde}{\tau} \leq \|{\mathcal{V}}\_{{\mathcal{S}}\_t, v}\|$ always holds. We refer to each $\tau=0,1,\ldots, {\widetilde}{\tau}$ as a diffusion step in this section. To simplify the exposition, we also define $S^{0:\tau} \stackrel{\Delta}{=} \bigcup\_{\tau'=0}^{\tau} S^{\tau'}$ for all τ ≥ 0 and $S^{0:-1} \stackrel{\Delta}{=} \emptyset$. Since ${{\bf w}}$ is random, $\left( {\mathcal{S}}^{\tau} \right )\_{\tau=0}^{{\widetilde}{\tau}}$ is a stochastic process, which we refer to as the diffusion process. Note that ${\widetilde}{\tau}$ is also random; in particular, it is a stopping time. Based on the shorthand notations defined above, we have the following lemma for the diffusion process $\left( {\mathcal{S}}^{\tau} \right )\_{\tau=0}^{{\widetilde}{\tau}}$ under any weight function w: [lemma:transition] For any weight function w : E → [0, 1], any step $\tau =0, 1, \ldots, {\widetilde}{\tau}$, any Sτ and S0 : τ − 1, we have $$h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) = \left \{ \begin{array}{ll} 1 & \text{if $v \in {\mathcal{S}}^\tau$} \\ 0 & \text{if ${\mathcal{S}}^\tau = \emptyset$} \\ {\mathbb{E}}\left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \middle \| ( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1} ) \right] & \text{otherwise} \end{array} \right.,$$ where the expectation is over Sτ + 1 under weight w. Note that the tuple (Sτ, S0 : τ − 1) in the conditional expectation means that Sτ is the source node set and nodes in S0 : τ − 1 have been removed. Notice that by definition, h(Sτ, S0 : τ − 1, w) = 1 if v ∈ Sτ and h(Sτ, S0 : τ − 1, w) = 0 if Sτ = ∅. Also note that in these two cases, ${\widetilde}{\tau}=\tau$. Otherwise, we prove that $h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) = {\mathbb{E}}\left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \middle \| ({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1})\right]$. Recall that by definition, h(Sτ, S0 : τ − 1, w) is the probability that v will be influenced conditioning on $$\text{source node set ${\mathcal{S}}^\tau$ and removed node set ${\mathcal{S}}^{0:\tau-1}$,}$$ that is $$\label{eqn:lemma6:def1} h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) = {\mathbb{E}}\left[ \mathbf{1} \left( \text{$v$ is influenced} \right) \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} )\right]$$ Let ${{\bf w}}(e)$, ∀e ∈ E(Sτ, S0 : τ) be any possible realization. Now we analyze the probability that v will be influenced conditioning on $$\text{source node set ${\mathcal{S}}^\tau$, removed node set ${\mathcal{S}}^{0:\tau-1}$, and ${{\bf w}}(e)$ for all $e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})$.} \label{eqn:conditions}$$ Specifically, conditioning on Equation [eqn:conditions], we can define a new weight function wʹ as $$w'(e) = \left \{ \begin{array}{ll} {{\bf w}}(e) & \text{if $e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})$} \\ w(e) & \text{otherwise} \end{array} \right. \label{eqn:w'}$$ then h(Sτ, S0 : τ − 1, wʹ) is the probability that v will be influenced conditioning on Equation [eqn:conditions]. That is, $$\label{eqn:lemma6:def2} h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w' \right) = {\mathbb{E}}\left[ \mathbf{1} \left( \text{$v$ is influenced} \right) \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} ), {{\bf w}}(e) \, \forall e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau}) \right],$$ for any possible realization of ${{\bf w}}(e)$, ∀e ∈ E(Sτ, S0 : τ). Notice that on the lefthand of Equation [eqn:lemma6:def2], wʹ encodes the conditioning on ${{\bf w}}(e)$ for all e ∈ E(Sτ, S0 : τ) (see Equation [eqn:w’]). From here to Equation [eqn:lemma6:eqn4], we focus on an arbitrary but fixed realization of ${{\bf w}}(e)$, ∀e ∈ E(Sτ, S0 : τ) (or equivalently, an arbitrary but fixed wʹ). Based on the definition of Sτ + 1, conditioning on Equation [eqn:conditions], Sτ + 1 is deterministic and all nodes in Sτ + 1 can also be treated as source nodes. Thus, we have h(Sτ, S0 : τ − 1, wʹ) = h(Sτ ∪ Sτ + 1, S0 : τ − 1, wʹ), conditioning on Equation [eqn:conditions]. On the other hand, conditioning on Equation [eqn:conditions], we can treat any edge e ∈ E(Sτ, S0 : τ) with ${{\bf w}}(e)=0$ as having been removed. Since nodes in S0 : τ − 1 have also been removed, and $v \notin {\mathcal{S}}^{\tau}$, then if there is a path from Sτ to v, then it must go through Sτ + 1, and the last node on the path in Sτ + 1 must be after the last node on the path in Sτ (note that the path might come back to Sτ for several times). Hence, conditioning on Equation [eqn:conditions], if nodes in Sτ + 1 are also treated as source nodes, then Sτ is irrelevant for influence on v and can be removed. So we have h(Sτ, S0 : τ − 1, wʹ) = h(Sτ ∪ Sτ + 1, S0 : τ − 1, wʹ) = h(Sτ + 1, S0 : τ, w). Note that in the last equation we change the weight function back to w since edges in E(Sτ, S0 : τ) have been removed. Thus, conditioning on Equation [eqn:conditions], we have $$\begin{aligned} \label{eqn:lemma6:eqn4} h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right)=& \, h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w' \right) \nonumber \\ =& \, {\mathbb{E}}\left[ \mathbf{1} \left( \text{$v$ is influenced} \right) \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} ), {{\bf w}}(e) \, \forall e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau}) \right].\end{aligned}$$ Notice again that Equation [eqn:lemma6:eqn4] holds for any possible realization of ${{\bf w}}(e)$, ∀e ∈ E(Sτ, S0 : τ). Finally, we have $$\begin{aligned} h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) \stackrel{(a)}{=} & \, {\mathbb{E}}\left[ \mathbf{1} \left( \text{$v$ is influenced} \right) \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} )\right] \nonumber \\ \stackrel{(b)}{=}& \, {\mathbb{E}}\left[ {\mathbb{E}}\left[ \mathbf{1} \left( \text{$v$ is influenced} \right) \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} ), {{\bf w}}(e) \, \forall e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau}) \right] \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} )\right] \nonumber \\ \stackrel{(c)}{=}& \, {\mathbb{E}}\left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \middle \| ({\mathcal{S}}^\tau,{\mathcal{S}}^{0:\tau-1} )\right],\end{aligned}$$ where (a) follows from Equation [eqn:lemma6:def1], (b) follows from the tower rule, and (c) follows from Equation [eqn:lemma6:eqn4]. This concludes the proof. Consider two weight functions U, w : E → [0, 1] s.t.U(e) ≥ w(e) for all e ∈ E. The following lemma bounds the difference h(Sτ, S0 : τ − 1, U) − h(Sτ, S0 : τ − 1, w) in a recursive way. [lemma:recursivebound] For any two weight functions w, U : E → [0, 1] s.t. U(e) ≥ w(e) for all e ∈ E, any step $\tau =0, 1, \ldots, {\widetilde}{\tau}$, any Sτ and S0 : τ − 1, we have h(Sτ, S0 : τ − 1, U) − h(Sτ, S0 : τ − 1, w) = 0 if v ∈ Sτ or Sτ = ∅; and otherwise $$\begin{aligned} h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, U \right) - h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) \leq & \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U(e) - w(e) \right] \nonumber \\ + & \, {\mathbb{E}}\left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \middle \| ( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1} ) \right], \nonumber\end{aligned}$$ where the expectation is over Sτ + 1 under weight w. Recall that the tuple (Sτ, S0 : τ − 1) in the conditional expectation means that Sτ is the source node set and nodes in S0 : τ − 1 have been removed. First, note that if v ∈ Sτ or Sτ = ∅, then h(Sτ, S0 : τ − 1, U) − h(Sτ, S0 : τ − 1, w) = 0 follows directly from Lemma [lemma:transition]. Otherwise, to simplify the exposition, we overload the notation and use w(Sτ + 1) to denote the conditional probability of Sτ + 1 conditioning on (Sτ, S0 : τ − 1) under the weight function w, and similarly for U(Sτ + 1). That is $$\begin{aligned} w( {\mathcal{S}}^{\tau+1}) \stackrel{\Delta}{=}& \, \mathrm{Prob} \left[ {\mathcal{S}}^{\tau+1} \middle \| ( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1} ) ; w \right] \nonumber \\ U( {\mathcal{S}}^{\tau+1}) \stackrel{\Delta}{=}& \, \mathrm{Prob} \left[ {\mathcal{S}}^{\tau+1} \middle \| ( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1} ) ; U \right],\end{aligned}$$ where the tuple (Sτ, S0 : τ − 1) in the conditional probability means that Sτ is the source node set and nodes in S0 : τ − 1 have been removed, and w and U after the semicolon indicate the weight function. Then from Lemma [lemma:transition], we have $$\begin{aligned} h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, U \right) =& \, \sum\_{{\mathcal{S}}^{\tau+1}} U( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \nonumber \\ h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) =& \, \sum\_{{\mathcal{S}}^{\tau+1}} w( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \nonumber\end{aligned}$$ where the sum is over all possible realization of Sτ + 1. Hence we have $$\begin{aligned} & \, h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, U \right) - h \left( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1}, w \right) \nonumber \\ = & \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - w( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \right] \nonumber \\ = & \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - w( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \right] \nonumber \\ +& \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ w( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - w( {\mathcal{S}}^{\tau+1}) h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \right] \nonumber \\ = & \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \nonumber \\ +& \, \sum\_{{\mathcal{S}}^{\tau+1}} w( {\mathcal{S}}^{\tau+1}) \left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \right],\end{aligned}$$ where the sum in the above equations is also over all the possible realizations of Sτ + 1. Notice that by definition, we have $$\begin{gathered} {\mathbb{E}}\left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \middle \| ( {\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau-1} ) \right] = \\ \sum\_{{\mathcal{S}}^{\tau+1}} w( {\mathcal{S}}^{\tau+1}) \left[ h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) - h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, w \right) \right],\end{gathered}$$ where the expectation in the lefthand side is over Sτ + 1 under weight w, or equivalently, over ${{\bf w}}(e)$ for all e ∈ E(Sτ, S0 : τ) under weight w. Thus, to prove Lemma [lemma:recursivebound], it is sufficient to prove that ∑Sτ + 1[U(Sτ + 1) − w(Sτ + 1)]h(Sτ + 1, S0 : τ, U) ≤ ∑e ∈ E(Sτ, S0 : τ)[U(e) − w(e)]. Notice that $$\begin{aligned} & \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \nonumber \\ \stackrel{(a)}{\leq} & \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \mathbf{1}\left[ U( {\mathcal{S}}^{\tau+1}) \geq w( {\mathcal{S}}^{\tau+1}) \right] \nonumber \\ \stackrel{(b)}{\leq} & \, \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] \mathbf{1}\left[ U( {\mathcal{S}}^{\tau+1}) \geq w( {\mathcal{S}}^{\tau+1}) \right] \nonumber \\ \stackrel{(c)}{=} & \, \frac{1}{2} \sum\_{{\mathcal{S}}^{\tau+1}} \left \| U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right \|,\end{aligned}$$ where (a) holds since $$\begin{gathered} \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) = \\ \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \mathbf{1}\left[ U( {\mathcal{S}}^{\tau+1}) \geq w( {\mathcal{S}}^{\tau+1}) \right] \\ + \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] h \left( {\mathcal{S}}^{\tau+1}, {\mathcal{S}}^{0:\tau}, U \right) \mathbf{1}\left[ U( {\mathcal{S}}^{\tau+1}) < w( {\mathcal{S}}^{\tau+1}) \right], \nonumber\end{gathered}$$ and the second term on the righthand side is non-positive. And (b) holds since 0 ≤ h(Sτ + 1, S0 : τ, U) ≤ 1 by definition. To prove (c), we define shorthand notations $$\begin{aligned} A^+=& \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] \mathbf{1}\left[ U( {\mathcal{S}}^{\tau+1}) \geq w( {\mathcal{S}}^{\tau+1}) \right] \nonumber \\ A^-=& \sum\_{{\mathcal{S}}^{\tau+1}} \left[ U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right] \mathbf{1}\left[ U( {\mathcal{S}}^{\tau+1}) < w( {\mathcal{S}}^{\tau+1}) \right] \nonumber\end{aligned}$$ Then we have A+ + A− = ∑Sτ + 1[U(Sτ + 1) − w(Sτ + 1)] = 0, since by definition ∑Sτ + 1U(Sτ + 1) = ∑Sτ + 1w(Sτ + 1) = 1. Moreover, we also have A+ − A− = ∑Sτ + 1∣U(Sτ + 1) − w(Sτ + 1)∣. And hence $A^+ = \frac{1}{2} \sum\_{{\mathcal{S}}^{\tau+1}} \left \| U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right \| $. Thus, to prove Lemma [lemma:recursivebound], it is sufficient to prove $$\begin{aligned} \frac{1}{2} \sum\_{{\mathcal{S}}^{\tau+1}} \left \| U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right \| \leq \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U(e) - w(e) \right].\end{aligned}$$ Let ${\widetilde}{{{\bf w}}} \in \{0, 1\}^{\|{\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})\|}$ be an arbitrary edge activation realization for edges in E(Sτ, S0 : τ). Also with a little bit abuse of notation, we use $w({\widetilde}{{{\bf w}}})$ to denote the probability of ${\widetilde}{{{\bf w}}}$ under weight w. Notice that $$w({\widetilde}{{{\bf w}}}) = \prod\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} w(e)^{{\widetilde}{{{\bf w}}}(e)} \left[ 1- w(e)\right]^{1 - {\widetilde}{{{\bf w}}}(e)},$$ and $U({\widetilde}{{{\bf w}}})$ is defined similarly. Recall that by definition Sτ + 1 is a deterministic function of source node set Sτ, removed nodes S0 : τ − 1, and ${\widetilde}{{{\bf w}}}$. Hence, for any possible realized Sτ + 1, let W(Sτ + 1) denote the set of ${\widetilde}{{{\bf w}}}$’s that lead to this Sτ + 1, then we have $$U( {\mathcal{S}}^{\tau+1}) =\sum\_{{\widetilde}{{{\bf w}}} \in \mathbf{W}({\mathcal{S}}^{\tau+1})} U({\widetilde}{{{\bf w}}}) \quad \text{ and } \quad w( {\mathcal{S}}^{\tau+1}) =\sum\_{{\widetilde}{{{\bf w}}} \in \mathbf{W}({\mathcal{S}}^{\tau+1})} w({\widetilde}{{{\bf w}}})$$ Thus, we have $$\begin{aligned} \frac{1}{2} \sum\_{{\mathcal{S}}^{\tau+1}} \left \| U( {\mathcal{S}}^{\tau+1}) - w( {\mathcal{S}}^{\tau+1}) \right \| = & \, \frac{1}{2} \sum\_{{\mathcal{S}}^{\tau+1}} \left \| \sum\_{{\widetilde}{{{\bf w}}} \in \mathbf{W}({\mathcal{S}}^{\tau+1})} [ U({\widetilde}{{{\bf w}}}) - w({\widetilde}{{{\bf w}}}) ] \right \| \nonumber \\ \leq & \, \frac{1}{2} \sum\_{{\mathcal{S}}^{\tau+1}} \sum\_{{\widetilde}{{{\bf w}}} \in \mathbf{W}({\mathcal{S}}^{\tau+1})} \left \| U({\widetilde}{{{\bf w}}}) - w({\widetilde}{{{\bf w}}}) \right \| \nonumber \\ = & \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}} } \left \| U({\widetilde}{{{\bf w}}}) - w({\widetilde}{{{\bf w}}}) \right \|\end{aligned}$$ Finally, we prove that $$\frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}} } \left \| U({\widetilde}{{{\bf w}}}) - w({\widetilde}{{{\bf w}}}) \right \| \leq \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U(e) - w(e) \right]$$ by mathematical induction. Without loss of generality, we order the edges in E(Sτ, S0 : τ) as 1, 2, …, ∣E(Sτ, S0 : τ)∣. For any k = 1, …, ∣E(Sτ, S0 : τ)∣, we use ${\widetilde}{{{\bf w}}}\_k \in \{0, 1\}^{k}$ to denote an arbitrary edge activation realization for edges 1, …, k. Then, we prove $$\label{eqn:induction} \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_k} \left \| U({\widetilde}{{{\bf w}}}\_k) - w({\widetilde}{{{\bf w}}}\_k) \right \| \leq \sum\_{e =1}^k \left[U(e) - w(e) \right]$$ for all k = 1, …, ∣E(Sτ, S0 : τ)∣ by mathematical induction. Notice that when k = 1, we have $$\frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_1} \left \| U({\widetilde}{{{\bf w}}}\_1) - w({\widetilde}{{{\bf w}}}\_1) \right \| =\frac{1}{2} \left[ \|U(1) - w(1)\| +\|(1-U(1)) - (1-w(1))\| \right] = U(1)-w(1).$$ Now assume that the induction hypothesis holds for k, we prove that it also holds for k + 1. Note that $$\begin{aligned} \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_{k+1}} \left \| U({\widetilde}{{{\bf w}}}\_{k+1}) - w({\widetilde}{{{\bf w}}}\_{k+1}) \right \| = & \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_{k}} \left[ \left \| U({\widetilde}{{{\bf w}}}\_{k}) U(k+1) - w({\widetilde}{{{\bf w}}}\_{k}) w(k+1) \right \| \right. \nonumber \\ +& \left. \left \| U({\widetilde}{{{\bf w}}}\_{k}) (1-U(k+1)) - w({\widetilde}{{{\bf w}}}\_{k}) (1-w(k+1)) \right \| \right] \nonumber \\ \stackrel{(a)}{\leq} & \, \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_{k}} [ \left \| U({\widetilde}{{{\bf w}}}\_{k}) U(k+1) - w({\widetilde}{{{\bf w}}}\_{k}) U(k+1) \right \| \nonumber \\ +& \, \left \| w({\widetilde}{{{\bf w}}}\_{k}) U(k+1) - w({\widetilde}{{{\bf w}}}\_{k}) w(k+1) \right \| \nonumber \\ +& \, \left \| U({\widetilde}{{{\bf w}}}\_{k}) (1-U(k+1)) - w({\widetilde}{{{\bf w}}}\_{k}) (1-U(k+1)) \right \| \nonumber \\ +& \, \left \| w({\widetilde}{{{\bf w}}}\_{k}) (1-U(k+1)) - w({\widetilde}{{{\bf w}}}\_{k}) (1-w(k+1)) \right \| ] \nonumber \\ =& \, \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_{k}} [ U(k+1) \left \| U({\widetilde}{{{\bf w}}}\_{k}) - w({\widetilde}{{{\bf w}}}\_{k}) \right \| + w({\widetilde}{{{\bf w}}}\_{k}) \left \| U(k+1) - w(k+1) \right \| \nonumber \\ +& \, (1-U(k+1)) \left \| U({\widetilde}{{{\bf w}}}\_{k}) - w({\widetilde}{{{\bf w}}}\_{k}) \right \| + w({\widetilde}{{{\bf w}}}\_{k}) \left \| U(k+1) - w(k+1) \right \| ] \nonumber \\ =& \, \frac{1}{2} \sum\_{{\widetilde}{{{\bf w}}}\_{k}} \left \| U({\widetilde}{{{\bf w}}}\_{k}) - w({\widetilde}{{{\bf w}}}\_{k}) \right \| + \left [ U(k+1) - w(k+1)\right ] \nonumber \\ \stackrel{(b)}{\leq} & \, \sum\_{e =1}^k \left[U(e) - w(e) \right] + \left [ U(k+1) - w(k+1)\right ] \nonumber \\ =& \, \sum\_{e =1}^{k+1} \left[U(e) - w(e) \right],\end{aligned}$$ where (a) follows from the triangular inequality and (b) follows from the induction hypothesis. Hence, we have proved Equation [eqn:induction] by induction hypothesis. As we have proved above, this is sufficient to prove Lemma [lemma:recursivebound]. Finally, we prove the following lemma: [lemma:cumulativebound] For any two weight functions w, U : E → [0, 1] s.t. U(e) ≥ w(e) for all e ∈ E, we have $$\textstyle f({\mathcal{S}}\_t, U, v) -f({\mathcal{S}}\_t, w, v) \leq {\mathbb{E}}\left[ \sum\_{\tau=0}^{{\widetilde}{\tau}-1} \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U(e) - w(e) \right] \middle \| {\mathcal{S}}\_t \right],$$ where ${\widetilde}{\tau}$ is the stopping time when Sτ = ∅ or v ∈ Sτ, and the expectation is under the weight function w. Recall that the diffusion process $\left( {\mathcal{S}}^{\tau} \right )\_{\tau=0}^{{\widetilde}{\tau}}$ is a stochastic process. Note that by definition, if we treat the pair (Sτ, S0 : τ − 1) as the state of the diffusion process at diffusion step τ, and assume that ${{\bf w}}(e) \sim \mathrm{Bern} \left( w(e) \right)$ are independently sampled for all e ∈ ESt, v, then the sequence $({\mathcal{S}}^{0}, {\mathcal{S}}^{0:-1}), ({\mathcal{S}}^{0}, {\mathcal{S}}^{0:-1}), \ldots, ({\mathcal{S}}^{{\widetilde}{\tau}}, {\mathcal{S}}^{0:{\widetilde}{\tau}-1})$ follows a Markov chain, specifically, * For any state (Sτ, S0 : τ − 1) s.t. $v \notin {\mathcal{S}}^{\tau}$ and Sτ ≠ ∅, its transition probabilities to the next state (Sτ + 1, S0 : τ) depend on w(e)’s for e ∈ E(Sτ, S0 : τ). * Any state (Sτ, S0 : τ − 1) s.t. v ∈ Sτ or Sτ = ∅ is a terminal state and the state transition terminates once visiting such a state. Recall that by definition of the stopping time ${\widetilde}{\tau}$, the state transition terminates at ${\widetilde}{\tau}$. We define h(Sτ, S0 : τ − 1, U) − h(Sτ, S0 : τ − 1, w) as the ``value" at state (Sτ, S0 : τ − 1). Also note that the states in this Markov chain is topologically sortable in the sense that it will never revisit a state it visits before. Hence, we can compute h(Sτ, S0 : τ − 1, U) − h(Sτ, S0 : τ − 1, w) via a backward induction from the terminal states, based on a valid topological order. Thus, from Lemma [lemma:recursivebound], we have $$\begin{aligned} f({\mathcal{S}}\_t, U, v) -f({\mathcal{S}}\_t, w, v) \stackrel{(a)}{=}& \, h({\mathcal{S}}^0, \emptyset, U)- h({\mathcal{S}}^0, \emptyset, w) \nonumber \\ \stackrel{(b)}{\leq}& \, {\mathbb{E}}\left[ \sum\_{\tau=0}^{{\widetilde}{\tau}-1} \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U(e) - w(e) \right] \middle \| {\mathcal{S}}^0 \right],\end{aligned}$$ where (a) follows from the definition of h, and (b) follows from the backward induction. Since S0 = St by definition, we have proved Lemma [lemma:cumulativebound]. Finally, we prove Theorem [theorem:graph] based on Lemma [lemma:cumulativebound]. Recall that the favorable event at round t − 1 is defined as $$\xi\_{t-1}= \left \{ \|x\_{e}{^\mathsf{\scriptscriptstyle T}}({\overline}{\theta}\_{\tau-1} - \theta^\)\| \leq c \sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{\tau-1}^{-1} x\_{e}}, \, \forall e \in {\mathcal{E}}, \, \forall \tau \leq t \right \}.$$ Also, based on Algorithm [alg:imb], we have $$0 \leq {\overline}{w}(e) \leq U\_t(e) \leq 1,\forall e \in {\mathcal{E}}.$$ Thus, from Lemma [lemma:cumulativebound], we have $$\textstyle f({\mathcal{S}}\_t, U\_t, v) -f({\mathcal{S}}\_t, {\overline}{w}, v) \leq {\mathbb{E}}\left[ \sum\_{\tau=0}^{{\widetilde}{\tau}-1} \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U\_t(e) - {\overline}{w}(e) \right] \middle \| {\mathcal{S}}\_t, {\mathcal{H}}\_{t-1} \right],$$ where the expectation is based on the weight function ${\overline}{w}$. Recall that O*t(e) is the event that edge e is observed at round t. Recall that by definition, all edges in E(Sτ, S0 : τ) are observed at round t (since they are going out from an influenced node in Sτ, see Definition [def:Ot]) and belong to ESt, v, so we have $$\begin{aligned} \textstyle f({\mathcal{S}}\_t, U\_t, v) -f({\mathcal{S}}\_t, {\overline}{w}, v) &\leq \, {\mathbb{E}}\left[ \sum\_{\tau=0}^{{\widetilde}{\tau}-1} \sum\_{e \in {\mathcal{E}}({\mathcal{S}}^\tau, {\mathcal{S}}^{0:\tau})} \left[U\_t(e) - {\overline}{w}(e) \right] \middle \| {\mathcal{S}}\_t, {\mathcal{H}}\_{t-1} \right] \nonumber \\ &\leq {\mathbb{E}}\left[ \sum\_{e \in {\mathcal{E}}\_{{\mathcal{S}}\_t, v}} \mathbf{1}\left(O\_t(e)\right)\left[U\_t(e) - {\overline}{w}(e) \right] \middle \| {\mathcal{S}}\_t, {\mathcal{H}}\_{t-1} \right].\end{aligned}$$ This completes the proof for Theorem [theorem:graph]. Proof of Lemma [lemma:worst] ---------------------------- To simplify the exposition, we define $z\_{t,e}=\sqrt{x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{-1} x\_{e}}$ for all t = 1, 2…, n and all e ∈ E, and use Eto denote the set of edges observed at round t. Recall that $$\begin{aligned} {{\bf M}}\_t = {{\bf M}}\_{t-1} + \frac{1}{\sigma^2} \sum\_{e \in {\mathcal{E}}} x\_{e} x\_{e}{^\mathsf{\scriptscriptstyle T}}\mathbf{1} \left \{ O\_t(e) \right \} = {{\bf M}}\_{t-1} + \frac{1}{\sigma^2} \sum\_{e \in {\mathcal{E}}\_t^o} x\_{e} x\_{e}{^\mathsf{\scriptscriptstyle T}}\label{eqn:M\_update}.\end{aligned}$$ Thus, for all (t, e) such that e ∈ Eto (i.e., edge e is observed at round t), we have that $$\begin{aligned} \det \left[{{\bf M}}\_t \right] \geq & \det \left[ {{\bf M}}\_{t-1} + \frac{1}{\sigma^2} x\_{e} x\_{e}{^\mathsf{\scriptscriptstyle T}}\right] = \det \left[ {{\bf M}}\_{t-1}^{\frac{1}{2}} \left ( {{\bf I}}+ \frac{1}{\sigma^2} {{\bf M}}\_{t-1}^{- \frac{1}{2}} x\_{e} x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{- \frac{1}{2}} \right) {{\bf M}}\_{t-1}^{\frac{1}{2}} \right] \nonumber \\ = & \det \left[ {{\bf M}}\_{t-1} \right] \det \left[{{\bf I}}+ \frac{1}{\sigma^2} {{\bf M}}\_{t-1}^{- \frac{1}{2}} x\_{e} x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{- \frac{1}{2}} \right] \nonumber \\ = &\det \left[ {{\bf M}}\_{t-1} \right] \left( 1 + \frac{1}{\sigma^2} x\_{e}{^\mathsf{\scriptscriptstyle T}}{{\bf M}}\_{t-1}^{- 1} x\_{e} \right) = \det \left[ {{\bf M}}\_{t-1} \right] \left( 1 + \frac{z\_{t,e}^2}{\sigma^2} \right). \nonumber\end{aligned}$$ Thus, we have $$\left( \det \left[{{\bf M}}\_t \right] \right)^{\|{\mathcal{E}}\_t^o\|} \geq \left( \det \left[ {{\bf M}}\_{t-1} \right] \right)^{\|{\mathcal{E}}\_t^o\|} \prod\_{e \in {\mathcal{E}}\_t^o} \left( 1 + \frac{z\_{t,e}^2}{\sigma^2} \right).$$ Notice that when the feature matrix ${{\	arxiv_0000731
Analysis of vibration impact on stability of dewetting thin liquid film ======================================================================= Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description is reasonable and the amplitude equation holds. The linear and nonlinear analyses of the amplitude equation and the numerical computations show that such vibration stabilizes the film against dewetting and rupture. Introduction ============ Studies of stability, dewetting and rupture of thin liquid films on solid substrates are of great importance for micro- and nanotechnologies. Since the first works appeared in 1960s and 1970s, these subjects continue to attract enormous attention, see for instance Refs. . Refs. and are the reviews of recent results in this extremely diverse field. The subject of this paper is the general theoretical investigation of the impact of the vibration on the stability of a thin dewetting liquid film. Our interest in studying vibration impacts stems from the large body of works in fluid mechanics of macroscopic fluid layers, including Refs. (the experiment) and Refs. (the theory and numerical modeling), where the vibration is shown to drastically affect the stability characteristics and the dynamics of fluid surfaces and interfaces. In this paper we assume that the external influences on the film are the vibration, the gravity, and the long-range molecular attraction by the planar substrate, typified by van der Waals forces. Other effects (such as, for instance, the thermocapillarity and the evaporation) can be easily included by simply adding the corresponding terms to the final evolution equation for the film thickness. The theory we develop is based on the standard longwave lubrication approximation, as discussed by Oron, Davis, and Bankoff, and on the time-averaging method. The general discussion of the averaging methods can be found in Refs. . The key idea is the separation of the dynamics onto fast pulsations and slow relaxation processes. This approach works well when the vibration frequency is high in a certain sense, i.e. when there exists a large difference in the characteristic times (such as the viscous relaxation time and the vibration period, see below). The first transparent explanation of such separation of the time scales was given by Kapitza in his pioneering work on a pendulum with an oscillating point of support. The paper by Blekhman contains other examples in mechanics. Many examples of the successful application of the averaging method can be found in thermal vibrational convection, dynamics of inclusions in fluids, dynamics of granular materials, motion of disperse fluids, and filtration of inclusions in porous media. Interestingly, the vibration of the solid plate (on which the fluid system is located) often is capable of complete suppression of instabilities. For example, Wolf experimentally investigated the damping of the Rayleigh-Taylor instability in the horizontal two-layer system by a vertical high frequency vibration. The theoretical analysis of this situation (in the linear approximation) by the averaging technique was performed by Cherepanov (summary of this paper can be found in Ref. ). The longwave instability in this system was also analyzed by Lapuerta, Mancebo, and Vega. After the analysis of the linear longwave instability at the moderate vibration frequency, they proceed to the averaged description at high frequency. The generalization of the latter analysis to the nonisothermal situation was developed by Thiele, Vega, and Knobloch. They account for the Marangoni effect and perform a detailed investigation of the corresponding amplitude equation. Another widespread vibration-induced phenomenon is the parametric excitation, which emerges when the frequency of the vibration is comparable to one of the eigenfrequencies of the system (for instance, to the frequency of the capillary-gravity waves). Faraday was first to observe parametric waves on the surface of vertically oscillating horizontal layer. Linear and nonlinear analyses of parametric instability were performed, for instance, by Benjamin & Ursell, Kumar & Tuckerman, Lyubimov & Cherepanov, and by Mancebo and Vega. To the best of our knowledge the latter paper is the most detailed study to-date of the linear aspects of Faraday instability. It must be noted that the situations termed ``the averaged motion“ and ``the parametric instability” are often closely connected, although they operate within different intervals of the vibration frequencies. Indeed, in the studies of the averaged dynamics one has to ensure stability of the pulsatile motion (periodic in time). Most fluid systems have an eigenfrequency spectrum unbounded from above and thus the eigenfrequency is an increasing function of the mode number. Thus, even the high-frequency vibration is capable of parametric excitation of the higher modes. For the pulsatile motion to be stable, a window of parameters such as the amplitude and the frequency of the vibration must be chosen, where the parametric instability does not emerge. It is also worth noting that most papers, where the averaging method is employed, deal with the vibration of “inviscid” frequency, i.e. the vibration period Tp = 2π/ω (here ω is the dimensional frequency) is assumed small compared to the characteristic time of viscous relaxation, $\tau\_v=\hat H\_0^2/\nu$ (here ν is the kinematic viscosity and $\hat H\_0$ is the mean fluid layer thickness). It is clear that this assumption is quite reasonable for macroscopic layers, but for thin films (of thickness $100-1000 \, \AA$) it requires extremely large frequencies, $100 \,{\rm MHz}$ and higher. Now, we make a very important point, as follows. A thin film allows for averaged description even when the viscosity is large [i.e., τv = O(Tp) and even τv ≪ Tp]: one needs only to assume that the period of the vibration is small compared to the characteristic time of the film evolution, τl = O(k2) (here k is the wavenumber). Due to the lubrication approximation, τl ≫ τv. Thus the condition τl ≫ Tp is much milder than the usual inviscid approximation τv ≫ Tp. Therefore, the averaging procedure can be applied even to ultrathin films. To the best of our knowledge the only paper developing similar approximation is Ref. , where the linear stability problem is studied for “moderate” (“finite”) non-dimensional frequencies ωτv = O(1). However, by assuming the amplitude of the vibration “finite” (which means that it is of the order of the fluid layer thickness), the authors obtain that the impact of the vibration at moderate frequency is small. Thus, they focus on the high-frequency, small-amplitude case ωτv ≫ 1. In this paper we assume large vibration amplitude and develop the nonlinear amplitude equation for the thickness of the film. The outline of the paper is as follows. The mathematical formulation of the problem is presented in Sec. [sec:first]. We state and discuss assumptions and governing equations. In particular, the separation of the processes onto fast pulsatile and slow averaged motions is performed in Sec. [ssec:splitting]. The pulsatile problem is solved in Sec. [sec:puls]. The amplitude equation [Eq. ([h-avt])] governing the averaged dynamics of the film thickness is obtained in Sec. [sec:aver]. This equation is the main result of this paper. It can be used to study impacts of the vertical vibration (in the frequency range for which the averaged description is applicable) on the dynamics of a film, in the presence of the surface tension and wetting interactions with the substrate. Two limiting cases of “low” and “high” vibration frequency are analyzed in Sec. [sec:limit]. These cases correspond to the different ratios of the inertial and viscous forces in the oscillatory motion. (The viscosity dominates at low frequencies, while the inertia force dominates at high frequencies.) In Sec. [sec:3D] the 3D generalization of the theory is presented. Conditions of parametric instability of the oscillatory motion are analyzed in Sec. [sec:stab], where the Faraday instability and the shear flow instability are discussed. We show that for any admissible vibration frequency there exists a finite range of vibration amplitudes for which such instabilities are not present. In Sec. [sec:layer] we address the averaged behavior of the system within the framework of the obtained amplitude equation. Results of linear and weakly nonlinear analyses of the equilibrium state with the flat surface are presented, as well as the results of direct numerical simulations. In particular, we show that the vibration influence is stabilizing, i.e. it can delay or completely suppress the film rupture by intermolecular attractive forces. Finally, Sec. [sec:conc] summarizes the results. Formulation of the problem ========================== Governing equations ------------------- We consider a three-dimensional (3D), laterally unbounded thin liquid film of unperturbed thickness $\hat H\_0$ on a planar, horizontal substrate. The Cartesian reference frame is chosen such that the x − and y − axes are in the substrate plane and the z − axis is normal to the substrate (Fig. [fig:1]). [fig:1] The substrate-film system is subjected to the vertical harmonic vibration of the amplitude $\hat b$ and the frequency ω. Thus, in the reference frame of the substrate the acceleration of gravity is modulated, $$\label{gt} g(t)=g\_0+\hat b\omega^2 \cos\omega t.$$ Since $\hat H\_0$ is small, the intermolecular interaction of the film surface with the substrate has to be taken into account. Thus, we prescribe the potential energy $\phi(\hat H)$ to the unit length of the film layer. In this paper we consider only the van der Waals attractive potential, $$\phi(\hat H)=\frac{A^\prime}{6\pi \hat H^3},$$ where A′ > 0 is the Hamaker constant. The model presented in this paper can be readily extended to incorporate other models of wetting interactions – it is only necessary to replace $\phi(\hat H)$ with an appropriate function. We scale the time, the length, the velocity and the pressure by $\hat H\_0^2/\nu, \ \hat H\_0, \ \nu/\hat H\_0, \ \rho (\nu/\hat H\_0)^2$, respectively (here, ν is the kinematic viscosity and ρ is the density of the liquid). Then, the liquid motion is governed by the following non-dimensional problem: [baseeq] $$\begin{aligned} \nabla \cdot {\bf v} &=& 0, \\ \nonumber {\bf v}\_t + {\bf v}\cdot \nabla {\bf v} &=& - {\bf \nabla} p + \nabla ^2 {\bf v} \\ &&-\left(G\_0 +B \Omega^2\cos\Omega t \right) {\bf e\_z},\end{aligned}$$ [basebcs] $$\begin{aligned} {\bf v} &=& 0 \ {\rm at} \ z = 0, \\ \nonumber h\_t + {\bf v} \cdot {\bf \nabla} h &=& w,\\ \left(p-\phi - Ca K\right){\bf n} &=& {\bf n}\cdot {\bf T} \ {\rm at} \ z = h(x,y,t).\end{aligned}$$ Here, ${\bf v}$ is the fluid velocity, w is its component normal to the substrate, p is the pressure in the liquid, $\bf T$ is the viscous stress tensor, $h=\hat H/\hat H\_0$ is the dimensionless thickness of the layer, ${\bf e}\_z$ is the unit vector directed along the z − axis, ${\bf n}= \left({\bf e}\_z-\nabla h\right)/\sqrt{1+\left(\nabla h\right)^2}$ is the normal unit vector to the free surface, $K=\nabla \cdot {\bf n}$ is the mean curvature of the free surface, ϕ = A/h3 [where $A=A'/(6\pi\rho\nu^2\hat H\_0)$ is the non-dimensional Hamaker constant], $Ca=\sigma \hat H\_0/(\rho \nu^2)$ is the capillary number (where σ is the surface tension), $G\_0=g\_0 \hat H\_0^3/\nu^2$ is the Galileo number, $B=\hat b/\hat H\_0$ is the non-dimensional amplitude, and $\Omega=\omega \hat H\_0^2/\nu$ is the non-dimensional frequency. We consider the nonlinear evolution of the large-scale perturbations. As it is usually done, we introduce small parameter ε, which is of the order of the ratio of the mean thickness $\hat H\_0$ to the perturbation wavelength, i.e. ε ≪ 1 for long waves. Below for the sake of simplicity and for more transparent presentation of ideas, we consider the 2D model, assuming that ${\bf v}=u{\bf e}\_x+ w{\bf e}\_z$, where ${\bf e}\_x$ is the unit vector directed along the x − axis, and all fields are independent of y. The theory extends trivially in three dimensions at the replacement of the x-derivative by the 2D gradient. We derive the 3D analogue in Sec. [sec:3D]. Next we introduce the conventional stretched coordinates and the time: $$\label{coords} X=\epsilon x, \ Z=z, \ T=\epsilon^2 t, \ \tau=\Omega t,$$ such that ∂/∂t = ε2∂/∂T + Ω∂/∂τ. Now we rescale the velocity components as follows: u = εU, w = ε2W. To this end, both the pressure p and the surface position h remain unscaled. Throughout this paper we assume that the capillary number is large: Ca = Cε− 2. This is quite realistic and widely used assumption. Substituting the expansions ([coords])-([Ca]) into the problem ([baseeq]) and ([basebcs]) we arrive at the following set of equations and boundary conditions: [perturbeq] $$\begin{aligned} W\_Z &=& -U\_X, \\ \nonumber \Omega U\_\tau &=&- p\_X + U\_{ZZ} \\ && + \epsilon^2 \left(U\_{XX} - U\_T - U U\_X - W U\_Z \right), \\ p\_Z &=& - G\_0 - B \Omega^2\cos\tau + \epsilon^2 \left( W\_{ZZ}- \Omega W\_\tau\right),\end{aligned}$$ [perturbbcs] $$\begin{aligned} U &=& 0, \ W = 0 \ \ {\rm at} \ Z = 0, \\ \nonumber U\_Z &=&- \epsilon^2 \left(W\_X - 4h\_X U\_X\right), \\ \nonumber \Omega h\_\tau &=& \epsilon^2 \left(W - h\_T - U h\_X \right),\\ \nonumber p &=& \phi - C h\_{XX} \\ &&+\epsilon^2 \left(\frac{3}{2} C h\_X^2h\_{XX} + 2 W\_Z\right) \ {\rm at} \ Z = h.\end{aligned}$$ All terms of order ε4 have been omitted. Separation of the pulsating and averaged dynamics ------------------------------------------------- In this paper we consider the case of moderate vibration frequency: Ω ≫ ε2, i.e. the vibration period is small compared to the characteristic time of the surface evolution. This assumption makes possible the averaging of the dynamics of the film over the vibration period. The main purpose is the rigorous asymptotic analysis (in powers of ε) which results in the set of equations and boundary conditions, where the dependence on τ is averaged out. This set is then used to derive the amplitude equation for the thickness of the film. As for Ω, this quantity is not an asymptotic parameter. This means that, generally speaking, we assume Ω neither large nor small. Therefore Ω will enter the amplitude equation as a finite parameter. However, within the framework of the equation set ([perturbeq]) and ([perturbbcs]) we will consider three qualitatively different cases: (i) Ω ≪ 1, i.e. the vibration is of “low” frequency; this is the quasi-Stokes approximation with week influence of fluid inertia. Note that in this case, due to Eqs. ([coords]), the condition ε2 ≪ Ω must be retained to legalize the averaging procedure. (ii) Ω = O(1). This assumption means that the vibration period is comparable to the time of the momentum relaxation across the layer, $\hat H\_0^2/\nu$. (iii) Ω ≫ 1, i.e. the vibration is of “high” frequency; viscosity is negligible except for the thin boundary layer near the rigid wall. As usual, the boundary layer in the vicinity of the free surface does not play an important role (see Ref. , for example). Since the van der Waals interaction is important for films of thickness $100-1000 \ \AA$, let us estimate the typical values of Ω. Taking $\hat H\_0=10^{-5} \ {\rm cm}$ and $\nu=10^{-2} \ {\rm cm^2 s^{-1}}$ (viscosity of water), we obtain Ω ≈ 10− 4 for $\omega/(2\pi)=1 \ {\rm kHz}$ (typical for mechanical vibrator), and Ω = 10− 2 ÷ 1 for $n=0.1 \div 10 \ {\rm MHz}$ (typical for ultrasound). Therefore the case (i) can be easily achieved either mechanically or by means of ultrasound irradiation of the substrate from below, the case (ii) can be reached only using the ultrasound, and the case (iii) seems unrealistic. Nevertheless we shall consider this case below, since our results, upon neglecting ϕ, can as well be applied to the description of macroscopic fluid layers. Besides, this limiting case was studied in detail by many authors and therefore it allows for the verification of our results. We represent each field f = {U, W, p, h} as the sum of the average part $\bar f(T)$ and the pulsation $\tilde f(\tau)$, where τ and T can now be termed the “fast” and the “slow” times, respectively. Thus, we write [splitting] $$\begin{aligned} U &=& \bar U + \tilde U, \ W = \bar W + \tilde W, \\ p &=& \bar p + \tilde p, \ h = \bar h + \tilde h.\end{aligned}$$ We now assume that B = ε− 1b, i.e. the amplitude of the vibration is large compared to the mean film thickness. This assumption seems surprising as it is customary to impose the small-amplitude high frequency vibration. However, large amplitudes are permitted when the large-scale dynamics is considered. Indeed, it is shown below that due to the longwave approximation the impact of the vibration becomes non-negligible only at large amplitudes. Also, it will be made clear momentarily that in some sense the pulsatile motion is still small-amplitude. The assumption of large vibration amplitude means that the oscillating part of the pressure field $\tilde p$ (which is forced by the inertia force BΩ2cosτ) is of order ε− 1, which in turn leads to ε− 1 scaling for the velocities of the pulsation $\tilde U$ and $\tilde W$. The pulsation of the surface height $\tilde h$ obviously has the same order as $\tilde w=\epsilon^2 \tilde W$ [see Eq. ([velo])], i.e. it is of the order ε. Therefore, it is convenient to redefine the pulsation parts of all fields, rewriting Eqs. ([splitting]) as follows [rescaling] $$\begin{aligned} U &=& \bar U + \epsilon^{-1} \tilde U, \ W = \bar W + \epsilon^{-1}\tilde W, \\ \label{pres\_scal} p &=& \bar p + \epsilon^{-1}\tilde p, \ h = \bar h + \epsilon \tilde h,\end{aligned}$$ where $\tilde U, \tilde W, \tilde p$ and $\tilde h$ are O(1) quantities. Accounting for the initial scaling ([velo]) one can conclude that the full components of the velocity field are: $$\label{rescaled\_velo} u=\epsilon \bar U + \tilde U, \ w=\epsilon^2 W + \epsilon \tilde W,$$ while the pressure field and the surface deviation are given by Eqs. ([presscal]). Note that the pulsations of the fluid velocity and the pressure are larger than their mean parts, while the opposite is true for the pulsation of the film height. Moreover, the scaling ([rescaledvelo]) means that the pulsation remains “small-amplitude”. Indeed, the typical horizontal (vertical) displacement of the fluid particle during one period is O(1) [O(ε)], which is small in comparison with the respective characteristic lengthscale, O(ε− 1) [O(1)]. Substitution of the expansions ([rescaling]) in the equation sets ([perturbeq]) and ([perturbbcs]) allows to separate fast pulsations from background slow (averaged) motions. Keeping terms of zeroth and first orders in ε, we obtain the following sets. (i) For the pulsations: [pulsexp] $$\begin{aligned} \tilde W\_Z &=& -\tilde U\_X, \\ \nonumber \Omega \tilde U\_{\tau} &=& - \tilde p\_X + \tilde U\_{ZZ} - \epsilon \left(\tilde U \tilde U\_X + \tilde W \tilde U\_Z \right)\_p , \\ \tilde p\_Z &=& - b\Omega^2\cos\tau , \label{puls\_p\_z} \\ \tilde U &=& 0, \ \tilde W = 0 \ {\rm at} \ Z = 0, \\ \nonumber \Omega \tilde h\_{\tau} &=& - \tilde U \bar h\_X + \tilde W- \epsilon \left[\tilde U\_Z \tilde h \bar h\_X +\left(\tilde U \tilde h\right)\_X\right]\_p , \\ \tilde U\_Z &=&- \epsilon \left(\tilde U\_{ZZ} \tilde h\right)\_p , \ \tilde p = -\epsilon (\tilde p\_Z\tilde h)\_p \ {\rm at} \ Z = \bar h,\end{aligned}$$ where the subscript “p” denotes the pulsating part of the corresponding expression: $$f\_p=f-\bar f.$$ (It is obvious that in general the term squared with respect to pulsations contains both the averaged and the pulsating components, for instance $\cos^2\tau=\frac{1}{2}+\frac{1}{2}\cos2\tau$.) (ii) For the averaged parts: [averexp] $$\begin{aligned} \bar W\_Z &=& -\bar U\_X, \ \bar p\_Z = -G\_0, \\ \bar U\_{ZZ} &=& \bar p\_X + \langle \tilde U \tilde U\_X + \tilde W \tilde U\_Z \rangle, \\ \bar U &=& 0, \ \bar W = 0 \ {\rm at} \ Z = 0, \\ \nonumber \bar p &=&- \langle\tilde p\_Z \tilde h\rangle -\frac{\epsilon}{2}\langle\tilde p\_{ZZ} \tilde h^2\rangle + \phi(\bar h) - C \bar h\_{XX}, \\ \nonumber \bar h\_T &=& - \bar U \bar h\_X - \langle \tilde U \tilde h\_X \rangle + \bar W + \langle \tilde W\_Z \tilde h \rangle - \frac{\epsilon}{2}\langle \tilde U\_{Z} \tilde h^2 \rangle\_X,\\ \bar U\_Z &=&-\langle \tilde U\_{ZZ} \tilde h \rangle-\frac{\epsilon}{2}\langle \tilde U\_{ZZZ} \tilde h^2 \rangle \label{aver\_dyn} \ {\rm at} \ Z = \bar h.\end{aligned}$$ In the system ([averexp]) the angular brackets denote averaging in time τ. Note that the boundary conditions at the free surface have been shifted at the mean position $\bar h$. This leads to the following expansion in powers of ε of the arbitrary field F: $$\label{shift} F(Z=\bar h + \epsilon \tilde h) \approx F(\bar h) + \epsilon F\_Z (\bar h) \tilde h + \frac{1}{2}\epsilon^2 F\_{ZZ} (\bar h) \tilde h^2.$$ As we have noted, the ε2-terms in Eqs. ([averexp]) have been omitted. On the other hand, the third term in Eq. ([shift]) has to be taken into account for some fields, because it produces a correction of order ε. For instance: $$\begin{aligned} \nonumber \left< p(Z=\bar h + \epsilon \tilde h)\right> &\approx& \left< \epsilon^{-1} \left( \tilde p+ \epsilon \tilde p\_Z \tilde h + \frac{1}{2}\epsilon^2 \tilde p\_{ZZ} \tilde h^2 \right) +\bar p + \epsilon \bar p\_Z \tilde h \right>\_{Z=\bar h}\\ &=&\bar p (Z=\bar h) +\left< \tilde p\_Z \tilde h + \frac{1}{2}\epsilon \tilde p\_{ZZ} \tilde h^2 \right>\_{Z=\bar h}.\end{aligned}$$ However, all terms cubic with respect to pulsations vanish after the averaging. Moreover, in further analysis we will disregard the ε-terms in Eqs. ([pulsexp]) and ([averexp]) (i.e., use these boundary value problems in the leading zeroth order). It is also important to emphasize that the zeroth order of the problem governing the pulsations [problem ([pulsexp])] is linear. This is because all nonlinear terms ($\tilde U \tilde U\_X,\ \tilde W \tilde U\_Z$, etc.) are small due to the longwave approximation, despite the scaling ([rescaledvelo]). Pulsatile motion ================ Equations. ([pulsexp]) become in the zeroth order in ε: [pulsfull] $$\begin{aligned} \tilde p\_Z&=&-b \Omega^2 \cos \tau,\\ \Omega \tilde U\_{\tau}&=&-\tilde p\_X + \tilde U\_{ZZ}, \ \tilde W\_Z=-\tilde U\_X, \\ \tilde U&=&\tilde W=0 \ {\rm at} \ Z = 0, \\ \nonumber \tilde p&=&0, \ \tilde U\_Z=0, \\ \Omega \tilde h\_{\tau} &=&-\tilde U \bar h\_X + \tilde W \ {\rm at} \ Z = \bar h.\end{aligned}$$ According to the conventional method for the solution of linear problems, we need to separate the general solution of the homogeneous problem from the particular solution of the nonhomogeneous one. The former solution corresponds to the gravity-capillary waves damped by viscosity and thus it is not of interest. (In fact, the gravity-capillary waves are completely damped at times of order τ. Thus the omitted solution represents fast relaxation of the initial conditions, which has no effect on slow dynamics with the characteristic time scale T.) The latter solution corresponds to the periodically forced motion, which can be represented in the complex form as follows: [pulscomplex] $$\begin{aligned} \tilde p &=& b \Omega {\rm Re} \left[q(X,Z)\exp{\left(i \tau\right)}\right],\\ \tilde U &=& b \Omega {\rm Re} \left[I(X,Z)\exp{\left(i\tau\right)}\right],\\ \tilde W &=& b \Omega {\rm Re} \left[K(X,Z)\exp{\left(i\tau\right)}\right],\\ \tilde h &=& b {\rm Re} \left[H(X)\exp{\left(i \tau\right)}\right].\end{aligned}$$ The set of equations and boundary conditions governing the amplitudes of pulsations reads: [puls] $$\begin{aligned} \label{puls\_u} q\_Z&=&-\Omega, \ I\_{ZZ}+\alpha^2 I=q\_X, \ K\_Z=-I\_X, \\ \label{puls\_noslip} I&=&K=0 \ {\rm at} \ Z=0,\\ q&=&0, \ i H=K - I h\_X, \ I\_Z=0 \ {\rm at} \ Z=h,\end{aligned}$$ where α2 = − iΩ. Hereafter we omit the bar over $\bar h$. At α = 0 this problem coincides with the conventional equations for the thin film in the absence of the vibration [cf. Eqs. (2.22)-(2.24) in Ref. at $\Phi=\beta\_0=\Sigma=\Pi\_0=\bar C^{-1}=0$]. However, the term α2I originating from the inertia of the fluid drastically complicates the solution. The solution of the boundary value problem ([puls]) is: [solpuls] $$\begin{aligned} q&=&\Omega (h-Z), \ I=i h\_X\left(1-\frac{\cos\alpha (h-Z)}{\cos\alpha h}\right), \\ \nonumber K&=&-i\left[h\_{XX}\left(Z+\frac{\sin\alpha (h-Z)-\sin\alpha h}{\alpha \cos\alpha h}\right) \right.\\ && - \left. h\_X^2\frac{1-\cos\alpha Z}{\cos^2\alpha h}\right],\\ H&=&-\left[h h\_{XX}f(\alpha h)-h\_X^2\tan^2\alpha h\right] \\ &=& -\left[h f(\alpha h)h\_X \right]\_X,\end{aligned}$$ where $$\label{f\_definition} f(y)\equiv 1-\frac{\tan y}{y}.$$ Now, if we set h = 1 + ξ and linearize Eqs. ([solpuls]) with respect to ξ, then we arrive at Eqs. (2.30)-(2.32) in Ref. . However, we have considerably expanded the domain of validity for this solution, as will be explained in Secs. [sec:aver] and [sec:layer]. Next, we proceed to the analysis of the limiting cases for the pulsatile motion, i.e. Ω ≪ 1 (low frequency) and Ω ≫ 1 (high frequency). In the former case the viscous term IZZ dominates in Eq. ([pulsu]) and the inertial term − iΩI exerts a week impact only. Thus the solution of the problem ([puls]) simplifies to: [solpulslf] $$\begin{aligned} q&=&\Omega (h-Z), \\ \nonumber I&=&-\frac{\Omega h\_X}{2}Z(2h-Z) \\ &&\times \left[1-\frac{i\Omega}{12}\left(4h^2+2hZ-Z^2\right)\right], \\ \nonumber K&=&\frac{\Omega h\_{XX}}{6}Z^2\left[ 3h-Z -\frac{i\Omega }{20}\left(20h^3-5hZ^2+Z^3\right)\right] \\&& +\frac{\Omega h\_X^2}{2}Z^2\left[1-i\Omega \left(h^2-\frac{Z^2}{12}\right)\right],\\ H&=&-\frac{i\Omega}{3}\left[h^3 h\_X \left(1-\frac{2}{5}i\Omega h^2\right)\right]\_X.\end{aligned}$$ Equations ([solpulslf]) can also be obtained by expanding Eqs. ([solpuls]) in powers of the small parameter α and keeping all terms of order α4. The terms proportional to Ω2 originate from the evolution of $\tilde U$ in τ. These terms are small corrections, but they must be retained since they govern the solution of the averaged problem. In the Ω ≫ 1 case the amplitudes of the pulsations are [see Eqs. ([solpuls])]: $$\label{high\_freq\_puls} q=\Omega(h-Z), \ I=ih\_X, \ K= - i h\_{XX}Z, \ H=-\left(hh\_X\right)\_X.$$ Equations ([highfreqpuls]) constitute the solution to the vibration problem for an inviscid film. The longitudinal component of the velocity, I, is uniform across the liquid layer; it is determined by the pulsations of the pressure gradient only. The transversal component, K, is the linear function of Z; it vanishes at values of X corresponding to the extremum of I. Note that H is real, i.e the surface deformation is in phase or in antiphase with the vertical motion of the substrate. Of course, I given by Eq. ([highfreqpuls]) is inconsistent with the no-slip condition ([pulsnoslip]). In order to vanish I at the rigid wall one has to take the boundary layer into account. Introducing the “fast” coordinate $\eta=Z/\sqrt{\Omega}$ near the wall we arrive at $$\begin{aligned} q^{(i)}\_\eta&=&0, \ iI^{(i)}-I\_{\eta\eta}^{(i)}=-\Omega^{-1}q\_X, \ K^{(i)}\_\eta=0,\\ I^{(i)}&=&K^{(i)}=0 \ {\rm at} \ \eta=0,\\ I^{(i)} &\to& ih\_X, \ q^{(i)} \to q(Z=0)=\Omega h \ {\rm at} \ \eta \to \infty.\end{aligned}$$ The solution of this problem is well-known (see, for example, Ref. ): [sol-bound] $$\begin{aligned} q^{(i)}&=&q(Z=0)=\Omega h, \\ I^{(i)}&=&ih\_X\left[1-\exp\left(-\beta \eta \right)\right], \ K^{(i)}=0,\end{aligned}$$ where $\beta=\sqrt{i}=(1+i)/\sqrt{2}$. ∣I(i)(η)∣ increases from zero at the rigid wall to the maximal value 1.069∣hX∣ at η ≈ 2.284 and then decays to ∣I∣ = ∣hX∣. Of course, the solution ([sol-bound]) matches the solution ([solpuls]) at Ω ≫ 1, Z → 0. Examples of the distribution of Ir and Ii across the layer are given in Fig. [fig:Vz](a) and Fig. [fig:Vz](b), respectively, for different values of Ω. (Here and below we use the subscripts “r” and “i” for the real and the imaginary parts of complex variables.) Since I is proportional to hX, the value of the latter derivative only rescales the longitudinal velocity. Thus we set hX = 1 for these sketches and for Fig. [fig:Vm]. [fig:Vz] [fig:Vm] One can see immediately that at Ω < Ωc ≈ 19.74 the intensity of the oscillations is maximal at the free surface and monotonously decreases to the rigid wall. At higher values of Ω there exists a maximum in the inner part of the layer. With increase of Ω the Z-coordinate of this maximum tends to zero. At Ω = 100 the velocity profile agrees well with the asymptotic formula ([sol-bound]). The dependence of the maximal amplitude of the pulsation velocity and of ∣I(Z = h)∣ on the frequency of the vibration is shown in Fig. [fig:Vm] by the solid and dashed lines, respectively. At large Ω the solid line reaches the asymptotical value 1.069, the dashed line tends to unity. To make more clear the behavior of the pulsation velocity we show in Figs. [fig:Om1] and [fig:Om10] the isolines of the pulsation streamfunction at four progressive time moments. The streamfunction Ψ is defined as $$\Psi=i h\_X\left[Z+\frac{\sin\alpha (h-Z)-\sin\alpha h}{\alpha \cos\alpha h}\right],$$ so that I = ΨZ, K = − ΨX. Again for illustration purposes only we set h = 1 + acoskX with a = 0.1, k = 1 in these figures. We point out that these figures present the streamfunctions of the time-periodic motion, i.e. one must not be concerned that the isolines are open curves. Each fluid particle oscillates near its mean position with the small (on the scale of the figure) amplitude, and its instantaneous pulsation velocity is tangential to the momentary isoline at the point. [fig:Om1] [fig:Om10] It must be emphasized that within the framework of the longwave approximation there is no need for the stability analysis of the solution ([solpuls]). Indeed, the leading order of Eqs. ([pulsexp]) [given by Eqs. ([pulsfull])] is the linear problem. As was mentioned above, the homogeneous problem originating from Eqs. ([pulsfull]) obviously has only the decaying solutions: at finite Ω, the perturbations decay due to the viscosity. Thus, stability of the oscillatory motion is evident except for the limiting case Ω ≫ 1, which has been studied in detail. However, stability with respect to perturbations of finite or small wavelength must be addressed. We briefly discuss this issue in Sec. [sec:stab]. Also, in the Appendix [app:p-flow] we discuss the reduction of the flow ([solpuls]) to the oscillatory Poiseuille flow. The analysis in Sec. [sec:stab] is partially based on this result. Averaged motion =============== In this section we solve the leading order of the averaged problem ([averexp]): [aver0] $$\begin{aligned} \bar p\_Z&=&-G\_0, \ \bar W\_Z=-\bar U\_X, \\ \bar U\_{ZZ}&=&\bar p\_X+\langle{\tilde U \tilde U\_X + \tilde W \tilde U\_Z}\rangle, \\ \bar U&=&\bar W=0 \ {\rm at} \ Z = 0, \\ \nonumber \bar p&=&\phi( h)-C h\_{XX} + b \Omega^2 \langle{\tilde h \cos \tau}\rangle, \\ \nonumber h\_T &=& \bar W -\bar U h\_X+ \langle{\tilde h \tilde W\_Z - \tilde U \tilde h\_X }\rangle, \\ \bar U\_Z&=&-\langle \tilde U\_{ZZ} \tilde h\rangle \ {\rm at} \ Z = h.\end{aligned}$$ Note that the averaged term in the boundary condition for the pressure results when $\tilde p\_Z$ in the corresponding boundary condition ([averdyn]) is replaced by − bΩ2cosτ [see Eqs. ([pulsfull])]. Substituting the forms ([pulscomplex]), accounting for the obvious equality $$\label{aver\_LL} \langle {{\rm Re} \left( B e^{i \tau} \right){\rm Re} \left( D e^{i \tau} \right)}\rangle=\frac{1}{2}{\rm Re} \left(B D^\\right)$$ for the calculation of averages in Eqs. ([aver0]), and noting that $$\langle \tilde U\tilde h\_X - \tilde W\_Z \tilde h\rangle= \langle \tilde U \tilde h\_X + \tilde U\_X \tilde h\rangle = \langle \tilde U \tilde h\rangle\_X,$$ we obtain (the overbars are omitted): [aver] $$\begin{aligned} \label{aver\_velo} p\_Z&=&-G\_0, \ W\_Z=-U\_X, \\ U\_{ZZ}&=&p\_X+\frac{1}{2} b^2 \Omega^2{\rm Re} \left(I^\ I\_X+ K^\I\_Z\right),\\ U&=&W=0 \ {\rm at} \ Z = 0, \\ \nonumber p&=&\phi-C h\_{XX} + \frac{1}{2} b^2 \Omega^2{\rm Re} H, \\ \nonumber h\_T &=&- U h\_X +W - \frac{1}{2}b^2 \Omega {\rm Re} \left(I^\ H\right)\_X, \\ U\_Z&=&-\frac{1}{2}b^2 \Omega {\rm Re} \left(I\_{ZZ}^\* H\right) \ {\rm at} \ Z = h.\end{aligned}$$ The evolutionary equation for h can be rewritten in the form $$\label{int\_aver} h\_T =- \partial\_X \int\_0^h U dZ - \frac{1}{2} b^2 \Omega {\rm Re} \left(I^\* H\right)\_X \ {\rm at} \ Z = h.$$ Analytical integration of this set of equations is performed in Appendix [app-aver]. It results in the following nonlinear equation for h: [h-avt] $$\begin{aligned} h\_T&=&\left(\frac{1}{3}h^3 \Pi\_X-\frac{1}{2}b^2 \Omega^2Q\right)\_X, \\ \nonumber \Pi & \equiv& p(Z=0)\\ &=& \phi(h)-C h\_{XX} + G\_0 h +\frac{1}{2}b^2 \Omega^2{\rm Re} H.\\ Q&=&Q\_1(\gamma)h^2h\_X^3+Q\_2(\gamma)h^3h\_Xh\_{XX},\end{aligned}$$ where $\gamma=\sqrt{2\Omega}h$ and [coeff] $$\begin{aligned} Q\_1&=&3\frac{2\sinh\gamma \sin\gamma-\gamma\left(\sinh\gamma\cos\gamma+\sin\gamma\cosh\gamma\right)} {\gamma^2\left(\cos\gamma+\cosh\gamma\right)^2}, \\ Q\_2&=&-\frac{1}{3}+\frac{11\left(\sinh\gamma-\sin\gamma\right)- 3\gamma\left(\cosh\gamma-\cos\gamma\right)}{\gamma^3\left(\cos\gamma+\cosh\gamma\right)}.\end{aligned}$$ Also note that the function f(αh) [see Eq. ([fdefinition])] can be expressed in terms of γ: [fgam] $$\begin{aligned} f\_r(\alpha h)&=&1-\frac{\sinh\gamma+\sin\gamma}{\gamma\left(\cos\gamma+\cosh\gamma\right)},\\ f\_i(\alpha h) &=&\frac{\sinh\gamma-\sin\gamma}{\gamma\left(\cos\gamma+\cosh\gamma\right)}.\end{aligned}$$ The dependence of functions fr, i(αh) and Q1, 2 on the parameter γ is given in Figs. [fig:Qf](a) and [fig:Qf](b), respectively. One can see that 0 ≤ fr(αh) < 1 and the coefficient Q2 is negative for all values of γ. [fig:Qf] One can immediately see that along with the regular contributions due to the wetting potential, surface tension and gravity, the expression for Π contains the nonlinear contribution due to the vibration. The nonlinear Q-term is entirely due to the vibration. The obtained equations of slow motion allow simplification in the limits of low and high frequency Ω. For these limiting cases, considered next, the solution to the boundary value problem ([aver]) is not very cumbersome and can be presented in detail. Frequency-based analysis of Eqs. ([aver]) and ([h-avt]) ======================================================= Low vibration frequency, Ω ≪ 1 ------------------------------ We look for the solution of the averaged problem ([aver]) in the form of the power series in Ω, using the corresponding solution ([solpulslf]) of the pulsatile problem. It is clear that W can be easily expressed via U by means of the continuity equation, the last relation in Eqs. ([avervelo]). However, the corresponding expression involves different combinations of derivatives of h and is difficult to understand. Thus, for the sake of brevity, here we present only the longitudinal component of the averaged velocity: [solslowfreq] $$\begin{aligned} \nonumber U&=&\frac{1}{2}Z(Z-2h)\Pi\_X+\frac{\left(b \Omega^2\right)^2}{720}Z\left[75h\_X^3 hZ^3\right. \\ && \left. + h\_X h\_{XX}\left(Z^5-6hZ^4+15h^2Z^3-48h^5\right)\right], \label{Usols\_lowfreq}\\ \Pi&=&\phi(h)-C h\_{XX} +G\_0 h - \frac{\left(b\Omega^2\right)^2}{15}(h^5 h\_X)\_X.\end{aligned}$$ (Also note that the pressure p = Π − G0Z.) The terms proportional to Ω4 are retained in this solution, while the higher order terms have been omitted. The first term in Eq. ([Usolslowfreq]) is conventional, while the second one represents the impact of the vibration (as well as the last term in Π). Substituting Eqs. ([solslowfreq]) in Eq. ([intaver]), we obtain the evolutionary equation for h: $$\label{h\_aver\_low} h\_T =\frac{1}{3}\left(h^3\Pi\_X\right)\_X + \left(b \Omega^2\right)^2 \left(\frac{2}{63}h^7h\_Xh\_{XX}-\frac{1}{15}h^6 h\_X^3\right)\_X.$$ Equation ([haverlow]) can be obtained also from Eq. ([h-avt]) by noticing that at small γ the coefficients Q1, 2 (and fr) are proportional to γ4: $$\begin{aligned} \label{small} f\_r(\alpha h)\approx& Q\_1\approx \frac{1}{30} \gamma^4, \ Q\_2 \approx -\frac{1}{63} \gamma^4.\end{aligned}$$ Equation ([haverlow]) makes it clear that it is necessary to provide large vibration amplitude in order to gain the finite impact of the vibration on the dynamics of the film height. In other words, the rescaled acceleration bΩ2 has to be finite. Moreover, we show in Sec. [sec:stab] that such values of the amplitude do not cause the parametric instability. It is important to recognize that one needs only γ ≪ 1 in order to obtain Eq. ([haverlow]). However, this limit can be reached not only for small Ω, but also for small local values of h. This means that Eq. ([haverlow]) can be used (independently of the value of Ω) near rupture, i.e. in the close vicinity of the point Xr, such that h(Xr, T) → 0. Note that the general Eq. ([h-avt]) should be applied far away from this point. However, since the vibration terms in Eq. ([haverlow]) are proportional to h9, they are negligible at small h in comparison with the term originating from the surface tension (which is of order h4) and with the dominant van der Waals term. In other words, only the competition of the surface tension and the van der Waals interaction governs the behavior of the film near rupture, and the vibration does not provide a noticeable impact. (Of course, in the very close vicinity of the rupture point only the van der Waals interaction contributes to the film dynamics – see Refs. .) High vibration frequency, Ω ≫ 1 ------------------------------- Here we use the ‘inviscid’’ solution for the pulsations given by Eqs. ([highfreqpuls]). We do not consider the impact of the boundary layer, i.e. the solution given by Eq. ([sol-bound]), for the obvious reason. It is known from Rayleigh and Schlichting, that the boundary layer can produce an independent averaged motion. However, the intensity of this flow is rather small in comparison with the volumetric sources under consideration. Indeed, estimating the longitudinal component of the averaged velocity generated at the external border of the boundary layer, one obtains US ∝ b2Ω(∣I∣2)X, whereas the dominant contribution in the Eq. ([avervelo]) is proportional to b2Ω2(∣I∣2)X. A similar situation exists in many problems of thermal vibrational convection. Now one can see that in the set ([aver]) [hfmean] $$\begin{aligned} {\rm Re}H&=&H=-\left(h h\_X\right)\_X, \\ {\rm Re} \left(I^\* I\_X+ K^\I\_Z\right)&=&h\_X h\_{XX}=\frac{1}{2}\left(h\_X^2\right)\_X.\end{aligned}$$ The corresponding terms are proportional to Ω2. Other averaged terms in Eqs. ([aver]) are proportional to Ω, and thus they can be safely neglected. The calculation gives [u-hifr] $$\begin{aligned} p&=&\Pi-GZ, \\ \Pi&=&\phi(h)-C h\_{XX} +G\_0 h - \frac{b^2\Omega^2}{2}(h h\_X)\_X, \\ U&=&\frac{1}{2}Z(Z-2h)\left(\Pi+\frac{b^2 \Omega^2}{4} h\_X^2\right)\_X,\end{aligned}$$ which in view of Eq. ([intaver]) results in the following equation governing the evolution of the thin film thickness: $$\begin{aligned} \label{h\_t-hf} \nonumber h\_T&=&\frac{1}{3}\left\{h^3 \left[\phi(h)-C h\_{XX} + G\_0 h \right.\right.\\ && - \left.\left. \frac{b^2\Omega^2}{4} \left(2hh\_{XX}+h\_X^2\right)\right]\_X\right\}\_X.\end{aligned}$$ Since for γ* ≫ 1 $$\begin{aligned} \label{large} f\_r(\alpha h)&\approx 1-\gamma^{-1}, \ Q\_1 = e.s.t., \ Q\_2 \approx -\frac{1}{3} -\frac{3}{\gamma^2},\end{aligned}$$ Eq. ([ht-hf]) also follows directly from Eq. ([h-avt]) in the high frequency approximation. Here “e.s.t.” denotes the exponentially small term. Obviously, at finite b and large Ω the vibration determines the film dynamics at large. One has to assume that b ≪ 1 to retain the competition of the surface tension and gravity in the evolution of the film. Such small amplitude, high frequency vibration has been subject of many papers. Equation ([ht-hf]) coincides with the equation derived, in the high frequency approximation, by Lapuerta et al. (see also Ref. ). Formally, we obtain that the same equation remains valid at larger amplitudes, but in fact finite values of b cannot be reached for Ω ≫ 1 due to the instability of the pulsatile motion (see Ref. and Sec. [sec:stab]). Finally, we recall from Sec. [ssec:splitting] that the high frequency limit is equivalent to the approximation of thick fluid layer. Thus the Eq. ([ht-hf]), strictly speaking, can be applied only with ϕ = 0. 3D case ======= In this section we generalize the theory to the 3D case. The starting point is the 3D analogue of Eqs. ([perturbeq]) and ([perturbbcs]): [perturbeq3D] $$\begin{aligned} W\_Z &=& -{\bf \nabla} \cdot {\bf U}, \\ \nonumber \Omega {\bf U}\_\tau &=& -{\bf \nabla}p + {\bf U}\_{ZZ} + \epsilon^2 \left(\nabla^2{\bf U} - {\bf U}\_T\right) \\ && -\epsilon^2 \left({\bf U} \cdot {\bf \nabla} {\bf U} + W {\bf U}\_Z \right), \\ p\_Z &=&- G\_0 - B \Omega^2\cos\tau + \epsilon^2 \left( W\_{ZZ}- \Omega W\_\tau\right),\end{aligned}$$ [perturbbcs3D] $$\begin{aligned} {\bf U} &=& 0, \ W = 0 \ \ {\rm at} \ Z = 0, \\ \nonumber {\bf U}\_Z &=&- \epsilon^2 \left[{\bf \nabla} W - 2 {\bf \nabla}h {\bf \nabla} \cdot {\bf U} - {\bf \nabla}h \cdot \left({\bf \nabla U}+{\bf \nabla U}^T\right)\right],\\ \nonumber \Omega h\_\tau &=& \epsilon^2 \left(W - h\_T - {\bf U}\cdot {\bf \nabla} h \right),\\ \nonumber p &=& \phi - C \nabla^2 h \\ && +\epsilon^2 \left[\frac{3}{2} C \left(\nabla h\right)^2 \nabla^2 h + 2 W\_Z\right] \ {\rm at} \ Z = h.\end{aligned}$$ Here ∇ = (∂X, ∂Y, 0) is the 2D gradient, ${\bf U}$ is the projection of the velocity onto the X − Y plane, i.e. ${\bf v} = \epsilon {\bf U} + \epsilon^2 W {\bf e\_z}$, $\left({\bf \nabla U}\right)$ is second-order tensor [i.e., $\left({\bf \nabla U}\right)\_{jl}=\partial U\_l/\partial x\_j,\ j,l=1,2$], and other notations are unchanged. It is easy to see that in the leading order (ε0) the only difference between the systems ([perturbeq]) and ([perturbbcs]) (2D case) and ([perturbeq3D]) and ([perturbbcs3D]) (3D case) is the replacement of U by ${\bf U}$ and ∂X by ${\bf \nabla}$. Less evident is that only same changes are warranted in the solution as well. This can be easily checked by repeating the analysis quite similar to the one presented in Secs. [sec:first]-[sec:limit]. Here we show the results only. The solution of the problem for the pulsations is [cf. Eqs. ([solpuls])]: [solpuls3D] $$\begin{aligned} q&=&\Omega (h-Z), \ {\bf I}=i \left[1-\frac{\cos\alpha (h-Z)}{\cos\alpha h}\right]{\bf \nabla}h, \\ \nonumber K&=&-i\left[Z+\frac{\sin\alpha (h-Z)-\sin\alpha h}{\alpha \cos\alpha h}\right]\nabla^2 h \\ && +i\frac{1-\cos\alpha Z}{\cos^2\alpha h}\left({\bf \nabla}h\right)^2,\\ H&=&-{\bf \nabla} \cdot \left[h f(\alpha h){\bf \nabla}h \right],\end{aligned}$$ while the averaged dynamics of the free surface is governed by the following equation: [h-avt3D] $$\begin{aligned} h\_T&=&{\bf \nabla}\cdot \left[\frac{1}{3}h^3 {\bf \nabla} \Pi-\frac{b^2\Omega^2}{2}{\bf Q}(h)\right], \\ \nonumber \Pi& \equiv& p(Z=0)\\ &=& \phi(h)-C \nabla^2h + G\_0 h +\frac{b^2\Omega^2}{2}{\rm Re} H.\\ {\bf Q}&=&Q\_1(\gamma)h^2\left(\nabla h\right)^2{\bf \nabla}h +Q\_2(\gamma)h^3 \nabla^2 h {\bf \nabla}h,\end{aligned}$$ with Q1, 2 and f given by Eqs. ([coeff])-([fgam]), respectively. In the limiting cases Eq. ([h-avt3D]) simplifies as follows: (i) Low frequency, Ω ≪ 1 $$\begin{aligned} \label{h\_aver\_low\_3D} \nonumber h\_T &=&\frac{1}{3}{\bf \nabla}\cdot\left(h^3\ {\bf \nabla}\Pi\right) \\ && + \left(b \Omega^2\right)^2 {\bf \nabla}\cdot \left[\frac{2h^7}{63} \nabla^2h {\bf \nabla}h-\frac{h^6}{15} \left(\nabla h\right)^2 {\bf \nabla}h\right],\\ \Pi&=&\phi(h)-C \nabla^2h +G\_0 h - \frac{b^2\Omega^4}{15}{\bf \nabla}\cdot (h^5 {\bf \nabla}h).\end{aligned}$$ (ii) High frequency, Ω ≫ 1 $$\begin{aligned} \label{h\_t-hf\_3D} h\_T&=&\frac{1}{3}{\bf \nabla}\cdot \left\{h^3 {\bf \nabla}\left[\Pi+ \frac{b^2\Omega^2}{4}\left({\bf \nabla}h\right)^2\right]\right\}, \\ \Pi&=&\phi(h)-C \nabla^2h +G\_0 h - \frac{b^2\Omega^2}{2}{\bf \nabla}\cdot\left(h {\bf \nabla}h\right).\end{aligned}$$ Omitting the van der Waals interaction potential ϕ(h) one immediately reduces the last equation to Eqs. (3.11) and (3.12) in Ref. [see also Eq. (3.19) in Ref. at Ma = 0]. Stability of the pulsatile flow =============================== Analysis of the deformable mode ------------------------------- In this section we analyze stability of the pulsatile motion given by Eqs. ([solpuls]). It was mentioned already in Sec. [sec:puls] that there is no any doubt about its stability within the framework of the longwave approximation, Eqs. ([pulsfull]). However, the question whether this motion is stable with respect to perturbations with shorter wavelength is quite reasonable, especially in view of unusual scaling ([amplit]): the amplitude of the vibration is large (of order ε− 1) and one can expect the emergence of the parametric instability. Therefore, we return to the governing equations ([baseeq]) and ([basebcs]) and study the stability of the “base state”: $$\begin{aligned} \label{stab\_basic} p\_0&=&\epsilon^{-1}b\Omega{\rm Re} \left[ q(\epsilon x,z)\exp\left(i\Omega t\right)\right] + O(1),\\ u\_0&=&b\Omega{\rm Re} \left[I(\epsilon x,z)\exp\left(i\Omega t\right)\right] + O(\epsilon),\\ w\_0&=&O(\epsilon), h\_0=h\_0(\epsilon x,\epsilon^2 t)+O(\epsilon).\end{aligned}$$ [See Eqs. ([presscal]) and ([rescaledvelo]) for scalings and Eqs. ([pulscomplex]) for the pulsation velocity and pressure.] We restrict analysis to the 2D base state since the stability problem for the 3D base state [h0 = h0(εx, εy, ε2t)] admits reduction to the one with the 2D base state. This will be shown below. Introducing small perturbations and linearizing the problem near the base state we obtain: [perteq] $$\begin{aligned} u\_x + w\_z &=& 0, \\ u\_t + u\_0 u\_x + w u\_{0z} &=& - p\_x + \nabla ^2 u, \\ w\_t + u\_0 w\_x &=& - p\_z + \nabla ^2 w,\end{aligned}$$ [pertbcs] $$\begin{aligned} u &=& 0, \ w = 0 \ \ {\rm at} \ z = 0, \\ \nonumber \xi\_t&=& w -u\_0 \xi\_x,\ \ u\_z + w\_x =-u\_{0zz}\xi,\\ \nonumber p &=& \left[\epsilon^{-1} b\Omega^2 \cos\Omega t + G\_0+ \phi'(h\_0)\right]\xi \\ &&- \epsilon^{-2}C \xi\_{xx} - 2w\_z =0 \ {\rm at} \ z = h\_0,\end{aligned}$$ where ξ is the perturbation to the surface deflection, and the obvious notations {p, u, w} are used for other perturbations. Again, we will show below that the 3D perturbations do not warrant the consideration. Note that for this analysis one can safely neglect variation of h0 on the time scale T = ε2t and on the length scale X = εx. Thus the unperturbed surface is assumed to be locally flat. Same approximation is also appropriate for the velocity components, thus we consider the plane-parallel flow with vanishing transversal component w0 and the longitudinal component nearly constant in x. The obtained problem is quite similar to the problem governing the Faraday instability (see, for example, Ref. ). The only difference is the presence of the base flow u0 in Eqs. ([perteq]) and ([pertbcs]). Presenting all fields in the form of normal perturbations $\left(u,w,p,\xi\right)=\left(\hat u(t,z),\hat w(t,z),\hat p(t,z),\hat \xi(t,z)\right)e^{ikx}$, where k is the wavenumber of the perturbation, we obtain the following problem for the amplitudes: [perteqk] $$\begin{aligned} ik \hat u + \hat w\_z &=& 0, \\ \hat u\_t + ik u\_0 \hat u + \hat w u\_{0z} &=& - ik\hat p + \hat u\_{zz}-k^2\hat u, \\ \hat w\_t + ik u\_0 \hat w &=& - \hat p\_z + \hat w\_{zz}-k^2\hat w,\end{aligned}$$ [pertbcsk] $$\begin{aligned} \hat u &=& 0, \ \hat w = 0 \ \ {\rm at} \ z = 0, \\ \nonumber \hat \xi\_t&=& \hat w - ik u\_0 \hat \xi,\ \ \hat u\_z + ik \hat w =-u\_{0zz}\hat \xi,\\ \nonumber \hat p &=& \left[\epsilon^{-1} b\Omega^2 \cos\Omega t + G\_0+ \phi'(h\_0)\right]\hat \xi \\ &&+ \epsilon^{-2}Ck^2 \hat \xi - 2\hat w\_z =0 \ {\rm at} \ z = h\_0.\end{aligned}$$ This problem contains terms of different orders with respect to small parameter ε, which simplifies the analysis. Indeed, the term ε− 2Cξxx prevails in the boundary condition and produces the stabilizing effect, since it suppresses deviations of the surface. Instability may occur when this term is comparable to the potentially destabilizing term ε− 1bΩ2cosΩt in the same boundary condition. This is only possible for long waves with $k=\sqrt{\epsilon}K$. It can be shown easily that for other k the perturbations decay, except for the case considered in Sec. [ssec:nondef]. Choosing the following scalings for the perturbations (see Ref. ): $$\begin{aligned} \label{pert\_sca} \hat p=\hat P, \ \hat u = \sqrt{\epsilon} \hat U, \ \hat w=\epsilon \hat W, \ \hat \xi=\epsilon \hat \Xi,\end{aligned}$$ we obtain in the leading order [pertMancebo] $$\begin{aligned} \hat U\_t &=&- iK\hat P + \hat U\_{zz}, \ iK \hat U + \hat W\_z = 0, \ \hat P\_z = 0,\\ \hat u &=& 0, \ \hat w = 0 \ \ {\rm at} \ z = 0, \\ \nonumber \hat \Xi\_t&=& \hat W, \ \hat U\_z =0,\\ \hat P &=& b\Omega^2 \cos\Omega t \hat \Xi + C K^2 \hat \Xi =0 \ {\rm at} \ z = h\_0.\end{aligned}$$ It can be seen that in Eqs. ([perteqk]) and ([pertbcsk]) the terms containing base flow u0 are of low order, and thus they dropped out of Eqs. ([pertMancebo]). This means that within the framework of scaling ([pertsca]) Faraday instability prevails over the instability due to shear flow (see Sec. [ssec:nondef] for the opposite case). Next, it is evident that there is no preferential direction in the x − y plane for the problem ([pertMancebo]). This allows the reduction of the stability with respect to 3D perturbations to 2D problem under consideration: one needs only to choose the x-axis in the direction of perturbations’ wavevector. Moreover, since the velocity of the base state also dropped out from the leading order of the stability problem, the symmetry properties of the base state are inessential. Therefore, even for 3D base state the stability is governed by the same Eqs. ([pertMancebo]). Thus, we showed that the problem under consideration completely reduces to the analysis of Faraday instability. Such analysis was performed in detail by Mancebo and Vega. Equations ([pertMancebo]) can be rewritten in the form of Eqs. (2.11)-(2.13) in their paper. Indeed, the case considered here corresponds to the case B.1.2 (Long-wave limit) in their paper. Introducing the parameters as in Ref. , we obtain: $$\begin{aligned} \label{par\_Mancebo} \tilde \omega\_{MV}=\Omega h\_0^2, \ \tilde a\_{MV}=\frac{b}{\sqrt{C}h\_0^{3/2}}, \ \gamma\_{MV} = \epsilon \frac{G\_0}{\sqrt{C}}h\_0^{5/2} \ll 1.\end{aligned}$$ Here the subscript “MV” is used to mark the parameters used by Mancebo and Vega. Note that these parameters must be calculated using the local thickness $\hat H=\hat H\_0 h\_0$ instead of the mean value $\hat H\_0$, which causes the appearance of h0 in Eqs. ([parMancebo]). The critical value of the acceleration $\tilde a\_{MV,c} \tilde \omega\_{MV}^2$ as a function of $\tilde \omega\_{MV}$ is presented in Fig. 4 of Ref. . Thus, the critical value of the amplitude is $$\label{b\_c} b\_c=\frac{\sqrt{C}\Phi\_{MV}(\Omega h\_0^2)}{\Omega^2 h\_0^{5/2}},$$ where Φ(μ) is the function given in Ref. (Fig. 4 there). Due to Eq. ([parMancebo]) we are interested only in the line corresponding to γMV = 0. At μ → 0, ΦMV(μ) ≈ 8.5, then it decreases to approximately 5.5 at μ ≈ 6 and after that grows. At large $\tilde \omega\_{MV}$ the asymptotic formula $$\Phi\_{MV}(\mu) \approx \sqrt{\mu}$$ holds. It means that $$b\_c\left(\Omega h\_0\right)^{3/2}C^{-1/2} \to 1 \ {\rm at} \ \Omega \to \infty.$$ Using Eqs. ([h-avt]), ([haverlow]), or ([ht-hf]) one has to ensure that b < bc. Value of bc can be extracted from the mentioned figure at given Ω. Note that the stability condition should be valid at any X, therefore the value of h0, which minimizes ΦMV(Ωh02)h0− 5/2 (at each time moment) must be substituted in Eq. ([bc]) to determine whether the layer is stable or not. For most of cases, except for 3 < Ω < 8, this means that the maximum value of h0 should be used. In the opposite case the minimum value of ΦMV(μ) ( ≈ 5.5) and again the maximum of h0 can be used in order to estimate bc. For the case of low frequency one can take the acceleration bΩ2 up to $$\label{b\_c\_smOm} b\_c \Omega^2 \approx 8.5\sqrt{C}h\_0^{-5/2}$$ in Eqs. ([haverlow]). In the opposite limiting case, Ω ≫ 1, the parameter bΩ, which enters Eq. ([ht-hf]), should be small: $$\label{b\_c\_largeOm} b\Omega < b\_c\Omega = \sqrt{\frac{C}{\Omega}h\_0^3} \ll 1.$$ The inequality ([bclargeOm]) holds in the limiting case 1 ≪ Ω ≪ ε− 1, where the Faraday instability is caused by the longwave perturbations and the dissipation is negligible beyond the boundary layers near the rigid wall and free surface. Similar to Eq. ([bclargeOm]) the stability bound bcΩ is small even for Ω = O(ε− 1); the perturbations with moderate wavelength are critical in this case. Thus, in this case bΩ must also be small to prevent the parametric instability. For very large frequency, i.e. Ω ≫ ε− 1, the volume dissipation prevails, which leads to another limitation on the amplitude. Indeed, it follows from Ref. that the inequality $$\tilde b \sqrt{\frac{\omega}{\nu}} < A\_{MV}\left(\frac{\omega\rho^2\nu^3}{\sigma^2}\right)$$ (in dimensional form) or $$b \sqrt{\Omega} < \epsilon A\_{MV}\left(\Omega\epsilon^4C^{-2}\right)$$ (in dimensionless form) must hold true to prevent the Faraday instability. Here AMV(μ) is the function obtained by Mancebo and Vega, which has the following asymptotics: $$A\_{MV}\left(\mu\right)\approx \left(256\mu\right)^{1/6} \ {\rm at} \ \mu \ll 1.$$ This gives the following critical velocity: $$b\_c\sqrt{\Omega} =\epsilon\left(\frac{256\Omega\epsilon^4}{C^2}\right)^{1/6}=\left(\frac{16\sqrt{\Omega}\epsilon^5}{C}\right)^{1/3}.$$ Multiplying this relation by $\sqrt{\Omega}$ we obtain: $$b\_c\Omega =\left(\frac{16\Omega^2\epsilon^5}{C}\right)^{1/3},$$ i.e. the vibration is non-negligible in Eq. ([ht-hf]) at high frequency only if Ω2ε5 is finite or large. This provides, of course, the usual limitation on the amplitude and the frequency of the vibration in the case of inviscid pulsatile motion (see also Refs. ). It must be emphasized that advective terms, being proportional to u0, are not important for short waves, since the advective term iku0u remains small in comparison with ut because of the dispersion relation ω2 = Cak3 at large k. Analysis of the nondeformable mode ---------------------------------- The above stability analysis deals with the capillary waves, which are based on the surface deviations. If b < bc, where bc is given by Eq. ([bc]), the perturbations of this type decay. However, there is also a mode, which corresponds to the nondeformable surface. Indeed, it is obvious that the finite-amplitude plane-parallel flow with the profile u0 becomes unstable at certain intensity of the motion. Let us analyze this mode in detail. Setting $\hat \xi=0$ in Eqs. ([perteqk]) and ([pertbcsk]) gives: [ndfmperteq] $$\begin{aligned} ik \hat u + \hat w\_z &=& 0, \\ \hat u\_t+ik u\_0 \hat u + \hat w u\_{0z} &=& - ik \hat p + \nabla ^2 \hat u, \\ \hat w\_t +ik u\_0 \hat w &=& - \hat p\_z + \nabla^2 \hat w,\\ \hat u &=& 0, \ \hat w = 0 \ \ {\rm at} \ z = 0, \\ \label{bc\_fs\_pert} \hat w&=&0, \ \hat u\_z =0 \ {\rm at} \ z = h\_0.\end{aligned}$$ This problem is one of stability for the periodic in time, plane-parallel flow. In view of Eqs. ([solpuls]) and ([stabbasic]) this problem is characterized only by the parameters b, Ω and “local” values of h0 and h0X. Due to well-known Squire theorem there is no need in analysis of 3D perturbations – 2D one are critical. Moreover, the base state should not necessarily be 2D in the entire layer. It is sufficient that the flow is locally 2D at any point in the X − Y plane. Note that according to the analysis in Appendix [app:p-flow] the problem ([ndfmperteq]) for the base flow ([stabbasic]), but posed on the interval 0 < z < 2h0 and with the no-slip condition at z = 2h0 instead of Eq. ([bcfspert]), is the conventional stability problem of the oscillatory Poiseuille flow (i.e., the flow which arises due to the periodically oscillating longitudinal pressure gradient Px = P1cosΩt). The more general problem, where the pressure gradient equals P0 + P1cosΩt, has been investigated already, see Refs. and references therein. Due to symmetry the latter problem can be split into problems for “even” and for “odd” perturbations, meaning that w is even or odd function of the coordinate z − h0. For the odd mode both w and uz vanish at z = h0, which coincides with the boundary condition ([bcfspert]). Therefore, the problem under consideration is the particular case of the stability problem for the oscillatory Poiseuille flow. However, to the best of our knowledge there is no detailed analysis of the flow stability for the particular case we need, i.e. P0 = 0. It is easy to consider two limiting cases, i.e. the high frequency and the low frequency. The first case, in view of Eq. ([highfreqpuls]) and ([sol-bound]) is reduced to the stability of a Stokes layer. The latter is known to be stable. At Ω → 0 we have u0 ∝ bΩ2 from Eqs. ([pulscomplex]) and ([solpulslf]). On the other hand in this case we can “freeze” the evolution of the flow and assume that $$\label{u0\_om0} u\_0=-\frac{b \Omega^2}{2} h\_X z(2h\_0-z).$$ This means that we assume the frequency so low that the perturbations either grow or decay before the flow in fixed point of the layer will change itself. In view of the above-mentioned symmetry properties, the problem ([ndfmperteq]) with u0 given by Eq. ([u0om0]) is identical to the stability problem for the stationary Poiseuille flow in the entire layer (h0 being the half of the layer thickness), but only for the odd perturbations. If we introduce the Reynolds number based on the velocity of the flow at z = h0, we obtain: $$Re=\frac{b\Omega^2 h\_X h\_0^3}{2}.$$ [It follows from Eq. ([ndfmperteq]) that the viscosity is equal to unity.] It is known that the critical value of the Reynolds number is 5772, i.e. the flow remains stable for Re < Rec = 5772. Thus the flow forced by low frequency vibration is stable if $$b\Omega^2 <b\_{c1}\Omega^2 =\frac{11542}{h\_X h\_0^3}.$$ Of course, this limitation is less severe than Eq. ([bcsmOm]) for any reasonable value of hX. The product hXh03 should be maximized over the longitudinal coordinate X at each moment of time. Thus, the pulsatile flow is stable at low frequency up to finite value of the acceleration, i.e. the vibration can produce the finite impact in this limiting case. The complete analysis of the problem ([ndfmperteq]) will be performed elsewhere. This research will provide the threshold value of the amplitude Rec(Ωh02) = bc1Ω∣I(αh0)∣h0hX. (We again define the Reynolds number via the velocity at the center of the layer, z = h0.) Recall that due to the minimization of bc1, the maximal value of ∣I(αh0)∣h0hX should be used in Eq. ([bcPois]). To conclude, we showed in this section that there exists an upper bound b = bc below which the pulsatile flow is stable. The further analysis as well as the results of Secs. [sec:puls]-[sec:3D] are based on the assumption b < bc. Evolution of the perturbations to the flat layer ================================================ Linear stability analysis ------------------------- The amplitude equation ([h-avt]) has the obvious solution h0 = 1 corresponding to the equilibrium state. It follows from Eqs. ([solpuls]) and ([aver]) that [basic] $$\begin{aligned} h\_0&=&1, \ p\_0=G\_0(1-Z)+\phi(1), \ q\_0=\Omega(1-Z), \\ U\_0&=&W\_0=I\_0=K\_0=H\_0=0.\end{aligned}$$ Thus in the reference frame of the substrate the fluid is motionless – vibration only adds the oscillatory component to the pressure field. Let in Eq. ([h-avt]) h = 1 + ξ, where ξ is small perturbation. Linearizing with respect to ξ we obtain: $$\label{xi\_t\_aver} \xi\_T=\frac{1}{3}\left\{ \left[\phi'(1)+ G\_0\right]\xi-\left[C+\frac{b^2 \Omega^2 f\_r(\alpha)}{2}\right] \xi\_{XX} \right\}\_{XX},$$ where α2 = − iΩ and f(y) is given by Eq. ([fdefinition]). The linear stability of the layer without accounting for van der Waals attraction, and for the opposite direction of the gravity field was studied by Lapuerta et al.. In this setup the Rayleigh-Taylor instability emerges. The equations governing the dynamics of small perturbations derived in Ref. [see Eqs. (2.35) there] can be obtained from Eq. ([xitaver]). However, even in this case there is an important difference in the interpretation of the results. Lapuerta et al. address the case of finite vibration amplitude B and consequently, they find that the influence of the vibration is small. To gain a finite impact of the vibration they proceed to the detailed analysis in the limit of high vibration frequencies, Ω ≫ 1. Conversely, in this paper large vibration amplitude is considered and we show that the Eq. ([xitaver]) remains valid even in this case. Thus, we extend the domain of applicability of the results obtained by Lapuerta et al. by showing that the finite impact of the vibration is possible even at moderate frequencies, which is most important for thin films. The typical stability curves are shown in Ref. : Figure 4 there presents the dependence of the dimensionless amplitude of the vibration AL on the dimensionless frequency ωL for different values of the parameter αL, which is proportional to the surface tension. The results of our linear stability analysis (for ϕ = 0) can be extracted from this figure when the following substitutions are made: $$A\_L=k\frac{b\Omega^2}{3A-G\_0}, \ \omega\_L=\Omega, \ \alpha\_L=\frac{k^2C}{3A-G\_0}.$$ [Recall that ϕ(h) = Ah− 3.] However, to avoid the recalculation we present the results of stability analysis below. Seeking the solution in the form of a plane wave ξ = ξ0exp( − λT + ikX), where λ is the decay rate and k is the wavenumber, we obtain $$\label{decay} \lambda=\frac{1}{3}k^2\left\{ \phi^{\prime}(1) +G\_0+k^2\left[C+\frac{1}{2}b^2 \Omega^2 f\_r(\alpha)\right]\right\}.$$ The stability criteria (λ = 0) is $$\label{stab0} \phi'(1)+ G\_0 + \left[C+\frac{1}{2}b^2 \Omega^2f\_r(\alpha)\right] k^2=0.$$ Thus the vibration and the surface tension do not damp the longwave instability: the perturbations with small k grow at ϕʹ(1) + G0 < 0. However, in confined cavities with large aspect ratios the spectrum of the wavenumbers is discrete and bounded from below. Therefore the impact of the vibration and the surface tension becomes determinative in this case. From Eq. ([stab0]) one can see that the critical value of the wavenumber, kc (i.e the value that corresponds to vanishing growth rate of the perturbation) becomes smaller due to vibration: $$\label{kc} k\_c^2=-\frac{\phi'(1)+ G\_0} {C+\frac{1}{2}b^2 \Omega^2f\_r(\alpha)}.$$ Again we note that fr grows monotonically from zero (at Ω → 0) to unity (at Ω → ∞), see Fig. [fig:Qf](a). Thus the vibration leads to the stabilization of the thin film. This stabilization effect is obviously augmented with the increase of the frequency, even when this increase is accompanied by the decrease of b to keep fixed the power of the vibration, b2Ω2. At large Ω Eq. ([stab0]) reduces to $$\label{stab\_hf} \phi'(1)+ G\_0 + \left(C+\frac{b^2 \Omega^2}{2}\right) k^2=0.$$ A similar equation (without the first term and with negative G0) is the well-known result on the suppression of the Rayleigh-Taylor instability. [!t] [fig:decay] [fig:kc] Introduction of the rescaled wavenumber $\tilde k=k\sqrt{C/3A}$ and the growth rate $\tilde\lambda=\lambda C/\left(3A^2\right) $ results in the following expressions for $\tilde\lambda$: $$\label{decay1} \tilde \lambda=\tilde k^2\left\{\tilde G\_0-1 +\tilde k^2\left[1+V f\_r(\alpha)\right]\right\},$$ and for the critical wavenumber: $$\label{kc\_resc} \tilde k\_c^2=\frac{1- \tilde G\_0} {1+V f\_r(\alpha)}.$$ Here $$\label{resc\_par} \tilde G\_0= \frac{G\_0}{3A}, \ V=\frac{b^2 \Omega^2}{2C}=\frac{B^2 \Omega^2}{2Ca}$$ are the rescaled Galileo number and the vibrational parameter, respectively. The Galileo number usually is quite small for thin films. We take $\tilde G\_0=3.3\cdot 10^{-4}$. This corresponds to A′ = 6π ⋅ 10− 21J (value for water) and $\hat H\_0=1000 \AA$. Dependence of $\tilde \lambda(\tilde k)$ is shown in Fig. [fig:decay] for various values of Ω and V. Clearly, $\tilde k\_c$ decreases with V. Moreover, as it is shown in Fig. [fig:kc], kc also decreases with growth of Ω. Thus the vibration stabilizes the film. And, stabilization is more pronounced for larger values of Ω. Weakly nonlinear analysis ------------------------- Let consider the behavior of perturbations near the stability threshold determined in the previous subsection. For this purpose it is convenient to rescale the time and the coordinate as follows: $$\tilde X = \sqrt{\frac{3A}{C}} X, \ \tilde T= \frac{3A^2}{C} T.$$ Substituting these relations into Eqs. ([h-avt]) one can obtain [h-avt-resc] $$\begin{aligned} h\_{\tilde T}&=&\left(h^3 \tilde\Pi\_{\tilde X}-3V Q\right)\_{\tilde X}, \\ \tilde \Pi& \equiv& \frac{1}{3h^3}- h\_{\tilde X \tilde X} + \tilde G\_0 h +V{\rm Re} H,\\ Q&=&Q\_1(\gamma)h^2h\_{\tilde X}^3+Q\_2(\gamma)h^3h\_{\tilde X} h\_{\tilde X \tilde X},\end{aligned}$$ where Q1, 2 are given by Eqs. ([coeff]) and the rescaled parameters defined by Eq. ([rescpar]) are used. Below we use only the rescaled coordinate and the time. Thus we omit the tildes above X and T. In Sec. [ssec:linear] it has been shown that the growth rate is real at the stability threshold. Consequently, the branching solution is stationary, i.e. one can omit the left-hand side of the amplitude equation to study direction of branching only. This also allows us to integrate Eq. ([h-avt-resc]) once. Besides, seeking the solution with fixed wavenumber $\tilde k$ we introduce the variable $\zeta=\tilde k X$. The surface deflection h now is a 2π-periodic function of ζ, and it solves the following equation: $$\label{h-av\_stat} h^3 \left(\frac{1}{3h^3}- \tilde k^2 h\_{\zeta\zeta} + \tilde G\_0 h +V \tilde k^2{\rm Re} H\right)\_\zeta-3V \tilde k^2 Q=const.$$ Note that X must be replaced with ζ in the expressions for Q1, 2 and H. Next, we expand h near the base solution ([basic]) as follows: h = 1 + δh1 + δ2h2 + …. The wavenumber is assumed close to the critical value k0, i.e. $$\tilde k=\tilde k\_0+\delta^2 \tilde k\_2 + \ldots.$$ Substituting these expansions into Eq. ([h-avstat]), we arrive at order zero to $$\hat L h\_1\equiv (1-\tilde G\_0)h\_1^\prime+\tilde k\_0^2\left[1+V f\_r(\alpha)\right]h\_1^{\prime\prime\prime}=0,$$ where the prime denotes the derivative with respect to ζ. Solution of this equation is h1 = acosζ. The wavenumber is given by the expression $$\tilde k\_0^2=\frac{1-\tilde G\_0}{1+V f\_r(\alpha)},$$ which obviously coincides with Eq. ([kcresc]). Recall that we are interested in the case $\tilde G\_0 < 1$ which holds true for the thin film. The equation at the second order in δ has the form: $$\begin{aligned} \nonumber \hat L h\_2&=&2(h\_1^2)^\prime+V \tilde k\_0^2F\_1 \left(h\_1 h\_1^{\prime\prime\prime}+3h\_1^\prime h\_1^{\prime\prime}\right) \\ && -3V\tilde k\_0^2Q\_2h\_1^\prime h\_1^{\prime\prime}.\end{aligned}$$ The solution of this equation is $$\begin{aligned} h\_2&=&C\_1 a^2 \cos 2\zeta, \\ \nonumber C\_1&=&-\frac{1}{3(1-\tilde G\_0)}\left[1+V \tilde k\_0^2\left(F\_1+\frac{3}{4}Q\_2\right)\right].\end{aligned}$$ Hereafter $$F\_1=\left(\frac{dF}{dh}\right)\_{h=1}, \ F\_2=\left(\frac{d^2F}{dh^2}\right)\_{h=1}, \ F\equiv hf\_r(\alpha h).$$ At the third order we need only the solvability condition. (We do not present here the corresponding equation for h3.) This condition couples the correction to the wavenumber k2 to the amplitude of the perturbation a, as follows: $$\tilde k\_2=\frac{\tilde k\_0a^2}{2\left(1-\tilde G\_0\right)}\left[\frac{5}{2}-2C\_1+V\tilde k\_0^2\left(\frac{9}{4}Q\_1+3Q\_2C\_1-\frac{3}{4}Q\_2^\prime-\frac{1}{2}C\_1F\_1-\frac{1}{8}F\_2\right)\right].$$ Our numerical simulations show that the expression in the square brackets is always positive. Thus $\tilde k\_2>0$, i.e. a small amplitude solution exists at $\tilde k>\tilde k\_0$. This means that the subcritical bifurcation takes place. To summarize, the solution emerging at $\tilde k = \tilde k\_0$ is unstable and a finite amplitude excitation (probably leading to rupture) is expected. Nonlinear evolution of perturbations ------------------------------------ In this Section we analyze the finite-amplitude deflections of the free surface, starting from stationary solutions hs of the amplitude equation. For this purpose we look for the periodic (with respect to ζ) solutions of Eq. ([h-avstat]). Due to evident symmetry properties of the solution one can integrate Eq. ([h-avstat]) over half of the period, vanishing all the odd derivatives at ζ = 0, π. This means that const. = 0 in Eq. ([h-avstat]). We also have two boundary conditions: $$\label{bc\_zeta} h\_s^{\prime}=0 \ {\rm at} \ \zeta=0,\pi.$$ Besides, the stationary solution conserves the liquid volume: ∫0π(hs − 1)dζ = 0. This provides the third boundary condition for the third order ODE, Eq. ([h-avstat]), completing the problem statement. [!t] [fig:shapestat] [!t] [fig:hstat] The shooting method is applied to numerically integrate this boundary value problem. Some results are presented in Figs. [fig:shapestat] and [fig:hstat]. It is clearly seen that only the subcritical (and, consequently, unstable) solution branch is present, i.e. there is no bifurcation except the inverse pitchfork bifurcation studied in Sec. [ssec:weakly]. Therefore, the lower branches in Fig. [fig:hstat] are the boundaries of domains of attraction: the initial perturbation with min[h(ζ, T = 0)] > hm decays and the free surface becomes flat, while in the opposite case the free surface is attracted to the solid, which leads to rupture. These predictions are in good agreement with the results of the numerical simulation of Eqs. ([h-avt-resc]), as shown in Fig. [fig:evol]. [fig:evol] The upper branches in Figs. [fig:hstat] are of importance for the stability of the oscillatory flow (see Sec. [sec:stab]). Indeed, Eq. ([bc]) requires the maximal value of the surface deviation h0, which can be extracted from Fig. [fig:hstat]. It is important to point out that the increase of the vibration amplitude amplifies the film stability: at fixed $\tilde k$ and with Ω increasing, the initial deviation of the flat surface decays at larger values of V. The only exception is provided by Ω = 2. In this case hm is nonmonotonic function of V starting from certain value of $\tilde k$. Also it is interesting that the maximal deviation of the surface decreases with growth of V: the surface tends to become flat as it is shown in Fig. [fig:shapestat]. In some sense this tendency is reminiscent of the known averaged behavior : the free surface/interface tries to orient normally to the vibration axis. But it is necessary to keep in mind that the surfaces shown in Fig. [fig:shapestat] correspond to unstable states. Summary ======= In this paper the impacts of the vertical vibration on the dynamics of the thin liquid film are analyzed. The set of equations governing the averaged dynamics of the fluid flow and the nonlinear, fourth-order amplitude equation ([h-avt]) [or ([h-avt3D]) in 3D case] describing the averaged evolution of the film thickness are obtained in the lubrication approximation. We use the paradoxical scaling (at least at the first glance), assuming (i) the vibration period is comparable to the characteristic time of the momentum relaxation across the layer and (ii) the vibration amplitude is large in comparison with the mean layer thickness. The first condition (termed ``moderate frequency“ or ``finite frequency”) allows us to consider ultrathin liquid layers within the framework of the averaging method. The second condition warrants that the impact of the vibration is not vanishingly small. Using the results from Refs. we prove that these assumptions do not necessarily lead to the parametric instability. Indeed, we show that the stability problem for the pulsatile flow can be separated into the problems for the deformable and non-deformable modes. The former problem reduces to the analysis of Faraday instability, while the latter problem reduces to the stability analysis of the oscillatory Poiseuille flow. Analyzing well known results obtained for these two problems we deduce that there exists a window of stability: below certain threshold intensity of the vibration [Eqs. ([bc]) and ([bcPois])] the pulsatile flow is stable, while the averaged effects are well pronounced. The analyses of the averaged dynamics of the thin film demonstrate the strong stabilizing impact of the vibration. First, the vibration damps the short-wavelength instability. In other words, it decreases the critical cut-off wavenumber kc, such that instability occurs at k < kc only [see Eqs. ([decay]), ([kc]) and Figs. [fig:decay] and [fig:kc]]. In this sense the vibration acts in a way similar to the surface tension. Therefore, in order to prevent a longwave instability one can use a cavity of horizontal size L < Lc ∼ kc− 1, which is larger in presence of the vibration. Second, the vibration augments the domain of attraction of the flat undeformed surface, i.e. larger initial surface deflections decay (see Fig. [fig:hstat]) or, in other words, larger initial distortions of a flat surface are admissible without occurrence of dewetting. Thus, the vertical vibration of moderate frequency is the effective method of control of the thin film instability. This is especially important since the standard (high-frequency) approximation cannot be applied to thin films. Acknowledgments =============== S. S. was partially supported by the Fund “Perm Hydrodynamics”. A. A. A. acknowledges CRDF for the financial support within the framework of grant Y3-MP-09-01. Reduction of pulsatile flow to the oscillatory Poiseuille flow ============================================================== In this section we show that the pulsatile flow given by Eqs. ([pulscomplex]) and ([solpuls]) can be reduced to the well- known oscillatory Poiseuille flow. Such a transformation is useful for the stability analysis carried out in Sec. [ssec:nondef]. Throughout this section we omit the tildes over the velocity and the pressure field, with the understanding that only the oscillatory components are involved. In order to obtain the oscillatory Poiseuille flow we return to the unscaled coordinates x and z, setting locally h = h0 + O(ε), hX = const + O(ε). These equations mean that the layer thickness changes slowly with x. Thus one can assume the constant thickness h0. In view of the scaling ([presscal]) and Eqs. ([pulscomplex]) and ([solpuls]) the longitudinal pressure gradient is bΩhXcosτ. Thus the pressure gradient is spatially uniform but oscillates in time. This means that some kind of the so-called oscillatory Poiseuille flow is under consideration. Neglecting hXX according to Eq. ([hxconst]) we arrive at the following expressions for the amplitudes of pulsations [solhx] $$\begin{aligned} I&=&i h\_X\left(1-\frac{\cos\alpha (h\_0-z)}{\cos\alpha h\_0}\right), \\ K&=&i h\_X^2\frac{1-\cos\alpha Z}{\cos^2\alpha h\_0}, \ H=h\_X^2\tan^2\alpha h\_0,\end{aligned}$$ which along with the scalings ([presscal]) and ([rescaledvelo]) gives: $$\begin{aligned} \label{sol\_hx\_t}u&=&b \Omega h\_X {\rm Re} \left[i \left(1-\frac{\cos\alpha (h\_0-z)}{\cos\alpha h\_0}\right)e^{i\tau}\right], \\ \label{wh} w&=&O(\epsilon), \ h-h\_0=O(\epsilon).\end{aligned}$$ Thus this case corresponds to oscillatory 1D flow in a locally flat layer under the spatially uniform longitudinal gradient of pressure. Note that the second relation in Eqs. ([wh]) justifies the first assumption in Eqs. ([hxconst]). Separating the real and imaginary parts in Eq. ([solhxt]) we arrive at the following expressions for the x-component of the pulsation velocity: $$\begin{aligned} \label{puls\_real} u &=& -b \Omega h\_X \left\{\left[1-\frac{a\_S(h\_0)a\_S(z)+b\_S(h\_0)b\_S(z)}{a\_S^2(h\_0)+b\_S^2(h\_0)} \right]\sin\tau -\frac{a\_S(h\_0)b\_S(z)-a\_S(z)b\_S(h\_0)} {a\_S^2(h\_0)+b\_S^2(h\_0)}\cos \tau\right\},\\ \nonumber a\_S(z)&\equiv &\cos\alpha\_rz\cosh\alpha\_rz, \ b\_S(z)\equiv \sin\alpha\_rz\sinh\alpha\_rz, \ \alpha\_r=\sqrt{\Omega/2}.\end{aligned}$$ Such flow, but in a gap between two rigid boundaries, is well studied – see, for instance, the recent papers by Singer et al. and Straatman et al., who address the stability problem for the flow. (Usually the modulated Poiseuille flow is considered, i.e. the pressure gradient oscillates about a non-zero mean value.) Due to symmetry the oscillatory Poiseuille flows in a layer with the free nondeformable surface at z = h0 and in a layer with the upper rigid wall at z = 2h0 are identical. Indeed, in the latter case in the plane of symmetry z = h0 the “no-stress” condition uz = 0 is obviously held. Thus, introducing variable yS = z − h0 in Eq. ([pulsreal]) we obtain the velocity profile of the oscillatory Poiseuille flow [cf. oscillatory part of Eq. (2.2) in Ref. ]. Solution of the set of averaged equations ========================================= Solution of the problem for the averaged fields can be represented as the sum of two solutions. The first one is conventional (see, for example, Ref. ); we mark it by the subscript “c”: [aver-sol1] $$\begin{aligned} \nonumber p\_c &\equiv & \Pi-G\_0Z\\ &=&\phi-C h\_{XX} +G\_0(h-Z) + \frac{b^2\Omega^2}{2}{\rm Re}H, \\ \label{Uc} U\_c&=&\frac{1}{2}Z(Z-2h)\Pi\_X, \\ W\_c&=&-\frac{1}{3}\left[Z^2(Z-3h)\Pi\_X\right]\_X.\end{aligned}$$ Note that this solution leads to the first term in the evolution equation for the film thickness, Eq. ([h-avt]). This part of the solution coincides with Eqs. (2.47) in Ref. up to the averaged correction to the pressure field. The second part is the solution of the nonhomogeneous boundary value problem with the remaining vibration-generated terms at the right-hand sides. Using subscript “v” for this part of solution, we rewrite it in the following form: $$\label{aver-sol2} p\_v=0, \ U\_v=\frac{1}{2}b^2\Omega^2 U^{(v)}, \ W\_v=\frac{1}{2}b^2\Omega^2W^{(v)},$$ where U(v) and W(v) solve the following boundary value problem: $$\begin{aligned} \label{aver-v} U^{(v)}\_{ZZ}&=&{\rm Re} \left(I^\* I\_X+ K^\I\_Z\right), \ W^{(v)}\_Z=-U\_X^{(v)}, \\ U^{(v)}&=&W^{(v)}=0 \ {\rm at} \ Z = 0, \\ U\_Z^{(v)}&=&-\Omega^{-1} {\rm Re} \left(I\_{ZZ}^\ H\right) \ {\rm at} \ Z = h.\end{aligned}$$ This set of equations can be easily integrated. First, $$U\_Z^{(v)}=-\Omega^{-1}{\rm Re} \left[I\_{ZZ}^\(h) H\right]-\int\_Z^h{\rm Re} \left(I^\ I\_X+ K^\I\_Z\right) dZ.$$ Accounting for Eq. ([puls]) of the pulsatile motion and integrating by parts, one can rewrite the last expression as $$U\_Z^{(v)}=-h\_X{\rm Re} H + {\rm Re} \left(I^\K\right) -\partial\_X \int\_Z^h \|I\|^2 dZ.$$ After one more integration we arrive at the following solution: $$\begin{aligned} \nonumber U^{(v)}&=&-Z h\_X{\rm Re} H + \int\_0^Z{\rm Re}\left(I^\K\right)dZ \\&&-\partial\_X \int\_0^Z d\zeta \int\_\zeta^h \|I(X,\xi)\|^2 d \xi.\end{aligned}$$ Evaluation of these integrals leads to the cumbersome formulas, which we do not present here. We also do not present the expression for W(v), as it is not needed in order to obtain the evolution equation for film thickness h. Indeed, this part of the solution results in the term $$-\partial\_X\int\_0^h U^{(v)}dZ-\Omega^{-1}{\rm Re}\{I^\H\}$$ at the right-hand side of such an equation. This term translates to the term Q(h) in Eq. ([h-avt]). 99 U. Thiele, M. Mertig, and W. Pompe, Phys. Rev. Lett. 80, 2869 (1998). M. P. Ida and M. J. Miksis, SIAM J. Appl. Math. 58, 456 (1998). A. Ghatak, R. Khanna, and A. Sharma, J. Colloid Interface Sci. 212 483 (1999). L. M. Pismen, B. Y. Rubinstein, and I. Bazhlekov, Phys. Fluids 12, 480 (2000). T. P. Witelski and A. J. Bernoff, Physica D 147, 155 (2000). R. Seemann, S. Herminghaus, and K. Jacobs, Phys. Rev. Lett. 87, 196101 (2001). A. A. Golovin, B. Y. Rubinstein, and L. M. Pismen, Langmuir 17, 3930 (2001). K. B. Glasner and T. P. Witelski, Phys. Rev. E 67, 016302 (2003). M. Bestehorn, A. Pototsky, and U. Thiele, Eur. Phys. J. B 33, 457 (2003). T. Yi and H. Wong, J. Colloid and Interface Sci. 313, 579 (2007). A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys. 69, 931 (1997). R. Seemann, S. Herminghaus, and K. Jacobs, J. Phys.: Condensed Matter 13, 4925 (2001). G. H. Wolf, Z. Phys. 227, 299 (1969). G. H. Wolf, Phys. Rev. Lett. 24, 444 (1970). V. Kozlov, A. Ivanova, and P. Evesque, Europhys. Lett. 42, 413 (1998). A.A. Ivanova, V. G. Kozlov, and P. Evesque, Izv. RAN. Mekh. Zhidk i Gaza 3, 28 (2001) [Fluid Dynamics 36, 362 (2001)]. G. Z. Gershuni and D. V. Lyubimov, Thermal Vibrational Convection (Wiley, New York, 1998). D. V. Lyubimov, T. P. Lyubimova, and A. A. Cherepanov, Dynamics of interfaces in vibration fields (Fizmatlit, Moscow, 2004) (in Russian). V. Lapuerta, F. J. Mancebo, and J. M. Vega, Phys. Rev. E 64, 016318. U. Thiele, J. M. Vega, and E. Knobloch, J. Fluid Mech. 546, 61 (2006). M.V. Khenner, D.V. Lyubimov, T.S. Belozerova, and B. Roux, Europ. J. Mech. B/Fluids 18 1085 (1999). H. Schlichting, Boundary-Layer Theory, 8th Ed. (Springer, 2000). J. A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems (Springer-Verlag, New York, 1985). A. H. Nayfeh, Perturbaton methods (Wiley, New York, 1973). P. L. Kapitza, Zh. Eksp. Teor. Fiz. 21, 588 (1951) (in Russian). I. I. Blekhman, Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications (World Scientific, Singapore, 2000). Z. C. Feng and Y. H. Su, Phys. Fluids 9, 519 (1997). Z. Zapryanov and S. Tabakova, Dynamics of Bubbles, Drops and Rigid Particles (Kluwer Academic Publishers, Dordrecht, 1999). P. Evesque and J. Rajchenbach, Phys. Rev. Lett. 62, 44 (1989). A. V. Straube, D. V. Lyubimov, and S. V. Shklyaev, Phys. Fluids 18, 053303 (2006). D. V. Lyubimov and G. A. Sedelnikov, Izv. RAN. Mekh. Zhidk. i Gaza 1, 6 (2006) [Fluid Dynamics 41, 3 (2006)]. A. A. Cherepanov, in Some problems of stability of a liquid surface, Sverdlovsk, USSR, 1984 (in Russian). M. Faraday, Phylos. Trans. R. Soc. London 121, 299 (1831). T. B. Benjamin and F. Ursell, Proc. Roy. Soc. A 255, 505 (1954). K. Kumar and L. S. Tuckerman, J. Fluid Mech. 279, 49 (1994). D. V. Lyubimov and A. A. Cherepanov, in Some problems of stability of a liquid surface, Sverdlovsk, USSR, 1984 (in Russian). F. J. Mancebo and J. M. Vega, J. Fluid Mech. 467, 307 (2002). I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, Zh. Eksp. Theor. Fiz. 37, 229 (1959) [Sov. Phys. JETP 10, 161 (1960)]. C. C. Mei and L. F. Liu, J. Fluid Mech. 59, 239 (1973). B. A. Singer, J. H. Ferziger, and H. L. Reed, J. Fluid Mech. 208, 45 (1989). A. G. Straatman, R. E. Khayat, E. Haj-Qasem, and D. A. Steinman, Phys. Fluids 14, 1938 (2002). J. W. S. Rayleigh, The Theory of Sound, 2nd edition (Dover Publ., New York, 1945). H. B. Squire, Proc. R. Soc. London, Ser. A 142, 621 (1933). S. H. Davis, Annu. Rev. Fluid Mech. 7, 57 (1976). C. von Kerczek and S. H. Davis, J. Fluid Mech. 62, 753 (1974). S. A. Orszag, J. Fluid Mech. 50, 689 (1971). D. V. Lyubimov, A. A. Cherepanov, T. P. Lyubimova, and B. Roux, Micrograv. Quart. 6, 69 (1996). Analysis of vibration impact on stability of dewetting thin liquid film ======================================================================= Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description is reasonable and the amplitude equation holds. The linear and nonlinear analyses of the amplitude equation and the numerical computations show that such vibration stabilizes the film against dewetting and rupture. Introduction ============ Studies of stability, dewetting and rupture of thin liquid films on solid substrates are of great importance for micro- and nanotechnologies. Since the first works appeared in 1960s and 1970s, these subjects continue to attract enormous attention, see for instance Refs. . Refs. and are the reviews of recent results in this extremely diverse field. The subject of this paper is the general theoretical investigation of the impact of the vibration on the stability of a thin dewetting liquid film. Our interest in studying vibration impacts stems from the large body of works in fluid mechanics of macroscopic fluid layers, including Refs. (the experiment) and Refs. (the theory and numerical modeling), where the vibration is shown to drastically affect the stability characteristics and the dynamics of fluid surfaces and interfaces. In this paper we assume that the external influences on the film are the vibration, the gravity, and the long-range molecular attraction by the planar substrate, typified by van der Waals forces. Other effects (such as, for instance, the thermocapillarity and the evaporation) can be easily included by simply adding the corresponding terms to the final evolution equation for the film thickness. The theory we develop is based on the standard longwave lubrication approximation, as discussed by Oron, Davis, and Bankoff, and on the time-averaging method. The general discussion of the averaging methods can be found in Refs. . The key idea is the separation of the dynamics onto fast pulsations and slow relaxation processes. This approach works well when the vibration frequency is high in a certain sense, i.e. when there exists a large difference in the characteristic times (such as the viscous relaxation time and the vibration period, see below). The first transparent explanation of such separation of the time scales was given by Kapitza in his pioneering work on a pendulum with an oscillating point of support. The paper by Blekhman contains other examples in mechanics. Many examples of the successful application of the averaging method can be found in thermal vibrational convection, dynamics of inclusions in fluids, dynamics of granular materials, motion of disperse fluids, and filtration of inclusions in porous media. Interestingly, the vibration of the solid plate (on which the fluid system is located) often is capable of complete suppression of instabilities. For example, Wolf experimentally investigated the damping of the Rayleigh-Taylor instability in the horizontal two-layer system by a vertical high frequency vibration. The theoretical analysis of this situation (in the linear approximation) by the averaging technique was performed by Cherepanov (summary of this paper can be found in Ref. ). The longwave instability in this system was also analyzed by Lapuerta, Mancebo, and Vega. After the analysis of the linear longwave instability at the moderate vibration frequency, they proceed to the averaged description at high frequency. The generalization of the latter analysis to the nonisothermal situation was developed by Thiele, Vega, and Knobloch. They account for the Marangoni effect and perform a detailed investigation of the corresponding amplitude equation. Another widespread vibration-induced phenomenon is the parametric excitation, which emerges when the frequency of the vibration is comparable to one of the eigenfrequencies of the system (for instance, to the frequency of the capillary-gravity waves). Faraday was first to observe parametric waves on the surface of vertically oscillating horizontal layer. Linear and nonlinear analyses of parametric instability were performed, for instance, by Benjamin & Ursell, Kumar & Tuckerman, Lyubimov & Cherepanov, and by Mancebo and Vega. To the best of our knowledge the latter paper is the most detailed study to-date of the linear aspects of Faraday instability. It must be noted that the situations termed ``the averaged motion“ and ``the parametric instability” are often closely connected, although they operate within different intervals of the vibration frequencies. Indeed, in the studies of the averaged dynamics one has to ensure stability of the pulsatile motion (periodic in time). Most fluid systems have an eigenfrequency spectrum unbounded from above and thus the eigenfrequency is an increasing function of the mode number. Thus, even the high-frequency vibration is capable of parametric excitation of the higher modes. For the pulsatile motion to be stable, a window of parameters such as the amplitude and the frequency of the vibration must be chosen, where the parametric instability does not emerge. It is also worth noting that most papers, where the averaging method is employed, deal with the vibration of “inviscid” frequency, i.e. the vibration period Tp = 2π/ω (here ω is the dimensional frequency) is assumed small compared to the characteristic time of viscous relaxation, $\tau\_v=\hat H\_0^2/\nu$ (here ν is the kinematic viscosity and $\hat H\_0$ is the mean fluid layer thickness). It is clear that this assumption is quite reasonable for macroscopic layers, but for thin films (of thickness $100-1000 \, \AA$) it requires extremely large frequencies, $100 \,{\rm MHz}$ and higher. Now, we make a very important point, as follows. A thin film allows for averaged description even when the viscosity is large [i.e., τv = O(Tp) and even τv ≪ Tp]: one needs only to assume that the period of the vibration is small compared to the characteristic time of the film evolution, τl = O(k2) (here k is the wavenumber). Due to the lubrication approximation, τl ≫ τv. Thus the condition τl ≫ Tp is much milder than the usual inviscid approximation τv ≫ Tp. Therefore, the averaging procedure can be applied even to ultrathin films. To the best of our knowledge the only paper developing similar approximation is Ref. , where the linear stability problem is studied for “moderate” (“finite”) non-dimensional frequencies ωτv = O(1). However, by assuming the amplitude of the vibration “finite” (which means that it is of the order of the fluid layer thickness), the authors obtain that the impact of the vibration at moderate frequency is small. Thus, they focus on the high-frequency, small-amplitude case ωτv ≫ 1. In this paper we assume large vibration amplitude and develop the nonlinear amplitude equation for the thickness of the film. The outline of the paper is as follows. The mathematical formulation of the problem is presented in Sec. [sec:first]. We state and discuss assumptions and governing equations. In particular, the separation of the processes onto fast pulsatile and slow averaged motions is performed in Sec. [ssec:splitting]. The pulsatile problem is solved in Sec. [sec:puls]. The amplitude equation [Eq. ([h-avt])] governing the averaged dynamics of the film thickness is obtained in Sec. [sec:aver]. This equation is the main result of this paper. It can be used to study impacts of the vertical vibration (in the frequency range for which the averaged description is applicable) on the dynamics of a film, in the presence of the surface tension and wetting interactions with the substrate. Two limiting cases of “low” and “high” vibration frequency are analyzed in Sec. [sec:limit]. These cases correspond to the different ratios of the inertial and viscous forces in the oscillatory motion. (The viscosity dominates at low frequencies, while the inertia force dominates at high frequencies.) In Sec. [sec:3D] the 3D generalization of the theory is presented. Conditions of parametric instability of the oscillatory motion are analyzed in Sec. [sec:stab], where the Faraday instability and the shear flow instability are discussed. We show that for any admissible vibration frequency there exists a finite range of vibration amplitudes for which such instabilities are not present. In Sec. [sec:layer] we address the averaged behavior of the system within the framework of the obtained amplitude equation. Results of linear and weakly nonlinear analyses of the equilibrium state with the flat surface are presented, as well as the results of direct numerical simulations. In particular, we show that the vibration influence is stabilizing, i.e. it can delay or completely suppress the film rupture by intermolecular attractive forces. Finally, Sec. [sec:conc] summarizes the results. Formulation of the problem ========================== Governing equations ------------------- We consider a three-dimensional (3D), laterally unbounded thin liquid film of unperturbed thickness $\hat H\_0$ on a planar, horizontal substrate. The Cartesian reference frame is chosen such that the x − and y − axes are in the substrate plane and the z − axis is normal to the substrate (Fig. [fig:1]). [fig:1] The substrate-film system is subjected to the vertical harmonic vibration of the amplitude $\hat b$ and the frequency ω. Thus, in the reference frame of the substrate the acceleration of gravity is modulated, $$\label{gt} g(t)=g\_0+\hat b\omega^2 \cos\omega t.$$ Since $\hat H\_0$ is small, the intermolecular interaction of the film surface with the substrate has to be taken into account. Thus, we prescribe the potential energy $\phi(\hat H)$ to the unit length of the film layer. In this paper we consider only the van der Waals attractive potential, $$\phi(\hat H)=\frac{A^\prime}{6\pi \hat H^3},$$ where A′ > 0 is the Hamaker constant. The model presented in this paper can be readily extended to incorporate other models of wetting interactions – it is only necessary to replace $\phi(\hat H)$ with an appropriate function. We scale the time, the length, the velocity and the pressure by $\hat H\_0^2/\nu, \ \hat H\_0, \ \nu/\hat H\_0, \ \rho (\nu/\hat H\_0)^2$, respectively (here, ν is the kinematic viscosity and ρ is the density of the liquid). Then, the liquid motion is governed by the following non-dimensional problem: [baseeq] $$\begin{aligned} \nabla \cdot {\bf v} &=& 0, \\ \nonumber {\bf v}\_t + {\bf v}\cdot \nabla {\bf v} &=& - {\bf \nabla} p + \nabla ^2 {\bf v} \\ &&-\left(G\_0 +B \Omega^2\cos\Omega t \right) {\bf e\_z},\end{aligned}$$ [basebcs] $$\begin{aligned} {\bf v} &=& 0 \ {\rm at} \ z = 0, \\ \nonumber h\_t + {\bf v} \cdot {\bf \nabla} h &=& w,\\ \left(p-\phi - Ca K\right){\bf n} &=& {\bf n}\cdot {\bf T} \ {\rm at} \ z = h(x,y,t).\end{aligned}$$ Here, ${\bf v}$ is the fluid velocity, w is its component normal to the substrate, p is the pressure in the liquid, $\bf T$ is the viscous stress tensor, $h=\hat H/\hat H\_0$ is the dimensionless thickness of the layer, ${\bf e}\_z$ is the unit vector directed along the z − axis, ${\bf n}= \left({\bf e}\_z-\nabla h\right)/\sqrt{1+\left(\nabla h\right)^2}$ is the normal unit vector to the free surface, $K=\nabla \cdot {\bf n}$ is the mean curvature of the free surface, ϕ = A/h3 [where $A=A'/(6\pi\rho\nu^2\hat H\_0)$ is the non-dimensional Hamaker constant], $Ca=\sigma \hat H\_0/(\rho \nu^2)$ is the capillary number (where σ is the surface tension), $G\_0=g\_0 \hat H\_0^3/\nu^2$ is the Galileo number, $B=\hat b/\hat H\_0$ is the non-dimensional amplitude, and $\Omega=\omega \hat H\_0^2/\nu$ is the non-dimensional frequency. We consider the nonlinear evolution of the large-scale perturbations. As it is usually done, we introduce small parameter ε, which is of the order of the ratio of the mean thickness $\hat H\_0$ to the perturbation wavelength, i.e. ε ≪ 1 for long waves. Below for the sake of simplicity and for more transparent presentation of ideas, we consider the 2D model, assuming that ${\bf v}=u{\bf e}\_x+ w{\bf e}\_z$, where ${\bf e}\_x$ is the unit vector directed along the x − axis, and all fields are independent of y. The theory extends trivially in three dimensions at the replacement of the x-derivative by the 2D gradient. We derive the 3D analogue in Sec. [sec:3D]. Next we introduce the conventional stretched coordinates and the time: $$\label{coords} X=\epsilon x, \ Z=z, \ T=\epsilon^2 t, \ \tau=\Omega t,$$ such that ∂/∂t = ε2∂/∂T + Ω∂/∂τ. Now we rescale the velocity components as follows: u = εU, w = ε2W. To this end, both the pressure p and the surface position h remain unscaled. Throughout this paper we assume that the capillary number is large: Ca = Cε− 2. This is quite realistic and widely used assumption. Substituting the expansions ([coords])-([Ca]) into the problem ([baseeq]) and ([basebcs]) we arrive at the following set of equations and boundary conditions: [perturbeq] $$\begin{aligned} W\_Z &=& -U\_X, \\ \nonumber \Omega U\_\tau &=&- p\_X + U\_{ZZ} \\ && + \epsilon^2 \left(U\_{XX} - U\_T - U U\_X - W U\_Z \right), \\ p\_Z &=& - G\_0 - B \Omega^2\cos\tau + \epsilon^2 \left( W\_{ZZ}- \Omega W\_\tau\right),\end{aligned}$$ [perturbbcs] $$\begin{aligned} U &=& 0, \ W = 0 \ \ {\rm at} \ Z = 0, \\ \nonumber U\_Z &=&- \epsilon^2 \left(W\_X - 4h\_X U\_X\right), \\ \nonumber \Omega h\_\tau &=& \epsilon^2 \left(W - h\_T - U h\_X \right),\\ \nonumber p &=& \phi - C h\_{XX} \\ &&+\epsilon^2 \left(\frac{3}{2} C h\_X^2h\_{XX} + 2 W\_Z\right) \ {\rm at} \ Z = h.\end{aligned}$$ All terms of order ε4 have been omitted. Separation of the pulsating and averaged dynamics ------------------------------------------------- In this paper we consider the case of moderate vibration frequency: Ω ≫ ε2, i.e. the vibration period is small compared to the characteristic time of the surface evolution. This assumption makes possible the averaging of the dynamics of the film over the vibration period. The main purpose is the rigorous asymptotic analysis (in powers of ε) which results in the set of equations and boundary conditions, where the dependence on τ is averaged out. This set is then used to derive the amplitude equation for the thickness of the film. As for Ω, this quantity is not an asymptotic parameter. This means that, generally speaking, we assume Ω neither large nor small. Therefore Ω will enter the amplitude equation as a finite parameter. However, within the framework of the equation set ([perturbeq]) and ([perturbbcs]) we will consider three qualitatively different cases: (i) Ω ≪ 1, i.e. the vibration is of “low” frequency; this is the quasi-Stokes approximation with week influence of fluid inertia. Note that in this case, due to Eqs. ([coords]), the condition ε2 ≪ Ω must be retained to legalize the averaging procedure. (ii) Ω = O(1). This assumption means that the vibration period is comparable to the time of the momentum relaxation across the layer, $\hat H\_0^2/\nu$. (iii) Ω ≫ 1, i.e. the vibration is of “high” frequency; viscosity is negligible except for the thin boundary layer near the rigid wall. As usual, the boundary layer in the vicinity of the free surface does not play an important role (see Ref. , for example). Since the van der Waals interaction is important for films of thickness $100-1000 \ \AA$, let us estimate the typical values of Ω. Taking $\hat H\_0=10^{-5} \ {\rm cm}$ and $\nu=10^{-2} \ {\rm cm^2 s^{-1}}$ (viscosity of water), we obtain Ω ≈ 10− 4 for $\omega/(2\pi)=1 \ {\rm kHz}$ (typical for mechanical vibrator), and Ω = 10− 2 ÷ 1 for $n=0.1 \div 10 \ {\rm MHz}$ (typical for ultrasound). Therefore the case (i) can be easily achieved either mechanically or by means of ultrasound irradiation of the substrate from below, the case (ii) can be reached only using the ultrasound, and the case (iii) seems unrealistic. Nevertheless we shall consider this case below, since our results, upon neglecting ϕ, can as well be applied to the description of macroscopic fluid layers. Besides, this limiting case was studied in detail by many authors and therefore it allows for the verification of our results. We represent each field f = {U, W, p, h} as the sum of the average part $\bar f(T)$ and the pulsation $\tilde f(\tau)$, where τ and T can now be termed the “fast” and the “slow” times, respectively. Thus, we write [splitting] $$\begin{aligned} U &=& \bar U + \tilde U, \ W = \bar W + \tilde W, \\ p &=& \bar p + \tilde p, \ h = \bar h + \tilde h.\end{aligned}$$ We now assume that B = ε− 1b, i.e. the amplitude of the vibration is large compared to the mean film thickness. This assumption seems surprising as it is customary to impose the small-amplitude high frequency vibration. However, large amplitudes are permitted when the large-scale dynamics is considered. Indeed, it is shown below that due to the longwave approximation the impact of the vibration becomes non-negligible only at large amplitudes. Also, it will be made clear momentarily that in some sense the pulsatile motion is still small-amplitude. The assumption of large vibration amplitude means that the oscillating part of the pressure field $\tilde p$ (which is forced by the inertia force BΩ2cosτ) is of order ε− 1, which in turn leads to ε− 1 scaling for the velocities of the pulsation $\tilde U$ and $\tilde W$. The pulsation of the surface height $\tilde h$ obviously has the same order as $\tilde w=\epsilon^2 \tilde W$ [see Eq. ([velo])], i.e. it is of the order ε. Therefore, it is convenient to redefine the pulsation parts of all fields, rewriting Eqs. ([splitting]) as follows [rescaling] $$\begin{aligned} U &=& \bar U + \epsilon^{-1} \tilde U, \ W = \bar W + \epsilon^{-1}\tilde W, \\ \label{pres\_scal} p &=& \bar p + \epsilon^{-1}\tilde p, \ h = \bar h + \epsilon \tilde h,\end{aligned}$$ where $\tilde U, \tilde W, \tilde p$ and $\tilde h$ are O(1) quantities. Accounting for the initial scaling ([velo]) one can conclude that the full components of the velocity field are: $$\label{rescaled\_velo} u=\epsilon \bar U + \tilde U, \ w=\epsilon^2 W + \epsilon \tilde W,$$ while the pressure field and the surface deviation are given by Eqs. ([presscal]). Note that the pulsations of the fluid velocity and the pressure are larger than their mean parts, while the opposite is true for the pulsation of the film height. Moreover, the scaling ([rescaledvelo]) means that the pulsation remains “small-amplitude”. Indeed, the typical horizontal (vertical) displacement of the fluid particle during one period is O(1) [O(ε)], which is small in comparison with the respective characteristic lengthscale, O(ε− 1) [O(1)]. Substitution of the expansions ([rescaling]) in the equation sets ([perturbeq]) and ([perturbbcs]) allows to separate fast pulsations from background slow (averaged) motions. Keeping terms of zeroth and first orders in ε, we obtain the following sets. (i) For the pulsations: [pulsexp]	arxiv_0000732
Phase transition for the frog modelon biregular trees ===================================================== [1](#fn1) We study the frog model with death on the biregular tree Td1, d2. Initially, there is a random number of active and inactive particles located on the vertices of the tree. Each active particle moves as a discrete-time independent simple random walk on Td1, d2 and has a probability of death (1 − p) before each step. When an active particle visits a vertex which has not been visited previously, the inactive particles placed there are activated. We prove that this model undergoes a phase transition: for values of p below a critical probability pc, the system dies out almost surely, and for p > pc, the system survives with positive probability. We establish explicit bounds for pc in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies pc(Td1, d2) = 1/2 + Θ(1/d1 + 1/d2) as d1, d2 → ∞. Introduction ============ This paper addresses the issue of phase transition for the frog model with death, a discrete-time growing system of simple random walks on a rooted graph G, which is described as follows. Initially there is an independent random number of particles at each vertex of G. All particles are inactive at time zero, except for those that might be placed at ${\varnothing}$, the root of G. Each active particle moves as a discrete-time independent simple random walk (SRW) on the vertices of G, and has a probability of death (1 − p) before each step. When an active particle visits an inactive particle, that particle becomes active and starts to walk, performing exactly the same dynamics, independently of everything else. In the literature, the particles are often referred to as frogs, whose possible states are described as sleeping (inactive) and awake (active). This process can be thought as a model for describing rumor (or infection) spreading. One can think of every awake frog as an informed (or infected) agent, which shares the rumor with (or infects) a sleeping frog at the first time they meet. In the last decades, there has been a growing interest in understanding the behavior of stochastic processes on more general graph structures than the d-dimensional integer lattice Zd and the homogeneous tree Td of degree (d + 1). In particular, the study of phase transitions and critical phenomena for stochastic systems on nonhomogeneous trees and nonamenable quasi-transitive graphs has become an important research area. Among the models studied, are percolation, the contact process, and branching random walks. For the frog model, most of the work has involved studying the process on Zd, Td and recently on d-ary trees. As far as we know, only considers another kind of tree, proving the recurrence of the process (without death) on a 3, 2-alternating tree (in which the generations of vertices alternate between having 2 and 3 children). The first published paper dealing with the frog model (with p = 1, G = Zd) is due to, where it was referred to as the egg model. They proved that, starting from the one-particle-per-vertex initial configuration, almost surely infinitely many frogs will visit the origin for all d ≥ 3 (that is, although each frog is individually transient, the process is recurrent). exhibits the critical rate at which the frog model with Bernoulli(α/∣∣x∣∣2) sleeping frogs at each x ∈ Zd \ {0} changes from transience to recurrence. A similar result is obtained by for the model on d-ary trees, with Poisson(μ) sleeping frogs at each vertex. More precisely, the authors prove that the model undergoes a phase transition between transience and recurrence, as the initial density μ of frogs increases. For the model starting with one frog per vertex, establish that there is a phase transition in the dimension of the tree, by proving recurrence for d = 2 and transience for d ≥ 5. Based on simulations, they conjecture that the model is recurrent for d = 3, and transient for d = 4. In, for the frog model without death on Zd, it is proved that, starting from the one-particle-per-vertex initial configuration, the set of the original positions of all awake frogs, rescaled by the elapsed time, converges to a nonempty compact convex set. prove the same statement in the case of random initial configuration; these results are known as shape theorems. For a continuous-time version of the frog model, a limiting shape result is stated by. We refer to for a survey on some results for the model and its variations. Regarding the frog model with death, the existence of phase transition as p varies was first studied by. As we will detail later, the occurrence of phase transition means that there is a nontrivial value pc of p separating the regimes of extinction and survival of the process. obtain lower bounds on pc for general graphs and bounded degree graphs, and establish the existence of phase transition on Zd and Td, for d ≥ 2, under rather broad conditions. prove that the critical probability for the frog model on a homogeneous tree of degree (d + 1) is at most (d + 1)/(2d); that result is an improvement of the upper bound stated by, namely, (d + 1)/(2d − 2). Further improvements on the upper bound for this critical probability were recently obtained by, using Renewal Theory, and by. For more details on the subject, see these papers and references therein. The aim of the present paper is to deepen the study of the critical phenomenon of the frog model, going beyond Zd and homogeneous trees. We consider the model on a specific class of nonhomogeneous trees, namely, biregular trees. In the main results, we prove a sufficient condition for the existence of phase transition, and present explicit bounds for the critical probability, in the case of random initial configuration. Since there is more than one parameter measuring the size of such trees, bounds on the critical parameter are harder to get at. The asymptotic behavior of the critical probability for large values of the dimension of the tree is also derived. We hope that our study will stimulate further works regarding the occurrence of phase transition on other nonhomogeneous trees and nonamenable graphs. Formal description of the model and main results ================================================ We start off with some basic definitions and notation of Graph Theory. Let G = (V, E) be an infinite connected locally finite graph, with vertex-set V and edge-set E. A vertex ${\varnothing}\in {\mathbb{V}}$ is fixed and called the root of G. We denote an unoriented edge with endpoints x and y by xy. Vertices x and y are said to be neighbors if they belong to a common edge xy; we denote this by x ∼ y. The degree of a vertex is the number of its neighbors. A path of length n from x to y is a sequence x = x0, …, xn = y of vertices such that xi ∼ xi + 1 for all i = 0, …, n − 1. The graph distance ${\textnormal{dist}}(x, y)$ between x and y is the minimal length of a path connecting the two vertices; the level of x is ${\textnormal{dist}}({\varnothing}, x)$. A tree is a connected graph without loops or cycles, where by a cycle in a graph we mean a sequence of vertices x0, …, xn, n ≥ 3, with no repetitions besides xn = x0. A graph is bipartite if its vertex-set V can be partitioned into two subsets V1 and V2, in such a form that every edge joins a vertex of V1 to a vertex of V2. For d1 ≥ 1 and d2 ≥ 1, we denote by Td1, d2 the (d1, d2)-biregular tree, which is the bipartite tree where the degree of a vertex is (d1 + 1) or (d2 + 1), according to whether the level of the vertex is even or odd. In this case, the class V1 (resp. V2) is the set of vertices at even (resp. odd) distance from the root. From now on, a vertex x ∈ Vi will be called a type i* vertex. Notice that T1, 1 is isomorphic to Z. See Figure [Fg: bitree] for an illustration of T2, 4. [Fg: bitree] Now we describe the frog model in a formal way, keeping the notation of and, whenever possible. We write N = {1, 2, …} and N0 = N ∪ {0}. Let η* be a random variable assuming values in N0, and define $\rho\_{k}:={\mathds{P}}[\eta=k]$, k ∈ N0. We suppose that ${\mathds{P}}[\eta \geq 1] > 0$, that is, ρ0 < 1. For s ∈ [0, 1], let $\varphi(s) := {\mathds{E}}[s^\eta]$ be the probability generating function of η. To define the frog model, let {η(x) : x ∈ V}, {(Snx(k))n ∈ N0; k ∈ N, x ∈ V} and {(Ξpx(k)); k ∈ N, x ∈ V} be independent sets of random objects defined as follows. For each x ∈ V, η(x) has the same law as η, and gives the initial number of frogs at vertex x. If η(x) ≥ 1, then for each k ∈ {1, …, η(x)}, (Snx(k))n ∈ N0 is a discrete-time SRW on G starting from x, and Ξpx(k) is a random variable whose law is given by ${\mathds{P}}[\Xi\_p^x(k)=j]=(1-p)p^{j-1}$, j ∈ N, where p ∈ [0, 1] is a fixed parameter. These random objects describe respectively the trajectory and the lifetime of the k-th frog placed initially at x. Thus, the k-th frog at vertex x, whenever it is awoken, follows the SRW (Snx(k))n ∈ N0, and disappears at the instant it reaches a total of (Ξpx(k) − 1) jumps. At the moment the frog disappears, it is not able to awake other frogs (first the frog decides whether or not to survive, and only after that it is allowed to jump). There is no interaction between awake frogs. At time n = 0, only the frogs that might be placed at the root of G are awake. We call this model the frog model on G, with survival parameter p and initial configuration ruled by η, and denote it by ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}.$ A particular realization of the frog model survives if for every instant of time there is at least one awake frog. Otherwise, we say that it dies out. A coupling argument shows that ${\mathds{P}}[{\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}} \text{ survives}]$ is a nondecreasing function of p, and therefore we define the critical probability as $${\ensuremath{{p\_c}({\mathbb{G}}, \eta)}} = \inf\left\{p:{\mathds{P}}[{\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}} \text{ survives}]>0\right\}.$$ As usual, we say that ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}$ exhibits phase transition if pc(G, η) ∈ (0, 1). For G = Td1, d2 and η ≡ 1 (one-particle-per-vertex initial configuration), we simply write pc(Td1, d2). As proved by, if d1 = d2 = 1, then under the condition ${\mathds{E}}\log (\eta \vee 1)< \infty$, we have that the frog model dies out almost surely for every p < 1, that is, pc(Z, η) = 1. The picture is quite different for higher degrees. Indeed, prove that the frog model on Td exhibits phase transition for every d ≥ 2, provided that ρ0 < 1 and ${\mathds{E}}\eta^\delta<\infty$ for some δ > 0. Here we extend this result to the case where the process lives on Td1, d2 (assuming d1 ≥ 2 or d2 ≥ 2). We begin by showing a sufficient condition to guarantee the almost sure extinction of the frog model on bounded degree trees for small enough p. [T: SCE] Let G be an infinite tree of bounded degree, and suppose that there exists δ > 0 such that ${\mathds{E}}\eta^\delta<\infty$. Then pc(G, η) > 0. This result is an extension of Theorem 1.2 in, which is stated for a homogeneous tree Td, d ≥ 2. Note, however, that one can not prove Theorem [T: SCE] by using a direct coupling and the corresponding result for homogeneous trees. As shown by, in general, the critical parameter of the frog model is not a monotonic function of the graph. A longstanding open question is whether, for a class of graphs (for example, trees), G1 ⊂ G2 in this class implies that pc(G2, η) ≤ pc(G1, η). This problem was first raised by (for Zd), and discussed again in, and. Under the condition that the mean number of frogs placed initially at every vertex is finite, we derive a nontrivial lower bound for the critical probability on Td1, d2. [T: LB] Suppose that ${\mathds{E}}\eta<\infty$. If d1 ≥ 2 or d2 ≥ 2, then $${\ensuremath{{p\_c}({\mathbb{T}}\_{{d\_1,d\_2}}, \eta)}}\geq \sqrt{\frac{(d\_1+1)(d\_2+1)}{[d\_1({\mathds{E}}\eta+1) + 1][d\_2({\mathds{E}}\eta+1) +1]}}.$$ The proofs of Theorems [T: SCE] and [T: LB] are given in Section [S: Extinction]. To present an upper bound for pc(Td1, d2, η), we need the following definition. [D: tudo] Let q = 1 − ρ0 > 0, κ = (d1 + 1)(d2 + 1), Δ = κ2 − 2κ(d1 + d2)p2 + (d2 − d1)2p4. We also define the functions α, β, f : [0, 1] → [0, 1] given by $$\begin{aligned} \alpha(p)&=\alpha^{({d\_1,d\_2})}(p)=\left\{ \begin{array}{cl} \dfrac{\kappa+p^2(d\_2-d\_1)-\sqrt{\Delta}}{2d\_2(d\_1+1)p} & \text{if } 0 < p \leq 1, \\[0.5cm] 0 & \text{if } p = 0, \end{array} \right. \label{F: Prob-alpha}\\[0.2cm] \beta(p)&=\beta^{({d\_1,d\_2})}(p)=\left\{ \begin{array}{cl} \dfrac{\kappa+p^2(d\_1-d\_2)-\sqrt{\Delta}}{2d\_1(d\_2+1)p} & \text{if } 0 < p \leq 1, \\[0.5cm] 0 & \text{if } p = 0, \end{array} \right. \label{F: Prob-beta}\\[0.25cm] f^{({d\_1,d\_2}, q)}(p)&=\alpha(p)\beta(p)[1+q(1-\alpha(p))][1+q(1-\beta(p))]-\frac{1}{d\_1d\_2}. \label{F: Function-f}\end{aligned}$$ [T: UBA] Suppose that ρ0 < 1. If d1 ≥ 2 or d2 ≥ 2, then pc(Td1, d2, η) ≤ p̃(d1, d2, q), where p̃(d1, d2, q) is the unique root of the function f(d1, d2, q) in the interval (0, 1). [C: UB bi-re] Suppose that η ≡ 1. If d1 ≥ 2 or d2 ≥ 2, then [C: UBH] $${\ensuremath{{p\_c}({\mathbb{T}}\_{{d\_1,d\_2}})}} \leq \frac{1}{2}\sqrt{\frac{(d\_1 + 1)(d\_2 + 1)}{d\_1 d\_2}}.$$ The proofs of Theorem [T: UBA] and Corollary [C: UB bi-re] are presented in Section [S: Survival]. In the case of d1 = d2 = d and η ≡ 1 in Theorem [T: UBA] (also in Corollary [C: UB bi-re]), we obtain the upper bound for the critical probability pc(Td) that is proved in, namely, (d + 1)/(2d). From Theorems [T: SCE] and [T: UBA], we conclude: [T: PT] Suppose that ρ0 < 1 and ${\mathds{E}}\eta^\delta<\infty$ for some δ > 0. If d1 ≥ 2 or d2 ≥ 2, then 0 < pc(Td1, d2, η) < 1. For comparison, we include a result proved by, that establishes a lower bound for the critical probability of the frog model on bounded degree graphs. This result also appears in. [] [P: LB Alves] Suppose that G is a graph with maximum degree (D + 1), and ${\mathds{E}}\eta<\infty$. Then, $${\ensuremath{{p\_c}({\mathbb{G}}, \eta)}} \geq \dfrac{D+1}{D({\mathds{E}}\eta+1)+1}.$$ Notice that, for G = Td1, d2 with d1 = d2, the lower bound given in Theorem [T: LB] equals the one provided by Proposition [P: LB Alves]. However, for d1 ≠ d2, the lower bound stated in Theorem [T: LB] is better than that of Proposition [P: LB Alves]. For a numerical illustration, we consider η ≡ 1, and compute the bounds established in Theorems [T: LB], [T: UBA] and Proposition [P: LB Alves] for some values of the pair (d1, d2). The resulting values are given in Table [NV]. [ht] P1.8cmP4cmP4cmP3.6cm (d1, d2) & LB Proposition [P: LB Alves] &LB Theorem [T: LB] & UB Theorem [T: UBA] (1, 2) & 0.6000 & 0.6325 & 0.8588 (1, 3) & 0.5714 & 0.6172 & 0.8039 (1, 4) & 0.5556 & 0.6086 & 0.7749 (2, 2) & 0.6000 & 0.6000 & 0.7500 (2, 3) & 0.5714 & 0.5855 & 0.7063 (2, 4) & 0.5556 & 0.5774 & 0.6828 (3, 100) & 0.5025 & 0.5359 & 0.5771 (3, 1000) & 0.5002 & 0.5347 & 0.5743 (4, 10000) & 0.5000 & 0.5271 & 0.5572 [NV] Now we present a result concerning the asymptotic behavior of pc(Td1, d2) as d1, d2 → ∞. The lower and upper bounds given in Theorem [T: LB] with η ≡ 1 and Corollary [C: UB bi-re] are, respectively, $$\frac{1}{2} + \frac{1}{8} \left(\frac{1}{d\_1}+\frac{1}{d\_2}\right) + O\left(\frac{1}{d\_1^2}+\frac{1}{d\_2^2}\right) \text{ and } \; \frac{1}{2} + \frac{1}{4} \left(\frac{1}{d\_1}+\frac{1}{d\_2}\right) + O\left(\frac{1}{d\_1^2}+\frac{1}{d\_2^2}\right).$$ As a consequence, we find the correct order of magnitude for the critical probability. For the frog model on Td1, d2 starting from the one-particle-per-vertex initial configuration, we have $${\ensuremath{{p\_c}({\mathbb{T}}\_{{d\_1,d\_2}})}}= \frac{1}{2} +\Theta\left(\frac{1}{d\_1}+\frac{1}{d\_2}\right) \quad \text{as } \, d\_1, d\_2 \to \infty.$$ Survival of the process ======================= In brief, the key idea to prove Theorem [T: UBA] is to describe ${\ensuremath{\textnormal{FM}({\mathbb{T}}\_{{d\_1,d\_2}},p, \eta)}}$ as a percolation model which dominates suitably defined Galton–Watson branching processes. In Subsections [SS: FM Perc] and [SS: BT], we describe the frog model on a graph G as a particular percolation model, and we compute its parameters when G = Td1, d2. Then, in Subsection [SS: SBP], we define a sequence of Galton–Watson branching processes which are dominated by the frog model on Td1, d2. Finding p such that each one of these branching processes is supercritical leads us to obtain a sequence of upper bounds for the critical probability, which converges to the upper bound p̃(d1, d2, q) stated in Theorem [T: UBA]. Subsection [SS: Proofs] is devoted to the finalization of the proof. The technique to prove Theorem [T: UBA] is similar to that used by to establish an upper bound for the critical probability of the frog model on homogeneous trees. However, here we have to deal with the computations related to the fact that the degrees of the vertices of the tree alternate. Also to overcome this issue, we construct the sequence of approximating branching processes in a new manner, leading directly to the desired upper bound. An analogous approach is used by to derive an improved upper bound for the critical parameter on homogeneous trees. seen as percolation -------------------- In the bond percolation model, each edge of an infinite locally finite graph $\bar {\mathbb{G}}=(\bar {\mathbb{V}}, \bar {\mathbb{E}})$ is randomly assigned the value 1 (open) or 0 (closed), according to some probability measure on the product space $\{0, 1\}^{\bar {\mathbb{E}}}$. Then, one studies the connectivity properties of the random subgraph of $\bar{\mathbb{G}}$ which arises by removing closed edges. For a fundamental reference on the subject, we refer to . Next, we describe the frog model on an infinite graph G = (V, E) as a particular bond percolation model. Indeed, for every x ∈ V and 1 ≤ k ≤ η(x), we define the virtual set of vertices visited by the k-th frog located originally at x by Rxk : = {Snx(k) : 0 ≤ n < Ξpx(k)} ⊂ V. The set Rxk becomes real in the case when x is actually visited (and thus the sleeping frogs placed there are awoken). We define the range of x by $${\mathcal{R}}\_x:= \left\{ \begin{array}{cl} \bigcup\_{k=1}^{\eta(x)}{\mathcal{R}}\_x^k &\text{if } \eta(x) \geq 1, \\[0.2cm] \{x\} &\text{otherwise}. \end{array} \right.$$ Now let ${\overset{\rightarrow}{\mathbb{G}}}=({\mathbb{V}}, {\overset{\,\rightarrow}{\mathbb{E}}})$ be the oriented graph with vertex-set V and edge-set ${\overset{\,\rightarrow}{\mathbb{E}}}:=\{{\overset{\,\rightarrow}{xy}}:(x, y) \in {\mathbb{V}}\times {\mathbb{V}}, x \neq y\}$. That is, for every pair of distinct vertices x and y, an oriented edge is drawn from x to y. Then, we introduce the following notations: for x, y ∈ V distinct, [x → y] : = [y ∈ Rx], $[{\ensuremath{x \nrightarrow y}}]:=[y \notin {\mathcal{R}}\_x]$. This defines an oriented dependent long range anisotropic percolation model on ${\overset{\rightarrow}{\mathbb{G}}}$: ${\overset{\,\rightarrow}{xy}}$ is declared to be open if [x → y], and closed otherwise. In general, the probability that an oriented edge is open depends on its orientation. For instance, when G = Td1, d2, an edge emanating from a type 1 vertex to a type 2 vertex has probability of being open different from an edge emanating from a type 2 vertex to a type 1 vertex. This feature gives the anisotropy of the oriented percolation model. Notice also that the cluster of the root has infinite size if and only if there exists an infinite sequence of distinct vertices ${\varnothing}=x\_0, x\_1, x\_2, \dots$ such that xj → xj + 1 for all j ≥ 1. Of course, this event is equivalent to the survival of ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}$. Biregular trees --------------- Now let us consider G = Td1, d2. Our next purpose is to obtain an explicit formula for the probability that an oriented edge is open. Toward this end, let α(d1, d2)(p) (resp. β(d1, d2)(p)) denote the probability that a type 1 (resp. type 2) frog ever visits a type 2 (resp. type 1) neighbor vertex. Avoiding a cumbersome notation, we sometimes write α(p) or simply α (resp. β(p), β). With a typical abuse of language, we call the percolation model associated to ${\ensuremath{\textnormal{FM}({\mathbb{T}}\_{{d\_1,d\_2}},p, \eta)}}$ the anisotropic percolation model on the biregular tree Td1, d2, with parameters α, β and ruled by η. We denote it by ${\ensuremath{\textnormal{APM}({\mathbb{T}}\_{{d\_1,d\_2}},\alpha, \beta, \eta)}}$. The following lemma about hitting probabilities of SRWs on Td1, d2 provides a formula for α and β. Recall that V1 and V2 are respectively the set of vertices at even and odd distance from the root of Td1, d2. [L: Prob open edge] Let x ∼ y be a pair of neighbor vertices, and suppose (x, y) ∈ Vi × Vj, with i, j ∈ {1, 2}, i ≠ j. Then, $${\mathds{P}}[y\in{\mathcal{R}}\_x^1]= \begin{cases} \alpha(p) &\text{if } i=1, j=2, \\[0.2cm] \beta(p) &\text{if } i=2, j=1, \end{cases}$$ where α and β are given in Definition [D: tudo]. Let τij : = τxy be the first time when the simple random walk on Td1, d2 starting from x visits y. Suppose first that p < 1, so conditioning on the lifetime of the frog located at x, we have $${\mathds{P}}[y\in{\mathcal{R}}\_x^1]={\mathds{E}}[p^{\tau\_{ij}}].$$ Now by conditioning on the first jump of the frog at x, we have $$\begin{aligned} {\mathds{E}}[p^{\tau\_{12}}]&=\frac{1}{d\_1+1}p+\frac{d\_1}{d\_1+1}p{\mathds{E}}[p^{\tau\_{12}}]{\mathds{E}}[p^{\tau\_{21}}]\\ {\mathds{E}}[p^{\tau\_{21}}]&=\frac{1}{d\_2+1}p+\frac{d\_2}{d\_2+1}p{\mathds{E}}[p^{\tau\_{12}}]{\mathds{E}}[p^{\tau\_{21}}]\end{aligned}$$ By right-continuity of the probability generating functions of τ12 and τ21, it follows that $\lim\_{p\to 0^+} {\mathds{E}}[p^{\tau\_{12}}]=\lim\_{p\to 0^+}{\mathds{E}}[p^{\tau\_{21}}]=0$. Therefore, and are the only possible solutions for the previous system of equations. This concludes the proof for p < 1. Finally, if p = 1, then $${\mathds{P}}[y\in{\mathcal{R}}\_x^1]={\mathds{P}}[\tau\_{ij}<\infty]=\lim\_{p\to 1^-} {\mathds{E}}[p^{\tau\_{ij}}]=\frac{d\_j+1}{d\_j(d\_i+1)}.\qedhere$$ If d1 ≤ d2, then α(d1, d2)(p) ≥ β(d1, d2)(p). [L: Prob long open edge] Let x and y be two vertices of Td1, d2 with (x, y) ∈ Vi × Vj, i, j ∈ {1, 2}, and let $k = {\textnormal{dist}}(x, y) \geq 1$. Let πη(i, j, k) denote the probability that the oriented edge ${\overset{\,\rightarrow}{xy}}$ is open. Then, $$\label{F: prob open edge aleator} \pi\_{\eta}(i, j, k)= \begin{cases} 1-\varphi(1-\alpha^n\beta^{n-1}) &\text{if } i=1, j=2, k=2n-1, \\[0.1cm] 1-\varphi(1-\alpha^{n-1}\beta^n) &\text{if } i=2, j=1, k=2n-1, \\[0.1cm] 1-\varphi(1-\alpha^n\beta^n) &\text{if } i=j, k=2n, \end{cases}$$ where $\varphi(s)={\mathds{E}}(s^\eta)$, s ∈ [0, 1], is the probability generating function of η. By conditioning on the initial number of frogs at vertex x, it is enough to consider η ≡ 1, that is, φ(s) ≡ s. In this case, πη(i, j, k) is the probability that a type i frog ever visits a type j vertex at distance k. So, we need to prove that for η ≡ 1, $$\label{F: prob open edge} \pi\_{\eta}(i, j, k)= \begin{cases} \alpha^n\beta^{n-1} &\text{if } i=1, j=2, k=2n-1, \\[0.1cm] \alpha^{n-1}\beta^n &\text{if } i=2, j=1, k=2n-1, \\[0.1cm] \alpha^n\beta^n &\text{if } i=j, k=2n. \end{cases}$$ Formula follows from the fact that if (x, y) ∈ V1 × V2 with k = 2n − 1, then the first time when the frog starting from x visits y is equal in distribution to the sum of 2n − 1 independent random variables, such that n of them are independent copies of τ12 and n − 1 of them are independent copies of τ21. Using this fact and Lemma [L: Prob open edge], we obtain that πη(1, 2, 2n − 1) = αnβn − 1. The other cases are analogous. A sequence of branching processes dominated by the frog model ------------------------------------------------------------- The central idea is as follows. For each n ≥ 1, we define a Galton–Watson branching process whose survival implies that the cluster of the root in ${\ensuremath{\textnormal{APM}({\mathbb{T}}\_{{d\_1,d\_2}},\alpha, \beta, \eta)}}$ has infinite size. For each branching process, we find a sufficient condition which guarantees that the process is supercritical. We get in this manner a sequence of upper bounds for the critical probability, which converges to the upper bound given in the statement of Theorem [T: UBA]. This technique of using embedded branching processes is very similar to that used for the contact process on homogeneous trees; see the paper by. Let us now carry out this plan. [D: Sets] 1. We define a partial order on the set V as follows: for x, y ∈ V, we say that x ≼ y if x belongs to the path connecting $ {\varnothing}$ and y; x ≺ y if x ≼ y and x ≠ y. 2. For any vertex $ x \neq {\varnothing}$, let V+(x) = {y ∈ V : x ≼ y}. Also define $ {\mathbb{V}}^+({\varnothing}) = {\mathbb{V}}\setminus {\mathbb{V}}^+(z) $, where z is a fixed vertex neighbor to $ {\varnothing}$. 3. For x ∈ V and k ∈ N, define $ L\_k (x) = \{ y \in {\mathbb{V}}^+(x) : {\textnormal{dist}}(x, y) = k \} $. [D: k Cam] For x ∈ V and y ∈ Lk(x), k ∈ N, consider x0 = x ≺ x1 ≺ ⋯ ≺ xk − 1 ≺ xk = y, the path connecting x and y. For each ℓ = 1, 2, …, n − 1, we denote by $[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_{\ell} }}]$ the event that $[{\ensuremath{x\_0 \rightarrow x\_{\ell}}}] \cap [{\ensuremath{x\_0 \nrightarrow x\_{\ell+1}}}]$. We define the event $[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{k}}}]$ inductively on k by: * If k = 1, then $$[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_k}}]:=[{\ensuremath{x\_0 \rightarrow x\_k}}].$$ * If k ≥ 2, then $$[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{k}}}]:=[{\ensuremath{x\_0 \rightarrow x\_{k}}}]\cup\bigcup\_{\ell=1}^{k-1}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_{\ell} }}, {\ensuremath{x\_{\ell} \stackrel {o}{\rightarrow} x\_{k}}}].$$ We denote the complement of $[{\ensuremath{x \stackrel {o}{\rightarrow} y}}]$ by $[{\ensuremath{x \stackrel {o}{\nrightarrow} y}}]$. Now we construct a sequence of Galton–Watson branching processes embedded in ${\ensuremath{\textnormal{APM}({\mathbb{T}}\_{{d\_1,d\_2}},\alpha, \beta, \eta)}}$. Starting from the root, the potential direct descendants of a vertex x are vertices located in Lk(x), where k = 2n is an even number. Roughly speaking, a vertex y ∈ L2n(x) is said to be a child of vertex x if and only if $[{\ensuremath{x \stackrel {o}{\rightarrow} y}}]$. More formally, let n ∈ N be fixed. Define ${\mathcal{Y}}\_{0, n}:=\{{\varnothing}\}$, and for ℓ ∈ N define $${\mathcal{Y}}\_{\ell, n}:=\bigcup\_{x\in {\mathcal{Y}}\_{\ell-1, n}}\{y\in L\_{2n}(x): {\ensuremath{x \stackrel {o}{\rightarrow} y}}\}.$$ Let Yℓ, n = ∣Yℓ, n∣ be the cardinality of Yℓ, n. [P: BPdom] For every n ∈ N, {Yℓ, n}ℓ ∈ N0 is a Galton–Watson branching process whose survival implies the occurrence of percolation. In addition, its mean number of offspring per individual satisfies ${\mathds{E}}[Y\_{1, n}]\geq (d\_1 d\_2)^n \, \phi^{({d\_1,d\_2}, q)}\_n(p)$, where ϕn(d1, d2, q)(p) : = q[α(p)β(p)(1 + q(1 − β(p))]n[1 + q(1 − α(p))]n − 1. Since the first claim in Lemma [P: BPdom] is clear, to prove it, we have to compute ${\mathds{E}}[Y\_{1, n}]$. To accomplish this, we show a recursive formula for the probability of the event $[{\ensuremath{x \stackrel {o}{\rightarrow} y}}]$. Let us define the domain $\mathcal{A}\_{{d\_1,d\_2}}:= \left[0, \frac{d\_2+1}{d\_2(d\_1+1)}\right]\times\left[0, \frac{d\_1+1}{d\_1(d\_2+1)}\right] \subseteq [0, 1]^2$. In formulas that appear from this point on, we assume that a summation of the form ∑10 equals 0. [L: Prob cam open] For (x, y) ∈ Vi × Vj, i, j ∈ {1, 2}, with ${\textnormal{dist}}(x, y)=k \geq 1$, define $$\nu\_{\eta}(i, j, k) := {\mathds{P}}[{\ensuremath{x \stackrel {o}{\rightarrow} y}}].$$ For every n ≥ 1, there exists functions Kn, Fn, Kn⋆, Fn⋆ with domain Ad1, d2, not depending on d1, d2, such that $$\begin{aligned} \nu\_{\eta}(1, 2, 2n-1)&=K\_n(\alpha(p), \beta(p)), &\nu\_{\eta}(1, 1, 2n)&=F\_n(\alpha(p), \beta(p)), \\[0.2cm] \nu\_{\eta}(2, 1, 2n-1)&=K\_n^{\star}(\alpha(p), \beta(p)), &\nu\_{\eta}(2, 2, 2n)&=F\_n^{\star}(\alpha(p), \beta(p)).\end{aligned}$$ For n ≥ 1 and (a, b) ∈ Ad1, d2, we define the following sequence of functions recursively: $$\begin{aligned} K\_n(a, b) &= [1-\varphi(1-a^n b^{n-1})] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^{\ell+1}b^\ell)-\varphi(1-a^\ell b^\ell)]K\_{n-\ell}(a, b) \\ &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^\ell b^\ell)-\varphi(1-a^\ell b^{\ell-1})]F\_{n-\ell}^{\star}(a, b), \end{aligned}\\[0.2cm] K^{\star}\_n(a, b) &= [1-\varphi(1-b^n a^{n-1})] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-b^{\ell+1}a^\ell)-\varphi(1-a^\ell b^\ell)]K^{\star}\_{n-\ell}(a, b) \\ &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^\ell b^\ell)-\varphi(1-b^\ell a^{\ell-1})] F\_{n-\ell}(a, b), \end{aligned}\\[0.2cm] F\_n(a, b) &= [1-\varphi(1-a^n b^n)] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^{\ell+1}b^\ell)-\varphi(1-a^\ell b^\ell)] F\_{n-\ell}(a, b) \\ &+ \sum\_{\ell=1}^{n} [\varphi(1-a^\ell b^\ell)-\varphi(1-a^\ell b^{\ell-1})]K\_{n+1-\ell}^{\star}(a, b), \end{aligned}\\[0.2cm] F\_n^{\star}(a, b) &= [1-\varphi(1-b^n a^n)] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-b^{\ell+1}a^\ell)-\varphi(1-a^\ell b^\ell)]F\_{n-\ell}^{\star}(a, b)\\ &+ \sum\_{\ell=1}^{n} [\varphi(1-a^\ell b^\ell)-\varphi(1-b^\ell a^{\ell-1})]K\_{n+1-\ell}(a, b). \end{aligned}\end{aligned}$$ Next, using that ${\mathds{P}}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_\ell }}]={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_\ell}}]-{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{\ell+1}}}]$, we obtain, for every n ≥ 1, $$\begin{aligned} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{2n-1}}}]&={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n-1}}}]+\sum\_{\ell=1}^{2n-2} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_\ell }}]{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n-1}}}]\\ &={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n-1}}}]+\sum\_{\ell=1}^{2n-2}\{{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_\ell}}]-{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{\ell+1}}}]\}{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n-1}}}], \\ \intertext{and} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{2n}}}]&={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n}}}]+\sum\_{\ell=1}^{2n-1} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_\ell }}]{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n}}}]\\ &={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n}}}]+\sum\_{\ell=1}^{2n-1}\{{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_\ell}}]-{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{\ell+1}}}]\}{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n}}}].\end{aligned}$$ We split the proof into two cases: * If (x0, x2n − 1) ∈ Vi × Vj with i ≠ j, then $$\begin{aligned} \nu\_\eta(i, j, 2n-1)=\pi\_\eta(i, j, 2n-1) &+ \sum\_{\ell=1}^{n-1}[\pi\_\eta(i, i, 2\ell)-\pi\_\eta(i, j, 2\ell+1)]\nu\_\eta(i, j, 2(n-\ell)-1)]\\ &+ \sum\_{\ell=1}^{n-1} [\pi\_\eta(i, j, 2\ell-1)-\pi\_\eta(i, i, 2\ell)]\nu\_\eta(j, j, 2(n-\ell)).\end{aligned}$$ * If (x0, x2n) ∈ Vi × Vi, then $$\begin{aligned} \nu\_\eta(i, i, 2n)=\pi\_\eta(i, i, 2n) &+ \sum\_{\ell=1}^{n-1} [\pi\_\eta(i, i, 2\ell)-\pi\_\eta(i, j, 2\ell+1)]\nu\_\eta(i, i, 2(n-\ell))\\ &+ \sum\_{\ell=1}^{n} [\pi\_\eta(i, j, 2\ell-1)-\pi\_\eta(i, i, 2\ell)]\nu\_\eta(j, i, 2(n-\ell)+1).\end{aligned}$$ Using Equation , the result follows by induction on n. [Proof of Lemma [P: BPdom]] From Lemma [L: Prob cam open], the Galton–Watson branching process has mean number of progeny per individual given by $${\mathds{E}}[Y\_{1, n}] = (d\_1d\_2)^n \, \nu\_{\eta}(1, 1, 2n) = (d\_1d\_2)^n \, F\_n(\alpha(p), \beta(p)).$$ To obtain a lower bound for this expected value (not depending on the function φ), we truncate the initial configuration of the frog model. We consider the modified initial configuration η′ by $$\eta^{\prime}(x) = {\mathds{1}}\_{\{\eta(x) \geq 1 \}}, \, x \in {\mathbb{V}}.$$ Since the initial condition η′ is dominated by η in the usual stochastic order, it follows that νη(1, 1, 2n) ≥ νη′(1, 1, 2n). But for the restricted frog model with initial configuration ruled by η′, νη′(1, 1, 2n) = Fn(q)(α(p), β(p)), where Fn(q)(a, b) is given by the formulas stated in the proof of Lemma [L: Prob cam open] with the choice φ(s) = 1 − q(1 − s). Using these formulas and induction on n, we have that Fn(q) satisfies Fn + 1(q)(a, b) = ab[1 + q(1 − a)][1 + q(1 − b)]Fn(q)(a, b), n ≥ 1, with initial condition F1(q)(a, b) = q[ab(1 + q(1 − b))]. Consequently, for every n ≥ 1, Fn(q)(a, b) = q[ab(1 + q(1 − b)]n[1 + q(1 − a)]n − 1. The result follows by noting that ϕn(d1, d2, q)(p) given in is simply Fn(q)(α(p), β(p)). We underline that in the case when the random variable η has Bernoulli distribution, the functions Kn, Fn, Kn⋆ and Fn⋆ defined in Lemma [L: Prob cam open] are polynomial in (a, b), but this does not hold in general. Of course, by dealing only with this initial distribution, we could derive Equation from Definition [D: k Cam] in a more direct way. Instead, we prefer to consider the general initial configuration in order to establish Lemma [L: Prob cam open], for future reference. Proofs of Theorem [T: UBA] and Corollary [C: UB bi-re] ------------------------------------------------------ From Lemma [P: BPdom], it follows that, by solving for each n ≥ 1 the equation in p (d1d2)n ϕn(d1, d2, q)(p) = 1, we obtain a sequence of upper bounds for the critical probability pc(Td1, d2, η). So, for every n ≥ 1, we define the function $$\label{F: f\_n} f^{({d\_1,d\_2}, q)}\_n(p) = {\left[ \phi^{({d\_1,d\_2}, q)}\_n(p) \right]}^{1 / n}-\frac{1}{d\_1d\_2}.$$ Notice that, for p ∈ [0, 1], limn → ∞fn(d1, d2, q)(p) = f(d1, d2, q)(p), where f(d1, d2, q)(p) is defined in . To prove the upper bound presented in Theorem [T: UBA], we use the following result of Real Analysis, whose proof is included for the sake of completeness. [L: Conv] Let {fn} be a sequence of increasing, continuous real-valued functions defined on [0, 1], such that fn(0) < 0 and fn(1) > 0 for every n. Suppose that {fn} converges pointwise as n → ∞ to an increasing, continuous function f defined on [0, 1], and let $\tilde r\_n$ be the unique root of fn in [0, 1]. Then, there exists $\tilde r = \lim\_{n \to \infty} \tilde r\_n$ and $f (\tilde r) = 0$. [Proof of Lemma [L: Conv]] First, we observe that, under the stated conditions, the sequence {fn} converges uniformly to f on [0, 1] (see, e.g., ). Let ${\underaccent{\bar}{r}}= \liminf\_{n \to \infty} \tilde r\_n$ and ${\bar{r}}= \limsup\_{n \to \infty} \tilde r\_n$. Then, there exists a subsequence $\{\tilde r\_{n\_k}\}$ such that $\tilde r\_{n\_k} \to {\underaccent{\bar}{r}}$ as k → ∞. Consequently, $f\_{n\_k} (\tilde r\_{n\_k}) \to f({\underaccent{\bar}{r}})$ as k → ∞ (by applying ). From this, it follows that $f({\underaccent{\bar}{r}}) = 0$. Analogously, f(r̄) = 0. But since f is an increasing, continuous function with f(0) ≤ 0 ≤ f(1), we conclude that f has a unique root in the interval [0, 1]. Hence, ${\underaccent{\bar}{r}}= {\bar{r}}$, and the proof is complete. [Proof of Theorem [T: UBA]] It is straightforward to prove that there exists a positive integer N = N(d1, d2) such that {fn(d1, d2, q)}n ≥ N and f(d1, d2, q) defined in and satisfy the conditions of Lemma [L: Conv]. Therefore, by defining p̃n(d1, d2, q) as the unique root of fn(d1, d2, q) in the interval [0, 1], it follows that limn → ∞p̃n(d1, d2, q) = p̃(d1, d2, q), where the limit is the unique root of f(d1, d2, q)(p) in (0, 1). Since fn(d1, d2, q)(p) > 0 for every p > p̃n(d1, d2, q), from Lemma [P: BPdom], we have that pc(Td1, d2, η) ≤ p̃n(d1, d2, q). Taking n → ∞, the results follows. [Proof of Corollary [C: UB bi-re]] If η ≡ 1, then the function f(d1, d2, q)(p) simplifies to $$f^{({d\_1,d\_2})}(p) = \alpha(p)\beta(p)[2-\alpha(p)][2-\beta(p)]-\frac{1}{d\_1d\_2}.$$ The upper bound p̃(d1, d2) for pc(Td1, d2) is the unique root in (0, 1) of this continuous and increasing function. Let $$\bar{p} = \bar{p}({d\_1,d\_2}) = \frac{1}{2}\sqrt{\frac{(d\_1 + 1)(d\_2 + 1)}{d\_1 d\_2}}$$ denote the upper bound given in the statement of Corollary [C: UB bi-re]. The result follows from the fact that f(d1, d2)(p̄) ≥ 0. This inequality can be verified with the help of a symbolic computation software. For instance, in Mathematica, one can define the function f(d1, d2)(p) and p̄(d1, d2), and then use the command Reduce, to show that the statement f(d1, d2)(p̄(d1, d2)) ≥ 0 ∧ (d1 ≥ 2 ∧ d2 ≥ 1) ∨ (d1 ≥ 1 ∧ d2 ≥ 2) is equivalent to $$(1 \leq d\_1 < 2 \land d\_2 \geq 2) \lor (d\_1 \geq 2 \land d\_2 \geq 1). \qedhere$$ Extinction of the process ========================= To prove Theorem [T: SCE], we define a coupled percolation process that dominates the frog model. This percolation model is defined in a simple manner, not taking into account the trajectories of the frogs, so a coupling between the process on two graphs G1 and G2, with G1 ⊂ G2, is easily constructed. The end of the proof relies on the idea employed to establish Theorem 1.2 in. [Proof of Theorem [T: SCE]] Let G be an infinite tree of bounded degree, and let Δ denote the maximum degree of G. Recall that the extinction of the frog model on G is equivalent to the finiteness of the cluster of the root in the percolation model on ${\overset{\rightarrow}{\mathbb{G}}}$ that is defined in Subsection [SS: FM Perc]. Given a realization of ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}$, we define, for every x ∈ V, $${\mathcal{B}}\_x:= \left\{ \begin{array}{cl} \bigcup\_{k=1}^{\eta(x)} \{ y \in {\mathbb{V}}: {\textnormal{dist}}(x,y) < \Xi\_p^x(k) \} &\text{if } \eta(x) \geq 1, \\[0.2cm] \{x\} &\text{otherwise}. \end{array} \right.$$ Then, we consider the following oriented percolation model on ${\overset{\rightarrow}{\mathbb{G}}}$: the oriented edge ${\overset{\,\rightarrow}{xy}}$ from vertex x to vertex y is declared to be open if [y ∈ Bx], and closed otherwise. We call this model the disk-percolation model on G, with parameter p and initial configuration ruled by η. study this model with η ≡ 1 on general graphs and spherically symmetric trees. Since Rx ⊂ Bx for every vertex x, we have that the frog model on G dies out if the cluster of the root in the disk-percolation model on G has finite size. Now let d = max{Δ − 1, 2}. Viewing G as a subgraph of Td, we conclude that there is a natural coupling between the disk-percolation model on G and on Td, in such a way that the cluster of the root on G is finite, whenever it is finite on Td. In addition, the disk-percolation model on Td is dominated by a Galton–Watson branching process whose family size is distributed as ∣Bx \ {x}∣ (where x is a fixed vertex of Td). Consequently, to finish the proof, it suffices to show that this branching process is subcritical for small enough p > 0. The remainder of the proof follows the argument used to prove Theorem 1.2 in, which we adapt here to our context. For vertices x and y with ${\textnormal{dist}}(x,y) = k \geq 1$, by conditioning on η(x), we have $$\label{F: DP} {\mathds{P}}[y \in {\mathcal{B}}\_x] = \sum\_{i=1}^{\infty} \rho\_{i} \left[1 - (1 - p^k)^i\right].$$ Let k̂(i, p) = ⌊logi/log(1/p)⌋, where ⌊x⌋ denotes the largest integer which is less than or equal to x. By using elementary calculus, one proves that there exists a constant β̂ such that 1 − (1 − pk)i ≤ β̂ pk − k̂(i, p) − 1 for every k ≥ k̂(i, p) + 1. Given a vertex x of Td, let sk(Td) = (d + 1) dk − 1 denote the cardinality of the set of vertices that are at distance k from x. Using and, we obtain that, for some positive constants C1 and C2, $$\begin{aligned} {\mathds{E}}\|{\mathcal{B}}\_x\setminus\{x\}\| &= \sum\_{k=1}^\infty s\_k({\mathbb{T}}\_d) \sum\_{i=1}^{\infty} \rho\_{i} \left[1 - (1 - p^k)^i\right]\\ &\leq \sum\_{i=1}^\infty \rho\_i \, \Bigl[ \sum\_{k=1}^{\hat{k}(i,p)} s\_k({\mathbb{T}}\_d) + \sum\_{k=1}^\infty \hat{\beta} \, s\_{\hat{k} + k}({\mathbb{T}}\_d) \, p^{k-1} \Bigr]\\ &\leq C\_1 \displaystyle\sum\_{i=1}^\infty \rho\_i \, i^{\frac{\log d}{\log(1/p)}} + C\_2.\end{aligned}$$ For some p0 > 0, the last series converges uniformly in [0, p0]. Hence, there exists a small enough p > 0 such that ${\mathds{E}}\|{\mathcal{B}}\_x\setminus\{x\}\|<1$, completing the proof. Our strategy to show Theorem [T: LB] is to compare the frog model on Td1, d2 with a suitable subcritical multitype Galton–Watson branching process. The method is inspired by the idea of comparing the frog model with an ordinary branching process that is used by to prove Proposition [P: LB Alves]. [Proof of Theorem [T: LB]] Consider the multitype Galton–Watson branching process with two types, defined as follows. It starts with zero particles of type 1, and a random number of particles of type 2, which is given by the sum of (d1 + 2) independent copies of η. For i = 1, 2 and k1, k2 ∈ N0, let p(i)(k1, k2) denote the probability that a type i particle produces k1 particles of type 1, and k2 particles of type 2. Recall that $\rho\_k := {\mathds{P}}[\eta = k]$, and suppose that the progeny distribution is given by $$\begin{aligned} {3} \mathbf{p}^{(1)}(0, 0) &= 1-p, \qquad &\mathbf{p}^{(1)}(0, 1) &= \frac{p (1 + d\_1 \rho\_0)}{d\_1+1}, \qquad &\mathbf{p}^{(1)}(0, k) &= \frac{pd\_1 \rho\_{k-1}}{d\_1+1}, \; k=2, 3, \dots \\[0.2cm] \mathbf{p}^{(2)}(0, 0) &= 1-p, &\mathbf{p}^{(2)}(1, 0) &= \frac{p (1 + d\_2 \rho\_0)}{d\_2+1}, &\mathbf{p}^{(2)}(k, 0) &= \frac{pd\_2 \rho\_{k-1}}{d\_2+1}, \; k=2, 3, \dots\end{aligned}$$ Notice that, in the frog model on Td1, d2, at time n = 1, the number of awake frogs placed on type 1 and type 2 vertices is stochastically smaller than the initial configuration of the multitype Galton–Watson process just defined. Furthermore, in the frog model, every vertex with at least one awake frog at time n ≥ 1 has at least one neighbor vertex whose original frogs have been awoken prior to time n. Thus, the multitype Galton–Watson process dominates the frog model on Td1, d2, in the sense that the frog model becomes extinct if this process does. Now let M : = (mi, j)i, j ∈ {1, 2} be the first moment matrix, i.e., mi, j is the expected number of type j offspring of a single type i particle in one generation. Since the number of types is finite, it is well known (see ) that the multitype Galton–Watson process dies out almost surely if and only if the largest eigenvalue λ(M) of the matrix M is less than 1. In our case, the first moment matrix is given by $$M = \begin{bmatrix} 0 & \frac{p}{d\_1+1}(1+d\_{1}({\mathds{E}}\eta+1)) \\[0.2cm] \frac{p}{d\_2+1}(1+d\_{2}({\mathds{E}}\eta+1)) & 0 \end{bmatrix}.$$ An elementary calculation shows that if $$p<\sqrt{\frac{(d\_1+1) (d\_2+1)}{[d\_1({\mathds{E}}\eta+1)+1] [d\_2({\mathds{E}}\eta+1)+1]}},$$ then λ(M) < 1, therefore the multitype Galton–Watson process dies out almost surely. Consequently, the same happens to the frog model. Acknowledgements ================ The authors are grateful to two anonymous referees, who carefully read the paper and made many valuable questions and suggestions that helped us to improve it. --- 1. The authors are thankful to the National Council for Scientific and Technological Development – CNPq (Jaime Utria’s grant No. 140887/2017-2), and the São Paulo Research Foundation – FAPESP (Grant No. 2017/10555-0).[↩](#fnref1) Phase transition for the frog modelon biregular trees ===================================================== [1](#fn1) We study the frog model with death on the biregular tree Td1, d2. Initially, there is a random number of active and inactive particles located on the vertices of the tree. Each active particle moves as a discrete-time independent simple random walk on Td1, d2 and has a probability of death (1 − p) before each step. When an active particle visits a vertex which has not been visited previously, the inactive particles placed there are activated. We prove that this model undergoes a phase transition: for values of p below a critical probability pc, the system dies out almost surely, and for p > pc, the system survives with positive probability. We establish explicit bounds for pc in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies pc(Td1, d2) = 1/2 + Θ(1/d1 + 1/d2) as d1, d2 → ∞. Introduction ============ This paper addresses the issue of phase transition for the frog model with death, a discrete-time growing system of simple random walks on a rooted graph G, which is described as follows. Initially there is an independent random number of particles at each vertex of G. All particles are inactive at time zero, except for those that might be placed at ${\varnothing}$, the root of G. Each active particle moves as a discrete-time independent simple random walk (SRW) on the vertices of G, and has a probability of death (1 − p) before each step. When an active particle visits an inactive particle, that particle becomes active and starts to walk, performing exactly the same dynamics, independently of everything else. In the literature, the particles are often referred to as frogs, whose possible states are described as sleeping (inactive) and awake (active). This process can be thought as a model for describing rumor (or infection) spreading. One can think of every awake frog as an informed (or infected) agent, which shares the rumor with (or infects) a sleeping frog at the first time they meet. In the last decades, there has been a growing interest in understanding the behavior of stochastic processes on more general graph structures than the d-dimensional integer lattice Zd and the homogeneous tree Td of degree (d + 1). In particular, the study of phase transitions and critical phenomena for stochastic systems on nonhomogeneous trees and nonamenable quasi-transitive graphs has become an important research area. Among the models studied, are percolation, the contact process, and branching random walks. For the frog model, most of the work has involved studying the process on Zd, Td and recently on d-ary trees. As far as we know, only considers another kind of tree, proving the recurrence of the process (without death) on a 3, 2-alternating tree (in which the generations of vertices alternate between having 2 and 3 children). The first published paper dealing with the frog model (with p = 1, G = Zd) is due to, where it was referred to as the egg model. They proved that, starting from the one-particle-per-vertex initial configuration, almost surely infinitely many frogs will visit the origin for all d ≥ 3 (that is, although each frog is individually transient, the process is recurrent). exhibits the critical rate at which the frog model with Bernoulli(α/∣∣x∣∣2) sleeping frogs at each x ∈ Zd \ {0} changes from transience to recurrence. A similar result is obtained by for the model on d-ary trees, with Poisson(μ) sleeping frogs at each vertex. More precisely, the authors prove that the model undergoes a phase transition between transience and recurrence, as the initial density μ of frogs increases. For the model starting with one frog per vertex, establish that there is a phase transition in the dimension of the tree, by proving recurrence for d = 2 and transience for d ≥ 5. Based on simulations, they conjecture that the model is recurrent for d = 3, and transient for d = 4. In, for the frog model without death on Zd, it is proved that, starting from the one-particle-per-vertex initial configuration, the set of the original positions of all awake frogs, rescaled by the elapsed time, converges to a nonempty compact convex set. prove the same statement in the case of random initial configuration; these results are known as shape theorems. For a continuous-time version of the frog model, a limiting shape result is stated by. We refer to for a survey on some results for the model and its variations. Regarding the frog model with death, the existence of phase transition as p varies was first studied by. As we will detail later, the occurrence of phase transition means that there is a nontrivial value pc of p separating the regimes of extinction and survival of the process. obtain lower bounds on pc for general graphs and bounded degree graphs, and establish the existence of phase transition on Zd and Td, for d ≥ 2, under rather broad conditions. prove that the critical probability for the frog model on a homogeneous tree of degree (d + 1) is at most (d + 1)/(2d); that result is an improvement of the upper bound stated by, namely, (d + 1)/(2d − 2). Further improvements on the upper bound for this critical probability were recently obtained by, using Renewal Theory, and by. For more details on the subject, see these papers and references therein. The aim of the present paper is to deepen the study of the critical phenomenon of the frog model, going beyond Zd and homogeneous trees. We consider the model on a specific class of nonhomogeneous trees, namely, biregular trees. In the main results, we prove a sufficient condition for the existence of phase transition, and present explicit bounds for the critical probability, in the case of random initial configuration. Since there is more than one parameter measuring the size of such trees, bounds on the critical parameter are harder to get at. The asymptotic behavior of the critical probability for large values of the dimension of the tree is also derived. We hope that our study will stimulate further works regarding the occurrence of phase transition on other nonhomogeneous trees and nonamenable graphs. Formal description of the model and main results ================================================ We start off with some basic definitions and notation of Graph Theory. Let G = (V, E) be an infinite connected locally finite graph, with vertex-set V and edge-set E. A vertex ${\varnothing}\in {\mathbb{V}}$ is fixed and called the root of G. We denote an unoriented edge with endpoints x and y by xy. Vertices x and y are said to be neighbors if they belong to a common edge xy; we denote this by x ∼ y. The degree of a vertex is the number of its neighbors. A path of length n from x to y is a sequence x = x0, …, xn = y of vertices such that xi ∼ xi + 1 for all i = 0, …, n − 1. The graph distance ${\textnormal{dist}}(x, y)$ between x and y is the minimal length of a path connecting the two vertices; the level of x is ${\textnormal{dist}}({\varnothing}, x)$. A tree is a connected graph without loops or cycles, where by a cycle in a graph we mean a sequence of vertices x0, …, xn, n ≥ 3, with no repetitions besides xn = x0. A graph is bipartite if its vertex-set V can be partitioned into two subsets V1 and V2, in such a form that every edge joins a vertex of V1 to a vertex of V2. For d1 ≥ 1 and d2 ≥ 1, we denote by Td1, d2 the (d1, d2)-biregular tree, which is the bipartite tree where the degree of a vertex is (d1 + 1) or (d2 + 1), according to whether the level of the vertex is even or odd. In this case, the class V1 (resp. V2) is the set of vertices at even (resp. odd) distance from the root. From now on, a vertex x ∈ Vi will be called a type i* vertex. Notice that T1, 1 is isomorphic to Z. See Figure [Fg: bitree] for an illustration of T2, 4. [Fg: bitree] Now we describe the frog model in a formal way, keeping the notation of and, whenever possible. We write N = {1, 2, …} and N0 = N ∪ {0}. Let η* be a random variable assuming values in N0, and define $\rho\_{k}:={\mathds{P}}[\eta=k]$, k ∈ N0. We suppose that ${\mathds{P}}[\eta \geq 1] > 0$, that is, ρ0 < 1. For s ∈ [0, 1], let $\varphi(s) := {\mathds{E}}[s^\eta]$ be the probability generating function of η. To define the frog model, let {η(x) : x ∈ V}, {(Snx(k))n ∈ N0; k ∈ N, x ∈ V} and {(Ξpx(k)); k ∈ N, x ∈ V} be independent sets of random objects defined as follows. For each x ∈ V, η(x) has the same law as η, and gives the initial number of frogs at vertex x. If η(x) ≥ 1, then for each k ∈ {1, …, η(x)}, (Snx(k))n ∈ N0 is a discrete-time SRW on G starting from x, and Ξpx(k) is a random variable whose law is given by ${\mathds{P}}[\Xi\_p^x(k)=j]=(1-p)p^{j-1}$, j ∈ N, where p ∈ [0, 1] is a fixed parameter. These random objects describe respectively the trajectory and the lifetime of the k-th frog placed initially at x. Thus, the k-th frog at vertex x, whenever it is awoken, follows the SRW (Snx(k))n ∈ N0, and disappears at the instant it reaches a total of (Ξpx(k) − 1) jumps. At the moment the frog disappears, it is not able to awake other frogs (first the frog decides whether or not to survive, and only after that it is allowed to jump). There is no interaction between awake frogs. At time n = 0, only the frogs that might be placed at the root of G are awake. We call this model the frog model on G, with survival parameter p and initial configuration ruled by η, and denote it by ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}.$ A particular realization of the frog model survives if for every instant of time there is at least one awake frog. Otherwise, we say that it dies out. A coupling argument shows that ${\mathds{P}}[{\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}} \text{ survives}]$ is a nondecreasing function of p, and therefore we define the critical probability as $${\ensuremath{{p\_c}({\mathbb{G}}, \eta)}} = \inf\left\{p:{\mathds{P}}[{\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}} \text{ survives}]>0\right\}.$$ As usual, we say that ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}$ exhibits phase transition if pc(G, η) ∈ (0, 1). For G = Td1, d2 and η ≡ 1 (one-particle-per-vertex initial configuration), we simply write pc(Td1, d2). As proved by, if d1 = d2 = 1, then under the condition ${\mathds{E}}\log (\eta \vee 1)< \infty$, we have that the frog model dies out almost surely for every p < 1, that is, pc(Z, η) = 1. The picture is quite different for higher degrees. Indeed, prove that the frog model on Td exhibits phase transition for every d ≥ 2, provided that ρ0 < 1 and ${\mathds{E}}\eta^\delta<\infty$ for some δ > 0. Here we extend this result to the case where the process lives on Td1, d2 (assuming d1 ≥ 2 or d2 ≥ 2). We begin by showing a sufficient condition to guarantee the almost sure extinction of the frog model on bounded degree trees for small enough p. [T: SCE] Let G be an infinite tree of bounded degree, and suppose that there exists δ > 0 such that ${\mathds{E}}\eta^\delta<\infty$. Then pc(G, η) > 0. This result is an extension of Theorem 1.2 in, which is stated for a homogeneous tree Td, d ≥ 2. Note, however, that one can not prove Theorem [T: SCE] by using a direct coupling and the corresponding result for homogeneous trees. As shown by, in general, the critical parameter of the frog model is not a monotonic function of the graph. A longstanding open question is whether, for a class of graphs (for example, trees), G1 ⊂ G2 in this class implies that pc(G2, η) ≤ pc(G1, η). This problem was first raised by (for Zd), and discussed again in, and. Under the condition that the mean number of frogs placed initially at every vertex is finite, we derive a nontrivial lower bound for the critical probability on Td1, d2. [T: LB] Suppose that ${\mathds{E}}\eta<\infty$. If d1 ≥ 2 or d2 ≥ 2, then $${\ensuremath{{p\_c}({\mathbb{T}}\_{{d\_1,d\_2}}, \eta)}}\geq \sqrt{\frac{(d\_1+1)(d\_2+1)}{[d\_1({\mathds{E}}\eta+1) + 1][d\_2({\mathds{E}}\eta+1) +1]}}.$$ The proofs of Theorems [T: SCE] and [T: LB] are given in Section [S: Extinction]. To present an upper bound for pc(Td1, d2, η), we need the following definition. [D: tudo] Let q = 1 − ρ0 > 0, κ = (d1 + 1)(d2 + 1), Δ = κ2 − 2κ(d1 + d2)p2 + (d2 − d1)2p4. We also define the functions α, β, f : [0, 1] → [0, 1] given by $$\begin{aligned} \alpha(p)&=\alpha^{({d\_1,d\_2})}(p)=\left\{ \begin{array}{cl} \dfrac{\kappa+p^2(d\_2-d\_1)-\sqrt{\Delta}}{2d\_2(d\_1+1)p} & \text{if } 0 < p \leq 1, \\[0.5cm] 0 & \text{if } p = 0, \end{array} \right. \label{F: Prob-alpha}\\[0.2cm] \beta(p)&=\beta^{({d\_1,d\_2})}(p)=\left\{ \begin{array}{cl} \dfrac{\kappa+p^2(d\_1-d\_2)-\sqrt{\Delta}}{2d\_1(d\_2+1)p} & \text{if } 0 < p \leq 1, \\[0.5cm] 0 & \text{if } p = 0, \end{array} \right. \label{F: Prob-beta}\\[0.25cm] f^{({d\_1,d\_2}, q)}(p)&=\alpha(p)\beta(p)[1+q(1-\alpha(p))][1+q(1-\beta(p))]-\frac{1}{d\_1d\_2}. \label{F: Function-f}\end{aligned}$$ [T: UBA] Suppose that ρ0 < 1. If d1 ≥ 2 or d2 ≥ 2, then pc(Td1, d2, η) ≤ p̃(d1, d2, q), where p̃(d1, d2, q) is the unique root of the function f(d1, d2, q) in the interval (0, 1). [C: UB bi-re] Suppose that η ≡ 1. If d1 ≥ 2 or d2 ≥ 2, then [C: UBH] $${\ensuremath{{p\_c}({\mathbb{T}}\_{{d\_1,d\_2}})}} \leq \frac{1}{2}\sqrt{\frac{(d\_1 + 1)(d\_2 + 1)}{d\_1 d\_2}}.$$ The proofs of Theorem [T: UBA] and Corollary [C: UB bi-re] are presented in Section [S: Survival]. In the case of d1 = d2 = d and η ≡ 1 in Theorem [T: UBA] (also in Corollary [C: UB bi-re]), we obtain the upper bound for the critical probability pc(Td) that is proved in, namely, (d + 1)/(2d). From Theorems [T: SCE] and [T: UBA], we conclude: [T: PT] Suppose that ρ0 < 1 and ${\mathds{E}}\eta^\delta<\infty$ for some δ > 0. If d1 ≥ 2 or d2 ≥ 2, then 0 < pc(Td1, d2, η) < 1. For comparison, we include a result proved by, that establishes a lower bound for the critical probability of the frog model on bounded degree graphs. This result also appears in. [] [P: LB Alves] Suppose that G is a graph with maximum degree (D + 1), and ${\mathds{E}}\eta<\infty$. Then, $${\ensuremath{{p\_c}({\mathbb{G}}, \eta)}} \geq \dfrac{D+1}{D({\mathds{E}}\eta+1)+1}.$$ Notice that, for G = Td1, d2 with d1 = d2, the lower bound given in Theorem [T: LB] equals the one provided by Proposition [P: LB Alves]. However, for d1 ≠ d2, the lower bound stated in Theorem [T: LB] is better than that of Proposition [P: LB Alves]. For a numerical illustration, we consider η ≡ 1, and compute the bounds established in Theorems [T: LB], [T: UBA] and Proposition [P: LB Alves] for some values of the pair (d1, d2). The resulting values are given in Table [NV]. [ht] P1.8cmP4cmP4cmP3.6cm (d1, d2) & LB Proposition [P: LB Alves] &LB Theorem [T: LB] & UB Theorem [T: UBA] (1, 2) & 0.6000 & 0.6325 & 0.8588 (1, 3) & 0.5714 & 0.6172 & 0.8039 (1, 4) & 0.5556 & 0.6086 & 0.7749 (2, 2) & 0.6000 & 0.6000 & 0.7500 (2, 3) & 0.5714 & 0.5855 & 0.7063 (2, 4) & 0.5556 & 0.5774 & 0.6828 (3, 100) & 0.5025 & 0.5359 & 0.5771 (3, 1000) & 0.5002 & 0.5347 & 0.5743 (4, 10000) & 0.5000 & 0.5271 & 0.5572 [NV] Now we present a result concerning the asymptotic behavior of pc(Td1, d2) as d1, d2 → ∞. The lower and upper bounds given in Theorem [T: LB] with η ≡ 1 and Corollary [C: UB bi-re] are, respectively, $$\frac{1}{2} + \frac{1}{8} \left(\frac{1}{d\_1}+\frac{1}{d\_2}\right) + O\left(\frac{1}{d\_1^2}+\frac{1}{d\_2^2}\right) \text{ and } \; \frac{1}{2} + \frac{1}{4} \left(\frac{1}{d\_1}+\frac{1}{d\_2}\right) + O\left(\frac{1}{d\_1^2}+\frac{1}{d\_2^2}\right).$$ As a consequence, we find the correct order of magnitude for the critical probability. For the frog model on Td1, d2 starting from the one-particle-per-vertex initial configuration, we have $${\ensuremath{{p\_c}({\mathbb{T}}\_{{d\_1,d\_2}})}}= \frac{1}{2} +\Theta\left(\frac{1}{d\_1}+\frac{1}{d\_2}\right) \quad \text{as } \, d\_1, d\_2 \to \infty.$$ Survival of the process ======================= In brief, the key idea to prove Theorem [T: UBA] is to describe ${\ensuremath{\textnormal{FM}({\mathbb{T}}\_{{d\_1,d\_2}},p, \eta)}}$ as a percolation model which dominates suitably defined Galton–Watson branching processes. In Subsections [SS: FM Perc] and [SS: BT], we describe the frog model on a graph G as a particular percolation model, and we compute its parameters when G = Td1, d2. Then, in Subsection [SS: SBP], we define a sequence of Galton–Watson branching processes which are dominated by the frog model on Td1, d2. Finding p such that each one of these branching processes is supercritical leads us to obtain a sequence of upper bounds for the critical probability, which converges to the upper bound p̃(d1, d2, q) stated in Theorem [T: UBA]. Subsection [SS: Proofs] is devoted to the finalization of the proof. The technique to prove Theorem [T: UBA] is similar to that used by to establish an upper bound for the critical probability of the frog model on homogeneous trees. However, here we have to deal with the computations related to the fact that the degrees of the vertices of the tree alternate. Also to overcome this issue, we construct the sequence of approximating branching processes in a new manner, leading directly to the desired upper bound. An analogous approach is used by to derive an improved upper bound for the critical parameter on homogeneous trees. seen as percolation -------------------- In the bond percolation model, each edge of an infinite locally finite graph $\bar {\mathbb{G}}=(\bar {\mathbb{V}}, \bar {\mathbb{E}})$ is randomly assigned the value 1 (open) or 0 (closed), according to some probability measure on the product space $\{0, 1\}^{\bar {\mathbb{E}}}$. Then, one studies the connectivity properties of the random subgraph of $\bar{\mathbb{G}}$ which arises by removing closed edges. For a fundamental reference on the subject, we refer to . Next, we describe the frog model on an infinite graph G = (V, E) as a particular bond percolation model. Indeed, for every x ∈ V and 1 ≤ k ≤ η(x), we define the virtual set of vertices visited by the k-th frog located originally at x by Rxk : = {Snx(k) : 0 ≤ n < Ξpx(k)} ⊂ V. The set Rxk becomes real in the case when x is actually visited (and thus the sleeping frogs placed there are awoken). We define the range of x by $${\mathcal{R}}\_x:= \left\{ \begin{array}{cl} \bigcup\_{k=1}^{\eta(x)}{\mathcal{R}}\_x^k &\text{if } \eta(x) \geq 1, \\[0.2cm] \{x\} &\text{otherwise}. \end{array} \right.$$ Now let ${\overset{\rightarrow}{\mathbb{G}}}=({\mathbb{V}}, {\overset{\,\rightarrow}{\mathbb{E}}})$ be the oriented graph with vertex-set V and edge-set ${\overset{\,\rightarrow}{\mathbb{E}}}:=\{{\overset{\,\rightarrow}{xy}}:(x, y) \in {\mathbb{V}}\times {\mathbb{V}}, x \neq y\}$. That is, for every pair of distinct vertices x and y, an oriented edge is drawn from x to y. Then, we introduce the following notations: for x, y ∈ V distinct, [x → y] : = [y ∈ Rx], $[{\ensuremath{x \nrightarrow y}}]:=[y \notin {\mathcal{R}}\_x]$. This defines an oriented dependent long range anisotropic percolation model on ${\overset{\rightarrow}{\mathbb{G}}}$: ${\overset{\,\rightarrow}{xy}}$ is declared to be open if [x → y], and closed otherwise. In general, the probability that an oriented edge is open depends on its orientation. For instance, when G = Td1, d2, an edge emanating from a type 1 vertex to a type 2 vertex has probability of being open different from an edge emanating from a type 2 vertex to a type 1 vertex. This feature gives the anisotropy of the oriented percolation model. Notice also that the cluster of the root has infinite size if and only if there exists an infinite sequence of distinct vertices ${\varnothing}=x\_0, x\_1, x\_2, \dots$ such that xj → xj + 1 for all j ≥ 1. Of course, this event is equivalent to the survival of ${\ensuremath{\textnormal{FM}({\mathbb{G}},p, \eta)}}$. Biregular trees --------------- Now let us consider G = Td1, d2. Our next purpose is to obtain an explicit formula for the probability that an oriented edge is open. Toward this end, let α(d1, d2)(p) (resp. β(d1, d2)(p)) denote the probability that a type 1 (resp. type 2) frog ever visits a type 2 (resp. type 1) neighbor vertex. Avoiding a cumbersome notation, we sometimes write α(p) or simply α (resp. β(p), β). With a typical abuse of language, we call the percolation model associated to ${\ensuremath{\textnormal{FM}({\mathbb{T}}\_{{d\_1,d\_2}},p, \eta)}}$ the anisotropic percolation model on the biregular tree Td1, d2, with parameters α, β and ruled by η. We denote it by ${\ensuremath{\textnormal{APM}({\mathbb{T}}\_{{d\_1,d\_2}},\alpha, \beta, \eta)}}$. The following lemma about hitting probabilities of SRWs on Td1, d2 provides a formula for α and β. Recall that V1 and V2 are respectively the set of vertices at even and odd distance from the root of Td1, d2. [L: Prob open edge] Let x ∼ y be a pair of neighbor vertices, and suppose (x, y) ∈ Vi × Vj, with i, j ∈ {1, 2}, i ≠ j. Then, $${\mathds{P}}[y\in{\mathcal{R}}\_x^1]= \begin{cases} \alpha(p) &\text{if } i=1, j=2, \\[0.2cm] \beta(p) &\text{if } i=2, j=1, \end{cases}$$ where α and β are given in Definition [D: tudo]. Let τij : = τxy be the first time when the simple random walk on Td1, d2 starting from x visits y. Suppose first that p < 1, so conditioning on the lifetime of the frog located at x, we have $${\mathds{P}}[y\in{\mathcal{R}}\_x^1]={\mathds{E}}[p^{\tau\_{ij}}].$$ Now by conditioning on the first jump of the frog at x, we have $$\begin{aligned} {\mathds{E}}[p^{\tau\_{12}}]&=\frac{1}{d\_1+1}p+\frac{d\_1}{d\_1+1}p{\mathds{E}}[p^{\tau\_{12}}]{\mathds{E}}[p^{\tau\_{21}}]\\ {\mathds{E}}[p^{\tau\_{21}}]&=\frac{1}{d\_2+1}p+\frac{d\_2}{d\_2+1}p{\mathds{E}}[p^{\tau\_{12}}]{\mathds{E}}[p^{\tau\_{21}}]\end{aligned}$$ By right-continuity of the probability generating functions of τ12 and τ21, it follows that $\lim\_{p\to 0^+} {\mathds{E}}[p^{\tau\_{12}}]=\lim\_{p\to 0^+}{\mathds{E}}[p^{\tau\_{21}}]=0$. Therefore, and are the only possible solutions for the previous system of equations. This concludes the proof for p < 1. Finally, if p = 1, then $${\mathds{P}}[y\in{\mathcal{R}}\_x^1]={\mathds{P}}[\tau\_{ij}<\infty]=\lim\_{p\to 1^-} {\mathds{E}}[p^{\tau\_{ij}}]=\frac{d\_j+1}{d\_j(d\_i+1)}.\qedhere$$ If d1 ≤ d2, then α(d1, d2)(p) ≥ β(d1, d2)(p). [L: Prob long open edge] Let x and y be two vertices of Td1, d2 with (x, y) ∈ Vi × Vj, i, j ∈ {1, 2}, and let $k = {\textnormal{dist}}(x, y) \geq 1$. Let πη(i, j, k) denote the probability that the oriented edge ${\overset{\,\rightarrow}{xy}}$ is open. Then, $$\label{F: prob open edge aleator} \pi\_{\eta}(i, j, k)= \begin{cases} 1-\varphi(1-\alpha^n\beta^{n-1}) &\text{if } i=1, j=2, k=2n-1, \\[0.1cm] 1-\varphi(1-\alpha^{n-1}\beta^n) &\text{if } i=2, j=1, k=2n-1, \\[0.1cm] 1-\varphi(1-\alpha^n\beta^n) &\text{if } i=j, k=2n, \end{cases}$$ where $\varphi(s)={\mathds{E}}(s^\eta)$, s ∈ [0, 1], is the probability generating function of η. By conditioning on the initial number of frogs at vertex x, it is enough to consider η ≡ 1, that is, φ(s) ≡ s. In this case, πη(i, j, k) is the probability that a type i frog ever visits a type j vertex at distance k. So, we need to prove that for η ≡ 1, $$\label{F: prob open edge} \pi\_{\eta}(i, j, k)= \begin{cases} \alpha^n\beta^{n-1} &\text{if } i=1, j=2, k=2n-1, \\[0.1cm] \alpha^{n-1}\beta^n &\text{if } i=2, j=1, k=2n-1, \\[0.1cm] \alpha^n\beta^n &\text{if } i=j, k=2n. \end{cases}$$ Formula follows from the fact that if (x, y) ∈ V1 × V2 with k = 2n − 1, then the first time when the frog starting from x visits y is equal in distribution to the sum of 2n − 1 independent random variables, such that n of them are independent copies of τ12 and n − 1 of them are independent copies of τ21. Using this fact and Lemma [L: Prob open edge], we obtain that πη(1, 2, 2n − 1) = αnβn − 1. The other cases are analogous. A sequence of branching processes dominated by the frog model ------------------------------------------------------------- The central idea is as follows. For each n ≥ 1, we define a Galton–Watson branching process whose survival implies that the cluster of the root in ${\ensuremath{\textnormal{APM}({\mathbb{T}}\_{{d\_1,d\_2}},\alpha, \beta, \eta)}}$ has infinite size. For each branching process, we find a sufficient condition which guarantees that the process is supercritical. We get in this manner a sequence of upper bounds for the critical probability, which converges to the upper bound given in the statement of Theorem [T: UBA]. This technique of using embedded branching processes is very similar to that used for the contact process on homogeneous trees; see the paper by. Let us now carry out this plan. [D: Sets] 1. We define a partial order on the set V as follows: for x, y ∈ V, we say that x ≼ y if x belongs to the path connecting $ {\varnothing}$ and y; x ≺ y if x ≼ y and x ≠ y. 2. For any vertex $ x \neq {\varnothing}$, let V+(x) = {y ∈ V : x ≼ y}. Also define $ {\mathbb{V}}^+({\varnothing}) = {\mathbb{V}}\setminus {\mathbb{V}}^+(z) $, where z is a fixed vertex neighbor to $ {\varnothing}$. 3. For x ∈ V and k ∈ N, define $ L\_k (x) = \{ y \in {\mathbb{V}}^+(x) : {\textnormal{dist}}(x, y) = k \} $. [D: k Cam] For x ∈ V and y ∈ Lk(x), k ∈ N, consider x0 = x ≺ x1 ≺ ⋯ ≺ xk − 1 ≺ xk = y, the path connecting x and y. For each ℓ = 1, 2, …, n − 1, we denote by $[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_{\ell} }}]$ the event that $[{\ensuremath{x\_0 \rightarrow x\_{\ell}}}] \cap [{\ensuremath{x\_0 \nrightarrow x\_{\ell+1}}}]$. We define the event $[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{k}}}]$ inductively on k by: * If k = 1, then $$[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_k}}]:=[{\ensuremath{x\_0 \rightarrow x\_k}}].$$ * If k ≥ 2, then $$[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{k}}}]:=[{\ensuremath{x\_0 \rightarrow x\_{k}}}]\cup\bigcup\_{\ell=1}^{k-1}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_{\ell} }}, {\ensuremath{x\_{\ell} \stackrel {o}{\rightarrow} x\_{k}}}].$$ We denote the complement of $[{\ensuremath{x \stackrel {o}{\rightarrow} y}}]$ by $[{\ensuremath{x \stackrel {o}{\nrightarrow} y}}]$. Now we construct a sequence of Galton–Watson branching processes embedded in ${\ensuremath{\textnormal{APM}({\mathbb{T}}\_{{d\_1,d\_2}},\alpha, \beta, \eta)}}$. Starting from the root, the potential direct descendants of a vertex x are vertices located in Lk(x), where k = 2n is an even number. Roughly speaking, a vertex y ∈ L2n(x) is said to be a child of vertex x if and only if $[{\ensuremath{x \stackrel {o}{\rightarrow} y}}]$. More formally, let n ∈ N be fixed. Define ${\mathcal{Y}}\_{0, n}:=\{{\varnothing}\}$, and for ℓ ∈ N define $${\mathcal{Y}}\_{\ell, n}:=\bigcup\_{x\in {\mathcal{Y}}\_{\ell-1, n}}\{y\in L\_{2n}(x): {\ensuremath{x \stackrel {o}{\rightarrow} y}}\}.$$ Let Yℓ, n = ∣Yℓ, n∣ be the cardinality of Yℓ, n. [P: BPdom] For every n ∈ N, {Yℓ, n}ℓ ∈ N0 is a Galton–Watson branching process whose survival implies the occurrence of percolation. In addition, its mean number of offspring per individual satisfies ${\mathds{E}}[Y\_{1, n}]\geq (d\_1 d\_2)^n \, \phi^{({d\_1,d\_2}, q)}\_n(p)$, where ϕn(d1, d2, q)(p) : = q[α(p)β(p)(1 + q(1 − β(p))]n[1 + q(1 − α(p))]n − 1. Since the first claim in Lemma [P: BPdom] is clear, to prove it, we have to compute ${\mathds{E}}[Y\_{1, n}]$. To accomplish this, we show a recursive formula for the probability of the event $[{\ensuremath{x \stackrel {o}{\rightarrow} y}}]$. Let us define the domain $\mathcal{A}\_{{d\_1,d\_2}}:= \left[0, \frac{d\_2+1}{d\_2(d\_1+1)}\right]\times\left[0, \frac{d\_1+1}{d\_1(d\_2+1)}\right] \subseteq [0, 1]^2$. In formulas that appear from this point on, we assume that a summation of the form ∑10 equals 0. [L: Prob cam open] For (x, y) ∈ Vi × Vj, i, j ∈ {1, 2}, with ${\textnormal{dist}}(x, y)=k \geq 1$, define $$\nu\_{\eta}(i, j, k) := {\mathds{P}}[{\ensuremath{x \stackrel {o}{\rightarrow} y}}].$$ For every n ≥ 1, there exists functions Kn, Fn, Kn⋆, Fn⋆ with domain Ad1, d2, not depending on d1, d2, such that $$\begin{aligned} \nu\_{\eta}(1, 2, 2n-1)&=K\_n(\alpha(p), \beta(p)), &\nu\_{\eta}(1, 1, 2n)&=F\_n(\alpha(p), \beta(p)), \\[0.2cm] \nu\_{\eta}(2, 1, 2n-1)&=K\_n^{\star}(\alpha(p), \beta(p)), &\nu\_{\eta}(2, 2, 2n)&=F\_n^{\star}(\alpha(p), \beta(p)).\end{aligned}$$ For n ≥ 1 and (a, b) ∈ Ad1, d2, we define the following sequence of functions recursively: $$\begin{aligned} K\_n(a, b) &= [1-\varphi(1-a^n b^{n-1})] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^{\ell+1}b^\ell)-\varphi(1-a^\ell b^\ell)]K\_{n-\ell}(a, b) \\ &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^\ell b^\ell)-\varphi(1-a^\ell b^{\ell-1})]F\_{n-\ell}^{\star}(a, b), \end{aligned}\\[0.2cm] K^{\star}\_n(a, b) &= [1-\varphi(1-b^n a^{n-1})] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-b^{\ell+1}a^\ell)-\varphi(1-a^\ell b^\ell)]K^{\star}\_{n-\ell}(a, b) \\ &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^\ell b^\ell)-\varphi(1-b^\ell a^{\ell-1})] F\_{n-\ell}(a, b), \end{aligned}\\[0.2cm] F\_n(a, b) &= [1-\varphi(1-a^n b^n)] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-a^{\ell+1}b^\ell)-\varphi(1-a^\ell b^\ell)] F\_{n-\ell}(a, b) \\ &+ \sum\_{\ell=1}^{n} [\varphi(1-a^\ell b^\ell)-\varphi(1-a^\ell b^{\ell-1})]K\_{n+1-\ell}^{\star}(a, b), \end{aligned}\\[0.2cm] F\_n^{\star}(a, b) &= [1-\varphi(1-b^n a^n)] \begin{aligned}[t] &+ \sum\_{\ell=1}^{n-1} [\varphi(1-b^{\ell+1}a^\ell)-\varphi(1-a^\ell b^\ell)]F\_{n-\ell}^{\star}(a, b)\\ &+ \sum\_{\ell=1}^{n} [\varphi(1-a^\ell b^\ell)-\varphi(1-b^\ell a^{\ell-1})]K\_{n+1-\ell}(a, b). \end{aligned}\end{aligned}$$ Next, using that ${\mathds{P}}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_\ell }}]={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_\ell}}]-{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{\ell+1}}}]$, we obtain, for every n ≥ 1, $$\begin{aligned} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{2n-1}}}]&={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n-1}}}]+\sum\_{\ell=1}^{2n-2} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_\ell }}]{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n-1}}}]\\ &={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n-1}}}]+\sum\_{\ell=1}^{2n-2}\{{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_\ell}}]-{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{\ell+1}}}]\}{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n-1}}}], \\ \intertext{and} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {o}{\rightarrow} x\_{2n}}}]&={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n}}}]+\sum\_{\ell=1}^{2n-1} {\mathds{P}}[{\ensuremath{x\_0 \stackrel {}{\leadsto} x\_\ell }}]{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n}}}]\\ &={\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{2n}}}]+\sum\_{\ell=1}^{2n-1}\{{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_\ell}}]-{\mathds{P}}[{\ensuremath{x\_0 \rightarrow x\_{\ell+1}}}]\}{\mathds{P}}[{\ensuremath{x\_\ell \stackrel {o}{\rightarrow} x\_{2n}}}].\end{aligned}$$ We split the proof into two cases: * If (x0, x2n − 1) ∈ Vi × Vj with i ≠ j, then $$\begin{aligned} \nu\_\eta(i, j, 2n-1)=\pi\_\eta(i, j, 2n-1) &+ \sum\_{\ell=1}^{n-1}[\pi\_\eta(i, i, 2\ell)-\pi\_\eta(i, j, 2\ell+1)]\nu\_\eta(i, j, 2(n-\ell)-1)]\\ &+ \sum\_{\ell=1}^{n-1} [\pi\_\eta(i, j, 2\ell-1)-\pi\_\eta(i, i, 2\ell)]\nu\_\eta(j, j, 2(n-\ell)).\end{aligned}$$ * If (x0, x2n) ∈ Vi × Vi, then $$\begin{aligned} \nu\_\eta(i, i, 2n)=\pi\_\eta(i, i, 2n) &+ \sum\_{\ell=1}^{n-1} [\pi\_\eta(i, i, 2\ell)-\pi\_\eta(i, j, 2\ell+1)]\nu\_\eta(i, i, 2(n-\ell))\\ &+ \sum\_{\ell=1}^{n} [\pi\_\eta(i, j, 2\ell-1)-\pi\_\eta(i, i, 2\ell)]\nu\_\eta(j, i, 2(n-\ell)+1).\end{aligned}$$ Using Equation , the result follows by induction on n. [Proof of Lemma [P: BPdom]] From Lemma [L: Prob cam open], the Galton–Watson branching process has mean number of progeny per individual given by $${\mathds{E}}[Y\_{1, n}] = (d\_1d\_2)^n \, \nu\_{\eta}(1, 1, 2n) = (d\_1d\_2)^n \, F\_n(\alpha(p), \beta(p)).$$ To obtain a lower bound for this expected value (not depending on the function φ), we truncate the initial configuration of the frog model. We consider the modified initial configuration η′ by $$\eta^{\prime}(x) = {\mathds{1}}\_{\{\eta(x) \geq 1 \}}, \, x \in {\mathbb{V}}.$$ Since the initial condition η′ is dominated by η in the usual stochastic order, it follows that νη(1, 1, 2n) ≥ νη′(1, 1, 2n). But for the restricted frog model with initial configuration ruled by η′, νη′(1, 1, 2n) = Fn(q)(α(p), β(p)), where Fn(q)(a, b) is given by the formulas stated in the proof of Lemma [L: Prob cam open] with the choice φ(s) = 1 − q(1 − s). Using these formulas and induction on n, we have that Fn(q) satisfies Fn + 1(q)(a, b) = ab[1 + q(1 − a)][1 + q(1 − b)]Fn(q)(a, b), n ≥ 1, with initial condition F1(q)(a, b) = q[ab(1 + q(1 − b))]. Consequently, for every n ≥ 1, Fn(q)(a, b) = q[ab(1 + q(1 − b)]n[1 + q(1 − a)]n − 1. The result follows by noting that ϕn(d1, d2, q)(p) given in is simply Fn(q)(α(p), β(p)). We underline that in the case when the random variable η has Bernoulli distribution, the functions Kn, Fn, Kn⋆ and Fn⋆ defined in Lemma [L: Prob cam open] are polynomial in (a, b), but this does not hold in general. Of course, by dealing only with this initial distribution, we could derive Equation from Definition [D: k Cam] in a more direct way. Instead, we prefer to consider the general initial configuration in order to establish Lemma [L: Prob cam open], for future reference. Proofs of Theorem [T: UBA] and Corollary [C: UB bi-re] ------------------------------------------------------ From Lemma [P: BPdom], it follows that, by solving for each n ≥ 1 the equation in p (d1d2)n ϕn(d1, d2, q)(p) = 1, we obtain a sequence of upper bounds for the critical probability pc(Td1, d2, η). So, for every n ≥ 1, we define the function $$\label{F: f\_n} f^{({d\_1,d\_2}, q)}\_n(p) = {\left[ \phi^{({d\_1,d\_2}, q)}\_n(p) \right]}^{1 / n}-\frac{1}{d\_1d\_2}.$$ Notice that, for p ∈ [0, 1], limn → ∞fn(d1, d2, q)(p) = f(d1, d2, q)(p), where f(d1, d2, q)(p) is defined in . To prove the upper bound presented in Theorem [T: UBA], we use the following result of Real Analysis, whose proof is included for the sake of completeness. [L: Conv] Let {fn} be a sequence of increasing, continuous real-valued functions defined on [0, 1], such that fn(0) < 0 and fn(1) > 0 for every n. Suppose that {fn} converges pointwise as n → ∞ to an increasing, continuous function f defined on [0, 1], and let $\tilde r\_n$ be the unique root of fn in [0, 1]. Then, there exists $\tilde r = \lim\_{n \to \infty} \tilde r\_n$ and $f (\tilde r) = 0$. [Proof of Lemma [L: Conv]] First, we observe that, under the stated conditions, the sequence {fn} converges uniformly to f on [0, 1] (see, e.g., ). Let ${\underaccent{\bar}{r}}= \liminf\_{n \to \infty} \tilde r\_n$ and ${\bar{r}}= \limsup\_{n \to \infty} \tilde r\_n$. Then, there exists a subsequence $\{\tilde r\_{n\_k}\}$ such that $\tilde r\_{n\_k} \to {\underaccent{\bar}{r}}$ as k → ∞. Consequently, $f\_{n\_k} (\tilde r\_{n\_k}) \to f({\underaccent{\bar}{r}})$ as k → ∞ (by applying ). From this, it follows that $f({\underaccent{\bar}{r}}) = 0$. Analogously, f(r̄) = 0. But since f is an increasing, continuous function with f(0) ≤ 0 ≤ f(1), we conclude that f has a unique root in the interval [0, 1]. Hence, ${\underaccent{\bar}{r}}= {\bar{r}}$, and the proof is complete. [Proof of Theorem [T: UBA]] It is straightforward to prove that there exists a positive integer N = N(d1, d2) such that {fn(d1, d2, q)}n ≥ N and f(d1, d2, q) defined in and satisfy the conditions of Lemma [L: Conv]. Therefore, by defining p̃n(d1, d2, q) as the unique root of fn(d1, d2, q) in the interval [0, 1], it follows that limn → ∞p̃n(d1, d2, q) = p̃(d1, d2, q), where the limit is the unique root of f(d1, d2, q)(p) in (0, 1). Since fn(d1, d2, q)(p) > 0 for every p > p̃n(d1, d2, q), from Lemma [P: BPdom], we have that pc(Td1, d2, η) ≤ p̃n(d1, d2, q). Taking n → ∞, the results follows. [Proof of Corollary [C: UB bi-re]] If η ≡ 1, then the function f(d1, d2, q)(p) simplifies to $$f^{({d\_1,d\_2})}(p) = \alpha(p)\beta(p)[2-\alpha(p)][2-\beta(p)]-\frac{1}{d\_1d\_2}.$$ The upper bound p̃(d1, d2) for pc(Td1, d2) is the unique root in (0, 1) of this continuous and increasing function. Let $$\bar{p} = \bar{p}({d\_1,d\_2}) = \frac{1}{2}\sqrt{\frac{(d\_1 + 1)(d\_2 + 1)}{d\_1 d\_2}}$$ denote the upper bound given in the statement of Corollary [C: UB bi-re]. The result follows from the fact that f(d1, d2)(p̄) ≥ 0. This inequality can be verified with the help of a symbolic computation software. For instance, in Mathematica, one can define the function f(d1, d2)(p) and p̄(d1, d2), and then use the command Reduce, to show that the statement f(d1, d2)(p̄(d1, d2)) ≥ 0 ∧ (d1 ≥ 2 ∧ d2 ≥ 1) ∨ (d1 ≥ 1 ∧ d2 ≥ 2) is equivalent to $$(1 \leq d\_1 < 2 \land d\_2 \geq 2) \lor (d\_1 \geq 2 \land d\_2 \geq 1). \qedhere$$ Extinction of the process ========================= To prove Theorem [T: SCE], we define a coupled percolation process that dominates the frog model. This percolation model is defined in a simple manner, not taking into account the trajectories of the frogs, so a coupling between the process	arxiv_0000735
IMSc/2010/06/10 10 + 1 to 3 + 1 in an Early Universe with mutually BPS Intersecting Branes Samrat Bhowmick and S. Kalyana Rama Institute of Mathematical Sciences, C. I. T. Campus, Tharamani, CHENNAI 600 113, India. email: samrat, [email protected] ABSTRACT > > We assume that the early universe is homogeneous, anisotropic, and is dominated by the mutually BPS 22ʹ55ʹ intersecting branes of M theory. The spatial directions are all taken to be toroidal. Using analytical and numerical methods, we study the evolution of such an universe. We find that, asymptotically, three spatial directions expand to infinity and the remaining spatial directions reach stabilised values. Any stabilised values can be obtained by a fine tuning of initial brane densities. We give a physical description of the stabilisation mechanism. Also, from the perspective of four dimensional spacetime, the effective four dimensional Newton’s constant G4 is now time varying. Its time dependence will follow from explicit solutions. We find in the present case that, asymptotically, G4 exhibits characteristic log periodic oscillations. > > > PACS numbers: 11.25.Yb, 98.80.Cq, 11.25.-w I. Introduction In early universe, temperatures and densities reach Planckian scales. Its description then requires a quantum theory of gravity. A promising candidate for such a theory is string/M theory. When the temperatures and densities reach string/M theory scales, the appropriate description is expected to be given in terms of highly energetic and highly interacting string/M theory excitations –. [1](#fn1) In this context, one of us have proposed in an earlier work an entropic principle according to which the final spacetime configuration that emerges from such high temperature string/M theory phase is the one that has maximum entropy for a given energy. This principle implies, under certain assumptions, that the number of large spacetime dimensions is 3 + 1 . High densities and high temperatures also arise near black hole singularities. Therefore, it is reasonable to expect that the string/M theory configurations which describe such regions of black holes will describe the early universe also. Consider the case of black holes. Various properties of a class of black holes have been successfully described using mutually BPS intersecting configurations of string/M theory branes. [2](#fn2) Black hole entropies are calculated from counting excitations of such configurations, and Hawking radiation is calculated from interactions between them. In the extremal limit, such brane configurations consist of only branes and no antibranes. In the near extremal limit, they consist of a small number of antibranes also. It is the interaction between branes and antibranes which give rise to Hawking radiation. String theory calculations are tractable and match those of Bekenstein and Hawking in the extremal and near extremal limits. But they are not tractable in the far extremal limit where the numbers of branes and antibranes are comparable. However, even in the far extremal limit, black hole dynamics is expected to be described by mutually BPS intersecting brane configurations where they now consist of branes, antibranes, and other excitations living on them, all at non zero temperature and in dynamical equilibrium with each other –. For the sake of brevity, we will refer to such far extremal configurations also as brane configurations even though they may now consist of branes and antibranes, left moving and right moving waves, and other excitations. The entropy S of N stacks of mutually BPS intersecting brane configurations, in the limit where S ≫ 1 , is expected to be given by $$\label{sn} S \sim \prod\_I \sqrt{n\_I + \bar{n}\_I} \sim {\cal E}^{\frac{N}{2}}$$ where nI and n̄I , I = 1, ⋯, N , denote the numbers of branes and antibranes of Ith type, ${\cal E}$ is the total energy, and the second expression applies for the charge neutral case where nI = n̄I for all I . The proof for this expression is given by comparing it in various limits with the entropy of the corresponding black holes, see also –. For N ≤ 4 and when other calculable factors omitted here are restored, this expression matches that for the corresponding black holes in the extremal and near extremal limit and, in the models based on that of Danielsson et al, matches upto a numerical factor in the far extremal limit – also. However, no such proof exists for N > 4 since no analogous object, black hole or otherwise, is known whose entropy is $\propto {\cal E}^\* \;$ with \* > 2 . Note that, in the limit of large ${\cal E} \;$, the entropy $S({\cal E})$ is $\ll {\cal E}$ for radiation in a finite volume and is $ \sim {\cal E}$ for strings in the Hagedorn regime. In comparison, the entropy given in ([sn]) is much larger when N > 2 . This is because the branes in the mutually BPS intersecting brane configurations form bound states, become fractional, and support very low energy excitations which lead to a large entropy. Thus, for a given energy, such brane configurations are highly entropic. Another novel consequence of fractional branes is the following. According to the ‘fuzz ball’ picture for black holes, the fractional branes arising from the bound states formed by intersecting brane configurations have non trivial transverse spatial extensions due to quantum dynamics. The size of their transverse extent is of the order of Schwarzschild radius of the black holes. Therefore, essentially, the region inside the ‘horizon’ of the black hole is not empty but is filled with fuzz ball whose fuzz arise from the quantum dynamics of fractional strings/branes. Chowdhury and Mathur have recently extended the fuzz ball picture to the early universe. They have argued that the early universe is filled with fractional branes arising from the bound states of the intersecting brane configurations, and that the brane configurations with highest N are entropically favorable, see equation ([sn]). However, as mentioned below equation ([sn]) and noted also in, the entropy expression in ([sn]) is proved in various limits for N ≤ 4 only and no proof exists for N > 4. Also, we are not certain of the existence of any system whose entropy $S({\cal E})$ is parametrically larger than ${\cal E}^2$ for large ${\cal E} \;$. See related discussions in. Therefore, in the following we will assume that N ≤ 4 . Then, a homogeneous early universe in string/M theory may be taken to be dominated by the maximum entropic N = 4 brane configurations distributed uniformly in the common transverse space. Such N = 4 mutually BPS intersecting brane configurations in the early universe may then provide a concrete realisation of the entropic principle proposed earlier by one of us to determine the number (3 + 1) of large spacetime dimensions. Indeed, in further works, using M theory symmetries and certain natural assumptions, we have shown that these configurations lead to three spatial directions expanding and the remaining seven spatial directions stabilising to constant sizes. In this paper, we assume that the early universe in M theory is homogeneous and anisotropic and that it is dominated by N = 4 mutually BPS intersecting brane configurations. [3](#fn3) In this context, it is natural to assume that all spatial directions are on equal footing to begin with. Therefore we assume that the ten dimensional space is toroidal. We then present a thorough analysis of the evolution of such an universe. The corresponding energy momentum tensor TAB has been calculated in under certain assumptions. However, general relations among the components of TAB may be obtained using U duality symmetries of M theory which are, therefore, valid more generally. [4](#fn4) We show in this paper that these U duality relations alone imply, under a technical assumption, that the N = 4 mutually BPS intersecting brane configurations with identical numbers of branes and antibranes will asymptotically lead to an effective (3 + 1) dimensional expanding universe. In order to proceed further, and to obtain the details of the evolution, we make further assumptions about TAB . We then analyse the evolution equations in D dimensions in general, and then specialise to the eleven dimensional case of interest here. We are unable to solve explicitly the relevant equations. However, applying the general analysis mentioned above, we describe the qualitative features of the evolution of the N = 4 brane configuration. In the asymptotic limit, three spatial directions expand as in the standard FRW universe and the remaining seven spatial directions reach constant, stabilised values. These values depend on the initial conditions and can be obtained numerically. Also, we find that any stabilised values may be obtained, but requires a fine tuning of the initial brane densities. Using the analysis given here, we present a physical description of the mechanism of stabilisation of the seven brane directions. The stabilisation is due, in essence, to the relations among the components of TAB which follow from U duality symmetries, and to each of the brane directions in the N = 4 configuration being wrapped by, and being transverse to, just the right number and kind of branes. This mechanism is very different from the ones proposed in string theory or in brane gas models – to obtain large 3 + 1 dimensional spacetime. (See section I A below also.) In the asymptotic limit, the eleven dimensional universe being studied here can also be considered from the perspective of four dimensional spacetime. One then obtains an effective four dimensional Newton’s constant G4 which is now time varying. Its precise time dependence will follow from explicit solutions of the eleven dimensional evolution equations. We find that, in the case of N = 4 brane configuration, G4 has a characteristic asymptotic time dependence : the fractional deviation δG4 of G4 from its asymptotic value exhibits log periodic oscillations given by $$\label{fG4} \delta G\_4 \; \propto \; \frac{1}{t^\alpha} \; \; Sin (\omega \; ln \; t + \phi ) \; \;.$$ The proportionality constant and the phase angle ϕ depend on initial conditions and matching details of the asymptotics, but the exponents α and ω depend only on the configuration parameters. Such log periodic oscillations seem to be ubiquitous and occur in a variety of physical systems. But, to our knowledge, this is the first time it appears in a cosmological context. One expects such a behaviour to leave some novel imprint in the late time universe, but its implications are not clear to us. Since we are unable to solve the evolution equations explicitly, we analyse them using numerical methods. We present the results of the numerical analysis of the evolution. We illustrate the typical evolution of the scale factors showing stabilisation and the log periodic oscillations mentioned above. By way of illustration, we choose a few sets of initial values and present the resulting values for the sizes of the stabilised directions and ratios of the string/M theory scales to the effective four dimensional scale. We also discuss critically the implications of our assumptions. As we will explain, many important dynamical questions must be answered before one understands how our known 3 + 1 dimensional universe may emerge from M theory. Until these questions are answered and our assumptions justified, our assumptions are to be regarded conservatively as amounting to a choice of initial conditions which are fine tuned and may not arise naturally. The organisation of this paper is as follows. In section II, we describe the U duality symmetries of M theory and their consequences, and present our ansatzes for the energy momentum tenosr TAB and for the equations of state. In section III, we present a general analysis of D dimensional evolution equations. In section IV, we specialise to the eleven dimensional case of N = 4 intersecting brane configurations and describe the various results mentioned above. In section V, we discuss the stabilised values of the brane directions, their ranges, and the necessity of fine tuning. In section VI, we present the four dimensional perspective and the time variations of G4 . In section VII, we present the results of numerical analysis. In section VIII, we conclude by presenting a brief summary, a few comments on the assumptions made, and by mentioning a few issues which may be studied further. In Appendix A, we highlight the points related to U duality symmetries in the black hole case. In Appendices B – D, we present certain results required in the text of the paper. A. Intersecting brane vs Brane gas models In this subsection, we note that the branes and antibranes in the mutually BPS intersecting brane configurations considered here and in – are different from those in the string/brane gas models – in many important aspects. These differences are explained in detail in section 2.6 of and section 6 of. Briefly, the differences are the following. (1) In brane gas models, the branes can intersect each other arbitrarily. In the brane configurations here, the intersections must follow specific rules. Consequently, U duality symmetries of M theory imply certain relations among the components of the energy momentum tensor TAB which turn out to be crucial elements in our case. (2) The branes in brane gas models support excitations on their surfaces and, at high energies, have $S \sim {\cal E} \;$ where S is the entropy and ${\cal E}$ the energy. Here, the intersecting branes form bound states, become fractional, support very low energy excitations and, hence, are highly entropic. At high energies, $S \sim {\cal E}^{\frac{N} {2}} \;$ which, for N > 2 , vastly exceeds the entropy in brane gas models. Such intersecting brane configurations are, therefore, the entropically favourable ones. (3) In brane gas models, the branes are assumed to annihilate if they intersect each other. Here, the intersections are necessary for formation of bound states, and thereby of fractional branes leading to high entropic excitations. Also, the intersecting configurations here consist of branes, antibranes, and large number of low energy excitations living on them. All these constituents are at non zero temperature and in dynamical equilibrium with each other –. The excitations are long–lived and non interacting to the leading order, hence the branes and antibranes here are metastable and do not immediately annihilate. Note that, in string/M theory description, the Hawking evaporation of black holes is due to the annihilation of branes and antibranes. Also, large black holes consist of stacks of large numbers of branes and antibranes, and have a long life time. This implies that such stacks of branes and antibranes do not immediately annihilate and have a long life time. Hence, we assume that the mutually BPS intersecting brane configurations describing the early universe also consists of stacks of large numbers of branes and antibranes having a long life time and, in particular, that the brane antibrane annihilation effects are negligible during the evolution of the universe atleast until the brane directions are stabilised resulting in an effective 3 + 1 dimensional universe. II. U duality symmetries and Equations of state In this paper, we assume that the early universe in M theory is homogeneous and anisotropic and that it is dominated by N = 4 mutually BPS intersecting brane configurations. To be specific, we consider 22ʹ55ʹ configurations. [5](#fn5) We study the consequent evolution of such an universe dictated by a (10 + 1) dimensional effective action given, in the standard notation, by $$\label{s11} S\_{11} = \frac{1}{16 \pi G\_{11}} \; \int d^{11} x \; \sqrt{-g} \; R + S\_{br}$$ where Sbr is the action for the fields corresponding to the branes. The corresponding equations of motion are given, in the standard notation and in units where 8πG11 = 1 , by [6](#fn6) $$\label{r11} R\_{A B} - \frac{1}{2} \; g\_{A B} R = T\_{A B} \; \; \;, \; \; \; \; \sum\_A \nabla\_A T^A\_{\; \; \; B} = 0$$ where A = (0, i) with i = 1, 2, ⋯, 10 and TAB is the energy momentum tensor corresponding to the action Sbr . For black hole case, TAB is obtained from the action for higher form gauge fields. With a suitable ansatz for the metric, equations of motion ([r11]) are solved to obtain black hole solutions. For cosmological case, TAB is often determined using equations of state of the dominant constituent of the universe. Such equations of state may be obtained if the underlying physics is known; or, one may assume a general ansatz for them and proceed. [7](#fn7) TAB for intersecting branes in the early universe has been calculated in assuming that the branes and antibranes in the intersecting brane configurations are non interacting and that their numbers are all equal, i.e. nI = n̄I for I = 1, 2, ⋯, N and n1 = ⋯ = nN . However, general relations among the components of TAB may be obtained using U duality symmetries of M theory, involving chains of dimensional reduction and uplifting and T and S dualities of string theory, using which 2 branes and 5 branes or 22ʹ55ʹ and 55ʹ5ʺW configurations can be interchanged. Such relations are valid more generally, for example even when nI and n̄I are all different. These general relations on the equations of state are sufficient to show, under a technical assumption, that the N = 4 mutually BPS intersecting brane configurations with identical numbers of branes and antibranes, i.e. with n1 = ⋯ = n4 and n̄1 = ⋯ = n̄4 , will asymptotically lead to an effective (3 + 1) dimensional expanding universe. To obtain the details of the evolution, however, we need further assumptions and an ansatz of the type p = wρ . We now present the details. Let the line element ds be given by ds2 = − dt2 + ∑ie2λi(dxi)2 where eλi are scale factors and, due to homogeneity, λi are functions of the physical time t only. (Parametrising the scale factors as eλi turns out to be convenient for our purposes.) It then follows that TAB depends on t only and that it is of the form TBA = diag( − ρ, pi) . We assume that ρ > 0 . From equations ([r11]) one now obtains $$\begin{aligned} \Lambda\_t^2 - \sum\_i (\lambda^i\_t)^2 & = & 2 \; \rho \label{t21} \\ \lambda^i\_{t t} + \Lambda\_t \lambda^i\_t & = & p\_i + \frac{1}{9} \; (\rho - \sum\_j p\_j) \label{t22} \\ \rho\_t + \rho \Lambda\_t + \sum\_i p\_i \lambda^i\_t & = & 0 \label{t23}\end{aligned}$$ where Λ = ∑iλi and the subscripts t denote time derivatives. Note, from equation ([t21]), that Λt cannot vanish. Hence, if Λt > 0 at an initial time t0 then it follows that eΛ increases monotonically for t > t0 . We assume the evolution to be such that eΛ → ∞ eventually. In the context of early universe in M theory, it is natural to assume that all spatial directions are on equal footing to begin with. Therefore we assume that the ten dimensional space is toroidal. Further, we assume that the early universe is homogeneous and is dominated by the 22ʹ55ʹ configuration where, with no loss of generality, we take two stacks each of 2 branes and 5 branes to be along (x1, x2) , (x3, x4) , (x1, x3, x5, x6, x7) , and (x2, x4, x5, x6, x7) directions respectively, and take these intersecting branes to be distributed uniformly along the common transverse space directions (x8, x9, x10) . Note that the total brane charges must vanish, i.e. nI = n̄I for all I , since the common transverse space is compact. We denote this 22ʹ55ʹ configuration as (12, 34, 13567, 24567) . The meaning of this notation is clear and, below, such a notation will be used to denote other configurations also. A. U duality relations We now describe the relations which follow from U duality symmetries, involving chains of dimensional reduction and uplifting and T and S dualities of string theory. See for more details. Let ↓ k and ↑ k denote dimensional reduction and uplifting along kth direction between M theory and type IIA string theory, Ti denote T duality along ith direction in type IIA and IIB string theories, and S denote S duality in type IIB string theory. Then U dualities of the type ↑ jTiSTi ↓ j interchange i and j directions, and U dualities of the type ↑ kTiTj ↓ k transform one mutually BPS N intersecting brane configuration to another. For example, the U duality ↑ 5T3T4 ↓ 5 transforms a 2 brane configuration (12) to the 5 brane configuration (12345) . Similarly, the U duality ↑ 5T1T2 ↓ 5 transforms the 22ʹ55ʹ configuration (12, 34, 13567, 24567) to the W55ʹ5ʺ configuration (5, 12345, 23567, 14567) ; whereas ↑ 6T4T5 ↓ 6 transforms it to the $\tilde{5'} 2' 5 \tilde{2} \;$ configuration (12456, 35, 13467, 27) . The U dualities transform the corresponding gravitational fields also. In the case of metric of the form given in equation ([ds]), the U duality ↑ kTiTj ↓ k transforms the λis in the scale factors to λʹis given by $$\begin{aligned} & & \lambda'^i = \lambda^j - 2 \lambda \; \;, \; \; \; \lambda'^j = \lambda^i - 2 \lambda \; \;, \; \; \; \lambda'^k = \lambda^k - 2 \lambda \nonumber \\ & & \lambda'^l = \lambda^l + \lambda \; \;, \; \; \; l \ne i, j, k \; \;, \; \; \; \lambda = \frac{\lambda^i + \lambda^j + \lambda^k}{3} \; \;. \label{uduality}\end{aligned}$$ Consider the 2 brane configuration (12) with scale factors eλi and the 5 brane configuration (12345) with scale factors eλʹi . By inspection, or by using U dualities ↑ jTiSTi ↓ j for appropriate (i, j), these scale factors may be expected to obey the ‘obvious’ symmetries: $$\begin{aligned} 2 & : & \lambda^1 = \lambda^2 \; \;, \; \; \; \lambda^3 = \cdots = \lambda^{10} \label{obv2} \\ 5 & : & \lambda'^1 = \cdots = \lambda'^5 \; \;, \; \; \; \lambda'^6 = \cdots = \lambda'^{10} \label{obv5} \; \;. \end{aligned}$$ Now, these two configurations are related by the U duality ↑ 5T3T4 ↓ 5 . Hence the corresponding λis and λʹis must obey the relations of the type given in ([uduality]). Combined with the obvious symmetry relations above, it is straightforward to show that λ∥ + 2λ⊥ = 2λʹ∥ + λʹ⊥ = 0 where the superscripts ∥ and ⊥ denote spatial directions along and transverse to the branes respectively. Similarly, the obvious symmetry relations for the 22ʹ55ʹ configuration (12, 34, 13567, 24567) are 22ʹ55ʹ : λ5 = λ6 = λ7 , λ8 = λ9 = λ10 . Proceeding as in the case of 2 and 5 branes above, and using the U duality ↑ 5T1T2 ↓ 5 which relates the 22ʹ55ʹ and W55ʹ5ʺ configurations, one obtains two more relations given by λ1 + λ4 + λ5 = λ2 + λ3 + λ5 = 0 . In general, for an N intersecting brane configuration, the U duality symmetries will lead to 10 − N relations among the λis, leaving only N of them independent. These relations are of the form ∑iciλi = 0 where ci are constants. Clearly, such a relation can be violated by constant scaling of xi coordinates. Hence, we interpret it as implying a relation among the components of TAB . In view of equation ([t22]), we interpret a U duality relation ∑iciλi = 0 as implying that $$\label{weak} \sum\_i c\_i f^i = 0 \; \; \;, \; \; \; \; f^i = p\_i + \frac{1}{9} \; (\rho - \sum\_j p\_j) \; \;.$$ Substituting equation ([weak]) in equation ([t22]), it follows upon an integration that ∑iciλti = c e− Λ where c is an integration constant. If ∑iciλti = 0 initially at t = t0 then c = 0 . In such cases then ∑iciλti = 0 for all t and, hence, ∑iciλi = v where v is another integration constant. In general ∑iciλti ≠ 0 initially at t = t0 and, hence, c ≠ 0 . Let the evolution be such that eΛ ∼ tβ → ∞ in the limit t → ∞ . Then it follows from equation ([K]) that ∑iciλti → 0 in this limit. If, furthermore, β > 1 then equation ([K]) also gives $$\label{L} \sum\_i c\_i \lambda^i = v + {\cal O}(t^{1 - \beta}) \to v$$ where v is an integration constant. If β ≤ 1 then ∑iciλi is a function of t . We will see later that, in the solutions we obtain with further assumptions, β turns out to be > 1 for N > 1 . Note that, as can be seen from the above steps, the U duality relations follow as long as the directions involved in the U duality operations are isometry directions. Since none of the common transverse directions are involved in obtaining the relations above, it follows that they are valid even if the common transverse directions are not compact. Thus the U duality relations are applicable in such cases also. Similarly, the time dependence of λis played no role in obtaining the U duality relations here. Hence, these relations may be expected to arise for the black hole case also. They indeed arise as we point out in Appendix A. B. A general result We now consider a general result for the 22ʹ55ʹ configuration that follows from the U duality relations alone. The λis for this configuration obey the relations given in equations ([obv2255]) and ([2255reln]). Note that a suitable U duality, for example ↑ 6T4T5 ↓ 6 , can transform 2 branes and 5 branes into each other. Hence, we will refer to two types of branes as being identical if they have identical numbers of branes and antibranes, i.e. Ith type is identical to Jth type if nI = nJ and n̄I = n̄J . Consider the case when 2 and 2ʹ branes in the 22ʹ55ʹ configurations are identical. This will enhance the obvious symmetry relations. It is easy to see that we now have one more independent relation λ1 = λ3 . If, instead, 5 and 5ʹ branes are identical, then the extra independent relation is λ1 = λ2 . Similarly, if 2 and 5ʹ branes are identical then, after a few steps involving U duality ↑ 6T4T5 ↓ 6 which interchanges 2 and 5ʹ branes, it follows that the extra independent relation is λ2 = λ5 . Now if all the four types of branes in the 22ʹ55ʹ configuration are identical, i.e. if n1 = ⋯ = n4 and n̄1 = ⋯ = n̄4 , then, we have three extra independent relations λ1 = λ2 = λ3 = λ5 . Combined with equations ([obv2255]) and ([2255reln]), we get λ1 = ⋯ = λ7 = 0 which is to be interpreted as f1 = ⋯ = f7 = 0 , see equation ([weak]). Hence, as described earlier, it follows for i = 1, ⋯, 7 that if λti = 0 initially at t = t0 then λti = 0 and λi = vi for all t where vi are constants. Or, if eΛ ∼ tβ → ∞ in the limit t → ∞ with β > 1 , it then follows for i = 1, ⋯, 7 that λti → 0 and λi → vi in this limit. Obtaining the values of the asymptotic constants vi , however, requires knowing the details of evolution. It also follows similarly that $e^{\lambda^i} \sim e^{ \frac{\Lambda} {3}} \to \infty \;$ for i = 8, 9, 10 . It is straightforward to show that same results are obtained for the equivalent 55ʹ5ʺW configuration also. Thus, assuming either that λt1 = ⋯ = λt7 = 0 initially at t = t0 or that eΛ ∼ tβ → ∞ in the limit t → ∞ with β > 1 , we obtain that the N = 4 mutually BPS intersecting brane configurations with identical numbers of branes and antibranes, i.e. with n1 = ⋯ = n4 and n̄1 = ⋯ = n̄4 , will asymptotically lead to an effective (3 + 1) dimensional expanding universe with the remaining seven spatial directions reaching constant sizes. This result follows as a consequence of U duality symmetries alone, which imply relations of the type given in equation ([weak]) among the components of the energy momentum tensor TAB . This result is otherwise independent of the details of the equations of state, and also of the ansatzes for TAB we make in the following in order to proceed further. *C. Ansatz for TAB*** The dynamics underlying the general result given above may be understood in more detail, and the asymptotic constants vi can be obtained, if an explicit solution for the evolution is available. In the following, we will make a few assumptions which enable us to obtain such details. Consider now the case of 2 branes or 5 branes only. From the U duality relations given by equations ([obv2]), ([obv5]), ([25reln]), and ([weak]), it follows easily that p∥ = − ρ + 2p⊥ where p∥ is the pressure along the brane directions and p⊥ is the pressure along the transverse directions. For the case of waves, one obtains p∥ = ρ . We write these U duality relations for the N = 1 configurations in the form p∥ = z (ρ − p⊥) + p⊥ where z = − 1 for 2 and 5 branes and = + 1 for waves. (A similar relation may be obtained in the black hole case also, see Appendix A.) In general, ρ , p∥ , and p⊥ are functions of the numbers n and n̄ of branes and antibranes, satisfying the U duality relations ([z]). If n = n̄ then p∥ and p⊥ may be thought of as functions of ρ satisfying equation ([z]). Consider now mutually BPS N intersecting brane configuration. In the black hole case, it turns out that the energy momentum tensors TB(I)A of the Ith type of branes are mutually non interacting and seperately conserved –. That is, TBA = ∑ITB(I)A , ∑A∇ATB(I)A = 0 . We assume that this is the case in the context of early universe also where TBA = diag( − ρ, pi) , TB(I)A = diag( − ρI, piI) , ρI > 0 , and (ρI, piI) satisfy the U duality relations in ([z]) for all I . Equations ([tabI]) now give $$\begin{aligned} & & \rho = \sum\_I \rho\_I \; \;, \; \; \; \; p\_i = \sum\_I p\_{i I} \label{add} \\ & & (\rho\_I)\_t + \rho\_I \Lambda\_t + \sum\_i p\_{i I} \lambda^i\_t = 0 \; \;. \label{t23I}\end{aligned}$$ We have verified explicitly for a variety of mutually BPS N intersecting brane configurations that equations ([z]) and ([add]) are sufficient to satisfy all the relations of the type ∑icifi = 0 implied by U duality symmetries. See for more details. To solve the evolution equations ([t21]), ([t22]), ([add]), and ([t23I]), we need the functions ρI , p∥ I , and p⊥ I . To proceed further, we assume that nI = n̄I for all I . This is necessary if, as is the case here, the common transverse directions are compact and hence the nett charges must vanish. Then p∥ I and p⊥ I may be thought of as functions of ρI satisfying equation ([z]). It is natural to expect that p⊥ I(ρI) is the same function for waves, 2 branes, and 5 branes since they can all be transformed into each other by U duality operations which do not involve the transverse directions. We assume that this is the case. We further assume that this function p⊥(ρ) is given by p⊥ = (1 − u) ρ where u is a constant. Such a parametrisation of the equation of state, instead of the usual one p = wρ , leads to elegant expressions as will become clear in the following, see also. The results of correspond to the case where u = 1 . Here, we assume only that 0 < u < 2 . The constant u is arbitrary otherwise. It now follows that piI in equation ([add]) are of the form piI = (1 − uiI) ρI and that the constants uiI can be obtained in terms of u using equations ([z]) and ([prho]). Thus, for 2 branes, 5 branes, and waves, we have u⊥ = u , u∥ = (1 − z) u , and hence $$\begin{aligned} 2 & : & u\_i = (\; \; 2, \; \; 2, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1) \; u \nonumber \\ 5 & : & u\_i = (\; \; 2, \; \; 2, \; \; 2, \; \; 2, \; \; 2, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1) \; u \nonumber \\ W & : & u\_i = (\; \; 0, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1) \; u \label{25W}\end{aligned}$$ where the I superscripts have been omitted since N = 1 . Similarly, uiI for the 22ʹ55ʹ configuration are given by $$\begin{aligned} 2 & : & u\_i^1 = (\; \; 2, \; \; 2, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1) \; u \nonumber \\ 2' & : & u\_i^2 = (\; \; 1, \; \; 1, \; \; 2, \; \; 2, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1, \; \; 1) \; u \nonumber \\ 5 & : & u\_i^3 = (\; \; 2, \; \; 1, \; \; 2, \; \; 1, \; \; 2, \; \; 2, \; \; 2, \; \; 1, \; \; 1, \; \; 1) \; u \nonumber \\ 5' & : & u\_i^4 = (\; \; 1, \; \; 2, \; \; 1, \; \; 2, \; \; 2, \; \; 2, \; \; 2, \; \; 1, \; \; 1, \; \; 1) \; u \; \;. \label{I} \end{aligned}$$ This completes our ansatz for the energy momentum tensor TAB for the intersecting brane configurations in the early universe. III. General Analysis: Evolution equations The evolution of the universe can now be analysed. In this section, we first present the analysis in a general form which is applicable to a D dimensional homogeneous, anisotropic universe. We specialise to the intersecting brane configurations in the next section. The D dimensional line element ds is given by equation ([ds]), now with i = 1, 2, ⋯, D − 1 . The total energy momentum tensor TAB of the dominant consituents of the universe is given by equation ([tabdiag]). The equations of motion for the evolution of the universe is given, in units where 8πGD = 1 , by equations ([t21]) – ([t23]) with 9 in equation ([t22]) now replaced by D − 2 . Defining $$\label{Gij} G\_{i j} = 1 - \delta\_{i j} \; \;, \; \; \; G^{i j} = \frac{1}{D - 2} - \delta\_{i j} \; \;,$$ the equations ([t21]) and ([t22]), with 9 replaced by D − 2 , may be written compactly as $$\begin{aligned} \sum\_{i, j} G\_{i j} \lambda^i\_t \; \lambda^j\_t & = & 2 \; \rho \label{a1} \\ \lambda^i\_{t t} + \Lambda\_t \lambda^i\_t & = & \sum\_j G^{i j} \; (\rho - p\_j) \label{a2}\end{aligned}$$ where i, j, ⋯ run from 1 to D − 1 . Let the universe be dominated by N types of mutually non interacting and seperately conserved matter labelled by I = 1, ⋯, N . Then the corresponding energy momentum tensors TAB(I) and their components ρI and piI satisfy equations ([tabI]) – ([t23I]). Further, let the equations of state be given by piI = (1 − uiI) ρI where uiI are constants. Equations ([t23I]), ([a1]), and ([a2]) may now be simplified and cast in various useful forms as follow. Using piI = (1 − uiI) ρI , equation ([t23I]) can be integrated to give ρI = elI − 2Λ , lI = ∑iuiIλi + l0I where l0I are integration constants. Further using equations ([add]) and ([lI]), equations ([a1]) and ([a2]) become $$\begin{aligned} \sum\_{i, j} G\_{i j} \lambda^i\_t \; \lambda^j\_t & = & 2 \; \sum\_J e^{l^J - 2 \Lambda} \label{b1} \\ \lambda^i\_{t t} + \Lambda\_t \lambda^i\_t & = & \sum\_J u^{i J} \; e^{l^J - 2 \Lambda} \label{b2}\end{aligned}$$ where uiJ = ∑jGij ujJ . Let the initial conditions at an initial time t0 be given, with no loss of generality, by (λi, λti, lI, ltI, ρI)t = t0 = (0, ki, l0I, KI, ρI0) where ρI0 = el0I , KI = ∑iuiIki , ∑i, jGijkikj = 2 ∑Jel0J . Equations ([b1]) and ([b2]) may now be solved for the D − 1 variables λi with the above initial conditions. Or, instead, these equations may be manipulated so that one needs to solve for N variables lI only, see equations ([tau]), ([c3]), ([c5]), and ([c6]) below. We now perform these manipulations. First define a variable τ(t) as follows: dτ = e− Λ dt , τ(t0) = 0 . Then, for λi(t) or equivalently λi(τ(t)) , we have λti = e− Λ λτi , λtti + Λtλti = e− 2Λ λττi where the subscripts τ denote τ–derivatives. Note that the initial values at τ(t0) = 0 remain unchanged since Λ = 0 , and hence λti = λτi at t = t0 . Equations ([b1]) and ([b2]) now become $$\begin{aligned} \sum\_{i, j} G\_{i j} \lambda^i\_\tau \; \lambda^j\_\tau & = & 2 \; \sum\_J e^{l^J} \label{c1} \\ \lambda^i\_{\tau \tau} & = & \sum\_J u^{i J} \; e^{l^J} \; \;. \label{c2}\end{aligned}$$ Also, from lI = ∑iuiIλi + l0I , it follows that $$\label{c3} l^I\_{\tau \tau} = \sum\_J {\cal G}^{I J} \; e^{l^J}$$ where $$\label{cG^IJ} {\cal G}^{I J} = \sum\_i u^I\_i \; u^{i J} = \sum\_{i, j} G^{i j} \; u^I\_i \; u^J\_j \; \;.$$ We assume that ${\cal G}\_{I J} \; $ exists such that $\sum\_J {\cal G}\_{I J} \; {\cal G}^{J K} = \delta\_I^{\; \; K} \; $, i.e. that the matrix ${\cal G}$ formed by ${\cal G}^{I J} \; $ is invertible. [8](#fn8) Then, from equation ([c3]), we have $$\label{c4} \sum\_J {\cal G}\_{I J} \; l^J\_{\tau \tau} = e^{l^I} \; \;.$$ Substituting this expression for elI into equation ([c2]), then integrating it twice and incorporating the initial conditions given in equation ([ic]), we get $$\label{c5} \lambda^i = \sum\_I u^i\_I \; (l^I - l^I\_0) + \; L^i \; \tau \; \; \;, \; \;\; \; u^i\_I = \sum\_{j, J} {\cal G}\_{I J} \; G^{i j} \; u^J\_j$$ where Li are integration constants. It follows from λτi(0) that Li are related to initial values ki and KI by ki = ∑IuIi KI + Li . Using this expression for ki in the relation KI = ∑iuiIki , or substituting the expression for λi given in equation ([c5]) into the equation ([lI]) for lI , leads to the following N constraints on Li : ∑iuiILi = 0 , I = 1, 2, ⋯, N . Now, using equations ([c5]) and ([uiLi]), equation ([c1]) may be written in terms of lI as follows: $$\label{c6} \sum\_{I, J} {\cal G}\_{I J} \; l^I\_\tau \; l^J\_\tau = 2 \; ( E + \sum\_I e^{l^I} ) \; \;, \; \; \; 2 E = - \; \sum\_{i, j} G\_{i j} L^i L^j \; \;.$$ One may now solve equations ([c3]) and ([c6]) for N variables lI(τ) . Then λi(τ) are obtained from equation ([c5]) and t(τ) from equation ([tau]). Inverting t(τ) then gives τ(t) , and thereby λi(t) . *A. N* = 1 case** Consider the N = 1 case. Note that we are still considering the general D dimensional universe, not the eleven dimensional one. We assume here that ${\cal G}^{1 1} = {\cal G} > 0 \;$. Now, as shown in Appendix B, it follows in general that if ∑iuiLi = 0 and ∑i, jGijuiuj > 0 then E ≥ 0 and E vanishes if and only if Li all vanish. Since ∑iui1Li = 0 , see equation ([uiLi]), and we assume that ${\cal G}^{1 1} = \sum\_{i, j} G^{i j} u^1\_i u^1\_j > 0 \;$, we have E ≥ 0 . We further assume that E > 0 , equivalently that Lis do not all vanish. Omitting the I labels, equations ([c3]) and ([c6]) for l(τ) become $$\label{1c1} l\_{\tau \tau} = {\cal G} \; e^l \; \; \;, \; \; \; (l\_\tau)^2 = 2 \; {\cal G} \; (E + e^l) \; \;.$$ The initial values are l0 = l(0) and K = lτ(0) obeying $K^2 = 2 \; {\cal G} \; (E + e^{l\_0}) \;$. We take K > 0 with no loss of generality. Then the solution for l(τ) is given by $$\label{1c2} e^l = \frac{E}{Sinh^2 \; \alpha (\tau\_\infty - \tau)}$$ where $$\label{1alpha} 2 \alpha^2 = {\cal G} E \; \;, \; \; \; Sinh^2 \; \alpha \tau\_\infty = E \; e^{- l\_0} \; \;, \; \; \; K = 2 \alpha \; Coth \; \alpha \tau\_\infty \; \;.$$ The sign of α is immaterial but, just to be definite, we take it to be positive. The sign of τ∞ is same as that of K , hence τ∞ > 0 . Also, λi(τ) and t(τ) may now be obtained but are not needed here for our purposes. Note that el → 4E e2α(τ − τ∞) and vanishes in the limit τ → − ∞ , whereas $e^l \to \frac{2} {{\cal G}} \; (\tau\_\infty - \tau)^{- 2} \;$ and diverges in the limit τ → τ∞ . The value of τ∞ depends on the initial values l0 and K, or equivalently E , as given in equations ([1alpha]). It is finite and can be evaluated exactly. However, if el0 ≪ E then τ∞ may be approximated in a way that will be useful later on. From the exact solution given above, we have Sinh2 ατ∞ = E e− l0 and K = 2α Coth ατ∞ . In the limit el0 ≪ E , we then have e2ατ∞ ≃ 4 E e− l0 and K ≃ 2α . It, therefore, follows that $$\label{tauexact} \tau\_\infty \simeq \; \frac{1}{K} \; (ln \; E - l\_0 + ln \; 4) \; \;.$$ In the limit el0 ≪ E , the evolution of l(τ) may also be thought of as follows. Consider E to be fixed and el0 to be very small so that el0 ≪ E . It then follows from equations ([1c1]) that, at initial times, lττ is very small and that $l\_\tau \simeq \sqrt{2 {\cal G} E} = 2 \alpha \;$ is independent of el . Hence, l(τ) evolves as if there is no ‘force’, i.e. l(τ) ≃ l0 + Kτ where K = lτ(0) > 0 is the initial ‘velocity’. Once el becomes of ${\cal O} (E) \;$ then it affects lτ . But, from then on, el evolves quickly and diverges soon after. This suggests that one may well approximate τ∞ by the time τa required for l , which starts from l0 with a velocity K and evolves freely with no force, to reach ln E – namely, to reach a value where el = el0 + Kτa = E . In other words, if el0 ≪ E then $$\label{tauapprox} \tau\_\infty \simeq \tau\_a = \; \frac{1}{K} \; (ln \; E - l\_0) \; \;.$$ A comparison with equation ([tauexact]) shows that the exact τ∞ which follows from solving the evolution equations is indeed well approximated by τa in equation ([tauapprox]) in the limit el0 ≪ E . Note that τa is calculated using only the initial values, requiring no knowledge of the exact solution. *B. N* > 1 case** When N > 1 , the equations of motion can be solved if ${\cal G}^{I J} \;$ are of certain form –. For example, if ${\cal G}^{I J} \propto \delta^{I J} \;$ then the solutions are similar to those in the N = 1 case described above. For general forms of ${\cal G}^{I J} \;$, we are unable to obtain explicit solutions. Nevertheless, the general evolution can still be analysed if one assumes suitable asymptotic forms for the scale factors eλi . It follows from equations ([Gij]) and ([a1]) that Λt cannot vanish. With no loss of generality, let Λt > 0 initially at t = t0 . Then eΛ decreases monotonically for t < t0 , equivalently τ < 0 , and increases monotonically for t > t0 , equivalently τ > 0 . Further features of the evolution depend on the structure of ${\cal G}^{I J} \;$ and uiI . In the cases of interest here, it turns out that eΛ and also all elI vanish in the limit τ → − ∞ , and diverge in the limit τ → τ∞ where τ∞ is finite. We assume such a behaviour and analyse the asymptotic solutions. *1. Asymptotic evolution: eΛ → 0* We assume that (eΛ, elI) → 0 in the limit τ → − ∞ . Then, equations ([c2]) and ([c3]) can be solved since their right hand sides depend only on elIs which now vanish. Hence, in the limit τ → − ∞ , we write elI = ec̃Iτ = tb̃I , eλi = ec̃iτ = tb̃i which are valid upto multiplcative constants and where (c̃I, c̃i, b̃I, b̃i) are constants. Also, the equalities in the asymptotic expressions here and in the following are valid only upto the leading order. Equation ([c5]) now implies that c̃i = ∑IuIic̃I + Li . Also, eΛ = ec̃τ where c̃ = ∑ic̃i . Then it follows from equation ([tau]) that t ∼ ec̃τ . Hence, $$\label{bc} \tilde{b}^I = \frac {\tilde{c}^I} {\tilde{c}} \; \; \;, \; \; \; \; \tilde{b}^i = \frac {\tilde{c}^i} {\tilde{c}} \; \; \;, \; \; \; \; \sum\_i \tilde{b}^i = 1 \; \;.$$ Furthermore, equation ([c1]) implies that (∑ib̃i)2 − ∑i(b̃i)2 = 0 . Thus the evolution is of Kasner type in the limit τ → − ∞ . The constants c̃Is in equations ([0l]) must be such that the resulting ∑ib̃i = ∑i(b̃i)2 = 1 , but are otherwise arbitrary. In an actual evolution, however, c̃Is can be determined in terms of the initial values l0I and KI with no arbitrariness, but this requires complete solution for lI(τ) . *2. Asymptotic evolution: eΛ → ∞* We assume that eΛ → ∞ in the limit τ → τ∞ where τ∞ is finite. Whether this limit is reached at a finite or infinite physical time t depends on the values of uiI , see below. Λ(τ) may be obtained in terms of lI(τ) using equation ([c5]). Hence, in the limit eΛ → ∞ , some or all of the elIs diverge. Consider the following ansatz in the limit τ → τ∞ : elI = ecI (τ∞ − τ)− 2γI , eλi = eci (τ∞ − τ)− 2γi , where cI and γI are constants, and some or all of the γIs must be > 0 so that some or all of the elIs diverge. Equation ([c5]) now implies that γi = ∑IuIi γI , ci = ∑IuIi (cI − l0I) + Li τ∞ . Also, eΛ = ec (τ∞ − τ)− 2γ where c = ∑ici and γ = ∑iγi . For the ansatz in equations ([asymptau]) to be consistent, it is necessary that γ > 0 so that eΛ → ∞ in the limit τ → τ∞ . Now t(τ) follows from equation ([tau]) and is given by $$\label{ttauasymp} t - t\_s = \frac{1}{2 \gamma - 1} \; e^c \; (\tau\_\infty - \tau)^{- (2 \gamma - 1)} \; \; \;, \; \; \; \; \gamma = \sum\_{i, I} u^i\_I \; \gamma^I$$ where ts is a finite constant. If 2γ < 1 then t → ts which means that eΛ → ∞ at a finite physical time ts . If 2γ > 1 then t → ∞ in the limit eΛ → ∞. Which case is realised, i.e. whether 2γ < 1 or > 1 , depends on the values of uiI . Using equation ([ttauasymp]), the asymptotic behaviour of elI and eλi can be obtained in terms of t. For example, let 2γ > 1 and elI = ebI + 2b tβI , eλi = ebi tβi , eΛ = eb tβ in the limit t → ∞ . It then follows that $$\label{betagamma} \beta^I = \; \frac{2 \; \gamma^I}{2 \gamma - 1} \; \;, \; \; \; \beta^i = \; \frac{2 \; \gamma^i}{2 \gamma - 1} \; \;, \; \; \; \beta = \; \frac{2 \; \gamma}{2 \gamma - 1} \; \;.$$ Note that, in this case, we have eΛ ∼ tβ in the limit t → ∞ with β > 1 . See the discussion below equation ([weak]) for the relevance of this feature. To obtain the values of γI , and thereby γi , in equation ([asymptau]), consider equation ([c4]) from which it follows that $$\label{asymp2tau} 2 \; \sum\_J {\cal G}\_{I J} \; \gamma^J = \; e^{c^I} \; (\tau\_\infty - \tau)^{2 (1 - \gamma^I) } \; \;.$$ The left hand side in the above equation is a constant but the right hand side depends on τ . This is consistent if γI = 1 in which case the right hand side becomes a positive constant, or if γI < 1 in which case the right hand side vanishes in the limit τ → τ∞ . Thus, there are two possibilities: $$\begin{aligned} {\bf (i)} \; \; \; \gamma^I = 1 & \Longrightarrow & 2 \; \sum\_{J} {\cal G}\_{I J} \; \gamma^J \; = \; e^{c^I} > 0 \label{eqtau} \\ {\bf (ii)} \; \; \; \gamma^I \ne 1 & \Longrightarrow & \sum\_{J} {\cal G}\_{I J} \; \gamma^J = 0 \; \; \;, \; \; \; \gamma^I < 1 \; \;. \label{neqtau}\end{aligned}$$ For a given ${\cal G}\_{I J} \;$, the possible consistent solutions for (γI, ecI) are to be obtained as follows. Assume that some I’s are of type (i) and the remaining ones are of type (ii). Then solve equations ([eqtau]) and ([neqtau]) for ecI in type (i) and for γI in type (ii). Such a solution is consistent if the resulting (ecI, γI) satisfy ecI > 0 for Is in type (i) and γI < 1 for Is in type (ii). Also, some or all of the γIs must be > 0 as required in equation ([asymptau]). (It is further necessary that the resulting γ > 0 so that eΛ → ∞ , but calculating γ requires uiI .) Consider an example, which will be useful later, where ${\cal G}^{I J} \;$ and ${\cal G}\_{I J} \;$ are given by $$\label{ab} {\cal G}^{I J} = a \; (b - \delta^{I J}) \; \; \;, \; \; \; \; \; \; {\cal G}\_{I J} = \frac{1}{a} \; ( \frac{b}{N b - 1} - \delta\_{I J} )$$ with a > 0 and Nb > 1 . It is then easy to show that the only possibility is the one given in (i). Also $\sum\_J {\cal G}\_{I J} = \frac{1}{a (N b - 1)} > 0 \;$, and thus γI = 1 for all I is a consistent solution as required by equation ([eqtau]). In the N = 1 case, we get ${\cal G}^{1 1} = {\cal G} = a \; (b - 1) > 0 \;$, and el in the limit τ → τ∞ obtained as described above agrees with that obtained from the explicit solution, see below equation ([1alpha]). Thus ecI and γI, and thereby γi = ∑IuIi γI and γ = ∑i, IuIi γI , are all determined ultimately by uiI . The constants ci are given by equation ([cigammai]) and they depend on uiI , on the initial values l0I and Li , and also on τ∞ . But determining τ∞ , and hence determining ci when Li do not all vanish, requires complete solution for lI(τ) . *C. Deviations from elI* → ∞ Asymptotics** We consider the deviations of lI(τ) from its asymptotic behaviour given in equation ([asymptau]), which will turn out to be of interest. Let the deviations sI(τ) for I = 1, 2, ⋯, N be defined, in the limit τ → τ∞ , by elI = ecI (τ∞ − τ)− 2γI esI(τ) where cI and γI are determined as described earlier. For the purpose of illustration, and also for later use, we now assume that all the Is are of type (i), namely that γI = 1 and $e^{c^I} = 2 \; \sum\_{J} {\cal G}\_{I J} > 0 \;$ for all I . It then follows straightforwardly from equation ([c3]) that $$\label{si2} (\tau\_\infty - \tau)^2 \; s^I\_{\tau \tau} = 2 \; \sum\_{K, L} {\cal G}^{I K} \; {\cal G}\_{K L} \; (e^{s^K} - 1) \; \;.$$ Consider the example of ${\cal G}^{I J} \;$ given in equation ([ab]). Then $\sum\_J {\cal G}\_{I J} = \frac{1}{a (N b - 1)} \;$ and, for any σK , one has $$\label{sigma} \sum\_{K, L} {\cal G}^{I K} \; {\cal G}\_{K L} \; \sigma^K = - \; \frac{1}{N b - 1} \; (\sigma^I - b \; \sum\_K \sigma^K ) \; \;.$$ In equation ([si2]), σK = 2 (esK − 1) . It now follows easily that, up to the leading order in {sK} , the difference sI − sJ obeys the equation $$\label{sisj} (\tau\_\infty - \tau)^2 \; (s^I - s^J)\_{\tau \tau} + \frac{2}{N b - 1} \; (s^I - s^J) = 0 \; \;.$$ The solutions to these equations are of the form $$\label{sisjsoln} (s^I - s^J) \; \sim \; (\tau\_\infty - \tau)^{ \frac{1}{2} \; (1 \pm \sqrt{\Delta})} \; \; \;, \; \; \; \; \Delta = 1 - \frac{8}{N b - 1} \; \;.$$ Note that sI − sJ = lI − lJ since γI and cI are same for all I , see equation ([si]). Hence, equations ([sisj]) and ([sisjsoln]) can be used to understand in more detail the behaviour of lIs as they all diverge in the limit τ → τ∞ as given in equation ([asymptau]). We will discuss these features in sections VI and VII. IV. Intersecting Branes We now analyse the evolution of the universe dominated by mutually BPS N intersecting brane configurations of M theory. The number of spacetime dimensions D = 11 . The equations of state are assumed to be given by piI = (1 − uiI) ρI where, as a consequence of U duality symmetries, uiI are parametrised in terms of one constant u . The indices i, j, ⋯ run from 1 to 10 and the indices I, J, ⋯ from 1 to N . For 2 branes, 5 branes, and waves, N = 1 and the corresponding uiI are given in equations ([25W]). For 22ʹ55ʹ configuration, N = 4 and the corresponding uiI are given in equations ([I]). A. Evolution Equations The evolution of λi describing the scale factors is given by the equations described earlier which, for ease of reference, we summarise below: $$\begin{aligned} \lambda^i\_{\tau \tau} & = & \sum\_J u^{i J} \; e^{l^J} \label{d2} \\ l^I\_{\tau \tau} & = & \sum\_J {\cal G}^{I J} \; e^{l^J} \label{d3} \\ \lambda^i & = & \sum\_J u^i\_J \; (l^J - l^J\_0) + \; L^i \; \tau \label{d5} \end{aligned}$$ where $$\label{defn} u^{i I} = \sum\_j G^{i j} \; u^I\_j \; \; \;, \; \; \; \; {\cal G}^{I J} = \sum\_{i, j} G^{i j} \; u^I\_i \; u^J\_j \; \; \;, \; \; \; \; u^i\_I = \sum\_{j, J} {\cal G}\_{I J} \; G^{i j} \; u^J\_j$$ with Gij and ${\cal G}\_{I J} \;$ as defined earlier, and Li are arbitrary constants satisfying the constraints ∑iuiILi = 0 for all I . Also, lτI obey the constraint $$\label{d6} \sum\_{I, J} {\cal G}\_{I J} \; l^I\_\tau \; l^J\_\tau = 2 \; ( E + \sum\_J e^{l^J} )$$ where 2E = − ∑i, jGijLiLj . Equations ([d3]) and ([d6]) are to be solved for lI(τ) with initial conditions lI(0) = l0I = ln ρI0 and lτI(0) = KI where ρI0 are initial densities and $$\label{d6ic} \sum\_{I, J} {\cal G}\_{I J} \; K^I \; K^J = 2 \; ( E + \sum\_J e^{l^J\_0} ) \; \;.$$ Then λi(τ) follow from equation ([d5]) and the physical time t(τ) from dt = eΛ dτ . Inverting t(τ) then gives τ(t) , and thereby λi(t) . We can now calculate ${\cal G}^{I J} \;$ for the mutually BPS intersecting brane configurations. As explained in footnote 2, in the BPS configurations two stacks of 2 branes intersect at a point; two stacks of 5 branes intersect along three common spatial directions; a stack of 2 branes intersects a stack of 5 branes along one common spatial direction; and, waves, if present, will be along a common intersection direction. With these rules given, it is now straightforward to calculate ${\cal G}^{I J} \;$ using equations ([25W]) and ([defn]). It turns out because of the BPS intersection rules that the resulting ${\cal G}^{I J} \;$ are given by $$\label{G^IJ} {\cal G}^{I J} = 2 u^2 \; (1 - \delta^{I J}) \; \;.$$ The corresponding ${\cal G}\_{I J} \;$ exists for N > 1 , and is given by $$\label{G\_IJ} {\cal G}\_{I J} = \frac{1}{2 u^2} \; ( \frac{1}{N - 1} - \delta\_{I J} ) \; \;.$$ Note that, for N > 1 , the above ${\cal G}^{I J} \;$ is a special case of the example considered earlier in equation ([ab]), now with a = 2u2 and b = 1 , It is also straightforward to calculate uiI and uIi for the 22ʹ55ʹ configuration using the definitions in equation ([defn]) and the uiI in equation ([I]). They are given by $$\begin{aligned} 2 & : & u^{i 1} \; \propto \; (-2, -2, \; \; 1, \; \; 1, \; \; 1, \; \; \; 1, \; \; \; 1, \; \; 1, \; \; 1, \; \; 1) \nonumber \\ 2' & : & u^{i 2} \; \propto \; (\; \; 1, \; \; 1, -2, -2, \; \; 1, \; \; \; 1, \; \; \; 1, \; \; 1, \; \; 1, \; \; 1) \nonumber \\ 5 & : & u^{i 3} \; \propto \; (-1, \; \; 2, -1, \; \; 2, -1, - 1, -1, \; \; 2, \; \; 2, \; \; 2) \nonumber \\ 5' & : & u^{i 4} \; \propto \; (\; \; 2, -1, \; \; 2, -1, -1, -1, -1, \; \; 2, \; \; 2, \; \; 2) \label{I2} \end{aligned}$$ where the proportionality constant is $\frac{u}{3} \;$, and by $$\begin{aligned} 2 & : & u\_1^i \; \propto \; (\; \; 2, \; \; 2, -1, -1, -1, -1, -1, \; \; 1, \; \; 1, \; \; 1) \nonumber \\ 2' & : & u\_2^i \; \propto \; (-1, -1, \; \; 2, \; \; 2, -1, -1, - 1, \; \; 1, \; \; 1, \; \; 1) \nonumber \\ 5 & : & u\_3^i \; \propto \; (\; \; 1, -2, \; \; 1, -2, \; \; \; 1, \; \; 1, \; \; 1, \; \; \; 0, \; \; 0, \; \; 0) \nonumber \\ 5' & : & u\_4^i \; \propto \; (-2, \; \; 1, -2, \; \; 1, \; \; \; 1, \; \; 1, \; \; 1, \; \; \; 0, \; \; 0, \; \; 0) \label{I3}\end{aligned}$$ where the proportionality constant is $\frac{1}{6 u} \;$. We are unable to solve equations ([d3]), ([d6]), and ([GIJ]) for N > 1 .[9](#fn9) However, applying the general analysis described in section III and making further use of the explicit forms of uiI and ${\cal G}^{I J} \;$ given in equations ([I]) and ([GIJ]), one can understand the qualitative features of the evolution of the 22ʹ55ʹ configuration. We first make several remarks which will lead to an immediate understanding of the evolution of this configuration. (1) Let ui = ∑IuiI . It can then be checked that ∑i, jGijuiuj > 0 . Also, ∑iuiLi = 0 since ∑iuiILi = 0 for all I . Hence, as shown in Appendix B, it follows that E given in equation ([d6]) is ≥ 0 and that it vanishes if and only if Li all vanish. (2) The constraints ∑iuiILi = 0 imply that $$\begin{aligned} & & L^1 - L^4 = L^2 - L^3 = L^5 + L^6 + L^7 = 0 \nonumber \\ & & L^8 + L^9 + L^{10} = - 3 \; (L^1 + L^2) \; \;. \label{LI4}\end{aligned}$$ Thus, for example, we may take (L1, L2, L6, L7, L8, L9) to be independent. The remaining Lis are then determined by the above equations. Also, we have L ≡ ∑iLi = − (L1 + L2) . Using equations ([LI4]), ([LLc]), and the Schwarz inequality ([sch]) in Appendix B, we write E as $$\begin{aligned} 2 E & = & \sum\_i (L^i)^2 - (\sum\_i L^i)^2 \nonumber \\ & = & 3 (L)^2 + \sum\_{i = 5}^7 (L^i)^2 + 2 \sigma\_2^2 + \sigma\_3^2 \nonumber \\ & = & 3 (L^1)^2 + (L^1 + 2 L^2)^2 + \sum\_{i = 5}^7 (L^i)^2 + \sigma\_3^2 \label{E>}\end{aligned}$$ where the first line is the definition of E , σ2 = 0 if and only if L1 = L2 , and σ3 = 0 if and only if L8 = L9 = L10 . See the Schwarz inequality given in equation ([sch]). It is easy to show that the above expressions for E imply that (Li)2 for all i are bounded above by E as follows: E ≥ ci(Li)2 ≥ 0 where ci are constants of ${\cal O}(1) \;$. In particular, note the inequality 2E ≥ 3(L)2 which is required in Appendix C. (3) It follows from equations ([d5]), ([I3]), and ([LLc]) that $$\label{d5ex} \Lambda\_\tau = \sum\_i \lambda^i\_\tau = \frac{1}{6 u} \; (2 l^1\_\tau + 2 l^2\_\tau + l^3\_\tau + l^4\_\tau) + L \; \;.$$ Using the explicit form of ${\cal G}\_{I J} \;$ given in equation ([GIJ]) with N = 4 , equation ([d6]) becomes (∑IlτI)2 − 3 ∑I(lτI)2 = 12u2 (E + ∑IelI) > 0 where the inequality follows since E ≥ 0 and elI > 0 . We show in Appendix C that this inequality implies that none of (Λτ, lτI) may vanish, and that they must all have same sign. Hence, for all τ throughout the evolution, (Λτ, lτI) must all be non vanishing, and be all positive or all negative. B. Asymptotic evolution With no loss of generality, let Λt > 0 initially at t = t0 . Then it follows from the above result that (Λτ, lτI) must all be positive and non vanishing for all τ . Hence, (Λ, lI) are all monotonically increasing functions for all τ throughout the evolution. Equation ([d3]) may be written, using equation ([GIJ]), as lττI = 2u2 ∑J ≠ I elJ . In the past, τ and all lI decrease continuously. Hence, the right hand side in equation ([e3]) becomes more and more negligible. The asymptotic solution in the limit τ → − ∞ is then given by lI = c̃Iτ + d̃I where c̃I > 0 . Thus elI → 0 in this limit. Similarly, in the future, τ and all lI increase continuously. However, the right hand side in equation ([e3]) increases exponentially now. It is then obvious that all elI → ∞ within a finite interval of τ , i.e. at a finite value τ∞ of τ . In this context, see equations ([1c1]) and ([1c2]), and the general analysis given in section III B.2. We now analyse the corresponding asymptotic solutions. *1. Asymptotic evolution: eΛ → 0* It follows from the above discussion that eΛ → 0 in the limit τ → − ∞ . Also, in this limit, we have elI = ec̃Iτ = tb̃I , eλi = ec̃iτ = tb̃i upto multiplcative constants where (c̃I, c̃i, b̃I, b̃i) are constants. The evolution is then of Kasner type and is similar to that described in section III B.1. The constants c̃Is are determined by the initial values l0I and KI , but obtaining the exact dependence in the general case requires complete solution for lI(τ) . However, if the initial values l0I are large and negative then we have elI ≪ 1 for all τ < 0 and, hence, c̃I = KI to a good approximation. *2. Asymptotic evolution: eΛ → ∞* It follows from the above discussion that eΛ → ∞ in the limit τ → τ∞ where τ∞ is finite. Also, elI → ∞ in this limit and τ∞ depends on the initial values l0I and KI . Although solutions for lI(τ) are not known, their asymptotic forms in the limit τ → τ∞ , and hence those of λi(τ) , may be obtained following the analysis given in section III B.2. $\; {\cal G}^{I J} \;$ in equation ([GIJ]) is a special case of the example ([ab]) where, now, N = 4 , a = 2u2 , and b = 1 . Hence, it can be shown to correspond to the possibility (i) given in equation ([eqtau]). Therefore, we have γI = 1 and $e^{c^I} = 2 \sum\_J {\cal G}\_{I J} = \frac{1}{3 u^2} \;$. It then follows from equation ([asymptau]) that elI and eλi are given in the limit τ → τ∞ by $$\begin{aligned} e^{l^I} & = & \frac{1}{3 u^2} \; \frac{1}{(\tau\_\infty - \tau)^2} \label{lItau} \\ e^{\lambda^i} & = & e^{v^i} \; \left( \frac{1}{3 u^2} \; \; \frac{1}{(\tau\_\infty - \tau)^2} \right)^{\sum\_I u^i\_I} \label{lambdaitau}\end{aligned}$$ where, since ρI0 = el0I , we have vi = − ∑JuJi l0J + Li τ∞ , evi = eLi τ∞ ∏J(ρJ0)− uJi . Also, since $\gamma = \sum\_{i, I} u^i\_I = \frac{1}{u} \;$, we have from equation ([ttauasymp]) that the physical time t is given in this limit by $$\label{ttau} t - t\_s = A \; \left( \tau\_\infty - \tau \right)^{- \; \frac{2 - u}{u}}$$ where ts and A are finite constants. Clearly, t → ∞ in the limit τ → τ∞ since it is assumed that 0 < u < 2 . In this limit, the scale factors eλi may be written in terms of t as eλi = evi (B t)βi where B is a constant and $\beta^i = \frac{2 u}{2 - u} \; \sum\_J u^i\_J \;$. Using equation ([I3]) for uIi , the exponents βi are given by βi ∝ (0, 0, 0, 0, 0, 0, 0, 1, 1, 1) where the proportionality constant is $\frac{2}{3 (2 - u)} \;$. Note that $\beta = \sum\_i \beta^i = \frac{2}{2 - u} > 1 \;$. Hence, we have eΛ ∼ tβ in the limit t → ∞ with β > 1 . See the discussion below equation ([weak]) for the relevance of this feature. Thus, asymptotically in the limit t → ∞ , we obtain that $e^{\lambda^i} \to t^{\frac{2}{3 (2 - u)}} \; $ for the common transverse directions i = 8, 9, 10 . Hence, these directions continue to expand, their expansion being precisely that of a (3 + 1) – dimensional homogeneous, isotropic universe containing a perfect fluid whose equation of state is p = (1 − u) ρ . Also, eλi → evi for the brane directions i = 1, ⋯, 7 . Hence, these directions cease to expand or contract. Their sizes are thus stabilised and are given by evi . Note that this result is in accord with the general result described in section II B since, in the limit τ → τ∞ , the brane densities ρI ∝ elI all become equal and hence the four types of branes all become identical; and, t → ∞ and eΛ ∼ tβ → ∞ with β > 1 . C. Mechanism of stabilisation Using the asymptotic solutions, we can now give a physical interpretation of the dynamical mechanism underlying the stabilisation of the brane directions seen above for the 22ʹ55ʹ configuration. We first study the stabilisation process. Consider equation ([d2]) for λττ1 , for example. Using the values of uiI given in equation ([I2]), we have λττ1 ∝ ( − 2el1 + el2 − el3 + 2el4) . In the 22ʹ55ʹ configuration, x1 direction is wrapped by 2 branes and 5 branes and is transverse to 2ʹ branes and 5ʹ branes. Thus, from the above equation for λ1 and from similar equations for λ2, ⋯, λ7 , we see that 2 brane and 5 brane directions ‘contract with a force’ proportional to 2ρ(2) and ρ(5) respectively, whereas the directions transverse to them ‘expand with a force’ proportional to ρ(2) and 2ρ(5) respectively, where ρ( \* ) ∝ el( \* ) are the time dependent densities of the corresponding branes. When ρI ∝ elI all become equal, the forces of expansion cancel the forces of contraction resulting in vanishing nett force for the x1 direction. Then, using equation ([fttau]), one has λττ1 = e2Λ (λtt1 + Λtλt1) = 0 . Now, as described earlier in the context of equations ([K]) and ([L]), the transient ‘velocity’ λt1 is damped and λ1 reaches a constant value in the expanding universe here since we have eΛ ∼ tβ in the limit t → ∞ with β > 1 . The result is the stabilisation of the x1 direction. The stabilised size ev1 of x1 direction is given by $$\label{v1} e^{v^1} = e^{ L^1 \; \tau\_\infty } \; \; \left( \frac{\rho\_{2 0} \; \rho\_{4 0}^2} {\rho\_{3 0} \; \rho\_{1 0}^2} \right)^{\frac{1}{6 u}} \; \;,$$ see equation ([vi]). Note that ev1 can be interpreted as arising from the imbalance among the initial brane densities ρI0 , and from the parts L1 of λt1(0) which indicate the transients. The above analysis can be similarly applied to the stabilisation of other brane directions (x2, ⋯, x7) in the 22ʹ55ʹ configuration. Thus, three conditions need to be satisfied for stabilisation: (1) the time dependent brane densities ρI ∝ elI all become equal; (2) the forces of expansion and contraction for each of the brane directions be just right so that the nett force vanishes; (3) the universe be expanding as eΛ ∼ tβ in the limit t → ∞ with β > 1 so that the transient velocities are damped and the corresponding scale factors reach constant values. For any mutually BPS N > 1 intersecting brane configurations with the equations of state as assumed here, it is straightforward to show using the earlier analysis that the evolution equations ensure that elI all become equal asymptotically even if they were unequal initally, and that eΛ ∼ tβ in the limit t → ∞ with β > 1 . Thus conditions (1) and (3) are satisfied. Condition (2) requires the brane configuration to be such that each of the brane directions is wrapped by, and is transverse to, just the right number and kind of branes. This condition is satisfied for the N = 4 configurations 22ʹ55ʹ and 55ʹ5ʺW , both of which result in the stabilisation of seven brane directions and the expansion of the remaining three spatial directions. To our knowledge, the only other configurations which satisfy the condition (2) are the N = 3 configurations 22ʹ2ʺ and 25W , both of which result in the stabilisation of six brane directions and the expansion of the remaining four spatial directions. However it is the N = 4 configurations that are entropically favourable, see equation ([sn]). Note that, as described in section II B and upto certain technical assumptions regarding the equality of brane densities and the asymptotic behaviour of eΛ , the stabilisation here follows essentially as a consequence of U duality symmetries. In particular, it is independent of the ansatz for energy momentum tensors, or of the assumptions about equations of state, as long as the components of the energy momentum tensors obey the U duality constraints of the type given in equation ([weak]). Obtaining the details of the stabilisation, however, requires further assumptions e.g. of the type made here. Note also that the present mechanism of stabilisation of seven brane directions, and the consequent emergence of three large spatial directions, is very different from the ones proposed in string theory or in brane gas models –. V. Stabilised sizes of brane directions We thus see for the 22ʹ55ʹ configuration that, asymptotically in the limit eΛ → ∞ , the initial (10 + 1) – dimensional universe effectively becomes (3 + 1) – dimensional. Also, if vs = min{v1, ⋯, v7} then a dimensional reduction of the (10 + 1) – dimensional M theory along the corresponding xs direction gives type IIA string theory with its dilaton now stabilised. Using the standard relations, one can obtain the string coupling constant gs , the string scale Ms , and the four dimensional Planck scale M4 in terms of the M theory scale M11 and the stabilsed values evi . Defining vc = ∑i = 17vi and assuming, with no loss of generality, that the coordinate sizes of all spatial directions are of ${\cal O} (M\_{11}^{- 1}) \;$, we obtain gs2 = e3vs , M42 = evc − vs Ms2 = evc M112 where the equalities are valid upto numerical factors of ${\cal O} (1) \;$ only and $$\label{vc} e^{v^c} = e^{ L^c \; \tau\_\infty } \; \; \left( \frac{\rho\_{1 0} \; \rho\_{2 0}} {\rho\_{3 0} \; \rho\_{4 0}} \right)^{\frac{1}{6 u}} \; \; \;, \; \; \; \; \; \; L^c = \sum\_{i = 1}^7 L^i$$ as follows from equations ([I3]), ([vi]), and ρI0 = el0I . Also, note that $g\_s = (\frac {M\_s} {M\_{11}})^3 \;$. Since we have an asymptotically 3 + 1 dimensional universe evolving from a 10 + 1 dimensional one, it is of interest to study the resulting ratios $\frac{M\_{11}}{M\_4} \;$ and $\frac{M\_s}{M\_4} \;$, and study their dependence on the initial values (l0I, KI, Li) . In particular, one may like to know the generic values of these ratios and to know whether arbitrarily small values are possible. Setting M4 = 1019 GeV , one then knows the generic scales of M11 and Ms and, for example, whether M11 = 10− 15 M4 = 10 TeV is possible. In view of the relations between (M11, Ms, M4) given in equation ([411]), this requires studying the stabilised values evc and evc − vs , their dependence on (l0I, KI, Li) , and knowing whether they can be arbitrarily large. Note that if Li = 0 for all i then vi are all determined in terms of l0I only, see equation ([vi]). It is then obvious from equations ([vi]) and ([vc]) that any values for evc and evc − vs , no matter how large, may be obtained by fine tuning ρI0 correspondingly. [10](#fn10) This statement remains true even when Lis do not all vanish. In this case, however, one may question the necessity of fine tuning since, for example, the relation evc ∝ eLcτ∞ suggests that large values such as 1030 ∼ e70 may be obtained by tuning Lis, or τ∞ , or both to within a couple of orders of magnitude only. It turns out, as we explain below, that fine tuning is still necessary to obtain such large values. Consider first the possibility of tuning Li . Note that equations ([d3]) and ([d6]) are invariant under the scaling $$\label{scaling} (E, \; e^{l^I}, \; \tau) \; \; \; \longrightarrow \; \; \; (\sigma^2 E, \; \sigma^2 e^{l^I}, \; \frac{\tau}{\sigma})$$ where σ is a positive constant. The initial values scale correspondingly as (el0I, KI, Li) → (σ2el0I, σKI, σLi) . It then follows from equation ([d5]) that λi, and hence evi , remain invariant.[11](#fn11) This scaling property merely reflects the choice of a scale for time. For example, using this scaling, one may set ∑Jel0J = 1 or, when E > 0 as is the case here, set E = 1 . The corresponding σ then provides a natural time scale for evolution. We set E = 1 using the above scaling. With E = 1 , the value of τ∞ now depends only on l0I and KI . Since 2E = ∑i(Li)2 − (∑iLi)2 , it is still plausible to have a range of non zero measure where Li are large and E = 1 , and thereby obtain large values for evc and evc − vs . However, Lis are further constrained by ∑iuiILi = 0 , I = 1, ⋯, 4 , and consequently their magnitudes are all bounded from above. For example, with E = 1 , we obtain $(L^c)^2 \le \frac{8}{3} \;$. See remark (2) in section IV A. Thus, large values of evi cannot be obtained by tuning Li alone. Consider now the possibility of tuning τ∞ . Obtaining the dependence of τ∞ on (l0I, KI) requires explicit solutions which are not available. Hence, we obtain τ∞ numerically. We will present the numerical results in the next section. Here we point out that an approximate expression for τ∞ can be given in the limit when el0I ≪ E for all I . The reasoning involved is analogous to that used in obtaining τa in equation ([tauapprox]). Using similar reasoning and setting E = 1 now, we have that if el0I ≪ 1 for all I then $$\label{tauaI} \tau\_\infty \simeq \tau\_a = \; min \; \{\tau\_I \} \; \;, \; \; \; \; \tau\_I = - \; \frac{l^I\_0}{K^I} \; \;.$$ Note that τa can be calculated easily and requires no knowledge of explicit solutions. Our numerical results show that τa given above indeed provides a good approximation to τ∞ when el0I ≪ 1 for all I . Note also that KI must satisfy equation ([d6ic]) with E = 1 . It then follows from an analysis similar to that given in Appendix C that KI are all positive, cannot be too small, and are of ${\cal O}(1) \;$ generically. Hence, in the limit el0I ≪ 1 for all I, τa in equation ([tauaI]) are of ${\cal O}(min \{ - l^I\_0 \}) \;$. This indicates that large values of τ∞ , and hence of evi , cannot be obtained by tuning KI alone; a tuning of l0I , which translates to fine tuning of ρI0 = el0I , is required. Our numerical analysis also supports this conclusion. We thus find that, even when Lis do not all vanish, a fine tuning of ρI0 = el0I is necessary to obtain large values for evc − vs and evc . VI. Time varying Newton’s constant The evolution of the eleven dimensional early universe which is dominated by the 22ʹ55ʹ configuration described here can also be considered from the perspective of four dimensional spacetime. Indeed, in general, let the eleven dimensional line element ds be given by ds2 = gμν dxμdxν + ∑i = 17e2λi(dxi)2 where xμ = (x0, x8, x9, x10), with x0 = t, describes the four dimensional spacetime, and the fields gμν and λi , i = 1, ⋯, 7, depend on xμ only. Also, let Λc = ∑i = 17λi . It is then straightforward to show that the gravitational part of the eleven dimensional action S11 given in equation ([s11]) becomes $$\label{s4} S\_4 = \frac{V\_7}{16 \pi G\_{11}} \; \int d^4 x \; \sqrt{-g\_{(4)}} \; e^{\Lambda^c} \; \{ \; R\_{(4)} + (\nabla\_{(4)} \Lambda^c)^2 - \sum\_{i = 1}^7 (\nabla\_{(4)} \lambda^i)^2 \; \}$$ where V7 is the coordinate volume of the seven dimensional space and the subscripts (4) indicate that the corresponding quantities are with respect to the four dimensional metric gμν . The action S4 describes four dimensional spacetime in which the effective Newton’s constant G4 is spacetime dependent and is given by $$\label{tGN4} G\_4 (x^\mu) = e^{- \Lambda^c (x^\mu)} \; \; \frac{G\_{11}}{V\_7} \; \;.$$ In the case of early universe, the fields gμν and λi depend on t only. Then G4 is time dependent and we have, for G4 and its fractional time derivative, $$\label{GN4} G\_4 (t) = e^{- \Lambda^c (t)} \; \; \frac{G\_{11}}{V\_7} \; \; \;, \; \; \; \; \frac{(G\_4)\_t}{G\_4} = - \; \Lambda^c\_t \; \;.$$ For the four dimensional spacetime arising from the 22ʹ55ʹ configuration, the gμν fields are just the scale factors (eλ8, eλ9, eλ10) for (x8, x9, x10) directions, and all λi(τ) are given in equation ([d5]) in terms of lI(τ) , I = 1, ⋯, 4 . Then, using equation ([I3]) and the definitions of Λc , Lc , and vc , we have $$\label{Lambdac} \Lambda^c = - \; \frac{1}{6 u} (l^1 + l^2 - l^3 - l^4) - L^c (\tau\_\infty - \tau) + v^c \; \;.$$ In the limit t → ∞ , we have from the results given earlier that τ → τ∞ and the fields lI all become equal. Then Λc → vc where evc is given in equation ([vc]), and the three dimensional scale factors evolve as in the standard FRW case, namely $e^{\lambda^8} = e^{\lambda^9} = e^{\lambda^{10}} \sim t^{\frac{2}{3 (2 - u)}} \;$ as given in equations ([lambdait]) and ([betai]). It thus follows that the effective Newton’s constant G4 varies with time in the early universe and, in the case of 22ʹ55ʹ configuration, approaches a constant value $= e^{- v^c} \; \frac{G\_{11}}{V\_7} \;$ as the four dimensional universe expands to large size. The precise time dependence of G4 will follow from explicit solutions to equations ([d3]) and ([d6]). The consequences of a such a time dependent G4 are clearly interesting, and are likely to be important too. But their study is beyond the scope of the present paper. However, we like to point out here a characteristic feature of the time dependence of G4 which arises in the case of 22ʹ55ʹ configuration. Consider the behaviour of the differences lI − lJ in the limit τ → τ∞ which, in our case, vanish to the leading order. These quantities have been analysed in section III C and, for the example of the ${\cal G}^{I J} \;$ given in equation ([ab]), they are given by equations ([sisj]) and ([sisjsoln]) to the non trivial leading order. The case of 22ʹ55ʹ configuration corresponds to N = 4 , a = 2u2 , and b = 1 . Noting that sI − sJ = lI − lJ and that Δ < 0 in our case, equation ([sisjsoln]) now gives $$\label{liljsoln} (l^I - l^J) \; \sim \; (\tau\_\infty - \tau)^{ \frac{1}{2} (1 \; \pm \; i \; \sqrt{\frac{5}{3}})}$$ to the leading order. Clearly, Λc(τ) given in equation ([Lambdac]) will also have the same form as above to the non trivial leading order. Thus, taking the real part and writing in terms of t using equation ([ttau]), we have $$\label{lilj} \Lambda^c = v^c + \frac{b}{t^\alpha} \; \; Sin (\omega \; ln \; t + \phi )$$ to the non trivial leading order in the limit t → ∞ where b and ϕ are constants, $ \alpha = \frac{u}{2 (2 - u)} \;$, and $\omega = \sqrt{ \frac{5}{3}} \; \frac{u}{2 (2 - u)} \;$. Correspondingly, the time varying Newton’s constant is given by $$\label{G4} G\_4 \; \propto \; e^{- \Lambda^c} \; = \; e^{- v^c} \; (1 - \frac{b}{t^\alpha} \; \; Sin (\omega \; ln \; t + \phi ) \; )$$ to the leading order in the limit t → ∞ . Note that the constants b and ϕ depend on the details of matching. The constants α and ω arise as real and imaginary parts of an exponent on time variable, see equation ([liljsoln]). They do not depend on the initial values (l0I, KI, Li) and thus are independent of the details of evolution, but depend only on the configuration parameters N and u . The amplitude of time variation of G4 is dictated by α , and it vanishes in the limit t → ∞ . Hence, the time variation of G4 in equation ([G4]) is unlikely to contradict any late time observations. The time variation of G4 has log periodic oscillations also: G4 has an oscillatory behaviour where the nth and (n + 1)th nodes occur at times tn and tn + 1 which are related by $ln \; t\_{n + 1} = \frac {\pi} {\omega} + ln \; t\_n \;$, i.e. by $t\_{n + 1} = e^{\frac {\pi} {\omega}} \; t\_n \;$. The characteristic signatures and observational consequences of such log periodic variations of G4 are not clear to us. Log periodic behaviour occurs in many physical systems with ‘discrete self similarity’ or ‘discrete scale symmetry’: for example, in quantum mechanical systems with strongly attractive $\frac{1}{r^2} \;$ potentials near zero energy ; in Choptuik scaling and brane – black hole merger transitions ; and in a variety of dynamical systems. Algebraically, the log periodicity arises when an exponent on an independent variable becomes complex for certain values of system parameters. The relevant equations and solutions can often be cast in a form given in equations ([sisj]) and ([sisjsoln]). But we are not aware of a physical reason which explains the ubiquity of the log periodicity. To our knowledge, this is the first time a log periodic behaviour appears in a cosmological context. One expects such a behaviour to leave some novel imprint in the universe. But it is not clear to us which effects to look for, or which observables are sensitive to the log periodic variations of G4 . VII. Numerical results We are unable to solve explicitly the equations ([d3]) – ([d6]) describing the early universe evolution. Hence, we have analysed these equations numerically. In this section, we briefly describe our procedure and present a few illustrative results. We have analysed both the $u = \frac{2}{3} \;$ and u = 1 cases which would correspond to four dimensional universe dominated by radiation and pressureless dust respectively. The results are qualitatively the same and, hence, we take $u = \frac{2}{3} \;$ in the following. Note that ω in equation ([G4]) is then determined and, for $u = \frac{2}{3} \;$, the nth and (n + 1)th nodes in the log periodic oscillations occur at times tn and tn + 1 related by $ln (\frac{t\_{n + 1}}{t\_n}) = 4 \pi \sqrt{ \frac {3} {5} } \simeq 9.734 \;$. We proceed as follows. We start at an initial time τ = 0 and choose a set of initial values l0I = ln ρI0 . For each set of l0I , we further choose numerous arbitrary sets of (KI, Li) such that KI > 0 , E = 1 , and equations ([d6ic]) and ([LI4]) are satisfied. [12](#fn12) For each set of initial values (l0I, KI, Li) , we then numerically analyse the evolution for τ > 0 and obtain the value of τ∞ ; the evolution of lI, (λ1, ⋯, λ10), and t ; the stabilised values (v1, ⋯, v7) ; and the resulting values for $(g\_s, \frac {M\_{11}} {M\_4}, \frac {M\_s} {M\_4}) \;$. For a few sets of initial values, we have analysed the evolution for τ < 0 also. We find that the numerical results we have obtained confirm the asymptotic features described in this paper: (1) eλi and lI all vanish in the limit τ → − ∞ . In this limit, the evolution of the scale factors eλi is of Kasner type. (2) lI and the physical time t all diverge in the limit τ → τ∞ where τ∞ is finite. In this limit, the scale factors (eλ8, eλ9, eλ10) evolve as in the standard FRW case and (eλ1, ⋯, eλ7) reach constant values. (3) τa given in equation ([tauaI]) provides a good approximation to τ∞ when el0I ≪ 1 for all I . (4) Any values for the ratios $\frac{M\_{11}}{M\_4} \;$ and $\frac{M\_s}{M\_4} \;$ can be obtained, but a corresponding fine tuning of ρI0 = el0I is necessary. (5) The log periodic oscillations of lI − lJ , equivalently of (λ1, ⋯, λ7) , can also be seen in the limit τ → τ∞ . They can be matched to solutions of the type given in equation ([liljsoln]). To illustrate the values of τ∞ and the ratios $(\frac {M\_{11}} {M\_4}, \; \frac{M\_s}{M\_4}) \;$ one obtains, and to give an idea of their dependence on the initial values l0I , we tabulate these quantities in Table I for a few sets of initial values (l0I, KI, Li) . We have also tabulated the values of τa as given by equation ([tauaI]). The value of gs follows from $g\_s = (\frac {M\_s} {M\_{11}})^3 \;$ and, hence, is not tabulated. \|\|c\|\|c\|\|c\|c\|\|c\|c\|\| & & & & & & − (l01, l02,	arxiv_0000742
Lattice QCD and Three-particle Decays of Resonances =================================================== Most strong-interaction resonances have decay channels involving three or more particles, including many of the recently discovered X, Y and Z resonances. In order to study such resonances from first principles using lattice QCD, one must understand finite-volume effects for three particles in the cubic box used in calculations. Here we review efforts to develop a three-particle quantization condition that relates finite-volume energies to infinite-volume scattering amplitudes. We describe in detail the three approaches that have been followed, and present new results on the relationship between the corresponding results. We show examples of the numerical implementation of all three approaches and point out the important issues that remain to be resolved. finite-volume quantum field theory, lattice QCD, three-particle quantization condition, resonances INTRODUCTION ============ One of the striking features of the strong interaction is the abundance of resonances. These are very short-lived (having widths Γ ∼ 100 MeV), and are not asymptotic states, but are manifested through the behavior of the scattering amplitudes of particles that are stable under the strong interactions, e.g. pions, kaons and nucleons. Examples of resonances include the rho, decaying via ρ → ππ, as well as the a1(1260) → πππ and the Roper resonance, N(1440) → Nπ, Nππ, where in each case we have shown the final states with the highest branching fractions. The lightest such resonances form patterns when arranged according to basic properties such as spin, charge and strangeness, for example the flavor SU(3) decuplet of J = 3/2 baryons (Δ, Σ\, Ξ\, and Ω) and the nonet of JP = 1− mesons (ρ, ω, K\, and ϕ). This regularity, together with that of the particles stable under strong decay, was crucial in determining the underlying degrees of freedom, the quarks and gluons. This culminated in the formulation of quantum chromodynamics (QCD) in the early 1970s . While the lightest resonances can be categorized in quark-model language as quark-antiquark and three-quark states, this does not hold for higher-lying states. Examples include the f0(980) and a0(980), long considered to have a tetraquark ($q q \bar q\bar q$) component , and pentaquark candidates such as the P*c(4450)+ . Other resonances that do not fit into the simple quark-model classification are the recently discovered X, Y and Z states, which contain open or hidden charm and bottom quarks. For a recent summary of the experimental and theoretical status of the many states whose classification is unclear, see Ref. and the Particle Data Group listings and reviews . This situation, and in particular the multitude of X, Y and Z resonances, has led to a resurgence of interest in hadron spectroscopy, and a renewed appreciation of the need to extract predictions from first-principles QCD. In particular, many of the new states lie close to thresholds, and involve decays to multiple channels, and it is crucial to disentangle kinematical effects such as threshold cusps from truly resonant behavior. The latter is identified as a pole in the analytic continuation of a scattering amplitude to complex values of the center-of-mass energy. A crucial tool in disentangling the underlying properties of such resonances is first-principles calculations based in lattice QCD (LQCD). With this method, unlike with the quark model or other approximate approaches, one can systematically remove all sources of uncertainty in the calculations. The major such sources are the statistical errors inherent in a Monte Carlo calculation, the need to work at nonzero lattice spacing, the use of larger-than-physical quark masses, and the need to work with a finite space-time volume, with spatial box length[1](#fn1) L and Euclidean time extent Lt. Over the last decade or so, an increasing number of LQCD results for single-particle quantities have controlled all of these errors, in some cases at subpercent precision. One indication of this improved level of control is the increasingly common use of physical light-quark masses in calculations. For a review of results for well-controlled single-particle quantities using LQCD, see Ref. . [] Using LQCD to calculate the properties of resonances is, however, more challenging than for single-particle properties. A resonance is observed experimentally by studying the scattering of the decay products, e.g. two pions in the case of the ρ resonance. By measuring scattering rates for various kinematics and then performing a partial-wave analysis, one can in principle determine the scattering amplitudes projected to any given angular-momentum component and search for resonances. Lattice calculations cannot, however, reproduce this setup. The use of a finite volume does not allow the consideration of states with well-separated decay products (so one cannot approach the in- and out-states needed for a theoretical description of scattering in quantum field theory) and the need to use Euclidean time (in order to avoid a numerically intractable sign problem) makes real-time processes such as scattering inaccessible. Thus one is forced to use an indirect approach. The indirect approach that is by now widely applied was first introduced in seminal work by Lüscher . The essential observation is that the volume dependence of the energies of multiparticle states is governed by infinite-volume scattering amplitudes. By determining, in a lattice calculation, the finite-volume spectrum as a function of L, one is thus doing something analogous to a scattering experiment. Crudely speaking, the multi-particle finite-volume state in a large enough box contains particles that are almost moving freely, and thus approximate a scattering state. In the case of resonances that only decay to channels containing two particles, this has been placed on a rigorous footing by the derivation of so-called two-particle quantization conditions, i.e. equations that are satisfied only at the energies of finite-volume states and yet depend on infinite-volume scattering quantities. The simplest cases were worked out in Refs. (and will be reviewed in subsequent sections), and extensions to arbitrary spins, noncubic boxes, moving frames, and multiple two-particle channels have been subsequently derived. We do not review this literature here, as it is not the topic of this work, but point the interested reader to the comprehensive review provided in Ref. . [] We now come to the essential phenomenological motivation for the present review: Most resonances have some decay channels that involve three or more stable hadrons. Examples noted above are the a1(1260), with a dominant decay into three pions, the Roper resonance, which decays both to two- and three-particle channels, and many of the X, Y and Z resonances. For such resonances, the two-particle formalism simply does not apply. Thus, in order to address many pressing questions in hadron spectroscopy using LQCD, a three-particle quantization condition is needed. This is an equation that, given information about two- and three-particle scattering, predicts the finite-volume spectrum or, conversely, provides constraints on the scattering amplitudes given the spectrum calculated using LQCD.[2](#fn2) The major purpose of this review is to explain the theoretical progress that has been made over the last five or so years in deriving three-particle quantization conditions. Another motivation for this review is that advances in algorithms, methods, and the speed of computers, have allowed LQCD calculations to determine the finite-volume spectrum in the energy regime in which states have a significant three-particle component. Many examples can be found in Ref. , but we note in particular the spectra determined in Refs. and . At present, these calculations use heavier-than-physical quarks (so that $m\_\pi \gtrsim 230\;$MeV), but, nevertheless, their interpretation requires a quantization condition that can account for three, and in some cases more, particles. Indeed, the need for such a quantization condition becomes more pressing as the quark masses are lowered to their physical values, for then the multi-pion thresholds drop rapidly. Although motivated by results from LQCD, the derivations of the quantization conditions are in fact based in finite-volume continuum quantum field theory (QFT). In particular, we assume in the following that calculations have controlled the errors associated with nonzero lattice spacing and unphysical quark masses by appropriate extrapolations (or, in the latter case, possibly interpolations). As noted above, there are also errors associated with the finite extent of the lattice in the Euclidean time direction, Lt. Lattice calculations are mostly done with boundary conditions in the temporal direction chosen to give the system a thermal interpretation, with temperature T = 1/Lt. By choosing Lt large enough one works at very low temperatures, and then it is possible to extract the spectrum with controlled errors from the dependence of correlators on the Euclidean time separation. Thus the only issue we address here is the impact of working in a finite, cubic box, of length L. We further assume that, as in most lattice calculations, the spatial boundary conditions are periodic. [] Within this box one can fix all internal quantum numbers of the overall state. For example, one can set Q = ∣e∣, B = S = 0 (B being baryon number, and S strangeness), choose odd G-parity,[3](#fn3) and also fix the total three-momentum $\vec P$ to one of the allowed finite-volume values $2\pi \vec n/L$ (with $\vec n$ a vector of integers). Then the lightest finite-volume state corresponds to a π+ with momentum $\vec P$, and the excitations correspond to interacting π+π+π− and π+π0π0 states. If instead one projects onto Q = 2∣e∣ and even G-parity, while keeping B = S = 0, then the available states are approximately described as interacting π+π+, π+π+π+π− and π+π+π0π0 states, etc. Similarly, if we take Q = 3∣e∣ and odd G-parity, then the lightest state consists of π+π+π+. It is important to keep in mind, however, that we do not need to keep track of the individual particle components. We simply create the state with an operator having the requisite quantum numbers. Then the QCD dynamics, encoded in terms of quarks and gluons interacting via the QCD Lagrangian, lead to a low-lying spectrum of multi-hadron finite-volume states. The only output is a set of L-dependent energy levels, and this is all the input needed by the quantization conditions. It will be useful for the more technical discussion of the subsequent sections to describe some general features of the dependence of energy levels on L. For single, stable particles, a key result is that the energies, and other properties, depend on volume as (MπL)− nexp( − αMπL), where n and α depend on the quantity . These corrections arise from virtual pions propagating to the neighboring cells of the periodic system. They drop off rapidly with L, and it has been found that using $M\_\pi L \gtrsim 4$ is usually sufficient for finite-volume uncertainties to become subleading to other sources. Throughout this review, we assume that such exponentially-suppressed dependence can be neglected. The weakness of such dependence is crucial for the above-described successes of LQCD in attaining subpercent precision for certain single-particle quantities. [] The volume dependence for multiple-particle states is, however, quite different. In this case the asymptotic behavior exhibits power-law scaling of the form L− n. We describe the origin of this behavior below, but for now note only that multiparticle finite-L effects fall off much more slowly than the exponentially suppressed terms and cannot be ignored. Indeed, the quantization conditions allow this dependence to be used as a tool rather than an unwanted artifact. In this way, an apparent source of systematic uncertainty in LQCD calculations has become a powerful window into resonance physics. This review is organized as follows. Three approaches have been followed in the development of the three-particle quantization condition, and we discuss them in turn.[4](#fn4) The first uses a diagrammatic, all-orders analysis in a generic effective QFT. We refer to this as the RFT approach, with the R emphasizing that this is a relativistic approach. It was developed by us in Refs. for the simplest case of three identical scalars with a Z2 symmetry forbidding 2 ↔ 3 transitions, and subsequently generalized in collaboration with Briceño in Refs. . This is the topic of the following section, Sec. [sec:RFT]. We first describe the derivation of the two-particle quantization condition in Sec. [sec:QC2], providing two simple examples in order to motivate the general derivation. We then, in Sec. [sec:QC3], provide a description of the derivation of the three-particle quantization condition. Here we are able to use results from Ref. to simplify and shorten the derivation given in the original work, Ref. . Our hope is that this will make this rather technical derivation more accessible. We close Sec. [sec:RFT] with brief discussions of how the three-particle quantization condition can be truncated and thus made practical, and of two analytic checks of the formalism. Two alternate approaches have subsequently been followed, both of which greatly simplify the derivation of the quantization condition. These are described in Sec. [sec:alt]. To date, these only consider the case in which the interaction between particle pairs (denoted “dimers”) is purely s-wave. This is in contrast to the RFT approach, for which all waves are included in the dimer interactions. The two alternate approaches are that based on non-relativistic effective field theory (NREFT) , and that using a finite-volume implementation of unitarity constraints, which we refer to as the finite-volume unitarity (FVU) approach . For both approaches we describe the derivation of first the two- and then the three-particle quantization conditions, and then describe the relation of the results to those in the RFT approach. These latter two sections, Secs. [sec:NREFTtoRFT] and [sec:FVUtoRFT], present new results, which we think illuminate the similarities and differences between the approaches. A key question for all approaches is whether they can be implemented in practice. This has been addressed over the last year or so in all three approaches by showing how, in the simplest approximation of s-wave dimer interactions and a local three-particle interaction, the quantization conditions can be used to predict the finite-volume spectrum. Sec. [sec:num] gives examples of the results from all three approaches. We close with a list of summary points and issues for future work. In order to keep this review within bounds, there are many topics that we do not cover. As already noted, we do not discuss, except in passing, the two-particle formalism, or the extensive numerical results from LQCD calculations using the two-particle quantization condition to determine two-particle scattering amplitudes. For a recent, thorough review of both of these topics, we refer the reader again to Ref. . We also do not discuss the approach to two- and three-particle systems developed by the HALQCD collaboration, and reviewed in Ref. . In the two-particle case, this approach uses the Bethe-Salpeter amplitude in order to extract a potential that can then be inserted into the Schrödinger equation to determine bound-state energies and scattering amplitudes. It has been successfully applied to many two-baryon systems—for a recent example see Ref. . The extension to three particles is, however, so far restricted to the nonrelativistic domain, and thus is not directly applicable to most of the resonances of interest in QCD. We also do not discuss the impressive recent progress in simulating multiple interacting nucleons by discretizing the truncated pionless chiral EFT, and working in a finite volume. For a review see Ref. , and for an example of recent work see Ref. . This is a powerful method for studying bound states and near-threshold behavior, but does not apply to the resonances of interest, where dynamical pions are essential. Finally, we do not discuss recent ideas for determining the finite-volume multi-particle spectrum in NRQM using a variational approach , as again this is restricted to the nonrelativistic regime. RELATIVISTIC QUANTIZATION CONDITIONS USING A DIAGRAMMATIC APPROACH IN QFT ========================================================================= In this section we review the derivations of the two- and three-particle quantization conditions using an all-orders diagrammatic analysis in a generic relativistic field theory. We refer to this as the RFT approach. We begin with two particles as this allows us to explain the general strategy and to introduce notation that will aid in the explication of the more complicated case of three particles. We will derive these results in the simplest setting, namely for identical, spinless particles, which is the only setting considered to date for the three-particle quantization condition. Two-particle quantization condition in the RFT approach ------------------------------------------------------- The two-particle quantization condition provides the relation between the spectrum of a field theory in a finite box and the infinite-volume two-to-two scattering amplitude, ${\mathcal M}\_2$. We begin with some kinematical notation for ${\mathcal M}\_2$. We use $P^\mu=(E,\vec P)$ to denote the total 4-momentum of the scattering pair, so that $P^\mu P\_\mu = E^2 - \vec P^2 = s$. Denoting the 4-momentum of one of the incoming particles as k, and that of one of the outgoing as kʹ, we write the dependence of the amplitude as ${\mathcal M}\_2(P;k', k)$. We use this notation also for off-shell amplitudes, for which kμkμ ≠ m2, with m the physical mass of the particle.[5](#fn5) A special role is played by the c.m. (center-of-momentum) frame. We denote quantities in this frame with a superscript \* , e.g. the total energy is $E^\=\sqrt{s}$, and the incoming 3-momentum $\vec k$ boosted to this frame is $\vec k^\$. When the two incoming particles, with momenta k and P − k, are each on shell, this implies a constraint on the magnitude of the CM-frame momentum, $k^\\equiv \|\vec k^\\|$. In this case one finds that the latter is equal to $q^\=\sqrt{E^{\2}/4 - m^2}$, meaning that for fixed E\* only angular degrees of freedom remain. This allows one to decompose the on-shell amplitude into angular-momentum components, ${\mathcal M}\_2^{(\ell)}(s)$, in the standard way. [] Unitarity provides an important constraint on ${\mathcal M}\_2^{(\ell)}(s)$, one that will play a central role in Sec. [sec:FVU2]. Specifically, it implies that, for s ≥ 4m2, $$\mathcal M\_2^{(\ell)}(s)^{-1} = \mathcal K\_2^{(\ell)}(s)^{-1} - i \rho(s) \,,\quad \mathcal K\_2^{(\ell)}(s)^{-1} = \frac{q^\* \cot \delta^{(\ell)}(q^\)}{16 \pi \sqrt{s}} \,,\quad \rho(s) = \frac{q^\}{16\pi\sqrt{s}}\,, \label{eq:unitarity}$$ [] where the K matrix, $\mathcal K\_2^{(\ell)}(s)$, is a real function that is meromorphic (analytic up to poles) for s > 0. When we discuss the three-particle case we need to consider ${\mathcal M}\_2^{(\ell)}(s)$ below threshold such that q\* 2 < 0. To make sense of this is useful to note that ${\mathcal M}\_2^{(\ell)}(s)$ has a branch cut along the real axis in the complex s plane, running from s = 4m2 to ∞. The conventions established above correspond to real energies just above the cut. To remain on the same Riemann sheet for s < 4m2, one must analytically continue the phase-space factor as $\rho(s) = i \|q^\\|/(16\pi\sqrt{s})$. Turning now to the finite volume, we open with only a brief comment on our set up: In this review we restrict attention to periodic, cubic volumes, with length L* in each of the three spatial directions. ### Example: leading term in the threshold expansion[sec:ex1] To gain intuition into the general approach, we discuss two simple examples of the derivation of the quantization condition. The first concerns the lowest two-particle energy for $\vec P=0$, which, as was already shown over 60 years ago , satisfies $$E\_0(L) = 2 m + {4 \pi a}/{ (m L^3)} + \mathcal O(1/L^4) \,. \label{eq:HYth}$$ [] Here a is the scattering length, defined by $$q^\* \cot \delta^{0}(q^\) = - {1}/{a} + \mathcal O(q^{\2}) \,. \label{eq:a}$$ It is convenient here, and in the following, to introduce a finite-volume correlator that is closely related to the two-to-two scattering amplitude, ${\mathcal M}\_2$. This object, called ${\mathcal M}\_{2L}$, will have poles at the energies of the finite-volume states. It is defined by calculating exactly the same set of Feynman diagrams as for ${\mathcal M}\_2$, but with a sum rather than an integral over the allowed three-dimensional momentum modes. For example, in λϕ4 theory, the leading-order and next-to-leading-order contributions are shown in Figure [fig:1plus2](a) and given by $$i \mathcal M\_{2L} \equiv - i \lambda - \frac{\lambda^2}{2} \int \frac{d \ell^0}{2 \pi} \frac{1}{L^3} \sum\_{\vec \ell} \frac{i}{\ell^2 - m^2 + i \epsilon} \frac{i}{(P - \ell)^2 - m^2 + i \epsilon} + \cdots + \mathcal O(\lambda^3) \,, \label{eq:M2Lth}$$ where the ellipses represents the t- and u-channel diagrams. Some choice of UV regularization is implicit here, but need not be specified as it will play no role in the final result. [fig:1plus2] Our aim is now to pull out the power-law volume dependence. If all sums were replaced by integrals, then we would simply obtain the second-order expression for ${\mathcal M}\_2$. For the t- and u- channel contributions, it can be shown, as discussed below, that the sum-integral difference has an exponentially-suppressed volume dependence, scaling as e− mL. We neglect such dependence throughout this review as it is numerically small in practice. Thus the only source of power-law volume dependence is the s-channel loop shown explicitly in Eq. ([eq:M2Lth]). This can be simplified by doing the ℓ0 integral, and then replacing the sum with the integral plus sum-integral difference. One then obtains $$i \mathcal M\_{2L} = i \mathcal M\_2 - \frac{\lambda^2}{2} \bigg [ \frac{1}{L^3} \sum\_{\vec \ell} - \int \! \frac{d^3 \vec \ell}{(2 \pi)^3} \bigg ] \frac{i}{2 \omega\_{ \ell} 2 \omega\_{ P - \ell} (E - \omega\_{ \ell} - \omega\_{ P - \ell} + i \epsilon)} + \mathcal O(e^{- m L},\lambda^3) \,, \label{eq:M2LM2}$$ where $\omega\_{ \ell} \equiv \sqrt{m^2 + \vec \ell^2}$. We next set $\vec P=0$ and consider the weak coupling limit, $\vert \mathcal M\_2 \vert \ll 1$. Then the two-particle energy will be very close to the lowest-lying non-interacting state, $E=2m + {\mathcal O}(\lambda)$. In this regime, the dominant volume-dependent contribution to ${\mathcal M}\_{2L}$ comes from the $\vec \ell=0$ contribution to the sum, leading to $$\label{eq:M2Lnearpole} i \mathcal M\_{2L} = i \mathcal M\_2 - \frac{\lambda^2}{2} \frac{1}{L^3} \frac{i [1 + \mathcal O(1/L)]}{4 m^2 (E - 2 m + i \epsilon)} + \dots$$ At this stage we identify a problem with the truncated expression. While the leading-order term contains no poles, the next-to-leading order result contains a pole at the non-interacting level E = 2m. Both results are incorrect for the interacting theory. The problem arises because, for $E - 2 m = \mathcal O(\lambda)$, the first two terms are effectively of the same order. More importantly, some of the terms we have neglected are of this order as well. To resolve the issue we must include all diagrams with any number of s-channel bubbles, as shown in Figure [fig:1plus2](b) Doing so leads to a geometric series, and summing this leads to a shift in the pole in $\mathcal M\_{2L}$. To see this explicitly, we work to all orders in λ/(E − 2m) while only working to leading-order in the scattering amplitude (so that ${\mathcal M}\_2=-\lambda$). We find $$\label{eq:M2LnearpoleSUM} \mathcal M\_{2L} = \mathcal M\_2 \sum\_{n=0}^\infty \bigg ( \frac{(-1)}{2 L^3} \frac{[1 + \mathcal O(1/L)]} {4 m^2 (E - 2 m + i \epsilon)} \mathcal M\_2 \bigg )^n = \frac{{\mathcal M}\_2 (E - 2 m)}{ E - 2 m +{\mathcal M}\_2/(8m^2 L^3) + \mathcal O(1/L^4) } \,.$$ [] If we now make the replacement $\mathcal M\_2 = -32 \pi m a$, valid at leading order in λ for all s (and to all orders in λ for s = 4m2), then we find that the pole is indeed shifted to the position given in Eq. ([eq:HYth]). ### Example: quantization condition in 1 + 1 dimensions[sec:1plus1] Our second example shows how the quantization condition emerges for general values of E and L. Working in 1 + 1 dimensions keeps the essential features of the derivation while simplifying the calculation. For simplicity, we restrict to the c.m. frame, $\vec P=0$, and drop the argument P in ${\mathcal M}\_2$ and related quantities. It is instructive to first show how the unitarity relation arises diagrammatically. To this end, we expand ${\mathcal M}\_2$ in powers of the Bethe-Salpeter (BS) kernel, as shown in Figures [fig:1plus2](c)-(d). This kernel is the sum of all diagrams that are two-particle irreducible in the s-channel. Thus it contains loops with three or more particles, and single-particle poles, but no two-particle loops. We focus on the kinematic region m < E < 3m, within which the BS kernel has no on-shell cuts, and thus is real. Thus the only sources of imaginary contributions are the two-particle loops, which can have on-shell cuts. Performing the time component integrals, the expression for ${\mathcal M}\_2$ can be brought into the form[6](#fn6) $$\begin{aligned} \hspace{-10pt} i \mathcal M\_2(p',p) & = \sum\_{n=0}^\infty \prod\_{j=0}^{n-1} \left ( \frac12 \int\_{k\_{j+1}} i B\_2(k\_j, k\_{j+1}) \frac{i}{(2 \omega\_{k\_j})^2 (E - 2 \omega\_{k\_j} + i \epsilon)} \right ) i B\_2(k\_n, p) \bigg \vert\_{k^0=p'} \,. \label{eq:M21dim}\end{aligned}$$ Here we have left the momenta pʹ and p general. We now set these on-shell, via pʹ = p = q\, and drop the dependence on the left-hand side. We are interested in extracting the imaginary part. Thus we use the identity relating the i*ε prescription to a principal value (PV) prescription plus an imaginary δ-function term: $$\begin{aligned} {\mathcal M}\_2 & = \sum\_{n=0}^\infty \left ( - B\_2 \otimes \left[\mathcal I\_{{{\rm PV}}} - \frac12 \int\_k \frac{i}{(2 \omega\_{k})^2} \delta(E - 2 \omega\_{k})\right] \otimes \right )^n B\_2 \,,\end{aligned}$$ where ⊗ indicates integration of adjacent quantities over the common momentum, and ${\mathcal I}\_{{\rm PV}}$ is the pole term with the PV prescription. Thus factors of B2 adjacent to ⊗ are evaluated off shell. By contrast, those adjacent to the delta-function can be set on shell, i.e. with 2ωkj = E. Performing the integral we find $$\begin{aligned} {\mathcal M}\_2 & = \sum\_{n=0}^\infty \left ( [ - B\_2 \otimes \mathcal I\_{{{\rm PV}}} \otimes ] \ + B\_2 \, \frac{i }{8 p E} \right )^n B\_2 \,.\end{aligned}$$ The sum on the right-hand side with the imaginary term dropped leads precisely to the K matrix, ${\mathcal K}\_2$, where we stress that the external arguments of the B2s are on shell. Reorganizing the full right-hand side, keeping the imaginary term, thus gives $$\begin{aligned} {\mathcal M}\_2 & = \sum\_{n=0}^\infty \left ( \sum\_{m=0}^\infty [ - B\_2 \otimes \mathcal I\_{{{\rm PV}}} \otimes ]^m B\_2 \frac{i}{8q^\E} \right )^n \sum\_{p=0}^\infty [ - B\_2 \otimes \mathcal I\_{{{\rm PV}}} \otimes ]^p B\_2\,, \\ & = \sum\_{n=0}^\infty \left ( - \mathcal K\_2 \left[ - \frac{i }{8 q^\ E} \right ] \right )^n \mathcal K\_2 = \frac1{{\mathcal K}\_2^{-1} - i/(8 q^\* E)}\,, \end{aligned}$$ [] where again it is crucial that the B2s next to the i/(8q\*E) terms are on shell, so that it is the on-shell ${\mathcal K}\_2$ that appears. The final result is simply the unitarity relation, Eq. ([eq:unitarity]), but written in one spatial dimension. We now argue that a similar analysis can be applied to the finite-volume correlator ${\mathcal M}\_{2L}$ introduced above. We recall that $\mathcal M\_{2L}$ is a real function of energy whose poles give the finite-volume spectrum of the theory. The first step is to note that ${\mathcal M}\_{2L}$ is given by Eq. ([eq:M21dim]), except that the integrals are replaced by finite-volume sums over kj = 2πnj/L, with nj an arbitrary integer, and the iε is dropped. This is shown diagrammatically in Figure [fig:1plus2](e)-(f). The BS kernels in ${\mathcal M}\_{2L}$ are, strictly speaking, the finite-volume versions, but these differ from those in infinite volume only by exponentially-suppressed terms, a difference we neglect. This holds in our kinematic regime because the loops inside B2 have integrands that cannot go on shell, and are thus nonsingular. One can then use the Poisson summation formula to show that the sum-integral difference is exponentially suppressed. This result holds in any number of dimensions, and was first derived in Ref. . [] We now follow analogous steps to the demonstration of unitarity shown above, except that here we separate the sum over finite-volume modes into the principal-value integral and the residual sum-integral difference (instead of separating the iε-integral into a PV integral and an imaginary part): $$\begin{aligned} \mathcal M\_{2L} & = \sum\_{n=0}^\infty \left ([ - B\_2 \otimes \mathcal I\_{{{\rm PV}}} \otimes ] \ - B\_2 \otimes \, \frac12 \bigg [ \frac{1}{L} \sum\_k - {{\rm PV}}\int \frac{dk}{2 \pi} \bigg ] \frac{ 1}{(2 \omega\_k)^2 (E-2\omega\_k)} \otimes \right )^n B\_2 \,, \\ & = \sum\_{n=0}^\infty \left ([ - B\_2 \otimes \mathcal I\_{{{\rm PV}}} \otimes ] \ - B\_2 \, \frac{ 1}{4E L} \frac{L^2}{4 \pi^2} \bigg [ \sum\_n - {{\rm PV}}\int dn \bigg ]\frac{1 }{ x^2 - n^2 } \right )^n B\_2 + \mathcal O(e^{- m L}) \,,\end{aligned}$$ where in the second line we have rearranged the expression and introduced x ≡ q\*L/(2π). Unlike in our derivation of the unitarity relation, here we have no Dirac delta function to project B2 to its on-shell value. Nonetheless, we note that the factors of the BS kernels in the L-dependent terms can, in fact be evaluated at k = q\. This is justified because the difference is exponentially suppressed $$\begin{gathered} \bigg [ \frac{1}{L} \sum\_k - {{\rm PV}}\int \frac{dk}{2 \pi} \bigg ]\frac{ \omega\_k}{(2 \omega\_k)^2 (q^{\2} - k^2)} [ B\_2(p'', k) B\_2(k, p') - B\_2(p'', q^\) B\_2(q^\, p') ] = \\ \bigg [ \frac{1}{L} \sum\_k - \int \frac{dk}{2 \pi} \bigg ]\frac{ \omega\_k}{(2 \omega\_k)^2 (q^{\2} - k^2)} \big [ b(k) (q^{\2} - k^2) \big ] \propto \sum\_{n'\ne 0} \int \frac{dk}{2 \pi} \frac{b(k) e^{i L k n'}}{\sqrt{k^2 + m^2}} = \mathcal O(e^{- m L}) \,.\end{gathered}$$ [] The first line here shows the difference between on- and off-shell BS kernels and, on the second line, we have used the result that this must equal a smooth function (denoted b(k)) times q\* 2 − k2. Canceling the pole, and then using the Poisson summation formula, we have rewritten the result as a series of Fourier transforms with respect to integer multiples of L. This leads to our conclusion that kernels can be placed on shell up to terms that we neglect. In this way we have an effective delta function, in place of the true factor of δ(E − 2ω) that appeared in the unitarity derivation. Following the remaining steps exactly as above, and using ${\mathcal K}\_2^{-1}= q^\\cot \delta(q^\)/(8q^{\2} E)$, we find $$\mathcal M\_{2L} = \frac{8 q^\ E}{ \cot \delta(q^\) + \cot \phi(q^\,L) } \,,$$ where $$\cot \phi(q^\, L) \equiv \frac{x }{\pi} \bigg [ \sum\_n - {{\rm PV}}\int dn \bigg ]\frac{1 }{ x^2 - n^2 } = \frac{x }{\pi} \frac{\pi \cot (\pi x)}{x} = \cot \frac{q^\ L}{2}\,, \label{eq:1dsum}$$ and the second equality follows by noting that the principal-value integral is identically zero and the sum can be evaluated analytically. This implies that the finite-volume spectrum is given by all solutions to $$\cot \delta(q^\) + \cot \frac{q^\ L}{2} = 0 \,,$$ equivalently e2iδ(q\) + i*q\*L = 1 , [] which is the well-known result for 1+1-dimensional theories . These simplified cases illustrate the key steps in the derivation of the quantization conditions: [Key steps] 1. Demonstrate that power-like finite-volume dependence arises only from s-channel diagrams. 2. Sum contributions from all Feynman diagrams to identify the shift in the finite-volume energies. ### General derivation of two-particle quantization condition This was first presented in the seminal work of Lüscher , but the approach followed here is more closely based on the derivation of Ref. . We begin with the BS equation for the two-particle scattering amplitude $$\pmb {\mathcal M}\_2= \textbf B\_2 + \textbf B\_2 \otimes \mathcal I \otimes \pmb {\mathcal M}\_2 \,, \label{eq:M2}$$ shown diagrammatically in Figure [fig:1plus2](c). Here we have introduced a boldface notation in which coordinate dependence, and factors of i, are suppressed, e.g. $\textbf B\_2 \equiv i B\_2(P; k'; k)$. The symbol $\otimes{\mathcal I}\otimes$ here indicates an integral over the two-particle loop, now in 3 + 1 dimensions, $$\label{eq:BSM2} \textbf B\_2 \otimes \mathcal I \otimes \pmb {\mathcal M}\_s \equiv \frac12 \int \frac{d^4 k}{(2 \pi)^4} i B\_2(P;k'';k') \frac{i z(k')}{k'^2 - m^2 + i \epsilon} \frac{i z(P-k')}{(P-k')^2 - m^2 + i \epsilon} i \mathcal M\_2(P;k';k) \,,$$ where z(k)/(k2 − m2 + iε) is the fully dressed propagator, normalized so that z = 1 on shell. To determine the finite-volume spectrum, we again use ${\mathcal M}\_{2L}$, although, in fact, any finite-volume correlator would suffice [a different choice was made in Ref. ]. We now use the result described above that the finite- and infinite-volume versions of the BS kernel differ by terms scaling as e− mL if m < E\* < 3m (or 0 < E\* < 4m is there is a $\mathbb Z\_2$ symmetry separating even- and odd-particle number sectors). This result allows us to write the BS equation for the finite-volume correlator [shown in Figure [fig:1plus2](e)]: $$\begin{aligned} \pmb {\mathcal M}\_{2L}&= \textbf B\_2 + \textbf B\_2 \otimes \mathcal S \otimes \pmb {\mathcal M}\_{2L} \,, \label{eq:MLinitdecom} \\ \textbf B\_2 \otimes \mathcal S \otimes \pmb {\mathcal M}\_{2L} &\equiv \frac12\int \frac{d k'^0}{2 \pi} \frac{1}{L^3} \sum\_{\vec k'} i B\_2(P;k'';k') \frac{i z(k')}{k'^2 - m^2 + i \epsilon} \frac{i z(P-k')}{(P-k')^2 - m^2 + i \epsilon} i \mathcal M\_{2L}(P;k';k) \,. \label{eq:BSML}\end{aligned}$$ Here the finite-volume momentum $\vec k'$ is summed over the values $2\pi \vec n/L$, with $\vec n$ a vector of integers. We next replace the sum with an integral and a sum-integral difference. All power-like L dependence lies in the latter quantity. As in the 1 + 1-dimensional analysis above, and as shown in detail in Refs. , the sum-integral difference picks out the on shell values of the quantities on either side of ${\mathcal S}$. Specifically, it is found that $$\begin{aligned} \label{eq:SisIplusF} \textbf B\_2 \otimes \mathcal S \otimes \pmb {\mathcal M}\_{2L} &= \textbf B\_2 \otimes \mathcal I \otimes \pmb {\mathcal M}\_{2L} + \textbf B\_2 \textbf F\_2^{{i\epsilon}}\pmb {\mathcal M}\_{2L} \\ \textbf B\_2 \textbf F\_2^{{i\epsilon}}\pmb {\mathcal M}\_{2L} & \equiv i B\_{2;\ell''' m'''; \ell'' m'' }(P) \, i F\_{2;\ell'' m''; \ell' m'}^{i\epsilon}(P,L) \, i \mathcal M\_{2L; \ell' m'; \ell m}(P) \,, \\ i F\_{2;\ell'' m''; \ell' m'}^{i\epsilon}(P,L) & \equiv \frac{1}{2} \bigg[ \frac{1}{L^3} \sum\_{\vec k} - \int \frac{d^3 \vec k}{(2 \pi)^3}\bigg] \frac{i \mathcal Y\_{\ell'' m''}(\vec k^\) \mathcal Y^\\_{\ell' m'}(\vec k^\) } {2 \omega\_{k} 2 \omega\_{ P - k} \ (E - \omega\_{ \ell} - \omega\_{ P - k} + i \epsilon)} \,. \label{eq:Fdef} \end{aligned}$$ [] where the product in the second line is now a matrix product with indices given by the angular momentum in the c.m. frame of the two on-shell particles. The quantity F2i*ε is a matrix of geometric functions that encodes how angular momentum-states mix due to the reduced symmetry of the finite-volume system.[7](#fn7) It is the generalization to three-spatial dimensions of the simple one-dimensional sum evaluated in Eq. ([eq:1dsum]). We have introduced $\mathcal Y\_{\ell m}(\vec k^\) = \sqrt{4 \pi} (k^\/q^\)^\ell Y\_{\ell m}(\hat k^\)$, where the prefactor multiplying the spherical harmonic removes spurious singularities near $\vec k^\* = \vec 0$. At this stage the derivation proceeds by substituting Eq. ([eq:SisIplusF]) into Eq. ([eq:MLinitdecom]) and rearranging $$\begin{aligned} \label{eq:M2LwithF} \pmb {\mathcal M}\_{2L} & = \textbf B\_2 + \textbf B\_2 \big [ \otimes \mathcal I \otimes + \textbf F\_2^{{i\epsilon}}\big ] \pmb {\mathcal M}\_{2L} \,, \\ & = \sum\_{n=0}^\infty \textbf B\_2 \Big [ \big [ \otimes \mathcal I \otimes + \textbf F\_2^{{i\epsilon}}\big ] \textbf B\_2 \Big ]^n \,, \\ & = \sum\_{n=0}^\infty \Big (\sum\_{n'=0}^\infty \textbf B \_2 \big [ \otimes \mathcal I \otimes \textbf B\_2 \big ]^{n'} \Big ) \bigg [ \textbf F\_2^{{i\epsilon}}\Big (\sum\_{n'=0}^\infty \textbf B\_2 \big [ \otimes \mathcal I \otimes \textbf B\_2 \big ]^{n'} \Big ) \bigg ]^n \,, \\ & = \sum\_{n=0}^\infty \pmb {\mathcal M}\_2 \big[ \textbf F\_2^{{i\epsilon}}\pmb {\mathcal M}\_2 \big ]^n = \pmb {\mathcal M}\_2 \frac{1}{1 - \textbf F\_2^{{i\epsilon}}\pmb {\mathcal M}}\_2 \label{eq:M2Lfinal} \,.\end{aligned}$$ In the first line we have substituted the identity for $\mathcal S$, Eq. ([eq:SisIplusF]). Then we have iteratively substituted the expression for $\pmb {\mathcal M}\_{2L}$ to display all volume dependence. In the third line we have regrouped into separate sums over $\mathcal I$ and $\textbf F\_2^{{i\epsilon}}$. In the final step, we have identified the sums involving $\mathcal I$ with ${\mathcal M}\_2$, using Eq. ([eq:M2]). Alternatively, one can formally solve Eq. ([eq:M2LwithF]) directly for $ \pmb {\mathcal M}\_{2L}^{-1} $ $${\pmb{\mathcal M}}\_{2L}^{-1} = \big [ \textbf B\_2^{-1} - \otimes \, \mathcal I \otimes \! \big ] - \textbf F\_2^{{i\epsilon}}\,,$$ and identify the square-bracketed contribution as ${\pmb {\mathcal M}}\_2^{-1}$. But since the final step is difficult to justify without an all-orders expansion in $\textbf B\_2$, it is not clear to us that this more direct line adds anything to the derivation. We deduce that, for fixed values of $\vec P$ and L, the finite-volume energy spectrum in the region E\* < 3m is given (up to e− mL corrections) by all solutions in E to the quantization condition [] $$\det\_{\ell' m'; \ell m} \big [\mathcal M\_2^{-1}(E^\) + F\_2^{{i\epsilon}}(E, \vec P, L) \big ] = 0\,, \label{eq:QC2}$$ where $\mathcal M\_{2;\ell' m'; \ell m} = \delta\_{\ell' \ell} \delta\_{m'm} \mathcal M\_2^{(\ell)}$. The extensions to multiple coupled two-particle channels, including different species and particles with spin, are known, as reviewed in Ref. . In the following it will be useful to express the quantization condition in terms of the K matrix. To this end we introduce a version of F2 in which the PV prescription is used in the integral in Eq. ([eq:Fdef]) rather than the i*ε prescription. We denote this simply as F2 without a superscript. It is straightforward so show [see, e.g., Ref. ] that the quantization condition ([eq:QC2]) can be exactly rewritten as [] $$\det\_{\ell' m'; \ell m} \big [\mathcal K\_2^{-1}(E^\) + F\_2(E, \vec P, L) \big ] = 0\,, \label{eq:QC2K}$$ where $\mathcal K\_{2;\ell' m'; \ell m} = \delta\_{\ell' \ell} \delta\_{m'm} \mathcal K\_2^{(\ell)}$. Three-particle quantization condition in the RFT approach --------------------------------------------------------- We now turn to the derivation of the three-particle quantization condition, presented in Refs. . As is discussed in detail in those publications, the quantization condition depends on an intermediate quantity, referred to as a divergence-free three-particle K matrix and denoted by ${\mathcal K\_{\text{df},3}}$. This is a non-standard object that encodes the short-distance part of the the three-particle scattering amplitude. It has the same degrees of freedom as the standard scattering amplitude but should be easier to extract from the finite-volume spectrum, since it is a smooth, real function. Most importantly, its relation to the standard $\textbf 3 \to \textbf 3$ scattering amplitude, $\mathcal M\_3$, is known and depends only on the on-shell $\textbf 2 \to \textbf 2$ scattering amplitude. Thus, the envisioned work-flow is summarized by $$E\_n(L) \ \ \Longrightarrow \ \ {\mathcal K\_{\text{df},3}}, \mathcal M\_2 \ \ \Longrightarrow \ \ \mathcal M\_3 \,.$$ [] In this section we sketch the derivation of the quantization condition for three identical relativistic scalar particles. Following Refs. , we restrict attention to theories with a $\mathbb Z\_2$ symmetry that decouples the even- and odd-particle-number sectors. In addition we assume that the two-particle K matrices have no poles in the kinematic region of interest. An example of such a theory is the 3π+ system in QCD in the isospin limit, where G-parity plays the role of the $\mathbb Z\_2$ symmetry, and the 2π+ subsystem is not resonant. The generalizations to include $\textbf 2 \to \textbf 3$ scattering and to describe systems with sub-channel resonances (generating K-matrix poles) have been worked out in Refs. and, respectively. The derivation we present is a simplified version of that given in Ref. , based on the improved approach introduced in Ref. . We focus on the important steps, pointing the reader to Refs. for detailed justifications. We begin in Sec. [sec:prelim] by setting up the strategy of the derivation, based on a skeleton-expansion of a finite-volume correlation function, C*L In Sec. [sec:decom] we summarize how the L dependence is isolated for diagrams of all topologies. We then, in Sec. [sec:combine], combine results to reach a closed form for CL, from which immediately follows the quantization condition in terms of ${\mathcal K\_{\text{df},3}}$. Finally, in Sec. [sec:KtoM] we review the relation between ${\mathcal K\_{\text{df},3}}$ and $\mathcal M\_3$. ### Preliminaries[sec:prelim] As in the two-particle case, the derivation presented here is carried out to all perturbative orders in a generic, relativistic quantum field theory. By “generic” we mean that no assumptions about the vertices or power-counting scheme are required. Consider the finite-volume correlation function $$C\_L(E, \vec P) \equiv \int\_L d^4 x \, e^{- i E t + i \vec P \cdot \vec x} \langle \sigma(x) \sigma^\dagger(0) \rangle\_L \,,$$ [] where σ†(0) has odd-particle quantum numbers. In the region m < E\* < 5m (where m is the physical particle mass and $E^\* = \sqrt{E^2 - \vec P^2}$), all power-like L-dependence arises from intermediate three-particle states. [fig:3plus4] With this in mind, in Ref. we construct a skeleton expansion in terms of BS kernels and fully dressed propagators, as shown in Figure [fig:3plus4]. The expansion is entirely motivated by the goal of displaying all important L-dependence, equivalently all on-shell or long-distance intermediate states. The final structure can be expressed in terms of two kernels: B2, already used above, and B3, which contains all 3-particle irreducible diagrams in three-to-three scattering (as well as one-particle propagation that introduces no singularities in the kinematic window considered). Both B2 and B3 have only exponentially suppressed L dependence, and thus, under the approximation guiding the derivation, can be replaced by their infinite-volume counterparts. As we explain in more detail in the following, the relevant volume-dependence, generated by sums over $\vec k = 2 \pi \vec n/L$ in the loops of the skeleton expansion, can be expressed in terms of two finite-volume geometric functions, denoted F and G. The first is closely related to the zeta-function of Eq. ([eq:Fdef]) and is defined in terms of the PV version of that quantity:[8](#fn8) $$F\_{k' \ell' m'; k \ell m} (P, L) \equiv \delta\_{k' k} H(\vec k) F\_{2; \ell' m, \ell m}(P-k, L) \,. \label{eq:F}$$ [] Here we display the indices that all matrices in the three-particle quantization condition share: $\vec k$, ℓ, and m. These can be understood by separating the three particles into a dimer [called “scattering pair” in Ref. ] and a spectator.[9](#fn9) The spectator is constrained to have one of the discrete finite-volume momenta, $\vec k = 2 \pi \vec n/L$, while the dimer is decomposed into angular momentum states in its c.m. frame (as was done in the two-particle quantization condition), leading to the ℓm indices. The δkʹk in Eq. ([eq:F]) thus represents a situation in which the spectator does not interact. The argument of F2 gives the four-momentum flowing through the dimer, $(P-k)^\mu = (E - \omega\_k, \vec P - \vec k)$. A new feature in Eq. ([eq:F]) is the appearance of the cutoff function $H(\vec k)$. For fixed P, as $\|\vec k\|$ increases, the dimer c.m. energy, $$E\_{2,k}^\* = \sqrt{(E-\omega\_k)^2 - (\vec P-\vec k)^2}\,, \label{eq:E2kstar}$$ passes below threshold E2, k\* = 2m and eventually drops to zero. For technical reasons explained in Ref. the formalism requires that E2, k\* 2 ≥ 0. The cutoff function $H(\vec k)$ accomplishes this by smoothly varying from unity at and above threshold to zero when E2, k\* = 0. An explicit example of the cutoff is given in Ref. , but will not be needed here. One can think of it as a soft cutoff at $\|\vec k\|\sim m$. The presence of $H(\vec k)$ implies that the index $\vec k$ runs over a finite number of values. The full matrix space remains infinite dimensional, however, due to the angular-momentum degrees of freedom. If these are truncated, as is common practice in the two-particle sector, then F reduces to a finite-dimensional matrix. The locations in the diagrams of the skeleton expansion at which an F appears is shown in Figure [fig:3plus4]. A quantization condition based only on F would predict three-particle energies of the form $E\_n(L) = \omega\_k + E\_2(L, \vec P - \vec k)$ where the second term is an interacting two-particle level with the indicated momentum. Of course, for three identical particles, this cannot be the correct spectrum. It properly encodes the interactions of two, but neglects the third which enters as a non-interacting constituent, albeit with the proper relativistic energy. This motivates the appearance of the second geometric function, G, that encodes the volume effects of an exchange in the scattering pair. The explicit definition is[10](#fn10) $$G\_{p\,\ell' m';k\,\ell m} \equiv \left(\frac{k^\}{q\_p^\}\right)^{\ell'} \frac{4\pi Y\_{\ell'm'}(\hat k^\)H(\vec k) H(\vec p) Y^\\_{\ell m}(\hat p^\)} {(P-p -k)^2 - m^2} \left(\frac{p^\}{q\_k^\}\right)^{\ell} \frac1{2\omega\_k L^3} \,, \label{eq:G}$$ [] where $\vec k^\$ is $\vec k$ boosted to the dimer c.m. frame when $\vec p$ is the spectator momentum, and $q\_p^\=\sqrt{E\_{2,p}^{\2}/4 - m^2}$ is the momentum of each of the two dimer constituents when $\vec p$ is the spectator momentum and all particles are on shell. The same definitions hold for $\vec p^{\,\}$ and q*k\, with the roles $\vec k$ and $\vec p$ exchanged. Examples of the locations in skeleton-expansion diagrams that lead to factors of G* are shown in Figure [fig:3plus4]. At this stage we are left to explain three more-technical aspects of the notation. First, at various intermediate stages we consider sums over three-to-three diagrams in which an incoming or outgoing particle is singled-out by the property of being unscattered by the outermost two-to-two insertion. We use the superscript (u) to denote such unsymmetrized quantities. Given that we consider identical particles, the final result cannot (and, as we prove in Ref. , does not) depend on these incomplete objects. Second, we find it convenient to introduce a bold-faced notation for the BS kernels as well as other building blocks defined via partial diagrammatic sums. This simply denotes that factors of i and 1/(2ωL3) have been absorbed to simplify expressions, and is taken over from Ref. . Third, it is convenient to begin the derivation by studying only the part of $C\_L(E, \vec P)$ that survives when we send B3 → 0. This is indicated by a [B2] superscript, CL → CL[B2]. With these preliminaries established, we are now ready to jump into the derivation. ### Decomposing to the level of ${\mathbf K\_{{\mathrm{df}}}}$[sec:decom] We begin with Eq. (174) of Ref. , expressed in boldface notation as in Eq. (49) of Ref. , $$\begin{aligned} C^{[B\_2]}\_{L} &= C\_{L,0F} - \frac23 {{\pmb \sigma^\}}{{\mathbf F}}{{\pmb \sigma^{\dagger\}}}+ {{\mathbf A}}'^{(u)}\_{L,3} {{\mathbf F}}\_{33}^{(0)} \sum\_{n=0}^\infty \left( {\mathbf K\_{L,33}}^{(u,u)} {{\mathbf F}}\_{33}^{(0)} \right)^n {{\mathbf A}}^{(u)}\_{L,3} \,, \label{eq:Cno3MLres} \\ {{\mathbf F}}^{(0)}\_{33} & \equiv {{\mathbf F}}\frac{1}{1 - {{\mathbf K\_2}}{{\mathbf F}}} \,. \label{eq:F33zerodef}\end{aligned}$$ [] To obtain this result we have applied the method that led to the two-particle quantization condition for diagrams in which adjacent factors of B2 are attached to the same dimer. This explains the appearance of ${{\mathbf K\_2}}\equiv i [2 \omega L^3] \mathcal K\_2$, which is diagonal in spectator indices,[11](#fn11) as well as ${{\mathbf F}}\equiv i F/(2 \omega L^3)$. [fig:5plus6] The other quantities in Eq. ([eq:Cno3MLres]) involve diagrams in which particles pairwise scatter via ${{\mathbf K\_2}}$, with the remaining loops (containing exchange propagators) involving sums over the finite-volume momenta. These quantities are finite-volume objects, as indicated by the subscript L. They are shown Figure [fig:5plus6]. Specifically, ${\mathbf K\_{L,33}}^{(u,u)}$ is sum of all fully connected three-to-three scattering diagrams involving alternating pairwise scattering, while ${{\mathbf A}}^{(u)}\_{L,3}$, ${{\mathbf A}}'^{(u)}\_{L,3} $ and CL, 0F are defined similarly, but with additional endcap factors ${{\pmb \sigma^{\dagger\}}}$ and ${{\pmb \sigma^\}}$. The remaining task is to decompose the volume dependence of CL, 0F, ${{\mathbf A}}'^{(u)}\_{L,3} $, ${{\mathbf A}}^{(u)}\_{L,3}$ and ${\mathbf K\_{L,33}}^{(u,u)}$, expressing each object as matrix products of infinite-volume quantities and finite-volume geometric functions. To this end one can show that $$\begin{aligned} C\_{L,0F} &= {C}^{[B\_2]}\_{\infty } + 2 {{\mathbf A}}'^{(s)}\_3 {{{\mathbf F}}} ({{\mathbf A}}\_{L,3}^{(u)} - {{\pmb \sigma^{\dagger\}}}) \,, \label{eq:CL0Ffinal} \\ {{\mathbf A}}'^{(u)}\_{L,3} & = {{\mathbf A}}'^{(u)}\_3 + 2 {{\mathbf A}}'^{(s)}\_3 {{\mathbf F}}\left( {\mathbf K\_{L,33}}^{(u,u)}+ {{\mathbf K\_2}}\right) \,, \label{eq:ApL3final} \\ {{\mathbf A}}^{(u)}\_{L,3} & = {{\mathbf A}}^{(u)}\_3 + \left( {\mathbf K\_{L,33}}^{(u,u)} + {{\mathbf K\_2}}\right) {{\mathbf F}}\, 2 {{\mathbf A}}^{(s)}\_3 \,, \label{eq:AL3final} \end{aligned}$$ [] thereby reducing all unfactorized L-dependence to that of ${\mathbf K\_{L,33}}^{(u,u)}$. The derivation of the result for C*L, 0F is sketched in Figure [fig:5plus6]; those for the other quantities is similar. The idea is to move from left to right, replacing a given momentum sum with an integral plus a sum-integral difference. The term with the difference leads to a factor of ${{\mathbf F}}$ whereas that with the integral is further decomposed, by applying the same prescription to the next sum in the chain. In this way all contributions are cast into the same form: an ${{\mathbf F}}$-cut with an infinite-volume expression to the left and remaining L dependence to the right. At this stage it only remains to decompose $ {\mathbf K\_{L,33}}^{(u,u)} $. One finds $${\mathbf K\_{L,33}}^{(u,u)} = {\mathbf K\_{L,33}}^{(0)} + [1+ {{\mathbf T}}{{\mathbf G}}] {\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)} \frac1{1 - {{ \mathbf G\_{\mathbf K} }}{\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)} } [1+ {{\mathbf G}}{{\mathbf T}}] \,, \label{eq:KLmatrix3}$$ [] where $$\begin{gathered} \label{eq:KL330} {\mathbf K\_{L,33}}^{(0)} \equiv \frac1{1 - {{\mathbf K\_2}}{{\mathbf G}}} {{\mathbf K\_2}}{{\mathbf G}}{{\mathbf K\_2}}\,,\qquad {{\mathbf T}}\equiv {{\mathbf K\_2}}\frac1{1- {{\mathbf G}}{{\mathbf K\_2}}} \,, \qquad {{ \mathbf G\_{\mathbf K} }}\equiv \frac1{1- {{\mathbf G}}{{\mathbf K\_2}}} {{\mathbf G}}\,.\end{gathered}$$ This decomposition, explained in Ref. , leads to the first appearance of the second geometric function ${{\mathbf G}}=i(2\omega L^3)^{-1} G$, as well of the intermediate infinite-volume quantity ${\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)}= i{\mathcal K\_{\text{df},3}}^{(u,u)}$. Here the subscript “df” stands for divergence free and indicates that kinematic singularities present in the three-to-three scattering amplitude are absent from this quantity. ### Combining and symmetrizing[sec:combine] We now substitute all decompositions into Eq. ([eq:Cno3MLres]) and organize terms by the number ${\mathbf K\_{\mathrm{df}, 3 3}}$ factors that they contain. Beginning with the ${\mathbf K\_{{\mathrm{df}}}}$-independent contributions, note that $ {\mathbf K\_{L,33}}^{(u,u)} \longrightarrow {\mathbf K\_{L,33}}^{(0)}$ in the limit of ${\mathbf K\_{\mathrm{df}, 3 3}}\to 0$. This can be used to show that the factor appearing between ${{\mathbf A}}'^{(u)}\_{L,3}$ and ${{\mathbf A}}^{(u)}\_{L,3}$ in Eq. ([eq:Cno3MLres]) reduces as $${{\mathbf F}}\_{33}^{(0)} \sum\_{n=0}^\infty \left( {\mathbf K\_{L,33}}^{(u,u)} {{\mathbf F}}\_{33}^{(0)} \right)^n \ \ \ \xrightarrow{{\mathbf K\_{{\mathrm{df}}}}\to 0}\ \ \ {{\mathbf F}}\frac1{1- {{\mathbf T}}{{\mathbf F}}} \equiv {{\mathbf Z}}\,. \label{eq:Zdef}$$ It is then straightforward to reduce the two endcaps as well as CL, 0F to reach $$\begin{aligned} {{\mathbf A}}'^{(u) }\_{L,3} & \ \ \ \xrightarrow{{\mathbf K\_{{\mathrm{df}}}}\to 0}\ \ \ {{\mathbf A}}'\_3 - 2 {{\mathbf A}}'^{(s)}\_3 (1- {{\mathbf F}}{{\mathbf T}}) \,, \label{eq:ApL3noKdf\_v2} \\[5pt] {{\mathbf A}}^{(u)}\_{L,3} & \ \ \ \xrightarrow{{\mathbf K\_{{\mathrm{df}}}}\to 0}\ \ \ {{\mathbf A}}\_3 - (1 - {{\mathbf T}}{{\mathbf F}}) 2 {{\mathbf A}}^{(s)}\_3 \,, \label{eq:AL3noKdf\_v2} \\[5pt] C\_{L,0F} - {C}^{[B\_2] }\_{\infty } & \ \ \ \xrightarrow{{\mathbf K\_{{\mathrm{df}}}}\to 0}\ \ \ 2 {{\mathbf A}}'^{(s)}\_3 {{{\mathbf F}}} \left ( {{\mathbf A}}\_3-(1-{{\mathbf T}}{{\mathbf F}}) 2 {{\mathbf A}}^{(s)}\_3 - {{\pmb \sigma^{\dagger\}}}\right) \,. \label{eq:CL0FloKdf}\end{aligned}$$ Here we have expressed the endcaps in terms of ${{\mathbf A}}'\_3 \equiv {{\mathbf A}}'^{(u) }\_{3} + {{\mathbf A}}'^{(s) }\_{3} + {{\mathbf A}}'^{(t) }\_{3} $ and similarly for the unprimed object. This combines the three possible choices of momentum assignment for the external particle that is not attatched to the outermost $\bf 2 \to \bf 2$ insertion. Although we are forced to consider unysmmetrized quantities at various stages of the derivation, it is crucial that the final result should only depend on objects that have exchange symmetry with respect to the momenta of the three identical particles. This is proven explicitly for all systems considered so far in Refs. . Substituting the four relations [Eqs. ([eq:Zdef])-([eq:CL0FloKdf])] into Eq. ([eq:Cno3MLres]) leads to a complicated expression. However, the important part is given by the term containing the two symmetrized endcap factors ${{\mathbf A}}'\_3$ and ${{\mathbf A}}\_3$: $$C\_L^{[B\_2]} - C\_\infty^{[B\_2]} \supset {{\mathbf A}}'\_3 {{\mathbf Z}}{{\mathbf A}}\_3 = {{\mathbf A}}'\_3 \bigg [ {{\mathbf F}}+ {{\mathbf F}}\frac1{1- {\bf K}\_2 (\bf F + \bf G)} {\bf K}\_2 {{\mathbf F}}\bigg] {{\mathbf A}}\_3 \,,$$ where in the equality we have given an alternative, expanded expression for ${{\mathbf Z}}$. Remarkably, this simple result almost captures the full volume dependence at leading order in ${\mathbf K\_{\mathrm{df}, 3 3}}$. The set of remaining contributions modifies the result in two minor ways. First, many of the additional terms can be proven to have only exponentially-suppressed L* dependence and are absorbed into a redefinition of C∞[B2]. Second, the one additional volume cut that survives (after significant reshuffling) is a term that corrects the numerical factor multiplying $ {{\mathbf A}}'\_3 {{\mathbf F}}{{\mathbf A}}\_3$. One finds $$\begin{aligned} C\_L^{[B\_2]} - C\_\infty^{[B\_2]} &= {{\mathbf A}}'\_3 {{\mathbf F}}\_{33} {{\mathbf A}}\_3 + \mathcal O({\mathbf K\_{{\mathrm{df}}}}) \,, \\ {{\mathbf F}}\_{33} &\equiv \frac13 \mathbf F + \mathbf F \frac1{1- {\bf K}\_2 (\bf F + \bf G)} {\bf K}\_2 \mathbf F \,. \end{aligned}$$ [] As we will see below, ${{\mathbf F}}\_{33}$, which combines geometric functions together with factors of ${\mathcal K}\_2$, is the three-particle analog of ${{\mathbf F}}\_2$. It is the central object entering the three-particle quantization condition. We now continue the pattern by considering the next order in ${\mathbf K\_{\mathrm{df}, 3 3}}$. For example, from the decomposition of ${\mathbf K\_{L,33}}^{(u,u)}$ in Eq. ([eq:KLmatrix3]) together with the ${\mathbf K\_{{\mathrm{df}}}}$-independent result [Eq. ([eq:Zdef])] it is straightforward to show that $$\begin{aligned} {{\mathbf F}}\_{33}^{(0)} \sum\_{n=0}^\infty \left( {\mathbf K\_{L,33}}^{(u,u)} {{\mathbf F}}\_{33}^{(0)} \right)^n & = {{\mathbf Z}}+ {{\mathbf Z}}(1 + {{\mathbf T}}{{\mathbf G}}) \, {\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)} \, (1 + {{\mathbf G}}{{\mathbf T}}) {{\mathbf Z}}+ \mathcal O({\mathbf K\_{{\mathrm{df}}}}^2) \,. \label{eq:intermediate}\end{aligned}$$ The next step is to identify the $\mathcal O({\mathbf K\_{{\mathrm{df}}}})$ contributions to ${{\mathbf A}}'^{(u) }\_{L,3}$, ${{\mathbf A}}^{(u)}\_{L,3}$ and CL, 0F − C∞[B2] and assemble all terms to identify the corresponding contribution to CL[B2] − C∞[B2]. However, for the purposes of this review, we think it more instructive to consider a single contribution to CL[B2], given by sandwiching the ${\mathcal O}({\mathbf K\_{{\mathrm{df}}}})$ part of Eq. ([eq:intermediate]) between symmetrized, infinite-volume endcaps, $$\begin{gathered} {{\mathbf A}}'\_3 {{\mathbf Z}}(1 + {{\mathbf T}}{{\mathbf G}}) \, {\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)} \, (1 + {{\mathbf G}}{{\mathbf T}}) {{\mathbf Z}}{{\mathbf A}}\_3 = \\ {{\mathbf A}}'\_3 \bigg [ {{\mathbf F}}+ {{\mathbf F}}\frac1{1- {{\mathbf K\_2}}({{\mathbf F}}+ {{\mathbf G}})} {{\mathbf K\_2}}({{\mathbf F}}+ {{\mathbf G}}) \bigg ] {\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)} \bigg [ {{\mathbf F}}+ ({{\mathbf F}}+ {{\mathbf G}}) {{\mathbf K\_2}}\frac1{1- ({{\mathbf F}}+ {{\mathbf G}}) {{\mathbf K\_2}}} {{\mathbf F}}\bigg ] {{\mathbf A}}\_3 \,.\end{gathered}$$ As above, the inclusion of all other terms leads to only minor modifications to this key result. Again various L-independent terms are absorbed, not only into C∞[B2] but also into $ {{\mathbf A}}'\_3$ and ${{\mathbf A}}\_3$. A new feature that arises here is that the unsymmetrized K matrix, $ {\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)}$, can be replaced with a symmetrized form, up to a correction of the symmetry factor on the isolated ${{\mathbf F}}$ term (${{\mathbf F}}\to {{\mathbf F}}/3$) and a modification of the cuts multiplying $ {\mathbf K\_{\mathrm{df}, 3 3}}^{(u,u)}$. The upshot is that the square-bracketed factors are replaced with ${{\mathbf F}}\_{33}$, the same three-particle volume cut that appeared at the previous order: $$C\_L^{[B\_2]} - C\_\infty^{[B\_2]} = {{\mathbf A}}'\_3 {{\mathbf F}}\_{33} {{\mathbf A}}\_3 + {{\mathbf A}}'\_3 {{\mathbf F}}\_{33} {\mathbf K\_{\mathrm{df}, 3 3}}^{[B\_2]} {{\mathbf F}}\_{33} {{\mathbf A}}\_3 + \mathcal O({\mathbf K\_{{\mathrm{df}}}}^2) \,.$$ [] At this stage, we see the structure emerging and can assign a physical interpretation to the pattern. Within the finite-volume correlator, the three-particle state is created with an insertion of $ {{\mathbf A}}\_3 $. This is equal to a matrix element of σ† and measures the probability amplitude to create a three-particle state from the vacuum.[12](#fn12) The three particles then propagate and rescatter in the box, leading to a pattern of Feynman diagrams that translates into a set of nested geometric series. We organize these in powers of the short-distance three-to-three interaction, ${\mathbf K\_{\mathrm{df}, 3 3}}$. Then at leading order the volume affects arise due to a single insertion of ${{\mathbf F}}\_{33}$. This, in turn, is equal to a factor of ${{\mathbf F}}$ together with a sum over all terns of the form ${{\mathbf F}}{{\mathbf K\_2}}{{\mathbf F}}{{\mathbf K\_2}}{{\mathbf G}}\cdots {{\mathbf F}}{{\mathbf K\_2}}{{\mathbf F}}$ where either ${{\mathbf F}}$ or ${{\mathbf G}}$ can appear between any two factors of ${{\mathbf K\_2}}$. The sum over all such terms represents the fact that any particle pair can scatter (inducing a factor of ${{\mathbf K\_2}}$) and then propagate between consecutive re-scattering events (giving ${{\mathbf F}}$), or alternatively exchange the scattering pair (leading to ${{\mathbf G}}$). Finally the term that is linear in ${\mathbf K\_{\mathrm{df}, 3 3}}$ is governed by this same structure, together with the observation that a short-distance three-particle interaction may arise anywhere in the series of two-particle scattering events. From the physical intuition it is perhaps not so surprising that the same pattern continues to all orders in ${\mathbf K\_{{\mathrm{df}}}}$. This is proven explicitly in Refs. and. In addition, the inclusion of the three-particle BS kernels turns out to only modify the definition of the divergence-free K matrix. This can be accommodated by replacing ${\mathbf K\_{\mathrm{df}, 3 3}}^{[B\_2]}$ with ${\mathbf K\_{\mathrm{df}, 3 3}}$. Putting all this together leads to the main result of Ref. $$C\_L^{ } - C\_\infty^{ } = \sum\_{n=0}^\infty {{\mathbf A}}'\_3 {{\mathbf F}}\_{33} \Big [ {\mathbf K\_{\mathrm{df}, 3 3}}^{ } {{\mathbf F}}\_{33} \Big]^n {{\mathbf A}}\_3 = {{\mathbf A}}'\_3 {{\mathbf F}}\_{33} \frac{1}{1- {\mathbf K\_{\mathrm{df}, 3 3}}^{ } {{\mathbf F}}\_{33} } {{\mathbf A}}\_3 \,.$$ Thus the poles in the finite-volume correlator occur whenever the matrix appearing between $ {{\mathbf A}}'\_3$ and $ {{\mathbf A}}\_3$ has a divergent eigenvalue. We deduce that, for fixed values of $\vec P$ and L, the finite-volume energy spectrum in the region m < E\* < 5m is given (up to e− mL corrections) by all solutions in E to the quantization condition $$\det\_{\vec k' \ell' m'; \vec k \ell m} \big [\mathcal K\_{\text{df},3}(E^\) + F\_3(E, \vec P, L)^{-1} \big ] = 0 \,. \label{eq:QC}$$ [] Here we have returned to the non-bold notation, using ${{\mathbf F}}\_{33} = i F\_3$ and ${\mathbf K\_{\mathrm{df}, 3 3}}=i{\mathcal K\_{\text{df},3}}$, that we will use for the remainder of the discussion. ### Relating $\mathcal K\_{\mathrm{df},3}$ to $\mathcal M\_{3}$[sec:KtoM] In the previous sections we have related $\mathcal K\_{\text{df},3}(E^\) $ to the finite-volume energy spectrum. This object can, in principle, be constrained by calculating finite-volume energies, for example from the Euclidean-time decay of correlators calculated numerically using LQCD, and then applying Eq. ([eq:QC]). Of course, this is only of interest if $\mathcal K\_{\text{df},3}(E^\) $ can be related to infinite-volume observables. Indeed, as was shown in Ref. , this modified K matrix is related to the three-to-three scattering amplitude via an integral equation that depends only on known functions as well as the on-shell two-particle scattering amplitude. The relation has a number of desirable features. The equations are defined at fixed energy, meaning that $\mathcal K\_{\text{df},3}(E^\) $ is only required for E\* where the physical scattering amplitude is to be determined. In addition once the quantization condition is fixed, there is no ambiguity or scheme dependence in the relation between $\mathcal K\_{\text{df},3}$ and $\mathcal M\_3$. Finally the $\mathcal K\_{\text{df},3}$ to $\mathcal M\_3$ relation manifestly encodes the complicated unitarity constraints of the three-particle scattering amplitude . The relation is derived in a slightly round-about way. In particular we show in Ref. that one can define an alternative finite-volume correlation function, $\mathcal M\_{3L}$, that becomes the physical scattering amplitude in a carefully constructed L → ∞ limit. This new correlator differs from CL only in the interpolating fields. As a result it admits a similar skeleton expansion and a similar decomposition into geometric functions. In Ref. we show that $$\mathcal M\_{3L}[\mathcal K\_{\text{df},3}, \mathcal K\_2] \equiv \mathcal S \bigg[ \mathcal D\_L^{(u,u)} - \mathcal L\_L^{(u)} \frac{1}{1 + \mathcal K\_{\text{df},3} F\_3 } \mathcal K\_{\text{df},3} \mathcal R\_L^{(u)} \bigg ] \,, \label{eq:M3Lres}$$ where $ \mathcal D\_L^{(u,u)} $, $ \mathcal L\_L^{(u)} $ and $\mathcal R\_L^{(u)} $ are known explicitly, and are closely related to F3, while ${\mathcal S}$ indicates symmetrization. As an aside, we observe from Eq. ([eq:M3Lres]) that we can use ${\mathcal M}\_{3L}$ instead of CL in order to derive the quantization condition, as its poles lie at the same positions. Indeed, the unsymmetrized quantity in square braces is very similar to the dimer-particle scattering amplitude used in the alternative approaches to deriving the quantization condition discussed below. We now obtain ${\mathcal M}\_3$ by taking a judiciously chosen L → ∞ limit of ${\mathcal M}\_{3L}$. This requires that factors of iε in the pole terms be first put back in, so the L → ∞ limit is well defined. Thus we have $$\mathcal M\_{3}(\mathcal K\_{\text{df},3}, \mathcal K\_2) = \lim\_{\epsilon \to 0} \lim\_{L \to \infty} \mathcal M\_{3,L}(\mathcal K\_{\text{df},3}, \mathcal K\_2) \,,$$ which gives a set of purely infinite-volume integral equations, given explicitly in Ref. . These map the real, short-distance three-body quantity ${\mathcal K\_{\text{df},3}}$ to the complex, unitary, relativistic three-to-three scattering amplitude. Truncating the quantization condition ------------------------------------- The quantization condition Eq. ([eq:QC]) is a formal result, because the determinant runs over an infinite-dimensional matrix space. This is exactly as in the two-particle case and, in direct analogy to that case, truncating $\mathcal K\_2$ and $\mathcal K\_{\text{df},3}$ to vanish above some ℓmax is sufficient to reduce all matrices in the quantization condition to have finite dimension . Specifically, the matrices then have dimension $[(2 \ell\_{\text{max}} + 1) \times N\_{\vec k}]$ where $N\_{\vec k}$ is the number of finite-volume momenta, $\vec k$, for which the cutoff function, $H(\vec k)$, is non-zero. The simplest approximation is to take ℓmax = 0: the s-wave approximation. This is the approximation used from the beginning in the alternative approaches described below. It assumes that $\mathcal K\_2$ and the dimer within $\mathcal K\_{\text{df},3}$ are both dominated by s-wave interactions. Given this truncation, $\mathcal K\_2$ reduces to a single function of the two-particle CM energy, whereas $\mathcal K\_{\text{df},3}$ retains dependence on the spectator momenta $\vec k'^\$ and $\vec k^\$. This is the form of the RFT quantization condition that is used in the comparison with the results from alternate approaches; it is given explicitly in Eq. ([eq:F3s]) below. The s-wave approximation is well motivated at energies close to the three-particle threshold, since higher waves are then suppressed by factors of (qk\)ℓ. To study this systematically, one expands ${\mathcal K}\_2$ and ${\mathcal K\_{\text{df},3}}$ about threshold in powers of s* − 9m2 and related quantities. For ${\mathcal K}\_2$ this leads to the effective range expansion, with the leading term being the scattering length. The expansion of ${\mathcal K\_{\text{df},3}}$ is constrained because it is relativistically invariant, symmetric under initial and final particle interchange, and time reversal invariant . Using these symmetries, one can show that the first two terms in the expansion are not only pure s-wave but also isotropic, i.e. independent of the spectator momenta . At third order one must include d-wave contributions in both ${\mathcal K}\_2$ and ${\mathcal K\_{\text{df},3}}$ . The s-wave approximation plus isotropic ${\mathcal K\_{\text{df},3}}$ is referred to as the isotropic approximation. In this most extreme truncation ${\mathcal K\_{\text{df},3}}$ depends only on E\, so we can write $$\mathcal K\_{\text{df},3} = \delta\_{\ell,0} \delta\_{m,0} \vert 1 \rangle \mathcal K^{\text{iso}}\_{\text{df},3}(E^\) \langle 1 \vert \,,$$ where \|1⟩ is an unnormalized vector with unit entry for every active spectator momentum. If one further restricts to the A1 irreducible representation (irrep) of the cubic group, then, as shown in Ref. , all nontrivial solutions to the quantization condition satisfy $$\mathcal K^{\text{iso}}\_{\text{df},3}(E^\) = -1/F\_3^{\text{iso}}(E, \vec P, L) \,,\quad F\_3^{\text{iso}} \equiv \langle 1 \vert F\_{3,s} \vert 1 \rangle\,,$$ where F3, s* is the form of F3 after s-wave truncation [see Eq. ([eq:F3s])]. Thus, in this approximation, we recover a one-to-one correspondence between finite-volume energy levels and the value of $\mathcal K^{\text{iso}}\_{\text{df},3}(E^\) $. This approximation has been studied numerically in Ref. , and some of the results are shown in Sec. [sec:num] below. Analytic investigations of the RFT quantization condition --------------------------------------------------------- In this subsection we describe two analytic investigations of the three-particle quantization condition, Eq. ([eq:QC]). The main aim is to check the formalism by comparing to known results in special limits, but a side-benefit is that some new analytic results are obtained. ### 1/L* expansion Without interactions, the lowest energy state of three particles with $\vec P=0$ has energy 3m. Turning on interactions, this level will shift to E0(L). For sufficiently large L, E0(L) − 3m can be expanded in powers of 1/L, the so-called threshold expansion. The leading term scales as 1/L3, corresponding to the probability of two particles to overlap, while three-particle interactions enter first at 1/L6. This expansion was worked out previously, using NRQM, up to ${\mathcal O}(1/L^7)$ . We determined the threshold expansion, starting from Eq. ([eq:QC]), in Ref. . We found[13](#fn13) $$\begin{gathered} E\_0(L) = 3m + \frac{12 \pi a}{m L^3} \bigg \{ 1 - \mathcal I \frac{a}{\pi L} + \left(\frac{a}{\pi L}\right)^{\!2} \left({\cal I}^2+{\cal J}\right) + \frac{64\pi^2 a^2 }{mL^3} \mathcal C\_3+\frac{3\pi a}{m^2 L^3} + \frac{6\pi ra^2}{L^3} \\ + \left(\frac{a}{\pi L}\right)^{\!3} \bigg (\! -{\cal I}^3 + {\cal I}{\cal J} + 15 {\cal K} +\frac{16 \pi^3}{3} (3\sqrt 3 - 4\pi) \, \log \! \bigg (\frac{mL}{2\pi} \bigg ) + \mathcal C' \bigg ) \bigg \} - \frac{ \mathcal M\_{3,\mathrm{thr}}}{48 m^3 L^6} + \mathcal O \! \left(\frac{1}{L^7} \right) \,, \label{eq:thresh\_res}\end{gathered}$$ where we have introduced the following geometric constants: $\mathcal I=-8.914$, $\mathcal J=16.532$, $\mathcal K=8.402$, $\mathcal C\_3 = -0.05806$, $\mathcal C' = 2052$, whose origin is explained in Ref. . The scattering parameters that enter here are the scattering length, a, the effective range, r, and the threshold three-to-three scattering amplitude $\mathcal M\_{3,\mathrm{thr}}$. The latter has a somewhat subtle definition, due to the fact that the full scattering amplitude diverges at threshold and thus requires a subtraction. Again see Ref. for details. Note that the leading term is three times that found in Eq. ([eq:HYth]) for the two-particle system, as expected since there are three pairs of particles that can interact. The result of Eq. agrees with the results of Refs. for the terms through $\mathcal O(1/L^5)$. This provides a strong check on the formalism. Relativistic effects enter at $\mathcal O(1/L^6)$ and here no general check is available. However the logL term is universal and its coefficient also agrees with earlier work. To check the remaining 1/L6 terms, in Refs. we calculated E0(L) in λϕ4 theory up to ${\mathcal O}(\lambda^4)$, finding complete agreement with Eq. ([eq:threshres]). Working at this order checks all the terms entering at 1/L6. We also note that, for weakly interacting systems, the threshold expansion might provide a partial alternative to the full quantization condition. By fitting the L dependence of the threshold state at many volumes, one could in principle extract $\mathcal M\_{3,\mathrm{thr}}$ and thereby determine the near-threshold scattering amplitude. This approach has been followed successfully in λϕ4 theory in the recent work of Ref. . ### Volume-dependence of a three-body bound state Our second analytic result concerns the volume-dependence of a non-relativistic three-scalar Efimov bound state in the unitarity limit for two-particle interactions, i.e. with 1/a → 0. This was studied in Ref. using non-relativistic quantum mechanics in a finite-volume. The authors found that, when the infinite volume system contains a bound state, then the lowest lying state in finite volume has energy $$E\_B(L) = 3m - \frac{\kappa^2}{m} + \Delta E(L) \,,$$ where κ is the binding momentum, and $$\label{eq:MRRresult} \Delta E(L) = c \vert A \vert^2 \frac{\kappa^2}{m} \frac{1}{(\kappa L)^{3/2}} e^{- 2 \kappa L/\sqrt{3} } + \cdots \,.$$ Here the ellipsis indicates terms suppressed by additional factors of κ2/m2 or 1/(κL) as well as faster-decaying exponentials. The result depends on c ≃ − 96.351, a known geometric constant, and \|A\|2. The latter is a normalization correction that arises because the asymptotic wave-function is not a strict solution to the Schrödinger equation. It is expected to be close to unity when the pairwise interactions of the theory are well-described by a short-range potential. The generic relativistic quantization condition presented above, Eq. ([eq:QC]), holds for systems with a three-particle bound state, as long as there are no bound dimers. Indeed, in our numerical implementation in the isotropic approximation, we have found that the formalism can support such bound states . Assuming the existence of such a state (i.e. that there is a subthreshold pole in $\mathcal M\_3$), in Ref. we were able to provide another check on the quantization condition by reproducing the coefficient, power-law envelope and exponential decay given by Eq. ([eq:MRRresult]). In addition we extended the result to non-zero momentum in the finite-volume frame, and found that the moving-frame energy is shifted according to $$\Delta E (\vec P, L) = f\_3 \! \big [\vec n \big] \, \Delta E(L) + \cdots \,,\qquad f\_3\! \big [{\vec n} \big] = \frac{1}{6} \sum\_{\hat s } e^{i (2\pi/3) \hat s \cdot \vec n }\,, \label{eq:DEP}$$ where $\vec n = L \vec P/(2 \pi)$ and the sum runs over the six unit vectors pointing along the finite-volume axes. An interesting corollary is that the leading-order volume shift vanishes for $\vec n = (0,1,1)$. ALTERNATIVE APPROACHES ======================= As noted in the Introduction, two other approaches for deriving the three-particle quantization condition have been used, and in this section we describe them, quote the resulting form for the quantization condition, and explain in what ways these approaches agree with and differ from the generic RFT approach described in the previous section. In particular, we present new results concerning the analytic relation between the three approaches in the s-wave approximation, which is the only case that has been worked out explicitly within the new approaches. We discuss the two other approaches in historical order, beginning with that based on non-relativistic effective field theory (NREFT), introduced in Refs. , and then describing the approach based on extending the form of amplitudes required to satisfy unitarity to finite volume . We refer to the latter as the “finite-volume unitarity” (FVU) approach. NREFT approach -------------- NREFT is applicable when the momenta of all particles are in the non-relativistic domain, $\|\vec p \, \| \ll m$. In this regime there is no particle creation, which greatly simplifies the diagrammatic analysis. Examples where it is directly applicable in a three-particle context include three pions close to threshold in an isospin-symmetric world (so that G-parity forbids three-to-two transitions) and three nucleons close to threshold. The first calculations using NREFT to determine finite-volume properties of three-particle systems were numerical calculations for the triton and Efimov states . This approach was then used in Refs. to develop the three-particle quantization condition, which we review in the following. The authors consider a theory of identical scalars (i.e. the same setup as used in the REFT analysis presented earlier) for which the NREFT Lagrangian is[14](#fn14) $$\begin{aligned} {\mathcal L}&= \psi^\dagger \left(i \partial\_0 + \frac{\nabla^2}{2m}\right) \psi + {\mathcal L}\_2 + {\mathcal L}\_3\,, \label{eq:L} \\ {\mathcal L}\_2 &= - \tfrac12 C\_0 \psi^\dagger\psi^\dagger \psi \psi + \tfrac14 C\_2 \left(\psi^\dagger\overleftrightarrow{\nabla}^2 \psi^\dagger \psi \psi + \textrm{h.c.}\right) + \dots\,, \label{eq:L2} \\ {\mathcal L}\_3 &= - \tfrac16 D\_0\psi^\dagger\psi^\dagger \psi^\dagger\psi\psi \psi + \tfrac1{12} D\_2 \left(\psi^\dagger\psi^\dagger\overleftrightarrow{\nabla}^2 \psi^\dagger \psi \psi \psi + \textrm{h.c.}\right) + \dots\,, \label{eq:L3}\end{aligned}$$ [] where h.c. indicates hermitian conjugate, $\overleftrightarrow{\nabla}= (\overrightarrow{\nabla}-\overleftarrow{\nabla})/2$ is the Galilean-invariant derivative, and derivatives act only on adjacent objects.The field ψ only destroys particles, since there is no corresponding antiparticle in the theory. NREFT is an expansion in $\|\vec p \,\|^2/m^2$, and thus interaction terms are classified by the number of derivatives, with the constraint from Bose symmetry that only an even number is allowed. Thus the ellipses in Eqs. ([eq:L2]) and ([eq:L3]) indicate terms with four or more derivatives, at which order arise the first terms that lead to d-wave interactions between pairs of particles. The effects of virtual particle-antiparticle pairs in the underlying relativistic theory are subsumed into the a priori unknown (dimensionful) constants Ci and Di. We refer to these generically as LECs—low-energy constants. This theory has been extensively studied in infinite volume, e.g. in its application to the three nucleon system (where the fields become fermionic and have a spin index). For a review, see Ref. . The LECs are to be determined by solving the two- and three-particle scattering and bound-state problems (involving the solution of integral equations) and comparing to experimental results for phase shifts and bound-state energies. One must choose a regularization, a topic with an extensive literature given the subtleties that arise in the presence of large scattering lengths . For numerical applications, however, the standard choice is a hard cutoff, $\|\vec p \,\| < \Lambda$, and this is what is used in Refs. . The overall strategy employed in this approach is as follows. The NREFT is to be solved in finite volume, working first with only the leading order couplings C0 and D0, and including higher-order terms as needed. These LECs are to be determined by comparing the theoretically predicted spectrum to that obtained in a lattice calculation. In a second step, the NREFT is then solved in infinite-volume, predicting the three-particle scattering amplitudes and bound-state properties. This approach makes use of a crucial property of the LECs, namely that they are expected to be volume-independent, since they arise from integrating out short-distance physics. Thus the values obtained in finite volume can be applied unchanged in the infinite-volume calculations. This two-step strategy is similar to that used in the RFT analysis presented earlier, where the intermediate, cutoff-dependent quantity ${\mathcal K\_{\text{df},3}}$ was needed, and infinite-volume quantities were obtained by solving integral equations. In the NREFT approach the intermediate quantities are the LECs, which are also cutoff dependent, and thus not directly physical. What the NREFT approach provides in addition is a systematic power-counting scheme, valid as long as one works in the NR regime. [] An important technical point discussed in Ref. concerns a class of higher-order terms in ${\mathcal L}\_2$ and ${\mathcal L}\_3$ that lead to vanishing on-shell contributions to physical scattering amplitudes at tree level. An example from ${\mathcal L}\_2$ is a term leading to a vertex proportional to $(\vec k^2-\vec p^2)^2$, where $\vec k$ and $\vec p$ are the relative momenta between the two particles in initial and final states, respectively. It is argued that one can choose the regularization such that these terms do not contribute to physical quantities even when included in loop diagrams, and thus that they can be dropped from the beginning. In this way the intermediate quantities in the NREFT approach provide a complete description of the physical amplitudes. This is the analog here of the result in the RFT formalism that the intermediate quantity ${\mathcal K\_{\text{df},3}}$ is an on-shell amplitude. [] The restriction to “physical” LECs is implemented in Ref. using auxiliary dimer fields. These are simply a technical device in which a composite field is introduced for two of the particles for each choice of their relative angular momentum (which here is constrained to be even). Integrating out the infinite tower of dimer fields leads back to the original Lagrangian, Eq. ([eq:L]), but with only the physical LECs. The angular momentum of the dimer corresponds exactly to the indices ℓ and m in the RFT approach. Details of the implementation of the dimer fields can be found in Ref. . [] The NREFT quantization condition has been worked out explicitly so far only for the case of an s-wave dimer. This is the analog of truncating the RFT formalism to ℓ = 0, an approximation described in Sec. [sec:trunc] above. This approximation requires that the two-particle scattering amplitude vanish for all higher waves, and that the three-particle interaction does not couple to higher waves in a pair. A further restriction introduced to simplify the derivation is that $\vec P=0$. [fig:NREFT] ### Two-particle quantization condition The first step in the derivation of the three-particle quantization condition is to determine the dimer propagator in finite volume, denoted $\tau\_L(\vec k)$ (with dependence on the total energy E kept implicit). Here $\vec k$ is the spectator momentum, $\vec k$, which determines the dimer momentum to be $\vec P-\vec k=-\vec k$. τL is proportional to the finite-volume scattering amplitude, ${\mathcal M}\_{2L}$, discussed in Sec. [sec:QC2]. If we denote the s-wave component of the latter quantity, evaluated in the NR regime,[15](#fn15) by ${\mathcal M}\_{2L,s}^{{\rm NR}}(P-k)$, where the argument denotes the dimer four-momentum, then the precise relation is $$32 \pi m \tau\_L(\vec k) = {\mathcal M}\_{2L,s}^{{\rm NR}}(P-k)\,. \label{eq:tauLtoM2L}$$ The dimer propagator is given by the diagrams shown in Figure [fig:NREFT](a). Summing the geometric series leads to $$\tau\_L(\vec k)^{-1} = f\_\tau(q\_{k}^{\2}) + \frac{4\pi}{L^3} \sum\_{\vec a} \frac1{m E\_{\rm NR} - \vec k^2 - \vec a^2 - \vec k \cdot \vec a}\,. \label{eq:tauL}$$ [] Here f*τ arises from the vertices in ${\mathcal L}\_2$, and thus is a known function of the Ci. It is proportional to the inverse of the BS kernel. It depends on the squared relative c.m. momentum of the particles in the dimer, $$q\_{k}^{\2} = \tfrac14 \left[(E-\omega\_k)^2 - \vec k^2 - 4 m^2\right]\,. \label{eq:q2kstar}$$ Finally, $E\_{\rm NR}=E-3m$ is the nonrelativistic energy, while $\vec a$ is summed over the allowed finite-volume values. We see here how the NR limit simplifies the analysis compared to that outlined in Sec. [sec:QC2]. Here there are only s-channel loops, and these contain only two particles. All the other loops in RFT collapse to the point-like interactions parametrized by the LECs. In particular, the BS kernel required in Sec. [sec:QC2] is replaced here by the function 1/[32πmf*τ]. The nontrivial effort required to show that this kernel has only exponentially suppressed volume dependence in the RFT approach is replaced here by the assumed general result from EFT that LECs are independent of volume.[16](#fn16) With this result in hand, one simply determines the volume dependence implicitly by calculating the dimer propagator with the loop integrals replaced by sums. [] The function fτ can be determined by considering the dimer propagator in infinite volume, $\tau(\vec k)$, obtained from Eq. ([eq:tauL]) by changing the sum into an integral with an iε prescription for the pole. Regulating the ultraviolet divergence in some manner, the integral evaluates to $$I\_{\epsilon}(q\_k^{\2}) \equiv 4\pi\,\int\_{\vec a} \frac1{m E\_{\text{NR}} - \vec k^2 - \vec a^2 - \vec k \cdot \vec a + i \epsilon} = 4\pi \int\_{\vec a^\} \frac1{\vec q\_{k}^{\2} - a^{\2} + i \epsilon} = \sqrt{-q\_{k}^{\2}} + I\_{{\rm PV}}(q\_k^{\2})\,. \label{eq:INR}$$ Here $\int\_{\vec a}\equiv \int d^3a/(2\pi)^3$, and $\vec a^\$ is the result of boosting $\vec a$ to the dimer c.m. frame. $I\_{{\rm PV}}$ is the same integral except defined using the principal-value (PV) pole prescription. Thus $I\_{{\rm PV}}$ is a real, analytic function of q*k\* 2, which in fact evaluates to a constant. The square-root in Eq. ([eq:INR]) is defined to have a negative imaginary part above threshold. Using these results we obtain $$\tau(\vec k)^{-1} = f\_\tau(q\_{k}^{\2}) + I\_{{\rm PV}}+ \sqrt{-q\_{k}^{\2}}\,,$$ which, when compared to the known form of the scattering amplitude [see Eq. ([eq:unitarity]) above][17](#fn17) $$\tau(\vec k)^{-1} = {32\pi m}\left[{\mathcal M}\_{2,s}^{{\rm NR}}(P-k)\right]^{-1} = 32\pi m \left[{\mathcal K}\_{2,s}^{{\rm NR}}(q\_{k}^\)\right]^{-1} + \sqrt{-q\_{k}^{\2}}\,,$$ leads to the conclusion that $$f\_\tau(q\_{k}^{\2}) + I\_{{\rm PV}}= 32\pi m\left[ {\mathcal K}\_{2,s}^{{\rm NR}}(q\_{k}^\)\right]^{-1}\,. \label{eq:ftaures}$$ This shows explicitly how the LECs Ci, contained in fτ, are related to the physical quantity ${\mathcal K}\_{2,s}$, albeit in a cutoff dependent manner.[18](#fn18) Combining Eqs. ([eq:tauL]) and ([eq:ftaures]) we obtain the final result for τL, $$\begin{aligned} \left[{32\pi m} \tau\_L(\vec k)\right]^{-1} &= {\mathcal K}\_{2,s}^{{\rm NR}}(q\_{k}^\)^{-1} + \frac1{8m} \left[\frac1{L^3}\sum\_{\vec a} - {{\rm PV}}\,\int\_{\vec a}\right] \frac{1}{m E\_{\rm NR} - \vec k^2 - \vec a^2 -\vec k\cdot \vec a}\,, \\ &= {\mathcal K}\_{2,s}^{{\rm NR}}(q\_{k}^\)^{-1} + F\_{2,s}^{{\rm NR}}(P-k,L)\,. \label{eq:tauLfinal}\end{aligned}$$ The first equality is our preferred way of writing Eq. (3.2) of Ref. , as it shows that the volume-dependence of τL arises from a sum-integral difference, just as in the analysis in Sec. [sec:QC2].[19](#fn19) To obtain the second equality, we note that the sum-integral difference is simply the s-wave component of F2(P − k, L), Eq. ([eq:Fdef]), evaluated in the NR regime, and with the iε pole prescription replaced by the PV prescription. Thus we call it $F\_{2,s}^{{\rm NR}}$, which we emphasize is a real quantity. The two-particle quantization condition can now be obtained as an intermediate result. Energy levels are given by the positions of poles in ${\mathcal M}\_{2L}$, leading to the algebraic result $$\tau\_L^{-1} = 0 \ \ \Rightarrow \ \ ({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + F\_{2,s}^{{\rm NR}}= 0\,.$$ This is equivalent to the result derived in Sec. [sec:QC2], Eq. ([eq:QC2K]), when one keeps only the s-wave component and works in the NR regime. [] ### Three-particle quantization condition This is derived in Ref. by considering a quantity, $\widetilde {\mathcal M}\_{3L}$, the particle-dimer scattering amplitude. This is closely related to the s-wave restriction of the finite-volume three-particle amplitude, ${\mathcal M}\_{3L}$, introduced in Sec. [sec:KtoM].[20](#fn20) To obtain ${\mathcal M}\_{3L}$ from $\widetilde {\mathcal M}\_{3L}$, one adds vertices at each end connecting the dimer to two particles, and then symmetrizes. This implies that, as for ${\mathcal M}\_{3L}$, the poles of $\widetilde {\mathcal M}\_{3L}$ occur at the energies of three-particle finite-volume states, so that it is a good quantity to consider to derive the quantization condition. In NREFT, $\widetilde {\mathcal M}\_{3L}$ satisfies $$\widetilde{\mathcal M}\_{3L;pk} = Z\_{pk} + \frac{8\pi}{L^3} \sum\_{\vec q} Z\_{pq}\tau\_L(\vec q) \widetilde{\mathcal M}\_{3L;qk}\,, \label{eq:M3Lt}$$ [] which is shown schematically in Figure [fig:NREFT](b). Here, the subscripts p, q and k are shorthands for the corresponding spectator momenta, while the kernel Zpq, shown in Figure [fig:NREFT](c), is $$Z\_{pq} = Z^0\_{pq} + \frac{H\_0(\Lambda)}{\Lambda^2} + \dots \,, \qquad Z^0\_{pq} = \frac1{\vec p^{\,2} + \vec q^{\,2} +\vec p \cdot \vec q - m E\_{\text{NR}}} \,. \label{eq:Z}$$ Here H0 is a dimensionless constant proportional to D0/C02, Λ is the hard cutoff on the sums, and the ellipsis represents contributions from higher order terms in ${\mathcal L}\_3$, proportional to D2, etc. To use Eq. ([eq:M3Lt]) one assumes that ${\mathcal K}\_{2,s}$ has been determined using the two-particle quantization condition, and thus that τL is known. Since the sum over $\vec q$ is cut off, Eq. ([eq:M3Lt]) is a finite matrix equation for $\widetilde {\mathcal M}\_{3L}$, which can be solved numerically for a given choice of the LECs. One then adjusts the LECs until the finite-volume spectrum determined from a calculation in the underlying theory (i.e. lattice QCD if considering the 3π+ system) matches that given by Eq. ([eq:M3Lt]). In practice, one should project onto irreps (irreducible representations) of the symmetry group of the cubic lattice, as described in Ref. , which allows a more explicit formula for the quantization condition to be given. We will, instead, provide an alternative explicit formula in the following, one that shows more clearly the relationship to the RFT result derived in the previous section. Once one has determined the LECs as just described, a second step is required to predict the infinite-volume scattering amplitude ${\mathcal M}\_3$ and, from this, the properties of any bound states and three-particle resonances. In the NREFT approach this step is simple: one uses Eq. ([eq:M3Lt]) but with τL replaced by τ and the sum replaced by an integral with an iε pole-prescription. This leads to the standard NREFT integral equation for three-particle scattering, reviewed, for example, in Ref. . This step is the analog of the integral equation relating ${\mathcal K\_{\text{df},3}}$ to ${\mathcal M}\_3$ described in Sec. [sec:KtoM]. We now comment briefly on the derivation of Eq. ([eq:M3Lt]). This is conceptually just as straightforward as for two particles. This is because, in NREFT, all loops in a three-particle amplitude contain three particles, in constrast to a generic QFT in which (with a Z2 symmetry) one can have five, seven, etc. This allows one to calculate all the diagrams explicitly, without introducing auxiliary objects such as the BS kernels. In finite volume one simply replaces the integrals in these loops with momentum sums (after the time-component integral is done). LECs are again volume-independent, and summing all diagrams leads to Eq. ([eq:M3Lt]). ### Relation to RFT approach It has been shown in Ref. that the NREFT quantization condition described implicitly above, and the RFT quantization condition of Eq. ([eq:QC]), are algebraically equivalent when only the s-wave dimer is included in the latter and if only the leading order, momentum-independent, two- and three-particle interaction terms are kept, i.e. if the only nonvanishing LECs are C0 and D0. Here we give an alternative derivation of this result that is more direct and explicit. We begin by noting that Z0 is simply related to the switch factor G given in Eq. ([eq:G]). Denoting the s-wave (ℓʹ = ℓ = 0) part of G in the NR regime by $G\_s^{{\rm NR}}$, one can easily show that $$Z^0\_{pq} = - 4 m L^3G\_{s,pk}^{{\rm NR}}\,. \label{eq:GtoZ}$$ Here we have used the fact that the cutoff functions $H(\vec k)$ are unity to all orders in the NR expansion. To simplify subsequent manipulations we introduce the definitions $$\overline {\mathcal M}\_{3L} \equiv \frac{\widetilde {\mathcal M}\_{3L}}{4 m L^3}\ \ {\rm and} \ \ \overline H \equiv \frac1{4 m L^3} \left(\frac{H\_0(\Lambda)}{\Lambda^2} + \dots \right) = \overline H\_0 + \dots \,. \label{eq:Hbar}$$ In terms of these quantities Eq. ([eq:M3Lt]) can be rewritten as $$\overline {\mathcal M}\_{3L} = ( -G\_s^{{\rm NR}}+ \overline H) + (-G\_s^{{\rm NR}}+\overline H) {\mathcal M}\_{2L,s}^{{\rm NR}}\overline {\mathcal M}\_{3L}\,, \label{eq:M3Lbar}$$ where we have also used Eq. ([eq:tauLtoM2L]). This is a matrix equation, in which all spectator-momentum indices are implicit. Solving, we find $$\begin{aligned} \overline {\mathcal M}\_{3L} &= \frac 1{1 - (-G\_s^{{\rm NR}}+ \overline H) {\mathcal M}\_{2L,s}^{{\rm NR}}} (-G\_s^{{\rm NR}}+ \overline H)\,,\end{aligned}$$ which has a pole whenever $$\det\left[1 - (-G\_s^{{\rm NR}}+\overline H) {\mathcal M}\_{2L,s}^{{\rm NR}}\right] = 0 \,.$$ This can be rewritten using Eqs. ([eq:tauLtoM2L]) and ([eq:tauLfinal]) as $$\det\left[({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + F\_{s}^{{\rm NR}}+ G\_s^{{\rm NR}}-\overline H \right] = 0 \,. \label{eq:QC3NR}$$ [] Here, as in Sec. [sec:QC3], we have elevated ${\mathcal K}\_{2,s}^{{\rm NR}}$ into a diagonal matrix with entries ${\mathcal K}\_{2,s}^{{\rm NR}}(q\_k^\)$, while, following Eq. ([eq:F]), we denote the matrix form of $F\_{2,s}^{{\rm NR}}$ by $F\_{s}^{{\rm NR}}$. Equation ([eq:QC3NR]) is the NREFT quantization condition of Refs. expressed in notation similar to that of Ref. . Note that the expansion of ${\mathcal K}\_{2,s}$ and $\overline H$ in powers of momentum has not been truncated at this stage, i.e. all the C*i and Di are still included. The determinant is finite-dimensional because of the hard cutoff applied to the implicit spectator-momentum indices. We now compare this to the quantization condition of Ref. , given in Eq. ([eq:QC]). In the s-wave approximation, this has the form $$\det\left[F\_{3,s}^{-1} + {\mathcal K\_{\text{df},3,s}}\right] = 0\,, \quad F\_{3,s} = \frac{F\_{s}}{2\omega L^3} \left[ \frac13 - \frac1{{\mathcal K}\_{2,s}^{-1} + F\_{s} + G\_s} F\_{2,s}\right]\,. \label{eq:F3s}$$ Here the subscript s indicates keeping only ℓʹ = ℓ = 0 contributions, so all quantities are matrices with only spectator-momentum indices. Note that, at this stage, the NR limit has not been taken, so the sums are cut off by the functions $H(\vec k)$ in Fs [see Eq. ([eq:F])] and Gs [see Eq. ([eq:G])]. By straightforward algebraic manipulations, Eq. ([eq:F3s]) can be written in a form that looks similar to the NREFT result, Eq. ([eq:QC3NR]): $$\det\left[{\mathcal K}\_{2,s}^{-1} + F\_{s} + G\_s - \overline H^R\right]=0\,, \quad \overline H^R = -({\mathcal K}\_{2,s}^{-1} + G\_s - 2F\_{s}) {\mathcal K\_{\text{df},3,s}}\frac{F\_{s}}{6\omega L^3} \,. \label{eq:HSQC2}$$ [] Thus for the two quantization conditions to agree three conditions must be satisfied: (i) we must consider Eq. ([eq:HSQC2]) in the NR regime (so that $G\_s\to G\_s^{{\rm NR}}$, etc.), (ii) we must use a hard cutoff in this equation, and (iii) we must demonstrate $\overline H = \overline H^R$. We consider these requirements for equivalence in turn. The spectator momenta in Eq. ([eq:HSQC2]) run up to a smooth (rather than hard) cutoff at ΛR ∼ m. In principle, one could reduce ΛR into the NR regime so that $G\_{s} \to G\_{s}^{{\rm NR}}$ etc. Then, setting aside the issue of $\overline H$ vs. $\overline H^R$, the only difference between the results would be that between a hard and a smooth cutoff. In practice, however, reducing ΛR into the NR regime is problematic. Present calculations using LQCD are done with mπL ∼ 4 − 6 and $m\_\pi \approx m\_\pi^{\rm phys}$. With these parameters, even the first excited state lies outside the NR regime: $${E\_1}/{m\_\pi} = \sqrt{1 + [{2\pi}/({m\_\pi L})]^2} \sim 1.5-1.9 \,. \label{eq:relkin}$$ Thus a practical quantization condition should have its cutoff at a relativistic energy and include relativistic kinematics. In this regard, we note that Ref. argue that the NREFT quantization condition can be “relativized” by including the correct kinematical factors. Indeed, we see that this can be accomplished in the present instance by replacing each of the NR quantities by their relativistic counterparts, as introduced in the RFT approach. [] The final requirement for the agreement between the two quantization conditions, $\overline H=\overline H^R$, is, at first sight, more problematic. $L^3 \overline H$ is an infinite-volume quantity, while $L^3\overline H^R$ is not, since it contains Gs and Fs. In addition, $\overline H\_{pq}$ is, by construction, a smooth function of $\vec p$ and $\vec q$, while the presence of Gs and Fs implies that there are singularities in $\overline H^R$.[21](#fn21) It turns out that these issues can be resolved if one (a) takes the NR limit of Eq. ([eq:HSQC2]), (b) assumes that ${\mathcal K}\_{2,s}^{{\rm NR}}$ is independent of momentum, which is equivalent to keeping only the leading C0 term in the NR expansion of ${\mathcal L}\_2$, i.e. keeping only the scattering length in the effective range approximation, and (c) assumes that ${\mathcal K\_{\text{df},3,s}}$ and $\overline H$ are independent of spectator momenta. For ${\mathcal K\_{\text{df},3,s}}$, this is the isotropic limit that holds near threshold, as discussed in Sec. [sec:trunc]. For $\overline H$ this means keeping only the leading D0 term in the NR expansion of ${\mathcal L}\_3$. In other words, we consistently keep only the leading-order terms in the NR limit. In this combined limit, ${\mathcal K\_{\text{df},3,s}}= \| 1\rangle {\mathcal K^{\rm iso}\_{\text{df},3}}\langle 1 \|$, with ∣1⟩ the unnormalized isotropic vector having a unit entry in all positions. This allows us to use the following identity $$G\_s^{{\rm NR}}\|1\rangle = 2 F\_{s}^{{\rm NR}}\|1\rangle + I\_s^{{\rm NR}}\|1\rangle \,, \ \ {\rm with} \ \ I^{{\rm NR}}\_{s;pk} = \delta\_{pk} \frac{I\_{{\rm PV}}}{16 \pi m}\,, \label{eq:GFid}$$ together with its transpose. Recall that $I\_{{\rm PV}}$ is a momentum-independent, regularization-dependent constant. Denoting the leading contribution to $\overline H^R$ in the NR limit by $\overline H^{{\rm NR}}$, we then find $$\overline H^{{\rm NR}}= - ([{\mathcal K}\_{2,s}^{{\rm NR}}]^{-1} + I\_s^{{\rm NR}}) {\mathcal K\_{\text{df},3,s}}\frac{[{\mathcal K}\_{2,s}^{{\rm NR}}]^{-1}+F\_{s}^{{\rm NR}}+G\_s^{{\rm NR}}- I\_s^{{\rm NR}}-[{\mathcal K}\_{2,s}^{{\rm NR}}]^{-1}}{18m L^3} \,.$$ Inserting this result into the NR form of Eq ([eq:HSQC2]), the matrix inside the determinant can be written $$\begin{gathered} \left[1 + \frac{({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + I\_s^{{\rm NR}}}{18 m L^3} {\mathcal K\_{\text{df},3,s}}\right] \left[({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + F\_{s}^{{\rm NR}}+ G\_s^{{\rm NR}}\right] -\\ \frac1{18 mL^3}\left[({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + I\_s^{{\rm NR}}\right] {\mathcal K\_{\text{df},3,s}}\left[({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + I\_s^{{\rm NR}}\right] \,.\end{gathered}$$ This allows the leading NR term in the relativistic quantization condition to be written in exactly the form of the NR quantization condition, Eq. ([eq:QC3NR]), with $$\overline H = \overline H\_0 = \left[1 + \frac{({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + I\_s^{{\rm NR}}}{18 m L^3} {\mathcal K\_{\text{df},3,s}}\right]^{-1} [({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + I\_s^{{\rm NR}}] \frac{{\mathcal K\_{\text{df},3,s}}}{18 mL^3} [({\mathcal K}\_{2,s}^{{\rm NR}})^{-1} + I\_s^{{\rm NR}}]\,. \label{eq:Hbarres}$$ Using the fact that ${\mathcal K}\_{2,s}^{{\rm NR}}$ and $I\_s^{{\rm NR}}$ are proportional to the identity, it is simple to show that this result is isotropic, as required for complete equivalence. The overall factor of 1/(mL3) also matches that in $\overline H\_0$ [see Eq. ([eq:Hbar])]. Furthermore, since for any constant x, we have $$\frac1{1 + x \|1\rangle \langle 1 \|} \|1\rangle = \|1\rangle \frac1{1 +x N}\,, \quad N = \langle 1 \| 1\rangle \propto (L\Lambda)^3\,,$$ we see that the L3 in the first factor in Eq. ([eq:Hbarres]) cancels, so that the right-hand side is a volume-independent constant, as required to match $\overline H\_0$.[22](#fn22) The presence of the cutoff dependent quantity $I^{{\rm NR}}$ on the right-hand side is not an issue, because both H0 and ${\mathcal K\_{\text{df},3,s}}$ are cutoff dependent. Indeed, this result allows one to map the cutoff dependence of H0 known from NREFT to that of ${\mathcal K\_{\text{df},3,s}}$. ### Summary The derivation of the three-particle quantization condition is dramatically simplified by using NREFT, compared to the RFT derivation described earlier. This is partly due to the fact that, so far, only the s-wave dimer has been included in the former. But the primary reasons for the simplicity are (a) that the number of diagrams is sufficiently small that one can straightforwardly include them all in a simple and explicit integral equation; and (b) that the two-step approach using intermediate quantities is embraced from the beginning. No attempt is made to explicitly find all sources of power-law volume dependence by focusing on sum-integral differences. Instead, one simply uses the same NREFT in separate finite- and infinite-volume calculations. Given point (a) above, this can be done without further approximation. An additional advantage of the NREFT approach is that it is valid also in the presence of subchannel resonances, since any volume-dependence they introduce is included automatically. The simplicity of the NREFT approach is not available in a generic RFT, since one cannot solve the three-particle scattering problem in a generic theory. Thus one is forced to keep track of finite-volume effects explicitly, leading to a more complicated derivation, and with additional considerations required for subchannel resonances. Of course, if both approaches are carried out correctly, they should agree when we take the NR limit of the relativistic approach. This is indeed the case, as first discussed in Ref. , and as shown explicitly earlier in this section. The main drawback with the NREFT approach is simply that most three-body systems of interest in nuclear and particle physics are relativistic. We have already commented on the kinematics of three pions [see Eq. ([eq:relkin])]: for the volumes used in lattice QCD calculations the sum over spectator momenta necessarily lies in the relativistic domain. This conclusion appears unavoidable for applications of the three-particle quantization condition to lattice QCD. A further comment on the NREFT approach is that, once interaction terms quartic in derivatives are included, i.e. once C4 and D4 are nonzero, then one must also include d-wave (ℓ = 2) dimers, as they enter at the same power in the NR expansion. As one goes further into the relativistic domain, many higher-order terms will be needed, and thus many higher-order dimers must be included. In summary, for the NREFT approach to have broad utility, it is necessary both that the formalism be explicitly extended to include higher waves (which we expect to be relatively straightforward) and that the kinematics somehow be relativized. This second step is claimed also to be straightforward in Ref. . We think, however, that it will be important at each stage to check that the results agree with those from the RFT approach. Finite-volume unitarity (FVU) approach -------------------------------------- The third approach that has been used to derive a three-particle quantization condition aims to maintain relativistic invariance but to avoid much of the work of the diagrammatic RFT approach by using the constraints arising from unitarity . The starting point is a representation of ${\mathcal M}\_3$ in terms of dimers[23](#fn23) that is explicitly unitary in the s-channel . Then, by a judicious replacement of integrals with sums, a representation of a quantity similar to ${\mathcal M}\_{3L}$ is obtained, and from this follows the quantization condition. As in the NREFT approach, the formalism has been worked out so far only for s-wave dimers, and for $\vec P=0$. ### Two-particle quantization condition The discussion begins by considering the two-particle subsystems. The infinite-volume s-wave scattering amplitude is written as $${\mathcal M}\_{2,s} \equiv - v(q\_1,q\_2) \frac1{D(s\_{12})} v(p\_1,p\_2)\,, \label{eq:MDM2s}$$ where pi and qi are the initial and final momenta, respectively, of the scattering pair, and s12 = (p1 + p2)2. The s-channel cut lies in D, which is thus complex, while v is a smooth, real function. By relativistic invariance, it can depend only on s12, but it is convenient to write it as a function of Q2 = − (p1 − p2)2 = s12 − 4m2 (using the mostly-minus metric). The form used in Ref. is $$v(p\_1,p\_2) = \lambda(s\_{12}) f(Q^2)\,,\quad f(Q^2) = \frac{1}{1 + \exp\left[Q^2/4-(1-\Lambda/2)^2\right]}\,, \label{eq:v}$$ where λ is a smooth function and Λ a parameter. Unitarity determines the imaginary part of the denominator D, and the full quantity can be reconstructed using an appropriately subtracted dispersion relation. The result can be written[24](#fn24) $$D(s\_{12}) = s\_{12} - M\_0^2 - \frac{\lambda(s\_{12})^2}2 \int\_{\vec q} \frac1{2\omega\_q} \frac{f(4 \vec q^{\,2})^2}{s\_{12} - 4\omega\_q^2+ i \epsilon} \,, \label{eq:D}$$ with M0 an additional parameter. Thus we see that the introduction of the form factor in Eq. ([eq:MDM2s]) leads to its appearance as a convergence factor in the loop integral in D. The claim is that this form for D, when inserted into Eq. ([eq:MDM2s]), gives the most general unitary result for ${\mathcal M}\_{2,s}$. In applications, the form for λ(s12) must be tuned so as to match the known phase shift. The next step is to claim that the finite-volume amplitude ${\mathcal M}\_{2L}$ is obtained simply by replacing the integral in Eq. ([eq:D]) with the finite-volume momentum sum. Although the integral is frame-invariant, the sum depends on the choice of frame. Anticipating the three-particle application, we label the frame by the spectator momentum, $\vec k$ (so that the dimer momentum is $-\vec k$): $$\begin{aligned} {\mathcal M}\_{2L,s}(\vec k) &= - v\_{{\rm on}}(\vec k) \frac1{D\_L(\vec k)} v\_{{{\rm on}}}(\vec k)\,, \label{eq:MDM2L} \end{aligned}$$ Here $v\_{{\rm on}}$ is the same vertex function as appearing in Eq. ([eq:MDM2L]), but the new subscript is used to emphasize that external particles are on shell. The expression for $D\_L(\vec k)$ for general $\vec k$ is given in Ref. , but here we show only the form for the dimer at rest: $$\begin{aligned} D\_L(\vec k=0) &= E\_{2,k}^{\2} - M\_0^2 - \frac{\lambda(E\_{2,k}^{\2})^2}2 \frac1{L^3}\sum\_{\vec q} \frac1{2\omega\_q} \frac{f(4 \vec q^{\,2})^2}{E\_{2,k}^{\*2} - 4\omega\_q^2+ i \epsilon}\,, \label{eq:DL}\end{aligned}$$ where $E\_{	arxiv_0000748
A model-theoretic approach to rigidity of strongly ergodic, distal actions ========================================================================== We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other distal, ergodic system of the group as a factor. Introduction ============ The theory of measure-preserving dynamical systems was initiated by Halmos and von Neumann, who characterized the ergodic systems of the group of integers $\Z$ that can be represented as translations on a compact group in terms of their spectrum. This was extended by Mackey to general locally compact groups. Later, inspired by the work of Furstenberg in topological dynamics, Furstenberg and Zimmer considered the relative notion of a of a system and defined a system to be one obtained via a transfinite tower of compact extensions, starting from the trivial system. The notion of a distal system was central to Furstenberg’s ergodic-theoretic proof of Szemerédi’s theorem, and led to the Furstenberg–Zimmer structure theorem for general ergodic actions of locally compact groups. In the literature, compact systems are often referred to as or having (or ). Similarly, distal systems are also known as systems with. In this paper, we will use the terminology “compact” and “distal”. The notion of of measure-preserving systems of countable groups was introduced by Kechris as a weakening of the notion of a : a system $\cX$ is in another system $\cY$ (notation: $\cX <\_w \cY$) if $\cX$ is a factor of an ultrapower of $\cY$. Two systems $\cX$ and $\cY$ are if $\cX <\_w \cY$ and $\cY <\_w \cX$. It turns out that weak equivalence is a equivalence relation (the set of equivalence classes is compact ) and that many notions in ergodic theory are invariants of weak equivalence. Free actions of infinite amenable groups are all weakly equivalent, but non-amenable groups usually admit uncountably many classes of weak equivalence . By a result of Tucker-Drob , weak equivalence classes of free actions always contain uncountably many isomorphism classes (and they are not even classifiable by countable structures). This is why rigidity results that allow to recover the isomorphism class from the weak equivalence class of a system in special situations are particularly interesting. Two rigidity results about weak equivalence have appeared in the literature. The first one, due to Abért and Elek , states that if a system $\cX$ is weakly contained in a strongly ergodic system $\cY$, then $\cX$ is a factor of $\cY$. This was then generalized by Ioana and Tucker-Drob to distal systems. From this, they were able to conclude that for compact, strongly ergodic systems, weak equivalence implies isomorphism, because compact ergodic systems are, i.e., every endomorphism of the system is an automorphism. However, this is not true in general for distal systems (see Parry and Walters ). Nevertheless it turns out to be true for strongly ergodic, distal systems and we were able to prove the following. [th:i:distal-rigidity] Let Γ be a countable group. 1. Let $\cX$ be a distal, strongly ergodic, probability measure-preserving Γ-system. Then $\cX$ is coalescent and $\Aut(\Gamma\actson\cX)$ is compact. 2. Let $\cX$ and $\cY$ be two distal, strongly ergodic, probability measure-preserving Γ-systems. If they are weakly equivalent, then they are isomorphic. In the course of the proof of Theorem [th:i:distal-rigidity], we also give a new proof of the rigidity result of Ioana and Tucker-Drob mentioned above. The methods we use come from continuous logic, a relatively new branch of model theory, suitable for studying metric structures. While some model-theoretic concepts, such as ultraproducts, have been used before in ergodic theory, to our knowledge, this is the first application of continuous logic to dynamics of countable groups. The main model-theoretic notion behind the proof is that of : if M is a metric structure, $A \sub M$, and b ∈ M, we say that b is in the of A (notation: $b \in \acl^M(A)$) if it belongs to a compact set definable over A. Our main model-theoretic result can be stated as follows. [th:i:acl] Let $\cX$ be a strongly ergodic system, $\cZ$ be a factor of $\cX$ and $\cY$ be a distal extension of $\cZ$ in $\cX$. Then $\cY \sub \acl^\cX(\cZ)$. A slight strengthening of Theorem [th:i:acl] (replacing $\acl$ by $\operatorname{\acl\_\exists}$) easily implies Theorem [th:i:distal-rigidity]. We provide two rather different proofs of Theorem [th:i:acl], one based on the theory of compact extensions, and another using model-theoretic stability theory. Another feature of our approach is that it relies, in part, on the existence of certain canonical universal systems. Those systems are usually non-separable, i.e., they cannot be realized on a standard probability space. Working on a standard space is an assumption often made in the literature, either out of habit, or for the more essential reason that many important results fail in the non-separable setting (we give several examples in this paper). We depart from this tradition and make no separability assumptions throughout the paper unless specifically stated. A posteriori, somewhat surprisingly, it turns out that all strongly ergodic, distal systems are separable (Corollary [c:distal-is-in-Xalg]). It is possible to carry out the stability-theoretic proof presented in Section [sec:two-line-proof] by only using ergodic theoretic results in a separable situation. The following is our main result regarding universal systems. [th:i:universal-distal] Let Γ be a countable group and α be an ordinal. Then there exists a unique ergodic Γ-system $\cD\_\alpha(\Gamma)$ of distal rank at most α which contains every ergodic Γ-system of distal rank at most α as a factor. Moreover, $\cD\_{\omega\_1}(\Gamma) = \cD\_{\omega\_1+1}(\Gamma)$, that is, $\cD\_{\omega\_1}(\Gamma)$ is the universal ergodic, distal Γ-system. An analogous result bounding the rank of a topological distal minimal system for $\Gamma = \Z$ was proved by Beleznay and Foreman . For α = 1, the existence of a universal system is well-known: the maximal ergodic, compact system of Γ is a translation on a compact group known as the. For many groups (for example, $\Z$), the Bohr compactification is not metrizable, i.e., the corresponding dynamical system is not separable. However, it is a result of Wang that if Γ has property (T), then its Bohr compactification is a metrizable group. We recover this result as a consequence of Theorem [th:i:acl] and prove something more: the hierarchy of distal systems for property (T) groups collapses. [th:i:propT] Let Γ have property (T) and $\cZ$ be an ergodic Γ-system. Then every ergodic, distal extension of $\cZ$ is a compact extension. This theorem had been previously proved by Chifan and Peterson (unpublished) but our proof is different and independent from theirs. We would like to thank them for telling us about their result. For $\Gamma = \Z$, Beleznay and Foreman have proved that there exist separable ergodic, distal systems of rank α for every α < ω1. This implies that the tower of universal distal systems $(\cD\_\alpha(\Z))\_{\alpha \leq \omega\_1}$ is proper. [q:rank-distal] Let Γ be a countable group and suppose that $\cD\_2(\Gamma)$ is not strongly ergodic. Is it true that $\cD\_\alpha(\Gamma) \subsetneq \cD\_{\alpha+1}(\Gamma)$ for all α < ω1? In Theorem [th:propT] we prove that if $\cD\_2(\Gamma)$ is strongly ergodic, then every distal system of Γ is compact, so the condition that $\cD\_2(\Gamma)$ is not strongly ergodic is necessary. Until recently, it was an open question whether strongly ergodic, distal, non-compact systems exist. It was resolved by Glasner and Weiss who, using results of Bourgain and Gamburd on compact systems with spectral gap, constructed examples for Γ a free group. Finally, we would like to mention that the assumption that Γ is countable is made throughout the paper for simplicity. All results with appropriate modifications (for example replacing ω1 by ∣Γ∣+) hold for arbitrary discrete groups Γ. On the other hand, for the moment, our methods do not allow us to treat non-discrete, locally compact groups. The paper is organized as follows. In Section [sec:model-theor-fram], we describe how to treat probability measure-preserving dynamical systems in a model-theoretic framework and study the connection between weak containment and the existential theory. In Section [sec:basic-facts-cpct-exts], we develop some of the theory of compact extensions without separability assumptions. In Section [sec:universal-distal-systems], we consider universal systems and prove Theorem [th:i:universal-distal]. In Sections [sec:comp-extens-algebr] and [sec:two-line-proof] we give two proofs of Theorem [th:i:acl] and derive Theorem [th:i:distal-rigidity] from it. Finally, in Section [sec:groups-with-property-T], we consider groups with property (T). In an appendix, we include a proof of the fact that the distal rank does not increase on factors. Acknowledgments --------------- We are grateful to Matt Foreman, Eli Glasner, François Le Maître and Robin Tucker-Drob for useful discussions. Research was partially supported by the ANR projects AGRUME (ANR-17-CE40-0026) and GAMME (ANR-14-CE25-0004). A model-theoretic framework for ergodic theory ============================================== The language and the basic theory --------------------------------- In this section, we explain how to formalize probability measure-preserving (pmp) group actions within the framework of continuous logic. The structures of interest are measure-preserving group actions $\Gamma \actson (X, \cX, \mu)$ where Γ is a discrete group and $(X, \cX, \mu)$ is a probability space. As continuous logic deals with metric structures, it is natural to consider the measure algebra $\MALG(X, \cX, \mu)$ obtained by taking the quotient of $\cX$ by the equivalence relation of having symmetric difference of measure 0. We will abuse notation and write again $\cX$ for this quotient. It becomes naturally a complete metric space if endowed with the metric $d\_\mu(a, b) = \mu(a \sdiff b)$. By duality, a measure-preserving action $\Gamma \actson (X, \mu)$ is nothing but an action $\Gamma \actson \cX$ by isometries that also preserves the algebra structure. The signature that we consider will consist of the language of Boolean algebras $\set{\cap, \cup, \sminus, \emptyset, X}$ (here $\sminus$ is set difference and X stands for the maximum element of the Boolean algebra), a [0, 1]-valued predicate for the measure μ, and a unary function symbol for every element γ ∈ Γ. For a metric structure $\cX$, it is natural to measure its size using the (or just ), i.e., the smallest cardinal of a dense subset of $\cX$. We will denote the density of $\cX$ by $\|\cX\|$. Note that if $\cX$ is an infinite measure algebra, then $\|\cX\| = \|L^2(\cX)\|$ and $\|\cX\| \leq \aleph\_0$ iff $\cX$ is isomorphic to the measure algebra of a, i.e., one that can be realized by a Borel measure on the interval [0, 1]. [rem:Linfty] Sometimes it will be useful to consider the richer von Neumann algebra $L^\infty(X, \cX, \mu)$. It is a metric structure in the language that includes sorts for all $\nm{\cdot}\_\infty$-balls and symbols for addition, multiplication, the adjoint, and the L2-norm (which defines the metric). Note that the $\nm{\cdot}\_\infty$-norm is not part of the language. It follows from that the structures $\cX$ and $L^\infty(X, \cX, \mu)$ are bi-interpretable, so we can use them interchangeably. We will denote the latter structure simply by $L^\infty(\cX)$. (Many alternative but equivalent presentations of the structure $L^\infty(\cX)$ are possible; in fact, the language chosen in is quite different from the one proposed here, for instance in that it is one-sorted—only [0, 1]-valued functions are considered—and that the L1-norm is used instead of the L2-norm.) [df:MPAGamma] The theory PMPΓ includes the following axioms: * The universe is a probability measure algebra (see \Section 16 for the precise axioms). Each γ is an automorphism. * For every tuple γ1, γ2, …, γn of elements of Γ such that γ1γ2⋯γn = 1Γ, the automorphism γ1γ2⋯γn is equal to the identity. [rem:only-generators] If Γ is generated by a subset $S \sub \Gamma$, then we can include in the language only symbols for the elements of S and replace the third group of axioms by the relations that define the group. For example, if $\Gamma = \bF\_n$ is the free group on n generators, we can include in the language n function symbols and the third group of axioms will be empty. The models of PMPΓ are exactly the pmp actions of Γ. If $\cX \models {\mathrm{PMP}}\_\Gamma$ and $\cY$ is a substructure of $\cX$, then $\cY$ is a closed, Γ-invariant subalgebra of $\cX$, or, which is the same, a of $\cX$. We will use the words substructure and factor interchangeably and we will use the notation $\cY \sub \cX$. Dually, if $\iota \colon \cY \to \cX$ is an embedding, there exist spatial representations (X, μ) and (Y, ν) of $\cX$ and $\cY$ and a Γ-equivariant, measurable map π: X → Y such that π\*μ = ν and ι(a) = π− 1(a) for all $a \in \cY$. The fact that substructures of models of PMPΓ are again models implies that PMPΓ admits a (that is, with axioms of the form $\sup\_{\bar a} \phi(\bar a) \leq 0$, where ϕ is quantifier-free), something that can also be verified directly (if we work only with symbols for a generating set S, as in Remark [rem:only-generators], then S should be symmetric). Often one is interested in, i.e., actions for which non-identity elements of the group do not fix any points in X. [df:FREEGamma] The theory FPMPΓ includes all axioms of PMPΓ and, in addition, for every element γ ∈ Γ of finite order n, the axiom * $\inf\_{\set{a : \mu(a) = 1/n}} \max\_{i < j < n} \mu(\gamma^i a \cap \gamma^j a) = 0$, and for every element γ ∈ Γ of infinite order, an axiom of the previous form for every n > 1. It is clear that any action satisfying those axioms is free and, conversely, any free action satisfies the axioms (this follows from Rokhlin’s lemma for γ of infinite order). If Γ is infinite, the freeness of the action implies that the measure space is non-atomic, so separable models of FPMPΓ are, up to isomorphism, exactly the free, measure-preserving actions $\Gamma \actson ([0, 1], \lambda)$, where λ is the Lebesgue measure. Let us mention a few known facts about the model theory of FPMPΓ. If Γ is finite, we need to add non-atomicity as an axiom but once we do, the theory that we obtain is *ω-categorical, i.e., it has a unique separable model up to isomorphism. Indeed, given $\Gamma\actson X\_1$ and $\Gamma\actson X\_2$, we may choose fundamental domains A*i ⊆ Xi (i.e., measurable subsets meeting every orbit in exactly one point), then extend any measure-preserving isomorphism A1 ≃ A2 to an equivariant pmp isomorphism X1 ≃ X2. Conversely, if Γ is infinite, the theory FPMPΓ is not ω-categorical. If the group Γ is infinite and amenable, FPMPΓ is a complete, stable theory that eliminates quantifiers. This was proved in for $\Gamma = \Z$ and was generalized in to all amenable Γ. The existential theory and weak containment ------------------------------------------- The notion of for measure-preserving actions was introduced by Kechris , who was motivated by a similar notion for unitary representations. This notion has a strong model-theoretic flavor and it turns out that it can be expressed naturally in terms of the existential theories of the actions. [df:wcon] Let Γ be a discrete group and let $\cX$ and $\cY$ be two measure-preserving actions of Γ. We say that $\cX$ is in $\cY$ (notation: $\cX <\_w \cY$) if for every ε > 0 and finitely many $a\_0,\dots,a\_{n-1}\in\cX$ and γ0, …, γk − 1 ∈ Γ there are $b\_0,\dots,b\_{n-1} \in \cY$ such that ∣μ(ai ∩ γlaj) − μ(bi ∩ γlbj)∣ ≤ ε for every i, j < n and l < k. In the literature, the symbol ≺ is often used to represent weak containment; however, this conflicts with the well-established notation for an elementary substructure in model theory. Recall that an is a sentence of the form $\inf\_{\bar a} \phi(\bar a)$, where ϕ is quantifier-free. The existential theory of a structure M, denoted by $\Th\_\exists(M)$, is given by the set of statements of the form ϕ ≤ 0 true in M, where ϕ is an inf-sentence. It is clear that if $\cX$ and $\cY$ are Γ-systems with $\Th\_\exists(\cX) \sub \Th\_\exists(\cY)$, then $\cX <\_w \cY$. The converse is also easy to see. On the other hand, by the compactness theorem for continuous logic, the condition $\Th\_\exists(M) \sub \Th\_\exists(N)$ is equivalent to saying that M embeds in an elementary extension of N. We note these observations down; the equivalence between the first and third item below was first proved by Conley, Kechris and Tucker-Drob in (see also Carderi ). [l:wcon-existential] Let $\cX$ and $\cY$ be two pmp actions of Γ. Then the following are equivalent: 1. $\cX$ is weakly contained in $\cY$. 2. $\Th\_\exists(\cX) \sub \Th\_\exists(\cY)$ or, which is the same, $\cY \models \Th\_\exists(\cX)$. 3. There is an elementary extension (equivalently, an ultrapower) $\cY'$ of $\cY$ such that $\cX\subseteq\cY'$. Thus two systems are if they have the same existential theories. Alternatively, two systems $\cX$ and $\cY$ are weakly equivalent iff for all inf-sentences ϕ, we have that $\phi^\cX = \phi^\cY$. The space W of weak equivalence classes of actions of Γ can be identified with the space of existential theories consistent with PMPΓ. The topology on W is the weakest topology that makes evaluation of inf-sentences on W continuous; it follows from the compactness theorem that this topology is compact as W is a quotient of the space of all complete theories compatible with PMPΓ. This recovers a result of Abért and Elek . Note that W can also be viewed as the Gelfand space of the abstract C\-algebra generated by the inf-sentences modulo PMPΓ. Strongly ergodic actions ------------------------ A system $\cX$ is ergodic* if every invariant set is trivial or, which is the same, if $\cX$ does not contain any trivial two-point system as a factor. In the same vein, a system $\cX$ is if almost invariant sets are close to ∅ or X or, more precisely, if it does not weakly contain any trivial two-point system. Equivalently, $\cX$ is strongly ergodic if for every ε > 0 there are δ > 0 and finitely many γ0, …, γn − 1 ∈ Γ such that, whenever ε < μ(a) < 1 − ε, we have maxi < ndμ(a, γia) ≥ δ. If Γ is finitely generated, then in the previous definition one can replace the γi by the generators of Γ. We have the following additional easy equivalences, which indicate why model-theoretic methods are likely to be useful for the study of these actions. Let $\cX$ be a pmp Γ-system. The following are equivalent. 1. $\cX$ is strongly ergodic. 2. Every model of the (universal) first-order theory of $\cX$ is ergodic. 3. Every elementary extension $\cX'$ of $\cX$ is ergodic. A group Γ has property (T) if every ergodic pmp Γ-system is strongly ergodic. (This is not the standard definition but it is equivalent by.) Below we give a characterization of property (T) in terms of the theory PMPΓ. Recall that in continuous logic, a set S in a given structure is 0-definable if the distance function to S is a definable predicate. Similarly, if S denotes a family of sets, one in each model of a given theory T, then S is uniformly 0-definable if there is a common formula that gives in each model of T the distance function to the corresponding set in S. [p:T-definable] Let Γ be a countable group. Then the following are equivalent: 1. [i:Tdef:1] Γ has property (T). 2. [i:Tdef:2] The algebra of invariant sets $\set{a \in \cX : \gamma a = a \text{ for all } \gamma \in \Gamma}$ is uniformly 0-definable in the models of PMPΓ. 3. [i:Tdef:3] The class of ergodic Γ-actions is axiomatizable. We use the equivalences of definability of sets given in \Proposition 9.19, which also hold uniformly. Then the implication follows from the well-known fact that for a property (T) group Γ with a finite generating set Q, for every $\eps > 0$ there exists δ* > 0 such that for any pmp action $\Gamma \actson \cX$, if $a \in \cX$ is (Q, δ)-invariant (i.e., if maxγ ∈ Qdμ(a, γa) ≤ δ), then there exists $a' \in \cX$ which is Γ-invariant and satisfies $d\_\mu(a, a') \leq \eps$. If P(a) denotes the distance of a to the algebra of invariant sets, then the condition $\sup\_a \big(\min(\mu(a), 1-\mu(a)) - P(a)\big) \leq 0$ axiomatizes the class of ergodic systems within the models of PMPΓ. If $\cX$ is a an ergodic Γ-system which is not strongly ergodic, then any saturated elementary extension of $\cX$ is not ergodic but is elementarily equivalent to $\cX$. Some basic facts about compact extensions and distal systems ============================================================ First we recall some of the theory of compact systems and compact extensions, as developed by Mackey , Furstenberg , and Zimmer (see also Glasner ). In the above references, the authors work under the assumption that the systems are separable and make use of a variety of decomposition results that are only valid for separable spaces. As we work in a more general setting, we prove all results that we need. Let $\cZ\subseteq\cX$ be an extension of measure algebras. A Hilbert subspace $M\subseteq L^2(\cX)$ is a $\cZ$-module if it is closed under multiplication by functions of $L^\infty(\cZ)$. The $\cZ$-module generated by a set $S\subseteq L^2(\cX)$ is the closed span of $L^\infty(\cZ)S$. Given $f,g\in L^2(\cX)$, we define $\langle f,g\rangle\_\cZ\coloneqq \E(f\bar g\|\cZ)\in L^1(\cZ)$ (the conditional expectation of $f\bar g$ relative to $\cZ$). A set $S\subseteq L^2(\cX)$ is $\cZ$-orthonormal if $\langle f,g\rangle\_\cZ=0$ for distinct f, g ∈ S, and $\langle f,f\rangle\_\cZ$ is a characteristic function for each f ∈ S. If $f \in L^2(\cX)$, its (denoted by $\supp\_\cZ f$) is the support of the function $\ip{f, f}\_\cZ$, which is an element of $\cZ$. We will write $\chi\_\cZ(f)$ for the characteristic function of $\supp\_\cZ f$. Say that S is a $\cZ$-basis for a $\cZ$-module M if it is $\cZ$-orthonormal, generates M, and each element of S has full $\cZ$-support. The following lemma summarizes several properties of the conditional inner product $\ip{\cdot, \cdot}\_\cZ$. [l:inner-prod-Z] Let $\set{e\_i : i < n}$ be a finite, generating, $\cZ$-orthonormal set for a $\cZ$-module $M \sub L^2(\cX)$, and let f, g ∈ M. Then the following statements hold: 1. [i:ip:1] (Cauchy–Schwarz) $\|\ip{f, g}\_\cZ\|^2 \leq \ip{f, f}\_\cZ \ip{g, g}\_\cZ$. 2. [i:ip:2] $\supp \ip{f, g}\_\cZ \sub \supp\_\cZ f \cap \supp\_\cZ g$. In particular, $$\ip{f, g}\_\cZ = \ip{f, g}\_\cZ \chi\_\cZ(f) \chi\_\cZ(g).$$ 3. [i:ip:4] $f = \sum\_{i < n} \ip{f, e\_i}\_\cZ e\_i$. [i:ip:1] is standard and [i:ip:2] follows from [i:ip:1]. For [i:ip:4], first we check that $\ip{f, e\_i}\_\cZ e\_i \in L^2(\cX)$. Using [i:ip:1], $$\int \|\ip{f, e\_i}\_\cZ e\_i\|^2 \leq \int \ip{f, f}\_\cZ \|e\_i\|^2 \leq \nm{f}^2.$$ We also have $\ip{e\_i, e\_i}\_\cZ e\_i = e\_i$. This implies that the identity holds when f = ∑i < naiei with $a\_i \in L^\infty(\cZ)$, and as it is a closed condition, it must hold for all f ∈ M. [l:os-size] Let $\set{e\_i : i < n}$ and $\set{f\_j : j < m}$ be two generating, $\cZ$-orthonormal sets for a $\cZ$-module M. Let $a\_{ij} = \ip{e\_i, f\_j}\_\cZ$ and A = (aij)i < n, j < m. Then the following hold: 1. The entries of A are in $L^\infty(\cZ)$. 2. $AA^\* = \diag \big(\chi\_\cZ(e\_0), \ldots, \chi\_\cZ(e\_{n-1}) \big)$ and $A^\A = \diag \big(\chi\_\cZ(f\_0), \ldots, \chi\_\cZ(f\_{m-1}) \big)$. 3. $\sum\_{i < n} \ip{e\_i, e\_i}\_\cZ = \sum\_{j < m} \ip{f\_j, f\_j}\_\cZ = \sum\_{i, j} \|a\_{ij}\|^2$. This follows easily from Lemma [l:inner-prod-Z] and standard linear algebra. In the context of Γ-systems, we will be mainly interested in $\cZ$-modules that are Γ-invariant and finitely generated. Let $\cX$ be a Γ-system and let $\cZ$ be an ergodic factor thereof. Then, every finitely generated Γ-invariant $\cZ$-module $M\subseteq L^2(\cX)$ admits a finite $\cZ$-basis. Let f1, …, f*n be generators for M. We use Gram–Schmidt orthogonalization. Set $$h\_1 = f\_1, \quad h\_2 = f\_2 - P\_{\langle f\_1 \rangle\_{L^\infty(\cZ)}} f\_2, \quad \ldots, \quad h\_n = f\_n - P\_{\langle f\_1, \ldots, f\_{n-1} \rangle\_{L^\infty(\cZ)}} f\_n,$$ where $P\_{\langle F \rangle\_{L^\infty(\cZ)}}$ denotes the orthogonal projection onto the $\cZ$-module generated by F. Then $\langle h\_i,h\_j\rangle\_\cZ=0$ for i ≠ j and $M = \boplus\_{i = 1}^n M\_{h\_i}$ where Mh is the $\cZ$-module generated by h. Let $S\_i = \supp\_\cZ h\_i$. We define $e\_i=\big(\langle h\_i,h\_i\rangle\_\cZ\big)^{-1/2}h\_i$ with the convention that ei vanishes outside Si. Then $\int \|e\_i\|^2=\int\_{S\_i}\frac{\|h\_i\|^2}{\langle h\_i,h\_i\rangle\_\cZ}=\mu(S\_i)$, so $e\_i \in L^2(\cX)$ and $\set{e\_i : i < n}$ is a generating $\cZ$-orthonormal system for M. (Note that if $h\in L^2(\cX)$ and $g\in L^0(\cZ)$ is any $\cZ$-measurable function such that $gh\in L^2(\cX)$, then gh ∈ Mh.) By Lemma [l:os-size], the function $\sum\_{i<n}\bOne\_{S\_i}$ is Γ-invariant, so by the ergodicity assumption it must be constant, say equal to k. Denote by $\binom{n}{k}$ the set of all k-element subsets of n. For $\alpha \in \binom{n}{k}$, let Aα = ⋂i ∈ αSi. For i < k, define σi(α) for $\alpha \in \binom{n}{k}$ to be the i-th element of α. Finally, for i < k, define $e\_i' = \sum\_{\alpha \in \binom{n}{k}} \bOne\_{A\_\alpha} e\_{\sigma\_i(\alpha)}$. Then one easily checks that $\set{e\_i' : i < k}$ is a $\cZ$-basis for M. Suppose now that $\cZ\subseteq\cX$ is an extension of Γ-systems and that M is a Γ-invariant $\cZ$-module admitting a $\cZ$-basis {e0, …, en − 1}. Let X be the probability space associated to $\cX$. If γ ∈ Γ, the set {γei}i < n is again a $\cZ$-basis for M. Thus, if for x ∈ X we define $$u\_\gamma(x)=\big(\langle\gamma e\_i,e\_j\rangle\_{\cZ}(x)\big)\_{ij} \in \C^{n\times n},$$ by Lemma [l:os-size], uγ(x) is a unitary matrix almost surely and the function uγ is $\cZ$-measurable. Now consider the function $\theta\colon X\to \R^+$ defined by θ(x) = ∑i < n∣ei(x)∣2 = ⟨E(x), E(x)⟩, where E = (e1, …, en) and $\ip{\cdot, \cdot}$ denotes the usual inner product on $\C^n$. We have, almost surely, θ(γ− 1x) = ⟨(γE)(x), (γE)(x)⟩ = ⟨uγ(x)E(x), uγ(x)E(x)⟩ = ⟨E(x), E(x)⟩. That is, θ is Γ-invariant. Hence, if we further assume that $\cX$ is ergodic, θ must be constant, and this implies that each ei is bounded. Thus we have proved the following. [p:exist-basis] Let $\cX$ be an ergodic Γ-system, $\cZ \sub \cX$ be a factor, and $M \sub L^2(\cX)$ be a finitely generated, Γ-invariant $\cZ$-module. Then M admits a finite $\cZ$-basis and the elements of any such basis are in $L^\infty(\cX)$. Recall that a system $\cX$ is called if $L^2(\cX)$ decomposes as a sum of finite-dimensional representations of Γ. An extension $\cX \supseteq \cZ$ is if $L^2(\cX)$ decomposes as a direct sum of finitely generated, Γ-invariant $\cZ$-modules. A factor generated by compact factors is a compact factor as well. More generally, the following holds. [p:factor-generated-by-compact-factors] Let $\cX$ be an ergodic system. Suppose $\cZ\_i\subseteq\cY\_i\subseteq\cX$, i ∈ I, are factors such that each extension $\cZ\_i\subseteq\cY\_i$ is compact. Then the factor $\cY \sub \cX$ generated by all the $\cY\_i$ is a compact extension of the factor $\cZ$ generated by all the $\cZ\_i$. By hypothesis, we have $L^2(\cY\_i)=\bigoplus\_{M\in\mathcal{M}\_i}M$ where each M is a finitely generated Γ-invariant $\cZ\_i$-module. If $\{e\_0,\dots,e\_{n-1}\}\subset L^\infty(\cX)$ is a $\cZ\_{i\_1}$-basis for M1 ∈ Mi1 and $\{f\_0, \dots, f\_{m-1}\} \subset L^\infty(\cX)$ is a $\cZ\_{i\_2}$-basis for M2 ∈ Mi2, we denote by M1M2 the $\cZ$-module generated by the products {epfq : p < n, q < m} (which are in $L^\infty(\cX)$ by Proposition [p:exist-basis]). Similarly, we define the product modules M1M2⋯Mk over $\cZ$ for any given Mj ∈ Mij. These are clearly Γ-invariant. Then it is easy to check that $$L^2(\cY)=\sum\{M\_1\cdots M\_k: k \in \N, M\_j\in\mathcal{M}\_{i\_j}\text{ for some }i\_j \in I\}.$$ A standard orthogonalization procedure using Zorn’s lemma then yields $L^2(\cY)=\bigoplus\_s M'\_s$ for some appropriate finitely generated Γ-invariant $\cZ$-modules Mʹs. [c:maximal-cpct-extension] Let $\cX$ be an ergodic system and $\cZ \sub \cX$ be a factor of $\cX$. Then there exists a factor $\cY$ of $\cX$ such that $\cZ \sub \cY \sub \cX$, the extension $\cY \supseteq \cZ$ is compact, and every compact extension of $\cZ$ in $\cX$ is contained in $\cY$. This $\cY$ is called. Next we check that a factor of a compact extension is also compact. [l:factor-of-compact-is-compact] Suppose we have $\cZ\subseteq \cY\subseteq \cX$ and the extension $\cZ\subseteq \cX$ is compact. Then the extensions $\cZ\subseteq \cY$ and $\cY\subseteq\cX$ are compact. The projection of a finitely generated Γ-invariant $\cZ$-module in $L^2(\cX)$ to $L^2(\cY)$ is again a finitely generated Γ-invariant $\cZ$-module. This readily implies that $\cZ\subseteq \cY$ is compact. For the extension $\cY \sub \cX$, this is clear. Suppose that N is a finitely generated Γ-invariant $\cZ$-module and suppose that E = (e0, …, en − 1) is a $\cZ$-basis for N. Define the $m\_{E} \colon \Gamma \to M\_n(L^\infty(\cZ))$ by $$m\_E(\gamma) = \big( \ip{\gamma \cdot e\_i, e\_j}\_\cZ \big)\_{i,j < n}.$$ [l:dim-at-most-n] Suppose $\cX \supseteq \cZ$ is a compact extension with $\cX$ ergodic. Let E = (e1, …, en) be a $\cZ$-orthonormal basis for a finitely generated, Γ-invariant module $M \sub L^2(\cX)$. Then the set $\{E'\in L^2(\cX)^{\oplus n} : m\_{E'} = m\_E \}$ spans a subspace of $L^2(\cX)^{\oplus n}$ of dimension at most n. Let Eʹ = (eʹ1, …, eʹn) be a tuple with mEʹ = mE. We claim that the function $x \mapsto \ip{E(x), E'(x)}$ is Γ-invariant and therefore constant. (Here, again, $\ip{\cdot, \cdot}$ denotes the usual inner product in $\C^n$.) Indeed, given γ ∈ Γ and i, j < n, the conditional expectations $\ip{\gamma e\_i,e\_j}\_\cZ$ and $\ip{\gamma e'\_i,e'\_j}\_\cZ$ are equal. Hence, the $\cZ$-measurable map uγ: X → U(n) given by $u\_\gamma(x)=(\ip{\gamma e\_i,e\_j}\_\cZ(x))\_{ij}$ satisfies (γE)(x) = uγ(x)E(x) and also (γEʹ)(x) = uγ(x)Eʹ(x). Thus ⟨E(γ− 1x), Eʹ(γ− 1x)⟩ = ⟨uγ(x)E(x), uγ(x)Eʹ(x)⟩ = ⟨E(x), Eʹ(x)⟩, as desired. Now consider n + 1 tuples E0, …, En with mEi = mE. Consider the matrix $\bigl( \ip{E\_i(x), E\_j(x)} \bigr)\_{i, j \leq n}$, which, by the previous claim, does not depend on x. It has rank at most n, so there exist constants λ0, …, λn not all equal to 0 such that for all j and almost every x, $\sum\_{j = 0}^n \lambda\_i \ip{E\_i(x), E\_j(x)} = 0$. This implies that the function ∑iλiEi ∈ L2(X)⊕ n is orthogonal to each Ej, so equal to 0. Hence the Ei are linearly dependent. If $\cX$ is a Γ-system, we will denote by $\Aut(\cX)$ the group of automorphisms of $\cX$, i.e., the group of all automorphisms of the measure algebra that commute with the action of Γ. We equip $\Aut(\cX)$ with the pointwise convergence topology on $\cX$, which makes it into a topological group. $\Aut(\cX)$ can also be viewed as a closed subgroup of the unitary group $U(L^2(\cX))$ equipped with the strong operator topology. If $\cZ \sub \cX$ is a factor, we will denote $$\Aut\_\cZ(\cX) = \set{\sigma \in \Aut(\cX) : \sigma(a) = a \text{ for all } a \in \cZ}.$$ $\Aut\_\cZ(\cX)$ is a closed subgroup of $\Aut(\cX)$. [p:compact-aut-group] If $\cX \supseteq \cZ$ is a compact extension with $\cX$ ergodic, then $\Aut\_\cZ(\cX)$ is compact. Let $M \sub L^2(\cX)$ be a finitely generated, Γ-invariant $\cZ$-module. Let E = (e1, …, en) be a $\cZ$-basis for M. Then, by Lemma [l:dim-at-most-n], the orbit $\Aut\_\cZ(\cX) \cdot E$ is contained in a finite-dimensional subspace of $L^2(\cX)^{\oplus n}$. This implies that the orbit of every $f \in L^2(\cX)$ is contained in a finite-dimensional subspace of $L^2(\cX)$, so $\Aut\_{\cZ}(\cX)$, being a closed subgroup of a product of finite-dimensional unitary groups, is compact. By an isomorphism between Γ-invariant $\cZ$-modules we mean a linear isometry that commutes with the action of Γ and multiplication by elements of $L^\infty(\cZ)$. Note that an isomorphism between $\cZ$-modules preserves the conditional inner product $\ip{\cdot, \cdot}\_\cZ$. Note also that if E is a $\cZ$-basis for a module M and Eʹ is a $\cZ$-basis for a module Mʹ, then there is an isomorphism between M and Mʹ that maps E to Eʹ iff mE = mEʹ. We have the following consequence of the previous lemma. [p:comp-ext-size-bound] Suppose $\cX \supseteq \cZ$ is a compact extension with $\cX$ ergodic, and let $L^2(\cX)=\bigoplus\_{i\in I}M\_i$ be a decomposition into finitely generated, Γ-invariant $\cZ$-modules. Then, for each i ∈ I, the number of $\cZ$-modules appearing in the decomposition that are isomorphic to Mi is bounded by the size of a $\cZ$-basis of Mi and is therefore finite. In particular, $\|\cX\| \leq (\|\cZ\| + \aleph\_0)^{\aleph\_0}$. Suppose for contradiction that $M\_0, \ldots, M\_n \sub L^2(\cX)$ are isomorphic invariant submodules of $L^2(\cX)$ that are pairwise orthogonal. Let $E \in L^2(\cX)^{\oplus n}$ be a $\cZ$-basis of M0 and let fi: M0 → Mi be isomorphisms. Then f0(E), …, fn(E) are mutually orthogonal $\cZ$-bases of size n that contradict Lemma [l:dim-at-most-n]. For the second assertion, denote $\kappa = \|\cZ\| + \aleph\_0$ and observe that there are at most κℵ0 isomorphism classes of finitely generated, Γ-invariant $\cZ$-modules. Indeed, each such module is determined by its matrix coefficients $\ip{\gamma \cdot e\_i, e\_j}\_\cZ \in L^\infty(\cZ)$ for some $\cZ$-basis (e0, …, en − 1). As the cardinal of $L^\infty(\cZ)$ is κℵ0, this proves the claim. As each module can appear only finitely many times in the decomposition of $L^2(\cX)$ and has density character $\|\cZ\|$, this shows that $\|L^2(\cX)\| \leq \kappa^{\aleph\_0} \cdot \kappa = \kappa^{\aleph\_0}$. We recall now that an extension is distal if it can be obtained as a tower of compact extensions. More precisely, $\cZ\subseteq\cX$ is distal if there is an ordinal number α and intermediate extensions $\{\cX\_\beta\}\_{\beta\leq\alpha}$ such that * $\cZ=\cX\_0 \subseteq \cX\_\beta \subseteq \cX\_{\beta'}\subseteq \cX\_\alpha=\cX$ for β ≤ βʹ ≤ α, * each extension $\cX\_\beta\subseteq\cX\_{\beta+1}$ is compact, * if λ ≤ α is a limit ordinal, then $\cX\_\lambda$ is generated by the factors $\cX\_\beta$, β < λ. We will say that $\{\cX\_\beta\}\_{\beta\leq\alpha}$ is a distal tower for the extension $\cZ\subseteq\cX$. The minimal length α of such a tower is the rank of the distal extension, which we denote by $\operatorname{rk}(\cX/\cZ)$. If $\cZ$ is the trivial system, then $\cX$ is a distal system, and we denote the rank simply by $\operatorname{rk}(\cX)$. As for compact extensions, we have the following. [p:factor-generated-by-distal-factors] Let $\cX$ be an ergodic system and let $\cZ\_i\subseteq\cY\_i\subseteq \cX$, i ∈ I, be factors such that the extensions $\cZ\_i\subseteq\cY\_i$ are distal. Then the factor $\cY \sub \cX$ generated by all the $\cY\_i$ is a distal extension of the factor $\cZ$ generated by all the $\cZ\_i$. Moreover, $\operatorname{rk}(\cY/\cZ)\leq\sup\_{i\in I}\operatorname{rk}(\cY\_i/\cZ\_i)$. In particular, given an extension $\cZ\subseteq \cX$ with $\cX$ ergodic, there is a largest intermediate distal extension of $\cZ$ in $\cX$. Let $\alpha=\sup\_{i\in I}\operatorname{rk}(\cY\_i/\cZ\_i)$ and, for each i ∈ I, choose a distal tower $\{\cY\_{i,\beta}\}\_{\beta\leq\alpha}$ for the extension $\cZ\_i\subseteq \cY\_i$. Given β ≤ α, let $\cY\_\beta$ be the factor generated by all the $\cY\_{i,\beta}$ for i ∈ I. Then, using Proposition [p:factor-generated-by-compact-factors], it is clear that $\{\cY\_\beta\}\_{\beta\leq\alpha}$ is a distal tower for $\cZ\subseteq \cY$. [c:bound-on-distal-rank] If $\cZ\subseteq \cX$ is an ergodic, distal extension, then $\operatorname{rk}(\cX/\cZ)\leq (\|\cZ\|+\aleph\_0)^+$. It is enough to apply the previous proposition to the extensions $\cZ\subseteq \cW$ where $\cW$ varies over the factors of $\cX$ generated by $\cZ$ together with an additional element of $\cX$. Indeed, each such factor $\cW$ has density character at most $\kappa=\|\cZ\|+\aleph\_0$, hence $\operatorname{rk}(\cW/\cZ) < \kappa^+$. On the other hand, these factors generate $\cX$. A deeper result of the Furstenberg–Zimmer theory is that factors of distal extensions are again distal. We will need this fact for arbitrary (not necessarily separable) systems. The following lemma will be useful to transfer this result (and some others) from the standard theory to the non-separable setting. [l:transfer-lemma] Let $\cZ\subseteq\cX$ be a distal extension with $\cX$ ergodic and let $\cY$ be a separable factor of $\cX$. Then there are separable factors $\cZ' \subseteq \cY' \subseteq \cX$ with $\cZ' \subseteq \cZ$, $\cY \subseteq \cY'$ and such that the extension $\cZ' \subseteq \cY'$ is distal; moreover, $\operatorname{rk}(\cY'/\cZ')\leq \operatorname{rk}(\cX/\cZ)$. We proceed by induction on the distal rank $\alpha=\operatorname{rk}(\cX/\cZ)$. Let $(\cX\_\beta)\_{\beta\leq\alpha}$ be a distal tower for the extension $\cZ\subseteq\cX$. In the base case α = 0, we take $\cZ'=\cY'=\cY$. Suppose next that α = β + 1. There are finitely generated Γ-invariant $\cX\_\beta$-modules Mi such that $L^2(\cX\_\alpha)=\bigoplus\_{i\in I} M\_i$ and we can choose a countable subset J ⊆ I such that ⨁i ∈ JMi contains $L^2(\cY)$. Fix some Mi and a corresponding $\cX\_\beta$-basis e1, …, en. Since Γ is countable and $\cY$ is separable, we can find a separable factor $\cW \subseteq \cX\_\beta$ such that the $\cW$-module Mʹi generated by e1, …, en is Γ-invariant and contains the intersection $L^2(\cY)\cap M\_i$; moreover, since J is countable, we can choose $\cW$ which works for every i ∈ J. Then we consider the sum M = ⨁i ∈ JMʹi. By adding all the product modules (as defined in the proof of Proposition [p:factor-generated-by-compact-factors]) if necessary, we may assume that M defines a separable factor $\cY\_0\subseteq\cX$ (i.e., $M=L^2(\cY\_0)$) which contains $\cY$ and is a compact extension of $\cW$. Now we can apply the inductive hypothesis to get a distal extension $\cZ'\subseteq \cY\_1$ of separable factors of $\cX\_\beta$ such that $\cZ'\subseteq \cZ$ and $\cW \subseteq \cY\_1$, and moreover $\operatorname{rk}(\cY\_1/\cZ')\leq \beta$. Let $\cY'$ be the factor generated by $\cY\_0$ and $\cY\_1$. Then $\cY'$ is separable, contains $\cY$, and is a compact extension of $\cY\_1$, hence also a distal extension of $\cZ'$ with $\operatorname{rk}(\cY'/\cZ')\leq \alpha$. The pair $\cZ',\cY'$ is as required. Finally, if α is a limit ordinal, we can choose countably many βn < α such that $\cY$ is generated by the factors $\cY\cap\cX\_{\beta\_n}$. By the inductive hypothesis, we have separable distal extensions $\cZ\_n \subseteq\cY\_n$, of rank less than α, with $\cZ\_n\subseteq \cZ$ and $\cY\cap\cX\_{\beta\_n} \subseteq \cY\_n$. If $\cZ'$ and $\cY'$ are the factors generated respectively by the $\cZ\_n$ and the $\cY\_n$, then they are separable, $\cZ'\subseteq\cZ$, $\cY\subseteq\cY'$, and the extension $\cZ'\subseteq\cY'$ is distal of rank at most α by Proposition [p:factor-generated-by-distal-factors]. The following proposition states that factors of distal systems are distal; see also Proposition [p:rank-of-factors] for a refinement. [p:factor-of-distal-is-distal] Suppose $\cZ\subseteq \cY\subseteq \cX$ and $\cX$ is ergodic. Then, the extension $\cZ\subseteq \cX$ is distal if and only if the extensions $\cZ\subseteq \cY$ and $\cY\subseteq\cX$ are distal. The only difficulty is in proving that if $\cZ\subseteq \cX$ is distal, then so is $\cZ\subseteq \cY$. So assume the former extension is distal. If $\cX\_1$ and $\cX\_2$ are factors of $\cX$, we will denote by $\cX\_1\cX\_2$ the factor generated by $\cX\_1$ and $\cX\_2$. We prove first that if $\cW\subseteq\cX$ is a separable factor, then $\cZ\cW$ is a distal extension of $\cZ$. By Lemma [l:transfer-lemma], there are separable factors $\cZ' \subseteq \cW' \subseteq \cX$ with $\cZ' \subseteq \cZ$, $\cW \subseteq \cW'$ such that the extension $\cZ' \subseteq \cW'$ is distal. Then, by \Theorem 10.38, $\cZ'\cW$ is also a distal extension of $\cZ'$. Now applying Proposition [p:factor-generated-by-distal-factors] to the two distal extensions $\cZ'\cW \supseteq \cZ'$ and $\cZ \supseteq \cZ$ yields that the extension $\cZ\cW \supseteq \cZ$ is distal. Hence, since for each separable factor $\cW$ of $\cY$ the extension $\cZ\subseteq \cZ\cW$ is distal, the extension $\cZ\subseteq \cY$ is distal as well, again by Proposition [p:factor-generated-by-distal-factors]. We end this section with a well-known amalgamation result which is usually stated and proved for separable systems in the literature. [p:ergodic-joinings] Let $i\_1 \colon \cZ \to \cX\_1$ and $i\_2 \colon \cZ \to \cX\_2$ be embeddings of ergodic systems. Then there exists an ergodic system $\cY$ and embeddings $j\_1 \colon \cX\_1 \to \cY$ and $j\_2 \colon \cX\_2 \to \cY$ such that j1 ∘ i1 = j2 ∘ i2. If $\cY$ is moreover generated by $j\_1(\cX\_1)$ and $j\_2(\cX\_2)$, we say it is an. For simplicity of notation, we will identify $\cZ$ with its images $i\_1(\cZ) \sub \cX\_1$ and $i\_2(\cZ) \sub \cX\_2$. Let X1 and X2 be the Stone spaces of the Boolean algebras $\cX\_1$ and $\cX\_2$ and let μ1 and μ2 be the corresponding measures on X1 and X2. We identify $\cX\_i$ with the algebra of clopen sets of X*i. Denote by πi: X1 × X2 → Xi the projection maps. Let PΓ(X1 × X2) denote the compact, convex set of Radon probability measures on X1 × X2 that are invariant under the diagonal action of Γ. We consider $$\begin{gathered} J\_\cZ(\cX\_1, \cX\_2) \coloneqq \set[\big]{\nu \in P\_\Gamma(X\_1 \times X\_2) : (\pi\_1)\_\* \nu = \mu\_1, (\pi\_2)\_\\nu = \mu\_2 \And \\ \nu\big( (a \times X\_2) \sdiff (X\_1 \times a) \big) = 0 \text{ for all } a \in \cZ}, \end{gathered}$$ the set of joinings of $\cX\_1$ and $\cX\_2$ over $\cZ$, which is still compact and convex. Next we check that it is non-empty. For $a\_1\in \cX\_1$, $a\_2\in\cX\_2$, we define $$\lambda(a\_1 \times a\_2) \coloneqq \int\_Z \mu\_1(a\_1\|\cZ) \mu\_2(a\_2\|\cZ) \ud z.$$ It is easy to see that λ* extends to a finitely additive Γ-invariant measure on the Boolean algebra generated by the clopen rectangles. Thus λ defines a Γ-invariant linear functional on the dense subspace of C(X1 × X2) consisting of step functions over clopen rectangles. As it is also continuous for the sup norm, it extends to a linear functional on all of C(X1 × X2), i.e., to a Radon measure on X1 × X2. One readily checks that this measure belongs to $J\_{\cZ}(\cX\_1, \cX\_2)$. By the Krein–Milman theorem, there exists an extreme point λ0 of $J\_\cZ(\cX\_1, \cX\_2)$. We check that the system $\Gamma\actson (X\_1 \times X\_2, \cB, \lambda\_0)$, where $\cB$ is the Borel σ-algebra, is ergodic. For this, it is enough to see that λ0 is an extreme point of PΓ(X1 × X2). Suppose that λ0 = tλ1 + (1 − t)λ2 with λ1, λ2 ∈ PΓ(X1 × X2) and 0 < t < 1. Then μ1 = (π1)\*λ0 = t(π1)\*λ1 + (1 − t)(π1)\*λ2, which by the ergodicity of μ1 implies that (π1)\*λ1 = (π1)\*λ2 = μ1; similarly, (π2)\*λ1 = (π2)\*λ2 = μ2. The last condition in the definition of $J\_\cZ(\cX\_1, \cX\_2)$ is obviously satisfied because of positivity. The pmp system induced by the action of Γ on $(X\_1\times X\_2,\cB,\lambda)$, where λ is as in the proof of the previous proposition, is called the, and is denoted by $\cX\_1\otimes\_\cZ\cX\_2$. Universal ergodic distal systems ================================ Next we construct the largest ergodic compact extension and the largest ergodic distal extension of an ergodic system. We start by noting that we can amalgamate ergodic compact extensions. [l:compact-ext-prop] Let $\cZ$ be an ergodic system. Then the family of compact, ergodic extensions of $\cZ$ has the following properties: 1. [i:l:compact-ext-prop:1] It is directed: if $i\_1 \colon \cZ \to \cX\_1$ and $i\_2 \colon \cZ \to \cX\_2$ are compact, ergodic extensions of $\cZ$, then there exists a compact, ergodic extension $j \colon \cZ \to \cX\_3$ and embeddings $j\_1 \colon \cX\_1 \to \cX\_3$ and $j\_2 \colon \cX\_2 \to \cX\_3$ such that j1 ∘ i1 = j2 ∘ i2. 2. [i:l:compact-ext-prop:2] It is closed under direct limits. [i:l:compact-ext-prop:1] This follows from Propositions [p:ergodic-joinings] and [p:factor-generated-by-compact-factors]. [i:l:compact-ext-prop:2] Let $\cX=\varinjlim \cX\_\alpha$ be a direct limit of ergodic compact extensions of $\cZ$, and let us identify the factors $\cX\_\alpha$ with subsets of $\cX$. Then $\cX$ is generated by the $\cX\_\alpha$, hence is also a compact extension of $\cZ$. Ergodicity of $\cX$ follows readily from the consideration of the zero-mean invariant vectors in $L^2(\cX)$: their projection on each $L^2(\cX\_\alpha)$ is invariant, thus equal to zero; since the spaces $L^2(\cX\_\alpha)$ generate $L^2(\cX)$, this is enough. Next we show how to amalgamate the set of all ergodic compact extensions of a given system $\cZ$. If $\cZ$ is the trivial one-point system, this construction yields the well-known Bohr compactification of the group. We are grateful to Ehud Hrushovski for supplying the simple uniqueness argument below, which is more general than the one we originally had. A system $\cX$ is said to be coalescent over a factor $\cZ$ if every $\cZ$-embedding $\cX \to \cX$ (i.e., a self-embedding fixing $\cZ$ pointwise) is an isomorphism. [th:absolute-max-compact] Let $\cZ$ be any ergodic Γ-system. Then there exists a unique compact, ergodic extension $i\_0 \colon \cZ \to \widehat{\cZ}$ with the following property: if $i \colon \cZ \to \cX$ is any compact, ergodic extension, then there exists an embedding $j \colon \cX \to \widehat{\cZ}$ such that i0 = j ∘ i. Moreover $\|\widehat \cZ\| \leq (\|\cZ\| + \aleph\_0)^{\aleph\_0}$ and $\widehat \cZ$ is coalescent over $\cZ$. First we show existence. By Proposition [p:comp-ext-size-bound], there exists a set $\set{\cX\_\alpha : \alpha < \kappa}$ of representatives of all $\cZ$-isomorphism classes of compact, ergodic extensions of $\cZ$. We use Lemma [l:compact-ext-prop] to define another sequence $(\cX'\_\alpha)\_{\alpha \leq \kappa}$ of ergodic compact extensions of $\cZ$ with the property that $\cX\_\alpha \sub \cX'\_{\alpha}$ and $\cX'\_\alpha \subseteq \cX'\_{\alpha'}$ for α < αʹ ≤ κ. We set $\cX'\_0=\cX\_0$. For $\cX'\_{\alpha+1}$ we take any amalgam of $\cX\_\alpha$ and $\cX'\_\alpha$ over $\cZ$. Finally, if λ ≤ κ is a limit ordinal, we let $\cX'\_\lambda$ be the direct limit of the $\cX'\_\alpha$ for α < λ. The last system $\widehat \cZ \coloneqq \cX'\_\kappa$ is an ergodic compact extension of $\cZ$ that contains any other such extension as a factor. The bound on the density of $\widehat \cZ$ follows from Proposition [p:comp-ext-size-bound]. Uniqueness follows from coalescence. To prove coalescence, suppose that $\phi \colon \widehat{\cZ} \to \widehat{\cZ}$ is a $\cZ$-embedding which is not surjective. Let $\kappa = \|\widehat{\cZ}\|$ and consider the directed system $(\cZ\_\alpha : \alpha < \kappa^+)$ where each $\cZ\_\alpha$ is equal to $\widehat{\cZ}$, each map $\cZ\_\alpha \to \cZ\_{\alpha+1}$ is equal to ϕ, and for limit λ one takes the direct limit $\cZ'\_\lambda=\varinjlim\_{\alpha < \lambda} \cZ\_\alpha$ and a $\cZ$-embedding $\cZ'\_\lambda\to\widehat{\cZ}=\cZ\_\lambda$, and then defines the maps $\cZ\_\alpha\to\cZ\_\lambda$ for α < λ by the composition $\cZ\_\alpha\to\cZ'\_\lambda\to\cZ\_\lambda$. Then the limit $\varinjlim\_{\alpha<\kappa^+} \cZ\_\alpha$ is an ergodic, compact extension of $\cZ$ of density κ+, which contradicts the maximality of $\widehat{\cZ}$. The uniqueness of the construction has the following direct consequence. [c:autZ-to-authatZ] Every automorphism of $\cZ$ extends to an automorphism of $\widehat{\cZ}$. If $i\_0 \colon \cZ \to \widehat \cZ$ is as in the theorem and $\phi \colon \cZ \to \cZ$ is an automorphism, then $i = i\_0 \circ \phi\colon \cZ \to \widehat \cZ$ is also a universal, compact, ergodic extension of $\cZ$. By uniqueness, there is an isomorphism $\psi \colon \widehat \cZ \to \widehat \cZ$ such that ψ ∘ i0 = i = i0 ∘ ϕ. We can iterate the construction above and define an ergodic distal extension $\cZ\subseteq\cD\_\alpha(\cZ)$ for each ordinal number α in the natural way: * $\cD\_0(\cZ)=\cZ$; * $\cD\_{\alpha+1}(\cZ)=\widehat{\cD\_\alpha(\cZ)}$; * $\cD\_\lambda(\cZ)=\varinjlim\_{\beta<\lambda} \cD\_\beta(\cZ)$ if λ is a limit ordinal. Note that the tower $\cD\_\alpha(\cZ)$ is unique up to $\cZ$-isomorphism. More precisely, if α is an ordinal and $(\cD\_\beta)\_{\beta\leq\alpha}$, $(\cD'\_\beta)\_{\beta\leq\alpha}$ are two towers constructed as above, then there is an isomorphism $\cD\_\alpha\to\cD'\_\alpha$ that restricts to a $\cZ$-isomorphism $\cD\_\beta\to\cD'\_\beta$ for each β < α. This follows from Theorem [th:absolute-max-compact] and an easy induction using Corollary [c:autZ-to-authatZ]. Clearly, each $\cD\_\alpha(\cZ)$ is a distal extension of $\cZ$ of distal rank at most α. [th:universal-distal-systems] Let $\cZ$ be an ergodic system. Then the system $\cD\_\alpha(\cZ)$ is universal for the ergodic, distal extensions of $\cZ$ of rank at most α, i.e., any such system is a factor of $\cD\_\alpha(\cZ)$. Moreover, $\cD\_\alpha(\cZ)$ is coalescent over $\cZ$ and hence unique with this universal property. Let $(\cX\_\beta)\_{\beta\leq\alpha}$ be a distal tower over $\cZ$ of length α with $\cX\_\alpha$ ergodic. We construct by induction a sequence of embeddings $i\_\beta \colon \cX\_\beta \to \cD\_\beta(\cZ)$. We start with $i\_0 = \id$. For the successor step, suppose that iβ has already been constructed. Apply Proposition [p:ergodic-joinings] to find an ergodic joining $\cY$ of $\cX\_\beta \to \cX\_{\beta+1}$ and $\cX\_\beta \xrightarrow{i\_\beta} \cD\_\beta(\cZ)$ over $\cX\_\beta$. By Proposition [p:factor-generated-by-compact-factors], $\cY$ is a compact extension of $\cD\_\beta(\cZ)$, so by the universality of $\cD\_{\beta+1}$, there exists an embedding $\cY \to \cD\_{\beta+1}(\cZ)$ over $\cD\_\beta(\cZ)$. Now composing the embeddings $\cX\_{\beta +1} \to \cY$ and $\cY \to \cD\_{\beta+1}(\cZ)$ yields iβ + 1. If λ is a limit ordinal, one can just take iλ = ⋃α < λiβ. For the moreover assertion, we want to show that each self-embedding of $\cD\_\alpha(\cZ)$ fixing $\cZ$ is an isomorphism. We proceed by induction. Suppose this holds for every α < αʹ and let $\phi\colon \cD\_{\alpha'}(\cZ)\to \cD\_{\alpha'}(\cZ)$ be a $\cZ$-embedding. We show first that $\phi(\cD\_\alpha(\cZ))\subseteq \cD\_\alpha(\cZ)$ for each α < αʹ. Indeed, by Proposition [p:factor-generated-by-distal-factors], the system $\cW$ generated by $\phi(\cD\_\alpha(\cZ))$ and $\cD\_\alpha(\cZ)$ inside $\cD\_{\alpha'}(\cZ)$ is a distal extension of $\cZ$ of rank at most α, hence a factor of $\cD\_\alpha(\cZ)$ by the universal property we just proved. Thus we have embeddings $\cD\_\alpha(\cZ)\to\cW\to\cD\_\alpha(\cZ)$, and the inductive hypothesis implies they are surjective; hence $\phi(\cD\_\alpha(\cZ))\subseteq \cD\_\alpha(\cZ)$. Again by the inductive hypothesis, we actually have equality for each α < αʹ. Now if αʹ is of the form α + 1, the fact that ϕ is surjective follows from Theorem [th:absolute-max-compact]. The limit case is clear. The proof of Theorem [th:universal-distal-systems] implies that if α < αʹ and $\cY$ is a distal extension of $\cZ$ of rank at most α such that $\cY \sub \cD\_{\alpha'}(\cZ)$, then $\cY \sub \cD\_\alpha(\cZ)$. Thus within $\cD\_{\alpha'}(\cZ)$, the system $\cD\_\alpha(\cZ)$ can be defined as the maximal intermediate distal extension of $\cZ$ of rank at most α. [th:distal-god] Let $\cZ$ be an ergodic system and let $\kappa = \|\cZ\| + \aleph\_0$. Then we have $\cD\_{\kappa^++1}(\cZ)=\cD\_{\kappa^+}(\cZ)$, i.e., $\cD\_{\kappa^+}(\cZ)$ is the largest ergodic, distal extension of $\cZ$. In particular, every ergodic, distal extension of $\cZ$ has density at most κℵ0 + κ+. The bound on the rank follows from Corollary [c:bound-on-distal-rank]. Then, the density bound follows from Theorem [th:absolute-max-compact]. In particular, if we define $\cD\_\alpha(\Gamma)\coloneqq\cD\_\alpha(1)$ where 1 is the trivial 1-point system, then $\cD\_{\omega\_1}(\Gamma)$ is the largest ergodic, distal Γ-system. It is natural to ask what the actual distal rank of $\cD\_{\omega\_1}(\Gamma)$ is. For $\Gamma=\Z$ this is answered by a construction of Beleznay and Foreman. [th:rank-of-Domega1Z] We have $\cD\_\alpha(\Z)\subsetneq \cD\_{\alpha+1}(\Z)$ for all α < ω1. That is, $\operatorname{rk}(\cD\_{\omega\_1}(\Z))=\omega\_1$. In examples are built of ergodic, distal systems of arbitrarily high countable rank. Since the rank of a distal system is at least as large as the rank of any of its factors (see the Appendix), the result follows. On the other hand, there are groups Γ with no non-trivial compact systems (for example, finitely generated, simple, infinite groups) for which $\cD\_{\omega\_1}(\Gamma)=1$. We will show in Section [sec:groups-with-property-T] that for groups with property (T), all ergodic distal systems are compact, i.e., $\cD\_{\omega\_1}(\Gamma) = \cD\_1(\Gamma)$. The systems $\cD\_\alpha(\Gamma)$ give examples where many classical results in ergodic theory valid in the separable setting fail. We will mention two. Zimmer proved that if $(X, \cX, \mu)$ is any separable $\Z$-system and K is a non-trivial, compact, metrizable group, then there exists a cocycle $\rho \colon \Z \times X \to K$ which is not a coboundary. However, we have the following. [p:Zimmer-fail] Let $\cX = \cD\_{\omega\_1}(\Gamma)$ and let X be the probability space associated to $\cX$. Suppose that K is a compact group and ρ: Γ × X → K is a cocycle. Then ρ is a coboundary. Let L ≤ K be the of ρ. Then X × ρL is an ergodic, compact extension of $\cD\_{\omega\_1}(\Gamma)$, so it must be trivial. This means that L is trivial and ρ is a coboundary. Recall that an ergodic system $\cX$ is called if whenever $\cX\_1$ and $\cX\_2$ are two isomorphic copies of $\cX$ inside a larger ergodic system, then $\cX\_1 = \cX\_2$. If $\cX$ is not weakly mixing (for example, distal), 2-simplicity implies simplicity (which has a more complicated definition; see \Chapter 12 for more details). [p:Dalpha-simple] For every ordinal α* ≤ ω1, $\cD\_\alpha(\Gamma)$ is a simple system. Let $\cX = \cD\_\alpha(\Gamma)$ and suppose that $\cX\_1$ and $\cX\_2$ are two copies of $\cX$ in an ergodic system $\cY$. Then by Proposition [p:factor-generated-by-distal-factors], the system generated by $\cX\_1$ and $\cX\_2$ is distal of rank at most α. Now by universality and coalescence of $\cX$, we must have that $\cX\_1 = \cX\_2$. Proposition [p:Dalpha-simple] should be contrasted with a theorem of Veech (see \Theorem 12.1) that states that a separable, ergodic, simple system is either compact or weakly mixing. Combining the two, we have the following (see also Theorem [th:polish-AutD2] for a generalization). [c:from-Veech] Suppose that $\cD\_2(\Gamma)$ is a separable system. Then $\cD\_2(\Gamma) = \cD\_1(\Gamma)$ and every distal Γ-system is compact. We conclude this section with a simple observation for future reference. [p:inclusion-Dalpha] Let $\cX \sub \cY$ be an extension of ergodic systems. Then for every ordinal α, $\cD\_\alpha(\cX)$ is a factor of $\cD\_\alpha(\cY)$. Let $\cW$ be an ergodic joining of $\cY$ and $\cD\_\alpha(\cX)$ over $\cX$, which exists by Proposition [p:ergodic-joinings]. Then by Proposition [p:factor-generated-by-distal-factors], the extension $\cY \sub \cW$ is distal of rank at most α* and by the universal property of $\cD\_\alpha(\cY)$, $\cW$ is a factor of $\cD\_\alpha(\cY)$. Compact extensions and algebraic closure ======================================== Recall that an element of a structure M is algebraic over a subset A* ⊆ M* if it belongs to a compact A-definable subset of M. We say a ∈ M is algebraic if it is algebraic over ∅. The set $\acl^M(A)$ of all elements algebraic over A is the algebraic closure of A* in M, and forms a substructure of M. The substructure $\acl^M(\emptyset)\subseteq M$ is a fundamental model-theoretic invariant of M: it only depends on its first-order theory and it is a common substructure of all models of this theory. In the case of a Γ-system $\cX$, this defines a canonical factor $$\cX^{\operatorname{alg}} \coloneqq \acl^\cX(\emptyset) \subseteq\cX$$ and one may ask whether this is related to any natural notion in ergodic theory. If Γ is an infinite amenable group and the action $\Gamma \actson \cX$ is free, then $\acl^\cX(A)$ is just the factor generated by A* in $\cX$ (this follows from quantifier elimination). Hence $\cX^{\operatorname{alg}}$ is always the trivial factor. On the other hand, as we show below, for strongly ergodic actions the situation is completely different: if $\cX$ is strongly ergodic, then $\cX^{\operatorname{alg}}$ contains all distal factors of $\cX$. We will actually prove a stronger result by showing that algebraicity is witnessed by quantifier-free formulas (cf. Theorem [th:main-lemma]). Given a system $\cX$, a factor $\cZ$, and a tuple $F=(f\_i)\_{i\in I} \subset L^\infty(\cX)$, the is given by the joint conditional distribution over $\cZ$ of the tuple of random variables (γ ⋅ fi)γ ∈ Γ, i ∈ I. If F and Fʹ have the same quantifier-free type over $\cZ$, we will write $F \equiv\_\cZ F'$. In other words, if F = (fi) and Fʹ = (fʹi), we have that $F \equiv\_\cZ F'$ iff for every \* -polynomial P(x0, …, xn − 1) with coefficients from $L^\infty(\cZ)$, indices i0, …, in − 1 ∈ I and γ0, …, γn − 1 ∈ Γ, $$\E \big(P(\gamma\_0 \cdot f\_{i\_0}, \ldots, \gamma\_{n-1} \cdot f\_{i\_{n-1}})\big) = \E \big(P(\gamma\_0 \cdot f\_{i\_0}', \ldots, \gamma\_{n-1} \cdot f\_{i\_{n-1}}')\big).$$ In particular, the system generated by $\cZ$ and F is isomorphic to the one generated by $\cZ$ and Fʹ. The following is one of our main results. [th:main-lemma] Suppose $\cZ\subseteq\cX$ is a distal extension with $\cX$ strongly ergodic and $f\_0 \in L^\infty(\cX)$. Then the set $\set{f \in L^\infty(\cX) : f \equiv\_\cZ f\_0}$ is compact in the L2-norm. It is clear that the set is closed, so it is enough to prove that it is totally bounded. Suppose for contradiction that there exist $\eps > 0$ and $\set{f\_i : i < \omega} \subseteq L^\infty(\cX)$ such that $f\_i \equiv\_\cZ f\_0$ for all i and $\nm{f\_i - f\_j}\_2 \geq \eps$ for all i ≠ j. Since $\cX$ is a distal extension of $\cZ$, by Proposition [p:factor-of-distal-is-distal], the factor generated by $\cZ$ and f0 is a distal extension of $\cZ$. In view of Theorem [th:distal-god], let κ be a cardinal larger than the density of any ergodic, distal extension of $\cZ$. It follows from the compactness theorem that there exists an elementary extension $\cX'$ of $\cX$ and a sequence $(f'\_i)\_{i<\kappa}\subseteq L^\infty(\cX')$ such that $f\_i'\equiv\_\cZ f\_0$ for all i < κ and $\nm{f\_i' - f\_j'} \geq \eps$ for i ≠ j. The first condition ensures that for every i < κ, the factor generated by fʹi over $\cZ$ is a distal extension of $\cZ$. Hence, by Proposition [p:factor-generated-by-distal-factors], the factor $\cZ'$ generated by all the fʹi over $\cZ$ is also a distal extension of $\cZ$ and, moreover, $\|\cZ'\|\geq\kappa$ since (fʹi)i < κ is separated. On the other hand, $\cX'$ is ergodic since $\cX$ is strongly ergodic, and hence so is $\cZ'\subseteq \cX'$. This contradicts the choice of κ. [rem:after-main-lemma] If in Theorem [th:main-lemma] one assumes that the extension $\cZ \sub \cX$ is compact rather than distal, one does not need the strong ergodicity assumption on $\cX$ and the result follows easily from Lemma [l:dim-at-most-n]. Thus a more direct approach to Theorem [th:distal-strongly-erg-acl] below and many of the applications would use Lemma [l:dim-at-most-n] and then rely on the transitivity of the algebraic closure operator. Theorem [th:main-lemma] has the following model-theoretic consequence. [th:distal-strongly-erg-acl] Let $\cX$ be a strongly ergodic system and $\cZ \sub \cX$ be a factor. Then any distal extension of $\cZ$ in $\cX$ is contained in $\acl^\cX(\cZ)$. If $f, g \in L^\infty(\cX)$, we will write $\tp^\cX(f/\cZ) = \tp^\cX(g/\cZ)$ if f and g have the same type over $\cZ$, i.e., for any first-order formula ϕ(v) with parameters from $\cZ$, we have $\phi^\cX(f) = \phi^\cX(g)$. Let $\cX'$ be a sufficiently saturated elementary extension of $\cX$. Recall that for $f \in L^\infty(\cX)$, $$f \in \acl^\cX(\cZ) \iff \set{g \in L^\infty(\cX') : \tp^{\cX'}(g/\cZ) = \tp^{\cX'}(f/\cZ)} \text{ is compact}.$$ Let now $\cZ \sub \cY \sub \cX$ with $\cZ \sub \cY$ distal and let $f \in L^\infty(\cY)$. Note that $$\label{eq:1} \set{g \in L^\infty(\cX') : \tp^{\cX'}(g/\cZ) = \tp^{\cX'}(f/\cZ)} \sub \set{g \in L^\infty(\cX') : g \equiv\_\cZ f}.$$ Call the set on the right-hand side of A and let $\cW$ be the factor of $\cX'$ generated by A and $\cZ$. By Propositions [p:factor-of-distal-is-distal] and [p:factor-generated-by-distal-factors], the extension $\cZ \sub \cW$ is distal. Furthermore, $\cW$ is strongly ergodic as a factor of the strongly ergodic system $\cX'$. So by Theorem [th:main-lemma], A is compact and thus $f \in \acl^{\cX}(\cZ)$. These results have a number of corollaries. [c:Aut(distal)-is-compact] Let $\cZ\subseteq \cX$ be a distal extension with $\cX$ strongly ergodic. Then $\Aut\_\cZ(\cX)$ is compact. [c:distal-is-in-Xalg] Let $\cX$ be a strongly ergodic system and let $\cD$ be its maximal distal factor. Then $\cD \sub \cX^{\operatorname{alg}}$. In particular, $\cD$ is separable. The algebraic closure of the empty set is always separable if the language is countable. [rem:Bohr-T] Corollary [c:distal-is-in-Xalg] implies that any distal, ergodic system of a property (T) group is separable. In particular, the Bohr compactification of a property (T) group Γ is second countable (or, which is equivalent, Γ admits only countably many finite-dimensional unitary representations). This is a result of Wang . [c:Robin] Suppose $\cX$ is distal, strongly ergodic and the action $\Aut(\cX)\actson \cX$ is ergodic. Then $\cX$ is compact. By Corollary [c:Aut(distal)-is-compact], $K=\Aut(\cX)$ is compact. Let $L^2(\cX)=\bigoplus\_i M\_i$ be a decomposition into irreducible representations of K. For each i, let Ei denote the subspace generated by all the subrepresentations of $K \actson L^2(\cX)$ isomorphic to Mi. The ergodicity assumption and Lemma [l:dim-at-most-n] (which works for actions of arbitrary groups) imply that the Ei are finite-dimensional. They are also Γ-invariant and generate $L^2(\cX)$. This shows that $\cX$ is a compact system, as required. In, Gaboriau, Ioana and Tucker-Drob prove a cocycle superrigidity result for compact actions of product groups and countable targets. Corollary [c:Robin] above implies the following parallel result. We are grateful to Robin Tucker-Drob for suggesting this corollary. [c:Robin2] Let Γ and Λ be countable groups and $\Gamma \times \Lambda \actson X$ be a distal action such that the action $\Gamma \actson X$ is strongly ergodic and the action $\Lambda \actson X$ is ergodic. Then the original action $\Gamma \times \Lambda \actson X$ is compact. The following is the main result of (for separable systems). [Ioana–Tucker-Drob] [c:full-Ioana-TD] Let $\cX$ be a distal system and $\cY$ be strongly ergodic. If $\cX <\_w \cY$, then $\cX$ is a factor of $\cY$. By Lemma [l:wcon-existential], the condition $\cX <\_w \cY$ says that $\cX$ embeds as a factor of some elementary extension $\cY'$ of $\cY$. Thus, by Theorem [th:distal-strongly-erg-acl], $\cX \sub \cY'^{\operatorname{alg}}$. As the algebraic closure does not change with elementary extensions, this implies that $\cX \sub \cY^{\operatorname{alg}} \sub \cY$. The previous corollaries all follow from Theorem [th:distal-strongly-erg-acl]. Next we have an application that uses the full power of the Theorem [th:main-lemma]. Note that every ergodic, compact system is coalescent but this fails in general for ergodic, distal systems; see for an example of a non-coalescent ergodic $\Z$-action of distal rank 2. [th:distal-coalescent] Every strongly ergodic, distal system is coalescent. More generally, if $\cZ\subseteq \cX$ is a distal extension with $\cX$ strongly ergodic, then $\cX$ is coalescent over $\cZ$. Suppose $\phi\colon L^\infty(\cX)\to L^\infty(\cX)$ is a Γ-embedding that is the identity on $L^\infty(\cZ)$, but $f\in L^\infty(\cX)$ is not in the image of ϕ. Let $\epsilon=d(f,\phi(L^\infty(\cX)))>0$. Then for all i < j < ω, $\phi^i(f) \equiv\_\cZ \phi^j(f)$ and d(ϕi(f), ϕj(f)) ≥ ε. This contradicts Theorem [th:main-lemma]. [rem:Ioana] After reading a preliminary version of this paper, Adrian Ioana pointed out to us that it is possible to use \Lemma 2.6 (which implies that the semigroup of endomorphisms of a strongly ergodic, distal system is compact) instead of Theorem [th:main-lemma] in the proof above. [c:distal-wequiv] Let $\cX$ and $\cY$ be two strongly ergodic, distal systems which are weakly equivalent. Then they are isomorphic. By Corollary [c:full-Ioana-TD], $\cX$ and $\cY$ are bi-embeddable, which combined with Corollary [th:distal-coalescent] yields that they are isomorphic. Proof by model-theoretic stability ================================== In this section, we give a simple proof of Theorem [th:distal-strongly-erg-acl] that only uses basic stability theory. It requires more familiarity with model-theoretic notions than the previous sections. We will make use of the stable independence relation, i.e., the ternary relation $\Findep[T]{}$ that can be defined for any (possibly unstable) complete theory T* and that corresponds to non-forking restricted to stable formulas. We refer the reader to \Section 2.3, where this relation was considered in the classical, discrete setting, or to the more recent \Section 1, where the general metric case is discussed. We recall some terminology. If $\phi(\bar u,\bar v)$ is a stable formula, $\bar a$ is a $\bar u$-tuple from a model M ⊨ T, and B ⊆ M is any subset, the *ϕ-canonical base* of $\bar a$ over B, denoted by $\operatorname{Cb}^M\_\phi(\bar a/B)$, is the canonical parameter of the definition of the non-forking extensions of the ϕ-type of $\bar a$ over $\acl^{M^{\mathrm{meq}}}(B)$. The ϕ-canonical base is a metric imaginary element of M; we denote by Mmeq the expansion of M by all the metric imaginary sorts. We always have $\operatorname{Cb}^M\_\phi(\bar a/B)\in\acl^{M^{\mathrm{meq}}}(B)$. Then, given $\bar a,B,\bar c$ from M, we have that $\bar a\Findep[T]{B}\bar c$ (in words, $\bar a$ is stably independent from $\bar c$ over B*) iff for every stable formula ϕ* we have $\operatorname{Cb}^M\_\phi(\bar a/B\bar c)\in\acl^{M^{\mathrm{meq}}}(B)$, where $B\bar c$ is the union of B and $\bar c$. We also recall that non-forking extensions compatible with a given complete type always exist. [l:indep-existence-prop] For any small $\bar a,B,\bar c$ in a sufficiently saturated model M ⊨ T, there exists $\bar a'$ in M such that $\tp^M(\bar a'/B)=\tp^M(\bar a/B)$ and $\bar a'\Findep[T]{B} \bar c$. See Ben Yaacov \Corollary 2.4. If T* = APA is the theory of atomless probability measure algebras, which is stable, the relation $\Findep[T]{}$ coincides with the usual notion of independence in probability theory, and we shall denote it simply by $\Findep{}$. We recall that if $\cA$ and $\cC$ are subalgebras of some ambient model of APA generated respectively by the tuples $\bar a$ and $\bar c$, and if $\cB$ is a subalgebra of $\cC$, then for any formula ϕ and $e \in \cA$, the ϕ-canonical base of e over $\cC$ is definable from $\E(\chi\_e\|\cC) \in L^\infty(\cC)$, and one has $\bar a\Findep{\cB}\bar c$ iff $\E(\chi\_e\|\cC)\in L^\infty(\cB)$ for every $e \in \cA$. Given a Γ-system $\cX$, we will denote its complete first-order theory in the language of PMPΓ by $T^\cX$. In the proof of the next lemma, we will also need to consider the reduct of $\cX$ to the language of APA (i.e., forget the action of Γ). We denote this reduct by $\cX\|\_{\mathrm{APA}}$. We also write $\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}\coloneqq(\cX\|\_{\mathrm{APA}})^{\mathrm{meq}}$. Note that the imaginary sorts of $\cX\|\_{\mathrm{APA}}$ are also imaginary sorts of $\cX$, so $\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}$ is naturally included in $\cX^{\mathrm{meq}}$. The following is an instance of a general fact about independence and reducts when the reduct enjoys weak elimination of imaginaries. [l:Xmu-weak-elimination] Let $\cX$ be an atomless Γ-system, and let $\bar a,B,\bar c$ be taken from $\cX$. If $\bar a\Findep[T^\cX]{B} \bar c$, then $\bar a\Findep{\acl^\cX(B)}\bar c$. Suppose $\bar a\Findep[T^\cX]{B} \bar c$, and fix a formula $\phi(\bar u,\bar v)$ in the language of APA. Note that ϕ is stable. Let $e\coloneqq\operatorname{Cb}^\cX\_\phi(\bar a/B\bar c)=\operatorname{Cb}^\cX\_\phi(\bar a/\acl^\cX(B)\bar c)$ and $e'\coloneqq\operatorname{Cb}\_\phi^{\cX\|\_{\mathrm{APA}}}(\bar a/\acl^\cX(B)\bar c)$. Observe that both e and eʹ belong to $\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}$; in fact, up to fixing a uniform definition of ϕ-types, we can see e and eʹ as elements of the same corresponding sort of canonical parameters. Since APA has weak elimination of imaginaries, there is a real tuple $\bar b\subset\cX$ such that $e\in\acl^{\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}}(\bar b)$ and $\bar b\subset\acl^{\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}}(e)$. On the other hand, by the independence hypothesis, $e\in\acl^{\cX^{\mathrm{meq}}}(B)$. Hence $$\bar b\subset\acl^{\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}}(e)\cap\cX\subseteq\acl^{\cX^{\mathrm{meq}}}(B)\cap\cX=\acl^\cX(B),$$ and thus $e\in\acl^{\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}}(\acl^\cX(B))$. It follows that the extensions of ϕ-types defined by e are non-forking over $\acl^\cX(B)$ ($\subseteq\acl^\cX(B)\bar c$) in the sense of $\cX\|\_{\mathrm{APA}}$. Hence eʹ = e, and in particular, $e'\in\acl^{\cX\|\_{\mathrm{APA}}^{\mathrm{meq}}}(\acl^\cX(B))$. We conclude that $\bar a\Findep{\acl^\cX(B)}\bar c$. Now recall that an extension $\cZ\subseteq\cX$ of Γ-systems is weakly mixing if the relatively independent self-joining $\cX\otimes\_\cZ \cX$ is ergodic. [th:main-via-stability] Let $\cZ\subseteq\cX$ be an extension of Γ-systems with $\cX$ strongly ergodic. Then the extension $\acl^\cX(\cZ)\subseteq \cX$ is weakly mixing. We denote $\cB = \acl^\cX(\cZ)$. If $\cX$ is not atomless, then by ergodicity it must be finite and, in that case, $\cB=\cX$; so we may assume $\cX$ is atomless. Let $\cX'$ be a sufficiently saturated elementary extension of $\cX$, which must be ergodic by the assumption of strong ergodicity of $\cX$. Let $\bar a$ be a tuple enumerating the set $\cX$. By Lemma [l:indep-existence-prop], we can find a tuple $\bar a'$ in $\cX'$ with $\bar a' \equiv\_\cB \bar a$ and $\bar a' \Findep[T^\cX]{\cB} \bar a$. Hence, by Lemma [l:Xmu-weak-elimination], $\bar a' \Findep{\cB} \bar a$. The two conditions on $\bar a$ and $\bar a'$ say that the structure $\cW$ generated by $\bar a$ and $\bar a'$ inside $\cX'$ is isomorphic to the relatively independent self-joining $\cX\otimes\_\cB \cX$. But then, since $\cW\subseteq\cX'$, we conclude that $\cX\otimes\_\cB \cX$ is ergodic as required. We point out that Theorem [th:distal-strongly-erg-acl] is a direct consequence of this result, modulo the following simple remark. Let $\cX$ be an ergodic Γ-system and let $\cZ\subseteq \cY\subseteq\cX$ be factors. If the extension $\cY\subseteq\cX$ is weakly mixing, then $\cY$ contains the maximal distal extension of $\cZ$ in $\cX$. By induction, it suffices to show that $\cY$ contains $\cZ\_1$, the maximal compact extension of $\cZ$ in $\cX$. Let $\cW$ be the factor generated by $\cY$ and $\cZ\_1$. Then the extension $\cY\subseteq\cW$ is clearly weakly mixing and it is also compact by Proposition [p:factor-generated-by-compact-factors]. So it is enough to note that a compact, weakly mixing extension must be trivial or, which is the same, that a weakly mixing extension $\cY\subseteq\cW$ does not admit any finitely generated Γ-invariant submodule. Indeed, if e0, …, en − 1 is a $\cY$-basis for a non-trivial invariant $\cY$-module in $L^2(\cW)$, then the function $\sum\_{i<n}e\_i\otimes \conj{e\_i} \in L^2(\cW\otimes\_\cY \cW)$ is Γ-invariant but it is not constant. We thus see that one can obtain Theorem [th:distal-strongly-erg-acl] and its various ergodic-theoretic consequences by almost purely model-theoretic means. We may point out that this is also true of our result about coalescence, Theorem [th:distal-coalescent]. Indeed, using a somewhat lengthy (but elementary and purely model-theoretic) compactness argument, which we omit, one can improve the conclusion of Theorem [th:distal-strongly-erg-acl] to say that the maximal distal extension of $\cZ$ in $\cX$ is contained in $\operatorname{\acl\_\exists}^\cX(\cZ)$, the existential algebraic closure of $\cZ$. The set $\operatorname{\acl\_\exists}^\cX(\cZ)$ is the union of all compact subsets $A \sub \cX$ over $\cZ$, i.e., such that the distance function d( ⋅ , A) is a uniform limit of inf-formulas with parameters from $\cZ$. (This notion has the feature of inducing a closure operator $\operatorname{\acl\_\exists}\colon\mathcal{P}(\cX)\to\mathcal{P}(\cX)$, with properties similar to the ones of the usual algebraic closure.) Since every $\cZ$-embedding $\cX\to\cX$ must restrict to an isometry on every existentially $\cZ$-definable compact set, the condition $\cX=\operatorname{\acl\_\exists}^\cX(\cZ)$ readily implies that $\cX$ is coalescent over $\cZ$, and Theorem [th:distal-coalescent] follows. This alternative route (using compactness to pass from $\acl$ to $\acl\_\exists$) was our original approach to the coalescence result, before we realized that we could prove the stronger Theorem [th:main-lemma]; the proof of the latter requires more ergodic theory, but shows that algebraicity of strongly ergodic distal factors can be witnessed by quantifier-free formulas. Groups with property (T) ======================== Next, we consider actions of groups with property (T). Here we will be mostly interested in separable systems because, by Theorem [th:distal-strongly-erg-acl], if $\cZ$ is a separable system of a property (T) group (for example, the trivial one), then any distal extension of $\cZ$ is also separable. Thus we will make free use of the classical theory for separable systems, as presented, for example, in. Nevertheless, we state Theorem [th:propT] in full generality and prove it by a reduction to the separable case. It will be useful to consider the representation of ergodic compact extensions as skew-products. More precisely, recall that if $\cZ \sub \cX$ is an ergodic compact extension with $\cX$ separable, then there exists a compact, metrizable group K, a closed subgroup L ≤ K, and a cocycle ρ: Γ × Z → K such that X ≅ Z × ρK/L with the action of Γ given by γ ⋅ (z, kL) = (γ ⋅ z, ρ(γ, z)kL). Moreover, K can be chosen so that the system X ≅ Z × ρK is also an ergodic compact extension of $\cZ$. Our result about property (T) groups follows from the following more general theorem. [th:propT] Let $\cZ$ be a Γ-system and suppose that $\cD\_2(\cZ)$ is strongly ergodic. Then $\cD\_2(\cZ) = \cD\_1(\cZ)$. In particular, every distal extension of $\cZ$ is a compact extension. We write $\cD\_i$ instead of $\cD\_i(\cZ)$. We assume first that $\cZ$ is separable. As noted above, by Theorem [th:distal-strongly-erg-acl], both systems $\cD\_1$ and $\cD\_2$ are separable. We represent $\cD\_1$ and $\cZ$ as usual actions on probability spaces, by $\Gamma\actson D\_1$ and $\Gamma\actson Z$, respectively. Hence we can write D1 ≅ Z × ρ1K/L for appropriate K, L and ρ1 as above. In fact, we must have $L = \set{1\_{K}}$, otherwise the skew-product Z × ρ1K would be a proper extension of $\cD\_1$ which is still an ergodic compact extension of $\cZ$, contradicting the maximality of $\cD\_1$. Thus D1 ≅ Z × ρ1K. Note that K embeds as a subgroup of $\Aut\_\cZ(\cD\_1)$ by acting on Z × ρK on the right. Moreover, if we denote $$\cD\_1^K = \set{a \in \cD\_1 : k \cdot a = a \text{ for all } k \in K},$$ then, clearly, $\cD\_1^K = \cZ$. (We actually have $K=\Aut\_\cZ(\cD\_1)$, see \Proposition 6.16.) Now let $K\_2 = \Aut\_\cZ(\cD\_2)$, and consider the corresponding factor of invariant elements $\cD\_2^{K\_2} \subseteq \cD\_2$. We note that as by Corollary [c:autZ-to-authatZ], the restriction map $\Aut\_\cZ(\cD\_2) \to \Aut\_\cZ(\cD\_1)$ is surjective, we have $\cD\_1 \cap \cD\_2^{K\_2} = \cD\_1^K = \cZ$. Let $\cC$ be the maximal compact extension of $\cZ$ in $\cD\_2^{K\_2}$. Then $\cC \sub \cD\_1$, for $\cD\_1$ is the maximal intermediate compact extension of $\cZ$ within $\cD\_2$. Thus $\cC \sub \cZ$. As the extension $\cZ \sub \cD\_2^{K\_2}$ is distal (as an intermediate extension of the distal extension $\cZ \sub \cD\_2$), this implies that $\cZ = \cD\_2^{K\_2}$. Finally, by our hypothesis on $\cD\_2$ and Corollary [c:Aut(distal)-is-compact], the group K2 is compact, i.e., $\cZ = \cD\_2^{K\_2} \sub \cD\_2$ is a. By \Theorems 3.29 and 9.14, it must be a compact extension, so $\cD\_2 = \cD\_1$. We conclude that $\cD\_\alpha = \cD\_1$ for all α. By Theorem [th:universal-distal-systems], this means that every distal extension of $\cZ$ is a factor of $\cD\_1$, and thus a compact extension of $\cZ$ as per Lemma [l:factor-of-compact-is-compact]. Now suppose $\cZ$ is an arbitrary ergodic system with $\cD\_2(\cZ)$ strongly ergodic. Let $\cY\subseteq \cD\_2$ be any separable factor. By Lemma [l:transfer-lemma], there are separable factors $\cZ'\subseteq \cY'\subseteq \cD\_2$ with $\cZ'\sub \cZ$ and $\cY\subseteq\cY'$ such that the extension $\cZ'\subseteq\cY'$ is distal. As $\cD\_2(\cZ')$ is a factor of $\cD\_2(\cZ)$ (Proposition [p:inclusion-Dalpha]), it is also strongly ergodic and by what we have just shown in the separable case, the extension $\cZ' \sub \cY'$ is compact. Hence the system generated by $\cY$ and $\cZ$ is a compact extension of $\cZ$, and by Proposition [p:factor-generated-by-compact-factors], we can conclude that so is $\cD\_2$. So again $\cD\_2 = \cD\_1$ and we deduce as before that every ergodic distal extension of $\cZ$ is actually a compact extension. [c:T-distal-implies-compact] Suppose that Γ has property (T). Then the following hold: 1. If $\cZ$ is any ergodic Γ-system, then every distal extension of $\cZ$ is a compact extension. 2. [i:cTdic:2] Every ergodic, distal Γ-system is compact. If one is just interested in item [i:cTdic:2] of the last corollary, one can also obtain it as a consequence of Corollaries [c:from-Veech] and [c:distal-is-in-Xalg]. A variant of the previous argument in the case $\cZ=1$ (see below) shows that the conclusion $\cD\_2(\Gamma)=\cD\_1(\Gamma)$ can be obtained by assuming only that $\Aut(\cD\_2(\Gamma))$ is compact. On the other hand, note that in the exact sequence $$\label{eq:exact-sequence} 1 \to \Aut\_{\cD\_1(\Gamma)}(\cD\_2(\Gamma)) \to \Aut(\cD\_2(\Gamma)) \to \Aut(\cD\_1(\Gamma)) \to 1$$ both groups to the sides of $\Aut(\cD\_2(\Gamma))$ are always compact. If $\Aut(\cD\_2(\Gamma))$ is Polish, this implies that $\Aut(\cD\_2(\Gamma))$ is also compact. Therefore we obtain the following strengthening of Corollary [c:from-Veech]. [th:polish-AutD2] Suppose the group $\Aut(\cD\_2(\Gamma))$ is either compact or Polish. Then, $\cD\_2(\Gamma)=\cD\_1(\Gamma)$. We write $\cD\_i$ instead of $\cD\_i(\Gamma)$ and Ki instead of $\Aut(\cD\_i)$. In either case of the statement we have that K2 is compact. Indeed, K1 is clearly compact and $\Aut\_{\cD\_1}(\cD\_2)$ is compact by Proposition [p:compact-aut-group]. If K2 is Polish, then the map K2 → K1 in must be open and we can apply \Theorem 5.25 to conclude that K2 is compact. Now, as in the proof of Theorem [th:propT], the surjectivity of the restriction map K2 → K1 gives $\cD\_1\cap\cD\_2^{K\_2}=\cD\_1^{K\_1}=1$, hence the factor of invariant elements $\cD\_2^{K\_2}$ must be trivial. In other words, the action $K\_2\actson \cD\_2$ is ergodic. As in the proof of Corollary [c:Robin], this implies that $\cD\_2$ is compact, as required. [rem:D1-separable] In view of Theorem [th:polish-AutD2], one might ask whether it is enough to assume that $\cD\_1(\Gamma)$ is separable in order to conclude that $\cD\_2(\Gamma) = \cD\_1(\Gamma)$. The answer is negative, as shown by the following example. Let Γ be the Grigorchuk group (see for details on this group). First, any unitary representation of Γ factors through a finite group \p. 224. On the other hand, Γ is residually finite and finitely generated, so its profinite completion, which by the previous remark is equal to its Bohr compactification, is an infinite, metrizable, compact group. In other words, the probability space of $\cD\_1(\Gamma)$ is standard and non-atomic. The group Γ is amenable, so by, the action $\Gamma \actson \cD\_1(\Gamma)$ is orbit equivalent to an action of $\Z$, and thus, by Zimmer , there exist non-trivial cocycles $\Gamma \times \cD\_1(\Gamma) \to K$ for any compact, metrizable K. Hence $\cD\_1(\Gamma) \subsetneq \cD\_2(\Gamma)$ Distal rank and factors ======================= It is a folklore result that the distal rank of a factor cannot be larger than the distal rank of the original system, at least for actions of $\Z$. We have used this to deduce Theorem [th:rank-of-Domega1Z]. Since we could not find a proof in the literature, we include one here. We thank Eli Glasner, Matt Foreman and Jean-Paul Thouvenot for some helpful hints. We use again the convention that if $\cX$ and $\cY$ are factors of a system $\cW$, then $\cX\cY$ denotes the factor generated by them inside $\cW$. We also recall that we write $\cX\Findep{\cZ} \cY$ to say that $\cX$ and $\cY$ are relatively independent over $\cZ$. [p:Furst:max-comp-factor-of-a-product] Let $\cX$ and $\cY$ be separable Γ-systems inside some larger system, with a common factor $\cZ$. Assume $\cX$ is ergodic and $\cX\Findep{\cZ} \cY$. Let $\cY\_1$ be the maximal compact extension of $\cZ$ in $\cY$. Then $\cX\cY\_1$ is the maximal compact extension of $\cX$ inside $\cX\cY$. For $\Z$-actions this is exactly Theorem 7.4 in Furstenberg’s article, and its proof also works for actions of arbitrary countable groups. See also Glasner’s book (Chapter 9, Sections 3 and 4, and particularly Theorem 9.21), where a variant of this theorem is proved for arbitrary Γ-actions along the same lines as in (this variant corresponds precisely to Theorem 7.1 in ). [p:rank-of-factors] Suppose Γ is a countable group and $\cZ\subseteq \cY\subseteq \cX$ are Γ-systems with $\cX$ ergodic and the extensions $\cZ \sub \cY$ and $\cY \sub \cX$ distal. Then $\operatorname{rk}(\cY/\cZ)\leq \operatorname{rk}(\cX/\cZ)$. If this holds in the separable case, then the general case can be deduced from it exactly as in the proof of Proposition [p:factor-of-distal-is-distal] (note that the bounds for the distal rank that appear in Lemma [l:transfer-lemma] and Proposition [p:factor-generated-by-distal-factors] then become relevant). So we assume $\cX$ is separable. For each ordinal η, let $\cX\_\eta$ and $\cY\_\eta$ be the maximal distal intermediate extensions of $\cZ$ of rank at most η inside $\cX$ and $\cY$, respectively. If $\alpha=\operatorname{rk}(\cX/\cZ)$ and $\beta=\operatorname{rk}(\cY/\cZ)$, then $(\cX\_\eta)\_{\eta\leq\alpha}$ and $(\cY\_\eta)\_{\eta\leq\beta}$ are the corresponding distal towers for $\cX$ and $\cY$ over $\cZ$ and each $\cX\_{\eta+1}$ is the maximal compact extension of $\cX\_\eta$ inside $\cX$, and similarly for $\cY\_{\eta+1}$. We observe first that by the maximality of the $\cX\_\eta$, we have $\cY\_\eta\subseteq \cX\_\eta$ for every η. Next we show by induction that $\cX\_\eta \Findep{\cY\_\eta} \cY$ for each η. Thus, in particular, $\cX\Findep{\cY\_\alpha} \cY$, which implies $\cY\_\alpha=\cY$ and hence β ≤ α, as required. For η = 0, the claim becomes $\cZ\Findep{\cZ} \cY$ and holds trivially. Suppose inductively that $\cX\_\eta \Findep{\cY\_\eta} \cY$. By Proposition [p:Furst:max-comp-factor-of-a-product], $\cX\_\eta\cY\_{\eta+1}$ is the maximal compact extension of $\cX\_\eta$ inside $\cX\_\eta\cY$. On the other hand, the projection to $\cX\_\eta\cY$ of any finitely generated Γ-invariant module over $\cX\_\eta$ is again a finitely generated Γ-invariant module over $\cX\_\eta$. It follows that the projection of $L^2(\cX\_{\eta+1})$ to $L^2(\cX\_\eta\cY)$ is contained in $L^2(\cX\_\eta\cY\_{\eta+1})$ or, in other words, that $$\cX\_{\eta+1} \Findep{\cX\_\eta \cY\_{\eta+1}} \cX\_\eta\cY.$$ This together with the induction hypothesis implies that $\cX\_{\eta+1} \Findep{\cY\_{\eta+1}} \cY$. Finally, suppose η is a limit ordinal and $\cX\_\xi \Findep{\cY\_\xi} \cY$ for every ξ < η. We can write any $f\in L^2(\cX\_\eta)$ as f = ∑ξ < ηfξ for certain $f\_\xi\in L^2(\cX\_\xi)$. Hence, $$\E(f\|\cY)=\sum\_{\xi<\eta} \E(f\_\xi\|\cY)= \sum\_{\xi<\eta} \E(f\_\xi\|\cY\_\xi) \in L^2(\cY\_\eta).$$ This shows that $\cX\_\eta \Findep{\cY\_\eta} \cY$ and concludes the proof. A model-theoretic approach to rigidity of strongly ergodic, distal actions ========================================================================== We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other distal, ergodic system of the group as a factor. Introduction ============ The theory of measure-preserving dynamical systems was initiated by Halmos and von Neumann, who characterized the ergodic systems of the group of integers $\Z$ that can be represented as translations on a compact group in terms of their spectrum. This was extended by Mackey to general locally compact groups. Later, inspired by the work of Furstenberg in topological dynamics, Furstenberg and Zimmer considered the relative notion of a of a system and defined a system to be one obtained via a transfinite tower of compact extensions, starting from the trivial system. The notion of a distal system was central to Furstenberg’s ergodic-theoretic proof of Szemerédi’s theorem, and led to the Furstenberg–Zimmer structure theorem for general ergodic actions of locally compact groups. In the literature, compact systems are often referred to as or having (or ). Similarly, distal systems are also known as systems with. In this paper, we will use the terminology “compact” and “distal”. The notion of of measure-preserving systems of countable groups was introduced by Kechris as a weakening of the notion of a : a system $\cX$ is in another system $\cY$ (notation: $\cX <\_w \cY$) if $\cX$ is a factor of an ultrapower of $\cY$. Two systems $\cX$ and $\cY$ are if $\cX <\_w \cY$ and $\cY <\_w \cX$. It turns out that weak equivalence is a equivalence relation (the set of equivalence classes is compact ) and that many notions in ergodic theory are invariants of weak equivalence. Free actions of infinite amenable groups are all weakly equivalent, but non-amenable groups usually admit uncountably many classes of weak equivalence . By a result of Tucker-Drob , weak equivalence classes of free actions always contain uncountably many isomorphism classes (and they are not even classifiable by countable structures). This is why rigidity results that allow to recover the isomorphism class from the weak equivalence class of a system in special situations are particularly interesting. Two rigidity results about weak equivalence have appeared in the literature. The first one, due to Abért and Elek , states that if a system $\cX$ is weakly contained in a strongly ergodic system $\cY$, then $\cX$ is a factor of $\cY$. This was then generalized by Ioana and Tucker-Drob to distal systems. From this, they were able to conclude that for compact, strongly ergodic systems, weak equivalence implies isomorphism, because compact ergodic systems are, i.e., every endomorphism of the system is an automorphism. However, this is not true in general for distal systems (see Parry and Walters ). Nevertheless it turns out to be true for strongly ergodic, distal systems and we were able to prove the following. [th:i:distal-rigidity] Let Γ be a countable group. 1. Let $\cX$ be a distal, strongly ergodic, probability measure-preserving Γ-system. Then $\cX$ is coalescent and $\Aut(\Gamma\actson\cX)$ is compact. 2. Let $\cX$ and $\cY$ be two distal, strongly ergodic, probability measure-preserving Γ-systems. If they are weakly equivalent, then they are isomorphic. In the course of the proof of Theorem [th:i:distal-rigidity], we also give a new proof of the rigidity result of Ioana and Tucker-Drob mentioned above. The methods we use come from continuous logic, a relatively new branch of model theory, suitable for studying metric structures. While some model-theoretic concepts, such as ultraproducts, have been used before in ergodic theory, to our knowledge, this is the first application of continuous logic to dynamics of countable groups. The main model-theoretic notion behind the proof is that of : if M is a metric structure, $A \sub M$, and b ∈ M, we say that b is in the of A (notation: $b \in \acl^M(A)$) if it belongs to a compact set definable over A. Our main model-theoretic result can be stated as follows. [th:i:acl] Let $\cX$ be a strongly ergodic system, $\cZ$ be a factor of $\cX$ and $\cY$ be a distal extension of $\cZ$ in $\cX$. Then $\cY \sub \acl^\cX(\cZ)$. A slight strengthening of Theorem [th:i:acl] (replacing $\acl$ by $\operatorname{\acl\_\exists}$) easily implies Theorem [th:i:distal-rigidity]. We provide two rather different proofs of Theorem [th:i:acl], one based on the theory of compact extensions, and another using model-theoretic stability theory. Another feature of our approach is that it relies, in part, on the existence of certain canonical universal systems. Those systems are usually non-separable, i.e., they cannot be realized on a standard probability space. Working on a standard space is an assumption often made in the literature, either out of habit, or for the more essential reason that many important results fail in the non-separable setting (we give several examples in this paper). We depart from this tradition and make no separability assumptions throughout the paper unless specifically stated. A posteriori, somewhat surprisingly, it turns out that all strongly ergodic, distal systems are separable (Corollary [c:distal-is-in-Xalg]). It is possible to carry out the stability-theoretic proof presented in Section [sec:two-line-proof] by only using ergodic theoretic results in a separable situation. The following is our main result regarding universal systems. [th:i:universal-distal] Let Γ be a countable group and α be an ordinal. Then there exists a unique ergodic Γ-system $\cD\_\alpha(\Gamma)$ of distal rank at most α which contains every ergodic Γ-system of distal rank at most α as a factor. Moreover, $\cD\_{\omega\_1}(\Gamma) = \cD\_{\omega\_1+1}(\Gamma)$, that is, $\cD\_{\omega\_1}(\Gamma)$ is the universal ergodic, distal Γ-system. An analogous result bounding the rank of a topological distal minimal system for $\Gamma = \Z$ was proved by Beleznay and Foreman . For α = 1, the existence of a universal system is well-known: the maximal ergodic, compact system of Γ is a translation on a compact group known as the. For many groups (for example, $\Z$), the Bohr compactification is not metrizable, i.e., the corresponding dynamical system is not separable. However, it is a result of Wang that if Γ has property (T), then its Bohr compactification is a metrizable group. We recover this result as a consequence of Theorem [th:i:acl] and prove something more: the hierarchy of distal systems for property (T) groups collapses. [th:i:propT] Let Γ have property (T) and $\cZ$ be an ergodic Γ-system. Then every ergodic, distal extension of $\cZ$ is a compact extension. This theorem had been previously proved by Chifan and Peterson (unpublished) but our proof is different and independent from theirs. We would like to thank them for telling us about their result. For $\Gamma = \Z$, Beleznay and Foreman have proved that there exist separable ergodic, distal systems of rank α for every α < ω1. This implies that the tower of universal distal systems $(\cD\_\alpha(\Z))\_{\alpha \leq \omega\_1}$ is proper. [q:rank-distal] Let Γ be a countable group and suppose that $\cD\_2(\Gamma)$ is not strongly ergodic. Is it true that $\cD\_\alpha(\Gamma) \subsetneq \cD\_{\alpha+1}(\Gamma)$ for all α < ω1? In Theorem [th:propT] we prove that if $\cD\_2(\Gamma)$ is strongly ergodic, then every distal system of Γ is compact, so the condition that $\cD\_2(\Gamma)$ is not strongly ergodic is necessary. Until recently, it was an open question whether strongly ergodic, distal, non-compact systems exist. It was resolved by Glasner and Weiss who, using results of Bourgain and Gamburd on compact systems with spectral gap, constructed examples for Γ a free group. Finally, we would like to mention that the assumption that Γ is countable is made throughout the paper for simplicity. All results with appropriate modifications (for example replacing ω1 by ∣Γ∣+) hold for arbitrary discrete groups Γ. On the other hand, for the moment, our methods do not allow us to treat non-discrete, locally compact groups. The paper is organized as follows. In Section [sec:model-theor-fram], we describe how to treat probability measure-preserving dynamical systems in a model-theoretic framework and study the connection between weak containment and the existential theory. In Section [sec:basic-facts-cpct-exts], we develop some of the theory of compact extensions without separability assumptions. In Section [sec:universal-distal-systems], we consider universal systems and prove Theorem [th:i:universal-distal]. In Sections [sec:comp-extens-algebr] and [sec:two-line-proof] we give two proofs of Theorem [th:i:acl] and derive Theorem [th:i:distal-rigidity] from it. Finally, in Section [sec:groups-with-property-T], we consider groups with property (T). In an appendix, we include a proof of the fact that the distal rank does not increase on factors. Acknowledgments --------------- We are grateful to Matt Foreman, Eli Glasner, François Le Maître and Robin Tucker-Drob for useful discussions. Research was partially supported by the ANR projects AGRUME (ANR-17-CE40-0026) and GAMME (ANR-14-CE25-0004). A model-theoretic framework for ergodic theory ============================================== The language and the basic theory --------------------------------- In this section, we explain how to formalize probability measure-preserving (pmp) group actions within the framework of continuous logic. The structures of interest are measure-preserving group actions $\Gamma \actson (X, \cX, \mu)$ where Γ is a discrete group and $(X, \cX, \mu)$ is a probability space. As continuous logic deals with metric structures, it is natural to consider the measure algebra $\MALG(X, \cX, \mu)$ obtained by taking the quotient of $\cX$ by the equivalence relation of having symmetric difference of measure 0. We will abuse notation and write again $\cX$ for this quotient. It becomes naturally a complete metric space if endowed with the metric $d\_\mu(a, b) = \mu(a \sdiff b)$. By duality, a measure-preserving action $\Gamma \actson (X, \mu)$ is nothing but an action $\Gamma \actson \cX$ by isometries that also preserves the algebra structure. The signature that we consider will consist of the language of Boolean algebras $\set{\cap, \cup, \sminus, \emptyset, X}$ (here $\sminus$ is set difference and X stands for the maximum element of the Boolean algebra), a [0, 1]-valued predicate for the measure μ, and a unary function symbol for every element γ ∈ Γ. For a metric structure $\cX$, it is natural to measure its size using the (or just ), i.e., the smallest cardinal of a dense subset of $\cX$. We will denote the density of $\cX$ by $\|\cX\|$. Note that if $\cX$ is an infinite measure algebra, then $\|\cX\| = \|L^2(\cX)\|$ and $\|\cX\| \leq \aleph\_0$ iff $\cX$ is isomorphic to the measure algebra of a, i.e., one that can be realized by a Borel measure on the interval [0, 1]. [rem:Linfty] Sometimes it will be useful to consider the richer von Neumann algebra $L^\infty(X, \cX, \mu)$. It is a metric structure in the language that includes sorts for all $\nm{\cdot}\_\infty$-balls and symbols for addition, multiplication, the adjoint, and the L2-norm (which defines the metric). Note that the $\nm{\cdot}\_\infty$-norm is not part of the language. It follows from that the structures $\cX$ and $L^\	arxiv_0000751
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The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.[↩](#fnref1) 2. This is an instance of a more general fact that f induces a simplical map $f'\colon \operatorname{\text{\normalfont \sffamily Hom}}(\rel K,\rel H) \to \operatorname{\text{\normalfont \sffamily Hom}}(\rel K,\rel G)$ for each $\rel K$. This map is defined by fʹ(g): x ↦ f(g(x)).[↩](#fnref2) 3. We use notation [ \* ] both for sets {1, …, n} and for homotopy classes; the meaning will always be clear from the context.[↩](#fnref3) 4. This is in fact the composition of two adjoint pairs: taking sym and sub as functions from digraphs to graphs and the inclusion function ι from graphs to digraphs, we have $\operatorname{sym}\rel D \to \rel G$ if and only if $\rel D \to \iota \rel G$ and $\iota \rel G \to \rel D$ if and only if $\rel G \to \operatorname{sub}\rel D$ for all graphs $\rel G$ and digraphs $\rel D$.[↩](#fnref4) Topology and adjunction in promise constraint satisfaction ========================================================== ### 29 September 2022 [1](#fn1) The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a c-colouring of a graph that is promised to be k-colourable, where c ≥ k. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems. Introduction ============ In this paper we investigate the complexity of finding an approximate solution to fully satisfiable instances of constraint satisfaction problems. For example, for the classical problem of k-colouring a graph, one natural approximation version is the approximate graph colouring problem: The goal is to find a c-colouring of a given k-colourable graph, where c ≥ k ≥ 3. There is a huge gap in our understanding of the complexity of this problem. For k = 3, the best known efficient algorithm uses roughly c = O(n0.199) colours where n is the number of vertices of the graph. It has been long conjectured the problem is -hard for any fixed constants c ≥ k ≥ 3, but, say for k = 3, the state-of-the-art has only recently been improved from c = 4 to c = 5. Graph colouring problems naturally generalise to graph homomorphism problems and further to constraint satisfaction problems (CSPs). In a graph homomorphism problem, one is given two graphs and needs to decide whether there is a homomorphism (an edge-preserving map) from the first graph to the second. The CSP is a generalisation that uses arbitrary relational structures in place of graphs. One particularly important case that attracted much attention is when the second graph/structure is fixed; this is the so-called non-uniform CSP . This is also the only case we consider in this article. For graph homomorphisms, this is known as the $\rel H$-colouring problem: decide whether a given graph has a homomorphism to a fixed graph $\rel H$. The vs. -complete dichotomy of $\rel H$-colouring given in was one of the base cases that supported the Feder-Vardi dichotomy conjecture for CSPs. The study of the complexity of the CSP and the complete resolution of the CSP dichotomy conjecture was greatly influenced by the algebraic approach (see survey ). This approach has also made important contributions to the study of approximability of CSPs (e.g. ). Brakensiek and Guruswami suggested that perhaps progress on approximate graph colouring and similar open problems can be made by looking at a broader picture, by extending it to promise graph homomorphism and further to the promise constraint satisfaction problem (PCSP). Promise graph homomorphism is an approximation version of the graph homomorphism problem in the following sense: in $\operatorname{PCSP}(\rel H,\rel G)$, we fix (not one but) two graphs $\rel H$ and $\rel G$ such that there is a homomorphism from $\rel H$ to $\rel G$ (we write $\rel H\rightarrow \rel G$ to denote this). The goal is then to find a $\rel G$-colouring for a given graph when an $\rel H$-colouring is guaranteed to exist (but not given as part of input). The promise is that the input graph is always $\rel H$-colourable and hence $\rel G$-colourable as well. The PCSP is a natural generalisation of this to arbitrary relational structures, or in other words, a generalisation of the decision CSP to the promise setting. Brakensiek and Guruswami proposed a conjecture that $\operatorname{PCSP}(\rel H,\rel G)$ is NP-hard for all non-bipartite loopless graphs $\rel H$ and $\rel G$ such that $\rel H\rightarrow\rel G$. This would generalise the approximate graph colouring conjecture and greatly extend the Hell-Nešetřil dichotomy for $\rel H$-colouring . Given the huge success of the algebraic approach to the CSP, it is natural to investigate what it can do for PCSPs. This investigation was started by Austrin, Håstad, and Guruswami, with an application to a promise version of SAT. It was further developed by Brakensiek, Guruswami and others and applied to a range of problems, including versions of approximate graph and hypergraph colouring. A recent paper describes a general abstract algebraic theory for PCSPs, which shows, in particular, how algebraic properties precisely capture the power of gadget reductions in PCSPs. However, the algebraic theory of PCSPs is still very young and much remains to be done both in further developing it and in applying it to specific problems. We note that the aforementioned -hardness of 5-colouring a given 3-colourable graph was proved in by applying this abstract theory. The gist of the algebraic theory is that the complexity of $\operatorname{PCSP}(\rel H,\rel G)$ depends only on (certain properties of) polymorphisms, which are multi-variable functions that can be defined as homomorphisms from direct powers $\rel H^n$ into $\rel G$. However, the analysis of polymorphisms is in general a highly non-trivial task, and powerful tools are needed to conduct it. For resolving the CSP dichotomy conjecture, the structural theory of finite universal algebras provided such a tool. However, it is not clear how much this theory can be applied to the promise setting. In this paper, we show that algebraic topology gives a very useful tool to analyse polymorphisms and pinpoint the complexity of PCSPs. We do this by explaining how general PCSPs are naturally equipped with a topological structure, called homomorphism complexes, and how polymorphisms of a given PCSP can be understood through the continuous maps they induce. Homomorphism complexes (as well as several related constructions) have been actively studied in topological combinatorics , though mainly to give obstructions to the existence of homomorphisms and mostly for the case of graphs. However, methods of algebraic topology can also be used to obtain important information about polymorphisms: for example, to identify “influential” variables. We demonstrate how this new methodology can be applied to resolve a significant part of the Brakensiek-Guruswami conjecture. We also show that the simple notion of adjunction, which is a certain form of homomorphism duality, provides a powerful tool to reason about reductions between PCSPs. We observe that adjunctions always give rise to reductions between PCSPs. Moreover, we prove that many reductions between PCSPs work because of the presence of adjunction. This includes, in particular, all gadget reductions (that are captured by the algebraic approach) and all reductions satisfying very mild technical conditions. We demonstrate how adjunction can be applied by significantly improving the state-of-the-art in approximate graph colouring — via reductions that provably cannot be explained via the algebraic approach from . Related work ------------ The notion of PCSP was coined in, though one of the main examples of problems of this form, approximate graph colouring, has been around for a long time . The complexity landscape of PCSPs (beyond CSPs) is largely unknown, even for the Boolean case (see ) or for graph colouring and homomorphisms. Most notable examples of PCSPs studied before are related to graph and hypergraph colouring. We already mentioned some results concerning colouring 3-colourable graphs with a constant number of colours. Without additional complexity-theoretic assumptions, the strongest known -hardness results for colouring k-colourable graphs are as follows. For any k ≥ 3, it is -hard to colour a given k-colourable graph with 2k − 1 colours. For large enough k, it is -hard to colour a given k-colourable graph with 2Ω(k1/3) colours. The only earlier result about promise graph homomorphisms (with $\rel H\ne \rel G$) that involves more than approximate graph colouring is the -hardness of 3-colouring for graphs that admit a homomorphism to $\rel C\_5$, the five-element cycle. Under stronger assumptions (Khot’s 2-to-1 Conjecture for k ≥ 4 and its non-standard variant for k = 3), Dinur, Mossel, and Regev showed that finding a c-colouring of a k-colourable graph is NP-hard for all constants c ≥ k ≥ 3 . It was shown in that the above result for k = 2d still holds if one assumes the d-to-1 Conjecture of Khot for any fixed d ≥ 2 instead of the 2-to-1 Conjecture (which is the strongest in the family of d-to-1 conjectures). A variant of Khot’s 2-to-1 Conjecture with imperfect completeness has recently been proved , which implies hardness for approximate colouring variants for the weaker promise that most but not all of the graph is guaranteed to be k-colourable. A colouring of a hypergraph is an assignment of colours to its vertices that leaves no edge monochromatic. It is known that, for any constants c ≥ k ≥ 2, it is -hard to find a c-colouring of a given 3-uniform k-colourable hypergraph. Further variants of approximate hypergraph colouring, e.g. relating to strong or rainbow colourings, were studied in, but most complexity classifications related to them are still open in full generality. Some results are also known for colourings with a super-constant number of colours. For graphs, conditional hardness can be found in , and for hypergraphs, -hardness results were obtained in . An accessible exposition of the algebraic approach to the CSP can be found in, where many ideas and results leading to (but not including) the resolution of the Feder-Vardi conjecture are presented. The volume contains surveys concerning many aspects of the complexity and approximability of CSPs. The first link between the algebraic approach and PCSPs was found by Austrin, Håstad, and Guruswami, where they studied a promise version of (2k + 1)-SAT called (2 + ε)-SAT. They use a notion of polymorphism (which is the central concept in the algebraic theory of CSP) suitable for PCSPs. In, it was shown that the complexity of a PCSP is fully determined by its polymorphisms — in the sense that two PCSPs with the same set of polymorphisms have the same complexity. They also use polymorphisms to prove several hardness and tractability results. The algebraic theory of PCSP was lifted to an abstract level in, where it was shown that abstract properties of polymorphisms determine the complexity of PCSP. The topological methods that we develop in this paper originate in topological combinatorics, specifically in Lovász’s celebrated proof that gives a tight lower bound on the chromatic number of Kneser graphs. We refer to for an approachable introduction, and to for an in-depth technical reference. The modern view of this method is to assign a topological space to a graph in such a way that combinatorial properties of the graph (e.g. the chromatic number) are influenced by topological properties of the resulting space (e.g. topological connectivity). An intermediate step in the construction of the topological space is to assign a certain abstract simplicial complex to a graph (we introduce these below). In our proof, we use so-called homomorphism complexes that give a simplicial structure on the set of homomorphisms between two graphs (or other structures). We remark that restricting those complexes to vertices and edges (so called 1-skeletons) gives graphs of homomorphisms which have been used in CSP research before (see, e.g., ). We remark that three earlier results on the complexity of approximate hypergraph colouring were based on results from topological combinatorics using the Borsuk-Ulam theorem or similar . Their use of topology seems different from ours, and it remains to be seen whether they are all occurrences of a common pattern. Topological methods and adjunction (including some specific cases that we use in this paper) have also been actively used in research around Hedetniemi’s conjecture about the chromatic number of graph products (recently disproved by Shitov ). A few ideas in this paper are inspired by this line of research. A survey on adjunction and graph homomorphisms can be found in (see also ), which also discusses several specific constructions that we use in this paper. Our contributions ----------------- We first describe our methodological contributions related to topology and adjunction and then specific applications to promise graph homomorphism and approximate graph colouring. For simplicity, we will present the general theory for the case of graphs, which is what our applications are about; nevertheless, it generalises immediately to arbitrary relational structures. We do not assume that the reader is familiar with topological combinatorics or algebraic topology and provide the necessary definitions and explanations here and in later sections. It will be clear to the reader familiar with category theory that much of what we do in this paper can be naturally expressed in category-theoretic language. However, we avoid using this language, for the benefit of the readers less familiar with category theory. ### Topological analysis of polymorphisms As we mentioned before, the complexity of a problem $\operatorname{PCSP}(\rel H,\rel G)$ is completely determined by certain abstract properties of polymorphisms from $\rel H$ to $\rel G$. Our first contribution is the introduction of topology as a tool to analyse polymorphisms. The basis for such analysis is the fact that the set of all homomorphisms from a graph $\rel H$ to another graph $\rel G$ can be made into an abstract simplicial complex denoted by $\operatorname{\text{\normalfont \sffamily Hom}}(\rel H,\rel G)$. An abstract simplicial complex $\cox K$ is a downwards closed family of non-empty subsets of a vertex set $V(\cox K)$ — subsets in the family are called faces (or simplices), their elements are vertices. A simplical complex describes a topological space: the geometric realisation of $\cox K$, denoted ${\lvert \mathord{\cox K}\rvert}$, is the subspace of ${\mathbb{R}}^{V(\cox K)}$ obtained by identifying vertices with affinely independent points and, for each face, adding to the space the convex hull of the vertices in the face. Thus a pair $\{u,v\} \in \cox K$ becomes an edge, a triple (i.e., 3-element face) becomes a filled triangle, a quadruple becomes a filled tetrahedron, and so on. The resulting space can be analysed by using algebraic topology and the algebraic structures (groups, rings) that it associates with a topological space. The vertex set of the complex $\operatorname{\text{\normalfont \sffamily Hom}}(\rel H,\rel G)$ is the set of all homomorphisms from $\rel H$ to $\rel G$. A non-empty set {h1, …, hℓ} of such homomorphisms is a face if every function h: V(H) → V(G) satisfying h(v) ∈ {h1(v), …, hℓ(v)} for all v is a homomorphism $\rel H \to \rel G$. For example, if two homomorphisms h1, h2 differ at only one vertex v ∈ V(H), then they are connected by a line in ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel H,\rel G)}\rvert}$. Note the definition generalises in a straightforward way from graphs to arbitrary relational structures. There are several ways to use this notion for analysis of polymorphisms. One is to directly use the topological structure of ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel H^n,\rel G)}\rvert}$ — for example, by looking at various connectivity properties in this space and asking when polymorphisms (as points in this space) belong to the same component. Another one, and this is what we use in the paper, goes as follows. Any (say, n-ary) polymorphism f from $\rel H$ to $\rel G$, i.e., a homomorphism from $\rel H^n$ to $\rel G$, induces in a natural way a continuous map f̃ from the space ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel H^n)}\rvert}$ to ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel G)}\rvert}$, where $\rel K\_2$ is the two-element clique. One can then obtain information about f from algebraic invariants of f̃. As an important example, suppose that $\rel H, \rel G$ are (possibly different) odd cycles. It is well known and not hard to check that ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel H)}\rvert}$ is topologically equivalent to the circle ${\mathcal S}^1$ (we do this later in Example [ex:cycle]) and ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel H^n)}\rvert}$ to the n-torus ${\mathcal T}^n = {\mathcal S}^1 \times \dots \times {\mathcal S}^1$. A homomorphism f from $\rel H$ to $\rel G$ induces a continuous map f̃ from ${\mathcal S}^1$ to ${\mathcal S}^1$, and the main algebraic invariant of such a map is its degree, or winding number, which is an integer that intuitively measures how many times f̃ wraps the domain circle around the range circle (and in which direction). The degree of f̃ will be bounded because it arises from a discrete map f. Similarly, when analysing a homomorphism f from $\rel H^n$ to $\rel G$, we study f̃, which is now a continuous map from ${\mathcal T}^n$ to ${\mathcal S}^1$. Each of the n variables of f̃ corresponds to a circle in ${\mathcal T}^n$ and thus to a degree of f̃ restricted to that circle. We show that the number of variables whose degrees are non-zero is bounded, again because f̃ arises from a discrete function f. In this way, we obtain that each polymorphism f has a bounded number of coordinates (independent of n) that are “important”, or “influential”, and we can then use this information, together with the previously developed algebraic theory , to show that $\operatorname{PCSP}(\rel H,\rel G)$ is -hard. ### Adjunction We use symbols Λ, Γ for functions from the class of all (finite) graphs to itself. It will be convenient to write $\Lambda \rel H$ instead of $\Lambda(\rel H)$ for the image of $\rel H$ under Λ. The definitions and general properties again extend to all relational structures. Adjunction is a certain duality property between functions, best introduced with a concrete example. [ex:walk-power] For a graph $\rel G$ and an odd integer k one can consider the following functions: $\Lambda\_k \rel G$ is defined to be the graph obtained by subdividing each edge of $\rel G$ into a path of k edges, and $\Gamma\_k \rel G$ is the graph obtained by taking the k-th power of the adjacency matrix (with zeroes on the diagonal; equivalently, the vertex set remains unchanged and two vertices are adjacent if and only if there is a walk of length exactly k in $\rel G$). For example, $\Gamma\_3 \rel G$ has loops if $\rel G$ has triangles. Two functions Λ, Γ are called adjoint if $$\Lambda \rel H \to \rel G \text{ if and only if } \rel H \to \Gamma \rel G$$ for all graphs $\rel G,\rel H$. In this case Λ is also called the left adjoint to Γ, and Γ is the right adjoint to Λ. For example, it is well known and easy to check that Λk, Γk are adjoint, for any fixed odd k . Adjoint functions are always monotone with respect to the homomorphism preorder, i.e., $\rel H \to \rel G$ implies both $\Lambda\rel H \to \Lambda \rel G$ and $\Gamma\rel H \to \Gamma \rel G$ (see Lemma [lem:adj-technical]). Adjoint functions give us a way to reduce one PCSP to another. Indeed, consider any function Λ. We can always attempt to use it as a reduction between some PCSPs: if an instance graph $\rel I$ is guaranteed to be $\rel H$-colourable, then $\Lambda \rel I$ is guaranteed to be $\Lambda \rel H$-colourable if Λ is monotone. On the other hand if we find $\Lambda \rel I$ to be $\rel G$-colourable, this may imply that $\rel I$ is $\rel X$-colourable for some graph $\rel X$. In such a case Λ would be a reduction from $\operatorname{PCSP}(\rel H,\rel X)$ to $\operatorname{PCSP}(\Lambda \rel H, \rel G)$. What is the best possible $\rel X$? It is a graph $\rel X$ such that for any instance $\rel I$, $\Lambda \rel I \to \rel G$ holds if and only if $\rel I \to \rel X$. If such an $\rel X$ exists, it is essentially unique (since we just defined what homomorphisms $\rel X$ admits). The function that assigns to a graph $\rel G$ this best possible $\rel X$ is exactly the right adjoint to Λ. In this way, adjoints help us identify the best possible reduction a function gives, even though the proof that the reduction works might not need to mention the right adjoint. ### Applications Our applications of the above methodologies aim towards resolving the Brakensiek-Guruswami conjecture mentioned earlier: [Brakensiek and Guruswami ] [conj:main] [conj:bg] Let $\rel H$ and $\rel G$ be any non-bipartite loopless graphs with $\rel H\to \rel G$. Then $\operatorname{PCSP}(\rel H,\rel G)$ is -hard. We remark that the Hell-Nešetřil theorem confirms Conjecture [conj:main] for the case $\rel H = \rel G$. We also remark that Conjecture [conj:bg] covers all graphs: As discussed in Section [sec:preliminaries], if either $\rel H$ or $\rel G$ is bipartite or contains a loop then $\operatorname{PCSP}(\rel H, \rel G)$ can be easily solved in polynomial time. It is not hard to see that the conjecture is equivalent to the statement that $\operatorname{PCSP}(\rel C\_k,\rel K\_c)$ is -hard for all k ≥ 3 odd and c ≥ 3, where $\rel C\_k$ is a cycle on k vertices and Kc is a clique on c vertices. This is because we have a chain of homomorphisms $$\label{eq:chain} \dots \to \rel C\_k \to \dots \to \rel C\_5 \to \rel C\_3 = \rel K\_3 \to \rel K\_4 \to \dots \to \rel K\_c \to \dots$$ and, for each $(\rel H, \rel G)$ with a homomorphism $\rel H \to \rel G$, the problem $\operatorname{PCSP}(\rel H, \rel G)$ admits a trivial reduction from $\operatorname{PCSP}(\rel C\_k, \rel K\_c)$, where the promise is strengthened by requiring the input graph to be $\rel C\_k$-colourable, for an odd cycle $\rel C\_k$ in $\rel H$, and the goal is weakened to that of finding a $\rel K\_c$-colouring, where c is the chromatic number of $\rel G$ (so we have $\rel C\_k \to \rel H$ and $\rel G \to \rel K\_c$). The chain has a natural middle point $\rel K\_3$. The right half corresponds to the classical approximate graph colouring: find a c-colouring of a 3-colourable graph. Our applications make progress on the right half and show hardness for all of the left half. For the left half, we use the topological analysis of polymorphisms, as described above, to confirm Conjecture [conj:bg] for $\rel G = \rel K\_3$: [thm:K3] $\operatorname{PCSP}(\rel H,\rel K\_3)$ is -hard for every non-bipartite 3-colourable $\rel H$. Equivalently, $\operatorname{PCSP}(\rel C\_k,\rel K\_3)$ is -hard for all odd k ≥ 3. We in fact prove a more general result which covers other graphs $\rel G$ with similar topological properties to $\rel K\_3$ — namely that ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel G)}\rvert}$ maps to the circle ${\mathcal S}^1$ via a $\mathbb Z\_2$-map (see Definition [def:z2-map]). theoremmaintheoremtopology [thm:main-s1] Let $\rel H,\rel G$ be non-bipartite loopless graphs such that $\rel H \to \rel G$, and there is a $\mathbb Z\_2$-map from ${{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}}$ to ${\mathcal S}^1$. Then $\operatorname{PCSP}(\rel H,\rel G)$ is -hard. We give two specific classes of graphs $\rel G$ satisfying the assumptions of Theorem [thm:main-s1]: certain circular cliques and all square-free graphs. For positive integers p, q such that p ≥ 2q, the circular clique $\rel K\_{p/q}$ is the graph that has the same vertex set as the cycle $\rel C\_p$ and two vertices in it are connected by an edge if and only if they are at distance at least q in $\rel C\_p$ (see Fig. [fig:circular-cliques]). It well known that $\rel K\_{n/1}$ is isomorphic to $\rel K\_n$, $\rel K\_{(2n+1)/n}$ is isomorphic to $\rel C\_{2n+1}$, and that $\rel K\_{p/q}\to \rel K\_{p'/q'}$ if and only if p/q ≤ pʹ/qʹ (see, e.g., Theorem 6.3 in ), thus circular cliques refine the homomorphism order on odd cycles and cliques described above. The circular chromatic number of $\rel G$, $\chi\_c(\rel G)$, is defined as $\inf\{p/q \mid \rel G\to \rel K\_{p/q}\}$. Note that we always have $\chi(\rel G) = \lceil \chi\_c(\rel G)\rceil$ and also $\chi\_c(\rel G)\le 2+\frac{1}{n}$ if and only if $\rel G\to \rel C\_{2n+1}$. $$\begin{array}{c@{\qquad}c@{\qquad}c} \begin{tikzpicture}[scale = 1.3, baseline={([yshift=-.5ex]current bounding box.center)}] \draw [gray] (0,0) circle (1cm); \foreach \i/\c in {0/0,72/1,144/2,-144/3,-72/4} \node (\c) [circle,fill,inner sep=1.5,label={\i:$\c$}] at (\i:1) {}; \foreach \i/\c in {0/0,72/1,144/2,-144/3,-72/4} { \draw [thick] (\i:1) -- (72+\i:1); \draw [thick] (\i:1) -- (144+\i:1); } \end{tikzpicture} & \begin{tikzpicture}[scale = 1.3, baseline={([yshift=-.5ex]current bounding box.center)}] \draw [gray] (0,0) circle (1cm); \foreach \i/\c in {0/0,72/1,144/2,-144/3,-72/4} \node (\c) [circle,fill,inner sep=1.5,label={\i:$\c$}] at (\i:1) {}; \foreach \i/\c in {0/0,72/1,144/2,-144/3,-72/4} { \draw [thick] (\i:1) -- (144+\i:1); } \end{tikzpicture} & \begin{tikzpicture}[scale = 1.3, baseline={([yshift=-.5ex]current bounding box.center)}] \draw [gray] (0,0) circle (1cm); \foreach \i/\c in {0/0,51.42/1,102.85/2,154.28/3,205.71/4,257.14/5,308.57/6} \node (\c) [circle,fill,inner sep=1.5,label={\i:$\c$}] at (\i:1) {}; \foreach \i/\j in {0/2,1/3,2/4,3/5,4/6,5/0,6/1,0/3,1/4,2/5,3/6,4/0,5/1,6/2} \draw [thick] (\i) -- (\j); \end{tikzpicture} \\ \rel K\_{5/1} \simeq \rel K\_5 & \rel K\_{5/2} \simeq \rel C\_5 & \rel K\_{7/2} \end{array}$$ [fig:circular-cliques] The fact that circular cliques $\rel K\_{p/q}$ with 2 < p/q < 4 satisfy the topological condition of Theorem [thm:main-s1] is folklore, though we prove it later for completeness. The theorem in this case can be viewed as -hardness of colouring $(2+\eps)$-colourable graphs with $4-\eps$ colours: [thm:circular-cliques] $\operatorname{PCSP}(\rel K\_{p/q},\rel K\_{p'/q'})$ is -hard for all 2 < p/q ≤ pʹ/qʹ < 4. A graph is said to be square-free if it does not contain the 4-cycle $\rel C\_4$ as a subgraph. This includes all graphs of girth at least 5 and thus graphs of arbitrarily high chromatic number. Again, it will be a simple observation that square-free graphs satisfy the condition of Theorem [thm:main-s1]. Therefore, we confirm Conjecture [conj:bg] for square-free graphs $\rel G$. [thm:square-free] $\operatorname{PCSP}(\rel H,\rel G)$ is -hard for all non-bipartite loopless graphs $\rel H, \rel G$ such that $\rel H\rightarrow \rel G$ and $\rel G$ is square-free. Since the key assumption of Theorem [thm:main-s1] is topological, this raises a question whether topology is in some sense necessary to settle Conjecture [conj:bg]. Using adjointness, we argue that it is indeed the case, proving the following (see Theorem [thm:topoOnly] for a formal statement). [][thm:topoOnlyInformal] For any graph $\rel G$, the property that $\operatorname{PCSP}(\rel H, \rel G)$ is -hard for all non-bipartite $\rel G$-colourable graphs $\rel H$ depends only on the topology (and $\mathbb Z\_2$-action) of ${\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel G)}\rvert}$. Returning to the right half of the chain (the classical colouring problem), we first show that, to prove -hardness of c-colouring k-colourable graphs for all constants c ≥ k ≥ 3, it is enough to prove it for any fixed k (and all c ≥ k). [thm:conditional] Suppose there is an integer k such that $\operatorname{PCSP}(\rel K\_k,\rel K\_c)$ is -hard for all c ≥ k. Then $\operatorname{PCSP}(\rel K\_3,\rel K\_c)$ is -hard for all c ≥ 3. Following the reasoning in , the above theorem implies -hardness of all problems $\operatorname{PCSP}(\rel K\_k,\rel K\_c)$ with c ≥ k ≥ 3 if the d-to-1 conjecture of Khot holds for any fixed d ≥ 2. (The paper used an earlier version of Theorem [thm:conditional] with 4 in place of 3). Furthermore, we strengthen the best known asymptotic hardness: Huang showed that $\operatorname{PCSP}(\rel K\_k, \rel K\_c)$ is NP-hard for all sufficiently large k and c = 2Ω(k1/3). We improve this in two ways, using Huang’s result as a black-box. First, we improve the asymptotics from sub-exponential c = 2Ω(k1/3) to single-exponential $c = {{\binom{k}{\lfloor k/2 \rfloor}}} \in \Theta ( 2^k / \sqrt k )$. Second, we show the claim holds for k starting as low as 4. theoremmaintheoremasymptotics[thm:asymp] For all k ≥ 4 and $c = {{\binom{k}{\lfloor k/2 \rfloor}}} - 1$, $\operatorname{PCSP}(\rel K\_k, \rel K\_c)$ is -hard. In comparison, the previous best result relevant for all integers k was obtained in where -hardness of $\operatorname{PCSP}(\rel K\_k, \rel K\_{2k-1})$ is proved for all k ≥ 3. For k = 3, 4 we obtain no new results and for k = 5 the two bounds coincide: ${{\binom{k}{\lfloor k/2 \rfloor}}}-1 = 9 = 2k-1$. However, already for k = 6 we improve the bound from 2k − 1 = 11 to ${{\binom{k}{\lfloor k/2 \rfloor}}}-1 = 19$, and, for larger k, the improvement is even more dramatic. ### The organisation of the paper Section [sec:preliminaries] briefly describes the algebraic framework of : minions (sets of polymorphisms of a PCSP), minion homomorphism (which provide log-space reductions between corresponding problems), and a condition on minions that guarantees -hardness. Section [sec:topology] details the topological method and its application: Theorem [thm:main-s1]. The bulk of its content is devoted to expounding standard definitions with examples and then proving these definitions behave well when identifying variables of polymorphisms. Section [sec:adjunction] introduces adjunction in a wider context, in particular relating it to gadget reductions and minion homomorphisms. Adjoint functions that give reductions for approximate graph colouring are presented in Section [sec:righthard]. Finally Section [subsec:secondMainProof] uses another adjoint function to prove Theorem [thm:topoOnlyInformal]: that whether a graph $\rel G$ satisfies the Brakensiek-Guruswami conjecture for all $\rel H$ depends only on the topology of $\rel G$. Preliminaries ============= Promise graph homomorphism problems ----------------------------------- The approximate graph colouring problem and promise graph homomorphism problem are special cases of the PCSP, and we use the theory of PCSPs. However, we will not need the general definitions, so we define everything only for digraphs. For general definitions, see, e.g. . A digraph $\rel H$ is a pair $\rel H=(V(H),E(H))$, where V(H) is a set of vertices and E(H) ⊆ {(u, v) ∣ u, v ∈ V(H)} is a set of (directed) edges. Unless stated otherwise, our digraphs are finite and can have loops. We view undirected graphs as digraphs where each (non-loop) edge is presented in both directions. A homomorphism from a digraph $\rel H=(V(H),E(H))$ to another digraph $\rel G=(V(G),E(G))$ is a map h: V(H) → V(G) such that (h(u), h(v)) ∈ E(G) for every (u, v) ∈ E(H). In this case we write $h\colon \rel H\to \rel G$, and simply $\rel H\to \rel G$ to indicate that a homomorphism exists. We now define formally the promise (di)graph homomorphism problem. Fix two digraphs $\rel H$ and $\rel G$ such that $\rel H \rightarrow \rel G$. * The search variant of $\operatorname{PCSP}(\rel H,\rel G)$ is, given an input digraph $\rel I$ that maps homomorphically to $\rel H$, find a homomorphism $h\colon \rel I\to \rel G$. * The decision variant of $\operatorname{PCSP}(\rel H,\rel G)$ requires, given an input digraph $\rel I$ such that either $\rel I\to \rel H$ or $\rel I\not\to\rel G$, to output yes in the former case, and no in the latter case. We remark that the (decision) problem $\operatorname{PCSP}(\rel H,\rel H)$ is nothing else but the constraint satisfaction problem $\operatorname{CSP}(\rel H)$, also known as $\rel H$-colouring. There is an obvious reduction from the decision variant of each PCSP to the search variant, but it is not known whether the two variants are equivalent for each PCSP. The hardness results in this paper hold for the decision (and hence also for the search) version of $\operatorname{PCSP}(\rel H,\rel G)$. It is obvious that if at least one of $\rel H, \rel G$ is undirected and bipartite then the problem can be solved in polynomial time by using an algorithm for 2-colouring. If one of the graphs contains a loop, the problem is trivial. Recall that Brakensiek and Guruswami conjectured (see Conjecture [conj:main]) that, for undirected graphs, the problem is -hard in all the other cases. All applications in this paper concern undirected graphs, even though some proofs use digraphs. We remark that, as shown in Theorem F.3 of the arXiv version of (generalising the corresponding result for CSPs ), a complexity classification of all problems $\operatorname{PCSP}(\rel H,\rel G)$ for digraphs is equivalent to such a classification for all PCSPs (for arbitrary relational structures). Two (di)graphs $\rel H\_1$ and $\rel H\_2$ are called homomorphically equivalent if both $\rel H\_1\rightarrow \rel H\_2$ and $\rel H\_2\rightarrow \rel H\_1$. The binary relation $\rel H\_1\rightarrow \rel H\_2$ defines a preorder on the class of all digraphs (or all graphs), called the homomorphism preorder. We will use this preorder in Section [sec:adjunction]. We also define digraph powers, which are essential for the notion of polymorphisms. [def:nth-power] The *n-th direct (or tensor) power* of a digraph $\rel H$ is the digraph $\rel H^n$ whose vertices are all n-tuples of vertices of $\rel H$ (i.e., V(Hn) = V(H)n), and whose edges are defined as follows: we have an edge from (u1, …, un) to (v1, …, vn) in $\rel H^n$ if and only if (ui, vi) is an edge of $\rel H$ for all i ∈ {1, …, n}. Polymorphisms ------------- We use the notions of polymorphisms, minions and minion homomorphisms. We introduce these notions in the special case of digraphs below. General definitions and more insights can be found in. An n-ary polymorphism from a digraph $\rel H$ to a digraph $\rel G$ is a homomorphism from $\rel H^n$ to $\rel G$. To spell this out, it is a mapping f: V(H)n → V(G) such that, for all tuples (u1, v1), …, (un, vn) of edges of $\rel H$, we have (f(u1, …, un), f(v1, …, vn)) ∈ E(G). We denote the set of all polymorphisms from $\rel H$ to $\rel G$ by $\operatorname{Pol}(\rel H,\rel G)$. The n-ary polymorphisms from a digraph $\rel H$ to the k-clique $\rel K\_k$ are the k-colourings of $\rel H^n$. The set of all polymorphisms between any two digraphs has a certain algebraic structure, which we now describe. We denote by [n] the set {1, 2, …, n}. [def:minor] An n-ary function f: An → B is called a minor of an m-ary function g: Am → B if there is a map π: [m] → [n] such that f(x1, …, xn) = g(xπ(1), …, xπ(m)) for all x1, …, xn ∈ A. In this case, we write f = gπ. Alternatively, one can say that f is a minor of g if it is obtained from g by identifying variables, permuting variables, and introducing inessential variables. For sets A, B, let $\clo O(A,B) = \{f\colon A^n\rightarrow B\mid n\ge 1\}$. A (function) minion $\clo M$ on a pair of sets (A, B) is a non-empty subset of $\clo O(A,B)$ that is closed under taking minors. For fixed n ≥ 1, let $\clo M^{(n)}$ denote the set of n-ary functions from $\clo M$. It is easy to see that $\operatorname{Pol}(\rel H,\rel G)$ is a minion whenever $\rel H\rightarrow\rel G$. An important notion in our analysis of polymorphisms is that of an essential coordinate. [def:essential] A coordinate i of a function f: An → B is called essential if f depends on it, that is, if there exist a1, …, an and bi in A such that f(a1, …, ai − 1, ai, ai + 1, …, an) ≠ f(a1, …, ai − 1, bi, ai + 1, …, an). A coordinate of f that is not essential is called inessential. A minion $\clo M$ is said to have essential arity at most k*, if each function $f\in \clo M$ has at most k* essential variables. It is said to have bounded essential arity if it has essential arity at most k for some k. It is well known (see, e.g., ), and not hard to check, that the minion $\operatorname{Pol}(\rel K\_3,\rel K\_3)$ has essential arity at most 1. However for any odd k > 3, the minion $\operatorname{Pol}(\rel C\_k,\rel K\_3)$ does not have bounded essential arity. Indeed, fix a homomorphism $h\colon \rel C\_k\to\rel K\_3$ such that h(0) = h(2) = 0 and h(1) = 1 and define the following function from $\rel C\_k^n$ to $\rel K\_3$: $$f(x\_1,\ldots,x\_n) = \begin{cases} 2 & \text{if }x\_1=\ldots=x\_n=1,\\ h(x\_1) & \text{otherwise.} \end{cases}$$ It is easy to check that $f\in \operatorname{Pol}(\rel C\_k,\rel K\_3)$. By using Definition [def:essential] with a1 = … = an = 1 and bi = 0, one can verify that every coordinate i of f is essential. [def:minion-homomorphism] Let $\clo M$ and $\clo N$ be two minions (not necessarily on the same pairs of sets). A mapping $\xi\colon \clo M \to \clo N$ is called a minion homomorphism if 1. it preserves arities, i.e., maps n-ary functions to n-ary functions for all n, and 2. it preserves taking minors, i.e., for each π: {1, …, m} → {1, …, n} and each $g\in \clo M^{(m)}$ we have ξ(g)π = ξ(gπ), i.e., ξ(g)(xπ(1), …, xπ(m)) = ξ(g(xπ(1), …, xπ(m))). We refer to for examples of minion homomorphisms. Our proof of Theorem [thm:main-s1] is based on the following result. It is a special case of a result in (that generalised ). We remark that the proof of this theorem is by a reduction from Gap Label Cover, which is a common source of inapproximability results. [] [thm:bounded-arity] Let $\rel H, \rel G$ be digraphs such that $\rel H \to \rel G$. Assume that there exists a minion homomorphism $\xi\colon\operatorname{Pol}(\rel H,\rel G) \rightarrow \clo M$ for some minion $\clo M$ on a pair of (possibly infinite) sets such that $\clo M$ has bounded essential arity and does not contain a constant function (i.e., a function without essential variables). Then $\operatorname{PCSP}(\rel H,\rel G)$ is -hard. To prove Theorem [thm:main-s1], we will use Theorem [thm:bounded-arity] with the minion $\clo M=\clo Z\_{\leq N}$, for some constant N > 0. The set $\clo Z\_{\leq N}$ is defined to consist of all linear functions f on $\mathbb Z$ of the form f(x1, …, xn) = ∑i = 1ncixi for some $c\_i\in \mathbb Z$ such that ∑i = 1nci is odd and $\sum\_{i=1}^n {\lvert \mathord{c\_i}\rvert} \leq N$. It easy to see that $\clo Z\_{\leq N}$ is indeed a minion and that all functions in it have between 1 and N non-zero coefficients, meaning that it has bounded essential arity and contains no constant function. Topology ======== All graphs in this section are assumed to be undirected and loopless. Simplicial complexes -------------------- An (abstract) simplicial complex is a family of non-empty sets $\cox K$ that is downwards closed, i.e., if $\sigma\_1 \in \cox K$, σ2 ≠ ∅ and σ2 ⊆ σ1, then $\sigma\_2 \in \cox K$. Each $\sigma \in \cox K$ is called a face. The elements in these sets are vertices of $\cox K$. We denote the set of all vertices of $\cox K$ by $V(\cox K)$, i.e., $V(\cox K) {:=}\Union\_{\sigma \in \cox K} \sigma$. A simplicial map between complexes $\cox K$ and $\cox K'$ is a function $f \colon V(\cox K) \to V(\cox K')$ that preserves faces, i.e., if $\sigma \in \cox K$ then $f(\sigma) {:=}\{f(v) \mid v\in \sigma\} \in \cox K'$. Two simplicial complexes $\cox K$ and $\cox K'$ are isomorphic if there are simplicial maps $\alpha\colon \cox K \to \cox K'$ and $\beta\colon \cox K' \to \cox K$ such that both αβ and βα are identity maps. We will use the following notion of a product of simplicial complexes (see also and ). Let $\cox K\_1,\dots,\cox K\_n$ be simplicial complexes. We define the product $\cox K\_1\times \dots \times \cox K\_n$ to be the simplicial complex with vertices $$V( \cox K\_1 \times \dots \times \cox K\_n ) = V(\cox K\_1) \times \dots \times V(\cox K\_n),$$ so that $\sigma \subseteq V(\cox K\_1\times \dots\times \cox K\_n)$ is a face if there are faces $\sigma\_1 \in \cox K\_1$, …, $\sigma\_n \in \cox K\_n$ such that σ ⊆ σ1 × … × σn. ### From graphs to simplicial complexes. As mentioned before, there are several ways to assign a simplicial complex to a graph. For our use, the most convenient is the homomorphism complex. Our definition of this complex is slightly different from that in , but the difference is superficial (as we explain in Appendix [app:hom-and-box]). The vertices of such a complex are homomorphisms, while faces are determined by multihomomorphisms defined below. [def:multimorphism] A multihomomorphism from $\rel K$ to $\rel G$ is a mapping f: V(K) → 2V(G) such that, for each (u, v) ∈ E(K), we have f(u) × f(v) ⊆ E(G). [def:homcomplex] Let $\rel K$ and $\rel G$ be two graphs. We define a simplicial complex $\operatorname{\text{\normalfont \sffamily Hom}}(\rel K,\rel G)$ as follows. Its vertices are homomorphisms from $\rel K$ to $\rel G$, and σ = {f1, …, fℓ} is a face if the mapping u ↦ {f1(u), …, fℓ(u)} is a multihomomorphism from $\rel K$ to $\rel G$. We work almost exclusively with complexes ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$, where $\rel K\_2$ is the two-element clique. Such complexes (with our definition) appeared before, e.g. in , where they are called box complexes (which is not the traditional use of this name) and in , where these complexes appear under the name $\mathsf{B\_{edge}}(\rel G)$. The complex ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$ can be also described in the following way. The vertices of ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$ are all (oriented) edges of $\rel G$. The faces are directed bipartite subgraphs that can be extended to a complete directed bipartite subgraph of $\rel G$ (with all edges directed from one part to the other); more precisely, σ is a face if there are U, V ⊆ V(G) such that σ ⊆ U × V ⊆ E(G). The complexes ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$ have an additional structure obtained from the automorphism of $\rel K\_2$ that switches the two vertices. The group $\mathbb Z\_2$ then acts on the vertices of ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$ by reversing the direction of edges, i.e., − (a, b) = (b, a). [fig:box-k3] [ex:cycle] Let us consider the complex ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}$. Its vertices are all oriented edges of the k-cycle which means pairs of the form (i, i + 1) and (i + 1, i) where the addition is considered modulo k. It is not hard to see that the only directed complete bipartite subgraphs of $\rel C\_k$ are either two outgoing edges from a single vertex, or two incoming edges to a single vertex. The only non-trivial faces of ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}$ are therefore of the form {(i − 1, i), (i + 1, i)} or {(i, i − 1), (i, i + 1)}. The resulting complex can be drawn as a graph (see Fig. [fig:box-k3] for such a drawing of ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_5})}$). The exact structure depends on the parity of k. If k is odd, the complex is a single 2k-cycle where (i, j) is opposite to (j, i). The $\mathbb Z\_2$-action acts as the central reflection. If k is even, the complex consists of two disjoint k-cycles such that one contains all edges of the form (2i, 2i ± 1) and the other all edges of the from (2i ± 1, 2i). The $\mathbb Z\_2$-action in this case switches the two parts. A slightly more complicated example is ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel K\_4})}$. See Fig. [fig:box-k4] for graphical representations of this complex. There are two types of maximal directed complete bipartite subgraphs of $\rel K\_4$: either all three in/outgoing edges of a single vertex, or 4 directed edges from a two-element subset of $\rel K\_4$ to its complement. These, and all their non-empty subsets, are the faces of ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel K\_4})}$. In the pictures, the in/outgoing edges correspond to the triangular faces, and the faces containing 4 edges correspond to tetrahedrons that are represented as tetragons. Naturally, all subsets of these tetragons are also faces, nevertheless they are omitted from the picture for better readability. Also note that the outer face of the left diagram forms such a tetrahedron (corresponding to the bipartite subgraph {1, 3} × {0, 2}). The $\mathbb Z\_2$-symmetry of this complex is given by reversing edges; this corresponds to the antipodality on the spherical representation. [fig:box-k4] A $\mathbb Z\_2$-(simplicial) complex is a simplicial complex $\cox K$ with a simplicial map ${-}\colon \cox K\to \cox K$ such that − ( − v) = v for each $v\in V(\cox K)$. We also write − σ for the face { − v ∣ v ∈ σ}. We say that a simplicial map f from one $\mathbb Z\_2$-complex $\cox K$ to another $\mathbb Z\_2$-complex $\cox K'$ is a $\mathbb Z\_2$-simplicial map, if f( − v) = − f(v) for each $v\in V(\cox K)$ (note that the first − is taken in $\cox K$, while the second is taken in $\cox K'$). The map v ↦ − v can be also viewed as an action of the group $\mathbb Z\_2$ on $\cox K$ by simplicial maps. We remark that a product $\cox K\_1\times \dots \times \cox K\_n$ of $\mathbb Z\_2$-complexes is also $\mathbb Z\_2$-complex with the action defined component-wise, as − (v1, …, vn) = ( − v1, …, − vn). Every graph homomorphism $f\colon \rel H\to \rel G$ induces a $\mathbb Z\_2$-simplicial map $f'\colon {\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel H})} \to {\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$ defined by fʹ((a, b)) = (f(a), f(b)).[2](#fn2) Topological spaces ------------------ The spaces assigned to $\mathbb Z\_2$-complexes inherit the $\mathbb Z\_2$ symmetry. [def:z2-map] A $\mathbb Z\_2$-space is a topological space ⊤X with a distinguished continuous function − : ⊤X → ⊤X such that − ( − x) = x for each x. A Z2-map between two $\mathbb Z\_2$-spaces ⊤X and ⊤Y is a continuous function f: ⊤X → ⊤Y which preserves the action of $\mathbb Z\_2$, i.e., f( − x) = − f(x) for each x ∈ ⊤X (note that the first − is taken in ⊤X, while the second is taken in ⊤Y). As is the case for $\mathbb Z\_2$-complexes, $\mathbb Z\_2$-spaces are topological spaces with an action of the group $\mathbb Z\_2$ by continuous functions. Prime examples of $\mathbb Z\_2$-spaces are spheres: We define ${\mathcal S}^n$ as a subspace of $\mathbb R^{n+1}$ consisting of all unit vectors, i.e., ${\mathcal S}^n = \{ (x\_1,\dots,x\_{n+1})\in \mathbb R^{n+1} \mid x\_1^2 + \dots + x\_{n+1}^2 = 1 \}$, with antipodality as the chosen $\mathbb Z\_2$-action, i.e., − (x1, …, xn + 1) = ( − x1, …, − xn + 1). Other common $\mathbb Z\_2$-spaces are toruses. An n-torus ${\mathcal T}^n$ is defined as the n-th power ${\mathcal S}^1 \times \dots \times {\mathcal S}^1$, and is therefore naturally equipped with a $\mathbb Z\_2$-action defined to act coordinatewise. A $\mathbb Z\_2$-complex $\cox K$ is free if − σ ≠ σ for each $\sigma \in \cox K$ (equivalently, $\{-v,v\} \not \in \cox K$ for all vertices v of $\cox K$). Note that, for a loopless undirected graph $\rel G$, the complex ${\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}$ is always a free $\mathbb Z\_2$-complex. To ease a technical annoyance in the proofs below, we rephrase the definition of a geometric realisation (see also ) of a free $\mathbb Z\_2$-complex. [def:geom] Let $\cox K$ be a free $\mathbb Z\_2$-simplicial complex. Let v1, − v1, …, vn, − vn be all vertices of $\cox K$. We define ${\lvert \mathord{\cox K}\rvert}$, a geometric realisation of $\cox K$, as a subspace of $\mathbb R^n$. First, we identify the canonical unit vectors with v1, …, vn, so that v1 = (1, 0, …, 0), etc., and − v1, …, − vn with their opposites, so − v1 = ( − 1, 0…, 0), etc. Second, for each face $\sigma\subseteq V(\cox K)$, we define $\Delta^\sigma \subseteq \mathbb R^n$ to be the convex hull of σ, i.e., Δσ = {∑v ∈ σλvv ∣ ∑v ∈ σλv = 1, λv ≥ 0}. Finally, we set $${\lvert \mathord{\cox K}\rvert} = \bigcup\_{\sigma\in \cox K} \Delta^\sigma = \{ \sum\_{v\in \sigma} \lambda\_v v \mid \sigma \in \cox K, \sum\_{v\in \sigma} \lambda\_v = 1, \lambda\_v \geq 0 \}.$$ The action of $\mathbb Z\_2$ on ${\lvert \mathord{\cox K}\rvert}$ maps a point ∑v ∈ σλvv to the point ∑− v ∈ − σλv( − v) which can be equivalently described as reversing the sign of a vector, i.e., as − (x1, …, xn) = ( − x1, …, − xn). With the above definition, we can view $V(\cox K)$ as a subset of ${\lvert \mathord{\cox K}\rvert}$ — this will be useful in the technical proofs below. Also note that − v has two meanings that result in the same object: either it is a $\mathbb Z\_2$-counterpart of $v \in V(\cox K)$, or the opposite vector to $v\in {\lvert \mathord{\cox K}\rvert}$. Note that the geometric realisation of a free $\mathbb Z\_2$-complex is a free $\mathbb Z\_2$-space, i.e., a $\mathbb Z\_2$-space ⊤X such that − x ≠ x for all x ∈ ⊤X. To express abstractly what it means for two $\mathbb Z\_2$-spaces to be the same, we use the notion of $\mathbb Z\_2$-homeomorphism which is an analogue of the notion of homeomorphism. We remark that this is a strong notion of equivalence of topological spaces, akin to isomorphism, and that we will also use weaker notions of topological equivalence (see also Appendix [app:hom-and-box]). Two $\mathbb Z\_2$-topological spaces ⊤X and ⊤Y are $\mathbb Z\_2$-homeomorphic if there are $\mathbb Z\_2$-maps f: ⊤X → ⊤Y and g: ⊤Y → ⊤X such that fg is the identity on ⊤Y and gf is the identity on ⊤X. [ex:hom-k3] It is not hard to see that the geometric representation ${{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}\rvert}}$ of the homomorphism complex of an odd cycle $\rel C\_k$ is $\mathbb Z\_2$-homeomorphic to ${\mathcal S}^1$ (see Fig. [fig:box-k3sq] on page ). Let us define one such $\mathbb Z\_2$-homeomorphism $f\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}\rvert}} \to {\mathcal S}^1$. Choose k points on the circle in a regular pattern. Let us denote these vectors x1, …, xk. Note that since k is odd, $-x\_i \notin \{x\_1,\dots,x\_k\}$ for all i. We first define a map $f\_0\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}\rvert}} \to \mathbb R^2$ as follows: f0(v) = xb − xa for $v\in V(\cox K)$, v = (a, b), and extend it linearly. Note that the image of ${{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}\rvert}}$ forms a regular 2k-gon centred in the origin. We project the polygon onto ${\mathcal S}^1$ by putting $f(x) = f\_0(x)/{\lvert \mathord{f\_0(x)}\rvert}$. It is clear that f is continuous and f( − x) = − f(x), and therefore it is a $\mathbb Z\_2$-map. It is also not hard to see that it is 1-to-1 and therefore invertible, and that the inverse is a $\mathbb Z\_2$-map. While there is always a continuous function between two topological spaces ⊤X and ⊤Y (simply map everything to a single point), there might not be a $\mathbb Z\_2$-map between two $\mathbb Z\_2$-spaces. In particular, the Borsuk-Ulam theorem (see also ) states that there is no $\mathbb Z\_2$-map from a sphere ${\mathcal S}^m$ to a sphere ${\mathcal S}^n$ of smaller dimension (i.e., if m > n). Every $\mathbb Z\_2$-simplicial map $f\colon \cox K \to \cox K'$ induces a $\mathbb Z\_2$-map ${\lvert \mathord{f}\rvert}\colon {\lvert \mathord{\cox K}\rvert} \to {\lvert \mathord{\cox K'}\rvert}$ defined as a piece-wise linear extension of f: $${\lvert \mathord{f}\rvert}\colon \sum\_{v\in \sigma} \lambda\_v v \mapsto \sum\_{v\in \sigma} \lambda\_v f(v).$$ (Here, we use that $v\in V(\cox K)$ is also a point in ${\lvert \mathord{\cox K}\rvert}$, and therefore a vector in $\mathbb R^n$.) Consequently, every graph homomorphism $\rel H \to \rel G$ induces a $\mathbb Z\_2$-map from ${\lvert \mathord{{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel H})}}\rvert}$ to ${\lvert \mathord{{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}}\rvert}$. ### The fundamental group We briefly recall the definition of the fundamental group assigned to a topological space ⊤X, denoted ${{\pi\_1}(\mathord{\top X})}$. For more details, see. The elements of the group are homotopy classes of maps $f\colon {\mathcal S}^1 \to \top X$ defined as follows. Intuitively, two maps are homotopic if one can be continuously transformed into the other. We say that two continuous maps f, g: ⊤X → ⊤Y are homotopic if there is a continuous map h: ⊤X × [0, 1] → ⊤Y such that h(x, 0) = f(x) and h(x, 1) = g(x) for each x ∈ ⊤X. Any such map h is called a homotopy. The homotopy class of f: ⊤X* → ⊤Y* is the set of all continuous maps g: ⊤X → ⊤Y that are homotopic to f. We denote such a class by [f].[3](#fn3) Formally, the fundamental group is defined relative to a point x0 ∈ ⊤X, but the choice of the point is irrelevant if the space ⊤X is path connected (see ), i.e., if any two points in ⊤X are connected by a path. Fix one such choice x0 ∈ ⊤X. The elements of ${{\pi\_1}(\mathord{\top X})}$ are all homotopy classes of maps $\ell\colon {\mathcal S}^1 \to \top X$ such that ℓ((1, 0)) = x0. The group operation is given by so-called loop composition: seeing maps $\ell\_1,\ell\_2\colon {\mathcal S}^1 \to {\top X}$ as closed walks originating in x0, the product ℓ1 ⋅ ℓ2 is the closed walk that follows first ℓ1 and then ℓ2. While this product is not a group operation as is, it induces a group operation on the homotopy classes defined as [ℓ1] ⋅ [ℓ2] = [ℓ1 ⋅ ℓ2] (see for a proof). Any map f: ⊤X → ⊤Y induces a group homomorphism $f\_\\colon {{\pi\_1}(\mathord{\top X})}\to {{\pi\_1}(\mathord{\top Y})}$ defined by f\([ℓ]) = [fℓ] for each $\ell\colon {\mathcal S}^1 \to \top X$. The fundamental groups of many spaces are described in the literature. For example: [] [lem:fg-s1] ${{\pi\_1}(\mathord{{\mathcal S}^1})}$ is isomorphic to $\mathbb Z$. We also define $\mathbb Z\_2$-homotopy which is a strengthening of homotopy, restricting it to $\mathbb Z\_2$-maps. [def:z2-homotopy] Let f, g: ⊤X → ⊤Y be $\mathbb Z\_2$-maps. A homotopy h from f to g is a $\mathbb Z\_2$-homotopy if the map ht: x ↦ h(x, t) is a $\mathbb Z\_2$-map for each t ∈ [0, 1]. We say that f and g are $\mathbb Z\_2$-homotopic, if there is a $\mathbb Z\_2$-homotopy between them. Polymorphisms of complexes, spaces, and groups ---------------------------------------------- A polymorphism from one graph to another is defined as a homomorphism from a power. In the same way, we can define polymorphisms of any objects as long as we have a notion of a homomorphism and of a power. 1. Let $\cox K,\cox K'$ be two $\mathbb Z\_2$-simplicial complexes. An n-ary polymorphism from $\cox K$ to $\cox K'$ is a $\mathbb Z\_2$-simplicial map from the n-th power of $\cox K$ to $\cox K'$, i.e., $f\colon V(\cox K)^n \to V(\cox K')$ such that f( − v1, …, − vn) = − f(v1, …, vn) for all $v\_i\in V(\cox K)$ and $$f(\sigma\_1 \times \dots \times \sigma\_n) = \{ f( a\_1, \dots, a\_n ) \colon a\_i \in \sigma\_i \} \text{ is in } \cox K'$$ for all $\sigma\_1,\dots,\sigma\_n \in \cox K$. We denote by $\operatorname{Pol}(\cox K,\cox K')$ the set of all polymorphisms from $\cox K$ to $\cox K'$. 2. Let ⊤X, ⊤Y be two $\mathbb Z\_2$-spaces. An n-ary polymorphism from ⊤X to ⊤Y is a $\mathbb Z\_2$-map from ⊤Xn to ⊤Y, i.e., a continuous map f: ⊤Xn → ⊤Y such that f( − x1, …, − xn) = − f(x1, …, xn) for all xi ∈ X. Again, Pol(⊤X, ⊤Y) denotes the set of all polymorphisms from ⊤X to ⊤Y. 3. Let $\gr H$, $\gr G$ be two groups. An n-ary polymorphism from $\gr H$ to $\gr G$ is a group homomorphism from $\gr H^n$ to $\gr G$, i.e., a mapping f: Hn → G such that f(g1 ⋅ h1, …, gn ⋅ hn) = f(g1, …, gn) ⋅ f(h1, …, hn) for all gi, hi ∈ H. We denote the set of all polymorphisms from $\gr H$ to $\gr G$ by $\operatorname{Pol}(\gr H,\gr G)$. In all the cases above, it is easy to see that polymorphisms are closed under taking minors, and therefore Pol( − , − ) is always a minion. This allows us to talk about minion homomorphisms between minions of polymorphisms of different objects (graphs, simplicial complexes, topological spaces, or groups). By definition, $\operatorname{Pol}(\mathbb Z,\mathbb Z)$ consists of all group homomorphisms from $\mathbb Z^n$ to $\mathbb Z$ for all n > 0. It is straightforward to check that such an n-ary polymorphism in $\operatorname{Pol}(\mathbb Z,\mathbb Z)$ is a linear function, i.e., of the form (x1, …, xn) ↦ ∑i = 1ncixi for some $c\_1,\dots,c\_n \in \mathbb Z$, and conversely, any such function is a group homomorphism from $\mathbb Z^n$ to $\mathbb Z$. Proofs of Theorems [thm:K3] and [thm:main-s1] --------------------------------------------- We recall the statement of Theorem [thm:main-s1]. Theorem [thm:K3] is a direct corollary of the above and Example [ex:hom-k3]. In the rest of the section we prove Theorem [thm:main-s1] by using Theorem [thm:bounded-arity]. We show that there is a minion homomorphism from $\operatorname{Pol}(\rel H,\rel G)$ to the minion $\clo Z\_{\leq N}$ for some N. Recall that $\clo Z\_{\leq N}\subset \operatorname{Pol}(\mathbb Z,\mathbb Z)$ is defined to consist of all linear functions f on $\mathbb Z$ of the form f(x1, …, xn) = ∑i = 1ncixi for some $c\_i\in \mathbb Z$ such that ∑i = 1Nci is odd and $\sum\_{i=1}^N {\lvert \mathord{c\_i}\rvert} \leq N$. This is achieved in two steps. In the first step, we provide a minion homomorphism from $\operatorname{Pol}(\rel H,\rel G)$ to $\operatorname{Pol}(\mathbb Z,\mathbb Z)$. This is achieved by following the constructions described above, i.e., the transformations $$\text{graph} \buildrel {{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\ast})}}\over\longrightarrow \text{{$\mathbb Z\_2$}-complex} \buildrel {{\lvert \mathord{\ast}\rvert}}\over\longrightarrow \text{{$\mathbb Z\_2$}-space} \buildrel {{{\pi\_1}(\mathord{\ast})}}\over\longrightarrow \text{group},$$ and showing that pushing a polymorphism through this sequence of constructions preserves minors. This essentially follows from the well-known facts that these constructions behave well with respect to products. A detailed proof is presented in Section [sec:minion-homomorphism]. When we push a polymorphism $f\in \operatorname{Pol}(\rel H,\rel G)$ through these constructions, we first obtain a Z2-simplicial map $f'\in \operatorname{Pol}({\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel H})},{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})})$, which in turn induces a Z2-map ${\lvert \mathord{f'}\rvert} \in \operatorname{Pol}({{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel H})}\rvert}},{{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}})$. Then, by composing with the assumed Z2-map $ s \colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}} \to {\mathcal S}^1 $ (and assuming without loss of generality that $\rel H$ is an odd cycle), we obtain a polymorphism g of ${\mathcal S}^1$, and then finally a polymorphism g\* of the group $\pi\_1({\mathcal S}^1) \simeq \mathbb Z$. As discussed before, g\* can be described more concretely as a linear function whose coefficients ci are the winding numbers of maps ${\mathcal S}^1 \to {\mathcal S}^1$ defined by t ↦ s ∘ g(x0, …, t, …, x0) where $x\_0\in {\mathcal S}^1$ is an arbitrary (but fixed) point. In the second step, we use the discrete structure of the graphs $\rel H$ and $\rel G$, as well as the action of Z2, to show that the image of $\operatorname{Pol}(\rel H,\rel G)$ under the constructed minion homomorphism is contained in $\clo Z\_{\leq N}$ for some N. This is described in Section [sec:bounding-arity]. We note that there are several ways to present the proof of Theorem [thm:K3]. These presentations would look different on the surface, but in fact they use the same underlying topological concepts, just hidden to various extent. For example, the proof that appeared in the conference version hides topology in a more direct combinatorial approach. Yet another version of the proof can be given in the language of recolourings: the required minion homomorphism would map two polymorphisms f and fʹ from $\operatorname{Pol}(\rel H,\rel G)$ to the same function if and only if f can be recoloured to fʹ by changing one output value at a time, that is, if there is a sequence of polymorphisms $f\_0,\ldots,f\_n\in \operatorname{Pol}(\rel H,\rel G)$ such that f0 = f, fn = fʹ and, for each i ∈ {0, …, n − 1}, there is a unique tuple t̄i with fi(t̄i) ≠ fi + 1(t̄i). However, as shown in, recolourability is inherently a topological notion, so the resulting proof would have the same essence as the one presented here. We chose the current presentation because we believe that it reflects the ‘true essence’ of the proof. ### A minion homomorphism [fig:box-k3sq] As mentioned above, the required minion homomorphism is obtained as a composition of three mappings. The first one is a minion homomorphism from polymorphisms of graphs to polymorphisms of simplicial complexes. We implicitly use that for any graphs $\rel H\_1$ and $\rel H\_2$, there is a natural isomorphism of $\mathbb Z\_2$-simplicial complexes $$\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2, \rel H\_1) \times \operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2, \rel H\_2) \simeq \operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2, \rel H\_1 \times \rel H\_2)$$ given by the $\mathbb Z\_2$-simplicial map: ((a, b), (aʹ, bʹ)) ↦ ((a, aʹ), (b, bʹ)). See Fig. [fig:box-k3sq] for an example. [lem:mu1] For graphs $\rel H$, $\rel G$, the mapping $$\mu\_1\colon \operatorname{Pol}(\rel H,\rel G) \to \operatorname{Pol}(\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel H),\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel G))$$ defined as μ1(f)((u1, v1), …, (un, vn)) : = (f(u1, …, un), f(v1, …, vn)) is a minion homomorphism. Let us first check that μ1(f) is indeed a simplicial map. Assume that σ1, …, σn are faces in $\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel H)$, i.e., σi is a subset of edges of some complete directed bipartite subgraph of $\rel H$. We may assume without loss of generality that σi is the set of all edges of a complete directed bipartite subgraph of $\rel H$ which gives σi = Ui × Vi for some Ui, Vi ⊆ V(G), which form a bipartition of some complete bipartite subgraph of $\rel G$. By definition, μ1(f)(σ1 × … × σn) = {(f(u1, …, un), f(v1, …, vn)) : ui ∈ Ui, vi ∈ Vi} = f(U) × f(V), where U = U1 × ⋯ × Un and V = V1 × ⋯ × Vn. By the definition of graph product, all edges between U and V are present in $\rel H^n$. Consequently, f(U) × f(V) is a complete directed bipartite subgraph of $\rel G$, since f is a polymorphism. This implies that μ1(f)(σ1 × … × σn) is a face of $\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\rel G)$. It is straightforward that the mapping μ1 preserves both the $\mathbb Z\_2$-action and taking minors. The next step is from $\mathbb Z\_2$-simplicial complexes to $\mathbb Z\_2$-spaces. The map that we construct will not be a minion homomorphism, it will satisfy a weaker condition that will be sufficient later. Assume that ⊤X, ⊤Y are $\mathbb Z\_2$-spaces and let $\clo M$ be a minion. We say that a mapping $\xi\colon \clo M \to \operatorname{Pol}(\top X,\top Y)$ preserves minors up to $\mathbb Z\_2$-homotopy if for all n, m > 0, $f\in \clo M^{(n)}$, and π: [n] → [m], we have that ξ(fπ) is $\mathbb Z\_2$-homotopic to ξ(f)π. We recall that the points in the geometric representation of $\cox K$ can be viewed as convex combinations of vertices of $\cox K$, more precisely $ {\lvert \mathord{\cox K}\rvert} = \{ \sum\_{v\in \sigma} \lambda\_v v \mid \sigma \in \cox K, \lambda\_v \geq 0, \sum\_{v\in \sigma} \lambda\_v = 1 \} $ and the representation of each point is unique (as we identified v ∈ V(K) with affinely independent vectors in RV(K), namely the basis vectors). This is used in the following lemma. [lem:mu2] Let $\cox K$, $\cox K'$ be two $\mathbb Z\_2$-simplicial complexes. Let $\mu\_2\colon \operatorname{Pol}(\cox K,\cox K') \to \operatorname{Pol}({\lvert \mathord{\cox K}\rvert},{\lvert \mathord{\cox K'}\rvert})$ be the linear extension, i.e., μ2 takes $f \in \operatorname{Pol}(\cox K,\cox K')$ to μ2(f): (∑v ∈ σ1λ1, vv, …, ∑v ∈ σnλn, vv) ↦ ∑v1 ∈ σ1, …, vn ∈ σnλ1, v1⋯λn, vnf(v1, …vn) for a point in $\Delta^{\sigma\_1}\times\dots\times \Delta^{\sigma\_n} \subseteq {\lvert \mathord{\cox K}\rvert}^n$. Then μ2 preserves minors up to $\mathbb Z\_2$-homotopy. Let $f \colon \cox K^n \to \cox K'$ be a simplicial map and pick any π: [n] → [m]. Then $$\begin{aligned} \mu\_2(f^\pi)(\sum\_{v\in \sigma\_1} \lambda\_{1,v} v, \dots, \sum\_{v\in \sigma\_m} \lambda\_{m,v} v) &= \sum\_{v\_1 \in \sigma\_1,\dots,v\_m\in \sigma\_m} \lambda\_{1,v\_1}\cdots\lambda\_{m,v\_m} f^\pi(v\_1,\dots v\_m)\\ &= \sum\_{v\_1 \in \sigma\_1,\dots,v\_m\in \sigma\_m} \lambda\_{1,v\_1}\cdots\lambda\_{m,v\_m} f(v\_{\pi(1)},\dots v\_{\pi(n)}). \end{aligned}$$ On the other hand, if we take the induced map first and then the minor, we obtain: $$\begin{gathered} \mu\_2(f)(\sum\_{v\in \sigma\_{\pi(1)}} \lambda\_{\pi(1),v} v, \dots, \sum\_{v\in \sigma\_{\pi(n)}} \lambda\_{\pi(n),v} v) \\ = \sum\_{v\_1 \in \sigma\_{\pi(1)},\dots,v\_n\in \sigma\_{\pi(n)}} \lambda\_{\pi(1),v\_1}\cdots\lambda\_{\pi(n),v\_n} f(v\_1,\dots v\_n). \end{gathered}$$ Both points lie in $\Delta^\sigma \subseteq {\lvert \mathord{\cox K'}\rvert}$ for $\sigma = \{f(v\_1,\dots,v\_n) \mid v\_i\in \sigma\_{\pi(i)}\} \in \cox K'$. We can thus continuously move from one to the other. Formally, we define a homotopy $h\colon {\lvert \mathord{\cox K}\rvert}^m \times [0,1] \to {\lvert \mathord{\cox K'}\rvert}$ by $$\textstyle h(x\_1,\dots,x\_m,t) = t \mu\_2(f)(x\_{\pi(1)},\dots,x\_{\pi(n)}) + (1-t) \mu\_2(f^\pi)(x\_1,\dots,x\_m).$$ It is clear that $h\_t\colon \tup x \mapsto h(\tup x,t)$ is a well-defined $\mathbb Z\_2$-map, and therefore h is the required Z2-homotopy. (As a side note, observe that the homotopy is constant on vertices: for any vertices v1, …, vm of $\cox K$ and each t ∈ [0, 1], ht(v1, …, vm) is equal to fπ(v1, …, vm) = f(vπ(1), …, vπ(n)).) [lem:r] Let $\rel H$ be a non-bipartite graph. Then there is a $\mathbb Z\_2$-map $r\colon {\mathcal S}^1 \to {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel H})}\rvert}}$. Since the graph $\rel H$ is not bipartite, there is a homomorphism $h\colon \rel C\_k \to \rel H$ for some odd k. This induces a $\mathbb Z\_2$-map ${\lvert \mathord{h'}\rvert}\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}\rvert}} \to {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel H})}\rvert}}$, and since ${{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel C\_k})}\rvert}}$ is $\mathbb Z\_2$-homeomorphic to ${\mathcal S}^1$ (see Example [ex:hom-k3]), the claim follows. [lem:mu] Let $\rel H,\rel G$ be two graphs such that $\rel H$ is non-bipartite, $\rel H \to \rel G$, and there is a $\mathbb Z\_2$-map $s\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}} \to {\mathcal S}^1$. Then $\mu\colon \operatorname{Pol}({\rel H}, {\rel G}) \to \operatorname{Pol}( {\mathcal S}^1, {\mathcal S}^1 )$ defined as μ(f)(x1, …, xn) : = s(μ2μ1(f)(r(x1), …, r(xn))), where μ1, μ2, and r are from Lemmas [lem:mu1], [lem:mu2], and [lem:r], respectively, preserves minors up to $\mathbb Z\_2$-homotopy. Assume that f is a polymorphism from $\rel H$ to $\rel G$ of arity n and let π: [n] → [m]. We want to prove that μ(f)π is $\mathbb Z\_2$-homotopic to μ(fπ). From Lemmas [lem:mu1] and [lem:mu2], we have that μ2μ1(f)π and μ2μ1(fπ) are $\mathbb Z\_2$-homotopic; let hʹ be a $\mathbb Z\_2$-homotopy that witnesses this fact. We define a $\mathbb Z\_2$-homotopy $h\colon {\mathcal T}^m \times [0,1] \to {\mathcal S}^1$ by h(x1, …, xm, t) = shʹ(r(x1), …, r(xm), t). Indeed, for t = 0, h(x1, …, xm, 0) = s(μ2μ1(f)(r(xπ(1)), …, r(xπ(n)))) = μ(f)π(x1, …, xm) while for t = 1, $$h(x\_1,\dots,x\_m,1) = s\big(\mu\_2\mu\_1(f^\pi)\big(r(x\_1),\dots,r(x\_m)\big)\big) = \mu(f^\pi)(x\_1,\dots,x\_m).$$ This concludes the proof. The final step is from $\mathbb Z\_2$-spaces to the fundamental groups. Recall that, for a continuous function f : ⊤X → ⊤Y, we have a group homomorphism $f\_\\colon {{\pi\_1}(\mathord{\top X})}\to {{\pi\_1}(\mathord{\top Y})}$ defined as f\([ℓ]) = [fℓ]. We will also need a group homomorphism from ${{\pi\_1}(\mathord{\top X})}^n$ to ${{\pi\_1}(\mathord{\top X^n})}$, that is guaranteed to exist for any path connected space ⊤X by. One such homomorphism is the mapping $e\_n\colon {{\pi\_1}(\mathord{\top X})}^n \to {{\pi\_1}(\mathord{\top X^n})}$ defined as en([ℓ1], …, [ℓn]) : = [t ↦ (ℓ1(t), …, ℓn(t))] for $\ell\_1,\dots,\ell\_n\colon {\mathcal S}^1 \to \top X$. [lem:nu] Let ⊤X, ⊤Y be two path connected $\mathbb Z\_2$-spaces. Then the mapping ν(f) : = f\en* is a minion homomorphism from Pol(⊤X, ⊤Y) to $\operatorname{Pol}({{\pi\_1}(\mathord{\top X})},{{\pi\_1}(\mathord{\top Y})})$. Moreover, if f and g are homotopic then ν(f) = ν(g). Assume that f ∈ Pol(⊤X, ⊤Y) is of arity n, π: [n] → [m], and $\ell\_1,\dots,\ell\_m\colon {\mathcal S}^1 \to \top X$. To simplify notation, let ℓ(t) = (ℓ1(t), …, ℓm(t)) go from ${\mathcal S}^1$ to ⊤Xn and ℓπ(t) = (ℓπ(1)(t), …, ℓπ(n)(t)) go from ${\mathcal S}^1$ to ⊤Xm. Using the definitions of f\* and en, we get ν(fπ)(ℓ1, …, ℓm) = [fπℓ] = [fℓπ] = ν(f)π(ℓ1, …, ℓm). Finally, since f\* = g\* if f and g are homotopic, we also get that ν(f) = ν(g). We recall that ${{\pi\_1}(\mathord{{\mathcal S}^1})} \simeq \mathbb Z$ (Lemma [lem:fg-s1]). In the following statement we identify the two isomorphic groups to obtain a minion homomorphism to $\operatorname{Pol}(\mathbb Z,\mathbb Z)$, the minion of all linear functions on $\mathbb Z$. Let $\rel H,\rel G$ be two graphs such that $\rel H$ is non-bipartite, $\rel H \to \rel G$, and there is a $\mathbb Z\_2$-map $s\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}} \to {\mathcal S}^1$. The mapping νμ is a minion homomorphism from $\operatorname{Pol}(\rel H,\rel G)$ to $\operatorname{Pol}(\mathbb Z,\mathbb Z)$ assuming μ is as in Lemma [lem:mu] and ν is as in Lemma [lem:nu]. Clearly, $\nu\mu\colon \operatorname{Pol}(\rel H,\rel G) \to \operatorname{Pol}(\mathbb Z,\mathbb Z)$ is a well-defined mapping that preserves arities. We need to show that it also preserves minors. This follows from the facts that μ preserves minors up to $\mathbb Z\_2$-homotopy (Lemma [lem:mu]) and that ν is a minion homomorphism that is constant on $\mathbb Z\_2$-homotopy classes (Lemma [lem:nu]). More precisely, assume $f\in \operatorname{Pol}(\rel H,\rel G)$ is of arity n and π: [n] → [m]. Then μ(fπ) and μ(f)π are $\mathbb Z\_2$-homotopic, and therefore ν(μ(fπ)) = ν(μ(f)π) = ν(μ(f))π where the second equality follows from minor preservation by ν. We remark that, for any (n-ary) function $f\in \operatorname{Pol}(\rel H,\rel G)$, the coefficients of the linear function νμ(f) = ∑i = 1ncixi can be naturally thought of as the degrees of f at the corresponding coordinates. Such degrees can be defined in a combinatorial way (see ) — the intuitions in that approach are still topological, but the technical proofs become somewhat ad-hoc. ### Bounding essential arity To finish the analysis of polymorphisms from $\rel H$ to $\rel G$ necessary for applying Theorem [thm:bounded-arity], we need to bound the essential arity of functions in the image of νμ (defined above) and show that none of these functions is a constant function. We achieve this by proving that the image of νμ is contained in the minion $\clo Z\_{\leq N}$ for some N. Recall that this minion is defined to be the set of all functions $f\colon \mathbb Z^n \to \mathbb Z$ of the form f(x1, …, xn) = c1x1 + … + cnxn for some c1, …, $c\_n\in \mathbb Z$ with ∑i = 1n\|ci\| ≤ N and ∑i = 1nci odd. The oddness of the sum of coefficients follows from a well-known fact about $\mathbb Z\_2$-maps on ${\mathcal S}^1$. We recall that the degree of a map $f\colon {\mathcal S}^1 \to {\mathcal S}^1$ is the integer df such that the induced map f\* on ${{\pi\_1}(\mathord{{\mathcal S}^1})} = \mathbb Z$ is x ↦ df ⋅ x. [] [lem:odd] The degree of any $\mathbb Z\_2$-map $f\colon {\mathcal S}^1\to {\mathcal S}^1$ is odd. [lem:odd-2] Let $\rel H,\rel G$ be two graphs such that $\rel H$ is non-bipartite, $\rel H \to \rel G$, and there is a $\mathbb Z\_2$-map $s\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}} \to {\mathcal S}^1$. Let μ be as in Lemma [lem:mu], ν as in Lemma [lem:nu], and let $\clo H = \operatorname{Pol}(\rel H,\rel G)$. If $f \in \operatorname{Pol}(\rel H,\rel G)$ and νμ(f): (x1, …, xn) ↦ ∑i = 1ncixi, then ∑i = 1nci is odd. Consider the (unique) unary minor h(x) : = f(x, …, x) of f. Since μ(h) is a $\mathbb Z\_2$-map, by Lemma [lem:odd] it has an odd degree, i.e., νμ(h): x ↦ dhx for some odd dh. Finally, since νμ preserves minors, we get that dh = ∑i ∈ [n]ci which we wanted to show to be odd. The bound on the sum of absolute values of coefficients is given by the discrete structure of the involved graphs. The key here is that there are only finitely many polymorphisms of a fixed arity between two given finite graphs. [lem:bound] Let $\clo M$ be a minion on finite sets A, B. Assume that $\xi\colon \clo M \to \operatorname{Pol}(\mathbb Z,\mathbb Z)$ is a minion homomorphism. Then there exists N such that for all $f\in \clo M$, if ξ(f): (x1, …, xn) ↦ ∑i = 1ncixi, then ∑i = 1n\|ci\| ≤ N. We first consider binary functions. There are only finitely many functions $f\in \clo M^{(2)}$, so clearly the sum of the absolute values of the coefficients of ξ(f) is bounded by some N. We argue that the same N provides a bound for all other arities as well. Let $f\in \clo M$ and ξ(f)(x1, …, xn) = c1x1 + … + cnxn. Let σ: [n] → {0, 1} be defined as $$\sigma(i) = \begin{cases} 0 & \text{if $c\_i \leq 0$, and} \\ 1 & \text{if $c\_i > 0$.} \end{cases}$$ I.e., σ− 1(1) is the set of all coordinates of ξ(f) with positive coefficients, and σ− 1(0) of those with negative or zero coefficients. Now, let g = fσ, that is, g is the minor of f defined by g(x0, x1) = f(xσ(1), …, xσ(n)), so g is obtained by identifying all variables of f that induce positive coefficients on ξ(f), and also all those that induce negative coefficients. We let ξ(g)(x0, x1) = c+x1 + c−x0. Since ξ preserves minors, we get that c+ = ∑ci > 0ci and c− = ∑ci < 0ci. Note that c+ ≥ 0 and c− ≤ 0. Finally, ∑i ∈ [n]\|ci\| = ∑ci > 0ci − ∑ci < 0ci = c+ − c− = \|c+\| + \|c−\| ≤ N where the last inequality follows from the definition of N. We can now finish the proof of Theorem [thm:main-s1]. [Proof of Theorem [thm:main-s1]] We assume that $\rel H,\rel G$ are two graphs such that $\rel H$ is non-bipartite, $\rel H \to \rel G$, and there is a $\mathbb Z\_2$-map $s\colon {{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}} \to {\mathcal S}^1$. Let μ be as in Lemma [lem:mu], ν as in Lemma [lem:nu], and N be the bound obtained from Lemma [lem:bound] for $\clo M = \operatorname{Pol}(\rel H,\rel G)$ and ξ = νμ. We claim that $\nu\mu(f) \in \clo Z\_{\leq N}$ for each $f\in \operatorname{Pol}(\rel H,\rel G)$. Assume νμ(f): (x1, …, xn) ↦ ∑i = 1ncixi. We have that ∑i = 1nci is odd from Lemma [lem:odd-2], and ∑i = 1n\|ci\| ≤ N by the choice of N. This concludes that there is a minion homomorphism from $\operatorname{Pol}(\rel H,\rel G)$ to a minion of bounded essential arity, namely $\clo Z\_{\leq N}$, and thus Theorem [thm:main-s1] follows from Theorem [thm:bounded-arity]. Proofs of Corollaries [thm:circular-cliques] and [thm:square-free] ------------------------------------------------------------------ To show that Theorem [thm:main-s1] implies Corollaries [thm:circular-cliques] and [thm:square-free], we need the following facts about the structure of ${{\lvert \mathord{\operatorname{\text{\normalfont \sffamily Hom}}(\rel K\_2,\mathord{\rel G})}\rvert}}$ for the relevant graphs $\rel G$. These facts seem to be folklore, but we include proofs for completeness. [scale = 1.2, baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1cm); i/in 0/0,51.42/1,102.85/2,205.71/4,257.14/5 () [circle,fill=red,inner sep=1.5,label=i:$\c$] at (i:1) ; i/in 154.28/3,308.57/6 () [circle,draw,fill=white,inner sep=1.5,label=i:$\c$] at (i:1) ; i/in 1/3,3/5,4/6,6/1,0/3,3/6,6/2 (i) – (); i/in 4/2,5/0,4/1,5/2,4/0,5/1 (i) – (); [scale = 1.2, baseline=([yshift=-.5ex]current bounding box.center)] (-1.22,0.97) – (-1.9,0.43) – (-1.9,-0.43) – (-1.22,-0.97) – cycle; (1.22,-0.97) – (1.9,-0.43) – (1.9,0.43) – (1.22,0.97) – cycle; (-1.52,-0.35) – (-1.52,-1.22) – (-0.85,-1.76) – (0.0,-1.56) – cycle; (1.52,0.35) – (1.52,1.22) – (0.85,1.76) – (0.0,1.56) – cycle; (-0.68,-1.41) – (0.0,-1.95) – (0.85,-1.76) – (1.22,-0.97) – cycle; (0.68,1.41) – (0.0,1.95) – (-0.85,1.76) – (-1.22,0.97) – cycle; (0.68,-1.41) – (1.52,-1.22) – (1.9,-0.43) – (1.52,0.35) – cycle; (-0.68,1.41) – (-1.52,1.22) – (-1.9,0.43) – (-1.52,-0.35) – cycle; (1.52,-0.35) – (1.9,0.43) – (1.52,1.22) – (0.68,1.41) – cycle; (-1.52,0.35) – (-1.9,-0.43) – (-1.52,-1.22) – (-0.68,-1.41) – cycle; (1.22,0.97) – (0.85,1.76) – (0.0,1.95) – (-0.68,1.41) – cycle; (-1.22,-0.97) – (-0.85,-1.76) – (0.0,-1.95) – (0.68,-1.41) – cycle; (0.0,1.56) – (-0.85,1.76) – (-1.52,1.22) – (-1.52,0.35) – cycle; (0.0,-1.56) – (0.85,-1.76) – (1.52,-1.22) – (1.52,-0.35) – cycle; (1.9,0.43) – (1.52,1.22) – (0.85,1.76) – (0.0,1.95) – (0.68,1.41) – (1.22,0.97) – cycle; (-1.22,0.97) – (-1.22,0.97); (-1.22,0.97) – (-1.9,0.43); (-1.22,0.97) – (-1.9,-0.43); (-1.22,0.97) – (-1.22,-0.97); (-1.22,0.97) – (-0.85,1.76); (-1.9,0.43) – (-1.22,0.97); (-1.9,0.43) – (-1.9,0.43); (-1.9,0.43) – (-1.9,-0.43); (-1.9,0.43) – (-1.22,-0.97); (-1.9,0.43) – (-1.52,-0.35); (-1.9,0.43) – (-0.68,1.41); (-1.9,0.43) – (-1.52,1.22); (-1.9,-0.43) – (-1.22,0.97); (-1.9,-0.43) – (-1.9,0.43); (-1.9,-0.43) – (-1.9,-0.43); (-1.9,-0.43) – (-1.22,-0.97); (-1.9,-0.43) – (-1.52,-1.22); (-1.9,-0.43) – (-0.68,-1.41); (-1.9,-0.43) – (-1.52,0.35); (-1.22,-0.97) – (-1.22,0.97); (-1.22,-0.97) – (-1.9,0.43); (-1.22,-0.97) – (-1.9,-0.43); (-1.22,-0.97) – (-1.22,-0.97); (-1.22,-0.97) – (-0.85,-1.76); (-1.22,-0.97) – (0.0,-1.95); (-1.22,-0.97) – (0.68,-1.41); (-1.52,-0.35) – (-1.9,0.43); (-1.52,-0.35) – (-1.52,-0.35); (-1.52,-0.35) – (-1.52,-1.22); (-1.52,-0.35) – (-0.85,-1.76); (-1.52,-0.35) – (0.0,-1.56); (-1.52,-0.35) – (-0.68,1.41); (-1.52,-0.35) – (-1.52,1.22); (-1.52,-1.22) – (-1.9,-0.43); (-1.52,-1.22) – (-1.52,-0.35); (-1.52,-1.22) – (-1.52,-1.22); (-1.52,-1.22) – (-0.85,-1.76); (-1.52,-1.22) – (0.0,-1.56); (-1.52,-1.22) – (-0.68,-1.41); (-1.52,-1.22) – (-1.52,0.35); (-0.85,-1.76) – (-1.22,-0.97); (-0.85,-1.76) – (-1.52,-0.35); (-0.85,-1.76) – (-1.52,-1.22); (-0.85,-1.76) – (-0.85,-1.76); (-0.85,-1.76) – (0.0,-1.56); (-0.85,-1.76) – (0.0,-1.95); (-0.85,-1.76) – (0.68,-1.41); (0.0,-1.56) – (-1.52,-0.35); (0.0,-1.56) – (-1.52,-1.22); (0.0,-1.56) – (-0.85,-1.76); (0.0,-1.56) – (0.0,-1.56); (0.0,-1.56) – (0.85,-1.76); (0.0,-1.56) – (1.52,-1.22); (0.0,-1.56) – (1.52,-0.35); (1.22,-0.97) – (1.22,-0.97); (1.22,-0.97) – (-0.68,-1.41); (1.22,-0.97) – (0.0,-1.95); (1.22,-0.97) – (0.85,-1.76); (1.22,-0.97) – (1.9,-0.43); (-0.68,-1.41) – (-1.9,-0.43); (-0.68,-1.41) – (-1.52,-1.22); (-0.68,-1.41) – (1.22,-0.97); (-0.68,-1.41) – (-0.68,-1.41); (-0.68,-1.41) – (0.0,-1.95); (-0.68,-1.41) – (0.85,-1.76); (-0.68,-1.41) – (-1.52,0.35); (0.0,-1.95) – (-1.22,-0.97); (0.0,-1.95) – (-0.85,-1.76); (0.0,-1.95) – (1.22,-0.97); (0.0,-1.95) – (-0.68,-1.41); (0.0,-1.95) – (0.0,-1.95); (0.0,-1.95) – (0.85,-1.76); (0.0,-1.95) – (0.68,-1.41); (0.85,-1.76) – (0.0,-1.56); (0.85,-1.76) – (1.22,-0.97); (0.85,-1.76) – (-0.68,-1.41); (0.85,-1.76) – (0.0,-1.95); (0.85,-1.76) – (0.85,-1.76); (0.85,-1.76) – (1.52,-1.22); (0.85,-1.76) – (1.52,-0.35); (1.9,-0.43) – (1.22,-0.97); (1.9,-0.43) – (1.9,-0.43); (1.9,-0.43) – (1.52,0.35); (1.9,-0.43) – (0.68,-1.41); (1.9,-0.43) – (1.52,-1.22); (1.52,0.35) – (1.9,-0.43); (1.52,0.35) – (1.52,0.35); (1.52,0.35) – (0.68,-1.41); (1.52,0.35) – (1.52,-1.22); (1.52,0.35) – (0.0,1.56); (0.68,-1.41) – (-1.22,-0.97); (0.68,-1.41) – (-0.85,-1.76); (0.68,-1.41) – (0.0,-1.95); (0.68,-1.41) – (1.9,-0.43); (0.68,-1.41) – (1.52,0.35); (0.68,-1.41) – (0.68,-1.41); (0.68,-1.41) – (1.52,-1.22); (1.52,-1.22) – (0.0,-1.56); (1.52,-1.22) – (0.85,-1.76); (1.52,-1.22) – (1.9,-0.43); (1.52,-1.22) – (1.52,0.35); (1.52,-1.22) – (0.68,-1.41); (1.52,-1.22) – (1.52,-1.22); (1.52,-1.22) – (1.52,-0.35); (1.9,0.43) – (1.22,-0.97); (1.9,0.43) – (1.9,-0.43); (1.9,0.43) – (1.52,-0.35); (1.52,1.22) – (1.52,0.35); (1.52,1.22) – (1.52,-0.35); (1.52,1.22) – (0.0,1.56); (0.68,1.41) – (-1.22,0.97); (0.68,1.41) – (1.52,-0.35); (0.68,1.41) – (-0.85,1.76); (1.52,-0.35) – (0.0,-1.56); (1.52,-0.35) – (0.85,-1.76); (1.52,-0.35) – (1.52,-1.22); (1.52,-0.35) – (1.52,-0.35); (1.22,0.97) – (1.22,-0.97); (1.22,0.97) – (1.9,-0.43); (1.22,0.97) – (-0.68,1.41); (0.85,1.76) – (1.52,0.35); (0.85,1.76) – (-0.68,1.41); (0.85,1.76) – (0.0,1.56); (0.0,1.95) – (-1.22,0.97); (0.0,1.95) – (-0.68,1.41); (0.0,1.95) – (-0.85,1.76); (-0.68,1.41) – (-1.9,0.43); (-0.68,1.41) – (-1.52,-0.35); (-0.68,1.41) – (-0.68,1.41); (-0.68,1.41) – (-1.52,1.22); (0.0,1.56) – (1.52,0.35); (0.0,1.56) – (0.0,1.56); (0.0,1.56) – (-0.85,1.76); (0.0,1.56) – (-1.52,1.22); (0.0,1.56) – (-1.52,0.35); (-0.85,1.76) – (-1.22,0.97); (-0.85,1.76) – (0.0,1.56); (-0.85,1.76) – (-0.85,1.76); (-0.85,1.76) – (-1.52,1.22); (-0.85,1.76) – (-1.52,0.35); (-1.52,1.22) – (-1.9,0.43); (-1.52,1.22) – (-1.52,-0.35); (-1.52,1.22) – (-0.68,1.41); (-1.52,1.22) – (0.0,1.56); (-1.52,1.22) – (-0.85,1.76); (-1.52,1.22) – (-1.52,1.22); (-1.52,1.22) – (-1.52,0.35); (-1.52,	arxiv_0000753
Instantaneous support propagation for Λ-Fleming-Viot processes ============================================================== For a probability-measure-valued neutral Fleming-Viot process Zt with Lévy mutation and resampling mechanism associated to a general Λ-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time t > 0, with probability one the closed support S(Zt) of the Fleming-Viot process satisfies S(ν \* Zt) ⊆ S(Zt), where ν is the Lévy measure of the mutation process. To show this result, we apply Donnelly-Kurtz’s lookdown particle representation for Fleming-Viot processes. Introduction and main results ============================= Introduction ------------ In this work, we study generalized Fleming-Viot processes, also called Λ-Fleming-Viot processes, a class of probability measure-valued Markov processes which model the evolving distribution of genetic types in a population subject to random reproduction, mutation and genetic (allele) drift. The models considered are selectively neutral, meaning that no genetic type has a reproductive advantage over another. In the context of measure-valued Markov processes, it is natural to view mutation as a spatial motion on the space of genetic types and to study the measure’s support properties vis-à-vis the mutation/motion. It is such a perspective that we take here, in particular the case in which type space is Rd and the mutation operator is the generator of a Lévy process with jumps. The classical Fleming-Viot model was introduced by Fleming and Viot in. The model and its variants, both with and without mutation, have been studied in depth. For a survey of some classical results, see. One feature of the model is its connection to Kingman’s coalescent, an exchangeable integer partition-valued Markov process arising in mathematical population genetics. The coalescent describes the merging of ancestral lines backwards in time of samples from the Fleming-Viot process. The connection was made rigorous in via a construction of the Fleming-Viot process explicitly carrying versions of Kingman’s coalescent. The more general Λ-Fleming-Viot processes dual to more general coalescents can be found in Donnelly and Kurtz who identified them as a class of measure-valued dual processes to the Λ-coalescents, which themselves are a class of exchangeable integer partition-valued Markov processes generalizing Kingman’s coalescent by allowing multiple collisions. The Λ-coalescents were introduced independently by Pitman and Sagitov. The class of Λ-Fleming-Viot processes allows more general reproduction mechanisms in which each individual can possibly give birth to a large number of children that is comparable to the size of the whole population. Finally, we note that Fleming-Viot-type processes (both classical and Λ-) arise as the infinite particle limits of individual-based Markov models for the distribution of genetic types in a population of fixed size, such as the Moran process and its variants such as Cannings population models with non-overlapping generations; see Donnelly and Kurtz. We refer to Bertoin and Le Gall for an alternative representation of the mutationless Λ-Fleming-Viot process via a flow of bridges. An even more general class of Fleming-Viot processes that is dual to Ξ-coalescent involving simultaneous multiple collisions can be found in Birkner et al., where the lookdown particle representation is presented, and the pathwise convergence of the empirical measures of approximating particle systems to the limiting Ξ-Fleming-Viot process is also proved. The support properties of measure-valued Markov processes have been studied extensively. However, the majority of the literature concerns Dawson-Watanabe superprocesses, which, roughly speaking, are the spatial analogues of continuous state branching processes. They are infinitely divisible and their Laplace functionals have a useful representation in terms of solutions to semi-linear partial differential equations. These tools and others (for example the historical process and the Brownian snake) have led to very precise results concerning the supports of these processes, especially for super-Brownian motion. See for an introduction to Dawson-Watanabe superprocesses and a survey of some classical results concerning their supports. The literature on support properties of Fleming-Viot processes is sparser. Compact support and other properties for the classical Fleming-Viot process with Brownian mutation were established in a pioneering work of Dawson and Hochberg by introducing an infinite particle system representation. A refinement of this representation, the so-called lookdown particle construction, was proposed in for Fleming-Viot processes and other measure-valued processes. The lookdown representation encodes a genealogy of the Fleming-Viot process, and often plays the role of the historical process for superprocesses. Using the lookdown representation, more recent work of Liu and Zhou has established, among other properties, the compact support property, the one-sided modulus of continuity and Hausdorff dimension of the support process for the Λ-Fleming-Viot process with Brownian mutation with the associated Λ-coalescent coming down from infinity. For a measure-valued process, instantaneous support propagation occurs if the (closed) support of the random measure can reach arbitrarily far away within arbitrarily small time. Instantaneous support propagation for the super-Lévy process with binary branching and jump type spatial motion was first proved by Perkins in 1990. Surprisingly, similar instantaneous support propagation for the closely related Kingman-Fleming-Viot process with jump type mutation remained an unsolved problem. The goal of this paper is to provide an answer to this problem. Our main result, Theorem [thmmain], establishes a version of instantaneous support propagation when the mutation is a Lévy process with jumps. It states that the support of the measure obtained by convolving the Λ-Fleming-Viot process with the jump measure of the mutation process is contained in the support of the Λ-Fleming-Viot process itself. In many cases, this implies that the closed support is Rd. The analogous statement for Dawson-Watanabe superprocesses is known. We show that it holds for Λ-Fleming-Viot processes in complete generality: the mutation process can be any Lévy process and we require no conditions on the re-sampling mechanism/ancestral coalescent. Statement of main result ------------------------ We now introduce some notation. Let Mf(E) and M1(E) denote respectively the spaces of finite measures and probability measures on a Polish space E. When E = Rd, we simply write Mf and M1. Let B = B(Rd) denote the space of Borel functions from Rd into R, and let B+, Bb and Bb+ denote, respectively, the subspaces of non-negative, bounded, and non-negative and bounded Borel functions. For μ ∈ Mf and ϕ ∈ B, we will use the notations ⟨ϕ, μ⟩ and μ(ϕ) interchangeably (as appropriate) to denote the integral of ϕ with respect to μ, that is $$\langle \phi, \mu \rangle := \int\_{{\mathbb{R}}^d} \phi(x) \mu(dx) =: \mu(\phi). \nonumber$$ In order to define the Λ-Fleming-Viot process and state our main result, we introduce the objects and quantities necessary to describe the re-sampling mechanism and mutation. We begin with the former. Let Λ ∈ Mf([0, 1]). For 2 ≤ k ≤ n ∈ N, we define the rates λn, k by $$\lambda\_{n,k} = \int\_0^1 x^{k-2}(1-x)^{n-k} \Lambda(dx). \nonumber$$ These are of course the merger rates for the Λ-coalescent, which as we have noted encodes the genealogy of the Λ-Fleming-Viot process. This is described in greater detail in Sections [scoalescents] and [slookdown]. Now we introduce the mutation process. Let (Wt : t ≥ 0) be a Lévy process on Rd. We denote its law and expectation by PxW and ExW, respectively, when W0 = x. By the Lévy-Khintchine formula, for all t > 0 and all ξ ∈ Rd, $$\log {\mathbb{E}}^W\_0(e^{i \langle W\_t, \xi\rangle}) = -t \Psi(\xi), \nonumber$$ where the characteristic exponent Ψ is given by $$\Psi(\xi) = i \langle a, \xi \rangle + \frac 1 2 \langle \xi, Q \xi \rangle + \int\_{{\mathbb{R}}^d} \left( 1 - e^{i \langle x, \xi\rangle} + i \langle x, \xi\rangle 1\_{\{\|x\| < 1 \}}\right) \nu(dx). \nonumber$$ In the above, a ∈ Rd, Q ∈ Rd × d is a symmetric positive semidefinite matrix, and ν is a σ-finite measure on Rd, called the Lévy measure of Wt, satisfying ν({0}) = 0 and ∫Rd(1 ∧ ∣x∣2)ν(dx) < ∞; see e.g. Bertoin for details. The generator of Wt is the integro-differential operator A, which satisfies $$\label{eq\_Levy\_generator} A \phi (x) = a \cdot \nabla \phi (x)+\frac{1}{2}\nabla \cdot (Q \nabla \phi) (x) +\int\_{{\mathbb{R}}^d}\left(\phi(x+y)-\phi(x)-\nabla \phi (x) y1\_{\{ \|y\| < 1\}}\right)\nu(dy) \nonumber$$ for all ϕ ∈ Cb2, the space of bounded twice differentiable functions with bounded derivatives up to order two; see Schilling. To define the generator of the Λ-Fleming-Viot process, we introduce an appropriate class of test functions. For n ∈ N and ϕ1, …, ϕn ∈ Bb, let Fϕ1, …, ϕn denote the function on Mf defined by $$F\_{\phi\_1,\dots,\phi\_n}(\mu) = \prod\_{i=1}^n \langle \phi\_i, \mu \rangle. \nonumber$$ Let G(Mf) denote the class containing all such functions for all n ∈ N and ϕi ∈ Bb. For Fϕ1, …, ϕn ∈ G(Mf), we define $$\label{e\_generator} {\mathcal{A}}^{\Lambda,A} F\_{\phi\_1, \dots, \phi\_n} (\mu) := \sum\_{i = 1}^n \langle A \phi\_i, \mu \rangle \prod\_{j \neq i} \langle \phi\_i, \mu \rangle \,\,\, + \sum\_{J \subset [n] : \#J \geq 2} \lambda\_{n,\# J} (F^J\_{\phi\_1,\dots,\phi\_n}(\mu)- F\_{\phi\_1,\dots,\phi\_n}(\mu) ), \nonumber$$ where #J denotes the cardinality of the set J. In the above, for J ⊆ [n], the function Fϕ1, …, ϕnJ is equal to Fϕ1ʹ, …, ϕnʹ where ϕiʹ is defined as follows: if i ∉ J, ϕiʹ = ϕi; if i ∈ J, then ϕiʹ = ϕminJ. That is, the functions corresponding to indices in J are all replaced by the function corresponding to the lowest index in J. The Λ-Fleming-Viot process with mutation operator A, or the (Λ, A)-Fleming-Viot process, is the M1-valued Markov process with generator AΛ, A. We denote it by (Zt)t ≥ 0 and we write PμZ and EμZ to denote, respectively, the law and expectation associated to the process with initial measure Z0 = μ ∈ M1. (Zt)t ≥ 0 is a M1-valued strong Markov process with càdlàg paths. We denote the standard (right-continuous) filtration generated by Zt by (FtZ)t ≥ 0, and will occasionally write (Ft)t ≥ 0. We refer to Remark 1.1 of for a discussion of different constructions of Zt and the form of the generator on more general functions. Let S(μ) denote the closed (topological) support of a measure μ ∈ Mf. We denote the convolution of measures μ and ν by μ \* ν; for k ∈ N, the k-fold convolution of ν with itself is denoted ν(k). Our main result is the following. [thmmain] Let Zt be a (Λ, A)-Fleming-Viot process, where Λ ∈ Mf([0, 1]) and A is the generator of a Lévy process with non-degenerate Lévy measure ν. Then for any t > 0, with probability one, $$S( \nu^{(k)} \* Z\_t) \subseteq S(Z\_t) \quad \text{ for all } \, k \in {\mathbb{N}}. \nonumber$$ If S(ν) = Rd, then S(Zt) = Rd almost surely for all t > 0. The case in which S(ν) = Rd includes many important examples, such as the case when the mutation process is an α-stable process with any index α ∈ (0, 2). Our theorem of course implies that S(Zt) is unbounded whenever the mutation process has jumps. In contrast, it is known that if a Λ-coalescent comes down from infinity, then under an additional mild condition on the speed of coalescing, almost surely the support for the Λ-Fleming-Viot process with Brownian mutation remains compact at all strictly positive times; see Liu and Zhou. Further discussion. ------------------- As previously noted, the analogous statement concerning the supports of Dawson-Watanabe superprocesses with jump type spatial motion has been known to hold for some time. For binary branching, Perkins first gave a proof in, and Perkins and Evans gave an alternate proof shortly thereafter; see also Section III.2 of Perkins. Theorem [thmmain] establishes that instantaneous support propagation holds in full generality for Λ-Fleming-Viot processes with jump type mutation. The situation is somewhat different when one has Brownian mutation, as is indicated by the compact support theorems mentioned earlier. However, Birkner and Blath pointed out in that if the Λ-coalescent does not come down from infinity, then the Λ-Fleming-Viot process with Brownian mutation has support equal to all of Rd a.s. for all t > 0. The results of Liu and Zhou imply that staying infinite is a nearly optimal condition on the Λ-coalescent in order for this to occur, although whether or not there exist Λ-coalescents which come down from infinity for which the Brownian Λ-Fleming-Viot process has unbounded support remains unknown. For super-Brownian motion, the analogous case to the ``staying infinite" regime was recently studied by Mamin and Mytnik. It is generally thought that Fleming-Viot processes should have similar path properties to their Dawson-Watanabe counterparts, and our result is a verification of this heuristic for the support propagation property. The heuristic is justified in part by the explicit connection between certain sub-families of the two classes of processes. Building on work of Etheridge and March, Perkins proved that a binary-branching super-Brownian motion conditioned to have a constant total mass equal to 1 is a classical Fleming-Viot process with Brownian mutation. Another perspective is that the super-Brownian motion normalized by its total mass is equal in law to a Fleming-Viot process with (randomly) time-inhomogeneous Brownian mutation. This result was generalized in Birkner et al. to a similar relation between a motionless Alpha-stable branching superprocess and mutationless Fleming-Viot process that is dual to a Beta coalescent. When the Dawson-Watanabe superprocess has a spatial motion, the corresponding Λ-Fleming-Viot process has time-inhomogeneous mutation. (The one exception to this rule is when the superprocess has Neveu’s branching mechanism, which is associated to the Bolthausen-Sznitman, or Beta(1, 1) coalescent, which stays infinite. With Brownian motion/mutation, both Neveu’s superprocess and the Λ-Fleming-Viot process whose ancestral coalescent is the Beta(1, 1) coalescent have instantaneous support propagation.) The explicit connection between certain Λ-Fleming-Viot processes and Dawson-Watanabe superprocesses is a strong indication that they should share path properties, provided the associated time change is well-behaved. Blath has proposed using this connection as a means of deducing path properties of Fleming-Viot processes from known properties of the associated Dawson-Watanabe superprocess. Such an approach is necessarily limited to Fleming-Viot processes which are dual to a Beta coalescent, as the correspondence only holds in this case; see. To prove Theorem [thmmain], we take a different approach which is modelled after the original proof of instantaneous propagation of Perkins. Our argument has a similar structure to Perkins’, but where the original proof uses the branching property we make use of an ancestral decomposition. This is achieved using a lookdown construction. Other modifications are required to accommodate the general reproduction mechanism. Because we do not rely explicitly on the representation as a normalized, time-inhomogeneous Dawson-Watanabe superprocess, we are able to give a general proof which holds for reproduction mechanisms that are associated to general Λ-coalescents instead of only Beta coalescents. While we do not explicitly rely upon a representation of the Λ-Fleming-Viot process as a Dawson-Watanabe superprocess, our approach is still informed by the heuristic that small-time behaviour of these processes is similar. It should be pointed out that, although the theorem of Birkner et al. only holds for the Beta coalescents, there is a weaker connection between general Λ-coalescents and continuous-state branching processes. Berestycki et al. constructed a small-time coupling between the Λ-coalescent and an associated continuous-state branching process via two lookdown representations defined via coupled point processes. They used this coupling to give an alternative proof of the speed of coming down from infinity of the Λ-coalescent. The approach we develop here appears to be a viable alternative to the program proposed by Blath, mentioned above, for the study of support properties of generalized Fleming-Viot processes. As noted, the difficulty of analysing Fleming-Viot processes is in part because they are not infinitely divisible, and it is infinite divisibility/the branching property which underpins many of the proofs of support properties for Dawson-Watanabe superprocesses. Our argument substitutes the ancestral representation for the branching property. This acts as an approximate version of infinite divisibility: the decomposition of the process as the sum of a large, but not arbitrarily large, collection of identically distributed and nearly independent processes. This approximation becomes better as the time scale is taken to zero and appears to be a suitable surrogate for infinite divisibility for certain arguments. It would be interesting to see if this could be used to prove for Fleming-Viot processes some of the more precise support theorems known to hold for Dawson-Watanabe superprocesses, for example the exact Hausdorff measure function results in. We conclude by noting that instantaneous propagation as stated here can also be a useful tool in the proof of more subtle support properties. In a recent paper, the first author used the instantaneous support propagation of the Dawson-Watanabe superprocess with α-stable motion and (1 + β)-stable branching mechanism as a tool in the proof of new properties concerning the behaviour of the density over fractal sets. Organization of the paper. -------------------------- The remainder of the paper is organized as follows. Section [scoalescents] defines the Λ-coalescents and states some results which we will use. In Section [slookdown] we introduce the lookdown construction for the Λ-Fleming-Viot process with mutation and state the properties of this model relevant to our proof. We prove Theorem [thmmain] in Section [spf]. Λ-coalescents ============= We now give an overview of Λ-coalescents and state several results which we will use. There are several distinct (but related) classes of processes called ``coalescents" in probability theory; in our setting, a coalescent is an exchangeable integer partition-valued Markov process in which the only allowable transitions involve the merging, or coalescence, of blocks, and hence the partition can only become coarser. Such coalescents are models for stochastic coagulation and have applications in numerous areas. For a broad introduction to models of stochastic coagulation and fragmentation, see the book of Bertoin. A detailed account of the theory of random exchangeable integer partitions and the Λ-coalescents can be found in the notes of Berestycki. The Λ-coalescents form a general class of coalescing partition-valued Markov processes in which only one subset of blocks merges at any given time. Coalescents wherein multiple subsets of blocks can merge at the same time have also been studied and are referred to as Ξ-coalescents; see, for example Schweinsberg. For n ∈ N, let [n] : = {1, 2, …, n}, and write Pn and P∞ to denote the sets of partitions of [n] and N, respectively. We refer to the elements of a partition as its blocks. Let {λb, k : 1 ≤ k ≤ b ∈ N} be an array of non-negative rates. The rate λb, k is the rate at which a given subset of k blocks out of a total of b blocks merges. Because a partition in P∞ may have infinitely many blocks, one defines the dynamics on P∞ via projections onto Pn for all n ∈ N, with the requirement that the projections are consistent and the projected process on Pn is itself a Markov process. A structure theorem due to Pitman states that this is satisfied if and only if the array of rates satisfies $$\lambda\_{b,k} = \int\_0^1 x^{k - 2} (1-x)^{b-k} \Lambda(dx) \nonumber$$ for all 2 ≤ k ≤ b ∈ N for some Borel measure Λ ∈ Mf([0, 1]). Given Λ and its corresponding array of rates, the Λ-coalescent, denoted (Π(t) : t ≥ 0), is the Markov process on P∞ such that for all n ∈ N, its projection onto Pn, denoted Πn(t), is a Markov process governed by the rates {λb, k : 2 ≤ k ≤ b ∈ [n]} as described above. We write PΠ to denote the law of Π(t) and unless otherwise noted we always assume that Π(0) is the singleton partition of N. The partition Π(t) is easily seen to be exchangeable because the merger rates do not depend on which integers are in the blocks. Kingman’s coalescent, in which all mergers are binary, corresponds to the measure Λ = δ0. The well-studied Beta coalescents arise when Λ is chosen to be the distribution of a Beta(2 − α, α) random variable for α ∈ (0, 2). We write #A to denote the cardinality of a countable set A. Then Nt : = #Π(t) ∈ N ∪ { + ∞} is the number of blocks in Π(t). We will write Π(t) = (πi(t) : i = 1, …, Nt), where the sequence is infinite if Nt = ∞, and we use the convention that the blocks are ordered by their minimum elements, so that min{j : j ∈ πi(t)} < min{j : j ∈ πi + 1(t)} for i = 1, …, Nt − 1. For i ≤ Nt we define the asymptotic frequency of πi(t) to be $$\|\pi\_i(t)\| := \lim\_{n \to \infty} n^{-1} \#(\pi\_i(t)\cap [n]). \nonumber$$ It follows from Kingman’s theory of exchangeable partitions that the above limit exists for all t > 0 almost surely. This result was later reproved by Aldous using de Finetti’s theorem. The importance of the Λ-coalescents in our work is because they encode the genealogies of generalized Fleming-Viot processes. Indeed, we observe that given Λ ∈ Mf([0, 1]), the rates λb, k defined above are the same as those given in the introduction to define the generator of Zt. The correspondence is more transparent in the non-spatial setting, and to illustrate it we briefly consider a Λ-Fleming-Viot process without mutation. This is the measure-valued Markov process on M1(E) for some compact Polish space E (typically E = [0, 1]) with generator A0ΛFϕ1, …, ϕn(μ) : = ∑J ⊂ [n] : #J ≥ 2λn, #J(Fϕ1, …, ϕnJ(μ) − Fϕ1, …, ϕn(μ)), where Fϕ1, …, ϕn, Fϕ1, …, ϕnJ ∈ G(Mf(E)) are as defined in the introduction for n ∈ N, ϕi ∈ Bb(E) for i ∈ [n], and J ⊂ [n]. The process generated by A0Λ is a measure-valued dual process to the Λ-coalescent in the following way: one can consider the ranked mass coalescent associated to Π(t), which is the vector of asymptotic frequencies of blocks in Π(t) listed in decreasing order. (Special care must be taken if Π(t) contains singleton blocks, in which case we say Π(t) has dust.) Provided the initial state of the mutationless Λ-Fleming-Viot process has no atoms, the process of ranked vectors of its atoms’ masses is equal in distribution to the ranked mass coalescent associated to (Π(t) : t ≥ 0), which follows from the lookdown representation. See Theorem 3.1 of for a related result. Unlike Dawson-Watanabe superprocesses, there is no useful expression for the Laplace functional of Fleming-Viot processes. The study of Fleming-Viot processes often relies on analysis of the moments, which can be expressed using a dual process involving the coalescent. We now resume our discussion of the number of blocks. A Λ-coalescent comes down from infinity if, when the initial partition has infinitely many blocks, Π(t) has finitely many blocks a.s. for all t > 0. On the other hand, it stays infinite if, with probability one, Π(t) has infinitely many blocks for all t > 0. Given Λ({1}) = 0, the Λ-coalescent either comes down from infinity or stays infinite; see. Kingman’s coalescent and the Beta coalescent with α ∈ (1, 2) come down from infinity, while the Beta coalescents with α ∈ (0, 1] stay infinite. A necessary and sufficient condition on the merger rates for Π(t) to come down from infinity was first given by Schweinsberg. We review a different but equivalent criterion below. Let Λ ∈ Mf([0, 1]). We assume throughout that Λ({1}) = 0. We define ψΛ : R+ → R by ψΛ(u) = Λ({0})u2 + ∫(0, 1](e− ux − 1 + ux)x− 2Λ(dx). Then ψΛ is the Laplace exponent of a spectrally positive (one-dimensional) Lévy process and hence is also the branching mechanism of a continuous state branching process (CSBP). We refer to Chapter 12 of Kyprianou and Li for introductions on continuous-state branching and the Lamperti transform which maps between spectrally positive Lévy processes and CSBPs. It was observed by Bertoin and Le Gall that the Λ-coalescent comes down from infinity if and only if $$\int\_1^\infty \psi\_\Lambda(u)^{-1} du < \infty. \nonumber$$ One can in fact also obtain information about the speed of coming down from infinity (i.e. the asymptotic behaviour of Nt as t ↓ 0) by analysing ψΛ(u). For t > 0, we define v(t) by v(t) : = inf{r > 0 : ∫r∞ψΛ(u)− 1du > t}, with the convention that inf∅ = ∞. Then either v(t) = ∞ for all t, or v(t) is finite for all t and satisfies $$\int\_{v(t)}^\infty \psi\_\Lambda(u)^{-1} du = t. \nonumber$$ A convergence in probability version of the following result is originally due to Bertoin and Le Gall. The version we state now was proved by Berestycki et al.. [theoremblocks] Let Λ ∈ Mf([0, 1]) be such that the Λ-coalescent comes down from infinity. Then $$\lim\_{t\to 0}\frac{N\_t}{v(t)} = 1 \nonumber$$ almost surely and in Lp for all p ≥ 1. A corollary of this result is a global bound on the speed of coming down from infinity. It asserts, essentially, that Kingman’s coalescent comes down from infinity the fastest. Let Λ ∈ MF([0, 1]). From, we deduce that $$\psi\_\Lambda(u) \leq (\Lambda(\{0\}) + \frac 1 2 \Lambda((0,1])) u^2.$$ It then follows from the definition of v(t) that there is a constant cΛ > 0 such that v(t) ≥ cΛt− 1 for all t > 0. Theorem [theoremblocks] then implies that for all ε ∈ (0, cΛ), $$\liminf\_{t\to 0}\frac{N\_t}{t^{-1}} \geq c\_\Lambda - {\epsilon}\nonumber$$ almost surely. The proof of our main result does not require detailed information on the Λ-coalescent associated to the Fleming-Viot process. The argument is structured to allow maximum flexibility (and in fact complete generality) of the reproduction mechanism, and hence of the ancestral coalescent. The fact that Nt is at least of order t− 1 for small t is essentially the only property we use, and is used to prove a mild lemma (Lemma [lemmaspecialsequence]). The lookdown construction ========================= In this section we introduce the countable representation, or ``lookdown“ construction, of the Λ-Fleming Viot process. The idea of using empirical measures of particle systems to approximate the Fleming-Viot process first appeared in Dawson and Hochberg to study the support of the Fleming-Viot process. Donnelly and Kurtz first proposed a lookdown representation that provides nested exchangeable particle approximations to the classical Fleming-Viot process, and further extended the representation to more general measure-valued processes. Such a particle representation encodes the genealogy of the measure-valued process. The version of the lookdown construction used here was originally termed the ``modified” lookdown construction (see ) but is now a standard version, see Birkner and Blath and Blath, and in the sequel we omit the qualifer ``modified." Let Λ ∈ MF([0, 1]). We will assume that Λ({1}) = 0. We consider a (Rd)∞-valued process X(t) = (X1(t), X2(t), …). For n ∈ N, we define the empirical measure $$Z^{(n)}\_t := n^{-1} \sum\_{i = 1}^n \delta\_{X\_t^i}. \nonumber$$ If the vector (X1(t), X2(t), …) is exchangeable, then by de Finetti’s Theorem the empirical measures converge weakly. We can then define $$Z\_t := \lim\_{n \to \infty} n^{-1} Z^{(n)}\_t. \nonumber$$ The lookdown construction is defined in such a way that if the initial vector (X1(0), X2(0), …) is exchangeable, then so is (X1(t), X2(t), …), and Zt is defined for all t > 0. We next follow Birkner and Blath and Birkner et al. to present details of the lookdown construction. Let (X1(0), X2(0), …) ∈ (Rd)∞ be an exchangeable random vector. In order to define the process (X1(t), X2(t), …), we introduce several families of Poisson point processes. First, let {Nij(t) : 1 ≤ i < j ∈ N} be a family of independent Poisson processes on [0, ∞) with rate Λ({0}). At each jump time of the process Nij, the particle at level j looks down (since j > i) to the particle at level i and takes its type (or location). A jump of Nij therefore corresponds to a reproduction event for the particle at level i. The levels of other particles are shifted up to account for the birth. That is, supposing that ΔNij(t) = 1, we have $$X\_k(t) = \begin{cases} X\_k(t-) & \text{ if } k < j,\\ X\_i(t-) & \text{ if } j = k,\\ X\_{k-1}(t-) & \text{ if } j < k. \end{cases}$$ The dynamics described above govern the binary mergers and hence the ``Kingman" component of the Λ-coalescent embedded in the model. We now describe the dynamics arising from multiple mergers. We define Λ0 = Λ − Λ({0})δ0. Let N be a Poisson point process on [0, ∞) × (0, 1] with intensity dt ⊗ u− 2Λ0(du). Let {(ti, ui) : i = 1, 2, …} be an enumeration of the points in N. For each point (ti, ui), we associate a sequence {Yik : k = 1, 2, …} of independent Bernoulli(ui) random variables. If Yik = 1, then the particle at level k at time ti takes part in the birth event at this time. All participating levels take the location of the lowest participating level, and all the other particle positions retain their original order and are shifted upward accordingly. If the jump occurs at time t = ti, and j is the lowest index such that Yij = 1, then $$X\_k(t) = \begin{cases} X\_k(t-) & \text{ if } k \leq j, \\ X\_j(t-) & \text{ if } k > j\text{ and } Y\_{ik} = 1, \\ X\_{k - J\_{ik}} (t-) & \text{ if } k > j \text{ and } Y\_{ik} = 0, \end{cases} \nonumber$$ where Jik = #{l < k : Yil = 1} − 1. Let (W1(t), W2(t), …) be a sequence of independent mutation processes. Between the jump times, the increments of Xi(t) are those of Wi(t) for each i = 1, 2, …. These dynamics are well-defined, as one can show that for any i ∈ N, the number of lookdown events involving level i up to time t is finite. One can rigorously realize the dynamics of (X1(t), X2(t), …) as a countable system of stochastic differential equations driven by {Nij : 1 ≤ i < j ∈ N}, N, {Yik : i, k ∈ N} and (W1(t), W2(t), …). Rather than include this here, we refer to for details. Hereafter, we will always choose to realize the lookdown process by choosing the initial particle positions of (X1(0), X2(0), …) to be i.i.d. samples from a probability measure μ ∈ M1. We denote by LμA, Λ a probability measure which realizes the above construction with this choice of initial positions. The following is a consequence of Theorems 1.1 and 3.2 of. Under LμA, Λ, (X1(t), X2(t), …) is exchangeable for all t > 0, and $$Z\_t = \lim\_{n \to \infty} n^{-1} \sum\_{i=1}^n \delta\_{X\_i(t)} \nonumber$$ exists in M1(Rd) for all t > 0 almost surely. Furthermore, $${\mathcal{L}}^{A,\Lambda}\_\mu(Z\_\cdot \in \cdot) = {\mathbb{P}}^Z\_\mu(Z\_\cdot \in \cdot). \nonumber$$ Thus we can realize the Λ-Fleming-Viot process with mutation as the process of (limiting) empirical measures of the particle system under LμA, Λ. As we now discuss, this construction of Zt encodes its genealogy. Fix t > 0 and consider the process (X1(s), X2(s), …) running forward to time s. We consider time in reverse, backwards from t. For the individual which is at level i at time t, we denote the level of its ancestor at time t − s by Lit(s), for s ∈ [0, t]. (Note that Lit(0) = i for all i ∈ N.) For fixed t > 0, the collection of ancestor processes, (Lit(s) : s ∈ [0, t]) for i ∈ N, satisfy a natural system of stochastic diferential equations driven by the lookdown construction’s Poisson point processes; see Section 5 of. We emphasize that s ∈ [0, t] is the time before the reference time t; given 0 ≤ s ≤ t, the ancestor level Lit(s) corresponds to an individual in the population at (forward) time t − s. This is the convention used e.g. by Birkner and Blath. To recover the ancestral coalescent, we define blocks which consist of all individuals with a given common ancestor. With t fixed as above and s ∈ [0, t], the blocks in the ancestral partition at time s before t are the classes defined by the equivalence relation i ∼ j if Lit(s) = Ljt(s). The collection of such equivalence classes is an integer partition and we denote it by Πt(s). (Again, our convention is that s denotes the time before t and thus Πt(s) corresponds to the forward time process at time t − s.) Given t > 0, (Πt(s) : s ∈ [0, t]) has the law of a Λ-coalescent started at the singleton partition running to time t. Thus the ancestral coalescent of the Λ-Fleming-Viot process is a Λ-coalescent. We define the number of blocks Nst : = #Πt(s) and write Πt(s) = (π1t(s), π2t(s), …, πNstt(s)), where the blocks are again ordered by their minimum elements and the sequence is infinite if Nst = + ∞. The following lemma is proved in. [lemmalabels] Given 0 ≤ s ≤ t, for all i = 1, …, Nst, it holds that Ljt(s) = i for all j ∈ πit(s). In other words, the label of a block is equal to the level of its ancestor. Next, we recall that each block has an asymptotic frequency $$\|\pi^t\_i(s)\| := \lim\_{n \to \infty} n^{-1} \sum\_{j=1}^n 1(j \in \pi^t\_i(s)). \nonumber$$ For ϕ ∈ Bb(Rd), n ≥ 1 and 0 ≤ s ≤ t, we define Zi, s(n)(t, ϕ) = n− 1∑j = 1n1(j ∈ πit(s))ϕ(Xj(t)) and Zi, s(t, ϕ) = liminfn → ∞Zi, s(n)(t, ϕ). If ϕ ≡ 1, then the limit exists and is equal to the asymptotic frequency ∣πit(s)∣. If ϕ(x) = 1B(x) for a Borel set B ⊆ Rd, we will denote the above quantity by Zi, s(t, B). Intuitively, Zi, s(t, ⋅ ) is the measure associated to the cluster at time t whose common ancestor at time s before t is the level i individual. We end the section by introducing several σ-algebras associated to the lookdown model. To encode the behaviour of the Fleming-Viot process and ancestral coalescent under LμA, Λ, we can use either the natural filtrations associated to Zt and ΠT(t) or the filtrations generated by the auxiliary processes, that is, the point and mutation processes. Because our needs are relatively modest, we opt for the former. Let FtZ = σ(Zs : s ≤ t). Then (Zt)t ≥ 0 is a Markov process with respect to the filtration (FtZ)t ≥ 0 and we have $${\mathcal{L}}^{A,\Lambda}\_\mu(f(Z\_{t + \cdot}) \, \| \, {\mathcal{F}}^Z\_t) = {\mathcal{L}}^{A,\Lambda}\_{Z\_t}(f(Z\_\cdot)) \nonumber$$ for suitable functions f. For 0 ≤ s ≤ t, we define Ft, sΠ = σ(Πt(s) : s ≤ t). (Ft, sΠ)s ≤ t is then the natural filtration generated by the ancestral coalescent backwards from time T. The σ-algebra Ft, tΠ thus encodes the entire ancestral coalescent from time T back to time 0. For this special case we adopt the simplified notation FtΠ : = Ft, tΠ. We will simply write LμA, Λ( ⋅ ∣ X(s)) when conditioning on the vector X(s) of particle positions at time s. The proof of our main result uses the following basic properties of the lookdown representation. [lemmaancestorlaw] Given any 0 ≤ s ≤ t, i, j ∈ N with i < j, and μ ∈ M1, for any Borel set B ⊂ Rd, $${\mathcal{L}}^{A,\Lambda}\_\mu( X\_j(t) \in B, j \in \pi^t\_i(s)\, \| \, {\mathcal{F}}^\Pi\_{t,s}, X(t-s)) = {\mathbb{P}}^W\_{X\_i(t-s)}(W\_s \in B) 1(j \in\pi^t\_i(s)) \nonumber$$ [lemmaindepancestors] For any 0 < s ≤ t and i, j ∈ N with i ≠ j, Zi, s(t, ⋅ ) and Zj, s(t, ⋅ ) are conditionally independent given Ft, sΠ ∨ Ft − sZ. Both results are elementary and we omit the proofs. Proof of main result ==================== We begin with the statements and proofs of several lemmas. First, we fix notation which will be in effect for the remainder of the paper. We continue to write (Ws)s ≥ 0 to denote a copy of the mutation process, i.e. the Lévy process with generator A given by. Let (Ts)s > 0 denote the corresponding semigroup, which is the strongly continuous contraction semigroup defined as $$T\_s\phi(x) = {\mathbb{E}}^W\_x(\phi(W\_s)) \nonumber$$ for x ∈ Rd and s > 0. Let Λ ∈ Mf([0, 1]). We define $$\sigma = \sigma(\Lambda) = \Lambda([0,1]). \nonumber$$ We note that σ = λ2, 2, the rate at which the blocks containing a given pair of distinct indices merge in the Λ-coalescent. Let (Zt)t ≥ 0 be the Fleming-Viot process with mutation process (Ws)s ≥ 0 and ancestral coalescent encoded by Λ. The following lemma contains standard first and second moment formulae for Zt. [lemmaFVmoments]Let μ ∈ M1. (a) For ϕ ∈ Bb, $${\mathbb{E}}^Z\_\mu(\langle \phi, Z\_t \rangle) = \langle T\_t \phi, \mu \rangle. \nonumber$$ (b) For ϕ, ψ ∈ Bb, $$\begin{aligned} {\mathbb{E}}^Z\_\mu(\langle \phi, Z\_t \rangle \langle \psi, Z\_t \rangle) = \exp(-\sigma t)\langle T\_t \phi, \mu \rangle \langle T\_t \psi, \mu \rangle + \Big\langle \int\_0^t \sigma \exp(-\sigma s)) T\_{t-s}(T\_s\phi T\_s \psi) ds, \mu \Big\rangle. \nonumber\end{aligned}$$ We omit the proof. These formulae (as well as those for higher moments) can be seen by constructing a coalescing function-valued dual process to Zt. See, for example, Sections 1.12 and 2.8 of the book of Etheridge, where the argument is written in full for the case Λ = δ0. Suppose that ϕ ∈ Bb+ with ϕ(x) ≤ M for all x for some M > 0. Then Lemma [lemmaFVmoments](b) implies that EμZ(∣⟨ϕ, Zt⟩∣2) ≤ exp( − σt)∣⟨Ttϕ, μ⟩∣2 + M(1 − exp( − σt))⟨Ttϕ, μ⟩. The series of lemmas which follow use the lookdown representation of Zt to bound below the probability that the mass of Zt on a given Borel set exceeds some positive value. We consider a general Borel set B, but the reader may think of B as being an open ball. For the remainder of the work, we will denote the lookdown measure LμA, Λ simply by Lμ. Unless otherwise stated there are no additional assumptions on either the generator A or the measure Λ. We recall the definitions and of the cluster measures Zi, s(n)(t, ⋅ ) and Zi, s(t, ⋅ ), for i ∈ [Nst], consisting of the individuals in the coalescent block πit(s). Recall also from Lemma [lemmalabels] that the ancestor of πit(s) is the individual on level i. In the interest of simplifying our statements, hereafter it is implicit that i ∈ [Nst] whenever we discuss πit(s), and hence πit(s) is a block in the partition Πt(s). [lemmalocalclustermass] For any Borel set B ⊆ Rd, 0 < s ≤ t, and μ ∈ M1, we have $${\mathcal{L}}\_\mu \left(Z\_{i,s}(t,B) \geq \frac{p\|\pi\_i^t(s)\|}{2} \, \bigg\|\, {\mathcal{F}}^\Pi\_{t,s}, X(t-s) \right) \geq \frac p 2, \nonumber$$ where p = PXi(t − s)W(Ws ∈ B). We first observe that p is measurable with respect to X(t − s) and so the statement makes sense. Without loss of generality we assume that ∣πi(t)∣ > 0, as the result holds trivially if ∣πit(s)∣ = 0. By Lemma [lemmaancestorlaw], for j ∈ N (and assuming i ∈ [Nst]), $${\mathcal{L}}\_\mu (X\_j(t) \in B, j \in \pi^t\_i(s) \| {\mathcal{F}}^\Pi\_{t,s}, X(t-s)) = {\mathbb{P}}^W\_{X\_i(t-s)}(W\_{s} \in B) 1(j \in \pi^t\_i(s)). \nonumber$$ It follows that $${\mathcal{L}}\_\mu (Z^{(n)}\_{i,s}(t,B) \, \| \, ) = {\mathbb{P}}^W\_{X\_i(t-s)}(W\_s \in B) \, Z^{(n)}\_{i,s}(t,1). \nonumber$$ Taking the limit infimum of both sides, and applying Fatou’s Lemma and the fact that Zi(n)(t, 1) converges to πi(t), we obtain that Lμ(Zi, s(t, B) ∣ Ft, sΠ, X(t − s)) ≥ PXi(t − s)W(Ws ∈ B)∣πit(s)∣. Let E = {Zi, s(t, B) ≥ p∣πit(s)∣/2}. Then $$Z\_{i,s}(t,B) \leq \frac{p \|\pi^t\_i(s)\|}{2} 1\_{E^c} + \|\pi^t\_i(s)\| 1\_{E}. \nonumber$$ Taking the conditional expectation of both sides of the above and applying gives $$p \|\pi\_i^t(s)\| \leq \frac{p \|\pi^t\_i(s)\|}{2}(1 - {\mathcal{L}}\_\mu(E \, \| \, {\mathcal{F}}^\Pi\_{t,s}, X(t-s)))+ \|\pi^t\_i(s)\|{\mathcal{L}}\_\mu (E \, \| \, {\mathcal{F}}^\Pi\_{t,s}, X(t-s)). \nonumber$$ Rearranging the above, we have $${\mathcal{L}}\_\mu(E \, \| \, {\mathcal{F}}^\Pi\_{t,s}, X(t-s)) \geq \frac{p}{ 2}\left(1 - \frac p 2 \right)^{-1} \geq \frac p 2, \nonumber$$ which proves the result. For B ⊂ Rd and ε > 0, we define the ε-enlargement Bε : = {x ∈ Rd : d(x, B) < ε}, where d(x, B) : = infy ∈ B∣x − y∣. Equivalently, Bε = {x + y : x ∈ B, y ∈ B(0, ε)}, so Bε is clearly open. For t, ε > 0, we define p(t, ε) = P0W(∣Wt∣ < ε). To state the next lemma, we introduce some notation. If (Π(s))s ≥ 0 is a Λ-coalescent, for s > 0 and b ∈ [0, 1], we denote by Ns(b) the number of blocks in Π(s) with asymptotic frequency greater than or equal to b. That is, recalling that Π(s) = (π1(s), …, πNs(s)), $$N\_t(b) = \# \{ i \in [N\_t] : \|\pi\_i(t)\| \geq b\}. \nonumber$$ Note that Nt(0) = Nt, the number of blocks in the coalescent. If (Πt(s))s ∈ [0, t] is the ancestral coalescent backwards from time t in the lookdown construction, we denote the analogous quantity by Nst(b). That is, for 0 ≤ s ≤ t, $$N^t\_s(b) := \# \{ i \in [N^t\_s] : \|\pi\_i^t(s)\| \geq b \}. \nonumber$$ [lemmaclusterestimate] For any μ ∈ M1, t, ε > 0, b ∈ (0, 1/2] and Borel B ⊂ Rd, we have $${\mathcal{L}}\_\mu ( Z(t, B\_\epsilon) \geq b\cdot p(t,\epsilon)\, \| \, {\mathcal{F}}^\Pi\_t) \geq 1 - \left(1 - \frac{\mu(B) \cdot p(t,\epsilon)}{2} \right)^{N^t\_t(2b)}$$ We define the event E = {Z(t, Bε) ≥ b ⋅ p(t, ε)}. We condition on FtΠ, with respect to which Ntt(2b) is obviously measurable. To simplify notation we will write N = Ntt(2b). Suppose that the N blocks of Πt(t) with mass at least 2b are {πi1t(t), …, πiNt(t)} for some indices i1, …, iN. We hereafter omit the time parameter and write πij for j ∈ [N]. For each j ∈ [N], let xij = Xij(0), that is, xij is the location of the ancestor of πij, and define the events $$E^1\_j = \{x\_{i\_j} \in B\} \nonumber$$ and $$E^2\_j = \{Z\_{i\_j,t}(t,B(x\_{i\_j},\epsilon)) \geq p(t,\epsilon) \|\pi\_{i\_j}\| / 2 \}, \nonumber$$ where B(xij, ε) is the open ball of radius ε centered at xij. The events E11, …, EN1, E12, …, EN2 are pairwise independent given FtΠ. To see this, we first note that E11, …, EN1 are independent because the vector of initial particle positions (from which xij are taken) is a vector of i.i.d. samples from μ. By translation invariance of the spatial motions, the event Ej2 is independent of the value of xij, and hence of Ej1, and it is trivially independent of Ejʹ1 for jʹ ≠ j. Finally, E12, …, EN2 are independent by Lemma [lemmaindepancestors]. Next, we argue that if Ej1 ∩ Ej2 occurs for any j ∈ [N], then E occurs. To see this, we first note that on Ej1 we have B(xij, ε) ⊆ Bε. Hence, if Ej2 occurs as well, we have $$\begin{aligned} Z\_{i\_j, t}(t, B\_\epsilon) \geq p(t,\epsilon) \|\pi\_{i\_j}\|/2 \geq p(t,\epsilon) b. \nonumber\end{aligned}$$ The claim then follows because Z(t, Bε) ≥ Zij, t(t, Bε). We therefore obtain that $$\begin{aligned} \label{e\_clusterlemma1} {\mathcal{L}}\_\mu (E \, \| \, {\mathcal{F}}^{\Pi}\_t) &\geq {\mathcal{L}}\_\mu ( \cup\_{j=1}^N E^1\_j \cap E^2\_j \, \| \, {\mathcal{F}}^\Pi\_t ) \nonumber \\ &= 1 - \prod\_{j=1}^N (1 - {\mathcal{L}}\_\mu(E^1\_j \cap E^2\_j \, \| \, {\mathcal{F}}^\Pi\_t)) \nonumber \\ &= 1 - (1- {\mathcal{L}}\_\mu (E^1\_1 \cap E^2\_1 \, \| \, {\mathcal{F}}^\Pi\_t))^N,\end{aligned}$$ where the second line uses conditional independence and the third line follows because the events have the same probability for each j ∈ [N]. (In the case of Ej2, this is due to translation invariance of the spatial motion.) The point xi1 is a random variable with distribution μ, and hence Lμ(E11 ∣ FtΠ) = μ(B). To bound the probability of E12 below, we apply Lemma [lemmalocalclustermass] directly. By translation invariance, this yields Lμ(E12 ∣ FtΠ) ≥ p(t, ε)/2, where we recall the definition of p(t, ε) from. Applying independence again and using these bounds, we obtain from that $$\begin{aligned} {\mathcal{L}}\_\mu(E \, \| \, {\mathcal{F}}^\Pi\_t) &\geq 1 - \left(1 - \frac{ \mu(B) p(t, \epsilon)}{2}\right)^N, \nonumber\end{aligned}$$ completing the proof. [lemmamasslwrbd] Given any 0 < δ < t0, μ ∈ M1 and B ⊆ Rd, define m = Tt0 − δμ(B). Then for any ε > 0 and b ∈ (0, 1/2], we have $$\label{e\_lemma\_masslwrdbd} {\mathbb{P}}^Z\_\mu \left( Z(t\_0,B\_\epsilon) \geq b \cdot p(\delta,\epsilon) \right) \geq \frac{{\mathbb{E}}^\Pi(1 - e^{-\theta})}{8(\sigma+1)} \left[ 1 \wedge \frac{m}{ t\_0-\delta} \right],$$ where θ = m + σ(t0 − δ)p(δ, ε)Nδ(2b), and Π is the ancestral coalescent of Z. We begin with the conditional probability of the desired event given Ft0 − δ and apply the Markov property at time t0 − δ, yielding $$\begin{aligned} \label{e\_masslemmaaux00} {\mathbb{P}}^Z\_\mu(Z(t\_0,B\_{\epsilon}) \geq b \cdot p(\delta,{\epsilon}) \, \| \, {\mathcal{F}}\_{t\_0 - \delta}) &= {\mathbb{E}}^Z\_{Z\_{t\_0 - \delta}} (Z(\delta,\Sigma\_{\epsilon}) \geq b \cdot p(\delta,{\epsilon}))\nonumber \\ &= {\mathcal{L}}\_{Z\_{t\_0 - \delta}}(Z(\delta,B\_{\epsilon}) \geq b \cdot p(\delta,{\epsilon})).\end{aligned}$$ The second line simply realizes the Fleming-Viot process via a lookdown construction. We estimate the quantity above by conditioning on the ancestral coalescent. Let μ̃ ∈ M1 and let Y denote a version of the conditional probability of {Z(δ, Bε) ≥ b ⋅ p(δ, ε)} given FδΠ with respect to the measure ${\mathcal{L}}\_{\tilde \mu}$. Then by Lemma [lemmaclusterestimate] we have $$Y \geq 1 - \left( 1 - \frac{\tilde{\mu}(B) \cdot p(\delta,{\epsilon})}{2}\right)^{N^\delta\_\delta(2b)}, \nonumber$$ where we recall that Nδδ(2b) denotes the number of blocks in Πδ(δ) with asymptotic frequency at least 2b. It then follows that $$\begin{aligned} {\mathcal{L}}\_\nu(Z(\delta,B\_{\epsilon}) \geq b \cdot p(\delta,{\epsilon})) &= {\mathcal{L}}\_{\tilde{\mu}}(Y) \nonumber \\ & \geq {\mathcal{L}}\_{\tilde{\mu}} \bigg( 1 - \left( 1 - \frac{\tilde{\mu}(B) \cdot p(\delta,{\epsilon})}{2}\right)^{N^\delta\_\delta(2b)} \bigg) \nonumber \\ & = {\mathbb{E}}^\Pi \bigg(1 - \left( 1 - \frac{\tilde{\mu}(B) \cdot p(\delta,{\epsilon})}{2}\right)^{N\_\delta(2b)} \bigg). \nonumber\end{aligned}$$ In the last line, the expectation under Lμ̃ can be written as an expectation with respect to EΠ because the only randomness is through the ancestral coalescent (via Nδδ(2b)). Because this holds for any μ̃ ∈ M1, we may apply it in to obtain $${\mathbb{P}}^Z\_\mu(Z(t\_0,B\_{\epsilon}) \geq b \cdot p(\delta,{\epsilon}) \, \| \, {\mathcal{F}}\_{t\_0 -\delta}) \geq {\mathbb{E}}^\Pi \bigg(1 - \left( 1 - \frac{Z(t\_0 - \delta,B) \cdot p(\delta,{\epsilon})}{2}\right)^{N\_\delta(2b)} \bigg). \nonumber$$ Finally, we take the expectation of both sides under EμZ to obtain $$\label{e\_masslemmaaux11} {\mathbb{P}}^Z\_\mu(Z(t\_0,B\_{\epsilon}) \geq b \cdot p(\delta,{\epsilon})) \geq {\mathbb{E}}^Z\_\mu \otimes {\mathbb{E}}^\Pi \bigg(1 - \left( 1 - \frac{Z(t\_0 - \delta,B) \cdot p(\delta,{\epsilon})}{2}\right)^{N\_\delta(2b)} \bigg),$$ where EμZ ⊗ EΠ denotes the expectation with respect to the product of the two measures. To simplify notation we will hereafter write N = Nδ(2b). The elementary inequality (1 − x)n ≤ e− nx for all x ∈ [0, 1] and n ∈ N implies that $$\label{e\_masslemmaaux1} 1 - \left(1 - \frac{ Z(t\_0 -\delta, B) p(\delta, \epsilon)}{2}\right)^N \geq 1 - \exp \left(- \frac{N Z(t\_0 - \delta, B) p(\delta, \epsilon)}{2}\right).$$ The next step is the approximation 1 − e− x ≈ x, but we need to keep track of constants. For any θ > 0, 1 − e− x ≥ xθ− 1(1 − e− θ) for all x ∈ [0, θ]. Let θ > 0. By restricting to the event where the exponent has absolute value less than θ, it follows that $$\begin{aligned} \label{e\_masslemmaaux2} &1 - \exp \left(- \frac{N Z(t\_0 - \delta, B) p(\delta, \epsilon)}{2}\right) \nonumber \\ &\hspace{1 cm} \geq \frac{(1-e^{-\theta}) N p(\delta,\epsilon) }{2 \theta} Z(t\_0-\delta,B) 1\left(Z(t\_0-\delta,B) \leq \frac{2 \theta}{N p(\delta, \epsilon)}\right)\end{aligned}$$ To obtain a lower bound on the terms involving Z(t0 − δ, B), we proceed as follows: $$\begin{aligned} &Z(t\_0-\delta,B) 1\left(Z(t\_0-\delta,B) \leq \frac{2 \theta}{N p(\delta, \epsilon)}\right) \nonumber \\ &\hspace{1 cm}= Z(t\_0-\delta,B)\left[1 - 1 \left(\frac{Z(t\_0-\delta,B) \, N p(\delta,{\epsilon}) }{2 \theta} > 1 \right)\right] \nonumber \\ &\hspace{1 cm}\geq Z(t\_0 -\delta,B) - Z(t-\delta,B)^2 \frac{\, N p(\delta,\epsilon)}{2 \theta}. \nonumber\end{aligned}$$ From Lemma [lemmaFVmoments](a) and the definition of m, we have m = EμZ(Z(t0 − δ, B)) = Tt0 − δμ(B). For the second moment, the estimate implies EμZ(Z(t0 − δ, B)2) ≤ e− σ(t0 − δ)m2 + (1 − e− σ(t0 − δ))m ≤ m2 + σ(t0 − δ)m. Given N, taking the expectation on both sides of the previous inequality, this moment estimate yields $$\begin{aligned} \label{e\_masslemmaaux3} &{\mathbb{E}}^Z\_\mu \left(Z(t\_0-\delta,B) 1\left(Z(t\_0-\delta,B) \leq \frac{2 \theta}{N p(\delta,{\epsilon})}\right)\right) \nonumber \\ &\hspace{1 cm}\geq m - \left[e^{-\sigma (t\_0-\delta)} m^2 + (1-e^{-\sigma (t\_0-\delta)}) m \right] \frac{N p(\delta,{\epsilon})}{2\theta} \nonumber \\ &\hspace{1 cm}\geq m - m [m + \sigma(t\_0-\delta)] \frac{N p(\delta,{\epsilon})}{2\theta}.\end{aligned}$$ We now choose the value of θ to be θ = [m + σ(t0 − δ)]p(δ, ε)N and observe the cancellations this choice will cause in the last expression of and in. We combine, and to obtain $$\begin{aligned} {\mathbb{E}}^Z\_\mu \otimes {\mathbb{E}}^\Pi \bigg( 1 - \left(1 - \frac{ Z(t\_0 -\delta, B) p(\delta, \epsilon)}{2}\right)^{N} \bigg) &\geq {\mathbb{E}}^\Pi \bigg( \frac{(1 - e^{-\theta})(m - m/2)}{2[m+ \sigma(t\_0 - \delta)]} \bigg) \nonumber \\ &\geq \frac{ {\mathbb{E}}^\Pi(1 - e^{-\theta})}{8(\sigma + 1)} \left[ 1 \wedge \frac{m}{t\_0 - \delta}\right]. \nonumber\end{aligned}$$ By, this completes the proof. Recall that (Ts)s > 0 is the semigroup associated to Ws, the Markov process with generator A, which has form Aϕ(x) = ∇ ⋅ (Q∇ϕ)(x) + a ⋅ ∇ϕ(x) + ∫Rd(ϕ(x + y) − ϕ(x) − ∇ϕ(x)y1{∣y∣ < 1})ν(dy), where a, Q and ν are as described in the introduction. We write Ts\* to denote the adjoint of Ts. It is standard that (Ts\)s* > 0 is a Markov semigroup and its generator, A\, generates a Lévy process we denote W*s\* which is the time reversal of Ws. In particular, we have A\*ϕ(x) = ∇ ⋅ (Q∇ϕ)(x) − b ⋅ ∇ϕ(x) + ∫Rd[ϕ(x − y) − ϕ(x) + ∇ϕ(x)y1(0, 1](∣y∣)]ν(dy). For details, see Section II.1 of Bertoin. We recall that Lévy processes are Feller, and in particular (Ts)s > 0 and (Ts\)s* > 0 are strongly continuous contraction semigroups on the Banach space C0 of continuous functions which vanish at infinity equipped with the topology of uniform convergence. (See Proposition I.5 of.) We hereafter adopt the convention that ∞ ⋅ 0 : = 0. The convolution of two measures μ and ν is denoted μ \* ν. [lemmaliminfgenerator] Given any t, ε > 0, bounded B ⊂ Rd, and positive, strictly decreasing sequences (δn)n ≥ 1 and (εn)n ≥ 1 converging to 0, with probability one we have $$\liminf\_{n \to \infty} {\epsilon}\_n^{-1} T\_{{\epsilon}\_n} Z\_{t-\delta\_n} (B\_\epsilon) \geq (\nu \* Z\_t)(B\_{\epsilon}) + \infty \cdot Z\_t (B\_{\epsilon}). \nonumber$$ First we recall that t → Zt is càdlàg in the weak topology, and hence the (weak) left limit Zt − = lims ↑ tZs exists for all t > 0. Furthermore, the process has no fixed time discontinuities, so given t > 0 we have Zt − = Zt almost surely. In particular, if U ⊆ Rd is open and t > 0 is fixed, then lims ↑ tZs(U) = Zt(U) almost surely (in the weak topology). We use this fact several times in what follows. Let t, ε, B, (δn)n ≥ 1 and (εn)n ≥ 1 be as in the statement of the lemma. We decompose εn− 1TεnZt − δn(Bε) by writing $$\begin{aligned} \label{e\_liminfsemigroup1} {\epsilon}\_n^{-1} T\_{{\epsilon}\_n} Z\_{t-\delta\_n} (B\_\epsilon) &= {\epsilon}\_n^{-1} \langle 1\_{B\_{\epsilon}}, T\_{{\epsilon}\_n} Z\_{t-\delta\_n} \rangle \nonumber \\ &= {\epsilon}\_n^{-1} \langle T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}}, Z\_{t-\delta\_n} \rangle \nonumber \\ &= \int\_{B\_\epsilon^c} \epsilon\_n^{-1} T\_{\epsilon\_n}^\* 1\_{B\_{\epsilon}}(x) Z\_{t-\delta\_n}(dx) + \int\_{B\_{{\epsilon}}} \epsilon\_n^{-1} T\_{\epsilon\_n}^\* 1\_{B\_{\epsilon}}(x) Z\_{t-\delta\_n}(dx).\end{aligned}$$ Consider the second term in. For εʹ ∈ (0, ε) and x ∈ Bεʹ, we have B(x, ε − εʹ) ⊂ Bε, and hence $$T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}}(x) = {\mathbb{P}}^{W^\}\_x(W^\\_{{\epsilon}\_n} \in B\_{\epsilon}) \geq {\mathbb{P}}^{W^\}\_0(W\_{{\epsilon}\_n}^\ \in B(0,{\epsilon}- {\epsilon}')). \nonumber$$ Since the right hand side of the above converges to 1 as εn → 0, it follows that Tεn\1B*ε → 1 uniformly on Bεʹ. Consequently, $T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}} (x) \geq \frac 1 2$ for all x ∈ Bεʹ for sufficiently large n. We conclude that $$\begin{aligned} \liminf\_{n \to \infty} {\epsilon}\_n^{-1} \int\_{B\_{{\epsilon}}} T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}} (x) Z\_{t-\delta\_n} (dx) \geq \infty \cdot \int\_{B\_{{\epsilon}'}} Z\_{t}(dx) \,\, \text{ a.s.}, \nonumber\end{aligned}$$ for every εʹ ∈ (0, ε), since Zt − δn → Zt weakly and hence liminfn → ∞Zt − δn(Bεʹ) ≥ Zt(Bεʹ). By inner regularity (continuity of probability) of Zt, taking εʹ ↑ ε gives $$\begin{aligned} \label{e\_liminfsemigroup2} \liminf\_{n \to \infty} {\epsilon}\_n^{-1} \int\_{B\_{{\epsilon}}} T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}} (x) Z\_{t-\delta\_n} (dx) \geq \infty \cdot Z\_t(B\_{{\epsilon}}) \,\, \text{ a.s.}\end{aligned}$$ We now consider the first term in the right hand side of. Since B is bounded, for εʹ ∈ (0, ε) we can choose a non-negative function φ ∈ Cc∞ satisfying $$1\_{B\_{{\epsilon}'}} \leq \varphi \leq 1\_{B\_{\epsilon}} \nonumber$$ such that φ, Dφ and D2φ vanish on Bεc. (Dφ and D2φ denote the gradient and Hessian of φ, respectively.) Note that we also have Tεn\1B*εʹ ≤ Tεn\*φ ≤ Tεn\1B*ε. Since φ is in the domain of A\* (viewed as an operator on C0), we have $${\epsilon}\_n^{-1} (T\_{{\epsilon}\_n}^\\varphi - \varphi) \to A^\ \varphi \nonumber$$ uniformly as n → ∞. However, because φ(x) = 0 for x ∈ Bεc, this implies $${\epsilon}\_n^{-1} T\_{{\epsilon}\_n}^\\varphi\to A^\ \varphi \quad \text{uniformly on} \,\, B\_{\epsilon}^c. \nonumber$$ It follows from, the above, and the weak convergence of Zt − δn to Zt that $$\begin{aligned} \liminf\_{n\to \infty} \int\_{B\_{\epsilon}^c} {\epsilon}\_n^{-1} T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}}(x) Z\_{t-\delta\_n}(dx) & \geq \int\_{B\_{\epsilon}^c} A^\* \varphi(x) Z\_t(dx). \nonumber\end{aligned}$$ Because φ, Dφ and D2φ vanish on Bεc, we obtain from that $$A^\\varphi(x) = \int\_{{\mathbb{R}}^d} \varphi(x-y) \nu(dy) \nonumber$$ for all x* ∈ Bεc. We thus obtain that $$\begin{aligned} \liminf\_{n\to \infty} \int\_{B\_{\epsilon}^c} {\epsilon}\_n^{-1} T\_{{\epsilon}\_n}^\* 1\_{B\_{\epsilon}}(x) Z\_{t-\delta\_n}(dx) &\geq \int\_{B\_{\epsilon}^c} \int\_{{\mathbb{R}}^d} \varphi(x-y) \nu(dy) Z\_t(dx) \nonumber \\& \geq \int\_{B\_{\epsilon}^c} \int\_{{\mathbb{R}}^d} 1\_{B\_{{\epsilon}'}}(x-y) \nu(dy) Z\_t(dx). \nonumber\end{aligned}$$ Taking εʹ ↑ ε and applying monotone convergence, we obtain $$\begin{aligned} \liminf\_{n\to \infty} \int\_{B\_{\epsilon}^c} {\epsilon}\_n^{-1} T\_{{\epsilon}\_n} 1\_{B\_{\epsilon}} Z\_{t-\delta\_n}(dx) &\geq \int\_{B\_{\epsilon}^c} \int\_{{\mathbb{R}}^d} 1\_{B\_{{\epsilon}}}(x - y) \nu(dy) Z\_t(dx). \nonumber\end{aligned}$$ Combined with and, this yields $$\begin{aligned} \liminf\_{n \to \infty} \epsilon\_n^{-1} \langle T\_{\epsilon\_n} 1\_{B\_\epsilon}, Z\_{t-\delta\_n}\rangle \geq \int\_{B\_{\epsilon}^c}\int\_{{\mathbb{R}}^d} 1\_{B\_{\epsilon}}(x-y) \nu(dy) Z\_t(dx) + \infty \cdot Z\_t(B\_{\epsilon}). \nonumber\end{aligned}$$ To obtain the desired result, we observe that $$\int\_{B\_{\epsilon}} \int\_{{\mathbb{R}}^d} 1\_{B\_{\epsilon}}(x-y) \nu(dy) Z\_t(dx) > 0 \Rightarrow Z\_t(B\_{\epsilon}) > 0, \nonumber$$ and hence $$\begin{aligned} &\int\_{B\_{\epsilon}^c}\int\_{{\mathbb{R}}^d} 1\_{B\_{\epsilon}}(x-y) \nu(dy) Z\_t(dx) + \infty \cdot Z\_t(B\_{\epsilon}) \nonumber \\ &\hspace{1 cm}= \int\_{B\_{\epsilon}^c}\int\_{{\mathbb{R}}^d} 1\_{B\_{\epsilon}}(x-y) \nu(dy) Z\_t(dx) + \infty \cdot Z\_t(B\_{\epsilon}) + \int\_{B\_{\epsilon}} \int\_{{\mathbb{R}}^d} 1\_{B\_{\epsilon}}(x-y) \nu(dy) Z\_t(dx) \nonumber \\ &\hspace{1 cm}= (\nu \* Z\_t)(B\_{\epsilon}) + \infty \cdot Z\_t(B\_{\epsilon}), \nonumber\end{aligned}$$ which completes the proof. The last lemma we need to prove our main result is the following result that allows us to handle general Λ-coalescents. In order to state it, we review the notion of dust. An exchangeable integer partition is said to have dust if a positive proportion of the integers belong to singleton blocks. In fact, all blocks with asymptotic frequency equal to zero are singleton blocks, and an exchangeable partition has singleton blocks if and only if a positive proportion of the integers belong to singleton blocks. (See Proposition 2.8 of Bertoin.) Thus, for Π(t), having no dust implies that limb → 0+Nt(b) = Nt. (Note that this holds if Nt < ∞ or Nt = + ∞.) Hereafter, we say that the Λ-coalescent has no dust if Π(t) has no dust a.s. for all t > 0, in which case we have limb → 0+Nt(b) = Nt a.s. for all t > 0. Note that any Λ-coalescent that comes down from infinity does not have dust; a necessary and sufficient condition for a Λ-coalescent to have no dust is given in Theorem 8 of. Recall the function v(t) defined in. [lemmaspecialsequence] Let Λ ∈ Mf([0, 1]) be such that the Λ-coalescent a.s. has no dust. Then for any a ∈ (0, 1), there exist positive sequences (δn)n ≥ 1 and (bn)n ≥ 1 ⊂ [0, 1/2] and a constant c > 0 such that δn → 0 and bn → 0 as n → ∞ and the following hold: * δn + 1 ≤ aδn for all n ∈ N. * PΠ(Nδn/2(2bn) ≥ cδn− 1) ≥ 1/4 for all n ∈ N. * n → bnv(δn + 1) is increasing and bnv(δn + 1) ≥ n for all n ∈ N. We assume that v(δ) < ∞ for all δ > 0, as (under the assumption that Πt has no dust) the lemma is trivial otherwise. Theorem [theoremblocks] implies that Nδ/v(δ) → 1 in distribution as δ → 0. Hence there exists δ0 > 0 such that if 0 < δ ≤ δ0, then Nδ ≥ v(δ)/2 with probability at least 1/2. By, there is a constant c > 0 such that v(δ) ≥ cδ− 1. Hence $${\mathbb{P}}^\Pi( N\_\delta \geq c \delta^{-1} ) \geq \frac 1 2 \nonumber$$ for any δ ≤ δ0. We also have $$\lim\_{b \to 0^+} N\_\delta(b) = N\_\delta \nonumber$$ almost surely for any δ > 0, since Π(t) does not have dust. Consequently, $$\lim\_{b \to 0^+} {\mathbb{P}}^\Pi ( N\_\delta(b) \geq c \delta^{-1}/2 ) \geq 1/2. \nonumber$$ It follows that, given δ ≤ δ0 there exists b\(δ) > 0 such that PΠ(N*δ(2b\(δ)) ≥ c*δ− 1/2) ≥ 1/4. Now fix δ1 < δ0 and choose b1 = b\(δ1/2). Since lims* → 0+v(s) = ∞, we can choose δ2 ∈ (0, aδ1] to be sufficiently small so that v(δ2)b1 ≥ 2. We then choose b2 = b\(δ2/2). We continue the construction sequentially, always taking b*n = b\(δ*n/2) and choosing δn + 1 ∈ (0, aδn] such that v(δn + 1)bn ≥ n + 1, which guarantees that δn + 1 ≤ aδn and v(δn + 1)bn → ∞. This also implies that v(δn) → ∞, and hence δn → 0 as n → ∞. The fact that bn = b\(δ*n/2) for all n ensures that Nδn/2(2bn) ≥ cδn− 1 with probability at least 1/4 for all n by, and the proof is complete. We now prove Theorem [thmmain] under the assumption that Π(t) does not have dust. (This is the important case, as indeed the proof is very simple when Π(t) has dust. We discuss this afterwards.) The idea of the proof is as follows. For an open ball B, we use Lemmas [lemmamasslwrbd] and [lemmaliminfgenerator] to show that Zt − δn + 1(B) ≥ bn infinitely often a.s. on {ν \* Zt(B) > 0}, where δn and bn are chosen as in Lemma [lemmaspecialsequence]. We then estimate Zt(B) conditionally given Zt − δn + 1; the corresponding ancestral representation of Zt has order v(δn + 1) ancestors, so when Zt − δn + 1(B) ≥ bn, the expected number of ancestors originating in B is of order at least bnv(δn + 1), which goes to infinity as n → ∞. Each cluster originating in B locally generates mass (conditionally) independently with some positive probability. Therefore, the probability that a cluster from time t − δn + 1 originates in B and generates mass in B at time t converges can be shown to converge to 1 using the estimates developed before. [Proof of Theorem [thmmain]] We fix t > 0 and μ ∈ M1. Let B ⊂ Rd be bounded. For the time being we allow an arbitrary set, but we will consider Bε for ε > 0 and the reader may think of Bε as an open ball. By Lemma [lemmaspecialsequence], there exists a constant c0 > 0 and positive sequences (δn)n ≥ 1 and (bn)n ≥ 1 ⊂ [0, 1/2], both converging to zero, satisfying the following: $$\begin{aligned} & \delta\_{n+1} \leq \frac{\delta\_n}{3} \text{ for all } n \in {\mathbb{N}}, \label{e\_sequence1} \\& {\mathbb{P}}^\Pi(N(\delta\_n /2, 2 b\_n) \geq c\_0 \delta\_n^{-1}) \geq \frac 1 4, \label{e\_sequence2} \\& b\_n v(\delta\_{n+1}) \text{ is increasing and } b\_n v(\delta\_{n+1}) \geq n \text{ for all } n \in {\mathbb{N}}. \label{e\_sequence3}\end{aligned}$$ Without loss of generality we assume that δ1 < t. Let ε > 0. For n ∈ N we define the event In by $$I\_n := \{Z\_{t-\delta\_{n+1}}(B\_{\epsilon}) \geq b\_n /2\}. \nonumber$$ First, we estimate the probability of In conditional on Ft − δn. Let μn : = Zt − δn under PμZ. By the Markov property, $${\mathbb{P}}^Z\_\mu(I\_{n} \, \| \, {\mathcal{F}}\_{t-\delta\_n}) = {\mathbb{P}}^Z\_{\mu\_n} ( Z\_{\delta\_n - \delta\_{n+1}}(B\_{\epsilon}) \geq b\_n/2). \nonumber$$ We let εʹ ∈ (0, ε) and define $$A\_n := \{ Z\_{\delta\_n - \delta\_{n+1}}(B\_{\epsilon}) \geq b\_n \cdot p(\delta\_n / 2, {\epsilon}- {\epsilon}') \} \nonumber.$$ By the definition of p(δn/2, ε − εʹ) (see ) and the continuity in probability of Lévy processes, lims → 0+p(s, r) = 1 for any r > 0. Hence there exists n0(ε − εʹ) ∈ N such that $$\label{e\_pf\_locallevy} n \geq n\_0({\epsilon}-{\epsilon}') \Rightarrow p(\delta\_n/2,{\epsilon}-{\epsilon}') \geq \frac 1 2.$$ This implies that PμZ(In ∣ Ft − δn) ≥ PμnZ(An) for all n ≥ n0(ε − εʹ). We now apply Lemma [lemmamasslwrbd] to estimate PμnZ(An). In the notation of that lemma we take t0 = δn − δn + 1, δ = δn/2 and b = bn. Observe that in this case, t0 − δ = δn − δn + 1 − δn/2 = δn/2 − δn + 1 > 0 by. We define $$m\_n = T\_{\delta\_n/2 - \delta\_{n+1}} \mu\_n (B\_{{\epsilon}'}). \nonumber$$ Finally, we observe that Bε = (Bεʹ)ε − εʹ. Applying Lemma [lemmamasslwrbd] with these parameters, we obtain $$\label{e\_pf\_smallmeanbd1} {\mathbb{P}}^Z\_{\mu\_n}( A\_n ) \geq \frac{{\mathbb{E}}^\Pi(1 - e^{-\theta\_n})}{8(\sigma + 1)} \left[ 1 \wedge m\_n (\delta\_n / 2 - \delta\_{n+1} )^{-1} \right],$$ where θn = mn + σ(δn/2 − δn + 1)p(δn/2, ε − εʹ)Nδn/2(2bn) and we recall that Nδn/2(2bn) denotes the number of blocks in the Λ-coalescent at time δn/2 with at asymptotic frequency at least 2bn. By and, if n ≥ n0(ε − εʹ) then $$\begin{aligned} \theta\_n &= m\_n+ \sigma (\delta\_n / 2 - \delta\_{n+1}) p(\delta\_n / 2,{\epsilon}- {\epsilon}') N\_{\delta\_n/2}(2b\_n) \nonumber \\ &\geq \frac{\sigma \delta\_n}{12} N\_{\delta\_n/2}(2b\_n). \nonumber\end{aligned}$$ If Nδn/2(2bn) ≥ c0δn− 1, θn is bounded below by a constant (and hence so is 1 − e− θn). Thus it follows from that EΠ(1 − e− θn) ≥ c1 > 0 for some c1 > 0, for all n ≥ n0(ε − εʹ). Combined with, this implies that $${\mathbb{P}}^Z\_{\mu\_n}( A\_n) \geq c\_2 \left[ 1 \wedge m\_n (\delta\_n / 2 - \delta\_{n+1} )^{-1} \right] \nonumber$$ for a constant c2 > 0, for all n ≥ n0(ε − εʹ). Returning to, recalling that μn = Zt − δn, and writing mn in terms of its definition, we have shown that for sufficiently large n, $$\label{e\_pf\_condbound1} {\mathbb{P}}^Z\_\mu(I\_{n} \, \| \, {\mathcal{F}}\_{t-\delta\_n}) \geq c\_2 \left[ 1 \wedge \frac{T\_{\delta\_n/2 - \delta\_{n+1}} Z\_{t - \delta\_n} (B\_{{\epsilon}'})}{\delta\_n / 2 - \delta\_{n+1}} \right].$$ Observe that the main term in the above is the quantity from Lemma [lemmaliminfgenerator] with εn = δn/2 − δn + 1. By Lemma [lemmaliminfgenerator], $$\liminf\_{n \to \infty} (\delta\_n / 2 - \delta\_{n+1} )^{-1} T\_{\delta\_n/2 - \delta\_{n+1}} Z\_{t - \delta\_n} (B\_{{\epsilon}'}) \geq \nu \* Z\_t(B\_{\epsilon'}) + \infty \cdot Z\_t (B\_{\epsilon'}). \nonumber$$ Taking only the first term in the above, implies that $$\liminf\_{ n \to \infty} {\mathbb{P}}^Z\_\mu(I\_{n} \, \| \, {\mathcal{F}}\_{t-\delta\_n}) \geq c\_2 \left[ 1 \wedge \nu \* Z\_t(B\_{\epsilon'}) \right]. \nonumber$$ As this holds for all εʹ ∈ (0, ε) and limεʹ ↑ εZt(Bεʹ) = Zt(Bε) (by inner regularity of the Borel measure Zt), taking εʹ ↑ ε implies that liminfn → ∞PμZ(In ∣ Ft − δn) ≥ c2[1 ∧ ν \* Zt(Bε)]. If ν \* Zt(Bε) > 0, the right hand side above is positive and bounded below. The extended (second) Borel-Cantelli lemma for conditionally independent events then implies that In occurs infinitely often. That is, $$\text{$I\_n$ occurs infinitely often almost surely on $\{\nu \* Z\_t(B\_{\epsilon}) > 0\}$.} \nonumber$$ We now specialize our argument to open balls and complete the proof. Let x0 ∈ Rd and r > 0. If we take B = {x0} and ε = rʹ ∈ (0, r), $$\label{e\_pf\_ballmassio} \text{$I\_n(x\_0,r')$ occurs infinitely often almost surely on $\{\nu \* Z\_t(B(x\_0,r')) > 0\}$,}$$ where In(x0, rʹ) = {Zt − δn + 1(B(x0, rʹ)) ≥ bn/2}. We define n1 = inf{n : In(x0, rʹ)occurs} and nk = inf{n > nk − 1 : In(x0, rʹ)occurs} for all k ≥ 2, with the convention that inf∅ = ∞. Then on {ν \* Zt(B(x0, r)) > 0}, nk < ∞ for all k almost surely. For k ∈ N we define the random times $$\tau\_k = \begin{cases} t - \delta\_{n\_k + 1} &\text{ if } n\_k < \infty, \\ t &\text{ if } n\_k = \infty. \end{cases} \nonumber$$ Indeed, it is straightforward to show that τk is a (Fs)-stopping time. By the previous discussion, is equivalent to $$\label{e\_stoppingtimes\_as} n\_k < \infty \text{ and } \tau\_k = t - \delta\_{n\_k + 1} \,\, \text{ for all $k \in {\mathbb{N}}$ a.s. on $\{ \nu \* Z\_t(B(x\_0,r')) > 0\}$.}$$ We apply the strong Markov property at τk and use the lookdown representation to obtain that, when nk < ∞, PμZ(Zt(B(x0, rʹ)) > 0 ∣ Fτk) = Lν̃k(Zδnk + 1(B(x0, rʹ)) > 0), where we define μ̃k : = Zτk = Zt − δnk + 1. We estimate the right-hand side by conditioning on the ancestral coalescent. Omitting the time dependence of the blocks, we write Πδnk + 1(δnk + 1) = {πi : i ∈ [Nδnk + 1]}, and recall from Lemma [lemmalabels] that the ancestor of πi is the level i individual. For i ∈ [Nδnk + 1], by Lemma [lemmalocalclustermass], and again using the fact that all blocks have positive asymptotic frequency when Π(t) comes down from infinity, we have $$\label{e\_pf\_goodclusterbd} {\mathcal{L}}\_{\tilde{\mu}\_k} (Z\_{i,\delta\_{n\_k+1}}(\delta\_{n\_k+1}, B(x\_0,r)) > 0 \, \| \, {\mathcal{F}}^\Pi\_{\delta\_{n\_k+1}}, X(0)) \geq {\mathbb{P}}^W\_{X\_i(0)}(W\_{\delta\_{n\_k+1}} \in B(x\_0,r))/2. \nonumber$$ If y ∈ B(x0, r − rʹ), we have PyW(Wδnk + 1 ∈ B(x0, r)) ≥ p(δnk + 1, r − rʹ) by spatial homogeneousness of the Lévy process. The initial particle positions are i.i.d. with distribution μ̃k, and hence Xi(0) ∈ B(x0, rʹ) with probability μ̃k(B(x0, rʹ)). Hence, for each i ∈ [Nδnk + 1], $${\mathcal{L}}\_{\tilde{\mu}\_k} (Z\_{i, \delta\_{n\_k}+1}(\delta\_{n\_k+1}, B(x\_0,r)) > 0 \, \| \, {\mathcal{F}}^\Pi\_{\delta\_{n\_k+1}}) \geq \frac{p(\delta\_{n\_k+1},r-r') \tilde{\mu}\_k(B(x\_0,r'))}{2}. \nonumber$$ By definition, Z(δnk + 1, B(x0, rʹ)) ≥ Zi, δnk + 1(δnk + 1, B(x0, rʹ)) for each i ∈ [Nδnk + 1], and hence for the former to be positive we simply require that Zi, δnk + 1(δnk + 1, B(x0, rʹ)) is positive for some i ∈ [Nδnk + 1]. By Lemma [lemmaindepancestors], the ancestral clusters {Zi, δnk + 1(δnk + 1, ⋅ ) : i ∈ [Nδnk + 1]} are conditionally independent given Fδnk + 1Π, and hence $${\mathcal{L}}\_{\tilde{\mu}\_k} (Z(\delta\_{n\_k+1}, B(x\_0,r)) > 0 \, \| \, {\mathcal{F}}^\Pi\_{\delta\_{n\_k+1}}) \geq 1 - \left(1 - \frac{p(\delta\_{n\_k+1},r-r') \tilde{\mu}\_k (B(x\_0,r'))}{2} \right)^{N\_{\delta\_{n\_k+1}}}. \nonumber$$ By definition of nk and μ̃k, μ̃k(B(x0, rʹ)) ≥ bnk/2 on {nk < ∞}. Thus, for sufficiently large k and nk < ∞, $$\begin{aligned} {\mathcal{L}}\_{\tilde{\mu}\_k} (Z(\delta\_{n\_k+1}, B(x\_0,r)) > 0 \, \| \, {\mathcal{F}}^\Pi\_{\delta\_{n\_k+1}}) &\geq 1 - \left(1 - \frac{p(\delta\_{n\_k+1}, r - r') b\_{n\_k}}{4} \right)^{N\_{\delta\_{n\_k+1}}} \nonumber \\ &\geq 1 - e^{- b\_{n\_k} N\_{\delta\_{n\_k+1}}/8} \nonumber \\ &\geq (1 - e^{- b\_{n\_k} v(\delta\_{n\_k+1})/16}) 1(N\_{\delta\_{n\_k+1}} \geq v(\delta\_{n\_k+1}) / 2) \nonumber\end{aligned}$$ The second inequality implicitly uses the fact that p(δn + 1, r − rʹ) ≥ 1/2 for sufficiently large n (c.f. ). Taking the expectation of the above, we obtain $$\begin{aligned} \label{e\_pf\_clustersbd} {\mathcal{L}}\_{\tilde{\mu}\_k} (Z(\delta\_{n\_k+1}, B(x\_0,r)) > 0) = (1 - e^{-b\_{n\_k} v(\delta\_{n\_k+1}) / 16}) {\mathbb{P}}^\Pi(N\_{\delta\_{n\_k+1}} \geq v(\delta\_{n\_k+1})/2).\end{aligned}$$ By Theorem [theoremblocks], $$\frac{N\_s}{v(s)} \to 1 \nonumber$$ in distribution as s ↓ 0. In particular, there exists a non-decreasing function $s \to \tilde{{\epsilon}}(s)$ such that $\lim\_{s \to 0^+} \tilde{{\epsilon}}(s) = 0$ and $$\sup\_{u \in (0,s]}{\mathbb{P}}^\Pi(N\_u \geq v(u)/2) \geq 1 - \tilde{{\epsilon}}(s). \nonumber$$ By, for nk < ∞ we have $$\begin{aligned} {\mathcal{L}}\_{Z\_{\tau\_k}} (Z(\delta\_{n\_k+1}, B(x\_0,r)) > 0) \geq (1 - e^{-b\_{n\_k} v(\delta\_{n\_k+1}) / 16}) (1 - \tilde{{\epsilon}}(\delta\_{n\_k+1})). \nonumber\end{aligned}$$ By, we have bnkv(δnk + 1) ≥ nk ≥ k on {nk < ∞}. Combining this with the above and, we obtain that on the event {nk < ∞}, $$\begin{aligned} {\mathbb{P}}(Z\_t(B(x\_0,r) > 0 \, \| \, {\mathcal{F}}\_{\tau\_k}) \geq (1 - e^{-k / 16}) (1 - \tilde{{\epsilon}}(\delta\_{k+1})). \nonumber\end{aligned}$$ We now compute $$\begin{aligned} {\mathbb{P}}^Z\_\mu( Z\_t(B(x\_0,r)> 0, n\_k < \infty) &= {\mathbb{E}}^Z\_\mu( P(Z\_t(B(x\_0,r) >0, n\_k < \infty \, \| \, {\mathcal{F}}\_{\tau\_k})) \nonumber \\& ={\mathbb{E}}^Z\_\mu( P(Z\_t(B(x\_0,r) >0 \, \| \, {\mathcal{F}}\_{\tau\_k}) 1\_{\{n\_k < \infty\}}) \nonumber \\ &\geq (1 - e^{-k / 16}) (1 - \tilde{{\epsilon}}(\delta\_{k+1})) {\mathbb{P}}^Z\_\mu(n\_k < \infty).\end{aligned}$$ The second line uses the fact that {nk < ∞} ∈ Fτk, and the third line uses the bound for the conditional probability derived above. Let E = ∩ k = 1∞{nk < ∞} and note that, since {nk + 1 < ∞} ⊆ {nk < ∞}, 1E = limk → ∞1{nk < ∞} a.s. Taking k → ∞ in the above, it follows that $${\mathbb{P}}^Z\_\mu( Z\_t(B(x\_0,r)> 0, \mathcal{E}) \geq {\mathbb{P}}^Z\_\mu(\mathcal{E}), \nonumber$$ and hence the two sides are equal. By, {ν \* Zt(B(x0, rʹ)) > 0} ⊆ E, and therefore $${\mathbb{P}}^Z\_\mu(Z\_t(B(x\_0,r)) = 0, \nu \* Z\_t(B(x\_0,r')) > 0 ) = 0. \nonumber$$ Regularity of ν \* Zt implies that ν \* Zt(B(x0, r)) > 0 if and only if ν \* Zt(B(x0, rʹ)) > 0 for some rʹ < r. We thus deduce that $${\mathbb{P}}^Z\_\mu(Z\_t(B(x\_0,r)) = 0, \nu \* Z\_t(B(x\_0,r)) > 0 ) = 0. \nonumber$$ To complete the argument, we observe that the implication holds simultaneously for the countable collection of open balls with rational centres and radii almost surely. In particular, the event that there exists x0 ∈ Qd and r ∈ Q+ such that ν \* Zt(B(x0, r)) > 0 but Zt(B(x0, r)) = 0 has probability zero. On the complement of this event, an elementary topological argument yields that S(ν \* Zt) ⊆ S(Zt). Hence S(ν \* Zt) ⊆ S(Zt) almost surely. Iterating the previous procedure, it follows that for k ∈ N, the same is true of the k-fold convolution of ν with Zt, and the proof is complete. We conclude by remarking on the case that Π(t) has dust, i.e. when a positive proportion of the integers belong to singleton blocks in Π(t). This case is much easier and we just sketch the proof. First, we claim that without loss of generality we may assume the mutation process Wt is a pure jump Lévy process. If Wt has a Brownian component, then the argument of Birkner and Blath when Π(t) does not come down from infinity applies, and it follows that S(Zt) = Rd almost surely. If Wt has a drift, then there is a driftless Lévy process Ŵt and a ∈ Rd such that Wt = at + Ŵt. One can construct coupled Fleming-Viot processes associated to mutation processes Wt and Ŵt, denoted Zt and Ẑt with the obvious correspondence, such that Zt = Ẑt + at for all t > 0, where Ẑt + at is the measure Ẑt shifted by at ∈ Rd. Thus S(Zt) = S(Ẑt + at), and it follows that if S(ν \* Ẑt) ⊆ S(Ẑt), then S(ν \* Zt) ⊆ S(Zt), so it suffices to consider Ẑt. Let Z0 = μ ∈ M1. Suppose that Π(t) has dust and assume in addition that Λ({1}) = 0. Under this additional assumption, it is not hard to see that for all t > 0 there exists α(t) ∈ (0, 1] such that a subset of the integers with frequency α(t) belong to singleton blocks. The individuals in singleton blocks evolve conditionally as independent copies of Wt and exchangeability implies that their initial positions are uniformly distributed from μ. The law of large numbers then implies that Zt has a component which is the heat flow associated to Wt, started from initial measure α(t)μ. In particular, it follows that S(Ttμ) ⊆ S(Zt). On the other hand, it is clear from the lookdown construction that S(Zt) ⊆ S(Ttμ). Thus we must have S(Zt) = S(Ttμ). The claim then follows from the observation that, for any measure μ, S(Ttμ) = S(ν \* Ttμ), which is a consequence of Wt being a pure jump process with Lévy measure ν. Now suppose that Λ({1}) = c > 0. Then there is a rate c Poisson process with jump times 0 < t1 < t2 < ⋯ such that at time tk, Zt jumps to δxk, where xk is a sample from Ztk −. Thus with probability one there is some k\* ∈ {0} ∪ N such that tk\* < t < tk\* + 1 (we define t0 = 0). The same reasoning as above then implies S(Zt) = S(Tt − tk\δx*k), and the result follows from the same argument. 99 Aldous, D.J., Changeability and related topics. École d’été de probabilités de Saint-Flour, XIII, Lecture Notes in Math. 1117, 1985. Berestycki, J., Berestycki, N., and Limic, V., The Λ-coalescent speed of coming down from infinity. Ann. Probab. 38, (2010) 207-233. Berestycki, J., Berestycki, N., and Limic, V., A small-time coupling between Λ-coalescents and branching processes. Ann. Probab. 24, (2014) 449-475. 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Scalable Scheduling for Industrial Time-Sensitive Networking: A Hyper-flow Graph Based Scheme ============================================================================================= Industrial Time-Sensitive Networking (TSN) provides deterministic mechanisms for real-time and reliable flow transmission. Increasing attention has been paid to efficient scheduling for time-sensitive flows with stringent requirements such as ultra-low latency and jitter. In TSN, the fine-grained traffic shaping protocol, cyclic queuing and forwarding (CQF), eliminates uncertain delay and frame loss by cyclic traffic forwarding and queuing. However, it inevitably causes high scheduling complexity. Moreover, complexity is quite sensitive to flow attributes and network scale. The problem stems in part from the lack of an attribute mining mechanism in existing frame-based scheduling. For time-critical industrial networks with large-scale complex flows, a so-called hyper-flow graph based scheduling scheme is proposed to improve the scheduling scalability in terms of schedulability, scheduling efficiency and latency & jitter. The hyper-flow graph is built by aggregating similar flow sets as hyper-flow nodes and designing a hierarchical scheduling framework. The flow attribute-sensitive scheduling information is embedded into the condensed maximal cliques, and reverse maps them precisely to congestion flow portions for re-scheduling. Its parallel scheduling reduces network scale induced complexity. Further, this scheme is designed in its entirety as a comprehensive scheduling algorithm GH2. It improves the three criteria of scalability along a Pareto front. Extensive simulation studies demonstrate its superiority. Notably, GH2 is verified its scheduling stability with a runtime of less than 100 ms for 1000 flows and near 1/430 of the SOTA FITS method for 2000 flows. Industrial Time-Sensitive Networking, scalable scheduling, hyper-flow graph, hierarchical framework Introduction ============ a basic infrastructure of the Industrial Internet of Things, device networking in factory automation is incorporating more and more sensors, controllers and actuators to build ubiquitous networked cyber-physical systems. Massive network applications require ultra-low latency and jitter, the most critical quality of service (QoS) metrics. For example, industrial automation and power grid systems require tight deterministic latency in the range of milliseconds to microseconds. The jitter is also bounded within a few microseconds. These drive the industrial network evolution to ultra-low latency and ultra-high reliability. Time-Sensitive Networking (TSN), an Ethernet extension under development from IEEE 802.1 TSN Task Group, empowers Ethernet bridge networks determinism by a capability collection governed via a series of standards. It integrates mechanisms such as traffic shaping, bandwidth policing and clock synchronization to control flow forwarding in a precise manner with bounded low latency & jitter and extremely low frame loss. Specifically, IEEE 802.1Qbv builds a gate-operation mechanism named time-aware shaper during the forwarding process. Inside the output port of TSN supported network devices, this shaper works to manage traffic forwarding queues and time through a gate control list (GCL). It records sequential gate states and state durations. Following this, TSN devices transmit flows in queues orderly and cyclically. Similarly, IEEE 802.1Qci constructs an en-queue policing mechanism for flow caching inside every TSN device. The flow forwarding between these devices is synchronized with IEEE 802.1AS. Moreover, a ping-pong gate configuration model, called cyclic queuing and forwarding (CQF), is given as IEEE 802.1Qch to implement the transmission determinism with these mechanisms. Nevertheless, further flow scheduling is desired for the real-time transmission of massive flows with ultra-low latency and jitter limits. It is considered as the deterministic scheduling problem (DSP). Since the advent of TSN, related research for DSP has been widely conducted and discussed. This problem is first formalized with systematic constraints & goals and proved to be NP-hard. It is solved by Satisfiability Model Theory or Integer Linear Programming, which show a high complexity and are hardly adaptable to large-scale complex industrial flow scenarios. Thus, the rapid schedulability assessment, and even further, the efficient determinism scheme design is desired to achieve the scalable scheduling capability. With such a vision, the flow incremental or partitioning scheduling methods are proposed in works. Typically, Quan et al. designs a flow injection-time scheduling (FITS) method that allocates flows incrementally with reasonable slot resources by injection offset. Guo et al. co-designs the flow incremental ordering and injection offset selection by a mapping score metric-based scheduling (MSS) method. In addition, Atallah et al. proposes an iterated flow partition scheduling method, where a degree of conflict-aware stream partitioning (DASP) strategy is adopted to optimize scheduling partitioning. Compared to global scheduling, these above ways segment flows into serial and interrelated scheduling blocks to downsize the network scale of each scheduling step. It facilitates the balance between scheduling optimality and efficiency. However, the complexity of these serial patterns is still high and difficult to improve. And even more so, their frame-based patterns during each block scheduling lack in attribute mining mechanism that inevitably possesses flow attribute induced high complexity. To further cope with the complexity dilemma, the correlation between TSN mechanisms and flow attributes is deeply analyzed to optimize the key scheduling processes, like constraint satisfiability and goal optimality checking. A divisibility theory-based flow sequence analysis method is developed to metric the scheduling space and elaborate flow confluences on time slots under the CQF model. Its building flow graph converts the frame-based scheduling into the equivalent flow graph-based pattern for efficient scheduling. Even so, its flow scale induced exponential graph complexity restricts the scheduling efficiency. Therefore, the current DSP is desired to schedule with better scalability, which means the faster scheduling with superior schedulability and latency & jitter performances. This paper focuses on improving the scheduling efficiency and scalability under the QoS and determinism assurances for large-scale complex flow scenarios. With the CQF-configured network, we holistically consider the scheduling complexity and optimality and deeply mine flow attribute-driven scheduling features. A hyper-flow graph based scheme is developed to innovate traditional serial and frame-based scheduling patterns. With this scheme, the critical causes of scheduling complexity, network scale (including device and flow scale) and flow attributes, are significantly suppressed in their impact on scheduling. Meanwhile, the schedulability and QoS performances are well-guaranteed and even enhanced. The contributions of the scheme are summarized as follows: * A hyper-flow graph based methodology is proposed to design and optimize the flow scheduling processes systematically. Instead of traditional frame-based patterns, the attribute-driven hyper-flow graph maps/embeds their flow attribute-sensitive redundant scheduling information into equivalent and less maximal hyper-flow cliques. Also, the re-scheduling information is precisely reverse mapped without violent retrieve by introducing the conflict clique. These ways suppress flow attribute induced complexity and drive the Pareto-optimized scheduling. * A hierarchical scheduling framework is constructed to balance scheduling efficiency and optimality for large-scale complex flows. With the unified design by hyper-flow graphs, four progressive phases are closely interlinked, including lightweight flow partitioning, parallel partition scheduling, parallel flow synthesizing and precise flow re-scheduling. These two parallel phases reduce the flow and device scale induced scheduling complexity, respectively, and the last phase improves the schedulability and optimality with precise flow fine-tuning. * A hyper-flow graph based hierarchical (GH2) algorithm designs the scheduling scheme in its entirety. Guided by the graph, an attribute-driven partitioning strategy, a parallel hyper-flow graph based scheduling (HFG) method, a parallel Bron-Kerbosch synthesizing method and a conflict clique based re-scheduling (CCR) method are designed comprehensively and interlinked hierarchically. It improves the schedulability, scheduling efficiency and QoS performances along a Pareto front. The rest of this paper is organized as follows. First, Section II establishes the CQF-configured network model and formulates its corresponding scheduling problem. Then, the hyper-flow graph based scheme is illustrated in Section III, including the hyper-flow graph based methodology, hierarchical framework and comprehensive flow scheduling algorithm. Section IV evaluates the performances of GH2, and Section V makes the final conclusion. Network Model and Problem Formulation ===================================== Network and Flow Model ---------------------- For the target network with the determinism demands, we model it as a directed graph N = < V,E > , where V denotes a set of network nodes νκ, consisting of terminal hosts Vh and TSN switches Vs. E denotes the directed physical links ɛι. In this network, the time-sensitive applications communicate through information flows F. Each of them fi is characterized by the attribute tuple {li, pi, bi, di, ji, Ri}. These denote the frame length, flow period, baseline time of generation, maximum allowable latency, maximum allowable jitter and transmission route, respectively. The flow route Ri is predefined as the following form of link association: $$R\_i = [ \varepsilon\_{i, 0}, \varepsilon\_{i, 1} \cdots, \varepsilon\_{i, h\_i} ], \nonumber$$ which connects the links between the source and destination host nodes with the hop count hi. [CQF] [CQF2] To guarantee the deterministic transmission of target flows, that is, the bounded latency and jitter, all the output ports of TSN switches are implemented with two statically configured Ping-Pong queues from IEEE 802.1Qch, called the CQF model. As shown in Fig. [CQF], it performs en-queue and de-queue operations alternately with a predetermined duration T. The transmission and reception control between both nodes of link ɛι are precisely aligned based on IEEE 802.1AS. In this way, the transmission process is divided into numbered time slots s ∈ S with length T, and the target flows are transmitted and queued along their route Ri slot after slot like Fig. [CQF2]. The end-to-end determinism of flow transmission is assured with the bounded latency [hi ⋅ T − T, hi ⋅ T + T] and jitter [0, 2 ⋅ T]. Considering the slotted transmission behavior, partial flow attributes are converted into the slot expression as p̊i, b̊i, d̊i, j̊i, given by $ \frac{p\_i}{T} $, $ \lfloor \frac{b\_i}{T} \rfloor $, $ \lfloor \frac{d\_i}{T} \rfloor $, $ \lfloor \frac{j\_i}{T} \rfloor $, respectively. [injection] As analyzed in, unscheduled flow injection into the network is easy to cause chaotic flow confluences on the same slot. It further results in slot congestion and overflow due to excessive occupancy beyond the slot capacity. Further, the potential frame loss occurs and the deterministic mechanism is broken. To avoid this, the injection delay oi is introduced to regulate the postponement between flow generation and injection. As shown in Fig. refinjection, the offset design of flow injection allows slot resources to be allocated to target flows without overflow, which facilitates flow scheduling more flexibly and effectively. It is worth noting that the injection offset is symbolized as oi under a general assumption that all frames of each flow fi share the same delay. This assumption reduces the variable scale to boost the scheduling efficiency and eliminates the additional flow jitter to improve QoS performances, but it is not necessary to obey. Hyper-flow Graph Model ---------------------- Given the generation periodicity and transmission determinism, target flows are kept forwarding periodically on every link by the uniform injection offset among each flow. Based on this, we cluster the frames of each flow with a unified forwarding slot representation. For any flow fi ∈ F on link ɛι ∈ Ri, its frame forwarding slots are modeled as a periodic sequence $$\aleph(k\_i^{\varepsilon\_\iota}) = q\_i^{\varepsilon\_\iota} +k\_i^{\varepsilon\_\iota} \cdot \mathring{p\_i},$$ where the baseline forwarding slot qiɛι is given as $ \mathring{b\_i} +o\_i +h\_{i}^{\varepsilon\_\iota} $. hiɛι is the passed hop number among Ri, that is, the passed slots after flow injecting. The sequence count ki accumulates every periodic frame. The flow attribute integrated slot positioning parameters $ (q\_i^{\varepsilon\_\iota}, \mathring{p\_i}) $ are called as the feature tuple of flow fi. [graph] In order to efficiently guide flow offset selection, we build a correlation graph on each link ɛι ∈ E to characterize flow confluence relationships. Considering the specific forwarding slots of target flows are dominated by their feature tuples, we aggregate the flow set with the same $ (q\_i^{\varepsilon\_\iota}, \mathring{p\_i}) $ as a hyper-flow ϕ and weight it the frame length sum among the flow set. Then, we take the hyper-flows Φɛι as graphical nodes and connect their undirected edges following the adjacency matrix Ψɛι as $$\Psi\_{\phi\_{\alpha}, \phi\_{\beta}}^{\varepsilon\_\iota} = \begin{cases} 1 & \text{if\ } \gcd(\mathring{p}\_{\alpha}, \mathring{p}\_{\beta}) \| (q\_{\alpha}^{\varepsilon\_\iota} -q\_{\beta}^{\varepsilon\_\iota}). \\ 0 & \text{otherwise.} \end{cases}$$ Like Fig. [graph], the flow aggregated hyper-flows are correlated by the relationship between their forwarding slot parameters. The hyper-flow graph is desired to elaborate the flow confluences on slots and efficiently extract the key scheduling information. Problem Formulation for CQF-configured network ---------------------------------------------- With the above models, we construct the optimal scheduling problem for the CQF-configured network with variable flow injection offset. To meet flow QoS demands and network determinism requirements, the CQF-based DSP is demonstrated by the following constraints and goal to regulate the scheduling selection of flow offsets. ### Latency Constraint In the CQF model, flow transmission latency is locked as (1) within the switching network. Affected by the injection offset oi, the worst-case latency ϑi is extended to (oi + hi + 1) ⋅ T. In order to bound oi into a QoS allowed range, we construct the latency constraint as $$\begin{aligned} & \forall f\_i \in \mathcal{F}: \\ & 0 \leq o\_i < \mathring{d}\_i -h\_i. \end{aligned}$$ ### Jitter Constraint Besides the latency, the flow jitter also needs to be within a tolerable limitation j̊i as follows. It is produced by the possible offset fluctuation $ \tilde{o\_i} $ within a flow and the endogenous jitter 2 ⋅ T from the CQF model. $$\begin{aligned} & \forall f\_i \in \mathcal{F}: \\ & \tilde{o\_i} + 2 \leq \mathring{j}\_i. \end{aligned}$$ ### Capacity Constraint During the flow forwarding on the common links, there exit potential flow confluences sharing the same slot sɛι. Constrained by the limited slot capacity Λ, the target flows need to be allocated reasonable slots without overflow. Thus, the constraint is demonstrated as: $$\Delta\_{i,s}^{\varepsilon\_\iota} = \begin{cases} 1 \leftarrow s^{\varepsilon\_\iota} \equiv q\_i^{\varepsilon\_\iota} \bmod \mathring{p}\_i, \\ 0 \leftarrow \text{otherwise}, \end{cases}$$ $$\begin{aligned} & \forall \varepsilon\_\iota \in \mathcal{E}, s^{\varepsilon\_\iota} \in \{1,2,\cdots,C\}: \\ & \sum\_{f\_i \in \mathcal{F}^{\varepsilon\_\iota}} \Delta\_{i,s}^{\varepsilon\_\iota} \cdot l\_i \leq \Lambda, \end{aligned}$$ where the boolean variable Δi, sɛι in (6) works to verify whether flow fi is forwarded at slot sɛι of link ɛι. It is confirmed by the congruence of checked slot sɛι and flow forwarding basetime qiɛι in relation to period p̊i. The constraint in (7) restricts the slot occupancy filled by flow confluences on each slot sɛι ∈ S to no more than the available slot capacity Λ. It is given as $ \gamma \cdot \min \{(T -\delta^{\varepsilon\_\iota}) \cdot \Gamma, \mho^{\varepsilon\_\iota}\} $, where δɛι is the slot misalignment owing to the clock synchronization errors, Γ is the link bandwidth, $ \mho^{\varepsilon\_\iota} $ is the physical queue depth, and γ is the capacity distribution factor used to reserve resources for deterministic transmission of aperiodic time-sensitive flows. Moreover, the capacity constraint covers slots S within a bounded scheduling view C given as the least common multiple of all flow periods, that is, lcm(P̊n), where P̊n is {p̊1, p̊2, ⋯, p̊n}. With this range, the global determinism of flow scheduling is assured by cyclically repeating the flow forwarding in these C slots. [!ht] [method] Among the feasible scheduling space for the above constraints (4)-(7), we aim to enhance the real-time transmission ability of flows F and the load balancing ability of slots S. The former improves latency performances, and the latter helps cope with burst flows and schedulability. Thus, a composite optimization goal is expressed as $$\min\_{\{o\_i\}}{ (1-\rho) \cdot \frac{1}{n} \sum\_{f\_i}{\frac{\vartheta\_i}{\mathring{d}\_i}} + \rho \cdot \left[ \frac{\zeta\_s^{\varepsilon\_\iota}}{\Lambda} \right]^{\max\_{s, \varepsilon\_\iota}} },$$ where the optimization value symbolized as ℑ is the combined weight of the former transmission average real-time rate ϑi/d̊i in target flows and the latter maximum slot occupancy rate ζsɛι/Λ among all slots on links E. ζsɛι is denoted as the slot occupancy of slot sɛι on link ɛι, and ρ ∈ [0, 1] is the weight factor working to balance these two goals. The complexity of general constraint-based optimization problems relies on intensive computation for constraint satisfiability and goal optimality. Considering the formulation of the above CQF-based DSP, the scale of scheduling variables, constraints and goal components expands with the growing network scale including device and flow scale. It causes an increasingly high scheduling complexity. So do the complicating flow attributes, especially the flow periods. According to the TSN profile standard for industrial automation, the flow periods of time-sensitive network systems are complex with large range spans. It inflates the scheduling view C to increase the scheduling complexity. So far, these complexity dilemmas still need to be properly addressed. Thus, we work on exploring a systematic scheme to solve these problems uniformly while well-balancing the scheduling optimality (8), that is, the proposed hyper-flow graph based scheme. Hyper-flow Graph Based Scheme ============================= In this section, we elaborate on the hyper-flow graph based scheme including the hyper-flow graph based methodology and four-phase hierarchical framework. It is graphically depicted as Fig. [method]. A comprehensive scheduling algorithm GH2 designs the scheme in its entirety for scalable flow scheduling in CQF-configured TSN. Thus, we build the scheme following the GH2 as Algo. 1 hierarchically. [!t] $ \{ \mathcal{F}\_{\varsigma} \} \leftarrow $ Attribute-driven flow partitioning for F O, Φ, L ← Parallel HFG scheduling($ \{ \mathcal{F}\_{\varsigma} \}, \mathcal{A} $) Υ ← Parallel Bron-Kerbosch synthesizing(Φ, L) $ \dot{\mathcal{O}} \leftarrow $ Precise CCR re-scheduling(Υ, Λ) $ \mathcal{O} \leftarrow \mathcal{O} \cup \dot{\mathcal{O}} $ Given the complexity analysis of the above DSP, we first explore a divide-and-conquer hierarchical design to reduce the network scale induced complexity. Specifically, the first flow division phase divides target flows F into separate partitions $ \mathcal{F}\_{\varsigma} $ in line 1, where an attribute A-driven lightweight strategy is designed to accelerate flow partitioning. For more than one flow partition, the second partition conquering phase spreads them over multiple parallel scheduling sub-problems to reduce the flow scale induced complexity. With the flow scheduling of optimized slot allocations, every partition obtains a local flow offset solution that is aggregated as a preliminary scheduling solution O in line 3. Considering the partitions share a common network, the global scheduling information, that is, the slot occupancy by flow confluences, needs to be synthesized from these sub-problems. The third flow synthesis phase works on it link-parallelly in line 4 to suppress the device scale induced complexity. Unusually, when only one partition exists, the synthesis phase is skipped to line 6. For the specific scheduling and synthesizing processes, the hyper-flow graph based methodology is explored by building a hyper-flows Φ and weights L based graph. It maps/embeds one-by-one slot occupancy within flow attribute-sensitive bound C into equivalent and less maximal hyper-flow cliques Υ. With the condensed scheduling information, a parallel HFG flow scheduling method in line 3&6 and a parallel Bron-Kerbosch flow synthesizing method in line 4 are proposed to improve the scheduling efficiency further. By these three phases, GH2 fast-tracks a preliminary scheduling solution and its global maximal hyper-flow cliques under the optimized QoS performances. After the flow synthesis of scheduling information, there may be some overflow slots where their occupancy exceeds the slot capacity Λ or other desired values. It violates the capacity constraint (7) and affects the scheduling optimality (8). Hence, the last flow fine-tuning phase in line 7 is desired to re-schedule the overflow flows inside these slots. Meanwhile, the above preliminary solution is rectified as slightly as possible to maintain QoS performances. For this vision, the assumption that all frames of each flow share the same delay is broken. We aim to precisely position the overflow slots and re-scheduling overflow flow portions inside them. By extending the hyper-flow graph based methodology, the conflict clique is first introduced and proved equivalent to overflow slots. Then, the re-scheduling information, that is, the overflow slots and their flow portions, is precisely reverse mapped from the maximal hyper-flow cliques without violent retrieve. It drives an efficient CCR re-scheduling method in line 7 to precisely fine-tune flows for schedulability improvement and load-balancing. After this phase, the supplement solution $ \dot{\mathcal{O}} $ is given and constitutes the complete scheduling solution together with O in line 8. The specific processes of the above hyper-flow graph based scheme, that is, GH2 scheduling algorithm, are demonstrated in phases as follows. Lightweight Flow Division Phase ------------------------------- For the DSP with large-scale complex flows, we divide the target flows F into multiple partitions $ \mathcal{F}\_{\varsigma} $ for the subsequent separate scheduling. Distinguishing from SOTA partitioning strategies, like spectral clustering that constructs a similarity graph to cluster flows, we design a lightweight strategy to reduce time costs and maintain scheduling performances. It partitions flows via their sorted typical attitudes. Considering that load balancing for every partition scheduling sub-problem helps to reduce the slot overflow, we view this as the partition principle of the CQF-based DSP. Then, an intuitive law is observed that the flows with similar attributes facilitate the balance of slot occupancy. For instance, the flows with similar even the same flow period pi could be evenly distributed over different slots with fewer flow confluences. The similar frame length l̊i could bring approximate occupancy to balance the slots. Thus, we divide flows based on the similarity of flow attributes as follows. $$\begin{aligned} & \Theta \leftarrow \{ f\_{\alpha}, f\_{\beta}, \cdots \in \mathcal{F} \mid f\_{\alpha} \prec f\_{\beta} \prec \cdots \}, \\ & \{ \mathcal{F}\_1, \mathcal{F}\_2,..., \mathcal{F}\_{\varsigma},... \} \leftarrow \text{sequentially slice } \Theta, \end{aligned}$$ where we refer flow attitude li, pi or the bandwidth consumption depicted by li/pi as a potential partitioning bases. Target flows are first sorted by one of the above basis and then batch sliced into one-by-one partition $ \mathcal{F}\_{\varsigma} $ with a specified scale Ξ. Parallel Partition Conquering Phase ----------------------------------- For the above flow partitions, we schedule them as separate sub-problems to cut their interlink. This way reduces the flow scale in each sub-problem and the scheduling complexity. Due to the separability, a parallel computing structure is adopted to further reduce the complexity compared to the serial patterns. Each conquering unit corresponds to one flow partition and deploys the HFG scheduling method. [!t] $ \Theta\_{\varsigma} \leftarrow \{ f\_{\alpha}, f\_{\beta}, \cdots \in \mathcal{F}\_{\varsigma} \mid f\_{\alpha} \prec f\_{\beta} \prec \cdots \} $ o ← 0, ōε ← min{pε, dε − hε} and ℑε ← ∞ ϕɛι ← get $ (\mathring{p\_{\epsilon}}, q\_{\epsilon}^{\varepsilon\_\iota}) $ by (2) $ \zeta\_{\varpi}^{\varepsilon\_\iota} \leftarrow l\_{\epsilon} + \sum\_{\phi \in \varpi\_{\kappa}^{\varepsilon\_\iota}} \ell\_{\phi}^{\varepsilon\_\iota} $ $ \bar{\zeta}\_o \leftarrow \max{\{ \max{\{ \zeta\_{\varpi}^{\varepsilon\_\iota} \}, \bar{\zeta}} \} }$ ℑo ← get the optimization value by (11) oε ← o, ζ̄ε ← ζ̄o and ℑε ← ℑo {ϕɛι}ε ← {ϕɛι} and {Υ̃ϕɛι}ε ← {Υ̃ϕɛι} break o ← o + 1 $ \mathcal{O}\_{\varsigma} \vartriangleleft (f\_{\epsilon}, o\_{\epsilon}) $ and ζ̄ ← max{ζ̄ε, ζ̄} ℓϕɛι ← ℓϕɛι + lε $ \mathcal{O} \vartriangleleft \mathcal{O}\_{\varsigma} $ Before the scheduling process, the slot expressions of flow attributes are given with predetermined slot length T satisfying conditions T ∣ gcd(Pn), j̊i ≥ 2 and d̊i − hi > 0. They participate in the parallel HFG scheduling method shown as Algo. 2, which distributes flow partitions to their respective conquering units in line 1. For every partition, the corresponding flows are scheduled incrementally to get their optimal injection offset in lines 2-29, which reduces device nodes involved in each scheduling to their own routes. In line 2, flows are first sorted to determine the scheduling order as $ \Theta\_{\varsigma} $. During each flow fε scheduling, its feasible offset o is traversed below the bound ōε given in line 4. It is the smaller of flow period pε and latency constraint (4). The optimal offset oε is filtered with the greedy policy preferring a lower transmission real-time rate and slot occupancy rate expressed as $$\min\_{o\_{\epsilon}}{ (1-\rho) \cdot \frac{1}{n\_{\epsilon}} \sum\_{f\_i}{\frac{\vartheta\_i}{\mathring{d}\_i}} + \rho \cdot \frac{\bar{\zeta}\_{\epsilon}}{\Lambda} }.$$ It is designed to match the global optimization goal (8), where nε counts the flows that have been scheduled in the current partition and ζ̄ε is the global maximum slot occupancy among all slots S of the whole network in fε scheduling. Further, the constant part in (10) during offset searching of each flow fε is removed to simplify the local goal as $$\min\_{o\_{\epsilon}}{ (1-\rho) \cdot \frac{o\_{\epsilon}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} + \rho \cdot \frac{\bar{\zeta}\_{\epsilon}}{\Lambda} }.$$ This optimal value is symbolized as ℑε and obtained in lines 5-23. Specifically, with increasing offset o in lines 5&23, its corresponding optimization value ℑo is given in lines 6-17 and filtered in lines 18-22. In this key process, the critical cause of scheduling complexity is obtaining maximum slot occupancy ζ̄o under each offset and recurring offset searching. We first discuss the resolutions for the former as follows. Since retrieving the slot occupancy slot-by-slot within the bound C is computationally complex, the complexity is quite sensitive to flow periods that determine the bound. It exponentially expands as the multiplicity of periods. With the built hyper-flow graph (3) on each link ɛι ∈ E, we prove the equivalence between hyper-flow cliques and flow confluences on any slot under the support of following Lemma. For n hyper-flow sequences formed as ℵ(ki) = qi + ki ⋅ p̊i, if any sequence pair ℵ(kα) and ℵ(kβ) satisfies that gcd(p̊α, p̊β)∣(qα − qβ), there exist z ∈ Z satisfying p̊i∣(z − qi), i ∈ {1, 2, ..., n}, and vice versa. [1] It means that each hyper-flow clique indicates the flow confluences on a certain slot sɛι, and the flow confluences on every slot sɛι are carried by a hyper-flow clique. That is, the occupancy of any slot is given by the combined weight of its corresponding clique. Thus, the maximum slot occupancy ζ̄o is equal to the maximum combined weight among all hyper-flow cliques. The obtaining of maximum slot occupancy ζ̄o is converted to a maximal clique enumeration problem (MCEP) as shown in Fig. [fig13](a). Compared to violently retrieve, the hyper-flow graph based pattern maps/embeds the redundant slot occupancy information into less maximal flow cliques Υɛι. Essentially, it merges the identical or contained flow confluences on different slots. [fig13] In this way, we design a dynamic graph and clique update method to obtain the maximum slot occupancy ζ̄o under offset o of flow fε for incremental scheduling. As shown in lines 6-16 of Algo. 2, for each link ɛι along route Rε, the feature tuple of flow fε is extracted in line 7 and checked its existence in the hyper-flow nodes $ \Phi\_{\varsigma}^{\varepsilon\_\iota} $. It covers hyper-flows that have been scheduled on link ɛι. If it exists, the weight ℓϕɛι of corresponding ϕɛι increases by lε (without real update in the pseudo-code during the offset search), and so do all combined weights $ \zeta\_{\varpi}^{\varepsilon\_\iota} $ of maximal cliques $ \varpi\_{\kappa}^{\varepsilon\_\iota} \in \Upsilon\_{\phi}^{\varepsilon\_\iota} $ that contains ϕɛι among the maximal hyper-flow clique set $ \Upsilon\_{\varsigma}^{\varepsilon\_\iota} $ in lines 9-10. They imply the potential maximum occupancy of the slots containing fε. If not, the tuple extends the graph as a new hyper-flow ϕɛι. Its neighbors $ \varkappa\_{\phi}^{\varepsilon\_\iota} $ are given by (3) in line 12, where the neighbors mean its connected nodes by (3) in the graph. The potential maximum slot occupancy containing fε is obtained based on the following theorem. In the graph with maximal cliques $ \varpi\_{\kappa} \in \Upsilon $, if a new node ϕ joins with the neighbors $ \varkappa\_{\phi} $, the new maximal cliques $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $ compared to Υ are included in the clique set $$\tilde{\Upsilon}\_{\phi} = \{ \tilde{\varpi}\_{\kappa} \mid \tilde{\varpi}\_{\kappa} = (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \},\ \varpi\_{\kappa} \in \Upsilon\_{\varkappa} \},$$ where $ \Upsilon\_{\varkappa} $ is $ \{ \varpi\_{\kappa} \in \Upsilon \mid (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \neq \varnothing \} $ as the subset of Υ. Specially, it is modified to set $ \{ \varnothing \} $ when empty. Furthermore, for any clique $ \tilde{\varpi}\_{\kappa} \in \tilde{\Upsilon}\_{\phi} - \breve{\Upsilon} $, there exits a new maximal clique $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $ satisfying that $ \tilde{\varpi}\_{\kappa} \subseteq \breve{\varpi}\_{\kappa} $. [1] It is deduced and verified step by step as follows. Considering that $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \subseteq \varkappa\_{\phi} $, any element $ \phi\_j \in (\varpi\_{\kappa} \cap \varkappa\_{\phi}) $ is connected with ϕ. Moreover, since the subset $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) $ of clique $ \varpi\_{\kappa} \in \Upsilon $ must be a clique, we prove that $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $ is a clique and Υ̃ϕ is the clique set. For any new maximal clique $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $, it is easy to deduced that $ \phi \in \breve{\varpi}\_{\kappa} $. Otherwise, $ \breve{\varpi}\_{\kappa} $ is constituted by the original nodes of the graph and $ \breve{\varpi}\_{\kappa} \in \Upsilon $, which violates the definition of $ \breve{\varpi}\_{\kappa} $. Considering the full connection feature between nodes of the clique, the remaining elements of $ \breve{\varpi}\_{\kappa} $ except ϕ only exist in its neighbors, that is, $ \breve{\varpi}\_{\kappa} \subseteq (\varkappa\_{\phi} \cup \{ \phi \}) $. In addition, the fully connected remaining elements must be included in a certain maximal clique $ \varpi\_{\kappa} \in \Upsilon $, that is, $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cup \{ \phi \}) $. Hence, there exists $ \varpi\_{\kappa} \in \Upsilon $ satisfying that $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cup \{ \phi \}) \cap (\varkappa\_{\phi} \cup \{ \phi \}) $, that is, $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $. At this time, the maximal clique $ \varpi\_{\kappa} \in \Upsilon\_{\varkappa} $ also exists if the above existing $ \varpi\_{\kappa} $ satisfies the condition $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \neq \varnothing $. Otherwise, all these existing $ \varpi\_{\kappa} $ satisfies that $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) = \varnothing $ and $ \breve{\varpi}\_{\kappa} \subseteq \{ \phi \} $, which means the neighbors $ \varkappa\_{\phi} $ are the empty set $ \varnothing $. Thus, $ \Upsilon\_{\varkappa} $ is the set $ \{ \varnothing \} $. There exists $ \varpi\_{\kappa} = \varnothing $ satisfying that $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $. Since the maximal clique cannot be the proper subset of any other clique, for any $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $, there exists $ \varpi\_{\kappa} \in \Upsilon\_{\varkappa} $ satisfying the condition $ \breve{\varpi}\_{\kappa} = (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $. Hence, Ῠ ⊆ Υ̃ϕ holds. Similarly, for any clique $ \tilde{\varpi}\_{\kappa} \in \tilde{\Upsilon}\_{\phi} - \breve{\Upsilon} $, since node $ \phi \in \tilde{\varpi}\_{\kappa} $, we have that $ \phi \notin \varpi\_{\kappa} $, $ \forall \varpi\_{\kappa} \in \Upsilon $. Thus, there must exists the new maximal clique $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $ satisfying the condition $ \tilde{\varpi}\_{\kappa} \subseteq \breve{\varpi}\_{\kappa} $. Otherwise, $ \tilde{\varpi}\_{\kappa} $ is verified as a new maximal clique, which conflicts with $ \tilde{\varpi}\_{\kappa} \in \tilde{\Upsilon}\_{\phi} - \breve{\Upsilon} $. According to this theorem, the maximum occupancy of the slots containing fε is implied in the combined weights $ \zeta\_{\varpi}^{\varepsilon\_\iota} $ of cliques $ \tilde{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \tilde{\Upsilon}\_{\phi}^{\varepsilon\_\iota} $ given by (12). These values are obtained in lines 13-15. Based on this, the maximum slot occupancy ζ̄o under each offset is measured by comparing $ \max{\{ \zeta\_{\varpi}^{\varepsilon\_\iota} \}} $ and the previous maximum occupancy ζ̄ before flow fε scheduling in line 16. Then, the optimization value ℑo for offset o during flow fε scheduling is given by (11) in line 17, which works as the filtering basis for the optimal value ℑε and offset oε in lines 18-20. Under each offset o, the computational complexity to get ζ̄o is less than min{(m − ρ) ⋅ 3ρ/3, lcm(P̊n)/p̊ε} on each link, where m is the neighbor number of flow fε in the flow graph. ρ are the degeneracy of the neighbor built sub-graphs. Compared to the above hyper-flow graph based pattern, the traditional frame-based scheduling builds a slot occupancy list covering all slots S of the whole network as shown in Fig. [fig13](b). Due to the uncertainty of which slots have current maximum occupancy, it puts all periodic frames of flow fε into their specific slots sɛι ∈ S within the bound C along every link ɛι ∈ Rε during each offset search. The occupancy ζsɛι of these slots is measured one-by-one. This flow period-sensitive pattern owns the computational complexity as constant lcm(P̊n)/p̊ε on each link. For the large-scale complex flows, it almost exponentially expands and severely impacts scheduling efficiency. In our previous work, a flow graph-based scheduling pattern is explored to obtain the key scheduling information, that is, maximum slot occupancy ζ̄o. Distinguished from the hyper-flow nodes, this graph is built by treating each flow as a node and connects edges with a similar matrix Ψɛι $$\Psi\_{f\_{\alpha}, f\_{\beta}}^{\varepsilon\_\iota} = \begin{cases} 1 & \text{if\ } \gcd(\mathring{p}\_{\alpha}, \mathring{p}\_{\beta}) \| (q\_{\alpha}^{\varepsilon\_\iota} -q\_{\beta}^{\varepsilon\_\iota}). \\ 0 & \text{otherwise.} \end{cases}$$ As shown in Fig. [fig13](c), it is easy to prove that the hyper-flow graph can be shrunk from the flow graph with the following node merging principles. 1. The merged flow nodes are interconnected in pairs with edges by (13); 2. The merged flow nodes have the same neighbors $ \varkappa\_{\phi} $ except for each other; 3. After merging, the new node inherits the weight as the sum of the merged flow nodes. Thus, these two patterns share the same number and combined weights of maximal cliques. Their computational complexity is similar, but the hyper-flow graph pattern has smaller m and ρ. Since the complexity of MCEP rises exponentially along with the node scale, the smaller hyper-flow graph facilitates the large-scale flow scheduling. It also curbs the exponential trend of MECP due to its slower growth. To sum up, the hyper-flow graph based scheduling pattern suppresses the sensitivity of scheduling complexity to flow attributes, while weakening the impact of flow scale on it. Since the recurring offset searching accumulates the above complexity, we further improve offset filtering efficiency with an early-break strategy. It provides a breaking condition for offset traversal before reaching bound ōε. Specifically, the optimization values under offsets behind current o are compared with current optimal value ℑε in lines 21-22. Compared to this optimal ℑε, if all values ℑõ under offset õ > o satisfy that $$\Im\_{\epsilon} \leq \inf\_{\tilde{o} > o}{\Im\_{\tilde{o}}} = \inf\_{\tilde{o} > o}{ \{ (1-\rho) \cdot \dfrac{\tilde{o}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} + \rho \cdot \frac{\bar{\zeta}\_{\tilde{o}}}{\Lambda} \} }.$$ It means that the current value ℑε and its offset oε is optimal for flow fε scheduling. Thus, we break the offset search from its traversal. Based on this, we deflate (14) as $$\Im\_{\epsilon} \leq \inf\_{\tilde{o} > o}{ \{ (1-\rho) \cdot \dfrac{\tilde{o}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} \} } + \inf\_{\tilde{o} > o}{ \{ \rho \cdot \frac{\bar{\zeta}\_{\tilde{o}}}{\Lambda} \} }.$$ It is easy to prove that (14) must hold if the condition (15) is satisfied. Further, we have that $$\begin{aligned} & \inf\_{\tilde{o} > o}{ \{ (1-\rho) \cdot \dfrac{\tilde{o}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} \} } = (1-\rho) \cdot \dfrac{o+1}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}}, \\ & \inf\_{\tilde{o} > o}{ \{ \rho \cdot \frac{\bar{\zeta}\_{\tilde{o}}}{\Lambda} \} } = \rho \cdot \frac{\max{ \{ \bar{\zeta}, l\_{\epsilon} \} }}{\Lambda}, \end{aligned} \nonumber$$ which builds the early-break threshold under offset o as $$\check{\Im}\_o = (1-\rho) \cdot \dfrac{o+1}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} + \rho \cdot \frac{\max{ \{ \bar{\zeta}, l\_{\epsilon} \} }}{\Lambda}.$$ If the condition $ \Im\_{\epsilon} \leq \check{\Im}\_o $ is satisfied, the offset traversal ends immediately in line 22. Unusually, when weight factor ρ is 1, the early-break condition is converted into ζ̄ε ≤ max{ζ̄, lε} introduced in work. After the flow fε scheduling by offset search, the optimal offset oε is stored and the global maximum slot occupancy ζ̄ is updated in line 24, where the symbol $ \vartriangleleft $ represents the store operation. Then, the hyper-flow graph and its maximal cliques on each link ɛι ∈ Rε are updated in lines 25-29. Specifically, if there exists the hyper-flow ϕɛι with the same feature tuple as optimal offset oε, this hyper-flow adds frame length lε to its weight ℓϕɛι in line 27. Otherwise, the tuple is extended as a new hyper-flow with weight lε, and stored in line 29 with its corresponding potential maximal cliques Υ̃ϕɛι. Due to Theorem 1, they need to be reduced to maximal hyper-flow cliques Ῠϕɛι and then stored. In the incremental way, every flow partition is scheduled and their flow injection offsets $ \mathcal{O}\_{\varsigma} $ are aggregated as a preliminary solution O in line 30. Link-parallel Flow Synthesis Phase ---------------------------------- Due to the network sharing of sub-problems in the above partition scheduling, their common slot occupancy affects load balancing and causes potential overflow. Hence, the global slot occupancy information, that is, maximal hyper-flow cliques, needs to be synthesized from these sub-problems. The complete hyper-flow graphs are built link-by-link that accumulates the device scale induced scheduling complexity. To deal with this, we adopt a link-parallel structure that spreads the hyper-flow graph synthesis for each link ɛι ∈ E over multiple computing units as shown in Fig. [linkparallel]. [linkparallel] In the second phase, each scheduling sub-problem identifies its hyper-flows on each link. Hence, the graph and maximal cliques synthesis could be separated across links. The pseudo-code is shown as Algo. 3. Available computing units are first allocated to every link containing target flows in line 1. For any link ɛι ∈ E, the hyper-flows $ \Phi\_{\varsigma}^{\varepsilon\_\iota} $ and their corresponding weights $ \{ \ell\_{\phi}^{\varepsilon\_\iota} \}\_{\varsigma}^{\varepsilon\_\iota} $ from sub-problems are synthesized by (3) in lines 2-3. It forms a whole hyper-flow graph with nodes Φɛι and weights Lɛι, where the same nodes are merged and new edges are connected. Meanwhile, the neighbors are accumulated as Nɛι for each hyper-flow in the synthesized graph in line 4. Based on this, line 5 adopts a Bron-Kerbosch Degeneracy method to enumerate the global maximal cliques Υɛι on each link. Also, their combined weights are counted and aggregated in lines 6-8. After this, the global slot occupancy information is given in line 8 and stored as Υ. [!h] Φɛι ← merge the same ϕɛι from all partitions Lɛι ← sum up weights ℓϕɛι for the same ϕɛι Nɛι ← get neighbors $ \varkappa\_{\phi}^{\varepsilon\_\iota} $ for each ϕɛι ∈ Φɛι Υɛι ← Bron-Kerbosch Degeneracy(Φɛι, Nɛι) $ \zeta\_{\varpi}^{\varepsilon\_\iota} \leftarrow \sum\_{\phi \in \varpi\_{\kappa}^{\varepsilon\_\iota}} \ell\_{\phi}^{\varepsilon\_\iota} $ $ \Upsilon \vartriangleleft \Upsilon^{\varepsilon\_\iota} $ Precise Flow Fine-tuning Phase ------------------------------ In this phase, global hyper-flow graphs and their maximal cliques are involved in overflow avoidance to improve schedulability and load-balancing. The potential overflow slots are desired to precisely fine-tune the flows inside them, while the optimized QoS performances are maintained. To handle this, the conflict clique is first defined and proved its superiority for flow fine-tuning. It is an extension of the above hyper-flow graph based methodology and drives the design of an efficient CCR re-scheduling method shown in Fig. [CCR], including the following overflow checking, conflict clique detecting, sub-flow backtracking and sub-flow re-scheduling. [!t] [CCR] Limited by available slot capacity Λ in constraint (7), the global slot occupancy needs to be checked for overflow. According to the maximal hyper-flow cliques from the synthesis phase, we give the overflow maximal cliques $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \hat{\Upsilon}^{\varepsilon\_\iota} $ on each link ɛι ∈ E as $$\hat{\Upsilon}^{\varepsilon\_\iota} = \{ \varpi\_{\kappa}^{\varepsilon\_\iota} \mid \varpi\_{\kappa}^{\varepsilon\_\iota} \in \Upsilon^{\varepsilon\_\iota} \text{ and } \zeta\_{\varpi}^{\varepsilon\_\iota} > \Lambda \},$$ which are aggregated as Υ̂. With further analysis of Lemma 1, we deduce that the flow confluences on overflow slots $ \hat{\mathcal{S}} $ must be carried by the sub-cliques in these cliques Υ̂. To avoid slot overflow, a naive way is to backtrack and re-schedule the flows $ \dot{f}\_i \in \dot{\mathcal{F}} $ satisfying that $$\forall \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \hat{\Upsilon},\ \text{the combined weight of } \{ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} - \dot{\mathcal{F}} \} \leq \Lambda. \nonumber$$ With different flow offsets re-filtered by Algo. 2, the hyper-flow graphs are modified to eliminate the overflow maximal cliques Υ̂, that is, the overflow slots. Since these flows occupy not only overflow slots but also others, the whole flow re-scheduling impacts current slot allocation and QoS performances more. To focus on overflow avoidance while maintaining the QoS, we aim to precisely position the overflow slots $ \hat{s}^{\varepsilon\_\iota} \in \hat{\mathcal{S}} $ and fine-tune the flow portions inside them. It breaks the assumption that all frames of each flow share the same delay and makes the preliminary solution be rectified as slightly as possible. Between the overflow slots $ \hat{\mathcal{S}} $, flow portions and given overflow maximal cliques Υ̂, we introduce the conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ to build their correlation. This concept is first given by us and defined as In a hyper-flow graph with weighted nodes, the conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ is a clique with the features as 1. Its combined weight exceeds slot capacity Λ; 2. The weight of its any proper sub-clique is below Λ. [1] Under such conditions, these conflict cliques $ \dot{\Upsilon} $ have the advance in re-scheduling shown as the following theorem. For all conflict cliques $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \dot{\Upsilon} $ under the above definition, they satisfy that 1. The confluence slots $ \dot{s}^{\varepsilon\_\iota} \in \dot{\mathcal{S}} $ of flows in conflict cliques are equivalent to overflow slots $ \hat{\mathcal{S}} $; 2. The slot overflow would be avoided by backtracking and re-scheduling the flow portions $\dot{\mathcal{F}} $ given as $$\{ f\_i^{\dot{\varpi}} \mid \text{ flows } \{ f\_i \} \text{ of any one node in } \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota},\ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \dot{\Upsilon} \}, \nonumber$$ where $ f\_i^{\dot{\varpi}} $ represents the flow fi portion that occupies the confluence slots $ \dot{s}^{\varepsilon\_\iota} $ of clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $. [2] The propositions are verified step by step as follows. For any conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \dot{\Upsilon} $, it is easy to get that the corresponding flow confluence slots overflow due to feature 1). Conversely, for any overflow slot $ \hat{s}^{\varepsilon\_\iota} \in \hat{\mathcal{S}} $ and its corresponding flow confluences as the clique $ \varpi $ by Lemma 1, there must exist a sub-clique satisfying all features of conflict cliques. Moreover, the flow confluence slots of this sub-clique must cover the confluence slots of $ \varpi $, that is, the overflow slot ŝɛι. Thus, proposition 1) holds. During the backtracking for re-scheduling following proposition 2), each conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \dot{\Upsilon} $ takes out one node with the flows set {fi}. It means that their flow portions $ f\_i^{\dot{\varpi}} $ in the confluence slots $ \dot{s}^{\varepsilon\_\iota} $ of conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ would be re-scheduled outside these slots. In this way, there is no conflict clique satisfying the features 1)&2) among the network. For any overflow slot, its flow confluences would be below the slot capacity. Otherwise, the overflow flow confluences imply the existence of conflict cliques by proposition 1), which violates the above inferences. Thus, proposition 2) holds. Based on this theorem, we design the following methods to precisely position overflow slots and fine-tune their flow portions, including recursive conflict clique detection, Euclidean-based slot positioning and low-cost sub-flow re-scheduling. ### Recursive Conflict Clique Detection For the Theorem 2-based re-scheduling, all conflict cliques $ \dot{\Upsilon} $ need to be detected from the overflow maximal cliques Υ̂. The detection pseudo-code is shown as Algo. 4, where a specific recursive process is illustrated independently. In lines 1-3, each overflow maximal clique $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \hat{\Upsilon}^{\varepsilon\_\iota} $ on every link ɛι ∈ E is sorted its hyper-flows in ascending order of their weights. The ordered queue Θκɛι works for the ConflictCliqueDetection process along with the combined weight ζ̂κɛι of $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} $, the slot capacity Λ and temporary variable ω initialized to $ \varnothing $. $$\begin{aligned} & \textbf{proc } \text{ConflictCliqueDetection} (\theta, \zeta, \Lambda, \omega) \\ & \ \text{1:} \ \dot{\varpi} \leftarrow \omega \cup \theta \text{ and get the first element } \phi \in \dot{\varpi} \text{;} \\ & \ \text{2:} \ \textbf{if } \zeta -\ell\_{\phi} > \Lambda \ \textbf{then} \\ & \ \text{3:} \ \quad \text{report } \dot{\varpi} \text{ as a conflict clique and }\textbf{return;}\\ & \ \text{4:} \ \textbf{for } \phi\_i \text{ in } \theta \textbf{ do} \\ & \ \text{5:} \ \quad \textbf{if } \zeta -\ell\_{\phi\_i} > \Lambda \ \textbf{then} \\ & \ \text{6:} \ \quad \quad \tilde{\theta} \leftarrow \{ \phi\_{i+1}, \phi\_{i+2}, \cdots \in \theta \} \text{ and } \tilde{\omega} \leftarrow \{ \cdots, \phi\_i \in \theta \} \\ & \ \text{7:} \ \quad \quad \text{ConflictCliqueDetection} (\tilde{\theta}, \zeta-\ell\_{\phi\_i}, \Lambda, \{ \omega, \tilde{\omega} \}) \end{aligned} \nonumber$$ [!t] $ \Theta\_{\kappa}^{\varepsilon\_\iota} \leftarrow \{ \phi\_1^{\varepsilon\_\iota}, \phi\_2^{\varepsilon\_\iota}, \cdots \in \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} \ \| \ \ell\_{\phi\_1}^{\varepsilon\_\iota} < \ell\_{\phi\_2}^{\varepsilon\_\iota} < \cdots \} $ $ \dot{\Upsilon}^{\varepsilon\_\iota} \vartriangleleft $ ConflictCliqueDetection($ \Theta\_{\kappa}^{\varepsilon\_\iota}, \Lambda, \hat{\zeta}\_{\kappa}^{\varepsilon\_\iota}, \varnothing $) $ \dot{\Upsilon} \vartriangleleft \dot{\Upsilon}^{\varepsilon\_\iota} $ [fig5] During the process, conflict cliques implied in every maximal clique $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ are detected with the iteration of cliques. In each iteration, the given clique $ \dot{\varpi} $ satisfying feature 1) of the conflict clique is composed of the reserved part ω and flexible part θ in line 1 and checked if the feature 2) holds in line 2. Specifically, feature 2) is checked by comparing slot capacity Λ and the sub-clique removing the hyper-flow node with the minimum weight, that is, the first element $ \phi \in \dot{\varpi} $. The checking passed clique is verified as a conflict clique and gathered into set $ \dot{\Upsilon} $. At this time, its corresponding iteration ends in line 3. If feature 2) is not satisfied, it means that there exist conflict cliques among the proper sub-cliques of $ \dot{\varpi} $. Hence, by popping one hyper-flow ϕi from the flexible part θ of clique $ \dot{\varpi} $, the sub-cliques of $ \dot{\varpi} $ are filtered by feature 1) and then traversed for the next round of iterations in lines 4-7. In this way, the clique iterations start from maximal clique $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ and continue round by round with the scale of checked cliques decreasing. The iterations end until all conflict cliques are detected. In the same round, they shared the same clique scale. It is worth noting that in order to avoid iterating and checking the duplicate cliques, we distinguish the flexible part θ and reserved part ω for each detection process, which varies with rounds of iterations in lines 6-7. The former is traversed for clique iteration rather than the latter. It makes the sub-cliques unique like Fig. [fig5] in each round of iterations. In this way, all conflict cliques $ \dot{\Upsilon} $ are obtained from every overflow maximal clique $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \hat{\Upsilon} $ in line 4 of Algo. 4. The conflict cliques on links ɛι ∈ E are stored as $ \dot{\Upsilon}^{\varepsilon\_\iota} $ and then aggregated as set $ \dot{\Upsilon} $ in line 5. On each link, the same conflict cliques $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ may be implied in multiple overflow maximal cliques $ \hat{\varpi}\_{\kappa}^{\varepsilon\_\iota} $. They are de-duplicated into one clique. ### Euclidean-based Slot Positioning Based on Theorem 2 and given conflict cliques $ \dot{\Upsilon} $, we aim to position the confluence slots $ \dot{s}^{\varepsilon\_\iota} $ of every conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ and backtracking their specific flow portions $ \dot{\mathcal{F}} $. Hence, a precise clique-aware slot positioning method is designed under the guidance of the following theorem. For n sequences as ℵ(ki) = qi + ki ⋅ p̊i, where i ∈ {1, 2, ..., n}, there are two equivalent propositions as 1. For any sequence pair ℵ(kα) and ℵ(kβ), it satisfies that gcd(p̊α, p̊β)∣(qα − qβ); 2. There exist a unique z ∈ [0, lcm(P̊n)) that satisfies the conditions p̊i∣(z − qi), i ∈ {1, 2, ..., n}. [3] As an extension of Lemma 1, this theorem only needs to be verified its sufficiency from proposition 1) to 2). From Lemma 1, if proposition 1) holds, there must exist zʹ ∈ ( − ∞, ∞) satisfying that p̊i ∣ (zʹ − qi), i ∈ {1, 2, ..., n}. Then, we set value z to $ z' \bmod \operatorname{lcm}(\mathring{P}^n) $. Through the divisibility theory, we have that $ \mathring{p}\_i \mid ((z' \bmod \text{lcm}(P^n)) -q\_i) $, that is, p̊i ∣ (z − qi), where i ∈ {1, 2, ..., n} and z ∈ [0, lcm(P̊n)). The existence of z ∈ [0, lcm(P̊n)) is proved. Moreover, to verify the uniqueness of the above value z, we assume that there exits two different z(1), z(2) ∈ [0, lcm(P̊n)). They both satisfy the conditions in proposition 2). That is, $$\mathring{p}\_i \| (z^{(1)} -q\_i),\ \mathring{p}\_i \| (z^{(2)} -q\_i),\ i \in \{1,2,...,n\}. \nonumber$$ By the divisibility theory, we deduce that p̊i ∣ (z(1) − z(2)), i ∈ {1, 2, ..., n}, which are further converted to lcm(Pn) ∣ (z(1) − z(2)). Then, owing to the definition z(1), z(2) ∈ [0, lcm(P̊n)), the relationship ∣z(1) − z(2)∣ < lcm(P̊n) is confirmed. Hence, there must be (z(1) − z(2)) = 0, that is, z(1) = z(2). It violates the above assumption. The uniqueness of z holds. [!t] $ \dot{q} \leftarrow 0 $ and $ \dot{p} \leftarrow 1 $ (qɛι, p̊) ← ϕɛι $ x, y, \gcd(\dot{p}, \mathring{p}) \leftarrow $ EEA($ \dot{p}, \mathring{p} $) $ (\dot{q}, \dot{p}) \leftarrow $ update $ \dot{q} $ and $ \dot{p} $ by (21) {fi} ← choose $ \phi^{\varepsilon\_\iota} \in \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ $ \dot{f}\_{\imath} \leftarrow $ get a sub-flow of fi sharing attributes except $ \dot{p}\_{\imath} \leftarrow \dot{p} $ and $ \dot{b}\_{\imath} \leftarrow \dot{q}-(h\_{i}^{\varepsilon\_\iota}+o\_i) $ $ \dot{\mathcal{F}} \vartriangleleft \dot{f}\_{\imath} $ Due to the periodicity of sequences, it is similarly deduced that the unique z exists in each interval [k ⋅ lcm(P̊n), (k + 1) ⋅ lcm(P̊n)), k ∈ Z. These z are periodically aligned with period lcm(P̊n) and form a sequence as z + k ⋅ lcm(P̊n). It formalizes the confluence slots $ \dot{s}^{\varepsilon\_\iota} $ of conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \dot{\Upsilon} $. Further, the specific z is deduced and flow portions are backtracked by the Euclidean-based method shown in Algo. 5. With the conflict cliques $ \dot{\Upsilon} $ traversal in lines 1-2, each of them $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ is positioned its confluence slots via the processes in lines 3-7. Specifically, the hyper-flows ϕɛι in clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ participate in confluences one by one. The initial confluence slots is formalized as $ \dot{q} + \dot{k} \cdot \dot{p} = 0 + \dot{k} \cdot 1 $ in line 3. With the confluence of ϕɛι in turn, the values of $ \dot{q} $ and $ \dot{p} $ are updated in lines 4-7. Within each turn, we aim to get the confluence slots $ \ddot{q} +\ddot{k} \cdot \ddot{p} $ of the existing $ \dot{q} + \dot{k} \cdot \dot{p} $ and newly joined ϕɛι with qiɛι + kiɛι ⋅ p̊i, which are expressed as $$\{ \ddot{q} +\ddot{k} \cdot \ddot{p} \mid \dot{q} + \dot{k} \cdot \dot{p} = q\_i^{\varepsilon\_\iota} + k\_i^{\varepsilon\_\iota} \cdot \mathring{p}\_i,\ \dot{k}, k\_i^{\varepsilon\_\iota} \in \mathbb{Z} \}. \nonumber$$ Based on the above inferences, period $ \ddot{p} $ is given as $ \operatorname{lcm}(\dot{p}, \mathring{p}\_i) $. Then, considering the uniqueness of $ \ddot{q} $ from Theorem 1, only a pair of specific x̃, ỹ satisfying $$\dot{q} +\tilde{x} \cdot \dot{p} = q\_i^{\varepsilon\_\iota} +\tilde{y} \cdot \mathring{p}\_i$$ is required to get the value $ \ddot{q} $ with $ (\dot{q} +\tilde{x} \cdot \dot{p}) \bmod \ddot{p} $. By the extended Euclidean algorithm (EEA), a pair of feasible x, y satisfying that $ x \cdot \dot{p} +y \cdot \mathring{p}\_i = \gcd(\dot{p}, \mathring{p}\_i) $ is given to convert (18) into $$\dfrac{\tilde{x} \cdot \dot{p} -\tilde{y} \cdot \mathring{p}\_i}{x \cdot \dot{p} + y \cdot \mathring{p}\_i} = \dfrac{q\_i^{\varepsilon\_\iota} -\dot{q}}{\gcd(\dot{p}, \mathring{p}\_i)}.$$ Further, the feasible x̃ and ỹ in (19) are obtained by enhancing the above equiproportional relation as $$\dfrac{\tilde{x} \cdot \dot{p}}{x \cdot \dot{p}} = \dfrac{-\tilde{y} \cdot \mathring{p}\_i}{y \cdot \mathring{p}\_i} = \dfrac{q\_i^{\varepsilon\_\iota} -\dot{q}}{\gcd(\dot{p}, \mathring{p}\_i)},$$ which gives the specific value x̃ as $ x \cdot (q\_i^{\varepsilon\_\iota} -\dot{q}) / \gcd(\dot{p}, \mathring{p}\_i) $ and ỹ satisfying (18), (19). Thus, the baseline $ \ddot{q} $ of confluence slots is obtained. It and period $ \ddot{p} $ together identify the confluence slots of the existing $ \dot{q} + \dot{k} \cdot \dot{p} $ and newly joined qiɛι + kiɛι ⋅ p̊i as the tuple $$(\ddot{q}, \ddot{p}) = ((\dot{q}\_i +\tilde{x} \cdot \dot{p}\_i) \bmod \ddot{p}, \operatorname{lcm}(\dot{p}, \mathring{p}\_i)).$$ With the above way performed cyclically in lines 6-7, the confluence slots $ \dot{s}^{\varepsilon\_\iota} $ of conflict clique $ \dot{\varpi}\_{\kappa}^{\varepsilon\_\iota} $ are precisely positioned without violently retrieving all occupied slots of flows. And then, we follow Theorem 2 to choose any one hyper-flow ϕɛι with flows fi in this conflict clique. The re-scheduled flow portions $ \{ f\_i^{\dot{\varpi}} \} $ are backtracked in line 8. Since they are the sub-sequences of flows fi, these flow portions are identified as sub-flows $ \dot{f}\_{\imath} \in \dot{\mathcal{F}} $ in lines 9-10, where each of them inherits the flow fi attributes except the period $ \dot{p}\_{\imath} $ and baseline time of generation $ \dot{b}\_{\imath} $. All backtracked sub-flows are aggregated as $ \dot{\mathcal{F}} $ without duplication in line 11. ### Low-cost sub-flow re-scheduling After backtracking the sub-flows $ \dot{\mathcal{F}} $, their fine-tuning is needed to avoid slot overflow and rectify the preliminary solution O as slightly as possible. Hence, we design a low-cost HFG method as a variant of Algo. 2 to re-schedule sub-flows $ \dot{\mathcal{F}} $. It follows the incremental and hyper-flow graph based pattern, but with a lower computation cost to reduce the complexity. As shown in Algo. 6, the sub-flows $ \dot{f}\_{\imath} \in \dot{\mathcal{F}} $ are re-scheduled one by one in line 1. Each of them is allocated with the feasible slots by the offset $ \dot{o} $ search in lines 3-7. Relative to the whole flow, the new offset of sub-flow $ \dot{f}\_{\imath} $ causes the flow jitter limited by $ \dot{j}\_{\imath} $ in jitter constraint (5). Hence, the offset search range is bounded from $ (o\_i-\dot{j}\_{\imath})^+ $ to bound $ \bar{o}\_{\imath} $ in line 2. For each offset $ \dot{o} $, its corresponding maximum slot occupancy ζ̄o is obtained in line 4 based on the hyper-flow graphs and maximal cliques Υ. The specific processes follow lines 6-16 of Algo. 2. To avoid slot overflow, the feasible offset is filtered with condition ζ̄o ≤ Λ. The first found among them is given as the optimal offset $ \dot{o}\_{\imath} $ in lines 5-6 to reduce scheduling complexity. If there is no feasible offset, the whole scheduling fails and stops in lines 8-9. Otherwise, the sub-flows with optimal offsets are stored as the supplemental solution $ \dot{\mathcal{O}} $ in line 10. The hyper-flows Φ, their weights L and maximal cliques Υ are updated via lines 24-29 of Algo. 2. With this low-cost method, slot overflow is avoided and then the schedulability and load-balancing are improved. At the end of GH2, the supplemental solution $ \dot{\mathcal{O}} $ and preliminary O are aggregated as a complete scheduling solution. [!t] $ \dot{o} \leftarrow (o\_i-\dot{j}\_{\imath})^+ $ and $ \bar{o}\_{\imath} \leftarrow \min \{ o\_i+\dot{j}\_{\imath}, \bar{o}\_i \} $ ζ̄o, {ϕɛι}, {Υ̃ϕɛι} ← follow line 6-16 in Algo. 2 $ \dot{o}\_{\imath} \leftarrow \dot{o} $ and break $ \dot{o} \leftarrow \dot{o} +1 $ Scheduling fails and break $ \dot{\mathcal{O}} \vartriangleleft (\dot{f}\_{\imath}, \dot{o}\_{\imath}) $ and update Φ, L and Υ following line 24-29 in Algo. 2 With the flow graph, the CCR re-scheduling methods also apply by only adjusting the process object from hyper-flow to flow nodes. [1] The flow fine-tuning phase is actually the decomposition and re-scheduling of flows resulting in slot overflow. For these sub-flows that fail in the above re-scheduling due to slot overflow, their further decomposition as a new re-scheduling round could better balance the load. Ignoring the constraints of latency and jitter, the CCR re-scheduling round by round tends to balance the slot loads ideally. [2] The scheduling procedure of GH2 is also suitable for online and dynamic network scenarios, where the time-sensitive applications or network structures evolve over time. Once that occurs, GH2 could cleanup silent application flows from previous scheduling solution O and slot occupancy Υ. Then, newly joined flows are scheduled by Algo. 1 to update the current solution. In this way, the on-the-fly network system is re-configured efficiently without affecting in-transit flows. [3] Simulation and Evaluation ========================= In this section, we conduct a comprehensive analysis of GH2 scheduling algorithm for its scalability in terms of schedulability, scheduling efficiency, real-time transmission ability and load balancing ability. Simulation Setup ---------------- All the simulations are run in the Intel (R) Eight-Core (TM) i7-9700HQ CPU with 3.0 GHz and 16 GB of RAM. ### Network Parameters Compared to CEV network with more switches, simulation networks are constructed with 8 TSN switches configured statically by the CQF model. It facilitates the network simulation with more intensive flow confluence to validate scheduling performances. The specific switch topologies are networked as the linear, ring and tree forms shown in Fig. [fig6], respectively. These topologies could assemble almost all structures of industrial networks. With the typical settings of 125 us slot length T, < 2 us synchronization errors δɛι, 1 Gbit/s link bandwidth Γ, 1 Mb queue depth $ \mho^{\varepsilon\_\iota} $ and 80% distribution factor γ, the available slot capacity Λ is given to 12.3 KB. [fig6] ### Flow Attributes Referring to the TSN profile standard for industrial automation, we generate simulation flow cases that close to actual industrial scenarios with standardized attributes illustrated in Table [tab2]. [!h] [simulation1] [!t] p70pt\|p90pt Attribute& Value range of flow cases a frame length li& 64 Bytes $ \thicksim $ 1500 Bytes flow period pi& Type 1: 1 ms $ \thicksim $ 200 ms Type 2: 1 ms $ \thicksim $ 400 ms Type 3: 1 ms $ \thicksim $ 600 ms Type 4: 1 ms $ \thicksim $ 800 ms basetime bi& 0 ms $ \thicksim $ pi ms allowable latency di& 0.1 ⋅ pi ms $ \thicksim $ 0.5 ⋅ pi ms allowable jitter ji& 500 us $ \thicksim $ 0.1 ⋅ pi ms hop count hi& 1 hop $ \thicksim $ 6 hops [tab2] Simulation Results ------------------ To obtain accurate and stable scheduling performances, we repeat 20 tests for each simulation with different flow cases. The final results are given by averaging. Except for specific parameter evaluations, the optimization weight factor ρ is set as 0.5 to balance the average real-time transmission rate and global maximum slot occupancy rate (abbreviated as real-time rate and slot occupancy rate later). The partition scale Ξ is set as 500 flows to balance the scheduling time and optimality. Moreover, the normal simulations adopt the linear topology and flow cases with 1ms $ \thicksim $ 200ms period interval. ### Sorting Strategy Comparison In incremental scheduling of the HFG method for each partition, different flow orders are compared to filter the optimal sorting strategy, including the random, down or up order by flow period, frame length and offset bound, respectively. As shown in Fig. [simulation1](a)&(b), the length descent sorting reflects the fastest scheduling efficiency and best optimization capability. Separately, both the real-time rate and slot occupancy rate of it rank at a relatively low echelon compared to the other sorting strategies. Thus, we adopt the down order of length as the optimal sorting strategy during the following simulations. ### Partition Strategy Comparison This simulation works to compare the different partition methods and filter the optimal one. The flow cases are partitioned by the flow period, frame length, bandwidth consumption illustrated in (9) and the SOTA randomly generating, spectral cluster, respectively. These flow partitions are scheduled via the first three phases of GH2. As reflected in Fig. [simulation1](c), the period-based flow partitioning shows relatively superior performances under different flow scales in terms of the time cost and optimal goal. Fig. [simulation1](d) shows that the period-based partitioning is superior at slot occupancy rate, while length-based partitioning benefits the real-time rate. Synthesizing the above results, we validate the benefits of attribute-driven flow partitioning and adopt the period-based strategy for GH2. ### Re-scheduling method Comparison In the flow fine-tuning phase, the systematic CCR re-scheduling method can be performed with the hyper-flow or flow graph based pattern. To evaluate their re-scheduling capability and then filter the optimal way, we decrease the available slot capacity Λ in gradient under 2000 flows. It starts from the slot occupancy rate after the first three phases for GH2. As shown in Fig. [simulation2], with the decreasing of slot capacity, the re-scheduling time grows due to the increasing conflict cliques and re-scheduled sub-flows, while the real-time rate remains almost invariant owing to precisely fine-tuning. Comparing the above two ways, we adopt the faster hyper-flow graph based CCR method for GH2. [simulation2] ### Algorithm Comparison Adopting the above strategies, our proposed GH2 algorithm is compared its scheduling performances with SOTAs as DASP, FITS, FGS, MSS for DSP. They pursue the same optimization goal (8). As shown in Fig. [simulation1](e)&(f), our proposed HFG method reflects its Pareto optimization including higher scheduling efficiency, more balanced slot loads and lower transmission latency, while its slot occupancy rate and real-time rate are equal to these of FITS and FGS since their same offset filtering condition. Furthermore, GH2 improves the scheduling efficiency and schedulability at different flow scales, especially in large-scale complex flow scenarios. Under 1000 flows, its runtime is reduced to nearly 1/300, 1/200 of the SOTA FITS, FGS algorithms, respectively. Under 2000 flows, the ratios are 1/430, 1/1000. Besides, when the flow scale grows to 2500, the SOTA algorithms are hard to get a feasible solution. GH2 improves 80% schedulability of FITS and FGS methods to 100%. Although the load balancing ability is sacrificed due to the separate partition scheduling that restricts the global optimality, GH2 can improve it with the CCR re-scheduling method, where the results are marked as GH2 \. In this way, GH2 constructs the Pareto front of scheduling efficiency and load balancing, and verifies its better scalability in various flow scenarios. Moreover, three scheduling patterns shown in Fig. [fig13] and adopted in FITS, FGS, HFG are compared for their computing complexity with the scheduling information scale (abbreviated as Sch-Info scale), respectively. Fig. [simulation3] reflects that the hyper-flow graph based and flow graph-based scheduling patterns compress the slot occupancy information of the frame-based pattern exponentially. It is because of the desensitization of flow attributes. Compared to the flow graph-based pattern, the hyper-flow way shrinks the graph scale significantly. These explain and confirm the scheduling efficiency improvement of our HFG and GH2 algorithms. Then, the scheduling efficiency stability of GH2 compared to SOTAs is evaluated under different flow scales. As shown in Fig. [simulation4], GH2 reflects the least time fluctuation and significant stability improvement. When more re-scheduling occurs as GH2 \, its stability is relatively reduced due to the scale uncertainty of re-scheduled sub-flows, but still superior to these SOTA algorithms. [simulation3] [simulation4] [!t] [simulation5] ### Partition Scale & Slot length Evaluation In GH2, the key parameters containing partition scale Ξ and slot length T are evaluated under 1000 flows for their influences on scheduling performances. From Fig. [simulation5](a)&(b), we conclude that both the increasing partition scale and slot length contribute to improving load balancing while raising the transmission latency. Compared to the partition scale, the slot length shows a greater impact. It allows them to work together for better performances. For the scheduling efficiency, it is influenced conjointly by these two parameters. With their synergizing, GH2 consistently schedules 1000 flows within 100 ms. ### Optimization Weight Factor Evaluation Just like above, we evaluate the influences of optimization weight factor ρ for GH2 at different flow scales. Besides different weight factors, another local optimization goal from is evaluated with the expression as $ \min\_{o\_{\epsilon}}{ \frac{\tilde{\zeta}\_{\epsilon}}{\Lambda} } $ and marked as Nob. ζ̃ε is the maximum occupancy among slots containing current scheduled flow fε and not applied for the early-break strategy. The simulation results are shown in Fig. [simulation5](c)&(d). As the weight factor grows, the scheduling time and real-time rate increase while the slot occupancy rate decreases. Hence, the factor ρ is conformed for its effect on balancing the above two optimization goals, which extend the Pareto front constructed by GH2. Specially, when flows reach a certain scale, slot overflow occurs and a relatively higher time cost is consumed by the re-scheduling. The smaller the factor ρ, the earlier this occurs. Moreover, the Nob-directed scheduling shows the highest time cost, real-time rate and lowest slot occupancy, which suits the scenarios with stronger load balancing preferences. ### Network Topology Influence To verify the universality of GH2, we evaluate its performances in linear, ring and tree topologies, respectively, where different simulation flow cases share the same attributes except their topologies. As shown in Fig. [simulation5](e)&(f), GH2 manifests the low scheduling complexity and superior real-time transmission & load balancing abilities in all these topologies. Among them, the ring topology shows better performances since its relatively short route from the source to the destination host. ### Period Attribute Influence Considering the period influence discussed in Section III, we evaluate the scheduling stability of GH2 under different flow period intervals. Four types of flow cases as shown in Table [tab2] are scheduled at different flow scales. Fig. [simulation6](a)&(b) reflect that all of them have low scheduling runtime and similar optimization performances, where the extension of period interval somewhat raises the scheduling efficiency and improves both the real-time transmission and load balancing abilities. It further confirms the scalability for various flow scenarios and desensitization for flow attributes of GH2. [simulation6] Conclusion ========== In this paper, we investigate the scalable scheduling for the CQF-based DSP with large-scale complex industrial flows. Considering its high complexity and limited optimality, we deeply mine the attribute-driven scheduling features and develop a hyper-flow graph based scheduling scheme. The hyper-flow graph is built by taking the similar flow sets as the hyper-flow nodes and proved its equivalence with flow attribute-sensitive scheduling and re-scheduling information. It embeds the redundant scheduling information in less maximal cliques, and precisely reverse maps them to overflow flow portions. Under the systematical guidance of the above methodology, a “flow division-conquering-synthesis-fine tuning” hierarchical framework is built to optimize and balance the scheduling complexity and optimality. Its parallel scheduling reduces the device and flow scale induced complexity, while the precise fine-tuning improves the schedulability and load balancing. Further, this scheme is refined as a comprehensive scheduling algorithm GH2, including the attribute-driven lightweight partitioning, parallel HFG scheduling, parallel Bron-Kerbosch synthesizing and precise CCR re-scheduling methods. Simulation results demonstrate the superiority of GH2 in terms of the scheduling efficiency, schedulability and QoS performances along a Pareto front. This scheme can also be extended into the online and dynamic network or other time-related scheduling scenarios. Scalable Scheduling for Industrial Time-Sensitive Networking: A Hyper-flow Graph Based Scheme ============================================================================================= Industrial Time-Sensitive Networking (TSN) provides deterministic mechanisms for real-time and reliable flow transmission. Increasing attention has been paid to efficient scheduling for time-sensitive flows with stringent requirements such as ultra-low latency and jitter. In TSN, the fine-grained traffic shaping protocol, cyclic queuing and forwarding (CQF), eliminates uncertain delay and frame loss by cyclic traffic forwarding and queuing. However, it inevitably causes high scheduling complexity. Moreover, complexity is quite sensitive to flow attributes and network scale. The problem stems in part from the lack of an attribute mining mechanism in existing frame-based scheduling. For time-critical industrial networks with large-scale complex flows, a so-called hyper-flow graph based scheduling scheme is proposed to improve the scheduling scalability in terms of schedulability, scheduling efficiency and latency & jitter. The hyper-flow graph is built by aggregating similar flow sets as hyper-flow nodes and designing a hierarchical scheduling framework. The flow attribute-sensitive scheduling information is embedded into the condensed maximal cliques, and reverse maps them precisely to congestion flow portions for re-scheduling. Its parallel scheduling reduces network scale induced complexity. Further, this scheme is designed in its entirety as a comprehensive scheduling algorithm GH2. It improves the three criteria of scalability along a Pareto front. Extensive simulation studies demonstrate its superiority. Notably, GH2 is verified its scheduling stability with a runtime of less than 100 ms for 1000 flows and near 1/430 of the SOTA FITS method for 2000 flows. Industrial Time-Sensitive Networking, scalable scheduling, hyper-flow graph, hierarchical framework Introduction ============ a basic infrastructure of the Industrial Internet of Things, device networking in factory automation is incorporating more and more sensors, controllers and actuators to build ubiquitous networked cyber-physical systems. Massive network applications require ultra-low latency and jitter, the most critical quality of service (QoS) metrics. For example, industrial automation and power grid systems require tight deterministic latency in the range of milliseconds to microseconds. The jitter is also bounded within a few microseconds. These drive the industrial network evolution to ultra-low latency and ultra-high reliability. Time-Sensitive Networking (TSN), an Ethernet extension under development from IEEE 802.1 TSN Task Group, empowers Ethernet bridge networks determinism by a capability collection governed via a series of standards. It integrates mechanisms such as traffic shaping, bandwidth policing and clock synchronization to control flow forwarding in a precise manner with bounded low latency & jitter and extremely low frame loss. Specifically, IEEE 802.1Qbv builds a gate-operation mechanism named time-aware shaper during the forwarding process. Inside the output port of TSN supported network devices, this shaper works to manage traffic forwarding queues and time through a gate control list (GCL). It records sequential gate states and state durations. Following this, TSN devices transmit flows in queues orderly and cyclically. Similarly, IEEE 802.1Qci constructs an en-queue policing mechanism for flow caching inside every TSN device. The flow forwarding between these devices is synchronized with IEEE 802.1AS. Moreover, a ping-pong gate configuration model, called cyclic queuing and forwarding (CQF), is given as IEEE 802.1Qch to implement the transmission determinism with these mechanisms. Nevertheless, further flow scheduling is desired for the real-time transmission of massive flows with ultra-low latency and jitter limits. It is considered as the deterministic scheduling problem (DSP). Since the advent of TSN, related research for DSP has been widely conducted and discussed. This problem is first formalized with systematic constraints & goals and proved to be NP-hard. It is solved by Satisfiability Model Theory or Integer Linear Programming, which show a high complexity and are hardly adaptable to large-scale complex industrial flow scenarios. Thus, the rapid schedulability assessment, and even further, the efficient determinism scheme design is desired to achieve the scalable scheduling capability. With such a vision, the flow incremental or partitioning scheduling methods are proposed in works. Typically, Quan et al. designs a flow injection-time scheduling (FITS) method that allocates flows incrementally with reasonable slot resources by injection offset. Guo et al. co-designs the flow incremental ordering and injection offset selection by a mapping score metric-based scheduling (MSS) method. In addition, Atallah et al. proposes an iterated flow partition scheduling method, where a degree of conflict-aware stream partitioning (DASP) strategy is adopted to optimize scheduling partitioning. Compared to global scheduling, these above ways segment flows into serial and interrelated scheduling blocks to downsize the network scale of each scheduling step. It facilitates the balance between scheduling optimality and efficiency. However, the complexity of these serial patterns is still high and difficult to improve. And even more so, their frame-based patterns during each block scheduling lack in attribute mining mechanism that inevitably possesses flow attribute induced high complexity. To further cope with the complexity dilemma, the correlation between TSN mechanisms and flow attributes is deeply analyzed to optimize the key scheduling processes, like constraint satisfiability and goal optimality checking. A divisibility theory-based flow sequence analysis method is developed to metric the scheduling space and elaborate flow confluences on time slots under the CQF model. Its building flow graph converts the frame-based scheduling into the equivalent flow graph-based pattern for efficient scheduling. Even so, its flow scale induced exponential graph complexity restricts the scheduling efficiency. Therefore, the current DSP is desired to schedule with better scalability, which means the faster scheduling with superior schedulability and latency & jitter performances. This paper focuses on improving the scheduling efficiency and scalability under the QoS and determinism assurances for large-scale complex flow scenarios. With the CQF-configured network, we holistically consider the scheduling complexity and optimality and deeply mine flow attribute-driven scheduling features. A hyper-flow graph based scheme is developed to innovate traditional serial and frame-based scheduling patterns. With this scheme, the critical causes of scheduling complexity, network scale (including device and flow scale) and flow attributes, are significantly suppressed in their impact on scheduling. Meanwhile, the schedulability and QoS performances are well-guaranteed and even enhanced. The contributions of the scheme are summarized as follows: * A hyper-flow graph based methodology is proposed to design and optimize the flow scheduling processes systematically. Instead of traditional frame-based patterns, the attribute-driven hyper-flow graph maps/embeds their flow attribute-sensitive redundant scheduling information into equivalent and less maximal hyper-flow cliques. Also, the re-scheduling information is precisely reverse mapped without violent retrieve by introducing the conflict clique. These ways suppress flow attribute induced complexity and drive the Pareto-optimized scheduling. * A hierarchical scheduling framework is constructed to balance scheduling efficiency and optimality for large-scale complex flows. With the unified design by hyper-flow graphs, four progressive phases are closely interlinked, including lightweight flow partitioning, parallel partition scheduling, parallel flow synthesizing and precise flow re-scheduling. These two parallel phases reduce the flow and device scale induced scheduling complexity, respectively, and the last phase improves the schedulability and optimality with precise flow fine-tuning. * A hyper-flow graph based hierarchical (GH2) algorithm designs the scheduling scheme in its entirety. Guided by the graph, an attribute-driven partitioning strategy, a parallel hyper-flow graph based scheduling (HFG) method, a parallel Bron-Kerbosch synthesizing method and a conflict clique based re-scheduling (CCR) method are designed comprehensively and interlinked hierarchically. It improves the schedulability, scheduling efficiency and QoS performances along a Pareto front. The rest of this paper is organized as follows. First, Section II establishes the CQF-configured network model and formulates its corresponding scheduling problem. Then, the hyper-flow graph based scheme is illustrated in Section III, including the hyper-flow graph based methodology, hierarchical framework and comprehensive flow scheduling algorithm. Section IV evaluates the performances of GH2, and Section V makes the final conclusion. Network Model and Problem Formulation ===================================== Network and Flow Model ---------------------- For the target network with the determinism demands, we model it as a directed graph N = < V,E > , where V denotes a set of network nodes νκ, consisting of terminal hosts Vh and TSN switches Vs. E denotes the directed physical links ɛι. In this network, the time-sensitive applications communicate through information flows F. Each of them fi is characterized by the attribute tuple {li, pi, bi, di, ji, Ri}. These denote the frame length, flow period, baseline time of generation, maximum allowable latency, maximum allowable jitter and transmission route, respectively. The flow route Ri is predefined as the following form of link association: $$R\_i = [ \varepsilon\_{i, 0}, \varepsilon\_{i, 1} \cdots, \varepsilon\_{i, h\_i} ], \nonumber$$ which connects the links between the source and destination host nodes with the hop count hi. [CQF] [CQF2] To guarantee the deterministic transmission of target flows, that is, the bounded latency and jitter, all the output ports of TSN switches are implemented with two statically configured Ping-Pong queues from IEEE 802.1Qch, called the CQF model. As shown in Fig. [CQF], it performs en-queue and de-queue operations alternately with a predetermined duration T. The transmission and reception control between both nodes of link ɛι are precisely aligned based on IEEE 802.1AS. In this way, the transmission process is divided into numbered time slots s ∈ S with length T, and the target flows are transmitted and queued along their route Ri slot after slot like Fig. [CQF2]. The end-to-end determinism of flow transmission is assured with the bounded latency [hi ⋅ T − T, hi ⋅ T + T] and jitter [0, 2 ⋅ T]. Considering the slotted transmission behavior, partial flow attributes are converted into the slot expression as p̊i, b̊i, d̊i, j̊i, given by $ \frac{p\_i}{T} $, $ \lfloor \frac{b\_i}{T} \rfloor $, $ \lfloor \frac{d\_i}{T} \rfloor $, $ \lfloor \frac{j\_i}{T} \rfloor $, respectively. [injection] As analyzed in, unscheduled flow injection into the network is easy to cause chaotic flow confluences on the same slot. It further results in slot congestion and overflow due to excessive occupancy beyond the slot capacity. Further, the potential frame loss occurs and the deterministic mechanism is broken. To avoid this, the injection delay oi is introduced to regulate the postponement between flow generation and injection. As shown in Fig. refinjection, the offset design of flow injection allows slot resources to be allocated to target flows without overflow, which facilitates flow scheduling more flexibly and effectively. It is worth noting that the injection offset is symbolized as oi under a general assumption that all frames of each flow fi share the same delay. This assumption reduces the variable scale to boost the scheduling efficiency and eliminates the additional flow jitter to improve QoS performances, but it is not necessary to obey. Hyper-flow Graph Model ---------------------- Given the generation periodicity and transmission determinism, target flows are kept forwarding periodically on every link by the uniform injection offset among each flow. Based on this, we cluster the frames of each flow with a unified forwarding slot representation. For any flow fi ∈ F on link ɛι ∈ Ri, its frame forwarding slots are modeled as a periodic sequence $$\aleph(k\_i^{\varepsilon\_\iota}) = q\_i^{\varepsilon\_\iota} +k\_i^{\varepsilon\_\iota} \cdot \mathring{p\_i},$$ where the baseline forwarding slot qiɛι is given as $ \mathring{b\_i} +o\_i +h\_{i}^{\varepsilon\_\iota} $. hiɛι is the passed hop number among Ri, that is, the passed slots after flow injecting. The sequence count ki accumulates every periodic frame. The flow attribute integrated slot positioning parameters $ (q\_i^{\varepsilon\_\iota}, \mathring{p\_i}) $ are called as the feature tuple of flow fi. [graph] In order to efficiently guide flow offset selection, we build a correlation graph on each link ɛι ∈ E to characterize flow confluence relationships. Considering the specific forwarding slots of target flows are dominated by their feature tuples, we aggregate the flow set with the same $ (q\_i^{\varepsilon\_\iota}, \mathring{p\_i}) $ as a hyper-flow ϕ and weight it the frame length sum among the flow set. Then, we take the hyper-flows Φɛι as graphical nodes and connect their undirected edges following the adjacency matrix Ψɛι as $$\Psi\_{\phi\_{\alpha}, \phi\_{\beta}}^{\varepsilon\_\iota} = \begin{cases} 1 & \text{if\ } \gcd(\mathring{p}\_{\alpha}, \mathring{p}\_{\beta}) \| (q\_{\alpha}^{\varepsilon\_\iota} -q\_{\beta}^{\varepsilon\_\iota}). \\ 0 & \text{otherwise.} \end{cases}$$ Like Fig. [graph], the flow aggregated hyper-flows are correlated by the relationship between their forwarding slot parameters. The hyper-flow graph is desired to elaborate the flow confluences on slots and efficiently extract the key scheduling information. Problem Formulation for CQF-configured network ---------------------------------------------- With the above models, we construct the optimal scheduling problem for the CQF-configured network with variable flow injection offset. To meet flow QoS demands and network determinism requirements, the CQF-based DSP is demonstrated by the following constraints and goal to regulate the scheduling selection of flow offsets. ### Latency Constraint In the CQF model, flow transmission latency is locked as (1) within the switching network. Affected by the injection offset oi, the worst-case latency ϑi is extended to (oi + hi + 1) ⋅ T. In order to bound oi into a QoS allowed range, we construct the latency constraint as $$\begin{aligned} & \forall f\_i \in \mathcal{F}: \\ & 0 \leq o\_i < \mathring{d}\_i -h\_i. \end{aligned}$$ ### Jitter Constraint Besides the latency, the flow jitter also needs to be within a tolerable limitation j̊i as follows. It is produced by the possible offset fluctuation $ \tilde{o\_i} $ within a flow and the endogenous jitter 2 ⋅ T from the CQF model. $$\begin{aligned} & \forall f\_i \in \mathcal{F}: \\ & \tilde{o\_i} + 2 \leq \mathring{j}\_i. \end{aligned}$$ ### Capacity Constraint During the flow forwarding on the common links, there exit potential flow confluences sharing the same slot sɛι. Constrained by the limited slot capacity Λ, the target flows need to be allocated reasonable slots without overflow. Thus, the constraint is demonstrated as: $$\Delta\_{i,s}^{\varepsilon\_\iota} = \begin{cases} 1 \leftarrow s^{\varepsilon\_\iota} \equiv q\_i^{\varepsilon\_\iota} \bmod \mathring{p}\_i, \\ 0 \leftarrow \text{otherwise}, \end{cases}$$ $$\begin{aligned} & \forall \varepsilon\_\iota \in \mathcal{E}, s^{\varepsilon\_\iota} \in \{1,2,\cdots,C\}: \\ & \sum\_{f\_i \in \mathcal{F}^{\varepsilon\_\iota}} \Delta\_{i,s}^{\varepsilon\_\iota} \cdot l\_i \leq \Lambda, \end{aligned}$$ where the boolean variable Δi, sɛι in (6) works to verify whether flow fi is forwarded at slot sɛι of link ɛι. It is confirmed by the congruence of checked slot sɛι and flow forwarding basetime qiɛι in relation to period p̊i. The constraint in (7) restricts the slot occupancy filled by flow confluences on each slot sɛι ∈ S to no more than the available slot capacity Λ. It is given as $ \gamma \cdot \min \{(T -\delta^{\varepsilon\_\iota}) \cdot \Gamma, \mho^{\varepsilon\_\iota}\} $, where δɛι is the slot misalignment owing to the clock synchronization errors, Γ is the link bandwidth, $ \mho^{\varepsilon\_\iota} $ is the physical queue depth, and γ is the capacity distribution factor used to reserve resources for deterministic transmission of aperiodic time-sensitive flows. Moreover, the capacity constraint covers slots S within a bounded scheduling view C given as the least common multiple of all flow periods, that is, lcm(P̊n), where P̊n is {p̊1, p̊2, ⋯, p̊n}. With this range, the global determinism of flow scheduling is assured by cyclically repeating the flow forwarding in these C slots. [!ht] [method] Among the feasible scheduling space for the above constraints (4)-(7), we aim to enhance the real-time transmission ability of flows F and the load balancing ability of slots S. The former improves latency performances, and the latter helps cope with burst flows and schedulability. Thus, a composite optimization goal is expressed as $$\min\_{\{o\_i\}}{ (1-\rho) \cdot \frac{1}{n} \sum\_{f\_i}{\frac{\vartheta\_i}{\mathring{d}\_i}} + \rho \cdot \left[ \frac{\zeta\_s^{\varepsilon\_\iota}}{\Lambda} \right]^{\max\_{s, \varepsilon\_\iota}} },$$ where the optimization value symbolized as ℑ is the combined weight of the former transmission average real-time rate ϑi/d̊i in target flows and the latter maximum slot occupancy rate ζsɛι/Λ among all slots on links E. ζsɛι is denoted as the slot occupancy of slot sɛι on link ɛι, and ρ ∈ [0, 1] is the weight factor working to balance these two goals. The complexity of general constraint-based optimization problems relies on intensive computation for constraint satisfiability and goal optimality. Considering the formulation of the above CQF-based DSP, the scale of scheduling variables, constraints and goal components expands with the growing network scale including device and flow scale. It causes an increasingly high scheduling complexity. So do the complicating flow attributes, especially the flow periods. According to the TSN profile standard for industrial automation, the flow periods of time-sensitive network systems are complex with large range spans. It inflates the scheduling view C to increase the scheduling complexity. So far, these complexity dilemmas still need to be properly addressed. Thus, we work on exploring a systematic scheme to solve these problems uniformly while well-balancing the scheduling optimality (8), that is, the proposed hyper-flow graph based scheme. Hyper-flow Graph Based Scheme ============================= In this section, we elaborate on the hyper-flow graph based scheme including the hyper-flow graph based methodology and four-phase hierarchical framework. It is graphically depicted as Fig. [method]. A comprehensive scheduling algorithm GH2 designs the scheme in its entirety for scalable flow scheduling in CQF-configured TSN. Thus, we build the scheme following the GH2 as Algo. 1 hierarchically. [!t] $ \{ \mathcal{F}\_{\varsigma} \} \leftarrow $ Attribute-driven flow partitioning for F O, Φ, L ← Parallel HFG scheduling($ \{ \mathcal{F}\_{\varsigma} \}, \mathcal{A} $) Υ ← Parallel Bron-Kerbosch synthesizing(Φ, L) $ \dot{\mathcal{O}} \leftarrow $ Precise CCR re-scheduling(Υ, Λ) $ \mathcal{O} \leftarrow \mathcal{O} \cup \dot{\mathcal{O}} $ Given the complexity analysis of the above DSP, we first explore a divide-and-conquer hierarchical design to reduce the network scale induced complexity. Specifically, the first flow division phase divides target flows F into separate partitions $ \mathcal{F}\_{\varsigma} $ in line 1, where an attribute A-driven lightweight strategy is designed to accelerate flow partitioning. For more than one flow partition, the second partition conquering phase spreads them over multiple parallel scheduling sub-problems to reduce the flow scale induced complexity. With the flow scheduling of optimized slot allocations, every partition obtains a local flow offset solution that is aggregated as a preliminary scheduling solution O in line 3. Considering the partitions share a common network, the global scheduling information, that is, the slot occupancy by flow confluences, needs to be synthesized from these sub-problems. The third flow synthesis phase works on it link-parallelly in line 4 to suppress the device scale induced complexity. Unusually, when only one partition exists, the synthesis phase is skipped to line 6. For the specific scheduling and synthesizing processes, the hyper-flow graph based methodology is explored by building a hyper-flows Φ and weights L based graph. It maps/embeds one-by-one slot occupancy within flow attribute-sensitive bound C into equivalent and less maximal hyper-flow cliques Υ. With the condensed scheduling information, a parallel HFG flow scheduling method in line 3&6 and a parallel Bron-Kerbosch flow synthesizing method in line 4 are proposed to improve the scheduling efficiency further. By these three phases, GH2 fast-tracks a preliminary scheduling solution and its global maximal hyper-flow cliques under the optimized QoS performances. After the flow synthesis of scheduling information, there may be some overflow slots where their occupancy exceeds the slot capacity Λ or other desired values. It violates the capacity constraint (7) and affects the scheduling optimality (8). Hence, the last flow fine-tuning phase in line 7 is desired to re-schedule the overflow flows inside these slots. Meanwhile, the above preliminary solution is rectified as slightly as possible to maintain QoS performances. For this vision, the assumption that all frames of each flow share the same delay is broken. We aim to precisely position the overflow slots and re-scheduling overflow flow portions inside them. By extending the hyper-flow graph based methodology, the conflict clique is first introduced and proved equivalent to overflow slots. Then, the re-scheduling information, that is, the overflow slots and their flow portions, is precisely reverse mapped from the maximal hyper-flow cliques without violent retrieve. It drives an efficient CCR re-scheduling method in line 7 to precisely fine-tune flows for schedulability improvement and load-balancing. After this phase, the supplement solution $ \dot{\mathcal{O}} $ is given and constitutes the complete scheduling solution together with O in line 8. The specific processes of the above hyper-flow graph based scheme, that is, GH2 scheduling algorithm, are demonstrated in phases as follows. Lightweight Flow Division Phase ------------------------------- For the DSP with large-scale complex flows, we divide the target flows F into multiple partitions $ \mathcal{F}\_{\varsigma} $ for the subsequent separate scheduling. Distinguishing from SOTA partitioning strategies, like spectral clustering that constructs a similarity graph to cluster flows, we design a lightweight strategy to reduce time costs and maintain scheduling performances. It partitions flows via their sorted typical attitudes. Considering that load balancing for every partition scheduling sub-problem helps to reduce the slot overflow, we view this as the partition principle of the CQF-based DSP. Then, an intuitive law is observed that the flows with similar attributes facilitate the balance of slot occupancy. For instance, the flows with similar even the same flow period pi could be evenly distributed over different slots with fewer flow confluences. The similar frame length l̊i could bring approximate occupancy to balance the slots. Thus, we divide flows based on the similarity of flow attributes as follows. $$\begin{aligned} & \Theta \leftarrow \{ f\_{\alpha}, f\_{\beta}, \cdots \in \mathcal{F} \mid f\_{\alpha} \prec f\_{\beta} \prec \cdots \}, \\ & \{ \mathcal{F}\_1, \mathcal{F}\_2,..., \mathcal{F}\_{\varsigma},... \} \leftarrow \text{sequentially slice } \Theta, \end{aligned}$$ where we refer flow attitude li, pi or the bandwidth consumption depicted by li/pi as a potential partitioning bases. Target flows are first sorted by one of the above basis and then batch sliced into one-by-one partition $ \mathcal{F}\_{\varsigma} $ with a specified scale Ξ. Parallel Partition Conquering Phase ----------------------------------- For the above flow partitions, we schedule them as separate sub-problems to cut their interlink. This way reduces the flow scale in each sub-problem and the scheduling complexity. Due to the separability, a parallel computing structure is adopted to further reduce the complexity compared to the serial patterns. Each conquering unit corresponds to one flow partition and deploys the HFG scheduling method. [!t] $ \Theta\_{\varsigma} \leftarrow \{ f\_{\alpha}, f\_{\beta}, \cdots \in \mathcal{F}\_{\varsigma} \mid f\_{\alpha} \prec f\_{\beta} \prec \cdots \} $ o ← 0, ōε ← min{pε, dε − hε} and ℑε ← ∞ ϕɛι ← get $ (\mathring{p\_{\epsilon}}, q\_{\epsilon}^{\varepsilon\_\iota}) $ by (2) $ \zeta\_{\varpi}^{\varepsilon\_\iota} \leftarrow l\_{\epsilon} + \sum\_{\phi \in \varpi\_{\kappa}^{\varepsilon\_\iota}} \ell\_{\phi}^{\varepsilon\_\iota} $ $ \bar{\zeta}\_o \leftarrow \max{\{ \max{\{ \zeta\_{\varpi}^{\varepsilon\_\iota} \}, \bar{\zeta}} \} }$ ℑo ← get the optimization value by (11) oε ← o, ζ̄ε ← ζ̄o and ℑε ← ℑo {ϕɛι}ε ← {ϕɛι} and {Υ̃ϕɛι}ε ← {Υ̃ϕɛι} break o ← o + 1 $ \mathcal{O}\_{\varsigma} \vartriangleleft (f\_{\epsilon}, o\_{\epsilon}) $ and ζ̄ ← max{ζ̄ε, ζ̄} ℓϕɛι ← ℓϕɛι + lε $ \mathcal{O} \vartriangleleft \mathcal{O}\_{\varsigma} $ Before the scheduling process, the slot expressions of flow attributes are given with predetermined slot length T satisfying conditions T ∣ gcd(Pn), j̊i ≥ 2 and d̊i − hi > 0. They participate in the parallel HFG scheduling method shown as Algo. 2, which distributes flow partitions to their respective conquering units in line 1. For every partition, the corresponding flows are scheduled incrementally to get their optimal injection offset in lines 2-29, which reduces device nodes involved in each scheduling to their own routes. In line 2, flows are first sorted to determine the scheduling order as $ \Theta\_{\varsigma} $. During each flow fε scheduling, its feasible offset o is traversed below the bound ōε given in line 4. It is the smaller of flow period pε and latency constraint (4). The optimal offset oε is filtered with the greedy policy preferring a lower transmission real-time rate and slot occupancy rate expressed as $$\min\_{o\_{\epsilon}}{ (1-\rho) \cdot \frac{1}{n\_{\epsilon}} \sum\_{f\_i}{\frac{\vartheta\_i}{\mathring{d}\_i}} + \rho \cdot \frac{\bar{\zeta}\_{\epsilon}}{\Lambda} }.$$ It is designed to match the global optimization goal (8), where nε counts the flows that have been scheduled in the current partition and ζ̄ε is the global maximum slot occupancy among all slots S of the whole network in fε scheduling. Further, the constant part in (10) during offset searching of each flow fε is removed to simplify the local goal as $$\min\_{o\_{\epsilon}}{ (1-\rho) \cdot \frac{o\_{\epsilon}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} + \rho \cdot \frac{\bar{\zeta}\_{\epsilon}}{\Lambda} }.$$ This optimal value is symbolized as ℑε and obtained in lines 5-23. Specifically, with increasing offset o in lines 5&23, its corresponding optimization value ℑo is given in lines 6-17 and filtered in lines 18-22. In this key process, the critical cause of scheduling complexity is obtaining maximum slot occupancy ζ̄o under each offset and recurring offset searching. We first discuss the resolutions for the former as follows. Since retrieving the slot occupancy slot-by-slot within the bound C is computationally complex, the complexity is quite sensitive to flow periods that determine the bound. It exponentially expands as the multiplicity of periods. With the built hyper-flow graph (3) on each link ɛι ∈ E, we prove the equivalence between hyper-flow cliques and flow confluences on any slot under the support of following Lemma. For n hyper-flow sequences formed as ℵ(ki) = qi + ki ⋅ p̊i, if any sequence pair ℵ(kα) and ℵ(kβ) satisfies that gcd(p̊α, p̊β)∣(qα − qβ), there exist z ∈ Z satisfying p̊i∣(z − qi), i ∈ {1, 2, ..., n}, and vice versa. [1] It means that each hyper-flow clique indicates the flow confluences on a certain slot sɛι, and the flow confluences on every slot sɛι are carried by a hyper-flow clique. That is, the occupancy of any slot is given by the combined weight of its corresponding clique. Thus, the maximum slot occupancy ζ̄o is equal to the maximum combined weight among all hyper-flow cliques. The obtaining of maximum slot occupancy ζ̄o is converted to a maximal clique enumeration problem (MCEP) as shown in Fig. [fig13](a). Compared to violently retrieve, the hyper-flow graph based pattern maps/embeds the redundant slot occupancy information into less maximal flow cliques Υɛι. Essentially, it merges the identical or contained flow confluences on different slots. [fig13] In this way, we design a dynamic graph and clique update method to obtain the maximum slot occupancy ζ̄o under offset o of flow fε for incremental scheduling. As shown in lines 6-16 of Algo. 2, for each link ɛι along route Rε, the feature tuple of flow fε is extracted in line 7 and checked its existence in the hyper-flow nodes $ \Phi\_{\varsigma}^{\varepsilon\_\iota} $. It covers hyper-flows that have been scheduled on link ɛι. If it exists, the weight ℓϕɛι of corresponding ϕɛι increases by lε (without real update in the pseudo-code during the offset search), and so do all combined weights $ \zeta\_{\varpi}^{\varepsilon\_\iota} $ of maximal cliques $ \varpi\_{\kappa}^{\varepsilon\_\iota} \in \Upsilon\_{\phi}^{\varepsilon\_\iota} $ that contains ϕɛι among the maximal hyper-flow clique set $ \Upsilon\_{\varsigma}^{\varepsilon\_\iota} $ in lines 9-10. They imply the potential maximum occupancy of the slots containing fε. If not, the tuple extends the graph as a new hyper-flow ϕɛι. Its neighbors $ \varkappa\_{\phi}^{\varepsilon\_\iota} $ are given by (3) in line 12, where the neighbors mean its connected nodes by (3) in the graph. The potential maximum slot occupancy containing fε is obtained based on the following theorem. In the graph with maximal cliques $ \varpi\_{\kappa} \in \Upsilon $, if a new node ϕ joins with the neighbors $ \varkappa\_{\phi} $, the new maximal cliques $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $ compared to Υ are included in the clique set $$\tilde{\Upsilon}\_{\phi} = \{ \tilde{\varpi}\_{\kappa} \mid \tilde{\varpi}\_{\kappa} = (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \},\ \varpi\_{\kappa} \in \Upsilon\_{\varkappa} \},$$ where $ \Upsilon\_{\varkappa} $ is $ \{ \varpi\_{\kappa} \in \Upsilon \mid (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \neq \varnothing \} $ as the subset of Υ. Specially, it is modified to set $ \{ \varnothing \} $ when empty. Furthermore, for any clique $ \tilde{\varpi}\_{\kappa} \in \tilde{\Upsilon}\_{\phi} - \breve{\Upsilon} $, there exits a new maximal clique $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $ satisfying that $ \tilde{\varpi}\_{\kappa} \subseteq \breve{\varpi}\_{\kappa} $. [1] It is deduced and verified step by step as follows. Considering that $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \subseteq \varkappa\_{\phi} $, any element $ \phi\_j \in (\varpi\_{\kappa} \cap \varkappa\_{\phi}) $ is connected with ϕ. Moreover, since the subset $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) $ of clique $ \varpi\_{\kappa} \in \Upsilon $ must be a clique, we prove that $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $ is a clique and Υ̃ϕ is the clique set. For any new maximal clique $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $, it is easy to deduced that $ \phi \in \breve{\varpi}\_{\kappa} $. Otherwise, $ \breve{\varpi}\_{\kappa} $ is constituted by the original nodes of the graph and $ \breve{\varpi}\_{\kappa} \in \Upsilon $, which violates the definition of $ \breve{\varpi}\_{\kappa} $. Considering the full connection feature between nodes of the clique, the remaining elements of $ \breve{\varpi}\_{\kappa} $ except ϕ only exist in its neighbors, that is, $ \breve{\varpi}\_{\kappa} \subseteq (\varkappa\_{\phi} \cup \{ \phi \}) $. In addition, the fully connected remaining elements must be included in a certain maximal clique $ \varpi\_{\kappa} \in \Upsilon $, that is, $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cup \{ \phi \}) $. Hence, there exists $ \varpi\_{\kappa} \in \Upsilon $ satisfying that $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cup \{ \phi \}) \cap (\varkappa\_{\phi} \cup \{ \phi \}) $, that is, $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $. At this time, the maximal clique $ \varpi\_{\kappa} \in \Upsilon\_{\varkappa} $ also exists if the above existing $ \varpi\_{\kappa} $ satisfies the condition $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \neq \varnothing $. Otherwise, all these existing $ \varpi\_{\kappa} $ satisfies that $ (\varpi\_{\kappa} \cap \varkappa\_{\phi}) = \varnothing $ and $ \breve{\varpi}\_{\kappa} \subseteq \{ \phi \} $, which means the neighbors $ \varkappa\_{\phi} $ are the empty set $ \varnothing $. Thus, $ \Upsilon\_{\varkappa} $ is the set $ \{ \varnothing \} $. There exists $ \varpi\_{\kappa} = \varnothing $ satisfying that $ \breve{\varpi}\_{\kappa} \subseteq (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $. Since the maximal clique cannot be the proper subset of any other clique, for any $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $, there exists $ \varpi\_{\kappa} \in \Upsilon\_{\varkappa} $ satisfying the condition $ \breve{\varpi}\_{\kappa} = (\varpi\_{\kappa} \cap \varkappa\_{\phi}) \cup \{ \phi \} $. Hence, Ῠ ⊆ Υ̃ϕ holds. Similarly, for any clique $ \tilde{\varpi}\_{\kappa} \in \tilde{\Upsilon}\_{\phi} - \breve{\Upsilon} $, since node $ \phi \in \tilde{\varpi}\_{\kappa} $, we have that $ \phi \notin \varpi\_{\kappa} $, $ \forall \varpi\_{\kappa} \in \Upsilon $. Thus, there must exists the new maximal clique $ \breve{\varpi}\_{\kappa} \in \breve{\Upsilon} $ satisfying the condition $ \tilde{\varpi}\_{\kappa} \subseteq \breve{\varpi}\_{\kappa} $. Otherwise, $ \tilde{\varpi}\_{\kappa} $ is verified as a new maximal clique, which conflicts with $ \tilde{\varpi}\_{\kappa} \in \tilde{\Upsilon}\_{\phi} - \breve{\Upsilon} $. According to this theorem, the maximum occupancy of the slots containing fε is implied in the combined weights $ \zeta\_{\varpi}^{\varepsilon\_\iota} $ of cliques $ \tilde{\varpi}\_{\kappa}^{\varepsilon\_\iota} \in \tilde{\Upsilon}\_{\phi}^{\varepsilon\_\iota} $ given by (12). These values are obtained in lines 13-15. Based on this, the maximum slot occupancy ζ̄o under each offset is measured by comparing $ \max{\{ \zeta\_{\varpi}^{\varepsilon\_\iota} \}} $ and the previous maximum occupancy ζ̄ before flow fε scheduling in line 16. Then, the optimization value ℑo for offset o during flow fε scheduling is given by (11) in line 17, which works as the filtering basis for the optimal value ℑε and offset oε in lines 18-20. Under each offset o, the computational complexity to get ζ̄o is less than min{(m − ρ) ⋅ 3ρ/3, lcm(P̊n)/p̊ε} on each link, where m is the neighbor number of flow fε in the flow graph. ρ are the degeneracy of the neighbor built sub-graphs. Compared to the above hyper-flow graph based pattern, the traditional frame-based scheduling builds a slot occupancy list covering all slots S of the whole network as shown in Fig. [fig13](b). Due to the uncertainty of which slots have current maximum occupancy, it puts all periodic frames of flow fε into their specific slots sɛι ∈ S within the bound C along every link ɛι ∈ Rε during each offset search. The occupancy ζsɛι of these slots is measured one-by-one. This flow period-sensitive pattern owns the computational complexity as constant lcm(P̊n)/p̊ε on each link. For the large-scale complex flows, it almost exponentially expands and severely impacts scheduling efficiency. In our previous work, a flow graph-based scheduling pattern is explored to obtain the key scheduling information, that is, maximum slot occupancy ζ̄o. Distinguished from the hyper-flow nodes, this graph is built by treating each flow as a node and connects edges with a similar matrix Ψɛι $$\Psi\_{f\_{\alpha}, f\_{\beta}}^{\varepsilon\_\iota} = \begin{cases} 1 & \text{if\ } \gcd(\mathring{p}\_{\alpha}, \mathring{p}\_{\beta}) \| (q\_{\alpha}^{\varepsilon\_\iota} -q\_{\beta}^{\varepsilon\_\iota}). \\ 0 & \text{otherwise.} \end{cases}$$ As shown in Fig. [fig13](c), it is easy to prove that the hyper-flow graph can be shrunk from the flow graph with the following node merging principles. 1. The merged flow nodes are interconnected in pairs with edges by (13); 2. The merged flow nodes have the same neighbors $ \varkappa\_{\phi} $ except for each other; 3. After merging, the new node inherits the weight as the sum of the merged flow nodes. Thus, these two patterns share the same number and combined weights of maximal cliques. Their computational complexity is similar, but the hyper-flow graph pattern has smaller m and ρ. Since the complexity of MCEP rises exponentially along with the node scale, the smaller hyper-flow graph facilitates the large-scale flow scheduling. It also curbs the exponential trend of MECP due to its slower growth. To sum up, the hyper-flow graph based scheduling pattern suppresses the sensitivity of scheduling complexity to flow attributes, while weakening the impact of flow scale on it. Since the recurring offset searching accumulates the above complexity, we further improve offset filtering efficiency with an early-break strategy. It provides a breaking condition for offset traversal before reaching bound ōε. Specifically, the optimization values under offsets behind current o are compared with current optimal value ℑε in lines 21-22. Compared to this optimal ℑε, if all values ℑõ under offset õ > o satisfy that $$\Im\_{\epsilon} \leq \inf\_{\tilde{o} > o}{\Im\_{\tilde{o}}} = \inf\_{\tilde{o} > o}{ \{ (1-\rho) \cdot \dfrac{\tilde{o}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} + \rho \cdot \frac{\bar{\zeta}\_{\tilde{o}}}{\Lambda} \} }.$$ It means that the current value ℑε and its offset oε is optimal for flow fε scheduling. Thus, we break the offset search from its traversal. Based on this, we deflate (14) as $$\Im\_{\epsilon} \leq \inf\_{\tilde{o} > o}{ \{ (1-\rho) \cdot \dfrac{\tilde{o}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} \} } + \inf\_{\tilde{o} > o}{ \{ \rho \cdot \frac{\bar{\zeta}\_{\tilde{o}}}{\Lambda} \} }.$$ It is easy to prove that (14) must hold if the condition (15) is satisfied. Further, we have that $$\begin{aligned} & \inf\_{\tilde{o} > o}{ \{ (1-\rho) \cdot \dfrac{\tilde{o}}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} \} } = (1-\rho) \cdot \dfrac{o+1}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}}, \\ & \inf\_{\tilde{o} > o}{ \{ \rho \cdot \frac{\bar{\zeta}\_{\tilde{o}}}{\Lambda} \} } = \rho \cdot \frac{\max{ \{ \bar{\zeta}, l\_{\epsilon} \} }}{\Lambda}, \end{aligned} \nonumber$$ which builds the early-break threshold under offset o as $$\check{\Im}\_o = (1-\rho) \cdot \dfrac{o+1}{n\_{\epsilon} \cdot \mathring{d}\_{\epsilon}} + \rho \cdot \frac{\max{ \{ \bar{\zeta}, l\_{\epsilon} \} }}{\Lambda}.$$ If the condition $ \Im\_{\epsilon} \leq \check{\Im}\_o $ is satisfied, the offset traversal ends immediately in line 22. Unusually, when weight factor ρ is 1, the early-break condition is converted into ζ̄ε ≤ max{ζ̄, lε} introduced in work. After the flow fε scheduling by offset search, the optimal offset oε is stored and the global maximum slot occupancy ζ̄ is updated in line 24, where the symbol $ \vartriangleleft $ represents the store operation. Then, the hyper-flow graph and its maximal cliques on each link ɛι ∈ Rε are updated in lines 25-29. Specifically, if there exists the hyper-flow ϕɛι with the same feature tuple as optimal offset oε, this hyper-flow adds frame length lε to its weight ℓϕɛι in line 27. Otherwise, the tuple is extended as a new hyper-flow with weight lε, and stored in line 29 with its corresponding potential maximal cliques Υ̃ϕɛι. Due to Theorem 1, they need to be reduced to maximal hyper-flow cliques Ῠϕɛι and then stored. In the incremental way, every flow partition is scheduled and their flow injection offsets $ \mathcal{O}\_{\varsigma} $ are aggregated as a preliminary solution O in line 30. Link-parallel Flow	arxiv_0000763
The Sparsest Additive Spanner via Multiple Weighted BFS Trees ============================================================= Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known ( + 6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing ( + 2) and ( + 4)-all-pairs spanners, the sparsest case of ( + 6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a ( + 6)-spanner with the essentially-optimal sparsity of Õ(n4/3) edges. We then show a distributed implementation of our algorithm, answering an open problem in . A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in ∣S∣ + D − 1 rounds, where S is the set of sources and D is the diameter of the network graph. Keywords: Distributed graph algorithms, congest model, weighted BFS trees, additive spanners Introduction ============ A spanner of a graph G is a spanning subgraph H of G that approximately preserves distances. Spanners find many applications in distributed computing , and thus their distributed construction is the center of many research papers. We focus on spanners that approximately preserve distances between all pairs of nodes, and where the stretch is only by an additive factor (purely-additive all-pairs spanners). Out of the abundant research on distributed constructions of spanners, only two papers discuss the construction of purely additive spanners in the congest model: the construction of ( + 2)-spanners is discussed in , and the construction of ( + 4)-spanners and ( + 8)-spanners in , along with other types of additive spanners and lower bounds. However, the distributed construction of ( + 6)-spanners remained elusive, stated explicitly as an open question in . This is especially important since additive factors greater then 6 cannot yield essentially sparser spanners . In this paper, we give a distributed algorithm for constructing a ( + 6)-spanner, with an optimal number of edges up to sub-polynomial factors; our spanner is even sparser than the ( + 8)-spanner presented in . Several sequential algorithms building ( + 6)-spanners were presented, but none of them seems to be appropriate for a distributed setting. Thus, to achieve our result we also present a new, simple sequential algorithm for constructing ( + 6)-spanners, a result that could be of independent interest. As a key ingredient, we provide a distributed construction of multiple weighted BFS trees. Constructing a breadth-first search (BFS) tree is a central task in many computational settings. In the classic synchronous distributed setting, constructing a BFS tree from a given source is straightforward. Due to its importance, this task has received much attention in additional distributed settings, such as the asynchronous setting (see, e.g., and references therein). Moreover, at the heart of many distributed applications lies a graph structure that represents the edges of multiple BFS trees , which are rooted at the nodes of a given subset S ⊆ V, where G = (V, E) is the underlying communication graph. Such a structure is used in distance computation and estimation , routing table construction , spanner construction , and more. When the bandwidth is limited, constructing multiple BFS trees efficiently is a non-trivial task. Indeed, distributed constructions of multiple BFS trees in the congest model , where in each round of communication every node can send O(logn)-bit messages to each of its neighbors, have been given in . They showed that it is possible to build BFS trees from a set of sources S in O(∣S∣ + D) rounds, where D is the diameter of the graph G; it is easy to show that this is asymptotically tight. In some cases, different edges of the graph may have different attributes, which can be represented using edge weights. The existence of edge weights has been extensively studied in various tasks, such as finding or approximating lightest paths , finding a minimum spanning tree (MST) in the graph , finding a maximum matching , and more. However, as far as we are aware, no study addresses the problem of constructing multiple weighted BFS (WBFS) trees, where the goal is not to find the lightest paths from the sources to the nodes, but rather the lightest shortest paths. That is, the path in a WBFS tree from the source s to a node v is the lightest among all shortest paths from s to v in G. Thus, we provide an algorithm that constructs multiple WBFS trees from a set of source nodes S in the congest model. Our algorithm completes in ∣S∣ + D − 1 rounds, which implies that no overhead is needed for incorporating the existence of weights. While our multiple WBFS algorithm can be used for graphs that have initial edge weights, we actually use it to construct ( + 6)-spanners in unweighted graphs, to which we artificially add edge weights to distinguish “desired” edges from “undesired” ones. More generally, this algorithm can be used to find consistent shortest paths in an unweighted graph: a set of shortest paths is consistent if, given four nodes s, t, u and v such that u and v lie on the shortest (s, t)-path, the shortest (u, v)-path is a subpath of the shortest (s, t)-path. Such structures have many applications, but achieving them is non-trivial, especially in the distributed setting. Our WBFS algorithm can be used for this goal, by assigning the edges of an unweighted graph random weights of polynomial size. The isolation lemma then guarantees that with high probability, the lightest shortest paths derived from this weight function will constitute a set of consistent shortest paths. Our contribution ---------------- At a high level, our approach for building multiple WBFS trees is to generalize the algorithm of Lenzen et al. in order to handle weights. In , the messages are pairs consisting of a source node and a distance, which are prioritized by the distance traversed so far. When incorporating weights into this framework it makes sense to use triplets instead of pairs, where each triplet also contains the weight of the respective path. However, it may be that a node v needs to send multiple messages that correspond to the same source and the same distance but contain different weights, since congestion over edges may cause the respective messages to arrive at v in different rounds and, in the worst case, in a decreasing order of weights. The challenge in generalizing this framework therefore lies in guaranteeing that despite the need to consider weights, we can carefully choose a total order to prioritize triplets, such that not too many messages need to be sent, allowing us to handle congestion. Our construction and its proof appear in Section [sec: wbfs algorithm], giving the following. [thm: weighted bfs] Given a weighted graph G = (V, E, w) and a set of nodes S ⊆ V, there exists an algorithm for the congest model that constructs a WBFS tree rooted at s, for every s ∈ S, in ∣S∣ + D − 1 rounds. Our multiple WBFS trees construction can have many applications, as it allows consistent tie-breaking. Here, we present one such application: pinning down the question of constructing ( + 6)-spanners in the congest model. The construction of additive spanners in the congest model was studied beforehand , but the + 6 case remained unresolved, for reasons we describe below. Naturally, the quality of a spanner is measured by its sparsity, which is the motivation for allowing some stretch in the distances to begin with, and different spanners present different tradeoffs between stretch and sparsity. The properties of our ( + 6)-spanner construction algorithm are summarized in the following theorem.[1](#fn1) [thm: 6ap] There exists an algorithm for the congest model that constructs a ( + 6)-spanner with O(n4/3log4/3n) edges in $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds and succeeds w.h.p. Previous distributed algorithms for spanners similar to ours, i.e., purely additive all-pairs spanners, construct a ( + 2)-spanner with $\tilde O(n^{3/2})$ edges in $\tilde O(n^{1/2}+D)$ rounds , a ( + 4)-spanner with $\tilde O(n^{7/5})$ edges in $\tilde O(n^{3/5}+D)$ rounds , and a ( + 8)-spanner with $\tilde O(n^{15/11})$ edges in $\tilde O(n^{7/11}+D)$ rounds . Hence, our algorithm is currently the best non-trivial spanner construction algorithm in terms of density, sparser even than the previous ( + 8)-spanner. The parameters of our algorithm are tuned to achieve the best sparsity possible, and interestingly, one can change the algorithm and achieve a worst sparsity of $\tilde O(n^{15/11})$ edges in $\tilde O(n^{7/11}+D)$ rounds. These are the same parameters of the ( + 8)-spanner algorithm, but with a better stretch. The option of getting even sparser spanners by allowing more stretch was essentially ruled out , while the question of improving the running time remains open for all stretch parameters. Other spanner construction algorithms ------------------------------------- Previous distributed spanner construction algorithms all build upon known sequential algorithms, and present distributed implementations of them, or of a slight variant of them . For example, many sequential algorithms start in a clustering phase, where stars around high-degree nodes are added to the spanner one by one. Implementing this directly in the distributed setting will take too long; instead, we use a classical approach of choosing cluster centers at random, which yields almost as good results, and can be implemented in a constant time. Similar methods are used for implementing other parts of the construction. However, the approach of finding a distributed implementation for a sequential algorithm fails for all known ( + 6)-spanner algorithms, as described next. Thus, we introduce a new sequential algorithm for the problem, and then present its distributed implementation. There are three known approaches for the design of sequential ( + 6)-spanner algorithms. The first, presented by Baswana et al. , is based on measuring the quality of paths in terms of cost and value, and adding to the spanner only paths which are “affordable”. This approach was later extended by Kavitha to other families of additive spanners. The second approach, presented by Woodruff , uses a subroutine that finds almost-shortest paths between pairs of nodes, and obtains a faster algorithm at the expense of a slightly worst sparsity guarantee. The third approach, presented by Knudsen , is based on repeatedly going over pairs of nodes, and adding a shortest path between a pair of nodes to the spanner if their current distance in the spanner is too large. Unfortunately, direct implementation in the congest model of the known sequential algorithms is highly inefficient. We are not aware of fast distributed algorithms that allow the computation of the cost and value of paths needed for the algorithm of . Similarly, for , the almost-shortest paths subroutine seems too costly for the congest model. The algorithm of needs repeated updates of the distances in the spanner between pairs of nodes after every addition of a path to it, which is a sequential process in essence, and thus we do not find it suitable for an efficient distributed implementation. A different approach for the distributed construction of ( + 6)-spanners could be to adapt a distributed algorithm with different stretch guarantees to construct a ( + 6)-spanner. This approach does not seem to work: the distributed algorithms for constructing ( + 2)-spanners and ( + 4)-spanner are both very much tailored for achieving the desired stretch, and it is not clear how to change them in order to construct sparser spanners with higher stretch. The ( + 8)-spanner construction algorithm starts with clustering, and then constructs a ( + 4)-pairwise spanner between the cluster centers. Replacing the ( + 4)-pairwise spanner by a ( + 2)-pairwise spanner will indeed yield a ( + 6)-all-pairs spanner, as desired. However, even using the sparsest ( + 2)-pairwise spanners , the resulting ( + 6)-spanner may have Õ(n5/3) edges, denser than our new ( + 6)-spanner and than the known ( + 8)-spanner . Thus, we start by presenting a new sequential algorithm for the construction of ( + 6)-spanners, an algorithm that is more suitable for a distributed implementation, and then discuss its distributed implementation. Our construction starts with a clustering phase, and then adds paths that minimize the number of additional edges that need to be added to the spanner. To implement our construction in the congest model, we assign weights to the edges and use our WBFS algorithm to find shortest paths with as few edges as possible that are not yet in the spanner. Note that although the graph and the spanner we construct for it are both unweighted, the ability of our multiple WBFS algorithm to handle weights is crucial for our solution. A ( + 6)-spanner must contain $n^{4/3}/2^{O(\sqrt{\log n})}$ edges . The best sequential algorithms construct a spanner with O(n4/3) edges. Our distributed algorithm constructs a spanner with O(n4/3log4/3n) edges, which is slightly denser than optimal but still sparser than the O(n4/3log3n) edges in the fast sequential construction of . Related work ------------ A natural approach for building multiple (unweighted) BFS trees in the congest model is to start the construction simultaneously from all sources, sending pairs composed of a source id and the distance the message have traversed so far. The main challenge in this approach appears when multiple such pairs should be sent on one edge in a single round. In this case, the algorithm needs to locally prioritize the message, in a way that will not compromise the correctness of the constructed BFS trees. One approach for prioritizing the messages is by source id, i.e., have a queue at each node sorted by source id. However, this alone does not work and might create incorrect BFS trees. Attempts to overcome the problems of this approach by having a different queue for each outgoing edge are also insufficient . An elegant way to resolve this is to sort messages not by source id, but by the distance they traversed so far, with a preference to messages that traversed smaller distances. Such an algorithm was suggested by Lenzen and Peleg , finishing in (∣S∣ + D − 1) rounds when executed form a set S of sources. We base our multiple WBFS construction algorithm on this algorithm. A variant of the Lenzen-Peleg algorithm, with a reduced message complexity, was recently suggested in . Spanners were first introduced in 1989 , and since then have been a topic for wide research due to their abundant applications. Prime examples for the need for sparse spanners can be found in synchronizing distributed networks , information dissemination , compact routing schemes , and more. Distributed constructions of various spanners have been widely studied . Lower bounds were given in . However, obtaining an efficient and sparse ( + 6)-all-pairs spanner has remained an open question . Several lower bounds for the time complexity of spanner construction in the congest model were presented in , but these are applicable only to pairwise spanners with a bounded number of pairs, and not to all-pairs spanners. A lower bound from states that the construction of a spanner with $\tilde O(n^{4/3})$ edges, such as the one we build, must take $\tilde\Omega(n^{3/8})$ rounds. This lower bound does not take into account the bandwidth restrictions at all (it is proven for the local model), and so we believe that a higher lower bound for the congest model should apply, but this is left as an intriguing open question. Preliminaries ============= All graphs in this work are simple, connected and undirected. A graph can be unweighted, G = (V, E), or weighted G = (V, E, w) with w : E → {0, …, W}, in which case we assume W ∈ poly(n). Given a path ρ in a weighted graph G, we use ∣ρ∣ to denote the length of ρ, which is the number of edges in it, and w(ρ) to denote the weight of the path, which is the sum of its edge-weights. The distance between two nodes u, v in a graph G, denoted δG(u, v), is the minimum length of a path in G connecting u and v. The diameter of a graph (weighted or unweighted) is D = maxu, v ∈ V{δG(u, v)}. We consider the congest model of computation, where the nodes of a graph communicate synchronously by exchanging O(logn)-bit messages along the edges. The goal is to distributively solve a problem while minimizing the number of communication rounds. ##### WBFS trees: We are interested in a weighted BFS tree, which consists of all lightest shortest paths from the root, formally defined as follows. Given a connected, weighted graph G = (V, E, w) and a node s ∈ V, a weighted BFS tree (WBFS) for G rooted at s is a spanning tree Ts of G satisfying the following properties: 1. For each v ∈ V, the path from s to v in T is a shortest path in G between s and v. 2. For each v ∈ V, no shortest path from s to v in G is lighter than the path from s to v in T. We emphasize that this is different than requiring a subgraph containing all lightest paths from the root. One may wonder if a WBFS tree always exists, but this is easily evident by the following refinement of a (sequential) BFS search, returning a WBFS tree: go over the nodes in an order of non-decreasing distances from the source s, starting with w(s) = 0; each node v chooses as a parent a neighbor u that was already processed and minimizes w(v) = w(u) + w(u, v), and adds the edge {u, v} to the tree. Each node has a single parent, so this is indeed a tree; the node ordering guarantees that this is indeed a BFS tree, assuring (i); and the parent choice guarantees the paths are lightest among the shortest, assuring (ii). In fact, this can be seen as an algorithm implementing a consistent tie-breaking strategy between paths of equal lengths, and thus, it is not surprising its output is a tree. ##### Spanners: Given a graph G = (V, E), a subgraph H = (V, Eʹ) of G is called an (α, β)-spanner if for every u, v ∈ V it holds that δH(u, v) ≤ αδG(u, v) + β. The parameters α and β are called the stretch parameters. When α = 1, such a spanner is called a purely additive spanner. In this paper we focus on purely additive ( + 6)-spanners, i.e., α = 1 and β = 6. For completeness, we mention that when β = 0, such a spanner is called a multiplicative spanner. In addition, while sometimes the stretch parameters need to be guaranteed only for some subset of all the pairs of nodes of the graph (such as in pairwise spanners), we emphasize that our construction provides the promise of a + 6 stretch for all pairs. Multiple Weighted BFS Trees =========================== In the congest model, the problem of finding a WBFS tree requires each node to know its parent in the WBFS tree, and the unweighted and weighted distances to the source within the tree. This allows the node to send messages to the source node through the lightest among all shortest paths. When there are multiple sources, each node should know the parent leading to each of the sources in S. We define data structures for representing multiple WBFS trees. Given a node v ∈ V, the *S-proximity-list* (or proximity list for short) of v, noted ${{\textnormal{\texttt{PL}}}^\ast\_v}$, is an ascending lexicographically ordered list of triples (d(s, v), s, w(s, v)), where d(s, v) and w(s, v) are the length and weight of the path from s to v in Ts. Two different triples are ordered such that (d(s, v), s, w(s, v)) < (d(t, v), s, w(t, v)) if d(s, v) < d(t, v), or d(s, v) = d(t, v) and s < t, where s and t may be compared by any predefined order on the node identifiers. Note that Ts contains a single path from s to v, so ${{\textnormal{\texttt{PL}}}^\ast\_v}$ cannot contain two triplets with d(s, v) = d(t, v), s = t and w(s, v) ≠ w(t, v). The *S-path-map* (or path-map for short) of v is a mapping from each source s ∈ S to the parent of v in Ts, noted by ${{\textnormal{\texttt{PM}}}^\ast\_v}$. The list ${{\textnormal{\texttt{PM}}}^\ast\_v}$ is sorted with respect to the order of ${{\textnormal{\texttt{PL}}}^\ast\_v}$, such that the first records of ${{\textnormal{\texttt{PM}}}^\ast\_v}$ belong to sources closest to v. Algorithm [alg: parallel weighted bfs], which constructs multiple WBFS trees from a set S in the congest model, is based on carefully extending the distributed Bellman-Ford-based algorithm of Lenzen et al. . The heart of the algorithm is a loop (Line [line: for loop]), and each iteration of it takes a single round in the congest model. We show that ∣S∣ + D − 1 iterations of the loop suffice in order to construct the desired WBFS trees. The algorithm builds the WBFS trees by gradually updating the proximity list and the path map of each node. Each round is composed of two phases: updating the neighbors about changes in the proximity list, and receiving updates from other nodes. The path map is only used by the current node, and therefore changes to it are not sent. Ideally, each node would update its neighbors regarding all the changes made to its proximity list. However, due to bandwidth restrictions, a node cannot send the entire list in each round. Therefore, at each round each node sends to all of its neighbors the lexicographically smallest triplet in its proximity list that it has not yet sent, while maintaining a record noting which triplets have been sent and which are waiting. Each triplet is only sent once, though a node may send multiple triplets regarding a single source. A node uses the messages received in the current round in order to update its proximity list and path map for the next round. A triplet (ds, s, ws) received by a node v from a neighbor u represents the length ds and weight ws of some path ρ from s to u in the graph. The node v then considers the extended path ρʹ = ρ ∘ v from s to v, compares it to its currently known best path from s to v, and updates the proximity list and path map in case a shorter path has been found, or a lighter path with the same length. [t] Lv ← () [alg: parallel weighted bfs] To prove correctness, we generalize the proof of to handle weights, and show that our algorithm solves the weighted (S, d, k)-detection problem: each node should learn which are the sources from S closest to it, but at most k of them and only up to distance d. This is formally defined as follows. [def:weighted sdk detection] Given a weighted graph G = (V, E, w), a subset S ⊆ V of source nodes, and a node v ∈ V, let ${{\textnormal{\texttt{PL}}}^\ast\_v}$ denote the S-proximity-list and let ${{\textnormal{\texttt{PM}}}^\ast\_v}$ denote the path map of the node v. The weighted (S, d, k)-detection problem requires that each node v ∈ V learns the first min{k, λvd} entries of ${{\textnormal{\texttt{PL}}}^\ast\_v}$ and ${{\textnormal{\texttt{PM}}}^\ast\_v}$, where λvd is the number of sources s ∈ S such that d(s, v) ≤ d. Given a node v, $\operatorname{\textnormal{\texttt{PL}}}\_v$ is a variable in Algorithm [alg: parallel weighted bfs] holding the proximity list of v, and we denote by $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ the state of the list $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the beginning of round r of the algorithm, and by $\operatorname{\textnormal{\texttt{PL}}}\_v^{(\infty)}$ the value of $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the end of the algorithm. Recall that ${{\textnormal{\texttt{PL}}}^\ast\_v}$ is the true proximity list, so our goal is proving $\operatorname{\textnormal{\texttt{PL}}}\_v^{(\infty)} = {{\textnormal{\texttt{PL}}}^\ast\_v}$, i.e., proving that the algorithm obtains the correct values of the proximity list. We use similar notations for the path map ${{\textnormal{\texttt{PM}}}^\ast\_v}$. Since the records of $\operatorname{\textnormal{\texttt{PM}}}\_v$ are updated under the same conditions as the records of $\operatorname{\textnormal{\texttt{PL}}}\_v$, the correctness of $\operatorname{\textnormal{\texttt{PM}}}\_v$ at the end of the algorithm with respect to ${{\textnormal{\texttt{PM}}}^\ast\_v}$ immediately follows, and we omit the details. We start by showing that if there was no bound on the number of rounds, then the values of $\operatorname{\textnormal{\texttt{PL}}}\_v$ would have eventually converged to the true values of ${{\textnormal{\texttt{PL}}}^\ast\_v}$. [lem: eventual distances are correct] Given a graph G = (V, E, w) and a set S ⊆ V, if we let the for loop in Line [line: for loop] of Algorithm [alg: parallel weighted bfs] to run forever, then there exists a round $r\_0\in {\mathop{\mathbb{N}}}$ such that no node v ∈ V sends messages or modifies $\operatorname{\textnormal{\texttt{PL}}}\_v$ after round r0. Moreover, $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)} = {{\textnormal{\texttt{PL}}}^\ast\_v}$, i.e., for every $(d\_s,s,w\_s)\in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$, it holds that ds = d(v, s) and ws = min{w(ρ) ∣ ρconnects vwith s, and ∣ρ∣ = ds}. In each iteration of the algorithm, each node v ∈ V sends to all of its neighbors the first triplet (ds, s, ws) such that $\operatorname{\textnormal{\texttt{sent}}}\_v(d\_s,s,w\_s) = \operatorname{\textnormal{\texttt{FALSE}}}$. Each triplet received is sent at most once. Therefore, if we show the existence of a round r0 where for each v ∈ V, all messages in $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$ have been sent in previous rounds, it implies that no message is sent in round r0, and hence $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0 + 1)} = \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. This claim is applied inductively, concluding that for any round r > r0, it holds that $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)} = \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. We prove the existence of the round r0 by showing that the number of messages that are sent by each node v ∈ V is finite. Each triple (ds, s, ws) is inserted into $\operatorname{\textnormal{\texttt{PL}}}\_v$ when a message is received from another node u ∈ V, and such a message implies that a path from s to v through u of length ds and weight ws exists. Thus, the number of messages (ds, s, ws) sent by a node v is upper bounded by the number of paths from s to v of length ds. Furthermore, the algorithm does not insert a message into $\operatorname{\textnormal{\texttt{PL}}}\_v$ if it has already inserted a lexicographically smaller message from the same source. As the graph is finite, the number of paths is bounded, and eventually no node adds further triplets to its lists or sends additional messages. It remains to show that for all s ∈ S and v ∈ V it holds that $(d(s,v), s, w(s,v)) \in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. First, we show that if a triplet (ds, s, ws) is added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ in some round r, then there exists a path ρ from s to v such that ds = ∣ρ∣ and ws = w(ρ). For a source s ∈ S, we insert the triplet (0, s, 0) into $\operatorname{\textnormal{\texttt{PL}}}\_s$ at the beginning of the algorithm, so the claim is true at initialization. Assume there exists a round where a triplet (ds, s, ws) is inserted into $\operatorname{\textnormal{\texttt{PL}}}\_v$ but no corresponding path exists, and let r be the first such round. This implies that there exists a node u ∈ V that is a neighbor of v, which sends the message (ds − 1, s, ws − w(u, v)) to v in round r. The triplet (ds − 1, s, ws − w(u, v)) must have been inserted into $\operatorname{\textnormal{\texttt{PL}}}\_u$ in some round rʹ < r, and by the minimality of r there exists a path ρ from s to u where ∣ρ∣ = ds − 1 and w(ρ) = ws − w(u, v). Since u is a neighbor of v, the path ρʹ = ρ ∘ v is valid, satisfying ∣ρʹ∣ = ds and w(ρʹ) = ws, contradicting the assumption. To complete the proof, we claim that the correct triplet (d(s, v), s, w(s, v)) is indeed added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ at some round of the algorithm, and is not removed. Consider the path from s to v in the WBFS tree Ts, denoted by ρ = (v0 = s, v1, …, vd(s, v) = v). At initialization, the triplet (0, s, 0) is added to $\operatorname{\textnormal{\texttt{PL}}}\_s$. From then, at each round there exists some i ≤ d(s, v) such that $(d(s,v\_i), s, w(s,v\_i)) \in \operatorname{\textnormal{\texttt{PL}}}\_{v\_i}$ and $\operatorname{\textnormal{\texttt{sent}}}\_{v\_i}(d(s,v\_i), s, w(s,v\_i))=\operatorname{\textnormal{\texttt{FALSE}}}$. Since we proved that this message is eventually sent, this implies that in the beginning of the next round, the triplet (d(s, vi + 1), s, w(s, vi + 1)) is added to $\operatorname{\textnormal{\texttt{PL}}}\_{v\_{i+1}}$. By the definition of a WBFS tree, all other paths from s to v must be longer or not lighter, implying the triplet cannot be discarded for a lexicographically smaller triplet. This concludes that for any source s and node v, it holds that $(d(s,v), s, w(s,v)) \in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. Lemma [lem: eventual distances are correct] shows that without the limit on the number of rounds, the algorithm would compute the right values; however, it does not bound the number of rounds needed for this to occur. Next, we show that ∣S∣ + D − 1 rounds suffice. We cannot apply the claims of directly, since the existence of weights restricts the number of viable solutions even further, causing more updates to the proximity list and an increase in the number of messages sent. However, we do use a similar technique: we bound the number of rounds in which the k smallest entries of $\operatorname{\textnormal{\texttt{PL}}}\_v$ can change. For an entry $(d\_s,s,w\_s) \in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$, let ℓv(r)(ds, s, ws) denote the index of the entry in the lexicographically ordered list $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ at the beginning of round r. For completeness, we define ℓv(r)(ds, s, ws) = − ∞ if (ds, s, ws) did not appear in $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the beginning of round r, and $\operatorname{\textnormal{\texttt{PL}}}\_v=\infty$ if the triplet was removed from $\operatorname{\textnormal{\texttt{PL}}}\_v$ before the beginning of this round. Note that a removed triplet is never returned to the list, since the lexicographical order is transitive. [helper lemma] For a triplet (ds, s, ws), the following holds: 1. ℓv(r)(ds, s, ws) is non-decreasing with r. 2. When a triplet (ds, s, ws) is sent from a node u to a node v at round r, it causes the addition of a new triplet (dʹs, s, wʹs) to $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the end of round r. This triplet satisfies dʹs = ds + 1, wʹs = ws + w(u, v), and ℓu(r)(ds, s, ws) ≤ ℓv(r + 1)(dʹs, s, wʹs). Part (i) follows from the fact that the number of triplets below (ds, s, ws) cannot decrease. To prove part (ii), we show that all the triplets below (ds, s, ws) in $\operatorname{\textnormal{\texttt{PL}}}\_u$ are sent from u to v and added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ before (ds, s, ws) is sent and added. Part (i) is a consequence of the method used by our algorithm for managing the list $\operatorname{\textnormal{\texttt{PL}}}\_v$. According to our algorithm, triplets are not removed from $\operatorname{\textnormal{\texttt{PL}}}\_v$ when they are sent. The only case in which a triplet (dt, t, wt) is removed from $\operatorname{\textnormal{\texttt{PL}}}\_v$ is when a lexicographically smaller triplet (dʹt, t, wʹt) is added to the list instead. When this happens in round r, it holds that ℓv(r)(dt, t, wt) ≥ ℓv(r + 1)(dʹt, t, wʹt), since the new triplet is lexicographically smaller. Hence, for every other triplet $(d\_s,s,w\_s)\in \operatorname{\textnormal{\texttt{PL}}}\_v$, the number of lexicographically smaller triplets in $\operatorname{\textnormal{\texttt{PL}}}\_v$ cannot decrease throughout the algorithm. We now turn to prove part (ii) of the lemma. Note that the only claim which does not follows trivially from the algorithm is the inequality ℓu(r)(ds, s, ws) ≤ ℓv(r + 1)(dʹs, s, wʹs). By the fact that the triplet (ds, s, ws) is sent by the node u in round r, we conclude that the ℓu(r)(ds, s, ws) − 1 triplets preceding it in the list $\operatorname{\textnormal{\texttt{PL}}}\_u^{(r)}$ have already been sent by u in earlier rounds, and arrived at the node v. For each such triplet (dt, t, wt), either dt ≤ ds, or t < s and dt = ds. Therefore, when added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ as (dt + 1, t, wt + w(u, v)) it is lexicographically smaller than (dʹs, s, wʹs). At round r, either (dt + 1, t, wt + w(u, v)) is in $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ or it was replaced by a lexicographically smaller triplet containing t. Thus, there are at least ℓu(r)(ds, s, ws) − 1 triplets smaller than (dʹs, s, wʹs) in $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r+1)}$, and hence ℓu(r)(ds, s, ws) ≤ ℓv(r + 1)(dʹs, s, wʹs). Lemma [helper lemma] implies that as the algorithm progresses, messages at higher indexes of the proximity list are sent and updated. This can be used to obtain an upper bound on the round in which a triplet at a certain index of the proximity list can be sent or received, as formalized by the next lemma. [lem: first entries stabilize] In round $r\in {\mathop{\mathbb{N}}}$ of Algorithm [alg: parallel weighted bfs], a node v ∈ V can: 1. send a message (ds, s, ws) only if ds + ℓv(r)(ds, s, ws) ≥ r 2. add to $\operatorname{\textnormal{\texttt{PL}}}\_v$ a triplet (ds, s, ws) only if ds + ℓv(r + 1)(ds, s, ws) > r Part (i), when put in words, is rather intuitive: while a triplet might need to wait before being sent, the waiting time is bounded from above by the distance the triplet has traversed from its source, plus the number of triplets that were to be sent before it. Part (ii) is complementary to part (i): the time before a triplet is added, is, once more, bounded by the distance it traversed plus the number of lexicographically smaller triplets. We start by showing that, for a given round r, if Lemma [lem: first entries stabilize](i) holds for all nodes then Lemma [lem: first entries stabilize](ii) holds as well. Consider a triplet (dʹs, s, wʹs) that is added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ as a result of a message (ds, s, ws) sent from u to v in round r, where dʹs = ds + 1 and wʹs = ws + w(u, v). Lemma [lem: first entries stabilize](i) implies that ds + ℓu(r)(ds, s, ws) ≥ r, and by Lemma [helper lemma](ii) we have that ℓv(r + 1)(dʹs, s, wʹs) ≥ ℓu(r)(ds, s, ws). As dʹs > ds, we conclude dʹs + ℓv(r + 1)(dʹs, s, wʹs) > ds + ℓu(r)(ds, s, ws) ≥ r, which implies Lemma [lem: first entries stabilize](ii). Next, we prove by induction that both parts of the lemma hold. In round 1, Lemma [lem: first entries stabilize](i) holds trivially, since by definition ℓv(1)(ds, s, ws) ≥ 1. Assume that Lemma [lem: first entries stabilize] holds at round r − 1; we show the lemma holds at round r. Since Lemma [lem: first entries stabilize](i) implies Lemma [lem: first entries stabilize](ii), it is sufficient to show that every message (ds, s, ws) sent by some node v ∈ V in round r satisfies ds + ℓv(r)(ds, s, ws) ≥ r. Observe that if (ds, s, ws) is sent by a node v in round r, then the triplet must have been added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ in some round rʹ ≤ r − 1. If rʹ = r − 1, according to the induction hypothesis, Lemma [lem: first entries stabilize](ii) holds and ds + ℓv(r)(ds, s, ws) > r − 1, implying ds + ℓv(r)(ds, s, ws) ≥ r, since all the terms are integers. Otherwise rʹ < r − 1. In this case, in round r − 1 the triplet (ds, s, ws) appeared in $\operatorname{\textnormal{\texttt{PL}}}\_v$ and was not yet sent. Since (ds, s, ws) is sent in round r, a different triplet (dt, t, wt) with t ≠ s must have been sent in round r − 1, implying: ds + ℓv(r − 1)(ds, s, ws) > dt + ℓv(r − 1)(dt, t, wt). By Lemma [helper lemma](i), we have that ℓv(r)(ds, s, ws) ≥ ℓv(r − 1)(ds, s, ws), and combined with the induction hypothesis for Lemma [lem: first entries stabilize](i) in round r − 1 we conclude: ds + ℓv(r)(ds, s, ws) ≥ ds + ℓv(r − 1)(ds, s, ws) > dt + ℓv(r − 1)(dt, t, wt) ≥ r − 1. This gives that ds + ℓv(r)(ds, s, ws) ≥ r, since all the terms are integers. Lemma [lem: eventual distances are correct] implies that eventually, the lists $\operatorname{\textnormal{\texttt{PL}}}\_v$ converge to contain the correct values, and Lemma [lem: first entries stabilize] restricts the number of rounds in which specific list entries may change. From this, we conclude that the algorithm solves the weighted (S, d, k)-detection problem. [lem: first r entries are correct] Given an instance of the weighted (S, d, k)-detection problem, for every v ∈ V and round r of an execution of Algorithm [alg: parallel weighted bfs] with r ≥ min{d, D} + min{k, ∣S∣}, the truncation of $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ to the first min{k, λvd} entries, where λvd is the number sources s ∈ S such that d(s, v) ≤ d, solves weighted (S, d, k)-detection problem. This lemma says that the truncated list is correct at the beginning of the relevant round. To prove it, we use Lemma [lem: first entries stabilize](ii) to show that the values in the truncated list cannot change at round r or later, and Lemma [lem: eventual distances are correct] to deduce they are correct. Assume w.l.o.g that d ≤ D, as D bounds the distance to any source, and k ≤ ∣S∣, as otherwise v needs to learn about all sources. By Lemma [lem: eventual distances are correct], there is a round r0 when all entries of $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$ are correct, and let (ds, s, ws) be a triplet in one of the first min{k, λvd} entries of $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. Since (ds, s, ws) is one of the first λvd entries and $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)} = {{\textnormal{\texttt{PL}}}^\ast\_v}$, we have ds ≤ d. Let r be the round when (ds, s, ws) is inserted to the list $\operatorname{\textnormal{\texttt{PL}}}\_v$. By Lemma [lem: first entries stabilize](ii), r < ds + ℓv(r + 1)(ds, s, ws). By Lemma [helper lemma](i), when the triplet is inserted to the list, it is already placed in one of the first min{k, λvd} entries, i.e., ℓv(r + 1)(ds, s, ws) ≤ min{k, λvd} ≤ k. Hence, r < ds + ℓv(r + 1)(ds, s, ws) ≤ d + k. Since this claim holds for any of the first min{k, λvd} entries, these were all correct at the beginning of round d + k, and in all the succeeding rounds. The construction of multiple WBFS trees is an instance of the (S, D, ∣S∣)-detection problem. Lemma [lem: first r entries are correct] shows that after ∣S∣ + D − 1 rounds of Algorithm [alg: parallel weighted bfs] on such an instance, all the entries of the list $\operatorname{\textnormal{\texttt{PL}}}\_v^{({\ensuremath{\left\|S\right\|}}+D)}$ are correct, yielding the main result of this section. thm: weighted bfs Given a weighted graph G = (V, E, w) and a set of nodes S ⊆ V, there exists an algorithm for the congest model that constructs a WBFS tree rooted at s, for every s ∈ S, in ∣S∣ + D − 1 rounds. A ( + 6)-Spanner Construction ============================= In this section we discuss the distributed construction of ( + 6)-spanners. First, we present a template for constructing a ( + 6)-spanner and analyze the stretch and sparsity of the constructed spanner. Then, we provide an implementation of our template in the congest model and analyze its running time. A cluster Ci around a cluster center ci ∈ V is a subset of the set of neighbors of ci in G. A node belonging to a cluster is clustered, while the other nodes are unclustered. Our algorithm starts by randomly choosing cluster centers, and adding edges between them to their neighbors, where each neighbor arbitrarily chooses a single center to connect to. Then, additional edges are added, to connect each unclustered node to all its neighbors. Next, shortest paths between clusters are added to the spanner. In order to find these shortest paths in the congest model, we use the WBFS construction algorithm to build WBFS trees from random sources. At the heart of our algorithm stands the path-hitting framework of Woodruff : a shortest path in the graph which has many edges between clustered nodes, must go through many clusters. This fact is used in order to show that a path with many missing edges (edges not in H) is more likely to have an adjacent source of a WBFS tree, and thus it is well approximated by a path within the spanner. Woodruff’s algorithm starts with a similar clustering step. However, in order to add paths between clusters, it uses an involved subroutine that finds light almost-shortest paths between pairs of nodes. This subroutine seems too global to be implemented efficiently in a distributed setting, so in our construction it is replaced by only considering lightest shortest paths, which we do using the WBFS trees defined earlier. Our algorithm constructs a ( + 6)-spanner with O(n4/3log4/3n) edges in Õ(n2/3 + D) rounds, as stated next. thm: 6ap There exists an algorithm for the congest model that constructs a ( + 6)-spanner with O(n4/3log4/3n) edges in $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds and succeeds w.h.p. Lemmas [lemma: 6ap size] and [lemma: 6ap stretch] analyze the size and stretch of Algorithm ${\tt 6AP}$ given below. The number of rounds of its distributed implementation is analyzed in Lemma [lemma: 6ap complexity], giving Theorem [thm: 6ap]. We use c > 2 to denote a constant that can be chosen according to the desired exponent of 1/n in the failure probability. ### Algorithm ${\tt 6AP}$ Input: a graph G = (V, E), a constant c > 2; Output: a subgraph H of G; Initialization: n ← ∣V∣; H ← (V, ∅); k ← 1 #### Clustering. Pick each node as a cluster center w.p. $\frac{c}{n^{1/3}\log^{1/3}n}$, and denote the set of selected nodes by C = {c1, c2, …}. For each ci, initialize a cluster Ci ← ∅. For each node v ∈ V, choose a neighbor ci of v which is a cluster center, if such a neighbor exists, add the edge (v, ci) to H, and add v to Ci. If none of the neighbors of v is a cluster center, add to H all the edges adjacent to v. Let H0 ← H. #### Path Buying. While $k\leq \frac{8cn^{2/3}}{\log^{1/3} n}$ do: 1. Sk ← ∅ 2. Add each cluster center ci ∈ C to Sk w.p. $\frac{8c^2\log n}{k}$, independently of the other centers 3. For each pair (ci, cj) ∈ C × Sk: 1. A ← ∅ /\* A is a set of paths \/ 2. For each v* ∈ Cj: 1. Among all the shortest paths from ci to v, let Pv be a path with minimum ∣Pv \ H0∣ 2. If ∣Pv \ H0∣ < 2k, add Pv to A 3. If A ≠ ∅, add to H one of the shortest among the paths of A 4. k ← 2k [lemma: 6ap size] Algorithm ${\tt 6AP}$ outputs a subgraph H of G with O(n4/3log4/3n) edges, with probability at least 1 − O(n− c + 1). The algorithm starts with H = (V, ∅) and only adds edges from G, so H is indeed a subgraph of G over the same node set. In the first part of the clustering phase, each node adds to H at most one edge, connecting it to a single cluster center, for a total of O(n) edges. Then, the probability that a node of degree at least n1/3log4/3n is left unclustered is at most $\left(1-\frac{c}{n^{1/3}\log^{1/3}n}\right)^{n^{1/3}\log^{4/3}n}$, which is O(n− c). A union bound implies that all nodes of degree at least n1/3log4/3n are clustered w.p. 1 − O(n− c + 1), and thus the total number of edges added to H by unclustered nodes in the second part of the clustering phase is O(n4/3log4/3n), w.p. 1 − O(n− c + 1). We start the analysis of the path buying phase by bounding the size of C. A node v ∈ V is added to C w.p. $\frac{c}{n^{1/3}\log^{1/3}n}$, so ${\mathop{\mathbb{E}}}[{\ensuremath{\left\|{\ensuremath{\mathcal{C}}}\right\|}}] = \frac{cn^{2/3}}{\log^{1/3} n}$. A Chernoff bound implies that $$\Pr\left[{\ensuremath{\left\|{\ensuremath{\mathcal{C}}}\right\|}} > \frac{4cn^{2/3}}{\log^{1/3} n}\right] \leq \exp\left(- \frac{cn^{2/3}}{\log^{1/3} n}\right) =o(n^{-c}).$$ Similarly, for each value of k, we have ${\mathop{\mathbb{E}}}[{\ensuremath{\left\|S\_k\right\|}}] = \frac{8c^2 n^{2/3}\log^{2/3} n}{k}$, and $$\Pr\left[{\ensuremath{\left\|S\_k\right\|}} > \frac{32c^2 n^{2/3}\log^{2/3} n}{k}\right] \leq \exp\left(-\frac{8c^2 n^{2/3}\log^{2/3} n}{k}\right) =O(n^{-c}),$$ where the last equality follows since $k\leq \frac{n^{2/3}}{\log^{1/3}n}$. A union bound implies that ${\ensuremath{\left\|{\ensuremath{\mathcal{C}}}\right\|}} = O\left(\frac{n^{2/3}}{\log^{1/3} n}\right)$ and ${\ensuremath{\left\|S\_k\right\|}} = O\left(\frac{n^{2/3}\log^{2/3} n}{k}\right)$ for all k, w.p. at least 1 − O(n− c + 1). Finally, for each k, for each (ci, cj) ∈ C × Sk we add at most one path with less than 2k missing edges to H. Thus, for each value of k we add less than ∣C∣ ⋅ ∣Sk∣ ⋅ 2k = O(n4/3log1/3n) edges to H, w.p. at least 1 − O(n− c + 1). Summing over all O(logn) values of k, and adding the number of edges contributed by the clustering phase, we conclude that H has at most O(n4/3log4/3n) edges, w.p. at least 1 − O(n− c + 1). [lemma: 6ap stretch] The graph H constructed by Algorithm ${\tt 6AP}$ satisfies δH(x, y) ≤ δG(x, y) + 6 for each pair (x, y) ∈ V × V, with probability at least 1 − O(n− c + 2). [fig: +6 stretch] Consider a shortest path ρ in G between two nodes x, y ∈ V (see Figure [fig: +6 stretch]). Let xʹ and yʹ be the first and last clustered nodes on ρ, respectively. If all nodes of ρ are unclustered, then ρ is fully contained in H0 and we are done. Let c1 and c3 be the centers of the clusters containing xʹ and yʹ, respectively. Let σ be a shortest path in G between c1 and c3, and denote by kʹ the number of edges of σ \ H0. Let k be the largest power of 2 such that k ≤ kʹ. An edge can be in σ \ H0 only if it connects two clustered nodes. Hence, kʹ, the number of edges in σ \ H0, is smaller than the number of clustered nodes in σ. On the other hand, σ cannot contain more than three nodes of the same cluster: the distance between every two nodes in a cluster is at most two, so a shortest path cannot traverse more than three nodes of the same cluster. Thus, the number of clusters intersecting σ is at least kʹ/3. As kʹ/3 ≥ k/3, the probability that none of the centers of these clusters is chosen to Sk is at most $\left( 1-\frac{8c^2 \log n}{k} \right)^{k/3} = O\left(n^{-c^2}\right)$. For each pair of nodes, a cluster center on a shortest path between them is chosen to Sk, for the appropriate value of k, with similar probability. A union bound implies that this claim holds for all pairs in V × V w.p. at least 1 − O(n− c2 + 2). Let w be a node on σ in a cluster C2 such that c2 ∈ Sk, if such a cluster exists. Denote by σ[c1, w] the sub-path of σ from c1 to w. As there are kʹ < 2k edges in σ \ H0, there are also less than 2k edges in σ[c1, w] \ H0. Thus, in step 3(b) of the path-buying phase for k, either the path σ[c1, w] or some other path between c1 and w of length at most δG(c1, w) is added to A. In step 3(c), a path from c1 to some node w1 ∈ C2 is added to H, and this is a shortest path in A, so δH(c1, w1) ≤ δG(c1, w). Similarly, a shortest path from c3 to some w3 ∈ C2 is added to H, and δH(c3, w3) ≤ δG(c3, w). The path σ is a shortest path from c1 to c3 in G, so ∣σ∣ ≤ δG(xʹ, yʹ) + 2. As δG(c1, w) + δG(c3, w) = ∣σ∣, we conclude δH(c1, w1) + δH(c3, w3) ≤ ∣σ∣ ≤ δG(xʹ, yʹ) + 2. Consider the path from x to y in H composed of the sub-path of ρ from x to xʹ, the edge (xʹ, c1), the path from c1 to w1, the edges (w1, c2) and (c2, w3), the path from w3 to c3, the edge (c3, yʹ), and finally, the sub-path of ρ from yʹ to y. This is a path from x to y in H, implying $$\begin{split} \delta\_H(x,y) & \leq \delta\_H (x,x') + 1 +\delta\_H (c\_1,w\_1) +2 + \delta\_H (w\_3,c\_3) +1 +\delta\_H (y',y)\\ & \leq \delta\_G (x,x') +4 +\delta\_G(x',y') +2 + \delta\_G (y',y) = \delta\_G(x,y) +6, \end{split}$$ as desired. We now discuss the implementation of Algorithm ${\tt 6AP}$ in the congest model. [lemma: 6ap complexity] Algorithm ${\tt 6AP}$ can be implemented in $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds in the congest model, with probability at least 1 − o(n− c). For the clustering phase, each node decides locally w.p. $\frac{c}{n^{1/3}\log^{1/3}n}$ to become a cluster center, and notifies its neighbors. Each node with a neighbor that is a cluster center now joins a cluster by sending a message to such a neighbor and adding the appropreate edge to the spanner. A node with no neighboring cluster centers notifies all its neighbors and adds all its edges to the spanner. This is done in a constant number of rounds. Before the path buying phase, the nodes construct a single BFS tree, along which they compute an upper bound Dʹ on D, satisfying D ≤ Dʹ < 2D, and count the number of cluster centers, ∣C∣. The nodes mark the edges of H0 with weight 0 and the other edges with weight 1. Then, they construct a WBFS tree rooted at each cluster center by executing Algorithm [alg: parallel weighted bfs] for ∣C∣ + Dʹ many rounds. By the proof of Lemma [lemma: 6ap size], we have ${\ensuremath{\left\|{\ensuremath{\mathcal{C}}}\right\|}}\in O\left(\frac{n^{2/3}}{\log^{1/3}n}\right)$ w.p. at least 1 − o(n− c), and thus the construction of the WBFS trees takes $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds with the same probability. Each node v now knows about a “good” path to each cluster center ci, i.e., a shortest path from ci to v, with a minimal number of edges not in H after the clustering phase. A node v in a cluster Cj notifies its neighbor cj about all the distances to other cluster centers in C and the number of missing edges in each such path. That is, each v ∈ Cj sends ∣C∣ messages to cj, which takes $O\left(\frac{n^{2/3}}{\log^{1/3}n}\right)$ rounds. Each cluster center cj decides locally to join each set Sk w.p. $\frac{8c^2\log n}{k}$. For each other center ci ∈ C, cj locally constructs the list A: for each v ∈ Cj, A contains the shortest path from ci to v ∈ Cj found by the WBFS algorithm, and the number of missing edges in it. Then, cj chooses from A a path from ci to some v ∈ Cj with a minimal number of missing edges, and if it has at most 2k missing edges, cj sends a “buy ci” message to v. Finally, all nodes simultaneously execute a “buy” phase, where “buy ci” messages are sent up the WBFS tree. To avoid congestion, we assume that during the execution of Algorithm [alg: parallel weighted bfs], each node keeps a record of the messages it got in each round and the WBFS source each message referred to. Each node v then sends messages in reversed order: if v has a message “buy ci”, and it got a message from u regarding ci in the r-before-last round of Algorithm [alg: parallel weighted bfs], then it sends the message “buy ci” to u in round r of the “buy” phase. Then, u adds “buy ci” to its list of messages, and adds the edge (u, v) to the spanner. This parts takes $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds, just like the execution of Algorithm [alg: parallel weighted bfs]. Discussion and Open Questions ============================= While we present an application of WBFS trees, our algorithm also solves the weighted (S, d, k)-detection problem, a result that could be of independent interest. The question of finding the lightest paths between all pairs of nodes in a graph, or computing their weights, is a fundamental question in many computational models. In the congest model, a randomized algorithm for exactly computing these distances in Õ(n) rounds was recently presented . The exact time complexity of computing these distances deterministically in the congest model is still open , and we hope our study of lightest shortest paths could facilitate future research on it. While this paper settles the question of constructing sparse ( + 6)-spanners fast, the study of spanner construction in distributed environments still lags behind the study of sequential spanner construction algorithms. In the field of purely additive spanners, we still do not have fast algorithms, e.g., for the construction of sparse ( + 0)-pairwise spanners (a.k.a. pairwise preservers) and ( + 6)-pairwise spanners. A more intriguing question is proving time lower bounds for the construction of spanners in the congest model: while Ω(D) rounds are known to be necessary , lower bounds that depend on other parameters of the graph or the spanners exist only for pairwise spanners . Finding a lower bound for the construction of all-pairs spanners in the congest model, which does not depend on D, is still an open question. Such a lower bound could show that the n3/2 term in the time bound of our construction is inevitable, or motivate the design of faster algorithms for the problem. #### Acknowledgements We thank Shiri Chechik and Pierre Fraigniaud for discussions regarding ( + 6)-spanners, and the reviewers of TCS journal for discussions on consistent shortest paths. This project has received funding from the European Union’s Horizon 2020 Research And Innovation Program under grant agreement no.755839, and from the Israel Science Foundation (grant 1696/14). Ami Paz was supported by the Fondation Sciences Mathématiques de Paris (FSMP). --- 1. We use w.h.p. to indicate a probability that is at least 1 − 1/nc for some constant c ≥ 1 of choice.[↩](#fnref1) The Sparsest Additive Spanner via Multiple Weighted BFS Trees ============================================================= Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known ( + 6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing ( + 2) and ( + 4)-all-pairs spanners, the sparsest case of ( + 6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a ( + 6)-spanner with the essentially-optimal sparsity of Õ(n4/3) edges. We then show a distributed implementation of our algorithm, answering an open problem in . A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in ∣S∣ + D − 1 rounds, where S is the set of sources and D is the diameter of the network graph. Keywords: Distributed graph algorithms, congest model, weighted BFS trees, additive spanners Introduction ============ A spanner of a graph G is a spanning subgraph H of G that approximately preserves distances. Spanners find many applications in distributed computing , and thus their distributed construction is the center of many research papers. We focus on spanners that approximately preserve distances between all pairs of nodes, and where the stretch is only by an additive factor (purely-additive all-pairs spanners). Out of the abundant research on distributed constructions of spanners, only two papers discuss the construction of purely additive spanners in the congest model: the construction of ( + 2)-spanners is discussed in , and the construction of ( + 4)-spanners and ( + 8)-spanners in , along with other types of additive spanners and lower bounds. However, the distributed construction of ( + 6)-spanners remained elusive, stated explicitly as an open question in . This is especially important since additive factors greater then 6 cannot yield essentially sparser spanners . In this paper, we give a distributed algorithm for constructing a ( + 6)-spanner, with an optimal number of edges up to sub-polynomial factors; our spanner is even sparser than the ( + 8)-spanner presented in . Several sequential algorithms building ( + 6)-spanners were presented, but none of them seems to be appropriate for a distributed setting. Thus, to achieve our result we also present a new, simple sequential algorithm for constructing ( + 6)-spanners, a result that could be of independent interest. As a key ingredient, we provide a distributed construction of multiple weighted BFS trees. Constructing a breadth-first search (BFS) tree is a central task in many computational settings. In the classic synchronous distributed setting, constructing a BFS tree from a given source is straightforward. Due to its importance, this task has received much attention in additional distributed settings, such as the asynchronous setting (see, e.g., and references therein). Moreover, at the heart of many distributed applications lies a graph structure that represents the edges of multiple BFS trees , which are rooted at the nodes of a given subset S ⊆ V, where G = (V, E) is the underlying communication graph. Such a structure is used in distance computation and estimation , routing table construction , spanner construction , and more. When the bandwidth is limited, constructing multiple BFS trees efficiently is a non-trivial task. Indeed, distributed constructions of multiple BFS trees in the congest model , where in each round of communication every node can send O(logn)-bit messages to each of its neighbors, have been given in . They showed that it is possible to build BFS trees from a set of sources S in O(∣S∣ + D) rounds, where D is the diameter of the graph G; it is easy to show that this is asymptotically tight. In some cases, different edges of the graph may have different attributes, which can be represented using edge weights. The existence of edge weights has been extensively studied in various tasks, such as finding or approximating lightest paths , finding a minimum spanning tree (MST) in the graph , finding a maximum matching , and more. However, as far as we are aware, no study addresses the problem of constructing multiple weighted BFS (WBFS) trees, where the goal is not to find the lightest paths from the sources to the nodes, but rather the lightest shortest paths. That is, the path in a WBFS tree from the source s to a node v is the lightest among all shortest paths from s to v in G. Thus, we provide an algorithm that constructs multiple WBFS trees from a set of source nodes S in the congest model. Our algorithm completes in ∣S∣ + D − 1 rounds, which implies that no overhead is needed for incorporating the existence of weights. While our multiple WBFS algorithm can be used for graphs that have initial edge weights, we actually use it to construct ( + 6)-spanners in unweighted graphs, to which we artificially add edge weights to distinguish “desired” edges from “undesired” ones. More generally, this algorithm can be used to find consistent shortest paths in an unweighted graph: a set of shortest paths is consistent if, given four nodes s, t, u and v such that u and v lie on the shortest (s, t)-path, the shortest (u, v)-path is a subpath of the shortest (s, t)-path. Such structures have many applications, but achieving them is non-trivial, especially in the distributed setting. Our WBFS algorithm can be used for this goal, by assigning the edges of an unweighted graph random weights of polynomial size. The isolation lemma then guarantees that with high probability, the lightest shortest paths derived from this weight function will constitute a set of consistent shortest paths. Our contribution ---------------- At a high level, our approach for building multiple WBFS trees is to generalize the algorithm of Lenzen et al. in order to handle weights. In , the messages are pairs consisting of a source node and a distance, which are prioritized by the distance traversed so far. When incorporating weights into this framework it makes sense to use triplets instead of pairs, where each triplet also contains the weight of the respective path. However, it may be that a node v needs to send multiple messages that correspond to the same source and the same distance but contain different weights, since congestion over edges may cause the respective messages to arrive at v in different rounds and, in the worst case, in a decreasing order of weights. The challenge in generalizing this framework therefore lies in guaranteeing that despite the need to consider weights, we can carefully choose a total order to prioritize triplets, such that not too many messages need to be sent, allowing us to handle congestion. Our construction and its proof appear in Section [sec: wbfs algorithm], giving the following. [thm: weighted bfs] Given a weighted graph G = (V, E, w) and a set of nodes S ⊆ V, there exists an algorithm for the congest model that constructs a WBFS tree rooted at s, for every s ∈ S, in ∣S∣ + D − 1 rounds. Our multiple WBFS trees construction can have many applications, as it allows consistent tie-breaking. Here, we present one such application: pinning down the question of constructing ( + 6)-spanners in the congest model. The construction of additive spanners in the congest model was studied beforehand , but the + 6 case remained unresolved, for reasons we describe below. Naturally, the quality of a spanner is measured by its sparsity, which is the motivation for allowing some stretch in the distances to begin with, and different spanners present different tradeoffs between stretch and sparsity. The properties of our ( + 6)-spanner construction algorithm are summarized in the following theorem.[1](#fn1) [thm: 6ap] There exists an algorithm for the congest model that constructs a ( + 6)-spanner with O(n4/3log4/3n) edges in $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds and succeeds w.h.p. Previous distributed algorithms for spanners similar to ours, i.e., purely additive all-pairs spanners, construct a ( + 2)-spanner with $\tilde O(n^{3/2})$ edges in $\tilde O(n^{1/2}+D)$ rounds , a ( + 4)-spanner with $\tilde O(n^{7/5})$ edges in $\tilde O(n^{3/5}+D)$ rounds , and a ( + 8)-spanner with $\tilde O(n^{15/11})$ edges in $\tilde O(n^{7/11}+D)$ rounds . Hence, our algorithm is currently the best non-trivial spanner construction algorithm in terms of density, sparser even than the previous ( + 8)-spanner. The parameters of our algorithm are tuned to achieve the best sparsity possible, and interestingly, one can change the algorithm and achieve a worst sparsity of $\tilde O(n^{15/11})$ edges in $\tilde O(n^{7/11}+D)$ rounds. These are the same parameters of the ( + 8)-spanner algorithm, but with a better stretch. The option of getting even sparser spanners by allowing more stretch was essentially ruled out , while the question of improving the running time remains open for all stretch parameters. Other spanner construction algorithms ------------------------------------- Previous distributed spanner construction algorithms all build upon known sequential algorithms, and present distributed implementations of them, or of a slight variant of them . For example, many sequential algorithms start in a clustering phase, where stars around high-degree nodes are added to the spanner one by one. Implementing this directly in the distributed setting will take too long; instead, we use a classical approach of choosing cluster centers at random, which yields almost as good results, and can be implemented in a constant time. Similar methods are used for implementing other parts of the construction. However, the approach of finding a distributed implementation for a sequential algorithm fails for all known ( + 6)-spanner algorithms, as described next. Thus, we introduce a new sequential algorithm for the problem, and then present its distributed implementation. There are three known approaches for the design of sequential ( + 6)-spanner algorithms. The first, presented by Baswana et al. , is based on measuring the quality of paths in terms of cost and value, and adding to the spanner only paths which are “affordable”. This approach was later extended by Kavitha to other families of additive spanners. The second approach, presented by Woodruff , uses a subroutine that finds almost-shortest paths between pairs of nodes, and obtains a faster algorithm at the expense of a slightly worst sparsity guarantee. The third approach, presented by Knudsen , is based on repeatedly going over pairs of nodes, and adding a shortest path between a pair of nodes to the spanner if their current distance in the spanner is too large. Unfortunately, direct implementation in the congest model of the known sequential algorithms is highly inefficient. We are not aware of fast distributed algorithms that allow the computation of the cost and value of paths needed for the algorithm of . Similarly, for , the almost-shortest paths subroutine seems too costly for the congest model. The algorithm of needs repeated updates of the distances in the spanner between pairs of nodes after every addition of a path to it, which is a sequential process in essence, and thus we do not find it suitable for an efficient distributed implementation. A different approach for the distributed construction of ( + 6)-spanners could be to adapt a distributed algorithm with different stretch guarantees to construct a ( + 6)-spanner. This approach does not seem to work: the distributed algorithms for constructing ( + 2)-spanners and ( + 4)-spanner are both very much tailored for achieving the desired stretch, and it is not clear how to change them in order to construct sparser spanners with higher stretch. The ( + 8)-spanner construction algorithm starts with clustering, and then constructs a ( + 4)-pairwise spanner between the cluster centers. Replacing the ( + 4)-pairwise spanner by a ( + 2)-pairwise spanner will indeed yield a ( + 6)-all-pairs spanner, as desired. However, even using the sparsest ( + 2)-pairwise spanners , the resulting ( + 6)-spanner may have Õ(n5/3) edges, denser than our new ( + 6)-spanner and than the known ( + 8)-spanner . Thus, we start by presenting a new sequential algorithm for the construction of ( + 6)-spanners, an algorithm that is more suitable for a distributed implementation, and then discuss its distributed implementation. Our construction starts with a clustering phase, and then adds paths that minimize the number of additional edges that need to be added to the spanner. To implement our construction in the congest model, we assign weights to the edges and use our WBFS algorithm to find shortest paths with as few edges as possible that are not yet in the spanner. Note that although the graph and the spanner we construct for it are both unweighted, the ability of our multiple WBFS algorithm to handle weights is crucial for our solution. A ( + 6)-spanner must contain $n^{4/3}/2^{O(\sqrt{\log n})}$ edges . The best sequential algorithms construct a spanner with O(n4/3) edges. Our distributed algorithm constructs a spanner with O(n4/3log4/3n) edges, which is slightly denser than optimal but still sparser than the O(n4/3log3n) edges in the fast sequential construction of . Related work ------------ A natural approach for building multiple (unweighted) BFS trees in the congest model is to start the construction simultaneously from all sources, sending pairs composed of a source id and the distance the message have traversed so far. The main challenge in this approach appears when multiple such pairs should be sent on one edge in a single round. In this case, the algorithm needs to locally prioritize the message, in a way that will not compromise the correctness of the constructed BFS trees. One approach for prioritizing the messages is by source id, i.e., have a queue at each node sorted by source id. However, this alone does not work and might create incorrect BFS trees. Attempts to overcome the problems of this approach by having a different queue for each outgoing edge are also insufficient . An elegant way to resolve this is to sort messages not by source id, but by the distance they traversed so far, with a preference to messages that traversed smaller distances. Such an algorithm was suggested by Lenzen and Peleg , finishing in (∣S∣ + D − 1) rounds when executed form a set S of sources. We base our multiple WBFS construction algorithm on this algorithm. A variant of the Lenzen-Peleg algorithm, with a reduced message complexity, was recently suggested in . Spanners were first introduced in 1989 , and since then have been a topic for wide research due to their abundant applications. Prime examples for the need for sparse spanners can be found in synchronizing distributed networks , information dissemination , compact routing schemes , and more. Distributed constructions of various spanners have been widely studied . Lower bounds were given in . However, obtaining an efficient and sparse ( + 6)-all-pairs spanner has remained an open question . Several lower bounds for the time complexity of spanner construction in the congest model were presented in , but these are applicable only to pairwise spanners with a bounded number of pairs, and not to all-pairs spanners. A lower bound from states that the construction of a spanner with $\tilde O(n^{4/3})$ edges, such as the one we build, must take $\tilde\Omega(n^{3/8})$ rounds. This lower bound does not take into account the bandwidth restrictions at all (it is proven for the local model), and so we believe that a higher lower bound for the congest model should apply, but this is left as an intriguing open question. Preliminaries ============= All graphs in this work are simple, connected and undirected. A graph can be unweighted, G = (V, E), or weighted G = (V, E, w) with w : E → {0, …, W}, in which case we assume W ∈ poly(n). Given a path ρ in a weighted graph G, we use ∣ρ∣ to denote the length of ρ, which is the number of edges in it, and w(ρ) to denote the weight of the path, which is the sum of its edge-weights. The distance between two nodes u, v in a graph G, denoted δG(u, v), is the minimum length of a path in G connecting u and v. The diameter of a graph (weighted or unweighted) is D = maxu, v ∈ V{δG(u, v)}. We consider the congest model of computation, where the nodes of a graph communicate synchronously by exchanging O(logn)-bit messages along the edges. The goal is to distributively solve a problem while minimizing the number of communication rounds. ##### WBFS trees: We are interested in a weighted BFS tree, which consists of all lightest shortest paths from the root, formally defined as follows. Given a connected, weighted graph G = (V, E, w) and a node s ∈ V, a weighted BFS tree (WBFS) for G rooted at s is a spanning tree Ts of G satisfying the following properties: 1. For each v ∈ V, the path from s to v in T is a shortest path in G between s and v. 2. For each v ∈ V, no shortest path from s to v in G is lighter than the path from s to v in T. We emphasize that this is different than requiring a subgraph containing all lightest paths from the root. One may wonder if a WBFS tree always exists, but this is easily evident by the following refinement of a (sequential) BFS search, returning a WBFS tree: go over the nodes in an order of non-decreasing distances from the source s, starting with w(s) = 0; each node v chooses as a parent a neighbor u that was already processed and minimizes w(v) = w(u) + w(u, v), and adds the edge {u, v} to the tree. Each node has a single parent, so this is indeed a tree; the node ordering guarantees that this is indeed a BFS tree, assuring (i); and the parent choice guarantees the paths are lightest among the shortest, assuring (ii). In fact, this can be seen as an algorithm implementing a consistent tie-breaking strategy between paths of equal lengths, and thus, it is not surprising its output is a tree. ##### Spanners: Given a graph G = (V, E), a subgraph H = (V, Eʹ) of G is called an (α, β)-spanner if for every u, v ∈ V it holds that δH(u, v) ≤ αδG(u, v) + β. The parameters α and β are called the stretch parameters. When α = 1, such a spanner is called a purely additive spanner. In this paper we focus on purely additive ( + 6)-spanners, i.e., α = 1 and β = 6. For completeness, we mention that when β = 0, such a spanner is called a multiplicative spanner. In addition, while sometimes the stretch parameters need to be guaranteed only for some subset of all the pairs of nodes of the graph (such as in pairwise spanners), we emphasize that our construction provides the promise of a + 6 stretch for all pairs. Multiple Weighted BFS Trees =========================== In the congest model, the problem of finding a WBFS tree requires each node to know its parent in the WBFS tree, and the unweighted and weighted distances to the source within the tree. This allows the node to send messages to the source node through the lightest among all shortest paths. When there are multiple sources, each node should know the parent leading to each of the sources in S. We define data structures for representing multiple WBFS trees. Given a node v ∈ V, the *S-proximity-list* (or proximity list for short) of v, noted ${{\textnormal{\texttt{PL}}}^\ast\_v}$, is an ascending lexicographically ordered list of triples (d(s, v), s, w(s, v)), where d(s, v) and w(s, v) are the length and weight of the path from s to v in Ts. Two different triples are ordered such that (d(s, v), s, w(s, v)) < (d(t, v), s, w(t, v)) if d(s, v) < d(t, v), or d(s, v) = d(t, v) and s < t, where s and t may be compared by any predefined order on the node identifiers. Note that Ts contains a single path from s to v, so ${{\textnormal{\texttt{PL}}}^\ast\_v}$ cannot contain two triplets with d(s, v) = d(t, v), s = t and w(s, v) ≠ w(t, v). The *S-path-map* (or path-map for short) of v is a mapping from each source s ∈ S to the parent of v in Ts, noted by ${{\textnormal{\texttt{PM}}}^\ast\_v}$. The list ${{\textnormal{\texttt{PM}}}^\ast\_v}$ is sorted with respect to the order of ${{\textnormal{\texttt{PL}}}^\ast\_v}$, such that the first records of ${{\textnormal{\texttt{PM}}}^\ast\_v}$ belong to sources closest to v. Algorithm [alg: parallel weighted bfs], which constructs multiple WBFS trees from a set S in the congest model, is based on carefully extending the distributed Bellman-Ford-based algorithm of Lenzen et al. . The heart of the algorithm is a loop (Line [line: for loop]), and each iteration of it takes a single round in the congest model. We show that ∣S∣ + D − 1 iterations of the loop suffice in order to construct the desired WBFS trees. The algorithm builds the WBFS trees by gradually updating the proximity list and the path map of each node. Each round is composed of two phases: updating the neighbors about changes in the proximity list, and receiving updates from other nodes. The path map is only used by the current node, and therefore changes to it are not sent. Ideally, each node would update its neighbors regarding all the changes made to its proximity list. However, due to bandwidth restrictions, a node cannot send the entire list in each round. Therefore, at each round each node sends to all of its neighbors the lexicographically smallest triplet in its proximity list that it has not yet sent, while maintaining a record noting which triplets have been sent and which are waiting. Each triplet is only sent once, though a node may send multiple triplets regarding a single source. A node uses the messages received in the current round in order to update its proximity list and path map for the next round. A triplet (ds, s, ws) received by a node v from a neighbor u represents the length ds and weight ws of some path ρ from s to u in the graph. The node v then considers the extended path ρʹ = ρ ∘ v from s to v, compares it to its currently known best path from s to v, and updates the proximity list and path map in case a shorter path has been found, or a lighter path with the same length. [t] Lv ← () [alg: parallel weighted bfs] To prove correctness, we generalize the proof of to handle weights, and show that our algorithm solves the weighted (S, d, k)-detection problem: each node should learn which are the sources from S closest to it, but at most k of them and only up to distance d. This is formally defined as follows. [def:weighted sdk detection] Given a weighted graph G = (V, E, w), a subset S ⊆ V of source nodes, and a node v ∈ V, let ${{\textnormal{\texttt{PL}}}^\ast\_v}$ denote the S-proximity-list and let ${{\textnormal{\texttt{PM}}}^\ast\_v}$ denote the path map of the node v. The weighted (S, d, k)-detection problem requires that each node v ∈ V learns the first min{k, λvd} entries of ${{\textnormal{\texttt{PL}}}^\ast\_v}$ and ${{\textnormal{\texttt{PM}}}^\ast\_v}$, where λvd is the number of sources s ∈ S such that d(s, v) ≤ d. Given a node v, $\operatorname{\textnormal{\texttt{PL}}}\_v$ is a variable in Algorithm [alg: parallel weighted bfs] holding the proximity list of v, and we denote by $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ the state of the list $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the beginning of round r of the algorithm, and by $\operatorname{\textnormal{\texttt{PL}}}\_v^{(\infty)}$ the value of $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the end of the algorithm. Recall that ${{\textnormal{\texttt{PL}}}^\ast\_v}$ is the true proximity list, so our goal is proving $\operatorname{\textnormal{\texttt{PL}}}\_v^{(\infty)} = {{\textnormal{\texttt{PL}}}^\ast\_v}$, i.e., proving that the algorithm obtains the correct values of the proximity list. We use similar notations for the path map ${{\textnormal{\texttt{PM}}}^\ast\_v}$. Since the records of $\operatorname{\textnormal{\texttt{PM}}}\_v$ are updated under the same conditions as the records of $\operatorname{\textnormal{\texttt{PL}}}\_v$, the correctness of $\operatorname{\textnormal{\texttt{PM}}}\_v$ at the end of the algorithm with respect to ${{\textnormal{\texttt{PM}}}^\ast\_v}$ immediately follows, and we omit the details. We start by showing that if there was no bound on the number of rounds, then the values of $\operatorname{\textnormal{\texttt{PL}}}\_v$ would have eventually converged to the true values of ${{\textnormal{\texttt{PL}}}^\ast\_v}$. [lem: eventual distances are correct] Given a graph G = (V, E, w) and a set S ⊆ V, if we let the for loop in Line [line: for loop] of Algorithm [alg: parallel weighted bfs] to run forever, then there exists a round $r\_0\in {\mathop{\mathbb{N}}}$ such that no node v ∈ V sends messages or modifies $\operatorname{\textnormal{\texttt{PL}}}\_v$ after round r0. Moreover, $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)} = {{\textnormal{\texttt{PL}}}^\ast\_v}$, i.e., for every $(d\_s,s,w\_s)\in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$, it holds that ds = d(v, s) and ws = min{w(ρ) ∣ ρconnects vwith s, and ∣ρ∣ = ds}. In each iteration of the algorithm, each node v ∈ V sends to all of its neighbors the first triplet (ds, s, ws) such that $\operatorname{\textnormal{\texttt{sent}}}\_v(d\_s,s,w\_s) = \operatorname{\textnormal{\texttt{FALSE}}}$. Each triplet received is sent at most once. Therefore, if we show the existence of a round r0 where for each v ∈ V, all messages in $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$ have been sent in previous rounds, it implies that no message is sent in round r0, and hence $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0 + 1)} = \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. This claim is applied inductively, concluding that for any round r > r0, it holds that $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)} = \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. We prove the existence of the round r0 by showing that the number of messages that are sent by each node v ∈ V is finite. Each triple (ds, s, ws) is inserted into $\operatorname{\textnormal{\texttt{PL}}}\_v$ when a message is received from another node u ∈ V, and such a message implies that a path from s to v through u of length ds and weight ws exists. Thus, the number of messages (ds, s, ws) sent by a node v is upper bounded by the number of paths from s to v of length ds. Furthermore, the algorithm does not insert a message into $\operatorname{\textnormal{\texttt{PL}}}\_v$ if it has already inserted a lexicographically smaller message from the same source. As the graph is finite, the number of paths is bounded, and eventually no node adds further triplets to its lists or sends additional messages. It remains to show that for all s ∈ S and v ∈ V it holds that $(d(s,v), s, w(s,v)) \in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. First, we show that if a triplet (ds, s, ws) is added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ in some round r, then there exists a path ρ from s to v such that ds = ∣ρ∣ and ws = w(ρ). For a source s ∈ S, we insert the triplet (0, s, 0) into $\operatorname{\textnormal{\texttt{PL}}}\_s$ at the beginning of the algorithm, so the claim is true at initialization. Assume there exists a round where a triplet (ds, s, ws) is inserted into $\operatorname{\textnormal{\texttt{PL}}}\_v$ but no corresponding path exists, and let r be the first such round. This implies that there exists a node u ∈ V that is a neighbor of v, which sends the message (ds − 1, s, ws − w(u, v)) to v in round r. The triplet (ds − 1, s, ws − w(u, v)) must have been inserted into $\operatorname{\textnormal{\texttt{PL}}}\_u$ in some round rʹ < r, and by the minimality of r there exists a path ρ from s to u where ∣ρ∣ = ds − 1 and w(ρ) = ws − w(u, v). Since u is a neighbor of v, the path ρʹ = ρ ∘ v is valid, satisfying ∣ρʹ∣ = ds and w(ρʹ) = ws, contradicting the assumption. To complete the proof, we claim that the correct triplet (d(s, v), s, w(s, v)) is indeed added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ at some round of the algorithm, and is not removed. Consider the path from s to v in the WBFS tree Ts, denoted by ρ = (v0 = s, v1, …, vd(s, v) = v). At initialization, the triplet (0, s, 0) is added to $\operatorname{\textnormal{\texttt{PL}}}\_s$. From then, at each round there exists some i ≤ d(s, v) such that $(d(s,v\_i), s, w(s,v\_i)) \in \operatorname{\textnormal{\texttt{PL}}}\_{v\_i}$ and $\operatorname{\textnormal{\texttt{sent}}}\_{v\_i}(d(s,v\_i), s, w(s,v\_i))=\operatorname{\textnormal{\texttt{FALSE}}}$. Since we proved that this message is eventually sent, this implies that in the beginning of the next round, the triplet (d(s, vi + 1), s, w(s, vi + 1)) is added to $\operatorname{\textnormal{\texttt{PL}}}\_{v\_{i+1}}$. By the definition of a WBFS tree, all other paths from s to v must be longer or not lighter, implying the triplet cannot be discarded for a lexicographically smaller triplet. This concludes that for any source s and node v, it holds that $(d(s,v), s, w(s,v)) \in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. Lemma [lem: eventual distances are correct] shows that without the limit on the number of rounds, the algorithm would compute the right values; however, it does not bound the number of rounds needed for this to occur. Next, we show that ∣S∣ + D − 1 rounds suffice. We cannot apply the claims of directly, since the existence of weights restricts the number of viable solutions even further, causing more updates to the proximity list and an increase in the number of messages sent. However, we do use a similar technique: we bound the number of rounds in which the k smallest entries of $\operatorname{\textnormal{\texttt{PL}}}\_v$ can change. For an entry $(d\_s,s,w\_s) \in \operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$, let ℓv(r)(ds, s, ws) denote the index of the entry in the lexicographically ordered list $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ at the beginning of round r. For completeness, we define ℓv(r)(ds, s, ws) = − ∞ if (ds, s, ws) did not appear in $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the beginning of round r, and $\operatorname{\textnormal{\texttt{PL}}}\_v=\infty$ if the triplet was removed from $\operatorname{\textnormal{\texttt{PL}}}\_v$ before the beginning of this round. Note that a removed triplet is never returned to the list, since the lexicographical order is transitive. [helper lemma] For a triplet (ds, s, ws), the following holds: 1. ℓv(r)(ds, s, ws) is non-decreasing with r. 2. When a triplet (ds, s, ws) is sent from a node u to a node v at round r, it causes the addition of a new triplet (dʹs, s, wʹs) to $\operatorname{\textnormal{\texttt{PL}}}\_v$ at the end of round r. This triplet satisfies dʹs = ds + 1, wʹs = ws + w(u, v), and ℓu(r)(ds, s, ws) ≤ ℓv(r + 1)(dʹs, s, wʹs). Part (i) follows from the fact that the number of triplets below (ds, s, ws) cannot decrease. To prove part (ii), we show that all the triplets below (ds, s, ws) in $\operatorname{\textnormal{\texttt{PL}}}\_u$ are sent from u to v and added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ before (ds, s, ws) is sent and added. Part (i) is a consequence of the method used by our algorithm for managing the list $\operatorname{\textnormal{\texttt{PL}}}\_v$. According to our algorithm, triplets are not removed from $\operatorname{\textnormal{\texttt{PL}}}\_v$ when they are sent. The only case in which a triplet (dt, t, wt) is removed from $\operatorname{\textnormal{\texttt{PL}}}\_v$ is when a lexicographically smaller triplet (dʹt, t, wʹt) is added to the list instead. When this happens in round r, it holds that ℓv(r)(dt, t, wt) ≥ ℓv(r + 1)(dʹt, t, wʹt), since the new triplet is lexicographically smaller. Hence, for every other triplet $(d\_s,s,w\_s)\in \operatorname{\textnormal{\texttt{PL}}}\_v$, the number of lexicographically smaller triplets in $\operatorname{\textnormal{\texttt{PL}}}\_v$ cannot decrease throughout the algorithm. We now turn to prove part (ii) of the lemma. Note that the only claim which does not follows trivially from the algorithm is the inequality ℓu(r)(ds, s, ws) ≤ ℓv(r + 1)(dʹs, s, wʹs). By the fact that the triplet (ds, s, ws) is sent by the node u in round r, we conclude that the ℓu(r)(ds, s, ws) − 1 triplets preceding it in the list $\operatorname{\textnormal{\texttt{PL}}}\_u^{(r)}$ have already been sent by u in earlier rounds, and arrived at the node v. For each such triplet (dt, t, wt), either dt ≤ ds, or t < s and dt = ds. Therefore, when added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ as (dt + 1, t, wt + w(u, v)) it is lexicographically smaller than (dʹs, s, wʹs). At round r, either (dt + 1, t, wt + w(u, v)) is in $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ or it was replaced by a lexicographically smaller triplet containing t. Thus, there are at least ℓu(r)(ds, s, ws) − 1 triplets smaller than (dʹs, s, wʹs) in $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r+1)}$, and hence ℓu(r)(ds, s, ws) ≤ ℓv(r + 1)(dʹs, s, wʹs). Lemma [helper lemma] implies that as the algorithm progresses, messages at higher indexes of the proximity list are sent and updated. This can be used to obtain an upper bound on the round in which a triplet at a certain index of the proximity list can be sent or received, as formalized by the next lemma. [lem: first entries stabilize] In round $r\in {\mathop{\mathbb{N}}}$ of Algorithm [alg: parallel weighted bfs], a node v ∈ V can: 1. send a message (ds, s, ws) only if ds + ℓv(r)(ds, s, ws) ≥ r 2. add to $\operatorname{\textnormal{\texttt{PL}}}\_v$ a triplet (ds, s, ws) only if ds + ℓv(r + 1)(ds, s, ws) > r Part (i), when put in words, is rather intuitive: while a triplet might need to wait before being sent, the waiting time is bounded from above by the distance the triplet has traversed from its source, plus the number of triplets that were to be sent before it. Part (ii) is complementary to part (i): the time before a triplet is added, is, once more, bounded by the distance it traversed plus the number of lexicographically smaller triplets. We start by showing that, for a given round r, if Lemma [lem: first entries stabilize](i) holds for all nodes then Lemma [lem: first entries stabilize](ii) holds as well. Consider a triplet (dʹs, s, wʹs) that is added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ as a result of a message (ds, s, ws) sent from u to v in round r, where dʹs = ds + 1 and wʹs = ws + w(u, v). Lemma [lem: first entries stabilize](i) implies that ds + ℓu(r)(ds, s, ws) ≥ r, and by Lemma [helper lemma](ii) we have that ℓv(r + 1)(dʹs, s, wʹs) ≥ ℓu(r)(ds, s, ws). As dʹs > ds, we conclude dʹs + ℓv(r + 1)(dʹs, s, wʹs) > ds + ℓu(r)(ds, s, ws) ≥ r, which implies Lemma [lem: first entries stabilize](ii). Next, we prove by induction that both parts of the lemma hold. In round 1, Lemma [lem: first entries stabilize](i) holds trivially, since by definition ℓv(1)(ds, s, ws) ≥ 1. Assume that Lemma [lem: first entries stabilize] holds at round r − 1; we show the lemma holds at round r. Since Lemma [lem: first entries stabilize](i) implies Lemma [lem: first entries stabilize](ii), it is sufficient to show that every message (ds, s, ws) sent by some node v ∈ V in round r satisfies ds + ℓv(r)(ds, s, ws) ≥ r. Observe that if (ds, s, ws) is sent by a node v in round r, then the triplet must have been added to $\operatorname{\textnormal{\texttt{PL}}}\_v$ in some round rʹ ≤ r − 1. If rʹ = r − 1, according to the induction hypothesis, Lemma [lem: first entries stabilize](ii) holds and ds + ℓv(r)(ds, s, ws) > r − 1, implying ds + ℓv(r)(ds, s, ws) ≥ r, since all the terms are integers. Otherwise rʹ < r − 1. In this case, in round r − 1 the triplet (ds, s, ws) appeared in $\operatorname{\textnormal{\texttt{PL}}}\_v$ and was not yet sent. Since (ds, s, ws) is sent in round r, a different triplet (dt, t, wt) with t ≠ s must have been sent in round r − 1, implying: ds + ℓv(r − 1)(ds, s, ws) > dt + ℓv(r − 1)(dt, t, wt). By Lemma [helper lemma](i), we have that ℓv(r)(ds, s, ws) ≥ ℓv(r − 1)(ds, s, ws), and combined with the induction hypothesis for Lemma [lem: first entries stabilize](i) in round r − 1 we conclude: ds + ℓv(r)(ds, s, ws) ≥ ds + ℓv(r − 1)(ds, s, ws) > dt + ℓv(r − 1)(dt, t, wt) ≥ r − 1. This gives that ds + ℓv(r)(ds, s, ws) ≥ r, since all the terms are integers. Lemma [lem: eventual distances are correct] implies that eventually, the lists $\operatorname{\textnormal{\texttt{PL}}}\_v$ converge to contain the correct values, and Lemma [lem: first entries stabilize] restricts the number of rounds in which specific list entries may change. From this, we conclude that the algorithm solves the weighted (S, d, k)-detection problem. [lem: first r entries are correct] Given an instance of the weighted (S, d, k)-detection problem, for every v ∈ V and round r of an execution of Algorithm [alg: parallel weighted bfs] with r ≥ min{d, D} + min{k, ∣S∣}, the truncation of $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r)}$ to the first min{k, λvd} entries, where λvd is the number sources s ∈ S such that d(s, v) ≤ d, solves weighted (S, d, k)-detection problem. This lemma says that the truncated list is correct at the beginning of the relevant round. To prove it, we use Lemma [lem: first entries stabilize](ii) to show that the values in the truncated list cannot change at round r or later, and Lemma [lem: eventual distances are correct] to deduce they are correct. Assume w.l.o.g that d ≤ D, as D bounds the distance to any source, and k ≤ ∣S∣, as otherwise v needs to learn about all sources. By Lemma [lem: eventual distances are correct], there is a round r0 when all entries of $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$ are correct, and let (ds, s, ws) be a triplet in one of the first min{k, λvd} entries of $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)}$. Since (ds, s, ws) is one of the first λvd entries and $\operatorname{\textnormal{\texttt{PL}}}\_v^{(r\_0)} = {{\textnormal{\texttt{PL}}}^\ast\_v}$, we have ds ≤ d. Let r be the round when (ds, s, ws) is inserted to the list $\operatorname{\textnormal{\texttt{PL}}}\_v$. By Lemma [lem: first entries stabilize](ii), r < ds + ℓv(r + 1)(ds, s, ws). By Lemma [helper lemma](i), when the triplet is inserted to the list, it is already placed in one of the first min{k, λvd} entries, i.e., ℓv(r + 1)(ds, s, ws) ≤ min{k, λvd} ≤ k. Hence, r < ds + ℓv(r + 1)(ds, s, ws) ≤ d + k. Since this claim holds for any of the first min{k, λvd} entries, these were all correct at the beginning of round d + k, and in all the succeeding rounds. The construction of multiple WBFS trees is an instance of the (S, D, ∣S∣)-detection problem. Lemma [lem: first r entries are correct] shows that after ∣S∣ + D − 1 rounds of Algorithm [alg: parallel weighted bfs] on such an instance, all the entries of the list $\operatorname{\textnormal{\texttt{PL}}}\_v^{({\ensuremath{\left\|S\right\|}}+D)}$ are correct, yielding the main result of this section. thm: weighted bfs Given a weighted graph G = (V, E, w) and a set of nodes S ⊆ V, there exists an algorithm for the congest model that constructs a WBFS tree rooted at s, for every s ∈ S, in ∣S∣ + D − 1 rounds. A ( + 6)-Spanner Construction ============================= In this section we discuss the distributed construction of ( + 6)-spanners. First, we present a template for constructing a ( + 6)-spanner and analyze the stretch and sparsity of the constructed spanner. Then, we provide an implementation of our template in the congest model and analyze its running time. A cluster Ci around a cluster center ci ∈ V is a subset of the set of neighbors of ci in G. A node belonging to a cluster is clustered, while the other nodes are unclustered. Our algorithm starts by randomly choosing cluster centers, and adding edges between them to their neighbors, where each neighbor arbitrarily chooses a single center to connect to. Then, additional edges are added, to connect each unclustered node to all its neighbors. Next, shortest paths between clusters are added to the spanner. In order to find these shortest paths in the congest model, we use the WBFS construction algorithm to build WBFS trees from random sources. At the heart of our algorithm stands the path-hitting framework of Woodruff : a shortest path in the graph which has many edges between clustered nodes, must go through many clusters. This fact is used in order to show that a path with many missing edges (edges not in H) is more likely to have an adjacent source of a WBFS tree, and thus it is well approximated by a path within the spanner. Woodruff’s algorithm starts with a similar clustering step. However, in order to add paths between clusters, it uses an involved subroutine that finds light almost-shortest paths between pairs of nodes. This subroutine seems too global to be implemented efficiently in a distributed setting, so in our construction it is replaced by only considering lightest shortest paths, which we do using the WBFS trees defined earlier. Our algorithm constructs a ( + 6)-spanner with O(n4/3log4/3n) edges in Õ(n2/3 + D) rounds, as stated next. thm: 6ap There exists an algorithm for the congest model that constructs a ( + 6)-spanner with O(n4/3log4/3n) edges in $O\left(\frac{n^{2/3}}{\log^{1/3}n} +D\right)$ rounds and succeeds w.h.p. Lemmas [lemma: 6ap size] and [lemma: 6ap stretch] analyze the size and stretch of Algorithm ${\tt 6AP}$ given below. The number of rounds of its distributed implementation is analyzed in Lemma [lemma: 6ap complexity], giving Theorem [thm: 6ap]. We use c > 2 to denote a constant that can be chosen according to the desired exponent of 1/n in the failure probability. ### Algorithm ${\tt 6AP}$ Input: a graph G = (V, E), a constant c > 2; Output: a subgraph H of G; Initialization: n ← ∣V∣; H ← (V, ∅); k ← 1 #### Clustering. Pick each node as a cluster center w.p. $\frac{c}{n^{1/3}\log^{1/3}n}$, and denote the set of selected nodes by C = {c1, c2, …}. For each ci, initialize a cluster Ci ← ∅. For each node v ∈ V, choose a neighbor ci of v which is a cluster center, if such a neighbor exists, add the edge (v, ci) to H, and add v to Ci. If none of the neighbors of v is a cluster center, add to H all the edges adjacent to v. Let H0 ← H. #### Path Buying. While $k\leq \frac{8cn^{2/3}}{\log^{1/3} n}$ do: 1. Sk ← ∅ 2. Add each cluster center ci ∈ C to Sk w.p. $\frac{8c^2\log n}{k}$, independently of the other centers 3. For each pair (ci, cj) ∈ C × Sk: 1. A ← ∅ /\* A is a set of paths \/ 2. For each v* ∈ Cj: 1. Among all the shortest paths from ci to v, let Pv be a path with minimum ∣Pv \ H0∣ 2. If ∣Pv \ H0∣ < 2k, add Pv to A 3. If A ≠ ∅, add to H one of the shortest among the paths of A 4. k ← 2k [lemma: 6ap size] Algorithm ${\tt 6AP}$ outputs a subgraph H of G with O(n4/3log4/3n) edges, with probability at least 1 − O(n− c + 1). The algorithm starts with H = (V, ∅) and only adds edges from G, so H is indeed a subgraph of G over the same node set. In the first part of the clustering phase, each node adds to H at most one edge, connecting it to a single cluster center, for a total of O(n) edges. Then, the probability that a node of degree at least n1/3log4/3n is left unclustered is at most $\left(1	arxiv_0000769
Finsler Gravitational Waves of (α, β)-Type and their Observational Signature ================================================================================ We introduce a new class of (α, β)-type exact solutions in Finsler gravity closely related to the well-known pp-waves in general relativity. Our class contains most of the exact solutions currently known in the literature as special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, and we investigate the physical effect of such waves. More precisely, we compute the Finslerian correction to the radar distance along an interferometer arm at the moment a Finslerian gravitational wave passes a detector. We come to the remarkable conclusion that the effect of a Finslerian gravitational wave on an interferometer is indistinguishable from that of standard gravitational wave in general relativity. Along the way we also physically motivate a modification of the Randers metric and prove that it has some very interesting properties. Introduction ============ Even though in the general theory of relativity (GR) the geometry of spacetime is modelled by a metric of Lorentzian signature, there is no clear physical principle, nor experimental evidence, that tells us that this spacetime geometry should necessarily be. In fact, as suggested already in 1985 by Tavakol and Van den Bergh, the axiomatic approach by Ehlers, Pirani and Schild (EPS) is compatible with Finsler geometry, a natural extension of geometry. This was originally overlooked due to too restrictive differentiability assumptions, as recently pointed out in and then worked out in detail in. Other axiomatic approaches also allow for types of geometry more general than the type used in GR, see e.g.. This indicates that such types of geometries should not a priori be excluded from our theories and motivates the study of extensions of general relativity based on more general spacetime geometries. In this regard Finsler geometry is the natural candidate as it provides the most general geometric framework that is still compatible with the clock postulate in the usual sense, namely that the proper time interval measured by an observer between two events can be defined as the length of its worldline connecting these events, in this case the Finslerian length rather than the length. We remark that Weyl geometry, another generalization of Lorentzian geometry, is also compatible with the clock postulate, but in that case the definition of proper time has to be revised. Further motivation for the study of Finsler spacetime geometry comes from quantum gravity phenomenology. Inspired by various approaches to quantum gravity, a generic feature of phenomenological or effective quantum gravity models is the presence of Planck-scale modified dispersion relations (MDR), related to departure from (local) Lorentz symmetry, which may manifest either in the sense of Lorentz invariance violation (LIV) or in the sense of deformed Lorentz symmetry. It turns out that such MDRs generically induce a Finsler geometry on spacetime. The mathematical details of this were investigated in ; see e.g. for applications to specific quantum gravity phenomenology models. Here we consider the (action-based) approach to Finsler gravity outlined in. Structurally the theory is completely analogous to general relativity, but Einstein’s field equation is replaced by Pfeifer and Wohlfarth’s field equation. For spacetimes the latter reduces to the former. Although any solution to the field equations of GR is a solution in Finsler gravity, not many exact, properly Finslerian solutions are known as of yet. To the best of our knowledge the only ones currently known in the literature are the (m-Kropina type) Finsler pp-waves and their generalization as Very General Relativity (VGR) spacetimes, and the Randers pp-waves. Here we introduce a large class of exact vacuum solutions that contains most of the aforementioned solutions as special cases, the only exception being those solutions in that are not of pp-wave type. Namely, we prove that any Finsler metric constructed from a metric α and a 1-form β that is covariantly constant with respect to α, is an exact vacuum solution in Finsler gravity if α is a vacuum solution in general relativity. We classify all such solutions, leading to two possibilities: either α is flat Minkowski space, or α is a pp-wave. Our solutions are (α, β)-metrics of Berwald type. The natural question that arises is whether and how such spacetimes can be physically distinguished from their general relativistic counterparts. To answer this question we consider the versions of our exact solutions, which may be interpretted as Finslerian gravitational waves, and we study their physical effect. More precisely, we ask the question what would be observed in an interferometer experiment when such a Finslerian gravitational wave would pass the earth, and what would be the difference with a classical general relativistic gravitational wave. The relevant observable measured in interferometer experiments is essentially the radar distance, Although at first sight the expression for the Finsler radar length looks different from the corresponding expression GR, we show that this is nothing but a coordinate artifact. Remarkably, when the two expressions are interpreted correctly in terms of observable quantities, it becomes clear that there is in fact no observational difference between the Finsler and GR case, at least as far as radar distance measurements are concerned. We discuss the significance of this. To the best of our knowledge this is the first time an explicit expression for the Finslerian Radar length has been obtained in the case of finite spacetime separations, and as such our work may be seen as a proof of concept. In contrast, the radar length for infinitesimal separations has been studied in. We do point out that our results rely on the assumption that the amplitude of the gravitational wave, as well as the parameter λ that characterized the departure from (pseudo)-Riemannian geometry, are sufficiently small, so that a certain perturbative expansion is valid. This nevertheless seems physically justified. We argue in a heuristic manner that up to first order in λ, any physically viable (α, β)-metric can be equivalently described by a slightly modified version of a standard Randers metric. Indeed, the causal structure of the standard Randers metric does not in general have a straightforward physical interpretation. We therefore propose to modify the Randers metric slightly, only changing some relative signs in different subsets of the tangent bundle. We then prove that these modified Randers metrics have the nice property that their causal structure is completely equivalent to the causal structure of some auxiliary metric. This analysis is done in full generality, i.e. not just for our exact solutions. In the special case, however, that the defining 1-form of the Randers metric is covariantly constant (as is the case for our solutions) we prove that not only the causal structure, but also the affine structure of the Finsler metric coincides with that of the auxilliary (pseudo)-Riemannian metric, i.e. the timelike, spacelike and null geodesics of the Finsler metric can be understood, respectively, as the timelike, spacelike and null geodesics of the auxiliary (pseudo)-Riemannian metric. This leads to the particularly nice property that the existence of radar neighborhoods is guaranteed, i.e. that given an observer and any event in spacetime, there is (at least locally) exactly one future pointing light ray and one past pointing light ray that connect the event to the worldline of the observer. This is of essential importance in our work, because without this property the notion of radar distance would not even make sense. Structure of this article ------------------------- The paper is organized as follows. We begin in Section [sec:Finslergravity] with a discussion of Finsler geometry and the core ideas behind Finsler gravity. Then in Section [sec:abmetrics] we introduce (α, β)-metrics, and in particular Randers metrics and discuss their relevance to Finsler gravity. We then introduce our new solutions to the field equations and show that after linearization these solutions may be interpretted as Finslerian gravitational waves. Next, in section [sec:Randers] we propose our modification of the standard Randers metric and prove that it has very satisfactory properties with respect to its causal structure, affine structure, Lorentzian signature, etc. We conclude in section [sec:discussion]. Finsler Gravity =============== Finsler spacetime geometry -------------------------- A note about causal structure and physical interpretation --------------------------------------------------------- Given a Finsler spacetime geometry, it is natural to postulate, in analogy with GR, that matter travels along timelike geodesics and light travels on null geodesics. The generalization of the notion of null direction is mathematically straightforward. A vector yu at a point xμ is said to be null (or lightlike) if F(x, y)2 = gμν(x, y)yμyν = 0. However, the structure of the light cone, composed of such null vectors, may be non-trivial. In GR it is always the case that the light cone separates the tangent space at each point into three connected components, that we may interpret as forward-pointing timelike vectors, backward-pointing timelike vectors, and spacelike vectors, respectively. It is then a consequence that a timelike vector is one that has positive (or negative, depending on the convention) Riemannian norm. For a generic Finsler spacetime geometry these properties of the lightcone structure are by no means guaranteed and as such it is not obvious in general how to even define what one means a by timelike vector. It certainly does not suffice to define them as positive length vectors. We do not discuss this issue any further in its full generality here. Only in the specific case of the Randers metric, in Section [sec:Randers], will we dive into the details. We argue that the causal structure of the standard Randers metric does not have a straightforward physical interpretation, but we prove that, by modifying the definition only slightly, the causal structure of such a modified Randers metric has exactly the desirable properties mentioned above in the case of GR, allowing for a straightforward physical interpretation. This will be exploited in Section [sec:radardistance], where we compute the radar distance for a Finslerian gravitational wave of (modified) Randers type passing an interferometer. It is worth mentioning that in the ideal case the (forward and backward) timelike cones should be contained in the subbundle $\mathcal A$. This statement is essentially the condition that geometry is well-defined for all timelike infinitesimal spacetime separations. This property is satisfied by our modified Randers metrics (up to a set of measure zero). It can be argued that it is not strictly necessary for spacelike vectors to be contained in $\mathcal A$, as it would not be possible, not even in principle, to perform any physical experiment that probes such directions. Whether the lightcone should be contained in $\mathcal A$ is a more delicate question, which we will not further explore here. The field equations ------------------- In the context of Finsler gravity, arguably the simplest and cleanest proposal for a vacuum field equation was the one by Rutz. The Rutz equation, Ric = 0, can be derived from the geodesic deviation equation in complete analogy to the way Einstein’s vacuum field equation, Rμν = 0 (to which it reduces in the classical setting), can be derived by considering geodesic deviation. However, it turns out that Rutz’s equation is not variational, i.e. it cannot be obtained by extremizing an action functional. In fact, its variational completion (i.e. the variational equation that is as close as possible to it, in a well-defined sense ) turns out to be the field equation that was proposed by Pfeifer and Wohlfarth in using a Finsler extension of the Einstein-Hilbert action. This is again in complete analogy to the situation in GR, where the vacuum Einstein equation in the form $R\_{\mu\nu} - \frac{1}{2}g\_{\mu\nu}R = 0$ is also precisely the variational completion of the equation Rμν = 0. While in the GR case the completed equation happens to be equivalent to the former, this is not true any longer in the Finsler setting. Although several other proposals have been made as well, we consider the Pfeifer-Wohlfarth equation[1](#fn1) to be by far the most promising, and from here onwards we will refer to it simply as the vacuum field equation in Finsler gravity. We do not show the field equation in full generality here, as its general form is not required for our present purposes. In the case of Berwald spacetimes it can be expressed relatively simply as $$\begin{aligned} \label{eq:BEFEs} \left( F^2 g^{\mu\nu} - 3 y^\mu y^\nu \right) R\_{\mu\nu} = 0\,,\end{aligned}$$ where Rμν is the Finsler Ricci tensor and since we are in a Berwald setting, Rμν = Rμν(x) only depends on x. Clearly the vanishing of the Finsler Ricci tensor is a sufficient condition for a Berwald spacetime to be a solution to Eq.. (α, β)-Metrics ================== (α, β)-metrics – basic definitions -------------------------------------- An important class of Finsler geometries is given by the so-called (α, β)-metrics. Here $\alpha = \sqrt{\|a\_{\mu\nu}\dot x^\mu\dot x^\nu\|}$ and $\beta = b\_\mu\dot x^\nu$ are scalar variables defined in terms of a metric aμν on M and a 1-form bμ on M, and an (α, β)-metric is simply a Finsler metric that is constructed only from α and β, i.e. F = F(α, β). Due to homogeneity it follows that any such F can be written in the standard form F = αϕ(β/α) for some function ϕ, at least whenever α ≠ 0. Well-known examples of (α, β)-metrics are: * Pseudo-Riemannian Finsler metrics F = α; * Randers metrics F = α + β; * Kropina metrics $F = \frac{\alpha^2}{\beta}$; * Exact (α, β)-metric solutions in Finsler gravity ---------------------------------------------------- From the physical viewpoint, (α, β)-metrics allow us to deform a GR spacetime α into a Finsler spacetime by the 1-form β. And it turns out, as we will prove below, that these types of metrics can be used to generalize some of the vacuum solutions to Einstein’s field equations to properly Finslerian vacuum solutions in Finsler gravity. This procedure is possible whenever such a solution admits a covariantly constant vector field, or equivalently, 1-form. Namely: if the Lorentzian metric α solves the classical Einstein equations and the 1-form β is covariantly constant with respect to α then any (α, β)-metric constructed from the given α and β is a solution to the Finslerian field equations. To see why this is true, we first recall the following well-known result (see e.g. section 6.3.2. in ): [prop:coincidingspray] Let F be an (α, β)-metric. If β is covariantly constant with respect to α then F is of Berwald type and the affine connection of F coincides with the Levi-Civita connection of α. If the affine connection of F is the same as the connection of α, the associated curvature tensors and (affine) Ricci tensors are also the same. So if α happens be a vacuum solution to Einstein gravity, i.e. its Ricci tensor vanishes, then it follows that the affine Ricci tensor of F vanishes as well, which implies, by eq., that F is a vacuum solution to Pfeifer and Wohlfarth’s field equation in Finsler gravity. We may summarize this result in the following theorem. [theor:(alpha,beta)solutions] Let F be any (α, β)-metric such that α solves the classical vacuum Einstein equations and β is covariantly constant with respect to α. Then F is a vacuum solution to the field equation in Finsler gravity. In this way (α, β)-metrics provide a mechanism to Finslerize any vacuum solution to Einstein’s field equations, as long as the solution admits a covariantly 1-form, or equivalently a covariantly constant vector field. The theorem generalizes some of the results obtained in for Randers metrics and in for m-Kropina metrics (i.e. VGR spacetimes) to arbitrary Finsler spacetimes with (α, β)-metric. In particular, all pp-wave type solutions in Finsler gravity currently known in the literature are of this type. Let’s investigate this type of solution in some more detail. It turns out that if a vacuum solution α to Einstein’s field equations admits a covariantly constant 1-form β, then either α is flat, or β is necessarily null (see also ). We remark that this result assumes that the spacetime dimension is 1 + 3 and generally is not true in higher dimensions. This leads to two classes of solutions. First class of solutions The first of these possibilities, where α is flat, leads to a class of solutions that can always be written in suitable coordinates in the following way. Right below Eq. we have used the notation $\alpha = \sqrt{\|A\|} = \sqrt{\|a\_{ij}{\text{d}}x^i {\text{d}}x^j\|}$. This should be understood pointwise, i.e. $$\begin{aligned} \alpha = \alpha(y) = \sqrt{\|a\_{ij}{\text{d}}x^i {\text{d}}x^j\|}(y) = \sqrt{\|a\_{ij}{\text{d}}x^i(y) {\text{d}}x^j(y)\|} = \sqrt{\|a\_{ij} y^i y^j\|}.\end{aligned}$$ In other words, we sometimes write α for the function $\sqrt{\|a\_{ij}{\text{d}}x^i {\text{d}}x^j\|}:y\mapsto \sqrt{\|a\_{ij} y^i y^j\|}$, and at other times we write α for its value $\sqrt{\|a\_{ij} y^i y^j\|}$ at y. It should always be clear from context what is meant. Second class of solutions The second possibility, that β is null, leads to a class of solutions that seems to be more interesting. In this case α is CCNV spacetime metric, meaning that it admits a covariantly constant null vector (CCNV), namely in this case β, or rather its vector equivalent via the isomorphism induced by α. CCNV metrics are also known as pp-waves (plane-fronted gravitational waves with parallel rays) and have been studied in detail in (see section 24.5 in for a summary). It is an elementary result that by choosing suitable coordinates (u, v, x1, x2), such α and β can always be expressed in the form $$\begin{aligned} A &= -2{\text{d}}u \left({\text{d}}v + H(u,x)\, {\text{d}}u + \,W\_a(u,x)\,{\text{d}}x^a\right) +h\_{ab}(u,x) {\text{d}}x^a {\text{d}}x^b, \label{eq:original\_pp\_wave\_metric} \\ \beta &= {\text{d}}u,\label{eq:original\_1form}\end{aligned}$$ where xa = x1, x2 and hab is a two-dimensional Riemannian metric. This holds irrespective of whether α is a solution to Einstein’s field equations or not. If α is additionally assumed to be a vacuum solution, as in Theorem [theor:(alpha,beta)solutions], it turns out that the expression for A can be simplified even more without changing the form of β*. To see this, we first consider only the metric A. Since A* is a vacuum solution to Einstein’s field equations, it follows that the functions Wa can be eliminated and hab may be chosen as δab, by a suitable coordinate transformation (section 24.5 in ). The metric then takes the form $$\begin{aligned} A = -2{\text{d}}u \left({\text{d}}v + H(u,x)\, {\text{d}}u\right) +\delta\_{ab} {\text{d}}x^a {\text{d}}x^b. \label{eq:reduced\_pp\_wave\_metric}\end{aligned}$$ We are, however, not only interested in the transformation behaviour of A alone, but also in that of β, because an (α, β)-metric is composed of both. To see why we may assume without loss of generality that the form of β = du remains invariant we use the fact that any coordinate transformation $$\begin{aligned} (u,v,x^1,x^2)\mapsto (\bar u,\bar v,\bar x^1,\bar x^2)\end{aligned}$$ that leaves the generic form of the metric invariant, but in general changing the expressions for the metric functions $H, W\_a, h\_{ab}\mapsto \bar H, \bar W\_a, \bar h\_{ab}$, has the specific property that $u = \phi(\bar u)$ for some function ϕ depending on $\bar u$ alone (see section 31.2 in ). This applies in particular to the transformation that relates and. We can therefore express the 1-form as $\beta = {\text{d}}u = \phi'(\bar u){\text{d}}\bar u$, or equivalently $\bar b\_\mu = \phi'(\bar u)\delta\_\mu^u$. However, since β is covariantly constant with respect to A, we must have $\bar \nabla\_\mu\bar b\_\nu=0$. All Christoffel symbols $\bar \Gamma^u\_{\mu\nu}$ of the metric with upper index u vanish identically, however. Hence $$\begin{aligned} \bar \nabla\_{\bar u} \bar b\_{\bar u} = \partial\_{\bar u} b\_{\bar u} - \bar \Gamma^u\_{uu}\phi'(\bar u) = \phi''(\bar u)\stackrel{!}{=}0.\end{aligned}$$ It follows that $\phi'(\bar u)= C =$ constant, i.e. $\beta = C {\text{d}}\bar u$. In this case it is easily seen that scaling $\bar u$ by C and scaling $\bar v$ by 1/C leaves the metric invariant and brings the 1-form back into its original form, proving that we may assume without loss of generality that the 1-form remains invariant under the coordinate transformation. Finally, the metric is a vacuum solution to Einstein’s field equations if and only if (∂x12 + ∂x22)H = 0. We may therefore characterize the second class of solutions in the following way. Note that when H = 0 the geometries in Class 2 are also contained in Class 1. It is not the case, however, that Class 1 is a subset of Class 2 because in Class 1 the 1-form β need not be null, necessarily. The preceding line of argument shows that these two classes of solutions in fact exhaust all possibilities, which we encapsulate in the following theorem. [theor:(alpha,beta)solutions2] Any vacuum solution of the type of Theorem [theor:(alpha,beta)solutions] must belong to one of the two classes introduced above. Before we move on to (α, β)-type solutions of plane-wave type, we end this section by noting that for specific types of (α, β)-metrics, stronger results have been obtained than the ones derived above: * For Randers metrics of Berwald type any vacuum solution to must be of the type described in theorem [theor:(alpha,beta)solutions], that is, α is necessarily a vacuum solution in Einstein gravity and β is necessarily covariantly constant. Any such solution is therefore either of Class 1 or Class 2 in the terminology introduced above. * For m-Kropina metrics some vacuum solutions of a more general type than the one in theorem [theor:(alpha,beta)solutions] have been obtained in the context of Very General Relativity (VGR). * Any pseudo-Riemannian Finsler metric F = α is trivially a vacuum solution in Finsler gravity if and only if it is a vacuum solution in Einstein gravity. To the best of our knowledge this list comprises all exact solutions in Finsler gravity currently known in the literature. Plane wave solutions in Brinkman and Rosen coordinates ------------------------------------------------------ Eq. expresses the pp-wave metric in Brinkmann form. For the description of the physical effects of (plane) gravitational waves in general relativity, it is sometimes more convenient to use a different coordinate system, known as Rosen coordinates. This remains true in the Finsler case. When we compute the effect on the radar distance of a passing Randers gravitational wave in section [sec:radardistance], our starting point will be the expression for the gravitational wave in Rosen coordinates. Therefore we briefly review the relation between the two coordinate systems here. Rosen coordinates can be introduced for the subclass of pp-waves known as plane waves. These can be characterized by the property that the curvature tensor does not change (i.e. is covariantly constant) along the Euclidean ‘wave surfaces’ given in Brinkmann coordinates by du = dv = 0, i.e. $$\begin{aligned} \label{eq:invariance\_of\_Riemann\_curvature} \nabla\_{\partial\_{x^1}} R^\rho{}\_{\sigma\mu\nu} = \nabla\_{\partial\_{x^2}} R^\rho{}\_{\sigma\mu\nu} = 0.\end{aligned}$$ We note that ∇∂vRρσμν = 0 always holds, identically, so The conditions are equivalent to the statement that ∂a∂b∂cH = 0 in Brinkmann coordinates, i.e. that H(u, x) is a second order polynomial in xa. In that case there always exists a coordinate transformation that removes the linear and constant terms (section 24.5 in ) so that the metric can be written as $$\begin{aligned} \label{eq:plane\_wave\_brinkmann} A = -2{\text{d}}u {\text{d}}v + A\_{ab}(u)x^ax^b\, {\text{d}}u^2+\delta\_{ab}\, {\text{d}}x^a {\text{d}}x^b\end{aligned}$$ This is the standard expression for a plane-wave metric in Brinkmann form. Moreover, an argument very similar to the one given in the previous subsection, shows that we may assume without loss of generality that the 1-form β = du remains unchanged under this transformation. Any such plane wave metric can also be written in Rosen form $$\begin{aligned} \label{eq:Rosen\_form} {\text{d}}s^2 = -2{\text{d}}U{\text{d}}V + h\_{ij}(U){\text{d}}y^i {\text{d}}y^j,\end{aligned}$$ where hij is a two-dimensional Riemannian metric. And conversely, any metric of Rosen form can be cast in the form. The two coordinate systems are related via $$\begin{aligned} U = u,\quad V = v - \dfrac{1}{2}\dot E\_{ai} E^i{}\_b x^a x^b, \quad x^a = E^a{}\_iy^i,\end{aligned}$$ where $A\_{ab} = \ddot E\_{ai}E^i{}\_b$ and Eai is a vielbein for hij in the sense that hij = EaiEbjδab, satisfying the additional symmetry condition $\dot E\_{ai} E^i{}\_b = \dot E\_{bi} E^i{}\_a$. Such a vielbein can always be chosen. For details we recommend the lecture notes by Matthias Blau and references therein (see also the Appendix of ). Note that we have momentarily labelled the y-coordinates by indices i, j, k, … so as to distinguish them from indices a, b, c, … in order that we may apply the usual notation with regards to the vielbein indices: Eia represents the (matrix) inverse of Eai and indices a, b, c… are raised and lowered with δab, whereas indices i, j, k, … are raised and lowered with hij. The dot that appears sometimes above the vielbein represents a U-derivative. Since the vielbein depends only on U, this derivative is equivalent to a u-derivative, and moreover the raising and lowering of the a, b, c, … indices commutes with taking such a derivative of the vielbein. It is again the case that, after relabeling U, V ↦ u, v, the 1-form β = du = dU remains unchanged under this transformation, which in this case is easy to see. After also relabelling y ↦ x, we conclude that we can express any Class 2 solution of plane-wave type in Rosen coordinates as follows, $$\begin{aligned} F = \alpha\, \phi(\beta/\alpha), \qquad A = -2{\text{d}}u{\text{d}}v + h\_{ij}(u){\text{d}}x^i {\text{d}}x^j, \qquad \beta = {\text{d}}u \label{eq:ab\_plane\_wave},\end{aligned}$$ where $\alpha = \sqrt{\|A\|}$. And conversely, for any choice of ϕ, hij(u), this is a vacuum solution to the field equations in Finsler gravity if A is a vacuum solution to Einstein’s field equation. The resulting geometry is of Berwald type with affine connection identical to the Levi-Civita connection of α. Linearized gravitational wave solutions --------------------------------------- The exact vacuum field equation for plane-wave metrics does not have a particularly nice expression in Rosen coordinates. The field equation, however, turns out to be very simple. So let’s consider the scenario that the pseudo-Riemannian metric α is very close to the Minkowski metric. In this case we may write hij(u) = δij + ɛfij(u) with ɛ ≪ 1. The field equations (i.e. to first order in ɛ) for α then simply read[2](#fn2) $$\begin{aligned} f\_{11}''(u) + f\_{22}''(u)=0.\end{aligned}$$ Hence f11 and − f22 must be equal up to an affine function of u. Here we will focus on the case where f11 = − f22, which can always be achieved by means of the transverse traceless gauge[3](#fn3). Conventionally one writes the subscripts as $f\_{11} = -f\_{22} \eqqcolon f\_+$ and $f\_{12} \eqqcolon f\_\times$, denoting the plus and cross polarization of the gravitational wave, so we will stick to that notation from here onwards. That brings us to the following expression that describes Finslerian gravitational waves of (α, β)-type: $$\begin{aligned} \label{eq:ab\_grav\_waves} F = \alpha\, \phi(\beta/\alpha), \qquad\left\{\begin{array}{ll} A = -2{\text{d}}u {\text{d}}v + (1+\varepsilon f\_+(u)) {\text{d}}x^2 + (1-\varepsilon f\_+(u)){\text{d}}y^2 + 2\varepsilon f\_\times (u) {\text{d}}x\,{\text{d}}y \\ \beta = {\text{d}}u \end{array}\right.\end{aligned}$$ Note that if we substitute $u= (t-z)/\sqrt{2}$ and $v = (t+z)/\sqrt{2}$, then A reduces to the standard expression for a gravitational wave metric in GR, i.e. $$\begin{aligned} F = \alpha\, \phi(\beta/\alpha), \qquad\left\{\begin{array}{ll} A = -{\text{d}}t^2 + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y+ {\text{d}}z^2 \\ \beta = \frac{1}{\sqrt{2}}\left({\text{d}}t - {\text{d}}z\right) \end{array}\right.,\end{aligned}$$ for any choice of the function ϕ. Linearized (α, β)-metrics are Randers metrics ------------------------------------------------- It is natural to linearlize not only in ɛ, characterizing the departure from flatness, but to also use a perturbative expansion in the ‘size’ of the 1-form, characterizing the departure from GR and pseudo-Riemannian geometry. The physical intuition here is that, seeing how well GR works in most regimes, the most interesting class of Finsler spacetimes constists of those ones that are very close to GR spacetimes. The purpose of this section is to highlight that any (α, β)-metric is perturbatively equivalent to a Randers metric, to first order, so that from the physics point of view, Randers metrics are actually quite a bit more general than they might seem at first glance. After pointing this out we will turn our focus exclusively to Randers metrics for the remainder of the article. So consider an (α, β)-metric constructed form a pseudo-Riemannian metric α and a 1-form β such that β ≪ 1. To see what happens in such a scenario, we replace β with λβ and expand to first order in λ. Then we obtain $$\begin{aligned} F = \alpha \phi\left(\frac{\lambda\beta}{\alpha}\right) \approx \alpha \left(\phi(0)+ \lambda \phi'(0)\frac{\beta}{\alpha}\right) = \alpha \phi(0) + \lambda\phi'(0)\beta = \tilde\alpha + \tilde\beta.\end{aligned}$$ Hence to first order in λ, any (α, β)-metric is indeed equivalent to a Randers metric[4](#fn4). Consequently, by replacing du by λ du in, which technically can be achieved by a coordinate transformation that scales u by λ and v by 1/λ, it follows that to first order in λ the Finsler metric of the (α, β)-type gravitational waves takes the form, $$\begin{aligned} \label{eq:metric\_naive\_Randers\_grav\_wave0} F = \alpha + \beta, \qquad\left\{\begin{array}{ll} A = -2{\text{d}}u {\text{d}}v + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y \\ \beta = \lambda\,{\text{d}}u \end{array}\right..\end{aligned}$$ The parameter λ then characterizes the departure from GR and pseudo-Riemannian geometry. We will assume without loss of generality that λ > 0. Finally, replacing also u and v by t and z, according to $u= (t-z)/\sqrt(2)$ and $v = (t+z)/\sqrt{2}$, we can write the metric in the following way, which we will take as the starting point for the calculation of the radar distance in Section [sec:radardistance]. $$\begin{aligned} \label{eq:metric\_naive\_Randers\_grav\_wave} F = \alpha + \beta, \qquad\left\{\begin{array}{ll} A = -{\text{d}}t^2 + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y+ {\text{d}}z^2 \\ \beta = \frac{\lambda}{\sqrt{2}}\left({\text{d}}t - {\text{d}}z\right) \end{array}\right. \end{aligned}$$ Modified Randers Metrics ======================== Motivated by the argument above we will now turn our focus to the simplest properly Finslerian (α, β)-metric, the Randers metric, conventionally defined as F = α + β. We will argue that in order to have a physically causal structure, the conventional definition must be modified slightly. It might seem to the reader that modifying the Randers metric would be in conflict with the spirit of the previous section, since to first order any (α, β)-metric should reduce to a Randers metric. It is important to note, however, that there is in principle the possibility that to different regions of the tangent bundle could correspond different Randers metrics. More precisely, we could define one (α, β)-metric F1 on a conic subbundle $\mathcal A\_1\subset TM\setminus 0$ and another (α, β)-metric, F2, on a different conic subbundle $\mathcal A\_2\subset TM\setminus 0$. If the two subbundles do not overlap then this defines a perfectly valid (α, β)-type Finsler spacetime on the union $\mathcal A = \mathcal A\_1\cup\mathcal A\_2$. To first order in the deviation from geometry this Finsler metric would reduce to a certain Randers metric on $\mathcal A\_1$ and to a different Randers metric on $\mathcal A\_2$. our modification of the Randers metric, introduced below, is therefore completely consistent with the previous results. After a heuristic argument that motivates the desired modification, we show that our proposed version of the modified Randers metric has a very satisfactory causal structure. As a result of this a clear (future and past) timelike cone can be identified and within these timelike cones the signature of the Fundamental tensor is Lorentzian everywhere. The only constraint is that b2 ≡ aμνbμbν > − 1, which, interestingly, is in some sense the opposite of the condition b2 < 1 that appears in the well-known positive definite case, see e.g.. In one were to adopt the opposite signature convention to ours, however, the constraint in the Lorentzian case would also turn out to be b2 < 1, matching the positive definite case. Motivation and definition ------------------------- First of all, let us review why the definition of a Randers metric is not as clear in Lorentzian signature as it is in Euclidean signature. The original definition of a Randers metric, in positive definite Finsler geometry, is just F = α + β, with $\alpha = \sqrt{a\_{ij}y^i y^j}$ a Riemannian metric and β = biyi any 1-form[5](#fn5). This is well-defined as long as α is positive-definite, because in that case A ≡ aijyiyj is always positive. If we allow aij to be a Lorentzian metric, however, the quantity A can become negative, in which case $\sqrt{A}$ is ill-defined, as we want F to be a real function. One way to remedy this, at least at a technical level, is to restrict the conic subbundle $\mathcal A\subset TM\setminus 0$ to those vectors for which aijyiyj > 0. This was the approach in e.g., where it was shown that if $\mathcal A$ is defined as the forward timecone[6](#fn6) corresponding to α, then under certain conditions on the 1-form β, such a Randers metric satisfies all axioms of a Finsler spacetime. The fact that $\mathcal A$ is restricted in this way, however, leads to issues when it comes to the physical interpretation. Here we take a different approach. The obvious first alternative to restricting $\mathcal A$ to vectors with positive norm is to simply replace A by ∣A∣ and define $\alpha = \sqrt{\|A\|}$, as we have done throughout this article. In that case there’s no need to restrict $\mathcal A$ to the timecone anymore. This leads to a Randers metric of the form $F = \sqrt{\|A\|} +\beta$. An undesirable consequence of this definition, however, is that light rays can only propagate into one half of the tangent space, namely the half given by β < 0, which follows immediately from the null condition F = 0 (see also ). In fact, the light cone separates the tangent space into only two connected components[7](#fn7) and there is consequently not a straightforward interpretation in terms of timelike, spacelike and lightlike directions, at least not in the conventional way[8](#fn8). We therefore take the viewpoint that outside of the half plane β < 0 in each tangent space, this version of the Randers metric cannot be valid, and we need to modify it in that region. It is possible to remove the condition β ≤ 0, extending the lightcone to the other half-plane β > 0, by changing F to $F={\text{sgn}}(A)\sqrt{\|A\|}+{\text{sgn}}(\beta)\beta = {\text{sgn}}(A)\sqrt{\|A\|}+\|\beta\|$. The result of this is that, under some mild assumptions (details will follow below) the single lightcone (from the β < 0 half space) is mirrored to the complementary (β ≥ 0) half space, whereas in the original half space intersected with the original cone of definition consisting of α-timelike vectors, F reduces to the standard Randers metric with an overall minus sign, F = − (α + β). This minus sign is not of any relevence, though, as the geometry is essentially determined by F2. In particular, F is now reversible, i.e. invariant under y → − y. Notice also that we could have chosen a minus sign instead of a plus sign in the modified definition of F, but it turns out that in that case the resulting Finsler metric would not be guaranteed to have Lorentzian signature everywhere inside of the timelike cones[9](#fn9). The present metric does have this property as long as b2 > − 1, and we discuss this in detail below. Motivated by the preceding heuristic argument we define the modified Randers metric as follows, $$\begin{aligned} F = {\text{sgn}}(A)\alpha + \|\beta\|,\label{eq:modified\_randers\_metric}\end{aligned}$$ where we recall for completeness that $\alpha = \sqrt{\|A\|}$, A = aijyiyj β = biyi. Both α and A will sometimes be referred to as the (pseudo-)Riemannian metric, by a slight abuse of language, but it should always be clear from context what is meant. Causal structure ---------------- Next we will show that the modified Randers metric indeed has very nice properties. By definition, the light cone is given by $$\begin{aligned} F = 0 \qquad \Leftrightarrow \qquad {\text{sgn}}(A)\leq 0 \,\&\, \|A\| = \|\beta\|^2 \qquad \Leftrightarrow \qquad A = -\beta^2.\end{aligned}$$ It therefore follows that $$\begin{aligned} F=0 \qquad \Leftrightarrow\qquad (a\_{\mu\nu}+ b\_\mu b\_\nu){\text{d}}x^\mu {\text{d}}x^\nu = 0,\end{aligned}$$ meaning that the light cone of F is just the light cone of the auxilliary Lorentzian metric $\tilde a\_{\mu\nu}(x) = a\_{\mu\nu}+b\_\mu b\_\nu$. Indeed, the matrix determinant lemma guarantees that as long as b2 = aμνbμbν > − 1 the metric aμν + bμbν has Lorentzian signature, provided that aμν has Lorentzian signature. (For a proof see appendix [sec:proofofsignature].) This shows that as long as b2 > − 1 the light cone separates the tangent space at each point into three connected components, which we can naturally interpret in the usual manner as the forward time cone, backward timecone, and the remainder consisting of spacelike vectors. Coincidentally we note that $$\begin{aligned} F<0 \qquad \Leftrightarrow\qquad (a\_{\mu\nu}+ b\_\mu b\_\nu) y^\mu y^\nu < 0,\end{aligned}$$ and hence it also follows that $$\begin{aligned} F>0 \qquad \Leftrightarrow\qquad (a\_{\mu\nu}+ b\_\mu b\_\nu)y^\mu y^\nu > 0.\end{aligned}$$ This leads to the additional convenience that F-timelike vectors are precisely given by F < 0, and F-spacelike vectors by F > 0, in addition to the null vectors being given, by definition, by F = 0. We summarize these results in the following proposition. As long as b2 > − 1, the causal structure of the modified Randers metric F = sgn(A)α + ∣β∣ is identical to the causal structure of the Lorentzian metric aμν + bμbν, with null vectors given by F = 0, timelike vectors given by F < 0, and spacelike vectors by F > 0. As a result of these nice features of the causal structure of the modified Randers metric, it is possible to define time orientations in the usual manner, by means of a nowhere vanishing timelike vector field T. Such T selects one of the two timelike cones as the ‘forward’ one, namely the one that contains T. Then another timelike vector y is future oriented (i.e. lies in the same forward cone as T) if and only if (aμν + bμbν)Tμyν < 0. In the special case that β is covariantly constant with respect to α we have even more satisfactory results. In that case not only the causal structure but also the affine structure of F can be understood in terms of aμν + bμbν. If β is covariantly constant with respect to α and satisfies b2 > − 1 then the causal structure and the affine structure of the modified Randers metric F = sgn(A)α + ∣β∣ are identical to those of the Lorentzian metric $\tilde a\_{\mu\nu} = a\_{\mu\nu} + b\_\mu b\_\nu$. In other words, the timelike, spacelike and null geodesics of F coincide with the timelike, spacelike and null geodesics of $\tilde a\_{\mu\nu}$. The discussion above indicates that the causal structures coincide. It remains to show that also the affine structures concide in the case of a covariantly constant 1-form. This is again a result of the properties of $\tilde a\_{\mu\nu}$. It can be shown (see Appendix [sec:proofofsignature]) that the Christoffel symbols of $\tilde a\_{\mu\nu}$ can be expressed in terms of the Christoffel symbols of aμν as $$\begin{aligned} \widetilde\Gamma^\rho\_{\mu\nu} &= \Gamma^\rho\_{\mu\nu} +\frac{1}{1+b^2}b^\rho \nabla\_{(\mu}b\_{\nu)} - \left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)\left( b\_\mu\nabla\_{[\lambda} b\_{\nu]} + b\_\nu\nabla\_{[\lambda} b\_{\mu]}\right).\end{aligned}$$ Hence it follows immediately that if bμ is covariantly constant then $\widetilde\Gamma^\rho\_{\mu\nu} = \Gamma^\rho\_{\mu\nu}$ and the affine structure of $\tilde a\_{\mu\nu}$ is the same as that of aμν. We also known, by Prop. [prop:coincidingspray], that the affine structure of F is the same as that of aμν. Hence the affine structure of F is the same as that of $\tilde a\_{\mu\nu}$. From this it follows immediately that the existence of radar neighborhoods is guaranteed. This is of essential importance in our work, because what it essentially says is that the radar distance, calculated in Section [sec:radardistance], is a well-defined notion. Regularity and signature ------------------------ Given an (α, β)-metric of the form F = αϕ(s), with s = β/α and $\alpha = \sqrt{\|A\|}$, it can be shown that the determinant of the fundamental tensor is given by $$\begin{aligned} \label{eq:ab\_det\_formula\_main\_text} \det g\_{ij} = \phi^{n+1}(\phi-s\phi')^{n-2}(\phi-s\phi' + ({\text{sgn}}(A) b^2-s^2)\phi'')\det a\_{ij}.\end{aligned}$$ The proof can be found in Appendix [sec:abdeterminant]. Because of the appearance of sgn(A) the expression is slightly different from the well–known positive definite analogue, to which it reduces when A > 0, i.e. sgn(A) = 1. For a modified Randers metric of the form F = sgn(A)α + ∣β∣ the function ϕ is given by ϕ(s) = sgn(A) + ∣s∣, so this reduces to $$\begin{aligned} \frac{\det g}{\det a} = {\text{sgn}}(A)^{n-1}\left({\text{sgn}}(A)+\|s\|\right)^{n+1} = \left({\text{sgn}}(A)\frac{F}{\alpha}\right)^{n+1}.\end{aligned}$$ Assuming the spacetime dimension n is even, this means that g has Lorentzian signature[10](#fn10) if and only if sgn(A)F > 0. Let us see what this entails. First note that F < 0 trivially implies A < 0. Hence F < 0 implies Lorentzian signature. Before we move on, we should point out that this is a very satisfactory result. It means that within the entire timelike cone of F, the signature of the fundamental tensor is Lorentzian. Similarly, A > 0 implies F > 0. Hence A > 0 also implies Lorentzian signature. What remains is the region where A ≤ 0 and F ≥ 0. Equivalently, A ≤ 0 and A + β2 ≥ 0. In this region, the determinant of the fundamental tensor is either undefined, is positive, or vanishes, so in any case the signature is not Lorentzian. But as this region lies outside the timelike cone, this is not a problem, as argued in section [sec:Finslerspacetimesinterpretation]. It is helpful to think in terms of both the light cone of the metric aij and the light cone of the metric aij + bibj (i.e. that of F). As mentioned previously, as long as b2 > − 1, the latter metric is Lorentzian, provided the former is. That means its light cone is just a conventional one that we’re familiar with from GR, just like the light cone of aij. The only region where the signature is not Lorentzian, is precisely the region in between these two lightcones. Note that since F < 0 implies A < 0, the F-lightcone can never reach outside of the aij-light cone. The details depend on the causal character of the 1-form β and are listed below. These properties can be checked easily by noting we may always choose coordinates such that at a given point x ∈ M the metric A has the form of the Minkowski metric and the 1-form β has only one component (in the timelike or spacelike case) or two components (in the null case)[11](#fn11). * If β is null it is easily seen that the two lightcones intersect only for yμ that are multiples of bμ. Thus their intersection spans a single line in the tangent space. * If β is timelike and b2 > − 1 then the light cones do not intersect (apart from the trivial intersection in the origin). * If β is spacelike (and assuming dimM > 2), then aij induces a Lorentzian metric on the (dimM − 1)-dimensional hypersurface defined by β = 0. In this case the two light cones intersect along the light cone of this induced Lorentzian metric. * If b2 = − 1 there is only a single cone, namely the one corresponding to α. The ‘light cone’ corresponding to F = 0 is now in fact a line, consisting all of multiples of bμ. This case therefore does not have a viable physical interpretation. * If b2 < − 1 there is only a single cone, namely the one corresponding to α. The ‘light cone’ corresponding to F = 0 is now non-existent, as F = 0 has no solutions. This case therefore does not have a viable physical interpretation either. [b]0.24![Null 1-form with \rho=0.6](figL1 "fig:") [fig:L1] [b]0.24![Null 1-form with \rho=1](figL2 "fig:") [fig:L2] [b]0.24![Null 1-form with \rho=1.4](figL3 "fig:") [fig:L3] [b]0.24![Timelike 1-form with \rho=0.65](figT1 "fig:") [fig:T1] [b]0.24![Timelike 1-form with \rho=0.8](figT2 "fig:") [fig:T2] [b]0.24![Timelike 1-form with \rho=0.9](figT3 "fig:") [fig:T3] [b]0.24![Spacelike 1-form with \rho=0.8](figS1 "fig:") [fig:S1] [b]0.24![Spacelike 1-form with \rho=1.4](figS2 "fig:") [fig:S2] [b]0.24![Spacelike 1-form with \rho=2](figS3 "fig:") [fig:S3] [b]0.24![Timelike 1-form with b^2=-1.](figLwithNormEqualToOne "fig:") [fig:T5] [b]0.24![Timelike 1-form with b^2<-1.](figLwithNormGreaterThanOne "fig:") [fig:T4] [fig:lightconesandsignatures] To get a better idea, Fig. [fig:lightconesandsignatures] displays the lightcones and the regions (in green) where the signature of the fundamental tensor is Lorentzian, for the modified Randers metric $$\begin{aligned} F = {\text{sgn}}(A)\alpha + \|\beta\|,\qquad A = -({\text{d}}x^0)^2+({\text{d}}x^1)^2+({\text{d}}x^2)^2,\qquad \beta =\left\{\begin{array}{ll} \rho \,{\text{d}}x^0 & \text{if \textit{timelike}}\\ \rho\,({\text{d}}x^0+{\text{d}}x^1) &\text{if \textit{null}}\\ \rho \,{\text{d}}x^1 & \text{if \textit{spacelike}} \end{array}\right.,\end{aligned}$$ for a number of representative values of the parameter ρ. In each subfigure, the inner lightcone is that of F and the outer lightcone that of A. Note that for any aij and bi, it is always possible at any given point x ∈ M, to choose coordinates in such a way that A and β have the above form (or rather their analog in the relevant spacetime dimensionality). The following proposition summarizes these results. As long as b2 > − 1, the signature of the fundamental tensor of F = sgn(A)α + ∣β∣ is Lorentzian within the entire timelike cone, which is given by F < 0. Immediately outside of the timelike cone there is a region that does not have Lorentzian signature, and further away (namely when A > 0) the signature becomes Lorentzian again. When b2 ≤ − 1 the timelike cone is the empty set, so this case is not physically interesting. Since we only require Lorentzian signature within the timelike cone, these results are very satisfactory. Finally, regarding the regularity of F, clearly F is smooth everywhere except when A = 0 or β = 0. In particular, the set where F is not smooth has measure zero. Radar Distance for a Finsler Gravitational Wave =============================================== Now we are finally in the position to analyze the physical effects of a passing Finslerian gravitation wave of (α, β)-type. We have seen in Section [sec:linabmetricisranders] that, to first order, an (α, β)-metrics is equivalent to a Randers metric. And we have argued in section [sec:Randers] that this should not be the standard Randers metric but rather our modified Randers metric. Thus our starting point will be the gravitational wave solution of modified-Randers type. That is, we are interested in the solution but with the conventional Randers metric F = α + β replaced by the modified Randers metric F = sgn(A)α + ∣β∣, where $\alpha = \sqrt{\|A\|}$, A = aijyiyj. Note that this modification does not change any of the results pertaining to classification of solutions to the field equations, by the argument given at the beginning of section [sec:Randers] that a modified Randers metric is ‘locally’ equivalent to a standard Randers metric, in a certain precise sense. The relevant Finsler metric is therefore given by $$\begin{aligned} \label{eq:metric\_Randers\_grav\_wave} F = {\text{sgn}}(A)\alpha + \|\beta\|, \qquad\left\{\begin{array}{ll} A = -{\text{d}}t^2 + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z)){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y+ {\text{d}}z^2 \\ \beta = \frac{\lambda}{\sqrt{2}}\left({\text{d}}t - {\text{d}}z\right) \end{array}\right. \end{aligned}$$ Since actual gravitational wave measurements are done with interferometers, that effectively measure the radar distance, the aim of this section is to compute that radar distance during the passing of a gravitational wave of the form. The setup is as follows. A light ray is emitted from some spacetime location with coordinates (t0, x0, y0, z0), travels to another location in spacetime with coodinates (t0 + Δt, x0 + Δx, y0 + Δy, z0 + Δz), where it is reflected and after which it travels back to the original (spatial) location, with spacetime coordinates (t0 + Δttot, x0, y0, z0), being received there again. We are interested in the amount of proper time that passes between emission and reception of the light ray, as measured by an ‘inertial’ observer[12](#fn12) located at spatial coordinates (x0, y0, z0). Because light travels forwards and backwards during this time interval, one half of the time interval is usually called the radar distance between the two spacetime points (sometimes the value is multiplied by the velocity of light, c, which we have set to 1, so that it has the dimensions of distance). In other words, the radar distance can be expressed as R = Δτ/2. Finslerian null geodesics ------------------------- Radar distance -------------- Before we move on, let us summarize in what ways the Finslerian parameter λ has entered our analysis so far: 1. The null trajectories are altered due to the fact the Finsler metric induces a modified null condition or MDR. As a result, it takes a larger coordinate time interval for a light ray to travel a given spatial coordinate distance. This effect works in all spatial directions, even the direction parallel to the propagation direction of the light ray. This effect enters at order λ2. 2. The ratio of proper time and coordinate time is altered with the result that less proper time is experienced per unit coordinate time. This effect enters at order λ. There is, however, a third way in which the parameter enters. Namely in the relation between the coordinate distance and radar distance in the absence of the wave. For a gravitational wave in GR these conveniently coincide; in the case of our Randers waves they don’t. The formula for the radar distance derived above refers merely to coordinates. In order to make sense of the result, we would like to express the right hand side in terms of measurable quantities, like the radar distances in the various directions in the absence of the wave. Employing Eq. we write $$\begin{aligned} \Delta X = \left(1 - \frac{\lambda }{\sqrt{2}}\right)\Delta x + \frac{\lambda^2}{4}\Delta x, \\ \Delta Y = \left(1 - \frac{\lambda }{\sqrt{2}}\right)\Delta y + \frac{\lambda^2}{4}\Delta y, \\ \Delta Z = \left(1 - \frac{\lambda }{\sqrt{2}}\right)\Delta z + \frac{\lambda^2}{2}\Delta z,\end{aligned}$$ for the radar distance in the x, y and z direction in the absence of the wave, and $$\begin{aligned} R\_0 = \left(1 - \frac{\lambda }{\sqrt{2}}\right)\Delta \ell + \frac{\lambda^2}{4}\left(\Delta\ell + \frac{\Delta z^2}{\Delta\ell} \right),\end{aligned}$$ for the radar distance in the relevant direction in the absence of the wave. Eliminating the coordinate distances in favour of the physical radar distances by virtue of the inverse transformations, valid to second order in λ, $$\begin{aligned} \Delta x &= \Delta X\left(1 + \frac{\lambda }{\sqrt{2}} + \frac{\lambda^2}{4}\right)\\ \Delta y &= \Delta Y\left(1 + \frac{\lambda }{\sqrt{2}} + \frac{\lambda^2}{4}\right)\\ \Delta z &= \Delta Z\left(1 + \frac{\lambda }{\sqrt{2}}\right) \\ \Delta \ell &= R\_0\left(1 + \frac{\lambda }{\sqrt{2}} + \frac{3}{4}\lambda^2\right) - \frac{\Delta z^2}{4 R\_0}\lambda^2 \\ &= R\_0\left(1 + \frac{\lambda }{\sqrt{2}} + \frac{\lambda^2}{4}\right) - \frac{\Delta Z^2}{4 R\_0}\lambda^2\end{aligned}$$ we can express the radar distance in the presence of the wave as $$\begin{aligned} \boxed{ R = R\_0 + \varepsilon\left(\frac{\Delta X^2 - \Delta Y^2 }{4R\_0}\right)\bar f\_{+,\text{tot}} + \varepsilon\left(\frac{\Delta X\Delta Y }{2R\_0}\right)\bar f\_{\times,\text{tot}} + \mathcal O(\varepsilon^2, \lambda^3, \varepsilon\lambda^2). }\end{aligned}$$ This is a remarkable result. By expressing the radar distance in terms of the physical observables ΔX, ΔY and R0 rather than merely coordinates, all dependence on λ has disappeared to the desired order and the expression is identical to its GR counterpart, Eq.! We must conclude, therefore, that the effect of a Randers gravitational wave on interferometer experiments is virtually indistinguishable from that of a conventional GR gravitational wave. It is important to remark that by no means this implies that all phenomena in such a Finsler spacetime are identical to their GR counterparts. It might be possible to detect the presence of a non-vanishing λ by some other means. This is a very interesting and important questions, however it is beyond the scope of this article and something to explore in future work. Our results pertain merely to gravitational wave effects as observed by interferometers. Discussion ========== The main aim of this paper was to study the physical effect of Finslerian gravitational waves and, in particular, to investigate the question if and how such waves can be distinguished, observationally, from the classical gravitational waves of general relativity. To this effect we have derived an expression for the radar distance at the moment a Finsler gravitational passes, say, the earth. This radar distance is the main observable that is measured by interferometers. Remarkably, we have found that the expression for the radar distance is indistinguishable from its non-Finslerian counterpart, leading us to conclude that interferometer experiments would not be able to distinguish between a general relativistic and a Finslerian gravitational wave, at least not with regards to the radar distance. This is on the one hand disappointing, since indicates means we cannot use such measurements to test the Finslerian character of our spacetime. On the other hand, though, it means that the current gravitational wave measurements are all compatible with the idea that spacetime has a Finslerian nature. To the best of our knowledge this is the first time an explicit expression for the Finslerian Radar length has been obtained for the case of finite spacetime separations, and as such our work may be seen as a proof of concept. Repeating the analysis for other Finsler spacetime geometries may lead to additional insight as to the observational signature of Finsler gravity. The other parts of the article, leading up to the calculation of the radar length, were more mathematical in nature. We have introduced a class of exact solutions to the field equation in Finsler gravity that have a close resemblance to the well-known general relativistic pp-waves, and that generalize all of the pp-wave-type solutions currently known in the literature. These solutions are (α, β)-metrics, where α is a classical pp-wave and β is its defining covariantly constant null 1-form. Consequently our solutions are of Berwald type. Their linearized versions, we have shown, may be interpreted as Finslerian gravitational waves of modified Randers type. Indeed, along the way we have introduced a small modification to the standard definition the Randers metric, motivated by the observation that the physical interpretation of the causal structure of the standard Randers metric is not immediately obvious. In contrast, we have shown that our modified Randers metrics have the nice property that their causal structure is completely equivalent to the causal structure of some auxiliary metric, hence leading to a perfectly clear physical interpretation. We stress that this auxilliary metric is different from the defining metric α. In the special case that the defining 1-form of the Randers metric is covariantly constant (which is the case, for example, for our solutions) we have even more satifactory results. In this case not only the causal structure, but also the affine structure of the Randers metric coincides with that of the auxilliary (pseudo)-Riemannian metric, i.e. the timelike, spacelike and null geodesics of the Finsler metric can be understood, respectively, as the timelike, spacelike and null geodesics of the auxiliary (pseudo)-Riemannian metric. A particularly nice consequence of this is the guaranteed existence of radar neighborhoods, i.e. that given an observer and any event in spacetime, there is (at least locally) exactly one future pointing light ray and one past pointing light ray that connect the worldline of the observer to the event. This is of essential importance in our work, because without this property it would have not been possible to perform the calculation of the radar distance in the last part of the article, simply because the notion of radar distance would not even make sense in that case. Let us now point out some of the limitations of our investigation. First of all, it is by no means expected that the Finslerian gravitational waves discussed here should be only possible ones. Although being much larger than even the complete class of all Lorentzian (i.e. non-Finslerian) geometries, the class of (α, β)-metrics of Berwald type, to which we have restricted our analysis, is still quite restrictive in the large scheme of (Finsler geometric) things. So even though our results suggest that there is no observable difference between the Finslerian gravitational waves discussed in this article and their GR counterparts, there might be more general types of Finslerian gravitational waves that could be distinguished observationally from the general relativistic ones by means of interferometer experiments. Furthermore, radar distance experiments are by no means the only way of probing our spacetime geometry. It might be possible to detect the Finslerian character of spacetime in some other way. We have not explored this possibility here, but we plan to investigate this in the future. Moreover, we have assumed in our calculations that the amplitude of the gravitational waves as well as the Finslerian deviation from general relativity are sufficienty small such that a perturbative approach to first order in the former and second order in the latter is valid. It would be of interest to repeat the calculation to higher order in perturbation theory. We expect that this would in principle be a straightforward, yet possibly tedious, exercise. S.H. wants to thank Rick Sengers and Nicky van den Berg for fruitful discussions and for their input with regards to the figures. S.H. also wants to thank Luc Florack for fruitful discussions, in particular his suggestions with regards to perturbation theory. We would like to acknowledge networking support by the COST Action CA18108, supported by COST (European Cooperation in Science and Technology). Some Properties of the Metric aμν + bμbν ====================================================== Proof of Lorentzian signature ----------------------------- Here we prove that if aμν has Lorentzian signature and aμνbμbν > − 1 then $\tilde a\_{\mu\nu} = a\_{\mu\nu}+b\_\mu b\_\nu$ also has Lorentzian signature. We write b2 = aμνbμbν. First, the matrix determinant lemma says that $$\begin{aligned} \det \tilde a= (1+b^2)\det a.\end{aligned}$$ As long as b2 > − 1 this implies that $\det \tilde a$ has the same sign as deta, so assuming aμν is Lorentzian, $\tilde a$ has negative determinant. In 4D this immediately implies that $\tilde a$ is Lorentzian (although the signs of the eigenvalues might be flipped with respect to aμν). However, let’s assume the dimensionality is arbitrary. Consider the family of 1-forms bμ(η) = ηbμ, where η ∈ [0, 1]. For any η we have $$\begin{aligned} \det \widetilde {a^{(\eta)}} = \left[1+\left(b^{(\eta)}\right)^2\right]\det a = \left[1+\eta^2 b^2\right]\det a,\end{aligned}$$ $\det \widetilde {a^{(\eta)}}$ has the same sign for all values of η. Now since each of the n eigenvalues of $\widetilde {a^{(\eta)}}$ can be expressed as continuous function of η, it follows that the respective signs of the n eigenvalues cannot change when we change η. To see why, suppose that the k-th eigenvalue is positive for some η1 and negative for some η2. By the intermediate value theorem, there must exist some η between η1 and η2 for which the eigenvalue vanishes. In that case the determinant vanishes for that value of η, which is a contradiction, as the determinant never vanishes as we have just seen. This argument proves that $\widetilde {a^{(\eta)}}$ has the same signature for all values of η, because the signs of the eigenvalues remain unchanged. In particular, $\tilde a = \widetilde {a^{(1)}}$ has the same signature as $a = \widetilde {a^{(0)}}$. Therefore, if aμν is Lorentzian and b2 > − 1 then $\tilde a$ is Lorentzian as well. Affine structure ---------------- Here we derive an explicit formula for the Christoffel symbols of the metric $\tilde a\_{\mu\nu} = a\_{\mu\nu}+b\_\mu b\_\nu$, where it is again assumed that b2 > − 1. The Christoffel symbols of $\tilde a\_{\mu\nu}$ can be expressed as $$\begin{aligned} \widetilde\Gamma^\rho\_{\mu\nu} &= \Gamma^\rho\_{\mu\nu} +\frac{1}{1+b^2}b^\rho \nabla\_{(\mu}b\_{\nu)} - \left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)\left( b\_\mu\nabla\_{[\lambda} b\_{\nu]} + b\_\nu\nabla\_{[\lambda} b\_{\mu]}\right).\end{aligned}$$ where ∇ is the covariant derivative corresponding to aμν We prove this below, but first we point out the following immediate consequence. If bμ is covariantly constant with respect to aμν, the affine structure of $\tilde a\_{\mu\nu}$ is the same as the affine structure of aμν, i.e. $\widetilde\Gamma^\rho\_{\mu\nu} = \Gamma^\rho\_{\mu\nu}$. As long as b2 > − 1 the formula for the determinant displayed above shows that $\tilde a\_{\mu\nu} = a\_{\mu\nu}+b\_\mu b\_\nu$ is invertible as a matrix. It can be easily checked that its inverse is given by $$\begin{aligned} \tilde a^{\mu\nu} =a^{\mu\nu} - \frac{1}{1+b^2}b^\mu b^\nu.\end{aligned}$$ Unless otherwise specified (as in the case of $\widetilde\Gamma$ below!) indices are raises and lowered with aμν. Denoting Γλμν = aλρΓμνρ and $\widetilde\Gamma\_{\lambda\mu\nu}=\tilde a\_{\lambda\rho}\widetilde\Gamma^\rho\_{\mu\nu}$ we first note that we can express the latter as $$\begin{aligned} \widetilde\Gamma\_{\lambda\mu\nu} =\frac{1}{2} \left( \partial\_\mu \tilde a\_{\lambda\nu} + \partial\_\nu \tilde a\_{\mu\lambda} - \partial\_\lambda \tilde a\_{\mu\nu}\right) = \Gamma\_{\lambda\mu\nu} + b\_\lambda\partial\_{(\mu} b\_{\nu)} - b\_\mu\partial\_{[\lambda} b\_{\nu]} - b\_\nu\partial\_{[\lambda} b\_{\mu]},\end{aligned}$$ where (μ, ν) denotes symmetrization and [μ, ν] denotes anti-symmetrization. Therefore it follows that $$\begin{aligned} \widetilde\Gamma^\rho\_{\mu\nu} &= \tilde a^{\rho\lambda}\widetilde\Gamma\_{\lambda\mu\nu} = \left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)\left(\Gamma\_{\lambda\mu\nu} + b\_\lambda\partial\_{(\mu} b\_{\nu)} -b\_\mu\partial\_{[\lambda} b\_{\nu]} -b\_\nu\partial\_{[\lambda} b\_{\mu]}\right) \\ &=\Gamma^\rho\_{\mu\nu} - \frac{1}{1+b^2}b^\rho b\_\lambda\Gamma^\lambda\_{\mu\nu} +\left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)b\_\lambda\partial\_{(\mu} b\_{\nu)} - \left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)\left( b\_\mu\partial\_{[\lambda} b\_{\nu]} + b\_\nu\partial\_{[\lambda} b\_{\mu]}\right).\end{aligned}$$ The second and third term add up to $$\begin{aligned} -\frac{1}{1+b^2}b^\rho & b\_\lambda\Gamma^\lambda\_{\mu\nu} +\left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)b\_\lambda\partial\_{(\mu} b\_{\nu)} \\ &= -\frac{1}{1+b^2}b^\rho b\_\lambda\Gamma^\lambda\_{\mu\nu} +b^\rho\partial\_{(\mu} b\_{\nu)} - \frac{b^2}{1+b^2} b^\rho \partial\_{(\mu} b\_{\nu)} =\\ &=-\frac{1}{1+b^2}b^\rho b\_\lambda\Gamma^\lambda\_{\mu\nu} \ + \frac{1}{1+b^2} b^\rho \partial\_{(\mu} b\_{\nu)}\\ &=\frac{1}{1+b^2}b^\rho \left(\partial\_{(\mu}b\_{\nu)}-b\_\lambda\Gamma^\lambda\_{\mu\nu} \right)\\ &=\frac{1}{1+b^2}b^\rho \nabla\_{(\mu} b\_{\nu)}.\end{aligned}$$ This shows that $$\begin{aligned} \widetilde\Gamma^\rho\_{\mu\nu} &= \Gamma^\rho\_{\mu\nu} +\frac{1}{1+b^2}b^\rho \nabla\_{(\mu}b\_{\nu)} - \left(a^{\rho\lambda} - \frac{1}{1+b^2}b^\rho b^\lambda\right)\left( b\_\mu\partial\_{[\lambda} b\_{\nu]} + b\_\nu\partial\_{[\lambda} b\_{\mu]}\right).\end{aligned}$$ Finally, we may replace all partial derivatives with covariant ones because $$\begin{aligned} \nabla\_{[\lambda} b\_{\nu]}= \nabla\_\lambda b\_\nu - \nabla\_\nu b\_\lambda = \partial\_\lambda b\_\nu - \Gamma^\mu\_{\lambda\nu} b\_\mu- \partial\_\nu b\_\lambda+\Gamma^\mu\_{\nu\lambda} b\_\mu = \partial\_\lambda b\_\nu- \partial\_\nu b\_\lambda = \partial\_{[\lambda} b\_{\nu]}.\end{aligned}$$ That yields the desired formula. Determinant of a Not Necessarily Positive Definite (α, β)-Metric ==================================================================== Here we derive the formula Eq. for the determinant of a not necessarily positive definite (α, β)-metric, generalizing the well-known result from the positive definite case. More precisely, we consider Finsler metrics the form F = αϕ(s), where s = β/α, $\alpha = \sqrt{\|A\|}=\sqrt{\|a\_{ij}y^i y^j\|}$, A = aijyiyj = sgn(A)α2, and where aij is assumed to be a metric, i.e. not necessarily Riemannian/positive definite. In complete analogy with the positive definite case, it can be shown by direct calculation that the fundamental tensor $g\_{ij}\equiv \tfrac{1}{2}\bar\partial\_i\bar\partial\_j F^2$ is given by $$\begin{aligned} \label{eq:alpha\_beta\_metric\_fundamental\_tensor} g\_{ij} = {\text{sgn}}(A)\rho a\_{ij} + \rho\_0 b\_i b\_j + \rho\_1(b\_i\alpha\_j + \alpha\_i b\_j) + \rho\_2\alpha\_i\alpha\_j,\end{aligned}$$ where we have defined αi = aijyj/α, and with coefficients given by $$\begin{aligned} \rho &= \phi(\phi-s\phi'),\\ \rho\_0&= \phi \phi'' + \phi'\phi',\\ \rho\_1 &= -(s\rho\_0 - \phi \phi') = -\left[s(\phi \phi'' + \phi'\phi') - \phi\phi '\right],\\ \rho\_2 &= -s\rho\_1 = s\left[s(\phi \phi'' + \phi'\phi') - \phi\phi '\right].\end{aligned}$$ The only difference here with the positive definite case is the factor sign(A) appearing in the first term in Eq.. Denoting dimM = n can write this in matrix notation as $$\begin{aligned} g = {\text{sgn}}(A)\rho \left(a + UWV^T\right),\end{aligned}$$ in terms of the three matrices $$\begin{aligned} W = \frac{{\text{sgn}}(A)}{\rho}\mathbb I\_{4\times 4},\qquad U = (\vec b, \vec b, \vec \alpha, \vec \alpha), \qquad V = (\rho\_0 \vec b, \rho\_1 \vec\alpha, \rho\_1 \vec b, \rho\_2 \vec\alpha).\end{aligned}$$ U and V are both n × 4 matrices. It is a well-known result (one of the matrix determinant lemmas, ) that assuming a is an invertible matrix the determinant of the expression in brackets is equal to $$\begin{aligned} \det \left(a + UWV^T\right) = \det \left(\mathbb I\_{4\times 4} + WV^Ta^{-1}U\right)\det a.\end{aligned}$$ It follows that $$\begin{aligned} \det g = {\text{sgn}}(A)^n\rho^n\det \left(\mathbb I\_{4\times 4} + WV^Ta^{-1}U\right)\det a.\end{aligned}$$ The matrix product $WV^Ta^{-1}U = \tfrac{{\text{sgn}}(A)}{\rho}V^Ta^{-1}U$ can be evaluated by explicit computation and reads $$\begin{aligned} WV^Ta^{-1}U = \frac{{\text{sgn}}(A)}{\rho}\left( \begin{array}{cccc} b^2 \text{$\rho\_0 $} & b^2 \text{$\rho\_0$} & {\text{sgn}}(A) s \text{$\rho\_0$} & {\text{sgn}}(A) s \text{$\rho\_0$} \\ {\text{sgn}}(A) s \rho & {\text{sgn}}(A) s \rho & {\text{sgn}}(A) \rho & {\text{sgn}}(A) \rho \\ b^2 \text{$\rho\_1$} & b^2 \text{$\rho\_1$} & {\text{sgn}}(A) s \text{$\rho\_1$} & {\text{sgn}}(A) s \text{$\rho\_1$} \\ {\text{sgn}}(A) s \text{$\rho\_2$} & {\text{sgn}}(A) s \text{$\rho\_2$} & {\text{sgn}}(A) \text{$\rho\_2$} & {\text{sgn}}(A) \text{$\rho\_2$} \\ \end{array} \right).\end{aligned}$$ Hence we obtain $$\begin{aligned} \det g&= {\text{sgn}}(A)^n\rho^n \det\left(\mathbb I\_{4\times 4}+\frac{{\text{sgn}}(A)}{\rho}\left( \begin{array}{cccc} b^2 \text{$\rho\_0 $} & b^2 \text{$\rho\_0$} & {\text{sgn}}(A) s \text{$\rho\_0$} & {\text{sgn}}(A) s \text{$\rho\_0$} \\ {\text{sgn}}(A) s \rho & {\text{sgn}}(A) s \rho & {\text{sgn}}(A) \rho & {\text{sgn}}(A) \rho \\ b^2 \text{$\rho\_1$} & b^2 \text{$\rho\_1$} & {\text{sgn}}(A) s \text{$\rho\_1$} & {\text{sgn}}(A) s \text{$\rho\_1$} \\ {\text{sgn}}(A) s \text{$\rho\_2$} & {\text{sgn}}(A) s \text{$\rho\_2$} & {\text{sgn}}(A) \text{$\rho\_2$} & {\text{sgn}}(A) \text{$\rho\_2$} \\ \end{array} \right)\right)\det a \\ &= {\text{sgn}}(A)^n\rho^n \det\left(\mathbb I\_{4\times 4}+\frac{1}{\rho}\left( \begin{array}{cccc} {\text{sgn}}(A) b^2 \text{$\rho\_0 $} & {\text{sgn}}(A) b^2 \text{$\rho\_0$} & s \text{$\rho\_0$} & s \text{$\rho\_0$} \\ s \rho & s \rho & \rho & \rho \\ {\text{sgn}}(A) b^2 \text{$\rho\_1$} & {\text{sgn}}(A) b^2 \text{$\rho\_1$} & s \text{$\rho\_1$} & s \text{$\rho\_1$} \\ s \text{$\rho\_2$} & s \text{$\rho\_2$} & \text{$\rho\_2$} & \text{$\rho\_2$} \\ \end{array} \right)\right)\det a \\ &= \phi^{n+1}(\phi-s\phi')^{n-2}(\phi-s\phi' + ({\text{sgn}}(A) b^2-s^2)\phi'')\det a\_{ij}.\end{aligned}$$ Some useful identities that we have used are: αi = sgn(A)yi/α so that αiαi = sgn(A) and αibi = sgn(A)s. We conclude that $$\begin{aligned} \boxed{ \det g\_{ij} = \phi^{n+1}(\phi-s\phi')^{n-2}(\phi-s\phi' + ({\text{sgn}}(A) b^2-s^2)\phi'')\det a\_{ij}.}\end{aligned}$$ In the case that α is positive definite, sign(A) = 1 everywhere, so the formula reduces to the standard result (see e.g. ). --- 1. In the positive definite setting a similar field equation has been obtained by Chen and Shen.[↩](#fnref1) 2. The full vacuum field equation for F is more complicated in general, but as discussed extensively above, if the vacuum field equation for α is satisfied then so is the field equation for F. In the case of Randers metrics, to which we will turn momentarily, the field equation for F is even equivalent to the field equation for α. Hence for our present purposes the field equations for α suffice.[↩](#fnref2) 3. We leave open the question whether the form of the 1-form β = du always remains invariant under such a transformation to the transverse traceless gauge.[↩](#fnref3) 4. Actually this is not true for all (α, β)-metrics but only those which allow an expansion around s = β/α = 0. This excludes Kropina metrics, for instance, because they are not well-behaved in the limit β → 0.[↩](#fnref4) 5. In order to satisfy all the axioms of a Finsler space, the 1-form must satisfy ∣b∣2 < 1, see e.g..[↩](#fnref5) 6. We note that the signature convention in is the opposite as the one employed here, so in that case the condition aijyiyj > 0 precisely select the timelike, not spacelike, vectors.[↩](#fnref6) 7. This can be checked easily in suitable coordinates adapted to β.[↩](#fnref7) 8. We note that in the approach by Javaloyes and Sánchez a single, future pointing (by definition) cone is sufficient, though.[↩](#fnref8) 9. In case one employs the opposite signature convention ( + , − , − , − ) the converse would be true. In that case the preferable choice would be F = sgn(A)α − ∣β∣ rather than F = sgn(A)α + ∣β∣.[↩](#fnref9) 10. The argument is the same as in the positive definitive case, using the same methods as those employed in Appendix [sec:proofofsignature].[↩](#fnref10) 11. We recall that this can be seen as follows. First, since aij is Lorentzian, it is always possible to choose coordinates such that A is just the Minkowski metric at a given point x ∈ M. Writing bμ = (b0, b1, …bn − 1) in these coordinates, we may do a spatial rotation on the coordinates b1, …, bn − 1, such that they are transformed into (b1, 0, …, 0), leaving the metric at x unchanged. Then bμ = (b0, b1, 0, …, 0). Now we separate the three cases. If b2 = 0, it follows that b1 = ± b0 and by applying if necessary a spatial reflection in the x1 direction we may choose either sign. If b2 < 0 then we may go to the local rest frame by a Lorentz transformation, making b1 = 0. If on the other hand b2 > 0 we may perform a Lorentz transformation making b0 = 0.[↩](#fnref11) 12. In this context, we say that an observer is intertial if it would considered an intertial observer in the absence of the wave (i.e. when f+ = f× = 0). In other words, thinking of the gravitational wave as having a finite duration as it passes the Earth, an observer is inertial precisely if it is inertial before and after the wave passes.[↩](#fnref12) Finsler Gravitational Waves of (α, β)-Type and their Observational Signature ================================================================================ We introduce a new class of (α, β)-type exact solutions in Finsler gravity closely related to the well-known pp-waves in general relativity. Our class contains most of the exact solutions currently known in the literature as special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, and we investigate the physical effect of such waves. More precisely, we compute the Finslerian correction to the radar distance along an interferometer arm at the moment a Finslerian gravitational wave passes a detector. We come to the remarkable conclusion that the effect of a Finslerian gravitational wave on an interferometer is indistinguishable from that of standard gravitational wave in general relativity. Along the way we also physically motivate a modification of the Randers metric and prove that it has some very interesting properties. Introduction ============ Even though in the general theory of relativity (GR) the geometry of spacetime is modelled by a metric of Lorentzian signature, there is no clear physical principle, nor experimental evidence, that tells us that this spacetime geometry should necessarily be. In fact, as suggested already in 1985 by Tavakol and Van den Bergh, the axiomatic approach by Ehlers, Pirani and Schild (EPS) is compatible with Finsler geometry, a natural extension of geometry. This was originally overlooked due to too restrictive differentiability assumptions, as recently pointed out in and then worked out in detail in. Other axiomatic approaches also allow for types of geometry more general than the type used in GR, see e.g.. This indicates that such types of geometries should not a priori be excluded from our theories and motivates the study of extensions of general relativity based on more general spacetime geometries. In this regard Finsler geometry is the natural candidate as it provides the most general geometric framework that is still compatible with the clock postulate in the usual sense, namely that the proper time interval measured by an observer between two events can be defined as the length of its worldline connecting these events, in this case the Finslerian length rather than the length. We remark that Weyl geometry, another generalization of Lorentzian geometry, is also compatible with the clock postulate, but in that case the definition of proper time has to be revised. Further motivation for the study of Finsler spacetime geometry comes from quantum gravity phenomenology. Inspired by various approaches to quantum gravity, a generic feature of phenomenological or effective quantum gravity models is the presence of Planck-scale modified dispersion relations (MDR), related to departure from (local) Lorentz symmetry, which may manifest either in the sense of Lorentz invariance violation (LIV) or in the sense of deformed Lorentz symmetry. It turns out that such MDRs generically induce a Finsler geometry on spacetime. The mathematical details of this were investigated in ; see e.g. for applications to specific quantum gravity phenomenology models. Here we consider the (action-based) approach to Finsler gravity outlined in. Structurally the theory is completely analogous to general relativity, but Einstein’s field equation is replaced by Pfeifer and Wohlfarth’s field equation. For spacetimes the latter reduces to the former. Although any solution to the field equations of GR is a solution in Finsler gravity, not many exact, properly Finslerian solutions are known as of yet. To the best of our knowledge the only ones currently known in the literature are the (m-Kropina type) Finsler pp-waves and their generalization as Very General Relativity (VGR) spacetimes, and the Randers pp-waves. Here we introduce a large class of exact vacuum solutions that contains most of the aforementioned solutions as special cases, the only exception being those solutions in that are not of pp-wave type. Namely, we prove that any Finsler metric constructed from a metric α and a 1-form β that is covariantly constant with respect to α, is an exact vacuum solution in Finsler gravity if α is a vacuum solution in general relativity. We classify all such solutions, leading to two possibilities: either α is flat Minkowski space, or α is a pp-wave. Our solutions are (α, β)-metrics of Berwald type. The natural question that arises is whether and how such spacetimes can be physically distinguished from their general relativistic counterparts. To answer this question we consider the versions of our exact solutions, which may be interpretted as Finslerian gravitational waves, and we study their physical effect. More precisely, we ask the question what would be observed in an interferometer experiment when such a Finslerian gravitational wave would pass the earth, and what would be the difference with a classical general relativistic gravitational wave. The relevant observable measured in interferometer experiments is essentially the radar distance, Although at first sight the expression for the Finsler radar length looks different from the corresponding expression GR, we show that this is nothing but a coordinate artifact. Remarkably, when the two expressions are interpreted correctly in terms of observable quantities, it becomes clear that there is in fact no observational difference between the Finsler and GR case, at least as far as radar distance measurements are concerned. We discuss the significance of this. To the best of our knowledge this is the first time an explicit expression for the Finslerian Radar length has been obtained in the case of finite spacetime separations, and as such our work may be seen as a proof of concept. In contrast, the radar length for infinitesimal separations has been studied in. We do point out that our results rely on the assumption that the amplitude of the gravitational wave, as well as the parameter λ that characterized the departure from (pseudo)-Riemannian geometry, are sufficiently small, so that a certain perturbative expansion is valid. This nevertheless seems physically justified. We argue in a heuristic manner that up to first order in λ, any physically viable (α, β)-metric can be equivalently described by a slightly modified version of a standard Randers metric. Indeed, the causal structure of the standard Randers metric does not in general have a straightforward physical interpretation. We therefore propose to modify the Randers metric slightly, only changing some relative signs in different subsets of the tangent bundle. We then prove that these modified Randers metrics have the nice property that their causal structure is completely equivalent to the causal structure of some auxiliary metric. This analysis is done in full generality, i.e. not just for our exact solutions. In the special case, however, that the defining 1-form of the Randers metric is covariantly constant (as is the case for our solutions) we prove that not only the causal structure, but also the affine structure of the Finsler metric coincides with that of the auxilliary (pseudo)-Riemannian metric, i.e. the timelike, spacelike and null geodesics of the Finsler metric can be understood, respectively, as the timelike, spacelike and null geodesics of the auxiliary (pseudo)-Riemannian metric. This leads to the particularly nice property that the existence of radar neighborhoods is guaranteed, i.e. that given an observer and any event in spacetime, there is (at least locally) exactly one future pointing light ray and one past pointing light ray that connect the event to the worldline of the observer. This is of essential importance in our work, because without this property the notion of radar distance would not even make sense. Structure of this article ------------------------- The paper is organized as follows. We begin in Section [sec:Finslergravity] with a discussion of Finsler geometry and the core ideas behind Finsler gravity. Then in Section [sec:abmetrics] we introduce (α, β)-metrics, and in particular Randers metrics and discuss their relevance to Finsler gravity. We then introduce our new solutions to the field equations and show that after linearization these solutions may be interpretted as Finslerian gravitational waves. Next, in section [sec:Randers] we propose our modification of the standard Randers metric and prove that it has very satisfactory properties with respect to its causal structure, affine structure, Lorentzian signature, etc. We conclude in section [sec:discussion]. Finsler Gravity =============== Finsler spacetime geometry -------------------------- A note about causal structure and physical interpretation --------------------------------------------------------- Given a Finsler spacetime geometry, it is natural to postulate, in analogy with GR, that matter travels along timelike geodesics and light travels on null geodesics. The generalization of the notion of null direction is mathematically straightforward. A vector yu at a point xμ is said to be null (or lightlike) if F(x, y)2 = gμν(x, y)yμyν = 0. However, the structure of the light cone, composed of such null vectors, may be non-trivial. In GR it is always the case that the light cone separates the tangent space at each point into three connected components, that we may interpret as forward-pointing timelike vectors, backward-pointing timelike vectors, and spacelike vectors, respectively. It is then a consequence that a timelike vector is one that has positive (or negative, depending on the convention) Riemannian norm. For a generic Finsler spacetime geometry these properties of the lightcone structure are by no means guaranteed and as such it is not obvious in general how to even define what one means a by timelike vector. It certainly does not suffice to define them as positive length vectors. We do not discuss this issue any further in its full generality here. Only in the specific case of the Randers metric, in Section [sec:Randers], will we dive into the details. We argue that the causal structure of the standard Randers metric does not have a straightforward physical interpretation, but we prove that, by modifying the definition only slightly, the causal structure of such a modified Randers metric has exactly the desirable properties mentioned above in the case of GR, allowing for a straightforward physical interpretation. This will be exploited in Section [sec:radardistance], where we compute the radar distance for a Finslerian gravitational wave of (modified) Randers type passing an interferometer. It is worth mentioning that in the ideal case the (forward and backward) timelike cones should be contained in the subbundle $\mathcal A$. This statement is essentially the condition that geometry is well-defined for all timelike infinitesimal spacetime separations. This property is satisfied by our modified Randers metrics (up to a set of measure zero). It can be argued that it is not strictly necessary for spacelike vectors to be contained in $\mathcal A$, as it would not be possible, not even in principle, to perform any physical experiment that probes such directions. Whether the lightcone should be contained in $\mathcal A$ is a more delicate question, which we will not further explore here. The field equations ------------------- In the context of Finsler gravity, arguably the simplest and cleanest proposal for a vacuum field equation was the one by Rutz. The Rutz equation, Ric = 0, can be derived from the geodesic deviation equation in complete analogy to the way Einstein’s vacuum field equation, Rμν = 0 (to which it reduces in the classical setting), can be derived by considering geodesic deviation. However, it turns out that Rutz’s equation is not variational, i.e. it cannot be obtained by extremizing an action functional. In fact, its variational completion (i.e. the variational equation that is as close as possible to it, in a well-defined sense ) turns out to be the field equation that was proposed by Pfeifer and Wohlfarth in using a Finsler extension of the Einstein-Hilbert action. This is again in complete analogy to the situation in GR, where the vacuum Einstein equation in the form $R\_{\mu\nu} - \frac{1}{2}g\_{\mu\nu}R = 0$ is also precisely the variational completion of the equation Rμν = 0. While in the GR case the completed equation happens to be equivalent to the former, this is not true any longer in the Finsler setting. Although several other proposals have been made as well, we consider the Pfeifer-Wohlfarth equation[1](#fn1) to be by far the most promising, and from here onwards we will refer to it simply as the vacuum field equation in Finsler gravity. We do not show the field equation in full generality here, as its general form is not required for our present purposes. In the case of Berwald spacetimes it can be expressed relatively simply as $$\begin{aligned} \label{eq:BEFEs} \left( F^2 g^{\mu\nu} - 3 y^\mu y^\nu \right) R\_{\mu\nu} = 0\,,\end{aligned}$$ where Rμν is the Finsler Ricci tensor and since we are in a Berwald setting, Rμν = Rμν(x) only depends on x. Clearly the vanishing of the Finsler Ricci tensor is a sufficient condition for a Berwald spacetime to be a solution to Eq.. (α, β)-Metrics ================== (α, β)-metrics – basic definitions -------------------------------------- An important class of Finsler geometries is given by the so-called (α, β)-metrics. Here $\alpha = \sqrt{\|a\_{\mu\nu}\dot x^\mu\dot x^\nu\|}$ and $\beta = b\_\mu\dot x^\nu$ are scalar variables defined in terms of a metric aμν on M and a 1-form bμ on M, and an (α, β)-metric is simply a Finsler metric that is constructed only from α and β, i.e. F = F(α, β). Due to homogeneity it follows that any such F can be written in the standard form F = αϕ(β/α) for some function ϕ, at least whenever α ≠ 0. Well-known examples of (α, β)-metrics are: * Pseudo-Riemannian Finsler metrics F = α; * Randers metrics F = α + β; * Kropina metrics $F = \frac{\alpha^2}{\beta}$; * Exact (α, β)-metric solutions in Finsler gravity ---------------------------------------------------- From the physical viewpoint, (α, β)-metrics allow us to deform a GR spacetime α into a Finsler spacetime by the 1-form β. And it turns out, as we will prove below, that these types of metrics can be used to generalize some of the vacuum solutions to Einstein’s field equations to properly Finslerian vacuum solutions in Finsler gravity. This procedure is possible whenever such a solution admits a covariantly constant vector field, or equivalently, 1-form. Namely: if the Lorentzian metric α solves the classical Einstein equations and the 1-form β is covariantly constant with respect to α then any (α, β)-metric constructed from the given α and β is a solution to the Finslerian field equations. To see why this is true, we first recall the following well-known result (see e.g. section 6.3.2. in ): [prop:coincidingspray] Let F be an (α, β)-metric. If β is covariantly constant with respect to α then F is of Berwald type and the affine connection of F coincides with the Levi-Civita connection of α. If the affine connection of F is the same as the connection of α, the associated curvature tensors and (affine) Ricci tensors are also the same. So if α happens be a vacuum solution to Einstein gravity, i.e. its Ricci tensor vanishes, then it follows that the affine Ricci tensor of F vanishes as well, which implies, by eq., that F is a vacuum solution to Pfeifer and Wohlfarth’s field equation in Finsler gravity. We may summarize this result in the following theorem. [theor:(alpha,beta)solutions] Let F be any (α, β)-metric such that α solves the classical vacuum Einstein equations and β is covariantly constant with respect to α. Then F is a vacuum solution to the field equation in Finsler gravity. In this way (α, β)-metrics provide a mechanism to Finslerize any vacuum solution to Einstein’s field equations, as long as the solution admits a covariantly 1-form, or equivalently a covariantly constant vector field. The theorem generalizes some of the results obtained in for Randers metrics and in for m-Kropina metrics (i.e. VGR spacetimes) to arbitrary Finsler spacetimes with (α, β)-metric. In particular, all pp-wave type solutions in Finsler gravity currently known in the literature are of this type. Let’s investigate this type of solution in some more detail. It turns out that if a vacuum solution α to Einstein’s field equations admits a covariantly constant 1-form β, then either α is flat, or β is necessarily null (see also ). We remark that this result assumes that the spacetime dimension is 1 + 3 and generally is not true in higher dimensions. This leads to two classes of solutions. First class of solutions The first of these possibilities, where α is flat, leads to a class of solutions that can always be written in suitable coordinates in the following way. Right below Eq. we have used the notation $\alpha = \sqrt{\|A\|} = \sqrt{\|a\_{ij}{\text{d}}x^i {\text{d}}x^j\|}$. This should be understood pointwise, i.e. $$\begin{aligned} \alpha = \alpha(y) = \sqrt{\|a\_{ij}{\text{d}}x^i {\text{d}}x^j\|}(y) = \sqrt{\|a\_{ij}{\text{d}}x^i(y) {\text{d}}x^j(y)\|} = \sqrt{\|a\_{ij} y^i y^j\|}.\end{aligned}$$ In other words, we sometimes write α for the function $\sqrt{\|a\_{ij}{\text{d}}x^i {\text{d}}x^j\|}:y\mapsto \sqrt{\|a\_{ij} y^i y^j\|}$, and at other times we write α for its value $\sqrt{\|a\_{ij} y^i y^j\|}$ at y. It should always be clear from context what is meant. Second class of solutions The second possibility, that β is null, leads to a class of solutions that seems to be more interesting. In this case α is CCNV spacetime metric, meaning that it admits a covariantly constant null vector (CCNV), namely in this case β, or rather its vector equivalent via the isomorphism induced by α. CCNV metrics are also known as pp-waves (plane-fronted gravitational waves with parallel rays) and have been studied in detail in (see section 24.5 in for a summary). It is an elementary result that by choosing suitable coordinates (u, v, x1, x2), such α and β can always be expressed in the form $$\begin{aligned} A &= -2{\text{d}}u \left({\text{d}}v + H(u,x)\, {\text{d}}u + \,W\_a(u,x)\,{\text{d}}x^a\right) +h\_{ab}(u,x) {\text{d}}x^a {\text{d}}x^b, \label{eq:original\_pp\_wave\_metric} \\ \beta &= {\text{d}}u,\label{eq:original\_1form}\end{aligned}$$ where xa = x1, x2 and hab is a two-dimensional Riemannian metric. This holds irrespective of whether α is a solution to Einstein’s field equations or not. If α is additionally assumed to be a vacuum solution, as in Theorem [theor:(alpha,beta)solutions], it turns out that the expression for A can be simplified even more without changing the form of β*. To see this, we first consider only the metric A. Since A* is a vacuum solution to Einstein’s field equations, it follows that the functions Wa can be eliminated and hab may be chosen as δab, by a suitable coordinate transformation (section 24.5 in ). The metric then takes the form $$\begin{aligned} A = -2{\text{d}}u \left({\text{d}}v + H(u,x)\, {\text{d}}u\right) +\delta\_{ab} {\text{d}}x^a {\text{d}}x^b. \label{eq:reduced\_pp\_wave\_metric}\end{aligned}$$ We are, however, not only interested in the transformation behaviour of A alone, but also in that of β, because an (α, β)-metric is composed of both. To see why we may assume without loss of generality that the form of β = du remains invariant we use the fact that any coordinate transformation $$\begin{aligned} (u,v,x^1,x^2)\mapsto (\bar u,\bar v,\bar x^1,\bar x^2)\end{aligned}$$ that leaves the generic form of the metric invariant, but in general changing the expressions for the metric functions $H, W\_a, h\_{ab}\mapsto \bar H, \bar W\_a, \bar h\_{ab}$, has the specific property that $u = \phi(\bar u)$ for some function ϕ depending on $\bar u$ alone (see section 31.2 in ). This applies in particular to the transformation that relates and. We can therefore express the 1-form as $\beta = {\text{d}}u = \phi'(\bar u){\text{d}}\bar u$, or equivalently $\bar b\_\mu = \phi'(\bar u)\delta\_\mu^u$. However, since β is covariantly constant with respect to A, we must have $\bar \nabla\_\mu\bar b\_\nu=0$. All Christoffel symbols $\bar \Gamma^u\_{\mu\nu}$ of the metric with upper index u vanish identically, however. Hence $$\begin{aligned} \bar \nabla\_{\bar u} \bar b\_{\bar u} = \partial\_{\bar u} b\_{\bar u} - \bar \Gamma^u\_{uu}\phi'(\bar u) = \phi''(\bar u)\stackrel{!}{=}0.\end{aligned}$$ It follows that $\phi'(\bar u)= C =$ constant, i.e. $\beta = C {\text{d}}\bar u$. In this case it is easily seen that scaling $\bar u$ by C and scaling $\bar v$ by 1/C leaves the metric invariant and brings the 1-form back into its original form, proving that we may assume without loss of generality that the 1-form remains invariant under the coordinate transformation. Finally, the metric is a vacuum solution to Einstein’s field equations if and only if (∂x12 + ∂x22)H = 0. We may therefore characterize the second class of solutions in the following way. Note that when H = 0 the geometries in Class 2 are also contained in Class 1. It is not the case, however, that Class 1 is a subset of Class 2 because in Class 1 the 1-form β need not be null, necessarily. The preceding line of argument shows that these two classes of solutions in fact exhaust all possibilities, which we encapsulate in the following theorem. [theor:(alpha,beta)solutions2] Any vacuum solution of the type of Theorem [theor:(alpha,beta)solutions] must belong to one of the two classes introduced above. Before we move on to (α, β)-type solutions of plane-wave type, we end this section by noting that for specific types of (α, β)-metrics, stronger results have been obtained than the ones derived above: * For Randers metrics of Berwald type any vacuum solution to must be of the type described in theorem [theor:(alpha,beta)solutions], that is, α is necessarily a vacuum solution in Einstein gravity and β is necessarily covariantly constant. Any such solution is therefore either of Class 1 or Class 2 in the terminology introduced above. * For m-Kropina metrics some vacuum solutions of a more general type than the one in theorem [theor:(alpha,beta)solutions] have been obtained in the context of Very General Relativity (VGR). * Any pseudo-Riemannian Finsler metric F = α is trivially a vacuum solution in Finsler gravity if and only if it is a vacuum solution in Einstein gravity. To the best of our knowledge this list comprises all exact solutions in Finsler gravity currently known in the literature. Plane wave solutions in Brinkman and Rosen coordinates ------------------------------------------------------ Eq. expresses the pp-wave metric in Brinkmann form. For the description of the physical effects of (plane) gravitational waves in general relativity, it is sometimes more convenient to use a different coordinate system, known as Rosen coordinates. This remains true in the Finsler case. When we compute the effect on the radar distance of a passing Randers gravitational wave in section [sec:radardistance], our starting point will be the expression for the gravitational wave in Rosen coordinates. Therefore we briefly review the relation between the two coordinate systems here. Rosen coordinates can be introduced for the subclass of pp-waves known as plane waves. These can be characterized by the property that the curvature tensor does not change (i.e. is covariantly constant) along the Euclidean ‘wave surfaces’ given in Brinkmann coordinates by du = dv = 0, i.e. $$\begin{aligned} \label{eq:invariance\_of\_Riemann\_curvature} \nabla\_{\partial\_{x^1}} R^\rho{}\_{\sigma\mu\nu} = \nabla\_{\partial\_{x^2}} R^\rho{}\_{\sigma\mu\nu} = 0.\end{aligned}$$ We note that ∇∂vRρσμν = 0 always holds, identically, so The conditions are equivalent to the statement that ∂a∂b∂cH = 0 in Brinkmann coordinates, i.e. that H(u, x) is a second order polynomial in xa. In that case there always exists a coordinate transformation that removes the linear and constant terms (section 24.5 in ) so that the metric can be written as $$\begin{aligned} \label{eq:plane\_wave\_brinkmann} A = -2{\text{d}}u {\text{d}}v + A\_{ab}(u)x^ax^b\, {\text{d}}u^2+\delta\_{ab}\, {\text{d}}x^a {\text{d}}x^b\end{aligned}$$ This is the standard expression for a plane-wave metric in Brinkmann form. Moreover, an argument very similar to the one given in the previous subsection, shows that we may assume without loss of generality that the 1-form β = du remains unchanged under this transformation. Any such plane wave metric can also be written in Rosen form $$\begin{aligned} \label{eq:Rosen\_form} {\text{d}}s^2 = -2{\text{d}}U{\text{d}}V + h\_{ij}(U){\text{d}}y^i {\text{d}}y^j,\end{aligned}$$ where hij is a two-dimensional Riemannian metric. And conversely, any metric of Rosen form can be cast in the form. The two coordinate systems are related via $$\begin{aligned} U = u,\quad V = v - \dfrac{1}{2}\dot E\_{ai} E^i{}\_b x^a x^b, \quad x^a = E^a{}\_iy^i,\end{aligned}$$ where $A\_{ab} = \ddot E\_{ai}E^i{}\_b$ and Eai is a vielbein for hij in the sense that hij = EaiEbjδab, satisfying the additional symmetry condition $\dot E\_{ai} E^i{}\_b = \dot E\_{bi} E^i{}\_a$. Such a vielbein can always be chosen. For details we recommend the lecture notes by Matthias Blau and references therein (see also the Appendix of ). Note that we have momentarily labelled the y-coordinates by indices i, j, k, … so as to distinguish them from indices a, b, c, … in order that we may apply the usual notation with regards to the vielbein indices: Eia represents the (matrix) inverse of Eai and indices a, b, c… are raised and lowered with δab, whereas indices i, j, k, … are raised and lowered with hij. The dot that appears sometimes above the vielbein represents a U-derivative. Since the vielbein depends only on U, this derivative is equivalent to a u-derivative, and moreover the raising and lowering of the a, b, c, … indices commutes with taking such a derivative of the vielbein. It is again the case that, after relabeling U, V ↦ u, v, the 1-form β = du = dU remains unchanged under this transformation, which in this case is easy to see. After also relabelling y ↦ x, we conclude that we can express any Class 2 solution of plane-wave type in Rosen coordinates as follows, $$\begin{aligned} F = \alpha\, \phi(\beta/\alpha), \qquad A = -2{\text{d}}u{\text{d}}v + h\_{ij}(u){\text{d}}x^i {\text{d}}x^j, \qquad \beta = {\text{d}}u \label{eq:ab\_plane\_wave},\end{aligned}$$ where $\alpha = \sqrt{\|A\|}$. And conversely, for any choice of ϕ, hij(u), this is a vacuum solution to the field equations in Finsler gravity if A is a vacuum solution to Einstein’s field equation. The resulting geometry is of Berwald type with affine connection identical to the Levi-Civita connection of α. Linearized gravitational wave solutions --------------------------------------- The exact vacuum field equation for plane-wave metrics does not have a particularly nice expression in Rosen coordinates. The field equation, however, turns out to be very simple. So let’s consider the scenario that the pseudo-Riemannian metric α is very close to the Minkowski metric. In this case we may write hij(u) = δij + ɛfij(u) with ɛ ≪ 1. The field equations (i.e. to first order in ɛ) for α then simply read[2](#fn2) $$\begin{aligned} f\_{11}''(u) + f\_{22}''(u)=0.\end{aligned}$$ Hence f11 and − f22 must be equal up to an affine function of u. Here we will focus on the case where f11 = − f22, which can always be achieved by means of the transverse traceless gauge[3](#fn3). Conventionally one writes the subscripts as $f\_{11} = -f\_{22} \eqqcolon f\_+$ and $f\_{12} \eqqcolon f\_\times$, denoting the plus and cross polarization of the gravitational wave, so we will stick to that notation from here onwards. That brings us to the following expression that describes Finslerian gravitational waves of (α, β)-type: $$\begin{aligned} \label{eq:ab\_grav\_waves} F = \alpha\, \phi(\beta/\alpha), \qquad\left\{\begin{array}{ll} A = -2{\text{d}}u {\text{d}}v + (1+\varepsilon f\_+(u)) {\text{d}}x^2 + (1-\varepsilon f\_+(u)){\text{d}}y^2 + 2\varepsilon f\_\times (u) {\text{d}}x\,{\text{d}}y \\ \beta = {\text{d}}u \end{array}\right.\end{aligned}$$ Note that if we substitute $u= (t-z)/\sqrt{2}$ and $v = (t+z)/\sqrt{2}$, then A reduces to the standard expression for a gravitational wave metric in GR, i.e. $$\begin{aligned} F = \alpha\, \phi(\beta/\alpha), \qquad\left\{\begin{array}{ll} A = -{\text{d}}t^2 + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y+ {\text{d}}z^2 \\ \beta = \frac{1}{\sqrt{2}}\left({\text{d}}t - {\text{d}}z\right) \end{array}\right.,\end{aligned}$$ for any choice of the function ϕ. Linearized (α, β)-metrics are Randers metrics ------------------------------------------------- It is natural to linearlize not only in ɛ, characterizing the departure from flatness, but to also use a perturbative expansion in the ‘size’ of the 1-form, characterizing the departure from GR and pseudo-Riemannian geometry. The physical intuition here is that, seeing how well GR works in most regimes, the most interesting class of Finsler spacetimes constists of those ones that are very close to GR spacetimes. The purpose of this section is to highlight that any (α, β)-metric is perturbatively equivalent to a Randers metric, to first order, so that from the physics point of view, Randers metrics are actually quite a bit more general than they might seem at first glance. After pointing this out we will turn our focus exclusively to Randers metrics for the remainder of the article. So consider an (α, β)-metric constructed form a pseudo-Riemannian metric α and a 1-form β such that β ≪ 1. To see what happens in such a scenario, we replace β with λβ and expand to first order in λ. Then we obtain $$\begin{aligned} F = \alpha \phi\left(\frac{\lambda\beta}{\alpha}\right) \approx \alpha \left(\phi(0)+ \lambda \phi'(0)\frac{\beta}{\alpha}\right) = \alpha \phi(0) + \lambda\phi'(0)\beta = \tilde\alpha + \tilde\beta.\end{aligned}$$ Hence to first order in λ, any (α, β)-metric is indeed equivalent to a Randers metric[4](#fn4). Consequently, by replacing du by λ du in, which technically can be achieved by a coordinate transformation that scales u by λ and v by 1/λ, it follows that to first order in λ the Finsler metric of the (α, β)-type gravitational waves takes the form, $$\begin{aligned} \label{eq:metric\_naive\_Randers\_grav\_wave0} F = \alpha + \beta, \qquad\left\{\begin{array}{ll} A = -2{\text{d}}u {\text{d}}v + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y \\ \beta = \lambda\,{\text{d}}u \end{array}\right..\end{aligned}$$ The parameter λ then characterizes the departure from GR and pseudo-Riemannian geometry. We will assume without loss of generality that λ > 0. Finally, replacing also u and v by t and z, according to $u= (t-z)/\sqrt(2)$ and $v = (t+z)/\sqrt{2}$, we can write the metric in the following way, which we will take as the starting point for the calculation of the radar distance in Section [sec:radardistance]. $$\begin{aligned} \label{eq:metric\_naive\_Randers\_grav\_wave} F = \alpha + \beta, \qquad\left\{\begin{array}{ll} A = -{\text{d}}t^2 + (1+\varepsilon f\_+(t-z)) {\text{d}}x^2 + (1-\varepsilon f\_+(t-z){\text{d}}y^2 + 2\varepsilon f\_\times (t-z) {\text{d}}x\,{\text{d}}y+ {\text{d}}z^2 \\ \beta = \frac{\lambda}{\sqrt{2}}\left({\text{d}}t - {\text{d}}z\right) \end{array}\right. \end{aligned}$$ Modified Randers Metrics ======================== Motivated by the argument above we will now turn our focus to the simplest properly Finslerian (α, β)-metric, the Randers metric, conventionally defined as F = α + β. We will argue that in order to have a physically causal structure, the conventional definition must be modified slightly. It might seem to the reader that modifying the Randers metric would be in conflict with the spirit of the previous section, since to first order any (α, β)-metric should reduce to a Randers metric. It is important to note, however, that there is in principle the possibility that to different regions of the tangent bundle could correspond different Randers metrics. More precisely, we could define one (α, β)-metric F1 on a conic subbundle $\mathcal A\_1\subset TM\setminus 0$ and another (α, β)-metric, F2, on a different conic subbundle $\mathcal A\_2\subset TM\setminus 0$. If the two subbundles do not overlap then this defines a perfectly valid (α, β)-type Finsler spacetime on the union $\mathcal A = \mathcal A\_1\cup\mathcal A\_2$. To first order in the deviation from geometry this Finsler metric would reduce to a certain Randers metric on $\mathcal A\_1$ and to a different Randers metric on $\mathcal A\_2$. our modification of the Randers metric, introduced below, is therefore completely consistent with the previous results. After a heuristic argument that motivates the desired modification, we show that our proposed version of the modified Randers metric has a very satisfactory causal structure. As a result of this a clear (future and past) timelike cone can be identified and within these timelike cones the signature of the Fundamental tensor is Lorentzian everywhere. The only constraint is that b2 ≡ aμνbμbν > − 1, which, interestingly, is in some sense the opposite of the condition b2 < 1 that appears in the well-known positive definite case, see e.g.. In one were to adopt the opposite signature convention to ours, however, the constraint in the Lorentzian case would also turn out to be b2 < 1, matching the positive definite case. Motivation and definition ------------------------- First of all, let us review why the definition of a Randers metric is not as clear in Lorentzian signature as it is in Euclidean signature. The original definition of a Randers metric, in positive definite Finsler geometry, is just F = α + β, with $\alpha = \sqrt{a\_{ij}y^i y^j}$ a Riemannian metric and β = biyi any 1-form[5](#fn5). This is well-defined as long as α is positive-definite, because in that case A ≡ aijyiyj is always positive. If we allow ai*	arxiv_0000772
Hidden Markov trees for high-dimensional distributions ====================================================== The Pólya tree (PT) process is a general-purpose Bayesian nonparametric model that has found wide application in a range of inference problems. It has a simple analytic form and the posterior computation boils down to beta-binomial conjugate updates along a partition tree over the sample space. Recent development in PT models shows that performance of these models can be substantially improved by (i) allowing the partition tree to adapt to the structure of the underlying distributions and (ii) incorporating latent state variables that characterize local features of the underlying distributions. However, important limitations of the PT remain, including (i) the sensitivity in the posterior inference with respect to the choice of the partition tree, and (ii) the lack of scalability with respect to dimensionality of the sample space. We consider a modeling strategy for PT models that incorporates a flexible prior on the partition tree along with latent states with Markov dependency. We introduce a hybrid algorithm combining sequential Monte Carlo (SMC) and recursive message passing for posterior sampling that can scale up to 100 dimensions. Moreover, we investigate the large sample properties of the tree structures and latent states under the posterior model. We carry out extensive numerical experiments in density estimation and two-group comparison, which show that flexible partitioning can substantially improve the performance of PT models in both inference tasks. We demonstrate an application to a mass cytometry data set with 19 dimensions and over 200,000 observations. Introduction ============ The Pólya tree (PT) is a stochastic process that generates random probability measures and is introduced as a prior for Bayesian nonparametric inference. While the PT generalizes the Dirichlet process (DP) as it yields the DP under certain hyperparameters, the statistical properties and practical applications of the PT are very different. While the DP is most frequently used as a mixing distribution that induces clustering structures, the PT is often adopted for directly modeling probability densities. The PT is defined generatively on a recursive partition—or a partition tree—over the sample space through coarse-to-fine sequential probability assignment at each tree split. In a classical (univariate) PT, the tree is dyadic and the conditional probability assigned to the two children nodes at each tree split arises from independent beta priors, which leads to analytic simplicity and ease in computing the posterior. Obtaining the posterior is straightforward from beta-binomial conjugacy and incurs a computational budget that scales only linearly with the sample size,. The PT has been applied in various contexts beyond the original application of density estimation. A far-from-exhaustive list includes survival analysis, imputing missing values, goodness-of-fit tests, two-group comparison, density regression, ANOVA, testing independence, and hierarchical modeling. The PT has also been utilized in semi-parametric analyses such as in (generalized) linear models. Early developments of the PT are based on an a priori fixed partition tree on the sample space. The resulting inference can be sensitive to the choice of the partition points defining the tree. In particular, the resulting process, both a priori and a posteriori, can be jumpy at these points. In hypothesis testing and model choice, this sensitivity is also reflected in the sometimes substantial change in the marginal likelihood/Bayes factor when the partition points are slightly varied. To remedy the issue, modified the PT model so that observations are generated from the PT model with slightly different partition points. and proposed a mixture of PTs by defining partition points along quantiles of a parametric model endowed with a hyperprior to allow model averaging on the partition points. This strategy does not allow individual partition points to adapt to local features of the distribution but only the whole set of points to the global structure of the distribution, and is most effective when the underlying density is close to the specified parametric model. took a different approach by modeling the probability assignments within each level of the tree in a correlated manner to smooth out the random measure over the boundaries of partitioning. While these approaches alleviate the sensitivity to partition points in low-dimensional settings, they are not easily applicable (though in principle possible) to problems with even just a handful of dimensions. Moreover, Bayesian inference with these models generally require MCMC, whose effectiveness can (in fact often does) still suffer from the sensitivity with respect to the partition points. noted that such a “symmetric” partition scheme is undesirable as the dimensionality increases, in which case due to the exponential growth of the partition blocks, the vast majority of the blocks are barely, if at all, populated by data. As such they propose to incorporate adaptivity into the partitioning strategy with respect to the structure of the underlying distribution through adopting a Bayesian CART-like prior However, in order to maintain the analytic simplicity of the posterior and achieving MCMC-free exact Bayesian inference with a linear computational budget, the Bayesian CART-like prior has to be restricted to only divide at the middle point (or otherwise a pre-determined fixed point) on one of the dimensions at each tree split. Not only does this hamper the model’s ability to adapt to distributional structures, but it makes the model suffer from the same sensitivity with respect to the partition points. Moreover, even with this restriction, the inference algorithm (based on recursive message passing) is only computationally practical for up to about 10 dimensions on continuous sample spaces. In a different vein, recent developments have demonstrated that aside from enhancing the partitioning strategy, the PT can also be substantially improved by adopting more flexible priors (as opposed to independent betas) on the probability assignment at each tree split. One strategy for enriching the PT in this regard is by introducing latent state variables at each tree split and adopt priors on the probability assignment given these states. When the latent states are discrete and modeled with Markov dependency, analytical simplicity is preserved and exact Bayesian inference can proceed through recursive message passing that maintains the linear computational budget. Given these developments, we are motivated by the following questions: Is it possible to incorporate into the PT a very flexible partition tree prior, such as the general Bayesian CART (i.e., without the restriction to partition at middle points), that will (i) enhance its adaptivity to distributional structures in multivariate settings; (ii) resolve its sensitivity to the choice of partition points; and (iii) allow a tractable form of the joint posterior and a posterior inference algorithm that is scalable to moderately high-dimensional problems (e.g., up to 100 dimensions)? Moreover, should such a strategy exist, can the resulting model and inference algorithm be made compatible with incorporating (possibly Markov dependent) latent states on the tree nodes? The goals of making the partition tree prior more flexible while enhancing the computational scalability appear at odds with each other. Large tree spaces are well known to be very hard to compute over. In moderate to high dimensional settings exact inference involving flexible tree structures is beyond reach and even the most advanced MCMC approaches tailor-made for trees encounter substantial difficulty due to the pervasive multi-modality of distributions in such spaces. Recent advances in sequential Monte Carlo (SMC) for regression tree models, however, suggest that efficient inference is possible in moderately high-dimensional settings (up to about 100 dimensions). Moreover, once the partition tree is sampled, the conditional posterior for the rest of the model can be computed analytically through recursive message passing. We will therefore exploit a hybrid strategy that uses a new SMC sampler to efficiently sample from the marginal posterior of the partition tree structure, recursive message passing to compute the exact conditional posterior of the latent state variables given the tree. Beyond the methodological development, we will also investigate the theoretical properties of the posterior on the partition tree and the latent states. Previous theoretical literature on the PT and related models have mostly focused on establishing the posterior consistency and the contraction rate of the random measure induced under these models. In multivariate settings, however, the partition tree itself is highly informative about the underlying distribution. Moreover, in applications involving model choice and hypothesis testing, it is often the latent states, not the random measures, that are of direct interest. As such, we focus on studying the asymptotic behavior of the marginal posterior on the partition tree and latent states, establishing consistency results on their convergence toward the trees and states that most closely characterize the underlying truth. The rest of the paper is organized as follows. In Section [sec: method] we describe a flexible prior on the partition tree structure that relaxes the restriction of “dividing in the middle” on partition points and present a general form of PT models that adopt this prior along with latent states associated with the tree nodes with a Markov dependency structure. In Section [sec: posterior inference], we present our hybrid computational strategy that can work effectively up to 100 dimensions consisting of an SMC algorithm for sampling on the marginal posterior of the partition tree and a recursive message passing algorithm for obtaining the exact conditional posterior of the latent states and the random measure given the sampled trees. In Section [sec: theory] we investigate the asymptotic properties of the tree structures and latent states identified under the posterior model. In Section [sec: experiments], we carry out extensive numerical experiments to examine the performance of our method in the context of two important applications of PTs—density estimation and the two-group comparison, followed by an application to a data set from mass cytometry in Section [sec:mass]. In Section [sec: conclusion] we conclude with a brief discussion. Method ====== [sec: method] We first review the PT model in Section [sec: plain vanilla PT]. Then we introduce a new class of PT models that incorporates both the flexible partition prior and latent states with Markov dependency. Pólya trees ----------- Without loss of generality, we consider a continuous sample space represented as a d-dimensional rectangle Ω = (0, 1]d. We use μ to denote the Lebesgue measure on Ω. A (dyadic) recursive partitioning T on Ω is a sequence of partitions of Ω such that the partition blocks at each level of the partitioning are obtained by dividing each block in the previous level into two children blocks. Formally, we can write T = ⋃k = 0∞Ak, where Ak is a partition of Ω in the kth level. More specifically, A0 = {Ω}, and A ∈ Ak (k = 0, 1, 2, …) is divided into Al and Ar, which satisfy Al, Ar ∈ Ak + 1, Al ∪ Ar = A, and Al ∩ Ar = ∅. (Throughout the paper, a subscript “l” or “r” on a node indicates the left or right child node.) For example, when d = 1 and if the tree is recursively divided at the middle point of each node, then nodes in level k are of the form (l/2k, (l + 1)/2k] for some l ∈ {0, …, 2k − 1}. Another common strategy is to define the tree based on the quantiles of a probability measure F so that A ∈ Ak is of the form $A=(F^{-1}(\frac{l}{2^k}), F^{-1}(\frac{l+1}{2^k})]$ for l ∈ 0, …, 2k − 1. Given a partition tree T, we can define a random measure Q by putting a prior on the conditional probability θ(A) = Q(Al∣A) = 1 − Q(Ar∣A) at each A ∈ T. Under the PT prior, the parameters θ(A) follow independent beta distributions Beta(αl(A), αr(A)), where αl(A) and αr(A) are hyperparameters. with a simple conjugate update on the conditional proabilities: $$\theta(A) \mid x\_1,\dots, x\_n \sim {\rm Beta}(\alpha\_l(A) + n(A\_l), \alpha\_r(A) + n(A\_r)),$$ where n(A) represents the number of observations in a set A ⊂ Ω. Though the tree needs to be infinitely deep to ensure full support of the PT, for practical purposes, one typically sets a sufficiently large maximum depth (or resolution) of T and compute the posteriors of θ(A)’s defined on this finite tree structure. We shall refer to a node in the deepest level as a “leaf” or “terminal node”. On a leaf, the conditional distribution can be set to a baseline, such as the uniform distribution μ( ⋅ ∣A). In Section [sec: posterior inference] when we present inference algorithms, we shall adopt this practical strategy and assume T is finite and use N(T) and L(T) to denote the collection of the non-terminal nodes and the leaf nodes, respectively. Incorporating flexible partition points --------------------------------------- We incorporate a Bayesian-CART like prior on T by randomizing both the dimension in which to divide a node and the location to divide. Our prior relaxes the “always-divide-in-the-middle” restriction imposed in. suppose we have a node A in the rectanglar form, A = (a1, b1] × ⋯ × (ad, bd]. We divide A into two rectangular children by cutting along a randomly chosen dimension at a random location. The dimension to divide D(A) ∈ {1, 2, …, d}, and the (relative) location to divide L(A) ∈ (0, 1) are given independent priors of the following forms: $$\begin{aligned} D(A) &\sim \mathrm{Mult}(\lambda\_1(A), \dots, \lambda\_d(A)) \quad \text{and} \quad L(A) \sim \sum^{N\_L-1}\_{l=1} \beta\_l(A) \delta\_{l/N\_L} (\cdot), \label{prior of d and l}\end{aligned}$$ where δx( ⋅ ) represents the unit point mass at x, and NL − 1 is the total number of grid points along (0, 1). Both and {βl(A)}l = 1, …, NL − 1 sum to 1. In the above, we have adopted a uniform grid over (0, 1), but it does not have to be as such. With D(A) = j and L(A) = l/NL, the two children nodes Al and Ar are $$\begin{aligned} A\_l &= (a\_1,b\_1] \times \cdots \times (a\_j, a\_j + {l}/{N\_L} \cdot (b\_j - a\_j)] \times \cdots \times (a\_d, b\_d], \\ A\_r &= (a\_1,b\_1] \times \cdots \times (a\_j + {l}/{N\_L} \cdot (b\_j - a\_j), b\_j] \times \cdots \times (a\_d, b\_d]. \end{aligned}$$ In principle one could adopt a continuous prior on the partition location L(A). A discretized prior is helpful, however, because it posterior computation. In practice, as long as the grid is dense enough, the discrete prior will be practically just as flexible. Indeed we have verified in extensive numerical experiments that when NL is large enough (more than 30 to 50) over a uniform grid, posterior inference no longer improves in any noticeable way. For the prior on D(A), we set λj(A) = 1/d for all nodes A as a default choice. When L(A) is given a weak prior widely spread over (0, 1), the resulting inference can be sensitive to the “tail” behavior of the distribution in the node, This issue can be effectively addressed by making the prior of L(A) depend on the sample size n(A) so that it encourages more balanced divisions at large sample sizes. More specifically, we adopt the following prior with an exponentially decaying tail $$\begin{aligned} P(L(A) = l/N\_L) = \beta\_l &\propto \exp \left[ - \eta n(A) f(\|l/N\_L - 0.5\|) \right],\ l = 1,\dots,N\_L-1, \label{prior of L} \end{aligned}$$ where η ≥ 0 is a hyperparameter and is an increasing function. In the following, we shall use a function f(x) = x, and so our prior on L(A) is a (discretized) Laplace distribution. Another generalization of the prior on L(A) is to incorporate a spike-and-slab set-up with a spike at the middle point 1/2. In particular, one can adopt a dependent spike prior among the nodes such that once a node A is divided exactly at the middle point, so are its descendants. This generalization will substantially reduce the amount of computation in regions of the sample space where the data are either sparse or lack interesting structure, e.g., close to the uniform distribution. We implement the spike-and-slab in our software but defer the details of this generalization to Supplementary Materials A to avoid distracting the reader from the main ideas. Given the tree prior, our PT model now consists of the two components—tree generation and conditional probability assignment. Figures [fig: HMPT graphical representation](a) and [fig: HMPT graphical representation](b) present a graphical model representation for each. [b]0.33![Partition tree generation (\mathcal{A}_0: All potential nodes)](net_tree.PNG "fig:") [b]0.33![PT without latent states](net15.PNG "fig:") [b]0.33![PT with latent states](net2.PNG "fig:") [fig: HMPT graphical representation] Hidden Markov Pólya tree models ------------------------------- ### General framework Next we extend the above model to accommodate two recent developments in the PT literature: (i) incorporating latent state variables along the tree structure and (ii) joint modeling of. As in recent literature, we consider incorporating discrete state variables that follow a Markov process along the tree structure. Because the description in this section always pertains to the model given the randomly generated partition tree T, for brevity we shall not keep stating “given T”. Next we specify a prior on Q in terms of a joint prior on the conditional probability on each A ∈ T, θg(A) = Qg(Al ∣ A) = 1 − Qg(Ar ∣ A). We use latent variable modeling to incorporate dependency among the tree nodes. Specifically, let {V(A) : A ∈ T} denote a collection of latent state variables, one for each A, and without loss of generality, assume that V(A) takes discrete values from {1, …, I}. (In practice, the number of states I can differ among A.) Joint priors of θg(A) for all g and A are then defined conditionally on these latent states. Existing literature has exploited these latent states to characterize both the within- structure of each distribution Qg and the between- relationship among the Qg. An example of within-sample structures is the smoothness of each underlying distribution, which is explored in the context of density estimation. An example of between- structures is the difference between two (or more) distributions. Dependent modeling of the latent states over the partition tree is desirable as a priori one would expect interesting structures (both within- and between-) to exhibit themselves in a correlated manner over the sample space., functions tend to have similar smoothness over adjacent locations, and two- difference tend to be clustered in space. A computationally efficient strategy for modeling such dependency over the tree is by a hidden Markov process along the tree, which starts from the root node,, and sequentially generates the latent states in a coarse-to-fine fashion according to (possibly node-specific) transition matrices ${\bm {{\bm \xi}}}(A)$ whose (i, iʹ)th element is $$\begin{aligned} {{\bm \xi}}\_{i,i'}(A) &= P(V(A) = i' \mid V(A^p) = i) \quad \text{for } i,i'\in\{1,\dots,I\}, \end{aligned}$$ where Ap denotes A’s parent. (We shall use superscript “p” to indicate the parent of a node in T.) Given the V(A)’s, {θg(A)}g = 1, …, G can then be modeled as conditionally independent. The specific choices of these conditional priors are problem-dependent. We give two examples below. Figure [fig: HMPT graphical representation](c) presents a graphical model representation for the latent state modeling given T, which along with our generalized prior on the partition tree T presented in Figure [fig: HMPT graphical representation](a) forms the most general version of the model we consider in this work. [subsec: examples] An example of within- structures that the latent state V(A) can characterize is the smoothness of the density functions for the random measures. For example, proposed the adaptive Pólya tree (APT) model which incorporates latent states to allow different levels of local smoothness in the underlying distribution. This is achieved by modeling the θ(A)’s as Beta(m(A)ν(A), (1 − m(A))ν(A)), where m(A) is the prior mean and ν(A) the precision parameter which characterizes the smoothness of the random measure with larger ν(A) corresponding to more smoothness, and then modeling the precision parameters conditional on the latent state V(A) with a hyperprior ν(A) ∣ V(A) = i ∼ Fi with the (conditional) hyperprior Fi given V(A) = i. In two-group comparison, we are interested in testing and identifying differences between two measures Q = {Qg}g = 1, 2 based on an i.i.d. sample from each. The “global” testing problem can be formulated as testing the following null and alternative hypotheses: H0 : Q1 = Q2 vs H1 : Q1 ≠ Q2. Noting that two-group differences may exist in parts of the sample space and not others, the coupling OPT and the multi-resolution scanning (MRS) model are PT-based models that allow the measures to differ on some nodes A ∈ T and not others. This more “local” perspective on two-group comparison enables these models to not only test for H0 vs H1, but to identify regions on which the two measures differ. To achieve this, these models incorporate state variables that characterize whether the conditional probabilities on each A are equal: $$\begin{aligned} V(A) = 1 \Leftrightarrow Q\_1(A\_l \mid A) \neq Q\_2(A\_l \mid A), \ V(A) = 2 \Leftrightarrow Q\_1(A\_l \mid A) = Q\_2(A\_l \mid A). \label{eq:2sample\_state}\end{aligned}$$ When V(A) = 1, the two corresponding conditional probabilities are given independent beta priors, whereas if V(A) = 2, they are tied and given a single beta prior. Markov dependency among the states on different nodes are incorporated to induce the desired spatial correlation of differences. Additional latent states can be further incorporated to reflect more complex relationships between the distributions. In fact, the MRS also incorporates an additional state V(A) = 3, which introduces the same coupled prior as V(A) = 2, but works as an absorbing state that once V(A) = 3, all descendants of A will remain in that state, corresponding to the case that the conditional distributions Q1( ⋅ ∣A), and Q2( ⋅ ∣A) are completely equal. Bayesian inference ================== [sec: posterior inference] In sum, the models we consider consisting of the following components: (i) the partition tree T defined by the dimension and location variables D’s and L’s, which follow the priors given in Eq. ; (ii) the latent state variables V(A) given T which follow a Markov prior; (iii) the conditional probabilities along the given tree T, {θg(A)}g = 1G, whose joint prior are specified independently across the nodes on T given the latent states; and finally (iv) given the random measures Qg defined by T and θg(A)’s, we observe an i.i.d. sample ${{\bm x}}\_g$ from each Qg, independently across g. Formally, we have the following full hierarchical model: $$\begin{aligned} T\mid {\bm \lambda},\eta &\sim p(T\mid {\bm \lambda},\eta)\\ \{V(A): A\in T\}\mid {{\bm \xi}},T &\sim {\rm Markov}({{{\bm \xi}}})\\ (\theta\_1(A),\dots,\theta\_G(A)) \mid V(A), T &{\stackrel{\mathrm{ind}}{\sim}}p(\theta\_1(A),\dots,\theta\_G(A) \mid V(A)) \text{ for $A\in T$}\\ {{\bm x}}\_{g}=(x\_{g,1},x\_{g,2},\ldots,x\_{g,n\_g})\mid Q\_{g} &{\stackrel{\mathrm{iid}}{\sim}}Q\_{g} \text{ for $g=1,2,\ldots,G$.}\end{aligned}$$ The key to Bayesian inference is the ability to either compute or sample from the joint posterior $(T,{\mathbf{V}},{{\bm \theta}})$ given all data ${{\bm x}}=({{\bm x}}\_1,\ldots,{{\bm x}}\_G)$, where V and ${{\bm \theta}}$ represent the totality of all latent states and conditional probabilities given T respectively. While in some problems such as density estimation one may mainly be interested in just the marginal posterior of the Qg’s, in others such as two-group comparison where one wants to characterize the between-group relationships among the distributions, the latent states (along with T), are often of prime interest. In multivariate and even high-dimensional problems, the tree structure T is also of great interest as it sheds light on the underlying structures in the distributions. To this end, we shall take advantage of recent developments in sequential Monte Carlo (SMC) sampling for tree-based models and advances in message passing algorithms for PT models with Markov dependency. We introduce a hybrid algorithm that combines these two computational strategies to effectively sample from the joint posterior in high-dimensional spaces. Overall, the hybrid algorithm consists of two stages: 1. Sampling from the marginal posterior of the partition tree We design an SMC sampler—that is, a particle filter—to sample a collection of tree structures T1, …, TM by growing each tree from coarse to fine scales. It uses one-step look-ahead message passing to construct proposal distributions for D(A) and L(A), one node at a time. 2. Computing the conditional posterior given the sampled trees Given each tree sampled by the SMC, we analytically compute the exact conditional posteriors of V(A)’s and θ(A)’s SMC to sample from tree posterior --------------------------------- In the SMC stage to sample the trees, each particle stores a realized form of a finite tree structure, and one node of each tree is divided at each step of the SMC sampling. Suppose Tt is the finite tree obtained after dividing the sample space t times in a particle, and for this tree we define the target distribution $$\begin{aligned} \pi\_t(T\_t) = P(T\_t \mid {\mathbf{x}}) \propto P(T\_t)P({\mathbf{x}}\mid T\_t). \end{aligned}$$ Here P(Tt) is the joint prior of the variables D(A)’s and L(A)’s for the non-leaf nodes of Tt, and P(x ∣ Tt) is the marginal likelihood given the tree Tt under the hierarchical model, in which V and ${\bm \theta}$ are integrated out. To sample from this target distribution, we sequentially construct a set of M particles {Ttm, Wtm}m = 1M, where Ttm is a realized tree and Wtm is the associated importance weight for the mth particle. Examples of generated trees are given in Figure [fig: SMC], where the sample space has been divided three times, and in the next step, new partition boundaries will be added in the gray nodes. Otherwise a node is bisected along a boundary whose dimension and location are randomly drawn from a proposal distribution. For each particle, a finite tree Tt is formed by a sequence of decisions {Js}s = 1t, where Js = (Ds, Ls) correspond to all of the variables D(A) and L(A) at the sth step of the SMC. \| \| \| --- \| \| image \| [fig: SMC] More specifically, suppose at the current step t, we are to divide At ∈ Tt − 1, into At, l and At, r with decision Jt. Let Mi(At ∣ Jt) be the marginal likelihood on the node At under the decision Jt evaluated based on the observations in At. That is, $$\begin{aligned} \hspace{-1em} M\_i (A\_t \mid J\_t) &=\!\! \int \left[ \prod^G\_{g=1} \theta\_g(A\_t)^{n\_g(A\_{t,l})} (1 - \theta\_g(A\_t))^{n\_g(A\_{t,r})} \right] d P(\theta\_1(A\_t),\dots, \theta\_G(A\_t) \| V(A\_t) = i), \label{def of MA}\end{aligned}$$ where ng(A) is the number of observations of the gth group included in A. For example, if the {θg(At)}g = 1G follow independent beta priors written as Beta(αli(At), αri(At)) given V(At) = i, then the marginal likelihood has the following expression $ M\_i(A\_t \mid J\_t)= \prod\_{g=1}^{G} \frac{B(\alpha^i\_l(A) + n\_g(A\_{t,l}), \alpha^i\_r(A) + n\_g(A\_{t,r}))}{B(\alpha^i\_l(A\_t), \alpha^i\_r(A\_t))},$ where B( ⋅ , ⋅ ) is the beta function. [prop: components to make SMC] For every possible decision Jt and states i = 1, …, I, let ${\bf \varphi}\_i(A\_t)$ be a function defined recursively: $$\begin{aligned} {\bm \varphi}\_i(A\_t) = \begin{cases} \frac{{{\bm \xi}}\_{1, i}(\Omega) M\_i(\Omega \mid J\_t)}{\sum^I\_{j=1} {{\bm \xi}}\_{1, j}(\Omega) M\_j(\Omega \mid J\_t)} & \text{if $A\_t = \Omega$}\\ \frac{\sum^I\_{j=1} {\bf \varphi}\_j(A^p\_t) {{\bm \xi}}\_{j, i}(A\_t) M\_i(A\_t \mid J\_t)} {\sum^I\_{k=1} \sum^I\_{j=1} {\bf \varphi}\_j(A^p\_t) {{\bm \xi}}\_{j, k}(A\_t) M\_i(A\_t \mid J\_t)} & \text{Otherwise}, \end{cases} \label{def of varphi} \end{aligned}$$ where Atp is At’s parent node. Also, let h(Jt ∣ At) be a function of Jt defined as $$\begin{aligned} h(J\_t \mid A\_t) = \sum^I\_{i=1} \left\{ \sum^I\_{j=1} {\bm \varphi}\_j(A^p\_t) {{\bm \xi}}\_{j,i}(A\_t) \right\} M\_i(A\_t \mid J\_t) \frac{ \mu(A\_{t,l})^{-n(A\_{t,l})} \mu(A\_{t,r})^{-n(A\_{t,r})}} { \mu(A\_{t})^{-n(A\_{t})} }, \label{def: h in the SMC property} \end{aligned}$$ where n(A) denotes the total number of observations included in a node A. Then the target distribution πt(Tt) can be expressed in terms of πt − 1(Tt − 1) as $$\begin{aligned} \pi\_t(T\_t) &= C\_t \pi\_{t-1}(T\_{t-1}) \pi\_t(J\_t \mid T\_{t-1}) w\_t(T\_{t-1}), \end{aligned}$$ where Ct is a constant and $$\begin{aligned} \pi\_t(J\_t \mid T\_{t-1}) &= \frac{ P(J\_t) h(J\_t \mid A\_t)}{\sum\_{j\_t} P(j\_t) h(j\_t \mid A\_t)},\ w\_t(T\_{t-1}) = \sum\_{j\_t} P(j\_t) h(j\_t \mid A\_t) \end{aligned}$$ The summation over jt is taken over all possible decisions. [cor: proposal and incremental weight] Let h(Jt ∣ At) be the function defined in Proposition [prop: components to make SMC]. Then the proposal distribution πt(Dt ∣ Tt − 1) is given by πt(Dt ∣ Tt − 1) = πt(Dt ∣ Tt − 1)πt(Lt ∣ Dt, Tt − 1), where 1. πt(Dt ∣ Tt − 1) is ${\rm Mult}(\tilde{\lambda}\_1(A\_t),\dots,\tilde{\lambda}\_d(A\_t))$ with $$\begin{aligned} \tilde{\lambda}\_j(A\_t) &\propto \sum^{N\_L-1}\_{l=1} \pi\_t((j, l/N\_L) \mid T\_{t-1}) \propto \lambda\_j(A\_t) \sum\_{l=1}^{N\_L-1} \beta\_l(A\_t) h( (j, l/N\_L) \mid A\_t). \end{aligned}$$ 2. Given D(At) = j, the conditional posterior of L(At) is $$\begin{aligned} \pi\_t(L\_t =l/N\_L \mid D\_t = j, T\_{t-1}) = \sum^{N\_L-1}\_{l=1} \tilde{\beta}\_l (A\_t) \delta\_{l/N\_L} (\cdot), \end{aligned}$$ for j = 1, 2, …, I and l = 1, …, NL − 1 with $$\begin{aligned} \tilde{\beta}\_l(A\_t) &\propto \beta(A\_t) h(j, l/N\_L \mid T\_{t-1}). \end{aligned}$$ We also have an analytical expression of the incremental weight: $$\begin{aligned} w\_t(T\_{t-1}) &= \sum^d\_{j=1} \sum\_{l=1}^{N\_L-1} \lambda\_j(A\_t) \beta\_l(A\_t) h( (j, l/N\_L) \mid A\_t). \end{aligned}$$ We summarize the algorithm in updating the particle system from {Tt − 1m, Wt − 1m}m = 1M to {Ttm, Wtm}m = 1M below. All operations are repeated for m = 1, …, M. 1. Choosing the current node From Tt − 1m, choose the, which is denoted by At. 2. Obtaining the information of the parent node Locate At’s parent node, Atp, and fetch the values of ${\bm \varphi}\_i(A^p\_t)$ for i = 1, …, I. 3. Computing the necessary quantities For all possible Jt = (Dt, Lt), compute Mi(A ∣ Jt) (i = 1, …, I) and h(Jt ∣ At). 4. Dividing the current node Compute the parameters λ̃j(At) for j = 1, …, d and sample $$\begin{aligned} D^m\_t \sim {\rm Mult}(\tilde{\lambda}\_1(A\_t),\dots,\tilde{\lambda}\_d(A\_t)). \end{aligned}$$ Given Dtm, compute the parameters β̃l(At) for l = 1, …, NL − 1 and sample $$\begin{aligned} L^m\_t \sim \sum^{N\_L-1}\_{l=1} \tilde{\beta}\_l(A\_t) \delta\_{l/N\_L}(\cdot). \end{aligned}$$ Divide the current node At with Jtm = (Dtm, Ltm) to update the tree Ttm. 5. Updating the importance weight Compute the incremental weight wt(Tt − 1m) and update the importance weights $$\begin{aligned} W^m\_t &= \frac{W^m\_{t-1} w\_t(T^m\_{t-1})}{\sum^M\_{m'=1} W^{m'}\_{t-1} w\_t(T^{m'}\_{t-1})}. \end{aligned}$$ If the effective sample size 1/∑m = 1M(Wtm)2 is less than some prespecified threshold (e.g., M/10), resample the particles. 6. Computing the information on the current node for its descendants Given Jtm, compute ${\bm \varphi}\_i(A\_t)$ for i = 1, …, I. A common technique in SMC is to resample the particles according to the importance weights {Wtm}m = 1M when the effective sample size of the particles drops below a level. In sampling trees, however, the importance weights are affected by the choice of nodes to divide in multiple steps, and so the standard resampling scheme can be too “short-sighted” and often results in sacrificing promising trees prematurely. To address this issue we follow the strategy proposed in by resampling the particles according to weights atm ∝ (Wtm)κ for some κ ∈ (0, 1], and compute the new importance weights proportional to Wtm/atm. We generally recommend using a moderate choice of κ such as 0.5, which we have found satisfactory in a variety of numerical experiments, and will be our default choice in all of our later examples. Posterior computation given sampled tree structures --------------------------------------------------- The second stage of our inference strategy is to compute the posterior distributions of the latent states V(A) and the conditional probabilities θg(A) given each sampled tree. We shall focus on Given both the tree and the latent states, the posterior of θg(A) boils down to the corresponding posterior of standard PT models Now suppose the SMC algorithm has produced a collection of finite trees {Tm}m = 1M along with the importance weights {Wm}m = 1M. Given each tree Tm, it is possible to analytically calculate the exact posterior of {V(A)}A ∈ Tm with recursive message passing (a form of dynamic programming), which we describe below. For A ∈ N(Tm), let ϕA(i) be the marginal likelihood on A given that V(A) = i. Specifically, $$\begin{aligned} \phi\_A(i) &= \int q({\mathbf{x}}\mid A) P(dq \mid V(A) = i), \text{ where } q({\mathbf{x}}\mid A) = \prod^G\_{g=1} \prod\_{z \in x\_{g}(A)} q\_g(z \mid A). \label{def of marginal likelihood}\end{aligned}$$ In Eq. , taking the integration with respect to P(dq ∣ V(A) = i) is equivalent to integrating out θg(A) as well as the θg(Aʹ) and V(Aʹ) terms for all descendants Aʹ of A. Another useful quantity is the marginal likelihood on a node A given the state of its parent node V(Ap) = i, which we denote as ΦA(i) and is given by $$\begin{aligned} \label{eq:Phi} \Phi\_A(i) &= \begin{cases} \prod\_{x \in {\mathbf{x}}(A)} \mu(x \mid A) & \text{if } $A$ \text{ is a leaf node},\\ \sum\_{i'=1}^{I} {{\bm \xi}}\_{i,i'}(A) \phi\_A(i') & \text{if } $A$ \text{ is a non-leaf node}. \end{cases}\end{aligned}$$ Note that the ΦA(i) and ϕA(i) terms are related by $$\begin{aligned} \label{eq:phi} \phi\_A(i) &= M\_i(A \mid J(A)) \Phi\_{A\_l}(i) \Phi\_{A\_r}(i),\end{aligned}$$ where Mi is the marginal likelihood defined in ([def of MA]) given under the decision J(A) = (D(A), L(A)) to divide A into Al and Ar. By iteratively computing Eqs. and in a bottom-up fashion (i.e., starting from the leaves all the way to the root), we can compute the pair {(ϕA(i), ΦA(i)) : A, i} for all nodes in the tree, and these pairs are the “messages” passed along the tree from leaf to root. Given the values of {(ϕA(i), ΦA(i)) : A, i}, we can now obtain the posterior Markov transition probability matrices of the latent states given the tree T, $\tilde{{{\bm \xi}}}(A) = (\tilde{{{\bm \xi}}}\_{i,i'}(A))\_{I\times I}$ where the (i, iʹ)th element $\tilde{{{\bm \xi}}}\_{i,i'}(A)=P(V(\Omega) = i' \mid V(A^p) =i,{\mathbf{x}}, T)$ and the posterior marginal probabilities of the latent states given the tree T, $$\begin{aligned} \tilde{{{\bm \gamma}}}(A) & = (\tilde{{{\bm \gamma}}}\_i(A))^I\_{i=1} = (P(V(\Omega) = i \mid {\mathbf{x}}, T))^I\_{i=1}.\end{aligned}$$ Specifically, by the Bayes’ theorem, $\tilde{{{\bm \xi}}}(A) = D^{-1}\_1(A) {{\bm \xi}}(A) D\_2(A),$ where D1(A) and D2(A) are diagonal matrices with D1(A)i, i = ΦA(i) and D2(A)i, i = ϕA(i). After computing these transition matrices, we can compute $\tilde{{{\bm \gamma}}}(A)$ (the feedback “message”) in the top-down manner (i.e., starting from the root and down to the leaves) as $\tilde{{{\bm \gamma}}}(\Omega) = \tilde{{{\bm \xi}}}\_{1,\cdot} (\Omega)$ and $\tilde{{{\bm \gamma}}}(A) = \tilde{{{\bm \gamma}}}(A^p) \tilde{{{\bm \xi}}}(A) \text{ for } A \neq \Omega.$ Now that we have completely described our inference algorithm, we provide two specific examples to demonstrate how one may use the output of the algorithm—namely the sampled trees along with the conditional posterior given the trees—to carry out inference. The first example is density estimation which involves learning the within-sample distributional features while the second is two-group comparison whose focus is on learning the between-sample structures. The inference strategies for these quintessential examples are generalizable to a variety of other tasks. ### Example 1: Density estimation We consider the problem of estimating an unknown density based on a set of i.i.d. observations from that density. This corresponds to G = 1 and so we can drop the subscript g to simplify the notation. We shall use the posterior mean density, also called the predictive density—E[q( ⋅ ) ∣ x]—as an estimate for the density q = dQ/dμ. Specifically, for a given tree T, we define for all A ∈ T and i ∈ {1, 2, …, I}, the following quantity $$\begin{aligned} e\_A(i) := {\mathbb{E}}[Q(A) I\_{\{V(A) = i\}} \mid {\mathbf{x}}].\end{aligned}$$ All of these eA(i) terms can be computed together by a single top-down (i.e., root-to-leaf) recursion on the tree as given in the following proposition. [prop: computing predictive density] For the root node, $e\_\Omega(i) = \tilde{{{\bm \gamma}}}\_{1,i}(\Omega)$. For a non-root node A, eA(i) can be computed recursively as $$\begin{aligned} e\_A(i') &= \sum^I\_{i=1} \tilde{{{\bm \xi}}}\_{i,i'}(A){\mathbb{E}}[ \vartheta(A^p)\mid V(A^p) = i, T^m, {\mathbf{x}}]\, e\_{A^p}(i), \text{ where} \\ \vartheta(A^p) &= \begin{cases} \theta(A^p) & \text{ if } A \text{ is the left child of $A^p$},\\ 1-\theta(A^p) & \text{ if } A \text{ is the right child of $A^p$}. \end{cases}\end{aligned}$$ Now with a recipe for obtaining all ei(A)’s for a given tree, we can obtain the conditional predictive measure given the tree T, $$\begin{aligned} {\mathbb{E}}[Q(B) \mid {\mathbf{x}}, T] = \sum\_{A\in \mathcal{L}(T)} \frac{\mu(B \cap A)}{\mu(A)} \sum^I\_{i=1} e\_A(i) \quad \text{for any Borel set $B$.}\end{aligned}$$ Now, given an SMC sample of M trees and weights, the posterior predictive density can be computed by $$\begin{aligned} {\mathbb{E}}[q(x) \mid {\mathbf{x}}] &\approx \sum^M\_{m=1} W^m \frac{{\mathbb{E}}[Q(B^m(x)) \mid {\mathbf{x}}, T^m]}{\mu(B^m(x))}, \end{aligned}$$ where Bm(x) ∈ L(Tm) the leaf node to which x belongs. ### Example 2: Two-group comparison If we are interested in comparing two groups of observations using generalizations to the PT models described in Section [sec: HMM General framework], we shall compute the posterior probability of the two hypotheses H0 and H1. For example, when V(A) is defined as in Eq , the posterior probability of the “global” null hypothesis H0 : Q1 = Q2 is given by $$\begin{aligned} P(H\_0\mid {{\bm x}}) &= \sum\_{T \in {\mathcal{T}}} P(V(A) \neq 1 \text{ for all } A \in \mathcal{N}(T) \mid T,x) P(T \mid {{\bm x}})\\ &\approx \sum^M\_{m=1} W^m P(V(A)\neq 1 \text{ for all } A \in \mathcal{N}(T^m) \mid T^m, {\mathbf{x}}),\end{aligned}$$ where the sum over T in the first row is over all finite trees with maximum resolution K and the quantity P(V(A) ≠ 1for all A ∈ N(Tm) ∣ Tm, x) again is available analytically by message passing (details given in Supplementary Materials [sec: extra MRS algorithm]). In addition to testing the existence of any difference between groups, it is usually of interest to detect where and how the underlying distributions differ. To this end, we can compute the “posterior marginal alternative probability” (PMAP) on each node A, along any sampled tree Tm: $$\begin{aligned} P(\theta\_1(A) \neq \theta\_2(A) \mid T^m, {\mathbf{x}}) = P(V(A) =1 \mid T^m, {\mathbf{x}}) = \tilde{{{\bm \gamma}}}\_i(A).\end{aligned}$$ Reporting the PMAPs along a representative tree such as the MAP among the sampled trees can be a particularly useful visualizing tool to help understand the nature of the underlying difference. One can also report on each A the estimated magnitude of the difference using a notion of “effect size” based on the log-odds ratio, $\mathrm{eff}(A) = \left\| \log \left[ \frac{\theta\_1(A)}{1 - \theta\_1(A)} \right] - \log \left[ \frac{\theta\_2(A)}{1 - \theta\_2(A)} \right] \right\|.$ In particular, one can report the posterior expected effect size E[eff(A) ∣ x, T], which can be computed using a standard Monte Carlo (not MCMC) sample from the exact posterior given the representative tree. We will demonstrate this using a mass cytometry data set in Section [sec:mass] Theoretical Properties ====================== Next we investigate the theoretical properties of the proposed model. Previous theoretical analysis on the PT had mostly focused on establishing the marginal posterior consistency and contraction of the random measures Qg with respect to an unknown fixed truth. We shall take a different perspective and instead investigate the asymptotic behavior of the marginal posterior of the partition tree T and the latent states as these are often critical quantities of practical importance in applications. We note that once given the tree and the latent states, our model reduces to standard PTs and thus the posterior consistency of the random measures Qg’s will follow from previous results once we establish the posterior consistency of the tree and the latent states. As the sample size increases, the two key theoretical questions of interest here are: * What tree structures does the marginal posterior of T concentrate around? * How does the posterior of the latent states given the tree behave? These two questions have broad relevance in inference using PT models, and previously several authors have investigated the second question in the two-group comparison context for their variants of the PT model. In addressing the second question more generally, we aim to provide results that encompass these previous analyses as special cases. According to our limited knowledge, we are not aware of previous studies on the first question. We will address each of the two questions in turn. Throughout this section, we consider finite PTs with maximum depth of the trees set to some value K. We use TK to denote this collection of trees. Also, Finally, we consider models that satisfy Assumption [assumption: sample size and true measures] and Assumption [assumption on state variables] described below. The models discussed in Section [subsec: examples] all meet this requirement. [assumption: sample size and true measures] * There exists ζg ∈ (0, 1] such that $ \zeta\_g = \lim\_{n \to \infty} \frac{n\_g}{n}$ for g ∈ {1, …, G}, * The sampling distribution Pg satisfies Pg ≪ μ, and Additionally, given the tree T and the latent states, the parameters {θg(A)}g = 1G are given one of the following priors (the model can adopt a mix of these priors for different combinations of A and V(A) values): Prior A : θg(A) independently follow a beta prior. Prior B : θ1(A) = ⋯ = θG(A) and follow a beta prior. Prior C : Establishing the theoretical properties also requires a condition on the latent states. In particular, under some states, the support of the prior of the parameters {θg(A)}g = 1G needs to include the true conditional probabilities. To describe this requirement, given a tree T ∈ TK, let Si(A ∣ T) be the support of the prior on (θ1(A), …, θG(A)) under the state V(A) = i. Then, let τ(A ∣ T) denote the collection of That is, $$\begin{aligned} \tau(A \mid T) := \{i \in \{1,\dots,I\}: (P\_1(A\_l \mid A),\dots, P\_G(A\_l \mid A)) \in S\_i(A \mid T) \}.\end{aligned}$$ The next assumption states that the prior for the latent states must give positive probability for all the latent states to all simultaneously be feasible. [assumption on state variables] For every T ∈ TK, P(V(A) ∈ τ(A ∣ T)for all A ∈ T) > 0. With these assumptions, we next derive asymptotic properties for the marginal posteriors for the tree and the state variables. In the following, we use the notation xn instead of x for the data to indicate the total sample size. In order to describe the posterior convergence of the partition trees, we introduce a notion for “tree-based approximation for probability measures”. Let T be a finite tree and H a probability measure. Then the “tree-based approximation of H under T”, denoted by H∣T, is defined as $H \|\_T (B) = \sum\_{A \in \mathcal{L}(T)} H(A) \frac{\mu(B \cap A)}{\mu(A)}.$ for any B ∈ B(Ω). The following theorem then characterizes the trees the posterior concentrates on as the sample size grows. [thm: characterizign the post of trees] Let TMK be the collection of trees under which the tree-based approximation of the measures Pg minimizes the Kullback-Leibler divergence from the Pg’s plus a penalty term on unbalanced splits. That is, $$\begin{aligned} {\mathcal{T}}^K\_M & = {\mathop{\rm arg~min}\limits}\_{T \in {\mathcal{T}}^K} \sum^G\_{g=1} \zeta\_g \left\{ {\rm KL}(P\_g \|\| P\_g\|\_T) + \eta B\_g(T) \right\}, \label{def of TKM}\end{aligned}$$ where $$B\_g(T) = \sum\_{A \in \mathcal{N}(T)} P\_g(A) f\left( \left\| \frac{\mu(A\_l)}{\mu(A)} - 0.5 \right\| \right).$$ Then the marginal posterior of T concentrates on TMK. That is, as n → ∞, $$\begin{aligned} P(T \in {\mathcal{T}}^K\_M \mid {\mathbf{x}}\_n ) \xrightarrow{p} 1.\end{aligned}$$ For the state variables, it is desirable that their posterior distribution concentrates on a collection of. Moreover, when multiple configurations of the states are feasible, it is desirable that the posterior concentrates around such configurations that provide the most parsimonious representation of the true distributions. For example, if the true conditional distribution on a node is uniform, a model that introduces a possible non-uniform structure on this node is feasible but redundant. and showed that, in quite general settings of multi-resolution inference, the posterior probability of such redundant models tends to concentrate its mass on 0. By adapting their techniques, we show that the same property holds in the case of our model. To formally describe the results, we need to define the complexity of the model specified by the latent states. Given the state V(A) = i, the complexity of the {θg(A)}g = 1G, in other words, the number of free parameters of the prior distribution under the ith state is denoted by Ci(A). For example, for two-group comparison, $$\begin{aligned} C\_i(A) = \begin{cases} 1 & \text{ if } \theta\_1(A) = \theta\_2(A)\\ 2 & \text{ if } \theta\_1(A) \neq \theta\_2(A). \end{cases}\end{aligned}$$ Next we introduce the complexity of a combination of states on the tree T. Given a tree T, let V denote a combination of the state variables {V(A)}A ∈ N(T) and let v = {v(A)}A ∈ N(T) (v(A) ∈ {1, …, I}) be one of the possible realizations of V. Then we define the model complexity under v as follows: $$\begin{aligned} C({\mathbf{v}}) = \sum\_{A \in \mathcal{N}(T)} C\_{{\mathbf{v}}(A)}(A). \label{def: complexity}\end{aligned}$$ The next theorem shows that the posterior distribution of the states given the tree will concentrate on those that are feasible and most parsimonious. [thm: posterior of state variables] For T ∈ TK, let VT = {v : v(A) ∈ τ(A ∣ T)for all A ∈ N(T)}. Then $P \left(\{{\mathbf{V}}\in {\mathcal{V}}\_T\} \cap \left\{ C({\mathbf{V}}) = \min\_{{\mathbf{v}}\in {\mathcal{V}}\_T} C({\mathbf{v}}) \right\} \mid T, {\mathbf{x}}\_n\right) \xrightarrow{p} 1$. Remark: Consistency results for several existing models are special cases of this theorem. For example, we derive the consistency of the MRS model for two-group comparison as a corollary in Supplementary Materials C. Experiments =========== In this section, we carry out simulation studies to examine the performance of our model and inference algorithm. In particular, we are interested in (i) understanding how the model with the flexible tree prior compares to those with a “divide-in-the-middle” restriction, and (ii) verifying the linear scalability of our inference algorithm with respect to increasing dimensionality. We again consider the two quintessential examples—(i) density estimation and (ii) the two-group comparison—for inferring within-group and between-group structures respectively. We shall consider both low-dimensional settings where the underlying structure is easy to interpret and software for existing PT models are available, and high-dimensional settings for which existing implementation of PT models is not applicable and we use our SMC algorithm to carry out inference for both our model and the earlier models with fixed partitioning points (which are special cases of our model). Throughout the experiments, the parameters NL and M are fixed to 32 and 1000 respectively. We note that larger NL values can also be adopted at a linear computational cost but did not lead to noticeable change in performance in our examples. Details such as the settings of hyper-parameters and simulated data sets are provided in Supplementary Materials [sec: Details of the experiemnts] unless explicitly described in this section. Density estimation ------------------ We first consider 2D examples to observe what kind of tree structures are obtained under the flexible model and how prior specification in Eq. influences the performance. After that, we move to higher dimensional cases to examine the scalability of our new SMC method and the effect of incorporating the flexible partition. For this task we compare our model with the APT model which also incorporates a prior on the dimension to divide but restricts partitioning at middle points. Its posterior computation is implemented by the apt function in the R package PTT. ### Two-dimensional cases Simulated data are generated from the three scenarios with the densities visualized in the first row of Figure [fig: estimeted densities and trees]. (Details on the simulation settings are provided Supplementary Materials [sec: Details of the experiemnts].1.2.) Also presented in Figure [fig: estimeted densities and trees] are examples of The results for the first scenario confirms that our more flexible model is much more effective in capturing the discontinuous boundaries of the true density. For the second scenario, our model tends to draw the boundaries that surround the true clusters. In the trees given under the different values of η, however, we can see that fewer nodes were divided inside the clusters when η = 0.01. In contrast, when η = 0.1, the representative tree draws outlines of the clusters and divides regions inside of the clusters at the same time. A similar phenomenon is observed in the third scenario—under our model with flexible partitioning points, partition lines are formed around the region with high density, when η = 0.1 for the boundaries were also drawn within the high probability region. The quantitative comparison based on the KL divergence is provided in Supplementary Materials [sec: additional figures], which is consistent with the explanation above. \| \| \| \| \| --- \| --- \| --- \| \| image \| image \| image \| \| image \| image \| image \| \| image \| image \| image \| \| image \| image \| image \| [fig: estimeted densities and trees] ### Higher-dimensional cases We generate d-dimensional i.i.d. observations from a density with independent pairs of margins, i.e., f(x1, x2, …, xd) = ∏j = 1d/2fj(x2j − 1, x2j) where $$\begin{aligned} f\_j(x\_{2j-1},x\_{2j})=& p\_j {\rm Beta}(x\_{2j-1}\mid 0.25, 1) \times {\rm Beta}(x\_{2j}\mid 0.25, 1) \\ + &(1-p\_j) {\rm Beta}(x\_{2j-1}\mid 50/j, 50/j) \times {\rm Beta}(x\_{2j}\mid 50/j, 50/j),\end{aligned}$$ with pj = 0.25 + 0.7/j. We consider two different situations: (i), and the sample size n changes from 5,000 to 50,000; and (ii) the sample size n = 10, 000 and the dimensionality changes from 10 to 100. For our method, the maximum depth K is set to 15. cc![image](wall_time.png) [fig: wall time] Figure [fig: wall time] presents the computational time for five different data sets. To obtain the result, we used a singe-core environment using The computational time is linear in both the sample size and the dimensionality. \| \| \| \| --- \| --- \| \| image \| image \| [fig: multi pred] Because in the high-dimensional settings we cannot obtain the KL divergence between the estimated density and the true density through numerical approximation, we compare the models based on predictive scores, which is in essence an empirical estimate of the KL divergence. Specifically, for each training set ${{\bm x}}$, we can generate a new test set denoted by x\* = (x1\, …, x*n\) from the same true model and compute $$\begin{aligned} \frac{1}{n} \sum^{n}\_{i = 1} \log \hat{p}(x\_i^\ \mid {\mathbf{x}}), \label{definition of D}\end{aligned}$$ where p̂ is We repeat this computation for 50 test/training set pairs and take the average. The results, given in Figure [fig: multi pred], show that our model substantially outperforms the competitors by this criteria both when d = 6 with varying sample size and when n is fixed with varying dimensionality. Two-group comparison -------------------- Next we consider the two-group comparison problem, evaluate the performance of our model, and compare it to the original MRS with the “divide-in-the-middle” restriction. We use three scenarios (“Local location shift”, “Local dispersion difference”, and “Correlation”) to generate 50-dimensional data sets. (Details of the scenarios are provided in Supplementary Materials [sec: Details of the experiemnts].2.2.) The first two scenarios involve two-group difference that lies in only parts of the sample space, or “local” differences. which will help demonstrate the usefulness of inferring the partition tree in identifying the nature of the differences. The sample size is n1 = n2 = 2, 000 in all scenarios. The original algorithm for inference under the MRS model by message passing, which is implemented by the mrs function in the R package MRS, is not scalable beyond about 10 dimensions even with fixed partition locations. Hence we compute the posterior for both our model and the original MRS in all scenarios with our SMC and message passing hybrid algorithm. We compare the performance using receiver operating characteristic (ROC) curves computed based on 200 simulated data sets under each scenario. \| \| \| --- \| \| image \| [ROC] \| \| \| --- \| \| image \| [fig: twosampleproblem samplewindows] Figure [ROC] presents the ROC curves. For the location shift and dispersion differences, the model with flexible partitioning results in substantially higher sensitivity. For the correlation scenario, the model with fixed partitioning locations performed slightly better. This is not surprising since in this scenario the difference exists smoothly over entire ranges of the dimensions without natural “optimal” division points, and so the performance gap is the cost for searching over more possible partition locations, none of which improves the model fit than the middle point. It is worth noting again that while the model with fixed partitioning performs well here, it is only with our new computational algorithm that it can be fit to data of such dimensionality. To demonstrate the posterior model can help understand the nature of the differences, we present under each scenario the node with the highest PMAP, or $P(V(A)=1\mid {{\bm x}})=P(\theta\_1(A)\neq \theta\_2(A)\mid {{\bm x}})$, in Figure [fig: twosampleproblem samplewindows]. In the location shift and dispersion difference scenarios the boundaries are away from the middle point to characterize the difference, which partly explains the sensitivity gain in adopting the flexible tree prior. Application to a mass cytometry data set ======================================== Finally, we apply our model for two-group comparison to a mass cytometry data set collected by. The data set records 19 different measurements including physical measurements and biomarkers on single cells in blood samples from a group of HIV patients as well as in reference samples from healthy donors. For demonstration, we compare the sample from an individual patient sample (Patient #1) and to that from a healthy donor to identify differences in immune cell profiles from these samples. The sample sizes are 29, 226 for the health donor and and 228, 498 for the patient, with each observation corresponding to a cell. We set η = 0.1 and the maximum depth K to 25. Given the large sample sizes, the posterior probability for the global alternative P(Q1 ≠ Q2 ∣ x) is almost 1 and so is of less interest. Our focus is instead on identifying the cell subsets on which the samples differ and on quantifying such differences. To this end, we identify a representative tree and report the “effect size” (i.e., the posterior expected log-odds ratio between the two samples) on each node in a representative tree—the MAP among the sampled trees. . We note that the nodes on which there is significant evidence for two-group differences, as well as those with large estimated effect sizes tend to be nested or clustered in subbranches of the tree, which is consistent with our intuition that there is spatial correlation in the two-group differences, and justifies the hidden Markov structure embedded in the MRS model. \| \| \| --- \| \| image \| [fig: truncated tree for FC] Figure [fig: node with large difference (FC)] in Supplementary Materials [sec: additional figures] presents the 20 nodes with the largest values of estimated eff(A). In this figure, many of the nodes are in very deep levels of the tree. We adopted a spike-and-slab with higher spike probability in very deep tree levels to further speed up the computation (details given in Supplementary Materials [sec: refixing]) and that explains why many of the very deep, small nodes plotted have partition lines in the middle under the MAP tree. Concluding Remarks ================== We have proposed a general framework for the PT model that incorporates a flexible prior on the partition tree and can accommodate latent state variables with Markov dependency along the partition tree. We have proposed a sampling algorithm that combines SMC and recursive message passing that can scale up to moderately high-dimensional ( ∼ 100-dim) problems. Our numerical experiments confirm that our sampling algorithm scales linearly in the sample size and the flexible partitioning tree prior can result in substantial gain in performance in some settings. Though we have mainly used two inference tasks—namely density estimation and two-group comparison—to demonstrate our model and algorithm, our approach can be readily applied to other PT models with a hidden Markov structure. One notable limitation of our model—and in fact all CART-like models—is that we only consider trees in which the node boundaries are all parallel to the axes. This could lead to inefficiency in inference. For example, when there is a strong correlation between several variables, drawing boundaries slanted according to the correlation structure would be more effective in characterizing the underlying distribution. Such trees will need to be represented by more than just the D(A) and L(A) used in our model, and how to efficiently compute the posterior distribution is an open problem. Software ======== An R package for our method is available at <https://github.com/MaStatLab/SMCMP>. Acknowledgment ============== LM’s research is partly supported by NSF grants DMS-2013930 and DMS-1749789. NA is partly supported by a fellowship from the Nakajima Foundation. Spike-and-slab type prior for L(A) ====================================== Introducing an auxiliary variable --------------------------------- The location variable L(A) follows a spike-and-slab type prior which is expressed with an auxiliary variable R(A) as $$\begin{aligned} L(A) \mid R(A) &\sim {\mathbf{1}}\_{\{R(A)=1\}} \delta\_{1/2}(\cdot) + {\mathbf{1}}\_{\{R(A)=0\}} \sum^{N\_L-1}\_{l=1,l/N\_L\neq 1/2} \hat{\beta}\_l(A) \delta\_{l/N\_L}(\cdot),\end{aligned}$$ where 1 is the indicator function and the sum of the parameters βl(A) is 1. Under this prior, L(A) follows the prior degenerated at 1/2 if R(A) = 1 and otherwise follows the distribution on grid points other than the middle point. R(A) follows an asymmetric hidden Markov process $$\begin{aligned} P(R(A) = 1 \mid R(A^p) = 0) &= r(A), \\ P(R(A) = 1 \mid R(A^p) = 1) &= 1,\end{aligned}$$ where r(A) ∈ [0, 1]. R(A) = 1 is the absorbing state, so once A is divided at the middle point, L(Aʹ) = 1/2 for every A’s descendant node Aʹ. In the estimation we especially set the parameters as follows: $$\begin{aligned} r(A) &= \beta\_{N\_L/2},\ \hat{\beta}\_l = \frac{\beta\_l(A)}{ \sum^{N\_L-1}\_{l=1,l/N\_L\neq 1/2} \beta\_l(A) },\end{aligned}$$ where {βl(A)}l = 1, …, NL is given in ([prior of L]). Under this setting the prior of L(A) satisfies $$\begin{aligned} P(L(A) =l \mid R(A^p) = 0) &= \beta\_l(A),\ l=1,\dots,N\_L-1.\end{aligned}$$ Hence, L(A) follows the same prior as defined in ([prior of d and l]) unless A’s parent node is divided at the middle point, so the spike-and-slab prior can be seen as a natural extension. SMC algorithm ------------- In the SMC algorithm, we sample values of R(A) in addition to D(A) and L(A). If R(A) = 1, which is equivalent to L(A) = 1/2, is sampled, we conclude there is no interesting structure on the node A so fix L to 1/2 for all the subsequent nodes. Hence, we need to generalize the SMC algorithm discussed in Section [subsec: smc algorithm] to sample from the joint posterior distribution of the finite trees and the auxiliary variables R. To describe this joint posterior, let Tt denote the finite tree structure, which is determined by the sequence of decisions J1 : t dividing the nodes A1 : t, and let R1 : t be a sequence of the re-fixing variables for A1 : t. Then the target distribution we want to sample from in the SMC is defined as $$\begin{aligned} \pi\_t(T\_t, R\_{1:t}) &= P(T\_t, R\_{1:t} \mid {\mathbf{x}}) \propto P(T\_t, R\_{1:t}) P({\mathbf{x}}\mid T\_t, R\_{1:t}).\end{aligned}$$ The prior P(Tt, R1 : t) = P(J1 : t, R1 : t) have a Markov chain structure on the tree, and its transition probability is decomposed as $$\begin{aligned} P(J\_t, R\_t \mid J^p\_t, R^p\_t) &= P(R\_t \mid R^p\_t) P(J\_t). \end{aligned}$$ where Jtp = J(Atp) and Rtp = R(Atp) (Atp is the parent node of At). On the other hand, because R1 : t are conditionally independent of the observations given Tt, the likelihood only depends on Tt as follows: $$\begin{aligned} P({\mathbf{x}}\mid T\_t, R\_{1:t}) = P({\mathbf{x}}\mid T\_t).\end{aligned}$$ The likelihood has the same form as in the original case without the auxiliary variables R’s. Thus we can obtain the following proposition as a generalization of Proposition [prop: components to make SMC]. [prop: SMC] Let h(Jt ∣ At) be a function of Jt defined as $$\begin{aligned} h(J\_t \mid A\_t) = \sum^I\_{i=1} \left\{ \sum^I\_{j=1} {\bm \varphi}\_j(A^p\_t) {{\bm \xi}}\_{j,i}(A\_t) \right\} M\_i(A\_t \mid J\_t) \frac{\mu({\mathbf{x}}(A\_{t,l}) \mid A\_{t,l})\mu({\mathbf{x}}(A\_{t,r}) \mid A\_{t,r})} { \mu({\mathbf{x}}(A\_t) \mid A\_t) }. \end{aligned}$$ Then the target distribution πt(Tt, R1 : t) is expressed with πt(Tt − 1, R1 : t − 1) as $$\begin{aligned} \pi\_t(T\_t, R\_t) &= C \pi\_t(T\_{t-1}, R\_{1:t}) \pi\_t(R\_t \mid T\_{t-1}, R^p\_t) \pi\_t(J\_t \mid T\_{t-1}, R\_t) w\_t(T\_{t-1}, R^p\_t), \end{aligned}$$ where C is a constant and $$\begin{aligned} \pi\_t(R\_t \mid T\_{t-1}, R^p\_t) &= \frac{P(R\_t \mid R^p\_t) \sum\_{j\_t} P(j\_t \mid R\_t) h(j\_t \mid A\_t) }{ \sum\_{i=0,1} P(R\_t = i\mid R^p\_t) \left\{ \sum\_{j\_t} P(j\_t \mid R\_t=i) h(j\_t \mid A\_t) \right\} } , \\ \pi\_t(J\_t \mid T\_{t-1}, R\_t) &= \frac{ P(J\_t \mid R\_t) h(J\_t \mid A\_t)}{\sum\_{j\_t} P(j\_t \mid R\_t) h(j\_t \mid A\_t)},\\ w\_t(T\_{t-1}, R^p\_t) &= \sum\_{i=0,1} P(R\_t = i\mid R^p\_t) \left\{ \sum\_{j\_t} P(j\_t \mid R\_t=i) h(j\_t \mid A\_t) \right\}. \end{aligned}$$ The summation with jt is taken over all possible decisions. Its proof is essentially the same as Proposition [prop: components to make SMC] and Corollary [cor: proposal and incremental weight] so it is omitted in this material. The conditional posteriors πt(Rt ∣ Rtp, Tt − 1) and πt(Jt ∣ Rt, Tt − 1) are analytically obtained as follows. First, if Rtp = 0, πt(Rt ∣ Rtp, Tt − 1) is Bernoulli(r̃(At)), where $$\begin{aligned} \tilde{r}(A\_t) &= r(A\_t) \sum^J\_{j=1} \lambda\_j(A\_t) h((j, 1/2) \mid A\_t) \\ & \times \left[ r(A\_t) \sum^J\_{j=1} \lambda\_j(A\_t) h((j, 1/2) \mid A\_t) + (1-r(A\_t)) \sum^J\_{j=1} \sum^{N\_L-1}\_{l=1} \lambda\_j(A\_t) \beta\_l(A\_t) h((j, l/N\_L) \mid A\_t) \right]^{-1}, \end{aligned}$$ If Rtp = 1, then Rt is fixed to 1. Second, if Rt = 0, the posterior of Dt and Lt is the same distribution given in Section [subsec: smc algorithm]. On the other hand, if Rt = 1, Lt is fixed to 1/2, and the conditional posterior πt(Jt ∣ Lt, Rt, Tt − 1) is Mult(λ̂1(At), …, λ̂d(At)), where $$\begin{aligned} \hat{\lambda}\_j(A\_t) & \propto \lambda\_j(A\_t) h(j, 1/2 \mid A\_t). \end{aligned}$$ After sampling (Rt, Jt), the incremental weight wt(Tt − 1, Rtp) (Rtp = 0, 1) is computed as $$\begin{aligned} &w\_t(T\_{t-1}, 0) \\ &= r(A\_t) \sum^J\_{j=1} \lambda\_j(A\_t) h((j, 1/2) \mid A\_t) + (1-r(A\_t)) \sum^J\_{j=1} \sum^{N\_L-1}\_{l=1} \lambda\_j(A\_t) \beta\_l(A\_t) h((j, l/N\_L) \mid A\_t), \\ &w\_t(T\_{t-1}, 1) = \sum^J\_{j=1} \lambda\_j(A\_t) h((j, 1/2) \mid A\_t), \end{aligned}$$ with which we update the importance weight Wt as Wt ∝ Wt − 1wt(Tt − 1, Rtp). The procedure to update the particle system {Tt − 1m, Wt − 1m}m = 1M to obtain {Ttm, Wtm}m = 1M is described in the following algorithm. The operations involving the index m is repeated for m = 1, …, M. 1. Choosing the current node From Tt − 1m, choose the oldest note from the leaf nodes, which is denoted by At. 2. Obtaining the information of the parent node Pick up At’s parent node, which is denoted by Atp, and load the values of ${\bm \varphi}\_i(A^p\_t)$ for i = 1, …, I and Rtm, p = R(Atp). 3. Computing the necessary quantities If Rtm, p = 0, compute Mi(A ∣ j, l/NL) (i = 1, …, I) and h(j, l/NL ∣ At) for j = 1, …, d and l = 1, …, NL − 1. If Rtm, p = 1, compute Mi(A ∣ j, 1/2) (i = 1, …, I) and h(j, 1/2 ∣ At) for j = 1, …, d. 4. Deciding whether to fix the partition or not If Rtm, p = 0, compute the parameter r̃(At) and draw Rtm ∼ Bernoulli(r̃(At)). If Rtm, p = 1, set Rtm to be 1. 5. Dividing the current node Sample Jtm = (Dtm, Ltm) as follows: * If Rtm = 0, compute the parameters λ̃j(At) for j = 1, …, d and sample $$\begin{aligned} D^m\_t \sim {\rm Mult}(\tilde{\lambda}\_1(A\_t),\dots,\tilde{\lambda}\_d(A\_t)). \end{aligned}$$ Given Dtm, compute the parameters β̃l(At) for l = 1, …, NL − 1 and sample $$\begin{aligned} L^m\_t \sim \sum^{N\_L-1}\_{l=1} \tilde{\beta}\_l(A\_t) \delta\_{l/N\_L}(\cdot).\end{aligned}$$ * If Rtm = 1, compute the parameters λ̂j(At) for j = 1, …, d and sample $$\begin{aligned} D^m\_t \sim {\rm Mult}(\hat{\lambda}\_1(A\_t),\dots,\hat{\lambda}\_d(A\_t)),\end{aligned}$$ and set Ltm = 1/2. Divide the current node At with Jtm = (Dtm, Ltm) to update the tree Ttm. 6. Storing the information of the state’s posterior Given Jtm, compute ${\bm \varphi}\_i(A\_t)$ for i = 1, …, I and store them to the memory. 7. Updating the importance weight Compute the incremental weight wt(Tt − 1m, Rtm, p) and update the importance weights as $$\begin{aligned} W^m\_t &= \frac{W^m\_{t-1} w\_t(T^m\_{t-1}, R^{m,p}\_t)}{\sum^M\_{m'=1} W^{m'}\_{t-1} w\_t(T^{m'}\_{t-1}, R^{m',p}\_t)}. \end{aligned}$$ If the effective sample size 1/∑m = 1M(Wtm)2 is less than some prespecified threshold (M/10, say), resample the particles. Additional algorithm for the MRS model ====================================== To describe the algorithm proposed in, we keep using the same notations in Section [subsubsection: Comparing the two hypotheses]. Given the tree structure T, we compute functions ψ̃(A) for A ∈ N(T) in the bottom-up (from the leaf nodes to the root node) manner as follows: $$\begin{aligned} \tilde{\psi}(A) = \begin{cases} \tilde{{{\bm \xi}}}\_{2,2} (A) + \tilde{{{\bm \xi}}}\_{2,3} (A) & \text{ if } A \in \mathcal{L}(T), \\ \tilde{{{\bm \xi}}}\_{2,2} (A)\tilde{\psi}(A\_l) \tilde{\psi}(A\_r) + \tilde{{{\bm \xi}}}\_{2,3} (A) & \text{ if } A \in \mathcal{N}(T) \setminus \{ \Omega \} ,\\ \tilde{{{\bm \xi}}}\_{1,2} (A)\tilde{\psi}(A\_l) \tilde{\psi}(A\_r) + \tilde{{{\bm \xi}}}\_{1,3} (A) & \text{ if } A = \Omega. \end{cases}\end{aligned}$$ Recall that only the first row of ${{\bm \xi}}(\Omega)$ is meaningful as the initial distribution. Then we obtain ψ̃(Ω) = P(H0 ∣ T, x). Proofs ====== Posterior computation --------------------- [lem: conditinal post of the state given the tree] For the finite tree Tt, let As be a node whose children nodes are leaf nodes. Then we have $$\begin{aligned} \pi\_t(V\_s = i \mid T\_t) = \frac{\pi\_t(T\_t, V\_s = i)}{\pi\_t(T\_t)} = {\bm \varphi}\_i(A\_s), \end{aligned}$$ where ${\bm \varphi}\_i(A\_s)$ is defined in ([def of varphi]). (Proof) Suppose that As belongs to the kth layer of Tt. Then there is a sub-sequence {ρ(l)}l = 1k such that Aρ(l) belongs to the lth layer and $$\begin{aligned} \Omega = A\_{\rho(1)} \supset A\_{\rho(2)} \supset \cdots \supset A\_{\rho(k)} = A\_s.\end{aligned}$$ By the definition of πt(Tt, V1 : t), for a sequence {vl}l = 1k such that vl ∈ {1, …, I}, we obtain the expression of the conditional posterior of {Vρ(l)}l = 1k as $$\begin{aligned} \pi\_t(\{V\_{\rho(l)}\}^k\_{l=1} = \{v\_l\}^k\_{l=1} \mid T\_t) &\propto P(\{V\_{\rho(l)}\}^k\_{l=1} = \{v\_l\}^k\_{l=1}) \prod^k\_{l=1} M\_{v\_l} (A\_{\rho(l)} \mid J\_{\rho(l)}) \\ &= \prod^k\_{l=1} {{\bm \xi}}\_{v\_{l-1}, v\_l} (A\_{\rho(l)}) M\_{v\_l} (A\_{\rho(l)} \mid J\_{\rho(l)}),\end{aligned}$$ where v0 = 1. We show that for every k = 1, …, K $$\begin{aligned} \pi\_t(V\_{\rho(k)} = v\_k \mid T\_t) &\propto \sum^I\_{v\_1=1} \cdots \sum^I\_{v\_{k-1}=1} \left\{ \prod^{k}\_{l=1} {{\bm \xi}}\_{v\_{l-1}, v\_l} (A\_{\rho(l)}) M\_{v\_l} (A\_{\rho(l)} \mid J\_{\rho(l)}) \right\} \notag \\ &\propto {\bm \varphi}\_{v\_{k}}(A\_s) \label{conclusion of induction}\end{aligned}$$ holds by induction. First, if k = 1, which is equivalent to s = 1, ρ(1) = 1, and As = Ω, the posterior of V(Ω) is written as $$\begin{aligned} \pi\_t(V(\Omega) = v\_1 \mid T\_t) & \propto {{\bm \xi}}\_{1, v\_1}(\Omega) M\_{v\_1}(A\_1 \mid J\_1) \propto {\bm \varphi}\_{v\_1}(\Omega). \end{aligned}$$ Second, assume that ([conclusion of induction]) holds for k = k̄. Then, if k = k̄ + 1, we have $$\begin{aligned} &\pi\_t(V\_{\bar{k}} = v\_{\bar{k}}, V\_{\bar{k}+1} = v\_{\bar{k}+1} \mid T\_t) \\ &\propto \sum^I\_{v\_1=1} \cdots \sum^I\_{v\_{\bar{k}-1}=1} \pi\_t(\{V\_{\rho(l)}\}^{\bar{k}+1}\_{l=1} = \{v\_l\}^{\bar{k}+1}\_{l=1} \mid T\_t)\\ &\propto \sum^I\_{v\_1=1} \cdots \sum^I\_{v\_{\bar{k}-1}=1} \left\{ \prod^{\bar{k}}\_{l=1} {{\bm \xi}}\_{v\_{l-1}, v\_l} (A\_{\rho(l)}) M\_{v\_l} (A\_{\rho(l)} \mid J\_{\rho(l)}) \right\} {{\bm \xi}}\_{v\_{\bar{k}}, v\_{\bar{k}+1}} (A\_s) M\_{v\_{\bar{k}+1}} (A\_{s} \mid J\_{s})\\ &\propto {\bm \varphi}\_{v\_{\bar{k}}}(A\_{\rho(\bar{k})}) {{\bm \xi}}\_{v\_{\bar{k}}, v\_{\bar{k}+1}} (A\_s) M\_{v\_{\bar{k}+1}} (A\_{s} \mid J\_{s}),\end{aligned}$$ from which we obtain $$\begin{aligned} \pi\_t(V\_{\bar{k}+1} = v\_{\bar{k}+1} \mid T\_t) &= \sum^I\_{v\_{\bar{k}}=1} \pi\_t(V\_{\bar{k}} = v\_{\bar{k}}, V\_{\bar{k}+1} = v\_{\bar{k}+1} \mid T\_t)\\ &\propto \sum^I\_{v\_{\bar{k}}=1} {\bm \varphi}\_{v\_{\bar{k}}}(A\_{\rho(\bar{k})}) {{\bm \xi}}\_{v\_{\bar{k}}, v\_{\bar{k}+1}} (A\_s) M\_{v\_{\bar{k}+1}} (A\_{s} \mid J\_{s})\\ &\propto {\bm \varphi}\_{v\_{\bar{k}+1}} (A\_s). \qed \end{aligned}$$ ### Proof of Proposition [prop: components to make SMC] Let the finite tree Tt consist of a sequence of decisions J1 : t = {Js}s = 1t, which sequentially divides nodes A1 : t = {As}s = 1t. To derive the proposition for the marginal posterior of Tt, we first consider the joint posterior of Tt and a sequence of the state variables V1 : t which are defined for the nodes A1 : t. From the structure of the model, the joint posterior is written as $$\begin{aligned} \pi\_t(T\_t, V\_{1:t}) &= P(J\_{1:t}, V\_{1:t} \mid {\mathbf{x}}) \notag \\ &= \frac{1}{Z\_t} P(J\_{1:t})P(V\_{1:t}) \prod^t\_{s=1, A\_s \in \mathcal{N}(T\_t)} M\_{V\_s}(A\_s \mid J\_s) \prod^t\_{s=1, A\_s \in \mathcal{L}(T\_t)} \mu({\mathbf{x}}(A\_s) \mid A\_s), \label{def of joint post}\end{aligned}$$ where Zt is the normalizing constant, and As, l and As, r are the children nodes of As. For Tt − 1 and Tt, since At is divided into At, l and At, r, we have $$\begin{aligned} \mathcal{N}(T\_t) &= \mathcal{N}(T\_{t-1}) \cup \{A\_t\}, \\ \mathcal{L}(T\_t) &= \mathcal{L}(T\_{t-1}) \setminus \{A\_t\} \cup \{A\_{t,l}, A\_{t,r}\}.\end{aligned}$$ With the expression of the joint posterior in ([def of joint post]), we obtain $$\begin{aligned} \pi\_t(T\_t, V\_{1:t}) &= \frac{Z\_t}{Z\_{t-1}} \pi\_{t-1} (T\_{t-1}, V\_{1:t-1}) P(J\_t) P(V\_t \mid V\_{1:t-1}) M\_{V\_t}(A\_t \mid J\_t) \frac{\mu({\mathbf{x}}(A\_{s,l}) \mid A\_{t,l})\mu({\mathbf{x}}(A\_{s,r}) \mid A\_{s,r})} { \mu({\mathbf{x}}(A\_s) \mid A\_s) }. \label{relationship between pi\_t and pi\_t-1}\end{aligned}$$ Let Atp denote the parent node of At and Vtp = V(Atp). Then, since the state variables follow the hidden Markov process, $P(V\_t \mid V\_{1:t-1}) = {{\bm \xi}}\_{V^p\_t, V\_t}(A\_t)$. Integrating out V1 : t − 1 \ Vtp in ([relationship between pit and pit-1]) gives $$\begin{aligned} \pi\_t(T\_t, V^p\_t, V\_t) &= \frac{Z\_t}{Z\_{t-1}} \pi\_{t-1} (T\_{t-1}, V^p\_t) P(J\_t) {{\bm \xi}}\_{V^p\_t, V\_t}(A\_t) M\_{V\_t}(A\_t \mid J\_t) \frac{\mu({\mathbf{x}}(A\_{s,l}) \mid A\_{t,l})\mu({\mathbf{x}}(A\_{s,r}) \mid A\_{s,r})} { \mu({\mathbf{x}}(A\_s) \mid A\_s) }.\end{aligned}$$ Because At is a leaf node of Tt − 1, by Lemma [lem: conditinal post of the state given the tree], we have $$\begin{aligned} \pi\_{t-1} (T\_{t-1}, V^p\_t=j) &= \pi\_{t-1} (T\_{t-1}) \pi\_{t-1}( V^p\_t=j \mid T\_{t-1})\\ &= \pi\_{t-1} (T\_{t-1}) {\bm \varphi}\_{j}(A^p\_t).\end{aligned}$$ Hence, we obtain the expression of the marginal distribution of Tt as $$\begin{aligned} \pi\_t(T\_t) &= \sum^I\_{i=1} \sum^I\_{j=1} \pi\_t(T\_t, V^p\_t=j, V\_t=i) \\ &= \frac{Z\_t}{Z\_{t-1}}\pi\_{t-1} (T\_{t-1}) P(J\_t) \sum^I\_{i=1} \left\{ \sum^I\_{j=1} {\bm \varphi}\_j(A^p\_t) {{\bm \xi}}\_{j,i}(A\_t) \right\} M\_i(A\_t \mid J\_t) \frac{\mu({\mathbf{x}}(A\_{t,l}) \mid A\_{t,l})\mu({\mathbf{x}}(A\_{t,r}) \mid A\_{t,r})} { \mu({\mathbf{x}}(A\_t) \mid A\_t) },\end{aligned}$$ which completes the proof. ### Proof of Corollary [cor: proposal and incremental weight] ### Proof of Proposition [prop: computing predictive density] In this discussion, we suppress x and Tm in the expectation for simplicity. First, when A = Ω, by the definition $e\_\Omega(i) = \tilde{{{\bm \gamma}}}\_{1,i}(\Omega)$. Next, if A is not the root node, we can decompose eA(iʹ) as $$\begin{aligned} e\_A(i') &= \sum^I\_{i=1} {\mathbb{E}}[ Q(A) I[V(A) = i']I[V(A^p) = i]].\end{aligned}$$ For the summand, because θ(Ap) and V(A) are conditionally independent given V(Ap), we obtain $$\begin{aligned} &{\mathbb{E}}[ Q(A) I[V(A) = i']I[V(A^p) = i]] \\ &= {\mathbb{E}}[{\mathbb{E}}[ \vartheta(A^p) I[V(A) = i']\mid V(A^p)]I[V(A^p) = i] Q(A^p)] \\ &= {\mathbb{E}}[{\mathbb{E}}[ \vartheta(A^p) I[V(A) = i']\mid V(A^p) = i]I[V(A^p) = i] Q(A^p)] \\ &={\mathbb{E}}[{\mathbb{E}}[ \vartheta(A^p)\mid V(A^p) = i] P(V(A) = i'\mid V(A^p) = i) I[V(A^p) = i] Q(A^p)]\\ &= \tilde{{{\bm \xi}}}\_{i,i'}{\mathbb{E}}[ \vartheta(A^p)\mid V(A^p) = i] e\_{A^p}(i).\end{aligned}$$ Therefore, we obtain $$\begin{aligned} e\_A(i') &= \sum^I\_{i=1} \tilde{{{\bm \xi}}}\_{i,i'}(A){\mathbb{E}}[ \vartheta(A^p)\mid V(A^p) = i] e\_{A^p}(i).\end{aligned}$$ Asymptotic properties of the tree posteriors -------------------------------------------- [lem: limit of log of sum] [lem: asymptotic behavior for a single node] For T ∈ TK and A ∈ N(T), if i ∈ τ(A ∣ T), then $$\begin{aligned} &\frac{\log M\_i(A \mid j\_A)}{n} \xrightarrow{p} \sum^G\_{g=1} \zeta\_g P\_g(A) \left[ P\_g(A\_l \mid A) \log P\_g(A\_l \mid A) + P\_g(A\_r \mid A) \log P\_g(A\_r \mid A) \right],\end{aligned}$$ where jA is the splitting rule that divides A into Al and Ar. (Proof) By the result of, since the parameter θ(A) follow the beta distribution, which belongs to a continuous exponential family, the log of the marginal likelihood is written as $$\begin{aligned} \log M\_i(A\mid T) &= \hat{l}\_A(i,T) - \frac{r\_i}{2} \log n(A) + \mathcal{O}\_p(1), \notag \\ \hat{l}\_A(i,T) &= \log \left[ \prod^G\_{g=1} \hat{\theta}\_g(A)^{n\_g(A\_l)} (1 - \hat{\theta}\_g(A))^{n\_g(A\_r)} \right] \notag \\ &= \sum^G\_{g=1} \left[ n\_g(A\_l) \log \hat{\theta}\_g(A) + n\_g(A\_r) \log (1-\hat{\theta}\_g(A)) \right] \label{def of likelihood with MLE} ,\end{aligned}$$ where the definition of θ̂g(A) (the MLE) and ri (the number of parameters) depend on which type of priors in Assumption [assumption: sample size and true measures] is introduced by the state i: $$\begin{aligned} \hat{\theta}\_g(A) = \begin{cases} \frac{n\_g(A\_l)}{n\_g(A)} & \text{(Prior A)}, \\ \frac{n(A\_l)}{n(A)} & \text{(Prior B)}, \\ \frac{P\_g(A\_l)}{P\_g(A)} & \text{(Prior C)}, \end{cases} \ \ \ r\_i = \begin{cases} G & \text{(Prior A)}, \\ 1 & \text{(Prior B)}, \\ 0 & \text{(Prior C)}. \end{cases} \label{def of MLE}\end{aligned}$$ Notice that, for Prior C, the constant c(A) and the true measures Pg must satisfy c(A) = Pg(Al ∣ A) for every g = 1, …, G because if this does not hold, the state i is not included in the set of feasible states τ(A ∣ T). Since i ∈ τ(A ∣ T), the law of large numbers gives $ \hat{\theta}\_g(A) \xrightarrow{p} P\_g(A\_l \mid A) $. Hence, we obtain the limit of l̂A(i, T)/n as $$\begin{aligned} \frac{\hat{l}\_A(i,T)}{n} &= \sum^G\_{g=1} \frac{n\_g(\Omega)}{n(\Omega)} \frac{n\_g(A)}{n\_g(\Omega)} \left[ \frac{n\_g(A\_l)}{n\_g(A)} \log \hat{\theta}\_g(A) + \frac{n\_g(A\_r)}{n\_g(A)} \log (1-\hat{\theta}\_g(A)) \right] \\ &\xrightarrow{p} \sum^G\_{g=1} \zeta\_g P\_g(A) \left[ P\_g(A\_l \mid A) \log P\_g(A\_l \mid A) + P\_g(A\_r \mid A) \log P\_g(A\_r \mid A) \right].\end{aligned}$$ [lem: comparison of log ML] For T ∈ TK, A ∈ N(T), i ∈ τ(A ∣ T) and j ∈ {1, …, I}, we have $$\begin{aligned} \frac{\log M\_i(A \mid T) - \log M\_j(A \mid T)}{n} \xrightarrow{p} c\_{i,j},\end{aligned}$$ where ci, j = 0 if j ∈ τ(A ∣ T) and ci, j > 0 if j ∈ {1, …, I} \ τ(A ∣ T). (Proof) If j ∈ τ(A ∣ T), obtaining the result $$\begin{aligned} \frac{\log M\_i(A \mid T) - \log M\_j(A \mid T)}{n} \xrightarrow{p} 0\end{aligned}$$ is straightforward from the proof of Proposition [lem: asymptotic behavior for a single node]. Hence we consider the case of j ∈ {1, …, I} \ τ(A ∣ T). Under the state j, for every g, the estimator θ̂g(A) is defined as in ([def of MLE]), and there exists Cg ∈ (0, 1) such that $\hat{\theta}\_g(A) \xrightarrow{p} C\_g$. By the definition of τ(A ∣ T), there exists g\* such that Cg\* ≠ Pg\(A*l ∣ A). As in the proof of Proposition [lem: asymptotic behavior for a single node], for the difference of the marginal likelihoods, we obtain $$\begin{aligned} &\frac{\log M\_i(A \mid T) - \log M\_j(A \mid T)}{n} \xrightarrow{p} \sum^G\_{g=1} \zeta\_g P\_g(A) \Lambda\_{g}, \\ &\Lambda\_{g} = P\_g(A\_l \mid A) \log \frac{P\_g(A\_l \mid A)}{C\_g} + P\_g(A\_r \mid A) \log \frac{P\_g(A\_r \mid A)}{1-C\_g}.\end{aligned}$$ Because Λg is the KL divergence of the two discrete distributions, Λg ≥ 0 for all g and Λg\* > 0. By Assumption [assumption: sample size and true measures], this result implies that $$\begin{aligned} \sum^G\_{g=1} \zeta\_g P\_g(A) \Lambda\_{g} > 0.\end{aligned}$$ ### Proof of Theorem [thm: characterizign the post of trees] and Theorem [thm: posterior of state variables] In this proof, we modify the notation for the marginal likelihood defined in Eq. and use Mi(A ∣ T) to represent the likelihood on A of the tree T under the ith state to reflect its dependency on the tree structure. Let T ∈ TK and V denote a set of a combination of the states for all of the non-leaf nodes of T. Notice that an element of V does not need to satisfy P(V = v) > 0, where V is the totality of the state variables. In the following proof, for v ∈ V, v(A) denotes a state on a node A. Let l(v, T) denote the log of the joint likelihood function $$\begin{aligned} &l({\mathbf{v}}, T) = \log P({\mathbf{x}}\_n \mid T, {\mathbf{v}}) = \sum\_{A \in \mathcal{N}(T)} l\_A ({\mathbf{v}}(A), T) + \sum\_{A\in \mathcal{L}(T)} \log \mu({\mathbf{x}}\_n(A) \mid A), \label{non-asymptotic decomposition of the likelihood}\end{aligned}$$ where lA(v(A), T) = logMA(v(A) ∣ T). By, this likelihood lA has the following expression $$\begin{aligned} l\_A({\mathbf{v}}(A), T) &= \hat{l}\_{A}({\mathbf{v}}(A), T) - \frac{r\_{{\mathbf{v}}(A)}}{2} \log n (A) + {\mathcal{O}\_p(1)},\end{aligned}$$ where l̂A and ri are defined in ([def of likelihood with MLE]). Let $\bar{{\mathbf{v}}} \in {\mathcal{V}}$ be a collection of states such that, for all A ∈ N(T), θg(A) is fixed to μ(Al)/μ(A). For $\bar{{\mathbf{v}}}$, we have $$\begin{aligned} l\_A(\bar{{\mathbf{v}}}, T) &= \sum\_{A \in \mathcal{N}(T)} \left\{ n(A\_l) \log \left( \frac{\mu(A\_l)}{\mu(A)} \right) + n(A\_r) \log \left( \frac{\mu(A\_r)}{\mu(A)} \right) \right\} + \sum\_{A\in \mathcal{L}(T)} \log \mu({\mathbf{x}}\_n(A) \mid A)\\ &= \log \mu({\mathbf{x}}) = 0.\end{aligned}$$ Hence l(v, T) is rewritten as $$\begin{aligned} &l({\mathbf{v}}, T) = l({\mathbf{v}}, T) - l(\bar{{\mathbf{v}}}, T) = \sum\_{A \in \mathcal{N}(T)} \left\{ \hat{l}\_{A}({\mathbf{v}}(A), T) - \hat{l}\_{A}(\bar{{\mathbf{v}}}(A), T) \right\} - \frac{C({\mathbf{v}})}{2} \log n + {\mathcal{O}\_p(1)}.\end{aligned}$$ For the part inside of the braces, when v is replaced with vT ∈ VT, where VT = {v : v(A) ∈ τ(A ∣ T)for all A ∈ N(T)}, the definition of l̂A gives $$\begin{aligned} \frac{\hat{l}({\mathbf{v}}\_T(A), T) - \hat{l}(\bar{{\mathbf{v}}}(A), T)}{n} &= \sum^G\_{g=1} \frac{n\_g(A)}{n} \left[ \frac{n\_g(A\_l)}{n\_g(A)} \log \frac{\hat{\theta}\_g(A)}{\mu(A\_l \mid A)} + \frac{n\_g(A\_r)}{n\_g(A)} \log \frac{1- \hat{\theta}\_g(A)}{\mu(A\_r \mid A)} \right]\\ & \xrightarrow{p} \sum^G\_{g=1} \zeta\_g P\_g(A) \left[ P\_g(A\_l \mid A) \log \frac{P\_g(A\_l \mid A)}{\mu(A\_l \mid A)} + P\_g(A\_r \mid A) \log \frac{P\_g(A\_r \mid A)}{\mu(A\_r \mid A)} \right].\end{aligned}$$ For all A ∈ L(T), there exists an unique sequence of nodes $$\begin{aligned} \Omega = B\_{A,0} \supset B\_{A,1} \supset \cdots \supset B\_{A,K} = A, \label{path of nodes}\end{aligned}$$ where BA, k ∈ T (k = 0, …, K) is a node in the kth level. With this sequence, we obtain the limit of the scaled log-likelihood as $$\begin{aligned} &\frac{l({\mathbf{v}}\_T, T)}{n} \nonumber \\ &\xrightarrow{p} \sum\_{A \in \mathcal{N}(T)} \sum^G\_{g=1} \zeta\_g P\_g(A) \left[ P\_g(A\_l \mid A) \log \frac{P\_g(A\_l \mid A)}{\mu(A\_l \mid A)} + P\_g(A\_r \mid A) \log \frac{P\_g(A\_r \mid A)}{\mu(A\_r \mid A)} \right] \nonumber \\ &= \sum^G\_{g=1} \zeta\_g\sum\_{A \in \mathcal{N}(T)} \left[ P\_g(A\_l) \log \frac{P\_g(A\_l \mid A)}{\mu(A\_l \mid A)} + P\_g(A\_r) \log \frac{P\_g(A\_r \mid A)}{\mu(A\_r \mid A)} \right] \nonumber \\ &= \sum^G\_{g=1} \zeta\_g \sum\_{A \in \mathcal{L}(T)} P\_g(A) \left[ \log \frac{P\_g(B\_{A,1} \mid B\_{A,0})}{\mu(B\_{A,1} \mid B\_{A,0})} + \cdots + \log \frac{P\_g(B\_{A,K} \mid B\_{A,K-1})}{\mu(B\_{A,K} \mid B\_{A,K-1})} \right] \nonumber \\ &= \sum^G\_{g=1} \zeta\_g \sum\_{A \in \mathcal{L}(T)} P\_g(A) \log \frac{P\_g(A)}{\mu(A)} = \sum^G\_{g=1} \zeta\_g KL(P\_g \|\_T \|\| \mu). \label{limit of the best state variables}\end{aligned}$$ Because Pg∣T admits the density function $$\begin{aligned} p\_g\|\_T(x) = \sum\_{A \in \mathcal{L}(T)} {\mathbf{1}}\_A (x) \frac{P\_g(A)}{\mu(A)},\ x \in \Omega\end{aligned}$$ the KL divergence in ([limit of the best state variables]) is rewritten as $$\begin{aligned} KL(P\_g\|\_{T} \|\| \mu) &= \sum\_{A \in \mathcal{L}(T)} P\_g(A) \log \frac{P\_g(A)}{\mu(A)} \\ &= \int p\_g \sum\_{A \in \mathcal{L}(T)} \mathbf{1}\_A \log \frac{P\_g(A)}{\mu(A)} d \mu \\ &= \int p\_g \log p\_g \|\_{T} d\mu \\ &= \int p\_g(x) \log \frac{p\_g(x)}{\mu(x)} d \mu(x) - \int p\_g \log \frac{p\_g}{p\_g\|\_{T}} d \mu\\ &= KL(P\_g \|\| \mu) - KL(P\_g \|\| P\_g \|\_T).\end{aligned}$$ Because KL(Pg∣∣μ) is independent of T, we obtain another expression of TMK in ([def of TKM]) as $$\begin{aligned} {\mathcal{T}}^K\_M &= {\mathop{\rm arg~max}\limits}\_{T \in {\mathcal{T}}^K} \sum^G\_{g=1} \zeta\_g KL(P\_g\|\_{T} \|\| \mu)\end{aligned}$$ By Lemma [lem: comparison of log ML] and ([non-asymptotic decomposition of the likelihood]), for v ∈ V \ VT, we can show that $$\begin{aligned} \operatorname\{p-lim}\_{n \to \infty} \frac{l({\mathbf{v}}, T) - l({\mathbf{v}}\_T, T)}{n} = \operatorname\{p-lim}\_{n \to \infty} \sum\_{A \in \mathcal{N}(T)} \frac{l\_A({\mathbf{v}}, T) - l\_A({\mathbf{v}}\_T, T)}{n} > 0 \label{plim of the difference of l},\end{aligned}$$ and p-limn → ∞l(v, T)/n exists. Hence, for TM ∈ TMK, vʹ ∈ VTM, T ∈ TK \ TMK and v ∈ V, we have $$\begin{aligned} \operatorname\{p-lim}\_{n \to \infty} \frac{l({\mathbf{v}}', T\_M) - l({\mathbf{v}}, T)}{n} \geq \sum^G\_{g=1} \zeta\_g KL(P\_g \|\_{T\_M} \|\| \mu) - \sum^G\_{g=1} \zeta\_g KL(P\_g \|\_T \|\| \mu) > 0.\end{aligned}$$ Hence, for such TM* and vʹ, we obtain $$\begin{aligned} \frac{P({\mathbf{x}}\_n \mid T)}{P({\mathbf{x}}\_n \mid T\_M)} = \frac{ P(T) \sum\_{{\mathbf{v}}\in {\mathcal{V}}} \exp(l({\mathbf{v}}, T)) P({\mathbf{v}})}{ P(T\_M) \sum\_{{\mathbf{v}}\in {\mathcal{V}}} \exp(l({\mathbf{v}}, T\_M)) P({\mathbf{v}})} \leq \sum\_{{\mathbf{v}}\in {\mathcal{V}}} \frac{ \exp( l({\mathbf{v}}, T)) P(T) P({\mathbf{v}})}{ \exp( l({\mathbf{v}}', T\_M)) P(T\_M) P({\mathbf{v}}')} \xrightarrow{p} 0.\end{aligned}$$ This result implies $p(T \in {\mathcal{T}}^K\_M \mid {\mathbf{x}}\_n ) \xrightarrow{p} 1$, which completes the proof of Theorem [thm: characterizign the post of trees]. To prove Theorem [thm: posterior of state variables], we fix T ∈ TK and define a set ST as $$\begin{aligned} {\mathcal{S}}\_T = \left\{ {\mathbf{v}}\in {\mathcal{V}}\_T \mid {\mathbf{v}}\in {\mathop{\rm arg~min}\limits}\_{{\mathbf{v}}' \in {\mathcal{V}}\_T } C({\mathbf{v}}' ) \right\}.\end{aligned}$$ Then we want to show $P({\mathbf{V}}\in {\mathcal{S}}\_T \mid T, {\mathbf{x}}\_n) \xrightarrow{p} 1$. The result ([plim of the difference of l]) implies $$\begin{aligned} p({\mathbf{V}}\in {\mathcal{V}}\_T \mid T, {\mathbf{x}}\_n) \xrightarrow{p} 1,\end{aligned}$$ so we only need to compare the elements of VT. Let v ∈ VT \ ST and vʹ ∈ ST. For the difference of the log likelihoods, we have $$\begin{aligned} l({\mathbf{v}}', T) - l({\mathbf{v}},T) &= \sum\_{A \in \mathcal{N}(T)} \left[ \hat{l}\_A({\mathbf{v}}'(A), T) - \hat{l}\_A({\mathbf{v}}(A), T) \right] + \frac{C({\mathbf{v}}) - C({\mathbf{v}}')}{2} \log n + {\mathcal{O}\_p(1)},\end{aligned}$$ where l̂A and C is defined in ([def of MLE]) and ([def: complexity]), respectively. If v(A) and vʹ(A) introduce the same type of the prior (e.g., Prior A and Prior A), because the corresponding estimators θ̂g(A) have the same form, $$\begin{aligned} \hat{l}\_A({\mathbf{v}}'(A), T) - \hat{l}\_A({\mathbf{v}}(A), T) = 0.\end{aligned}$$ On the other hand, if v(A) and vʹ(A) introduce different types of the prior (e.g., Prior A and Prior B), because they are the maximized log-likelihood under the two nested hypotheses, $$\begin{aligned} -2 [\hat{l}\_A({\mathbf{v}}'(A), T) - \hat{l}\_A({\mathbf{v}}(A), T)]\end{aligned}$$ weakly converges to the χ2 distribution. Hence, we obtain $$\begin{aligned} \frac{l({\mathbf{v}}', T) - l({\mathbf{v}},T)}{\log n} \xrightarrow{p} \frac{C({\mathbf{v}}) - C({\mathbf{v}}')}{2} > 0,\end{aligned}$$ which implies $P({\mathbf{V}}\in {\mathcal{S}}\_T\mid T, {\mathbf{x}}\_n) \xrightarrow{p} 1$. ### Consistency for the MRS model ============================= To describe the consistency, for a possible node A, we define a variable Z(A) as follows: $$\begin{aligned} Z(A) = \begin{cases} 1 & \text{ if } V(A) = 1,\\ 0 & \text{ if } V(A)\in \{2,3\}. \end{cases}\end{aligned}$$ Hence, θ1(A) = θ2(A) if Z(A) = 0 and θ1(A) ≠ θ2(A) with probability one if Z(A) = 1. Then we can obtain the following consistency result. [cor: consistency for MRS] Let Z = {Z(A)}A ∈ N(T) and z = {z(A)}A ∈ N(T) be a collection of Z(A) on T ∈ TK and one of its realizations, respectively. If P(Z = z) > 0 for any possible z, then $$\begin{aligned} P \left( Z(A) = {\mathbf{1}}\_{\{ P\_1(A\_l \mid A) \neq P\_2(A\_l \mid A) \}} \text{ for all } A \in \mathcal{N}(T) \mid T, {\mathbf{x}}\_n \right) \xrightarrow{p} 1 , \end{aligned}$$ where 1 is the indicator function, and $$\begin{aligned} P \left( Z(A) = {\mathbf{1}}\_{\{ P\_1(A\_l \mid A) \neq P\_2(A\_l \mid A) \}} \text{ for all } A \in \mathcal{N}(T) \mid {\mathbf{x}}\_n \right) \xrightarrow{p} 1, \end{aligned}$$ where T is random. (Proof) In this case, VT in Theorem [thm: posterior of state variables] is written as $$\begin{aligned} {\mathcal{V}}\_T = \left\{ {\mathbf{v}}\mid {\mathbf{v}}(A) = 1 \text{ if } P\_1(A\_l \mid A) \neq P\_2(A\_l \mid A) \right\}.\end{aligned}$$ We additionally define $\tilde{{\mathcal{V}}}\_T$ as $$\begin{aligned} \tilde{{\mathcal{V}}}\_T = \left\{ {\mathbf{v}}\mid {\mathbf{v}}(A) = 2 \text{ if } P\_1(A\_l \mid A) = P\_2(A\_l \mid A) \right\}.\end{aligned}$$ Then, under the condition that v ∈ VT, the complexity C(v) is minimized if and only if ${\mathbf{v}}\in \tilde{{\mathcal{V}}}\_T$. Hence, by Theorem [thm: posterior of state variables] we obtain $$\begin{aligned} & P \left( Z(A) = {\mathbf{1}}\_{\{ P\_1(A\_l \mid A) \neq P\_2(A\_l \mid A) \}} \text{ for all } A \in \mathcal{N}(T) \mid T, {\mathbf{x}}\_n \right)\\ &= P \left( {\mathbf{V}}\in {\mathcal{V}}\_T \cap \tilde{{\mathcal{V}}}\_T \mid T, {\mathbf{x}}\_n \right)\xrightarrow{p} 1.\end{aligned}$$ We can show the second result by using Theorem [thm: characterizign the post of trees] as follows: $$\begin{aligned} P \left( {\mathbf{V}}\in {\mathcal{V}}\_T \cap \tilde{{\mathcal{V}}}\_T \mid {\mathbf{x}}\_n \right) &= \sum\_{T \in {\mathcal{T}}^K} P \left( {\mathbf{V}}\in {\mathcal{V}}\_T \cap \tilde{{\mathcal{V}}}\_T \mid T, {\mathbf{x}}\_n \right) P(T \mid {\mathbf{x}}\_n)\\ &\geq {\mathop{\rm arg~min}\limits}\_{T\_M \in {\mathcal{T}}^K\_M} P \left( {\mathbf{V}}\in {\mathcal{V}}\_T \cap \tilde{{\mathcal{V}}}\_T	arxiv_0000775
Two dusty Seyfert galaxies and one ULIRG with a Seyfert nucleus (Mrk273) also lie above the relation defined by the SSGSS and galaxies, further supporting the idea that dust along the line-of-sight to a deeply buried AGN can lead to stronger silicate absorption, relative to nebular line absorption, than normal Hiiregions. We are not aware of any models which simultaneously predict the strength of silicate absorption and attenuation of the optical emission lines. Qualitatively Figure [fig:dust] suggests a two component model is required to describe the dust distribution in galaxies/AGN. The lower branch is consistent with silicate absorption arising in the clumpy distribution of stellar birthclouds, with the densest clouds causing some silicate absorption, and the Balmer emission lines originating from the same star formation regions but further out in the clouds. The upwards scatter of some sources from this branch is consistent with the dust being concentrated in a dense shell-like distribution obscuring a single nuclear source, most likely an AGN in the cases studied here, but also conceivably a strong nuclear starburst. Galaxies with an AGN -------------------- [fig:oiiioiv] As explained in Section [sec:nir], a dust curve and accurate Balmer decrement that adequately corrects Hii region emission lines for dust attenuation, will not necessarily do the same for lines from the NLR of AGN. Particularly in composite objects, a higher fraction of Balmer lines will arise from within the Hii regions of the galaxy compared to lines such as which are stronger in NLRs. Therefore a form of differential extinction might arise between the Balmer and higher ionisation lines, and a corresponding bias in the correction of AGN NLR lines for dust using the measured Balmer decrement. Additionally, the form of the dust curve may not be the same for light originating from Hii regions and from an AGN NLR, due to the different relative geometry between the light source and the dust cloud, and the different balance between dense and diffuse dust attenuating the lines. The mid-IR line at 25.89 μm has been shown to be an accurate indicator of AGN power, and the / ratio a possible measure of AGN attenuation. In this section we test the accuracy of dust-corrected as a measure of AGN luminosity in our sources by comparing to the mid-IR line. In Figure [fig:oiiioiv] we plot dust-uncorrected and dust-corrected vs. for the Seyferts, the 3 galaxies and the 2 ULIRGs with measured lines. Note that the correlation coefficient is similar for the Seyferts whether the dust correction is performed or not. These Seyferts have very small Balmer decrements so the dust correction has only a small effect. Because some of these objects are classified as composite AGN-starformation by their optical line ratios, and is also produced by starformation, in the right hand panel we apply a correction for this following the method of, based on the expected /Hα ratio for metal-rich starforming galaxies[14](#fn14). is also produced in Hii regions, although the contamination is expected to be much smaller than for due to the higher ionisation level. Clearly, there is a tight correlation between dust-corrected and for the Seyfert sample. All of the dusty galaxies and ULIRGs with measured fall within a factor of 3 of the relation defined by the Seyferts. Only the upper limit for Arp220 is marginally inconsistent with the relation defined by the Seyfert galaxies. Taking the remainder of the galaxies and stacking the SH spectra in luminosity units shows a clear line, which we fit together with the neighbouring Feii line using a double Gaussian. The corresponding stacked - luminosity also lies exactly on the Seyfert relation, once contamination of from star formation has been accounted for. Errors are estimated from the standard deviation of the luminosity on removing one spectrum at a time from the stack. This strong correlation between and dust corrected using the observed Balmer decrement and dust-curve appropriate for Hii regions, for even dusty composite galaxies, shows that there is no evidence for significant differential extinction between lines emitted from the NLR and Hii regions of a galaxy. Although slightly surprising, this may imply that the majority of dust attenuation suffered by NLR emission lines comes from the diffuse dust and Hii regions in the intervening galaxy, through which they must pass on their way to the observer. Unfortunately neither of the two ULIRGs with extreme silicate absorption (08572 and 15250) have measured, so we cannot test directly whether optical and mid-IR NLR lines agree even in the presence of extreme, dense obscuration of the central light source. Such a study may be possible with the large samples of ULIRGs now available in archives. Discussion ========== The potential of dust attenuation and complicated dust geometries to render useless optical observations of extreme galaxies is sometimes used to justify rest-frame IR observations. In the previous sections we have used the correlation between the unattenuated mid-IR and attenuated optical nebular emission lines to argue that a single dust curve can accurately correct optical emission lines for dust attenuation in a large range of different galaxies. The correction is accurate to within a factor of 2 in most cases, and to at least a factor of 4 for even the most heavily obscured ULIRGs. Here, we discuss the implications of our work for deriving SFRs and black hole accretion rates from the optical emission lines Hα,, and. We present useful relations for dust corrections at the wavelengths of each of these lines. Star formation rates -------------------- [fig:atten] The left and right panels of Fig. [fig:atten] show the fractional attenuation (in flux units) as a function of observed Balmer decrement at Hα and, for each of the different dust curves discussed in Section [sec:results]. Using the best-fit μ0.4 dust curve, the recommended correction at Hα is: $${\rm L[H\alpha]\_{corr}} ={\rm L[H\alpha]\_{obs}}\left(\frac{\rm (H\alpha/H\beta)\_{obs}}{B}\right)^{2.67\_{-0.32}^{+0.40}}$$ and at [15](#fn15): $${\rm L[OII]\_{corr}} ={\rm L[OII]\_{obs}}\left(\frac{\rm (H\alpha/H\beta)\_{obs}}{B}\right)^{4.89\_{-0.29}^{+0.36}}$$ where B is the unattenuated ratio as discussed above and errors are propagated from the errors on the best-fit μ. While for Hiigalaxies and strong Seyferts the choice of B is clear, for composite galaxies the unknown B may cause small systematic uncertainties. However as Hii regions have relatively stronger Hαemission, a value of B = 2.86 is probably close to reality. To derive a star formation rate from optical nebular emission lines, two steps are required. This paper has focused on the first step: to correct the line for dust attenuation. The second step is to convert the dust-corrected line luminosity into a star formation rate. While our results show that dust-corrected Hα luminosity traces [Ne] luminosity, this is not enough to confirm the reliability of Hα as an indicator of star formation rate in extremely dusty objects. This is because dense dust clouds can have high optical depths even in the mid-IR, as indicated by the strong Silicate absorption in some galaxies, and thus even the neon lines may be attenuated in some objects. In addition, the most extreme compact, dusty galaxies may even have a non-negligible fraction of their ionizing photons absorbed by dust and not gas (and thus not traced by Hα). Encouragingly, Ho & Keto (2007) have already shown how [Ne] luminosity correlates strongly with both the total infrared luminosity and Brackett-α luminosity in star-forming galaxies. Their galaxy sample includes a handful of ULIRGs, showing that these relations hold even at extreme luminosities. This result provides circumstantial evidence that the neon luminosity is tracing star-formation even in the dustiest of galaxies. In contrast, Farrah et al. (2007) find [Ne] luminosity in ULIRGs to be deficient by 0.4dex from the Ho & Keto relation, which they ascribe to extinction of the [Ne] lines. In a companion paper we plan to address this issue directly by comparing the star formation rates derived from the full SEDs, with that derived from the emission lines. Black hole accretion rates -------------------------- As with star formation rate, the derivation of black hole accretion rates (BHAR) from nebular emission line luminosities requires two steps: correction for dust emission discussed in this paper, and then conversion into BHAR via a bolometric correction. For the best-fit μ0.4 dust curve, the relevant correction for dust attenuation at is: $${\rm L[OIII]\_{corr}} ={\rm L[OIII]}\left(\frac{\rm (H\alpha/H\beta)\_{obs}}{B}\right)^{3.55\_{-0.32}^{+0.40}}$$ where B is the unattenuated ratio as discussed above. The correction caused by the different dust curves is presented in the central panel of Figure [fig:atten]. The exponent for the μ0.4 dust curve is larger than that used in some AGN literature and can lead to significant differences in estimated bolometric AGN luminosity and accretion rates as a fraction of $L\_{\rm Edd}$. Using a smaller exponent than appropriate will cause an underestimate in $\_{\rm corr}$ which will vary with dust content. For example, for the relatively dust-free sample of Seyfert galaxies studied in this work, the difference between the μ0.4 dust curve exponent of 3.55, and 2.94 (from a MW extinction law) used in e.g. and leads to differences ranging from 5 to 30%. The potential increase in the under- (or over-)estimation of the line luminosity with increasing dust content because of an inappropriate dust-curve has important implications for understanding whether the bolometric correction for changes systematically with, due to the correlations between SFR of the host galaxy and the luminosity of the central AGN and the SFR and dust content of the host. Our results do not explicitly constrain which dust-curve is most appropriate for Seyferts, however, we would like to emphasise the importance of including the errors caused by the dust-attenuation correction when studying emission line trends. AGN detection ------------- Even in the two ULIRGs in our sample where there is evidence for a heavily buried central nucleus, we would like to emphasise that the optical emission line ratios still indicate the possible presence of an AGN, albeit not conclusively. This shows that even where the AGN is deeply buried, some radiation may still escape to ionise the NLR clouds. This has important implications for using optical emission line ratios to identify AGN in objects such as ULIRGs, and will become increasingly important at high-z. Unfortunately neither of the two ULIRGs with strong silicate absorption in our sample have measured, and therefore we cannot directly verify that their position on the BPT diagram, and therefore their correction of for contamination from star formation, is correct. The specific study of a larger sample of ULIRGs to compare the bolometric AGN luminosities derived from mid-IR lines, mid-IR continuum and dust- and star-formation corrected high-quality optical emission lines would now be possible with samples in the SDSS and Spitzer archives. Conclusions =========== The purpose of this paper was to verify, or otherwise, the accuracy of fundamental quantities derived from optical emission lines, particularly in the case of significant dust attenuation of the lines. In particular, optical emission lines provide powerful measures of star formation rate and black hole accretion rate. We compared two “dusty” galaxy samples, including 6 ULIRGs, to a sample of “ordinary” galaxies from the SSGSS all with high quality SDSS optical spectra and mid-IR spectra from the Spitzer Space Telescope IRS spectrograph. The results are extremely encouraging for using rest-frame optical spectra to study dusty galaxies with an AV < 4.5. Our results are promising even for galaxies as dusty as ULIRGs, although a much larger sample should be studied covering the full range of this diverse population, in order to verify our conclusions. Our results have important implications for future high-redshift spectroscopic surveys, where galaxies have higher star formation rates and dust contents may be correspondingly higher. We have compared several different dust curves used in the literature to correct optical emission lines for dust attenuation. Our results favour a dust curve which lies close to the MW extinction curve, indicating that a significant fraction of the dust which attenuates nebular emission lines has a screen-like geometry. This is in agreement with the model for emission line attenuation presented by and and based upon the 2-component CF00 dust model. In this model, the screen-like extinction arises from the dense stellar birth clouds which surround the hottest stars responsible for ionising the gas. We find a best-fit dust-curve with on average 40% of the optical depth at 5500Å arising from diffuse ISM dust. The μ0.4 dust curve is not significantly different from the Calzetti dust-curve in the optical wavelength regime, however, we focus our results and analysis on the former which was intended for emission lines and allows greater flexibility, rather than the latter which was derived from galaxy continua. Particular conclusions with regard to measuring star formation from optical emission lines in the presence of dust: * The MW, Calzetti, μ0.3 and best-fit μ0.4 dust curves all allow accurate correction of Hα (to better than a factor of 2), and therefore accurate calculation of star formation rates, for galaxies with observed Hα/Hβ ratios of as much as 10 (τV < 4.5). * For only 2/6 ULIRGs with τV < 6 does the optical Hα differ from the mid-IR [Ne] lines by more than a factor of 2, but then only by a factor of 4. In these two cases (08572 and 15250) we have argued that the observed excess of [Ne] is in fact caused by a buried AGN, rather than uncertainties in the dust correction. * The majority of galaxies in our samples favour a dust curve close to the MW screen-like dust extinction curve. This suggests that much of the extra dust in the unusually dusty objects is located in dense clouds, rather than in the diffuse ISM. * There is tentative evidence for a range in the shape of the dust curves in ordinary starforming galaxies. For the CF00 dust model, this range translates into a variation in the fraction of optical depth arising in the diffuse ISM of between around 30% and 90%, with the remaining fraction arising in dense birth clouds. However, the variation in the intrinsic [Ne]/Hα ratio with metallicity, the issue of relative aperture correction between the IRS and SDSS data, and the large statistical errors on the Hβ emission line, prevent us from quantifying this scatter more precisely. Particular conclusions with regard to identifying AGN and measuring their accretion rates from optical emission lines in the presence of dust: * The best-fit μ0.4 dust curve and measured Balmer decrement allows accurate correction of for dust attenuation, to within a factor of 3. This is slightly surprising, and suggests that the effects of variations in the relative geometry of the source and dust, and differential extinction between the Balmer and higher ionisation lines, are minimal. * The method to separate the contributions to optical emission lines from both star formation and AGN presented in tightens the relation between and for composite objects. This supports the use of the position of galaxies on the optical BPT line ratio diagram to measure the amount of originating from the AGN, and therefore the estimates of black hole accretion rates from corrected for dust attenuation and star formation contamination. * All ULIRGs in our sample are classified as AGN or composite objects from their optical emission line ratios. While the composite line ratios may be caused by shocks, we have argued that other observational results favour the presence of a buried AGN contributing significantly to the mid-IR continuum. It is therefore possible that sufficient ionising flux escapes from the central source to ionise the NLR, even in the case of heavily buried nuclei such as in ULIRGS 08572 and 15250. Unfortunately these two ULIRGs do not have measured preventing us from verifying our corrections to for dust attenuation and star formation contamination in such extreme cases. In the case of ULIRGs, whose properties vary widely, clearly a much larger sample should be studied before firm conclusions can be drawn. Acknowledgements ================ We would like to thank Jarle Brinchmann for his help in understanding aperture bias issues and Stephane Charlot for comments on an early draft. We thank the anonymous referee for a careful reading of the manuscript and comments which improved its clarity. The function fitting performed in this paper used the IDL MPFIT software http://purl.com/net/mpfit (Markwardt 2009). The ITERA package was used to investigate intrinsic emission line ratios http://www.strw.leidenuniv.nl/ brent/itera.html. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. 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H., Schiminovich D., Meisenheimer K., 2009,, 707, 1566 Sample information ================== ccccccc Name & RA (deg) & Dec (deg) & specobjid & rowindex & z PSB01&181.225&52.395&248485859366535168&284940&0.0632 PSB02&169.600&56.036&256085720727289856&296570&0.0680 PSB03&173.251&60.275&268189274979958784&315471&0.0644 PSB04&167.291&45.857&404988882665340928&491844&0.0636 PSB05&136.004&1.458&132797726868897792&105860&0.0535 PSB06&120.199&37.729&213582458531086336&232315&0.0416 PSB07&339.726&13.194&208235580251701248&223368&0.0628 PSB08&128.311&32.583&262560681922920448&306604&0.0558 PSB09&239.431&27.465&391758540508758016&472114&0.0316 PSB10&139.864&33.791&448335754997792768&536057&0.0193 PSB11&158.162&12.178&450306505485320192&539391&0.0329 ccccccc Name & RA (deg) & Dec (deg) & specobjid & rowindex & z IRAS 08572+3915&135.106&39.065&337714854909444096&394321&0.0579 IRAS 12112+0305&183.441&2.811&146028589375029248&126942&0.0730 IRAS 15250+3609&231.748&35.977&394574695615168512&476759&0.0553 Arp220&233.738&23.504&609061542893191168&755340&0.0181 Mrk273&206.176&55.887&372055066565148672&439991&0.0373 UGC5101&143.965&61.353&137019791402598400&111495&0.0393 ccccccc Name & RA (deg) & Dec (deg) & specobjid & rowindex & z 2MASXJ08244333+2959238&126.180&29.990&339966523504328704&397741&0.0250 CGCG064-017&149.812&12.988&491120230488080384&603606&0.0340 Mrk0609&51.356&-6.144&129701503752470528&101843&0.0340 NCG0291&13.375&-8.768&185153160128495616&190581&0.0190 NGC5695&219.342&36.568&389226545811554304&467780&0.0140 2MASXJ15522564+2753435&238.107&27.895&391758541012074496&472216&0.0740 2MASXJ12384342+0927362&189.681&9.460&347285137836212224&409636&0.0830 2MASXJ08035923+2345201&120.997&23.756&356292212926971904&415287&0.0290 2MASXJ16164729+3716209&244.197&37.273&297464197648744448&366718&0.1520 2MASXJ11570483+5249036&179.270&52.818&248485858699640832&284823&0.0360 2MASXJ12183945+4706275&184.664&47.108&408648327781941248&498207&0.0940 SBS1133+572&173.954&56.952&369240319515951104&435171&0.0510 SBS1609+490&242.716&48.911&175301007746531328&174339&0.0450 IC0486&120.087&26.614&261716087635181568&305068&0.0270 CGCG242-028&170.755&47.052&405833433537839104&493622&0.0250 Mrk1457&176.840&52.450&247922896524017664&283914&0.0490 2MASXJ13463217+6423247&206.634&64.390&170234565783191552&166011&0.0240 CGCG218-007&200.952&43.301&387537584101785600&464960&0.0270 2MASXJ10181928+3722419&154.580&37.378&401892408733728768&486221&0.0490 2MASXJ11110693+0228477&167.779&2.480&143495710086529024&122473&0.0350 galaxies ========= While three of the samples studied in this paper have been presented elsewhere, the sample has not. Here we summarise its selection criteria and the mid-IR observations. Selection --------- The galaxies were selected to have: * Observed Hα/Hβ flux ratios greater than 8 * Strong Balmer absorption lines compared to their 4000Å break strength [${\rm PC2}>0.05\times {\rm PC1}+0.4$, ]. Although they were not selected on a single Balmer absorption line equivalent width, which have considerably larger errors than the spectral indices used here, most have HδA > 5Å and Dn(4000) < 1.4. * Accurately measured emission lines: a signal-to-noise > 3σ for all four BPT lines, together with a Hβ flux $\rm >40\times10^{-17} erg/s/cm^2$. * A contribution to their emission lines by an AGN, i.e. they lie above the empirically derived demarcation line of on the BPT diagram. * z < 0.07 so that the SDSS spectrum, taken through a 3fibre, samples only light coming from the central few kpc of the galaxy. Mid-IR observations ------------------- IRS observations in both high and low resolution were carried out using staring mode during program 40330 (PI Timothy Heckman). We used SPICE to extract the 1D spectra from the low-resolution basic calibrated data (BCD) pipeline products, using the point-source and optimal extraction options. The two 1D spectra (i.e. one for each nod) were combined using a weighted mean. For the high-resolution spectra, we first identified and interpolated over additional rogue pixels using a custom IDL routine and the rogue pixel masks suitable for our campaign numbers (41-46). This procedure was repeated for the dedicated off-source background observations. All 3 science exposures were combined using a simple arithmetic mean, resulting in two 2D spectra. These were sky-subtracted and the 1D spectra were extracted using SPICE with the point-source and optimal extraction options. For each nod, the edges of each order were inspected and trimmed and the orders combined using a weighted mean where wavelengths were covered by more than one order. The 1D spectra of each nod were subsequently combined using a weighted mean. --- 1. Throughout this paper we use the term “effective attenuation” to encompass all effects relating to the dust i.e. absorption and scattering, and geometrical effects. We note that this differs from some other papers which apply the term to “single-line-of-sight” attenuation.[↩](#fnref1) 2. Commonly used MW extinction curves include, and [↩](#fnref2) 3. The difference in the intrinsic Balmer decrements between NLR and Hii regions (due to different gas temperatures), will be small relative to the difference caused by attenuation.[↩](#fnref3) 4. These line ratio predictions can be obtained using the publicly available ITERA software package [↩](#fnref4) 5. http://www.mpa-garching.mpg.de/SDSS/DR7/[↩](#fnref5) 6. http://isc.astro.cornell.edu/SmartDoc/LineFitting[↩](#fnref6) 7. We found that PAHfit does not fit the continuum well for ULIRGs with strong Silicate absorption, and therefore the emission line measurements were unreliable in these cases.[↩](#fnref7) 8. http://www.astro.columbia.edu/ssgss/[↩](#fnref8) 9. Petrosian magnitudes and radii are estimates of total magnitudes suitable for all types of galaxies. They are derived by the SDSS photometry pipelines, with details found here: http://www.sdss.org/dr7/algorithms/photometry.html[↩](#fnref9) 10. Courtesy of Jarle Brinchmann, publicly available at http://www.mpa-garching.mpg.de/SDSS/DR7/[↩](#fnref10) 11. The short-low (SL) IRS module covering the wavelength range 5.2-14.5μm has a slit aperture of 3.6 × 57. The long-low (LL) IRS module covering the wavelength range 14.0-38.0μm has a slit aperture of 10.6 × 168. At high resolution, the short-high (SH) module covers 9.9-19.6μm with a slit aperture of 4.7 × 11.3 and the long-high (LH) module covers 18.7-37.2 with a slit aperture of 11.1 × 22.3.[↩](#fnref11) 12. Face-on galaxy orientations can also cause increased scattering into the line-of-sight and thus greyer dust curves.[↩](#fnref12) 13. Allowing the intercept to be non-zero results in the same fitted parameters within the errors.[↩](#fnref13) 14. We note that such a correction would not be appropriate for the ULIRGs if shocks were a significant contributor to the total Hα luminosity.[↩](#fnref14) 15. Note that this is an extrapolation of the dust-curve beyond the wavelength region directly tested in this paper.[↩](#fnref15) Optical vs. infrared studies of dusty galaxies and AGN: (I) Nebular emission lines ================================================================================== Optical nebular emission lines are commonly used to estimate the star formation rate of galaxies and the black hole accretion rate of their central active nucleus. The accuracy of the conversion from line strengths to physical properties depends upon the accuracy to which the lines can be corrected for dust attenuation. For studies of single galaxies with normal amounts of dust, most dust corrections result in the same derived properties within the errors. However, for statistical studies of populations of galaxies, or for studies of galaxies with higher dust contents such as might be found in some classes of “transition” galaxies, significant uncertainty arises from the dust attenuation correction. In this paper we compare the strength of the predominantly unobscured mid-IR + emission lines to the optical Hα emission lines in four samples of galaxies: (i) ordinary star forming galaxies (80 galaxies), (ii) optically selected dusty galaxies (11), (iii) ULIRGs (6), (iv) Seyfert 2 galaxies (20). We show that a single dust attenuation curve applied to all samples can correct the Hα luminosity for dust attenuation to a factor better than 2. Similarly, we compare and luminosities to find that can be corrected to a factor better than 3. This shows that the total dust attenuation suffered by the AGN narrow line region is not significantly different to that suffered by the starforming Hii regions in the galaxy. We provide explicit dust attenuation corrections, together with errors, for, and Hα. The best-fit average attenuation curve is slightly greyer than the Milky-Way extinction law, indicating either that external galaxies have slightly different typical dust properties to the Milky Way, or that there is a significant contribution from scattering. Finally, we uncover an intriguing correlation between Silicate absorption and Balmer decrement, two measures of dust in galaxies which probe entirely different regimes in optical depth. Type keywords here. Introduction ============ In the last few decades large optical spectroscopic surveys have dramatically improved our knowledge of the global distribution of galaxy physical properties such as stellar mass, star formation rates, star formation histories, and both gas phase and stellar metallicities. Furthermore, at low to intermediate redshifts, the sheer statistics of current optical spectroscopic surveys allow us to identify and study significant numbers of rare objects which represent transient phases in the lifetime of galaxies: starburst, post-starburst and “green-valley” galaxies, major mergers, and powerful active galactic nuclei (AGN). These transient phases, although rare, are globally important. For example, black hole growth occurs primarily in rare, high accretion rate events, and it is still debated whether the infrequent gas-rich galaxy major mergers are responsible for the morphological transformation of galaxies and the build-up of the red sequence. In the near future, the James Webb Space Telescope (JWST) will allow rest-frame optical spectroscopic observations at high-redshift, where galaxies are more active in both star formation and black hole growth. There is an obvious benefit in using the information provided in the redshift survey to study the properties of the galaxies in detail. For example, when star formation is the dominant mechanism of excitation, Hα is a very simple and reliable measure of star formation rate in galaxies. However, it must be accurately corrected for dust attenuation. In ordinary galaxies, the tight relations between dust-corrected Hα and other indicators of star formation rate from other wavelength regimes provide circumstantial evidence that dust attenuation in the optical can be corrected for accurately. suggests that errors on SFRs caused by the assumed dust curve (attenuation as a function of wavelength) can be ± 10-20% for typical star-forming galaxies in the local Universe. However, knowing the “correct” dust curve is crucial. find that the use of different dust curves in spectral-energy-distribution (SED) fitting of high redshift K-selected galaxies can alter parameters such as star formation rates and star formation timescales by factors of 2-3. The large errors reported in this study result from the fact that the “correct” dust-curve to use at high redshift is largely unconstrained and there is a large range in the steepness of dust-curves observed along lines-of-sight to stars in the Magellanic clouds and Milky Way. With the complex physical processes that occur during transient phases in the life-cycle of a galaxy, it is uncertain how far rest-frame optical studies alone can help us in understanding their true impact. As dust content increases, and galaxy geometries become more complicated, corrections for dust attenuation become more uncertain. How much physical information can we really infer from optical data alone when faced with extreme star formation, dust production, AGN activity and complicated spatial geometries of disturbed galaxies? In this paper, we compare emission line fluxes in the optical to those in the mid infra-red where dust attenuation is negligible, except in small and well known wavelength bands. To test the recovery of Hαluminosity from optical data in the presence of significant dust, we compare and contrast a low-z “representative” sample of galaxies with (i) dusty galaxies selected from the Sloan Digital Sky Survey (SDSS) based on their Balmer emission line ratios and strong Balmer absorption lines indicating a possible recent decrease in star formation; (ii) local Ultra-luminous Infrared Galaxies (ULIRGs) selected from the Infrared Astronomy Satellite (IRAS) Bright Galaxy Sample (BGS) based on their total IR luminosities (L$\_{\rm TIR}>10^{12}$). The control sample of galaxies arise from the Spitzer-SDSS-GALEX Spectroscopic Survey (SSGSS) and were chosen to be “representative” of the ordinary galaxies within the SDSS and GALEX surveys. For the AGN in these samples, we further compare the / line ratio to a sample of Seyfert 2 galaxies, to test the recovery of black hole accretion rates from optical emission line. In a companion paper we will address the effect of dust on the optical continuum, and thus estimations of star formation histories, and compare the star formation rates derived from the multi-wavelength continuum with those from Hα luminosity. Understanding our selected “unusual” galaxies is crucial for understanding galaxy evolution. The galaxies show optical emission lines indicating obscured AGN, and the correction of their emission for dust attenuation using different dust-curves results in them contributing between an insignificant ∼ 1% and a much more significant ∼ 10% to the total luminosity (or black hole accretion rate) of all narrow-line AGN in the SDSS. This has important implications for the starburst-AGN connection, for example, as the strong Balmer absorption lines suggest this accretion is happening after the starburst has decayed. Low-z ULIRGs are usually associated with major gas-rich mergers. Some models of galaxy evolution have used gas-rich major mergers as the driving force to change the observational properties of the galaxy population over cosmic time. The mergers lead to powerful starbursts and QSO driven outflows, resulting in the population of red, quiescent ellipticals seen in the local universe. ULIRGs become increasingly common at earlier epochs, and dominate the star formation rate density beyond a z ∼ 1. Understanding the basic properties of ULIRGs at low-redshift, and the limitations of the optical wavelength regime for identifying and understanding the properties of their power sources, is crucial in an era when near-infrared (NIR) ground-based spectrographs and the JWST will soon allow us to study galaxies with rest-frame optical spectra at increasingly early epochs. The outline of this paper is as follows. After briefly reviewing the topic of dust extinction and attenuation in Section [sec:review], we present the spectral analysis methods and four samples in detail in Section [sec:samples]. In Section [sec:results] we compare the data with a variety of different dust curves commonly used in the literature. We conclude in Section [sec:disc] by discussing the implications of our work for measuring star formation rates, black hole accretion rates, and identifying AGN in optical spectroscopic surveys. Finally, we provide relations for correcting the, and Hα emission lines using a measured Hα/Hβ flux ratio (Balmer decrement), assuming a single dust curve which provides a good fit to all of the samples studied in this paper. Throughout the paper we quote line luminosities in erg/s and masses in solar masses. Optical parameters for all samples have been extracted from the SDSS-DR7, and may differ from those in earlier releases due to updates in the SDSS reduction pipeline. All optical wavelengths are quoted as measured in a vacuum, in keeping with SDSS convention, optical wavelengths are given in Å and mid-IR wavelengths in μm. Review of dust attenuation laws =============================== Dust grains in galaxies which intercept and scatter stellar and nuclear light complicate the analysis of integrated galaxy spectra. Optical light reaching the observer is effected by dust in three different ways: (i) absorption; (ii) scattering out of the line-of-sight; (iii) scattering into the line-of-sight. The first two combined are usually termed “extinction” and all three combined are termed “attenuation”. The final “effective attenuation” observed in the integrated light of galaxies results from the integration of attenuated light over all lines-of-sight to light sources throughout the entire galaxy[1](#fn1). In this section, we introduce the basics of the effects of dust on studying the integrated light from galaxies. gives an extensive recent review of the literature, although note that the terminology differs in places from that adopted here. The effective attenuation Aλ, in magnitudes, at a given wavelength λ is given by: $$\begin{aligned} A\_\lambda &=& -2.5\log \left(\frac{I\_{\lambda,obs}}{I\_{\lambda,intr}}\right)\\ &=& -2.5\log \left[\exp(-\tau\_\lambda)\right]\end{aligned}$$ where Iλ, obs is the observed luminosity, Iλ, intr is the intrinsic luminosity of the source, and τλ is the effective optical depth of the dust. Observationally we measure the dimming and reddening of light from sources, thus deriving an effective attenuation or extinction curve kλ: $$\begin{aligned} k\_\lambda &=& \frac{A\_\lambda}{\rm E(B-V)}\\ &=& \frac{\rm E(\lambda-V)}{\rm E(B-V)}+R\_V\end{aligned}$$ where the colour excess is defined by E(λ − V) ≡ Aλ − AV and RV is the total-to-selective extinction in the V band. In the case of extinction, RV is a function of the grain composition, size and shape. In the case of effective attenuation observed in galaxies it is additionally a function of the local geometry of the dust and light sources and global geometry of the galaxy (bulge and disk, and inclination). In this work we will use the term “dust curve” to mean either “effective attenuation” or “extinction” curve, depending on the particular curve under discussion. The effective attenuation suffered by the ultra-violet (UV) to optical stellar continua of starburst galaxies, where the unattenuated stellar continuum is relatively well understood, has been measured empirically from the ratio of heavily attenuated to less attenuated spectra. The dust-curve is found to be “greyer” (i.e. flatter) than extinction curves measured along single lines-of-sight towards stars in the Milky Way[2](#fn2), as expected from geometrical effects. Similarly, fit the ratio of far infra-red (FIR) to UV luminosities and UV slope in starburst galaxies to find an effective attenuation curve that is well fit by the shallow powerlaw τλ ∝ λ− 0.7. Dust curves derived from stellar continua are not necessarily suitable for the correction of nebular emission lines, however. In starburst galaxies, nebular emission lines are found to be more attenuated than the stellar continua, indicating that the light passes through a greater quantity, and possibly different distribution or composition, of dust. Any difference in the distribution or composition of dust will result in a different shape for the dust attenuation curve and therefore errors in the correction of nebular emission lines using a dust-curve derived from the stellar continua. The stronger attenuation of nebular emission lines compared to the stellar continuum, can be successfully reproduced by models in which dust is distributed primarily in 2 components: diffuse interstellar dust, and stellar birthclouds which disperse after a finite time ( ∼ 107 years). This allows the emission lines, which are ionised by the hottest stars still enveloped in their dense birthclouds, to experience greater attenuation than the UV stellar continua, which has an additional contribution from cooler stars around which the birthclouds have already dispersed. In Wild et al. (2007) we presented explicitly the form of the dust-curve for the correction of nebular emission lines in the CF00 dust model: $$\label{eqn:w07} \frac{\tau\_\lambda}{\tau\_V} = \frac{A\_\lambda}{A\_V} =(1-\mu)(\lambda/\lambda\_V)^{-n} + \mu(\lambda/\lambda\_V)^{-m}.$$ We reiterate that we use τV to refer to the final “effective attenuation” suffered by light, including geometrical effects, and that this model is strictly an “angle averaged” model, averaging over all sight-lines through a galaxy. The first term describes the slope of the screen-like extinction suffered as the light leaves the stellar birth-clouds and the exponent is set to match the extinction observed along lines-of-sight in the MW (n = 1.3). The second term describes the additional, much greyer attenuation caused by the intervening diffuse ISM, where CF00 derive a best-fit exponent of m = 0.7. The fraction of the total effective optical depth at 5500Åcaused by dust in the diffuse ISM is controlled by the parameter μ. CF00 showed that the typical value for μ in their starburst dataset was about 0.3. The resulting model attenuation curve, labelled μ0.3 throughout the paper, naturally lies between the pure continuum τλ ∝ λ− 0.7 attenuation (equivalent to μ = 1), and pure screen-like attenuation (equivalent to μ = 0). Recently, applied and extended the two component CF00 dust model to a sample of local galaxies with multi-wavelength photometry and optical Hα and Hβ emission line measurements, finding an excellent match to the data. We note that an additional complication arises in the case of nebular emission from the narrow line region (NLR) of an AGN, where a different dust geometry could lead to a different effective attenuation curve than that found for nebular emission from Hiiregions. Additionally, for galaxies with contributions to nebular emission from both star formation and an AGN NLR, the use of the Balmer decrement for the correction of NLR emission lines may be somewhat inappropriate. In general it is reasonably assumed that the Hydrogen recombination lines are dominated by the emission from Hiiregions in all but the strongest AGN. However, this is not the case for higher ionization and higher excitation lines where emission from the NLR can dominate. Thus, while the Balmer decrement will show how much dust is in Hii regions, there is no reason to believe that light from the NLR would pass through the same amount of dust[3](#fn3). Therefore the extinction correction applied to these other lines, such as []λ5007Å, may be incorrect. We investigate these effects in Section [sec:bhar]. Attenuation of nebular emission lines ------------------------------------- In order to determine the amount of attenuation suffered by optical nebular emission lines, ideally one would observe a pair of lines whose intrinsic ratio is independent of physical conditions and where one of the pair lies at a wavelength long enough that attenuation is negligible. Hydrogen recombination line ratios that include one line in the NIR wavelength range have been shown to be a powerful tool for constraining the dust attenuation curves for nebular emission lines in galaxies. The data are well fit by a screen-like MW extinction curve, which has since been favoured for the correction of emission lines for dust attenuation. These results suggest that a substantial fraction of the attenuation of emission lines must arise from dust in a foreground screen-like geometry (e.g. the birthclouds in the CF00 model). However, the samples studied to date have been small, containing only local starburst galaxies. Extending this study to a larger sample of galaxies, including more ordinary starforming galaxies and dustier galaxies is the purpose of this paper. Sample sizes using ground based NIR observations are restricted primarily by the difficulty resulting from contamination of observations by OH sky lines. Hubble Space Telescope observations of NIR Paschen-α line have been used to reveal the detailed distribution of star formation in galaxies relatively unimpeded by dust. The detection of dust lanes in some galaxies in this sample indicate some level of extinction still exists even at 1.875μm. Uncertainties in the form of dust-curve in the NIR regime therefore cause a small additional source of uncertainty when using NIR emission lines to constrain dust-curves in the optical. With the launch of the Spitzer Space Telescope, with the Infrared Spectrograph on board, high quality, moderate resolution mid-IR spectra are available for the first time for a large number of low-z galaxies. Extinction in the mid-IR is negligible, except in small wavelength regions centered on 9.7 and 18μm where strong absorption can arise from the stretching and bending modes of amorphous silicate grains. While H recombination lines are found in the Mid-IR, they unfortunately tend to be too weak to be easily measured. Two of the strongest and easily observable emission lines in the mid-IR arise from neon; [Neii] 12.8μm and [Neiii] 15.5μm. In typical Hii regions, over 95% of neon is in these two states (Ne+ and Ne2 +), thus we expect that the sum of these two lines will trace very well the ionized gas, and thus Hα, in galaxies. Indeed, have already suggested that these mid-IR Neon lines could provide a useful new indicator of star formation rate in galaxies. They have shown that [Neii]+[Neiii]luminosity correlates strongly with both total infrared luminosity in galaxies and the NIR Brα emission line in giant Hiiregions. have shown that [Neii]luminosity correlates strongly with that of Paα in star-forming regions in LIRGS. Neon has the added benefit of being an abundant element, especially as it is not depleted onto dust grains. Therefore in this work we use the sum of these two lines as a tracer of the unextinguished emission from ionized gas in these galaxies. Although these lines lie on the edge of the silicate absorption features, in general the extinction caused by this is minimal as we verify in Section [sec:samples]. Combining Spitzer spectra with optical SDSS spectra provides a new and extremely powerful dataset, in which both optical and mid-IR emission lines have been measured much more accurately than was possible in the past. Starburst/Hii region modelling suggests that for typical twice Solar metallicity Hii regions, the ratio L([Neii]+[Neiii])/L(Hα) should be between 0.5-1.0, demonstrating how luminous the mid-IR Ne lines are. The range is caused by ionization effects, arising from different stellar models, the mean age of the ionizing stellar population, and ionizing flux densities. There is one major disadvantage of using the [Ne]-to-Hα ratio to constrain the optical dust-curve in galaxies, in that the intrinsic ratio is directly dependent upon the metallicity of the gas, greatly expanding the range of possible values. For Solar metallicity Hii regions the models predict L([Neii]+[Neiii])/L(Hα) should be smaller by a factor of 0.6. However, we expect that this is not a dominant concern in the samples studied in this paper as we are exploring massive, dusty, and therefore metal-rich, galaxies. We discuss the impact of variations in the intrinsic ratio where relevant. We refer the reader to for further details on the formation of the neon lines in Hii regions. In AGN the ionizing spectrum is much harder than found in star-forming regions, meaning that neon starts to populate higher ionization states. The sum of the mid-IR neon lines still traces the ionized gas very well, but the ionisation is offset to higher values due to the harder radiation field. In addition, the harder radiation field of AGN also create partially (or X-ray) ionized regions, which have a higher L([[Neii]])/L(Hα) ratio. AGN photoionization models suggest L([Neii]+[Neiii])/L(Hα) ∼ 2, a factor of 2-4 higher than Hiiregions [4](#fn4). Again, we will indicate the effect of this result where relevant. The Samples =========== [fig:z] In this section we give the methodology used to measure the nebular emission lines, followed by the selection criteria of our four samples of galaxies. All samples have SDSS optical 5-band imaging and fibre spectroscopy, together with Spitzer IRS spectroscopy. The SDSS specobjid, SDSS-MPA rowindex, RA and Dec of the, ULIRG and Seyfert samples are given in Tables [tab:psb], [tab:ul] and [tab:sey] to allow easy identification of the objects in the catalogues used in this paper. The equivalent information for the SSGSS sample is available online at http://www.astro.columbia.edu/ssgss. For a quick visualisation of the samples: Figure [fig:z] presents their distributions in redshift, stellar mass and Balmer decrement (Hα/Hβ flux ratio), Figure [fig:bpt] shows their positions on the BPT diagram, and in Figure [fig:spoon] we place all our samples on the diagram of equivalent width vs. the strength of the 9.7μm silicate absorption. We discuss these figures further in Section [sec:comparison]. The raw optical emission-line measurements for all SDSS-DR7 galaxies are provided on the SDSS-MPA web pages[5](#fn5) and details are given in and ; errors have been rescaled according to the information provided on the web site. Note that all broad emission line objects have been identified by the SDSS automated pipeline and are not included within the SDSS-MPA sample. Briefly, spectral synthesis models [Bruzual & Charlot (2003); 2008 version using the MILES spectral library, Sánchez-Blázquez et al. (2006)] are used to fit the stellar continuum, including stellar absorption features, and emission lines are measured from the residual of the data and model fit. Therefore underlying stellar absorption is accurately accounted for when measuring optical emission line fluxes. Intervening Galactic extinction is corrected for prior to the calculation of the emission line fluxes. The quality of the fits obtained is excellent. For all samples, IRS spectroscopy was carried out in staring mode and 1D spectra were extracted using the standard Spitzer IRS Custom Extraction (SPICE). For the galaxies the “optimal extraction” option was chosen, which weights the extraction by the object profile and the signal-to-noise of each pixel and is suitable for faint sources. Individual 1D spectra were combined using custom algorithms. To maximise consistency between the heterogeneous samples in this paper, the final low-resolution 1D spectra of all samples were reanalysed using the same custom algorithms and analysis packages where appropriate. To obtain and fluxes for the SSGSS and Seyfert samples we used the mid-IR spectral fitting package PAHfit which fits Gaussian profiles with widths set according to the instrument module. PAHfit was run in the rest-frame using a screen-like dust model, and although dust attenuation is tiny at the wavelengths of the Ne lines, for complete consistency between samples we removed the correction applied by the code. Our final results are unchanged whether the minimal dust correction is applied or not. Where high resolution spectroscopy is available, we use the high-resolution measurements of [Neii] and [Neiii] due to the smaller error bars. For the sample, the [Neii] and [Neiii] emission line fluxes were measured interactively from the short-high (SH) IRS spectra using the 2-step baseline and Gaussian line-fit module from the SMART IDEA package [6](#fn6). For the ULIRG sample, fluxes were taken from high resolution spectroscopy, given in Table 4 of. To ensure consistency between the four samples, where possible[7](#fn7) we verified that [Neii] and [Neiii] fluxes measured with PAHfit or directly from the low-resolution spectra for the and ULIRG samples were equal within the errors to those measured from the high-resolution spectra (with and without the minimal correction for attenuation applied in PAHfit). For the Seyfert sample, fluxes were found to match within the errors those measured in the high-resolution mode, given in Table 6 of. was measured from high-resolution (LH) IRS spectroscopy in all cases (measurement from the low-resolution spectra is not possible due to a nearby FeII line). For the sample, line fluxes were measured interactively using the SMART IDEA package as described above. For the ULIRG and Seyfert samples line fluxes were taken from the tables in Armus et al. (2007) and respectively. One of the major components of a star-forming galaxy spectrum in the mid-IR is the broad emission features generally accepted to result from the vibrational modes of polycyclic aromatic hydrocarbons (PAHs). While we do not study the PAH features in detail in this paper, PAHfit outputs their fluxes and equivalent widths which we use to provide further clarification of the nature of our objects where necessary. To visualise the different dust and AGN properties of our different samples in Section [sec:comparison], we measure the equivalent width of and the strength of the 9.7μm silicate absorption. We attempted in both cases to follow closely the methods as described in, and we used the ULIRG sample to verify that our results are consistent with those given in over the full range of the measurements. The SSGSS sample ---------------- Our control sample is the Spitzer-SDSS-GALEX Spectroscopic Survey, a Spitzer Legacy survey (P.I. D. Schiminovich) which targeted 100 “ordinary” galaxies from the SDSS spectroscopic survey and GALEX surveys for follow-up mid-IR spectroscopy with the Infrared Spectrograph using the Spitzer Space Telescope. The galaxies are magnitude limited to the SDSS spectroscopic magnitude limit of r < 17.7 and additionally were selected to have GALEX-NUV fluxes > 1.5mJy for the faint sample (40 galaxies) and > 5mJy for the bright sample (60 galaxies). They were chosen such that their mass, colour, star formation rate and redshift distribution are representative of galaxies in the SDSS survey. A description of the survey, together with data tables containing SDSS and mid-IR measurements and derived parameters, are available at the team website[8](#fn8). In this paper we make use of the IRS low resolution spectroscopy of 80 SSGSS galaxies. The IRS spectra of the remaining galaxies are either incomplete due to problems during observations or reductions, or suffer from considerable noise at short wavelengths, 5 of these caused by a solar flare. A further 4 galaxies with median per-pixel-SNR < 10 in their SDSS spectra are removed, as the spectra were of insufficient SNR to allow accurate stellar continuum fitting around the Balmer emission lines. Further details of the SSGSS can be found in and. The dusty Balmer-strong sample ------------------------------ Our first sample of galaxies with significant dust contents are 12 galaxies selected from the SDSS-DR4 to have an observed Hα/Hβflux ratios greater than 8 (where a ratio close to 3 indicates little dust) and optical line ratios that indicated the possible presence of an AGN. Additionally, they were selected to have strong Balmer absorption lines, which might indicate a recent sharp decline in star formation rate (“post-starburst”). However, the strong Balmer absorption lines might alternatively be caused by unusual dust properties rather than star formation history. This is the subject of a future paper. Both high- and low-resolution IRS spectra were obtained for all of these 12 galaxies during Spitzer Cycle-4 (P.I. T. Heckman). Further observational details of this sample are provided in Appendix [app:psb]. One of the 12 galaxies was subsequently discovered to have double peaked narrow Hα and lines, together with significant underlying broad emission which was not accounted for in the original emission line measurements. Its IRS spectrum is dominated by emission from the AGN. Attempts to robustly remeasure the Hβ line accounting for the double peak were unsuccessful due to significant stellar absorption, and its Ne/Hα and PAH/Hα ratios suggests that it is not in fact unusually dusty. We therefore remove it from all further discussion, leaving 11 galaxies. The ULIRGS ---------- Our second sample of dusty galaxies is composed of the 10 ULIRGs ($\rm L\_{IR}>10^{12}L\_\odot$) from the IRAS Bright Galaxy Sample ; the Spitzer IRS low- and high-resolution observations of these galaxies have been presented in detail in. Of the 10 BGS ULIRGs, 6 are found in the SDSS spectroscopic sample [IRAS15327+2340 (Arp220), IRAS 08572+3915, IRAS 15250+3609,IRAS 09320+6134 (UGC 5101),IRAS 12112+0305], 3 do not lie within the SDSS footprint (IRAS 05189-2524, IRAS 22491-1808, IRAS 14348-1445), and 1 is optically too faint to have been targeted [IRAS 12540+5708 (Mrk 231) has an SDSS r-band magnitude of 19.4]. All 6 of the ULIRGs with SDSS spectra are late stage mergers. The Seyfert sample ------------------ In Section [sec:bhar] we compare the high ionisation emission lines in the dusty galaxies with those of a sample of relatively dust-free (low Balmer decrement) Seyfert galaxies. This is composed of the top 20 λ5007 flux emitters out of all SDSS-DR4 main-sample galaxies classified as Seyfert galaxies based on their position on the BPT diagram. They have fluxes greater than 4 × 10− 14erg/s/cm2, corresponding to a luminosity of ∼ 1041erg/s at the median redshift of the sample. Both high- and low-resolution IRS spectra were obtained, with high signal-to-noise ratio. Further details of the Seyfert sample, along with a study of their X-ray properties can be found in and. Sample comparison ----------------- [fig:bpt] [fig:spoon] The samples that we will compare and contrast in this paper were targeted for different purposes and thus any conclusions drawn must take account of potential systematics caused by their different selections. The redshift distribution of the 4 samples is shown in the left panel of Figure [fig:z]. The black line shows the redshift distribution of the SDSS spectroscopic galaxy catalog with r-band extinction corrected petrosian magnitude[9](#fn9) in the range 14.5 < r < 17.77. The Seyfert and ULIRG sample are flux limited in nebular emission and IR flux respectively, and lie predominantly at low redshifts. The SSGSS galaxies were selected to be representative of the full SDSS sample, and therefore their redshift distribution tracks closely that of the full SDSS survey. The sample is limited to z < 0.07 to ensure that the stellar population sampled by the optical fibre through which the spectrum is taken lies within the central region of the galaxy close to the AGN. The stellar mass distribution of the 4 samples is shown in the central panel of Figure [fig:z]. The stellar masses[10](#fn10) have been estimated from stellar population model fits to the optical 5-band SDSS photometry, using a Bayesian analysis similar to that described in and. Most samples cover the full range in stellar mass probed by the SDSS, although the galaxies do not cover the low-mass end of the distribution. Note that the histograms are not corrected for incompleteness effects due to sample or survey flux limits, and therefore are not suitable for drawing conclusions about the underlying distributions of galaxy mass. To aid the reader, the position of the galaxy mass bimodality is marked by a downward arrow. The right hand panel of Figure [fig:z] shows the Balmer decrement distributions of the 4 samples, showing how the and ULIRG samples are significantly dustier on average than the SSGSS and Seyfert samples. In Figure [fig:bpt] we show the position of the four samples on the BPT diagram, with the underlying grayscale showing the distribution of spectroscopically observed galaxies in the SDSS. Only those galaxies with SNR > 3 in all four emission lines are plotted. This is the most commonly used optical diagram for diagnosing the presence of an AGN. Throughout this paper, we distinguish 4 classes of galaxies: (i) AGN-dominated, with emission-line ratios which place them above the stringent theoretical starburst criterion of, or have /Hα > 0.0; (ii) star-formation dominated, which lie below the observationally determined demarcation line of, or have /Hα < − 0.5; (iii) composite objects, which lie between the AGN-dominated and star-forming dominated samples; we term them “composite” to indicate that the emission may be caused by a combination of both AGN and starformation ; and (iv) unclassified objects with non-existent or weak (SNR < 3σ) emission lines which do not allow classification into the first three samples. We note that the BPT diagram does not provide a firm demarcation between the samples and that the demarcation lines are not motivated by robust theoretical predictions; we use the lines only to provide a convenient broad classification. present up-to-date modelling of the “maximum starburst” demarcation line, placing it much closer to the observed star forming branch in the SDSS than the line. Shocks, high metallicity, and ionisation by old stars can also cause galaxies to lie in the “composite” class. We note that most of the galaxies have detectable molecular H2 lines, as do the majority of ULIRGs. In the case of ULIRGs these lines are thought to arise primarily from photodissociation regions (PDR) associated with star formation. The H2/PAH flux ratio of the galaxies is exactly that expected for star forming galaxies (Ogle et al. in prep.), indicating that shocks are similarly not a dominant excitation mechanism in the galaxies. This suggests that the high /Hα line ratios in these samples are in general caused by a central AGN, rather than shocks. Signal-to-noise ratio limits placed on emission lines may preferentially cause very high-metallicity galaxies and LINERs caused by shocks and/or ionisation by old stars to be “unclassified” in the SSGSS sample. However, none of these caveats affect the results of this paper which do not rely on a precise identification of weak AGN, or on the precise positioning of pure AGN or starforming galaxies on the BPT diagram. Where possible we include unclassified SSGSS galaxies in the figures. Out of the 80 SSGSS galaxies, 4 are AGN-dominated, 21 are composite objects, 35 are star-forming and 20 are unclassified. Out of the 11 galaxies 2 are AGN-dominated and the remainder composite objects. 3 ULIRGs are AGN-dominated (Arp220, Mrk273 and UGC5101) and 3 are composite objects (08572, 12112, 15250). All Seyferts clearly lie in the AGN dominated region by definition. Figure [fig:bpt] supports our assumption that the intrinsic L([Neii]+[Neiii])/L(Hα) ratio will not vary greatly between our samples. Most galaxies in the samples lie in the solar and greater region of this line ratio diagnostic diagram. Finally, in Figure [fig:spoon] we follow and plot the equivalent width (EQW) of the 6.2μm PAH feature against the strength of the 9.7μm silicate absorption. We plot only those objects with mean SNR-per-pixel > 2 in the wavelength regions used to define the continuum for the measurement of the silicate absorption at 9.7μm and with total flux/error > 3 in the 6.2μm PAH feature. The lower branch is interpreted as a mixing between AGN and starformation activity, and the upper branch as an evolution from a dust obscured, to dust unobscured nuclear starburst or AGN (see Rowan-Robinson & Efstathiou 2009). Because of its focus on both dust and AGN activity, Fig. [fig:spoon] is particularly useful for characterising our 4 different samples: Galaxies in the SSGSS occupy the bottom right corner of the diagram as expected for “ordinary” star-forming galaxies or weak narrow-line AGN. They have strong EQW() and weak silicate absorption. The composite galaxies in this sample have a lower mean EQW() than the star-forming galaxies, strengthening the argument that these galaxies contain an AGN. The large scatter, and significant number of objects with apparent silicate emission, is indicative of the generally lower quality of these spectra compared to our other samples. SSGSS galaxies with optical emission lines which are too weak to classify them on the optical BPT diagram do not differ in mid-IR properties from their classified counterparts. The galaxies generally have stronger Silicate absorption than SSGSS galaxies, placing them between the ULIRG and SSGSS samples and therefore, within the common interpretaion of the diagram, at the end of their evolution from a dusty starburst. M82 is the prototype galaxy for this region of the diagram. The ULIRGS are a mixed group, as has been discussed at length in the literature. Although our sample is small, it is interesting to note that the ULIRGs with particularly strong silicate dust absorption (08572, 15250), show optical emission line ratios indicative of a mixture of star formation and AGN and/or shocks. Three out of the four ULIRGs with weaker silicate absorption (Arp220, Mrk273, UGC5101), have AGN-dominated optical emission (i.e. implying little contribution from star formation, except possibly in the case of extreme shocks). This is consistent with the interpretation of the upper branch as an evolution from dusty starburst galaxies in the top-left to galaxies undergoing less intense, less dusty star formation in the bottom-right, with dust-obscured AGN activity occurring throughout the evolution. The Seyferts are generally found to have very weak PAH equivalent widths and little Silicate absorption, and therefore lie in the region of the diagram commonly interpreted as being occupied by strong AGN which heat the dust to high temperatures. 2/20 show some silicate absorption together with stronger EQW(). The median (mean) BPTy values of the 4 Seyferts with EQW() > 0.2 are 0.78 (0.75), compared to 1.01 (0.99) for the remainder of the sample. This suggests that emission from star formation contributes to the optical emission lines even in the Seyfert branch of the BPT diagram. These results are consistent with the interpretation of the lower branch of Fig. [fig:spoon] as mixing between AGN and starformation activity. Aperture Corrections -------------------- “Aperture bias” describes the effects caused by the fact that a fixed aperture or slit width may probe only a small fraction of a galaxy’s total light, and this fraction is a function of galaxy radius and redshift. The physical size probed by the SDSS 3 fibre varies from under 1kpc for the majority of the ULIRG sample to around 10kpc at the maximum redshift of the SSGSS galaxies. Fortunately the results in this paper rely upon line ratios or equivalent widths rather than absolute luminosities, and therefore aperture bias in this usual sense is less relevant. However, comparing line strengths from the IRS spectra with those from the SDSS spectra leads to other aperture related issues. The finite (and different) slit and fibre apertures may mean that the IRS and SDSS spectra sample a different fraction of the same galaxy, and this will cause measured line ratios to differ from the true values. There are two important issues: (i) light is lost from the slit/fibre due to the PSF; (ii) light is lost from the outer regions of extended sources, depending on the size of the aperture and the accurate positioning of the aperture on the source. In this paper the precise flux calibration between the two datasets is relatively unimportant, but it is still important that we minimise any additional scatter in the observed relations. To help the reader to understand the limitations of our data, we outline here the flux calibration procedures of the mid-IR and optical spectra. The IRS PSF increases substantially from the blue to the red wavelength extremes of the instrument, and therefore each IRS slit has a different width designed so as not to loose too much of the light at the reddest ends[11](#fn11). The spectral extraction procedure accounts for the varying PSF along the slit by normalising the flux at each wavelength according to the fraction of light missed from a calibration stellar point-source. The SDSS DR7 spectra have been spectrophotometrically calibrated using the PSF magnitudes of stars observed on the same plate as the galaxies. To aid aperture corrections, the SDSS-MPA emission line measurements have been recalibrated to the 3 fibre magnitudes. Therefore, to match the stellar PSF-normalised mid-IR spectra, we have removed this recalibration, returning the flux level of the SDSS emission lines to the stellar PSF-normalised values. The use of stellar PSF spectrophotometric calibration for both optical and mid-IR spectra allows good relative calibration for point sources, but inaccuracies will still arise in the case of extended sources. For centrally concentrated sources, such as circumnuclear starbursts in ULIRGs or AGN NLR emission, the point-source approximation is a good one. However, for extended sources, the measured mid-IR to optical line ratios may depend on the relative size of the apertures, and the relative surface brightness distribution of the sources. Both effects are difficult to correct for accurately, especially in the case of radial differences within the sources (for example, in AGN vs. starforming nebular emission). We have therefore not attempted any further aperture correction, and will caution the reader at points in the text where this effect may be relevant. Results ======= [fig:neha] [fig:neharesid] In Figure [fig:neha] we plot Hα vs. + (from now on [Ne]) luminosities for all of the samples, firstly with Hα uncorrected for dust attenuation, and then corrected with five different dust-curves selected from the literature to cover the full range in possible greyness. The panels are ordered by increasing greyness of the dust-curve used, from the screen-like extinction-curves (SMC and MW), to galaxy continuum derived attenuation-curves (Calzetti and λ− 0.7). The SMC curve was evaluated from the tabulated results of with an RV = 3.1; the MW curve was taken from with an RV = 3.1; the μ0.3 curve is from ; the Calzetti curve is from with RV = 4.05; the λ− 0.7 curve is from CF00 as discussed in Section [sec:review]. In each panel a maximum-likelihood best-fit linear relation is fit to the SSGSS-starforming sample allowing for errors in both x and y axis quantities and fixing the intercept to be zero (dashed line). Twice and half the best-fit [Ne] luminosity for each Hα luminosity is indicated as dotted lines to aid comparison between the panels of the scatter between the two quantities. The best-fit gradient (g, equal to the best-fit intrinsic [Ne]/Hα ratio) and Pearson’s correlation coefficient (r) are given in the bottom right of each panel, both evaluated using the SSGSS-starforming sample only. In panel a it is immediately evident that without correcting Hαfor dust attenuation the scatter between Hα and [Ne] emission luminosities is large, even for the SSGSS galaxies with normal levels of dust. For the dusty and ULIRG samples, there is a clear excess of [Ne] for their observed Hα luminosity as expected for dusty galaxies. As we move through the dust laws, from least to most grey [panels b to f], we see steady trends. The best-fit intrinsic [Ne]/Hα ratio monotonically decreases, and the correlation coefficient is maximal for the MW, μ0.3 and Calzetti dust-curves. Focussing on the dusty and ULIRG samples, we notice that these move from lying above the relation defined by the SSGSS-starforming galaxies in panels a and b, to lying below the relation in panel f. This is caused by their under- and over-correction for dust attenuation using the different dust-curves. In Figure [fig:neharesid] we show the samples more clearly by plotting the residual of the [Ne] luminosity from the linear fit using the MW, μ0.3 and Calzetti dust curves. The mean and variance of each sample in these figures is informative for comparing the different dust curves. For the dusty ULIRGs the mean is 2.0, 1.6, 1.5 for the MW, μ0.3 and Calzetti dust-curves respectively i.e. they are significantly offset for all dust-curves (see below for further discussion). For the sample, the mean is 1.1, 0.9 and 0.8, suggesting perhaps that a dust curve in between the MW and μ0.3 is most appropriate but any offset between this extreme dusty sample and ordinary starforming galaxies is very small. For the SSGSS starforming sample, the mean is 1 by definition for all dust-curves. The variance is ∼ 0.15 for all dust-curves, i.e. the scatter is less than a factor of 2 and there is no strong trend with Hα luminosity for the starformation dominated galaxies. From these figures we can identify the expected offset of the AGN caused by a higher intrinsic [Ne]/Hα ratio. For the Seyfert sample and μ0.3 dust curve the mean offset is 1.26, i.e. about a 25% enhancement in [Ne] for their Hα luminosity. The magnitude of this offset does not correlate with position on the BPT diagram or PAH equivalent width, although the size of the sample is too small to rule out the possibility that those with a smaller offset have a greater contribution to star formation in their lines. Averaged over the samples, the offset is seen only for the Seyfert galaxies, and the three SSGSS-AGN which lie in or close to the Seyfert region on the BPT diagram. The SSGSS composite sample have a mean of 1 for all dust-curves, which suggests that the AGN in general does not contribute significantly to the Ne/Hα line ratio in these composite galaxies. However, the trend upwards at low Hα luminosities is driven by composite objects, suggesting there may be a small effect at low star formation rates. The observed average enhancement in the [Ne]/Hα ratio of the ULIRGs noted above, could be due to small number statistics, the dust curve (i.e. a shallower dust curve would be more appropriate), or the presence of deeply buried AGN. We note that the ULIRGs are only offset by at most a factor of 3, i.e. dust corrected Hα luminosity tracks mid-IR [Ne] luminosity to better than a factor of 3 even in extreme dusty and disturbed objects. We will return to these points below. The maximum likelihood fits provide a measure of the dust-free ratio of [Ne] to Hα luminosity in our IRS/SDSS datasets, when star formation is the dominant excitation mechanism for the line emission: $${\rm L[Ne]} = g\times{\rm L(H\alpha)\_{corr}}$$ where L[Ne] = L()+L() and g = 0.89, 0.78, 0.75 for the MW, μ0.3 and Calzetti dust-curves respectively. While it is encouraging that the measured intrinsic ratio lies exactly at the center of the range predicted by the models for twice Solar metallicity Hii regions (see Section [sec:nir]), a precise comparison between models and data would only be possible after much greater care has been taken to perform precise aperture corrections. In the case of the AGN, the emission line source is point-like and therefore the relative calibration between the mid-IR and optical spectra should be the most accurate. The fact that the average [Ne]/Hα ratio is a little lower than predicted may be due to contamination by star formation in a subset of the Seyfert sample (see Section [sec:comparison]). In the following subsection we study the differences between the dust curves in more detail. Comparison of common dust attenuation laws ------------------------------------------ [fig:dustlaws] In Fig. [fig:dustlaws] we plot the observed [Ne]/Hα vs. Hα/Hβluminosity ratios for all of our samples. In the left-hand panel we use a logarithmic scale to allow the dustiest ULIRGs to appear. In the right-hand panel we focus on the less extreme objects, and only include galaxies where their optical emission line ratios indicate starformation is a significant emission line source (classified as composite or starforming in the optical BPT diagram, Fig. [fig:bpt]). Overplotted as black lines are the predicted line ratios from the different dust curves presented in the previous subsection. In each case the intrinsic [Ne]/Hα ratio is set from the fit to the SSGSS-starforming galaxies as described above. Although there are several factors which can move galaxies within this diagram, dust has the strongest effect, increasing both the Balmer decrement and observed [Ne]/Hα ratio by an order of magnitude over the range probed by our samples. The different dust-curves predict different relations between these two line ratios, even when we allow the intrinsic [Ne]/Hα ratio to be a free parameter fit by the data. As shown in the previous section, the MW, μ0.3 and Calzetti dust-curves provide the best fits to the SSGSS-starforming galaxies. Figure [fig:dustlaws] shows that the μ0.3 and Calzetti attenuation curves also pass through the center of the sample. The scatter of the SSGSS-starforming galaxies about the best-fit dust-curves is large, and this may have several origins. Firstly, the errors on the line flux measurements, particularly in Hβ are significant. Secondly, we expect some variation due to different metallicities. The grey shaded region shows the range predicted by the theoretical models for twice Solar Hii regions, and the arrow in the left hand panel shows how this region moves for Solar Hiiregions. The 3 galaxies with the lowest measured [Ne]/Hα ratio lie significantly to the left of the 2Z⊙ prediction, but within the Z⊙ prediction. These 3 galaxies also have some of the highest /Hβ and [Neiii]/[Neii] ratios of the sample, indicative of harder ionisation perhaps resulting from lower metallicity star formation. The single Seyfert galaxy with a [Ne]/Hα ratio less than 1 has the lowest /Hα ratio of the whole sample, unusually low for Seyfert galaxies, and likely associated with accretion of low-metallicity gas. While low metallicities can explain the scatter of SSGSS-starforming galaxies to the left of the dust-curve predictions, the scatter to the right is unlikely to be caused by metallicity variations as scatter in this direction requires significant and unlikely neon abundance increases (i.e. greater than twice solar). This leads us to the third, and most interesting, cause for scatter in the line ratio relation: intrinsic differences in the type or distribution of dust in galaxies[12](#fn12). In a 2-component dust model, increasing the fraction of dust in the diffuse ISM relative to the dense birthclouds increases the contribution from scattering and makes the dust-curve greyer. In the CF00 emission line dust model given in Eqn. [eqn:w07] this relative amount of diffuse vs. dense dust is controlled by the parameter μ. By increasing μ to 0.9 (i.e. 90% of the effective optical depth from the dust comes from the ISM, and 10% from the stellar birthclouds), we can describe the lower envelope of the SSGSS galaxies. While this qualitative result is suggestive, a further detailed study including orientation, metallicity and ionisation effects would be required to confirm that these galaxies do indeed have a significantly different dust geometry. It is evident that the 3 ULIRGs classified as composite galaxies by their optical line ratios also appear to favour a greyer dust curve than the dusty galaxies. However, as we discuss further below, it is also plausible that the high Ne/Hα ratio observed in these ULIRGs, as well as in some SSGSS-composite galaxies, is simply due to a deeply buried AGN. We summarise the results up to this point by noting that a single appropriate dust curve, combined with an accurately measured Balmer decrement, can recover the intrinsic optical emission line luminosities to better than a factor of two in the majority of ordinary starforming and dusty galaxies. Our data shows that this remains true for galaxies with dust contents up to an AV of 4.4. However, while the use of a MW, μ0.3 or Calzetti dust-curves for large samples of galaxies will cause the least overall bias in the final dust-corrected emission line luminosities, the effect of a greyer dust-curve in individual galaxies should not be understated. A galaxy with a very typical measured Hα/Hβ ratio of 4.5 would have its Hα luminosity, and thus star-formation rate, corrected by a factor of around 2.5 using the average dust curve, whereas a factor of 7 would be more appropriate if the majority of its dust were in the diffuse ISM. Fitting the dust curve ---------------------- Throughout the remainder of the paper we will focus on results using the CF00 μ dust-curve as it allows greater flexibility than the traditional dust-curves in the calculation of errors, and the discussion of variations in dust-curves within the sample as presented in the previous section. Additionally, it is the only dust-curve specifically motivated and justified for the correction of nebular emission lines. The MW dust-curve assumes a screen-like dust distribution, and the Calzetti dust-curve is intended for correcting the continuum light from galaxies. Rather than assuming μ = 0.3 as derived by CF00, we can fit the [Ne] vs. Hα luminosities of the SSGSS-starforming galaxies to obtain the best-fit μ and intrinsic [Ne]/Hα ratio together with errors. We use a maximum-likelihood linear fit, accounting for errors on both quantities, and fix the intercept to be zero[13](#fn13). Formal, statistical errors on the derived μ are at the level of 5%. The errors obtained from bootstrap resampling with replacement of the data are more significant, and we quote these errors below, and use them to obtain errors on the dust correction formulae presented in Section [sec:sfr] and [sec:bolcorrn]. We find μ = 0.4 ± 0.2, i.e. on average 40% of the optical depth at 5500Å arises in the ISM of the galaxies. This is consistent within the errors with the μ = 0.3 derived by CF00 for their sample of local starburst galaxies. The intrinsic [Ne]/Hα ratio fit at the same time is g = 0.6 ± 0.1. The resulting curve is shown as the solid gray line in Figure [fig:dustlaws] and given explicitly as: $$\label{eqn:w10a} \frac{\tau\_\lambda}{\tau\_V} = \frac{A\_\lambda}{A\_V} = 0.6(\lambda/5500)^{-1.3} + 0.4(\lambda/5500)^{-0.7}$$ where λ is in Å. Given a measured Hα/Hβ flux ratio, the optical depth at any wavelength can be obtained from: $$\label{eqn:w10b} \tau\_\lambda = 3.22\ln\left(\frac{2.86}{f(H\alpha)/f(H\beta)} \right) \frac{\tau\_\lambda}{\tau\_V}$$ where τλ/τV is given in Eqn. [eqn:w10a] Aλ = 1.086τλ is the attenuation in magnitudes, and a Case B recombination ratio of 2.86 is assumed. Dense vs. diffuse dust ---------------------- [fig:dust] Extinction in the mid-IR continuum is dominated by the stretching and bending modes of amorphous silicate grains with two strong features at 9.7μm and 18μm. Importantly for this section, it is evident from the many theoretical models that any significant silicate absorption, as found in the ULIRGs and many of the galaxies, implies optical extinction at levels of τV ∼ 50 − 100. It is therefore impossible for ionising UV continuum and associated line radiation to pass through these clouds directly. In Figure [fig:dust] we plot the Balmer decrement [Hα/(B × Hβ) where B is the unattenuated ratio] against the 9.7μm Silicate absorption strength for all four samples. For Hiiregions we set B = 2.86 for Case B recombination at 10,000K and electron density of 102 − 104/cm− 3. For AGN we set B = 3.1. Surprisingly, we see a correlation between these two measures of dust content for most galaxies in our samples. However, it is clear that they do not trace the same dust because the τV measured from the Balmer emission lines (upper axis) is an order of magnitude lower than that indicated by the mid-IR silicate absorption strengths (right-hand axis, from zero-age starburst models of Rowan-Robinson & Efstathiou 2009). The dotted line is the best-fit squared relation to the SSGSS-starforming, composite and galaxies: $${\rm Sil}\_{9.7{\ensuremath{\mu\mbox{m}}}} = -0.17\left(\frac{f({\ensuremath{\mbox{H}\alpha}})}{B f({\ensuremath{\mbox{H}\beta}})} -1\right)^2$$ In the case of starforming galaxies without a strong AGN, the mid-IR continuum predominantly arises from dust grains situated within the Hii regions re-radiating the light from young and hot stars. It is this dust continuum that is absorbed by the Silicate grains along the line-of-sight between the light source and the observer. Balmer emission lines also come from ionised gas within Hii regions. Thus, the observed correlation between the Silicate absorption strength and Balmer decrement implies either: (i) a correlation between the amount of dust in dense stellar birth clouds which cause the Silicate absorption, and the amount of dust in more diffuse stellar birth clouds which cause the absorption of the optical emission lines; or (ii) a self-similar structure for Hii regions in which the Balmer lines that reach the observer have passed through a fixed fraction of the dense dust clouds. There are two notable exceptions where the optical Balmer decrement fails to predict the presence of an extremely significant attenuation in the mid-IR: the ULIRGs 15250 and 08572. find that both of these ULIRGs have significant contributions from AGN to their bolometric luminosity (53 and 87% respectively), using a spectral decomposition between 5 and 8μm. Models have shown that extremely strong silicate absorption, as observed in these two ULIRGs, requires a steep temperature gradient within the obscuring cloud, which can only be obtained where the light source responsible for heating the dust that causes the mid-IR continuum is deeply buried in a smooth thick shell of dust that is both geometrically and optically thick. This is again consistent with the source being an AGN, although a nuclear starburst may suffice. A clumpy distribution of dust leads to much shallower absorption or emission. Our emission line analysis is also consistent with these galaxies harbouring a buried AGN, they have an increased [Ne]/Hα ratio and lie in the composite region of the BPT diagram. While the position of these galaxies on the BPT diagram is not in itself conclusive proof that these two ULIRGs harbour an AGN, if their position is caused by the AGN then it indicates that ionising radiation from the AGN escapes to ionise the NLR, despite the dense dust cloud that presumably surrounds the central continuum source. Two dusty Seyfert galaxies and one ULIRG with a Seyfert nucleus (Mrk273) also lie above the relation defined by the SSGSS and galaxies, further supporting the idea that dust along the line-of-sight to a deeply buried AGN can lead to stronger silicate absorption, relative to nebular line absorption, than normal Hiiregions. We are not aware of any models which simultaneously predict the strength of silicate absorption and attenuation of the optical emission lines. Qualitatively Figure [fig:dust] suggests a two component model is required to describe the dust distribution in galaxies/AGN. The lower branch is consistent with silicate absorption arising in the clumpy distribution of stellar birthclouds, with the densest clouds causing some silicate absorption, and the Balmer emission lines originating from the same star formation regions but further out in the clouds. The upwards scatter of some sources from this branch is consistent with the dust being concentrated in a dense shell-like distribution obscuring a single nuclear source, most likely an AGN in the cases studied here, but also conceivably a strong nuclear starburst. Galaxies with an AGN -------------------- [fig:oiiioiv] As explained in Section [sec:nir], a dust curve and accurate Balmer decrement that adequately corrects Hii region emission lines for dust attenuation, will not necessarily do the same for lines from the NLR of AGN. Particularly in composite objects, a higher fraction of Balmer lines will arise from within the Hii regions of the galaxy compared to lines such as which are stronger in NLRs. Therefore a form of differential extinction might arise between the Balmer and higher ionisation lines, and a corresponding bias in the correction of AGN NLR lines for dust using the measured Balmer decrement. Additionally, the form of the dust curve may not be the same for light originating from Hii regions and from an AGN NLR, due to the different relative geometry between the light source and the dust cloud, and the different balance between dense and diffuse dust attenuating the lines. The mid-IR line at 25.89 μm has been shown to be an accurate indicator of AGN power, and the / ratio a possible measure of AGN attenuation. In this section we test the accuracy of dust-corrected as a measure of AGN luminosity in our sources by comparing to the mid-IR line. In Figure [fig:oiiioiv] we plot dust-uncorrected and dust-corrected vs. for the Seyferts, the 3 galaxies and the 2 ULIRGs with measured lines. Note that the correlation coefficient is similar for the Seyferts whether the dust correction is performed or not. These Seyferts have very small Balmer decrements so the dust correction has only a small effect. Because some of these objects are classified as composite AGN-starformation by their optical line ratios, and is also produced by starformation, in the right hand panel we apply a correction for this following the method of, based on the expected /Hα ratio for metal-rich starforming galaxies[14](#fn14). is also produced in Hii regions, although the contamination is expected to be much smaller than for due to the higher ionisation level. Clearly, there is a tight correlation between dust-corrected and for the Seyfert sample. All of the dusty galaxies and ULIRGs with measured fall within a factor of 3 of the relation defined by the Seyferts. Only the upper limit for Arp220 is marginally inconsistent with the relation defined by the Seyfert galaxies. Taking the remainder of the galaxies and stacking the SH spectra in luminosity units shows a clear line, which we fit together with the neighbouring Feii line using a double Gaussian. The corresponding stacked - luminosity also lies exactly on the Seyfert relation, once contamination of from star formation has been accounted for. Errors are estimated from the standard deviation of the luminosity on removing one spectrum at a time from the stack. This strong correlation between and dust corrected using the observed Balmer decrement and dust-curve appropriate for Hii regions, for even dusty composite galaxies, shows that there is no evidence for significant differential extinction between lines emitted from the NLR and Hii regions of a galaxy. Although slightly surprising, this may imply that the majority of dust attenuation suffered by NLR emission lines comes from the diffuse dust and Hii regions in the intervening galaxy, through which they must pass on their way to the observer. Unfortunately neither of the two ULIRGs with extreme silicate absorption (08572 and 15250) have measured, so we cannot test directly whether optical and mid-IR NLR lines agree even in the presence of extreme, dense obscuration of the central light source. Such a study may be possible with the large samples of ULIRGs now available in archives. Discussion ========== The potential of dust attenuation and complicated dust geometries to render useless optical observations of extreme galaxies is sometimes used to justify rest-frame IR observations. In the previous sections we have used the correlation between the unattenuated mid-IR and attenuated optical nebular emission lines to argue that a single dust curve can accurately correct optical emission lines for dust attenuation in a large range of different galaxies. The correction is accurate to within a factor of 2 in most cases, and to at least a factor of 4 for even the most heavily obscured ULIRGs. Here, we discuss the implications of our work for deriving SFRs and black hole accretion rates from the optical emission lines Hα,, and. We present useful relations for dust corrections at the wavelengths of each of these lines. Star formation rates -------------------- [fig:atten] The left and right panels of Fig. [fig:atten] show the fractional attenuation (in flux units) as a function of observed Balmer decrement at Hα and, for each of the different dust curves discussed in Section [sec:results]. Using the best-fit μ0.4 dust curve, the recommended correction at Hα is: $${\rm L[H\alpha]\_{corr}} ={\rm L[H\alpha]\_{obs}}\left(\frac{\rm (H\alpha/H\beta)\_{obs}}{B}\right)^{2.67\_{-0.32}^{+0.40}}$$ and at [15](#fn15): $${\rm L[OII]\_{corr}} ={\rm L[OII]\_{obs}}\left(\frac{\rm (H\alpha/H\beta)\_{obs}}{B}\right)^{4.89\_{-0.29}^{+0.36}}$$ where B is the unattenuated ratio as discussed above and errors are propagated from the errors on the best-fit μ. While for Hiigalaxies and strong Seyferts the choice of B is clear, for composite galaxies the unknown B may cause small systematic uncertainties. However as Hii regions have relatively stronger Hαemission, a value of B = 2.86 is probably close to reality. To derive a star formation rate from optical nebular emission lines, two steps are required. This paper has focused on the first step: to correct the line for dust attenuation. The second step is to convert the dust-corrected line luminosity into a star formation rate. While our results show that dust-corrected Hα luminosity traces [Ne] luminosity, this is not enough to confirm the reliability of Hα as an indicator of star formation rate in extremely dusty objects. This is because dense dust clouds can have high optical depths even in the mid-IR, as indicated by the strong Silicate absorption in some galaxies, and thus even the neon lines may be attenuated in some objects. In addition, the most extreme compact, dusty galaxies may even have a non-negligible fraction of their ionizing photons absorbed by dust and not gas (and thus not traced by Hα). Encouragingly, Ho & Keto (2007) have already shown how [Ne] luminosity correlates strongly with both the total infrared luminosity and Brackett-α luminosity in star-forming galaxies. Their galaxy sample includes a handful of ULIRGs, showing that these relations hold even at extreme luminosities. This result provides circumstantial evidence that the neon luminosity is tracing star-formation even in the dustiest of galaxies. In contrast, Farrah et al. (2007) find [Ne] luminosity in ULIRGs to be deficient by 0.4dex from the Ho & Keto relation, which they ascribe to extinction of the [Ne] lines. In a companion paper we plan to address this issue directly by comparing the star formation rates derived from the full SEDs, with that derived from the emission lines. Black hole accretion rates -------------------------- As with star formation rate, the derivation of black hole accretion rates (BHAR) from nebular emission line luminosities requires two steps: correction for dust emission discussed in this paper, and then conversion into BHAR via a bolometric correction. For the best-fit μ0.4 dust curve, the relevant correction for dust attenuation at is: $${\rm L[OIII]\_{corr}} ={\rm L[OIII]}\left(\frac{\rm (H\alpha/H\beta)\_{obs}}{B}\right)^{3.55\_{-0.32}^{+0.40}}$$ where B is the unattenuated ratio as discussed above. The correction caused by the different dust curves is presented in the central panel of Figure [fig:atten]. The exponent for the μ0.4 dust curve is larger than that used in some AGN literature and can lead to significant differences in estimated bolometric AGN luminosity and accretion rates as a fraction of $L\_{\rm Edd}$. Using a smaller exponent than appropriate will cause an underestimate in $\_{\rm corr}$ which will vary with dust content. For example, for the relatively dust-free sample of Seyfert galaxies studied in this work, the difference between the μ0.4 dust curve exponent of 3.55, and 2.94 (from a MW extinction law) used in e.g. and leads to differences ranging from 5 to 30%. The potential increase in the under- (or over-)estimation of the line luminosity with increasing dust content because of an inappropriate dust-curve has important implications for understanding whether the bolometric correction for changes systematically with, due to the correlations between SFR of the host galaxy and the luminosity of the central AGN and the SFR and dust content of the host. Our results do not explicitly constrain which dust-curve is most appropriate for Seyferts, however, we would like to emphasise the importance of including the errors caused by the dust-attenuation correction when studying emission line trends. AGN detection ------------- Even in the two ULIRGs in our sample where there is evidence for a heavily buried central nucleus, we would like to emphasise that the optical emission line ratios still indicate the possible presence of an AGN, albeit not conclusively. This shows that even where the AGN is deeply buried, some radiation may still escape to ionise the NLR clouds. This has important implications for using optical emission line ratios to identify AGN in objects such as ULIRGs, and will become increasingly important at high-z. Unfortunately neither of the two ULIRGs with strong silicate absorption in our sample have measured, and therefore we cannot directly verify that their position on the BPT diagram, and therefore their correction of for contamination from star formation, is correct. The specific study of a larger sample of ULIRGs to compare the bolometric AGN luminosities derived from mid-IR lines, mid-IR continuum and dust- and star-formation corrected high-quality optical emission lines would now be possible with samples in the SDSS and Spitzer archives. Conclusions =========== The purpose of this paper was to verify, or otherwise, the accuracy of fundamental quantities derived from optical emission lines, particularly in the case of significant dust attenuation of the lines. In particular, optical emission lines provide powerful measures of star formation rate and black hole accretion rate. We compared two “dusty” galaxy samples, including 6 ULIRGs, to a sample of “ordinary” galaxies from the SSGSS all with high quality SDSS optical spectra and mid-IR spectra from the Spitzer Space Telescope IRS spectrograph. The results are extremely encouraging for using rest-frame optical spectra to study dusty galaxies with an AV < 4.5. Our results are promising even for galaxies as dusty as ULIRGs, although a much larger sample should be studied covering the full range of this diverse population, in order to verify our conclusions. Our results have important implications for future high-redshift spectroscopic surveys, where galaxies have higher star formation rates and dust contents may be correspondingly higher. We have compared several different dust curves used in the literature to correct optical emission lines for dust attenuation. Our results favour a dust curve which lies close to the MW extinction curve, indicating that a significant fraction of the dust which attenuates nebular emission lines has a screen-like geometry. This is in agreement with the model for emission line attenuation presented by and and based upon the 2-component CF00 dust model. In this model, the screen-like extinction arises from the dense stellar birth clouds which surround the hottest stars responsible for ionising the gas. We find a best-fit dust-curve with on average 40% of the optical depth at 5500Å arising from diffuse ISM dust. The μ0.4 dust curve is not significantly different from the Calzetti dust-curve in the optical wavelength regime, however, we focus our results and analysis on the former which was intended for emission lines and allows greater flexibility, rather than the latter which was derived from galaxy continua. Particular conclusions with regard to measuring star formation from optical emission lines in the presence of dust: * The MW, Calzetti, μ0.3 and best-fit μ0.4 dust curves all allow accurate correction of Hα (to better than a factor of 2), and therefore accurate calculation of star formation rates, for galaxies with observed Hα/Hβ ratios of as much as 10 (τV < 4.5). * For only 2/6 ULIRGs with τV < 6 does the optical Hα differ from the mid-IR [Ne] lines by more than a factor of 2, but then only by a factor of 4. In these two cases (08572 and 15250) we have argued that the observed excess of [Ne] is in fact caused by a buried AGN, rather than uncertainties in the dust correction. * The majority of galaxies in our samples favour a dust curve close to the MW screen-like dust extinction curve. This suggests that much of the extra dust in the unusually dusty objects is located in dense clouds, rather than in the diffuse ISM. * There is tentative evidence for a range in the shape of the dust curves in ordinary starforming galaxies. For the CF00 dust model, this range translates into a variation in the fraction of optical depth arising in the diffuse ISM of between around 30% and 90%, with the remaining fraction arising in dense birth clouds. However, the variation in the intrinsic [Ne]/Hα ratio with metallicity, the issue of relative aperture correction between the IRS and SDSS data, and the large statistical errors on the Hβ emission line, prevent us from quantifying this scatter more precisely. Particular conclusions with regard to identifying AGN and measuring their accretion rates from optical emission lines in the presence of dust: * The best-fit μ0.4 dust curve and measured Balmer decrement allows accurate correction of for dust attenuation, to within a factor of 3. This is slightly surprising, and suggests that the effects of variations in the relative geometry of the source and dust, and differential extinction between the Balmer and higher ionisation lines, are minimal. * The method to separate the contributions to optical emission lines from both star formation and AGN presented in tightens the relation between and for composite objects. This supports the use of the position of galaxies on the optical BPT line ratio diagram to measure the amount of originating from the AGN, and therefore the estimates of black hole accretion rates from corrected for dust attenuation and star formation contamination. * All ULIRGs in our sample are classified as AGN or composite objects from their optical emission line ratios. While the composite line ratios may be caused by shocks, we have argued that other observational results favour the presence of a buried AGN contributing significantly to the mid-IR continuum. It is therefore possible that sufficient ionising flux escapes from the central source to ionise the NLR, even in the case of heavily buried nuclei such as in ULIRGS 08572 and 15250. Unfortunately these two ULIRGs do not have measured preventing us from verifying our corrections to for dust attenuation and star formation contamination in such extreme cases. In the case of ULIRGs, whose properties vary widely, clearly a much larger sample should be studied before firm conclusions can be drawn. Acknowledgements ================ We would like to thank Jarle Brinchmann for his help in understanding aperture bias issues and Stephane Charlot for comments on an early draft. We thank the anonymous referee for a careful reading of the manuscript and comments which improved its clarity. The function fitting performed in this paper used the IDL MPFIT software http://purl.com/net/mpfit (Markwardt 2009). The ITERA package was used to investigate intrinsic emission line ratios http://www.strw.leidenuniv.nl/ brent/itera.html. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. 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H., Schiminovich D., Meisenheimer K., 2009,, 707, 1566 Sample information ================== ccccccc Name & RA (deg) & Dec (deg) & specobjid & rowindex & z PSB01&181.225&52.395&248485859366535168&284940&0.0632 PSB02&169.600&56.036&256085720727289856&296570&0.0680 PSB03&173.251&60.275&268189274979958784&315471&0.0644 PSB04&167.291&45.857&404988882665340928&491844&0.0636 PSB05&136.004&1.458&132797726868897792&105860&0.0535 PSB06&120.199&37.729&213582458531086336&232315&0.0416 PSB07&339.726&13.194&208235580251701248&223368&0.0628 PSB08&128.311&32.583&262560681922920448&306604&0.0558 PSB09&239.431&27.465&391758540508758016&472114&0.0316 PSB10&139.864&33.791&448335754997792768&536057&0.0193 PSB11&158.162&12.178&450306505485320192&539391&0.0329 ccccccc Name & RA (deg) & Dec (deg) & specobjid & rowindex & z IRAS 08572+3915&135.106&39.065&337714854909444096&394321&0.0579 IRAS 12112+0305&183.441&2.811&146028589375029248&126942&0.0730 IRAS 15250+3609&231.748&35.977&394574695615168512&476759&0.0553 Arp220&233.738&23.504&609061542893191168&755340&0.0181 Mrk273&206.176&55.887&372055066565148672&439991&0.0373 UGC5101&143.965&61.353&137019791402598400&111495&0.0393 ccccccc Name & RA (deg) & Dec (deg) & specobjid & rowindex & z 2MASXJ08244333+2959238&126.180&29.990&339966523504328704&397741&0.0250 CGCG064-017&149.812&12.988&491120230488080384&603606&0.0340 Mrk0609&51.356&-6.144&129701503752470528&101843&0.0340 NCG0291&13.375&-8.768&185153160128495616&190581&0.0190 NGC5695&219.342&36.568&389226545811554304&467780&0.0140 2MASXJ15522564+2753435&238.107&27.895&391758541012074496&472216&0.0740 2MASXJ12384342+0927362&189.681&9.460&347285137836212224&409636&0.0830 2MASXJ08035923+2345201&120.997&23.756&356292212926971904&415287&0.0290 2MASXJ16164729+3716209&244.197&37.273&297464197648744448&366718&0.1520 2MASXJ11570483+5249036&179.270&52.818&248485858699640832&284823&0.0360 2MASXJ12183945+4706275&184.664&47.108&408648327781941248&498207&0.0940 SBS1133+572&173.954&56.952&369240319515951104&435171&0.0510 SBS1609+490&242.716&48.911&175301007746531328&174339&0.0450 IC0486&120.087&26.614&261716087635181568&305068&0.0270 CGCG242-028&170.755&47.052&405833433537839104&493622&0.0250 Mrk1457&176.840&52.450&247922896524017664&283914&0.0490 2MASXJ13463217+6423247&206.634&64.390&170234565783191552&166011&0.0240 CGCG218-007&200.952&43.301&387537584101785600&464960&0.0270 2MASXJ10181928+3722419&154.580&37.378&401892408733728768&486221&0.0490 2MASXJ11110693+0228477&167.779&2.480&143495710086529024&122473&0.0350 galaxies ========= While three of the samples studied in this paper have been presented elsewhere, the sample has not. Here we summarise its selection criteria and the mid-IR observations. Selection --------- The galaxies were selected to have: * Observed Hα/Hβ flux ratios greater than 8 * Strong Balmer absorption lines compared to their 4000Å break strength [${\rm PC2}>0.05\times {\rm PC1}+0.4$, ]. Although they were not selected on a single Balmer absorption line equivalent width, which have considerably larger errors than the spectral indices used here, most have HδA > 5Å and Dn(4000) < 1.4. * Accurately measured emission lines: a signal-to-noise > 3σ for all four BPT lines, together with a Hβ flux $\rm >40\times10^{-17} erg/s/cm^2$. * A contribution to their emission lines by an AGN, i.e. they lie above the empirically derived demarcation line of on the BPT diagram. * z < 0.07 so that the SDSS spectrum, taken through a 3fibre, samples only light coming from the central few kpc of the galaxy. Mid-IR observations ------------------- IRS observations in both high and low resolution were carried out using staring mode during program 40330 (PI Timothy Heckman). We used SPICE to extract the 1D spectra from the low-resolution basic calibrated data (BCD) pipeline products, using the point-source and optimal extraction options. The two 1D spectra (i.e. one for each nod) were combined using a weighted mean. For the high-resolution spectra, we first identified and interpolated over additional rogue pixels using a custom IDL routine and the rogue pixel masks suitable for our campaign numbers (41-46). This procedure was repeated for the dedicated off-source background observations. All 3 science exposures were combined using a simple arithmetic mean, resulting in two 2D spectra. These were sky-subtracted and the 1D spectra were extracted using SPICE with the point-source and optimal extraction options. For each nod, the edges of each order were inspected and trimmed and the orders combined using a weighted mean where wavelengths were covered by more than one order. The 1D spectra of each nod were subsequently combined using a weighted mean. --- 1. Throughout this paper we use the term “effective attenuation” to encompass all effects relating to the dust i.e. absorption and scattering, and geometrical effects. We note that this differs from some other papers which apply the term to “single-line-of-sight” attenuation.[↩](#fnref1) 2. Commonly used MW extinction curves include, and [↩](#fnref2) 3. The difference in the intrinsic Balmer decrements between NLR and Hii regions (due to different gas temperatures), will be small relative to the difference caused by attenuation.[↩](#fnref3) 4. These line ratio predictions can be obtained using the publicly available ITERA software package [↩](#fnref4) 5. http://www.mpa-garching.mpg.de/SDSS/DR7/[↩](#fnref5) 6. http://isc.astro.cornell.edu/SmartDoc/LineFitting[↩](#fnref6) 7. We found that PAHfit does not fit the continuum well for ULIRGs with strong Silicate absorption, and therefore the emission line measurements were unreliable in these cases.[↩](#fnref7) 8. http://www.astro.columbia.edu/ssgss/[↩](#fnref8) 9. Petrosian magnitudes and radii are estimates of total magnitudes suitable for all types of galaxies. They are derived by the SDSS photometry pipelines, with details found here: http://www.sdss.org/dr7/algorithms/photometry.html[↩](#fnref9) 10. Courtesy of Jarle Brinchmann, publicly available at http://www.mpa-garching.mpg.de/SDSS/DR7/[↩](#fnref10) 11. The short-low (SL) IRS module covering the wavelength range 5.2-14.5μm has a slit aperture of 3.6 × 57. The long-low (LL) IRS module covering the wavelength range 14.0-38.0μm has a slit aperture of 10.6 × 168. At high resolution, the short-high (SH) module covers 9.9-19.6μm with a slit aperture of 4.7 × 11.3 and the long-high (LH) module covers 18.7-37.2 with a slit aperture of 11.1 × 22.3.[↩](#fnref11) 12. Face-on galaxy orientations can also cause increased scattering into the line-of-sight and thus greyer dust curves.[↩](#fnref12) 13. Allowing the intercept to be non-zero results in the same fitted parameters within the errors.[↩](#fnref13) 14. We note that such a correction would not be appropriate for the ULIRGs if shocks were a significant contributor to the total Hα luminosity.[↩](#fnref14) 15. Note that this is an extrapolation of the dust-curve beyond the wavelength region directly tested in this paper.[↩](#fnref15)	arxiv_0000781
. The first is breadth-first, in which halos are ordered and accessed according to the simulation snapshot; this is the default halo ordering in the trees. An alternate method is depth-first, in which halos are ordered and accessed first along main progenitor branches (i.e., following the most-massive progenitor line) followed by the next-most-massive progenitor branches. For an illustration, see Figure [MergerTrees]. The following fields will always exist in current and later versions of Consistent Trees, although more ordering fields may be added at a later date: * Breadth\_first\_ID: Unique ID (across all simulation snapshots) corresponding to the breadth-first order of halos within a tree. Sorting on this ID always recovers the original order in which the halos were printed in the tree file. * Depth\_first\_ID: Unique ID (across all simulation snapshots) corresponding to the depth-first ordering of halos within a tree. Sorting on this ID reorganizes the tree into depth-first order. This ordering has the useful property that full merger histories are contiguous, even for halos not at the last simulation snapshot; see Last\_progenitor\_depthfirst\_ID and Last\_mainleaf\_depthfirst\_ID below. * Tree\_root\_ID: Halo ID of the final descendent halo (i.e., the descendent halo at the last simulation snapshot). * Orig\_halo\_ID: Original halo ID from halo finder, except for phantom halos. This allows cross-referencing halos in the merger trees with other data (e.g., particles) saved by the halo finder. * Snap\_num: Snapshot number from which the halo originated—necessary as not all halo finders generate unique IDs. * Next\_coprogenitor\_depthfirst\_ID: Depth-first ID of next “coprogenitor”—i.e., the next halo that shares the same descendent halo. * Last\_progenitor\_depthfirst\_ID: Depth-first ID of last progenitor. When a merger tree is sorted by depth-first ID, then the halos between this halo’s Depth\_first\_ID and this halo’s Last\_progenitor\_depthfirst\_ID correspond to its full merger history. * Last\_mainleaf\_depthfirst\_ID: Depth-first ID of last progenitor on this halo’s main progenitor branch. When a merger tree is sorted by depth-first ID, then the halos between this halo’s Depth\_first\_ID and this halo’s Last\_mainleaf\_depthfirst\_ID correspond to its most-massive progenitor history. Note that to better understand the last two fields, it is helpful to refer to Figure [MergerTrees]. Additional Halo Properties -------------------------- These fields are halo-finder dependent. With the current version of Consistent Trees and Rockstar, they include: * Tidal\_Force: Strongest tidal force from any nearby halo, in dimensionless units (Rhalo/Rhill). The Hill radius is an upper bound on the spatial extent of newly infalling material that can remain gravitationally bound to a secondary halo. It is given by Rhill = D(Msec/3Mprim)1/3 = Rsec(D/31/3Rprim), where the primary (secondary) halo is the larger-virial-radius (smaller-virial-radius) halo of a pair whose centers are separated by distance D. * Tidal\_ID: Halo ID of halo exerting strongest tidal force. * Rs\_Klypin: NFW scale radius in units of comoving kpc/h, determined using Vmax and Mvir, Equation ([RsKlypin]). * Mvir\_all: Halo mass, including unbound particles (M⊙/h). * M200m–M2500c: Mass (M⊙/h) enclosed within specified overdensities. These include 200m, 200c, 500c, and 2500c, where ρc is critical density and ρm = ΩMρc is the mean matter density. * Xoff: Offset of halo center (defined as the density peak) from the center of mass within the halo radius (comoving kpc/h). * Voff: Offset of halo center (defined as the density peak) from average particle velocity (physical km/s). * Spin\_Bullock: Bullock spin parameter, Equation ([lambdaB]). * b\_to\_a, c\_to\_a: Ratio of second and third largest shape ellipsoid axes (B and C) to largest shape ellipsoid axis (A) (dimensionless), calculated according to the method in. Additional subscripts (e.g., `500c`) indicate that only particles within a specified halo radius are considered. * A[x],A[y],A[z]: Largest shape ellipsoid axis (comoving kpc/h). Additional subscripts (e.g., `500c`) indicate that only particles within a specified halo radius are considered. * T/∣U∣: ratio of kinetic to potential energies for halo particles. * M\_pe\_: Pseudo-evolution corrected masses (very experimental). Additional Catalog Fields ------------------------- These fields are not in the trees, but are provided for convenience in the halo catalogs. Their order and content may change in future Consistent Trees versions. * Macc, Vacc: Halo’s (or its main progenitor’s) mass and vmax at the last timestep that it was a distinct halo. * Mpeak, Vpeak: Peak mass and peak vmax along the halo’s main progenitor branch. * Halfmass\_Scale: Scale factor at which the halo mass in the main progenitor branch first reaches half of Mpeak. * Acc\_Rate\_: Halo mass accretion rates in M⊙/h/yr. These include: Inst (averaged since the previous snapshot); 100Myr (averaged over past 100 Myr); X\Tdyn: averaged over past X virial dynamical times (i.e., $X/\sqrt{G\rho\_\mathrm{vir}}$); Mpeak: averaged growth of Mpeak, averaged from the current halo’s redshift (z) to z* + 0.5. * Mpeak\_Scale: Scale at which Mpeak was reached along the main progenitor branch. * Acc\_Scale: Last scale at which the halo (or its main progenitor) was distinct. * First\_Acc\_Scale: Last scale at which the halo (or its main progenitor) and all earlier progenitor halos were all distinct.[6](#fn6) * First\_Acc\_(Mvir∣Vmax): Mvir and vmax at First\_Acc\_Scale for the main progenitor. * Vmax@Mpeak: Main progenitor’s vmax at Mpeak\_Scale. * Tidal\_Force\_Tdyn: Dimensionless tidal force averaged over the past virial dynamical time. --- 1. [email protected][↩](#fnref1) 2. <https://www.cosmosim.org/cms/simulations/multidark-project/> with explanations in and at <https://www.cosmosim.org/cms/documentation/introduction/whats-different/> for users of the earlier site <http://www.multidark.org/>.[↩](#fnref2) 3. <http://hipacc.ucsc.edu/Bolshoi/MergerTrees.html>[↩](#fnref3) 4. <http://code.google.com/p/rockstar>[↩](#fnref4) 5. <https://bitbucket.org/pbehroozi/consistent-trees>[↩](#fnref5) 6. In theory. In practice, flybys often happen for very early progenitors, so these are ignored if halos at least double their mass following the flyby.[↩](#fnref6) Halo and Subhalo Demographics with Planck Cosmological Parameters: Bolshoi-Planck and MultiDark-Planck Simulations ================================================================================================================== ### Released 2016 Xxxxx XX -0.5in [firstpage] We report and provide fitting functions for the abundance of dark matter halos and subhalos as a function of mass, circular velocity, and redshift from the new Bolshoi-Planck and MultiDark-Planck ΛCDM cosmological simulations, based on the Planck parameters. We also report halo mass accretion rates and concentrations. We show that the higher cosmological matter density of the Planck parameters compared with the WMAP parameters leads to higher abundance of massive halos at high redshifts. We find that the median halo spin parameter is nearly independent of redshift, leading to predicted evolution of galaxy sizes that is consistent with observations, while the significant decrease with redshift in median predicts more decrease in galaxy sizes than is observed. Using the Tully-Fisher and Faber-Jackson relations between galaxy velocity and mass, we show that a simple model of how galaxy velocity is related to halo maximum circular velocity leads to increasing overprediction of cosmic stellar mass density as redshift increases beyond, implying that such velocity-mass relations must change at. By making a realistic model of how observed galaxy velocities are related to halo circular velocity, we show that recent optical and radio observations of the abundance of galaxies are in good agreement with our ΛCDM simulations. Our halo demographics are based on updated versions of the Rockstar and Consistent Trees codes, and this paper includes appendices explaining all of their outputs. This paper is an introduction to a series of related papers presenting other analyses of the Bolshoi-Planck and MultiDark-Planck simulations. Cosmology: Large Scale Structure - Dark Matter - Galaxies: Halos - Methods: Numerical Introduction ============ ![Observational constraints on \sigma_8 and \Omega_{\rm M} compared to values assumed in cosmological N-body simulations. The observations plotted are as follows: WMAP5+BAO+SN, WMAP7+BAO+H_0, WMAP9+eCMB+BAO+H_0, Planck13+WP+highL+BAO, and Planck15+TT,TE,EE+lowP+lensing+ext. ](s8-OmM "fig:") [s8-OmM.pdf] In the ΛCDM standard modern theory of structure formation in the universe, galaxies populate dark matter halos and subhalos. The demographics of these halos as a function of redshift are thus an important input to the prediction of the properties and distribution of galaxies. A number of large cosmological simulations have now been run, although many cover large volumes but with resolution too low to identify all dark matter halos that host most galaxies. The mass resolution required to do this is $\lesssim 10^8 h^{-1} M\_\odot$, and the force resolution should be $\lesssim 1 h^{-1}$ kpc. High-resolution cosmological dark matter simulations that are particularly useful for studying galaxy hosts include the Millennium simulations, Bolshoi, MultiDark, Jubilee, DarkSky, Q Continuum, ν2GC, and Bolshoi-Planck and MultiDark-Planck simulations. Figure 1 shows the WMAP5/7/9 and Planck 2013 and Planck 2015 cosmological parameters σ8 and ΩM, and the cosmological parameters adopted for these simulations. The Millennium simulations used the first-year (WMAP1) parameters ; the Bolshoi, Q Continuum, and Jubilee simulations used the WMAP5/7 cosmological parameters; while the ν2GC and Bolshoi-Planck simulations used the Planck 2013 parameters, and the DarkSky simulations used parameters between WMAP9 and Planck 2013. In this paper we use the Rockstar halo finder and Consistent Trees to analyze results for the recent Bolshoi-Planck (BolshoiP), Small MultiDark-Planck (SMDPL) and MultiDark-Planck (MDPL) simulations based on the 2013 Planck cosmological parameters and compatible with the Planck 2015 parameters. The BolshoiP, SMDPL and MDPL simulations are not the largest of the new high-resolution simulations, but they do have the advantage that they have been analyzed in great detail, and all of these analyses are being made publicly available. In addition, in this paper we show the effects of the change from the WMAP5/7 to the Planck 2013 cosmological parameters. In this paper we focus on the scaling relations of several basic halo properties, updating their scaling relations as a function of redshift for the Planck cosmological parameters as well as the redshift evolution of halo/subhalo number densities. For the majority of these halo properties we report fitting functions that can be very useful not only to gain insight about the halo/subhalo population but also for the galaxy-halo connection and thus for galaxy evolution. Indeed, techniques such as subhalo abundance matching and halo occupation distribution models require as inputs the halo/subhalo number densities. Furthermore, simplified prescriptions for the evolution of dark matter halo properties are ideal tools for people interested in understanding average properties of halos and the galaxies that they host. Here we analyze all dark matter halos and subhalos found by Rockstar, and do not just focus on those that satisfy some criteria for being ``relaxed“ or otherwise ``good,” in contrast to some earlier studies of dark matter halo properties. The reason is that all sufficiently massive halos are expected to host galaxies or, for the more massive ones, groups or clusters of galaxies. This paper is an introduction to a series of papers presenting additional analyses of the Bolshoi-Planck and MultiDark-Planck simulations. The statistics and physical meaning of halo concentration are discussed in detail in, which is also an overview of the Bolshoi-Planck and MultiDark-Planck simulations, including BigMultiDark simulations in $(2.5 h^{-1} {\rm Gpc})^3$ volumes that we do not discuss here since they are mainly useful for statistics of galaxy clusters. The Stellar Halo Accretion Rate Co-evolution (SHARC) assumption—i.e., that the star formation rate of central galaxies on the main sequence of star formation is proportional to their host halo’s mass accretion rate—was explored in, which used abundance matching based on the Bolshoi-Planck simulation. That paper showed that SHARC is remarkably consistent with the observed galaxy star formation rate out to z ∼ 4 and that the ∼ 0.3 dex dispersion in the halo mass accretion rate is consistent with the observed small dispersion of the star formation rate about the main sequence. The clustering properties of halos and subhalos is the subject of Rodriguez-Puebla et al. 2016b (in prep.). How properties of dark matter halos vary with the density of their environment on length scales from 0.5 to 16 h− 1 Mpc is discussed in Lee et al. (2016a, in prep.), which shows among other things that halos in low-density regions experience lower tidal forces and have lower spin parameters, and that a large fraction of lower-mass halos in high-density regions are ``stripped," i.e. their mass at z = 0 is less than that of their progenitors at higher redshifts. Another paper (Lee et al., 2016b, in prep.) studies the causes of halo stripping and properties of such stripped halos. Further papers comparing with observations are also in preparation, along with mock galaxy catalogs based on Bolshoi-Planck. This paper is organized as follows: §2 discusses the simulations and how we define the halo mass. §3 describes the key scaling relations for distinct halos (i.e., those that are not subhalos) and gives figures and fitting formulas for maximum halo circular velocity (§3.1), halo mass accretion rates (§3.2) and mass growth (§3.3). §4 discusses halo (§4.1) and subhalo (§4.2) number densities, and the number of subhalos as a function of their host halo mass (§4.3). §5 presents the halo and subhalo velocity functions. §4 and §5 also compare the Planck cosmology halo mass and velocity functions with those from the WMAP5/7 cosmological parameters. §6 discusses the dependence of halo concentration and spin on mass and redshift. §7 discusses the evolution of the Tully-Fisher and Faber-Jackson relations between halo circular velocity $V\_{\rm max}$ and the stellar mass of the central galaxies in these halos. §8 compares the halo velocity function with the galaxy velocity function from optical and radio observations. §9 summarizes and briefly discusses the key results in this paper. Appendix A is an overview of the Consistent Trees merger tree information and halo catalogs and Appendix B summarizes the Rockstar and Consistent Trees fields. The simulations =============== 16.cm l \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| l \| l \|l Simulation & box & particles & mp & ε & ΩM & ΩB & ΩΛ & σ8 & ns & H0 & Code & Ref. BolshoiP & 0.25 & 20483 & 1.5 × 108 & 1.0 & 0.307 & 0.048 & 0.693 & 0.823 & 0.96 & 67.8 & ART & a SMDPL & 0.4 & 38403 & 9.6 × 107 & 1.5 & 0.307 & 0.048 & 0.693 & 0.829 & 0.96 & 67.8 & GADGET-2 & a MDPL & 1.0 & 38403 & 1.5 × 109 & 5 & 0.307 & 0.048 & 0.693 & 0.829 & 0.96 & 67.8 & GADGET-2 & a Bolshoi & 0.25 & 20483 & 1.3 × 108 & 1.0 & 0.270 & 0.047 & 0.730 & 0.820 & 0.95 & 70.0 & ART & b MultiDark & 1.0 & 20483 & 8.7 × 109 & 7.0 & 0.270 & 0.047 & 0.730 & 0.820 & 0.95 & 70.0 & ART & c [simtable] The cosmological parameter values for the Bolshoi-Planck and MultiDark-Planck simulations are $\Omega\_{\Lambda,0}=0.693,\Omega\_{\rm M,0}=0.307,\Omega\_{\rm B,0}=0.048, h=0.678, n\_s=0.96$ and σ8 = 0.823. The parameters are the same for the MultiDark-Planck simulations except for σ8 = 0.829. Simulation volumes, resolutions and other parameters of the Bolshoi and MultiDark simulations with WMAP7/9 parameters, and the new Bolshoi-Planck and the $(400 h^{-1} {\rm Mpc})^3$ and $(1 h^{-1} {\rm Gpc})^3$ MultiDark-Planck simulations are summarized in Table 1. The details about the number and redshift distribution of the saved timesteps of these simulations are given in Appendix A. Outputs from these simulations are available online at the CosmoSim website.[2](#fn2) Entire Rockstar and Consistent Trees outputs are downloadable from another website.[3](#fn3) ![image](f1a) [f1a] In this paper we define halo masses by using spherical overdensities according to the redshift-dependent virial overdensity $\Delta\_{\rm vir}(z)$ given by the spherical collapse model, for which give the fitting formula $\Delta\_{\rm vir}(z) = (18\pi^2 + 82x - 39x^2)/\Omega(z)$, where Ω(z) is the ratio of mean matter density $\rho\_{\rm m}$ to critical density $\rho\_{\rm c}$ at redshift z, and x ≡ Ω(z) − 1. Figure [f1a] shows the redshift dependence of $\Delta\_{\rm vir}$ for the cosmology of the Bolshoi-Plank simulation. The value of the virial overdensity at z = 0 is $\Delta\_{\rm vir} = 333$, while for large z it asymptotes to 18π2 = 178. The virial radius $R\_{\rm vir}$ of a halo of virial mass $\mvir$ is defined as the radius within which the mean density is $\Delta\_{\rm vir}$ times the mean matter density $\rho\_{\rm m} = \Omega\_M \rho\_{\rm c}$ at that redshift. Then the virial halo mass is $$\mvir = \frac{4\pi}{3} \Delta\_{\rm vir} \rho\_{\rm m} R\_{\rm vir}^3. \label{Mvir}$$ Another common choice employed to define halos is the radius $R\_{\rm 200c}$ enclosing 200 times critical density, with corresponding halo mass $$M\_{\rm 200c} = \frac{4\pi}{3} 200 \rho\_{\rm c} R\_{\rm 200c}^3,$$ which was used in. Yet another common choice is $R\_{\rm 200m}$ enclosing 200 times mean density, with corresponding halo mass $M\_{\rm 200m}$. Although we use $M\_{\rm vir}$ in this paper, the Rockstar/Consistent Trees analyses of the Bolshoi-Planck and MultiDark-Planck simulations described in the Appendices include outputs for both $M\_{\rm vir}$ and $M\_{\rm 200m}$. We note that argued that much of the mass evolution of dark matter halos is an artifact caused by the changing radius of the halo as the mean cosmic matter density $\rho\_{\rm m}$ declines as the universe expands. They call this phenomenon ``pseudoevolution," since the dark matter distribution in the interior of most halos hardly changes at low redshift. Recently proposed that the best physically-based definition of halo radius is the “splashback radius” $R\_{\rm sp} \approx 2 R\_{\rm 200m}$, where there is typically a sharp drop in the density. Using this definition, there is actually more halo mass increase than for $R\_{\rm vir}$ or $R\_{\rm 200m}$. This is discussed further in, where we argue that for purposes like predicting galaxy star formation, the $R\_{\rm vir}$ definition used here works fine. Basic scaling relations for distinct halos ========================================== ![Left Panel: Maximum halo circular velocity, v, as a function of at z = 0, 2, 4, 6 and 8. Medians are shown as the solid lines for the BolshoiP and MDPL simulations, filled circles are the medians of the SMDPL simulation. At z=0 the grey band is the 68\% range of the maximum circular velocity. The dotted lines show the fits to the simulation. A single power law is able to reproduce the results from the simulation. The slopes are approximately independent of redshift with a value of \sim1/3. Right Panel: The highest maximum circular velocity reached along the main progenitor branch,, as a function of at z = 0, 2, 4, 6 and 8. Similarly to the v panel, medians are shown as the solid lines. At z=0 the grey band is the 1\sigma (68\%) range of the maximum circular velocity. The dotted lines show the fits to the simulation. Also, the slopes are approximately independent of redshift with a value of \sim1/3. ](f1 "fig:") [f1] Galaxies form and evolve in dark matter halos, and it is expected that visible galaxies are hosted by all halos in the mass range $\mvir\grtsim10^{10.2}h^{-1}\msun$ where halos can be resolved with at least $\grtsim 100$ particles in the BolshoiP and SMDPL simulations. Therefore, the statistical properties of dark matter halos, which can be studied in great detail in high resolution numerical N − body simulations, can provide hints on the nature of galaxy properties and spatial distributions. In this Section, we report dark matter halo velocity and mass and their scaling relations. Dependence on mass and redshift of halo concentration and spin are discussed in §[C-lambda]. Maximum halo circular velocity ------------------------------ As usual, the circular velocity is defined as $V\_{\rm circ} \equiv \sqrt{G M(<r)/r}$, where M( < r) is the halo mass enclosed by a sphere of radius r. Dark matter halos have circular velocity that grows from 0 at r = 0 to a maximum value v at a radius $R\_{\rm max}$ that is usually considerably less than $R\_{\rm vir}$. (§[concentrations] shows that for the NFW radial halo mass distribution, $R\_{\rm max} = 2.1626\times\ R\_s$.) Because v characterizes the inner halo, it may correlate better with the properties of the central galaxy than does. The left panel of Figure [f1] shows the medians of the maximum halo circular velocity, v, as a function of at z = 0, 2, 4, 6 and 8, solid lines. The grey band at z = 0 shows the 68% range of the maximum circular velocity, i.e., the halo distribution between the 16th and 84th percentiles. We find that the 68% range of the distribution is approximately independent of redshift and halo mass, with a value of ∼ 0.05 dex. In general, the v– relation follows a power law-fit at all redshifts and over the mass range where we can resolve distinct halos in the Bolshoi-Planck simulations, $\mvir\sim10^{10.2}\msun$. To a good approximation, the v– slope is given by α ∼ 1/3, as expected from spherical collapse. In reality, however, the slope depends slightly on redshift as we will quantify below. Distinct halos can lose mass due to stripping events as a result of interactions with other halos. In consequence, the maximum halo circular velocity v can significantly decrease (this will be discussed in more detail in Lee et al 2016b). This reduction in v can introduce an extra source of uncertainty when relating galaxies to dark matter halos, since it is expected that stripping would affect halos more significantly than the central galaxies deep inside them. Therefore, in the case of stripped halos, the correlation between the present v of the halo and galaxy stellar mass/luminosity is not trivial. Indeed, and found that the highest maximum circular velocity reached along the halo’s main progenitor branch,, is a better halo proxy for galaxy stellar mass/luminosity. For these reasons we find it useful to report the – relation in this paper. The right panel of Figure [f1] shows the redshift evolution of the highest maximum circular velocity reached along the main progenitor branch,, as a function of at z = 0, 2, 4, 6 and 8. As for v–, medians are shown as the solid lines. The grey band at z = 0 is the 68% of the distribution. We find that the slope depends more on redshift than for the v– relation. The differences in the slopes, especially at lower redshifts, are a consequence of tidal stripping events, as mentioned above. On the other hand, we find that the scatter is independent of redshift and mass and of the order of ∼ 0.05 dex, similarly to v–. ![Halo mass accretion rates as a function of at z = 0, 2, 4, 6 and 8. Medians are shown as the solid lines for the BolshoiP and MDPL simulations, and filled circles are the same for the SMDPL simulation. At z=0 the grey band is the 68\% range of halo mass accretion rates for the BolshoiP and MDPL simulations. The dotted lines show the fits to the simulations when using a power law, Equation ([dmhdt-mh]). Upper Panel: Instantaneous halo mass accretion rates. The long dashed line shows the fits using a double power law, Equation ([dmhdt-mh-dbl]), for the BolshoiP and MDPL simulations. Bottom Left Panel: Halo mass accretion rates averaged over a dynamical time. A single power law reproduces the results from the simulations. The slopes are approximately independent of redshift with a value of \sim1.1. Bottom Right Panel: Mass accretion rates of averaged from the current halo’s redshift, z, to z+0.5. As for the dynamical time averaged accretion rates, a single power law again reproduces the results from the simulations. The slopes are approximately independent of redshift with a value of \sim1.1. ](f2 "fig:") [f2] The dotted lines in Figure [f1] show power-law fits to the BolshoiP, SMDPL and MDPLsimulations both for the v– (left panel) and – (right panel) relationships. We motivate the power law-fits based on the well known results for isothermal dark matter halo profiles. For a singular isothermal sphere the circular velocity (assumed to be independent of radius inside the halo) evolves with redshift as $${v\_\mathrm{max}}\ \propto\ \left[ \mvir E(z)\right]^{1/3}, \label{iso\_shpere}$$ where E(z) is the expansion rate H/H0 for a flat universe: $$E(z) = \sqrt{ \Omega\_{\Lambda,0}+\Omega\_{\rm m,0}(1+z)^3}. \label{E(z)}$$ Of course this is a simplification since the mass profile of dark matter halos is markedly different from an isothermal sphere as previous studies based on high-resolution N-body simulations have shown. However, the general structure of Equation ([isoshpere]) can be useful to suggest fitting functions for redshift evolution. Based on this, we assume the following parametric form for the redshift evolution both for the v– and – relationships $$V(\mvir,z) = \beta(z) \left[ \mvirtw E(z) \right] ^ { \alpha(z) }, \label{vmax-mh}$$ where $\mvirtw \equiv \mvir/(10^{12}h^{-1}\msun)$. Our fitting functions for v–, with a representing the scale factor a = 1/(1 + z), are α(z) = 0.346 − 0.059a + 0.025a2, and logβ(z) = 2.209 + 0.060a − 0.021a2, while the fitting functions for – are α(z) = 0.346 − 0.080a + 0.042a2, and logβ(z) = 2.205 + 0.150a − 0.063a2. The cold baryonic mass, Mb, in galaxies—i.e., the star and cold gas mass— is mostly atomic and molecular hydrogen, atomic helium, and a smaller amount of heavier components such as CO molecules. Abundance matching estimates of the baryonic mass in galaxies have shown that the $M\_b/\mvir$ ratio depends on halo mass, and it peaks around $M\_b/\mvir\sim0.05$ for Milky Way mass galaxies. This implies that cold gas and stars in galaxies represent only $\lesssim30\%$ of the cosmic baryon fraction, $f\_{\rm c,b} = \Omega\_{\rm B,0} / \Omega\_{\rm M,0} = 0.156$. If one assumes, as a crude approximation, that the the ratio $M\_b/\mvir$ is constant with mass, then the v– relation implies that the relation of maximum circular velocity and cold baryon mass is a power law with vmax ∝ Mb1/3 and a scatter of ∼ 0.05 dex. Observations have shown that the baryonic Tully-Fisher relation at z ∼ 0 has a power law of slope ∼ 0.3 and intrinsic scatter of 0.05 dex, in rough agreement with this expectation. Of course, this is an oversimplified picture and the dependence of Mb with halo mass should be taken into account; for more details see, e.g., and. Halo mass accretion rates ------------------------- Halo mass accretion is responsible for controlling the rate at which the baryonic mass, Mb, is deposited in galaxies. In the past, instantaneous halo mass accretion rates, $d\mvir / dt$, have been studied in great detail based on the Extended Press-Schechter formalism or high-resolution N − body simulations or either both. In addition to instantaneous halo mass accretion rates $d\mvir / dt$ calculated between stored simulation timesteps, in this paper we extend the above work by also studying halo mass accretion rates averaged over the dynamical time $d\mdyn / dt$, defined as $$\big\langle\frac{d\mvir}{dt}\big\rangle\_{\rm dyn} = \frac{\mvir(t) - \mvir(t-t\_{\rm dyn})}{t\_{\rm dyn}}, \label{tdyn}$$ where the dynamical time of the halo is $t\_{\rm dyn}(z) = [G \Delta\_{\rm vir}(z) \rho\_{\rm crit}(z)]^{-1/2}$. The ratio of the dynamical time to the Hubble time tH = H− 1 is $t\_{\rm dyn}/t\_H = [(8\pi)/(3 \Delta\_{\rm vir})]^{1/2}$, which equals 0.16 at a = 0 and asymptotes to 0.22 at high redshift. We also report halo mass accretion rates of the maximum mass reached along the main progenitor branch averaged from the current halo’s redshift, z, to z + 0.5, $d\mpeak / dt$. For more details the reader is referred to Appendix B4. Figure [f2] shows the medians of halo mass accretion rates at z = 0, 2, 4, 6 and 8. The upper panel shows the instantaneous halo mass accretion rate, $d\mvir / dt$, while the left and right bottom panels show halo mass accretion rates averaged over a dynamical time, $d\mdyn / dt$, and halo accretion rates, $d\mpeak / dt$, respectively. The grey band at z = 0 in all the panels shows the distribution of halo mass accretion rates between the 16th and 84th percentiles, i.e., the 68% of the distribution. We find that the dispersion is roughly independent of redshift and in slightly dependent on halo mass. The dispersion for $d\mvir / dt$ is roughly ∼ 0.35 dex, while for $d\mdyn / dt$ is ∼ 0.3 dex, and for $d\mpeak / dt$ is ∼ 0.25 dex. The $d\mvir / dt - \mvir$ relations follow a power law-fit, especially at high redshifts. As a crude estimation, the slope for the $d\mvir / dt - \mvir$ relations is α ∼ 1.1, consistent with previous studies based on the Press-Schechter formalism and the Millennium and Millennium-II high resolution N − body simulations. c c c c c c c $d\mvir/dt$ [h− 1/yrs] & α0 & α1 & α2 & β0 & β1 & β2 Instantaneous & 0.975 & 0.300 & -0.224 & 2.677 & -1.708 & 0.661 Dynamical averaged & 1.000 & 0.329 & -0.206 & 2.730 & -1.828 & 0.654 Peak & 0.997 & 0.328 & -0.200 & 2.711 & -1.739 & 0.672 [T2] The dotted lines in Figure [f4] show power-law fits to the simulations for the halo mass accretion rates, given by $$\frac{d\mvir}{dt} = \beta(z) \mvirtw^ { \alpha(z) } E(z), \label{dmhdt-mh}$$ where logβ(z) = β0 + β1a + β2a2, and α(z) = α0 + α1a + α2a2. Table [T2] lists the best fit parameters for the $d\mvir / dt - \mvir$ relations. Power-law fits can provide an accurate description for both $d\mdyn / dt$ and $d\mpeak / dt$ for the three simulations. As can be observed in the upper panel of Figure [f2], however, a power-law fit is a poor description of the instantaneous halo mass accretion rates, especially for the BolshoiPand MDPL simulations at low masses and low redshifts. In order to find a better description of the instantaneous halo mass accretion rates for the BolshoiP and MDPL simulations we use a double power-law fit $$\frac{d\mvir}{dt} = \beta(z)[\mvirtw^{\alpha(z)} + \mvirtw^{ \gamma(z) }] E(z), \label{dmhdt-mh-dbl}$$ where the normalization is given by logβ(z) = 2.437 − 1.857 × a + 0.685 × a2, and the powers α(z) and γ(z) are given respectively by α(z) = 1.120 − 0.609 × a + 0.097 × a2, and γ(z) = 0.917 + 0.845 × a − 0.532 × a2. The dashed lines in Figure [f2] show this fit to the simulations. Finally, based on the above definitions of halo accretion rates, the rate at which the cosmological baryonic inflow material is accreted into the dark matter halo is calculated as $dM\_{\rm c,b} / dt = f\_{\rm c,b}\times d\mvir / dt$, where the cosmic baryon fraction is $f\_{\rm c,b} \equiv \Omega\_{\rm B,0} / \Omega\_{M,0} = 0.156$ for our cosmology. The rate $dM\_{\rm c,b} / dt$ is an important quantity; it equals the star formation rate plus the gas outflow rate if the galaxy is in “equilibrium” in bathtub model terms. Galaxies can be divided into two main groups: star-forming and quiescent. Star-forming galaxies are typically blue young disk galaxies, while many quiescent galaxies are red old spheroids. These properties are partially determined by the mass of the dark matter halo in which they reside but, due to complexity of the galaxy formation process, a dependence on other halo and/or environmental properties is expected. For example, star-forming galaxies at a given redshift are known to show a tight dependence of star formation rates on stellar mass, which is known as the ``main sequence" of galaxy formation. The slopes and dispersions of halo mass accretion rates reported above are very similar to the observed dispersion and slope of the star formation rates on the main sequence. This naturally suggests that the halo mass accretion rate is controlling not only the baryon fraction that is entering the galaxies, but also their star formation efficiency. The galaxy stellar-to-halo mass relation is known to be nearly independent of redshift from z = 0 out to z ∼ 4, so the galaxy star formation rate is determined on average by the mass accretion rate of the halo in which it resides: $d M\_\ast / dt = (d M\_\ast / d \mvir) (d \mvir / dt)$. A recent paper by some of us,, made the stronger assumption that this is true halo by halo for star-forming galaxies, which we called Stellar-Halo Accretion Rate Coevolution (SHARC). We showed that the SHARC assumption predicts galaxy star formation rates on the main sequence that are in good agreement with observations up to z ∼ 4, and that in addition it approximately matches the small observed dispersion of ∼ 0.3 dex of the galaxy star formation rates about the main sequence. Halo assembly ------------- ![Median halo mass growth for progenitors z = 0 with masses of \mvir=10^{11},10^{12},10^{13}, and 10^{14} h^{-1}, solid lines. Fits to simulations are shown with the dotted lines. The shaded area shows the dispersion around the medians. ](f4 "fig:") [f4] ![Median v growth for progenitors with masses of \mvir=10^{11},10^{12},10^{13}, and 10^{14}h^{-1}, solid lines. Fits to simulations are shown with the dotted lines. The shaded area shows the dispersion around the medians. ](f5 "fig:") [f5] Figure [f4] presents the medians of the halo mass growth for progenitors at z = 0 with masses of $\mvir=10^{11},10^{12},10^{13}, 10^{14}$ and 1015h− 1, for the BolshoiP (black solid line) SMDPL (red solid line) and MDPL (blue solid line) simulations. In order to avoid resolution effects and thus obtain reliable statistics we require that at every redshift at least 90% of the halos can be resolved with at least 100 particles. The first thing to note is that the three simulations agree with each other at all redshifts. From the figure it is evident that high mass halos assembled more rapidly at later epochs than lower mass halos. This is consistent with the fact the slopes obtained for halo mass accretion rates are slightly greater than 1. For the Planck cosmology we find that 1012h− 1 halos formed half of their mass by z ∼ 1.2. Progenitors of $\mvir = 10^{13}, 10^{14}$, and 1015h− 1 halos reached the mass of $10^{12}h^{-1}\msun$ at z ∼ 2.5, 3.9, and z ∼ 5 respectively. Theoretically, the characteristic mass of 1012h− 1 is expected to mark a transition above which the formation of stars in galaxies becomes increasingly inefficient. The reasons for this are that at halo masses above 1012h− 1 the efficiency at which the virial shocks can heat the gas increases, and the gas can be kept from cooling by energy emitted from accretion onto supermassive black holes in these high-mass halos. Thus central galaxies in massive halos are expected to become passive systems roughly at the epoch when the halo reached the mass of 1012h− 1. Note that Figure 2 of is similar to Figure [f4], including also a comparison to warm dark matter cosmologies. In order to characterize the growth of dark matter halos we use the fitting function from $$\mvir (\mvir(0), z) = M\_{\rm 13} (z) 10 ^{f(M\_{\rm vir}(0), z)} \label{AMH}$$ where $$M\_{\rm 13} (z) = 10^{13.6}h^{-1}\msun(1+z)^{2.755}(1+\frac{z}{2})^{-6.351}\exp{(-0.413 z)}$$ $$f(\mvir,z) = \log\left(\frac{\mvir(0)}{M\_{\rm 13} (0)}\right)\frac{g(\mvir(0),1)}{g(\mvir(0,a))}$$ $$g(\mvir(0),a) = 1 + \exp[-3.676 (a-a\_0(\mvir(0)))]$$ $$a\_0(\mvir(0)) = 0.592 - \log\left[0.113\left(\frac{10^{15.7}h^{-1}\msun}{\mvir(0)}\right)+1\right].$$ As is emphasized in, as opposed to other previous descriptions, the above parameterization avoids the problem that progenitor histories of halos with different masses cross. Figure [f5] presents the medians of the maximum circular velocity growth for the progenitors of halos of the same masses $\mvir = 10^{11}$ to 1015h− 1 described above. The maximum circular velocity is more directly connected to the central potential depth of the halo than the virial circular velocity or mass, and presumably more connected to the formation of the host galaxy. We find that for halos of $\mvir=10^{11}h^{-1}\msun$, v is practically constant after z ∼ 2, while for halos of $\mvir=10^{12}h^{-1}\msun$ and $\mvir=10^{13}h^{-1}\msun$, v is constant since z ∼ 1 and z ∼ 0.5, respectively. This is consistent with the fact that the interiors of halos hardly change at low redshifts. The dotted line in Figure [f5] shows the growth of v when combining Equations ([vmax-mh]) and ([AMH]). This simple prescription reproduces accurately the redshift dependence of halo v growth. Halo and subhalo number densities ================================= Distinct halo mass function --------------------------- ![image](f6) ![Left Panel: The halo mass function from z=0 to z=9. Right Panel: Cumulative halo mass function. The various solid lines show the fits to the simulations, Equation ([Tinker]). ](f7 "fig:") [f6] c c c c χi & χ0, i & χ1, i & χ2, i A & 0.144 & -0.011 & 0.003 a & 1.351 & 0.068 & 0.006 b & 3.113 & -0.077 & -0.013 c & 1.187 & 0.009 & 0.000 [T3] The comoving number density of distinct halos at the mass range $\log\mvir$ and $\log\mvir+d\log\mvir$, i.e., the halo mass function $(d\nh/d\log\mvir)$, are presented in the left panel of Figure [f6] both for the Bolshoi-Planck and MD-Planck simulations. The right panel of the same figure shows the cumulative comoving number density, $\nh(>\mvir)$. We compare the measured halo mass function from the simulations to the analytical fitting formula reported in, which we find provides accurate fits to the results of the Bolshoi-Planck and MD-Planck simulations at low redshifts. At high redshifts, however, it tends to underestimate $d\nh/d\log\mvir$. Theoretically, the comoving number density of halos at the mass range and $\mvir+d\mvir$ is given by $$\frac{d\nh}{d\mvir}=f(\sigma)\frac{\rhom}{M\_{\rm vir}^2}\left\|\frac{d\ln\sigma^{-1}}{d\ln \mvir}\right\|. \label{hmf}$$ where σ is the amplitude of the perturbations and f(σ) is the halo multiplicity function. The cumulative number density of halos above the mass is simply: $$\nh(>\mvir) = \int\_{M\_{\rm vir}}^{\infty}\frac{d\nh}{d\log\mvir}d\log\mvir. \label{chmf}$$ In this paper we use the parametrization given in : $$f(\sigma) = A \left[ \left(\frac{\sigma}{b}\right)^{-a} + 1 \right] e^{-c / \sigma^2}. \label{Tinker}$$ The amplitude of the perturbations is given by $$\sigma^2(\mvir, a) = \frac{D^2(a)}{2\pi^2}\int P(k) k^2 W^2(k,\mvir) dk, \label{sigma\_mvir}$$ where P(k) is the power spectrum of perturbations and $W^2(k,\mvir)$ is a window function defined to be the Fourier transform of the real-space top-hat filter of a sphere of mass $\mvir$ and D(a) is the linear growth-rate factor of the perturbations given by the expression $$D(a) = \frac{g(a)}{g(1)},$$ where to a good approximation g(a) is given by : $$g(a) =\frac{ \frac{5}{2}\Omega\_{\rm m}(a)a}{\Omega\_{\rm m}(a) - \Omega\_{\Lambda}(a) + [1 + \frac{1}{2} \Omega\_{\rm m}(a) ] / [1 + \frac{1}{70} \Omega\_{\Lambda}(a) ] }.$$ Figure [f6a] shows the amplitude of perturbations, $\sigma(\mvir)$, as a function of for a = 1. The red solid line is calculated based on the transfer function from CAMB. Following, we find that the amplitude of perturbations σ is well given by $$\sigma(\mvir) = \frac{17.111y^{0.405}}{1+1.306y^{0.22}+6.218y^{0.317}}, \label{sigma\_mvir\_fit}$$ with $y = 1 / \mvirtw$. Note that the above fit is only valid for the Bolshoi-Planck cosmology studied in this paper. The dashed line in Figure [f6a] shows the best fitting model for $\sigma(\mvir)$ given by the above equations. ![Amplitude of linear perturbations, \sigma(\mvir), as a function of. The red solid line shows the numerical solution to Equation ([sigmamvir]). The dashed black line shows the fit to the amplitude of perturbations given by Equation ([sigmamvirfit]). ](f6a "fig:") [f6a] ![Characteristic halo mass M_{\rm C} as a function of redshift. The red solid line shows the numerical solution to Equation ([MPSequation]). The dashed black line shows our numerical fit to M_{\rm C} given by Equation ([sigmamvirfit]). ](MPS "fig:") [fMPS] The characteristic mass $M\_{\rm C}(z)$ of halos just collapsing at redshift z is given by ν = 1 in $$\nu = \frac{\delta\_c}{\sigma(M\_{\rm C},z)}. \label{MPS\_equation}$$ where δc = 1.686 for uniform-density spherical collapse. Figure [fMPS] shows the Characteristic halo mass $M\_{\rm C}$ as a function of redshift, red solid line. Note the strong dependence with redshift. For our cosmology, at z = 0 we find that $M\_{\rm C} \sim 5\times10^{12}h^{-1}\msun$ while at z = 1, 2 and 5 we find that $M\_{\rm C} \sim 1.5\times10^{11},$ 6.3 × 109 and 7.6 × 108h− 1, respectively. We find that to good accuracy the redshift dependence of $M\_{\rm C}$ is given by the following fitting function, $$\log M\_{\rm C} ( a ) = 12.68 - 0.084 y^{0.01} - 5.33 y^{1.92} - 8.22 y^{7.8}, \label{MPS}$$ where y ≡ z/(1 + z) = 1 − a. Next, we update the best fitting parameters to the formulae for the virial halo mass definition for the Planck cosmology. In order to find the best fitting parameters to the redshift evolution of the halo mass function we will assume the following redshift dependence for the parameters χi = A, a, b and c in Equation ([Tinker]), χi = χ0, i + χ1, iz + χ2, iz2. Table [T3] lists the best fit parameters to the redshift evolution of the halo mass function. In the left panel of Figure [f6] we present the best fits to the resulting halo mass functions from the simulations. For completeness, in the right panel of the same Figure we also show the resulting cumulative halo mass function using the best fit parameters from table [T3]. Figure [f11] shows the ratio of the number densities $n\_{\rm BP}$ and $n\_{\rm B}$ between the Bolshoi-Planck and the Bolshoi simulations as a function of from z = 0 to z = 8. The different cosmological parameters imply that at z = 0, on average, there are ∼ 12% more Milky-Way mass halos in the Bolshoi-Planck than in the Bolshoi simulation. This fraction increases to higher masses, ∼ 25% for $\mvir\sim3\times10^{13}h^{-1}\msun$. This fraction also increases with redshift, and we find that at z = 2, 4 and 6 there are ∼ 25, 40 and 60% more Milky-Way mass halos in the Bolshoi-Planck than in the Bolshoi simulation. At z = 8, there are about 3 times as many $\mvir = 10^{11}h^{-1} \msun$ halos in Bolshoi-Planck as in Bolshoi. In the cold dark matter cosmology it is predicted that the number density of dark matter halos is a strong function of halo mass at low masses $ d\nh / d\mvir \propto M\_{\rm vir} ^ {-1.8}$. In contrast, the observed galaxy stellar mass function, as well as the luminosity function, has a slope that is flatter. Recent analysis have found slopes between α ∼ 1.4 − 1.6 meaning that, for some reason, the star formation efficiency in low mass halos has been suppressed. Nevertheless, measurements of the baryonic mass have found slopes as steep as α ∼ 1.9. ![The ratio of the number densities of distinct halos between the Bolshoi-Planck n_{\rm BP} and the Bolshoi n_{\rm B} simulations as a function of from z=0 to z=8 are shown as filled circles, and the ratio of the fitting functions are shown as solid lines. ](f11 "fig:") [f11] Subhalo mass function --------------------- ![image](f16a) ![Subhalo mass (left panel) and (right panel) functions are shown for redshifts from z=0 to 9, along with fitting functions Equations ([f16fita]) to ([f16fitc]). ](f16 "fig:") [f16] c c c Parameter & & C0 & -0.1213 & -0.0863 C1 & 0.0113 & 0.0087 C2 & -0.0168 & -0.0113 C3 & -0.0032 & -0.0039 C4 & 0.0005 & 0.0004 $ \alpha\_{{\rm sub},1}$ & 0.1810 & 0.0724 $ \alpha\_{{\rm sub},2}$ & 0.2138 & 0.2206 $M\_{0} [h^{-1}\msun]$ & 11.1416 & 11.9046 $M\_{1} [h^{-1}\msun]$ & -0.6595 & -0.6364 $M\_{2} [h^{-1}\msun]$ & -0.0015 & -0.02069 $M\_{3} [h^{-1}\msun]$ & 0.0183 & 0.0220 $M\_{4} [h^{-1}\msun]$ & -0.00164 & -0.0012 [T4] ![image](f14) ![The cumulative subhalo mass function for different host halos as a function of \mu = \macc / \mvir and \mpeak / \mvir is shown as solid curves. The dotted curve is the fitting function Equation ([nsubmvir]). ](f14a "fig:") [f14] Subhalos can lose a significant fraction of their mass due to tidal striping. Since tidal stripping affects the dark matter more than the stars of the central galaxy deep inside the halo, this means that the correlation between galaxy stellar mass and present subhalo mass is not trivial. Therefore in approaches for connecting galaxies to dark matter (sub)halos, such as the abundance matching technique, it has been shown that the mass the subhalo had when it was still a distinct halo correlates better with the stellar mass of the galaxy it hosts. This comes from the fact that when assuming identical stellar-to-halo mass relations for central and satellite galaxies, the observed two-point correlation function is reproduced. Note, however, that while some authors have shown that this is true for most of the scales, others have argued that on the very small scales abundance matching fails in reproduce the observed clustering of galaxies. Therefore it is useful to report the subhalo mass function when subhalos were accreted for the first time into a bigger distinct halo, i.e., the comoving number density of subhalos at the mass range $\log\macc$ and $\log\macc+d\log\macc$. In addition, we also present results for $\mpeak$ in the Bolshoi-Planck simulations. Similarly to $\vpeak$, $\mpeak$ is defined as the maximum mass reached along the main progenitor assembly mass. Figure [f16] shows the redshift evolution of the subhalo mass function, $d \nsub / d \log \msub$, derived from the Bolshoi-Planck and MD-Planck simulations. The solid circles in the left panel show the resulting subhalo mass function for while the results for are shown in the right panel of the same figure. The solid lines in Figure [f16] show our best fitting models to the redshift evolution of the subhalo mass function. Previous reports of the mean number of subhalos above some mass $\msub$ at a given host of mass have found that this is nearly independent of host mass and scales as $\Nsub \propto (\msub / M\_{\rm vir})^\alpha$ with α ∼ − 1, see also below. This implies that to a good approximation the subhalo mass function is $d \nsub / d \log \msub\propto M\_{\rm vir}^{\alpha} d\nh / d\log\mvir$. Here we generalized the fitting model proposed in for the redshift evolution of the subhalo mass function for both and : $$\frac{d\nsub}{d\log\msub}=C\_{\rm sub}(z) G(\mvir,z)\frac{d\nh}{d\log\mvir},$$ where $$\log C\_{\rm sub}(z) = C\_0 + C\_1 a + C\_2 a^2 + C\_3 a^3 + C\_4 a^4, \label{f16fita}$$ and $$G(\mvir,z) = X^{\alpha\_{{\rm sub},1}} \exp ( - X^{\alpha\_{{\rm sub},2} }), \label{f16fitb}$$ where $X = \mvir / M\_{\rm cut}(z)$. The fitting function for $ M\_{\rm cut}(z)$ is given by $$\log(M\_{\rm cut}(z)) = M\_0 + M\_1 z + M\_2 z^2 + M\_3 z^3 + M\_4 z^4. \label{f16fitc}$$ Number of subhalos as a function of their host halo mass -------------------------------------------------------- ![Comparison between the measured subhalo mass function from the BolshoiP, SMDPL and MDPL simulations and by computing d \nsub / d \log \msub when using Equation ([nsubHOD]) for and. ](f14b "fig:") [suhmfcomp] ![Maximum circular velocity function of distinct halos from z=0 to z=9. The different solid lines show the fits to the simulation, Equation ([Vmax-fcn]). Upper Panel: The product E(z)\times V^{3}_{\rm max} \times\ d\nh/d\log{v_\mathrm{max}} is shown, to split the evolution of the velocity function. Bottom Panel: Same but for the product E(z) \times\ d\nh/d\log{v_\mathrm{max}}. ](f12 "fig:") [f12] Characterizing the number of subhalos in hosts of different masses is relevant for several reasons. The predicted abundance of satellites in Milky Way mass galaxies has been a very active topic since N − body numerical simulations could resolve subhalos in galactic halos. Subhalos are the natural sites for satellite galaxies, thus, using statistical approaches that connect the stellar mass of central/satellite galaxies to dark matter halos/subhalos allows prediction of the abundance of satellite galaxies as function of the stellar mass of their host. Previous studies have used these predictions in order to make direct comparisons to what is observed from large galaxy groups catalogues. These studies have found that in order to reconcile the observed abundance of satellite galaxies in clusters of different masses, the galaxy stellar-to-halo mass relation of central and satellite galaxies should be different, especially at lower masses. Figure [f14] shows the mean cumulative number of subhalos for various host halos with masses $\mvir=10^{11}, 10^{12}, 10^{13}$, and 1014, solid lines. In the left panel of this figure we present the results when defining subhalo mass at the time when they first became subhalos, i.e., at their time of first accretion,. The right panel of the same figure presents the results when defining the mass of subhalos when they reached the maximum mass over their main branch,. Following, in this paper we parametrise the mean cumulative number of subhalos at a given host halo mass,, as $$\Nsub=\mu\_0\left(\frac{\mu}{\mu\_1}\right)^a\exp\left[-\left(\frac{\mu}{\mu\_{\rm cut}}\right)^b\right], \label{nsub\_mvir}$$ $$\mu\_0 = (\mvirtw)^c$$ where $\mu=\msub/\mvir$ and μ0 is a normalisation term that depends on. We use the above functional form for both and. For we find that a = − 0.777, b = 1.210, μ1 = 0.030, $\mu\_{\rm cut}= 0.199$ and c = 0.102. For we find that a = − 0.749, b = 1.088, μ1 = 0.042, $\mu\_{\rm cut}= 0.199$ and c = 0.118. It is interesting to see the consistency of Equation ([nsubmvir]) with the values reported in the literature based on the spatial clustering of galaxies. Previous studies have concluded that the mean occupation number of satellite galaxies above some stellar mass M\* increases roughly proportionally to halo mass, i.e., $N\_{\rm sat}\propto \mvir$. A recent study to redshift z = 1.2 found $N\_{\rm sat}\propto M\_{\rm vir}^\alpha$ with favored values of α ≈ 0.95 to 1.2. Ignoring the exponential term in Equation ([nsubmvir]) for $\msub \lesssim 0.1\mvir$, we find that mean occupation number of subhalos (above some mass $M\_{\rm sub}$) is $N\_{\rm sub} \propto M\_{\rm vir}^{c-a} \approx M\_{\rm vir}^{0.9}$, which is consistent with observations. Note that in Equation ([nsubmvir]) the parameter μ1 gives the typical fractional mass of the most massive subhalo relative to the host halo for Milky Way sized halos (i.e., $\mvir\sim10^{12}h^{-1}\msun$), and thus $\mu\_1\times\mvir$ gives the typical mass of the most massive subhalo. When defining subhalo masses at the time of their first accretion, we find that the typical mass of the most massive subhalo is ∼ 3% of its host halo mass. For Milky Way mass halos, the most massive subhalo typically has a mass of $\macc\sim3\times10^{10}h^{-1}\msun$, which corresponds to a stellar mass of $\sim10^{7.9}\msun$ based on abundance matching results. This is more than an order of magnitude lower than the LMC. When defining subhalo masses as $\mpeak$, we find that the typical mass of the most massive subhalo is ∼ 4% of its host halo, which is also more than an order of magnitude less than the LMC. Indeed, only a small fraction of Milky Way mass galaxies have a satellite as massive as the LMC, which has a total mass ∼ 1011M⊙ including its dark matter halo. Note that the number of subhalos of mass between $\log\msub$ and $\log\msub+d\log\msub$ residing in halos of mass $\mvir$, usually referred as the conditional subhalo mass function, can be obtained by simply differentiating $$\Phisub =\frac{d\Nsub}{d\log\msub}.$$ We can therefore use this definition to infer the subhalo mass function: $$\frac{d\nsub}{d\log\msub} = \int \Phisub \frac{d\nh}{d\log\mvir}d\log\mvir. \label{nsub\_HOD}$$ Figure [suhmfcomp] shows the comparison between the measured subhalo mass function from the BolshoiP, SMDPL and MDPL simulations and by computing $d \nsub / d \log \msub$ when using Equation ([nsubHOD]) both for and. We find that Equation ([nsubHOD]) provides an accurate prescription for the subhalo mass function. Halo and subhalo velocity function ================================== c c c c c c χi & χ0, i & χ1, i & χ2, i & α1 & α2 log(A/E(z)) & 4.785 & -0.207 & 0.011 & 0.897 & 1.856 a & -1.120 & 0.394 & 0.306 & 0.081 & 0.554 b & 1.883 & -0.146 & 0.005 & 1 & 2 logV0 & 2.941 & -0.169 & 0.002 & 1 & 2 [T5] ![The ratio of the distinct halo number densites between the Bolshoi-Planck n_{\rm BP} and the Bolshoi n_{\rm B} simulations as a function of v at z=0, 1,2,4,6 and z=8. ](f10 "fig:") [f10] The comoving number density of distinct halos with maximum circular velocity between logvmax and logvmax + dlogvmax—i.e., the maximum circular velocity function $(d\nh/d\log{v\_\mathrm{max}})$—is recognized as a sensitive probe of dark matter. Figure [f12] shows the maximum circular velocity function $d\nh/d\log{v\_\mathrm{max}}$ from z = 0 to z = 8 for the Bolshoi-Planck and MD-Planck simulations. The upper panel of this figure shows the product $E(z)\times V^{3}\_{\rm vmax} \times\ d\nh/d\log{v\_\mathrm{max}}$ while the bottom panel shows the product $E(z) \times\ d\nh/d\log{v\_\mathrm{max}}$. Recall that E(z) is the expansion rate, Equation ([E(z)]). In this paper we parametrize the velocity function as $$\frac{d\nh}{d{v\_\mathrm{max}}} = A V\_{\rm max}^{-2} \left[ \left(\frac{{v\_\mathrm{max}}}{V\_0}\right)^{-a} + 1 \right] \exp\left[{-\left(\frac{{v\_\mathrm{max}}}{V\_0}\right)^{b}} \right]. \label{Vmax-fcn}$$ We assume the following redshift dependence for the parameters χi = log(A/E(z)), a, b and logV0 χi = χ0, i + χ1, izα1 + χ2, izα2. Table [T5] lists the best fit parameters for the velocity function. Figure [f10] shows the ratio of the number densities of distinct halos between the Bolshoi-Planck, $n\_{\rm BP}$, and the Bolshoi $n\_{\rm B}$ simulations as a function of v at z = 0, 1, 2, 4, 6 and z = 8. At z = 0, on average, there are ∼ 25% more halos with vmax = 200 km/s in the Bolshoi-Planck than in the Bolshoi simulation, and this stays practically constant for most vmax values up to z = 2. This fraction increases at z = 4, 6 and more drastically at z = 8 where we find that there are ∼ 60, 78 and 258% more vmax = 200 km/s halos in the Bolshoi-Planck than in the Bolshoi simulation. ![Redshift evolution of the subhalo maximum circular velocity function, with circular velocity V_{\rm acc} measured at accretion. ](f17a "fig:") [f17a] ![Redshift evolution of the subhalo circular velocity function, as a function of the subhalo’s peak circular velocity V_{\rm peak}. ](f17 "fig:") [f17] c c c Parameter & & C0 & -0.6768 & -0.5800 C1 & 1.3098 & 1.5905 C2 & -1.1288 &-1.1360 C3 & 0.0090 & -0.0378 C4 & 0.214820 & 0.18092 $ \alpha\_{{\rm sub},1}$ & 1.1375 & 1.1583 $ \alpha\_{{\rm sub},2}$ & 0.5200 & 0.5806 V0 [km s− 1] & 0.2595 & 0.5410 V1 [km s− 1] & 3.5144 & 3.4335 V2 [km s− 1] & -2.8817 & -3.0026 V3 [km s− 1] & -0.3910 & -0.3687 V4 [km s− 1] & 0.8729 & 0.9450 [T6] Figure [f17a] shows the redshift evolution of the subhalo maximum circular velocity function, $d\nsub / d\log\vsub$, at the time at the time of their accretion,. Similarly, Figure [f17] shows $d\nsub / d\log\vsub$ for. Recall that $\vpeak$ is defined as the maximum circular velocity reached along the main progenitor assembly. found that is a better proxy for galaxy stellar mass/luminosity than alternatives such as or. The solid lines in Figures [f17a] and [f17] show the best fits to $d\nsub / d\log\vsub$. We motivate the fitting functional form for $d\nsub / d\log\vsub$ using the same arguments as for $d\nsub / d\log\msub$. As we will show later, the mean number of subhalos above some maximum circular velocity is nearly independent of host maximum velocity. To be specific, we parameterize $d\nsub / d\log\vsub$ using the following functional form: $$\frac{d\nsub}{d\log\vsub}=C\_{\rm sub}(z) G(\vsub,z)\frac{d\nh}{d\log{v\_\mathrm{max}}}, \label{Vsubfcn}$$ where $$\log C\_{\rm sub}(z) = C\_0 + C\_1 a + C\_2 a^2 + C\_3 a^3 + C\_4 a^4,$$ and $$G(\mvir,z) = X^{\alpha\_{{\rm sub},1}} \exp ( - X^{\alpha\_{{\rm sub},2} }),$$ where $X = {v\_\mathrm{max}}/ V\_{\rm cut}(z)$. The function fit form for $ V\_{\rm cut}(z)$ is given by $$\log(V\_{\rm cut}(z)) = V\_0 + V\_1 z + V\_2 z^2 + V\_3 z^3 + V\_4 z^4.$$ ![image](f15a) ![Mean cumulative number of subhalos of maximum circular velocity V_{\rm sub} for host halos with maximum velocities {v_\mathrm{max}}= 200, 500, 1000 and 1580 km /s as a function of V_{\rm sub}/V_{\rm max} for (left panel) V_{\rm sub} = V_{\rm acc}, and (right panel) V_{\rm sub} = V_{\rm peak}. The dotted curve is the fitting function Equation ([vsubmvir]). ](f15 "fig:") [f15] Figure [f15] shows the mean cumulative number of subhalos of maximum circular velocity for various host halos with maximum circular velocities vmax = 200, 500, 1000 and vmax = 1580 km/s for the BolshoiP, SMDPL and MDPL simulations (dashed, solid and long dashed lines respectively). In the left panel of this figure we present the results when defining $\vsub = \vacc$. The right panel of the same figure presents the results when defining $\vsub = \vpeak$. Analogously to the mean cumulative number of subhalos at a given host halo mass,, we parameterize the mean cumulative number of subhalos above some maximum circular velocity at a given host halo maximum circular velocity as: $$\Nvsub=\left(\frac{\mu}{\mu\_1}\right)^a\exp\left[-\left(\frac{\mu}{\mu\_{\rm cut}}\right)^b\right], \label{vsub\_mvir}$$ where $\mu=\vsub/{v\_\mathrm{max}}$. We use the above functional form for both and. The resulting best fitting parameters when using are a = − 3.0881, b = 7.545, μ1 = 0.356 and $\mu\_{\rm cut}= 0.736$, while for are a = − 3.045, b = 8.850, μ1 = 0.416 and $\mu\_{\rm cut}= 0.738$. Similarly to Equation ([nsubmvir]), the parameter μ1 in Equation ([vsubmvir]) gives the typical fractional maximum circular velocity of the most massive subhalo relative to the host halo. For host halos with velocities of ∼ 200 km/s (MW-sized halo) the typical most massive subhalo has a velocity of $\vacc\sim 71.2$ km/s and of $\vpeak\sim 83.2$ km/s. These values are more consistent with the velocity of the LMC. See also which compared the number of satellites as massive as the LMC/SMC in the Bolshoi simulation with the number observed in MW-size galaxies in SDSS. The number of subhalos of maximum circular velocity between $\log\vsub$ and $\log\vsub+d\log\vsub$ residing in halos of maximum circular velocity vmax, referred as the conditional subhalo maximum circular velocity function, is given by $$\Vhisub =\frac{d\Nvsub}{d\log\vsub}.$$ Using this definition we can thus derive the maximum circular velocity function as: $$\frac{d\nsub}{d\log\vsub} = \int \Vhisub \frac{d\nh}{d\log{v\_\mathrm{max}}}d\log{v\_\mathrm{max}}. \label{nvmax\_HOD}$$ Halo concentration and spin =========================== Halo concentrations ------------------- ![Halo concentration as a function of at z = 0,1, 2, 4, and 6. The left panel in the figure shows halo concentrations calculated by finding the scale radius, R_s assuming a NFW profile in the simulation. For comparison we added the halo concentration-mass relations reported in for all halos selected by mass and from for relaxed halos. The right panel shows Klypin halo concentrations from determining the scale radius, R_s using the {v_\mathrm{max}} and \mvir relationship from the NFW formulae (see text). Solid lines in the left panel show the resulting Klypin concentrations by solving Equation ([RsKlypin]) and using the best fitting values for the {v_\mathrm{max}}-\mvir relation from Section [vmaxsection]. For comparison we added the halo concentration-mass relations reported in for all halos selected by v. ](f3 "fig:") [f3] High resolution N − body simulations have shown that the density profile of dark matter halos can be well described by the profile, $$\rho\_{\rm NFW}(r) = \frac{4\rho\_s}{(r/R\_s)(1+r/R\_s)^2}.$$ The scale radius Rs is the radius where the logarithmic slope of the density profile is -2. The NFW profile is completely characterized by two parameters, for example ρs and Rs, or more usefully the halo mass,, and its concentration parameter,. The concentration parameter is defined as the ratio between the virial radius $R\_{\rm vir}$ and the scale radius Rs: $$\cvir = \frac{R\_{\rm vir}}{R\_s}.$$ Figure [f3] shows halo concentrations,, as a function of for redshifts z = 0, 1, 2, 4, and 6. The left panel of the figure shows halo concentrations calculated by finding the best scale radius, Rs assuming a NFW profile for each halo in the simulation. Instead, the right panel shows halo concentrations calculated by determining the scale radius, Rs using the vmax and $\mvir$ relationship from the NFW formulae, see and ;. For the NFW profile, the radius at which the circular velocity is maximized is $R\_{\rm max} = 2.1626 R\_s$, and it can be shown that $$\frac{\cvir}{f(\cvir)} = V\_{\rm max}^2 \frac{R\_{\rm vir}}{G M\_{\rm vir}} \frac{2.1626}{f(2.1626)} \label{RsKlypin}$$ where $$f(x) \equiv \ln(1+x) - \frac{x}{1+x}.$$ The Klypin concentration $c\_{\rm vir,K}$ can be found be solving Equation ([RsKlypin]) numerically. It is more robust than determining Rs by fitting the NFW profile, especially for halos with few particles, since halo profiles are not well determined both at distances comparable to the simulation force resolution and also at large distances near $R\_{\rm vir}$. Figure [f3] shows that at high redshifts NFW concentrations are systematically lower than Klypin concentrations. Fitting functions for $c\_{\rm vir,K}$ are given in for all halos and for relaxed halos, for both Bolshoi-Planck/MultiDark-Planck and Bolshoi/MultiDark simulations; fitting functions are also given there for concentrations of halos defined by the 200c overdensity criterion. Key processes that drive the evolution of halo concentration are also discussed there. discusses the relation between halo concentration, the slope of the fluctuation power spectrum and the peak height. The solid lines in the left panel of Figure [f3] show the resulting Klypin concentrations by solving Equation ([RsKlypin]) and using the best fitting values for the ${v\_\mathrm{max}}-\mvir$ relation from Section [vmaxsection], see Equation ([vmax-mh]). At z = 0 and z = 1 the resulting concentrations are in very good agreement with what is found in the simulation with an accuracy of ∼ 3% for halos above $\mvir = 10^{10}h^{-1}\msun$. However, at higher redshifts z = 2, 4, 6, our predicted Klypin concentrations have an accuracy of ∼ 10%. Wouldn’t it be interesting to also compute concentrations from Equation [RsKlypin] by using and v at instead of and v: $$\frac{\cvir}{f(\cvir)} = V\_{\rm max}^2(\mpeak) \frac{R\_{\rm vir}(\mpeak)}{G \mpeak} \frac{2.1626}{f(2.1626)}.$$ Halo Spin --------- ![Spin parameter as a function of at z = 0, 2, 4,6, and 8. Medians are shown as the solid lines. At z = 0 the grey area is the 68\% range of the distribution. The left panel of this figure shows the spin parameter calculated using Equation ([lambdaP]) while the right panel shows the spin parameter calculated using Equation ([lambdaB]). ](f3a "fig:") [f3a] ![image](lambda_distr_log) [lambdadistrb] c c c c c Simulation & $\sigma\_{\rm P}$ & $\log\lambda\_{\rm 0, P}$ & $\sigma\_{\rm B}$ & $\log\lambda\_{\rm 0, B}$ BolshoiP & 0.248 & -1.423 & 0.268 & -1.459 SMDPL & 0.249 & -1.435 & 0.268 & -1.471 MDPL & 0.250 & –1.438 & 0.271 & –1.443 [Tspin1] c c c c c c c Simulation & $\alpha\_{\rm P}$ & $\beta\_{\rm P}$ & $\log\lambda\_{\rm 0, P}$ & $\alpha\_{\rm B}$ & $\beta\_{\rm B}$ & $\log\lambda\_{\rm 0, B}$ BolshoiP & 4.126 & 0.610 & -2.919 & 3.488 & 0.6042 & -2.878 SMDPL & 4.090 & 0.612 & -2.917 & 4.121 & 0.611 & -2.916 MDPL & 4.047 & 0.612 & -2.914 & 3.468 & 0.591 & -2.907 [Tspin2] The left panel of Figure [f3a] shows the medians for the spin parameter $\lambda\_{\rm P}$ as a function of at z = 0, 2, 4, 6, and 8. The spin parameter for every halo in the simulations was calculated using the definition : $$\lambda\_{\rm P} = \frac{J \| E \|^{1/2}} { G M\_{\rm vir}^{5/2}}, \label{lambdaP}$$ where J and E are the total angular momentum and the total energy of a halo of mass. As others have found, the spin parameter $\lambda\_{\rm P}$ correlates only weakly with halo mass especially at z = 0. The median value for Milky Way mass halos (i.e., with $\mvir \sim 10^{12} h^{-1}\msun$) at z = 0 is $\lambda\_{\rm P} \sim0.036$, and it decreases a factor of ∼ 1.8 at z = 6, that is, $\lambda\_{\rm P} \sim0.02$. For Milky Way mass halos, the dispersion is approximately ∼ 0.24 dex at z = 0 and it decreases to ∼ 0.16 dex at z = 6. Note that the dispersion is not symmetric, meaning that the distribution of $\lambda\_{\rm P}$ is not a lognormal distribution. This is consistent with previous findings based on high resolution N − body simulations. The right panel of Figure [f3a] shows the spin distribution calculated using the alternative definition : $$\lambda\_{\rm B} = \frac{J}{\sqrt{2} M\_{\rm vir} V\_{\rm vir} R\_{\rm vir}}, \label{lambdaB}$$ which can be obtained from Equation ([lambdaP]) by assuming all particles to be in circular orbits. Similarly to $\lambda\_{\rm P}$, the spin parameter $\lambda\_{\rm B}$ correlates only weakly with halo mass especially at z = 0. We found that the median value for Milky Way mass halos at z = 0 is $\lambda\_{\rm P} \sim0.035$ and it decreases to $\lambda\_{\rm P} \sim0.027$ at z = 6. For Milky Way mass halos, the dispersion of $\lambda\_{\rm B}$ is slightly larger than of $\lambda\_{\rm P}$; we find that it is ∼ 0.27 dex at z = 0 and it decreases to ∼ 0.2 dex at z = 6. The spin parameter $\lambda\_{\rm B}$ slightly increases at high redshifts especially for low mass halos, $\mvir\lesssim10^{12}\msun$. In contrast, the value of the spin parameter $\lambda\_{\rm P}$ shows a systematic decrease as redshift increases. This was previously noted over the interval z = 0 − 2 by, who attribute the different evolution of the two spin parameters mainly to different effects of minor mergers on $\lambda\_{\rm P}$ and $\lambda\_{\rm B}$. Figure [lambdadistrb] quantifies in more detail the distribution of halo spins separately for the BolshoiP, SMDPL and MDPLsimulations. In order to avoid resolution effects and to obtain reliable statistics, we calculate the distribution of halo spins in the halo mass range $10^{11}-10^{14}h^{-1}\msun$ for the BolshoiP (upper left panel) and SMDPL (bottom left panel) simulations, while for the MDPL (bottom right panel) simulation we do the same but for the mass range $10^{12}-10^{14}h^{-1}\msun$. In all the panels the grey filled circles show the distribution for $\lambda\_{\rm P}$ while the red filled circles show the distribution for $\lambda\_{\rm B}$. As anticipated from the $\lambda-\mvir$ relationship, the logλ distributions are asymmetrical. This is more evident for $\lambda\_{\rm P}$ than for $\lambda\_{\rm B}$. In order to quantify this we try to fit all the distributions using a lognormal probability distribution: $$P(\log\lambda) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{\log^2(\lambda/\lambda\_0)}{2\sigma^2}\right). \label{logn}$$ The best fit parameters of P(logλ) both for λP and for λB are listed in Table [Tspin1]. We find that while the lognormal distribution gives a fairly good description for P(logλB) this is not the case for P(logλP) for all the simulations. In particular, the distribution has too many halos with low values of λP. In order to provide a more accurate description of the halo distribution we propose to use a Schechter-like function given by P(logλ) = Af(λ), where $$f(\lambda) = \left(\frac{\lambda} {\lambda\_0}\right)^{-\alpha} \exp\left[-\left(\frac{\lambda} {\lambda\_0}\right)^\beta\right],$$ A = [∫− ∞∞f(λ)dλ]− 1. The best fit parameters for BolshoiP, SMDPL and MDPL simulations for both λP and λB are listed in Table [Tspin1] for the log-normal distribution and Table [Tspin2] for the Schechter-like distribution. We find that a Schechter-like function gives a more accurate prescription for the distribution of $\lambda\_{\rm P}$ than for $\lambda\_{\rm B}$. In particular, this distribution has some problems in reproducing the tail of high $\lambda\_{\rm B}$ that declines more like a lognormal distribution. It is thought that the angular momentum of galaxies is related the angular momentum of dark matter halos and thus to their spin parameter. Under this assumption, the scale length of disk galaxies, $R\_{\rm d}$, can be obtained in terms of λ and $R\_{\rm vir}$. Specifically the relation is given by $R\_{\rm d} \propto \lambda \times R\_{\rm vir}\propto \lambda \times M\_{\rm vir}^{1/3}$. As before, if we assume for simplicity that the $M\_b/\mvir$ ratio is constant, the relation between a galaxy’s radius and its baryonic mass is given by $R\_{\rm d}\propto \lambda \times M\_b^{1/3}$. Note that the scatter of the size–mass relation is just the resulting scatter of the $\lambda-\mvir$ relation. Indeed, the dispersion of the spin parameter, either $\lambda\_{\rm P}$ or $\lambda\_{\rm B}$, is very similar to the observed dispersion of disk galaxy scale lengths at least at low redshifts where reliable measurements can be obtained. Note that the different redshift evolution of the $\lambda\_{\rm P}-\mvir$ and $\lambda\_{\rm B}-\mvir$ relations leads to different predictions of the $R\_{\rm d} - M\_{\rm b}$ relation and its evolution. In particular, models of galaxy formation calculating galaxy sizes based on the spin parameter $\lambda\_{\rm B}$ will result in more extend galaxies (and potentially in larger numbers of low surface brightness galaxies) at high redshifts compared to those models using $\lambda\_{\rm P}$. Is also possible that galaxy star formation rates could be affected since more extended galaxies presumably have lower gas surface densities than more compact disks, and thus lower SFRs according to the Kennicutt-Schmidt law. Two recent papers have discussed the evolution of galaxy sizes out to redshift z ∼ 8 using Hubble Space Telescope images, mainly from the CANDELS survey. finds that the median effective radius re evolves with redshift as re ∝ (1 + z)− 1.3, with no evolution in the slope, the median Sérsic index (n ∼ 1.5), or the standard deviation of the log-normal distribution. They find that the ratio of the effective radius to the virial radius of the halos is nearly constant at $r\_e/R\_{\rm vir} = 0.01 - 0.035$. This is just what one would expect from the lack of redshift evolution in $\lambda\_{\rm B}$, while the factor of ∼ 2 decline in $\lambda\_{\rm P}$ from z = 0 to 8 would predict a corresponding decline in the ratio $r\_e/R\_{\rm vir}$. The other recent paper,, finds a slower decline of effective radius with redshift, and in fact cannot reject the possibility that there is no size evolution. This is possibly consistent with the modest increase with redshift of $\lambda\_{\rm B}$ for lower mass halos, and inconsistent with the expected decrease in $r\_e/R\_{\rm vir}$ from the decline in $\lambda\_{\rm P}$. The radii of these high-redshift galaxies are being measured in rest-frame UV, which is typically rather clumpy. It will be very interesting to see what sort of galaxy size evolution with higher redshifts is revealed by James Webb Space Telescope at rest-frame optical wavelengths. On the evolution of vmax − M\* ================================== ![Evolution of the velocity function v for fixed {v_\mathrm{max}}= 100, 190 and 450 km /s for the BolshoiP, SMDPL, and MDPL simulations. The solid lines are the fits to Equation ([Vmax-fcn]). For low velocities the velocity function is practically constant after redshift z\sim4, while for high velocity halos it is nearly constant after redshift z\sim1. ](f20 "fig:") [Vmaxcte] ![Cosmic stellar mass density since z\sim9. Filled circles show the observations compiled in. The solid curves show the predicted cosmic stellar mass density using fits to the Tully-Fisher and Faber-Jackson velocity-to-stellar-mass relations as described in the text, assuming that these relations are independent of redshift and that the v of dark matter halos is the same as the V_{\rm max,g} of galaxies. The solid black line shows the predicted cosmic stellar mass density for a range of stellar masses \log(M_\ast/M_\odot) = 8.5 - 11.25. Comparing the red curve, for \log(M_\ast/M_\odot) = 7 - 11.25, to the black curve shows that including lower stellar masses increases \rho_\ast more at high redshifts; comparing the blue curve, for \log(M_\ast/M_\odot) = 7 - 12, to the black and red curves shows that including higher stellar masses increases \rho_\ast more at low redshifts. Clearly, all of these predictions are inconsistent with observations at z\grtsim 1—they produce too much stellar mass density at early redshifts, and the wider stellar mass range represented by the blue curve exceeds the observed stellar mass density even at z=0. Since the stellar mass function is evolving, velocity-mass relations like Tully-Fisher must also evolve. The dotted lines show the predictions when using a model in which the maximum circular velocity-to-stellar mass relation evolves with redshift as described in the text. ](rho_star_multi "fig:") [TFfig] Early determinations of the evolution in the maximum circular velocity and the stellar mass/luminosity relations—the Tully-Fisher relation for spiral galaxies and the Faber-Jackson relation for ellipticals—have found only a weak evolution from z ∼ 0.85 to z ∼ 0. This result has been further supported and generalized in from z ∼ 1 to z ∼ 0, based on fairly large samples of galaxies from AEGIS and DEEP2 and adopting the indicator $S\_{0.5} ^2= 0.5 V^2\_{\rm max} + \sigma\_g^2$ which accounts for disordered motions. On the other hand, observations indicate that at $z \lesssim 1.5$ the number density of star forming galaxies at a fixed velocity evolves very little, while the number density of quiescent galaxies evolves more rapidly. Figure [Vmaxcte] shows that the comoving number density of low circular velocity halos is nearly constant since z ∼ 4 but high mass/velocity halos have more evolution. Halos of a given circular velocity at high redshift are lower in mass but denser than halos of the same circular velocity at lower redshift. To what extent is the nearly constant comoving number density of halos as a function of their circular velocity consistent with the weak evolution of the Tully-Fisher relation? This is particularly interesting if the galaxy stellar mass function evolves with redshift, as was first pointed out in. In this section we investigate the above question assuming that the Tully-Fisher and Faber-Jackson relations do not evolve with redshift and that there is a one-to-one correspondence between the maximum circular velocity of halos and galaxies, i.e., ${v\_\mathrm{max}}= V\_{\rm max,g}$. To do so, our first step is to convert the Tully-Fisher and Faber-Jackson relations into circular velocities. Arguments based on the Jeans equation in virialized systems result in the relation Vc = Kσ, where typical values for K are $\sqrt{2}-\sqrt{3}$. While there is an extensive discussion in the literature of what is the right value for K, following here we assume that K = 1.54 which is a value halfway between different groups. The next step is to derive an average maximum circular velocity-to-stellar mass relation for all galaxies: $\langle\log V\_{\rm max,g} \rangle= \langle\log V\_{\rm max,g}\rangle(\log M\_\)$. The method is to use the average Tully-Fisher and Faber-Jackson relations for local galaxies and take into account the observed fraction of disk and elliptical galaxies. For simplicity, we assume that all disk galaxies are star-forming systems while ellipticals correspond to quiescent galaxies. Then the average maximum circular velocity is given by $$\langle\log V\_{\rm max,g}\rangle =f\_{\rm SF} \langle\log V\_{\rm max,TF}\rangle + f\_{\rm Q}\langle\log V\_{\rm max,FJ}\rangle. \label{Vmax\_gal}$$ where $\langle\log V\_{\rm max,FJ}\rangle = \langle \log(1.54\sigma)\rangle$ and $f\_{\rm SF} =1 - f\_{\rm Q} $. Note that the above equation depends on stellar mass. We take the fraction of quiescent galaxies $f\_{\rm Q}$ from, and we use the fits for the Tully-Fisher and Faber-Jackson relations reported in. The fraction of quiescent galaxies $f\_{\rm Q}$ has been taken from. We assume for simplicity that the maximum circular velocity of dark matter halos, v, corresponds to the maximum circular velocity of galaxies, $V\_{\rm max,g}$, i.e., ${v\_\mathrm{max}}= V\_{\rm max,g}$. In this way, we can then solve Equation ([Vmaxgal]), $\langle\log V\_{\rm max,g} \rangle= \langle\log V\_{\rm max,g}\rangle(\log M\_\)$, for M\* and thus transform velocities into stellar mass. The black solid line in Figure [TFfig] shows the predicted cosmic stellar mass density for galaxies with stellar mass $M\_{\} = 10^{8.5} - 10^{11.25} \msun$ since z* = 9 assuming that $\langle\log V\_{\rm max,g}\rangle(\log M\_\)$ is independent of redshift, and the red and blue curves are the same for wider ranges of stellar masses. For comparison, we plot a recent compilation from of the evolution of the observed stellar mass density. Our simple model represented by the black curve seems to be roughly consistent with the observational evidence of the weak evolution of the maximum circular velocity since z* ∼ 1. In contrast, at high redshifts the model produces far too many stars. We thus conclude that a strong evolution of the Tully-Fisher and Faber-Jackson relations is required to higher redshifts in order to reconcile the predicted cosmic stellar mass density with observations. Based only on theoretical arguments, is it possible to derive a simple model for the redshift evolution for $\langle\log V\_{\rm max,g}\rangle(\log M\_\)$? In Section 3.1, we found that the evolution of the maximum circular velocity of dark matter halos is well described by ${v\_\mathrm{max}}\ \propto\ \left[ \mvir E(z)\right]^{\alpha}$. In other words, the zero point of the maximum circular velocity evolves with E(z)α. If we adopt the same reasoning, we can assume that the zero point of the Tully-Fisher and Faber-Jackson relations evolve with redshift as $E(z)^{\alpha\_{\rm TF}}$ and $E(z)^{\alpha\_{\rm FJ}}$ respectively, where $\alpha\_{\rm TF}=0.259$ and $\alpha\_{\rm FJ}=0.37$ are their corresponding slopes at z* = 0. The dotted line in Figure [TFfig] shows the predicted cosmic star formation rate density based on this simple evolutionary model. Despite the simplicity of this model, the predictions are much more consistent with observations at high redshifts than the non-evolving model that led to the solid black line in the figure. Nevertheless, the above models are very simple and they ignore the fact that the v of dark matter halos is not the $V\_{\rm max,g}$ of galaxies, as we discuss in the next section. Observed Velocity Function of Nearby Galaxies ============================================= ![Comparison of the predicted Local Volume 3D velocity function dN/d \log V from the BolshoiP simulation with the observed Local Volume optical velocity function of galaxies within \sim10 Mpc and the HI radio velocity function from the ALFALFA survey. ](vf_local_group "fig:") [V-function] Previous studies have considered that the observed distribution of galaxy velocities is a strong test for galaxy formation models and cosmology. The reason is simply because the comparison, at a first order, between the theoretical halo+subhalo velocity function and the observed velocity function of galaxies is more direct than the stellar mass/luminosity and halo+subhalo mass functions. In this section, we compare the local volume galaxy velocity function derived from optical galaxy observations in and the HI radio galaxy velocity function based on the ALFALFA survey from to compare with the theoretical halo+subhalo velocity function from ΛCDM with the Planck cosmological parameters. In the past, a number of works have studied the velocity function of halos+subhalos from high resolution N-body simulations to conclude that it differs from the observed galaxy velocity function by overpredicting the number of low velocities objects. Actually, these differences are not surprising since a more careful comparison between the theoretical halo+subhalo and galaxy velocity distributions (1) must include the effects of the baryons on the velocity profile of the halo/subhalo, and (2) consistently compare the radii at which galaxies and halo/subhalo velocities (usually v) are measured. Indeed, including the effects of the baryons are important since they could increase the maximum circular velocity as a result of their gravitational effect over the halo. For item (2) we note that local disk galaxies (which are the most extended objects in the local universe) typically have scale lengths between $R\_{\rm d}\sim1- 10$ kpc in the stellar mass range $10^9-10^{11.5}\msun$. This would imply that vmax of galaxies would be observed between R(vmax) ∼ 2 − 20 kpc, for the ideal disk $R({v\_\mathrm{max}})\sim2.2\times R\_{\rm d}$. This is actually markedly different from dark matter halos. Assuming that all halos follow a NFW profile, their maximum rotational velocity is reached at ∼ 2.16 × Rs. Based on the concentrations obtained in Section [concentrations], this would imply that v is reached between R(vmax) ∼ 10 − 300 kpc for halos between 1011 − 1014. Clearly using v of halos would result in an overestimation of the true maximum circular velocity of the galaxy. Although a more proper modeling of these effects requires sophisticated structural and dynamical models of galaxies like those described in recent papers, in this paper we follow a more empirical approach based on abundance matching. Our goal is to derive a correlation between the maximum circular velocity of galaxies, $V\_{\rm max,g}$, and dark matter halo/subhalos vmax without further modeling of galaxy formation. We summarize our algorithm as follows: 1. To each halo/subhalo in the BolshoiP simulation a galaxy with stellar mass M\* is assigned randomly from the probability distribution function $P(M\_{\}\|\vpeak)$. This probability distribution function is assumed to be lognormal with mean $\langle\log M\_{\} \rangle= \langle\log M\_{\} \rangle(\log \vpeak)$ obtained from abundance matching with halo/subhalo property. The scatter of the distribution is assumed to be constant with with a value of 0.15 dex. 2. The next step is to define $P\_V(V\_{\rm max,g}\|M\_\)$ as the lognormal probability distribution that a galaxy of mass M\* has an observed velocity of $V\_{\rm max,g}$. The mean of this distribution, $\langle\log V\_{\rm max,g} \rangle= \langle\log V\_{\rm max,g}\rangle(\log M\_\)$, is given by Equation ([Vmaxgal]). We asume that the scatter of the distribution is constant with M\ with a value of 0.08 dex. 3. $V\_{\rm max,g}$ assignment is based on the argument that at a fixed M\, larger $\vpeak$ corresponds to larger $V\_{\rm max,g}$. This assumption is reasonable since the halo contributes to the total velocity of the galaxy, $V\_{\rm max,g}$. More formally, we obtain galaxy velocities by solving the following equation for $V\_{\rm max,g}$ for a given M\: $$\int^\infty\_{V\_{\rm max,g}} P\_V(V\_{\rm max,g}'\|M\_\) dV\_{\rm max,g}' = \int^\infty\_{V\_{\rm peak}} P(V\_{\rm peak}'\|M\_\)dV\_{\rm peak}'.$$ In the last equation $P(V\_{\rm peak}\|M\_\)$ is the inverse of the distribution function $P(M\_{\}\|\vpeak)$. A few comments are necessary here: In (i) we use a galaxy stellar mass function that has been corrected for low surface brightness incompleteness in the SDSS and measured over the range between $\log M\_{\} / \msun =10^{7.6}-10^{12.2}$ (Rodríguez-Puebla et al. in prep). In (ii) we assume that the mean relation given by Equation ([Vmaxgal]) is valid in the same mass regime. The above procedure ensures that each halo/subhalo in the BolshoiP simulation will host a galaxy with stellar mass M\ and galaxy velocity $V\_{\rm max,g}$. Recall that the BolshoiP simulation is complete for halos with $V\_{\rm max} \grtsim 50$ km/s. The Local Volume is a sample of galaxies in a sphere of ∼ 10 Mpc centered on the Milky Way. showed that the luminosity function of the Local Volume is consistent with the luminosity function of the local galaxies in the SDSS. This reflects the fact that the Local Volume is not biased to extreme environments such as voids or clusters. In order to define Local Volume analogs in the BolshoiP, we have not carried out an exhaustive search to find similar environments to the Local Volume. Instead, Local Volume analogs are selected by centering spheres of 10 Mpc on galaxies with stellar masses in the bin $\log (M\_{\} /\msun)\in[10.64,10.84]$ corresponding to the mass of the Milky-Way, where we have used abundance matching to assign stellar masses to halos in BolshoiP. The upper panel of Figure [V-function] shows the predicted velocity function. The black solid line in the figure shows the median velocity function from all the Local Volume analogs found in the simulation, while the dashed curve shows the average. In order to get a sense of the most common configurations of the Local Volume under our definition, we also present the 68% range of the distribution as the grey shaded area. The width of the distribution as a function of mass at low velocities is of the order of ∼ 0.25 dex. For comparison, the green solid line shows the fitting function to the observed Local Volume velocity function reported in. Taking the fairly large dispersion into account, we see that the predicted velocity function is consistent with observations for galaxies above ∼ 50 km/s. We also find agreement when comparing with the HI radio galaxy velocity function based on the ALFALFA survey from. In the same figure we show predictions of the velocity function decomposed into centrals and satellites. The bottom panel in the same Figure [V-function] shows the contributions from centrals and satellites. We find that approximately ∼ 80% of the galaxies in the Local Volume are centrals. Summary & Discussion ==================== This paper presents many results, both graphically and with fitting functions, from the Bolshoi-Planck and MultiDark-Planck simulations of the large scale structure of the universe, based on the Planck cosmological parameters summarized in Table 1. Figure [s8-OmM.pdf] shows the WMAP5/7/9 and Planck constraints on the key cosmological parameters σ8 and $\Omega\_{\rm M}$, and the values adopted for these parameters in many cosmological simulations. The Bolshoi-Planck and MultiDark-Planck simulations have been analyzed using Rockstar and Consistent Trees to identify and characterize all dark matter halos in all stored time steps, and to construct merger trees of these halos. In this paper we use the virial radius $R\_{\rm vir}$ and virial mass $M\_{\rm vir}$, Equation ([Mvir]), to describe these dark matter halos. It is useful to characterize dark matter halos by their maximum or peak circular velocity, in addition to their virial mass. Figure [f1] shows the relations between $V\_{\rm max}$ and $M\_{\rm vir}$ and between $V\_{\rm peak}$ and $M\_{\rm vir}$. The slopes are roughly given by $V \propto M\_{\rm vir}^{1/3}$, and we give accurate fitting functions in Equation ([vmax-mh]). Cosmological simulations also allow determination of the mass accretion rates of the halos as a function of their virial mass and redshift. Figure [f2] shows these accretion rates three ways: Instantaneous accretion rates (i.e., between stored time steps), accretion rates averaged over the halo’s dynamical time, Equation ([tdyn]), and accretion rates of the maximum mass $M\_{\rm peak}$ along the main progenitor branch. We give power-law fitting functions in the form of Equation ([dmhdt-mh]) for all of these, and a better double-power-law fit, Equation ([dmhdt-mh-dbl]), for the instantaneous accretion rates. uses abundance matching with the Bolshoi-Planck simulation to explore the possibility that the halo mass accretion rate plays a large part in determining the star formation rate of the central galaxy in the halo, at least when this galaxy lies on the main sequence of star formation. This Stellar-Halo Accretion Rate Coevolution (SHARC) assumption predicts star formation rates on the main sequence that are in remarkably good agreement with observations for the redshift range z* = 0 to 4. The paper also shows that the ∼ 0.3 dex dispersion in the halo mass accretion rates leads to similar small dispersions in the predicted star formation rates, in rough agreement with observations. Figures [f4] and [f5] show the median halo mass growth and the maximum circular velocity growth as redshift decreases, and also the dispersions of these. It is remarkable how little change there is with redshift in the circular velocity of halos of $M\_{\rm vir}= 10^{11} M\_\odot$ at low redshifts. The Rockstar analysis of the Bolshoi-Planck simulation reliably finds all dark matter halos and subhalos with $\grtsim 100$ particles, which corresponds to $M\_{\rm vir} \grtsim 2\times10^{10} h^{-1}\msun$ or $V\_{\rm max} \grtsim 50$ km/s. All such halos are expected to host visible galaxies, so the number densities of dark matter halos and subhalos as a function of mass and redshift predict the corresponding abundances of central and satellite galaxies. We find that the formula Equation ([Tinker]) is a good approximation to the number density of distinct dark matter halos, shown in Figure [f6], and we give a fitting function Equation ([sigmamvirfit]) for the amplitude of fluctuations, shown in Figure [f6a]. The characteristic mass $M\_{\rm C}(z)$ of 1σ halos just collapsing at redshift z separates abundant halos with $M\_{\rm vir} < M\_{\rm C}$ from relatively rare halos with $M\_{\rm vir} > M\_{\rm C}$. The clustering properties of halos are also different below and above $M\_{\rm C}$, and halo properties such as their triaxial shapes scale with $M\_{\rm C}$. We plot $M\_{\rm C}(z)$ in Figure [fMPS] and give a fitting function Equation ([MPS]) for the Bolshoi-Planck cosmological parameters. The maximum circular velocity of dark matter halos is related to observable internal velocities of their central galaxies or the relative velocities of galaxies in groups and clusters, so it is useful to determine the velocity function of distinct halos. This is plotted in Figure [f12], with a corresponding fitting function Equation ([Vmax-fcn]). The Planck cosmological parameters, especially the higher $\Omega\_{\rm M}$ compared with WMAP5/7, result in a greater abundance of halos especially at high masses and redshifts. This is shown in Figure [f11]. At z = 0 there are ∼ 12% more $10^{12} h^{-1}\msun$ halos in the Bolshoi-Planck than in the Bolshoi simulation, and ∼ 25% more for $\mvir\sim3\times10^{13}h^{-1}\msun$. At z = 8 there are about 3 times as many $\mvir = 10^{11}h^{-1} \msun$ halos in Bolshoi-Planck as in Bolshoi. Similarly, there are more dark matter halos as a function of $V\_{\rm max}$ with the Planck parameters, as shown in Figure [f10]. At z = 0 to 2 there are ∼ 25% more halos with vmax = 200 km/s in the Bolshoi-Planck than in the Bolshoi simulation. This fraction increases at z = 4, 6 and 8, with ∼ 60, 78 and 258% more vmax = 200 km/s halos in the Bolshoi-Planck simulation. Figure [f16] characterizes the abundance of subhalos as a function of the mass they had at accretion, and also as a function of their peak mass along their major progenitor track, with corresponding fitting function given by Equation ([f16fitb]). We also plot the redshift evolution of the subhalo maximum circular velocity at accretion in Figure [f17a] and the subhalo peak circular velocity function in Figure [f17], with fitting functions Equation ([Vsubfcn]). It is also useful to know the number of subhalos with a given accreted mass or peak mass compared to the virial mass of the host halo. This is shown in Figure [f14], with fitting function Equation ([nsubmvir]). It is also useful to know the corresponding numbers of subhalos characterized by their circular velocities. Figure [f15] shows the number of subhalos as a function of $V\_{\rm sub}$ divided by the maximum circular velocity of the host halo, for $V\_{\rm sub}$ equal to the subhalo’s velocity at accretion $V\_{\rm acc}$ or its peak circular velocity $V\_{\rm peak}$, with fitting function Equation ([vsubmvir]). We calculate the concentration of dark matter halos two ways, by finding the best scale radius Rs assuming a NFW profile or by using Equation ([RsKlypin]). Figure [f3] shows the resulting concentrations as a function of $M\_{\rm vir}$ and redshift from the Bolshoi-Planck, SmallMultiDark-Planck (SMDPL), and MultiDark-Planck (MDPL) simulations. The halo spin parameter λ is a dimensionless way of characterizing the angular momentum of each dark matter halo. We calculate the halo spin parameter using both the definition Equation ([lambdaP]) and the definition Equation ([lambdaB]). The results as a function of $M\_{\rm vir}$ and redshift are shown in Figure [f3a]. The value of the Peebles spin parameter $\lambda\_{\rm P}$ shows a systematic decrease as redshift increases, as was previously noted by, but $\lambda\_{\rm B}$ is less dependent on redshift. Since a smaller value of λ is expected to lead to the cooling baryons becoming rotationally supported at a smaller radius, any redshift dependance could have implications for galaxy sizes as a function of redshift. The latest measurements of galaxy size evolution from HST images appear to favor the evolution expected from $\lambda\_{\rm B}$. The Tully-Fisher and Faber-Jackson relations relate rotation velocity V and velocity dispersion σ of galaxies to their stellar masses. When using V = 1.54σ and by taking into account the fraction of disk and elliptical galaxies generalize these relations to apply to observed galaxies that have comparable values of V and σ. But ΛCDM simulations show that the cumulative comoving number density of halos with galaxy-scale circular velocities are nearly constant out to rather high redshifts z ∼ 4 (see Figure [Vmaxcte]), while the stellar mass density decreases with increasing redshift. This implies that these stellar mass-velocity relations must change at redshifts $z \grtsim 1$. In Figure [TFfig] we show this, and also show a simple model of how these relations might change up to redshift z ∼ 9. In Section [LocalVolume] we compare the abundance of dark matter halos with the observed abundance of galaxies in optical and radio surveys. We show that when we take into account effects of baryons on the observed velocities and the effects of the radii where these velocities are measured, and we compare with the rather wide predicted velocity distribution for volumes of the simulation centered on Milky Way mass galaxies, the agreement between theory and observations is good for low-mass galaxies down to the ∼ 50 km/s completeness limit of our simulations, contrary to some claims in the literature. ΛCDM and observations are also in good agreement at higher masses. There are large catalogs of galaxy cluster observations using X-ray and optical surveys, and recently smaller catalogs of cluster Sunyaev-Zel’dovich detections. The key to constructing the cluster mass function from these observations is to obtain a reliable mass calibration from gravitational lensing. We have shown that the Tinker mass function, Equation ([Tinker]), is an excellent fit to the abundance of dark matter halos in our simulations, and find that the predicted and observed cluster abundance are in good agreement with the Planck cosmological parameters. Acknowledgments =============== We thank NASA Advanced Supercomputing (NAS) for access to their Pleiades supercomputer where our Bolshoi, MultiDark, and Bolshoi-Planck cosmological simulations were run, with Anatoly Klypin and Joel Primack as leaders. The MultiDark-Planck simulations were run in the SuperMUC supercomputer at the Leibniz-Rechenzentrum (LRZ) in Munich, thanks to the computing time provided by the Gauss Centre for Supercomputing (<www.gauss-centre.eu>) and the Partnership for Advanced Supercomputing in Europe (PRACE, <www.prace-ri.eu>). Stefan Gottloeber, Anatoly Klypin, Francisco Prada, and Gustavo Yepes were the leaders of the MultiDark-Planck simulation project, and we are very grateful to them for making the outputs of these simulations available to us. We also thank the Leibniz Institute for Astrophysics Potsdam (AIP) and the Spanish MultiDark Consolider project for supporting the CosmoSim database (<www.cosmosim.org>). ARP has been supported by a UC-MEXUS Fellowship. PB was partially supported by a Giacconi Fellowship from the Space Telescope Science Institute (STScI). The remainder of support for PB through program number HST-HF2-51353.001-A was provided by NASA through a Hubble Fellowship grant from STScI, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. AK, CL, and JP acknowledge support from NSF grant AST-1010033 and the CANDELS grant HST GO-12060 from STScI. We thank the anonymous Referee for a useful report that improved the presentation of this paper. We thank Kristin Riebe for giving us permission to use Fig. [MergerTrees], and we thank Vladimir Avila-Reese, Avishai Dekel, Sandra Faber, Stefan Gottloeber, Susan Kassin, David Koo, Gerard Lemson, Francisco Prada, Joe Silk, Rachel Somerville, and Frank van den Bosch for helpful discussions. Merger Tree Overview ==================== The publicly available[4](#fn4) Rockstar (Robust Overdensity Calculation using K-Space Topologically Adaptive Refinement) halo finder identifies dark matter halos based on adaptive hierarchical refinement of friends-of-friends groups of particles in six phase-space dimensions plus time. For halo masses, Rockstar calculates spherical overdensities using all the particles including any substructures in the halo. Before calculating the halo mass and $V\_{\rm max}$, the code performs an unbinding procedure using a modified Barnes-Hut method to accurately calculate particle potentials. describes many halo properties that are calculated by Rockstar, and compares results with the BDM halo finder that was also used to analyze the Bolshoi simulation. Results from BDM analyses of all the simulations in Table 1 are available online at the CosmoSim website.1 Those for the Rockstar analyses of the Bolshoi-Planck, SMDPL, and MDPL simulations can be downloaded in bulk from the UCSC Hyades system.2 Complete particle data was saved for 178 timesteps of the Bolshoi-Planck simulation, 117 timesteps of SMDPL, and 126 timesteps of MDPL. For scale factor a = 0.06 − 0.09 the scale factor change between timesteps was Δa ∼ 0.0021, for a = 0.09 − 0.14 Δa ∼ 0.0035, for a = 0.14 − 0.7 Δa ∼ 0.005, and for a = 0.7 − 1 Δa = 0.007. Thus the time interval between stored timesteps ranges from ∼ 10 Myr at z ∼ 15 to ∼ 100 Myr at z = 0. The cadence of these timesteps is shown in Figure [f2a]. ![image](f2a) [f2a] Consistent Trees generates merger trees and halo catalogs in a way that ensures consistency of halo mass, position, and velocity across time steps. This allows it to repair inconsistencies in halo catalogs—e.g., when a halo disappears for a few time steps, Consistent Trees can regenerate its expected properties by gravitational evolution from the surrounding time steps. The Consistent Trees code is publicly available,[5](#fn5) and Consistent Trees outputs used in this paper are available online.2 In these Appendixes we describe how these outputs are organized. Consistent Trees generates merger tree information in tree files (`tree_.dat`), which each contain halos and their full progenitor histories for cubical subvolumes of the simulation. It also generates catalogs (`hlist_.list`), which each contain all the halos for a single snapshot of the simulation, along with frequently-used information from their merger histories (e.g., peak mass, half-mass assembly time, mass accretion rate, etc.). These enable the user to avoid walking the trees for many common applications (e.g., abundance matching). Tree File Layout ---------------- Tree files contain header information (lines beginning with a `#` character), a single line with the total number of trees in the file, and then the actual merger trees. Each merger tree corresponds to the progenitor history of a single halo at the last snapshot of the simulation, containing all halos that fully merge (a.k.a., are disrupted) into the final halo. Specifically, subhalos that remain distinguishable from their host halo at the last simulation snapshot have separate merger trees in the tree files. Each tree file contains the merger histories for halos whose centers lie in a cubical subvolume of the simulation. For example, the Bolshoi simulation has 125 (=5 × 5 × 5) tree files, each corresponding to a (50 Mpc/h)3 subvolume of the total (250 Mpc/h)3 simulation volume. The merger trees’ basic format is a single header line with the final halo’s ID (`#tree XYZ`), also known as the tree root ID, followed by single lines for each halo in the full merger history, ordered by snapshot: ``` #tree XYZ halo XYZ @ snapshot N progenitor 1 @ snapshot N-1 progenitor 2 @ snapshot N-1 ... progenitor P @ snapshot N-1 progenitor 1 @ snapshot N-2 ... progenitor Q @ snapshot N-2 ... progenitor 1 @ snapshot 1 ... progenitor Z @ snapshot 1 ``` The format of the halo lines is described in Appendix B. Note here that the order of progenitors within a snapshot is not guaranteed; also, progenitors may not exist at all snapshots, especially if the final halo is close to the mass resolution of the simulation. Flyby halos (i.e., halos that pass in and out of the virial radius of a host, also called spashback halos) are similarly not guaranteed to be included in the merger history. For some applications (e.g., semi-analytic models, SAMs), it is necessary to process merger histories including all flyby halos and subhalos. These are also known as forests. Consistent Trees provides a `forests.list` file; each line of the file contains a tree root ID and the corresponding forest ID. All merger trees with the same forest ID belong to the same forest. To aid with collating forests from the tree files, Consistent Trees also provides a `locations.dat` file; each line of the file contains a tree root ID, a tree filename, and the byte offset in the tree file at which the given merger tree may be found. Catalog File Layout ------------------- Catalog files (`hlist_.list`) contain all the halos at a given snapshot of the simulation. The number in the filename corresponds to the scale factor a; e.g., `hlist_1.00000.list` corresponds to the simulation output at a = 1.0. These files contain header information (lines beginning with a `#` character), followed by a single line for each halo in the given snapshot. The halo line format follows that of the tree files, but includes several additional fields at the end. All fields are described in Appendix B. Alternate Formats ----------------- The above two formats are the only ones generated by default. Converters also exist for the Sussing, Galacticus, and Irate formats; please contact Peter Behroozi (`[email protected]`) for details. Merger Tree Fields ================== Each halo line contains several fields separated by a space character (`` "), giving information about the halo’s identification, properties, order within the merger tree, and so on. Please note that the ordering of some fields may change in new versions of Consistent Trees. However, the first header line in both the tree files and the catalogs always lists the field order, so it is always a good idea to double-check this line with what your code is expecting. Halo Identification and Properties ---------------------------------- [MergerTrees] The following fields are always guaranteed to exist, and will always exist in the following order at the beginning of the line: * Scale: Halo’s scale factor. * ID: Halo ID, guaranteed to be unique across all snapshots of the entire simulation. Because Consistent Trees adds and deletes halos, this may be different from the ID returned by the halo finder. * Desc\_Scale: Scale factor of descendent halo, if applicable. * Descid: Halo ID of descendent halo, if applicable. * Num\_prog: Number of progenitor halos—i.e., number of halos at the immediately preceding snapshot that fully merge into this halo. * Pid: Parent halo ID. For distinct halos (those that are not subhalos), this is `-1`. Otherwise, it is the halo ID of the smallest host halo (ordered by vmax) that contains this halo’s center within its radius. * Upid: Über-parent halo ID. For distinct halos, this is `-1`. Otherwise, it is the halo ID of the largest host halo (ordered by vmax) that contains this halo’s center within its radius. Note that it is geometrically possible for a subhalo to be within the virial radius of more than one distinct halo. Hence, there is no guarantee that the parent halo is a subhalo of the über-parent halo if the two IDs are different. * Phantom: Zero (i.e., ignorable) for most halos. This is nonzero only when the halo finder did not return an appropriate progenitor halo at this snapshot and Consistent Trees had to interpolate halo properties from nearby snapshots. * SAM\_Mvir: Do not use. For historical reasons, this was needed by certain semi-analytical models. * Mvir: Halo mass, in units of M⊙/h. * Rvir: Halo radius, in units of comoving kpc/h. * Rs: NFW scale radius, in units of comoving kpc/h. * Vrms: Halo particle velocity dispersion, in units of physical (i.e., non-comoving) km/s. * mmp?: `1` if the halo is the most-massive progenitor of its descendent halo; `0` if not. * scale\_of\_last\_MM: scale factor of the halo’s last major merger. This is typically defined as a mass ratio greater than 0.3:1, although it is user-adjustable. The exact definition always appears in the header lines. * Vmax: Maximum halo circular velocity (i.e., maximum of $\sqrt{GM(<R)/R}$, where M( < R) is the mass enclosed within radius R), in units of physical km/s. * X/Y/Z: Halo position, in units of comoving Mpc/h. * VX/VY/VZ: Halo velocity, in units of physical km/s. * JX/JY/JZ Halo angular momentum, in units of (M⊙/h) × (Mpc/h) × km/s (physical). * Spin: Dimensionless halo spin parameter. (Peebles spin, Equation ([lambdaP]), for Rockstar). Halo Ordering, Cross-Referencing, and Tree Walking -------------------------------------------------- When walking trees, there are two main approaches. The first is breadth-first, in which halos are ordered and accessed according to the simulation snapshot; this is the default halo ordering in the trees. An alternate method is depth-first, in which halos are ordered and accessed first along main progenitor branches (i.e., following the most-massive progenitor line) followed by the next-most-massive progenitor branches. For an illustration, see Figure [MergerTrees]. The following fields will always exist in current and later versions of Consistent Trees, although more ordering fields may be added at a later date: * Breadth\_first\_ID: Unique ID (across all simulation snapshots) corresponding to the breadth-first order of halos within a tree. Sorting on this ID always recovers the original order in which the halos were printed in the tree file. * Depth\_first\_ID: Unique ID (across all simulation snapshots) corresponding to the depth-first ordering of halos within a tree. Sorting on this ID reorganizes the tree into depth-first order. This ordering has the useful property that full merger histories are contiguous, even for halos not at the last simulation snapshot; see Last\_progenitor\_depthfirst\_ID and Last\_mainleaf\_depthfirst\_ID below. * Tree\_root\_ID: Halo ID of the final descendent halo (i.e., the descendent halo at the last simulation snapshot). * Orig\_halo\_ID: Original halo ID from halo finder, except for phantom halos. This allows cross-referencing halos in the merger trees with other data (e.g., particles) saved by the halo finder. * Snap\_num: Snapshot number from which the halo originated—necessary as not all halo finders generate unique IDs. * Next\_coprogenitor\_depthfirst\_ID: Depth-first ID of next “coprogenitor”—i.e., the next halo that shares the same descendent halo. * Last\_progenitor\_depthfirst\_ID: Depth-first ID of last progenitor. When a merger tree is sorted by depth-first ID, then the halos between this halo’s Depth\_first\_ID and this halo’s Last\_progenitor\_depthfirst\_ID correspond to its full merger history. * Last\_mainleaf\_depthfirst\_ID: Depth-first ID of last progenitor on this halo’s main progenitor branch. When a merger tree is sorted by depth-first ID, then the halos between this halo’s Depth\_first\_ID and this halo’s Last\_mainleaf\_depthfirst\_ID correspond to its most-massive progenitor history. Note that to better understand the last two fields, it is helpful to refer to Figure [MergerTrees]. Additional Halo Properties -------------------------- These fields are halo-finder dependent. With the current version of Consistent Trees and Rockstar, they include: * Tidal\_Force: Strongest tidal force from any nearby halo, in dimensionless units (Rhalo/Rhill). The Hill radius is an upper bound on the spatial extent of newly infalling material that can remain gravitationally bound to a secondary halo. It is given by Rhill = D(Msec/3Mprim)1/3 = Rsec(D/31/3Rprim), where the primary (secondary) halo is the larger-virial-radius (smaller-virial-radius) halo of a pair whose centers are separated by distance D. * Tidal\_ID: Halo ID of halo exerting strongest tidal force. * Rs\_Klypin: NFW scale radius in units of comoving kpc/h, determined using Vmax and Mvir, Equation ([RsKlypin]). * Mvir\_all: Halo mass, including unbound particles (M⊙/h). * M200m–M2500c: Mass (M⊙/h) enclosed within specified overdensities. These include 200m, 200c, 500c, and 2500c, where ρc is critical density and ρm = ΩMρc is the mean matter density. * Xoff: Offset of halo center (defined as the density peak) from the center of mass within the halo radius (comoving kpc/h). * Voff: Offset of halo center (defined as the density peak) from average particle velocity (physical km/s). * Spin\_Bullock: Bullock spin parameter, Equation ([lambdaB]). * b\_to\_a, c\_to\_a: Ratio of second and third largest shape ellipsoid axes (B and C) to largest shape ellipsoid axis (A) (dimensionless), calculated according to the method in. Additional subscripts (e.g., `500c`) indicate that only particles within a specified halo radius are considered. * A[x],A[y],A[z]: Largest shape ellipsoid axis (comoving kpc/h). Additional subscripts (e.g., `500c`) indicate that only particles within a specified halo radius are considered. * T/∣U∣: ratio of kinetic to potential energies for halo particles. * M\_pe\_: Pseudo-evolution corrected masses (very experimental). Additional Catalog Fields ------------------------- These fields are not in the trees, but are provided for convenience in the halo catalogs. Their order and content may change in future Consistent Trees versions. * Macc, Vacc: Halo’s (or its main progenitor’s) mass and vmax at the last timestep that it was a distinct halo. * Mpeak, Vpeak: Peak mass and peak vmax along the halo’s main progenitor branch. * Halfmass\_Scale: Scale factor at which the halo mass in the main progenitor branch first reaches half of Mpeak. * Acc\_Rate\_: Halo mass accretion rates in M⊙/h/yr. These include: Inst (averaged since the previous snapshot); 100Myr (averaged over past 100 Myr); X\Tdyn: averaged over past X virial dynamical times (i.e., $X/\sqrt{G\rho\_\mathrm{vir}}$); Mpeak: averaged growth of Mpeak, averaged from the current halo’s redshift (z) to z* + 0.5. * Mpeak\_Scale: Scale at which Mpeak was reached along the main progenitor branch. * Acc\_Scale: Last scale at which the halo (or its main progenitor) was distinct. * First\_Acc\_Scale: Last scale at which the halo (or its main progenitor) and all earlier progenitor halos were all distinct.[6](#fn6) * First\_Acc\_(Mvir∣Vmax): Mvir and vmax at First\_Acc\_Scale for the main progenitor. * Vmax@Mpeak: Main progenitor’s vmax at Mpeak\_Scale. * Tidal\_Force\_Tdyn: Dimensionless tidal force averaged over the past virial dynamical time. --- 1. [email protected][↩](#fnref1) 2. <https://www.cosmosim.org/cms/simulations/multidark-project/> with explanations in and at <https://www.cosmosim.org/cms/documentation/introduction/whats-different/> for users of the earlier site <http://www.multidark.org/>.[↩](#fnref2) 3. <http://hipacc.ucsc.edu/Bolshoi/MergerTrees.html>[↩](#fnref3) 4. <http://code.google.com/p/rockstar>[↩](#fnref4) 5. <https://bitbucket.org/pbehroozi/consistent-trees>[↩](#fnref5) 6. In theory. In practice, flybys often happen for very early progenitors, so these are ignored if halos at least double their mass following the flyby.[↩](#fnref6)	arxiv_0000782
Stable Dinner Party Seating Arrangements ======================================== A group of n agents with numerical preferences for each other are to be assigned to the n seats of a dining table. We study two natural topologies: circular (cycle) tables and panel (path) tables. For a given seating arrangement, an agent’s utility is the sum of their preference values towards their (at most two) direct neighbors. An arrangement is envy-free if no agent strictly prefers someone else’s seat, and it is stable if no two agents strictly prefer each other’s seats. Recently, it was shown that for both paths and cycles it is NP-hard to decide whether an envy-free arrangement exists, even for symmetric binary preferences. In contrast, we show that, if agents come from a bounded number of classes, the problem is solvable in polynomial time for arbitrarily-valued possibly asymmetric preferences, including outputting an arrangement if possible. We also give simpler proofs of the previous hardness results if preferences are allowed to be asymmetric. For stability, it is known that deciding the existence of stable arrangements is NP-hard for both topologies, but only if sufficiently-many numerical values are allowed. As it turns out, even constructing unstable instances can be challenging in certain cases, e.g., binary values. We propose a complete characterization of the existence of stable arrangements based on the number of distinct values in the preference matrix and the number of agent classes. We also ask the same question for non-negative values and give an almost-complete characterization, the most interesting outstanding case being that of paths with two-valued non-negative preferences, for which we experimentally find that stable arrangements always exist and prove it under the additional constraint that agents can only swap seats when sitting at most two positions away. Similarly to envy-freeness, we also give a polynomial-time algorithm for determining a stable arrangement assuming a bounded number of classes. We moreover consider the swap dynamics and exhibit instances where they do not converge, despite a stable arrangement existing. Introduction ============ Your festive dinner table is ready, and the guests are arriving. As soon as your guests take their assigned seats, two of them are unhappy about their neighbors and rather want to switch seats. Alas, right after the switch, two other guests become upset, and then pandemonium ensues! Could you have prevented all the social awkwardness by seating your guests “correctly” from the get-go? In this paper, we study the difficulty of finding a stable seating arrangement; i.e. one where no two guests would switch seats. We focus on two natural seating situations: a round table (cycle), and an expert panel (path). In either case, we assume guests only care about having the best possible set of direct left and right neighbors. In certain cases, not even a stable arrangement might make the cut, as a single guest envying the seat of another could potentially lead to trouble. Therefore, we are also interested in finding envy-free arrangements. Formally, n guests have to be assigned bijectively to the n seats of a dining table: either a path or a cycle graph. Guests express their preferences for the other guests numerically, with higher numbers corresponding to a greater desire to sit next to the respective guest. The utility of a guest g for a given seating arrangement is the sum of g’s preference values towards g’s neighbors. A guest g envies another guest if g’s utility would strictly increase if they swapped places. Two guests want to swap places whenever they envy each other. Our goal is to compute a stable (no two guests want to swap) respectively envy-free (no guest envies another) seating arrangement. Besides the table topology, we conduct our study in terms of two natural parameters. The first parameter is the number of numerical values guests can choose from when expressing their preferences for other guests. For instance, an example of two-valued preferences would be when all preference values are either zero or one (i.e., binary, also known as approval preferences), in which case every guest has a list of “favorite” guests they want to sit next to, and is indifferent towards the others. In contrast, if the values used were ± 1; i.e., every guest either likes or dislikes every other guest; then the preferences are still two-valued, but no longer binary. Increasing the number of allowed values allows for finer-grained preferences. It is also interesting to distinguish the special case of non-negative preferences; i.e., no guest can lose utility by gaining a neighbor. Our second parameter is the number of different guest classes. In particular, dinner party guests can often be put into certain categories, e.g., charmer, entertainer, diva, politico, introvert, outsider. Each class has its own preferences towards other classes, e.g., outsiders would prefer to sit next to a charmer, but not next to an introvert. [t] [h]0.45 c\|cc & & Bounded & Unbounded EF & Poly1 ctrerr & NP-hard2 ctrerr STA & Poly1 ctrerr & NP-hard3 ctrerr [tab:summaryrescomputationalcycle] [h]0.45 c\|cc & & Bounded & Unbounded EF & Poly1 ctrerr & NP-hard2 ctrerr STA & Poly1 ctrerr & NP-hard4 ctrerr [tab:summaryrescomputationalpath] [tab:summaryrescomputational] [t] [h]0.45 \| \| ≤ 2 \| 3 \| 4 \| ≥ 5 \| \| --- \| --- \| --- \| --- \| --- \| \| ≤ 2 \| S \| S3 ctrerr \| U4 ctrerr \| U \| \| 3 \| S \| U2 ctrerr \| U \| U \| \| ≥ 4 \| S1 ctrerr \| U \| U \| U \| [tab:summaryrescycle] [h]0.45 \| \| ≤ 2 \| 3 \| ≥ 4 \| \| --- \| --- \| --- \| --- \| \| ≤ 2 \| S \| U5 ctrerr \| U5 ctrerr \| \| 3 \| S \| U2 ctrerr \| U \| \| ≥ 4 \| S1 ctrerr \| U \| U \| [tab:summaryrespath] [tab:summaryres] Our Contribution. We study the existence and computational complexity of finding stable/envy-free arrangements on paths and cycles. Some of our results can be surprising. For instance, six people with binary preferences can always be stably seated at a round table, while for five (or seven) guests some preferences are inherently unstable, so we better invite (or uninvite) another guest. However, even for six people with binary preferences, for which a stable arrangement always exists, the swap dynamics might still never converge to one. Our computational results are exhibited in Table [tab:summaryrescomputational]. In summary, if the number of guest classes is bounded by a constant (which can be arbitrary), then both stable and envy-free arrangements can be computed in polynomial time whenever they exist with no assumption on the preference values, while dropping this assumption makes the two problems difficult even for very constrained preference values. For envy-freeness, this already happens for binary preferences, arguably the most prominent case. For stability, on the other hand, this requires more contrived values (four non-negative values for cycles and six values including negatives for paths), so it would be interesting to have a finer-grained understanding of stability in terms of the values allowed when expressing one’s preferences. For instance, is it still hard to find stable arrangements for binary preferences? As we show, it turns out to be surprisingly difficult to even construct unstable binary preferences, setting aside the computational considerations. To this end, we conduct a fine-grained study aiming to answer for which combinations of our two parameters, i.e., number of preference values and guest classes, do stable arrangements always exist and for which combinations this is not the case. Our results are exhibited in Table [tab:summaryres]. Notably, for cycles we give a full characterization, either showing guaranteed stability with arbitrary values or giving unstable instances with (simple) non-negative values. Similarly, for paths, we close all cases, with the notable exception of two-valued preferences with three or more classes being permitted, where all the unstable instances we found require negative values. We conjecture that for non-negative values stability on paths is guaranteed in the two-valued case. We support this conjecture with experimental evidence, as well as a partial result under the additional assumption of guests only being willing to swap seats if they are separated by at most one seat (see Table [tab:summaryres] for more details). Appendix. In the appendix, we supply the proofs omitted from the main text, as well as supporting material. Moreover, we show that stability is a highly fragile notion, being non-monotonic with respect to adding/removing guests. We also give evidence that knowledge about stability on paths is unlikely to transfer to computing cycle-stable arrangements. Finally, we use probabilistic tools to study the expected number of stable arrangements of Erds-Rényi binary preferences. Related Work ============ The algorithmic study of stability in collective decision-making has its roots in the seminal paper of Gale and Shapley, introducing the now well-known Stable Marriage and Stable Roommates problems. Classically, the former is presented as follows: an equal number of men and women want to form couples such that no man and woman from different couples strictly prefer each other over their current partners, in which case the matching is called stable. The authors give the celebrated Gale-Shapley deferred acceptance algorithm showing that a stable matching always exists and can be computed in linear time. Irving later extended the algorithm to also handle preferences with ties; i.e., a man (woman) being indifferent between two women (men). The Stable Roommates problem is the non-bipartite analog of Stable Marriage: an even number of students want to allocate themselves into identical two-person rooms in a dormitory. A matching is stable if no two students allocated to different rooms prefer each other over their current roommates. In this setting, stable matchings might no longer exist, but a polynomial-time algorithm for computing one if any exist is known. However, when ties are allowed, the problem becomes NP-hard. The seating arrangement problem that we study is, in fact, well-connected with Stable Roommates. Instead of one table with n seats, the latter considers n/2 tables with two seats each. However, there is another more subtle difference: in Stable Roommates, two people unhappy with their current roommates can choose to move into any free room. This is not possible if there are exactly n/2 rooms. Instead, the stability notion that we study corresponds to the distinct notion of exchange-stability in the Stable Roommates model, where unhappy students can agree to exchange roommates. Surprisingly, under exchange-stability, finding a stable roommate allocation becomes NP-hard even without ties. One can also see our problem through the lens of coalition formation. In particular, hedonic games consider the formation of coalitions under the assumption that individuals only care about members in their own coalition. Then, fixing the sizes of the coalitions allows one to generalize from tables of size two and study stability more generally. Bilò et al. successfully employ this approach to show a number of attractive computational results concerning exchange-stability. The main drawback of this approach, however, is that it assumes that any two people sitting at the same table can communicate, which is not the case for larger tables. Our approach takes the topology of the dining table into account. Some previous works have also considered the topology of the dining table. Perhaps closest to our paper is the model of Bodlaender et al. , in which n individuals are to be assigned to the n vertices of an undirected seating graph. The authors prove a number of computational results regarding both envy-freeness and exchange-stability, among other notions. However, we found some of the table topologies considered to be rather unnatural, especially in hardness proofs (e.g., trees or unions of cliques and independent sets). Bullinger and Suksompong also conduct an algorithmic study of a similar problem, but with a few key differences: (i) individuals are seated in the nodes of a graph, but there may be more seats than people; (ii) for the stability notion, they principally consider jump-stability, where unhappy people can choose to move to a free seat; (iii) individuals now contribute to everyone’s utility, although their contribution decreases with distance. Last but not least, studying stability in the context of Schelling games has recently been a popular area of research. In Schelling games, individuals belong to a fixed number of classes. However, unlike in our model, agents from one class only care about sitting next to others of their own class. This additional assumption often allows for stronger results; e.g., in the authors prove the existence of exchange-stable arrangements on regular and almost regular topological graphs such as cycles and paths, and show that the swap dynamics are guaranteed to converge in polynomial time on such topologies. Overall, it seems that exchange stability has been studied in the frameworks of both hedonic and Schelling games. However, both approaches present some shortcomings: on the one hand, Schelling games inherently consider a topology on which agents evolve, but, being historically motivated by the study of segregation (e.g., ethnic, racial), they usually restrict themselves to very simple preferences. On the other hand, works on hedonic games are accustomed to considering diverse preferences. However, while multiple works have introduced topological considerations, their analysis is usually constrained to graphs that can be interpreted as non-overlapping coalitions, e.g., with multiple fully connected components. One notable exception is the very recent work of Ceylan, Chen and Roy, appearing in IJCAI’23, also building on the model of Bodlaender et al. . In comparison to us, they also prove the NP-hardness of deciding the existence of envy-free arrangements for binary preferences for both topologies, in their case for symmetric preferences, but at the expense of more complex proofs. They moreover show that hardness holds for stability, although the presented proofs requires four non-negative values for cycles and six values including negatives for paths. In our work, we aim to understand stability under more natural preference values, such as approval preferences. Preliminaries ============= We write [n] = {1, …, n}. Given an undirected graph G = (V(G), E(G)), we write NG(v) for the set of neighbors of vertex v ∈ V(G). When clear from context, oftentimes we will simply write V, E and N(v) respectively. The model we describe next is similar to the one in. A group of n agents (guests) A has to be seated at a dining table represented by an undirected graph G = (V, E), where vertices correspond to seats. We will be interested in the cases of G being a cycle or a path. We assume that ∣V∣ = n and that no two agents can be seated in the same place, from which it also follows that all the seats have to be occupied. Agents have numerical preferences over each other, corresponding to how much utility they gain from being seated next to other agents. In particular, each agent i ∈ A has a preference over the other agents expressed as a function pi : A \ {i} → R, where pi(j) denotes the utility gained by agent i when sitting next to j. Note that we do not assume symmetry; i.e., it might be that pi(j) ≠ pj(i). We denote by P = (pi)i ∈ A the collection of agent preferences, or preference profile, of the agents. A number of different interpretations can be associated to P. In particular, we will usually see P as a matrix P = (pij)i, j ∈ A, where pij = pi(j). Note that the diagonal entries are not defined, but, for convenience, we will oftentimes use the convention that pii = 0. Using the matrix notation, we say that the preferences in P are binary when P ∈ {0, 1}n × n and k-valued if there exists Γ ⊆ R, ∣Γ∣ = k, such that P ∈ Γn × n (disregarding diagonal entries, since they are undefined). Note that binary preferences are two-valued, but two-valued preferences are not necessarily binary. Moreover, we will often represent binary preferences as a directed graph, where a directed edge between two agents signifies that the first agent approves of the second. Finally, when pij ≥ 0 for any two agents i, j ∈ A we say that the preferences are non-negative. We define a class of agents to be a subset of indistinguishable agents C ⊆ A. More formally, all agents in C share a common preference function pC : A → R and no agent in A discriminates between two agents in C. Note that this implies that the lines and columns of the preference matrix corresponding to agents in C are identical, if we adopt the convention that diagonal entries inside a class are all equal but not necessarily zero. We say that preference profile P has *k classes, or is a k-class profile, if A can be partitioned into k* classes C1 ∪ … ∪ Ck = A. We define an arrangement of the agents on G to be a bijection π : A → V(G), i.e., an assignment of each agent to a unique vertex of the seating graph (and vice-versa). For a given arrangement π, we define for each agent i ∈ A their utility Ui(π) = ∑v ∈ NG(π(i))pi(π− 1(v)) to be the sum of agent i’s preferences towards their graph neighbors in the arrangement. We say that agent i envies agent j whenever Ui(π) < Ui(πʹ), where πʹ is π with π(i) and π(j) swapped. We further say that (i, j) is a blocking pair if both i envies j and j envies i; i.e., they would both strictly increase their utility by exchanging seats. An arrangement is envy-free if no agent envies another, and it is stable if it induces no blocking pairs. Note that envy-freeness implies stability, but the converse is not necessarily true. By extension, we call preference profile P stable (respectively envy-free) on G if there exists a stable (respectively envy-free) arrangement π on G. Profile P is unstable if it is not stable. A few preliminary observations follow. Note that, for symmetric preferences; i.e., pij = pji for any two agents i, j ∈ A; a stable arrangement always exists, in fact on any graph G, not just cycles and paths. This is because swaps in that case strictly increase the total sum of agent utilities, and hence the swap dynamics will converge to a stable arrangement. Hence, the interesting case is the asymmetric one. Observe that envy-free arrangements need not exist for symmetric preferences (deciding existence is NP-hard for both paths and cycles ). Moreover, note that, for cycles, and in fact any regular graph G, agents have the same number of neighbors, so transforming the preferences of an agent i by adding/subtracting/multiplying by a positive constant the values pij does not inherently change the preferences. This implies that for cycles the two-valued case coincides with the binary case and the non-negative case coincides with the unconstrained case. It also shows that for cycles the definition of k-valued preferences: P ∈ Γn × n where ∣Γ∣ = k; can be restated equivalently to require that every row of P consists of at most k different values. Envy-Freeness ============= It is relatively easy to construct preferences with no envy-free arrangements: for paths, even if all agents like each other, the agents sitting at the endpoints will envy the others; for cycles, add an agent despised by everyone, agents sitting next to them will envy their peers. We now show that, furthermore, it is NP-hard to decide whether envy-free arrangements exist, for both paths and cycles, even under binary preferences. This has also been recently shown in in a stronger form, using only symmetric binary preferences, but we found the construction rather involved and the correctness argument based on careful counting delicate. [thnphardcycle] For binary preferences, deciding whether an envy-free arrangement on a cycle exists is NP-hard. We proceed by reduction from Hamiltonian Cycle on directed graphs. Let G = (V, E) be a directed graph[1](#fn1) such that, without loss of generality, V = [n]. If G has any vertices with no outgoing edges, then map the input instance to a canonical no-instance, unless n = 1, in which case we map it to a canonical yes-instance. Hence, from now on, assume that all vertices have outgoing edges. For each vertex v ∈ V introduce three agents xv, yv, zv such that agent xv only likes yv and dislikes[2](#fn2) everyone else, agent yv only likes zv and dislikes everyone else, and agent zv likes agent xu for all u ∈ V such that (v, u) ∈ E, and dislikes everyone else. We claim that the so-constructed preference profile P has an envy-free arrangement on a cycle precisely when G has a Hamiltonian cycle. To show this, first assume without loss of generality that 1 → 2 → … → n → 1 is a Hamiltonian cycle in G. Then, arranging agents around the cycle in the order x1, y1, z1, x2, y2, z2, …, xn, yn, zn is an envy-free arrangement. To see why, notice that in this arrangement all agents get utility 1, so envy could only potentially stem from an agent being able to swap places with another agent to get utility 2. To prove this is not possible, first notice that agents (xi)i ∈ [n] and (yi)i ∈ [n] each only like one other agent, so they can never get a utility of more than 1 in any arrangement. Moreover, no agent zi can get to a utility of 2 by a single swap because any two agents they like are seated at least three positions from each other on the cycle. Conversely, assume that an envy-free arrangement π exists. First, if xi is not sitting next to yi in π, then xi could improve by swapping to a place next to yi (also similarly for yi and zi). Therefore, in arrangement π agent xi is seated next to yi and yi is seated next to zi. Moreover, consider the other neighbor of zi in π. Since zi does not like yi, it follows that if zi also does not like their other neighbor, then zi could strictly improve their utility by swapping next to some agent they like, which is always possible because all vertices in G have outgoing edges. Therefore, the other neighbor of zi has to be some agent that they like, hence being of the form xj, where j ≠ i. Note that, by construction, (i, j) ∈ E. Putting together what we know, we get that under π the agents are arranged around in the cycle in some order xσ1, yσ1, zσ1, xσ2, yσ2, zσ2, …, xσn, yσn, zσn, where σ is a permutation of the n agents such that (σi, σi + 1) ∈ E holds for all i ∈ [n].[3](#fn3) Therefore, a Hamiltonian cycle σ1 → σ2 → … → σn → σ1 exists in G. A similar proof can be used to show hardness for the case of paths. We outline the changes required in the appendix. thnphardpath For binary preferences, deciding whether an envy-free arrangement on a path exists is NP-hard. Proving matching hardness results for stability under binary preferences would be highly desirable but for the time being is left open. We in fact conjecture that all instances with binary preferences are stable on a path (see Section [sec:binarypref]). Moreover, one might ask how does the number of agent classes affect the computational complexity of our problems. In the next section, we address this question, showing that limiting the number of agent classes renders the problems that we consider polynomial-time solvable, even for arbitrary preference values. Polynomial Solvability for k-Class Preferences ================================================ In this section, we show that deciding whether envy-free and stable arrangements exist for a given preference profile can be achieved in polynomial time, for both paths and cycles, assuming that the number of agent classes is bounded by a number k. Note that preferences in this case are not constrained to being binary, and can in fact be arbitrary. By extension, our algorithms can also be used to construct such arrangements whenever they exist. We begin with the case of paths. For simplicity, we assume that n ≥ 3, as for n ≤ 2 any arrangement is both stable and envy-free. Assume that the agent classes are identified by the numbers 1, …, k and that n1, n2, …, nk are the number of agents of each class in our preference profile, where n1 + … + nk = n. For ease of writing, we will see arrangements as sequences s = (si)i ∈ [n], where si ∈ [k] and for any agent class j ∈ [k] the number of values j in s is nj. Moreover, for brevity, we lift agent preferences to class preferences, in order to give meaning to statements such as “class a likes class b.” To simplify the treatment of agents sitting at the ends of the path, we introduce two agents of a dummy class 0 with preference values 0 from and towards the other agents. We require the dummy agents to sit at the two ends of the path; i.e., s0 = sn + 1 = 0. In order to use a common framework for stability and envy-freeness, we define the concept of compatible triples of agent classes, as follows. First, for envy-freeness, let a, b, c, d, e, f be agent classes, then we say that triples (a, b, c) and (d, e, f) are long-range compatible if pb(a) + pb(c) ≥ pb(d) + pb(f) and pe(d) + pe(f) ≥ pe(a) + pe(c); intuitively, neither b wants to swap with e, nor vice-versa. Furthermore, for a, b, c, d agent classes, we say that triples (a, b, c) and (b, c, d) are short-range compatible if pb(a) ≥ pb(d) and pc(d) ≥ pc(a); intuitively, if a, b, c, d are consecutive in the arrangement, then neither b wants to swap with c, nor vice-versa. For stability, we keep the same definitions but use “or” instead of “and.” Note that long-range and short-range compatibility do not imply each other. We call an arrangement s compatible if for all 1 ≤ i < j ≤ n the triplets (si − 1, si, si + 1) and (sj − 1, sj, sj + 1) are long-range compatible when j − i > 1 and short-range compatible when j − i = 1. Note that arrangement s is envy-free (resp. stable) if and only if it is compatible. In the following, we explain how to decide the existence of a compatible arrangement. Deciding whether compatible arrangements exist can be achieved in polynomial time. We first present a nondeterministic algorithm (i.e., with guessing) that solves the problem in polynomial time. The algorithm builds a compatible arrangement s one element at a time. Initially, the algorithm sets s0 ← 0 and guesses the values of s1 and s2. Then, at step i, for 3 ≤ i ≤ n + 1, the algorithm will guess si (except for i = n + 1, where we enforce that si ← 0) and check whether (si − 3, si − 2, si − 1) is short-range conflicting with (si − 2, si − 1, si), rejecting if so. Moreover, the algorithm will check whether (si − 2, si − 1, si) is long-range conflicting with any (sj − 2, sj − 1, sj) for 2 ≤ j ≤ i − 2, again rejecting if so. At the end, the algorithm checks whether for each i ∈ [k] value i occurs in s exactly ni times, accepting if so, and rejecting otherwise. Alone, this algorithm only shows containment in NP, which is not a very attractive result. Next, we show how the same algorithm can be implemented with only a constant number of variables, explaining afterward why this implies our result. First, to simulate the check at the end of the algorithm without requiring knowledge of the whole of s, it is enough to maintain throughout the execution counts (xj)j ∈ [k] such that at step i in the algorithm xj gives the number of positions 1 ≤ ℓ ≤ i such that sℓ = j. To simulate the short-range compatibility check, it is enough that at step i we have knowledge of si − 3, …, si. Finally, for the long-range compatibility check, a more insightful idea is required. In particular, we make the algorithm maintain throughout the execution counts ma, b, c for each triple (a, b, c) of agent classes, such that at step i value ma, b, c gives the number of positions 2 ≤ ℓ ≤ i such that (sℓ − 2, sℓ − 1, sℓ) = (a, b, c). Using this information, to check at step i whether (si − 2, si − 1, si) long range conflicts with any (sj − 2, sj − 1, sj) for 2 ≤ j ≤ i − 2, it is enough to temporarily decrease by one the values msi − 3, si − 2, si − 1 and msi − 2, si − 1, si and then check whether there exists a triple (a, b, c) of agent classes such that ma, b, c > 0 and (a, b, c) long-range conflicts with (si − 2, si − 1, si). In total, at step i, the algorithm only needs to know the values si − 3, …, si, as well as (xj)j ∈ [k] and the counts ma, b, c for all triples (a, b, c) of agent classes. Since k + 1 bounds the total number of agent classes, this is only a constant number of variables. As each variable can be represented with O(logn) bits, it follows that our nondeterministic algorithm uses only logarithmic space, implying containment in the corresponding complexity class NL. It is well known that NL ⊆ P, from which our conclusion follows. For readers less familiar with this result, we give a short overview of how our algorithm can be converted into a deterministic polynomial-time algorithm, as follows. Since our NL algorithm uses only logarithmic space, it follows that the space of algorithm states that can be reached depending on the nondeterministic choices is of at most polynomial size, since 2O(logn) is polynomial. Therefore, building a graph with vertices being states and oriented edges corresponding to transitions between states, the problem reduces to deciding whether an accepting state can be reached from the initial state, which can be done with any efficient graph search algorithm. [thm:algokclasspath] Fix k ≥ 1. Then, for k-class preferences, there are polynomial-time algorithms computing an envy-free/stable arrangement on a path or reporting the nonexistence thereof. For the case of cycles, a similar approach can be used, although with rather tedious, yet minor tweaks, presented in the appendix. thm:algokclasscycle Fix k ≥ 1. Then, for k-class preferences, there are polynomial-time algorithms computing an envy-free/stable arrangement on a cycle or reporting the nonexistence thereof. A Fine-Grained Analysis of Stability ==================================== Previously, we showed that deciding whether envy-free arrangements exist is NP-hard for binary preferences on both topologies. For stability, in it is shown that similar hardness results hold, but this time more than two values are needed in the proofs, namely four values for cycles and six for paths, the latter also requiring negative numbers. It is unclear whether hardness is retained without assuming this level of preference granularity, and a first step towards understanding the difficulty of the problem constrained to fewer/simpler allowed values is being able to construct instances where no stable arrangements exist; after all, a problem where the answer is always “yes” cannot be NP-hard. In this section, we conduct a fine-grained analysis of the conditions allowing for unstable instances. In particular, for both topologies we consider how different constraints on the number of agent classes as well as the number of different values allowed in the preferences influence the existence of unstable preferences The non-negativity of the values needed is also taken into account. Table [tab:summaryres] summarizes our results. Two-Class Preferences --------------------- As a warm-up, note that when all agents come from a single class, any arrangement on any given seating graph is stable. In the following, we extend this result to two classes of agents for cycles and paths. We begin with cycles: [thm:2classcycle] Two-class preferences always induce a stable arrangement on a cycle. Suppose there are two classes of agents, say and. Without loss of generality, preferences can be assumed to be binary, since for cycles one can normalize the preference values as described towards the end of the preliminaries section. First, note that any blocking pair must consist of one and one. Moreover, note that any arrangement is stable whenever one of the classes likes the two classes equally. Now, suppose this is not the case, meaning that each class has a preferred class to sit next to. There are only two cases to consider: If one class, say, prefers its own class, then sit all together and give the remaining seats to : all but two, say $\textcolor{blue}{B\_1}$ and $\textcolor{blue}{B\_2}$, get maximum utility, and neither $\textcolor{blue}{B\_1}$ nor $\textcolor{blue}{B\_2}$ can improve since no has more than one neighbor. Hence, no is part of a blocking pair, so the arrangement is stable. If both classes prefer the opposite class, we may assume there are at least as many as. Then we alternate between and for as long as there are without a seat, then seat all the remaining next to each other. Every has maximum utility, and hence cannot be part of a blocking pair, so the arrangement is stable. The following extends the result to the case of paths. The proof is largely similar, but the case analysis becomes more involved, because preferences can no longer be assumed to be binary, so we present it in the appendix. thm:2classpathstable Two-class preferences always induce a stable arrangement on a path. Note that a path of n is equivalent to a cycle of n + 1 where an agent with null preferences is added. This explains why the case of paths is harder to study than that of cycles, as it corresponds to having one more class of agents and potentially one more value (zero). Three-Class Three-Valued Preferences ------------------------------------ We now consider the case of three-valued preferences with three agent classes, exhibiting unstable non-negative preferences both for paths and for cycles. We begin with the case of cycles. [thm:unstternarycycle] For n ≥ 4, there exist three-class three-valued non-negative preferences such that all arrangements on a cycle are unstable. Consider three classes of agents: Alice, Bob, and n − 2 of Bob’s friends. The story goes as follows: Alice and Bob broke up. Alice does not want to hear about Bob and would hence prefer to sit next to any of his friends rather than Bob. On the other hand, Bob wants to win her back, so he would above all want to sit next to Alice. Finally, Bob’s friends prefer first Bob, then the other friends, and finally Alice. Used preference values can be arbitrary, so to get the required conclusion, we make sure that they are non-negative. To show that these preferences are unstable on a cycle, there are two cases: Alice and Bob can either sit next to each other, or separately.[4](#fn4) In the first case, Alice and her second neighbor, who is one of Bob’s friends, would exchange seats. After the switch, Alice is better as she no longer sits next to Bob, and the friend is better because he sits next to Bob. In the second case, Bob and one of Alice’s neighbors would exchange seats. Bob is better because he now sits next to Alice. To see that the neighbor, who is one of Bob’s friends, is also better, distinguish two sub-cases: if the friend sits right between Alice and Bob, then he is better because he now no longer sits next to Alice, while if this is not the case, he is better because before he was sitting next to Alice and a friend, while now he is sitting next to two friends. $$\mathcal{P} =\left[\begin{array}{ c c \| c c c} \* & 0 & 1 & \ldots & 1\\ 1 & \* & 0 & \ldots & 0\\ \hline 0 & 2 & 1 & \ldots & 1\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 2 & 1 & \ldots & 1 \end{array} \right]$$ [fig:expleprefternaryunst] It is possible to use the same construction for paths by allowing negative preference values, but otherwise, the proof of Theorem [thm:unstternarycycle] does not directly transfer to paths; e.g., for n = 5, path arrangement (F, F, B, A, F) is stable. Negative preferences turn out to not be necessary for n large enough. The main trick here is to use three copies of Bob to ensure that at least one of them does not sit at either end of the path. Formally, we have the following, proven in the appendix: thm:unstternpath For n ≥ 12, there exist three-class three-valued non-negative preferences such that all arrangements on a path are unstable. Two-Valued Preferences ---------------------- It remains to study what happens for two-valued preferences with three or more classes of agents. For cycles, we show that three classes always yield a stable arrangement, while four classes allow for a counterexample with binary values. For paths, we exhibit a counterexample with already three classes, but using some negative values. For non-negative values, we conjecture that a stable arrangement always exists, but have not been able to prove it. Instead, we gather both experimental and theoretical evidence to support it. [t] \| n \| 3 \| 4 \| 5 \| 6 \| 7 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Cycle \| 0 \| 0 \| 1 \| 0 \| 2 \| \| Path \| 0 \| 0 \| 0 \| 0 \| 0 \| [tab:summarybinresults] 0.32 [fig:5unst] 0.32 [fig:7unstable1] 0.32 [fig:7unstable2] [fig:57unstable] ### Two-Valued Preferences on Cycles. Cycles being regular graphs, it is sufficient to study binary preferences. We exhausted all binary preferences for n ≤ 7 using a Z3 Python solver (see Appendix [app:Z3solver]). Unstable preferences were found for n = 5 and n = 7, with one and, respectively, two non-isomorphic families of unstable preferences (see Table [tab:summarybinresults]). Examples of such preferences from each family P5, P7(1) and P7(2) are illustrated in Figure [fig:57unstable]. Analyzing why P5 is unstable turns out to be quite complex (see Appendix [sec:p5stabilityanalysis]), and, as of our current understanding, its instability seems to be more of a “small size artifact” than anything else. In contrast, with their highly regular structure, the two instances with n = 7 seem more promising. In particular, profile P7(2) consists of only four classes, denoted by A, B, C, D in Figure [fig:7unstable2]. In the following, we show that P7(2) can be extended to unstable preferences of any n ≥ 7. thm:binunstcycle For n ≥ 7, there exist four-class binary preferences such that all arrangements on a cycle are unstable. For n ≥ 7, we consider four classes A, B, C and D, as well as their respective members a, b1, b2, c and d1, …, dn − 4. Similarly to Figure [fig:7unstable2], we suppose that: (i) a only likes c; (ii) b1 and b2 both only like a and c; disliking each other; (iii) c only likes members of D; (iv) members of D all like each other, as well as b1 and b2, only disliking a and c. We show in the appendix why such preferences induce no stable arrangements on a cycle. This result shows that four classes of agents are sufficient to make all arrangements unstable on a cycle for two-valued preferences; we show in the following it is also necessary, as three classes of agents with two-valued preferences always induce a stable arrangement on a cycle. thm:3class2valuedcyclestable Three-class two-valued preferences always induce a stable arrangement on a cycle. We consider three classes: ,, and, each containing r, g, and b agents respectively. Without loss of generality, assume that r, g, b ≥ 1. Since the seating graph is regular, recall that we may assume the preferences to be binary. Note moreover that any blocking pair must have two agents of different colors. A principled case distinction now allows us to relatively quickly exhaust over all the possibilities for the preferences. We present the case distinction here and delegate the proofs themselves to specialized lemmas in the appendix. First, whenever at least one class likes its own kind, Lemmas [lem:3classsta:2classnarciss], [lem:3classsta:1classlikesitself+other] and [lem:3classsta:1classonlynarciss] together show the existence of a stable arrangement. Note that the proofs are constructive and all the stable arrangements presented intuitively seat the self-liking class consecutively. Then, if no class likes itself, Lemmas [lem:3classsta:nonarcisswithparia] and [lem:3classsta:nonarciss1likeall] show the existence of a stable arrangement whenever one class likes or is disliked by every other class. Note this time that intuitively all constructions present an alternation of two classes. This only leaves us to handle the case where each class likes and is liked by exactly another class: this case is treated separately in Lemma [lem:3classsta:maxutilarg], where we show that an arrangement maximizing utilitarian social welfare is always stable. The proof also shows that a modified variant of the swap dynamics converges. ### Two-Valued Preferences on Paths. We now consider the more complex case of paths, where the endpoints and the associated loss of regularity artificially introduce an implicit comparison with zero, hence giving rise to a change of behavior between positive and negative preferences. In general, if we allow for negative values in the preferences, then there exist three-class two-valued preferences such that no arrangement on a path is stable: thm:3class2valunstpath For n ≥ 3, there exist three-class two-valued preferences such that all arrangements on a path are unstable. Consider a variant of “Alice, Bob and Friends” where the preferences of Bob towards Alice, Alice towards Friends, and Friends towards Bob and themselves are all one; the preferences of Alice towards Bob, Bob towards Friends, and Friends towards Alice are all minus one. We show in the appendix why all arrangements on a path are unstable. For non-negative preferences, on the other hand, exhaustion for n ≤ 7 using a similar solver[5](#fn5) yields no unstable instances (see Table [tab:summarybinresults]). Surprised by the outcome, we also wrote C++ code to test all k-class instances with at most b agents per class for (k, b) ∈ {(4, 10), (5, 4)}, also leading to no unstable instances. We conjecture that two-valued instances with non-negative preferences always induce a stable arrangement on a path. In the following, we show that this is true under the additional assumption that two agents are only willing to swap seats when they are at most two positions away on the path, no matter how much their utilities would increase otherwise. This can be thought of as a practical constraint: once the agents are seated, each agent knows which other agents they envy, but finding out whether envy is reciprocal would be too cumbersome if the other agent is seated too far away. For this setup, we prove that the swap dynamics always converge, so a stable arrangement can be found by starting with an arbitrary arrangement and swapping blocking pairs until the arrangement becomes stable. This can be seen as a generalization of a result from, where agents only have preference for others of their own kind and swaps are only with adjacent agents. This is stated below and proven next. [thm:close-swaps-two-valued-is-stable] Two-valued non-negative preferences always induce a stable arrangement on a path assuming that agents are only willing to exchange seats with other agents sitting at distance at most two on the path. Moreover, the swap dynamics converge in this case. To begin showing this, note that no local swap could occur with an agent seated at either endpoint, since the preferences are non-negative. Since the two endpoints are the only irregularities, removing them from consideration, we can restrict our analysis from non-negative two-valued to binary without loss of generality. For any arrangement π, define the utilitarian social welfare W(π) = ∑i ∈ AUi(π). Moreover, to each arrangement π we associate a sequence S(π) of length n − 1 with elements in {0, 1, 2, 3}, constructed as follows. Let πi and πi + 1 be the agents sitting at positions i and i + 1 on the path: if they do not like each other, then S(π)i = 0, if they both like each other, then S(π)i = 3, if only πi likes πi + 1, then S(π)i = 1, otherwise S(π)i = 2. To prove that the swap dynamics converge, we define the potential Φ(π) = (W(π), S(π)), where sequences are compared lexicographically, and prove that swapping blocking pairs always strictly increases the potential. The following two lemmas show this for swaps at distances one and two, respectively. Let π be an arrangement where a and b form a blocking pair and sit in adjacent seats. Let πʹ be π with a and b’s seats swapped. Then, Φ(πʹ) > Φ(π). When n ≤ 2, there are no blocking pairs, so assume n ≥ 3. First, notice that swapping the places of a and b keeps a and b adjacent, from which the swap changes the utility of any agent by at most one. Since a and b’s utilities have to increase, they have to each change by exactly one. Moreover, note that neither a nor b can be seated at the ends of the table, as otherwise swapping would make one of them lose a neighbor while keeping the other one, hence not increasing their utility. Hence, assume that x is the other neighbor of a and y is the other neighbor of b; i.e., x, a, b, y are seated consecutively in this order on the path, at positions say i, …, i + 3. If either Ux(πʹ) ≥ Ux(π) or Uy(πʹ) ≥ Uy(π), it follows that W(πʹ) − W(π) = Ux(πʹ) − Ux(π) + Uy(πʹ) − Uy(π) + 2 ≥ 1, so Φ(πʹ) > Φ(π). Otherwise, we know that Ux(πʹ) − Ux(π) = Uy(πʹ) − Uy(π) = − 1, from which W(πʹ) = W(π). Together with Ua(πʹ) − Ua(π) = Ub(πʹ) − Ub(π) = 1, this means that preferences satisfy a → y → b → x → a, where an arrow u → v indicates that agent u likes v but not the other way around. Therefore, S(π)i = 1 and S(πʹ)i = 2. Since S(π) and S(πʹ) only differ at positions i, …, i + 2, this means that S(πʹ) > S(π), so Φ(πʹ) > Φ(π), as required. Let π be an arrangement where a and b form a blocking pair and sit two seats away. Let πʹ be π with a and b’s seats swapped. Then, Φ(πʹ) > Φ(π). The same argument works, except that now we consider five agents x, a, z, b, y seated at positions i, …, i + 4. This is because agent z remains a common neighbor to a and b when swapping places, and can, essentially, be ignored. Therefore, since the potential is upper-bounded, we get that the swap dynamics have to converge. One might now rightfully ask whether convergence is guaranteed to take polynomial time. While we could neither prove nor disprove this, in Appendix [sec:exponentialconvergence] we give evidence of why exponential time might be required. Moreover, note that for cycles convergence is not guaranteed even for swaps at distance at most two; e.g., P5 in Figure [fig:5unst], where any two agents are seated at most two seats away anyway. For paths, on the other hand, one could still hope that the result generalizes beyond distance at most two when the preferences are non-negative. This is however not the case, even when stable arrangements exist, as we show next (details in the appendix). 0.45 [fig:4noncvpref] 0.45 [fig:4noncvexple] [fig:4noncv] lem:4noncv Consider profile P4 in Figure [fig:4noncv]. A path stable arrangement exists, yet the swap dynamics started from certain arrangements cannot converge. This generalizes to n ≥ 4 agents by adding n − 4 dummy agents to P4 liking nobody and being liked by nobody and seating them at positions 5, …, n on the path. Hence, non-convergence for distance ≤ 3 is not a small-n artifact. Conclusions and Future Work =========================== We studied envy-freeness and exchange-stability on paths and cycles. For both topologies, we showed that finding envy-free/stable arrangements can be achieved in polynomial time when the number of agent classes is bounded, while for envy-freeness the problem becomes NP-hard without this restriction, even for binary preferences. For stability, it is known that for sufficiently many values the problem is also NP-hard. However, it would be interesting to see, for instance, if the same can be said about binary preferences. For cycles at least, we believe this to be the case, but were unable to prove it. In part, this is because of the difficulty of constructing unstable instances in the first place. Moreover, for both topologies, we gave a full characterization of the pairs (k, v) such that k-class v-valued unstable preferences exist. For paths, the characterization requires negative values in the two-valued case, and we are still unsure whether two-valued non-negative preferences that are unstable on a path exist. We, however, partially answer this in the negative by showing that the swap dynamics are guaranteed to converge if agents can only swap places with other agents seated at most two positions away from them. Without this assumption, convergence might not be guaranteed even when stable arrangements exist, so a different approach would be required to prove existence. It would also be interesting to know if unstable preferences are exceptions or the norm. We give a probabilistic treatment of this question for random preference digraphs of average degree $O(\sqrt{n})$ in the appendix. As an avenue for future research, it would be attractive to consider other kinds of tables commonly used in practice, the most relevant being the one shaped as a 2 × n grid, with guests on either side facing each other. It would additionally be interesting to consider non-additive utilities or to enforce additional constraints on the arrangement, such as certain people coming “in groups” and hence having to sit consecutively at the table. It would also be worth investigating replacing addition by taking minimum in the agents’ utilities (and also the more general lexicographical variant). ### Acknowledgements We thank Edith Elkind for the useful discussions regarding this work. We thank Giovanna Varricchio and Martin Bullinger for pointing out relevant literature. A preliminary version of this paper was presented at COMSOC’23; we thank the workshop attendees for their interesting questions and remarks. We thank the anonymous reviewers for their constructive feedback and useful suggestions contributing to improving the paper. Omitted Proofs ============== In this appendix, we provide the proofs omitted from the main text of the paper. Proofs Omitted From Section [sec:envy-free] ------------------------------------------- In this section, we prove Theorem [thnphardpath], restated below for convenience. We proceed similarly as for Theorem [thnphardcycle], this time reducing from Hamiltonian Path on directed graphs. To make the reduction work, we require the additional stipulation that the input graph G has a vertex with no outgoing edges, which we assume without loss of generality to be vertex n. Note that this preserves NP-hardness. Moreover, in the preprocessing step, will now only check that vertices v ∈ V \ {n} have outgoing edges. Otherwise, the construction of preference profile P stays the same. To show that if G has a Hamiltonian path, then there exists an envy-free arrangement on a path, the same argument as before can be used, except for the treatment of zn, who gets a utility of zero, but they still can not envy another agent because they approve of nobody (all other agents retain utility 1, as before). To show that an envy-free arrangement on a path implies the existence of a Hamiltonian path, the argument stays similar but again requires minor tweaks. In particular, for agent zn, and only for them, it holds that they do not necessarily need a second neighbor other than yn, because they like no other agents, so they will be happy with a utility of 0, obtained by sitting at one of the ends of the path. The analysis for the other agents stays the same, as they do require a second neighbor to get a utility of 1, and hence cannot sit at the ends. Proofs Omitted From Section [sec:kclassesalgo] ---------------------------------------------- In this section, we prove Theorem [thm:algokclasscycle], restated below for convenience. The proof idea stays similar to that for paths, first designing a nondeterministic logarithmic space algorithm deciding whether a compatible arrangement exists and then lifting this to one running in deterministic polynomial time. Since cycles no longer have endpoints, the definition of compatible arrangements needs adjusting. First, instead of introducing a dummy agent class 0 and placing it at positions 0 and n + 1 in s, we now make s0 stand for sn, and sn + 1 stand for s1. Similarly, sn + 2 = s2, etc. Moreover, in order for arrangement s to be compatible, we now require that for all 1 ≤ i < j ≤ n such that 1 < j − i < n − 1, the triplets (si − 1, si, si + 1) and (sj − 1, sj, sj + 1) are long-range compatible and, additionally, for all 1 ≤ i ≤ n the triplets (si − 1, si, si + 1) and (si, si + 1, si + 2) are short-range compatible. Note, therefore, that the pair (i, j) = (1, n) is the only pair for which the required check differs from the path case: previously, the check was for long-range compatibility, while now it is for short-range compatibility, and technically with the roles of i and j reversed. This time, instead of beginning the algorithm by guessing the values of s1 and s2, we instead begin it by guessing the values of s0, s1 and s2. The algorithm then proceeds as before. However, when the value i = n is reached, naturally, no guessing takes place, as sn has already been guessed, but all other computations execute as before. However, when i = n + 1 is reached, it is trickier; not only do we not have to guess sn + 1 = s1, but also the check performed has to be altered. In particular, instead of checking whether (sn − 1, sn, s1) is long-range compatible with (sn, s1, s2), the check now has to be for short-range compatibility. Checking for short-range compatibility can easily be incorporated, so it remains to show how to ensure that the two triplets are not also tested for long-range compatibility like in the previous implementation. This is done as follows: instead of temporarily decreasing the values msi − 3, si − 2, si − 1 and msi − 2, si − 1, si by one and then checking (si − 2, si − 1, si) against the counts in m, we now do the same but also decrease msn, s1, s2 by one. We stress that these rather tedious modifications are only applied for i = n + 1. The modified algorithm needs to store s0, s1 and s2 throughout its execution in addition to the state it already stored, but this does not impact the logarithmic space-bound, completing the proof. Proofs Omitted From Section [sec:2classes] ------------------------------------------ In this section, we prove Theorem [thm:2classpathstable], restated below for convenience. The introduction of agents sitting at the two endpoints calls for a more careful analysis: * If one class, say, likes everyone equally. Then either prefer to be on the endpoint or in the middle of the path (depending on the sign of their constant preference). If they prefer endpoints, seating two on the endpoints makes the arrangement stable; otherwise, sitting all in the middle ensures stability. Those two solutions are not available if and only if there is a single agent in one of the two classes: in that case, ensuring that this agent gets maximal utility stabilizes the whole arrangement. * If and prefer the same class, say. The reasoning from the previous proof still holds, as long as no sits at an endpoint. This would happen if and only if there is a unique : ensuring maximum utility for that one agent would then give stability. * If and both strictly prefer the opposite kind, alternate and starting from the most numerous class, say. Suppose: + Preferences for the opposite kind are positive for both. Whenever there is strictly more of one class than the other, the reasoning from the previous proof still holds. In case of equality, only the and at the endpoints would want to swap: it is not the case, as they would exchange a “different colour” neighbor for a “same colour” one. + Preferences for the opposite kind are negative for both. Now at most two agents have maximum utility: the extremal one, and the extremal one if there are as many as. If both have maximum utility, the others cannot improve, and the arrangement is stable. The arrangement is still stable if there are strictly more than, as the extremal could only agree to swap between two. As only are sitting between two, this cannot happen. + Preference of for is negative, but for is positive. Then non-extremal would envy the extremal, which would envy them back if and only if the preference of towards is strictly positive. In this latter case, proceed to the exchange: in the middle can only improve by switching with an extremal, which would never be accepted. This new arrangement is therefore stable. * If and both strictly prefer their own kind, sit all on one side and all on the other. Let $\textcolor{blue}{B\_{\mathit{out}}}$ be the extremal, $\textcolor{blue}{B\_{\mathit{in}}}$ be the only with both a and a neighbor $\textcolor{Red}{R\_{\mathit{in}}}$, and let $\textcolor{Red}{R\_{\mathit{out}}}$ be the extremal. As previously, $(\textcolor{blue}{B\_{\mathit{in}}}, \textcolor{Red}{R\_{\mathit{in}}})$ is never a blocking pair. Suppose: + Preferences for their own kind are positive for both. We verify that $(\textcolor{blue}{B\_{\mathit{in}}}, \textcolor{Red}{R\_{\mathit{out}}})$ is not a blocking pair, as $\textcolor{blue}{B\_{\mathit{in}}}$ would lose his only neighbor. By symmetry, the only remaining possible blocking pair is $(\textcolor{blue}{B\_{\mathit{out}}}, \textcolor{Red}{R\_{\mathit{out}}})$: it is also not a blocking pair since both would only gain an “opposite colour” neighbor. + Preferences for their own kind are negative for both. Both $\textcolor{blue}{\textcolor{blue}{B\_{\mathit{out}}}} $ and $\textcolor{Red}{R\_{\mathit{out}}}$ have maximum utility, hence $\textcolor{blue}{B\_{\mathit{out}}}$ an $\textcolor{Red}{R\_{\mathit{out}}}$ are not part of a blocking pair. Other except $\textcolor{blue}{B\_{\mathit{in}}}$ can only improve their utility by moving to an endpoint, i.e., by switching with $\textcolor{Red}{R\_{\mathit{out}}}$ or $\textcolor{blue}{B\_{\mathit{out}}}$, both being impossible. The only remaining possibility $(\textcolor{blue}{B\_{\mathit{in}}}, \textcolor{Red}{R\_{\mathit{in}}})$ is also not a blocking pair. Therefore, the arrangement is stable. + Preference of for is negative, but that of for is positive. Note that all but $\textcolor{Red}{R\_{\mathit{in}}}$ and $\textcolor{Red}{R\_{\mathit{out}}}$ have maximum utility, hence cannot be part of a blocking pair. Moreover, since $\textcolor{blue}{B\_{\mathit{out}}}$ has maximum utility, $\textcolor{Red}{R\_{\mathit{in}}}$ could only switch with having at least one blue neighbor, so it cannot improve. If dislike, $\textcolor{Red}{R\_{\mathit{out}}}$ is also unable to improve, and the arrangement is stable. On the contrary, suppose strictly like, and consider the arrangement obtained after exchanging $\textcolor{Red}{R\_{\mathit{out}}}$ and $\textcolor{blue}{B\_{\mathit{in}}}$: now both extremal have the second highest utility, and cannot improve since no is sitting between two, so the arrangement is stable. Hence, it is always possible to sit two classes in a stable manner on a path. Proofs Omitted From Section [sec:3-val3-class] ---------------------------------------------- In this section, we prove Theorem [thm:unstternpath], restated below for convenience. $$\mathcal{P} =\left[\begin{array}{ c c c c\| c c c} 0 & 0 & 0 & 0 & 1 & \ldots & 1\\ 1 & 0 & 0 & 0 & 0 & \ldots & 0\\ 1 & 0 & 0 & 0 & 0 & \ldots & 0\\ 1 & 0 & 0 & 0 & 0 & \ldots & 0\\ \hline 0 & 3 & 3 & 3 & 1 & \ldots & 1\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 3 & 3 & 3 & 1 & \ldots & 1\\ \end{array} \right]$$ [fig:expleprefunst3valuespath] We consider the same instance of preferences as in the proof of Theorem [thm:unstternarycycle], with the following modifications: we add two copies of Bob, and we suppose there are at least eight friends. Alice likes everyone but the Bobs, the Bobs only like Alice, the friends like the Bobs the most and Alice the least. We furthermore suppose that friends would rather be next to one Bob than between two other friends. Figure [fig:expleprefunst3valuespath] displays a possible instance of such preferences. Suppose one of the Bobs is sitting next to Alice. Since there are at least eight different friends, at least one of them is neither sitting beside a Bob nor at an endpoint of the table. Indeed, at most five friends are sitting next to Bobs (since Alice sits next to one of them), and two more friends can be sitting at an endpoint. Hence, Alice and this friend would both agree to switch places since it is always worth it for a friend to move beside a Bob, even if this means sitting at an endpoint. Now, suppose no Bob is sitting next to Alice. Since there are three Bobs, at least one of them is not sitting at an endpoint, say B1. The only case where one of Alice’s neighbors would not agree to switch with B1 is if it was already sitting next to another Bob, say B2; moreover, the only reason for him not to switch with B2 is if B2 is sitting at the endpoint of the table. Hence, the seating arrangement is of the form (B2, F, A, F, …, B1, …). Alice is not sitting at an endpoint, therefore possesses a second friend as a neighbor, and that second friend would agree to swap with at least one of the two remaining Bobs. Proofs Omitted From Section [sec:2valuedcycle] ---------------------------------------------- In this section, we complete the proof of Theorem [thm:binunstcycle]. Then, we prove the lemmas used in the proof of Theorem [thm:3class2valuedcyclestable]. Subsequently, we complete the proof of Theorem [thm:3class2valunstpath]. Finally, we prove Lemma [lem:4noncv]. ### Proving Theorem [thm:binunstcycle]. We now complete the proof of Theorem [thm:binunstcycle], restated below for convenience. [continued] We now show that all arrangements on a cycle are unstable for our preference profile, by considering every possible local arrangement around c and showing that they all induce a blocking pair. * If the local arrangement around c consists of (b1, c, b2), then c has utility 0 and would agree to swap with anyone having a neighbor in D. Since D contains strictly more than two agents, one of the di neighbors of a has another member of D as a neighbor. Performing a swap with c would increase its utility from 1 to 2, while c would improve to utility 1, so it is a blocking pair. * If it consists of (b1, c, a), c has again utility 0 and would switch with whoever has a neighbor in D. In particular, if the local arrangement is (b1, c, a, b2), then (b1, c) is a blocking pair, as b1 second neighbor is in D. If it is (b2, b1, c, a), then (b2, c) is a blocking pair, as b2’s second neighbor is in D. Otherwise, the local arrangement must be (di, b1, c, a, dj), and (b2, c) forms once again a blocking pair. The same reasoning naturally holds for local arrangements of the form (b2, c, a). * If the local arrangement around c consists of (di, c, a), then at least one member of B, say b1, is no neighbor of a and has utility 0. In that case, both di and b1 can increase their utility by exchanging seats. * At last, if it consists of (di, c, b1) or (di, c, dj), then agent a has utility 0, and di can always increase its utility by switching with a. Indeed, if a was his neighbor, he would exchange c for a member of D ∪ B while retaining a as a neighbor; otherwise he would exchange c for a second neighbor in D ∪ B. Since moving close to c would always increase a’s utility, (di, a) is a blocking pair. The same reasoning of course holds for local arrangements of the form (di, c, b2). ### Proving Theorem [thm:3class2valuedcyclestable]. We now prove the Lemmas used to prove Theorem [thm:3class2valuedcyclestable], restated below for convenience. [lem:3classsta:2classnarciss] Suppose at least two classes like their own kind. Then, there exists a stable arrangement. Suppose like and like. Then, consider the arrangement where each class sits among themselves. Let $\textcolor{Red}{R\_1}$ and $\textcolor{blue}{B\_1}$ be the neighboring and respectively, $\textcolor{Red}{R\_2}$ (resp. $\textcolor{blue}{B\_2}$) the (resp. Blue) with a neighbor (we can potentially have $\textcolor{Red}{R\_1} =\textcolor{Red}{R\_2}$). Note that all but $\textcolor{Red}{R\_1}$ and $\textcolor{Red}{R\_2}$ have maximum utility, $\textcolor{Red}{R\_1}$ can only improve its utility by switching with a, whereas $\textcolor{Red}{R\_2}$ cannot improve by switching with a. By symmetry, this also applies to $\textcolor{blue}{B\_1}$ and $\textcolor{blue}{B\_2}$, hence a blocking pair must include $\textcolor{Red}{R\_1}$ (or $\textcolor{blue}{B\_1}$) and a. This constitutes a blocking pair only if g > 1, like but dislike themselves. If this is the case, separate the and by seating one between them: all and inner now have maximum utility, and both extremal are now in $\textcolor{blue}{B\_2}$’s previous case. Hence the arrangement is stable. [lem:3classsta:1classlikesitself+other] Suppose precisely one class likes its own kind, and that they like at least another class. Then, there exists a stable arrangement. Suppose that like and, dislike and dislike. We start building a stable arrangement by seating all consecutively (on a path). Call the extremal the two (or unique if r = 1) at the endpoints of the path, and note that all but the extremal ones already have maximum utility. We then complete the arrangement starting from the neighbor of one of the extremal. * If g < b, start with a and alternate a and a until all are seated, giving the potential remaining seats to : have maximum utility and no would accept a swap with a (not even a neighbor), hence the arrangement is stable. * If g ≥ b and like, start with a and alternate a and a until all are depleted, giving the potential remaining seats to. Note that an extremal would never accept a swap with a unless it sits between two, in which case the already has maximum utility. Moreover, a would never accept a swap with a, even if it is a neighbor. Finally, since an exchange with a also cannot improve an extremal utility, the arrangement is stable. * If g ≥ b and dislike, start with a and then alternate two and two until all are depleted, give the remaining seats to. In case there is a unique, seat him between two. Note that no swap with a or a would give an extremal two neighbors or, hence would never be profitable. Furthermore, a swap with a would always give zero utility to a, and would never be agreed upon. Hence the arrangement is stable. [lem:3classsta:1classonlynarciss] Suppose precisely one class likes its own, and that they dislike both other classes. Then, there exists a stable arrangement. Suppose only like their own kind, but they also dislike both and. Seat all together and alternate and starting from the most numerous class, say. Note that no would agree to exchange with a seating between two, even if they are neighbors and the has a neighbor. Furthermore, in the case where g = b, the same reasoning can be applied to all sitting between two, and the arrangement is stable because the and with a neighbor would not exchange seats. Hence the arrangement is stable. [lem:3classsta:nonarcisswithparia] Suppose no class likes itself, and one class is disliked by the two others. Then, there exists a stable arrangement. Suppose no class likes, even themselves. Suppose without loss of generality that b ≥ g, start by seating all together, then alternate between a and a starting from until everyone is seated. No would accept to exchange seats with a sitting between two, even if they are neighbors. Furthermore, no such would benefit from switching with a. By symmetry, it also holds for, and a blocking pair would have to contain one of the / agents with a neighbor: those would neither exchange seats between them nor with any, hence the arrangement is stable. [lem:3classsta:nonarciss1likeall] Suppose no class likes itself, but one class likes the two others. Then, there exists a stable arrangement. Suppose dislike their own kind but like the two other classes. Using Lemma [lem:3classsta:nonarcisswithparia], we may assume that at least one class, say, like. * If r > b, start by alternating and starting with until all are depleted, then alternate and starting with until one of them is depleted; seat the remaining agents altogether. Note that all have maximum utility, and if two are neighbors, then all also have maximum utility. Otherwise, all are seated between two, and no can improve by swapping with a. Hence the arrangement is stable. * If r < b, start by alternating and starting with until all are depleted, then alternate and starting with until one of them is depleted; seat the remaining agents altogether. Note that all have maximum utility. Suppose no has a neighbor, i.e. every seats between two : no, not even a neighbor, would profit from switching with a, hence the arrangement is stable. Otherwise, no two are neighbors: sitting between two have maximum utility, and neither with a and a neighbor, nor with two neighbors can improve by exchanging with a, even if it is a neighbor. Hence the arrangement is still stable. * If r = b, alternate between and until depletion, and then seat all together. All have maximum utility, the same holds for all except the one with a neighbor. This extremal cannot improve by switching with a, and the arrangement is stable. We are now left with a single case to handle, without loss of generality the case where only like, only like, and only like, and no class likes their own kind. Also without loss of generality, assume n ≥ 4. For brevity in what follows, and consistently with the notation of Section [sec:kclassesalgo], denote by 1, by 2, and by 3, and write n1, n2 and n3 for r, g, and b, respectively, where n1 + n2 + n3 = n. An arrangement π can be seen as a sequence s = (si)i ∈ [n] consisting of n1 ones, n2 twos and n3 threes. Since s represents an arrangement on a cycle, one can assume that s0 = sn and sn + 1 = s1, etc. We will again lift agent preferences to class preferences. As in Section [sec:binarypref], let W(s) = W(π) be the utilitarian social welfare induced by s/π. Similarly in spirit to Section [sec:kclassesalgo], it will be useful to think in terms of long/short-range blocking triples. Namely, let a, b, c, d, e, f be agent classes, then we say that triples (a, b, c) and (d, e, f) are long-range blocking if pb(a) + pb(c) < pb(d) + pb(f) and pe(d) + pe(f) < pe(a) + pe(c); intuitively, neither b wants to swap with e, nor vice-versa. Furthermore, for a, b, c, d agent classes, we say that triples (a, b, c) and (b, c, d) are short-range blocking if pb(a) < pb(d) and pc(d) < pc(a); intuitively, if a, b, c, d are consecutive in the arrangement, then neither b wants to swap with c, nor vice-versa. An arrangement s is stable if the following two conditions hold: * For all 1 ≤ i < j ≤ n such that 1 < j − i < n − 1, the triplets (si − 1, si, si + 1) and (sj − 1, sj, sj + 1) are not long-range blocking; * For all 1 ≤ i ≤ n the triplets (si − 1, si, si + 1) and (si, si + 1, si + 2) are not short-range blocking. Before proving our main assertion, the following observation will be instrumental: [prop:adjacentseats] Given an arrangement s, the utilitarian social welfare W(s) is equal to n minus the number of pairs of agents from the same class seated in adjacent seats. Formally, W(s) = n − ∣{i ∈ [n] ∣ si = si + 1}∣. When two agents belonging to different classes are seated next to each other, by construction of the preferences, exactly one of them will like the other. Moreover, agents do not like agents from the same class as themselves. We now prove our assertion, in fact in a stronger form, as follows: [lem:3classsta:maxutilarg] Assume 0 only likes 1, 1 only likes 2, and 2 only likes 0, then any arrangement s maximizing the utilitarian social welfare W(s) is stable. Let s be any arrangement maximizing egalitarian social welfare. Assume for a contradiction that s is not stable. This can be either because of short-range or long-range blocking triples. We tackle the two cases separately: 1. Assume 1 ≤ i ≤ n is such that (si − 1, si, si + 1) and (si, si + 1, si + 2) are short-range blocking. In other words, write (si − 1, si, si + 1, si + 2) = (a, b, c, d), then pb(a) < pb(d) and pc(d) < pc(a). By symmetry, without loss of generality, assume b = 1 and c = 2. From this it follows that a = 3 and d = 2. Hence, before b and c swap seats we have (3, 1, 2, 2) while afterward, we have (3, 2, 1, 2), meaning that the swap would increase social welfare (recall Proposition [prop:adjacentseats]), contradicting the maximality of W(s). 2. Assume 1 ≤ i < j ≤ n are such that 1 < j − i < n − 1 and (si − 1, si, si + 1) and (sj − 1, sj, sj + 1) are long-range blocking. In other words, write (si − 1, si, si + 1) = (a, b, c) and (sj − 1, sj, sj + 1) = (d, e, f), then pb(a) + pb(c) < pb(d) + pb(f) and pe(d) + pe(f) < pe(a) + pe(c). By symmetry, without loss of generality, assume b = 1 and e = 2. The conditions on the preferences require that there are more threes among a, c than d, f but more twos among d, f than a, c. This implies that at least one of a and c is 3, say c = 3, and at least one of d and f is 2, say f = 2. So far, we know that (a, b, c, d, e, f) = (a, 1, 3, d, 2, 2). Looking at the conditions one more time, we get that (a, 1, 3) and (d, 2, 2) are long-range blocking if and only if d = 3 implies a = 3 and a = 2 implies d = 2. We can now list all the possibilities for (a, b, c, d, e, f) and write what would happen after b and e would swap places: 1. (1, 1, 3, 1, 2, 2) → (1, 2, 3, 1, 1, 2), the latter has higher welfare; 2. (1, 1, 3, 2, 2, 2) → (1, 2, 3, 2, 1, 2), the latter has higher welfare; 3. (2, 1, 3, 2, 2, 2) → (2, 2, 3, 2, 1, 2), the latter has higher welfare; 4. (3, 1, 3, 1, 2, 2) → (3, 2, 3, 1, 1, 2), the latter has the same welfare; 5. (3, 1, 3, 2, 2, 2) → (3, 2, 3, 2, 1, 2), the latter has higher welfare; 6. (3, 1, 3, 3, 2, 2) → (3, 2, 3, 3, 1, 2), the latter has higher welfare.Hence, in all cases except (a, b, c, d, e, f) = (3, 1, 3, 1, 2, 2), we get a contradiction of the maximality of W(s). We now need to handle this last case, which indeed requires further insight. For this case, we will now show that we can construct an arrangement sʹ such that W(sʹ) > W(s), again contradicting the maximality of W(s). The technique we will use, however, is different from the usual swapping of the seats of b and e. Namely, let x be the agent sitting to the left of 3, 1, 3. For illustration, if we were to write s in full, it would be …, x, (3, 1), 3, …, 1, 2, 2, …. We now take the two bracketed agents (3, 1) and reseat them between agents 2, 2. The seating arrangement sʹ obtained as a result looks as follows: …, x, 3, …, 1, 2, (3, 1), 2, …. Comparing the social welfares W(s) and W(sʹ) notice that the moved pair (3, 1) has “broken” the two adjacent twos 2, 2. Moreover, irrespective of the value of x, in s agent x is sitting next to a 3, and the same is true in sʹ. As a result, by Proposition [prop:adjacentseats], we get that W(sʹ) − W(s) = 1, contradicting the maximality of W(s). ### Proving Theorem [thm:3class2valunstpath]. We now complete the proof of Theorem [thm:3class2valunstpath], restated below for convenience. [continued] Suppose Alice sits next to Bob. On one hand, if Alice is seated at an endpoint of the path, then both Alice and Bob would improve by exchanging seats: after the exchange, Alice gets utility zero instead of minus one, whereas Bob gets utility one instead of zero. On the other hand, if Alice is not extremal, there is a Friend at one endpoint of the path not a neighbor of Bob: swapping seats with him would increase Alice’s utility from zero to one, while the Friend would improve from one (respectively minus one if its only neighbor is Alice) to two (respectively zero). Now suppose Bob is sitting away from Alice: exchanging with a neighbor of Alice would increase its utility from at most minus one to at least zero, whereas the Friend would improve from at most zero to at least one. Hence no arrangement on a path is stable. ### Proving Lemma [lem:4noncv]. We now prove Lemma [lem:4noncv], restated below for convenience. Let π\* be the arrangement (a, b, c, d). Since each agent approves of exactly one other agent, they can never get utility strictly greater than one. Since only d does not achieve utility one in π\, it is stable. Moreover, consider arrangements π1 = (a, d, b, c) and π2 = (c, d, b, a). The only blocking pair in π1 is (a, c) as both b* and d have utility one. Exchanging them leads to arrangement π2. Similarly, in π2, the only blocking pair is (b, d). Exchanging them gives π1 back, up to reversal of the seat numbers. Hence, the swap dynamics cannot converge. See Figure [fig:4noncvexple] for an illustration. Stability Analysis of Profile P5 ================================ [fig:proof5unst] In this appendix, we briefly analyze the stability of profile P5 from Figure [fig:5unst], which is the only preference profile with n = 5 that is unstable on a cycle. Despite our best efforts, there does not seem to be an easy explanation for the emergence of instability in this case. [lem:5unst] Binary preferences P5 induce no stable arrangements on a cycle. Figure [fig:proof5unst] provides a pictorial proof of the instability of P5. Each of the twelve possible cyclic arrangements is displayed and a blocking pair is shown in red. The outgoing edges of the nodes participating in the blocking pairs are shown in blue for effortless verification. Evidence for Exponential Convergence ==================================== In this appendix, we revisit the potential function used to prove convergence of the swap dynamics in the proof of Theorem [thm:close-swaps-two-valued-is-stable]. In essence, we show that each increase in potential might be very small, leading to an exponential number of increases. Note, however, that we do not exhibit preference instances where such exponential behavior can be observed.[6](#fn6) Moreover, we also do not exclude the possibility that the function can be shown to increase enough with each swap on average when the dynamics are carried out. What we show, instead, is that there is an exponentially long chain of potential values such that between any two consecutive values in the chain there are preferences for which the transition could occur by exchanging a blocking pair. The potential argument alone cannot guarantee polynomial time convergence of swap dynamics. We first consider the auxiliary potential sequence S(π) with values in {1, 2}, and study the effect of exchanges at distances one and two that keep the social welfare constant. To simplify notation, we map S(π) ∈ {1, 2}n − 1 to Sb(π) ∈ {0, 1}n − 1 by subtracting one. Performing an exchange at distance one while keeping the social welfare constant corresponds to the following modification of a subsequence of Sb(π): $0x1 \to 1\overline{x}0$, where x ∈ {0, 1} (see Figure [fig:exchangedist1potential]). Indeed, an exchange at distance one modifies the utility of at most four people, hence modifies a subsequence of length at most three of the auxiliary potential, the latter being defined not on vertices but edges; we call this operation f3. $$f\_3: 0x1 \to 1\overline{x}0$$ Similarly, we define the operation corresponding to exchanging at distance two that keeps the social welfare constant: it maps the subsequence of Sb(π) of length four 0xy1 to $1\overline{y}\overline{x}0$. For the same reason as above, we call this operator f4. $$f\_4:0xy1 \to 1\overline{y}\overline{x}0$$ In the following, we denote by “Apply fi at position j” the application of fi on the subsequence of indices [j, j + i]. Based on those two operators, we further define the operator f8 mapping the sequence of length 8 (0, 0, 0, 0, 0, 0, 0, 1) to (1, 0, 0, 0, 0, 0, 0, 0). It consists of the following operations: 1. Apply f3 at position 6: (0, 0, 0, 0, 0, 0, 0, 1) → (0, 0, 0, 0, 0, 1, 1, 0); 2. Apply f4 at position 3: (0, 0, 0, 0, 0, 1, 1, 0) → (0, 0, 1, 1, 1, 0, 1, 0); 3. Apply f4 at position 2: (0, 0, 1, 1, 1, 0, 1, 0) → (0, 1, 0, 0, 0, 0, 1, 0); 4. Apply f4 at position 4: (0, 1, 0, 0, 0, 0, 1, 0) → (0, 1, 0, 1, 1, 1, 0, 0); 5. Apply f4 at position 3: (0, 1, 0, 1, 1, 1, 0, 0) → (0, 1, 1, 0, 0, 0, 0, 0); 6. Apply f3 at position 1: (0, 1, 1, 0, 0, 0, 0, 0) → (1, 0, 0, 0, 0, 0, 0, 0). For k ≥ 3, we then recursively define the operators f3k − 1 mapping the sequence of length 3k − 1 (0, …, 0, 1) to (1, 0, …, 0) through the following operations: 1. Apply f3 at position 3(k − 1): (0, …, 0, 1) → (0, …, 0, 1, 1, 0); 2. Apply f3(k − 1) − 1 at position 2: (0, …, 0, 1, 1, 0) → (0, 1, 0…, 0, 1, 0); 3. Apply f3(k − 1) − 1 at position 3: (0, 1, 0…, 0, 1, 0) → (0, 1, 1, 0, …, 0); 4. Apply f3 at position 1: (0, 1, 1, 0, …, 0) → (1, 0, …, 0). Note that this is well-defined since f8 is defined separately. Noting that operator f3k − 1 contains more than 2k − 1 uses of f3 and f4 for all k ≥ 3 concludes the proof. [fig:exchangedist1potential] Non-Monotonicity of Stability ============================= In this section, we show that stability is non-monotonic, i.e., adding agents to a given instance can both introduce or destroy stability. For instance, consider preferences P5 in Figure [fig:5unst]. On a cycle, all binary preferences with either four or six agents possess a stable arrangement, implying that adding a fifth agent could destroy stability while adding a sixth agent would restore it. We now show that this phenomenon can occur for all values n ≥ 7: [thm:nonmonotonicsta] For all n ≥ 7, adding an agent can both destroy stability or restore it. We first prove that for n ≥ 7 adding an agent can break stability. Consider the unstable preferences of size n + 1 in Theorem [thm:binunstcycle]: we show that removing one agent creates a stable arrangement. Indeed, remove agent a and consider sitting c between b1 and b2: everyone but c has maximum utility (since b1 and b2 love only one person, they cannot get utilities higher than 1), hence c cannot exchange seats with anyone and the arrangement is stable. We now prove that for n ≥ 7 adding an agent can create stability. Consider the unstable preferences of size n in Theorem [thm:binunstcycle], and add one extra agent b3 ∈ B; i.e., which only likes a and c and is only liked by the members of D. Consider placing b1, c, b2, a, b3 consecutively on the cycle in this order: agent b2 and all members of D have utility 2, hence only b1, b3 or c can be part of a blocking pair. However, b1 and b3 both have utility one and would therefore only exchange for a seat with utility 2: there exists only one such seat, currently held by b2, who has maximum utility and would hence not agree to swap places. As a result, neither b1 nor b3 are part of a blocking pair, so the arrangement is stable. Blockwise-Diagonal Preferences ============================== In this section, we show that, even when the preferences of the agents are, in a certain sense, highly decomposable, knowledge about the stability of the subparts is unlikely to help us to find stable arrangements for the instance as a whole. In essence, we show why non-monotonicity can make reasoning about stability rather challenging, even in reasonably simple cases. To make the previous statements precise, we introduce the concept of agent components, as follows. We say that a set of agents C ⊆ A is isolated if for all a ∈ C and b ∈ A \ C, it holds that pa(b) = pa(b) = 0. Set C is called a component if none of its proper subsets are isolated. Note also that components and classes are two different notions: agents in the same component may not have the exact same preferences, but instead are limited to only caring about agents in their component. Assume that the set of agents A is partitioned into components A = C1 ∪ … ∪ Ck (this partition always exists and is unique).[7](#fn7) The preference matrix can then be represented, after a potential reordering of the agents, as a blockwise-diagonal matrix. When the partition into components is non-trivial; i.e., k > 1; intuitively, finding an arrangement that is stable for such preferences should be easier than for general ones: first, find a stable arrangement on a path for each component, and then join all those paths to obtain a stable arrangement on a cycle (or on a path). While this method is indeed guaranteed to produce a stable arrangement whenever each component admits a path stable arrangement (at least for non-negative preference values), we will actually show that there are many instances where a stable arrangement exists but can not be produced by this approach. Before showing this, we need a technical lemma for cycles, stated next. [lem:0utilstability] Let π be an arrangement where each agent sits between two agents from different components. If for any two distinct components Ci and Cj there is at most one pair of agents (a, b) ∈ Ci × Cj such that a and b are neighbors in π, then π is stable on a cycle. First, notice that in such an arrangement all agents get utility zero. Let a ≠ b be two agents. We want to show that they do not form a blocking pair. Assume i, j ∈ [k] are such that a ∈ Ci and b ∈ Cj. If i = j, then by assumption a would also have utility zero when seating in b’s seat, so (a, b) is not a blocking pair. Now, assume i ≠ j. Let aℓ and ar denote the two neighbors of a on π; we define bℓ and br similarly. Suppose a and b wanted to switch place: this means a would have strictly positive utility at b’s seat, and b would have strictly positive utility at a’s seat. Assuming a and b are not neighbors, this means that $\{a\_\ell, a\_r\} \cap \mathcal{C}\_j \neq \varnothing$ and $\{b\_\ell, b\_r\} \cap \mathcal{C}\_i \neq \varnothing.$ Let aʹ and bʹ be such that aʹ ∈ {aℓ, ar} ∩ Cj and bʹ ∈ {bℓ, br} ∩ Ci. Since (a, aʹ) and (bʹ, b) are both in Ci × Cj and are distinct pairs of adjacent agents in π, this contradicts the hypothesis. Furthermore, if a and b are neighbors, then b envying a implies that a’s second neighbor is also in Cj, from which the pairs of agents formed by a and its neighbors similarly contradict the hypothesis. Therefore, the arrangement is stable. Armed as such, we now show that any preference profile inducing no stable arrangements on a path can be used to construct preferences whose components all resemble this profile and yet the joint profile admits an arrangement that is stable on a cycle. In other words, even when a profile decomposes non-trivially into components, knowledge about the stability of the components does not necessarily help in resolving the stability of the instance as a whole. [thm:eulertourcomponents] Let Ppath ∈ {0, 1}ℓ × ℓ be a one-component preference profile such that no stable arrangement on a path exists. Then, there exists a preference profile P whose components all resemble Ppath admitting a stable arrangement on a cycle. We construct a larger blockwise-diagonal preference matrix P ∈ {0, 1}k × k by copying Ppath a number k = 2ℓ + 1 of times: $$\mathcal{P} =\left[\begin{array}{ c c c c} \mathcal{P}\_{\mathit{path}}^{(1)} & & & 0 \\ & \mathcal{P}\_{\mathit{path}}^{(2)} & & \\ & & \ddots & \\ 0 & & & \mathcal{P}\_{\mathit{path}}^{(k)} \end{array}\right]$$ Note that Ppath(1), …, Ppath(k) naturally gives a partition into components C1, …, Ck; moreover, the choice of Ppath immediately gives that all C1, …, Ck are unstable on paths. We now construct a cycle stable arrangement for P. Since k is odd, graph Kk, which is the undirected clique graph with k vertices, possesses an Euler tour T where each vertex is visited ℓ times. We construct the arrangement π (more precisely π− 1) on the cycle by replacing every occurrence of i in T by an agent ai ∈ Ci, without repetition, starting for example from seat one. For the π we have just constructed, it then holds that the conditions to apply Lemma [lem:0utilstability] are satisfied since the Euler Tour traverses each edge in Kk precisely once. By the lemma, π is a cycle stable arrangement of P. Stability of Random Binary Preferences ====================================== In this chapter, we conduct a study of stability using probabilistic tools. In particular, we employ the Lovász Local Lemma: [Lovász Local Lemma][lem:LLL] Let A1, A2, …, Ak be a sequence of events such that each event occurs with probability at most p and is independent of all but at most d other events. If epd < 1, then the probability that none of the events occurs $P \left(\cap\_{i=1}^k \overline{A\_i}\right) $ is greater than or equal to $ \left(1-\frac{1}{d+1}\right)^k.$ We now give a lower bound on the expected number of arrangements stable on a cycle when the preference graph is sampled from the Erds-Rényi model G(n, p) with average node degree either $O(\sqrt{n})$ or $n - O(\sqrt{n}).$ Suppose a binary preference graph P is drawn at random using the Erds-Rényi model G(n, p) with p ≤ Cn− 1/2, where C = (96e)− 1/2. Then, the expected number of stable arrangements on a cycle is at least: $$\frac{1}{2}(n-1)! \exp \left( -\frac{n(n-1)}{2n-3} \right).$$ The same results holds for p ≥ 1 − Cn− 1/2. Let S = ∑πSπ be the random variable counting the stable arrangements on a cycle for preference P, where Sπ = 1 if arrangement π is stable and 0 otherwise. Since all permutations of P follow the same distribution, all Sπ have the same expectation and we only need to consider the identity arrangement π = id. For i, j ∈ [n], event Lij corresponds to the $i^\textsuperscript{th}$ agent liking the $j^\textsuperscript{th}$ one, event Eij to the $i^\textsuperscript{th}$ agent envying the $j^\textsuperscript{th}$ one, and event Bij to agents (i, j) forming a blocking pair. Finally, let the random variable Ui denote the utility of the $i^\textsuperscript{th}$ agent. First, note that events Eij and Ekl are independent for all i ≠ k; from which Pr(Eij) = Pr(Eji). We therefore get: $$\label{equ:Bij\_E\_ij2} \begin{aligned} Pr(B\_{ij}) &= Pr(E\_{ij}\cap E\_{ji}) \\ &= Pr(E\_{ij})Pr(E\_{ji})\\ &= Pr(E\_{ij})^2 \end{aligned}$$ By symmetry, we only have to calculate (Pr(E1j))2 ≤ j ≤ n. If j ∈ {2, 3}: $$Pr(E\_{1j}) = Pr\left(E\_{1j} \mid \overline{L\_{1n}}\right)Pr\left(\overline{L\_{1n}}\right) = p(1-p)$$ If j ∈ {n − 1, n}: $$Pr(E\_{1j}) = Pr\left(E\_{1j} \mid \overline{L\_{12}}\right)Pr\left(\overline{L\_{12}}\right) = p(1-p)$$ If 3 < j < n − 1: $$\begin{aligned} Pr(E\_{1j}) &= Pr\left(E\_{1j} \mid U\_1 = 0\right) Pr(U\_1=0) + Pr\left(E\_{1j} \mid U\_1 = 1\right) Pr(U\_1=1) \\ &= (2p(1-p)+p^2)(1-p)^2 + p^2 2p(1-p) \end{aligned}$$ Together with Equation, we subsequently get that: $$\label{equ:Bij} Pr(B\_{ij}) = \left\{ \begin{array}{ll} p^2(1-p)^2 & \mbox{if } \|i-j\| \leq 2\\ \left(2p^3(1-p) + 2p(1-p)^3 + p^2(1-p)^2 \right)^2 & \mbox{otherwise.} \end{array} \right.$$ Now, consider the family of events (Bij)1 ≤ i < j ≤ n. Note that each even Bij is independent of events Bkl where $\{i,j\}\cap\{k,l\} =\varnothing$, but dependent of events with which it shares an index, so Bij depends on at most d = 2(n − 2) other events. Therefore, when $Pr(B\_{ij}) < \frac{1}{2e(n-2)}$, Lemma [lem:LLL] gives: $$\begin{aligned} \mathbb{E}[S\_\pi] = Pr(S\_\pi=1) & \geq \left(1-\frac{1}{2n-3}\right)^{\frac{n(n-1)}{2}} \\ &\geq \exp \left( -\frac{n(n-1)}{2n-3} \right)\\ \end{aligned}$$ By linearity of expectation, we finally get: $$\mathbb{E}[S] \geq \frac{1}{2}(n-1)! \exp \left( -\frac{n(n-1)}{2n-3} \right)$$ It is only left to verify that $Pr(B\_{ij}) \leq \frac{1}{2e(n-2)}.$ Note that Bij has at most six terms all strictly smaller than 8p2. Hence, for $p \leq \dfrac{1}{\sqrt{96en}} \leq \dfrac{1}{\sqrt{96e(n-2)}},$ we have: $$\begin{aligned} Pr(B\_{ij}) & < 6\times 8 p^2 \\ & \leq \frac{48}{96e(n-2)} = \frac{1}{2e(n-2)} \end{aligned}$$ Comparison with 6 × 8(1 − p)2 instead gives the result for $p \geq 1-\dfrac{1}{\sqrt{96en}}.$ In practice, our result implies that, for random preferences of average out-degree at most $O(\sqrt{n}),$ a naive approach sampling arrangements uniformly at random on average determines a stable arrangement using exponentially fewer samples than the theoretically required (n − 1)!/2. Z3 Solver for Non-Negative Two-Valued Preferences ================================================= In this section, we describe the Z3 solver[8](#fn8) employed in Section [sec:binarypref] to check whether all non-negative two-valued preferences for n ≤ 7 are stable on paths and cycles. We only describe the case of cycles, as for paths it is enough to add one additional agent with null preferences from and toward all other agents. We note that for cycles it is enough to consider the binary case. For paths it is enough to consider the cases Γ ∈ {{0, 1}, {1, 2}, {1, 3}, {2, 3}}. For clarity, we describe here only the binary case, but the necessary modifications for the other cases are straightforward. Listing [lst:Z3solver] shows the main body of the solver. In line 8, we introduce a function associating the boolean “i likes j” to each pair (i, j). In line 14, we define an array of n integers encoding the index of the agent placed in each seat; lines 17 to 24 constrain this array to be a permutation representing one of the (n − 1)!/2 cycles. In lines 15, 16, and 26 we implement the main constraint: all arrangements on a cycle must induce a blocking pair. Note that all simultaneous permutations of rows and columns of a solution are themselves solutions, as this corresponds to relabeling the agents. Therefore, it is desirable for efficiency to implement some kind of symmetry breaking. We do this in Listing [lst:SymBreak] by requiring that the agents are sorted by the number of agents they approve of, breaking ties by the number of agents that approve them. Finally, Listing [lst:isBlocking] shows how to check whether agent-pair (i, j) is a blocking. Note that adjacent and non-adjacent seats require different treatments, and so lead to different logical expressions. Finally, note that Z3 returns either “Unsatisfiable” when no solutions exist, or “Satisfiable” and one solution otherwise. Finding all possible solutions is therefore rather tedious: after finding a solution, to get another one, we need to add a constraint that “the preferences are not these ones.” This detail is omitted for brevity. [h] [lst:Z3solver] [lst:SymBreak] [lst:isBlocking] --- 1. In this proof G is not the seating graph but rather an arbitrary graph.[↩](#fnref1) 2. Technically, is indifferent to everyone else, but found this formulation reads better.[↩](#fnref2) 3. Assuming that addition is performed with wrap-around using $n + 1 \equiv 1 \pmod n$.[↩](#fnref3) 4. The swap dynamics here will exhibit so-called run-and-chase behaviour, which is common to many classes of hedonic games.[↩](#fnref4) 5. One can show that it suffices to try the cases Γ ∈ {{0, 1}, {1, 2}, {1, 3}, {2, 3}}.[↩](#fnref5) 6. We did not manage to construct such preferences.[↩](#fnref6) 7. This is because the components correspond to the connected components of the undirected graph with vertex set A and edges (a, b) for any two distinct agents such that either ua(b) ≠ 0 or ub(a) ≠ 0.[↩](#fnref7) 8. <https://github.com/Z3Prover/z3>[↩](#fnref8) Stable Dinner Party Seating Arrangements ======================================== A group of n agents with numerical preferences for each other are to be assigned to the n seats of a dining table. We study two natural topologies: circular (cycle) tables and panel (path) tables. For a given seating arrangement, an agent’s utility is the sum of their preference values towards their (at most two) direct neighbors. An arrangement is envy-free if no agent strictly prefers someone else’s seat, and it is stable if no two agents strictly prefer each other’s seats. Recently, it was shown that for both paths and cycles it is NP-hard to decide whether an envy-free arrangement exists, even for symmetric binary preferences. In contrast, we show that, if agents come from a bounded number of classes, the problem is solvable in polynomial time for arbitrarily-valued possibly asymmetric preferences, including outputting an arrangement if possible. We also give simpler proofs of the previous hardness results if preferences are allowed to be asymmetric. For stability, it is known that deciding the existence of stable arrangements is NP-hard for both topologies, but only if sufficiently-many numerical values are allowed. As it turns out, even constructing unstable instances can be challenging in certain cases, e.g., binary values. We propose a complete characterization of the existence of stable arrangements based on the number of distinct values in the preference matrix and the number of agent classes. We also ask the same question for non-negative values and give an almost-complete characterization, the most interesting outstanding case being that of paths with two-valued non-negative preferences, for which we experimentally find that stable arrangements always exist and prove it under the additional constraint that agents can only swap seats when sitting at most two positions away. Similarly to envy-freeness, we also give a polynomial-time algorithm for determining a stable arrangement assuming a bounded number of classes. We moreover consider the swap dynamics and exhibit instances where they do not converge, despite a stable arrangement existing. Introduction ============ Your festive dinner table is ready, and the guests are arriving. As soon as your guests take their assigned seats, two of them are unhappy about their neighbors and rather want to switch seats. Alas, right after the switch, two other guests become upset, and then pandemonium ensues! Could you have prevented all the social awkwardness by seating your guests “correctly” from the get-go? In this paper, we study the difficulty of finding a stable seating arrangement; i.e. one where no two guests would switch seats. We focus on two natural seating situations: a round table (cycle), and an expert panel (path). In either case, we assume guests only care about having the best possible set of direct left and right neighbors. In certain cases, not even a stable arrangement might make the cut, as a single guest envying the seat of another could potentially lead to trouble. Therefore, we are also interested in finding envy-free arrangements. Formally, n guests have to be assigned bijectively to the n seats of a dining table: either a path or a cycle graph. Guests express their preferences for the other guests numerically, with higher numbers corresponding to a greater desire to sit next to the respective guest. The utility of a guest g for a given seating arrangement is the sum of g’s preference values towards g’s neighbors. A guest g envies another guest if g’s utility would strictly increase if they swapped places. Two guests want to swap places whenever they envy each other. Our goal is to compute a stable (no two guests want to swap) respectively envy-free (no guest envies another) seating arrangement. Besides the table topology, we conduct our study in terms of two natural parameters. The first parameter is the number of numerical values guests can choose from when expressing their preferences for other guests. For instance, an example of two-valued preferences would be when all preference values are either zero or one (i.e., binary, also known as approval preferences), in which case every guest has a list of “favorite” guests they want to sit next to, and is indifferent towards the others. In contrast, if the values used were ± 1; i.e., every guest either likes or dislikes every other guest; then the preferences are still two-valued, but no longer binary. Increasing the number of allowed values allows for finer-grained preferences. It is also interesting to distinguish the special case of non-negative preferences; i.e., no guest can lose utility by gaining a neighbor. Our second parameter is the number of different guest classes. In particular, dinner party guests can often be put into certain categories, e.g., charmer, entertainer, diva, politico, introvert, outsider. Each class has its own preferences towards other classes, e.g., outsiders would prefer to sit next to a charmer, but not next to an introvert. [t] [h]0.45 c\|cc & & Bounded & Unbounded EF & Poly1 ctrerr & NP-hard2 ctrerr STA & Poly1 ctrerr & NP-hard3 ctrerr [tab:summaryrescomputationalcycle] [h]0.45 c\|cc & & Bounded & Unbounded EF & Poly1 ctrerr & NP-hard2 ctrerr STA & Poly1 ctrerr & NP-hard4 ctrerr [tab:summaryrescomputationalpath] [tab:summaryrescomputational] [t] [h]0.45 \| \| ≤ 2 \| 3 \| 4 \| ≥ 5 \| \| --- \| --- \| --- \| --- \| --- \| \| ≤ 2 \| S \| S3 ctrerr \| U4 ctrerr \| U \| \| 3 \| S \| U2 ctrerr \| U \| U \| \| ≥ 4 \| S1 ctrerr \| U \| U \| U \| [tab:summaryrescycle] [h]0.45 \| \| ≤ 2 \| 3 \| ≥ 4 \| \| --- \| --- \| --- \| --- \| \| ≤ 2 \| S \| U5 ctrerr \| U5 ctrerr \| \| 3 \| S \| U2 ctrerr \| U \| \| ≥ 4 \| S1 ctrerr \| U \| U \| [tab:summaryrespath] [tab:summaryres] Our Contribution. We study the existence and computational complexity of finding stable/envy-free arrangements on paths and cycles. Some of our results can be surprising. For instance, six people with binary preferences can always be stably seated at a round table, while for five (or seven) guests some preferences are inherently unstable, so we better invite (or uninvite) another guest. However, even for six people with binary preferences, for which a stable arrangement always exists, the swap dynamics might still never converge to one. Our computational results are exhibited in Table [tab:summaryrescomputational]. In summary, if the number of guest classes is bounded by a constant (which can be arbitrary), then both stable and envy-free arrangements can be computed in polynomial time whenever they exist with no assumption on the preference values, while dropping this assumption makes the two problems difficult even for very constrained preference values. For envy-freeness, this already happens for binary preferences, arguably the most prominent case. For stability, on the other hand, this requires more contrived values (four non-negative values for cycles and six values including negatives for paths), so it would be interesting to have a finer-grained understanding of stability in terms of the values allowed when expressing one’s preferences. For instance, is it still hard to find stable arrangements for binary preferences? As we show, it turns out to be surprisingly difficult to even construct unstable binary preferences, setting aside the computational considerations. To this end, we conduct a fine-grained study aiming to answer for which combinations of our two parameters, i.e., number of preference values and guest classes, do stable arrangements always exist and for which combinations this is not the case. Our results are exhibited in Table [tab:summaryres]. Notably, for cycles we give a full characterization, either showing guaranteed stability with arbitrary values or giving unstable instances with (simple) non-negative values. Similarly, for paths, we close all cases, with the notable exception of two-valued preferences with three or more classes being permitted, where all the unstable instances we found require negative values. We conjecture that for non-negative values stability on paths is guaranteed in the two-valued case. We support this conjecture with experimental evidence, as well as a partial result under the additional assumption of guests only being willing to swap seats if they are separated by at most one seat (see Table [tab:summaryres] for more details). Appendix. In the appendix, we supply the proofs omitted from the main text, as well as supporting material. Moreover, we show that stability is a highly fragile notion, being non-monotonic with respect to adding/removing guests. We also give evidence that knowledge about stability on paths is unlikely to transfer to computing cycle-stable arrangements. Finally, we use probabilistic tools to study the expected number of stable arrangements of Erds-Rényi binary preferences. Related Work ============ The algorithmic study of stability in collective decision-making has its roots in the seminal paper of Gale and Shapley, introducing the now well-known Stable Marriage and Stable Roommates problems. Classically, the former is presented as follows: an equal number of men and women want to form couples such that no man and woman from different couples strictly prefer each other over their current partners, in which case the matching is called stable. The authors give the celebrated Gale-Shapley deferred acceptance algorithm showing that a stable matching always exists and can be computed in linear time. Irving later extended the algorithm to also handle preferences with ties; i.e., a man (woman) being indifferent between two women (men). The Stable Roommates problem is the non-bipartite analog of Stable Marriage: an even number of students want to allocate themselves into identical two-person rooms in a dormitory. A matching is stable if no two students allocated to different rooms prefer each other over their current roommates. In this setting, stable matchings might no longer exist, but a polynomial-time algorithm for computing one if any exist is known. However, when ties are allowed, the problem becomes NP-hard. The seating arrangement problem that we study is, in fact, well-connected with Stable Roommates. Instead of one table with n seats, the latter considers n/2 tables with two seats each. However, there is another more subtle difference: in Stable Roommates, two people unhappy with their current roommates can choose to move into any free room. This is not possible if there are exactly n/2 rooms. Instead, the stability notion that we study corresponds to the distinct notion of exchange-stability in the Stable Roommates model, where unhappy students can agree to exchange roommates. Surprisingly, under exchange-stability, finding a stable roommate allocation becomes NP-hard even without ties. One can also see our problem through the lens of coalition formation. In particular, hedonic games consider the formation of coalitions under the assumption that individuals only care about members in their own coalition. Then, fixing the sizes of the coalitions allows one to generalize from tables of size two and study stability more generally. Bilò et al. successfully employ this approach to show a number of attractive computational results concerning exchange-stability. The main drawback of this approach, however, is that it assumes that any two people sitting at the same table can communicate, which is not the case for larger tables. Our approach takes the topology of the dining table into account. Some previous works have also considered the topology of the dining table. Perhaps closest to our paper is the model of Bodlaender et al. , in which n individuals are to be assigned to the n vertices of an undirected seating graph. The authors prove a number of computational results regarding both envy-freeness and exchange-stability, among other notions. However, we found some of the table topologies considered to be rather unnatural, especially in hardness proofs (e.g., trees or unions of cliques and independent sets). Bullinger and Suksompong also conduct an algorithmic study of a similar problem, but with a few key differences: (i) individuals are seated in the nodes of a graph, but there may be more seats than people; (ii) for the stability notion, they principally consider jump-stability, where unhappy people can choose to move to a free seat; (iii) individuals now contribute to everyone’s utility, although their contribution decreases with distance. Last but not least, studying stability in the context of Schelling games has recently been a popular area of research. In Schelling games, individuals belong to a fixed number of classes. However, unlike in our model, agents from one class only care about sitting next to others of their own class. This additional assumption often allows for stronger results; e.g., in the authors prove the existence of exchange-stable arrangements on regular and almost regular topological graphs such as cycles and paths, and show that the swap dynamics are guaranteed to converge in polynomial time on such topologies. Overall, it seems that exchange stability has been studied in the frameworks of both hedonic and Schelling games. However, both approaches present some shortcomings: on the one hand, Schelling games inherently consider a topology on which agents evolve, but, being historically motivated by the study of segregation (e.g., ethnic, racial), they usually restrict themselves to very simple preferences. On the other hand, works on hedonic games are accustomed to considering diverse preferences. However, while multiple works have introduced topological considerations, their analysis is usually constrained to graphs that can be interpreted as non-overlapping coalitions, e.g., with multiple fully connected components. One notable exception is the very recent work of Ceylan, Chen and Roy, appearing in IJCAI’23, also building on the model of Bodlaender et al. . In comparison to us, they also prove the NP-hardness of deciding the existence of envy-free arrangements for binary preferences for both topologies, in their case for symmetric preferences, but at the expense of more complex proofs. They moreover show that hardness holds for stability, although the presented proofs requires four non-negative values for cycles and six values including negatives for paths. In our work, we aim to understand stability under more natural preference values, such as approval preferences. Preliminaries ============= We write [n] = {1, …, n}. Given an undirected graph G = (V(G), E(G)), we write NG(v) for the set of neighbors of vertex v ∈ V(G). When clear from context, oftentimes we will simply write V, E and N(v) respectively. The model we describe next is similar to the one in. A group of n agents (guests) A has to be seated at a dining table represented by an undirected graph G = (V, E), where vertices correspond to seats. We will be interested in the cases of G being a cycle or a path. We assume that ∣V∣ = n and that no two agents can be seated in the same place, from which it also follows that all the seats have to be occupied. Agents have numerical preferences over each other, corresponding to how much utility they gain from being seated next to other agents. In particular, each agent i ∈ A has a preference over the other agents expressed as a function pi : A \ {i} → R, where pi(j) denotes the utility gained by agent i when sitting next to j. Note that we do not assume symmetry; i.e., it might be that pi(j) ≠ pj(i). We denote by P = (pi)i ∈ A the collection of agent preferences, or preference profile, of the agents. A number of different interpretations can be associated to P. In particular, we will usually see P as a matrix P = (pij)i, j ∈ A, where pij = pi(j). Note that the diagonal entries are not defined, but, for convenience, we will oftentimes use the convention that pii = 0. Using the matrix notation, we say that the preferences in P are binary when P ∈ {0, 1}n × n and k-valued if there exists Γ ⊆ R, ∣Γ∣ = k, such that P ∈ Γn × n (disregarding diagonal entries, since they are undefined). Note that binary preferences are two-valued, but two-valued preferences are not necessarily binary. Moreover, we will often represent binary preferences as a directed graph, where a directed edge between two agents signifies that the first agent approves of the second. Finally, when pij�	arxiv_0000787
Star Clusters in the Tidal Tails of Interacting Galaxies: Cluster Populations Across a Variety of Tail Environments =================================================================================================================== We have searched for compact stellar structures within 17 tidal tails in 13 different interacting galaxies using F606W- and F814W- band images from the Wide Field Planetary Camera 2 (WFPC2) on the Hubble Space Telescope (HST). The sample of tidal tails includes a diverse population of optical properties, merging galaxy mass ratios, content, and ages. Combining our tail sample with Knierman et al. (2003), we find evidence of star clusters formed in situ with MV < -8.5 and V-I < 2.0 in 10 of 23 tidal tails; we are able to identify cluster candidates to MV = -6.5 in the closest tails. Three tails offer clear examples of ``beads on a string" star formation morphology in V-I color maps. Two tails present both tidal dwarf galaxy (TDG) candidates and cluster candidates. Statistical diagnostics indicate that clusters in tidal tails may be drawn from the same power-law luminosity functions (with logarithmic slopes ≈ -2 – -2.5) found in quiescent spiral galaxies and the interiors of interacting systems. We find that the tail regions with the largest number of observable clusters are relatively young ($\lesssim$ 250 Myr old) and bright ($V \lesssim$ 24 mag arcsec− 2), probably attributed to the strong bursts of star formation in interacting systems soon after periapse. Otherwise, we find no statistical difference between cluster-rich and cluster-poor tails in terms of many observable characteristics, though this analysis suffers from complex, unresolved gas dynamics and projection effects. Introduction ============ Galaxy interactions and mergers are commonly observed phenomena, from intermediate (z ∼ 1) to low redshifts. The complex gravitational potential of interacting galaxies drastically affects their morphologies, producing tidal tails and other disturbed features (; ). Encounters between galaxies are also agents of photometric evolution, often localized a few kiloparsecs from the nucleus or strewn across tidally distorted features (; ; ; ; ). The intensity of these events may range from relatively rare global starbursts (e.g. NGC 6240 and Arp 220), to more frequent local concentrations of star-forming behavior that have little impact on the integrated optical colors of the host galaxies. Tidal tails are physically interesting environments for star formation. They are prevalent for ∼ 500 Myr–1 Gyr from their inception at the galaxies’ initial encounter, before dispersing into the intergalactic medium. They can host alternating sequences of tidally compressive and extensive regions that may shape any observed characteristics of emerging cluster populations, and are bereft of the familiar periodic density waves of spiral arms that can trigger new generations of clusters. Moreover, the low stellar and gaseous masses and densities of tidal tails (e.g. ) compared to their progenitor galaxies ensures their star formation histories are easier to disentangle than the often heavily extincted, star-forming engines of interacting galaxy interiors. Tail internal extinctions are low (AV $\lesssim$ 0.5; e.g. ), simplifying the photometric analysis often made arduous and convoluted by intricate extinction maps and inconvenient galaxy orientations. The resulting low stellar densities present challenges in studying the sparse stellar populations within tidal debris. One way to circumvent this issue is to focus on the clustered stellar component of tidal tails. With the observational resources of HST, star clusters are luminous tracers of the tidal tail star formation history (; ) out to distances ≈ 10–70 Mpc. Thus they allude to the overall star formation capacity of their tidal homes and the interplay of physical properties that allow such clusters to form and survive. Star clusters have been observed in tidal debris. For instance, populations of 40–50 compact clusters are found in the tidal tails of the Tadpole and Mice mergers (; ). also find clusters in the tails of NGC 6872, which appear to have luminosity and mass functions similar to those of clusters within the main bodies of mergers. It is difficult to assess the prevalence of star clusters across a variety of tidal tails, however, because studies have different resolution and completeness limits, and there are strongly age-dependent effects of cluster evolution (i.e. fading). Furthermore, the connection between star formation in clusters and many observed features of galaxy mergers – e.g. age, multiwavelength brightness and colors, and dynamical properties – is not straightforward. For instance,, hereafter K03, do not find a clear relationship between the number of star clusters they detect in tidal tails and various properties of a galaxy merger. They identify dozens of compact clusters brighter than MV = -8.5, with V-I colors ranging from 0.2 to 0.9 in one system (NGC 3256), but do not find such an excess in three other galaxies (NGC 4038, NGC 3921, and NGC 7252). Their sample of interactions is limited to relatively old ( ≈ 400 – 730 Myr), optically bright major mergers, however. It is uncertain what this implies for merging galaxies in the full spectrum of interaction ages, mass ratios, and other observable characteristics. Tidal tails do not appear to consistently have star clusters, unlike in the disks of gas-rich mergers (; ; ). The single tidal tail with a confirmed cluster excess from K03 (NGC 3256W) has 0.11 ± 0.03 cluster candidates kpc− 2 for sources with V-I < 0.7 and MV < -8.5. Additionally, clusters found in tidal debris often have ages similar to young clusters found within the interacting galaxies. In the eastern tail of NGC 3256, K03 infer young ages from their broadband colors, and spectroscopically verify that several of these clusters are $\lesssim$ 200 Myr old. These ages are also younger than the 400 Myr age of the tail they inhabit, strongly suggesting in situ formation. identify clusters in the tidal bridge of the NGC 7714/15 system, whose ages are similar to clusters they find in the interacting galaxies. Unlike these interior clusters, however, they do not find many bridge objects with masses above their age-averaged mass completeness limit of 104.8. The dependence of clustered star formation on properties is also not apparent. find that a threshold of column density N$\_{{\mbox{\scriptsize{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}}}} \sim$ 1020.6 cm− 2 is a necessary but not sufficient condition for generating clusters at the MV < -8.5 level. It is true that projection effects may confuse the translation from column density to volume density and pressure, but this questions how the complex interplay of content, tail densities, pressures, etc. shapes the star-forming environments of tidal tails. K03 also find that some systems devoid of star clusters seem to prefer harboring larger star- or cluster-forming complexes whose luminosities and sizes are consistent with tidal dwarf galaxies (TDGs). TDGs are potentially self-gravitating dwarf galaxy-sized accumulations of tidal debris (; ; ). NGC 7252 and NGC 3921 were presented in this context; they both show 7–8 clusters associated with their tidal dwarfs, and no statistical in-tail cluster excess anywhere else. The canonical example of a TDG is in NGC 4038/9 (; ), which shows evidence for star formation 2 Myr ago, or hundreds of Myr after the main tail formed. Through numerical simulations, and assert that many TDGs may live long enough to become independent from their host tails. Through Hα Fabry-Perot observations and modelling, find that the TDG candidate of NGC 7252 is a real condensation of matter and not a chance projection effect. explore the SEDs and star-forming properties of this TDG (and others) in depth. It is possible that the apparent mutual exclusivity of compact clusters vs. TDGs identified in the K03 tails stems from small sample statistics, but it is also plausible that the outcome is affected by certain dynamical properties. For instance, the tail velocity dispersion and the resulting ambient pressure may conceivably encourage or discourage the formation of TDGs (; ) or other bound structures. K03 posit that physical conditions in the tail may spur different modes of star formation, e.g. either TDG or tail cluster formation, but not both. Modeling work by suggests that variations in dark matter halo structure and other interaction parameters are key in deciding tail lengths and kinematic properties encouraging either star cluster formation along the tail or concentrated in kinematically distinct TDGs at the tail tip. This implies a top-down scenario where several gross tail and merger properties have the potential to help determine the cluster population in tidal debris. Observationally, this hypothesis is limited to a small interaction sample (K03), and requires follow-up with additional interactions. Fundamentally, there may be environmental factors that dictate how effectively stars and star clusters form in interacting pair of galaxies. Evidence now suggests that the majority of star formation occurs in clusters (; ; although see ). While the formation efficiency of bound clusters appears to scale directly with the total star and cluster formation rate of its environment, star clusters (observed and inferred) typically constitute $\lesssim$ 3–10% of the stellar complement of evolved galaxies. Thus, most clusters must fall prey to a variety of disruption mechanisms and contribute to the dominant field star component. It is unknown how processes determining star cluster formation and survival, and the luminous subset of these clusters we observe, relate to the physical conditions within tidal tails and the large-scale observable properties (luminosity, color, mass, TDGs, etc.) they prescribe. Cumulatively, these recent insights into the mechanics of clustered star formation and galaxy interactions merit a re-examination of clusters in tidal debris, specifically in terms of the characterization of these clusters and their relationships to their tail environments. In this paper, we extend the sample of merging galaxies of K03 with HST WFPC2 F606W- and F814W- band observations of twelve additional galaxies. The seventeen tidal tails of these otherwise isolated systems were selected to test environmental extremes for star formation across a broad range of interactions. An overview of the observations and data reduction is provided in §2, along with object detection and photometry. In §3, the properties of star cluster candidates of the tails are discussed, and §4 follows with a discussion of star cluster populations in the context of the observed tidal tail environment. §5 offers conclusions based on current data. Lastly, the appendices highlight observational details of the individual tidal tails and our efforts in optimizing the number of clusters detected at faint magnitudes across the sample. Observations and Reductions =========================== Observations ------------ Twelve interacting galaxies with seventeen tidal tails were selected for observation with the HST WFPC2 in the Cycle 16 program 11134 (P.I. K. Knierman). This sample reflects an expansion of the K03 sample of six tidal tails (from four individual galaxies), as well as the parameter space of interaction characteristics – interaction age, progenitor galaxy mass ratios, and optical properties – they define. The `` Rogues Gallery“ was used for selection guidance. The tidal tails selected in this study are displayed in, as SDSS r-band images with HST WFPC2 footprints overlaid. In systems where there are two obvious, separated tails or debris regions, they are indicated by an additional letter after their names, corresponding to their orientations (E for ``East,” W for ``West," etc.). All other tails are indicated by their primary catalog designations. [htbp] [alltails] WFPC2 images were taken from 2007 September to 2008 August in both F606W and F814W (``V606“ and ``I814” hereafter); a log of these observations is given in. Exposure times, t$\_{\rm{exp}}$, range from 900 to 2100 s in and, representing the total integration time of at least two separate exposures in each filter. This was meant to facilitate the identification and removal of cosmic rays. Images at two different dither positions were obtained for AM 1054-325/ESO 376- G 028, NGC 1614N/S, NGC 2444, NGC 2535, and NGC 6872E/W, with an offset of 2.5 and 5.5 pixels in the WF and PC chips, respectively. Integration times were selected to equal image depths of K03, allowing for direct comparison of analyses and results. In all cases, a gain of 7 e− 1 ADU− 1 was used. [htbp] llrl Merger/Tail & Filter & t$\_{\rm{exp}}$ (s) & Date NGC 1487E & F606W & 1000 & 2008 Aug 9 & F814W & 900 & 2008 Aug 9 NGC 1487W & F606W & 1000 & 2008 Aug 31 & F814W & 900 & 2008 Aug 31 NGC 4747 & F606W & 1000 & 2008 Jan 1 & F814W & 900 & 2008 Jan 1 NGC 520 & F606W & 1000 & 2007 Sept 23 & F814W & 900 & 2007 Sept 23 NGC 2992 & F606W & 1000 & 2007 Dec 28 & F814W & 900 & 2007 Dec 28 NGC 2993 & F606W & 1000 & 2007 Dec 3 & F814W & 900 & 2007 Dec 3 NGC 2782E & F606W & 1000 & 2007 Oct 8 & F814W & 900 & 2007 Oct 8 NGC 2782W & F606W & 1000 & 2007 Nov 8 & F814W & 900 & 2007 Nov 8 MCG-03-13-063 & F606W & 1000 & 2007 Nov 24 & F814W & 900 & 2007 Nov 24 AM 1054-325 & F606W & 1900 & 2008 Feb 24 & F814W & 1900 & 2008 Feb 24 NGC 2444 & F606W & 1900 & 2007 Oct 13 & F814W & 1900 & 2007 Oct 13 NGC 2535 & F606W & 1900 & 2007 Dec 9 & F814W & 1900 & 2007 Dec 9 NGC 1614N/S & F606W & 1900 & 2007 Nov 15 & F814W & 1900 & 2007 Nov 15 NGC 6872E & F606W & 2100 & 2008 Feb 23 & F814W & 2100 & 2008 Feb 23 NGC 6872W & F606W & 1900 & 2008 May 16 & F814W & 1900 & 2008 May 16 Pairs of images were reduced, averaged with the IRAFtask GCOMBINE, and cleaned for hot pixels with COSMICRAYS. Moreover, the low bias problem of WF4 has only been corrected for NGC 1487E/W. According to, this problem is manifested in image streaking whose photometric corrections are most pronounced in faint objects. Given this work’s preferential avoidance of this chip for tail placement, this issue would ordinarily only affect the tails of NGC 4747, NGC 2782, NGC 2535, and NGC 6872. However, our images for these systems employed deep exposures and are clean of image streaking (WF4 mean biases are on par with those of the WF2 and WF3 chips; $\gtrsim$ 300 DN). Thus, this will not have an appreciable effect on the data analysis. In addition, the Planetary Camera chip (PC) was not analyzed because source detection was compromised by a greater readnoise contribution from the larger number of pixels across each source. Neutral hydrogen data for NGC 2782, NGC 2992/3, NGC 2444, and NGC 2535 were taken from the VLA archives. The data were reduced in AIPS, using standard procedures. The neutral hydrogen data for NGC 6872 and NGC 1487 were obtained from the ATCA archivesand calibrated using the standard procedures of the ATNF version of Miriad. An upcoming publication that focuses on the distribution of these tails on kpc scales (Mullan et al. 2011; in preparation) will document the reduction procedures in greater detail. data for NGC 520 and NGC 1614 were obtained from and, respectively. Tail Definitions and Properties ------------------------------- Following K03, contiguous regions bounded by SAO DS9 contours of one count above the average background in V606-band images were defined as ``in tail.“ We applied this criterion to histogram-equalized images smoothed with a Gaussian kernel with a 5–7 pixel (0.5–0.7) FWHM. The tail surface brightness limit ranges between 24.7 – 25.8 mag arcsec− 2 in V. All other regions were designated ``out of tail.” Where galaxy interiors are imaged (e.g. AM 1054-325 E/W, MCG -03-13-063, NGC 2782E, and NGC 6872W), the inner extent of the tail was defined by DS9 contours set to where radial light profiles (measured by IRAF PVEC) showed a change in scale length as the disk connects to the tail. This method isolates ``anti-truncated" tails that may be the result of gravitational interactions. For AM 1054-325, radial light profiles from the inner galactic region were azimuthally-averaged into an ellipse (center at 10:46:58.877, -33:08:23.60; position angle 64.95∘, major axis 5.989, minor axis 5.661). This system’s tail may extend into this interior, but this superposition of tidal debris and host galaxy is unfit for our purposes. NGC 2444 is one final exception – it has no optical tail evident in WFPC2 images. Consequently, its in-tail definition relies on its debris. To be clear, all other tail definitions rely on said optically delineated tails in order to allow direct comparison to K03. Tail regions were converted from SAO DS9 contours to IMAGEJ polygons for optical photometry. Regions were masked to exclude obvious foreground stars and resolved background galaxies. Mean count measurements and errors from IMAGEJ were converted to WFPC2 and Johnson-Cousins magnitudes using the zeropoints and filter transformations of and. Corrections for Galactic extinction were made using AV values from, transformed to the appropriate bandpasses according to the prescription. For this extinction treatment, we used an effective cluster temperature T$\_{\rm{eff}}$ = 15,000 K, representing an aggregation of late B – early A stars, of ages 10 – 300 Myr for the input spectrum. presents a phenomenological summary of the tails. Tails are ordered by increasing luminosity distance (these values are provided in §3–4); we separate our sample from that of K03 in order to isolate any systematic errors that may accrue from the different bandpasses used in the observations. Many of these data from that paper are reproduced here for convenience. [htbp] 1.0@lrrrrrrl\|rrr & Tail & Interaction & Mass & Galaxy & SDSS Tail & Tail & M$\_{\mbox{\tiny{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}}}^{{\rm tot~d}}$ & TDG & Tail Vc & Tail & ⟨M$\_{\mbox{\tiny{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}}}$/A$\_{{\rm tail}}\rangle^{d}$ & Agea & Ratioa& SFRb & Lengthc & Lengthc, d & (109 ) & Candidatese & (mag & V − Ic& (106 & (Myr) & & ( yr− 1) & (kpc) & (kpc) & & & arcsec− 2) & & kpc− 2) NGC 1487E & 500 & 0.25 & 0.12 ± 0.01 & 9 & 19 & 0.614 ± 0.12 & no & 24.02 & 0.84 & 25.5 ± 5.1 NGC 1487W & 500 & 0.25 & 0.12 ± 0.01 & 12 & 17 & 0.541 ± 0.11 & no & 24.57 & 1.10 & 33.2 ± 6.6 NGC 4747 & 320f & 0.05 & 0.27 ± 0.02 & 22 & - & - & no & 24.64 & 0.94 & - NGC 520 & 300 & 0.05 & 6.5 ± 0.5 & 112 & 227 & 1.12 ± 0.22 & yes & 24.21 & 1.04 & 3.81 ± 0.76 NGC 2992 & 100 & 1 & 3.3 ± 0.2 & 23 & 33 & 1.43 ± 0.28 & yes & 23.47 & 1.14 & 6.44 ± 1.3 NGC 2993 & 100 & 1 & 5.0 ± 0.3 & 37 & 55 & 0.727 ± 0.15 & no & 24.78 & 0.73 & 5.77 ± 1.2 NGC 2782E & 200 & 0.25 & 3.7 ± 0.3 & 37 & 17 & 2.41 ± 0.48 & no & 23.81 & 1.02 & 7.08 ± 1.4 NGC 2782W & 200 & 0.25 & 3.7 ± 0.3 & 76 & 58 & 3.14 ± 0.63 & no & 25.49 & 1.05 & 5.17 ± 1.0 MCG -03-13-063 & 100f & 0.25 & 0.43 ± 0.03 & 41 & - & - & no & 23.91 & 1.01 & - ESO 376- G 028 & 85f & 1 & - & 10 & - & - & no & 22.76 & 1.33 & - AM 1054-325 & 85f & 1 & 0.64 ± 0.04 & 16 & - & - & yes & 22.65 & 0.84 & - NGC 2444 & 100 & 0.50 & - & 175 & 233 & 5.20 ± 1.0 & no & - & - & 1.40 ± 0.28 NGC 2535 & 100 & 0.30 & 2.9 ± 0.2 & 25 & 79 & 4.45 ± 0.89 & no & 24.37 & 1.02 & 6.84 ± 1.4 NGC 6872E & 150 & 0.20 & 2.9 ± 0.2 & 38 & 21 & 3.52 ± 0.50 & no & 24.06 & 0.75 & 2.94 ± 0.59 NGC 6872W & 150 & 0.20 & 2.9 ± 0.2 & 21 & 21 & 3.52 ± 0.50 & no & 23.57 & 1.16 & 1.68 ± 0.34 NGC 1614N & 750 & 1 & 35. ± 3. & 21 & 21 & 0.768 ± 0.16 & no & 22.27 & 0.88 & 2.56 ± 0.52 NGC 1614S & 750 & 1 & 35. ± 3. & 21 & 21 & 0.964 ± 0.19 & no & 23.00 & 0.68 & 2.89 ± 0.59 NGC 4038 & 420f & 1 & 5.2 ± 0.4 & 78 & 85 & 0.200 ± 0.040 & yes & 24.54 & 1.05 & 1.09 ± 0.22 NGC 3256E & 400f & 1 & 33. ± 2. & 51 & 77 & 1.40 ± 0.28 & no & 24.04 & 0.76 & 1.51 ± 0.30 NGC 3256W & 400f & 1 & 33. ± 2. & 46 & 57 & 2.20 ± 0.44 & no & 23.75 & 0.70 & 2.96 ± 0.59 NGC 7252E & 730f & 1 & 5.4 ± 0.4 & 40 & 66 & 1.50 ± 0.30 & yes & 25.42 & 0.89 & 0.593 ± 0.12 NGC 7252W & 730f & 1 & 5.4 ± 0.4 & 94 & 156 & 2.20 ± 0.44 & yes & 24.46 & 1.09 & 0.573 ± 0.11 NGC 3921S & 460f & 1 & 1.9 ± 0.1 & 43 & 53 & 4.30 ± 0.86 & yes & 24.72 & 0.94 & 0.872 ± 0.17 aWhen available, ages and mass ratios are adopted from the literature. We adopt errors in ages of ± 50 Myr and 25% for mass ratio. See Appendix A for references. The mass ratio is defined as the mass of the perturbing galaxy divided by that of the main galaxy. bGalaxy SFRs based on IRAS IR fluxes. See K03 and for the details of the calculation. IRAS measurements are unavailable for ESO 376- G 028 and NGC 2444. cTail photometry and length measurements were performed with IMAGEJ and corrected to standard V and I filters following ; see §2.3. No optical tail detection or photometry is possible for NGC 2444, so no values are given. Errors are ≈ 0.1 for V and 0.14 for V-I. d measurements were performed as per §2.2. Dashes indicate where data are unavailable. data and maps for K03 tails were obtained from and references therein. eBased on visual inspection of SDSS and WFPC2 images. fThese tails lacked age estimates in the literature, so these values were obtained by dividing the projected length of the tail by the estimated escape velocity. Similarly-calculated ages of K03 tails are reproduced from that publication. The first eight parameters listed pertain to ``global properties" of the interacting systems. Interaction ages (measured from periapse) and mass ratios are supplied by the literature; see Appendix A for specific references for each system. In cases where reliable ages were not found, they were estimated by dividing the projected length of the tail by an estimate of the escape or rotation velocity. This was demonstrated to provide acceptable accuracy ( ± 50 Myr) in K03. Mass ratios are defined as the mass of the perturbing galaxy divided by the mass of the primary system. Table 2 also lists host galaxy SFRs based on IRAS IR fluxes. K03 and have details of the calculation. Tail lengths were measured with freehand lines in IMAGEJ, tracing the tail curvature evident in SDSS r-band images and our maps. Note that in many cases the debris evidently extends beyond the optical tails (NGC 2782 is an interesting counterexample). Furthermore, we calculated the total mass of entire tidal tails,. In every channel of the data cubes, we measured the total flux (Jy km s− 1) and summed the flux in each channel to get the total flux F$\_{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}^{\rm{tot}}$. The total mass is then (e.g. ): $${\hbox{M$\_{\mbox{\scriptsize{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}}}^{{\rm tot}}$}}~[{\hbox{$\hbox{M}\_{\odot}$}}] = 2.36 \times 10^5 ~(D[\mathrm{Mpc}])^2~\rm{F}\_{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}^{\rm{tot}}[\mathrm{Jy~km~s}^{-1}]~,$$ where D is the distance to the tail in Mpc. We additionally indicate in which of these tails have visually identified TDG candidates. The last three columns of list WFPC2 FOV-scale properties of the tidal tails. We include the WFPC2-imaged optical tail V-band surface brightness and V − I color; for the K03 sample, tail photometry was conducted identically to our sample and corrected from the F555W/F814W bands to Johson-Cousins magnitudes (; ). Lastly, lists the average observed surface density within the optical tail areas (A$\_{\rm{tail}}$) of the WFPC2 FOVs. is expressed as: $$\begin{aligned} \nonumber {\hbox{$\langle$M$\_{\mbox{\scriptsize{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}}}$/A$\_{{\rm tail}}\rangle$}}~[{\hbox{$\hbox{M}\_{\odot}$}}~\mathrm{kpc}^{-2}] = 2.36 \times 10^5 ~(D[\mathrm{Mpc}])^2 \times \\ \rm{F}\_{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}^{\rm{tail}}[\mathrm{Jy~km~s}^{-1}]/\rm{A}\_{\rm{tail}}[\mathrm{kpc}^{2}]~.~\end{aligned}$$ Here, F$\_{{\hbox{{\rm H}\kern 0.1em{\sc i}}}}^{\rm{tail}}$ is the flux for the optical tail region within the WFPC2 FOV, and A$\_{\rm{tail}}$ is the area of the optical tail region measured from the images. This is similar to the area measured from the HST images, modulo differences in pixel sizes. The uncertainty in this quantity represents the uncertainties in measuring the flux of HI and assumes a 20% error in measuring the distance. Thus, we have a ``global" gas-richness parameter and a WFPC2-scale density metric directly comparable to the optical counterparts and the populations of star clusters they may contain. Given the paucity of numerical modeling for many of these interactions, inclinations remain highly uncertain and therefore no attempts at adjusting projection-dependent quantities (e.g. , tail lengths, and optical properties) have been made. Object Detection ---------------- Sources were detected using the DAOFIND algorithm in the IRAF package DAOPHOT. The criteria for detection were chosen to match K03 and provide a source list for photometry and analysis. These include the following: * At least 2 counts per object in and, along with a count threshold 2σ above the local background. A sky annulus with 5–8 pixel radii was used for both measuring noise around each object and determining its sky background. * A signal-to-noise ratio S/N > 3. * An error in V606-band magnitude < 0.25. * Detections in at least one dither position, accounting for the pixel offsets. For every in- and out-of-tail region, obvious background galaxies or foreground stars were removed manually. Photometry and Source Selection ------------------------------- Aperture photometry was performed on the source lists and cleaned images using the PHOT task in the package APPHOT. In light of potential source crowding, we used apertures with radii of 2, 5, and 8 for object, inner background, and outer background annuli, respectively. Most importantly, these apertures were selected for consistency with K03. Such a small object aperture is not problematic. Even the closest systems in our sample would not likely have individual clusters marginally resolved by WFPC2; NGC 1487E/W and NGC 4747 images have a pixel scale of ≈ 5–10 pc, compared to probable reff values ∼ 4 pc ( and references therein) to ∼ 10 pc. Thus, before diffraction, an object aperture with a 2-pixel radius will encompass > 1–2 effective radii of a cluster, or 50–100% of its light, and is therefore acceptable. Corrections for Galactic extinction were made with the procedure documented in §2.2. Nonoptimal charge transfer efficiency was corrected following, with the formulation of. Uncertainties for all values as well as photometric errors were propagated into the final values of all magnitudes and colors. Zeropoints were obtained from. Moreover, the concentration index ΔV – the difference in V606 magnitude between apertures of 0.5 pixel and 3 pixel radii – was calculated, with annuli of 6–9 pixel radii for background subtraction. Similar work can be found in,, and K03. The latter paper employs cutoff values for this structural parameter of 2.4 – 3, depending on distance, to eliminate extended objects from their source lists and concentrate on point sources. Because clusters would be largely unresolved and there would be no physical link between ΔV and the size of single clusters, no restrictions on ΔV have been made here. Furthermore, we do not wish to exclude marginally resolved cluster complexes or young clusters embedded in nebular emission (Hα is present in the bandpass); these would appear more diffuse in ΔV than individual compact clusters. We retain ΔV data in Appendix A for reference. Aperture corrections of 0.33 in V606 and 0.36 in I814, based on the brightest 5–10 point sources in each chip on all images, were applied (this includes the 0.1 mag correction between 0.5$\arcsec$ and infinite apertures as per ). We calculate uncertainties of 0.008 and 0.01 for these values, respectively. Errors in using stellar aperture corrections vs. cluster candidate aperture corrections for the most cluster-deficient tails – a few hundredths of a magnitude – are negligible for this investigation. Objects detected and measured with the above methodology were defined as star cluster candidates (SCCs) if they met the following color-magnitude criteria: * V-I < 2.0. When translated into WFPC2 bandpasses, this color limit ≈ 1.43. This criterion allows for populations of old, metal poor globular clusters (V-I ∼ 1) akin to those observed in the Galaxy, as well as metal-rich clusters with V-I < 1.5 (;, ; ). This also includes globular clusters with V-I ∼ 1.2–1.3 and solar metallicity. This color limit also allows clusters with redder colors from the stochastic populating of the stellar IMF at these low cluster masses, additional extinction (even if overall A$\_V \lesssim$ 0.5 mag), and uncertainties in metallicities. This V-I limit is also needed to account for problems in modeling binarity in stellar populations, mass segregation, and other dynamical effects than can influence a cluster’s integrated magnitudes and colors (see for a review). Furthermore, suggests that real clusters, subject to a variety of age- and environment-dependent disruption mechanisms, will fade and redden more rapidly with time than these traditional photometric models indicate, necessitating a more lenient color cutoff. * MV < -8.5 ( ≈ -8.6 in the WFPC2 system). To compare the SCC quantity across all tidal debris regions, a constant magnitude limit relatively unaffected by completeness is required. Most studies typically use such a bright magnitude cutoff (MV -8 – -9: K03;, ). This limit protects against a level of contamination from single main sequence and post main sequence stars prohibitive to statistical in-tail SCC identification ; Appendix B has more on this subject. We find median errors in and of 0.08 and 0.15, respectively, for all sources. This includes contributions from photometric errors, CTE, and aperture correction uncertainties. Finally, despite identical absolute magnitude limits, K03 used a different V-I < 0.7 threshold to select SCCs. This color distinction was arbitrary, based on a comparison of the in-tail SCC colors with photometry reported for clusters within their host galaxies. The followup work of employ an equally valid V-I < 2.0 criterion that also avoids selecting individual, luminous stars. In order to compare the results of those two benchmark works and combine with our own analyses, we chose the latter color and opt to recalculate many quantities published in K03 from their original source lists and their tails. We present them alongside our own in §3. Completeness ------------ Completeness curves are provided in for optically-defined in-tail and out-of-tail regions, in both and. The completeness fraction for each tail was determined using 10,000 artificial stars, sampled 100 at a time with the ADDSTAR procedure in the package DAOPHOT. K03 and find similar completeness fractions for their samples derived from WFPC2 data; ∼ 50% is reached at V ≈ 25.7 ( ≈ 25.5). This corresponds roughly to a visual estimate for completeness from the color-magnitude diagrams of this project’s sample (§3 and Figure Set 3). Completeness limits for out-of-tail regions are typically lower by 0.1–0.5 mag, i.e. by typical AV values for tidal tails. Beyond the tidal debris, detections are 50% complete at V ≈ 25.7–26.0, or ≈ 25.6–25.9. Translated to absolute magnitudes, these tail regions are 50% complete to MV = -4.6 for the closest debris (NGC 1487E/W), to -8.6 for the most distant tails (NGC 1614N/S). The optically bright merger AM 1054-325, has the brightest absolute 50% completeness limit of MV =-8.7. In §3.2, we discuss the ramifications of using a magnitude limit encumbered by the incompleteness of these tails. [htbp] Summary of the Tail Sample and Results ====================================== In Appendix A, we summarize the results from this investigation for each tail of the sample and compare them to studies in the literature. As an example of this work, introduces contours on an SDSS r-band image (panel a), and a -band mosaic of NGC 1487E, with in-tail and out-of-tail sources indicated (b). A background-subtracted map of the tail region is also provided (c). This map was smoothed with a 12 pixel (1.2$\arcsec$) Gaussian kernel, corresponding to a linear size of ≈ 70–380 pc, depending on distance. This was done to circumvent the typically noisy, low signal registered by the in-tail pixels. Color-magnitude diagrams (CMDs) for in-tail and out-of-tail sources are displayed in this Figure (d – e), as well as plots of color vs. ΔV (f). Population synthesis models of are overlaid in the CMDs. Evolutionary tracks are shown for a simple stellar population (SSP) given a zero-age burst of star formation, a initial mass function, solar metallicity and total stellar mass of 104, 105, and 106. Numbers indicate the age of the burst (log yr), and one magnitude of extinction in is shown as the labeled AV606 vector. Estimating ages and masses of putative clusters will inevitably be complicated by aforementioned stochastic effects, metallicity-age-extinction degeneracies, and model inadequacies. We do not expect extinctions exceeding 1 mag, so internal extinction is partially mitigated compared to galaxy interiors. Thus, with all these uncertainties in mind, ages and masses of detected clusters must be treated qualitatively. ``Beads on a String" Morphology ------------------------------- When smoothed with a 12-pixel ( ≈ 1.2) Gaussian kernel, three of the color maps in the show evidence for ``beads on a string" star formation, particularly for the bluest (and possibly youngest) objects we detect in the tails. NGC 1487E, MCG -03-13-063, and AM 1054-325 all exhibit this morphology when their spatial resolution is sufficiently downgraded, and our compact objects blend to clumps of ∼ 105 – 106. These beads are all separated by ∼ 3 kpc, adjusting for possible tail inclinations. We also compared these separations with those measured for a set of GALEX-imaged tidal tails presented in. These approaches are heuristically compatible, as both employ images with > 1$\arcsec$ resolution and the identification of young blue clumps of star formation within them. In the GALEX sample, we remeasure clump separations of ∼ 1–7 kpc, with reasonable agreement between the GALEX-imaged beads of NGC 2535 (3–4 kpc; ) and its optical counterpart (our study). This type of star formation is typically attributed to gravitational instabilities within the host environment (; ), scaling with the local Jeans mass. Within spiral arms of galaxies, these clumps are separated by the Jeans wavelength (the minimum radius necessary for collapse), which for that environment is ∼ 1–4 kpc. This is also similar to the clump separations observed in interacting galaxies at intermediate redshifts. The masses estimated here are at the low end of the observable mass range for the clumps of that study, which range from 106 to a few 108. Star-forming clumps with separations ∼ 3 kpc, while consistent with gravitational collapse scales of spiral galaxies and interacting pairs at intermediate redshifts, are not pronounced in all tails’ color maps. It is possible a combination of metallicity, age/star formation history, orientation, and smoothing effects contribute to this observed morphology in some tails but not in others. There may be more physical motivations as well. This formation morphology is pronounced in small subsets of this sample and the tidal debris of interactions surveyed by, implying that not all tails share the physical conditions necessary for promoting these structures. Certain ISM velocity dispersions and/or mass densities within the host environment are required to produce these beads, but determining these local quantities in the cluster-forming regions is beyond the scope of this work. A Quantitative Comparison of SCCs --------------------------------- In this set of interacting pairs, several have visually striking abundances of in-tail sources relative to regions beyond the tail. To quantitatively compare these systems, the relative areas of in-tail and out-of-tail regions on the three chips must be considered, along with foreground and background object contamination by non-cluster objects. Distance-related selection effects must also be acknowledged. presents the tails, their host galaxy luminosity distances in Mpc (from the NASA Extragalactic Database), their completeness limits (§2.5) and their pixel sizes (in pc). We also include the number of SCCs detected within (N$\_{\rm{in}}^{\rm{SCC}}$) and outside (N$\_{\rm{out}}^{\rm{SCC}}$) of the debris, and the area of the in-tail and out-of-tail regions in kpc2, A$\_{\rm{in}}$, and A$\_{\rm{out}}$. The in-tail excesses are then quantified as the difference between N$\_{\rm{in}}^{\rm{SCC}}$/A$\_{\rm{in}}$ and N$\_{\rm{out}}^{\rm{SCC}}$/A$\_{\rm{out}}$, with uncertainties from Poisson statistics of N$\_{\rm{in}}^{\rm{SCC}}$ and N$\_{\rm{out}}^{\rm{SCC}}$. Errors of 2.5σ are recorded. If there are local clusters outside the optically-defined tail, this method for background subtraction is systematically too large and gives a lower limit to actual cluster surface density. Thus, SCC excesses at the 2.5σ level are statistically significant. [htbp] 1.0@lrrrrrrrrrr Tail & Distance & MV, 50% & Pixel Size & N$\_{\rm{in}}^{\rm{SCC}}$ & N$\_{\rm{out}}^{\rm{SCC}}$ & A$\_{\rm{in}}$ & A$\_{\rm{out}}$ & N$\_{\rm{in}}^{\rm{SCC}}$/A$\_{\rm{in}}$ & N$\_{\rm{out}}^{\rm{SCC}}$/A$\_{\rm{out}}$ & $\Sigma\_{{\rm SCC}}^a$ & (Mpc) & & (pc) & & & (kpc2) & (kpc2) & (kpc− 2) & (kpc− 2) &(kpc− 2) NGC 1487E & 10.8 & -4.6 & 5.22 & 0 & 1 & 16.49 & 40.93 & 0 & 0.024 & -0.024 ± 0.061 NGC 1487W & 10.8 & -4.5 & 5.22 & 2 & 0 & 16.39 & 41.03 & 0.122 & 0 & 0.122 ± 0.216 NGC 4747 & 20.2 & -6 & 9.75 & 7 & 5 & 43.66 & 68.01 & 0.160 & 0.074 & 0.087 ± 0.172 NGC 520 & 27.2 & -6.4 & 13.13 & 0 & 4 & 26.40 & 375.97 & 0 & 0.011 & -0.011 ± 0.013 NGC 2992 & 36.6 & -7.2 & 17.67 & 37 & 15 & 214.38 & 198.58 & 0.173 & 0.076 & 0.097 ± 0.086 NGC 2993 & 36.6 & -7.3 & 17.67 & 15 & 7 & 142.10 & 270.86 & 0.106 & 0.026 & 0.080 ± 0.072 NGC 2782E & 38.1 & -7.5 & 18.40 & 87 & 4 & 336.79 & 165.83 & 0.258 & 0.024 & 0.234 ± 0.076 NGC 2782W & 38.1 & -7.3 & 18.40 & 10 & 7 & 235.91 & 266.71 & 0.042 & 0.026 & 0.016 ± 0.042 MCG -03-13-063 & 46.2 & -7.8 & 22.31 & 88 & 26 & 230.18 & 632.29 & 0.382 & 0.041 & 0.341 ± 0.104 ESO 376- G 028 & 52.9 & -8.1 & 25.54 & 4 & 60 & 80.97 & 1036.73 & 0.049 & 0.058 & -0.008 ± 0.065 AM 1054-325 & 52.9 & -8.7 & 25.54 & 180 & 60 & 136.28 & 981.42 & 1.321 & 0.061 & 1.260 ± 0.247 NGC 2444 & 58.2 & -8.2 & 28.10 & 27 & 15 & 716.94 & 543.56 & 0.038 & 0.028 & 0.010 ± 0.025 NGC 2535 & 59.8 & -8.4 & 28.88 & 44 & 49 & 295.03 & 995.44 & 0.149 & 0.049 & 0.100 ± 0.059 NGC 6872E & 62.6 & -8.3 & 30.23 & 163 & 63 & 588.99 & 999.53 & 0.277 & 0.063 & 0.214 ± 0.058 NGC 6872W & 62.6 & -8.5 & 30.23 & 195 & 36 & 788.90 & 799.62 & 0.247 & 0.045 & 0.202 ± 0.048 NGC 1614N & 65.6 & -8.6 & 31.68 & 33 & 28 & 362.63 & 1386.52 & 0.091 & 0.020 & 0.071 ± 0.041 NGC 1614S & 65.6 & -8.6 & 31.68 & 43 & 28 & 288.50 & 1460.65 & 0.149 & 0.019 & 0.130 ± 0.058 NGC 4038 & 13.8 & -5.1 & 6.66 & 3 & 3 & 76.92 & 76.92 & 0.039 & 0.039 & 0.000 ± 0.080 NGC 3256E & 42.8 & -7.8 & 20.67 & 49 & 55 & 290.32 & 333.33 & 0.169 & 0.165 & 0.004 ± 0.082 NGC 3256W & 42.8 & -7.8 & 20.67 & 62 & 58 & 205.88 & 428.57 & 0.301 & 0.135 & 0.166 ± 0.105 NGC 7252E & 62.2 & -8.1 & 30.03 & 23 & 32 & 700 & 1181.82 & 0.033 & 0.027 & 0.006 ± 0.021 NGC 7252W & 62.2 & -8.1 & 30.03 & 31 & 29 & 684.21 & 1250 & 0.045 & 0.023 & 0.022 ± 0.023 NGC 3921S & 84.5 & -8.8 & 40.80 & 14 & 30 & 692.31 & 2333.33 & 0.020 & 0.013 & 0.007 ± 0.015 aErrors of 2.5σ are quoted. Completeness may also have an effect on AM 1054-435 and NGC 1614N/S. Since the 50% completeness limit of these systems is approximately the MV selection criterion (MV, 50% = -8.6 – -8.7), these tails will have a systematic underrepresentation of in-tail and out-of-tail SCCs. We estimate, using a power-law LF with a logarithmic slope α = 2 – 2.5 for the in-tail sources (see §3.3), and extrapolating from the corresponding distribution for the out-of-tail sources, that the values quoted for these two tails may be underestimated by ≈ 25%. No corrections to these excesses have been made. outlines these results, including the interacting pairs of K03 to the right of the vertical dotted line, and their identically calculated excesses. Note that values for K03 tails are different than those originally published, because of our V-I < 2.0 criterion and the different luminosity distances we use here (for consistency, H0 = 73 km s− 1 Mpc− 1; we use the distance derived by for NGC 4038). The horizontal line marks where ; any points within 2.5σ of this level imply the host tail is most likely devoid of star clusters within the limit of our broadband observations (for clarity, only 1σ errors are shown). Overplotted boxes indicate tails with statistically insignificant SCC excesses, including tails whose SCCs reside primarily in TDG candidates (NGC 2992 and NGC 7252E/W; see §3.2.2). Black circles show the results of the analysis performed with the and V-I < 2.0 criteria. Blue triangles and red asterisks in show the results of similar calculations, but with fainter MV limits by one and two magnitudes (MV < -7.5 – -6.5), respectively. As the MV threshold increases, NGC 1487 through NGC 520 enjoy increases in. More distant systems experience similar changes in SCC excess as the magnitude limit is increased, but only until their completeness limits are reached. [htbp] To further explore the differences in source populations between in- and out-of-tail regions, we performed Kolmogorov-Smirnov (K–S) tests on the distributions of and for all in-tail vs. out-of-tail sources (N$\_{\rm{in}}$, and N$\_{\rm{out}}$); these are presented in, along with each tail’s for comparison. The results of K03 are also shown. PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$}}}}}$ and PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$-$I\_{814}$}}}}}$ values list the probabilities that the in-tail and out-of-tail objects are drawn from the same distribution of and, respectively. This provides a measure of the independence of in-tail sources from the out-of-tail sources. Note that in-tail and out-of-tail populations are likely to be independent in terms of color and magnitude distributions if their K–S probabilities are < 0.02, i.e. that the populations are distinct at the ≈ 2.5σ level. In three cases (AM 1054-325 and NGC 1614N/S), tails with statistically distinguishable populations in color and magnitude also have statistically significant SCC populations. This largely confirms that these tails have clusters. Tails with = 0 all have PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$}}}}}$ and PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$-$I\_{814}$}}}}}$ values > 0.02, reaffirming that they are devoid of star clusters at the MV < -8.5 level. Other tails with significant SCC excess but unfavorable K–S statistics still likely contain clusters, however. MCG -03-13-063, NGC 2535 and NGC 3256W have distinct source population differences in and not in, as well as SCC excesses. The similarities in the magnitude distributions may be due to a slight contamination of foreground sources for tails at low galactic latitudes (∣b∣ < 30∘. in these cases). NGC 2993 and NGC 6872E/W conversely show significant source distribution differences in magnitude but not in color. For NGC 2993, its tail extends beyond its optical debris, so contamination of the out-of-tail sources by faint, similarly-aged clusters within the tail but beyond the optical debris may account for its high PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$-$I\_{814}$}}}}}$. Sources within NGC 6872E/W debris may have variable extinction and metallicities, and/or may be contaminated by red ( $\lesssim$ 1) foreground sources (b ≈ -32.5∘). Lastly, NGC 2782E has a nonzero SCC excess, but shows insignificant source differences between its in-tail and out-of-tail regions in both and. It is probable that the out-of-tail regions of NGC 2782E are undersampled – with only 8 sources detected, it is difficult to determine their independence in color and magnitude from the ample in-tail objects. In summary, by combining K03 with this study, cluster candidate excesses at MV < -8.5 exist at the 2.5σ level in 10 out of 23 tails (NGC 2993, NGC 2782E, MCG -03-13-063, AM 1054-325, NGC 2535, NGC 6872E/W, and NGC 1614N/S, and NGC 3256W). Our detection of these SCCs in the tidal debris regions are all complete to at least ≈ 50% to that magnitude. Many of the tails with excesses also have and distributions that differ between tail and out of tail sources, indicating the in tail sources are real and associated with the tail. The tails with statistically nonzero cluster candidate excesses and source distributions indistinguishable from out-of-tail objects in color or magnitude can be explained by foreground contamination or poor statistics from undersampled out-of-tail areas. These tails are still likely to have clusters. [htbp] 1.0@lrrrr\|r Tail & N$\_{\rm{in}}$ & N$\_{\rm{out}}$ & PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$}}}}}$ & PKS$\_{\mbox{\scriptsize{{\hbox{$V\_{606}$-$I\_{814}$}}}}}$ & a (kpc− 2) NGC 1487E & 122 & 45 & 0.341 & 0.0263 & -0.024 ± 0.061 NGC 1487W & 121 & 67 & 0.593 & 0.396 & 0.122 ± 0.216 NGC 4747 & 117 & 45 & 0.878 & 0.646 & 0.087 ± 0.172 NGC 520 & 21 & 66 & 1.37 × 10− 3 & 0.757 & -0.011 ± 0.013 NGC 2992 & 104 & 45 & 0.0187 & 0.660 & 0.097 ± 0.086 NGC 2993 & 37 & 43 & 0.0172 & 0.652 & 0.080 ± 0.072 NGC 2782E & 187 & 8 & 0.875 & 0.963 & 0.234 ± 0.076 NGC 2782W & 49 & 31 & 0.0588 & 0.249 & 0.016 ± 0.042 MCG -03-13-063 & 141 & 48 & 0.324 & 0.0617 & 0.341 ± 0.104 ESO 376- G 028 & 4 & 82 & 0.793 & 0.0735 & -0.008 ± 0.065 AM 1054-325 & 184 & 82 & 2.36 × 10− 10 & 9.22 × 10− 25 & 1.260 ± 0.247 NGC 2444 & 34 & 16 & 0.545 & 0.241 & 0.010 ± 0.025 NGC 2535 & 45 & 59 & 0.0904 & 1.16 × 10− 4 & 0.100 ± 0.059 NGC 6872E & 180 & 93 & 7.26 × 10− 5 & 0.209 & 0.214 ± 0.058 NGC 6872W & 219 & 38 & 5.17 × 10− 3 & 0.119 & 0.202 ± 0.048 NGC 1614N & 33 & 28 & 1.94 × 10− 3 & 3.33 × 10− 4 & 0.071 ± 0.041 NGC 1614S & 43 & 28 & 2.61 × 10− 4 & 1.27 × 10− 6 & 0.130 ± 0.058 NGC 4038 & 44 & 26 & 0.270 & 0.890 & 0 ± 0.080 NGC 3256E & 70 & 60 & 0.200 & 1.00 × 10− 3 & 0.004 ± 0.082 NGC 3256W & 106 & 84 & 0.0300 & 2.20 × 10− 6 & 0.166 ± 0.105 NGC 7252E & 25 & 40 & 0.950 & 0.910 & 0.006 ± 0.021 NGC 7252W & 33 & 34 & 0.0500 & 0.900 & 0.022 ± 0.023 NGC 3921S & 14 & 30 & 0.980 & 0.0500 & 0.007 ± 0.015 aErrors of 2.5σ are quoted. ### and Global Tail Characteristics At the MV < -8.5 limit, several tails in our sample have comparable/larger SCC excesses to that of NGC 3256W in K03. NGC 6872E/W for example has an average excess of ≈ 0.2 kpc− 2, compared to ≈ 0.16 kpc− 2 for NGC 3256W. According to, both of these systems have long tidal tails that are rich in ( $\gtrsim$ 2 × 109 ), but have drastically different merging mass ratios. The tidal debris of MCG -03-13-063 and AM 1054-325 have drastically higher SCC densities ( ≈ 0.34 and 1.3 kpc− 2, respectively), and are both younger mergers by ≈ 300 Myr. They also have different merging mass ratios, and their total masses are unknown. NGC 1614S has a close correspondence of cluster excess ( ≈ 0.13 kpc− 2) to NGC 3256W and it is a product of a major merger, but it is a low, previously merged system with debris > 200 Myr older than NGC 3256. It appears that the surface density of clusters depends on the complex interplay of the global tail parameters of, if they are are indeed important. That is not to say there are no trends in the combined samples at all, but disentangling statistical influences of single parameters is difficult for this finite sample size. For instance, there may be an overall preference for cluster formation at young interaction ages, as the highest values belong to tails younger than 250 Myr. However, K–S tests demonstrate that tails with significant SCC excesses and tails without them have a probability of 0.715 of sharing a common age distribution. Age alone apparently does not direct cluster formation. Combined with other parameters like WFPC2 tail surface brightness and color, age may still have a role; this is addressed in §4.1. Furthermore, minor mergers appear as well suited for star cluster formation as major mergers; approximately 60% of either type of merger has a nonzero SCC density for MV < -8.5. K–S tests reveal a probability of 0.608 that tails with both statistically zero and nonzero SCC excesses are drawn from the same distribution of interacting mass ratios. By its qualitative nature, the distribution of mass ratios is very coarse, so such a high value is not unexpected. In terms of host galaxy SFR, no outstanding distinction is seen for cluster-abundant tails. From the discussion of individual systems in Appendix A, it is unlikely that these host galaxy SFRs correlate to in a systematic way, owing to different host galaxy orientations and/or the localized nature of star formation. K–S tests performed on the combined SCC-harboring and SCC-deficient samples also indicate no difference between these two tail populations (we find a probability of 0.787). This is in opposition to K03, who speculated on a connection between the SFR of NGC 3256 (33.4 yr− 1, the highest of their selection of tails) and the cluster candidate excess observed for its western tail. NGC 1614 has a similarly high IR-derived SFR ( ≈ 35 yr− 1), but is superceded by other systems in the sample with larger SCC excesses in their tails and lower IRAS-derived SFRs, e.g. AM 1054-325 and NGC 6872E/W. Projection effects and ULIRGS in our sample are probable culprits for this behavior. Likewise, most tail lengths cannot be accurately de-projected from their measurements in, so comparing to them is prohibitively qualitative. For, we calculate K–S probabilities of 0.886 for the SCC-differentiated tail samples. As in K03, overall gas richness is not an immediately crucial characteristic for forming massive star clusters. In short, the solution set of global properties from the present dataset that are linked to an SCC density at MV < -8.5 are not unique. There appears to be no single global property that universally regulates a tail’s capacity for forming star clusters at the MV < -8.5 level. We address the potential importance of WFPC2 FOV-scale properties in §4. ### and TDG candidates K03 suggested that the formation of tidal dwarf galaxies and star clusters in tidal tails are mutually exclusive. In our sample, the optical condensation present in the tidal tail of NGC 2992 is likely a true TDG, so further care must be taken to disentangle the tail sources from those attributed to this object. High resolution WFPC2 images of NGC 2992 in reveal more structure within what may be a smaller TDG than ground-based observations would indicate (contrast the SDSS image with the -band mosaic of that tail). Ignoring the SCCs attributable to the TDG candidate, we count NinSCC = 17. Replacing the previous value (NinSCC = 25) and subtracting an estimate of the TDG area ( ∼ 10–20 kpc2) from Ain in Table 4, we find ≈ 0.01 ± 0.07 kpc− 2. This measurement is obviously not significant at our required 2.5σ level. At MV < -7.5 and MV < -6.5, is still statistically insignificant; we calculate values < 0.1 ± 0.3 for both cases. The high inclination of NGC 2992 may also artificially inflate by providing a smaller projected tail area, so this must be interpreted even more conservatively. Furthermore, while our NGC 520 pointing focuses on the TDG candidate at the tip of the galaxy’s tail (see Appendix A.4 for an alternative explanation), ACS images for NGC 520 at similar depths show evidence for clusters at the base of the tail (Chandar et al. 2011, in preparation). Although our SCCs in Figure 3 and in later Figures are flagged as belonging to the TDG candidate, star clusters likely exist in this system’s tidal tail. Without deep imaging of the tail’s optical extent, however, it is unknown whether they exist beyond the tail base. For now, this system may be seen as a tentative counterexample. NGC 2782 makes a stronger case against TDG-tail cluster candidate mutual exclusivity. There is kinematic evidence for a tidal dwarf in NGC 2782E ; this is visible as an optical condensation in the upper left corner in chip WF3 in ). This kpc-sized agglomeration has blue colors ( ∼ 0.4) and emission line properties ([OIII], Hα+ [NII], and [SII]) consistent with a ∼ 108, metal-poor region. However, it contains few SCCs and the rest of the tail retains a substantial cluster candidate excess beyond the 2.5σ level (0.196 ± 0.07 kpc− 2). Moreover, AM 1054-325 has both a TDG candidate and the highest of the tails in this sample. Unfortunately, there are no high-resolution kinematic data present to evaluate rotations or masses for its TDG candidate and confirm or deny TDG status, so this inference is not completely proven. No TDG-masked calculations of have been performed for this system, but even an in-tail SCC decrease of a factor of ∼ 5 would leave a statistically significant ( ≈ 0.2 ± 0.11 kpc− 2). Though not absolute, there do seem to be counterexamples of the exclusivity of SCC and TDG formation in tidal tails. TDG formation appears favorable in tails with high velocity dispersions, and high Jeans masses. These objects may preferentially arise from interactions with extended dark matter halos, which allow efficient gas removal from host disks and may prevent strong episodes of cluster formation along the tails. Young tails with TDG candidates and SCCs may simply be still gas-rich enough along the debris for measurable SCC formation to occur. This cannot be confirmed without additional examples of these types of tails, with concordant or CO observations. A Statistical Diagnostic of Tail Cluster Populations ---------------------------------------------------- The variations in SCC quantities we see in the tail sample and for different magnitude limits must depend of the logistics of star cluster formation in tidal tails, as well as the details of their evolution. Each generation of star clusters is imprinted with the physical mechanisms that dictate how they form. What we observe forms the basis for constructing the clusters’ luminosity functions (LFs), which encode their underlying, heavily age-dependent mass functions (MFs). These MFs are shaped over time from cluster initial mass functions (CIMFs) by conditions in the tidal tail environment and their own internal physics. One method to tackle the LF is to plot the the brightest cluster MV vs. number of clusters observed brighter than MV = -9 in each tail. Linear relationships between these two varaibles arise if the luminosities of the clusters are drawn from a power-law distribution LF with a slope ≈ -2.0 (; ; ). This size-of-sample effect connects a heightened local star formation rate (introducing large numbers of clusters) with an increasing probability of producing high-mass, luminous clusters (brightest MV) In effect, the actual LF may not be a uniform power law, but appears as one for particular luminosity ranges. For the luminosities spanned by our SCC populations, suggests that a more accurate LF slope is ≈ -2.5, while offers ≈ -2.2 for populations of more strongly disrupting clusters. These values are strictly for spiral galaxies. We present our results in for two test cases – one where we count all in-tail SCCs in defining the number of clusters (Figure 5a), and a second where we statistically subtract out-of-tail SCCs to estimate the cluster number (Figure 5b). The latter method subtracts the number of out-of-tail SCCs per unit area (N$\_{\rm{out}}^{\rm{SCC}}$/A$\_{\rm{out}}$) from the number density within the tail (N$\_{\rm{in}}^{\rm{SCC}}$/A$\_{\rm{in}}$) and multiplies that by the area of the tail A$\_{\rm{in}}$. The ``true" number of clusters should fall between these bracketing situations. In both cases, NGC 3256E/W and NGC 7252E/W have been removed, as these tails have unreliable, anomalously luminous brightest SCCs (M$\_V \lesssim$ -14) that are not likely clusters. These tails exhibit a degree of contamination in their CMDs (K03) that prevents a safe determination of this quantity. The slope we obtain for the resulting trend is -2.33 ± 0.36 for the first case, and -1.81 ± 0.44 for the second. These are in rough agreement within errors with the value -2.3 ± 0.2 derived from clusters in a variety of starbursts and quiescent spirals by. Our uncertainties are higher given the nature of this study, but ultimately this indicates that the underlying luminosity functions of clusters in our tidal tail sample are similar to those elsewhere in the local universe. The apparent similarity in LFs between our tidal tail clusters and elsewhere in the local universe is insightful. For LF slopes ≈ -2 – -2.5, this implies that the underlying mechanisms at work in tidal tails are extremely similar in effect, if not in form, to those shaping the cluster populations observed in starburst and quiescent galaxies. Of course, our picture is incomplete, as star cluster LFs in tidal debris may be complicated by complex formation and dissolution histories from multiple sequences of extensive and compressive tides, and/or their passage through the ISM (e.g. ). There will also be stochastic effects at detectable masses and contributions from clusters born at different ages, as well as systematic effects and unseen variables in this study, all of which may contribute to the trends or scatter witnessed in. Discussion ========== SCC Excess, Interaction Age, and Optical Tail Characteristics ------------------------------------------------------------- Though the dependence of on the properties in is generally not straightforward, the most SCC-rich tails are all younger than 250 Myr. Since the differences cannot be statistically attributed to age alone, there may be other factors at work. To elucidate this, we plot a tail V vs. V-I CMD in (panel a), along with vs. tail V (panel b). Each graph presents the combined tail sample of this work and K03, color-coded by the progenitor system’s interaction age. Boxed data points again indicate tails with no evidence of SCCs at the MV= -8.5 level. An extinction vector of 1 magnitude in V is shown in (a), and a histogram of the surface brightnesses of cluster-rich and cluster-empty tails is displayed in (b). We also note that NGC 2444 had no optical tail and was eliminated from this Figure. \| \| \| --- \| \| \| \| \| \| \| shows that clusters are found in tails of a variety of surface brightness, V-I colors, and ages. As was seen in §3.2.1, the tail sample is simply too phenomenologically diverse for distinguishable tail populations in terms of age. For example, NGC 2992 and NGC 2782W are young tails devoid of MV < -8.5 clusters while the ≈ 750 Myr old debris of NGC 1614N/S have two of the highest SCC excess of the sample ( ∼ 0.1 kpc− 2). In terms of color, multiwavelength observations of other tidal tails that better circumvent the metallicity-extinction degeneracy show that blue colors typically imply young star formation and/or subsolar metallicity, rather than extinction effects. provide a similar perspective on extinction. Thus, one might expect blue tail colors to correlate with clustered star formation here. However, K–S tests reveal that tails with statistically consistent with zero have a 33.1% probability of sharing the same V-I distribution as those with > 0. There may be a number of explanations for this behavior. For instance, small variations of even 0.5 magnitudes of internal extinction may erase any trends of with tail V-I, which may be already unclear with our finite, 23-tail sample size. It is also possible that the blue colors expected of star formation are localized phenomena in most moderately star-forming tails, much as it is inferred in for galaxy interiors. Comparing excesses to tail surface brightness, we find a K–S probability of 0.0102 that cluster-bearing and cluster-devoid tails belong to the same tail V distribution. Evidently, tails with cluster excesses tend to be brighter. This is presented in to demonstrate that it is not simply a trivial relationship of having more luminous clusters contributing to an overall more luminous tail. The accompanying histogram reflects the statistically significant difference between cluster-rich and cluster-poor tails in terms of surface brightness, but it is also a reminder of the small sample size and the difficulties in interpretation, especially given variations in internal extinctions and projection effects. This could still be a promising result. Drawing upon the work of and, reason that the R-band surface brightness roughly traces the midplane pressure of galaxies. This pressure regulates the pressure within molecular clouds, which is important for producing clustered star formation and other bound structures. Although many of the underlying assumptions of this deduction – hydrostatic equilibrium of thin gas layers embedded in thin stellar disks, and the relative contributions of both of these components and their dynamics – are invalid for tidal tails, the general preference for cluster-forming tails to be brighter on average in V (which of course is not the same as R) may present a parallel paradigm. On more local scales tail surface brightness may offer a more accurate diagnostic of ambient pressure and the star clusters that form, but it is likely that the contribution of the gas potential and dynamics also needs to be more thoroughly addressed. Interestingly, the most cluster-rich tails ( $\gtrsim$ 0.2 kpc− 2) are all young ($\lesssim$ 250 Myr), and brighter than the median (V $\lesssim$ 24 mag arcsec− 2). Though we determined the age distributions for cluster-rich and cluster-devoid tails were indistinguishable, interaction age may be an additional ``bonus" for tails like AM 1054-325, MCG -03-13-063, NGC 2782E, and NGC 6872E, whose images capture strong bursts of star formation relatively near periapse for their interactions. This is typically seen in models of star-forming tidal debris (;, ). Thus, should higher surface brightness loosely trace higher ambient pressures and thus star formation capacity, SCC detection is contingent on strong overall star formation, which is typically seen at younger dynamical ages for a variety of interacting systems. SCC Excess and -------------- Although in §3.2 there was no clear relationship between tail gas richness and cluster candidate excess, the content of tidal tails warrant further investigation. In we explore the complex relationship between the optical/ properties of the tidal tails and their measurements. Boxed points are the same as in the previous Figures, and we plot points whose ``color-coded" data are either missing or unreliable in black. This includes the abnormally high densities of the NGC 1487E/W tails, which exhibit a superposition of disk and tail gas ( > 107 kpc− 2), and the brightest cluster candidate magnitudes of NGC 3256E/W and NGC 7252E/W (§3.3). [htbp] \| \| \| \| --- \| --- \| \| \| \| \| \| \| first offers a tail CMD whose points correspond to their values. As in, tails with and without clusters span a range ages and with no obvious relationship to integrated V or V-I. Examining the interplay of,, and reveals a different picture, however. shows the variation of with, with data points colored according to a scale of. Despite an overall increase of with tail richness, there is too much scatter for to be a useful diagnostic of cluster candidate excess. But between the boxed and unboxed data sets, tails lacking SCCs (boxed) appear generally less -dense on WFPC2 FOV scales than SCC-containing tails (unboxed). Without the aberrant NGC 1487E/W tails, K–S tests compute that tails without SCCs have a probability of 0.0216 of being drawn from the same distribution as those with SCCs. Thus, while is not a direct measure of the sub-kpc scales that are important in star formation (e.g. ; ; ), and the differences in populations are borderline distinct, it may be a useful zeroth order diagnostic. Cluster-deficient tails overall appear to have a different distribution of surface densities than tails with SCC excesses. Many of the boxed points in are bluer – have lower – than the normal points, although there is scatter (e.g. the -dense and highly inclined NGC 2992 whose SCCs are largely confined to its TDG). It is therefore interesting to consider whether there is some cutoff value or transition range of that would account for the different distributions. According to, it is possible to estimate a critical gas surface density for star formation, whose component we would observe here. Varying the input parameters of the models (see their Equation 25) and estimating the contribution of molecular gas yields plausible critical surface densities $\Sigma\_{\mbox{\scriptsize{{\hbox{{\rm H}\kern 0.1em{\sc i}}}\rm{, crit}}}}$ ≈ 2.7–3.4 × 106 kpc− 2. Here we have assumed mean metallicities $\lesssim$ Z⊙, and -ionizing UV radiation fields of 104–106 photons cm− 2 s− 1, which are reasonable estimates for tidal tails given their range of luminosities, ages, and sizes. We estimate that H2 contributes no more than 10% to the total gas density from of. We will hereafter refer to 2.7–3.4 × 106 kpc− 2 (2.7–3.4 pc− 2) as the fiducial critical gas density for star formation in tidal tails. Again, it is crucial to stress that a threshold density technically applies to sub-kpc scales, not the mean values quoted here. However, tails like NGC 2444, NGC 7252E/W, and NGC 3921S, with $\lesssim$ 1.7 × 106 kpc− 2, on average may be unable to generate significant SCC excesses on WFPC2 FOV scales. This does not prohibit more localized concentrations of cluster-forming activity, but may just imply an overall suppression of cluster formation. To further visualize the role of WFPC2 FOV-scale density, relates to interaction age. declines with respect to the interaction age (again, the anomalous NGC 1487E/W debris should be overlooked). Across the sample, there appears to be an exponential decline in with time; we plot example curves of ∝ e$^{-t/t\_{\rm{dep}}}$ for two depletion timescales $t\_{\rm{dep}}$. We include t$\_{\rm{dep}}$ = 100 Myr (blue dashed lines) and 1 Gyr (violet dotted lines). There are necessarily strong inter-tail differences in depletion timescales from star formation, orientations, and time-dependent gas dispersal. There is also a distance-related bias – clumps of emission at different places along a tail can influence measurements depending on the extent of coverage and placement of the WFPC2 FOV, and therefore distance. But in general, an average depletion timescale of ∼ 500 Myr for these tidal tails may be reasonable. This is on par with typical tidal tail lifetimes. In addition, red dashed lines in highlight our estimate of $\Sigma\_{\mbox{\scriptsize{{\hbox{{\rm H}\kern 0.1em{\sc i}}}\rm{, crit}}}}$. Evidently the notion of a critical surface density does not successfully translate to a critical for cluster formation. There are several tails (e.g. NGC 6872E/W) around or below this limit with an ample supply of clusters. Conversely, several tails above the $\Sigma\_{\mbox{\scriptsize{{\hbox{{\rm H}\kern 0.1em{\sc i}}}\rm{, crit}}}}$ limits are statistically lacking in clusters. Each point in glosses over the particular dynamics of each interaction that obtain varying quantities of the gas, the true three-dimensional densities and pressures of the tail ISM, and the multiple generations of stars and clusters that form a complex recycling and feedback network with said ISM. Thus there are clearly more complicated and local physics at work in determining cluster formation thresholds. The points plotted in are colored according to the corresponding scale of the brightest cluster candidate MV. Between the tails with and without significant cluster excesses, we find a K–S probability of 0.0224 for their brightest MV values being drawn from the same distribution. Like the K–S results for, the differences in the brightest MV distributions are marginal, but probably real. From the size-of-sample effect (e.g. ; §3.3), tails with cluster candidates likely have strong local star and cluster formation rates, and therefore have more luminous brightest cluster magnitudes. This does not seem to conspicuously relate to or interaction age, again obfuscating any additional attempts to connect overall tail star and cluster formation with dynamical age and large-scale content. Lastly, compares to, using the colors of the plotted points to again represent the brightest cluster candidate magnitudes for each tail. Since most of these tails are presumably produced from the -dominated gas of their progenitor galaxies’ outer regions, may approximately represent the total mean gas density. If tidal tails behave like spiral galaxies, the area-normalized tail star formation rate ΣSFR may possibly then scale with by a Schmidt-Kennicutt law; $\Sigma\_{\rm{SFR}} \propto$ $^{\rm N}$ for some index N. If we assume from the size-of-sample effect that the total number of clusters produced a tail scales roughly with the tail SFR, then ∝ $^{\rm N}$. N=1 implies a constant cluster formation efficiency; curves of constant cluster formation efficiency ε are indicated on the plot. Steeper curves for a certain average surface density indicate higher cluster formation efficiency. We avoid reporting efficiencies directly, as they require a conversion of to the total cluster mass density with a currently unreliable estimate of cluster population masses. In reality, the translation from to $\Sigma\_{\rm{SFR}}$ would include a treatment of how the luminosity evolution of clusters from earlier episodes and their appropriate timescales affects the measured SCC count, but this is an acceptable approximation given our other uncertainties. In comparing sub-kpc surface density to SFR density for a set of 18 nearby normal and -dominated galaxies and multiple resolution scales within them, demonstrate that neither the total gas density nor the surface density is the critical quantity in determining the SFR. Many galaxies can span about an order of magnitude in star formation efficiency for constant values of surface density. Employing similar methods to study regions beyond R25 in galaxy disks, however, regain a Kennicutt-Schmidt scaling of the SFR to density, reporting a suppression of star formation efficiency relative to the galaxy interiors. Our data do not show such a scaling, but rather tails spread across different cluster formation efficiencies and values for limited ranges of. The Schmidt-Kennicutt law applies strictly to rotating disks, and not tidal tails, so it is either possible no similar relationship would be observed between and any variety of gas for these environments at any scale, or a more directed, local study akin to and would be more illuminating. For now, it is interesting to note that tails with the most luminous brightest cluster magnitudes tend to lie around curves of higher ε, implying that some of our more cluster-rich tails (NGC 6872E/W and NGC 1614N/S) have generally higher star formation efficiencies (with respect to measured reservoirs) than other tails. But ultimately, the scatter in all the plots of implies that cluster formation in tidal tails is determined less by richness and mean density on WFPC2 FOV scales, and more by other variables that set star formation efficiencies and gas consumption timescales. While cluster-containing tails tend to have higher observed densities (often combined with young dynamical ages and bright, star-forming environments) than those without them, the number of cluster candidates observed in these tails cannot be well predicted by our current properties. Complex gas dynamics, star formation histories, and tail orientations do not allow a determination of critical gas properties required to produce SCCs. In an upcoming work, we will concentrate on densities on more localized scales, and may better uncover the role of neutral hydrogen in cluster formation within tidal tails. Cluster Candidates at Fainter Magnitudes ---------------------------------------- Of the 23 tails examined, 10 have nonzero SCC excesses at MV < -8.5, the range of which has few obvious dependences on many observable parameters. Provided the implications of and, fainter clusters are more numerous, and in the closest tails, detectable. While we cannot infer LFs and MFs from the current data, it is interesting to ascertain if star clusters exist to the faintest possible limit in each tail. In short, do star clusters of some magnitude form in all tidal tails? Ideally, to detect the maximum number of cluster candidates in each tail, changing the magnitude criterion to the completeness limits of the tails is desired. The generally higher values for MV < -6.5 in are illustrative of this concept. On the other hand, extending MV to fainter limits risks contaminating source lists with bright, high-mass main sequence stars from current bursts of star formation, as well as post main sequence stars above a certain mass around the red supergiant stage (e.g. ). The combination of these two effects is a strongly age-dependent function of the tails’ star formation histories and the minimum single stellar mass that the new MV or completeness limit provides. The blue and red (MV < -7.5 and -6.5) points in cannot be interpreted without this caveat. Appendix B explores these topics in further detail for the interested reader. Combining the extended magnitude limits with a treatment of single stellar contamination, we find that the NGC 520 TDG candidate probably has clusters to MV = -6.5, when there were no clusters brighter than MV = -8.5. NGC 1487E/W and NGC 4747 are also complete to fainter magnitudes, but their statistics are too poor for interpretation or their potential contamination was not negligible. For more distant tails, it cannot be determined for certain whether they have cluster excesses at fainter magnitudes than MV = -8.5. Some tails have insightful clues to their dearth of cluster candidates, however. We offer an explanation of some of these tails below. NGC 1487E/W, NGC 4038, NGC 3256E, NGC 7252E/W, and NGC 3921: It may be no coincidence that these mergers, which have no significant SCC excess at MV < -8.5, are all > 400 Myr old. A current low tail SFR and significant fading for any clusters formed near periapse may combine to generate an apparent paucity of star clusters. It is not necessarily true that there are no clusters that were formed in the these tails, but rather that the few that are there do not create a statistically significant excess above the background level at the magnitude limit of the WFPC2 images. Their cluster formation events may have been too weak, poorly timed, and/or poorly placed along the tails (the K03 tail pointing is a small fraction of the full length of the NGC 4038 debris region; see their ). provide a useful illustration of this age effect with their model of compressive and extensive tides in NGC 4038/9 (their ). If some fraction of cluster formation occurs where compressive tides arise in tidal tails, the nature of the tidal field may be a useful tracer of the star formation history. Their simulations show that, while the galaxy interiors always enjoy compressive tides along with the observationally obvious signatures of continuous nuclear star formation, these tides in the tails are sporadic at best. They preferentially occur at early ages ($\lesssim$ 100 Myr), and will only occur at later times at particular points along the tails. NGC 2444: This system, whose tail definition was assigned based on its debris and had no detectable optical debris in WFPC2 images, may similarly be devoid of significant star formation. This tail has the lowest recorded, 1.34 × 106 M⊙ kpc− 2, of the tails in the sample; this system’s -dependent tail definition warrants caution in comparing its mean density to those of the rest of the sample, however. Even if this medium were gravitationally bound, the average free fall timescale (1/$\sqrt{G\rho}$) would be $\gtrsim$ 1 Gyr, longer than the estimated tail age (100 Myr). So overall, star formation may be suppressed. Localized concentrations of on sub-WFPC2 FOV scales (kpc2, as opposed to the current ≈ 400 kpc2) may reveal more reasonable collapse timescales, however. Numerical models of suggest that high-velocity encounters (V ∼ 1000 km s− 1) between galaxies generally produce asymmetric, low-mass, gas-dominated tails without much star formation, as opposed to the optically bright (and star-forming), clumpy tails generated by slower mergers. Their model of NGC 4254 reproduces the 250 kpc-long stream with a rapid encounter velocity and mass ratio identical to that of NGC 2444/5. Their prototypes of these kinds of collisions produce average surface densities ∼ 2 pc− 2 or 2 × 106 kpc− 2, comparable to our result for that quantity. posit that the diffuse plume of NGC 2444 is a result of a recent ( ∼ 100 Myr), low impact parameter collision between NGC 2444 and a low surface brightness, -rich companion (not NGC 2445, which still interacts with the system). That unseen companion has since disrupted and its stellar component dispersed enough to fall below our optical tail detection threshold; they cite the velocity structure of the bloated bubble structure to the north of our WFPC2 FOV (online ) as kinematic evidence of this event. This model agrees with the timescales at play in this tail. To produce a ∼ 100 kpc long streamer in 100 Myr (the time required for an expanding shell in the tail to reach its current size from its velocity maps), collision speeds $\gtrsim$ 1000 km s− 1 are needed, rather than the tidal removal of disk gas comparable to orbital speeds at the outer disk radii. This suggests that merger dynamics play a dominant role in determining the SCC-harboring capacity of tidal tails. High-speed galaxy encounters may produce low-density tidal gas whose conditions, i.e. low metallicities, H2 deficiencies, and high velocity dispersions, are unfavorable for star formation in detectable quantities. NGC 2782W: It is interesting that the eastern tail of NGC 2782 has a profusion of in-tail sources ( ≈ 0.2 kpc− 2), while the western -rich, optically faint tail does not ( ≈ 0 kpc− 2). Optical colors and measurements are similar for these tails (V-I ≈ 1 and ≈ 5–8 × 106 kpc− 2). The main difference, aside from tail length, is optical brightness. The eastern tail is ≈ 1.5 magnitudes arcsec− 2 brighter in V than the western tail. The physical reason for the difference between the two tails has been explored. predicts that the eastern tail is predominantly the consumed, optically bright merger remnant of NGC 2782’s companion, while the western tail is the metal-poor debris left behind from the outer regions NGC 2782 itself. Therefore, the eastern tail has an ample molecular hydrogen supply from a merging galaxy interior to fuel its observed SCC density. The absence of CO emission and inferred lack of H2 in the western tail makes the apparent absence of young cluster candidates understandable. A low metallicity of this tail, and differences in the survivability of H2 and CO at low tail extinctions, may account for a nondetection of CO and an inaccurate translation to molecular hydrogen. But if the absence of cluster candidates in this tail is a question of gas availability, projection effects may be to blame. Theoretical models require large relative inclinations between the two galaxies, resulting in an extended plume with a sizable component along our line of sight. Thus, this system’s gas surface density may represent the same physical density as a ``flatter" tail with a lower surface density by as much as a factor of ∼ 3. Using Gemini optical spectroscopy, Torres-Flores et al. (2011, in preparation) confirm cluster status for several objects in NGC 2782W we label as SCCs, and assign them masses ∼ 105 and ages consistent with in situ formation. Since there are still only a few clusters in the ∼ 105 range, so this system suffers from a ``weak" cluster formation history compared to many of the other tails studied here, perhaps for reasons proposed above. NGC 2782W therefore has clusters, but they are effectively hidden in the statistics of our study. In summary, certain merger dynamics like encounter velocity, or variations in molecular gas content may account for some of the dispersion in and the range of cluster magnitudes possible across tidal debris. This could be especially pertinent for tails that have no demonstrable SCC excess even at the faintest MV limits. Variations in completeness (distance) and limited prior observations and theoretical work make this difficult to assess across the entire tail sample. Finding SCCs may be optimal in the closest, youngest tails that are actively star-forming. Disentangling other, potentially more promising effects provided by dark matter halos, merger geometries, etc. require further efforts in high resolution modeling. Conclusions =========== We have presented a survey of the star cluster candidates in 17 tidal tails of 12 interacting galaxies observed with HST/WFPC2 in and. We combined this sample with the six tails studied by. The 23 tails span the vast parameter space of interaction characteristics like ages, mass ratios, average surface densities, richness, tail lengths and surface brightnesses, and host galaxy star formation rates. Star cluster candidate (SCC) densities in the tail regions were computed by subtracting a background estimated from out-of-tail sources. A significant excess ( > 2.5σ) was detected at MV < − 8.5 and V-I < 2.0 for 10 of 23 tails. This color and magnitude range was selected to ensure completeness, exclude single stellar contaminants, and allow for stochastic and systematic effects of cluster evolution. We have excluded tails whose SCCs were largely constrained to tidal dwarf galaxy (TDG) candidates. In some cases cluster formation seems confined to these structures, as was seen in. We also identify several cases with definite in-tail SCC excesses and TDG candidates, but cannot extrapolate the significance of this with the paucity of multiwavelength and kinematic data for several of these systems. We also find three cases of ``beads on a string" morphology (with separations of ≈ 3 kpc) for clumps of sources/SCCs, characteristic of large-scale star formation by gravitational instabilities also seen in spiral galaxies and tidal tails of other studies. For some nearer tails, fainter sources could be detected, and in the NGC 520 TDG candidate a significant cluster excess was found for MV < − 7.5 and MV < − 6.5, where none was detected at MV < − 8.5. Given the limitations of our exposure times and broadband coverage for this project, it cannot be conclusively determined if the in-tail cluster candidate excesses at fainter magnitudes are significant for other tails, or if more distant tails have faint clusters that are currently undetectable. To the extent that we can determine them with small numbers, the cluster luminosity function in the tail environment may follow the same power-law slope as in quiescent and starburst galaxies. Combined with the emergent consensus on the universality of clustered star formation, we therefore contend that cluster populations are likely to exist to some level in most, if not all tails. This level may not always be detectable by HST. A complex combination of various factors, including environmental differences in star formation, gas supply, tail ages, and projection effects appear to influence the populations we observe. Of the global properties we considered (interaction age, progenitor mass ratios and star formation rates, and total tail masses), no parameter was individually responsible in shaping the cluster populations. However, we do find statistical differences in the populations or tails with and without cluster excesses in terms of V-band surface brightness and mean density. The tails that have the highest excesses of star cluster candidates at MV < -8.5 tend to be the youngest ($\lesssim$ 250 Myr), probably tracing the strongest episodes of star formation that are typically triggered close to periapse. Young tails also tend to have larger mean observed surface densities than older tails. It is important to note, however, that there are exceptions to all of these trends. We conclude that there is no single explanation for a tail to have a cluster excess, though there are several important influential factors like local gas density and local star formation activity at the time of observation. Quantities like magnitude of the brightest observed SCC and the overall tail surface brightness appear to loosely trace SCC excess. Merger dynamics not addressed here (e.g. encounter speeds and orientations; dark matter halo structure) may also be important in imprinting the resulting tidal field with a dynamical setup conducive to star and cluster formation. To disentangle the full assortment of interaction variables would be aided by a larger, deeper sample, more accurate cluster age-dating, and by a higher resolution, local study of the relationship between gas and star cluster formation. We wish to thank Bärbel Koribalski for sharing data and helpful comments for NGC 6872. This projected was supported by a grant from the Space Telescope Science Institute (grant no. HST-GO-11134.05-A). Facilities:,, A. Notes on Individual Systems ============================== Below we present observations of each tail. Note that interaction ages without citations were calculated by dividing the projected length of the tail by an estimate of the escape or rotation velocity. All observed and measured characteristics quoted are recorded in Tables 2–4. In the WFPC2 images and color maps, the second, third, and fourth chips (WF2, WF3, and WF4), are located in the lower left, lower right, and upper right parts of the mosaics when positioned as in Figure 3.1. NGC 1487 (Figures 3.1–3.2) -------------------------- NGC 1487 is the optically bright remnant of a ∼ 500 Myr-old merger between two comparably massed galaxies, whose eastern tail features a long ``spine" curving into the bottom right corner of WF3, surrounded by an extended plume that reaches into the other chips. There is one non-tail source and zero in-tail sources meeting the cluster selection MV and V-I criteria. The in-tail color distribution of detected objects is very broad, peaking at ∼ 0.5–0.8. The bluest sources of this distribution are concentrated in the blue knots evident in this tail’s color map, which trace the spine to the collection of sources at the tail’s tip. These may be either very young ($\lesssim$ 10 Myr) or metal-poor. There is considerable scatter in the concentration index plot. At this distance, the largest sources may be marginally resolved (1 pixel = 5.22 pc), so these concentration indices may loosely correspond to a range of physical sizes. For most in-tail sources, the V-I colors indicate ages ∼ 100 Myr, so many faint sources detected in this study may be clusters ∼ 104. Thus, we see evidence of star formation around the time of perigalacticon, with more recent, superimposed populations across the tail. These regions in particular paint a high-resolution ``beads on a string" across the color map. Considering possible projection effects, these young clumps are separated by ∼ 2–3 kpc. K–S tests reveal a 34.1% and 2.63% probability that all the in-tail and out-of-tail sources detected originate from the same distribution of and, respectively. The lack of SCCs is strange, given the high average densities measured for this tail ( = 2.55 ± 0.51 × 107 kpc− 2). Overall, the tail has one of the lowest total masses, however; = 6.14 ± 1.2 × 108. Why the debris with the highest content should also be devoid of cluster candidates $\gtrsim$ 105 M⊙ is not easily explained. In pc− 2, = 25.5 ± 5.1, well above the saturation threshold of 9 pc− 2 of. Above this limit, most of the gas available in the tail may be molecular, so inclination effects or the superposition of disk and tail gas may be responsible in artificially increasing the observed average density. If the true surface density of NGC 1487E is significantly lower and in the expected -dominated regime of typically low-Z, H2 poor outskirts of galactic disks, the overall SFR of the tail may be relatively low (c.f. the host galaxy IRAS SFR of 0.12 yr− 1). This may imply an underpopulation of the high-mass end of a star-forming generation’s cluster initial mass function (CIMF). This would then result in zero objects observed with photometric characteristics of ∼ 105 clusters. It is easy to check on the logical consistency behind this reasoning. If we reasonably assume that these clusters are drawn from a power-law luminosity function with a slope of -2 – -2.5 (§3.3), then it is possible to analytically estimate how many clusters we should observe brighter than a certain magnitude. Using the number of sources we observe between adjacent, complete luminosity bins (of size 1 mag) to normalize the LF, we calculate that there should be $\lesssim$ 1–3 sources brighter than MV = -7.5, and no objects brighter than MV = -8.5. With the tail age limits, this verifies that there should not be clusters present with masses $\gtrsim$ 104 in this tail. The western, also -dense yet -poor (= 33.2 ± 6.6 × 106 kpc− 2; = 5.41 ± 1.1 × 108 ), tail is a patchy assembly of diffuse light stretching across WF3 and WF4. Similar star-formation hotspots as in the eastern tail are observed, but are not identified as SCCs. There are two cluster candidates found within the tail, and zero beyond it. One SCC has a magnitude and color consistent with a ∼ 105 M⊙ cluster from the same ∼ 100–300 Myr generation as most of the sources fainter than the MV selection limit. The color map shows most of the cluster and field light is produced from this generation of star formation, with scattered bursts of younger and less massive ($\lesssim$ 100 Myr; ∼ 104 ) objects. This is qualitatively similar to the eastern tail. There is a 59.3% and 39.6% probability of in-tail and out-of-tail sources being drawn from the same and distribution. It is possible that the true tail boundary extends to the contours pictured in the online Figure, with several out-of-tail sources belonging to the in-tail source list. However, if this were true, the out-of-tail CMD still indicates that there would still be only two SCCs counted in this tail. The concentration index plot shows no useful distinction between in-tail and out-of-tail detections. In the galaxy interior, investigate several clusters whose discrepancies between photometric and dynamical masses offer insight into their current disruption. Their clusters are ≈ 8.5 Myr old and between 105 and 106. performed WFPC2 B- and I- band photometry on 560 clusters similar to this study, and while their age and mass estimates for clusters are also limited to two bandpasses, they suspect a similar, roughly bimodal age distribution in the NGC 1487 clusters, peaked at 15 Myr and 500 Myr. NGC 4747 (Figure 3.3) --------------------- This system is a ∼ 320 Myr old minor merger interacting with NGC 4725 (; ), featuring a long, optically faint tail extending through the third and fourth WFPC2 CCD chips. There is a concentration of sources in the third chip, with estimated ages $\lesssim$ 50 Myr (and corresponding masses $\lesssim$ 105 M⊙). Of the sources shown in the CMDs, 7 and 5 cluster candidates were detected in and outside the tidal regions, respectively, with ∼ 0–0.7 and 0.3–0.7. Population differences may be seen in the ΔV diagrams; in-tail sources show a slight preference for ΔV $\lesssim$ 2.0, compared to the more even dispersion for objects beyond the tail. Given the distribution of colors, SSP models posit ages of ∼ 30–300 Myr for ∼ 105 M⊙ cluster candidates in this image. The and K–S probabilities for all sources are 0.878 and 0.646, respectively. was not detected in this tail, and the interior of the merger has an IRAS-derived SFR of ≈ 0.27 yr− 1. NGC 520 (Figure 3.4) -------------------- NGC 520 appears to be a minor merger remnant. WFPC2 images present an optically faint, -defined ( = 1.12 ± 0.22 109 ) curved tail extending to the north. The HST pointing here is situated on the tidal dwarf candidate in this system (UGC 957), which may either be a genuine TDG or a small satellite galaxy engaged in a 3-body interaction with the other galaxies. This object coincident with bright emission ( = 3.81 ± 0.76 × 106 kpc− 2), but no CO has been detected at this location. Within the host galaxy, infrared emission indicates an SFR of ≈ 6.5 yr− 1. Given the TDG candidate’s distance from the galactic interior (112 kpc), the tail/galaxy SFRs may not correlate well. GALEX observations have revealed that no current star formation has occurred within NGC 520, and the youngest regions are ∼ 300 Myr old. Moving out to the tail, they find massive 108–109 M⊙ clumps of ages $\gtrsim$ 300 Myr. Optical spectroscopy of the NGC 520 nucleus confirm a 300 Myr old episode of star formation, with little current activity. This matches dynamical simulations of this interaction that place the age of this interaction at 300 Myr. The distribution of colors detected in the TDG is extremely broad in the CMD and color map and lacks an obvious peak. We detect 0 and 4 SCCs in and beyond the tail, respectively. Given the above age constraint, many sources fainter than MV = -8.5 may be ∼ 104 M⊙ clusters $\lesssim$ 10 Myr old, or $\lesssim$ 5 × 104 clusters with ages ∼ 150–500 Myr. From the ΔV diagram in the online Figure, NGC 520 may contain a marginally distinct population of compact objects, whereas the external sources are generally more extended and fainter (M$\_V \lesssim$ -7). K–S tests estimate a probability of 1.37 × 10− 3 and 0.757 for all detections originating from the same and distributions. NGC 2992 (Figure 3.5) --------------------- NGC 2992 is a peculiar Sa galaxy with an unusual optical condensation at the edge of its tail in WF2. With a combination of Hα, CO, optical, and data, asserts that this is a genuine TDG. Sources detected within outline prominent CO emission; with standard conversion factors, estimate that the TDG contains 3.5 × 107 of H2, contributing to a total of ≈ 2 × 109 of stellar and gaseous material. There appears to be a deficit of sources in the middle of the tail on WF3, which is visibly smoother and undisturbed. The color map is remarkably homogeneous, with colors consistent with ages of ∼ 100 Myr – 1 Gyr. find that this tail has a high metallicity within the TDG (12 + log(O/H) = 8.6), so redder colors are expected. The embedded SCC population with similar would be mostly composed of ∼ 105 M⊙ clusters. It is also possible from the CMDs that lower mass, $\lesssim$ 10 Myr old clusters contribute to the tail light. Of the sources indicated, 37 and 15 are SCCs in and out of the tail, respectively. Between these two regions, there do not appear to be distinctions in terms of concentration index. We find K–S and probabilities of 0.0187 and 0.660 for the entire source list. The location of SCCs coincides with the brightest contours; this tail has a mean density of = 6.44 ± 1.3 × 106 kpc− 2. It also has an interior IRAS-derived SFR of ≈ 3.4 yr− 1. This galaxy is in the early phases of merging with its similarly-massed companion NGC 2993. They both exhibit moderate masses ( = 1.43 ± 0.29 × 109 for NGC 2992), optical brightnesses, and short (apparent) tails relative to their disks. 100 Myr have elapsed since perigalacticon, and the final merger should occur in ∼ 700 Myr. Given this constraint, SCCs that form in situ must be $\lesssim$ 5 × 105, as mentioned above. Prior studies have noted difficulties in isolating central starbursts in this AGN-powered Seyfert galaxy (e.g. ; ), so the star cluster populations of that system remain unresolved. NGC 2993 (Figure 3.6) --------------------- The interacting counterpart to NGC 2992, NGC 2993 features an optically faint, moderately -rich (= 7.27 ± 1.5 × 108 ; = 5.77 ± 1.2 × 106 kpc− 2) tail that curves through the second and third chips of the -band mosaic. The interior of this galaxy supports an infrared-derived SFR of ≈ 5.0 yr− 1. The color map exhibits a strong regional variation of, from the blue spine to the redder outlying plume. In the color-magnitude diagram for the debris, in-tail objects are roughly distributed across the tail and its color gradient, and mixed with the out-of-tail sources in the concentration index diagram. There are 15 in-tail SCCs and 7 out-of-tail counterparts, the former of which show a degeneracy of possible high masses/old ages or lower masses and young ages akin to NGC 2992. From Hα equivalent widths, find groups of 5–7 Myr old starbursts in the main body of this system, which are systematically younger than the estimated ∼ 10 Myr – 700 Myr age range of objects detected here. If tail populations of NGC 2992/3 are around the same age, then our detected SCCs may have masses ∼ 104. K–S tests reveal a 1.72% and 65.2% probability that all the in-tail and out-of-tail sources detected originate from the same and distribution, respectively. NGC 2782 (Figures 3.7–3.8) -------------------------- NGC 2782 is an SABa-type, ∼ 200 Myr old minor merger. The eastern tail is short compared to the face-on optical disk, and rich in ( = 2.41 ± 0.48 × 109 ; = 7.08 ± 1.4 × 106 kpc− 2). The aftermath of the interaction is evident in the disrupted appearance of the main disk in WF4, highlighted by ripples and density inhomogeneities that continue down into the tail in the other two chips. The detected sources seem loosely correlated with the optically clumpy parts of the tidal debris, and appear only partially correlated with the bluer regions in the color map. We find 87 potential clusters in the tail, and 4 beyond. For all the in-tail objects, the CMD reveals a broad distribution of colors representative of a correspondingly broad range of putative cluster ages. SCCs may have a spread of ages and masses, from $\lesssim$ 10 Myr, ∼ 104 clusters, to those aged nearly 1 Gyr and 105 in mass. Given the age constraint and 100 Myr old cluster estimates for this tail of, our SCCs are more likely closer to the former than the latter. Contours of emission for the eastern tail abruptly end halfway across the third chip; this is also where the bluest (or conversely metal-poor) regions of the tail are. As seen in the online Figure, many of these sources tend to be photometrically extended (ΔV $\gtrsim$ 2), similar to the few external sources with generally high concentration indices and more limited spread of colors. Blue colors ( $\lesssim$ 0) and extended profiles may indicate young ages and concordant nebular emission. The majority of fainter in-tail SCCs appear more concentrated, with ΔV < 2.0. We determine K–S probabilities of 0.875 and 0.963 for and, respectively, for all sources. The western tail of NGC 2782 is markedly different, barely tracing WF3 to the PC and down into WF2. The optical extent of the tail does not perfectly match the tail illustrated on the SDSS image; the latter extends further into WF4 and WF2 than the optical boundary we employ here. This tail is also more -rich on average than its eastern companion ( = 3.14 ± 0.63 × 109 ; = 5.17 ± 1.0 × 106 kpc− 2). Its relative paucity of SCCs is in stark contrast to NGC 2782E. With a common progenitor galaxy IRAS SFR of ≈ 3.7 yr− 1, it is once more apparent that this infrared diagnostic may not correlate with tail SFR in all cases and cannot be interpreted too deeply. In terms of concentrations and colors, in-tail and non-tail sources are evenly dispersed about the cutoff, and do not show immediately recognizable differences in ΔV. The ages and masses of these SCCs are similar to those of the eastern tail. Assuming the inference that the tails are host to population of young clusters of ages < 100 Myr, our sources (if they are clusters) are probably between 104 and 105. 10 SCCs are found in the tail, and 7 are located outside it. Here, K–S tests anticipate a 5.65% and 2.49% chance that all in-tail and out-of-tail sources came from a single distribution of and. MCG-03-13-063 (Figure 3.9) -------------------------- This SBb-type, optically faint galaxy with unknown appears slightly perturbed by its ∼ 100 Myr old minor merger, with thin filamentary tails drawn from the main disk, enriched with classic ``beads on a string" hotspots of star formation. Assuming an approximately face-on orientation, the separation between these clumps is about 3 kpc. The tidal debris and galaxy interior (WF2-WF3) show an extended distribution in, down to very blue ( < -0.25) colors indicative of young ages and nebular emission not tracked by the SSP models. These sources overall are uniformly distributed in the ΔV diagram. Since any clusters would be unresolved at this distance, some may be crowded cluster complexes rather than individual clusters. CMDs show these objects as potentially very young (several < 10 Myr and others $\lesssim$ 100 Myr), and massive. With in situ age constraints, these SCCs would be ∼ 5 × 104–105 . In total, 88 cluster candidates are found in the tails, and 26 are beyond them. The probability that all in-tail and out-of-tail sources originate from one population is 0.324 for and 0.0617 for. In spite of this overwhelming evidence for clustered star formation in this tail, the IRAS-derived SFR is one of the lowest of the tail sample, at ≈ 0.43 yr− 1. This system is aligned nearly face-on, and may register lower IR SFRs than galaxies with higher inclinations. ESO 376- G 028 (Figure 3.10) and AM 1054-325 (Figure 3.11) ---------------------------------------------------------- AM 1054-325 is an optically bright spiral in the early stages (age ≈ 85 Myr) of merging with a nearby early type galaxy ESO 376- G 028, as displayed in the online Figure. The latter system features four SCCs embedded in a diffuse plume of debris, but the number of equivalent objects per unit area beyond its tidal debris makes it statistically insignificant. The general color of this debris is relatively red ( > 0.8); though data do not exist for this galaxy, on morphological and photometric grounds it stands to reason there is simply insufficient gaseous ``fuel" to trigger obvious star formation in what is probably a dry merging system. For ESO 376- G 028, we find and K–S probabilities of 0.445 and 0.291. AM 1054-325 is the more remarkable of the two, with large clumps of star-forming debris pulled out from the disk in WF3. colors in the tidal debris evidenced in the CMDs and color map highlight young and blue episodes of star formation, and some evince possible nebular emission in their blue ( < 0) colors. The bluest sources are closer to the ∼ 1–5 Myr estimates of star formation regions within the galaxy by, with some leeway in cluster mass. The host galaxy IR SFR is ≈ 0.64 yr− 1. The tail appears to have a base color consistent with an age $\lesssim$ 100 Myr. ``Beads" of newer ($\lesssim$ 10 Myr) and/or metal-poor bursts are strewn across the tail. These clumps in the smoothed color map are separated by ∼ 2–3 kpc, depending on tail orientation. The ΔV diagram shows the strongest disconnect between in-tail and out-of-tail sources thus far, with the former having typically bluer colors than the latter. The wide range of concentration indices may again be symptomatic of cluster complexes that are unresolved at this distance. 180 sources qualify as cluster candidates inside the tail, with 60 outside it. For AM 1054-325 sources, we determine K–S probabilities of 2.36 × 10− 10 () and 9.22 × 10− 25 (). The above age restrictions allow these SCCs masses ∼ 5 × 104–5 × 105. The tail’s orientation is unknown, however, and may negatively affect the number of SCCs measured. If the debris extends from the center-right of the galaxy disk and curls northward towards the observer, we may witness a superposition of disk-bound and in-tail clusters that combine to give unusually high SCC counts and exaggerate this tail’s cluster bearing capacity. Or if what we identify as a tail is partially a spiral arm pulled by the interaction, we would be observing not tidal debris, but galactic cluster candidates shaped by different arm-related triggering processes like density waves. NGC 2444 (Figure 3.12) ---------------------- NGC 2444 is a peculiar S0-type galaxy featuring a long tidal tail visible in Figure 3.12. suggest that the debris is a result from a collision between NGC 2444 and a low surface brightness, -rich companion, and not NGC 2445, which still interacts with the system. That companion has since disrupted and its stellar component dispersed enough to fall below our optical tail detection threshold. This model makes intuitive sense – collision speeds $\gtrsim$ 1000 km s− 1 are needed to create a ∼ 100 kpc long plume in 100 Myr. Though an rich ( = 5.20 ± 1.0 × 109 ) tail through the WFPC2 field of view is visible in the SDSS image of this tail’s online Figure, any optical counterpart is too faint to be contoured by the 1 count + background technique utilized here. As such, we make an exception for this tail and follow the extent (average = 1.40 ± 0.28 × 106 kpc− 2) illustrated in the online Figure to determine the tidal tail boundary. There are 27 sources meeting the SCC selection criteria within this tail, and 15 outside it. In all, they appear to span an age range of ∼ 10–500 Myr, depending on masses between 104–105 . They are also evenly dispersed among the out-of-tail objects in the ΔV diagram. K–S tests dictate a 54.5% and 24.1% chance that all in-tail and out-of-tail sources came from a single distribution of and. In contrast, and find circumnuclear star-forming regions with ages < 30–69 Myr. These are approximately timed with the last passage of the interacting galaxies that is thought to have produced the long plume and merger morphology characteristic of collisional ring galaxies. investigate the H2 and PAH properties of star-forming knots within the circumnuclear ring, determining ages of 2–7.5 Myr, contrasted with a nucleus age of about 500 Myr. NGC 2535 (Figure 3.13) ---------------------- An -rich ( = 4.45 ± 0.89 × 109 ), optically bright Sc galaxy ∼ 100 Myr past perigalacticon in its minor merger with NGC 2536, NGC 2535 possesses two main tidal tails; the long northern tail is illustrated in its online Figure. It contains a ``backbone" of clumps from WF3 to WF2, surrounded by more diffuse emission and anchored to an outer spiral arm in WF4. The optical tail coincides with the emission seen in Figure 3.13, ( = 6.84 ± 1.4 × 106 kpc− 2). The innermost limit of the tail boundary corresponds to the terminus of the hinge clumps seen at the rightmost side of WF3. These star-forming hotspots accumulate at the low shear tips of spiral arms, marking the end of the galaxy proper. It is a convenient check on the validity of our tail/galaxy distinction. These hinge clumps also appear as pronounced blue condensations in the color map. A number of sources are evident, aside from the SCCs selected. The in-tail objects are very blue overall in the CMD ( < 0.5), as opposed to the out-of-tail sources that have redder (0.2–2) colors. Both are scattered in terms of ΔV, however. We determine that the cluster candidates (41 in the tail and 49 outside) can be fit with ages of < 10 Myr for the brightest objects, to ∼ 100 Myr for sources located around the evolutionary tracks. Masses would then range from 104–105 . Using optical/UV colors and Hα data, constrain the ages of star-forming regions within interior features to ∼ 9 Myr for the ∼ 106–109 clumps (complexes) they study. They contend that clumps in the tidal regions of this interaction have similar ages, but lower masses (typically by a factor of ∼ 10). The K–S probabilities of NGC 2535 sources are 0.0904 () and 1.16 × 10− 4 (). Our galaxy IR SFR calculation estimates a rate of ≈ 2.9 yr− 1 in NGC 2535’s disk, similar to the 2.0 ± 0.8 yr− 1 and 4.9 ± 2.0 yr− 1 SFRs found for the clumps and total system by. NGC 6872 (Figures 3.14–3.15) ---------------------------- NGC 6872 is a large, optically bright SABc galaxy well endowed with in its combined eastern and western tail pointings ( = 3.52 ± 0.50 × 109 ), though its tremendous size translates this to a low of ≈ 1.5–3 × 106 kpc− 2. The galaxy bears long, thin tidal tails from its ∼ 150 Myr old interaction with a smaller companion, IC 4970 (; ). The eastern pointing features an intricate stream of tangled hotspots, from irregular clumps to cluster candidates. Sources in all chips have a large spread in – likely from a combination of age and extinction effects – with several extremely blue objects featuring possible nebular emission. There is a wide variety of concentration indices around ∼ 0–1, which is not dissimilar to the out-of-tail sources. Contours of on the SDSS image are roughly aligned with the most populated regions in the optical mosaic and color map. The proliferation of SCCs straddling the WF2-WF3 divide are uniformly blue and young or metal-poor. This may be a hotspot of the most recent episode of star formation or evidence of a metallicity gradient across the tail. The latter scenario is observed in the opposite tail by Trancho et al. (2011, in preparation); they find a gradual decline in metallicity from supersolar at the tail base to subsolar at the tip. What we witness in the color map of this tail may be the eastern tail’s equivalent gradient, portraying a sequence of cluster formation events beginning at the tail base and proceeding towards the tip, using the metal-poor ISM drawn from the galactic outskirts as fuel. By the time star formation has ignited at terminus of the tail, the base may have endured multiple generations of star formation and concordant metal enrichment. The western FOV of the tail, dominated by the galaxy body in WF4, has in-tail sources congregating in clumps around the WF2-WF3 divide, a number of objects spread throughout the tidal debris in all chips, and another assemblage east of IC 4970 coincident with a locally dense region. Both tail and external sources have a broad distribution of colors similar to that of the eastern FOV. In either case, there are significant numbers of extremely blue objects ( ∼ -0.75–0.0), as well as those with more moderate colors. Tail sources with ∼ 0.5–1.5 seem more concentrated at ΔV = 1.5–2, and the bluest objects tend to have concentration index values $\gtrsim$ 2. Out-of-tail sources are more evenly dispersed in –ΔV space. Source ages in both eastern and western FOVs range from $\lesssim$ 10 Myr for a few of the bluest cluster candidates, to 100–300 Myr for dimmer and/or redder hotspots. The former correspond well with the 104–106 M⊙, < 100 Myr old clusters detected and confirmed to be spatially coincident with regions of Hα emission (; ). It is unclear whether these SCCs form a MF with a logarithmic slope of -1.85 ± 0.11 as the former study suggested with their ground-based, lower resolution work. Many of the sources they pinpoint may be agglomerations of the higher resolution SCCs we define, which may decrease the MF slope to values closer to ≈ -2. We postulate a mass range of a few 104–106 for these objects. In the eastern pointing, there are 163 SCCs within the optical tail boundary, and 63 outside of it. NGC 6872W sports 195 SCCs in the tail, and 36 beyond. We find K–S probabilities of 7.26 × 10− 5 and 0.209 for all sources in and for the NGC 6872E FOV. For the western pointing, K–S tests predict a 0.517% and 11.9% chance that all in-tail and out-of-tail sources came from a single distribution of and. Our IR SFR of ≈ 2.9 yr− 1 for the central galaxy is significantly lower than U-band estimate of the local tail SFR of ≈ 16.5 yr− 1 for the combined debris regions, but the latter value is likely an overestimate (Bastian 2010, private communication). It is likely the ``true SFR" for the tail is closer to those quoted by ; ∼ 3 yr− 1. NGC 1614 (Figures 3.16–3.17) ---------------------------- NGC 1614 is ≈ 750 Myr into its merger with a similarly-massed companion ( and references therein), a peculiar SBc galaxy with a nearly linear plume of southern debris and a northern disky tail. Both tails are optically bright and contain comparable amounts of ( = 7.68 ± 1.6 and 9.64 ± 1.9 × 108 for northern and southern tails, respectively). IRAS-derived SFRs are problematic for its tidal tails; it is a heavily dust-obscured ULIRG and presents a nuclear cloud-cloud collision triggered SFR of ≈ 35. yr− 1. Its content is unevenly distributed across the optical debris – only the northern and southern tips of the northern tail present emission ( = 2.56 ± 0.52 × 106 kpc− 2), as does the optically bright ``stub" in the inner southern tail ( = 2.89 ± 0.59 × 106 kpc− 2). The color map of NGC 1614N displays a diffuse debris whose is generally consistent with its 750 Myr age. Individual SCCs are typically bluer (younger), with -estimated ages of ∼ 10 Myr to 300 Myr. One SCC may be younger than 10 Myr, depending on whether it falls in the ∼ 5 × 104–105 range of its companions. These objects show no particular ΔV preference in the concentration index diagram, but do approximately coincide with the regions of emission. In all, there are 33 and 28 SCCs in and outside the tail, respectively. The southern tail shows a greater diversity in SCC demographics than the northern debris. Sources populate the in-tail CMD around the 105–106 SSP tracks, with corresponding ages $\lesssim$ 300 Myr. The bluest and brightest of these are concentrated in the -rich stub forming the base of the tail. Sources outside the tail are largely redder, possibly older sources of the aforementioned age range. Like the northern tail, SCCs here do not display any preference for a particular range of ΔV. We find 38 SCCs in this northern tail. This system exhibits a ∼ 300 pc circumnuclear ring of massive regions in the central galaxy, whose photometric properties imply a series of current (5–8 Myr old), massive (5.6 × 108 ) bursts, over a base population 1 Gyr or older. The bluest sources in the southern tail may be commensurate with this activity; the ages and masses of the remaining SCCs are not well enough constrained to correlate their formation with intragalactic star formation. In the northern tail, and K–S probabilities for all sources are 1.94 × 10− 3 and 3.33 × 10− 4, respectively. All southern tail sources show and K–S probabilities of 2.61 × 10− 4 and 1.27 × 10− 6. B. SCC Detection at Fainter Magnitudes ====================================== SCC Detection Biases -------------------- To understand the presence or absence of cluster candidates at fainter luminosities, several key issues must be addressed – the rapid color and magnitude evolution of clusters as they age, the overall level of cluster production in the studied environment, and completeness limits of detected cluster candidates. Thus cluster detection is heavily contingent on age, size-of-sample effects, and distance. more clearly illustrates the combination of these three influences. The top panel of shows the color-magnitude evolution of model clusters from the population synthesis models of. The blue track follows a theoretical population with a metallicity 0.4 Z⊙, the black shows Z⊙, and the red 2.5 Z⊙. The tracks are repeated for masses of 106, 105, and 104 ; the 105 models are outlined in bold for emphasis. The vertical dotted line in the CMDs indicates where MV = -8.5. For reasonable metallicities, these cutoff values allow the detection of 105 clusters with log age (yr) ∼ 6–8.5. Our objective is to observe clusters born within a tail environment that typically disperses before log age(yr) ∼ 8.5–9, this is a suitable result. Shifting the model on the x-axis by 1 mag corresponds to a factor of 2.5 increase in cluster mass, so our color-magnitude criteria is sensitive to most appropriately-aged clusters $\gtrsim$ 5 × 105. Stochastic effects and cluster disruption will necessarily introduce some uncertainty. This also submits an age (or more accurately, star formation history) bias into the analysis – for tails with old ($\gtrsim$ 300 Myr) tidal debris, if they did not experience sufficiently recent episodes of cluster formation, clusters with masses $\lesssim$ 5 × 104 would have faded below our detection threshold. Given the apparent preference for star formation in tidal debris to occur soon after their host galaxies’ initial encounter (;, ), cluster detection is favored in young (in our case, $\lesssim$ 250 Myr old) tails. These young tails have a larger mass range of detectable clusters than old tails for populations born at early dynamical ages. This would be less of an issue if all tails produced ample quantities of high-mass clusters. But depending on the debris, more massive clusters may be buried in the statistics of background subtraction, or may simply be absent if the cluster formation events of a tail were not powerful enough to populate the $\gtrsim$ 5 × 105 end of the observed cluster mass function. The middle panel of considers the observational implications, displaying the MV606 distributions of sources for three tails. These histograms have been background-subtracted; i.e. the number of background sources was subtracted from the number of in-tail sources for each bin. The background-subtracted numbers of cluster candidates $N\_{\rm{SCC}}$ are also presented. We plot model LFs suggested by §3.3 for these tails – a power-law with a slope of -2 (purple lines), and -2.5 (yellow lines). Note that this choice of LF is motivated by statistical diagnostics (); we do not infer the LF from the individual distributions, but apply these LFs to them, normalized to the total number of SCCs of each. V606 distributions are	arxiv_0000791
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However, the diagram above clearly shows that F18 is not a constant map. We challenge the reader to find a nilpotent CA with same alphabet and radius but such that F19 is not a constant map. The attracting configuration x in the definitions above must be uniform, because it must attract in particular uniform configurations. Thus, these definitions can be expressed as properties of traces: * F is nilpotent if and only if there is t0 and q ∈ Q such that TxF(t) = q for all t ≥ t0 and all x ∈ QG. * F is asymptotically nilpotent if and only if there is q ∈ Q such that all x ∈ QG there is tx ∈ N such that TxF(t) = q for all t ≥ tx. Note that the state q in the above definitions must be quiescent, i.e. such that the uniform configuration $\overline{q}$ is a fixed point of F. The property of nilpotency is also well-known as a property of the limit set. The limit set of a CA F is the set ΩF = ⋂tFt(QG) (see ). Nilpotency is equivalent to ΩF being a singleton. Asymptotic nilpotency can be reformulated through the action of the CA on measures. [lem:asynilmeasure] A CA F on QG is asymptotically nilpotent if and only there is q ∈ Q such that for any μ ∈ M(QG) it holds ${F^t(\mu)\to\_t\delta\_{\overline{q}}}$. The property on measures implies asymptotic nilpotency by the same argument as the first part of the proof of Lemma [lem:convergencemeasure] with the additional fact that ${F^\omega(\delta\_c)=\delta\_{\overline{q}}}$ which implies that ${F^\omega(c)=\overline{q}}$. Conversely, if we suppose that F is asymptotically nilpotent, we can use the same reasoning as in part two of the proof of Lemma [lem:convergencemeasure]. The additional fact in this case is that ${F^\omega(c)=\overline{q}}$ for all c ∈ QG so that F− t([u]) ∩ Xt = ∅ for all pattern u except the uniform one: z ∈ Bn ↦ q. We deduce that Ft(μ) converges towards ${\delta\_{\overline{q}}}$. It turns out that, under some conditions on G, nilpotency and asymptotic nilpotency are equivalent. One might believe that this can be easily proved by a standard compacity argument, but it is not the case. [][theo:asnilimpnil] If G = Zd then nilpotency is equivalent to asymptotic nilpotency. [sketch] Let q be the quiescent state involved in the asymptotic nilpotency property. First, we remark that for any ball Bn there is a uniform bound Tn such that in the orbit of any configuration c there is some time step t ≤ Tn for which the cells inside Bn are all in state q: ∀z ∈ Bn, Ft(c)z = q. Indeed, otherwise we would obtain by compacity a configuration whose orbit would not converge to $\overline{q}$. Call finite any configuration which is everywhere q except on a finite region, and call mortal any configuration c such that there is t with ${F^t(c)=\overline{q}}$. Let us first establish the theorem for G = Z as in. (-1,0)–(6,0); (-.5,-.5)– node[midway,sloped,above] time (-.5,4); (2,0)–(2.5,1)–(2.7,0)–cycle; (1.5,0)–(1.8,0)–(2.5,1.5)–(3.5,0)–(4,0)–(2.3,2)–cycle; (.5,0)–(1,0)–(2.3,2.5)–(4.2,0)–(5,0)–(2.3,3)–cycle; (-.5,0)–(.2,0)–(2.3,3.2)–(5.2,0)–(5.5,0)–(2.3,4)–cycle; (2.3,-.25) node c0; (3.5,-.25) node c1; (4.75,-.25) node c2; (5.2,-.25) node[right] c3 ⋯; [fig:nested] There are two key arguments: * first, we show nilpotency on finite mortal configurations, i.e. there is a uniform T such that for all finite mortal c it holds ${F^T(c)=\overline{q}}$. This is proved using the remark above about times Tn: if there is no uniform time bound on the mortality of finite mortal configuration, then for arbitrary large Bn and arbitrary large t we can find a finite mortal configuration c such that c is uniformly q on Bn but Ft(c)0 ≠ q. From there, we can produce an infinite nested space-time diagram as in Figure [fig:nested] by taking a well-chosen sequence (ci) of such configurations and considering their superposition c: it is sufficient that the non-q portions of space-time diagrams are disjoint enough among the ci (taking into account the radius of the CA). The orbit of this configuration c gives a contradiction with the convergence hypothesis. * second, it can be shown that if the CA is not nilpotent, then we can extract finite mortal configurations that die arbitrarily late: indeed, by enclosing a finite configuration alive at time t with blocking words (whose existence is granted by Lemma [lem:convnotsens], see ), we obtain a mortal configuration still alive at time t − b where b is a constant depending only on the choice of blocking words. This contradicts the previous item, so we deduce that the CA is nilpotent. To extend the result to higher dimensions, it is enough to show that all finite configurations are mortal (this together with asymptotic nilpotency implies nilpotency, by the nested construction argument). To simplify we restrict to dimension 2 and consider any asymptotically nilpotent CA F. The idea is to reduce to the one-dimensional case by remarking that if a finite configuration c is periodized vertically to form a configuration cʹ, then cʹ is mortal because the action of F on cʹ can be seen as the action of a one-dimensional asymptotically nilpotent CA on a one-dimensional configuration. The key step of the proof is then to show that no finite configuration can spread too much in the horizontal direction. This step is proved by contradiction using again a nested construction of an infinite space-time diagram, but this time a vertical periodization is applied at each step to guarantee mortality. Of course, this argument on the bounded horizontal spreading of finite configurations is also valid vertically, so we deduce that the orbit of any finite configuration remains inside a finite support forever. This is enough to conclude mortality since F is supposed asymptotically nilpotent. The above result has been greatly generalized by in at least two ways: the class of dynamical systems considered (for instance, one can consider CA defined over a subshift rather than the whole space QG) and the class of groups G considered. To obtain these results, a notion of tiered dynamical systems was developed which goes way beyond the scope of this tutorial. The authors also coined the term nil-rigidity and used it with various classes of dynamical systems. Let us stick to the simplest setting and say that a group G is nilrigid if any asymptotically nilpotent CA on QG is nilpotent. Despite the many results obtained about nilrigidity in they left some interesting open questions (see section 11), some of which where already asked in. [] Is there any finitely generated group G which is not nilrigid? In particular, is the free group with 2 or more generators nilrigid? Variations with partial convergence and measures ------------------------------------------------ Let us now consider variations of these strong notions of nilpotency. We can first consider a similar property of orbits but restricted to a subset of initial configurations. Let us begin by two properties related to dense subsets of configurations (at least when G = Z2). A configuration is q-finite for some state q if it is everywhere q except on a finite region of G. A configuration x ∈ QG is totally periodic if its orbit by translation OG(x) is finite. When G = Zd a totally periodic configuration is a configuration with d non-colinear periods. [def:nilper] * F is nilpotent on periodic configurations if there is a configuration c such that for any totally periodic configuration x there is t with Ft(x) = c. * F is nilpotent on finite configurations if there is a state q such that for any q-finite configuration x there is t with ${F^t(x)=\overline{q}}$. Note again that the attracting configuration c must be uniform in the first definition. Nilpotency on finite configurations is also sometimes called eroder property (see Section [sec:related]). An asymptotically nilpotent CA must be nilpotent on periodic configurations and nilpotent on finite configurations, but the converse is far from being true as we will see below. As a second (seemingly unrelated) variation around nilpotency, we can change the point of view and switch from configurations to measures. A measure μ ∈ M(QG) is invariant for a CA on QG if F(μ) = μ. The following definition is classical in dynamical system theory and can be defined as a rigidity property of invariant measures. [def:uniqergo] F is uniquely ergodic if it possesses a unique invariant measure. The invariant measures of F are exactly the limit points of Cesaro mean sequences (un)n ∈ N of the form () $$u\_n= \frac{1}{n}\sum\_{t=0}^{n-1}F^t\mu$$ for μ ∈ M(QG). Orbit-wise unique ergodicity is equivalent to the convergence of all such Cesaro mean sequences to the same limit. Therefore we could also name this property Cesaro-nilpotency to stress the dynamical side of it but it is best known as unique ergodicity. Note that by Lemma [lem:asynilmeasure], any asymptotically nilpotent CA is also uniquely ergodic. Nilpotency over periodic configurations and unique ergodicity are both implied by asymptotic nilpotency. The next lemma shows that the three properties actually form a hierarchy when G = Zd. A uniquely ergodic CA with G = Zd is nilpotent on periodic configurations. [lem:uniqergonilperio] Consider any uniquely ergodic CA F on QG. Suppose first that there are two disjoint temporal cycles of totally periodic configurations, i.e. two sequences of totally periodic configurations (c0, …, ck − 1) and (d0, …, dl − 1) with ${F(c\_i)= c\_{i+1\bmod k}}$ and ${F(d\_j)=d\_{j+1\bmod l}}$ for 0 ≤ i < k and 0 ≤ j < l. Then we can define two measures ${\mu\_c = \frac{1}{k}\sum\_{i=0}^{k-1}\delta\_{c\_i}}$ and ${\mu\_d = \frac{1}{l}\sum\_{j=0}^{l-1}\delta\_{d\_j}}$ that are both invariant under F because (ci) and (dj) are cyclic orbits. Moreover since the two cycles are disjoint there must exist a cylinder [u] such that ci ∈ [u] for some i but dj ∉ [u] whatever j. Thus μc([u]) > 0 and μd([u]) = 0 which proves that μc and μd are two distinct invariant measures: a contradiction. We have shown that F has the following synchronization property: there is a cycle of totally periodic configurations (c0, …, c*k − 1) such that the orbit of any totally periodic configuration eventually enters this cycle (ci).* We claim that k = 1 which concludes the proof since it means that F is nilpotent on periodic configurations. We are actually going to show the stronger claim that no CA F (not necessarily uniquely ergodic) with G = Zd can have the synchronization property above with k ≥ 2. The argument we give is due to G. Richard (see for more background on the synchronization problem). First note that if this synchronization property holds for some CA F with G = Zd and d > 1 then it must also hold for some CA with G = Z. Indeed, considering F on configurations which are constant in d − 1 directions and periodic in the last one, we actually define a one-dimensional CA which has the synchronization property as F does. We can thus suppose without loss of generality that d = 1. Clearly, each configuration ci in the attracting cycle must be uniform because any uniform configuration has an orbit eventually entering this cycle. However, if the synchronization property holds with k ≥ 2 then F has no quiescent state. Now consider f : Qm → Q the local map of F and let w ∈ QN be a semi-infinite word verifying wn + m = f(wn, …, wn + m − 1) for all n ∈ N. There must exist n1 and n2 with n1 + m < n2 such that w[n1, n1 + m − 1] = w[n2, n2 + m − 1]. Consider the word w = w[n1, n2 − 1] of length at least m + 1. By construction and since F has no quiescent state w contains at least two distinct letters from Q. Moreover, if c is a periodic Z-configuration of period w then F(c) = σp(c) for some p. Therefore the orbit of c is a cycle of non-uniform configurations, which contradicts the synchronization property. Interestingly, we can also capture unique ergodicity as a property of traces of orbits on configurations: it is the property that there is some attracting state whose frequency goes to 1 when considering larger and larger prefixes of any trace. To be more precise, for any q ∈ Q and any word w ∈ QN we denote by dq(w) the (inferior) asymptotic density of occurrences of q in w: $$d\_q(w)=\liminf\_n \frac{\#\{0\leq i<n : w\_i=q\}}{n}.$$ We then get this equivalent formulation of unique ergodicity. (0,0) rectangle (3,1); (0,1)–node[midway,above] uniform freezing time (3,1); (4,0)–(7,0)–(7,1.5) to[out=225,in=315] (4,1.5) – cycle; (7,1.5) to[out=225,in=315] (4,1.5); (5.5,1.5) node non-uniform freezing time; (8,0) rectangle (11,2); (9.5,2.5) node no freezing time; (-.25,-.25)– node[midway,below] space +(1,0); (-.25,-.25)– node[midway,sloped,above] time +(0,1); (1.5,-1) node Nilpotent; [shift=(4,0)] (-.25,-.25)– node[midway,below] space +(1,0); (-.25,-.25)– node[midway,sloped,above] time +(0,1); (1.5,-1) node Asymptotically nilpotent; [shift=(8,0)] (-.25,-.25)– node[midway,below] space +(1,0); (-.25,-.25)– node[midway,sloped,above] time +(0,1); (1.5,-1) node Uniquely ergodic; [fig:nilorbits] [lem:traceuniqergo] A CA F on QG with G = Zd is uniquely ergodic if and only if there is q ∈ Q such that for any x ∈ QG it holds dq(TxF) = 1. Suppose first that F is uniquely ergodic, then by Lemma [lem:uniqergonilperio] it is nilpotent on periodic configurations and has a (unique) quiescent state q and its unique invariant measure is $\delta\_{\overline{q}}$. Consider any x ∈ QG and any ε > 0. If there were infinitely many n ∈ N such that ${\frac{\#\{0\leq i<n : {\mathcal{T}^{F}\_{x}}(i)\neq q\}}{n}>\epsilon}$, then we could extract a limit point μ from the sequence ${\bigl(\frac{1}{n}\sum\_{t=0}^{n-1}F^t(\delta\_x)\bigr)\_n}$ which would be an invariant measure such that μ([q]) < 1 − ε (recall that Ft(δx) = δFt(x)) hence different from $\delta\_{\overline{q}}$: a contradiction. We deduce that dq(TxF) = 1. Suppose now that dq(TxF) = 1 for all x ∈ QG for some q ∈ Q. Then q must be a quiescent state and thus the measure $\delta\_{\overline{q}}$ is invariant and also ergodic. If it were not the unique invariant measure then there would be another invariant ergodic measure μ (see ), which would necessarily verify μ([g ↦ q]) < 1 for some g ∈ G in order to differ from ${\delta\_{\overline{q}}}$ (otherwise the Choquet’s decomposition theorem on invariant measures would be contradicted). Note that μ is not necessarily translation-invariant, but translating it by g gives another F-invariant ergodic measure so we can actually suppose g = 0G so μ([q]) < 1. Then by the ergodic Theorem (see ), we would have some x ∈ QG with $$\frac{1}{n}\sum\_{t=0}^{n-1}f\bigl(F^t(x)\bigr)\to\_n \mu([q])$$ where f is the characteristic map of [q]. But then dq(TxF) < 1 which contradicts the hypothesis. Uniquely ergodic CA form a strictly larger class than nilpotent CA. The example in the following result to show this separation is highly non-trivial and we will not present it here, but it is quite remarkable that such behaviors are possible with CA. [] [theo:nilpernotue] There exists a uniquely ergodic CA on Z which is not nilpotent. The CA from the above theorem cannot be convergent because then it would be asymptotically nilpotent by Lemma [lem:traceuniqergo] and this would contradict Theorem [theo:asnilimpnil]. From Lemma [lem:convergencemeasure] it is also non-convergent when starting from some measures. However, there is convergence in Cesaro mean starting from any μ since otherwise, a Cesaro mean sequence would have several limit points and hence produce several invariant measure for the CA as already explained above. There exists a CA on Z which is nilpotent over periodic configurations but not uniquely ergodic. [Sketch of proof] Such a CA can be obtained using the classical construction of using Wang tiles, which formalizes some tiling of the plane Z2. Roughly, a Wang tileset is a finite set of tile types X together with vertical and horizontal constraints specifying which pairs of adjacent tile types is allowed horizontally (resp. vertically). A valid tiling is a configuration of XZ2 that satisfies the constraints everywhere. A tileset is aperiodic if it admits valid tilings, but no periodic ones. A tileset is NE-deterministic when the constraints have the following property: when fixing a tile at the north of a position and another one at the east, then there is at most one possible tile type that can be placed at the north-east without violating the local horizontal and vertical constraints. Taking a NE-deterministic aperiodic Wang tileset T (whose existence is proven in ) one can transform it into a one-dimensional CA F with alphabet T ∪ {s}, where s is a spreading state, and neighborhood {0, 1} as follows: if a cell and its right neighbor hold a pair of state from T that represent two tiles (one at the north, one at the east) that can be uniquely completed to t according to the NE-deterministic rule, then the cell turns into state t, otherwise it turns into state s (configurations of the CA represent diagonals of a tiling). One obtains a CA F that admits ${\delta\_{\overline{s}}}$ as invariant measure. Because the tileset has no valid periodic tiling, F is nilpotent on periodic configurations (the orbit of a periodic configuration without occurrence of s would give a periodic tiling of T, which is impossible, and any occurrence of s in a periodic configuration forces its orbit to converge towards $\overline{s}$). On the other hand, because the tileset admits a valid (aperiodic) tiling, it means that there is a configuration c whose orbit under F has no occurrence of s. Taking any limit point of the sequence of Cesaro means $$\mu\_n=\frac{1}{n}\sum\_{t=0}^{n-1}F^t(\delta\_c)$$ gives an invariant measure such that μ([s]) = 0 since μn([s]) = 0 for all n ∈ N. We thus get a second invariant measure for F. So far we considered properties concerning all initial configurations, or a meager set of initial configuration. Natural variants of nilpotency have been considered in the literature to capture the behavior on most or a large set of initial configurations. Two notions in particular emerged in literature that use different approach to define precisely most or large set: one is topological, the other uses measure theory. Recall that a set in a topological space is comeager (intuitively, large) if it is a countable intersection of sets with dense topological interior (for instance a countable intersection of dense open sets). Consider a CA F on QG and μ ∈ M(QG). Then F is: * generically nilpotent if there is q ∈ Q and a comeager set B ⊆ QG such that ${F^t(x)\to\_t\overline{q}}$ for all x ∈ B; * *μ-nilpotent* if there is q ∈ Q such that ${F^t(\mu)\to\_t\delta\_{\overline{q}}}$. Generic nilpotency and μ-nilpotency both specify a behavior on most configurations, where most is to be understood either topologically or according to a measure. It turns out that for well-behaved measures the topological version is stronger. [][theo:genmunil] Let F be a generically nilpotent CA on QG and μ ∈ M(QG) a full-support translation-ergodic measure. Then F is μ-nilpotent. [fig:nildiag] [scale=.7,node distance=1cm,>=stealth] =[draw, rectangle] (-2,1.25) rectangle (3,1.3); (-2,-1.25) rectangle (3,-1.3); (0,2.5) node ; (0,0) node ; (0,-2.5) node ; (12.5,2.5) node[nilprop] \| \| \| --- \| \| nilpotent over \| \| periodic \| \| configurations \| ; (8,2.5) node[nilprop] \| \| \| --- \| \| nilpotent over \| \| finite \| \| configurations \| ; (7.75,.75) – (7.75,1.55); (8.25,1.55) – node[thin,cross out,draw=black,minimum size=2mm] (8.25,.75); (9.85,2.5) – node[thin,cross out,draw=black,minimum size=2mm] (10.65,2.5); (12.25,.75) – node[midway] (UEnilper) (12.25,1.55); (12.75,1.55) – node[pos=0.4] (Nilpernoteu) (12.75,.75); (UEnilper)++(-.3,0) node[left] Lem. [lem:uniqergonilperio]; (Nilpernoteu) node[cross out,draw=black,minimum size=2mm] ; (Nilpernoteu)++(.3,0) node[right] Thm. [theo:nilpernotue]; (3,0) node[nilprop] nilpotent; (4.3,.25) – (6,.25); (6,-.25) – node[midway] (Nil) (4.3,-.25); (Nil)++(0,-.3) node Thm. [theo:asnilimpnil]; (11,-.25) – node[midway] (UEnotnil) (10,-.25); (10,.25) – node[midway,above] Lem. [lem:asynilmeasure] (11,.25); (UEnotnil) node[cross out,draw=black,minimum size=2mm] ; (UEnotnil)++(0,-.3) node Thm. [theo:nilpernotue]; (8,0) node[nilprop] \| \| \| --- \| \| asymptotically \| \| nilpotent \| ; (12.5,0) node[nilprop] \| \| \| --- \| \| uniquely \| \| ergodic \| ; (8,-2.5) node[nilprop] \| \| \| --- \| \| generically \| \| nilpotent \| ; (12.5,-2.5) node[nilprop] μ-nilpotent; (7.75,-0.75) – (7.75,-1.55); (8.25,-1.55) – node[pos=0.4] (Gennoasnil) (8.25,-0.75); (Gennoasnil) node[cross out,draw=black,minimum size=2mm] ; (10,-2.25) – node[midway] (Genmunil) (11,-2.25); (Genmunil)++(0,.3) node Thm. [theo:genmunil]; (11,-2.75) – node[midway] (Munotgen) (10,-2.75); (Munotgen) node[cross out,draw=black,minimum size=2mm] ; (Munotgen)++(0,-.3) node Thm. [theo:classicbootstrap]; [fig:nilimp] There are ways to characterize the previous variants of nilpotency by defining variants of limit sets, i.e. sets of (typical) asymptotic configurations. Each time, the nilpotency variant is equivalent to having such asymptotic sets being singletons (see ). The implication diagram of Figure [fig:nilimp] is not complete, in particular concerning the link between unique ergodicity and μ-nilpotency. What are the other implications or counter-examples in the diagram of Figure [fig:nilimp]? In particular, for what measure μ does unique ergodicity imply μ-nilpotency? Said differently, for what measure μ the convergence in Cesaro mean implied by unique ergodicity is actually a simple convergence? The undecidability garden ------------------------- We conclude this section by undecidability results related to the notions presented above. Of course, an undecidability result is a negative result and this might appear repulsive to some. However, behind almost any undecidability result there is a positive one: a construction technique (a reduction) providing examples with rich and complex behaviors. Each of the results below relies on a non-trivial and nice construction idea that is of independent interest. A presentation of each technique would take too much space for this tutorial, but we invite the reader to dig into the corresponding references. The following theorem follows from the undecidability of the domino problem on deterministic sets of Wang tiles. Several proof techniques are known and we suggest to read for a survey. [] The set of 1D nilpotent CA is Σ10-complete. As for the existence of a uniquely ergodic CA which is not nilpotent (Theorem [theo:nilpernotue]), the key tool in the following result is a self-simulating CA inspired from in which the only orbits than can survive produce sparser and sparser computations. [] The set of 1D uniquely ergodic CA is Π20-complete. The two following results use a construction technique introduced in which has proven to be useful when one wants to control the behavior of a CA on most initial configuration: it relies on a special precursor state that generates well-behaved zones that grow in a random context until they meet another well-behaved zone. Each zone can hold a properly initialized Turing computation. Then, a global process of merging or destruction of meeting zones is used to ensure that a least some Turing computations can go on forever. [] The set of 1D generically nilpotent CA is Σ20-complete. Recall from Theorem [theo:genmunil] that a generically nilpotent CA is necessarily μ-nilpotent for μ the uniform Bernoulli measure (which is ergodic and has full support). The converse is false and the following theorem illustrates how the weaker property of μ-nilpotence is strictly harder to decide. [] Let μ denote the uniform Bernoulli measure. The set of 1D μ-nilpotent CA is Π30-complete. Proving Convergence =================== Bounded change CA ----------------- Proving convergence of a CA F means proving that there exists a time bound for each trace TcF after which the trace is constant. This has been defined above as the freezing time ζF(c, 0). In general this time bound depends on the initial configuration (see Example [exa:zigzag]), but, even if there is a global bound that doesn’t depend on initial configuration like in nilpotent CA, it is generally uncomputable. Instead of bounding directly the time of the last change in traces, a way to tackle the problem is to bound the number of changes that can occur in a trace. The following definition is useful. [def:boundedchange] A CA F is k-change for some k ∈ N if for any x ∈ QG, there are at most k state changes in TxF, i.e. #{t ∈ N : TxF(t + 1) ≠ TxF(t)} ≤ k. A CA is bounded-change if it is k-change for some k. A bounded-change CA is necessarily convergent: a trace which is not ultimately constant cannot exist since it would contain infinitely many changes. However the converse is false as shown by the following example (it is used extensively as a basic building block in ). [exa:zigzag] Let us consider the state set ${Q=\{{\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}}},{\tikz[scale=.3]{{\draw[fill=white!50!gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\to$};}}},{\tikz[scale=.3]{{\draw[fill=gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\leftarrow$};}}},{\tikz[baseline,scale=.3]{{\draw[fill=white!50!red] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $<$};}}},{\tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}}},e\}}$ where ${\tikz[baseline,scale=.3]{{\draw[fill=white!50!red] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $<$};}}}$ and ${\tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}}}$ represent an active head moving left or right respectively, ${\tikz[scale=.3]{{\draw[fill=white!50!gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\to$};}}}$ and ${\tikz[scale=.3]{{\draw[fill=gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\leftarrow$};}}}$ are states indicating the position of the head (to the right and to the left respectively), ${\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}}}$ is a blank state and e is an error state. Let Σ be the set of configurations whose patterns of length two all appear in the space-time diagram of Figure [fig:convnotbc], i.e. configurations made of zones with a unique active head (possibly none if the zone is infinite) in a blank background. Define the CA F : QZ → QZ of radius 2, such that: * e is a spreading state and any cell turns into state e if there is some pattern not in L2 in its neighborhood, * in the Σ-valid zones with a unique head, the only cells that change their states in one step are the one around the head, and they update in such a way that the head makes zigzag inside the zone and reduces the size of the zone by one unit at each bounce on a border. Precisely, all the transitions appear in the space-time diagram of Figure [fig:convnotbc]. First, it is clear that F cannot be k-change whatever the value of k is, because in a Σ-valid finite zone of size n, the orbit of the central cells contains at least n changes. On the other hand, if we consider any initial configuration c ∈ QZ and any position z ∈ Z there are only 3 possible cases: * either c ∉ Σ and in this case the spreading state e will appear somewhere and reach position z at some step, * or c ∈ Σ and z belongs to a finite valid zone, so after some number of zigzags of the head, the zone will shrink letting z outside in state ${\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}}}$, * or c ∈ Σ and z belongs to an infinite valid zone, and after at most two passages of the head (at most one bounce on the boundary if any), the head will move towards infinity and cell z will remain unchanged forever. In any case, we have that the trace at z starting from c is eventually constant. Hence F is convergent. [scale=.3,baseline=(current bounding box.center)] (5,0) rectangle +(1,1); (5,0)+(.5,.5) node B;(5,1) rectangle +(1,1); (5,1)+(.5,.5) node B;(5,2) rectangle +(1,1); (5,2)+(.5,.5) node B;(5,3) rectangle +(1,1); (5,3)+(.5,.5) node B;(5,4) rectangle +(1,1); (5,4)+(.5,.5) node B;(5,5) rectangle +(1,1); (5,5)+(.5,.5) node B;(5,6) rectangle +(1,1); (5,6)+(.5,.5) node B;(5,7) rectangle +(1,1); (5,7)+(.5,.5) node B;(5,8) rectangle +(1,1); (5,8)+(.5,.5) node B;(5,9) rectangle +(1,1); (5,9)+(.5,.5) node B;(5,10) rectangle +(1,1); (5,10)+(.5,.5) node B;(5,11) rectangle +(1,1); (5,11)+(.5,.5) node B;(5,12) rectangle +(1,1); (5,12)+(.5,.5) node B;(5,13) rectangle +(1,1); (5,13)+(.5,.5) node B;(5,14) rectangle +(1,1); (5,14)+(.5,.5) node B;(5,15) rectangle +(1,1); (5,15)+(.5,.5) node B;(5,16) rectangle +(1,1); (5,16)+(.5,.5) node B;(5,17) rectangle +(1,1); (5,17)+(.5,.5) node B;(5,18) rectangle +(1,1); (5,18)+(.5,.5) node B;(5,19) rectangle +(1,1); (5,19)+(.5,.5) node B; (6,0) rectangle +(1,1); (6,0)+(.5,.5) node B;(6,1) rectangle +(1,1); (6,1)+(.5,.5) node B;(6,2) rectangle +(1,1); (6,2)+(.5,.5) node B;(6,3) rectangle +(1,1); (6,3)+(.5,.5) node B;(6,4) rectangle +(1,1); (6,4)+(.5,.5) node B;(6,5) rectangle +(1,1); (6,5)+(.5,.5) node B;(6,6) rectangle +(1,1); (6,6)+(.5,.5) node B;(6,7) rectangle +(1,1); (6,7)+(.5,.5) node B;(6,8) rectangle +(1,1); (6,8)+(.5,.5) node B;(6,9) rectangle +(1,1); (6,9)+(.5,.5) node B;(6,10) rectangle +(1,1); (6,10)+(.5,.5) node B;(6,11) rectangle +(1,1); (6,11)+(.5,.5) node B;(6,12) rectangle +(1,1); (6,12)+(.5,.5) node B;(6,13) rectangle +(1,1); (6,13)+(.5,.5) node B;(6,14) rectangle +(1,1); (6,14)+(.5,.5) node B;(6,15) rectangle +(1,1); (6,15)+(.5,.5) node B;(6,16) rectangle +(1,1); (6,16)+(.5,.5) node B;(6,17) rectangle +(1,1); (6,17)+(.5,.5) node B;(6,18) rectangle +(1,1); (6,18)+(.5,.5) node B;(6,19) rectangle +(1,1); (6,19)+(.5,.5) node B; (7,0) rectangle +(1,1);(7,0)+(.5,.5) node → ;(7,1) rectangle +(1,1);(7,1)+(.5,.5) node → ;(7,2) rectangle +(1,1); (7,2)+(.5,.5) node < ;(7,3) rectangle +(1,1); (7,3)+(.5,.5) node B;(7,4) rectangle +(1,1); (7,4)+(.5,.5) node B;(7,5) rectangle +(1,1); (7,5)+(.5,.5) node B;(7,6) rectangle +(1,1); (7,6)+(.5,.5) node B;(7,7) rectangle +(1,1); (7,7)+(.5,.5) node B;(7,8) rectangle +(1,1); (7,8)+(.5,.5) node B;(7,9) rectangle +(1,1); (7,9)+(.5,.5) node B;(7,10) rectangle +(1,1); (7,10)+(.5,.5) node B;(7,11) rectangle +(1,1); (7,11)+(.5,.5) node B;(7,12) rectangle +(1,1); (7,12)+(.5,.5) node B;(7,13) rectangle +(1,1); (7,13)+(.5,.5) node B;(7,14) rectangle +(1,1); (7,14)+(.5,.5) node B;(7,15) rectangle +(1,1); (7,15)+(.5,.5) node B;(7,16) rectangle +(1,1); (7,16)+(.5,.5) node B;(7,17) rectangle +(1,1); (7,17)+(.5,.5) node B;(7,18) rectangle +(1,1); (7,18)+(.5,.5) node B;(7,19) rectangle +(1,1); (7,19)+(.5,.5) node B; (8,0) rectangle +(1,1);(8,0)+(.5,.5) node → ;(8,1) rectangle +(1,1); (8,1)+(.5,.5) node < ;(8,2) rectangle +(1,1);(8,2)+(.5,.5) node ← ;(8,3) rectangle +(1,1); (8,3)+(.5,.5) node > ;(8,4) rectangle +(1,1);(8,4)+(.5,.5) node → ;(8,5) rectangle +(1,1);(8,5)+(.5,.5) node → ;(8,6) rectangle +(1,1);(8,6)+(.5,.5) node → ;(8,7) rectangle +(1,1);(8,7)+(.5,.5) node → ;(8,8) rectangle +(1,1);(8,8)+(.5,.5) node → ;(8,9) rectangle +(1,1);(8,9)+(.5,.5) node → ;(8,10) rectangle +(1,1);(8,10)+(.5,.5) node → ;(8,11) rectangle +(1,1); (8,11)+(.5,.5) node < ;(8,12) rectangle +(1,1); (8,12)+(.5,.5) node B;(8,13) rectangle +(1,1); (8,13)+(.5,.5) node B;(8,14) rectangle +(1,1); (8,14)+(.5,.5) node B;(8,15) rectangle +(1,1); (8,15)+(.5,.5) node B;(8,16) rectangle +(1,1); (8,16)+(.5,.5) node B;(8,17) rectangle +(1,1); (8,17)+(.5,.5) node B;(8,18) rectangle +(1,1); (8,18)+(.5,.5) node B;(8,19) rectangle +(1,1); (8,19)+(.5,.5) node B; (9,0) rectangle +(1,1); (9,0)+(.5,.5) node < ;(9,1) rectangle +(1,1);(9,1)+(.5,.5) node ← ;(9,2) rectangle +(1,1);(9,2)+(.5,.5) node ← ;(9,3) rectangle +(1,1);(9,3)+(.5,.5) node ← ;(9,4) rectangle +(1,1); (9,4)+(.5,.5) node > ;(9,5) rectangle +(1,1);(9,5)+(.5,.5) node → ;(9,6) rectangle +(1,1);(9,6)+(.5,.5) node → ;(9,7) rectangle +(1,1);(9,7)+(.5,.5) node → ;(9,8) rectangle +(1,1);(9,8)+(.5,.5) node → ;(9,9) rectangle +(1,1);(9,9)+(.5,.5) node → ;(9,10) rectangle +(1,1); (9,10)+(.5,.5) node < ;(9,11) rectangle +(1,1);(9,11)+(.5,.5) node ← ;(9,12) rectangle +(1,1); (9,12)+(.5,.5) node > ;(9,13) rectangle +(1,1);(9,13)+(.5,.5) node → ;(9,14) rectangle +(1,1);(9,14)+(.5,.5) node → ;(9,15) rectangle +(1,1);(9,15)+(.5,.5) node → ;(9,16) rectangle +(1,1); 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(14,0)+(.5,.5) node B;(14,1) rectangle +(1,1); (14,1)+(.5,.5) node B;(14,2) rectangle +(1,1); (14,2)+(.5,.5) node B;(14,3) rectangle +(1,1); (14,3)+(.5,.5) node B;(14,4) rectangle +(1,1); (14,4)+(.5,.5) node B;(14,5) rectangle +(1,1); (14,5)+(.5,.5) node B;(14,6) rectangle +(1,1); (14,6)+(.5,.5) node B;(14,7) rectangle +(1,1); (14,7)+(.5,.5) node B;(14,8) rectangle +(1,1); (14,8)+(.5,.5) node B;(14,9) rectangle +(1,1); (14,9)+(.5,.5) node B;(14,10) rectangle +(1,1); (14,10)+(.5,.5) node B;(14,11) rectangle +(1,1); (14,11)+(.5,.5) node B;(14,12) rectangle +(1,1); (14,12)+(.5,.5) node B;(14,13) rectangle +(1,1); (14,13)+(.5,.5) node B;(14,14) rectangle +(1,1); (14,14)+(.5,.5) node B;(14,15) rectangle +(1,1); (14,15)+(.5,.5) node B;(14,16) rectangle +(1,1); (14,16)+(.5,.5) node B;(14,17) rectangle +(1,1); (14,17)+(.5,.5) node B;(14,18) rectangle +(1,1); (14,18)+(.5,.5) node B;(14,19) rectangle +(1,1); (14,19)+(.5,.5) node B; [c]0.5 $$\begin{aligned} \Sigma &= \bigl\{c\in Q^{\mathbb{Z}}: \forall z\in{\mathbb{Z}}, c\_zc\_{z+1}\in L\_2\bigr\}\\ L\_2 &= \bigl\{\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}{\draw[fill=white] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $B$};}},\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}{\draw[fill=white!50!blue] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $>$};}},\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}{\draw[fill=white!50!red] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $<$};}},\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}{\draw[fill=white!50!gray] (1,0) rectangle +(1,1);\draw (1,0)+(.5,.5) node {\tiny $\to$};}},\\ &\tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}{\draw[fill=white] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $B$};}},\tikz[baseline,scale=.3]{{\draw[fill=white!50!red] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $<$};}{\draw[fill=white] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $B$};}},\tikz[baseline,scale=.3]{{\draw[fill=white!50!gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\to$};}{\draw[fill=white!50!blue] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $>$};}},\tikz[baseline,scale=.3]{{\draw[fill=white!50!gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\to$};}{\draw[fill=white!50!red] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $<$};}},\\ & \tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}{\draw[fill=gray] (1,0) rectangle +(1,1);\draw (1,0)+(.5,.5) node {\tiny $\leftarrow$};}},\tikz[baseline,scale=.3]{{\draw[fill=white!50!red] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $<$};}{\draw[fill=gray] (1,0) rectangle +(1,1);\draw (1,0)+(.5,.5) node {\tiny $\leftarrow$};}},\tikz[baseline,scale=.3]{{\draw[fill=gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\leftarrow$};}{\draw[fill=gray] (1,0) rectangle +(1,1);\draw (1,0)+(.5,.5) node {\tiny $\leftarrow$};}},\tikz[baseline,scale=.3]{{\draw[fill=gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\leftarrow$};}{\draw[fill=white] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $B$};}}\bigr\} \end{aligned}$$ [fig:convnotbc] To use a metaphor inspired from physics, each cell of a bounded-change CA can be seen as a system that evolves according to information received from its neighbors, and for which each state change has a fixed energy cost. Then, in each orbit and at each cell, the instantaneous energy defined as the number of remaining potential state changes is non-increasing with time, and strictly decreasing at each state change. The number k in Definition [def:boundedchange] can be interpreted as the cell-wise capacity, i.e. the maximal energy an individual cell can hold initially. As seen in the previous section, a nilpotent CA F is such that Ft is a constant map for some t, so it is always bounded-change. Nilpotency for CA with a spreading state is actually Turing reducible to the property of bounded-change as follows: consider F with a spreading state and add a {0, 1} component to states that constantly exchanges 0 and 1, except if there is a spreading state in the neighborhood in which case the additional component turns into 0. This new CA Fʹ is bounded-change if and only if F is nilpotent: first, if F is nilpotent then Fʹ also because the spreading state of F forces 0 everywhere on the second component; conversely, if F is not nilpotent then it must possess an orbit without any occurrence of the spreading state (by compacity), and then the corresponding orbit in Fʹ is constantly changing the state on the second layer. By undecidability of nilpotency we deduce the following theorem. [theo:bcundecidable] It is undecidable to determine whether a 1D CA is bounded-change or not. Despite the general undecidability, this approach turns out to be very fruitful to analyze some majority CA. A majority cellular automaton consists in taking in each cell the state which has the majority of occurrences among the neighboring cells. The majority CA with neighborhood V ⊆ G is defined on alphabet Q = {0, 1} by $$F(x)\_z = \begin{cases} 1 & \text{ if $\displaystyle\sum\_{i\in V} x\_{z+i} >\#V/2$, or $\displaystyle\sum\_{i\in V} x\_{z+i} =\#V/2$ and $x\_z=1$}\\ 0 & \text{ otherwise.} \end{cases}$$ A subset S ⊆ Zd is symmetric if ${x\in S\implies -x\in S}$. The following result concerning majority CA is a specific case of a much more general result from which deals with infinite graphs with some growth condition. A similar qualitative behavior for majority evolution rules was established before in the settings of automata networks (the lattice of cells is finite, the evolution rule is possibly non-uniform), see. It is based on a decreasing energy function in both case. Note however that do not state their main theorem as a bounded change property but only as a convergence result, although their proof technique allows to establish bounded-changedness as shown below. .2 ![image](maj0.png) c .2 ![image](maj2.png) F2(c) .2 ![image](maj4.png) F4(c) .2 ![image](maj6.png) F6(c) 2em .2 ![image](maj8.png) F8(c) ⋯ .2 ![image](maj_infty.png) $\bigl(F^2\bigr)^\omega(c)$ [fig:maj2D] [] [theo:majboundedchange] Let F be any majority CA with G = Zd and symmetric neighborhood S with 0 ∈ S, then F2 is bounded-change. Denote by r the radius of the CA, i.e. the maximum ∥u∥∞ for u ∈ S. Let us fix any x ∈ QG. The core of the argument is to consider the following “energy” function for any x ∈ QG and any n, t ∈ N: $$E\_n^t = \sum\_{u\in{B\_{n}}}\sum\_{v\in{B\_{n}}}a(u,v)\bigl\|F^{t+1}(x)\_u-F^t(x)\_v\bigr\|$$ where a(u, v) = 1 if v ∈ u + S, and 0 else. Because S is symmetric a(u, v) = a(v, u) so we can write Ent − 1 as follows: $$E\_n^{t-1} = \sum\_{u\in{B\_{n}}}\sum\_{v\in{B\_{n}}}a(u,v)\bigl\|F^{t-1}(x)\_u-F^t(x)\_v\bigr\|.$$ For any t > 0, we will first give an upper bound on Δnt = Ent − Ent − 1 which intuitively means that the energy Ent essentially decreases with time, but only up to a “controlled perturbation” of the border of the ball Bn. Denoting $$\Sigma\_{n,t}^+(u) = \sum\_{v\in{B\_{n}}}a(u,v)\bigl\|F^{t+1}(x)\_u-F^t(x)\_v\bigr\|,\text{ and}$$ $$\Sigma\_{n,t}^-(u) = \sum\_{v\in{B\_{n}}}a(u,v)\bigl\|F^{t-1}(x)\_u-F^t(x)\_v\bigr\|$$ we can rewrite Δnt as Δnt = ∑0 ≤ i ≤ n − r∑u ∈ Ci, tΣn, t+ − Σn, t− + ∑n − r < i ≤ n∑u ∈ Ci, tΣn, t+ − Σn, t− where Ci, t denote the cells that have different states between step t + 1 and t − 1 (i.e. Ft + 1(x)u ≠ Ft − 1(x)u) and belong to the sphere of radius i (i.e. ∥u∥∞ = i). This expression for Δnt holds because Σn, t+(u) = Σn, t−(u) when Ft + 1(x)u = Ft − 1(x)u. We first claim that ${\Sigma\_{n,t}^+(u)-\Sigma\_{n,t}^-(u)\leq \frac{\#S -1}{2}}$ for any u ∈ Bn: this holds because actually ${\Sigma\_{n,t}^+(u)\leq \frac{\#S -1}{2}}$ must hold to ensure that a majority of neighbors of u are in state Ft + 1(x)u at time t. We now claim that Σn, t+(u) − Σn, t−(u) ≤ − 1 for any u ∈ ⋃0 ≤ i ≤ n − rCi, t. Indeed, we can split the neighbors of any such u into two sets Su+ = {v ∈ u + S : Ft(x)v = Ft + 1(x)u}, and Su− = {v ∈ u + S : Ft(x)v = Ft − 1(x)u}, and then #Su+ = Σn, t−(u) because u + S ⊆ Bn, and similarly #Su− = Σn, t+(u). By the majority rule we must have #Su+ > #Su− (recall that S is of odd cardinality by hypothesis) which proves the claim. Let ${s=\frac{(\#S -1)}{2}}$. From the two claims we can rewrite Equation ([eq:deltatwosums]) above as: Δnt ≤ − #⋃0 ≤ i ≤ n − rCi, t + s#⋃n − r < i ≤ nCi, t. We are now going to deduce a bound on the number of changes in TxF2 from the above inequality in two steps: using a weighted sum spatially and then summing over time. Spatially, the trick is to use a telescoping sum to attenuate the perturbation of the second term in the upper bound of Equation ([eq:deltafinal]) and put more weight on the central cell. Choose n = kr and consider the sum $$TE\_n^t = E\_n^t + \frac{1}{s} E\_{n-r}^t + (1+\frac{1}{s})\frac{1}{s} E\_{n-2r}^t + (1+\frac{1}{s})^2\frac{1}{s} E\_{n-3r}^t + \cdots + (1+\frac{1}{s})^{k-1}\frac{1}{s} E\_0^t.$$ The weights are chosen so that ${TE\_n^t-TE\_n^{t-1} = \Delta\_n^t + \frac{1}{s} \Delta\_{n-r}^t + \cdots}$ can be nicely upper-bounded. Indeed Equation ([eq:deltafinal]) gives $$\Delta\_n^t + \frac{1}{s} \Delta\_{n-r}^t\leq -(1+\frac{1}{s})\#\bigcup\_{0\leq i\leq n-2r}C\_{i,t} + s\#\bigcup\_{n-r< i\leq n}C\_{i,t}$$ so the changes in the annulus between spheres of radii n − 2r and n − r are exactly compensated. Then, when adding ${(1+\frac{1}{s})\frac{1}{s} \Delta\_{n-2r}^t}$ upper-bounded by Equation ([eq:deltafinal]), the terms involving annulus from n − 3r to n − 2r are exactly compensated again, giving: $$\Delta\_n^t + \frac{1}{s} \Delta\_{n-r}^t + (1+\frac{1}{s})\frac{1}{s} \Delta\_{n-2r}^t \leq -(1+\frac{1}{s})^2\#\bigcup\_{0\leq i\leq n-3r}C\_{i,t} + s\#\bigcup\_{n-r< i\leq n}C\_{i,t}.$$ Repeating this argument, and by the identity $$1 + \frac{1}{s} + (1+\frac{1}{s})\frac{1}{s} + \cdots + (1+\frac{1}{s})^{k-1}\frac{1}{s} = (1+\frac{1}{s})^k,$$ we get $$TE\_n^t - TE\_n^{t-1}\leq - (1+\frac{1}{s})^kC\_{0,t} + s\#\bigcup\_{n-r< i\leq n}C\_{i,t} \leq - (1+\frac{1}{s})^kC\_{0,t} + s\#({B\_{n}}\setminus{B\_{n-r}}).$$ By summing over time the successive differences we then obtain $$TE\_n^t-TE\_n^0\leq - (1+\frac{1}{s})^k\sum\_{i< t}C\_{0,i} + (t-1)s\#({B\_{n}}\setminus{B\_{n-r}})$$ so $$\sum\_{0\leq i<t}C\_{0,i} \leq \frac{TE\_n^0}{(1+\frac{1}{s})^k} + \frac{(t-1)s\#({B\_{n}}\setminus{B\_{n-r}})}{(1+\frac{1}{s})^k}.$$ When n grows, the first term of this upper bound converges to a constant independent of t and of x (from the definition of TEn0 because En0 is at most the square of the size of Bn), and the second term vanishes. This concludes the proof because it shows that ∑i = 0∞C0, i is bounded independently of x, and it counts the number of state changes in the trace TxF2 and the trace TF(x)F2. Let us show two counter-examples to Theorem [theo:majboundedchange] when hypothesis on either G or the symmetry of neighborhood are removed: 1. let G be the free group with 2 generators, V the symmetric neighborhood made of all generators and their inverse and element 0G, and denote by F the associated majority CA. Define the configuration cb for any b : N → {0, 1} by cgb = b(∥g∥G). Because we are in the free group with 2 generators, if g ∈ Bn \ Bn − 1 then g + V contains 3 elements in Bn + 1 \ Bn and V is of size 5. Therefore we have F(cb)g = b(∥g∥G + 1), and thus F(cb) = cbʹ with bʹ(n) = b(n + 1) for all n ∈ N. By choosing b such that b(i + 1)⋯b(i + 2k) = 0k1k for arbitrarily large k and for some i, we deduce that there is no p > 0 such that Fp is convergent. 2. for G = Z2, let us consider the non-symmetric neighborhood V = {(0, 0), (0, 1), (1, 0)} and let F be the associated majority CA. Define the configuration cb for any b : N → {0, 1} by c(x, y)b = b(x + y). Again, for any (x, y) ∈ Z2 there are two elements zʹ in (x, y) + V such that czʹb = b(x + y + 1, so F(cb)(x, y) = b(x + y + 1). We can conclude as above that there is no p > 0 such that Fp is convergent. Freezing CA ----------- The argument of Theorem [theo:majboundedchange] to prove the bounded-change property is not immediate and the property is generally undecidable, even when G = Z (Theorem [theo:bcundecidable]). In some cases, however, the structure of the local rule of a CA directly implies the bounded-change property. The so called freezing CA, introduced in, are an example. [def:freezing] A CA F on QG is freezing if there is some partial order (Q, ≤ ) on its state set such that for any c ∈ QG and any z ∈ G it holds: F(c)z ≤ cz. Clearly, in a freezing CA the number of changes at any cell in any orbit is bounded by the depth of the partial order, so all freezing CA are bounded-change. The freezing property can be tested efficiently by looking at the local transition table. Indeed, given any CA F, we can compute the canonical state change graph (Q, → ) defined by q1 → q2 if some transition changes state q1 into q2. Then one can check that F is freezing if and only if this graph is acyclic. Going back to our earlier metaphor, the partial order involved in the freezing property can be seen as a local non-increasing energy that serves as a certificate for the bounded change property. Of course, not all bounded-change CA are freezing: for instance F2, where F is some CA from the hypothesis of Theorem [theo:majboundedchange], is a bounded-change CA but it cannot be freezing since F and hence F2 commute with the transformation that permutes states 0 and 1. Thus, if transition 0 → 1 exists in the canonical state change graph of F2, then transition 1 → 0 also exists, which prevent any freezing order to satisfy Definition [def:freezing]. There are many example of freezing CA studied in the literature. A freezing CA F for some order ≤ can also be monotone: x ≤ y ⇒ F(x) ≤ F(y) where ≤ here denotes the cellwise extension of ≤ to configurations. Among these examples, the classical bootstrap percolation CA has been the starting point of a rich branch of percolation theory, which can be viewed as the study of the qualitative behavior of monotone freezing CA initialized on random Bernoulli configurations. This CA, denoted FB in the sequel, is defined on {0, 1}Z2 as follows: $$F\_B(c)\_z = \begin{cases} 0&\text{ if $c\_z=0$ or ${\#\bigl\{z'\in z+N:c\_{z'}=0\bigr\}\geq 2}$,}\\ 1&\text{ else.} \end{cases}$$ where N = {(1, 0), (0, 1), ( − 1, 0), (0, − 1)}. FB is not nilpotent on periodic configurations (it admits both $\overline{0}$ and $\overline{1}$ as fixed points), and a classical result of shows that it is μ-nilpotent for any full-support Bernoulli measure[3](#fn3). Actually, much more is known on this CA started from random initial configurations. A qualitatively very different behavior is obtained by a similar CA, denoted FO, and defined on {0, 1}Z2 as follows: $$F\_O(c)\_z = \begin{cases} 0&\text{ if $c\_z=0$ or $c\_{z+(1,0)}=0$ or $c\_{z+(1,1)}=0$,}\\ 1&\text{ else.} \end{cases}$$ FO, contrary to FB, is μ-nilpotent for some Bernoulli measures but not all. As a third example of monotone freezing CA with 2 states, let’s consider FM defined as follows: $$F\_M(c)\_z = \begin{cases} 0&\text{ if $c\_z=0$ or ${\#\bigl\{z'\in z+N:c\_{z'}=0\bigr\}\geq 3}$,}\\ 1&\text{ else.} \end{cases}$$ A classification of behaviors of all monotone freezing CA with 2 states started from random Bernoulli configurations is given in. The following theorem focuses on examples FB and FO to illustrate this classification. .24 ![image](b0.05_000400.png) FB with μ([0]) = 0.05 .24 ![image](d0.29_000400.png) FO with μ([0]) = 0.29 .24 ![image](d0.35_000400.png) FO with μ([0]) = 0.35 .24 ![image](freezing_majority.png) FM with μ([0]) = 0.8 [fig:bootstrap] [theo:classicbootstrap] FB is generically nilpotent, and μ-nilpotent for any translation-ergodic measure μ of full support. However, there is a threshold 0 < θ < 1 such that FO is μ-nilpotent for Bernoulli measure μ when μ([0]) > θ and not μ-nilpotent if μ([0]) < θ. In particular FO is not generically nilpotent. Finally, FM is not μ-nilpotent as soon as μ has full support. It is enough to show generic nilpotency of FB and the μ-nilpotency follows by Theorem [theo:genmunil]. For any n ∈ N consider the set of configurations having an annulus of 0 of radius n around position (0, 0): $$A\_n = \{c\in\{0,1\}^{{\mathbb{Z}}^2} : \forall z, \\|z\\|=n\implies c\_z=0\}.$$ It is easy to show by induction that if ∥z∥ ≤ n and c ∈ An then FBω(c)z = 0 (the annulus of radius n − 1 is first turned into 0 starting from the corner, and so on). An is open (it’s a union of cylinders) and for any m ∈ N the union Um = ⋃n ≥ mAn is open and dense (any cylinder intersects An for large enough n). Therefore the intersection I = ⋂m ≥ 0Um is a comeager set. From the remark above, it holds FBω(c)z = 0 for any z ∈ Z2 and any c ∈ I. Therefore I is included in the realm of attraction of ${\overline{0}}$ and we deduce that FB is generically nilpotent. Concerning FO, the claim on Bernoulli measures for which it is μ-nilpotent is enough to show that it is not generically nilpotent by Theorem [theo:genmunil]. One can show that FOω(c)(0, 0) = 1 if and only if there is a directed path (πn)n ∈ N in N2 starting from π0 = (0, 0) and such that πn + 1 ∈ πn + {(0, 1), (1, 1)} for all n ∈ N, and verifying c(πn) = 1 for all n ∈ N. Indeed, if such path exists, it is easy to check that Ft(c)πn = 1 for all t ≥ 1 and all n ∈ N by induction, so in particular FOω(c)(0, 0) = 1. On the contrary, if no such infinite path exists, then there is a finite bound B on the longest finite directed path with the same properties (otherwise we could extract an infinite one by compacity). One can then check that FOB + 1(c)(0, 0) = 0. It is well-known from oriented percolation theory that when c is a μ-random configuration for Bernoulli μ, there is a threshold 0 < θ < 1 such that an infinite path (πn)n ∈ N exists: * with positive probability when μ([0]) < θ, * with probability 0 when μ([0]) > θ. We deduce that FOt(μ)([1]) → 0 so FO is μ-nilpotent in the second case, while FOt(μ)([1]) ≥ ε > 0 for all t ≥ 0 so FO is not μ-nilpotent in the first case. Finally, FM admits a finite obstacle: precisely, if a 2 by 2 square of 1s appear in some configuration, it will never disappear. Thus if μ gives a probability larger than ε > 0 to a cylinder made of this pattern and also to the cylinder [0], then so will Ft(μ) for any t ≥ 1. We deduce that FM is not μ-nilpotent. Although μ-nilpotency is well understood for monotone freezing CA with 2 states by bootstrap percolation theory, it remains hard for freezing CA in general. [] The set of 2D freezing CA with 2 states which are μ-nilpotent for all full-support Bernoulli measure μ is recursively inseparable from the set of 2D freezing CA with 2 states which are μ-nilpotent for no full-support Bernoulli measure μ. A measure μ is said limit-computable if there is a computable map ϕ such that limn → ∞ϕ(u, n) − μ([u]) = 0 for each cylinder [u]. One can check that if μʹ = limt → ∞Ft(μ) for some CA F and μ a computable measure, then μʹ is limit computable. There is a remarkable converse to this observation. [] Let G = Zd for some d ≥ 1. Let μ be any translation-invariant and limit-computable measure on some alphabet. Then there exists a CA F (on a possibly different alphabet) such that μ = limt → ∞Ft(μ0) where μ0 is the uniform Bernoulli measure on the alphabet of F. Interestingly, the construction in the above theorem relies on CA that are not convergent, although the particular orbit starting from μ0 (and actually many other initial measures) is itself convergent. We have no idea about whether it is possible to realize the same limit measures with convergent CA. What are the limit measures Fω(μ0) where F is convergent (or bounded change, or freezing) and μ0 is the uniform Bernoulli measure (or some computable measure)? Computational Power of Convergent Cellular Automata =================================================== The convergence property dramatically restricts the possible dynamics of CA as we have seen so far. One can therefore legitimately ask whether the obvious fact that general CA are computationally universal still holds for convergent, bounded-change and freezing CA. This section tackles this question and outlines an answer in three steps: yes, even freezing CA in dimension 1 are computationally universal, but there is some loss of complexity, and for a convergent CA F, it is interesting to study the map c ↦ Fω(c) from a computational point of view. Embeddings of computationally universal systems ----------------------------------------------- Let’s start by the obvious observation that the space-time diagrams of any 1D CA can be grown by a 2D freezing CA. [ex:freezing2Dsimu1D] Any 1D CA F with states Q and neighborhood V can be simulated by a 2D freezing CA Fʹ with states Q ∪ { \* } as follow. Let Vʹ = {(v, − 1) : v ∈ V}. A cell in a state from Q never changes. A cell in state \* looks at cells in its Vʹ neighborhood: if they are all in a state from Q then it updates to the state given by applying F on them, otherwise it stays in \* . Starting from a all- \* configuration except on one horizontal line where it is in a Q-configuration c0, this 2D freezing CA will compute step by step the space-time diagram of F on configuration c0. In dimension 1, it is also possible to embed any Turing machine computation (with a polynomial slowdown) inside a convergent CA using the zigzag trick of Example [exa:zigzag]: by adding more states, one can make the head actually compute like a Turing head and do one transition at each zigzag (see for details). For 1D freezing CA, the embedding of computationally universal systems is much more constrained but still possible using Minsky machine. [c]0.45 [scale=1] = [draw,circle,minimum size=15,fill=white] (qz) at (0,0) q0; (qu) at (1.5,0) q1; (qd) at (3,0) q2; (qh) at (4.5,0) h; (qz) edge node[above] + 1 (qu); (qu) edge node[above] − 1 (qd); (qd) edge[loop above] node[above] ≠ 0 : − 1 (qd); (qd) edge node[above] = 0 : 0 (qh); \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| step \| 0 \| 1 \| 2 \| 3 \| \| state \| q0 \| q1 \| q2 \| h \| \| counter \| 0 \| 1 \| 0 \| 0 \| [c]0.5 [scale=.7] (2.5,1.5) node t = 0; (3.5,1.5) node t = 1; (4.5,1.5) node t = 2; (5.5,1.5) node t = 3; (2,2) rectangle ++(1,1) node[pos=.5] q0, 0;(3,2) rectangle ++(1,1) node[pos=.5] b;(4,2) rectangle ++(1,1) node[pos=.5] ⋯; (2,3) rectangle ++(1,1) node[pos=.5] $\#\_{\-1}$;(3,3) rectangle ++(1,1) node[pos=.5] q0;(4,3) rectangle ++(1,1) node[pos=.5] b;(5,3) rectangle ++(1,1) node[pos=.5] ⋯; (2,4) rectangle ++(1,1) node[pos=.5] #0;(3,4) rectangle ++(1,1) node[pos=.5] q1, + 1;(4,4) rectangle ++(1,1) node[pos=.5] b;(5,4) rectangle ++(1,1) node[pos=.5] ⋯; (2,5) rectangle ++(1,1) node[pos=.5] #1;(3,5) rectangle ++(1,1) node[pos=.5] 1;(4,5) rectangle ++(1,1) node[pos=.5] q1;(5,5) rectangle ++(1,1) node[pos=.5] b; (2,6) rectangle ++(1,1) node[pos=.5] #1;(3,6) rectangle ++(1,1) node[pos=.5] $\#\_{\-1}$;(4,6) rectangle ++(1,1) node[pos=.5] $q\_2,\-1$;(5,6) rectangle ++(1,1) node[pos=.5] b; (2,7) rectangle ++(1,1) node[pos=.5] #1;(3,7) rectangle ++(1,1) node[pos=.5] #0;(4,7) rectangle ++(1,1) node[pos=.5] $\#\_{\-1}$;(5,7) rectangle ++(1,1) node[pos=.5] q2; (2,8) rectangle ++(1,1) node[pos=.5] #1;(3,8) rectangle ++(1,1) node[pos=.5] #1;(4,8) rectangle ++(1,1) node[pos=.5] #0;(5,8) rectangle ++(1,1) node[pos=.5] h, 0; (2,9) rectangle ++(1,1) node[pos=.5] #1;(3,9) rectangle ++(1,1) node[pos=.5] #1;(4,9) rectangle ++(1,1) node[pos=.5] #1;(5,9) rectangle ++(1,1) node[pos=.5] $\#\_{\-1}$; A k-counter Minsky machine is a 4-tuple M = (QM, q0, h, τ) where q0, h ∈ QM are the initial and halting states and τ : QM × {0, 1}k → QM × { − 1, 0, 1}k is its transition map, which verifies τ(h, ⋅ ) = (h, (0, …, 0)). A configuration of M is an element of QM × Nk. M transforms any configuration c = (q, (χi)1 ≤ i ≤ k) in one step into configuration M(c) = (qʹ, (max(0, χi + δi))1 ≤ i ≤ k) where (qʹ, (δi)1 ≤ i ≤ k) = τ(q, (min(1, χi))1 ≤ i ≤ k). M halts on input (χi)1 ≤ i ≤ k ∈ Nk if there is a time t such that Mt(q0, (χi)1 ≤ i ≤ k) ∈ (h, Nk). [ex:minskyfreezing] Any k-counter Minsky machine M can be embedded into a 1D freezing CA F in the following sense. To simplify the exposition, let’s take k = 1. A configuration of M at some step of the evolution is a pair (q, n) where q is a finite state and n ∈ N the current value of the counter. This configuration will be encoded into the trace of some cell z ∈ Z of F: the state q will appear at cell z at some time, and after the value n will be encoded by the time interval between specific state changes occurring at z. Successive configurations of an orbit of M are then encoded into successive cells of F through the temporal traces. In order to allow F to correctly produce an orbit where the trace at position z + 1 encodes the configuration which is the image by M of the configuration encoded into the trace at position z, the key point is to shift temporally the time interval containing the encoded configuration from position to position z + 1. This technical condition allows to implement all the operations on a counter locally (zero test, increment and decrement). Concretely, if the configuration of M at time 0 is encoded into the time interval [0, Δ0] at cell 0, then the configuration at step t of machine M is encoded into the time interval [2t, 2t + Δt] at cell t. See Figure [fig:minsky] for an example with details. From the embedding of Minsky machines as in the above example, one expects undecidability results to be transferred to 1D freezing CA. There are many ways to state such results, depending on the additional details implemented. Let us mention the following one which essentially relies on a Minsky machine embedding but requires a more technical construction than the above example. A freezing CA is always k-change for k no smaller than the cardinal of its alphabet, but determining the minimal k for which it is k-change is undecidable. [] There exists a constant k such that the following problem is undecidable: given a freezing 1D CA F, decide whether F is k-change. The intuition for the construction in the proof of the above theorem is as follows: * first, the construction is such that the largest sequence of state changes that can possibly occur in some orbit at a given cell is unique, no other sequence of state changes can be as long as this one; * second, if this sequence appears in some orbit, it enforces the presence of a correct halting computation of the Minsky machine; conversely, if the simulated Minsky machine halts, there exists an orbit where this special sequence of state changes occurs at some cell; * third, the length of this sequence, denoted L, is independent of the number of states of the Minsky machine being simulated (intuitively, only the initial and halting states of the machine appear in it). With these ingredients, one has a reduction from the halting problem of Minsky machines to the problem of whether a freezing CA is (L − 1)-change. Dimension 1: complexity gap between bounded change and convergent CA -------------------------------------------------------------------- There is a strong information flow bottleneck in bounded change CA that becomes critical in 1D. It is not the case for convergent CA. This can be formulated using communication complexity or classical computational complexity of the prediction problem (see also which generalizes the result to other G). The prediction problem associated to a CA asks for the state of a cell after a given time starting from a given input, a possible formalization is as follows: given an input pattern u ∈ QBtr where r is the radius of the considered CA, the problem is to determine the value of cell 0G after t steps starting from a configuration c ∈ [u] (note that by choice of the domain of u, this does not depend on c). The prediction problem has communication complexity at most log(n) and is in class NL for 1D bounded change CA, while it can be $\Omega(\sqrt{n})$ and P-complete for 1D convergent CA. Instead of presenting the above results and their formal setting, we are going to illustrate this difference between bounded change and convergent CA through the problem of recognizing palindromes. Before stating the result, let’s formalize the notion of language recognition by CA under time constraints (but without space constraint). Say a language L ⊆ {0, 1}\* is recognized by a CA F on QZ in time τ : N → N if {0, 1, B, A, R} ⊆ Q (where B is a blank state, A an accepting state and R a rejecting state) and for any n ∈ N and any u ∈ {0, 1}n, the orbit of configuration cu defined by $$c^u(z)= \begin{cases} u\_z&\text{ if $0\leq z<n$},\\ B&\text{ else,} \end{cases}$$ is such that: * there is t ≤ τ(n) such that Ft(cu)0 ∈ {A, R}, * for the minimal such time t it holds Ft(cu)0 = A ⇔ u ∈ L. Finally, PAL is the language of palindromes on alphabet {0, 1}, i.e. words u such that ui = un − 1 − i for 0 ≤ i < n = ∣u∣. The next theorem states that PAL is hard to recognize with bounded change CA, while it is easily recognized by convergent CA. Note that limitations of bounded change (or bounded communication) CA were established previously in different settings: in language recognition is considered with a space limitation (only the cells initially containing the input can be modified), and in the focus is on real-time recognition (i.e. τ(n) = n ). [theo:palrec] Let G = Z. Suppose that a k-change CA can recognize PAL in time τ(n), then the recognition time is at least exponential, i.e. τ(n) ≥ αn for some α > 1 and large enough n. However, there exists a convergent CA that recognizes PAL in quadratic time. Suppose by contradiction that a k-change CA F on QZ and of radius r recognizes PAL in time τ(n) ≤ αn with ${\alpha<2^{\frac{1}{4rk}}}$. For n ∈ N and u ∈ {0, 1}n denote by Tzu, n the prefix of the trace of length τ(n) and of width 2r starting at position z in the orbit of cu: Tzu, n : t ∈ {0, …, τ(n)} ↦ Ft(cu)z, …, Ft(cu)z + 2r. By the k-change property, there are at most (∣Q∣τ(n))2rk ≤ ∣Q∣2rkα2rkn such prefixes (for all choices of u and z) because for each of the 2r columns of the trace, there are at most k changes and each change can be described by a new state and a time step between 1 and τ(n). On the other hand, there are ${2^{\frac{n}{2}}}$ words in PAL of even length n, so for large enough even n and by choice of α there are u ≠ v in PAL such that Tn/2u, n = Tn/2v, n Then, consider the word w = u0⋯un/2 − 1vn/2⋯vn − 1. By the Equality ([eq:eqtrace]) above, we have that TcuF(t) = TcwF(t) for all 0 ≤ t ≤ τ(n). So in particular F must accept w in time at most τ(n) because it does for u. However, w is not a palindrome, which yields the desired contradiction. It is not difficult to adapt Example [exa:zigzag] to make a convergent CA that recognizes PAL in quadratic time. Roughly, using extra states, any input word BuB is turned in one step into a segment of state with two components: the binary component that keeps the information of u and the zigzag component of the form ${\tikz[baseline,scale=.3]{{\draw[fill=white] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $B$};}{\draw[fill=white!50!blue] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $>$};}{\draw[fill=gray] (2,0) rectangle +(1,1);\draw (2,0)+(.5,.5) node {\tiny $\leftarrow$};}}\cdots\tikz[baseline,scale=.3]{{\draw[fill=gray] (0,0) rectangle +(1,1);\draw (0,0)+(.5,.5) node {\tiny $\leftarrow$};}{\draw[fill=white] (1,0) rectangle +(1,1); \draw (1,0)+(.5,.5) node {\tiny $B$};}}}$, where the head state $\tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}}$ holds an extra bit, initialized to u0. Then, with the zigzag movement of the head, this bit held inside the right-moving head $\tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}}$ is compared to the bit in the binary component when the head bounces on the right border (i.e. u0 is compared to un − 1 at the end of the first zig). If the comparison fails, an error state e appears which is interpreted as a rejecting state. If not, the head turns into $\tikz[baseline,scale=.3]{{\draw[fill=white!50!red] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $<$};}}$ (which does not hold any additional information) and goes back to the left boundary as in the rule of Example [exa:zigzag]. Once the left boundary is reached again, the head becomes $\tikz[baseline,scale=.3]{{\draw[fill=white!50!blue] (0,0) rectangle +(1,1); \draw (0,0)+(.5,.5) node {\tiny $>$};}}$ and copies the bit from the binary component at this position (i.e. u1 at the start of the second zig). When the zone is shrunk down to a single position, an accepting state is generated that spreads to the left to reach the position which initially was holding value u0 (the accepting state spreads over any other state except e, which is not present in the orbit of cu if u is in PAL). The additional mechanism on top of the rule of Example [exa:zigzag] does not compromise the convergent property: the analysis is the same as in Example [exa:zigzag] with the presence of two spreading states instead of one (rejecting state e spreading over the accepting state which spreads over any other state). Note that the above theorem does not say anything about the existence of a bounded-change CA recognizing PAL in super-exponential time. It would be surprising that such a CA exists because the information to compare between the two halves of a palindrome is too dense to be dealt with and transmitted within the bounded change constraint, however we have no proof of this impossibility. Is there a bounded change CA recognizing PAL (with arbitrary time and space) or any language with linear communication complexity in the sense of? We don’t think that the quadratic time recognition of PAL by convergent CA is optimal in Theorem [theo:palrec], however we have no idea on how to recognize PAL in real-time (i.e. τ(n) = n) with a convergent CA. More generally, we don’t know whether the recognition time of languages by convergent CA can always be as good as the recognition time for general CA. Complexity of limit configurations ---------------------------------- Any convergent CA F on QG induces a map Fω : QG → QG. We say that a configuration c ∈ QG is computable if it is computable as a map. For any t, the map Ft transforms computable configurations into computable configurations, however there is no reason to expect that the same holds for Fω. Let us first make the following observation: the computability of the limit configuration is linked to the computability of freezing times. [lem:freezetime] Let F be a convergent CA over QG and c ∈ QG be any computable configuration. Then the map z ↦ Fω(c)z is Turing reducible to the map z ↦ ζF(c, z). Moreover, if F is freezing, both maps are Turing-equivalent. To compute Fω(c)z from ζF(c, z), it is sufficient to compute FζF(c, z)(c)z. Reciprocally, if F is supposed freezing, ζF(c, z) can be computed from Fω(c)z as it is the first time step t such that Ft(c)z = Fω(c)z. For bounded change 1D CA, it is impossible to produce uncomputable limits from computable initial configurations. The argument of the following proof is due to G. Richard. [] [thm:computablelimits] For any 1D bounded-change CA F and any computable configuration c, Fω(c) is computable. [sketch] We restrict without loss of generality to F of radius 1 and fix a computable configuration c. If we know the number of state changes in the orbit of c at cells z1 and z2, with z1 < z2, then we can compute the value Fω(c)z for all z ∈ [z1, z2]. Indeed, we can compute Ft(c) on cells [z1, z2] for increasing values of t until the correct number of changes is observed at z1 and z2. From that time on, the evolution of cells in the segment [z1, z2] is independent of the context since cells z1 and z2 no longer change and F has radius 1. So we can compute the evolution until this set of cells reaches a fixed point, which is then the value they have in Fω(c). The number of changes at any cell is bounded, and there is a maximal value L for which there are infinitely many positions z < 0 with exactly L changes. Moreover, there is a limit position zL to the left of which no cell has more than L changes. The same is true for positions z > 0 giving the corresponding constants R and zR. Then the algorithm to compute Fω(c)z given z is the following: compute larger and larger portions of the space-time diagram around position z until finding z1 ≤ z ≤ z2: 1. z1 ≤ zL and zR ≤ z2, 2. the state of z1 has changed L times and the state of z2 has changed R times. Then it is sufficient to apply the algorithm of the above to compute Fω(c)[z1, z2] and therefore obtain Fω(c)z. Interestingly, the above limitation for bounded change CA disappears when considering convergent CA. [] There exists a 1D convergent CA F and a computable configuration c such that Fω(c) is uncomputable. ![image](random_freezing_22_000220.png)![image](random_freezing_23_000220.png)![image](random_freezing_25_000220.png)![image](random_freezing_28_000220.png)![image](random_freezing_29_000220.png)![Examples of configurations obtained after some time from a finite seed by randomly chosen 2D freezing CA. All CA have the same neighborhood and the same number of states, with the same freezing order. All finite seed have the same size (the non-uniform portion of the configuration is a small centered square), and all example configurations were obtained after the same number of steps. The difference in the uniform background comes from the fact that not all examples have the same quiescent states.](random_freezing_30_000220.png "fig:") [fig:randomfreezing] In 2D, it is easy to find a freezing CA F and a computable configuration c such that Fω(c) is uncomputable: let c(i, j) be 1 if i, j ∈ N and Turing machine i halts in at most j steps on the empty tape, and 0 otherwise. Then, if F is the freezing CA that spreads state 1 vertically, we get that Fω(c)(i, 0) = 1 for i ∈ N if and only if machine i halts. A more interesting question is to ask whether Fω(c) can be uncomputable for some finite or eventually periodic configuration c (i.e. a configuration which is totally periodic up to a finite region of Z2). The behavior of freezing CA from finite configuration is quite rich (see Figure [fig:randomfreezing]) and it is actually possible to realize uncomputable limits with a freezing 1-change CA and slightly more with a 2-change CA. [theo:limitconfhard] For a configuration c ∈ QZ2 and q ∈ Q, define the set of cells of c in state q: χq(c) = {z ∈ Z2 : c(z) = q}. The following holds: * There exists a 1-change freezing CA F1, a state q of F1, and a finite configuration c1 such that χq(F1ω(c1)) is not computable. * For any 1-change freezing CA F, any state q and any finite configuration c, χq(Fω(c)) is either recursively enumerable or co-recursively enumerable. * There exists a 2-change freezing CA F2, a state q of F2, and a finite configuration c2 such that χq(F2ω(c2)) is neither recursively enumerable, nor co-recursively enumerable. The first item is the main result of. The second item is. The third item is. Recall from Lemma [lem:freezetime] that in order to produce an uncomputable limit configuration from a computable one by a freezing CA, the freezing times of cells must be uncomputable. We stress that for this reason it is not sufficient for a 2D freezing CA to be computationally universal in order to produce uncomputable limit configurations from eventually periodic initial ones. For instance, in the orbits described in the simple embedding of Example [ex:freezing2Dsimu1D], the freezing times are computable by construction since cells are frozen progressively line by line. We don’t have a general understanding of the ingredients behind the capacity of freezing CA to produce uncomputable limit configuration starting from simple (computable) initial ones. ![Configuration reached from a random finite initial configuration (center) by the CA game of life without death after a few hundreds of steps. Black represents state 0, white represents state 1.](lwd_random_finite.png "fig:") [fig:lwd] Concretely, one can consider the game of life without death CA, which is a CA on the space on configurations {0, 1}Z2 defined by[4](#fn4): $$F(c)\_z = \begin{cases} 0\text{ if $c\_z=0$ or $Z\_8(c,z)=3$},\\ 1\text{ else,} \end{cases}$$ where Z8(c, z) counts the number of occurrences of state 0 among the 8 neighbors of cell z in configuration c. The prediction problem of this CA was shown to be P-complete in, however its ability to produce complex limit configurations from simple initial ones has not been established to our knowledge (see Figure [fig:lwd]). Let F be the game of life without death CA. Is there a finite or eventually periodic configuration c such that Fω(c) is uncomputable? More generally, for which freezing CA is this possible? What are sufficient or necessary conditions to achieve this behavior? Some related works and further reading ====================================== What was presented so far is of course not an exhaustive view of cold dynamics in cellular automata, and it also does not tackle other dynamical systems. Let us finish this tutorial by pointing out some related research directions that does not exactly fit in the formalism or classes seen so far. #### Randomization and measure rigidity in CA. There have been much work on CA considered as dynamical systems acting on probability measures. Let us mention two (linked) aspects that extends what we presented above: recall that unique ergodicity (Definition [def:uniqergo]) is both the property of having a unique invariant measure, and the property that any initial measure converges in Cesaro mean towards this unique invariant measure under the CA. If one considers a surjective CA on G = Zd, then the uniform Bernoulli measure is an invariant measure. It can never be the only one (a CA always admits some linear combination of dirac measures as an invariant measure), but a natural question is to ask under which additional conditions it becomes unique, a phenomenon called measure rigidity (see and references therein). Note that the famous Furstenberg conjecture can be formulated as measure rigidity question on a particular CA. Another much studied phenomenon is called randomization and studied since the 80s : it is the property that a large class of initial measures (including Bernoulli measures of full support) converge in Cesaro mean towards the uniform Bernoulli measure (see and references therein). More recently, a characterization of randomization was obtained for Abelian CA and an example was shown where the convergence is not only in Cesaro mean but actually a direct convergence (in the topology on measures). Recall that this difference in the kind of convergence plays also a role in the links between μ-nilpotency and unique ergodicity as discussed in Section [sec:nil]. These two phenomena are now rather well understood on CA with an algebraic structure, but the generalization to other kind of CA remains largely open. #### Probabilistic CA, noise and ergodicity. A probabilistic CA is a CA whose local evolution rule is probabilistic, thus randomness can be introduced at each time step (contrary to deterministic CA acting on probability measures where randomness only appears at the initial time step). In this context, the property of unique ergodicity has been much studied, in particular by asking if the unique invariant measure is also attractive which is then called ergodicity and which can be seen as an analog of asymptotic nilpotency. Among the many examples of probabilistic CA, a particularly interesting family are those obtained from a deterministic one by adding a (small) noise at each transition. In that setting, ergodicity intuitively means that the noise will progressively erase any information from the initial configuration, and non-ergodicity is a form of resistance to noise. It appears that in many cases the resulting noisy CA is ergodic as a probabilistic CA. In fact, proving that a particular noisy CA is not ergodic turns out to be difficult. A famous example in dimension 2 is a non-symmetric majority rule which was proven to be non-ergodic in. The result in is in fact more general: any monotone deterministic CA which is nilpotent over finite configurations yields a non-ergodic probabilistic CA when adding a small enough noise. Thus in dimension 2, very simple rules can produce CA resistant to noise. In dimension 1, the only such example known is extremely complicated (recall that the construction technique involved in this result inspired the construction of Theorem [theo:nilpernotue]). #### Self-assembly tilings. In the field of DNA computing, several models have been proposed that describe how a spatially extended structure (an assembly) grows according to some local rules. For instance, the aTAM model, usually considered on the grid Z2, uses the formalism of Wang tiles with glues to define the local process of growth: an assembly is a (usually finite) set of occupied positions of Z2 and the aTAM rules determine what tiles can be added on an empty position in the neighborhood of the assembly. A key parameter of the model is temperature which determines the amount of glues necessary to allow a tile to be added. We skip the details of the definition, but they are inspired by biochemistry and one of the strengths of the model is that it is both theoretically interesting and realistic enough to allow in vitro experiments. We refer to for a global overview (including many other self-assembly models). Any aTAM system can be seen as a 1-change (asynchronous and non-deterministic) process acting on QZ2 where Q is the set of tiles plus a special symbol representing an empty position. Note that in the case of a deterministic aTAM system (called directed), one can even define a freezing (1-change) CA F that produces the same limit configurations (see proof of Theorem [theo:limitconfhard]). Important progress have been accomplished recently in understanding the computational power of aTAM depending on the temperature parameter. #### Automata networks. We worked with G infinite, but in some sense we already considered the case of a finite space of cells when introducing nilpotency on periodic configurations (Definition [def:nilper]): indeed, the orbit of a totally periodic configuration on Zd can be seen as an orbit of configurations on a finite space of cell which is a torus (because since CA commute with translations, periods are preserved under iterations). One can go much further by considering automata networks which are non-uniform CA on a space of cells which can be an arbitrary graph. Actually, Theorem [theo:majboundedchange] has its roots in the automata network settings. In this context, a particularly fruitful approach is to analyze how the graph of cells influences the dynamics of the network (see and references therein for an overview). Following this approach, many results focused on nilpotency or convergence towards a unique attractor. More recently, freezing automata networks where also considered. Acknowledgment ============== We thank the anonymous referees for their careful reading, their many suggestions and corrections, and in particular, for having corrected us on the fact that the bootstrap percolation CA FB is generically nilpotent (Theorem [theo:classicbootstrap]). --- 1. I2M, Université Aix-Marseille, CNRS[↩](#fnref1) 2. This notion will be presented in Section [sec:nil].[↩](#fnref2) 3. doesn’t use the notion of μ-nilpotence but the property ${\mu(\{c:F^\omega(c)=\overline{0}\})=1}$. It can be shown that the two are equivalent for freezing CA in general.[↩](#fnref3) 4. For the reader familiar with life-like CA, note that we use the somewhat unconventional representation of alive cells by 0 to stay consistent with Definition [def:freezing].[↩](#fnref4)	arxiv_0000799
Asymmetry between the electron- and hole-doped Motttransition in the periodic Anderson model ============================================================================================= We study the doping driven Mott metal-insulator transition (MIT) in the periodic Anderson model set in the Mott-Hubbard regime. A striking asymmetry for electron or hole driven transitions is found. The electron doped MIT at larger U is similar to the one found in the single band Hubbard model, with a first order character due to coexistence of solutions. The hole doped MIT, in contrast, is second order and can be described as the delocalization of Zhang-Rice singlets. INTRODUCTION ============ The nature of the Mott transition, i.e. the metal-insulator transition driven by electronic correlations, is a central problem in physics of strongly correlated electrons systems. The relevance of the problem was initially emphasized by Mott in the 40’s, trying to explain why some materials with odd electrons per the unit cell, like NiO, are insulators. In a Mott metal-insulator transition (MIT), a metallic system with a partially filled electron band suddenly opens an insulating gap. In practice the transition is usually driven by temperature, applied pressure, or chemical doping. The origin of the mechanism is in the correlation effects due to the strong on-site Coulomb repulsion experienced by electrons occupying rather localized orbitals, such as d in transition metal oxides or f in heavy fermion compounds. The classical example of an experimental system exhibiting a Mott transition is vanadium oxide V2O3, which has received continuous attention since the pioneering work of McWhan in the 70’s. That compound has a finite temperature first order metal-insulator transition that terminates at a high temperature 2nd order critical point, in analogy with the finite liquid-gas transition line in water. From a theoretical perspective, it is considered that the Hubbard model, which contains a tight binding band plus an interaction term that describes local Coulomb repulsion between electrons occupying the same site, is a minimal Hamiltonian that may capture the basic physics of the Mott MIT. The most significant work on this model was initiated by Hubbard in the 60’s, where, starting from the insulating state at large interaction values, he described how the system may close the correlation gap as the bandwidth is increased to values of the order of the Coulomb repulsion energy. Later, in the 70’s Brinkman and Rice, using a variational approach, started from the metallic state and showed how it can be destroyed by increasing correlation effects when the interaction strength becomes of the order of the bandwidth. Finally in the 90’s the theoretical development of the dynamical mean-field theory (DMFT) allowed to get new insight on this problem. In the scenario for the Mott transition realized in the DMFT solution of the Hubbard model, for low temperatures and moderate interaction, the half filled Mott insulator can be driven to a correlated metallic state through a first order transition. This transition can occur as a function of correlation strength, temperature or doping. The solution of the Hubbard model within DMFT provided not only a connection between the approaches of Hubbard and Brinkman-Rice by showing how the system evolves from a metal to an insulator, but also produced a detailed description of the basic experimental phenomenology observed in the V2O3 compound. In addition, it was later shown that the MIT can be described in terms of a Ginzburg-Landau scenario with theoretical predictions for the critical behavior of observables near the 2nd order critical point, that were eventually confirmed by experiments. The physics of the Mott transitions, especially those driven by doping at low temperatures, became of unparalleled interest in condensed matter physics with the discovery of the high temperature cuprate superconductors in the 80’s and, in smaller but also significant measure, by the discovery of the non-Fermi liquid behavior in heavy fermion systems. In those systems, the effect of strong correlations is undisputed, since the active electronic degrees of freedom involve the localized d and f orbitals. Therefore, these systems are identified as doped Mott insulators, however their phase diagrams and the evolution of their physical properties cannot be associated to the DMFT scenario for the Mott transition that was so successfully applied to V2O3. Besides the Hubbard model, the periodic Anderson model (PAM) is another minimal Hamiltonian that is often investigated in the context of strongly correlated electron systems. That model contains two orbitals per unit cell, one local with on-site Coulomb repulsion and the other non-interacting and itinerant. At each lattice site, the two orbitals are hybridized. This model is more realistic than the Hubbard, since it describes with greater detail the actual situation in real compounds. For instance, in transition metal oxides where the overlaps between neighboring oxygen p − orbitals provide itineracy to the electrons, while the localized d − orbitals of the transition metal experience the stronger correlation effects. Despite the higher degree of detail included in the PAM, it is often assumed in theoretical approaches that the physical behavior of the PAM would result qualitatively similar to that of the Hubbard model at low frequencies. That statement can be mathematically justified in certain parameter regimes, however its general validity is less evident The DMFT is a theoretical approach that is mathematically exact in the limit of large lattice dimensionality which has been extensively used to study the Mott transition in the Hubbard model, and, to a lesser degree, has also been employed to investigate the physics of the PAM. Therefore, in the light of the previous discussion, a natural question to address is whether within DMFT the Mott transition scenario of the PAM is indeed qualitatively similar to that of the Hubbard model or, if contrary to usual expectations, it brings about new physical behaviors. This issue has been the focus of our recent investigations, where we showed that in fact a different type of doping driven Mott MIT is realized in the PAM, even in a parameter regime whether it might be expected that the identification with the Hubbard model may hold. The present study extends and provides further details to that work. In particular, we present new data that illustrate the different behavior of physical quantities on the two sides of the transition, we provide new comparisons of the Green functions to results using a T = 0 numerical technique, and we extend the discussion of the physical origin of two transitions. We should clarify that there is a qualitatively different insulator state that can be realized in the PAM, namely, the Kondo insulator. It is obtained in the particle-hole symmetric case, where the total occupation is even (ie, 2 electrons per site). The Kondo insulator is qualitatively different from a Mott insulator, because it is due to band hybridization effect and correlations merely serve to reduce the hybridization gap. This insulator has a temperature-driven crossover to a metallic state that has been investigated within DMFT using QMC techniques. In contrast, the Mott insulator state that we consider here corresponds to a state with odd total number of particles and its physical origin is entirely due to strong correlation effects. The paper is organized as follows. In Sec. [sec:methodology] we introduce the PAM and justify the choice of the parameter regime. We also summarize the DMFT equations and provide details on the numerical techniques we use to solve the associated impurity problem. In Sec. [sec:results] we present the results and discuss the Mott transitions found in the PAM. In Sec. [sec:discussion] we present a discussion of the physical origin of the different scenario for the MIT found in the PAM with respect to the HM. In Sec.[sec:conclusions] we present the conclusions. METHODOLOGY =========== Model ----- The Hamiltonian of the PAM is given by Here piσ and piσ+ operators destroy and create electrons at p orbitals on site i with spin σ. The p orbitals have site energy εp and overlap via the hopping term tij = t to form a band. diσ and diσ+ operators destroy and create electrons at d orbitals on site i with spin σ. The local d orbitals have site energy εd and are hybridized to the p orbitals with the constant on-site amplitude tpd. U is the energy cost of double occupation of the d orbitals at each site and μ is the chemical potential. Δ0 = εd − εp, the difference between the d level and the center of the p band, defines the bare charge-transfer energy. Fig. [fig1] shows a schematic representation of the Hamiltonian. The insulator solutions of the PAM have been studied in literature mostly in the symmetric regime Δ0 = 0, μ = 0, where the system is a Kondo (or renormalized-band) insulator. In the present work, we shall focus on a different and less explored parameter regime, where the system is in a Mott insulator state. A key difference between these insulator states is that the former is realized at an even total electron occupation (ntot = 2), while the latter is realized at an odd occupation (ntot = 1 or ntot = 3). Yet, the Mott insulator state may still be classified as a charge-transfer insulator or Mott-Hubbard insulator, according to the Zaanen-Sawatzky-Allen scheme. [fig1] The action associated with the Hamiltonian ([PAMHam]) reads where ψkσ = {dσ, pkσ}, ψkσ+ = {dσ+, pkσ+} and the inverse matrix propagator Ĝ0− 1 is given by: $$\hat{G}\_{0\sigma}^{-1} (k,i\omega\_n)= \left( \begin{array}{cc} i\omega\_n+\mu-\epsilon\_d & t\_{pd} \\ t\_{pd} & i\omega\_n+\mu-\epsilon\_p -\epsilon\_k \end{array} \right),$$ where εk is the Fourier transform of the hopping term t. The lattice Green’s function Ĝ is then written using the Dyson equation $\hat{G}^{-1}=\hat{G}\_0^{-1}-\hat{\Sigma}$, where $$\hat{\Sigma}\_{\sigma}(k,i\omega\_n)= \left( \begin{array}{cc} \Sigma\_{\sigma}(k,i\omega\_n) & 0 \\ 0 & 0 \end{array} \right)$$ and Σσ(k, iωn) is the d − electron self-energy. Here we are interested in a magnetically disordered state, thus the spin index can be dropped. In this case, the lattice Green’s functions for the p and d electrons are explicitly given by: The local Green’s functions are then obtained performing the integration over momenta, $$G\_{\alpha}(i\omega\_n) = \frac{1}{N} \sum\_{k} G\_{\alpha}(k, i\omega\_n) =\int \rho\_0(\epsilon) G\_{\alpha}(\epsilon,i\omega\_n) d\epsilon,$$ where α = pp, dd and ρ0(ε) = ∑kδ(ε − εk) is the free (U = 0 and tpd = 0) density of states of the p electrons. The PAM has some simply solvable limits such as of vanishing hybridization, tpd = 0, or of vanishing correlation strength, U = 0. For the latter case, the PAM describes two hybridized one-particle bands, that are obtained diagonalizing the Hamiltonian and read: $$\begin{aligned} E\_{\pm} (k)=\frac{1}{2}\left(\epsilon\_d +\epsilon\_p +\epsilon\_k -2\mu \pm \sqrt{(\epsilon\_k-\Delta\_0)^2+4t\_{pd}^2}\,\right). \label{poles\_u0}\end{aligned}$$ [fig2] From the many interesting parameter regimes that this model has, we shall focus our study on a particular one where the low energy physics of the PAM is a priori expected to correspond to that of the Hubbard model. Thus, we consider the case where the localized d orbital is near the Fermi energy and with an occupation close to one, while the p orbital band is well beneath in energy and almost fully occupied. By virtue of the hybridization term, the d electrons acquire a finite dispersion and form a narrow band that crosses the Fermi energy. It has a bandwidth ∼ tpd2/Δ, where Δ is the distance between the two hybridized bands, Δ ≈ E+ − E− > Δ0, and is subject to strong correlation effects when the on-site Coulomb term is turned on. For reference, the solution for the non-interacting case in the chosen parameter regime is shown in Fig. [fig2]. We set the units adopting a model semicircular density of states for the p-electrons with half-bandwidth D = 2t = 1 As we shall see in the next section, this density of states is actually realized in a Bethe lattice in the limit of infinite spatial dimensions. In the upper panels we show the single particle dispersion E± and the resulting density of states for Δ0 = 1, tpd = 0.9 and μ = 0.529, that gives a total occupation per site ntot equal to 3. In the lower panel we plot the total occupation ntot = np + nd together with the partial occupation. The plateaux in the curves signal incompressible states that correspond to insulators. These are observed at ntot = 2 and ntot = 4. In the first case, ntot equal to 2, one has a hybridization gap insulator, since the band E− is full and E+ is empty. For ntot = 4 the state is of a full band insulator. For 2 < ntot < 4 the system is metallic, since the lower band with mostly p character is full and the narrow band with mainly d character is partially filled. We shall now focus on the effects of correlation on such a metallic case, realized for ntot = 3, where the lower electron band with mostly p character is close to being full (i.e. is occupied by two electrons) and the narrow d electron band is half-filled (i.e. with an occupation nd close to 1). In the chosen parameter regime, there is a single band crossing the Fermi energy which has mostly d electron character. So correlations effects will affect it strongerly. For values of the interaction U larger than its bandwidth ∼ tpd2/Δ one may expect that a correlation gap would open and the system becomes a Mott insulator. This is the regime where the identification of the low energy physics of the PAM with the one band Hubbard model may hold. DMFT and the limit of infinite dimensions ----------------------------------------- To go beyond this qualitative discussion we need to obtain reliable solutions of the model Hamiltonian in the strongly interacting regime. We thus recur to the DMFT formulation where exact numerical methods can be used to solve the problem. The DMFT solution becomes exact in the limit of large spatial dimensionality or, equivalently, large lattice connectivity z. For this limit to remain physical one is required to rescale the hopping t amplitude as $t/\sqrt{z}$, so that the density of states ρ0(ε) = ∑kδ(ε − εk) gives a finite value for the mean kinetic energy. As is well known in DMFT, the specific lattice structure is not essential, and several lattice types could be used. For instance, the free (i.e., tpd = 0, U = 0) density of states of the hypercubic lattice, which is the generalization of the square lattice to the limit of high z, reads: $$\rho\_0^{\rm hyper}(\epsilon)= \frac{1}{t\sqrt{\pi}} \exp{\left(-\frac{\epsilon^2}{2t^2}\right)}, \label{dos\_HCL}$$ where ε denotes the noninteracting single particle energy. Another lattice type which is often adopted is the Bethe lattice, whose density of states reads, $$\rho\_0(\epsilon)=\frac{1}{2\pi t^2} \sqrt{4t^2-\epsilon^2}. \label{dos\_BL}$$ In the following we shall adopt this type of lattice structure, as it is better suited for some of the numerical methods that we shall employ. As unit of energy we set the half-bandwidth of the Bethe lattice semi-circular DOS, D = 2t = 1. The key mathematical simplification arising from the z → ∞ limit is the locality of the self-energy, i.e. its k independence. Thus, there is no longer need to keep the momentum label in the single particle energies of the band structure εk, and the energy ε itself is simply kept as the quantum number. Mean-field equations -------------------- In the limit of large lattice connectivity z → ∞, the PAM can be exactly mapped onto a single impurity Anderson model supplemented with a self consistency condition. The derivation of the DMFT equations has already been presented in detail elsewhere, so here we shall just briefly summarize the main steps and the final expressions. A direct way to derive the DMFT equations is to apply the cavity method. The key idea is to focus on a given (any) site of the lattice and to integrate out the degrees of freedom on all the other lattice sites in order to obtain the local effective action at the selected site. In doing that, one shall also obtain a self-consistency condition which restores the translational invariance that was (temporarily) broken with the selection of a given lattice site. After integration of all sites other than the one selected, the local effective action is obtained, where ψσ = {dσ, pσ}, ψσ+ = {dσ+, pσ+} correspond to the two atomic orbitals of the given (arbitrary) site of the lattice. The local inverse propagator reads where G̃pp is the cavity Green’s function that encodes the information of the propagation of electrons in the lattice, restricted not to return to the local site. These two equations define the so called associated impurity problem of the model. In order to restore the translational invariance and to obtain a closed set of equations, one has to relate the local inverse propagator ${\cal G}\_0$ to the Green’s function of the original lattice. In the Bethe lattice this relation is simple : the p − electron cavity Green’s function G̃pp becomes the local p − electron Green’s function, G̃pp = Gpp. From this relation and ([cavity]), one obtains the self-consistency condition for the impurity problem. It can be casted only in terms of $[{\cal G}\_0]\_{dd}$ and reads, In practice an iterative procedure is implemented to solve the set of DMFT equations: given an ansatz for $[{\cal G}\_0]\_{dd}$, and the fact that the interactions are local and only act on the d orbital, the impurity many-body problem ([Seff]) can be solved to produce a local d − electron Green’s function $G\_{dd}=-\langle dd^+ \rangle\_{S\_{\rm eff}}$. This defines a local self-energy $\Sigma=[{\cal G}\_0^{-1}]\_{dd}-G\_{dd}^{-1}$, that allows for the calculation of the local p − electron Green’s function Gpp ([latticeGFfull]) The obtained Gpp and Σ are then used as input to the self-consistency condition Eq. [self] to produce a new $[{\cal G}\_0]\_{dd}$. This process is iterated until convergence is reached. At the self-consistent point, the Green’s functions Gdd, Gpp and the self-energy Σ correspond to the local propagators of the original lattice model. Moreover, since in the z → ∞ limit Σ is local, then all the k dependent (or ε dependent) propagators of the original lattice can also be computed from this self-energy. It is useful rewrite the local Green’s functions in terms of the Hilbert transform of the density of states D̃(ξ) = ∫− ∞∞dερ0(ε)/(ξ − ε). For the conduction electron Green’s function Gpp we get and for the d − electron Green’s function Gdd we obtain This expression has a transparent physical interpretation: there are two processes that a d − electron can undergo: either remain fluctuating at the local site (first term), or fluctuate for some time, then jump to the p site and propagate, and then return and fluctuate some more time (second term). Numerical solution ------------------ For the solution of the associated impurity many-body problem ([Seff]), one may use a variety of techniques. Here we shall employ two numerical methods which are both a priori exact: Hirsch-Fye quantum Monte Carlo (QMC) and exact diagonalization (ED). The interest of using different techniques is that they have complementary range of applicability and that they allow for a crosscheck of the numerical results. The first method, QMC, is a finite temperature calculation and is exact in the statistical sense. The other method is formulated at T = 0 and rely in a finite size representation of the local site environment (i.e., the cavity Green’s function) by a bath of non-interacting atomic sites connected to the local impurity. In the limit of large number of atomic sites in the bath, this approach also becomes a priori exact. Both methods have already been well documented in the literature, so here we shall limit ourselves to briefly provide the relevant technical details. To implement the QMC, it is useful to first perform the integration on the non-interacting local p − site in the action ([self]), so that in the many-body problem the interacting d − orbital is the only explicit degree of freedom. We then solve the impurity problem using the standard Hirsch-Fye QMC algorithm, where the imaginary time interval [0, β] is discretized in L time-slices of width Δτ = β/L (where β is the inverse temperature). We set UΔτ < 1 to limit the systematic errors introduced by the Trotter decomposition. The precision of the calculations then basically depends on two remaining factors, the statistical error and the criterion for the convergence of the solution of the DMFT equations. For the former, we typically perform 105 sweeps. When required, we may do up to 106 sweeps, such as near the Mott transition, or to compute the analytic continuation of data to the real axis using the maximum entropy method. The quality of the convergence is controlled by monitoring the behavior with iteration number of Gdd(iω1), the imaginary part of the d − electron Green’s function at the first Matsubara frequency, which shows the largest variations. We stop the DMFT iterations when the fluctuations of this quantity become of the order of the QMC statistical error and remains stable for a few more iterations. In generic regions of the parameter space we have studied, the solution converge in less than 20-30 iterations, but hundreds may be necessary close to a phase boundary. The ED algorithm is based on the representation of the cavity Green’s function by finite number auxiliary atomic sites. They conform the “bath” or environment of the local impurity. In general, an infinite number of sites may be required to faithfully represent the dynamic environment, however, this is not possible to do in practice. Therefore, one has to adopt a strategy to best represent the environment with a finite number of auxiliary sites. One may use two different “geometries” to represent the bath: either the “chain” geometry, as described in Ref. or the “star” geometry, as described in Refs. . In both cases, the effective impurity problem consists of a central impurity site, composed of an explicit d − orbital and a p − orbital, where, by virtue of ([cavity]) only the latter is connected to the bath. In practice, less than 10 sites can be dealt with in this method and the ED is performed at T = 0 using the Lanczos technique, which is convenient to obtain the Green functions. In the case of the chain, parameters of the auxiliary atomic sites can be obtained in terms of a continued fraction expansion of the computed Green’s functions. On the other hand, in the case of the star geometry, the parameters are obtained by a χ2 minimization of the difference between the computed local Green function and a finite size parametrization of the cavity. The ED method can be substantially improved by supplementing it with the Density Matrix Renormalization group (DMRG) technique. Several ED-DMRG procedures for the solution of the DMFT equations have been proposed recently. The ED-DMRG method that we use here is in essence identical to the ED with the linear chain. Since the linear geometry is perfectly adapted for the DMRG procedure, we can “grow” the bath to contain a higher number of auxiliary sites with respect to the standard ED. In practice, we use up to 40 sites. Nevertheless, we point out that despite the large number of sites the spectral functions on the real frequency axis do not become continuous but still show a discrete multipole structure. We also observe that the number of poles is roughly similar to the number of sites, but the spectral weight remains rather concentrated in a relatively small number of poles, emphasizing the discrete nature of the Green’s functions. However, this does not prevent that the agreement on the Mastubara imaginary frequency axis is usually excellent (e.g. see Fig.[fig3]). RESULTS ======= The Mott insulating state ------------------------- In this section we shall present our results for the Mott-Hubbard regime. In the Hubbard model, the system undergoes a Mott metal-insulator transition when the density is n = 1 and the interaction strength U becomes of the order of the bandwidth. Here, a similar phenomenon is expected as U is increased to a value of the order of the effective bandwidth at the Fermi energy ∼ tpd2/Δ and keeping nd at about 1. As described before, the Mott state is realized by setting a value of the interaction larger than the effective bandwidth U > tpd2/Δ and keeping the total occupation ntot = np + nd = 3. Similarly to the Hubbard model case, in the PAM there is an on-site Coulomb interaction acting on d − orbitals. This interaction punishes the double occupation of d sites, and consequently favors the tendency to localization and to magnetic moment formation of the d electrons. Notice that we have chosen in the definition of our Hamiltonian ([PAMHam]) and ([Seff]) the so called non-magnetic form for the interaction. This is motivated by the fact that we are interested in the Mott physics of a paramagnetic correlated state at nd ≈ 1, i.e. nd ↑ = nd ↓ ∼ 1/2. Therefore the interaction term $U\left(n\_{d\uparrow}-\tfrac{1}{2}\right)\left(n\_{d\downarrow}-\tfrac{1}{2}\right)$ approximately cancels at the Hartree-Fock level. This allows to obtain some immediate physical insight. The cancellation implies that for low values of U, where first order perturbation holds, the interacting density of states of the model remains essentially identical as in the non-interacting case (see Fig. [fig2], upper panels). Therefore, the position of the correlated narrow band remains approximately fixed at the Fermi energy. Since the position remains unrenormalized, at higher values of U one would expect that the narrow band splits, forming a lower Hubbard band and an upper Hubbard band, below and above the Fermi energy respectively and both carrying half of the spectral intensity of the narrow band. At the Fermi energy a large charge gap would then open and the system becomes a Mott insulator. [fig3] ### Opening of the Mott gap This scenario is in fact borne out in the actual model solution that is shown in Fig. [fig3]. The data corresponds to a T = 0 calculation using the ED-DMRG method with 40 sites in the bath. The values of the interaction are U = 0.5 (upper panel) and U = 2 (lower panel). This latter value of U is sufficient to drive the system to the Mott insulating state. In fact one may estimate that the critical U should be about twice the effective bandwidth tpd2/Δ (we shall present later on the full phase diagram). The insets of the figure contain a comparison of the results for the Green’s functions in Matsubara frequency from ED-DMRG at T = 0 and QMC at the low temperature T = 1/128. The agreement is very satisfactory. The ED-DMRG method also provides the propagators in real frequency, we thus plot the more intuitive DOS in the main panels of the figure. In the weakly correlated case (upper panel), the DOS resembles the non interacting one: at lower energies $-3\lesssim \omega\lesssim-1.5$ there is a band with dominant p − character, while at the Fermi energy there is a narrower band with mainly d − character. In the Mott insulating state shown in the lower panel, the DOS consists of three features: similarly as before, there is a p − like band at high (negative) energies. However, the main qualitative difference is that now the narrow band at EF is splitted in a lower Hubbard band and an upper Hubbard band respectively below and above the Fermi energy. The Mott-Hubbard character of the transition in this parameter regime is seen from the fact that both lower and upper Hubbard band have dominant d character. Moreover, one also observes that the p component is not negligible, especially in the lower Hubbard band. We should also mention that the apparent multiple peak structure of the main three features appearing in the DOS are merely due to the discreteness of the finite number of sites used to describe the environment in the ED technique. Though we are using as many as 40 auxiliary sites in the environment, the discretization effect still remains rather noticeable. Nevertheless, the splitting of the narrow band at the Fermi energy with the consequent opening of a large Mott gap $\Delta\_{\rm M}$ that signals the Mott insulator state is clearly observed. [fig4] The transition from metallic to insulating state with the consequent opening of the Mott gap, can be also observed from the behavior of the partial p, d and total particle occupation np, nd and ntot respectively, as a function of the chemical potential. In Fig. [fig4] we show these quantities and their first derivative κ = ∂n/∂μ proportional to the charge compressibility, for U = 2. The plateaux observed in the occupations, with the respective vanishing of the compressibility for ntot = 0 and ntot = 4, correspond to the completely empty or completely full band insulators. The case ntot = 2 corresponds to the hybridization band insulator, similar to the one already discussed in the non-interacting case. However, in contrast to the results for the non-interacting case, the strong value of the interaction creates additional plateaux in the n(μ) curves. The new plateaux occur when the total number of particles is exactly ntot = 3, which is odd and signals the Mott state. An important aspect that we should mention is that the Mott insulator state occurs where the correlated d site has an occupation close to one, but not exactly unity. This can be seen is Fig. [fig4] (top panel), where the Mott plateau occurs at nd = 1 + ν (and np = 2 − ν), with ν ≈ 0.13. The specific value of ν depends on the hybridization and therefore one can consider it as a measure of the mix-valence character of the Mott insulating state. It is the total number of particles exactly equal to three (or one hole) what is required for the onset of the Mott insulator state. This implies that in this model where the mixed p − d valence is explicitly included through the hybridization, the Mott localization occurs for a “composite” object which has a mixed p and d character. ### Size of the Mott gap [fig5] The size of the Mott gap is naively expected of be of order U, since it should mostly reflect the energetic cost to doubly occupy the d orbitals. However our results show that the size of the Mott gap $\Delta\_{\rm M}$ may be substantially smaller than the bare value U. In Fig. [fig5] we plot $\Delta\_{\rm M}$ as a function of U, for several values of the bare position of the p − band εp, which amount to increase the charge transfer energy Δ0. As the energy of the p − orbitals is shifted down to larger (negative) energies, the effective bandwidth of the narrow band at the Fermi energy decreases. In addition, the p − electron band becomes essentially full with np → 2 as εp → − ∞ (and keeping tpd fixed). This implies a decrease in the mix-valence character of the electrons at the Fermi energy ν, as shown in the lower panel of Fig. [fig5]. In this limit the size of the Mott gap approaches the “bare” value $\Delta\_{\rm M} \approx U$. However, it is interesting to observe that the smaller values of εp lead to a substantial renormalization of the size of the expected Mott gap. This effect can be thought as due to an effective screening that the p − electrons provide, or, in more naive terms, because the electrons only “feel” the repulsive term U during the time they spend on the d − orbital, but not when they visit the p site. So as the mixed p − d character is increased, the effect of the U is renormalized downwards. [fig6] For completeness we show the behavior of the imaginary part of the Green’s functions at low Matsubara frequency. In Fig. [fig6] we present numerical results for several insulating states obtained varying the chemical potential within the Mott plateau. The data were obtained with both QMC and ED-DMRG, so also serve to illustrate the good agreement between the two methods. Note that the imaginary part of both the p and the d electron Green’s functions goes to zero at ωn → 0. By analytic continuation this implies that the p and d electron DOS vanish at the Fermi level, consistent with the insulating character of the solutions. Accordingly, when the chemical potential μ is varied within the Mott plateau, the p − and d − electron DOS simply experience a rigid shift in energy. However, as we shall see later, there are dramatic changes in the DOS lineshapes as the system turns metallic upon doping. Doped Mott insulator -------------------- So far we have shown that the model does have a Mott insulator state which similarly as in the Hubbard model case develop two incoherent Hubbard bands above and below the Fermi energy. However, unlike in the Hubbard model case, the size of the gap, i.e. the separation between the Hubbard bands, may be substantially smaller than U if the hybridization is relatively high. In the following section we shall proceed to dope this Mott insulator with δ carriers, with δ = ntot − 3. As was already reported in Ref. , we will observe that the insulator to metal transitions that can be obtained by either particle, δ > 0, or hole doping, δ < 0, are qualitatively different. The former will essentially reproduce the known scenario for the Mott MIT that is realized in the DMFT solution of one band Hubbard model. This was to be expected since we have tuned the parameters of the model to the regime where the identification of the low energy physics of the PAM and the one band Hubbard model, was expected to hold. However we shall see that, rather surprisingly, the hole doping insulator to metal transition bears out a qualitatively different scenario. ### particle doping (δ > 0) In this section we shall first describe the MIT driven by particle doping and demonstrate that it realizes the same first order transition scenario as the one in the single band Hubbard model. The metallization of the Mott insulator is most directly seen from the changes that take place in the density of states. [!ht] [fig7] In Fig. [fig7] we show the evolution of that quantity for the p − and d − electron components as a function of doping. The data are obtained from analytical continuation of QMC results at U = 2 and low temperature. To compute the analytical continuation using the MEM, we used over 106 sweeps, so to minimize the uncertainty due to statistical errors. In the left top panel we see the DOS for the case where the chemical potential μ is close to the upper energy edge of the Mott gap, therefore still in the insulator state with δ = 0. The left bottom panel shows the metallic state that is obtained when the chemical potential enters the upper Hubbard band, doping the parent Mott insulator with particles. We observe a broad peak at the Fermi energy and a strong transfer of spectral weight from the lower to the upper Hubbard band. The d orbital character remains dominant in the DOS upon doping. In the left middle panel we show the DOS in the region of small doping, very near the transition. The data reveal that both the p and d electron components of the DOS show a narrow quasiparticle peak at the Fermi energy, flanked by the incoherent upper Hubbard band at higher energies. This coherent peak carries a small fraction of the spectral intensity which is of order δ. In addition, from the enhancement of the slope of the self-energy, one observes that the quasiparticles acquire a heavy mass. All these features are consistent with the MIT scenario found in the one band Hubbard model. The right panels of Fig. [fig7] show a comparison of QMC and ED-DMRG results for the DOS at high doping. The discrete peaks in the ED-DMRG data are due to the finite number of sites in the effective bath. Nevertheless, the agreement in the distribution of the spectral weight of the two methods is very satisfactory. The comparison of the data also allows for a non-trivial benchmark of the numerical results and is a useful illustration of the advantages and disadvantages of the different numerical techniques. ED-DMRG is exact but has the drawback of discrete poles structures that persist even for many effective atoms in the bath. QMC, in contrast, produce smooth spectra but their numerical precision cannot be guaranteed due to the uncertainties in the process of analytic continuation of data to the real frequency axis. [fig7b] In Fig. [fig7b] we present the d − electron density of states ρd(ω) for several values of particle doping. The data show the presence of the quasiparticle peak evolving in between the two Hubbard bands. One may see that though the data show a reasonable systematic evolution some details are beyond the precision of the technique, such as the precise size of the gap and the form of the bandedges. The most substantial confirmation that the particle doping driven MIT scenario in the PAM and in the HM are in fact qualitatively analogous comes, however, from the observation of the hysteresis effect in the particle number n(μ) curve. The hysteresis is a hallmark of the first order nature of this doping driven transition and it was observed and studied in detail in the Hubbard model. We also find it here in the PAM, and it is most clearly appreciated in the behavior of nd versus μ. There is a strong dependence of the nd(μ) curves as the temperature is lowered, signaling strong correlation effects been active with very low energies. In Fig. [fig8] we show the occupation of the d electrons as a function of the chemical potential μ, obtained from QMC. [fig8] The main panel shows a detail of the QMC data at the low temperature T = 1/64, where the hysteresis loop can be clearly seen. These results were verified using the T = 0 ED technique. In order to observe the hysteresis cycle, we use as a seed for the iterative procedure the converged solution from the previous set of parameters. Thus, the solutions can be continuously “followed” in parameter space, until it shows a sudden jump. The discontinuous jump occur at the approximate position of the spinodal lines. Similarly as in previous studies on the Hubbard model, the hysteresis defines a region of parameters where two solutions of the DMFT equations can be stabilized. The true physical transition should occur where the free energy of the solutions cross. The precise determination of that line is beyond the scope of our present study and probably would require further refinement of the numerical techniques due to the very low energy scales involved. One key point that provides further support to our results is the critical slowing down phenomenon observed in the QMC calculation at the phase boundaries of the coexistence region. This phenomenon is characterized by an enhancement of the number of iterations required to achieve self-consistency and also by an enhancement of the statistical Monte Carlo fluctuations that reveal the shallowness of the energy landscape when two solutions merge. The root mean square deviation of the lowest frequency component of the d − electron Green’s function is plotted in the inset of Fig. [fig8] as a function of μ for the low temperature T = 1/64. As we approach the phase boundaries of the coexistence region, we can see that the root mean square deviation increases. Starting with an insulating solution, if we increase the chemical potential, the RMS increases until we reach a critical value of μ, where the insulating solution disappears (open circles). Similarly, starting form a metallic solution at low doping, if we reduce the chemical potential, the RMS grows until a critical value of μ, where the metallic solution disappears (open squares). Evidently, the coexistence of solutions can also be observed from the behavior of other quantities, such as the double occupancy, or the low frequency part of the Green’s functions. The latter is shown in Fig. [fig9] for both p − and d − electron components. These results support the claim that the MIT scenario for δ > 0 in the PAM is completely analogous to the one found in the Hubbard model investigations. [fig9] ### hole doping (δ < 0) [!ht] [fig10] In the previous section we showed that upon particle doping, δ > 0, the Mott MIT in the PAM realizes the same physical scenario as the one observed in the single band Hubbard model. While in the latter the particle-hole symmetry implies an identical transition for δ < 0, we shall see that this is not the case in the PAM. A key point to appreciate is that we shall keep all model parameters fixed, with the obvious exception of the chemical potential, which controls the occupation. Therefore, if the chosen model parameters led, for δ > 0, to the identification of the PAM with the HM physics, then one may also expect that this would be the case for δ < 0 as well. Rather surprisingly this turns not to be the case. In this section we shall describe the main physical behavior of the model for the hole doping driven MIT, and in the next we shall argue about the origin of this unexpected result. We begin by showing the effect of hole doping in the DOS. In Fig.[fig10] we plot the change in the p and d components of the DOS as the system evolves from Mott insulator to a hole-doped metallic state (left panels, top to bottom). In the insulator state we observe that the chemical potential is located within the correlation gap, and the lower and upper Hubbard bands can be well appreciated. As we already discussed before, in this Mott insulator state the DOS at low frequencies has mostly d electron character, since the d orbital was initially located at the Fermi energy. Upon hole doping, the chemical potential moves within the lower Hubbard band. The metallization produces a wide and strong quasiparticle peak at the Fermi energy. At small doping, the low frequency part of the spectrum has a characteristic three peaks structure: the lower Hubbard band around ω ∼ − 0.4, the quasiparticle peak that crosses the Fermi level, and the upper Hubbard band at ω ∼ 0.8 An interesting aspect to appreciate is that the quasiparticle peak has, as before, a larger d-character, but, in addition, now it also has a substantial p-electron component. We shall see that this aspect will be consistent with the interpretation of the quasiparticle peak now emerging not from mere delocalization of d-electrons as before, but from the delocalization of a composite object that involves a p-hole and a d-electron spin. We can also observe a transfer of spectral weight, with an increase of the relative intensity of the lower Hubbard band plus quasiparticle, at the expense of a decrease of the upper Hubbard band. The structure that is seen below ω ≈ − 1.5 corresponds to the fully filled band which remains with predominantly p − electron character. The results shown were obtained with high quality QMC data (at least 106 sweeps to reduce the statistical errors) and performed the analytic continuation to the real axis using the maximum entropy method. In the right panels of Fig. [fig10] we present a comparison of QMC and ED-DMRG results for the DOS at heavy doping δ = − 0.28. Similarly as before (cf. Fig. [fig7]), the apparent multi-peak substructures of the ED-DMRG results are not physical and only due to the discrete number of poles that result from a finite number of atomic sites in the auxiliary bath. Compared to the standard ED, this method produces much smoother spectra with greater detail due to the dramatic increase in the number of poles. However, as is also evident from the data, the discreteness due the finite size representation of the bath remains a shortcoming of the method. The overall comparison of the two methods remains, nevertheless, very satisfactory and serves to illustrate their relative advantages and disadvantages. [fig10b] To complete this study, we show in Fig. [fig10b] the detailed evolution of the quasi-particle peak at the Fermi level plus the Hubbard bands as a function of doping. The data correspond to the d − electron density of states obtained from analytic continuation of QMC data. The emergence of a quasiparticle peak at the Fermi energy for the hole doped case may seem, at first sight, similar to the metallic state obtained from particle doping. However, this will not turn to be the case and the physical origin of the two quasiparticle excitations will be shown to be qualitatively different. The reason for this unexpected asymmetry will be discussed in the next section. [fig11] In order to fully underpin the nature of the order of the hole doped transition, we now look for hysteresis effects. Thus, as we did before, we continuously follow the solutions in parameter space. First, we start from the insulator and lower the chemical potential till we obtain a significantly doped metal; and then, we start from the metal and increase μ until we reach again the insulator. Neither our QMC numerical simulations nor the ED-DMRG studies showed any indication of hysteresis effects. The QMC data down to T = 1/128 are shown in Fig. [fig11]. Compared to the results for particle doping (inset), the present ones show a negligible temperature dependence. Thus, up to our current numerical capacity we have to conclude that the metal-insulator transition in the hole doped case is of second order. As no evidence of coexistence has been found, neither in finite or zero temperature calculations, we are led to the conclusion that the hole doping driven transition at T = 0 is continuous, i.e. has a second order character. Thus, it is qualitatively different from the particle doped case, and, consequently, also qualitatively different from the Hubbard model scenario. Of course, we cannot rule out the eventual existence of tiny energy scales (which may modify our proposed scenario) which remain beyond the numerical precision of our methodology. This issue cannot be resolved nor by ED or ED-DMRG data due to the finite frequency cut-off set by the finite size of the clusters to diagonalize. Resolving this issue would probably require NRG study. However this is not fully clear since NRG method requires a good separation of energy scales, which is not the case here. DISCUSSION ========== Our results for the doping driven MIT, that arise from either particle or hole doping of the Mott insulator, show a qualitative asymmetry, thus questioning the expected mapping of the PAM onto the Hubbard model. In this section we shall address in more detail this issue from the perspective of the physical nature of the two MIT taking place in the PAM. Phase diagram ------------- [fig12] To begin, we map out the phase diagram in U − μ parameter space to explore the respective ranges of the first and the second order transitions. Our results are summarized in Fig. [fig12], that was constructed using finite T quantum Monte Carlo and T = 0 ED data. The latter were obtained using the “star-bath” geometry, which we found better suited than the “chain-bath” geometry to study the possible coexistence between the metallic and the insulating regions. The phase diagram in the U − μ plane shows a central V-shaped Mott insulator region with the correlated metallic phases for particle and hole doping, respectively to the right and to the left. A threshold value for the strength of the interaction U (tip of the V-shape boundary) is required to obtain a Mott insulator state. This threshold depends on the value of the “bandstructure” parameters Δ0 and tpd. This feature is analogous to the existence of critical value of the ratio U/D in the one band Hubbard model. The central V-shaped Mott insulating region shows a remarkable asymmetry comparing the hole and the electron sides. In contrast, in the one band Hubbard model, due to the particle-hole symmetry, the V-shape onset of the Mott insulator is symmetric respect to the tip and behaves as Uc ∼ Uc1 ± 2μ, where Uc1 is the value of the interaction at which the insulator disappears. In the PAM, the transition line to the particle doped side behaves as well like $U\_c\sim {\rm const}+2\mu$. However, the transition line to the hole doped metal is almost vertical. We also mapped out the order of the transition along the boundary lines. Upon hole doping, both QMC calculations down to the low temperature T = 1/128, and ED calculations at T = 0 show no trace of coexistent solutions along all the transition line, indicating a second order transition. On the other hand, upon particle doping, the dashed line in the phase diagram displays the region of the parameter space where, for sufficient low temperatures, an insulating state coexists with a metallic state. In fact within ED method at T = 0 we find that the DMFT equations have two different solutions all along the transition line. Within QMC, and down to the low temperature T = 1/128, we found a coexistence region only for a sufficiently large strength of the repulsive interaction U (approximately U ≈ 2 for our specific choice of parameters). However, for smaller values of U, the phase boundary remains strongly dependent on temperature. This suggests that the temperature below which there is a coexistence between metallic and insulating state decreases rapidly approaching the tip of the V-shape. This should be expected, since the two boundary lines, to holes and particle doped metal, eventually merge at the tip. Similarly, we also expect the T = 0 width of the coexistent region to become narrower as one approaches the tip of the V. However, obtaining detailed and precise numerical data there turned out to be beyond our methods’ capabilities. We note that at the Mott insulator state (δ = 0) the occupation of the conduction p − electrons is almost saturated at np = 2 − ν (with ν ≈ 0.1). On the other hand, the occupation of the non dispersive d − electrons which carry strong magnetic moments due to the on-site repulsion is close to one, with nd = 1 + ν (cf Fig. [fig5]). Therefore, there is only a small number ∼ ν of p − holes available to screen a number of order one of d magnetic moments. Therefore, a natural issue to consider is whether the physics associated to the “exhaustion problem” of Nozières (see also related works on the Kondo lattice model and on the PAM ) may play a role in the different transitions at δ greater or smaller than 0. An important point to realize is, however, that the exhaustion situation is even more extreme on the particle side than on the hole-doped side. In fact, while the number of d − electron remain always of order one on both sides, the number of available p − holes is substantially smaller for δ > 0 with respect to the δ < 0 case. Nevertheless, the Hubbard-like first-order transition scenario takes place only on the particle-doped side. In other words, the PAM metal-insulator transition scenario is analogous to the one in the Hubbard model when the PAM is even deeper in the exhaustion limit (δ > 0, np ≈ 2). This implies that while exhaustion should play a role, it is not obviously responsible for the failure of the mapping of the PAM onto the HM for δ < 0. Nature of doped carriers ------------------------ We now address the issue of the physical nature of the metallic states in this system. Although the low frequency part of the DOS of both, particle and hole doped metals, have a similar three peaks structure with dominant d character, they realize physically different states. For instance, their charge compressibility has a very contrasting temperature dependence. In Fig. [fig13] we show the derivative of the total occupation with respect to the chemical potential, κ = ∂ntot/∂μ (an observable proportional to the compressibility) as a function of doping. In the Mott insulator (δ = 0) κ is zero, indicating the incompressible Mott state. As expected, upon doping, k increases, indicating that the system becomes compressible. For small particle doping, the compressibility rapidly increases with the temperature, similarly as in the one band Hubbard model case. Upon hole doping, κ is much less dependent on temperature. [fig13] In the PAM the physics of the metallic states is usually discussed in terms of the screening between the p conduction electrons (or holes) and the magnetic moment of the local d − electrons. In fact, at each lattice site a p − and a d − electron may form a local singlet, which is the underlying idea in the argument of Zhang and Rice (ZR) in the context of high temperature cuprate superconductors. However, if on the particle side of the Mott MIT we have argued that there are essentially no available holes, then the question is, what is screening the d − moments so to produce a normal (but heavy) Fermi liquid metal, analogous to the one in the doped Mott insulator in the Hubbard model? [fig14] The answer to this question is that, similarly to the one band Hubbard model case, the d − electrons are screening themselves. In fact, the local ZR singlet formation does not take place for δ > 0 simply because there are no holes available for screening. Nevertheless, despite the high filling of the p − orbitals, the strong hybridization tpd still allows for delocalization of the d − electrons, through charge fluctuations across the p − sites, with an effective amplitude $t\_{\rm eff}\sim t\_{pd}^2/\Delta$. The key physical point in this process is that since the p − orbitals are almost full, they have a negligible local magnetic moment, so these charge fluctuations take place without significant magnetic p − d coupling. Therefore the magnetic phase coherence of the d − electrons is preserved and, in consequence, a superexchange mechanism between neighboring d sites occurs. Thus, from the point of view of the d − electrons, they have strong magnetic moments, they delocalize keeping their quantum mechanical phase through essentially non-magnetic p-sites, and therefore also experience antiferromagnetic correlations with nearest neighbor d − sites. These physical ingredients are evidently also realized for the carriers in the single band Hubbard model. Therefore we can now rationalize the underlying mechanism for the mapping of the PAM onto the one band Hubbard model at δ > 0, cf. Fig. [fig14a]. On the other hand, the situation is very different as the chemical potential is lowered to dope holes into the system. There, the number of available holes become more significant and they can lock with the robust d − magnetic moments to form local ZR singlets. However, when the d − electrons of these singlets want to delocalize, i.e. hop to the neighboring sites and eventually form a quasiparticle band, they completely loose the information of their spin phase, thus the antiferromagnetic correlations between neighboring d do not build up. In consequence, Hubbard model like physics does not take place and the nature of both the MIT and the ensuing correlated metallic state becomes fundamentally different. That is the key physical reason why the mapping of the PAM with a one band HM is no longer valid, cf. Fig. [fig14b]. In order to substantiate the previous qualitative discussion, we show in Fig. [fig15] the d and p local moment formation as a function of doping for a large strength of the interaction U and for a value slightly above the tip. The local moment formation is defined as ⟨(mαz)2⟩ = ⟨(nα ↑ − nα ↓)2⟩ = nα − 2⟨nα ↑nα ↓⟩, where α = p, d. Notice that the difference between the particle occupation n and the moment directly measures the double occupancies of the sites. In the Mott insulator (δ = 0) the local moment of the d electrons, ⟨(mdz)2⟩, is large because the d − sites are predominantly single occupied due to the effect of U. On the other hand, the p − band is almost fully occupied, and thus the local moment of the p − electron, ⟨(mpz)2⟩, is significantly smaller. As one dopes the Mott insulator with particles or holes, the relative distribution of the local moments among both p and d sites is strikingly asymmetric. Upon particle doping, δ > 0, ⟨(mpz)2⟩ becomes even smaller, since the occupation of the p band get saturated. On the other hand, the d moment decreases more rapidly (and linearly) with the doping, since the charge fluctuations between the single occupied and double occupied d states increase. Therefore these results support the view that for the particle doped side of the transition the p − sites are magnetically inert and, consequently, the d-electrons screen themselves as they form a heavy mass metal. This heavy metal paramagnetic state is physically analogous to the one realized in the Hubbard model case. For the hole doped metal, δ < 0, the p − electron local moment, ⟨(mpz)2⟩, increases linearly with the doping because holes are mostly added to p − sites. In contrast, the opposite behavior occurs for the d local moment, which slightly linearly decreases with hole doping. Thus, the increase of the magnetic character of p − electrons is consistent with our argument for the formation of local singlets in the hole doped case. [fig15] To fully underpin our hypothesis we compute the magnetic moment correlation between the d − and p − sites, ⟨mpzmdz⟩ = ⟨(nd ↑ − nd ↓)(np ↑ − np ↓)⟩ The results are shown in the bottom panel of Fig. [fig15]. They illustrate that in fact on the particle doped side the magnetic correlations are negligible, however on the hole doped side they increasingly grow as the chemical potential moves into the lower Hubbard band. The growth of the expectation value is commensurate with the increase in hole doping and signals that the doped p − holes bind magnetically to the local d − electron magnetic moments. This represents the formation of the equivalent to Zhang Rice singlets in the present model, that only occurs at δ < 0. Formation of Zhang-Rice like singlets ------------------------------------- The origin of the two different MIT scenarios and the resulting correlated metallic states can also be argued from an energetic point of view. The doping introduces new states inside the Mott gap ΔM, which is renormalized by the hybridization tpd. These states are a mixture of p and d states. An estimate of the energy gain of the Zhang Rice singlet formation is $$E\_{\rm ZR} \sim \nu(1-\nu) \frac{t\_{pd}^2}{\Delta \pm \Delta\_M} \label{energy\_ZR}$$ This results from the magnetic energy gain from the hopping of a p − hole (2 − np = ν) on a d − site singly occupied (1 − ν). We should compare the above binding energy with the delocalization energy of a d − electron between two neighboring d − sites, which is proportional to $$E\_{\rm SE} \sim (1-\nu)^2 \frac{t\_{pd}^4}{\Delta^2U} = (1-\nu)^2 \frac{t\_{\rm eff}^2}{U} \label{energy\_SE}$$ This results from the virtual hopping of a d − electron to its nearest neighbor d site (i.e. through two p sites). A priori this gain is of order one, since the d site are approximatively all singly occupied, and 1 − ν electrons participate in the superexchange process. In fact, in the region of parameter we are investigating, the parameters tpd, Δ and U are of order one. Therefore, the energy gain in the delocalization of a d − electron ([energySE]) is of order (1 − ν)2. Upon particle doping, the energy gain of singlet formation ([energyZR]) is of order ν(1 − ν), thus for small but finite ν, it is much smaller than the spin exchange energy gain ([energySE]). Thus we can understand that for particle doping the spin exchange energy gain dominates on the energy gain of singlet formation. This imply that the metallic state realized upon particle doping, is due to the delocalization of mostly d electrons. Therefore the d electrons play the same role as the single type of carriers in the single band Hubbard model and produce qualitatively similar MIT scenario. On the other hand, upon hole doping, the energy gain of Zhang Rice singlet formation can be substantial. In fact, in our region of parameters, Δ is of the same order of ΔM, and the partial cancellation of the denominator in ([energyZR]) explains that for a small but finite ν (which is controlled by the hybridization), the energy gain of singlet formation in the hole doped transition case dominates on the spin exchange energy gain. Thus we can understand that, upon hole doping, the ensuing metallic state is due to the delocalization of these composite objects. The nature of this metallic state has been studied in detail in our recent work where we found that it severely deviated from the Fermi liquid paradigm. The physical reason is that the delocalized holes undergo a strong magnetic scattering from the local magnetic moments as they hop from site to site. Thus no coherent behavior for hole propagation is realized, at least down to very low temperature scales, leading to the observed non-Fermi liquid character. Restoring the mapping of PAM onto Hubbard model ----------------------------------------------- [fig16] We have shown that the new physics in the PAM with respect to the Hubbard model comes from the local coupling between the p and d electrons. Thus we expect that disfavoring this binding may restore the validity of the mapping on the hole doped side of the MIT. To test this hypothesis, we lower the energy position of the p − band, εp (i.e. we increase the charge transfer energy Δ0). We already noticed in Sec. [subsec:mottinsulator] that upon increasing the bare charge transfer energy Δ0 = εd − εp, the Mott gap approaches the bare value U and the mix-valence character of the electrons is in fact decreased. In Fig. [fig16] we show the phase diagram in the plane U − μ for different values of the position of the p band. Upon increasing Δ0, the threshold value of U to obtain the Mott insulator region becomes smaller. This results from the fact that the interaction U is competing with a decreasing bandwidth ∼ tpd2/Δ. In addition, as expected, the V-shaped boundaries of the Mott insulator become more symmetric at larger Δ0. To verify whether the character of the transition can be modified on the hole doped side, we observe the temperature behavior of the particle number, n. Upon increasing Δ0, we find that n versus μ curves becomes more temperature dependent, which is a first indication of the possible realization of the Hubbard model scenario also in the hole doped side. However, due to the reduction of the effective bandwidth teff ∼ tpd2/Δ, the temperature below which we may observe the hysteresis cycle in the n versus μ curves should be extremely low. With our current numerical capabilities we were only able to obtain evidence of a small hysteresis at Δ0 = 3. [fig17] To complete this study we also computed the local moment of p − and d − electrons upon increasing Δ0. In Fig. [fig17] we plot the moments (top panel) and the magnetic binding between the p − and d − electrons (bottom panel) as a function of doping for several values of the position of the p − band. In the Mott insulator (δ = 0), upon decreasing the position of the p band, ⟨(mpz)2⟩ is decreased and ⟨(mdz)2⟩ is increased. This is because the mix-valence character of the electrons is reduced, since the p band become essentially full with np → 2 as εp → − ∞. Thus the doping has just a small effect on ⟨(mpz)2⟩. On the other hand, upon hole doping, the d local moments decreases more rapidly when the p − band is deep in energy. As a result, the magnetic correlation between the p − and d − electrons, ⟨mpzmdz⟩, shown in the bottom panel of Fig. [fig17], is expectedly punished by higher values of Δ0. This is fully consistent with the mentioned recovery of the mapping of the PAM onto the Hubbard model. Therefore we can understand that for lowering in energy the position of the p − band the mapping of the PAM onto the HM should hold. CONCLUSIONS =========== In this paper, using DMFT, we considered the doping-driven Mott transitions of the periodic Anderson model when it is set in the Mott-Hubbard parameter regime. We discussed the transitions with respect to reference case of the well understood scenario realized in the single band Hubbard model. In contrast to the latter, the PAM has a qualitatively different metal-insulator transition for particle or hole doping. Upon particle doping of the Mott insulator, the metallic state is reached through a first order transition, that is analogous to that of the Hubbard model. However, upon hole doping the Mott insulator, there is a continuous (i.e. second order) insulator-metal transition through a quantum critical line in the parameter space U − μ. We argued that the hole doped metal has delocalized Zhang Rice singlets that fail to build substantial superexchange as compared to the Hubbard model (and particle doped case). In fact, we discussed the qualitative differences between these two transitions, showing that it is not due to the physics of the “exhaustion”, but indeed is related to the magnetic interaction that develops between the two species of electrons in the model. Our results on the magnetic correlation between the d − and the p − electrons (see lower panel of Fig. [fig15]) show that upon particle doping the p − electrons permit the charge fluctuations and the delocalization of the d − electrons without magnetic p − d coupling. On the hole doped case, in contrast, the system favors the formation of singlet pairs p − d. Upon increasing the charge transfer energy, we could recover the mapping of the PAM to the Hubbard model for the hole doped case. This signifies that a substantial mix-valence character was the key ingredient for the realization of the second order transition in this model. Our findings may be important looking at the present effort to apply DMFT calculation in regard to real materials. Those studies usually carry the implicit assumption of the Hubbard model as the underlying low energy Hamiltonian of complex systems. Our work indicates that the Hubbard model scenario may be questionable when the hybridization of the correlated band with another band is high. In particular, our work is relevant for the analysis of the metal-insulator transitions of transition-metal oxides, that usually have oxygen orbital mediating the delocalization of the d correlated electrons of the transition-metal. Therefore the role of the oxygens band and their hybridization with the localised band should be explicitly considered in the investigation of the Mott transition. ACKNOWLEDGMENTS =============== We acknowledge M. 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Phys. 78, 865 (2006). Asymmetry between the electron- and hole-doped Motttransition in the periodic Anderson model ============================================================================================= We study the doping driven Mott metal-insulator transition (MIT) in the periodic Anderson model set in the Mott-Hubbard regime. A striking asymmetry for electron or hole driven transitions is found. The electron doped MIT at larger U is similar to the one found in the single band Hubbard model, with a first order character due to coexistence of solutions. The hole doped MIT, in contrast, is second order and can be described as the delocalization of Zhang-Rice singlets. INTRODUCTION ============ The nature of the Mott transition, i.e. the metal-insulator transition driven by electronic correlations, is a central problem in physics of strongly correlated electrons systems. The relevance of the problem was initially emphasized by Mott in the 40’s, trying to explain why some materials with odd electrons per the unit cell, like NiO, are insulators. In a Mott metal-insulator transition (MIT), a metallic system with a partially filled electron band suddenly opens an insulating gap. In practice the transition is usually driven by temperature, applied pressure, or chemical doping. The origin of the mechanism is in the correlation effects due to the strong on-site Coulomb repulsion experienced by electrons occupying rather localized orbitals, such as d in transition metal oxides or f in heavy fermion compounds. The classical example of an experimental system exhibiting a Mott transition is vanadium oxide V2O3, which has received continuous attention since the pioneering work of McWhan in the 70’s. That compound has a finite temperature first order metal-insulator transition that terminates at a high temperature 2nd order critical point, in analogy with the finite liquid-gas transition line in water. From a theoretical perspective, it is considered that the Hubbard model, which contains a tight binding band plus an interaction term that describes local Coulomb repulsion between electrons occupying the same site, is a minimal Hamiltonian that may capture the basic physics of the Mott MIT. The most significant work on this model was initiated by Hubbard in the 60’s, where, starting from the insulating state at large interaction values, he described how the system may close the correlation gap as the bandwidth is increased to values of the order of the Coulomb repulsion energy. Later, in the 70’s Brinkman and Rice, using a variational approach, started from the metallic state and showed how it can be destroyed by increasing correlation effects when the interaction strength becomes of the order of the bandwidth. Finally in the 90’s the theoretical development of the dynamical mean-field theory (DMFT) allowed to get new insight on this problem. In the scenario for the Mott transition realized in the DMFT solution of the Hubbard model, for low temperatures and moderate interaction, the half filled Mott insulator can be driven to a correlated metallic state through a first order transition. This transition can occur as a function of correlation strength, temperature or doping. The solution of the Hubbard model within DMFT provided not only a connection between the approaches of Hubbard and Brinkman-Rice by showing how the system evolves from a metal to an insulator, but also produced a detailed description of the basic experimental phenomenology observed in the V2O3 compound. In addition, it was later shown that the MIT can be described in terms of a Ginzburg-Landau scenario with theoretical predictions for the critical behavior of observables near the 2nd order critical point, that were eventually confirmed by experiments. The physics of the Mott transitions, especially those driven by doping at low temperatures, became of unparalleled interest in condensed matter physics with the discovery of the high temperature cuprate superconductors in the 80’s and, in smaller but also significant measure, by the discovery of the non-Fermi liquid behavior in heavy fermion systems. In those systems, the effect of strong correlations is undisputed, since the active electronic degrees of freedom involve the localized d and f orbitals. Therefore, these systems are identified as doped Mott insulators, however their phase diagrams and the evolution of their physical properties cannot be associated to the DMFT scenario for the Mott transition that was so successfully applied to V2O3. Besides the Hubbard model, the periodic Anderson model (PAM) is another minimal Hamiltonian that is often investigated in the context of strongly correlated electron systems. That model contains two orbitals per unit cell, one local with on-site Coulomb repulsion and the other non-interacting and itinerant. At each lattice site, the two orbitals are hybridized. This model is more realistic than the Hubbard, since it describes with greater detail the actual situation in real compounds. For instance, in transition metal oxides where the overlaps between neighboring oxygen p − orbitals provide itineracy to the electrons, while the localized d − orbitals of the transition metal experience the stronger correlation effects. Despite the higher degree of detail included in the PAM, it is often assumed in theoretical approaches that the physical behavior of the PAM would result qualitatively similar to that of the Hubbard model at low frequencies. That statement can be mathematically justified in certain parameter regimes, however its general validity is less evident The DMFT is a theoretical approach that is mathematically exact in the limit of large lattice dimensionality which has been extensively used to study the Mott transition in the Hubbard model, and, to a lesser degree, has also been employed to investigate the physics of the PAM. Therefore, in the light of the previous discussion, a natural question to address is whether within DMFT the Mott transition scenario of the PAM is indeed qualitatively similar to that of the Hubbard model or, if contrary to usual expectations, it brings about new physical behaviors. This issue has been the focus of our recent investigations, where we showed that in fact a different type of doping driven Mott MIT is realized in the PAM, even in a parameter regime whether it might be expected that the identification with the Hubbard model may hold. The present study extends and provides further details to that work. In particular, we present new data that illustrate the different behavior of physical quantities on the two sides of the transition, we provide new comparisons of the Green functions to results using a T = 0 numerical technique, and we extend the discussion of the physical origin of two transitions. We should clarify that there is a qualitatively different insulator state that can be realized in the PAM, namely, the Kondo insulator. It is obtained in the particle-hole symmetric case, where the total occupation is even (ie, 2 electrons per site). The Kondo insulator is qualitatively different from a Mott insulator, because it is due to band hybridization effect and correlations merely serve to reduce the hybridization gap. This insulator has a temperature-driven crossover to a metallic state that has been investigated within DMFT using QMC techniques. In contrast, the Mott insulator state that we consider here corresponds to a state with odd total number of particles and its physical origin is entirely due to strong correlation effects. The paper is organized as follows. In Sec. [sec:methodology] we introduce the PAM and justify the choice of the parameter regime. We also summarize the DMFT equations and provide details on the numerical techniques we use to solve the associated impurity problem. In Sec. [sec:results] we present the results and discuss the Mott transitions found in the PAM. In Sec. [sec:discussion] we present a discussion of the physical origin of the different scenario for the MIT found in the PAM with respect to the HM. In Sec.[sec:conclusions] we present the conclusions. METHODOLOGY =========== Model ----- The Hamiltonian of the PAM is given by Here piσ and piσ+ operators destroy and create electrons at p orbitals on site i with spin σ. The p orbitals have site energy εp and overlap via the hopping term tij = t to form a band. diσ and diσ+ operators destroy and create electrons at d orbitals on site i with spin σ. The local d orbitals have site energy εd and are hybridized to the p orbitals with the constant on-site amplitude tpd. U is the energy cost of double occupation of the d orbitals at each site and μ is the chemical potential. Δ0 = εd − εp, the difference between the d level and the center of the p band, defines the bare charge-transfer energy. Fig. [fig1] shows a schematic representation of the Hamiltonian. The insulator solutions of the PAM have been studied in literature mostly in the symmetric regime Δ0 = 0, μ = 0, where the system is a Kondo (or renormalized-band) insulator. In the present work, we shall focus on a different and less explored parameter regime, where the system is in a Mott insulator state. A key difference between these insulator states is that the former is realized at an even total electron occupation (ntot = 2), while the latter is realized at an odd occupation (ntot = 1 or ntot = 3). Yet, the Mott insulator state may still be classified as a charge-transfer insulator or Mott-Hubbard insulator, according to the Zaanen-Sawatzky-Allen scheme. [fig1] The action associated with the Hamiltonian ([PAMHam]) reads where ψkσ = {dσ, pkσ}, ψkσ+ = {dσ+, pkσ+} and the inverse matrix propagator Ĝ0− 1 is given by: $$\hat{G}\_{0\sigma}^{-1} (k,i\omega\_n)= \left( \begin{array}{cc} i\omega\_n+\mu-\epsilon\_d & t\_{pd} \\ t\_{pd} & i\omega\_n+\mu-\epsilon\_p -\epsilon\_k \end{array} \right),$$ where εk is the Fourier transform of the hopping term t. The lattice Green’s function Ĝ is then written using the Dyson equation $\hat{G}^{-1}=\hat{G}\_0^{-1}-\hat{\Sigma}$, where $$\hat{\Sigma}\_{\sigma}(k,i\omega\_n)= \left( \begin{array}{cc} \Sigma\_{\sigma}(k,i\omega\_n) & 0 \\ 0 & 0 \end{array} \right)$$ and Σσ(k, iωn) is the d − electron self-energy. Here we are interested in a magnetically disordered state, thus the spin index can be dropped. In this case, the lattice Green’s functions for the p and d electrons are explicitly given by: The local Green’s functions are then obtained performing the integration over momenta, $$G\_{\alpha}(i\omega\_n) = \frac{1}{N} \sum\_{k} G\_{\alpha}(k, i\omega\_n) =\int \rho\_0(\epsilon) G\_{\alpha}(\epsilon,i\omega\_n) d\epsilon,$$ where α = pp, dd and ρ0(ε) = ∑kδ(ε − εk) is the free (U = 0 and tpd = 0) density of states of the p electrons. The PAM has some simply solvable limits such as of vanishing hybridization, tpd = 0, or of vanishing correlation strength, U = 0. For the latter case, the PAM describes two hybridized one-particle bands, that are obtained diagonalizing the Hamiltonian and read: $$\begin{aligned} E\_{\pm} (k)=\frac{1}{2}\left(\epsilon\_d +\epsilon\_p +\epsilon\_k -2\mu \pm \sqrt{(\epsilon\_k-\Delta\_0)^2+4t\_{pd}^2}\,\right). \label{poles\_u0}\end{aligned}$$ [fig2] From the many interesting parameter regimes that this model has, we shall focus our study on a particular one where the low energy physics of the PAM is a priori expected to correspond to that of the Hubbard model. Thus, we consider the case where the localized d orbital is near the Fermi energy and with an occupation close to one, while the p orbital band is well beneath in energy and almost fully occupied. By virtue of the hybridization term, the d electrons acquire a finite dispersion and form a narrow band that crosses the Fermi energy. It has a bandwidth ∼ tpd2/Δ, where Δ is the distance between the two hybridized bands, Δ ≈ E+ − E− > Δ0, and is subject to strong correlation effects when the on-site Coulomb term is turned on. For reference, the solution for the non-interacting case in the chosen parameter regime is shown in Fig. [fig2]. We set the units adopting a model semicircular density of states for the p-electrons with half-bandwidth D = 2t = 1 As we shall see in the next section, this density of states is actually realized in a Bethe lattice in the limit of infinite spatial dimensions. In the upper panels we show the single particle dispersion E± and the resulting density of states for Δ0 = 1, tpd = 0.9 and μ = 0.529, that gives a total occupation per site ntot equal to 3. In the lower panel we plot the total occupation ntot = np + nd together with the partial occupation. The plateaux in the curves signal incompressible states that correspond to insulators. These are observed at ntot = 2 and ntot = 4. In the first case, ntot equal to 2, one has a hybridization gap insulator, since the band E− is full and E+ is empty. For ntot = 4 the state is of a full band insulator. For 2 < ntot < 4 the system is metallic, since the lower band with mostly p character is full and the narrow band with mainly d character is partially filled. We shall now focus on the effects of correlation on such a metallic case, realized for ntot = 3, where the lower electron band with mostly p character is close to being full (i.e. is occupied by two electrons) and the narrow d electron band is half-filled (i.e. with an occupation nd close to 1). In the chosen parameter regime, there is a single band crossing the Fermi energy which has mostly d electron character. So correlations effects will affect it strongerly. For values of the interaction U larger than its bandwidth ∼ tpd2/Δ one may expect that a correlation gap would open and the system becomes a Mott insulator. This is the regime where the identification of the low energy physics of the PAM with the one band Hubbard model may hold. DMFT and the limit of infinite dimensions ----------------------------------------- To go beyond this qualitative discussion we need to obtain reliable solutions of the model Hamiltonian in the strongly interacting regime. We thus recur to the DMFT formulation where exact numerical methods can be used to solve the problem. The DMFT solution becomes exact in the limit of large spatial dimensionality or, equivalently, large lattice connectivity z. For this limit to remain physical one is required to rescale the hopping t amplitude as $t/\sqrt{z}$, so that the density of states ρ0(ε) = ∑kδ(ε − εk) gives a finite value for the mean kinetic energy. As is well known in DMFT, the specific lattice structure is not essential, and several lattice types could be used. For instance, the free (i.e., tpd = 0, U = 0) density of states of the hypercubic lattice, which is the generalization of the square lattice to the limit of high z, reads: $$\rho\_0^{\rm hyper}(\epsilon)= \frac{1}{t\sqrt{\pi}} \exp{\left(-\frac{\epsilon^2}{2t^2}\right)}, \label{dos\_HCL}$$ where ε denotes the noninteracting single particle energy. Another lattice type which is often adopted is the Bethe lattice, whose density of states reads, $$\rho\_0(\epsilon)=\frac{1}{2\pi t^2} \sqrt{4t^2-\epsilon^2}. \label{dos\_BL}$$ In the following we shall adopt this type of lattice structure, as it is better suited for some of the numerical methods that we shall employ. As unit of energy we set the half-bandwidth of the Bethe lattice semi-circular DOS, D = 2t = 1. The key mathematical simplification arising from the z → ∞ limit is the locality of the self-energy, i.e. its k independence. Thus, there is no longer need to keep the momentum label in the single particle energies of the band structure εk, and the energy ε itself is simply kept as the quantum number. Mean-field equations -------------------- In the limit of large lattice connectivity z → ∞, the PAM can be exactly mapped onto a single impurity Anderson model supplemented with a self consistency condition. The derivation of the DMFT equations has already been presented in detail elsewhere, so here we shall just briefly summarize the main steps and the final expressions. A direct way to derive the DMFT equations is to apply the cavity method. The key idea is to focus on a given (any) site of the lattice and to integrate out the degrees of freedom on all the other lattice sites in order to obtain the local effective action at the selected site. In doing that, one shall also obtain a self-consistency condition which restores the translational invariance that was (temporarily) broken with the selection of a given lattice site. After integration of all sites other than the one selected, the local effective action is obtained, where ψσ = {dσ, pσ}, ψσ+ = {dσ+, pσ+} correspond to the two atomic orbitals of the given (arbitrary) site of the lattice. The local inverse propagator reads where G̃pp is the cavity Green’s function that encodes the information of the propagation of electrons in the lattice, restricted not to return to the local site. These two equations define the so called associated impurity problem of the model. In order to restore the translational invariance and to obtain a closed set of equations, one has to relate the local inverse propagator ${\cal G}\_0$ to the Green’s function of the original lattice. In the Bethe lattice this relation is simple : the p − electron cavity Green’s function G̃pp becomes the local p − electron Green’s function, G̃pp = Gpp. From this relation and ([cavity]), one obtains the self-consistency condition for the impurity problem. It can be casted only in terms of $[{\cal G}\_0]\_{dd}$ and reads, In practice an iterative procedure is implemented to solve the set of DMFT equations: given an ansatz for $[{\cal G}\_0]\_{dd}$, and the fact that the interactions are local and only act on the d orbital, the impurity many-body problem ([Seff]) can be solved to produce a local d − electron Green’s function $G\_{dd}=-\langle dd^+ \rangle\_{S\_{\rm eff}}$. This defines a local self-energy $\Sigma=[{\cal G}\_0^{-1}]\_{dd}-G\_{dd}^{-1}$, that allows for the calculation of the local p − electron Green’s function Gpp ([latticeGFfull]) The obtained Gpp and Σ are then used as input to the self-consistency condition Eq. [self] to produce a new $[{\cal G}\_0]\_{dd}$. This process is iterated until convergence is reached. At the self-consistent point, the Green’s functions Gdd, Gpp and the self-energy Σ correspond to the local propagators of the original lattice model. Moreover, since in the z → ∞ limit Σ is local, then all the k dependent (or ε dependent) propagators of the original lattice can also be computed from this self-energy. It is useful rewrite the local Green’s functions in terms of the Hilbert transform of the density of states D̃(ξ) = ∫− ∞∞dερ0(ε)/(ξ − ε). For the conduction electron Green’s function Gpp we get and for the d − electron Green’s function Gdd we obtain This expression has a transparent physical interpretation: there are two processes that a d − electron can undergo: either remain fluctuating at the local site (first term), or fluctuate for some time, then jump to the p site and propagate, and then return and fluctuate some more time (second term). Numerical solution ------------------ For the solution of the associated impurity many-body problem ([Seff]), one may use a variety of techniques. Here we shall employ two numerical methods which are both a priori exact: Hirsch-Fye quantum Monte Carlo (QMC) and exact diagonalization (ED). The interest of using different techniques is that they have complementary range of applicability and that they allow for a crosscheck of the numerical results. The first method, QMC, is a finite temperature calculation and is exact in the statistical sense. The other method is formulated at T = 0 and rely in a finite size representation of the local site environment (i.e., the cavity Green’s function) by a bath of non-interacting atomic sites connected to the local impurity. In the limit of large number of atomic sites in the bath, this approach also becomes a priori exact. Both methods have already been well documented in the literature, so here we shall limit ourselves to briefly provide the relevant technical details. To implement the QMC, it is useful to first perform the integration on the non-interacting local p − site in the action ([self]), so that in the many-body problem the interacting d − orbital is the only explicit degree of freedom. We then solve the impurity problem using the standard Hirsch-Fye QMC algorithm, where the imaginary time interval [0, β] is discretized in L time-slices of width Δτ = β/L (where β is the inverse temperature). We set UΔτ < 1 to limit the systematic errors introduced by the Trotter decomposition. The precision of the calculations then basically depends on two remaining factors, the statistical error and the criterion for the convergence of the solution of the DMFT equations. For the former, we typically perform 105 sweeps. When required, we may do up to 106 sweeps, such as near the Mott transition, or to compute the analytic continuation of data to the real axis using the maximum entropy method. The quality of the convergence is controlled by monitoring the behavior with iteration number of Gdd(iω1), the imaginary part of the d − electron Green’s function at the first Matsubara frequency, which shows the largest variations. We stop the DMFT iterations when the fluctuations of this quantity become of the order of the QMC statistical error and remains stable for a few more iterations. In generic regions of the parameter space we have studied, the solution converge in less than 20-30 iterations, but hundreds may be necessary close to a phase boundary. The ED algorithm is based on the representation of the cavity Green’s function by finite number auxiliary atomic sites. They conform the “bath” or environment of the local impurity. In general, an infinite number of sites may be required to faithfully represent the dynamic environment, however, this is not possible to do in practice. Therefore, one has to adopt a strategy to best represent the environment with a finite number of auxiliary sites. One may use two different “geometries” to represent the bath: either the “chain” geometry, as described in Ref. or the “star” geometry, as described in Refs. . In both cases, the effective impurity problem consists of a central impurity site, composed of an explicit d − orbital and a p − orbital, where, by virtue of ([cavity]) only the latter is connected to the bath. In practice, less than 10 sites can be dealt with in this method and the ED is performed at T = 0 using the Lanczos technique, which is convenient to obtain the Green functions. In the case of the chain, parameters of the auxiliary atomic sites can be obtained in terms of a continued fraction expansion of the computed Green’s functions. On the other hand, in the case of the star geometry, the parameters are obtained by a χ2 minimization of the difference between the computed local Green function and a finite size parametrization of the cavity. The ED method can be substantially improved by supplementing it with the Density Matrix Renormalization group (DMRG) technique. Several ED-DMRG procedures for the solution of the DMFT equations have been proposed recently. The ED-DMRG method that we use here is in essence identical to the ED with the linear chain. Since the linear geometry is perfectly adapted for the DMRG procedure, we can “grow” the bath to contain a higher number of auxiliary sites with respect to the standard ED. In practice, we use up to 40 sites. Nevertheless, we point out that despite the large number of sites the spectral functions on the real frequency axis do not become continuous but still show a discrete multipole structure. We also observe that the number of poles is roughly similar to the number of sites, but the spectral weight remains rather concentrated in a relatively small number of poles, emphasizing the discrete nature of the Green’s functions. However, this does not prevent that the agreement on the Mastubara imaginary frequency axis is usually excellent (e.g. see Fig.[fig3]). RESULTS ======= The Mott insulating state ------------------------- In this section we shall present our results for the Mott-Hubbard regime. In the Hubbard model, the system undergoes a Mott metal-insulator transition when the density is n = 1 and the interaction strength U becomes of the order of the bandwidth. Here, a similar phenomenon is expected as U is increased to a value of the order of the effective bandwidth at the Fermi energy ∼ tpd2/Δ and keeping nd at about 1. As described before, the Mott state is realized by setting a value of the interaction larger than the effective bandwidth U > tpd2/Δ and keeping the total occupation ntot = np + nd = 3. Similarly to the Hubbard model case, in the PAM there is an on-site Coulomb interaction acting on d − orbitals. This interaction punishes the double occupation of d sites, and consequently favors the tendency to localization and to magnetic moment formation of the d electrons. Notice that we have chosen in the definition of our Hamiltonian ([PAMHam]) and ([Seff]) the so called non-magnetic form for the interaction. This is motivated by the fact that we are interested in the Mott physics of a paramagnetic correlated state at nd ≈ 1, i.e. nd ↑ = nd ↓ ∼ 1/2. Therefore the interaction term $U\left(n\_{d\uparrow}-\tfrac{1}{2}\right)\left(n\_{d\downarrow}-\tfrac{1}{2}\right)$ approximately cancels at the Hartree-Fock level. This allows to obtain some immediate physical insight. The cancellation implies that for low values of U, where first order perturbation holds, the interacting density of states of the model remains essentially identical as in the non-interacting case (see Fig. [fig2], upper panels). Therefore, the position of the correlated narrow band remains approximately fixed at the Fermi energy. Since the position remains unrenormalized, at higher values of U one would expect that the narrow band splits, forming a lower Hubbard band and an upper Hubbard band, below and above the Fermi energy respectively and both carrying half of the spectral intensity of the narrow band. At the Fermi energy a large charge gap would then open and the system becomes a Mott insulator. [fig3] ### Opening of the Mott gap This scenario is in fact borne out in the actual model solution that is shown in Fig. [fig3]. The data corresponds to a T = 0 calculation using the ED-DMRG method with 40 sites in the bath. The values of the interaction are U = 0.5 (upper panel) and U = 2 (lower panel). This latter value of U is sufficient to drive the system to the Mott insulating state. In fact one may estimate that the critical U should be about twice the effective bandwidth tpd2/Δ (we shall present later on the full phase diagram). The insets of the figure contain a comparison of the results for the Green’s functions in Matsubara frequency from ED-DMRG at T = 0 and QMC at the low temperature T = 1/128. The agreement is very satisfactory. The ED-DMRG method also provides the propagators in real frequency, we thus plot the more intuitive DOS in the main panels of the figure. In the weakly correlated case (upper panel), the DOS resembles the non interacting one: at lower energies $-3\lesssim \omega\lesssim-1.5$ there is a band with dominant p − character, while at the Fermi energy there is a narrower band with mainly d − character. In the Mott insulating state shown in the lower panel, the DOS consists of three features: similarly as before, there is a p − like band at high (negative) energies. However, the main qualitative difference is that now the narrow band at EF is splitted in a lower Hubbard band and an upper Hubbard band respectively below and above the Fermi energy. The Mott-Hubbard character of the transition in this parameter regime is seen from the fact that both lower and upper Hubbard band have dominant d character. Moreover, one also observes that the p component is not negligible, especially in the lower Hubbard band. We should also mention that the apparent multiple peak structure of the main three features appearing in the DOS are merely due to the discreteness of the finite number of sites used to describe the environment in the ED technique. Though we are using as many as 40 auxiliary sites in the environment, the discretization effect still remains rather noticeable. Nevertheless, the splitting of the narrow band at the Fermi energy with the consequent opening of a large Mott gap $\Delta\_{\rm M}$ that signals the Mott insulator state is clearly observed. [fig4] The transition from metallic to insulating state with the consequent opening of the Mott gap, can be also observed from the behavior of the partial p, d and total particle occupation np, nd and ntot respectively, as a function of the chemical potential. In Fig. [fig4] we show these quantities and their first derivative κ = ∂n/∂μ proportional to the charge compressibility, for U = 2. The plateaux observed in the occupations, with the respective vanishing of the compressibility for ntot = 0 and ntot = 4, correspond to the completely empty or completely full band insulators. The case ntot = 2 corresponds to the hybridization band insulator, similar to the one already discussed in the non-interacting case. However, in contrast to the results for the non-interacting case, the strong value of the interaction creates additional plateaux in the n(μ) curves. The new plateaux occur when the total number of particles is exactly ntot = 3, which is odd and signals the Mott state. An important aspect that we should mention is that the Mott insulator state occurs where the correlated d site has an occupation close to one, but not exactly unity. This can be seen is Fig. [fig4] (top panel), where the Mott plateau occurs at nd = 1 + ν (and np = 2 − ν), with ν ≈ 0.13. The specific value of ν depends on the hybridization and therefore one can consider it as a measure of the mix-valence character of the Mott insulating state. It is the total number of particles exactly equal to three (or one hole) what is required for the onset of the Mott insulator state. This implies that in this model where the mixed p − d valence is explicitly included through the hybridization, the Mott localization occurs for a “composite” object which has a mixed p and d character. ### Size of the Mott gap [fig5] The size of the Mott gap is naively expected of be of order U, since it should mostly reflect the energetic cost to doubly occupy the d orbitals. However our results show that the size of the Mott gap $\Delta\_{\rm M}$ may be substantially smaller than the bare value U. In Fig. [fig5] we plot $\Delta\_{\rm M}$ as a function of U, for several values of the bare position of the p − band εp, which amount to increase the charge transfer energy Δ0. As the energy of the p − orbitals is shifted down to larger (negative) energies, the effective bandwidth of the narrow band at the Fermi energy decreases. In addition, the p − electron band becomes essentially full with np → 2 as εp → − ∞ (and keeping tpd fixed). This implies a decrease in the mix-valence character of the electrons at the Fermi energy ν, as shown in the lower panel of Fig. [fig5]. In this limit the size of the Mott gap approaches the “bare” value $\Delta\_{\rm M} \approx U$. However, it is interesting to observe that the smaller values of εp lead to a substantial renormalization of the size of the expected Mott gap. This effect can be thought as due to an effective screening that the p − electrons provide, or, in more naive terms, because the electrons only “feel” the repulsive term U during the time they spend on the d − orbital, but not when they visit the p site. So as the mixed p − d character is increased, the effect of the U is renormalized downwards. [fig6] For completeness we show the behavior of the imaginary part of the Green’s functions at low Matsubara frequency. In Fig. [fig6] we present numerical results for several insulating states obtained varying the chemical potential within the Mott plateau. The data were obtained with both QMC and ED-DMRG, so also serve to illustrate the good agreement between the two methods. Note that the imaginary part of both the p and the d electron Green’s functions goes to zero at ωn → 0. By analytic continuation this implies that the p and d electron DOS vanish at the Fermi level, consistent with the insulating character of the solutions. Accordingly, when the chemical potential μ is varied within the Mott plateau, the p − and d − electron DOS simply experience a rigid shift in energy. However, as we shall see later, there are dramatic changes in the DOS lineshapes as the system turns metallic upon doping. Doped Mott insulator -------------------- So far we have shown that the model does have a Mott insulator state which similarly as in the Hubbard model case develop two incoherent Hubbard bands above and below the Fermi energy. However, unlike in the Hubbard model case, the size of the gap, i.e. the separation between the Hubbard bands, may be substantially smaller than U if the hybridization is relatively high. In the following section we shall proceed to dope this Mott insulator with δ carriers, with δ = ntot − 3. As was already reported in Ref. , we will observe that the insulator to metal transitions that can be obtained by either particle, δ > 0, or hole doping, δ < 0, are qualitatively different. The former will essentially reproduce the known scenario for the Mott MIT that is realized in the DMFT solution of one band Hubbard model. This was to be expected since we have tuned the parameters of the model to the regime where the identification of the low energy physics of the PAM and the one band Hubbard model, was expected to hold. However we shall see that, rather surprisingly, the hole doping insulator to metal transition bears out a qualitatively different scenario. ### particle doping (δ > 0) In this section we shall first describe the MIT driven by particle doping and demonstrate that it realizes the same first order transition scenario as the one in the single band Hubbard model. The metallization of the Mott insulator is most directly seen from the changes that take place in the density of states. [!ht] [fig7] In Fig. [fig7] we show the evolution of that quantity for the p − and d − electron components as a function of doping. The data are obtained from analytical continuation of QMC results at U = 2 and low temperature. To compute the analytical continuation using the MEM, we used over 106 sweeps, so to minimize the uncertainty due to statistical errors. In the left top panel we see the DOS for the case where the chemical potential μ is close to the upper energy edge of the Mott gap, therefore still in the insulator state with δ = 0. The left bottom panel shows the metallic state that is obtained when the chemical potential enters the upper Hubbard band, doping the parent Mott insulator with particles. We observe a broad peak at the Fermi energy and a strong transfer of spectral weight from the lower to the upper Hubbard band. The d orbital character remains dominant in the DOS upon doping. In the left middle panel we show the DOS in the region of small doping, very near the transition. The data reveal that both the p and d electron components of the DOS show a narrow quasiparticle peak at the Fermi energy, flanked by the incoherent upper Hubbard band at higher energies. This coherent peak carries a small fraction of the spectral intensity which is of order δ. In addition, from the enhancement of the slope of the self-energy, one observes that the quasiparticles acquire a heavy mass. All these features are consistent with the MIT scenario found in the one band Hubbard model. The right panels of Fig. [fig7] show a comparison of QMC and ED-DMRG results for the DOS at high doping. The discrete peaks in the ED-DMRG data are due to the finite number of sites in the effective bath. Nevertheless, the agreement in the distribution of the spectral weight of the two methods is very satisfactory. The comparison of the data also allows for a non-trivial benchmark of the numerical results and is a useful illustration of the advantages and disadvantages of the different numerical techniques. ED-DMRG is exact but has the drawback of discrete poles structures that persist even for many effective atoms in the bath. QMC, in contrast, produce smooth spectra but their numerical precision cannot be guaranteed due to the uncertainties in the process of analytic continuation of data to the real frequency axis. [fig7b] In Fig. [fig7b] we present the d − electron density of states ρd(ω) for several values of particle doping. The data show the presence of the quasiparticle peak evolving in between the two Hubbard bands. One may see that though the data show a reasonable systematic evolution some details are beyond the precision of the technique, such as the precise size of the gap and the form of the bandedges. The most substantial confirmation that the particle doping driven MIT scenario in the PAM and in the HM are in fact qualitatively analogous comes, however, from the observation of the hysteresis effect in the particle number n(μ) curve. The hysteresis is a hallmark of the first order nature of this doping driven transition and it was observed and studied in detail in the Hubbard model. We also find it here in the PAM, and it is most clearly appreciated in the behavior of nd versus μ. There is a strong dependence of the nd(μ) curves as the temperature is lowered, signaling strong correlation effects been active with very low energies. In Fig. [fig8] we show the occupation of the d electrons as a function of the chemical potential μ, obtained from QMC. [fig8] The main panel shows a detail of the QMC data at the low temperature T = 1/64, where the hysteresis loop can be clearly seen. These results were verified using the T = 0 ED technique. In order to observe the hysteresis cycle, we use as a seed for the iterative procedure the converged solution from the previous set of parameters. Thus, the solutions can be continuously “followed” in parameter space, until it shows a sudden jump. The discontinuous jump occur at the approximate position of the spinodal lines. Similarly as in previous studies on the Hubbard model, the hysteresis defines a region of parameters where two solutions of the DMFT equations can be stabilized. The true physical transition should occur where the free energy of the solutions cross. The precise determination of that line is beyond the scope of our present study and probably would require further refinement of the numerical techniques due to the very low energy scales involved. One key point that provides further support to our results is the critical slowing down phenomenon observed in the QMC calculation at the phase boundaries of the coexistence region. This phenomenon is characterized by an enhancement of the number of iterations required to achieve self-consistency and also by an enhancement of the statistical Monte Carlo fluctuations that reveal the shallowness of the energy landscape when two solutions merge. The root mean square deviation of the lowest frequency component of the d − electron Green’s function is plotted in the inset of Fig. [fig8] as a function of μ for the low temperature T = 1/64. As we approach the phase boundaries of the coexistence region, we can see that the root mean square deviation increases. Starting with an insulating solution, if we increase the chemical potential, the RMS increases until we reach a critical value of μ, where the insulating solution disappears (open circles). Similarly, starting form a metallic solution at low doping, if we reduce the chemical potential, the RMS grows until a critical value of μ, where the metallic solution disappears (open squares). Evidently, the coexistence of solutions can also be observed from the behavior of other quantities, such as the double occupancy, or the low frequency part of the Green’s functions. The latter is shown in Fig. [fig9] for both p − and d − electron components. These results support the claim that the MIT scenario for δ > 0 in the PAM is completely analogous to the one found in the Hubbard model investigations. [fig9] ### hole doping (δ < 0) [!ht] [fig10] In the previous section we showed that upon particle doping, δ > 0, the Mott MIT in the PAM realizes the same physical scenario as the one observed in the single band Hubbard model. While in the latter the particle-hole symmetry implies an identical transition for δ < 0, we shall see that this is not the case in the PAM. A key point to appreciate is that we shall keep all model parameters fixed, with the obvious exception of the chemical potential, which controls the occupation. Therefore, if the chosen model parameters led, for δ > 0, to the identification of the PAM with the HM physics, then one may also expect that this would be the case for δ < 0 as well. Rather surprisingly this turns not to be the case. In this section we shall describe the main physical behavior of the model for the hole doping driven MIT, and in the next we shall argue about the origin of this unexpected result. We begin by showing the effect of hole doping in the DOS. In Fig.[fig10] we plot the change in the p and d components of the DOS as the system evolves from Mott insulator to a hole-doped metallic state (left panels, top to bottom). In the insulator state we observe that the chemical potential is located within the correlation gap, and the lower and upper Hubbard bands can be well appreciated. As we already discussed before, in this Mott insulator state the DOS at low frequencies has mostly d electron character, since the d orbital was initially located at the Fermi energy. Upon hole doping, the chemical potential moves within the lower Hubbard band. The metallization produces a wide and strong quasiparticle peak at the Fermi energy. At small doping, the low frequency part of the spectrum has a characteristic three peaks structure: the lower Hubbard band around ω ∼ − 0.4, the quasiparticle peak that crosses the Fermi level, and the upper Hubbard band at ω ∼ 0.8 An interesting aspect to appreciate is that the quasiparticle peak has, as before, a larger d-character, but, in addition, now it also has a substantial p-electron component. We shall see that this aspect will be consistent with the interpretation of the quasiparticle peak now emerging not from mere delocalization of d-electrons as before, but from the delocalization of a composite object that involves a p-hole and a d-electron spin. We can also observe a transfer of spectral weight, with an increase of the relative intensity of the lower Hubbard band plus quasiparticle, at the expense of a decrease of the upper Hubbard band. The structure that is seen below ω ≈ − 1.5 corresponds to the fully filled band which remains with predominantly p − electron character. The results shown were obtained with high quality QMC data (at least 106 sweeps to reduce the statistical errors) and performed the analytic continuation to the real axis using the maximum entropy method. In the right panels of Fig. [fig10] we present a comparison of QMC and ED-DMRG results for the DOS at heavy doping δ = − 0.28. Similarly as before (cf. Fig. [fig7]), the apparent multi-peak substructures of the ED-DMRG results are not physical and only due to the discrete number of poles that result from a finite number of atomic sites in the auxiliary bath. Compared to the standard ED, this method produces much smoother spectra with greater detail due to the dramatic increase in the number of poles. However, as is also evident from the data, the discreteness due the finite size representation of the bath remains a shortcoming of the method. The overall comparison of the two methods remains, nevertheless, very satisfactory and serves to illustrate their relative advantages and disadvantages. [fig10b] To complete this study, we show in Fig. [fig10b] the detailed evolution of the quasi-particle peak at the Fermi level plus the Hubbard bands as a function of doping. The data correspond to the d − electron density of states obtained from analytic continuation of QMC data. The emergence of a quasiparticle peak at the Fermi energy for the hole doped case may seem, at first sight, similar to the metallic state obtained from particle doping. However, this will not turn to be the case and the physical origin of the two quasiparticle excitations will be shown to be qualitatively different. The reason for this unexpected asymmetry will be discussed in the next section. [fig11] In order to fully underpin the nature of the order of the hole doped transition, we now look for hysteresis effects. Thus, as we did before, we continuously follow the solutions in parameter space. First, we start from the insulator and lower the chemical potential till we obtain a significantly doped metal; and then, we start from the metal and increase μ until we reach again the insulator. Neither our QMC numerical simulations nor the ED-DMRG studies showed any indication of hysteresis effects. The QMC data down to T = 1/128 are shown in Fig. [fig11]. Compared to the results for particle doping (inset), the present ones show a negligible temperature dependence. Thus, up to our current numerical capacity we have to conclude that the metal-insulator transition in the hole doped case is of second order. As no evidence of coexistence has been found, neither in finite or zero temperature calculations, we are led to the conclusion that the hole doping driven transition at T = 0 is continuous, i.e. has a second order character. Thus, it is qualitatively different from the particle doped case, and, consequently, also qualitatively different from the Hubbard model scenario. Of course, we cannot rule out the eventual existence of tiny energy scales (which may modify our proposed scenario) which remain beyond the numerical precision of our methodology. This issue cannot be resolved nor by ED or ED-DMRG data due to the finite frequency cut-off set by the finite size of the clusters to diagonalize. Resolving this issue would probably require NRG study. However this is not fully clear since NRG method requires a good separation of energy scales, which is not the case here. DISCUSSION ========== Our results for the doping driven MIT, that arise from either particle or hole doping of the Mott insulator, show a qualitative asymmetry, thus questioning the expected mapping of the PAM onto the Hubbard model. In this section we shall address in more detail this issue from the perspective of the physical nature of the two MIT taking place in the PAM. Phase diagram ------------- [fig12] To begin, we map out the phase diagram in U − μ parameter space to explore the respective ranges of the first and the second order transitions. Our results are summarized in Fig. [fig12], that was constructed using finite T quantum Monte Carlo and T = 0 ED data. The latter were obtained using the “star-bath” geometry, which we found better suited than the “chain-bath” geometry to study the possible coexistence between the metallic and the insulating regions. The phase diagram in the U − μ plane shows a central V-shaped Mott insulator region with the correlated metallic phases for particle and hole doping, respectively to the right and to the left. A threshold value for the strength of the interaction U (tip of the V-shape boundary) is required to obtain a Mott insulator state. This threshold depends on the value of the “bandstructure” parameters Δ0 and tpd. This feature is analogous to the existence of critical value of the ratio U/D in the one band Hubbard model. The central V-shaped Mott insulating region shows a remarkable asymmetry comparing the hole and the electron sides. In contrast, in the one band Hubbard model, due to the particle-hole symmetry, the V-shape onset of the Mott insulator is symmetric respect to the tip and behaves as Uc ∼ Uc1 ± 2μ, where Uc1 is the value of the interaction at which the insulator disappears. In the PAM, the transition line to the particle doped side behaves as well like $U\_c\sim {\rm const}+2\mu$. However, the transition line to the hole doped metal is almost vertical. We also mapped out the order of the transition along the boundary lines. Upon hole doping, both QMC calculations down to the low temperature T = 1/128, and ED calculations at T = 0 show no trace of coexistent solutions along all the transition line, indicating a second order transition. On the other hand, upon particle doping, the dashed line in the phase diagram displays the region of the parameter space where, for sufficient low temperatures, an insulating state coexists with a metallic state. In fact within ED method at T = 0 we find that the DMFT equations have two different solutions all along the transition line. Within QMC, and down to the low temperature T = 1/128, we found a coexistence region only for a sufficiently large strength of the repulsive interaction U (approximately U ≈ 2 for our specific choice of parameters). However, for smaller values of U, the phase boundary remains strongly dependent on temperature. This suggests that the temperature below which there is a coexistence between metallic and insulating state decreases rapidly approaching the tip of the V-shape. This should be expected, since the two boundary lines, to holes and particle doped metal, eventually merge at the tip. Similarly, we also expect the T = 0 width of the coexistent region to become narrower as one approaches the tip of the V. However, obtaining detailed and precise numerical data there turned out to be beyond our methods’ capabilities. We note that at the Mott insulator state (δ = 0) the occupation of the conduction p − electrons is almost saturated at np = 2 − ν (with ν ≈ 0.1). On the other hand, the occupation of the non dispersive d − electrons which carry strong magnetic moments due to the on-site repulsion is close to one, with nd = 1 + ν (cf Fig. [fig5]). Therefore, there is only a small number ∼ ν of p − holes available to screen a number of order one of d magnetic moments. Therefore, a natural issue to consider is whether the physics associated to the “exhaustion problem” of Nozières (see also related works on the Kondo lattice model and on the PAM ) may play a role in the different transitions at δ greater or smaller than 0. An important point to realize is, however, that the exhaustion situation is even more extreme on the particle side than on the hole-doped side. In fact, while the number of d − electron remain always of order one on both sides, the number of available p − holes is substantially smaller for δ > 0 with respect to the δ < 0 case. Nevertheless, the Hubbard-like first-order transition scenario takes place only on the particle-doped side. In other words, the PAM metal-insulator transition scenario is analogous to the one in the Hubbard model when the PAM is even deeper in the exhaustion limit (δ > 0, np ≈ 2). This implies that while exhaustion should play a role, it is not obviously responsible for the failure of the mapping of the PAM onto the HM for δ < 0. Nature of doped carriers ------------------------ We now address the issue of the physical nature of the metallic states in this system. Although the low frequency part of the DOS of both, particle and hole doped metals, have a similar three peaks structure with dominant d character, they realize physically different states. For instance, their charge compressibility has a very contrasting temperature dependence. In Fig. [fig13] we show the derivative of the total occupation with respect to the chemical potential, κ = ∂ntot/∂μ (an observable proportional to the compressibility) as a function of doping. In the Mott insulator (δ = 0) κ is zero, indicating the incompressible Mott state. As expected, upon doping, k increases, indicating that the system becomes compressible. For small particle doping, the compressibility rapidly increases with the temperature, similarly as in the one band Hubbard model case. Upon hole doping, κ is much less dependent on temperature. [fig13] In the PAM the physics of the metallic states is usually discussed in terms of the screening between the p conduction electrons (or holes) and the magnetic moment of the local d − electrons. In fact, at each lattice site a p − and a d − electron may form a local singlet, which is the underlying idea in the argument of Zhang and Rice (ZR) in the context of high temperature cuprate superconductors. However, if on the particle side of the Mott MIT we have argued that there are essentially no available holes, then the question is, what is screening the d − moments so to produce a normal (but heavy) Fermi liquid metal, analogous to the one in the doped Mott insulator in the Hubbard model? [fig14] The answer to this question is that, similarly to the one band Hubbard model case, the d − electrons are screening themselves. In fact, the local ZR singlet formation does not take place for δ > 0 simply because there are no holes available for screening. Nevertheless, despite the high filling of the p − orbitals, the strong hybridization tpd still allows for delocalization of the d − electrons, through charge fluctuations across the p − sites, with an effective amplitude $t\_{\rm eff}\sim t\_{pd}^2/\Delta$. The key physical point in this process is that since the p − orbitals are almost full, they have a negligible local magnetic moment, so these charge fluctuations take place without significant magnetic p − d coupling. Therefore the magnetic phase coherence of the d − electrons is preserved and, in consequence, a superexchange mechanism between neighboring d sites occurs. Thus, from the point of view of the d − electrons, they have strong magnetic moments, they delocalize keeping their quantum mechanical phase through essentially non-magnetic p-sites, and therefore also experience antiferromagnetic correlations with nearest neighbor d − sites. These physical ingredients are evidently also realized for the carriers in the single band Hubbard model. Therefore we can now rationalize the underlying mechanism for the mapping of the PAM onto the one band Hubbard model at δ > 0, cf. Fig. [fig14a]. On the other hand, the situation is very different as the chemical potential is lowered to dope holes into the system. There, the number of available holes become more significant and they can lock with the robust d − magnetic moments to form local ZR singlets. However, when the d − electrons of these singlets want to delocalize, i.e. hop to the neighboring sites and eventually form a quasiparticle band, they completely loose the information of their spin phase, thus the antiferromagnetic correlations between neighboring d do not build up. In consequence, Hubbard model like physics does not take place and the nature of both the MIT and the ensuing correlated metallic state becomes fundamentally different. That is the key physical reason why the mapping of the PAM with a one band HM is no longer valid, cf. Fig. [fig14b]. In order to substantiate the previous qualitative discussion, we show in Fig. [fig15] the d and p local moment formation as a function of doping for a large strength of the interaction U and for a value slightly above the tip. The local moment formation is defined as ⟨(mαz)2⟩ = ⟨(nα ↑ − nα ↓)2⟩ = nα − 2⟨nα ↑nα ↓⟩, where α = p, d. Notice that the difference between the particle occupation n and the moment directly measures the double occupancies of the sites. In the Mott insulator (δ = 0) the local moment of the d electrons, ⟨(mdz)2⟩, is large because the d − sites are predominantly single occupied due to the effect of U. On the other hand, the p − band is almost fully occupied, and thus the local moment of the p − electron, ⟨(mpz)2⟩, is significantly smaller. As one dopes the Mott insulator with particles or holes, the relative distribution of the local moments among both p and d sites is strikingly asymmetric. Upon particle doping, δ > 0, ⟨(mpz)2⟩ becomes even smaller, since the occupation of the p band get saturated. On the other hand, the d moment decreases more rapidly (and linearly) with the doping, since the charge fluctuations between the single occupied and double occupied d states increase. Therefore these results support the view that for the particle doped side of the transition the p − sites are magnetically inert and, consequently, the d-electrons screen themselves as they form a heavy mass metal. This heavy metal paramagnetic state is physically analogous to the one realized in the Hubbard model case. For the hole doped metal, δ < 0, the p − electron local moment, ⟨(mpz)2⟩, increases linearly with the doping because holes are mostly added to p − sites. In contrast, the opposite behavior occurs for the d local moment, which	arxiv_0000800
Generalized tight p-frames and spectral bounds for Laplace-like operators. ============================================================================ We prove sharp upper bounds for sums of eigenvalues (and other spectral functionals) of Laplace-like operators, including bi-Laplacian and fractional Laplacian. We show that among linear images of a highly symmetric domain, our spectral functionals are maximal on the original domain. We exploit the symmetries of the domain, and the operator, avoiding necessity of finding good test functions for variational problems. This is especially important for fractional Laplacian, since exact solutions are not even known on intervals, making it hard to find good test functions. To achieve our goals we generalize tight p-fusion frames, to extract the best possible geometric results for domains with isometry groups admitting tight p-frames. Any such group generates a tight p-fusion frame via conjugation of arbitrary projection matrix. We show that generalized tight p-frames can also be obtained by conjugation of arbitrary rectangular matrix, with frame constant depending on the singular values of the matrix. Introduction ============ The dependence of the Laplace eigenvalues on the shape of the domain was extensively studied by many authors. In particular, shapes that maximize/minimize spectral functionals are of great interest (see monograph for a comprehensive overview). We find extremizing domains for spectral functionals of a broad family of Laplace-like operators, ranging from bi-Laplacian (and higher order operators) to fractional Laplacian (and other non-local operators). The important common feature of all treated operators is invariance under isometries of the Euclidean space. We derive bounds for sums of eigenvalues of such operators on linearly transformed highly symmetric domains in terms of the eigenvalues on the symmetric domain. We obtain sharp bounds based purely on the symmetries of the domain and the operator, regardless if the eigenvalues and eigenfunctions of the symmetric domain are known explicitly. See for bounds involving commonly used operators. This work extends and to more general operators. Our proofs are based on the theory of tight p-frames, recently studied by Bachoc, Ehler and Okoudjou (). In, we generalize certain aspects of this theory, to find an exact method of evaluating quadratic forms on transformed domains. This part of the paper may be of independent interest. As discovered by Bachoc and Ehler, existence of tight p-frames is strongly connected to the theory of invariant polynomials for irreducible representations of finite groups of symmetries (subgroups of orthogonal groups). We emphasize and exploit this connection even further in to generalize p-fusion frames involving subspace projection matrices, to arbitrary matrices. The frame constants for such generalizations are particularly hard to evaluate. We meet the challenge (in ) using a mixture of techniques spanning variety of fields of mathematics. We use probabilistic arguments to establish two combinatorial formulas involving symmetric polynomials. We use algebra of symmetric functions on abstract alphabet (including alphabet manipulations) to relate our combinatorial identities. Finally, cycle index of the symmetric group provides the simplest form of the generalized p-frame constants. As a corollary of that result we also get a formula for the moments of a generalized χ2-distribution (sum of squares of centered Gaussian random variables with arbitrary variances),. Main results ------------ In this section we present the eigenvalue estimates for a few commonly used Laplace-like operators. Most of these results are special cases of the general Fourier multiplier eigenvalue bound,. We present these special cases due to the importance of the operators as well as relative simplicity of the statements of the results. It is also worth noting that Hardy-Littlewood-Pólya majorization can be used to further generalize all results for sums of eigenvalues to sums of any concave increasing function of eigenvalues (see ). Particularly interesting generalizations involve products of eigenvalues, partial sums of the spectral zeta function and trace of the heat kernel (cf. ). Results for majorized sums can be added to virtually any bound in this paper. For example implies. Our examples naturally split into plate-related higher order multipliers, and fractional probability related multipliers. The common theme is that we transform a domain using an invertible linear transformation on Rd. Throughout this section T will denote the transformation as well as its defining matrix. Clamped plate with tension -------------------------- Consider the eigenvalue problem for the bi-Laplacian operator with tension and clamped boundary conditions $$\begin{aligned} \Delta^2u-\tau \Delta u&=\Gamma u &\text{ on }\Omega,\\ u=\frac{\partial u}{\partial n}&=0 &\text{ on }\partial\Omega.\end{aligned}$$ Its eigenvalues Γi correspond to frequencies of oscillations (energy levels) of a rigid plate Ω with clamped boundary. The constant τ corresponds to the ratio of the lateral tension to the flexural rigidity of the plate. Positive value of τ means that the plate is under tension, while negative τ indicates compression. Note that the Laplacian is a negative operator, hence − sign in front of τ ensures that the higher the tension, the higher the eigenvalues. The eigenvalues Γi were extensively studied both theoretically (e.g. Ashbaugh-Benguria, Kawohl-Levine-Velte, Nadirashvili, Payne, Szegö, Talenti ) and numerically (e.g. Kuttler-Sigillito, Leissa, McLaurin, Shibaoka, Wieners ). Abundance of numerical results clearly suggests theoretical difficulties in finding Γi. Indeed, one can explicitly find only the eigenvalues of balls. We would like to emphasize, that our results naturally split into weaker but more general bounds obtained using classical frames, and stronger bounds requiring higher order frames. Whenever we write that Ω admits a p-frame, we mean that the isometry group of Ω is rich enough to generate p-frames. Note that classical 1-frames are related to irreducibility of the isometry group of Ω via Schur’s lemma (see e.g. Vale-Waldron ). In we give a list of domains admitting p-frames. In order to state the first theorem we need to define appropriate matrix norms. The Schatten norm of order n of matrix T is a sum of the n-th powers of the singular values of T. Equivalently $$\begin{aligned} \\|T\\|\_{n}^{n}=\operatorname{Tr}\left(\sqrt{T^{\dagger}T}\right)^n, \end{aligned}$$ where † denotes the transpose and Tr is the trace of a matrix. In particular when n = 2 we obtain a more familiar Hilbert-Schmidt norm of T. We will use the notation ⋅ ∣τ, Ω to indicate that geometric/spectral quantity is evaluated on Ω, with tension parameter τ. [thmplate] Suppose Ω is a bounded domain in Rd with irreducible group of isometries (e.g. ball, regular polygon/polytope) and T is a linear transformation. Then for the linearly transformed domain T(Ω): $$\begin{aligned} \label{plateweaker} \Gamma\_1+\dots+\Gamma\_n \Big\|\_{\tau C(T^{-1})/\\|T^{-1}\\|\_2^2,T(\Omega)}\le \left.\frac{C(T^{-1})}{d}(\Gamma\_1+\dots+\Gamma\_n) \right\|\_{\tau,\Omega}, \end{aligned}$$ where $$\begin{aligned} C(T^{-1})&=\\|T^{-1}\\|\_4^4. \end{aligned}$$ Furthermore if Ω admits 2-frames (e.g. balls, regular polygons except for squares) then $$\begin{aligned} \label{platestronger} \Gamma\_1+\dots+\Gamma\_n \Big\|\_{\tau D(T^{-1})/\\|T^{-1}\\|\_2^2,T(\Omega)}\le \left.\frac{D(T^{-1})}d(\Gamma\_1+\dots+\Gamma\_n) \right\|\_{\tau,\Omega}, \end{aligned}$$ where $$\begin{aligned} D(T^{-1})&=\frac{\\|T^{-1}\\|\_2^4+2\\|T^{-1}\\|\_4^4}{d+2}\le C(T^{-1}). \end{aligned}$$ In both cases, equality holds when T is a scalar multiple of an orthogonal matrix. Note that we need to vary τ on the left of and to keep equality for all T of the form cId (simple scaling). We prove this result in. The inequality D(T− 1) ≤ C(T− 1) (with equality if and only if T is a scalar multiple of an orthogonal matrix) follows from quadratic-arithmetic mean inequality. However, that does not mean that is sharper than when τ > 0. We do have a smaller constant on the right, but we also have a smaller tension parameter on the left. The inequalities are also not comparable when τ < 0, since we might have negative eigenvalues. Finally, not every irreducible isometry group (symmetric domain Ω) admits 2-frames (see ). In particular, squares do not admit 2-frames. Therefore cannot be used to bound plate eigenvalues of rectangles. This somewhat disappointing limitation is not just an artifact of our method. In we show that is simply wrong for rectangles. In the same section we compare our bounds with known results for elliptical and triangular plates. One can view this theorem as a starting point for various natural simplifications. We state the most interesting corollaries for the 2-frame case, but similar results can be obtained for classical frames and/or more general Fourier multipliers. Analogous simplifications for Laplace eigenvalue problems were developed in and. In a slight expense of accuracy, we can avoid changing τ by fixing volume of the transformed domain. We get: [corstrongplate] Suppose τ = 0, or ∣detT∣ = 1 and τ > 0. If Ω admits 2-frames then $$\begin{aligned} \left.\Gamma\_1+\dots+\Gamma\_n \right\|\_{\tau,T(\Omega)}\le \left.\frac{D(T^{-1})}d(\Gamma\_1+\dots+\Gamma\_n) \right\|\_{\tau,\Omega}. \end{aligned}$$ The proof is essentially identical to the proof of and will be omitted. Instead of the constants involving matrix norms, we can also express the 2-frame case in a more geometric way. Let I2p(Ω) denote the polar 2p-moment of mass of Ω around the origin $$\begin{aligned} I\_{2p}(\Omega)=\int\_\Omega \|x\|^{2p} \,dx.\end{aligned}$$ See for details. So that I2 is the polar moment of inertia of the domain and I0 = V is its volume. Then gives the following equivalent restatement of. [plateiso] Suppose Ω admits 2-frames. If τ = 0 or ∣detT∣ = 1 and τ > 0, then $$\begin{aligned} (\Gamma\_1+\dots+\Gamma\_n)V^{4/d} \Big\|\_{\tau,T(\Omega)}\cdot \left.\frac{V^{1+4/d}}{I\_4}\right\|\_{T^{-1}(\Omega)} \end{aligned}$$ is maximal when T is a scalar multiple of an orthogonal matrix. Note that both factors above are scale invariant, and it is not possible to combine them into one simpler quantity on a single domain (see the discussion in ). When n = 1 and τ = 0, minimization of the first factor (evaluated on T(Ω)) reduces to the classical Rayleigh conjecture for the lowest plate eigenvalue, and balls should be the minimizers among all domains. However this is only known in dimensions 2 (Nadirashvili ) and 3 (Ashbaugh-Benguria ). In higher dimensions partial results were obtained by Talenti and Ashbaugh-Laugesen. The second, purely geometric factor in is maximal for balls, hence it can be viewed as a compensating factor, turning lower isoperimetric bound into a more general upper bound. Finally, a simple relation between moments of mass of T(Ω) and T− 1(Ω) in two dimensions, see, gives: [corplateiso] If planar Ω admits 2-frames then simplifies to the following: $$\begin{aligned} \left.(\Gamma\_1+\dots+\Gamma\_n)A^2\cdot \frac{A^3}{I\_4}\right\|\_{\tau,T(\Omega)} \end{aligned}$$ is maximal when T is a scalar multiple of an orthogonal matrix. In particular, among triangles, the quantity above is maximal for equilateral triangles. Similarly, among ellipses, the balls are maximizers. Note that square might be a maximizer among all parallelograms in the above corollary, even though does not hold, since the corollary is weaker than. Indeed, neither I4 nor eigenvalues for squares can be estimated using 2-frames, and there could be some case-specific cancellation. Finally, extends to a related plate buckling problem (see for detailed exposition). The goal is to find a critical value Λ of the tension parameter τ, that forces plate to buckle. The simplest, two-dimensional version of reads If planar Ω admits 2-frames, then the buckling eigenvalue Λ satisfies: $$\begin{aligned} \Lambda A \cdot\frac{AI\_2}{I\_4}\Big\|\_{T(\Omega)} \end{aligned}$$ is maximal when T is a scalar multiple of an orthogonal matrix. Again, the first (scale invariant) factor should be minimal for balls among all domains, but this Pólya-Szegö conjecture is still open. Bochner subordinators --------------------- We also prove a general result for sums of eigenvalues of the generators of arbitrary subordinated Brownian motions. More precisely, for any complete Bernstein function Ψ (see ), one defines a semigroup associated with a subordinated Brownian motion with generator Ψ( − Δ). Here we state theorems for the most often used cases, while proofs and general results are contained in. Perhaps the best known example of a subordinator is an α/2-stable subordinator Ψ(t) = tα/2 with 0 < α < 2. The resulting operator is the fractional Laplacian ( − Δ)α/2 on a domain Ω with Dirichlet boundary condition outside of Ω. This operator generates a semigroup of the symmetric α-stable process killed while exiting Ω. It is worth noting that this operator is not the same as the α/2-power of the Dirichlet Laplacian. Hence its spectrum is not just a power of the Dirichlet spectrum (see Chen-Song ). The spectral theory of the fractional Laplacian is of great interest. The smallest eigenvalue of this operator satisfies Rayleigh-Faber-Krahn isoperimetric inequality (Bañuelos et al. ). More precisely, among all domains with fixed area/volume, ball has the least smallest eigenvalue. Various spectral properties of the fractional Laplacian have been studied by e.g. Bañuelos-Kulczycki, Chen-Song, Frank-Geisinger, Yolcu-Yolcu. For a comprehensive overview see. Numerical results have also been obtained: for an interval by Kulczycki et al., for a ball by Dyda. Clearly, it is not possible to find exact eigenvalues for balls, or even intervals. As in the plate case, we prove bounds for sums of eigenvalues involving norms of the transformation T. Here we only state the simplified isoperimetric upper bound (cf. ) [thmfractional] (Fractional Laplacian) For any Ω with irreducible isometry group and 0 < α ≤ 2, the eigenvalues of ( − Δ)α/2 satisfy: $$\begin{aligned} \left.\left(\lambda\_1+\dots+\lambda\_n\right)^{2/\alpha} V^{2/d}\right\|\_{\alpha,T(\Omega)}\cdot \left.\frac{V^{1+2/d}}{I\_2}\right\|\_{T^{-1}(\Omega)} \end{aligned}$$ is maximal when T is a scalar multiple of an orthogonal matrix. The special case α = 2 reduces to (upper bound for the sums of Dirichlet eigenvalues of the classical Laplacian). As in the plate case we get a significantly simpler statement in two dimensions (cf. ). In dimension 2 $$\begin{aligned} \left.\left(\lambda\_1+\dots+\lambda\_n\right)^{2/\alpha}A\cdot \frac{A^2}{I\_2}\right\|\_{\alpha,T(\Omega)} \end{aligned}$$ is maximal when T is scalar multiple of an orthogonal matrix. In particular, disk is the extremizer among ellipses, square among parallelograms and eqiulateral among all triangles. We can also use a p-frame-like identity with p = 1/2 (even though p is an integer in true frames), available for the isometry group of the disk, to get the following improved result for ellipses, involving the isoperimetric ratio A/L2: [thmperimeter] Let E be an ellipse with perimeter L and area A. Then for α ≤ 1, $$\begin{aligned} \left.(\lambda\_1+\dots+\lambda\_n)^{2/\alpha} A\cdot \frac{A}{L^2}\right\|\_{\alpha,E} \end{aligned}$$ is maximal when E is a disk. Note that λ1A2/L2 for the Dirichlet Laplacian is a relatively well behaved, bounded quantity, but with degenerate domains as extremizers (Polya and Makai ), instead of balls. Therefore, it would be interesting to check what is the extremal domain in our case, among all convex domains, even just for the first eigenvalue. Fix mass m ≥ 0. Relativistic subordinator $\Psi(t)=\sqrt{m^2+t}-m$ leads to another important operator: the Klein-Gordon operator $\sqrt{m^2-\Delta}-m$ that models relativistic particles in quantum mechanics (see Harrell-Yolcu and Yolcu ). This example requires more care, as the relativistic constant must change with the linear transformation, as did the tension constant for plates. For that reason we use ⋅ ∣m, Ω to indicate that a quantity is evaluated on Ω with mass m. [thmrelativistic] (Klein-Gordon operator) Sums of eigenvalues of $\sqrt{m^2-\Delta}-m$ satisfy $$\begin{aligned} \lambda\_1+\dots+\lambda\_n \Big\|\_{md/\\|T^{-1}\\|\_{2},T(\Omega)}\le \left.\frac{\\|T^{-1}\\|\_{2}}{\sqrt{d}}\left(\lambda\_1+\dots+\lambda\_n\right)\right\|\_{m,\Omega}. \end{aligned}$$ Furthermore, if ∣detT∣ = 1, then $$\begin{aligned} \left.\left(\lambda\_1+\dots+\lambda\_n\right)^2 V^{2/d}\right\|\_{m,T(\Omega)}\cdot \left.\frac{V^{1+2/d}}{I\_2}\right\|\_{T^{-1}(\Omega)} \end{aligned}$$ is maximal when T is an orthogonal transformation. Note that as in the fractional Laplacian case, we also have a simplified two-dimensional version, without the inverse transformation T− 1. Finally, when m = 0 this last result reduces to the fractional Laplacian with α = 1. Our final example involves a generic complete Bernstein function Ψ, and it unifies and (additional assumption ∣detT∣ = 1 is irrelevant for the fractional Laplacian due to homogeneity of Ψ). For any concave Ψ (in particular any Bernstein function) there exists 0 < β ≤ 1 such that Ψ(t/c) ≤ c− βΨ(t) for any c > 1. Therefore [thmsubor] If ∣detT∣ = 1, then the eigenvalues of Ψ( − Δ) satisfy $$\begin{aligned} \lambda\_1+\dots+\lambda\_n \Big\|\_{\Psi,T(\Omega)}\le \left.\frac{\\|T^{-1}\\|\_{2}^{2\beta}}{d^\beta}\left(\lambda\_1+\dots+\lambda\_n\right)\right\|\_{\Psi,\Omega}. \end{aligned}$$ Note that for fractional Laplacian we have β = α/2 and for the relativistic operator β = 1/2. General approach ================ Now we develop the general context that will be used to handle all operators described in the main results section. Note that these are only examples of what we can work with, albeit the best known ones. Quadratic forms in frequency domain ----------------------------------- Define the Sobolev space Hβ(Rd) as the subset of L2(Rd), such that the Fourier transform of a function f ∈ L2(Rd) satisfies $$\begin{aligned} \\|f\\|\_\beta^2 = \int\_{{{{\mathbb{R}}}^d}}\|\xi\|^{2\beta} \|\widehat f\|^2 d\xi<\infty.\end{aligned}$$ Then ∥f∥β is the Sobolev seminorm and ∥f∥0 is the standard L2 norm. Note that when β is an integer, this coincides with the classical definitions involving derivatives of f. The subspace H0β(Ω) of Hβ(Rd) is a closure of Cc∞(Ω) in the Sobolev norm $\sqrt{\\|f\\|\_0^2+\\|f\\|\_\beta^2}$. See for a detailed discussion about various ways to define fractional Sobolev spaces, their equivalence and embedding properties. Consider a weakly defined self-adjoint linear operator A with the domain H0β(Ω) (possibly fractional Sobolev space). That is, for any u, v ∈ Cc∞(Ω) we define the symmetric quadratic form corresponding to A $$\begin{aligned} Q\_A(u,v)=\int\_\Omega (Au)(x) v(x) dx=\int\_\Omega u(x) (Av)(x) dx.\end{aligned}$$ This form can then be extended to u, v ∈ H0β(Ω). In many classical cases, including Dirichlet Laplacian, one can use Fourier transform to rewrite the quadratic form in the frequency domain. Note that the assumption that u ∈ H0β(Ω) (informally: function u is zero on the boundary (or outside Ω for fractional cases) as are all its derivatives up to order β) means that extending with u = 0 outside Ω gives a function in Hβ(Rd). Parseval’s theorem can now be used to justify a frequency domain formula for the quadratic form $$\begin{aligned} \label{Qfourier} Q\_A(u,v)=\int\_{{{{\mathbb{R}}}^d}}s(\xi) \widehat{u}(\xi) \overline{\widehat{v}(\xi)} d\xi,\end{aligned}$$ where s(ξ) is called the Fourier multiplier (or symbol) corresponding to A. In particular, classical Laplacian corresponds to s(ξ) = ∣ξ∣2, while powers of the Laplacian with Dirichlet boundary conditon can be defined as Fourier multipliers ∣ξ∣2β with the domain H0β(Ω). In this paper we only consider multipliers s(ξ) = f(∣ξ∣2) with at most polynomially growing bounded below function f (in particular any linear combination of power functions with positive coefficient for the highest power). Since ∣ξ∣2 is the Fourier multiplier for the Dirichlet Laplacian on Ω, we call these Laplace related multipliers. The multiplier only depends on the length of ξ, hence it is invariant under isometries in frequency domain. This corresponds to operators A invariant under isometries of physical space, just like the classical Laplacian. We are interested in the interaction of linear transformations with these operators. In particular, how does the spectrum of the operator change under such transformation? Consider an invertible linear map T : Rd → Rd. We will abuse the notation slightly, by using T to denote both the linear transformation, and the matrix that defines it. The Fourier transform almost commutes with T, since $$\begin{aligned} \widehat{u\circ T}(\xi)&=\int\_{{{{\mathbb{R}}}^d}}u(Tx) \exp(-2\pi i \xi\cdot x)dx=\int\_{{{{\mathbb{R}}}^d}}u(y)\exp(-2\pi i \xi\cdot T^{-1}y)\|\det T\|^{-1}dy \\&= \int\_{{{{\mathbb{R}}}^d}}u(y)\exp(-2\pi i T^{-\dagger}\xi\cdot y)\|\det T\|^{-1}dy=\|\det T\|^{-1}\widehat u(T^{-\dagger}\xi),\end{aligned}$$ where T− † denotes the transposed inverse of the matrix T. This allows us to evaluate the quadratic form on the linearly transformed functions $$\begin{aligned} Q\_A(u\circ T,u\circ T)&=\int\_{{{{\mathbb{R}}}^d}}f(\|\xi\|^2)\|\widehat{u\circ T}\|^2d\xi=\|\det T\|^{-2}\int\_{{{{{\mathbb{R}}}^d}}} f(\|\xi\|^2)\|\widehat u(T^{-\dagger}\xi)\|^2d\xi\nonumber \\&= \|\det T\|^{-2}\int\_{{{{{\mathbb{R}}}^d}}} f(\|T^{\dagger}\xi\|^2)\|\widehat u(\xi)\|^2\|\det T^{\dagger}\|d\xi\nonumber \\&= \|\det T\|^{-1}\int\_{{{{{\mathbb{R}}}^d}}}f(\|T^{\dagger}\xi\|^2)\|\widehat u\|^2d\xi.\label{QT}\end{aligned}$$ Note that f is bounded below and grows at most polynomially, hence f(∣ξ2∣) ≈ f(∣T†ξ∣2) and the last integral converges. Furthermore, taking f(t) = t2β implies that u ∘ T belongs to Hβ(Rd) whenever u ∈ Hβ(Rd). Finally, if Cc∞(Ω) ∋ un → u ∈ H0β(Ω) in the Hβ(Rd) norm, then un ∘ T → u ∘ T by a similar calculation. Hence u ∘ T is in the domain of the operator A if and only if u is in its domain. If T is an orthogonal matrix, we get QA(u ∘ T, v ∘ T) = QA(u, v) (showing invariance of QA under isometries). While for T = cId and f homogeneous of degree β we get QA(u ∘ T, u ∘ T) = c2β − dQA(u, u) (use with the particular choice of f and T). For general matrices T, one cannot immediately simplify the quadratic form. However, p-frames described in the next section allow us to simplify many interesting cases. Note also that L2 norm (equivalent to β = 0) satisfies $$\begin{aligned} \\|u\circ T\\|\_0^2=\|\det T\|^{-1}\\|u\\|\_0^2.\end{aligned}$$ From now on, we also assume that H0β(Ω) embeds compactly into L2(Ω). When β = 1, this is true for any bounded domain Ω (Rellich’s theorem), while H0β(Ω) with β > 1 imbeds isometrically into H01(Ω), which then imbeds compactly into L2. For the case β < 1 we refer the reader to. Boundedness below of f implies that for some c the operator A + c is elliptic. This, together with compact embedding implies discreteness of the spectrum (see Osborn-Babuška, or Blanchard-Brüning, cf. Section 4 of Laugesen ). Geometric averaging and eigenvalue bounds ----------------------------------------- In order to find the smallest eigenvalue one can minimize the Rayleigh quotient (see e.g. Bandle ) $$\begin{aligned} R[u]&=\frac{Q\_A(u,u)}{\\|u\\|\_0^2},\\ \lambda\_1&=\inf\_u R[u].\end{aligned}$$ Similarly one can find the sum of consecutive eigenvalues $$\begin{aligned} \sum\_{i=1}^n \lambda\_i &= \inf\left\{ \sum\_{i=1}^n R[u\_i]:\; u\_i \text{ mutually $L^2$-orthogonal}\right\}.\end{aligned}$$ Consider highly symmetric domain Ω (with irreducible symmetry group G) and its linear image T(Ω). Suppose ui are the orthonormal eigenfunctions for λi(Ω). Due to symmetry of Ω, for any isometry U ∈ G, functions ui ∘ U are also orthonormal eigenfunctions for Ω. Usually, these are not the same eigenfunctions, but they belong to the same eigenvalues, and are still orthogonal. Consider functions vi : T(Ω) → R defined by vi(x) = ui ∘ U ∘ T− 1(x). Linear transformations preserve L2 orthogonality, and zero boundary condition. More formally, ui ∘ U ∘ T− 1 ∈ H0β(T(Ω)). We get $$\begin{aligned} \lambda\_1\Big\|\_{T(\Omega))}\le R[v\_1]=\frac{Q\_A(u\circ U\circ T^{-1},u\circ U\circ T^{-1})}{\\|u\circ U\circ T^{-1}\\|\_0^2}=\frac{ \int\_{{{{{\mathbb{R}}}^d}}}f(\|T^{-\dagger}U^\dagger \xi\|^2)\|\widehat u\|^2d\xi }{\\|u\\|\_0^2}\end{aligned}$$ and similarly for sums of eigenvalues. But the left side is independent of U, hence we can average over the group G of symmetries of Ω (or any of its subgroups). Therefore $$\begin{aligned} \label{average1} \lambda\_1\Big\|\_{T(\Omega)}\le \frac{1}{\|G\|}\sum\_{U\in G}\frac{\int\_{{{{{\mathbb{R}}}^d}}}f(\|T^{-\dagger}U^\dagger \xi\|^2)\|\widehat u\|^2d\xi }{\\|u\\|\_0^2}\end{aligned}$$ Similarly, when the ui are orthonormal, $$\begin{aligned} \label{averagesum} \lambda\_1+\dots+\lambda\_n\Big\|\_{T(\Omega)}\le \frac{1}{\|G\|}\sum\_{U\in G}\sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}f(\|T^{-\dagger}U^\dagger \xi\|^2)\|\widehat u\_i\|^2d\xi.\end{aligned}$$ In the simplest case f(t) = t, the averaging of this type was used in to obtain sharp bounds for eigenvalues of the Laplacian, by separating ∣ξ∣2 from T. This required irreducibility of G as a subgroup of the orthogonal group on Rd, but ui did not need to be known explicitly. In we generalize the theory of p-frames so that optimal averaging can be applied to other power functions f, leading to the plate results described in. Note that averaging can always be performed if G is the full orthogonal group (we average with respect to Haar measure on this group). See for an application of this approach. Note also that the Haar measure averaging generalizes to nonlinear transformations of balls, assuming f(t) = t. A bound for general Fourier multipliers --------------------------------------- Now we introduce a classical 1-frame based, general approach that allows us to handle almost arbitrary multipliers f(∣ξ∣2). Recall the Schatten 1-norm $$\begin{aligned} \\|T\\|\_1 = \operatorname{Tr}\sqrt{T T^\dagger},\end{aligned}$$ where † denotes the transpose of a matrix. Assume M is symmetric and nonnegative definite, so that ∥M∥1 = TrM. Write the spectral decomposition M = VEV− 1 with orthonormal V and diagonal E. Define $$\begin{aligned} \Phi(M) = V\Phi(E)V^{-1},\end{aligned}$$ with entry-wise action of Φ on the diagonal matrix E. For nonnegative Φ we have $$\begin{aligned} \\|\Phi(M)\\|\_1 = \operatorname{Tr}\Phi(M) = \operatorname{Tr}\Phi(E).\end{aligned}$$ For arbitrary Φ the trace still makes sense (sum of the diagonal elements or eigenvalues is well defined), though the Schatten 1-norm is no longer equal to the trace. Write f = Φ1 − Φ2 and for arbitrary matrix T define $$\begin{aligned} \label{newmult} F[\Phi\_1,\Phi\_2,T](t) = \frac{1}{d}\operatorname{Tr}\Phi\_1(t T^{-1}T^{-\dagger})-\Phi\_2\left(\frac{t}{d}\operatorname{Tr}T^{-1}T^{-\dagger} \right)\end{aligned}$$ Note that this quantity strongly depends on the choice of Φi. However, we have the following weak form of invariance. For any linear function l $$\begin{aligned} F[\Phi\_1+l,\Phi\_2+l,T](t)=F[\Phi\_1,\Phi\_2,T](t).\end{aligned}$$ We will use this property in in the proof of the next theorem. Any f ∈ Cloc1, 1(0, ∞) can be decomposed as a difference of two convex functions, therefore the following theorem can be applied to virtually arbitrary multipliers. Due to a complicated statement of the result, we employ the notation ⋅ ∣f, Ω, which indicates that the quantities are evaluated on Ω and the operator has the Fourier multiplier f(∣ξ∣2). [thmgeneralmult] Suppose it is possible to write f(t) = Φ1(t) − Φ2(t), where Φi are convex. Let T be an invertible linear transformation and Ω be a domain with irreducible isometry group. Then $$\begin{aligned} \lambda\_1+\dots+\lambda\_n \Big\|\_{f,T(\Omega)}\le \lambda\_1+\dots+\lambda\_n\Big\|\_{F[\Phi\_1,\Phi\_2,T],\Omega}, \end{aligned}$$ with equality when T is a multiple of an orthogonal matrix. Furthermore, if f = Ψ ∘ Φ with concave Ψ and convex Φ then $$\begin{aligned} \lambda\_1+\dots+\lambda\_n \Big\|\_{f,T(\Omega)}\le \lambda\_1+\dots+\lambda\_n\Big\|\_{\Psi\circ F[\Phi,0,T],\Omega}. \end{aligned}$$ The proof is postponed to. Note that this result is very broad, but awkward to apply, in practice. The Fourier multiplier on the right is not the same as the multiplier on the left. Thus we compare eigenvalues of different operators. Cases listed in the previous section use special form of f to simplify the statement of this theorem. Also, presents a much simpler looking case Φ1(t) = 0 (or equivalently Φ(t) = t). Finally, if f ∈ C2(0, ∞) and fʺ(t) ≥ − c, then we can convexify f by taking Φ1(t) = f(t) + ct2 and Φ2(t) = ct2. This approach applies to any polynomial f(t) of degree at least 3 with positive leading term. Furthermore, higher order frames lead to improved results, as described in. As with all our results, Hardy-Littlewood-Pólya majorization (cf. ) implies: [thmgeneralmajorized] Let f and Ω be as in. For any concave increasing function ψ $$\begin{aligned} \psi(\lambda\_1)+\dots+\psi(\lambda\_n) \Big\|\_{f,T(\Omega)}\le \psi(\lambda\_1)+\dots+\psi(\lambda\_n)\Big\|\_{F[\Phi\_1,\Phi\_2,T],\Omega}, \end{aligned}$$ Generalized p-frames from finite symmetry groups ================================================== General setup ------------- Bachoc and Ehler considered the following generalization of the classical frames. A set of vectors {vi}i = 1N in Rd is called a tight p-frame, for an integer p, if $$\begin{aligned} \label{pframe} \frac{1}{N}\sum\_i \langle x, v\_i\rangle^{2p} = \frac{(1/2)\_p}{(d/2)\_p} \|x\|^{2p}\|v\_i\|^{2p},\end{aligned}$$ where ( ⋅ )p denotes the rising factorial. In the same paper, authors also consider a more general tight p-fusion frames. The inner products ⟨x, vi⟩ can be viewed as a projection of x onto vectors vi. Replace these rank 1 projections with arbitrary projections on subspaces Vi to obtain tight p-fusion frames. We are interested in a special case of fixed dimension Vi, say dim(Vi) = k. The set of the projections PVi (given as orthogonal matrices) is a tight p-fusion frame if for any x $$\begin{aligned} \label{PVframe} \frac{1}{N}\sum\_i \|P\_{V\_i}x\|^{2p} = \frac{(k/2)\_p}{(d/2)\_p} \|x\|^{2p}.\end{aligned}$$ Note that when p = 1, the definitions above reduce to classical tight frames and classical tight fusion frames (see Casazza et al.. Let G be a finite group of symmetries of a polytope in dimension d (subgroup of the orthogonal group O(d)). This group can be used to generate projections PVi using the orbit of a single subspace V. The orbit U(V), for U ∈ G, may contain the same subspace many times, resulting in Vi = Vj for some i and j. This is not explicitly forbidden in the definition of the tight p-fusion frame. In fact, the union of two tight p-fusion frames is again a tight p-fusion frame. One can even combine frames with different subspace dimensions, resulting in a new frame constant. Note also that different choices of V may result in different lengths of the orbits. Nevertheless, group G generates a tight p-fusion frame if $$\begin{aligned} \frac{1}{\|G\|} \sum\_{U\in G} \|P\_{U(V)}(x)\|^{2p}=\frac{1}{\|G\|} \sum\_{U\in G} \|UP\_{V}U^{-1}x\|^{2p}=\frac{(k/2)\_p}{(d/2)\_p} \|x\|^{2p}.\end{aligned}$$ The existence of tight p-fusion frames generated by G is equivalent to uniqueness of the invariant polynomials of degree 2p for a given group G. In particular, if G is not irreducible, then there exists an invariant hyperplane (orbit not spanning the whole Rd), hence also invariant polynomial of degree 1. Its square is also invariant. But ∣x∣2 is a different invariant of degree 2, since it is invariant even under the full orthogonal group. This shows necessity of the uniqueness for p = 1 (classical frames). The following lemma states sufficiency condition for existence of tight p-frames. The argument used in its proof will lead to generalized tight p-frames later in this section. [][framelemma] If a finite symmetry group G acting as isometries of Rd has unique invariant polynomial of degree 2p, then any orbit of G generates a tight q-frame for any q ≤ p. Group G cannot have fundamental invariant polynomials of degree 2q > 2 with q ≤ p. Otherwise we could get more than one invariant polynomial of degree 2p. Indeed, any subgroup of the orthogonal group has invariant quadratic ∣x∣2. If there is a fundamental invariant f of degree 2q < 2p, then ∣x∣2p − 2qf would be a 2p-degree invariant, different then ∣x∣2p. Hence there is a unique invariant polynomial of degree 2q, equal ∣x∣2q, for any q ≤ p. We want to prove the tight q-frame identity with q ≤ p for an orbit of G. That is $$\begin{aligned} \label{cdotG} F(v):=\frac{1}{\|G\|}\sum\_U \langle x,Uv\rangle^{2q} = \frac{(1/2)\_q}{(d/2)\_q} \|x\|^{2q}\|v\|^{2q}.\end{aligned}$$ Note that the right side is an invariant polynomial in v. The inner product on the left is linear in v, hence the left side is a polynomial of degree 2q in v. Suppose Uʹ ∈ G. $$\begin{aligned} F(U'v)=\frac{1}{\|G\|}\sum\_U \langle x, UU'v\rangle^{2q}=\frac{1}{\|G\|}\sum\_{UU'} \langle x, UU'v\rangle^{2p}=\frac{1}{\|G\|}\sum\_U \langle x, Uv\rangle^{2q}=F(v),\end{aligned}$$ using group property of G. Hence F(v) is an invariant polynomial of degree 2q. Therefore it must equal c∣v∣2q. The left side and the right side differ by a constant (in v, may still dependent on x), by uniqueness. However U is an isometry, hence ⟨x, Uv⟩ = ⟨U− 1x, v⟩. Therefore the same argument shows that both sides differ by a constant dependent only on v. We see that holds with an unknown constant (independent of x and v). This constant is the same for any tight p-frame and is computed in. We compute the same constant in, as a special case of the polynomial Fp from. Generalized tight p-frames ---------------------------- The following reduction is crucial in working with classical frames as well as p-frames : $$\begin{aligned} \langle x,Uy\rangle^2=\operatorname{Tr}(P\_x UP\_yU^{-1})=\operatorname{Tr}(P\_x P\_{Uy})=\langle P\_x,P\_{Uy}\rangle\_{HS}.\end{aligned}$$ where Px = xx† is a projection matrix, and ⟨ ⋅ , ⋅ ⟩HS is the Hilbert-Schmidt inner product of matrices. A tight 2-frame identity, using above notion, states that $$\begin{aligned} \frac{1}{\|G\|}\sum\_U \langle P\_x,P\_{Uy}\rangle\_{HS}^2 = \frac{3}{d(d+2)}\|P\_x\|\_{HS}^2\|P\_y\|\_{HS}^2.\end{aligned}$$ Therefore, matrices PUy form a tight frame in the space of rank 1 matrices equipped with Hilbert-Schmidt inner product. We will show that the same matrices form a tight frame in the space of all symmetric matrices. [theoremframes] Let G be a group of symmetries with unique invariant polynomial of degree 2p. Then for any symmetric matrix M, projections PUy formed by the orbit Uy satisfy $$\begin{aligned} \label{matrixframe} \frac{1}{\|G\|}\sum\_U \langle M,P\_{Uy}\rangle\_{HS}^p = F\_{p}(\sigma(M))\|P\_y\|\_{HS}^p=F\_{p}(\sigma(M))\|y\|^{2p}, \end{aligned}$$ where Fp(σ(M)) is a homogeneous, symmetric polynomial of degree p, and σ(M) is the multiset of eigenvalues of M (with repeated elements for multiple eigenvalues). Note that ⟨M, PUy⟩HS = ⟨U− 1MU, Py⟩HS. Hence orbit of M under conjugation with group G forms a tight-frame-like object. However, even full orthogonal group acting by conjugation on matrices cannot change the spectrum of the matrix. Hence any symmetric polynomial in σ(M) is invariant under this action. Therefore, polynomials Fp are not just ∣σ(M)∣p, as is the case for the vector based actions (cf. ). Note that the left side of is a polynomial of degree 2p in y. It is also invariant under G. As in, we get that holds up to a constant Fp(M) depending only on M. However, the left side is a polynomial of degree p in the entries mij of the matrix M. Therefore, the constant term Fp(M) is a polynomial in the entries mij of M. This polynomial is obviously homogeneous. We need to show that it only depends on eigenvalues as a multiset (counting multiplicities). That is Fp is a symmetric homogeneous polynomial. Start by writing the spectral decomposition of M $$\begin{aligned} M=VEV^{-1},\end{aligned}$$ with orthonormal V and diagonal E. For any vector y $$\begin{aligned} \langle VEV^{-1},P\_{y}\rangle\_{HS}=\operatorname{Tr}(VEV^{-1}P\_{y})=\operatorname{Tr}(EV^{-1}P\_{y}V)= \langle E,P\_{V^{-1}y}\rangle\_{HS}. \end{aligned}$$ Consider Haar measure μ on the orthogonal group, that is normalized to μ(O(d)) = 1. We have $$\begin{aligned} F\_{p}(M)\|y\|^{2p}&=\int\_{O(d)} F\_{p}(M)\|Wy\|^{2p}\,d\mu(W)=\int\_{O(d)} \frac{1}{\|G\|}\sum\_U \langle M,P\_{UWy}\rangle\_{HS}^p \,d\mu(W) \\&= \frac{1}{\|G\|}\sum\_U \int\_{O(d)} \langle E,P\_{V^{-1}UWy}\rangle\_{HS}^p \,d\mu(W) \end{aligned}$$ But Haar measure is invariant under actions of the group it is defined on, hence we can substitute V− 1UW → W to get $$\begin{aligned} \label{intOd} F\_{p}(M)\|y\|^{2p}=\int\_{O(d)} \langle E,P\_{Wy}\rangle\_{HS}^p \,d\mu(W) \end{aligned}$$ Therefore polynomial Fp(M) depends only on the eigenvalue matrix E. However, we can use permutation matrices (subgroup of O(d)) to permute the diagonal entries in E (spectral decomposition is unique up to the order of the diagonal entries in E). Hence Fp only depends on eigenvalues as a multiset (it is symmetric). For arbitrary matrix T, define its singular values s(T) as the multiset of eigenvalues of the matrix $\sqrt{T^\dagger T}$. [corframes] For any matrix T with d columns (possibly rectangular), and any group G with unique invariant polynomial of degree 2p $$\begin{aligned} \frac{1}{\|G\|}\sum\_U \|TUx\|^{2p} = F\_{p}(s^2(T))\|x\|^{2p}. \end{aligned}$$ Note that $$\begin{aligned} \|TUx\|^2=Tr(TUx(TUx)^\dagger)=Tr(TUxx^\dagger U^{-1}T^{\dagger})=Tr(T^\dagger T U P\_xU^{-1})=\langle T^\dagger T,P\_{Ux}\rangle\_{HS}, \end{aligned}$$ and T†T is symmetric. In the theorem and the corollary above the finite group G can be replaced with full orthogonal group O(d). This also means that the sum is replaced with the Haar measure integral. Finally, any restriction based on uniqueness of the invariant polynomials no longer applies. Hence G = O(d) can be used with arbitrary value of p in any dimension. Groups with unique invariant polynomial of degree 2p (admitting p-frames) ----------------------------------------------------------------------------- Existence of the invariant polynomials is a very well understood subject. In particular one can easily find a table of the degrees of fundamental invariants for reflection groups (see Benson and Grove ). In case a symmetry group is missing from a table (e.g. groups of rotations, without reflections) one can use Molien’s series to check the number of linearly independent invariants of a given degree. See also for SAGE implementation of a Molien series. Below we list most of the interesting examples. As we already pointed out, every subgroup of the orthogonal group has a quadratic fundamental invariant ∣x∣2. The full symmetry group of the tetrahedron, cube and octahedron (as well as their higher dimensional equivalents, Coxeter groups An, Cn and Dn) have a fundamental invariant of degree 4. Therefore the uniqueness fails in these cases, since ∣x∣4 is also invariant. Hence those groups do not admit tight p-frames with p > 1. Nevertheless, in we will construct non-tight p-frames for these groups, and use them to get bounds for Laplace-like operators. In dimension 3, only icosahedral group (Coxeter H3) does not have a fundamental invariant of degree 4, but it does have one of degree 6. Hence icosahedral symmetry admits tight 2-frames, but not 3-frames. The rotation group of the icosahedron (subgroup of index 2 of H3) does not have a degree 4 fundamental invariant either, as we check using SAGE in. In dimension 4, the group of symmetries of a 24-cell (Coxeter F4) admits 2-frames, while the group of symmetries of 120-cell (Coxeter H4) allows for 5-frames. Finally, Coxeter groups E6 and E7 admit 2-frames and E8 allows for 3-frames. All these examples show that in dimensions at least 3, there are few finite groups admitting higher order frames, possibly none. However, one can always get an infinite frame using full orthogonal group (symmetries of a ball) and Haar measure integral instead of summation. Dimension 2 is the most interesting, since the group of symmetries I2(n) of a regular n-gon has only two fundamental invariants. One of course of degree 2, the other of degree n. Therefore I2(n) admits p-frames for any p < n/2. Furthermore, if n is odd, we also get p-frames for any p < n. In particular symmetry of the equilateral triangle allows for 2-frames, while regular pentagon admits 4-frames. Somewhat surprisingly, symmetry of the square is not enough to allow even 2-frames. On the other hand, the fact that squares and cubes do not admit 2-frames is equivalent to Pythagorean theorem holding only with squares of the lengths of the sides of a right triangle (simplex). Polynomials Fp ================== In this section we derive a formula for the polynomials Fp used in the statement of and. We start with formula with unit vector y and diagonal matrix E with entries e1, …, ed. It is enough to consider positive ei, and for convenience we choose to work with matrix E2, instead of E. $$\begin{aligned} \label{oversphere} F\_{p}(E^2)=\int\_{O(d)} \langle E^2,P\_{Uy}\rangle\_{HS}^p,d\mu(U)=\int\_{O(d)} \|EUy\|^{2p}\,d\mu(U)=\frac{1}{\|S^{d-1}\|}\int\_{S^{d-1}} \|E\theta\|^{2p} dS(\theta),\end{aligned}$$ since Haar measure on O(d) induces a uniform measure on the sphere. In, authors used rotational invariance of the Laplacian, and the fact that the Laplacian restricted to a subspace is still the Laplacian to handle matrices E with 0’s and 1’s on the diagonal (subspace projections). In general one might try to turn the integral into an integral in spherical coordinates, however the calculations become very tedious in higher dimensions. Furthermore, we would like to also find a formula for Fp that would not require the singular value decomposition of the transformation. Such a formula would be easier to apply to symbolic matrices, for which singular value decomposition is not easy to find. Direct, polar coordinates based approach in two dimensions has an advantage when we take the whole orthogonal group as G. That is when Ω is a disk, and T(Ω) is an ellipse. In this case constants Fp can be evaluated even for non-integer values of p, albeit in terms of a rather complicated combination of hypergeometric functions 2F1. Note that we defined Fp as polynomials (integer values of p), however naturally extends to any real p. In particular, p = 1/2 leads to an elliptic integral of the first kind, that gives slightly better upper bounds for $\sqrt{-\Delta}$, than tight 1-frame (see and its proof in ). Effectively we are using 1/2-frame-like identity, that is available for the full orthogonal group. p-moments of the sums of squares of Gaussian random variables. ---------------------------------------------------------------- From the probabilistic point of view, the Laplacian is closely connected to Brownian motion and Gaussian random variables. We tackle the general matrix E in the integral above using Gaussian random variables. Note that $$\begin{aligned} \int\_0^\infty e^{-r^2/2} r^{2p+d-1} dr=2^{p-1+d/2}\Gamma(p+d/2).\end{aligned}$$ Therefore can be rewritten as $$\begin{aligned} \label{fpgaus} F\_p(E^2)= \frac{\Gamma(d/2)}{2^{p}\Gamma(p+d/2)}\int\_{{{{\mathbb{R}}}^d}}\frac{1}{(2\pi)^{d/2}}e^{-\|x\|^2/2} \|Ex\|^{2p}dx=\frac{1}{2^p (d/2)\_p}{{\bf E}}\left(\sum\_i e\_i^2X\_i^2\right)^p,\end{aligned}$$ where Xi are independent standard normal random variables. On the right we have a somewhat complicated 2p-moment of these random variables. Note that taking e1 = … = ek = 1 and ek + 1 = … = ed = 0, then integrating resulting k-dimensional Gaussian leads to a different proof of (constant in ). To find the general formula we first establish two probabilistic results. Then we use abstract algebra of symmetric functions to deduce an easy to calculate formula for Fp. By mλ(a1, …, an) we denote the monomial symmetric polynomial with exponent patern λ (see e.g. ), that is a sum of all possible monomials with the same exponent pattern. [lemgaus] Let Xi, …, Xn be independent standard normal random variables, and ai ∈ R. Then $$\begin{aligned} {{\bf E}}\left(\sum\_i a\_iX\_i^2\right)^p =\frac{p!}{2^p}\sum\_{\lambda}\left(\prod\_{k\in\lambda}\binom{2k}{k}\right)m\_\lambda(a\_1,\dots,a\_n), \label{GM} \end{aligned}$$ where we sum over all integer partitions λ of p. We do not know whether this is a new result, but we could not find it in any probability related book. $$\begin{aligned} {{\bf E}}\left(\sum\_i a\_iX\_i^2\right)^p &=\sum\_{k\_1+\dots+k\_p=p} \binom{p}{k\_1,\dots,k\_d}\prod\_{i=1}^n a\_i^{k\_i}{{\bf E}}X\_i^{2k\_i} \\&= \sum\_{k\_1+\dots+k\_p=p} \binom{p}{k\_1,\dots,k\_d}\prod\_{i=1}^n (2k\_i-1)!! a\_i^{k\_i} \end{aligned}$$ Note that multinomial coefficient, as well as the double factorial representing the Gaussian moment only depend on the multiset {ki}. Therefore the sum can be rewritten as a sum over all partitions λ of p, involving monomial symmetric polynomials mλ(a1, …, an). Since Xi are identically distributed, they have the same moments, and we can treat ai as variables of the polynomial. We get $$\begin{aligned} {{\bf E}}\left(\sum\_i a\_iX\_i^2\right)^p &= {p!}\sum\_{\lambda}\left(\prod\_{k\in\lambda}\frac{(2k-1)!!}{k!}\right)m\_\lambda(a\_1,\dots,a\_n) \\&= \frac{p!}{2^p}\sum\_{\lambda}\left(\prod\_{k\in\lambda}\binom{2k}{k}\right)m\_\lambda(a\_1,\dots,a\_n)\end{aligned}$$ We also need an auxiliary result for double sequences of standard normal random variables. The following lemma relies on the relation between Gaussian random variables and the χ2 distribution. This time we need complete homogenous polynomials hp, defined as the sum of all possible monomials of degree p. Again, the monograph provides an excellent source of information about these and other kinds of symmetric polynomials. [lemdouble] Let Xi, …, Xn, Y1, …, Yn be independent standard normal random variables, and ai ∈ R. Then $$\begin{aligned} \frac{p!}{2^p} \sum\_{\lambda}\left(\prod\_{k\in\lambda}\binom{2k}{k}\right)&m\_\lambda(a\_1,\dots,a\_n,a\_1,\dots,a\_n) \\&= {{\bf E}}\left(\sum\_i a\_i(X\_i^2+Y\_i^2)\right)^p = 2^p p! h\_p(a\_1,\dots,a\_n). \end{aligned}$$ Note that monomial symmetric polynomial mλ has 2n variables, but is evaluated at two copies of a1, …, an. As a result we get a polynomial with more than one exponent pattern. The first equality follows directly from with 2n standard normal random variables, with each factor ai repeated twice. To get the second formula, note that Xi2 + Yi2 has χ2(2) distribution, which also equals exponential distribution with λ = 1/2. Let Zi be exponential random variables with λ = 1, then $$\begin{aligned} {{\bf E}}\left(\sum\_i a\_i(X\_i^2+Y\_i^2)\right)^p &= 2^{p}{{\bf E}}\left(\sum\_i a\_i Z\_i\right)^p = 2^{p}\sum\_{k\_1+\dots+k\_p=p} \binom{p}{k\_1,\dots,k\_p}\prod\_{i=1}^n a\_i^{k\_i}{{\bf E}}Z\_i^{k\_i} \\&= 2^{p}\sum\_{k\_1+\dots+k\_p=p} \binom{p}{k\_1,\dots,k\_p}\prod\_{i=1}^n k\_i! a\_i^{k\_i}= {2^p p!} h\_p(a\_1,\dots,a\_n). \end{aligned}$$ In the last equality we cancel all ki! terms with the multinomial coefficient, and we get all possible monomials of combined degree p, each with coefficient p!. Therefore we get complete homogeneous polynomial hp of degree p in n variables. Algebra of symmetric functions ------------------------------ Consider the ring of symmetric functions over rational numbers, denoted by Λ in, defined on countably many variables {a1, a2, …}. Let An = {a1, …, an}. For any partition λ of p we define a λ indexed power sum basis element $$\begin{aligned} p\_\lambda(A\_n)=\prod\_{k\in\lambda} (a\_1^{k}+\dots+a\_n^{k})=\prod\_{k\in\lambda} p\_{k}(A\_n),\end{aligned}$$ where pk(An) is called a power sum polynomial of degree k. Then pλ(An) generates Λ. In the previous section we used monomial symmetric polynomials, which also form a basis for the same ring. Note that any symmetric function has the same expansion in monomial basis (or power sum basis), regardless of the number of variables in An. In other words, Λ can be treated as an abstract ring without ever specifying the variable set A (alphabet). Furthermore, one can define abstract algebraic alphabet operations over rational numbers, and these operation reduce to intuitive alphabet operations over natural numbers. See for even broader context. In particular, for any symmetric function S(A) we could try to define doubling operation $$\begin{aligned} S(2A)=s(a\_1,\dots,a\_n,a\_1,\dots,a\_n).\end{aligned}$$ Hence S(2A) would be a symmetric function of 2n variables, evaluated on two copies of A. However it is not immediately clear how one could make this precise. Expanding S in the power sum basis with rational coefficients qλ leads to an intuitive definition $$\begin{aligned} S(2A)&=\sum\_\lambda q\_\lambda p\_\lambda(2A)=\sum\_\lambda q\_\lambda \prod\_{k\in \lambda} p\_k(2A) \\&=\sum\_\lambda q\_\lambda \prod\_{k\in \lambda} 2p\_k(A),\end{aligned}$$ since power sum polynomials are simply sums of powers of all variables. It is clear that using power sum basis we can define S(qA) for any rational q. And for any natural number q we can interpret this transformation as taking qn variables evaluated on q copies of An. It is also clear that θq : S(A) → S(qA) is an automorphism of Λ with inverse θ1/q. From now on, let $$\begin{aligned} \label{Sgaus} S(A)={{\bf E}}\left(\sum\_i a\_iX\_i^2\right)^p \end{aligned}$$ implies that S(A) a symmetric function over rational numbers, expanded in monomial symmetric basis mλ. But gives $$\begin{aligned} S(2A)=2^pp!h\_p(A).\end{aligned}$$ Note that direct proof of this fact would be tedious, since monomial symmetric polynomials mλ do not interact in a simple way with automorphism θ. It is also not trivial to transform from monomial to power sum basis. Invertibility of θ gives $$\begin{aligned} \label{stoh} S(A)=2^p p! h\_p\left(\frac{1}{2}A\right).\end{aligned}$$ This provides a very concise formula for S(A). Unfortunately, action of θ on complete homogeneous polynomials is not simple. Hence we will rewrite this expression using power sum basis. Cycle index in power sum basis ------------------------------ For any finite group G of permutations we can define a so-called cycle index Z(G, c1, …, cp) in the following way. Consider a sequence of dummy variables c1, …, cp, and use ci to denote cycle of length i. Every permutation can now be represented using a monomial of degree equal to the number of cycles in this permutation. The cycle index is an average of those monomials over all elements of the group. Note that ci corresponds to cycles of length i, hence replacing ci with power sum polynomial pi(A) leads to a homogeneous symmetric polynomial in alphabet A (variables a1, …, an). We are interested in the cycle index of the symmetric group Sp. We get a polynomial, whose coefficients count number of permutations with the cycle lengths given by λ (denote these coefficients by qλ). We have $$\begin{aligned} \label{Zs} Z\left(S\_p,a\_1+\dots+a\_p,a\_1^2+\dots+a\_n^2,\dots,a\_1^p+\dots+a\_n^p\right)=\frac{1}{p!}\sum\_\lambda q\_\lambda p\_\lambda(A).\end{aligned}$$ Simple counting argument shows that if λ has jk cycles of length k then $$\begin{aligned} \label{sl} q\_\lambda=\frac{p!}{\prod\_{k=1}^p k^{j\_k} j\_k!}.\end{aligned}$$ However, Pólya’s enumeration theorem (generalized Burnside’s lemma, see or any modern enumerative combinatorics book) states that this cycle index is a generating function of colorings of p objects with n colors (monomial ∏i = 1naiki encodes ki elements of color i). But the colorings are considered the same if there is a permutation of objects that maps one coloring to the other. This last restriction clearly implies that there is exactly one coloring per monomial. Hence $$\begin{aligned} \label{Zh} Z\left(S\_p,a\_1+\dots+a\_p,a\_1^2+\dots+a\_n^2,\dots,a\_1^p+\dots+a\_n^p\right)= h\_p(A).\end{aligned}$$ Therefore s(A) defined by satisfies $$\begin{aligned} S(A)&=2^pp!h\_p\left( \frac{1}{2}A \right)=4^p\sum\_\lambda q\_\lambda p\_\lambda\left( \frac{1}{2}A \right)=2^pp!\sum\_\lambda q\_\lambda \prod\_{k\in \lambda} \frac{1}{2} p\_k(A)\nonumber \\&=2^pp! Z\left(S\_p,\frac{a\_1+\dots+a\_n}{2},\frac{a\_1^2+\dots+a\_n^2}{2},\dots,\frac{a\_1^p+\dots+a\_n^p}{2}\right)\label{sZ}\end{aligned}$$ To find S(A) one needs to replace ci in the abstract cycle index of Sp with pi(A)/2. Schatten norms and formulas for polynomials Fp. --------------------------------------------------- Let T be an arbitrary rectangular matrix, and s(T) the multiset of its singular values. The Schatten norm of T of order 2n equals $$\begin{aligned} \\|T\\|\_{2n}^{2n} = \sum\_{\sigma\in s^2(T)} \sigma^{2n}= p\_n(s^2(T)),\end{aligned}$$ where pn is the power sum polynomial of order n. This description fits perfectly into the framework of the power sum basis for symmetric polynomials discussed in the previous section. However, the same norm can also be calculated using matrix trace $$\begin{aligned} \\|T\\|\_{2n}^{2n} = \operatorname{Tr}\left((T^\dagger T)^n\right) = \operatorname{Tr}\left( (TT^\dagger)^n\right),\end{aligned}$$ where T† denotes the transposed matrix. This formula allows for calculating Schatten norms without finding singular values of T. One can also choose the order of multiplication that gives smaller matrices. This characterization can be useful when matrix T involves further unknowns, in which case finding singular values would be hard. Now we return to polynomials Fp(s2(T)). and provide a combinatorial formula: [lemmafpm] For any (possibly rectangular) matrix T $$\begin{aligned} F\_p(s^2(T))= \frac{p!}{4^p (d/2)\_p}\sum\_{\lambda}\left(\prod\_{k\_i\in\lambda}\binom{2k\_i}{k\_i}\right)m\_\lambda(s^2(T)). \end{aligned}$$ This result is easy to apply when p is small. In particular, when p = 2 the generalized tight 2-frame identity reads $$\begin{aligned} \frac{1}{\|G\|}\sum\_U \|TUx\|^4 &= \frac{1}{d(d+2)}\left(3\sum\_{i=1}^d s\_i^4 + 2\sum\_{i<j} s\_i^2s\_j^2 \right)\|x\|^4\nonumber \\&= \frac{\\|T\\|\_2^4+2\\|T\\|\_4^4}{d^2+2d}\|x\|^4,\label{schatten4} \end{aligned}$$ where si are singular values of T. Note that the second formula involving Schatten norms allows us to find the value of Fp without finding singular values of T. When T is an identity matrix, the Schatten p-norm equals $\sqrt[p]{d}$. Hence the T dependent term reduces to 1, as expected. For p = 3 we get $$\begin{aligned} \frac{1}{\|G\|}\sum\_U \|TUx\|^6 &= \frac{1}{ d(d+2)(d+4)}\left(15\sum\_{i=1}^d s\_i^6 + 9\sum\_{i<j} (s\_i^4s\_j^2+s\_j^4s\_i^2) + 6\sum\_{i<j<k} s\_i^2 s\_j^2 s\_k^2 \right)\|x\|^6 \\&= \frac{\\|T\\|\_2^6+6\\|T\\|\_4^4\\|T\\|\_2^2+8\\|T\\|\_6^6}{d^3+6d^2+8d}\|x\|^6,\end{aligned}$$ Alert reader may notice that coefficients in the numerator are the same as in the denominator. This is only true for p = 2 and p = 3, since the number of partitions of those integers equals p. For p = 4, we already have five partitions, including two of length two, 4 = 3 + 1 = 2 + 2, and it is no longer clear which one should be used. To obtain Schatten-type formula for Fp we must use the symmetric polynomial approach developed in the previous section. For arbitrary d and arbitrary p the following Schatten-type formula seems most convenient, especially when finding singular values is not practical. [thmcycle] For any (possibly rectangular) matrix T $$\begin{aligned} F\_p(s^2(T))&= \frac{p!}{(d/2)\_p} Z\left(S\_p,\frac{\\|T\\|\_2^2}{2},\frac{\\|T\\|\_4^2}{2},\dots,\frac{\\|T\\|\_{2p}^{2p}}{2}\right) \\&= \frac{p!}{(d/2)\_p}\sum\_{j\_1+2j\_2+\dots+pj\_p=p} \prod\_{k=1}^p \frac{\\|T\\|\_{2k}^{2kj\_k}}{(2k)^{j\_k}j\_k!}.\label{fpcycles} \end{aligned}$$ where Z(Sp, c1, …, cp) is the cycle index for the symmetric group Sp. The first formula follows from combined with and. To get the second formula, start with, decompose each λ into jk cycles of length k, and apply formula for qλ. A very restricted choice of partitions in dimension 2 (any monomial symmetric polynomial involving partition with 3 or more pieces equals 0) allows for a simpler singular value based formula for arbitrary value of p $$\begin{aligned} \frac{1}{\|G\|}\sum\_U \|TUx\|^{2p} &= \frac{1}{4^p}\sum\_{k=0}^p\binom{2k}{k}\binom{2p-2k}{p-k} s\_1^ks\_2^{p-k}.\end{aligned}$$ Unfortunately, it seems that the simplest form of this series is a hypergeometric function in s1/s2. Even taking s1 = s2 = 1 (identity matrix), leads to a nontrivial combinatorial identity $$\begin{aligned} \sum\_{k=0}^p\binom{2k}{k}\binom{2p-2k}{p-k}=4^p.\end{aligned}$$ A note about generalized χ2 distributions ------------------------------------------- Let X1, …, Xn be independent, centered Gaussian random variables with variances σi. provides a formula for the p-moment of ∑iXi2, a special case of the generalized χ2 distribution. On the other hand, gives a much simpler formula for a related χ2, with each Gaussian repeated twice. One can also think about this case as a sum of the squared magnitudes of complex gaussian random variables. The relation between this distribution and exponential random variables was the key to a simplicity of. Symmetric polynomial manipulations, in particular formula, allow us to find a concise formula for p-moments of the real case ∑iXi2 in terms of the complex (double) χ2. Combining this theoretical result with cycle index decomposition leads to another way of calculating the same moments. Finally, symmetric polynomial manipulations, including θ authomorphism, are easy to perform using SAGE computer algebra system, providing a convenient way of finding these moments (see section below). SAGE code --------- In this section we explore possibility to quickly evaluate polynomials Fp using SAGE Computer Algebra System. We used Sage Cell Webserver to perform all calculations and to provide direct links to these calculation. The cycle index of the symmetric group is a well known polynomial implemented in SAGE via GAP library. In particular, one can obtain the cycle index (with powersums as variables) using the [following SAGE code](https://sagecell.sagemath.org/?z=eJzLU7BVMOHlCq7MzU0tKcpMdi_KLy3QyNPUS65MzkmNz8xLSa3Q0NQryUgtSdQw1DfSBACOWw_t&lang=sage) ``` n = 4 # frame order cycle_index = SymmetricGroup(n).cycle_index() Z = cycle_index.theta(1/2) # apply theta automorphism print Z ``` The output ``` 1/384p[1,1,1,1] + 1/32p[2,1,1] + 1/32p[2,2] + 1/12p[3,1] + 1/8p[4] ``` is the expansion of the cycle index from in terms of powersum basis, after the application of the θ* automorphism. E.g. p[2, 1, 1] corresponds to ∥T∥44(∥T∥22)2 in the formula for Fp(s2(T)). Another way of obtaining the cycle index is to explicitly convert complete homogeneous polynomial to powersum basis using. In SAGE (Symmetrica package), [the following input](https://sagecell.sagemath.org/?z=eJzLU7BVMOHlCq7MzU0tKcpMdivNSy7JzM8r1ggM1NTLzMtKTS6JL87ILyrJSMxLKdaIVi9Q11HPUI_V5OUq0MiIzovV1CvJSC1J1DDUN9IEAAMqGR4=&lang=sage) produces the same cycle index Z as above ``` n = 4 # frame order SymmetricFunctions(QQ).inject_shorthands(["p","h"]) Z = p(h[n]).theta(1/2) # convert h to p, and apply theta authomorphism ``` Abstract power sum polynomials in Sage can be expanded in any number of variables (SAGE Symmetrica library). Using cycle index Z we just calculated, we obtain singular value based expansion for Fp given by (or with Schatten norms expressed using singular values) by [executing](https://sagecell.sagemath.org/?z=eJzLU7BVSAFiE16utMTkkvyizMQcjTxN_aLM4sy89HiEWIq-kU6eplZwZW5uaklRZrJ7UX5pAVClXnJlck5qfGZeSmqFhqZeSUZqSaKGob6Rpl5qRUFiXopGiiYAeG8hTg==&lang=sage) ``` d = 4 # dimension Fp = factorial(n)/rising_factorial(d/2,n)Z.expand(d) # expand in dimension d ``` Let us also point out that Molien series that counts invariant polynomials for permutation groups is also implemented in Sage. Unfortunately, the series uses the representation of the group as permutation matrices, hence in a too-high dimension. However, one can implement Molien’s series using Molien’s theorem and Coxeter groups acting as reflections (with the help of MPFI, GAP, MPFR, ginac, GMP and Maxima libraries). This allows one to quickly check if a given group of isometries admits higher order frames. In particular, to check uniqueness of the invariants for the group of rotations (no reflections) of a tetrahedron [use the following code](https://sagecell.sagemath.org/?z=eJxlj8EKwjAQRO8F_yH0lEiNFr3mIB5y6heUUkLdhkialE0q7d-bVkHQ2zK7s_NGipufIQJK9NNI6_yaF-emMMNoYQAXVTTeiRyht9Ctc852mRSVimjmt0dyDS5Q9reoNek9Ek2MI5JbEyJlxPRE82G7oozfIWlClE0yI4xBSN5595i06pa2syqENskI4Y3yhJSzyypvDTgRpoHWFhzFX1OCOYa1g-kXWh7mPX6iNiBcgda0hh0l93gH_H7lUS3WI52LU3FJ6gtfmGBx&lang=sage) ``` G=CoxeterGroup(["A",3],implementation="reflection") # tetrahedral group in R3 G=MatrixGroup(G.gens()) # as matrix group G=MatrixGroup([g for g in G.list() if g.matrix().det()==1]) # rotations only reps=G.conjugacy_class_representatives() # Molien's theorem M=sum([len(r.conjugacy_class())/simplify(1-xr).det() for r in reps])/G.order() M.taylor(x,0,4) # expand as series ``` This outputs ``` 2t^4 + t^3 + t^2 + 1 ``` We have exactly one invariant for degrees 0, 2 and 3, but two invariants of degree 4. Finally, note that we can also find all fundamental invariants in Sage and check their degrees (using PARI, MPFI, Singular, GAP, FLINT, MPFR, ginac, GMP and NTL libraries). In particular for the rotation group of the icosahedron [we have](https://sagecell.sagemath.org/?z=eJxljT0LgzAURXfB_xCcEpBAkY6ZhKaLk6NICfoMD2ISnmmx_77pB-3Q7X2ce65WbdghAWkK18iH6lzVzVjjGh2s4JNJGLyqCBYH03OuRFl0vepMItz7aCbg7X1yIYUVpxOCm_lR1E3dZE5_sLdbSwt-4-LvMXQ9t4ItgZhl6JmWDrfEBcOFWbm-SC7kDPmm1GHMAvQ3Q2h82pSW3-WSC4BMCpRrymKIOWMJgL_l8Sn_JceyeAAmzFQl&lang=sage) ``` G=CoxeterGroup(["H",3],implementation="reflection") # icosahedral group in R3 MS=MatrixSpace(CyclotomicField(5),3,3) # icosahedron uses 5th root of unity G=MatrixGroup([MS(g) for g in G.gens()]) # change the base ring for generators G=MatrixGroup([g for g in G.list() if g.matrix().det()==1]) # rotations only invariants=G.invariant_generators() print [p.degree() for p in invariants] ``` And we get the following degrees ``` [2, 6, 10, 15] ``` Note however, that the representation using reflections (generators satisfy M2 = Id) does not give the subgroup of the orthogonal group (generators should satisfy M*M† = Id). Instead, one gets a group that acts on a sheared regular polytope. Therefore the invariants will not be the same as the invariants we use in this paper, but the degrees coincide. We see that the group of rotations of icosahedron admits 2-frames, but not 3-frames, just like its full symmetry group. (Non-tight) p-frames for any irreducible groups of symmetries =============================================================== Let G be an irreducible group of symmetries. This group may or may not admit higher order frames, but for any matrix T we have the tight 1-frame identity $$\begin{aligned} \label{Tframeid} \frac{1}{\|G\|}\sum\_U \|TU x\|^2=\frac{1}{d}\\|T\\|\_2^2 \|x\|^2.\end{aligned}$$ We also have the inner product version of this identity, with p = 1. Note that if T has just one row, this reduces to classical tight frame identity for vectors. Note also that with many rows we are simply adding one tight frame identity per row. Upper bounds for eigenvalues obtained in rely on this identity. For a given multiplier f, a special case of its transformed version defined in reads $$\begin{aligned} F[\Phi\_1,0,T](t) = \frac{1}{d}\operatorname{Tr}\Phi\_1(t T^{-\dagger}T^{-1}).\end{aligned}$$ Recall that the trace of a function of a matrix means the trace of the spectrally defined function of the matrix (see the paragraph above ). [nontight] For any irreducible group of symmetries G acting on Rd, any convex function Φ and any concave function Ψ $$\begin{aligned} \frac{1}{\|G\|}\sum\_U \Psi(\Phi(\|T^{-1}U x\|^2))\le \Psi\circ F[\Phi,0,T](\|\xi\|^2),\\ \frac{1}{\|G\|}\sum\_U \Phi(\Psi(\|T^{-1}U x\|^2))\ge \Phi\circ F[\Psi,0,T](\|\xi\|^2). \end{aligned}$$ Furthermore, we get equalities if and only if T is a multiple of an orthogonal matrix, or both Ψ and Φ are linear. As an immediate corollary we get a bound for the constants Fp in tight p-frame identities. [cornontight] Polynomials Fp used in the tight p-frame identities ( and ) satisfy $$\begin{aligned} \frac1{d^p}{\\|T\\|\_{2}^{2p}}\le F\_{p}(s^2(T))\le \frac1d{\\|T\\|\_{2p}^{2p}}. \end{aligned}$$ Hence, orbits of any irreducible group G form (not necessarily tight) p-frames. [Proof of ] Note that we can assume that ∣x∣ = 1 by considering ∣x∣ a part of the matrix T. First consider the case Φ ≥ 0. Write the singular value decomposition of T− 1 = WSV with diagonal matrix S and orthogonal matrices W and V. If ∣v∣ = 1, then the inner product ⟨S2v, v⟩ can be interpreted as a sum over a probability measure given by vi2. Therefore Jensen inequality gives $$\begin{aligned} \Phi(\|T^{-1}Ux\|^2)&=\Phi(\langle S^2\, VUx,VUx\rangle)\le \langle \Phi(S^2)\, VUx,VUx\rangle \\&=\operatorname{Tr}(\Phi(S^2)VUxx^\dagger U^\dagger V^\dagger) =\operatorname{Tr}(V^\dagger \Phi(S^2) VP\_{Ux})=\langle V^\dagger \Phi(S^2) V,P\_{Ux}\rangle\_{HS} \end{aligned}$$ Note that we get equality if and only if S is a multiple of identity. Note also that we can replace Φ with Φ + l where l is linear (not necessarily nonnegative). Concavity of Ψ, with group averaging as a convex combination, gives $$\begin{aligned} \frac{1}{\|G\|}\sum\_U &\Psi(\Phi(\|T^{-1}Ux\|^{2}))\le \Psi\left(\frac{1}{\|G\|}\sum\_U \Phi(\|T^{-1}Ux\|^{2})\right)\le \Psi\left(\frac{1}{\|G\|} \sum\_U \langle V^\dagger \Phi(S^2) V,P\_{Ux}\rangle\_{HS}\right) \\&=\Psi\left(\frac{1}{d}\\|V^\dagger \Phi(S^2)V\\|\_{1}\right) \le \Psi\left(\frac{1}{d}\\|\Phi(T^{-\dagger}T^{-1})\\|\_{1}\right)=\Psi\left(\frac{1}{d}\operatorname{Tr}(\Phi(T^{-\dagger}T^{-1}))\right) \end{aligned}$$ using with p = 1. We also get the equality statement for the upper bound if and only if T is a multiple of an orthogonal matrix, or both Ψ and Φ are linear. Note that we need Φ ≥ 0, so that the Schatten 1-norm equals the trace of a matrix. However, as above, we can also handle the case Φ + l by applying tight frame identity separately to Φ and l. The latter gives equalities in the above calculation. Note that for general convex Φ there exists linear function l (any supporting line or affine minorant), such that Φ ≥ l. Then Φ1 = Φ − l is nonnegative and convex. Now repeat the proof with Φ = Φ1 + l. Hence the theorem holds for arbitrary convex functions Φ. Finally, all inequalities in the above proof come from convexity of Φ and concavity of Ψ. We can replace Φ and Ψ with − Φ and − Ψ to get the opposite inequalities, proving the lower bound in the theorem. In the above proof we used convexity/concavity to turn Φ-frame type expression into a 1-frame. We needed to do this to obtain valid 1-frame identity for G. However, straightforward modification of the above proof leads to non-tight frame identities based on p-frames whenever these are admissible for a given symmetry group G. provides a framework for handling almost arbitrary multipliers. Any function possessing left and right derivatives, such that those derivatives are of bounded variation on any closed bounded subinterval of (0, ∞), can be represented using a difference of two convex functions (class of DC functions, see or ). Hence any function f = Φ1 − Φ2, where Φi are convex, can be handled by. In particular any f ∈ C1, 1 can be written as a difference of nonnegative convex functions. [Proof of ] Start with with f = Φ1 − Φ2 $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{T(\Omega)}&\le \frac{1}{\|G\|}\sum\_{U\in G}\sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}(\Phi\_1-\Phi\_2)(\|T^{-\dagger}U^\dagger \xi\|^2)\|\widehat u\_i\|^2d\xi\end{aligned}$$ Now apply, first with Ψ(t) = t and Φ = Φ1, then with Ψ = − Φ2 and Φ(t) = t. On the right we get a sum of quadratic forms for the multiplier from. Choose ui to be the eigenfunctions for that multiplier to get the result. Similarly we get the second part of the theorem. Higher moments of mass ====================== As the first application of generalized p-frames, we find a relation between higher order moments mass of a highly symmetric domain Ω and its linearly transformed image. This section generalizes, where the second moment (the moment of inertia) is treated using classical tight frames. In we defined the 2p-moment of mass as $$\begin{aligned} I\_{2p}(\Omega)=\int\_\Omega \|x\|^{2p} dx.\end{aligned}$$ Note that rescaling the domain with a fixed scaling factor c, scales I2p by c2p + d. In particular volume (p = 0) scales like cd. Hence V1 + 2p/d/I2p is a scale invariant quantity. Moreover [lemmoments] If Ω admits p-frames, then $$\begin{aligned} F\_p(s^2(T)) = \frac{I\_{2p}(T(\Omega))}{I\_{2p}(\Omega)} \frac{V(\Omega)^{1+4p/d}}{V(T^{-1}(\Omega))^{2p/d}V(T(\Omega))^{1+2p/d}} \end{aligned}$$ Furthermore, for any Ω with irreducible isometry group $$\begin{aligned} \frac{1}{d^p} \\|T\\|\_{HS}^{2p}\le\frac{I\_{2p}(T(\Omega))}{I\_{2p}(\Omega)} \frac{V(\Omega)^{1+4p/d}}{V(T^{-1}(\Omega))^{2p/d}V(T(\Omega))^{1+2p/d}} \le \frac{1}{d}\\|T\\|\_{2p}^{2p}. \end{aligned}$$ Suppose Ω admits tight p-frames. Let T be an arbitrary linear transformation. Then $$\begin{aligned} I\_{2p}(T(\Omega))=\int\_{T(\Omega)} \|x\|^{2p} dx = \|T\| \int\_{\Omega} \|T x\|^{2p} dx\end{aligned}$$ But Ω is invariant under its isometry group G, hence $$\begin{aligned} I\_{2p}(T(\Omega)) = \|T\| \frac{1}{\|G\|}\sum\_U \int\_{\Omega} \|T Ux\|^{2p} dx=\|T\| F\_p(s^2(T)) I\_{2p}(\Omega),\end{aligned}$$ by. Similarly, if the group of symmetries of Ω is just irreducible, implies $$\begin{aligned} \frac{1}{d^p} \\|T\\|\_{HS}^{2p}\le\frac{I\_{2p}(T(\Omega))}{I\_{2p}(\Omega)} \|T\|^{-1} \le \frac{1}{d}\\|T\\|\_{2p}^{2p}.\end{aligned}$$ As in the proof of we finish the proof by noting that $$\begin{aligned} \|T\|=\frac{V(T^{-1}(\Omega))^{2p/d}V(T(\Omega))^{1+2p/d}}{V(\Omega)^{1+4p/d}}.\end{aligned}$$ Using Schatten norms based formula for Fp we get In two dimensions $$\begin{aligned} F\_p(s^2(T^{-1}))\|T\|^{2p}=F\_p(s^2(T)) \end{aligned}$$ Note that $$\begin{aligned} \\|T^{-1}\\|\_{k}^{k}=s\_1^{-k}+s\_2^{-k}=\frac{s\_1^k+s\_2^k}{(s\_1s\_2)^k}=\frac{\\|T\\|\_k^k}{\|T\|^k}. \end{aligned}$$ Combine this Schatten norm property and to get the result. This allows us to compare the higher moments of T(Ω) and T− 1(Ω) in dimension two. [momentstwo] Let A be the area of a two dimensional domain Ω. Then $$\begin{aligned} \left.\frac{A^{1+p}}{I\_{2p}}\right\|\_{T(\Omega)}= \left.\frac{A^{1+p}}{I\_{2p}}\right\|\_{T^{-1}(\Omega)} \end{aligned}$$ $$\begin{aligned} I\_{2p}(T(\Omega))=\|T\|F\_p(s^2(T))I\_{2p}(\Omega)=\|T\|^{2+2p}\|T^{-1}\|F\_p(s^2(T^{-1}))I\_{2p}(\Omega)=\|T\|^{2+2p}I\_{2p}(T^{-1}(\Omega)) \end{aligned}$$ Now divide by ∣T∣1 + p and note that ∣T∣− 1 = ∣T− 1∣. Bi-Laplacian and convex multipliers =================================== Plate problem ------------- In this section we apply the theory of p-matrix frames to the Fourier multipliers of the form Φ(∣ξ∣2) with convex Φ. Consider a quadratic form $$\begin{aligned} Q\_A(u,v)=\int\_{{{{{\mathbb{R}}}^d}}} \Phi(\|\xi\|^2) \widehat u(\xi)\overline{\widehat v(\xi)}\,d\xi. \end{aligned}$$ We get a weak formulation of the underlying operator A, with discrete spectrum for any domain Ω (due to Dirichlet boundary condition). As described in, the smallest eigenvalue can be obtained by minimizing Rayleigh quotient, and the sums via finite subspace minimization. Recall, but with convex multiplier Φ $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{T(\Omega)}&\le \frac{1}{\|G\|}\sum\_{U\in G}\sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}\Phi(\|T^{-\dagger}U^\dagger \xi\|^2)\|\widehat u\_i\|^2d\xi\nonumber \\&= \sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}\left(\frac{1}{\|G\|}\sum\_{U\in G}\Phi(\|T^{-\dagger}U^\dagger \xi\|^2)\right)\|\widehat u\_i\|^2d\xi\label{plateaverage}\end{aligned}$$ Bi-Laplacian is perhaps the most import example of such multipliers. In we described the differential equation leading to a vibrating plate problem. In terms of Fourier multipliers and quadratic forms we have $$\begin{aligned} Q\_A(u,u)=\int\_{{{{{\mathbb{R}}}^d}}} (\|\xi\|^4+\tau \|\xi\|^2) \|\widehat u\|^2\,d\xi.\end{aligned}$$ Hence bi-Laplacian can be weakly defined on H02(Ω). Since this is a subspace of H01(Ω), we clearly get compact embedding into L2, and discrete spectrum. The function Φ(t) = t2 + τt is convex, even for negative values of τ. Hence we can apply with Φ1 = Φ and Φ2 = 0. This leads to $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{\|\xi\|^4+\tau \|\xi\|^2,T(\Omega)}\le \lambda\_1+\dots+\lambda\_n\Big\|\_{(\|\xi\|^4\\|T\\|\_4^4+\tau\|\xi\|^2\\|T\\|^{2})/d,\Omega}\end{aligned}$$ Due to almost homogeneous nature of the multiplier, we can take ∥T∥44/d out of the quadratic form. Furthermore we can rescale τ so that we have no dependence on T on Ω and we obtain from. Note that the multiplier is not homogeneous, hence we must have some dependence of the multipliers in on transformation T. We chose to keep the operator on the symmetric domain Ω as simple as possible. Now we can use tight 2-frames to improve the result for domains Ω that admit tight 2-frames. We skip in favor of a direct approach based on. We have $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{T(\Omega)}&\le \sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}\left(\frac{1}{\|G\|}\sum\_{U\in G} (\|T^{-\dagger}U^\dagger \xi\|^4+\tau\|T^{-\dagger}U^\dagger \xi\|^2 )\right)\|\widehat u\_i\|^2d\xi \\&= \sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}\left( F\_2(s^2(T^{-1})) \|\xi\|^4+\tau \frac{\\|T^{-1}\\|\_2^2}{d}\|\xi\|^2 \right)\|\widehat u\_i\|^2d\xi\end{aligned}$$ using. Define D(T− 1) = d F2(s2(T− 1)) (or use directly) to get from. Note that we need to choose ui to be the eigenfunctions for the multiplier we have on the right of the inequality. Finally, D(T− 1) ≤ ∥T− 1∥44/d is a special case of. Higher order Laplacians ----------------------- When f(t) is a polynomial, then the operator f( − Δ) is a higher order differential operator. We can estimate its eigenvalues using and, if needed, the convexification procedure described just under that theorem. We can however improve the bounds when the symmetry group of Ω admits tight 2-frames. Assuming fʺ(t) ≥ − c, we define $$\begin{aligned} \Phi\_1(t)=f(t)+ct^2,\\ \Phi\_2(t)=ct^2.\end{aligned}$$ We handle Φ1 as in. However, we use tight 2-frame identity on − Φ2 (as in the plate problem). Note that this is possible even for negative terms, since 2-frames provide exact Rayleigh quotient evaluation for the multiplier ∣ξ∣4. As a consequence we obtain a stronger version of $$\begin{aligned} \lambda\_1+\dots+\lambda\_n \Big\|\_{f,T(\Omega)}\le \lambda\_1+\dots+\lambda\_n\Big\|\_{G[\Phi\_1,T],\Omega}. \end{aligned}$$ with $$\begin{aligned} G[\Phi\_1,T](t) = \frac{1}{d}\operatorname{Tr}\Phi\_1(t T^{-1}T^{-\dagger})-c F\_2(s^2(T^{-1}))t^2.\end{aligned}$$ We can also completely forgo the use of if Ω admits tight p-frames for 2p larger or equal to the degree of the polynomial f. Then each monomial in f can be exactly averaged using appropriate higher order tight frame, leading to further improvements. In particular, poly-Laplacian Δp with p > 1 corresponds to f(t) = tp. If Ω admits tight p-frames (in particular for disks), then the eigenvalues of the linearly transformed domain (e.g. ellipse) can be estimated by Fp(s2(T)) times the eigenvalues on Ω. Furthermore, Fp(s2(T)) can be rewritten as the 2p-moment of mass using, leading to an analog of with I4 replaced by I2p, and properly adjusted powers of the volume. Buckling problem ---------------- Finally, we tackle a related plate buckling problem. We want to bound the principal eigenvalue Λ in $$\begin{aligned} \Delta^2 u +\Lambda \Delta u = 0\text{ in }\Omega,\\ u=\frac{\partial u}{\partial \nu} = 0,\text{ on }\partial\Omega.\end{aligned}$$ The eigenvalue Λ corresponds to the critical compression level that forces a plate to buckle. See e.g. Bramble and Payne, Ashbaugh and Laugesen, or Henrot for known results and history of the problem. The approach described above cannot be directly used to find a buckling eigenvalue estimate, due to the more complicated form of the Rayleigh quotient. The lowest buckling eigenvalue equals $$\begin{aligned} \Lambda(\Omega)=\inf\_{u\in H\_0^2(\Omega)} \frac{\int\_{{{{{\mathbb{R}}}^d}}} \|\xi\|^4 \|\widehat u\|^2 d\xi}{\int\_{{{{{\mathbb{R}}}^d}}} \|\xi\|^2 \|\widehat u\|^2 d\xi}.\end{aligned}$$ Therefore the numerator is the same as for bi-Laplace eigenvalues, while the denominator is the quadratic form of the Laplacian. In order to apply our method we assume that u is the eigenfunction for Λ(Ω) and we rewrite a variational upper bound on T(Ω) as $$\begin{aligned} \Lambda(T(\Omega))\int\_{{{{{\mathbb{R}}}^d}}}\|T^{-\dagger}U^\dagger \xi\|^2\|\widehat u\|^2d\xi \le \int\_{{{{{\mathbb{R}}}^d}}}\|T^{-\dagger}U^\dagger \xi\|^4\|\widehat u\|^2d\xi.\end{aligned}$$ We average this inequality (each side) over the group G, as before. $$\begin{aligned} \label{buckineq} \Lambda(T(\Omega))\int\_{{{{{\mathbb{R}}}^d}}}\left(\frac{1}{G}\sum\_{U\in G}\|T^{-\dagger}U^\dagger \xi\|^2\right)\|\widehat u\|^2d\xi \le \int\_{{{{{\mathbb{R}}}^d}}}\left(\frac{1}{G}\sum\_{U\in G}\|T^{-\dagger}U^\dagger \xi\|^4\right)\|\widehat u\|^2d\xi.\end{aligned}$$ On the left, we can use the tight frame property for G. On the right, we either use 2-frame identity if the group G allows for it, or a bound from. In the 2-frame case we can also use to express the bound using moments of mass. We get: [thmbuckling] Suppose isomtery group G of the domain Ω admits 2-frames. For any linear transformation T $$\begin{aligned} \Lambda\Big\|\_{T(\Omega)}\le \frac{\\|T^{-1}\\|\_2^4+2\\|T^{-1}\\|\_4^4}{(d+2)\\|T^{-1}\\|\_2^2}\Lambda\Big\|\_\Omega. \end{aligned}$$ Equivalently $$\begin{aligned} \Lambda V^{2/d}\Big\|\_{T(\Omega)}\cdot\left.\frac{V^{2/d} I\_2}{I\_4}\right\|\_{T^{-1}(\Omega)} \end{aligned}$$ is maximal when T is a multiple of an orthogonal matrix. Note that similar results can be obtained for any Rayleigh quotient involving two quadratic forms with convex or concave multipliers. In case group G does not allow for appropriate higher order frame, we can use the lower bound from to simplify the left side of. Numerical comparisons --------------------- ### Plate problem We wish to compare our upper bound for the lowest plate eigenvalue without tension (τ = 0) in dimension 2 with other known results. The exact eigenvalue for a disk is known, hence we can easily get upper bounds for any ellipse. The disk eigenvalue can be written in terms of the lowest zero of a certain combination of Bessel functions, see e.g.. Numerically $$\begin{aligned} \lambda\_1(D) \approx \frac{104.36}{r^4},\end{aligned}$$ for a disk with radius r. Therefore Nadirashvili isoperimetric inequality applied to an ellipse with semiaxes a and b gives $$\begin{aligned} \lambda\_1\|\_{E(a,b)}\ge \frac{104.36}{a^2b^2}.\end{aligned}$$ Take a linear transformation with singular values a and b to transform a unit disk into an ellipse with semiaxes a and b (ratio r = a/b). Then inequality gives $$\begin{aligned} \label{ellipseupper} \Gamma\_1\|\_{E(a,b)}\le \frac{104.36}{a^2b^2} \frac{3r^2+2+3r^{-2}}{8}\;\left(\le \frac{104.36}{a^2b^2} \frac{r^2+r^{-2}}{2}\right).\end{aligned}$$ The inequality in the parentheses is the weaker upper bound (obtained from classical tight frames instead of 2-frames). A very comprehensive report by Leissa contains an extensive section devoted ellipses. Results from various sources, including Shibaoka and McNitt, are compared. Upper bounds (3.8) and (3.6) on page 38 of Leissa are especially relevant. The bounds are essentially variational with a trial functions given by (3.7) and (3.5). The bounds have the same dependence on semiaxes a and b, but differ by a multiplicative constant. This fact should not be a surprise, since both test functions are compositions of linear functions and test functions for disks. As we showed, the Rayleigh quotient for such functions splits into a constant depending on the transformation, and the Rayleigh quotient of the original test function on the disk. Therefore, one might immediately choose the exact eigenfunction for the disk and get the best possible result of this type. In fact Table 3.2 in this report lists upper bounds that seem to be coming from taking the eigenfunction for the disk composed with a linear transformation (instead of using ), exactly recovering our 2-frame based result for the first eigenvalue, without tension. It is clear, that one can avoid tedious trial function calculations for ellipses due to enough symmetry of the disk. In fact, without knowing the exact value for the disk, we would still get the same semiaxes dependence, and only need to somehow estimate the eigenvalue on the disk, separately. Or simply use the best known upper bound for the disk. It is also worth noting that classical tight 1-frames are not strong enough to recover the results from Leissa’s report. It is crucial to work with 2-frames, since we get exact values of the Rayleigh quotients. In we compare our bounds with bounds due to McLaurin. Note that 2-frame bound is quite close to his nearly exact values. However, McLaurin is using a collocation method, involving case by case optimization of linear combinations of certain exact solutions. Clearly, this cannot be achieved for arbitrary multipliers. \| a/b \| 1.1 \| 1.2 \| 2 \| 4 \| \| --- \| --- \| --- \| --- \| --- \| \| 1-frames, \| 106.262 \| 111.375 \| 221.765 \| 838.1 \| \| 2-frames, \| 105.786 \| 109.621 \| 192.414 \| 654.7 \| \| McLaurin (upper) \| 105.741 \| 109.440 \| 187.382 \| 603.2 \| \| McLaurin (lower) \| 105.741 \| 109.440 \| 187.380 \| 587.2 \| [plateellipse] Unfortunately, the stronger 2-frame result does not apply to rectangles, since the group of isometries of the square does not admit 2-frames. This is a fundamental limitation, not related to our method, as we now show. The exact eigenvalue of the square is not known, but Wieners proved a validated numerical bound 1294.93396 ± 0.00002, while McLaurin obtained slightly worse bound using collocation, namely 1294.94 ± 0.14. Taking 1295 as the eigenvalue of the square, if we falsely applied to the rectangle with sides 1 and 2 we would get the upper bound of around 597 for that rectangle. Kuttler and Sigillito give a lower bound for the same rectangle that equals 603.8. The same contradiction happens for 4 : 1 rectangle, also studied by Kuttler and Sigillito. The upper bound in is simply wrong for rectangles. One can of course use classical tight frames, but this leads to overestimation of the Rayleigh quotients. One can also check that, the 4th moment of mass I4 for rectangles is underestimated by the 2-frame averaging. In view of the rectangles example, it is somewhat surprising that is actually true for triangles. The lowest plate eigenvalues of the equilateral triangle and the right isosceles triangles have been numerically estimated by Kuttler and Sigillito. Rescaling to have area equal 1, we get 1839 for the equilateral triangle and 2216 for the right isosceles triangle. Taking their value for equilateral triangle and finding our stronger upper bound leads to 2389 as a bound for the right isosceles triangle. Leissa also lists a few values for isosceles triangles in Table 7.2. After rescaling to area 1, the equilateral triangle has 1845, the right isosceles triangle 2190, and the acute isosceles triangle with angle 30 degrees has an eigenvalue 2490. Our upper bound for this triangle is 2910, while trial function based bound due to Cox and Klein (equation (7.2) in Leissa’s report) is about 3400. The same bound seems better than our result for obtuse isosceles triangles (using Figure 7.2 from Leissa), indicating that their trial function is not just based on a linearly transformed trial function from equilateral triangle. Nevertheless, using our method we get easy to find bounds, that are not too far from the known values. Note also that a linear transformation between two triangles is easy to write down as a matrix transforming vertices. The appropriate Schatten norms are then easily calculated without finding singular values. ### Buckling problem As in the plate problem, we compare our upper bounds with already available results. The exact buckling eigenvalue for a ball is known, while other domains would require a numerical approach. Furthermore, the Faber-Krahn-type isoperimetric inequality is still an open conjecture. Therefore tight bounds are essential. Estimates are available for ellipses, by work of McLaurin, giving us some comparisons with our bounds. summarizes results using 2-frame based, and a weaker classical frame bounds. The 2-frame based bound is again quite tight, thanks to exactly evaluated Rayleigh quotient. The exact buckling eigenvalue of the unit disk equals j1, 12 ≈ 14.682 (square of the first zero of Bessel J1 function, see e.g. ). provides the following bound for an ellipse with semiaxes a and b (ratio r = a/b) $$\begin{aligned} \Lambda\|\_{E(a,b)}\le \frac{j\_{1,1}^2}{ab} \frac{3r^4+2r^2+3}{4r(r^2+1)}\;\left(\le \frac{14.682}{ab} \frac{r^4+1}{r(r^2+1)}\right).\end{aligned}$$ Note that $\frac{j\_{1,1}^2}{ab}$ represents the unscaled eigenvalue of a disk with area ab. Its numerator corresponds to the scale-invariant product ΛA that is conjectured to be minimal for the disk among all domains (Pólya-Szegö conjecture, see ). \| a/b \| 1.2 \| 1.4 \| 1.6 \| 2 \| 4 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 1-frames \| 15.41 \| 17.15 \| 19.47 \| 24.96 \| 55.49 \| \| 2-frames, \| 15.17 \| 16.34 \| 17.90 \| 21.66 \| 43.34 \| \| McLaurin (lower bound) \| 15.1 \| 16.1 \| 17.5 \| 20.8 \| 39.0 \| [bucklingellipse] Bochner’s subordinators (fractional order operators) ==================================================== In this section we discuss general fractional operators related to Bochner subordination of the Brownian semigroup. This ties our results to the potential theory of the subordinated Brownian motion (see for a broad overview). At the level of pseudodifferential operators we seek upper bounds for operators Ψ( − Δ), where the function Ψ is a complete Bernstein function. Weak formulations for such operators have the following frequency domain representation $$\begin{aligned} Q(u,u)=(\Psi(-\Delta)u,u)=\int\_{{{{{\mathbb{R}}}^d}}} \Psi(\|\xi\|^2) \|\widehat u\|^2\,d\xi, \end{aligned}$$ with the domain H0β(Ω) where β ≤ 1 depends on Ψ. See Chen-Song and Schilling-Song-Vondraček for a detailed account. Even though the above formulation uses complete Bernstein functions, in order to apply our method we only need Ψ to be concave. Therefore our results apply to any weakly defined operator with a multiplier Ψ(∣ξ∣2) with concave Ψ, as long as its definition makes sense and the spectrum is discrete. The quadratic form above fits the framework described in. Therefore we can estimate the eigenvalues of Ψ( − Δ) on T(Ω) using the eigenvalues of a slightly modified multiplier on Ω, via with Φ1 = 0 and Φ2 = − Ψ. Suppose ui are orthonormal eigenfunctions for Ψ( − Δ) on a domain Ω with irreducible isometry group. Then implies: [thmconcave] For any concave Ψ, linear transformation T and $c=\frac{\\|T^{-1}\\|\_2^2}{d}$ we have $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{\Psi(\|\xi\|^2/c),T(\Omega)}&\le \lambda\_1+\dots+\lambda\_n\Big\|\_{\Psi(\|\xi\|^2),\Omega} \end{aligned}$$ Note that unless Ψ is homogeneous we are working with a modified multiplier on T(Ω), since Ψ(∣ξ∣2/c) is not proportional to Ψ(∣ξ∣2). We observed the same phenomenon in the bi-Laplacian case with tension term. Finally, ∥T− 1∥22 is related to the second moment of mass I2 (polar moment of inertia) of T(Ω) (see or with p = 1). Take Ψ(t) = tα/2 (α/2-stable subordinator) to get. Note that homogeneity of Ψ allows us to restate the result using one operator and using geometric quantities. We can also derive a stronger result for ellipses as images of unit ball B (full orthogonal group as isometry group with integral over Haar measure as averaging). We plug the multiplier into to get $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{T(B)}\le \sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}\int\_{O(d)}\|T^{-\dagger}U^\dagger \xi\|^\alpha\,d\mu(U)\|\widehat u\_i\|^2d\xi\end{aligned}$$ Rotational invariance of Haar measure allows us to diagonalize T and arrive at with 2p = α (note that p is not an integer here). The eigenvalues of the matrix represent semiaxes of the ellipsoid. For simplicity we only look at ellipses (dimension 2) with semiaxes a and b. We get $$\begin{aligned} \lambda\_1+\dots+\lambda\_n\Big\|\_{E(a,b)}\le \sum\_{i=1}^n\int\_{{{{{\mathbb{R}}}^d}}}\left( \frac{1}{2\pi} \int\_0^{2\pi} (a^2\cos^2\theta+b^2\sin^2\theta)^\alpha\,d\theta\right) \|\xi\|^\alpha\|\widehat u\_i\|^2 \,d\xi\end{aligned}$$ Note that when α = 1 we recover the perimeter of the ellipse divided by the perimeter of the disk, inside the parentheses. For arbitrary α the resulting integral is a rather complicated combination of hypergeometric functions 2F1. At the slight expense of accuracy, for α ≤ 1 we can use Jensen inequality to get perimeter to the power α. Now take ui to be the eigenfunctions on the disk to get. To obtain this result we essentially used a 1/2-frame-like identity (we used with p = 1/2). This is only possible for ellipses. Indeed, linear transformation of a square could give a rhombus or a rectangle, and the perimeter would not be the same. Similarly for triangles. Now we switch to relativistic subordinator $\Psi\_m(t)=\sqrt{m^2+t}-m$, which is no longer homogeneous. Therefore the constant m must be rescaled as did τ in the plate with tension case. Note that $$\begin{aligned} \Psi\_m(t/c)=\frac{1}{\sqrt{c}}\left(\sqrt{(\sqrt{c}m)^2+t}-\sqrt{c} m\right).\end{aligned}$$ Therefore the mass constant needs to scale with $\sqrt{c}$ in order to have equality for eigenvalues of balls with arbitrary radii. However, Ψm(t) as a function of m is decreasing, and for c > 1 $$\begin{aligned} \Psi\_m(t/c)\ge \frac{\Psi\_m(t)}{\sqrt{c}}.\end{aligned}$$ Hence implies. Finally, when Ψ is any complete Bernstein function, then Ψ( − Δ) is a generator of a generic subordinated Brownian motion. Concavity of Ψ implies that $$\begin{aligned} f(t/c)\ge \frac{f(t)}{c}.\end{aligned}$$ Moreover, when Ψ satisfies Ψ(t/c) ≤ c− βΨ(t) we get. Related results --------------- Even though the spectrum of the fractional Laplacian was studied in many contexts, there are very few known bounds for eigenvalues. Furthermore, it is not even possible to find the exact eigenvalues for intervals, or balls. Kulczycki, Kwaśnicki, Małecki and Stós found very accurate numerical estimates for intervals, while Dyda found the best (so far) estimates for the first eigenvalue for balls in arbitrary dimension. The only known general bounds are based on isoperimetric inequality and inradius of the domain, by work of Bañuelos, Latała and Méndez-Hernández. In this case, the eigenvalues of a general domain are estimated by the eigenvalue of a ball. One can also find bounds for the eigenvalues of the fractional Laplacian by relating them to the eigenvalues of the Laplacian, using quadratic form comparability (Chen-Song ). We propose another approach, based on comparisons to ellipsoids, instead of balls. This gives sharper results for elongated domains, comparing to inradius based approach. Let E ⊂ Ω be a John ellipsoid of Ω (ellipsoid with maximal volume contained in Ω, see John or Ball ). Then dE ($\sqrt{d} E$ if Ω is centrally symmetric) contains Ω. Hence $$\begin{aligned} (\lambda\_1+\dots+\lambda\_n)c^{-\alpha/2}\Big\|\_{\alpha,E}\le \lambda\_1+\dots+\lambda\_n\Big\|\_{\alpha,\Omega}\le \lambda\_1+\dots+\lambda\_n\Big\|\_{\alpha,E},\end{aligned}$$ where c equals d for general domains, or $\sqrt{d}$ for centrally symmetric domains. It is also possible to find an optimal constant c, for a specific domain Ω, which will be no worse than the general case. Now we find T such that T− 1(E) is a unit disk, and use our bounds to get an upper bound on Ω. Let Ω be a plane domain with John ellipse E with semiaxes of length 1 and a > 1. There exists a linear transformation T− 1 that takes this ellipse to a unit disk D. It also transforms the original domain into Ωʹ with John ellipse D (the ellipse is a disk). (or ) gives the following bound for the lowest eigenvalue of the original domain $$\begin{aligned} \lambda\_1 \Big\|\_{\alpha,\Omega}\le \lambda\_1\Big\|\_{\alpha,E} \le \frac{\\|T^{-1}\\|\_{HS}^\alpha}{2^{\alpha/2}}\lambda\_1\Big\|\_{\alpha,D}= \left(\frac{1+1/a^2}{2}\right)^{\alpha/2}\lambda\_1\Big\|\_{\alpha,D}\end{aligned}$$ The domain Ω has inradius r, that satisfies 1 ≤ r2 ≤ a, since John ellipse has the largest volume of all ellipses inside Ω, and inradius is at least as large as the short semiaxis in John ellipse. Inradius bound from reads $$\begin{aligned} \lambda\_1\Big\|\_{\alpha,\Omega}\le \frac{1}{r^\alpha} \lambda\_1\Big\|\_{\alpha,D}.\end{aligned}$$ It is not clear if our bound is better in general then inradius based bound due to Bañuelos et. al.. If Ω is a rectangle (or ellipse), then r = 1 and our bound is certainly better. Applying the same linear transformation T to rotated squares leads to parallelograms. The maximal inradius is achieved when the image is a rhombus. Surprisingly, one checks that in this case both bounds give the same result. Hence our bound is at least as good on parallelograms as the inradius based bound. On the other hand, the discussion in shows that inradius based bound is better than our bound for all triangles. In general, inradius based bound should be better for polygons with inscribed circle and other nearly round domains. While our bound should be stronger on elongated domains. For α ≤ 1 we can also use to get stronger upper bounds for λ1∣α, E, however this involves the perimeter of ellipse E (elliptic integral). Acknowledgements ================ The work was partially supported by NCN grant 2012/07/B/ST1/03356. The author is grateful to Richard Laugesen for invaluable discussions on the spectral theory and the plate problems, as well as suggested improvements to some arguments. 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Math. 9 (1959), 707–713	arxiv_0000802
[alg:postsym] could be larger than the error resulting from the two-pass method of Section [sec:postsym] by a factor of $1/\tau\_{\rm min}$, where $\tau\_{\rm min}$ is the minimal singular value of the matrix $\mtx{Q}^{\adj}\mtx{\Omega}$. The situation can be improved by oversampling. Suppose that we seek a rank-k approximate eigenvalue decomposition. Pick a small oversampling parameter p. Draw an n × (k + p) random matrix $\mtx{\Omega}$, and form the sample matrix $\mtx{Y} = \mtx{A}\mtx{\Omega}$. Let $\mtx{Q}$ denote the n × k matrix formed by the k leading left singular vectors of $\mtx{Y}$. Now, the linear system has a coefficient matrix $\mtx{Q}^{\adj}\mtx{\Omega}$ of size k × (k + p), so it is overdetermined. An approximate solution of this system yields a k × k matrix $\mtx{B}$. Computational costs =================== So far, we have postponed a detailed discussion of the computational cost of randomized matrix approximation algorithms because it is necessary to account for both the first stage, where we compute an approximate basis for the range (§[sec:algorithm]), and the second stage, where we postprocess the basis to complete the factorization (§[sec:otherfactorizations]). We are now prepared to compare the cost of the two-stage scheme with the cost of traditional techniques. Choosing an appropriate algorithm, whether classical or randomized, requires us to consider the properties of the input matrix. To draw a nuanced picture, we discuss three representative computational environments in §[sec:generalmat]–[sec:outofcore]. We close with some comments on parallel implementations in §[sec:parallel]. For concreteness, we focus on the problem of computing an approximate SVD of an m × n matrix $\mtx{A}$ with numerical rank k. The costs for other factorizations are similar. General matrices that fit in core memory ---------------------------------------- Suppose that $\mtx{A}$ is a general matrix presented as an array of numbers that fits in core memory. In this case, the appropriate method for Stage A is to use a structured random matrix (§[sec:ailonchazelle]), which allows us to find a basis that captures the action of the matrix using $\bigO(mn\log(k) + k^2 m)$ flops. For Stage B, we apply the row-extraction technique (§[sec:postrows]), which costs an additional $\bigO(k^2(m+n))$ flops. The total number of operations $T\_{\rm random}$ for this approach satisfies $$T\_{\rm random} \sim mn \log(k) + k^2 (m+n).$$ As a rule of thumb, the approximation error of this procedure satisfies $$\label{eq:fastalg-err-heur} \norm{ \mtx{A} - \mtx{U\Sigma V}^\adj } \lesssim n \cdot \sigma\_{k+1},$$ where σk + 1 is the (k + 1)th singular value of $\mtx{A}$. The estimate, which follows from Theorem [thm:SRFT] and Lemma [lem:convertID], reflects the worst-case scenario; actual errors are usually smaller. This algorithm should be compared with modern deterministic techniques, such as rank-revealing QR followed by postprocessing (§[sec:partialdecomp]) which typically require $$T\_{\rm RRQR} \sim kmn$$ operations to achieve a comparable error. In this setting, the randomized algorithm can be several times faster than classical techniques even for problems of moderate size, say m, n ∼ 103 and k ∼ 102. See §[sec:numSRFT] for numerical evidence. In case row extraction is impractical, there is an alternative $\bigO(mn\log(k))$ technique described in . When the error is unacceptably large, we can use the direct method (§[sec:postSVD]) for Stage B, which brings the total cost to $\bigO( kmn )$ flops. Matrices for which matrix–vector products can be rapidly evaluated ------------------------------------------------------------------ In many problems in data mining and scientific computing, the cost $T\_{\rm mult}$ of performing the matrix–vector multiplication $\vct{x} \mapsto \mtx{A}\vct{x}$ is substantially smaller than the nominal cost $\bigO(mn)$ for the dense case. It is not uncommon that $\bigO(m + n)$ flops suffice. Standard examples include (i) very sparse matrices; (ii) structured matrices, such as Töplitz operators, that can be applied using the FFT or other means; and (iii) matrices that arise from physical problems, such as discretized integral operators, that can be applied via, e.g., the fast multipole method. Suppose that both $\mtx{A}$ and $\mtx{A}^\adj$ admit fast multiplies. The appropriate randomized approach for this scenario completes Stage A using Algorithm [alg:basic] with p constant (for the fixed-rank problem) or Algorithm [alg:adaptive2] (for the fixed-precision problem) at a cost of $(k+p) \, T\_{\rm mult} + \bigO(k^2 m)$ flops. For Stage B, we invoke Algorithm [alg:Atranspose], which requires $(k+p) \, T\_{\rm mult} + \bigO(k^2 (m + n))$ flops. The total cost $T\_{\rm sparse}$ satisfies $$\label{eq:sparsealg-cost} T\_{\rm sparse} = 2\,(k + p)\, T\_{\rm mult} + \bigO(k^2 (m + n)).$$ As a rule of thumb, the approximation error of this procedure satisfies $$\label{eq:sparsealg-error} \norm{ \mtx{A} - \mtx{U\Sigma V}^\adj } \lesssim \sqrt{kn} \cdot \sigma\_{k+1}.$$ The estimate follows from Corollary [cor:tail-spec-error-gauss] and the discussion in §[sec:postSVD]. Actual errors are usually smaller. When the singular spectrum of $\mtx{A}$ decays slowly, we can incorporate q iterations of the power method (Algorithm [alg:poweriteration]) to obtain superior solutions to the fixed-rank problem. The computational cost increases to, cf. , $$\label{eq:sparsealg-cost2} T\_{\rm sparse} = (2q+2)\,(k + p) \, T\_{\rm mult} + \bigO(k^2 (m + n)),$$ while the error improves to $$\label{eq:sparsealg-error2} \norm{ \mtx{A} - \mtx{U\Sigma V}^\adj } \lesssim (kn)^{1/2(2q+1)} \cdot \sigma\_{k+1}.$$ The estimate takes into account the discussion in §[sec:avg-power-method]. The power scheme can also be adapted for the fixed-precision problem (§[sec:powerscheme]). In this setting, the classical prescription for obtaining a partial SVD is some variation of a Krylov-subspace method; see §[sec:krylov]. These methods exhibit great diversity, so it is hard to specify a “typical” computational cost. To a first approximation, it is fair to say that in order to obtain an approximate SVD of rank k, the cost of a numerically stable implementation of a Krylov method is no less than the cost with p set to zero. At this price, the Krylov method often obtains better accuracy than the basic randomized method obtained by combining Algorithms [alg:basic] and [alg:Atranspose], especially for matrices whose singular values decay slowly. On the other hand, the randomized schemes are inherently more robust and allow much more freedom in organizing the computation to suit a particular application or a particular hardware architecture. The latter point is in practice of crucial importance because it is usually much faster to apply a matrix to k vectors simultaneously than it is to execute k matrix–vector multiplications consecutively. In practice, blocking and parallelism can lead to enough gain that a few steps of the power method (Algorithm [alg:poweriteration]) can be performed more quickly than k steps of a Krylov method. Any comparison between randomized sampling schemes and Krylov variants becomes complicated because of the fact that “basic” Krylov schemes such as Lanczos or Arnoldi are inherently unstable. To obtain numerical robustness, we must incorporate sophisticated modifications such as restarts, reorthogonalization procedures, etc. Constructing a high-quality implementation is sufficiently hard that the authors of a popular book on “numerical recipes” qualify their treatment of spectral computations as follows : > > You have probably gathered by now that the solution of eigensystems is a fairly complicated business. It is. It is one of the few subjects covered in this book for which we do not recommend that you avoid canned routines. On the contrary, the purpose of this chapter is precisely to give you some appreciation of what is going on inside such canned routines, so that you can make intelligent choices about using them, and intelligent diagnoses when something goes wrong. > > > Randomized sampling does not eliminate the difficulties referred to in this quotation; however it reduces the task of computing a partial spectral decomposition of a very large matrix to the task of computing a full decomposition of a small dense matrix. (For example, in Algorithm [alg:Atranspose], the input matrix $\mtx{A}$ is large and $\mtx{B}$ is small.) The latter task is much better understood and is eminently suitable for using canned routines. Random sampling schemes interact with the large matrix only through matrix–matrix products, which can easily be implemented by a user in a manner appropriate to the application and to the available hardware. The comparison is further complicated by the fact that there is significant overlap between the two sets of ideas. Algorithm [alg:poweriteration] is conceptually similar to a “block Lanczos method” with a random starting matrix. Indeed, we believe that there are significant opportunities for cross-fertilization in this area. Hybrid schemes that combine the best ideas from both fields may perform very well. General matrices stored in slow memory or streamed -------------------------------------------------- The traditional metric for numerical algorithms is the number of floating-point operations they require. When the data does not fit in fast memory, however, the computational time is often dominated by the cost of memory access. In this setting, a more appropriate measure of algorithmic performance is, which counts how many times the data needs to be cycled through fast memory. Flop counts become largely irrelevant. All the classical matrix factorization techniques that we discuss in §[sec:standardfactorizations]—including dense SVD, rank-revealing QR, Krylov methods, and so forth—require at least k passes over the the matrix, which is prohibitively expensive for huge data matrices. A desire to reduce the pass count of matrix approximation algorithms served as one of the early motivations for developing randomized schemes . Detailed recent work appears in . For many matrices, randomized techniques can produce an accurate approximation using just one pass over the data. For Hermitian matrices, we obtain a single-pass algorithm by combining Algorithm [alg:basic], which constructs an approximate basis, with Algorithm [alg:postsym], which produces an eigenvalue decomposition without any additional access to the matrix. Section [sec:onepass] describes the analogous technique for general matrices. For the huge matrices that arise in applications such as data mining, it is common that the singular spectrum decays slowly. Relevant applications include image processing (see §§[sec:graphlaplacian]–[sec:eigenfaces] for numerical examples), statistical data analysis, and network monitoring. To compute approximate factorizations in these environments, it is crucial to enhance the accuracy of the randomized approach using the power scheme, Algorithm [alg:poweriteration], or some other device. This approach increases the pass count somewhat, but in our experience it is very rare that more than five passes are required. Gains from parallelization -------------------------- As mentioned in §§[sec:fastmatvec]–[sec:outofcore], randomized methods often outperform classical techniques not because they involve fewer floating-point operations but rather because they allow us to reorganize the calculations to exploit the matrix properties and the computer architecture more fully. In addition, these methods are well suited for parallel implementation. For example, in Algorithm [alg:basic], the computational bottleneck is the evaluation of the matrix product $\mtx{A\Omega}$, which is embarrassingly parallelizable. Numerical examples ================== By this time, the reader has surely formulated a pointed question: Do these randomized matrix approximation algorithms actually work in practice? In this section, we attempt to address this concern by illustrating how the algorithms perform on a diverse collection of test cases. Section [sec:example1] starts with two examples from the physical sciences involving discrete approximations to operators with exponentially decaying spectra. Sections [sec:graphlaplacian] and [sec:eigenfaces] continue with two examples of matrices arising in “data mining.” These are large matrices whose singular spectra decay slowly; one is sparse and fits in RAM, one is dense and is stored out-of-core. Finally, §[sec:numSRFT] investigates the performance of randomized methods based on structured random matrices. Sections [sec:example1]–[sec:eigenfaces] focus on the algorithms for Stage A that we presented in §[sec:algorithm] because we wish to isolate the performance of the randomized step. Computational examples illustrating truly large data matrices have been reported elsewhere, for instance in. Two matrices with rapidly decaying singular values -------------------------------------------------- We first illustrate the behavior of the adaptive range approximation method, Algorithm [alg:adaptive2]. We apply it to two matrices associated with the numerical analysis of differential and integral operators. The matrices in question have rapidly decaying singular values and our intent is to demonstrate that in this environment, the approximation error of a bare-bones randomized method such as Algorithm [alg:adaptive2] is very close to the minimal error achievable by any method. We observe that the approximation error of a randomized method is itself a random variable (it is a function of the random matrix $\mtx{\Omega}$) so what we need to demonstrate is not only that the error is small in a typical realization, but also that it clusters tightly around the mean value. We first consider a 200 × 200 matrix $\mtx{A}$ that results from discretizing the following single-layer operator associated with the Laplace equation: $$\label{eq:int\_op\_laplace} [S \sigma](x) = {\rm const} \cdot \int\_{\Gamma\_1} \log \abs{x-y} \, \sigma(y) \idiff{A(y)}, \qquad x \in \Gamma\_2,$$ where Γ1 and Γ2 are the two contours in R2 illustrated in Figure [fig:contours](a). We approximate the integral with the trapezoidal rule, which converges superalgebraically because the kernel is smooth. In the absence of floating-point errors, we estimate that the discretization error would be less than 10− 20 for a smooth source σ. The leading constant is selected so the matrix $\mtx{A}$ has unit operator norm. We implement Algorithm [alg:adaptive2] in Matlab v6.5. Gaussian test matrices are generated using the randn command. For each number ℓ of samples, we compare the following three quantities: 1. The minimum rank-ℓ approximation error σℓ + 1 is determined using svd. 2. The actual error $e\_{\ell} = \norm{\big(\Id - \mtx{Q}^{(\ell)}(\mtx{Q}^{(\ell)})^{\adj} \big)\mtx{A}}$ is computed with norm. 3. A random estimator fℓ for the actual error eℓ is obtained from , with the parameter r set to 5. Note that any values less than 10− 15 should be considered numerical artifacts. Figure [fig:laplacefullrun] tracks a characteristic execution of Algorithm [alg:adaptive2]. We make three observations: (i) The error eℓ incurred by the algorithm is remarkably close to the theoretical minimum σℓ + 1. (ii) The error estimate always produces an upper bound for the actual error. Without the built-in 10 × safety margin, the estimate would track the actual error almost exactly. (iii) The basis constructed by the algorithm essentially reaches full double-precision accuracy. How typical is the trial documented in Figure [fig:laplacefullrun]? To answer this question, we examine the empirical performance of the algorithm over 2000 independent trials. Figure [fig:laplacestats] charts the error estimate versus the actual error at four points during the course of execution: ℓ = 25, 50, 75, 100. We offer four observations: (i) The initial run detailed in Figure [fig:laplacefullrun] is entirely typical. (ii) Both the actual and estimated error concentrate about their mean value. (iii) The actual error drifts slowly away from the optimal error as the number ℓ of samples increases. (iv) The error estimator is always pessimistic by a factor of about ten, which means that the algorithm never produces a basis with lower accuracy than requested. The only effect of selecting an unlucky sample matrix $\mtx{\Omega}$ is that the algorithm proceeds for a few additional steps. ccc (40,40) (-10,00) (11,29)Γ1 (31,09)Γ2 & & (a) && (b) [fig:contours] (125,87) (09,00) (62,00)ℓ (91,79.5)log10(fℓ) (91,76.75)log10(eℓ) (91,74)log10(σℓ + 1) (48,85)Approximation errors (8,30) [fig:laplacefullrun] (110,75) (10,03) (26,-2)$\footnotesize \log\_{10}(e\_{\ell})$ (4,13) (40,09)“y = x” (18,29)Minimal (18,26)error (29,73)ℓ = 25 (77,73)ℓ = 50 (29,34)ℓ = 75 (77,34)ℓ = 100 [fig:laplacestats] We next consider a matrix $\mtx{B}$ which is defined implicitly in the sense that we cannot access its elements directly; we can only evaluate the map $\vct{x} \mapsto \mtx{B}\vct{x}$ for a given vector $\vct{x}$. To be precise, $\mtx{B}$ represents a transfer matrix for a network of resistors like the one shown in Figure [fig:contours](b). The vector $\vct{x}$ represents a set of electric potentials specified on the red nodes in the figure. These potentials induce a unique equilibrium field on the network in which the potential of each black and blue node is the average of the potentials of its three or four neighbors. The vector $\mtx{B}\vct{x}$ is then the restriction of the potential to the blue exterior nodes. Given a vector $\vct{x}$, the vector $\mtx{B}\vct{x}$ can be obtained by solving a large sparse linear system whose coefficient matrix is the classical five-point stencil approximating the 2D Laplace operator. We applied Algorithm [alg:adaptive2] to the 1596 × 532 matrix $\mtx{B}$ associated with a lattice in which there were 532 nodes (red) on the “inner ring” and 1596 nodes on the (blue) “outer ring.” Each application of $\mtx{B}$ to a vector requires the solution of a sparse linear system of size roughly 140 000 × 140 000. We implemented the scheme in Matlab using the “backslash” operator for the linear solve. The results of a typical trial appear in Figure [fig:latticefullrun]. Qualitatively, the performance matches the results in Figure [fig:laplacestats]. (125,86) (09,00) (62,00)ℓ (91,79.5)log10(fℓ) (91,76.75)log10(eℓ) (91,74)log10(σℓ + 1) (48,85)Approximation errors (8,30) [fig:latticefullrun] A large, sparse, noisy matrix arising in image processing --------------------------------------------------------- Our next example involves a matrix that arises in image processing. A recent line of work uses information about the local geometry of an image to develop promising new algorithms for standard tasks, such as denoising, inpainting, and so forth. These methods are based on approximating a associated with the image. The dominant eigenvectors of this matrix provide “coordinates” that help us smooth out noisy image patches. We begin with a 95 × 95 pixel grayscale image. The intensity of each pixel is represented as an integer in the range 0 to 4095. We form for each pixel i a vector $\vct{x}^{(i)} \in \mathbb{R}^{25}$ by gathering the 25 intensities of the pixels in a 5 × 5 neighborhood centered at pixel i (with appropriate modifications near the edges). Next, we form the 9025 × 9025 $\widetilde{\mtx{W}}$ that reflects the similarities between patches: $$\widetilde{w}\_{ij} = \exp\big\{-\smnorm{}{ \vct{x}^{(i)} - \vct{x}^{(j)} }^2 / \sigma^2 \big\},$$ where the parameter σ = 50 controls the level of sensitivity. We obtain a sparse weight matrix $\mtx{W}$ by zeroing out all entries in $\widetilde{\mtx{W}}$ except the seven largest ones in each row. The object is then to construct the low frequency eigenvectors of the graph Laplacian matrix $$\mtx{L} = \Id - \mtx{D}^{-1/2} \mtx{W} \mtx{D}^{-1/2},$$ where $\mtx{D}$ is the diagonal matrix with entries dii = ∑jwij. These are the eigenvectors associated with the dominant eigenvalues of the auxiliary matrix $\mtx{A} = \mtx{D}^{-1/2} \mtx{W} \mtx{D}^{-1/2}$. The matrix $\mtx{A}$ is large, and its eigenvalues decay slowly, so we use the power scheme summarized in Algorithm [alg:poweriteration] to approximate it. Figure [fig:Meyer][left] illustrates how the approximation error eℓ declines as the number ℓ of samples increases. When we set the exponent q = 0, which corresponds with the basic Algorithm [alg:basic], the approximation is rather poor. The graph illustrates that increasing the exponent q slightly results in a tremendous improvement in the accuracy of the power scheme. Next, we illustrate the results of using the two-stage approach to approximate the eigenvalues of $\mtx{A}$. In Stage A, we construct a basis for $\mtx{A}$ using Algorithm [alg:poweriteration] with ℓ = 100 samples for different values of q. In Stage B, we apply the Hermitian variant of Algorithm [alg:Atranspose] described in §[sec:postsym] to compute an approximate eigenvalue decomposition. Figure [fig:Meyer][right] shows the approximate eigenvalues and the actual eigenvalues of $\mtx{A}$. Once again, we see that the minimal exponent q = 0 produces miserable results, but the largest eigenvalues are quite accurate even for q = 1. (125,86) (10,00) (38,00)ℓ (92,00)j (24,84)Approximation error eℓ (74,84)Estimated Eigenvalues λj (8,35) (86,77.5)“Exact” eigenvalues (87,75.0) λj for q = 3 (87,72.5) λj for q = 2 (87,70.0) λj for q = 1 (87,67.5)λj for q = 0 [fig:Meyer] Eigenfaces ---------- Our next example involves a large, dense matrix derived from the FERET databank of face images . A simple method for performing face recognition is to identify the principal directions of the image data, which are called. Each of the original photographs can be summarized by its components along these principal directions. To identify the subject in a new picture, we compute its decomposition in this basis and use a classification technique, such as nearest neighbors, to select the closest image in the database. We construct a data matrix $\mtx{A}$ as follows: The FERET database contains 7254 images, and each 384 × 256 image contains 98 304 pixels. First, we build a 98 304 × 7254 matrix $\widetilde{\mtx{A}}$ whose columns are the images. We form $\mtx{A}$ by centering each column of $\widetilde{\mtx{A}}$ and scaling it to unit norm, so that the images are roughly comparable. The eigenfaces are the dominant left singular vectors of this matrix. Our goal then is to compute an approximate SVD of the matrix $\mtx{A}$. Represented as an array of double-precision real numbers, $\mtx{A}$ would require 5.4GB of storage, which does not fit within the fast memory of many machines. It is possible to compress the database down to at 57MB or less (in JPEG format), but then the data would have to be uncompressed with each sweep over the matrix. Furthermore, the matrix $\mtx{A}$ has slowly decaying singular values, so we need to use the power scheme, Algorithm [alg:poweriteration], to capture the range of the matrix accurately. To address these concerns, we implemented the power scheme to run in a pass-efficient manner. An additional difficulty arises because the size of the data makes it expensive to calculate the actual error eℓ incurred by the approximation or to determine the minimal error σℓ + 1. To estimate the errors, we use the technique described in Remark [remark:bettererrorestimate]. Figure [fig:eigenfaces] describes the behavior of the power scheme, which is similar to its performance for the graph Laplacian in §[sec:graphlaplacian]. When the exponent q = 0, the approximation of the data matrix is very poor, but it improves quickly as q increases. Likewise, the estimate for the spectrum of $\mtx{A}$ appears to converge rapidly; the largest singular values are already quite accurate when q = 1. We see essentially no improvement in the estimates after the first 3–5 passes over the matrix. (125,88) (08,00) (24,86)Approximation error eℓ (74,86)Estimated Singular Values σj (8,40) (34,81.50)Minimal error (est) (35,78.75)q = 0 (35,76.00) q = 1 (35,73.25) q = 2 (35,70.50) q = 3 (37,00)ℓ (93,00)j [fig:eigenfaces] Performance of structured random matrices ----------------------------------------- Our final set of experiments illustrates that the structured random matrices described in §[sec:ailonchazelle] lead to matrix approximation algorithms that are both fast and accurate. First, we compare the computational speeds of four methods for computing an approximation to the ℓ dominant terms in the SVD of an n × n matrix $\mtx{A}$. For now, we are interested in execution time only (not accuracy), so the choice of matrix is irrelevant and we have selected $\mtx{A}$ to be a Gaussian matrix. The four methods are summarized in the following table; Remark [rem:fortran] provides more details on the implementation. \|l\|\|l\|l\| Method & Stage A & Stage B `direct` & Rank-revealing QR executed using column & Algorithm [alg:Atranspose] & pivoting and Householder reflectors & `gauss` & Algorithm [alg:basic] with a Gaussian random matrix & Algorithm [alg:Atranspose] `srft` & Algorithm [alg:basic] with the modified SRFT & Algorithm [alg:extractrows] `svd` & Full SVD with LAPACK routine dgesdd & Truncate to ℓ terms Table [tab:runtimes] lists the measured runtime of a single execution of each algorithm for various choices of the dimension n of the input matrix and the rank ℓ of the approximation. Of course, the cost of the full SVD does not depend on the number ℓ of components required. A more informative way to look at the runtime data is to compare the relative cost of the algorithms. The direct method is the best deterministic approach for dense matrices, so we calculate the factor by which the randomized methods improve on this benchmark. Figure [fig:speedup] displays the results. We make two observations: (i) Using an SRFT often leads to a dramatic speed-up over classical techniques, even for moderate problem sizes. (ii) Using a standard Gaussian test matrix typically leads to a moderate speed-up over classical methods, primarily because performing a matrix–matrix multiplication is faster than a QR factorization. Second, we investigate how the choice of random test matrix influences the error in approximating an input matrix. For these experiments, we return to the 200 × 200 matrix $\mtx{A}$ defined in Section [sec:example1]. Consider variations of Algorithm [alg:basic] obtained when the random test matrix $\mtx{\Omega}$ is drawn from the following four distributions: = = Gauss: The standard Gaussian distribution. Ortho: The uniform distribution on n × ℓ orthonormal matrices. SRFT: The SRFT distribution defined in . GSRFT: The modified SRFT distribution defined in . Intuitively, we expect that Ortho should provide the best performance. For each distribution, we perform 100000 trials of the following experiment. Apply the corresponding version of Algorithm [alg:basic] to the matrix $\mtx{A}$, and calculate the approximation error $e\_{\ell} = \norm{\mtx{A} - \mtx{Q}\_{\ell}\mtx{Q}\_{\ell}^{\adj}\mtx{A}}$. Figure [fig:SRFTerrors] displays the empirical probability density function for the error eℓ obtained with each algorithm. We offer three observations: (i) The SRFT actually performs slightly better than a Gaussian random matrix for this example. (ii) The standard SRFT and the modified SRFT have essentially identical errors. (iii) There is almost no difference between the Gaussian random matrix and the random orthonormal matrix in the first three plots, while the fourth plot shows that the random orthonormal matrix performs better. This behavior occurs because, with high probability, a tall Gaussian matrix is well conditioned and a (nearly) square Gaussian matrix is not. [rem:fortran] The running times reported in Table [tab:runtimes] and in Figure [fig:speedup] depend strongly on both the computer hardware and the coding of the algorithms. The experiments reported here were performed on a standard office desktop with a 3.2GHz Pentium IV processor and 2GB of RAM. The algorithms were implemented in Fortran 90 and compiled with the Lahey compiler. The Lahey versions of BLAS and LAPACK were used to accelerate all matrix–matrix multiplications, as well as the SVD computations in Algorithms [alg:Atranspose] and [alg:extractrows]. We used the code for the modified SRFT provided in the publicly available software package `id\underline{\mbox{ }}dist`. \|r\|\|r@er\|r@er\|r@er\|\|r@er\|r@er\|r@er\|\|r@er\|r@er\|r@er\| & & & & & & & & & & & & 10 & 1.08&-1 & 5.63&-2 & 9.06&-2 & 4.22&-1 & 2.16&-1 & 3.56&-1 & 1.70&0 & 8.94&-1 & 1.45&0 20 & 1.97&-1 & 9.69&-2 & 1.03&-1 & 7.67&-1 & 3.69&-1 & 3.89&-1 & 3.07&0 & 1.44&0 & 1.53&0 40 & 3.91&-1 & 1.84&-1 & 1.27&-1 & 1.50&0 & 6.69&-1 & 4.33&-1 & 6.03&0 & 2.64&0 & 1.63&0 80 & 7.84&-1 & 4.00&-1 & 2.19&-1 & 3.04&0 & 1.43&0 & 6.64&-1 & 1.20&1 & 5.43&0 & 2.08&0 160 & 1.70&0 & 9.92&-1 & 6.92&-1 & 6.36&0 & 3.36&0 & 1.61&0 & 2.46&1 & 1.16&1 & 3.94&0 320 & 3.89&0 & 2.65&0 & 2.98&0 & 1.34&1 & 7.45&0 & 5.87&0 & 5.00&1 & 2.41&1 & 1.21&1 640 & 1.03&1 & 8.75&0 & 1.81&1 & 3.14&1 & 2.13&1 & 2.99&1 & 1.06&2 & 5.80&1 & 5.35&1 1280 & & & & 7.97&1 & 6.69&1 & 3.13&2 & 2.40&2 & 1.68&2 & 4.03&2 & & & & & & & & & (130,105) (10,04) (27,00)ℓ (69,00)ℓ (111,00)ℓ (21, 102)n = 1024 (63, 102)n = 2048 (105,102)n = 4096 (21,96)$t^{\rm(direct)}/t^{\rm(gauss)}$ (21,93)$t^{\rm(direct)}/t^{\rm(srft)}$ (21,90)$t^{\rm(direct)}/t^{\rm(svd)}$ (3,40) [fig:speedup] (130,102) (08,00) (02,13) (36,55)e25 (101,55)e50 (36,03)e75 (101,03)e100 (33,100)ℓ = 25 (33,48)ℓ = 75 (98,100)ℓ = 50 (98,48)ℓ = 100 (48,42.5)Gauss (48,39.5)Ortho (48,36.5)SRFT (48,33.5)GSRFT [fig:SRFTerrors] Part III: Theory This part of the paper, §§[sec:lin-alg-prelim]–[sec:SRFTs], provides a detailed analysis of randomized sampling schemes for constructing an approximate basis for the range of a matrix, the task we refer to as Stage A in the framework of §[sec:framework]. More precisely, we assess the quality of the basis $\mtx{Q}$ that the proto-algorithm of §[sec:sketchofalgorithm] produces by establishing rigorous bounds for the approximation error $$\label{eq:basicerror} \triplenorm{ \mtx{A} - \mtx{QQ}^\adj \mtx{A} },$$ where $\triplenorm{\cdot}$ denotes either the spectral norm or the Frobenius norm. The difficulty in developing these bounds is that the matrix $\mtx{Q}$ is random, and its distribution is a complicated nonlinear function of the input matrix $\mtx{A}$ and the random test matrix $\mtx{\Omega}$. Naturally, any estimate for the approximation error must depend on the properties of the input matrix and the distribution of the test matrix. To address these challenges, we split the argument into two pieces. The first part exploits techniques from linear algebra to deliver a generic error bound that depends on the interaction between the test matrix $\mtx{\Omega}$ and the right singular vectors of the input matrix $\mtx{A}$, as well as the tail singular values of $\mtx{A}$. In the second part of the argument, we take into account the distribution of the random matrix to estimate the error for specific instantiations of the proto-algorithm. This bipartite proof is common in the literature on randomized linear algebra, but our argument is most similar in spirit to . Section [sec:lin-alg-prelim] surveys the basic linear algebraic tools we need. Section [sec:basicerr] uses these methods to derive a generic error bound. Afterward, we specialize these results to the case where the test matrix is Gaussian (§[sec:gaussians]) and the case where the test matrix is a subsampled random Fourier transform (§[sec:SRFTs]). Theoretical preliminaries ========================= We proceed with some additional background from linear algebra. Section [sec:psd] sets out properties of positive-semidefinite matrices, and §[sec:orthproj] offers some results for orthogonal projectors. Standard references for this material include . Positive semidefinite matrices ------------------------------ An Hermitian matrix $\mtx{M}$ is (briefly, ) when $\vct{u}^\adj \mtx{M} \vct{u} \geq 0$ for all $\vct{u} \neq \vct{0}$. If the inequalities are strict, $\mtx{M}$ is (briefly, ). The psd matrices form a convex cone, which induces a partial ordering on the linear space of Hermitian matrices: $\mtx{M} \psdle \mtx{N}$ if and only if $\mtx{N} - \mtx{M}$ is psd. This ordering allows us to write $\mtx{M} \psdge \mtx{0}$ to indicate that the matrix $\mtx{M}$ is psd. Alternatively, we can define a psd (resp., pd) matrix as an Hermitian matrix with nonnegative (resp., positive) eigenvalues. In particular, each psd matrix is diagonalizable, and the inverse of a pd matrix is also pd. The spectral norm of a psd matrix $\mtx{M}$ has the variational characterization $$\label{eq:norm\_psd} \norm{\mtx{M}} = \max\_{\vct{u} \neq \vct{0}} \frac{\vct{u}^\adj \mtx{M} \vct{u}}{\vct{u}^\adj \vct{u}},$$ according to the Rayleigh–Ritz theorem . It follows that $$\label{eq:psdle\_norm} \mtx{M} \psdle \mtx{N} \quad\Longrightarrow\quad \norm{\mtx{M}} \leq \norm{\mtx{N}}.$$ A fundamental fact is that conjugation preserves the psd property. [Conjugation Rule] [prop:conjugation] Suppose that $\mtx{M} \psdge \mtx{0}$. For every $\mtx{A}$, the matrix $\mtx{A}^\adj \mtx{M} \mtx{A} \psdge \mtx{0}$. In particular, $$\mtx{M} \psdle \mtx{N} \quad\Longrightarrow\quad \mtx{A}^\adj \mtx{M} \mtx{A} \psdle \mtx{A}^\adj \mtx{N} \mtx{A}.$$ Our argument invokes the conjugation rule repeatedly. As a first application, we establish a perturbation bound for the matrix inverse near the identity matrix. [Perturbation of Inverses] [prop:perturb-inv] Suppose that $\mtx{M} \psdge \mtx{0}$. Then $$\Id - (\Id + \mtx{M})^{-1} \psdle \mtx{M}$$ Define $\mtx{R} = \mtx{M}^{1/2}$, the psd square root of $\mtx{M}$ promised by . We have the chain of relations $$\Id - (\Id + \mtx{R}^2)^{-1} = (\Id + \mtx{R}^2)^{-1} \mtx{R}^2 = \mtx{R} (\Id + \mtx{R}^2)^{-1} \mtx{R} \psdle \mtx{R}^2.$$ The first equality can be verified algebraically. The second holds because rational functions of a diagonalizable matrix, such as $\mtx{R}$, commute. The last relation follows from the conjugation rule because $(\Id + \mtx{R}^2)^{-1} \psdle \Id$. Next, we present a generalization of the fact that the spectral norm of a psd matrix is controlled by its trace. [prop:spec-norm-sum] We have $\norm{\mtx{M}} \leq \norm{\mtx{A}} + \norm{\mtx{C}}$ for each partitioned psd matrix $$\label{eqn:M-matrix} \mtx{M} = \begin{bmatrix} \mtx{A} & \mtx{B} \\ \mtx{B}^\adj & \mtx{C} \end{bmatrix}.$$ The variational characterization of the spectral norm implies that $$\begin{aligned} \norm{ \mtx{M} } &= \sup\_{\normsq{\vct{x}} + \normsq{\vct{y}}=1} \begin{bmatrix} \vct{x} \\ \vct{y} \end{bmatrix}^\adj \begin{bmatrix} \mtx{A} & \mtx{B} \\ \mtx{B}^\adj & \mtx{C} \end{bmatrix} \begin{bmatrix} \vct{x} \\ \vct{y} \end{bmatrix} \\ &\leq \sup\_{\normsq{\vct{x}} + \normsq{\vct{y}}=1} \bigl(\norm{\mtx{A}} \normsq{\vct{x}} + 2 \norm{\mtx{B}} \norm{\vct{x}} \norm{\vct{y}} + \norm{\mtx{C}} \normsq{\vct{y}}\bigr).\end{aligned}$$ The block generalization of Hadamard’s psd criterion states that $\normsq{ \mtx{B} } \leq \norm{\mtx{A}} \norm{\mtx{C}}$. Thus, $$\norm{ \mtx{M} } \leq \sup\_{\normsq{\vct{x}} + \normsq{\vct{y}}=1} \bigl(\norm{\mtx{A}}^{1/2} \norm{\vct{x}} + \norm{\mtx{C}}^{1/2} \norm{\vct{y}}\bigr)^{2} = \norm{\mtx{A}} + \norm{\mtx{C}}.$$ This point completes the argument. Orthogonal projectors --------------------- An is an Hermitian matrix $\mtx{P}$ that satisfies the polynomial $\mtx{P}^2 = \mtx{P}$. This identity implies $\mtx{0} \psdle \mtx{P} \psdle \Id$. An orthogonal projector is completely determined by its range. For a given matrix $\mtx{M}$, we write $\mtx{P}\_{\mtx{M}}$ for the unique orthogonal projector with $\range(\mtx{P}\_{\mtx{M}}) = \range(\mtx{M})$. When $\mtx{M}$ has full column rank, we can express this projector explicitly: $$\label{eqn:orth-proj-form} \mtx{P}\_{\mtx{M}} = \mtx{M} (\mtx{M}^\adj \mtx{M})^{-1} \mtx{M}^\adj.$$ The orthogonal projector onto the complementary subspace, $\range(\mtx{P})^{\perp}$, is the matrix $\Id - \mtx{P}$. Our argument hinges on several other facts about orthogonal projectors. [prop:conj-proj] Suppose $\mtx{U}$ is unitary. Then $\mtx{U}^\adj \mtx{P}\_{\mtx{M}} \mtx{U} = \mtx{P}\_{\mtx{U}^\adj \mtx{M}}$. Abbreviate $\mtx{P} = \mtx{U}^\adj \mtx{P}\_{\mtx{M}} \mtx{U}$. It is clear that $\mtx{P}$ is an orthogonal projector since it is Hermitian and $\mtx{P}^{2} = \mtx{P}$. Evidently, $$\range(\mtx{P}) = \mtx{U}^\adj \range(\mtx{M}) = \range(\mtx{U}^\adj \mtx{M}).$$ Since the range determines the orthogonal projector, we conclude $\mtx{P} = \mtx{P}\_{\mtx{U}^\adj \mtx{M}}$. [prop:proj-range] Suppose $\range(\mtx{N}) \subset \range(\mtx{M})$. Then, for each matrix $\mtx{A}$, it holds that $ \norm{\mtx{P}\_{\mtx{N}} \mtx{A}} \leq \norm{\mtx{P}\_{\mtx{M}} \mtx{A}}$ and that $\norm{(\Id - \mtx{P}\_{\mtx{M}}) \mtx{A}} \leq \norm{(\Id - \mtx{P}\_{\mtx{N}}) \mtx{A}}$. The projector $\mtx{P}\_{\mtx{N}} \psdle \Id$, so the conjugation rule yields $\mtx{P}\_{\mtx{M}}\mtx{P}\_{\mtx{N}}\mtx{P}\_{\mtx{M}} \psdle \mtx{P}\_{\mtx{M}}$. The hypothesis $\range(\mtx{N}) \subset \range(\mtx{M})$ implies that $\mtx{P}\_{\mtx{M}} \mtx{P}\_{\mtx{N}} = \mtx{P}\_{\mtx{N}}$, which results in $$\mtx{P}\_{\mtx{M}} \mtx{P}\_{\mtx{N}} \mtx{P}\_{\mtx{M}} = \mtx{P}\_{\mtx{N}} \mtx{P}\_{\mtx{M}} = (\mtx{P}\_{\mtx{M}} \mtx{P}\_{\mtx{N}})^\adj = \mtx{P}\_{\mtx{N}}.$$ In summary, $\mtx{P}\_{\mtx{N}} \psdle \mtx{P}\_{\mtx{M}}$. The conjugation rule shows that $\mtx{A}^\adj \mtx{P}\_{\mtx{N}} \mtx{A} \psdle \mtx{A}^\adj \mtx{P}\_{\mtx{M}} \mtx{A}$. We conclude from that $$\normsq{ \mtx{P}\_{\mtx{N}} \mtx{A} } = \norm{ \mtx{A}^\adj \mtx{P}\_{\mtx{N}} \mtx{A} } \leq \norm{ \mtx{A}^\adj \mtx{P}\_{\mtx{M}} \mtx{A} } = \normsq{ \mtx{P}\_{\mtx{M}} \mtx{A} }.$$ The second statement follows from the first by taking orthogonal complements. Finally, we need a generalization of the scalar inequality $\abs{px}^q \leq \abs{p} \abs{x}^q$, which holds when $\abs{p} \leq 1$ and q ≥ 1. [prop:proj-power] Let $\mtx{P}$ be an orthogonal projector, and let $\mtx{M}$ be a matrix. For each positive number q, $$\label{eq:trivial} \norm{ \mtx{PM} } \leq \norm{ \mtx{P} {(\mtx{MM}^\adj)}^{q} \mtx{M} }^{1/(2q+1)}.$$ Suppose that $\mtx{R}$ is an orthogonal projector, $\mtx{D}$ is a nonnegative diagonal matrix, and t ≥ 1. We claim that $$\label{eqn:power-claim} \norm{ \mtx{RDR} }^t \leq \norm{\mtx{R} \mtx{D}^t \mtx{R}}.$$ Granted this inequality, we quickly complete the proof. Using an SVD $\mtx{M} = \mtx{U\Sigma V}^\adj$, we compute $$\begin{gathered} \norm{ \mtx{PM} }^{2(2q+1)} = \norm{ \mtx{PMM}^\adj \mtx{P} }^{2q+1} = \norm{ (\mtx{U}^\adj \mtx{PU}) \cdot \mtx{\Sigma}^2 \cdot (\mtx{U}^\adj \mtx{P} \mtx{U}) }^{2q+1} \\ \leq \smnorm{}{ (\mtx{U}^\adj \mtx{PU}) \cdot \mtx{\Sigma}^{2(2q+1)} \cdot (\mtx{U}^\adj \mtx{P} \mtx{U}) } = \smnorm{}{ \mtx{P} (\mtx{MM}^\adj)^{2(2q+1)} \mtx{P} } \\ = \smnorm{}{ \mtx{P} (\mtx{MM}^\adj)^q \mtx{M} \cdot \mtx{M}^\adj (\mtx{MM}^\adj)^q \mtx{P} } = \norm{ \mtx{P} (\mtx{MM}^\adj)^{q} \mtx{M} }^2.\end{gathered}$$ We have used the unitary invariance of the spectral norm in the second and fourth relations. The inequality applies because $\mtx{U}^\adj \mtx{PU}$ is an orthogonal projector. Take a square root to finish the argument. Now, we turn to the claim . This relation follows immediately from , but we offer a direct argument based on more elementary considerations. Let $\vct{x}$ be a unit vector at which $$\vct{x}^\adj (\mtx{RDR}) \vct{x} = \norm{\mtx{RDR}}.$$ We must have $\mtx{R}\vct{x} = \vct{x}$. Otherwise, $\norm{ \mtx{R}\vct{x} } < 1$ because $\mtx{R}$ is an orthogonal projector, which implies that the unit vector $\vct{y} = \mtx{R}\vct{x} / \norm{ \mtx{R}\vct{x} }$ verifies $$\vct{y}^\adj (\mtx{RDR}) \vct{y} = \frac{(\mtx{R}\vct{x})^\adj (\mtx{RDR}) (\mtx{R}\vct{x})}{ \normsq{\mtx{R}\vct{x}} } = \frac{\vct{x}^\adj (\mtx{RDR}) \vct{x}}{ \normsq{\mtx{R}\vct{x}} } > \vct{x}^\adj (\mtx{RDR}) \vct{x}.$$ Writing xj for the entries of $\vct{x}$ and dj for the diagonal entries of $\mtx{D}$, we find that $$\begin{gathered} \norm{\mtx{RDR}}^t = [\vct{x}^\adj (\mtx{RDR}) \vct{x}]^t = [\vct{x}^\adj \mtx{D} \vct{x}]^t = \left[ \sum\nolimits\_j d\_j x\_j^2 \right]^t \\ \leq \left[ \sum\nolimits\_j d\_j^t x\_j^2 \right] = \vct{x}^\adj \mtx{D}^t \vct{x} = (\mtx{R} \vct{x})^\adj \mtx{D}^t (\mtx{R} \vct{x}) \leq \norm{ \mtx{RD}^t\mtx{R} }.\end{gathered}$$ The inequality is Jensen’s, which applies because ∑xj2 = 1 and the function $z \mapsto \abs{z}^t$ is convex for t ≥ 1. Error bounds via linear algebra =============================== We are now prepared to develop a deterministic error analysis for the proto-algorithm described in §[sec:sketchofalgorithm]. To begin, we must introduce some notation. Afterward, we establish the key error bound, which strengthens a result from the literature . Finally, we explain why the power method can be used to improve the performance of the proto-algorithm. Setup ----- Let $\mtx{A}$ be an m × n matrix that has a singular value decomposition $\mtx{A} = \mtx{U\Sigma V}^{\adj}$, as described in Section [sec:SVD]. Roughly speaking, the proto-algorithm tries to approximate the subspace spanned by the first k left singular vectors, where k is now a fixed number. To perform the analysis, it is appropriate to partition the singular value decomposition as follows. $$\label{eq:mugpart} \begin{array}{@{}c@{}r@{}c@{}c@{}c@{}c@{}c} && k & n - k && n & \\ \mtx{A} = \mtx{U} &\left. \begin{array}{c} \\ \\ \end{array} \!\!\! \right[ & \begin{array}{c} \mtx{\Sigma}\_1 \\ \phantom{\mtx{\Sigma}\_2} \end{array} & \begin{array}{c} \phantom{\mtx{\Sigma}\_1} \\ \mtx{\Sigma}\_2 \end{array} & \left] \!\!\! \begin{array}{c} \\ \\ \end{array} \right. & \left[\begin{array}{c} \mtx{V}\_{1}^{\adj} \\ \mtx{V}\_{2}^{\adj}\end{array}\right]\, & \begin{array}{c} k \\ n - k \\ \end{array} \end{array}$$ The matrices $\mtx{\Sigma}\_1$ and $\mtx{\Sigma}\_2$ are square. We will see that the left unitary factor $\mtx{U}$ does not play a significant role in the analysis. Let $\mtx{\Omega}$ be an n × ℓ test matrix, where ℓ denotes the number of samples. We assume only that ℓ ≥ k. Decompose the test matrix in the coordinate system determined by the right unitary factor of $\mtx{A}$: $$\label{eq:def\_Omegaj} \mtx{\Omega}\_1 = \mtx{V}\_1^\adj\, \mtx{\Omega} \quad\text{and}\quad \mtx{\Omega}\_2 = \mtx{V}\_2^\adj\, \mtx{\Omega}.$$ The error bound for the proto-algorithm depends critically on the properties of the matrices $\mtx{\Omega}\_{1}$ and $\mtx{\Omega}\_{2}$. With this notation, the sample matrix $\mtx{Y}$ can be expressed as $$\label{eqn:X-struct} \begin{array}{@{}c@{}c@{}c} & \ell & \\ \mtx{Y} = \mtx{A}\mtx{\Omega} = \mtx{U} \left. \begin{array}{@{}c} \\ \\ \end{array} \right[ & \begin{array}{c} \mtx{\Sigma}\_1 \mtx{\Omega}\_1 \\ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \end{array} & \left] \begin{array}{c} k \\ n - k \end{array} \right. \end{array}$$ It is a useful intuition that the block $\mtx{\Sigma}\_1 \mtx{\Omega}\_1$ in reflects the gross behavior of $\mtx{A}$, while the block $\mtx{\Sigma}\_2 \mtx{\Omega}\_2$ represents a perturbation. A deterministic error bound for the proto-algorithm --------------------------------------------------- The proto-algorithm constructs an orthonormal basis $\mtx{Q}$ for the range of the sample matrix $\mtx{Y}$, and our goal is to quantify how well this basis captures the action of the input $\mtx{A}$. Since $\mtx{QQ}^\adj = \mtx{P}\_{\mtx{Y}}$, the challenge is to obtain bounds on the approximation error $$\triplenorm{ \mtx{A} - \mtx{QQ}^\adj \mtx{A} } =\triplenorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} }.$$ The following theorem shows that the behavior of the proto-algorithm depends on the interaction between the test matrix and the right singular vectors of the input matrix, as well as the singular spectrum of the input matrix. [Deterministic error bound] [thm:main-error-bd] Let $\mtx{A}$ be an m × n matrix with singular value decomposition $\mtx{A} = \mtx{U\Sigma V}^\adj$, and fix k ≥ 0. Choose a test matrix $\mtx{\Omega}$, and construct the sample matrix $\mtx{Y} = \mtx{A\Omega}$. Partition $\mtx{\Sigma}$ as specified in, and define $\mtx{\Omega}\_1$ and $\mtx{\Omega}\_2$ via. Assuming that $\mtx{\Omega}\_1$ has full row rank, the approximation error satisfies $$\label{eq:main-error-bd} \triplenorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} }^2 \leq \triplenorm{\mtx{\Sigma}\_2}^2 + \smtriplenorm{\mtx{\Sigma}\_2\mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv}^2,$$ where $\triplenorm{\cdot}$ denotes either the spectral norm or the Frobenius norm. Theorem [thm:main-error-bd] sharpens the result , which lacks the squares present in . This refinement yields slightly better error estimates than the earlier bound, and it has consequences for the probabilistic behavior of the error when the test matrix $\mtx{\Omega}$ is random. The proof here is different in spirit from the earlier analysis; our argument is inspired by the perturbation theory of orthogonal projectors . We establish the bound for the spectral-norm error. The bound for the Frobenius-norm error follows from an analogous argument that is slightly easier. Let us begin with some preliminary simplifications. First, we argue that the left unitary factor $\mtx{U}$ plays no essential role in the argument. In effect, we execute the proof for an auxiliary input matrix $\widetilde{\mtx{A}}$ and an associated sample matrix $\widetilde{\mtx{Y}}$ defined by $$\label{eqn:aux-matrices} \widetilde{\mtx{A}} = \mtx{U}^\adj \mtx{A} = \begin{bmatrix} \mtx{\Sigma}\_1\mtx{V}\_1^\adj \\ \mtx{\Sigma}\_2 \mtx{V}\_2^\adj \end{bmatrix} \quad\text{and}\quad \widetilde{\mtx{Y}} = \widetilde{\mtx{A}} \mtx{\Omega} = \begin{bmatrix} \mtx{\Sigma}\_1 \mtx{\Omega}\_1 \\ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \end{bmatrix}.$$ Owing to the unitary invariance of the spectral norm and to Proposition [prop:conj-proj], we have the identity $$\label{eqn:A-to-A0} \norm{(\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} = \smnorm{}{\mtx{U}^\adj (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{U} \widetilde{\mtx{A}}} = \smnorm{}{(\Id - \mtx{P}\_{\mtx{U}^\adj \mtx{Y}}) \widetilde{\mtx{A}} } = \smnorm{}{(\Id - \mtx{P}\_{\widetilde{\mtx{Y}}}) \widetilde{\mtx{A}} }.$$ In view of , it suffices to prove that $$\label{eqn:aux-error-bd} \smnorm{}{(\Id - \mtx{P}\_{\widetilde{\mtx{Y}}}) \widetilde{\mtx{A}} } \leq \smnorm{}{\mtx{\Sigma}\_2}^2 + \smnorm{}{\mtx{\Sigma}\_2\mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv}^2.$$ Second, we assume that the number k is chosen so the diagonal entries of $\mtx{\Sigma}\_1$ are strictly positive. Suppose not. Then $\mtx{\Sigma}\_2$ is zero because of the ordering of the singular values. As a consequence, $$\range(\widetilde{\mtx{A}}) = \range \begin{bmatrix} \mtx{\Sigma}\_1 \mtx{V}\_1^\adj \\ \mtx{0} \end{bmatrix} = \range \begin{bmatrix} \mtx{\Sigma}\_1 \mtx{\Omega}\_1 \\ \mtx{0} \end{bmatrix} = \range( \widetilde{\mtx{Y}} ).$$ This calculation uses the decompositions presented in , as well as the fact that both $\mtx{V}\_1^\adj$ and $\mtx{\Omega}\_1$ have full row rank. We conclude that $$\smnorm{}{ (\Id - \mtx{P}\_{\widetilde{\mtx{Y}}}) \widetilde{\mtx{A}} } = 0,$$ so the error bound holds trivially. (In fact, both sides are zero.) The main argument is based on ideas from perturbation theory. To illustrate the concept, we start with a matrix related to $\widetilde{\mtx{Y}}$: $$\begin{array}{@{}c@{}c@{}c} & \ell & \\ \mtx{W} = \left. \begin{array}{@{}c} \\ \\ \end{array} \right[ & \begin{array}{c} \mtx{\Sigma}\_1 \mtx{\Omega}\_1 \\ \mtx{0} \end{array} & \left] \begin{array}{c} k \\ n - k \end{array} \right. \end{array}$$ The matrix $\mtx{W}$ has the same range as a related matrix formed by “flattening out” the spectrum of the top block. Indeed, since $\mtx{\Sigma}\_1 \mtx{\Omega}\_1$ has full row rank, $$\begin{array}{@{}c@{}c@{}c} & k & \\ \range(\mtx{W}) = \range \left. \begin{array}{@{}c} \\ \\ \end{array} \right[ & \begin{array}{c} \Id \\ \mtx{0} \end{array} & \left] \begin{array}{c} k \\ n - k \end{array} \right. \end{array}$$ The matrix on the right-hand side has full column rank, so it is legal to apply the formula for an orthogonal projector, which immediately yields $$\label{eqn:IPW} \mtx{P}\_{\mtx{W}} = \begin{bmatrix} \Id & \mtx{0} \\ \mtx{0} & \mtx{0} \end{bmatrix} \quad\text{and}\quad \Id - \mtx{P}\_{\mtx{W}} = \begin{bmatrix} \mtx{0} & \mtx{0} \\ \mtx{0} & \Id \end{bmatrix}.$$ In words, the range of $\mtx{W}$ aligns with the first k coordinates, which span the same subspace as the first k left singular vectors of the auxiliary input matrix $\widetilde{\mtx{A}}$. Therefore, $\range(\mtx{W})$ captures the action of $\widetilde{\mtx{A}}$, which is what we wanted from $\range(\widetilde{\mtx{Y}})$. We treat the auxiliary sample matrix $\widetilde{\mtx{Y}}$ as a perturbation of $\mtx{W}$, and we hope that their ranges are close to each other. To make the comparison rigorous, let us emulate the arguments outlined in the last paragraph. Referring to the display , we flatten out the top block of $\widetilde{\mtx{Y}}$ to obtain the matrix $$\label{eqn:ZF-def} \mtx{Z} = \widetilde{\mtx{Y}} \cdot \mtx{\Omega}\_1^\psinv \mtx{\Sigma}\_1^{-1} = \begin{bmatrix} \Id \\ \mtx{F} \end{bmatrix} \quad\text{where}\quad \mtx{F} = \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv \mtx{\Sigma}\_1^{-1}.$$ Let us return to the error bound . The construction ensures that $\range(\mtx{Z}) \subset \range(\widetilde{\mtx{Y}})$, so Proposition [prop:proj-range] implies that the error satisfies $$\smnorm{}{ (\Id - \mtx{P}\_{\widetilde{\mtx{Y}}}) \widetilde{\mtx{A}} } \leq \smnorm{}{ (\Id - \mtx{P}\_{\mtx{Z}}) \widetilde{\mtx{A}} }.$$ Squaring this relation, we obtain $$\label{eqn:proj-norm-sq} \smnorm{}{(\Id - \mtx{P}\_{\widetilde{\mtx{Y}}}) \widetilde{\mtx{A}}}^2 \leq \smnorm{}{(\Id - \mtx{P}\_{\mtx{Z}})\widetilde{\mtx{A}}}^2 = \smnorm{}{ \widetilde{\mtx{A}}^\adj (\Id - \mtx{P}\_{\mtx{Z}}) \widetilde{\mtx{A}} } = \norm{ \mtx{\Sigma}^\adj (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{\Sigma} }.$$ The last identity follows from the definition $\widetilde{\mtx{A}} = \mtx{\Sigma} \mtx{V}^\adj$ and the unitary invariance of the spectral norm. Therefore, we can complete the proof of by producing a suitable bound for the right-hand side of. To continue, we need a detailed representation of the projector $\Id - \mtx{P}\_{\mtx{Z}}$. The construction ensures that $\mtx{Z}$ has full column rank, so we can apply the formula for an orthogonal projector to see that $$\mtx{P}\_{\mtx{Z}} = \mtx{Z} (\mtx{Z}^\adj \mtx{Z})^{-1} \mtx{Z}^\adj = \begin{bmatrix} \Id \\ \mtx{F} \end{bmatrix} (\Id + \mtx{F}^\adj\mtx{F})^{-1} \begin{bmatrix} \Id \\ \mtx{F} \end{bmatrix}^\adj.$$ Expanding this expression, we determine that the complementary projector satisfies $$\Id - \mtx{P}\_{\mtx{Z}} = \begin{bmatrix} \Id - (\Id + \mtx{F}^\adj \mtx{F})^{-1} & -(\Id + \mtx{F}^\adj \mtx{F})^{-1} \mtx{F}^\adj \\ - \mtx{F} (\Id + \mtx{F}^\adj \mtx{F})^{-1} & \Id - \mtx{F}(\Id + \mtx{F}^\adj\mtx{F})^{-1} \mtx{F}^\adj \end{bmatrix}. \label{eqn:block-proj}$$ The partitioning here conforms with the partitioning of $\mtx{\Sigma}$. When we conjugate the matrix by $\mtx{\Sigma}$, copies of $\mtx{\Sigma}\_1^{-1}$, presently hidden in the top-left block, will cancel to happy effect. The latter point may not seem obvious, owing to the complicated form of. In reality, the block matrix is less fearsome than it looks. Proposition [prop:perturb-inv], on the perturbation of inverses, shows that the top-left block verifies $$\Id - (\Id + \mtx{F}^\adj \mtx{F})^{-1} \psdle \mtx{F}^\adj \mtx{F}.$$ The bottom-right block satisfies $$\Id - \mtx{F}(\Id + \mtx{F}^\adj \mtx{F})^{-1} \mtx{F}^\adj \psdle \Id$$ because the conjugation rule guarantees that $\mtx{F}(\Id + \mtx{F}^\adj \mtx{F})^{-1} \mtx{F}^\adj \psdge \mtx{0}$. We abbreviate the off-diagonal blocks with the symbol $\mtx{B} = -(\Id + \mtx{F}^\adj \mtx{F})^{-1}\mtx{F}^\adj$. In summary, $$\Id - \mtx{P}\_{\mtx{Z}} \psdle \begin{bmatrix} \mtx{F}^\adj \mtx{F} & \mtx{B} \\ \mtx{B}^\adj & \Id \end{bmatrix}.$$ This relation exposes the key structural properties of the projector. Compare this relation with the expression for the “ideal” projector $\Id - \mtx{P}\_{\mtx{W}}$. Moving toward the estimate required by , we conjugate the last relation by $\mtx{\Sigma}$ to obtain $$\mtx{\Sigma}^\adj (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{\Sigma} \psdle \begin{bmatrix} \mtx{\Sigma}\_1^\adj \mtx{F}^\adj \mtx{F} \mtx{\Sigma}\_1 & \mtx{\Sigma}\_1^\adj \mtx{B} \mtx{\Sigma}\_2 \\ \mtx{\Sigma}\_2^\adj \mtx{B}^\adj \mtx{\Sigma}\_1 & \mtx{\Sigma}\_2^\adj \mtx{\Sigma}\_2 \end{bmatrix}.$$ The conjugation rule demonstrates that the matrix on the left-hand side is psd, so the matrix on the right-hand side is too. Proposition [prop:spec-norm-sum] results in the norm bound $$\norm{ \mtx{\Sigma}^\adj (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{\Sigma} } \leq \norm{ \mtx{\Sigma}\_1^\adj \mtx{F}^\adj \mtx{F} \mtx{\Sigma}\_1 } + \norm{ \mtx{\Sigma}\_2^\adj \mtx{\Sigma}\_2 } = \normsq{\mtx{F} \mtx{\Sigma}\_1} + \normsq{ \mtx{\Sigma}\_2 }.$$ Recall that $\mtx{F} = \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv \mtx{\Sigma}\_1^{-1}$, so the factor $\mtx{\Sigma}\_1$ cancels neatly. Therefore, $$\norm{ \mtx{\Sigma}^\adj(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{\Sigma}} \leq \smnorm{}{\mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }^2 + \normsq{\mtx{\Sigma}\_2}.$$ Finally, introduce the latter inequality into to complete the proof. Analysis of the power scheme ---------------------------- Theorem [thm:main-error-bd] suggests that the performance of the proto-algorithm depends strongly on the relationship between the large singular values of $\mtx{A}$ listed in $\mtx{\Sigma}\_1$ and the small singular values listed in $\mtx{\Sigma}\_2$. When a substantial proportion of the mass of $\mtx{A}$ appears in the small singular values, the constructed basis $\mtx{Q}$ may have low accuracy. Conversely, when the large singular values dominate, it is much easier to identify a good low-rank basis. To improve the performance of the proto-algorithm, we can run it with a closely related input matrix whose singular values decay more rapidly . Fix a positive integer q, and set $$\mtx{B} = (\mtx{AA}^\adj)^{q} \mtx{A} = \mtx{U}\mtx{\Sigma}^{2q+1} \mtx{V}^\adj.$$ We apply the proto-algorithm to $\mtx{B}$, which generates a sample matrix $\mtx{Z} = \mtx{B\Omega}$ and constructs a basis $\mtx{Q}$ for the range of $\mtx{Z}$. Section [sec:powerscheme] elaborates on the implementation details, and describes a reformulation that sometimes improves the accuracy when the scheme is executed in finite-precision arithmetic. The following result describes how well we can approximate the original matrix $\mtx{A}$ within the range of $\mtx{Z}$. [Power scheme] [thm:power-method] Let $\mtx{A}$ be an m × n matrix, and let $\mtx{\Omega}$ be an n × ℓ matrix. Fix a nonnegative integer q, form $\mtx{B} = {(\mtx{A}^\adj \mtx{A})}^q \mtx{A}$, and compute the sample matrix $\mtx{Z} = \mtx{B\Omega}$. Then $$\norm{ (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A} } \leq \norm{ (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{B} }^{1/(2q+1)}.$$ We determine that $$\norm{ (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A} } \leq \norm{ (\Id - \mtx{P}\_{\mtx{Z}}) {(\mtx{AA}^\adj)}^q \mtx{A} }^{1/(2q+1)} = \norm{ (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{B} }^{1/(2q+1)}$$ as a direct consequence of Proposition [prop:proj-power]. Let us illustrate how the power scheme interacts with the main error bound . Let σk + 1 denote the (k + 1)th singular value of $\mtx{A}$. First, suppose we approximate $\mtx{A}$ in the range of the sample matrix $\mtx{Y} = \mtx{A\Omega}$. Since $\norm{\mtx{\Sigma}\_2} = \sigma\_{k+1}$, Theorem [thm:main-error-bd] implies that $$\label{eq:simple} \norm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \leq \left( 1 + \smnorm{}{\mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv}^2 \right)^{1/2} \sigma\_{k+1}.$$ Now, define $\mtx{B} = (\mtx{AA}^\adj)^q \mtx{A}$, and suppose we approximate $\mtx{A}$ within the range of the sample matrix $\mtx{Z} = \mtx{B\Omega}$. Together, Theorem [thm:power-method] and Theorem [thm:main-error-bd] imply that $$\norm{ (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A} } \leq \norm{ (\Id - \mtx{P}\_{\mtx{Z}}) \mtx{B} }^{1/(2q+1)} \leq \left( 1 + \smnorm{}{\mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv}^2 \right)^{1/(4q+2)} \sigma\_{k+1}$$ because σk + 12q + 1 is the (k + 1)th singular value of $\mtx{B}$. In effect, the power scheme drives down the suboptimality of the bound exponentially fast as the power q increases. In principle, we can make the extra factor as close to one as we like, although this increases the cost of the algorithm. Analysis of truncated SVD ------------------------- Finally, let us study the truncated SVD described in Remark [rem:truncation]. Suppose that we approximate the input matrix $\mtx{A}$ inside the range of the sample matrix $\mtx{Z}$. In essence, the truncation step computes a best rank-k approximation $\widehat{\mtx{A}}\_{(k)}$ of the compressed matrix $\mtx{P}\_{\mtx{Z}} \mtx{A}$. The next result provides a simple error bound for this method; this argument was proposed by Ming Gu. [Analysis of Truncated SVD] [thm:truncation] Let $\mtx{A}$ be an m × n matrix with singular values σ1 ≥ σ2 ≥ σ3 ≥ …, and let $\mtx{Z}$ be an m × ℓ matrix, where ℓ ≥ k. Suppose that $\widehat{\mtx{A}}\_{(k)}$ is a best rank-k approximation of $\mtx{P}\_{\mtx{Z}} \mtx{A}$ with respect to the spectral norm. Then $$\smnorm{}{ \mtx{A} - \widehat{\mtx{A}}\_{(k)} } \leq \sigma\_{k+1} + \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A}}.$$ Apply the triangle inequality to split the error into two components. $$\label{eqn:rand-pca-1} \smnorm{}{ \mtx{A} - \widehat{\mtx{A}}\_k } \leq \smnorm{}{ \mtx{A} - \mtx{P}\_{\mtx{Z}} \mtx{A} } + \smnorm{}{ \mtx{P}\_{\mtx{Z}} \mtx{A} - \widehat{\mtx{A}}\_{(k)} }.$$ We have already developed a detailed theory for estimating the first term. To analyze the second term, we introduce a best rank-k approximation $\mtx{A}\_{(k)}$ of the matrix $\mtx{A}$. Note that $$\smnorm{}{ \mtx{P}\_{\mtx{Z}} \mtx{A} - \widehat{\mtx{A}}\_{(k)} } \leq \smnorm{}{ \mtx{P}\_{\mtx{Z}} \mtx{A} - \mtx{P}\_{\mtx{Z}} \mtx{A}\_{(k)} }$$ because $\widehat{\mtx{A}}\_{(k)}$ is a best rank-k approximation to the matrix $\mtx{P}\_{\mtx{Z}} \mtx{A}$, whereas $\mtx{P}\_{\mtx{Z}} \mtx{A}\_{(k)}$ is an undistinguished rank-k matrix. It follows that $$\label{eqn:rand-pca-2} \smnorm{}{ \mtx{P}\_{\mtx{Z}} \mtx{A} - \widehat{\mtx{A}}\_{(k)} } \leq \smnorm{}{ \mtx{P}\_{\mtx{Z}} (\mtx{A} - \mtx{A}\_{(k)}) } \leq \smnorm{}{ \mtx{A} - \mtx{A}\_{(k)} } = \sigma\_{k+1}.$$ The second inequality holds because the orthogonal projector is a contraction; the last identity follows from Mirsky’s theorem . Combine and to reach the main result. In the randomized setting, the truncation step appears to be less damaging than the error bound of Theorem [thm:truncation] suggests, but we currently lack a complete theoretical understanding of its behavior. Gaussian test matrices ====================== The error bound in Theorem [thm:main-error-bd] shows that the performance of the proto-algorithm depends on the interaction between the test matrix $\mtx{\Omega}$ and the right singular vectors of the input matrix $\mtx{A}$. Algorithm [alg:basic] is a particularly simple version of the proto-algorithm that draws the test matrix according to the standard Gaussian distribution. The literature contains a wealth of information about these matrices, which allows us to perform a very precise error analysis. We focus on the real case in this section. Analogous results hold in the complex case, where the algorithm even exhibits superior performance. Technical background -------------------- A is a random matrix whose entries are independent standard normal variables. The distribution of a standard Gaussian matrix is rotationally invariant: If $\mtx{U}$ and $\mtx{V}$ are orthonormal matrices, then $\mtx{U}^\adj \mtx{G} \mtx{V}$ also has the standard Gaussian distribution. Our analysis requires detailed information about the properties of Gaussian matrices. In particular, we must understand how the norm of a Gaussian matrix and its pseudoinverse vary. We summarize the relevant results and citations here, reserving the details for Appendix [app:gauss]. [Expected norm of a scaled Gaussian matrix] [prop:scaled-gauss] Fix matrices $\mtx{S}, \mtx{T}$, and draw a standard Gaussian matrix $\mtx{G}$. Then $$\begin{gathered} \left( \Expect \fnormsq{ \mtx{SGT} } \right)^{1/2} = \fnorm{\mtx{S}} \fnorm{\mtx{T}} \quad\text{and} \label{eqn:avg-fnormsq} \\ \Expect \norm{ \mtx{SGT} } \leq \norm{\mtx{S}} \fnorm{\mtx{T}} + \fnorm{\mtx{S}} \norm{\mtx{T}}. \label{eqn:avg-specnorm}\end{gathered}$$ The identity follows from a direct calculation. The second bound relies on methods developed by Gordon . See Propositions [prop:gauss-frob-expect] and [prop:gauss-spec-expect]. [Expected norm of a pseudo-inverted Gaussian matrix] [prop:gauss-inv-expect] Draw a k × (k + p) standard Gaussian matrix $\mtx{G}$ with k ≥ 2 and p ≥ 2. Then $$\begin{gathered} \left( \Expect \fnormsq{ \mtx{G}^\psinv } \right)^{1/2} = \sqrt{\frac{k}{p-1}} \quad\text{and} \label{eqn:avg-inv-fnormsq} \\ \Expect \norm{ \mtx{G}^\psinv } \leq \frac{\econst\sqrt{k+p}}{p} \label{eqn:avg-inv-specnorm}.\end{gathered}$$ The first identity is a standard result from multivariate statistics . The second follows from work of Chen and Dongarra . See Proposition [prop:inv-gauss-spec-expect] and [prop:inv-gauss-frob-expect]. To study the probability that Algorithm [alg:basic] produces a large error, we rely on tail bounds for functions of Gaussian matrices. The next proposition rephrases a well-known result on concentration of measure . See also and. [Concentration for functions of a Gaussian matrix] [prop:gauss-tail] Suppose that h is a Lipschitz function on matrices: L Draw a standard Gaussian matrix $\mtx{G}$. Then $$\Prob{ h(\mtx{G}) \geq \Expect h(\mtx{G}) + Lt } \leq \econst^{-t^2/2}.$$ Finally, we state some large deviation bounds for the norm of a pseudo-inverted Gaussian matrix. [Norm bounds for a pseudo-inverted Gaussian matrix] [prop:gauss-inv-tails] Let $\mtx{G}$ be a k × (k + p) Gaussian matrix where p ≥ 4. For all t ≥ 1, $$\begin{gathered} \Prob{ \smnorm{\rm F}{\mtx{G}^\psinv} \geq \sqrt{\frac{12 k}{p}} \cdot t } \leq 4 t^{-p} \quad\text{and} \label{eqn:tail-inv-fnormsq} \\ \Prob{ \smnorm{}{\mtx{G}^\psinv} \geq \frac{\econst\sqrt{k+p}}{p+1} \cdot t } \leq t^{-(p+1)}. \label{eqn:tail-inv-specnorm}\end{gathered}$$ Compare these estimates with Proposition [prop:gauss-inv-expect]. It seems that is new; we were unable to find a comparable analysis in the random matrix literature. Although the form of is not optimal, it allows us to produce more transparent results than a fully detailed estimate. The bound essentially appears in the work of Chen and Dongarra . See Propositions [prop:inv-gauss-spec-tail] and Theorem [thm:inv-gauss-frob-tail] for more information. Average-case analysis of Algorithm [alg:basic] ---------------------------------------------- We separate our analysis into two pieces. First, we present information about expected values. In the next subsection, we describe bounds on the probability of a large deviation. We begin with the simplest result, which provides an estimate for the expected approximation error in the Frobenius norm. All proofs are postponed to the end of the section. [Average Frobenius error] [thm:avg-frob-error-gauss] Suppose that $\mtx{A}$ is a real m × n matrix with singular values σ1 ≥ σ2 ≥ σ3 ≥ …. Choose a target rank k ≥ 2 and an oversampling parameter p ≥ 2, where k + p ≤ min{m, n}. Draw an n × (k + p) standard Gaussian matrix $\mtx{\Omega}$, and construct the sample matrix $\mtx{Y} = \mtx{A\Omega}$. Then the expected approximation error $$\Expect \fnorm{(\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} \leq \left( 1 + \frac{k}{p-1} \right)^{1/2} \left( \sum\nolimits\_{j>k} \sigma\_j^2 \right)^{1/2}.$$ This theorem predicts several intriguing behaviors of Algorithm [alg:basic]. The Eckart–Young theorem shows that (∑j > kσj2)1/2 is the minimal Frobenius-norm error when approximating $\mtx{A}$ with a rank-k matrix. This quantity is the appropriate benchmark for the performance of the algorithm. If the small singular values of $\mtx{A}$ are very flat, the series may be as large as $\sigma\_{k+1} \sqrt{\min\{m,n\} - k}$. On the other hand, when the singular values exhibit some decay, the error may be on the same order as σk + 1. The error bound always exceeds this baseline error, but it may be polynomially larger, depending on the ratio between the target rank k and the oversampling parameter p. For p small (say, less than five), the error is somewhat variable because the small singular values of a nearly square Gaussian matrix are very unstable. As the oversampling increases, the performance improves quickly. When p ∼ k, the error is already within a constant factor of the baseline. The error bound for the spectral norm is somewhat more complicated, but it reveals some interesting new features. [Average spectral error] [thm:avg-spec-error-gauss] Under the hypotheses of Theorem [thm:avg-frob-error-gauss], $$\Expect \norm{(\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} \leq \left(1 + \sqrt{\frac{k}{p-1}} \right) \sigma\_{k+1} + \frac{\econst\sqrt{k+p}}{p} \left(\sum\nolimits\_{j>k} \sigma\_{j}^2 \right)^{1/2}.$$ Mirsky has shown that the quantity σk + 1 is the minimum spectral-norm error when approximating $\mtx{A}$ with a rank-k matrix, so the first term in Theorem [thm:avg-spec-error-gauss] is analogous with the error bound in Theorem [thm:avg-frob-error-gauss]. The second term represents a new phenomenon: we also pay for the Frobenius-norm error in approximating $\mtx{A}$. Note that, as the amount p of oversampling increases, the polynomial factor in the second term declines much more quickly than the factor in the first term. When p ∼ k, the factor on the σk + 1 term is constant, while the factor on the series has order k− 1/2 We also note that the bound in Theorem [thm:avg-spec-error-gauss] implies $$\Expect \norm{(\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} \leq \left[ 1 + \sqrt{\frac{k}{p-1}} + \frac{\econst\sqrt{k+p}}{p} \cdot \sqrt{\min\{m,n\} - k} \right] \sigma\_{k+1},$$ so the average spectral-norm error always lies within a small polynomial factor of the baseline σk + 1. Let us continue with the proofs of these results. [Theorem [thm:avg-frob-error-gauss]] Let $\mtx{V}$ be the right unitary factor of $\mtx{A}$. Partition $\mtx{V} = [ \mtx{V}\_1 \ \| \ \mtx{V}\_2 ]$ into blocks containing, respectively, k and n − k columns. Recall that $$\mtx{\Omega}\_1 = \mtx{V}\_1^\adj \mtx{\Omega} \quad\text{and}\quad \mtx{\Omega}\_2 = \mtx{V}\_2^\adj \mtx{\Omega}.$$ The Gaussian distribution is rotationally invariant, so $\mtx{V}^\adj \mtx{\Omega}$ is also a standard Gaussian matrix. Observe that $\mtx{\Omega}\_1$ and $\mtx{\Omega}\_2$ are nonoverlapping submatrices of $\mtx{V}^\adj \mtx{\Omega}$, so these two matrices are not only standard Gaussian but also stochastically independent. Furthermore, the rows of a (fat) Gaussian matrix are almost surely in general position, so the k × (k + p) matrix $\mtx{\Omega}\_1$ has full row rank with probability one. Hölder’s inequality and Theorem [thm:main-error-bd] together imply that $$\Expect \fnorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} \leq \left( \Expect \fnormsq{(\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} \right)^{1/2} \leq \left( \smnorm{\rm F}{\mtx{\Sigma}\_2}^2 + \Expect \smnorm{\rm F}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }^2 \right)^{1/2}.$$ We compute this expectation by conditioning on the value of $\mtx{\Omega}\_1$ and applying Proposition [prop:scaled-gauss] to the scaled Gaussian matrix $\mtx{\Omega}\_2$. Thus, $$\begin{gathered} \Expect \smnorm{\rm F}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }^2 = \Expect \left( \Expect \left[ \smnorm{\rm F}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }^2 \ \big\vert \ \mtx{\Omega}\_1 \right] \right) = \Expect \left( \fnormsq{\mtx{\Sigma}\_2} \smnorm{\rm F}{\mtx{\Omega}\_1^\psinv}^2 \right) \\ = \fnormsq{\mtx{\Sigma}\_2} \cdot \Expect \smnorm{\rm F}{\mtx{\Omega}\_1^\psinv}^2 = \frac{k}{p-1} \cdot \fnormsq{\mtx{\Sigma}\_2},\end{gathered}$$ where the last expectation follows from relation of Proposition [prop:gauss-inv-expect]. In summary, $$\Expect \fnorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A}} \leq \left(1 + \frac{k}{p-1}\right)^{1/2} \fnorm{ \mtx{\Sigma}\_2 }.$$ Observe that $\fnormsq{\mtx{\Sigma}\_2} = \sum\_{j>k} \sigma\_j^2$ to complete the proof. [Theorem [thm:avg-spec-error-gauss]] The argument is similar to the proof of Theorem [thm:avg-frob-error-gauss]. First, Theorem [thm:main-error-bd] implies that $$\Expect \norm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \leq \Expect \left( \normsq{\mtx{\Sigma}\_2} + \smnorm{}{\mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }^2 \right)^{1/2} \leq \norm{\mtx{\Sigma}\_2} + \Expect \smnorm{}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }.$$ We condition on $\mtx{\Omega}\_1$ and apply Proposition [prop:scaled-gauss] to bound the expectation with respect to $\mtx{\Omega}\_2$. Thus, $$\begin{aligned} \Expect \smnorm{}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv } &\leq \Expect \left( \norm{\mtx{\Sigma}\_2} \smnorm{\rm F}{\mtx{\Omega}\_1^\psinv} + \fnorm{\mtx{\Sigma}\_2} \smnorm{}{\mtx{\Omega}\_1^\psinv} \right) \\ &\leq \norm{\mtx{\Sigma}\_2} \left( \Expect \smnorm{\rm F}{\mtx{\Omega}\_1^\psinv}^2 \right)^{1/2} + \fnorm{\mtx{\Sigma}\_2} \cdot \Expect \smnorm{}{\mtx{\Omega}\_1^\psinv}.\end{aligned}$$ where the second relation requires Hölder’s inequality. Applying both parts of Proposition [prop:gauss-inv-expect], we obtain $$\Expect \smnorm{}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv } \leq \sqrt{\frac{k}{p-1}} \norm{\mtx{\Sigma}\_2} + \frac{\econst\sqrt{k+p}}{p} \fnorm{\mtx{\Sigma}\_2}.$$ Note that $\norm{\mtx{\Sigma}\_2} = \sigma\_{k+1}$ to wrap up. Probabilistic error bounds for Algorithm [alg:basic] ---------------------------------------------------- We can develop tail bounds for the approximation error, which demonstrate that the average performance of the algorithm is representative of the actual performance. We begin with the Frobenius norm because the result is somewhat simpler. [Deviation bounds for the Frobenius error] [thm:tail-frob-error-gauss] Frame the hypotheses of Theorem [thm:avg-frob-error-gauss]. Assume further that p ≥ 4. For all u, t ≥ 1, $$\fnorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \leq \left( 1 + t \cdot \sqrt{12k/p} \right) \left( \sum\nolimits\_{j > k} \sigma\_j^2 \right)^{1/2} + ut \cdot \frac{\econst\sqrt{k+p}}{p+1} \cdot \sigma\_{k+1},$$ with failure probability at most $5 t^{-p} + 2 \econst^{-u^2/2}$. To parse this theorem, observe that the first term in the error bound corresponds with the expected approximation error in Theorem [thm:avg-frob-error-gauss]. The second term represents a deviation above the mean. An analogous result holds for the spectral norm. [Deviation bounds for the spectral error] [thm:tail-spec-error-gauss] Frame the hypotheses of Theorem [thm:avg-frob-error-gauss]. Assume further that p ≥ 4. For all u, t ≥ 1, $$\begin{gathered} \norm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \\ \leq \left[ \left( 1 + t \cdot \sqrt{12k/p} \right) \sigma\_{k+1} + t \cdot \frac{\econst\sqrt{k+p}}{p+1} \left( \sum\nolimits\_{j>k} \sigma\_j^2 \right)^{1/2} \right] + ut \cdot \frac{\econst\sqrt{k+p}}{p+1} \sigma\_{k+1},\end{gathered}$$ with failure probability at most $5 t^{-p} + \econst^{-u^2/2}$. The bracket corresponds with the expected spectral-norm error while the remaining term represents a deviation above the mean. Neither the numerical constants nor the precise form of the bound are optimal because of the slackness in Proposition [prop:gauss-inv-tails]. Nevertheless, the theorem gives a fairly good picture of what is actually happening. We acknowledge that the current form of Theorem [thm:tail-spec-error-gauss] is complicated. To produce more transparent results, we make appropriate selections for the parameters u, t and bound the numerical constants. [Simplified deviation bounds for the spectral error] [cor:tail-spec-error-gauss] Frame the hypotheses of Theorem [thm:avg-frob-error-gauss], and assume further that p ≥ 4. Then $$\norm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \leq \left( 1 + 17 \sqrt{1 + k/p} \right) \sigma\_{k+1} + \frac{8\sqrt{k+p}}{p+1} \left( \sum\nolimits\_{j > k} \sigma\_j^2 \right)^{1/2},$$ with failure probability at most $6\econst^{-p}$. Moreover, $$\norm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \leq \left( 1 + 8 \sqrt{(k + p) \cdot p \log p} \right) \sigma\_{k+1} + 3 \sqrt{k+p} \left( \sum\nolimits\_{j > k} \sigma\_j^2 \right)^{1/2},$$ with failure probability at most 6p− p. The first part of the result follows from the choices $t = \econst$ and $u = \sqrt{2p}$, and the second emerges when t = p and $u = \sqrt{2p\log p}$. Another interesting parameter selection is t = pc/p and $u = \sqrt{2c\log p}$, which yields a failure probability 6p− c. Corollary [cor:tail-spec-error-gauss] should be compared with . Although our result contains sharper error estimates, the failure probabilities are usually worse. The error bound presented in §[sec:prototheorem] follows after further simplification of the second bound from Corollary [cor:tail-spec-error-gauss]. We continue with a proof of Theorem [thm:tail-spec-error-gauss]. The same argument can be used to obtain a bound for the Frobenius-norm error, but we omit a detailed account. [Theorem [thm:tail-spec-error-gauss]] Since $\mtx{\Omega}\_1$ and $\mtx{\Omega}\_2$ are independent from each other, we can study how the error depends on the matrix $\mtx{\Omega}\_2$ by conditioning on the event that $\mtx{\Omega}\_1$ is not too irregular. To that end, we define a (parameterized) event on which the spectral and Frobenius norms of the matrix $\mtx{\Omega}\_1^\psinv$ are both controlled. For t ≥ 1, let $$E\_t = \left\{ \mtx{\Omega}\_1 : \smnorm{}{ \mtx{\Omega}\_1^\psinv } \leq \frac{\econst\sqrt{k+p}}{p+1} \cdot t \quad\text{and}\quad \smnorm{\rm F}{ \mtx{\Omega}\_1^\psinv } \leq \sqrt{\frac{12k}{p}} \cdot t \right\}.$$ Invoking both parts of Proposition [prop:gauss-inv-tails], we find that $$\Probe{E\_t^c} \leq t^{-(p+1)} + 4 t^{-p} \leq 5 t^{-p}.$$ Consider the function $h( \mtx{X} ) = \smnorm{}{ \mtx{\Sigma}\_2 \mtx{X} \mtx{\Omega}\_1^\psinv }$. We quickly compute its Lipschitz constant L with the lower triangle inequality and some standard norm estimates: $$\begin{gathered} \abs{ h(\mtx{X}) - h(\mtx{Y}) } \leq \smnorm{}{ \mtx{\Sigma}\_2 (\mtx{X} - \mtx{Y}) \mtx{\Omega}\_1^\psinv } \\ \leq \norm{ \mtx{\Sigma}\_2 } \norm{ \mtx{X} - \mtx{Y} } \smnorm{}{ \mtx{\Omega}\_1^\psinv } \leq \norm{ \mtx{\Sigma}\_2 } \smnorm{}{ \mtx{\Omega}\_1^\psinv } \fnorm{ \mtx{X} - \mtx{Y} }.\end{gathered}$$ Therefore, $L \leq \norm{\mtx{\Sigma}\_2} \smnorm{}{\mtx{\Omega}\_1^\psinv}$. Relation of Proposition [prop:scaled-gauss] implies that $$\Expect[ h(\mtx{\Omega}\_2) \; \| \; \mtx{\Omega}\_1 ] \leq \norm{\mtx{\Sigma}\_2} \smnorm{\rm F}{ \mtx{\Omega}\_1^\psinv } + \fnorm{\mtx{\Sigma}\_2} \smnorm{}{ \mtx{\Omega}\_1^\psinv }.$$ Applying the concentration of measure inequality, Proposition [prop:gauss-tail], conditionally to the random variable $h(\mtx{\Omega}\_2) = \smnorm{}{ \mtx{\Sigma}\_2 \mtx{\Omega}\_2 \mtx{\Omega}\_1^\psinv }$ results in $$\Prob{ \smnorm{}{\mtx{\Sigma}\_2\mtx{\Omega}\_2\mtx{\Omega}\_1^\psinv} > \norm{\mtx{\Sigma}\_2} \smnorm{\rm F}{ \mtx{\Omega}\_1^\psinv } + \fnorm{\mtx{\Sigma}\_2} \smnorm{}{ \mtx{\Omega}\_1^\psinv } + \norm{\mtx{\Sigma}\_2} \smnorm{}{\mtx{\Omega}\_1^\psinv} \cdot u \ \big\vert \ E\_t } \leq \econst^{-u^2/2}.$$ Under the event Et, we have explicit bounds on the norms of $\mtx{\Omega}\_1^\psinv$, so $$\begin{gathered} \Prob{ \smnorm{}{\mtx{\Sigma}\_2\mtx{\Omega}\_2\mtx{\Omega}\_1^\psinv} > \norm{\mtx{\Sigma}\_2} \sqrt{\frac{12k}{p}} \cdot t + \fnorm{\mtx{\Sigma}\_2} \frac{\econst\sqrt{k+p}}{p+1} \cdot t + \norm{\mtx{\Sigma}\_2} \frac{\econst\sqrt{k+p}}{p+1} \cdot ut \ \bigg\vert \ E\_t } \\ \leq \econst^{-u^2/2}.\end{gathered}$$ Use the fact $\Probe{E\_t^c} \leq 5 t^{-p}$ to remove the conditioning. Therefore, $$\begin{gathered} \Prob{ \smnorm{}{\mtx{\Sigma}\_2\mtx{\Omega}\_2\mtx{\Omega}\_1^\psinv} > \norm{\mtx{\Sigma}\_2} \sqrt{\frac{12k}{p}} \cdot t + \fnorm{\mtx{\Sigma}\_2} \frac{\econst\sqrt{k+p}}{p+1} \cdot t + \norm{\mtx{\Sigma}\_2} \frac{\econst\sqrt{k+p}}{p+1} \cdot ut } \\ \leq 5 t^{-p} + \econst^{-u^2/2}.\end{gathered}$$ Insert the expressions for the norms of $\mtx{\Sigma}\_2$ into this result to complete the probability bound. Finally, introduce this estimate into the error bound from Theorem [thm:main-error-bd]. Analysis of the power scheme ---------------------------- Theorem [thm:avg-spec-error-gauss] makes it clear that the performance of the randomized approximation scheme, Algorithm [alg:basic], depends heavily on the singular spectrum of the input matrix. The power scheme outlined in Algorithm [alg:poweriteration] addresses this problem by enhancing the decay of spectrum. We can combine our analysis of Algorithm [alg:basic] with Theorem [thm:power-method] to obtain a detailed report on the behavior of the performance of the power scheme using a Gaussian matrix. [Average spectral error for the power scheme] [cor:power-method-spec-gauss] Frame the hypotheses of Theorem [thm:avg-frob-error-gauss]. Define $\mtx{B} = {(\mtx{A}\mtx{A}^\adj)}^{q} \mtx{A}$ for a nonnegative integer q, and construct the sample matrix $\mtx{Z} = \mtx{B\Omega}$. Then $$\Expect \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A}} \leq \left[ \left( 1 + \sqrt{\frac{k}{p-1}} \right) \sigma\_{k+1}^{2q+1} + \frac{\econst\sqrt{k+p}}{p} \left( \sum\nolimits\_{j>k} \sigma\_j^{2(2q+1)} \right)^{1/2} \right]^{1/(2q+1)}.$$ By Hölder’s inequality and Theorem [thm:power-method], $$\Expect \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A}} \leq \left( \Expect \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A}}^{2q+1} \right)^{1/(2q+1)} \leq \left( \Expect \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{B}} \right)^{1/(2q+1)}.$$ Invoke Theorem [thm:avg-spec-error-gauss] to bound the right-hand side, noting that $\sigma\_{j}(\mtx{B}) = \sigma\_j^{2q+1}$. The true message of Corollary [cor:power-method-spec-gauss] emerges if we bound the series using its largest term σk + 14q + 2 and draw the factor σk + 1 out of the bracket: $$\label{eqn:power-method-weak-bd} \Expect \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A}} \leq \left[ 1 + \sqrt{\frac{k}{p-1}} + \frac{\econst\sqrt{k+p}}{p} \cdot \sqrt{ \min\{m,n\} - k } \right]^{1/(2q+1)} \sigma\_{k+1}.$$ In words, as we increase the exponent q, the power scheme drives the extra factor in the error to one exponentially fast. By the time q ∼ log(min{m, n}), $$\Expect \norm{(\Id - \mtx{P}\_{\mtx{Z}}) \mtx{A}} \sim \sigma\_{k+1},$$ which is the baseline for the spectral norm. In most situations, the error bound given by Corollary [cor:power-method-spec-gauss] is substantially better than the estimates discussed in the last paragraph. For example, suppose that the tail singular values exhibit the decay profile j j(1+)/(4q+2). Then the series in Corollary [cor:power-method-spec-gauss] is comparable with its largest term, which allows us to remove the dimensional factor min{m, n} from the error bound. To obtain large deviation bounds for the performance of the power scheme, simply combine Theorem [thm:power-method] with Theorem [thm:tail-spec-error-gauss]. We omit a detailed statement. We lack an analogous theory for the Frobenius norm because Theorem [thm:power-method] depends on Proposition [prop:proj-power], which is not true for the Frobenius norm. It is possible to obtain some results by estimating the Frobenius norm in terms of the spectral norm. SRFT test matrices ================== Another way to implement the proto-algorithm from §[sec:sketchofalgorithm] is to use a structured random matrix so that the matrix product in Step 2 can be performed quickly. One type of structured random matrix that has been proposed in the literature is the, or SRFT, which we discussed in §[sec:ailonchazelle]. In this section, we present bounds on the performance of the proto-algorithm when it is implemented with an SRFT test matrix. In contrast with the results for Gaussian test matrices, the results in this section hold for both real and complex input matrices. Construction and Properties --------------------------- Recall from §[sec:ailonchazelle] that an SRFT is a tall n × ℓ matrix of the form $\mtx{\Omega} = \sqrt{n/\ell} \cdot \mtx{DFR}^\adj$ where * $\mtx{D}$ is a random n × n diagonal matrix whose entries are independent and uniformly distributed on the complex unit circle; * $\mtx{F}$ is the n × n unitary discrete Fourier transform; and * $\mtx{R}$ is a random ℓ × n matrix that restricts an n-dimensional vector to ℓ coordinates, chosen uniformly at random. Up to scaling, an SRFT is just a section of a unitary matrix, so it satisfies the norm identity $\norm{\mtx{\Omega}} = \sqrt{n/\ell}$. The critical fact is that an appropriately designed SRFT approximately preserves the geometry of an entire subspace of vectors. [The SRFT preserves geometry] [thm:SRFT-spec-bd] Fix an n × k orthonormal matrix $\mtx{V}$, and draw an n × ℓ SRFT matrix $\mtx{\Omega}$ where the parameter ℓ satisfies $$4 \left[ \sqrt{k} + \sqrt{8\log(kn)} \right]^{2} \log(k) \leq \ell \leq n.$$ Then $$0.40 \leq \sigma\_{k}(\mtx{V}^\adj \mtx{\Omega}) \quad\text{and}\quad \sigma\_{1}(\mtx{V}^\adj \mtx{\Omega}) \leq 1.48$$ with failure probability at most $\bigO(k^{-1})$. In words, the kernel of an SRFT of dimension ℓ ∼ klog(k) is unlikely to intersect a fixed k-dimensional subspace. In contrast with the Gaussian case, the logarithmic factor log(k) in the lower bound on ℓ cannot generally be removed (Remark [rem:coupon]). Theorem [thm:SRFT-spec-bd] follows from a straightforward variation of the argument in , which establishes equivalent bounds for a real analog of the SRFT, called the (SRHT). We omit further details. For large problems, we can obtain better numerical constants . Fix a small, positive number ι. If k ≫ log(n), then sampling ℓ ≥ (1 + ι) ⋅ klog(k) coordinates is sufficient to ensure that $\sigma\_k(\mtx{V}^\adj \mtx{\Omega}) \geq \iota$ with failure probability at most $\bigO(k^{-\cnst{c}\iota})$. This sampling bound is essentially optimal because (1 − ι) ⋅ klog(k) samples are not adequate in the worst case; see Remark [rem:coupon]. [rem:coupon] The logarithmic factor in Theorem [thm:SRFT-spec-bd] is necessary when the orthonormal matrix $\mtx{V}$ is particularly evil. Let us describe an infinite family of worst-case examples. Fix an integer k, and let n = k2. Form an n × k orthonormal matrix $\mtx{V}$ by regular decimation of the n × n identity matrix. More precisely, $\mtx{V}$ is the matrix whose jth row has a unit entry in column (j − 1)/k when $j \equiv 1 \pmod{k}$ and is zero otherwise. To see why this type of matrix is nasty, it is helpful to consider the auxiliary matrix $\mtx{W} = \mtx{V}^\adj \mtx{DF}$. Observe that, up to scaling and modulation of columns, $\mtx{W}$ consists of k copies of a k × k DFT concatenated horizontally. Suppose that we apply the SRFT $\mtx{\Omega} = \mtx{DFR}^\adj$ to the matrix $\mtx{V}^\adj$. We obtain a matrix of the form $\mtx{X} = \mtx{V}^\adj \mtx{\Omega} = \mtx{WR}^\adj$, which consists of ℓ random columns sampled from $\mtx{W}$. Theorem [thm:SRFT-spec-bd] certainly cannot hold unless $\sigma\_k(\mtx{X}) > 0$. To ensure the latter event occurs, we must pick at least one copy each of the k distinct columns of $\mtx{W}$. This is the coupon collector’s problem in disguise. To obtain a complete set of k coupons (i.e., columns) with nonnegligible probability, we must draw at least klog(k) columns. The fact that we are sampling without replacement does not improve the analysis appreciably because the matrix has too many columns. Performance guarantees ---------------------- We are now prepared to present detailed information on the performance of the proto-algorithm when the test matrix $\mtx{\Omega}$ is an SRFT. [Error bounds for SRFT] [thm:SRFT] Fix an m × n matrix $\mtx{A}$ with singular values σ1 ≥ σ2 ≥ σ3 ≥ …. Draw an n × ℓ SRFT matrix $\mtx{\Omega}$, where $$4 \left[\sqrt{k} + \sqrt{8\log(kn)} \right]^2 \log(k) \leq \ell \leq n.$$ Construct the sample matrix $\mtx{Y} = \mtx{A\Omega}$. Then $$\begin{aligned} \norm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } &\leq \sqrt{1 + 7n/\ell} \cdot \sigma\_{k+1} \quad\text{and} \\ \fnorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } &\leq \sqrt{1 + 7n/\ell} \cdot \left( \sum\nolimits\_{j > k} \sigma\_j^2 \right)^{1/2}\end{aligned}$$ with failure probability at most $\bigO(k^{-1})$. As we saw in §[sec:gauss-avg-case], the quantity σk + 1 is the minimal spectral-norm error possible when approximating $\mtx{A}$ with a rank-k matrix. Similarly, the series in the second bound is the minimal Frobenius-norm error when approximating $\mtx{A}$ with a rank-k matrix. We see that both error bounds lie within a polynomial factor of the baseline, and this factor decreases with the number ℓ of samples we retain. The likelihood of error with an SRFT test matrix is substantially worse than in the Gaussian case. The failure probability here is roughly k− 1, while in the Gaussian case, the failure probability is roughly $\econst^{-(\ell - k)}$. This qualitative difference is not an artifact of the analysis; discrete sampling techniques inherently fail with higher probability. Matrix approximation schemes based on SRFTs often perform much better in practice than the error analysis here would indicate. While it is not generally possible to guarantee accuracy with a sampling parameter less than ℓ ∼ klog(k), we have found empirically that the choice ℓ = k + 20 is adequate in almost all applications. Indeed, SRFTs sometimes perform even better than Gaussian matrices (see, e.g., Figure [fig:SRFTerrors]). We complete the section with the proof of Theorem [thm:SRFT]. [Theorem [thm:SRFT]] Let $\mtx{V}$ be the right unitary factor of matrix $\mtx{A}$, and partition $\mtx{V} = [ \mtx{V}\_1 \ \| \ \mtx{V}\_2 ]$ into blocks containing, respectively, k and n − k columns. Recall that $$\mtx{\Omega}\_1 = \mtx{V}\_1^\adj \mtx{\Omega} \quad\text{and}\quad \mtx{\Omega}\_2 = \mtx{V}\_2^\adj \mtx{\Omega}.$$ where $\mtx{\Omega}$ is the conjugate transpose of an SRFT. Theorem [thm:SRFT-spec-bd] ensures that the submatrix $\mtx{\Omega}\_1$ has full row rank, with failure probability at most $\bigO(k^{-1})$. Therefore, Theorem [thm:main-error-bd] implies that $$\triplenorm{ (\Id - \mtx{P}\_{\mtx{Y}}) \mtx{A} } \leq \triplenorm{\mtx{\Sigma}\_2} \left[ 1 + \smnorm{}{ \mtx{\Omega}\_1^\psinv }^2 \cdot \normsq{ \mtx{\Omega}\_2 } \right]^{1/2},$$ where $\triplenorm{\cdot}$ denotes either the spectral norm or the Frobenius norm. Our application of Theorem [thm:SRFT-spec-bd] also ensures that the spectral norm of $\mtx{\Omega}\_1^\psinv$ is under control. $$\smnorm{}{ \mtx{\Omega}\_1^\psinv }^2 \leq \frac{1}{0.40^2} < 7.$$ We may bound the spectral norm of $\mtx{\Omega}\_2$ deterministically. $$\norm{\mtx{\Omega}\_2} = \norm{ \mtx{V}\_2^\adj \mtx{\Omega} } \leq \norm{ \mtx{V}\_2^\adj } \norm{ \mtx{\Omega} } = \sqrt{n/\ell}$$ since $\mtx{V}\_2$ and $\sqrt{\ell/n} \cdot \mtx{\Omega}$ are both orthonormal matrices. Combine these estimates to complete the proof. Acknowledgments =============== The authors have benefited from valuable discussions with many researchers, among them Inderjit Dhillon, Petros Drineas, Ming Gu, Edo Liberty, Michael Mahoney, Vladimir Rokhlin, Yoel Shkolnisky, and Arthur Szlam. In particular, we would like to thank Mark Tygert for his insightful remarks on early drafts of this paper. The example in Section [sec:graphlaplacian] was provided by François Meyer of the University of Colorado at Boulder. The example in Section [sec:eigenfaces] comes from the FERET database of facial images collected under the FERET program, sponsored by the DoD Counterdrug Technology Development Program Office. The work reported was initiated during the program Mathematics of Knowledge and Search Engines held at IPAM in the fall of 2007. Finally, we would like to thank the anonymous referees, whose thoughtful remarks have helped us to improve the manuscript dramatically. On Gaussian matrices ==================== This appendix collects some of the properties of Gaussian matrices that we use in our analysis. Most of the results follow quickly from material that is already available in the literature. One fact, however, requires a surprisingly difficult new argument. We focus on the real case here; the complex case is similar but actually yields better results. Expectation of norms -------------------- We begin with the expected Frobenius norm of a scaled Gaussian matrix, which follows from an easy calculation. [prop:gauss-frob-expect] Fix real matrices $\mtx{S}, \mtx{T}$, and draw a standard Gaussian matrix $\mtx{G}$. Then $$\left( \Expect \fnormsq{ \mtx{SGT} } \right)^{1/2} = \fnorm{\mtx{S}} \fnorm{\mtx{T}}.$$ The distribution of a Gaussian matrix is invariant under orthogonal transformations, and the Frobenius norm is also unitarily invariant. As a result, it represents no loss of generality to assume that $\mtx{S}$ and $\mtx{T}$ are diagonal. Therefore, $$\Expect \fnormsq{ \mtx{SGT} } = \Expect \left[ \sum\nolimits\_{jk} \abssq{s\_{jj} g\_{jk} t\_{kk}} \right] = \sum\nolimits\_{jk} \abssq{s\_{jj}} \abssq{t\_{kk}} = \fnormsq{ \mtx{S} } \fnormsq{\mtx{T} }.$$ Since the right-hand side is unitarily invariant, we have also identified the value of the expectation for general matrices $\mtx{S}$ and $\mtx{T}$. The literature contains an excellent bound for the expected spectral norm of a scaled Gaussian matrix. The result is due to Gordon , who established the bound using a sharp version of Slepian’s lemma. See and for additional discussion. [prop:gauss-spec-expect] Fix real matrices $\mtx{S}, \mtx{T}$, and draw a standard Gaussian matrix $\mtx{G}$. Then $$\Expect \norm{ \mtx{SGT} } \leq \norm{\mtx{S}} \fnorm{\mtx{T}} + \fnorm{\mtx{S}} \norm{\mtx{T}}.$$ Spectral norm of pseudoinverse ------------------------------ Now, we turn to the pseudoinverse of a Gaussian matrix. Recently, Chen and Dongarra developed a good bound on the probability that its spectral norm is large. The statement here follows from after an application of Stirling’s approximation. See also [prop:inv-gauss-spec-tail] Let $\mtx{G}$ be an m × n standard Gaussian matrix with n ≥ m ≥ 2. For each t > 0, $$\label{eqn:gauss-lower-bd-2} \Prob{ \smnorm{}{\mtx{G}^\psinv} > t } \leq \frac{1}{\sqrt{2\pi(n-m+1)}} \left[ \frac{\econst\sqrt{n}}{n-m+1}\right]^{n-m+1} t^{-(n-m+1)}.$$ We can use Proposition [prop:inv-gauss-spec-tail] to bound the expected spectral norm of a pseudo-inverted Gaussian matrix. [prop:inv-gauss-spec-expect] Let $\mtx{G}$ be a m × n standard Gaussian matrix with n − m ≥ 1 and m ≥ 2. Then $$\Expect \smnorm{}{\mtx{G}^\psinv} < \frac{\econst\sqrt{n}}{n-m}$$ Let us make the abbreviations p = n − m and $$C = \frac{1}{\sqrt{2\pi(p+1)}} \left[ \frac{\econst\sqrt{n}}{p+1}\right]^{p+1}.$$ We compute the expectation by way of a standard argument. The integral formula for the mean of a nonnegative random variable implies that, for all E > 0, $$\begin{gathered} \Expect \smnorm{}{\mtx{G}^\psinv} = \int\_0^{\infty} \Prob{ \smnorm{}{\mtx{G}^\psinv} > t }\idiff{t} \leq E + \int\_E^{\infty} \Prob{ \smnorm{}{\mtx{G}^\psinv} > t }\idiff{t} \\ \leq E + C \int\_E^\infty t^{-(p+1)} \idiff{t} = E + \frac{1}{p} C E^{-p},\end{gathered}$$ where the second inequality follows from Proposition [prop:inv-gauss-spec-tail]. The right-hand side is minimized when E = C1/(p + 1). Substitute and simplify. Frobenius norm of pseudoinverse ------------------------------- The squared Frobenius norm of a pseudo-inverted Gaussian matrix is closely connected with the trace of an inverted Wishart matrix. This observation leads to an exact expression for the expectation. [prop:inv-gauss-frob-expect] Let $\mtx{G}$ be an m × n standard Gaussian matrix with n − m ≥ 2. Then $$\Expect \fnormsq{\mtx{G}^\psinv} = \frac{m}{n-m-1}.$$ Observe that $$\smnorm{\rm F}{ \mtx{G}^\psinv }^2 = \trace \left[ (\mtx{G}^\psinv)^\adj \mtx{G}^\psinv \right] = \trace \left[ (\mtx{GG}^\adj)^{-1} \right].$$ The second identity holds almost surely because the Wishart matrix $\mtx{GG}^\adj$ is invertible with probability one. The random matrix $(\mtx{GG}^\adj)^{-1}$ follows the inverted Wishart distribution, so we can compute its expected trace explicitly using a formula from . On the other hand, very little seems to be known about the tail behavior of the Frobenius norm of a pseudo-inverted Gaussian matrix. The following theorem, which is new, provides an adequate bound on the probability of a large deviation. [thm:inv-gauss-frob-tail] Let $\mtx{G}$ be an m × n standard Gaussian matrix with n − m ≥ 4. For each t ≥ 1, $$\Prob{ \fnormsq{\mtx{G}^\psinv} > \frac{12 m}{n-m} \cdot t } \leq 4 t^{-(n-m)/2}.$$ Neither the precise form of Theorem [thm:inv-gauss-frob-tail] nor the constants are ideal; we have focused instead on establishing a useful bound with minimal fuss. The rest of the section is devoted to the rather lengthy proof. Unfortunately, most of the standard methods for producing tail bounds fail for random variables that do not exhibit normal or exponential concentration. Our argument relies on special properties of Gaussian matrices and a dose of brute force. ### Technical background We begin with a piece of notation. For any number q ≥ 1, we define the Lq norm of a random variable Z by $$\Expect^q(Z) = \left( \Expect \abs{Z}^q \right)^{1/q}.$$ In particular, the Lq norm satisfies the triangle inequality. We continue with a collection of technical results. First, we present a striking fact about the structure of Gaussian matrices . [prop:bidiag] For n ≥ m, an m × n standard Gaussian matrix is orthogonally equivalent with a random bidiagonal matrix $$\label{eqn:bidiag} \mtx{L} = \begin{bmatrix} X\_n \\ Y\_{m-1} & X\_{n-1} \\ & Y\_{m-2} & X\_{n-2} \\ && \ddots & \ddots \\ &&& Y\_1 & X\_{n-(m-1)} &&& \end{bmatrix}\_{m \times n},$$ where, for each j, the random variables Xj2 and Yj2 follow the χ2 distribution with j degrees of freedom. Furthermore, these variates are mutually independent. We also require the moments of a chi-square variate, which are expressed in terms of special functions. [prop:chisq-moment-exact] Let Ξ be a χ2 variate with k degrees of freedom. When 0 ≤ q < k/2, $$\Expect \left(\Xi^q \right) = \frac{ 2^q \Gamma(k/2+q)}{\Gamma(k/2)} \quad\text{and}\quad \Expect \left(\Xi^{-q} \right) = \frac{ \Gamma(k/2 - q)}{2^q \Gamma(k/2)}.$$ Recall that a χ2 variate with k degrees of freedom has the probability density function f(t) = tk/2 - 1 -t/2,. Bytheintegralformulaforexpectation, (q) = 0 tq f(t) =, where the second equality follows from Euler’s integral expression for the gamma function. The other calculation is similar. To streamline the proof, we eliminate the gamma functions from Proposition [prop:chisq-moment-exact]. The next result bounds the positive moments of a chi-square variate. [lem:chisq-moment] Let Ξ be a χ2 variate with k degrees of freedom. For q ≥ 1, $$\Expect^q (\Xi) \leq k+q.$$ Write q = r + θ, where r = ⌊q⌋. Repeated application of the functional equation zΓ(z) = Γ(z + 1) yields $$\Expect^q (\Xi) = \left[ \frac{2^\theta \Gamma(k/2 + \theta)}{\Gamma(k/2)} \cdot \prod\_{j=1}^{r} (k + 2(q-j)) \right]^{1/q}.$$ The gamma function is logarithmically convex, so $$\frac{2^\theta \Gamma(k/2 + \theta)}{\Gamma(k/2)} \leq \frac{2^\theta \cdot \Gamma(k/2)^{1-\theta} \cdot \Gamma(k/2 + 1)^\theta}{\Gamma(k/2)} = k^\theta \leq \left[ \prod\_{j=1}^r (k + 2(q-j)) \right]^{\theta/r}.$$ The second inequality holds because k is smaller than each term in the product, hence is smaller than their geometric mean. As a consequence, $$\Expect^q(\Xi) \leq \left[\prod\_{j=1}^r (k + 2(q-j)) \right]^{1/r} \leq \frac{1}{r} \sum\_{j=1}^r (k + 2(q-j)) \leq k + q.$$ The second relation is the inequality between the geometric and arithmetic mean. Finally, we develop a bound for the negative moments of a chi-square variate. [lem:inv-chisq-moment] Let Ξ be a χ2 variate with k degrees of freedom, where k ≥ 5. When 2 ≤ q ≤ (k − 1)/2, $$\Expect^q \left( \Xi^{-1} \right) < \frac{3}{k}.$$ We establish the bound for q = (k − 1)/2. For smaller values of q, the result follows from Hölder’s inequality. Proposition [prop:chisq-moment-exact] shows that $$\Expect^q \left(\Xi^{-1}\right) = \left[ \frac{\Gamma(k/2-q)}{2^q \Gamma(k/2)} \right]^{1/q} = \left[ \frac{\Gamma(1/2)}{2^q \Gamma(k/2)} \right]^{1/q}.$$ Stirling’s approximation ensures that $\Gamma(k/2) \geq \sqrt{2\pi} \cdot (k/2)^{(k-1)/2} \cdot \econst^{-k/2}$. Since the value $\Gamma(1/2) = \sqrt{\pi}$, $$\Expect^q \left(\Xi^{-1}\right) \leq \left[ \frac{\sqrt{\pi}}{2^q\sqrt{2\pi}\cdot (k/2)^{q}\cdot \econst^{-q-1/2}} \right]^{1/q} = \frac{\econst}{k} \left[ \frac{\econst}{2} \right]^{1/2q} < \frac{3}{k},$$ where we used the assumption q ≥ 2 to complete the numerical estimate. ### Proof of Theorem [thm:inv-gauss-frob-tail] Let $\mtx{G}$ be an m × n Gaussian matrix, where we assume that n − m ≥ 4. Define the random variable $$Z = \fnormsq{\mtx{G}^\psinv}.$$ Our goal is to develop a tail bound for Z. The argument is inspired by work of Szarek for square Gaussian matrices. The first step is to find an explicit, tractable representation for the random variable. According to Proposition [prop:bidiag], a Gaussian matrix $\mtx{G}$ is orthogonally equivalent with a bidiagonal matrix $\mtx{L}$ of the form . Making an analogy with the inversion formula for a triangular matrix, we realize that the pseudoinverse of $\mtx{L}$ is given by $$\mtx{L}^{\psinv} = \begin{bmatrix} X\_n^{-1} \\ \frac{-Y\_{m-1}}{X\_n X\_{n-1}} & X\_{n-1}^{-1} \\ & \frac{-Y\_{m-2}}{X\_{n-1} X\_{n-2}} & X\_{n-2}^{-1} \\ && \ddots & \ddots \\ &&& \frac{-Y\_1}{X\_{n-(m-2)} X\_{n-(m-1)}} & X\_{n-(m-1)}^{-1} \\ \\ \\ \end{bmatrix}\_{n \times m}.$$ Because $\mtx{L}^\psinv$ is orthogonally equivalent with $\mtx{G}^\psinv$ and the Frobenius norm is unitarily invariant, we have the relations $$Z = \smnorm{\rm F}{ \mtx{G}^\psinv }^2 = \smnorm{\rm F}{ \mtx{L}^{\psinv} }^2 \leq \sum\_{j=0}^{m-1} \frac{1}{X\_{n-j}^2} \left( 1 + \frac{Y\_{m-j}^2}{X\_{n-j+1}^2}\right),$$ where we have added an extra subdiagonal term (corresponding with j = 0) so that we can avoid exceptional cases later. We abbreviate the summands as $$W\_j = \frac{1}{X\_{n-j}^2} \left( 1 + \frac{Y\_{m-j}^2}{X\_{n-j+1}^2}\right), \quad j = 0, 1, 2, \dots, m - 1.$$ Next, we develop a large deviation bound for each summand by computing a moment and invoking Markov’s inequality. For the exponent q = (n − m)/2, Lemmas [lem:chisq-moment] and [lem:inv-chisq-moment] yield $$\begin{aligned} \Expect^q (W\_j) &= \Expect^q(X\_{n-j}^{-2}) \cdot \Expect^q\left[1+ \frac{Y\_{m-j}^2}{X\_{n-j+1}^2}\right] \\ &\leq \Expect^q(X\_{n-j}^{-2}) \left[1 + \Expect^q (Y\_{m-j}^2) \cdot \Expect^q(X\_{n-j+1}^{-2}) \right] \\ &\leq \frac{3}{n-j} \left[ 1 + \frac{3(m-j+q)}{n-j+1} \right] \\ &= \frac{3}{n-j} \left[1 + 3 - \frac{3(n-m+1-q)}{n-j+1}\right]\end{aligned}$$ Note that the first two relations require the independence of the variates and the triangle inequality for the Lq norm. The maximum value of the bracket evidently occurs when j = 0, so $$\Expect^q(W\_j) < \frac{12}{n - j}, \quad j = 0, 1, 2, \dots, m-1.$$ Markov’s inequality results in $$\Prob{ W\_j \geq \frac{12}{n - j} \cdot u } \leq u^{-q}.$$ Select u = t ⋅ (n − j)/(n − m) to reach $$\Prob{ W\_j \geq \frac{12}{n - m} \cdot t } \leq \left[ \frac{n-m}{n-j} \right]^{q} t^{-q}.$$ To complete the argument, we combine these estimates by means of the union bound and clean up the resulting mess. Since Z ≤ ∑j = 0m − 1Wj, $$\Prob{ Z \geq \frac{12m}{n-m} \cdot t } \leq t^{-q} \sum\_{j=0}^{m-1} \left[ \frac{n-m}{n-j} \right]^q.$$ To control the sum on the right-hand side, observe that $$\begin{gathered} \sum\_{j=0}^{m-1} \left[ \frac{n-m}{n-j} \right]^q < (n-m)^q \int\_0^m (n-x)^{-q} \idiff{x} \\ < \frac{(n-m)^q}{q-1} (n-m)^{-q + 1} = \frac{2(n-m)}{n-m-2} \leq 4,\end{gathered}$$ where the last inequality follows from the hypothesis n − m ≥ 4. Together, the estimates in this paragraph produce the advertised bound. It would be very interesting to find more conceptual and extensible argument that yields accurate concentration results for inverse spectral functions of a Gaussian matrix.	arxiv_0000804
Simulating radiative feedback and star cluster formation in GMCs: I. Dependence on gravitational boundedness ============================================================================================================ ### 16 June 2016 [firstpage] Radiative feedback is an important consequence of cluster formation in Giant Molecular Clouds (GMCs) in which newly formed clusters heat and ionize their surrounding gas. The process of cluster formation, and the role of radiative feedback, has not been fully explored in different GMC environments. We present a suite of simulations which explore how the initial gravitational boundedness, and radiative feedback, affect cluster formation. We model the early evolution ( < 5 Myr) of turbulent, 106 M⊙ clouds with virial parameters ranging from 0.5 to 5. To model cluster formation, we use cluster sink particles, coupled to a raytracing scheme, and a custom subgrid model which populates a cluster via sampling an IMF with an efficiency of 20% per freefall time. We find that radiative feedback only decreases the cluster particle formation efficiency by a few percent. The initial virial parameter plays a much stronger role in limiting cluster formation, with a spread of cluster formation efficiencies of 37% to 71% for the most unbound to the most bound model. The total number of clusters increases while the maximum mass cluster decreases with an increasing initial virial parameter, resulting in steeper mass distributions. The star formation rates in our cluster particles are initially consistent with observations but rise to higher values at late times. This suggests that radiative feedback alone is not responsible for dispersing a GMC over the first 5 Myr of cluster formation. Introduction ============ The modern paradigm for star cluster formation suggests that clusters are born from dense clumps (n > 104 cm− 3) which form in supersonically turbulent, filamentary molecular clouds. Recent observations show that young stellar clusters tend to form at the intersection of filaments in regions that are fed by higher than average accretion rates. These overdense regions may then fragment further, resulting in highly subclustered objects which undergo mergers. The question of how the high accretion flow onto dense, star-forming clumps is halted is of particular importance. The mass accretion history of these objects has implications for the final cluster mass and, hence, the observed cluster mass function. Moreover, the overall conversion of molecular gas to stars is inefficient. Understanding what processes limit the star formation efficiency in molecular clouds is key to a complete star formation theory as well as underpinning all discussions of simulations and theories of galaxy formation and evolution. Several mechanisms have been proposed to explain the low star formation efficiency observed in molecular clouds. For example, turbulent velocity fields have been shown to lower the overall star formation rate per freefall time. This cannot be the sole mechanism at work, however, since, given enough time, all of the gas will be converted to stars. The added pressure support via magnetic fields can also lower the overall star formation efficiency. While magnetic fields may play an important role in limiting cloud fragmentation, they also contribute to the dispersal of gas within individual star forming cores by means of hydromagnetic bipolar outflows and jets that are associated with young stars of all masses. The process of feedback from newly-formed stars can both limit the star formation efficiency and disperse the surrounding gas, thereby halting the star formation process. Feedback comes in various forms: protostellar jets, stellar winds, ionization and heating of the gas, and radiation pressure. Radiative feedback has been suggested as being the most important form of feedback in clusters which host massive star formation. The ionizing radiation released from newly-formed stars heats the gas to approximately 10,000 K and can drive the expansion of massive HII regions. The direct input of momentum via high energy photons interacting with surrounding dust grains can also drive strong outflows. Previous studies of radiative feedback on both small and large spatial scales have shown that the star formation efficiency can be significantly reduced. Studies which model the formation of individual stars, however, are typically limited to the clump scale or low mass molecular clouds. These studies therefore neglect the impact of radiative feedback on the global structure of massive GMC’s. Since cluster formation can be broadly distributed throughout a cloud, cluster interactions over the entire range of the GMC must be considered. There have been studies of radiative feedback in massive GMCs (106 M⊙). Due to numerical constraints, the supergiant molecular cloud regime ( > 106 M⊙) has not been examined nor do most models follow the cloud’s evolution up to the point of dispersal which is crucial to placing constraints on the lifetime of molecular clouds. Recent results suggest that radiation feedback alone is not effective in accounting for low star formation rates in galaxy formation simulations. The lifetimes of molecular clouds, and the processes responsible for destroying them, are challenging to measure. Estimates of GMC lifetimes range from a single freefall time, τff, up to tens of freefall times. Recent observations of the deuterium fraction in both massive, starless cores and Infrared Dark Clouds (IRDCs) indicates ages within a range of 6 to 10 local freefall times, corresponding to ∼ 3 Myr for IRDCs. We note that direct measurements of the lifetimes of molecular clouds out to a redshift of 4 are now possible with ALMA (Atacama Large Millimeter Array) and the method outlined in. Theoretical estimates suggest that radiative feedback, in particular radiation pressure, is the most important physical process responsible for destroying massive GMC’s but numerical simulations of star formation in GMC’s which include radiative feedback are required to validate these claims. The role of radiative feedback in different molecular cloud environments has also not been explored in detail. The observed clouds in the Milky Way and extragalactic sources show a wide range of properties. More specifically, molecular clouds have been observed to have a range of virial parameters, defined as the ratio of the cloud’s kinetic energy to gravitational potential energy. This ratio may vary from 0.5 up to 10 with the mean being approximately 1. Simulations have shown that dense, self-gravitating clumps can still fragment out of clouds that are globally unbound. Furthermore, the star formation efficiency is naturally lowered in unbound clouds. Radiative feedback may play a stronger role in lowering the star formation efficiency in unbound versus bound clouds since the added energy and momentum can more easily disperse the surrounding, unbound gas. To investigate these questions, we have chosen to simulate the formation of young stellar clusters in molecular clouds with a range of properties while including detailed radiative feedback. As described above, we focus on varying the initial virial parameter in turbulent, 106 M⊙ GMCs. We are focusing on this mass because the majority of star formation in the Milky Way is hosted in these clouds. In order to simulate the long term evolution of these clouds, we use sink particles to represent young clusters and combine this with a subgrid model to track star formation within the clusters. In Section 2, we describe the details of this subgrid model and the initial conditions used in our simulations. In Section 3, we describe the evolution of clouds with varying initial virial parameters, discuss the role of radiative feedback, and compare cluster properties across clouds. We find that radiative feedback has only a slight effect on the star formation efficiency. We find, in fact, that efficiencies are dominated by the gravitational binding of the cloud and that virial and sub virial initial cloud models do not show the low efficiencies that observations demand. In Section 4, we discuss how our results compare to recent observations of local star forming regions, followed by our conclusions in the final section. Numerical Methods ================= We perform numerical simulations using version 2.5 of the hydrodynamical code FLASH. FLASH is an adaptive mesh refinement code (AMR) which integrates the compressible gas dynamic equations on a Eulerian grid and includes modules to treat self-gravity, radiative transfer, and star formation. We include cooling via molecular lines and dust, and use a simple ideal gas law equation of state with an adiabatic index of 1.67. Radiative transfer is handled via a hybrid-characteristics raytracing scheme developed by and adapted for star formation simulations by. This scheme follows the propagation of both ionizing and non-ionizing radiation and makes use of the DORIC routines. The DORIC package is an iterative scheme used to calculate the ionization, heating, and cooling rates of a large number of species. For simplicity, however, hydrogen is used as the only gas component. have shown that the code can accurately reproduce the analytic solution for the expansion of a D-type ionization front from, and have shown the code accurately treats R-type fronts. The opacity to the non-ionizing radiation is represented by the Planck mean opacities from the dust model of. The Planck mean opacities are used because the raytracer has no frequency dependence apart from ionizing versus non-ionizing radiation. Radiation pressure is included by adopting a single UV opacity of 775 cm2 g− 1 from. The associated radiative force per unit mass is given by, $$F = \frac{L}{c}\frac{e^{-\tau}}{4\pi r^2}$$ where L is the source luminosity, c is the speed of light, and τ and r are the optical depth and distance between the source and the target cell. To represent the formation of star clusters, we make use of Lagragian sink particles (Banerjee et al. 2009; Federrath et al. 2010). In order to form a particle, a region of gas in the simulation volume must meet the following conditions: * At the highest level of refinement * The divergence is less than zero (ie. converging) * At a local gravitational minimum * The region is Jeans unstable * The region is gravitationally bound * Not within the radius of another particle. These particles were designed to represent stars but the above conditions also model the formation of a dense clump which hosts cluster formation. In order to examine the impact of radiative feedback from stellar clusters, we require a subgrid model for the radiative output of a cluster as it evolves over time. Below, we provide a brief summary of this model. A more detailed description can be found in Howard et al. (2014). Subgrid model for cluster formation: Cluster sinks -------------------------------------------------- One of the most important aspects of radiative feedback of a young forming cluster on its surrounding host GMC is the shutting off of the accretion flow into the cluster forming region. A cluster must therefore be assigned the correct, combined radiative output of all its member stars as star formation proceeds. Here, we provide a brief description of the subgrid model we use to form stars within a cluster particle. In order for cluster formation to begin, the host clump must reach sufficient density in order to fragment and collapse. The threshold density for cluster formation has been observationally measured by who found that the transition from starless to star forming clumps, in local star forming regions, occurs at a density of ∼ 104 cm− 3. This number has also been quoted by other authors who found that the SFRs in molecular clouds are well correlated to the dense ( > 104 cm− 3) gas mass. More recent work by, aimed at producing probability density functions for gas density from column density observations, found a density threshold of 5 × 103 cm− 3. Analytical models for the density threshold, which also account for environmental effects, find a threshold of 1.5 × 104 cm− 3 when adopting Solar neighbourhood parameters. We have therefore chosen to use a fiducial density threshold for cluster formation of 104 cm− 3. We report on several simulations in which we vary this value by an order of magnitude. Once this critical density threshold is reached, as well as the other conditions outlined above, a cluster, represented by a sink particle in this work, forms. Star formation then proceeds within this cluster as gas accretes onto this dense region. With this general framework in mind, we use our subgrid model for the formation, radiative feedback, and evolution of a cluster that addresses how the original gas reservoir (ie. clump) is divided into stars over time and how accreted gas is handled. We refer the reader to that paper for the details, which we now briefly summarize. To treat the conversion of reservoir mass into stars, we divide it into main sequence stars at a prescribed efficiency according to an IMF. We use a star formation efficiency of 20% per freefall time, where the freefall time is given by $$t\_{ff} = \sqrt{\frac{3\pi}{32 G n \mu m\_{H}}}$$ where n is the number density, μ is the mean molecular weight, and mH is the mass of hydrogen. As discussed above, our fiducial value for n is taken to be 104 cm− 3. The resulting freefall time is 0.36 Myr. We use the Chabrier IMF, given by $$\xi(log \ m) = \begin{cases} 0.093\times exp\{\frac{-(log \ m - log \ 0.2)^2}{2\times(0.55)^2}\}, & \mbox{\ m $\leq$ 1 \ M$\_{\odot}$ } \\ 0.041m^{-1.35 \pm 0.3}, & \mbox{\ m $>$ 1 \ M$\_{\odot}$}, \end{cases}$$ as the probability distribution from which we draw our stars. We chose to sample the Chabrier IMF every tenth of a freefall time, or 3.6 × 104 years. As shown in, this sampling frequency allows cluster properties to evolve smoothly over time while still accurately reproducing the IMF from which the stars are drawn. The masses of all stars formed in a cluster are recorded, meaning that its stellar content is known at all times. Using the analytic fits for the luminosity of main sequence stars provided by, we know the luminosity of each star within the cluster. The total luminosity, and ionizing photon rate, of each cluster is the sum of the stellar components. These quantities are then used by the raytracer to treat the radiative transfer. Gas accreted by a cluster is simply added to the gaseous reservoir. Since we are sampling an IMF in order to get our final stellar masses, we do not need to treat accretion onto individual stars. The accreted gas can then be used to form new stars at a later time. We do not include mass loss from the cluster particle because we assume that the reservoir gas remains gravitationally bound to the cluster over long timescales, even in the presence of stellar feedback. Earlier simulations by found that the inclusion of radiative feedback was not sufficient to unbind the majority of the clump mass. The reason for this is collimated ionized outflows which are released into low density regions perpendicular to the dense filaments out of which the stars form. We therefore assume our gas reservoir is in a similarly filamentary state, preventing its disruption via stellar feedback. Furthermore, recent numerical works, both with and without feedback mechanisms, and observational studies suggest that cluster formation proceeds via gas exhaustion rather than gas expulsion. This further justifies the use of cluster particles which do not lose mass. We note that, when performing a simulation, we apply a mass threshold below which particles do not radiate. This was implemented in order to reduce the computational time. The raytracing scheme is computationally expensive so by allowing only massive particles to radiate, the total time spent during the raytracing step can be greatly reduced. We have chosen a threshold of 1000 M⊙ contained in stars. Note that this is not the total mass of the cluster particle, but rather the stellar mass. We have chosen this threshold because it is approximately the mass at which the first O stars are expected to appear. We have verified that, at any given time, particles above this mass dominate the total luminosity budget so we are therefore confident that the application of a mass threshold should not impact our results significantly. We have performed tests of the raytracing scheme similar to, except with our cluster sink particles, as described at the beginning of this Section. These were low resolution simulations, with 128 grid cells along each simulation axis, where a cluster sink particle was placed at the center of a uniform medium with a number density of 103 cm− 3. We allowed the cluster particle to sample the IMF once at the beginning of the simulation, after which no new stars were added to the cluster, and compared the resulting HII region to the analytical expression for the expansion of a D-type ionization front from. We find that, during expansion, the size of the HII region does not differ by more than 15% from the analytical solution. This error is reduced to less than 9% if we instead compare to the analytical expression presented in, as was done in the radiative transfer code comparison project STARBENCH. The addition of radiation pressure produces an HII region that is ∼ 10% larger compared to the test without radiation pressure. Additionally, the central density inside the HII region is significantly lower when radiation pressure is included, differing by approximately two orders of magnitude compared to the test without radiation pressure. ### Cluster Sink Mergers ![image](mach_numbers.png) [fig:mach] The sink particle routines in FLASH allow for particle mergers. In order for a merger to occur, several conditions must be met. Firstly, the particles must be closer than an accretion radius, defined to be 2.5 cells at the highest level of refinement. Secondly, the particles must be approaching one another. Lastly, the particles must be gravitationally bound to one another. More specifically, the total gravitational potential energy of the particle pair, at a distance of an accretion radius, must be greater than the total kinetic energy. When all of these conditions are met, the smaller mass particle is merged to the more massive particle. The new position is taken to be the center of mass position and the resulting particle velocity is determined via momentum conservation. We have made custom modifications to the merging routines in order to better represent the merging of cluster sink particles. As mentioned above, cluster sink particles have their mass divided between fully-formed stars and the gaseous reservoir. When a cluster particle merger occurs, we combine the reservoir masses and the total stellar mass. Since we track the masses of all stars formed in each particle, the total stellar content remains the same after merging. We emphasize that, while we will be using merger to describe the joining of two cluster sink particles, this differs from the dynamical mergers associated with fully formed clusters that are devoid of gas. In our context, a merger is more aptly defined as the coalescence of a central stellar cluster as well as the envelope of surrounding gas. Recent studies indicate that gas poor clusters may form before feedback starts acting since the local freefall time in these dense regions is the highest. This means that the stellar cluster within our particles may be spatially segregated from the surrounding gas reservoir. Therefore, the merger of the stellar component in our particles may be thought of in the traditional way, namely a dynamical merger of gas poor stellar clusters. The reservoir can be interpreted as the cocoon of gas which surrounds the central cluster, which is also joined during the merger. We do not resolve the structure of the gas reservoir, however, so we cannot distinguish between the merger of gas rich clusters and gas poor clusters with a surrounding envelope of gas. \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:3d] Initial Conditions ------------------ We simulate the formation of stellar subclusters in collapsing GMC’s with different initial properties and examine the role of radiative feedback in these environments. The clouds have the same initial average of n = 100 cm− 3 and masses of 106 M⊙. The radius of all clouds is 33.8 pc, corresponding to a minimum cell size of 0.13 pc. This results in a particle accretion radius of 0.325 pc (ie. 2.5 cells at the highest level of refinement). We have chosen an initial density profile which is uniform in the inner half of the cloud and decreasing as r− 3/2 in the outer half of the cloud. A quadratic fit is applied in the transition from uniform to decreasing density to ensure a continuous and smooth function. We use outflow conditions at the boundary of the simulation volume. The total mass in our simulations is therefore not conserved but can decrease over time as gas escapes the domain. This is particularly relevant to the discussion in the next Section. The cloud is overlaid with a Kolomogorov turbulent velocity spectrum. The turbulence is not driven after it is initially imposed and its strength is determined via the virial parameter. The virial parameter relates the total kinetic energy to the total gravitational potential energy in a cloud. It is defined as, $$\label{virial} \alpha = 2\frac{E\_{kin}}{\|E\_{grav}\|} \approx \frac{5 \sigma^2 R}{G M}$$ where Ekin is the total kinetic energy, Egrav is the total gravitational potential energy, σ is the velocity dispersion, R is the cloud radius, and M is the cloud mass. Equation [virial] is exact for a uniform density cloud. The velocity dispersion can be expressed as σ = Ma × cs where Ma is the Mach Number and cs is the sound speed, which is taken to be 2.3 × 105 cm/s at 10 K. For a uniform density cloud without magnetic fields, a virial parameter of 1 corresponds to virial equilibrium and α > 2 represents gravitationally unbound clouds. We use the first relation in Equation [virial] since the kinetic and gravitational potential energies are easily measurable in our simulations. We have chosen to examine initial virial parameters, defined as α0 henceforth, ranging from 0.5 to 5, which covers both bound, virialized, and unbound clouds. This range of virial parameters is within the range of observed GMC’s. The corresponding Mach Numbers can be found in Figure [fig:mach]. The models are evolved to ∼ 5 Myr, at which point supernovae are expected to inject large amounts of energy into the surrounding gas. Since the physics of Supernovae are not handled in the present code, we do not evolve the models past this point. Results ======= Global Evolution and Gas Properties ----------------------------------- \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:slicecoldensalpha1] \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:slicecoldensalpha3] \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:slicetemp] A three-dimensional view showing the evolution of the virialized simulation (α0 = 1) at three different times can be seen in Figure [fig:3d]. Three different density contours are plotted with densities ranging from 1.0 × 10− 23 to 3.0 × 10− 20 g ⋅ cm− 3. These specific densities were chosen to highlight the sparse, low density gas, the intermediate density filaments, and the dense cores which form primarily in the center of the simulation. The bounding box has a side length of 80.2 pc and the location of the cluster particles are overplotted with black circles. The red regions in Figure [fig:3d] represent gas which has an ionization fraction greater than 95%, highlighting the HII regions produced by the cluster particles. The first panel in Figure [fig:3d] shows that the gas quickly breaks up into filaments. A total of 17 cluster particles have formed by this time but have not produced enough stars to produce HII regions. The second panel, plotted at 2.68 Myr, still shows a filamentary network of gas but one which is more centrally condensed compared to the previous panel due to the global collapse of the gas. Dense cores are also now visible, scattered throughout the central region of the box where filaments intersect. A total of 84 particles are present at this time, some of which have produced enough stars to begin feeding back on their environment to produce small, localized HII regions. The total number of cluster particles remains roughly the same with a total of 88 particles at 3.65 Myr, shown in the bottom panel of Figure [fig:3d], but the stellar content in these clusters has grown significantly, allowing larger HII regions to form which cover a large fraction of the box. One region in particular has expanded outward from the central region into low density gas and extends to the edge of the simulation volume. To give a complementary view to Figure [fig:3d], we show density slices through the center of the z-axis and column density projections in Figure [fig:slicecoldensalpha1], for α0 = 1, and Figure [fig:slicecoldensalpha3], for α0 = 3. The corresponding temperature slices are in Figure [fig:slicetemp]. We show an example of a bound and unbound (α0 = 3) simulation illustrate the effects that initial boundedness plays on the evolution of the cloud. The locations of the cluster particles are projected onto the viewing plane in all images. Like the three-dimensional images discussed above, both clouds quickly break up into filaments by ∼ 0.5 Myr. Since the unbound cloud has a higher Mach number, the filaments are more pronounced with more low density gas filling the voids between the filaments. The temperature profiles look similar at this point, with cold 10 K gas in the center surrounded by warmer ∼ 300 K gas. There are visible shocks, however, in only the unbound simulation, along the periphery of the cloud. Comparing the two simulations at the earliest times also shows that only the unbound cloud has formed cluster particles. This can again be attributed to its higher Mach number, leading to stronger density enhancements which fragment into particles. The middle panels of Figure [fig:slicecoldensalpha1], [fig:slicecoldensalpha3], and [fig:slicetemp], plotted at ∼ 2.5 Myr, show that both simulations have produced many clusters, totalling 81 and 107 for the bound and unbound simulation, respectively. The clusters are grouped more heavily towards the centre of the volume for the bound run, indicative of collapsing gas resulting in higher central densities and a more centrally condensed grouping of clusters. An HII region is also visible in the middle panel of Figure [fig:slicetemp]. The corresponding density slice shows that the HII region has disrupted the filaments, effectively smearing them out to a more uniform medium with density enhancements around the edges. The final panel, shown at ∼ 4 Myr, shows more marked differences between the initially bound and unbound clouds. The bound cloud is now even more centrally condensed due to global collapse, with one dense central core from which filaments radiate outwards. A large grouping of clusters remains in the center, allowing further accretion. The outskirts are devoid of dense gas, which has allowed a large HII region to extend northwards, away from the central condensation, where there is less gas to shield these regions from the ionizing photons produced by massive stars in the central grouping. The unbound simulation, on the other hand, has dense gas pervading a large fraction of the box and lacks a dense, central core. The number of clusters has grown slightly for a total of 87 and 140 for the bound and unbound run, respectively. This suggests that the majority of cluster formation occurs during the early evolution of the clouds. ![image](mass_components.png) [fig:masscomponents] ![image](alpha_vs_time.png) [fig:alpha] The above visualizations provide a qualitative description of the evolution of a small subset of the completed simulations. A quantitative comparison showing both the gas and cluster particle evolution for all simulated clouds, with radiative feedback included, can be seen in Figure [fig:masscomponents]. The total mass in cluster particles is shown by the dashed lines, the total gas mass present in the simulation volume is shown by the dotted lines, and their sum is the shown by the solid lines. Since outflow boundary conditions are present, the solid lines do not remain constant at 106 M⊙ because gas can leave the simulation volume. The total amount of mass lost from the simulation depends heavily on α0. From Figure [fig:masscomponents], it can be seen that the initially bound runs lose a negligible amount of mass. The unbound runs, on the other hand, lose a significant amount. For example, the case with an initial virial parameter of 5 loses ∼ 30% of the total mass by the end of the simulation. Clearly α0 plays an important role in the long term evolution of the cloud. However, the virial parameter does not remain constant over the course of the simulation because the initial supersonic turbulence is damped via shocks. As the turbulence is damped, the velocity dispersion of the gas decreases which results in a decreasing α. The loss of high velocity gas from the simulation volume and the formation of cluster particles further modifies α. In Figure [fig:alpha], we plot the global virial parameter as a function of time on the left, and the contributions to α by the gas and cluster particles on the right. The global virial parameter, and the contributions from particles and gas, are expressed via, $$\alpha = 2\frac{E\_{kin}}{\|E\_{grav}\|} = 2\frac{E\_{kin,p}}{\|E\_{grav}\|} + 2\frac{E\_{kin,g}}{\|E\_{grav}\|} = \alpha\_{p} + \alpha\_{g}$$ where Ekin is the kinetic energy, Egrav is the gravitational potential energy, and the subscripts g and p refer to gas and particles. We see from Figure [fig:alpha] that α initially decreases as the turbulence damps. The rate at which α decreases is greater for unbound simulations. All simulations tend towards a virial state (ie. α = 1), and cross at approximately 2.3 Myr. The tendency towards a virial state is likely due to gravitational collapse after the decay of the initial turbulence which, in turn, drives enough internal turbulence to maintain α at 1. Because the virial parameters intersect at similar times, the simulations with higher α0 decrease more rapidly compared to lower α0 simulations. This is attributed to the turbulent decay time, τdec, which is given by τdec ∼ L/(Mcs), where L is the box size, M is the turbulent Mach number, and cs is the sound speed. Measuring the turbulent decay times from Figure [fig:alpha] shows a relation that is roughly inversely proportional to the initial Mach number, as suggested by τdec. The relation is not exact, however, due to varying degrees of mass loss from bound and unbound models. The right hand panel shows that this early phase, characterized by a decreasing α, is dominated almost exclusively by the gas energetics. A significant number of cluster particles have formed by this time (eg. ∼ 10% of all mass is in the form of cluster particles by 2.3 Myr for the simulation with α0 = 0.5), but they contribute negligibly to the overall α. After approximately 3 Myr, however, particle dynamics begin to play a significant role in the energetics, leading to a rise in α. The cluster particle contribution even outweighs the gaseous contribution for the initially bound runs. The role of the particle contributions to the global energetics points towards interesting dynamical differences between the simulations. We find that, in the bound simulations, the cluster particles are more centrally condensed relative to unbound simulations. This tighter grouping leads to stronger two-body interactions which result in velocities in excess of 50 km/s. These large particle velocities are responsible for the dominant contribution to α. For comparison, the velocities in unbound simulations do not exceed 20 km/s. We leave the discussion of particle dynamics, and its connection to α0, to a later paper. ![image](boundedness.png) [fig:boundedness] Formation Efficiencies and Particle Evolution --------------------------------------------- The discussion so far has been focused on global properties, such as the total mass in cluster particles, rather than the properties of individual objects. We move now to a discussion of the cluster particles themselves. The top panel of Figure [fig:boundedness] shows the time evolution of the two most massive cluster particles that survive to the end of the simulation for all clouds. Since the particles are not allowed to lose mass, these curves are monotonically increasing as a function of time. All curves also show a series of discrete jumps in mass. These jumps are due to merging events, as described in Section [methods]. It can be concluded from these curves that merging plays a significant role in determining the mass of the most massive particles in these simulations. This behaviour has also been observed in cluster-scale simulations. An interesting trend is noted in the top panel of Figure [fig:boundedness], namely that the simulations with a lower α0 tend to produce more massive cluster particles. The maximum mass cluster particles, including the mass of gas and stars, range from 3.26 × 104 M⊙ to 3.27 × 105 M⊙ for simulations with an α0 of 5 and 0.5, respectively. This trend can be attributed to the central concentration of dense gas that forms as a result of gravitational collapse in the bound simulations, allowing the most massive particles to continue accreting strongly from their surroundings. This suggests that the distribution of cluster particle masses depends on α0, which is discussed further below. The solid lines in the bottom panel of Figure [fig:boundedness] show the total mass of all cluster particles (shown on the left y-axis). There is a spread between simulations in both the total mass in cluster particles, and the time at which cluster formation begins. There is a spread of ∼ 260 kyr between when the most unbound cloud (α0 = 5) and the most bound cloud (α0 = 0.5) begin to form particles. While the unbound simulations form the first cluster particles, the bound simulations have more mass contained in particles overall. This can be seen by the dashed lines in the bottom panel of [fig:boundedness]. We refer to these lines as the “dense clump formation efficiency”, hereafter εcl, and are calculated by dividing the total mass in cluster particles over the initial mass of the cloud, ie. $$\epsilon\_{cl} = \frac{\sum\_{i} M\_i}{M\_{total}} = \frac{\sum\_{i} M\_i}{10^6 M\_{\odot}}$$ where Mi is the mass of the ith particle and the total cloud mass of 106 M⊙ has been shown for clarity. We have chosen the nomenclature of “dense clump formation efficiency”, rather than a cluster formation efficiency, because the mass contained in these particles is not solely in stars, but also a reservoir of unused gas which may be used for future star formation. Since εcl closely follows the total mass contained in particles, we see a similar spread between simulations. The early evolution is similar, until about 2 Myr, when the dense clump formation efficiencies begin to diverge. The final efficiencies cover a range of 37%, for an α0 of 5, to 71%, for an α0 of 0.5 with the unbound runs ending up less efficient than the bound runs. This is consistent with earlier simulations which demonstrated that particle formation efficiencies, over a fixed time, are significantly reduced as the Mach number is increased. ![image](nstars_vs_time_m10_6.png) [fig:nstars] Our εcl values can be compared to the observationally measured “dense gas mass fractions” (DGMFs) in nearby molecular clouds. Recent work by has measured the fraction of dense gas, defined to be gas above an extinction of AV > 7 corresponding to gas at densities of n > 104 cm− 3, in local molecular clouds with masses ranging from 200 to 2 × 105 M⊙. We note that our simulations have greater masses than the clouds presented in, but the observations still provide a useful point of comparison. The observed clouds span a range of evolutionary stages from starless clouds to star forming clouds with well developed HII regions. They find that the DGMF covers a large range, with a maximum value of ∼ 0.8. The average DGMF over the entire mass range is found to be 0.39 with a cloud-to-cloud scatter of 0.28, and the higher mass clouds have DGMFs close to the maximum value. We see that our own DGMFs, as measured by the total mass in cluster particles, fall within the observed range. The simulation having the closest DGMF to the observational average is the unbound simulation α0 = 3. This suggests that initially unbound molecular clouds more closely reproduce the properties of locally observed clouds. This point can be elaborated further by examining the dash-dotted lines in Figure [fig:boundedness]. These lines represent the star formation efficiency of the clouds (εsf), defined as the total mass in stars within cluster particles divided by the initial cloud mass. These curves show similar trends to εcl in the sense that higher initial virial parameters correspond to lower efficiencies. The curves take longer to diverge, however, as a consequence of our subgrid model which converts the cluster mass to stars gradually over time. The final εsf ranges from 18% to 34%. These values are higher than the global εsf in observed molecular clouds which typically span a range of 1-5% (eg. ). Since the highest virial parameter produces the lowest efficiency which is close to the observed values, this again suggests that a high initial virial parameter, in combination with radiative feedback, is required to match observations. There are, however, other forms of feedback that this work neglects which could also play a role in reproducing observed εsf values. When discussing cluster particles properties up to this point, we have focused solely on mass. To fully understand the resulting mass distributions, however, the total number of cluster sink particles needs to be discussed. In Figure [fig:nstars], we show the total number of cluster particles as a function of time. The number of particles can either increase via formation, or decrease via mergers or particles leaving the simulation volume. It can be seen that a large burst of cluster formation occurs early but levels out to roughly constant values around 2 Myr. The final number of clusters ranges from 52 for α0 = 0.5 to 189 for α0 = 5. As noted earlier, we see that initially unbound simulations begin to form cluster particles earlier. These curves do not cross, however, which is in contrast to the mass curves discussed in Figure [fig:boundedness]. This means that while the bound simulations contain the most mass in cluster particles, they have the lowest number of particles overall. The bound simulations are therefore expected to have more massive particles, on average, than unbound simulations. This has will be discussed further in the following section. Radiative Feedback Effects -------------------------- ![image](rad_vs_hydro.png) [fig:radvshydro] The above discussion has been focused on the role that α0 plays in shaping cloud properties and the resulting efficiencies. We now move on to discuss the impact of radiative feedback on these properties. To do this, we have completed an additional five simulations which are identical to those presented above but with radiative feedback not included. We refer to the simulations with radiative feedback included as “RHD” (Radiation Hydrodynamics) and the simulations with radiative feedback turned off as “HD” (Hydrodynamics). The main effect of radiative feedback is the suppression of star, or cluster, formation via the heating and ionization of the gas surrounding the newly formed stars. The strength of this suppression, however, is not fully understood and depends on the environment being modelled. For example, found that ionization feedback had a negligible effect on global SFEs in initially gravitationally bound clouds. In contrast, later work by showed that ionization feedback can reduce the SFE by up to 50% in smaller, 104 M⊙ clouds. Since it is clear that the GMC environment is a crucial component in determining the role of radiative feedback, we examine whether this role differs in 106 M⊙ GMCs with various α0. We have also included a rudimentary treatment of radiation pressure in our radiative transfer scheme which was not included in these previous works. ![image](mass_components_alpha3.png) [fig:mcalpha3] We compare the dense clump formation efficiencies, εcl, and star formation efficiencies, εsf, between RHD and HD simulations in Figure [fig:radvshydro]. Solid lines represent the RHD simulations and dashed lines represent the HD simulations. We see that the inclusion of radiative feedback plays a small role in determining εcl and εsf in the early evolution of a GMC, regardless of initial boundedness. The largest difference in εcl occurs in the simulation with an α0 of 3, with a difference of ∼ 10% between the RHD and HD simulation. The other simulations show that radiative feedback does suppress εcl but only by < 3%. Since εsf closely mirrors the evolution of εcl but with a time delay, the difference in SFEs between RHD and HD simulations is less noticeable. Overall, these results suggest that radiative feedback plays a minimal role in the early evolution of a massive GMC. Comparing these results to those presented in Figure [fig:boundedness], it is clear that α0 is the major contributing factor to the final efficiencies. Visual examination of Figure [fig:slicetemp], however, shows that radiative feedback is producing large scale HII regions which are evidently not strongly suppressing the global efficiencies in the cloud. The filamentary and porous nature of the cloud is likely limiting the role of radiative feedback. ![image](clusters_hyd.png) [fig:clustershyd] It is possible that if we followed the evolution of the cloud over longer timescales, we would see a larger suppression in efficiencies due to the lower overall density of the cloud which can be more easily disrupted by radiative feedback. Since we did not follow the cloud evolution to > 5 Myr, at which point Supernovae are expected to begin injecting significant amounts of energy and momentum into the surrounding gas, we can only conclude that radiative feedback does not greatly suppress cluster and star formation in the early evolution of a GMC. To show in which ways radiative feedback is suppressing the formation of particles, we focus on the simulation which had the largest difference between the RHD and HD simulations, namely the simulation with an α0 of 3. The observed difference in efficiency could either be due to radiation unbinding gas which then leaves the simulation volume, or through the heating and ionization of gas locally which prevents future particle formation and limits the accretion rate onto existing particles. To see which of these scenarios is more likely, we examine how the mass is divided among gas and particles in Figure [fig:mcalpha3]. Figure [fig:mcalpha3] shows that the RHD simulation has comparably less mass contained in particles and more gas mass than the HD simulation. However, the sums of these components, shown by the solid lines, are nearly identical, indicating that the RHD and HD runs have lost the same amount of mass from the simulation volume. From this we can conclude that radiative feedback is not driving outflows from the GMC, but is instead suppressing the formation and accretion of cluster particles. We note that this may be a product of our large cloud masses (106 M⊙) which have correspondingly deep potential wells which can trap the gas even under the influence of radiative feedback. Outflows may be more relevant in the early evolution of less massive GMCs where the energy and momentum input from stars may be sufficient to unbind the cloud. While global efficiencies are not greatly influenced by the inclusion of radiative feedback, it is possible that the cluster properties formed in an RHD versus an HD simulation may differ. In Figure [fig:clustershyd], we plot cluster properties for HD and RHD simulations to examine differences that arise. On the left, we show the total number of particles in all simulations. On the right, we plot the maximum and median mass cluster particles as a function of α0. The left hand panel of Figure [fig:clustershyd] shows no clear trend in the number of cluster particles in RHD versus HD simulations. The simulations with an α0 of 0.5 and 5 show no difference between the number of particles with radiative feedback included. The simulations with an α0 of 1 and 2 show an excess of clusters in the HD simulations at intermediate times but the difference becomes negligible at late times. Only the simulations with an α0 of 3 have different numbers of clusters at late times, with the RHD simulation forming ∼ 30 more cluster particles than the HD simulation. The right hand panel of Figure [fig:clustershyd] shows the maximum and median mass cluster particles versus α0 in both HD (black) and RHD (red) simulations. The maximum mass curves (solid lines) indicate that, except for the α0 = 3, HD simulations tend to produce more massive cluster particles than their RHD counterparts. This effect has been observed in other simulations of radiative feedback in which the distribution of particle masses is shifted to lower values. The median mass clusters show the opposite trend, with the RHD median masses being greater than the HD median masses. This indicates that the mass distribution is not simply being shifted to lower masses since the same trend would be seen in the median mass clusters as in the maximum mass clusters. It should be noted that since our subgrid model only includes radiative feedback from massive clusters (M\* > 1000 M⊙), the median mass clusters in the RHD simulations are not outputting radiation which may account for the observed trend reversal. Cluster Formation Thresholds ---------------------------- ![image](eff_nthresh.png) [fig:effnthresh] As discussed above, recent studies suggest that the strength of radiative feedback in different GMC environments may vary. We have shown that this is not the case for 106 M⊙ clouds which have different initial virial parameters. Another parameter, however, which also affects a cluster particle’s local environment is the threshold density for formation. We have chosen a fiducial value of 104 cm− 3 (see Section [methods]) motivated by observations of cluster forming regions. If, for example, a higher threshold density were chosen, the global GMC environment would be denser at the time of cluster formation and the radiation released by these clusters would be propagating into denser gas. This may have an important impact on the strength of radiative feedback. To examine the role that radiative feedback plays in clouds with different formation thresholds, we have resimulated a bound (α0 = 0.5) and an unbound (α0 = 3) cloud with formation thresholds of 103 and 105 cm− 3. The results of these simulations are presented in Figure [fig:effnthresh]. The onset of cluster formation differs among simulations with different formation thresholds. As expected, simulations with higher thresholds begin to form clusters later since the cloud has to collapse to higher densities before particles form. Comparing HD simulations, shown by dashed lines, and RHD simulations, shown by solid lines, we see that lower formation thresholds mirror the results presented in Figure [fig:radvshydro] in the sense that radiative feedback plays a minor role in controlling εcl. In fact, the differences in efficiency between RHD and HD simulations are reduced even further in the case of a low formation threshold. For the case of a high formation threshold, we see that the strength of radiative feedback is enhanced. The final efficiencies between RHD and HD simulation differ by 21% and 10% compared to 2% and 8% at the same time for the fiducial formation threshold of 104 cm− 3. These comparisons were at the time corresponding to the end of the shortest running simulation. The higher density immediately surrounding particles efficiently couples the radiation to the gas rather than the radiation streaming to low density voids perpendicular to filaments. The impact of radiative feedback on filaments is significantly more pronounced. This clearly indicates that the strength of radiative feedback is not a constant, but instead varies depending on the GMC environment. This may have important consequences for star formation in clumps and cores which are the densest regions in a molecular cloud. Indeed, observations of cores show that the densest cores (n ∼ 106 cm− 3) have lifetimes of approximately a freefall time while less dense cores (n ∼ 103 − 4 cm− 3) have lifetimes that are roughly ten times as long. This may be due, in part, to the increased strength of radiative feedback in denser regions. ![image](nstars_nthresh.png) [fig:nstarsnthresh] The most significant difference between simulations with different particle thresholds is the number of cluster particles that form, as shown in Figure [fig:nstarsnthresh]. As the particle formation threshold increases, the number of particles decreases. For example, changing the formation threshold from 103 to 105 cm− 3 in the simulation with an α0 of 3 reduces the number of particles from 167 to 26. This trend is expected since the regions which form particles need to reach higher densities in the case of a higher threshold, and these regions become increasingly rare as the threshold increases. While the particle formation efficiency is reduced in the case of a high formation threshold, it is not reduced by the factor of ∼ 6 that is seen in the particle number (for α0 = 3). This is because the particles which form in the high threshold cases are, on average, more massive. Since a particle has a fixed size, and the regions out of which the particles form have higher densities for higher threshold cases, this result is expected. The combination of fewer particles which are, on average, more massive for higher threshold cases could have a significant impact on the cluster particle mass distribution. This is discussed briefly in the next Section. In the remainder of this Section, however, we will continue to focus on simulations with the fiducial value of the formation threshold because it is observationally motivated, and a full exploration of the formation threshold parameter space is beyond the scope of this paper. Cluster Properties ------------------ ![image](ECMF_evolution.png) [fig:ECMFevolution] We now move to discuss cluster properties in more detail with a focus on mass distributions and stellar content. In Figure [fig:ECMFevolution], we show representative snapshots of the normalized cluster particle mass function at 2, 3, and 4 Myr for the simulation with α0 = 3. The other simulations show similar evolution, and a comparison between simulations is made below. The early evolution is dominated by low mass particles with the maximum particle mass reaching 104 M⊙ by 2 Myr. While massive particles are present, the majority have masses on the order of Solar masses. As time progresses, as shown in the middle panel of Figure [fig:ECMFevolution], the distribution shifts to higher masses because of both ongoing gas accretion and particle mergers. This same trend is evident in moving from 3 to 4 Myr. The highest mass particles have grown to approximately 105 M⊙. The lowest bins have been depopulated, resulting in a turnover in the distribution just below 100 M⊙. ![image](ECMF_boundedness.png) [fig:ECMFboundedness] ![image](star_reservoir_comparison.png) [fig:starreservoir] We compare the final total (ie. stellar content and gas reservoir) cluster particle mass distributions across simulations in Figure [fig:ECMFboundedness]. In order to compare to the results presented in, who measured the clump mass function in W43 using JCMT, we fit the distributions with a line for masses greater than log(M) > 2.4 M⊙. We discuss the observational results and the connection to our simulations in Section [clumpobs]. The resulting slopes are shown in the upper right hand side of each panel. The fits range from -1.49 to -1.73 for simulations with an α0 of 1 and 5, respectively. The slope for all combined data, shown in the lower right panel, is -1.67. There is a slight trend towards steeper slopes as α0 increases. Since the lower α0 clouds have the most massive particles, more mass in cluster particles as a whole, and a smaller number of particles, this trend is easily understood. In other words, the mass distributions shift to higher masses as α0 decreases, which manifests as a steepening slope when fitting over a fixed range. The mass of individual cluster particles is of two types: the cumulative mass distribution of stars as determined by generations of random sampling of the IMF from the available gas reservoirs, and the current gas reservoir. The mass in each type depends on several factors. Firstly, the age of the particle determines how long it has been forming stars suggesting that older clusters may, on average, contain a larger fraction of mass in stars relative to younger clusters. Secondly, the accretion histories of individual clusters affect the division of mass. As mentioned in Section [methods], any accreted mass is added directly to the gas reservoir. Therefore, if a cluster is undergoing strong accretion, the gas reservoir may dominate over the stellar mass. ![image](star_reservoir_vs_age.png) [fig:reservoirage] We examine the division of particle mass into stars and the gas reservoir in Figure [fig:starreservoir]. We plot the gas mass fraction, defined as the ratio of the gas reservoir to the total mass, for each particle versus the total particle mass. The line at a gas mass fraction of 0.5 delineates clusters which are gas dominated from clusters which are stellar dominated. We show the gas mass fractions at 2 Myr, on the left, and 4 Myr, on the right. We only show three simulations, those with α0 of 0.5, 2, and 3, for readability. Figure [fig:starreservoir] shows that the majority of cluster particles are still dominated by the gas reservoir at 4 Myr. These clusters are either undergoing strong accretion, or have not had enough time to convert their mass into stars. High gas mass fractions within clusters at late times appears to contradict observations which suggest that massive clusters are completely devoid of gas by 4 Myr. Our subgrid model does not allow for gas expulsion from cluster sink particles which may explain the high gas mass fractions still present at late times. Even without gas expulsion from cluster particles, however, radiative feedback could conceivably halt further gas accretion which would naturally produce lower gas mass fractions at late times. Since this is not observed, it suggests that radiative feedback is not the only process responsible for producing gas free clusters by ∼ 4 Myr. Stellar winds and protostellar jets may play a role in clearing young, star-forming regions of gas. ![image](IMF_all.png) [fig:IMF] c\|c\|c\|c\|c\|c\| & & 0.5 & 1 & 2 & 3 & 5 & 3.27 × 105 & 2.22 × 105 & 1.78 × 105 & 1.57 × 105 & 3.26 × 104 & 368 & 485 & 313 & 290 & 318 & 1.50 × 105 & 1.13 × 105 & 8.58 × 104 & 8.70 × 104 & 1.26 × 104 & 144 & 107 & 103 & 91 & 97 [table:mass] [] The overall spread in gas mass fractions shows differences among the simulations. The unbound simulation (ie. virial parameter of 5) contains the largest number of stellar dominated clusters. This is likely tied to the dynamical histories of individual particles. The higher gas velocities present in the high α0 simulations can result in cluster particles drifting away from their local gas sources, thereby shutting off future accretion and producing stellar dominated clusters. This is less pronounced in the lower α0 simulations because of the global collapse of both gas and particles which facilitates ongoing accretion. The progression from a gas to a stellar dominated cluster particle is inherently a time driven process as cluster particles convert their mass into stars. This would suggest that the oldest clusters should have the lowest gas mass fractions. This may not be case, however, because ongoing accretion can supply the clusters with a fresh gas supply. To visualize this interplay, we plot the gas mass fraction as a function of cluster age, instead of mass, in Figure [fig:reservoirage]. We show the results from two simulations, with α0 values of 1 and 3, to show the difference between bound and unbound simulations. We plot the gas mass fractions of all particles present at 2, 3, and 4 Myr. We see from Figure [fig:reservoirage] that, generally speaking, older cluster particles have lower gas mass fractions (ie. a higher fraction of stars) than younger clusters. We do, however, see significant outliers with some old clusters having high gas mass fractions. This highlights the importance of the local environment on star formation, suggesting that, even within one GMC, there can be distinct subclustered regions which can have very different star formation histories in comparison to their neighbours. Stellar Content --------------- ![image](SFR_vs_time_smoothed.png) [fig:SFRall] The stellar mass distribution within individual cluster particles is partially determined by the subgrid model for star formation. A sufficiently massive particle with a large reservoir of mass to draw from will produce an IMF which is fully sampled. From, particles with masses of ∼ 5000 M⊙ are sufficiently massive to form O stars. For smaller clusters, however, the amount of mass available at sampling may not be sufficient to form massive stars. For this reason, we examine the stellar mass distribution within individual cluster particles, as well as the total cloud, in Figure [fig:IMF]. We show the stellar mass distribution in both the maximum mass cluster particle, in the left panel, and the median mass particle, in the middle panel, to illustrate the variations between different star forming regions. The right hand panel of Figure [fig:IMF] shows the stellar mass distribution of the entire cloud. For reference, the mass of the the maximum and median mass cluster for all simulations is shown in Table [table:mass]. ![image](age_spread_all.png) [fig:age] As expected, the most massive particles show a fully sampled IMF with stars covering the entire mass range. There is little cluster to cluster variation among the most massive particles. We see markedly different distributions for the median mass clusters, however, which vary in mass between 290 to 485 M⊙. Because these clusters have significantly smaller masses, they rarely form stars with masses greater than ∼ 5 M⊙. Moreover, the distributions show stronger variations among clusters. Because these particles form a small number of stars, the effects of randomly sampling an IMF become more apparent. Overall, these distributions highlight the importance that the accretion histories play in subclustered regions and how this impacts the type of stars that are able to form. Star Formation Rates -------------------- We conclude this Section with a discussion of the star formation rates within entire clouds and individual cluster particles, and relate this to the resulting stellar age spreads. First, in Figure [fig:SFRall], we plot the total star formation rates from our simulations. As a consequence of our subgrid model for star formation, which only forms new stars at prescribed time intervals after a particle is formed, there are large, instantaneous jumps in the star formation rate. Since all particles do not form at the same time, the times at which particles sample the IMF are staggered. This means that, in a given timestep, there may be no particles which form new stars resulting in a SFR of zero. In other timesteps, the opposite may be true resulting in a large, instantaneous SFR. To aid in readability, the SFR curves presented in Figure [fig:SFRall] have been smoothed using a sliding average window. While this helps significantly in interpreting the Figure, there are still some spikes in the SFR that remain. The curves show qualitatively similar trends across simulations. There is a rapid onset of star formation, beginning around 0.5 Myr, which rises up to a roughly constant value. We do not see a turnover in the SFR. We do not expect to see a complete halting of star formation as a consequence of our subgrid model in which dense gas is assumed to remain bound within the gravitational potential well of the cluster. As shown in, however, the SFR will decrease for particles which are no longer accreting. This suggests that radiative feedback is not sufficient to halt the accretion onto a large number of particles and create a turn over in the SFR. At late times, the variation in the SFR increases. This is likely due to the reason discussed above, namely that the formation of new stars in cluster particles is staggered. The variation becomes more pronounced at late times because there are more particles overall, and the particle masses increase as the simulation evolves meaning there is more mass out of which new stars can form. Additionally, the particle accretion rates become more variable at late times due to large velocities (see Figure [fig:alpha]) which can either disconnect a particle from its host filament or move it to a gas rich region where strong accretion is possible. Since the SFR is intimately tied to the accretion rate of particles, this is likely contributing to the variable SFR at late times. The final, approximately constant, SFRs cover roughly a range of 4 × 10− 2 to 2 × 10− 1 M⊙ ⋅ yr− 1, with variations of greater than an order of magnitude seen in some simulations. The evolution of the global SFR is directly tied to the resulting stellar age distributions within the simulations. We compare the age spreads in the entire cloud, as well as the maximum mass and median mass cluster particle, in Figure [fig:age]. The right panel, which shows the age spread of all stars in the respective simulations, mirrors the global SFR. The majority of stars are newly formed with a rapid decline in the amount of older stars. We do not see a turn over in the distribution since star formation is still occurring at high rates. The age distributions within the maximum mass clusters are also strongly peaked towards young ages. This is not surprising since these clusters have a significant gas reservoir to continue forming stars. The median mass clusters show more variations between simulations. Some clusters, the median cluster in the simulation α0 = 1 for example, are newly formed and as a consequence the age distribution is narrow, ∼ 1 Myr, and peaked at young ages. The older median mass clusters, like in the simulation with α0 = 2, show much broader distributions with age spreads up to ∼ 3 Myr which also have a turnover at intermediate ages. This shows that the formation of a stellar population in a molecular cloud is not a uniform process with distinct regions forming stars at the same time. Instead, different subcluster regions have separate star formation histories tied to their local environments. ![image](SFR_m1e6_all.png) [fig:SFRall] Comparison to Observations ========================== Mass Distributions ------------------ We refer back to Figure [fig:ECMFboundedness] and discuss the connection to observed clump mass function. We focus on the recent results presented in who measured masses for 1029 clumps in W43. We have chosen this dataset in particular because W43 is one of the most massive cloud complexes in the Milky Way, with a total mass of ∼ 7 × 106 M⊙ of which ∼ 8.4 × 106 M⊙ is contained in dense clumps, providing a good comparison to our simulated dataset. The results presented in for W43 show a clump mass distribution with a high mass slope of -1.87 ± 0.05, which is consistent with the results presented in. This slope is inconsistent with our results, which range from -1.49 to -1.73, and suggests that our simulations have either overproduced very massive particles, under produced low mass particles, or is a combination of both effects. There are examples of clump mass functions which have smaller slopes (eg. and discussions therein), but these studies have different cut off masses for the fits making comparisons difficult to make. Moreover, measured the slope of the clump mass function in NGC 7538 and M17 and found inconsistent results between the two regions, suggesting that there may not be a universal clump mass function. As a final remark, we note that the slopes of our particle mass functions are consistent with those measured for entire GMC’s by, who found a slope of -1.50 ± 0.36 The above discussion has focused on simulations which use a particle formation threshold of 104 cm− 3. As discussed in Section [thresholds], we have also completed a subset of simulations which increase and decrease the formation thresholds by an order of magnitude. In the discussion surrounding Figure [fig:nstarsnthresh], we noted that increasing the formation threshold significantly reduces the number of cluster particles while increasing the average particle mass. The will clearly impact the resulting particle mass distributions. We find that the resulting high mass slope, analyzed in the same way as above, differs dramatically for the high formation threshold of 105 cm− 3. For an α0 of 0.5, the resulting slopes are -1.58, -1.54, and -1.05 for low, fiducial, and high thresholds, respectively. The corresponding slopes for the α0 = 3 simulations are -1.81, -1.7, and -1.09. The combination of less particles and higher average masses, in the case of a high formation threshold, results in significantly shallower slopes which are less consistent with observations. This provides further justification for the choice of 104 cm− 3 as our formation threshold. Not only is this value observationally motivated, it results in particle mass distributions which are closer to measured values. Star Formation Rates in Molecular Clouds ---------------------------------------- The SFRs discussed earlier were measured globally for each simulation. In order to make comparisons to observations, we plot the SFR of individual cluster particles in Figure [fig:SFRall]. We again only show the simulations with an α0 of 0.5, in red, 2, in blue, and 5, in purple for readability. We have over plotted the observational results from in black crosses. These observations used 2MASS data to catalog the young stellar objects (YSOs) in 11 local star forming regions within 450 pc of the sun. The observed clouds are the Pipe Nebula, the Ophiuchus cloud, the Lupus cloud complex, Taurus, Perseus, the California clouds, RCrA, and the Orion cloud complex. How measurements of the YSO content in these clouds are used to estimate SFRs is described below. ![image](merged_fraction.png) [fig:merging] The black triangles in Figure [fig:SFRall] represent the dataset from who also measured SFRs in local regions, some of which are the same regions as in. The solid line represents the SFR-mass relation measured for extragalactic sources by. These observations measured the IR and HCN luminosity in the star forming cores of nearby galaxies, and used these measurements as proxies for the SFR. The resulting SFR-mass relation is given by, SFR(M⊙yr− 1) ∼ 1.2 × 10− 8Mdense(M⊙) where Mdense is the mass of dense gas (n > 104 cm− 3) traced by IR emission. We have extrapolated this relation to molecular cloud masses in Figure [fig:SFRall]. In order to make a comparison to the observations, we have measured the SFR in each cluster particle in an identical manner to. The following formula was used to estimate the SFR in each region, $$SFR = \frac{N\bar{M}}{\tau\_{YSO}} = 0.25N\times10^{-6} M\_{\odot} yr^{-1} \label{SFRmass}$$ where N is the total number of young stellar objects (YSOs) in each region, M̄ is the median mass of a YSO in M⊙, and τYSO is the average lifetime of a YSO. The median mass, M̄, was taken to be 0.5 M⊙, consistent with the IMF and therefore consistent with the stellar distributions within our cluster particles. An average YSO lifetime of 2 Myr was assumed. While we do not have prestellar evolution in our subgrid model, and therefore do not know the exact number of YSOs, we do know the formation times for all stars. When solving for the SFR via equation [SFRmass], we therefore take N to be the cumulative number of stars formed within the last 2 Myr to make the comparison to observations as consistent as possible. At early times, the SFRs agree well with the measured values, in particular the values. As time progresses, the SFRs in our simulations shift to higher values while following the same slope. This is consistent with the global picture presented earlier, where the global SFR rises rapidly to a roughly constant value producing age spreads which are heavily peaked at early times. While low to intermediate masses fall roughly in the range of the observations, the disparity between the observations and our results is particularly pronounced at high masses. At late times, the SFRs in the most massive cluster particles approach an order of magnitude greater than the observations. Since these massive clusters contain the most stars, this again suggests that radiative feedback is not the sole form of feedback necessary to halt the star forming process. The high SFRs of massive clusters in our simulations are not just a product of the local environment and accretion rate, but also a product of subcluster merging which can provide these massive clusters with fresh supplies of gas to continue forming stars at high rates. We show this in Figure [fig:merging] which plots the fraction of mass obtained via mergers. The percentage of surviving clusters which underwent at least one merging event varies from 9.6% to 15.2% for α0 values of 5 and 2, respectively, with no discernible trend as α0 changes. All values are shown in the legend of Figure [fig:merging]. Figure [fig:merging] shows that merging is not a significant form of mass accretion for the majority of cluster particles. This is particularly true for low mass clusters which grow solely via gas accretion, as shown by the large grouping of particles at zero. Above roughly 103 M⊙, however, merging becomes a significant source of mass accretion with some particles obtaining more than half of their total mass via merging. This highlights the role that subcluster merging plays in the build up of young stellar clusters. High mass clusters in particular are not formed solely via gravitational fragmentation but rather through the merging of multiple, lower mass subclusters. Based on our simulations, we can conclude than subcluster merging plays a significant role in the early development of clusters with mergers producing large clusters within ∼ 4 Myr. Conclusions =========== We have simulated a suite of turbulent, 106 M⊙ GMCs using FLASH which have initial virial parameters in the range of 0.5 to 5. Using these, we explore the role that radiative feedback and gravitational boundedness play in the formation of star clusters and the early evolution ( < 5 Myr) of a GMC. To do this, we have used sink particles to represent star clusters and implemented a subgrid model to populate clusters with stars gradually over time via randomly sampling the IMF. The radiative output of these evolving clusters was then coupled to a raytracing scheme to treat the radiative transfer. The main conclusions of this work are as follows: * The initial virial parameter, α0, strongly influences the dense clump formation efficiency, εcl, and the star formation efficiency, εsf, during the early evolution of the GMCs. Models that are more bound (ie. lower α0) have higher efficiencies in comparison to unbound models despite a delayed onset of cluster formation. This is tied to the global gas evolution of the cloud which becomes strongly centrally condensed in bound clouds, which can prolong gas accretion onto newly formed clusters, and the strong mass loss from unbound clouds. The final εcl values range from 37% to 71% and the final star formation efficiencies range from 19% to 33% for virial parameters of 5 and 0.5, respectively. * Radiative feedback does not strongly affect formation efficiencies over the range of time simulated. The inclusion of radiative feedback reduced εcl by only ∼ 1%, except in the case of an α0 = 3, in which εcl was reduced from 52% to 41%. This reduction was not due to radiation unbinding gas and driving outflows, but instead by suppressing cluster formation and accretion locally via heating and ionization. * The strength of radiative feedback is enhanced if a higher formation threshold for cluster particle formation is used. We have chosen a fiducial value of 104 cm− 3 and explored using thresholds that are an order of magnitude smaller and larger than this value for a bound (α0 = 0.5) and an unbound (α0 = 5) cloud. In both cases, the higher threshold simulations showed larger differences in εcl between runs with and without feedback. A higher threshold of 105 cm− 3 for the α0 = 0.5 model produced a difference in εcl between the HD and RHD simulations of 21% compared to 2% in the fiducial threshold simulations. The difference between the HD and RHD simulations for the α0 = 3 model was 10% compared to 8% with the fiducial threshold. Since the average density of the cloud is naturally higher when clusters form in the higher threshold case, this shows that the strength of radiative feedback can be enhanced in clouds with higher density. This may have important implications for Globular Cluster formation which is thought to occur in high density, high pressure environments. * Cluster properties are sensitive to the initial virial parameter of the cloud out of which they form. The total number of clusters formed increases as α0 increases, with the final number of clusters being 52 for the most bound simulation and 189 for the most unbound simulation. This, in combination with the result that bound clouds produce more massive clusters than unbound clouds, produces increasingly steeper cluster mass distributions as the initial virial parameter increases. This is a product of the mass distributions shifting to higher masses as α0 decreases, resulting in a steeper slope when fitting over a fixed range. The high mass slopes of our cluster mass distributions range from -1.54 to -1.73 which is only slightly shallower than the mass function presented in who measured a value of -1.87 for clumps in W43. We also find that there are more gas poor, star rich cluster particles formed in unbound simulations compared to bound simulations. While we do not have a prescription for mass loss from cluster particles, this highlights the role that dynamics plays in the early evolution of clusters. The gas poor clusters in bound simulations have been ejected from their host filaments effectively halting further gas accretion. In contrast, clusters formed in bound clusters are centrally condensed and continue to accrete, resulting in gas rich clusters. * We have compared the SFRs of our cluster particles to observations of local star forming regions. The SFRs in our simulations agree with observations at early times but are higher at late times, particularly for the most massive cluster particles. This is related to the global SFRs in our models which show a sharp rise in star formation which levels out to a constant value. We see no evidence for a turnover in the global SFRs which suggests that, at least in the early evolution of a GMC, radiative feedback alone is not responsible for halting star formation and dispersing the cloud. Other forms of feedback, such as protostellar jets, stellar winds, and supernovae, should be included to fully understand the cluster formation process. We plan to extend our work by exploring different cloud properties such as magnetic field strengths and metallicity. We will also be exploring radiative feedback in cloud with masses 104 − 7 M⊙. The highest mass clouds will probe the Supergiant Molecular Cloud regime in which Globular Cluster mass objects may form. We will also model clouds with a range of initial densities because, as our results show, the density structure of a cloud can alter the strength of radiative feedback. Acknowledgments =============== We thank an anonymous referee for a useful report which improved the quality of this paper. We thank Ralf Klessen and Mordecai Maclow for their invaluable input on numerics and theory, and Henrik Beuther and Jouni Kainulainen for interesting discussions of observations. C.S.H. and R.E.P gratefully acknowledge KITP, Santa Barbara, for their support and hospitality while participating as an affiliate (C.S.H) and an invited participant (R.E.P.) in the program “Gravity’s Loyal Opposition: The Physics of Star Formation and Feedback”. We thank Norm Murray, Andre Kravstov, and Christoph Federrath for interesting discussions during this time about simulations and theory. C.S.H. and R.E.P. also thank MPIA and the Institut für Theoretische Astrophysik (ITA) in the Zentrum für Astronomie der Universität Heidelberg for their generous support during R.E.P’s sabbatical leave (2015/16) and C.S.H.’s extended visit (Oct. - Nov., 2015). R.E.P. and W.E.H. are supported by Discovery Grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada. C.S.H. acknowledges financial support provided by the Natural Sciences and Engineering Research Council (NSERC) through a Postgraduate scholarship. The FLASH code was in part developed by the DOE-supported Alliances Center for Astrophysical Thermonuclear Flashes (ASCI) at the University of Chicago. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca) and Compute/Calcul Canada. [lastpage] --- 1. E-mail: [email protected][↩](#fnref1) Simulating radiative feedback and star cluster formation in GMCs: I. Dependence on gravitational boundedness ============================================================================================================ ### 16 June 2016 [firstpage] Radiative feedback is an important consequence of cluster formation in Giant Molecular Clouds (GMCs) in which newly formed clusters heat and ionize their surrounding gas. The process of cluster formation, and the role of radiative feedback, has not been fully explored in different GMC environments. We present a suite of simulations which explore how the initial gravitational boundedness, and radiative feedback, affect cluster formation. We model the early evolution ( < 5 Myr) of turbulent, 106 M⊙ clouds with virial parameters ranging from 0.5 to 5. To model cluster formation, we use cluster sink particles, coupled to a raytracing scheme, and a custom subgrid model which populates a cluster via sampling an IMF with an efficiency of 20% per freefall time. We find that radiative feedback only decreases the cluster particle formation efficiency by a few percent. The initial virial parameter plays a much stronger role in limiting cluster formation, with a spread of cluster formation efficiencies of 37% to 71% for the most unbound to the most bound model. The total number of clusters increases while the maximum mass cluster decreases with an increasing initial virial parameter, resulting in steeper mass distributions. The star formation rates in our cluster particles are initially consistent with observations but rise to higher values at late times. This suggests that radiative feedback alone is not responsible for dispersing a GMC over the first 5 Myr of cluster formation. Introduction ============ The modern paradigm for star cluster formation suggests that clusters are born from dense clumps (n > 104 cm− 3) which form in supersonically turbulent, filamentary molecular clouds. Recent observations show that young stellar clusters tend to form at the intersection of filaments in regions that are fed by higher than average accretion rates. These overdense regions may then fragment further, resulting in highly subclustered objects which undergo mergers. The question of how the high accretion flow onto dense, star-forming clumps is halted is of particular importance. The mass accretion history of these objects has implications for the final cluster mass and, hence, the observed cluster mass function. Moreover, the overall conversion of molecular gas to stars is inefficient. Understanding what processes limit the star formation efficiency in molecular clouds is key to a complete star formation theory as well as underpinning all discussions of simulations and theories of galaxy formation and evolution. Several mechanisms have been proposed to explain the low star formation efficiency observed in molecular clouds. For example, turbulent velocity fields have been shown to lower the overall star formation rate per freefall time. This cannot be the sole mechanism at work, however, since, given enough time, all of the gas will be converted to stars. The added pressure support via magnetic fields can also lower the overall star formation efficiency. While magnetic fields may play an important role in limiting cloud fragmentation, they also contribute to the dispersal of gas within individual star forming cores by means of hydromagnetic bipolar outflows and jets that are associated with young stars of all masses. The process of feedback from newly-formed stars can both limit the star formation efficiency and disperse the surrounding gas, thereby halting the star formation process. Feedback comes in various forms: protostellar jets, stellar winds, ionization and heating of the gas, and radiation pressure. Radiative feedback has been suggested as being the most important form of feedback in clusters which host massive star formation. The ionizing radiation released from newly-formed stars heats the gas to approximately 10,000 K and can drive the expansion of massive HII regions. The direct input of momentum via high energy photons interacting with surrounding dust grains can also drive strong outflows. Previous studies of radiative feedback on both small and large spatial scales have shown that the star formation efficiency can be significantly reduced. Studies which model the formation of individual stars, however, are typically limited to the clump scale or low mass molecular clouds. These studies therefore neglect the impact of radiative feedback on the global structure of massive GMC’s. Since cluster formation can be broadly distributed throughout a cloud, cluster interactions over the entire range of the GMC must be considered. There have been studies of radiative feedback in massive GMCs (106 M⊙). Due to numerical constraints, the supergiant molecular cloud regime ( > 106 M⊙) has not been examined nor do most models follow the cloud’s evolution up to the point of dispersal which is crucial to placing constraints on the lifetime of molecular clouds. Recent results suggest that radiation feedback alone is not effective in accounting for low star formation rates in galaxy formation simulations. The lifetimes of molecular clouds, and the processes responsible for destroying them, are challenging to measure. Estimates of GMC lifetimes range from a single freefall time, τff, up to tens of freefall times. Recent observations of the deuterium fraction in both massive, starless cores and Infrared Dark Clouds (IRDCs) indicates ages within a range of 6 to 10 local freefall times, corresponding to ∼ 3 Myr for IRDCs. We note that direct measurements of the lifetimes of molecular clouds out to a redshift of 4 are now possible with ALMA (Atacama Large Millimeter Array) and the method outlined in. Theoretical estimates suggest that radiative feedback, in particular radiation pressure, is the most important physical process responsible for destroying massive GMC’s but numerical simulations of star formation in GMC’s which include radiative feedback are required to validate these claims. The role of radiative feedback in different molecular cloud environments has also not been explored in detail. The observed clouds in the Milky Way and extragalactic sources show a wide range of properties. More specifically, molecular clouds have been observed to have a range of virial parameters, defined as the ratio of the cloud’s kinetic energy to gravitational potential energy. This ratio may vary from 0.5 up to 10 with the mean being approximately 1. Simulations have shown that dense, self-gravitating clumps can still fragment out of clouds that are globally unbound. Furthermore, the star formation efficiency is naturally lowered in unbound clouds. Radiative feedback may play a stronger role in lowering the star formation efficiency in unbound versus bound clouds since the added energy and momentum can more easily disperse the surrounding, unbound gas. To investigate these questions, we have chosen to simulate the formation of young stellar clusters in molecular clouds with a range of properties while including detailed radiative feedback. As described above, we focus on varying the initial virial parameter in turbulent, 106 M⊙ GMCs. We are focusing on this mass because the majority of star formation in the Milky Way is hosted in these clouds. In order to simulate the long term evolution of these clouds, we use sink particles to represent young clusters and combine this with a subgrid model to track star formation within the clusters. In Section 2, we describe the details of this subgrid model and the initial conditions used in our simulations. In Section 3, we describe the evolution of clouds with varying initial virial parameters, discuss the role of radiative feedback, and compare cluster properties across clouds. We find that radiative feedback has only a slight effect on the star formation efficiency. We find, in fact, that efficiencies are dominated by the gravitational binding of the cloud and that virial and sub virial initial cloud models do not show the low efficiencies that observations demand. In Section 4, we discuss how our results compare to recent observations of local star forming regions, followed by our conclusions in the final section. Numerical Methods ================= We perform numerical simulations using version 2.5 of the hydrodynamical code FLASH. FLASH is an adaptive mesh refinement code (AMR) which integrates the compressible gas dynamic equations on a Eulerian grid and includes modules to treat self-gravity, radiative transfer, and star formation. We include cooling via molecular lines and dust, and use a simple ideal gas law equation of state with an adiabatic index of 1.67. Radiative transfer is handled via a hybrid-characteristics raytracing scheme developed by and adapted for star formation simulations by. This scheme follows the propagation of both ionizing and non-ionizing radiation and makes use of the DORIC routines. The DORIC package is an iterative scheme used to calculate the ionization, heating, and cooling rates of a large number of species. For simplicity, however, hydrogen is used as the only gas component. have shown that the code can accurately reproduce the analytic solution for the expansion of a D-type ionization front from, and have shown the code accurately treats R-type fronts. The opacity to the non-ionizing radiation is represented by the Planck mean opacities from the dust model of. The Planck mean opacities are used because the raytracer has no frequency dependence apart from ionizing versus non-ionizing radiation. Radiation pressure is included by adopting a single UV opacity of 775 cm2 g− 1 from. The associated radiative force per unit mass is given by, $$F = \frac{L}{c}\frac{e^{-\tau}}{4\pi r^2}$$ where L is the source luminosity, c is the speed of light, and τ and r are the optical depth and distance between the source and the target cell. To represent the formation of star clusters, we make use of Lagragian sink particles (Banerjee et al. 2009; Federrath et al. 2010). In order to form a particle, a region of gas in the simulation volume must meet the following conditions: * At the highest level of refinement * The divergence is less than zero (ie. converging) * At a local gravitational minimum * The region is Jeans unstable * The region is gravitationally bound * Not within the radius of another particle. These particles were designed to represent stars but the above conditions also model the formation of a dense clump which hosts cluster formation. In order to examine the impact of radiative feedback from stellar clusters, we require a subgrid model for the radiative output of a cluster as it evolves over time. Below, we provide a brief summary of this model. A more detailed description can be found in Howard et al. (2014). Subgrid model for cluster formation: Cluster sinks -------------------------------------------------- One of the most important aspects of radiative feedback of a young forming cluster on its surrounding host GMC is the shutting off of the accretion flow into the cluster forming region. A cluster must therefore be assigned the correct, combined radiative output of all its member stars as star formation proceeds. Here, we provide a brief description of the subgrid model we use to form stars within a cluster particle. In order for cluster formation to begin, the host clump must reach sufficient density in order to fragment and collapse. The threshold density for cluster formation has been observationally measured by who found that the transition from starless to star forming clumps, in local star forming regions, occurs at a density of ∼ 104 cm− 3. This number has also been quoted by other authors who found that the SFRs in molecular clouds are well correlated to the dense ( > 104 cm− 3) gas mass. More recent work by, aimed at producing probability density functions for gas density from column density observations, found a density threshold of 5 × 103 cm− 3. Analytical models for the density threshold, which also account for environmental effects, find a threshold of 1.5 × 104 cm− 3 when adopting Solar neighbourhood parameters. We have therefore chosen to use a fiducial density threshold for cluster formation of 104 cm− 3. We report on several simulations in which we vary this value by an order of magnitude. Once this critical density threshold is reached, as well as the other conditions outlined above, a cluster, represented by a sink particle in this work, forms. Star formation then proceeds within this cluster as gas accretes onto this dense region. With this general framework in mind, we use our subgrid model for the formation, radiative feedback, and evolution of a cluster that addresses how the original gas reservoir (ie. clump) is divided into stars over time and how accreted gas is handled. We refer the reader to that paper for the details, which we now briefly summarize. To treat the conversion of reservoir mass into stars, we divide it into main sequence stars at a prescribed efficiency according to an IMF. We use a star formation efficiency of 20% per freefall time, where the freefall time is given by $$t\_{ff} = \sqrt{\frac{3\pi}{32 G n \mu m\_{H}}}$$ where n is the number density, μ is the mean molecular weight, and mH is the mass of hydrogen. As discussed above, our fiducial value for n is taken to be 104 cm− 3. The resulting freefall time is 0.36 Myr. We use the Chabrier IMF, given by $$\xi(log \ m) = \begin{cases} 0.093\times exp\{\frac{-(log \ m - log \ 0.2)^2}{2\times(0.55)^2}\}, & \mbox{\ m $\leq$ 1 \ M$\_{\odot}$ } \\ 0.041m^{-1.35 \pm 0.3}, & \mbox{\ m $>$ 1 \ M$\_{\odot}$}, \end{cases}$$ as the probability distribution from which we draw our stars. We chose to sample the Chabrier IMF every tenth of a freefall time, or 3.6 × 104 years. As shown in, this sampling frequency allows cluster properties to evolve smoothly over time while still accurately reproducing the IMF from which the stars are drawn. The masses of all stars formed in a cluster are recorded, meaning that its stellar content is known at all times. Using the analytic fits for the luminosity of main sequence stars provided by, we know the luminosity of each star within the cluster. The total luminosity, and ionizing photon rate, of each cluster is the sum of the stellar components. These quantities are then used by the raytracer to treat the radiative transfer. Gas accreted by a cluster is simply added to the gaseous reservoir. Since we are sampling an IMF in order to get our final stellar masses, we do not need to treat accretion onto individual stars. The accreted gas can then be used to form new stars at a later time. We do not include mass loss from the cluster particle because we assume that the reservoir gas remains gravitationally bound to the cluster over long timescales, even in the presence of stellar feedback. Earlier simulations by found that the inclusion of radiative feedback was not sufficient to unbind the majority of the clump mass. The reason for this is collimated ionized outflows which are released into low density regions perpendicular to the dense filaments out of which the stars form. We therefore assume our gas reservoir is in a similarly filamentary state, preventing its disruption via stellar feedback. Furthermore, recent numerical works, both with and without feedback mechanisms, and observational studies suggest that cluster formation proceeds via gas exhaustion rather than gas expulsion. This further justifies the use of cluster particles which do not lose mass. We note that, when performing a simulation, we apply a mass threshold below which particles do not radiate. This was implemented in order to reduce the computational time. The raytracing scheme is computationally expensive so by allowing only massive particles to radiate, the total time spent during the raytracing step can be greatly reduced. We have chosen a threshold of 1000 M⊙ contained in stars. Note that this is not the total mass of the cluster particle, but rather the stellar mass. We have chosen this threshold because it is approximately the mass at which the first O stars are expected to appear. We have verified that, at any given time, particles above this mass dominate the total luminosity budget so we are therefore confident that the application of a mass threshold should not impact our results significantly. We have performed tests of the raytracing scheme similar to, except with our cluster sink particles, as described at the beginning of this Section. These were low resolution simulations, with 128 grid cells along each simulation axis, where a cluster sink particle was placed at the center of a uniform medium with a number density of 103 cm− 3. We allowed the cluster particle to sample the IMF once at the beginning of the simulation, after which no new stars were added to the cluster, and compared the resulting HII region to the analytical expression for the expansion of a D-type ionization front from. We find that, during expansion, the size of the HII region does not differ by more than 15% from the analytical solution. This error is reduced to less than 9% if we instead compare to the analytical expression presented in, as was done in the radiative transfer code comparison project STARBENCH. The addition of radiation pressure produces an HII region that is ∼ 10% larger compared to the test without radiation pressure. Additionally, the central density inside the HII region is significantly lower when radiation pressure is included, differing by approximately two orders of magnitude compared to the test without radiation pressure. ### Cluster Sink Mergers ![image](mach_numbers.png) [fig:mach] The sink particle routines in FLASH allow for particle mergers. In order for a merger to occur, several conditions must be met. Firstly, the particles must be closer than an accretion radius, defined to be 2.5 cells at the highest level of refinement. Secondly, the particles must be approaching one another. Lastly, the particles must be gravitationally bound to one another. More specifically, the total gravitational potential energy of the particle pair, at a distance of an accretion radius, must be greater than the total kinetic energy. When all of these conditions are met, the smaller mass particle is merged to the more massive particle. The new position is taken to be the center of mass position and the resulting particle velocity is determined via momentum conservation. We have made custom modifications to the merging routines in order to better represent the merging of cluster sink particles. As mentioned above, cluster sink particles have their mass divided between fully-formed stars and the gaseous reservoir. When a cluster particle merger occurs, we combine the reservoir masses and the total stellar mass. Since we track the masses of all stars formed in each particle, the total stellar content remains the same after merging. We emphasize that, while we will be using merger to describe the joining of two cluster sink particles, this differs from the dynamical mergers associated with fully formed clusters that are devoid of gas. In our context, a merger is more aptly defined as the coalescence of a central stellar cluster as well as the envelope of surrounding gas. Recent studies indicate that gas poor clusters may form before feedback starts acting since the local freefall time in these dense regions is the highest. This means that the stellar cluster within our particles may be spatially segregated from the surrounding gas reservoir. Therefore, the merger of the stellar component in our particles may be thought of in the traditional way, namely a dynamical merger of gas poor stellar clusters. The reservoir can be interpreted as the cocoon of gas which surrounds the central cluster, which is also joined during the merger. We do not resolve the structure of the gas reservoir, however, so we cannot distinguish between the merger of gas rich clusters and gas poor clusters with a surrounding envelope of gas. \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:3d] Initial Conditions ------------------ We simulate the formation of stellar subclusters in collapsing GMC’s with different initial properties and examine the role of radiative feedback in these environments. The clouds have the same initial average of n = 100 cm− 3 and masses of 106 M⊙. The radius of all clouds is 33.8 pc, corresponding to a minimum cell size of 0.13 pc. This results in a particle accretion radius of 0.325 pc (ie. 2.5 cells at the highest level of refinement). We have chosen an initial density profile which is uniform in the inner half of the cloud and decreasing as r− 3/2 in the outer half of the cloud. A quadratic fit is applied in the transition from uniform to decreasing density to ensure a continuous and smooth function. We use outflow conditions at the boundary of the simulation volume. The total mass in our simulations is therefore not conserved but can decrease over time as gas escapes the domain. This is particularly relevant to the discussion in the next Section. The cloud is overlaid with a Kolomogorov turbulent velocity spectrum. The turbulence is not driven after it is initially imposed and its strength is determined via the virial parameter. The virial parameter relates the total kinetic energy to the total gravitational potential energy in a cloud. It is defined as, $$\label{virial} \alpha = 2\frac{E\_{kin}}{\|E\_{grav}\|} \approx \frac{5 \sigma^2 R}{G M}$$ where Ekin is the total kinetic energy, Egrav is the total gravitational potential energy, σ is the velocity dispersion, R is the cloud radius, and M is the cloud mass. Equation [virial] is exact for a uniform density cloud. The velocity dispersion can be expressed as σ = Ma × cs where Ma is the Mach Number and cs is the sound speed, which is taken to be 2.3 × 105 cm/s at 10 K. For a uniform density cloud without magnetic fields, a virial parameter of 1 corresponds to virial equilibrium and α > 2 represents gravitationally unbound clouds. We use the first relation in Equation [virial] since the kinetic and gravitational potential energies are easily measurable in our simulations. We have chosen to examine initial virial parameters, defined as α0 henceforth, ranging from 0.5 to 5, which covers both bound, virialized, and unbound clouds. This range of virial parameters is within the range of observed GMC’s. The corresponding Mach Numbers can be found in Figure [fig:mach]. The models are evolved to ∼ 5 Myr, at which point supernovae are expected to inject large amounts of energy into the surrounding gas. Since the physics of Supernovae are not handled in the present code, we do not evolve the models past this point. Results ======= Global Evolution and Gas Properties ----------------------------------- \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:slicecoldensalpha1] \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:slicecoldensalpha3] \| \| \| --- \| \| image \| \| image \| \| image \| \| \| [fig:slicetemp] A three-dimensional view showing the evolution of the virialized simulation (α0 = 1) at three different times can be seen in Figure [fig:3d]. Three different density contours are plotted with densities ranging from 1.0 × 10− 23 to 3.0 × 10− 20 g ⋅ cm− 3. These specific densities were chosen to highlight the sparse, low density gas, the intermediate density filaments, and the dense cores which form primarily in the center of the simulation. The bounding box has a side length of 80.2 pc and the location of the cluster particles are overplotted with black circles. The red regions in Figure [fig:3d] represent gas which has an ionization fraction greater than 95%, highlighting the HII regions produced by the cluster particles. The first panel in Figure [fig:3d] shows that the gas quickly breaks up into filaments. A total of 17 cluster particles have formed by this time but have not produced enough stars to produce HII regions. The second panel, plotted at 2.68 Myr, still shows a filamentary network of gas but one which is more centrally condensed compared to the previous panel due to the global collapse of the gas. Dense cores are also now visible, scattered throughout the central region of the box where filaments intersect. A total of 84 particles are present at this time, some of which have produced enough stars to begin feeding back on their environment to produce small, localized HII regions. The total number of cluster particles remains roughly the same with a total of 88 particles at 3.65 Myr, shown in the bottom panel of Figure [fig:3d], but the stellar content in these clusters has grown significantly, allowing larger HII regions to form which cover a large fraction of the box. One region in particular has expanded outward from the central region into low density gas and extends to the edge of the simulation volume. To give a complementary view to Figure [fig:3d], we show density slices through the center of the z-axis and column density projections in Figure [fig:slicecoldensalpha1], for α0 = 1, and Figure [fig:slicecoldensalpha3], for α0 = 3. The corresponding temperature slices are in Figure [fig:slicetemp]. We show an example of a bound and unbound (α0 = 3) simulation illustrate the effects that initial boundedness plays on the evolution of the cloud. The locations of the cluster particles are projected onto the viewing plane in all images. Like the three-dimensional images discussed above, both clouds quickly break up into filaments by ∼ 0.5 Myr. Since the unbound cloud has a higher Mach number, the filaments are more pronounced with more low density gas filling the voids between the filaments. The temperature profiles look similar at this point, with cold 10 K gas in the center surrounded by warmer ∼ 300 K gas. There are visible shocks, however, in only the unbound simulation, along the periphery of the cloud. Comparing the two simulations at the earliest times also shows that only the unbound cloud has formed cluster particles. This can again be attributed to its higher Mach number, leading to stronger density enhancements which fragment into particles. The middle panels of Figure [fig:slicecoldensalpha1], [fig:slicecoldensalpha3], and [fig:slicetemp], plotted at ∼ 2.5 Myr, show that both simulations have produced many clusters, totalling 81 and 107 for the bound and unbound simulation, respectively. The clusters are grouped more heavily towards the centre of the volume for the bound run, indicative of collapsing gas resulting in higher central densities and a more centrally condensed grouping of clusters. An HII region is also visible in the middle panel of Figure [fig:slicetemp]. The corresponding density slice shows that the HII region has disrupted the filaments, effectively smearing them out to a more uniform medium with density enhancements around the edges. The final panel, shown at ∼ 4 Myr, shows more marked differences between the initially bound and unbound clouds. The bound cloud is now even more centrally condensed due to global collapse, with one dense central core from which filaments radiate outwards. A large grouping of clusters remains in the center, allowing further accretion. The outskirts are devoid of dense gas, which has allowed a large HII region to extend northwards, away from the central condensation, where there is less gas to shield these regions from the ionizing photons produced by massive stars in the central grouping. The unbound simulation, on the other hand, has dense gas pervading a large fraction of the box and lacks a dense, central core. The number of clusters has grown slightly for a total of 87 and 140 for the bound and unbound run, respectively. This suggests that the majority of cluster formation occurs during the early evolution of the clouds. ![image](mass_components.png) [fig:masscomponents] ![image](alpha_vs_time.png) [fig:alpha] The above visualizations provide a qualitative description of the evolution of a small subset of the completed simulations. A quantitative comparison showing both the gas and cluster particle evolution for all simulated clouds, with radiative feedback included, can be seen in Figure [fig:masscomponents]. The total mass in cluster particles is shown by the dashed lines, the total gas mass present in the simulation volume is shown by the dotted lines, and their sum is the shown by the solid lines. Since outflow boundary conditions are present, the solid lines do not remain constant at 106 M⊙ because gas can leave the simulation volume. The total amount of mass lost from the simulation depends heavily on α0. From Figure [fig:masscomponents], it can be seen that the initially bound runs lose a negligible amount of mass. The unbound runs, on the other hand, lose a significant amount. For example, the case with an initial virial parameter of 5 loses ∼ 30% of the total mass by the end of the simulation. Clearly α0 plays an important role in the long term evolution of the cloud. However, the virial parameter does not remain constant over the course of the simulation because the initial supersonic turbulence is damped via shocks. As the turbulence is damped, the velocity dispersion of the gas decreases which results in a decreasing α. The loss of high velocity gas from the simulation volume and the formation of cluster particles further modifies α. In Figure [fig:alpha], we plot the global virial parameter as a function of time on the left, and the contributions to α by the gas and cluster particles on the right. The global virial parameter, and the contributions from particles and gas, are expressed via, $$\alpha = 2\frac{E\_{kin}}{\|E\_{grav}\|} = 2\frac{E\_{kin,p}}{\|E\_{grav}\|} + 2\frac{E\_{kin,g}}{\|E\_{grav}\|} = \alpha\_{p} + \alpha\_{g}$$ where Ekin is the kinetic energy, Egrav is the gravitational potential energy, and the subscripts g and p refer to gas and particles. We see from Figure [fig:alpha] that α initially decreases as the turbulence damps. The rate at which α decreases is greater for unbound simulations. All simulations tend towards a virial state (ie. α = 1), and cross at approximately 2.3 Myr. The tendency towards a virial state is likely due to gravitational collapse after the decay of the initial turbulence which, in turn, drives enough internal turbulence to maintain α at 1. Because the virial parameters intersect at similar times, the simulations with higher α0 decrease more rapidly compared to lower α0 simulations. This is attributed to the turbulent decay time, τdec, which is given by τdec ∼ L/(Mcs), where L is the box size, M is the turbulent Mach number, and cs is the sound speed. Measuring the turbulent decay times from Figure [fig:alpha] shows a relation that is roughly inversely proportional to the initial Mach number, as suggested by τdec. The relation is not exact, however, due to varying degrees of mass loss from bound and unbound models. The right hand panel shows that this early phase, characterized by a decreasing α, is dominated almost exclusively by the gas energetics. A significant number of cluster particles have formed by this time (eg. ∼ 10% of all mass is in the form of cluster particles by 2.3 Myr for the simulation with α0 = 0.5), but they contribute negligibly to the overall α. After approximately 3 Myr, however, particle dynamics begin to play a significant role in the energetics, leading to a rise in α. The cluster particle contribution even outweighs the gaseous contribution for the initially bound runs. The role of the particle contributions to the global energetics points towards interesting dynamical differences between the simulations. We find that, in the bound simulations, the cluster particles are more centrally condensed relative to unbound simulations. This tighter grouping leads to stronger two-body interactions which result in velocities in excess of 50 km/s. These large particle velocities are responsible for the dominant contribution to α. For comparison, the velocities in unbound simulations do not exceed 20 km/s. We leave the discussion of particle dynamics, and its connection to α0, to a later paper. ![image](boundedness.png) [fig:boundedness] Formation Efficiencies and Particle Evolution --------------------------------------------- The discussion so far has been focused on global properties, such as the total mass in cluster particles, rather than the properties of individual objects. We move now to a discussion of the cluster particles themselves. The top panel of Figure [fig:boundedness] shows the time evolution of the two most massive cluster particles that survive to the end of the simulation for all clouds. Since the particles are not allowed to lose mass, these curves are monotonically increasing as a function of time. All curves also show a series of discrete jumps in mass. These jumps are due to merging events, as described in Section [methods]. It can be concluded from these curves that merging plays a significant role in determining the mass of the most massive particles in these simulations. This behaviour has also been observed in cluster-scale simulations. An interesting trend is noted in the top panel of Figure [fig:boundedness], namely that the simulations with a lower α0 tend to produce more massive cluster particles. The maximum mass cluster particles, including the mass of gas and stars, range from 3.26 × 104 M⊙ to 3.27 × 105 M⊙ for simulations with an α0 of 5 and 0.5, respectively. This trend can be attributed to the central concentration of dense gas that forms as a result of gravitational collapse in the bound simulations, allowing the most massive particles to continue accreting strongly from their surroundings. This suggests that the distribution of cluster particle masses depends on α0, which is discussed further below. The solid lines in the bottom panel of Figure [fig:boundedness] show the total mass of all cluster particles (shown on the left y-axis). There is a spread between simulations in both the total mass in cluster particles, and the time at which cluster formation begins. There is a spread of ∼ 260 kyr between when the most unbound cloud (α0 = 5) and the most bound cloud (α0 = 0.5) begin to form particles. While the unbound simulations form the first cluster particles, the bound simulations have more mass contained in particles overall. This can be seen by the dashed lines in the bottom panel of [fig:boundedness]. We refer to these lines as the “dense clump formation efficiency”, hereafter εcl, and are calculated by dividing the total mass in cluster particles over the initial mass of the cloud, ie. $$\epsilon\_{cl} = \frac{\sum\_{i} M\_i}{M\_{total}} = \frac{\sum\_{i} M\_i}{10^6 M\_{\odot}}$$ where Mi is the mass of the ith particle and the total cloud mass of 106 M⊙ has been shown for clarity. We have chosen the nomenclature of “dense clump formation efficiency”, rather than a cluster formation efficiency, because the mass contained in these particles is not solely in stars, but also a reservoir of unused gas which may be used for future star formation. Since εcl closely follows the total mass contained in particles, we see a similar spread between simulations. The early evolution is similar, until about 2 Myr, when the dense clump formation efficiencies begin to diverge. The final efficiencies cover a range of 37%, for an α0 of 5, to 71%, for an α0 of 0.5 with the unbound runs ending up less efficient than the bound runs. This is consistent with earlier simulations which demonstrated that particle formation efficiencies, over a fixed time, are significantly reduced as the Mach number is increased. ![image](nstars_vs_time_m10_6.png) [fig:nstars] Our εcl values can be compared to the observationally measured “dense gas mass fractions” (DGMFs) in nearby molecular clouds. Recent work by has measured the fraction of dense gas, defined to be gas above an extinction of AV > 7 corresponding to gas at densities of n > 104 cm− 3, in local molecular clouds with masses ranging from 200 to 2 × 105 M⊙. We note that our simulations have greater masses than the clouds presented in, but the observations still provide a useful point of comparison. The observed clouds span a range of evolutionary stages from starless clouds to star forming clouds with well developed HII regions. They find that the DGMF covers a large range, with a maximum value of ∼ 0.8. The average DGMF over the entire mass range is found to be 0.39 with a cloud-to-cloud scatter of 0.28, and the higher mass clouds have DGMFs close to the maximum value. We see that our own DGMFs, as measured by the total mass in cluster particles, fall within the observed range. The simulation having the closest DGMF to the observational average is the unbound simulation α0 = 3. This suggests that initially unbound molecular clouds more closely reproduce the properties of locally observed clouds. This point can be elaborated further by examining the dash-dotted lines in Figure [fig:boundedness]. These lines represent the star formation efficiency of the clouds (εsf), defined as the total mass in stars within cluster particles divided by the initial cloud mass. These curves show similar trends to εcl in the sense that higher initial virial parameters correspond to lower efficiencies. The curves take longer to diverge, however, as a consequence of our subgrid model which converts the cluster mass to stars gradually over time. The final εsf ranges from 18% to 34%. These values are higher than the global εsf in observed molecular clouds which typically span a range of 1-5% (eg. ). Since the highest virial parameter produces the lowest efficiency which is close to the observed values, this again suggests that a high initial virial parameter, in combination with radiative feedback, is required to match observations. There are, however, other forms of feedback that this work neglects which could also play a role in reproducing observed εsf values. When discussing cluster particles properties up to this point, we have focused solely on mass. To fully understand the resulting mass distributions, however, the total number of cluster sink particles needs to be discussed. In Figure [fig:nstars], we show the total number of cluster particles as a function of time. The number of particles can either increase via formation, or decrease via mergers or particles leaving the simulation volume. It can be seen that a large burst of cluster formation occurs early but levels out to roughly constant values around 2 Myr. The final number of clusters ranges from 52 for α0 = 0.5 to 189 for α0 = 5. As noted earlier, we see that initially unbound simulations begin to form cluster particles earlier. These curves do not cross, however, which is in contrast to the mass curves discussed in Figure [fig:boundedness]. This means that while the bound simulations contain the most mass in cluster particles, they have the lowest number of particles overall. The bound simulations are therefore expected to have more massive particles, on average, than unbound simulations. This has will be discussed further in the following section. Radiative Feedback Effects -------------------------- ![image](rad_vs_hydro.png) [fig:radvshydro] The above discussion has been focused on the role that α0 plays in shaping cloud properties and the resulting efficiencies. We now move on to discuss the impact of radiative feedback on these properties. To do this, we have completed an additional five simulations which are identical to those presented above but with radiative feedback not included. We refer to the simulations with radiative feedback included as “RHD” (Radiation Hydrodynamics) and the simulations with radiative feedback turned off as “HD” (Hydrodynamics). The main effect of radiative feedback is the suppression of star, or cluster, formation via the heating and ionization of the gas surrounding the newly formed stars. The strength of this suppression, however, is not fully understood and depends on the environment being modelled. For example, found that ionization feedback had a negligible effect on global SFEs in initially gravitationally bound clouds. In contrast, later work by showed that ionization feedback can reduce the SFE by up to 50% in smaller, 104 M⊙ clouds. Since it is clear that the GMC environment is a crucial component in determining the role of radiative feedback, we examine whether this role differs in 106 M⊙ GMCs with various α0. We have also included a rudimentary treatment of radiation pressure in our radiative transfer scheme which was not included in these previous works. ![image](mass_components_alpha3.png) [fig:mcalpha3] We compare the dense clump formation efficiencies, εcl, and star formation efficiencies, εsf, between RHD and HD simulations in Figure [fig:radvshydro]. Solid lines represent the RHD simulations and dashed lines represent the HD simulations. We see that the inclusion of radiative feedback plays a small role in determining εcl and εsf in the early evolution of a GMC, regardless of initial boundedness. The largest difference in εcl occurs in the simulation with an α0 of 3, with a difference of ∼ 10% between the RHD and HD simulation. The other simulations show that radiative feedback does suppress εcl but only by < 3%. Since εsf closely mirrors the evolution of εcl but with a time delay, the difference in SFEs between RHD and HD simulations is less noticeable. Overall, these results suggest that radiative feedback plays a minimal role in the early evolution of a massive GMC. Comparing these results to those presented in Figure [fig:boundedness], it is clear that α0 is the major contributing factor to the final efficiencies. Visual examination of Figure [fig:slicetemp], however, shows that radiative feedback is producing large scale HII regions which are evidently not strongly suppressing the global efficiencies in the cloud. The filamentary and porous nature of the cloud is likely limiting the role of radiative feedback. ![image](clusters_hyd.png) [fig:clustershyd] It is possible that if we followed the evolution of the cloud over longer timescales, we would see a larger suppression in efficiencies due to the lower overall density of the cloud which can be more easily disrupted by radiative feedback. Since we did not follow the cloud evolution to > 5 Myr, at which point Supernovae are expected to begin injecting significant amounts of energy and momentum into the surrounding gas, we can only conclude that radiative feedback does not greatly suppress cluster and star formation in the early evolution of a GMC. To show in which ways radiative feedback is suppressing the formation of particles, we focus on the simulation which had the largest difference between the RHD and HD simulations, namely the simulation with an α0 of 3. The observed difference in efficiency could either be due to radiation unbinding gas which then leaves the simulation volume, or through the heating and ionization of gas locally which prevents future particle formation and limits the accretion rate onto existing particles. To see which of these scenarios is more likely, we examine how the mass is divided among gas and particles in Figure [fig:mcalpha3]. Figure [fig:mcalpha3] shows that the RHD simulation has comparably less mass contained in particles and more gas mass than the HD simulation. However, the sums of these components, shown by the solid lines, are nearly identical, indicating that the RHD and HD runs have lost the same amount of mass from the simulation volume. From this we can conclude that radiative feedback is not driving outflows from the GMC, but is instead suppressing the formation and accretion of cluster particles. We note that this may be a product of our large cloud masses (106 M⊙) which have correspondingly deep potential wells which can trap the gas even under the influence of radiative feedback. Outflows may be more relevant in the early evolution of less massive GMCs where the energy and momentum input from stars may be sufficient to unbind the cloud. While global efficiencies are not greatly influenced by the inclusion of radiative feedback, it is possible that the cluster properties formed in an RHD versus an HD simulation may differ. In Figure [fig:clustershyd], we plot cluster properties for HD and RHD simulations to examine differences that arise. On the left, we show the total number of particles in all simulations. On the right, we plot the maximum and median mass cluster particles as a function of α0. The left hand panel of Figure [fig:clustershyd] shows no clear trend in the number of cluster particles in RHD versus HD simulations. The simulations with an α0 of 0.5 and 5 show no difference between the number of particles with radiative feedback included. The simulations with an α0 of 1 and 2 show an excess of clusters in the HD simulations at intermediate times but the difference becomes negligible at late times. Only the simulations with an α0 of 3 have different numbers of clusters at late times, with the RHD simulation forming ∼ 30 more cluster particles than the HD simulation. The right hand panel of Figure [fig:clustershyd] shows the maximum and median mass cluster particles versus α0 in both HD (black) and RHD (red) simulations. The maximum mass curves (solid lines) indicate that, except for the α0 = 3, HD simulations tend to produce more massive cluster particles than their RHD counterparts. This effect has been observed in other simulations of radiative feedback in which the distribution of particle masses is shifted to lower values. The median mass clusters show the opposite trend, with the RHD median masses being greater than the HD median masses. This indicates that the mass distribution is not simply being shifted to lower masses since the same trend would be seen in the median mass clusters as in the maximum mass clusters. It should be noted that since our subgrid model only includes radiative feedback from massive clusters (M\* > 1000 M⊙), the median mass clusters in the RHD simulations are not outputting radiation which may account for the observed trend reversal. Cluster Formation Thresholds ---------------------------- ![image](eff_nthresh.png) [fig:effnthresh] As discussed above, recent studies suggest that the strength of radiative feedback in different GMC environments may vary. We have shown that this is not the case for 106 M⊙ clouds which have different initial virial parameters. Another parameter, however, which also affects a cluster particle’s local environment is the threshold density for formation. We have chosen a fiducial value of 104 cm− 3 (see Section [methods]) motivated by observations of cluster forming regions. If, for example, a higher threshold density were chosen, the global GMC environment would be denser at the time of cluster formation and the radiation released by these clusters would be propagating into denser gas. This may have an important impact on the strength of radiative feedback. To examine the role that radiative feedback plays in clouds with different formation thresholds, we have resimulated a bound (α0 = 0.5) and an unbound (α0 = 3) cloud with formation thresholds of 103 and 105 cm− 3. The results of these simulations are presented in Figure [fig:effnthresh]. The onset of cluster formation differs among simulations with different formation thresholds. As expected, simulations with higher thresholds begin to form clusters later since the cloud has to collapse to higher densities before particles form. Comparing HD simulations, shown by dashed lines, and RHD simulations, shown by solid lines, we see that lower formation thresholds mirror the results presented in Figure [fig:radvshydro] in the sense that radiative feedback plays a minor role in controlling εcl. In fact, the differences in efficiency between RHD and HD simulations are reduced even further in the case of a low formation threshold. For the case of a high formation threshold, we see that the strength of radiative feedback is enhanced. The final efficiencies between RHD and HD simulation differ by 21% and 10% compared to 2% and 8% at the same time for the fiducial formation threshold of 104 cm− 3. These comparisons were at the time corresponding to the end of the shortest running simulation. The higher density immediately surrounding particles efficiently couples the radiation to the gas rather than the radiation streaming to low density voids perpendicular to filaments. The impact of radiative feedback on filaments is significantly more pronounced. This clearly indicates that the strength of radiative feedback is not a constant, but instead varies depending on the GMC environment. This may have important consequences for star formation in clumps and cores which are the densest regions in a molecular cloud. Indeed, observations of cores show that the densest cores (n ∼ 106 cm− 3) have lifetimes of approximately a freefall time while less dense cores (n ∼ 103 − 4 cm− 3) have lifetimes that are roughly ten times as long. This may be due, in part, to the increased strength of radiative feedback in denser regions. ![image](nstars_nthresh.png) [fig:nstarsnthresh] The most significant difference between simulations with different particle thresholds is the number of cluster particles that form, as shown in Figure [fig:nstarsnthresh]. As the particle formation threshold increases, the number of particles decreases. For example, changing the formation threshold from 103 to 105 cm− 3 in the simulation with an α0 of 3 reduces the number of particles from 167 to 26. This trend is expected since the regions which form particles need to reach higher densities in the case of a higher threshold, and these regions become increasingly rare as the threshold increases. While the particle formation efficiency is reduced in the case of a high formation threshold, it is not reduced by the factor of ∼ 6 that is seen in the particle number (for α0 = 3). This is because the particles which form in the high threshold cases are, on average, more massive. Since a particle has a fixed size, and the regions out of which the particles form have higher densities for higher threshold cases, this result is expected. The combination of fewer particles which are, on average, more massive for higher threshold cases could have a significant impact on the cluster particle mass distribution. This is discussed briefly in the next Section. In the remainder of this Section, however, we will continue to focus on simulations with the fiducial value of the formation threshold because it is observationally motivated, and a full exploration of the formation threshold parameter space is beyond the scope of this paper. Cluster Properties ------------------ ![image](ECMF_evolution.png) [fig:ECMFevolution] We now move to discuss cluster properties in more detail with a focus on mass distributions and stellar content. In Figure [fig:ECMFevolution], we show representative snapshots of the normalized cluster particle mass function at 2, 3, and 4 Myr for the simulation with α0 = 3. The other simulations show similar evolution, and a comparison between simulations is made below. The early evolution is dominated by low mass particles with the maximum particle mass reaching 104 M⊙ by 2 Myr. While massive particles are present, the majority have masses on the order of Solar masses. As time progresses, as shown in the middle panel of Figure [fig:ECMFevolution], the distribution shifts to higher masses because of both ongoing gas accretion and particle mergers. This same trend is evident in moving from 3 to 4 Myr. The highest mass particles have grown to approximately 105 M⊙. The lowest bins have been depopulated, resulting in a turnover in the distribution just below 100 M⊙. ![image](ECMF_boundedness.png) [fig:ECMFboundedness] ![image](star_reservoir_comparison.png) [fig:starreservoir] We compare the final total (ie. stellar content and gas reservoir) cluster particle mass distributions across simulations in Figure [fig:ECMFboundedness]. In order to compare to the results presented in, who measured the clump mass function in W43 using JCMT, we fit the distributions with a line for masses greater than log(M) > 2.4 M⊙. We discuss the observational results and the connection to our simulations in Section [clumpobs]. The resulting slopes are shown in the upper right hand side of each panel. The fits range from -1.49 to -1.73 for simulations with an α0 of 1 and 5, respectively. The slope for all combined data, shown in the lower right panel, is -1.67. There is a slight trend towards steeper slopes as α0 increases. Since the lower α0 clouds have the most massive particles, more mass in cluster particles as a whole, and a smaller number of particles, this trend is easily understood. In other words, the mass distributions shift to higher masses as α0 decreases, which manifests as a steepening slope when fitting over a fixed range. The mass of individual cluster particles is of two types: the cumulative mass distribution of stars as determined by generations of random sampling of the IMF from the available gas reservoirs, and the current gas reservoir. The mass in each type depends on several factors. Firstly, the age of the particle determines how long it has been forming stars suggesting that older clusters may, on average, contain a larger fraction of mass in stars relative to younger clusters. Secondly, the accretion histories of individual clusters affect the division of mass. As mentioned in Section [methods], any accreted mass is added directly to the gas reservoir. Therefore, if a cluster is undergoing strong accretion, the gas reservoir may dominate over the stellar mass. ![image](star_reservoir_vs_age.png) [fig:reservoirage] We examine the division of particle mass into stars and the gas reservoir in Figure [fig:starreservoir]. We plot the gas mass fraction, defined as the ratio of the gas reservoir to the total mass, for each particle versus the total particle mass. The line at a gas mass fraction of 0.5 delineates clusters which are gas dominated from clusters which are stellar dominated. We show the gas mass fractions at 2 Myr, on the left, and 4 Myr, on the right. We only show three simulations, those with α0 of 0.5, 2, and 3, for readability. Figure [fig:starreservoir] shows that the majority of cluster particles are still dominated by the gas reservoir at 4 Myr. These clusters are either undergoing strong accretion, or have not had enough time to convert their mass into stars. High gas mass fractions within clusters at late times appears to contradict observations which suggest that massive clusters are completely devoid of gas by 4 Myr. Our subgrid model does not allow for gas expulsion from cluster sink particles which may explain the high gas mass fractions still present at late times. Even without gas expulsion from cluster particles, however, radiative feedback could conceivably halt further gas accretion which would naturally produce lower gas mass fractions at late times. Since this is not observed, it suggests that radiative feedback is not the only process responsible for producing gas free clusters by ∼ 4 Myr. Stellar winds and protostellar jets may play a role in clearing young, star-forming regions of gas. ![image](IMF_all.png) [fig:IMF] c\|c\|c\|c\|c\|c\| & & 0.5 & 1 & 2 & 3 & 5 & 3.27 × 105 & 2.22 × 105 & 1.78 × 105 & 1.57 × 105 & 3.26 × 104 & 368 & 485 & 313 & 290 & 318 & 1.50 × 105 & 1.13 × 105 & 8.58 × 104 & 8.70 × 104 & 1.26 × 104 & 144 & 107 & 103 & 91 & 97 [table:mass] [] The overall spread in gas mass fractions shows differences among the simulations. The unbound simulation (ie. virial parameter of 5) contains the largest number of stellar dominated clusters. This is likely tied to the dynamical histories of individual particles. The higher gas velocities present in the high α0 simulations can result in cluster particles drifting away from their local gas sources, thereby shutting off future accretion and producing stellar dominated clusters. This is less pronounced in the lower α0 simulations because of the global collapse of both gas and particles which facilitates ongoing accretion. The progression from a gas to a stellar dominated cluster particle is inherently a time driven process as cluster particles convert their mass into stars. This would suggest that the oldest clusters should have the lowest gas mass fractions. This may not be case, however, because ongoing accretion can supply the clusters with a fresh gas supply. To visualize this interplay, we plot the gas mass fraction as a function of cluster age, instead of mass, in Figure [fig:reservoirage]. We show the results from two simulations, with α0 values of 1 and 3, to show the difference between bound and unbound simulations. We plot the gas mass fractions of all particles present at 2, 3, and 4 Myr. We see from Figure [fig:reservoirage] that, generally speaking, older cluster particles have lower gas mass fractions (ie. a higher fraction of stars) than younger clusters. We do, however, see significant outliers with some old clusters having high gas mass fractions. This highlights the importance of the local environment on star formation, suggesting that, even within one GMC, there can be distinct subclustered regions which can have very different star formation histories in comparison to their neighbours. Stellar Content --------------- ![image](SFR_vs_time_smoothed.png) [fig:SFRall] The stellar mass distribution within individual cluster particles is partially determined by the subgrid model for star formation. A sufficiently massive particle with a large reservoir of mass to draw from will produce an IMF which is fully sampled. From, particles with masses of ∼ 5000 M⊙ are sufficiently massive to form O stars. For smaller clusters, however, the amount of mass available at sampling may not be sufficient to form massive stars. For this reason, we examine the stellar mass distribution within individual cluster particles, as well as the total cloud, in Figure [fig:IMF]. We show the stellar mass distribution in both the maximum mass cluster particle, in the left panel, and the median mass particle, in the middle panel, to illustrate the variations between different star forming regions. The right hand panel of Figure [fig:IMF] shows the stellar mass distribution of the entire cloud. For reference, the mass of the the maximum and median mass cluster for all simulations is shown in Table [table:mass]. ![image](age_spread_all.png) [fig:age] As expected, the most massive particles show a fully sampled IMF with stars covering the entire mass range. There is little cluster to cluster variation among the most massive particles. We see markedly different distributions for the median mass clusters, however, which vary in mass between 290 to 485 M⊙. Because these clusters have significantly smaller masses, they rarely form stars with masses greater than ∼ 5 M⊙. Moreover, the distributions show stronger variations among clusters. Because these particles form a small number of stars, the effects of randomly sampling an IMF become more apparent. Overall, these distributions highlight the importance that the accretion histories play in subclustered regions and how this impacts the type of stars that are able to form. Star Formation Rates -------------------- We conclude this Section with a discussion of the star formation rates within entire clouds and individual cluster particles, and relate this to the resulting stellar age spreads. First, in Figure [fig:SFRall], we plot the total star formation rates from our simulations. As a consequence of our subgrid model for star formation, which only forms new stars at prescribed time intervals after a particle is formed, there are large, instantaneous jumps in the star formation rate. Since all particles do not form at the same time, the times at which particles sample the IMF are staggered. This means that, in a given timestep, there may be no particles which form new stars resulting in a SFR of zero. In other timesteps, the opposite may be true resulting in a large, instantaneous SFR. To aid in readability, the SFR curves presented in Figure [fig:SFRall] have been smoothed using a sliding average window. While this helps significantly in interpreting the Figure, there are still some spikes in the SFR that remain. The curves show qualitatively similar trends across simulations. There is a rapid onset of star formation, beginning around 0.5 Myr, which rises up to a roughly constant value. We do not see a turnover in the SFR. We do not expect to see a complete halting of star formation as a consequence of our subgrid model in which dense gas is assumed to remain bound within the gravitational potential well of the cluster. As shown in, however, the SFR will decrease for particles which are no longer accreting. This suggests that radiative feedback is not sufficient to halt the accretion onto a large number of particles and create a turn over in the SFR. At late times, the variation in the SFR increases. This is likely due to the reason discussed above, namely that the formation of new stars in cluster particles is staggered. The variation becomes more pronounced at late times because there are more particles overall, and the particle masses increase as the simulation evolves meaning there is more mass out of which new stars can form. Additionally, the particle accretion rates become more variable at late times due to large velocities (see Figure [fig:alpha]) which can either disconnect a particle from its host filament or move it to a gas rich region where strong accretion is possible. Since the SFR is intimately tied to the accretion rate of particles, this is likely contributing to the variable SFR at late times. The final, approximately constant, SFRs cover roughly a range of 4 × 10− 2 to 2 × 10− 1 M⊙ ⋅ yr− 1, with variations of greater than an order of magnitude seen in some simulations. The evolution of the global SFR is directly tied to the resulting stellar age distributions within the simulations. We compare the age spreads in the entire cloud, as well as the maximum mass and median mass cluster particle, in Figure [fig:age]. The right panel, which shows the age spread of all stars in the respective simulations, mirrors the global SFR. The majority of stars are newly formed with a rapid decline in the amount of older stars. We do not see a turn over in the distribution since star formation is still occurring at high rates. The age distributions within the maximum mass clusters are also strongly peaked towards young ages. This is not surprising since these clusters have a significant gas reservoir to continue forming stars. The median mass clusters show more variations between simulations. Some clusters, the median cluster in the simulation α0 = 1 for example, are newly formed and as a consequence the age distribution is narrow, ∼ 1 Myr, and peaked at young ages. The older median mass clusters, like in the simulation with α0 = 2, show much broader distributions with age spreads up to ∼ 3 Myr which also have a turnover at intermediate ages. This shows that the formation of a stellar population in a molecular cloud is not a uniform process with distinct regions forming stars at the same time. Instead, different subcluster regions have separate star formation histories tied to their local environments. ![image](SFR_m1e6_all.png) [fig:SFRall] Comparison to Observations ========================== Mass Distributions ------------------ We refer back to Figure [fig:ECMFboundedness] and discuss the connection to observed clump mass function. We focus on the recent results presented in who measured masses for 1029 clumps in W43. We have chosen this dataset in particular because W43 is one of the most massive cloud complexes in the Milky Way, with a total mass of ∼ 7 × 106 M⊙ of which ∼ 8.4 × 106 M⊙ is contained in dense clumps, providing a good comparison to our simulated dataset. The results presented in for W43 show a clump mass distribution with a high mass slope of -1.87 ± 0.05, which is consistent with the results presented in. This slope is inconsistent with our results, which range from -1.49 to -1.73, and suggests that our simulations have either overproduced very massive particles, under produced low mass particles, or is a combination of both effects. There are examples of clump mass functions which have smaller slopes (eg. and discussions therein), but these studies have different cut off masses for the fits making comparisons difficult to make. Moreover, measured the slope of the clump mass function in NGC 7538 and M17 and found inconsistent results between the two regions, suggesting that there may not be a universal clump mass function. As a final remark, we note that the slopes of our particle mass functions are consistent with those measured for entire GMC’s by, who found a slope of -1.50 ± 0.36 The above discussion has focused on simulations which use a particle formation threshold of 104 cm− 3. As discussed in Section [thresholds], we have also completed a subset of simulations which increase and decrease the formation thresholds by an order of magnitude. In the discussion surrounding Figure [fig:nstarsnthresh], we noted that increasing the formation threshold significantly reduces the number of cluster particles while increasing the average particle mass. The will clearly impact the resulting particle mass distributions. We find that the resulting high mass slope, analyzed in the same way as above, differs dramatically for the high formation threshold of 105 cm− 3. For an α0 of 0.5, the resulting slopes are -1.58, -1.54, and -1.05 for low, fiducial, and high thresholds, respectively. The corresponding slopes for the α0 = 3 simulations are -1.81, -1.7, and -1.09. The combination of less particles and higher average masses, in the case of a high formation threshold, results in significantly shallower slopes which are less consistent with observations. This provides further justification for the choice of 104 cm− 3 as our formation threshold. Not only is this value observationally motivated, it results in particle mass distributions which are closer to measured values. Star Formation Rates in Molecular Clouds ---------------------------------------- The SFRs discussed earlier were measured globally for each simulation. In order to make comparisons to observations, we plot the SFR of individual cluster particles in Figure [fig:SFRall]. We again only show the simulations with an α0 of 0.5, in red, 2, in blue, and 5, in purple for readability. We have over plotted the observational results from in black crosses. These observations used 2MASS data to catalog the young stellar objects (YSOs) in 11 local star forming regions within 450 pc of the sun. The observed clouds are the Pipe Nebula, the Ophiuchus cloud, the Lupus cloud complex, Taurus, Perseus, the California clouds, RCrA, and the Orion cloud complex. How measurements of the YSO content in these clouds are used to estimate SFRs is described below. ![image](merged_fraction.png) [fig:merging] The black triangles in Figure [fig:SFRall] represent the dataset from who also measured SFRs in local regions, some of which are the same regions as in. The solid line represents the SFR-mass relation measured for extragalactic sources by. These observations measured the IR and HCN luminosity in the star forming cores of nearby galaxies, and used these measurements as proxies for the SFR. The resulting SFR-mass relation is given by, SFR(M⊙yr− 1) ∼ 1.2 × 10− 8Mdense(M⊙) where Mdense is the mass of dense gas (n > 104 cm− 3) traced by IR emission. We have extrapolated this relation to molecular cloud masses in Figure [fig:SFRall]. In order to make a comparison to the observations, we have measured the SFR in each cluster particle in an identical manner to. The following formula was used to estimate the SFR in each region, $$SFR = \frac{N\bar{M}}{\tau\_{YSO}} = 0.25N\times10^{-6} M\_{\odot} yr^{-1} \label{SFRmass}$$ where N is the total number of young stellar objects (YSOs) in each region, M̄ is the median mass of a YSO in M⊙, and τYSO is the average lifetime of a YSO. The median mass, M̄, was taken to be 0.5 M⊙, consistent with the IMF and therefore consistent with the stellar distributions within our cluster particles. An average YSO lifetime of 2 Myr was assumed. While we do not have prestellar evolution in our subgrid model, and therefore do not know the exact number of YSOs, we do know the formation times for all stars. When solving for the SFR via equation [	arxiv_0000806
Transforming floundering into success ===================================== We show how logic programs with “delays” can be transformed to programs without delays in a way which preserves information concerning floundering (also known as deadlock). This allows a declarative (model-theoretic), bottom-up or goal independent approach to be used for analysis and debugging of properties related to floundering. We rely on some previously introduced restrictions on delay primitives and a key observation which allows properties such as groundness to be analysed by approximating the (ground) success set. This paper is to appear in Theory and Practice of Logic Programming (TPLP). Floundering, delays, coroutining, program analysis, abstract interpretation, program transformation, declarative debugging Introduction ============ Constructs for delaying calls have long been a popular extension to conventional Prolog. Such constructs allow sound implementation of negation, more efficient versions of “generate and test” algorithms, more flexible modes and data-flow, a mechanism for coordinating concurrent execution and forms of constraint programming. They also introduce a new class of errors into logic programming: rather than computing the desired result, a computation may flounder (some calls are delayed and never resumed). Tools for locating possible bugs, either statically or dynamically, are desirable. Static analysis can also be used to improve efficiency and in the design of new languages where data and control flow are known more precisely at compile time. The core contribution of this paper is to show how a program with “delays” can be transformed into a program without delays whose (ground) success set contains much information about floundering and computed answers of the original program. Some technical results are given which extend known results about floundering, and these are used to establish the properties of two new program transformations. The main motivation we discuss is program analysis, though we also mention declarative debugging. Analysis of properties such as which goals flounder can be quite subtle, even for very simple programs. The term floundering was originally introduced in the context of negation, where negated calls delay until they are ground, and sometimes they never become ground. In this paper we don’t directly deal with negation but our approach can equally be used for analysing this form of delaying of negated calls. Subcomputations are also delayed in some other forms of resolution, for example, those which use tabling. For these computational models delaying is more determined by the overall structure of the computation (for example, recursion) rather than the instantiation state of variables in the call, and it is doubtful our methods could be adapted easily. This paper is structured as follows. In section [sec:delprim] delay declarations are described and the procedural semantics of Prolog with delays is discussed informally. In section [sec:examples] we give some sample programs which use delays. In section [sec:anal1] we discuss in more detail some properties of delaying code which, ideally, we would like to be able to analyse. In section [sec:ground] we briefly discuss an observation concerning computed answers which is important to our approach. In section [sec:sldf] we review a theoretical model of Prolog with delays and extend some previous results concerning floundering. In section [sec:elim] we give a program transformation that converts floundering into success. In section [sec:cfs] a more precise characterisation of floundering is provided, along with a second transformation. In section [sec:ddis] we briefly discuss declarative debugging of floundering and a related model-theoretic semantics. In section [sec:related] we discuss some related work and we conclude in section [sec:conc]. Delay declarations and their procedural meaning =============================================== Dozens of different control annotations have been proposed for logic programming languages. In the programs in this paper we use “delay” declarations of the form `:- delay A if C` where `A` is an atom p(V1, V2, …, VN), the Vi are distinct variables, p/N is a predicate and `C` is a condition consisting of `var/1`, `nonground/1` (with arguments the Vi), “`,`” and “`;`”. Procedurally, a call p(V1, V2, …, VN) delays if `C` holds (with the conventional meaning of `var` and `nonground`). The procedural semantics of Prolog with delays is typically difficult to describe precisely and, to our knowledge, is not done in any manuals for the various Prolog systems which support delays. Here we describe the procedural semantics of NU-Prolog, and where the imprecision lies; other systems we know of are very similar. By default, goals are executed left to right, as in standard Prolog. If the leftmost sub-goal delays (due to some delay annotation in the program), the next leftmost is tried. Thus the leftmost non-delaying subgoal is selected. Complexities arise when delayed goals become further instantiated and may be resumed. When a delayed goal becomes instantiated enough to be called (due to unification of another call with the head of a clause), the precise timing of when is it resumed can be difficult to predict. With a single call to resume, it is done immediately after the head unification is completed[1](#fn1). With multiple calls to resume, they are normally resumed in the order in which they were first delayed. It is as if they are inserted at the start of the current goal in this order. However, this is not always the case. Some calls may delay until multiple variables are instantiated to non-variable terms. This is implemented by initially delaying until one of those variables is instantiated. When that occurs, the call is effectively resumed but may immediately delay again if the other variables are not instantiated. Similarly, when delaying until some term is ground, the delaying occurs on one variable at a time and the call can be resumed and immediately delayed again multiple times. The order in which multiple calls are resumed depends on when they were most recently delayed. This depends on the order in which the variables are considered, which is not specified. In NU-Prolog, the code generated to delay calls is combined with the code for clause indexing and it is difficult to predict the order in which different variables are considered without understanding a rather complex part of the compiler. The situation is even worse in parallel logic programming systems. In Parallel NU-Prolog the default computation rule is exactly the same as for NU-Prolog. However, if an idle processor is available a call which is instantiated enough may delay and be (almost) immediately resumed on another processor. Even with total knowledge of the implementation, the precise execution of a program cannot be determined. Any program analysis based on procedural semantics must respect the fact that the computation rule is generally not known precisely but (we hope) not lose too much information. Example code with delays ======================== ``` :- delay append(As, Bs, Cs) if var(As), var(Cs). append([], As, As). append(A.As, Bs, A.Cs) :- append(As, Bs, Cs). append3(As, Bs, Cs, ABCs) :- append(Bs, Cs, BCs), append(As, BCs, ABCs). :- delay reverse(As, Bs) if var(As), var(Bs). reverse([], []). reverse(A.As, Bs) :- append(Cs, [A], Bs), reverse(As, Cs). ``` [figrev] We now present two small examples of code which uses delays. The first will be used later to explain our techniques. Figure gives a version of `append` which delays until the first or third argument is instantiated. This delays (most) calls to `append` which have infinite derivations. Delaying such calls allows `append` to be used more flexibly in other predicates. For example, `append3` can be used to `append` three lists together or to split one list into three. Without the delay declaration for `append`, the latter “backwards” mode would not terminate. With the delay declaration, the first call to `append` delays. The second call then does one resolution step, instantiating variable `BCs`. This allows the first call to resume, do one resolution step and delay again, et cetera. In a similar way, this version of `reverse` works in both forwards and backwards modes—if either argument is instantiated to a list it will compute the other argument. If the second argument is instantiated, no calls are delayed. However, if only the first argument is instantiated, all the calls to `append` initially delay and after the last recursive call to `reverse` succeeds, the multiple calls to `append` proceed in an interleaved fashion. For any given mode, the code for `append3` and `reverse` can be statically reordered to produce a version which works without delaying. The Mercury compiler does such reordering automatically, but without automatic reordering it requires some slightly tricky coding to produce such flexible versions of these predicates. ``` submaxtree(Tree, NewTree) :- submaxtree1(Tree, Max, Max, NewTree). submaxtree1(nil, _, 0, nil). submaxtree1(t(L, E, R), GMax, Max, t(NewL, NewE, NewR)) :- submaxtree1(L, GMax, MaxL, NewL), submaxtree1(R, GMax, MaxR, NewR), max3(E, MaxL, MaxR, Max), plus(NewE, GMax, E). % delays; later mode o,i,i max3(A, B, C, D) :-... :- delay plus(A, B, C) if var(A), var(B) ; var(A), var(C) ; var(B), var(C). ``` Figure is a variant of the `maxtree` program (see, for example) which takes a tree and constructs a new tree containing copies of a logic variable in each node, then binds the variable to a number (the maximum number in the original tree). The `submaxtree` program fills each node in the new tree with the original value minus the maximum. This is done by delaying a call to `plus` for each node until the maximum is known, then resuming all these delayed calls. We assume a version of `plus` which delays until two of its three arguments are instantiated; NU-Prolog has such a predicate built in. All calls to `plus` become sufficiently instantiated at the same time (when `GMax` becomes instantiated). In most systems they will be called in the order they were delayed. If `plus` only worked in the forward mode the calls would not be sufficiently instantiated and the computation would flounder. We also assume a predicate `max3/4` which calculates the maximum of three numbers. It is not possible to statically reorder the clause bodies to eliminate the delays. Even dynamic reordering clause bodies each time a clause instance is introduced (also known as a local computation rule) is insufficient. Without coroutining, two passes over the tree are necessary, doubling the amount of “boilerplate” traversal code—the first to compute `GMax` and the second to build the new tree. Analysis of code with delays ============================ Delays can be used to write concise and flexible code, the behaviour of which can be very subtle. For example, shows that when bugs are introduced to a four-clause permutation program with delays, a wide variety of counter-intuitive behaviour results. Even with such a tiny program, the combination of interleaved execution and backtracking makes understanding why it misbehaves very challenging. Another tiny example is the (arguably correct) definition of `reverse` in Figure [figrev]. Having first written and used equivalent code around twenty-five years ago, the author did not become fully aware of its floundering properties until the preparation of this paper. It was incorrectly thought that all calls to `reverse` which have an infinite number of solutions with different list lengths (such as `reverse([a,b,c\|Xs],Ys)`) would flounder. Section [secusingf] gives a precise characterisation of the actual behaviour. Automated methods of analysis of code with delays are highly desirable because manual analysis is just too complex to be reliable. Analysis of code with delays can address many different issues. It may be that we expect code to succeed or finitely fail for certain classes of goals but some such goals may actually flounder, typically with a computed answer less instantiated than expected—this is the main focus of the declarative debugging work of, also discussed in section [sec:ddis]. In section [sec:elim] we give a transformation which allows analysis of computed answers (of successful or floundered derivations) which can detect such cases. Conversely, we may expect certain goals to flounder when actually they succeed. This is particularly important if the goal has an infinite number of solutions and is part of a larger computation—success can result in non-termination where floundering would not. In section [sec:cfs] we give a further transformation which captures floundering precisely. Both methods reduce the problem of analysing a program with delays to analysing the success set of a program without delays. A deeper understanding of floundering can also help us in other ways. For example, although the declarative debugger of doesn’t rely on either of these transformations directly, it is based on the insights of this paper. Similarly, these insights may help us optimise code, either by simplifying delay annotations or, more significantly, eliminating them entirely (possibly with some reordering of code). They may also help our understanding of termination properties of code with delays. Semantics and computed answers ============================== ccc P1 & P2 & P3 & & `p(a).` &`p(X).` & `p(a).` & & `p(X).` `q(a).` &`q(a).` & `q(a).` Before proceeding further, with more technical material, we make an observation about computed answers which is fundamental to our work. The conventional approach to the semantics of logic programs is that the set of function symbols in the language is precisely that in the program—see, for example, the textbook which combines and refines some of the original work of van Emden, Kowalski, Apt and others. This means that, unlike in Prolog, new function symbols cannot occur in the goal (or the semantics of the program differs depending on the goal). An alternative is to define the (typically infinite) set of function symbols a priori and assume that both the program and goals use a subset of these function symbols. This approach has been examined in, where various results are given, and earlier in, where forms of equivalence of logic programs are explored. For example, Figure gives three programs with different sets of computed answers for `p(Y)`. They are all equivalent using the “Lloyd” declarative semantics but with extra function symbols programs P2 and P3 are equivalent but P1 is not. One advantage of the latter semantics is that the universal closure of a goal is true if and only if it succeeds with a computed answer substitution which is empty (or simply a renaming of variables)—see the discussion in. Another is that the model-theoretic and fixed-point semantics can capture information about (non-ground) computed answers and, as we will show later, floundering! Since the Lloyd semantics deals only with sets of ground atoms, this fact is somewhat surprising, and does not seem to have been exploited for program analysis until now. We partition the set of function symbols into program function symbols and extraneous function symbols. Programs and goals may only contain program function symbols. Program atoms are atoms containing only program function symbols. Substitutions in derivations (including computed answer substitutions) contain only program function symbols. [obspfs] This is a consequence of most general unifiers being used (indeed, programming with delays makes little sense without this). Non-ground computed answers can be identified in the success set by the presence of extraneous function symbols. For example, if ⋈ is an extraneous function symbol then an atom such as `p(\bowtie)` appears in the success set if and only if it is an instance of some non-ground computed answer. If we assume there are an infinite number of terms whose principal function symbol is an extraneous function symbol then computed answers can be captured more precisely—we make this assumption later for our analysis of floundering. Note this semantics cannot determine whether a variable exists in all computed answers (or derivations) of a goal—in both P2 and P3 of Figure the success set contains `p(\bowtie)` and `p(a)`. However, it does precisely capture groundness in all computed answers (or variables in some computed answer), a property which has attracted much more interest. For example, many consider it of interest that in all computed answers of `append`, if the third argument is ground the second argument is also ground. Using the semantics we suggest, this is equivalent to saying if ⋈ occurs in the second argument it also occurs in the third argument. If we can find a superset of the success set (for example, a model) which has this property, the groundness dependency must hold. Thus a small variation to the Lloyd semantics leads to significant additional precision while retaining the simple model-theoretic and fixed-point semantics and the relationship between them. SLDF resolution =============== A model of Prolog with delays, SLDF resolution, is presented in. Here we review the model and main results, concerning ground atoms, and extend these result to non-ground atoms. We define the non-ground flounder set, which approximates the floundering behaviour of a program. However, we first give discuss two important closure properties which hold for SLD resolution (where there is no floundering). [Closure properties] [clprops] If an atom A has a successful SLD derivation with computed answer A (an empty computed answer substitution) then, using the same program clauses in the derivation * any atom with A as an instance has a computed answer with A as an instance, and * any atom Aθ has a computed answer Aθ. Such properties allow computed answers to be captured precisely by the set of computed answers of maximally general atoms, and generally simplifies analysis. When delays are introduced (SLDF resolution), only closure property 2 holds for successful atoms—a less instantiated version of a successful atom may flounder rather than succeed. For floundered atoms only closure property 1 holds (see proposition [propGthca])—an instance of a floundered atom may succeed, loop, finitely fail or flounder with an even more instantiated floundered computed answer. The weaker closure properties (compared to SLD resolution) means it is harder to precisely characterise the behaviour of SLDF resolution using sets of atoms. We now review SLDF resolution, define the set of atoms we use to approximate its behaviour and show the relationship between the two. SLDF resolution is similar to SLD resolution (see ), but the computation (atom selection) rule is restricted to be safe: an atom may only be selected if it is in the “callable atom set”. It is desirable that this set is closed under instantiation and the results below and those in this paper rely on this property. This property seems quite intuitive and holds for most logic programming systems with flexible computation rules. Another restriction suggested in is that all ground atoms should be callable. While this is not required for our technical results, it is a pragmatic choice. SLDF derivations can be failed, successful, infinite or floundered, in which the last resolvent consists only of atoms which are not callable (we say it is immediately floundered). Given the assumption above, for a program P the following sets of ground atoms can be defined independently of the (safe, and also fair in the case of finite failure) computation rule: * The success set SS(P) (ground atoms with successful derivations). * The finite failure set FF(P) (ground atoms with finitely failed SLD trees). * The flounder set FS(P) (ground atoms with floundered derivations). Note that some atoms in FS(P) may also be in SS(P) and have infinite (fair) derivations. The fact that floundering is independent of the computation rule suggests it is a declarative property in some sense. However, it has not been fully exploited for analysis until now, perhaps due to the lack of non-procedural definitions of FS(P). Note also that the results above only refer to ground atoms. An atom such as `q(X)` may have floundered derivations but no instance may appear in FS because no ground instance flounders. However, FS can contain information about floundering of non-ground atoms and conjunctions. For example, if the program contains the definition `p :- q(X)`, FS will contain `p` if and only if `q(X)` flounders. Relying on the existence of such definitions is a problem unless we know a priori which goals we want to analyse, and FS gives us no information about substitutions in floundered derivations. Substitutions in floundered (sub)computations can influence termination and are very important for certain programming styles, particularly those associated with parallel programming. Most Prolog systems which support delays print variable bindings at the top level for both successful and floundered derivations. Thus we use the term(s) computed answer (substitution) for both successful and floundered derivations, explicitly adding the words “successful” or “floundered” where we feel it aids clarity. The analysis proposed in this paper can be seen as being based on the following generalisation of FS. The non-ground flounder set, NFS(P), of a program P is the set of program atoms which have floundered derivations with empty floundered computed answer substitutions. Successful derivations can be conservatively approximated by simply ignoring delays—the lack of closure property 1 for successful atoms and the fact that an atom may have both successful and floundered derivations prevents our approach being more precise for analysis of success. The key results of this section, propositions [propGthNFS] and [propGthca], show how NFS(P) contains much but not all information about computed answers of floundered derivations. The results in, and some we prove in this paper, rely on the notion of two derivations of the same goal using the same clause selection. Used in in the context of successful derivations, we formalise it here. We assign each clause in the program a unique positive integer and use zero for the top level goal. We annotate each atom used in a derivation with a superscript to indicate the sequence of clauses and atoms within those clauses used to introduce it. Annotations are lists of pairs ⟨c, a⟩, where c is a clause number and a is the number of an atom within the clause. We use these annotations for both SLD and SLDF resolution. The annotation si of an atom used in a derivation is as follows. If goal A1s1, A2s2, …Ansn is the top level goal, si = ⟨0, i⟩:nil. Applying substitutions to atoms does not change their annotation. If Ai is selected and resolved with a variant of clause number j, H ← B1, B2, …Bk, each Bm atom is annotated with ⟨j, m⟩:si. Two atoms in different derivations or goals are corresponding atoms if they have the same annotation. Two derivations with the same top level goal, or with one top level goal an instance of the other, have the same clause selection if all pairs of corresponding selected atoms in the two derivations are matched with the same clause. Although not explicitly stated, the proofs in can easily be adapted to show that (successful and floundered) computed answers of successful and floundered derivations are independent of the computation rule (see Lemma [lemmgiGth]). For any two successful or floundered derivations of the same goal with the same clause selection but a different (safe) computation rule, each selected atom in one has a corresponding atom selected in the other. Derivation length, computed answer substitutions and the last resolvent are all the same, up to renaming. [lemmgiGth] Suppose D is the SLD derivation G, G1α1, G2α2, …, GNαN, …, where αi is the composition of the most general unifiers used in the first i steps, and Dʹ is the SLD derivation Gθ, Gʹ1αʹ1, Gʹ2αʹ2, …, GʹNαʹN, …, where θ affects only variables in G. Suppose also that D and Dʹ use the same set of program clauses and corresponding set of selected atoms in the first N steps. Then GʹNαʹN, is a most general instance (m.g.i.) of GNθ and GNαN, and GθαʹN is a m.g.i of Gθ and GαN. Let Hi ← Bi, 1 ≤ i ≤ N, be the ith program clause variant used in the first N steps of D, Ci, 1 ≤ i ≤ N, be the atom in a body of one of these clauses or in G whose corresponding instance is selected and matched with Hi, and Ri, 1 ≤ i ≤ K be the atoms as above corresponding to those in GN (they are not selected in the first N steps). Let terms C and H be as follows, where the connectives and predicate symbols are mapped to function symbols. $$\begin{aligned} \nonumber C & = & f(H\_1, B\_1, H\_2, B\_2, \ldots, H\_N, B\_N, C\_1, C\_2, \ldots, C\_N, R\_1, R\_2, \ldots, R\_K) \\ \nonumber H & = & f(H\_1, B\_1, H\_2, B\_2, \ldots, H\_N, B\_N, H\_1, H\_2, \ldots, H\_N, R\_1, R\_2, \ldots, R\_K)\end{aligned}$$ A m.g.i. of C and H can be obtained by left to right unification of the arguments of C and a variant of H which shares no variables with C. The first 2N argument unifications yield a renaming substitution, resulting in the same variants of program clause heads and bodies in the instances of H and C. The next N are the same as the unifications in D, modulo the clause variants used, and the other unifications yield empty unifiers. Thus CαN is a m.g.i. of C and H. Other orders of the arguments Ci and Hi (or unification orders) correspond to different computation rules but result is the same most general instance, or a variant. So, by the same construction, CαʹN is a m.g.i. of H and Cθ, and must be an instance of CαN. CαN is an instance of H so CαʹN must be a m.g.i. of Cθ and CαN. The instances of the initial goals and Nth resolvents can be extracted from the arguments of CαN, Cθ and CαʹN so the result follows. We can now show something similar to the converse of closure property 1, for floundering. This allows us to infer certain information about program behaviour from NFS(P). [propGthNFS] If D is the floundered SLDF derivation G, G1α1, G2α2, …, GNαN with floundered computed answer Gθ, then Gθ has a floundered SLDF derivation Dʹ with a renaming (or empty) floundered computed answer substitution (Gθ ∈ NFS(P)). Let Dʹ be a derivation using the same selection and computation rule as D. Dʹ cannot flounder before N steps because the ith resolvent is an instance of Giαi and the callable atom set is closed under instantiation. Dʹ cannot fail before N steps because the ith resolvent is no more instantiated than GiαN, and αN is a unifier of all pairs of calls and clause heads in the first N steps. Consider the Nth resolvent, GʹNαʹN. By Lemma [lemmgiGth], GʹNαʹN is a m.g.i. of GNθ and GNαN and since GNαN is an instance of GNθ, GʹNαʹN must be a variant of GNαN, so it is immediately floundered. Similarly, GθαʹN is a variant of Gθ, so the floundered computed answer substitution is a renaming. [lemFiinst] If Gθ has a floundered SLDF derivation Dʹ with the last resolvent being Fʹ1, Fʹ2, …, Fʹk and G has a SLDF derivation D using the same clause selection rule then each Fʹi is an instance of all its corresponding atoms in D. If G is immediately floundered the result is trivial. We use induction on the length of Dʹ. For length 0, since the callable atom set is closed under instantiation and Gθ is immediately floundered, G must also be immediately floundered. Assume it is true for length N. Suppose the first selected atom in D is A. Aθ is also callable, so we can construct a derivation Dʺ using the same clause selection as Dʹ but with Aθ as the first selected atom. The lengths of Dʺ and Dʹ are equal and their last resolvents are variants due to the result stated earlier. The first resolvent in Dʺ (after selecting Aθ) is an instance of the first resolvent in D and has a derivation of length N so the result follows. We can now show that closure property 1 holds for floundering: [propGthca] If Gθ has a floundered SLDF derivation Dʹ with a renaming (or empty) floundered computed answer substitution (Gθ ∈ NFS(P)), then G has a floundered SLDF derivation D with a floundered computed answer with Gθ as an instance. From G we can construct a derivation D using the same clause selection as that used in Dʹ and any safe computation rule. The callable atom set is closed under instantiation so by Lemma [lemFiinst], any atom selected in D must have a corresponding atom selected in Dʹ and thus D cannot be successful or longer that Dʹ. D uses the variants of the same clauses used in Dʹ, which has a more (or equally) instantiated top level goal, Gθ, so D cannot be failed. It must therefore be floundered and have a computed answer with Gθ as an instance. From these propositions we know that an atom A will flounder if and only if it has an instance in NFS(P). Also, the maximally general instances of A in NFS(P) will be floundered computed answers. The imprecision of NFS(P) with respect to floundered computed answers is apparent when there are atoms in NFS(P) which are instances of other atoms in NFS(P). If NFS(P) = {p(f(X)), p(f(f(f(X))))} for example, we know p(Y) will have the first atom as a floundered computed answer. The second atom may also be a floundered computed answer (via a different floundered derivation) or it may only be returned for more instantiated goal such as p(f(f(X))). In practice, there is usually a single maximally general instance of a goal in NFS(P) and this is the only answer computed, even when there are an infinite number of instances. For example, the non-ground flounder set for `append` has an infinite number of instances of the atom `append(A, [B], C)`, including `append(Xs, [Y], Zs)`, `append([X1\|Xs], [Y], [X1\|Zs])` and `append([X1, X2\|Xs], [42], [X1, X2\|Zs])`, but only the first is computed. Converting floundering into success =================================== We now present a program transformation which converts a program P, with delays, into a program Pʹ, without delays. The success set of Pʹ is the union of the success set of P and a set isomorphic to the non-ground flounder set of P. Thus analysis of some properties of programs with delays can be reduced to analysis of programs without delays. The SF() transformation --------------------------- Type, groundness and other dependencies are of interest in programs with and without delays as they give us important information concerning correctness. In the version of naive reverse without delays, analysis can tell us that in all computed answers of `reverse/2` both arguments are lists. In the delaying version (Figure [figrev]) this is not the case, since there are floundered computed answers where both arguments are variables. This increases the flexibility of `reverse/2` since it can delay rather than computing an infinite number of answers (this is particularly important when `reverse/2` is called as part of a larger computation). In (successful and floundered) computed answers for the delaying version of `reverse/2`, the first argument of is a list if and only if the second argument is a list. This tells us that if either argument is a list in a call, the other argument will be instantiated to a list by the `reverse/2` computation (assuming it terminates). If the delay declaration for `append/3` was changed so it delayed if just the first argument was a variable, `reverse/2` would not work backwards. It would flounder rather than instantiate the first argument to a list and the “if” part of this dependency would not hold. This section shows how a program with delays can be very simply transformed into a program without delays which can be analysed to reveal information such as this. Analysis of success in a program without delays cannot give us information about (non-ground) delayed calls directly because success is closed under instantiation (closure property 2) whereas floundering is not. However, extraneous function symbols allow us to encode non-ground atoms using ground atoms, re-establishing this proposition and allowing analysis. The encoding uses an isomorphism between the (infinite) set of variables and the set of terms with extraneous principal function symbols (this set must also be infinite to avoid loss of precision in the encoding; it is sufficient to have a single extraneous function symbol with arity greater than zero). The encoded flounder set (EFS(P)) of a program P is the set of ground instances of atoms in NFS(P) such that distinct variables are replaced by distinct terms with extraneous principal function symbols. NFS(P) can be reconstructed from the atoms in EFS(P) by finding the set of most specific generalisations which contain only program function symbols. For example, the non-ground flounder set for append contains atoms such as `append(Xs, [Y], Zs)` whereas the encoded flounder set contains atoms such as `append(\bowtie, [\otimes(1)], \otimes(2))`, assuming ⋈ and ⊗ are extraneous function symbols. We introduce two new “builtin” predicates, `evar/1` and `enonground/1`, which are true if their argument is an encoded variable or non-ground term, respectively. For simplicity, our treatment assumes they are defined using an (infinite) set of facts: `evar(T)` for all terms T where the principal function symbol is not a program function symbol and `enonground(T)` for all terms T which have at least one extraneous function symbol. This can cause an infinite branching factor in SLD trees (for example, a call such as `evar(X)`). However, since in this paper we deal with single derivations but not SLD trees (or finite failure), it causes us no difficulties. ``` evar('VAR'(_)). enonground(A) :- evar(A). enonground([A\|B]) :- enonground(A). enonground([A\|B]) :- enonground(B). ``` [figevar] It is also possible to define `evar/1` and `enonground/1` in Prolog. Figure gives a definition which assumes `’VAR’/1` is the only extraneous function symbol of the original program and `’.’/2` is the only program function symbol with arity greater than zero for the original program (if there are other such function symbols, more clauses are needed for `enonground/1`). These definitions depart from our theoretical treatment in that they can involve deeper proof trees (due to recursive calls) and they can have non-ground computed answers. However, they can be useful for observing floundering behaviour, especially with a fair (or depth-bounded) search strategy—see section [secusingf]. We now define the SF() transformation: Given a program P (not defining predicates `evar/1` or `enonground/1`) containing delay declarations, SF(P) is the program with all clauses of P plus, for each delay declaration `:- delay A if C` in P, the clause `A :- C’` where `C’` is `C` with `var` replaced by `evar` and `nonground` replaced by `enonground`. These additional clauses introduced for delay declarations and those in the definitions of `evar/1` and `enonground/1` are referred to as delay clauses. ``` append_sf(As, Bs, Cs) :- evar(As), evar(Cs). append_sf([], As, As). append_sf(A.As, Bs, A.Cs) :- append_sf(As, Bs, Cs). reverse_sf(As, Bs) :- evar(As), evar(Bs). reverse_sf([], []). reverse_sf(A.As, Bs) :- append_sf(Cs, [A], Bs), reverse_sf(As, Cs). ``` To avoid possible confusion, the code in this paper uses “`_sf`” suffixes for the new predicate definitions; our theoretical treatment assumes the original predicate names are used for the new predicate definitions. For example, Figure shows the transformed version of `reverse` (from Figure [figrev]). Figures [figextrad] and [figextradu] give further examples. Immediately floundered atoms in NFS(P) have matching delay declarations with true right hand sides. Corresponding (encoded) atoms in EFS(P) have matching ground delay clause instances with successful bodies. We have described how `evar` and `enonground` behave. Some languages have delay conditions which cannot be expressed using var and nonground. For example, in NU-Prolog `X ~= Y` delays whereas `X ~= X` does not. To analyse such constructs we need additional primitives similar to `evar` and `enonground`. The key to designing such constructs is that the delay clauses should implement the encoding as defined above. Properties of SF() ---------------------- The following propositions show how successful derivations in SF(P) correspond to successful or floundered derivations in P: the success set of SF(P) is the union of the success set of P without delays and the encoded flounder set of P (Proposition [propsssfp]). Note that when we talk of successful derivations and/or SS(P) here, SLD resolution rather than SLDF resolution is used (delays are ignored when dealing with success). The lack of closure property 2 is problematic when dealing with success if delays are considered and SLDF resolution used. [propnodeliff] A goal G has a successful SLD derivation D with program P (ignoring delays) if and only if it has a successful derivation D with SF(P) which uses no delay clauses. SF(P) without delay clauses is the same as P without delays. We now deal with floundering, which is more complex. [lemimfl] A goal G is immediately floundered with program P if and only if it has a successful derivation D with SF(P) which uses only delay clauses. Follows from the way in which delay clauses implement the encoding of the flounder set. [lemsubsefs] A goal G which is immediately floundered with program P has a computed answer substitution θ in SF(P) such that all variables bound by θ are bound to distinct terms with extraneous principal function symbols (or are simply renamed). By Lemma [lemimfl], there is a derivation where all non-renaming substitutions are due to calls to `evar/1` and `enonground/1`. A call to `evar/1` binds its argument to a term with an extraneous principal function symbol. Multiple calls with distinct variables will have some of the infinite number of computed answers binding their arguments to distinct terms. Similarly, some computed answers to `enonground/1` will bind all distinct variables in its argument to distinct terms with extraneous principal function symbols. [lemnfsiffsfdel] Given a program P, a goal G has a floundered derivation D with an empty floundered computed answer substitution if and only if it has a successful derivation Dʹ with SF(P) in which delay clauses are selected and the successful computed answer, Gθ, is such that all variables bound by θ are bound to distinct terms with extraneous principal function symbols (or are simply renamed). (Only if) Derivation D can be reproduced with SF(P) since it has all the clauses of P and the computation rule is unrestricted. By Lemma [lemsubsefs] the last resolvent in D must have a successful derivation such that the computed answer substitution has the desired property. (If) By repeated application of the switching lemma to Dʹ we can construct a successful derivation Dʺ = G, G1, G2, …, Gn with SF(P) such that callable atoms are selected in preference to atoms which would delay in P. The derivation has a prefix D = G, G1, G2, …, Gm where only callable atoms are selected, except for Gm, which would be immediately floundered in P (a delay clause is used in Dʺ so an immediately floundered goal must be reached at some stage). Callable atoms are not matched with delay clauses (by Lemma [lemimfl], if a callable atom is resolved with a delay clause the resolvent cannot succeed). Variables bound by the computed answer substitution θ of Dʺ are bound to distinct terms with extraneous principal function symbols (or simply renamed), and all of the non-renaming bindings must be due to delay clauses. Thus D is a floundered SLDF derivation in P with an empty (or renaming) answer substitution. [lemsfallsubs] Given a program P, a goal G has a successful derivation with SF(P) with computed answer, Gθ, such that all variables bound by θ are bound to distinct terms with extraneous principal function symbols (or are simply renamed) if and only if there are successful derivations with all such computed answers. All such substitutions are due to delay clauses and the sets of `enonground/1` and `evar/1` atoms which succeed are closed under the operation of replacing one extraneous function symbol with another. [propfliffsfdel] Goal G has a floundered SLDF derivation D with program P if and only if it has a successful derivation Dʹ with SF(P) in which delay clauses are selected. Propositions [propGthNFS] and [propGthca] imply G flounders if and only if an instance flounders with an empty floundered computed answer substitution so by Lemma [lemnfsiffsfdel] it is sufficient to show that an instance of G has a successful derivation with SF(P) in which delay clauses are selected and all variables bound by the computed answer substitution are bound to distinct terms with extraneous principal function symbols (or are simply renamed) iff G has a successful derivation Dʹ with SF(P) in which delay clauses are selected. (Only if) By closure property 1. (If) Consider a derivation G, G1α1, G2α2, …, GNαN using the same clause selection as in Dʹ but with a computation rule such that atoms resolved with delay clauses are selected at the end, from Gmαm. By Lemma [lemmgiGth], Gαm has a derivation where the mth resolvent is a variant of Gmαm and the substitution at that point is a renaming substitution for Gαm. By Lemma [lemsubsefs], a computed answer substitution for Gmαm has the desired property. [propsssfp] For any program P, SS(SF(P)) = EFS(P) ∪ SS(P). The set of atoms in SS(SF(P)) with derivations which don’t use delay clauses is SS(P) by Proposition [propnodeliff]. The set of atoms in SS(SF(P)) with derivations which use delay clauses is EFS(P) by Lemmas [lemnfsiffsfdel] and [lemsfallsubs]. Note that although there is a bijection between successful SLD derivations with P and successful SLD derivations with SF(P) which don’t use delay clauses, there is not a bijection between floundered SLDF derivations with P and successful derivations with SF(P) which use delay clauses, even if multiple solutions to `evar/1` and `enonground/1` are ignored. SF(P) generally has additional derivations. This is unavoidable due to the imprecision of NFS(P) mentioned in Section [sec:sldf]. ``` p(X, Y) :- q(X), q(Y). p_sf(X, Y) :- q_sf(X), q_sf(Y). :- delay q(V) when var(V). q_sf(V) :- evar(V). q(a). q_sf(a). ``` [figextrad] For example, consider the definition of `p/2` in Figure. The success set, ignoring delays, is {`p(a,a)`} and NFS(P) is {`p(X,Y)`, `p(a,V)`, `p(V,a)`}. Thus the computed answers of `p_sf(X,Y)` encode all these four atoms, since SF(P) computes the union of the success set and the encoded flounder set. However, `p(X,Y)` only has one floundered derivation, with the empty answer substitution. The other two atoms in NFS(P) are only computed for more instantiated goals and the derivations in SF(P) correspond to these computations (the same atoms are selected, ignoring `evar/1` and `enonground/1`). ``` p :- q(X). p_sf :- q_sf(X) :- delay q(V) when var(V). q_sf(V) :- evar(V). q(a). q_sf(a). q(X) :- q(X). q_sf(X) :- q_sf(X). ``` [figextradu] It is also possible to have successful derivations in SF(P) which do not correspond to any SLD or SLDF derivation in P. For example, in Figure, the goal `p` has a single floundered SLDF derivation, where `q(X)` immediately flounders, whereas `p_sf` has an infinite number of derivations which use delay clauses and the derivations of `q(X)` have unbounded length. This is related to the fact that `p` has an infinite SLD tree. Analysis using SF(P) -------------------------- Type dependencies of SF(P) can be analysed in the same ways as any other Prolog program. The following set of atoms, where l(X) means X is a list, is a model of the transformed reverse program, showing these type dependencies hold (and thus they hold for computed answers in the original reverse program with delays): {append(A, B, C)∣(l(A) ∧ l(B)) ↔ l(C)} ∪ {reverse(A, B)∣l(A) ↔ l(B)} It is not necessary to consider the complex procedural semantics of Prolog with delays, or even the procedural semantics of Prolog without delays since bottom-up analysis can be used. Similarly, the SF() transformation makes it relatively easy to show that `submaxtree/2` can indeed compute a tree of integers when given a tree of integers as the first argument. Groundness in P can also be analysed by analysing SF(P) using specialised types. We can define the type ground to be the set of terms constructed from only program function symbols. The dependencies which hold for lists above also hold for type ground, indicating the corresponding groundness dependencies hold for computed answers of reverse with delays. Similarly, nonvar can be defined as the set of terms with a program principal function symbol. By extending the type/mode checker described in we have demonstrated it is possible to check non-trivial useful properties of P by checking models of SF(P). For more complicated cases it is necessary to support sub-types, as nonvar and list are both subtypes of ground. This approach to groundness analysis is not reliant on the SF() transformation—it can be applied to any logic program due to Observation [obspfs]. The analysis can be identical to conventional groundness analysis using Boolean functions because logic programs can be abstracted in an identical way. A unification X = f(Y1, Y2, …, YN) can be abstracted by X ↔ Y1 ∧ Y2 ∧ … ∧ YN, assuming f/N is a program function symbol. Calls evar(X) and enonground(X) can be abstracted as X ↔ False. New characterisations of the flounder set ========================================= We now present a second transformation, which allows us to capture the non-ground flounder set more precisely. The F() transformation ------------------------ The results here suggest a solution to the open problem posed in : how the flounder set can be defined inductively. Such a definition may be a very useful basis for analysis of floundering as an alternative to a purely model theoretic approach. The semantics of SF(P) captures both successful and floundered derivations of P. By defining a variant of the immediate consequence operator TP we can distinguish atoms with derivations which use delay clauses. An f-interpretation is a set of ground atoms, some of which may be flagged (to indicate floundering). If I is an f-interpretation, A(I) is the set of atoms in I and FA(I) is the set of atoms in I which are flagged. The union of two f-interpretations I and J is the f-interpretation K such that A(K) = A(I) ∪ A(J) and FA(K) = FA(I) ∪ FA(J). Given a program P, TPf is a mapping from f-interpretations to f-interpretations, defined as follows. A(TPf(I)) = TSF(P)(A(I)) and an atom A in this set is flagged if there is a ground instance of a clause in SF(P), A ← B1, B2, …, Bk, such that each Bi is in I and some Bi is flagged in I or if the predicate of A is `evar/1` or `enonground/1`. TPf ↑ n and TPf ↑ ω are defined in the same way as TP ↑ n and TP ↑ ω. [propflagiffsfdel] A ground atom other than `evar/1` or `enonground/1` is flagged in TPf ↑ n if and only if it has a proof tree of height ≤ n in SF(P) which uses a delay clause, and is flagged in TPf ↑ ω if and only if it has a successful derivation in SF(P) which uses a delay clause. A standard result is that TSF(P) ↑ ω (and TSF(P) ↑ n) contains exactly those ground atoms with proof trees in SF(P) (of height ≤ n, respectively). A(TPf ↑ n) = TSF(P) ↑ n since ∀ I A(TPf(I)) = TSF(P)(I). From the definition of TPf, these atoms are flagged if and only if they are derived using `evar/1` or `enonground/1`, that is, if a delay clause is used in the derivation. A ground atom A other than `evar/1` or `enonground/1` is flagged in TPf ↑ ω if and only if it is in the encoded flounder set of P. By Proposition [propflagiffsfdel] it is sufficient to show that A ∈ EFS(P) iff A has a successful derivation D in SF(P) which uses a delay clause. If: The decoded version of A (distinct terms with extraneous principal function symbols are replaced by distinct variables), B, has a successful derivation Dʹ in SF(P) using the same clause selection as that in D, with a computed answer Bθ, which has A as an instance. Since Bθ has A as an instance, any variables bound by θ must be bound to distinct terms with extraneous principal function symbols (or simply renamed). Thus Dʹ satisfies the condition of Lemma [lemnfsiffsfdel] so B is in NFS(P). Only if: A ∈ EFS(P), so A = Bγ, where B ∈ NFS(P). By Lemmas [lemnfsiffsfdel] and [lemsfallsubs], B has a derivation which uses delay clauses and has a computed answer with an instance Bγ. A has a successful derivation using the same clause selection. Thus we have an inductive/fixed-point characterisation of (a set isomorphic to) the non-ground flounder set. It may be practical to base floundering analysis on TPf. It is monotonic with respect to the set of atoms (A(TPf(I)) ⊆ A(TPf(Iʹ)) if A(I) ⊆ A(Iʹ)) and for a given set of atoms it is monotonic with respect to the flagged atoms in the set (FA(TPf(I)) ⊆ FA(TPf(Iʹ)) if A(I) = A(Iʹ) and FA(I) ⊆ FA(Iʹ)). Monotonicity is important for the structure of fixed-points, particularly the existence of a least fixed-point. Alternatively, the definition of TPf can be mirrored by a further transformation which produces a program whose success set is the encoded flounder set of P. An advantage is that it can then be analysed using standard techniques. A disadvantage is that the transformation increases the program size, which will affect analysis time. Given a Horn clause program P, F(P) is the program consisting of the predicate definitions in SF(P) (we assume each predicate has a sf subscript/postfix) plus the following new definitions. For each clause psf(X̄)`:-`B in SF(P) we add a clause pf(X̄)`:-`Bʹ. For delay clauses, Bʹ = B. For other clauses, Bʹ = B, D, where D is the disjunction of all calls in B, with “`_sf`” replaced by “`_f`”. If B is the empty conjunction (`true`) then Bʹ is the empty disjunction (`fail`). ``` append_f(As, Bs, Cs) :- evar(As), evar(Cs). append_f([], As, As) :- fail. append_f(A.As, Bs, A.Cs) :- append_sf(As, Bs, Cs), append_f(As, Bs, Cs). reverse_f(As, Bs) :- evar(As), evar(Bs). reverse_f([], []) :- fail. reverse_f(A.As, Bs) :- reverse_sf(As, Cs), append_sf(Cs, [A], Bs), (reverse_f(As, Cs) ; append_f(Cs, [A], Bs)). ``` Figure gives the new clauses generated for `reverse`. Note that we assume the original program consists of only Horn clauses but the transformed program contains disjunctions. These could be eliminated by further transformation. The transformation is designed so that TF(P) (extended to handle disjunctions) is essentially the same as TPf: flagged atoms correspond to the f subscripted predicates and the set of all atoms corresponds to the sf subscripted predicates. The success set of the f subscripted predicates in F(P) is the encoded non-ground flounder set of the corresponding predicates in P. Analysis using F(P) ----------------------- The transformation allows us to observe the floundering behaviour of the original program very clearly. If we define `evar` as in Figure [figevar] and run the goal `append_f(X,Y,Z)` using a fair search strategy, we get computed answers of the form `X = [A`1`,A`2`,...,A`N`\|'VAR'(B)]`, `Y = C`, `Z = [A`1`,A`2`,...,A`N`\|'VAR'(D)]`. Occurrences of `’VAR’/1` in answers correspond to variables in computed answers of floundered derivations of the original program and variables correspond to arbitrary terms. Thus a call to `append` flounders if and only if it has an instance such that the first and third arguments are “incomplete lists” (lists with a variable at the tail rather than nil) of the same “length” with pair-wise identical elements. For example, `append(X,[a],[a\|Z])` flounders (it also has a successful derivation) whereas `append([a,V\|X],Y,[V,b\|Z])` does not. Running `reverse_f` we discovered to our surprise (as mentioned in section [sec:examples]) that `reverse` flounders if and only if the first argument is an incomplete list and the second argument is a variable (rather than incomplete list). A call such as `reverse(X,[a\|Y])` returns an infinite number of answers rather than floundering! With a suitably expressive domain the transformed program can be analysed with established techniques to obtain precise information about the original program with delays. Powerful techniques have been developed to help construct domains. For example, we can start with a simple domain containing four types: lists, var (the complement of our type nonvar), incomplete lists (this is a supertype of var), and a “top” element (the universal type). Completing this domain using disjunction adds two additional elements: “list or var” and “list or incomplete list”. The Heyting completion of this domain introduces implications or dependencies such as X is a list if Y is a list. This domain can be used as a basis for interpretations of the program and to infer and express useful information about floundering. reversef(X, Y) = X ∈ il ∧ Y ∈ v reversesf(X, Y) = reversef(X, Y) ∨ X ∈ l ∧ Y ∈ l appendf(X, Y, Z) = X ∈ il ∧ Z ∈ il ∧ (X ∈ v ↔ Z ∈ v) appendsf(X, Y, Z) = appendf(X, Y, Z) ∨ $$\hspace{3cm} X \in l \wedge ( Y \in l \leftrightarrow Z \in l) \wedge ( Y \in il \leftrightarrow Z \in il )$$ [figfsnrev] For example, Figure gives the minimal model of the program for this domain, where v represents the type var, l the type list and il the set of incomplete lists (this was found using the system described in, with additional modifications and manual intervention). It expresses the fact that `reverse` flounders only if the first argument is an incomplete list and the second is a variable. The condition for `append` is somewhat more complex. It is possible to drop the last conjunct for appendsf and replace ↔ by → for appendf to obtain a simpler model. Further simplification does not seem possible without weakening the condition for `reverse`. We note that careful design of the types in the domain is crucial for the precision. The incomplete list type is able to make the important distinction between (encoded versions of) `[X]` and `[[]\|X]`. Analysis without this distinction must conclude that calls to `reverse/2` where both arguments are (complete) lists may flounder. To see this, consider the following instance of the recursive clause for `reverse/2`. ``` reverse([a,X], [X]) :- append([X], [a], [X]), reverse([X], [X]). ``` If we replace the two occurrences of `[X]` in `append` by `[[]\|X]` then the clause body flounders with an empty computed answer substitution. Thus any safe approximation to the set of floundering atoms must include the head of this clause. Inferring models is significantly more challenging than checking models. The domain is huge and the models can be quite complex, even for simple programs (see the condition for appendsf in Figure, for example). After some ad hoc attempts to find models for F(P), particularly minimum models within our abstract domain, a more systematic approach was developed. We use the relationship between predicates in P and their subscripted variants in minimum models. We first compute a model AP for P (the minimum model for P in our abstract domain). We use this as a starting point to compute a (larger) model ASF(P) for the `"_sf"` predicates. We then use ASF(P) \ AP as a starting point to compute a model for the `"_f"` predicates. This strategy may also be useful for automatic inference of precise floundering information since although there are three separate fixed-point calculations, each one is relatively simple and should converge quickly. Declarative debugging, inadmissibility and semantics ==================================================== Declarative debugging can be an attractive alternative to static analysis since more information is known at debug time than at static analysis time and hence bugs can potentially be located more easily and precisely. The F() transformation of Section [sec:cfs] potentially provides a mechanism for declarative debugging of incorrectly floundered computations—a floundered derivation of P corresponds to a successful derivation of F(P) and debugging of incorrect successful derivations is well understood. The main novel requirement is that the user must be able to determine which (encoded) atoms should flounder (that is, an intended interpretation for the `_f` predicates). It is also important for the debugger to understand the relationship between the `_f` and `_sf` predicates because their intended interpretations are not independent. In we propose a more practical approach which doesn’t use the transformations and encoding explicitly, but does use them to guide the design. It uses the three-valued debugging scheme of, where atoms can be correct, incorrect or inadmissible, meaning they should never occur. Atoms which have insufficiently instantiated “inputs” (and hence flounder) are considered inadmissible. The user effectively supplies a three-valued interpretation in the style of for SF(P) and the debugger finds a clause (possibly a delay clause) for which this interpretation is not a (three-valued) model. As well as model-theoretic semantics, provides a fixed-point semantics. This could also be applied to analysis of delays by using transformation and encoding, particularly if the user specifies intended modes in some way. Related work ============ The transformation-based method used to detect deadlocks in parallel logic programs bears superficial similarity to our work here. However, those transformations do not eliminate delays, and both the original code and transformed code have impure features such as pruning operators for committed choice non-determinism and nonvar checks. Our approach to analysis of floundering here is unusual in that it supports a declarative, “bottom-up” or “goal independent” approach. Analysis of logic programs with the conventional left to right computation rule has been done using both top-down and bottom-up methods. The top-down methods are based on the procedural semantics—SLD resolution—maintaining information about variables and substitutions to obtain approximations to the sets of calls and answers to procedures. The bottom-up methods (which are independent of the computation rule) are based on the fixed-point semantics (the immediate consequence operator, which is very closely related to the model theoretic semantics) to obtain approximations to the set of answers to procedures. An advantage of the bottom-up approach is its simplicity. Using the standard fixed point semantics (see also ) the domain contains sets of ground atoms and a clause can be treated as equivalent to the set of its ground instances. The disadvantage is lack of precision: the naive bottom-up approach obtains no information about calls or non-ground computed answers, both of which seem important for modeling systems with flexible computation rules. Two methods are used to re-gain this information. Non-ground computed answers can be captured by using a more complicated immediate consequence operator, such as the S-semantics, making the domain more complex by re-introducing variables. Calls can be captured by using the magic set (or similar) transformation, adding complexity to the program being analysed, but this assumes a left to right computation rule. Since there has been no known bottom-up method for approximating the instantiation states of calls in logic programs with delays, it is natural that most other work on analysis of such programs has been based on the top-down procedural semantics. The more recent approach of uses bottom-up analysis, and argues strongly for the practicality of bottom-up methods. A relatively standard bottom-up least fixed-point analysis is used to compute groundness dependencies for successful computed answers of all predicates using the Pos domain (positive Boolean functions). In addition, a novel greatest fixed-point computation is used to find sufficient conditions for predicates to be flounder-free, using the Mon domain (monotonic Boolean functions). However, this analysis assumes a local computation rule is used. Programs such as `submaxtree/2` (and examples given in ) have cyclic data-flow and do not work with a local computation rule, so the greatest fixed-point computation results in significant loss of precision. Our transformations make no assumptions about the computation rule other than it is safe with respect to the delay declarations, so (in this respect) it can be more precise. We have shown an alternative way the “Lloyd” semantics can be adapted to capture information about variables: simply change the set of function symbols rather than the immediate consequence operator. The extra function symbols allow us to encode and capture the behaviour of non-ground atoms. Furthermore, by encoding the non-ground flounder set it becomes closed under instantiation, allowing safe approximation by the success set of a (transformed) program without delays. Floundering information can then be obtained by a simple bottom-up analysis using sets of ground atoms. The complexity associated with variables does not magically disappear entirely. In practice it can re-emerge in the abstract domain of types used in the analysis. However, careful integration of type and instantiation information seems unavoidable if analysis of floundering is to be precise, so combining both in the type domain is probably a good idea. Using the procedural semantics has the advantage of being (strictly) more expressive than the declarative approach, so analysis of more properties is possible. Analysis of (for example) whether a particular sub-goal will ever delay (for a particular computation rule) is beyond the scope of our approach and can only be done with procedural information. A disadvantage is the additional complexity. Each (non-ground) atom has a set of computed answers and for each one there is a set of immediately floundered atoms. The analysis domain typically contains representations of sets of these triples. We believe that analysis of such things as computed answers and whether a computation flounders is likely to benefit from the declarative approach we have proposed, where the analysis domain can contain just sets of ground atoms. Expressive languages for defining such sets have been developed for type-related analysis. Conclusion ========== With an intuitive restriction on delay primitives, floundering is independent of the computation rule. However, the development of a declarative rather than procedural understanding of floundering has been hindered because it is not closed under instantiation. In this paper we have shown how non-ground atoms can be encoded by ground atoms, using function symbols which do not occur in the program or goal. Some may consider this to be a theoretical “hack”, but it has numerous advantages. This technique, along with two quite simple program transformations, allows floundering behaviour of a logic program with delays to be precisely captured by the success set of a logic program without delays. By simply executing the transformed program using a fair search strategy, the delaying behaviour can be exposed. Declarative debugging can be used to diagnose errors related to control as well as logic, and alternative semantic frameworks can be applied. 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In some systems it may occur after the head unification plus calls to certain built-in predicates at the start of the matching clause.[↩](#fnref1) Transforming floundering into success ===================================== We show how logic programs with “delays” can be transformed to programs without delays in a way which preserves information concerning floundering (also known as deadlock). This allows a declarative (model-theoretic), bottom-up or goal independent approach to be used for analysis and debugging of properties related to floundering. We rely on some previously introduced restrictions on delay primitives and a key observation which allows properties such as groundness to be analysed by approximating the (ground) success set. This paper is to appear in Theory and Practice of Logic Programming (TPLP). Floundering, delays, coroutining, program analysis, abstract interpretation, program transformation, declarative debugging Introduction ============ Constructs for delaying calls have long been a popular extension to conventional Prolog. Such constructs allow sound implementation of negation, more efficient versions of “generate and test” algorithms, more flexible modes and data-flow, a mechanism for coordinating concurrent execution and forms of constraint programming. They also introduce a new class of errors into logic programming: rather than computing the desired result, a computation may flounder (some calls are delayed and never resumed). Tools for locating possible bugs, either statically or dynamically, are desirable. Static analysis can also be used to improve efficiency and in the design of new languages where data and control flow are known more precisely at compile time. The core contribution of this paper is to show how a program with “delays” can be transformed into a program without delays whose (ground) success set contains much information about floundering and computed answers of the original program. Some technical results are given which extend known results about floundering, and these are used to establish the properties of two new program transformations. The main motivation we discuss is program analysis, though we also mention declarative debugging. Analysis of properties such as which goals flounder can be quite subtle, even for very simple programs. The term floundering was originally introduced in the context of negation, where negated calls delay until they are ground, and sometimes they never become ground. In this paper we don’t directly deal with negation but our approach can equally be used for analysing this form of delaying of negated calls. Subcomputations are also delayed in some other forms of resolution, for example, those which use tabling. For these computational models delaying is more determined by the overall structure of the computation (for example, recursion) rather than the instantiation state of variables in the call, and it is doubtful our methods could be adapted easily. This paper is structured as follows. In section [sec:delprim] delay declarations are described and the procedural semantics of Prolog with delays is discussed informally. In section [sec:examples] we give some sample programs which use delays. In section [sec:anal1] we discuss in more detail some properties of delaying code which, ideally, we would like to be able to analyse. In section [sec:ground] we briefly discuss an observation concerning computed answers which is important to our approach. In section [sec:sldf] we review a theoretical model of Prolog with delays and extend some previous results concerning floundering. In section [sec:elim] we give a program transformation that converts floundering into success. In section [sec:cfs] a more precise characterisation of floundering is provided, along with a second transformation. In section [sec:ddis] we briefly discuss declarative debugging of floundering and a related model-theoretic semantics. In section [sec:related] we discuss some related work and we conclude in section [sec:conc]. Delay declarations and their procedural meaning =============================================== Dozens of different control annotations have been proposed for logic programming languages. In the programs in this paper we use “delay” declarations of the form `:- delay A if C` where `A` is an atom p(V1, V2, …, VN), the Vi are distinct variables, p/N is a predicate and `C` is a condition consisting of `var/1`, `nonground/1` (with arguments the Vi), “`,`” and “`;`”. Procedurally, a call p(V1, V2, …, VN) delays if `C` holds (with the conventional meaning of `var` and `nonground`). The procedural semantics of Prolog with delays is typically difficult to describe precisely and, to our knowledge, is not done in any manuals for the various Prolog systems which support delays. Here we describe the procedural semantics of NU-Prolog, and where the imprecision lies; other systems we know of are very similar. By default, goals are executed left to right, as in standard Prolog. If the leftmost sub-goal delays (due to some delay annotation in the program), the next leftmost is tried. Thus the leftmost non-delaying subgoal is selected. Complexities arise when delayed goals become further instantiated and may be resumed. When a delayed goal becomes instantiated enough to be called (due to unification of another call with the head of a clause), the precise timing of when is it resumed can be difficult to predict. With a single call to resume, it is done immediately after the head unification is completed[1](#fn1). With multiple calls to resume, they are normally resumed in the order in which they were first delayed. It is as if they are inserted at the start of the current goal in this order. However, this is not always the case. Some calls may delay until multiple variables are instantiated to non-variable terms. This is implemented by initially delaying until one of those variables is instantiated. When that occurs, the call is effectively resumed but may immediately delay again if the other variables are not instantiated. Similarly, when delaying until some term is ground, the delaying occurs on one variable at a time and the call can be resumed and immediately delayed again multiple times. The order in which multiple calls are resumed depends on when they were most recently delayed. This depends on the order in which the variables are considered, which is not specified. In NU-Prolog, the code generated to delay calls is combined with the code for clause indexing and it is difficult to predict the order in which different variables are considered without understanding a rather complex part of the compiler. The situation is even worse in parallel logic programming systems. In Parallel NU-Prolog the default computation rule is exactly the same as for NU-Prolog. However, if an idle processor is available a call which is instantiated enough may delay and be (almost) immediately resumed on another processor. Even with total knowledge of the implementation, the precise execution of a program cannot be determined. Any program analysis based on procedural semantics must respect the fact that the computation rule is generally not known precisely but (we hope) not lose too much information. Example code with delays ======================== ``` :- delay append(As, Bs, Cs) if var(As), var(Cs). append([], As, As). append(A.As, Bs, A.Cs) :- append(As, Bs, Cs). append3(As, Bs, Cs, ABCs) :- append(Bs, Cs, BCs), append(As, BCs, ABCs). :- delay reverse(As, Bs) if var(As), var(Bs). reverse([], []). reverse(A.As, Bs) :- append(Cs, [A], Bs), reverse(As, Cs). ``` [figrev] We now present two small examples of code which uses delays. The first will be used later to explain our techniques. Figure gives a version of `append` which delays until the first or third argument is instantiated. This delays (most) calls to `append` which have infinite derivations. Delaying such calls allows `append` to be used more flexibly in other predicates. For example, `append3` can be used to `append` three lists together or to split one list into three. Without the delay declaration for `append`, the latter “backwards” mode would not terminate. With the delay declaration, the first call to `append` delays. The second call then does one resolution step, instantiating variable `BCs`. This allows the first call to resume, do one resolution step and delay again, et cetera. In a similar way, this version of `reverse` works in both forwards and backwards modes—if either argument is instantiated to a list it will compute the other argument. If the second argument is instantiated, no calls are delayed. However, if only the first argument is instantiated, all the calls to `append` initially delay and after the last recursive call to `reverse` succeeds, the multiple calls to `append` proceed in an interleaved fashion. For any given mode, the code for `append3` and `reverse` can be statically reordered to produce a version which works without delaying. The Mercury compiler does such reordering automatically, but without automatic reordering it requires some slightly tricky coding to produce such flexible versions of these predicates. ``` submaxtree(Tree, NewTree) :- submaxtree1(Tree, Max, Max, NewTree). submaxtree1(nil, _, 0, nil). submaxtree1(t(L, E, R), GMax, Max, t(NewL, NewE, NewR)) :- submaxtree1(L, GMax, MaxL, NewL), submaxtree1(R, GMax, MaxR, NewR), max3(E, MaxL, MaxR, Max), plus(NewE, GMax, E). % delays; later mode o,i,i max3(A, B, C, D) :-... :- delay plus(A, B, C) if var(A), var(B) ; var(A), var(C) ; var(B), var(C). ``` Figure is a variant of the `maxtree` program (see, for example) which takes a tree and constructs a new tree containing copies of a logic variable in each node, then binds the variable to a number (the maximum number in the original tree). The `submaxtree` program fills each node in the new tree with the original value minus the maximum. This is done by delaying a call to `plus` for each node until the maximum is known, then resuming all these delayed calls. We assume a version of `plus` which delays until two of its three arguments are instantiated; NU-Prolog has such a predicate built in. All calls to `plus` become sufficiently instantiated at the same time (when `GMax` becomes instantiated). In most systems they will be called in the order they were delayed. If `plus` only worked in the forward mode the calls would not be sufficiently instantiated and the computation would flounder. We also assume a predicate `max3/4` which calculates the maximum of three numbers. It is not possible to statically reorder the clause bodies to eliminate the delays. Even dynamic reordering clause bodies each time a clause instance is introduced (also known as a local computation rule) is insufficient. Without coroutining, two passes over the tree are necessary, doubling the amount of “boilerplate” traversal code—the first to compute `GMax` and the second to build the new tree. Analysis of code with delays ============================ Delays can be used to write concise and flexible code, the behaviour of which can be very subtle. For example, shows that when bugs are introduced to a four-clause permutation program with delays, a wide variety of counter-intuitive behaviour results. Even with such a tiny program, the combination of interleaved execution and backtracking makes understanding why it misbehaves very challenging. Another tiny example is the (arguably correct) definition of `reverse` in Figure [figrev]. Having first written and used equivalent code around twenty-five years ago, the author did not become fully aware of its floundering properties until the preparation of this paper. It was incorrectly thought that all calls to `reverse` which have an infinite number of solutions with different list lengths (such as `reverse([a,b,c\|Xs],Ys)`) would flounder. Section [secusingf] gives a precise characterisation of the actual behaviour. Automated methods of analysis of code with delays are highly desirable because manual analysis is just too complex to be reliable. Analysis of code with delays can address many different issues. It may be that we expect code to succeed or finitely fail for certain classes of goals but some such goals may actually flounder, typically with a computed answer less instantiated than expected—this is the main focus of the declarative debugging work of, also discussed in section [sec:ddis]. In section [sec:elim] we give a transformation which allows analysis of computed answers (of successful or floundered derivations) which can detect such cases. Conversely, we may expect certain goals to flounder when actually they succeed. This is particularly important if the goal has an infinite number of solutions and is part of a larger computation—success can result in non-termination where floundering would not. In section [sec:cfs] we give a further transformation which captures floundering precisely. Both methods reduce the problem of analysing a program with delays to analysing the success set of a program without delays. A deeper understanding of floundering can also help us in other ways. For example, although the declarative debugger of doesn’t rely on either of these transformations directly, it is based on the insights of this paper. Similarly, these insights may help us optimise code, either by simplifying delay annotations or, more significantly, eliminating them entirely (possibly with some reordering of code). They may also help our understanding of termination properties of code with delays. Semantics and computed answers ============================== ccc P1 & P2 & P3 & & `p(a).` &`p(X).` & `p(a).` & & `p(X).` `q(a).` &`q(a).` & `q(a).` Before proceeding further, with more technical material, we make an observation about computed answers which is fundamental to our work. The conventional approach to the semantics of logic programs is that the set of function symbols in the language is precisely that in the program—see, for example, the textbook which combines and refines some of the original work of van Emden, Kowalski, Apt and others. This means that, unlike in Prolog, new function symbols cannot occur in the goal (or the semantics of the program differs depending on the goal). An alternative is to define the (typically infinite) set of function symbols a priori and assume that both the program and goals use a subset of these function symbols. This approach has been examined in, where various results are given, and earlier in, where forms of equivalence of logic programs are explored. For example, Figure gives three programs with different sets of computed answers for `p(Y)`. They are all equivalent using the “Lloyd” declarative semantics but with extra function symbols programs P2 and P3 are equivalent but P1 is not. One advantage of the latter semantics is that the universal closure of a goal is true if and only if it succeeds with a computed answer substitution which is empty (or simply a renaming of variables)—see the discussion in. Another is that the model-theoretic and fixed-point semantics can capture information about (non-ground) computed answers and, as we will show later, floundering! Since the Lloyd semantics deals only with sets of ground atoms, this fact is somewhat surprising, and does not seem to have been exploited for program analysis until now. We partition the set of function symbols into program function symbols and extraneous function symbols. Programs and goals may only contain program function symbols. Program atoms are atoms containing only program function symbols. Substitutions in derivations (including computed answer substitutions) contain only program function symbols. [obspfs] This is a consequence of most general unifiers being used (indeed, programming with delays makes little sense without this). Non-ground computed answers can be identified in the success set by the presence of extraneous function symbols. For example, if ⋈ is an extraneous function symbol then an atom such as `p(\bowtie)` appears in the success set if and only if it is an instance of some non-ground computed answer. If we assume there are an infinite number of terms whose principal function symbol is an extraneous function symbol then computed answers can be captured more precisely—we make this assumption later for our analysis of floundering. Note this semantics cannot determine whether a variable exists in all computed answers (or derivations) of a goal—in both P2 and P3 of Figure the success set contains `p(\bowtie)` and `p(a)`. However, it does precisely capture groundness in all computed answers (or variables in some computed answer), a property which has attracted much more interest. For example, many consider it of interest that in all computed answers of `append`, if the third argument is ground the second argument is also ground. Using the semantics we suggest, this is equivalent to saying if ⋈ occurs in the second argument it also occurs in the third argument. If we can find a superset of the success set (for example, a model) which has this property, the groundness dependency must hold. Thus a small variation to the Lloyd semantics leads to significant additional precision while retaining the simple model-theoretic and fixed-point semantics and the relationship between them. SLDF resolution =============== A model of Prolog with delays, SLDF resolution, is presented in. Here we review the model and main results, concerning ground atoms, and extend these result to non-ground atoms. We define the non-ground flounder set, which approximates the floundering behaviour of a program. However, we first give discuss two important closure properties which hold for SLD resolution (where there is no floundering). [Closure properties] [clprops] If an atom A has a successful SLD derivation with computed answer A (an empty computed answer substitution) then, using the same program clauses in the derivation * any atom with A as an instance has a computed answer with A as an instance, and * any atom Aθ has a computed answer Aθ. Such properties allow computed answers to be captured precisely by the set of computed answers of maximally general atoms, and generally simplifies analysis. When delays are introduced (SLDF resolution), only closure property 2 holds for successful atoms—a less instantiated version of a successful atom may flounder rather than succeed. For floundered atoms only closure property 1 holds (see proposition [propGthca])—an instance of a floundered atom may succeed, loop, finitely fail or flounder with an even more instantiated floundered computed answer. The weaker closure properties (compared to SLD resolution) means it is harder to precisely characterise the behaviour of SLDF resolution using sets of atoms. We now review SLDF resolution, define the set of atoms we use to approximate its behaviour and show the relationship between the two. SLDF resolution is similar to SLD resolution (see ), but the computation (atom selection) rule is restricted to be safe: an atom may only be selected if it is in the “callable atom set”. It is desirable that this set is closed under instantiation and the results below and those in this paper rely on this property. This property seems quite intuitive and holds for most logic programming systems with flexible computation rules. Another restriction suggested in is that all ground atoms should be callable. While this is not required for our technical results, it is a pragmatic choice. SLDF derivations can be failed, successful, infinite or floundered, in which the last resolvent consists only of atoms which are not callable (we say it is immediately floundered). Given the assumption above, for a program P the following sets of ground atoms can be defined independently of the (safe, and also fair in the case of finite failure) computation rule: * The success set SS(P) (ground atoms with successful derivations). * The finite failure set FF(P) (ground atoms with finitely failed SLD trees). * The flounder set FS(P) (ground atoms with floundered derivations). Note that some atoms in FS(P) may also be in SS(P) and have infinite (fair) derivations. The fact that floundering is independent of the computation rule suggests it is a declarative property in some sense. However, it has not been fully exploited for analysis until now, perhaps due to the lack of non-procedural definitions of FS(P). Note also that the results above only refer to ground atoms. An atom such as `q(X)` may have floundered derivations but no instance may appear in FS because no ground instance flounders. However, FS can contain information about floundering of non-ground atoms and conjunctions. For example, if the program contains the definition `p :- q(X)`, FS will contain `p` if and only if `q(X)` flounders. Relying on the existence of such definitions is a problem unless we know a priori which goals we want to analyse, and FS gives us no information about substitutions in floundered derivations. Substitutions in floundered (sub)computations can influence termination and are very important for certain programming styles, particularly those associated with parallel programming. Most Prolog systems which support delays print variable bindings at the top level for both successful and floundered derivations. Thus we use the term(s) computed answer (substitution) for both successful and floundered derivations, explicitly adding the words “successful” or “floundered” where we feel it aids clarity. The analysis proposed in this paper can be seen as being based on the following generalisation of FS. The non-ground flounder set, NFS(P), of a program P is the set of program atoms which have floundered derivations with empty floundered computed answer substitutions. Successful derivations can be conservatively approximated by simply ignoring delays—the lack of closure property 1 for successful atoms and the fact that an atom may have both successful and floundered derivations prevents our approach being more precise for analysis of success. The key results of this section, propositions [propGthNFS] and [propGthca], show how NFS(P) contains much but not all information about computed answers of floundered derivations. The results in, and some we prove in this paper, rely on the notion of two derivations of the same goal using the same clause selection. Used in in the context of successful derivations, we formalise it here. We assign each clause in the program a unique positive integer and use zero for the top level goal. We annotate each atom used in a derivation with a superscript to indicate the sequence of clauses and atoms within those clauses used to introduce it. Annotations are lists of pairs ⟨c, a⟩, where c is a clause number and a is the number of an atom within the clause. We use these annotations for both SLD and SLDF resolution. The annotation si of an atom used in a derivation is as follows. If goal A1s1, A2s2, …Ansn is the top level goal, si = ⟨0, i⟩:nil. Applying substitutions to atoms does not change their annotation. If Ai is selected and resolved with a variant of clause number j, H ← B1, B2, …Bk, each Bm atom is annotated with ⟨j, m⟩:si. Two atoms in different derivations or goals are corresponding atoms if they have the same annotation. Two derivations with the same top level goal, or with one top level goal an instance of the other, have the same clause selection if all pairs of corresponding selected atoms in the two derivations are matched with the same clause. Although not explicitly stated, the proofs in can easily be adapted to show that (successful and floundered) computed answers of successful and floundered derivations are independent of the computation rule (see Lemma [lemmgiGth]). For any two successful or floundered derivations of the same goal with the same clause selection but a different (safe) computation rule, each selected atom in one has a corresponding atom selected in the other. Derivation length, computed answer substitutions and the last resolvent are all the same, up to renaming. [lemmgiGth] Suppose D is the SLD derivation G, G1α1, G2α2, …, GNαN, …, where αi is the composition of the most general unifiers used in the first i steps, and Dʹ is the SLD derivation Gθ, Gʹ1αʹ1, Gʹ2αʹ2, …, GʹNαʹN, …, where θ affects only variables in G. Suppose also that D and Dʹ use the same set of program clauses and corresponding set of selected atoms in the first N steps. Then GʹNαʹN, is a most general instance (m.g.i.) of GNθ and GNαN, and GθαʹN is a m.g.i of Gθ and GαN. Let Hi ← Bi, 1 ≤ i ≤ N, be the ith program clause variant used in the first N steps of D, Ci, 1 ≤ i ≤ N, be the atom in a body of one of these clauses or in G whose corresponding instance is selected and matched with Hi, and Ri, 1 ≤ i ≤ K be the atoms as above corresponding to those in GN (they are not selected in the first N steps). Let terms C and H be as follows, where the connectives and predicate symbols are mapped to function symbols. $$\begin{aligned} \nonumber C & = & f(H\_1, B\_1, H\_2, B\_2, \ldots, H\_N, B\_N, C\_1, C\_2, \ldots, C\_N, R\_1, R\_2, \ldots, R\_K) \\ \nonumber H & = & f(H\_1, B\_1, H\_2, B\_2, \ldots, H\_N, B\_N, H\_1, H\_2, \ldots, H\_N, R\_1, R\_2, \ldots, R\_K)\end{aligned}$$ A m.g.i. of C and H can be obtained by left to right unification of the arguments of C and a variant of H which shares no variables with C. The first 2N argument unifications yield a renaming substitution, resulting in the same variants of program clause heads and bodies in the instances of H and C. The next N are the same as the unifications in D, modulo the clause variants used, and the other unifications yield empty unifiers. Thus CαN is a m.g.i. of C and H. Other orders of the arguments Ci and Hi (or unification orders) correspond to different computation rules but result is the same most general instance, or a variant. So, by the same construction, CαʹN is a m.g.i. of H and Cθ, and must be an instance of CαN. CαN is an instance of H so CαʹN must be a m.g.i. of Cθ and CαN. The instances of the initial goals and Nth resolvents can be extracted from the arguments of CαN, Cθ and CαʹN so the result follows. We can now show something similar to the converse of closure property 1, for floundering. This allows us to infer certain information about program behaviour from NFS(P). [propGthNFS] If D is the floundered SLDF derivation G, G1α1, G2α2, …, GNαN with floundered computed answer Gθ, then Gθ has a floundered SLDF derivation Dʹ with a renaming (or empty) floundered computed answer substitution (Gθ ∈ NFS(P)). Let Dʹ be a derivation using the same selection and computation rule as D. Dʹ cannot flounder before N steps because the ith resolvent is an instance of Giαi and the callable atom set is closed under instantiation. Dʹ cannot fail before N steps because the ith resolvent is no more instantiated than GiαN, and αN is a unifier of all pairs of calls and clause heads in the first N steps. Consider the Nth resolvent, GʹNαʹN. By Lemma [lemmgiGth], GʹNαʹN is a m.g.i. of GNθ and GNαN and since GNαN is an instance of GNθ, GʹNαʹN must be a variant of GNαN, so it is immediately floundered. Similarly, GθαʹN is a variant of Gθ, so the floundered computed answer substitution is a renaming. [lemFiinst] If Gθ has a floundered SLDF derivation Dʹ with the last resolvent being Fʹ1, Fʹ2, …, Fʹk and G has a SLDF derivation D using the same clause selection rule then each Fʹi is an instance of all its corresponding atoms in D. If G is immediately floundered the result is trivial. We use induction on the length of Dʹ. For length 0, since the callable atom set is closed under instantiation and Gθ is immediately floundered, G must also be immediately floundered. Assume it is true for length N. Suppose the first selected atom in D is A. Aθ is also callable, so we can construct a derivation Dʺ using the same clause selection as Dʹ but with Aθ as the first selected atom. The lengths of Dʺ and Dʹ are equal and their last resolvents are variants due to the result stated earlier. The first resolvent in Dʺ (after selecting Aθ) is an instance of the first resolvent in D and has a derivation of length N so the result follows. We can now show that closure property 1 holds for floundering: [propGthca] If Gθ has a floundered SLDF derivation Dʹ with a renaming (or empty) floundered computed answer substitution (Gθ ∈ NFS(P)), then G has a floundered SLDF derivation D with a floundered computed answer with Gθ as an instance. From G we can construct a derivation D using the same clause selection as that used in Dʹ and any safe computation rule. The callable atom set is closed under instantiation so by Lemma [lemFiinst], any atom selected in D must have a corresponding atom selected in Dʹ and thus D cannot be successful or longer that Dʹ. D uses the variants of the same clauses used in Dʹ, which has a more (or equally) instantiated top level goal, Gθ, so D cannot be failed. It must therefore be floundered and have a computed answer with Gθ as an instance. From these propositions we know that an atom A will flounder if and only if it has an instance in NFS(P). Also, the maximally general instances of A in NFS(P) will be floundered computed answers. The imprecision of NFS(P) with respect to floundered computed answers is apparent when there are atoms in NFS(P) which are instances of other atoms in NFS(P). If NFS(P) = {p(f(X)), p(f(f(f(X))))} for example, we know p(Y) will have the first atom as a floundered computed answer. The second atom may also be a floundered computed answer (via a different floundered derivation) or it may only be returned for more instantiated goal such as p(f(f(X))). In practice, there is usually a single maximally general instance of a goal in NFS(P) and this is the only answer computed, even when there are an infinite number of instances. For example, the non-ground flounder set for `append` has an infinite number of instances of the atom `append(A, [B], C)`, including `append(Xs, [Y], Zs)`, `append([X1\|Xs], [Y], [X1\|Zs])` and `append([X1, X2\|Xs], [42], [X1, X2\|Zs])`, but only the first is computed. Converting floundering into success =================================== We now present a program transformation which converts a program P, with delays, into a program Pʹ, without delays. The success set of Pʹ is the union of the success set of P and a set isomorphic to the non-ground flounder set of P. Thus analysis of some properties of programs with delays can be reduced to analysis of programs without delays. The SF() transformation --------------------------- Type, groundness and other dependencies are of interest in programs with and without delays as they give us important information concerning correctness. In the version of naive reverse without delays, analysis can tell us that in all computed answers of `reverse/2` both arguments are lists. In the delaying version (Figure [figrev]) this is not the case, since there are floundered computed answers where both arguments are variables. This increases the flexibility of `reverse/2` since it can delay rather than computing an infinite number of answers (this is particularly important when `reverse/2` is called as part of a larger computation). In (successful and floundered) computed answers for the delaying version of `reverse/2`, the first argument of is a list if and only if the second argument is a list. This tells us that if either argument is a list in a call, the other argument will be instantiated to a list by the `reverse/2` computation (assuming it terminates). If the delay declaration for `append/3` was changed so it delayed if just the first argument was a variable, `reverse/2` would not work backwards. It would flounder rather than instantiate the first argument to a list and the “if” part of this dependency would not hold. This section shows how a program with delays can be very simply transformed into a program without delays which can be analysed to reveal information such as this. Analysis of success in a program without delays cannot give us information about (non-ground) delayed calls directly because success is closed under instantiation (closure property 2) whereas floundering is not. However, extraneous function symbols allow us to encode non-ground atoms using ground atoms, re-establishing this proposition and allowing analysis. The encoding uses an isomorphism between the (infinite) set of variables and the set of terms with extraneous principal function symbols (this set must also be infinite to avoid loss of precision in the encoding; it is sufficient to have a single extraneous function symbol with arity greater than zero). The encoded flounder set (EFS(P)) of a program P is the set of ground instances of atoms in NFS(P) such that distinct variables are replaced by distinct terms with extraneous principal function symbols. NFS(P) can be reconstructed from the atoms in EFS(P) by finding the set of most specific generalisations which contain only program function symbols. For example, the non-ground flounder set for append contains atoms such as `append(Xs, [Y], Zs)` whereas the encoded flounder set contains atoms such as `append(\bowtie, [\otimes(1)], \otimes(2))`, assuming ⋈ and ⊗ are extraneous function symbols. We introduce two new “builtin” predicates, `evar/1` and `enonground/1`, which are true if their argument is an encoded variable or non-ground term, respectively. For simplicity, our treatment assumes they are defined using an (infinite) set of facts: `evar(T)` for all terms T where the principal function symbol is not a program function symbol and `enonground(T)` for all terms T which have at least one extraneous function symbol. This can cause an infinite branching factor in SLD trees (for example, a call such as `evar(X)`). However, since in this paper we deal with single derivations but not SLD trees (or finite failure), it causes us no difficulties. ``` evar('VAR'(_)). enonground(A) :- evar(A). enonground([A\|B]) :- enonground(A). enonground([A\|B]) :- enonground(B). ``` [figevar] It is also possible to define `evar/1` and `enonground/1` in Prolog. Figure gives a definition which assumes `’VAR’/1` is the only extraneous function symbol of the original program and `’.’/2` is the only program function symbol with arity greater than zero for the original program (if there are other such function symbols, more clauses are needed for `enonground/1`). These definitions depart from our theoretical treatment in that they can involve deeper proof trees (due to recursive calls) and they can have non-ground computed answers. However, they can be useful for observing floundering behaviour, especially with a fair (or depth-bounded) search strategy—see section [secusingf]. We now define the SF() transformation: Given a program P (not defining predicates `evar/1` or `enonground/1`) containing delay declarations, SF(P) is the program with all clauses of P plus, for each delay declaration `:- delay A if C` in P, the clause `A :- C’` where `C’` is `C` with `var` replaced by `evar` and `nonground` replaced by `enonground`. These additional clauses introduced for delay declarations and those in the definitions of `evar/1` and `enonground/1` are referred to as delay clauses. ``` append_sf(As, Bs, Cs) :- evar(As), evar(Cs). append_sf([], As, As). append_sf(A.As, Bs, A.Cs) :- append_sf(As, Bs, Cs). reverse_sf(As, Bs) :- evar(As), evar(Bs). reverse_sf([], []). reverse_sf(A.As, Bs) :- append_sf(Cs, [A], Bs), reverse_sf(As, Cs). ``` To avoid possible confusion, the code in this paper uses “`_sf`” suffixes for the new predicate definitions; our theoretical treatment assumes the original predicate names are used for the new predicate definitions. For example, Figure shows the transformed version of `reverse` (from Figure [figrev]). Figures [figextrad] and [figextradu] give further examples. Immediately floundered atoms in NFS(P) have matching delay declarations with true right hand sides. Corresponding (encoded) atoms in EFS(P) have matching ground delay clause instances with successful bodies. We have described how `evar` and `enonground` behave. Some languages have delay conditions which cannot be expressed using var and nonground. For example, in NU-Prolog `X ~= Y` delays whereas `X ~= X` does not. To analyse such constructs we need additional primitives similar to `evar` and `enonground`. The key to designing such constructs is that the delay clauses should implement the encoding as defined above. Properties of SF() ---------------------- The following propositions show how successful derivations in SF(P) correspond to successful or floundered derivations in P: the success set of SF(P) is the union of the success set of P without delays and the encoded flounder set of P (Proposition [propsssfp]). Note that when we talk of successful derivations and/or SS(P) here, SLD resolution rather than SLDF resolution is used (delays are ignored when dealing with success). The lack of closure property 2 is problematic when dealing with success if delays are considered and SLDF resolution used. [propnodeliff] A goal G has a successful SLD derivation D with program P (ignoring delays) if and only if it has a successful derivation D with SF(P) which uses no delay clauses. SF(P) without delay clauses is the same as P without delays. We now deal with floundering, which is more complex. [lemimfl] A goal G is immediately floundered with program P if and only if it has a successful derivation D with SF(P) which uses only delay clauses. Follows from the way in which delay clauses implement the encoding of the flounder set. [lemsubsefs] A goal G which is immediately floundered with program P has a computed answer substitution θ in SF(P) such that all variables bound by θ are bound to distinct terms with extraneous principal function symbols (or are simply renamed). By Lemma [lemimfl], there is a derivation where all non-renaming substitutions are due to calls to `evar/1` and `enonground/1`. A call to `evar/1` binds its argument to a term with an extraneous principal function symbol. Multiple calls with distinct variables will have some of the infinite number of computed answers binding their arguments to distinct terms. Similarly, some computed answers to `enonground/1` will bind all distinct variables in its argument to distinct terms with extraneous principal function symbols. [lemnfsiffsfdel] Given a program P, a goal G has a floundered derivation D with an empty floundered computed answer substitution if and only if it has a successful derivation Dʹ with SF(P) in which delay clauses are selected and the successful computed answer, Gθ, is such that all variables bound by θ are bound to distinct terms with extraneous principal function symbols (or are simply renamed). (Only if) Derivation D can be reproduced with SF(P) since it has all the clauses of P and the computation rule is unrestricted. By Lemma [lemsubsefs] the last resolvent in D must have a successful derivation such that the computed answer substitution has the desired property. (If) By repeated application of the switching lemma to Dʹ we can construct a successful derivation Dʺ = G, G1, G2, …, Gn with SF(P) such that callable atoms are selected in preference to atoms which would delay in P. The derivation has a prefix D = G, G1, G2, …, Gm where only callable atoms are selected, except for Gm, which would be immediately floundered in P (a delay clause is used in Dʺ so an immediately floundered goal must be reached at some stage). Callable atoms are not matched with delay clauses (by Lemma [lemimfl], if a callable atom is resolved with a delay clause the resolvent cannot succeed). Variables bound by the computed answer substitution θ of Dʺ are bound to distinct terms with extraneous principal function symbols (or simply renamed), and all of the non-renaming bindings must be due to delay clauses. Thus D is a floundered SLDF derivation in P with an empty (or renaming) answer substitution. [lemsfallsubs] Given a program P, a goal G has a successful derivation with SF(P) with computed answer, Gθ, such that all variables bound by θ are bound to distinct terms with extraneous principal function symbols (or are simply renamed) if and only if there are successful derivations with all such computed answers. All such substitutions are due to delay clauses and the sets of `enonground/1` and `evar/1` atoms which succeed are closed under the operation of replacing one extraneous function symbol with another. [propfliffsfdel] Goal G has a floundered SLDF derivation D with program P if and only if it has a successful derivation Dʹ with SF(P) in which delay clauses are selected. Propositions [propGthNFS] and [propGthca] imply G flounders if and only if an instance flounders with an empty floundered computed answer substitution so by Lemma [lemnfsiffsfdel] it is sufficient to show that an instance of G has a successful derivation with SF(P) in which delay clauses are selected and all variables bound by the computed answer substitution are bound to distinct terms with extraneous principal function symbols (or are simply renamed) iff G has a successful derivation Dʹ with SF(P) in which delay clauses are selected. (Only if) By closure property 1. (If) Consider a derivation G, G1α1, G2α2, …, GNαN using the same clause selection as in Dʹ but with a computation rule such that atoms resolved with delay clauses are selected at the end, from Gmαm. By Lemma [lemmgiGth], Gαm has a derivation where the mth resolvent is a variant of Gmαm and the substitution at that point is a renaming substitution for Gαm. By Lemma [lemsubsefs], a computed answer substitution for Gmαm has the desired property. [propsssfp] For any program P, SS(SF(P)) = EFS(P) ∪ SS(P). The set of atoms in SS(SF(P)) with derivations which don’t use delay clauses is SS(P) by Proposition [propnodeliff]. The set of atoms in SS(SF(P)) with derivations which use delay clauses is EFS(P) by Lemmas [lemnfsiffsfdel] and [lemsfallsubs]. Note that although there is a bijection between successful SLD derivations with P and successful SLD derivations with SF(P) which don’t use delay clauses, there is not a bijection between floundered SLDF derivations with P and successful derivations with SF(P) which use delay clauses, even if multiple solutions to `evar/1` and `enonground/1` are ignored. SF(P) generally has additional derivations. This is unavoidable due to the imprecision of NFS(P) mentioned in Section [sec:sldf]. ``` p(X, Y) :- q(X), q(Y). p_sf(X, Y) :- q_sf(X), q_sf(Y). :- delay q(V) when var(V). q_sf(V) :- evar(V). q(a). q_sf(a). ``` [figextrad] For example, consider the definition of `p/2` in Figure. The success set, ignoring delays, is {`p(a,a)`} and NFS(P) is {`p(X,Y)`, `p(a,V)`, `p(V,a)`}. Thus the computed answers of `p_sf(X,Y)` encode all these four atoms, since SF(P) computes the union of the success set and the encoded flounder set. However, `p(X,Y)` only has one floundered derivation, with the empty answer substitution. The other two atoms in NFS(P) are only computed for more instantiated goals and the derivations in SF(P) correspond to these computations (the same atoms are selected, ignoring `evar/1` and `enonground/1`). ``` p :- q(X). p_sf :- q_sf(X) :- delay q(V) when var(V). q_sf(V) :- evar(V). q(a). q_sf(a). q(X) :- q(X). q_sf(X) :- q_sf(X). ``` [figextradu] It is also possible to have successful derivations in SF(P) which do not correspond to any SLD or SLDF derivation in P. For example, in Figure, the goal `p` has a single floundered SLDF derivation, where `q(X)` immediately flounders, whereas `p_sf` has an infinite number of derivations which use delay clauses and the derivations of `q(X)` have unbounded length. This is related to the fact that `p` has an infinite SLD tree. Analysis using SF(P) -------------------------- Type dependencies of SF(P) can be analysed in the same ways as any other Prolog program. The following set of atoms, where l(X) means X is a list, is a model of the transformed reverse program, showing these type dependencies hold (and thus they hold for computed answers in the original reverse program with delays): {append(A, B, C)∣(l(A) ∧ l(B)) ↔ l(C)} ∪ {reverse(A, B)∣l(A) ↔ l(B)} It is not necessary to consider the complex procedural semantics of Prolog with delays, or even the procedural semantics of Prolog without delays since bottom-up analysis can be used. Similarly, the SF() transformation makes it relatively easy to show that `submaxtree/2` can indeed compute a tree of integers when given a tree of integers as the first argument. Groundness in P can also be analysed by analysing SF(P) using specialised types. We can define the type ground to be the set of terms constructed from only program function symbols. The dependencies which hold for lists above also hold for type ground, indicating the corresponding groundness dependencies hold for computed answers of reverse with delays. Similarly, nonvar can be defined as the set of terms with a program principal function symbol. By extending the type/mode checker described in we have demonstrated it is possible to check non-trivial useful properties of P by checking models of SF(P). For more complicated cases it is necessary to support sub-types, as nonvar and list are both subtypes of ground. This approach to groundness analysis is not reliant on the SF() transformation—it can be applied to any logic program due to Observation [obspfs]. The analysis can be identical to conventional groundness analysis using Boolean functions because logic programs can be abstracted in an identical way. A unification X = f(Y1, Y2, …, YN) can be abstracted by X ↔ Y1 ∧ Y2 ∧ … ∧ YN, assuming f/N is a program function symbol. Calls evar(X) and enonground(X) can be abstracted as X ↔ False. New characterisations of the flounder set ========================================= We now present a second transformation, which allows us to capture the non-ground flounder set more precisely. The F() transformation ------------------------ The results here suggest a solution to the open problem posed in : how the flounder set can be defined inductively. Such a definition may be a very useful basis for analysis of floundering as an alternative to a purely model theoretic approach. The semantics of SF(P) captures both successful and floundered derivations of P. By defining a variant of the immediate consequence operator TP we can distinguish atoms with derivations which use delay clauses. An f-interpretation is a set of ground atoms, some of which may be flagged (to indicate floundering). If I is an f-interpretation, A(I) is the set of atoms in I and FA(I) is the set of atoms in I which are flagged. The union of two f-interpretations I and J is the f-interpretation K such that A(K) = A(I) ∪ A(J) and FA(K) = FA(I) ∪ FA(J). Given a program P, TPf is a mapping from f-interpretations to f-interpretations, defined as follows. A(TPf(I)) = TSF(P)(A(I)) and an atom A in this set is flagged if there is a ground instance of a clause in SF(P), A ← B1, B2, …, Bk, such that each Bi is in I and some Bi is flagged in I or if the predicate of A is `evar/1` or `enonground/1`. TPf ↑ n and TPf ↑ ω are defined in the same way as TP ↑ n and TP ↑ ω. [propflagiffsfdel] A ground atom other than `evar/1` or `enonground/1` is flagged in TPf ↑ n if and only if it has a proof tree of height ≤ n in SF(P) which uses a delay clause, and is flagged in TPf ↑ ω if and only if it has a successful derivation in SF(P) which uses a delay clause. A standard result is that TSF(P) ↑ ω (and TSF(P) ↑ n) contains exactly those ground atoms with proof trees in SF(P) (of height ≤ n, respectively). A(TPf ↑ n) = TSF(P) ↑ n since ∀ I A(TPf(I)) = TSF(P)(I). From the definition of TPf, these atoms are flagged if and only if they are derived using `evar/1` or `enonground/1`, that is, if a delay clause is used in the derivation. A ground atom A other than `evar/1` or `enonground/1` is flagged in TPf ↑ ω if and only if it is in the encoded flounder set of P. By Proposition [propflagiffsfdel] it is sufficient to show that A ∈ EFS(P) iff A has a successful derivation D in SF(P) which uses a delay clause. If: The decoded version of A (distinct terms with extraneous principal function symbols are replaced by distinct variables), B, has a successful derivation Dʹ in SF(P) using the same clause selection as that in D, with a computed answer Bθ, which has A as an instance. Since Bθ has A as an instance, any variables bound by θ must be bound to distinct terms with extraneous principal function symbols (or simply renamed). Thus Dʹ satisfies the condition of Lemma [lemnfsiffsfdel] so B is in NFS(P). Only if: A ∈ EFS(P), so A = Bγ, where B ∈ NFS(P). By Lemmas [lemnfsiffsfdel] and [lemsfallsubs], B has a derivation which uses delay clauses and has a computed answer with an instance Bγ. A has a successful derivation using the same clause selection. Thus we have an inductive/fixed-point characterisation of (a set isomorphic to) the non-ground flounder set. It may be practical to base floundering analysis on TPf. It is monotonic with respect to the set of atoms (A(TPf(I)) ⊆ A(TPf(Iʹ)) if A(I) ⊆ A(Iʹ)) and for a given set of atoms it is monotonic with respect to the flagged atoms in the set (FA(TPf(I)) ⊆ FA(TPf(Iʹ)) if A(I) = A(Iʹ) and FA(I) ⊆ FA(Iʹ)). Monotonicity is important for the structure of fixed-points, particularly the existence of a least fixed-point. Alternatively, the definition of TPf can be mirrored by a further transformation which produces a program whose success set is the encoded flounder set of P. An advantage is that it can then be analysed using standard techniques. A disadvantage is that the transformation increases the program size, which will affect analysis time. Given a Horn clause program P, F(P) is the program consisting of the predicate definitions in SF(P) (we assume each predicate has a sf subscript/postfix) plus the following new definitions. For each clause psf(X̄)`:-`B in SF(P) we add a clause pf(X̄)`:-`Bʹ. For delay clauses, Bʹ = B. For other clauses, Bʹ = B, D, where D is the disjunction of all calls in B, with “`_sf`” replaced by “`_f`”. If B is the empty conjunction (`true`) then Bʹ is the empty disjunction (`fail`). ``` append_f(As, Bs, Cs) :- evar(As), evar(Cs). append_f([], As, As) :- fail. append_f(A.As, Bs, A.Cs) :- append_sf(As, Bs, Cs), append_f(As, Bs, Cs). reverse_f(As, Bs) :- evar(As), evar(Bs). reverse_f([], []) :- fail. reverse_f(A.As, Bs) :- reverse_sf(As, Cs), append_sf(Cs, [A], Bs), (reverse_f(As, Cs) ; append_f(Cs, [A], Bs)). ``` Figure gives the new clauses generated for `reverse`. Note that we assume the original program consists of only Horn clauses but the transformed program contains disjunctions. These could be eliminated by further transformation. The transformation is designed so that TF(P) (extended to handle disjunctions) is essentially the same as TPf: flagged atoms correspond to the f subscripted predicates and the set of all atoms corresponds to the sf subscripted predicates. The success set of the f subscripted predicates in F(P) is the encoded non-ground flounder set of the corresponding predicates in P. Analysis using F(P) ----------------------- The transformation allows us to observe the floundering behaviour of the original program very clearly. If we define `evar` as in Figure [figevar] and run the goal `append_f(X,Y,Z)` using a fair search strategy, we get computed answers of the form `X = [A`1`,A`2`,...,A`N`\|'VAR'(B)]`, `Y = C`, `Z = [A`1`,A`2`,...,A`N`\|'VAR'(D)]`. Occurrences of `’VAR’/1` in answers correspond to variables in computed answers of floundered derivations of the original program and variables correspond to arbitrary terms. Thus a call to `append` flounders if and only if it has an instance such that the first and third arguments are “incomplete lists” (lists with a variable at the tail rather than nil) of the same “length” with pair-wise identical elements. For example, `append(X,[a],[a\|Z])` flounders (it also has a successful derivation) whereas `append([a,V\|X],Y,[V,b\|Z])` does not. Running `reverse_f` we discovered to our surprise (as mentioned in section [sec:examples]) that `reverse` flounders if and only if the first argument is an incomplete list and the second argument is a variable (rather than incomplete list). A call such as `reverse(X,[a\|Y])` returns an infinite number of answers rather than floundering! With a suitably expressive domain the transformed program can be analysed with established techniques to obtain precise information about the original program with delays. Powerful techniques have been developed to help construct domains. For example, we can start with a simple domain containing four types: lists, var (the complement of our type nonvar), incomplete lists (this is a supertype of var), and a “top” element (the universal type). Completing this domain using disjunction adds two additional elements: “list or var” and “list or incomplete list”. The Heyting completion of this domain introduces implications or dependencies such as X is a list if Y is a list. This domain can be used as a basis for interpretations of the program and to infer and express useful information about floundering. reversef(X, Y) =�	arxiv_0000808
Rubidium abundances of galactic disk stars ========================================== Introduction ============ Rubidium (Rb, Z = 37) is a neutron-capture element (in which both s- and r-processes are involved), whose abundances spectroscopically determined for various types of stars may provide us with useful information regarding mixing of nuclear-process products or galactic chemical evolution. However, published spectroscopic studies of stellar Rb abundances have been rather limited in number, which may presumably reflect the comparatively large technical difficulty of its abundance determination (i.e., weak and blended line feature has to be dealt in many cases). Those investigations on Rb abundances so far have focused mainly on metal-poor stars. Following Gratton & Sneden (1994) who determined the abundances of Rb (along with other neutron-rich elements) for 19 stars at − 2.8 < [Fe/H] < 0, Tomkin & Lambert (1999) conducted a more detailed Rb abundance analysis for 44 metal-deficient giants and dwarfs in the metallicity range of − 2.0 < [Fe/H] < 0.0. These previous studies revealed that the [Rb/Fe] ratio tends to be moderately supersolar ($0 \lesssim$ [Rb/Fe] $\lesssim 0.5$) at the metal-poor regime ([Fe/H] $\lesssim -1$). Successively, Rb abundances of globular cluster giants (Yong et al. 2006, 2008; D’Orazi et al. 2013) or AGB giants (García-Hernández et al. 2006; Shejeelammal et al. 2020) were also investigated. On the other hand, in contrast to such metal-poor halo stars or peculiar AGB stars, the behaviours of Rb abundances in galactic disk stars (mildly metal-deficient through metal-rich regime) seem to have been poorly investigated. What can be read from a small number of samples at $-0.5 \lesssim$ [Fe/H] $\lesssim 0.0$ in previous studies quoted above is that [Rb/Fe] tends to be slightly subsolar by 0.1–0.3 dex (see, e.g., Fig. 2 in Gratton & Sneden 1994 or Fig. 3 in Tomkin & Lambert 1999). In this context, the problem of reference Rb abundance should be mentioned, which might have something to do with this trend. That is, Gratton & Sneden (1994) as well as Tomkin & Lambert (1999) similarly remarked that the solar Rb abundance[1](#fn1) A⊙ ≃ 2.6, which they determined from the solar spectrum and used as the fiducial abundance for evaluation of [Rb/Fe] ( ≡ A\* − A⊙ − [Fe/H]), is larger than the solar-system Rb abundance ($A\_{\rm s.s.} \simeq 2.4$) well established from meteorites, though they could not find out the cause for this difference. If the latter $A\_{\rm s.s.}$ were used as the zero-point abundance, the subsolar tendency would be mitigated in the direction of [Rb/Fe] ∼ 0. Recently, an interesting finding has been reported by Abia et al. (2020), who examined the Rb abundances of 57 M dwarfs in the metallicity range $-0.3 \lesssim$ [Fe/H] $\lesssim +0.3$ observed with CARMENES. The result they obtained was somewhat surprising: The [Rb/Fe] ratios of many stars turned out to be subsolar by a few tenths dex and show an increasing tendency towards higher metallicity from [Rb/Fe] ∼ − 0.4 (at [Fe/H] ∼ − 0.3) to [Rb/Fe] ∼ + 0.2 (at [Fe/H] ∼ + 0.3). Such a trend is unusual for a neutron-capture element and can not be explained even by the state-of-the-art chemical evolution models. So, if this is real, a significant revision would be required on the theoretical side. Yet, given that reliability of abundance determinations in M dwarfs has not been firmly established, follow-up verification should be necessary. Abia et al. (2020) themselves remarked “Additional Rb abundance measurements in FGK dwarfs of near solar metallicity, as well as an evaluation of the impact of stellar activity on abundance determinations in M dwarfs, are urgently needed to confirm or disprove the main findings of this study.” This problem attracted the author’s interest and eventually motivated this study. Conveniently, observational data for Rb abundance determination are already available for a large number of FGK-type stars belonging to the disk population (field dwarfs as well as giants, Hyades cluster stars) in the relevant metallicity range ($-0.6 \lesssim$ [Fe/H] $\lesssim +0.3$), for which atmospheric parameters were also established in the previous papers. Accordingly, it was decided to determine the Rb abundances for these sample stars and check up their behaviours, while paying attention also to examining the credibility of resulting abundances. Specifically, the purpose of this investigation was to clarify the following points: * Do the Rb abundances resulting from various Hyades stars of F-, G-, and K-type agree with each other? This would serve as a reliability check for the adopted procedure of abundance determination. * Are the Rb abundances derived from dwarf and giant stars consistent with each other at the same [Fe/H]? * Is the solar photospheric Rb abundance determined from the spectrum of the Sun (or Moon) agree with the solar-system (meteorite) abundance? Or does it show an appreciable discrepancy such as found in previous studies? * Finally, how do the [Rb/Fe] ratios of disk stars (especially those of near-solar metallicity) behave itself with a change in [Fe/H]? Could the trend reported by Abia et al. (2020) be confirmed? Program stars and their parameters ================================== A total of 487 stars were selected from the samples of previous studies of the author, which are divided into 4 groups (group 1 — 47 Hyades FGK dwarfs, group 2 — 100 field FGK dwarfs, group 3 — 101 field GK dwarfs, and group 4 — 239 field GK giants) as summarised in Table 1, where the references to the original papers are also given for more detailed information. Regarding the basic observational data, the spectra obtained by using HIDES (HIgh-Dispersion Echelle Spectrograph) attached to the 188 cm reflector at Okayama Astrophysical Observatory were used in most cases, though those acquired by HDS (High-Dispersion Spectrograph) with 8.2 m Subaru Telescope were partly used for 13 (out of 47) group 1 stars and 17 (out of 101) group 3 stars. As to the reference solar spectrum, the Moon spectrum[2](#fn2) included in Okayama spectrum database (Takeda et al. 2005a) was employed along with Neckel’s (1994, 1999) solar disk-centre spectrum and Kurucz et al.’s (1984) solar flux spectrum. The atmospheric parameters [$T\_{\rm eff}$ (effective temperature), logg (surface gravity), $v\_{\rm t}$ (microturbulence), and [Fe/H] (metallicity)] of the program stars were spectroscopically determined in the same consistent manner from the equivalent widths of Fe i and Fe ii lines by using the TGVIT program (Takeda et al. 2005b). These stellar parameters are given in the online materials (tableE1,dat, tableE2.dat, tableE3.dat, and tableE4.dat) for each of the 4 groups. As seen from the parameter ranges for each group indicated in Table 1, while the temperature ranges of group 1 and group 2 stars widely extend from F- through K-type, those of group 3/group 4 are limited to a rather narrow span of ∼ 1000 K (i,e., ∼ 5500–4500 K corresponding to late/mid G–mid/early K). Similarly, the same atmospheric models were used as employed in the original references; they are based on Kurucz’s (1993a) ATLAS9 model atmosphere grids (computed with the convective overshooting effect), which were interpolated in terms of $T\_{\rm eff}$, logg, and [Fe/H] for each star. Abundance determination ======================= Outline of the method --------------------- Regarding determinations of Rb abundances and related quantities, the resonance Rb i line at 7800 Å (which comprises multiple components and is blended with the Si i line) is exclusively used in this study.[3](#fn3) The procedures are essentially the same as those adopted in the author’s previous papers (e.g., Takeda & Kawanomoto 2005, Takeda et al. 2015, Takeda & Honda 2020). First, by applying Takeda’s (1995) numerical algorithm, the best fit between the synthetic and observed spectra is accomplished in the specified wavelength region (7776–7804 Å) while varying the abundances of important elements (A(Ni), A(Si), A(Rb) and A(Fe) in the present case), $v\_{\rm M}$ (macrobroadening velocity[4](#fn4)), and Δλ (radial velocity or wavelength shift). Then, the equivalent width (W)[5](#fn5) of the relevant Rb i 7800 line is “inversely” evaluated from the best-fit solution of A(Rb) with the same atmospheric model/parameters as used in the spectrum-fitting analysis. From such evaluated W, the non-LTE abundance ($A^{\rm N}$) and LTE abundance ($A^{\rm L}$) are calculated, from which the non-LTE correction is derived as $\Delta \equiv A^{\rm N} - A^{\rm L}$. Besides, this W can be further used to estimate the abundance uncertainties due to ambiguities of atmospheric parameters by perturbing the standard values interchangeably. Adopted data of spectral lines ------------------------------ Regarding the atomic data (wavelengths, oscillator strengths) of the Rb i 7800 line, Abia et al.’s (2020) Table 1 was invoked. The gf values of the component lines were scaled by assuming the solar-system (meteoritic) isotope ratio of 85Rb/87Rb = 2.43 as done by them. These scaled data are listed in Table 2. As for the damping parameters, the default treatment of Kurucz’s (1993a) WIDTH9 program was applied, though they are essentially insignificant in the present case because this Rb i line is generally weak. Otherwise, the data compiled in the VALD database (Ryabchikova et al. 2015) were used for the lines (other than Rb i 7800) contained in the 7776–7804 Å region. The exceptions were the following three Si i lines ($\chi\_{\rm low} = 6.181$ eV) whose gf values were empirically adjusted: Si i 7798.837 (loggf = − 2.06), Si i 7799.180 (loggf = − 1.75), and Si i 7799.996 (loggf = − 0.69). The abundances of all elements other than Ni, Si, Rb, and Fe were fixed at the metallicity-scaled solar abundances in the spectrum-fitting analysis. Non-LTE calculations -------------------- The statistical-equilibrium calculations for neutral rubidium were carried out in the similar manner as previously done for the lighter alkali elements: Li (Takeda & Kawanomoto 2005), K (Takeda et al. 1996; 2002), and Na (Takeda et al. 2003). For the present study, Rb i model atom was constructed comprising 19 terms (up to 11s 2S at 31917 cm− 1) and 59 radiative transitions, while using Kurucz & Bell’s (1995) compilation of atomic data (gf values, levels, etc.). Regarding the treatment of the photoionisation cross section, the hydrogenic approximation was assumed, except for the edge values of the ground term (5s 2S) and the first excited term (5p 2P∘), for which Lowell et al. (2002) and Nadeem & Haq (2011) were invoked, respectively. As to the collisional rates, the recipe described in Sect. 3.1.3 of Takeda (1991) was basically followed, while the H i collision rates computed with the classical Drawin’s cross section (cf. Steenbock & Holweger 1984) was multiplied by a factor of k = 10− 3 (i.e., suppressed to a negligible level).[6](#fn6) Since FGK-type stars (dwarfs as well as giants) of population I in the galactic disk are concerned in this study, the calculations were done on a grid of 125 (5 × 5 × 5) model atmospheres resulting from combinations of five $T\_{\rm eff}$ values (4500, 5000, 5500, 6000, 6500 K), five logg values (1.0, 2.0, 3.0, 4.0, 5.0), and five [Fe/H] values (+0.5, 0.0, − 0.5, − 1.0, − 1.5). The microturbulent velocity was assumed to be $v\_{\rm t}$ = 2 km s− 1, and the metallicity-scaled Rb abundance ([Rb/Fe] = 0) was used as A(Rb) = 2.60 + [Fe/H], where 2.60 is Anders & Grevesse’s (1989) solar Rb abundance adopted in ATLAS9 models. The depth-dependent non-LTE departure coefficients to be used for each star were then evaluated by interpolating this grid in terms of $T\_{\rm eff}$, logg, and [Fe/H]. Results ------- The Rb abundances were successfully determined for almost all cases, except for 3 stars in group 2 (HD 120136, 124850, 016673; these are F-type dwarfs with $T\_{\rm eff} \gtrsim 6300$ K and the Rb i 7800 line is very weak), for which A(Rb) solution was not converged and thus had to be fixed in the fitting process. Representative examples of spectrum fitting in the whole 7796–7804 Å region are shown in Fig. 1 (for the disk-centre Sun, disk-integrated Sun, Moon, and 8 dwarfs/giants in the wide $T\_{\rm eff}$ range). The close-up display of the accomplished fit in the 7799–7801 Å region is presented for all program stars in Fig. 2 (groups 1, 2, and 3) and Fig. 3 (group 4). After the A(Rb) solution has been established, the related quantities were further derived as described in Sect. 3.1: W (equivalent width), Δ (non-LTE correction), $A^{\rm N}$ (non-LTE abundance), δT ± (abundance changes for $T\_{\rm eff}$ perturbations by ± 100 K), δg ± (abundance changes for logg perturbations by ± 0.1 dex), and δv ± (abundance changes for $v\_{\rm t}$ perturbations by ± 0.5 km s− 1). Each of them are plotted against $T\_{\rm eff}$ in Fig 4 (group 1), Fig. 5 (group 2), Fig. 6 (group 3), and Fig. 7 (group 4). These results of W, Δ, $A^{\rm N}$ (along with $v\_{\rm M}$ derived from spectrum fitting as by-product) for stars of each group are presented in the online tables (tableE1.dat, tableE2.dat, tableE3.dat, and tableE4.dat). An inspection of Fig. 4–7 suggests the following trends which generally hold for all program stars irrespective of dwarfs or giants. * The equivalent width of Rb i 7800 (W) decreases with an increase in $T\_{\rm eff}$, reflecting that the occupation number of the Rb i ground level is quite $T\_{\rm eff}$-sensitive ($\propto 10^{+5040 \chi^{\rm ION}/T\_{\rm eff}}$ where $\chi^{\rm ION}$ = 4.18 eV is the ionisation potential of Rb i). * The non-LTE correction (Δ) progressively increases as $T\_{\rm eff}$ is lowered: While it is negative (Δ ∼ − 0.1 dex: non-LTE line strengthening) for F-type stars, it becomes positive (Δ ∼ + 0.1 dex: non-LTE line weakening) for K-type stars. As such, the non-LTE effect on Rb abundance determination ($\|\Delta\| \lesssim 0.1$ dex) is insignificant as far as stars at 4500 $\lesssim T\_{\rm eff} \lesssim 6500$ K are concerned. * Among the various sources of abundance errors, most important is δT ± ( ∼ ± 0.1 dex for $\Delta T\_{\rm eff} = \pm 100$ K; tending to enhance toward lower $T\_{\rm eff}$), which can be naturally understood from the large $T\_{\rm eff}$-sensitivity of W mentioned above. Meanwhile, δg ± and δv ± are totally insignificant. The Rb abundances determined in this investigation are compared with those derived by Tomkin & Lambert (1999) in Fig. 8 for 6 stars in common, where a satisfactory match is observed for most cases. One exception showing a rather large deviation ( ∼ 0.3–0.4 dex) is the G5V star μ Cas (= HD 6582), for which Rb abundance determination is considerably difficult because the Rb line is very weak (W = 1.6 mÅ) due to its low metallicity ([Fe/H] = − 0.8). Discussion ========== Solar Rb abundance problem -------------------------- Let us first discuss the solar rubidium abundance to be used as the reference standard, for which appreciable discrepancy from the meteoritic composition was reported in previous work (cf. Sect. 1). The non-LTE Rb abundance determined from Kurucz et al.’s (1984) solar flux spectrum is $A^{\rm N}\_{\odot} = 2.44$ (W = 5.9 mÅ, Δ = − 0.05 dex), that from Neckel’s (1994, 1999) solar disk-centre spectrum (direction cosine μ = cosθ = 1) is $A^{\rm N}\_{\odot} = 2.48$ (Wμ = 1 = 5.7 mÅ, Δ = − 0.05 dex), while that derived from Takeda et al.’s (2005) Moon spectrum is $A^{\rm N}\_{\odot} = 2.56$ (W = 7.7 mÅ, Δ = − 0.06 dex), resulting in a Sun − Moon difference of ∼ 0.1 dex despite that a satisfactory fit appears to be accomplished similarly for both cases (cf. Fig. 1). This suggests that Rb abundance determination from a severely blended feature is a delicate matter influenced by the difference in the fitting region (which had to be narrowed in the latter case owing to the spectrum defect in the short-wavelength side) or in the spectrum resolving power. Accordingly, these spectroscopically established solar Rb abundances are higher than the solar-system abundance ($A\_{\rm s.s.} = 2.36$; cf. Lodders 2020) by ∼ 0.1–0.2 dex. As such, the discrepancy between A⊙ and $A\_{\rm s.s.}$ reported by Gratton & Sneden (1994) and Tomkin & Lambert (1999) has been confirmed in this study. In contrast, Abia et al. (2020) and Korotin (2020) claimed that this problem has been settled, since they obtained $A^{\rm N}\_{\odot} = 2.35 \pm 0.05$ which is in very good agreement with $A\_{\rm s.s.}$ of 2.36. This is mainly because they adopted a more enhanced non-LTE correction (Δ = − 0.12 dex) than the value ( − 0.05 dex) mentioned above, which reflects the fact that their non-LTE abundance corrections for FGK-type stars are systematically shifted downward by ∼ 0.1 dex compared to those derived in this study (see Appendix A where this difference is discussed more in detail). Yet, it is not clear whether this solar abundance issue is simply attributed to the non-LTE effect, because the situation appears to be more complex and some other $T\_{\rm eff}$-dependent problem may be involved, as described in the next two sections. Implication from Hyades dwarfs ------------------------------ Comparing the Rb abundances of group 1 stars (47 FGK dwarfs belonging to the Hyades cluster which distribute at 4500 K $\lesssim T\_{\rm eff} \lesssim 6300$ K) with each other makes a valuable test for checking the validity of abundance determination, because similar abundances should be observed for such cluster stars born from the gas of same chemical composition. An inspection of Fig. 4c suggests an interesting trend that (i) $A^{\rm N}$ values for Hyades dwarfs at $T\_{\rm eff} \lesssim 5500$ K (mid-G through K type) are almost similar irrespective of $T\_{\rm eff}$, while (ii) those at $T\_{\rm eff} \gtrsim 5500$ K (F through early G type) show an increasing tendency towards higher $T\_{\rm eff}$. Regarding the former lower $T\_{\rm eff}$ group, it is reasonable to regard that the abundances are reliably established. Actually, the mean abundance ($\langle A^{\rm N} \rangle = 2.56 \pm 0.10$) derived for 18 stars of $T\_{\rm eff} < 5500$ K is consistent with the expected primordial Rb abundance of this cluster ($A\_{\rm s.s.}$ + [Fe/H]$\_{\rm Hyades}$), where $A\_{\rm s.s.}$ is the solar system abundance (2.36) and [Fe/H]$\_{\rm Hyades}$ is the Hyades metallicity ( ≃ + 0.2; cf. Takeda & Honda 2020). As to the latter higher $T\_{\rm eff}$ group, it should be remarked that errors involved with the resulting abundances must be larger, especially for F-type stars ($T\_{\rm eff} \gtrsim 6000$ K), because the line feature becomes considerably weak (due to higher $T\_{\rm eff}$) and broad/shallow (due to larger rotational velocity).[7](#fn7) Accordingly, those data points showing prominent deviations (e.g., $A^{\rm N} \gtrsim 3$ or $A^{\rm N} \lesssim 2$) should not be seriously taken. Even so, a global trend is recognised from Fig. 4c that $A^{\rm N}$ gradually increases from ∼ 2.6 (at $T\_{\rm eff} \sim 5500$ K) to ∼ 2.9 (at $T\_{\rm eff} \sim 6300$ K), which means that Rb abundances in this $T\_{\rm eff}$ range suffer some $T\_{\rm eff}$-dependent systematic errors (i.e., overestimation with its extent growing towards higher $T\_{\rm eff}$). Abundance distribution of near-solar metallicity stars ------------------------------------------------------ In order to check on this problem further for other field stars, stars with metallicities around the solar value ( − 0.3 < [Fe/H] < + 0.3) were sorted out from each group (2, 3, and 4) and the distributions of their Rb abundances were examined. The resulting $A^{\rm N}$ vs. $T\_{\rm eff}$ plots and the histograms of $A^{\rm N}$ are displayed in Fig. 9, where the subset of especially close-to-solar-metallicity stars ( − 0.1 < [Fe/H] < + 0.1) are separately shown with the pink colour. Let us pay attention to Fig. 9a, which shows that the Rb abundances derived for group 2 stars of near-solar metallicity (most of them are F–G dwarfs of $T\_{\rm eff} \gtrsim 5500$ K) tend to systematically increase with $T\_{\rm eff}$, just like the case of Hyades dwarfs discussed in the previous section. It has thus been corroborated that the Rb abundances of $T\_{\rm eff} \gtrsim 5500$ K dwarfs are not so trustable in the sense that they tend to be overestimated (this effect begins at ∼ 5500 K and grows with increasing $T\_{\rm eff}$). The cause of this embarrassing effect is not clear, which is limited only to stars of $T\_{\rm eff} \gtrsim 5500$ K (while the lower $T\_{\rm eff}$ side is unaffected). Attributing it to improper non-LTE corrections seems unlikely, because adjustment only at the higher $T\_{\rm eff}$ side by changing the parameters of non-LTE calculations (e.g., collision cross section) is difficult. As another possibility, the problem may be related to the structure of model atmosphere affected by convection, because the strength of such weak (i.e., deep-forming) and T-sensitive Rb i 7800 line is sensitive to the condition of the lower photosphere, which critically depends upon the treatment of convection (i.e., mixing length, overshooting) especially in F-type stars. The models adopted in this study are based on Kurucz’s (1993a) standard ATLAS9 model atmosphere grid computed with the mixing length of l = 1.25Hp (Hp: pressure scale height) by including the convective overshooting effect (OVERWT=1). Although this “with overshooting” model is known to represent the solar observational data fairly well, it is not necessarily adequate for the higher-$T\_{\rm eff}$ F-type star Procyon (F5 IV–V), for which “no overshooting” model seems more suitable (cf. Takeda et al. 1996). So, with the consideration that no-overshooting model may be more preferable especially for F-type stars, Rb abundances were re-determined by using no-overshooting (OVERWT=0) ATLAS9 models for all 47 Hyades stars (group 1) in order to see how much abundance changes [$\delta\_{\rm os} \equiv A$(no overshooting) − A(with overshooting)] would result. Interestingly, as shown by red crosses in Fig. 4b, the use of no-overshooting models tends to decrease A ($\delta\_{\rm os} < 0$) at $T\_{\rm eff} \gtrsim 5500$ K with the extent of variation ($\|\delta\_{\rm os}\|$) increasing with $T\_{\rm eff}$ (up to $\delta\_{\rm os} \sim 0.04$ dex at $T\_{\rm eff} \sim 6300$ K) (which reflects that switching-off overshooting tends to strengthen the line because of the steeper T-gradient), while abundances are almost unaffected ($\delta\_{\rm os} \sim 0$) at $T\_{\rm eff} \lesssim 5500$ K. Accordingly, switching-off the convective overshooting tends to act in the direction of mitigating the $T\_{\rm eff}$-dependent trend (Fig. 4c) at least qualitatively, though still insufficient in the quantitative sense. In any event, it would be promising to improve the modelling of convective atmosphere (e.g., by using state-of-the-art 3D hydrodynamical simulations) toward resolving the $T\_{\rm eff}$-dependent systematic effect seen in the Rb abundances. [Rb/Fe] vs. [Fe/H] relation --------------------------- Before discussing the behaviour of [Rb/Fe], we turn to the problem of reference abundance already mentioned in Sect. 4.1, where the solar abundance ($A\_{\odot}^{\rm N}$) was shown to be by ∼ 0.1–0.2 dex higher than the solar-system abundance ($A\_{\rm s.s.}$). It is worth pointing out that this difference may be explained by the trend seen in the $A^{\rm N}$ vs. $T\_{\rm eff}$ relation described in Sect. 4.2 and Sect. 4.3. That is, since the Sun is in the $T\_{\rm eff}$ range ($\gtrsim 5500$ K) where Rb abundances suffer $T\_{\rm eff}$-dependent overestimation effect, $A\_{\odot}^{\rm N}$ is expected to be overestimated by ∼ 0.1 dex (as seen from the gradient observed in Fig. 4c), which may be the cause for such an apparent discrepancy (despite that the true photospheric Rb abundance is the same as $A\_{\rm s.s.}$). Accordingly, careful consideration is required regarding the fiducial Rb abundance adopted for evaluation of [Rb/Fe]. One possibility might be to use the solar abundance ($A\_{\odot}^{\rm N}$) as usual, while limiting the sample to those of mid-G through F type (like most of the group 2 stars). In this case, however, while a reasonable zero point is accomplished since the centre of Rb abundance distribution is well represented by $A\_{\odot}^{\rm N}$, a larger dispersion would result because abundance data of different $T\_{\rm eff}$-dependent overestimation are mixed (cf. Fig. 9d). Therefore, as an alternative approach, all objects at $T\_{\rm eff} \gtrsim 5500$ K were decided not to use for [Rb/Fe] analysis in this paper. That is, (i) the sample stars are restricted only to those of group 3 and group 4 (i.e., dwarfs and giants at $T\_{\rm eff} \lesssim 5500$ K), and (ii) [Rb/Fe] ratio is calculated by referring to $A\_{\rm s.s.}$ (instead of $A\_{\odot}^{\rm N}$) as [Rb/Fe] $\equiv A^{\rm N} - 2.36-[{\rm Fe}/{\rm H}]$, which is a reasonable choice because $A\_{\rm s.s.}$ well matches the centre of Rb abundance distribution of near-solar metallicity stars in group 3 and group 4 (cf. Fig. 9e and Fig. 9f). The resulting [Rb/Fe] vs. [Fe/H] relations obtained as such are illustrated in Fig. 10, from which the following characteristics are observed. * A similar tendency of gradually increasing [Rb/Fe] for a decrease in [Fe/H] is observed for both group 3 dwarfs (Fig. 10a) and group 4 giants (Fig. 10b), though a somewhat larger scatter is seen for the former. * A quantitative inspection of the mean ⟨[Rb/Fe]⟩ values at each metallicity bin in the [Fe/H] range of $-0.4 \lesssim$ [Fe/H] $\lesssim +0.2$ (cf. Fig. 10d and Fig. 10e) suggests that (1) d[Rb/Fe]/d[Fe/H] gradient is ∼ − 0.4 at $-0.4 \lesssim$ [Fe/H] $\lesssim 0.0$, (2) [Rb/Fe] appears rather flat ( ∼ 0) at $0.0 \lesssim$ [Fe/H] $\lesssim +0.2$, and (3) this trend almost equally holds for both dwarfs and giants (though ⟨[Rb/Fe]⟩ for the former tends to be slightly higher than the latter by $\lesssim 0.05$ dex). * This equality of [Rb/Fe] (at the same [Fe/H]) for group 3 and group 4 stars implies that initial Rb abundances are retained in the stellar surface for both dwarfs or giants, which is understandable since AGB stars are not included in group 4 (many are red clump giants). * Such a gentle increase in [Rb/Fe] from ∼ 0 (at [Fe/H] ∼ 0) to ∼ + 0.2 (at [Fe/H] ∼ − 0.5) is favourably (at least in the qualitative sense) compared with the theoretical prediction from the recent chemical evolution model (depicted by the solid line in Fig. 10a and Fig. 10b). * In summary, the overall behaviour of the [Rb/Fe] ratios in galactic disk stars (dwarfs as well as giants) around the solar metallicity, gradually increasing with a decrease in [Fe/H], is almost consistent with the result of theoretical simulation. Comparison with previous studies -------------------------------- In Fig. 10c are plotted the published results of [Rb/Fe] ratios against [Fe/H], which were reported in three previous papers (cf. Sect. 1): Gratton & Sneden (1994), Tomkin & Lambert (1999), and Abia et al. (2020). Regarding Gratton & Sneden (1994) and Tomkin & Lambert (1999), only a small number of disk stars ($-1 \lesssim$ [Fe/H] $\lesssim 0$) were included in their sample (since their interest was placed mainly on metal-poor halo stars of [Fe/H] $\lesssim -1$) which tend to show subsolar (negative) [Rb/Fe] ratios by a few tenths dex. This may be due to the fact (at least partly) that they used the solar abundance (2.6) as the reference abundance for the zero point of [Rb/H], which is by 0.2 dex higher than the meteoritic abundance that would have been more appropriate (note that most of their sample stars are at $T\_{\rm eff} \lesssim 5500$ K). In contrast, the consequence obtained in this study is in apparent conflict with Abia et al.’s (2020) [Rb/Fe] vs. [Fe/H] correlation derived from M dwarfs of near-solar metallicity, showing a controversial trend that [Rb/Fe] (being appreciably subsolar by several tenths dex) steeply increase towards higher metallicity within a narrow range of $-0.3 \lesssim$ [Fe/H] $\lesssim +0.3$. It is hard to understand why they obtained such a conclusion. It should be remarked here that their [Rb/Fe] analysis seems to be based on LTE abundances (while using the LTE solar abundance of 2.47 as the reference) despite that they also derived non-LTE Rb abundances by applying (negative) non-LTE corrections ranging from − 0.28 to − 0.13 dex (tending to decrease towards lower $T\_{\rm eff}$; cf. Fig. 4 therein). Since their non-LTE correction for the Sun is − 0.12 dex, non-LTE [Rb/Fe] would be shifted further downward (by ∼ 0.1 dex on the average), making the anomaly even more exaggerated, which may be the reason why they did not employ non-LTE abundances. It seems premature, however, to regard that the validity of their non-LTE correction results derived for M dwarfs is established. Since the non-LTE formation of the Rb i resonance line is determined by the delicate interplay between two opposite effects of line weakening and line strengthening (cf. Appendix A), its net result could be sensitive to detailed computational conditions. For example, Pavlenko & Magazzù (1996) showed in their line formation calculation for the Li i 6708 line in G–M dwarfs and subgiants that the non-LTE effect can either strengthen or weaken the line at $T\_{\rm eff} \lesssim 4000$ K. Accordingly, “positive” non-LTE corrections might be possible depending on the situation also for the case of Rb i line formation in M dwarfs, by which the apparent anomaly of subsolar [Rb/Fe] at issue could be mitigated. Summary and conclusion ====================== Spectroscopic studies of stellar rubidium abundances in context of galactic chemical evolution have been rather scarce, presumably due to the technical difficulty of its abundance determination where weak and blended Rb i line has to be generally invoked. Recently, Abia et al. (2020) determined the Rb abundances of 57 M-type dwarfs of $-0.3 \lesssim$ [Fe/H] $\lesssim +0.3$ and reported the anomalous trends of [Rb/Fe] ratios, being appreciably subsolar by several tenths dex and increasing towards higher [Fe/H], which can not be explained by any current theory of galactic nucleosynthesis. With an aim of examining whether such a tendency is actually observed in FGK-type stars, Rb abundances were determined by applying the spectrum fitting method to the Rb i 7800 line for ∼ 500 late-type dwarfs as well as giants (including Hyades cluster stars) belonging to the galactic disk population ($-0.6 \lesssim$ [Fe/H] $\lesssim +0.3$). The Rb abundance of the Sun determined from the solar spectrum turned out to be by ∼ 0.1–0.2 dex higher than the solar system abundance derived from meteorites, as already reported in the earlier work by Gratton & Sneden (1994) and Tomkin & Lambert (1999). This difference may be related to the $T\_{\rm eff}$-dependent effect seen in F–G dwarfs (see below). An inspection of the results derived for Hyades dwarfs revealed that they show a systematic $T\_{\rm eff}$-dependent trend at $\gtrsim 5500$ K, which means that the Rb abundances derived for early-G to F stars (including the Sun) are likely to suffer some overestimation increasing towards higher $T\_{\rm ef}$. This might be due to some imperfect treatment of surface convection in classical model atmospheres, because T-sensitive Rb i line is weak and forms deep in the photosphere where the structure is critically affected by how the convection is treated (especially in F-type stars). For this reason, in the present study of [Rb/Fe], the final samople was limited to only late-G and K stars of $T\_{\rm eff} \lesssim 5500$ K and the meteoritic abundance (instead of the solar abundance) was employed as the reference Rb abundance, The resulting [Rb/Fe] vs.[Fe/H] relations for field G–K dwarfs and giants turned out to be similar to each other, both showing a gradual increase of [Rb/Fe] with a decrease in [Fe/H] (with a d[Rb/Fe]/d[Fe/H] gradient of ∼ − 0.4 around the solar metallicity). This trend is almost consistent (at least qualitatively) with the theoretical prediction of chemical evolution models. Accordingly, this study could not confirm the anomalous behaviours of [Rb/Fe] derived by Abia et al. (2020). The reason why they obtained such results for M dwarfs is yet to be clarified. Acknowledgments =============== This research has made use of the SIMBAD database, operated by CDS, Strasbourg, France. 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[h] \| Group \| 1 \| 2 \| 3 \| 4 \| \| --- \| --- \| --- \| --- \| --- \| \| Number of stars \| 47 \| \100 \| 101 \| 239 \| \| Star type \| Hyades cluster stars \| field stars \| field stars \| $^{\S}$field stars \| \| Sp. class \| mid F–mid K \| early F-early K \| late G–mid K \| mid G–early K \| \| Lum. class \| dwarfs \| dwarfs (+subgiants) \| dwarfs \| giants \| \| $^{\dagger}T\_{\rm eff}$ \| [4525, 6307] \| [5150, 6968] \| [4495, 5482] \| [4494, 5624] \| \| †logg* \| [4.08, 4.71] \| [3.38, 4.72] \| [4.22, 4.73] \| [1.42, 3.49] \| \| †[Fe/H] \| [ − 0.02, +0.33] \| [ − 0.90, +0.46] \| [ − 1.27, +0.29] \| [ − 0.77, +0.19] \| \| #Obs. data \| TH20 \| \T05a \| TH20 \| T15 \| \| #Parameters \| TH20 \| T05b \| TH20 \| TSM08 \| \| Remarks \| validity check for A(Rb) \| not used for [Rb/Fe] \| adopted for [Rb/Fe] \| adopted for [Rb/Fe] \| $^{\S}$Including 4 Hyades giants. †Parenthesised values indicate [minimum, maximum]. #Previous papers of the author, which describe the observational data and atmospheric parameters of each star: TH20 ⋯ Takeda & Honda (2020), T05a ⋯ Takeda et al. (2005a), T05b ⋯ Takeda et al. (2005b), T15 ⋯ Takeda et al. (2015), TSM08 ⋯ Takeda et al. (2008). \Although the Okayama spectrum database published in T05a includes spectra of 160 FGK stars in total, many of those around ∼ 7800 Å are unfortunately damaged by serious bad columns in CCD. Only the data of 100 stars among them turned out to be somehow usable. [h] \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| λ \| $\chi\_{\rm low}$ \| loggf \| isotope \| \| (Å) \| (eV) \| (dex) \| \| \| 7800.183 \| 0.000 \| − 0.663 \| 87Rb \| \| 7800.186 \| 0.000 \| − 0.663 \| 87Rb \| \| 7800.188 \| 0.000 \| − 1.061 \| 87Rb \| \| 7800.317 \| 0.000 \| − 0.216 \| 87Rb \| \| 7800.322 \| 0.000 \| − 0.663 \| 87Rb \| \| 7800.325 \| 0.000 \| − 1.362 \| 87Rb \| \| 7800.233 \| 0.000 \| − 0.744 \| 85Rb \| \| 7800.234 \| 0.000 \| − 0.647 \| 85Rb \| \| 7800.235 \| 0.000 \| − 0.760 \| 85Rb \| \| 7800.292 \| 0.000 \| − 0.282 \| 85Rb \| \| 7800.295 \| 0.000 \| − 0.647 \| 85Rb \| \| 7800.296 \| 0.000 \| − 1.191 \| 85Rb \| \| 7947.507 \| 0.000 \| − 1.204 \| 87Rb \| \| 7947.524 \| 0.000 \| − 1.903 \| 87Rb \| \| 7947.651 \| 0.000 \| − 1.204 \| 87Rb \| \| 7947.668 \| 0.000 \| − 1.204 \| 87Rb \| \| 7947.563 \| 0.000 \| − 0.803 \| 85Rb \| \| 7947.570 \| 0.000 \| − 1.347 \| 85Rb \| \| 7947.626 \| 0.000 \| − 0.900 \| 85Rb \| \| 7947.634 \| 0.000 \| − 0.803 \| 85Rb \| These data are based on Abia et al.’s (2020) Table 1. The gf values are scaled by assuming 85Rb/87Rb = 2.43. Although the Rb i 7800 line is essentially invoked in the present study, the data for the Rb i 7947 line (the other line of the resonance doublet) are also shown because it is used in Appendix B. [fig:1] [fig:2] [fig:3] [fig:4] [fig:5] [fig:6] [fig:7] [fig:8] [fig:9] [fig:10] Non-LTE effect on the Rb I 7800 line ==================================== Physical mechanism of non-LTE line formation -------------------------------------------- As mentioned in Sect. 3.4, the non-LTE correction (Δ) for the Rb i 7800 line monotonically increases with a decrease in $T\_{\rm eff}$ from ∼ − 0.1 dex (F stars) to ∼ + 0.1 dex (K stars) while changing the sign at $T\_{\rm eff}\sim$ 5000–5500 K (Fig. 4b, Fig. 5b, Fig. 6b, and Fig. 7b). Although this extent of Δ is quantitatively insignificant, the physical reason for such a $T\_{\rm eff}$-dependent trend is worth explanation. In Fig. A1 are shown the $l\_{0}^{\rm NLTE}(\tau)/l\_{0}^{\rm LTE}(\tau)$ (the non-LTE-to-LTE line-centre opacity ratio; almost equal to $\simeq b\_{\rm 1}$) and $S\_{\rm L}(\tau)/B(\tau)$ (the ratio of the line source function to the Planck function; nearly equal to ≃ b2/b1) for the Rb i 5s 2S–5p 2P∘ transition (relevant to the Rb i 7800 line) computed on the models of different $T\_{\rm eff}$ and different logg while assuming [Rb/Fe] = [Fe/H] = 0, where b1 and b2 are the non-LTE departure coefficients for the ground and first-excited terms, respectively. Besides, the LTE and non-LTE equivalent widths ($W^{\rm L}$ and $W^{\rm N}$) for the Rb i 7800 line along with the corresponding non-LTE abundance correction (Δ) computed for logg = 2 and logg = 4 models are plotted against $T\_{\rm eff}$ in Fig. A2. The upper panels of Fig. A1 elucidate how the non-LTE effect influences the strength of Rb i 7800 line for models of different $T\_{\rm eff}$. In the present case of FGK stars where Rb i line is comparatively weak, two kinds of mechanisms are important both of which are caused by imbalance between J (local mean intensity) and B (Planck function at the local electron temperature): (i) overrecombination (J < B) and (ii) overionisation (J > B). As we go upwards from the deep optically thick layer (τ > 1, where J = B holds), J < B is first realized at the lower photosphere $\tau \lesssim 1$ (because of the dilution of radiation due to photon escape), but eventually the inequality J > B comes into effect at the higher optically thin region because J is almost stabilised while B(T) continues to decrease as T is lowered upwards. The net non-LTE effect on the number population of the ground level (lower level of the Rb i 7800 line) is determined by the interplay of these two mechanisms (i: line strengthening, ii: line weakening). In panels (a)–(c) of Fig. A1, the action of effect (i) is seen in the overpopulation bump at $10^{-1} \lesssim \tau \lesssim 1$, while that of effect (ii) is manifest in the systematic underpopulation observed at $\tau \lesssim 10^{-2}$. This implies that effect (i) is relatively more important at higher $T\_{\rm eff}$ (where the line is weak and deep-forming) while effect (ii) is more significant at lower $T\_{\rm eff}$ (line is stronger and its formation region is higher). Moreover, this $T\_{\rm eff}$-dependence of the relative importance between (i) and (ii) is further enhanced by the fact that the Planck function (B) around the ionisation edge becomes very T-sensitive as $T\_{\rm eff}$ is lowered (Wien region of the Planck function) which means that the inequality of J > B causing (ii) is realized more easily. What has been described above is sufficient for interpreting the non-LTE effect on the Rb i 7800 line in FGK stars and its dependence upon $T\_{\rm eff}$. Regarding F-type stars of higher $T\_{\rm eff}$ (cf. Fig. A1c), effect (i) is more important and acts in the direction of strengthening the line (Δ < 0). In contrast, effect (ii) is dominant in K-type stars of lower $T\_{\rm eff}$ (cf. Fig. A1a), where the line is weakened by the non-LTE effect (Δ > 0). For the case of G-type stars, both effects tend to compensate with each other, yielding Δ ∼ 0. As such, the trends of $W^{\rm L}$, $W^{\rm N}$, and Δ with a change of $T\_{\rm eff}$, which are displayed in Fig. A2, can be reasonably understood. Note also that the non-LTE effect tends to be enhanced as the surface gravity is decreased because of the lowered atmospheric density (implying less collision rates). Comparison with Korotin’s (2020) result --------------------------------------- Korotin (2020) has recently conducted non-LTE calculations for the Rb i resonance lines on an extensive model grid covering $T\_{\rm eff} = $ 3500–6500 K, logg = 1.0–5.0, and [Fe/H] from − 1.0 to +0.5, which were employed by Abia et al. (2020) for evaluating their non-LTE corrections applied to M dwarfs. Since his calculations also cover F-, G-, and K-type stars, it is worthwhile to compare the non-LTE corrections derived by both independent calculations with each other. By inspecting Fig. A2b along with Korotin’s (2020) Fig. 6a, the run of Δ at 4500 $\lesssim T\_{\rm eff} \lesssim 6500$ K for logg = 2 and logg = 4 can be compared with each other. This comparison suggests that, while the overall gradient (${\rm d}\Delta/{\rm d}T\_{\rm eff}$) as well as its logg-dependence is reasonably consistent, $\Delta\_{\rm Korotin}$ is systematically larger than $\Delta\_{\rm Takeda}$ by ∼ 0.1 dex. That is, as $T\_{\rm eff}$ is lowered from ∼ 6500 K (F-type star) to ∼ 4500 K (K-type star), $\Delta\_{\rm Takeda}$ increases from ∼ − 0.1 to ∼ + 0.1, while $\Delta\_{\rm Korotin}$ runs from ∼ − 0.2 to ∼ 0.0. The reason for this systematic difference can be understood by comparing the behaviour of b1(τ) in Korotin’s (2020) Fig. 3 or Fig. 7 with that of $l\_{0}^{\rm NLTE}(\tau)/l\_{0}^{\rm LTE}(\tau)$ (almost equivalent to b1) shown in the upper panels of Fig. A1. As can be recognised from this comparison done for the models of same $T\_{\rm eff}$, the overpopulation hump of b1 > 1 at the lower atmosphere is apparently more conspicuous while the underpopulation (b < 1) at the upper atmosphere is less pronounced in the results of Korotin (2020). This means that the overionisation effect (ii) is repressed relative to the overrecombination effect (i) in his calculation, which suggests that the evaluation of photoionisation rates may have been appreciably different. Unfortunately, Korotin (2020) did not describe any details about the calculation of photoionising radiation field (J), whereas the author followed the procedure described in Sect. 3.1.2 of Takeda (1991) by consulting Kurucz’s (1993a) ATLAS9 program along with the line opacity distribution function published by Kurucz (1993b). Finally, some comments regarding the non-LTE effect in M dwarfs may be in order. According to the calculation of Korotin (2020), δ( < 0) progressively increases as $T\_{\rm eff}$ is lowered until it approaches ∼ 0 at $T\_{\rm eff} \sim 4500$ K. However, it then shows a downturn and begins to ever decline with a decrease in $T\_{\rm eff}$ over the $T\_{\rm eff}$ range of ∼ 3500–4000 K corresponding to M-type stars. This is presumably due to the dilution effect of line source function ($S\_{\rm L}/B <1$) which becomes important as the line gets stronger and saturated (such as the case of strong K i resonance line at 7665/7699 Å). As such, the Δ values of M dwarfs calculated by Korotin (2020) remain always negative and this non-LTE effect (acting to strength the line) is enhanced towards lower $T\_{\rm eff}$, as shown in Fig. 4 of Abia et al. (2020). Yet, it might be possible that the overionisation effect (which acts in the direction of line weakening) was not sufficiently taken into account in Korotin’s (2020) calculation as mentioned above. This is the reason for having remarked the possibility of positive Δ in the last paragraph of Sect. 4.5. [fig:A1] [fig:A2] Comparison of the abundances derived from Rb I 7800 and 7947 lines ================================================================== As remarked in footnote 3 of Sect. 3.1, only the stronger line at 7800 Å was invoked in this investigation, while the weaker one of the doublet at 7947 Å was not used since reliable Rb abundance determination revealed to be generally more difficult for this line (i.e., because of the lower quality of observational data along with weaker line strength). Grevesse et al. (2015) reported in their comprehensive analysis of solar photospheric abundances of Cu through Th that the Rb abundace derived from Rb i 7800 is higher than that from Rb i 7947 by 0.11 dex. They thus suspected that the Rb i 7800 line might be blended with some other unknown line leading to a slight overestimation. It is interesting to examine whether such a difference is observed in the abundances derived from these two lines also for stars other than the Sun. Therefore, Rb abundances were determined from the Rb i 7947 line (by using the atomic data given in Table 2) for HIP 104214 (= 61 Cyg A) and ξ Boo B (K5 V dwarfs with $T\_{\rm eff}$ of 4523 K and 4495 K, respectively; lowest $T\_{\rm eff}$ stars in group 3) in order to compare them with those derived from Rb i 7800 line. Owing to sufficiently high S/N ratio (because these two stars are apparently bright) and comparatively large line strength (due to low $T\_{\rm eff}$), Rb abundances from this Rb i 7947 line could be satisfactorily established by the spectrum fitting in the 7945–7050 Å region (cf. Fig. B1), followed by evaluations of W, $A^{\rm N}$, and Δ as done in Sect. 3.4. The results obtained for two lines are presented in Table B1. Regarding ξ Boo B, $A^{\rm N}\_{7947}$ is appreciably larger than $A^{\rm N}\_{7800}$ by 0.15 dex (i.e., inversed inequality to that reported for the Sun). However, this should not be seriously taken, since it must be due to influence of a telluric line at ∼ 7947.7 Å (see the solar spectrum shown at the top of Fig. B1), which just contaminates the Rb i line at 7947.6 Å because of the stellar line shift ( ∼ + 0.1 Å) corresponding to the radial velocity of $V\_{\rm r}^{\rm topo} = +4.5$ km s− 1 (cf. Table B1). Meanwhile, Rb i 7947 line is essentially free from telluric blending for the case of HIP 104214 showing a considerably large radial velocity ($V\_{\rm r}^{\rm topo} = -82.0$ km s− 1). According to Table B1, $A^{\rm N}\_{7800}$ is slightly larger than $A^{\rm N}\_{7947}$ by +0.05 dex in HIP 104214. Although this trend (sign) of abundance difference is similar to what Grevesse et al. (2015) reported for the Sun, its extent is so insignificant that the abundances from these two lines may be regarded as practically equal. Therefore, care should be taken in interpreting the abundance discrepancy (0.11 dex) between Rb 7800/7947 lines found by Grevesse et al. (2015). Since the effect of telluric contamination is considerable in the solar spectrum (cf. Fig. B1), the problem might be on the side of the Rb i 7947 line (e.g., underestimation of the abundance due to inadequate removal of the telluric line) rather than the blending on the Rb i 7800 line suggested by Grevesse et al. (2015). [h] c r r@ c@ c r@ c@ c c lObject & $^{\}V\_{\rm r}^{\rm topo}$ &W* & Δ & $A^{\rm N}$ & W & Δ & $A^{\rm N}$ & $\delta A^{\rm N}\_{7800-7947}$ & Remark & (km s− 1) & (mÅ) & (dex) & (dex) & (mÅ) & (dex) & (dex) & (dex) & & & & & HIP 104214 & − 82.0 & 34.1 & +0.059 & 2.284 & 17.6 & +0.060 & 2.236 & + 0.048 & ξ Boo B& + 4.5 & 40.5 & +0.063 & 2.347 & 31.2 & +0.063 & 2.494 & − 0.147 & telluric contaminated (7947 line) \Topocentric radial velocity (stellar radial velocity relative to the observer). [fig:B1] --- 1. Throughout this paper, A* denotes the logarithmic number abundance in the usual normalisation of A(H) = 12.00.[↩](#fnref1) 2. Although this spectrum is damaged in the short-wavelength side of the important Rb i 7800 line like other spectra of many group 2 stars taken from Takeda et al.’s (2005a) database (cf. the note to Table 1), the whole profile of Si i+Rb i line feature turned out fortunately usable (cf. Fig. 1).[↩](#fnref2) 3. The other resonance line of the doublet at 7947 Å was not used, because it is weaker and the quality of observed spectra in this wavelength region is poor (e.g., contamination of telluric lines) compared to the Rb i 7800 line.[↩](#fnref3) 4. This $v\_{\rm M}$ (including instrumental broadening, rotational broadening, and macroturbulence) is the e-folding half-width of the Gaussian broadening function $f(v) \propto \exp[-(v/v\_{\rm M})^{2}]$.[↩](#fnref4) 5. It should be noted that this W is evaluated by integrating the synthesised line profile on the assumption that this Rb i 7800 line feature (comprising multiple components as listed in Table 2) is isolated, despite that it is actually blended with the neighbouring Si i line. Therefore, it is not so much a directly observable quantity in real spectra as rather a useful measure of line strength.[↩](#fnref5) 6. This k value was adopted by consulting the previous cases for the resonance lines of other alkali species; i.e., k = 10− 3 for Li i 6708 as well as K i 7699, k = 10− 1 for Na i 5890/5896. Actually, however, the choice of this correction factor does not matter much, because the resulting non-LTE effect turns out anyhow insignificant even by drastically reducing the H i collisions as such. For example, the ∣Δ∣ values are $\lesssim 0.1$ dex at most (cf. Sect. 3.4), which are further reduced by $\lesssim$ several hundredths dex if the H i collision effect is included with k = 1.[↩](#fnref6) 7. Note that the error bars attached to the data points in Fig. 4c represent only the errors due to ambiguities of atmospheric parameters and thus do not include those due to photometric noises or due to any systematic effect caused by imperfect fitting. The uncertainty of W (δW) due to random noise was evaluated by using the relation derived by Cayrel (1988) (as done in Sect. 4.2 of Takeda & Honda 2020), which depends on S/N, line width, and pixel size. For a typical S/N of ∼ 100, δW turned out to be ∼ 1–2 mÅin most cases ($\lesssim 3$ mÅ for the broad-line F-type stars of vsini ∼ 10–20 km s− 1). Accordingly, abundance results derived from very weak lines (i.e., W less than several mÅ) significantly suffer from this error and thus should not be seriously taken. Otherwise, the effect of δW on the abundance is practically insignificant.[↩](#fnref7) Rubidium abundances of galactic disk stars ========================================== Introduction ============ Rubidium (Rb, Z = 37) is a neutron-capture element (in which both s- and r-processes are involved), whose abundances spectroscopically determined for various types of stars may provide us with useful information regarding mixing of nuclear-process products or galactic chemical evolution. However, published spectroscopic studies of stellar Rb abundances have been rather limited in number, which may presumably reflect the comparatively large technical difficulty of its abundance determination (i.e., weak and blended line feature has to be dealt in many cases). Those investigations on Rb abundances so far have focused mainly on metal-poor stars. Following Gratton & Sneden (1994) who determined the abundances of Rb (along with other neutron-rich elements) for 19 stars at − 2.8 < [Fe/H] < 0, Tomkin & Lambert (1999) conducted a more detailed Rb abundance analysis for 44 metal-deficient giants and dwarfs in the metallicity range of − 2.0 < [Fe/H] < 0.0. These previous studies revealed that the [Rb/Fe] ratio tends to be moderately supersolar ($0 \lesssim$ [Rb/Fe] $\lesssim 0.5$) at the metal-poor regime ([Fe/H] $\lesssim -1$). Successively, Rb abundances of globular cluster giants (Yong et al. 2006, 2008; D’Orazi et al. 2013) or AGB giants (García-Hernández et al. 2006; Shejeelammal et al. 2020) were also investigated. On the other hand, in contrast to such metal-poor halo stars or peculiar AGB stars, the behaviours of Rb abundances in galactic disk stars (mildly metal-deficient through metal-rich regime) seem to have been poorly investigated. What can be read from a small number of samples at $-0.5 \lesssim$ [Fe/H] $\lesssim 0.0$ in previous studies quoted above is that [Rb/Fe] tends to be slightly subsolar by 0.1–0.3 dex (see, e.g., Fig. 2 in Gratton & Sneden 1994 or Fig. 3 in Tomkin & Lambert 1999). In this context, the problem of reference Rb abundance should be mentioned, which might have something to do with this trend. That is, Gratton & Sneden (1994) as well as Tomkin & Lambert (1999) similarly remarked that the solar Rb abundance[1](#fn1) A⊙ ≃ 2.6, which they determined from the solar spectrum and used as the fiducial abundance for evaluation of [Rb/Fe] ( ≡ A\* − A⊙ − [Fe/H]), is larger than the solar-system Rb abundance ($A\_{\rm s.s.} \simeq 2.4$) well established from meteorites, though they could not find out the cause for this difference. If the latter $A\_{\rm s.s.}$ were used as the zero-point abundance, the subsolar tendency would be mitigated in the direction of [Rb/Fe] ∼ 0. Recently, an interesting finding has been reported by Abia et al. (2020), who examined the Rb abundances of 57 M dwarfs in the metallicity range $-0.3 \lesssim$ [Fe/H] $\lesssim +0.3$ observed with CARMENES. The result they obtained was somewhat surprising: The [Rb/Fe] ratios of many stars turned out to be subsolar by a few tenths dex and show an increasing tendency towards higher metallicity from [Rb/Fe] ∼ − 0.4 (at [Fe/H] ∼ − 0.3) to [Rb/Fe] ∼ + 0.2 (at [Fe/H] ∼ + 0.3). Such a trend is unusual for a neutron-capture element and can not be explained even by the state-of-the-art chemical evolution models. So, if this is real, a significant revision would be required on the theoretical side. Yet, given that reliability of abundance determinations in M dwarfs has not been firmly established, follow-up verification should be necessary. Abia et al. (2020) themselves remarked “Additional Rb abundance measurements in FGK dwarfs of near solar metallicity, as well as an evaluation of the impact of stellar activity on abundance determinations in M dwarfs, are urgently needed to confirm or disprove the main findings of this study.” This problem attracted the author’s interest and eventually motivated this study. Conveniently, observational data for Rb abundance determination are already available for a large number of FGK-type stars belonging to the disk population (field dwarfs as well as giants, Hyades cluster stars) in the relevant metallicity range ($-0.6 \lesssim$ [Fe/H] $\lesssim +0.3$), for which atmospheric parameters were also established in the previous papers. Accordingly, it was decided to determine the Rb abundances for these sample stars and check up their behaviours, while paying attention also to examining the credibility of resulting abundances. Specifically, the purpose of this investigation was to clarify the following points: * Do the Rb abundances resulting from various Hyades stars of F-, G-, and K-type agree with each other? This would serve as a reliability check for the adopted procedure of abundance determination. * Are the Rb abundances derived from dwarf and giant stars consistent with each other at the same [Fe/H]? * Is the solar photospheric Rb abundance determined from the spectrum of the Sun (or Moon) agree with the solar-system (meteorite) abundance? Or does it show an appreciable discrepancy such as found in previous studies? * Finally, how do the [Rb/Fe] ratios of disk stars (especially those of near-solar metallicity) behave itself with a change in [Fe/H]? Could the trend reported by Abia et al. (2020) be confirmed? Program stars and their parameters ================================== A total of 487 stars were selected from the samples of previous studies of the author, which are divided into 4 groups (group 1 — 47 Hyades FGK dwarfs, group 2 — 100 field FGK dwarfs, group 3 — 101 field GK dwarfs, and group 4 — 239 field GK giants) as summarised in Table 1, where the references to the original papers are also given for more detailed information. Regarding the basic observational data, the spectra obtained by using HIDES (HIgh-Dispersion Echelle Spectrograph) attached to the 188 cm reflector at Okayama Astrophysical Observatory were used in most cases, though those acquired by HDS (High-Dispersion Spectrograph) with 8.2 m Subaru Telescope were partly used for 13 (out of 47) group 1 stars and 17 (out of 101) group 3 stars. As to the reference solar spectrum, the Moon spectrum[2](#fn2) included in Okayama spectrum database (Takeda et al. 2005a) was employed along with Neckel’s (1994, 1999) solar disk-centre spectrum and Kurucz et al.’s (1984) solar flux spectrum. The atmospheric parameters [$T\_{\rm eff}$ (effective temperature), logg (surface gravity), $v\_{\rm t}$ (microturbulence), and [Fe/H] (metallicity)] of the program stars were spectroscopically determined in the same consistent manner from the equivalent widths of Fe i and Fe ii lines by using the TGVIT program (Takeda et al. 2005b). These stellar parameters are given in the online materials (tableE1,dat, tableE2.dat, tableE3.dat, and tableE4.dat) for each of the 4 groups. As seen from the parameter ranges for each group indicated in Table 1, while the temperature ranges of group 1 and group 2 stars widely extend from F- through K-type, those of group 3/group 4 are limited to a rather narrow span of ∼ 1000 K (i,e., ∼ 5500–4500 K corresponding to late/mid G–mid/early K). Similarly, the same atmospheric models were used as employed in the original references; they are based on Kurucz’s (1993a) ATLAS9 model atmosphere grids (computed with the convective overshooting effect), which were interpolated in terms of $T\_{\rm eff}$, logg, and [Fe/H] for each star. Abundance determination ======================= Outline of the method --------------------- Regarding determinations of Rb abundances and related quantities, the resonance Rb i line at 7800 Å (which comprises multiple components and is blended with the Si i line) is exclusively used in this study.[3](#fn3) The procedures are essentially the same as those adopted in the author’s previous papers (e.g., Takeda & Kawanomoto 2005, Takeda et al. 2015, Takeda & Honda 2020). First, by applying Takeda’s (1995) numerical algorithm, the best fit between the synthetic and observed spectra is accomplished in the specified wavelength region (7776–7804 Å) while varying the abundances of important elements (A(Ni), A(Si), A(Rb) and A(Fe) in the present case), $v\_{\rm M}$ (macrobroadening velocity[4](#fn4)), and Δλ (radial velocity or wavelength shift). Then, the equivalent width (W)[5](#fn5) of the relevant Rb i 7800 line is “inversely” evaluated from the best-fit solution of A(Rb) with the same atmospheric model/parameters as used in the spectrum-fitting analysis. From such evaluated W, the non-LTE abundance ($A^{\rm N}$) and LTE abundance ($A^{\rm L}$) are calculated, from which the non-LTE correction is derived as $\Delta \equiv A^{\rm N} - A^{\rm L}$. Besides, this W can be further used to estimate the abundance uncertainties due to ambiguities of atmospheric parameters by perturbing the standard values interchangeably. Adopted data of spectral lines ------------------------------ Regarding the atomic data (wavelengths, oscillator strengths) of the Rb i 7800 line, Abia et al.’s (2020) Table 1 was invoked. The gf values of the component lines were scaled by assuming the solar-system (meteoritic) isotope ratio of 85Rb/87Rb = 2.43 as done by them. These scaled data are listed in Table 2. As for the damping parameters, the default treatment of Kurucz’s (1993a) WIDTH9 program was applied, though they are essentially insignificant in the present case because this Rb i line is generally weak. Otherwise, the data compiled in the VALD database (Ryabchikova et al. 2015) were used for the lines (other than Rb i 7800) contained in the 7776–7804 Å region. The exceptions were the following three Si i lines ($\chi\_{\rm low} = 6.181$ eV) whose gf values were empirically adjusted: Si i 7798.837 (loggf = − 2.06), Si i 7799.180 (loggf = − 1.75), and Si i 7799.996 (loggf = − 0.69). The abundances of all elements other than Ni, Si, Rb, and Fe were fixed at the metallicity-scaled solar abundances in the spectrum-fitting analysis. Non-LTE calculations -------------------- The statistical-equilibrium calculations for neutral rubidium were carried out in the similar manner as previously done for the lighter alkali elements: Li (Takeda & Kawanomoto 2005), K (Takeda et al. 1996; 2002), and Na (Takeda et al. 2003). For the present study, Rb i model atom was constructed comprising 19 terms (up to 11s 2S at 31917 cm− 1) and 59 radiative transitions, while using Kurucz & Bell’s (1995) compilation of atomic data (gf values, levels, etc.). Regarding the treatment of the photoionisation cross section, the hydrogenic approximation was assumed, except for the edge values of the ground term (5s 2S) and the first excited term (5p 2P∘), for which Lowell et al. (2002) and Nadeem & Haq (2011) were invoked, respectively. As to the collisional rates, the recipe described in Sect. 3.1.3 of Takeda (1991) was basically followed, while the H i collision rates computed with the classical Drawin’s cross section (cf. Steenbock & Holweger 1984) was multiplied by a factor of k = 10− 3 (i.e., suppressed to a negligible level).[6](#fn6) Since FGK-type stars (dwarfs as well as giants) of population I in the galactic disk are concerned in this study, the calculations were done on a grid of 125 (5 × 5 × 5) model atmospheres resulting from combinations of five $T\_{\rm eff}$ values (4500, 5000, 5500, 6000, 6500 K), five logg values (1.0, 2.0, 3.0, 4.0, 5.0), and five [Fe/H] values (+0.5, 0.0, − 0.5, − 1.0, − 1.5). The microturbulent velocity was assumed to be $v\_{\rm t}$ = 2 km s− 1, and the metallicity-scaled Rb abundance ([Rb/Fe] = 0) was used as A(Rb) = 2.60 + [Fe/H], where 2.60 is Anders & Grevesse’s (1989) solar Rb abundance adopted in ATLAS9 models. The depth-dependent non-LTE departure coefficients to be used for each star were then evaluated by interpolating this grid in terms of $T\_{\rm eff}$, logg, and [Fe/H]. Results ------- The Rb abundances were successfully determined for almost all cases, except for 3 stars in group 2 (HD 120136, 124850, 016673; these are F-type dwarfs with $T\_{\rm eff} \gtrsim 6300$ K and the Rb i 7800 line is very weak), for which A(Rb) solution was not converged and thus had to be fixed in the fitting process. Representative examples of spectrum fitting in the whole 7796–7804 Å region are shown in Fig. 1 (for the disk-centre Sun, disk-integrated Sun, Moon, and 8 dwarfs/giants in the wide $T\_{\rm eff}$ range). The close-up display of the accomplished fit in the 7799–7801 Å region is presented for all program stars in Fig. 2 (groups 1, 2, and 3) and Fig. 3 (group 4). After the A(Rb) solution has been established, the related quantities were further derived as described in Sect. 3.1: W (equivalent width), Δ (non-LTE correction), $A^{\rm N}$ (non-LTE abundance), δT ± (abundance changes for $T\_{\rm eff}$ perturbations by ± 100 K), δg ± (abundance changes for logg perturbations by ± 0.1 dex), and δv ± (abundance changes for $v\_{\rm t}$ perturbations by ± 0.5 km s− 1). Each of them are plotted against $T\_{\rm eff}$ in Fig 4 (group 1), Fig. 5 (group 2), Fig. 6 (group 3), and Fig. 7 (group 4). These results of W, Δ, $A^{\rm N}$ (along with $v\_{\rm M}$ derived from spectrum fitting as by-product) for stars of each group are presented in the online tables (tableE1.dat, tableE2.dat, tableE3.dat, and tableE4.dat). An inspection of Fig. 4–7 suggests the following trends which generally hold for all program stars irrespective of dwarfs or giants. * The equivalent width of Rb i 7800 (W) decreases with an increase in $T\_{\rm eff}$, reflecting that the occupation number of the Rb i ground level is quite $T\_{\rm eff}$-sensitive ($\propto 10^{+5040 \chi^{\rm ION}/T\_{\rm eff}}$ where $\chi^{\rm ION}$ = 4.18 eV is the ionisation potential of Rb i). * The non-LTE correction (Δ) progressively increases as $T\_{\rm eff}$ is lowered: While it is negative (Δ ∼ − 0.1 dex: non-LTE line strengthening) for F-type stars, it becomes positive (Δ ∼ + 0.1 dex: non-LTE line weakening) for K-type stars. As such, the non-LTE effect on Rb abundance determination ($\|\Delta\| \lesssim 0.1$ dex) is insignificant as far as stars at 4500 $\lesssim T\_{\rm eff} \lesssim 6500$ K are concerned. * Among the various sources of abundance errors, most important is δT ± ( ∼ ± 0.1 dex for $\Delta T\_{\rm eff} = \pm 100$ K; tending to enhance toward lower $T\_{\rm eff}$), which can be naturally understood from the large $T\_{\rm eff}$-sensitivity of W mentioned above. Meanwhile, δg ± and δv ± are totally insignificant. The Rb abundances determined in this investigation are compared with those derived by Tomkin & Lambert (1999) in Fig. 8 for 6 stars in common, where a satisfactory match is observed for most cases. One exception showing a rather large deviation ( ∼ 0.3–0.4 dex) is the G5V star μ Cas (= HD 6582), for which Rb abundance determination is considerably difficult because the Rb line is very weak (W = 1.6 mÅ) due to its low metallicity ([Fe/H] = − 0.8). Discussion ========== Solar Rb abundance problem -------------------------- Let us first discuss the solar rubidium abundance to be used as the reference standard, for which appreciable discrepancy from the meteoritic composition was reported in previous work (cf. Sect. 1). The non-LTE Rb abundance determined from Kurucz et al.’s (1984) solar flux spectrum is $A^{\rm N}\_{\odot} = 2.44$ (W = 5.9 mÅ, Δ = − 0.05 dex), that from Neckel’s (1994, 1999) solar disk-centre spectrum (direction cosine μ = cosθ = 1) is $A^{\rm N}\_{\odot} = 2.48$ (Wμ = 1 = 5.7 mÅ, Δ = − 0.05 dex), while that derived from Takeda et al.’s (2005) Moon spectrum is $A^{\rm N}\_{\odot} = 2.56$ (W = 7.7 mÅ, Δ = − 0.06 dex), resulting in a Sun − Moon difference of ∼ 0.1 dex despite that a satisfactory fit appears to be accomplished similarly for both cases (cf. Fig. 1). This suggests that Rb abundance determination from a severely blended feature is a delicate matter influenced by the difference in the fitting region (which had to be narrowed in the latter case owing to the spectrum defect in the short-wavelength side) or in the spectrum resolving power. Accordingly, these spectroscopically established solar Rb abundances are higher than the solar-system abundance ($A\_{\rm s.s.} = 2.36$; cf. Lodders 2020) by ∼ 0.1–0.2 dex. As such, the discrepancy between A⊙ and $A\_{\rm s.s.}$ reported by Gratton & Sneden (1994) and Tomkin & Lambert (1999) has been confirmed in this study. In contrast, Abia et al. (2020) and Korotin (2020) claimed that this problem has been settled, since they obtained $A^{\rm N}\_{\odot} = 2.35 \pm 0.05$ which is in very good agreement with $A\_{\rm s.s.}$ of 2.36. This is mainly because they adopted a more enhanced non-LTE correction (Δ = − 0.12 dex) than the value ( − 0.05 dex) mentioned above, which reflects the fact that their non-LTE abundance corrections for FGK-type stars are systematically shifted downward by ∼ 0.1 dex compared to those derived in this study (see Appendix A where this difference is discussed more in detail). Yet, it is not clear whether this solar abundance issue is simply attributed to the non-LTE effect, because the situation appears to be more complex and some other $T\_{\rm eff}$-dependent problem may be involved, as described in the next two sections. Implication from Hyades dwarfs ------------------------------ Comparing the Rb abundances of group 1 stars (47 FGK dwarfs belonging to the Hyades cluster which distribute at 4500 K $\lesssim T\_{\rm eff} \lesssim 6300$ K) with each other makes a valuable test for checking the validity of abundance determination, because similar abundances should be observed for such cluster stars born from the gas of same chemical composition. An inspection of Fig. 4c suggests an interesting trend that (i) $A^{\rm N}$ values for Hyades dwarfs at $T\_{\rm eff} \lesssim 5500$ K (mid-G through K type) are almost similar irrespective of $T\_{\rm eff}$, while (ii) those at $T\_{\rm eff} \gtrsim 5500$ K (F through early G type) show an increasing tendency towards higher $T\_{\rm eff}$. Regarding the former lower $T\_{\rm eff}$ group, it is reasonable to regard that the abundances are reliably established. Actually, the mean abundance ($\langle A^{\rm N} \rangle = 2.56 \pm 0.10$) derived for 18 stars of $T\_{\rm eff} < 5500$ K is consistent with the expected primordial Rb abundance of this cluster ($A\_{\rm s.s.}$ + [Fe/H]$\_{\rm Hyades}$), where $A\_{\rm s.s.}$ is the solar system abundance (2.36) and [Fe/H]$\_{\rm Hyades}$ is the Hyades metallicity ( ≃ + 0.2; cf. Takeda & Honda 2020). As to the latter higher $T\_{\rm eff}$ group, it should be remarked that errors involved with the resulting abundances must be larger, especially for F-type stars ($T\_{\rm eff} \gtrsim 6000$ K), because the line feature becomes considerably weak (due to higher $T\_{\rm eff}$) and broad/shallow (due to larger rotational velocity).[7](#fn7) Accordingly, those data points showing prominent deviations (e.g., $A^{\rm N} \gtrsim 3$ or $A^{\rm N} \lesssim 2$) should not be seriously taken. Even so, a global trend is recognised from Fig. 4c that $A^{\rm N}$ gradually increases from ∼ 2.6 (at $T\_{\rm eff} \sim 5500$ K) to ∼ 2.9 (at $T\_{\rm eff} \sim 6300$ K), which means that Rb abundances in this $T\_{\rm eff}$ range suffer some $T\_{\rm eff}$-dependent systematic errors (i.e., overestimation with its extent growing towards higher $T\_{\rm eff}$). Abundance distribution of near-solar metallicity stars ------------------------------------------------------ In order to check on this problem further for other field stars, stars with metallicities around the solar value ( − 0.3 < [Fe/H] < + 0.3) were sorted out from each group (2, 3, and 4) and the distributions of their Rb abundances were examined. The resulting $A^{\rm N}$ vs. $T\_{\rm eff}$ plots and the histograms of $A^{\rm N}$ are displayed in Fig. 9, where the subset of especially close-to-solar-metallicity stars ( − 0.1 < [Fe/H] < + 0.1) are separately shown with the pink colour. Let us pay attention to Fig. 9a, which shows that the Rb abundances derived for group 2 stars of near-solar metallicity (most of them are F–G dwarfs of $T\_{\rm eff} \gtrsim 5500$ K) tend to systematically increase with $T\_{\rm eff}$, just like the case of Hyades dwarfs discussed in the previous section. It has thus been corroborated that the Rb abundances of $T\_{\rm eff} \gtrsim 5500$ K dwarfs are not so trustable in the sense that they tend to be overestimated (this effect begins at ∼ 5500 K and grows with increasing $T\_{\rm eff}$). The cause of this embarrassing effect is not clear, which is limited only to stars of $T\_{\rm eff} \gtrsim 5500$ K (while the lower $T\_{\rm eff}$ side is unaffected). Attributing it to improper non-LTE corrections seems unlikely, because adjustment only at the higher $T\_{\rm eff}$ side by changing the parameters of non-LTE calculations (e.g., collision cross section) is difficult. As another possibility, the problem may be related to the structure of model atmosphere affected by convection, because the strength of such weak (i.e., deep-forming) and T-sensitive Rb i 7800 line is sensitive to the condition of the lower photosphere, which critically depends upon the treatment of convection (i.e., mixing length, overshooting) especially in F-type stars. The models adopted in this study are based on Kurucz’s (1993a) standard ATLAS9 model atmosphere grid computed with the mixing length of l = 1.25Hp (Hp: pressure scale height) by including the convective overshooting effect (OVERWT=1). Although this “with overshooting” model is known to represent the solar observational data fairly well, it is not necessarily adequate for the higher-$T\_{\rm eff}$ F-type star Procyon (F5 IV–V), for which “no overshooting” model seems more suitable (cf. Takeda et al. 1996). So, with the consideration that no-overshooting model may be more preferable especially for F-type stars, Rb abundances were re-determined by using no-overshooting (OVERWT=0) ATLAS9 models for all 47 Hyades stars (group 1) in order to see how much abundance changes [$\delta\_{\rm os} \equiv A$(no overshooting) − A(with overshooting)] would result. Interestingly, as shown by red crosses in Fig. 4b, the use of no-overshooting models tends to decrease A ($\delta\_{\rm os} < 0$) at $T\_{\rm eff} \gtrsim 5500$ K with the extent of variation ($\|\delta\_{\rm os}\|$) increasing with $T\_{\rm eff}$ (up to $\delta\_{\rm os} \sim 0.04$ dex at $T\_{\rm eff} \sim 6300$ K) (which reflects that switching-off overshooting tends to strengthen the line because of the steeper T-gradient), while abundances are almost unaffected ($\delta\_{\rm os} \sim 0$) at $T\_{\rm eff} \lesssim 5500$ K. Accordingly, switching-off the convective overshooting tends to act in the direction of mitigating the $T\_{\rm eff}$-dependent trend (Fig. 4c) at least qualitatively, though still insufficient in the quantitative sense. In any event, it would be promising to improve the modelling of convective atmosphere (e.g., by using state-of-the-art 3D hydrodynamical simulations) toward resolving the $T\_{\rm eff}$-dependent systematic effect seen in the Rb abundances. [Rb/Fe] vs. [Fe/H] relation --------------------------- Before discussing the behaviour of [Rb/Fe], we turn to the problem of reference abundance already mentioned in Sect. 4.1, where the solar abundance ($A\_{\odot}^{\rm N}$) was shown to be by ∼ 0.1–0.2 dex higher than the solar-system abundance ($A\_{\rm s.s.}$). It is worth pointing out that this difference may be explained by the trend seen in the $A^{\rm N}$ vs. $T\_{\rm eff}$ relation described in Sect. 4.2 and Sect. 4.3. That is, since the Sun is in the $T\_{\rm eff}$ range ($\gtrsim 5500$ K) where Rb abundances suffer $T\_{\rm eff}$-dependent overestimation effect, $A\_{\odot}^{\rm N}$ is expected to be overestimated by ∼ 0.1 dex (as seen from the gradient observed in Fig. 4c), which may be the cause for such an apparent discrepancy (despite that the true photospheric Rb abundance is the same as $A\_{\rm s.s.}$). Accordingly, careful consideration is required regarding the fiducial Rb abundance adopted for evaluation of [Rb/Fe]. One possibility might be to use the solar abundance ($A\_{\odot}^{\rm N}$) as usual, while limiting the sample to those of mid-G through F type (like most of the group 2 stars). In this case, however, while a reasonable zero point is accomplished since the centre of Rb abundance distribution is well represented by $A\_{\odot}^{\rm N}$, a larger dispersion would result because abundance data of different $T\_{\rm eff}$-dependent overestimation are mixed (cf. Fig. 9d). Therefore, as an alternative approach, all objects at $T\_{\rm eff} \gtrsim 5500$ K were decided not to use for [Rb/Fe] analysis in this paper. That is, (i) the sample stars are restricted only to those of group 3 and group 4 (i.e., dwarfs and giants at $T\_{\rm eff} \lesssim 5500$ K), and (ii) [Rb/Fe] ratio is calculated by referring to $A\_{\rm s.s.}$ (instead of $A\_{\odot}^{\rm N}$) as [Rb/Fe] $\equiv A^{\rm N} - 2.36-[{\rm Fe}/{\rm H}]$, which is a reasonable choice because $A\_{\rm s.s.}$ well matches the centre of Rb abundance distribution of near-solar metallicity stars in group 3 and group 4 (cf. Fig. 9e and Fig. 9f). The resulting [Rb/Fe] vs. [Fe/H] relations obtained as such are illustrated in Fig. 10, from which the following characteristics are observed. * A similar tendency of gradually increasing [Rb/Fe] for a decrease in [Fe/H] is observed for both group 3 dwarfs (Fig. 10a) and group 4 giants (Fig. 10b), though a somewhat larger scatter is seen for the former. * A quantitative inspection of the mean ⟨[Rb/Fe]⟩ values at each metallicity bin in the [Fe/H] range of $-0.4 \lesssim$ [Fe/H] $\lesssim +0.2$ (cf. Fig. 10d and Fig. 10e) suggests that (1) d[Rb/Fe]/d[Fe/H] gradient is ∼ − 0.4 at $-0.4 \lesssim$ [Fe/H] $\lesssim 0.0$, (2) [Rb/Fe] appears rather flat ( ∼ 0) at $0.0 \lesssim$ [Fe/H] $\lesssim +0.2$, and (3) this trend almost equally holds for both dwarfs and giants (though ⟨[Rb/Fe]⟩ for the former tends to be slightly higher than the latter by $\lesssim 0.05$ dex). * This equality of [Rb/Fe] (at the same [Fe/H]) for group 3 and group 4 stars implies that initial Rb abundances are retained in the stellar surface for both dwarfs or giants, which is understandable since AGB stars are not included in group 4 (many are red clump giants). * Such a gentle increase in [Rb/Fe] from ∼ 0 (at [Fe/H] ∼ 0) to ∼ + 0.2 (at [Fe/H] ∼ − 0.5) is favourably (at least in the qualitative sense) compared with the theoretical prediction from the recent chemical evolution model (depicted by the solid line in Fig. 10a and Fig. 10b). * In summary, the overall behaviour of the [Rb/Fe] ratios in galactic disk stars (dwarfs as well as giants) around the solar metallicity, gradually increasing with a decrease in [Fe/H], is almost consistent with the result of theoretical simulation. Comparison with previous studies -------------------------------- In Fig. 10c are plotted the published results of [Rb/Fe] ratios against [Fe/H], which were reported in three previous papers (cf. Sect. 1): Gratton & Sneden (1994), Tomkin & Lambert (1999), and Abia et al. (2020). Regarding Gratton & Sneden (1994) and Tomkin & Lambert (1999), only a small number of disk stars ($-1 \lesssim$ [Fe/H] $\lesssim 0$) were included in their sample (since their interest was placed mainly on metal-poor halo stars of [Fe/H] $\lesssim -1$) which tend to show subsolar (negative) [Rb/Fe] ratios by a few tenths dex. This may be due to the fact (at least partly) that they used the solar abundance (2.6) as the reference abundance for the zero point of [Rb/H], which is by 0.2 dex higher than the meteoritic abundance that would have been more appropriate (note that most of their sample stars are at $T\_{\rm eff} \lesssim 5500$ K). In contrast, the consequence obtained in this study is in apparent conflict with Abia et al.’s (2020) [Rb/Fe] vs. [Fe/H] correlation derived from M dwarfs of near-solar metallicity, showing a controversial trend that [Rb/Fe] (being appreciably subsolar by several tenths dex) steeply increase towards higher metallicity within a narrow range of $-0.3 \lesssim$ [Fe/H] $\lesssim +0.3$. It is hard to understand why they obtained such a conclusion. It should be remarked here that their [Rb/Fe] analysis seems to be based on LTE abundances (while using the LTE solar abundance of 2.47 as the reference) despite that they also derived non-LTE Rb abundances by applying (negative) non-LTE corrections ranging from − 0.28 to − 0.13 dex (tending to decrease towards lower $T\_{\rm eff}$; cf. Fig. 4 therein). Since their non-LTE correction for the Sun is − 0.12 dex, non-LTE [Rb/Fe] would be shifted further downward (by ∼ 0.1 dex on the average), making the anomaly even more exaggerated, which may be the reason why they did not employ non-LTE abundances. It seems premature, however, to regard that the validity of their non-LTE correction results derived for M dwarfs is established. Since the non-LTE formation of the Rb i resonance line is determined by the delicate interplay between two opposite effects of line weakening and line strengthening (cf. Appendix A), its net result could be sensitive to detailed computational conditions. For example, Pavlenko & Magazzù (1996) showed in their line formation calculation for the Li i 6708 line in G–M dwarfs and subgiants that the non-LTE effect can either strengthen or weaken the line at $T\_{\rm eff} \lesssim 4000$ K. Accordingly, “positive” non-LTE corrections might be possible depending on the situation also for the case of Rb i line formation in M dwarfs, by which the apparent anomaly of subsolar [Rb/Fe] at issue could be mitigated. Summary and conclusion ====================== Spectroscopic studies of stellar rubidium abundances in context of galactic chemical evolution have been rather scarce, presumably due to the technical difficulty of its abundance determination where weak and blended Rb i line has to be generally invoked. Recently, Abia et al. (2020) determined the Rb abundances of 57 M-type dwarfs of $-0.3 \lesssim$ [Fe/H] $\lesssim +0.3$ and reported the anomalous trends of [Rb/Fe] ratios, being appreciably subsolar by several tenths dex and increasing towards higher [Fe/H], which can not be explained by any current theory of galactic nucleosynthesis. With an aim of examining whether such a tendency is actually observed in FGK-type stars, Rb abundances were determined by applying the spectrum fitting method to the Rb i 7800 line for ∼ 500 late-type dwarfs as well as giants (including Hyades cluster stars) belonging to the galactic disk population ($-0.6 \lesssim$ [Fe/H] $\lesssim +0.3$). The Rb abundance of the Sun determined from the solar spectrum turned out to be by ∼ 0.1–0.2 dex higher than the solar system abundance derived from meteorites, as already reported in the earlier work by Gratton & Sneden (1994) and Tomkin & Lambert (1999). This difference may be related to the $T\_{\rm eff}$-dependent effect seen in F–G dwarfs (see below). An inspection of the results derived for Hyades dwarfs revealed that they show a systematic $T\_{\rm eff}$-dependent trend at $\gtrsim 5500$ K, which means that the Rb abundances derived for early-G to F stars (including the Sun) are likely to suffer some overestimation increasing towards higher $T\_{\rm ef}$. This might be due to some imperfect treatment of surface convection in classical model atmospheres, because T-sensitive Rb i line is weak and forms deep in the photosphere where the structure is critically affected by how the convection is treated (especially in F-type stars). For this reason, in the present study of [Rb/Fe], the final samople was limited to only late-G and K stars of $T\_{\rm eff} \lesssim 5500$ K and the meteoritic abundance (instead of the solar abundance) was employed as the reference Rb abundance, The resulting [Rb/Fe] vs.[Fe/H] relations for field G–K dwarfs and giants turned out to be similar to each other, both showing a gradual increase of [Rb/Fe] with a decrease in [Fe/H] (with a d[Rb/Fe]/d[Fe/H] gradient of ∼ − 0.4 around the solar metallicity). This trend is almost consistent (at least qualitatively) with the theoretical prediction of chemical evolution models. Accordingly, this study could not confirm the anomalous behaviours of [Rb/Fe] derived by Abia et al. (2020). The reason why they obtained such results for M dwarfs is yet to be clarified. Acknowledgments =============== This research has made use of the SIMBAD database, operated by CDS, Strasbourg, France. 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[h] \| Group \| 1 \| 2 \| 3 \| 4 \| \| --- \| --- \| --- \| --- \| --- \| \| Number of stars \| 47 \| \100 \| 101 \| 239 \| \| Star type \| Hyades cluster stars \| field stars \| field stars \| $^{\S}$field stars \| \| Sp. class \| mid F–mid K \| early F-early K \| late G–mid K \| mid G–early K \| \| Lum. class \| dwarfs \| dwarfs (+subgiants) \| dwarfs \| giants \| \| $^{\dagger}T\_{\rm eff}$ \| [4525, 6307] \| [5150, 6968] \| [4495, 5482] \| [4494, 5624] \| \| †logg* \| [4.08, 4.71] \| [3.38, 4.72] \| [4.22, 4.73] \| [1.42, 3.49] \| \| †[Fe/H] \| [ − 0.02, +0.33] \| [ − 0.90, +0.46] \| [ − 1.27, +0.29] \| [ − 0.77, +0.19] \| \| #Obs. data \| TH20 \| \T05a \| TH20 \| T15 \| \| #Parameters \| TH20 \| T05b \| TH20 \| TSM08 \| \| Remarks \| validity check for A(Rb) \| not used for [Rb/Fe] \| adopted for [Rb/Fe] \| adopted for [Rb/Fe] \| $^{\S}$Including 4 Hyades giants. †Parenthesised values indicate [minimum, maximum]. #Previous papers of the author, which describe the observational data and atmospheric parameters of each star: TH20 ⋯ Takeda & Honda (2020), T05a ⋯ Takeda et al. (2005a), T05b ⋯ Takeda et al. (2005b), T15 ⋯ Takeda et al. (2015), TSM08 ⋯ Takeda et al. (2008). \Although the Okayama spectrum database published in T05a includes spectra of 160 FGK stars in total, many of those around ∼ 7800 Å are unfortunately damaged by serious bad columns in CCD. Only the data of 100 stars among them turned out to be somehow usable. [h] \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| λ \| $\chi\_{\rm low}$ \| loggf \| isotope \| \| (Å) \| (eV) \| (dex) \| \| \| 7800.183 \| 0.000 \| − 0.663 \| 87Rb \| \| 7800.186 \| 0.000 \| − 0.663 \| 87Rb \| \| 7800.188 \| 0.000 \| − 1.061 \| 87Rb \| \| 7800.317 \| 0.000 \| − 0.216 \| 87Rb \| \| 7800.322 \| 0.000 \| − 0.663 \| 87Rb \| \| 7800.325 \| 0.000 \| − 1.362 \| 87Rb \| \| 7800.233 \| 0.000 \| − 0.744 \| 85Rb \| \| 7800.234 \| 0.000 \| − 0.647 \| 85Rb \| \| 7800.235 \| 0.000 \| − 0.760 \| 85Rb \| \| 7800.292 \| 0.000 \| − 0.282 \| 85Rb \| \| 7800.295 \| 0.000 \| − 0.647 \| 85Rb \| \| 7800.296 \| 0.000 \| − 1.191 \| 85Rb \| \| 7947.507 \| 0.000 \| − 1.204 \| 87Rb \| \| 7947.524 \| 0.000 \| − 1.903 \| 87Rb \| \| 7947.651 \| 0.000 \| − 1.204 \| 87Rb \| \| 7947.668 \| 0.000 \| − 1.204 \| 87Rb \| \| 7947.563 \| 0.000 \| − 0.803 \| 85Rb \| \| 7947.570 \| 0.000 \| − 1.347 \| 85Rb \| \| 7947.626 \| 0.000 \| − 0.900 \| 85Rb \| \| 7947.634 \| 0.000 \| − 0.803 \| 85Rb \| These data are based on Abia et al.’s (2020) Table 1. The gf values are scaled by assuming 85Rb/87Rb = 2.43. Although the Rb i 7800 line is essentially invoked in the present study, the data for the Rb i 7947 line (the other line of the resonance doublet) are also shown because it is used in Appendix B. [fig:1] [fig:2] [fig:3] [fig:4] [fig:5] [fig:6] [fig:7] [fig:8] [fig:9] [fig:10] Non-LTE effect on the Rb I 7800 line ==================================== Physical mechanism of non-LTE line formation -------------------------------------------- As mentioned in Sect. 3.4, the non-LTE correction (Δ) for the Rb i 7800 line monotonically increases with a decrease in $T\_{\rm eff}$ from ∼ − 0.1 dex (F stars) to ∼ + 0.1 dex (K stars) while changing the sign at $T\_{\rm eff}\sim$ 5000–5500 K (Fig. 4b, Fig. 5b, Fig. 6b, and Fig. 7b). Although this extent of Δ is quantitatively insignificant, the physical reason for such a $T\_{\rm eff}$-dependent trend is worth explanation. In Fig. A1 are shown the $l\_{0}^{\rm NLTE}(\tau)/l\_{0}^{\rm LTE}(\tau)$ (the non-LTE-to-LTE line-centre opacity ratio; almost equal to $\simeq b\_{\rm 1}$) and $S\_{\rm L}(\tau)/B(\tau)$ (the ratio of the line source function to the Planck function; nearly equal to ≃ b2/b1) for the Rb i 5s 2S–5p 2P∘ transition (relevant to the Rb i 7800 line) computed on the models of different $T\_{\rm eff}$ and different logg while assuming [Rb/Fe] = [Fe/H] = 0, where b1 and b2 are the non-LTE departure coefficients for the ground and first-excited terms, respectively. Besides, the LTE and non-LTE equivalent widths ($W^{\rm L}$ and $W^{\rm N}$) for the Rb i 7800 line along with the corresponding non-LTE abundance correction (Δ) computed for logg = 2 and logg = 4 models are plotted against $T\_{\rm eff}$ in Fig. A2. The upper panels of Fig. A1 elucidate how the non-LTE effect influences the strength of Rb i 7800 line for models of different $T\_{\rm eff}$. In the present case of FGK stars where Rb i line is comparatively weak, two kinds of mechanisms are important both of which are caused by imbalance between J (local mean intensity) and B (Planck function at the local electron temperature): (i) overrecombination (J < B) and (ii) overionisation (J > B). As we go upwards from the deep optically thick layer (τ > 1, where J = B holds), J < B is first realized at the lower photosphere $\tau \lesssim 1$ (because of the dilution of radiation due to photon escape), but eventually the inequality J > B comes into effect at the higher optically thin region because J is almost stabilised while B(T) continues to decrease as T is lowered upwards. The net non-LTE effect on the number population of the ground level (lower level of the Rb i 7800 line) is determined by the interplay of these two mechanisms (i: line strengthening, ii: line weakening). In panels (a)–(c) of Fig. A1, the action of effect (i) is seen in the overpopulation bump at $10^{-1} \lesssim \tau \lesssim 1$, while that of effect (ii) is manifest in the systematic underpopulation observed at $\tau \lesssim 10^{-2}$. This implies that effect (i) is relatively more important at higher $T\_{\rm eff}$ (where the line is weak and deep-forming) while effect (ii) is more significant at lower $T\_{\rm eff}$ (line is stronger and its formation region is higher). Moreover, this $T\_{\rm eff}$-dependence of the relative importance between (i) and (ii) is further enhanced by the fact that the Planck function (B) around the ionisation edge becomes very T-sensitive as $T\_{\rm eff}$ is lowered (Wien region of the Planck function) which means that the inequality of J > B causing (ii) is realized more easily. What has been described above is sufficient for interpreting the non-LTE effect on the Rb i 7800 line in FGK stars and its dependence upon $T\_{\rm eff}$. Regarding F-type stars of higher $T\_{\rm eff}$ (cf. Fig. A1c), effect (i) is more important and acts in the direction of strengthening the line (Δ < 0). In contrast, effect (ii) is dominant in K-type stars of lower $T\_{\rm eff}$ (cf. Fig. A1a), where the line is weakened by the non-LTE effect (Δ > 0). For the case of G-type stars, both effects tend to compensate with each other, yielding Δ ∼ 0. As such, the trends of $W^{\rm L}$, $W^{\rm N}$, and Δ with a change of $T\_{\rm eff}$, which are displayed in Fig. A2, can be reasonably understood. Note also that the non-LTE effect tends to be enhanced as the surface gravity is decreased because of the lowered atmospheric density (implying less collision rates). Comparison with Korotin’s (2020) result --------------------------------------- Korotin (2020) has recently conducted non-LTE calculations for the Rb i resonance lines on an extensive model grid covering $T\_{\rm eff} = $ 3500–6500 K, logg = 1.0–5.0, and [Fe/H] from − 1.0 to +0.5, which were employed by Abia et al. (2020) for evaluating their non-LTE corrections applied to M dwarfs. Since his calculations also cover F-, G-, and K-type stars, it is worthwhile to compare the non-LTE corrections derived by both independent calculations with each other. By inspecting Fig. A2b along with Korotin’s (2020) Fig. 6a, the run of Δ at 4500 $\lesssim T\_{\rm eff} \lesssim 6500$ K for logg = 2 and logg = 4 can be compared with each other. This comparison suggests that, while the overall gradient (${\rm d}\Delta/{\rm d}T\_{\rm eff}$) as well as its logg-dependence is reasonably consistent, $\Delta\_{\rm Korotin}$ is systematically larger than $\Delta\_{\rm Takeda}$ by ∼ 0.1 dex. That is, as $T\_{\rm eff}$ is lowered from ∼ 6500 K (F-type star) to ∼ 4500 K (K-type star), $\Delta\_{\rm Takeda}$ increases from ∼ − 0.1 to ∼ + 0.1, while $\Delta\_{\rm Korotin}$ runs from ∼ − 0.2 to ∼ 0.0. The reason for this systematic difference can be understood by comparing the behaviour of b1(τ) in Korotin’s (2020) Fig. 3 or Fig. 7 with that of $l\_{0}^{\rm NLTE}(\tau)/l\_{0}^{\rm LTE}(\tau)$ (almost equivalent to b1) shown in the upper panels of Fig. A1. As can be recognised from this comparison done for the models of same $T\_{\rm eff}$, the overpopulation hump of b1 > 1 at the lower atmosphere is apparently more conspicuous while the underpopulation (b < 1) at the upper atmosphere is less pronounced in the results of Korotin (2020). This means that the overionisation effect (ii) is repressed relative to the overrecombination effect (i) in his calculation, which suggests that the evaluation of photoionisation rates may have been appreciably different. Unfortunately, Korotin (2020) did not describe any details about the calculation of photoionising radiation field (J), whereas the author followed the procedure described in Sect. 3.1.2 of Takeda (1991) by consulting Kurucz’s (1993a) ATLAS9 program along with the line opacity distribution function published by Kurucz (1993b). Finally, some comments regarding the non-LTE effect in M dwarfs may be in order. According to the calculation of Korotin (2020), δ( < 0) progressively increases as $T\_{\rm eff}$ is lowered until it approaches ∼ 0 at $T\_{\rm eff} \sim 4500$ K. However, it then shows a downturn and begins to ever decline with a decrease in $T\_{\rm eff}$ over the $T\_{\rm eff}$ range of ∼ 3500–4000 K corresponding to M-type stars. This is presumably due to the dilution effect of line source function ($S\_{\rm L}/B <1$) which becomes important as the line gets stronger and saturated (such as the case of strong K i resonance line at 7665/7699 Å). As such, the Δ values of M dwarfs calculated by Korotin (2020) remain always negative and this non-LTE effect (acting to strength the line) is enhanced towards lower $T\_{\rm eff}$, as shown in Fig. 4 of Abia et al. (2020). Yet, it might be possible that the overionisation effect (which acts in the direction of line weakening) was not sufficiently taken into account in Korotin’s (2020) calculation as mentioned above. This is the reason for having remarked the possibility of positive Δ in the last paragraph of Sect. 4.5. [fig:A1] [fig:A2] Comparison of the abundances derived from Rb I 7800 and 7947 lines ================================================================== As remarked in footnote 3 of Sect. 3.1, only the stronger line at 7800 Å was invoked in this investigation, while the weaker one of the doublet at 7947 Å was not used since reliable Rb abundance determination revealed to be generally more difficult for this line (i.e., because of the lower quality of observational data along with weaker line strength). Grevesse et al. (2015) reported in their comprehensive analysis of solar photospheric abundances of Cu through Th that the Rb abundace derived from Rb i 7800 is higher than that from Rb i 7947 by 0.11 dex. They thus suspected that the Rb i 7800 line might be blended with some other unknown line leading to a slight overestimation. It is interesting to examine whether such a difference is observed in the abundances derived from these two lines also for stars other than the Sun. Therefore, Rb abundances were determined from the Rb i 7947 line (by using the atomic data given in Table 2) for HIP 104214 (= 61 Cyg A) and ξ Boo B (K5 V dwarfs with $T\_{\rm eff}$ of 4523 K and 4495 K, respectively; lowest $T\_{\rm eff}$ stars in group 3) in order to compare them with those derived from Rb i 7800 line. Owing to sufficiently high S/N ratio (because these two stars are apparently bright) and comparatively large line strength (due to low $T\_{\rm eff}$), Rb abundances from this Rb i 7947 line could be satisfactorily established by the spectrum fitting in the 7945–7050 Å region (cf. Fig. B1), followed by evaluations of W, $A^{\rm N}$, and Δ as done in Sect. 3.4. The results obtained for two lines are presented in Table B1. Regarding ξ Boo B, $A^{\rm N}\_{7947}$ is appreciably larger than $A^{\rm N}\_{7800}$ by 0.15 dex (i.e., inversed inequality to that reported for the Sun). However, this should not be seriously taken, since it must be due to influence of a telluric line at ∼ 7947.7 Å (see the solar spectrum shown at the top of Fig. B1), which just contaminates the Rb i line at 7947.6 Å because of the stellar line shift ( ∼ + 0.1 Å) corresponding to the radial velocity of $V\_{\rm r}^{\rm topo} = +4.5$ km s− 1 (cf. Table B1). Meanwhile, Rb i 7947 line is essentially free from telluric blending for the case of HIP 104214 showing a considerably large radial velocity ($V\_{\rm r}^{\rm topo} = -82.0$ km s− 1). According to Table B1, $A^{\rm N}\_{7800}$ is slightly larger than $A^{\rm N}\_{7947}$ by +0.05 dex in HIP 104214. Although this trend (sign) of abundance difference is similar to what Grevesse et al. (2015) reported for the Sun, its extent is so insignificant that the abundances from these two lines may be regarded as practically equal. Therefore, care should be taken in interpreting the abundance discrepancy (0.11 dex) between Rb 7800/7947 lines found by Grevesse et al. (2015). Since the effect of telluric contamination is considerable in the solar spectrum (cf. Fig. B1), the problem might be on the side of the Rb i 7947 line (e.g., underestimation of the abundance due to inadequate removal of the telluric line) rather than the blending on the Rb i 7800 line suggested by Grevesse et al. (2015). [h] c r r@ c@ c r@ c@ c c lObject & $^{\}V\_{\rm r}^{\rm topo}$ &W* & Δ & $A^{\rm N}$ & W & Δ & $A^{\rm N}$ & $\delta A^{\rm N}\_{7800-7947}$ & Remark & (km s− 1) & (mÅ) & (dex) & (dex) & (mÅ) & (dex) & (dex) & (dex) & & & & & HIP 104214 & − 82.0 & 34.1 & +0.059 & 2.284 & 17.6 & +0.060 & 2.236 & + 0.048 & ξ Boo B& + 4.5 & 40.5 & +0.063 & 2.347 & 31.2 & +0.063 & 2.494 & − 0.147 & telluric contaminated (7947 line) \Topocentric radial velocity (stellar radial velocity relative to the observer). [fig:B1] --- 1. Throughout this paper, A* denotes the logarithmic number abundance in the usual normalisation of A(H) = 12.00.[↩](#fnref1) 2. Although this spectrum is damaged in the short-wavelength side of the important Rb i 7800 line like other spectra of many group 2 stars taken from Takeda et al.’s (2005a) database (cf. the note to Table 1), the whole profile of Si i+Rb i line feature turned out fortunately usable (cf. Fig. 1).[↩](#fnref2) 3. The other resonance line of the doublet at 7947 Å was not used, because it is weaker and the quality of observed spectra in this wavelength region is poor (e.g., contamination of telluric lines) compared to the Rb i 7800 line.[↩](#fnref3) 4. This $v\_{\rm M}$ (including instrumental broadening, rotational broadening, and macroturbulence) is the e-folding half-width of the Gaussian broadening function $f(v) \propto \exp[-(v/v\_{\rm M})^{2}]$.[↩](#fnref4) 5. It should be noted that this W is evaluated by integrating the synthesised line profile on the assumption that this Rb i 7800 line feature (comprising multiple components as listed in Table 2) is isolated, despite that it is actually blended with the neighbouring Si i line. Therefore, it is not so much a directly observable quantity in real spectra as rather a useful measure of line strength.[↩](#fnref5) 6. This k value was adopted by consulting the previous cases for the resonance lines of other alkali species; i.e., k = 10− 3 for Li i 6708 as well as K i 7699, k = 10− 1 for Na i 5890/5896. Actually, however, the choice of this correction factor does not matter much, because the resulting non-LTE effect turns out anyhow insignificant even by drastically reducing the H i collisions as such. For example, the ∣Δ∣ values are $\lesssim 0.1$ dex at most (cf. Sect. 3.4), which are further reduced by $\lesssim$ several hundredths dex if the H i collision effect is included with k =�	arxiv_0000811
Musielak-Orlicz-Hardy Spaces Associated with Operators and Their Applications ================================================================================= 13.8cm Let X be a metric space with doubling measure and L a nonnegative self-adjoint operator in L2(X) satisfying the Davies-Gaffney estimates. Let φ : X × [0, ∞) → [0, ∞) be a function such that φ(x, ⋅ ) is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}\_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ) ∈ (0, 1] and it satisfies the uniformly reverse Hölder inequality of order 2/[2 − I(φ)]. In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ, L(X), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak-Orlicz $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{BMO}\_{\varphi,\,L}(\mathcal{X})$, which is further proved to be the dual space of Hφ, L(X) and hence whose φ-Carleson measure characterization is deduced. Characterizations of Hφ, L(X), including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize Hφ, L(X) in terms of the Littlewood-Paley gλ\-function g*λ, L\* and establish a Hörmander-type spectral multiplier theorem for L on Hφ, L(X). Moreover, for the Musielak-Orlicz-Hardy space Hφ, L(Rn) associated with the Schrödinger operator L : = − Δ + V, where 0 ≤ V ∈ Lloc1(Rn), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom and the molecule; finally, the authors show that the Riesz transform ∇L− 1/2 is bounded from Hφ, L(Rn) to the Musielak-Orlicz space Lφ(Rn) when i(φ) ∈ (0, 1], and from Hφ, L(Rn) to the Musielak-Orlicz-Hardy space Hφ(Rn) when $i(\varphi)\in(\frac{n}{n+1},1]$, where i(φ) denotes the uniformly critical lower type index of φ. Introduction[s1] ================ The real-variable theory of Hardy spaces on the n-dimensional Euclidean space $\rn$, initiated by Stein and Weiss, plays an important role in various fields of analysis (see, for example, ). It is well known that the Hardy space $H^p(\rn)$ when p ∈ (0, 1] is a suitable substitute of the Lebesgue space $L^p(\rn)$; for example, the classical Riesz transform is bounded on $H^p(\rn)$, but not on $L^p(\rn)$ when p ∈ (0, 1]. Moreover, the practicability of $H^p(\rn)$ with p ∈ (0, 1], as a substitute for $L^p(\rn)$ with p ∈ (0, 1], comes from its several equivalent real-variable characterizations, which were originally motivated by Fefferman and Stein in their seminal paper. Among these characterizations, a very important and useful characterization of the Hardy spaces $H^p(\rn)$ is their atomic characterizations, which were obtained by Coifman when n = 1 and Latter when n > 1. Moreover, a direct extension of the atomic characterization of the Hardy spaces is the molecular characterization established by Taibleson and Weiss. On the other hand, as a natural generalization of $L^p (\rn)$, the Orlicz space was introduced by Birnbaum-Orlicz in and Orlicz in, which has extensive applications in several branches of mathematics (see, for example, for more details). Moreover, the Orlicz-Hardy space, introduced and studied in, is also a suitable substitute of the Orlicz space in the study of the boundedness of operators (see, for example, ). Furthermore, weighted local Orlicz-Hardy spaces and their dual spaces were also studied in. All theories of these function spaces are intimately connected with the Laplace operator $\Delta:=\sum\_{i=1}^n\frac{\partial^2}{\partial x\_i^2}$ on $\rn$. Recall that the classical $\bbmo$ space (the space of functions with bounded mean oscillation) is originally introduced by John and Nirenberg to solve some problems in partial differential equations. Since Fefferman and Stein proved that $\bbmo(\rn)$ is the dual space of $H^1 (\rn)$, the space $\bbmo(\rn)$ plays an important role in not only partial differential equations but also harmonic analysis (see, for example, for more details). Moreover, the generalized space $\bbmo\_\rho(\rn)$ was introduced and studied in and it was proved therein to be the dual space of the Orlicz-Hardy space $H\_\Phi(\rn)$, where Φ denotes the Orlicz function on $(0,\fz)$ and ρ(t) : = t− 1/Φ− 1(t− 1) for all $t\in(0,\fz)$. Here and in what follows, Φ− 1 denotes the inverse function of Φ. Recently, Ky introduced a new Musielak-Orlicz-Hardy space, $H\_{\fai}(\rn)$, via the grand maximal function, which contains both the Orlicz-Hardy space in and the weighted Hardy space $H^p\_{\overline}(\rn)$ with ${\overline}\in A\_{\fz}(\rn)$ in as the spacial cases. Here, $\fai:\,\rn\times[0,\fz)\to[0,\fz)$ is a function such that $\fai(x,\cdot)$ is an Orlicz function of uniformly upper type 1 and lower type p for some p ∈ (0, 1] (see Section [s2] below for the definitions of uniformly upper or lower types), and $\fai(\cdot,t)$ is a Muckenhoupt weight, and $A\_q(\rn)$ with $q\in[1,\fz]$ denotes the class of Muckenhoupt’s weights (see, for example, for their definitions and properties). Moreover, the Musielak-Orlicz $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{BMO}\_{\fai}(\rn)$ was also introduced and further proved to be the dual space of $H\_{\fai}(\rn)$ in by using the atomic characterization of $H\_{\fai}(\rn)$ established in. Furthermore, some interesting applications of the spaces $H\_\fai(\rn)$ and $\mathop\mathrm{BMO}\_{\fai}(\rn)$ were given in. Moreover, the radial and the non-tangential maximal functions characterizations, the Littlewood-Paley function characterization and the molecular characterization of $H\_\fai(\rn)$ were obtained in. As an application of the Lusin area function characterization of $H\_\fai(\rn)$, the $\fai$-Carleson measure characterization of the space $\bbmo\_\fai(\rn)$ was obtained in. Furthermore, the local Musielak-Orlicz-Hardy space and its dual space were studied in. It is worth pointing out that Musielak-Orlicz functions are the natural generalization of Orlicz functions (see, for example, ) and the motivation to study function spaces of Musielak-Orlicz type is attributed to their extensive applications to many branches of physics and mathematics (see, for example, for more details). In recent years, the study of function spaces associated with different operators inspired great interests (see, for example, and their references). More precisely, Auscher, Duong and McIntosh initially studied the Hardy space $H^1\_L(\rn)$ associated with the operator L whose heat kernel satisfies a pointwise Poisson upper bound estimate. Based on this, Duong and Yan introduced the $\bbmo$-type space $\bbmo\_L(\rn)$ associated with L and proved that the dual space of $H^1\_L(\rn)$ is just $\bbmo\_{L^\ast}(\rn)$, where L\* denotes the adjoint operator of L in $L^2(\rn)$. Moreover, Yan further generalized these results to the Hardy space $H^p\_L(\rn)$ with p ∈ (0, 1] close to 1 and its dual space. Also, the Orlicz-Hardy space and its dual space associated with such an L were studied in. Moreover, Hofmann and Mayboroda and Hofmann et al. introduced the Hardy and Sobolev spaces associated with a second order divergence form elliptic operator L on $\rn$ with bounded measurable complex coefficients and these operators may not have the pointwise heat kernel bounds, and further established several equivalent characterizations for these spaces and studied their dual spaces. Meanwhile, the Orlicz-Hardy space and its dual space associated with L were independently studied in. Furthermore, Orlicz-Hardy spaces associated with a second order divergence form elliptic operator on the strongly Lipschitz domain of $\rn$ were studied in. It is worth pointing out that the strongly Lipschitz domain of $\rn$ is a special space of homogeneous type in the sense of Coifman and Weiss. Recall that the Hardy spaces on strongly Lipschitz domains associated with the Laplace operator having some boundary conditions were originally and systematically studied by Chang et al. in and Auscher et al.. On the other hand, the Hardy space associated with the Schrödinger operator − Δ + V was studied in, where the nonnegative potential V satisfies the reverse Hölder inequality (see, for example, for the definition of the reverse Hölder inequality). More generally, for nonnegative self-adjoint operators L satisfying the Davies-Gaffney estimates, Hofmann et al. studied the Hardy space $H^1\_L({{\mathcal X}})$ associated with L and its dual space on a metric measure space ${{\mathcal X}}$, which was extended to the Orlicz-Hardy space in. As a special case of this setting, several equivalent characterizations and some applications of the Hardy space $H^1\_L(\rn)$ and the Orlicz-Hardy space $H\_{\Phi,\,L}(\rn)$ associated with the Schrödinger operator L : = − Δ + V were, respectively, obtained in and, where $0\le V\in L^1\_{\loc}(\rn)$. Moreover, Song and Yan studied the weighted Hardy space $H^1\_{{\overline},\,L}(\rn)$ associated with the Schrödinger operator L, where ${\overline}\in A\_1(\rn)$. Very recently, some special Musielak-Orlicz-Hardy spaces associated with the Schrödinger operator L : = − Δ + V on $\rn$, where the nonnegative potential V satisfies the reverse Hölder inequality of order n/2, were studied by Ky and further applied to the study of commutators of singular integral operators associated with the operator L. Let ${{\mathcal X}}$ be a metric measure space, L a nonnegative self-adjoint operator on $L^2({{\mathcal X}})$ satisfying the Davies-Gaffmey estimates, and $E(\lz)$ the spectral resolution of L. For any bounded Borel function $m:\ [0,\fz)\to\cc$, by using the spectral theorem, it is well known that the operator $$\begin{aligned} \label{1.1} m(L):=\int\_0^\fz m(\lz)\,dE(\lz)\end{aligned}$$ is well defined and bounded on $L^2({{\mathcal X}})$. It is an interesting problem to find some sufficient conditions on m and L such that m(L) in is bounded on various function spaces on ${{\mathcal X}}$, which was extensively studied (see, for example, and their references). Specially, Duong and Yan proved that m(L) is bounded on the Hardy space $H^p\_L({{\mathcal X}})$, with $p\in(0,\fz)$, associated with L when ${{\mathcal X}}$ is a metric space with doubling measure and the function m satisfies a Hörmander-type condition. Throughout the whole paper, let ${{\mathcal X}}$ be a metric space with doubling measure μ and L a nonnegative self-adjoint operator in $L^2({{\mathcal X}})$ satisfying the Davies-Gaffney estimates. Let $\fai:\,{{\mathcal X}}\times[0,\fz)\to[0,\fz)$ be a growth function as in Definition [d2.3] below, which means that $\fai(x,\cdot)$ is an Orlicz function (see Section [s2.3] below), $\fai(\cdot,t)\in \aa\_{\infty}({{\mathcal X}})$ (the class of uniformly Muckenhoupt weights in Definition [d2.2] below), and its uniformly critical upper type index $I(\fai)\in(0,1]$ (see below). Moreover, we always assume that $\vz\in\rh\_{2/[2-I(\vz)]}({{\mathcal X}})$ (see Definition [d2.2] below). A typical example of such a $\fai$ is $$\label{1.2} \fai(x,t):={\overline}(x)\Phi(t)$$ for all $x\in{{\mathcal X}}$ and $t\in[0,\fz)$, where ${\overline}\in A\_{\fz}({{\mathcal X}})$ (the class of Muckenhoupt weights) and Φ is an Orlicz function on $[0,\fz)$ of upper type p1 ∈ (0, 1] and lower type p2 ∈ (0, 1] (see below for the definition of types). Let $x\_0\in{{\mathcal X}}$. Another typical and useful example of the growth function $\fai$ is $$\label{1.3} \fai(x,t):=\frac{t^{\az}}{[\ln(e+d(x,x\_0))]^{\bz}+[\ln(e+t)]^{\gz}}$$ for all $x\in{{\mathcal X}}$ and $t\in[0,\fz)$ with some $\az\in(0,1]$, $\bz\in[0,n)$ and $\gz\in [0,2\az(1+\ln2)]$ (see Section [s2.3] for more details). It is worth pointing out that such a function $\fai$ naturally appears in the study of the pointwise multiplier characterization for the BMO-type space on the metric space with doubling measure (see ). Motivated by, in this paper, we study the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ and its dual space. More precisely, for all $f\in L^2({{\mathcal X}})$ and $x\in{{\mathcal X}}$, define $$\label{1.4} S\_L(f)(x):=\lf\{\int\_{\bgz(x)}\lf\|t^2Le^{-t^2L}f(y)\r\|^2 \frac{d\mu(y)\,dt}{V(x,t)t}\r\}^{1/2},$$ here and in what follows, $\bgz(x):=\{(y,t)\in{{\mathcal X}}\times(0,\fz):\ d(x,y)<t\}$, d denotes the metric on ${{\mathcal X}}$, $B(x,t):=\{y\in{{\mathcal X}}:\ d(x,y)<t\}$, μ denotes the nonnegative Borel regular measure on ${{\mathcal X}}$ and V(x, t) : = μ(B(x, t)). The Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ is then defined to be the completion of the set $\{f\in H^2({{\mathcal X}}):\ S\_L(f)\in L^\fai({{\mathcal X}})\}$ with respect to the quasi-norm $$\\|f\\|\_{H\_{\fai,\,L}({{\mathcal X}})}:=\\|S\_L(f)\\|\_{L^\fai({{\mathcal X}})}:= \inf\lf\{\lz\in(0,\fz):\ \int\_{{{\mathcal X}}}\fai\lf(x,\frac{S\_L(f)(x)}{\lz}\r)\,d\mu(x)\le1\r\},$$ where $H^2({{\mathcal X}}):=\overline{R(L)}$ and $\overline{R(L)}$ denotes the closure of the range of L* in $L^2({{\mathcal X}})$. In this paper, we first establish the atomic decomposition of $H\_{\fai,\,L}({{\mathcal X}})$ and further obtain its molecular decomposition. Using the atomic and the molecular decompositions of $H\_{\fai,\,L}({{\mathcal X}})$, we then prove that its dual space is the Musielak-Orlicz ${\mathop\mathrm{BMO}}$-type space $\bbmo\_{\fai,\,L}({{\mathcal X}})$, which is characterized by the $\fai$-Carleson measure, and further establish the atomic and the molecular characterizations of $H\_{\fai,\,L}({{\mathcal X}})$. We also obtain another characterization of $H\_{\fai,\,L}({{\mathcal X}})$ via the Lusin area function associated with the Poisson semigroup of L. As applications, by using the atomic characterization, we prove that Littlewood-Paley functions g*L and gλ, L\* are bounded from Hφ, L(X) to the Musielak-Orlicz space Lφ(X); as a corollary, we characterize Hφ, L(X) in terms of the Littlewood-Paley $g^\ast\_\lz$-function gλ, L\. We further establish a Hörmander-type spectral multiplier theorem for L* on Hφ, L(X) by using the atomic and the molecular characterizations of Hφ, L(X). As further applications, we obtain several equivalent characterizations of the Musielak-Orlicz-Hardy space Hφ, L(Rn) associated with the Schrödinger operator L : = − Δ + V, where 0 ≤ V ∈ Lloc1(Rn), in terms of the Lusin-area function, the non-tangential maximal function, the radial maximal function, the atom and the molecule. Finally, we show that the Riesz transform ∇L− 1/2 is bounded from Hφ, L(Rn) to Lφ(Rn) when i(φ) ∈ (0, 1] and from Hφ, L(Rn) to the Musielak-Orlicz-Hardy space Hφ(Rn) when $i(\varphi)\in(\frac{n}{n+1},1]$, where $i(\vz)$ denotes the uniformly critical lower type index of $\vz$ (see below). The key step of the above approach is to establish the atomic (molecular) characterization of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$. To this end, we inherit a method used in. We first establish the atomic decomposition of the Musielak-Orlicz tent space $T\_\fai({{\mathcal X}}\times(0,\fz))$ associated with $\fai$, whose proof implies that if $f\in T\_\fai({{\mathcal X}}\times(0,\fz))\cap T^2\_2({{\mathcal X}}\times(0,\fz))$, then the atomic decomposition of f holds true in both $T\_\fai({{\mathcal X}}\times(0,\fz))$ and $T^2\_2({{\mathcal X}}\times(0,\fz))$. We point out that in this paper, by the assumptions on L, we only know that the Lusin area function SL as in is bounded on $L^2({{\mathcal X}})$ (see below). To prove that the atomic decomposition of $f\in T\_\fai({{\mathcal X}}\times(0,\fz))\cap T^2\_2({{\mathcal X}}\times(0,\fz))$ holds true in $T^2\_2 ({{\mathcal X}}\times(0,\fz))$ (see Corollary [c3.1] below), we need the additional assumption that $\fai(\cdot,t)$ for all $t\in [0,\fz)$ belongs to the uniformly reverse Hölder class $\rh\_{2/[2-I(\fai)]}({{\mathcal X}})$. Then by the fact that the operator πΨ, L in below is bounded from $T^2\_2({{\mathcal X}}\times(0,\fz))$ to $L^2({{\mathcal X}})$, we further obtain the $L^2({{\mathcal X}})$-convergence of the corresponding atomic decomposition for functions in $H\_{\fai,\,L}({{\mathcal X}})\cap L^2({{\mathcal X}})$, since for all $f\in H\_{\fai,\,L}({{\mathcal X}})\cap L^2({{\mathcal X}})$, $t^2Le^{-t^2L}f\in T^2\_2({{\mathcal X}}\times(0,\fz))\cap T\_\fai({{\mathcal X}}\times(0,\fz))$. This technique plays a fundamental role in the whole paper. We remark that the method used to obtain the atomic characterization of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ in this paper is different from that in, but more close to the method in. More precisely, in, the atomic characterization of the weighted Hardy space $H^1\_L(\rn)$, associated with the Schrödinger operator L, was established by using the Calderón reproducing formula associated with L and a subtle decomposition of all dyadic cubes in $\rn$. However, in this paper, we establish the atomic characterization of $H\_{\fai,\,L}({{\mathcal X}})$ by using the Calderón reproducing formula associated with L (see below), the atomic decomposition of the Musielak-Orlicz tent space established in Theorem [t3.1] below and some boundedness (see Proposition [p4.1] below) of the operator πΨ, L defined in below. Moreover, we also point out that the notion of atoms in our atomic decomposition of the Musielak-Orlicz tent space is different from that in. Since the weight also appears in the norm of atoms used by Bui and Duong when establishing the atomic decomposition of elements in the weighted tent space, Bui and Duong had to require the weight ${\overline}\in A\_1({{\mathcal X}})\cap RH\_{2/(2-p)}({{\mathcal X}})$ in order to obtain the atomic decomposition of the weighted Hardy space $H^p\_{{\overline},\,L}({{\mathcal X}})$ with p ∈ (0, 1] (see the proof of for the details). Instead of this, we do not use the weight in the norm of our $T\_\fai({{\mathcal X}}\times(0,\fz))$-atoms. Due to this subtle choice, we are able to relax the requirements on the growth function into $\fai\in\aa\_\fz({{\mathcal X}})\cap\rh\_{2/[2-I(\fai)]}({{\mathcal X}})$, which essentially improves the results of Bui and Duong even when $\vz$ is as in. Another important estimate, appeared in the approach of this paper, is that there exists a positive constant C such that, for any $\lz\in\cc$ and $(\fai,\,M)$-atom $\az$ adapted to the ball B (or any $(\fai,\,M,\,\epz)$-molecule $\az$ adapted to the ball B), $$\label{1.5} \int\_{{{\mathcal X}}}\fai(x,S\_L(\lz\az)(x))\,d\mu(x) \le C\fai\lf(B,\|\lz\|\\|\chi\_B\\|^{-1}\_{L^\fai({{\mathcal X}})}\r);$$ see Definitions [d4.2] and [d4.3] below for the notions of $(\fai,\,M)$-atoms and $(\fai,\,M,\,\epz)$-molecules. A main difficulty to prove is how to take $S\_L(\lz\az)(x)$ out of the position of the time variable of $\fai$. In, to obtain when $\vz$ is as in with ${\overline}\equiv 1$, it was assumed that Φ is a concave Orlicz function on $(0,\fz)$. In this case, Jensen’s inequality does the job. In the present setting, the spatial variable and the time variable of $\fai$ are combinative, so Jensen’s inequality does not work even when $\fai$ is concave about the time variable. To overcome this difficulty, we subtly use the properties of $\fai$ which are the uniformly upper p1 ∈ (0, 1] and lower type p2 ∈ (0, 1] (see the proof of below). Precisely, this paper is organized as follows. In Section [s2], we first recall some notions and notation on metric measure spaces and then describe some basic assumptions on the operator L studied in this paper. We also recall some notation, some examples and some basic properties concerning growth functions considered in this paper. In Section [s3], we first recall some notions about tent spaces and then study the Musielak-Orlicz tent space $T\_\fai({{\mathcal X}}\times(0,\fz))$ associated with growth function $\fai$. The main target of this section is to establish the atomic characterization for $T\_\fai({{\mathcal X}}\times(0,\fz))$ (see Theorem [t3.1] below). Assume further that $\fai\in\rh\_{2/[2-I(\fai)]}({{\mathcal X}})$. As a byproduct, we know that if $f\in T\_\fai({{\mathcal X}}\times(0,\fz))\cap T\_2^2({{\mathcal X}}\times(0,\fz))$, then the atomic decomposition of f holds true in both $T\_\fai({{\mathcal X}}\times(0,\fz))$ and $T\_2^2({{\mathcal X}}\times(0,\fz))$, which plays an important role in the remainder of this paper (see Corollary [c3.1] below). We point out that Theorem [t3.1] and Corollary [c3.1] completely cover by taking $\fai$ as in with ${\overline}\equiv1$ and Φ concave. In Section [s4], we first introduce the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ and prove that the operator πΨ, L in below maps the Musielak-Orlicz tent space $T\_\fai({{\mathcal X}}\times(0,\fz))$ continuously into $H\_{\fai,\,L}({{\mathcal X}})$ (see Proposition [p4.1] below). By this and the atomic decomposition of $T\_\fai({{\mathcal X}}\times(0,\fz))$, we conclude that, for each $f\in H\_{\fai,\,L}({{\mathcal X}})$, there exists an atomic decomposition of f holding true in $H\_{\fai,\,L}({{\mathcal X}})$ (see Corollary [c4.1] below). We should point out that to obtain the atomic decomposition of $H\_{\fai,\,L}({{\mathcal X}})$, we borrow some ideas from, and the estimate is very important for this procedure. Via this atomic decomposition of $H\_{\fai,\,L}({{\mathcal X}})$, we further prove that the dual space of $H\_{\fai,\,L}({{\mathcal X}})$ is just the Musielak-Orlicz ${\mathop\mathrm{BMO}}$-type space $\bbmo\_{\fai,\,L}({{\mathcal X}})$ (see Theorem [t4.1] below). As an application of this duality, we establish the $\fai$-Carleson measure characterization of the space $\bbmo\_{\fai,\,L}({{\mathcal X}})$ (see Theorem [t4.2] below). We remark that when $\fai$ is as in with ${\overline}\equiv1$ and Φ concave, the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ and the Musielak-Orlicz ${\mathop\mathrm{BMO}}$-type space $\bbmo\_{\fai,\,L}({{\mathcal X}})$ are respectively the Orlicz-Hardy space $H\_{\Phi,\,L}({{\mathcal X}})$ and the $\bbmo$-type space $\bbmo\_{\rho,\,L}({{\mathcal X}})$ introduced in. In Section [s5], by Proposition [p4.3] and Theorem [t4.1], we establish the equivalence between $H\_{\fai,\,L}({{\mathcal X}})$ and the atomic (resp. molecular) Musielak-Orlicz-Hardy space $H^M\_{\fai,\,\at}({{\mathcal X}})$ (resp. $H^{M,\,\epz}\_{\fai,\,\mol}({{\mathcal X}})$) (see Theorem [t5.1] below). We notice that the series in $H^M\_{\fai,\,\at}({{\mathcal X}})$ (resp. $H^{M,\,\epz}\_{\fai,\,\mol}({{\mathcal X}})$) is required to converge in the norm of $(\bbmo\_{\fai,\,L}({{\mathcal X}}))^\ast$, where $(\bbmo\_{\fai,\,L}({{\mathcal X}}))^\ast$ denotes the dual space of $\bbmo\_{\fai,\,L}({{\mathcal X}})$; while in Corollary [c4.1] below, the atomic decomposition holds true in $H\_{\fai,\,L}({{\mathcal X}})$. Applying its atomic characterization, we further characterize the Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ in terms of the Lusin area function associated with the Poisson semigroup of L (see Theorem [t5.2] below). Observe that Theorems [t5.1] and [t5.2] completely cover by taking $\fai$ as in with ${\overline}\equiv1$ and Φ concave. In Section [s6], we give some applications of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ to the boundedness of operators. More precisely, in Subsection [s6.1], we prove that the Littlewood-Paley g-function gL is bounded from $H\_{\fai,\,L}({{\mathcal X}})$ to the Musielak-Orlicz space $L^\fai({{\mathcal X}})$ (see Theorem [t6.1] below); in Subsection [s6.2], we show that the $g\_\lz^\ast$-function $g^\ast\_{\lz,\,L}$ is bounded from $H\_{\fai,\,L}({{\mathcal X}})$ to $L^\fai({{\mathcal X}})$ (see Theorem [t6.2] below). As a corollary, we characterize Hφ, L(X) in terms of the $g\_\lz^\ast$-function gλ, L\* (see Corollary [c6.1] below). Observe that when ${{\mathcal X}}:=\rn$ and L : = − Δ, $g^\ast\_{\lz,\,L}$ is just the classical Littlewood-Paley $g^\ast\_\lz$-function. Moreover, the range of $\lz$ in Theorem [t6.2] coincides with the corresponding result on the classical Littlewood-Paley $g^\ast\_\lz$-function on $\rn$ in the case that $\fai$ is as in with that ${\overline}\in A\_q(\rn)$, $q\in[1,\fz)$, and Φ(t) : = tp for all $t\in[0,\fz)$, p ∈ (0, 1] (see Remark [r6.2] below). Thus, in some sense, the range of $\lz$ in Theorem [t6.2] is sharp, which is attributed to the use of the unweighted norm in our definition of tent atoms, appearing in the atomic decomposition of the tent space $T\_\fai({{\mathcal X}}\times(0,\fz))$. Finally, in Subsection [s6.3], we establish a Hörmander-type spectral multiplier theorem for m(L) as in on Hφ, L(X) (see Theorem [t6.3] below). Let p ∈ (0, 1]. We remark that Theorem [t6.3] covers in the case that p ∈ (0, 1] by taking $\fai(x,t):=t^p$ for all $x\in\rn$ and $t\in[0,\fz)$. A typical example of the function m satisfying the condition of Theorem [t6.3] is $m(\lz)=\lz^{i\gz}$ for all $\lz\in\rr$ and some real value $\gz$, where i denotes the unit imaginary number (see Corollary [c6.2] below). As applications, in Section [s7], we study the Musielak-Orlicz-Hardy spaces $H\_{\fai,\,L}(\rn)$ associated with the Schrödinger operator L : = − Δ + V, where $0\le V\in L^1\_{\loc}(\rn)$. As an application of Theorems [t5.1] and [t5.2], we characterize $H\_{\fai,\,L}(\rn)$ in terms of the Lusin-area function associated with the Poisson semigroup of L, the atom and the molecule (see Theorem [t7.1] below). Moreover, characterizations of $H\_{\fai,\,L}(\rn)$, in terms of the non-tangential maximal functions associated with the heat semigroup and the Poisson semigroup of L, the radial maximal functions associated with the heat semigroup and the Poisson semigroup of L, are also established (see Theorem [t7.2] below). Observe that Theorem [t7.2] completely covers by taking $\fai$ as in with ${\overline}\equiv1$ and Φ satisfying that there exist $q\_1,\,q\_2\in(0,\fz)$ such that q1 < 1 < q2 and [Φ(tq2)]q1 is a convex function on $(0,\fz)$. Finally, we show that the Riesz transform ∇L− 1/2 associated with L is bounded from $H\_{\fai,\,L}(\rn)$ to $L^\fai(\rn)$ when $i(\fai)\in(0,1]$, and from $H\_{\fai,\,L}(\rn)$ to the Musielak-Orlicz-Hardy space $H\_{\fai}(\rn)$ introduced by Ky when $i(\fai)\in(\frac{n}{n+1},1]$ (see Theorems [t7.3] and [t7.4] below). We remark that the boundedness of ∇L− 1/2 from $H^1\_L(\rn)$ to the classical Hardy space $H^1(\rn)$ was first established in and that Theorems [t7.3] and [t7.4] are respectively when $\fai$ is as in with ${\overline}\equiv1$ and Φ concave. We also point out that when n = 1 and $\fai(x,t):=t$ for all $x\in\rn$ and $t\in[0,\fz)$, the Hardy space $H\_{\fai,\,L}(\rn)$ coincides with the Hardy space introduced by Czaja and Zienkiewicz ; if L : = − Δ + V with V belonging to the reverse Hölder class ${\mathop\mathrm{RH}}\_q(\rn)$ for some q ≥ n/2 and n ≥ 3, and $\fai(x,t):=t^p$ with $p\in(\frac{n}{n+1},1]$ for all $x\in\rn$ and $t\in[0,\fz)$, then the Hardy space $H\_{\fai,\,L}(\rn)$ coincides with the Hardy space introduced by Dziubański and Zienkiewicz. To prove Theorem [t7.2] below, we borrow some ideas from the proof of. To this end, via invoking the Caccioppoli inequality associated with L, the special differential structure of L itself and the divergence theorem, we first establish a weighted ``good-$\lz$ inequality" concerning the non-tangential maximal function ${{\mathcal N}}\_P (f)$, associated with the Poisson semigroup of L, and the truncated variant of the Lusin area function ${\widetilde}{S}\_P (f)$ in Lemma [l7.2] below, which is a suitable substitute, in the present setting, of a distribution inequality concerning the non-tangential maximal function ${{\mathcal N}}\_P (f)$ and the Lusin area function ${\widetilde}{S}\_P (f)$, appeared in the proof of (see also ). We then use the Moser type local boundedness estimate from (see also Lemma [l7.3] below), which is the substitute of the classical mean value property for harmonic functions in this setting. Moreover, a more delicate estimate in below than that used in the proof of is established, which leads us in Theorem [t7.2] below to remove the additional assumption, appeared in, that there exist $q\_1,\,q\_2\in(0,\fz)$ such that q1 < 1 < q2 and [Φ(tq2)]q1 is a convex function on $(0,\fz)$ even when $\fai$ is as in with ${\overline}\equiv1$. The proof of Theorem [t7.3] is a skillful application of the atomic characterization of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}(\rn)$, a Davies-Gaffney type estimate (see or Lemma [l7.4] below) and the $L^2(\rn)$-boundedness of the Riesz transform ∇L− 1/2. Furthermore, as an application of the atomic characterization of $H\_{\fai,\,L}(\rn)$ obtained in Theorem [t7.1] and the atomic characterization of the Musielak-Orlicz-Hardy space $H\_\fai(\rn)$ established by Ky (see also Lemma [l7.5] below), we obtain the boundedness of the Riesz transform ∇L− 1/2 from $H\_{\fai,\,L}(\rn)$ to $H\_\fai(\rn)$ in Theorem [t7.4] below. More precisely, for any given atom $\az$ as in Definition [d4.2] below, we prove that $$\nabla L^{-1/2}(\az)=\sum\_j b\_j$$ in $L^2(\rn)$, where, for each j, bj is a multiples of an atom introduced by Ky. Observe that the atom in Definition [d4.2] below is different from the atom in in that the norm of the atom in Definition [d4.2] is not weighted, but the atom introduced by Ky is weighted and, moreover, that, in the present setting, ∇L− 1/2 is known to be bounded on $L^p(\rn)$ only with p ∈ (1, 2]. Thus, in order to prove that, for each j, bj is a multiple of an atom as in, we need the assumption that $q(\fai)<2$ and $r(\fai)>2/[2-q(\fai)]$ (see below for the details), where $q(\fai)$ and $r(\fai)$ are, respectively, as in and below. We remark that there exist more applications of the results in this paper. For example, motivated by, in a forthcoming paper, we will apply the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}(\rn)$ and the Musielak-Orlicz $\bbmo$-type space $\bbmo\_{\fai,\,L}(\rn)$ associated with the Schrödinger operator L, introduced in this paper, to the study of pointwise multipliers on BMO-type space associated with the Schrödinger operator L and commutators of singular integral operators associated with the operator L. This is reasonable, since $\vz$ in naturally appears in the study of these problems in. Moreover, motivated by, in another forthcoming paper, we will further establish various maximal function characterizations of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}(\boz)$ on the strongly Lipschitz domain $\boz$ of $\rn$ associated with the Schrödinger operator L with some boundary conditions, which is a special case of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ introduced in this paper. After the first version of this paper was put on arXiv, we learned from Dr. Bui that, in, Bui and Duong also introduced the weighted Hardy space $H^p\_{L,\,{\overline}}({{\mathcal X}})$, with p ∈ (0, 1] and ${\overline}\in A\_1({{\mathcal X}})$ satisfying the reverse Hölder inequality of order 2/(2 − p), by the Lusin area function associated with the heat semigroup generated by L. Moreover, Bui and Duong established the atomic and the molecular characterizations of $H^p\_{L,\,{\overline}}({{\mathcal X}})$ and, as applications, obtained the boundedness on $H^p\_{L,\,{\overline}}({{\mathcal X}})$ of the generalized Riesz transforms associated with L and of the spectral multipliers of L. These results are partially overlapped with the results of this paper when $\vz$ is as in with Φ(t) : = tp for p ∈ (0, 1] and $t\in[0,\fz)$. As have observed above, the atomic decomposition of the weighted tent space obtained in and the Riesz transforms considered in are different from these in this paper. We also point out that, it is motivated by, in the present version of this paper, we replace the assumption in the first version that the growth function $\fai$ is of uniformly upper type 1 by the assumption that $\fai$ is of uniformly upper type p1 for some p1 ∈ (0, 1] and hence, in the main results of this paper, we improve the assumption in the first version that $\fai\in\rh\_2({{\mathcal X}})$ into the weaker assumption that $\fai\in\rh\_{2/[2-I(\fai)]}({{\mathcal X}})$. Finally we make some conventions on notation. Throughout the whole paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. We also use $C(\gz,\bz,\cdots)$ to denote a positive constant depending on the indicated parameters $\gz$, $\bz$, ⋯. The symbol $A\ls B$ means that A ≤ CB. If $A\ls B$ and $B\ls A$, then we write A ∼ B. The symbol $\lfz s\rfz$ for $s\in\rr$ denotes the maximal integer not more than s. For any given normed spaces $\mathcal A$ and $\mathcal B$ with the corresponding norms $\\|\cdot\\|\_{\mathcal A}$ and $\\|\cdot\\|\_{\mathcal B}$, the symbol ${\mathcal A}\subset{\mathcal B}$ means that for all $f\in \mathcal A$, then $f\in\mathcal B$ and $\\|f\\|\_{\mathcal B}\ls \\|f\\|\_{\mathcal A}$. For any measurable subset E of ${{\mathcal X}}$, we denote by $E^\complement$ the set ${{\mathcal X}}\setminus E$ and by χE its characteristic function. We also set $\nn:=\{1,\,2,\, \cdots\}$ and $\zz\_+:=\{0\}\cup\nn$. For any $\tz:=(\tz\_{1},\ldots,\tz\_{n})\in\zz\_+^{n}$, let $\|\tz\|:=\tz\_{1}+\cdots+\tz\_{n}$. For any subsets E, $F\subset{{\mathcal X}}$ and $z\in{{\mathcal X}}$, let $\dist(E, F):=\inf\_{x\in E,\,y\in F}d(x,y)$ and $\dist(z, E):=\inf\_{x\in E}d(z,x).$ Preliminaries[s2] ================= In Subsection [s2.1], we first recall some notions on metric measure spaces and then, in Subsection [s2.2], describe some basic assumptions on the operator L studied in this paper. In Subsection [s2.3], we recall some notions concerning growth functions considered in this paper and also give some specific examples of growth functions satisfying the assumptions of this paper. Subsection [s2.4] is devoted to recalling some properties of growth functions established in. Metric measure spaces[s2.1] --------------------------- Throughout the whole paper, we let ${{\mathcal X}}$ be a set, d a metric on ${{\mathcal X}}$ and μ a nonnegative Borel regular measure on ${{\mathcal X}}$. For all $x\in{{\mathcal X}}$ and $r\in(0,\fz)$, let $$B(x,r):=\{y\in{{\mathcal X}}:\ d(x,y)<r\}$$ and V(x, r) : = μ(B(x, r)). Moreover, we assume that there exists a constant $C\_1\in[1,\fz)$ such that, for all $x\in{{\mathcal X}}$ and $r\in(0,\fz)$, $$\label{2.1} V(x,2r)\le C\_1V(x,r)<\fz.$$ Observe that $({{\mathcal X}},\,d,\,\mu)$ is a space of homogeneous type in the sense of Coifman and Weiss. Recall that in the definition of spaces of homogeneous type in, d is assumed to be a quasi-metric. However, for simplicity, we always assume that d is a metric. Notice that the doubling property implies that the following strong homogeneity property that, for some positive constants C and n, $$\label{2.2} V(x,\lz r)\le C\lz^n V(x,r)$$ uniformly for all $\lz\in[1,\fz)$, $x\in{{\mathcal X}}$ and $r\in(0,\fz)$. There also exist constants $C\in(0,\fz)$ and N ∈ [0, n] such that, for all $x,\,y\in{{\mathcal X}}$ and $r\in(0,\fz)$, $$\label{2.3} V(x,r)\le C\lf[1+\frac{d(x,y)}{r}\r]^NV(y,r).$$ Indeed, the property with N = n is a simple corollary of the triangle inequality for the metric d and the strong homogeneity property. In the cases of Euclidean spaces and Lie groups of polynomial growth, N can be chosen to be 0. In what follows, to simplify the notation, for each ball $B\subset{{\mathcal X}}$, set $$\label{2.4} U\_0(B):=B\ \text{and}\ U\_j(B):=2^jB\setminus2^{j-1}B \ \text{for}\ j\in\nn.$$ Furthermore, for $p\in(0,\fz]$, the space of p-integrable functions on ${{\mathcal X}}$ is denoted by $L^p({{\mathcal X}})$ and the (quasi-)norm of $f\in L^p({{\mathcal X}})$ by $\\|f\\|\_{L^p({{\mathcal X}})}$. Assumptions on operators L[s2.2] ---------------------------------- Throughout the whole paper, as in, we always suppose that the considered operators L satisfy the following assumptions. L is a nonnegative self-adjoint operator in $L^2({{\mathcal X}})$. The operator L generates an analytic semigroup {e− tL}t > 0 which satisfies the Davies-Gaffney estimates, namely, there exist positive constants C2 and C3 such that, for all closed sets E and F in ${{\mathcal X}}$, $t\in(0,\fz)$ and f ∈ L2(E), $$\label{2.5} \lf\\|e^{-tL}f\r\\|\_{L^2(F)}\le C\_2\exp\lf\{-\frac{[\dist(E,F)]^2}{C\_3t}\r\}\\|f\\|\_{L^2(E)},$$ here and in what follows, $\dist(E,F):=\inf\_{x\in E,\,y\in F}d(x,y)$ and L2(E) is the set of all μ-measurable functions supported in E** such that $\\|f\\|\_{L^2(E)}:=\{\int\_E\|f(x)\|^2\,d\mu(x)\}^{1/2}<\fz$. Examples of operators satisfying Assumptions (A) and (B) include second order elliptic self-adjoint operators in divergence form on $\rn$ with bounded measurable coefficients, (degenerate) Schrödinger operators with nonnegative potential or with magnetic field, and Laplace-Beltrami operators on all complete Riemannian manifolds (see, for example, ). By Assumptions (A) and (B), we have the following results which were established in. [l2.1] Let L satisfy Assumptions (A) and (B). Then for every fixed $k\in\nn$, the family of operators, {(t2L)ke− t2L}t > 0, satisfies the Davies-Gaffney estimates with positive constants C2 and C3 only depending on n, N and k. In what follows, for any operator T, let KT denote its integral kernel. It is well known that if L satisfies Assumptions (A) and (B), and $T:=\cos(t\sqrt{L})$ with $t\in(0,\fz)$, then there exists a positive constant C4 such that $$\label{2.6} \supp K\_T\subset{{\mathcal D}}\_t:=\lf\{(x,y)\in{{\mathcal X}}\times{{\mathcal X}}:\ d(x,y)\le C\_4t\r\}$$ (see, for example,, and ). This observation plays a key role in obtaining the atomic characterization of the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ (see and Proposition [p4.2] below). [l2.2] Assume that L satisfies Assumptions (A) and (B). Let $\pz\in C^\fz\_c(\rr)$ be even and $\supp\pz\subset(-C\_4^{-1},C\_4^{-1})$, where C4 is as in. Let ${\widetilde}\Phi$ denote the Fourier transform of $\pz$. Then for every $\kappa\in\nn$ and $t\in(0,\fz)$, the kernel $K\_{(t^2L)^{\kappa}{\widetilde}\Phi(t\sqrt{L})}$ of $(t^2L)^{\kappa}{\widetilde}\Phi(t\sqrt{L})$ satisfies that $\supp K\_{(t^2L)^{\kappa}{\widetilde}\Phi(t\sqrt{L})}\subset\{(x,y) \in{{\mathcal X}}\times{{\mathcal X}}:\ d(x,y)\le t\}$. For any given $\dz\in (0,\fz)$, let ϕ be a measurable function from $\cc$ to $\cc$ satisfying that there exists a positive constant $C(\dz)$ such that, for all $z\in\cc$, $\|\phi(z)\|\le C(\dz)\frac{\|z\|^\dz}{1+\|z\|^{2\dz}}$. Then $\int\_0^{\fz}\|\phi(t)\|^2t^{-1}\,dt<\fz$. It was proved in that, for all $f\in L^2({{\mathcal X}})$, $$\label{2.7} \int\_0^\fz\lf\\|\phi(t\sqrt{L})f\r\\|^2\_{L^2({{\mathcal X}})}\,\frac{dt}{t}\le \lf\{\int\_0^\fz\|\phi(t)\|^2\,\frac{dt}{t}\r\}\\|f\\|^2\_{L^2({{\mathcal X}})},$$ which is often used in what follows. Growth functions[s2.3] ---------------------- We recall that a function $\Phi:[0,\fz)\to[0,\fz)$ is called an Orlicz function if it is nondecreasing, Φ(0) = 0, Φ(t) > 0 for all $t\in(0,\fz)$ and $\lim\_{t\to\fz}\Phi(t)=\fz$ (see, for example, ). The function Φ is said to be of upper type p** (resp. lower type p*) for some $p\in[0,\fz)$, if there exists a positive constant C* such that, for all $t\in[1,\fz)$ (resp. t ∈ [0, 1]) and $s\in[0,\fz)$, Φ(st) ≤ CtpΦ(s). If Φ is of both upper type p1 and lower type p2, then Φ is said to be of type (p1, p2). The function Φ is said to be of strictly lower type p** if, for all t ∈ [0, 1] and $s\in[0,\fz)$, Φ(st) ≤ tpΦ(s). Define $$\label{2.8} p\_{\Phi}:=\sup\{p\in[0,\fz):\,\Phi(st)\le t^p\Phi(s) \,\text{holds true for all}\ t\in[0,1]\,\text{and}\ s\in[0,\fz)\}.$$ It was proved in that Φ is also of strictly lower type pΦ; in other words, pΦ is attainable. For a given function $\fai:\,{{\mathcal X}}\times[0,\fz)\to[0,\fz)$ such that, for any $x\in{{\mathcal X}}$, $\fai(x,\cdot)$ is an Orlicz function, $\fai$ is said to be of uniformly upper type p** (resp. uniformly lower type p*) for some $p\in[0,\fz)$, if there exists a positive constant C* such that, for all $x\in{{\mathcal X}}$, $t\in[1,\fz)$ (resp. t ∈ [0, 1]) and $s\in[0,\fz)$, $$\label{2.9} \fai(x,st)\le Ct^p\fai(x,s);$$ $\fai$ is said to be of positive uniformly upper type (resp. uniformly lower type) if it is of uniformly upper type (resp. uniformly lower type) p for some $p\in(0,\fz)$. Moreover, let $$\label{2.10} I(\fai):=\inf\{p\in(0,\fz):\ \fai\ \text{is of uniformly upper type}\ p\}$$ and $$\label{2.11} i(\fai):=\sup\{p\in(0,\fz):\ \fai\ \text{is of uniformly lower type}\ p\}.$$ In what follows, $I(\fai)$ and $i(\fai)$ are, respectively, called the uniformly critical upper type index and the uniformly critical lower type index of $\fai$. Observe that $I(\fai)$ and $i(\fai)$ may not be attainable, namely, $\fai$ may not be of uniformly upper type $I(\fai)$ and uniformly lower type $i(\fai)$ (see below for some examples). Let $\fai:{{\mathcal X}}\times[0,\fz)\to[0,\fz)$ satisfy that $x\mapsto\fai(x,t)$ is measurable for all $t\in[0,\fz)$. Following Ky, $\fai(\cdot,t)$ is said to be uniformly locally integrable if, for all bounded subsets K of ${{\mathcal X}}$, $$\int\_{K}\sup\_{t\in(0,\fz)}\lf\{\fai(x,t) \lf[\int\_{K}\fai(y,t)\,d\mu(y)\r]^{-1}\r\}\,d\mu(x)<\fz.$$ [d2.2] Let $\fai:{{\mathcal X}}\times[0,\fz)\to[0,\fz)$ be uniformly locally integrable. The function $\fai(\cdot,t)$ is said to satisfy the uniformly Muckenhoupt condition for some $q\in[1,\fz)$, denoted by $\fai\in\aa\_q({{\mathcal X}})$, if, when $q\in (1,\fz)$, $$\aa\_q (\fai):=\sup\_{t\in(0,\fz)}\sup\_{B\subset{{\mathcal X}}}\frac{1}{\mu(B)}\int\_B \fai(x,t)\,d\mu(x) \lf\{\frac{1}{\mu(B)}\int\_B [\fai(y,t)]^{-q'/q}\,d\mu(y)\r\}^{q/q'}<\fz,$$ where 1/q + 1/qʹ = 1, or $$\aa\_1 (\fai):=\sup\_{t\in(0,\fz)} \sup\_{B\subset{{\mathcal X}}}\frac{1}{\mu(B)}\int\_B \fai(x,t)\,d\mu(x) \lf(\esup\_{y\in B}[\fai(y,t)]^{-1}\r)<\fz.$$ Here the first supremums are taken over all $t\in(0,\fz)$ and the second ones over all balls $B\subset{{\mathcal X}}$. The function $\fai(\cdot,t)$ is said to satisfy the uniformly reverse Hölder condition for some $q\in(1,\fz]$, denoted by $\fai\in \rh\_q({{\mathcal X}})$, if, when $q\in (1,\fz)$, $$\begin{aligned} \rh\_q (\fai):&&=\sup\_{t\in(0,\fz)}\sup\_{B\subset{{\mathcal X}}}\lf\{\frac{1} {\mu(B)}\int\_B [\fai(x,t)]^q\,d\mu(x)\r\}^{1/q}\\ \nonumber &&\hspace{6 em}\times\lf\{\frac{1}{\mu(B)}\int\_B \fai(x,t)\,d\mu(x)\r\}^{-1}<\fz,\end{aligned}$$ or $$\rh\_{\fz} (\fai):=\sup\_{t\in (0,\fz)}\sup\_{B\subset{{\mathcal X}}}\lf\{\esup\_{y\in B}\fai(y,t)\r\}\lf\{\frac{1}{\mu(B)}\int\_B \fai(x,t)\,d\mu(x)\r\}^{-1} <\fz.$$ Here the first supremums are taken over all $t\in(0,\fz)$ and the second ones over all balls $B\subset{{\mathcal X}}$. Recall that in Definition [d2.2], when ${{\mathcal X}}=\rn$, $\aa\_q(\rn)$ with $q\in[1,\fz)$ was introduced by Ky. Let $\aa\_{\fz}({{\mathcal X}}):=\cup\_{q\in[1,\fz)}\aa\_{q}({{\mathcal X}})$ and define the critical indices of $\fai$ as follows: $$\label{2.12} q(\fai):=\inf\lf\{q\in[1,\fz):\ \fai\in\aa\_{q}({{\mathcal X}})\r\}$$ and $$\label{2.13} r(\fai):=\sup\lf\{q\in(1,\fz]:\ \fai\in\rh\_{q}({{\mathcal X}})\r\}.$$ Observe that, if $q(\fai)\in(1,\fz)$, then $\fai\not\in\aa\_{q(\fai)}({{\mathcal X}})$, and there exists $\fai\not\in\aa\_1({{\mathcal X}})$ such that $q(\fai)=1$ (see, for example, ). Similarly, if $r(\fai)\in(1,\fz)$, then $\fai\not\in\rh\_{r(\fai)}({{\mathcal X}})$, and there exists $\fai\not\in\rh\_\fz({{\mathcal X}})$ such that $r(\fai)=\fz$ (see, for example, ). Now we introduce the notion of growth functions. [d2.3] A function $\fai:\ {{\mathcal X}}\times[0,\fz)\to[0,\fz)$ is called a growth function if the following hold true: 1. $\fai$ is a Musielak-Orlicz function, namely, 1. the function $\fai(x,\cdot):\ [0,\fz)\to[0,\fz)$ is an Orlicz function for all $x\in{{\mathcal X}}$; 2. the function $\fai(\cdot,t)$ is a measurable function for all $t\in[0,\fz)$. 2. $\fai\in \aa\_{\fz}({{\mathcal X}})$. 3. The function $\fai$ is of positive uniformly upper type p1 for some p1 ∈ (0, 1] and of uniformly lower type p2 for some p2 ∈ (0, 1]. [r2.1] By the definitions of the uniformly upper type and the uniformly lower type, we see that, if the growth function $\fai$ is of positive uniformly upper type p1 and of positive uniformly lower type p2, then p1 ≥ p2. Clearly, $\fai(x,t):={\overline}(x)\Phi(t)$ is a growth function if ${\overline}\in A\_{\fz}({{\mathcal X}})$ and Φ is an Orlicz function of upper type p1 for some p1 ∈ (0, 1] and of lower type p2 for some p2 ∈ (0, 1]. It is known that, for p ∈ (0, 1], if Φ(t) : = tp for all $t\in [0,\fz)$, then Φ is an Orlicz function of type (p, p); for $p\in[\frac{1}{2},1]$, if Φ(t) : = tp/ln(e + t) for all $t\in [0,\fz)$, then Φ is an Orlicz function of lower type q for q ∈ (0, p) and of upper type p; for $p\in(0,\frac{1}{2}]$, if Φ(t) : = tpln(e + t) for all $t\in [0,\fz)$, then Φ is an Orlicz function of lower type p and of upper type q for q ∈ (p, 1]. Recall that if an Orlicz function is of upper type p ∈ (0, 1), then it is also of upper type 1. Another typical and useful growth function is $\fai$ as in. It is easy to show that if $\fai$ is as in, then $\fai\in \aa\_1({{\mathcal X}})$, $\fai$ is of uniformly upper type $\az$, $I(\fai)=i(\fai)=\az$, $i(\fai)$ is not attainable, but $I(\fai)$ is attainable. Moreover, it is worth to point out that such function $\fai$ naturally appears in the study of the pointwise multiplier characterization for the BMO-type space on the metric space with doubling measure (see ). We also point out that when ${{\mathcal X}}=\rn$, a similar example of such $\fai$ is given by Ky replacing d(x, x0) by ∣x∣, where ∣ ⋅ ∣ denotes the Euclidean distance on $\rn$. Some basic properties on growth functions[s2.4] ----------------------------------------------- Throughout the whole paper, we always assume that $\fai$ is a growth function as in Definition [d2.3]. Let us now introduce the Musielak-Orlicz space. The Musielak-Orlicz space $L^{\fai}({{\mathcal X}})$ is defined to be the set of all measurable functions f such that $\int\_{{{\mathcal X}}}\fai(x,\|f(x)\|)\,d\mu(x)<\fz$ with Luxembourg norm $$\\|f\\|\_{L^{\fai}({{\mathcal X}})}:=\inf\lf\{\lz\in(0,\fz):\ \int\_{{{\mathcal X}}} \fai\lf(x,\frac{\|f(x)\|}{\lz}\r)\,d\mu(x)\le1\r\}.$$ In what follows, for any measurable subset E of ${{\mathcal X}}$ and $t\in[0,\fz)$, let $$\fai(E,t):=\int\_E\fai(x,t)\,d\mu(x).$$ The following Lemmas [l2.3] and [l2.5] on the properties of growth functions are, respectively,. [l2.3] (i) Let $\fai$ be a growth function. Then $\fai$ is uniformly σ-quasi-subadditive on ${{\mathcal X}}\times[0,\fz)$, namely, there exists a positive constant C such that, for all $(x,t\_j)\in{{\mathcal X}}\times[0,\fz)$ with $j\in\nn$, $\fai(x,\sum\_{j=1}^{\fz}t\_j)\le C\sum\_{j=1}^{\fz}\fai(x,t\_j).$ (ii) Let $\fai$ be a growth function and ${\widetilde}{\fai}(x,t):=\int\_0^t\frac{\fai(x,s)}{s}\,ds$ for all $(x,t)\in{{\mathcal X}}\times[0,\fz)$. Then ${\widetilde}{\fai}$ is a growth function, which is equivalent to $\fai$; moreover, ${\widetilde}{\fai}(x,\cdot)$ is continuous and strictly increasing. [l2.5] Let c be a positive constant. Then there exists a positive constant C such that (i) $\int\_{{\mathcal X}}\fai(x,\frac{\|f(x)\|}{\lz})\,d\mu(x)\le c$ for some $\lz\in(0,\fz)$ implies that $\\|f\\|\_{L^{\fai}({{\mathcal X}})}\le C\lz;$ (ii) $\sum\_j\fai(B\_j,\frac{t\_j}{\lz})\le c$ for some $\lz\in(0,\fz)$ implies that $$\inf\lf\{\az\in(0,\fz):\ \sum\_j\fai\lf(B\_j,\frac{t\_j}{\az}\r)\le1\r\}\le C\lz,$$ where {tj}j is a sequence of positive numbers and {Bj}j a sequence of balls. In what follows, for any given ball B : = B(x, t), with $x\in{{\mathcal X}}$ and $r\in(0,\fz)$, and $\lz\in(0,\fz)$, we write $\lz B$ for the $\lz$-dilated ball of B, namely, $\lz B:=B(x,\lz t)$. We have the following properties for $\aa\_\fz({{\mathcal X}})$, whose proofs are similar to those in, and we omit the details. In what follows, ${{\mathcal M}}$ denotes the Hardy-Littlewood maximal function on ${{\mathcal X}}$, namely, for all $x\in{{\mathcal X}}$, $${{\mathcal M}}(f)(x):=\sup\_{x\in B}\frac{1}{\mu(B)}\int\_B\|f(y)\|\,d\mu(y),$$ where the supremum is taken over all balls B ∋ x. [l2.6] (i) $\aa\_1({{\mathcal X}})\subset\aa\_p({{\mathcal X}})\subset\aa\_q({{\mathcal X}})$ for $1\le p\le q<\fz$. (ii) $\rh\_{\fz}({{\mathcal X}})\subset\rh\_p({{\mathcal X}})\subset\rh\_q({{\mathcal X}})$ for $1<q\le p\le\fz$. (iii) If $\fai\in\aa\_p({{\mathcal X}})$ with $p\in(1,\fz)$, then there exists some q ∈ (1, p) such that $\fai\in\aa\_q({{\mathcal X}})$. (iv) If $\fai\in\rh\_p({{\mathcal X}})$ with $p\in(1,\fz)$, then there exists some $q\in(p,\fz)$ such that $\fai\in\rh\_q({{\mathcal X}})$. (v) $\aa\_\fz({{\mathcal X}})=\cup\_{p\in[1,\fz)}\aa\_p({{\mathcal X}}) \subset\cup\_{q\in(1,\fz]}\rh\_q({{\mathcal X}})$. (vi) If $p\in(1,\fz)$ and $\fai\in \aa\_{p}({{\mathcal X}})$, then there exists a positive constant C such that, for all measurable functions f on ${{\mathcal X}}$ and $t\in[0,\fz)$, $$\int\_{{{\mathcal X}}}\lf[{{\mathcal M}}(f)(x)\r]^p\fai(x,t)\,d\mu(x)\le C\int\_{{{\mathcal X}}}\|f(x)\|^p\fai(x,t)\,d\mu(x).$$ (vii) If $\fai\in \aa\_{p}({{\mathcal X}})$ with $p\in[1,\fz)$, then there exists a positive constant C such that, for all balls $B\_1,\,B\_2\subset{{\mathcal X}}$ with B1 ⊂ B2 and $t\in[0,\fz)$, $\frac{\fai(B\_2,t)}{\fai(B\_1,t)}\le C[\frac{\mu(B\_2)}{\mu(B\_1)}]^p.$ (viii) If $\fai\in \rh\_{q}({{\mathcal X}})$ with $q\in(1,\fz]$, then there exists a positive constant C such that, for all balls $B\_1,\,B\_2\subset{{\mathcal X}}$ with B1 ⊂ B2 and $t\in[0,\fz)$, $\frac{\fai(B\_2,t)}{\fai(B\_1,t)}\ge C[\frac{\mu(B\_2)}{\mu(B\_1)}]^{(q-1)/q}.$ [r2.2] We remark that in the setting of the Euclidean space $\rn$, Lemma [l2.6](v) can be improved to $\aa\_\fz(\rn)=\cup\_{p\in[1,\fz)}\aa\_p(\rn)=\cup\_{q\in(1,\fz]}\rh\_q(\rn)$ (see, for example, ). However, in the present setting, the inverse inclusion in Lemma [l2.6](v) may not be true (see for a counterexample). Musielak-Orlicz tent spaces $T\_\fai({{\mathcal X}}\times(0,\fz))$[s3] ====================================================================== In this section, we study the Musielak-Orlicz tent space associated with the growth function. We first recall some notions as follows. For any $\nu\in(0,\fz)$ and $x\in{{\mathcal X}}$, let $\bgz\_{\nu}(x):=\{(y,t)\in{{\mathcal X}}\times(0,\fz):\ d(x,y)<\nu t\}$ be the cone of aperture ν* with vertex $x\in{{\mathcal X}}$. For any closed subset F* of ${{\mathcal X}}$, denote by ${{\mathcal R}}\_{\nu}F$ the union of all cones with vertices in F*, namely, ${{\mathcal R}}\_{\nu}F:=\cup\_{x\in F}\bgz\_{\nu}(x)$ and, for any open subset O* of ${{\mathcal X}}$, denote the tent over O** by Tν(O), which is defined as $T\_{\nu}(O):=[{{\mathcal R}}\_{\nu}(O^\complement)]^{\complement}$. It is easy to see that $T\_{\nu}(O)=\{(x,t)\in{{\mathcal X}}\times(0,\fz):\ d(x,O^\complement) \ge\nu t\}.$ In what follows, we denote $\bgz\_1(x)$ and T1(O) simply by $\bgz(x)$ and Ô, respectively. For all measurable functions g on ${{\mathcal X}}\times(0,\fz)$ and $x\in{{\mathcal X}}$, define $${{\mathcal A}}(g)(x):=\lf\{\int\_{\bgz(x)}\|g(y,t)\|^2 \frac{d\mu(y)}{V(x,t)}\frac{dt}{t}\r\}^{1/2}.$$ If ${{\mathcal X}}=\rn$, Coifman, Meyer and Stein introduced the tent space $T^p\_2(\rr^{n+1}\_+)$ for $p\in(0,\fz)$, here and in what follows, $\rr^{n+1}\_+:=\rn\times(0,\fz)$. The tent space $T^p\_2({{\mathcal X}}\times(0,\fz))$ on spaces of homogenous type was introduced by Russ. Recall that a measurable function g is said to belong to the tent space $T^p\_2({{\mathcal X}}\times(0,\fz))$ with $p\in(0,\fz)$, if $\\|g\\|\_{T^p\_2({{\mathcal X}}\times(0,\fz))}:=\\|{{\mathcal A}}(g)\\|\_{L^p({{\mathcal X}})}<\fz$. Moreover, Harboure, Salinas and Viviani and Jiang and Yang, respectively, introduced the Orlicz tent spaces $T\_{\bfai}(\rr^{n+1}\_+)$ and $T\_{\bfai}({{\mathcal X}}\times(0,\fz))$. Let $\fai$ be as in Definition [d2.3]. In what follows, we denote by $T\_\fai({{\mathcal X}}\times(0,\fz))$ the space of all measurable functions g* on ${{\mathcal X}}\times(0,\fz)$ such that ${{\mathcal A}}(g)\in L^{\fai}({{\mathcal X}})$* and, for any $g\in T\_\fai({{\mathcal X}}\times(0,\fz))$, define its quasi-norm by $$\\|g\\|\_{T\_\fai({{\mathcal X}}\times(0,\fz))}:=\\|{{\mathcal A}}(g)\\|\_{L^{\fai}({{\mathcal X}})}= \inf\lf\{\lz\in(0,\fz):\ \int\_{{{\mathcal X}}}\fai\lf(x,\frac{{{\mathcal A}}(g)(x)}{\lz}\r)\,d\mu(x)\le1\r\}.$$ A function a on ${{\mathcal X}}\times(0,\fz)$ is called a $T\_\fai({{\mathcal X}}\times(0,\fz))$-atom if (i) there exists a ball $B\subset{{\mathcal X}}$ such that $\supp a\subset\widehat{B}$; (ii) $\int\_{\widehat{B}}\|a(x,t)\|^2\frac{d\mu(x)\,dt}{t}\le \mu(B)\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-2}$. For functions in $T\_\fai({{\mathcal X}}\times(0,\fz))$, we have the following atomic decomposition. [t3.1] Let $\fai$ be as in Definition [d2.3]. Then for any $f\in T\_\fai({{\mathcal X}}\times(0,\fz))$, there exist $\{\lz\_j\}\_j\subset\cc$ and a sequence {aj}j of $T\_\fai({{\mathcal X}}\times(0,\fz))$-atoms such that, for almost every $(x,t)\in{{\mathcal X}}\times(0,\fz)$, $$\label{3.1} f(x,t)=\sum\_{j}\lz\_ja\_j(x,t).$$ Moreover, there exists a positive constant C such that, for all $f\in T\_\fai({{\mathcal X}}\times(0,\fz))$, $$\begin{aligned} \label{3.2} \blz(\{\lz\_j a\_j\}\_j)&&:=\inf\lf\{\lz\in(0,\fz):\ \sum\_j\fai\lf(B\_j,\frac{\|\lz\_j\|} {\lz\\|\chi\_{B\_j}\\|\_{L^\fai({{\mathcal X}})}}\r)\le1\r\}\\ &&\le C\\|f\\|\_{T\_\fai({{\mathcal X}}\times(0,\fz))},\nonumber\end{aligned}$$ where, for each j, $\widehat{B\_j}$ appears in the support of aj. We prove Theorem [t3.1] by borrowing some ideas from the proof of (see also and ). To this end, we first need some known facts as follows. Let F be a closed subset of ${{\mathcal X}}$ and $O:=F^\complement$. Assume that $\mu(O)<\fz$. For any fixed $\gz\in(0,1)$, we say that $x\in{{\mathcal X}}$ has the global $\gz$-density with respect to F** if, for all $r\in(0,\fz)$, $$\frac{\mu(B(x,r)\cap F)}{\mu(B(x,r))}\ge\gz.$$ Denote by $F\_\gz^{\ast}$ the set of all such x*. It is easy to prove that $F\_\gz^{\ast}$ with $\gz\in(0,1)$ is a closed subset of F. Let $\gz\in(0,1)$ and $O\_\gz^{\ast}:=(F\_\gz^{\ast})^\complement$. Then it is easy to see that $O\subset O\_\gz^{\ast}$. Indeed, from the definition of O\, we deduce that $O\_\gz^\ast=\{x\in{{\mathcal X}}:\ {\widetilde}{{{\mathcal M}}}(\chi\_O)(x)>1-\gz\}$, where ${\widetilde}{{{\mathcal M}}}$ denotes the centered Hardy-Littlewood maximal function on ${{\mathcal X}}$, which, together with the fact that ${\widetilde}{{{\mathcal M}}}$ is of weak type (1, 1) (see ), further implies that there exists a positive constant $C(\gz)$, depending on $\gz$, such that $\mu(O\_\gz^\ast)\le C(\gz)\mu(O)$. Recall that, for all $f\in L^1\_\loc({{\mathcal X}})$ and $x\in{{\mathcal X}}$, $${\widetilde}{{{\mathcal M}}}(f)(x):=\sup\_{r\in(0,\fz)}\frac{1}{\mu(B(x,r))} \int\_{B(x,r)}\|f(y)\|\,d\mu(y).$$ It is well known that there exists a positive constant C5 such that, for all $x\in{{\mathcal X}}$ and $f\in L^1\_{\loc}({{\mathcal X}})$, $$\label{3.3} C\_5^{-1}{\widetilde}{{{\mathcal M}}}(f)(x)\le{{\mathcal M}}(f)(x)\le C\_5{\widetilde}{{{\mathcal M}}}(f)(x).$$ The following Lemma [l3.1] was established in. [l3.1] Let $\ez\in(0,1)$. Then there exist $\gz\_0\in(0,1)$ and $C(\ez,\gz\_0)\in(0,\fz)$ such that, for any closed subset F of ${{\mathcal X}}$ whose complement has finite measure, $\gz\in[\gz\_0,1)$ and nonnegative measurable function H on ${{\mathcal X}}\times(0,\fz)$, $$\int\_{{{\mathcal R}}\_{1-\ez}(F^\ast\_\gz)}H(y,t)V(y,t)\,d\mu(y)\,dt\le C(\ez,\gz\_0)\int\_F\lf\{\int\_{\bgz(x)}H(y,t)\,d\mu(y)\,dt\r\}\,d\mu(x),$$ where $F^\ast\_\gz$ denotes the set of points in ${{\mathcal X}}$ with the global $\gz$-density with respect to F. To prove Theorem [t3.1], we need a covering lemma established in. [l3.2] Let $\boz$ be a proper open subset of finite measure of ${{\mathcal X}}$. For any $x\in{{\mathcal X}}$, define $r(x):=d(x,\boz^\complement)/10$. Then there exist a positive integer M and a sequence {xj}j of points in ${{\mathcal X}}$ such that, if rj : = r(xj), then (i) $\boz=\cup\_j B(x\_j,r\_j)$; (ii) B(xi, ri/4) ∩ B(xj, rj/4) = ∅ if i ≠ j; (iii) for each j, $\sharp\{i:\ B(x\_i,5r\_i)\cap B(x\_j,5r\_j)\neq\emptyset\}\le M$, where $\sharp E$ denotes the cardinality of the set E; Moreover, there exist nonnegative functions {ϕj}j on ${{\mathcal X}}$ such that (iv) for all j, $\supp\phi\_j\subset B(x\_j,2r\_j)$; (v) for all j and x ∈ B(xj, rj), ϕj(x) ≥ 1/M; (vi) $\sum\_j\phi\_j=\chi\_{\boz}$. Moreover, we also need the following Lemma [l3.3], whose proof is similar to that of. We omit the details. [l3.3] Let $f\in T\_\fai({{\mathcal X}}\times(0,\fz))$ and $\boz\_k:=\{x\in{{\mathcal X}}:\ {{\mathcal A}}(f)(x)>2^k\}$ for all $k\in\zz$. Then there exists a positive constant C such that, for all $\lz\in(0,\fz)$, $$\sum\_{k\in\zz}\fai\lf(\boz\_k,\frac{2^k}{\lz}\r)\le C\int\_{{{\mathcal X}}}\fai\lf(x,\frac{{{\mathcal A}}(f)(x)}{\lz}\r)\,d\mu(x).$$ Now we prove Theorem [t3.1] by using Lemmas [l3.2] and [l3.3]. [Proof of Theorem [t3.1]] Let $f\in T\_\fai({{\mathcal X}}\times(0,\fz))$. For any $k\in\zz$, let $O\_k:=\{x\in{{\mathcal X}}:\ {{\mathcal A}}(f)(x)>2^k\}$ and $F\_k:=O\_k^\complement$. Since $f\in T\_\fai({{\mathcal X}}\times(0,\fz))$, for each k, Ok is an open set of ${{\mathcal X}}$ with $\mu(O\_k)<\fz$. Let $\ez\in(0,1)$ and $\gz\_0$ be as in Lemma [l3.1]. Let $\gz\in[\gz\_0,1)$ such that $C\_5(1-\gz)\le1/2$. In what follows, we denote $(F\_{k,\,\gz})^\ast$ and $(O\_{k,\,\gz})^\ast$ simply by Fk\* and Ok\, respectively. We claim that $\supp f\subset \cup\_{k\in\zz} T\_{1-\ez}(O^{\ast}\_k)\cup E$, where $E\subset{{\mathcal X}}\times(0,\fz)$ satisfies $\int\_E\frac{d\mu(y)\,dt}{t}=0$. Indeed, let (x, t) be the Lebesgue point of f* and $(x,t)\not\in \cup\_{k\in\zz}T\_{1-\ez}(O^{\ast}\_k)$. Then there exists a sequence $\{y\_k\}\_{k\in\zz}$ of points such that $\{y\_k\}\_{k\in\zz}\subset B(x,(1-\ez)t)$ and for each k, $y\_k\not\in T\_{1-\ez}(O^{\ast}\_k)$, which implies that, for each $k\in\zz$, ${\widetilde}{{{\mathcal M}}}(\chi\_{O\_k})(y\_k)\le 1-\gz$. From this, and $C\_5(1-\gz)\le1/2$, we deduce that $\mu(B(x,t)\cap\{z\in{{\mathcal X}}:\ {{\mathcal A}}(f)(z)\le2^k\})\ge\mu(B(x,t))/2.$ Letting $k\to-\fz$, we then see that $\mu(B(x,t)\cap\{z\in{{\mathcal X}}:\ {{\mathcal A}}(f)(z)=0\})\ge\mu(B(x,t))/2$. Therefore, there exists y ∈ B(x, t) such that f = 0 almost everywhere in $\bgz(y)$, which, together with Lebesgue’s differentiation theorem (see ), implies that f(x, t) = 0. By this, we know that the claim holds true. If $O\_k^\ast={{\mathcal X}}$ for some $k\in\zz$, then $\mu({{\mathcal X}})<\fz$, which implies that ${{\mathcal X}}$ is a ball (see ). In this case, set Ik : = {1}, $B\_{k,\,1}:={{\mathcal X}}$ and ϕk, 1 ≡ 1. If Ok\* is a proper subset of ${{\mathcal X}}$, by Lemma [l3.2] with $\boz=O\_k^{\ast}$, we obtain a set Ik of indices and balls {Bk, j}j ∈ Ik : = {B(xk, j, 2rk, j)}j ∈ Ik and functions {ϕk j}j ∈ Ik satisfying that, for each j ∈ Ik, $\supp\phi\_{k\,j}\subset B(x\_{x,\,j},2r\_{k,\,j})$ and ∑j ∈ Ikϕk, j = χOk\. Thus, for each $(x,t)\in{{\mathcal X}}\times(0,\fz)$, we see that $$\lf(\chi\_{T\_{1-\ez}(O^{\ast}\_k)}-\chi\_{T\_{1-\ez} (O^{\ast}\_{k+1})}\r)(x,t) =\sum\_{j\in I\_k}\phi\_{k,\,j}(x)\lf(\chi\_{T\_{1-\ez}(O^{\ast}\_k)} -\chi\_{T\_{1-\ez}(O^{\ast}\_{k+1})}\r)(x,t).$$ From this, $\supp f\subset\{\cup\_{k\in\zz}T\_{1-\ez}(O^{\ast}\_k)\cup E\}$ and $\int\_E\frac{d\mu(y)\,dt}{t}=0$, we infer that $$f=\sum\_{k\in\zz} f\lf(\chi\_{T\_{1-\ez}(O^{\ast}\_k)} -\chi\_{T\_{1-\ez}(O^{\ast}\_{k+1})}\r)=\sum\_{k\in\zz}\sum\_{j\in I\_k}f\phi\_{k,\,j}\lf(\chi\_{T\_{1-\ez}(O^{\ast}\_k)} -\chi\_{T\_{1-\ez}(O^{\ast}\_{k+1})}\r)$$ almost everywhere on ${{\mathcal X}}\times(0,\fz)$. For each k* and j, let $$a\_{k,\,j}:=2^{-k}\\|\chi\_{B\_{k,\,j}}\\|\_{L^\fai({{\mathcal X}})}^{-1} f\phi\_{k,\,j}\lf(\chi\_{T\_{1-\ez}(O^{\ast}\_k)}- \chi\_{T\_{1-\ez}(O^{\ast}\_{k+1})}\r)$$ and $\lz\_{k\,j}:=2^{k}\\|\chi\_{B\_{k,\,j}}\\|\_{L^\fai({{\mathcal X}})}$. Then $f= \sum\_{k\in\zz}\sum\_{j\in I\_k}\lz\_{k,\,j}a\_{k,\,j}$ almost everywhere. Similar to the proof of, we see that, for each k and j, $\supp a\_{k,\,j}\subset\widehat{C(\ez) B\_{k,\,j}}$, where $C(\ez)\in(1,\fz)$ is a positive constant independent of k and j. By Lemma [l3.1], $\supp a\_{k,\,j}\subset(T\_{1-\ez}(O^{\ast}\_{k+1}))^\complement ={{\mathcal R}}\_{1-\ez}(F^\ast\_{k+1})$ and the definition of Fk + 1, we know that, for each k and j, $$\begin{aligned} \lf\\|a\_{k,\,j}\r\\|^2\_{T^2\_2({{\mathcal X}}\times(0,\fz))}&&=\int\_{{{\mathcal X}}\times(0,\fz)} \|a\_{k,\,j}(y,t)\|^2\, \frac{d\mu(y)\,dt}{t}\ls\int\_{{{\mathcal R}}\_{1-\ez}(F^\ast\_{k+1})}\|a\_{k,\,j}(y,t)\|^2\, \frac{d\mu(y)\,dt}{t}\\ &&\ls\int\_{F\_{k+1}}\lf\{\int\_{\bgz(x)} \|a\_{k,\,j}(y,t)\|^2\, \frac{d\mu(y)\,dt}{V(y,t)t}\r\}\,d\mu(x)\\ &&\ls\int\_{F\_{k+1}\cap(C(\ez) B\_{k,\,j})}\lf[{{\mathcal A}}(a\_{k,\,j})(x)\r]^2\,d\mu(x)\\ &&\ls2^{-2k} \lf\\|\chi\_{B\_{k,\,j}}\r\\|^{-2}\_{L^\fai({{\mathcal X}})}\int\_{F\_{k+1}\cap(C(\ez) B\_{k,\,j})}\lf[{{\mathcal A}}(f)(x)\r]^2\,d\mu(x)\\ &&\ls V\lf(C(\ez) B\_{k,\,j}\r)\lf\\|\chi\_{C(\ez) B\_{k,\,j}}\r\\|^{-2}\_{L^\fai({{\mathcal X}})},\end{aligned}$$ which implies that up to a harmless multiplicative constant, each ak, j is a $T\_\fai({{\mathcal X}}\times(0,\fz))$-atom. Moreover, by, Lemma [l2.5](i) and Lemma [l3.3], we know that, for all $\lz\in(0,\fz)$, $$\begin{aligned} \sum\_{k\in\zz}\sum\_{j\in I\_k}\fai\lf(C(\ez) B\_{k,\,j},\frac{\|\lz\_{k,\,j}\|}{\lz\\|\chi\_{C(\ez) B\_{k,\,j}}\\|\_{L^\fai({{\mathcal X}})}}\r)&&\ls\sum\_{k\in\zz}\sum\_{j\in I\_k}\fai\lf(B\_{k,\,j},\frac{2^k}{\lz}\r)\ls\sum\_{k\in\zz} \fai\lf(O^\ast\_k,\frac{2^k}{\lz}\r)\\ &&\ls\int\_{{{\mathcal X}}}\fai\lf(x,\frac{{{\mathcal A}}(f)(x)}{\lz}\r)\,d\mu(x),\end{aligned}$$ which implies that $\blz(\{\lz\_{k,\,j}a\_{k,\,j}\}\_{k\in\zz,\,j})\ls\\|f\\|\_{T\_\fai({{\mathcal X}}\times(0,\fz))}$. This finishes the proof of Theorem [t3.1]. [c3.1] Let $\fai$ be as in Definition [d2.3] with $\fai\in \rh\_{2/[2-I(\fai)]}({{\mathcal X}})$, where $I(\fai)$ is as in. If $f\in T\_\fai({{\mathcal X}}\times(0,\fz)) \cap T^2\_2({{\mathcal X}}\times(0,\fz)) $, then in Theorem [t3.1] holds true in both $T\_\fai({{\mathcal X}}\times(0,\fz))$ and $T^2\_2({{\mathcal X}}\times(0,\fz))$. By the uniformly upper type p1 property of $\fai$ with some $p\_1\in[I(\fai),1]$, Theorem [t3.1] and its proof, similar to the proof of, we can show Corollary [c3.1] and omit the details here. In what follows, let $T^b\_{\fai}({{\mathcal X}}\times(0,\fz))$ and $T^{p,\,b}\_2({{\mathcal X}}\times(0,\fz))$ with $p\in(0,\fz)$ denote, respectively, the set of all functions in $T\_\fai({{\mathcal X}}\times(0,\fz))$ and $T^p\_2({{\mathcal X}}\times(0,\fz))$ with bounded support. Here and in what follows, a function f on ${{\mathcal X}}\times(0,\fz)$ is said to have bounded support means that there exist a ball $B\subset{{\mathcal X}}$ and $0<c\_1<c\_2<\fz$ such that $\supp f\subset B\times(c\_1,c\_2)$. [p3.1] Let $\fai$ be as in Definition [d2.3]. Then $T^b\_{\fai}({{\mathcal X}}\times(0,\fz)) \subset T^{2,\,b}\_2({{\mathcal X}}\times(0,\fz))$ as sets. The proof of Proposition [p3.1] is an application of the uniformly lower type p2 property of $\fai$ for some p2 ∈ (0, 1], which is similar to that of. We omit the details. Musielak-Orlicz-Hardy spaces $H\_{\fai,\,L}({{\mathcal X}})$ and their duals[s4] ================================================================================ In this section, we always assume that the operator L satisfies Assumptions (A) and (B), and the growth function $\fai$ is as in Definition [d2.3]. We introduce the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ associated with L via the Lusin-area function and give its dual space via the atomic and molecular decomposition of $H\_{\fai,\,L}({{\mathcal X}})$. Let us begin with some notions. In order to introduce the Musielak-Orlicz-Hardy space associated with L, we follow the ideas appeared in and first define the $L^2({{\mathcal X}})$ adapted Hardy space $$\label{4.1} H^2({{\mathcal X}}):=H^2\_L({{\mathcal X}}):=\overline{R(L)},$$ where $\overline{R(L)}$ denotes the closure of the range of L* in $L^2({{\mathcal X}})$. Then $L^2({{\mathcal X}})$ is the orthogonal sum of $H^2({{\mathcal X}})$ and the null space N(L), namely, $L^2({{\mathcal X}})=\overline{R(L)}\bigoplus N(L)$. For all functions $f\in L^2({{\mathcal X}})$, let the Lusin-area function S*L(f) be as in. From, it follows that SL is bounded on $L^2({{\mathcal X}})$. Hofmann et al. introduced the Hardy space $H^1\_L({{\mathcal X}})$ associated with L as the completion of $\{f\in H^2({{\mathcal X}}):\ S\_L(f)\in L^1({{\mathcal X}})\}$ with respect to the norm $\\|f\\|\_{H^1\_L({{\mathcal X}})}:=\\|f\\|\_{L^1({{\mathcal X}})}$. The Orlicz-Hardy space $H\_{\bfai,\,L}({{\mathcal X}})$ was introduced in in a similar way. Following, we now introduce the Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ associated with L as follows. [d4.1] Let L satisfy Assumptions (A) and (B) and $\fai$ be as in Definition [d2.3]. A function $f\in H^2({{\mathcal X}})$ is said to be in ${\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$ if $S\_L (f)\in L^{\fai}({{\mathcal X}})$; moreover, define $$\\|f\\|\_{H\_{\fai,\,L}({{\mathcal X}})}:=\\|S\_L(f)\\|\_{L^\fai({{\mathcal X}})}:= \inf\lf\{\lz\in(0,\fz):\ \int\_{{{\mathcal X}}}\fai\lf(x,\frac{S\_L(f)(x)}{\lz}\r)\,d\mu(x)\le1\r\}.$$ The Musielak-Orlicz-Hardy space $H\_{\fai,\,L}({{\mathcal X}})$ is defined to be the completion of ${\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$ in the quasi-norm $\\|\cdot\\|\_{H\_{\fai,\,L}({{\mathcal X}})}$. [r4.1] (i) Notice that for $0\neq f\in L^2({{\mathcal X}})$, $\\|S\_L(f)\\|\_{L^\fai({{\mathcal X}})}=0$ holds true if and only if f ∈ N(L). Indeed, if f ∈ N(L), then t2Le− t2Lf = 0 almost everywhere and hence $\\|S\_L(f)\\|\_{L^\fai({{\mathcal X}})}=0$. Conversely, if $\\|S\_L(f)\\|\_{L^\fai({{\mathcal X}})}=0$, then t2Le− t2Lf = 0 almost everywhere on ${{\mathcal X}}\times(0,\fz)$. Hence, for all $t\in (0,\fz)$, (e− t2L − I)f = ∫0t − 2sLe− s2Lfds = 0, which further implies that Lf = Le− t2Lf = 0 almost everywhere and f ∈ N(L). Thus, in Definition [d4.1], it is necessary to use $\overline{R(L)}$ rather than $L^2({{\mathcal X}})$ to guarantee $\\|\cdot\\|\_{H\_{\fai,\,L}({{\mathcal X}})}$ to be a quasi-norm (see also and ). Moreover, we know that, if the kernels of the semigroup {e− tL}t > 0 satisfy the Gaussian upper bounded estimates, then N(L) = {0} and hence $H^2({{\mathcal X}})=L^2({{\mathcal X}})$ (see, for example, ). (ii) It is easy to see that $\\|\cdot\\|\_{H\_{\fai,\,L}({{\mathcal X}})}$ is a quasi-norm. (iii) From the Aoki-Rolewicz theorem in, it follows that there exists a quasi-norm ∥ ∣ ⋅ ∥ ∣ on ${\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$ and $\gz\in(0,1]$ such that, for all $f\in{\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$, $\\|\!\|f\\|\!\|\sim\\|f\\|\_{H\_{\fai,\,L}({{\mathcal X}})}$ and, for any sequence $\{f\_j\}\_j\subset{\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$, $$\lf\\|\!\lf\|\sum\_j f\_j\r\\|\!\r\|^{\gz}\le\sum\_j\\|\!\|f\_j\\|\!\|^{\gz}.$$ By the theorem of completion of Yosida, it follows that $({\widetilde}{H}\_{\fai,\,L}({{\mathcal X}}),\,\\|\!\|\cdot\\|\!\|)$ has a completion space $(H\_{\fai,\,L}({{\mathcal X}}),\\|\!\|\cdot\\|\!\|)$; namely, for any $f\in (H\_{\fai,\,L}({{\mathcal X}}),\\|\!\|\cdot\\|\!\|)$, there exists a Cauchy sequence $\{f\_k\}\_{k=1}^{\fz} \subset{\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$ such that $\lim\_{k\to\fz}\\|\!\|f\_k -f\\|\!\|=0$. Moreover, if $\{f\_k\}\_{k=1}^{\fz}$ is a Cauchy sequence in $({\widetilde}{H}\_{\fai,\,L}({{\mathcal X}}),\,\\|\!\|\cdot\\|\!\|)$, then there exists a unique $f\in H\_{\fai,\,L}({{\mathcal X}})$ such that $\lim\_{k\to\fz}\\|\!\|f\_k -f\\|\!\|=0$. Furthermore, by the fact that $\\|\!\|f\\|\!\|\sim\\|f\\|\_{H\_{\fai,\,L}({{\mathcal X}})}$ for all $f\in{\widetilde}{H}\_{\fai,\,L}({{\mathcal X}})$, we know that the spaces $(H\_{\fai,\,L}({{\mathcal X}}), \\|\cdot\\|\_{H\_{\fai,\,L}({{\mathcal X}})})$ and $(H\_{\fai,\,L}({{\mathcal X}}), \\|\!\|\cdot\\|\!\|)$ coincide with equivalent quasi-norms. (iv) If $\fai(x,t):=t$ for all $x\in{{\mathcal X}}$ and $t\in(0,\fz)$, the space $H\_{\fai,\,L}({{\mathcal X}})$ is just the space $H^1\_L({{\mathcal X}})$ introduced by Hofmann et al.. Moreover, if $\fai$ is as in with ${\overline}\equiv1$ and Φ concave on $(0,\fz)$, the space $H\_{\fai,\,L}({{\mathcal X}})$ is just the Orlicz-Hardy space $H\_{\bfai,\,L}({{\mathcal X}})$ introduced in. We now introduce $(\fai,M)$-atoms and $(\fai,\,M,\,\epz)$-molecules as follows. [d4.2] Let $M\in\nn$. A function $\az\in L^2({{\mathcal X}})$ is called a $(\fai,M)$-atom associated with the operator L if there exist a function $b\in {{\mathcal D}}(L^M)$ and a ball $B\subset{{\mathcal X}}$ such that (i) $\az=L^M b$; (ii) $\supp (L^kb)\subset B$, k ∈ {0, ⋯, M}; (iii) $\\|(r^2\_B L)^kb\\|\_{L^2({{\mathcal X}})}\le r^{2M}\_B[\mu(B)]^{1/2}\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}$, k ∈ {0, ⋯, M}. [d4.3] Let $M\in\nn$ and $\epz\in(0,\fz)$. A function $\bz\in L^2({{\mathcal X}})$ is called a $(\fai,\,M,\,\epz)$-molecule associated with the operator L if there exist a function $b\in {{\mathcal D}}(L^M)$ and a ball $B\subset{{\mathcal X}}$ such that (i) $\bz=L^M b$; (ii) for each k ∈ {0, ⋯, M} and $j\in\zz\_+$, there holds true $$\\|(r^2\_B L)^kb\\|\_{L^2(U\_j(B))}\le 2^{-j\epz}r^{2M}\_B[\mu(B)]^{1/2}\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1},$$ where Uj(B) with $j\in\zz\_+$ is as in. [r4.2] Let Φ be a concave Orlicz function on $(0,\fz)$ with pΦ ∈ (0, 1]. When $\fai(x,t)=\Phi(t)$ for all $x\in{{\mathcal X}}$ and $t\in[0,\fz)$, the $(\fai,\,M)$-atom is just the (Φ, M)-atom introduced in. However, the $(\fai,\,M,\,\epz)$-molecule is different from the $(\Phi,\,M,\,\epz)$-molecule in even when $\fai(x,t)=\Phi(t)$ for all $x\in{{\mathcal X}}$ and $t\in[0,\fz)$. More precisely, recall that $\bz$ is called a $(\Phi,\,M,\,\epz)$-molecule, introduced in, if (ii) of Definition [d4.3] is replaced by that, for each k ∈ {0, ⋯, M} and $j\in\zz\_+$, there holds true $$\lf\\|(r^2\_B L)^kb\r\\|\_{L^2(U\_j(B))}\le 2^{-j\epz}r^{2M}\_B[\mu(2^jB)]^{-1/2}[\rho(\mu(2^jB))]^{-1},$$ where Uj(B) with $j\in\zz\_+$ is as in and ρ is given by ρ(t) : = t− 1/Φ− 1(t− 1) for all $t\in(0,\fz)$. Let p2 be any lower type of Φ. Then for any $\epz\in(0,\fz)$, every $(\fai,\,M,\,\epz)$-molecule is a $(\Phi,\,M,\,\epz-n(1/p\_2-1/2))$-molecule when $\fai:=\Phi$. Indeed, by, we know that ρ is of upper type 1/p2 − 1, which, together with, implies that, for all $j\in\nn$, $[\rho(\mu(2^jB))]^{-1}\gs2^{-jn(1/p\_2-1)}[\rho(\mu(B))]^{-1}$. From this and, we further deduce that, for all $j\in\nn$, $[\mu(2^jB)]^{-1/2}[\rho(\mu(2^jB))]^{-1}\gs2^{-jn(1/p\_2-1/2)} [\mu(B)]^{-1/2}[\rho(\mu(B))]^{-1}$, which, together with the fact that $\\|\chi\_B\\|\_{L^\fai({{\mathcal X}})}=\mu(B)\rho(\mu(B))$, implies that the claim holds true. We point out that the notion of $(\fai,\,M,\,\epz)$-molecules is motivated by, which is convenient in applications (see, for example, for more details). Decompositions into atoms and molecules[s4.1] --------------------------------------------- Recall that a function f on ${{\mathcal X}}\times(0,\fz)$ is said to have bounded support, if there exist a ball $B\subset{{\mathcal X}}$ and $0<c\_1<c\_2<\fz$ such that $\supp f\subset B\times(c\_1,c\_2)$. In what follows, let $L^2\_b({{\mathcal X}}\times(0,\fz))$ denote the set of all functions $f\in L^2({{\mathcal X}}\times(0,\fz))$ with bounded support, $M\in\nn$ and $M>\frac{n}{2}[\frac{q(\fai)}{i(\fai)}-\frac{1}2]$, where n, $q(\fai)$ and $i(\fai)$ are respectively as in, and. Let ${\widetilde}\Phi$ be as in Lemma [l2.2] and $\Psi(t):=t^{2(M+1)}{\widetilde}\bfai(t)$ for all $t\in(0,\fz)$. For all $f\in L^2\_b({{\mathcal X}}\times(0,\fz))$ and $x\in{{\mathcal X}}$, define $$\label{4.2} \pi\_{\Psi,\,L}(f)(x):=C\_{\Psi}\int\_0^{\fz}\Psi(t\sqrt{L}) (f(\cdot,t))(x)\,\frac{dt}{t},$$ where CΨ is a positive constant such that $$\label{4.3} C\_{\Psi}\int\_0^\fz\Psi(t)t^2e^{-t^2}\,\frac{dt}{t}=1.$$ By and Hölder’s inequality, we easily see that, if $f\in L^2\_b({{\mathcal X}}\times(0,\fz))$, then $\pi\_{\Psi,\,L}(f)\in L^2({{\mathcal X}})$. Moreover, we have the following boundedness of πΨ, L. [p4.1] Let L satisfy Assumptions (A) and (B), πΨ, L be as in, $\fai$ as in Definition [d2.3] with $\fai\in\rh\_{2/[2-I(\fai)]}({{\mathcal X}})$ and $I(\fai)$ being as in, and $M\in\nn$ with $M>\frac{n}{2}[\frac{q(\fai)}{i(\fai)}-\frac{1}2]$, where n, $q(\fai)$ and $i(\fai)$ are, respectively, as in, and. Then (i) the operator πΨ, L, initially defined on the space $T^{2,\,b}\_2({{\mathcal X}}\times(0,\fz))$, extends to a bounded linear operator from $T^2\_2({{\mathcal X}}\times(0,\fz))$ to $L^2({{\mathcal X}})$; (ii) the operator πΨ, L, initially defined on the space $T^{b}\_{\fai}({{\mathcal X}})$, extends to a bounded linear operator from $T\_\fai({{\mathcal X}}\times(0,\fz))$ to $H\_{\fai,\,L}({{\mathcal X}})$. The conclusion (i) is just. We only need to show (ii) of this proposition. Let $f\in T^b\_{\fai}({{\mathcal X}}\times(0,\fz))$. Then by Proposition [p3.1], Corollary [c3.1] and (i), we know that $$\pi\_{\Psi,\,L}(f)=\sum\_j\lz\_j\pi\_{\Psi,\,L}(a\_j)=:\sum\_j\lz\_j\az\_j$$ in $L^2({{\mathcal X}})$, where $\{\lz\_j\}\_j$ and {aj}j satisfy and, respectively. Recall that, for each j, $\supp a\_j\subset\widehat{B\_j}$ and Bj is a ball of ${{\mathcal X}}$. Moreover, from, we deduce that SL is bounded on $L^2({{\mathcal X}})$, which implies that, for all $x\in{{\mathcal X}}$, $S\_L(\pi\_{\Psi,\,L}(f))(x)\le\sum\_j\|\lz\_j\|S\_L(\az\_j)(x)$. This, combined with Lemma [l2.3](i), yields that $$\label{4.4} \int\_{{{\mathcal X}}}\fai(x,S\_L(\pi\_{\Psi,\,L}(f))(x))\,d\mu(x)\ls \sum\_j\int\_{{{\mathcal X}}}\fai(x,\|\lz\_j\|S\_L(\az\_j)(x))\,d\mu(x).$$ We now show that $\az\_j=\pi\_{\Psi,\,L}(a\_j)$ is a multiple of a $(\fai,M)$-atom for each j. Let $$b\_j:=C\_{\Psi}\int\_0^\fz t^{2(M+1)} L{\widetilde}\Phi(t\sqrt{L})(a\_j(\cdot,t))\,\frac{dt}{t},$$ where CΨ is as in. Then for each j, from the definitions of $\az\_j$ and bj, it follows that $\az\_j=L^Mb\_j$. Moreover, by Lemma [l2.2], we know that, for each k ∈ {0, ⋯, M}, $\supp (L^k b\_j)\subset B\_j$. Furthermore, for any h ∈ L2(Bj), from Hölder’s inequality and, we infer that $$\begin{aligned} &&\lf\|\int\_{{{\mathcal X}}}(r^2\_{B\_j}L)^k b\_j(x)h(x)\,d\mu(x)\r\|\\ &&\hs=C\_{\Psi}\lf\|\int\_{{{\mathcal X}}}\int\_0^\fz t^{2(M+1)}(r^2\_{B\_j}L)^k L{\widetilde}\Phi(t\sqrt{L})(a\_j(\cdot,t))(x)h(x)\,\frac{d\mu(x)\,dt}{t}\r\|\\ &&\hs\ls r^{2M}\_{B\_j}\int\_{{{\mathcal X}}}\int\_0^{r\_B}\lf\|a\_j(y,t)(t^2L)^{k+1} {\widetilde}\Phi(t\sqrt{L})h(y)\r\|\,\frac{d\mu(y)\,dt}{t}\\ &&\hs\ls r^{2M}\_{B\_j}\\|a\_j\\|\_{T^2\_2({{\mathcal X}}\times(0,\fz))}\lf\{\int\_{{{\mathcal X}}}\int\_0^\fz\|(t^2L)^{k+1} {\widetilde}\Phi(t\sqrt{L})h(y)\|^2\,\frac{d\mu(y)\,dt}{t}\r\}^{1/2}\\ &&\hs\ls r^{2M}\_{B\_j}\\|a\_j\\|\_{T^2\_2({{\mathcal X}}\times(0,\fz))}\\|h\\|\_{L^2({{\mathcal X}})}\ls r^{2M}\_{B\_j}[V(B\_j)]^{1/2}\\|\chi\_{B\_j}\\|\_{L^\fai({{\mathcal X}})}^{-1} \\|h\\|\_{L^2({{\mathcal X}})},\end{aligned}$$ which implies that $\\|(r^2\_{B\_j}L)^k b\_j\\|\_{L^2({{\mathcal X}})}\ls r^{2M}\_{B\_j}[V(B\_j)]^{1/2} \\|\chi\_{B\_j}\\|\_{L^\fai({{\mathcal X}})}^{-1}$. Therefore, $\az\_j$ is a $(\fai,M)$-atom up to a harmless constant. We claim that, for any $\lz\in\cc$ and $(\fai,M)$-atom $\az$ supported in a ball $B\subset{{\mathcal X}}$, $$\label{4.5} \int\_{{{\mathcal X}}}\fai(x,S\_L(\lz \az)(x))\,d\mu(x)\ls\fai\lf(B,\frac{\|\lz\|}{\\|\chi\_B\\|\_{L^\fai({{\mathcal X}})}}\r).$$ If holds true, by, the facts that, for all $\lz\in(0,\fz)$, $$S\_L(\pi\_{\Psi,\,L}(f/\lz))=S\_L(\pi\_{\Psi,\,L}(f))/\lz$$ and $\pi\_{\Psi,\,L}(f/\lz)=\sum\_j\lz\_j\az\_j/\lz$, and $S\_L(\pi\_{\Psi,\,L}(f))\le\sum\_j\|\lz\_j\|S\_L(\az\_j)$, we see that, for all $\lz\in(0,\fz)$, $$\int\_{{{\mathcal X}}}\fai\lf(x,\frac{S\_L(\pi\_{\Psi,\,L}(f))(x)} {\lz}\r)\,d\mu(x)\ls\sum\_j \fai\lf(B\_j,\frac{\|\lz\_j\|}{\lz\\|\chi\_{B\_j}\\|\_{L^\fai({{\mathcal X}})}}\r),$$ which, together with, implies that $\\|\pi\_{\Psi,\,L}(f)\\|\_{H\_{\fai,\,L}({{\mathcal X}})}\ls\blz(\{\lz\_j\az\_j\}\_j) \ls\\|f\\|\_{T\_\fai({{\mathcal X}}\times(0,\fz))}$, and hence completes the proof of (ii). Now we prove. Write $$\label{4.6} \int\_{{{\mathcal X}}}\fai(x,S\_L(\lz\az)(x))\,d\mu(x)=\sum\_{k=0}^{\fz} \int\_{U\_k(B)}\fai(x,\|\lz\|S\_L(\az)(x))\,d\mu(x).$$ From the assumption $\fai\in\rh\_{2/[2-I(\fai)]}({{\mathcal X}})$, Lemma [l2.6](iv) and the definition of $I(\fai)$, we infer that, there exists $p\_1\in[I(\fai),1]$ such that $\fai$ is of uniformly upper type p1 and $\fai\in\rh\_{2/(2-p\_1)}({{\mathcal X}})$. For k ∈ {0, ⋯, 4}, by the uniformly upper type p1 property of $\fai$, Hölder’s inequality, $\fai\in\rh\_{2/(2-p\_1)}({{\mathcal X}})$, the $L^2({{\mathcal X}})$-boundedness of SL and, we conclude that $$\begin{aligned} \label{4.7} \hs\hs&&\int\_{U\_k(B)}\fai\lf(x,\|\lz\|S\_L(\az)(x)\r)\,d\mu(x)\\ \nonumber &&\hs\ls\int\_{U\_k(B)}\fai\lf(x,\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}\r) \lf\{1+\lf[S\_L(\az)(x)\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}\r]^{p\_1}\r\}\,d\mu(x)\\ \nonumber &&\hs\ls\fai\lf(U\_k(B),\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}\r) +\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{p\_1}\\ \nonumber &&\hs\hs\times \lf\{\int\_{U\_k(B)} \lf[\fai\lf(x,\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1} \r)\r]^{\frac{2}{2-p\_1}}\,d\mu(x)\r\} ^{\frac{2-p\_1}{2}}\lf\{\int\_{U\_k(B)} [S\_L(\az) (x)]^2\,d\mu(x)\r\}^{\frac{p\_1}{2}}\\ \nonumber &&\hs\ls\fai\lf(U\_k(B),\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}\r) \ls\fai\lf(B,\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}\r).\end{aligned}$$ From the assumption that $M>\frac{n}{2}[\frac{q(\fai)}{i(\fai)}-\frac12]$, it follows that, there exist $p\_2\in(0,i(\fai))$ and $q\_0\in(q(\fai),\fz)$ such that $M>\frac{n}{2}(\frac{q\_0}{p\_2}-\frac{1}{2})$. Moreover, by the definitions of $i(\fai)$ and $q(\fai)$, we know that $\fai$ is of uniformly lower type p2 and $\fai\in\aa\_{q\_0}({{\mathcal X}})$. When $k\in\nn$ with k ≥ 5, from the uniformly upper type p1 and lower type p2 properties of $\fai$, it follows that $$\begin{aligned} \label{4.8} &&\int\_{U\_k(B)}\fai\lf(x,\|\lz\|S\_L(\az)(x)\r)\,d\mu(x)\\ \nonumber &&\hs\ls \int\_{U\_k(B)}\fai\lf(x,\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}\r) \lf[S\_L(\az)(x)\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}\r]^{p\_1}\,d\mu(x)\\ \nonumber &&\hs\hs+\int\_{U\_k(B)}\fai\lf(x,\|\lz\|\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}^{-1}\r) \lf[S\_L(\az)(x)\\|\chi\_{B}\\|\_{L^\fai({{\mathcal X}})}\r]^{p\_2}\,d\mu(x)=:\mathrm{E}\_k+\mathrm{F}\_k.\end{aligned}$$ To estimate Ek and Fk, we first estimate $\int\_{U\_k(B)}[S\_L(\az)(x)]^2\,d\mu(x)$. Write $$\begin{aligned} \label{4.9} &&\int\_{U\_k(B)}[S\_L(\az)(x)]^2\,d\mu(x)\\ \nonumber &&\hs=\int\_{U\_k(B)}\int\_0^{\frac{d(x,x\_B)}{4}}\int\_{d(x,y)<t} \lf\|(t^2L)^{M+1}e^{-t^2L}b(y)\r\|^2\frac{d\mu(y)}{V(x,t)} \frac{dt}{t^{4M+1 }}\,d\mu(x)\\ \nonumber&&\hs\hs+ \int\_{U\_k(B)}\int\_{\frac{d(x,x\_B)}{4}}^\fz\int\_{d(x,y)<t}\cdots =:\mathrm{H}\_k+\mathrm{I}\_k.\end{aligned}$$ We first estimate the term Hk. Let $$G\_k(B):=\{y\in{{\mathcal X}}:\ \text{there exists}\ x\in U\_k(B)\ \text{such that}\ d(x,y)<d(x,x\_B)/4\}.$$ From x ∈ Uk(B), it follows that d(x, xB) ∈ [2k − 1rB, 2krB). Let z ∈ B and y ∈ Gk(B). Then d(y, z) ≥ d(x, xB) − d(y, x) − d(z, xB) ≥ 3d(x, xB)/4 − rB ≥ 2k − 2rB, which implies that $\dist(G\_k(B),B)\ge2^{k-2}r\_B$. By this, Fubini’s theorem, and, we know that $$\begin{aligned} \label{4.10} \qquad\ \mathrm{H}\_k &&\ls\int\_0^{2^{k+1}r\_B}\int\_{G\_k(B)}\lf\|(t^2L)^{M+1}e^{-t^2L}b(y)\r\|^2 \,d\mu(y)\,\frac{dt}{t^{4M+1}}\\ \nonumber &&\ls\\|b\\|^2\_{L^2(B)}\int\_0^{2^{k+1}r\_B} \exp\lf\{-\frac{[\dist(G\_k(B),B)]^2}{C\_3t^2}\r\}\,\frac{dt}{t^{4M+1}}\\ \nonumber &&\ls r^{4M}\_B \mu(B)\\|\chi\_B\\|\_{L^\fai({{\mathcal X}})}^{-2}\int\_0^{2^{k+1}r\_B} \lf[\frac{t}{2^k r\_B}\r]^{4M+1}\,\frac{dt}{t^{4M+1}} \ls2^{-4kM}\mu(B)\\|\chi\_B\\|\_{L^\fai({{\mathcal X}})}^{-2}.\end{aligned}$$ For Ik, from Lemma [l2.1], it follows that $$\begin{aligned} \mathrm{I}\_k&&\ls\int\_{2^{k-2}r\_B}^\fz \int\_{{{\mathcal X}}}\lf\|(t^2L)^{M+1}e^{-t^2L}b(y)\r\|^2 \,d\mu(y)\,\	arxiv_0000817
Electroweak Baryogenesis in Non-minimal Composite Higgs Models ============================================================== We address electroweak baryogenesis in the context of composite Higgs models, pointing out that modifications to the Higgs and top quark sectors can play an important rôle in generating the baryon asymmetry. Our main observation is that composite Higgs models that include a light, gauge singlet scalar in the spectrum [as in the model based on the symmetry breaking pattern SO(6) → SO(5)], provide all necessary ingredients for viable baryogenesis. In particular, the singlet leads to a strongly first-order electroweak phase transition and introduces new sources of CP violation in dimension-five operators involving the top quark. We discuss the amount of baryon asymmetry produced and the experimental constraints on the model. Motivation ========== It is well known that the baryon asymmetry observed in the universe cannot be explained by the Standard Model (SM) of particle physics. Indeed, neither the CP violation encapsulated in the CKM matrix nor the departure from thermal equilibrium during the electroweak phase transition (EWPhT) are large enough, given the LEP lower bound on the mass of the Higgs boson. This calls for physics beyond the Standard Model. In particular, new bosons could enhance the strength of the EWPhT, while new fermions, or new fermionic couplings, could provide additional sources of CP violation. Both ingredients are present in supersymmetric extensions of the SM and (perhaps not surprisingly given the added motivations coming from naturalness, gauge coupling unification, and Dark Matter) most studies of electroweak baryogenesis (EWBG) have concentrated on the minimal supersymmetric standard model (MSSM). Unfortunately, the LEP bound on the Higgs mass has pushed MSSM EWBG into fine-tuned territory. Furthermore, this and other bounds on sparticle masses from LEP and LHC have set strong constraints on some scenarios of low-energy supersymmetry, putting the original naturalness motivation under some pressure. Although non-minimal supersymmetric extensions of the MSSM can overcome some of these problems, the time is perhaps right to examine EWBG in non-supersymmetric models that address the hierarchy problem (and necessarily incorporate new degrees of freedom at the EW scale). Among the possibilities that have been studied, models in which the Higgs boson is composite and the SM fermions are partially composite seem to offer the most plausible alternative to supersymmetry. In such models the Higgs boson arises as a bound state of new, strongly-interacting dynamics and is naturally light, since it originates as a pseudo Nambu-Goldstone Boson (PNGB) of a broken global symmetry, similar to the pions of QCD. The pattern of symmetry breaking can be chosen to reduce the tension with the electroweak precision observables (EWPO) T and $Zb\bar b$, while the presence of a (composite) Higgs boson allows a mechanism for suppressing the contributions to S. Moreover, the assumption that the SM fermions are partially composite results in a suppression of dangerous contributions to flavor physics observables, provided the theory is approximately conformal in the ultraviolet. Such a theory would presumably give rise to a plethora of bosonic and fermionic resonances at or around the strong coupling scale and this exotic dynamics could certainly play a rôle in generating the observed baryon asymmetry (ref. , for example, pointed out that the radion, in the context of the AdS/CFT correspondence, could be relevant). Indeed it would seem that the connection between naturalness arguments and EWBG would be especially tight in such a theory, since the dominant radiative corrections (involving the mostly-composite Higgs and top quark sectors) shape the potential at both zero and non-zero temperatures and would be responsible both for stabilizing the weak scale and generating the required strongly first-order phase transition. Moreover, new, strong couplings between the top and Higgs sectors could provide the required source of CP-violation. Unfortunately, a quantitative demonstration of this link would seem to be out of the question if the strongly-coupled physics lies close to the EW scale, given our limited understanding of strongly-coupled dynamics. However, current data suggest that the strong coupling scale must be pushed up to a few TeV or so. Indeed, the suppression of the S parameter mentioned above is achieved in precisely this way and a similar hierarchy is also needed for sufficient suppression of contributions to flavor physics observables. Whilst this separation is bad news for naturalness (it requires a fine-tuning at the level of 10% or so), it at least offers the hope that we may be able to study EWBG in such models in an effective theory approach, in which the details of the strong dynamics at the strong coupling scale are integrated out. In such a framework, one only needs to specify the light degrees of freedom, together with the operators of low dimension that are present in the low-energy effective Lagrangian. The simplest such Lagrangian would contain only the SM degrees of freedom, together with additional operators beginning at dimension six. Such a Lagrangian would arise, for example, as the low energy limit of the ‘minimal composite Higgs model’, based on the coset SO(5)/SO(4). Clearly, since the SM on its own is deficient from the point of view of EWBG, the dimension-six (or higher) operators must have a large effect on both the EWPhT (via sextic and higher contributions to the Higgs potential ) and on CP-violating physics , but this conflicts with the need for a large strong-coupling scale (which suppresses the higher-dimension operators) and in any case might jeopardize the validity of the effective field theory expansion. However, there does not seem to be any compelling reason (pace Occam) to choose the composite Higgs model based on the coset SO(5)/SO(4) over one based on a larger coset, and featuring the same desirable properties. On the contrary, from the point of view of the EWPhT, we know that the most favorable case (at the renormalizable level) occurs when the Higgs sector is extended to include a gauge singlet scalar and that in this way, one may have a strongly first-order phase transition (for a recent comprehensive analysis, see and references therein). Such a scenario is realized in the composite Higgs model based on the global symmetry breaking pattern SO(6) → SO(5), where the coset is five-dimensional and the low-energy spectrum includes four degrees of freedom corresponding to the Higgs doublet plus one, real, singlet PNGB. What is more, the non-renormalizable operators in the low-energy effective Lagrangian of this model begin at dimension-five and include an operator coupling the singlet and the Higgs to a pair of top quarks that violates CP. In this article, we show that such a model can generate the baryon asymmetry. As we have argued, it suffices to study the low-energy effective theory of the SM plus a singlet, including the aforementioned dimension-five operator. The scenario offers a testable way to explain the origin of the baryon asymmetry and can also be compared with constraints on new, CP-violating physics coming from electric dipole moment (EDM) tests and from LEP. In Section [Sec:SingletModlel] we summarize the features of the composite Higgs model with a singlet that are relevant for baryogenesis (more details are given in Appendix [Sec:CH]) and in Section [sec:EWBG] we study how the baryon asymmetry arises in this scenario. In Section [sec:EDM], we study electric dipole moment and LEP constraints, while in Section [sec:SCPV] we quantify how much explicit CP violation is needed to obtain a sufficient net baryon asymmetry. In Section [sec:PT] we estimate the characteristics of the phase transition (such as the wall thickness and critical temperature) in a special case where the theory is approximately $\Z2$-symmetric: then the structure of the effective Lagrangian is simpler and allows for an analytical study. Finally, in Section [Sec:Concl] we present our conclusions. In Appendix [Sec:appendixTransport], we collect the transport equations used to calculate the baryon asymmetry. The SM plus a Singlet from a Composite Higgs[Sec:SingletModlel] =============================================================== We are interested in composite Higgs models that, in the low energy spectrum, include the SM and a further real, scalar degree of freedom, singlet under the SM gauge group. One example is the model based on the SO(6)/SO(5) coset of ref. , which we summarize in Appendix [Sec:CH]. In this Section, we highlight the features that play a rôle in EWBG: in particular, we concentrate on the effective scalar potential and on the couplings between the Higgs and top-quark sectors, which, from naturalness arguments, are expected to be mostly composite. The most general effective scalar potential at the renormalizable level is V = Veven + Vodd, with [VtreeEven] V&& -h2 \|H\|2+h \|H\|4 && - s2 s2 +s s4 +m s2\|H\|2, V&&m s \|H\|2 +13 s +3 s3 ,[VtreeOdd] where μh, s, m, 1, 3 have dimension of mass and λh, s, m are dimensionless[1](#fn1); H denotes the Higgs SU(2)L doublet with physical component $h/\sqrt{2}$. Veven(odd) denotes the part of the potential that is even (odd) with respect to the Z2 transformation s → − s. While this is an isometry of the coset space, whether or not it is a symmetry of the effective Lagrangian depends on how the SM fermions are coupled to the sigma model . Let us now consider the couplings between the singlet and the fermions. Lorentz invariance alone allows the singlet s to couple to a Dirac fermion F via s \|F(a+ib5)F , where a (b) is a dimensionless coefficient describing its (pseudo)scalar-like couplings. In the SM, however, the SU(2)L × U(1)Y gauge symmetry forbids such a term in the Lagrangian and s can interact with the SM fermions only at the non-renormalizable level, beginning at dimension five with the operator [topcoupling] H \|Q3(a+ib5) t + h.c. , where f is the analogue of the pion decay constant and is related to the mass mρ (of order the confinement scale Λ) and coupling gρ of the strong sector resonances via mρ = gρf, where $g\_{SM}\lesssim g\_\rho \lesssim 4\pi$ and gSM is a typical SM coupling. In eq. ([topcoupling]) we have written only the coupling between the singlet s and the third generation SU(2)L doublet, Q3, and singlet, t. Indeed, naturalness implies that the Higgs and top sectors be mostly composite, so that the strong dynamics is expected to influence mostly the interactions within and between these two sectors. Even in this case, interactions with the lighter fermions will be present in the mass eigenstate basis, but are expected to be of the order of the corresponding (small) Yukawa couplings. Finally, it is useful for what follows to consider how one may implement CP in this context: If Vodd vanishes, a = 0 and b ≠ 0, the singlet behaves as a pseudoscalar and CP is conserved; similarly for b = 0 and a ≠ 0 the singlet is scalar-like and CP is also conserved in the Lagrangian. Other non-trivial choices inevitably violate CP. Electroweak Baryogenesis[sec:EWBG] ================================== Two conditions need to be fulfilled during the EWPhT in order to create enough baryon/antibaryon asymmetry. First of all, CP violation must be present within the wall separating the broken from the unbroken phase. This sources an excess of left-handed versus right-handed fermions[2](#fn2) in front of the wall which is converted into a baryon versus antibaryon excess by non-perturbative electroweak (sphaleron) processes. For this excess to be conserved, these sphaleron processes must be quickly suppressed within the broken phase. This brings us to the second condition: that the EWPhT be strongly first-order (if $v\_c \equiv \left< h\right>\|\_{T\_c}$ is the value of the Higgs VEV in the broken phase at the critical temperature Tc, then this condition reads $v\_c/T\_c\gtrsim 1$ ). Neither of these conditions is fulfilled in the SM, as the CP violation encoded in the CKM matrix is too small and, anyway, the phase transition is really a crossover, given the lower bound on the Higgs mass from LEP. The strength of the EWPhT in the SM plus a singlet has been thoroughly studied. Many analyses concentrated on loop effects involving the singlet, which enhance the cubic term ETh3 in the Higgs potential at finite temperature, while reducing the quartic λhh4 (at a given Higgs mass) that enters the above condition $1\lesssim v\_c/T\_c\approx E/\lambda\_h$. LEP bounds on the Higgs mass, however, suggest that one singlet scalar is not enough, if it contributes only via loop effects. Furthermore, it was recently pointed out that magnetic fields generated during the EWPhT might increase the sphaleron rate within the broken phase, calling for even stronger phase transitions in order to have successful baryogenesis. The strongest phase transitions are achieved when the singlet contributes through tree-level effects, i.e. when the tree-level potential for H and s is such that a barrier separates the EW broken and unbroken phases (not necessarily with vanishing VEV $\left< s \right> $ along the singlet direction). Indeed, in the case of a barrier generated only at loop-level, the jump in the Higgs VEV is proportional to the critical temperature Tc (times a loop factor), and is hence constrained to be small at small temperature. In the case of a tree-level barrier, on the other hand, the Higgs VEV at the critical temperature depends on a combination of dimensionful parameters in the potential and its effect can be present even at small Tc (and is enhanced by a small Tc appearing in the denominator of vc/Tc). In what follows we will concentrate on this possibility, assuming that the transition is strongly first-order and relying on the analysis of, which studies strong phase transitions induced by tree-level effects in the SM plus a singlet. One important implication of scenarios with a tree-level barrier is that a strong transition is necessarily accompanied by a variation of the singlet VEV during the EWPhT. This can be understood by noticing that, were the singlet VEV constant, the potential would have the same shape as the SM potential at tree-level and would have, therefore, no tree-level barrier. When the EWPhT is strongly first-order, bubbles of the broken phase nucleate within a universe in the unbroken phase and expand. CP-violating interactions can then source EWBG within the wall separating the two phases. In the composite version of the SM plus a singlet outlined in the previous section, with non-vanishing, pseudoscalar couplings between singlet and fermions [b ≠ 0 in eq. ([topcoupling])], the source is provided by a variation in the VEV of s. Indeed, from eq. ([topcoupling]), we can write the top quark mass, which receives contributions from both h and s, as [topmass] mt= v \|mt\| eit, where yt is the top Yukawa and we defined the VEVs v < h >, w< s > , with v = 246 GeV. At vanishing temperature, the phase Θt can be absorbed in a redefinition of the top quark field and is thus unphysical; the only effect of a non-zero w is a shift between the top-mass and the Yukawa coupling compared to the relation that holds in the SM. However, at finite temperature and, in particular, during a phase transition, w may change: then the space-time-dependent complex phase in the quark mass cannot be rotated away by a simple field redefinition, since it would reappear in the kinetic term. For this CP violation to source EWBG, Θt, and therefore w, must change during the EWPhT. (As noted above, while this change is not guaranteed when the barrier is induced by loop effects, a change in w is always present when the barrier appears at tree-level). Then, assuming that we know how w and v change along the direction z perpendicular to the bubble wall, we may write mt(z) = ∣mt(z)∣ eiΘt(z), similarly to what has been done previously for the two-Higgs doublet model (2HDM). This space-dependence results in different dispersion relations for particles and anti-particles (in particular top quarks) in the bubble wall, which, in the WKB formalism, imply force terms in the transport equations. This, in turn, causes an excess of left-handed anti-quarks in front of the wall, in the symmetric phase, which biases sphaleron transitions and generates a net baryon asymmetry. We estimate the resulting asymmetry as follows. First, we approximate both scalar field solutions to have a kink profile of common extent along the z direction, so that [kinkansatz] w(z) wc +, and v(z), where Lw is the width of the wall, Δwc is the total change in ⟨s⟩ at the critical temperature, Tc, and vc is the Higgs VEV at Tc. The complex phase of the top mass then changes as t(z), with t. With this ansatz, we solve the transport equations numerically, using the method of, which we summarize for completeness in Appendix [Sec:appendixTransport]. The results are displayed in Figure [DeltaS], where we show, for a given strength of the phase transition and width of the wall (both are approximated analytically, for the $\Z2$ symmetric case, in Section [sec:PT]), the source of ΔΘt that is needed to reproduce the observed baryon asymmetry, (nB − nB̄)/nγ ≈ 6 × 10− 10. The final baryon asymmetry increases with ΔΘt or vc/Tc but decreases with LwTc or Lwvc. [DeltaS] [DeltaSZ2] In Figure [DeltaSZ2] we show the change in Θt taking place during a strong phase transition in the version of this model with a $\Z2$ symmetry, as described in Sections [sec:SCPV] and [sec:PT]. For f/b = 500 GeV, mh = 120 GeV and ms = 80, 130 GeV (upper and lower plots respectively), the shaded region shows a projection of the other parameters of the model onto the vc/Tc vs. ΔΘt plane. The lower and upper bounds on the shaded regions are due to the requirement of having the right EW minimum, eq. ([eq:ccbound]), while the unshaded region in the middle correspond to an instability of the vacuum at the critical temperature, eq. ([kappak]). The black lines are examples with fixed λm, and along the red lines a fixed amount of baryon asymmetry is reproduced (the solid line corresponding to the observed amount). To summarize, for a strong (tree-level) phase transition, as may occur in the SM plus a singlet, a non-vanishing pseudoscalar coupling i bγ5 between s and the top quark generates a baryon asymmetry. For a strong enough EWPhT with $b\Delta w\_c / f\gtrsim 0.1$, the observed value of (nB − nB̄)/nγ can be reproduced. In the composite version of the SM plus a singlet, these requirements are fulfilled for natural values of the parameters. We close this Section with a comparison of our EWBG scenario with previous studies of EWBG in non-supersymmetric models, such as the two-Higgs doublet model or the SM with a low cut-off. In the former, CP violation arises already at the level of renormalizable operators in the Higgs potential, through a complex phase between the two Higgs VEVs. Very strong phase transitions (induced by tree-level barriers) are not possible in that context since, contrary to the case with a singlet, the second Higgs doublet cannot acquire a VEV prior to the EWPhT by definition. (To circumvent this problem, ref. studies a 2HDM with an additional singlet: the two Higgs doublets violate CP; the singlet strengthens the EWPhT.) Although the non-supersymmetric 2HDM does not address the hierarchy problem, it is worth noting that it can also arise as the low-energy limit of composite Higgs models. The behaviour at finite temperature of other scenarios that address the hierarchy problem but lead only to a light single Higgs, such as the Minimal Composite Higgs or Little Higgs models, have been also analyzed. Refs. studied the temperature behaviour of a Higgs that arises as the PNGB of a broken global symmetry,[3](#fn3) parametrizing the deviations from the SM through effective operators. A strong EWPhT can result in this setting from the dimension-six operator h6, which stabilizes a Higgs potential with negative quartic coupling, as discussed in. This creates a large tree-level barrier but the reliability of the effective-theory description is not then obvious. Different dimension-six operators are responsible for sourcing CP violation, in a manner similar to our eq. ([topmass]), and for generating a complex mass for the top quark: mt ∼ yt(vh + ivh3/Λ2). Compared to the model proposed here, these operators (which would arise also in our model, in the limit of a heavy singlet) are dimension-six and hence generally smaller than the ones involving the singlet. Electric Dipole Moments and other Constraints[sec:EDM] ====================================================== The presence of a scalar that mixes with the Higgs and has pseudoscalar couplings to fermions induces an electric dipole moment (EDM) for the electron and for the neutron. The electron EDM receives the largest contribution from the two-loop Feynman diagram of Figure [BarrZeeE], [BarrZeeE] where the electron flips its chirality by coupling to the Higgs, which then mixes with the singlet. The singlet, in turn, couples to two photons through a top quark loop. We assume here that the singlet couples only to the top quark, which is the only coupling required for EWBG. If it couples also to light quarks, then other diagrams may give larger effects.[4](#fn4) Note that if these couplings were hierarchical and if light flavors couple to the singlet with a strength of order their Higgs Yukawa, as in the scenario of partial compositeness, then the diagram of Figure [BarrZeeE] dominates. This diagram, which receives opposite sign contributions from the two scalar mass eigenstates, gives [EEDM] &=& Z &&, where me, t are the electron and top quark masses, Z is the mixing between h and s, given by $$\label{Z} Z=\cos\theta\sin\theta=\frac{m\_{sh}^2}{\sqrt{(m\_h^2-m\_s^2)^2+4m\_{sh}^4}}\,$$ and $$G(z)\equiv z \int\_0^1dx \frac{1}{x(1-x)-z}\log\left[\frac{x(1-x)}{z}\right]\.$$ The angle θ measures the mixing between the (h, s) gauge eigenstates and the mass eigenstates (η1, η2), with masses m1, 2: $$\begin{aligned} \eta\_1&=s \sin\theta+h \cos\theta\,\\ \eta\_2&=s \cos\theta-h \sin\theta\.\end{aligned}$$ The mostly-doublet eigenstate is η1, so that θ ∈ [ − π/4, π/4]. Finally, ms2, mh2 and msh2 are the entries of the (squared) mass matrix for s and h at the electroweak minimum (their expression in terms of the parameters of the potential of eq. ([VtreeOdd]) can be found in ). The present bound on the electron EDM (at 90% C.L.) is $$\frac{d\_e}{e}<1.05 \times 10^{-27}\textrm{cm}\,$$ leading to the constraint $$\label{constraint} Z \, \frac{b}{y\_t}\, \frac{v}{f}\left\|G\left(m\_t^2/m\_1^2\right)-G\left(m\_t^2/m\_2^2\right)\right\|\lesssim 0.32\,$$ which is not very restrictive, since ∣Z∣ < 1/2 and v ≪ f, while the absolute value of the difference is smaller than ∼ 1 for $m\_s\gtrsim 40 {\,\mathrm{GeV}}$. A similar diagram, with an up or a down quark qu, d replacing the electron, and with a gluon replacing the photon, induces a chromoelectric dipole moment (CEDM) du, dC for the up and down quarks, defined via $$\mathcal{L}\supset -\frac{i}{2}d^{C}\_{u,d}\,g\_s\,\bar{q}\_{u,d}\,\sigma\_{\mu\nu}\tilde{G}^{\mu\nu}q\_{u,d}\,.$$ This diagram gives the largest contribution to the neutron EDM in our scenario, $$\label{errorCEDM} \frac{d\_n}{e}\approx(1.1\pm0.5)\,\frac{d^{C}\_d+0.5d^{C}\_u}{g\_s(1{\,\mathrm{GeV}})}.$$ The computation of the contribution is similar to that which leads to eq. ([EEDM]). In the end, one finds, && 14/23.Plugging in the numbers and comparing with the present bound on the neutron EDM (which, at 90% C.L., is dn/e < 2.9 × 10− 26cm), one obtains a stronger constraint: $$\label{constraint\_b} Z \, \frac{b}{y\_t}\, \frac{v}{f}\left\|G\left(m\_t^2/m\_1^2\right)-G\left(m\_t^2/m\_2^2\right)\right\| \lesssim (0.06-0.19),$$ where the uncertainty stems from uncertainties in the measured values of the up and down quark masses and from eq. ([errorCEDM]). Strong constraints on the presence of a singlet scalar that mixes with the Higgs also come from LEP, in the form of direct searches and electroweak precision observables (EWPO). The former apply only when one of the mass eigenstates is lighter than 114 GeV: in this case, direct searches constrain its mixing with the Higgs h, parametrized by the angle θ in eq. ([Z]). In addition, EWPO constrain the contribution of heavy states to the ρ parameter, and also lead to a bound on a combination of the mixing angle θ and the mass eigenvalues. Neglecting the contributions of heavy composite states, we can approximate the latter bound, at 95% C.L., as cos2θ lnm1 + sin2θ lnm2 < ln143 GeV. We illustrate the various bounds in Figure [fig:bounds], for a representative scale of new physics given by f/b = 500 GeV, as suggested by the composite Higgs model of, for Higgs masses (here we mean the eigenstate that is mostly EW doublet) m1 = 120, 140 GeV (upper and lower plots). This choice of Higgs masses avoids current LHC constraints, which might nevertheless become relevant for other mass values or in the future, when more luminosity is collected. EDM bounds disfavor large mixing angles and singlet masses much larger/smaller than mh. In both cases, a light singlet with large mixing with the Higgs is disfavored by direct searches, as illustrated by the uniform turquoise region. [fig:bounds] Spontaneous CP violation[sec:SCPV] ====================================== In previous sections we have argued that in composite Higgs models with a singlet coupled as a pseudoscalar to the top quark [eq. ([topcoupling])], the baryon asymmetry can be generated during the EWPhT. A necessary condition for EWBG is that the singlet VEV changes during the EWPhT, cf. Figure [DeltaS]. However, this condition is not sufficient, as we will show in this section. We consider a scenario in which the scalar potential is Z2-symmetric [$V^{\rm odd}=0$ in eq. ([Vtree])] and the singlet coupling to fermions is purely pseudoscalar-like [a = 0 in eq. ([topcoupling])], so that the singlet is a CP-odd field. This corresponds to a scenario without explicit CP violation, but where CP can be spontaneously broken by the VEV of s. However, as we will discuss below, the existence of the symmetry s → − s leads to a cancellation between positive and negative contributions to the baryon asymmetry. In this section we study how a small explicit breaking of this symmetry can lead to a sufficient net baryon asymmetry. We note in passing that the case Vodd = 0 has been discussed in several contexts before: in a Singlet-Majoron model, in the context of singlet dark matter, and in its own right . In the $\Z2$ symmetric case, a singlet VEV after the EWPhT produces dangerous cosmic domain walls and is therefore excluded. Since, for baryogenesis, the singlet VEV must change during the EWPhT, bΔwc ≠ 0, we are led to the case in which the singlet has a VEV prior to the EWPhT and none afterwards; this is fortunate since it is precisely the setting that produces the strongest phase transitions. In this case, there are in fact two transitions: first, the $\Z2$-breaking transition, at Ts, from the point (h, s) = (0, 0) to some point (h, s) = (0, ± wc) and later on, at Tc < Ts, the EWPhT from (h, s) = (0, wc) to (h, s) = (vc, 0). Note that this sequence is quite plausible, as the Higgs, unlike s, couples sizably to the gauge bosons and strongly to the top quark and therefore its mass receives larger finite-temperature corrections, which can delay its phase transition. Since the discrete $\Z2$-symmetry is broken spontaneously, albeit temporarily, at Ts, domain walls (DWs) will be produced: if they live long enough, they might come to dominate the energy density of the universe, leading to a period of late inflation and features in the cosmic microwave background which are in conflict with observations. The temperature at which electroweak DWs start to dominate, however, is TDW ∼ 10− 7 GeV, which, in an expanding universe, happens much later than the electroweak phase transition TEW ∼ 100 GeV, when the DWs disappear because the symmetry along the s direction is restored. Issues with the vanishing of the total baryon asymmetry arise because, during the first transition at Ts, different patches of the universe (distributed equally, on average) would end up in either the + wc or the − wc vacuum (denoted + and − in the following), forming a network of domains. While the + patches would produce a net baryon asymmetry, the − patches would produce a net antibaryon asymmetry (since the change in the singlet VEV during the EWPhT in different patches is equal and opposite in sign). In the following we discuss the possibility that a small explicit breaking of the Z2-symmetry in the scalar potential (Vodd ≠ 0) lifts the degeneracy between the ± vacua and biases one vacuum with respect to the other so that, at the time of the EWPhT, the whole universe lies in the − vacuum . We parametrize this explicit breaking by a small difference V( + wc) − V( − wc) = ΔV, which can be generated, for instance, through loop effects involving a small a ≠ 0 in eq. ([topcoupling]). The network dynamics depends mainly on three forces: the pressure difference between the + and − phases ΔV, the surface tension and the Hubble expansion. Friction is not very important, as discussed below. The surface tension σ and the pressure difference balance each other when the typical size of the domains ξ is of the order of the size of the critical bubble for the transition + → − , which scales as [eq:xicrit] cr ~/V . As long as $\xi<\xi\_{\rm cr}$ the dynamics of the domains is dominated by the surface tension: small-curvature structures will be smoothed-out or collapse, increasing the typical length scale ξ of the network and entering the so-called Kibble-regime, with ξ growing linearly with conformal time. When ξ reaches $\xi\_{\rm cr}$, the pressure difference ΔV becomes relevant and the energetically preferred − regions expand exponentially fast into the + regions. This behaviour has been confirmed by numerical simulations in different settings. In our model the temperature of the electroweak phase transition is significantly lower than the temperature of the first Z2-breaking phase transition (see next section) so that DWs will evolve for a period of time of order of the Hubble scale and all + regions will have vanished at the time of the electroweak phase transition as long as $\xi\_{\rm cr}\ll 1/H$. This leads to a very weak lower bound on ΔV (for more details see ) [boundDV] ~10-16. Notice that this bound is in principle independent of the nature of the Z2-breaking phase transition. Indeed, in a first-order transition the typical size of domains is given by the duration of the phase transition, which is of order ξ ∼ 1/(100H) . On the other hand, if the transition is second-order, the typical size of the domains is of order 1/TG , where $T\_G\simlt T\_c$ is the Ginzburg temperature below which the broken phase settles in the either the + or − vacuum and thermal transitions to the other vacuum are Boltzmann suppressed [5](#fn5). So, even if in most cases one expects the pressure difference to dominate over the surface tension right after the end of the Z2-breaking phase transition, if condition ([boundDV]) is satisfied the DW length scale will reach ξcr, and DWs will quickly disapear, before the EWPhT [6](#fn6). Notice that it is also possible that DWs be still around at the time of the EWPhT but that the volume occupied by the − phase, V−, is sufficiently larger than the volume V+ of the + phase, so that the suppression factor (V− − V+)/(V− + V+) on the net baryon asymmetry produced is not particularly small. Interestingly, this can happen in a time scale significantly shorter than the one needed for domain wall collapse, leading to a bound on ΔV even weaker than ([boundDV]). Finally, let us examine why friction is not important in our setting , so that one expects DWs to move with relativistic velocities. Friction is due to the plasma having different properties in the two phases. For example, during the EWPhT, W-bosons and top quarks get a mass from their couplings to the Higgs VEV. This change in mass implies a change in momentum when these particles cross the Higgs bubble wall and this produces a friction force on the wall. Now however, the properties of the particles in the + and − phases differ only by very small effects from the operators that induce the potential difference ΔV. This suppresses friction and both sides of the equation [eq:vterm] V vw Ffriction, that determines the terminal velocity of the wall, vw, vanish in the limit ΔV → 0. Quantum mechanical reflection effects of particles with an s-dependent mass that will feel the presence of the wall (as discussed in a similar context in ) are expected to be negligible too. To go beyond this rough estimate, consider the approach of ref. , which works in the limit of a weakly coupled plasma and small wall velocities. The friction due to the mass change of the particles in the plasma between the phases is Ffriction ~ (m2)2. Since the pressure difference as well as the mass difference are linear in ΔV, this small source of friction cannot lead to small terminal wall velocities according to ([eq:vterm]). Nevertheless, in some cases with small friction one might end up with terminal velocities that are subsonic, as discussed in the analysis of , which is valid for strongly interacting plasmas (when the free streaming length of the particles in the plasma is much shorter than the wall thickness). The hydrodynamic obstruction to a larger terminal velocity is due to the release of latent heat in front of the moving wall that increases the temperature and reduces, as a result, the pressure difference experienced by the wall. In principle, this effect can lead to small wall velocities when the pressure difference along the wall is much smaller than the latent heat. However, in the present case, both arise from the difference in potential ΔV so that the hydrodynamic obstruction (if present) has to occur at wall velocities that are of order of the speed of light. To summarize, even though sources of explicit CP violation are needed in our setup , already a very weak explicit breaking of CP in the scalar sector gives rise to substantial CP violation for electroweak baryogenesis. Essentially, this is due to the two-stage nature of the phase transition. In the composite Higgs model on which we focus (Ref. , for details see Appendix [Sec:CH]), a $\Z2$-preserving potential with a small explicit breaking could arise, for example, if the couplings of the top-quarks preserve the $\Z2$ [i.e. a = 0 in eq. ([topcoupling])], while lighter quarks also couple to the singlet, but with $\Z2$-violating couplings. In this case the leading terms in the potential, coming from loop effects involving the top quark Yukawa, will generate Veven, while a small Vodd would also be generated, proportional to the (smaller) light quark Yukawas. Alternatively, it might be possible that the same mechanism that is responsible for the CKM phase could generate a (presumably small) CP-violating contribution to the scalar potential via loop effects. Electroweak Phase Transition[sec:PT] ==================================== In this section we discuss the characteristics of the phase transition that are relevant for electroweak baryogenesis, namely the sphaleron washout parameter vc/Tc, the wall thickness Lw, and the change in singlet VEV during the EWPhT, Δwc. Motivated by its simplicity and by the discussion of the last section, we focus on a scalar potential with $\Z2$ symmetry, eq. ([VtreeEven]). We assume the existence of a small odd part in the potential that breaks CP explicitly, but this contribution will hardly influence the transition dynamics. Following , we parametrize the potential in the mean-field approximation as [eq:Vcrit] V(h,s,T) &=& 4 2 + 4 s2 h2 && + (T2 - Tc2) (ch h2 + cs s2 ) , where $$\label{kappak} \kappa\equiv \lambda\_m-2\lambda\_h\frac{v\_c^2}{w\_c^2}\.$$ By construction, this potential leads to a tree-level barrier and hence a strong EWPhT. Indeed, at the critical temperature T = Tc, the potential has degenerate minima at (h, s) = (vc, 0) and (h, s) = (0, ± wc), separated by a barrier parametrized by κ > 0. The two coefficients ch and cs are, in the mean-field approximation ch &=& 148 , cs &=& 112 . The condition for ensuring that the electroweak minimum is the global one at temperatures below Tc reads [eq:ccbound] > . This relation implies bounds on wc. Moreover, our low-energy effective theory approach also requires wc ≪ f, so that this parameter is typically restricted to a fixed range around vc. Given the form of the potential ([eq:Vcrit]), the critical temperature Tc can be expressed in terms of v0, the Higgs VEV in the EW breaking minimum at T = 0, as [eq:Tcrit] Tc2 = (v02 - vc2) . This can be compared with the temperature at which the Z2-symmetry breaks: Ts2 &=& Tc2 + &=& Tc2 > Tc2 , which we have taken to be higher than Tc2, as argued in the previous section. The Higgs mass is [eq:mh] mh2 = 2 h v02 , while for the singlet mass one finds [eq:ms] ms2 = 12 v02 + h (v02 - vc2) . Both contributions to the singlet mass are positive, due to eq. ([eq:ccbound]).[7](#fn7) Next, we determine the wall thickness in the thin-wall approximation. The tunneling path will extremize the Euclidean action S1 = -+ d , with boundary conditions &&h(-)=0 , h()=vc , h’()=0 , &&s(-)=wc , s()=0 , s’()=0 , and will proceed over (or close to) the saddle point with the potential value V= 14 (4 h vc2 + wc2) . [Vsad] We can estimate the parametric dependence of Lw by approximating S1(vc2+wc2)+Lw V , which leads to Lw2= . To estimate the numerical constant in this formula assume for simplicity that the tunneling of the fields proceeds along the path h() = vc , s() = wc , that passes a barrier of height V = κvc2wc2/16. Using the ansatz () = [1 + (/Lw)] , we obtain integral expressions for α and β that can be calculated to give α/β ≃ 2.7. In this way we arrive at the estimate [eq:Lw] L2w 2.7 (1+) . This result would become unreliable for κ ≪ 1, as the potential then develops a flat direction and loop effects (giving rise to cubic terms from the Daisy re-summed contributions) become important and strengthen the phase transition. We do not discuss this particular case here. Another pathological limit, λh → 0, is forbidden by the lower limit on the Higgs mass. To summarize, a scalar potential, its critical temperature and the wall width, are uniquely determined (in the $\Z2$ symmetric case) by the values of the Higgs and singlet masses at vanishing temperature, mh and ms, by the strength of the EWPhT, vc/Tc, and by the change in the singlet VEV, wc. So, it is always possible to find parameter choices that give the right baryon asymmetry, as long as bounds for the consistency of the potential (such as perturbativity of the couplings, stability of the high-T minimum and of the T = 0 EW breaking one) are respected: this is illustrated by the broad shaded regions of Figure [DeltaSZ2]. We give some explicit examples in Table [tab:examples]. In the first case, we consider a model with small Higgs-singlet mixing and a relatively light Higgs (mh ∼ 120 GeV). Hence, the experimental bounds of Figure [fig:bounds] are fulfilled independently of the singlet mass. In the second case, we consider a model where the Higgs is relatively heavy (mh ∼ 140 GeV). Besides this, we assume a sizable Higgs-singlet mixing, together with singlet and Higgs masses of similar size. This can be achieved by adding only a relatively small operator s∣H∣2 to the scalar potential (that will have a negligible impact on the characteristics of the phase transition). In both cases viable EWBG is possible with a strong coupling scale f as high as several TeV. Larger values of f can be compensated by increasing vc/Tc, which enhances the strength of the phase transition and the CP-violating sources in the Higgs wall. However, notice that, although in our simplified treatment, vc/Tc has been taken as a free parameter, for very large values of vc/Tc ($\gtrsim 4$) one expects that the phase transition will never be completed. \| \| λh \| κ \| wc/vc \| bwc/f \| \| --- \| --- \| --- \| --- \| --- \| \| S1 \| 0.12 \| 0.12 \| 1.0 \| 0.1 \| \| S2 \| 0.16 \| 0.48 \| 1.0 \| 0.15 \| 0.1 cm \| \| mh \| ms \| vc \| f/b \| Lwvc \| vc/Tc \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| S1 \| 120 GeV \| 81 GeV \| 188 GeV \| 1.88 TeV \| 7.1 \| 2.0 \| \| S2 \| 140 GeV \| 139.2 GeV \| 177.8 GeV \| 1.185 TeV \| 3.5 \| 1.5 \| Conclusions[Sec:Concl] ====================== Composite Higgs models have become very plausible options for natural electroweak symmetry breaking and it is tempting to explore whether such scenarios can also solve other problems that the Standard Model fails to address. One important and long-standing problem is the generation of the cosmological baryon asymmetry and here we have shown in some detail how EW baryogenesis can be successfully achieved in such a model. The required ingredients turn out to be simple, not least because naturalness itself implies modifications to the Higgs and top sectors, which are precisely the ones believed to be responsible for EWBG. Firstly, the electroweak phase transition can easily be strongly first-order, for realistic values of the Higgs mass, if the model contains in addition a scalar singlet. Moreover, if that scalar couples to the top quark it can lead to a non-trivial CP-violating phase along the bubbles of the electroweak transition, creating the seed for sphalerons to generate a non-zero baryon asymmetry. We have shown that a composite model, based on the coset SO(6)/SO(5) proposed previously , contains all these ingredients. Its low-energy effective theory features an additional pseudo-Goldstone field that is a singlet and has dimension-five, pseudoscalar couplings to top quarks that can break CP. Motivated by this prospect, we have calculated quantitatively the expected amount of asymmetry, showing that baryogenesis is successfully achieved in a broad region of the parameter space that passes other experimental constraints. Interestingly, the new complex phases and the mixing between the Higgs and the singlet predict contributions to the EDMs of neutron and electron, not far beyond the reach of current and future experiments. Acknowledgements ================ Three of us (JRE, TK and FR) are grateful to the organizers of the Workshop ’Electroweak Baryogenesis in the Era of the LHC’, at the Weizmann Institute, for stimulating discussions which shaped this article. JRE thanks T. Hiramatsu, M. Kawasaki and K. ’I. Saikawa for sharing unpublished results of numerical simulations, while FR thanks M. Pietroni and A. Riotto for interesting discussions. This work was supported by the Fondazione Cariparo Excellence Grant LHCosmo; the ERC starting grant Cosmo@LHC (204072); the Spanish Ministry MICINN under contracts FPA2010-17747 and FPA2008-01430; the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042); and the Generalitat de Catalunya grant 2009SGR894. The SO(6)/SO(5) Composite Higgs Model ============================================= In composite Higgs models, similarly to QCD, the hierarchy between the Plank and the TeV scale is due to the slow logarithmic running of an asymptotically free interaction that becomes strong and confines close to the electroweak scale. New (techni-) fermions are charged under this strong interaction but also under a global symmetry; here we are interested in the case SO(6) ≃ SU(4). As the strong interaction confines, the global symmetry is broken down to a subgroup, which we take to be SO(5). From Goldstone’s theorem, we expect five massless degrees of freedom [equivalent to the number of broken global symmetries in SO(6)/SO(5)], which are analogues of the pions in QCD. The embedding of the SM group SU(2)L in SO(5) is such that these five Nambu-Goldstone bosons (NGBs), may be identified as the four degrees of freedom belonging to the Higgs doublet plus one degree of freedom, corresponding to a real singlet s. From symmetry arguments alone, we can extract the leading kinetic part of the σ-model Lagrangian (in the unitary gauge) as kin&=& (h)2+ (s)2 && +, [kin]where f is the analogue of the pion decay constant fπ, h is the physical Higgs and the fields are defined such that ⟨h⟩2 + ⟨s⟩2 ≤ f2 . The Higgs identified above transforms as a ${\bf (2,2)}$ of SU(2)L × SU(2)R ∼ SO(4) ⊂ SO(5) and its contributions to the ρ-parameter are protected by the SO(4) symmetry, which is broken to the custodial SO(3) symmetry in the EW vacuum. The SO(6) symmetry is broken explicitly, both by the SM fermions, which do not come in a complete representation of SO(6), and by the gauging of SU(2)L ⊂ SO(6). Thus, loops involving SM fermions or gauge bosons communicate the explicit breaking to the (pseudo) NGBs and generate a potential for the Higgs and the singlet. In unitary gauge, the leading couplings between the W±, Z gauge bosons and the physical Higgs h are $$\mathcal{L}\_{WZh}=\frac{g^2}{4}h^2\left[W^{\mu+} W^-\_\mu+\frac{1}{2\cos^2\theta\_W}Z^\mu Z\_\mu\right]\,.$$ We will assume the partial compositeness scenario (which seems most favorable from the point of view of flavor physics), in which fermions couple linearly to operators of the strong sector and embed them into representations of the global SO(6). As shown in, among the possible low-dimensional representations, only the 6 of SO(6) can result in a realistic low-energy spectrum. The resulting couplings of the fermions to the NGBs in the σ-model Lagrangian are $$\begin{aligned} \label{fermionCH} \mathcal{L}\_{\rm f} =&\sum\_{j,k} \left\{\Pi\_1^{j}(q^2)\,\bar q^j\_L {\!\not\! p}q^j\_L\, \frac{{{ h}}^2}{f^2}\, \right.\nonumber\\ &+\Pi\_2^{j}(q^2)\, \bar q^j\_R {\!\not\! p}q^j\_R \frac{1}{f^2}\left\| r(h,s)+ i \epsilon^j s\right\|^2 \, \nonumber \\ &+\left. Y^{jk}(q^2) \,\bar q^j\_L q^k\_R\, \frac{{{ h}}}{f} \left[r(h,s) + i \epsilon^k s \right]\,\right\}+ h.c. \,,\end{aligned}$$ with r(h,s) , and where the sum is over fermion flavors j, k and εj are complex numbers measuring the superposition between different embeddings of the SU(2)L singlet quarks into representations of SO(6). Indeed, while the first four components of a 6 of SO(6) transform as a doublet under SU(2)L, both the fifth and the sixth transform as a singlet and can both accommodate the SU(2)L-singlet quarks. In eq. ([fermionCH]), Π(q2) and Y(q2) are q2-dependent form factors and depend on the details of the UV physics. Note that these form factors give rise to the SM Yukawas and, in particular, to interactions where s couples to the fermions as a CP-odd scalar (for a more detailed discussion on CP in composite Higgs models see ). Loops involving SM fermions and gauge bosons break the global symmetry SO(6) explicitly and generate a potential for h and s which can, in principle, be calculated if the form factors where known, or estimated using the arguments of. The top quark and the W-bosons, which have the largest couplings to the strong sector, give the largest contributions, inducing a potential of the form [pot] V( h,S) && h2+ h4 + 1f2 \| r(h,s) + i t s\|2 && -1 cm. Here α, λ, β, γ, and δ are parameters that depend on integrals over form factors, and εt is the ε parameter for the top quark and will be complex in general. This potential can be expanded for small s/f and h/f in order to obtain the low-energy effective potential. Notice that in the limit ε = ± 1, the potential does not depend on the singlet, which is, therefore, a massless NGB. In summary, we see that the composite Higgs model of ref. based on the SO(6)/SO(5) coset structure reduces, at low energy, to the SM plus a singlet. The setup predicts the right couplings to fermions [cf. eq. ([fermionCH]) versus eq. ([topcoupling])] and the right potential [cf. eq. ([pot]) versus eq. ([Vtree])-([VtreeOdd])] to provide all necessary ingredients to produce the observed baryon asymmetry through EWBG. Furthermore, small explicit CP violation (whose importance we discuss in Section [sec:SCPV]) can be naturally understood in this framework if εt is purely real or imaginary. In this case, the dominant contributions to the potential for s and h preserve CP and generate a Z2-symmetric potential of the form of $V^{\rm even}$ in eq. ([VtreeEven]). In this case, the potential would receive a subdominant Z2-violating contribution, $V^{\rm odd}$, from lighter quarks with complex ε. Transport Equations =================== In our analysis of the baryon asymmetry, we adopt a set of diffusion equations as in ref. . The equations describe the evolution of (the CP odd parts of) the chemical potentials and velocity fields, denoted μi and vi. Initially, the species under consideration are the left-handed quarks, the right-handed top and the Higgs. The right-handed bottom is irrelevant since it is only produced by a chiral flip of the left-handed bottom suppressed by the bottom mass. The dynamics of the Higgs typically has a relatively small impact on the final asymmetry and can be neglected . The diffusion equations for the left-handed quarks then read $$\begin{aligned} \label{eq1} && \hskip -1 cm (3\kappa\_t+3\kappa\_b)v\_{\rm w}\mu\_{q\_3}^\prime -(3K\_{1,t}+3K\_{1,b})v\_{q\_3}^\prime {\nonumber}\\ && \hskip -1 cm -6\Gamma\_y\left(\mu\_{q\_3}+\mu\_{t}\right) -6\Gamma\_m\left(\mu\_{q\_3}+\mu\_{t}\right) {\nonumber}\\ && \hskip -1 cm -6\Gamma\_{ss}\left[\left(2+9\kappa\_t+9\kappa\_b\right) \mu\_{q\_3}+\left(1-9\kappa\_t\right)\mu\_{t}\right] {\nonumber}\\ &=& 0 \, \end{aligned}$$ and $$\begin{aligned} && \hskip -1 cm -(K\_{1,t}+K\_{1,b})\mu\_{q\_3}^\prime +(K\_{2,t}+K\_{2,b})v\_{\rm w}v\_{q\_3}^\prime {\nonumber}\\ && \hskip -1 cm -\left(\frac{K\_{1,t}^2}{\kappa\_t D\_{Q}}+ \frac{K\_{1,b}^2}{\kappa\_bD\_{Q}}\right)v\_{q\_3}{\nonumber}\\ &=& K\_{4,t}v\_{\rm w}m^2\_t\Theta\_t'' +K\_{5,t}v\_{\rm w}(m^2\_t)'\Theta\_t' \, \end{aligned}$$ while the diffusion equations for the right-handed top are $$\begin{aligned} && \hskip -1 cm 3\kappa\_tv\_{\rm w}\mu\_{t}'-3K\_{1,t}v\_{t}' {\nonumber}\\ && \hskip -1 cm -6\Gamma\_y\left(\mu\_{q\_3}+\mu\_{t}\right) -6\Gamma\_m\left(\mu\_{q\_3}+\mu\_{t}\right){\nonumber}\\ && \hskip -1 cm 3\Gamma\_{ss}\left[\left(2+9\kappa\_t+9\kappa\_b\right) \mu\_{q\_3}+\left(1-9\kappa\_t\right)\mu\_{t}\right]{\nonumber}\\ &=& 0 \,\end{aligned}$$ and $$\begin{aligned} && \hskip -1 cm -K\_{1,t}\mu\_{t}'+K\_{2,t}v\_{\rm w}v\_{t}'-\frac{K\_{1,t}^2}{ \kappa\_t D\_Q}v\_{t} {\nonumber}\\ \label{eq4} &=& K\_{4,t}v\_{\rm w}m^2\_t\Theta\_t'' +K\_{5,t}v\_{\rm w}(m^2\_t)'\Theta\_t' \,\end{aligned}$$ where primes denote derivatives w.r.t. the coordinate z perpendicular to the wall. Compared to the implementation in we neglected some sub-leading source terms that contain derivatives acting on the chemical potentials and velocity fields (more precisely their CP-even parts). The final baryon asymmetry results from the sphalerons acting on the CP-odd component of the left-handed quark density B &=& = 0dz L(z) e-z , [eq:wsint] L &=& (1 + 2 t + 2 b) q3 - 2 t t , &=& , g\* = 106.75 . The functions Km, j and κi denote certain moments in momentum space defined by < X > && d3p f+’(m=0) , f(mi) &=& 1e 1 , namely i && < f’(mi)>, K1,i && < f’(mi)>, K2,i && < pz2 f’(mi)>, K3,i && < f’(mi)>, K4,i && < f’(mi)>, K5,i && < f’(mi)>. The momenta are normalized such that for massless bosons (fermions) one finds κi = 2(1). For the interaction rates and the quark diffusion constant we use the values ws = 1.0 10-6 T4 , && ss = 4.9 10-4 T4 , y = 4.2 10-3 T , && m = , DQ &=& . As long as the wall velocity is clearly subsonic ($v\_w \ll 1/\sqrt{3}$), the CP-odd particle densities are linear in the wall velocities. If at the same time the sphaleron process is not saturated (i.e. as long as the exponent in the integrand of ([eq:wsint]) is small) the final asymmetry will depend only weakly on the wall velocity. We assume that we are in this window (which might well be due to hydrodynamic obstructions ) and adopt the value vw = 0.01. In our setup, the functions Km, i, κi and Γy inherit a spatial dependence from the top mass. This precludes a straightforward numerical solution of the system of diffusion equations. We avoid this problem by matching the eigenvectors of the diffusion system with the correct sign [i.e. the ones that die off at infinity] far away from the wall, where all coefficients are constant, to numerical solutions in the wall. 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At even higher temperatures, the same mechanism that cuts off quadratic divergences in the Higgs potential also affects its finite temperature corrections and could lead to non-restoration of the EW symmetry.[↩](#fnref3) 4. A similar diagram arises in the 2HDM, except that there the singlet is replaced by the CP-odd neutral scalar A0. Diagrams with a Z boson instead of a photon give suppressed contributions to EDMs.[↩](#fnref4) 5. It has been suggested that the DW scale is instead governed by a non-equilibrium mechanism related to critical slowing down leading to ξZ ∼ Tc− 1(Mpl/Tc)1/4 (which, for Tc ≪ MPl, implies 1/TG ≪ ξZ ≪ 1/H). However, this analysis seems to require $(1-T\_G/T\_c)/(H\xi\_0)^2<{\cal{O}}(1)$ (where ξ0 is the field correlation length at T = 0), a condition that is grossly violated for Tc ≪ MPl.[↩](#fnref5) 6. If TG is high enough, it is possible to avoid the formation of DWs altogether. Indeed if the breaking is so big that, by the time the + minimum develops, at T+, thermal fluctuations to it are suppressed (T+ < TG). Assuming that the dominant $\Z2$-breaking comes from a tadpole, δV = μ13s, the condition for T+ < TG to happen is: 13> , which is much larger than ([boundDV]).[↩](#fnref6) 7. Very light singlets are disfavored: this is a consequence of our parametrization which, by construction, leads to tree level barriers. Light singlets can, of course, exist, but are generally not linked with very strong phase transitions (unless a cancellation in the parameters of the potential induces a flat direction ).[↩](#fnref7) Electroweak Baryogenesis in Non-minimal Composite Higgs Models ============================================================== We address electroweak baryogenesis in the context of composite Higgs models, pointing out that modifications to the Higgs and top quark sectors can play an important rôle in generating the baryon asymmetry. Our main observation is that composite Higgs models that include a light, gauge singlet scalar in the spectrum [as in the model based on the symmetry breaking pattern SO(6) → SO(5)], provide all necessary ingredients for viable baryogenesis. In particular, the singlet leads to a strongly first-order electroweak phase transition and introduces new sources of CP violation in dimension-five operators involving the top quark. We discuss the amount of baryon asymmetry produced and the experimental constraints on the model. Motivation ========== It is well known that the baryon asymmetry observed in the universe cannot be explained by the Standard Model (SM) of particle physics. Indeed, neither the CP violation encapsulated in the CKM matrix nor the departure from thermal equilibrium during the electroweak phase transition (EWPhT) are large enough, given the LEP lower bound on the mass of the Higgs boson. This calls for physics beyond the Standard Model. In particular, new bosons could enhance the strength of the EWPhT, while new fermions, or new fermionic couplings, could provide additional sources of CP violation. Both ingredients are present in supersymmetric extensions of the SM and (perhaps not surprisingly given the added motivations coming from naturalness, gauge coupling unification, and Dark Matter) most studies of electroweak baryogenesis (EWBG) have concentrated on the minimal supersymmetric standard model (MSSM). Unfortunately, the LEP bound on the Higgs mass has pushed MSSM EWBG into fine-tuned territory. Furthermore, this and other bounds on sparticle masses from LEP and LHC have set strong constraints on some scenarios of low-energy supersymmetry, putting the original naturalness motivation under some pressure. Although non-minimal supersymmetric extensions of the MSSM can overcome some of these problems, the time is perhaps right to examine EWBG in non-supersymmetric models that address the hierarchy problem (and necessarily incorporate new degrees of freedom at the EW scale). Among the possibilities that have been studied, models in which the Higgs boson is composite and the SM fermions are partially composite seem to offer the most plausible alternative to supersymmetry. In such models the Higgs boson arises as a bound state of new, strongly-interacting dynamics and is naturally light, since it originates as a pseudo Nambu-Goldstone Boson (PNGB) of a broken global symmetry, similar to the pions of QCD. The pattern of symmetry breaking can be chosen to reduce the tension with the electroweak precision observables (EWPO) T and $Zb\bar b$, while the presence of a (composite) Higgs boson allows a mechanism for suppressing the contributions to S. Moreover, the assumption that the SM fermions are partially composite results in a suppression of dangerous contributions to flavor physics observables, provided the theory is approximately conformal in the ultraviolet. Such a theory would presumably give rise to a plethora of bosonic and fermionic resonances at or around the strong coupling scale and this exotic dynamics could certainly play a rôle in generating the observed baryon asymmetry (ref. , for example, pointed out that the radion, in the context of the AdS/CFT correspondence, could be relevant). Indeed it would seem that the connection between naturalness arguments and EWBG would be especially tight in such a theory, since the dominant radiative corrections (involving the mostly-composite Higgs and top quark sectors) shape the potential at both zero and non-zero temperatures and would be responsible both for stabilizing the weak scale and generating the required strongly first-order phase transition. Moreover, new, strong couplings between the top and Higgs sectors could provide the required source of CP-violation. Unfortunately, a quantitative demonstration of this link would seem to be out of the question if the strongly-coupled physics lies close to the EW scale, given our limited understanding of strongly-coupled dynamics. However, current data suggest that the strong coupling scale must be pushed up to a few TeV or so. Indeed, the suppression of the S parameter mentioned above is achieved in precisely this way and a similar hierarchy is also needed for sufficient suppression of contributions to flavor physics observables. Whilst this separation is bad news for naturalness (it requires a fine-tuning at the level of 10% or so), it at least offers the hope that we may be able to study EWBG in such models in an effective theory approach, in which the details of the strong dynamics at the strong coupling scale are integrated out. In such a framework, one only needs to specify the light degrees of freedom, together with the operators of low dimension that are present in the low-energy effective Lagrangian. The simplest such Lagrangian would contain only the SM degrees of freedom, together with additional operators beginning at dimension six. Such a Lagrangian would arise, for example, as the low energy limit of the ‘minimal composite Higgs model’, based on the coset SO(5)/SO(4). Clearly, since the SM on its own is deficient from the point of view of EWBG, the dimension-six (or higher) operators must have a large effect on both the EWPhT (via sextic and higher contributions to the Higgs potential ) and on CP-violating physics , but this conflicts with the need for a large strong-coupling scale (which suppresses the higher-dimension operators) and in any case might jeopardize the validity of the effective field theory expansion. However, there does not seem to be any compelling reason (pace Occam) to choose the composite Higgs model based on the coset SO(5)/SO(4) over one based on a larger coset, and featuring the same desirable properties. On the contrary, from the point of view of the EWPhT, we know that the most favorable case (at the renormalizable level) occurs when the Higgs sector is extended to include a gauge singlet scalar and that in this way, one may have a strongly first-order phase transition (for a recent comprehensive analysis, see and references therein). Such a scenario is realized in the composite Higgs model based on the global symmetry breaking pattern SO(6) → SO(5), where the coset is five-dimensional and the low-energy spectrum includes four degrees of freedom corresponding to the Higgs doublet plus one, real, singlet PNGB. What is more, the non-renormalizable operators in the low-energy effective Lagrangian of this model begin at dimension-five and include an operator coupling the singlet and the Higgs to a pair of top quarks that violates CP. In this article, we show that such a model can generate the baryon asymmetry. As we have argued, it suffices to study the low-energy effective theory of the SM plus a singlet, including the aforementioned dimension-five operator. The scenario offers a testable way to explain the origin of the baryon asymmetry and can also be compared with constraints on new, CP-violating physics coming from electric dipole moment (EDM) tests and from LEP. In Section [Sec:SingletModlel] we summarize the features of the composite Higgs model with a singlet that are relevant for baryogenesis (more details are given in Appendix [Sec:CH]) and in Section [sec:EWBG] we study how the baryon asymmetry arises in this scenario. In Section [sec:EDM], we study electric dipole moment and LEP constraints, while in Section [sec:SCPV] we quantify how much explicit CP violation is needed to obtain a sufficient net baryon asymmetry. In Section [sec:PT] we estimate the characteristics of the phase transition (such as the wall thickness and critical temperature) in a special case where the theory is approximately $\Z2$-symmetric: then the structure of the effective Lagrangian is simpler and allows for an analytical study. Finally, in Section [Sec:Concl] we present our conclusions. In Appendix [Sec:appendixTransport], we collect the transport equations used to calculate the baryon asymmetry. The SM plus a Singlet from a Composite Higgs[Sec:SingletModlel] =============================================================== We are interested in composite Higgs models that, in the low energy spectrum, include the SM and a further real, scalar degree of freedom, singlet under the SM gauge group. One example is the model based on the SO(6)/SO(5) coset of ref. , which we summarize in Appendix [Sec:CH]. In this Section, we highlight the features that play a rôle in EWBG: in particular, we concentrate on the effective scalar potential and on the couplings between the Higgs and top-quark sectors, which, from naturalness arguments, are expected to be mostly composite. The most general effective scalar potential at the renormalizable level is V = Veven + Vodd, with [VtreeEven] V&& -h2 \|H\|2+h \|H\|4 && - s2 s2 +s s4 +m s2\|H\|2, V&&m s \|H\|2 +13 s +3 s3 ,[VtreeOdd] where μh, s, m, 1, 3 have dimension of mass and λh, s, m are dimensionless[1](#fn1); H denotes the Higgs SU(2)L doublet with physical component $h/\sqrt{2}$. Veven(odd) denotes the part of the potential that is even (odd) with respect to the Z2 transformation s → − s. While this is an isometry of the coset space, whether or not it is a symmetry of the effective Lagrangian depends on how the SM fermions are coupled to the sigma model . Let us now consider the couplings between the singlet and the fermions. Lorentz invariance alone allows the singlet s to couple to a Dirac fermion F via s \|F(a+ib5)F , where a (b) is a dimensionless coefficient describing its (pseudo)scalar-like couplings. In the SM, however, the SU(2)L × U(1)Y gauge symmetry forbids such a term in the Lagrangian and s can interact with the SM fermions only at the non-renormalizable level, beginning at dimension five with the operator [topcoupling] H \|Q3(a+ib5) t + h.c. , where f is the analogue of the pion decay constant and is related to the mass mρ (of order the confinement scale Λ) and coupling gρ of the strong sector resonances via mρ = gρf, where $g\_{SM}\lesssim g\_\rho \lesssim 4\pi$ and gSM is a typical SM coupling. In eq. ([topcoupling]) we have written only the coupling between the singlet s and the third generation SU(2)L doublet, Q3, and singlet, t. Indeed, naturalness implies that the Higgs and top sectors be mostly composite, so that the strong dynamics is expected to influence mostly the interactions within and between these two sectors. Even in this case, interactions with the lighter fermions will be present in the mass eigenstate basis, but are expected to be of the order of the corresponding (small) Yukawa couplings. Finally, it is useful for what follows to consider how one may implement CP in this context: If Vodd vanishes, a = 0 and b ≠ 0, the singlet behaves as a pseudoscalar and CP is conserved; similarly for b = 0 and a ≠ 0 the singlet is scalar-like and CP is also conserved in the Lagrangian. Other non-trivial choices inevitably violate CP. Electroweak Baryogenesis[sec:EWBG] ================================== Two conditions need to be fulfilled during the EWPhT in order to create enough baryon/antibaryon asymmetry. First of all, CP violation must be present within the wall separating the broken from the unbroken phase. This sources an excess of left-handed versus right-handed fermions[2](#fn2) in front of the wall which is converted into a baryon versus antibaryon excess by non-perturbative electroweak (sphaleron) processes. For this excess to be conserved, these sphaleron processes must be quickly suppressed within the broken phase. This brings us to the second condition: that the EWPhT be strongly first-order (if $v\_c \equiv \left< h\right>\|\_{T\_c}$ is the value of the Higgs VEV in the broken phase at the critical temperature Tc, then this condition reads $v\_c/T\_c\gtrsim 1$ ). Neither of these conditions is fulfilled in the SM, as the CP violation encoded in the CKM matrix is too small and, anyway, the phase transition is really a crossover, given the lower bound on the Higgs mass from LEP. The strength of the EWPhT in the SM plus a singlet has been thoroughly studied. Many analyses concentrated on loop effects involving the singlet, which enhance the cubic term ETh3 in the Higgs potential at finite temperature, while reducing the quartic λhh4 (at a given Higgs mass) that enters the above condition $1\lesssim v\_c/T\_c\approx E/\lambda\_h$. LEP bounds on the Higgs mass, however, suggest that one singlet scalar is not enough, if it contributes only via loop effects. Furthermore, it was recently pointed out that magnetic fields generated during the EWPhT might increase the sphaleron rate within the broken phase, calling for even stronger phase transitions in order to have successful baryogenesis. The strongest phase transitions are achieved when the singlet contributes through tree-level effects, i.e. when the tree-level potential for H and s is such that a barrier separates the EW broken and unbroken phases (not necessarily with vanishing VEV $\left< s \right> $ along the singlet direction). Indeed, in the case of a barrier generated only at loop-level, the jump in the Higgs VEV is proportional to the critical temperature Tc (times a loop factor), and is hence constrained to be small at small temperature. In the case of a tree-level barrier, on the other hand, the Higgs VEV at the critical temperature depends on a combination of dimensionful parameters in the potential and its effect can be present even at small Tc (and is enhanced by a small Tc appearing in the denominator of vc/Tc). In what follows we will concentrate on this possibility, assuming that the transition is strongly first-order and relying on the analysis of, which studies strong phase transitions induced by tree-level effects in the SM plus a singlet. One important implication of scenarios with a tree-level barrier is that a strong transition is necessarily accompanied by a variation of the singlet VEV during the EWPhT. This can be understood by noticing that, were the singlet VEV constant, the potential would have the same shape as the SM potential at tree-level and would have, therefore, no tree-level barrier. When the EWPhT is strongly first-order, bubbles of the broken phase nucleate within a universe in the unbroken phase and expand. CP-violating interactions can then source EWBG within the wall separating the two phases. In the composite version of the SM plus a singlet outlined in the previous section, with non-vanishing, pseudoscalar couplings between singlet and fermions [b ≠ 0 in eq. ([topcoupling])], the source is provided by a variation in the VEV of s. Indeed, from eq. ([topcoupling]), we can write the top quark mass, which receives contributions from both h and s, as [topmass] mt= v \|mt\| eit, where yt is the top Yukawa and we defined the VEVs v < h >, w< s > , with v = 246 GeV. At vanishing temperature, the phase Θt can be absorbed in a redefinition of the top quark field and is thus unphysical; the only effect of a non-zero w is a shift between the top-mass and the Yukawa coupling compared to the relation that holds in the SM. However, at finite temperature and, in particular, during a phase transition, w may change: then the space-time-dependent complex phase in the quark mass cannot be rotated away by a simple field redefinition, since it would reappear in the kinetic term. For this CP violation to source EWBG, Θt, and therefore w, must change during the EWPhT. (As noted above, while this change is not guaranteed when the barrier is induced by loop effects, a change in w is always present when the barrier appears at tree-level). Then, assuming that we know how w and v change along the direction z perpendicular to the bubble wall, we may write mt(z) = ∣mt(z)∣ eiΘt(z), similarly to what has been done previously for the two-Higgs doublet model (2HDM). This space-dependence results in different dispersion relations for particles and anti-particles (in particular top quarks) in the bubble wall, which, in the WKB formalism, imply force terms in the transport equations. This, in turn, causes an excess of left-handed anti-quarks in front of the wall, in the symmetric phase, which biases sphaleron transitions and generates a net baryon asymmetry. We estimate the resulting asymmetry as follows. First, we approximate both scalar field solutions to have a kink profile of common extent along the z direction, so that [kinkansatz] w(z) wc +, and v(z), where Lw is the width of the wall, Δwc is the total change in ⟨s⟩ at the critical temperature, Tc, and vc is the Higgs VEV at Tc. The complex phase of the top mass then changes as t(z), with t. With this ansatz, we solve the transport equations numerically, using the method of, which we summarize for completeness in Appendix [Sec:appendixTransport]. The results are displayed in Figure [DeltaS], where we show, for a given strength of the phase transition and width of the wall (both are approximated analytically, for the $\Z2$ symmetric case, in Section [sec:PT]), the source of ΔΘt that is needed to reproduce the observed baryon asymmetry, (nB − nB̄)/nγ ≈ 6 × 10− 10. The final baryon asymmetry increases with ΔΘt or vc/Tc but decreases with LwTc or Lwvc. [DeltaS] [DeltaSZ2] In Figure [DeltaSZ2] we show the change in Θt taking place during a strong phase transition in the version of this model with a $\Z2$ symmetry, as described in Sections [sec:SCPV] and [sec:PT]. For f/b = 500 GeV, mh = 120 GeV and ms = 80, 130 GeV (upper and lower plots respectively), the shaded region shows a projection of the other parameters of the model onto the vc/Tc vs. ΔΘt plane. The lower and upper bounds on the shaded regions are due to the requirement of having the right EW minimum, eq. ([eq:ccbound]), while the unshaded region in the middle correspond to an instability of the vacuum at the critical temperature, eq. ([kappak]). The black lines are examples with fixed λm, and along the red lines a fixed amount of baryon asymmetry is reproduced (the solid line corresponding to the observed amount). To summarize, for a strong (tree-level) phase transition, as may occur in the SM plus a singlet, a non-vanishing pseudoscalar coupling i bγ5 between s and the top quark generates a baryon asymmetry. For a strong enough EWPhT with $b\Delta w\_c / f\gtrsim 0.1$, the observed value of (nB − nB̄)/nγ can be reproduced. In the composite version of the SM plus a singlet, these requirements are fulfilled for natural values of the parameters. We close this Section with a comparison of our EWBG scenario with previous studies of EWBG in non-supersymmetric models, such as the two-Higgs doublet model or the SM with a low cut-off. In the former, CP violation arises already at the level of renormalizable operators in the Higgs potential, through a complex phase between the two Higgs VEVs. Very strong phase transitions (induced by tree-level barriers) are not possible in that context since, contrary to the case with a singlet, the second Higgs doublet cannot acquire a VEV prior to the EWPhT by definition. (To circumvent this problem, ref. studies a 2HDM with an additional singlet: the two Higgs doublets violate CP; the singlet strengthens the EWPhT.) Although the non-supersymmetric 2HDM does not address the hierarchy problem, it is worth noting that it can also arise as the low-energy limit of composite Higgs models. The behaviour at finite temperature of other scenarios that address the hierarchy problem but lead only to a light single Higgs, such as the Minimal Composite Higgs or Little Higgs models, have been also analyzed. Refs. studied the temperature behaviour of a Higgs that arises as the PNGB of a broken global symmetry,[3](#fn3) parametrizing the deviations from the SM through effective operators. A strong EWPhT can result in this setting from the dimension-six operator h6, which stabilizes a Higgs potential with negative quartic coupling, as discussed in. This creates a large tree-level barrier but the reliability of the effective-theory description is not then obvious. Different dimension-six operators are responsible for sourcing CP violation, in a manner similar to our eq. ([topmass]), and for generating a complex mass for the top quark: mt ∼ yt(vh + ivh3/Λ2). Compared to the model proposed here, these operators (which would arise also in our model, in the limit of a heavy singlet) are dimension-six and hence generally smaller than the ones involving the singlet. Electric Dipole Moments and other Constraints[sec:EDM] ====================================================== The presence of a scalar that mixes with the Higgs and has pseudoscalar couplings to fermions induces an electric dipole moment (EDM) for the electron and for the neutron. The electron EDM receives the largest contribution from the two-loop Feynman diagram of Figure [BarrZeeE], [BarrZeeE] where the electron flips its chirality by coupling to the Higgs, which then mixes with the singlet. The singlet, in turn, couples to two photons through a top quark loop. We assume here that the singlet couples only to the top quark, which is the only coupling required for EWBG. If it couples also to light quarks, then other diagrams may give larger effects.[4](#fn4) Note that if these couplings were hierarchical and if light flavors couple to the singlet with a strength of order their Higgs Yukawa, as in the scenario of partial compositeness, then the diagram of Figure [BarrZeeE] dominates. This diagram, which receives opposite sign contributions from the two scalar mass eigenstates, gives [EEDM] &=& Z &&, where me, t are the electron and top quark masses, Z is the mixing between h and s, given by $$\label{Z} Z=\cos\theta\sin\theta=\frac{m\_{sh}^2}{\sqrt{(m\_h^2-m\_s^2)^2+4m\_{sh}^4}}\,$$ and $$G(z)\equiv z \int\_0^1dx \frac{1}{x(1-x)-z}\log\left[\frac{x(1-x)}{z}\right]\.$$ The angle θ measures the mixing between the (h, s) gauge eigenstates and the mass eigenstates (η1, η2), with masses m1, 2: $$\begin{aligned} \eta\_1&=s \sin\theta+h \cos\theta\,\\ \eta\_2&=s \cos\theta-h \sin\theta\.\end{aligned}$$ The mostly-doublet eigenstate is η1, so that θ ∈ [ − π/4, π/4]. Finally, ms2, mh2 and msh2 are the entries of the (squared) mass matrix for s and h at the electroweak minimum (their expression in terms of the parameters of the potential of eq. ([VtreeOdd]) can be found in ). The present bound on the electron EDM (at 90% C.L.) is $$\frac{d\_e}{e}<1.05 \times 10^{-27}\textrm{cm}\,$$ leading to the constraint $$\label{constraint} Z \, \frac{b}{y\_t}\, \frac{v}{f}\left\|G\left(m\_t^2/m\_1^2\right)-G\left(m\_t^2/m\_2^2\right)\right\|\lesssim 0.32\,$$ which is not very restrictive, since ∣Z∣ < 1/2 and v ≪ f, while the absolute value of the difference is smaller than ∼ 1 for $m\_s\gtrsim 40 {\,\mathrm{GeV}}$. A similar diagram, with an up or a down quark qu, d replacing the electron, and with a gluon replacing the photon, induces a chromoelectric dipole moment (CEDM) du, dC for the up and down quarks, defined via $$\mathcal{L}\supset -\frac{i}{2}d^{C}\_{u,d}\,g\_s\,\bar{q}\_{u,d}\,\sigma\_{\mu\nu}\tilde{G}^{\mu\nu}q\_{u,d}\,.$$ This diagram gives the largest contribution to the neutron EDM in our scenario, $$\label{errorCEDM} \frac{d\_n}{e}\approx(1.1\pm0.5)\,\frac{d^{C}\_d+0.5d^{C}\_u}{g\_s(1{\,\mathrm{GeV}})}.$$ The computation of the contribution is similar to that which leads to eq. ([EEDM]). In the end, one finds, && 14/23.Plugging in the numbers and comparing with the present bound on the neutron EDM (which, at 90% C.L., is dn/e < 2.9 × 10− 26cm), one obtains a stronger constraint: $$\label{constraint\_b} Z \, \frac{b}{y\_t}\, \frac{v}{f}\left\|G\left(m\_t^2/m\_1^2\right)-G\left(m\_t^2/m\_2^2\right)\right\| \lesssim (0.06-0.19),$$ where the uncertainty stems from uncertainties in the measured values of the up and down quark masses and from eq. ([errorCEDM]). Strong constraints on the presence of a singlet scalar that mixes with the Higgs also come from LEP, in the form of direct searches and electroweak precision observables (EWPO). The former apply only when one of the mass eigenstates is lighter than 114 GeV: in this case, direct searches constrain its mixing with the Higgs h, parametrized by the angle θ in eq. ([Z]). In addition, EWPO constrain the contribution of heavy states to the ρ parameter, and also lead to a bound on a combination of the mixing angle θ and the mass eigenvalues. Neglecting the contributions of heavy composite states, we can approximate the latter bound, at 95% C.L., as cos2θ lnm1 + sin2θ lnm2 < ln143 GeV. We illustrate the various bounds in Figure [fig:bounds], for a representative scale of new physics given by f/b = 500 GeV, as suggested by the composite Higgs model of, for Higgs masses (here we mean the eigenstate that is mostly EW doublet) m1 = 120, 140 GeV (upper and lower plots). This choice of Higgs masses avoids current LHC constraints, which might nevertheless become relevant for other mass values or in the future, when more luminosity is collected. EDM bounds disfavor large mixing angles and singlet masses much larger/smaller than mh. In both cases, a light singlet with large mixing with the Higgs is disfavored by direct searches, as illustrated by the uniform turquoise region. [fig:bounds] Spontaneous CP violation[sec:SCPV] ====================================== In previous sections we have argued that in composite Higgs models with a singlet coupled as a pseudoscalar to the top quark [eq. ([topcoupling])], the baryon asymmetry can be generated during the EWPhT. A necessary condition for EWBG is that the singlet VEV changes during the EWPhT, cf. Figure [DeltaS]. However, this condition is not sufficient, as we will show in this section. We consider a scenario in which the scalar potential is Z2-symmetric [$V^{\rm odd}=0$ in eq. ([Vtree])] and the singlet coupling to fermions is purely pseudoscalar-like [a = 0 in eq. ([topcoupling])], so that the singlet is a CP-odd field. This corresponds to a scenario without explicit CP violation, but where CP can be spontaneously broken by the VEV of s. However, as we will discuss below, the existence of the symmetry s → − s leads to a cancellation between positive and negative contributions to the baryon asymmetry. In this section we study how a small explicit breaking of this symmetry can lead to a sufficient net baryon asymmetry. We note in passing that the case Vodd = 0 has been discussed in several contexts before: in a Singlet-Majoron model, in the context of singlet dark matter, and in its own right . In the $\Z2$ symmetric case, a singlet VEV after the EWPhT produces dangerous cosmic domain walls and is therefore excluded. Since, for baryogenesis, the singlet VEV must change during the EWPhT, bΔwc ≠ 0, we are led to the case in which the singlet has a VEV prior to the EWPhT and none afterwards; this is fortunate since it is precisely the setting that produces the strongest phase transitions. In this case, there are in fact two transitions: first, the $\Z2$-breaking transition, at Ts, from the point (h, s) = (0, 0) to some point (h, s) = (0, ± wc) and later on, at Tc < Ts, the EWPhT from (h, s) = (0, wc) to (h, s) = (vc, 0). Note that this sequence is quite plausible, as the Higgs, unlike s, couples sizably to the gauge bosons and strongly to the top quark and therefore its mass receives larger finite-temperature corrections, which can delay its phase transition. Since the discrete $\Z2$-symmetry is broken spontaneously, albeit temporarily, at Ts, domain walls (DWs) will be produced: if they live long enough, they might come to dominate the energy density of the universe, leading to a period of late inflation and features in the cosmic microwave background which are in conflict with observations. The temperature at which electroweak DWs start to dominate, however, is TDW ∼ 10− 7 GeV, which, in an expanding universe, happens much later than the electroweak phase transition TEW ∼ 100 GeV, when the DWs disappear because the symmetry along the s direction is restored. Issues with the vanishing of the total baryon asymmetry arise because, during the first transition at Ts, different patches of the universe (distributed equally, on average) would end up in either the + wc or the − wc vacuum (denoted + and − in the following), forming a network of domains. While the + patches would produce a net baryon asymmetry, the − patches would produce a net antibaryon asymmetry (since the change in the singlet VEV during the EWPhT in different patches is equal and opposite in sign). In the following we discuss the possibility that a small explicit breaking of the Z2-symmetry in the scalar potential (Vodd ≠ 0) lifts the degeneracy between the ± vacua and biases one vacuum with respect to the other so that, at the time of the EWPhT, the whole universe lies in the − vacuum . We parametrize this explicit breaking by a small difference V( + wc) − V( − wc) = ΔV, which can be generated, for instance, through loop effects involving a small a ≠ 0 in eq. ([topcoupling]). The network dynamics depends mainly on three forces: the pressure difference between the + and − phases ΔV, the surface tension and the Hubble expansion. Friction is not very important, as discussed below. The surface tension σ and the pressure difference balance each other when the typical size of the domains ξ is of the order of the size of the critical bubble for the transition + → − , which scales as [eq:xicrit] cr ~/V . As long as $\xi<\xi\_{\rm cr}$ the dynamics of the domains is dominated by the surface tension: small-curvature structures will be smoothed-out or collapse, increasing the typical length scale ξ of the network and entering the so-called Kibble-regime, with ξ growing linearly with conformal time. When ξ reaches $\xi\_{\rm cr}$, the pressure difference ΔV becomes relevant and the energetically preferred − regions expand exponentially fast into the + regions. This behaviour has been confirmed by numerical simulations in different settings. In our model the temperature of the electroweak phase transition is significantly lower than the temperature of the first Z2-breaking phase transition (see next section) so that DWs will evolve for a period of time of order of the Hubble scale and all + regions will have vanished at the time of the electroweak phase transition as long as $\xi\_{\rm cr}\ll 1/H$. This leads to a very weak lower bound on ΔV (for more details see ) [boundDV] ~10-16. Notice that this bound is in principle independent of the nature of the Z2-breaking phase transition. Indeed, in a first-order transition the typical size of domains is given by the duration of the phase transition, which is of order ξ ∼ 1/(100H) . On the other hand, if the transition is second-order, the typical size of the domains is of order 1/TG , where $T\_G\simlt T\_c$ is the Ginzburg temperature below which the broken phase settles in the either the + or − vacuum and thermal transitions to the other vacuum are Boltzmann suppressed [5](#fn5). So, even if in most cases one expects the pressure difference to dominate over the surface tension right after the end of the Z2-breaking phase transition, if condition ([boundDV]) is satisfied the DW length scale will reach ξcr, and DWs will quickly disapear, before the EWPhT [6](#fn6). Notice that it is also possible that DWs be still around at the time of the EWPhT but that the volume occupied by the − phase, V−, is sufficiently larger than the volume V+ of the + phase, so that the suppression factor (V− − V+)/(V− + V+) on the net baryon asymmetry produced is not particularly small. Interestingly, this can happen in a time scale significantly shorter than the one needed for domain wall collapse, leading to a bound on ΔV even weaker than ([boundDV]). Finally, let us examine why friction is not important in our setting , so that one expects DWs to move with relativistic velocities. Friction is due to the plasma having different properties in the two phases. For example, during the EWPhT, W-bosons and top quarks get a mass from their couplings to the Higgs VEV. This change in mass implies a change in momentum when these particles cross the Higgs bubble wall and this produces a friction force on the wall. Now however, the properties of the particles in the + and − phases differ only by very small effects from the operators that induce the potential difference ΔV. This suppresses friction and both sides of the equation [eq:vterm] V vw Ffriction, that determines the terminal velocity of the wall, vw, vanish in the limit ΔV → 0. Quantum mechanical reflection effects of particles with an s-dependent mass that will feel the presence of the wall (as discussed in a similar context in ) are expected to be negligible too. To go beyond this rough estimate, consider the approach of ref. , which works in the limit of a weakly coupled plasma and small wall velocities. The friction due to the mass change of the particles in the plasma between the phases is Ffriction ~ (m2)2. Since the pressure difference as well as the mass difference are linear in ΔV, this small source of friction cannot lead to small terminal wall velocities according to ([eq:vterm]). Nevertheless, in some cases with small friction one might end up with terminal velocities that are subsonic, as discussed in the analysis of , which is valid for strongly interacting plasmas (when the free streaming length of the particles in the plasma is much shorter than the wall thickness). The hydrodynamic obstruction to a larger terminal velocity is due to the release of latent heat in front of the moving wall that increases the temperature and reduces, as a result, the pressure difference experienced by the wall. In principle, this effect can lead to small wall velocities when the pressure difference along the wall is much smaller than the latent heat. However, in the present case, both arise from the difference in potential ΔV so that the hydrodynamic obstruction (if present) has to occur at wall velocities that are of order of the speed of light. To summarize, even though sources of explicit CP violation are needed in our setup , already a very weak explicit breaking of CP in the scalar sector gives rise to substantial CP violation for electroweak baryogenesis. Essentially, this is due to the two-stage nature of the phase transition. In the composite Higgs model on which we focus (Ref. , for details see Appendix [Sec:CH]), a $\Z2$-preserving potential with a small explicit breaking could arise, for example, if the couplings of the top-quarks preserve the $\Z2$ [i.e. a = 0 in eq. ([topcoupling])], while lighter quarks also couple to the singlet, but with $\Z2$-violating couplings. In this case the leading terms in the potential, coming from loop effects involving the top quark Yukawa, will generate Veven, while a small Vodd would also be generated, proportional to the (smaller) light quark Yukawas. Alternatively, it might be possible that the same mechanism that is responsible for the CKM phase could generate a (presumably small) CP-violating contribution to the scalar potential via loop effects. Electroweak Phase Transition[sec:PT] ==================================== In this section we discuss the characteristics of the phase transition that are relevant for electroweak baryogenesis, namely the sphaleron washout parameter vc/Tc, the wall thickness Lw, and the change in singlet VEV during the EWPhT, Δwc. Motivated by its simplicity and by the discussion of the last section, we focus on a scalar potential with $\Z2$ symmetry, eq. ([VtreeEven]). We assume the existence of a small odd part in the potential that breaks CP explicitly, but this contribution will hardly influence the transition dynamics. Following , we parametrize the potential in the mean-field approximation as [eq:Vcrit] V(h,s,T) &=& 4 2 + 4 s2 h2 && + (T2 - Tc2) (ch h2 + cs s2 ) , where $$\label{kappak} \kappa\equiv \lambda\_m-2\lambda\_h\frac{v\_c^2}{w\_c^2}\.$$ By construction, this potential leads to a tree-level barrier and hence a strong EWPhT. Indeed, at the critical temperature T = Tc, the potential has degenerate minima at (h, s) = (vc, 0) and (h, s) = (0, ± wc), separated by a barrier parametrized by κ > 0. The two coefficients ch and cs are, in the mean-field approximation ch &=& 148 , cs &=& 112 . The condition for ensuring that the electroweak minimum is the global one at temperatures below Tc reads [eq:ccbound] > . This relation implies bounds on wc. Moreover, our low-energy effective theory approach also requires wc ≪ f, so that this parameter is typically restricted to a fixed range around vc. Given the form of the potential ([eq:Vcrit]), the critical temperature Tc can be expressed in terms of v0, the Higgs VEV in the EW breaking minimum at T = 0, as [eq:Tcrit] Tc2 = (v02 - vc2) . This can be compared with the temperature at which the Z2-symmetry breaks: Ts2 &=& Tc2 + &=& Tc2 > Tc2 , which we have taken to be higher than Tc2, as argued in the previous section. The Higgs mass is [eq:mh] mh2 = 2 h v02 , while for the singlet mass one finds [eq:ms] ms2 = 12 v02 + h (v02 - vc2) . Both contributions to the singlet mass are positive, due to eq. ([eq:ccbound]).[7](#fn7) Next, we determine the wall thickness in the thin-wall approximation. The tunneling path will extremize the Euclidean action S1 = -+ d , with boundary conditions &&h(-)=0 , h()=vc , h’()=0 , &&s(-)=wc , s()=0 , s’()=0 , and will proceed over (or close to) the saddle point with the potential value V= 14 (4 h vc2 + wc2) . [Vsad] We can estimate the parametric dependence of Lw by approximating S1(vc2+wc2)+Lw V , which leads to Lw2= . To estimate the numerical constant in this formula assume for simplicity that the tunneling of the fields proceeds along the path h() = vc , s() = wc , that passes a barrier of height V = κvc2wc2/16. Using the ansatz () = [1 + (/Lw)] , we obtain integral expressions for α and β that can be calculated to give α/β ≃ 2.7. In this way we arrive at the estimate [eq:Lw] L2w 2.7 (1+) . This result would become unreliable for κ ≪ 1, as the potential then develops a flat direction and loop effects (giving rise to cubic terms from the Daisy re-summed contributions) become important and strengthen the phase transition. We do not discuss this particular case here. Another pathological limit, λh → 0, is forbidden by the lower limit on the Higgs mass. To summarize, a scalar potential, its critical temperature and the wall width, are uniquely determined (in the $\Z2$ symmetric case) by the values of the Higgs and singlet masses at vanishing temperature, mh and ms, by the strength of the EWPhT, vc/Tc, and by the change in the singlet VEV, wc. So, it is always possible to find parameter choices that give the right baryon asymmetry, as long as bounds for the consistency of the potential (such as perturbativity of the couplings, stability of the high-T minimum and of the T = 0 EW breaking one) are respected: this is illustrated by the broad shaded regions of Figure [DeltaSZ2]. We give some explicit examples in Table [tab:examples]. In the first case, we consider a model with small Higgs-singlet mixing and a relatively light Higgs (mh ∼ 120 GeV). Hence, the experimental bounds of Figure [fig:bounds] are fulfilled independently of the singlet mass. In the second case, we consider a model where the Higgs is relatively heavy (mh ∼ 140 GeV). Besides this, we assume a sizable Higgs-singlet mixing, together with singlet and Higgs masses of similar size. This can be achieved by adding only a relatively small operator s∣H∣2 to the scalar potential (that will have a negligible impact on the characteristics of the phase transition). In both cases viable EWBG is possible with a strong coupling scale f as high as several TeV. Larger values of f can be compensated by increasing vc/Tc, which enhances the strength of the phase transition and the CP-violating sources in the Higgs wall. However, notice that, although in our simplified treatment, vc/Tc has been taken as a free parameter, for very large values of vc/Tc ($\gtrsim 4$) one expects that the phase transition will never be completed. \| \| λh \| κ \| wc/vc \| bwc/f \| \| --- \| --- \| --- \| --- \| --- \| \| S1 \| 0.12 \| 0.12 \| 1.0 \| 0.1 \| \| S2 \| 0.16 \| 0.48 \| 1.0 \| 0.15 \| 0.1 cm \| \| mh \| ms \| vc \| f/b \| Lwvc \| vc/Tc \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| S1 \| 120 GeV \| 81 GeV \| 188 GeV \| 1.88 TeV \| 7.1 \| 2.0 \| \| S2 \| 140 GeV \| 139.2 GeV \| 177.8 GeV \| 1.185 TeV \| 3.5 \| 1.5 \| Conclusions[Sec:Concl] ====================== Composite Higgs models have become very plausible options for natural electroweak symmetry breaking and it is tempting to explore whether such scenarios can also solve other problems that the Standard Model fails to address. One important and long-standing problem is the generation of the cosmological baryon asymmetry and here we have shown in some detail how EW baryogenesis can be successfully achieved in such a model. The required ingredients turn out to be simple, not least because naturalness itself implies modifications to the Higgs and top sectors, which are precisely the ones believed to be responsible for EWBG. Firstly, the electroweak phase transition can easily be strongly first-order, for realistic values of the Higgs mass, if the model contains in addition a scalar singlet. Moreover, if that scalar couples to the top quark it can lead to a non-trivial CP-violating phase along the bubbles of the electroweak transition, creating the seed for sphalerons to generate a non-zero baryon asymmetry. We have shown that a composite model, based on the coset SO(6)/SO(5) proposed previously , contains all these ingredients. Its low-energy effective theory features an additional pseudo-Goldstone field that is a singlet and has dimension-five, pseudoscalar couplings to top quarks that can break CP. Motivated by this prospect, we have calculated quantitatively the expected amount of asymmetry, showing that baryogenesis is successfully achieved in a broad region of the parameter space that passes other experimental constraints. Interestingly, the new complex phases and the mixing between the Higgs and the singlet predict contributions to the EDMs of neutron and electron, not far beyond the reach of current and future experiments. Acknowledgements ================ Three of us (JRE, TK and FR) are grateful to the organizers of the Workshop ’Electroweak Baryogenesis in the Era of the LHC’, at the Weizmann Institute, for stimulating discussions which shaped this article. JRE thanks T. Hiramatsu, M. Kawasaki and K. ’I. Saikawa for sharing unpublished results of numerical simulations, while FR thanks M. Pietroni and A. Riotto for interesting discussions. This work was supported by the Fondazione Cariparo Excellence Grant LHCosmo; the ERC starting grant Cosmo@LHC (204072); the Spanish Ministry MICINN under contracts FPA2010-17747 and FPA2008-01430; the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042); and the Generalitat de Catalunya grant 2009SGR894. The SO(6)/SO(5) Composite Higgs Model ============================================= In composite Higgs models, similarly to QCD, the hierarchy between the Plank and the TeV scale is due to the slow logarithmic running of an asymptotically free interaction that becomes strong and confines close to the electroweak scale. New (techni-) fermions are charged under this strong interaction but also under a global symmetry; here we are interested in the case SO(6) ≃ SU(4). As the strong interaction confines, the global symmetry is broken down to a subgroup, which we take to be SO(5). From Goldstone’s theorem, we expect five massless degrees of freedom [equivalent to the number of broken global symmetries in SO(6)/SO(5)], which are analogues of the pions in QCD. The embedding of the SM group SU(2)L in SO(5) is such that these five Nambu-Goldstone bosons (NGBs), may be identified as the four degrees of freedom belonging to the Higgs doublet plus one degree of freedom, corresponding to a real singlet s. From symmetry arguments alone, we can extract the leading kinetic part of the σ-model Lagrangian (in the unitary gauge) as kin&=& (h)2	arxiv_0000819
Optimal Dithering Configuration Mitigating Rayleigh-Backscattering-Induced Distortion in Radioastronomic Optical Fiber Systems ================================================================================================================================ In the context of Radioastronomic applications where the Analog Radio-over-Fiber technology is used for the antenna downlink, detrimental nonlinearity effects arise because of the interference between the forward signal generated by the laser and the Rayleigh backscattered one which is re-forwarded by the laser itself toward the photodetector. The adoption of the so called dithering technique, which involves the direct modulation of the laser with a sinusoidal tone and takes advantage of the laser chirping phenomenon, has been proved to reduce such Rayleigh Back Scattering - induced nonlinearities. The frequency and the amplitude of the dithering tone should both be as low as possible, in order to avoid undesired collateral effects on the received spectrum as well as keep at low levels the global energy consumption. Through a comprehensive analysis of dithered Radio over Fiber systems, it is demonstrated that a progressive reduction of the dithering tone frequency affects in a peculiar fashion both the chirping characteristics of the field emitted by the laser and the spectrum pattern of the received signal at the fiber end. Accounting for the concurrent effects caused by such phenomena, optimal operating conditions are identified for the implementation of the dithering tone technique in radioastronomic systems. Nonlinearities; Rayleigh backscattering; Radioastronomy, RoF, Laser feedback, Dithering. Introduction ============ which adopt the Radio-over Fiber (RoF) technology in one of its most cost-efficient versions consist essentially in a radiofrequency (RF) signal which performs the direct intensity modulation (D-IM) of a laser source, propagates in a span of Standard Single Mode Fiber (SSMF) and is directly detected (DD) at the receiver end (D-IMDD RoF systems). When the described scheme is utilized within Radioastronomic scenarios, peculiar features have to be considered, which distinguish these systems e.g. from those which apply the D-IMDD RoF technology to the mobile network. One of them is related to the frequencies of the transmitted RF signals, which in the case of the mobile network (4G, 5G signals and beyond) can reach several tens of GHz, and do not fall below around 700MHz, while within Radioastronomic application can also belong to intervals ranging from few tens to few hundreds MHz. In addition to this, while RoF systems for the mobile network are typically designed assuming that at the optical transmitter’s side the RF powers of the signals range roughly in an interval of ± few units of dBm, in the case of signals received by radioastronomic antennas these powers can exhibit values of just -60dBm when they come from astronomical sources, while they arrive to a maximum of around -20dBm when they consist in undesired Radio Frequency Interference (RFI) signals coming from satellites, radio and/or television transmitters, etc.. As a consequence of the low power levels of the signals which directly modulate the laser source, in many RoF D-IMDD systems utilized for radioastronomic applications it is relatively straightforward to reduce to negligible values the nonlinearities which typically bring detriment to RoF systems in the mobile networks. Indeed, the spurious frequencies caused by the nonlinearity of the laser Power-Current curves typically in this case fall below the noise floor. Moreover, it is enough to operate in the vicinity of the second optical window (around the wavelength λ = 1310nm), to guarantee that, due to the low level of the RF power modulating the laser, the creation of nonlinearities generated by the combined effect of laser chirp and fiber chromatic dispersion is avoided. It has however been shown that in this kind of systems a primary cause of undesired nonlinear effects takes place when a generic RFI signal (with frequency and angular frequency given respectively by fRF and ωRF = 2πfRF) is transmitted in the RoF System together with the signals coming from sky sources, after that they have all been received by the Radioastronomic antenna. Indeed, the portion of such RFI signal which is reflected by Rayleigh Backscattering (RB) reaches the laser section, is partly reflected again and interacts at the receiver end with the transmitted RFI signal itself, generating nonlinear frequency terms (e.g. at frequencies 2fRF, 3fRF, ....). Although the RFI disturbance at frequency fRF can be filtered out at the signal post-processing stage, the same cannot be done for all generated spurious frequency terms. One countermeasure to the described problem could be represented by the introduction of a further optical isolator at the fiber input section, which would be added to the one embedded in the laser. However this solution would hardly meet the cost limitations (e.g. possibly 100USD or less for a single front-end receiver) imposed by the very high number of RoF downlinks which should be realized within radioastronomic facilities operating at frequencies of tens/hundreds of MHz. In a previous work, an appropriate cost-effective solution has been demonstrated, consisting in the introduction of a dithering tone as a further modulation current for the optical transmitter, which pushes the level of the nonlinear terms below the noise floor. This solution had actually already been proposed in the past, within telecommunication scenarios, with the aim to reduce noise effects in optical fiber systems,, and its application finalized to the reduction of the spurious terms constitutes an additional newly-evidenced advantage offered by this technique. In the application of the mentioned dithering tone technique it would however be desirable that both the frequency fd (or angular frequency ωd = 2πfd) and the amplitude Id of the current tone exhibit the lowest possible values. The first requirement is related to the fact that the distance from fRF of the possibly generated spurious frequency terms, fRF ± fd, fRF ± 2fd, ... should be lower than the resolution bandwidth of the reception filter, so that they can be removed from the received signal together with the RFI term at frequency fRF. The second requirement comes instead from the necessity to keep at low levels the energy consumption of the tone generator. To assist the designer in identifying the appropriate solution to the problem, a theoretical model of the laser behavior under current modulation in presence of optical feedback by RB is necessary. However, the models developed so far describing the effects produced by optical feedback under laser current modulation, are not referred to RB, since they consider only the reflection coming from a single external cavity, typically put on the back-facet of the laser. At the same time, the effects produced by RB feedback have been considered only with reference to the linewidth reduction and optical frequency shift and hopping, in all cases for continuous-wave operation, without considering the presence of a modulating current. In the present paper, a rigorous mathematical model based on the laser Rate Equations is developed to describe the effects of the feedback due to Rayleigh Backscattering on an optical field modulated by an RF signal. The model, which will be shown to be in agreement with experimental measurements, is utilized to describe the effects of the introduction of the dithering tone and to identify the optimal working point which allows keep both Id and ωd as low as possible. The paper is organized as follows: Section II provides a description of the model developed to study the impact of RB on the characteristics of the field emitted by a directly modulated laser. Section III describes the experimental results, which confirm the correctness of the theoretical model adopted and allow to identify the optimal operating conditions for the implementation of the dithering tone technique. Finally, conclusions will be drawn. Theoretical Model ================= The theoretical model presented in this work is based on the physical structure shown in Figure [fig:initialscheme], composed by a laser-based optical transmitter whose emitted field is coupled through a focusing lens and an isolator to an optical fiber and reaches finally an optical receiver based on a PIN photodetector. ![Scheme of the structure considered.](Figures/Fig1.png "fig:") [fig:initialscheme] The laser cavity, exhibits a length Lcav and power reflectivities at the facets Rl, 1 and Rl, 2, respectively. The injection current Iinj(t) can be written in complex notation as Iinj(t) = Ibias + ∑iIiejωit, where Ibias represents the bias current, and Ii represents the amplitude of the generic i − th sinusoidal RF tone which directly modulates the laser, with angular frequency ωi and frequency fi = ωi/(2π). For formal simplicity and without loss of generality, in the following only two modulating tones will be considered, namely a term with amplitude IRF and angular frequency ωRF = 2πfRF, and the so called dithering tone with amplitude Id and angular frequency ωd = 2πfd, resulting in: Iinj(t) = Ibias + Idejωdt + IRFejωRFt For this reason, instead of ∑i = d, RF…, the simplified notation ∑i… will be adopted throughout the paper. In principle, the electrical field emitted by the laser source and propagating into the fiber should be represented in vector form, to account for the evolution of its state of polarization (SOP). However, the phenomenon of Rayleigh backscattering, which takes place within the fiber in correspondence to the reflection sections zk, k = 1, …, K (see again Figure [fig:initialscheme]) can be regarded as substantially isotropic, while the change of the SOP of the field between zk and an adjacent section can be described by an appropriate Jones matrix, which is unitary. Consequently, it can be shown that for a given initial SOP and in absence of Polarization Dependent Losses (PDL), the electrical field exhibits at the fiber end section a corresponding final SOP on which the quantities which are of interest in the present work (e.g. powers received at different frequencies) do not depend. In presence of Polarization Dependent Losses (PDL), the quantities which are of interest in the present work can instead exhibit variations depending on the final SOP of the electric field. This could be appreciated for example in presence of changes of environmental quantities (e.g. temperature). Indeed, while it can be reasonably assumed that the devices located within the transmitter (single-mode laser, lens, isolator) still determine a stable SOP for the emitted field, the optical fiber can undergo local birefringence perturbations which can change in real time the SOP of the field impinging the detecting area of the PIN. In presence of PDL of this last device, the received powers would then exhibit a polarization-dependent fluctuating behavior. However the relative variations of the measured quantities can be typically considered as low (e.g.few tenths of dB or less), and, at the same time, on fiber connections of the order of few km length, like the ones considered in this work, they can be appreciated on time scales (e.g. tens of seconds or more) which are much longer than the ones of the phenomena investigated. For all these reasons, in the rest of the paper the electrical field will be described as a scalar quantity without losing the generality of the results that will be obtained. Re-transmitted and re-injected portions of the Rayleigh Backscattered field --------------------------------------------------------------------------- By virtue of the considerations developed above, the electrical field generated inside the laser can be expressed as E(t) = ∣E(t)∣ej[ω0t + θ(t)]ejϕ(t) where ∣E(t)∣ is its module, ω0 the optical angular frequency that the field would exhibit in absence of modulating tones and in absence of RB as well, θ(t) is the time-varying phase due to their presence, while ϕ(t) represents the laser phase noise contribution. The expressions of ∣E(t)∣ and θ(t) are given by: $$\begin{aligned} &&\|E(t)\|= E\_0\sqrt{1+ \zeta\_{RB}+\sum\_i m\_icos(\omega\_it)} \label{eq:E}\\ &&\theta(t)=\Delta \omega t\,\, +\sum\_{i} \|M\_i\|cos(\omega\_{i}t+ \angle M\_i) \label{eq:theta}\end{aligned}$$ In $E\_0 =\sqrt{\eta (I\_{bias}-I\_{th})}$ is the electrical field amplitude, with Ith threshold current and η power-current slope efficiency of the laser, ζRB represents a small increment to the unit number under square root, which derives from the presence of RB and will be further described in the next subsection, while mi = Ii/(Ibias − Ith) represents the optical modulation index at the considered frequency. In Δω represents a variation of the angular frequency due to RB and to direct modulation, whose presence determines the total optical angular frequency of the emitted field to be ωopt = ω0 + Δω. Moreover, the quantities ∣Mi∣ and ∠Mi are respectively module and phase of the phase modulation index due to the chirp effect at ωi. In the quantity ejϕ(t) is a random function whose Power Spectral Density, determined as Fourier Transform of its autocorrelation function, is given by: $$\mathcal{F\_{\xi}} \left\{ \left\langle e^{j\phi(t)-\phi(t-\xi)} \right \rangle \right\}= \mathcal{F\_{\xi}} \left\{ e^{-\|\xi\|\Delta\_{\Omega}} \right\} =\frac{2\Delta\_{\Omega}}{\Delta\_{\Omega}^2+\omega^2} \label{eq:phase\_noise}$$ where $\mathcal{F\_{\xi}}\{\cdots\}$, ⟨⋯⟩ indicate respectively the operations of Fourier Transformation with respect to the variable ξ, and time averaging. The quantity ΔΩ = 1/τcoh is the coherence angular frequency band of the laser source, reciprocal of its coherence time τcoh. Still referring to Fig. [fig:initialscheme], the field given by exits the laser right-hand-side facet, is focused by a coupling lens into an isolator, and is subsequently injected at the input section (z = 0) of the optical fiber of length L, propagating with a group velocity vg. Putting τ = z/vg, at the generic section z ∈ [0, L] of the fiber the resulting electric field can then be written as: $$\begin{aligned} &&E\_{TX}(t,z) =\sqrt{1-R\_{l,2}}\cdot \sqrt{\eta\_c} \cdot {{E}}(t-\tau) e^{-\alpha v\_g \tau} \label{eq:fiber\_field}\end{aligned}$$ where ηc is the power coupling coefficient between the laser and the optical fiber given by the coupling lens, while α is the attenuation constant of the fundamental mode of the optical fiber. As above mentioned, as ETX propagates along the fiber, the phenomenon of Rayleigh backscattering takes place. The inhomogeneity of the longitudinal profile of the refractive index of the optical fiber determines indeed the presence of many reflection points at the coordinates zk, k = 1, …, K, each one of which generates a back reflected electrical field. At the output facet of the laser cavity a Rayleigh Backscattered field is then present, given by: $$\label{eq:RB field of} E\_{RB}(t) =\frac{\sqrt{1-R\_{l,2}}\cdot \eta\_c}{\sqrt{\alpha\_{iso}}}\sum\_{k=1}^K \rho \_k {{E}}(t-2 \tau \_k)e^{-\alpha v\_g 2 \tau \_k}$$ where αiso is the isolator power attenuation in the backward direction, while τk = zk/vg. Then, ρk = ∣ρk∣ej∠ρk is the complex reflection coefficient at the k − th section due to RB. Note that the weak-feedback approximation, i.e. ∣ρk∣ < < 1, is taken, meaning that possible multiple reflections within a single fiber section are not considered. The different ρk’s are assumed to be complex random variables, with zero mean value and Gaussian distribution. The variance of both their real and imaginary parts is $\sigma^2\_{\rho}=\frac{1}{2} \alpha\_s \mathcal{S} d z\_k$, where S is the so-called backscattering factor or recapture factor, which depends on the characteristics of the fiber considered and exhibits typical values of the order of 10− 3 in case of the standard G.652 fiber, αs is the Rayleigh attenuation coefficient, which for the considered wavelengths can be assumed to coincide with α, and dzk = zk − zk − 1 is the length of the interval considered. The ρk’s are also assumed to be delta-correlated, namely: E[ρkρh\] = α*sS dzkδkh where E[ ⋅ ] represents the expected value operator, while δkh is the delta Kronecker function. Still referring to Fig. [fig:initialscheme], a portion of the field reported in, given by $E\_{RB1}(t)=\sqrt{1-R\_{l,2}}E\_{RB}(t)$, is re-injected into the laser cavity. As will be shown in Subsection [sub:rateeq], this fact has significant consequences on the phase modulation index due to the chirp effect Md at ωd, which also influences the identification of the optimal frequency and amplitude characteristics of the dithering tone itself. Another portion of this field, given by $E\_{RB2}(t,z)=\sqrt{R\_{l,2}}E\_{RB}(t,z)$ is instead reflected by the laser facet. As will be shown in Subsection [sub:finalspectrum], this reflected field adds to the one given by in input to the optical fiber, contributing to the final form of the power spectral density of the photocurrent detected at the receiving end of the system. The knowledge of the behavior of this quantity allows to put into evidence particular physical aspects which also contribute to determine the optimal frequency and amplitude characteristics of the dithering tone to be utilized. Re-injected portion of the RB field and effect on the Chirp term ---------------------------------------------------------------- The Rate Equations, expressed below in the so called Semi Classical form , describe the behaviors of S(t) = ∣E(t)∣2, θ(t) and N(t), which represent respectively the number of photons in the laser cavity, the previously introduced time varying phase θ(t) and the carrier density N(t), expressed in $\left[\frac{1}{m^3}\right]$: $$\begin{aligned} \frac{d S(t)}{dt}&=\left(G(N,S)-\frac{1}{\tau\_p}\right)S(t)+ \Gamma\_S(t)\label{eq:S}\\ \nonumber\\ \frac{d\theta(t)}{dt}&=\frac{\alpha\_H}{2}G\_N(N(t)-N\_{th})-\Gamma\_{\theta}(t)\label{eq:phi}\\ \nonumber\\ \frac{dN(t)}{dt} &= \frac{I\_{inj}(t)}{eV}-\frac{N(t)}{\tau\_s}-\frac{G(N,S)}{V}S(t)\label{eq:N}\end{aligned}$$ At the second side of () the quantity G(N, S) is the laser cavity gain which depends both from N, and S through the relation: $$\label{eq:gain} G(N,S) = \frac{g\_0}{1+\epsilon S}(N-N\_{tr})\simeq g\_0\cdot(1-\epsilon S)(N-N\_{tr})$$ where the transparency level Ntr is the density of carriers in the cavity for which G(N, S) = 0, ε < < 1/S is the gain compression factor and g0 represents the nominal gain-slope in absence of compression, multiplied by the volume of the laser cavity. The quantity τp represents the average photon’s lifetime, or inverse of the cavity losses. The term ΓS(t) at the right-hand-side of, which represents the RB contribution to dS(t)/dt, is given by: $$\begin{aligned} &&\Gamma\_S(t)=2\sum\_{k=1}^K\|C\_k\| \sqrt{S(t)S(t-2\tau\_k)}\cdot\nonumber\\ &&\cdot\cos[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:Gamma\_S}\end{aligned}$$ and will be derived in Appendix [app:1]. Within, Ck is the k-th compound-cavity coefficient defined as follows: $$C\_k=\frac{(1-R\_{l,2})\eta\_c}{\sqrt{\alpha\_{iso}}}\rho\_ke^{-\alpha v\_g 2 \tau\_k}\frac{1}{{2}\tau\_{cav}}\frac{1}{\sqrt{R\_{l,2}}}$$ where τcav = 2 ⋅ ncavLcav/c is the round-trip time of the laser cavity with refractive index ncav. By solving the laser rate equations in the reference steady state case where both external RF modulation and RB feedback are absent, it results that the carrier density is equal to the amount Nth which balances the cavity losses, and it results also that number of photons is $S\_0= \frac{\tau\_p}{e}\left(I\_{bias}-I\_{th}\right)$ with $I\_{th}=N\_{th}\frac{eV}{\tau\_s}$, where the relationship τp = 1/G(Nth, S0) has been exploited. At the second side of ([eq:phi]) αH is the linewidth enhancement factor of the laser source while it is $G\_N = \left. \frac{\partial G}{\partial N}\right\|\_{S\_0}=g\_0(1-\epsilon S\_0)$. Similarly to ([eq:S]), the last term at the right-hand-side of ([eq:phi]) represents the RB contribution to dθ(t)/dt: $$\begin{aligned} &&\Gamma\_{\theta}=\sum\_{k=1}^K\|C\_k\|\sqrt{\frac{S(t-2\tau\_k)}{S(t)}}\cdot \nonumber \\ &&\sin[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:Gamma\_PHI}\end{aligned}$$ and will also be derived in Appendix [app:1]. Finally, at the second side of ([eq:N]) the quantities e, V, τs represent respectively the electron charge, the volume of the active region and the carrier-lifetime. Assuming that partial RB field re-injection and RF modulation of the injection current determine only a perturbative effect on the laser emission characteristics, the behavior of G(N, S) can be represented by its expansion to the first order: G(N, S) ≃ G(Nth, S0) + GN(N − Nth) + GS(S − S0) where $G\_S = \left. \frac{\partial G}{\partial S}\right\|\_{N\_{th}}=-g\_0 \epsilon (N\_{th}-N\_{tr})$. Within the hypothesis taken, in presence of an injection current Iinj(t) given by the second side of ([eqIinj]) S(t), θ(t) and N(t) exhibit expressions given by: $$\begin{aligned} S(t) &= S\_0 + \Delta S +\sum\_{i} S\_{i}e^{j\omega\_{i}t}\label{eq:delta S}\\ \theta(t) &= \Delta \omega t + \sum\_{i} M\_{i}e^{j\omega\_{i}t}\label{eq:delta theta}\\ N(t) &= N\_{th} + \Delta N +\sum\_{i} N\_{i}e^{j\omega\_{i}t}\label{eq:delta N}\end{aligned}$$ where is re-written in complex notation and where (ΔS, Δω, ΔN), and (Si, Mi, Ni) ∀i, are the unknowns which, within the perturbative regime taken, respect the conditions: $$\begin{aligned} &&\Delta S, \, \|S\_{i}\|\_{\forall i} << S\_0 \label{eq:perturb\_1} \\ && \Delta \omega, \,\|\omega\_{i}M\_{i}\|\_{\forall i} << \omega\_{0} \label{eq:perturb\_2} \\ &&\Delta N, \,\|N\_{i}\|\_{\forall i} << N\_{th} \label{eq:perturb\_3}\end{aligned}$$ In line with,,, it will be assumed that the perturbation caused the generic i − th modulating tone can be determined separately from the one due to the other, namely that the expression of the triplet (Si, Mi, Ni) can be determined just considering Iinj(t) = Ibias + Iiejωit (i = d or i = RF). Inserting the relations,, with the summations composed by one element into,,, and considering the terms at DC and at ωi of the Jacobi-Anger expansions of and (detailed in Appendix [app:2]), it is possible to obtain two systems of three equations. The first one, which is omitted for brevity, represents the steady-state terms and allows to determine ΔS, Δω, ΔN, while the second one represents the modulating terms with angular frequency ωi, which allow to determine Si, Mi, Ni. The latter of the two systems assumes the following form: $$\overline{\overline{A}}\cdot\left(\begin{array}{c} S\_i\\ M\_i\\ N\_i \end{array}\right)=\left(\begin{array}{c} 0\\ 0\\ \frac{I\_i}{eV} \end{array}\right)$$ where the matrix $\overline{\overline{A}}$ is given by: $$\begin{aligned} &\overline{\overline{A}} =\label{eq:system}\\ &\left(\begin{array}{ccc} j\omega\_i-G\_SS\_0&+2S\_0\gamma\_{S,i}&-G\_NS\_0\\ \\ 0&j\omega\_i+\gamma\_{\theta,i}&-\frac{\alpha\_H}{2}G\_N\\ \\ \frac{1}{\tau\_pV}+\frac{G\_S S\_0}{V}&0&j\omega\_i+\frac{1}{\tau\_s}+\frac{G\_N S\_0}{V} \nonumber \end{array}\right)\end{aligned}$$ and where the elements γS, i and γθ, i, located in positions (1, 2) and (2, 2), are derived from ΓS and Γθ, as shown in Appendix [app:2]. As will be evidenced in the next Subsection, the higher values exhibited by the module ∣Md∣ of the phase modulation index due to the chirp effect at ωd, the more effective the dithering technique results in reducing to negligible levels the undesired RB-related nonlinearities. For this reason, the behavior of Mi, which is obtained by solving, will be analyzed in the following. [t] \| Symbol \| Physical meaning \| Order of Magnitude \| \| --- \| --- \| --- \| \| g0 \| Gain slope \| ∼ 10− 12 m3/sec \| \| Ntr \| Carriers Transparency Level \| ∼ 1. ⋅ 1024 m− 3 \| \| Nth \| Carriers Threshold Level \| ∼ 2. ⋅ 1024 m− 3 \| \| ε \| Gain compression factor \| ∼ 10− 8 \| \| τp \| Photon Lifetime \| ∼ 10− 12sec \| \| τs \| Carrier Lifetime \| ∼ 10− 9sec \| \| Rl, 1, Rl, 2 \| Laser mirrors reflectivities \| ∼ 0.3, …, 0.5 \| \| Lcav \| Laser cavity length \| ∼ few 100μm \| \| V \| Volume of Laser active region \| ∼ 10− 16m3 \| \| ncav \| Laser cavity refractive index \| ∼ 3.5, …, 5 \| \| αH \| Linewidth enhancement factor \| ∼ 3, …, 6 \| \| \| Number of photons in steady state \| \| \| S0 \| (assuming for Ibias − Ith \| ∼ 105 \| \| \| a value of few tens mA) \| \| \| GN \| ∂G/∂N in steady state \| ∼ 10− 12 m3 \| \| GS \| ∂G/∂S in steady state \| ∼ 104 \| [laser:parameters] In doing so, some simplifications can be taken, which allow to put this quantity in a more readable form. Indeed, considering modulating frequencies fi which can range from KHz to hundreds MHz, and assuming for Ibias a value such that Ibias − Ith ∼ few tens of mA, taking into account the typical orders of magnitude of Rate Equations parameters reported in Table [laser:parameters], the determinant of $\overline{\overline{A}}$ can be reduced to: $$det \left[ \overline{\overline{A}} \right] \simeq \frac{G\_N S\_0}{\tau\_p V} \left[j \omega\_i+\gamma\_{RB,i} \right] \label{eq:det\_M}$$ where it has been put: $$\begin{aligned} &&\gamma\_{RB,i}=\gamma\_{S,i}-\alpha\_H \gamma\_{\theta,i}\simeq \sqrt{1+\alpha\_H^2}\cdot \sum\_{k=1}^{K} \|C\_k\| \cdot \label{eq:gamma\_RB}\\ &&\cdot \frac{2J\_1(X\_{i,k})}{X\_{i,k}} \left(1-e^{-j\omega\_i2\tau\_k}\right)\cos\left(\psi\_k-\tan^{-1}\alpha\_H\right) \nonumber\end{aligned}$$ with: $$\begin{aligned} &&X\_{i,k}= 2 \|M\_i\|sin(\omega\_i \tau\_k +\angle M\_i) \label{eq:X\_i\_k}\\ && \psi\_k =\omega\_{opt} 2\tau\_k - \angle \rho\_k \label{eq:psi\_k}\end{aligned}$$ The resulting expected value of ∣Mi∣ is then given by the expression: $$\|M\_{i}\| = \mathbb{E}\left[ \frac{\frac{\tau\_p}{e}\frac{\alpha\_H}{2}G\_S\,I\_i} {\sqrt{\omega\_i^2 + \|\gamma\_{RB,i}\|^2 }} \right] \label{eq:theta\_i}$$ in which the statistical expectation can be straightforwardly applied through and utilizing the properties of the ρk’s specified in Section [sub:RB1RB2]. It can be at first noted that the value of ∣Mi∣ are proportional to the amplitude of the modulating current Ii. Secondly, regarding the dependence of the frequency fi (or angular frequency ωi) of the modulating current, it can be first observed that in case the RB effects were absent, the value of ∣Mi∣ would result inversely proportional to ωi. In the considered case, the dependence of ∣Mi∣ on ωi does not instead follow such simple pattern. Solving iteratively (since ∣Mi∣ is contained in γRB, i) it is indeed possible to obtain the curves reported in Fig.[fig:thetaimodule], which have been modelled in accordance with the orders of magnitude of the parameters reported in Table [laser:parameters] and Table [tab:parameters]. At the right hand side of the figure it can be noted that for frequencies greater than few hundred MHz it can effectively be assumed for all the reported curves $\|M\_i\|\simeq \frac{K\_{f,i}I\_i}{f\_i}$ where $K\_{f,i}=\frac{1}{2 \pi}\frac{\tau\_p}{e}\frac{\alpha\_H}{2}G\_S$, is the so called adiabatic chirp factor, expressed in [Hz/A]. Indeed, a constant value ∣Mi∣ = 10n rad (n = 1, …, 6 in the figure) corresponds to the linear relationship $I\_i=\frac{10^n \, rad}{K\_{f,i}}f\_i$. This is the behavior of ∣Mi∣ when i = RF is considered, since the undesired RFI signals which affect radioastronomic plants operating in the lower bandwidth of the radioastronomic spectrum exhibit frequencies of tens to hundreds of MHz. When instead the dithering tone is considered (i = d), it is however convenient to give lower values to fd. For example, a given constant value ∣Md∣ = 103 rad would be obtained with Id ≃ 10 mA if fd ≃ 2 MHz were chosen, but can be obtained with Id ≃ 0.1 mA if fd ≃ 20 kHz. Note that giving low values to fd fulfills also the requirement to keep the terms fRF ± fd, fRF ± 2fd, ... within the resolution bandwidth of the reception filter, as mentioned in the Introduction. Still considering as example the curve referred to the value ∣Md∣ = 103 rad, it can however also be noted that for frequencies lower than few kHz the value of Id to be utilized becomes practically independent from fd. This happens because, as can be shown with a detailed analysis of its composing terms, ∣γRB, d∣ is weakly dependent on the frequency. Going therefore from right to left in the horizontal axis of Fig.[fig:thetaimodule], sooner or later ωd reaches values lower than ∣γRB, d∣ so that the second side of becomes almost independent from fd. The same considerations apply for the other curves ∣Md∣ = constant, with the difference that the values of the frequencies at which the lines become independent from fd decrease for increasing values of ∣Md∣ (or increase for decreasing values of ∣Md∣). This behavior can be intuitively explained, since an increase of ∣Md∣ determines an increase of Xd, k (see ) which in turn brings to a reduction of the terms $\frac{2J\_1(X\_{d,k})}{X\_{d,k}}$ of the sum present in, i.e. a decrease of the “cutoff angular frequency” ∣γRB, d∣. ![Values of the modulating current amplitude I_i which, for a given modulating frequency f_i determine assigned values of the module of the phase modulation index due to the adiabatic chirp \|M_i\|. Continuous lines: realistic cases where the effect of RB is taken into account. Dashed lines: theoretical values of I_i in case no RB field re-injection were present. Note that the subscripts can be assumed as i=RF or i=d.](Figures/Fig2.png "fig:") [fig:thetaimodule] Still in Fig. [fig:thetaimodule] the behavior of ∣Md∣ in case of absence of RB is also reported, which puts into evidence the described effect of the RB field re-injection, namely the impossibility at a given low frequency to obtain high values of ∣Md∣ for dithering current amplitudes lower than a minimum “threshold” value. Note however that minimal increases of Id from this “threshold” value lead to very high variations of ∣Md∣. This can be appreciated in Fig. [fig:zoomthetaimodule], which focalizes the same quantities represented in Fig. [fig:thetaimodule] in limited ranges of fd and Id. ![Values of \|M_i\| which are reached in the interval f_i \in [1, 10] kHz in the vicinity of the values of I_i where the curves of Fig.[fig:thetaimodule] appear to exhibit an almost horizontal behavior. Note that the subscripts can be assumed as i=RF or i=d.](Figures/Fig3.png "fig:") [fig:zoomthetaimodule] Given the mentioned requirements for an effective application of the dithering tone technique (high ∣Md∣, Id, fd as low as possible), the practical indication is then to utilize a value of Id around such “threshold” value, while keeping fd as low as possible. A further physical phenomenon will be however highlighted in the next Section, which prevents the choice of arbitrarily low values for fd. Power Spectral Density at the Receiving End and effects of the Dithering Tone characteristics ----------------------------------------------------------------------------------------------- As mentioned at the end of Section II.A, at the final end z = L of the fiber connection, the field received by the PIN photodetector is given by the sum ETX(t, L) + ERB2(t, L), with the first addend given by with τ = τL = L/vg, and the second one given by: $$\begin{aligned} &&E\_{RB2}(t,L) =\frac{\sqrt{R\_{l,2}(1-R\_{l,2})}\cdot \eta\_c^{3/2}}{\sqrt{\alpha\_{iso}}} \times \nonumber\\ &&\times \sum\_{k=1}^K \rho \_k {{E}}(t-2 \tau \_k-\tau\_L)e^{-\alpha v\_g (2 \tau \_k+\tau\_L)} \label{eq:E\_RB2\_L}\end{aligned}$$ The current generated by the photo-detector can then be computed as: $$\begin{aligned} &i\_{out}(t)&=\mathcal{R}\left\|E\_{TX}(t,L) + E\_{RB2}(t,L)\right\|^2\nonumber\\ &&=\mathcal{R}\cdot \left\|E\_{TX}(t,L)\right\|^2+\mathcal{R}\cdot\left\|E\_{RB}(t,L)\right\|^2+\nonumber\\ &&+\mathcal{R}\cdot 2\Re {\it \bf e} \{E\_{TX}(t,L)\cdot E\_{RB}^\(t,L)\} \label{eq:i\_out\_00}\end{aligned}$$ where R is the Responsivity of the detector. At the right-end-side of Equation, the first addend would coincide with the total detected current if the RB effect were absent. Moreover, the term ∣ERB(t, L)∣2 is much smaller with respect to the others therefore the second added will be neglected in the following. In order to analyze the RB-induced spurious terms in the current power spectrum, only the last addend at the right hand side of has then to be considered, which will be named as ioutT*X, RB. From Equations (with τ = τL) and, exploiting the prostaferesis formula $\cos(u)-\cos(v)=-2\sin\left( \frac{u-v}{2}\right) \sin\left( \frac{u+v}{2} \right)$ with u = ωi(t − τL) + ∠Mi and v = ωi(t − τL − 2τk) + ∠Mi, its expression results to be: $$\begin{aligned} &i\_{out\_{TX,RB}}(t)\simeq \mathcal{R}\sqrt{\frac{R\_{l,2}}{\alpha\_{iso}}}(1-R\_{l,2})\eta\_c^2\|E(t-\tau\_L)\| \times \label{eq:i\_out\_TX\_RB\_no\_approx}\\ &\times 2\Re {\it \bf e} \left\{ \sum\_{k=1}^{k\_{max}}\rho^\\_k \|E(t- 2\tau\_k -\tau\_L)\| e^{-\alpha v\_g 2\tau\_k+j\omega\_{opt}2\tau\_k} \times \right.\nonumber\\ &\times \left.e^{j\left\{-\sum\_i X\_{i,k}sin\left[\omega\_i(t-\tau\_k-\tau\_L)+\angle M\_i\right] + \phi(t-\tau\_L)-\phi(t-2\tau\_k-\tau\_L)\right\}}\right\} \nonumber\end{aligned}$$ where X*i, k has been defined in. ![Qualitative representation of the behavior of PSD_{TX,RB} (\omega) in the vicinity of \omega= 2 \omega_{RF}, represented by with n_{RF}=2, for the k-th elementary RB reflection, when \omega_{d} is sufficiently greater than \Delta_{\Omega}. In this case the power of the second-harmonic distortion term P_{2\omega_{RF}} practically consists in the only element corresponding to n_{d}=0.](Figures/Fig4.png "fig:") [fig:PSD1] In line with the perturbative approach adopted, in both field modules can be assumed to be equal to E0. Collecting all the constants in a unique term $\Upsilon=\mathcal{R}\sqrt{\frac{R\_{l,2}}{\alpha\_{iso}}}(1-R\_{l,2})\eta\_c^2E\_0^2$, and exploiting the Jacobi-Anger expansion e− jusin(v) = ∑n = − ∞∞Jn(u)e− jnv, where u = Xi, k and v = ωi(t − τL − τk) + ∠Mi, taking advantage of and, after a lengthy but straightforward derivation, the expected value of the power spectral density PSDTX, RB of ioutTX, RB(t) assumes the following expression: $$\begin{aligned} \label{eq:PSD\_TX\_RB} & PSD\_{TX,RB} (\omega)= \nonumber \\ &={F\_{\xi}} \left \{ \mathbb{E} \left[ \left \langle i\_{out\_{TX,RB}}(t)i\_{out\_{TX,RB}}^{\ast}(t-\xi) \right \rangle \right] \right\} \simeq \nonumber \\ &\simeq \Upsilon^2 \sum\_{k=1}^{k\_{max}}{\sigma^2\_{\rho\_k}} e^{-4\alpha v\_g \tau\_k} \sum\_{\substack{n\_{d},n\_{RF}=\\=-\infty}}^{+\infty} {J\_{n\_{d}}^2(X\_{d,k}) J\_{n\_{RF}}^2(X\_{RF,k})} \times \nonumber \\ &\times \frac{2\Delta\_{\Omega}}{\Delta\_{\Omega}^2+[\omega-(n\_{d}\omega\_{d}+n\_{RF}\omega\_{RF})]^2}\end{aligned}$$ Focusing for example in the vicinity of ω = 2ωRF, i.e. considering nRF = 2 in, and referring to the generic contribution related to the k − th elementary RB reflection, the function PSDTX, RB(ω), as can be appreciated in Fig. [fig:PSD1], consists in the superposition of lorentzian functions centered in 2ωRF, 2ωRF ± ωd, 2ωRF ± 2ωd, … with amplitudes proportional respectively to J22(XRF, k)J02(Xd, k), J22(XRF, k)J± 12(Xd, k), J22(XRF, k)J± 22(Xd, k), .... The power of the undesired second-harmonic distortion term P2ωRF within an elementary bandwidth δΩ centered in 2ωRF can be directly evaluated as the integral of the Power Spectral Density multiplied by the load resistance RL: P2ωRF = RL ⋅ ∫2ωRF − δΩ/22ωRF + δΩ/2PSDTX, RB(ω)dω and results to be given by the last side of multiplied by δΩ. Note that if the dithering tone were absent, the contribution to P2ωRF coming from each elementary RB reflection would be just proportional to J22(XRF, k) and its impact could be reduced increasing ∣MRF∣, i.e. appropriately operating on ωRF and IRF, operations which would all result as not realizable, e.g. in the case where the modulating signal consists in an uncontrollable RFI disturbance. On the contrary, in presence of the dithering tone (see again the case represented in Fig. [fig:PSD1]) the contribution to P2ωRF coming from each elementary RB reflection is proportional to J22(XRF, k)J02(Xd, k). In this case, it is possible to reduce its impact reducing the value of J02(Xd, k), i.e. increasing ∣Md∣, and the previous Section II-B has been just devoted to determine how to obtain high values of ∣Md∣ while keeping Id and fd (or equivalently ωd) at their lowest possible levels. However, continuing to decrease ωd, the “tails” of the higher order terms of the sum in nd of (i.e. those corresponding to nd = ± 1, ± 2, …) result to contribute to the term at frequency 2ωRF and consequently prevent the reduction of the power at 2ωRF which is pursued with the dithering tone introduction. The phenomenon is qualitatively represented in Fig. [fig:PSD2] and its effects are illustrated in Fig. [fig:finalcomparison], which shows the simulated behavior of the values of P2ωRF normalized to its maximum, named in the figure P2ωRF, norm, covering a wide range of dithering frequencies, i.e, fd ∈ [100 − 4 × 106] Hz. The value of ωd cannot be reduced at will, for the insurgence of this additional phenomenon and an optimum working point can be identified for the application of the dithering tone technique through which the minimum value of Id can be applied t guarantee a given low level of P2ωRF, while keeping fd as low as possible. The considerations just developed with reference to the second harmonic term can be straightforwardly applied to the other spurious frequency terms (e.g. at frequencies 3fRF,fRF + fd, etc...). ![Same as Fig.[fig:PSD1] with the difference that \omega_{d} has been so decreased that the tails of the elements corresponding to n_{d} \not = 0 of the sum in n_{d} of contribute to the term at frequency 2f_{RF}.](Figures/Fig5.png "fig:") [fig:PSD2] ![Simulated behavior of the values of current I_i and frequency f_i which gives the normalized second-harmonic power P_{2\omega_{RF},norm}=cost. Both current and frequency axis are plotted in log-scale for a better view of the whole behavior.](Figures/Fig6.png "fig:") [fig:finalcomparison] Experimental Measurements and discussion ======================================== With the aim of confirm the theoretical results derived in previous section, measurements have been performed using the setup illustrated in Figure [fig:setup]. ![Experimental setup utilized. The setup is composed of two Signal Generators (SG1 and SG2), an RF passive coupler, a Distributed FeedBack Laser Diode (DFB-LD), a span of G.652 fiber, a Positive-Intrinsic-Negative PhotoDiode (PIN-PD) and a Spectrum Analyzer.](Figures/Fig7.png "fig:")[fig:setup] [hbpt] \| Symbol \| Physical meaning \| Value \| \| --- \| --- \| --- \| \| Ith \| Laser threshold current \| 9 mA \| \| Ibias \| Laser bias current \| 37 mA \| \| Popt \| Laser output power \| 6 dBm \| \| αiso \| Laser power isolation \| 20 dB \| \| ΔΩ \| Laser Coherence angular frequency band \| $3.14\cdot 10^6 \frac{rad}{s}$ \| \| α \| Fiber field attenuation factor \| 5− 5 $\frac{neper}{m}$ \| \| ηc \| Power coupling lens coefficient \| 0.4 \| \| S \| Fiber Recapture factor \| 10− 3 \| \| R \| PD Responsivity \| 1 $\frac{A}{W}$ \| \| RL \| Load Resistance \| 50 Ω \| [tab:parameters] The RoF link under test is composed of a DFB laser working at 1310nm connected to a span of G.652 fiber which is directly connected to a PIN Photodiode biased at VPD = 5 Volts. The characteristics of laser and photodiode are reported in Table [tab:parameters]. Two separate signal generators (SGs) are used to supply the laser. In particular, SG1 is used to supply the bias current Ibias and the dithering tone id(t) where both amplitude Id and frequency fd have been varied. The SG2 instead, is used to generate the RF tone iRF(t) at the frequency fRF = 70 MHz, simulating then a possible RFI falling in an interval of interest for low-frequency radioastronomic facilities. The three currents are then coupled using a passive coupler which is connected directly to the Laser. Indeed, in order to set the desired amplitudes of the three currents, the insertion loss of the coupler has been taken into account so that in the following discussion the values of Ibias and the amplitudes of id(t), iRF(t) are considered ones injected into the laser, i.e. after the coupler. After propagating in a standard ITU-T G.652 optical fiber of length L = 5 km, which is in the order of the maximum distances covered in SKAlow, the signal is detected by photodiode and through a spectrum analyzer the amplitude levels of the spurious tones generated at frequencies 2fRF and fRF + fd are evaluated. [Fig:confronti] .5 [fig:MeasHD2] .5 [fig:MeasIMD2] .5 [fig:simHD2] .5 [fig:simIMD2] As a preliminary test, the values of the dithering current amplitude Id which lead to a maximum of the intermodulation term at frequency ωRF + ωd, PωRF + ωd have been measured for varying values of fd. The behavior of this spurious component is described by considering nd = 1, nRF = 1 and it can be shown that the value of current Id which gives the maximum of PωRF + ωd is obtained when Md ≃ 0.92, which, with good approximation, corresponds to the value which gives the maximum of ∑k = 1kmaxJ12(Xd, k). Figure [Fig:confronti] shows the good agreement between simulated and measured values of Id. Note that this analysis have been performed considering fd = 1 kHz as the lowest value of dithering frequency. The reason behind this choice is that for lower frequencies it is not possible to identify experimentally the maximum of PωRF + ωd. After this preliminary test, an analysis of the behavior of the terms P2ωRF and PωRF + ωd as a function of Id and fd has been experimentally performed. The measurements have been conducted by fixing the value of certain frequency fd and sweeping the current Id for a range [0.001 − 0.1] mA. This sweeping has been then repeated for each frequency fd of interest, generating in this way the maps shown in Figures [fig:MeasHD2] and [fig:MeasIMD2]. Those maps represent the normalized values of P2ωRF and PωRF + ωd, named in the figures P2ωRF, norm and PωRF + ωd, norm respectively, measured at the output of the photodetector through the spectrum analyzer (see again Fig.[fig:setup]). An analogous map has been performed also by implementing the model shown in previous section, and the results are represented in Fig. [fig:simHD2] and [fig:simIMD2]. The maps show the different levels of harmonic and intermodulation distortion which offer a good reproduction of the experimental results, confirming the correctness of the theoretical study performed. In particular, both simulations and experimental results show that for the considered link, a frequency fd ≃ 10 kHz and an amplitude Id ≃ 0.04 mA constitute the optimum working point which allows to reduce the powers of both second harmonic distortion term P2ωRF and second order intermodulation product PωRF + ωd by at least 25 dB referred to their respective maximum values. Conclusion ========== An exhaustive study of the detrimental nonlinear effects produced by the Rayleigh Backscattering present within Analog Radio-over-Fiber link, and of the technique used to mitigate them, namely the dithering technique, has been shown. As a first important and novel aspect, a detailed description of the behavior of a directly modulated laser in the presence of optical feedback due to Rayleigh Backscattering has been presented, showing the significant changes produced on the laser spurious phase modulation (i.e. frequency chirp) characteristics and on the power spectrum of the signal received at the end of the fiber connection. Then, it has been shown that these changes impact significantly on the optimum characteristics of current and frequency of the sinusoidal dithering tone. The technique showed to be particularly effective on the typical scenarios of low frequency radioastronomic systems (for example, it has been applied in the demonstrators of SKA-low), which suffers to a relatively great degree the impact of RB due to their peculiar characteristics. Derivation of ΓS, Γθ ------------------------ The rate equation of the complex electrical field E(t) in presence of RBS can be written as follows : $$\begin{aligned} &\frac{dE(t)}{dt}=\label{eq:Kobayashi\_Rayleigh}\\ &=\left[j\omega(N) + \frac{1}{2}\left(G(N,S) -\frac{1}{\tau\_p}\right)\right]E(t)+\sum\_{k=1}^{K}C\_kE(t-2\tau\_k)\nonumber\end{aligned}$$ where the last term on the right-hand-side represents the RBS contribution. Putting E(t) = ∣E(t)∣ej[ω0t + θ(t)] and creating two equations, one for the real part and one for the imaginary part it results: $$\begin{aligned} && \|E(t)\|\frac{d\|E(t)\|}{dt}= \frac{1}{2}\left(G(N,S) -\frac{1}{\tau\_p}\right)\|E(t)\|^2+ \nonumber \\ && = +\sum\_{k=1}^K\|C\_k\| \|E(t)\|\|E(t-2\tau\_k)\|\cdot\nonumber\\ &&\cdot\cos[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:app1\_1}\end{aligned}$$ and: $$\begin{aligned} && \frac{d\theta(t)}{dt}= \left[\omega(N) -\omega\_0 \right]-\sum\_{k=1}^K\|C\_k\| \frac{\|E(t-2\tau\_k)\|}{\|E(t)\|}\cdot\nonumber\\ &&\cdot\sin[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:app1\_2}\end{aligned}$$ Exploiting the relationship S(⋯) = ∣E(⋯)∣2, $\|E(t)\|\frac{d\|E(\cdots)\|}{dt}=\frac{1}{2}\frac{dS(\cdots)}{dt}$ the expressions given by, can be obtained Derivation of γS, i, γθ, i -------------------------------------- Starting from and exploiting,, it is: $$\begin{aligned} &&\Gamma\_S(t)=2\sum\_{k=1}^K\|C\_k\| \left( \left[S\_0+\Delta S +S\_i\cos(\omega\_i t +\angle S\_i)\right]\cdot \right. \nonumber \\ && \left. \cdot\left[S\_0+\Delta S +\|S\_i\| \cos(\omega\_i (t-2\tau\_k) +\angle S\_i)\right]\right)^{\frac{1}{2}} \cdot\nonumber\\ &&\cdot\cos\left[(\omega\_{0}+\Delta \omega)\cdot 2\tau\_k -\angle \rho\_k + \right. \nonumber \\ &&\left. -\|M\_i\|\cdot 2\sin(\omega\_i 2\tau\_k)\sin\left(\omega\_i(t-2\tau\_k)+\angle M\_i\right)\right] \label{eq:gamma\_Si\_1}\end{aligned}$$ From assumption the terms ΔS and Si can be neglected under the square root. Putting then ψk = (ω0 + Δω) ⋅ 2τk − ∠ρk and exploiting the truncated Jacobi-Anger expansion e− jusin(v) = J0(u) + 2J1(u)sin(v), with u = 2∣Mi∣sin(ωiτk) and v = sin(ωi(t − τk) + ∠Mi), in order to focus on the terms at DC and at ωi, it results: $$\begin{aligned} &&\Gamma\_S(t)=2 S\_0\sum\_{k=1}^K\|C\_k\| \cos(\psi\_k)\cdot J\_0\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right]+ \nonumber \\ && + 2 S\_0 \sum\_{k=1}^K\|C\_k\|\sin(\psi\_k)\cdot \nonumber \\ &&\cdot 2J\_1\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right] \cdot \sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right] \label{eq:gamma\_Si\_2}\end{aligned}$$ Putting now Xi = 2∣Mi∣sin(ωiτk) and elaborating further the term written in the last line of it results: $$\begin{aligned} && 2J\_1\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right] \cdot \sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right]= \nonumber \\ &&= \frac{2 J\_1(X\_i)}{X\_i} \cdot 2\|M\_i\|\sin(\omega\_i \tau\_k) sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right] \nonumber \\ &&= \Re {\it \bf e} \left\{ 2\frac{J\_1(X\_i)}{X\_i} \cdot 2\|M\_i\| \frac{e^{j\omega\_i\tau\_k}-e^{-j\omega\_i\tau\_k}}{2j} \right.\cdot \nonumber \\ && \left. \cdot (-j)e^{j\left[\omega\_i(t-\tau\_k)+\angle M\_i\right]}\right\} \nonumber \\ &&= \Re {\it \bf e} \left\{ -2\frac{J\_1(X\_i)}{X\_i} \cdot \left[1-e^{-j2\omega\_i\tau\_k} \right] \right.\cdot \nonumber \\ && \left. \cdot M\_ie^{j\left[\omega\_i t\right]}\right\} \label{eq:gamma\_Si\_3}\end{aligned}$$ Therefore, the second addend of ΓS(t) reported in the two last lines of, which contributes to the system at angular frequency ωi in becomes: $$\begin{aligned} && - 2 S\_0\sum\_{k=1}^K\|C\_k\| \sin(\psi\_k)\cdot \frac{2 J\_1(X\_i)}{X\_i} \cdot \left[1-e^{-j2\omega\_i\tau\_k} \right] \cdot \nonumber \\ && \cdot M\_ie^{j\omega\_i t}= -2 S\_0 \cdot \gamma\_{S,i} \cdot M\_ie^{j\omega\_i t} \label{eq:gamma\_Si\_4}\end{aligned}$$ from which the expression of γS, i can be recognized. Proceeding in a similar fashion with Γθ, starting from and exploiting, and the condition, it is: $$\begin{aligned} && \Gamma\_{\theta}= -\sum\_{k=1}^K\|C\_k\| \cos(\psi\_k)\cdot 2J\_1\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right] \cdot \nonumber \\ && \cdot \sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right] + \nonumber \\ && +\sum\_{k=1}^K\|C\_k\|\sin(\psi\_k)\cdot J\_0\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right] \label{eq:gamma\_thetai\_1}\end{aligned}$$ The first addend of Γθ(t), reported in the first two lines of, contributes to the system at angular frequency ωi in and can be written as: $$\begin{aligned} && \sum\_{k=1}^K\|C\_k\| \cos(\psi\_k)\cdot \frac{2 J\_1(X\_i)}{X\_i} \cdot \left[1-e^{-j2\omega\_i\tau\_k} \right] \cdot \nonumber \\ && \cdot M\_ie^{j\omega\_i t}= \gamma\_{\theta,i} \cdot M\_ie^{j\omega\_i t} \label{eq:gamma\_theta\_2}\end{aligned}$$ and again, from the expression of γθ, i can be recognized. --- 1. J. Nanni, A. Giovannini and G. Tartarini are with the Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione “Guglielmo Marconi”, Università di Bologna, 40136 Bologna (BO), Italy (e-mail: [email protected]; [email protected]; [email protected]).[↩](#fnref1) 2. S. Rusticelli is with 3PSystem s.r.l, via Riviera delBrenta 170, 30032 Fiesso d’Artico (VE), Italy (e-mail:[email protected]).[↩](#fnref2) 3. F. Perini, J. Monari are with Institute of Radio Astronomy, National Institute for Astrophysics, Via Fiorentina 3513, 40059 Medicina (BO), Italy (e-mail: [email protected];[email protected];[email protected]).[↩](#fnref3) 4. E. Lenzi is with RF Optics di Enrico Lenzi, Via Sopra Castello 21, 40061, Minerbio (BO), Italy (e-mail:[email protected]).[↩](#fnref4) 5. R. Wayth is with ICRAR/Curtin University, Bentley Western Australia, 6102 (e-mail:[email protected]).[↩](#fnref5) Optimal Dithering Configuration Mitigating Rayleigh-Backscattering-Induced Distortion in Radioastronomic Optical Fiber Systems ================================================================================================================================ In the context of Radioastronomic applications where the Analog Radio-over-Fiber technology is used for the antenna downlink, detrimental nonlinearity effects arise because of the interference between the forward signal generated by the laser and the Rayleigh backscattered one which is re-forwarded by the laser itself toward the photodetector. The adoption of the so called dithering technique, which involves the direct modulation of the laser with a sinusoidal tone and takes advantage of the laser chirping phenomenon, has been proved to reduce such Rayleigh Back Scattering - induced nonlinearities. The frequency and the amplitude of the dithering tone should both be as low as possible, in order to avoid undesired collateral effects on the received spectrum as well as keep at low levels the global energy consumption. Through a comprehensive analysis of dithered Radio over Fiber systems, it is demonstrated that a progressive reduction of the dithering tone frequency affects in a peculiar fashion both the chirping characteristics of the field emitted by the laser and the spectrum pattern of the received signal at the fiber end. Accounting for the concurrent effects caused by such phenomena, optimal operating conditions are identified for the implementation of the dithering tone technique in radioastronomic systems. Nonlinearities; Rayleigh backscattering; Radioastronomy, RoF, Laser feedback, Dithering. Introduction ============ which adopt the Radio-over Fiber (RoF) technology in one of its most cost-efficient versions consist essentially in a radiofrequency (RF) signal which performs the direct intensity modulation (D-IM) of a laser source, propagates in a span of Standard Single Mode Fiber (SSMF) and is directly detected (DD) at the receiver end (D-IMDD RoF systems). When the described scheme is utilized within Radioastronomic scenarios, peculiar features have to be considered, which distinguish these systems e.g. from those which apply the D-IMDD RoF technology to the mobile network. One of them is related to the frequencies of the transmitted RF signals, which in the case of the mobile network (4G, 5G signals and beyond) can reach several tens of GHz, and do not fall below around 700MHz, while within Radioastronomic application can also belong to intervals ranging from few tens to few hundreds MHz. In addition to this, while RoF systems for the mobile network are typically designed assuming that at the optical transmitter’s side the RF powers of the signals range roughly in an interval of ± few units of dBm, in the case of signals received by radioastronomic antennas these powers can exhibit values of just -60dBm when they come from astronomical sources, while they arrive to a maximum of around -20dBm when they consist in undesired Radio Frequency Interference (RFI) signals coming from satellites, radio and/or television transmitters, etc.. As a consequence of the low power levels of the signals which directly modulate the laser source, in many RoF D-IMDD systems utilized for radioastronomic applications it is relatively straightforward to reduce to negligible values the nonlinearities which typically bring detriment to RoF systems in the mobile networks. Indeed, the spurious frequencies caused by the nonlinearity of the laser Power-Current curves typically in this case fall below the noise floor. Moreover, it is enough to operate in the vicinity of the second optical window (around the wavelength λ = 1310nm), to guarantee that, due to the low level of the RF power modulating the laser, the creation of nonlinearities generated by the combined effect of laser chirp and fiber chromatic dispersion is avoided. It has however been shown that in this kind of systems a primary cause of undesired nonlinear effects takes place when a generic RFI signal (with frequency and angular frequency given respectively by fRF and ωRF = 2πfRF) is transmitted in the RoF System together with the signals coming from sky sources, after that they have all been received by the Radioastronomic antenna. Indeed, the portion of such RFI signal which is reflected by Rayleigh Backscattering (RB) reaches the laser section, is partly reflected again and interacts at the receiver end with the transmitted RFI signal itself, generating nonlinear frequency terms (e.g. at frequencies 2fRF, 3fRF, ....). Although the RFI disturbance at frequency fRF can be filtered out at the signal post-processing stage, the same cannot be done for all generated spurious frequency terms. One countermeasure to the described problem could be represented by the introduction of a further optical isolator at the fiber input section, which would be added to the one embedded in the laser. However this solution would hardly meet the cost limitations (e.g. possibly 100USD or less for a single front-end receiver) imposed by the very high number of RoF downlinks which should be realized within radioastronomic facilities operating at frequencies of tens/hundreds of MHz. In a previous work, an appropriate cost-effective solution has been demonstrated, consisting in the introduction of a dithering tone as a further modulation current for the optical transmitter, which pushes the level of the nonlinear terms below the noise floor. This solution had actually already been proposed in the past, within telecommunication scenarios, with the aim to reduce noise effects in optical fiber systems,, and its application finalized to the reduction of the spurious terms constitutes an additional newly-evidenced advantage offered by this technique. In the application of the mentioned dithering tone technique it would however be desirable that both the frequency fd (or angular frequency ωd = 2πfd) and the amplitude Id of the current tone exhibit the lowest possible values. The first requirement is related to the fact that the distance from fRF of the possibly generated spurious frequency terms, fRF ± fd, fRF ± 2fd, ... should be lower than the resolution bandwidth of the reception filter, so that they can be removed from the received signal together with the RFI term at frequency fRF. The second requirement comes instead from the necessity to keep at low levels the energy consumption of the tone generator. To assist the designer in identifying the appropriate solution to the problem, a theoretical model of the laser behavior under current modulation in presence of optical feedback by RB is necessary. However, the models developed so far describing the effects produced by optical feedback under laser current modulation, are not referred to RB, since they consider only the reflection coming from a single external cavity, typically put on the back-facet of the laser. At the same time, the effects produced by RB feedback have been considered only with reference to the linewidth reduction and optical frequency shift and hopping, in all cases for continuous-wave operation, without considering the presence of a modulating current. In the present paper, a rigorous mathematical model based on the laser Rate Equations is developed to describe the effects of the feedback due to Rayleigh Backscattering on an optical field modulated by an RF signal. The model, which will be shown to be in agreement with experimental measurements, is utilized to describe the effects of the introduction of the dithering tone and to identify the optimal working point which allows keep both Id and ωd as low as possible. The paper is organized as follows: Section II provides a description of the model developed to study the impact of RB on the characteristics of the field emitted by a directly modulated laser. Section III describes the experimental results, which confirm the correctness of the theoretical model adopted and allow to identify the optimal operating conditions for the implementation of the dithering tone technique. Finally, conclusions will be drawn. Theoretical Model ================= The theoretical model presented in this work is based on the physical structure shown in Figure [fig:initialscheme], composed by a laser-based optical transmitter whose emitted field is coupled through a focusing lens and an isolator to an optical fiber and reaches finally an optical receiver based on a PIN photodetector. ![Scheme of the structure considered.](Figures/Fig1.png "fig:") [fig:initialscheme] The laser cavity, exhibits a length Lcav and power reflectivities at the facets Rl, 1 and Rl, 2, respectively. The injection current Iinj(t) can be written in complex notation as Iinj(t) = Ibias + ∑iIiejωit, where Ibias represents the bias current, and Ii represents the amplitude of the generic i − th sinusoidal RF tone which directly modulates the laser, with angular frequency ωi and frequency fi = ωi/(2π). For formal simplicity and without loss of generality, in the following only two modulating tones will be considered, namely a term with amplitude IRF and angular frequency ωRF = 2πfRF, and the so called dithering tone with amplitude Id and angular frequency ωd = 2πfd, resulting in: Iinj(t) = Ibias + Idejωdt + IRFejωRFt For this reason, instead of ∑i = d, RF…, the simplified notation ∑i… will be adopted throughout the paper. In principle, the electrical field emitted by the laser source and propagating into the fiber should be represented in vector form, to account for the evolution of its state of polarization (SOP). However, the phenomenon of Rayleigh backscattering, which takes place within the fiber in correspondence to the reflection sections zk, k = 1, …, K (see again Figure [fig:initialscheme]) can be regarded as substantially isotropic, while the change of the SOP of the field between zk and an adjacent section can be described by an appropriate Jones matrix, which is unitary. Consequently, it can be shown that for a given initial SOP and in absence of Polarization Dependent Losses (PDL), the electrical field exhibits at the fiber end section a corresponding final SOP on which the quantities which are of interest in the present work (e.g. powers received at different frequencies) do not depend. In presence of Polarization Dependent Losses (PDL), the quantities which are of interest in the present work can instead exhibit variations depending on the final SOP of the electric field. This could be appreciated for example in presence of changes of environmental quantities (e.g. temperature). Indeed, while it can be reasonably assumed that the devices located within the transmitter (single-mode laser, lens, isolator) still determine a stable SOP for the emitted field, the optical fiber can undergo local birefringence perturbations which can change in real time the SOP of the field impinging the detecting area of the PIN. In presence of PDL of this last device, the received powers would then exhibit a polarization-dependent fluctuating behavior. However the relative variations of the measured quantities can be typically considered as low (e.g.few tenths of dB or less), and, at the same time, on fiber connections of the order of few km length, like the ones considered in this work, they can be appreciated on time scales (e.g. tens of seconds or more) which are much longer than the ones of the phenomena investigated. For all these reasons, in the rest of the paper the electrical field will be described as a scalar quantity without losing the generality of the results that will be obtained. Re-transmitted and re-injected portions of the Rayleigh Backscattered field --------------------------------------------------------------------------- By virtue of the considerations developed above, the electrical field generated inside the laser can be expressed as E(t) = ∣E(t)∣ej[ω0t + θ(t)]ejϕ(t) where ∣E(t)∣ is its module, ω0 the optical angular frequency that the field would exhibit in absence of modulating tones and in absence of RB as well, θ(t) is the time-varying phase due to their presence, while ϕ(t) represents the laser phase noise contribution. The expressions of ∣E(t)∣ and θ(t) are given by: $$\begin{aligned} &&\|E(t)\|= E\_0\sqrt{1+ \zeta\_{RB}+\sum\_i m\_icos(\omega\_it)} \label{eq:E}\\ &&\theta(t)=\Delta \omega t\,\, +\sum\_{i} \|M\_i\|cos(\omega\_{i}t+ \angle M\_i) \label{eq:theta}\end{aligned}$$ In $E\_0 =\sqrt{\eta (I\_{bias}-I\_{th})}$ is the electrical field amplitude, with Ith threshold current and η power-current slope efficiency of the laser, ζRB represents a small increment to the unit number under square root, which derives from the presence of RB and will be further described in the next subsection, while mi = Ii/(Ibias − Ith) represents the optical modulation index at the considered frequency. In Δω represents a variation of the angular frequency due to RB and to direct modulation, whose presence determines the total optical angular frequency of the emitted field to be ωopt = ω0 + Δω. Moreover, the quantities ∣Mi∣ and ∠Mi are respectively module and phase of the phase modulation index due to the chirp effect at ωi. In the quantity ejϕ(t) is a random function whose Power Spectral Density, determined as Fourier Transform of its autocorrelation function, is given by: $$\mathcal{F\_{\xi}} \left\{ \left\langle e^{j\phi(t)-\phi(t-\xi)} \right \rangle \right\}= \mathcal{F\_{\xi}} \left\{ e^{-\|\xi\|\Delta\_{\Omega}} \right\} =\frac{2\Delta\_{\Omega}}{\Delta\_{\Omega}^2+\omega^2} \label{eq:phase\_noise}$$ where $\mathcal{F\_{\xi}}\{\cdots\}$, ⟨⋯⟩ indicate respectively the operations of Fourier Transformation with respect to the variable ξ, and time averaging. The quantity ΔΩ = 1/τcoh is the coherence angular frequency band of the laser source, reciprocal of its coherence time τcoh. Still referring to Fig. [fig:initialscheme], the field given by exits the laser right-hand-side facet, is focused by a coupling lens into an isolator, and is subsequently injected at the input section (z = 0) of the optical fiber of length L, propagating with a group velocity vg. Putting τ = z/vg, at the generic section z ∈ [0, L] of the fiber the resulting electric field can then be written as: $$\begin{aligned} &&E\_{TX}(t,z) =\sqrt{1-R\_{l,2}}\cdot \sqrt{\eta\_c} \cdot {{E}}(t-\tau) e^{-\alpha v\_g \tau} \label{eq:fiber\_field}\end{aligned}$$ where ηc is the power coupling coefficient between the laser and the optical fiber given by the coupling lens, while α is the attenuation constant of the fundamental mode of the optical fiber. As above mentioned, as ETX propagates along the fiber, the phenomenon of Rayleigh backscattering takes place. The inhomogeneity of the longitudinal profile of the refractive index of the optical fiber determines indeed the presence of many reflection points at the coordinates zk, k = 1, …, K, each one of which generates a back reflected electrical field. At the output facet of the laser cavity a Rayleigh Backscattered field is then present, given by: $$\label{eq:RB field of} E\_{RB}(t) =\frac{\sqrt{1-R\_{l,2}}\cdot \eta\_c}{\sqrt{\alpha\_{iso}}}\sum\_{k=1}^K \rho \_k {{E}}(t-2 \tau \_k)e^{-\alpha v\_g 2 \tau \_k}$$ where αiso is the isolator power attenuation in the backward direction, while τk = zk/vg. Then, ρk = ∣ρk∣ej∠ρk is the complex reflection coefficient at the k − th section due to RB. Note that the weak-feedback approximation, i.e. ∣ρk∣ < < 1, is taken, meaning that possible multiple reflections within a single fiber section are not considered. The different ρk’s are assumed to be complex random variables, with zero mean value and Gaussian distribution. The variance of both their real and imaginary parts is $\sigma^2\_{\rho}=\frac{1}{2} \alpha\_s \mathcal{S} d z\_k$, where S is the so-called backscattering factor or recapture factor, which depends on the characteristics of the fiber considered and exhibits typical values of the order of 10− 3 in case of the standard G.652 fiber, αs is the Rayleigh attenuation coefficient, which for the considered wavelengths can be assumed to coincide with α, and dzk = zk − zk − 1 is the length of the interval considered. The ρk’s are also assumed to be delta-correlated, namely: E[ρkρh\] = α*sS dzkδkh where E[ ⋅ ] represents the expected value operator, while δkh is the delta Kronecker function. Still referring to Fig. [fig:initialscheme], a portion of the field reported in, given by $E\_{RB1}(t)=\sqrt{1-R\_{l,2}}E\_{RB}(t)$, is re-injected into the laser cavity. As will be shown in Subsection [sub:rateeq], this fact has significant consequences on the phase modulation index due to the chirp effect Md at ωd, which also influences the identification of the optimal frequency and amplitude characteristics of the dithering tone itself. Another portion of this field, given by $E\_{RB2}(t,z)=\sqrt{R\_{l,2}}E\_{RB}(t,z)$ is instead reflected by the laser facet. As will be shown in Subsection [sub:finalspectrum], this reflected field adds to the one given by in input to the optical fiber, contributing to the final form of the power spectral density of the photocurrent detected at the receiving end of the system. The knowledge of the behavior of this quantity allows to put into evidence particular physical aspects which also contribute to determine the optimal frequency and amplitude characteristics of the dithering tone to be utilized. Re-injected portion of the RB field and effect on the Chirp term ---------------------------------------------------------------- The Rate Equations, expressed below in the so called Semi Classical form , describe the behaviors of S(t) = ∣E(t)∣2, θ(t) and N(t), which represent respectively the number of photons in the laser cavity, the previously introduced time varying phase θ(t) and the carrier density N(t), expressed in $\left[\frac{1}{m^3}\right]$: $$\begin{aligned} \frac{d S(t)}{dt}&=\left(G(N,S)-\frac{1}{\tau\_p}\right)S(t)+ \Gamma\_S(t)\label{eq:S}\\ \nonumber\\ \frac{d\theta(t)}{dt}&=\frac{\alpha\_H}{2}G\_N(N(t)-N\_{th})-\Gamma\_{\theta}(t)\label{eq:phi}\\ \nonumber\\ \frac{dN(t)}{dt} &= \frac{I\_{inj}(t)}{eV}-\frac{N(t)}{\tau\_s}-\frac{G(N,S)}{V}S(t)\label{eq:N}\end{aligned}$$ At the second side of () the quantity G(N, S) is the laser cavity gain which depends both from N, and S through the relation: $$\label{eq:gain} G(N,S) = \frac{g\_0}{1+\epsilon S}(N-N\_{tr})\simeq g\_0\cdot(1-\epsilon S)(N-N\_{tr})$$ where the transparency level Ntr is the density of carriers in the cavity for which G(N, S) = 0, ε < < 1/S is the gain compression factor and g0 represents the nominal gain-slope in absence of compression, multiplied by the volume of the laser cavity. The quantity τp represents the average photon’s lifetime, or inverse of the cavity losses. The term ΓS(t) at the right-hand-side of, which represents the RB contribution to dS(t)/dt, is given by: $$\begin{aligned} &&\Gamma\_S(t)=2\sum\_{k=1}^K\|C\_k\| \sqrt{S(t)S(t-2\tau\_k)}\cdot\nonumber\\ &&\cdot\cos[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:Gamma\_S}\end{aligned}$$ and will be derived in Appendix [app:1]. Within, Ck is the k-th compound-cavity coefficient defined as follows: $$C\_k=\frac{(1-R\_{l,2})\eta\_c}{\sqrt{\alpha\_{iso}}}\rho\_ke^{-\alpha v\_g 2 \tau\_k}\frac{1}{{2}\tau\_{cav}}\frac{1}{\sqrt{R\_{l,2}}}$$ where τcav = 2 ⋅ ncavLcav/c is the round-trip time of the laser cavity with refractive index ncav. By solving the laser rate equations in the reference steady state case where both external RF modulation and RB feedback are absent, it results that the carrier density is equal to the amount Nth which balances the cavity losses, and it results also that number of photons is $S\_0= \frac{\tau\_p}{e}\left(I\_{bias}-I\_{th}\right)$ with $I\_{th}=N\_{th}\frac{eV}{\tau\_s}$, where the relationship τp = 1/G(Nth, S0) has been exploited. At the second side of ([eq:phi]) αH is the linewidth enhancement factor of the laser source while it is $G\_N = \left. \frac{\partial G}{\partial N}\right\|\_{S\_0}=g\_0(1-\epsilon S\_0)$. Similarly to ([eq:S]), the last term at the right-hand-side of ([eq:phi]) represents the RB contribution to dθ(t)/dt: $$\begin{aligned} &&\Gamma\_{\theta}=\sum\_{k=1}^K\|C\_k\|\sqrt{\frac{S(t-2\tau\_k)}{S(t)}}\cdot \nonumber \\ &&\sin[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:Gamma\_PHI}\end{aligned}$$ and will also be derived in Appendix [app:1]. Finally, at the second side of ([eq:N]) the quantities e, V, τs represent respectively the electron charge, the volume of the active region and the carrier-lifetime. Assuming that partial RB field re-injection and RF modulation of the injection current determine only a perturbative effect on the laser emission characteristics, the behavior of G(N, S) can be represented by its expansion to the first order: G(N, S) ≃ G(Nth, S0) + GN(N − Nth) + GS(S − S0) where $G\_S = \left. \frac{\partial G}{\partial S}\right\|\_{N\_{th}}=-g\_0 \epsilon (N\_{th}-N\_{tr})$. Within the hypothesis taken, in presence of an injection current Iinj(t) given by the second side of ([eqIinj]) S(t), θ(t) and N(t) exhibit expressions given by: $$\begin{aligned} S(t) &= S\_0 + \Delta S +\sum\_{i} S\_{i}e^{j\omega\_{i}t}\label{eq:delta S}\\ \theta(t) &= \Delta \omega t + \sum\_{i} M\_{i}e^{j\omega\_{i}t}\label{eq:delta theta}\\ N(t) &= N\_{th} + \Delta N +\sum\_{i} N\_{i}e^{j\omega\_{i}t}\label{eq:delta N}\end{aligned}$$ where is re-written in complex notation and where (ΔS, Δω, ΔN), and (Si, Mi, Ni) ∀i, are the unknowns which, within the perturbative regime taken, respect the conditions: $$\begin{aligned} &&\Delta S, \, \|S\_{i}\|\_{\forall i} << S\_0 \label{eq:perturb\_1} \\ && \Delta \omega, \,\|\omega\_{i}M\_{i}\|\_{\forall i} << \omega\_{0} \label{eq:perturb\_2} \\ &&\Delta N, \,\|N\_{i}\|\_{\forall i} << N\_{th} \label{eq:perturb\_3}\end{aligned}$$ In line with,,, it will be assumed that the perturbation caused the generic i − th modulating tone can be determined separately from the one due to the other, namely that the expression of the triplet (Si, Mi, Ni) can be determined just considering Iinj(t) = Ibias + Iiejωit (i = d or i = RF). Inserting the relations,, with the summations composed by one element into,,, and considering the terms at DC and at ωi of the Jacobi-Anger expansions of and (detailed in Appendix [app:2]), it is possible to obtain two systems of three equations. The first one, which is omitted for brevity, represents the steady-state terms and allows to determine ΔS, Δω, ΔN, while the second one represents the modulating terms with angular frequency ωi, which allow to determine Si, Mi, Ni. The latter of the two systems assumes the following form: $$\overline{\overline{A}}\cdot\left(\begin{array}{c} S\_i\\ M\_i\\ N\_i \end{array}\right)=\left(\begin{array}{c} 0\\ 0\\ \frac{I\_i}{eV} \end{array}\right)$$ where the matrix $\overline{\overline{A}}$ is given by: $$\begin{aligned} &\overline{\overline{A}} =\label{eq:system}\\ &\left(\begin{array}{ccc} j\omega\_i-G\_SS\_0&+2S\_0\gamma\_{S,i}&-G\_NS\_0\\ \\ 0&j\omega\_i+\gamma\_{\theta,i}&-\frac{\alpha\_H}{2}G\_N\\ \\ \frac{1}{\tau\_pV}+\frac{G\_S S\_0}{V}&0&j\omega\_i+\frac{1}{\tau\_s}+\frac{G\_N S\_0}{V} \nonumber \end{array}\right)\end{aligned}$$ and where the elements γS, i and γθ, i, located in positions (1, 2) and (2, 2), are derived from ΓS and Γθ, as shown in Appendix [app:2]. As will be evidenced in the next Subsection, the higher values exhibited by the module ∣Md∣ of the phase modulation index due to the chirp effect at ωd, the more effective the dithering technique results in reducing to negligible levels the undesired RB-related nonlinearities. For this reason, the behavior of Mi, which is obtained by solving, will be analyzed in the following. [t] \| Symbol \| Physical meaning \| Order of Magnitude \| \| --- \| --- \| --- \| \| g0 \| Gain slope \| ∼ 10− 12 m3/sec \| \| Ntr \| Carriers Transparency Level \| ∼ 1. ⋅ 1024 m− 3 \| \| Nth \| Carriers Threshold Level \| ∼ 2. ⋅ 1024 m− 3 \| \| ε \| Gain compression factor \| ∼ 10− 8 \| \| τp \| Photon Lifetime \| ∼ 10− 12sec \| \| τs \| Carrier Lifetime \| ∼ 10− 9sec \| \| Rl, 1, Rl, 2 \| Laser mirrors reflectivities \| ∼ 0.3, …, 0.5 \| \| Lcav \| Laser cavity length \| ∼ few 100μm \| \| V \| Volume of Laser active region \| ∼ 10− 16m3 \| \| ncav \| Laser cavity refractive index \| ∼ 3.5, …, 5 \| \| αH \| Linewidth enhancement factor \| ∼ 3, …, 6 \| \| \| Number of photons in steady state \| \| \| S0 \| (assuming for Ibias − Ith \| ∼ 105 \| \| \| a value of few tens mA) \| \| \| GN \| ∂G/∂N in steady state \| ∼ 10− 12 m3 \| \| GS \| ∂G/∂S in steady state \| ∼ 104 \| [laser:parameters] In doing so, some simplifications can be taken, which allow to put this quantity in a more readable form. Indeed, considering modulating frequencies fi which can range from KHz to hundreds MHz, and assuming for Ibias a value such that Ibias − Ith ∼ few tens of mA, taking into account the typical orders of magnitude of Rate Equations parameters reported in Table [laser:parameters], the determinant of $\overline{\overline{A}}$ can be reduced to: $$det \left[ \overline{\overline{A}} \right] \simeq \frac{G\_N S\_0}{\tau\_p V} \left[j \omega\_i+\gamma\_{RB,i} \right] \label{eq:det\_M}$$ where it has been put: $$\begin{aligned} &&\gamma\_{RB,i}=\gamma\_{S,i}-\alpha\_H \gamma\_{\theta,i}\simeq \sqrt{1+\alpha\_H^2}\cdot \sum\_{k=1}^{K} \|C\_k\| \cdot \label{eq:gamma\_RB}\\ &&\cdot \frac{2J\_1(X\_{i,k})}{X\_{i,k}} \left(1-e^{-j\omega\_i2\tau\_k}\right)\cos\left(\psi\_k-\tan^{-1}\alpha\_H\right) \nonumber\end{aligned}$$ with: $$\begin{aligned} &&X\_{i,k}= 2 \|M\_i\|sin(\omega\_i \tau\_k +\angle M\_i) \label{eq:X\_i\_k}\\ && \psi\_k =\omega\_{opt} 2\tau\_k - \angle \rho\_k \label{eq:psi\_k}\end{aligned}$$ The resulting expected value of ∣Mi∣ is then given by the expression: $$\|M\_{i}\| = \mathbb{E}\left[ \frac{\frac{\tau\_p}{e}\frac{\alpha\_H}{2}G\_S\,I\_i} {\sqrt{\omega\_i^2 + \|\gamma\_{RB,i}\|^2 }} \right] \label{eq:theta\_i}$$ in which the statistical expectation can be straightforwardly applied through and utilizing the properties of the ρk’s specified in Section [sub:RB1RB2]. It can be at first noted that the value of ∣Mi∣ are proportional to the amplitude of the modulating current Ii. Secondly, regarding the dependence of the frequency fi (or angular frequency ωi) of the modulating current, it can be first observed that in case the RB effects were absent, the value of ∣Mi∣ would result inversely proportional to ωi. In the considered case, the dependence of ∣Mi∣ on ωi does not instead follow such simple pattern. Solving iteratively (since ∣Mi∣ is contained in γRB, i) it is indeed possible to obtain the curves reported in Fig.[fig:thetaimodule], which have been modelled in accordance with the orders of magnitude of the parameters reported in Table [laser:parameters] and Table [tab:parameters]. At the right hand side of the figure it can be noted that for frequencies greater than few hundred MHz it can effectively be assumed for all the reported curves $\|M\_i\|\simeq \frac{K\_{f,i}I\_i}{f\_i}$ where $K\_{f,i}=\frac{1}{2 \pi}\frac{\tau\_p}{e}\frac{\alpha\_H}{2}G\_S$, is the so called adiabatic chirp factor, expressed in [Hz/A]. Indeed, a constant value ∣Mi∣ = 10n rad (n = 1, …, 6 in the figure) corresponds to the linear relationship $I\_i=\frac{10^n \, rad}{K\_{f,i}}f\_i$. This is the behavior of ∣Mi∣ when i = RF is considered, since the undesired RFI signals which affect radioastronomic plants operating in the lower bandwidth of the radioastronomic spectrum exhibit frequencies of tens to hundreds of MHz. When instead the dithering tone is considered (i = d), it is however convenient to give lower values to fd. For example, a given constant value ∣Md∣ = 103 rad would be obtained with Id ≃ 10 mA if fd ≃ 2 MHz were chosen, but can be obtained with Id ≃ 0.1 mA if fd ≃ 20 kHz. Note that giving low values to fd fulfills also the requirement to keep the terms fRF ± fd, fRF ± 2fd, ... within the resolution bandwidth of the reception filter, as mentioned in the Introduction. Still considering as example the curve referred to the value ∣Md∣ = 103 rad, it can however also be noted that for frequencies lower than few kHz the value of Id to be utilized becomes practically independent from fd. This happens because, as can be shown with a detailed analysis of its composing terms, ∣γRB, d∣ is weakly dependent on the frequency. Going therefore from right to left in the horizontal axis of Fig.[fig:thetaimodule], sooner or later ωd reaches values lower than ∣γRB, d∣ so that the second side of becomes almost independent from fd. The same considerations apply for the other curves ∣Md∣ = constant, with the difference that the values of the frequencies at which the lines become independent from fd decrease for increasing values of ∣Md∣ (or increase for decreasing values of ∣Md∣). This behavior can be intuitively explained, since an increase of ∣Md∣ determines an increase of Xd, k (see ) which in turn brings to a reduction of the terms $\frac{2J\_1(X\_{d,k})}{X\_{d,k}}$ of the sum present in, i.e. a decrease of the “cutoff angular frequency” ∣γRB, d∣. ![Values of the modulating current amplitude I_i which, for a given modulating frequency f_i determine assigned values of the module of the phase modulation index due to the adiabatic chirp \|M_i\|. Continuous lines: realistic cases where the effect of RB is taken into account. Dashed lines: theoretical values of I_i in case no RB field re-injection were present. Note that the subscripts can be assumed as i=RF or i=d.](Figures/Fig2.png "fig:") [fig:thetaimodule] Still in Fig. [fig:thetaimodule] the behavior of ∣Md∣ in case of absence of RB is also reported, which puts into evidence the described effect of the RB field re-injection, namely the impossibility at a given low frequency to obtain high values of ∣Md∣ for dithering current amplitudes lower than a minimum “threshold” value. Note however that minimal increases of Id from this “threshold” value lead to very high variations of ∣Md∣. This can be appreciated in Fig. [fig:zoomthetaimodule], which focalizes the same quantities represented in Fig. [fig:thetaimodule] in limited ranges of fd and Id. ![Values of \|M_i\| which are reached in the interval f_i \in [1, 10] kHz in the vicinity of the values of I_i where the curves of Fig.[fig:thetaimodule] appear to exhibit an almost horizontal behavior. Note that the subscripts can be assumed as i=RF or i=d.](Figures/Fig3.png "fig:") [fig:zoomthetaimodule] Given the mentioned requirements for an effective application of the dithering tone technique (high ∣Md∣, Id, fd as low as possible), the practical indication is then to utilize a value of Id around such “threshold” value, while keeping fd as low as possible. A further physical phenomenon will be however highlighted in the next Section, which prevents the choice of arbitrarily low values for fd. Power Spectral Density at the Receiving End and effects of the Dithering Tone characteristics ----------------------------------------------------------------------------------------------- As mentioned at the end of Section II.A, at the final end z = L of the fiber connection, the field received by the PIN photodetector is given by the sum ETX(t, L) + ERB2(t, L), with the first addend given by with τ = τL = L/vg, and the second one given by: $$\begin{aligned} &&E\_{RB2}(t,L) =\frac{\sqrt{R\_{l,2}(1-R\_{l,2})}\cdot \eta\_c^{3/2}}{\sqrt{\alpha\_{iso}}} \times \nonumber\\ &&\times \sum\_{k=1}^K \rho \_k {{E}}(t-2 \tau \_k-\tau\_L)e^{-\alpha v\_g (2 \tau \_k+\tau\_L)} \label{eq:E\_RB2\_L}\end{aligned}$$ The current generated by the photo-detector can then be computed as: $$\begin{aligned} &i\_{out}(t)&=\mathcal{R}\left\|E\_{TX}(t,L) + E\_{RB2}(t,L)\right\|^2\nonumber\\ &&=\mathcal{R}\cdot \left\|E\_{TX}(t,L)\right\|^2+\mathcal{R}\cdot\left\|E\_{RB}(t,L)\right\|^2+\nonumber\\ &&+\mathcal{R}\cdot 2\Re {\it \bf e} \{E\_{TX}(t,L)\cdot E\_{RB}^\(t,L)\} \label{eq:i\_out\_00}\end{aligned}$$ where R is the Responsivity of the detector. At the right-end-side of Equation, the first addend would coincide with the total detected current if the RB effect were absent. Moreover, the term ∣ERB(t, L)∣2 is much smaller with respect to the others therefore the second added will be neglected in the following. In order to analyze the RB-induced spurious terms in the current power spectrum, only the last addend at the right hand side of has then to be considered, which will be named as ioutT*X, RB. From Equations (with τ = τL) and, exploiting the prostaferesis formula $\cos(u)-\cos(v)=-2\sin\left( \frac{u-v}{2}\right) \sin\left( \frac{u+v}{2} \right)$ with u = ωi(t − τL) + ∠Mi and v = ωi(t − τL − 2τk) + ∠Mi, its expression results to be: $$\begin{aligned} &i\_{out\_{TX,RB}}(t)\simeq \mathcal{R}\sqrt{\frac{R\_{l,2}}{\alpha\_{iso}}}(1-R\_{l,2})\eta\_c^2\|E(t-\tau\_L)\| \times \label{eq:i\_out\_TX\_RB\_no\_approx}\\ &\times 2\Re {\it \bf e} \left\{ \sum\_{k=1}^{k\_{max}}\rho^\\_k \|E(t- 2\tau\_k -\tau\_L)\| e^{-\alpha v\_g 2\tau\_k+j\omega\_{opt}2\tau\_k} \times \right.\nonumber\\ &\times \left.e^{j\left\{-\sum\_i X\_{i,k}sin\left[\omega\_i(t-\tau\_k-\tau\_L)+\angle M\_i\right] + \phi(t-\tau\_L)-\phi(t-2\tau\_k-\tau\_L)\right\}}\right\} \nonumber\end{aligned}$$ where X*i, k has been defined in. ![Qualitative representation of the behavior of PSD_{TX,RB} (\omega) in the vicinity of \omega= 2 \omega_{RF}, represented by with n_{RF}=2, for the k-th elementary RB reflection, when \omega_{d} is sufficiently greater than \Delta_{\Omega}. In this case the power of the second-harmonic distortion term P_{2\omega_{RF}} practically consists in the only element corresponding to n_{d}=0.](Figures/Fig4.png "fig:") [fig:PSD1] In line with the perturbative approach adopted, in both field modules can be assumed to be equal to E0. Collecting all the constants in a unique term $\Upsilon=\mathcal{R}\sqrt{\frac{R\_{l,2}}{\alpha\_{iso}}}(1-R\_{l,2})\eta\_c^2E\_0^2$, and exploiting the Jacobi-Anger expansion e− jusin(v) = ∑n = − ∞∞Jn(u)e− jnv, where u = Xi, k and v = ωi(t − τL − τk) + ∠Mi, taking advantage of and, after a lengthy but straightforward derivation, the expected value of the power spectral density PSDTX, RB of ioutTX, RB(t) assumes the following expression: $$\begin{aligned} \label{eq:PSD\_TX\_RB} & PSD\_{TX,RB} (\omega)= \nonumber \\ &={F\_{\xi}} \left \{ \mathbb{E} \left[ \left \langle i\_{out\_{TX,RB}}(t)i\_{out\_{TX,RB}}^{\ast}(t-\xi) \right \rangle \right] \right\} \simeq \nonumber \\ &\simeq \Upsilon^2 \sum\_{k=1}^{k\_{max}}{\sigma^2\_{\rho\_k}} e^{-4\alpha v\_g \tau\_k} \sum\_{\substack{n\_{d},n\_{RF}=\\=-\infty}}^{+\infty} {J\_{n\_{d}}^2(X\_{d,k}) J\_{n\_{RF}}^2(X\_{RF,k})} \times \nonumber \\ &\times \frac{2\Delta\_{\Omega}}{\Delta\_{\Omega}^2+[\omega-(n\_{d}\omega\_{d}+n\_{RF}\omega\_{RF})]^2}\end{aligned}$$ Focusing for example in the vicinity of ω = 2ωRF, i.e. considering nRF = 2 in, and referring to the generic contribution related to the k − th elementary RB reflection, the function PSDTX, RB(ω), as can be appreciated in Fig. [fig:PSD1], consists in the superposition of lorentzian functions centered in 2ωRF, 2ωRF ± ωd, 2ωRF ± 2ωd, … with amplitudes proportional respectively to J22(XRF, k)J02(Xd, k), J22(XRF, k)J± 12(Xd, k), J22(XRF, k)J± 22(Xd, k), .... The power of the undesired second-harmonic distortion term P2ωRF within an elementary bandwidth δΩ centered in 2ωRF can be directly evaluated as the integral of the Power Spectral Density multiplied by the load resistance RL: P2ωRF = RL ⋅ ∫2ωRF − δΩ/22ωRF + δΩ/2PSDTX, RB(ω)dω and results to be given by the last side of multiplied by δΩ. Note that if the dithering tone were absent, the contribution to P2ωRF coming from each elementary RB reflection would be just proportional to J22(XRF, k) and its impact could be reduced increasing ∣MRF∣, i.e. appropriately operating on ωRF and IRF, operations which would all result as not realizable, e.g. in the case where the modulating signal consists in an uncontrollable RFI disturbance. On the contrary, in presence of the dithering tone (see again the case represented in Fig. [fig:PSD1]) the contribution to P2ωRF coming from each elementary RB reflection is proportional to J22(XRF, k)J02(Xd, k). In this case, it is possible to reduce its impact reducing the value of J02(Xd, k), i.e. increasing ∣Md∣, and the previous Section II-B has been just devoted to determine how to obtain high values of ∣Md∣ while keeping Id and fd (or equivalently ωd) at their lowest possible levels. However, continuing to decrease ωd, the “tails” of the higher order terms of the sum in nd of (i.e. those corresponding to nd = ± 1, ± 2, …) result to contribute to the term at frequency 2ωRF and consequently prevent the reduction of the power at 2ωRF which is pursued with the dithering tone introduction. The phenomenon is qualitatively represented in Fig. [fig:PSD2] and its effects are illustrated in Fig. [fig:finalcomparison], which shows the simulated behavior of the values of P2ωRF normalized to its maximum, named in the figure P2ωRF, norm, covering a wide range of dithering frequencies, i.e, fd ∈ [100 − 4 × 106] Hz. The value of ωd cannot be reduced at will, for the insurgence of this additional phenomenon and an optimum working point can be identified for the application of the dithering tone technique through which the minimum value of Id can be applied t guarantee a given low level of P2ωRF, while keeping fd as low as possible. The considerations just developed with reference to the second harmonic term can be straightforwardly applied to the other spurious frequency terms (e.g. at frequencies 3fRF,fRF + fd, etc...). ![Same as Fig.[fig:PSD1] with the difference that \omega_{d} has been so decreased that the tails of the elements corresponding to n_{d} \not = 0 of the sum in n_{d} of contribute to the term at frequency 2f_{RF}.](Figures/Fig5.png "fig:") [fig:PSD2] ![Simulated behavior of the values of current I_i and frequency f_i which gives the normalized second-harmonic power P_{2\omega_{RF},norm}=cost. Both current and frequency axis are plotted in log-scale for a better view of the whole behavior.](Figures/Fig6.png "fig:") [fig:finalcomparison] Experimental Measurements and discussion ======================================== With the aim of confirm the theoretical results derived in previous section, measurements have been performed using the setup illustrated in Figure [fig:setup]. ![Experimental setup utilized. The setup is composed of two Signal Generators (SG1 and SG2), an RF passive coupler, a Distributed FeedBack Laser Diode (DFB-LD), a span of G.652 fiber, a Positive-Intrinsic-Negative PhotoDiode (PIN-PD) and a Spectrum Analyzer.](Figures/Fig7.png "fig:")[fig:setup] [hbpt] \| Symbol \| Physical meaning \| Value \| \| --- \| --- \| --- \| \| Ith \| Laser threshold current \| 9 mA \| \| Ibias \| Laser bias current \| 37 mA \| \| Popt \| Laser output power \| 6 dBm \| \| αiso \| Laser power isolation \| 20 dB \| \| ΔΩ \| Laser Coherence angular frequency band \| $3.14\cdot 10^6 \frac{rad}{s}$ \| \| α \| Fiber field attenuation factor \| 5− 5 $\frac{neper}{m}$ \| \| ηc \| Power coupling lens coefficient \| 0.4 \| \| S \| Fiber Recapture factor \| 10− 3 \| \| R \| PD Responsivity \| 1 $\frac{A}{W}$ \| \| RL \| Load Resistance \| 50 Ω \| [tab:parameters] The RoF link under test is composed of a DFB laser working at 1310nm connected to a span of G.652 fiber which is directly connected to a PIN Photodiode biased at VPD = 5 Volts. The characteristics of laser and photodiode are reported in Table [tab:parameters]. Two separate signal generators (SGs) are used to supply the laser. In particular, SG1 is used to supply the bias current Ibias and the dithering tone id(t) where both amplitude Id and frequency fd have been varied. The SG2 instead, is used to generate the RF tone iRF(t) at the frequency fRF = 70 MHz, simulating then a possible RFI falling in an interval of interest for low-frequency radioastronomic facilities. The three currents are then coupled using a passive coupler which is connected directly to the Laser. Indeed, in order to set the desired amplitudes of the three currents, the insertion loss of the coupler has been taken into account so that in the following discussion the values of Ibias and the amplitudes of id(t), iRF(t) are considered ones injected into the laser, i.e. after the coupler. After propagating in a standard ITU-T G.652 optical fiber of length L = 5 km, which is in the order of the maximum distances covered in SKAlow, the signal is detected by photodiode and through a spectrum analyzer the amplitude levels of the spurious tones generated at frequencies 2fRF and fRF + fd are evaluated. [Fig:confronti] .5 [fig:MeasHD2] .5 [fig:MeasIMD2] .5 [fig:simHD2] .5 [fig:simIMD2] As a preliminary test, the values of the dithering current amplitude Id which lead to a maximum of the intermodulation term at frequency ωRF + ωd, PωRF + ωd have been measured for varying values of fd. The behavior of this spurious component is described by considering nd = 1, nRF = 1 and it can be shown that the value of current Id which gives the maximum of PωRF + ωd is obtained when Md ≃ 0.92, which, with good approximation, corresponds to the value which gives the maximum of ∑k = 1kmaxJ12(Xd, k). Figure [Fig:confronti] shows the good agreement between simulated and measured values of Id. Note that this analysis have been performed considering fd = 1 kHz as the lowest value of dithering frequency. The reason behind this choice is that for lower frequencies it is not possible to identify experimentally the maximum of PωRF + ωd. After this preliminary test, an analysis of the behavior of the terms P2ωRF and PωRF + ωd as a function of Id and fd has been experimentally performed. The measurements have been conducted by fixing the value of certain frequency fd and sweeping the current Id for a range [0.001 − 0.1] mA. This sweeping has been then repeated for each frequency fd of interest, generating in this way the maps shown in Figures [fig:MeasHD2] and [fig:MeasIMD2]. Those maps represent the normalized values of P2ωRF and PωRF + ωd, named in the figures P2ωRF, norm and PωRF + ωd, norm respectively, measured at the output of the photodetector through the spectrum analyzer (see again Fig.[fig:setup]). An analogous map has been performed also by implementing the model shown in previous section, and the results are represented in Fig. [fig:simHD2] and [fig:simIMD2]. The maps show the different levels of harmonic and intermodulation distortion which offer a good reproduction of the experimental results, confirming the correctness of the theoretical study performed. In particular, both simulations and experimental results show that for the considered link, a frequency fd ≃ 10 kHz and an amplitude Id ≃ 0.04 mA constitute the optimum working point which allows to reduce the powers of both second harmonic distortion term P2ωRF and second order intermodulation product PωRF + ωd by at least 25 dB referred to their respective maximum values. Conclusion ========== An exhaustive study of the detrimental nonlinear effects produced by the Rayleigh Backscattering present within Analog Radio-over-Fiber link, and of the technique used to mitigate them, namely the dithering technique, has been shown. As a first important and novel aspect, a detailed description of the behavior of a directly modulated laser in the presence of optical feedback due to Rayleigh Backscattering has been presented, showing the significant changes produced on the laser spurious phase modulation (i.e. frequency chirp) characteristics and on the power spectrum of the signal received at the end of the fiber connection. Then, it has been shown that these changes impact significantly on the optimum characteristics of current and frequency of the sinusoidal dithering tone. The technique showed to be particularly effective on the typical scenarios of low frequency radioastronomic systems (for example, it has been applied in the demonstrators of SKA-low), which suffers to a relatively great degree the impact of RB due to their peculiar characteristics. Derivation of ΓS, Γθ ------------------------ The rate equation of the complex electrical field E(t) in presence of RBS can be written as follows : $$\begin{aligned} &\frac{dE(t)}{dt}=\label{eq:Kobayashi\_Rayleigh}\\ &=\left[j\omega(N) + \frac{1}{2}\left(G(N,S) -\frac{1}{\tau\_p}\right)\right]E(t)+\sum\_{k=1}^{K}C\_kE(t-2\tau\_k)\nonumber\end{aligned}$$ where the last term on the right-hand-side represents the RBS contribution. Putting E(t) = ∣E(t)∣ej[ω0t + θ(t)] and creating two equations, one for the real part and one for the imaginary part it results: $$\begin{aligned} && \|E(t)\|\frac{d\|E(t)\|}{dt}= \frac{1}{2}\left(G(N,S) -\frac{1}{\tau\_p}\right)\|E(t)\|^2+ \nonumber \\ && = +\sum\_{k=1}^K\|C\_k\| \|E(t)\|\|E(t-2\tau\_k)\|\cdot\nonumber\\ &&\cdot\cos[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:app1\_1}\end{aligned}$$ and: $$\begin{aligned} && \frac{d\theta(t)}{dt}= \left[\omega(N) -\omega\_0 \right]-\sum\_{k=1}^K\|C\_k\| \frac{\|E(t-2\tau\_k)\|}{\|E(t)\|}\cdot\nonumber\\ &&\cdot\sin[\omega\_{0}\cdot 2\tau\_k +\theta(t)-\theta(t-2\tau\_k)-\angle \rho\_k] \label{eq:app1\_2}\end{aligned}$$ Exploiting the relationship S(⋯) = ∣E(⋯)∣2, $\|E(t)\|\frac{d\|E(\cdots)\|}{dt}=\frac{1}{2}\frac{dS(\cdots)}{dt}$ the expressions given by, can be obtained Derivation of γS, i, γθ, i -------------------------------------- Starting from and exploiting,, it is: $$\begin{aligned} &&\Gamma\_S(t)=2\sum\_{k=1}^K\|C\_k\| \left( \left[S\_0+\Delta S +S\_i\cos(\omega\_i t +\angle S\_i)\right]\cdot \right. \nonumber \\ && \left. \cdot\left[S\_0+\Delta S +\|S\_i\| \cos(\omega\_i (t-2\tau\_k) +\angle S\_i)\right]\right)^{\frac{1}{2}} \cdot\nonumber\\ &&\cdot\cos\left[(\omega\_{0}+\Delta \omega)\cdot 2\tau\_k -\angle \rho\_k + \right. \nonumber \\ &&\left. -\|M\_i\|\cdot 2\sin(\omega\_i 2\tau\_k)\sin\left(\omega\_i(t-2\tau\_k)+\angle M\_i\right)\right] \label{eq:gamma\_Si\_1}\end{aligned}$$ From assumption the terms ΔS and Si can be neglected under the square root. Putting then ψk = (ω0 + Δω) ⋅ 2τk − ∠ρk and exploiting the truncated Jacobi-Anger expansion e− jusin(v) = J0(u) + 2J1(u)sin(v), with u = 2∣Mi∣sin(ωiτk) and v = sin(ωi(t − τk) + ∠Mi), in order to focus on the terms at DC and at ωi, it results: $$\begin{aligned} &&\Gamma\_S(t)=2 S\_0\sum\_{k=1}^K\|C\_k\| \cos(\psi\_k)\cdot J\_0\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right]+ \nonumber \\ && + 2 S\_0 \sum\_{k=1}^K\|C\_k\|\sin(\psi\_k)\cdot \nonumber \\ &&\cdot 2J\_1\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right] \cdot \sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right] \label{eq:gamma\_Si\_2}\end{aligned}$$ Putting now Xi = 2∣Mi∣sin(ωiτk) and elaborating further the term written in the last line of it results: $$\begin{aligned} && 2J\_1\left[2\|M\_i\|\sin(\omega\_i \tau\_k)\right] \cdot \sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right]= \nonumber \\ &&= \frac{2 J\_1(X\_i)}{X\_i} \cdot 2\|M\_i\|\sin(\omega\_i \tau\_k) sin\left[\omega\_i(t-\tau\_k)+\angle M\_i\right] \nonumber \\ &&= \Re {\it \bf e} \left\{ 2\frac{J\_1(X\_i)}{X\_i} \cdot 2\|M\_i\| \frac{e^{j\omega\_i\tau\_k}-e^{-j\omega\_i\tau\_k}}{2j} \right.\cdot \nonumber \\ && \left. \cdot (-j)e^{j\left[\omega\_i(t-\tau\_k)+\angle M\_i\right]}\right\} \nonumber \\ &&= \Re {\it \bf e} \left\{ -2\frac{J\_1(X\_i)}{X\_i} \cdot \left[1-e^{-j2\omega\_i\tau\_k}	arxiv_0000828
A nonstandard empirical likelihood for time series ================================================== , Standard blockwise empirical likelihood (BEL) for stationary, weakly dependent time series requires specifying a fixed block length as a tuning parameter for setting confidence regions. This aspect can be difficult and impacts coverage accuracy. As an alternative, this paper proposes a new version of BEL based on a simple, though nonstandard, data-blocking rule which uses a data block of every possible length. Consequently, the method does not involve the usual block selection issues and is also anticipated to exhibit better coverage performance. Its nonstandard blocking scheme, however, induces nonstandard asymptotics and requires a significantly different development compared to standard BEL. We establish the large-sample distribution of log-ratio statistics from the new BEL method for calibrating confidence regions for mean or smooth function parameters of time series. This limit law is not the usual chi-square one, but is distribution-free and can be reproduced through straightforward simulations. Numerical studies indicate that the proposed method generally exhibits better coverage accuracy than standard BEL. [class=AMS] Introduction ============ For independent, identically distributed data (i.i.d.), Owen introduced empirical likelihood (EL) as a general methodology for re-creating likelihood-type inference without a joint distribution for the data, as typically specified in parametric likelihood. However, the i.i.d. formulation of EL fails for dependent data by ignoring the underlying dependence structure. As a remedy, Kitamura proposed so-called blockwise empirical likelihood (BEL) methodology for stationary, weakly dependent processes, which has been shown to provide valid inference in various scenarios with time series (cf. ). Similarly to the i.i.d. EL version, BEL creates an EL log-ratio statistic having a chi-square limit for inference, but the BEL construction crucially involves blocks of consecutive observations in time, rather than individual observations. This data-blocking serves to capture the underlying time dependence and related concepts have also proven important in defining resampling methodologies for dependent data, such as block bootstrap and time subsampling methods. However, the coverage accuracy of BEL can depend crucially on the block length selection, which is a fixed value 1 ≤ b ≤ n for a given sample size n, and appropriate choices can vary with the underlying process (a point briefly illustrated at the end of this section). To advance the BEL methodology in a direction away from block selection with a goal of improved coverage accuracy, we propose an alternative version of BEL for stationary, weakly dependent time series, called an expansive block empirical likelihood (EBEL). The EBEL method involves a nonstandard, but simple, data-blocking rule where a data block of every possible length is used. Consequently, the method does not involve a block length choice in the standard sense. We investigate EBEL in the prototypical problem of inference about the process mean or a smooth function of means. For setting confidence regions for such parameters, we establish the limiting distribution of log-likelihood ratio statistics from the EBEL method. Because of the nonstandard blocking scheme, the justification of this limit distribution requires a new and substantially different treatment compared to that of standard BEL (which closely resembles that of EL for i.i.d. data in its large-sample development; cf. ). In fact, unlike with standard BEL or EL for i.i.d. data, the limiting distribution involved is nonstandard and not chi-square. However, the EBEL limit law is distribution-free, corresponding to a special integral of standard Brownian motion on [0, 1], and so can be easily approximated through simulation to obtain appropriate quantiles for calibrating confidence regions. In addition, we anticipate that the EBEL method may have generally better coverage accuracy than standard BEL methods, though formally establishing and comparing convergence rates is beyond the scope of this manuscript (and, in fact, optimal rates and block sizes for even standard BEL remain to be determined). Simulation studies, though, suggest that interval estimates from the EBEL method can perform much better than the standard BEL approach, especially when the later employs a poor block choice, and be less sensitive to the dependence strength of the underlying process. The rest of manuscript is organized as follows. We end this section by briefly recalling the standard BEL construction with overlapping blocks and its distributional features. In Section [sec2], we separately describe the EBEL method for inference on process means and smooth function model parameters, and establish the main distributional results in both cases. These results require introducing a new type of limit law based on Brownian motion, which is also given in Section [sec2]. Additionally, Section [sec2.1] describes how the usual EL theory developed by Owen, and often underlying many EL arguments including the time series extensions of BEL, fails here and requires new technical developments; consequently, the theory provided may be useful for future developments of EL (with an example given in Section [sec2.4]). Section [sec3] provides a numerical study of the coverage accuracy of the EBEL method and comparisons to standard BEL. Section [sec4] offers some concluding remarks and heuristic arguments on the expected performance of EBEL. Proofs of the main results appear in Section [sec5] and in supplementary materials, where the latter also presents some additional simulation summaries. To motivate what follows, we briefly recall the BEL construction, considering, for concreteness, inference about the mean EXt = μ ∈ Rd of a vector-valued stationary stretch X1, …, Xn. Upon choosing an integer block length 1 ≤ b ≤ n, a collection of maximally overlapping (OL) blocks of length b is given by $\{ (X\_i,\ldots,X\_{i+b-1})\dvtx i=1,\ldots,N\_b \equiv n-b+1\}$. For a given μ ∈ Rd value, each block in the collection provides a centered block sum Bi, μ ≡ ∑j = ii + b − 1(Xj − μ) for defining a BEL function $$\label{eqn:stan} L\_{{\mathrm{BEL}},n}(\mu) = \sup\Biggl\{ \prod \_{i=1}^{N\_b} p\_i \dvtx p\_i \geq0, \sum\_{i=1}^{N\_b}p\_i = 1, \sum\_{i=1}^{N\_b}p\_i B\_{i,\mu} = 0\_d \Biggr\}$$ and corresponding BEL ratio RBEL, n(μ) = Ln(μ)/Nb− Nb, where above 0d = (0, …, 0)′ ∈ Rd. The function LBEL, n(μ) assesses the plausibility of a value μ by maximizing a multinomial likelihood from probabilities {pi}i = 1Nb assigned to the centered block sums Bi, μ under a zero-expectation constraint. Without the linear mean constraint in ([eqn:stan]), the multinomial product is maximized when each pi = 1/Nb (i.e., the empirical distribution on blocks), defining the ratio RBEL, n(μ). Under certain mixing and moment conditions entailing weak dependence, and if the block b grows with the sample size n but at a smaller rate (i.e., b− 1 + b2/n → 0 as n → ∞), the log-EL ratio of the standard BEL has chi-square limit $$\label{eqn:stanl} -\frac{2}{b} \log R\_{{\mathrm{BEL}},n}(\mu\_0) \stackrel{d} {\rightarrow} \chi^2\_d,$$ at the true mean parameter μ0 ∈ Rd; cf. Kitamura. Here b− 1 represents an adjustment in ([eqn:stanl]) to account for OL blocks and, for i.i.d. data, a block length b = 1 above produces the EL distributional result of Owen. To illustrate the connection between block selection and performance, Figure [fig1] shows the coverage rate of nominal 90% BEL confidence intervals $\{\mu\in\mathbb{R}\dvtx-2/b \log R\_{{\mathrm{BEL}},n}(\mu\_0) \leq\chi ^2\_{1,0.9}\}$, as a function of the block size b, for estimating the mean of three different MA(2) processes based on samples of size n = 100. One observes that the coverage accuracy of BEL varies with the block length and that the best block size can depend on the underlying process. The EBEL method described next is a type of modification of the OL BEL version, without a particular fixed block length selection b. [fig1] Expansive block empirical likelihood ==================================== Mean inference -------------- Suppose X1, …, Xn represents a sample from a strictly stationary process $\{{X\_t}\dvtx t\in{\mathbb{Z}}\}$ taking values in Rd, and consider a problem about inference on the process mean EXt = μ ∈ Rd. While the BEL uses data blocks of a fixed length b for a given sample size n, the EBEL uses overlapping data blocks {(X1), (X1, X2), …, (X1, …, Xn)} that vary in length up to the longest block consisting of the entire time series. Hence this block collection, which constitutes a type of forward “scan” in the block subsampling language of McElroy and Politis , contains a data block of every possible length b for a given sample size n. This block sequence also appears in fixed-b asymptotic schemes and related self-normalization approaches; cf. Shao, Section 2; see also Section [sec4] here. In this sense, these blocks are interesting and novel to consider in a BEL framework. Other block schemes may be possible and potentially applied for practical gain (e.g., improved power), where the theoretical results of this paper could also directly apply. We leave this largely for future research, but we shall give one example of a modified, though related, blocking scheme in Section [sec2.4] while focusing the exposition on the block collection above [i.e., the alternative blocking incorporates a backward scan {(Xn), (Xn, Xn − 1), …, (Xn, …, X1)} with similar theoretical development]. Let $w\dvtx[0,1]\rightarrow[0,\infty)$ denote a nonnegative weighting function. To assess the likelihood of a given value of μ, we create centered block sums Ti, μ = w(i/n)∑j = 1i(Xj − μ), i = 1, …, n, and define a EBEL function $$\label{eqn:EBEL} L\_{n}(\mu) = \sup\Biggl\{ \prod \_{i=1}^{n} p\_i \dvtx p\_i \geq0, \sum\_{i=1}^{n}p\_i = 1, \sum\_{i=1}^{n} p\_i T\_{i,\mu} = 0\_d \Biggr\}$$ and ratio Rn(μ) = n− nLn(μ). After defining the block sums, the computation of Ln(μ) is analogous to the BEL version and essentially the same as that described by Owen for i.i.d. data. Namely, when the zero 0d vector lies in the interior convex hull of $\{T\_{i,\mu}\dvtx i=1,\ldots,n\} $, then Ln(μ) is the uniquely achieved maximum at probabilities pi = 1/[n(1 + λn, μ′Ti, μ)] > 0, i = 1, …, n, with a Lagrange multiplier λn, μ ∈ Rd satisfying $$\label{eqn:t0} \sum\_{i=1}^n \frac{T\_{i,\mu}}{n(1 + \lambda\_{n,\mu}^\prime T\_{i,\mu}) } = 0\_d;$$ see for these and other computational details. Regarding the weight function above in the EBEL formulation, more details are provided below and in Section [sec2.2]. The next section establishes the limiting distribution of the log-EL ratio from the EBEL method for setting confidence regions for the process mean μ parameter. However, it is helpful to initially describe how the subsequent developments of EL differ from previous ones with i.i.d. or weakly dependent data (cf. for BEL). The standard arguments for developing EL results, due to Owen (page 101), typically begin from algebraically re-writing ([eqn:t0]) to expand the Lagrange multiplier. If we consider the real-valued case d = 1 for simplicity, this becomes $$\lambda\_{n,\mu} = \frac{\sum\_{i=1}^n T\_{i,\mu}}{\sum\_{i=1}^n T\_{i,\mu }^2 } + \frac{ \lambda\_{n,\mu}^2}{\sum\_{i=1}^n T\_{i,\mu}^2} \sum \_{i=1}^n \frac{T\_{i,\mu}^3}{1 + \lambda\_{n,\mu}^\prime T\_{i,\mu}}.$$ In the usual independence or weak dependence cases of EL [e.g., where Bi, μ from ([eqn:stan]) replaces Ti, μ in the Lagrange multiplier above], the first right-hand side term dominates the second, which gives a substantive form for λn, μ as a ratio of sample means and consequently drives the large sample results (i.e., producing chi-square limits). However, in the EBEL case here, both terms on the right-hand side above have the same order, implying that the standard approach to developing EL results breaks down under the EBEL blocking scheme. The proofs here use a different EL argument than the standard one mentioned above, involving no asymptotic expansions of the Lagrange multiplier or Taylor expansions of the EL ratio based on these. The large sample results for the EBEL method require two mild assumptions stated below. Let Cd[0, 1] denote the metric space of all Rd-valued continuous functions on [0, 1] with the supremum metric ρ(g1, g2) ≡ sup0 ≤ t ≤ 1∥g1(t) − g2(t)∥, and let B(t) = (B1(t), …, Bd(t))′, 0 ≤ t ≤ 1, denote a Cd[0, 1]-valued random variable where B1(t), …, Bd(t) are i.i.d. copies of standard Brownian motion on [0, 1]. [((A.2))] The weight function $w\dvtx[0,1]\rightarrow[0,\infty)$ is continuous on [0, 1] and is strictly positive on an interval (0, c) for some c ∈ (0, 1]. Let EXt = μ0 ∈ Rd denote the true mean of the stationary process {Xt} and suppose d × d matrix Σ = ∑j = − ∞∞Cov(X0, Xj) is positive definite. For the empirical process Sn(t) on t ∈ [0, 1] defined by linear interpolation of $\{ S\_n(i/n) = \sum\_{j=1}^i (X\_j - \mu\_0) \dvtx i=0,\ldots , n \}$ with Sn(0) = 0, it holds that $ S\_n(\cdot)/n^{1/2} \stackrel{d}{\rightarrow} \Sigma^{1/2} B(\cdot)$ in Cd[0, 1]. Assumption (A.1) is used to guarantee that, in probability, the EBEL ratio Rn(μ0) positively exists at the true mean, which holds for uniformly weighted blocks w(t) = 1, t ∈ [0, 1], for example. Assumption (A.2) is a functional central limit theorem for weakly dependent data, which holds under appropriate mixing and moment conditions on {Xt}. Main distributional results --------------------------- To state the limit law for the log-EBEL ratio ([eqn:EBEL]), we require a result regarding a vector B(t) = (B1(t), …, Bd(t))′, 0 ≤ t ≤ 1, of i.i.d. copies B1(t), …, Bd(t) of standard Brownian motion on [0, 1]. Indeed, the limit distribution of − 2logRn(μ0) is a nonstandard functional of the vector of Brownian motion B( ⋅ ). Theorem [theorem1] identifies key elements of the limit law and describes some of its basic structural properties. [theorem1] Suppose that B(t) = (B1(t), …, Bd(t))′, 0 ≤ t ≤ 1, is defined on a probability space, and let f(t) = w(t)B(t), 0 ≤ t ≤ 1, where w( ⋅ ) satisfies assumption. Then, with probability 1 (w.p.1), there exists an Rd-valued random vector Yd satisfying the following: [(iii)] Yd is the unique minimizer of $$g\_d(a) \equiv-\int\_0^1 \log \bigl( 1 + a^\prime f(t)\bigr)\,dt\qquad \mbox{for $a\in\overline{K}\_d$},$$ where $\overline{K}\_d \equiv\{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) \geq0 \}$ is the closure of $K\_d \equiv\{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) > 0 \}$; the latter set is open, bounded and convex in Rd (w.p.1). On Kd, gd is also real-valued, strictly convex and infinitely differentiable (w.p.1). − ∞ < gd(Yd) < 0, Yd′∫01f(t) dt > 0, $ {0 \leq\int\_0^1 \frac{Y\_d^\prime f(t)}{ 1 + Y\_d^\prime f(t)}\,dt < \infty}$. If Yd ∈ Kd, then Yd is the unique solution to $ {\int\_0^1 \frac{f(t)}{1 + a^\prime f(t)}\,dt = 0\_d}$ for a ∈ Kd, and if $ {\int\_0^1 \frac{f(t)}{1 + a^\prime f(t)}\,dt = 0\_d}$ has a solution a ∈ Kd, then this solution is uniquely Yd. We use the subscript d in Theorem [theorem1] to denote the dimension of either the random vector Yd, the space Kd or the arguments of gd. The function gd is well defined and convex on $\overline {K}\_d$, though possibly gd(a) = + ∞ for some $a\in\partial K\_d = \{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) = 0 \}$ on the boundary of Kd; a minimizer of gd( ⋅ ) may also occur on $\partial K\_d \cap \{y \in\mathbb{R}^d\dvtx g\_d(y)\leq0 \}$. Importantly, the probability law of gd(Y1) is distribution-free, and because standard Brownian motion is fast and straightforward to simulate, the distribution of gd(Yd) can be approximately numerically. Parts (ii) and (iii) provide properties for characterizing and identifying the minimizer Yd. For example, considering the real-valued case d = 1, it holds that K1 = (m, M) where m = − [max0 ≤ t ≤ 1f(t)]− 1 < 0 < M = − [min0 ≤ t ≤ 1f(t)]− 1 and the derivative dg1(a)/da is strictly increasing on K1 by convexity. Because the derivative of g1 at 0 is − ∫01f(x) dx, parts (ii)–(iii) imply that if − ∫01f(x) dx < 0, then either Y1 = m or Y1 solves dg1(a)/da = 0 on m < a ≤ 0; alternatively, if − ∫01f(x) dx > 0, then Y1 = M or Y1 solves dg1(a)/da = 0 on 0 ≤ a < M. Additionally, while the weight function w( ⋅ ) influences the distribution of gd(Yd), the scale of w( ⋅ ) does not; defining f with w or cw, for a nonzero c ∈ R, produces the same minimized value gd(Yd). We may now state the main result on the large-sample behavior of the EBEL log-ratio evaluated at the true process mean EXt = μ0 ∈ Rd. Recall that, when Ln(μ0) > 0 in ([eqn:EBEL]), the EBEL log-ratio admits a representation ([eqn:t0]) at μ0 in terms of the Lagrange multiplier λn, μ0 ∈ Rd. [theorem2] Under assumptions, as n → ∞: [(ii)] $ n^{1/2} \Sigma^{1/2} \lambda\_{n,\mu\_0} \stackrel {d}{\rightarrow} Y\_d $; $ {-\frac{1}{n} \log R\_n(\mu\_0) \stackrel{d}{\rightarrow} -g\_d(Y\_d)}$, for Yd and gd(Yd) defined as in Theorem [theorem1], and Σ = ∑j = − ∞∞Cov(X0, Xj). From Theorem [theorem2](i), the Lagrange multiplier in the EBEL method has a limiting distribution which is not the typical normal one, as in the standard BEL case. This has a direct impact on the limit law of the EBEL ratio statistic. As Theorem [theorem2](ii) shows, the negative logarithm of the EBEL ratio statistic, scaled by the inverse of the sample size, has a nonstandard limit, given by the functional − gd(Yd) of the vector of Brownian motion B( ⋅ ) (cf. Theorem [theorem1]), that critically depends on the limit Yd of the scaled Lagrange multiplier. The distribution of − gd(Yd) is free of any population parameters so that quantiles of − gd(Yd), which are easy to compute numerically, can be used to calibrate the EBEL confidence regions. As − gd(Yd) is a strictly positive random variable, an approximate 100(1 − α)% confidence region for μ0 can be computed as $$\bigl\{\mu\in\mathbb{R}^d \dvtx- n^{-1} \log R\_n(\mu\_0) \leq a\_{d,1-\alpha}\bigr\},$$ where ad, 1 − α is the lower (1 − α) percentile of − gd(Yd). When d = 1, the confidence region is an interval; for d > 2, the region is guaranteed to be connected without voids in Rd. In contrast to the standard BEL ([eqn:stanl]), EBEL confidence regions do not require a similar fixed choice of block size. We next provide additional results that give the limit distribution of the log-EBEL ratio statistic under a sequence of local alternatives and that also show the size of a EBEL confidence region will be no larger than Op(n− 1/2) in diameter around the true mean EXt = μ0. Let $$\label{eqn:Gn} G\_n \equiv\bigl\{ \mu\in\mathbb{R}^d \dvtx R\_n(\mu)\geq R\_n(\mu\_0)>0\bigr\}$$ be the collection of mean parameter values which are at least as likely as μ0, and therefore elements of a EBEL confidence region whenever the true mean is. [co1] Suppose the assumptions of Theorem [theorem2] hold. For c ∈ Rd, define fc(t) = w(t)[B(t) + tΣ− 1/2c], t ∈ [0, 1], in terms of the vector of Brownian motion B(t). [(ii)] Then, as n → ∞, $-n^{-1}\log R\_n(\mu\_0 \pm n^{-1/2}c) \stackrel{d}{\rightarrow}$ $$-\min\biggl\{ -\int\_0^1 \log\bigl(1 + a^\prime f\_{c}(t)\bigr)\,dt\dvtx a \in\mathbb{R}^d, \min\_{0 \leq t \leq1} \bigl(1 + a^\prime f\_{c}(t)\bigr) \geq0 \biggr\};$$ $ \sup\{\\|\mu-\mu\_0\\|\dvtx\mu\in G\_n\} =O\_p(n^{-1/2})$, for Gn in ([eqn:Gn]). Hence along a sequence of local alternatives (n− 1/2 away from the true mean), the log-EBEL ratio converges to a random variable, defined as the optimizer of an integral involving Brownian motion; this resembles Theorem [theorem1] [involving f(t) = w(t)B(t) there], but the integrated function fc( ⋅ ) has an addition term w(t)tΣ− 1/2c under the alternative. With respect to Corollary [co1](i), the involved limit distribution can be described with similar properties as in Theorem [theorem1] upon replacing f(t) with fc(t) there. In particular, the limiting distribution under the scaled alternatives depends on Σ− 1/2c, similarly to the normal theory case (e.g., with standard BEL) where Σ− 1/2c determines the noncentrality parameter of a noncentral chi-square distribution. We note that Theorem [theorem2] remains valid for potentially negative-valued weight functions w( ⋅ ) as well. Simulations have shown that, with weight functions oscillating between positive and negative values on [0, 1] [e.g., w(t) = sin(2πt)], EBEL intervals for the process mean perform consistently well in terms of coverage accuracy. However, with weight functions w( ⋅ ) that vary in sign, a result as in Corollary [co1](ii) fails to hold. Hence, the weight functions w( ⋅ ) considered are nonnegative as stated in assumption (A.1). [re1] The EBEL results in Theorem [theorem2] also extend to certain parameters described by general estimating functions; for examples and similar EL results in the i.i.d. and time series cases, respectively, see and. Suppose θ ∈ Rp represents a parameter of interest and G( ⋅ ; ⋅ ) ∈ Rd × Rp → Rp is a vector of p estimating functions such that EG(Xt; θ0) = 0p holds at the true parameter value θ0. The previous process mean case corresponds to G(Xt; μ) = Xt − μ with Xt, μ ∈ Rd, d = p. A EBEL ratio statistic Rn(θ) for θ results by replacing Tu, i = w(i/n)∑j = 1i(Xj − μ) and 0d with Tθ, i = w(i/n)∑j = 1iG(Xj; θ) and 0p in ([eqn:EBEL]). Under the conditions of Theorem [theorem2] [substituting G(Xj; θ0) for Xj − μ0 in assumption (A.2)], $$-\frac{1}{n} \log R\_n(\theta\_0) \stackrel{d} { \rightarrow} -g\_p(Y\_p)$$ holds as n → ∞ with Yp and gp(Yp) as defined in Theorem [theorem1], generalizing Theorem [theorem2] and following by the same proof. The next section considers extensions of the EBEL approach to a different class of time series parameters. Smooth function model parameters -------------------------------- We next consider extending the EBEL method for inference on a broad class of parameters under the so-called “smooth function model;” cf.. For independent and time series data, respectively, Hall and La Scala and Kitamura have considered EL inference for similar parameters; see also, Section 4. If EXt = μ0 ∈ Rd again denotes the true mean of the process, the target parameter of interest is given by θ0 = H(μ0) ∈ Rp, based on a smooth function H(μ) = (H1(μ), …, Hp(μ))′ of the mean parameter μ, where $H\_i\dvtx\mathbb{R}^d\rightarrow \mathbb {R}$ for i = 1, …, p and p ≤ d. This framework allows a large variety of parameters to be considered such as sums, differences, products and ratios of means, which can be used, for example, to formulate parameters such as covariances and autocorrelations as functions of the m-dimensional moment structure (for a fixed m) of a time series. For a univariate stationary series U1, …, Un, for instance, one can define a multivariate series Xt based on transformations of (Ut, …, Ut + m − 1) and estimate parameters for the process {Ut} based on appropriate functions H of the mean of Xt. The correlations θ0 = H(μ0) of {Ut} at lags m and m1 < m, for example, can be formulated in ([eqn:H]) by H(x1, x2, x3, x4) = (x3 − x12, x4 − x12)′/[x2 − x12] and EXt = μ0 for Xt = (Ut, Ut2, UtUt + m1, UtUt + m)′ ∈ R4. and (Chapter 4) provide further examples of smooth function parameters. For inference on the parameter θ = H(μ), the EBEL ratio is defined as $$R\_n(\theta) = \sup\Biggl\{ \prod\_{i=1}^{n} p\_i \dvtx p\_i \geq0, \sum\_{i=1}^{n}p\_i = 1, \sum\_{i=1}^{n} p\_i T\_{i,\mu} = 0\_d, \mu\in\mathbb{R}^d, H(\mu)= \theta\Biggr\},$$ and its limit distribution is provided next. [theorem13] In addition to the assumptions of Theorem [theorem2], suppose H from ([eqn:H]) is continuously differentiable in a neighborhood of μ0 and that ∇μ0 has rank p ≤ d, where ∇μ ≡ [∂Hi(μ)/∂μj]i = 1, …, p; j = 1, …, d denotes the p × d matrix of first-order partial derivatives of H. Then, at the true parameter θ0 = H(μ0), as n → ∞, $$-\frac{1}{n} \log R\_{n}(\theta\_0) \stackrel{d} { \rightarrow} -g\_p(Y\_p)$$ with Yp and gp(Yp) as defined in Theorem [theorem1]. Theorem [theorem13] shows that the log-EBEL ratio statistic for the parameter θ0 = H(μ0) ∈ Rp under the smooth function model continues to have a limit of the same form as that in the case of the EBEL for the mean parameter μ0 ∈ Rd itself. The main difference is that the functional gp(Yp) is now defined in terms of a p-dimensional Brownian motion as in Theorem [theorem1], but with p ≤ d, where p denotes the dimension of the parameter θ0; see also Remark [re1]. It is interesting to note that, similarly to the traditional profile likelihood theory in a parametric set-up with i.i.d. observations, the limit law here does not depend on the function H as long as the matrix ∇μ0 of the first order partial derivatives of H at μ = μ0 has full rank p. Due to the nonstandard blocking, the proof of this EBEL result again requires a different development compared to the one for standard BEL (cf. ) that mimics the i.i.d. EL case (cf. ) involving expansion of Lagrange multipliers. Extensions to other data blocking --------------------------------- As mentioned in Section [sec2.1], other versions of EBEL may be possible with other data blocking schemes, which likewise involve no fixed block selection in the usual BEL sense and have a related theoretical development. We give one example here. Recall the EBEL function ([eqn:EBEL]) for the mean Ln(μ), μ ∈ Rd, involves centered block sums Ti, μ = w(i/n)∑j = 1i(Xj − μ), i = 1, …, n, based on blocks $\{ (X\_1), (X\_1,X\_2),\ldots,\break (X\_1,\ldots,X\_n)\}$. Reversed blocks for example, given by $\{ (X\_n), (X\_n,X\_{n-1}),\ldots,\break (X\_n,\ldots,X\_1)\}$, can also be additionally incorporated by defining further block sums Tn + i, μ = w(i/n) × ∑j = 1i(Xn − j + 1 − μ), i = 1, …, n, and a corresponding EBEL function $$\tilde{L}\_{n}(\mu) = \sup\Biggl\{ \prod \_{i=1}^{2n} p\_i \dvtx p\_i \geq0, \sum\_{i=1}^{2n}p\_i = 1, \sum\_{i=1}^{2n} p\_i T\_{i,\mu} = 0\_d \Biggr\}$$ and ratio R̃n(μ) = (2n)− 2nL̃n(μ). At the true mean μ0 ∈ Rd, the log-ratio − logR̃n(μ0) = ∑i = 12nlog[1 + λ̃n, μ0′Ti, μ0] can similarly be re-written in terms of a Lagrange multiplier λ̃n, μ0 ∈ Rd satisfying 0d = ∑i = 12nTi, μ0/[1 + λ̃n, μ0′Ti, μ0]. The EL distributional results of the previous subsections then extend in a natural manner, as described below for the mean inference case; cf. Theorem [theorem2]. For 0 ≤ t ≤ 1, recall f(t) = w(t)B(t) (cf. Theorem [theorem1]), for B(t) = (B1(t), …, Bd(t))′ denoting a vector of i.i.d. copies B1(t), …, Bd(t) of standard Brownian motion on [0, 1], and define additionally f̃(t) = w(t)[B(1) − B(1 − t)]. [theorem4] Under assumptions, as n → ∞, $$n^{1/2} \Sigma^{1/2} \tilde{\lambda}\_{n,\mu\_0} \stackrel{d} {\rightarrow} \tilde{Y}\_d,\qquad -\frac{1}{n} \log \tilde{R}\_n(\mu\_0) \stackrel{d} {\rightarrow} - \tilde{g}\_d(\tilde{Y}\_d)\in(0, \infty)$$ for a Rd-valued random vector Ỹd defined as the unique minimizer of $$\tilde{g}\_d(a) \equiv-\int\_0^1 \log\bigl( 1 + a^\prime f(t)\bigr)\,dt -\int\_0^1 \log\bigl( 1 + a^\prime\tilde{f}(t)\bigr)\,dt$$ for $a\in\overline{K}\_d\equiv\{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) \geq0, \min\_{0 \leq t \leq1} (1 + y^\prime\tilde{f}(t)) \geq0\}$. As in Theorem [theorem2], the limit law of the log EBEL ratio above is similarly distribution-free and easily simulated from Brownian motion. The main difference between Theorems [theorem2] and [theorem4] is that the reversed data blocks in the EL construction contribute a further integral component based on (reversed) Brownian motion in the limit. Straightforward analog versions of Theorem [theorem1] [regarding Ỹd and g̃d( ⋅ )] as well as Corollary [co1] and Theorem [theorem13] [with respect to R̃n( ⋅ )] also hold; we state these in the supplementary materials for completeness. Numerical studies ================= Here we summarize the results of a simulation study to investigate the performance of the EBEL method, considering the coverage accuracy of confidence intervals (CIs) for the process mean. We considered several real-valued ARMA processes, allowing a variety of dependence structures with ranges of weak and strong dependence, defined with respect to an underlying i.i.d. centered χ12-distributed innovation series; these processes appear in Table [tab2] in the following. Other i.i.d. innovation types (e.g., normal, Bernoulli, Pareto) produced qualitatively similar results. For each process, we generated 2000 samples of size n = 250, 500, 1000 for comparing the coverage accuracy of 90% CIs from various EL procedures. We applied the EBEL method with forward expansive data blocks, as in Section [sec2.1], as well as forward/backward data blocks, as in Section [sec2.4]; we denote these methods as EBEL1/EBEL2, respectively, in summarizing results. In addition to a constant weight w(t) = 1, we implemented these methods with several other choices of weight functions w(t) on [0, 1], each down-weighting the initial (smaller) data blocks in the EBEL construction and differing in their shapes. The resulting coverages were very similar across nonconstant weight functions and we provide results for two weight choices: linear w(t) = t and cosine-bell w(t) = [1 − cos(2πt)]/2. Additionally, for each weight function w(t), the limiting distribution of the EBEL ratio was approximated by 50,000 simulations to determine its 90th percentile for calibrating intervals, as listed in Table [tab1] with Monte Carlo error bounds. [tab1] @lcc@ $\bolds{w(t)$, $t\in[0,1]}$ & & w(t) = 1 &2.51 (0.03) &2.50 (0.03) w(t) = t &5.64 (0.09) &4.37 (0.06) w(t) = (1 − cos(2πt))/2 &7.00 (0.15) & 3.42 (0.09) For comparison, we also include coverage results for the standard BEL method with OL blocks (denoted as BEL). Kitamura (page 2093) considered a block order n1/3 for BEL as the method involves a block-based variance estimator in its asymptotic studentization mechanics (see Section [sec4]), which is asymptotically equivalent to the Bartlett kernel spectral density estimator at zero having n1/3 at its optimal block/lag order; cf.. Based this correspondence, we considered two data-driven block selection rules from the spectral kernel literature, which estimate the coefficient Ĉ in the theoretical optimal block length expression Cn1/3 known from spectral estimation. One block estimation approach (denoted FTK) is based on flat-top kernels and results in block estimates for BEL due to a procedure in Politis and White, page 60; we used a flat-top kernel bandwidth n1/5 for generally consistent estimation as described in. The second block estimation approach (denoted AAR) is due to Andrews, pages 834–835, producing block estimates for BEL based on bandwidth estimates for the Bartlett spectral kernel assuming an approximating AR(1) process. [tab2] @lcccccccccc@ & && && && & && && && Process & $\bolds{n}$ && 1 & $\bolds{t}$&& 1 &$\bolds{t}$&& FTK &AAR MA(2) &250&&90.6 &91.1 && 91.4 &91.4 &&93.7 &98.3 0.4, − 0.6&500&&91.0 &91.2 && 91.7 &91.5 && 93.4 &98.0 &1000&& 90.0& 90.0 && 90.4 &90.0 && 90.6 &96.6 MA(1)\&250&& 87.4 &89.4 && 90.2 &90.5 && 91.3 & 94.2 &500&& 89.4 & 90.8 &&90.4 &90.2 && 90.9 &92.7 &1000&&89.6 &89.8 && 90.8 & 90.2 && 91.3 &92.9 MA(3)&250&& 87.4 &88.5 && 90.4& 90.8 && 93.6 &92.7 − 1, − 1, − 1&500&& 87.8 &88.6 && 90.0 &90.2 &&93.4 & 92.0 &1000&& 89.7 &89.2 && 89.2 &89.8 && 92.2 &91.9 ARMA(1, 2)&250&& 84.4 &86.0 && 89.1& 89.8 && 93.8& 94.6 0.9, − 0.6, − 0.3&500&& 87.2 &88.7 && 90.4& 90.3 && 95.5 &95.2 &1000 && 89.4 &89.9 && 91.6& 91.6 && 95.6& 96.2 AR(1) &250&&89.2 &90.0&& 92.0& 91.4&& 95.8 &91.8 − 0.7 &500&& 89.4& 90.6 && 90.9& 90.8 && 95.2& 91.0 &1000&& 90.4& 90.2 && 90.4& 90.8 && 92.4 &92.0 AR(1) &250&&67.0 &70.5 && 79.0& 80.0 &&61.1 &76.4 0.9 &500&& 73.4 &77.0 && 82.4& 83.4 &&66.0& 81.4 &1000&& 77.4 &80.1 && 86.2& 87.2 && 74.6& 85.6 ARMA(1, 1) &250&&79.5 &81.8 && 86.3& 86.2 && 81.0 &80.2 0.7, − 0.5 &500 && 82.0 &84.6 && 86.3& 86.9&& 82.2 &82.0 &1000 && 85.0& 87.0 && 87.9& 89.0 && 85.4& 84.0 ARMA(2, 2) &250&&78.3 &81.0 && 84.0 &84.6 && 77.2& 73.0 0.3, 0.3, − 0.3, − 0.1 &500 && 81.5 &83.6 && 86.2 &87.2&& 81.0& 74.4 &1000 && 84.4 &85.4 && 88.4 &88.7 && 84.7& 75.3 ARMA(2, 2) &250&&81.2 &83.9 && 85.5& 86.2 && 79.4 &81.8 0.5, 0.3, 0.3, − 0.9 &500 && 84.2& 86.0 && 87.4 &88.0 && 82.8 &84.6 &1000 && 85.4& 86.2 && 88.0& 88.2 && 84.0& 85.5 MA(2) &250&&83.2 &85.0 && 87.4& 87.6 && 86.0 &79.2 0.1, 2 &500 && 84.6 &86.0 && 87.5 &88.6&& 86.8& 81.2 &1000 && 86.2 &87.2 && 89.2 &90.2 && 87.5 &80.4 Table [tab2] lists the realized coverage accuracy of 90% EL CIs for the mean. From the table, the linear weight function w(t) = t* generally produced slightly more accurate coverages for both EBEL1/EBEL2 methods than the constant weight w(t) = 1; additionally and interestingly, despite their shape differences, the coverage rates for both the linear and cosine-bell weight functions closely matched (to the extent that we defer the cosine-bell results to the supplementary materials ). For all sample sizes and processes in Table [tab2], the EBEL2 method with linear weight typically and consistently emerged as having the most accurate coverage properties, often exhibiting less sensitivity to the underlying dependence while most closely achieving the nominal coverage level. Additionally, linear weight-based EBEL1 generally performed similarly to, or somewhat better than, the best BEL method based on a data-driven block selection from among the FTK/AAR block rules and, at times, much better than the worst performer among the BEL methods with estimated blocks. Note as well that, while that the two block selection rules for BEL can produce similar coverages, their relative effectiveness often depends crucially on the underlying process, with no resulting clear best block selection for BEL. In the case of the strong positive AR(1) dependence model in Table [tab2], the AAR block selection for BEL performed well (i.e., better than EBEL1 or BEL/FTK approaches), but similar advantages in coverage accuracy did not necessarily carry over to other processes. In particular, for a process not approximated well by an AR(1) model, the BEL coverage rates from AAR block estimates may exhibit extreme over- or under-coverage under negative or positive dependence, respectively, and FTK block selections for BEL may prove better. Because of the blocking scheme in EBEL method and some of the method’s other connections to fixed-b asymptotics (see Section [sec4]), one might anticipate that there exist trade-offs in coverage accuracy (i.e., good size control properties) at the expense of power in testing, a phenomenon also associated with fixed-b asymptotics; cf.. This does seem to be the case. To illustrate, for various sample sizes n and processes, we approximated power curves for EBEL/BEL tests at the 10% level (based on the 90th percentile of the associated null limit law) along a sequence of local alternatives cn = μ0 + n− 1/2Σ1/2c, c = 0, 0.25, …, 5 where Σ = ∑k = − ∞∞Cov(X0, Xk); for example, with EBEL1, the power curves correspond to the rejection probabilities P( − n− 1logRn(cn) > q0.90) where q0.90 is a percentile from Table [tab1]. The alternative sequence cn was formulated to make power curves roughly comparable across processes with varying sample sizes, and so that the power curves can be plotted as a function of c = 0, 0.25, …, 5; for instance, by Corollary [co1](i), the asymptotic power curve of EBEL1 will be a function of c, as will the curves for BEL/EBEL2. Figures [fig2] and [fig3] display size adjusted power curves (APCs) for samples of size n = 500 based on 2000 simulations (curves are similar for n = 250, 1000 with additional results given in ). If a percentage α̂n denotes the actual size of the test for a given method and process (i.e., α̂n = 100% − coverage percentage in Table [tab2]), the APC is calibrated to have size 10% by vertically shifting the true power curve by 10% − α̂n; this allows the shapes of power curves to be more easily compared across methods. Figure [fig2] shows APCs for EBEL1/EBEL2 methods, where EBEL2 curves exhibit more power apparently as a result of combining two data block sets (i.e., forward/backward) in the EBEL construction rather than one; additionally, while EBEL2 power curves are quite similar across different weights, EBEL1 curves exhibit slightly more power for the constant weight function. Figure [fig3] shows APCs in comparing the linear weight-based EBEL2 method with BEL methods based on FTK/AAR block estimates. The APCs for EBEL2 generally tend to be smaller than those of BEL, though the APC of a block estimate-based BEL may not always dominate the associated curve of EBEL2. [fig2] [fig3] Conclusions =========== The proposed expansive block empirical likelihood(EBEL) is a type of variation on standard blockwise empirical likelihood (BEL) for time series which, instead of using a fixed block length b for a given sample size n, involves a nonstandard blocking scheme to capture the dependence structure. While the coverage accuracy of standard BEL methods can depend intricately on the block choice b (where the best b can vary with the underlying process), the EBEL method does not involve this type of block selection. As mentioned in the, we also anticipate that the EBEL method will generally have better rates of coverage accuracy compared to BEL. The simulations of Section [sec3] lend support to this notion, along with suggesting that the EBEL can be less sensitive to the strength of the underlying time dependence. While asymptotic coverage rates for BEL methods remain to be determined, we may offer the following heuristic based on analogs drawn to so-called “fixed-b asymptotics” (cf. Keifer, Vogelsang and Bunzel ; Bunzel et al. ; Kiefer and Vogelsang ), or related “self-normalization” (cf. Lobato ; Shao ) schemes. In asymptotic expansions of log-likelihood statistics from standard BEL formulations, the data blocks serve to provide a type of block-based variance estimator (cf. ) for purposes of normalizing scale and obtaining chi-square limits for log-BEL ratio statistics. Such variance estimators are consistent, requiring block sizes b which grow at a smaller rate than the sample size n (i.e., b− 1 + b/n → ∞ as n → ∞) and are known to have equivalences to variance estimators formulated as lag window estimates involving kernel functions and bandwidths b with similar behavior to block lengths b− 1 + b/n → ∞ (cf. ). That is, standard BEL intervals have parallels with normal theory intervals based on normalization with consistent lag window estimates. However, considering hypothesis testing with sample means, for example, there is some numerical and theoretical evidence (cf. ) that normalizing scale with inconsistent lag window estimates having fixed bandwidth ratios (e.g., b/n = C for some C ∈ (0, 1]) results in better size and lower power compared to normalization with consistent ones, though the former case requires calibrating intervals with nonnormal limit laws. Shao, Section 2.1, provides a nice summary of these points as well as the form of some of these distribution-free limit laws, which typically involve ratios of random variables defined by Brownian motion; cf.. While the EBEL method is not immediately analogous to normalizing with inconsistent variance estimators (as mentioned in Section [sec2.1], the usual EL expansions do not hold for EBEL), there are parallels in that the EBEL method does not use block lengths satisfying standard bandwidth conditions (cf. Section [sec2.1]), its blocking scheme itself appears in self-normalization literature (cf. Shao, Section 2) and confidence region calibration involves nonnormal limits based on Brownian motion. This heuristic in the mean case suggests that better coverage rates (and lower power) associated with fixed-b asymptotics over standard normal theory asymptotics may be anticipated to carry over to comparisons of EBEL to standard BEL formulations. Proofs of main results ====================== To establish Theorem [theorem1], we first require a lemma regarding a standard Brownian motion. For concreteness, suppose B(t) ≡ B(ω, t) = (B1(ω, t), …, Bd(ω, t))′, ω ∈ Ω, t ∈ [0, 1] is a random Cd[0, 1]-valued element defined on some probability space (Ω, F, P), where B1, …, Bd are again distributed as i.i.d. copies of standard Brownian motion on [0, 1]. In the following, we use the basic fact that each Bi( ⋅ ) is continuous on [0, 1] with probability 1 (w.p.1) along with the fact that increments of standard Brownian motion are independent; cf.. [le1] With probability 1, it holds that: [(iii)] min0 ≤ t < εa′B(t) < 0 < max0 ≤ t < εa′B(t) for all ε > 0 and a ∈ Rd,∥a∥ = 1. 0d is in the interior of the convex hull of B(t), 0 ≤ t ≤ 1. There exists a positive random variable M such that, for all a ∈ Rd, it holds that min0 ≤ t ≤ 1a′B(t) ≤ − M∥a∥ and M∥a∥ ≤ max0 ≤ t ≤ 1a′B(t). If assumption holds in addition,,, above hold upon replacing B(t) with f(t) = w(t)B(t), t ∈ [0, 1]. For real-valued Brownian motion, it is known thatmin0 ≤ t < εBi(t) < 0 < max0 ≤ t < εBi(t) holds for all ε > 0 w.p.1. (cf., Lemma 55); we modify the proof of this. Let {tn} ⊂ (0, 1) be a decreasing sequence where tn ↓ 0 as n → ∞. Pick and fix c1, …, cd ∈ { − 1, 1}, and define the event $A\_n \equiv A\_{n,c\_1,\ldots,c\_d} =\{\omega\in\Omega \dvtx c\_i B\_i(\omega,t\_n) >0, i=1,\ldots,d\}$. Then P(An) = 2− d for all n ≥ 1 by normality and independence. As the events Bn = ⋃k = n∞Ak, n ≥ 1, are decreasing, it holds that $$P \Biggl( \bigcap\_{n=1}^\infty B\_n \Biggr) = \lim\_{n\to\infty}P(B\_n) \geq\lim \_{n\to\infty}P(A\_n) = 2^{-d}.$$ Since ⋂n = 1∞Bn is a tail event generated by the independent random variables Bi(t1) − Bi(t2), Bi(t2) − Bi(t3), … for i = 1, …, d [i.e., increments of Brownian motion are independent and Bi(0) = 0), it follows from Kolmogorov’s 0–1 law that $1 =P ( \bigcap\_{n=1}^\infty B\_n ) = P ( \mbox{$An$ infinitely often (i.o.)} ]$. Hence $P ( \mbox{$An,c1,…,cd$}\break \mbox{i.o. for any $ci{1,-1}, i=1,…,d$} ) =1$ must hold, which implies part (i). For part (ii), if 0d is not in the interior convex hull of B(t), t ∈ [0, 1], then the supporting/separating hyperplane theorem would imply that, for some a ∈ Rd, ∥a∥ = 1, it holds that a′B(t) ≥ 0 for all t ∈ [0, 1], which contradicts part (i). To show part (iii), we use the events developed in part (i) and define $n\_{c\_1,\ldots,c\_d} = \min\{n \dvtx\mbox{$An,c1,…,cd$ holds}\} $. Define $M = \min\{ \|B\_i(t\_{n\_{c\_1,\ldots,c\_d}})\| \dvtx c\_1,\ldots,c\_d \in\{-1,1\}, i=1,\ldots,d\}>0$. For a = (a1, …, ad)′ ∈ Rd, let cia = max{ − sign(ai), 1}, i = 1, …, d. Then a′B(tnc1a, …, cda) = − ∑i = 1d∣aiBi(tnc1a, …, cda)∣ ≤ − M∥a∥, and likewise a′B(tn− c1a, …, − cda) = ∑i = 1d∣aiBi(tn− c1a, …, − cda)∣ ≥ M∥a∥. This estab-lishes (iii). Part (iv) follows from the fact that w(t) > 0 for t ∈ (0, c), and we may take the positive sequence {tn} ⊂ (0, c) in the proof of part (i). Then the results for B(t) imply the same hold upon substituting f(t) = w(t)B(t), t ∈ [0, 1]. Proof of Theorem [theorem1] The set $K\_d = \{a\in\mathbb {R}^d \dvtx\min\_{0 \leq t \leq1} (1+a^\prime f(t))>0\}$ is open, bounded and convex (w.p.1), where boundedness follows from Lemma [le1](iii), (iv). Likewise, the closure $\overline{K}\_d = \{a\in \mathbb {R}^d \dvtx\min\_{0 \leq t \leq1} (1+a^\prime f(t))\geq0\}$ is convex and bounded. Since min0 ≤ t ≤ 1(1 + a′f(t)) is a continuous function in a ∈ Rd, one may apply the dominated convergence theorem (DCT) [with the fact that min0 ≤ t ≤ 1(1 + a′f(t)] is bounded away from 0 on closed balls inside Kd around a) to show that partial derivatives of gd( ⋅ ) at a ∈ Kd (of all orders) exist, with first and second partial derivatives given by $$\frac{\partial g\_d(a)}{\partial a} = - \int\_0^1 \frac{f(t)}{ 1+a^\prime f(t)}\,dt, \qquad\frac{\partial^2 g\_d(a)}{\partial a\, \partial a^\prime} = \int\_0^1 \frac{f(t)f(t)^\prime}{ [1+a^\prime f(t)]^2}\,dt.$$ Because ∫01f(t)f(t)′ dt is positive definite by Lemma [le1](i), (iv) and the continuity of f, the matrix ∂2gd(a)/∂a ∂a′ is also positive definitive for all a ∈ Kd, implying gd is strictly convex on Kd. By Jensen’s inequality, it also holds that gd is convex on $\overline{K}\_d$. Note for $a\in\overline{K}\_d$, $g\_d(a) \geq- \int\_0^1 \log(1 + \sup\_{a\in\overline{K}\_d}\\|a\\|\cdot\sup\_{0 \leq t \leq1} \\|f(t)\\| ) > -\infty$ holds, so that $I \equiv\inf\_{a\in\overline{K}\_d}g\_d(a)$ exists. Additionally, 0d ∈ Kd with gd(0d) = 0 and ∂gd(0d)/∂a = − ∫01f(t) dt, where the components of ∫01f(t) dt are all nonzero (w.p.1) by normality and independence; by the continuity of partial derivatives on the open set Kd, there then exists ā ∈ Kd such that ā′∫01f(t) dt > 0 holds with the components of − ∫01f(t) and ∂gd(ā)/∂a having the same sign. By strict convexity, gd(0d) − gd(ā) > [∂gd(ā)/∂a]′(0d − ā) > 0 follows, implying I < 0 and I = infa ∈ K̃dgd(a) for the level set $\tilde{K}\_d \equiv\{a \in\overline{K}\_d\dvtx g\_d(a)\leq0\}$. Then, there exists a sequence an ∈ K̃d such that gd(an) < I + n− 1 for n ≥ 1. Since {an} is bounded, we may extract a subsequence such that ank → Yd ∈ K̃d, for some Yd ∈ K̃d. Pick δ ∈ (0, 1). Then, by the DCT, $$\begin{aligned} \underline{\lim}\, g\_d(a\_{n\_k}) &\geq& \underline{\lim} \int \_{\{t: a\_{n\_k}^\prime f(t)> -1 + \delta\}} -\log\bigl(1 + a\_{n\_k}^\prime f(t) \bigr)\,dt \\ & = & \int\_{\{t: Y\_d^\prime f(t)> -1 + \delta\}} -\log\bigl(1 +Y\_d^\prime f(t) \bigr)\,dt \\ &=& g\_d(Y\_d) + \int\_{\{t: Y\_d^\prime f(t) \leq-1 + \delta\}} \log \bigl(1 +Y\_d^\prime f(t) \bigr)\,dt.\end{aligned}$$ Note that because gd(Yd) ∈ ( − ∞, 0], it follows that − ∫{t : Yd′f(t) < 0}log(1 + Yd′f(t)) dt < ∞ and $\{ t\in[0,1]\dvtx Y\_d^\prime f(t)=-1\}$ has Lebesgue measure zero. Hence, the DCT yields $$\lim\_{\delta\rightarrow0}-\int\_{\{t: Y\_d^\prime f(t) \leq-1 + \delta \}} \log\bigl(1 +Y\_d^\prime f(t) \bigr)\,dt=0.$$ Consequently, $$I \geq\overline{\lim}\, g\_d(a\_{n\_k})\geq\underline{\lim}\, g\_d(a\_{n\_k}) \geq g\_d(Y\_d) \geq I,$$ establishing the existence of a minimizer Yd of gd on $\overline {K}\_d$ such that − ∞ < I = gd(Yd) < 0. For part (ii) of Theorem [theorem1], note yn = (1 − n− 1)Yd + n− 10d ∈ Kd, n ≥ 1, by convex geometry, as Kd is the convex interior of $\overline{K}\_d$. Then gd(yn) ≤ (1 − n− 1)gd(Yd) holds by convexity of gd and gd(0d) = 0, implying 0 ≤ n[gd(yn) − gd(Yd)] ≤ − gd(Yd) < ∞, from which it follows that gd(yn) → gd(Yd) and, by the mean value theorem, $$0 \leq n\bigl[g\_d(y\_n) - g\_d(Y\_d) \bigr]=\int\_0^1 \frac{Y\_d^\prime f(t)}{1 + c\_n Y\_d^\prime f(t)} \,dt \leq-g\_d(Y\_d)$$ holds for some (1 − n− 1) < cn < 1 [note cnYd ∈ Kd so min0 ≤ t ≤ 1(1 + cnYd′f(t)) > 0 for all n]; the latter implies 0 ≤ ∫{t : Yd′f(t) < 0} − Yd′f(t)/[1 + cnYd′f(t)] dt ≤ ∫{t : Yd′f(t) > 0}Yd′f(t) < ∞ so that Fatou’s lemma yields $$0 \leq\int\_{\{t: Y\_d^\prime f(t)<0\}} - \frac{Y\_d^\prime f(t)}{1 + Y\_d^\prime f(t)} \,dt <\infty$$ as n → ∞, and consequently ∫011/[1 + Yd′f(t)] dt < ∞. We may then apply the DCT to find $$\lim\_{n\to\infty} \int\_0^1 \frac{Y\_d^\prime f(t)}{1 + c\_n Y\_d^\prime f(t)} \,dt = \int\_0^1 \frac{Y\_d^\prime f(t)}{1 + Y\_d^\prime f(t)} \,dt \in[0,\infty).$$ Also by convexity and 0d ∈ Kd, 0 > gd(Yd) − gd(0d) > [∂gd(0d)/∂a]′(Yd − 0d) holds (w.p.1), implying Yd′∫01f(t) dt > 0 from ∂gd(0d)/∂a = − ∫01f(t) dt. This establishes part (ii) of Theorem [theorem1]. To show uniqueness of the minimizer, we shall construct sequences with the same properties in the proof of part (ii) above. Suppose x ∈ K̃d such that gd(x) = I = gd(Yd). Defining xn = (1 − n− 1)x + n− 10d ∈ Kd and yn = (1 − n− 1)Yd + n− 10d ∈ Kd for n ≥ 1, by convexity we have 0 ≥ gd(x) − gd(yn) > [∂gd(yn)/∂a]′(x − yn), so that taking limits yields 0 ≥ − ∫01(x − Yd)′f(t)/[1 + Yd′f(t)] dt, and, by symmetry, 0 ≥ − ∫01(Yd − x)′f(t)/[1 + x′f(t)] dt as well. Adding these terms gives $$0 \geq\int\_0^1 \frac{[(x-Y\_d)^\prime f(t)]^2 }{(1 + x^\prime f(t))(1 + Y\_d^\prime f(t))} \,dt,$$ implying that x = Yd by Lemma [le1](iv) and the continuity of f. Finally, to establish part (iii), if Yd ∈ Kd, then $0\_d = \partial g\_d(Y\_d)/\partial a = - \int\_0^1 f(t)/\break [1 + Y\_d^\prime f(t)]\,dt$ must hold. If there exists another $b \in\overline{K}\_d$ satisfying ∫01f(t)/[1 + b′f(t)] dt = 0d, then adding ∂gd(Yd)/∂a to this integral and multiplying by (Yd − b)′ yields 0 = ∫01[(b − Yd)′f(t)]2/[(1 + b′f(t))(1 + Yd′f(t))] dt, implying that b = Yd. Also, if 0d = ∫01f(t)/[1 + b′f(t)] dt = − ∂gd(b)/∂a holds for some b ∈ Kd, then strict convexity implies gd(a) − gd(b) > [∂gd(b)/∂a]′(a − b) = 0 for all $a \in\overline{K}\_d$, implying b = Yd is the unique minimizer of gd. Proof of Theorem [theorem2] Under assumption (A.2), we use Skorohod’s embedding theorem (cf., Theorem 1.1.04) to embed {Sn( ⋅ )} and {B( ⋅ )} in a larger probability space (Ω, F, P) such that sup0 ≤ t ≤ 1∥Σ− 1/2Sn(t)/n1/2 − B(t)∥ → 0 w.p.1 (P). Defining Tn(t) = w(t)Sn(t) and f(t) = w(t)B(t), t ∈ [0, 1], the continuity of w under assumption (A.1) then implies $$\label{eqn:Tn} \sup\_{0 \leq t \leq1}\biggl{\Vert}\frac{\Sigma ^{-1/2}T\_n(t)}{n^{1/2}} -f(t) \biggr{\Vert}\rightarrow0 \qquad\mbox{w.p.1.}$$ Note that Ti, μ0 = w(i/n)∑j = 1i(Xj − μ0) = Tn(i/n), i = 1, …, n. By ([eqn:Tn]) and Lemma [le1], 0d is in the interior convex hull of $\{T\_{i,\mu\_0}\dvtx i=1,\ldots ,n\}$ eventually (w.p.1) so that Ln(μ0) > 0 eventually (w.p.1). That is, by Lemma [le1](iv), there exists A ∈ F with P(A) = 1 and, for ω ∈ A, min0 ≤ t ≤a′f(ω, t) ≤ − M(ω) and max0 ≤ t ≤a′f(ω, t) ≥ M(ω) hold for some M(ω) > 0 and all a ∈ Rd, ∥a∥ = 1. Then, ([eqn:Tn]) implies $\min\_{1\leq i \leq n} a^\prime\Sigma ^{-1/2}T\_n(\omega, i/n) < 0 < a^\prime\max\_{1\leq i \leq n} \Sigma ^{-1/2}T\_n(\omega,\break i/n)$ holds for all a ∈ Rd, ∥a∥ = 1 eventually, implying 0d is in the interior convex hull of $\{\Sigma ^{-1/2}T\_n(i/n)\dvtx i=1,\ldots,n\}$. Hence, eventually (w.p.1) as in ([eqn:t0]), we can write $$\frac{1}{n} R\_n(\mu\_0) = -\frac{1}{n}\sum \_{i=1}^n \log\bigl(1 + \lambda \_{n,\mu\_0}^{\prime} T\_{i,\mu\_0}\bigr) = \frac{1}{n}\sum \_{i=1}^n \log\bigl(1 + \ell\_n^{\prime} T\_{i,n}\bigr),$$ where Ti, n ≡ Σ− 1/2Tn(i/n)/n1/2, i = 1, …, n and ℓn = n1/2Σ1/2λn, μ0 and $$\begin{aligned} \label{eqn:Tn2} \min\_{i=1,\ldots,n}\bigl(1 + \ell\_n^{\prime} T\_{i,n}\bigr)&>&0,\qquad \sum\_{i=1}^n \frac {1}{n(1 + \ell\_n^{\prime}T\_{i,n})} =1, \nonumber \\[-8pt] \\[-8pt] \nonumber \sum\_{i=1}^n \frac{T\_{i,n}}{n(1 + \ell\_n^{\prime} T\_{i,n})}& =&0\_d.\end{aligned}$$ From here, all considered convergence will be pointwise along some fixed ω ∈ A where P(A) = 1, and we suppress the dependence of terms f, Tn, etc. on ω. Then, ([eqn:Tn2]) [i.e., mini = 0, …, nℓn′(Σ− 1/2Tn(i/n)/n1/2) > − 1] with ([eqn:Tn]) and Lemma [le1](iv) implies that ∥ℓn∥ is bounded eventually. For any subsequence {nj} of {n}, we may extract a further subsequence {nk} ⊂ {nj} such that ℓnk → b for some $b \in\overline{K}\_d$. For simplicity, write nk ≡ k in the following. We will show below that k− 1logRk(μ0) → gd(Yd) and that ℓk → Yd, where $Y\_d\in\overline{K}\_d$ denotes the minimizer of gd(a) = − ∫01log(1 + a′f(t)) dt, $a\in\overline{K}\_d$ under Theorem [theorem1]. Since the subsequence {nj} is arbitrary, we then have n− 1logRn(μ0) → gd(Yd) and ℓn → Yd w.p.1, implying the distributional convergence in Theorem [theorem2]. Define Yε = (1 − ε)Yd + ε0d ∈ Kd (since 0d ∈ Kd, the interior of $\overline{K}\_d$) for ε ∈ (0, 1). From Yε ∈ Kd, min0 ≤ t ≤ 1(1 + Yε′f(t)) > δ holds for some δ > 0 (dependent on ε) so that min1 ≤ i ≤ k(1 + Yε′Ti, k) > δ holds eventually by ([eqn:Tn]). Then because $$g\_{d,k}(a) \equiv- \frac{1}{k} \sum \_{i=1}^k \log\bigl(1+ a^\prime T\_{i,k}\bigr)$$ is strictly convex on $a\in\{ y\in\mathbb{R}^d \dvtx\min\_{1\leq i \leq k} (1+y^\prime T\_{i,n})>0\}$ with a unique minimizer at ℓk by ([eqn:Tn2]) [i.e., ∂gd, k(ℓk)/∂a = 0d holds and strict convexity follows when k− 1∑i = 1kTi, kTi, k′ is positive definite, which holds eventually from k− 1∑i = 1kTi, kTi, k′ → ∫01f(t)f(t)′ dt by ([eqn:Tn]) and the DCT, with the latter matrix being positive definite w.p.1 by Lemma [le1](iv) and continuity of f], we have that $$g\_{d,k}(Y\_\epsilon) \geq g\_{d,k}(\ell\_k) = \frac{1}{k} \log R\_k(\mu\_0).$$ Define ḡd, k(a) ≡ − k− 1∑i = 1klog(1 + a′f(i/k)), a ∈ Kd. Then, by Taylor expansion [recalling min0 ≤ t ≤ 1(1 + Yε′f(t)) > δ, min1 ≤ i ≤ k(1 + Yε′Ti, k) > δ], $$\begin{aligned} &&\bigl\|g\_{d,k}(Y\_\epsilon) - \bar{g}\_{d,k}(Y\_\epsilon)\bigr\| \\ && \qquad\leq\frac{1}{k}\sum\_{i=1}^k \bigl\|Y\_\epsilon^\prime\bigl(T\_{i,k} -f(i/k)\bigr)\bigr\| \biggl( \frac{1}{1 + Y\_\epsilon^\prime T\_{i,k}} + \frac{1}{1 + Y\_\epsilon ^\prime f(i/k)} \biggr) \\ &&\qquad\leq \\|Y\_d\\| 2 \delta^{-1} \max\_{1 \leq i \leq k} \bigl\\|T\_{i,k} -f(i/k) \bigr\\| \rightarrow0\end{aligned}$$ from ([eqn:Tn]) and Theorem [theorem1]. Also, by the DCT, ḡd, k(Yε) → gd(Yε) as k → ∞. Hence, $g\_{d}(Y\_\epsilon) \geq\overline{\lim}\, g\_{d,k}(\ell \_k)$ holds and, since gd(Yε) ≤ (1 − ε)gd(Yd) by convexity and gd(0d) = 0, we have, letting ε → 0, that $$\label{eqn:g1} g\_{d}(Y\_d) \geq\overline{\lim}\, g\_{d,k}(\ell\_k).$$ Recalling $\ell\_k \rightarrow b \in\overline{K}\_d$, define bε = (1 − ε)b + ε0d ∈ Kd, so that min0 ≤ t ≤ 1(1 + bε′f(t)) > 0. Then, ḡd, k(bε) → gd(bε) by ([eqn:Tn]) and the DCT. And, by Taylor expansion and using ([eqn:Tn2]), $$\begin{aligned} &&\overline{\lim} \bigl\|g\_{d,k}(\ell\_k) - \bar{g}\_{d,k}(b\_\epsilon)\bigr\| \\ &&\qquad\leq \overline{\lim} \max\_{1 \leq i \leq k}\bigl\| \ell\_k^\prime T\_{i,k} - b\_\epsilon^\prime f(i/k)\bigr\| \Biggl( 1 + \frac{1}{k}\sum\_{i=1}^k \frac{1}{1 + b\_\epsilon^\prime f(i/k)} \Biggr) \\ &&\qquad\leq \epsilon\sup\_{0 \leq t \leq1} \bigl\|b^\prime f(t)\bigr\| \biggl( 1 + \int\_0^1 \frac{1}{1 + b\_\epsilon^\prime f(t)}\,dt \biggr)\equiv C( \epsilon),\end{aligned}$$ following from ([eqn:Tn]) and the DCT. Hence we have $$\label{eqn:g2} \underline{\lim}\,g\_{d,k}(\ell\_k) \geq g\_{d}(b\_\epsilon) - C(\epsilon).$$ We will show below that $$\label{eqn:b} \int\_0^1 \frac{1}{1 + b^\prime f(t)} \,dt <\infty$$ holds, in which case, limε → 0∫01[1 + bε′f(t)]− 1 dt = ∫01[1 + b′f(t)]− 1 dt < ∞ by the DCT and so that C(ε) → 0 as ε → 0 [noting sup0 ≤ t ≤ 1∣b′f(t)∣ < ∞ since f is continuous and $\overline{K}\_d$ is bounded by Theorem [theorem1]]. By Fatou’s lemma and the DCT, $\underline{\lim}\_{\epsilon\rightarrow0} g\_{d}(b\_\epsilon)\geq g\_{d}(b)$ holds also. Hence, by ([eqn:g1])–([eqn:g2]), we then have $$g\_{d}(Y\_d) \geq\overline{\lim}\, g\_{d,k}( \ell\_k) \geq\underline{\lim}\,g\_{d,k}(\ell\_k) \geq g\_{d}(b) \geq g\_{d}(Y\_d),$$ implying b = Yd by the uniqueness of the minimizer and limk → ∞k− 1 × logRk(μ0) = gd(Yd). To finally show ([eqn:b]), let $A = \{t\in[0,1]\dvtx1 + b^\prime f(t) \leq d\}$ for some 0 < d ≤ 1/2 chosen so that $\{t\in[0,1]\dvtx1 + b^\prime f(t)=d\}$ has Lebesgue measure zero (since f is continuous). Let Ac = [0, 1] \ A. Using the indicator function I( ⋅ ), define a simple function $$h\_k(t) \equiv\sum\_{i=1}^k \frac{\ell\_k^\prime T\_{i,k}}{1 + \ell \_k^\prime T\_{i,k}} \mathbb{I} \biggl(t \in\biggl(\frac{i-1}{k}, \frac {i}{k} \biggr] \biggr),\qquad t\in[0,1].$$ From ([eqn:Tn2]), note that $$\int\_A h\_k(t)\,dt + \int\_{A^c} h\_k(t) \,dt = \frac{1}{k} \sum\_{i=1}^k \frac {\ell\_k^\prime T\_{i,k}}{1 + \ell\_k^\prime T\_{i,k}} =0\_d.$$ From ([eqn:Tn]), I(t ∈ Ac)hk(t) → I(t ∈ Ac)b′f(t)/(1 + b′f(t)) [almost everywhere (a.e.) Lebesgue measure] and for large k, I(t ∈ Ac)∣hk(t)∣ ≤ 2C/d holds for t ∈ [0, 1], since eventually max1 ≤ i ≤ k∣ℓk′Ti, k∣ is bounded by a constant C > 0 and also 1 + b′f(t) + (ℓk′Ti, k − b′f(t)) > d/2 for t ∈ Ac, (i − 1)/k < t ≤ i/k. Then, by the DCT, ∫Achk(t) dt → ∫Acb′f(t)/(1 + b′f(t)) dt, and for δ ∈ (0, 1), note − I(t ∈ A)hk(t) ≥ h1, k(t) for $$h\_{1,k}(t) \equiv\sum\_{i=1}^k \frac{-\ell\_k^\prime T\_{i,k}}{1 + \ell \_k^\prime T\_{i,k} + \delta\mathbb{I}(\operatorname{sign}(\ell\_k^\prime T\_{i,k}) <0)} \mathbb{I} \biggl(t \in\biggl(\frac{i-1}{k}, \frac{i}{k} \biggr]\cap A \biggr).$$ Since ∣h1, k(t)∣ ≤ C/δ and h1, k(t) → − I(t ∈ A)b′f(t)/(1 + b′f(t) + δ) (a.e. Lebesgue measure), by the DCT $$\begin{aligned} 0&\leq&\int\_A \frac{-b^\prime f(t)}{1+b^\prime f(t) + \delta}\,dt = \lim \_{k\to\infty} \int\_A h\_{1,k}(t)\,dt \leq\lim\_{k\to\infty} \int\_A -h\_{k}(t)\,dt \\ &=& \int\_{A^c} \frac{b^\prime f(t)}{1+b^\prime f(t)}\,dt\end{aligned}$$ using ∫A − hk(t) dt = ∫Achk(t) dt. Letting δ → 0, Fatou’s lemma gives $$0\leq\int\_A \frac{-b^\prime f(t)}{1+b^\prime f(t) }\,dt \leq\int \_{A^c} \frac{b^\prime f(t)}{1+b^\prime f(t)}\,dt<\infty.$$ Because − b′f(t) ≥ 1/2 on A, ∫A[1 + b′f(t)]− 1 dt < ∞ holds, implying ([eqn:b]). Acknowledgements ================ The authors are very grateful to Editor Runze Li, two Associate Editors and two referees for constructive and insightful comments which improved the manuscript, especially the numerical studies. [id=suppA] 35 [mr] ()... [mr] ()... [mr] ()... [mr] ()... [mr], ()... [mr] ()... [mr] ()... [mr] ()., ed.,. [mr] ()... [mr] ()..,. [auto:STB\|2013/10/14\|10:36:11] ()... [mr] ()... [auto:STB\|2013/10/14\|10:36:11] ()... [mr], ()... [mr] ()... [mr] ()... [mr] ()..,. [mr] ()... [mr] ().. In.,. [mr] ()... [mr] ()... [auto:STB\|2013/10/14\|10:36:11], ().. [mr], ()... [mr] ()... [mr] ()... [mr] ()... [mr] ()... [mr], ()..,. [mr] ()... [mr] ()... [mr] ()... [mr], ()... [mr] ()..,. [mr] ()... A nonstandard empirical likelihood for time series ================================================== , Standard blockwise empirical likelihood (BEL) for stationary, weakly dependent time series requires specifying a fixed block length as a tuning parameter for setting confidence regions. This aspect can be difficult and impacts coverage accuracy. As an alternative, this paper proposes a new version of BEL based on a simple, though nonstandard, data-blocking rule which uses a data block of every possible length. Consequently, the method does not involve the usual block selection issues and is also anticipated to exhibit better coverage performance. Its nonstandard blocking scheme, however, induces nonstandard asymptotics and requires a significantly different development compared to standard BEL. We establish the large-sample distribution of log-ratio statistics from the new BEL method for calibrating confidence regions for mean or smooth function parameters of time series. This limit law is not the usual chi-square one, but is distribution-free and can be reproduced through straightforward simulations. Numerical studies indicate that the proposed method generally exhibits better coverage accuracy than standard BEL. [class=AMS] Introduction ============ For independent, identically distributed data (i.i.d.), Owen introduced empirical likelihood (EL) as a general methodology for re-creating likelihood-type inference without a joint distribution for the data, as typically specified in parametric likelihood. However, the i.i.d. formulation of EL fails for dependent data by ignoring the underlying dependence structure. As a remedy, Kitamura proposed so-called blockwise empirical likelihood (BEL) methodology for stationary, weakly dependent processes, which has been shown to provide valid inference in various scenarios with time series (cf. ). Similarly to the i.i.d. EL version, BEL creates an EL log-ratio statistic having a chi-square limit for inference, but the BEL construction crucially involves blocks of consecutive observations in time, rather than individual observations. This data-blocking serves to capture the underlying time dependence and related concepts have also proven important in defining resampling methodologies for dependent data, such as block bootstrap and time subsampling methods. However, the coverage accuracy of BEL can depend crucially on the block length selection, which is a fixed value 1 ≤ b ≤ n for a given sample size n, and appropriate choices can vary with the underlying process (a point briefly illustrated at the end of this section). To advance the BEL methodology in a direction away from block selection with a goal of improved coverage accuracy, we propose an alternative version of BEL for stationary, weakly dependent time series, called an expansive block empirical likelihood (EBEL). The EBEL method involves a nonstandard, but simple, data-blocking rule where a data block of every possible length is used. Consequently, the method does not involve a block length choice in the standard sense. We investigate EBEL in the prototypical problem of inference about the process mean or a smooth function of means. For setting confidence regions for such parameters, we establish the limiting distribution of log-likelihood ratio statistics from the EBEL method. Because of the nonstandard blocking scheme, the justification of this limit distribution requires a new and substantially different treatment compared to that of standard BEL (which closely resembles that of EL for i.i.d. data in its large-sample development; cf. ). In fact, unlike with standard BEL or EL for i.i.d. data, the limiting distribution involved is nonstandard and not chi-square. However, the EBEL limit law is distribution-free, corresponding to a special integral of standard Brownian motion on [0, 1], and so can be easily approximated through simulation to obtain appropriate quantiles for calibrating confidence regions. In addition, we anticipate that the EBEL method may have generally better coverage accuracy than standard BEL methods, though formally establishing and comparing convergence rates is beyond the scope of this manuscript (and, in fact, optimal rates and block sizes for even standard BEL remain to be determined). Simulation studies, though, suggest that interval estimates from the EBEL method can perform much better than the standard BEL approach, especially when the later employs a poor block choice, and be less sensitive to the dependence strength of the underlying process. The rest of manuscript is organized as follows. We end this section by briefly recalling the standard BEL construction with overlapping blocks and its distributional features. In Section [sec2], we separately describe the EBEL method for inference on process means and smooth function model parameters, and establish the main distributional results in both cases. These results require introducing a new type of limit law based on Brownian motion, which is also given in Section [sec2]. Additionally, Section [sec2.1] describes how the usual EL theory developed by Owen, and often underlying many EL arguments including the time series extensions of BEL, fails here and requires new technical developments; consequently, the theory provided may be useful for future developments of EL (with an example given in Section [sec2.4]). Section [sec3] provides a numerical study of the coverage accuracy of the EBEL method and comparisons to standard BEL. Section [sec4] offers some concluding remarks and heuristic arguments on the expected performance of EBEL. Proofs of the main results appear in Section [sec5] and in supplementary materials, where the latter also presents some additional simulation summaries. To motivate what follows, we briefly recall the BEL construction, considering, for concreteness, inference about the mean EXt = μ ∈ Rd of a vector-valued stationary stretch X1, …, Xn. Upon choosing an integer block length 1 ≤ b ≤ n, a collection of maximally overlapping (OL) blocks of length b is given by $\{ (X\_i,\ldots,X\_{i+b-1})\dvtx i=1,\ldots,N\_b \equiv n-b+1\}$. For a given μ ∈ Rd value, each block in the collection provides a centered block sum Bi, μ ≡ ∑j = ii + b − 1(Xj − μ) for defining a BEL function $$\label{eqn:stan} L\_{{\mathrm{BEL}},n}(\mu) = \sup\Biggl\{ \prod \_{i=1}^{N\_b} p\_i \dvtx p\_i \geq0, \sum\_{i=1}^{N\_b}p\_i = 1, \sum\_{i=1}^{N\_b}p\_i B\_{i,\mu} = 0\_d \Biggr\}$$ and corresponding BEL ratio RBEL, n(μ) = Ln(μ)/Nb− Nb, where above 0d = (0, …, 0)′ ∈ Rd. The function LBEL, n(μ) assesses the plausibility of a value μ by maximizing a multinomial likelihood from probabilities {pi}i = 1Nb assigned to the centered block sums Bi, μ under a zero-expectation constraint. Without the linear mean constraint in ([eqn:stan]), the multinomial product is maximized when each pi = 1/Nb (i.e., the empirical distribution on blocks), defining the ratio RBEL, n(μ). Under certain mixing and moment conditions entailing weak dependence, and if the block b grows with the sample size n but at a smaller rate (i.e., b− 1 + b2/n → 0 as n → ∞), the log-EL ratio of the standard BEL has chi-square limit $$\label{eqn:stanl} -\frac{2}{b} \log R\_{{\mathrm{BEL}},n}(\mu\_0) \stackrel{d} {\rightarrow} \chi^2\_d,$$ at the true mean parameter μ0 ∈ Rd; cf. Kitamura. Here b− 1 represents an adjustment in ([eqn:stanl]) to account for OL blocks and, for i.i.d. data, a block length b = 1 above produces the EL distributional result of Owen. To illustrate the connection between block selection and performance, Figure [fig1] shows the coverage rate of nominal 90% BEL confidence intervals $\{\mu\in\mathbb{R}\dvtx-2/b \log R\_{{\mathrm{BEL}},n}(\mu\_0) \leq\chi ^2\_{1,0.9}\}$, as a function of the block size b, for estimating the mean of three different MA(2) processes based on samples of size n = 100. One observes that the coverage accuracy of BEL varies with the block length and that the best block size can depend on the underlying process. The EBEL method described next is a type of modification of the OL BEL version, without a particular fixed block length selection b. [fig1] Expansive block empirical likelihood ==================================== Mean inference -------------- Suppose X1, …, Xn represents a sample from a strictly stationary process $\{{X\_t}\dvtx t\in{\mathbb{Z}}\}$ taking values in Rd, and consider a problem about inference on the process mean EXt = μ ∈ Rd. While the BEL uses data blocks of a fixed length b for a given sample size n, the EBEL uses overlapping data blocks {(X1), (X1, X2), …, (X1, …, Xn)} that vary in length up to the longest block consisting of the entire time series. Hence this block collection, which constitutes a type of forward “scan” in the block subsampling language of McElroy and Politis , contains a data block of every possible length b for a given sample size n. This block sequence also appears in fixed-b asymptotic schemes and related self-normalization approaches; cf. Shao, Section 2; see also Section [sec4] here. In this sense, these blocks are interesting and novel to consider in a BEL framework. Other block schemes may be possible and potentially applied for practical gain (e.g., improved power), where the theoretical results of this paper could also directly apply. We leave this largely for future research, but we shall give one example of a modified, though related, blocking scheme in Section [sec2.4] while focusing the exposition on the block collection above [i.e., the alternative blocking incorporates a backward scan {(Xn), (Xn, Xn − 1), …, (Xn, …, X1)} with similar theoretical development]. Let $w\dvtx[0,1]\rightarrow[0,\infty)$ denote a nonnegative weighting function. To assess the likelihood of a given value of μ, we create centered block sums Ti, μ = w(i/n)∑j = 1i(Xj − μ), i = 1, …, n, and define a EBEL function $$\label{eqn:EBEL} L\_{n}(\mu) = \sup\Biggl\{ \prod \_{i=1}^{n} p\_i \dvtx p\_i \geq0, \sum\_{i=1}^{n}p\_i = 1, \sum\_{i=1}^{n} p\_i T\_{i,\mu} = 0\_d \Biggr\}$$ and ratio Rn(μ) = n− nLn(μ). After defining the block sums, the computation of Ln(μ) is analogous to the BEL version and essentially the same as that described by Owen for i.i.d. data. Namely, when the zero 0d vector lies in the interior convex hull of $\{T\_{i,\mu}\dvtx i=1,\ldots,n\} $, then Ln(μ) is the uniquely achieved maximum at probabilities pi = 1/[n(1 + λn, μ′Ti, μ)] > 0, i = 1, …, n, with a Lagrange multiplier λn, μ ∈ Rd satisfying $$\label{eqn:t0} \sum\_{i=1}^n \frac{T\_{i,\mu}}{n(1 + \lambda\_{n,\mu}^\prime T\_{i,\mu}) } = 0\_d;$$ see for these and other computational details. Regarding the weight function above in the EBEL formulation, more details are provided below and in Section [sec2.2]. The next section establishes the limiting distribution of the log-EL ratio from the EBEL method for setting confidence regions for the process mean μ parameter. However, it is helpful to initially describe how the subsequent developments of EL differ from previous ones with i.i.d. or weakly dependent data (cf. for BEL). The standard arguments for developing EL results, due to Owen (page 101), typically begin from algebraically re-writing ([eqn:t0]) to expand the Lagrange multiplier. If we consider the real-valued case d = 1 for simplicity, this becomes $$\lambda\_{n,\mu} = \frac{\sum\_{i=1}^n T\_{i,\mu}}{\sum\_{i=1}^n T\_{i,\mu }^2 } + \frac{ \lambda\_{n,\mu}^2}{\sum\_{i=1}^n T\_{i,\mu}^2} \sum \_{i=1}^n \frac{T\_{i,\mu}^3}{1 + \lambda\_{n,\mu}^\prime T\_{i,\mu}}.$$ In the usual independence or weak dependence cases of EL [e.g., where Bi, μ from ([eqn:stan]) replaces Ti, μ in the Lagrange multiplier above], the first right-hand side term dominates the second, which gives a substantive form for λn, μ as a ratio of sample means and consequently drives the large sample results (i.e., producing chi-square limits). However, in the EBEL case here, both terms on the right-hand side above have the same order, implying that the standard approach to developing EL results breaks down under the EBEL blocking scheme. The proofs here use a different EL argument than the standard one mentioned above, involving no asymptotic expansions of the Lagrange multiplier or Taylor expansions of the EL ratio based on these. The large sample results for the EBEL method require two mild assumptions stated below. Let Cd[0, 1] denote the metric space of all Rd-valued continuous functions on [0, 1] with the supremum metric ρ(g1, g2) ≡ sup0 ≤ t ≤ 1∥g1(t) − g2(t)∥, and let B(t) = (B1(t), …, Bd(t))′, 0 ≤ t ≤ 1, denote a Cd[0, 1]-valued random variable where B1(t), …, Bd(t) are i.i.d. copies of standard Brownian motion on [0, 1]. [((A.2))] The weight function $w\dvtx[0,1]\rightarrow[0,\infty)$ is continuous on [0, 1] and is strictly positive on an interval (0, c) for some c ∈ (0, 1]. Let EXt = μ0 ∈ Rd denote the true mean of the stationary process {Xt} and suppose d × d matrix Σ = ∑j = − ∞∞Cov(X0, Xj) is positive definite. For the empirical process Sn(t) on t ∈ [0, 1] defined by linear interpolation of $\{ S\_n(i/n) = \sum\_{j=1}^i (X\_j - \mu\_0) \dvtx i=0,\ldots , n \}$ with Sn(0) = 0, it holds that $ S\_n(\cdot)/n^{1/2} \stackrel{d}{\rightarrow} \Sigma^{1/2} B(\cdot)$ in Cd[0, 1]. Assumption (A.1) is used to guarantee that, in probability, the EBEL ratio Rn(μ0) positively exists at the true mean, which holds for uniformly weighted blocks w(t) = 1, t ∈ [0, 1], for example. Assumption (A.2) is a functional central limit theorem for weakly dependent data, which holds under appropriate mixing and moment conditions on {Xt}. Main distributional results --------------------------- To state the limit law for the log-EBEL ratio ([eqn:EBEL]), we require a result regarding a vector B(t) = (B1(t), …, Bd(t))′, 0 ≤ t ≤ 1, of i.i.d. copies B1(t), …, Bd(t) of standard Brownian motion on [0, 1]. Indeed, the limit distribution of − 2logRn(μ0) is a nonstandard functional of the vector of Brownian motion B( ⋅ ). Theorem [theorem1] identifies key elements of the limit law and describes some of its basic structural properties. [theorem1] Suppose that B(t) = (B1(t), …, Bd(t))′, 0 ≤ t ≤ 1, is defined on a probability space, and let f(t) = w(t)B(t), 0 ≤ t ≤ 1, where w( ⋅ ) satisfies assumption. Then, with probability 1 (w.p.1), there exists an Rd-valued random vector Yd satisfying the following: [(iii)] Yd is the unique minimizer of $$g\_d(a) \equiv-\int\_0^1 \log \bigl( 1 + a^\prime f(t)\bigr)\,dt\qquad \mbox{for $a\in\overline{K}\_d$},$$ where $\overline{K}\_d \equiv\{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) \geq0 \}$ is the closure of $K\_d \equiv\{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) > 0 \}$; the latter set is open, bounded and convex in Rd (w.p.1). On Kd, gd is also real-valued, strictly convex and infinitely differentiable (w.p.1). − ∞ < gd(Yd) < 0, Yd′∫01f(t) dt > 0, $ {0 \leq\int\_0^1 \frac{Y\_d^\prime f(t)}{ 1 + Y\_d^\prime f(t)}\,dt < \infty}$. If Yd ∈ Kd, then Yd is the unique solution to $ {\int\_0^1 \frac{f(t)}{1 + a^\prime f(t)}\,dt = 0\_d}$ for a ∈ Kd, and if $ {\int\_0^1 \frac{f(t)}{1 + a^\prime f(t)}\,dt = 0\_d}$ has a solution a ∈ Kd, then this solution is uniquely Yd. We use the subscript d in Theorem [theorem1] to denote the dimension of either the random vector Yd, the space Kd or the arguments of gd. The function gd is well defined and convex on $\overline {K}\_d$, though possibly gd(a) = + ∞ for some $a\in\partial K\_d = \{ y \in\mathbb{R}^d \dvtx\min\_{0 \leq t \leq1} (1 + y^\prime f(t)) = 0 \}$ on the boundary of Kd; a minimizer of gd( ⋅ ) may also occur on $\partial K\_d \cap \{y \in\mathbb{R}^d\dvtx g\_d(y)\leq0 \}$. Importantly, the probability law of gd(Y1) is distribution-free, and because standard Brownian motion is fast and straightforward to simulate, the distribution of gd(Yd) can be approximately numerically. Parts (ii) and (iii) provide properties for characterizing and identifying the minimizer Yd. For example, considering the real-valued case d = 1, it holds that K1 = (m, M) where m = − [max0 ≤ t ≤ 1f(t)]− 1 < 0 < M = − [min0 ≤ t ≤ 1f(t)]− 1 and the derivative dg1(a)/da is strictly increasing on K1 by convexity. Because the derivative of g1 at 0 is − ∫01f(x) dx, parts (ii)–(iii) imply that if − ∫01f(x) dx < 0, then either Y1 = m or Y1 solves dg1(a)/da = 0 on m < a ≤ 0; alternatively, if − ∫01f(x) dx > 0, then Y1 = M or Y1 solves dg1(a)/da = 0 on 0 ≤ a < M. Additionally, while the weight function w( ⋅ ) influences the distribution of gd(Yd), the scale of w( ⋅ ) does not; defining f with w or cw, for a nonzero c ∈ R, produces the same minimized value gd(Yd). We may now state the main result on the large-sample behavior of the EBEL log-ratio evaluated at the true process mean EXt = μ0 ∈ Rd. Recall that, when Ln(μ0) > 0 in ([eqn:EBEL]), the EBEL log-ratio admits a representation ([eqn:t0]) at μ0 in terms of the Lagrange multiplier λn, μ0 ∈ Rd. [theorem2] Under assumptions, as n → ∞: [(ii)] $ n^{1/2} \Sigma^{1/2} \lambda\_{n,\mu\_0} \stackrel {d}{\rightarrow} Y\_d $; $ {-\frac{1}{n} \log R\_n(\mu\_0) \stackrel{d}{\rightarrow} -g\_d(Y\_d)}$, for Yd and gd(Yd) defined as in Theorem [theorem1], and Σ = ∑j = − ∞∞Cov(X0, Xj). From Theorem [theorem2](i), the Lagrange multiplier in the EBEL method has a limiting distribution which is not the typical normal one, as in the standard BEL case. This has a direct impact on the limit law of the EBEL ratio statistic. As Theorem [theorem2](ii) shows, the negative logarithm of the EBEL ratio statistic, scaled by the inverse of the sample size, has a nonstandard limit, given by the functional − gd(Yd) of the vector of Brownian motion B( ⋅ ) (cf. Theorem [theorem1]), that critically depends on the limit Yd of the scaled Lagrange multiplier. The distribution of − gd(Yd) is free of any population parameters so that quantiles of − gd(Yd), which are easy to compute numerically, can be used to calibrate the EBEL confidence regions. As − gd(Yd) is a strictly positive random variable, an approximate 100(1 − α)% confidence region for μ0 can be computed as $$\bigl\{\mu\in\mathbb{R}^d \dvtx- n^{-1} \log R\_n(\mu\_0) \leq a\_{d,1-\alpha}\bigr\},$$ where ad, 1 − α is the lower (1 − α) percentile of − gd(Yd). When d = 1, the confidence region is an interval; for d > 2, the region is guaranteed to be connected without voids in Rd. In contrast to the standard BEL ([eqn:stanl]), EBEL confidence regions do not require a similar fixed choice of block size. We next provide additional results that give the limit distribution of the log-EBEL ratio statistic under a sequence of local alternatives and that also show the size of a EBEL confidence region will be no larger than Op(n− 1/2) in diameter around the true mean EXt = μ0. Let $$\label{eqn:Gn} G\_n \equiv\bigl\{ \mu\in\mathbb{R}^d \dvtx R\_n(\mu)\geq R\_n(\mu\_0)>0\bigr\}$$ be the collection of mean parameter values which are at least as likely as μ0, and therefore elements of a EBEL confidence region whenever the true mean is. [co1] Suppose the assumptions of Theorem [theorem2] hold. For c ∈ Rd, define fc(t) = w(t)[B(t) + tΣ− 1/2c], t ∈ [0, 1], in terms of the vector of Brownian motion B(t). [(ii)] Then, as n → ∞, $-n^{-1}\log R\_n(\mu\_0 \pm n^{-1/2}c) \stackrel{d}{\rightarrow}$ $$-\min\biggl\{ -\int\_0^1 \log\bigl(1 + a^\prime f\_{c}(t)\bigr)\,dt\dvtx a \in\mathbb{R}^d, \min\_{0 \leq t \leq1} \bigl(1 + a^\prime f\_{c}(t)\bigr) \geq0 \biggr\};$$ $ \sup\{\\|\mu-\mu\_0\\|\dvtx\mu\in G\_n\} =O\_p(n^{-1/2})$, for Gn in ([eqn:Gn]). Hence along a sequence of local alternatives (n− 1/2 away from the true mean), the log-EBEL ratio converges to a random variable, defined as the optimizer of an integral involving Brownian motion; this resembles Theorem [theorem1] [involving f(t) = w(t)B(t) there], but the integrated function fc( ⋅ ) has an addition term w(t)tΣ− 1/2c under the alternative. With respect to Corollary [co1](i), the involved limit distribution can be described with similar properties as in Theorem [theorem1] upon replacing f(t) with fc(t) there. In particular, the limiting distribution under the scaled alternatives depends on Σ− 1/2c, similarly to the normal theory case (e.g., with standard BEL) where Σ− 1/2c determines the noncentrality parameter of a noncentral chi-square distribution. We note that Theorem [theorem2] remains valid for potentially negative-valued weight functions w( ⋅ ) as well. Simulations have shown that, with weight functions oscillating between positive and negative values on [0, 1] [e.g., w(t) = sin(2πt)], EBEL intervals for the process mean perform consistently well in terms of coverage accuracy. However, with weight functions w( ⋅ ) that vary in sign, a result as in Corollary [co1](ii) fails to hold. Hence, the weight functions w( ⋅ ) considered are nonnegative as stated in assumption (A.1). [re1] The EBEL results in Theorem [theorem2] also extend to certain parameters described by general estimating functions; for examples and similar EL results in the i.i.d. and time series cases, respectively, see and. Suppose θ ∈ Rp represents a parameter of interest and G( ⋅ ; ⋅ ) ∈ Rd × Rp → Rp is a vector of p estimating functions such that EG(Xt; θ0) = 0p holds at the true parameter value θ0. The previous process mean case corresponds to G(Xt; μ) = Xt − μ with Xt, μ ∈ Rd, d = p. A EBEL ratio statistic Rn(θ) for θ results by replacing Tu, i = w(i/n)∑j = 1i(Xj − μ) and 0d with Tθ, i = w(i/n)∑j = 1iG(Xj; θ) and 0p in ([eqn:EBEL]). Under the conditions of Theorem [theorem2] [substituting G(Xj; θ0) for Xj − μ0 in assumption (A.2)], $$-\frac{1}{n} \log R\_n(\theta\_0) \stackrel{d} { \rightarrow} -g\_p(Y\_p)$$ holds as n → ∞ with Yp and gp(Yp) as defined in Theorem [theorem1], generalizing Theorem [theorem2] and following by the same proof. The next section considers extensions of the EBEL approach to a different class of time series parameters. Smooth function model parameters -------------------------------- We next consider extending the EBEL method for inference on a broad class of parameters under the so-called “smooth function model;” cf.. For independent and time series data, respectively, Hall and La Scala and Kitamura have considered EL inference for similar parameters; see also, Section 4. If EXt = μ0 ∈ Rd again denotes the true mean of the process, the target parameter of interest is given by θ0 = H(μ0) ∈ Rp, based on a smooth function H(μ) = (H1(μ), …, Hp(μ))′ of the mean parameter μ, where $H\_i\dvtx\mathbb{R}^d\rightarrow \mathbb {R}$ for i = 1, …, p and p ≤ d. This framework allows a large variety of parameters to be considered such as sums, differences, products and ratios of means, which can be used, for example, to formulate parameters such as covariances and autocorrelations as functions of the m-dimensional moment structure (for a fixed m) of a time series. For a univariate stationary series U1, …, Un, for instance, one can define a multivariate series Xt based on transformations of (*	arxiv_0000829
right)\frac{1}{2k}$$ Since arc cosh(17) < 3.5255, it follows that $$\lambda\_{2k}<\frac{1}{2k}$$ for all k ≥ 3. This proves the proposition. [r.2kGamma6] In general, the first eigenvalue λ(Γ) of the Laplacian of H/Γ does not increase under finite covers: if Γʹ is a finite index subgroup of Γ, then λ(Γʹ) ≤ λ(Γ). Therefore, λ(Γ) ≤ λ2k < 1/k for any finite-index subgroup Γ of Γ6(2k). Γ6(2k), k ≥ 3, is not a congruence[37](#fn37) subgroup of SL(2, Z). Indeed, Selberg’s 3/16 theorem says that the first eigenvalue λ(Γ) of the Laplacian of H/Γ satisfies λ(Γ) ≥ 3/16 when Γ is congruence, but we know from Proposition [p.2kGamma6] that λ(Γ6(2k)) = λ2k < 1/6 for all k ≥ 3. Arithmetic Teichmüller curves S2k birational to H/Γ6(2k) ------------------------------------------------------------ In view of Remark [r.2kGamma6] and Ratner’s theorem [t.Ratner], the proof of Theorem [t.MaSc] is reduced to the following statement. [t.Z2k] For each k ≥ 3, there exists an explicit square-tiled surface Z2k (of genus 48k + 3 tiled by 192k unit squares) whose Veech group is contained in Γ6(2k). In particular, the hyperbolic surface H/Γ6(2k) is finitely covered by the arithmetic Teichmüller curve S2k generated by the SL(2, R)-orbit of Z2k. The construction of Z2k is based on the ideas of Ellenberg-McReynolds and Schmithüsen. Unfortunately, the implementation of these ideas is somewhat technical and, for this reason, we’ll just give a sketch of the construction of Z2k while referring to Sections 3 and 4 of for more details. We build Z2k with the aid of ramified covers of translation surfaces. For this sake, let us recall that if h : X1 → X2 is a finite covering of Riemann surfaces, then the ramification data rm(P, h) of a point P ∈ X2 consists of the ramification indices of all preimages of P counted with multiplicities. This notion is important for our purposes because any affine homeomorphism f̂ on X1 descending to an affine homeomorphism f of X2 under a translation covering h : X1 → X2 must respect the ramification data, i.e., rm(f(P), h) = rm(P, h). In particular, we can force such an affine homeomorphism f to respect certain partitions of the branch points of h by prescribing distinct ramification data to them and, as it turns out, this information can be used to put constraints on the linear part Df of f, that is, on the element Df of the Veech group of X1. The simplest example illustrating the ideas in the previous paragraph is the translation covering from E[2] : = C/(2Z ⊕ 2Zi) to E : = C/(Z ⊕ Zi) (given by the natural isogeny). Indeed, the affine homeomorphisms of E fixing the origin P : = (0, 0) correspond to SL(2, Z) and the affine homeomorphisms of E[2] fixing the 2-torsion points P, Q = (1, 0), S = (1, 1) correspond to the principal congruence subgroup Γ2 of level 2. Next, for each k ∈ N, one builds a translation covering q2k : Y2k → E[2] of degree [Γ2 : Γ6(2k)] = 24k such that: * q2k is ramified precisely over P, Q and S; * all affine homeomorphisms of E[2] fixing P, Q and S lift to affine homeomorphisms of Y2k: in particular, the image of the group Aff\*q2k(Y2k) of such lifts under the derivative homomorphism D is Γ2; * the fiber q2k− 1(R) of the 2-torsion point R = (0, 1) is bijectively mapped to Γ2/Γ6(2k) by a map θ which is equivariant with respect to the derivative homomorphism D : Aff\*q2k(Y2k) → Γ2, i.e., Df ⋅ θ(Ri) = θ(f(Ri)) for all Ri ∈ q2k− 1(R) and f ∈ Aff\*q2k(Y2k). * the Veech group of Y2k is SL(2, Z). Finally, the square-tiled surface Z2k is obtained from a (double) cover r2k : Z2k → Y2k such that the ramification data of P, Q and S with respect to r2k ∘ q2k are pairwise distinct and the ramification data of the point Rid : = θ− 1(id ⋅ Γ6(2k)) ∈ q2k− 1(R) is different from the ramification data of all points in q2k− 1(R). In this way, we have that any affine homeomorphism f̂ of Z2k descending to an affine homeomorphism f of Y2k has linear part Df̂ ∈ Γ6(2k). Indeed, our condition on the ramification data forces f ∈ Aff\*q2k(Y2k) to fix Rid. In particular, Df ⋅ Γ6(2k) = Df ⋅ θ(Rid) = θ(f(Rid)) = θ(Rid) = id ⋅ Γ6(2k), that is, Df̂ = Df ∈ Γ6(2k). Therefore, the proof of Theorem [t.Z2k] will be complete if we have that all affine homeomorphisms of Z2k descend to Y2k. Here, one exploits the action of Aff(Y2k) on q2k− 1({P, Q, R, S}) in order to detect a partition of q2k− 1({P, Q, R, S}) with the following property: if the ramification data of r on the atoms of this partition are distinct, then all affine homeomorphisms of Z2k descend to Y2k. Since it is not hard to produce a branched cover r with this feature (for any given partition of q2k− 1({P, Q, R, S})), this finishes our sketch of proof of Theorem [t.Z2k]. The first member Z6 of the family Z2k, k ≥ 3, is a square-tiled surface associated to the following pair of permutations σh and σv (on 576 unit squares). lcl σh &=& (1, 13, 193, 207, 243, 253)(2, 14, 194, 208, 244, 254)(3, 15, 195, 209, 245, 255) && (4, 16, 196, 210, 246, 256)(5, 17, 197, 211, 247, 257)(6, 18, 198, 212, 248, 258) && (7, 19, 199, 213, 249, 259)(8, 20, 200, 214, 250, 260)(9, 21, 201, 215, 251, 261) && (10, 22, 202, 216, 252, 262)(11, 23, 203, 205, 241, 263)(12, 24, 204, 206, 242, 264) && (25, 38, 266, 280, 220, 230, 26, 37, 265, 279, 219, 229)(27, 39, 267, 281, 221, 231) && (28, 40, 268, 282, 222, 232)(29, 41, 269, 283, 223, 233)(30, 42, 270, 284, 224, 234) && (31, 43, 271, 285, 225, 235)(32, 44, 272, 286, 226, 236)(33, 45, 273, 287, 227, 237) && (34, 46, 274, 288, 228, 238)(35, 47, 275, 277, 217, 239)(36, 48, 276, 278, 218, 240) && (49, 61, 433, 445, 337, 349)(50, 62, 434, 446, 338, 350)(51, 63, 435, 447, 339, 351) && (52, 64, 436, 448, 340, 352)(53, 65, 437, 449, 341, 353)(54, 66, 438, 450, 342, 354) && (55, 67, 439, 451, 343, 355)(56, 68, 440, 452, 344, 356)(57, 69, 441, 453, 345, 357) && (58, 70, 442, 454, 346, 358)(59, 71, 443, 455, 347, 359)(60, 72, 444, 456, 348, 360) && (73, 85, 361, 373, 457, 469)(74, 86, 362, 374, 458, 470)(75, 87, 363, 375, 459, 471) && (76, 88, 364, 376, 460, 472)(77, 89, 365, 377, 461, 473)(78, 90, 366, 378, 462, 474) && (79, 91, 367, 379, 463, 475)(80, 92, 368, 380, 464, 476)(81, 93, 369, 381, 465, 477) && (82, 94, 370, 382, 466, 478)(83, 95, 371, 383, 467, 479)(84, 96, 372, 384, 468, 480) && (97, 109, 385, 397, 481, 493)(98, 110, 386, 398, 482, 494)(99, 111, 387, 399, 483, 495) && (100, 112, 388, 400, 484, 496)(101, 113, 389, 401, 485, 497)(102, 114, 390, 402, 486, 498) && (103, 115, 391, 403, 487, 499)(104, 116, 392, 404, 488, 500)(105, 117, 393, 405, 489, 501) && (106, 118, 394, 406, 490, 502)(107, 119, 395, 407, 491, 503)(108, 120, 396, 408, 492, 504) && (121, 133, 505, 517, 409, 421)(122, 134, 506, 518, 410, 422)(123, 135, 507, 519, 411, 423) && (124, 136, 508, 520, 412, 424)(125, 137, 509, 521, 413, 425)(126, 138, 510, 522, 414, 426) && (127, 139, 511, 523, 415, 427)(128, 140, 512, 524, 416, 428)(129, 141, 513, 525, 417, 429) && (130, 142, 514, 526, 418, 430)(131, 143, 515, 527, 419, 431)(132, 144, 516, 528, 420, 432) && (145, 167, 539, 551, 299, 301)(146, 168, 540, 552, 300, 302)(147, 157, 529, 541, 289, 303) && (148, 158, 530, 542, 290, 304)(149, 159, 531, 543, 291, 305)(150, 160, 532, 544, 292, 306) && (151, 161, 533, 545, 293, 307)(152, 162, 534, 546, 294, 308)(153, 163, 535, 547, 295, 309) && (154, 164, 536, 548, 296, 310)(155, 165, 537, 549, 297, 311)(156, 166, 538, 550, 298, 312) && (169, 183, 315, 325, 553, 565)(170, 184, 316, 326, 554, 566)(171, 185, 317, 327, 555, 567) && (172, 186, 318, 328, 556, 568)(173, 187, 319, 329, 557, 569)(174, 188, 320, 330, 558, 570) && (175, 189, 321, 331, 559, 571)(176, 190, 322, 332, 560, 572)(177, 191, 323, 333, 561, 573) && (178, 192, 324, 334, 562, 574)(179, 181, 313, 335, 563, 575)(180, 182, 314, 336, 564, 576), σv &=& (1, 265, 289, 553, 433, 73)(2, 266, 290, 554, 434, 74)(3, 267, 291, 555, 435, 75) && (4, 268, 292, 556, 436, 76)(5, 269, 293, 557, 437, 77)(6, 270, 294, 558, 438, 78) && (7, 271, 295, 559, 439, 79)(8, 272, 296, 560, 440, 80)(9, 273, 297, 561, 441, 81) && (10, 274, 298, 562, 442, 82)(11, 275, 299, 563, 443, 83, 12, 276, 300, 564, 444, 84) && (13, 229, 157, 565, 493, 133)(14, 230, 158, 566, 494, 134)(15, 231, 159, 567, 495, 135) && (16, 232, 160, 568, 496, 136)(17, 233, 161, 569, 497, 137)(18, 234, 162, 570, 498, 138) && (19, 235, 163, 571, 499, 139)(20, 236, 164, 572, 500, 140)(21, 237, 165, 573, 501, 141) && (22, 238, 166, 574, 502, 142)(23, 239, 167, 575, 503, 143)(24, 240, 168, 576, 504, 144) && (25, 97, 505, 529, 169, 193, 26, 98, 506, 530, 170, 194)(27, 99, 507, 531, 171, 195) && (28, 100, 508, 532, 172, 196)(29, 101, 509, 533, 173, 197)(30, 102, 510, 534, 174, 198) && (31, 103, 511, 535, 175, 199)(32, 104, 512, 536, 176, 200)(33, 105, 513, 537, 177, 201) && (34, 106, 514, 538, 178, 202)(35, 107, 515, 539, 179, 203)(36, 108, 516, 540, 180, 204) && (37, 61, 469, 541, 325, 253)(38, 62, 470, 542, 326, 254)(39, 63, 471, 543, 327, 255) && (40, 64, 472, 544, 328, 256)(41, 65, 473, 545, 329, 257)(42, 66, 474, 546, 330, 258) && (43, 67, 475, 547, 331, 259)(44, 68, 476, 548, 332, 260)(45, 69, 477, 549, 333, 261) && (46, 70, 478, 550, 334, 262)(47, 71, 479, 551, 335, 263)(48, 72, 480, 552, 336, 264) && (49, 361, 145, 313, 385, 121)(50, 362, 146, 314, 386, 122)(51, 363, 147, 315, 387, 123) && (52, 364, 148, 316, 388, 124)(53, 365, 149, 317, 389, 125)(54, 366, 150, 318, 390, 126) && (55, 367, 151, 319, 391, 127)(56, 368, 152, 320, 392, 128)(57, 369, 153, 321, 393, 129) && (58, 370, 154, 322, 394, 130)(59, 371, 155, 323, 395, 131)(60, 372, 156, 324, 396, 132) && (85, 109, 421, 301, 181, 349)(86, 110, 422, 302, 182, 350)(87, 111, 423, 303, 183, 351) && (88, 112, 424, 304, 184, 352)(89, 113, 425, 305, 185, 353)(90, 114, 426, 306, 186, 354) && (91, 115, 427, 307, 187, 355)(92, 116, 428, 308, 188, 356)(93, 117, 429, 309, 189, 357) && (94, 118, 430, 310, 190, 358)(95, 119, 431, 311, 191, 359)(96, 120, 432, 312, 192, 360) && (205, 277, 397, 517, 445, 373)(206, 278, 398, 518, 446, 374)(207, 279, 399, 519, 447, 375) && (208, 280, 400, 520, 448, 376)(209, 281, 401, 521, 449, 377)(210, 282, 402, 522, 450, 378) && (211, 283, 403, 523, 451, 379)(212, 284, 404, 524, 452, 380)(213, 285, 405, 525, 453, 381) && (214, 286, 406, 526, 454, 382)(215, 287, 407, 527, 455, 383)(216, 288, 408, 528, 456, 384) && (217, 337, 457, 481, 409, 241)(218, 338, 458, 482, 410, 242)(219, 339, 459, 483, 411, 243) && (220, 340, 460, 484, 412, 244)(221, 341, 461, 485, 413, 245)(222, 342, 462, 486, 414, 246) && (223, 343, 463, 487, 415, 247)(224, 344, 464, 488, 416, 248)(225, 345, 465, 489, 417, 249) && (226, 346, 466, 490, 418, 250)(227, 347, 467, 491, 419, 251)(228, 348, 468, 492, 420, 252) --- --- Some finiteness results for algebraically primitive Teichmüller curves ====================================================================== --- --- Many applications of the dynamics of SL(2, R) on moduli spaces of translation surfaces to the investigation of translation flows and billiards rely on the features of the closure of certain SL(2, R)-orbits. For example, Delecroix-Hubert-Lelièvre exploited the properties of the closure of certain SL(2, R)-orbits of translation surfaces of genus five in order to confirm a conjecture of Hardy and Weber on the abnormal rate of diffusion of trajectories in Z2-periodic Ehrenfest wind-tree models. Partly motivated by potential further applications, the problem of classifying closures of SL(2, R)-orbits in moduli spaces of translation surfaces received a considerable attention in recent years. Some classification results for the closures of SL(2, R)-orbits in moduli spaces ------------------------------------------------------------------------------------ The quest of listing all SL(2, R)-orbit closures in moduli spaces of translation surfaces became a reasonable goal after the groundbreaking works of Eskin and Mirzakhani, Eskin, Mirzakhani and Mohammadi and Filip. Indeed, these results say that such SL(2, R)-orbit closures have many good properties including: they are affine in period coordinates, there are only countably many of them, and they are quasi-projective varieties with respect to the natural algebraic structure on moduli spaces. Despite the absence of a complete classification of SL(2, R)-orbit closures of translation surfaces, the current literature on the subject contains many papers. For this reason, instead of trying to give an exhaustive list of articles on this topic, we shall restrict ourselves to the discussion of the smallest possible SL(2, R)-orbit closures – namely, Teichmüller curves – while refereeing to the introduction of the paper of Apisa and the references therein for more details on higher-dimensional SL(2, R)-orbit closures. Arithmetic Teichmüller curves are always abundant: they form a dense subset in any connected component of any stratum of the moduli space of translation surfaces. This indicates that a complete classification of these objects is a challenging task and, indeed, we are able to list all arithmetic Teichmüller curves only in the case of the minimal stratum H(2) thanks to the works of Hubert and Lelièvre and McMullen. Non-arithmetic Teichmüller curves seem less abundant and we dispose of many partial results towards their classification. In fact, Calta and McMullen, obtained a complete classification of all SL(2, R)-orbit closures of translation surfaces of genus two: it follows from their results that the minimal stratum H(2) contains infinitely many non-arithmetic Teichmüller curves, but the principal stratum H(1, 1) contains just one non-arithmetic Teichmüller curve (generated by a regular decagon). In higher genera g ≥ 3, we have many results establishing the finiteness of algebraically primitive Teichmüller curves, i.e., Teichmüller curves whose trace field[38](#fn38) has the largest possible degree g over Q. For example: * Möller showed that H(g − 1, g − 1)hyp contains only finitely many algebraically primitive Teichmüller curves, and * Bainbridge and Möller established the finiteness of algebraically primitive Teichmüller curves in H(3, 1). The main theorem of this section (namely, Theorem [t.MW-A]) is a result due to Wright and the author showing the finiteness of algebraically primitive Teichmüller curves in the minimal stratum H(2g − 2) when g > 2 is a prime number. Before giving the precise statement of the main result of (and sketching its proof), let us mention that a recent work of Bainbridge, Habbeger and Möller proved the finiteness of algebraically primitive Teichmüller curves in all strata of the moduli space of translation surfaces of genus three: similarly to the work of Bainbridge and Möller mentionned above, Bainbridge, Habegger and Möller rely mostly on algebro-geometrical arguments, even though their treatment of the particular of the case H(2, 2)odd build upon the techniques of our joint work with Wright. Statement of the main results ----------------------------- The main result of our paper with A. Wright is: [t.MW-A] Let C be a connected component of a stratum H(k1, …, kσ) of the moduli space of translation surfaces of genus g ($=1+\sum\limits\_{j=1}^{\sigma} k\_j/2$). * If g ≥ 3, then the (countable) union A = ⋃Ci of all algebraically primitive Teichmüller curves Ci contained in C is not dense, i.e., $\overline{A}\neq\mathcal{C}$. * If g ≥ 3 is prime and C is a connected component of the minimal stratum H(2g − 2), then there are only finitely many algebraically primitive Teichmüller curves contained in C. Apisa recently improved item (b) for the hyperelliptic component C = H(2g − 2)hyp of the minimal stratum by removing the constraint “g is prime”. The technique of proof of this theorem is “flexible”: for example, it was used (beyond the context of algebraic primitivity) by Nguyen, Wright and the author (cf. ) to show that the hyperelliptic component H(4)hyp of the minimal stratum in genus 3 contains[39](#fn39) only finitely many non-arithmetic Teichmüller curves. A key idea in the proof of Theorem [t.MW-A] is the study of Hodge-Teichmüller planes: Let M be a translation surface. We say that P ⊂ H1(M, R) is a Hodge-Teichmüller plane if the (Gauss-Manin) parallel transport[40](#fn40) of P along the SL(2, R)-orbit of M respect the Hodge decomposition[41](#fn41), i.e., dimC((hP ⊗ C) ∩ H1, 0(hM)) = 1 for all h ∈ SL(2, R). [ex.HT-canonical] Any translation surface M = (X, ω) possesses a canonical Hodge-Teichmüller plane, namely its tautological plane spanR(Re(ω), Im(ω)) ⊂ H1(M, R). [ex.HT-Galois-conjugates] Let M be a Veech surface whose trace field k(M) = Q({tr(γ) : γ ∈ SL(M)}) associated to its Veech group SL(M) has degree k over Q. The k embeddings of k(M) can be used to construct k planes L1, …, Lk ⊂ H1(M, R) obtained from the tautological plane L1 by Galois conjugation. As it was observed by Möller, we have a decomposition H1(M, R) = L1 ⊕ … ⊕ Lk ⊕ M of variation of Hodge structures[42](#fn42) whose summands are symplectically orthogonal. By definition, this means that L1, …, Lk are symplectically orthogonal Hodge-Teichmüller planes. In fact, these planes are important for our purposes because of the following features highlighted in the next two theorems (compare with Theorems 1.2 and 1.3 in ). [t.MW-B] Suppose that M is an affine invariant submanifold in the moduli space of genus g translation surfaces containing a dense set of algebraically primitive Teichmüller curves. Then, every translation surface in M has g symplectically orthogonal Hodge-Teichmüller planes. [t.MW-C] Let C be a connected component of a stratum of translation surfaces of genus g ≥ 3. Then, there exists a translation surface MC ∈ C which does not have g − 1 symplectically orthogonal Hodge-Teichmüller planes. In other words, Theorem [t.MW-B] says that algebraically primitive Teichmüller curves support many Hodge-Teichmüller planes and, moreover, these planes pass to the closure of any sequence of algebraically primitive Teichmüller curves. On the other hand, Theorem [t.MW-C] says that the presence of many symplectically orthogonal Hodge-Teichmüller planes is not satisfied by all translation surfaces in any given stratum. Note that Theorems [t.MW-B] and [t.MW-C] trivially imply the item (a) of Theorem [t.MW-A]. Furthermore, these two theorems also imply immediately the item (b) of Theorem [t.MW-A] when they are combined with the following result of A. Wright (cf. ): [Wright][t.Wr] Let m ≥ 2 be a prime number. Denote by M an affine invariant submanifold of a connected component C the minimal stratum H(2m − 2). If M properly contains an algebraically primitive Teichmüller curve, then M = C. [Sketch of proof of Theorem [t.Wr]] Denote by k(M) the field of definition of M, i.e., the smallest extension of Q containing the coefficients of all affine equations in period coordinates defining M. The field of definition has the following three general properties: * the field of definition of a Teichmüller curve coincides with its trace field; * it has a hereditary property: if N and M are affine invariant submanifolds and N ⊂ M, then k(M) ⊂ k(N); * dimCp(TM) ⋅ degQ(k(M)) ≤ 2m whenever M is an affine invariant submanifold in a stratum of genus m translation surfaces whose tangent space TM ⊂ H1(M, div(ω), C) at a point (M, ω) ∈ M projects to a subspace p(TM) ⊂ H1(M, C) under the natural projection p : H1(M, div(ω), C) → H1(M, C). Let M be an affine invariant submanifold of a connected component C of H(2m − 2). Suppose that m ≥ 2 is a prime number and M properly contains an algebraically primitive Teichmüller curve C. Then, the first two properties above of the field of definition imply that the degree degQ(k(M)) divides the degree of the trace field of C, i.e., degQ(k(M)) divides m. Since m is a prime number, this means that degQ(k(M)) equals 1 or m. We affirm that degQ(k(M)) = 1: indeed, if degQ(k(M)) = m, then the third property of the field of definition would imply that dimCp(TM) ≤ 2, a contradiction with the fact that M properly contains a Teichmüller curve. Once we know that k(M) = Q, it is not hard to see that the tangent space to M has complex dimension at least 2m: in fact, since M is defined over Q, the space p(TM) at any point (M, ω) ∈ C contains the tangent space (tautological plane) to the algebraically primitive Teichmüller curve C and all of its m Galois conjugates. Because M ⊂ H(2m − 2) and the minimal stratum H(2m − 2) has complex dimension 2m ≤ dimC(TM), it follows that M is an open GL+(2, R)-invariant subset of H(2m − 2). By the ergodicity theorem of Masur and Veech, this implies that M is a connected component of the stratum H(2m − 2). In the sequel, we will discuss the proofs of Theorems [t.MW-B] and [t.MW-C]. More precisely, we will establish Theorem [t.MW-B] in Subsection [ss.t.MW-B] below by studying some continuity properties of Hodge-Teichmüller planes, and we will provide a sketch of proof of Theorem [t.MW-C] together with an elementary proof of a particular case of this theorem in Subsection [ss.t.MW-C] below. Proof of Theorem [t.MW-B] ------------------------- Recall that the most basic example of Hodge-Teichmüller plane associated to any given translation surface (M, ω) is the tautological plane L1 : = spanR(Re(ω), Im(ω)) ⊂ H1(M, R) (cf. Example [ex.HT-canonical]). If the SL(2, R)-orbit of a translation surface X = (M, ω) of genus g generates an algebraically Teichmüller curve C, then we have g symplectically orthogonal Hodge-Teichmüller planes L1, …, Lg (cf. Example [ex.HT-Galois-conjugates]). Therefore, the proof of Theorem [t.MW-B] is reduced to the following continuity property of Hodge-Teichmüller planes: [p.HT-continuity] Let C be a connected component of a stratum of the moduli space of translation surfaces. Suppose that Xn ∈ C is a sequence of translation surfaces converging to X ∈ C such that, for some fixed k ∈ N, each Xn possesses k symplectically orthogonal Hodge-Teichmüller planes, say Pn(1), …, Pn(k). Then, X possesses k symplectically orthogonal Hodge-Teichmüller planes. By definition, Xn = (Mn, ωn) converges to X = (M, ω) whenever we can find diffeomorphisms fn : Mn → M such that (fn)\(ω*n) → ω. By extracting an appropriate subsequence if necessary, we can assume that, for each 1 ≤ j ≤ k, (fn)\(P*n(j)) converges to a plane P(j) in the Grassmanian of planes of H1(M, R). We claim that P(j) is a Hodge-Teichmüller plane (for each 1 ≤ j ≤ k). In fact, given any h ∈ SL(2, R), we have that (ϕh ∘ fn ∘ ϕh− 1)\(hPn(j)) → h*P(j), where ϕh is the affine homeomorphism induced by h. On the other hand, we know that (hPn(j) ⊗ C) ∩ H1, 0(hMn) ≠ {0} for each n ∈ N and 1 ≤ j ≤ k (because Pn(j) are Hodge-Teichmüller planes) and, in general, H1, 0(N) varies continuously with N (see, e.g., ). Thus, it follows from that (hP(j) ⊗ C) ∩ H1, 0(hM) ≠ {0}, i.e., P(j) is a Hodge-Teichmüller plane. Finally, we affirm that P(j) are k pairwise distinct symplectically orthogonal planes. Indeed, the continuity of the symplectic intersection form implies that P(j) are mutually symplectically orthogonal. Nevertheless, this is not sufficient to obtain that P(j) are pairwise distinct. For this sake, we observe that, by definition, a Hodge-Teichmüller plane is Hodge-star[43](#fn43) invariant (because its complexification is the sum of its (1, 0) and (0, 1) parts). Hence, the symplectic orthogonal of a Hodge-Teichmüller plane coincides[44](#fn44) with its orthogonal for the Hodge inner product[45](#fn45) and, a fortiori, P(j) are mutually orthogonal with respect to the Hodge inner product. In particular, P(j) are pairwise distinct as it was claimed. Sketch of proof of Theorem [t.MW-C] ----------------------------------- Let X = (M, ω) be a translation surface. The group Aff(X) of affine homeomorphisms of X acts on H1(M, R) via symplectic matrices. Let Γ(X) be the image of the natural representation Aff(X) → Sp(H1(M, R)) and denote by $\overline{\Gamma(X)}$ the Zariski closure of Γ(X). The proof of Theorem [t.MW-C] starts with the following fact: [p.HT-mon-invariance] The set of Hodge-Teichmüller planes of X is $\overline{\Gamma(X)}$-invariant. For each h ∈ SL(2, R), let Hh1, 0 : = h− 1(H1, 0(hX)) ⊂ H1(X, C). By definition, a plane P ⊂ H1(X, R) is Hodge-Teichmüller if and only if PC : = P ⊗ C intersects Hh1, 0 non-trivially for all h ∈ SL(2, R). Given h ∈ SL(2, R), the condition that PC intersects H1, 0 non-trivially corresponds to a finite number of polynomial equations on the Grassmanian of planes in H1(X, R): indeed, if we form a matrix MP by listing a basis of P next to a basis of Hh1, 0, then PC ∩ Hh1, 0 ≠ {0} is equivalent to rank(MP) ≤ g + 1, i.e., all (g + 2) × (g + 2) minors of MP vanish. (Here, g is the genus of X.) In particular, the set P of Hodge-Teichmüller planes is a subvariety of the Grassmanian of planes in H1(X, R). It follows that the stabilizer of P contains the Zariski closure $\overline{\Gamma(X)}$ (because P is clearly Γ(M)-invariant). This proves the proposition. Given a translation surface X = (M, ω) (of genus g), let H1(X, R)⊥ be the ((2g − 2)-dimensional) symplectic orthogonal of the tautological plane spanR(Re(ω), Im(ω)). Denote by Γ⊥(X) the restriction of Γ(X) to H1(X, R)⊥ and let $\overline{\Gamma\_{\perp}(X)}$ its Zariski closure. We use Proposition [p.HT-mon-invariance] to show that a translation surface has few Hodge-Teichmüller planes when $\overline{\Gamma\_{\perp}(X)}$ is large: [p.few-HT-criterion] Let X = (M, ω) be a translation surface of genus g ≥ 3 such that $\overline{\Gamma\_{\perp}(X)} = \textrm{Sp}(H^1(X,\mathbb{R})^{\perp})$. Then, the sole Hodge-Teichmüller plane of X is its tautological plane. By contradiction, suppose that X has a Hodge-Teichmüller plane P0 which is not the tautological plane. Since the tautological plane is precisely the kernel of the natural projection π : H1(X, R) → H1(X, R)⊥, we have that P : = π(P0) is dimension ≥ 1 and, moreover, the complexification PC : = P ⊗ C intersects H1, 0(X) or H0, 1(X) (because the complexification of π respects H1, 0(X) and H0, 1(X)). Taking into account that P is a real subspace and H0, 1(X) is the complex conjugate of H1, 0(X), we have that PC intersects both H1, 0(X) and H0, 1(X) and, hence, P has dimension two. Note that the same argument applies to hP for all h ∈ SL(2, R). This means that P = π(P0) is a Hodge-Teichmüller plane whenever P0 is a non-tautological Hodge-Teichmüller plane. By Proposition [p.HT-mon-invariance], our hypothesis $\overline{\Gamma\_{\perp}(X)} = \textrm{Sp}(H^1(X,\mathbb{R}))^{\perp}$ implies that γ(P) is a Hodge-Teichmüller plane for all γ ∈ Sp(H1(X, R)⊥). This is a contradiction because Sp(H1(X, R)⊥) acts transitively on the set of symplectic planes in H1(X, R)⊥, but there are[46](#fn46) symplectic planes which are not Hodge-Teichmüller when g ≥ 3. This proposition allows us to establish some low-genus cases of Theorem [t.MW-C]: [p.MW-C-g3] Let C be a connected component of H(4). Then, there exists a square-tiled surface MC ∈ C such that $\overline{\Gamma\_{\perp}(M\_{\mathcal{C}})}\simeq \textrm{Sp}(4,\mathbb{R})$. In particular, MC has only one Hodge-Teichmüller plane. The minimal stratum H(4) of the moduli space of translation surfaces of genus 3 has two connected components H(4)hyp and H(4)odd. As it is explained in, these connected components are distinguished by the parity of the spin structure: M ∈ H(4)hyp, resp. H(4)odd, if and only if Φ(M) = 0, resp. 1, where Φ(M) ∈ Z/2Z is the so-called Arf invariant[47](#fn47)/parity of spin structure. From a direct inspection of the definitions, one can check that: * the square-tiled surface M\* associated to the permutations h\* = (1)(2, 3)(4, 5, 6), v\* = (1, 4, 2)(3, 5)(6) belongs to H(4)odd, and * the square-tiled surface M\* \* associated to the permutations h\* \* = (1)(2, 3)(4, 5, 6), v\* \* = (1, 2)(3, 4)(5)(6) belongs to H(4)hyp. We affirm that $\overline{\Gamma\_{\perp}(M\_{\ast})} \simeq \overline{\Gamma\_{\perp}(M\_{\ast\ast})} \simeq \textrm{Sp}(4,\mathbb{R})$. For the sake of exposition, we treat only the case of M\* (while referring to for the case of M\* \). By Poincaré duality, our task is equivalent to show that Aff(M\) acts on the annihilator H1⊥(M\, R) ⊂ H1(M, R) of the tautological plane in H1(M\, R) through a Zariski dense subgroup of Sp(H1⊥(M\, R)). In this direction, we shall compute the action of some elements of Aff(M\) and we will prove that they generate a Zariski dense group. Note that M\* decomposes into three horizontal, resp. vertical, cylinders whose waist curves σ0, σ1, σ2, resp. ζ0, ζ1, ζ2, have lengths 1, 2 and 3. Also, M\* decomposes into two cylinders in the slope 1 direction whose waist curves δ1 and δ2 are given by the property that δ1 intersects σ0 and δ2 intersects ζ2. See Figure [f.M-ast]. [f.M-ast] Let us consider Dehn multitwists A, B, C ∈ Aff(M\) in the horizontal, vertical and slope 1 directions with linear parts $$dA=\left(\begin{array}{cc} 1 & 6 \\ 0 & 1 \end{array}\right), \quad dB=\left(\begin{array}{cc} 1 & 0 \\ 6 & 1 \end{array}\right), \quad dC=\left(\begin{array}{cc} -2 & 3 \\ -3 & 4 \end{array}\right)$$ It is not hard to see that the actions of A, B* and C on H1(M\, R) are given by: A(σ*i) = σi ∀ i = 1, 2, 3, A(ζ0) = ζ0 + 2σ2, A(ζ1) = ζ1 + 3σ1 + 2σ2, A(ζ2) = ζ2 + 3σ1 + 2σ2 + 6σ0, B(σ0) = σ0 + 2ζ2, B(σ1) = σ1 + 3ζ1 + 2ζ2, B(σ2) = σ2 + 3ζ1 + 2ζ2 + 6ζ0, B(ζi) = ζi ∀ i = 1, 2, 3, C(σ0) = σ0 − δ1 C(σ1) = σ1 − δ1 − δ2 C(σ2) = σ2 − δ1 − 2δ2, C(ζ0) = ζ0 + δ2 C(ζ1) = ζ1 + δ1 + δ2 C(ζ2) = ζ2 + 2δ1 + δ2 Thus, we can get matrices for the actions of A, B and C on H1⊥(M\, R) after we fix a basis of this vector space. In this direction, we observe that H1⊥(M\, R) is the subspace of H1(M\, R) consisting of cycles with trivial intersection with σ* : = σ0 + σ1 + σ2 and ζ : = ζ0 + ζ1 + ζ2. In particular, the cycles $\overline{\sigma\_1}:=\sigma\_1-2\sigma\_0$, $\overline{\sigma\_2}:=\sigma\_2-3\sigma\_0$, $\overline{\zeta\_1}:=\zeta\_1-2\zeta\_0$, $\overline{\zeta\_2}:=\zeta\_2-3\zeta\_0$ form a basis of H1⊥(M\, R) because these cycles are linearly independent, H1⊥(M\, R) has dimension 2g − 2 and M\* has genus g = 3. Since δ1 = σ1 + σ0 + ζ2 and δ2 = σ2 + ζ1 + ζ0, we conclude that the matrices A\, B\, C\* of A, B, C with respect to the basis $\{\overline{\sigma\_1}, \overline{\sigma\_2}, \overline{\zeta\_1}, \overline{\zeta\_2}\}$ of H1⊥(M\, R) are: $$A\_{\ast} = \left(\begin{array}{cccc} 1 & 0 & 3 & 3 \\ 0 & 1 & -2 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \quad B\_{\ast}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 3 & 3 & 1 & 0 \\ -2 & -4 & 0 & 1 \end{array}\right), \quad C\_{\ast} = \left(\begin{array}{cccc} 2 & 2 & 1 & 2 \\ -1 & -1 & -1 & -2 \\ -1 & -2 & 0 & -2 \\ 1 & 2 & 1 & 3 \end{array}\right)$$ Once we computed these matrices, it suffices to check the Zariski closure G* of the group ⟨A\, B\, C\⟩ is Sp(4, R). As it turns out, this fact can be proved as follows. The Lie algebra g of G* contains $$\log A\_{\ast}=\left(\begin{array}{cccc} 0 & 0 & 3 & 3 \\ 0 & 0 & -2 & -4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right)\in\mathfrak{g}$$ and, a fortiori, g also contains the nine conjugates of logA\* by the matrices B\, B\2, A\*B\, A\2B\, B\A\*B\, C\, C\2, A\C\, B\C\* On the other hand, a direct computation reveals that logA\* and these nine conjugates are linearly independent. Since Sp(4, R) has dimension 10, this shows that G = Sp(4, R). In summary, we showed that $\overline{\Gamma\_{\perp}(M\_{\ast})} = \textrm{Sp}(H^1(M\_{\ast},\mathbb{R})^{\perp})\simeq \textrm{Sp}(4,\mathbb{R})$. In particular, M\* has only one Hodge-Teichmüller plane (by Proposition [p.few-HT-criterion]). This completes the proof of the proposition. At this point, the idea of the proof of Theorem [t.MW-C] can be explained as follows. If C were a connected component of a stratum of the moduli space of translation surfaces of genus g ≥ 3 such that all M ∈ C has (g − 1) Hodge-Teichmüller planes, then all translation surfaces in all “adjacent” strata to C would have “many” Hodge-Teichmüller planes thanks to a “continuity argument”. However, this is impossible because C is “adjacent” to a connected component of H(4), but Proposition [p.MW-C-g3] says that all connected components of H(4) contain some translation surfaces with few Hodge-Teichmüller planes. More concretely, we formalize this idea in in two steps. First, we use an elementary continuity argument (similar to Proposition [p.HT-continuity]) and the notion of adjacency[48](#fn48) of strata from to establish the following result (cf. ): [p.MW-prop-5-1] Let C be a connected component of H(k1, …, ks), s > 1, $\sum\limits\_{l=1}^s k\_l = 2g-2$. Suppose that all translation surfaces in C possess m ≥ 1 symplectically orthogonal Hodge-Teichmüller planes. Then, there exists a connected component Cʹ of the minimal stratum H(2g − 2) such that all translation surfaces in Cʹ also possess m ≥ 1 symplectically orthogonal Hodge-Teichmüller planes. Secondly, we use a sophisticated version of the previous continuity argument to move Hodge-Teichmüller planes across minimal strata (cf. ): [p.MW-prop-5-3] Let C be a connected component of H(2g − 2). Suppose that every translation surface in C has m ≥ 1 symplectically orthogonal Hodge-Teichmüller planes. Then, there exists a connected component Cʹ of H(2g − 4) such that every translation surface in Cʹ has (m − 1) symplectically orthogonal Hodge-Teichmüller planes. The basic idea behind the proof of this proposition is not difficult, but a complete argument (provided in Sections 5 and 6 of ) is somewhat technical partly because it requires a discussion of the so-called Deligne-Mumford compactification. For this reason, we will content ourselves with the outline of proof of this proposition. [Sketch of proof of Proposition [p.MW-prop-5-3]] In their study of connected components of strata of the moduli space of translation surfaces, Kontsevich and Zorich introduced a local surgery of Abelian differentials called bubbling a handle. This surgery increases the genus by one and it is defined in two steps, namely splitting a zero and gluing a torus. Roughly speaking, one splits a zero of order m by a certain (local) cutting and pasting operation which produces a pair of zeroes of orders mʹ and mʺ with mʹ + mʺ = m joined by a saddle connection with holonomy v ∈ R2. After splitting a zero, one can cut the saddle connection to obtain a slit. Then, one can bubble a handle by gluing a cylinder/torus/handle into this slit. In what follows, we will be interested in gluing a square torus/handle to the slit. See Figures [f.split-zero] and [f.square-slit] for an illustration of these procedures. [f.split-zero] [f.square-slit] The operation of bubbling a handle allows to understand the adjacencies of strata. For example, we claim that if C is a connected component of H(2g − 2), then there exists a connected component Cʹ ⊂ H(2g − 4) such that we can bubble a handle on every translation surface in Cʹ in order to obtain a translation surface in C (compare with ). In fact, Lemma 20 in says that C contains a translation surface X given by the suspension of an interval exchange transformation associated to a good standard permutation π, i.e., a permutation of the form $\pi=\left(\begin{array}{ccc} A & \pi'\_t & Z \\ Z & \pi'\_b & A \end{array}\right)$ such that the permutation $\pi'=\left(\begin{array}{c} \pi'\_t \\ \pi'\_b \end{array}\right)$ derived from π by erasing the letters A and Z is irreducible. Concretely, the irreducibility of πʹ means that we can use it to build (through suspension) a translation surface Xʹ belonging to H(2g − 4) (in this context). See Figure [f.MW-pre-bdry] for an illustration of X together with a “copy” of Xʹ inside it. [f.MW-pre-bdry] Denote by Cʹ the connected component of H(2g − 4) containing Xʹ. Observe that, by definition, X ∈ C is obtained from Xʹ ∈ Cʹ by bubbling a square handle. Since the operation of bubbling a handle can be performed in a continuous way (cf. ), we obtain the desired claim that one can bubble a (square) handle in any Mʹ ∈ Cʹ to obtain a translation surface in M ∈ C. By exploiting this relation between C ⊂ H(2g − 2) and Cʹ ⊂ H(2g − 4) through the operation of bubbling a square handle, one can complete the proof of the proposition along the following lines (cf. Propositions 5.5 and 5.6 in ). Suppose that every M ∈ C has m symplectically orthogonal Hodge-Teichmüller planes. Given Mʹ ∈ Cʹ, let us bubble a square handle to obtain M ∈ C. By degenerating the square handle (i.e., by letting the length of the sizes of the square tend to zero) in M to reach Mʹ, we have[49](#fn49) that M converges to a stable nodal curve[50](#fn50) of the form depicted in Figure [f.MW-bdry]. [f.MW-bdry] In this process, one of the m Hodge-Teichmüller planes of M might be “lost” (in case it converges to the plane H1(Y, R) associated to the degenerating square handle), but at least (m − 1) Hodge-Teichmüller planes of M will converge to (m − 1) (symplectically orthogonal) Hodge-Teichmüller planes of Mʹ. This completes our sketch of proof of the proposition. At this point, Theorem [t.MW-C] is a simple consequence of the following argument: by contradiction, suppose that C is a connected component of stratum of the moduli space of translation surfaces of genus g ≥ 3 such that every M ∈ C has (g − 1) symplectically orthogonal Hodge-Teichmüller planes. By Proposition [p.MW-prop-5-1], there would be a connected component Cʹ of H(2g − 2) such that every M ∈ Cʹ would also possess (g − 1) symplectically orthogonal Hodge-Teichmüller planes. By applying (g − 3) times Proposition [p.MW-prop-5-3], we would find a connected component Cʺ of H(4) such that every M ∈ Cʺ would have 2 Hodge-Teichmüller planes, a contradiction with Proposition [p.MW-C-g3]. Closing this subsection, let us give an elementary[51](#fn51) proof of the following particular[52](#fn52) cases of Theorem [t.MW-C]: [p.partial-MW-C-cover] For each d ≥ 1 odd, there exists a translation surface M\(d) in the odd connected component H(5d* − 1)odd of the minimal stratum H(5d − 1) which does not possess (5d − 1)/2 symplectically orthogonal Hodge-Teichmüller planes. We affirm that a square-tiled surface N of genus g covering the square-tiled surface M\* constructed in the proof of Proposition [p.MW-C-g3] can not have (g − 1) symplectically orthogonal Hodge-Teichmüller planes. In fact, if we denote by p : N → M\* the translation covering defining the translation structure of N, then H1(N, R) = E ⊕ H1(N, R)(p) where H1(N, R)(p) consists of all cycles projecting to zero under p and E ≃ H1(M\, R) is the symplectic orthogonal of H1(N, R)(p). By appplying an argument similar to the proof of Proposition [p.few-HT-criterion], one can check that if N* had (g − 1) symplectically orthogonal Hodge-Teichmüller planes, then two of them would be contained in E ≃ H1(M\, R) and, a fortiori, M\ would have two Hodge-Teichmüller planes, a contradiction with Proposition [p.MW-C-g3]. In view of the discussion in the previous paragraph, the proof of the proposition will be complete once we construct a degree d branched cover M\(d) ∈ H(5d* − 1)odd of M\. For technical reasons, it is desirable to “simplify” the geometry of M\ before studying its covers. In this direction, let us replace M\* by the square-tiled surface $\overline{M}\_{\ast}$ associated to the permutations $\overline{h}\_{\ast}=(1,2,3,4,5,6)$, $\overline{v}\_{\ast}=(1)(2,5,4)(3,6)$: there is no harm in doing so because $\overline{M}\_{\ast}$ belongs to the SL(2, Z)-orbit of M\* (indeed, $\overline{M}\_{\ast}=J\cdot T^2(M\_{\ast})$ where $J=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)$ and $T=\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right)$). Given d ≥ 3 an odd integer, let $\overline{M}\_{\ast}(d)$ be the square-tiled surface[53](#fn53) associated to the pair of permutations $$\overline{h}\_{\ast}(d):=(1\_1,2\_1,3\_1,4\_1,5\_1,6\_1)(1\_2,2\_2,3\_2,4\_2,5\_2,6\_2,1\_3,2\_3,3\_3,4\_3,5\_3,6\_3)\dots$$ (1d − 1, 2d − 1, 3d − 1, 4d − 1, 5d − 1, 6d − 1, 1d, 2d, 3d, 4d, 5d, 6d), $$\overline{v}\_{\ast}(d):=(1\_1,1\_2,\dots,1\_d)(2\_1,5\_1,4\_1)(3\_1,6\_1)\dots(2\_d,5\_d,4\_d)(3\_d,6\_d)$$ By definition, $\overline{M}\_{\ast}(d)$ is a degree d cover of $\overline{M}\_{\ast}$ belonging to the stratum H(5d − 1). Therefore, it remains only to verify that $\overline{M}\_{\ast}(d)\in \mathcal{H}(5d-1)^{\textrm{odd}}$ in order to complete the proof of the proposition. Since $\overline{M}\_{\ast}(d)$ is not hyperelliptic, we have that $\overline{M}\_{\ast}(d)\in \mathcal{H}(5d-1)^{\textrm{odd}}$ if and only if $\Phi(\overline{M}\_{\ast}(d))=1$, where the parity $\Phi(\overline{M}\_{\ast}(d))\in\mathbb{Z}\_2$ of the spin structure of $\overline{M}\_{\ast}(d)$ is defined as follows (see for more details). Let a1, b1, a2, b2, …, ag, bg (where g = (5d + 1)/2) be a canonical symplectic basis of $H\_1(\overline{M}\_{\ast}(d),\mathbb{Z}\_2)$. Then, $$\Phi(\overline{M}\_{\ast}(d))=\sum\limits\_{i=1}^{(5d+1)/2} \phi(a\_i)\phi(b\_i) \textrm{ (mod } 2),$$ where $\phi:H\_1(\overline{M}\_{\ast}(d),\mathbb{Z}\_2)\to\mathbb{Z}\_2$ satisfies the following properties: * ϕ(γ) = ind(γ) + 1 whenever γ is a simple smooth closed curve in $\overline{M}\_{\ast}(d)$ not passing through singular points whose Gauss map has degree ind(γ); * ϕ is a quadratic form representing the intersection form (., .) in the sense that ϕ(α + β) = ϕ(α) + ϕ(β) + (α, β). In other words, we need to build a canonical symplectic basis of $H\_1(\overline{M}\_{\ast}(d),\mathbb{R})$ in order to compute $\Phi(\overline{M}\_{\ast}(d))$. For this sake, we begin by fixing a basis of $H\_1(\overline{M}\_{\ast}(d),\mathbb{R})$. We think of it as (d + 1)/2 horizontal cylinders determined by the permutation $\overline{h}\_{\ast}(d)$ whose top and bottom boundaries are glued accordingly to the permutation $\overline{v}\_{\ast}(d)$. Using this geometrical representation, we define the following cycles in $H\_1(\overline{M}\_{\ast}(d),\mathbb{Z})$ (see Figure [f.M\d-cycles]). for each i = 1, …, d, let c23(i), c4(i), c5(i) and c6(i) be the homology classes of the vertical cycles within the horizontal cylinders connecting the middle of the bottom side of the squares 5i, 2i, 4i and 3i (resp.) to the top side of the squares 2i, 4i, 5i and 6i (resp.); * for each j = 2l, l = 1, …, (d − 1)/2, let c1(j) be the homology classes of the vertical cycles within the horizontal cylinders connecting the middles of the bottom side of the square 1j + 1 to the top side of the squares 1j; * let c1(1) be the homology class of the concatenation of the vertical cycles within the horizontal cylinders connecting the middles of the bottom side of the square 12l to the top side of the square 12l + 1 for l = 1, …, (d − 1)/2, the bottom side of the square 12 to the left side of the square 12, and the right side of the square 6d to the top side of the square 1d; * σ1 is the horizontal cycle connecting the left vertical side of the square 11 to the right vertical side of the square 61, and, for j = 2l, l = 1, …, (d − 1)/2, σj is the horizontal cycle connecting the left vertical side of the square 1j to the right vertical side of the square 6j + 1. [f.M\d-cycles] It is not hard to check that these 5d* + 1 cycles form a basis of $H\_1(\overline{M}\_{\ast}(d),\mathbb{Z})$. From this basis, we can produce a canonical basis of homology cycles of $\overline{M}\_{\ast}(d)$ using the orthogonalization procedure[54](#fn54) described in. More precisely, we start with the cycles a1 : = σ2 and b1 : = c1(2). Then, by induction, we use the cycles ai, bi to successively render the cycles c23(2), c4(2), c5(2), c6(2), …, σ1, c1(1), c23(1), c4(1), c5(1), c6(1) into a canonical basis a1, b1, a2, b2, …, ag, bg (where g = (5d + 1)/2). Using these cycles, we are ready to compute $\Phi(\overline{M}\_{\ast}(d))$ inductively. More concretely, we affirm that $\Phi(\overline{M}\_{\ast}(d+2))=\Phi(\overline{M}\_{\ast}(d))$ for each d ≥ 1 odd. Indeed, we note that the 10 cycles σd + 1, c1(d + 1), c23(d + 1), c4(d + 1), c5(d + 1), c6(d + 1), c23(d + 2), c4(d + 2), c5(d + 2), c6(d + 2) of $\overline{M}\_{\ast}(d+2)$ do not intersect the other 5d − 1 cycles of $\overline{M}\_{\ast}(d+2)$ defined above except for c1(1), and, moreover, they have trivial intersection with the cycle $$\begin{aligned} \widetilde{c}\_{1}^{(1)}&:=& c\_1^{(1)}-c\_1^{(d+1)}+c\_{23}^{(d+1)}-c\_4^{(d+1)}+2c\_5^{(d+1)}-2c\_6^{(d+1)} \\ &+& c\_4^{(d+2)}-c\_{23}^{(d+2)}+c\_6^{(d+2)}-c\_5^{(d+2)}\end{aligned}$$ satisfying $\phi(\widetilde{c}\_{1}^{(1)})=\phi(c\_1^{(1)})=1$. Thus, by replacing c1(1) by $\widetilde{c}\_1^{(1)}$ and by applying the orthogonalization procedure to these two sets of 10 and 5d − 1 cycles in an independent way, we deduce that $\Phi(\overline{M}\_{\ast}(d+2))$ differs from $\Phi(\overline{M}\_{\ast}(d))$ by a term of the form $$\Phi(\overline{M}\_{\ast}(d+2))-\Phi(\overline{M}\_{\ast}(d))=\sum\limits\_{i=1}^{5}\phi(a\_i^{(d+1)})\phi(b\_i^{(d+1)})$$ where ai(d + 1), bi(d + 1), i = 1, …, 5 is an orthogonalization of the 10 cycles σd + 1, c1(d + 1), c23(d + 1), c4(d + 1), c5(d + 1), c6(d + 1), c23(d + 2), c4(d + 2), c5(d + 2), c6(d + 2). On the other hand, by a direct computation, one can check that * a1(d + 1) = σd + 1, b1(d + 1) = c1(d + 1), * a2(d + 1) = c4(d + 1) − a1(d + 1) − b1(d + 1), b2(d + 1) = c23(d + 1) − a1(d + 1) − b1(d + 1), * a3(d + 1) = c6(d + 1) − a1(d + 1) − b1(d + 1) + a2(d + 1), b3(d + 1) = c5(d + 1) − a1(d + 1) − b1(d + 1) + a2(d + 1), * a4(d + 1) = c4(d + 2) − b1(d + 1) + b2(d + 1) − a2(d + 1) + 2b3(d + 1) − 2a3(d + 1), b4(d + 1) = c23(d + 2) − b1(d + 1) + b2(d + 1) − a2(d + 1) + 2b3(d + 1) − 2a3(d + 1), * a5(d + 1) = c5(d + 2) − c4(d + 2), b5(d + 1) = c6(d + 2) − c4(d + 2) is an orthogonalization of the 10 cycles σd + 1, c1(d + 1), c23(d + 1), c4(d + 1), c5(d + 1), c6(d + 1), c23(d + 2), c4(d + 2), c5(d + 2), c6(d + 2). Moreover, ϕ(ai(d + 1)) = ϕ(bi(d + 1)) = 1 for i = 1, 4, and ϕ(aj(d + 1)) = ϕ(bj(d + 1)) = 0 for j = 2, 3, 5. Hence, $$\Phi(\overline{M}\_{\ast}(d+2))-\Phi(\overline{M}\_{\ast}(d))=\sum\limits\_{i=1}^{5}\phi(a\_i^{(d+1)})\phi(b\_i^{(d+1)}) = 0,$$ as it was claimed. Therefore, $\Phi(\overline{M}\_{\ast}(d))=\Phi(\overline{M}\_{\ast}(1)):=\Phi(\overline{M}\_{\ast})=1\in\mathbb{Z}\_2$. --- --- Simplicity of Lyapunov exponents of arithmetic Teichmüller curves ================================================================= --- --- The circle of ideas developed for the study of Lyapunov exponents of the Kontsevich-Zorich cocycle over the SL(2, R)-action on moduli spaces of translation surfaces was fruitfully used in many contexts: * Zorich and Forni related the Lyapunov exponents of the KZ cocycle with respect to Masur-Veech measures to obtain a complete description of the deviations of ergodic averages of typical interval exchange transformations and translation flows; * Avila and Forni used the positivity of the second largest Lyapunov exponent of the KZ cocycle with respect to Masur-Veech measures to establish the weak mixing property for typical interval exchange transformations (not of rotation type) and translation flows (on higher genus surfaces); * Delecroix, Hubert and Lelièvre exploited the precise values of the Lyapunov exponents of the KZ cocycle over a certain closed SL(2, R)-invariant locus of translation surfaces of genus five to confirm a conjecture of Hardy and Weber on the abnormal rate of diffusion of typical trajectories in Z2-periodic Ehrenfest wind-tree models of Lorenz gases; * Kappes and Möller employed some invariants (inspired of the Lyapunov exponents of the KZ cocycle) the exact number (nine) of commensurability classes of the non-arithmetic lattices of PU(1, n) constructed by Deligne and Mostow. Kontsevich-Zorich conjecture and Veech’s question ------------------------------------------------- In some applications of the Lyapunov exponents of the KZ cocycles, it is important to know whether they have a qualitative property called simplicity, that is, if all of them appear with multiplicity one. For instance, the most complete picture for the deviations of ergodic averages of typical interval exchange transformations and translation flows depends on the simplicity of the Lyapunov exponents of the KZ cocycle with respect to Masur-Veech measures (see and ). In the case of Masur-Veech measures, the simplicity of the Lyapunov exponents of the KZ cocycle was conjectured by Kontsevich and Zorich after several numerical experiments. This conjecture was fully confirmed in a celebrated work of Avila and Viana after an important partial result of Forni. In the case of other SL(2, R)-invariant probability measures on moduli spaces of translation surfaces, Veech asked[55](#fn55) if the Lyapunov exponents of the KZ cocycle are always non-zero and/or simple. This question was negatively answered by Forni and the author with two examples called Eierlegende Wollmilchsau and Ornithorynque (see, e.g., ): loosely speaking, these are examples of arithmetic Teichmüller curves such that the KZ cocycle over them has many zero Lyapunov exponents. Of course, this answer to Veech’s question motivates the problem of finding criteria for the simplicity of the Lyapunov exponents of the KZ cocycle with respect to SL(2, R)-invariant probability measures in moduli spaces of translation surfaces. In the remainder of this section, we shall offer an answer to this problem in the context of the KZ cocycle over Teichmüller curves. Lyapunov exponents of Teichmüller curves and random products of matrices ------------------------------------------------------------------------ In this subsection, we will show that the Lyapunov exponents of the KZ cocycle with respect to SL(2, R)-invariant probability measures supported on Teichmüller curves can be computed via random products of matrices. The relevance of this fact for our purposes is explained by the vast literature (cf. Furstenberg, Guivarc’h-Raugi,, Goldsheid-Margulis, Avila-Viana, etc.) on the simplicity of Lyapunov exponents of random products of matrices. The analog of this statement for Masur-Veech measures is discussed in Avila-Viana paper in their reduction of the Kontsevich-Zorich conjecture to the study of “random” products of matrices in Rauzy-Veech monoids. More concretely, let X = (M, ω) be a translation surface whose SL(2, R)-orbit is closed in the moduli space of translation surfaces. Recall that, by a result of Smillie (see, e.g., ), SL(2, R)X is closed if and only if X is a Veech surface, i.e., the stabilizer SL(X) of X is a lattice of SL(2, R). We have a short exact sequence Aut(X) → Aff(X) → SL(X) where Aut(X) is the group of automorphisms of X and Aff(X) is the group of affine homeomorphisms of X. For the sake of exposition, we will assume that the group of automorphisms of X is trivial: Aut(X) = {Id}. In this case, the groups Aff(X) and SL(X) are isomorphic. Since the KZ cocycle GtKZ over the Teichmüller flow gt = diag(et, e− t) ∈ SL(2, R) is the quotient of the trivial cocycle SL(2, R)X × H1(X, R) → SL(2, R)X × H1(X, R) (η, [c]) ↦ (gt(η), [c]) by the diagonal action of the mapping class group and the stabilizer of SL(2, R)X in the mapping class group is precisely Aff(X), we have that the KZ cocycle is given by the actions on homology of the elements of Aff(X) appearing[56](#fn56) along the orbits of gt. In this setting, Eskin and the author proved that the Lyapunov exponents of the KZ cocycle are seen by random products of the matrices of actions on homology of affine homeomorphisms: [t.EM] Let X be a Veech surface of genus g ≥ 1. Denote by 1 = λ1 > λ2 ≥ … ≥ λg ≥ − λg ≥ … ≥ − λ2 > − λ1 = − 1 the Lyapunov exponents of the KZ cocycle with respect to the unique SL(2, R)-invariant probability measure on SL(2, R)X. Then, there exists a probability measure on (the countable group) Aff(X) assigning non-zero mass to every element and a constant λ > 0 such that the Lyapunov exponents of random products of matrices in Sp(2g, Z) with respect to the law ρ(ν) where ρ : Aff(X) → Sp(2g, Z) is the representation induced by the action of affine homeomorphisms on H1(X, Z) are precisely λ > λ ⋅ λ2 ≥ … ≥ λ ⋅ λg ≥ − λ ⋅ λg ≥ … ≥ − λ ⋅ λ2 > − λ Before discussing Lyapunov exponents, we need to “replace” the typical orbits of the geodesic flow gt on the hyperbolic plane H by certain random walks in SL(X) ⋅ i. In general, the random walk on SL(X) ⋅ i with a law ν of full support on SL(X) is tracked by a geodesic ray in H up to a sublinear error: for some λ > 0 and for νN-almost all sequences (γn)n ∈ N ⊂ SL(X), there exists a unit-speed geodesic ray {α(t) : t ∈ R} ⊂ H such that dH(γn…γ1 ⋅ i, α(λn)) = o(n) as n → ∞. Indeed, this is a direct consequence of Oseledets theorem saying that the random product γn…γ1 in SL(2, R) is “close” to the matrix $g\_{\lambda t} r\_{\theta(\overline{\gamma})}$, where λ > 0 is the top Lyapunov exponent associated to ν and $\theta(\underline{\gamma})\in [0,2\pi)$ is an angle depending on $\underline{\gamma}=(\gamma\_1,\gamma\_2,\dots)\in SL(X)^{\mathbb{N}}$. See, e.g., Lemma 4.1 of for more details. The distribution of the angles $\theta(\underline{\gamma})$ depend on ν. In particular, it is far from obvious that one can choose ν in such a way that a typical geodesic rays in H are tracked by random walks with law ν up to sublinear error: in concrete terms, this basically amounts to choose ν so that the distribution of angles $\theta(\underline{\gamma})\in [0,2\pi)$ is given by the Lebesgue measure. Fortunately, a profound theorem of Furstenberg ensures the existence of a probability measure ν on the lattice SL(X) with full support and the desired (Lebesgue) distribution of angles[57](#fn57). Once we know that a typical orbit of the geodesic flow is tracked by a random walk with sublinear error, let us come back to the discussion of Lyapunov exponents. Consider a typical geodesic ray gtrθ(ω) tracked by a random walk (γn…γ1)n ∈ N with sublinear error. Forni proved that the following general growth estimate for the KZ cocycle: $$\frac{d}{dt}\log\\|G\_t^{KZ}(\eta)\\|\leq 1$$ for all translation surface η, where ∥.∥ denotes the Hodge norm. From this estimate, we deduce: log∥(ρ(γn…γ1) ⋅ GλnKZ(rθω)− 1)± 1∥ ≤ dH(γn…γ1 ⋅ i, gλn(rθω)) = o(n) Hence, $$\lim\limits\_{n\to\infty}\frac{1}{n}\log\frac{\\|\rho(\gamma\_n\dots \gamma\_1)(w)\\|}{\\|G\_{\lambda n}^{KZ}(w)\\|}=0$$ for any vector w ∈ H1(X, R) − {0}. By definition of Lyapunov exponents, this equality means that λρ(ν)(w) = λ ⋅ λKZ(w) where λρ(ν)(w), resp. λKZ(w), is the Lyapunov exponent of w with respect to random products of matrices in Sp(2g, Z) with law ρ(ν), resp. KZ cocycle GtKZ. This proves the theorem. This theorem together with Avila-Viana criterion for the simplicity of Lyapunov exponents of random products of matrices yield the following statement (cf. ): [c.EM] Let X = (M, ω) be a Veech surface of genus g ≥ 1. Suppose that Aff(X) contains two elements ϕ and ψ whose actions on the annihilator H1(0)(X, R) of the tautological plane R ⋅ Re(ω) ⊕ R ⋅ Im(ω) ⊂ H1(X, R) are given by two matrices A and B (in Sp(H1(0)(X, R)) ≃ Sp(2g − 2, R)) such that: * A is pinching, i.e., all eigenvalues of A are real, simple and their moduli are distinct; * B is twisting with respect to A, i.e., for any 1 ≤ k ≤ g − 1, any isotropic A-invariant k-plane F and any coisotropic A-invariant (2g − 2 − k)-plane Fʹ, we have B(F) ∩ Fʹ = {0}. Then, the Lyapunov exponents of the KZ cocycle with respect to the Lebesgue measure on SL(2, R)X are simple, i.e., the Lyapunov spectrum has the form 1 = λ1 > λ2 > … > λg > − λg > … > − λ2 > − λ1 = − 1 By Theorem [t.EM], our task is equivalent to establish the simplicity of the Lyapunov spectra of the random products with law ρ0(ν) of matrices in Sp(H1(0)(X, R)), where ρ0 : Aff(X) → H1(0)(X, R) is the natural representation. By the simplicity criterion[58](#fn58) of Avila-Viana, a random product with law θ of matrices in Sp(2d, R) has simple Lyapunov spectrum whenever the support of η contains pinching and twisting matrices (in the sense of items (i) and (ii) above). Since ρ0(ν) gives positive mass to A and B (because ν assigns positive masses to ϕ and ψ), the proof of the corollary is complete. Theorem [t.EM], Corollary [c.EM] and its variants were applied by Eskin and the author to study the Lyapunov spectra of certain Teichmüller curves in genus four (of Prym type) and certain variations of Hodge structures of weight three associated to 14 families of Calabi-Yau threefolds (including mirror quintics). See Sections 3 and 4 of for more explanations. Galois-theoretical criterion for simplicity of exponents of origamis -------------------------------------------------------------------- From the practical point of view, Corollary [c.EM] is not quite easy to apply. Indeed, the verification of the pinching and twisting properties might be tricky: for example, Avila-Viana performed a somewhat long inductive procedure in order to establish the pinching and twisting properties in their context (of Rauzy-Veech monoids). As it turns out, Möller, Yoccoz and the author found an effective version of Corollary [c.EM] in the case of square-tiled surfaces thanks to some combinatorial arguments involving Galois theory. In the sequel, we will state and prove a Galois-theoretical simplicity criteria and we will discuss its applications to square-tiled surfaces of genus three. ### Galois-pinching matrices Let A ∈ Sp(2d, R) be a 2d × 2d symplectic matrix. The characteristic polynomial of A is a monic reciprocal polynomial P of degree 2d. Denote by $\widetilde{R}=\{\lambda\_i, \lambda\_i^{-1}: 1\leq i\leq d\}=P^{-1}(0)$ the set of roots of P and let $R=p(\widetilde{R})$ where p(λ) = λ + λ− 1. We say that a matrix A ∈ Sp(2d, Z) is Galois-pinching if its characteristic polynomial P is irreducible over Q, its eigenvalues are real ($\widetilde{R}\subset \mathbb{R}$), and the Galois group Gal of P is the largest possible, i.e., $Gal\simeq S\_d \rtimes (\mathbb{Z}/2\mathbb{Z})^d$ acts by the full permutation group on $R=p(\widetilde{R})$ and the subgroup fixing R pointwise acts by independent transpositions of each of the d pairs {λi, λi− 1}. For each $\lambda\in\widetilde{R}$, let us choose an eigenvector vλ of A associated to λ with coordinates in the field Q(λ) in such a way that vg.λ = g.vλ for all g ∈ Gal. The nomenclature “Galois-pinching” is justified by the following proposition: [p.Galois-pinching] A Galois-pinching matrix is pinching. By definition, all eigenvalues of a Galois-pinching matrix A are real and simple. Hence, our task is to show these eigenvalues have distinct moduli. Suppose that λ and − λ are eigenvalues of A. An element of the Galois group Gal fixing λ must also fix − λ, a contradiction with the fact that Gal is the largest possible. An important point about Galois-pinching matrices is the fact that they can be detected in an effective way. In order to illustrate this, let us consider the prototype P of characteristic polynomial of a matrix in Sp(4, Z), i.e., P(x) = x4 + ax3 + bx2 + ax + 1 is a monic reciprocal integral polynomial of degree four. The following elementary proposition characterizes the polynomials P with real, simple and positive roots (and, a fortiori, of distinct moduli). The polynomial P(x) = x4 + ax3 + bx2 + ax + 1 has real, simple and positive roots if and only if Δ1 : = a2 − 4b + 8 > 0, t : = − a − 4 > 0 and d : = b + 2 + 2a > 0 A simple calculation shows that λ is a root of P if and only if μ : = λ + λ− 1 − 2 is a root of the quadratic polynomial Q(y) : = y2 − ty + d of discriminant t2 − 4d = a2 − 4b + 8 : = Δ1. Since the roots λ of P are real, simple and positive if and only if the roots μ of Q have the same properties, the desired proposition follows. The next two propositions provide a criterion for the irreducibility of P over Q. The polynomial P(x) = x4 + ax3 + bx2 + ax + 1 ∈ Z[x] is a product of two reciprocal quadratic rational polynomials if and only if $\sqrt{\Delta\_1}\in\mathbb{Q}$. The quadratic polynomial Q(y) = y2 − ty + d with roots μ related to the roots λ of P via the formula μ = λ + λ− 1 − 2 is reducible over Q if and only if $\sqrt{\Delta\_1}\in\mathbb{Z}$. Let P(x) = x4 + ax3 + bx2 + ax + 1 be a monic reciprocal integral polynomial of degree four. Suppose that Δ1 : = a2 − 4b + 8 is not a square (i.e., $\sqrt{\Delta\_1}\notin\mathbb{Z}$) and P is reducible over Q. Then, Δ2 : = (b + 2 − 2a)(b + 2 + 2a) is a square, i.e., $\sqrt{\Delta\_2}\in\mathbb{Z}$. Since Δ1 is not a square, P has no rational root. Because P is reducible over Q (by assumption), the previous proposition implies that P = PʹPʺ where Pʹ, Pʺ ∈ Q[x] are monic irreducible quadratic polynomials which are not reciprocal. Thus, we can relabel the roots of P in such a way that Pʹ(x) = (x − λ1)(x − λ2), Pʺ(x) = (x − λ1− 1)(x − λ2− 1) Note that Pʹ ∈ Q[x] implies that λ1λ2, λ1 + λ2 ∈ Q and, a fortiori, λ12 + λ22 ∈ Q. Therefore, $$(\lambda\_1-\lambda\_1^{-1})(\lambda\_2-\lambda\_2^{-1}) = \lambda\_1\lambda\_2 - \frac{1-\lambda\_1^2-\lambda\_2^2}{\lambda\_1\lambda\_2}\in\mathbb{Q}$$ It follows that Δ2 = (b + 2 − 2a)(b + 2 + 2a) = (λ1 − λ1− 1)2(λ2 − λ2− 1)2 is a square. Furthermore, it is not hard to decide whether an irreducible monic reciprocal integral polynomial of degree four has the largest possible Galois group: [p.Galois-degree4] Let P(x) = x4 + ax3 + bx2 + ax + 1 ∈ Z[x] be irreducible over Q. The Galois group Gal of P is the largest possible (i.e., $Gal\simeq S\_2\rtimes(\mathbb{Z}/2\mathbb{Z})^2$ has order eight) if and only if $$\sqrt{\Delta\_1}, \sqrt{\Delta\_2}, \sqrt{\Delta\_1\Delta\_2}\notin\mathbb{Z},$$ (where Δ1 : = a2 − 4b + 8 and Δ2 : = (b + 2 − 2a)(b + 2 + 2a)). Moreover, in this case we have that the splitting field of P contains exactly three quadratic subfields, namely, $\mathbb{Q}(\sqrt{\Delta\_1})$, $\mathbb{Q}(\sqrt{\Delta\_2})$, $\mathbb{Q}(\sqrt{\Delta\_1\Delta\_2})$. The solution of this elementary exercise in Galois theory is explained in Lemmas 6.12 and 6.13 of. For the sake of completeness, let us sketch the proof of this proposition. Since P is irreducible, Gal acts transitively on the roots λ1, λ1− 1, λ2, λ2− 1 of P. Thus, if Gal can permute λi and λi− 1 independently for i = 1, 2, then Gal has order eight and, a fortiori, Gal is the largest possible. This reduces our task to show that if Gal can not permute λi and λi− 1 independently for i = 1, 2, then either $\sqrt{\Delta\_2}$ or $\sqrt{\Delta\_1\Delta\_2}$ is an integer. For this sake, we observe that if if Gal doesn’t permute λi and λi− 1 independently for i = 1, 2, then Gal permutes simultaneously λ1, λ1− 1 and λ2, λ2− 1. In this case, there are two possibilities: * either Gal is generated by the permutations (λ1, λ2)(λ1− 1, λ2− 1) and (λ1, λ1− 1)(λ2, λ2− 1), * or Gal is generated by the four cycle (λ1, λ2, λ1− 1, λ2− 1). These cases can be distinguished as follows. The expression (λ1 − λ1− 1)(λ2 − λ2− 1) is invariant in case (a) (i.e., Gal ≃ Z/2Z × Z/2Z is a Klein group) but not in case (b) (i.e., Gal ≃ Z/4Z is a cyclic group). Similarly, the expression (λ1 + λ1− 1 − λ2 − λ2− 1)(λ1 − λ1− 1)(λ2 − λ2− 1) is invariant in case (b) but not in case (a). Therefore: * (a) occurs if and only if (λ1 − λ1− 1)(λ2 − λ2− 1) ∈ Q; * (b) occurs if and only if (λ1 + λ1− 1 − λ2 − λ2− 1)(λ1 − λ1− 1)(λ2 − λ2− 1) ∈ Q. Since (λ1 − λ1− 1)2(λ2 − λ2− 1)2 = Δ2 and (λ1 + λ1− 1 − λ2 − λ2− 1)2(λ1 − λ1− 1)2(λ2 − λ2− 1)2 = Δ1Δ2, we conclude that * (a) occurs if and only if $\sqrt{\Delta\_2}\in\mathbb{Z}$; * (b) occurs if and only if $\sqrt{\Delta\_1\Delta\_2}\in\mathbb{Z}$. In any case, we show that either $\sqrt{\Delta\_2}$ or $\sqrt{\Delta\_1\Delta\_2}$ is an integer when Gal can not permute λi and λi− 1 independently for i = 1, 2. This completes our sketch of proof. In summary, the previous four propositions allow us to test whether a matrix A ∈ Sp(4, Z) is Galois-pinching by studying three discriminants Δ1, Δ2 and Δ1Δ2 naturally attached to its characteristic polynomial. ### Twisting with respect to Galois-pinching matrices I: statements of results After discussing the Galois-pinching property, let us study the twisting property with respect to Galois-pinching matrices. Our main result in this direction is: [t.MMY-twisting] Let A ∈ Sp(2d, Z) be a Galois-pinching matrix. Suppose that B ∈ Sp(2d, Z) has the property that A and B2 share no common proper invariant subspace. Then, there exists m ≥ 1 and, for any ℓ\, there are integers ℓi* ≥ ℓ\, 1 ≤ i* ≤ m − 1, such that the product BAℓ1…BAℓm − 1B is twisting with respect to A, i.e., for all 1 ≤ k ≤ d, for any A-invariant isotropic subspace F of dimension k and for any A-invariant coisotropic subspace Fʹ of dimension 2d − k, we have BAℓ1…BAℓm − 1B(F) ∩ Fʹ = {0}. This theorem constitutes the main ingredient in the simplicity criteria in. Before starting its somewhat long proof, let us make some comments on its statement and applicability. First, we observe that Theorem [t.MMY-twisting] becomes false if B2 is replaced by B in the hypothesis “A and B2 share no common proper invariant subspace”: for example, $A=\left(\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}\right)\in SL(2,\mathbb{Z})$ is Galois-pinching, $B=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)\in SL(2,\mathbb{Z})$ has no invariant subspaces, but BAℓ1…BAℓm − 1B is never twisting with respect to A (because B permutes the eigenspaces of A).[59](#fn59) Second, the condition “A and B2 have no common proper invariant subspace” might not be easy to check in general. For this reason, the following two propositions (cf. Proposition 4.7, Lemma 5.1 and Lemma 5.5 in ) are useful in certain applications of Theorem [t.MMY-twisting]. [p.unipotent-twisting] Let A ∈ Sp(2d, Z) be a Galois-pinching matrix. Suppose that B ∈ Sp(2d, Z) is unipotent and B ≠ Id. If A and B2 share a common proper invariant subspace, then (B − Id)(R2d) is a Lagrangian subspace of R2d. We begin by noticing that our assumptions imply that any subspace which is invariant under Bm for some m > 0 is also invariant under B. Indeed, if we write B = Id + N and Bm = Id + Nʹ, then the binomial formula says that Nʹ is nilpotent whenever N is nilpotent. Hence, B = (Id + Nʹ)1/m can be calculated by truncating the binomial series[60](#fn60) and, therefore, B is a polynomial function of Nʹ = Bm − Id. In particular, any subspace invariant under Bm is also invariant under B. In particular, our assumption that A and B2 share a common proper invariant subspace imply (in our current setting) that A and B share a common proper invariant subspace. Since A is Galois-pinching, any A-invariant subspace is spanned by eigenvectors. Denote by P the characteristic polynomial of A, let $\widetilde{R}=P^{-1}(0)$ and, for each $\lambda\in\widetilde{R}$, take vλ an eigenvector of A with eigenvalue λ whose coordinates belong to Q(λ) in such a way that vg.λ = g.vλ for all g in the Galois group Gal of P. Next, let us fix $R\_B\subset\widetilde{R}$ a proper subset of minimal cardinality such that the subspace E(RB) spanned by the vectors {vλ : λ ∈ RB} is B-invariant. Because B has integral entries, for all σ ∈ Gal the subspaces E(σ(RB)) are also B-invariant. Thus, the minimality of the cardinality of RB implies that, for each σ ∈ Gal, either σ(RB) ∩ RB = ∅ or σ(RB) = RB. This property together with the fact that Gal is the largest possible is a severe constraint on the proper subset $R\_B\subset \widetilde{R}$: * either RB = {λ} has cardinality one, * or RB has the form RB = {λ, λ− 1} The first possibility does not occur in our context: if B(vλ) = cvλ, then c = 1 (because B is unipotent); hence, using the action of Gal, we would deduce that B fixes all eigenvectors of A, so that B = Id, a contradiction. Therefore, B preserves some subspace E(λ, λ− 1) of the form E(λ, λ− 1) = Rvλ ⊕ Rvλ− 1. Using again the action of Gal, we deduce that B preserves all subspaces E(λ, λ− 1), $\lambda\in\widetilde{R}$, and the restrictions of B to such planes are Galois-conjugates. As B ≠ Id is unipotent, (B − Id)E(λ, λ− 1) is an one-dimensional subspace of E(λ, λ− 1) for all $\lambda\in\widetilde{R}$. Since the planes E(λ, λ− 1) are mutually symplectically orthogonal, it follows that (B − Id)(R2d) is a Lagrangian subspace. [p.Galois-twisting] Let A ∈ Sp(2d, Z) be a Galois-pinching matrix. Suppose that B ∈ Sp(2d, Z) has minimal polynomial of degree > 2 with no irreducible factor of even degree and a splitting field disjoint from the splitting field of the characteristic polynomial of A. Then, A and B2 do not share a common proper invariant subspace. We affirm that a B2-invariant subspace is also B-invariant. In fact, B and B2 have the same characteristic subspaces because λ and − λ can’t be both eigenvalues of B (thanks to our assumption that the minimal polynomial of B has no irreducible factor of even degree). Since an invariant subspace is the sum of its intersections with characteristic subspaces, it suffices to check that a B2-invariant subspace contained in a characteristic subspace of B is also B-invariant. Since B is unipotent up to scalar factors in its characteristic subspaces, the desired fact follows from the argument used in the beginning of the proof of Proposition [p.unipotent-twisting]. Let us now assume by contradiction that A and B2 share a common proper invariant subspace. From the discussion of the previous paragraph, this means that A and B share a common proper invariant subspace. By repeating the analysis in the proof of Proposition [p.unipotent-twisting], we have that all planes E(λ, λ− 1) = Rvλ ⊕ Rvλ− 1, $\lambda\in\widetilde{R}$, are invariant under A and B. Observe that E(λ, λ− 1) is defined over Q(λ + λ− 1) and the trace and determinant of B∣E(λ, λ− 1). From our hypothesis of disjointness of the splitting fields of A and B, the trace and determinant of B∣E(λ, λ− 1) are rational. Thus, the minimal polynomial of B has degree ≤ 2, a contradiction with our hypotheses. After these comments on the statement of Theorem [t.MMY-twisting], let us now prove this result. ### Twisting with respect to Galois-pinching matrices II: proof of Theorem [t.MMY-twisting] Consider the Galois-pinching matrix A and denote by $\widetilde{R}$ the set of its eigenvalues. For each $\lambda\in\widetilde{R}$, we select eigenvectors vλ of A associated to λ with coordinates in the field Q(λ) behaving coherently with respect to the Galois group Gal of the characteristic polynomial of A, i.e., vgλ = gvλ for all g ∈ Gal. The twisting propertyfor a matrix C with respect to A can be translated in terms of combinatorial properties of certain graphs naturally attached to its exterior powers ⋀kC. More concretely, for each 1 ≤ k ≤ d, let $\widetilde{R}\_k$ be the collection of all subsets of $\widetilde{R}$ with cardinality k. Note that, for each $\underline{\lambda}=\{\lambda\_1<\dots<\lambda\_k\}\in\widetilde{R}\_k$, we can associated a multivector $v\_{\underline{\lambda}} = v\_{\lambda\_1}\wedge\dots\wedge v\_{\lambda\_k}\in\bigwedge^k\mathbb{R}^{2d}$. By definition, the set $\{v\_{\underline{\lambda}}\vert \underline{\lambda}\in\widetilde{R}\_k\}$ is a basis of ⋀kR2d diagonalizing ⋀kA: indeed, $(\bigwedge^k A)(v\_{\underline{\lambda}}) = N(\underline{\lambda})v\_{\underline{\lambda}}$ where $N(\underline{\lambda}):=\prod\limits\_{\lambda\_i\in\underline{\lambda}}\lambda\_i$. In this setting, the A-invariant isotropic subspaces are easy to characterize. If p(λ) = λ + λ− 1 and R̂k is the collection of $\underline{\lambda}\in\widetilde{R}\_k$ such that $p\|\_{\underline{\lambda}}$ is injective, then the subspace generated by vλ1, …, vλk is isotropic if and only if $\underline{\lambda}=\{\lambda\_1,\dots,\lambda\_k\}\in\widehat{R}\_k$. Using this fact and an elementary (linear algebra) computation (cf. ), it is not hard to check that: A matrix C is twisting with respect to A if and only if for every 1 ≤ k ≤ d the coefficients $C^{(k)}\_{\underline{\lambda}, \underline{\lambda}'}$ of the matrix ⋀kC in the basis $\{v\_{\underline{\lambda}}\}$ satisfy the condition $$\label{e.condition-k} C^{(k)}\_{\underline{\lambda}, \underline{\lambda}'}\neq 0\, \quad \forall\, \underline{\lambda}, \underline{\lambda}'\in\widehat{R}\_k\,.$$ In other words, if Γk(C) is oriented graph with set of vertices Vert(Γk(C)) = R̂k and set of arrows $\{\underline{\lambda}\_0\to\underline{\lambda}\_1: C^{(k)}\_{\underline{\lambda}\_0,\underline{\lambda}\_1}\neq 0\}$, then C is twisting with respect to A if and only if Γk(C) is a complete graph for each 1 ≤ k ≤ d. In general, it is not easy to apply this lemma because the verification of the completeness of Γk(C) might be tricky. For this reason, we introduce the following (classical) notion: [d.mixing] The graph Γk(C) is mixing if there exists m ≥ 1 such that for all $\underline{\lambda}\_0,\underline{\lambda}\_1\in\widehat{R}\_k$ we can find an oriented path in Γk(C) of length m going from $\underline{\lambda}\_0$ to $\underline{\lambda}\_1$. In this definition, it is important to connect two arbitrary vertices by a path of length exactly m (and not only of length ≤ m). For instance, the graph in Figure [f.no-mixing] is not mixing because all paths connecting A to B have odd length while all paths connecting A to C have even length. [f.no-mixing] The relevance of this notion is explained by the following proposition (saying that if Γk(C) is mixing, then Γk(D) is complete for certain products D of the matrices C and A). [p.mixing/twisting] Let us assume that the graph Γk(C) is mixing with respect to an integer m ≥ 1. Then there exists a finite family of hyperplanes V1, …, Vt of Rm − 1 such that the following holds. For any $\underline{\ell}=(\ell\_1,\dots,\ell\_{m-1})\in \mathbb{Z}^{m-1}-(V\_1\cup\dots\cup V\_{m-1})$, the graph Γk(D(n)) associated to the matrix D(n) : = CAnℓ1…CAnℓm − 1C is complete for all sufficiently large $n\in \mathbb N$. Let us denote D : = D(n). By definition, $$D^{(k)}\_{\underline{\lambda}\_0,\underline{\lambda}\_m} = \sum\_{ \substack{\gamma \textrm{ path of length }m \\ \textrm{ in } \Gamma\_k(C) \textrm{ from } \underline{\lambda}\_0 \textrm{ to }\underline{\lambda}\_m}} C^{(k)}\_{\underline{\lambda}\_0,\underline{\lambda}\_1}N(\underline{\lambda}\_1)^{n\ell\_1} C^{(k)}\_{\underline{\lambda}\_1,\underline{\lambda}\_2}\dots N(\underline{\lambda}\_{m-1})^{n\ell\_{m-1}} C^{(k)}\_{\underline{\lambda}\_{m-1}, \underline{\lambda}\_m} \,.$$ Consider the linear forms: $$L\_\gamma(\underline{\ell}) = \sum\limits\_{i=1}^{n-1} \ell\_i\left(\sum\limits\_{\lambda\in\underline{\lambda}\_i} \log\|\lambda\|\right)\,.$$ Our assumption on C ensures that there are coefficients $c\_\gamma \not =0$ such that $$D^{(k)}\_{\underline{\lambda}\_0,\underline{\lambda}\_m} = \sum\limits\_{\gamma} c\_{\gamma} \exp(n L\_{\gamma}(\underline{\ell}))\,.$$ We want to prove that $D^{(k)}\_{\underline{\lambda}\_0,\underline{\lambda}\_m}\neq0$. Since $D^{(k)}\_{\underline{\lambda}\_0,\underline{\lambda}\_m}$ is a sum of exponentials with non-vanishing coefficients, our task is to show that $\underline{\ell}$ can be taken so that potential cancelations among these terms can be avoided. Here, the key idea is to prove that, for γ ≠ γʹ, the linear forms Lγ and Lγʹ are distinct. Indeed, suppose that this is the case and let us define $V(\gamma,\gamma')=\{\underline{\ell}: L\_{\gamma}(\underline{\ell}) = L\_{\gamma'}(\underline{\ell})\}$. By hypothesis, each V(γ,	arxiv_0000831

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